LMFDB - Embedded newform 2736.2.cg.a.2591.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1151,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))

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

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

chi := DirichletCharacter("2736.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.cg (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2591.2
Character	$\chi$	$=$	2736.2591
Dual form			2736.2.cg.a.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.27729 + 1.31480i) q^{5} -3.01718i q^{7} +O(q^{10})$	$q+(-2.27729 + 1.31480i) q^{5} -3.01718i q^{7} -0.730663 q^{11} +(2.20198 - 3.81393i) q^{13} +(-5.35027 + 3.08898i) q^{17} +(2.91110 - 3.24430i) q^{19} +(2.23308 - 3.86781i) q^{23} +(0.957376 - 1.65822i) q^{25} +(4.85101 + 2.80073i) q^{29} +6.37863i q^{31} +(3.96698 + 6.87101i) q^{35} -1.48920 q^{37} +(-3.54789 + 2.04837i) q^{41} +(-6.82401 + 3.93984i) q^{43} +(-1.28368 + 2.22340i) q^{47} -2.10340 q^{49} +(-9.22041 - 5.32341i) q^{53} +(1.66393 - 0.960673i) q^{55} +(-1.08092 - 1.87221i) q^{59} +(-4.26794 + 7.39228i) q^{61} +11.5806i q^{65} +(-8.40407 - 4.85209i) q^{67} +(5.00887 + 8.67561i) q^{71} +(-5.72788 - 9.92098i) q^{73} +2.20455i q^{77} +(1.78342 - 1.02966i) q^{79} +7.34096 q^{83} +(8.12276 - 14.0690i) q^{85} +(-2.22085 - 1.28221i) q^{89} +(-11.5073 - 6.64376i) q^{91} +(-2.36384 + 11.2157i) q^{95} +(-2.80967 - 4.86649i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 12 q^{13} - 12 q^{19} + 8 q^{25} + 16 q^{37} - 12 q^{43} + 16 q^{49} + 12 q^{55} - 60 q^{67} + 8 q^{73} + 12 q^{79} + 16 q^{85} - 12 q^{91} - 32 q^{97}+O(q^{100}) $ 24 * q + 12 * q^13 - 12 * q^19 + 8 * q^25 + 16 * q^37 - 12 * q^43 + 16 * q^49 + 12 * q^55 - 60 * q^67 + 8 * q^73 + 12 * q^79 + 16 * q^85 - 12 * q^91 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$	$1$

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
$5$	−2.27729	+	1.31480i	−1.01844	+	0.587995i	−0.913651	−	0.406500i	$-0.866749\pi$
$5$	−2.27729	+	1.31480i	−1.01844	+	0.587995i	−0.104786	+	0.994495i	$0.533416\pi$
$6$		0			0
$7$		−	3.01718i		−	1.14039i	−0.821510	−	0.570194i	$-0.806868\pi$
$7$		−	3.01718i		−	1.14039i	0.821510	−	0.570194i	$-0.193132\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.730663			−0.220303			−0.110152	−	0.993915i	$-0.535134\pi$
$11$	−0.730663			−0.220303			−0.110152	+	0.993915i	$0.535134\pi$
$12$		0			0
$13$	2.20198	−	3.81393i	0.610718	−	1.05779i	−0.380401	−	0.924821i	$-0.624214\pi$
$13$	2.20198	−	3.81393i	0.610718	−	1.05779i	0.991120	−	0.132973i	$-0.0424525\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.35027	+	3.08898i	−1.29763	+	0.749188i	−0.979994	−	0.199025i	$-0.936223\pi$
$17$	−5.35027	+	3.08898i	−1.29763	+	0.749188i	−0.317637	+	0.948213i	$0.602889\pi$
$18$		0			0
$19$	2.91110	−	3.24430i	0.667852	−	0.744294i
$19$	2.91110	−	3.24430i	0.667852	−	0.744294i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.23308	−	3.86781i	0.465630	−	0.806494i	−0.533600	−	0.845737i	$-0.679161\pi$
$23$	2.23308	−	3.86781i	0.465630	−	0.806494i	0.999230	+	0.0392426i	$0.0124945\pi$
$24$		0			0
$25$	0.957376	−	1.65822i	0.191475	−	0.331645i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.85101	+	2.80073i	0.900811	+	0.520083i	0.877463	−	0.479644i	$-0.159234\pi$
$29$	4.85101	+	2.80073i	0.900811	+	0.520083i	0.0233476	+	0.999727i	$0.492568\pi$
$30$		0			0
$31$			6.37863i			1.14564i	0.819682	+	0.572818i	$0.194150\pi$
$31$			6.37863i			1.14564i	−0.819682	+	0.572818i	$0.805850\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.96698	+	6.87101i	0.670542	+	1.16141i
$36$		0			0
$37$	−1.48920			−0.244823			−0.122411	−	0.992479i	$-0.539063\pi$
$37$	−1.48920			−0.244823			−0.122411	+	0.992479i	$0.539063\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.54789	+	2.04837i	−0.554087	+	0.319902i	−0.750769	−	0.660565i	$-0.770317\pi$
$41$	−3.54789	+	2.04837i	−0.554087	+	0.319902i	0.196682	+	0.980467i	$0.436983\pi$
$42$		0			0
$43$	−6.82401	+	3.93984i	−1.04065	+	0.600820i	−0.920018	−	0.391877i	$-0.871826\pi$
$43$	−6.82401	+	3.93984i	−1.04065	+	0.600820i	−0.120634	+	0.992697i	$0.538493\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.28368	+	2.22340i	−0.187244	+	0.324316i	−0.944330	−	0.328999i	$-0.893289\pi$
$47$	−1.28368	+	2.22340i	−0.187244	+	0.324316i	0.757087	+	0.653315i	$0.226622\pi$
$48$		0			0
$49$	−2.10340			−0.300485
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.22041	−	5.32341i	−1.26652	−	0.731226i	−0.292193	−	0.956359i	$-0.594385\pi$
$53$	−9.22041	−	5.32341i	−1.26652	−	0.731226i	−0.974328	+	0.225134i	$0.927718\pi$
$54$		0			0
$55$	1.66393	−	0.960673i	0.224365	−	0.129537i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.08092	−	1.87221i	−0.140724	−	0.243741i	0.787045	−	0.616895i	$-0.211610\pi$
$59$	−1.08092	−	1.87221i	−0.140724	−	0.243741i	−0.927770	+	0.373154i	$0.878276\pi$
$60$		0			0
$61$	−4.26794	+	7.39228i	−0.546453	+	0.946485i	0.452061	+	0.891987i	$0.350689\pi$
$61$	−4.26794	+	7.39228i	−0.546453	+	0.946485i	−0.998514	+	0.0544976i	$0.982644\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			11.5806i			1.43640i
$66$		0			0
$67$	−8.40407	−	4.85209i	−1.02672	−	0.592777i	−0.110677	−	0.993856i	$-0.535302\pi$
$67$	−8.40407	−	4.85209i	−1.02672	−	0.592777i	−0.916043	+	0.401079i	$0.868635\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.00887	+	8.67561i	0.594443	+	1.02961i	0.993625	+	0.112734i	$0.0359608\pi$
$71$	5.00887	+	8.67561i	0.594443	+	1.02961i	−0.399182	+	0.916872i	$0.630706\pi$
$72$		0			0
$73$	−5.72788	−	9.92098i	−0.670398	−	1.16116i	−0.977791	−	0.209580i	$-0.932790\pi$
$73$	−5.72788	−	9.92098i	−0.670398	−	1.16116i	0.307394	−	0.951582i	$-0.400543\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.20455i			0.251231i
$78$		0			0
