LMFDB - Embedded newform 5760.2.h.c.1151.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5760,2,Mod(1151,5760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5760, 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("5760.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5760 = 2^{7} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5760.h (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:	$45.9938315643$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1534132224.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1 $ x^8 + 18x^6 + 107x^4 + 210*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.6
Root			$-2.63019i$ of defining polynomial
Character	$\chi$	$=$	5760.1151
Dual form			5760.2.h.c.1151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -1.41421i q^{7} +O(q^{10})$	$q+1.00000i q^{5} -1.41421i q^{7} +3.84616 q^{11} +0.406868 q^{13} -4.43195i q^{17} +5.26038i q^{19} -0.992654 q^{23} -1.00000 q^{25} +8.26772i q^{29} +5.00735i q^{31} +1.41421 q^{35} -1.01773 q^{37} +7.68193i q^{41} -1.42460i q^{43} -6.66420 q^{47} +5.00000 q^{49} -1.78244i q^{53} +3.84616i q^{55} -8.45703 q^{59} +10.8431 q^{61} +0.406868i q^{65} -8.61087i q^{67} -5.40383 q^{71} +9.69232 q^{73} -5.43929i q^{77} +8.64951i q^{79} +12.4853 q^{83} +4.43195 q^{85} -3.19665i q^{89} -0.575398i q^{91} -5.26038 q^{95} +0.575398 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{11} + 8 q^{13} - 24 q^{23} - 8 q^{25} + 8 q^{37} + 8 q^{47} + 40 q^{49} - 8 q^{59} + 32 q^{61} - 32 q^{71} + 32 q^{83} + 8 q^{85} + 8 q^{95} + 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^11 + 8 * q^13 - 24 * q^23 - 8 * q^25 + 8 * q^37 + 8 * q^47 + 40 * q^49 - 8 * q^59 + 32 * q^61 - 32 * q^71 + 32 * q^83 + 8 * q^85 + 8 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5760\mathbb{Z}\right)^\times$.

$n$	$641$	$901$	$2431$	$3457$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	− 1.41421i	− 0.534522i	−0.963624	−	0.267261i	$-0.913881\pi$
$7$	− 1.41421i	− 0.534522i	0.963624	−	0.267261i	$-0.0861187\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.84616	1.15966	0.579831	−	0.814737i	$-0.303119\pi$
$11$	3.84616	1.15966	0.579831	+	0.814737i	$0.303119\pi$
$12$	0	0
$13$	0.406868	0.112845	0.0564224	−	0.998407i	$-0.482031\pi$
$13$	0.406868	0.112845	0.0564224	+	0.998407i	$0.482031\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.43195i	− 1.07491i	−0.843294	−	0.537453i	$-0.819387\pi$
$17$	− 4.43195i	− 1.07491i	0.843294	−	0.537453i	$-0.180613\pi$
$18$	0	0
$19$	5.26038i	1.20681i	0.797434	+	0.603406i	$0.206190\pi$
$19$	5.26038i	1.20681i	−0.797434	+	0.603406i	$0.793810\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.992654	−0.206983	−0.103491	−	0.994630i	$-0.533001\pi$
$23$	−0.992654	−0.206983	−0.103491	+	0.994630i	$0.533001\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.26772i	1.53528i	0.640883	+	0.767639i	$0.278568\pi$
$29$	8.26772i	1.53528i	−0.640883	+	0.767639i	$0.721432\pi$
$30$	0	0
$31$	5.00735i	0.899346i	0.893193	+	0.449673i	$0.148459\pi$
$31$	5.00735i	0.899346i	−0.893193	+	0.449673i	$0.851541\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.41421	0.239046
$36$	0	0
$37$	−1.01773	−0.167315	−0.0836573	−	0.996495i	$-0.526660\pi$
$37$	−1.01773	−0.167315	−0.0836573	+	0.996495i	$0.526660\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.68193i	1.19972i	0.800106	+	0.599858i	$0.204776\pi$
$41$	7.68193i	1.19972i	−0.800106	+	0.599858i	$0.795224\pi$
$42$	0	0
$43$	− 1.42460i	− 0.217250i	−0.994083	−	0.108625i	$-0.965355\pi$
$43$	− 1.42460i	− 0.217250i	0.994083	−	0.108625i	$-0.0346447\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.66420	−0.972073	−0.486037	−	0.873938i	$-0.661558\pi$
$47$	−6.66420	−0.972073	−0.486037	+	0.873938i	$0.661558\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.78244i	− 0.244837i	−0.992479	−	0.122418i	$-0.960935\pi$
$53$	− 1.78244i	− 0.244837i	0.992479	−	0.122418i	$-0.0390650\pi$
