LMFDB - Embedded newform 2736.2.d.b.2015.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2015,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2736.2015");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.d (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.6
Character	$\chi$	$=$	2736.2015
Dual form			2736.2.d.b.2015.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30935i q^{5} +4.59902i q^{7} +O(q^{10})$	$q-2.30935i q^{5} +4.59902i q^{7} +3.79484 q^{11} -2.59435 q^{13} +3.13956i q^{17} +1.00000i q^{19} -3.59769 q^{23} -0.333107 q^{25} -7.15929i q^{29} +6.83617i q^{31} +10.6208 q^{35} -5.26057 q^{37} +10.9660i q^{41} +5.05771i q^{43} -2.74636 q^{47} -14.1510 q^{49} +0.137719i q^{53} -8.76363i q^{55} +3.11627 q^{59} -1.56558 q^{61} +5.99127i q^{65} -2.53183i q^{67} +4.05844 q^{71} -12.9712 q^{73} +17.4526i q^{77} -11.5506i q^{79} -7.81563 q^{83} +7.25034 q^{85} +6.79089i q^{89} -11.9315i q^{91} +2.30935 q^{95} +15.2653 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 8 q^{25} - 32 q^{37} - 32 q^{49} + 8 q^{73} + 40 q^{85} + 16 q^{97}+O(q^{100}) $ 24 * q + 8 * q^25 - 32 * q^37 - 32 * q^49 + 8 * q^73 + 40 * q^85 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 2.30935i	− 1.03277i	−0.856355	−	0.516387i	$-0.827277\pi$
$5$	− 2.30935i	− 1.03277i	0.856355	−	0.516387i	$-0.172723\pi$
$6$	0	0
$7$	4.59902i	1.73827i	0.494577	+	0.869134i	$0.335323\pi$
$7$	4.59902i	1.73827i	−0.494577	+	0.869134i	$0.664677\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.79484	1.14419	0.572094	−	0.820188i	$-0.306131\pi$
$11$	3.79484	1.14419	0.572094	+	0.820188i	$0.306131\pi$
$12$	0	0
$13$	−2.59435	−0.719544	−0.359772	−	0.933040i	$-0.617145\pi$
$13$	−2.59435	−0.719544	−0.359772	+	0.933040i	$0.617145\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.13956i	0.761454i	0.924687	+	0.380727i	$0.124326\pi$
$17$	3.13956i	0.761454i	−0.924687	+	0.380727i	$0.875674\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.59769	−0.750170	−0.375085	−	0.926990i	$-0.622387\pi$
$23$	−3.59769	−0.750170	−0.375085	+	0.926990i	$0.622387\pi$
$24$	0	0
$25$	−0.333107	−0.0666215
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.15929i	− 1.32945i	−0.747090	−	0.664723i	$-0.768549\pi$
$29$	− 7.15929i	− 1.32945i	0.747090	−	0.664723i	$-0.231451\pi$
$30$	0	0
$31$	6.83617i	1.22781i	0.789379	+	0.613906i	$0.210403\pi$
$31$	6.83617i	1.22781i	−0.789379	+	0.613906i	$0.789597\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.6208	1.79524
$36$	0	0
$37$	−5.26057	−0.864832	−0.432416	−	0.901674i	$-0.642339\pi$
$37$	−5.26057	−0.864832	−0.432416	+	0.901674i	$0.642339\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.9660i	1.71260i	0.516481	+	0.856299i	$0.327242\pi$
$41$	10.9660i	1.71260i	−0.516481	+	0.856299i	$0.672758\pi$
$42$	0	0
$43$	5.05771i	0.771294i	0.922647	+	0.385647i	$0.126022\pi$
$43$	5.05771i	0.771294i	−0.922647	+	0.385647i	$0.873978\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.74636	−0.400598	−0.200299	−	0.979735i	$-0.564191\pi$
$47$	−2.74636	−0.400598	−0.200299	+	0.979735i	$0.564191\pi$
$48$	0	0
$49$	−14.1510	−2.02158
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.137719i	0.0189171i	0.999955	+	0.00945857i	$0.00301080\pi$
$53$	0.137719i	0.0189171i	−0.999955	+	0.00945857i	$0.996989\pi$
$54$	0	0
$55$	− 8.76363i	− 1.18169i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.11627	0.405704	0.202852	−	0.979209i	$-0.434979\pi$
$59$	3.11627	0.405704	0.202852	+	0.979209i	$0.434979\pi$
$60$	0	0
$61$	−1.56558	−0.200452	−0.100226	−	0.994965i	$-0.531957\pi$
$61$	−1.56558	−0.200452	−0.100226	+	0.994965i	$0.531957\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.99127i	0.743126i
$66$	0	0
$67$	− 2.53183i	− 0.309313i	−0.987968	−	0.154656i	$-0.950573\pi$
$67$	− 2.53183i	− 0.309313i	0.987968	−	0.154656i	$-0.0494270\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.05844	0.481648	0.240824	−	0.970569i	$-0.422582\pi$
$71$	4.05844	0.481648	0.240824	+	0.970569i	$0.422582\pi$
$72$	0	0
$73$	−12.9712	−1.51816	−0.759080	−	0.650998i	$-0.774351\pi$
$73$	−12.9712	−1.51816	−0.759080	+	0.650998i	$0.774351\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.4526i	1.98891i
$78$	0	0
$79$	− 11.5506i	− 1.29954i	−0.760130	−	0.649771i	$-0.774865\pi$
$79$	− 11.5506i	− 1.29954i	0.760130	−	0.649771i	$-0.225135\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.81563	−0.857877	−0.428938	−	0.903334i	$-0.641112\pi$
$83$	−7.81563	−0.857877	−0.428938	+	0.903334i	$0.641112\pi$
$84$	0	0
$85$	7.25034	0.786410
