LMFDB - Embedded newform 75.9.d.d.74.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(74,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.d (of order $2$, degree $1$, not 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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 3278 x^{18} + 4245491 x^{16} + 2854629536 x^{14} + 1117319469691 x^{12} + 266849787342054 x^{10} + \cdots + 89\!\cdots\!00 $ x^20 + 3278x^18 + 4245491x^16 + 2854629536x^14 + 1117319469691x^12 + 266849787342054x^10 + 38935163332201981x^8 + 3308855301815965532x^6 + 143709036159536320436x^4 + 2262672788809149782000*x^2 + 8995759125454185640000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{22}\cdot 5^{26} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			74.12
Root			$-15.0371i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.9.d.d.74.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.03414 q^{2} +(79.6728 + 14.6031i) q^{3} -246.794 q^{4} +(241.739 + 44.3080i) q^{6} -59.7348i q^{7} -1525.55 q^{8} +(6134.50 + 2326.94i) q^{9} +O(q^{10})$	\(q+3.03414 q^{2} +(79.6728 + 14.6031i) q^{3} -246.794 q^{4} +(241.739 + 44.3080i) q^{6} -59.7348i q^{7} -1525.55 q^{8} +(6134.50 + 2326.94i) q^{9} -21885.6i q^{11} +(-19662.8 - 3603.96i) q^{12} +38460.5i q^{13} -181.244i q^{14} +58550.5 q^{16} +36101.0 q^{17} +(18612.9 + 7060.28i) q^{18} +136807. q^{19} +(872.314 - 4759.23i) q^{21} -66404.0i q^{22} +235710. q^{23} +(-121545. - 22277.8i) q^{24} +116695. i q^{26} +(454772. + 274977. i) q^{27} +14742.2i q^{28} +896225. i q^{29} +68043.4 q^{31} +568191. q^{32} +(319598. - 1.74369e6i) q^{33} +109535. q^{34} +(-1.51396e6 - 574275. i) q^{36} -66517.7i q^{37} +415091. q^{38} +(-561643. + 3.06425e6i) q^{39} -3.76632e6i q^{41} +(2646.73 - 14440.2i) q^{42} +2.03993e6i q^{43} +5.40123e6i q^{44} +715178. q^{46} +5.93447e6 q^{47} +(4.66488e6 + 855021. i) q^{48} +5.76123e6 q^{49} +(2.87626e6 + 527187. i) q^{51} -9.49181e6i q^{52} +1.31066e7 q^{53} +(1.37984e6 + 834319. i) q^{54} +91128.3i q^{56} +(1.08998e7 + 1.99781e6i) q^{57} +2.71928e6i q^{58} +1.58410e7i q^{59} -3.51489e6 q^{61} +206453. q^{62} +(138999. - 366443. i) q^{63} -1.32650e7 q^{64} +(969706. - 5.29059e6i) q^{66} +1.39297e7i q^{67} -8.90950e6 q^{68} +(1.87797e7 + 3.44210e6i) q^{69} -1.51547e7i q^{71} +(-9.35847e6 - 3.54986e6i) q^{72} -3.93167e7i q^{73} -201824. i q^{74} -3.37631e7 q^{76} -1.30733e6 q^{77} +(-1.70411e6 + 9.29738e6i) q^{78} -3.15613e7 q^{79} +(3.22174e7 + 2.85492e7i) q^{81} -1.14275e7i q^{82} -4.88680e7 q^{83} +(-215282. + 1.17455e6i) q^{84} +6.18943e6i q^{86} +(-1.30877e7 + 7.14048e7i) q^{87} +3.33875e7i q^{88} -1.09263e8i q^{89} +2.29743e6 q^{91} -5.81718e7 q^{92} +(5.42120e6 + 993646. i) q^{93} +1.80060e7 q^{94} +(4.52694e7 + 8.29737e6i) q^{96} -1.16098e8i q^{97} +1.74804e7 q^{98} +(5.09265e7 - 1.34257e8i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9}+O(q^{10}) $ 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9	$ 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9} + 89268 q^{16} + 287868 q^{19} + 1346856 q^{21} + 2033718 q^{24} - 6028120 q^{31} - 9954292 q^{34} + 9001054 q^{36} + 15026564 q^{39} - 27877272 q^{46} - 17907092 q^{49} + 12418574 q^{51} + 17772544 q^{54} + 3040440 q^{61} + 9073996 q^{64} + 20930590 q^{66} - 22789956 q^{69} - 59058092 q^{76} - 17099792 q^{79} + 45225890 q^{81} + 273766584 q^{84} - 214448528 q^{91} - 712237192 q^{94} + 1050848002 q^{96} + 764842670 q^{99}+O(q^{100}) $ 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9 + 89268 * q^16 + 287868 * q^19 + 1346856 * q^21 + 2033718 * q^24 - 6028120 * q^31 - 9954292 * q^34 + 9001054 * q^36 + 15026564 * q^39 - 27877272 * q^46 - 17907092 * q^49 + 12418574 * q^51 + 17772544 * q^54 + 3040440 * q^61 + 9073996 * q^64 + 20930590 * q^66 - 22789956 * q^69 - 59058092 * q^76 - 17099792 * q^79 + 45225890 * q^81 + 273766584 * q^84 - 214448528 * q^91 - 712237192 * q^94 + 1050848002 * q^96 + 764842670 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$26$	$52$
$\chi(n)$	$-1$	$-1$

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 $

$2$	3.03414			0.189634			0.0948170	−	0.995495i	$-0.469773\pi$
$2$	3.03414			0.189634			0.0948170	+	0.995495i	$0.469773\pi$
$3$	79.6728	+	14.6031i	0.983614	+	0.180286i
$3$	79.6728	+	14.6031i	0.983614	+	0.180286i
$4$	−246.794			−0.964039
$5$		0			0
$5$		0			0
$6$	241.739	+	44.3080i	0.186527	+	0.0341882i
$7$		−	59.7348i		−	0.0248791i	−0.999923	−	0.0124396i	$-0.996040\pi$
$7$		−	59.7348i		−	0.0248791i	0.999923	−	0.0124396i	$-0.00395974\pi$
$8$	−1525.55			−0.372448
$9$	6134.50	+	2326.94i	0.934994	+	0.354663i
$10$		0			0
$11$		−	21885.6i		−	1.49482i	−0.664366	−	0.747408i	$-0.731298\pi$
$11$		−	21885.6i		−	1.49482i	0.664366	−	0.747408i	$-0.268702\pi$
$12$	−19662.8	−	3603.96i	−0.948243	−	0.173802i
$13$			38460.5i			1.34661i	0.739366	+	0.673304i	$0.235125\pi$
$13$			38460.5i			1.34661i	−0.739366	+	0.673304i	$0.764875\pi$
$14$		−	181.244i		−	0.00471792i
$15$		0			0
$16$	58550.5			0.893410
$17$	36101.0			0.432238			0.216119	−	0.976367i	$-0.430660\pi$
$17$	36101.0			0.432238			0.216119	+	0.976367i	$0.430660\pi$
$18$	18612.9	+	7060.28i	0.177307	+	0.0672561i
$19$	136807.			1.04977			0.524884	−	0.851174i	$-0.324109\pi$
$19$	136807.			1.04977			0.524884	+	0.851174i	$0.324109\pi$
$20$		0			0
$21$	872.314	−	4759.23i	0.00448534	−	0.0244715i
$22$		−	66404.0i		−	0.283468i
$23$	235710.			0.842300			0.421150	−	0.906991i	$-0.361627\pi$
$23$	235710.			0.842300			0.421150	+	0.906991i	$0.361627\pi$
$24$	−121545.	−	22277.8i	−0.366346	−	0.0671470i
$25$		0			0
$26$			116695.i			0.255363i
$27$	454772.	+	274977.i	0.855733	+	0.517417i
$28$			14742.2i			0.0239844i
$29$			896225.i			1.26714i	0.773685	+	0.633571i	$0.218412\pi$
$29$			896225.i			1.26714i	−0.773685	+	0.633571i	$0.781588\pi$
$30$		0			0
$31$	68043.4			0.0736782			0.0368391	−	0.999321i	$-0.488271\pi$
$31$	68043.4			0.0736782			0.0368391	+	0.999321i	$0.488271\pi$
$32$	568191.			0.541869
$33$	319598.	−	1.74369e6i	0.269493	−	1.47032i
$34$	109535.			0.0819670
$35$		0			0
$36$	−1.51396e6	−	574275.i	−0.901371	−	0.341909i
$37$		−	66517.7i		−	0.0354920i	−0.999843	−	0.0177460i	$-0.994351\pi$
$37$		−	66517.7i		−	0.0354920i	0.999843	−	0.0177460i	$-0.00564902\pi$
$38$	415091.			0.199071
$39$	−561643.	+	3.06425e6i	−0.242774	+	1.32454i
$40$		0			0
$41$		−	3.76632e6i		−	1.33285i	−0.745572	−	0.666425i	$-0.767824\pi$
$41$		−	3.76632e6i		−	1.33285i	0.745572	−	0.666425i	$-0.232176\pi$
$42$	2646.73	−	14440.2i	0.000850573	−	0.00464062i
$43$			2.03993e6i			0.596679i	0.954460	+	0.298339i	$0.0964327\pi$
$43$			2.03993e6i			0.596679i	−0.954460	+	0.298339i	$0.903567\pi$
$44$			5.40123e6i			1.44106i
$45$		0			0
$46$	715178.			0.159729
$47$	5.93447e6			1.21616			0.608080	−	0.793876i	$-0.291940\pi$
$47$	5.93447e6			1.21616			0.608080	+	0.793876i	$0.291940\pi$
$48$	4.66488e6	+	855021.i	0.878771	+	0.161069i
$49$	5.76123e6			0.999381
$50$		0			0
$51$	2.87626e6	+	527187.i	0.425156	+	0.0779263i
$52$		−	9.49181e6i		−	1.29818i
$53$	1.31066e7			1.66107			0.830534	−	0.556969i	$-0.188036\pi$
$53$	1.31066e7			1.66107			0.830534	+	0.556969i	$0.188036\pi$
$54$	1.37984e6	+	834319.i	0.162276	+	0.0981199i
$55$		0			0
$56$			91128.3i			0.00926619i
$57$	1.08998e7	+	1.99781e6i	1.03257	+	0.189258i
$58$			2.71928e6i			0.240293i
$59$			1.58410e7i			1.30730i	0.756799	+	0.653648i	$0.226762\pi$