$54$	0	0
$55$	3.84616i	0.518616i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.45703	−1.10101	−0.550506	−	0.834831i	$-0.685565\pi$
$59$	−8.45703	−1.10101	−0.550506	+	0.834831i	$0.685565\pi$
$60$	0	0
$61$	10.8431	1.38832	0.694160	−	0.719821i	$-0.255776\pi$
$61$	10.8431	1.38832	0.694160	+	0.719821i	$0.255776\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.406868i	0.0504657i
$66$	0	0
$67$	− 8.61087i	− 1.05198i	−0.850489	−	0.525992i	$-0.823694\pi$
$67$	− 8.61087i	− 1.05198i	0.850489	−	0.525992i	$-0.176306\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.40383	−0.641316	−0.320658	−	0.947195i	$-0.603904\pi$
$71$	−5.40383	−0.641316	−0.320658	+	0.947195i	$0.603904\pi$
$72$	0	0
$73$	9.69232	1.13440	0.567200	−	0.823580i	$-0.308027\pi$
$73$	9.69232	1.13440	0.567200	+	0.823580i	$0.308027\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.43929i	− 0.619865i
$78$	0	0
$79$	8.64951i	0.973146i	0.873640	+	0.486573i	$0.161753\pi$
$79$	8.64951i	0.973146i	−0.873640	+	0.486573i	$0.838247\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.4853	1.37044	0.685219	−	0.728337i	$-0.259707\pi$
$83$	12.4853	1.37044	0.685219	+	0.728337i	$0.259707\pi$
$84$	0	0
$85$	4.43195	0.480712
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.19665i	− 0.338845i	−0.985544	−	0.169422i	$-0.945810\pi$
$89$	− 3.19665i	− 0.338845i	0.985544	−	0.169422i	$-0.0541902\pi$
$90$	0	0
$91$	− 0.575398i	− 0.0603181i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.26038	−0.539703
$96$	0	0
$97$	0.575398	0.0584228	0.0292114	−	0.999573i	$-0.490700\pi$
$97$	0.575398	0.0584228	0.0292114	+	0.999573i	$0.490700\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.828427i	0.0824316i	0.999150	+	0.0412158i	$0.0131231\pi$
$101$	0.828427i	0.0824316i	−0.999150	+	0.0412158i	$0.986877\pi$
$102$	0	0
$103$	− 13.3388i	− 1.31431i	−0.753756	−	0.657155i	$-0.771760\pi$
$103$	− 13.3388i	− 1.31431i	0.753756	−	0.657155i	$-0.228240\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.84312	0.468202	0.234101	−	0.972212i	$-0.424785\pi$
$107$	4.84312	0.468202	0.234101	+	0.972212i	$0.424785\pi$
$108$	0	0
$109$	4.03547	0.386528	0.193264	−	0.981147i	$-0.438093\pi$
$109$	4.03547	0.386528	0.193264	+	0.981147i	$0.438093\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.7811i	1.29642i	0.761462	+	0.648210i	$0.224482\pi$
$113$	13.7811i	1.29642i	−0.761462	+	0.648210i	$0.775518\pi$
$114$	0	0
$115$	− 0.992654i	− 0.0925655i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.26772	−0.574561
$120$	0	0
$121$	3.79296	0.344814
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	0.803347i	0.0712855i	0.999365	+	0.0356428i	$0.0113478\pi$
$127$	0.803347i	0.0712855i	−0.999365	+	0.0356428i	$0.988652\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.8254	1.03319	0.516594	−	0.856230i	$-0.327200\pi$
$131$	11.8254	1.03319	0.516594	+	0.856230i	$0.327200\pi$
$132$	0	0
$133$	7.43929	0.645069
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.9172i	1.44534i	0.691195	+	0.722668i	$0.257085\pi$
$137$	16.9172i	1.44534i	−0.691195	+	0.722668i	$0.742915\pi$
$138$	0	0
$139$	− 17.1495i	− 1.45460i	−0.686320	−	0.727300i	$-0.740775\pi$
$139$	− 17.1495i	− 1.45460i	0.686320	−	0.727300i	$-0.259225\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.56488	0.130862
$144$	0	0
$145$	−8.26772	−0.686597
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.7530i	1.70015i	0.526660	+	0.850076i	$0.323444\pi$
$149$	20.7530i	1.70015i	−0.526660	+	0.850076i	$0.676556\pi$
$150$	0	0
$151$	− 6.35652i	− 0.517287i	−0.965973	−	0.258643i	$-0.916725\pi$
$151$	− 6.35652i	− 0.517287i	0.965973	−	0.258643i	$-0.0832754\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.00735	−0.402200
$156$	0	0
$157$	5.32541	0.425014	0.212507	−	0.977160i	$-0.431837\pi$
$157$	5.32541	0.425014	0.212507	+	0.977160i	$0.431837\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.40383i	0.110637i