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.79089i	0.719833i	0.932985	+	0.359916i	$0.117195\pi$
$89$	6.79089i	0.719833i	−0.932985	+	0.359916i	$0.882805\pi$
$90$	0	0
$91$	− 11.9315i	− 1.25076i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.30935	0.236935
$96$	0	0
$97$	15.2653	1.54995	0.774977	−	0.631990i	$-0.217761\pi$
$97$	15.2653	1.54995	0.774977	+	0.631990i	$0.217761\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.2579i	1.31921i	0.751611	+	0.659607i	$0.229277\pi$
$101$	13.2579i	1.31921i	−0.751611	+	0.659607i	$0.770723\pi$
$102$	0	0
$103$	8.88437i	0.875403i	0.899120	+	0.437701i	$0.144207\pi$
$103$	8.88437i	0.875403i	−0.899120	+	0.437701i	$0.855793\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.43618	0.332188	0.166094	−	0.986110i	$-0.446884\pi$
$107$	3.43618	0.332188	0.166094	+	0.986110i	$0.446884\pi$
$108$	0	0
$109$	3.31946	0.317947	0.158973	−	0.987283i	$-0.449182\pi$
$109$	3.31946	0.317947	0.158973	+	0.987283i	$0.449182\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.7272i	1.47949i	0.672885	+	0.739747i	$0.265055\pi$
$113$	15.7272i	1.47949i	−0.672885	+	0.739747i	$0.734945\pi$
$114$	0	0
$115$	8.30833i	0.774756i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−14.4389	−1.32361
$120$	0	0
$121$	3.40084	0.309167
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 10.7775i	− 0.963969i
$126$	0	0
$127$	15.6110i	1.38525i	0.721298	+	0.692625i	$0.243546\pi$
$127$	15.6110i	1.38525i	−0.721298	+	0.692625i	$0.756454\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.8470	1.12245	0.561224	−	0.827664i	$-0.310331\pi$
$131$	12.8470	1.12245	0.561224	+	0.827664i	$0.310331\pi$
$132$	0	0
$133$	−4.59902	−0.398786
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.5425i	1.24245i	0.783631	+	0.621226i	$0.213365\pi$
$137$	14.5425i	1.24245i	−0.783631	+	0.621226i	$0.786635\pi$
$138$	0	0
$139$	13.3660i	1.13369i	0.823823	+	0.566846i	$0.191837\pi$
$139$	13.3660i	1.13369i	−0.823823	+	0.566846i	$0.808163\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.84516	−0.823294
$144$	0	0
$145$	−16.5333	−1.37302
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.8887i	0.892041i	0.895023	+	0.446020i	$0.147159\pi$
$149$	10.8887i	0.892041i	−0.895023	+	0.446020i	$0.852841\pi$
$150$	0	0
$151$	3.35582i	0.273093i	0.990634	+	0.136546i	$0.0436002\pi$
$151$	3.35582i	0.273093i	−0.990634	+	0.136546i	$0.956400\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.7871	1.26805
$156$	0	0
$157$	−12.6151	−1.00679	−0.503397	−	0.864055i	$-0.667917\pi$
$157$	−12.6151	−1.00679	−0.503397	+	0.864055i	$0.667917\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 16.5459i	− 1.30400i
$162$	0	0
$163$	− 16.1689i	− 1.26645i	−0.773970	−	0.633223i	$-0.781732\pi$
$163$	− 16.1689i	− 1.26645i	0.773970	−	0.633223i	$-0.218268\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.5551	1.04892	0.524461	−	0.851435i	$-0.324267\pi$
$167$	13.5551	1.04892	0.524461	+	0.851435i	$0.324267\pi$
$168$	0	0
$169$	−6.26934	−0.482257
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 13.0128i	− 0.989348i	−0.869079	−	0.494674i	$-0.835287\pi$
$173$	− 13.0128i	− 0.989348i	0.869079	−	0.494674i	$-0.164713\pi$
$174$	0	0
$175$	− 1.53197i	− 0.115806i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.0752	0.827797	0.413899	−	0.910323i	$-0.364167\pi$
$179$	11.0752	0.827797	0.413899	+	0.910323i	$0.364167\pi$
$180$	0	0
$181$	9.68382	0.719793	0.359896	−	0.932992i	$-0.382812\pi$
$181$	9.68382	0.719793	0.359896	+	0.932992i	$0.382812\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.1485i	0.893176i
$186$	0	0
$187$	11.9141i	0.871247i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.06032	0.583224	0.291612	−	0.956537i	$-0.405808\pi$
$191$	8.06032	0.583224	0.291612	+	0.956537i	$0.405808\pi$
$192$	0	0
$193$	10.5894	0.762240	0.381120	−	0.924526i	$-0.375538\pi$
$193$	10.5894	0.762240	0.381120	+	0.924526i	$0.375538\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.3333i	1.16370i	0.813297	+	0.581849i	$0.197670\pi$
$197$	16.3333i	1.16370i	−0.813297	+	0.581849i	$0.802330\pi$
$198$	0	0
$199$	− 20.2999i	− 1.43902i	−0.694482	−	0.719511i	$-0.744366\pi$
$199$	− 20.2999i	− 1.43902i	0.694482	−	0.719511i	$-0.255634\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	32.9258	2.31093
$204$	0	0
$205$	25.3243	1.76873
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.79484i	0.262495i
$210$	0	0
$211$	24.5022i	1.68680i	0.537287	+	0.843400i	$0.319449\pi$
$211$	24.5022i	1.68680i	−0.537287	+	0.843400i	$0.680551\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.6800	0.796572