$79$	1.78342	−	1.02966i	0.200650	−	0.115845i	−0.396309	−	0.918117i	$-0.629709\pi$
$79$	1.78342	−	1.02966i	0.200650	−	0.115845i	0.596959	+	0.802272i	$0.296376\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.34096			0.805775			0.402888	−	0.915249i	$-0.368007\pi$
$83$	7.34096			0.805775			0.402888	+	0.915249i	$0.368007\pi$
$84$		0			0
$85$	8.12276	−	14.0690i	0.881037	−	1.52600i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.22085	−	1.28221i	−0.235410	−	0.135914i	0.377655	−	0.925946i	$-0.376730\pi$
$89$	−2.22085	−	1.28221i	−0.235410	−	0.135914i	−0.613065	+	0.790032i	$0.710064\pi$
$90$		0			0
$91$	−11.5073	−	6.64376i	−1.20630	−	0.696456i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.36384	+	11.2157i	−0.242525	+	1.15071i
$96$		0			0
$97$	−2.80967	−	4.86649i	−0.285278	−	0.494117i	0.687398	−	0.726281i	$-0.258753\pi$
$97$	−2.80967	−	4.86649i	−0.285278	−	0.494117i	−0.972677	+	0.232164i	$0.925419\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−16.8914	−	9.75226i	−1.68076	−	0.970387i	−0.961158	−	0.275999i	$-0.910991\pi$
$101$	−16.8914	−	9.75226i	−1.68076	−	0.970387i	−0.719601	−	0.694388i	$-0.755675\pi$
$102$		0			0
$103$			19.6453i			1.93571i	0.251503	+	0.967857i	$0.419075\pi$
$103$			19.6453i			1.93571i	−0.251503	+	0.967857i	$0.580925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.6114			−1.41253			−0.706267	−	0.707946i	$-0.749622\pi$
$107$	−14.6114			−1.41253			−0.706267	+	0.707946i	$0.749622\pi$
$108$		0			0
$109$	2.80401	+	4.85669i	0.268576	+	0.465187i	0.968494	−	0.249036i	$-0.0801137\pi$
$109$	2.80401	+	4.85669i	0.268576	+	0.465187i	−0.699918	+	0.714223i	$0.746780\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			5.55408i			0.522484i	0.965273	+	0.261242i	$0.0841320\pi$
$113$			5.55408i			0.522484i	−0.965273	+	0.261242i	$0.915868\pi$
$114$		0			0
$115$			11.7442i			1.09515i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.32002	+	16.1427i	0.854365	+	1.47980i
$120$		0			0
$121$	−10.4661			−0.951466
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	8.11294i		−	0.725644i
$126$		0			0
$127$	1.99683	+	1.15287i	0.177190	+	0.102301i	0.585972	−	0.810331i	$-0.300713\pi$
$127$	1.99683	+	1.15287i	0.177190	+	0.102301i	−0.408782	+	0.912632i	$0.634046\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.24465	+	7.35196i	0.370857	+	0.642343i	0.989698	−	0.143173i	$-0.0457306\pi$
$131$	4.24465	+	7.35196i	0.370857	+	0.642343i	−0.618840	+	0.785517i	$0.712397\pi$
$132$		0			0
$133$	−9.78865	−	8.78333i	−0.848784	−	0.761611i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.68866	−	3.86170i	−0.571450	−	0.329927i	0.186278	−	0.982497i	$-0.440357\pi$
$137$	−6.68866	−	3.86170i	−0.571450	−	0.329927i	−0.757728	+	0.652570i	$0.773691\pi$
$138$		0			0
$139$	−8.66344	−	5.00184i	−0.734824	−	0.424251i	0.0853606	−	0.996350i	$-0.472796\pi$
$139$	−8.66344	−	5.00184i	−0.734824	−	0.424251i	−0.820184	+	0.572099i	$0.806129\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.60890	+	2.78670i	−0.134543	+	0.233036i
$144$		0			0
$145$	−14.7296			−1.22322
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.897686	−	0.518279i	0.0735413	−	0.0424591i	−0.462778	−	0.886474i	$-0.653147\pi$
$149$	0.897686	−	0.518279i	0.0735413	−	0.0424591i	0.536320	+	0.844015i	$0.319814\pi$
$150$		0			0
$151$			21.5740i			1.75567i	0.478965	+	0.877834i	$0.341012\pi$
$151$			21.5740i			1.75567i	−0.478965	+	0.877834i	$0.658988\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.38660	−	14.5260i	−0.673628	−	1.16676i
$156$		0			0
$157$	−7.79444	−	13.5004i	−0.622064	−	1.07745i	−0.989101	−	0.147240i	$-0.952961\pi$
$157$	−7.79444	−	13.5004i	−0.622064	−	1.07745i	0.367036	−	0.930207i	$-0.380372\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−11.6699	−	6.73762i	−0.919717	−	0.530999i
$162$		0			0
$163$		−	18.2536i		−	1.42973i	−0.699261	−	0.714867i	$-0.746487\pi$
$163$		−	18.2536i		−	1.42973i	0.699261	−	0.714867i	$-0.253513\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.74585	−	3.02390i	0.135098	−	0.233997i	−0.790537	−	0.612414i	$-0.790198\pi$
$167$	1.74585	−	3.02390i	0.135098	−	0.233997i	0.925635	+	0.378418i	$0.123532\pi$
$168$		0			0
$169$	−3.19739	−	5.53805i	−0.245953	−	0.426004i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.9620	+	6.32893i	−0.833428	+	0.481180i	−0.855025	−	0.518587i	$-0.826458\pi$
$173$	−10.9620	+	6.32893i	−0.833428	+	0.481180i	0.0215969	+	0.999767i	$0.493125\pi$
$174$		0			0
$175$	−5.00317	−	2.88858i	−0.378204	−	0.218356i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.95556			0.146165			0.0730826	−	0.997326i	$-0.476716\pi$
$179$	1.95556			0.146165			0.0730826	+	0.997326i	$0.476716\pi$
$180$		0			0
$181$	−3.90914	+	6.77082i	−0.290564	+	0.503271i	−0.973943	−	0.226792i	$-0.927176\pi$
$181$	−3.90914	+	6.77082i	−0.290564	+	0.503271i	0.683379	+	0.730064i	$0.260510\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.39134	−	1.95799i	0.249336	−	0.143954i
$186$		0			0
$187$	3.90925	−	2.25700i	0.285872	−	0.165048i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.37771			0.0996879			0.0498439	−	0.998757i	$-0.484128\pi$
$191$	1.37771			0.0996879			0.0498439	+	0.998757i	$0.484128\pi$
$192$		0			0
$193$	2.67422	+	4.63188i	0.192494	+	0.333410i	0.946076	−	0.323944i	$-0.105009\pi$
$193$	2.67422	+	4.63188i	0.192494	+	0.333410i	−0.753582	+	0.657354i	$0.771676\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.1527i			1.50707i	0.657409	+	0.753534i	$0.271652\pi$
$197$			21.1527i			1.50707i	−0.657409	+	0.753534i	$0.728348\pi$
$198$		0			0
$199$	8.22547	+	4.74898i	0.583088	+	0.336646i	0.762360	−	0.647154i	$-0.224041\pi$
$199$	8.22547	+	4.74898i	0.583088	+	0.336646i	−0.179272	+	0.983800i	$0.557374\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.45033	−	14.6364i	0.593097	−	1.02727i
$204$		0			0