$59$			1.58410e7i			1.30730i	−0.756799	+	0.653648i	$0.773238\pi$
$60$		0			0
$61$	−3.51489e6			−0.253859			−0.126929	−	0.991912i	$-0.540512\pi$
$61$	−3.51489e6			−0.253859			−0.126929	+	0.991912i	$0.540512\pi$
$62$	206453.			0.0139719
$63$	138999.	−	366443.i	0.00882370	−	0.0232618i
$64$	−1.32650e7			−0.790653
$65$		0			0
$66$	969706.	−	5.29059e6i	0.0511051	−	0.278823i
$67$			1.39297e7i			0.691263i	0.938370	+	0.345632i	$0.112335\pi$
$67$			1.39297e7i			0.691263i	−0.938370	+	0.345632i	$0.887665\pi$
$68$	−8.90950e6			−0.416695
$69$	1.87797e7	+	3.44210e6i	0.828498	+	0.151854i
$70$		0			0
$71$		−	1.51547e7i		−	0.596367i	−0.954508	−	0.298184i	$-0.903619\pi$
$71$		−	1.51547e7i		−	0.596367i	0.954508	−	0.298184i	$-0.0963808\pi$
$72$	−9.35847e6	−	3.54986e6i	−0.348237	−	0.132094i
$73$		−	3.93167e7i		−	1.38448i	−0.721669	−	0.692238i	$-0.756625\pi$
$73$		−	3.93167e7i		−	1.38448i	0.721669	−	0.692238i	$-0.243375\pi$
$74$		−	201824.i		−	0.00673049i
$75$		0			0
$76$	−3.37631e7			−1.01202
$77$	−1.30733e6			−0.0371897
$78$	−1.70411e6	+	9.29738e6i	−0.0460382	+	0.251178i
$79$	−3.15613e7			−0.810301			−0.405151	−	0.914250i	$-0.632781\pi$
$79$	−3.15613e7			−0.810301			−0.405151	+	0.914250i	$0.632781\pi$
$80$		0			0
$81$	3.22174e7	+	2.85492e7i	0.748429	+	0.663215i
$82$		−	1.14275e7i		−	0.252754i
$83$	−4.88680e7			−1.02970			−0.514852	−	0.857279i	$-0.672153\pi$
$83$	−4.88680e7			−1.02970			−0.514852	+	0.857279i	$0.672153\pi$
$84$	−215282.	+	1.17455e6i	−0.00432405	+	0.0235914i
$85$		0			0
$86$			6.18943e6i			0.113151i
$87$	−1.30877e7	+	7.14048e7i	−0.228447	+	1.24638i
$88$			3.33875e7i			0.556741i
$89$		−	1.09263e8i		−	1.74145i	−0.491766	−	0.870727i	$-0.663648\pi$
$89$		−	1.09263e8i		−	1.74145i	0.491766	−	0.870727i	$-0.336352\pi$
$90$		0			0
$91$	2.29743e6			0.0335024
$92$	−5.81718e7			−0.812010
$93$	5.42120e6	+	993646.i	0.0724709	+	0.0132831i
$94$	1.80060e7			0.230625
$95$		0			0
$96$	4.52694e7	+	8.29737e6i	0.532990	+	0.0976912i
$97$		−	1.16098e8i		−	1.31140i	−0.755020	−	0.655702i	$-0.772373\pi$
$97$		−	1.16098e8i		−	1.31140i	0.755020	−	0.655702i	$-0.227627\pi$
$98$	1.74804e7			0.189517
$99$	5.09265e7	−	1.34257e8i	0.530155	−	1.39764i
$100$		0			0
$101$			1.46015e8i			1.40318i	0.712582	+	0.701589i	$0.247526\pi$
$101$			1.46015e8i			1.40318i	−0.712582	+	0.701589i	$0.752474\pi$
$102$	8.72700e6	+	1.59956e6i	0.0806239	+	0.0147775i
$103$			1.60527e8i			1.42626i	0.701032	+	0.713130i	$0.252723\pi$
$103$			1.60527e8i			1.42626i	−0.701032	+	0.713130i	$0.747277\pi$
$104$		−	5.86733e7i		−	0.501542i
$105$		0			0
$106$	3.97673e7			0.314995
$107$	−1.10586e8			−0.843654			−0.421827	−	0.906676i	$-0.638611\pi$
$107$	−1.10586e8			−0.843654			−0.421827	+	0.906676i	$0.638611\pi$
$108$	−1.12235e8	−	6.78626e7i	−0.824960	−	0.498810i
$109$	−1.53065e8			−1.08435			−0.542176	−	0.840265i	$-0.682400\pi$
$109$	−1.53065e8			−1.08435			−0.542176	+	0.840265i	$0.682400\pi$
$110$		0			0
$111$	971367.	−	5.29965e6i	0.00639869	−	0.0349104i
$112$		−	3.49750e6i		−	0.0222273i
$113$	−1.01707e8			−0.623791			−0.311895	−	0.950116i	$-0.600964\pi$
$113$	−1.01707e8			−0.623791			−0.311895	+	0.950116i	$0.600964\pi$
$114$	3.30714e7	+	6.06163e6i	0.195810	+	0.0358897i
$115$		0			0
$116$		−	2.21183e8i		−	1.22157i
$117$	−8.94953e7	+	2.35936e8i	−0.477592	+	1.25907i
$118$			4.80638e7i			0.247908i
$119$		−	2.15648e6i		−	0.0107537i
$120$		0			0
$121$	−2.64620e8			−1.23447
$122$	−1.06647e7			−0.0481402
$123$	5.50000e7	−	3.00073e8i	0.240294	−	1.31101i
$124$	−1.67927e7			−0.0710287
$125$		0			0
$126$	421744.	−	1.11184e6i	0.00167327	−	0.00441123i
$127$		−	4.85620e8i		−	1.86673i	−0.358928	−	0.933365i	$-0.616858\pi$
$127$		−	4.85620e8i		−	1.86673i	0.358928	−	0.933365i	$-0.383142\pi$
$128$	−1.85705e8			−0.691804
$129$	−2.97893e7	+	1.62527e8i	−0.107573	+	0.586902i
$130$		0			0
$131$			2.26001e8i			0.767407i	0.923456	+	0.383704i	$0.125352\pi$
$131$			2.26001e8i			0.767407i	−0.923456	+	0.383704i	$0.874648\pi$
$132$	−7.88749e7	+	4.30331e8i	−0.259802	+	1.41745i
$133$		−	8.17211e6i		−	0.0261173i
$134$			4.22648e7i			0.131087i
$135$		0			0
$136$	−5.50738e7			−0.160986
$137$	5.46283e8			1.55073			0.775364	−	0.631515i	$-0.217566\pi$
$137$	5.46283e8			1.55073			0.775364	+	0.631515i	$0.217566\pi$
$138$	5.69802e7	+	1.04438e7i	0.157111	+	0.0287967i
$139$	−2.15686e8			−0.577779			−0.288890	−	0.957362i	$-0.593286\pi$
$139$	−2.15686e8			−0.577779			−0.288890	+	0.957362i	$0.593286\pi$
$140$		0			0
$141$	4.72816e8	+	8.66618e7i	1.19623	+	0.219256i
$142$		−	4.59815e7i		−	0.113091i
$143$	8.41730e8			2.01293
$144$	3.59178e8	+	1.36244e8i	0.835333	+	0.316859i
$145$		0			0
$146$		−	1.19292e8i		−	0.262544i
$147$	4.59013e8	+	8.41320e7i	0.983005	+	0.180174i
$148$			1.64162e7i			0.0342157i
$149$			4.32442e8i			0.877371i	0.898641	+	0.438685i	$0.144556\pi$
$149$			4.32442e8i			0.877371i	−0.898641	+	0.438685i	$0.855444\pi$
$150$		0			0
$151$	−3.22279e8			−0.619904			−0.309952	−	0.950752i	$-0.600313\pi$
$151$	−3.22279e8			−0.619904			−0.309952	+	0.950752i	$0.600313\pi$
$152$	−2.08705e8			−0.390984
$153$	2.21461e8	+	8.40049e7i	0.404140	+	0.153299i
$154$	−3.96663e6			−0.00705242
$155$		0			0
$156$	1.38610e8	−	7.56239e8i	0.234043	−	1.27691i
$157$		−	5.59602e8i		−	0.921045i	−0.887648	−	0.460522i	$-0.847662\pi$
$157$		−	5.59602e8i		−	0.921045i	0.887648	−	0.460522i	$-0.152338\pi$
$158$	−9.57615e7			−0.153661
$159$	1.04424e9	+	1.91398e8i	1.63385	+	0.299466i
$160$		0			0
$161$		−	1.40801e7i		−	0.0209557i
$162$	9.77522e7	+	8.66225e7i	0.141927	+	0.125768i
$163$			4.67465e8i			0.662215i	0.943593	+	0.331107i	$0.107422\pi$
$163$			4.67465e8i			0.662215i	−0.943593	+	0.331107i	$0.892578\pi$
$164$			9.29504e8i			1.28492i
$165$		0			0
$166$	−1.48273e8			−0.195267
$167$	4.54837e8			0.584776			0.292388	−	0.956300i	$-0.405550\pi$
$167$	4.54837e8			0.584776			0.292388	+	0.956300i	$0.405550\pi$
$168$	−1.33076e6	+	7.26044e6i	−0.00167056	+	0.00911435i
$169$	−6.63477e8			−0.813353
$170$		0			0
$171$	8.39240e8	+	3.18341e8i	0.981526	+	0.372313i
$172$		−	5.03442e8i		−	0.575222i
$173$	8.28969e8			0.925452			0.462726	−	0.886501i	$-0.346871\pi$
$173$	8.28969e8			0.925452			0.462726	+	0.886501i	$0.346871\pi$
$174$	−3.97099e7	+	2.16652e8i	−0.0433214	+	0.236356i
$175$		0			0
$176$		−	1.28141e9i		−	1.33548i
$177$	−2.31328e8	+	1.26209e9i	−0.235686	+	1.28587i
$178$		−	3.31519e8i		−	0.330239i
$179$		−	1.84168e8i		−	0.179392i	−0.995969	−	0.0896959i	$-0.971410\pi$
$179$		−	1.84168e8i		−	0.179392i	0.995969	−	0.0896959i	$-0.0285895\pi$
$180$		0			0
$181$	−1.14139e9			−1.06345			−0.531727	−	0.846916i	$-0.678457\pi$
$181$	−1.14139e9			−1.06345			−0.531727	+	0.846916i	$0.678457\pi$
$182$	6.97072e6			0.00635319
$183$	−2.80041e8	−	5.13283e7i	−0.249699	−	0.0457670i
$184$	−3.59587e8			−0.313713
$185$		0			0
$186$	1.64487e7	+	3.01486e6i	0.0137429	+	0.00251893i
$187$		−	7.90091e8i		−	0.646116i
$188$	−1.46459e9			−1.17243
$189$	1.64257e7	−	2.71657e7i	0.0128729	−	0.0212899i
$190$		0			0
$191$			1.62451e9i			1.22064i	0.792153	+	0.610322i	$0.208960\pi$
$191$			1.62451e9i			1.22064i	−0.792153	+	0.610322i	$0.791040\pi$
$192$	−1.05686e9	−	1.93710e8i	−0.777698	−	0.142543i
$193$		−	1.50247e9i		−	1.08287i	−0.840743	−	0.541434i	$-0.817882\pi$
$193$		−	1.50247e9i		−	1.08287i	0.840743	−	0.541434i	$-0.182118\pi$
$194$		−	3.52257e8i		−	0.248687i
$195$		0			0