$216$	0	0
$217$	−31.4397	−2.13427
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 8.14512i	− 0.547900i
$222$	0	0
$223$	18.0046i	1.20567i	0.797864	+	0.602837i	$0.205963\pi$
$223$	18.0046i	1.20567i	−0.797864	+	0.602837i	$0.794037\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.2658	−1.14597	−0.572984	−	0.819566i	$-0.694214\pi$
$227$	−17.2658	−1.14597	−0.572984	+	0.819566i	$0.694214\pi$
$228$	0	0
$229$	−24.7433	−1.63508	−0.817542	−	0.575869i	$-0.804664\pi$
$229$	−24.7433	−1.63508	−0.817542	+	0.575869i	$0.804664\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 10.9532i	− 0.717566i	−0.933421	−	0.358783i	$-0.883192\pi$
$233$	− 10.9532i	− 0.717566i	0.933421	−	0.358783i	$-0.116808\pi$
$234$	0	0
$235$	6.34231i	0.413727i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.2463	−0.921515	−0.460758	−	0.887526i	$-0.652422\pi$
$239$	−14.2463	−0.921515	−0.460758	+	0.887526i	$0.652422\pi$
$240$	0	0
$241$	20.5815	1.32577	0.662886	−	0.748720i	$-0.269331\pi$
$241$	20.5815	1.32577	0.662886	+	0.748720i	$0.269331\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	32.6797i	2.08783i
$246$	0	0
$247$	− 2.59435i	− 0.165075i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.69527	0.296363	0.148182	−	0.988960i	$-0.452658\pi$
$251$	4.69527	0.296363	0.148182	+	0.988960i	$0.452658\pi$
$252$	0	0
$253$	−13.6527	−0.858336
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 27.6888i	− 1.72718i	−0.504193	−	0.863591i	$-0.668210\pi$
$257$	− 27.6888i	− 1.72718i	0.504193	−	0.863591i	$-0.331790\pi$
$258$	0	0
$259$	− 24.1935i	− 1.50331i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.6358	1.21080	0.605398	−	0.795923i	$-0.293014\pi$
$263$	19.6358	1.21080	0.605398	+	0.795923i	$0.293014\pi$
$264$	0	0
$265$	0.318041	0.0195371
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 31.1742i	− 1.90072i	−0.311146	−	0.950362i	$-0.600713\pi$
$269$	− 31.1742i	− 1.90072i	0.311146	−	0.950362i	$-0.399287\pi$
$270$	0	0
$271$	9.39410i	0.570651i	0.958431	+	0.285326i	$0.0921017\pi$
$271$	9.39410i	0.570651i	−0.958431	+	0.285326i	$0.907898\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.26409	−0.0762275
$276$	0	0
$277$	−2.91806	−0.175329	−0.0876645	−	0.996150i	$-0.527940\pi$
$277$	−2.91806	−0.175329	−0.0876645	+	0.996150i	$0.527940\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 27.8881i	− 1.66367i	−0.555025	−	0.831834i	$-0.687291\pi$
$281$	− 27.8881i	− 1.66367i	0.555025	−	0.831834i	$-0.312709\pi$
$282$	0	0
$283$	30.2624i	1.79891i	0.437011	+	0.899456i	$0.356037\pi$
$283$	30.2624i	1.79891i	−0.437011	+	0.899456i	$0.643963\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−50.4328	−2.97695
$288$	0	0
$289$	7.14318	0.420187
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.4890i	1.02172i	0.859665	+	0.510859i	$0.170673\pi$
$293$	17.4890i	1.02172i	−0.859665	+	0.510859i	$0.829327\pi$
$294$	0	0
$295$	− 7.19656i	− 0.419000i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.33367	0.539780
$300$	0	0
$301$	−23.2605	−1.34071
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.61548i	0.207022i
$306$	0	0
$307$	− 13.4669i	− 0.768595i	−0.923209	−	0.384298i	$-0.874444\pi$
$307$	− 13.4669i	− 0.768595i	0.923209	−	0.384298i	$-0.125556\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.70465	0.323481	0.161741	−	0.986833i	$-0.448289\pi$
$311$	5.70465	0.323481	0.161741	+	0.986833i	$0.448289\pi$
$312$	0	0
$313$	−22.2278	−1.25639	−0.628195	−	0.778056i	$-0.716206\pi$
$313$	−22.2278	−1.25639	−0.628195	+	0.778056i	$0.716206\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.48644i	− 0.139653i	−0.997559	−	0.0698263i	$-0.977756\pi$
$317$	− 2.48644i	− 0.139653i	0.997559	−	0.0698263i	$-0.0222445\pi$
$318$	0	0
$319$	− 27.1684i	− 1.52114i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.13956	−0.174690
$324$	0	0
$325$	0.864198	0.0479371
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 12.6306i	− 0.696346i
$330$	0	0
$331$	19.9408i	1.09605i	0.836463	+	0.548024i	$0.184620\pi$
$331$	19.9408i	1.09605i	−0.836463	+	0.548024i	$0.815380\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.84690	−0.319450
$336$	0	0
$337$	9.29494	0.506327	0.253164	−	0.967423i	$-0.418529\pi$
$337$	9.29494	0.506327	0.253164	+	0.967423i	$0.418529\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	25.9422i	1.40485i
$342$	0	0
$343$	− 32.8878i	− 1.77577i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−35.0583	−1.88203	−0.941015	−	0.338366i	$-0.890126\pi$
