LMFDB - Embedded newform 76.11.c.b.37.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,11,Mod(37,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("76.37");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	76.c (of order $2$, degree $1$, 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:	$48.2871512032$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 604871 x^{12} + 143853611883 x^{10} + \cdots + 10\!\cdots\!00 $ x^14 + 604871x^12 + 143853611883x^10 + 17017320063766869x^8 + 1043743496089962538800x^6 + 31381033749119511427838016x^4 + 379176566191191014686305546240x^2 + 1060823187210043470011170317926400
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{35}\cdot 3^{6}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.9
Root			$142.542i$ of defining polynomial
Character	$\chi$	$=$	76.37
Dual form			76.11.c.b.37.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+142.542i q^{3} +3674.44 q^{5} -13444.2 q^{7} +38730.8 q^{9} +O(q^{10})$	$q+142.542i q^{3} +3674.44 q^{5} -13444.2 q^{7} +38730.8 q^{9} -100674. q^{11} -620510. i q^{13} +523762. i q^{15} +698070. q^{17} +(-682231. - 2.38026e6i) q^{19} -1.91636e6i q^{21} -1.05232e7 q^{23} +3.73592e6 q^{25} +1.39377e7i q^{27} +3.25817e6i q^{29} +2.10556e7i q^{31} -1.43502e7i q^{33} -4.93999e7 q^{35} -7.96901e7i q^{37} +8.84486e7 q^{39} -2.25906e8i q^{41} +2.27907e8 q^{43} +1.42314e8 q^{45} -1.47784e8 q^{47} -1.01729e8 q^{49} +9.95042e7i q^{51} +1.13727e7i q^{53} -3.69920e8 q^{55} +(3.39286e8 - 9.72465e7i) q^{57} -4.79625e8i q^{59} +1.12598e9 q^{61} -5.20704e8 q^{63} -2.28003e9i q^{65} -1.88202e9i q^{67} -1.49999e9i q^{69} +1.27797e8i q^{71} -1.59310e9 q^{73} +5.32525e8i q^{75} +1.35348e9 q^{77} -5.62792e9i q^{79} +3.00309e8 q^{81} +4.12884e9 q^{83} +2.56502e9 q^{85} -4.64425e8 q^{87} -6.15474e9i q^{89} +8.34225e9i q^{91} -3.00130e9 q^{93} +(-2.50682e9 - 8.74613e9i) q^{95} +7.20642e9i q^{97} -3.89918e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 5062 q^{5} - 39624 q^{7} - 383056 q^{9}+O(q^{10}) $ 14 * q + 5062 * q^5 - 39624 * q^7 - 383056 * q^9	$ 14 q + 5062 q^{5} - 39624 q^{7} - 383056 q^{9} + 59746 q^{11} - 1287896 q^{17} + 4622604 q^{19} - 11539958 q^{23} - 417808 q^{25} - 77028922 q^{35} + 14606790 q^{39} + 71330230 q^{43} - 169810130 q^{45} - 394833110 q^{47} - 265204650 q^{49} - 612557042 q^{55} + 319590486 q^{57} - 1396847538 q^{61} - 3080375402 q^{63} - 7242583772 q^{73} - 922511302 q^{77} - 9163778710 q^{81} - 9328943264 q^{83} - 13063772714 q^{85} + 11593996398 q^{87} + 16259309364 q^{93} + 11710957630 q^{95} - 20299698578 q^{99}+O(q^{100}) $ 14 * q + 5062 * q^5 - 39624 * q^7 - 383056 * q^9 + 59746 * q^11 - 1287896 * q^17 + 4622604 * q^19 - 11539958 * q^23 - 417808 * q^25 - 77028922 * q^35 + 14606790 * q^39 + 71330230 * q^43 - 169810130 * q^45 - 394833110 * q^47 - 265204650 * q^49 - 612557042 * q^55 + 319590486 * q^57 - 1396847538 * q^61 - 3080375402 * q^63 - 7242583772 * q^73 - 922511302 * q^77 - 9163778710 * q^81 - 9328943264 * q^83 - 13063772714 * q^85 + 11593996398 * q^87 + 16259309364 * q^93 + 11710957630 * q^95 - 20299698578 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			142.542i			0.586592i	0.956022	+	0.293296i	$0.0947521\pi$
$3$			142.542i			0.586592i	−0.956022	+	0.293296i	$0.905248\pi$
$4$		0			0
$5$	3674.44			1.17582			0.587911	−	0.808925i	$-0.299950\pi$
$5$	3674.44			1.17582			0.587911	+	0.808925i	$0.299950\pi$
$6$		0			0
$7$	−13444.2			−0.799916			−0.399958	−	0.916534i	$-0.630975\pi$
$7$	−13444.2			−0.799916			−0.399958	+	0.916534i	$0.630975\pi$
$8$		0			0
$9$	38730.8			0.655910
$10$		0			0
$11$	−100674.			−0.625105			−0.312553	−	0.949900i	$-0.601184\pi$
$11$	−100674.			−0.625105			−0.312553	+	0.949900i	$0.601184\pi$
$12$		0			0
$13$		−	620510.i		−	1.67121i	−0.549328	−	0.835607i	$-0.685116\pi$
$13$		−	620510.i		−	1.67121i	0.549328	−	0.835607i	$-0.314884\pi$
$14$		0			0
$15$			523762.i			0.689728i
$16$		0			0
$17$	698070.			0.491648			0.245824	−	0.969314i	$-0.420941\pi$
$17$	698070.			0.491648			0.245824	+	0.969314i	$0.420941\pi$
$18$		0			0
$19$	−682231.	−	2.38026e6i	−0.275527	−	0.961293i
$19$	−682231.	−	2.38026e6i	−0.275527	−	0.961293i
$20$		0			0
$21$		−	1.91636e6i		−	0.469224i
$22$		0			0
$23$	−1.05232e7			−1.63496			−0.817480	−	0.575957i	$-0.804630\pi$
$23$	−1.05232e7			−1.63496			−0.817480	+	0.575957i	$0.804630\pi$
$24$		0			0
$25$	3.73592e6			0.382558
$26$		0			0
$27$			1.39377e7i			0.971343i
$28$		0			0
$29$			3.25817e6i			0.158848i	0.996841	+	0.0794242i	$0.0253082\pi$
$29$			3.25817e6i			0.158848i	−0.996841	+	0.0794242i	$0.974692\pi$
$30$		0			0
$31$			2.10556e7i			0.735459i	0.929933	+	0.367729i	$0.119865\pi$
$31$			2.10556e7i			0.735459i	−0.929933	+	0.367729i	$0.880135\pi$
$32$		0			0
$33$		−	1.43502e7i		−	0.366682i
$34$		0			0
$35$	−4.93999e7			−0.940559
$36$		0			0
$37$		−	7.96901e7i		−	1.14920i	−0.818434	−	0.574600i	$-0.805158\pi$
$37$		−	7.96901e7i		−	1.14920i	0.818434	−	0.574600i	$-0.194842\pi$
$38$		0			0
$39$	8.84486e7			0.980321
$40$		0			0
$41$		−	2.25906e8i		−	1.94988i	−0.222463	−	0.974941i	$-0.571410\pi$
$41$		−	2.25906e8i		−	1.94988i	0.222463	−	0.974941i	$-0.428590\pi$
$42$		0			0
$43$	2.27907e8			1.55030			0.775150	−	0.631777i	$-0.217674\pi$
$43$	2.27907e8			1.55030			0.775150	+	0.631777i	$0.217674\pi$
$44$		0			0
$45$	1.42314e8			0.771234
$46$		0			0
$47$	−1.47784e8			−0.644373			−0.322187	−	0.946676i	$-0.604418\pi$
$47$	−1.47784e8			−0.644373			−0.322187	+	0.946676i	$0.604418\pi$
$48$		0			0
$49$	−1.01729e8			−0.360135
$50$		0			0
$51$			9.95042e7i			0.288397i
$52$		0			0
$53$			1.13727e7i			0.0271946i	0.999908	+	0.0135973i	$0.00432829\pi$
$53$			1.13727e7i			0.0271946i	−0.999908	+	0.0135973i	$0.995672\pi$
$54$		0			0
$55$	−3.69920e8			−0.735013
$56$		0			0