$196$	−1.42184e9			−0.963442
$197$	1.00297e9			0.665924			0.332962	−	0.942940i	$-0.391952\pi$
$197$	1.00297e9			0.665924			0.332962	+	0.942940i	$0.391952\pi$
$198$	1.54518e8	−	4.07355e8i	0.100535	−	0.265041i
$199$	−6.24974e8			−0.398520			−0.199260	−	0.979947i	$-0.563854\pi$
$199$	−6.24974e8			−0.398520			−0.199260	+	0.979947i	$0.563854\pi$
$200$		0			0
$201$	−2.03418e8	+	1.10982e9i	−0.124625	+	0.679936i
$202$			4.43031e8i			0.266090i
$203$	5.35358e7			0.0315254
$204$	−7.09845e8	−	1.30107e8i	−0.409867	−	0.0751240i
$205$		0			0
$206$			4.87061e8i			0.270467i
$207$	1.44596e9	+	5.48484e8i	0.787545	+	0.298732i
$208$			2.25188e9i			1.20307i
$209$		−	2.99410e9i		−	1.56921i
$210$		0			0
$211$	3.84677e9			1.94073			0.970367	−	0.241636i	$-0.0776839\pi$
$211$	3.84677e9			1.94073			0.970367	+	0.241636i	$0.0776839\pi$
$212$	−3.23463e9			−1.60133
$213$	2.21306e8	−	1.20742e9i	0.107516	−	0.586596i
$214$	−3.35533e8			−0.159985
$215$		0			0
$216$	−6.93776e8	−	4.19490e8i	−0.318716	−	0.192711i
$217$		−	4.06455e6i		−	0.00183305i
$218$	−4.64422e8			−0.205630
$219$	5.74147e8	−	3.13247e9i	0.249601	−	1.36179i
$220$		0			0
$221$			1.38846e9i			0.582055i
$222$	2.94727e6	−	1.60799e7i	0.00121341	−	0.00662020i
$223$			3.37297e9i			1.36393i	0.731383	+	0.681967i	$0.238875\pi$
$223$			3.37297e9i			1.36393i	−0.731383	+	0.681967i	$0.761125\pi$
$224$		−	3.39408e7i		−	0.0134812i
$225$		0			0
$226$	−3.08595e8			−0.118292
$227$	−2.59338e9			−0.976705			−0.488352	−	0.872646i	$-0.662402\pi$
$227$	−2.59338e9			−0.976705			−0.488352	+	0.872646i	$0.662402\pi$
$228$	−2.69000e9	−	4.93046e8i	−0.995434	−	0.182452i
$229$	2.81270e9			1.02278			0.511389	−	0.859349i	$-0.329131\pi$
$229$	2.81270e9			1.02278			0.511389	+	0.859349i	$0.329131\pi$
$230$		0			0
$231$	−1.04159e8	−	1.90911e7i	−0.0365803	−	0.00670476i
$232$		−	1.36724e9i		−	0.471945i
$233$	−3.25957e9			−1.10595			−0.552977	−	0.833196i	$-0.686508\pi$
$233$	−3.25957e9			−1.10595			−0.552977	+	0.833196i	$0.686508\pi$
$234$	−2.71542e8	+	7.15862e8i	−0.0905676	+	0.238762i
$235$		0			0
$236$		−	3.90946e9i		−	1.26028i
$237$	−2.51458e9	−	4.60894e8i	−0.797024	−	0.146086i
$238$		−	6.54308e6i		−	0.00203927i
$239$		−	1.97409e9i		−	0.605028i	−0.953145	−	0.302514i	$-0.902174\pi$
$239$		−	1.97409e9i		−	0.605028i	0.953145	−	0.302514i	$-0.0978260\pi$
$240$		0			0
$241$	4.48773e8			0.133033			0.0665163	−	0.997785i	$-0.478812\pi$
$241$	4.48773e8			0.133033			0.0665163	+	0.997785i	$0.478812\pi$
$242$	−8.02895e8			−0.234098
$243$	2.14994e9	+	2.74507e9i	0.616597	+	0.787279i
$244$	8.67453e8			0.244730
$245$		0			0
$246$	1.66878e8	−	9.10464e8i	0.0455678	−	0.248612i
$247$			5.26165e9i			1.41362i
$248$	−1.03803e8			−0.0274413
$249$	−3.89345e9	−	7.13626e8i	−1.01283	−	0.185641i
$250$		0			0
$251$			2.31424e9i			0.583060i	0.956562	+	0.291530i	$0.0941644\pi$
$251$			2.31424e9i			0.583060i	−0.956562	+	0.291530i	$0.905836\pi$
$252$	−3.43042e7	+	9.04359e7i	−0.00850639	+	0.0224253i
$253$		−	5.15865e9i		−	1.25908i
$254$		−	1.47344e9i		−	0.353995i
$255$		0			0
$256$	2.83238e9			0.659464
$257$	5.64145e9			1.29318			0.646589	−	0.762839i	$-0.276195\pi$
$257$	5.64145e9			1.29318			0.646589	+	0.762839i	$0.276195\pi$
$258$	−9.03850e7	+	4.93129e8i	−0.0203994	+	0.111296i
$259$	−3.97342e6			−0.000883010
$260$		0			0
$261$	−2.08547e9	+	5.49789e9i	−0.449408	+	1.18477i
$262$			6.85721e8i			0.145526i
$263$	3.58501e9			0.749320			0.374660	−	0.927162i	$-0.377760\pi$
$263$	3.58501e9			0.749320			0.374660	+	0.927162i	$0.377760\pi$
$264$	−4.87562e8	+	2.66008e9i	−0.100372	+	0.547619i
$265$		0			0
$266$		−	2.47954e7i		−	0.00495272i
$267$	1.59558e9	−	8.70527e9i	0.313959	−	1.71292i
$268$		−	3.43777e9i		−	0.666405i
$269$			6.68399e9i			1.27652i	0.769822	+	0.638259i	$0.220345\pi$
$269$			6.68399e9i			1.27652i	−0.769822	+	0.638259i	$0.779655\pi$
$270$		0			0
$271$	−3.71996e9			−0.689702			−0.344851	−	0.938658i	$-0.612070\pi$
$271$	−3.71996e9			−0.689702			−0.344851	+	0.938658i	$0.612070\pi$
$272$	2.11373e9			0.386166
$273$	1.83042e8	+	3.35496e7i	0.0329535	+	0.00604000i
$274$	1.65750e9			0.294071
$275$		0			0
$276$	−4.63471e9	−	8.49490e8i	−0.798704	−	0.146394i
$277$			3.64024e9i			0.618317i	0.951011	+	0.309158i	$0.100047\pi$
$277$			3.64024e9i			0.618317i	−0.951011	+	0.309158i	$0.899953\pi$
$278$	−6.54421e8			−0.109567
$279$	4.17412e8	+	1.58333e8i	0.0688887	+	0.0261309i
$280$		0			0
$281$		−	1.06642e9i		−	0.171041i	−0.996336	−	0.0855207i	$-0.972745\pi$
$281$		−	1.06642e9i		−	0.171041i	0.996336	−	0.0855207i	$-0.0272554\pi$
$282$	1.43459e9	+	2.62944e8i	0.226846	+	0.0415784i
$283$		−	4.50485e9i		−	0.702319i	−0.936316	−	0.351160i	$-0.885787\pi$
$283$		−	4.50485e9i		−	0.702319i	0.936316	−	0.351160i	$-0.114213\pi$
$284$			3.74009e9i			0.574921i
$285$		0			0
$286$	2.55393e9			0.381720
$287$	−2.24980e8			−0.0331601
$288$	3.48557e9	+	1.32215e9i	0.506645	+	0.192181i
$289$	−5.67248e9			−0.813170
$290$		0			0
$291$	1.69539e9	−	9.24982e9i	0.236427	−	1.28992i
$292$			9.70312e9i			1.33469i
$293$	7.62525e9			1.03463			0.517313	−	0.855796i	$-0.326932\pi$
$293$	7.62525e9			1.03463			0.517313	+	0.855796i	$0.326932\pi$
$294$	1.39271e9	+	2.55268e8i	0.186411	+	0.0341671i
$295$		0			0
$296$			1.01476e8i			0.0132189i
$297$	6.01803e9	−	9.95295e9i	0.773443	−	1.27916i
$298$			1.31209e9i			0.166379i
$299$			9.06552e9i			1.13425i
$300$		0			0
$301$	1.21855e8			0.0148448
$302$	−9.77840e8			−0.117555
$303$	−2.13228e9	+	1.16334e10i	−0.252972	+	1.38019i
$304$	8.01010e9			0.937873
$305$		0			0
$306$	6.71945e8	+	2.54883e8i	0.0766387	+	0.0290707i
$307$		−	5.35640e9i		−	0.603003i	−0.953466	−	0.301501i	$-0.902512\pi$
$307$		−	5.35640e9i		−	0.603003i	0.953466	−	0.301501i	$-0.0974878\pi$
$308$	3.22641e8			0.0358523
$309$	−2.34419e9	+	1.27896e10i	−0.257134	+	1.40289i
$310$		0			0
$311$		−	8.46968e9i		−	0.905369i	−0.891671	−	0.452685i	$-0.850466\pi$
$311$		−	8.46968e9i		−	0.905369i	0.891671	−	0.452685i	$-0.149534\pi$
$312$	8.56814e8	−	4.67466e9i	0.0904207	−	0.493324i
$313$		−	3.96394e9i		−	0.413000i	−0.978447	−	0.206500i	$-0.933793\pi$
$313$		−	3.96394e9i		−	0.413000i	0.978447	−	0.206500i	$-0.0662073\pi$
$314$		−	1.69791e9i		−	0.174661i
$315$		0			0
$316$	7.78914e9			0.781162
$317$	1.64274e9			0.162679			0.0813393	−	0.996686i	$-0.474080\pi$
$317$	1.64274e9			0.162679			0.0813393	+	0.996686i	$0.474080\pi$
$318$	3.16837e9	+	5.80728e8i	0.309833	+	0.0567890i
$319$	1.96144e10			1.89414
$320$		0			0
$321$	−8.81067e9	−	1.61490e9i	−0.829830	−	0.152099i
$322$		−	4.27210e7i		−	0.00397391i
$323$	4.93885e9			0.453750
$324$	−7.95106e9	−	7.04578e9i	−0.721514	−	0.639365i
$325$		0			0
$326$			1.41836e9i			0.125578i
$327$	−1.21951e10	−	2.23523e9i	−1.06658	−	0.195493i
$328$			5.74570e9i			0.496418i
$329$		−	3.54494e8i		−	0.0302570i
$330$		0			0
$331$	7.10076e9			0.591552			0.295776	−	0.955257i	$-0.404422\pi$
$331$	7.10076e9			0.591552			0.295776	+	0.955257i	$0.404422\pi$
$332$	1.20603e10			0.992675
$333$	1.54783e8	−	4.08053e8i	0.0125877	−	0.0331848i
$334$	1.38004e9			0.110893
$335$		0			0
$336$	5.10745e7	−	2.78656e8i	0.00400725	−	0.0218630i
$337$			2.32757e10i			1.80461i	0.431099	+	0.902305i	$0.358126\pi$
$337$			2.32757e10i			1.80461i	−0.431099	+	0.902305i	$0.641874\pi$
$338$	−2.01308e9			−0.154239
$339$	−8.10331e9	−	1.48525e9i	−0.613569	−	0.112460i
$340$		0			0
$341$		−	1.48917e9i		−	0.110135i