$205$	5.38639	−	9.32950i	0.376202	−	0.651601i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.12703	+	2.37049i	−0.147130	+	0.163970i
$210$		0			0
$211$	24.6313	−	14.2209i	1.69569	−	0.979006i	0.745929	−	0.666025i	$-0.232006\pi$
$211$	24.6313	−	14.2209i	1.69569	−	0.979006i	0.949759	−	0.312981i	$-0.101328\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.3602	−	17.9444i	0.706558	−	1.22379i
$216$		0			0
$217$	19.2455			1.30647
$218$		0			0
$219$		0			0
$220$		0			0
$221$			27.2074i			1.83017i
$222$		0			0
$223$	−15.0576	+	8.69352i	−1.00833	+	0.582161i	−0.910703	−	0.413061i	$-0.864459\pi$
$223$	−15.0576	+	8.69352i	−1.00833	+	0.582161i	−0.0976300	+	0.995223i	$0.531126\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−26.9603			−1.78942			−0.894708	−	0.446652i	$-0.852616\pi$
$227$	−26.9603			−1.78942			−0.894708	+	0.446652i	$0.852616\pi$
$228$		0			0
$229$	−0.645395			−0.0426489			−0.0213245	−	0.999773i	$-0.506788\pi$
$229$	−0.645395			−0.0426489			−0.0213245	+	0.999773i	$0.506788\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.3758	+	11.1866i	−1.26935	+	0.732862i	−0.974866	−	0.222794i	$-0.928482\pi$
$233$	−19.3758	+	11.1866i	−1.26935	+	0.732862i	−0.294488	+	0.955655i	$0.595149\pi$
$234$		0			0
$235$		−	6.75110i		−	0.440393i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.37564			0.347721			0.173861	−	0.984770i	$-0.444376\pi$
$239$	5.37564			0.347721			0.173861	+	0.984770i	$0.444376\pi$
$240$		0			0
$241$	−0.0588490	+	0.101929i	−0.00379080	+	0.00656585i	−0.867915	−	0.496713i	$-0.834540\pi$
$241$	−0.0588490	+	0.101929i	−0.00379080	+	0.00656585i	0.864124	+	0.503279i	$0.167873\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.79005	−	2.76554i	0.306025	−	0.176684i
$246$		0			0
$247$	−5.96338	−	18.2466i	−0.379441	−	1.16100i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.16780	+	8.95089i	−0.326188	+	0.564975i	−0.981752	−	0.190165i	$-0.939098\pi$
$251$	−5.16780	+	8.95089i	−0.326188	+	0.564975i	0.655564	+	0.755140i	$0.272431\pi$
$252$		0			0
$253$	−1.63163	+	2.82607i	−0.102580	+	0.177673i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.7958	+	10.8518i	1.17245	+	0.676914i	0.954256	−	0.298991i	$-0.0966501\pi$
$257$	18.7958	+	10.8518i	1.17245	+	0.676914i	0.218194	+	0.975905i	$0.429983\pi$
$258$		0			0
$259$			4.49319i			0.279193i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.5325	+	23.4391i	0.834453	+	1.44531i	0.894475	+	0.447117i	$0.147549\pi$
$263$	13.5325	+	23.4391i	0.834453	+	1.44531i	−0.0600228	+	0.998197i	$0.519117\pi$
$264$		0			0
$265$	27.9968			1.71983
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.93328	−	2.84823i	0.300787	−	0.173660i	−0.342009	−	0.939697i	$-0.611107\pi$
$269$	4.93328	−	2.84823i	0.300787	−	0.173660i	0.642796	+	0.766037i	$0.277774\pi$
$270$		0			0
$271$	−2.59990	+	1.50105i	−0.157933	+	0.0911824i	−0.576883	−	0.816827i	$-0.695731\pi$
$271$	−2.59990	+	1.50105i	−0.157933	+	0.0911824i	0.418951	+	0.908009i	$0.362398\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.699519	+	1.21160i	−0.0421826	+	0.0730624i
$276$		0			0
$277$	1.08132			0.0649703			0.0324851	−	0.999472i	$-0.489658\pi$
$277$	1.08132			0.0649703			0.0324851	+	0.999472i	$0.489658\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.6773	−	9.05127i	−0.935227	−	0.539954i	−0.0467662	−	0.998906i	$-0.514892\pi$
$281$	−15.6773	−	9.05127i	−0.935227	−	0.539954i	−0.888461	+	0.458952i	$0.848225\pi$
$282$		0			0
$283$	1.62115	−	0.935971i	0.0963674	−	0.0556377i	−0.451042	−	0.892503i	$-0.648947\pi$
$283$	1.62115	−	0.935971i	0.0963674	−	0.0556377i	0.547409	+	0.836865i	$0.315614\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.18032	+	10.7046i	0.364813	+	0.631875i
$288$		0			0
$289$	10.5836	−	18.3313i	0.622564	−	1.07831i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.51284i		−	0.0883812i	−0.999023	−	0.0441906i	$-0.985929\pi$
$293$		−	1.51284i		−	0.0883812i	0.999023	−	0.0441906i	$-0.0140709\pi$
$294$		0			0
$295$	4.92316	+	2.84239i	0.286637	+	0.165490i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.83438	−	17.0337i	−0.568737	−	0.985082i
$300$		0			0
$301$	11.8872	+	20.5893i	0.685169	+	1.18675i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	22.4459i		−	1.28525i
$306$		0			0
$307$	0.715534	−	0.413114i	0.0408377	−	0.0235776i	−0.479442	−	0.877573i	$-0.659161\pi$
$307$	0.715534	−	0.413114i	0.0408377	−	0.0235776i	0.520280	+	0.853996i	$0.325828\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	26.4749			1.50126			0.750628	−	0.660725i	$-0.229751\pi$
$311$	26.4749			1.50126			0.750628	+	0.660725i	$0.229751\pi$
$312$		0			0
$313$	−5.36483	+	9.29215i	−0.303238	+	0.525223i	−0.976867	−	0.213846i	$-0.931401\pi$
$313$	−5.36483	+	9.29215i	−0.303238	+	0.525223i	0.673630	+	0.739069i	$0.264734\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.5225	−	8.96192i	−0.871830	−	0.503351i	−0.00387421	−	0.999992i	$-0.501233\pi$
$317$	−15.5225	−	8.96192i	−0.871830	−	0.503351i	−0.867956	+	0.496641i	$0.834567\pi$
$318$		0			0
$319$	−3.54446	−	2.04639i	−0.198452	−	0.114576i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.55360	+	26.3502i	−0.309010	+	1.46617i
$324$		0			0
$325$	−4.21624	−	7.30274i	−0.233875	−	0.405083i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.70839	+	3.87309i	0.369846	+	0.213531i
$330$		0			0
$331$			17.6283i			0.968936i	0.874809	+	0.484468i	$0.160987\pi$
$331$			17.6283i			0.968936i	−0.874809	+	0.484468i	$0.839013\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	25.5180			1.39420
$336$		0			0
$337$	11.5924	+	20.0786i	0.631477	+	1.09375i	0.987250	+	0.159178i	$0.0508843\pi$
$337$	11.5924	+	20.0786i	0.631477	+	1.09375i	−0.355773	+	0.934572i	$0.615782\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	4.66063i		−	0.252387i
$342$		0			0