$57$	3.39286e8	−	9.72465e7i	0.563887	−	0.161622i
$58$		0			0
$59$		−	4.79625e8i		−	0.670876i	−0.942062	−	0.335438i	$-0.891116\pi$
$59$		−	4.79625e8i		−	0.670876i	0.942062	−	0.335438i	$-0.108884\pi$
$60$		0			0
$61$	1.12598e9			1.33316			0.666579	−	0.745435i	$-0.267758\pi$
$61$	1.12598e9			1.33316			0.666579	+	0.745435i	$0.267758\pi$
$62$		0			0
$63$	−5.20704e8			−0.524673
$64$		0			0
$65$		−	2.28003e9i		−	1.96505i
$66$		0			0
$67$		−	1.88202e9i		−	1.39396i	−0.717090	−	0.696981i	$-0.754526\pi$
$67$		−	1.88202e9i		−	1.39396i	0.717090	−	0.696981i	$-0.245474\pi$
$68$		0			0
$69$		−	1.49999e9i		−	0.959054i
$70$		0			0
$71$			1.27797e8i			0.0708318i	0.999373	+	0.0354159i	$0.0112756\pi$
$71$			1.27797e8i			0.0708318i	−0.999373	+	0.0354159i	$0.988724\pi$
$72$		0			0
$73$	−1.59310e9			−0.768471			−0.384236	−	0.923235i	$-0.625535\pi$
$73$	−1.59310e9			−0.768471			−0.384236	+	0.923235i	$0.625535\pi$
$74$		0			0
$75$			5.32525e8i			0.224406i
$76$		0			0
$77$	1.35348e9			0.500031
$78$		0			0
$79$		−	5.62792e9i		−	1.82899i	−0.404594	−	0.914497i	$-0.632587\pi$
$79$		−	5.62792e9i		−	1.82899i	0.404594	−	0.914497i	$-0.367413\pi$
$80$		0			0
$81$	3.00309e8			0.0861278
$82$		0			0
$83$	4.12884e9			1.04818			0.524092	−	0.851662i	$-0.324405\pi$
$83$	4.12884e9			1.04818			0.524092	+	0.851662i	$0.324405\pi$
$84$		0			0
$85$	2.56502e9			0.578091
$86$		0			0
$87$	−4.64425e8			−0.0931792
$88$		0			0
$89$		−	6.15474e9i		−	1.10220i	−0.834440	−	0.551099i	$-0.814209\pi$
$89$		−	6.15474e9i		−	1.10220i	0.834440	−	0.551099i	$-0.185791\pi$
$90$		0			0
$91$			8.34225e9i			1.33683i
$92$		0			0
$93$	−3.00130e9			−0.431414
$94$		0			0
$95$	−2.50682e9	−	8.74613e9i	−0.323970	−	1.13031i
$96$		0			0
$97$			7.20642e9i			0.839191i	0.907711	+	0.419595i	$0.137828\pi$
$97$			7.20642e9i			0.839191i	−0.907711	+	0.419595i	$0.862172\pi$
$98$		0			0
$99$	−3.89918e9			−0.410013
$100$		0			0
$101$	1.66147e10			1.58083			0.790415	−	0.612571i	$-0.209865\pi$
$101$	1.66147e10			1.58083			0.790415	+	0.612571i	$0.209865\pi$
$102$		0			0
$103$		−	7.05322e8i		−	0.0608417i	−0.999537	−	0.0304209i	$-0.990315\pi$
$103$		−	7.05322e8i		−	0.0608417i	0.999537	−	0.0304209i	$-0.00968476\pi$
$104$		0			0
$105$		−	7.04155e9i		−	0.551724i
$106$		0			0
$107$		−	1.43261e10i		−	1.02143i	−0.859750	−	0.510715i	$-0.829381\pi$
$107$		−	1.43261e10i		−	1.02143i	0.859750	−	0.510715i	$-0.170619\pi$
$108$		0			0
$109$			3.03800e10i			1.97449i	0.159201	+	0.987246i	$0.449108\pi$
$109$			3.03800e10i			1.97449i	−0.159201	+	0.987246i	$0.550892\pi$
$110$		0			0
$111$	1.13592e10			0.674111
$112$		0			0
$113$			8.27269e9i			0.449008i	0.974473	+	0.224504i	$0.0720762\pi$
$113$			8.27269e9i			0.449008i	−0.974473	+	0.224504i	$0.927924\pi$
$114$		0			0
$115$	−3.86668e10			−1.92242
$116$		0			0
$117$		−	2.40329e10i		−	1.09617i
$118$		0			0
$119$	−9.38498e9			−0.393277
$120$		0			0
$121$	−1.58022e10			−0.609244
$122$		0			0
$123$	3.22010e10			1.14379
$124$		0			0
$125$	−2.21558e10			−0.726002
$126$		0			0
$127$			5.06096e10i			1.53184i	0.642934	+	0.765922i	$0.277717\pi$
$127$			5.06096e10i			1.53184i	−0.642934	+	0.765922i	$0.722283\pi$
$128$		0			0
$129$			3.24863e10i			0.909394i
$130$		0			0
$131$	−2.81402e10			−0.729408			−0.364704	−	0.931123i	$-0.618830\pi$
$131$	−2.81402e10			−0.729408			−0.364704	+	0.931123i	$0.618830\pi$
$132$		0			0
$133$	9.17204e9	+	3.20006e10i	0.220398	+	0.768954i
$134$		0			0
$135$			5.12134e10i			1.14213i
$136$		0			0
$137$	1.60174e10			0.331887			0.165944	−	0.986135i	$-0.446933\pi$
$137$	1.60174e10			0.331887			0.165944	+	0.986135i	$0.446933\pi$
$138$		0			0
$139$	3.02980e9			0.0583903			0.0291951	−	0.999574i	$-0.490706\pi$
$139$	3.02980e9			0.0583903			0.0291951	+	0.999574i	$0.490706\pi$
$140$		0			0
$141$		−	2.10654e10i		−	0.377984i
$142$		0			0
$143$			6.24691e10i			1.04468i
$144$		0			0
$145$			1.19719e10i			0.186778i
$146$		0			0
$147$		−	1.45007e10i		−	0.211252i
$148$		0			0
$149$	−1.23792e11			−1.68562			−0.842810	−	0.538212i	$-0.819100\pi$
$149$	−1.23792e11			−1.68562			−0.842810	+	0.538212i	$0.819100\pi$
$150$		0			0
$151$			7.78731e9i			0.0991979i	0.998769	+	0.0495989i	$0.0157943\pi$
$151$			7.78731e9i			0.0991979i	−0.998769	+	0.0495989i	$0.984206\pi$
$152$		0			0
$153$	2.70368e10			0.322477
$154$		0			0
$155$			7.73675e10i			0.864769i
$156$		0			0
$157$	1.14897e11			1.20451			0.602257	−	0.798302i	$-0.294268\pi$
$157$	1.14897e11			1.20451			0.602257	+	0.798302i	$0.294268\pi$
$158$		0			0
$159$	−1.62108e9			−0.0159521
$160$		0			0
$161$	1.41475e11			1.30783
$162$		0			0
$163$	7.41213e10			0.644177			0.322089	−	0.946710i	$-0.395615\pi$
$163$	7.41213e10			0.644177			0.322089	+	0.946710i	$0.395615\pi$
$164$		0			0
$165$		−	5.27291e10i		−	0.431152i
$166$		0			0
$167$		−	3.88841e10i		−	0.299357i	−0.988735	−	0.149678i	$-0.952176\pi$
$167$		−	3.88841e10i		−	0.299357i	0.988735	−	0.149678i	$-0.0478238\pi$
$168$		0			0
$169$	−2.47174e11			−1.79296
$170$		0			0
$171$	−2.64234e10	−	9.21893e10i	−0.180721	−	0.630522i
$172$		0			0
$173$		−	8.36984e10i		−	0.540116i	−0.962844	−	0.270058i	$-0.912957\pi$
$173$		−	8.36984e10i		−	0.540116i	0.962844	−	0.270058i	$-0.0870428\pi$
$174$		0			0
$175$	−5.02264e10			−0.306014
$176$		0			0
$177$	6.83667e10			0.393530
$178$		0			0
$179$			1.41047e11i			0.767536i	0.923430	+	0.383768i	$0.125374\pi$
$179$			1.41047e11i			0.767536i	−0.923430	+	0.383768i	$0.874626\pi$
$180$		0			0
$181$		−	1.10005e11i		−	0.566262i	−0.959081	−	0.283131i	$-0.908627\pi$
$181$		−	1.10005e11i		−	0.566262i	0.959081	−	0.283131i	$-0.0913732\pi$
$182$		0			0
$183$			1.60499e11i			0.782019i
$184$		0			0
$185$		−	2.92817e11i		−	1.35126i
$186$		0			0
$187$	−7.02774e10			−0.307332
$188$		0			0
$189$		−	1.87381e11i		−	0.776993i
$190$		0			0
