LMFDB - Embedded newform 75.9.f.d.7.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(7,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.7");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 192 x^{9} + 27713 x^{8} - 24384 x^{7} + 18432 x^{6} - 2072064 x^{5} + 128589064 x^{4} + \cdots + 846810000 $ x^12 - 192x^9 + 27713x^8 - 24384x^7 + 18432x^6 - 2072064x^5 + 128589064x^4 - 223046400x^3 + 184320000x^2 + 558720000*x + 846810000
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{18}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.5
Root			$1.66670 + 1.66670i$ of defining polynomial
Character	$\chi$	$=$	75.7
Dual form			75.9.f.d.43.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(15.2792 + 15.2792i) q^{2} +(-33.0681 + 33.0681i) q^{3} +210.909i q^{4} -1010.51 q^{6} +(-1776.62 - 1776.62i) q^{7} +(688.952 - 688.952i) q^{8} -2187.00i q^{9} +O(q^{10})$	\(q+(15.2792 + 15.2792i) q^{2} +(-33.0681 + 33.0681i) q^{3} +210.909i q^{4} -1010.51 q^{6} +(-1776.62 - 1776.62i) q^{7} +(688.952 - 688.952i) q^{8} -2187.00i q^{9} +4477.64 q^{11} +(-6974.37 - 6974.37i) q^{12} +(18402.3 - 18402.3i) q^{13} -54290.9i q^{14} +75046.1 q^{16} +(40628.0 + 40628.0i) q^{17} +(33415.7 - 33415.7i) q^{18} -161124. i q^{19} +117499. q^{21} +(68414.9 + 68414.9i) q^{22} +(-208665. + 208665. i) q^{23} +45564.7i q^{24} +562344. q^{26} +(72320.0 + 72320.0i) q^{27} +(374706. - 374706. i) q^{28} -1.08625e6i q^{29} +1.33249e6 q^{31} +(970274. + 970274. i) q^{32} +(-148067. + 148067. i) q^{33} +1.24153e6i q^{34} +461259. q^{36} +(778701. + 778701. i) q^{37} +(2.46185e6 - 2.46185e6i) q^{38} +1.21706e6i q^{39} -3.79519e6 q^{41} +(1.79530e6 + 1.79530e6i) q^{42} +(3.70334e6 - 3.70334e6i) q^{43} +944377. i q^{44} -6.37648e6 q^{46} +(3.40201e6 + 3.40201e6i) q^{47} +(-2.48163e6 + 2.48163e6i) q^{48} +547985. i q^{49} -2.68698e6 q^{51} +(3.88121e6 + 3.88121e6i) q^{52} +(-3.98610e6 + 3.98610e6i) q^{53} +2.20999e6i q^{54} -2.44802e6 q^{56} +(5.32806e6 + 5.32806e6i) q^{57} +(1.65971e7 - 1.65971e7i) q^{58} -4.53528e6i q^{59} -4.88819e6 q^{61} +(2.03595e7 + 2.03595e7i) q^{62} +(-3.88548e6 + 3.88548e6i) q^{63} +1.04383e7i q^{64} -4.52471e6 q^{66} +(-1.35112e7 - 1.35112e7i) q^{67} +(-8.56883e6 + 8.56883e6i) q^{68} -1.38003e7i q^{69} +4.27542e7 q^{71} +(-1.50674e6 - 1.50674e6i) q^{72} +(3.74805e7 - 3.74805e7i) q^{73} +2.37959e7i q^{74} +3.39825e7 q^{76} +(-7.95509e6 - 7.95509e6i) q^{77} +(-1.85957e7 + 1.85957e7i) q^{78} -6.47257e7i q^{79} -4.78297e6 q^{81} +(-5.79876e7 - 5.79876e7i) q^{82} +(-4.04525e7 + 4.04525e7i) q^{83} +2.47817e7i q^{84} +1.13168e8 q^{86} +(3.59203e7 + 3.59203e7i) q^{87} +(3.08488e6 - 3.08488e6i) q^{88} -9.02203e7i q^{89} -6.53878e7 q^{91} +(-4.40094e7 - 4.40094e7i) q^{92} +(-4.40630e7 + 4.40630e7i) q^{93} +1.03960e8i q^{94} -6.41702e7 q^{96} +(-1.44242e7 - 1.44242e7i) q^{97} +(-8.37278e6 + 8.37278e6i) q^{98} -9.79261e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2268 q^{6}+O(q^{10}) $ 12 * q + 2268 * q^6	$ 12 q + 2268 q^{6} - 88920 q^{11} - 485796 q^{16} + 79704 q^{21} - 4055976 q^{26} + 5658696 q^{31} + 5065092 q^{36} + 1798056 q^{41} - 10882464 q^{46} + 4329936 q^{51} + 7268040 q^{56} + 33649848 q^{61} - 141361848 q^{66} + 335506464 q^{71} - 395386536 q^{76} - 57395628 q^{81} + 489958560 q^{86} - 216875664 q^{91} - 710311356 q^{96}+O(q^{100}) $ 12 * q + 2268 * q^6 - 88920 * q^11 - 485796 * q^16 + 79704 * q^21 - 4055976 * q^26 + 5658696 * q^31 + 5065092 * q^36 + 1798056 * q^41 - 10882464 * q^46 + 4329936 * q^51 + 7268040 * q^56 + 33649848 * q^61 - 141361848 * q^66 + 335506464 * q^71 - 395386536 * q^76 - 57395628 * q^81 + 489958560 * q^86 - 216875664 * q^91 - 710311356 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	15.2792	+	15.2792i	0.954951	+	0.954951i	0.999028	−	0.0440768i	$-0.0140346\pi$
$2$	15.2792	+	15.2792i	0.954951	+	0.954951i	−0.0440768	+	0.999028i	$0.514035\pi$
$3$	−33.0681	+	33.0681i	−0.408248	+	0.408248i
$3$	−33.0681	+	33.0681i	−0.408248	+	0.408248i
$4$			210.909i			0.823864i
$5$		0			0
$5$		0			0
$6$	−1010.51			−0.779715
$7$	−1776.62	−	1776.62i	−0.739952	−	0.739952i	0.232617	−	0.972568i	$-0.425271\pi$
$7$	−1776.62	−	1776.62i	−0.739952	−	0.739952i	−0.972568	+	0.232617i	$0.925271\pi$
$8$	688.952	−	688.952i	0.168201	−	0.168201i
$9$		−	2187.00i		−	0.333333i
$10$		0			0
$11$	4477.64			0.305829			0.152915	−	0.988239i	$-0.451134\pi$
$11$	4477.64			0.305829			0.152915	+	0.988239i	$0.451134\pi$
$12$	−6974.37	−	6974.37i	−0.336341	−	0.336341i
$13$	18402.3	−	18402.3i	0.644314	−	0.644314i	−0.307299	−	0.951613i	$-0.599425\pi$
$13$	18402.3	−	18402.3i	0.644314	−	0.644314i	0.951613	+	0.307299i	$0.0994251\pi$
$14$		−	54290.9i		−	1.41324i
$15$		0			0
$16$	75046.1			1.14511
$17$	40628.0	+	40628.0i	0.486441	+	0.486441i	0.907181	−	0.420740i	$-0.138230\pi$
$17$	40628.0	+	40628.0i	0.486441	+	0.486441i	−0.420740	+	0.907181i	$0.638230\pi$
$18$	33415.7	−	33415.7i	0.318317	−	0.318317i
$19$		−	161124.i		−	1.23636i	−0.786036	−	0.618181i	$-0.787870\pi$
$19$		−	161124.i		−	1.23636i	0.786036	−	0.618181i	$-0.212130\pi$
$20$		0			0
$21$	117499.			0.604168
$22$	68414.9	+	68414.9i	0.292052	+	0.292052i
$23$	−208665.	+	208665.i	−0.745656	+	0.745656i	−0.973660	−	0.228004i	$-0.926780\pi$
$23$	−208665.	+	208665.i	−0.745656	+	0.745656i	0.228004	+	0.973660i	$0.426780\pi$
$24$			45564.7i			0.137336i
$25$		0			0
$26$	562344.			1.23058
$27$	72320.0	+	72320.0i	0.136083	+	0.136083i
$28$	374706.	−	374706.i	0.609620	−	0.609620i
$29$		−	1.08625e6i		−	1.53581i	−0.640561	−	0.767907i	$-0.721298\pi$
$29$		−	1.08625e6i		−	1.53581i	0.640561	−	0.767907i	$-0.278702\pi$
$30$		0			0
$31$	1.33249e6			1.44284			0.721420	−	0.692498i	$-0.243490\pi$
$31$	1.33249e6			1.44284			0.721420	+	0.692498i	$0.243490\pi$
$32$	970274.	+	970274.i	0.925325	+	0.925325i
$33$	−148067.	+	148067.i	−0.124854	+	0.124854i
$34$			1.24153e6i			0.929055i
$35$		0			0
$36$	461259.			0.274621
$37$	778701.	+	778701.i	0.415493	+	0.415493i	0.883647	−	0.468154i	$-0.155081\pi$
$37$	778701.	+	778701.i	0.415493	+	0.415493i	−0.468154	+	0.883647i	$0.655081\pi$
$38$	2.46185e6	−	2.46185e6i	1.18066	−	1.18066i
$39$			1.21706e6i			0.526080i
$40$		0			0
$41$	−3.79519e6			−1.34307			−0.671535	−	0.740973i	$-0.734365\pi$
$41$	−3.79519e6			−1.34307			−0.671535	+	0.740973i	$0.734365\pi$
$42$	1.79530e6	+	1.79530e6i	0.576951	+	0.576951i
$43$	3.70334e6	−	3.70334e6i	1.08323	−	1.08323i	0.0870210	−	0.996206i	$-0.472265\pi$
$43$	3.70334e6	−	3.70334e6i	1.08323	−	1.08323i	0.996206	−	0.0870210i	$-0.0277347\pi$
$44$			944377.i			0.251962i
$45$		0			0
$46$	−6.37648e6			−1.42413
$47$	3.40201e6	+	3.40201e6i	0.697179	+	0.697179i	0.963801	−	0.266622i	$-0.0859076\pi$
$47$	3.40201e6	+	3.40201e6i	0.697179	+	0.697179i	−0.266622	+	0.963801i	$0.585908\pi$
$48$	−2.48163e6	+	2.48163e6i	−0.467490	+	0.467490i
$49$			547985.i			0.0950570i
$50$		0			0
$51$	−2.68698e6			−0.397177
$52$	3.88121e6	+	3.88121e6i	0.530827	+	0.530827i
$53$	−3.98610e6	+	3.98610e6i	−0.505178	+	0.505178i	−0.913042	−	0.407865i	$-0.866274\pi$
$53$	−3.98610e6	+	3.98610e6i	−0.505178	+	0.505178i	0.407865	+	0.913042i	$0.366274\pi$
$54$			2.20999e6i			0.259905i
$55$		0			0
$56$	−2.44802e6			−0.248921
$57$	5.32806e6	+	5.32806e6i	0.504742	+	0.504742i
$58$	1.65971e7	−	1.65971e7i	1.46663	−	1.46663i
$59$		−	4.53528e6i		−	0.374279i	−0.982333	−	0.187140i	$-0.940078\pi$
$59$		−	4.53528e6i		−	0.374279i	0.982333	−	0.187140i	$-0.0599217\pi$
$60$		0			0
$61$	−4.88819e6			−0.353044			−0.176522	−	0.984297i	$-0.556485\pi$
$61$	−4.88819e6			−0.353044			−0.176522	+	0.984297i	$0.556485\pi$
$62$	2.03595e7	+	2.03595e7i	1.37784	+	1.37784i
$63$	−3.88548e6	+	3.88548e6i	−0.246651	+	0.246651i
$64$			1.04383e7i			0.622169i
$65$		0			0
$66$	−4.52471e6			−0.238459
$67$	−1.35112e7	−	1.35112e7i	−0.670492	−	0.670492i	0.287337	−	0.957829i	$-0.407230\pi$