$162$	0	0
$163$	− 0.378615i	− 0.0296554i	−0.999890	−	0.0148277i	$-0.995280\pi$
$163$	− 0.378615i	− 0.0296554i	0.999890	−	0.0148277i	$-0.00471997\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.1703	−1.32867	−0.664337	−	0.747433i	$-0.731286\pi$
$167$	−17.1703	−1.32867	−0.664337	+	0.747433i	$0.731286\pi$
$168$	0	0
$169$	−12.8345	−0.987266
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.7383i	0.968476i	0.874936	+	0.484238i	$0.160903\pi$
$173$	12.7383i	0.968476i	−0.874936	+	0.484238i	$0.839097\pi$
$174$	0	0
$175$	1.41421i	0.106904i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.1452	0.982517	0.491258	−	0.871014i	$-0.336537\pi$
$179$	13.1452	0.982517	0.491258	+	0.871014i	$0.336537\pi$
$180$	0	0
$181$	−7.70701	−0.572858	−0.286429	−	0.958101i	$-0.592468\pi$
$181$	−7.70701	−0.572858	−0.286429	+	0.958101i	$0.592468\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.01773i	− 0.0748253i
$186$	0	0
$187$	− 17.0460i	− 1.24653i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.1863	0.809410	0.404705	−	0.914447i	$-0.367374\pi$
$191$	11.1863	0.809410	0.404705	+	0.914447i	$0.367374\pi$
$192$	0	0
$193$	13.8952	1.00020	0.500099	−	0.865968i	$-0.333297\pi$
$193$	13.8952	1.00020	0.500099	+	0.865968i	$0.333297\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.5354i	1.03561i	0.855499	+	0.517804i	$0.173250\pi$
$197$	14.5354i	1.03561i	−0.855499	+	0.517804i	$0.826750\pi$
$198$	0	0
$199$	18.7144i	1.32663i	0.748342	+	0.663313i	$0.230850\pi$
$199$	18.7144i	1.32663i	−0.748342	+	0.663313i	$0.769150\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	11.6923	0.820640
$204$	0	0
$205$	−7.68193	−0.536530
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.2323i	1.39949i
$210$	0	0
$211$	3.49263i	0.240442i	0.992747	+	0.120221i	$0.0383604\pi$
$211$	3.49263i	0.240442i	−0.992747	+	0.120221i	$0.961640\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.42460	0.0971571
$216$	0	0
$217$	7.08146	0.480721
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.80322i	− 0.121297i
$222$	0	0
$223$	27.9350i	1.87066i	0.353772	+	0.935332i	$0.384899\pi$
$223$	27.9350i	1.87066i	−0.353772	+	0.935332i	$0.615101\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.5061	0.830056	0.415028	−	0.909809i	$-0.363772\pi$
$227$	12.5061	0.830056	0.415028	+	0.909809i	$0.363772\pi$
$228$	0	0
$229$	−3.65685	−0.241652	−0.120826	−	0.992674i	$-0.538554\pi$
$229$	−3.65685	−0.241652	−0.120826	+	0.992674i	$0.538554\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.361010i	0.0236506i	0.999930	+	0.0118253i	$0.00376419\pi$
$233$	0.361010i	0.0236506i	−0.999930	+	0.0118253i	$0.996236\pi$
$234$	0	0
$235$	− 6.66420i	− 0.434724i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.36387	−0.476329	−0.238165	−	0.971225i	$-0.576546\pi$
$239$	−7.36387	−0.476329	−0.238165	+	0.971225i	$0.576546\pi$
$240$	0	0
$241$	23.7425	1.52939	0.764694	−	0.644394i	$-0.222890\pi$
$241$	23.7425	1.52939	0.764694	+	0.644394i	$0.222890\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.00000i	0.319438i
$246$	0	0
$247$	2.14028i	0.136183i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.279424	−0.0176371	−0.00881854	−	0.999961i	$-0.502807\pi$
$251$	−0.279424	−0.0176371	−0.00881854	+	0.999961i	$0.502807\pi$
$252$	0	0
$253$	−3.81791	−0.240030
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.5741i	0.909106i	0.890720	+	0.454553i	$0.150201\pi$
$257$	14.5741i	0.909106i	−0.890720	+	0.454553i	$0.849799\pi$
$258$	0	0
$259$	1.43929i	0.0894334i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.0073	−0.678742	−0.339371	−	0.940653i	$-0.610214\pi$
$263$	−11.0073	−0.678742	−0.339371	+	0.940653i	$0.610214\pi$
$264$	0	0
$265$	1.78244	0.109494
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.04599i	0.429601i	0.976658	+	0.214801i	$0.0689102\pi$
$269$	7.04599i	0.429601i	−0.976658	+	0.214801i	$0.931090\pi$
$270$	0	0
$271$	1.49263i	0.0906706i	0.998972	+	0.0453353i	$0.0144356\pi$