$343$		−	14.7739i		−	0.797718i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.11621	+	1.93333i	0.0599211	+	0.103786i	0.894430	−	0.447208i	$-0.147582\pi$
$347$	1.11621	+	1.93333i	0.0599211	+	0.103786i	−0.834509	+	0.550995i	$0.814248\pi$
$348$		0			0
$349$	−31.8456			−1.70466			−0.852329	−	0.523007i	$-0.824810\pi$
$349$	−31.8456			−1.70466			−0.852329	+	0.523007i	$0.824810\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	10.5384i		−	0.560901i	−0.959868	−	0.280451i	$-0.909516\pi$
$353$		−	10.5384i		−	0.560901i	0.959868	−	0.280451i	$-0.0904839\pi$
$354$		0			0
$355$	−22.8133	−	13.1713i	−1.21080	−	0.699059i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−17.3642	−	30.0756i	−0.916446	−	1.58733i	−0.804770	−	0.593587i	$-0.797711\pi$
$359$	−17.3642	−	30.0756i	−0.916446	−	1.58733i	−0.111677	−	0.993745i	$-0.535622\pi$
$360$		0			0
$361$	−2.05098	−	18.8890i	−0.107946	−	0.994157i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	26.0881	+	15.0620i	1.36551	+	0.788380i
$366$		0			0
$367$	−13.0717	−	7.54694i	−0.682337	−	0.393947i	0.118398	−	0.992966i	$-0.462224\pi$
$367$	−13.0717	−	7.54694i	−0.682337	−	0.393947i	−0.800735	+	0.599019i	$0.795557\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−16.0617	+	27.8197i	−0.833882	+	1.44433i
$372$		0			0
$373$	31.2927			1.62027			0.810136	−	0.586242i	$-0.199393\pi$
$373$	31.2927			1.62027			0.810136	+	0.586242i	$0.199393\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.3636	−	12.3343i	1.10028	−	0.635249i
$378$		0			0
$379$		−	13.6109i		−	0.699146i	−0.936909	−	0.349573i	$-0.886327\pi$
$379$		−	13.6109i		−	0.699146i	0.936909	−	0.349573i	$-0.113673\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−14.2735	−	24.7223i	−0.729339	−	1.26325i	−0.957163	−	0.289550i	$-0.906494\pi$
$383$	−14.2735	−	24.7223i	−0.729339	−	1.26325i	0.227824	−	0.973702i	$-0.426839\pi$
$384$		0			0
$385$	−2.89853	−	5.02040i	−0.147723	−	0.255863i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.3036	+	6.52612i	0.573114	+	0.330887i	0.758392	−	0.651799i	$-0.225985\pi$
$389$	11.3036	+	6.52612i	0.573114	+	0.330887i	−0.185278	+	0.982686i	$0.559319\pi$
$390$		0			0
$391$			27.5918i			1.39538i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.70757	+	4.68965i	−0.136233	+	0.235962i
$396$		0			0
$397$	3.22502	+	5.58590i	0.161859	+	0.280348i	0.935535	−	0.353233i	$-0.114918\pi$
$397$	3.22502	+	5.58590i	0.161859	+	0.280348i	−0.773676	+	0.633581i	$0.781584\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	27.8231	−	16.0637i	1.38942	−	0.802181i	0.396169	−	0.918178i	$-0.370339\pi$
$401$	27.8231	−	16.0637i	1.38942	−	0.802181i	0.993250	+	0.115996i	$0.0370061\pi$
$402$		0			0
$403$	24.3277	+	14.0456i	1.21185	+	0.699661i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.08810			0.0539353
$408$		0			0
$409$	−13.7369	+	23.7930i	−0.679247	+	1.17649i	0.295961	+	0.955200i	$0.404360\pi$
$409$	−13.7369	+	23.7930i	−0.679247	+	1.17649i	−0.975208	+	0.221290i	$0.928973\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.64881	+	3.26134i	−0.277960	+	0.160480i
$414$		0			0
$415$	−16.7175	+	9.65186i	−0.820631	+	0.473791i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.7384			−0.817723			−0.408861	−	0.912597i	$-0.634074\pi$
$419$	−16.7384			−0.817723			−0.408861	+	0.912597i	$0.634074\pi$
$420$		0			0
$421$	5.37905	+	9.31679i	0.262159	+	0.454072i	0.966815	−	0.255476i	$-0.0822322\pi$
$421$	5.37905	+	9.31679i	0.262159	+	0.454072i	−0.704657	+	0.709549i	$0.748899\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			11.8293i			0.573803i
$426$		0			0
$427$	22.3039	+	12.8771i	1.07936	+	0.623169i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	10.9070	−	18.8915i	0.525373	−	0.909972i	−0.474191	−	0.880422i	$-0.657259\pi$
$431$	10.9070	−	18.8915i	0.525373	−	0.909972i	0.999563	−	0.0295500i	$-0.00940743\pi$
$432$		0			0
$433$	−12.6446	+	21.9010i	−0.607659	+	1.05250i	0.383967	+	0.923347i	$0.374558\pi$
$433$	−12.6446	+	21.9010i	−0.607659	+	1.05250i	−0.991625	+	0.129149i	$0.958776\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.04762	−	18.5044i	−0.289297	−	0.885185i
$438$		0			0
$439$	27.0701	−	15.6289i	1.29198	−	0.745927i	0.312978	−	0.949760i	$-0.398673\pi$
$439$	27.0701	−	15.6289i	1.29198	−	0.745927i	0.979006	+	0.203833i	$0.0653401\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.86303	+	10.1551i	−0.278561	+	0.482482i	−0.971027	−	0.238968i	$-0.923191\pi$
$443$	−5.86303	+	10.1551i	−0.278561	+	0.482482i	0.692466	+	0.721450i	$0.256524\pi$
$444$		0			0
$445$	6.74337			0.319667
$446$		0			0
$447$		0			0
$448$		0			0
$449$			19.3531i			0.913329i	0.889639	+	0.456664i	$0.150956\pi$
$449$			19.3531i			0.913329i	−0.889639	+	0.456664i	$0.849044\pi$
$450$		0			0
$451$	2.59231	−	1.49667i	0.122067	−	0.0704755i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	34.9408			1.63805
$456$		0			0
$457$	−7.43069			−0.347593			−0.173796	−	0.984782i	$-0.555603\pi$
$457$	−7.43069			−0.347593			−0.173796	+	0.984782i	$0.555603\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	30.0829	−	17.3684i	1.40110	−	0.808925i	0.406594	−	0.913609i	$-0.366716\pi$
$461$	30.0829	−	17.3684i	1.40110	−	0.808925i	0.994506	+	0.104684i	$0.0333832\pi$
$462$		0			0
$463$		−	8.95687i		−	0.416261i	−0.978101	−	0.208130i	$-0.933262\pi$
$463$		−	8.95687i		−	0.416261i	0.978101	−	0.208130i	$-0.0667378\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.282420			−0.0130689			−0.00653443	−	0.999979i	$-0.502080\pi$
$467$	−0.282420			−0.0130689			−0.00653443	+	0.999979i	$0.502080\pi$
$468$		0			0
$469$	−14.6397	+	25.3566i	−0.675996	+	1.17086i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.98605	−	2.87870i	0.229259	−	0.132363i
$474$		0			0
$475$	−2.59276	−	7.93327i	−0.118964	−	0.364004i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.93957	+	10.2876i	−0.271386	+	0.470054i	−0.969217	−	0.246208i	$-0.920815\pi$