$67$	−1.35112e7	−	1.35112e7i	−0.670492	−	0.670492i	−0.957829	+	0.287337i	$0.907230\pi$
$68$	−8.56883e6	+	8.56883e6i	−0.400761	+	0.400761i
$69$		−	1.38003e7i		−	0.608826i
$70$		0			0
$71$	4.27542e7			1.68246			0.841232	−	0.540675i	$-0.181831\pi$
$71$	4.27542e7			1.68246			0.841232	+	0.540675i	$0.181831\pi$
$72$	−1.50674e6	−	1.50674e6i	−0.0560670	−	0.0560670i
$73$	3.74805e7	−	3.74805e7i	1.31982	−	1.31982i	0.405902	−	0.913917i	$-0.366957\pi$
$73$	3.74805e7	−	3.74805e7i	1.31982	−	1.31982i	0.913917	−	0.405902i	$-0.133043\pi$
$74$			2.37959e7i			0.793552i
$75$		0			0
$76$	3.39825e7			1.01859
$77$	−7.95509e6	−	7.95509e6i	−0.226299	−	0.226299i
$78$	−1.85957e7	+	1.85957e7i	−0.502381	+	0.502381i
$79$		−	6.47257e7i		−	1.66176i	−0.556451	−	0.830880i	$-0.687837\pi$
$79$		−	6.47257e7i		−	1.66176i	0.556451	−	0.830880i	$-0.312163\pi$
$80$		0			0
$81$	−4.78297e6			−0.111111
$82$	−5.79876e7	−	5.79876e7i	−1.28257	−	1.28257i
$83$	−4.04525e7	+	4.04525e7i	−0.852379	+	0.852379i	−0.990426	−	0.138047i	$-0.955917\pi$
$83$	−4.04525e7	+	4.04525e7i	−0.852379	+	0.852379i	0.138047	+	0.990426i	$0.455917\pi$
$84$			2.47817e7i			0.497752i
$85$		0			0
$86$	1.13168e8			2.06886
$87$	3.59203e7	+	3.59203e7i	0.626993	+	0.626993i
$88$	3.08488e6	−	3.08488e6i	0.0514408	−	0.0514408i
$89$		−	9.02203e7i		−	1.43795i	−0.695035	−	0.718976i	$-0.744611\pi$
$89$		−	9.02203e7i		−	1.43795i	0.695035	−	0.718976i	$-0.255389\pi$
$90$		0			0
$91$	−6.53878e7			−0.953523
$92$	−4.40094e7	−	4.40094e7i	−0.614320	−	0.614320i
$93$	−4.40630e7	+	4.40630e7i	−0.589037	+	0.589037i
$94$			1.03960e8i			1.33154i
$95$		0			0
$96$	−6.41702e7			−0.755525
$97$	−1.44242e7	−	1.44242e7i	−0.162932	−	0.162932i	0.620932	−	0.783864i	$-0.286754\pi$
$97$	−1.44242e7	−	1.44242e7i	−0.162932	−	0.162932i	−0.783864	+	0.620932i	$0.786754\pi$
$98$	−8.37278e6	+	8.37278e6i	−0.0907748	+	0.0907748i
$99$		−	9.79261e6i		−	0.101943i
$100$		0			0
$101$	−1.42466e8			−1.36907			−0.684535	−	0.728980i	$-0.739995\pi$
$101$	−1.42466e8			−1.36907			−0.684535	+	0.728980i	$0.739995\pi$
$102$	−4.10550e7	−	4.10550e7i	−0.379285	−	0.379285i
$103$	−7.31307e7	+	7.31307e7i	−0.649756	+	0.649756i	−0.952934	−	0.303178i	$-0.901952\pi$
$103$	−7.31307e7	+	7.31307e7i	−0.649756	+	0.649756i	0.303178	+	0.952934i	$0.401952\pi$
$104$		−	2.53565e7i		−	0.216749i
$105$		0			0
$106$	−1.21809e8			−0.964840
$107$	−5.10438e7	−	5.10438e7i	−0.389411	−	0.389411i	0.485067	−	0.874477i	$-0.338795\pi$
$107$	−5.10438e7	−	5.10438e7i	−0.389411	−	0.389411i	−0.874477	+	0.485067i	$0.838795\pi$
$108$	−1.52529e7	+	1.52529e7i	−0.112114	+	0.112114i
$109$			5.91324e7i			0.418909i	0.977818	+	0.209454i	$0.0671688\pi$
$109$			5.91324e7i			0.418909i	−0.977818	+	0.209454i	$0.932831\pi$
$110$		0			0
$111$	−5.15004e7			−0.339249
$112$	−1.33329e8	−	1.33329e8i	−0.847328	−	0.847328i
$113$	−5.36434e7	+	5.36434e7i	−0.329005	+	0.329005i	−0.852208	−	0.523203i	$-0.824737\pi$
$113$	−5.36434e7	+	5.36434e7i	−0.329005	+	0.329005i	0.523203	+	0.852208i	$0.324737\pi$
$114$			1.62817e8i			0.964009i
$115$		0			0
$116$	2.29101e8			1.26530
$117$	−4.02457e7	−	4.02457e7i	−0.214771	−	0.214771i
$118$	6.92955e7	−	6.92955e7i	0.357419	−	0.357419i
$119$		−	1.44361e8i		−	0.719886i
$120$		0			0
$121$	−1.94310e8			−0.906469
$122$	−7.46877e7	−	7.46877e7i	−0.337140	−	0.337140i
$123$	1.25500e8	−	1.25500e8i	0.548306	−	0.548306i
$124$			2.81035e8i			1.18870i
$125$		0			0
$126$	−1.18734e8			−0.471079
$127$	1.38129e8	+	1.38129e8i	0.530969	+	0.530969i	0.920861	−	0.389892i	$-0.127488\pi$
$127$	1.38129e8	+	1.38129e8i	0.530969	+	0.530969i	−0.389892	+	0.920861i	$0.627488\pi$
$128$	8.89015e7	−	8.89015e7i	0.331184	−	0.331184i
$129$			2.44925e8i			0.884452i
$130$		0			0
$131$	2.43973e8			0.828432			0.414216	−	0.910179i	$-0.364056\pi$
$131$	2.43973e8			0.828432			0.414216	+	0.910179i	$0.364056\pi$
$132$	−3.12288e7	−	3.12288e7i	−0.102863	−	0.102863i
$133$	−2.86256e8	+	2.86256e8i	−0.914848	+	0.914848i
$134$		−	4.12880e8i		−	1.28058i
$135$		0			0
$136$	5.59815e7			0.163640
$137$	2.63122e8	+	2.63122e8i	0.746920	+	0.746920i	0.973900	−	0.226979i	$-0.0728850\pi$
$137$	2.63122e8	+	2.63122e8i	0.746920	+	0.746920i	−0.226979	+	0.973900i	$0.572885\pi$
$138$	2.10858e8	−	2.10858e8i	0.581399	−	0.581399i
$139$		−	1.08062e8i		−	0.289477i	−0.989470	−	0.144739i	$-0.953766\pi$
$139$		−	1.08062e8i		−	0.289477i	0.989470	−	0.144739i	$-0.0462341\pi$
$140$		0			0
$141$	−2.24996e8			−0.569244
$142$	6.53251e8	+	6.53251e8i	1.60667	+	1.60667i
$143$	8.23988e7	−	8.23988e7i	0.197050	−	0.197050i
$144$		−	1.64126e8i		−	0.381704i
$145$		0			0
$146$	1.14535e9			2.52073
$147$	−1.81208e7	−	1.81208e7i	−0.0388069	−	0.0388069i
$148$	−1.64235e8	+	1.64235e8i	−0.342310	+	0.342310i
$149$		−	2.55974e8i		−	0.519340i	−0.965697	−	0.259670i	$-0.916386\pi$
$149$		−	2.55974e8i		−	0.519340i	0.965697	−	0.259670i	$-0.0836138\pi$
$150$		0			0
$151$	3.02108e8			0.581105			0.290553	−	0.956859i	$-0.406161\pi$
$151$	3.02108e8			0.581105			0.290553	+	0.956859i	$0.406161\pi$
$152$	−1.11007e8	−	1.11007e8i	−0.207957	−	0.207957i
$153$	8.88535e7	−	8.88535e7i	0.162147	−	0.162147i
$154$		−	2.43095e8i		−	0.432209i
$155$		0			0
$156$	−2.56688e8			−0.433419
$157$	1.07682e8	+	1.07682e8i	0.177233	+	0.177233i	0.790148	−	0.612916i	$-0.210004\pi$
$157$	1.07682e8	+	1.07682e8i	0.177233	+	0.177233i	−0.612916	+	0.790148i	$0.710004\pi$
$158$	9.88958e8	−	9.88958e8i	1.58690	−	1.58690i
$159$		−	2.63625e8i		−	0.412476i
$160$		0			0
$161$	7.41439e8			1.10350
$162$	−7.30800e7	−	7.30800e7i	−0.106106	−	0.106106i
$163$	−7.20616e8	+	7.20616e8i	−1.02083	+	1.02083i	−0.0210510	+	0.999778i	$0.506701\pi$
$163$	−7.20616e8	+	7.20616e8i	−1.02083	+	1.02083i	−0.999778	+	0.0210510i	$0.993299\pi$
$164$		−	8.00442e8i		−	1.10651i
$165$		0			0
$166$	−1.23616e9			−1.62796
$167$	4.94776e8	+	4.94776e8i	0.636125	+	0.636125i	0.949597	−	0.313472i	$-0.101492\pi$
$167$	4.94776e8	+	4.94776e8i	0.636125	+	0.636125i	−0.313472	+	0.949597i	$0.601492\pi$
$168$	8.09513e7	−	8.09513e7i	0.101622	−	0.101622i
$169$			1.38445e8i			0.169718i
$170$		0			0
$171$	−3.52378e8			−0.412120
$172$	7.81068e8	+	7.81068e8i	0.892432	+	0.892432i
$173$	3.67823e8	−	3.67823e8i	0.410634	−	0.410634i	−0.471325	−	0.881959i	$-0.656224\pi$
$173$	3.67823e8	−	3.67823e8i	0.410634	−	0.410634i	0.881959	+	0.471325i	$0.156224\pi$
$174$			1.09767e9i			1.19750i
$175$		0			0
$176$	3.36030e8			0.350209
$177$	1.49973e8	+	1.49973e8i	0.152799	+	0.152799i
$178$	1.37850e9	−	1.37850e9i	1.37317	−	1.37317i
$179$			1.47156e9i			1.43340i	0.697383	+	0.716699i	$0.254348\pi$
$179$			1.47156e9i			1.43340i	−0.697383	+	0.716699i	$0.745652\pi$
$180$		0			0
$181$	−1.08757e9			−1.01331			−0.506654	−	0.862149i	$-0.669118\pi$
$181$	−1.08757e9			−1.01331			−0.506654	+	0.862149i	$0.669118\pi$
$182$	−9.99074e8	−	9.99074e8i	−0.910568	−	0.910568i
$183$	1.61643e8	−	1.61643e8i	0.144130	−	0.144130i
$184$			2.87521e8i			0.250840i
$185$		0			0
$186$	−1.34650e9			−1.12500
$187$	1.81918e8	+	1.81918e8i	0.148768	+	0.148768i
$188$	−7.17516e8	+	7.17516e8i	−0.574381	+	0.574381i
$189$		−	2.56971e8i		−	0.201389i
$190$		0			0
$191$	4.78708e8			0.359698			0.179849	−	0.983694i	$-0.442439\pi$
$191$	4.78708e8			0.359698			0.179849	+	0.983694i	$0.442439\pi$
$192$	−3.45174e8	−	3.45174e8i	−0.253999	−	0.253999i
$193$	−1.58182e9	+	1.58182e9i	−1.14006	+	1.14006i	−0.151624	+	0.988438i	$0.548450\pi$
$193$	−1.58182e9	+	1.58182e9i	−1.14006	+	1.14006i	−0.988438	+	0.151624i	$0.951550\pi$
$194$		−	4.40782e8i		−	0.311184i
$195$		0			0
$196$	−1.15575e8			−0.0783141
$197$	1.05458e9	+	1.05458e9i	0.700185	+	0.700185i	0.964450	−	0.264265i	$-0.0851293\pi$
$197$	1.05458e9	+	1.05458e9i	0.700185	+	0.700185i	−0.264265	+	0.964450i	$0.585129\pi$
$198$	1.49623e8	−	1.49623e8i	0.0973507	−	0.0973507i
$199$			2.81932e9i			1.79776i	0.438190	+	0.898882i	$0.355620\pi$