$342$	2.54637e9	+	9.65893e8i	0.186131	+	0.0706032i
$343$		−	6.88505e8i		−	0.0497428i
$344$		−	3.11201e9i		−	0.222232i
$345$		0			0
$346$	2.51521e9			0.175497
$347$	−1.85680e9			−0.128070			−0.0640351	−	0.997948i	$-0.520397\pi$
$347$	−1.85680e9			−0.128070			−0.0640351	+	0.997948i	$0.520397\pi$
$348$	3.22996e9	−	1.76223e10i	0.220232	−	1.20156i
$349$	−1.48512e10			−1.00106			−0.500529	−	0.865720i	$-0.666861\pi$
$349$	−1.48512e10			−1.00106			−0.500529	+	0.865720i	$0.666861\pi$
$350$		0			0
$351$	−1.05757e10	+	1.74907e10i	−0.696758	+	1.15234i
$352$		−	1.24352e10i		−	0.809994i
$353$	−1.49807e10			−0.964791			−0.482396	−	0.875954i	$-0.660233\pi$
$353$	−1.49807e10			−0.964791			−0.482396	+	0.875954i	$0.660233\pi$
$354$	−7.01881e8	+	3.82937e9i	−0.0446941	+	0.243845i
$355$		0			0
$356$			2.69654e10i			1.67883i
$357$	3.14914e7	−	1.71813e8i	0.00193874	−	0.0105775i
$358$		−	5.58793e8i		−	0.0340188i
$359$		−	2.40350e10i		−	1.44699i	−0.690327	−	0.723497i	$-0.742534\pi$
$359$		−	2.40350e10i		−	1.44699i	0.690327	−	0.723497i	$-0.257466\pi$
$360$		0			0
$361$	1.73251e9			0.102011
$362$	−3.46313e9			−0.201667
$363$	−2.10830e10	−	3.86428e9i	−1.21424	−	0.222557i
$364$	−5.66991e8			−0.0322976
$365$		0			0
$366$	−8.49683e8	−	1.55737e8i	−0.0473514	−	0.00867898i
$367$			6.35979e9i			0.350573i	0.984517	+	0.175287i	$0.0560852\pi$
$367$			6.35979e9i			0.350573i	−0.984517	+	0.175287i	$0.943915\pi$
$368$	1.38009e10			0.752519
$369$	8.76400e9	−	2.31045e10i	0.472712	−	1.24621i
$370$		0			0
$371$		−	7.82921e8i		−	0.0413259i
$372$	−1.33792e9	−	2.45226e8i	−0.0698648	−	0.0128054i
$373$		−	9.55992e9i		−	0.493877i	−0.969031	−	0.246939i	$-0.920575\pi$
$373$		−	9.55992e9i		−	0.493877i	0.969031	−	0.246939i	$-0.0794246\pi$
$374$		−	2.39725e9i		−	0.122526i
$375$		0			0
$376$	−9.05333e9			−0.452957
$377$	−3.44693e10			−1.70634
$378$	4.98378e7	−	8.24245e7i	0.00244114	−	0.00403728i
$379$	−1.47270e9			−0.0713767			−0.0356883	−	0.999363i	$-0.511362\pi$
$379$	−1.47270e9			−0.0713767			−0.0356883	+	0.999363i	$0.511362\pi$
$380$		0			0
$381$	7.09157e9	−	3.86907e10i	0.336544	−	1.83614i
$382$			4.92900e9i			0.231476i
$383$	−3.53991e10			−1.64512			−0.822559	−	0.568680i	$-0.807454\pi$
$383$	−3.53991e10			−1.64512			−0.822559	+	0.568680i	$0.807454\pi$
$384$	−1.47956e10	−	2.71187e9i	−0.680468	−	0.124722i
$385$		0			0
$386$		−	4.55870e9i		−	0.205349i
$387$	−4.74679e9	+	1.25139e10i	−0.211620	+	0.557891i
$388$			2.86522e10i			1.26424i
$389$			1.00866e10i			0.440502i	0.975443	+	0.220251i	$0.0706877\pi$
$389$			1.00866e10i			0.440502i	−0.975443	+	0.220251i	$0.929312\pi$
$390$		0			0
$391$	8.50936e9			0.364074
$392$	−8.78904e9			−0.372218
$393$	−3.30033e9	+	1.80062e10i	−0.138352	+	0.754833i
$394$	3.04316e9			0.126282
$395$		0			0
$396$	−1.25684e10	+	3.31338e10i	−0.511090	+	1.34738i
$397$		−	1.16090e10i		−	0.467339i	−0.972316	−	0.233670i	$-0.924927\pi$
$397$		−	1.16090e10i		−	0.467339i	0.972316	−	0.233670i	$-0.0750734\pi$
$398$	−1.89626e9			−0.0755729
$399$	1.19338e8	−	6.51095e8i	0.00470857	−	0.0256893i
$400$		0			0
$401$		−	1.62151e10i		−	0.627108i	−0.949570	−	0.313554i	$-0.898480\pi$
$401$		−	1.62151e10i		−	0.627108i	0.949570	−	0.313554i	$-0.101520\pi$
$402$	−6.17198e8	+	3.36735e9i	−0.0236331	+	0.128939i
$403$			2.61698e9i			0.0992157i
$404$		−	3.60357e10i		−	1.35272i
$405$		0			0
$406$	1.62435e8			0.00597828
$407$	−1.45578e9			−0.0530540
$408$	−4.38788e9	−	8.04249e8i	−0.158349	−	0.0290235i
$409$	8.05756e9			0.287945			0.143973	−	0.989582i	$-0.454012\pi$
$409$	8.05756e9			0.287945			0.143973	+	0.989582i	$0.454012\pi$
$410$		0			0
$411$	4.35239e10	+	7.97744e9i	1.52532	+	0.279574i
$412$		−	3.96171e10i		−	1.37497i
$413$	9.46257e8			0.0325244
$414$	4.38726e9	+	1.66418e9i	0.149345	+	0.0566498i
$415$		0			0
$416$			2.18529e10i			0.729685i
$417$	−1.71843e10	−	3.14969e9i	−0.568312	−	0.104165i
$418$		−	9.08451e9i		−	0.297575i
$419$		−	1.33632e10i		−	0.433567i	−0.976220	−	0.216783i	$-0.930443\pi$
$419$		−	1.33632e10i		−	0.433567i	0.976220	−	0.216783i	$-0.0695565\pi$
$420$		0			0
$421$	−4.39642e10			−1.39949			−0.699747	−	0.714391i	$-0.746704\pi$
$421$	−4.39642e10			−1.39949			−0.699747	+	0.714391i	$0.746704\pi$
$422$	1.16716e10			0.368029
$423$	3.64050e10	+	1.38092e10i	1.13710	+	0.431327i
$424$	−1.99948e10			−0.618662
$425$		0			0
$426$	6.71474e8	−	3.66347e9i	0.0203888	−	0.111238i
$427$			2.09961e8i			0.00631578i
$428$	2.72919e10			0.813315
$429$	6.70630e10	+	1.22919e10i	1.97995	+	0.362902i
$430$		0			0
$431$		−	1.47674e10i		−	0.427950i	−0.976839	−	0.213975i	$-0.931359\pi$
$431$		−	1.47674e10i		−	0.427950i	0.976839	−	0.213975i	$-0.0686412\pi$
$432$	2.66271e10	+	1.61000e10i	0.764521	+	0.462266i
$433$			1.57125e8i			0.00446986i	0.999998	+	0.00223493i	$0.000711401\pi$
$433$			1.57125e8i			0.00446986i	−0.999998	+	0.00223493i	$0.999289\pi$
$434$		−	1.23324e7i		−	0.000347608i
$435$		0			0
$436$	3.77756e10			1.04536
$437$	3.22467e10			0.884219
$438$	1.74204e9	−	9.50436e9i	0.0473328	−	0.258242i
$439$	−6.17824e10			−1.66344			−0.831719	−	0.555197i	$-0.812643\pi$
$439$	−6.17824e10			−1.66344			−0.831719	+	0.555197i	$0.812643\pi$
$440$		0			0
$441$	3.53423e10	+	1.34061e10i	0.934416	+	0.354443i
$442$			4.21279e9i			0.110377i
$443$	−8.26926e9			−0.214710			−0.107355	−	0.994221i	$-0.534238\pi$
$443$	−8.26926e9			−0.214710			−0.107355	+	0.994221i	$0.534238\pi$
$444$	−2.39727e8	+	1.30792e9i	−0.00616859	+	0.0336550i
$445$		0			0
$446$			1.02341e10i			0.258648i
$447$	−6.31501e9	+	3.44539e10i	−0.158177	+	0.862995i
$448$			7.92379e8i			0.0196708i
$449$			7.95461e10i			1.95719i	0.205788	+	0.978597i	$0.434024\pi$
$449$			7.95461e10i			1.95719i	−0.205788	+	0.978597i	$0.565976\pi$
$450$		0			0
$451$	−8.24281e10			−1.99236
$452$	2.51008e10			0.601358
$453$	−2.56769e10	−	4.70628e9i	−0.609746	−	0.111760i
$454$	−7.86870e9			−0.185216
$455$		0			0
$456$	−1.66281e10	−	3.04775e9i	−0.384578	−	0.0704888i
$457$		−	2.02591e10i		−	0.464467i	−0.972660	−	0.232234i	$-0.925397\pi$
$457$		−	2.02591e10i		−	0.464467i	0.972660	−	0.232234i	$-0.0746034\pi$
$458$	8.53413e9			0.193953
$459$	1.64177e10	+	9.92693e9i	0.369881	+	0.223648i
$460$		0			0
$461$		−	3.97783e10i		−	0.880729i	−0.897819	−	0.440364i	$-0.854849\pi$
$461$		−	3.97783e10i		−	0.880729i	0.897819	−	0.440364i	$-0.145151\pi$
$462$	−3.16032e8	−	5.79251e7i	−0.00693687	−	0.00127145i
$463$		−	3.84754e10i		−	0.837258i	−0.908157	−	0.418629i	$-0.862511\pi$
$463$		−	3.84754e10i		−	0.837258i	0.908157	−	0.418629i	$-0.137489\pi$
$464$			5.24745e10i			1.13208i
$465$		0			0
$466$	−9.89001e9			−0.209726
$467$	−2.20915e10			−0.464470			−0.232235	−	0.972660i	$-0.574604\pi$
$467$	−2.20915e10			−0.464470			−0.232235	+	0.972660i	$0.574604\pi$
$468$	2.20869e10	−	5.82275e10i	0.460417	−	1.21379i
$469$	8.32089e8			0.0171980
$470$		0			0
$471$	8.17194e9	−	4.45850e10i	0.166051	−	0.905953i
$472$		−	2.41662e10i		−	0.486900i
$473$	4.46450e10			0.891925
$474$	−7.62958e9	−	1.39842e9i	−0.151143	−	0.0277028i
$475$		0			0
$476$			5.32207e8i			0.0103670i
$477$	8.04025e10	+	3.04984e10i	1.55309	+	0.589119i
$478$		−	5.98968e9i		−	0.114734i
$479$			2.39823e10i			0.455564i	0.973712	+	0.227782i	$0.0731473\pi$
$479$			2.39823e10i			0.455564i	−0.973712	+	0.227782i	$0.926853\pi$
$480$		0			0
$481$	2.55830e9			0.0477938
$482$	1.36164e9			0.0252275
$483$	2.05613e8	−	1.12180e9i	0.00377800	−	0.0206123i
$484$	6.53066e10			1.19008