$191$	−8.15777e10			−0.320926			−0.160463	−	0.987042i	$-0.551299\pi$
$191$	−8.15777e10			−0.320926			−0.160463	+	0.987042i	$0.551299\pi$
$192$		0			0
$193$			2.34114e11i			0.874260i	0.899398	+	0.437130i	$0.144005\pi$
$193$			2.34114e11i			0.874260i	−0.899398	+	0.437130i	$0.855995\pi$
$194$		0			0
$195$	3.25000e11			1.15268
$196$		0			0
$197$	−2.49999e11			−0.842573			−0.421286	−	0.906928i	$-0.638421\pi$
$197$	−2.49999e11			−0.842573			−0.421286	+	0.906928i	$0.638421\pi$
$198$		0			0
$199$	1.91073e10			0.0612258			0.0306129	−	0.999531i	$-0.490254\pi$
$199$	1.91073e10			0.0612258			0.0306129	+	0.999531i	$0.490254\pi$
$200$		0			0
$201$	2.68267e11			0.817687
$202$		0			0
$203$		−	4.38034e10i		−	0.127065i
$204$		0			0
$205$		−	8.30079e11i		−	2.29272i
$206$		0			0
$207$	−4.07571e11			−1.07239
$208$		0			0
$209$	6.86828e10	+	2.39630e11i	0.172233	+	0.600910i
$210$		0			0
$211$		−	6.99843e11i		−	1.67336i	−0.547695	−	0.836678i	$-0.684494\pi$
$211$		−	6.99843e11i		−	1.67336i	0.547695	−	0.836678i	$-0.315506\pi$
$212$		0			0
$213$	−1.82164e10			−0.0415494
$214$		0			0
$215$	8.37433e11			1.82288
$216$		0			0
$217$		−	2.83075e11i		−	0.588305i
$218$		0			0
$219$		−	2.27083e11i		−	0.450779i
$220$		0			0
$221$		−	4.33160e11i		−	0.821649i
$222$		0			0
$223$			1.76028e11i			0.319195i	0.987182	+	0.159598i	$0.0510197\pi$
$223$			1.76028e11i			0.319195i	−0.987182	+	0.159598i	$0.948980\pi$
$224$		0			0
$225$	1.44695e11			0.250924
$226$		0			0
$227$			2.49490e11i			0.413927i	0.978349	+	0.206964i	$0.0663582\pi$
$227$			2.49490e11i			0.413927i	−0.978349	+	0.206964i	$0.933642\pi$
$228$		0			0
$229$	8.06015e11			1.27987			0.639935	−	0.768429i	$-0.278961\pi$
$229$	8.06015e11			1.27987			0.639935	+	0.768429i	$0.278961\pi$
$230$		0			0
$231$			1.92927e11i			0.293314i
$232$		0			0
$233$	−1.14918e12			−1.67344			−0.836718	−	0.547633i	$-0.815529\pi$
$233$	−1.14918e12			−1.67344			−0.836718	+	0.547633i	$0.815529\pi$
$234$		0			0
$235$	−5.43023e11			−0.757668
$236$		0			0
$237$	8.02213e11			1.07287
$238$		0			0
$239$	1.26201e12			1.61836			0.809179	−	0.587562i	$-0.199912\pi$
$239$	1.26201e12			1.61836			0.809179	+	0.587562i	$0.199912\pi$
$240$		0			0
$241$		−	1.90034e11i		−	0.233747i	−0.993147	−	0.116874i	$-0.962713\pi$
$241$		−	1.90034e11i		−	0.233747i	0.993147	−	0.116874i	$-0.0372873\pi$
$242$		0			0
$243$			8.65815e11i			1.02187i
$244$		0			0
$245$	−3.73798e11			−0.423455
$246$		0			0
$247$	−1.47697e12	+	4.23331e11i	−1.60653	+	0.460464i
$248$		0			0
$249$			5.88532e11i			0.614856i
$250$		0			0
$251$	−1.41300e12			−1.41832			−0.709161	−	0.705046i	$-0.750926\pi$
$251$	−1.41300e12			−1.41832			−0.709161	+	0.705046i	$0.750926\pi$
$252$		0			0
$253$	1.05941e12			1.02202
$254$		0			0
$255$			3.65623e11i			0.339103i
$256$		0			0
$257$			8.99856e11i			0.802615i	0.915943	+	0.401308i	$0.131444\pi$
$257$			8.99856e11i			0.802615i	−0.915943	+	0.401308i	$0.868556\pi$
$258$		0			0
$259$			1.07137e12i			0.919263i
$260$		0			0
$261$			1.26191e11i			0.104190i
$262$		0			0
$263$	−2.12781e12			−1.69104			−0.845520	−	0.533943i	$-0.820710\pi$
$263$	−2.12781e12			−1.69104			−0.845520	+	0.533943i	$0.820710\pi$
$264$		0			0
$265$			4.17882e10i			0.0319760i
$266$		0			0
$267$	8.77308e11			0.646540
$268$		0			0
$269$		−	2.18874e12i		−	1.55393i	−0.629542	−	0.776966i	$-0.716757\pi$
$269$		−	2.18874e12i		−	1.55393i	0.629542	−	0.776966i	$-0.283243\pi$
$270$		0			0
$271$	1.54167e12			1.05474			0.527369	−	0.849636i	$-0.323178\pi$
$271$	1.54167e12			1.05474			0.527369	+	0.849636i	$0.323178\pi$
$272$		0			0
$273$	−1.18912e12			−0.784174
$274$		0			0
$275$	−3.76109e11			−0.239139
$276$		0			0
$277$	1.02563e12			0.628912			0.314456	−	0.949272i	$-0.398178\pi$
$277$	1.02563e12			0.628912			0.314456	+	0.949272i	$0.398178\pi$
$278$		0			0
$279$			8.15499e11i			0.482395i
$280$		0			0
$281$			8.76266e11i			0.500154i	0.968226	+	0.250077i	$0.0804560\pi$
$281$			8.76266e11i			0.500154i	−0.968226	+	0.250077i	$0.919544\pi$
$282$		0			0
$283$	−9.86407e11			−0.543406			−0.271703	−	0.962381i	$-0.587587\pi$
$283$	−9.86407e11			−0.543406			−0.271703	+	0.962381i	$0.587587\pi$
$284$		0			0
$285$	1.24669e12	−	3.57327e11i	0.663031	−	0.190038i
$286$		0			0
$287$			3.03712e12i			1.55974i
$288$		0			0
$289$	−1.52869e12			−0.758282
$290$		0			0
$291$	−1.02722e12			−0.492263
$292$		0			0
$293$		−	2.74962e12i		−	1.27331i	−0.771149	−	0.636655i	$-0.780318\pi$
$293$		−	2.74962e12i		−	1.27331i	0.771149	−	0.636655i	$-0.219682\pi$
$294$		0			0
$295$		−	1.76236e12i		−	0.788831i
$296$		0			0
$297$		−	1.40316e12i		−	0.607192i
$298$		0			0
$299$			6.52973e12i			2.73237i
$300$		0			0
$301$	−3.06403e12			−1.24011
$302$		0			0
$303$			2.36829e12i			0.927302i
$304$		0			0
$305$	4.13735e12			1.56756
$306$		0			0
$307$			3.90097e12i			1.43048i	0.698881	+	0.715238i	$0.253682\pi$
$307$			3.90097e12i			1.43048i	−0.698881	+	0.715238i	$0.746318\pi$
$308$		0			0
$309$	1.00538e11			0.0356893
$310$		0			0
$311$	2.25163e12			0.773917			0.386958	−	0.922097i	$-0.373526\pi$
$311$	2.25163e12			0.773917			0.386958	+	0.922097i	$0.373526\pi$
$312$		0			0
$313$	1.22399e12			0.407434			0.203717	−	0.979030i	$-0.434698\pi$
$313$	1.22399e12			0.407434			0.203717	+	0.979030i	$0.434698\pi$
$314$		0			0
$315$	−1.91330e12			−0.616922
$316$		0			0
$317$			4.42175e12i			1.38133i	0.723174	+	0.690666i	$0.242682\pi$
$317$			4.42175e12i			1.38133i	−0.723174	+	0.690666i	$0.757318\pi$
$318$		0			0
$319$		−	3.28012e11i		−	0.0992970i
$320$		0			0
$321$	2.04207e12			0.599163
$322$		0			0
$323$	−4.76245e11	−	1.66159e12i	−0.135462	−	0.472618i
$324$		0			0
$325$		−	2.31818e12i		−	0.639337i
$326$		0			0
$327$	−4.33042e12			−1.15822
$328$		0			0
$329$	1.98683e12			0.515444
$330$		0			0
$331$			6.85182e11i			0.172451i	0.996276	+	0.0862256i	$0.0274806\pi$