$271$	1.49263i	0.0906706i	−0.998972	+	0.0453353i	$0.985564\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.84616	−0.231932
$276$	0	0
$277$	27.5385	1.65463	0.827314	−	0.561740i	$-0.189868\pi$
$277$	27.5385	1.65463	0.827314	+	0.561740i	$0.189868\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1.77205i	− 0.105712i	−0.998602	−	0.0528558i	$-0.983168\pi$
$281$	− 1.77205i	− 0.105712i	0.998602	−	0.0528558i	$-0.0168324\pi$
$282$	0	0
$283$	7.46007i	0.443455i	0.975109	+	0.221728i	$0.0711696\pi$
$283$	7.46007i	0.443455i	−0.975109	+	0.221728i	$0.928830\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.8639	0.641275
$288$	0	0
$289$	−2.64216	−0.155421
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 20.5208i	− 1.19884i	−0.800437	−	0.599418i	$-0.795399\pi$
$293$	− 20.5208i	− 1.19884i	0.800437	−	0.599418i	$-0.204601\pi$
$294$	0	0
$295$	− 8.45703i	− 0.492387i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.403879	−0.0233569
$300$	0	0
$301$	−2.01469	−0.116125
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.8431i	0.620875i
$306$	0	0
$307$	24.1776i	1.37989i	0.723862	+	0.689944i	$0.242365\pi$
$307$	24.1776i	1.37989i	−0.723862	+	0.689944i	$0.757635\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.1421	−1.48238	−0.741192	−	0.671293i	$-0.765739\pi$
$311$	−26.1421	−1.48238	−0.741192	+	0.671293i	$0.765739\pi$
$312$	0	0
$313$	28.7322	1.62404	0.812021	−	0.583629i	$-0.198368\pi$
$313$	28.7322	1.62404	0.812021	+	0.583629i	$0.198368\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.3137i	0.747772i	0.927475	+	0.373886i	$0.121975\pi$
$317$	13.3137i	0.747772i	−0.927475	+	0.373886i	$0.878025\pi$
$318$	0	0
$319$	31.7990i	1.78040i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	23.3137	1.29721
$324$	0	0
$325$	−0.406868	−0.0225690
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.42460i	0.519595i
$330$	0	0
$331$	− 31.7811i	− 1.74685i	−0.486960	−	0.873424i	$-0.661894\pi$
$331$	− 31.7811i	− 1.74685i	0.486960	−	0.873424i	$-0.338106\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.61087	0.470462
$336$	0	0
$337$	4.03547	0.219826	0.109913	−	0.993941i	$-0.464943\pi$
$337$	4.03547	0.219826	0.109913	+	0.993941i	$0.464943\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.2591i	1.04294i
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.435121	0.0233585	0.0116793	−	0.999932i	$-0.496282\pi$
$347$	0.435121	0.0233585	0.0116793	+	0.999932i	$0.496282\pi$
$348$	0	0
$349$	−22.6629	−1.21312	−0.606558	−	0.795039i	$-0.707450\pi$
$349$	−22.6629	−1.21312	−0.606558	+	0.795039i	$0.707450\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 19.2812i	− 1.02623i	−0.858319	−	0.513116i	$-0.828491\pi$
$353$	− 19.2812i	− 1.02623i	0.858319	−	0.513116i	$-0.171509\pi$
$354$	0	0
$355$	− 5.40383i	− 0.286805i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.66103	0.457112	0.228556	−	0.973531i	$-0.426600\pi$
$359$	8.66103	0.457112	0.228556	+	0.973531i	$0.426600\pi$
$360$	0	0
$361$	−8.67155	−0.456397
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.69232i	0.507319i
$366$	0	0
$367$	23.8595i	1.24546i	0.782438	+	0.622729i	$0.213976\pi$
$367$	23.8595i	1.24546i	−0.782438	+	0.622729i	$0.786024\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.52075	−0.130871
$372$	0	0
$373$	23.5177	1.21770	0.608850	−	0.793285i	$-0.291631\pi$
$373$	23.5177	1.21770	0.608850	+	0.793285i	$0.291631\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.36387i	0.173248i
$378$	0	0
$379$	− 23.6408i	− 1.21435i	−0.794569	−	0.607174i	$-0.792303\pi$
$379$	− 23.6408i	− 1.21435i	0.794569	−	0.607174i	$-0.207697\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−33.2143	−1.69717	−0.848587	−	0.529057i	$-0.822546\pi$
$383$	−33.2143	−1.69717	−0.848587	+	0.529057i	$0.822546\pi$
$384$	0	0
$385$	5.43929	0.277212
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.46456i	0.277064i	0.990358	+	0.138532i	$0.0442384\pi$