$479$	−5.93957	+	10.2876i	−0.271386	+	0.470054i	0.697831	+	0.716263i	$0.254149\pi$
$480$		0			0
$481$	−3.27918	+	5.67971i	−0.149518	+	0.258972i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.7969	+	7.38828i	0.581076	+	0.335484i
$486$		0			0
$487$			6.04436i			0.273896i	0.990578	+	0.136948i	$0.0437294\pi$
$487$			6.04436i			0.273896i	−0.990578	+	0.136948i	$0.956271\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.76957	−	11.7252i	−0.305506	−	0.529153i	0.671868	−	0.740671i	$-0.265492\pi$
$491$	−6.76957	−	11.7252i	−0.305506	−	0.529153i	−0.977374	+	0.211519i	$0.932159\pi$
$492$		0			0
$493$	−34.6056			−1.55856
$494$		0			0
$495$		0			0
$496$		0			0
$497$	26.1759	−	15.1127i	1.17415	−	0.677896i
$498$		0			0
$499$	22.1361	−	12.7803i	0.990950	−	0.572125i	0.0853915	−	0.996347i	$-0.472786\pi$
$499$	22.1361	−	12.7803i	0.990950	−	0.572125i	0.905558	+	0.424222i	$0.139453\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.234869	+	0.406806i	−0.0104723	+	0.0181386i	−0.871214	−	0.490903i	$-0.836667\pi$
$503$	−0.234869	+	0.406806i	−0.0104723	+	0.0181386i	0.860742	+	0.509042i	$0.170000\pi$
$504$		0			0
$505$	51.2889			2.28233
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.39853	+	3.11684i	0.239285	+	0.138152i	0.614848	−	0.788645i	$-0.289217\pi$
$509$	5.39853	+	3.11684i	0.239285	+	0.138152i	−0.375563	+	0.926797i	$0.622551\pi$
$510$		0			0
$511$	−29.9334	+	17.2821i	−1.32418	+	0.764514i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−25.8296	−	44.7382i	−1.13819	−	1.97140i
$516$		0			0
$517$	0.937937	−	1.62455i	0.0412504	−	0.0714478i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	24.7521i		−	1.08441i	−0.840247	−	0.542204i	$-0.817590\pi$
$521$		−	24.7521i		−	1.08441i	0.840247	−	0.542204i	$-0.182410\pi$
$522$		0			0
$523$	16.9312	+	9.77523i	0.740350	+	0.427441i	0.822196	−	0.569204i	$-0.192749\pi$
$523$	16.9312	+	9.77523i	0.740350	+	0.427441i	−0.0818469	+	0.996645i	$0.526082\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−19.7035	−	34.1274i	−0.858297	−	1.48661i
$528$		0			0
$529$	1.52669	+	2.64430i	0.0663778	+	0.114970i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			18.0419i			0.781481i
$534$		0			0
$535$	33.2743	−	19.2109i	1.43858	−	0.830562i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.53688			0.0661979
$540$		0			0
$541$	6.43658	−	11.1485i	0.276730	−	0.479311i	−0.693840	−	0.720129i	$-0.744083\pi$
$541$	6.43658	−	11.1485i	0.276730	−	0.479311i	0.970570	+	0.240819i	$0.0774159\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.7711	−	7.37341i	−0.547055	−	0.315842i
$546$		0			0
$547$	−5.74630	−	3.31763i	−0.245694	−	0.141852i	0.372097	−	0.928194i	$-0.378639\pi$
$547$	−5.74630	−	3.31763i	−0.245694	−	0.141852i	−0.617791	+	0.786342i	$0.711972\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	23.2082	−	7.58493i	0.988703	−	0.323129i
$552$		0			0
$553$	−3.10666	−	5.38089i	−0.132109	−	0.228819i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−24.8855	−	14.3676i	−1.05443	−	0.608776i	−0.130545	−	0.991442i	$-0.541673\pi$
$557$	−24.8855	−	14.3676i	−1.05443	−	0.608776i	−0.923887	+	0.382666i	$0.875006\pi$
$558$		0			0
$559$			34.7018i			1.46773i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−35.7237			−1.50558			−0.752788	−	0.658263i	$-0.771291\pi$
$563$	−35.7237			−1.50558			−0.752788	+	0.658263i	$0.771291\pi$
$564$		0			0
$565$	−7.30248	−	12.6483i	−0.307218	−	0.532116i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	23.2818i		−	0.976023i	−0.872837	−	0.488011i	$-0.837722\pi$
$569$		−	23.2818i		−	0.976023i	0.872837	−	0.488011i	$-0.162278\pi$
$570$		0			0
$571$		−	23.1334i		−	0.968102i	−0.875040	−	0.484051i	$-0.839165\pi$
$571$		−	23.1334i		−	0.968102i	0.875040	−	0.484051i	$-0.160835\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.27580	−	7.40590i	−0.178313	−	0.308847i
$576$		0			0
$577$	−14.6533			−0.610023			−0.305012	−	0.952349i	$-0.598660\pi$
$577$	−14.6533			−0.610023			−0.305012	+	0.952349i	$0.598660\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	22.1490i		−	0.918897i
$582$		0			0
$583$	6.73701	+	3.88962i	0.279019	+	0.161091i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.1867	−	22.8401i	−0.544275	−	0.942712i	−0.998652	−	0.0519029i	$-0.983471\pi$
$587$	−13.1867	−	22.8401i	−0.544275	−	0.942712i	0.454377	−	0.890810i	$-0.349862\pi$
$588$		0			0
$589$	20.6942	+	18.5688i	0.852690	+	0.765116i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	32.8529	+	18.9676i	1.34911	+	0.778908i	0.988123	−	0.153664i	$-0.0491072\pi$
$593$	32.8529	+	18.9676i	1.34911	+	0.778908i	0.360985	+	0.932572i	$0.382441\pi$
$594$		0			0
$595$	−42.4488	−	24.5078i	−1.74023	−	1.00472i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.55133	−	2.68698i	0.0633856	−	0.109787i	−0.832591	−	0.553888i	$-0.813144\pi$
$599$	1.55133	−	2.68698i	0.0633856	−	0.109787i	0.895977	+	0.444101i	$0.146477\pi$
$600$		0			0
$601$	−43.4637			−1.77292			−0.886460	−	0.462805i	$-0.846843\pi$
$601$	−43.4637			−1.77292			−0.886460	+	0.462805i	$0.846843\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	23.8344	−	13.7608i	0.969008	−	0.559457i
$606$		0			0
$607$			2.19694i			0.0891708i	0.999006	+	0.0445854i	$0.0141967\pi$
$607$			2.19694i			0.0891708i	−0.999006	+	0.0445854i	$0.985803\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.65326	+	9.79173i	0.228706	+	0.396131i
$612$		0			0
$613$	2.82963	+	4.90107i	0.114288	+	0.197952i	0.917495	−	0.397748i	$-0.130208\pi$
$613$	2.82963	+	4.90107i	0.114288	+	0.197952i	−0.803207	+	0.595700i	$0.796875\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−17.6435	−	10.1865i	−0.710302	−	0.410093i	0.100871	−	0.994900i	$-0.467837\pi$
$617$	−17.6435	−	10.1865i	−0.710302	−	0.410093i	−0.811173	+	0.584807i	$0.801170\pi$
$618$		0			0
$619$			40.3661i			1.62245i	0.584735	+	0.811224i	$0.301199\pi$