$485$		0			0
$486$	6.52323e9	+	8.32894e9i	0.116928	+	0.149295i
$487$		−	4.42866e10i		−	0.787329i	−0.919254	−	0.393664i	$-0.871207\pi$
$487$		−	4.42866e10i		−	0.787329i	0.919254	−	0.393664i	$-0.128793\pi$
$488$	5.36213e9			0.0945492
$489$	−6.82645e9	+	3.72442e10i	−0.119388	+	0.651364i
$490$		0			0
$491$		−	3.70873e10i		−	0.638116i	−0.947735	−	0.319058i	$-0.896634\pi$
$491$		−	3.70873e10i		−	0.638116i	0.947735	−	0.319058i	$-0.103366\pi$
$492$	−1.35737e10	+	7.40562e10i	−0.231652	+	1.26387i
$493$			3.23546e10i			0.547707i
$494$			1.59646e10i			0.268071i
$495$		0			0
$496$	3.98398e9			0.0658249
$497$	−9.05262e8			−0.0148371
$498$	−1.18133e10	−	2.16524e9i	−0.192067	−	0.0352038i
$499$	−4.56047e10			−0.735541			−0.367771	−	0.929917i	$-0.619879\pi$
$499$	−4.56047e10			−0.735541			−0.367771	+	0.929917i	$0.619879\pi$
$500$		0			0
$501$	3.62381e10	+	6.64204e9i	0.575194	+	0.105427i
$502$			7.02173e9i			0.110568i
$503$	−3.73919e10			−0.584125			−0.292063	−	0.956399i	$-0.594342\pi$
$503$	−3.73919e10			−0.584125			−0.292063	+	0.956399i	$0.594342\pi$
$504$	−2.12050e8	+	5.59026e8i	−0.00328637	+	0.00866383i
$505$		0			0
$506$		−	1.56521e10i		−	0.238765i
$507$	−5.28610e10	−	9.68884e9i	−0.800026	−	0.146636i
$508$			1.19848e11i			1.79960i
$509$		−	4.63948e9i		−	0.0691192i	−0.999403	−	0.0345596i	$-0.988997\pi$
$509$		−	4.63948e9i		−	0.0691192i	0.999403	−	0.0345596i	$-0.0110029\pi$
$510$		0			0
$511$	−2.34857e9			−0.0344445
$512$	5.61342e10			0.816861
$513$	6.22158e10	+	3.76187e10i	0.898321	+	0.543168i
$514$	1.71170e10			0.245230
$515$		0			0
$516$	7.35182e9	−	4.01106e10i	0.103704	−	0.565796i
$517$		−	1.29879e11i		−	1.81793i
$518$	−1.20559e7			−0.000167449
$519$	6.60462e10	+	1.21055e10i	0.910288	+	0.166846i
$520$		0			0
$521$			5.90088e10i			0.800877i	0.916324	+	0.400439i	$0.131142\pi$
$521$			5.90088e10i			0.800877i	−0.916324	+	0.400439i	$0.868858\pi$
$522$	−6.32760e9	+	1.66814e10i	−0.0852230	+	0.224673i
$523$			6.02622e9i			0.0805449i	0.999189	+	0.0402725i	$0.0128226\pi$
$523$			6.02622e9i			0.0805449i	−0.999189	+	0.0402725i	$0.987177\pi$
$524$		−	5.57758e10i		−	0.739811i
$525$		0			0
$526$	1.08774e10			0.142096
$527$	2.45643e9			0.0318465
$528$	1.87126e10	−	1.02094e11i	0.240768	−	1.31360i
$529$	−2.27518e10			−0.290531
$530$		0			0
$531$	−3.68610e10	+	9.71764e10i	−0.463649	+	1.22231i
$532$			2.01683e9i			0.0251781i
$533$	1.44854e11			1.79483
$534$	4.84121e9	−	2.64130e10i	0.0595373	−	0.324828i
$535$		0			0
$536$		−	2.12505e10i		−	0.257460i
$537$	2.68943e9	−	1.46732e10i	0.0323417	−	0.176452i
$538$			2.02802e10i			0.242071i
$539$		−	1.26088e11i		−	1.49389i
$540$		0			0
$541$	4.00361e10			0.467372			0.233686	−	0.972312i	$-0.424921\pi$
$541$	4.00361e10			0.467372			0.233686	+	0.972312i	$0.424921\pi$
$542$	−1.12869e10			−0.130791
$543$	−9.09374e10	−	1.66678e10i	−1.04603	−	0.191725i
$544$	2.05123e10			0.234217
$545$		0			0
$546$	5.55377e8	+	1.01794e8i	0.00624909	+	0.00114539i
$547$			7.70856e10i			0.861041i	0.902581	+	0.430521i	$0.141670\pi$
$547$			7.70856e10i			0.861041i	−0.902581	+	0.430521i	$0.858330\pi$
$548$	−1.34819e11			−1.49496
$549$	−2.15621e10	−	8.17894e9i	−0.237356	−	0.0900342i
$550$		0			0
$551$			1.22610e11i			1.33020i
$552$	−2.86493e10	−	5.25109e9i	−0.308573	−	0.0565579i
$553$			1.88531e9i			0.0201596i
$554$			1.10450e10i			0.117254i
$555$		0			0
$556$	5.32299e10			0.557002
$557$	−1.00834e10			−0.104758			−0.0523789	−	0.998627i	$-0.516680\pi$
$557$	−1.00834e10			−0.104758			−0.0523789	+	0.998627i	$0.516680\pi$
$558$	1.26649e9	+	4.80405e8i	0.0130636	+	0.00495531i
$559$	−7.84565e10			−0.803492
$560$		0			0
$561$	1.15378e10	−	6.29487e10i	0.116485	−	0.635529i
$562$		−	3.23566e9i		−	0.0324352i
$563$	−1.17038e11			−1.16492			−0.582458	−	0.812861i	$-0.697909\pi$
$563$	−1.17038e11			−1.16492			−0.582458	+	0.812861i	$0.697909\pi$
$564$	−1.16688e11	−	2.13876e10i	−1.15321	−	0.211371i
$565$		0			0
$566$		−	1.36684e10i		−	0.133184i
$567$	1.70538e9	−	1.92450e9i	0.0165002	−	0.0186202i
$568$			2.31192e10i			0.222116i
$569$		−	2.10738e10i		−	0.201045i	−0.994935	−	0.100522i	$-0.967949\pi$
$569$		−	2.10738e10i		−	0.201045i	0.994935	−	0.100522i	$-0.0320514\pi$
$570$		0			0
$571$	−1.66672e11			−1.56790			−0.783949	−	0.620826i	$-0.786798\pi$
$571$	−1.66672e11			−1.56790			−0.783949	+	0.620826i	$0.786798\pi$
$572$	−2.07734e11			−1.94054
$573$	−2.37229e10	+	1.29429e11i	−0.220065	+	1.20064i
$574$	−6.82621e8			−0.00628829
$575$		0			0
$576$	−8.13739e10	−	3.08668e10i	−0.739256	−	0.280415i
$577$			9.88790e10i			0.892074i	0.895014	+	0.446037i	$0.147165\pi$
$577$			9.88790e10i			0.892074i	−0.895014	+	0.446037i	$0.852835\pi$
$578$	−1.72111e10			−0.154205
$579$	2.19407e10	−	1.19706e11i	0.195225	−	1.06512i
$580$		0			0
$581$			2.91912e9i			0.0256181i
$582$	5.14405e9	−	2.80653e10i	0.0448346	−	0.244612i
$583$		−	2.86846e11i		−	2.48299i
$584$			5.99795e10i			0.515646i
$585$		0			0
$586$	2.31361e10			0.196200
$587$	−4.94023e10			−0.416097			−0.208049	−	0.978118i	$-0.566711\pi$
$587$	−4.94023e10			−0.416097			−0.208049	+	0.978118i	$0.566711\pi$
$588$	−1.13282e11	−	2.07633e10i	−0.947656	−	0.173695i
$589$	9.30879e9			0.0773450
$590$		0			0
$591$	7.99097e10	+	1.46465e10i	0.655012	+	0.120056i
$592$		−	3.89465e9i		−	0.0317089i
$593$	−5.39077e10			−0.435945			−0.217973	−	0.975955i	$-0.569944\pi$
$593$	−5.39077e10			−0.435945			−0.217973	+	0.975955i	$0.569944\pi$
$594$	1.82596e10	−	3.01987e10i	0.146671	−	0.242573i
$595$		0			0
$596$		−	1.06724e11i		−	0.845820i
$597$	−4.97934e10	−	9.12658e9i	−0.391990	−	0.0718474i
$598$			2.75061e10i			0.215092i
$599$		−	1.84260e11i		−	1.43128i	−0.698469	−	0.715640i	$-0.746135\pi$
$599$		−	1.84260e11i		−	1.43128i	0.698469	−	0.715640i	$-0.253865\pi$
$600$		0			0
$601$	−4.09269e9			−0.0313698			−0.0156849	−	0.999877i	$-0.504993\pi$
$601$	−4.09269e9			−0.0313698			−0.0156849	+	0.999877i	$0.504993\pi$
$602$	3.69724e8			0.00281509
$603$	−3.24137e10	+	8.54519e10i	−0.245165	+	0.646327i
$604$	7.95365e10			0.597611
$605$		0			0
$606$	−6.46964e9	+	3.52975e10i	−0.0479722	+	0.261730i
$607$			1.45907e11i			1.07478i	0.843333	+	0.537392i	$0.180590\pi$
$607$			1.45907e11i			1.07478i	−0.843333	+	0.537392i	$0.819410\pi$
$608$	7.77323e10			0.568836
$609$	4.26535e9	+	7.81790e8i	0.0310088	+	0.00568357i
$610$		0			0
$611$			2.28243e11i			1.63769i
$612$	−5.46553e10	−	2.07319e10i	−0.389607	−	0.147786i
$613$		−	9.26923e10i		−	0.656450i	−0.944600	−	0.328225i	$-0.893550\pi$
$613$		−	9.26923e10i		−	0.656450i	0.944600	−	0.328225i	$-0.106450\pi$
$614$		−	1.62521e10i		−	0.114350i
$615$		0			0
$616$	1.99440e9			0.0138512
$617$	2.27799e11			1.57185			0.785925	−	0.618322i	$-0.212187\pi$
$617$	2.27799e11			1.57185			0.785925	+	0.618322i	$0.212187\pi$
$618$	−7.11262e9	+	3.88055e10i	−0.0487613	+	0.266036i
$619$	−1.05797e11			−0.720627			−0.360313	−	0.932831i	$-0.617330\pi$
$619$	−1.05797e11			−0.720627			−0.360313	+	0.932831i	$0.617330\pi$
$620$		0			0
$621$	1.07194e11	+	6.48148e10i	0.720784	+	0.435820i
$622$		−	2.56982e10i		−	0.171689i
$623$	−6.52679e9			−0.0433259
$624$	−3.28845e10	+	1.79414e11i	−0.216897	+	1.18336i
$625$		0			0
$626$		−	1.20272e10i		−	0.0783187i
$627$	4.37231e10	−	2.38548e11i	0.282905	−	1.54350i
$628$			1.38106e11i			0.887923i
$629$		−	2.40135e9i		−	0.0153410i
$630$		0			0
$631$	2.99698e11			1.89045			0.945227	−	0.326414i	$-0.105840\pi$