$389$	5.46456i	0.277064i	−0.990358	+	0.138532i	$0.955762\pi$
$390$	0	0
$391$	4.39939i	0.222487i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.64951	−0.435204
$396$	0	0
$397$	18.6493	0.935983	0.467991	−	0.883733i	$-0.344978\pi$
$397$	18.6493	0.935983	0.467991	+	0.883733i	$0.344978\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 13.2113i	− 0.659743i	−0.944026	−	0.329872i	$-0.892995\pi$
$401$	− 13.2113i	− 0.659743i	0.944026	−	0.329872i	$-0.107005\pi$
$402$	0	0
$403$	2.03733i	0.101487i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.91437	−0.194028
$408$	0	0
$409$	4.84920	0.239778	0.119889	−	0.992787i	$-0.461746\pi$
$409$	4.84920	0.239778	0.119889	+	0.992787i	$0.461746\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.9600i	0.588515i
$414$	0	0
$415$	12.4853i	0.612878i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5.22060	−0.255043	−0.127522	−	0.991836i	$-0.540702\pi$
$419$	−5.22060	−0.255043	−0.127522	+	0.991836i	$0.540702\pi$
$420$	0	0
$421$	13.3137	0.648870	0.324435	−	0.945908i	$-0.394826\pi$
$421$	13.3137	0.648870	0.324435	+	0.945908i	$0.394826\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.43195i	0.214981i
$426$	0	0
$427$	− 15.3345i	− 0.742088i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.1671	−1.01958	−0.509791	−	0.860298i	$-0.670277\pi$
$431$	−21.1671	−1.01958	−0.509791	+	0.860298i	$0.670277\pi$
$432$	0	0
$433$	−35.9262	−1.72650	−0.863251	−	0.504775i	$-0.831575\pi$
$433$	−35.9262	−1.72650	−0.863251	+	0.504775i	$0.831575\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.22173i	− 0.249789i
$438$	0	0
$439$	0.927802i	0.0442816i	0.999755	+	0.0221408i	$0.00704821\pi$
$439$	0.927802i	0.0442816i	−0.999755	+	0.0221408i	$0.992952\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.5061	−0.784227	−0.392113	−	0.919917i	$-0.628256\pi$
$443$	−16.5061	−0.784227	−0.392113	+	0.919917i	$0.628256\pi$
$444$	0	0
$445$	3.19665	0.151536
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16.7695i	0.791401i	0.918380	+	0.395700i	$0.129498\pi$
$449$	16.7695i	0.791401i	−0.918380	+	0.395700i	$0.870502\pi$
$450$	0	0
$451$	29.5460i	1.39126i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.575398	0.0269751
$456$	0	0
$457$	−17.6169	−0.824084	−0.412042	−	0.911165i	$-0.635184\pi$
$457$	−17.6169	−0.824084	−0.412042	+	0.911165i	$0.635184\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.2885i	0.758631i	0.925267	+	0.379315i	$0.123840\pi$
$461$	16.2885i	0.758631i	−0.925267	+	0.379315i	$0.876160\pi$
$462$	0	0
$463$	− 13.0519i	− 0.606573i	−0.952899	−	0.303286i	$-0.901916\pi$
$463$	− 13.0519i	− 0.606573i	0.952899	−	0.303286i	$-0.0980839\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.665514	0.0307963	0.0153982	−	0.999881i	$-0.495098\pi$
$467$	0.665514	0.0307963	0.0153982	+	0.999881i	$0.495098\pi$
$468$	0	0
$469$	−12.1776	−0.562310
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 5.47925i	− 0.251936i
$474$	0	0
$475$	− 5.26038i	− 0.241363i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	39.1316	1.78797	0.893985	−	0.448098i	$-0.147898\pi$
$479$	39.1316	1.78797	0.893985	+	0.448098i	$0.147898\pi$
$480$	0	0
$481$	−0.414083	−0.0188806
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.575398i	0.0261275i
$486$	0	0
$487$	− 40.7528i	− 1.84669i	−0.383976	−	0.923343i	$-0.625445\pi$
$487$	− 40.7528i	− 1.84669i	0.383976	−	0.923343i	$-0.374555\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.3627	1.23486	0.617432	−	0.786625i	$-0.288173\pi$
$491$	27.3627	1.23486	0.617432	+	0.786625i	$0.288173\pi$
$492$	0	0
$493$	36.6421	1.65028
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.64216i	0.342798i
$498$	0	0
$499$	− 30.6597i	− 1.37252i	−0.727358	−	0.686259i	$-0.759252\pi$
$499$	− 30.6597i	− 1.37252i	0.727358	−	0.686259i	$-0.240748\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.0930	−1.38637	−0.693184	−	0.720761i	$-0.743793\pi$