$199$			2.81932e9i			1.79776i	−0.438190	+	0.898882i	$0.644380\pi$
$200$		0			0
$201$	8.93578e8			0.547455
$202$	−2.17677e9	−	2.17677e9i	−1.30740	−	1.30740i
$203$	−1.92986e9	+	1.92986e9i	−1.13643	+	1.13643i
$204$		−	5.66710e8i		−	0.327220i
$205$		0			0
$206$	−2.23476e9			−1.24097
$207$	4.56351e8	+	4.56351e8i	0.248552	+	0.248552i
$208$	1.38102e9	−	1.38102e9i	0.737812	−	0.737812i
$209$		−	7.21455e8i		−	0.378115i
$210$		0			0
$211$	1.35002e9			0.681100			0.340550	−	0.940226i	$-0.389387\pi$
$211$	1.35002e9			0.681100			0.340550	+	0.940226i	$0.389387\pi$
$212$	−8.40704e8	−	8.40704e8i	−0.416198	−	0.416198i
$213$	−1.41380e9	+	1.41380e9i	−0.686863	+	0.686863i
$214$		−	1.55982e9i		−	0.743737i
$215$		0			0
$216$	9.96499e7			0.0457785
$217$	−2.36734e9	−	2.36734e9i	−1.06763	−	1.06763i
$218$	−9.03497e8	+	9.03497e8i	−0.400038	+	0.400038i
$219$			2.47882e9i			1.07763i
$220$		0			0
$221$	1.49530e9			0.626842
$222$	−7.86885e8	−	7.86885e8i	−0.323966	−	0.323966i
$223$	9.94179e8	−	9.94179e8i	0.402018	−	0.402018i	−0.476926	−	0.878944i	$-0.658249\pi$
$223$	9.94179e8	−	9.94179e8i	0.402018	−	0.402018i	0.878944	+	0.476926i	$0.158249\pi$
$224$		−	3.44762e9i		−	1.36939i
$225$		0			0
$226$	−1.63926e9			−0.628367
$227$	8.75761e7	+	8.75761e7i	0.0329824	+	0.0329824i	0.723406	−	0.690423i	$-0.242576\pi$
$227$	8.75761e7	+	8.75761e7i	0.0329824	+	0.0329824i	−0.690423	+	0.723406i	$0.742576\pi$
$228$	−1.12374e9	+	1.12374e9i	−0.415839	+	0.415839i
$229$		−	2.09791e9i		−	0.762859i	−0.924398	−	0.381430i	$-0.875432\pi$
$229$		−	2.09791e9i		−	0.762859i	0.924398	−	0.381430i	$-0.124568\pi$
$230$		0			0
$231$	5.26120e8			0.184772
$232$	−7.48375e8	−	7.48375e8i	−0.258326	−	0.258326i
$233$	−2.74740e8	+	2.74740e8i	−0.0932178	+	0.0932178i	−0.752178	−	0.658960i	$-0.770997\pi$
$233$	−2.74740e8	+	2.74740e8i	−0.0932178	+	0.0932178i	0.658960	+	0.752178i	$0.270997\pi$
$234$		−	1.22985e9i		−	0.410192i
$235$		0			0
$236$	9.56532e8			0.308355
$237$	2.14036e9	+	2.14036e9i	0.678411	+	0.678411i
$238$	2.20573e9	−	2.20573e9i	0.687456	−	0.687456i
$239$		−	5.48652e8i		−	0.168153i	−0.996459	−	0.0840767i	$-0.973206\pi$
$239$		−	5.48652e8i		−	0.168153i	0.996459	−	0.0840767i	$-0.0267941\pi$
$240$		0			0
$241$	5.12166e9			1.51825			0.759123	−	0.650947i	$-0.225628\pi$
$241$	5.12166e9			1.51825			0.759123	+	0.650947i	$0.225628\pi$
$242$	−2.96890e9	−	2.96890e9i	−0.865633	−	0.865633i
$243$	1.58164e8	−	1.58164e8i	0.0453609	−	0.0453609i
$244$		−	1.03096e9i		−	0.290860i
$245$		0			0
$246$	3.83508e9			1.04721
$247$	−2.96504e9	−	2.96504e9i	−0.796605	−	0.796605i
$248$	9.18023e8	−	9.18023e8i	0.242687	−	0.242687i
$249$		−	2.67537e9i		−	0.695964i
$250$		0			0
$251$	−5.04785e8			−0.127178			−0.0635889	−	0.997976i	$-0.520255\pi$
$251$	−5.04785e8			−0.127178			−0.0635889	+	0.997976i	$0.520255\pi$
$252$	−8.19483e8	−	8.19483e8i	−0.203207	−	0.203207i
$253$	−9.34329e8	+	9.34329e8i	−0.228043	+	0.228043i
$254$			4.22100e9i			1.01410i
$255$		0			0
$256$	5.38889e9			1.25470
$257$	1.17418e9	+	1.17418e9i	0.269155	+	0.269155i	0.828760	−	0.559605i	$-0.189047\pi$
$257$	1.17418e9	+	1.17418e9i	0.269155	+	0.269155i	−0.559605	+	0.828760i	$0.689047\pi$
$258$	−3.74226e9	+	3.74226e9i	−0.844608	+	0.844608i
$259$		−	2.76692e9i		−	0.614890i
$260$		0			0
$261$	−2.37563e9			−0.511938
$262$	3.72772e9	+	3.72772e9i	0.791112	+	0.791112i
$263$	−3.37767e9	+	3.37767e9i	−0.705982	+	0.705982i	−0.965688	−	0.259706i	$-0.916374\pi$
$263$	−3.37767e9	+	3.37767e9i	−0.705982	+	0.705982i	0.259706	+	0.965688i	$0.416374\pi$
$264$			2.04022e8i			0.0420012i
$265$		0			0
$266$	−8.74755e9			−1.74727
$267$	2.98341e9	+	2.98341e9i	0.587041	+	0.587041i
$268$	2.84963e9	−	2.84963e9i	0.552395	−	0.552395i
$269$			5.72780e9i			1.09390i	0.837165	+	0.546951i	$0.184212\pi$
$269$			5.72780e9i			1.09390i	−0.837165	+	0.546951i	$0.815788\pi$
$270$		0			0
$271$	−6.27185e9			−1.16284			−0.581418	−	0.813605i	$-0.697502\pi$
$271$	−6.27185e9			−1.16284			−0.581418	+	0.813605i	$0.697502\pi$
$272$	3.04897e9	+	3.04897e9i	0.557029	+	0.557029i
$273$	2.16225e9	−	2.16225e9i	0.389274	−	0.389274i
$274$			8.04059e9i			1.42654i
$275$		0			0
$276$	2.91062e9			0.501590
$277$	−3.19244e8	−	3.19244e8i	−0.0542254	−	0.0542254i	0.679474	−	0.733700i	$-0.262208\pi$
$277$	−3.19244e8	−	3.19244e8i	−0.0542254	−	0.0542254i	−0.733700	+	0.679474i	$0.762208\pi$
$278$	1.65111e9	−	1.65111e9i	0.276437	−	0.276437i
$279$		−	2.91416e9i		−	0.480947i
$280$		0			0
$281$	−8.35532e9			−1.34010			−0.670051	−	0.742315i	$-0.733728\pi$
$281$	−8.35532e9			−1.34010			−0.670051	+	0.742315i	$0.733728\pi$
$282$	−3.43777e9	−	3.43777e9i	−0.543601	−	0.543601i
$283$	−1.34940e7	+	1.34940e7i	−0.00210375	+	0.00210375i	−0.708158	−	0.706054i	$-0.750474\pi$
$283$	−1.34940e7	+	1.34940e7i	−0.00210375	+	0.00210375i	0.706054	+	0.708158i	$0.250474\pi$
$284$			9.01726e9i			1.38612i
$285$		0			0
$286$	2.51798e9			0.376346
$287$	6.74263e9	+	6.74263e9i	0.993807	+	0.993807i
$288$	2.12199e9	−	2.12199e9i	0.308442	−	0.308442i
$289$		−	3.67448e9i		−	0.526750i
$290$		0			0
$291$	9.53964e8			0.133033
$292$	7.90499e9	+	7.90499e9i	1.08735	+	1.08735i
$293$	5.13714e9	−	5.13714e9i	0.697029	−	0.697029i	−0.266740	−	0.963769i	$-0.585946\pi$
$293$	5.13714e9	−	5.13714e9i	0.697029	−	0.697029i	0.963769	+	0.266740i	$0.0859464\pi$
$294$		−	5.53744e8i		−	0.0741173i
$295$		0			0
$296$	1.07298e9			0.139773
$297$	3.23823e8	+	3.23823e8i	0.0416181	+	0.0416181i
$298$	3.91109e9	−	3.91109e9i	0.495944	−	0.495944i
$299$			7.67982e9i			0.960874i
$300$		0			0
$301$	−1.31589e10			−1.60307
$302$	4.61598e9	+	4.61598e9i	0.554927	+	0.554927i
$303$	4.71108e9	−	4.71108e9i	0.558920	−	0.558920i
$304$		−	1.20917e10i		−	1.41577i
$305$		0			0
$306$	2.71523e9			0.309685
$307$	−2.16091e9	−	2.16091e9i	−0.243267	−	0.243267i	0.574933	−	0.818200i	$-0.305028\pi$
$307$	−2.16091e9	−	2.16091e9i	−0.243267	−	0.243267i	−0.818200	+	0.574933i	$0.805028\pi$
$308$	1.67780e9	−	1.67780e9i	0.186439	−	0.186439i
$309$		−	4.83659e9i		−	0.530524i
$310$		0			0
$311$	−6.44045e9			−0.688454			−0.344227	−	0.938887i	$-0.611859\pi$
$311$	−6.44045e9			−0.688454			−0.344227	+	0.938887i	$0.611859\pi$
$312$	8.38493e8	+	8.38493e8i	0.0884873	+	0.0884873i
$313$	−1.04720e10	+	1.04720e10i	−1.09107	+	1.09107i	−0.0956506	+	0.995415i	$0.530493\pi$
$313$	−1.04720e10	+	1.04720e10i	−1.09107	+	1.09107i	−0.995415	+	0.0956506i	$0.969507\pi$
$314$			3.29059e9i			0.338497i
$315$		0			0
$316$	1.36513e10			1.36907
$317$	−1.01318e10	−	1.01318e10i	−1.00334	−	1.00334i	−0.999994	−	0.00334628i	$-0.998935\pi$
$317$	−1.01318e10	−	1.01318e10i	−1.00334	−	1.00334i	−0.00334628	−	0.999994i	$-0.501065\pi$
$318$	4.02799e9	−	4.02799e9i	0.393894	−	0.393894i
$319$		−	4.86385e9i		−	0.469697i
$320$		0			0
$321$	3.37584e9			0.317953
$322$	1.13286e10	+	1.13286e10i	1.05379	+	1.05379i
$323$	6.54615e9	−	6.54615e9i	0.601417	−	0.601417i
$324$		−	1.00877e9i		−	0.0915405i
$325$		0			0
$326$	−2.20209e10			−1.94968
$327$	−1.95540e9	−	1.95540e9i	−0.171019	−	0.171019i
$328$	−2.61471e9	+	2.61471e9i	−0.225906	+	0.225906i
$329$		−	1.20882e10i		−	1.03176i
$330$		0			0
$331$	2.15436e10			1.79476			0.897378	−	0.441262i	$-0.145469\pi$
$331$	2.15436e10			1.79476			0.897378	+	0.441262i	$0.145469\pi$
$332$	−8.53180e9	−	8.53180e9i	−0.702244	−	0.702244i
$333$	1.70302e9	−	1.70302e9i	0.138498	−	0.138498i
$334$			1.51196e10i			1.21494i
$335$		0			0
$336$	8.81785e9			0.691840
$337$	4.14231e9	+	4.14231e9i	0.321161	+	0.321161i	0.849213	−	0.528051i	$-0.177077\pi$
$337$	4.14231e9	+	4.14231e9i	0.321161	+	0.321161i	−0.528051	+	0.849213i	$0.677077\pi$
$338$	−2.11533e9	+	2.11533e9i	−0.162073	+	0.162073i
$339$		−	3.54777e9i		−	0.268631i
$340$		0			0
$341$	5.96643e9			0.441262
$342$	−5.38406e9	−	5.38406e9i	−0.393555	−	0.393555i
$343$	−9.26832e9	+	9.26832e9i	−0.669614	+	0.669614i
$344$		−	5.10284e9i		−	0.364400i
$345$		0			0