$619$			40.3661i			1.62245i	−0.584735	+	0.811224i	$0.698801\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.86866	+	6.70072i	−0.154995	+	0.268459i
$624$		0			0
$625$	15.4537	+	26.7667i	0.618150	+	1.07067i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.96762	−	4.60011i	0.317690	−	0.183418i
$630$		0			0
$631$	32.0420	+	18.4995i	1.27557	+	0.736452i	0.976031	−	0.217631i	$-0.0698330\pi$
$631$	32.0420	+	18.4995i	1.27557	+	0.736452i	0.299541	+	0.954083i	$0.403166\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.06315			−0.240609
$636$		0			0
$637$	−4.63163	+	8.02222i	−0.183512	+	0.317852i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.37740	+	4.83669i	−0.330887	+	0.191038i	−0.656235	−	0.754557i	$-0.727852\pi$
$641$	−8.37740	+	4.83669i	−0.330887	+	0.191038i	0.325348	+	0.945594i	$0.394519\pi$
$642$		0			0
$643$	31.6351	−	18.2645i	1.24757	−	0.720282i	0.276942	−	0.960887i	$-0.410679\pi$
$643$	31.6351	−	18.2645i	1.24757	−	0.720282i	0.970623	+	0.240604i	$0.0773457\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−30.0388			−1.18095			−0.590474	−	0.807057i	$-0.701059\pi$
$647$	−30.0388			−1.18095			−0.590474	+	0.807057i	$0.701059\pi$
$648$		0			0
$649$	0.789791	+	1.36796i	0.0310020	+	0.0536970i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	35.2628i		−	1.37994i	−0.723838	−	0.689970i	$-0.757624\pi$
$653$		−	35.2628i		−	1.37994i	0.723838	−	0.689970i	$-0.242376\pi$
$654$		0			0
$655$	−19.3326	−	11.1617i	−0.755389	−	0.436124i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.8773	−	27.5002i	0.618490	−	1.07126i	−0.371272	−	0.928524i	$-0.621078\pi$
$659$	15.8773	−	27.5002i	0.618490	−	1.07126i	0.989761	−	0.142732i	$-0.0455886\pi$
$660$		0			0
$661$	10.1533	−	17.5861i	0.394919	−	0.684020i	−0.598172	−	0.801368i	$-0.704106\pi$
$661$	10.1533	−	17.5861i	0.394919	−	0.684020i	0.993091	+	0.117348i	$0.0374394\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	33.8399	+	7.13213i	1.31226	+	0.276572i
$666$		0			0
$667$	21.6654	−	12.5085i	0.838889	−	0.484333i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.11842	−	5.40127i	0.120385	−	0.208514i
$672$		0			0
$673$	−36.6513			−1.41280			−0.706402	−	0.707811i	$-0.749683\pi$
$673$	−36.6513			−1.41280			−0.706402	+	0.707811i	$0.749683\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	23.6921i		−	0.910562i	−0.890348	−	0.455281i	$-0.849539\pi$
$677$		−	23.6921i		−	0.910562i	0.890348	−	0.455281i	$-0.150461\pi$
$678$		0			0
$679$	−14.6831	+	8.47728i	−0.563485	+	0.325328i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.59785			−0.175932			−0.0879659	−	0.996123i	$-0.528037\pi$
$683$	−4.59785			−0.175932			−0.0879659	+	0.996123i	$0.528037\pi$
$684$		0			0
$685$	20.3094			0.775981
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−40.6062	+	23.4440i	−1.54697	+	0.893146i
$690$		0			0
$691$		−	33.3184i		−	1.26749i	−0.773542	−	0.633745i	$-0.781517\pi$
$691$		−	33.3184i		−	1.26749i	0.773542	−	0.633745i	$-0.218483\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	26.3056			0.997828
$696$		0			0
$697$	12.6548	−	21.9187i	0.479334	−	0.830231i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.9031	−	9.18164i	0.600651	−	0.346786i	−0.168647	−	0.985677i	$-0.553940\pi$
$701$	15.9031	−	9.18164i	0.600651	−	0.346786i	0.769297	+	0.638891i	$0.220606\pi$
$702$		0			0
$703$	−4.33521	+	4.83141i	−0.163506	+	0.182220i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−29.4244	+	50.9645i	−1.10662	+	1.91672i
$708$		0			0
$709$	−19.5671	+	33.8913i	−0.734860	+	1.27281i	0.219925	+	0.975517i	$0.429419\pi$
$709$	−19.5671	+	33.8913i	−0.734860	+	1.27281i	−0.954785	+	0.297298i	$0.903915\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	24.6714	+	14.2440i	0.923949	+	0.533442i
$714$		0			0
$715$		−	8.46151i		−	0.316443i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.0577	+	22.6166i	0.486969	+	0.843455i	0.999888	−	0.0149819i	$-0.00476905\pi$
$719$	13.0577	+	22.6166i	0.486969	+	0.843455i	−0.512919	+	0.858437i	$0.671436\pi$
$720$		0			0
$721$	59.2736			2.20746
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.28849	−	5.36271i	0.344966	−	0.199166i
$726$		0			0
$727$	35.3018	−	20.3815i	1.30927	−	0.755907i	0.327295	−	0.944922i	$-0.393863\pi$
$727$	35.3018	−	20.3815i	1.30927	−	0.755907i	0.981974	+	0.189015i	$0.0605295\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	24.3402	−	42.1584i	0.900254	−	1.55929i
$732$		0			0
$733$	23.0259			0.850482			0.425241	−	0.905080i	$-0.360189\pi$
$733$	23.0259			0.850482			0.425241	+	0.905080i	$0.360189\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.14055	+	3.54525i	0.226190	+	0.130591i
$738$		0			0
$739$	−6.72833	+	3.88460i	−0.247505	+	0.142897i	−0.618621	−	0.785689i	$-0.712309\pi$
$739$	−6.72833	+	3.88460i	−0.247505	+	0.142897i	0.371116	+	0.928587i	$0.378975\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.661917	+	1.14647i	0.0242834	+	0.0420600i	0.877912	−	0.478823i	$-0.158936\pi$
$743$	0.661917	+	1.14647i	0.0242834	+	0.0420600i	−0.853628	+	0.520883i	$0.825603\pi$
$744$		0			0
$745$	−1.36286	+	2.36055i	−0.0499314	+	0.0864838i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			44.0851i			1.61084i
$750$		0			0
$751$	−10.5737	−	6.10474i	−0.385841	−	0.222765i	0.294516	−	0.955647i	$-0.404842\pi$
$751$	−10.5737	−	6.10474i	−0.385841	−	0.222765i	−0.680356	+	0.732881i	$0.738175\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−28.3654	−	49.1303i	−1.03232	−	1.78804i
$756$		0			0
$757$	10.3695	+	17.9606i	0.376888	+	0.652788i	0.990608	−	0.136735i	$-0.0436610\pi$
$757$	10.3695	+	17.9606i	0.376888	+	0.652788i	−0.613720	+	0.789524i	$0.710328\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			41.6236i			1.50886i	0.656383	+	0.754428i	$0.272086\pi$
$761$			41.6236i			1.50886i	−0.656383	+	0.754428i	$0.727914\pi$