$503$	−31.0930	−1.38637	−0.693184	+	0.720761i	$0.743793\pi$
$504$	0	0
$505$	−0.828427	−0.0368645
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 7.70701i	− 0.341607i	−0.985305	−	0.170804i	$-0.945364\pi$
$509$	− 7.70701i	− 0.341607i	0.985305	−	0.170804i	$-0.0546364\pi$
$510$	0	0
$511$	− 13.7070i	− 0.606363i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.3388	0.587777
$516$	0	0
$517$	−25.6316	−1.12728
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13.4142i	0.587687i	0.955853	+	0.293844i	$0.0949345\pi$
$521$	13.4142i	0.587687i	−0.955853	+	0.293844i	$0.905065\pi$
$522$	0	0
$523$	12.4853i	0.545943i	0.962022	+	0.272972i	$0.0880065\pi$
$523$	12.4853i	0.545943i	−0.962022	+	0.272972i	$0.911993\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.1923	0.966712
$528$	0	0
$529$	−22.0146	−0.957158
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.12553i	0.135382i
$534$	0	0
$535$	4.84312i	0.209386i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	19.2308	0.828330
$540$	0	0
$541$	33.2546	1.42973	0.714863	−	0.699265i	$-0.246489\pi$
$541$	33.2546	1.42973	0.714863	+	0.699265i	$0.246489\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.03547i	0.172860i
$546$	0	0
$547$	34.9306i	1.49352i	0.665091	+	0.746762i	$0.268393\pi$
$547$	34.9306i	1.49352i	−0.665091	+	0.746762i	$0.731607\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−43.4913	−1.85279
$552$	0	0
$553$	12.2323	0.520168
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12.7089i	− 0.538495i	−0.963071	−	0.269247i	$-0.913225\pi$
$557$	− 12.7089i	− 0.538495i	0.963071	−	0.269247i	$-0.0867749\pi$
$558$	0	0
$559$	− 0.579625i	− 0.0245155i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.3137	0.982556	0.491278	−	0.871003i	$-0.336530\pi$
$563$	23.3137	0.982556	0.491278	+	0.871003i	$0.336530\pi$
$564$	0	0
$565$	−13.7811	−0.579776
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.8976i	0.624539i	0.949994	+	0.312270i	$0.101089\pi$
$569$	14.8976i	0.624539i	−0.949994	+	0.312270i	$0.898911\pi$
$570$	0	0
$571$	− 2.73962i	− 0.114650i	−0.998356	−	0.0573249i	$-0.981743\pi$
$571$	− 2.73962i	− 0.114650i	0.998356	−	0.0573249i	$-0.0182571\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.992654	0.0413965
$576$	0	0
$577$	13.5875	0.565656	0.282828	−	0.959171i	$-0.408727\pi$
$577$	13.5875	0.565656	0.282828	+	0.959171i	$0.408727\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 17.6569i	− 0.732530i
$582$	0	0
$583$	− 6.85555i	− 0.283928i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.0649	−1.24091	−0.620455	−	0.784242i	$-0.713052\pi$
$587$	−30.0649	−1.24091	−0.620455	+	0.784242i	$0.713052\pi$
$588$	0	0
$589$	−26.3405	−1.08534
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 22.4259i	− 0.920920i	−0.887681	−	0.460460i	$-0.847685\pi$
$593$	− 22.4259i	− 0.920920i	0.887681	−	0.460460i	$-0.152315\pi$
$594$	0	0
$595$	− 6.26772i	− 0.256952i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9.60221	−0.392336	−0.196168	−	0.980570i	$-0.562850\pi$
$599$	−9.60221	−0.392336	−0.196168	+	0.980570i	$0.562850\pi$
$600$	0	0
$601$	44.8613	1.82993	0.914965	−	0.403534i	$-0.132218\pi$
$601$	44.8613	1.82993	0.914965	+	0.403534i	$0.132218\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.79296i	0.154206i
$606$	0	0
$607$	− 5.46437i	− 0.221792i	−0.993832	−	0.110896i	$-0.964628\pi$
$607$	− 5.46437i	− 0.221792i	0.993832	−	0.110896i	$-0.0353721\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.71145	−0.109693
$612$	0	0
$613$	8.20586	0.331431	0.165716	−	0.986174i	$-0.447007\pi$
$613$	8.20586	0.331431	0.165716	+	0.986174i	$0.447007\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19.4881i	− 0.784563i	−0.919845	−	0.392281i	$-0.871686\pi$
$617$	− 19.4881i	− 0.784563i	0.919845	−	0.392281i	$-0.128314\pi$
$618$	0	0
$619$	− 19.0932i	− 0.767422i	−0.923453	−	0.383711i	$-0.874646\pi$
$619$	− 19.0932i	− 0.767422i	0.923453	−	0.383711i	$-0.125354\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.52075	−0.181120