$346$	1.12401e10			0.784271
$347$	1.35085e10	+	1.35085e10i	0.931730	+	0.931730i	0.997814	−	0.0660836i	$-0.0210504\pi$
$347$	1.35085e10	+	1.35085e10i	0.931730	+	0.931730i	−0.0660836	+	0.997814i	$0.521050\pi$
$348$	−7.57592e9	+	7.57592e9i	−0.516557	+	0.516557i
$349$		−	7.36977e9i		−	0.496766i	−0.968662	−	0.248383i	$-0.920101\pi$
$349$		−	7.36977e9i		−	0.496766i	0.968662	−	0.248383i	$-0.0798992\pi$
$350$		0			0
$351$	2.66170e9			0.175360
$352$	4.34454e9	+	4.34454e9i	0.282991	+	0.282991i
$353$	1.28656e10	−	1.28656e10i	0.828576	−	0.828576i	−0.158743	−	0.987320i	$-0.550744\pi$
$353$	1.28656e10	−	1.28656e10i	0.828576	−	0.828576i	0.987320	+	0.158743i	$0.0507443\pi$
$354$			4.58295e9i			0.291831i
$355$		0			0
$356$	1.90283e10			1.18468
$357$	4.77376e9	+	4.77376e9i	0.293892	+	0.293892i
$358$	−2.24843e10	+	2.24843e10i	−1.36882	+	1.36882i
$359$		−	2.68477e9i		−	0.161633i	−0.996729	−	0.0808165i	$-0.974247\pi$
$359$		−	2.68477e9i		−	0.161633i	0.996729	−	0.0808165i	$-0.0257528\pi$
$360$		0			0
$361$	−8.97733e9			−0.528589
$362$	−1.66172e10	−	1.66172e10i	−0.967660	−	0.967660i
$363$	6.42545e9	−	6.42545e9i	0.370064	−	0.370064i
$364$		−	1.37909e10i		−	0.785573i
$365$		0			0
$366$	4.93957e9			0.275273
$367$	−2.10778e10	−	2.10778e10i	−1.16188	−	1.16188i	−0.984064	−	0.177817i	$-0.943096\pi$
$367$	−2.10778e10	−	2.10778e10i	−1.16188	−	1.16188i	−0.177817	−	0.984064i	$-0.556904\pi$
$368$	−1.56595e10	+	1.56595e10i	−0.853860	+	0.853860i
$369$			8.30009e9i			0.447690i
$370$		0			0
$371$	1.41636e10			0.747614
$372$	−9.29330e9	−	9.29330e9i	−0.485286	−	0.485286i
$373$	2.95974e9	−	2.95974e9i	0.152904	−	0.152904i	−0.626510	−	0.779414i	$-0.715517\pi$
$373$	2.95974e9	−	2.95974e9i	0.152904	−	0.152904i	0.779414	+	0.626510i	$0.215517\pi$
$374$			5.55913e9i			0.284132i
$375$		0			0
$376$	4.68764e9			0.234532
$377$	−1.99895e10	−	1.99895e10i	−0.989547	−	0.989547i
$378$	3.92631e9	−	3.92631e9i	0.192317	−	0.192317i
$379$			8.01600e9i			0.388509i	0.980951	+	0.194254i	$0.0622287\pi$
$379$			8.01600e9i			0.388509i	−0.980951	+	0.194254i	$0.937771\pi$
$380$		0			0
$381$	−9.13531e9			−0.433534
$382$	7.31429e9	+	7.31429e9i	0.343494	+	0.343494i
$383$	1.37328e10	−	1.37328e10i	0.638210	−	0.638210i	−0.311904	−	0.950114i	$-0.600967\pi$
$383$	1.37328e10	−	1.37328e10i	0.638210	−	0.638210i	0.950114	+	0.311904i	$0.100967\pi$
$384$			5.87961e9i			0.270411i
$385$		0			0
$386$	−4.83380e10			−2.17741
$387$	−8.09920e9	−	8.09920e9i	−0.361076	−	0.361076i
$388$	3.04220e9	−	3.04220e9i	0.134234	−	0.134234i
$389$			1.23902e10i			0.541102i	0.962706	+	0.270551i	$0.0872059\pi$
$389$			1.23902e10i			0.541102i	−0.962706	+	0.270551i	$0.912794\pi$
$390$		0			0
$391$	−1.69553e10			−0.725436
$392$	3.77535e8	+	3.77535e8i	0.0159887	+	0.0159887i
$393$	−8.06773e9	+	8.06773e9i	−0.338206	+	0.338206i
$394$			3.22262e10i			1.33729i
$395$		0			0
$396$	2.06535e9			0.0839872
$397$	−3.17968e10	−	3.17968e10i	−1.28003	−	1.28003i	−0.940647	−	0.339385i	$-0.889781\pi$
$397$	−3.17968e10	−	3.17968e10i	−1.28003	−	1.28003i	−0.339385	−	0.940647i	$-0.610219\pi$
$398$	−4.30771e10	+	4.30771e10i	−1.71678	+	1.71678i
$399$		−	1.89319e10i		−	0.746970i
$400$		0			0
$401$	−1.45051e10			−0.560976			−0.280488	−	0.959857i	$-0.590496\pi$
$401$	−1.45051e10			−0.560976			−0.280488	+	0.959857i	$0.590496\pi$
$402$	1.36532e10	+	1.36532e10i	0.522793	+	0.522793i
$403$	2.45209e10	−	2.45209e10i	0.929642	−	0.929642i
$404$		−	3.00474e10i		−	1.12793i
$405$		0			0
$406$	−5.89736e10			−2.17047
$407$	3.48675e9	+	3.48675e9i	0.127070	+	0.127070i
$408$	−1.85120e9	+	1.85120e9i	−0.0668057	+	0.0668057i
$409$		−	4.43639e9i		−	0.158539i	−0.996853	−	0.0792696i	$-0.974741\pi$
$409$		−	4.43639e9i		−	0.158539i	0.996853	−	0.0792696i	$-0.0252588\pi$
$410$		0			0
$411$	−1.74019e10			−0.609858
$412$	−1.54239e10	−	1.54239e10i	−0.535311	−	0.535311i
$413$	−8.05749e9	+	8.05749e9i	−0.276949	+	0.276949i
$414$			1.39454e10i			0.474710i
$415$		0			0
$416$	3.57105e10			1.19240
$417$	3.57341e9	+	3.57341e9i	0.118179	+	0.118179i
$418$	1.10233e10	−	1.10233e10i	0.361082	−	0.361082i
$419$			4.69093e10i			1.52196i	0.648777	+	0.760979i	$0.275281\pi$
$419$			4.69093e10i			1.52196i	−0.648777	+	0.760979i	$0.724719\pi$
$420$		0			0
$421$	2.56171e10			0.815457			0.407728	−	0.913103i	$-0.366321\pi$
$421$	2.56171e10			0.815457			0.407728	+	0.913103i	$0.366321\pi$
$422$	2.06273e10	+	2.06273e10i	0.650417	+	0.650417i
$423$	7.44020e9	−	7.44020e9i	0.232393	−	0.232393i
$424$			5.49245e9i			0.169943i
$425$		0			0
$426$	−4.32036e10			−1.31184
$427$	8.68448e9	+	8.68448e9i	0.261235	+	0.261235i
$428$	1.07656e10	−	1.07656e10i	0.320822	−	0.320822i
$429$			5.44954e9i			0.160891i
$430$		0			0
$431$	5.60467e10			1.62420			0.812102	−	0.583515i	$-0.198323\pi$
$431$	5.60467e10			1.62420			0.812102	+	0.583515i	$0.198323\pi$
$432$	5.42733e9	+	5.42733e9i	0.155830	+	0.155830i
$433$	4.29249e9	−	4.29249e9i	0.122112	−	0.122112i	−0.643410	−	0.765522i	$-0.722481\pi$
$433$	4.29249e9	−	4.29249e9i	0.122112	−	0.122112i	0.765522	+	0.643410i	$0.222481\pi$
$434$		−	7.23422e10i		−	2.03907i
$435$		0			0
$436$	−1.24716e10			−0.345124
$437$	3.36209e10	+	3.36209e10i	0.921901	+	0.921901i
$438$	−3.78744e10	+	3.78744e10i	−1.02908	+	1.02908i
$439$		−	3.37587e10i		−	0.908925i	−0.890766	−	0.454463i	$-0.849831\pi$
$439$		−	3.37587e10i		−	0.908925i	0.890766	−	0.454463i	$-0.150169\pi$
$440$		0			0
$441$	1.19844e9			0.0316857
$442$	2.28469e10	+	2.28469e10i	0.598603	+	0.598603i
$443$	−1.41248e10	+	1.41248e10i	−0.366749	+	0.366749i	−0.866290	−	0.499541i	$-0.833502\pi$
$443$	−1.41248e10	+	1.41248e10i	−0.366749	+	0.366749i	0.499541	+	0.866290i	$0.333502\pi$
$444$		−	1.08619e10i		−	0.279495i
$445$		0			0
$446$	3.03806e10			0.767815
$447$	8.46459e9	+	8.46459e9i	0.212020	+	0.212020i
$448$	1.85449e10	−	1.85449e10i	0.460375	−	0.460375i
$449$			7.48403e10i			1.84141i	0.390260	+	0.920705i	$0.372385\pi$
$449$			7.48403e10i			1.84141i	−0.390260	+	0.920705i	$0.627615\pi$
$450$		0			0
$451$	−1.69935e10			−0.410750
$452$	−1.13139e10	−	1.13139e10i	−0.271055	−	0.271055i
$453$	−9.99015e9	+	9.99015e9i	−0.237235	+	0.237235i
$454$			2.67619e9i			0.0629932i
$455$		0			0
$456$	7.34155e9			0.169796
$457$	−1.71094e10	−	1.71094e10i	−0.392256	−	0.392256i	0.483235	−	0.875491i	$-0.339462\pi$
$457$	−1.71094e10	−	1.71094e10i	−0.392256	−	0.392256i	−0.875491	+	0.483235i	$0.839462\pi$
$458$	3.20544e10	−	3.20544e10i	0.728493	−	0.728493i
$459$			5.87644e9i			0.132392i
$460$		0			0
$461$	8.67176e10			1.92001			0.960006	−	0.279981i	$-0.0903281\pi$
$461$	8.67176e10			1.92001			0.960006	+	0.279981i	$0.0903281\pi$
$462$	8.03870e9	+	8.03870e9i	0.176448	+	0.176448i
$463$	5.15336e10	−	5.15336e10i	1.12142	−	1.12142i	0.129888	−	0.991529i	$-0.458538\pi$
$463$	5.15336e10	−	5.15336e10i	1.12142	−	1.12142i	0.991529	−	0.129888i	$-0.0414617\pi$
$464$		−	8.15189e10i		−	1.75868i
$465$		0			0
$466$	−8.39564e9			−0.178037
$467$	−6.73199e9	−	6.73199e9i	−0.141539	−	0.141539i	0.632787	−	0.774326i	$-0.281911\pi$
$467$	−6.73199e9	−	6.73199e9i	−0.141539	−	0.141539i	−0.774326	+	0.632787i	$0.781911\pi$
$468$	8.48820e9	−	8.48820e9i	0.176942	−	0.176942i
$469$			4.80085e10i			0.992264i
$470$		0			0
$471$	−7.12166e9			−0.144710
$472$	−3.12459e9	−	3.12459e9i	−0.0629542	−	0.0629542i
$473$	1.65822e10	−	1.65822e10i	0.331283	−	0.331283i
$474$			6.54060e10i			1.29570i
$475$		0			0
$476$	3.04472e10			0.593088
$477$	8.71759e9	+	8.71759e9i	0.168393	+	0.168393i
$478$	8.38298e9	−	8.38298e9i	0.160578	−	0.160578i
$479$		−	2.02194e10i		−	0.384084i	−0.981387	−	0.192042i	$-0.938489\pi$
$479$		−	2.02194e10i		−	0.384084i	0.981387	−	0.192042i	$-0.0615111\pi$
$480$		0			0
$481$	2.86597e10			0.535416
$482$	7.82549e10	+	7.82549e10i	1.44985	+	1.44985i
$483$	−2.45180e10	+	2.45180e10i	−0.450502	+	0.450502i
$484$		−	4.09817e10i		−	0.746807i
$485$		0			0