$631$	2.99698e11			1.89045			0.945227	+	0.326414i	$0.105840\pi$
$632$	4.81483e10			0.301795
$633$	3.06483e11	+	5.61748e10i	1.90893	+	0.349886i
$634$	4.98429e9			0.0308494
$635$		0			0
$636$	−2.57712e11	−	4.72358e10i	−1.57509	−	0.288697i
$637$			2.21580e11i			1.34577i
$638$	5.95129e10			0.359194
$639$	3.52641e10	−	9.29665e10i	0.211509	−	0.557600i
$640$		0			0
$641$		−	7.79955e10i		−	0.461995i	−0.972954	−	0.230998i	$-0.925801\pi$
$641$		−	7.79955e10i		−	0.461995i	0.972954	−	0.230998i	$-0.0741989\pi$
$642$	−2.67328e10	−	4.89983e9i	−0.157364	−	0.0288430i
$643$			1.10441e11i			0.646081i	0.946385	+	0.323041i	$0.104705\pi$
$643$			1.10441e11i			0.646081i	−0.946385	+	0.323041i	$0.895295\pi$
$644$			3.47488e9i			0.0202021i
$645$		0			0
$646$	1.49852e10			0.0860463
$647$	−1.12591e11			−0.642521			−0.321261	−	0.946991i	$-0.604107\pi$
$647$	−1.12591e11			−0.642521			−0.321261	+	0.946991i	$0.604107\pi$
$648$	−4.91492e10	−	4.35533e10i	−0.278751	−	0.247014i
$649$	3.46689e11			1.95417
$650$		0			0
$651$	5.93552e7	−	3.23834e8i	0.000330472	−	0.00180301i
$652$		−	1.15368e11i		−	0.638401i
$653$	1.11018e11			0.610578			0.305289	−	0.952260i	$-0.401247\pi$
$653$	1.11018e11			0.610578			0.305289	+	0.952260i	$0.401247\pi$
$654$	−3.70018e10	−	6.78201e9i	−0.202261	−	0.0370721i
$655$		0			0
$656$		−	2.20520e11i		−	1.19078i
$657$	9.14877e10	−	2.41188e11i	0.491022	−	1.29448i
$658$		−	1.07559e9i		−	0.00573775i
$659$			1.20450e11i			0.638653i	0.947645	+	0.319326i	$0.103457\pi$
$659$			1.20450e11i			0.638653i	−0.947645	+	0.319326i	$0.896543\pi$
$660$		0			0
$661$	−3.14150e10			−0.164563			−0.0822813	−	0.996609i	$-0.526221\pi$
$661$	−3.14150e10			−0.164563			−0.0822813	+	0.996609i	$0.526221\pi$
$662$	2.15447e10			0.112178
$663$	−2.02759e10	+	1.10622e11i	−0.104936	+	0.572518i
$664$	7.45506e10			0.383512
$665$		0			0
$666$	4.69634e8	−	1.23809e9i	0.00238705	−	0.00629297i
$667$			2.11249e11i			1.06731i
$668$	−1.12251e11			−0.563747
$669$	−4.92559e10	+	2.68734e11i	−0.245897	+	1.34158i
$670$		0			0
$671$			7.69253e10i			0.379472i
$672$	4.95641e8	−	2.70415e9i	0.00243047	−	0.0132603i
$673$		−	3.66345e11i		−	1.78579i	−0.450265	−	0.892895i	$-0.648671\pi$
$673$		−	3.66345e11i		−	1.78579i	0.450265	−	0.892895i	$-0.351329\pi$
$674$			7.06218e10i			0.342215i
$675$		0			0
$676$	1.63742e11			0.784104
$677$	2.51302e11			1.19630			0.598152	−	0.801382i	$-0.295902\pi$
$677$	2.51302e11			1.19630			0.598152	+	0.801382i	$0.295902\pi$
$678$	−2.45866e10	−	4.50645e9i	−0.116354	−	0.0213263i
$679$	−6.93506e9			−0.0326266
$680$		0			0
$681$	−2.06622e11	−	3.78715e10i	−0.960701	−	0.176086i
$682$		−	4.51835e9i		−	0.0208854i
$683$	2.45748e11			1.12929			0.564646	−	0.825333i	$-0.309013\pi$
$683$	2.45748e11			1.12929			0.564646	+	0.825333i	$0.309013\pi$
$684$	−2.07119e11	−	7.85647e10i	−0.946230	−	0.358925i
$685$		0			0
$686$		−	2.08902e9i		−	0.00943293i
$687$	2.24095e11	+	4.10742e10i	1.00602	+	0.184392i
$688$			1.19439e11i			0.533079i
$689$			5.04087e11i			2.23681i
$690$		0			0
$691$	−1.44391e11			−0.633326			−0.316663	−	0.948538i	$-0.602562\pi$
$691$	−1.44391e11			−0.633326			−0.316663	+	0.948538i	$0.602562\pi$
$692$	−2.04584e11			−0.892171
$693$	−8.01981e9	−	3.04208e9i	−0.0347721	−	0.0131898i
$694$	−5.63381e9			−0.0242864
$695$		0			0
$696$	1.99659e10	−	1.08931e11i	0.0850848	−	0.464212i
$697$		−	1.35968e11i		−	0.576109i
$698$	−4.50606e10			−0.189835
$699$	−2.59699e11	−	4.76000e10i	−1.08783	−	0.199388i
$700$		0			0
$701$		−	2.23367e11i		−	0.925009i	−0.886617	−	0.462505i	$-0.846951\pi$
$701$		−	2.23367e11i		−	0.925009i	0.886617	−	0.462505i	$-0.153049\pi$
$702$	−3.20883e10	+	5.30694e10i	−0.132129	+	0.218522i
$703$		−	9.10007e9i		−	0.0372583i
$704$			2.90312e11i			1.18188i
$705$		0			0
$706$	−4.54536e10			−0.182957
$707$	8.72218e9			0.0349098
$708$	5.70903e10	−	3.11477e11i	0.227211	−	1.23963i
$709$	8.44074e9			0.0334038			0.0167019	−	0.999861i	$-0.494683\pi$
$709$	8.44074e9			0.0334038			0.0167019	+	0.999861i	$0.494683\pi$
$710$		0			0
$711$	−1.93613e11	−	7.34414e10i	−0.757627	−	0.287384i
$712$			1.66686e11i			0.648602i
$713$	1.60385e10			0.0620591
$714$	9.55494e7	−	5.21305e8i	0.000367650	−	0.00200585i
$715$		0			0
$716$			4.54516e10i			0.172941i
$717$	2.88279e10	−	1.57281e11i	0.109078	−	0.595115i
$718$		−	7.29257e10i		−	0.274399i
$719$		−	3.06358e11i		−	1.14634i	−0.819436	−	0.573171i	$-0.805713\pi$
$719$		−	3.06358e11i		−	1.14634i	0.819436	−	0.573171i	$-0.194287\pi$
$720$		0			0
$721$	9.58903e9			0.0354841
$722$	5.25668e9			0.0193447
$723$	3.57550e10	+	6.55348e9i	0.130853	+	0.0239839i
$724$	2.81687e11			1.02521
$725$		0			0
$726$	−6.39689e10	−	1.17248e10i	−0.230262	−	0.0422044i
$727$			2.35218e10i			0.0842040i	0.999113	+	0.0421020i	$0.0134054\pi$
$727$			2.35218e10i			0.0842040i	−0.999113	+	0.0421020i	$0.986595\pi$
$728$	−3.50484e9			−0.0124779
$729$	1.31205e11	+	2.50103e11i	0.464559	+	0.885542i
$730$		0			0
$731$			7.36433e10i			0.257907i
$732$	6.91123e10	+	1.26675e10i	0.240719	+	0.0441212i
$733$			6.21715e10i			0.215365i	0.994185	+	0.107683i	$0.0343430\pi$
$733$			6.21715e10i			0.215365i	−0.994185	+	0.107683i	$0.965657\pi$
$734$			1.92965e10i			0.0664806i
$735$		0			0
$736$	1.33928e11			0.456416
$737$	3.04860e11			1.03331
$738$	2.65912e10	−	7.01022e10i	0.0896423	−	0.236323i
$739$	3.85080e10			0.129114			0.0645570	−	0.997914i	$-0.479437\pi$
$739$	3.85080e10			0.129114			0.0645570	+	0.997914i	$0.479437\pi$
$740$		0			0
$741$	−7.68365e10	+	4.19210e11i	−0.254856	+	1.39046i
$742$		−	2.37549e9i		−	0.00783679i
$743$	−1.81380e10			−0.0595160			−0.0297580	−	0.999557i	$-0.509474\pi$
$743$	−1.81380e10			−0.0595160			−0.0297580	+	0.999557i	$0.509474\pi$
$744$	−8.27031e9	−	1.51586e9i	−0.0269917	−	0.00494727i
$745$		0			0
$746$		−	2.90062e10i		−	0.0936558i
$747$	−2.99781e11	−	1.13713e11i	−0.962767	−	0.365198i
$748$			1.94990e11i			0.622881i
$749$			6.60582e9i			0.0209894i
$750$		0			0
$751$	4.66054e11			1.46513			0.732566	−	0.680696i	$-0.238323\pi$
$751$	4.66054e11			1.46513			0.732566	+	0.680696i	$0.238323\pi$
$752$	3.47466e11			1.08653
$753$	−3.37951e10	+	1.84382e11i	−0.105117	+	0.573506i
$754$	−1.04585e11			−0.323581
$755$		0			0
$756$	−4.05376e9	+	6.70433e9i	−0.0124100	+	0.0205243i
$757$			1.01684e11i			0.309649i	0.987942	+	0.154825i	$0.0494812\pi$
$757$			1.01684e11i			0.309649i	−0.987942	+	0.154825i	$0.950519\pi$
$758$	−4.46837e9			−0.0135354
$759$	7.53324e10	−	4.11004e11i	0.226994	−	1.23845i
$760$		0			0
$761$		−	1.11642e9i		−	0.00332880i	−0.999999	−	0.00166440i	$-0.999470\pi$
$761$		−	1.11642e9i		−	0.00332880i	0.999999	−	0.00166440i	$-0.000529796\pi$
$762$	2.15168e10	−	1.17393e11i	0.0638202	−	0.348195i
$763$			9.14331e9i			0.0269777i
$764$		−	4.00920e11i		−	1.17675i
$765$		0			0
$766$	−1.07406e11			−0.311970
$767$	−6.09251e11			−1.76041
$768$	2.25663e11	+	4.13615e10i	0.648658	+	0.118892i
$769$	−5.25128e11			−1.50162			−0.750810	−	0.660519i	$-0.770336\pi$
$769$	−5.25128e11			−1.50162			−0.750810	+	0.660519i	$0.770336\pi$
$770$		0			0
$771$	4.49470e11	+	8.23828e10i	1.27199	+	0.233141i
$772$			3.70800e11i			1.04393i
$773$	1.88899e11			0.529067			0.264534	−	0.964376i	$-0.414782\pi$
$773$	1.88899e11			0.529067			0.264534	+	0.964376i	$0.414782\pi$
$774$	−1.44024e10	+	3.79690e10i	−0.0401303	+	0.105795i
$775$		0			0
$776$			1.77113e11i			0.488430i
$777$	−3.16573e8	−	5.80244e7i	−0.000868541	−	0.000159194i