$331$			6.85182e11i			0.172451i	−0.996276	+	0.0862256i	$0.972519\pi$
$332$		0			0
$333$		−	3.08646e12i		−	0.753772i
$334$		0			0
$335$		−	6.91539e12i		−	1.63905i
$336$		0			0
$337$		−	6.51397e12i		−	1.49864i	−0.662210	−	0.749318i	$-0.730381\pi$
$337$		−	6.51397e12i		−	1.49864i	0.662210	−	0.749318i	$-0.269619\pi$
$338$		0			0
$339$	−1.17920e12			−0.263385
$340$		0			0
$341$		−	2.11974e12i		−	0.459739i
$342$		0			0
$343$	5.16531e12			1.08799
$344$		0			0
$345$		−	5.51163e12i		−	1.12768i
$346$		0			0
$347$	−5.53965e12			−1.10112			−0.550561	−	0.834795i	$-0.685586\pi$
$347$	−5.53965e12			−1.10112			−0.550561	+	0.834795i	$0.685586\pi$
$348$		0			0
$349$	−2.95261e11			−0.0570269			−0.0285134	−	0.999593i	$-0.509077\pi$
$349$	−2.95261e11			−0.0570269			−0.0285134	+	0.999593i	$0.509077\pi$
$350$		0			0
$351$	8.64849e12			1.62332
$352$		0			0
$353$	8.07031e11			0.147237			0.0736184	−	0.997286i	$-0.476545\pi$
$353$	8.07031e11			0.147237			0.0736184	+	0.997286i	$0.476545\pi$
$354$		0			0
$355$			4.69583e11i			0.0832857i
$356$		0			0
$357$		−	1.33775e12i		−	0.230693i
$358$		0			0
$359$	1.38059e12			0.231522			0.115761	−	0.993277i	$-0.463069\pi$
$359$	1.38059e12			0.231522			0.115761	+	0.993277i	$0.463069\pi$
$360$		0			0
$361$	−5.20019e12	+	3.24777e12i	−0.848170	+	0.529724i
$362$		0			0
$363$		−	2.25248e12i		−	0.357377i
$364$		0			0
$365$	−5.85374e12			−0.903586
$366$		0			0
$367$	8.82010e11			0.132478			0.0662389	−	0.997804i	$-0.478900\pi$
$367$	8.82010e11			0.132478			0.0662389	+	0.997804i	$0.478900\pi$
$368$		0			0
$369$		−	8.74952e12i		−	1.27895i
$370$		0			0
$371$		−	1.52896e11i		−	0.0217534i
$372$		0			0
$373$			6.07629e12i			0.841578i	0.907158	+	0.420789i	$0.138247\pi$
$373$			6.07629e12i			0.841578i	−0.907158	+	0.420789i	$0.861753\pi$
$374$		0			0
$375$		−	3.15813e12i		−	0.425867i
$376$		0			0
$377$	2.02172e12			0.265470
$378$		0			0
$379$		−	5.44584e12i		−	0.696416i	−0.937417	−	0.348208i	$-0.886790\pi$
$379$		−	5.44584e12i		−	0.696416i	0.937417	−	0.348208i	$-0.113210\pi$
$380$		0			0
$381$	−7.21399e12			−0.898567
$382$		0			0
$383$			1.14332e13i			1.38732i	0.720304	+	0.693658i	$0.244002\pi$
$383$			1.14332e13i			1.38732i	−0.720304	+	0.693658i	$0.755998\pi$
$384$		0			0
$385$	4.97328e12			0.587948
$386$		0			0
$387$	8.82704e12			1.01686
$388$		0			0
$389$	1.43362e12			0.160948			0.0804740	−	0.996757i	$-0.474357\pi$
$389$	1.43362e12			0.160948			0.0804740	+	0.996757i	$0.474357\pi$
$390$		0			0
$391$	−7.34591e12			−0.803825
$392$		0			0
$393$		−	4.01115e12i		−	0.427865i
$394$		0			0
$395$		−	2.06795e13i		−	2.15057i
$396$		0			0
$397$	5.68143e12			0.576109			0.288055	−	0.957614i	$-0.406992\pi$
$397$	5.68143e12			0.576109			0.288055	+	0.957614i	$0.406992\pi$
$398$		0			0
$399$	−4.56143e12	+	1.30740e12i	−0.451062	+	0.129284i
$400$		0			0
$401$		−	4.13819e12i		−	0.399107i	−0.979887	−	0.199553i	$-0.936051\pi$
$401$		−	4.13819e12i		−	0.399107i	0.979887	−	0.199553i	$-0.0639491\pi$
$402$		0			0
$403$	1.30652e13			1.22911
$404$		0			0
$405$	1.10347e12			0.101271
$406$		0			0
$407$			8.02270e12i			0.718371i
$408$		0			0
$409$			2.65398e12i			0.231889i	0.993256	+	0.115945i	$0.0369895\pi$
$409$			2.65398e12i			0.231889i	−0.993256	+	0.115945i	$0.963010\pi$
$410$		0			0
$411$			2.28316e12i			0.194682i
$412$		0			0
$413$			6.44817e12i			0.536644i
$414$		0			0
$415$	1.51712e13			1.23248
$416$		0			0
$417$			4.31874e11i			0.0342513i
$418$		0			0
$419$	−3.51190e12			−0.271939			−0.135970	−	0.990713i	$-0.543415\pi$
$419$	−3.51190e12			−0.271939			−0.135970	+	0.990713i	$0.543415\pi$
$420$		0			0
$421$			9.81413e12i			0.742064i	0.928620	+	0.371032i	$0.120996\pi$
$421$			9.81413e12i			0.742064i	−0.928620	+	0.371032i	$0.879004\pi$
$422$		0			0
$423$	−5.72379e12			−0.422651
$424$		0			0
$425$	2.60793e12			0.188084
$426$		0			0
$427$	−1.51379e13			−1.06641
$428$		0			0
$429$	−8.90446e12			−0.612803
$430$		0			0
$431$			1.02379e13i			0.688376i	0.938901	+	0.344188i	$0.111846\pi$
$431$			1.02379e13i			0.688376i	−0.938901	+	0.344188i	$0.888154\pi$
$432$		0			0
$433$		−	1.37814e13i		−	0.905425i	−0.891657	−	0.452712i	$-0.850456\pi$
$433$		−	1.37814e13i		−	0.905425i	0.891657	−	0.452712i	$-0.149544\pi$
$434$		0			0
$435$	−1.70650e12			−0.109562
$436$		0			0
$437$	7.17923e12	+	2.50478e13i	0.450475	+	1.57168i
$438$		0			0
$439$		−	2.94372e12i		−	0.180540i	−0.995917	−	0.0902700i	$-0.971227\pi$
$439$		−	2.94372e12i		−	0.180540i	0.995917	−	0.0902700i	$-0.0287730\pi$
$440$		0			0
$441$	−3.94006e12			−0.236216
$442$		0			0
$443$	−1.16926e13			−0.685320			−0.342660	−	0.939460i	$-0.611328\pi$
$443$	−1.16926e13			−0.685320			−0.342660	+	0.939460i	$0.611328\pi$
$444$		0			0
$445$		−	2.26152e13i		−	1.29599i
$446$		0			0
$447$		−	1.76455e13i		−	0.988771i
$448$		0			0
$449$			1.42546e13i			0.781130i	0.920575	+	0.390565i	$0.127720\pi$
$449$			1.42546e13i			0.781130i	−0.920575	+	0.390565i	$0.872280\pi$
$450$		0			0
$451$			2.27428e13i			1.21888i
$452$		0			0
$453$	−1.11002e12			−0.0581887
$454$		0			0
$455$			3.06531e13i			1.57187i
$456$		0			0
$457$	2.70636e12			0.135770			0.0678850	−	0.997693i	$-0.478375\pi$
$457$	2.70636e12			0.135770			0.0678850	+	0.997693i	$0.478375\pi$
$458$		0			0
$459$			9.72950e12i			0.477559i
$460$		0			0
$461$	2.87097e13			1.37887			0.689437	−	0.724346i	$-0.257858\pi$
$461$	2.87097e13			1.37887			0.689437	+	0.724346i	$0.257858\pi$
$462$		0			0
$463$	−1.27736e13			−0.600353			−0.300177	−	0.953884i	$-0.597046\pi$
$463$	−1.27736e13			−0.600353			−0.300177	+	0.953884i	$0.597046\pi$
$464$		0			0
$465$	−1.10281e13			−0.507267
$466$		0			0
$467$	−1.80446e13			−0.812388			−0.406194	−	0.913787i	$-0.633144\pi$
$467$	−1.80446e13			−0.812388			−0.406194	+	0.913787i	$0.633144\pi$
$468$		0			0
$469$			2.53023e13i			1.11505i
$470$		0			0