$762$		0			0
$763$	14.6535	−	8.46022i	0.530494	−	0.306281i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.52066			−0.343771
$768$		0			0
$769$	−1.74896	+	3.02929i	−0.0630692	+	0.109239i	−0.895836	−	0.444385i	$-0.853422\pi$
$769$	−1.74896	+	3.02929i	−0.0630692	+	0.109239i	0.832767	+	0.553624i	$0.186756\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.20851	−	0.697735i	−0.0434672	−	0.0250958i	0.478109	−	0.878301i	$-0.341322\pi$
$773$	−1.20851	−	0.697735i	−0.0434672	−	0.0250958i	−0.521576	+	0.853205i	$0.674656\pi$
$774$		0			0
$775$	10.5772	+	6.10675i	0.379944	+	0.219361i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.68272	+	17.4734i	−0.131947	+	0.626051i
$780$		0			0
$781$	−3.65979	−	6.33895i	−0.130958	−	0.226825i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	35.5005	+	20.4962i	1.26707	+	0.731541i
$786$		0			0
$787$		−	12.6125i		−	0.449586i	−0.974407	−	0.224793i	$-0.927829\pi$
$787$		−	12.6125i		−	0.449586i	0.974407	−	0.224793i	$-0.0721706\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	16.7577			0.595834
$792$		0			0
$793$	18.7958	+	32.5552i	0.667458	+	1.15607i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	13.6012i		−	0.481780i	−0.970552	−	0.240890i	$-0.922561\pi$
$797$		−	13.6012i		−	0.481780i	0.970552	−	0.240890i	$-0.0774393\pi$
$798$		0			0
$799$		−	15.8610i		−	0.561123i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.18515	+	7.24889i	0.147691	+	0.255808i
$804$		0			0
$805$	35.4344			1.24890
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	8.53063i		−	0.299921i	−0.988692	−	0.149960i	$-0.952085\pi$
$809$		−	8.53063i		−	0.299921i	0.988692	−	0.149960i	$-0.0479146\pi$
$810$		0			0
$811$	45.3271	+	26.1696i	1.59165	+	0.918939i	0.993024	+	0.117908i	$0.0376189\pi$
$811$	45.3271	+	26.1696i	1.59165	+	0.918939i	0.598624	+	0.801030i	$0.295714\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	23.9998	+	41.5688i	0.840675	+	1.45609i
$816$		0			0
$817$	−7.08334	+	33.6084i	−0.247815	+	1.17581i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	26.3083	+	15.1891i	0.918165	+	0.530103i	0.883049	−	0.469280i	$-0.155487\pi$
$821$	26.3083	+	15.1891i	0.918165	+	0.530103i	0.0351160	+	0.999383i	$0.488820\pi$
$822$		0			0
$823$	−31.4836	−	18.1770i	−1.09745	−	0.633612i	−0.161899	−	0.986807i	$-0.551762\pi$
$823$	−31.4836	−	18.1770i	−1.09745	−	0.633612i	−0.935550	+	0.353195i	$0.885095\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.2908	−	31.6806i	0.636033	−	1.10164i	−0.350263	−	0.936652i	$-0.613908\pi$
$827$	18.2908	−	31.6806i	0.636033	−	1.10164i	0.986295	−	0.164989i	$-0.0527590\pi$
$828$		0			0
$829$	13.1047			0.455147			0.227573	−	0.973761i	$-0.426921\pi$
$829$	13.1047			0.455147			0.227573	+	0.973761i	$0.426921\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	11.2537	−	6.49735i	0.389919	−	0.225120i
$834$		0			0
$835$			9.18175i			0.317748i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8.94545	+	15.4940i	0.308831	+	0.534911i	0.978107	−	0.208103i	$-0.0667289\pi$
$839$	8.94545	+	15.4940i	0.308831	+	0.534911i	−0.669276	+	0.743014i	$0.733396\pi$
$840$		0			0
$841$	1.18823	+	2.05807i	0.0409733	+	0.0709679i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14.5628	+	8.40784i	0.500976	+	0.289238i
$846$		0			0
$847$			31.5782i			1.08504i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.32550	+	5.75994i	−0.113997	+	0.197448i
$852$		0			0
$853$	−5.15772	−	8.93344i	−0.176597	−	0.305875i	0.764116	−	0.645079i	$-0.223176\pi$
$853$	−5.15772	−	8.93344i	−0.176597	−	0.305875i	−0.940713	+	0.339204i	$0.889842\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.34720	+	4.81926i	−0.285135	+	0.164623i	−0.635746	−	0.771898i	$-0.719307\pi$
$857$	−8.34720	+	4.81926i	−0.285135	+	0.164623i	0.350611	+	0.936521i	$0.385974\pi$
$858$		0			0
$859$	−38.1615	−	22.0326i	−1.30205	−	0.751741i	−0.321298	−	0.946978i	$-0.604119\pi$
$859$	−38.1615	−	22.0326i	−1.30205	−	0.751741i	−0.980756	+	0.195237i	$0.937452\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	51.0749			1.73861			0.869306	−	0.494275i	$-0.164566\pi$
$863$	51.0749			1.73861			0.869306	+	0.494275i	$0.164566\pi$
$864$		0			0
$865$	16.6425	−	28.8257i	0.565862	−	0.980102i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.30308	+	0.752331i	−0.0442038	+	0.0255211i
$870$		0			0
$871$	−37.0111	+	21.3684i	−1.25407	+	0.724040i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−24.4782			−0.827515
$876$		0			0
$877$	5.91775	+	10.2498i	0.199828	+	0.346112i	0.948473	−	0.316859i	$-0.102628\pi$
$877$	5.91775	+	10.2498i	0.199828	+	0.346112i	−0.748644	+	0.662972i	$0.769295\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	17.4075i		−	0.586475i	−0.956040	−	0.293237i	$-0.905267\pi$
$881$		−	17.4075i		−	0.586475i	0.956040	−	0.293237i	$-0.0947326\pi$
$882$		0			0
$883$	−33.2311	−	19.1860i	−1.11831	−	0.645659i	−0.177344	−	0.984149i	$-0.556751\pi$
$883$	−33.2311	−	19.1860i	−1.11831	−	0.645659i	−0.940970	+	0.338490i	$0.890084\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.0878	−	19.2046i	0.372292	−	0.644828i	−0.617626	−	0.786472i	$-0.711905\pi$
$887$	11.0878	−	19.2046i	0.372292	−	0.644828i	0.989918	+	0.141644i	$0.0452387\pi$
$888$		0			0
$889$	3.47842	−	6.02480i	0.116662	−	0.202065i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.47645	+	10.6372i	0.116335	+	0.355959i
$894$		0			0
$895$	−4.45338	+	2.57116i	−0.148860	+	0.0859443i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17.8649	+	30.9428i	−0.595826	+	1.03200i
$900$		0			0
$901$	65.7756			2.19130
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	20.5589i		−	0.683400i
$906$		0			0
$907$	−3.09253	+	1.78547i	−0.102686	+	0.0592856i	−0.550463	−	0.834859i	$-0.685549\pi$