$486$	4.83324e9			0.0866349
$487$	5.31073e10	+	5.31073e10i	0.944145	+	0.944145i	0.998521	−	0.0543755i	$-0.0173168\pi$
$487$	5.31073e10	+	5.31073e10i	0.944145	+	0.944145i	−0.0543755	+	0.998521i	$0.517317\pi$
$488$	−3.36773e9	+	3.36773e9i	−0.0593824	+	0.0593824i
$489$		−	4.76588e10i		−	0.833504i
$490$		0			0
$491$	2.28584e10			0.393296			0.196648	−	0.980474i	$-0.436994\pi$
$491$	2.28584e10			0.393296			0.196648	+	0.980474i	$0.436994\pi$
$492$	2.64691e10	+	2.64691e10i	0.451730	+	0.451730i
$493$	4.41323e10	−	4.41323e10i	0.747083	−	0.747083i
$494$		−	9.06071e10i		−	1.52144i
$495$		0			0
$496$	9.99983e10			1.65221
$497$	−7.59582e10	−	7.59582e10i	−1.24494	−	1.24494i
$498$	4.08776e10	−	4.08776e10i	0.664612	−	0.664612i
$499$			5.00108e10i			0.806606i	0.915066	+	0.403303i	$0.132138\pi$
$499$			5.00108e10i			0.806606i	−0.915066	+	0.403303i	$0.867862\pi$
$500$		0			0
$501$	−3.27226e10			−0.519394
$502$	−7.71272e9	−	7.71272e9i	−0.121449	−	0.121449i
$503$	−1.48953e10	+	1.48953e10i	−0.232689	+	0.232689i	−0.813814	−	0.581125i	$-0.802613\pi$
$503$	−1.48953e10	+	1.48953e10i	−0.232689	+	0.232689i	0.581125	+	0.813814i	$0.302613\pi$
$504$			5.35381e9i			0.0829738i
$505$		0			0
$506$	−2.85516e10			−0.435541
$507$	−4.57810e9	−	4.57810e9i	−0.0692873	−	0.0692873i
$508$	−2.91326e10	+	2.91326e10i	−0.437446	+	0.437446i
$509$		−	5.21786e10i		−	0.777358i	−0.921373	−	0.388679i	$-0.872932\pi$
$509$		−	5.21786e10i		−	0.777358i	0.921373	−	0.388679i	$-0.127068\pi$
$510$		0			0
$511$	−1.33178e11			−1.95320
$512$	5.95792e10	+	5.95792e10i	0.866992	+	0.866992i
$513$	1.16525e10	−	1.16525e10i	0.168247	−	0.168247i
$514$			3.58812e10i			0.514060i
$515$		0			0
$516$	−5.16569e10			−0.728668
$517$	1.52330e10	+	1.52330e10i	0.213218	+	0.213218i
$518$	4.22764e10	−	4.22764e10i	0.587190	−	0.587190i
$519$			2.43264e10i			0.335281i
$520$		0			0
$521$	9.76185e9			0.132489			0.0662447	−	0.997803i	$-0.478898\pi$
$521$	9.76185e9			0.132489			0.0662447	+	0.997803i	$0.478898\pi$
$522$	−3.62978e10	−	3.62978e10i	−0.488876	−	0.488876i
$523$	−3.56960e10	+	3.56960e10i	−0.477104	+	0.477104i	−0.904204	−	0.427100i	$-0.859535\pi$
$523$	−3.56960e10	+	3.56960e10i	−0.477104	+	0.477104i	0.427100	+	0.904204i	$0.359535\pi$
$524$			5.14562e10i			0.682515i
$525$		0			0
$526$	−1.03216e11			−1.34836
$527$	5.41366e10	+	5.41366e10i	0.701856	+	0.701856i
$528$	−1.11119e10	+	1.11119e10i	−0.142972	+	0.142972i
$529$		−	8.77137e9i		−	0.112007i
$530$		0			0
$531$	−9.91866e9			−0.124760
$532$	−6.03741e10	−	6.03741e10i	−0.753710	−	0.753710i
$533$	−6.98402e10	+	6.98402e10i	−0.865359	+	0.865359i
$534$			9.11685e10i			1.12119i
$535$		0			0
$536$	−1.86171e10			−0.225555
$537$	−4.86618e10	−	4.86618e10i	−0.585182	−	0.585182i
$538$	−8.75163e10	+	8.75163e10i	−1.04462	+	1.04462i
$539$			2.45368e9i			0.0290712i
$540$		0			0
$541$	−5.06153e10			−0.590871			−0.295436	−	0.955363i	$-0.595465\pi$
$541$	−5.06153e10			−0.590871			−0.295436	+	0.955363i	$0.595465\pi$
$542$	−9.58290e10	−	9.58290e10i	−1.11045	−	1.11045i
$543$	3.59638e10	−	3.59638e10i	0.413681	−	0.413681i
$544$			7.88406e10i			0.900232i
$545$		0			0
$546$	6.60750e10			0.743476
$547$	1.06937e11	+	1.06937e11i	1.19448	+	1.19448i	0.975796	+	0.218683i	$0.0701762\pi$
$547$	1.06937e11	+	1.06937e11i	1.19448	+	1.19448i	0.218683	+	0.975796i	$0.429824\pi$
$548$	−5.54948e10	+	5.54948e10i	−0.615361	+	0.615361i
$549$			1.06905e10i			0.117681i
$550$		0			0
$551$	−1.75021e11			−1.89882
$552$	−9.50776e9	−	9.50776e9i	−0.102405	−	0.102405i
$553$	−1.14993e11	+	1.14993e11i	−1.22962	+	1.22962i
$554$		−	9.75559e9i		−	0.103565i
$555$		0			0
$556$	2.27913e10			0.238490
$557$	−9.95647e10	−	9.95647e10i	−1.03439	−	1.03439i	−0.999387	−	0.0350030i	$-0.988856\pi$
$557$	−9.95647e10	−	9.95647e10i	−1.03439	−	1.03439i	−0.0350030	−	0.999387i	$-0.511144\pi$
$558$	4.45261e10	−	4.45261e10i	0.459281	−	0.459281i
$559$		−	1.36300e11i		−	1.39588i
$560$		0			0
$561$	−1.20314e10			−0.121468
$562$	−1.27663e11	−	1.27663e11i	−1.27973	−	1.27973i
$563$	1.21355e11	−	1.21355e11i	1.20788	−	1.20788i	0.236166	−	0.971713i	$-0.424109\pi$
$563$	1.21355e11	−	1.21355e11i	1.20788	−	1.20788i	0.971713	−	0.236166i	$-0.0758909\pi$
$564$		−	4.74538e10i		−	0.468980i
$565$		0			0
$566$	−4.12355e8			−0.00401796
$567$	8.49754e9	+	8.49754e9i	0.0822169	+	0.0822169i
$568$	2.94556e10	−	2.94556e10i	0.282992	−	0.282992i
$569$			1.62563e11i			1.55086i	0.631436	+	0.775428i	$0.282466\pi$
$569$			1.62563e11i			1.55086i	−0.631436	+	0.775428i	$0.717534\pi$
$570$		0			0
$571$	7.54958e10			0.710197			0.355098	−	0.934829i	$-0.384447\pi$
$571$	7.54958e10			0.710197			0.355098	+	0.934829i	$0.384447\pi$
$572$	1.73787e10	+	1.73787e10i	0.162343	+	0.162343i
$573$	−1.58300e10	+	1.58300e10i	−0.146846	+	0.146846i
$574$			2.06044e11i			1.89807i
$575$		0			0
$576$	2.28285e10			0.207390
$577$	1.95276e10	+	1.95276e10i	0.176175	+	0.176175i	0.789686	−	0.613511i	$-0.210243\pi$
$577$	1.95276e10	+	1.95276e10i	0.176175	+	0.176175i	−0.613511	+	0.789686i	$0.710243\pi$
$578$	5.61432e10	−	5.61432e10i	0.503021	−	0.503021i
$579$		−	1.04616e11i		−	0.930857i
$580$		0			0
$581$	1.43738e11			1.26144
$582$	1.45758e10	+	1.45758e10i	0.127040	+	0.127040i
$583$	−1.78483e10	+	1.78483e10i	−0.154498	+	0.154498i
$584$		−	5.16445e10i		−	0.443990i
$585$		0			0
$586$	1.56983e11			1.33126
$587$	−1.15917e11	−	1.15917e11i	−0.976322	−	0.976322i	0.0234041	−	0.999726i	$-0.492550\pi$
$587$	−1.15917e11	−	1.15917e11i	−0.976322	−	0.976322i	−0.999726	+	0.0234041i	$0.992550\pi$
$588$	3.82185e9	−	3.82185e9i	0.0319716	−	0.0319716i
$589$		−	2.14696e11i		−	1.78387i
$590$		0			0
$591$	−6.97457e10			−0.571699
$592$	5.84385e10	+	5.84385e10i	0.475786	+	0.475786i
$593$	−8.32677e10	+	8.32677e10i	−0.673377	+	0.673377i	−0.958493	−	0.285116i	$-0.907968\pi$
$593$	−8.32677e10	+	8.32677e10i	−0.673377	+	0.673377i	0.285116	+	0.958493i	$0.407968\pi$
$594$			9.89553e9i			0.0794865i
$595$		0			0
$596$	5.39874e10			0.427866
$597$	−9.32297e10	−	9.32297e10i	−0.733934	−	0.733934i
$598$	−1.17342e11	+	1.17342e11i	−0.917588	+	0.917588i
$599$			1.05049e11i			0.815991i	0.912984	+	0.407995i	$0.133772\pi$
$599$			1.05049e11i			0.815991i	−0.912984	+	0.407995i	$0.866228\pi$
$600$		0			0
$601$	−6.20830e10			−0.475855			−0.237928	−	0.971283i	$-0.576468\pi$
$601$	−6.20830e10			−0.475855			−0.237928	+	0.971283i	$0.576468\pi$
$602$	−2.01057e11	−	2.01057e11i	−1.53086	−	1.53086i
$603$	−2.95489e10	+	2.95489e10i	−0.223497	+	0.223497i
$604$			6.37174e10i			0.478752i
$605$		0			0
$606$	1.43963e11			1.06748
$607$	8.59912e10	+	8.59912e10i	0.633431	+	0.633431i	0.948927	−	0.315496i	$-0.102171\pi$
$607$	8.59912e10	+	8.59912e10i	0.633431	+	0.633431i	−0.315496	+	0.948927i	$0.602171\pi$
$608$	1.56334e11	−	1.56334e11i	1.14404	−	1.14404i
$609$		−	1.27634e11i		−	0.927890i
$610$		0			0
$611$	1.25209e11			0.898405
$612$	1.87400e10	+	1.87400e10i	0.133587	+	0.133587i
$613$	−9.83687e10	+	9.83687e10i	−0.696651	+	0.696651i	−0.963687	−	0.267036i	$-0.913956\pi$
$613$	−9.83687e10	+	9.83687e10i	−0.696651	+	0.696651i	0.267036	+	0.963687i	$0.413956\pi$
$614$		−	6.60340e10i		−	0.464616i
$615$		0			0
$616$	−1.09613e10			−0.0761274
$617$	2.02120e11	+	2.02120e11i	1.39466	+	1.39466i	0.814498	+	0.580166i	$0.197012\pi$
$617$	2.02120e11	+	2.02120e11i	1.39466	+	1.39466i	0.580166	+	0.814498i	$0.302988\pi$
$618$	7.38993e10	−	7.38993e10i	0.506625	−	0.506625i
$619$		−	1.10074e11i		−	0.749758i	−0.927074	−	0.374879i	$-0.877684\pi$
$619$		−	1.10074e11i		−	0.749758i	0.927074	−	0.374879i	$-0.122316\pi$
$620$		0			0
$621$	−3.01813e10			−0.202942
$622$	−9.84051e10	−	9.84051e10i	−0.657440	−	0.657440i
$623$	−1.60288e11	+	1.60288e11i	−1.06401	+	1.06401i
$624$			9.13352e10i			0.602421i
$625$		0			0
$626$	−3.20007e11			−2.08383
$627$	2.38572e10	+	2.38572e10i	0.154365	+	0.154365i
$628$	−2.27111e10	+	2.27111e10i	−0.146016	+	0.146016i