$347$	−35.0583	−1.88203	−0.941015	+	0.338366i	$0.890126\pi$
$348$	0	0
$349$	23.8860	1.27859	0.639293	−	0.768963i	$-0.279227\pi$
$349$	23.8860	1.27859	0.639293	+	0.768963i	$0.279227\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9.56498i	− 0.509092i	−0.967061	−	0.254546i	$-0.918074\pi$
$353$	− 9.56498i	− 0.509092i	0.967061	−	0.254546i	$-0.0819261\pi$
$354$	0	0
$355$	− 9.37236i	− 0.497433i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.9286	−1.36846	−0.684229	−	0.729267i	$-0.739861\pi$
$359$	−25.9286	−1.36846	−0.684229	+	0.729267i	$0.739861\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	29.9550i	1.56792i
$366$	0	0
$367$	3.37940i	0.176403i	0.996103	+	0.0882017i	$0.0281120\pi$
$367$	3.37940i	0.176403i	−0.996103	+	0.0882017i	$0.971888\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.633372	−0.0328830
$372$	0	0
$373$	−14.7767	−0.765106	−0.382553	−	0.923933i	$-0.624955\pi$
$373$	−14.7767	−0.765106	−0.382553	+	0.923933i	$0.624955\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.5737i	0.956595i
$378$	0	0
$379$	29.3039i	1.50524i	0.658456	+	0.752619i	$0.271210\pi$
$379$	29.3039i	1.50524i	−0.658456	+	0.752619i	$0.728790\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−33.1414	−1.69345	−0.846724	−	0.532033i	$-0.821428\pi$
$383$	−33.1414	−1.69345	−0.846724	+	0.532033i	$0.821428\pi$
$384$	0	0
$385$	40.3042	2.05409
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 33.4708i	− 1.69704i	−0.529167	−	0.848518i	$-0.677495\pi$
$389$	− 33.4708i	− 1.69704i	0.529167	−	0.848518i	$-0.322505\pi$
$390$	0	0
$391$	− 11.2952i	− 0.571220i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−26.6744	−1.34213
$396$	0	0
$397$	−18.0295	−0.904877	−0.452439	−	0.891796i	$-0.649446\pi$
$397$	−18.0295	−0.904877	−0.452439	+	0.891796i	$0.649446\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 37.2539i	− 1.86037i	−0.367091	−	0.930185i	$-0.619646\pi$
$401$	− 37.2539i	− 1.86037i	0.367091	−	0.930185i	$-0.380354\pi$
$402$	0	0
$403$	− 17.7354i	− 0.883465i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−19.9630	−0.989531
$408$	0	0
$409$	17.3135	0.856096	0.428048	−	0.903756i	$-0.359201\pi$
$409$	17.3135	0.856096	0.428048	+	0.903756i	$0.359201\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	14.3318i	0.705222i
$414$	0	0
$415$	18.0490i	0.885992i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.6232	1.05636	0.528182	−	0.849131i	$-0.322874\pi$
$419$	21.6232	1.05636	0.528182	+	0.849131i	$0.322874\pi$
$420$	0	0
$421$	−5.80161	−0.282753	−0.141377	−	0.989956i	$-0.545153\pi$
$421$	−5.80161	−0.282753	−0.141377	+	0.989956i	$0.545153\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.04581i	− 0.0507292i
$426$	0	0
$427$	− 7.20015i	− 0.348440i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.8641	1.53484	0.767420	−	0.641145i	$-0.221540\pi$
$431$	31.8641	1.53484	0.767420	+	0.641145i	$0.221540\pi$
$432$	0	0
$433$	14.8475	0.713526	0.356763	−	0.934195i	$-0.383880\pi$
$433$	14.8475	0.713526	0.356763	+	0.934195i	$0.383880\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.59769i	− 0.172101i
$438$	0	0
$439$	− 22.2748i	− 1.06312i	−0.847021	−	0.531560i	$-0.821606\pi$
$439$	− 22.2748i	− 1.06312i	0.847021	−	0.531560i	$-0.178394\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.5796	0.787719	0.393860	−	0.919171i	$-0.371140\pi$
$443$	16.5796	0.787719	0.393860	+	0.919171i	$0.371140\pi$
$444$	0	0
$445$	15.6825	0.743424
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.82426i	0.0860923i	0.999073	+	0.0430461i	$0.0137062\pi$
$449$	1.82426i	0.0860923i	−0.999073	+	0.0430461i	$0.986294\pi$
$450$	0	0
$451$	41.6142i	1.95953i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−27.5540	−1.29175
$456$	0	0
$457$	−19.0435	−0.890819	−0.445409	−	0.895327i	$-0.646942\pi$
$457$	−19.0435	−0.890819	−0.445409	+	0.895327i	$0.646942\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 4.05904i	− 0.189048i	−0.995523	−	0.0945241i	$-0.969867\pi$
$461$	− 4.05904i	− 0.189048i	0.995523	−	0.0945241i	$-0.0301329\pi$
$462$	0	0
$463$	− 35.6638i	− 1.65744i	−0.559665	−	0.828719i	$-0.689070\pi$
$463$	− 35.6638i	− 1.65744i	0.559665	−	0.828719i	$-0.310930\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.3568	1.31220	0.656098	−	0.754676i	$-0.272206\pi$
$467$	28.3568	1.31220	0.656098	+	0.754676i	$0.272206\pi$
$468$	0	0
$469$	11.6440	0.537669
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19.1932i	0.882505i
$474$	0	0
$475$	− 0.333107i	− 0.0152840i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0.592436	0.0270691	0.0135345	−	0.999908i	$-0.495692\pi$