$471$			1.63777e13i			0.706558i
$472$		0			0
$473$	−2.29443e13			−0.969101
$474$		0			0
$475$	−2.54876e12	−	8.89245e12i	−0.105405	−	0.367751i
$476$		0			0
$477$			4.40472e11i			0.0178372i
$478$		0			0
$479$	3.04788e13			1.20871			0.604353	−	0.796717i	$-0.293432\pi$
$479$	3.04788e13			1.20871			0.604353	+	0.796717i	$0.293432\pi$
$480$		0			0
$481$	−4.94485e13			−1.92056
$482$		0			0
$483$			2.01661e13i			0.767162i
$484$		0			0
$485$			2.64796e13i			0.986740i
$486$		0			0
$487$		−	3.78070e13i		−	1.38015i	−0.723736	−	0.690077i	$-0.757577\pi$
$487$		−	3.78070e13i		−	1.38015i	0.723736	−	0.690077i	$-0.242423\pi$
$488$		0			0
$489$			1.05654e13i			0.377869i
$490$		0			0
$491$	−1.98817e11			−0.00696700			−0.00348350	−	0.999994i	$-0.501109\pi$
$491$	−1.98817e11			−0.00696700			−0.00348350	+	0.999994i	$0.501109\pi$
$492$		0			0
$493$			2.27443e12i			0.0780976i
$494$		0			0
$495$	−1.43273e13			−0.482102
$496$		0			0
$497$		−	1.71812e12i		−	0.0566595i
$498$		0			0
$499$	5.35593e12			0.173114			0.0865569	−	0.996247i	$-0.472414\pi$
$499$	5.35593e12			0.173114			0.0865569	+	0.996247i	$0.472414\pi$
$500$		0			0
$501$	5.54260e12			0.175600
$502$		0			0
$503$	−9.06235e12			−0.281450			−0.140725	−	0.990049i	$-0.544943\pi$
$503$	−9.06235e12			−0.281450			−0.140725	+	0.990049i	$0.544943\pi$
$504$		0			0
$505$	6.10498e13			1.85878
$506$		0			0
$507$		−	3.52327e13i		−	1.05173i
$508$		0			0
$509$			3.41085e13i			0.998330i	0.866507	+	0.499165i	$0.166360\pi$
$509$			3.41085e13i			0.998330i	−0.866507	+	0.499165i	$0.833640\pi$
$510$		0			0
$511$	2.14179e13			0.614712
$512$		0			0
$513$	3.31754e13	−	9.50874e12i	0.933746	−	0.267631i
$514$		0			0
$515$		−	2.59167e12i		−	0.0715391i
$516$		0			0
$517$	1.48780e13			0.402801
$518$		0			0
$519$	1.19305e13			0.316827
$520$		0			0
$521$		−	2.08430e13i		−	0.542965i	−0.962443	−	0.271482i	$-0.912486\pi$
$521$		−	2.08430e13i		−	0.542965i	0.962443	−	0.271482i	$-0.0875138\pi$
$522$		0			0
$523$		−	7.71932e13i		−	1.97274i	−0.164534	−	0.986371i	$-0.552612\pi$
$523$		−	7.71932e13i		−	1.97274i	0.164534	−	0.986371i	$-0.447388\pi$
$524$		0			0
$525$		−	7.15936e12i		−	0.179505i
$526$		0			0
$527$			1.46983e13i			0.361587i
$528$		0			0
$529$	6.93104e13			1.67309
$530$		0			0
$531$		−	1.85763e13i		−	0.440034i
$532$		0			0
$533$	−1.40177e14			−3.25867
$534$		0			0
$535$		−	5.26404e13i		−	1.20102i
$536$		0			0
$537$	−2.01051e13			−0.450230
$538$		0			0
$539$	1.02415e13			0.225122
$540$		0			0
$541$	−6.20381e13			−1.33867			−0.669333	−	0.742962i	$-0.733420\pi$
$541$	−6.20381e13			−1.33867			−0.669333	+	0.742962i	$0.733420\pi$
$542$		0			0
$543$	1.56803e13			0.332165
$544$		0			0
$545$			1.11630e14i			2.32165i
$546$		0			0
$547$		−	4.21854e13i		−	0.861441i	−0.902485	−	0.430721i	$-0.858259\pi$
$547$		−	4.21854e13i		−	0.861441i	0.902485	−	0.430721i	$-0.141741\pi$
$548$		0			0
$549$	4.36101e13			0.874431
$550$		0			0
$551$	7.75527e12	−	2.22282e12i	0.152700	−	0.0437670i
$552$		0			0
$553$			7.56627e13i			1.46304i
$554$		0			0
$555$	4.17386e13			0.792635
$556$		0			0
$557$	7.00622e13			1.30680			0.653398	−	0.757014i	$-0.273343\pi$
$557$	7.00622e13			1.30680			0.653398	+	0.757014i	$0.273343\pi$
$558$		0			0
$559$		−	1.41419e14i		−	2.59088i
$560$		0			0
$561$		−	1.00175e13i		−	0.180278i
$562$		0			0
$563$		−	2.23889e13i		−	0.395813i	−0.980221	−	0.197906i	$-0.936586\pi$
$563$		−	2.23889e13i		−	0.395813i	0.980221	−	0.197906i	$-0.0634142\pi$
$564$		0			0
$565$			3.03975e13i			0.527954i
$566$		0			0
$567$	−4.03741e12			−0.0688950
$568$		0			0
$569$		−	3.75699e13i		−	0.629911i	−0.949106	−	0.314955i	$-0.898010\pi$
$569$		−	3.75699e13i		−	0.629911i	0.949106	−	0.314955i	$-0.101990\pi$
$570$		0			0
$571$	2.75973e13			0.454660			0.227330	−	0.973818i	$-0.427000\pi$
$571$	2.75973e13			0.454660			0.227330	+	0.973818i	$0.427000\pi$
$572$		0			0
$573$		−	1.16282e13i		−	0.188252i
$574$		0			0
$575$	−3.93137e13			−0.625467
$576$		0			0
$577$	−7.09765e13			−1.10978			−0.554888	−	0.831925i	$-0.687239\pi$
$577$	−7.09765e13			−1.10978			−0.554888	+	0.831925i	$0.687239\pi$
$578$		0			0
$579$	−3.33710e13			−0.512834
$580$		0			0
$581$	−5.55089e13			−0.838459
$582$		0			0
$583$		−	1.14493e12i		−	0.0169995i
$584$		0			0
$585$		−	8.83074e13i		−	1.28890i
$586$		0			0
$587$	5.62896e13			0.807677			0.403838	−	0.914830i	$-0.367676\pi$
$587$	5.62896e13			0.807677			0.403838	+	0.914830i	$0.367676\pi$
$588$		0			0
$589$	5.01177e13	−	1.43648e13i	0.706992	−	0.202639i
$590$		0			0
$591$		−	3.56353e13i		−	0.494246i
$592$		0			0
$593$	−2.70906e13			−0.369442			−0.184721	−	0.982791i	$-0.559138\pi$
$593$	−2.70906e13			−0.369442			−0.184721	+	0.982791i	$0.559138\pi$
$594$		0			0
$595$	−3.44846e13			−0.462424
$596$		0			0
$597$			2.72359e12i			0.0359145i
$598$		0			0
$599$			3.12822e13i			0.405661i	0.979214	+	0.202830i	$0.0650140\pi$
$599$			3.12822e13i			0.405661i	−0.979214	+	0.202830i	$0.934986\pi$
$600$		0			0
$601$		−	4.94460e13i		−	0.630607i	−0.948991	−	0.315304i	$-0.897894\pi$
$601$		−	4.94460e13i		−	0.630607i	0.948991	−	0.315304i	$-0.102106\pi$
$602$		0			0
$603$		−	7.28923e13i		−	0.914313i
$604$		0			0
$605$	−5.80643e13			−0.716362
$606$		0			0
$607$		−	9.69590e12i		−	0.117664i	−0.998268	−	0.0588322i	$-0.981262\pi$
$607$		−	9.69590e12i		−	0.117664i	0.998268	−	0.0588322i	$-0.0187377\pi$
$608$		0			0
$609$	6.24381e12			0.0745355
$610$		0			0
$611$			9.17013e13i			1.07689i
$612$		0			0
$613$	4.35421e13			0.503045			0.251522	−	0.967851i	$-0.419069\pi$
$613$	4.35421e13			0.503045			0.251522	+	0.967851i	$0.419069\pi$
$614$		0			0
$615$	1.18321e14			1.34489
$616$		0			0
$617$	4.09277e13			0.457712			0.228856	−	0.973460i	$-0.426502\pi$
$617$	4.09277e13			0.457712			0.228856	+	0.973460i	$0.426502\pi$
$618$		0			0
$619$	1.14064e14			1.25514			0.627572	−	0.778558i	$-0.284049\pi$