$629$			6.32742e10i			0.404226i
$630$		0			0
$631$	2.25703e11			1.42371			0.711854	−	0.702328i	$-0.247856\pi$
$631$	2.25703e11			1.42371			0.711854	+	0.702328i	$0.247856\pi$
$632$	−4.45929e10	−	4.45929e10i	−0.279510	−	0.279510i
$633$	−4.46427e10	+	4.46427e10i	−0.278058	+	0.278058i
$634$		−	3.09611e11i		−	1.91628i
$635$		0			0
$636$	5.56010e10			0.339824
$637$	1.00842e10	+	1.00842e10i	0.0612466	+	0.0612466i
$638$	7.43159e10	−	7.43159e10i	0.448537	−	0.448537i
$639$		−	9.35035e10i		−	0.560821i
$640$		0			0
$641$	−8.36281e9			−0.0495359			−0.0247679	−	0.999693i	$-0.507885\pi$
$641$	−8.36281e9			−0.0495359			−0.0247679	+	0.999693i	$0.507885\pi$
$642$	5.15803e10	+	5.15803e10i	0.303629	+	0.303629i
$643$	9.94984e10	−	9.94984e10i	0.582066	−	0.582066i	−0.353405	−	0.935471i	$-0.614976\pi$
$643$	9.94984e10	−	9.94984e10i	0.582066	−	0.582066i	0.935471	+	0.353405i	$0.114976\pi$
$644$			1.56376e11i			0.909134i
$645$		0			0
$646$	2.00040e11			1.14865
$647$	1.64550e11	+	1.64550e11i	0.939035	+	0.939035i	0.998245	−	0.0592109i	$-0.0188584\pi$
$647$	1.64550e11	+	1.64550e11i	0.939035	+	0.939035i	−0.0592109	+	0.998245i	$0.518858\pi$
$648$	−3.29523e9	+	3.29523e9i	−0.0186890	+	0.0186890i
$649$		−	2.03074e10i		−	0.114466i
$650$		0			0
$651$	1.56567e11			0.871718
$652$	−1.51984e11	−	1.51984e11i	−0.841025	−	0.841025i
$653$	−6.64536e10	+	6.64536e10i	−0.365482	+	0.365482i	−0.865826	−	0.500344i	$-0.833207\pi$
$653$	−6.64536e10	+	6.64536e10i	−0.365482	+	0.365482i	0.500344	+	0.865826i	$0.333207\pi$
$654$		−	5.97539e10i		−	0.326629i
$655$		0			0
$656$	−2.84814e11			−1.53797
$657$	−8.19699e10	−	8.19699e10i	−0.439940	−	0.439940i
$658$	1.84698e11	−	1.84698e11i	0.985278	−	0.985278i
$659$		−	5.17280e10i		−	0.274273i	−0.990552	−	0.137137i	$-0.956210\pi$
$659$		−	5.17280e10i		−	0.274273i	0.990552	−	0.137137i	$-0.0437900\pi$
$660$		0			0
$661$	−1.89964e11			−0.995095			−0.497547	−	0.867437i	$-0.665766\pi$
$661$	−1.89964e11			−0.995095			−0.497547	+	0.867437i	$0.665766\pi$
$662$	3.29169e11	+	3.29169e11i	1.71391	+	1.71391i
$663$	−4.94466e10	+	4.94466e10i	−0.255907	+	0.255907i
$664$			5.57396e10i			0.286742i
$665$		0			0
$666$	5.20416e10			0.264517
$667$	2.26663e11	+	2.26663e11i	1.14519	+	1.14519i
$668$	−1.04353e11	+	1.04353e11i	−0.524081	+	0.524081i
$669$			6.57512e10i			0.328246i
$670$		0			0
$671$	−2.18876e10			−0.107971
$672$	1.14006e11	+	1.14006e11i	0.559052	+	0.559052i
$673$	1.70060e11	−	1.70060e11i	0.828978	−	0.828978i	−0.158397	−	0.987375i	$-0.550633\pi$
$673$	1.70060e11	−	1.70060e11i	0.828978	−	0.828978i	0.987375	+	0.158397i	$0.0506326\pi$
$674$			1.26583e11i			0.613387i
$675$		0			0
$676$	−2.91992e10			−0.139825
$677$	−2.18049e11	−	2.18049e11i	−1.03801	−	1.03801i	−0.999249	−	0.0387581i	$-0.987660\pi$
$677$	−2.18049e11	−	2.18049e11i	−1.03801	−	1.03801i	−0.0387581	−	0.999249i	$-0.512340\pi$
$678$	5.42072e10	−	5.42072e10i	0.256530	−	0.256530i
$679$			5.12529e10i			0.241123i
$680$		0			0
$681$	−5.79195e9			−0.0269300
$682$	9.11624e10	+	9.11624e10i	0.421384	+	0.421384i
$683$	1.05222e11	−	1.05222e11i	0.483530	−	0.483530i	−0.422727	−	0.906257i	$-0.638927\pi$
$683$	1.05222e11	−	1.05222e11i	0.483530	−	0.483530i	0.906257	+	0.422727i	$0.138927\pi$
$684$		−	7.43197e10i		−	0.339531i
$685$		0			0
$686$	−2.83225e11			−1.27890
$687$	6.93738e10	+	6.93738e10i	0.311436	+	0.311436i
$688$	2.77921e11	−	2.77921e11i	1.24042	−	1.24042i
$689$			1.46706e11i			0.650986i
$690$		0			0
$691$	−3.02900e11			−1.32858			−0.664290	−	0.747475i	$-0.731266\pi$
$691$	−3.02900e11			−1.32858			−0.664290	+	0.747475i	$0.731266\pi$
$692$	7.75773e10	+	7.75773e10i	0.338307	+	0.338307i
$693$	−1.73978e10	+	1.73978e10i	−0.0754329	+	0.0754329i
$694$			4.12800e11i			1.77951i
$695$		0			0
$696$	4.94947e10			0.210922
$697$	−1.54191e11	−	1.54191e11i	−0.653324	−	0.653324i
$698$	1.12604e11	−	1.12604e11i	0.474388	−	0.474388i
$699$		−	1.81703e10i		−	0.0761120i
$700$		0			0
$701$	−3.79525e11			−1.57170			−0.785849	−	0.618419i	$-0.787773\pi$
$701$	−3.79525e11			−1.57170			−0.785849	+	0.618419i	$0.787773\pi$
$702$	4.06687e10	+	4.06687e10i	0.167460	+	0.167460i
$703$	1.25467e11	−	1.25467e11i	0.513700	−	0.513700i
$704$			4.67388e10i			0.190277i
$705$		0			0
$706$	3.93154e11			1.58250
$707$	2.53108e11	+	2.53108e11i	1.01305	+	1.01305i
$708$	−3.16307e10	+	3.16307e10i	−0.125886	+	0.125886i
$709$			2.85486e11i			1.12980i	0.825160	+	0.564899i	$0.191085\pi$
$709$			2.85486e11i			1.12980i	−0.825160	+	0.564899i	$0.808915\pi$
$710$		0			0
$711$	−1.41555e11			−0.553920
$712$	−6.21574e10	−	6.21574e10i	−0.241865	−	0.241865i
$713$	−2.78045e11	+	2.78045e11i	−1.07586	+	1.07586i
$714$			1.45879e11i			0.561305i
$715$		0			0
$716$	−3.10366e11			−1.18092
$717$	1.81429e10	+	1.81429e10i	0.0686483	+	0.0686483i
$718$	4.10213e10	−	4.10213e10i	0.154352	−	0.154352i
$719$			2.09894e11i			0.785389i	0.919669	+	0.392695i	$0.128457\pi$
$719$			2.09894e11i			0.785389i	−0.919669	+	0.392695i	$0.871543\pi$
$720$		0			0
$721$	2.59851e11			0.961577
$722$	−1.37167e11	−	1.37167e11i	−0.504777	−	0.504777i
$723$	−1.69364e11	+	1.69364e11i	−0.619822	+	0.619822i
$724$		−	2.29378e11i		−	0.834829i
$725$		0			0
$726$	1.96352e11			0.706787
$727$	−7.80140e10	−	7.80140e10i	−0.279277	−	0.279277i	0.553543	−	0.832820i	$-0.313275\pi$
$727$	−7.80140e10	−	7.80140e10i	−0.279277	−	0.279277i	−0.832820	+	0.553543i	$0.813275\pi$
$728$	−4.50490e10	+	4.50490e10i	−0.160384	+	0.160384i
$729$			1.04604e10i			0.0370370i
$730$		0			0
$731$	3.00919e11			1.05385
$732$	3.40921e10	+	3.40921e10i	0.118743	+	0.118743i
$733$	−6.83172e10	+	6.83172e10i	−0.236654	+	0.236654i	−0.815463	−	0.578809i	$-0.803518\pi$
$733$	−6.83172e10	+	6.83172e10i	−0.236654	+	0.236654i	0.578809	+	0.815463i	$0.303518\pi$
$734$		−	6.44106e11i		−	2.21908i
$735$		0			0
$736$	−4.04925e11			−1.37995
$737$	−6.04982e10	−	6.04982e10i	−0.205056	−	0.205056i
$738$	−1.26819e11	+	1.26819e11i	−0.427522	+	0.427522i
$739$		−	1.68021e11i		−	0.563358i	−0.959509	−	0.281679i	$-0.909109\pi$
$739$		−	1.68021e11i		−	0.563358i	0.959509	−	0.281679i	$-0.0908914\pi$
$740$		0			0
$741$	1.96097e11			0.650425
$742$	2.16409e11	+	2.16409e11i	0.713935	+	0.713935i
$743$	4.02895e11	−	4.02895e11i	1.32201	−	1.32201i	0.409872	−	0.912143i	$-0.365573\pi$
$743$	4.02895e11	−	4.02895e11i	1.32201	−	1.32201i	0.912143	−	0.409872i	$-0.134427\pi$
$744$			6.07146e10i			0.198153i
$745$		0			0
$746$	9.04450e10			0.292031
$747$	8.84695e10	+	8.84695e10i	0.284126	+	0.284126i
$748$	−3.83682e10	+	3.83682e10i	−0.122564	+	0.122564i
$749$			1.81371e11i			0.576290i
$750$		0			0
$751$	1.15049e11			0.361678			0.180839	−	0.983513i	$-0.442119\pi$
$751$	1.15049e11			0.361678			0.180839	+	0.983513i	$0.442119\pi$
$752$	2.55307e11	+	2.55307e11i	0.798348	+	0.798348i
$753$	1.66923e10	−	1.66923e10i	0.0519201	−	0.0519201i
$754$		−	6.10848e11i		−	1.88994i
$755$		0			0
$756$	5.41975e10			0.165917
$757$	−1.32371e11	−	1.32371e11i	−0.403096	−	0.403096i	0.476227	−	0.879323i	$-0.342004\pi$
$757$	−1.32371e11	−	1.32371e11i	−0.403096	−	0.403096i	−0.879323	+	0.476227i	$0.842004\pi$
$758$	−1.22478e11	+	1.22478e11i	−0.371007	+	0.371007i
$759$		−	6.17930e10i		−	0.186197i
$760$		0			0
$761$	2.75577e11			0.821683			0.410841	−	0.911707i	$-0.365235\pi$
$761$	2.75577e11			0.821683			0.410841	+	0.911707i	$0.365235\pi$
$762$	−1.39580e11	−	1.39580e11i	−0.414004	−	0.414004i
$763$	1.05056e11	−	1.05056e11i	0.309972	−	0.309972i
$764$			1.00964e11i			0.296342i
$765$		0			0
$766$	4.19652e11			1.21892
$767$	−8.34594e10	−	8.34594e10i	−0.241154	−	0.241154i
$768$	−1.78200e11	+	1.78200e11i	−0.512228	+	0.512228i
$769$		−	6.27339e10i		−	0.179390i	−0.995969	−	0.0896948i	$-0.971411\pi$
$769$		−	6.27339e10i		−	0.179390i	0.995969	−	0.0896948i	$-0.0285892\pi$
$770$		0			0
$771$	−7.76559e10			−0.219764
$772$	−3.33621e11	−	3.33621e11i	−0.939257	−	0.939257i