$907$	−3.09253	+	1.78547i	−0.102686	+	0.0592856i	0.447778	+	0.894145i	$0.352216\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	33.2094			1.10028			0.550138	−	0.835074i	$-0.314575\pi$
$911$	33.2094			1.10028			0.550138	+	0.835074i	$0.314575\pi$
$912$		0			0
$913$	−5.36377			−0.177515
$914$		0			0
$915$		0			0
$916$		0			0
$917$	22.1822	−	12.8069i	0.732521	−	0.422921i
$918$		0			0
$919$		−	12.5957i		−	0.415493i	−0.978183	−	0.207746i	$-0.933387\pi$
$919$		−	12.5957i		−	0.415493i	0.978183	−	0.207746i	$-0.0666128\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	44.1176			1.45215
$924$		0			0
$925$	−1.42572	+	2.46943i	−0.0468775	+	0.0811942i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−33.2298	+	19.1853i	−1.09024	+	0.629448i	−0.933639	−	0.358215i	$-0.883386\pi$
$929$	−33.2298	+	19.1853i	−1.09024	+	0.629448i	−0.156596	+	0.987663i	$0.550052\pi$
$930$		0			0
$931$	−6.12320	+	6.82406i	−0.200680	+	0.223649i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.93500	+	10.2797i	−0.194095	+	0.336183i
$936$		0			0
$937$	−28.0371	+	48.5617i	−0.915932	+	1.58644i	−0.110402	+	0.993887i	$0.535214\pi$
$937$	−28.0371	+	48.5617i	−0.915932	+	1.58644i	−0.805530	+	0.592555i	$0.798119\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−11.3546	−	6.55556i	−0.370148	−	0.213705i	0.303375	−	0.952871i	$-0.401887\pi$
$941$	−11.3546	−	6.55556i	−0.370148	−	0.213705i	−0.673523	+	0.739166i	$0.735220\pi$
$942$		0			0
$943$			18.2968i			0.595824i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.3590	+	42.1910i	0.791560	+	1.37102i	0.925001	+	0.379965i	$0.124064\pi$
$947$	24.3590	+	42.1910i	0.791560	+	1.37102i	−0.133441	+	0.991057i	$0.542603\pi$
$948$		0			0
$949$	−50.4506			−1.63770
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−13.6474	+	7.87933i	−0.442083	+	0.255237i	−0.704481	−	0.709723i	$-0.748820\pi$
$953$	−13.6474	+	7.87933i	−0.442083	+	0.255237i	0.262398	+	0.964960i	$0.415487\pi$
$954$		0			0
$955$	−3.13746	+	1.81141i	−0.101526	+	0.0586159i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.6515	+	20.1809i	−0.376245	+	0.651675i
$960$		0			0
$961$	−9.68697			−0.312483
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.1800	−	7.03210i	−0.392086	−	0.226371i
$966$		0			0
$967$	42.7997	−	24.7104i	1.37635	−	0.794633i	0.384628	−	0.923072i	$-0.374330\pi$
$967$	42.7997	−	24.7104i	1.37635	−	0.794633i	0.991718	+	0.128438i	$0.0409964\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8.95664	−	15.5134i	−0.287432	−	0.497848i	0.685764	−	0.727824i	$-0.259468\pi$
$971$	−8.95664	−	15.5134i	−0.287432	−	0.497848i	−0.973196	+	0.229977i	$0.926135\pi$
$972$		0			0
$973$	−15.0915	+	26.1392i	−0.483810	+	0.837984i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	46.2005i		−	1.47808i	−0.673660	−	0.739042i	$-0.735279\pi$
$977$		−	46.2005i		−	1.47808i	0.673660	−	0.739042i	$-0.264721\pi$
$978$		0			0
$979$	1.62269	+	0.936863i	0.0518615	+	0.0299423i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24.8820	−	43.0970i	−0.793614	−	1.37458i	−0.923716	−	0.383079i	$-0.874864\pi$
$983$	−24.8820	−	43.0970i	−0.793614	−	1.37458i	0.130102	−	0.991501i	$-0.458470\pi$
$984$		0			0
$985$	−27.8115	−	48.1709i	−0.886147	−	1.53485i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			35.1920i			1.11904i
$990$		0			0
$991$	12.8014	−	7.39089i	0.406650	−	0.234779i	−0.282699	−	0.959209i	$-0.591230\pi$
$991$	12.8014	−	7.39089i	0.406650	−	0.234779i	0.689349	+	0.724429i	$0.257897\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−24.9757			−0.791784
$996$		0			0
$997$	−1.77321	+	3.07130i	−0.0561582	+	0.0972689i	−0.892738	−	0.450576i	$-0.851218\pi$
$997$	−1.77321	+	3.07130i	−0.0561582	+	0.0972689i	0.836580	+	0.547845i	$0.184552\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.cg.a.2591.2	yes	24
3.2	odd	2	inner	2736.2.cg.a.2591.11	yes	24
4.3	odd	2		2736.2.cg.b.2591.2	yes	24
12.11	even	2		2736.2.cg.b.2591.11	yes	24
19.11	even	3		2736.2.cg.b.1151.11	yes	24
57.11	odd	6		2736.2.cg.b.1151.2	yes	24
76.11	odd	6	inner	2736.2.cg.a.1151.11	yes	24
228.11	even	6	inner	2736.2.cg.a.1151.2	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.a.1151.2	✓	24	228.11	even	6	inner
2736.2.cg.a.1151.11	yes	24	76.11	odd	6	inner
2736.2.cg.a.2591.2	yes	24	1.1	even	1	trivial
2736.2.cg.a.2591.11	yes	24	3.2	odd	2	inner
2736.2.cg.b.1151.2	yes	24	57.11	odd	6
2736.2.cg.b.1151.11	yes	24	19.11	even	3
2736.2.cg.b.2591.2	yes	24	4.3	odd	2
2736.2.cg.b.2591.11	yes	24	12.11	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.cg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.27729 + 1.31480i) q^{5} -3.01718i q^{7} +O(q^{10})\)	\(q+(-2.27729 + 1.31480i) q^{5} -3.01718i q^{7} -0.730663 q^{11} +(2.20198 - 3.81393i) q^{13} +(-5.35027 + 3.08898i) q^{17} +(2.91110 - 3.24430i) q^{19} +(2.23308 - 3.86781i) q^{23} +(0.957376 - 1.65822i) q^{25} +(4.85101 + 2.80073i) q^{29} +6.37863i q^{31} +(3.96698 + 6.87101i) q^{35} -1.48920 q^{37} +(-3.54789 + 2.04837i) q^{41} +(-6.82401 + 3.93984i) q^{43} +(-1.28368 + 2.22340i) q^{47} -2.10340 q^{49} +(-9.22041 - 5.32341i) q^{53} +(1.66393 - 0.960673i) q^{55} +(-1.08092 - 1.87221i) q^{59} +(-4.26794 + 7.39228i) q^{61} +11.5806i q^{65} +(-8.40407 - 4.85209i) q^{67} +(5.00887 + 8.67561i) q^{71} +(-5.72788 - 9.92098i) q^{73} +2.20455i q^{77} +(1.78342 - 1.02966i) q^{79} +7.34096 q^{83} +(8.12276 - 14.0690i) q^{85} +(-2.22085 - 1.28221i) q^{89} +(-11.5073 - 6.64376i) q^{91} +(-2.36384 + 11.2157i) q^{95} +(-2.80967 - 4.86649i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 12 q^{13} - 12 q^{19} + 8 q^{25} + 16 q^{37} - 12 q^{43} + 16 q^{49} + 12 q^{55} - 60 q^{67} + 8 q^{73} + 12 q^{79} + 16 q^{85} - 12 q^{91} - 32 q^{97}+O(q^{100}) \) 24 * q + 12 * q^13 - 12 * q^19 + 8 * q^25 + 16 * q^37 - 12 * q^43 + 16 * q^49 + 12 * q^55 - 60 * q^67 + 8 * q^73 + 12 * q^79 + 16 * q^85 - 12 * q^91 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more