$778$			3.06043e10i			0.0835342i
$779$		−	5.15257e11i		−	1.39918i
$780$		0			0
$781$	−3.31670e11			−0.891459
$782$	2.58186e10			0.0690408
$783$	−2.46441e11	+	4.07578e11i	−0.655641	+	1.08434i
$784$	3.37323e11			0.892857
$785$		0			0
$786$	−1.00137e10	+	5.46333e10i	−0.0262363	+	0.143142i
$787$		−	6.60029e11i		−	1.72054i	−0.509842	−	0.860268i	$-0.670296\pi$
$787$		−	6.60029e11i		−	1.72054i	0.509842	−	0.860268i	$-0.329704\pi$
$788$	−2.47528e11			−0.641977
$789$	2.85627e11	+	5.23523e10i	0.737042	+	0.135091i
$790$		0			0
$791$			6.07547e9i			0.0155194i
$792$	−7.76909e10	+	2.04816e11i	−0.197455	+	0.520550i
$793$		−	1.35184e11i		−	0.341848i
$794$		−	3.52233e10i		−	0.0886234i
$795$		0			0
$796$	1.54240e11			0.384189
$797$	−5.73881e11			−1.42229			−0.711146	−	0.703044i	$-0.751824\pi$
$797$	−5.73881e11			−1.42229			−0.711146	+	0.703044i	$0.751824\pi$
$798$	3.62090e8	−	1.97551e9i	0.000892904	−	0.00487157i
$799$	2.14240e11			0.525671
$800$		0			0
$801$	2.54248e11	−	6.70272e11i	0.617629	−	1.62825i
$802$		−	4.91990e10i		−	0.118921i
$803$	−8.60469e11			−2.06954
$804$	5.02022e10	−	2.73897e11i	0.120143	−	0.655485i
$805$		0			0
$806$			7.94029e9i			0.0188147i
$807$	−9.76071e10	+	5.32532e11i	−0.230138	+	1.25560i
$808$		−	2.22753e11i		−	0.522611i
$809$			2.07320e11i			0.484002i	0.970276	+	0.242001i	$0.0778038\pi$
$809$			2.07320e11i			0.484002i	−0.970276	+	0.242001i	$0.922196\pi$
$810$		0			0
$811$	−6.06207e11			−1.40132			−0.700660	−	0.713495i	$-0.747111\pi$
$811$	−6.06207e11			−1.40132			−0.700660	+	0.713495i	$0.747111\pi$
$812$	−1.32123e10			−0.0303917
$813$	−2.96380e11	−	5.43231e10i	−0.678400	−	0.124343i
$814$	−4.41704e9			−0.0100608
$815$		0			0
$816$	1.68407e11	+	3.08671e10i	0.379838	+	0.0696201i
$817$			2.79076e11i			0.626374i
$818$	2.44478e10			0.0546042
$819$	1.40936e10	+	5.34598e9i	0.0313246	+	0.0118821i
$820$		0			0
$821$		−	1.56657e11i		−	0.344808i	−0.985026	−	0.172404i	$-0.944846\pi$
$821$		−	1.56657e11i		−	0.344808i	0.985026	−	0.172404i	$-0.0551535\pi$
$822$	1.32058e11	+	2.42047e10i	0.289252	+	0.0530167i
$823$			3.65723e11i			0.797173i	0.917131	+	0.398587i	$0.130499\pi$
$823$			3.65723e11i			0.797173i	−0.917131	+	0.398587i	$0.869501\pi$
$824$		−	2.44892e11i		−	0.531208i
$825$		0			0
$826$	2.87108e9			0.00616772
$827$	−5.22532e11			−1.11710			−0.558549	−	0.829472i	$-0.688642\pi$
$827$	−5.22532e11			−1.11710			−0.558549	+	0.829472i	$0.688642\pi$
$828$	−3.56855e11	−	1.35362e11i	−0.759224	−	0.287990i
$829$	−7.56111e11			−1.60091			−0.800456	−	0.599391i	$-0.795409\pi$
$829$	−7.56111e11			−1.60091			−0.800456	+	0.599391i	$0.795409\pi$
$830$		0			0
$831$	−5.31589e10	+	2.90028e11i	−0.111474	+	0.608185i
$832$		−	5.10177e11i		−	1.06470i
$833$	2.07986e11			0.431971
$834$	−5.21395e10	−	9.55659e9i	−0.107771	−	0.0197533i
$835$		0			0
$836$			7.38925e11i			1.51278i
$837$	3.09442e10	+	1.87103e10i	0.0630489	+	0.0381224i
$838$		−	4.05460e10i		−	0.0822189i
$839$		−	4.63197e11i		−	0.934799i	−0.884046	−	0.467400i	$-0.845191\pi$
$839$		−	4.63197e11i		−	0.934799i	0.884046	−	0.467400i	$-0.154809\pi$
$840$		0			0
$841$	−3.02974e11			−0.605649
$842$	−1.33394e11			−0.265391
$843$	1.55730e10	−	8.49642e10i	0.0308363	−	0.168239i
$844$	−9.49359e11			−1.87094
$845$		0			0
$846$	1.10458e11	+	4.18990e10i	0.215633	+	0.0817942i
$847$			1.58070e10i			0.0307126i
$848$	7.67399e11			1.48401
$849$	6.57849e10	−	3.58914e11i	0.126618	−	0.690811i
$850$		0			0
$851$		−	1.56789e10i		−	0.0298949i
$852$	−5.46170e10	+	2.97983e11i	−0.103650	+	0.565501i
$853$		−	6.78685e11i		−	1.28195i	−0.767561	−	0.640976i	$-0.778530\pi$
$853$		−	6.78685e11i		−	1.28195i	0.767561	−	0.640976i	$-0.221470\pi$
$854$			6.37051e8i			0.00119769i
$855$		0			0
$856$	1.68704e11			0.314217
$857$	1.02265e12			1.89584			0.947921	−	0.318506i	$-0.103181\pi$
$857$	1.02265e12			1.89584			0.947921	+	0.318506i	$0.103181\pi$
$858$	2.03479e11	+	3.72953e10i	0.375465	+	0.0688185i
$859$	1.64678e11			0.302457			0.151228	−	0.988499i	$-0.451677\pi$
$859$	1.64678e11			0.302457			0.151228	+	0.988499i	$0.451677\pi$
$860$		0			0
$861$	−1.79248e10	−	3.28541e9i	−0.0326168	−	0.00597829i
$862$		−	4.48062e10i		−	0.0811539i
$863$	−9.97986e11			−1.79921			−0.899604	−	0.436707i	$-0.856145\pi$
$863$	−9.97986e11			−1.79921			−0.899604	+	0.436707i	$0.856145\pi$
$864$	2.58397e11	+	1.56239e11i	0.463695	+	0.280373i
$865$		0			0
$866$			4.76740e8i			0.000847637i
$867$	−4.51942e11	−	8.28359e10i	−0.799846	−	0.146603i
$868$			1.00311e9i			0.00176713i
$869$			6.90738e11i			1.21125i
$870$		0			0
$871$	−5.35744e11			−0.930861
$872$	2.33508e11			0.403865
$873$	2.70153e11	−	7.12201e11i	0.465106	−	1.22616i
$874$	9.78411e10			0.167678
$875$		0			0
$876$	−1.41696e11	+	7.73075e11i	−0.240625	+	1.31282i
$877$			2.30212e10i			0.0389160i	0.999811	+	0.0194580i	$0.00619407\pi$
$877$			2.30212e10i			0.0389160i	−0.999811	+	0.0194580i	$0.993806\pi$
$878$	−1.87457e11			−0.315444
$879$	6.07525e11	+	1.11352e11i	1.01767	+	0.186528i
$880$		0			0
$881$		−	4.23415e10i		−	0.0702851i	−0.999382	−	0.0351425i	$-0.988811\pi$
$881$		−	4.23415e10i		−	0.0702851i	0.999382	−	0.0351425i	$-0.0111885\pi$
$882$	1.07233e11	+	4.06759e10i	0.177197	+	0.0672145i
$883$			8.49512e11i			1.39742i	0.715406	+	0.698709i	$0.246242\pi$
$883$			8.49512e11i			1.39742i	−0.715406	+	0.698709i	$0.753758\pi$
$884$		−	3.42664e11i		−	0.561124i
$885$		0			0
$886$	−2.50901e10			−0.0407162
$887$	2.62546e11			0.424142			0.212071	−	0.977254i	$-0.431979\pi$
$887$	2.62546e11			0.424142			0.212071	+	0.977254i	$0.431979\pi$
$888$	−1.48187e9	+	8.08488e9i	−0.00238318	+	0.0130023i
$889$	−2.90084e10			−0.0464426
$890$		0			0
$891$	6.24817e11	−	7.05097e11i	0.991384	−	1.11876i
$892$		−	8.32429e11i		−	1.31489i
$893$	8.11876e11			1.27668
$894$	−1.91606e10	+	1.04538e11i	−0.0299958	+	0.163653i
$895$		0			0
$896$			1.10930e10i			0.0172115i
$897$	−1.32385e11	+	7.22275e11i	−0.204488	+	1.11566i
$898$			2.41354e11i			0.371150i
$899$			6.09822e10i			0.0933608i
$900$		0			0
$901$	4.73162e11			0.717977
$902$	−2.50098e11			−0.377820
$903$	9.70848e9	+	1.77946e9i	0.0146016	+	0.00267631i
$904$	1.55160e11			0.232330
$905$		0			0
$906$	−7.79072e10	−	1.42795e10i	−0.115629	−	0.0211934i
$907$		−	8.67841e11i		−	1.28236i	−0.767389	−	0.641182i	$-0.778445\pi$
$907$		−	8.67841e11i		−	1.28236i	0.767389	−	0.641182i	$-0.221555\pi$
$908$	6.40032e11			0.941582
$909$	−3.39769e11	+	8.95730e11i	−0.497655	+	1.31196i
$910$		0			0
$911$		−	7.78413e11i		−	1.13015i	−0.825039	−	0.565075i	$-0.808847\pi$
$911$		−	7.78413e11i		−	1.13015i	0.825039	−	0.565075i	$-0.191153\pi$
$912$	6.38187e11	+	1.16973e11i	0.922505	+	0.169085i
$913$			1.06951e12i			1.53922i
$914$		−	6.14690e10i		−	0.0880787i
$915$		0			0
$916$	−6.94157e11			−0.985998
$917$	1.35001e10			0.0190924
$918$	4.98136e10	+	3.01197e10i	0.0701419	+	0.0424112i
$919$	3.08965e11			0.433159			0.216579	−	0.976265i	$-0.430510\pi$
$919$	3.08965e11			0.433159			0.216579	+	0.976265i	$0.430510\pi$
$920$		0			0
$921$	7.82202e10	−	4.26759e11i	0.108713	−	0.593122i
$922$		−	1.20693e11i		−	0.167016i
$923$	5.82857e11			0.803073
$924$	2.57057e10	+	4.71157e9i	0.0352648	+	0.00646365i
$925$		0			0
$926$		−	1.16740e11i		−	0.158772i
$927$	−3.73537e11	+	9.84752e11i	−0.505842	+	1.33355i
$928$			5.09227e11i			0.686625i
$929$			1.36481e12i			1.83235i	0.400781	+	0.916174i	$0.368739\pi$