$479$	0.592436	0.0270691	0.0135345	+	0.999908i	$0.495692\pi$
$480$	0	0
$481$	13.6478	0.622284
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 35.2529i	− 1.60075i
$486$	0	0
$487$	4.24395i	0.192312i	0.995366	+	0.0961560i	$0.0306548\pi$
$487$	4.24395i	0.192312i	−0.995366	+	0.0961560i	$0.969345\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.0212	1.58048	0.790242	−	0.612794i	$-0.209955\pi$
$491$	35.0212	1.58048	0.790242	+	0.612794i	$0.209955\pi$
$492$	0	0
$493$	22.4770	1.01231
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.6649i	0.837233i
$498$	0	0
$499$	− 1.05457i	− 0.0472090i	−0.999721	−	0.0236045i	$-0.992486\pi$
$499$	− 1.05457i	− 0.0472090i	0.999721	−	0.0236045i	$-0.00751424\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.31650	0.147876	0.0739378	−	0.997263i	$-0.476443\pi$
$503$	3.31650	0.147876	0.0739378	+	0.997263i	$0.476443\pi$
$504$	0	0
$505$	30.6172	1.36245
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 0.537677i	− 0.0238321i	−0.999929	−	0.0119161i	$-0.996207\pi$
$509$	− 0.537677i	− 0.0238321i	0.999929	−	0.0119161i	$-0.00379309\pi$
$510$	0	0
$511$	− 59.6547i	− 2.63897i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	20.5171	0.904093
$516$	0	0
$517$	−10.4220	−0.458359
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.61888i	0.289979i	0.989433	+	0.144989i	$0.0463148\pi$
$521$	6.61888i	0.289979i	−0.989433	+	0.144989i	$0.953685\pi$
$522$	0	0
$523$	1.83537i	0.0802551i	0.999195	+	0.0401275i	$0.0127764\pi$
$523$	1.83537i	0.0802551i	−0.999195	+	0.0401275i	$0.987224\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.4625	−0.934923
$528$	0	0
$529$	−10.0566	−0.437245
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 28.4496i	− 1.23229i
$534$	0	0
$535$	− 7.93535i	− 0.343075i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−53.7009	−2.31306
$540$	0	0
$541$	14.3842	0.618426	0.309213	−	0.950993i	$-0.399934\pi$
$541$	14.3842	0.618426	0.309213	+	0.950993i	$0.399934\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 7.66581i	− 0.328367i
$546$	0	0
$547$	− 16.4349i	− 0.702708i	−0.936243	−	0.351354i	$-0.885721\pi$
$547$	− 16.4349i	− 0.702708i	0.936243	−	0.351354i	$-0.114279\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.15929	0.304996
$552$	0	0
$553$	53.1214	2.25895
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22.9982i	− 0.974465i	−0.873272	−	0.487232i	$-0.838007\pi$
$557$	− 22.9982i	− 0.974465i	0.873272	−	0.487232i	$-0.161993\pi$
$558$	0	0
$559$	− 13.1215i	− 0.554979i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.7827	0.833740	0.416870	−	0.908966i	$-0.363127\pi$
$563$	19.7827	0.833740	0.416870	+	0.908966i	$0.363127\pi$
$564$	0	0
$565$	36.3197	1.52798
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 29.0120i	− 1.21625i	−0.793843	−	0.608123i	$-0.791923\pi$
$569$	− 29.0120i	− 1.21625i	0.793843	−	0.608123i	$-0.208077\pi$
$570$	0	0
$571$	21.6241i	0.904942i	0.891779	+	0.452471i	$0.149457\pi$
$571$	21.6241i	0.904942i	−0.891779	+	0.452471i	$0.850543\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.19842	0.0499775
$576$	0	0
$577$	−26.4589	−1.10150	−0.550748	−	0.834671i	$-0.685658\pi$
$577$	−26.4589	−1.10150	−0.550748	+	0.834671i	$0.685658\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 35.9443i	− 1.49122i
$582$	0	0
$583$	0.522621i	0.0216448i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.52568	0.145520	0.0727602	−	0.997349i	$-0.476819\pi$
$587$	3.52568	0.145520	0.0727602	+	0.997349i	$0.476819\pi$
$588$	0	0
$589$	−6.83617	−0.281680
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 8.39120i	− 0.344585i	−0.985046	−	0.172293i	$-0.944883\pi$
$593$	− 8.39120i	− 0.344585i	0.985046	−	0.172293i	$-0.0551175\pi$
$594$	0	0
$595$	33.3445i	1.36699i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.9278	1.05938	0.529690	−	0.848191i	$-0.322308\pi$
$599$	25.9278	1.05938	0.529690	+	0.848191i	$0.322308\pi$
$600$	0	0
$601$	25.6855	1.04773	0.523867	−	0.851800i	$-0.324489\pi$
$601$	25.6855	1.04773	0.523867	+	0.851800i	$0.324489\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 7.85374i	− 0.319300i
$606$	0	0
$607$	− 10.0949i	− 0.409738i	−0.978789	−	0.204869i	$-0.934323\pi$
$607$	− 10.0949i	− 0.409738i	0.978789	−	0.204869i	$-0.0656769\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.12502	0.288247
$612$	0	0
$613$	6.72814	0.271747	0.135874	−	0.990726i	$-0.456616\pi$
$613$	6.72814	0.271747	0.135874	+	0.990726i	$0.456616\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 0.556691i	− 0.0224115i	−0.999937	−	0.0112058i	$-0.996433\pi$