$619$	1.14064e14			1.25514			0.627572	+	0.778558i	$0.284049\pi$
$620$		0			0
$621$		−	1.46669e14i		−	1.58811i
$622$		0			0
$623$			8.27454e13i			0.881665i
$624$		0			0
$625$	−1.17894e14			−1.23621
$626$		0			0
$627$	−3.41572e13	+	9.79017e12i	−0.352489	+	0.101031i
$628$		0			0
$629$		−	5.56293e13i		−	0.565002i
$630$		0			0
$631$	1.34690e13			0.134645			0.0673224	−	0.997731i	$-0.478554\pi$
$631$	1.34690e13			0.134645			0.0673224	+	0.997731i	$0.478554\pi$
$632$		0			0
$633$	9.97569e13			0.981577
$634$		0			0
$635$			1.85962e14i			1.80118i
$636$		0			0
$637$			6.31240e13i			0.601863i
$638$		0			0
$639$			4.94968e12i			0.0464593i
$640$		0			0
$641$			2.86888e13i			0.265108i	0.991176	+	0.132554i	$0.0423177\pi$
$641$			2.86888e13i			0.265108i	−0.991176	+	0.132554i	$0.957682\pi$
$642$		0			0
$643$	−1.27320e14			−1.15835			−0.579177	−	0.815202i	$-0.696626\pi$
$643$	−1.27320e14			−1.15835			−0.579177	+	0.815202i	$0.696626\pi$
$644$		0			0
$645$			1.19369e14i			1.06929i
$646$		0			0
$647$	−6.17093e13			−0.544289			−0.272144	−	0.962256i	$-0.587733\pi$
$647$	−6.17093e13			−0.544289			−0.272144	+	0.962256i	$0.587733\pi$
$648$		0			0
$649$			4.82857e13i			0.419368i
$650$		0			0
$651$	4.03500e13			0.345095
$652$		0			0
$653$	3.86921e13			0.325879			0.162940	−	0.986636i	$-0.447902\pi$
$653$	3.86921e13			0.325879			0.162940	+	0.986636i	$0.447902\pi$
$654$		0			0
$655$	−1.03400e14			−0.857654
$656$		0			0
$657$	−6.17019e13			−0.504048
$658$		0			0
$659$			1.69465e14i			1.36349i	0.731589	+	0.681746i	$0.238779\pi$
$659$			1.69465e14i			1.36349i	−0.731589	+	0.681746i	$0.761221\pi$
$660$		0			0
$661$		−	1.56768e13i		−	0.124236i	−0.998069	−	0.0621182i	$-0.980214\pi$
$661$		−	1.56768e13i		−	0.124236i	0.998069	−	0.0621182i	$-0.0197856\pi$
$662$		0			0
$663$	6.17434e13			0.481973
$664$		0			0
$665$	3.37022e13	+	1.17584e14i	0.259149	+	0.904153i
$666$		0			0
$667$		−	3.42862e13i		−	0.259711i
$668$		0			0
$669$	−2.50913e13			−0.187237
$670$		0			0
$671$	−1.13357e14			−0.833364
$672$		0			0
$673$			1.59142e14i			1.15269i	0.817208	+	0.576343i	$0.195521\pi$
$673$			1.59142e14i			1.15269i	−0.817208	+	0.576343i	$0.804479\pi$
$674$		0			0
$675$			5.20702e13i			0.371595i
$676$		0			0
$677$			2.07430e14i			1.45858i	0.684207	+	0.729288i	$0.260148\pi$
$677$			2.07430e14i			1.45858i	−0.684207	+	0.729288i	$0.739852\pi$
$678$		0			0
$679$		−	9.68844e13i		−	0.671282i
$680$		0			0
$681$	−3.55628e13			−0.242806
$682$		0			0
$683$		−	6.02727e13i		−	0.405525i	−0.979228	−	0.202762i	$-0.935008\pi$
$683$		−	6.02727e13i		−	0.405525i	0.979228	−	0.202762i	$-0.0649919\pi$
$684$		0			0
$685$	5.88552e13			0.390240
$686$		0			0
$687$			1.14891e14i			0.750761i
$688$		0			0
$689$	7.05685e12			0.0454480
$690$		0			0
$691$	2.01520e14			1.27917			0.639586	−	0.768720i	$-0.279106\pi$
$691$	2.01520e14			1.27917			0.639586	+	0.768720i	$0.279106\pi$
$692$		0			0
$693$	5.24213e13			0.327976
$694$		0			0
$695$	1.11328e13			0.0686566
$696$		0			0
$697$		−	1.57698e14i		−	0.958656i
$698$		0			0
$699$		−	1.63807e14i		−	0.981624i
$700$		0			0
$701$	−5.97003e13			−0.352684			−0.176342	−	0.984329i	$-0.556427\pi$
$701$	−5.97003e13			−0.352684			−0.176342	+	0.984329i	$0.556427\pi$
$702$		0			0
$703$	−1.89683e14	+	5.43671e13i	−1.10472	+	0.316635i
$704$		0			0
$705$		−	7.74035e13i		−	0.444442i
$706$		0			0
$707$	−2.23371e14			−1.26453
$708$		0			0
$709$	−1.14740e14			−0.640446			−0.320223	−	0.947342i	$-0.603758\pi$
$709$	−1.14740e14			−0.640446			−0.320223	+	0.947342i	$0.603758\pi$
$710$		0			0
$711$		−	2.17974e14i		−	1.19965i
$712$		0			0
$713$		−	2.21571e14i		−	1.20245i
$714$		0			0
$715$			2.29539e14i			1.22836i
$716$		0			0
$717$			1.79890e14i			0.949316i
$718$		0			0
$719$	−8.45866e13			−0.440207			−0.220104	−	0.975476i	$-0.570640\pi$
$719$	−8.45866e13			−0.440207			−0.220104	+	0.975476i	$0.570640\pi$
$720$		0			0
$721$			9.48248e12i			0.0486683i
$722$		0			0
$723$	2.70878e13			0.137114
$724$		0			0
$725$			1.21722e13i			0.0607688i
$726$		0			0
$727$	1.57082e14			0.773490			0.386745	−	0.922187i	$-0.373599\pi$
$727$	1.57082e14			0.773490			0.386745	+	0.922187i	$0.373599\pi$
$728$		0			0
$729$	−1.05682e14			−0.513290
$730$		0			0
$731$	1.59095e14			0.762202
$732$		0			0
$733$	5.51460e13			0.260612			0.130306	−	0.991474i	$-0.458404\pi$
$733$	5.51460e13			0.260612			0.130306	+	0.991474i	$0.458404\pi$
$734$		0			0
$735$		−	5.32819e13i		−	0.248395i
$736$		0			0
$737$			1.89470e14i			0.871373i
$738$		0			0
$739$	1.47906e14			0.671063			0.335532	−	0.942029i	$-0.391084\pi$
$739$	1.47906e14			0.671063			0.335532	+	0.942029i	$0.391084\pi$
$740$		0			0
$741$	−6.03424e13	−	2.10531e14i	−0.270104	−	0.942376i
$742$		0			0
$743$			6.84573e13i			0.302326i	0.988509	+	0.151163i	$0.0483019\pi$
$743$			6.84573e13i			0.302326i	−0.988509	+	0.151163i	$0.951698\pi$
$744$		0			0
$745$	−4.54865e14			−1.98199
$746$		0			0
$747$	1.59913e14			0.687514
$748$		0			0
$749$			1.92603e14i			0.817058i
$750$		0			0
$751$			3.29926e14i			1.38107i	0.723298	+	0.690536i	$0.242625\pi$
$751$			3.29926e14i			1.38107i	−0.723298	+	0.690536i	$0.757375\pi$
$752$		0			0
$753$		−	2.01412e14i		−	0.831976i
$754$		0			0
$755$			2.86140e13i			0.116639i
$756$		0			0
$757$	2.98865e14			1.20225			0.601126	−	0.799155i	$-0.294719\pi$
$757$	2.98865e14			1.20225			0.601126	+	0.799155i	$0.294719\pi$
$758$		0			0
$759$			1.51010e14i			0.599510i
$760$		0			0
$761$	3.74870e14			1.46878			0.734392	−	0.678726i	$-0.237467\pi$
$761$	3.74870e14			1.46878			0.734392	+	0.678726i	$0.237467\pi$
$762$		0			0
$763$		−	4.08434e14i		−	1.57943i
$764$		0			0
$765$	9.93453e13			0.379176
$766$		0			0
$767$	−2.97612e14			−1.12118
$768$		0			0
$769$	3.68559e14			1.37049			0.685244	−	0.728313i	$-0.259695\pi$