$929$			1.36481e12i			1.83235i	−0.400781	+	0.916174i	$0.631261\pi$
$930$		0			0
$931$	7.88175e11			1.04912
$932$	8.04443e11			1.06618
$933$	1.23684e11	−	6.74803e11i	0.163225	−	0.890534i
$934$	−6.70288e10			−0.0880793
$935$		0			0
$936$	1.36529e11	−	3.59931e11i	0.177878	−	0.468939i
$937$		−	9.26920e11i		−	1.20250i	−0.799062	−	0.601249i	$-0.794670\pi$
$937$		−	9.26920e11i		−	1.20250i	0.799062	−	0.601249i	$-0.205330\pi$
$938$	2.52468e9			0.00326133
$939$	5.78859e10	−	3.15818e11i	0.0744578	−	0.406232i
$940$		0			0
$941$		−	4.30760e11i		−	0.549385i	−0.961532	−	0.274692i	$-0.911424\pi$
$941$		−	4.30760e11i		−	0.549385i	0.961532	−	0.274692i	$-0.0885761\pi$
$942$	2.47948e10	−	1.35277e11i	0.0314889	−	0.171799i
$943$		−	8.87758e11i		−	1.12266i
$944$			9.27497e11i			1.16795i
$945$		0			0
$946$	1.35459e11			0.169139
$947$	−5.20879e11			−0.647645			−0.323823	−	0.946118i	$-0.604968\pi$
$947$	−5.20879e11			−0.647645			−0.323823	+	0.946118i	$0.604968\pi$
$948$	6.20582e11	+	1.13746e11i	0.768362	+	0.140832i
$949$	1.51214e12			1.86435
$950$		0			0
$951$	1.30881e11	+	2.39891e10i	0.160013	+	0.0293286i
$952$			3.28982e9i			0.00400520i
$953$	−1.01657e12			−1.23243			−0.616217	−	0.787576i	$-0.711336\pi$
$953$	−1.01657e12			−1.23243			−0.616217	+	0.787576i	$0.711336\pi$
$954$	2.43953e11	+	9.25363e10i	0.294518	+	0.111717i
$955$		0			0
$956$			4.87194e11i			0.583271i
$957$	1.56274e12	+	2.86432e11i	1.86311	+	0.341487i
$958$			7.27658e10i			0.0863903i
$959$		−	3.26321e10i		−	0.0385807i
$960$		0			0
$961$	−8.48261e11			−0.994572
$962$	7.76226e9			0.00906333
$963$	−6.78388e11	−	2.57327e11i	−0.788811	−	0.299213i
$964$	−1.10754e11			−0.128249
$965$		0			0
$966$	6.23860e8	−	3.40370e9i	0.000716438	−	0.00390879i
$967$			1.25609e12i			1.43653i	0.695769	+	0.718266i	$0.255064\pi$
$967$			1.25609e12i			1.43653i	−0.695769	+	0.718266i	$0.744936\pi$
$968$	4.03691e11			0.459777
$969$	3.93492e11	+	7.21227e10i	0.446315	+	0.0818045i
$970$		0			0
$971$		−	3.77457e11i		−	0.424610i	−0.977203	−	0.212305i	$-0.931903\pi$
$971$		−	3.77457e11i		−	0.424610i	0.977203	−	0.212305i	$-0.0680971\pi$
$972$	−5.30592e11	−	6.77467e11i	−0.594423	−	0.758968i
$973$			1.28839e10i			0.0143746i
$974$		−	1.34372e11i		−	0.149304i
$975$		0			0
$976$	−2.05798e11			−0.226800
$977$	−4.90276e11			−0.538100			−0.269050	−	0.963126i	$-0.586710\pi$
$977$	−4.90276e11			−0.538100			−0.269050	+	0.963126i	$0.586710\pi$
$978$	−2.07124e10	+	1.13004e11i	−0.0226400	+	0.123521i
$979$	−2.39128e12			−2.60315
$980$		0			0
$981$	−9.38978e11	−	3.56174e11i	−1.01386	−	0.384580i
$982$		−	1.12528e11i		−	0.121008i
$983$	8.66068e11			0.927551			0.463775	−	0.885953i	$-0.346494\pi$
$983$	8.66068e11			0.927551			0.463775	+	0.885953i	$0.346494\pi$
$984$	−8.39052e10	+	4.57776e11i	−0.0894970	+	0.488284i
$985$		0			0
$986$			9.81685e10i			0.103864i
$987$	5.17672e9	−	2.82435e10i	0.00545490	−	0.0297612i
$988$		−	1.29854e12i		−	1.36279i
$989$			4.80831e11i			0.502582i
$990$		0			0
$991$	5.45022e10			0.0565093			0.0282546	−	0.999601i	$-0.491005\pi$
$991$	5.45022e10			0.0565093			0.0282546	+	0.999601i	$0.491005\pi$
$992$	3.86616e10			0.0399240
$993$	5.65737e11	+	1.03693e11i	0.581859	+	0.106648i
$994$	−2.74669e9			−0.00281362
$995$		0			0
$996$	9.60880e11	+	1.76119e11i	0.976409	+	0.178965i
$997$			7.54906e10i			0.0764033i	0.999270	+	0.0382017i	$0.0121629\pi$
$997$			7.54906e10i			0.0764033i	−0.999270	+	0.0382017i	$0.987837\pi$
$998$	−1.38371e11			−0.139484
$999$	1.82908e10	−	3.02504e10i	0.0183642	−	0.0303717i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.d.d.74.12		20
3.2	odd	2	inner	75.9.d.d.74.10		20
5.2	odd	4		75.9.c.f.26.6	yes	10
5.3	odd	4		75.9.c.e.26.5	✓	10
5.4	even	2	inner	75.9.d.d.74.9		20
15.2	even	4		75.9.c.f.26.5	yes	10
15.8	even	4		75.9.c.e.26.6	yes	10
15.14	odd	2	inner	75.9.d.d.74.11		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.c.e.26.5	✓	10	5.3	odd	4
75.9.c.e.26.6	yes	10	15.8	even	4
75.9.c.f.26.5	yes	10	15.2	even	4
75.9.c.f.26.6	yes	10	5.2	odd	4
75.9.d.d.74.9		20	5.4	even	2	inner
75.9.d.d.74.10		20	3.2	odd	2	inner
75.9.d.d.74.11		20	15.14	odd	2	inner
75.9.d.d.74.12		20	1.1	even	1	trivial

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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.03414 q^{2} +(79.6728 + 14.6031i) q^{3} -246.794 q^{4} +(241.739 + 44.3080i) q^{6} -59.7348i q^{7} -1525.55 q^{8} +(6134.50 + 2326.94i) q^{9} +O(q^{10})\)	\(q+3.03414 q^{2} +(79.6728 + 14.6031i) q^{3} -246.794 q^{4} +(241.739 + 44.3080i) q^{6} -59.7348i q^{7} -1525.55 q^{8} +(6134.50 + 2326.94i) q^{9} -21885.6i q^{11} +(-19662.8 - 3603.96i) q^{12} +38460.5i q^{13} -181.244i q^{14} +58550.5 q^{16} +36101.0 q^{17} +(18612.9 + 7060.28i) q^{18} +136807. q^{19} +(872.314 - 4759.23i) q^{21} -66404.0i q^{22} +235710. q^{23} +(-121545. - 22277.8i) q^{24} +116695. i q^{26} +(454772. + 274977. i) q^{27} +14742.2i q^{28} +896225. i q^{29} +68043.4 q^{31} +568191. q^{32} +(319598. - 1.74369e6i) q^{33} +109535. q^{34} +(-1.51396e6 - 574275. i) q^{36} -66517.7i q^{37} +415091. q^{38} +(-561643. + 3.06425e6i) q^{39} -3.76632e6i q^{41} +(2646.73 - 14440.2i) q^{42} +2.03993e6i q^{43} +5.40123e6i q^{44} +715178. q^{46} +5.93447e6 q^{47} +(4.66488e6 + 855021. i) q^{48} +5.76123e6 q^{49} +(2.87626e6 + 527187. i) q^{51} -9.49181e6i q^{52} +1.31066e7 q^{53} +(1.37984e6 + 834319. i) q^{54} +91128.3i q^{56} +(1.08998e7 + 1.99781e6i) q^{57} +2.71928e6i q^{58} +1.58410e7i q^{59} -3.51489e6 q^{61} +206453. q^{62} +(138999. - 366443. i) q^{63} -1.32650e7 q^{64} +(969706. - 5.29059e6i) q^{66} +1.39297e7i q^{67} -8.90950e6 q^{68} +(1.87797e7 + 3.44210e6i) q^{69} -1.51547e7i q^{71} +(-9.35847e6 - 3.54986e6i) q^{72} -3.93167e7i q^{73} -201824. i q^{74} -3.37631e7 q^{76} -1.30733e6 q^{77} +(-1.70411e6 + 9.29738e6i) q^{78} -3.15613e7 q^{79} +(3.22174e7 + 2.85492e7i) q^{81} -1.14275e7i q^{82} -4.88680e7 q^{83} +(-215282. + 1.17455e6i) q^{84} +6.18943e6i q^{86} +(-1.30877e7 + 7.14048e7i) q^{87} +3.33875e7i q^{88} -1.09263e8i q^{89} +2.29743e6 q^{91} -5.81718e7 q^{92} +(5.42120e6 + 993646. i) q^{93} +1.80060e7 q^{94} +(4.52694e7 + 8.29737e6i) q^{96} -1.16098e8i q^{97} +1.74804e7 q^{98} +(5.09265e7 - 1.34257e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9}+O(q^{10}) \) 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9	\( 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9} + 89268 q^{16} + 287868 q^{19} + 1346856 q^{21} + 2033718 q^{24} - 6028120 q^{31} - 9954292 q^{34} + 9001054 q^{36} + 15026564 q^{39} - 27877272 q^{46} - 17907092 q^{49} + 12418574 q^{51} + 17772544 q^{54} + 3040440 q^{61} + 9073996 q^{64} + 20930590 q^{66} - 22789956 q^{69} - 59058092 q^{76} - 17099792 q^{79} + 45225890 q^{81} + 273766584 q^{84} - 214448528 q^{91} - 712237192 q^{94} + 1050848002 q^{96} + 764842670 q^{99}+O(q^{100}) \) 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9 + 89268 * q^16 + 287868 * q^19 + 1346856 * q^21 + 2033718 * q^24 - 6028120 * q^31 - 9954292 * q^34 + 9001054 * q^36 + 15026564 * q^39 - 27877272 * q^46 - 17907092 * q^49 + 12418574 * q^51 + 17772544 * q^54 + 3040440 * q^61 + 9073996 * q^64 + 20930590 * q^66 - 22789956 * q^69 - 59058092 * q^76 - 17099792 * q^79 + 45225890 * q^81 + 273766584 * q^84 - 214448528 * q^91 - 712237192 * q^94 + 1050848002 * q^96 + 764842670 * q^99

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