$617$	− 0.556691i	− 0.0224115i	0.999937	−	0.0112058i	$-0.00356698\pi$
$618$	0	0
$619$	− 26.7322i	− 1.07446i	−0.843437	−	0.537228i	$-0.819471\pi$
$619$	− 26.7322i	− 1.07446i	0.843437	−	0.537228i	$-0.180529\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−31.2315	−1.25126
$624$	0	0
$625$	−26.5546	−1.06218
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 16.5158i	− 0.658530i
$630$	0	0
$631$	21.2556i	0.846173i	0.906089	+	0.423087i	$0.139053\pi$
$631$	21.2556i	0.846173i	−0.906089	+	0.423087i	$0.860947\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	36.0512	1.43065
$636$	0	0
$637$	36.7127	1.45461
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 31.0043i	− 1.22460i	−0.790627	−	0.612298i	$-0.790245\pi$
$641$	− 31.0043i	− 1.22460i	0.790627	−	0.612298i	$-0.209755\pi$
$642$	0	0
$643$	− 6.40497i	− 0.252587i	−0.991993	−	0.126294i	$-0.959692\pi$
$643$	− 6.40497i	− 0.252587i	0.991993	−	0.126294i	$-0.0403082\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.4798	0.962402	0.481201	−	0.876610i	$-0.340201\pi$
$647$	24.4798	0.962402	0.481201	+	0.876610i	$0.340201\pi$
$648$	0	0
$649$	11.8258	0.464201
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 29.2771i	− 1.14570i	−0.819660	−	0.572851i	$-0.805838\pi$
$653$	− 29.2771i	− 1.14570i	0.819660	−	0.572851i	$-0.194162\pi$
$654$	0	0
$655$	− 29.6682i	− 1.15923i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−37.3893	−1.45648	−0.728241	−	0.685321i	$-0.759662\pi$
$659$	−37.3893	−1.45648	−0.728241	+	0.685321i	$0.759662\pi$
$660$	0	0
$661$	−32.4893	−1.26369	−0.631844	−	0.775095i	$-0.717702\pi$
$661$	−32.4893	−1.26369	−0.631844	+	0.775095i	$0.717702\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.6208i	0.411856i
$666$	0	0
$667$	25.7569i	0.997311i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.94114	−0.229355
$672$	0	0
$673$	29.4508	1.13525	0.567623	−	0.823289i	$-0.307863\pi$
$673$	29.4508	1.13525	0.567623	+	0.823289i	$0.307863\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.9281i	1.34240i	0.741277	+	0.671199i	$0.234220\pi$
$677$	34.9281i	1.34240i	−0.741277	+	0.671199i	$0.765780\pi$
$678$	0	0
$679$	70.2054i	2.69423i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.8549	−0.415350	−0.207675	−	0.978198i	$-0.566590\pi$
$683$	−10.8549	−0.415350	−0.207675	+	0.978198i	$0.566590\pi$
$684$	0	0
$685$	33.5838	1.28317
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 0.357291i	− 0.0136117i
$690$	0	0
$691$	− 2.25569i	− 0.0858105i	−0.999079	−	0.0429053i	$-0.986339\pi$
$691$	− 2.25569i	− 0.0858105i	0.999079	−	0.0429053i	$-0.0136614\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	30.8669	1.17085
$696$	0	0
$697$	−34.4283	−1.30407
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 30.3459i	− 1.14615i	−0.819504	−	0.573074i	$-0.805751\pi$
$701$	− 30.3459i	− 1.14615i	0.819504	−	0.573074i	$-0.194249\pi$
$702$	0	0
$703$	− 5.26057i	− 0.198406i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−60.9735	−2.29315
$708$	0	0
$709$	8.93931	0.335723	0.167861	−	0.985811i	$-0.446314\pi$
$709$	8.93931	0.335723	0.167861	+	0.985811i	$0.446314\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 24.5944i	− 0.921068i
$714$	0	0
$715$	22.7359i	0.850276i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−13.6161	−0.507796	−0.253898	−	0.967231i	$-0.581713\pi$
$719$	−13.6161	−0.507796	−0.253898	+	0.967231i	$0.581713\pi$
$720$	0	0
$721$	−40.8594	−1.52168
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.38481i	0.0885697i
$726$	0	0
$727$	− 22.4147i	− 0.831316i	−0.909521	−	0.415658i	$-0.863551\pi$
$727$	− 22.4147i	− 0.831316i	0.909521	−	0.415658i	$-0.136449\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−15.8790	−0.587305
$732$	0	0
$733$	−35.4301	−1.30864	−0.654320	−	0.756217i	$-0.727045\pi$
$733$	−35.4301	−1.30864	−0.654320	+	0.756217i	$0.727045\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.60792i	− 0.353912i
$738$	0	0
$739$	− 43.4104i	− 1.59688i	−0.602075	−	0.798439i	$-0.705659\pi$
$739$	− 43.4104i	− 1.59688i	0.602075	−	0.798439i	$-0.294341\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.78436	−0.248894	−0.124447	−	0.992226i	$-0.539716\pi$
$743$	−6.78436	−0.248894	−0.124447	+	0.992226i	$0.539716\pi$
$744$	0	0
$745$	25.1459	0.921276
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15.8031i	0.577432i
$750$	0	0
$751$	− 7.13866i	− 0.260493i	−0.991482	−	0.130247i	$-0.958423\pi$
$751$	− 7.13866i	− 0.260493i	0.991482	−	0.130247i	$-0.0415769\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.74977	0.282043