$769$	3.68559e14			1.37049			0.685244	+	0.728313i	$0.259695\pi$
$770$		0			0
$771$	−1.28267e14			−0.470808
$772$		0			0
$773$			3.17802e12i			0.0115149i	0.999983	+	0.00575744i	$0.00183266\pi$
$773$			3.17802e12i			0.0115149i	−0.999983	+	0.00575744i	$0.998167\pi$
$774$		0			0
$775$			7.86619e13i			0.281356i
$776$		0			0
$777$	−1.52715e14			−0.539232
$778$		0			0
$779$	−5.37714e14	+	1.54120e14i	−1.87441	+	0.537244i
$780$		0			0
$781$		−	1.28658e13i		−	0.0442774i
$782$		0			0
$783$	−4.54114e13			−0.154296
$784$		0			0
$785$	4.22184e14			1.41629
$786$		0			0
$787$		−	2.41267e14i		−	0.799143i	−0.916702	−	0.399571i	$-0.869159\pi$
$787$		−	2.41267e14i		−	0.799143i	0.916702	−	0.399571i	$-0.130841\pi$
$788$		0			0
$789$		−	3.03302e14i		−	0.991951i
$790$		0			0
$791$		−	1.11219e14i		−	0.359169i
$792$		0			0
$793$		−	6.98682e14i		−	2.22799i
$794$		0			0
$795$	−5.95657e12			−0.0187569
$796$		0			0
$797$		−	3.75892e14i		−	1.16888i	−0.811436	−	0.584441i	$-0.801314\pi$
$797$		−	3.75892e14i		−	1.16888i	0.811436	−	0.584441i	$-0.198686\pi$
$798$		0			0
$799$	−1.03163e14			−0.316805
$800$		0			0
$801$		−	2.38378e14i		−	0.722943i
$802$		0			0
$803$	1.60383e14			0.480375
$804$		0			0
$805$	5.19843e14			1.53778
$806$		0			0
$807$	3.11987e14			0.911524
$808$		0			0
$809$	4.11426e14			1.18727			0.593635	−	0.804735i	$-0.297692\pi$
$809$	4.11426e14			1.18727			0.593635	+	0.804735i	$0.297692\pi$
$810$		0			0
$811$		−	3.26931e14i		−	0.931863i	−0.884821	−	0.465932i	$-0.845719\pi$
$811$		−	3.26931e14i		−	0.931863i	0.884821	−	0.465932i	$-0.154281\pi$
$812$		0			0
$813$			2.19752e14i			0.618701i
$814$		0			0
$815$	2.72355e14			0.757438
$816$		0			0
$817$	−1.55485e14	−	5.42478e14i	−0.427149	−	1.49029i
$818$		0			0
$819$			3.23102e14i			0.876840i
$820$		0			0
$821$	−4.65421e14			−1.24776			−0.623879	−	0.781521i	$-0.714444\pi$
$821$	−4.65421e14			−1.24776			−0.623879	+	0.781521i	$0.714444\pi$
$822$		0			0
$823$	−1.91692e14			−0.507698			−0.253849	−	0.967244i	$-0.581697\pi$
$823$	−1.91692e14			−0.507698			−0.253849	+	0.967244i	$0.581697\pi$
$824$		0			0
$825$		−	5.36113e13i		−	0.140277i
$826$		0			0
$827$			5.88159e14i			1.52043i	0.649669	+	0.760217i	$0.274907\pi$
$827$			5.88159e14i			1.52043i	−0.649669	+	0.760217i	$0.725093\pi$
$828$		0			0
$829$		−	3.37215e14i		−	0.861260i	−0.902529	−	0.430630i	$-0.858291\pi$
$829$		−	3.37215e14i		−	0.861260i	0.902529	−	0.430630i	$-0.141709\pi$
$830$		0			0
$831$			1.46195e14i			0.368915i
$832$		0			0
$833$	−7.10141e13			−0.177060
$834$		0			0
$835$		−	1.42877e14i		−	0.351990i
$836$		0			0
$837$	−2.93466e14			−0.714383
$838$		0			0
$839$		−	5.22058e14i		−	1.25577i	−0.778308	−	0.627883i	$-0.783922\pi$
$839$		−	5.22058e14i		−	1.25577i	0.778308	−	0.627883i	$-0.216078\pi$
$840$		0			0
$841$	4.10092e14			0.974767
$842$		0			0
$843$	−1.24904e14			−0.293387
$844$		0			0
$845$	−9.08228e14			−2.10820
$846$		0			0
$847$	2.12448e14			0.487343
$848$		0			0
$849$		−	1.40604e14i		−	0.318757i
$850$		0			0
$851$			8.38592e14i			1.87890i
$852$		0			0
$853$	−1.72713e14			−0.382455			−0.191228	−	0.981546i	$-0.561247\pi$
$853$	−1.72713e14			−0.382455			−0.191228	+	0.981546i	$0.561247\pi$
$854$		0			0
$855$	−9.70912e13	−	3.38745e14i	−0.212495	−	0.741382i
$856$		0			0
$857$		−	1.20667e14i		−	0.261027i	−0.991447	−	0.130514i	$-0.958337\pi$
$857$		−	1.20667e14i		−	0.261027i	0.991447	−	0.130514i	$-0.0416626\pi$
$858$		0			0
$859$	4.82148e14			1.03090			0.515448	−	0.856921i	$-0.327626\pi$
$859$	4.82148e14			1.03090			0.515448	+	0.856921i	$0.327626\pi$
$860$		0			0
$861$	−4.32917e14			−0.914932
$862$		0			0
$863$			7.41275e14i			1.54855i	0.632849	+	0.774275i	$0.281885\pi$
$863$			7.41275e14i			1.54855i	−0.632849	+	0.774275i	$0.718115\pi$
$864$		0			0
$865$		−	3.07545e14i		−	0.635080i
$866$		0			0
$867$		−	2.17903e14i		−	0.444802i
$868$		0			0
$869$			5.66584e14i			1.14331i
$870$		0			0
$871$	−1.16781e15			−2.32961
$872$		0			0
$873$			2.79111e14i			0.550434i
$874$		0			0
$875$	2.97867e14			0.580740
$876$		0			0
$877$			6.61714e14i			1.27548i	0.770253	+	0.637738i	$0.220130\pi$
$877$			6.61714e14i			1.27548i	−0.770253	+	0.637738i	$0.779870\pi$
$878$		0			0
$879$	3.91935e14			0.746913
$880$		0			0
$881$	9.08409e14			1.71160			0.855799	−	0.517308i	$-0.173066\pi$
$881$	9.08409e14			1.71160			0.855799	+	0.517308i	$0.173066\pi$
$882$		0			0
$883$	1.28286e14			0.238987			0.119493	−	0.992835i	$-0.461873\pi$
$883$	1.28286e14			0.238987			0.119493	+	0.992835i	$0.461873\pi$
$884$		0			0
$885$	2.51210e14			0.462722
$886$		0			0
$887$			6.35108e14i			1.15672i	0.815781	+	0.578361i	$0.196308\pi$
$887$			6.35108e14i			1.15672i	−0.815781	+	0.578361i	$0.803692\pi$
$888$		0			0
$889$		−	6.80405e14i		−	1.22535i
$890$		0			0
$891$	−3.02333e13			−0.0538389
$892$		0			0
$893$	1.00823e14	+	3.51763e14i	0.177542	+	0.619432i
$894$		0			0
$895$			5.18269e14i			0.902486i
$896$		0			0
$897$	−9.30759e14			−1.60278
$898$		0			0
$899$	−6.86025e13			−0.116827
$900$		0			0
$901$			7.93891e12i			0.0133702i
$902$		0			0
$903$		−	4.36752e14i		−	0.727438i
$904$		0			0
$905$		−	4.04206e14i		−	0.665824i
$906$		0			0
$907$		−	1.08752e14i		−	0.177175i	−0.996068	−	0.0885873i	$-0.971765\pi$
$907$		−	1.08752e14i		−	0.177175i	0.996068	−	0.0885873i	$-0.0282352\pi$
$908$		0			0
$909$	6.43501e14			1.03688
$910$		0			0
$911$		−	6.35413e14i		−	1.01266i	−0.862339	−	0.506331i	$-0.831002\pi$
$911$		−	6.35413e14i		−	1.01266i	0.862339	−	0.506331i	$-0.168998\pi$
$912$		0			0
$913$	−4.15666e14			−0.655225
$914$		0			0
$915$			5.89746e14i			0.919516i
$916$		0			0
$917$	3.78322e14			0.583465
$918$		0			0
$919$	1.97942e14			0.301967			0.150984	−	0.988536i	$-0.451756\pi$
$919$	1.97942e14			0.301967			0.150984	+	0.988536i	$0.451756\pi$
$920$		0			0
$921$	−5.56051e14			−0.839105
$922$		0			0