$773$	−2.75539e11	+	2.75539e11i	−0.771730	+	0.771730i	−0.978409	−	0.206679i	$-0.933735\pi$
$773$	−2.75539e11	+	2.75539e11i	−0.771730	+	0.771730i	0.206679	+	0.978409i	$0.433735\pi$
$774$		−	2.47499e11i		−	0.689620i
$775$		0			0
$776$	−1.98752e10			−0.0548106
$777$	9.14968e10	+	9.14968e10i	0.251028	+	0.251028i
$778$	−1.89312e11	+	1.89312e11i	−0.516726	+	0.516726i
$779$			6.11496e11i			1.66052i
$780$		0			0
$781$	1.91438e11			0.514546
$782$	−2.59064e11	−	2.59064e11i	−0.692756	−	0.692756i
$783$	7.85577e10	−	7.85577e10i	0.208998	−	0.208998i
$784$			4.11241e10i			0.108851i
$785$		0			0
$786$	−2.46537e11			−0.645940
$787$	2.59480e11	+	2.59480e11i	0.676402	+	0.676402i	0.959184	−	0.282782i	$-0.0912573\pi$
$787$	2.59480e11	+	2.59480e11i	0.676402	+	0.676402i	−0.282782	+	0.959184i	$0.591257\pi$
$788$	−2.22420e11	+	2.22420e11i	−0.576858	+	0.576858i
$789$		−	2.23386e11i		−	0.576432i
$790$		0			0
$791$	1.90608e11			0.486895
$792$	−6.74663e9	−	6.74663e9i	−0.0171469	−	0.0171469i
$793$	−8.99537e10	+	8.99537e10i	−0.227471	+	0.227471i
$794$		−	9.71660e11i		−	2.44474i
$795$		0			0
$796$	−5.94622e11			−1.48111
$797$	−2.57815e11	−	2.57815e11i	−0.638961	−	0.638961i	0.311338	−	0.950299i	$-0.399223\pi$
$797$	−2.57815e11	−	2.57815e11i	−0.638961	−	0.638961i	−0.950299	+	0.311338i	$0.899223\pi$
$798$	2.89265e11	−	2.89265e11i	0.713320	−	0.713320i
$799$			2.76434e11i			0.678273i
$800$		0			0
$801$	−1.97312e11			−0.479317
$802$	−2.21627e11	−	2.21627e11i	−0.535705	−	0.535705i
$803$	1.67824e11	−	1.67824e11i	0.403639	−	0.403639i
$804$			1.88464e11i			0.451028i
$805$		0			0
$806$	7.49320e11			1.77553
$807$	−1.89407e11	−	1.89407e11i	−0.446584	−	0.446584i
$808$	−9.81522e10	+	9.81522e10i	−0.230279	+	0.230279i
$809$		−	4.40964e11i		−	1.02946i	−0.857353	−	0.514729i	$-0.827892\pi$
$809$		−	4.40964e11i		−	1.02946i	0.857353	−	0.514729i	$-0.172108\pi$
$810$		0			0
$811$	3.14945e11			0.728033			0.364016	−	0.931393i	$-0.381405\pi$
$811$	3.14945e11			0.728033			0.364016	+	0.931393i	$0.381405\pi$
$812$	−4.07026e11	−	4.07026e11i	−0.936262	−	0.936262i
$813$	2.07398e11	−	2.07398e11i	0.474726	−	0.474726i
$814$			1.06550e11i			0.242691i
$815$		0			0
$816$	−2.01648e11			−0.454813
$817$	−5.96696e11	−	5.96696e11i	−1.33926	−	1.33926i
$818$	6.77846e10	−	6.77846e10i	0.151397	−	0.151397i
$819$			1.43003e11i			0.317841i
$820$		0			0
$821$	2.96960e11			0.653621			0.326810	−	0.945090i	$-0.394026\pi$
$821$	2.96960e11			0.653621			0.326810	+	0.945090i	$0.394026\pi$
$822$	−2.65887e11	−	2.65887e11i	−0.582385	−	0.582385i
$823$	−1.54849e11	+	1.54849e11i	−0.337526	+	0.337526i	−0.855436	−	0.517909i	$-0.826711\pi$
$823$	−1.54849e11	+	1.54849e11i	−0.337526	+	0.337526i	0.517909	+	0.855436i	$0.326711\pi$
$824$			1.00767e11i			0.218579i
$825$		0			0
$826$	−2.46224e11			−0.528945
$827$	−4.19074e10	−	4.19074e10i	−0.0895919	−	0.0895919i	0.660890	−	0.750482i	$-0.270179\pi$
$827$	−4.19074e10	−	4.19074e10i	−0.0895919	−	0.0895919i	−0.750482	+	0.660890i	$0.770179\pi$
$828$	−9.62486e10	+	9.62486e10i	−0.204773	+	0.204773i
$829$			6.56756e10i			0.139055i	0.997580	+	0.0695274i	$0.0221491\pi$
$829$			6.56756e10i			0.139055i	−0.997580	+	0.0695274i	$0.977851\pi$
$830$		0			0
$831$	2.11136e10			0.0442749
$832$	1.92088e11	+	1.92088e11i	0.400872	+	0.400872i
$833$	−2.22635e10	+	2.22635e10i	−0.0462396	+	0.0462396i
$834$			1.09198e11i			0.225710i
$835$		0			0
$836$	1.52162e11			0.311516
$837$	9.63658e10	+	9.63658e10i	0.196346	+	0.196346i
$838$	−7.16737e11	+	7.16737e11i	−1.45340	+	1.45340i
$839$			7.98678e11i			1.61185i	0.592020	+	0.805924i	$0.298331\pi$
$839$			7.98678e11i			1.61185i	−0.592020	+	0.805924i	$0.701669\pi$
$840$		0			0
$841$	−6.79697e11			−1.35872
$842$	3.91409e11	+	3.91409e11i	0.778721	+	0.778721i
$843$	2.76295e11	−	2.76295e11i	0.547094	−	0.547094i
$844$			2.84732e11i			0.561134i
$845$		0			0
$846$	2.27361e11			0.443848
$847$	3.45215e11	+	3.45215e11i	0.670743	+	0.670743i
$848$	−2.99141e11	+	2.99141e11i	−0.578485	+	0.578485i
$849$		−	8.92440e8i		−	0.00171770i
$850$		0			0
$851$	−3.24976e11			−0.619630
$852$	−2.98184e11	−	2.98184e11i	−0.565882	−	0.565882i
$853$	−1.87598e11	+	1.87598e11i	−0.354350	+	0.354350i	−0.861725	−	0.507375i	$-0.830616\pi$
$853$	−1.87598e11	+	1.87598e11i	−0.354350	+	0.354350i	0.507375	+	0.861725i	$0.330616\pi$
$854$			2.65384e11i			0.498934i
$855$		0			0
$856$	−7.03334e10			−0.130999
$857$	−4.78106e11	−	4.78106e11i	−0.886342	−	0.886342i	0.107828	−	0.994170i	$-0.465610\pi$
$857$	−4.78106e11	−	4.78106e11i	−0.886342	−	0.886342i	−0.994170	+	0.107828i	$0.965610\pi$
$858$	−8.32648e10	+	8.32648e10i	−0.153643	+	0.153643i
$859$		−	2.92378e11i		−	0.536996i	−0.963280	−	0.268498i	$-0.913473\pi$
$859$		−	2.92378e11i		−	0.536996i	0.963280	−	0.268498i	$-0.0865273\pi$
$860$		0			0
$861$	−4.45932e11			−0.811440
$862$	8.56349e11	+	8.56349e11i	1.55104	+	1.55104i
$863$	8.69978e10	−	8.69978e10i	0.156843	−	0.156843i	−0.624323	−	0.781166i	$-0.714625\pi$
$863$	8.69978e10	−	8.69978e10i	0.156843	−	0.156843i	0.781166	+	0.624323i	$0.214625\pi$
$864$			1.40340e11i			0.251842i
$865$		0			0
$866$	1.31172e11			0.233222
$867$	1.21508e11	+	1.21508e11i	0.215045	+	0.215045i
$868$	4.99294e11	−	4.99294e11i	0.879584	−	0.879584i
$869$		−	2.89819e11i		−	0.508215i
$870$		0			0
$871$	−4.97272e11			−0.864015
$872$	4.07394e10	+	4.07394e10i	0.0704609	+	0.0704609i
$873$	−3.15458e10	+	3.15458e10i	−0.0543106	+	0.0543106i
$874$			1.02740e12i			1.76074i
$875$		0			0
$876$	−5.22806e11			−0.887819
$877$	−3.39966e11	−	3.39966e11i	−0.574695	−	0.574695i	0.358742	−	0.933437i	$-0.383206\pi$
$877$	−3.39966e11	−	3.39966e11i	−0.574695	−	0.574695i	−0.933437	+	0.358742i	$0.883206\pi$
$878$	5.15807e11	−	5.15807e11i	0.867979	−	0.867979i
$879$			3.39751e11i			0.569122i
$880$		0			0
$881$	2.22272e11			0.368961			0.184480	−	0.982836i	$-0.440940\pi$
$881$	2.22272e11			0.368961			0.184480	+	0.982836i	$0.440940\pi$
$882$	1.83113e10	+	1.83113e10i	0.0302583	+	0.0302583i
$883$	−2.34413e11	+	2.34413e11i	−0.385602	+	0.385602i	−0.873116	−	0.487513i	$-0.837904\pi$
$883$	−2.34413e11	+	2.34413e11i	−0.385602	+	0.385602i	0.487513	+	0.873116i	$0.337904\pi$
$884$			3.15372e11i			0.516432i
$885$		0			0
$886$	−4.31633e11			−0.700454
$887$	7.39337e11	+	7.39337e11i	1.19439	+	1.19439i	0.975820	+	0.218574i	$0.0701404\pi$
$887$	7.39337e11	+	7.39337e11i	1.19439	+	1.19439i	0.218574	+	0.975820i	$0.429860\pi$
$888$	−3.54813e10	+	3.54813e10i	−0.0570620	+	0.0570620i
$889$		−	4.90806e11i		−	0.785783i
$890$		0			0
$891$	−2.14164e10			−0.0339810
$892$	2.09682e11	+	2.09682e11i	0.331208	+	0.331208i
$893$	5.48145e11	−	5.48145e11i	0.861965	−	0.861965i
$894$			2.58665e11i			0.404937i
$895$		0			0
$896$	−3.15889e11			−0.490120
$897$	−2.53957e11	−	2.53957e11i	−0.392275	−	0.392275i
$898$	−1.14350e12	+	1.14350e12i	−1.75846	+	1.75846i
$899$		−	1.44742e12i		−	2.21593i
$900$		0			0
$901$	−3.23894e11			−0.491478
$902$	−2.59648e11	−	2.59648e11i	−0.392246	−	0.392246i
$903$	4.35139e11	−	4.35139e11i	0.654451	−	0.654451i
$904$			7.39154e10i			0.110678i
$905$		0			0
$906$	−3.05284e11			−0.453096
$907$	−9.79482e10	−	9.79482e10i	−0.144733	−	0.144733i	0.631027	−	0.775760i	$-0.282633\pi$
$907$	−9.79482e10	−	9.79482e10i	−0.144733	−	0.144733i	−0.775760	+	0.631027i	$0.782633\pi$
$908$	−1.84706e10	+	1.84706e10i	−0.0271730	+	0.0271730i
$909$			3.11573e11i			0.456357i
$910$		0			0
$911$	−6.30284e10			−0.0915088			−0.0457544	−	0.998953i	$-0.514569\pi$
$911$	−6.30284e10			−0.0915088			−0.0457544	+	0.998953i	$0.514569\pi$
$912$	3.99850e11	+	3.99850e11i	0.577987	+	0.577987i
$913$	−1.81132e11	+	1.81132e11i	−0.260682	+	0.260682i
$914$		−	5.22836e11i		−	0.749171i
$915$		0			0
$916$	4.42468e11			0.628492
$917$	−4.33449e11	−	4.33449e11i	−0.613000	−	0.613000i
$918$	−8.97874e10	+	8.97874e10i	−0.126428	+	0.126428i