$756$	0	0
$757$	23.5454	0.855773	0.427886	−	0.903832i	$-0.359258\pi$
$757$	23.5454	0.855773	0.427886	+	0.903832i	$0.359258\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.72201i	0.316173i	0.987425	+	0.158086i	$0.0505324\pi$
$761$	8.72201i	0.316173i	−0.987425	+	0.158086i	$0.949468\pi$
$762$	0	0
$763$	15.2663i	0.552677i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.08470	−0.291922
$768$	0	0
$769$	6.35770	0.229265	0.114632	−	0.993408i	$-0.463431\pi$
$769$	6.35770	0.229265	0.114632	+	0.993408i	$0.463431\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 24.5282i	− 0.882217i	−0.897454	−	0.441109i	$-0.854585\pi$
$773$	− 24.5282i	− 0.882217i	0.897454	−	0.441109i	$-0.145415\pi$
$774$	0	0
$775$	− 2.27718i	− 0.0817987i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.9660	−0.392897
$780$	0	0
$781$	15.4011	0.551096
$782$	0	0
$783$	0	0
$784$	0	0
$785$	29.1327i	1.03979i
$786$	0	0
$787$	4.30870i	0.153588i	0.997047	+	0.0767942i	$0.0244684\pi$
$787$	4.30870i	0.153588i	−0.997047	+	0.0767942i	$0.975532\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−72.3300	−2.57176
$792$	0	0
$793$	4.06167	0.144234
$794$	0	0
$795$	0	0
$796$	0	0
$797$	44.2034i	1.56576i	0.622170	+	0.782882i	$0.286251\pi$
$797$	44.2034i	1.56576i	−0.622170	+	0.782882i	$0.713749\pi$
$798$	0	0
$799$	− 8.62235i	− 0.305037i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−49.2235	−1.73706
$804$	0	0
$805$	−38.2102	−1.34673
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.7382i	0.940065i	0.882649	+	0.470032i	$0.155758\pi$
$809$	26.7382i	0.940065i	−0.882649	+	0.470032i	$0.844242\pi$
$810$	0	0
$811$	29.0534i	1.02020i	0.860114	+	0.510102i	$0.170392\pi$
$811$	29.0534i	1.02020i	−0.860114	+	0.510102i	$0.829608\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−37.3397	−1.30795
$816$	0	0
$817$	−5.05771	−0.176947
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 43.7000i	− 1.52514i	−0.646905	−	0.762571i	$-0.723937\pi$
$821$	− 43.7000i	− 1.52514i	0.646905	−	0.762571i	$-0.276063\pi$
$822$	0	0
$823$	− 14.4618i	− 0.504107i	−0.967713	−	0.252053i	$-0.918894\pi$
$823$	− 14.4618i	− 0.504107i	0.967713	−	0.252053i	$-0.0811058\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.2509	1.50398	0.751991	−	0.659174i	$-0.229094\pi$
$827$	43.2509	1.50398	0.751991	+	0.659174i	$0.229094\pi$
$828$	0	0
$829$	41.4620	1.44004	0.720018	−	0.693955i	$-0.244134\pi$
$829$	41.4620	1.44004	0.720018	+	0.693955i	$0.244134\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 44.4280i	− 1.53934i
$834$	0	0
$835$	− 31.3034i	− 1.08330i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−33.3799	−1.15240	−0.576200	−	0.817309i	$-0.695465\pi$
$839$	−33.3799	−1.15240	−0.576200	+	0.817309i	$0.695465\pi$
$840$	0	0
$841$	−22.2554	−0.767429
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.4781i	0.498062i
$846$	0	0
$847$	15.6405i	0.537416i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.9259	0.648771
$852$	0	0
$853$	−0.425628	−0.0145732	−0.00728661	−	0.999973i	$-0.502319\pi$
$853$	−0.425628	−0.0145732	−0.00728661	+	0.999973i	$0.502319\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	52.3294i	1.78754i	0.448526	+	0.893770i	$0.351949\pi$
$857$	52.3294i	1.78754i	−0.448526	+	0.893770i	$0.648051\pi$
$858$	0	0
$859$	16.2052i	0.552913i	0.961026	+	0.276456i	$0.0891601\pi$
$859$	16.2052i	0.552913i	−0.961026	+	0.276456i	$0.910840\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.4619	−1.24118	−0.620590	−	0.784136i	$-0.713107\pi$
$863$	−36.4619	−1.24118	−0.620590	+	0.784136i	$0.713107\pi$
$864$	0	0
$865$	−30.0512	−1.02177
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 43.8327i	− 1.48692i
$870$	0	0
$871$	6.56847i	0.222564i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	49.5660	1.67564
$876$	0	0
$877$	−25.0495	−0.845861	−0.422930	−	0.906162i	$-0.638999\pi$
$877$	−25.0495	−0.845861	−0.422930	+	0.906162i	$0.638999\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	40.7100i	1.37155i	0.727811	+	0.685777i	$0.240538\pi$
$881$	40.7100i	1.37155i	−0.727811	+	0.685777i	$0.759462\pi$
$882$	0	0
$883$	− 21.5108i	− 0.723895i	−0.932199	−	0.361947i	$-0.882112\pi$
$883$	− 21.5108i	− 0.723895i	0.932199	−	0.361947i	$-0.117888\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.1483	1.41520	0.707601	−	0.706612i	$-0.249777\pi$
$887$	42.1483	1.41520	0.707601	+	0.706612i	$0.249777\pi$
$888$	0	0
$889$	−71.7952	−2.40793
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2.74636i	− 0.0919034i