$923$	7.92993e13			0.118375
$924$		0			0
$925$		−	2.97716e14i		−	0.439636i
$926$		0			0
$927$		−	2.73177e13i		−	0.0399067i
$928$		0			0
$929$	−3.60545e14			−0.521052			−0.260526	−	0.965467i	$-0.583896\pi$
$929$	−3.60545e14			−0.521052			−0.260526	+	0.965467i	$0.583896\pi$
$930$		0			0
$931$	6.94029e13	+	2.42142e14i	0.0992268	+	0.346195i
$932$		0			0
$933$			3.20951e14i			0.453973i
$934$		0			0
$935$	−2.58230e14			−0.361368
$936$		0			0
$937$	1.15701e15			1.60191			0.800956	−	0.598723i	$-0.204325\pi$
$937$	1.15701e15			1.60191			0.800956	+	0.598723i	$0.204325\pi$
$938$		0			0
$939$			1.74470e14i			0.238997i
$940$		0			0
$941$			1.28289e15i			1.73876i	0.494141	+	0.869382i	$0.335483\pi$
$941$			1.28289e15i			1.73876i	−0.494141	+	0.869382i	$0.664517\pi$
$942$		0			0
$943$			2.37725e15i			3.18798i
$944$		0			0
$945$		−	6.88522e14i		−	0.913605i
$946$		0			0
$947$	−2.80412e14			−0.368168			−0.184084	−	0.982910i	$-0.558932\pi$
$947$	−2.80412e14			−0.368168			−0.184084	+	0.982910i	$0.558932\pi$
$948$		0			0
$949$			9.88532e14i			1.28428i
$950$		0			0
$951$	−6.30284e14			−0.810278
$952$		0			0
$953$		−	8.49726e14i		−	1.08097i	−0.841353	−	0.540486i	$-0.818240\pi$
$953$		−	8.49726e14i		−	1.08097i	0.841353	−	0.540486i	$-0.181760\pi$
$954$		0			0
$955$	−2.99753e14			−0.377352
$956$		0			0
$957$	4.67554e13			0.0582468
$958$		0			0
$959$	−2.15341e14			−0.265482
$960$		0			0
$961$	3.76291e14			0.459100
$962$		0			0
$963$		−	5.54861e14i		−	0.669966i
$964$		0			0
$965$			8.60239e14i			1.02797i
$966$		0			0
$967$	6.34594e14			0.750522			0.375261	−	0.926919i	$-0.377553\pi$
$967$	6.34594e14			0.750522			0.375261	+	0.926919i	$0.377553\pi$
$968$		0			0
$969$	2.36846e14	−	6.78849e13i	0.277234	−	0.0794610i
$970$		0			0
$971$		−	1.21074e15i		−	1.40267i	−0.712833	−	0.701334i	$-0.752588\pi$
$971$		−	1.21074e15i		−	1.40267i	0.712833	−	0.701334i	$-0.247412\pi$
$972$		0			0
$973$	−4.07332e13			−0.0467073
$974$		0			0
$975$	3.30437e14			0.375030
$976$		0			0
$977$			7.47203e14i			0.839394i	0.907664	+	0.419697i	$0.137864\pi$
$977$			7.47203e14i			0.839394i	−0.907664	+	0.419697i	$0.862136\pi$
$978$		0			0
$979$			6.19621e14i			0.688990i
$980$		0			0
$981$			1.17664e15i			1.29509i
$982$		0			0
$983$		−	1.34406e15i		−	1.46437i	−0.681104	−	0.732187i	$-0.738500\pi$
$983$		−	1.34406e15i		−	1.46437i	0.681104	−	0.732187i	$-0.261500\pi$
$984$		0			0
$985$	−9.18608e14			−0.990716
$986$		0			0
$987$			2.83207e14i			0.302355i
$988$		0			0
$989$	−2.39830e15			−2.53468
$990$		0			0
$991$			3.59667e14i			0.376298i	0.982140	+	0.188149i	$0.0602488\pi$
$991$			3.59667e14i			0.376298i	−0.982140	+	0.188149i	$0.939751\pi$
$992$		0			0
$993$	−9.76671e13			−0.101158
$994$		0			0
$995$	7.02088e13			0.0719906
$996$		0			0
$997$	−1.18633e15			−1.20429			−0.602145	−	0.798387i	$-0.705687\pi$
$997$	−1.18633e15			−1.20429			−0.602145	+	0.798387i	$0.705687\pi$
$998$		0			0
$999$	1.11070e15			1.11627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.11.c.b.37.9	yes	14
4.3	odd	2		304.11.e.d.113.6		14
19.18	odd	2	inner	76.11.c.b.37.6	✓	14
76.75	even	2		304.11.e.d.113.9		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.11.c.b.37.6	✓	14	19.18	odd	2	inner
76.11.c.b.37.9	yes	14	1.1	even	1	trivial
304.11.e.d.113.6		14	4.3	odd	2
304.11.e.d.113.9		14	76.75	even	2

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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	76.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+142.542i q^{3} +3674.44 q^{5} -13444.2 q^{7} +38730.8 q^{9} +O(q^{10})\)	\(q+142.542i q^{3} +3674.44 q^{5} -13444.2 q^{7} +38730.8 q^{9} -100674. q^{11} -620510. i q^{13} +523762. i q^{15} +698070. q^{17} +(-682231. - 2.38026e6i) q^{19} -1.91636e6i q^{21} -1.05232e7 q^{23} +3.73592e6 q^{25} +1.39377e7i q^{27} +3.25817e6i q^{29} +2.10556e7i q^{31} -1.43502e7i q^{33} -4.93999e7 q^{35} -7.96901e7i q^{37} +8.84486e7 q^{39} -2.25906e8i q^{41} +2.27907e8 q^{43} +1.42314e8 q^{45} -1.47784e8 q^{47} -1.01729e8 q^{49} +9.95042e7i q^{51} +1.13727e7i q^{53} -3.69920e8 q^{55} +(3.39286e8 - 9.72465e7i) q^{57} -4.79625e8i q^{59} +1.12598e9 q^{61} -5.20704e8 q^{63} -2.28003e9i q^{65} -1.88202e9i q^{67} -1.49999e9i q^{69} +1.27797e8i q^{71} -1.59310e9 q^{73} +5.32525e8i q^{75} +1.35348e9 q^{77} -5.62792e9i q^{79} +3.00309e8 q^{81} +4.12884e9 q^{83} +2.56502e9 q^{85} -4.64425e8 q^{87} -6.15474e9i q^{89} +8.34225e9i q^{91} -3.00130e9 q^{93} +(-2.50682e9 - 8.74613e9i) q^{95} +7.20642e9i q^{97} -3.89918e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 5062 q^{5} - 39624 q^{7} - 383056 q^{9}+O(q^{10}) \) 14 * q + 5062 * q^5 - 39624 * q^7 - 383056 * q^9	\( 14 q + 5062 q^{5} - 39624 q^{7} - 383056 q^{9} + 59746 q^{11} - 1287896 q^{17} + 4622604 q^{19} - 11539958 q^{23} - 417808 q^{25} - 77028922 q^{35} + 14606790 q^{39} + 71330230 q^{43} - 169810130 q^{45} - 394833110 q^{47} - 265204650 q^{49} - 612557042 q^{55} + 319590486 q^{57} - 1396847538 q^{61} - 3080375402 q^{63} - 7242583772 q^{73} - 922511302 q^{77} - 9163778710 q^{81} - 9328943264 q^{83} - 13063772714 q^{85} + 11593996398 q^{87} + 16259309364 q^{93} + 11710957630 q^{95} - 20299698578 q^{99}+O(q^{100}) \) 14 * q + 5062 * q^5 - 39624 * q^7 - 383056 * q^9 + 59746 * q^11 - 1287896 * q^17 + 4622604 * q^19 - 11539958 * q^23 - 417808 * q^25 - 77028922 * q^35 + 14606790 * q^39 + 71330230 * q^43 - 169810130 * q^45 - 394833110 * q^47 - 265204650 * q^49 - 612557042 * q^55 + 319590486 * q^57 - 1396847538 * q^61 - 3080375402 * q^63 - 7242583772 * q^73 - 922511302 * q^77 - 9163778710 * q^81 - 9328943264 * q^83 - 13063772714 * q^85 + 11593996398 * q^87 + 16259309364 * q^93 + 11710957630 * q^95 - 20299698578 * q^99

Introduction

L-functions

Modular forms

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