$919$			1.20904e12i			1.69503i	0.530772	+	0.847515i	$0.321902\pi$
$919$			1.20904e12i			1.69503i	−0.530772	+	0.847515i	$0.678098\pi$
$920$		0			0
$921$	1.42914e11			0.198626
$922$	1.32498e12	+	1.32498e12i	1.83352	+	1.83352i
$923$	7.86774e11	−	7.86774e11i	1.08403	−	1.08403i
$924$			1.10964e11i			0.152227i
$925$		0			0
$926$	1.57479e12			2.14180
$927$	1.59937e11	+	1.59937e11i	0.216585	+	0.216585i
$928$	1.05396e12	−	1.05396e12i	1.42113	−	1.42113i
$929$		−	1.37970e12i		−	1.85234i	−0.377106	−	0.926170i	$-0.623081\pi$
$929$		−	1.37970e12i		−	1.85234i	0.377106	−	0.926170i	$-0.376919\pi$
$930$		0			0
$931$	8.82934e10			0.117525
$932$	−5.79453e10	−	5.79453e10i	−0.0767988	−	0.0767988i
$933$	2.12974e11	−	2.12974e11i	0.281060	−	0.281060i
$934$		−	2.05719e11i		−	0.270325i
$935$		0			0
$936$	−5.54547e10			−0.0722496
$937$	−1.57203e11	−	1.57203e11i	−0.203940	−	0.203940i	0.597746	−	0.801686i	$-0.296063\pi$
$937$	−1.57203e11	−	1.57203e11i	−0.203940	−	0.203940i	−0.801686	+	0.597746i	$0.796063\pi$
$938$	−7.33533e11	+	7.33533e11i	−0.947564	+	0.947564i
$939$		−	6.92576e11i		−	0.890851i
$940$		0			0
$941$	5.08269e11			0.648239			0.324120	−	0.946016i	$-0.394932\pi$
$941$	5.08269e11			0.648239			0.324120	+	0.946016i	$0.394932\pi$
$942$	−1.08813e11	−	1.08813e11i	−0.138191	−	0.138191i
$943$	7.91925e11	−	7.91925e11i	1.00147	−	1.00147i
$944$		−	3.40355e11i		−	0.428592i
$945$		0			0
$946$	5.06727e11			0.632717
$947$	4.22518e11	+	4.22518e11i	0.525346	+	0.525346i	0.919181	−	0.393835i	$-0.128852\pi$
$947$	4.22518e11	+	4.22518e11i	0.525346	+	0.525346i	−0.393835	+	0.919181i	$0.628852\pi$
$948$	−4.51421e11	+	4.51421e11i	−0.558918	+	0.558918i
$949$		−	1.37945e12i		−	1.70076i
$950$		0			0
$951$	6.70077e11			0.819224
$952$	−9.94581e10	−	9.94581e10i	−0.121086	−	0.121086i
$953$	5.77256e11	−	5.77256e11i	0.699837	−	0.699837i	−0.264538	−	0.964375i	$-0.585219\pi$
$953$	5.77256e11	−	5.77256e11i	0.699837	−	0.699837i	0.964375	+	0.264538i	$0.0852194\pi$
$954$			2.66396e11i			0.321613i
$955$		0			0
$956$	1.15716e11			0.138536
$957$	1.60838e11	+	1.60838e11i	0.191753	+	0.191753i
$958$	3.08937e11	−	3.08937e11i	0.366782	−	0.366782i
$959$		−	9.34936e11i		−	1.10537i
$960$		0			0
$961$	9.22646e11			1.08179
$962$	4.37898e11	+	4.37898e11i	0.511297	+	0.511297i
$963$	−1.11633e11	+	1.11633e11i	−0.129804	+	0.129804i
$964$			1.08020e12i			1.25083i
$965$		0			0
$966$	−7.49232e11			−0.860415
$967$	−6.14848e11	−	6.14848e11i	−0.703173	−	0.703173i	0.261918	−	0.965090i	$-0.415645\pi$
$967$	−6.14848e11	−	6.14848e11i	−0.703173	−	0.703173i	−0.965090	+	0.261918i	$0.915645\pi$
$968$	−1.33870e11	+	1.33870e11i	−0.152469	+	0.152469i
$969$			4.32937e11i			0.491055i
$970$		0			0
$971$	−3.74017e11			−0.420741			−0.210370	−	0.977622i	$-0.567467\pi$
$971$	−3.74017e11			−0.420741			−0.210370	+	0.977622i	$0.567467\pi$
$972$	3.33582e10	+	3.33582e10i	0.0373712	+	0.0373712i
$973$	−1.91986e11	+	1.91986e11i	−0.214199	+	0.214199i
$974$			1.62288e12i			1.80323i
$975$		0			0
$976$	−3.66839e11			−0.404275
$977$	5.84649e11	+	5.84649e11i	0.641678	+	0.641678i	0.950968	−	0.309290i	$-0.100091\pi$
$977$	5.84649e11	+	5.84649e11i	0.641678	+	0.641678i	−0.309290	+	0.950968i	$0.600091\pi$
$978$	7.28189e11	−	7.28189e11i	0.795956	−	0.795956i
$979$		−	4.03974e11i		−	0.439767i
$980$		0			0
$981$	1.29323e11			0.139636
$982$	3.49259e11	+	3.49259e11i	0.375579	+	0.375579i
$983$	−1.87740e11	+	1.87740e11i	−0.201068	+	0.201068i	−0.800457	−	0.599390i	$-0.795410\pi$
$983$	−1.87740e11	+	1.87740e11i	−0.201068	+	0.201068i	0.599390	+	0.800457i	$0.295410\pi$
$984$		−	1.72927e11i		−	0.184451i
$985$		0			0
$986$	1.34861e12			1.42686
$987$	3.99734e11	+	3.99734e11i	0.421213	+	0.421213i
$988$	6.25355e11	−	6.25355e11i	0.656295	−	0.656295i
$989$			1.54552e12i			1.61543i
$990$		0			0
$991$	−3.37718e11			−0.350155			−0.175077	−	0.984555i	$-0.556018\pi$
$991$	−3.37718e11			−0.350155			−0.175077	+	0.984555i	$0.556018\pi$
$992$	1.29288e12	+	1.29288e12i	1.33510	+	1.33510i
$993$	−7.12405e11	+	7.12405e11i	−0.732706	+	0.732706i
$994$		−	2.32116e12i		−	2.37772i
$995$		0			0
$996$	5.64261e11			0.573380
$997$	−5.07145e11	−	5.07145e11i	−0.513277	−	0.513277i	0.402252	−	0.915529i	$-0.368228\pi$
$997$	−5.07145e11	−	5.07145e11i	−0.513277	−	0.513277i	−0.915529	+	0.402252i	$0.868228\pi$
$998$	−7.64126e11	+	7.64126e11i	−0.770270	+	0.770270i
$999$			1.12631e11i			0.113083i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.f.d.7.5	yes	12
5.2	odd	4	inner	75.9.f.d.43.2	yes	12
5.3	odd	4	inner	75.9.f.d.43.5	yes	12
5.4	even	2	inner	75.9.f.d.7.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.f.d.7.2	✓	12	5.4	even	2	inner
75.9.f.d.7.5	yes	12	1.1	even	1	trivial
75.9.f.d.43.2	yes	12	5.2	odd	4	inner
75.9.f.d.43.5	yes	12	5.3	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(15.2792 + 15.2792i) q^{2} +(-33.0681 + 33.0681i) q^{3} +210.909i q^{4} -1010.51 q^{6} +(-1776.62 - 1776.62i) q^{7} +(688.952 - 688.952i) q^{8} -2187.00i q^{9} +O(q^{10})\)	\(q+(15.2792 + 15.2792i) q^{2} +(-33.0681 + 33.0681i) q^{3} +210.909i q^{4} -1010.51 q^{6} +(-1776.62 - 1776.62i) q^{7} +(688.952 - 688.952i) q^{8} -2187.00i q^{9} +4477.64 q^{11} +(-6974.37 - 6974.37i) q^{12} +(18402.3 - 18402.3i) q^{13} -54290.9i q^{14} +75046.1 q^{16} +(40628.0 + 40628.0i) q^{17} +(33415.7 - 33415.7i) q^{18} -161124. i q^{19} +117499. q^{21} +(68414.9 + 68414.9i) q^{22} +(-208665. + 208665. i) q^{23} +45564.7i q^{24} +562344. q^{26} +(72320.0 + 72320.0i) q^{27} +(374706. - 374706. i) q^{28} -1.08625e6i q^{29} +1.33249e6 q^{31} +(970274. + 970274. i) q^{32} +(-148067. + 148067. i) q^{33} +1.24153e6i q^{34} +461259. q^{36} +(778701. + 778701. i) q^{37} +(2.46185e6 - 2.46185e6i) q^{38} +1.21706e6i q^{39} -3.79519e6 q^{41} +(1.79530e6 + 1.79530e6i) q^{42} +(3.70334e6 - 3.70334e6i) q^{43} +944377. i q^{44} -6.37648e6 q^{46} +(3.40201e6 + 3.40201e6i) q^{47} +(-2.48163e6 + 2.48163e6i) q^{48} +547985. i q^{49} -2.68698e6 q^{51} +(3.88121e6 + 3.88121e6i) q^{52} +(-3.98610e6 + 3.98610e6i) q^{53} +2.20999e6i q^{54} -2.44802e6 q^{56} +(5.32806e6 + 5.32806e6i) q^{57} +(1.65971e7 - 1.65971e7i) q^{58} -4.53528e6i q^{59} -4.88819e6 q^{61} +(2.03595e7 + 2.03595e7i) q^{62} +(-3.88548e6 + 3.88548e6i) q^{63} +1.04383e7i q^{64} -4.52471e6 q^{66} +(-1.35112e7 - 1.35112e7i) q^{67} +(-8.56883e6 + 8.56883e6i) q^{68} -1.38003e7i q^{69} +4.27542e7 q^{71} +(-1.50674e6 - 1.50674e6i) q^{72} +(3.74805e7 - 3.74805e7i) q^{73} +2.37959e7i q^{74} +3.39825e7 q^{76} +(-7.95509e6 - 7.95509e6i) q^{77} +(-1.85957e7 + 1.85957e7i) q^{78} -6.47257e7i q^{79} -4.78297e6 q^{81} +(-5.79876e7 - 5.79876e7i) q^{82} +(-4.04525e7 + 4.04525e7i) q^{83} +2.47817e7i q^{84} +1.13168e8 q^{86} +(3.59203e7 + 3.59203e7i) q^{87} +(3.08488e6 - 3.08488e6i) q^{88} -9.02203e7i q^{89} -6.53878e7 q^{91} +(-4.40094e7 - 4.40094e7i) q^{92} +(-4.40630e7 + 4.40630e7i) q^{93} +1.03960e8i q^{94} -6.41702e7 q^{96} +(-1.44242e7 - 1.44242e7i) q^{97} +(-8.37278e6 + 8.37278e6i) q^{98} -9.79261e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2268 q^{6}+O(q^{10}) \) 12 * q + 2268 * q^6	\( 12 q + 2268 q^{6} - 88920 q^{11} - 485796 q^{16} + 79704 q^{21} - 4055976 q^{26} + 5658696 q^{31} + 5065092 q^{36} + 1798056 q^{41} - 10882464 q^{46} + 4329936 q^{51} + 7268040 q^{56} + 33649848 q^{61} - 141361848 q^{66} + 335506464 q^{71} - 395386536 q^{76} - 57395628 q^{81} + 489958560 q^{86} - 216875664 q^{91} - 710311356 q^{96}+O(q^{100}) \) 12 * q + 2268 * q^6 - 88920 * q^11 - 485796 * q^16 + 79704 * q^21 - 4055976 * q^26 + 5658696 * q^31 + 5065092 * q^36 + 1798056 * q^41 - 10882464 * q^46 + 4329936 * q^51 + 7268040 * q^56 + 33649848 * q^61 - 141361848 * q^66 + 335506464 * q^71 - 395386536 * q^76 - 57395628 * q^81 + 489958560 * q^86 - 216875664 * q^91 - 710311356 * q^96

Introduction

L-functions

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