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.51055i	0.179847i
$630$	0	0
$631$	− 11.0993i	− 0.441857i	−0.975290	−	0.220928i	$-0.929091\pi$
$631$	− 11.0993i	− 0.441857i	0.975290	−	0.220928i	$-0.0709087\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.803347	−0.0318798
$636$	0	0
$637$	2.03434	0.0806034
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.74261i	0.266317i	0.991095	+	0.133159i	$0.0425120\pi$
$641$	6.74261i	0.266317i	−0.991095	+	0.133159i	$0.957488\pi$
$642$	0	0
$643$	41.7338i	1.64582i	0.568172	+	0.822910i	$0.307651\pi$
$643$	41.7338i	1.64582i	−0.568172	+	0.822910i	$0.692349\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−41.2853	−1.62309	−0.811546	−	0.584289i	$-0.801374\pi$
$647$	−41.2853	−1.62309	−0.811546	+	0.584289i	$0.801374\pi$
$648$	0	0
$649$	−32.5271	−1.27680
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 22.1482i	− 0.866727i	−0.901219	−	0.433363i	$-0.857327\pi$
$653$	− 22.1482i	− 0.866727i	0.901219	−	0.433363i	$-0.142673\pi$
$654$	0	0
$655$	11.8254i	0.462056i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10.5490	−0.410931	−0.205465	−	0.978664i	$-0.565871\pi$
$659$	−10.5490	−0.410931	−0.205465	+	0.978664i	$0.565871\pi$
$660$	0	0
$661$	23.0207	0.895402	0.447701	−	0.894183i	$-0.352243\pi$
$661$	23.0207	0.895402	0.447701	+	0.894183i	$0.352243\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.43929i	0.288483i
$666$	0	0
$667$	− 8.20699i	− 0.317776i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	41.7044	1.60998
$672$	0	0
$673$	41.9766	1.61808	0.809039	−	0.587754i	$-0.199988\pi$
$673$	41.9766	1.61808	0.809039	+	0.587754i	$0.199988\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 16.0441i	− 0.616624i	−0.951285	−	0.308312i	$-0.900236\pi$
$677$	− 16.0441i	− 0.616624i	0.951285	−	0.308312i	$-0.0997641\pi$
$678$	0	0
$679$	− 0.813735i	− 0.0312283i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−44.0769	−1.68656	−0.843278	−	0.537477i	$-0.819377\pi$
$683$	−44.0769	−1.68656	−0.843278	+	0.537477i	$0.819377\pi$
$684$	0	0
$685$	−16.9172	−0.646374
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 0.725217i	− 0.0276286i
$690$	0	0
$691$	31.3524i	1.19270i	0.802725	+	0.596350i	$0.203383\pi$
$691$	31.3524i	1.19270i	−0.802725	+	0.596350i	$0.796617\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.1495	0.650517
$696$	0	0
$697$	34.0459	1.28958
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 22.5147i	− 0.850367i	−0.905107	−	0.425184i	$-0.860210\pi$
$701$	− 22.5147i	− 0.850367i	0.905107	−	0.425184i	$-0.139790\pi$
$702$	0	0
$703$	− 5.35366i	− 0.201917i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.17157	0.0440615
$708$	0	0
$709$	−16.1214	−0.605450	−0.302725	−	0.953078i	$-0.597896\pi$
$709$	−16.1214	−0.605450	−0.302725	+	0.953078i	$0.597896\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 4.97056i	− 0.186149i
$714$	0	0
$715$	1.56488i	0.0585232i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	37.9894	1.41677	0.708383	−	0.705828i	$-0.249425\pi$
$719$	37.9894	1.41677	0.708383	+	0.705828i	$0.249425\pi$
$720$	0	0
$721$	−18.8639	−0.702528
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 8.26772i	− 0.307055i
$726$	0	0
$727$	36.9746i	1.37131i	0.727926	+	0.685656i	$0.240485\pi$
$727$	36.9746i	1.37131i	−0.727926	+	0.685656i	$0.759515\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.31376	−0.233523
$732$	0	0
$733$	32.8877	1.21473	0.607367	−	0.794422i	$-0.292226\pi$
$733$	32.8877	1.21473	0.607367	+	0.794422i	$0.292226\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 33.1188i	− 1.21995i
$738$	0	0
$739$	− 13.1095i	− 0.482242i	−0.970495	−	0.241121i	$-0.922485\pi$
$739$	− 13.1095i	− 0.482242i	0.970495	−	0.241121i	$-0.0775150\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	52.4187	1.92306	0.961529	−	0.274705i	$-0.0885801\pi$
$743$	52.4187	1.92306	0.961529	+	0.274705i	$0.0885801\pi$