$894$	0	0
$895$	− 25.5765i	− 0.854927i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	48.9421	1.63231
$900$	0	0
$901$	−0.432376	−0.0144045
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 22.3634i	− 0.743383i
$906$	0	0
$907$	− 52.5907i	− 1.74624i	−0.487502	−	0.873122i	$-0.662092\pi$
$907$	− 52.5907i	− 1.74624i	0.487502	−	0.873122i	$-0.337908\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.5308	−0.746479	−0.373240	−	0.927735i	$-0.621753\pi$
$911$	−22.5308	−0.746479	−0.373240	+	0.927735i	$0.621753\pi$
$912$	0	0
$913$	−29.6591	−0.981573
$914$	0	0
$915$	0	0
$916$	0	0
$917$	59.0837i	1.95111i
$918$	0	0
$919$	56.8296i	1.87464i	0.348474	+	0.937318i	$0.386700\pi$
$919$	56.8296i	1.87464i	−0.348474	+	0.937318i	$0.613300\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.5290	−0.346567
$924$	0	0
$925$	1.75233	0.0576164
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.0678i	1.18335i	0.806178	+	0.591674i	$0.201533\pi$
$929$	36.0678i	1.18335i	−0.806178	+	0.591674i	$0.798467\pi$
$930$	0	0
$931$	− 14.1510i	− 0.463781i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	27.5139	0.899801
$936$	0	0
$937$	29.9378	0.978026	0.489013	−	0.872277i	$-0.337357\pi$
$937$	29.9378	0.978026	0.489013	+	0.872277i	$0.337357\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 22.9100i	− 0.746845i	−0.927661	−	0.373423i	$-0.878184\pi$
$941$	− 22.9100i	− 0.746845i	0.927661	−	0.373423i	$-0.121816\pi$
$942$	0	0
$943$	− 39.4522i	− 1.28474i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.93818	0.160469	0.0802347	−	0.996776i	$-0.474433\pi$
$947$	4.93818	0.160469	0.0802347	+	0.996776i	$0.474433\pi$
$948$	0	0
$949$	33.6518	1.09238
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 24.9065i	− 0.806800i	−0.915024	−	0.403400i	$-0.867828\pi$
$953$	− 24.9065i	− 0.806800i	0.915024	−	0.403400i	$-0.132172\pi$
$954$	0	0
$955$	− 18.6141i	− 0.602339i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−66.8815	−2.15972
$960$	0	0
$961$	−15.7332	−0.507524
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 24.4546i	− 0.787221i
$966$	0	0
$967$	− 18.8893i	− 0.607440i	−0.952761	−	0.303720i	$-0.901771\pi$
$967$	− 18.8893i	− 0.607440i	0.952761	−	0.303720i	$-0.0982288\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	23.1607	0.743261	0.371631	−	0.928381i	$-0.378799\pi$
$971$	23.1607	0.743261	0.371631	+	0.928381i	$0.378799\pi$
$972$	0	0
$973$	−61.4708	−1.97066
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 7.54010i	− 0.241229i	−0.992699	−	0.120615i	$-0.961513\pi$
$977$	− 7.54010i	− 0.241229i	0.992699	−	0.120615i	$-0.0384865\pi$
$978$	0	0
$979$	25.7704i	0.823624i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	58.3065	1.85969	0.929844	−	0.367954i	$-0.119942\pi$
$983$	58.3065	1.85969	0.929844	+	0.367954i	$0.119942\pi$
$984$	0	0
$985$	37.7193	1.20184
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 18.1961i	− 0.578601i
$990$	0	0
$991$	21.7726i	0.691629i	0.938303	+	0.345814i	$0.112397\pi$
$991$	21.7726i	0.691629i	−0.938303	+	0.345814i	$0.887603\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−46.8796	−1.48618
$996$	0	0
$997$	11.8416	0.375028	0.187514	−	0.982262i	$-0.439957\pi$
$997$	11.8416	0.375028	0.187514	+	0.982262i	$0.439957\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.d.b.2015.6	yes	24
3.2	odd	2	inner	2736.2.d.b.2015.20	yes	24
4.3	odd	2	inner	2736.2.d.b.2015.5	✓	24
12.11	even	2	inner	2736.2.d.b.2015.19	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.d.b.2015.5	✓	24	4.3	odd	2	inner
2736.2.d.b.2015.6	yes	24	1.1	even	1	trivial
2736.2.d.b.2015.19	yes	24	12.11	even	2	inner
2736.2.d.b.2015.20	yes	24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.30935i q^{5} +4.59902i q^{7} +O(q^{10})\)	\(q-2.30935i q^{5} +4.59902i q^{7} +3.79484 q^{11} -2.59435 q^{13} +3.13956i q^{17} +1.00000i q^{19} -3.59769 q^{23} -0.333107 q^{25} -7.15929i q^{29} +6.83617i q^{31} +10.6208 q^{35} -5.26057 q^{37} +10.9660i q^{41} +5.05771i q^{43} -2.74636 q^{47} -14.1510 q^{49} +0.137719i q^{53} -8.76363i q^{55} +3.11627 q^{59} -1.56558 q^{61} +5.99127i q^{65} -2.53183i q^{67} +4.05844 q^{71} -12.9712 q^{73} +17.4526i q^{77} -11.5506i q^{79} -7.81563 q^{83} +7.25034 q^{85} +6.79089i q^{89} -11.9315i q^{91} +2.30935 q^{95} +15.2653 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 8 q^{25} - 32 q^{37} - 32 q^{49} + 8 q^{73} + 40 q^{85} + 16 q^{97}+O(q^{100}) \) 24 * q + 8 * q^25 - 32 * q^37 - 32 * q^49 + 8 * q^73 + 40 * q^85 + 16 * q^97

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