$744$	0	0
$745$	−20.7530	−0.760331
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 6.84920i	− 0.250264i
$750$	0	0
$751$	22.0904i	0.806090i	0.915180	+	0.403045i	$0.132048\pi$
$751$	22.0904i	0.806090i	−0.915180	+	0.403045i	$0.867952\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.35652	0.231338
$756$	0	0
$757$	−38.6954	−1.40641	−0.703203	−	0.710989i	$-0.748248\pi$
$757$	−38.6954	−1.40641	−0.703203	+	0.710989i	$0.748248\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 29.4142i	− 1.06626i	−0.846032	−	0.533132i	$-0.821015\pi$
$761$	− 29.4142i	− 1.06626i	0.846032	−	0.533132i	$-0.178985\pi$
$762$	0	0
$763$	− 5.70701i	− 0.206608i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.44089	−0.124243
$768$	0	0
$769$	−2.06259	−0.0743790	−0.0371895	−	0.999308i	$-0.511841\pi$
$769$	−2.06259	−0.0743790	−0.0371895	+	0.999308i	$0.511841\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.72779i	0.134079i	0.997750	+	0.0670397i	$0.0213554\pi$
$773$	3.72779i	0.134079i	−0.997750	+	0.0670397i	$0.978645\pi$
$774$	0	0
$775$	− 5.00735i	− 0.179869i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−40.4099	−1.44783
$780$	0	0
$781$	−20.7840	−0.743709
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.32541i	0.190072i
$786$	0	0
$787$	− 15.4054i	− 0.549144i	−0.961567	−	0.274572i	$-0.911464\pi$
$787$	− 15.4054i	− 0.549144i	0.961567	−	0.274572i	$-0.0885362\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	19.4895	0.692965
$792$	0	0
$793$	4.41172	0.156665
$794$	0	0
$795$	0	0
$796$	0	0
$797$	49.8507i	1.76580i	0.469557	+	0.882902i	$0.344414\pi$
$797$	49.8507i	1.76580i	−0.469557	+	0.882902i	$0.655586\pi$
$798$	0	0
$799$	29.5354i	1.04489i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	37.2782	1.31552
$804$	0	0
$805$	−1.40383	−0.0494783
$806$	0	0
$807$	0	0
$808$	0	0
$809$	52.4410i	1.84373i	0.387513	+	0.921864i	$0.373334\pi$
$809$	52.4410i	1.84373i	−0.387513	+	0.921864i	$0.626666\pi$
$810$	0	0
$811$	23.1284i	0.812150i	0.913840	+	0.406075i	$0.133103\pi$
$811$	23.1284i	0.812150i	−0.913840	+	0.406075i	$0.866897\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.378615	0.0132623
$816$	0	0
$817$	7.49394	0.262180
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 20.0581i	− 0.700033i	−0.936744	−	0.350016i	$-0.886176\pi$
$821$	− 20.0581i	− 0.700033i	0.936744	−	0.350016i	$-0.113824\pi$
$822$	0	0
$823$	− 53.6212i	− 1.86912i	−0.355809	−	0.934559i	$-0.615795\pi$
$823$	− 53.6212i	− 1.86912i	0.355809	−	0.934559i	$-0.384205\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	55.1565	1.91798	0.958990	−	0.283440i	$-0.0914757\pi$
$827$	55.1565	1.91798	0.958990	+	0.283440i	$0.0914757\pi$
$828$	0	0
$829$	−41.8284	−1.45276	−0.726380	−	0.687294i	$-0.758799\pi$
$829$	−41.8284	−1.45276	−0.726380	+	0.687294i	$0.758799\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 22.1597i	− 0.767789i
$834$	0	0
$835$	− 17.1703i	− 0.594201i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.57282	0.330491	0.165245	−	0.986253i	$-0.447158\pi$
$839$	9.57282	0.330491	0.165245	+	0.986253i	$0.447158\pi$
$840$	0	0
$841$	−39.3552	−1.35708
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 12.8345i	− 0.441519i
$846$	0	0
$847$	− 5.36405i	− 0.184311i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.01026	0.0346312
$852$	0	0
$853$	3.67042	0.125673	0.0628363	−	0.998024i	$-0.479985\pi$
$853$	3.67042	0.125673	0.0628363	+	0.998024i	$0.479985\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 46.6536i	− 1.59366i	−0.604205	−	0.796829i	$-0.706509\pi$
$857$	− 46.6536i	− 1.59366i	0.604205	−	0.796829i	$-0.293491\pi$
$858$	0	0
$859$	− 34.9546i	− 1.19263i	−0.802749	−	0.596317i	$-0.796630\pi$
$859$	− 34.9546i	− 1.19263i	0.802749	−	0.596317i	$-0.203370\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.3332	−1.61124	−0.805620	−	0.592433i	$-0.798168\pi$
$863$	−47.3332	−1.61124	−0.805620	+	0.592433i	$0.798168\pi$
$864$	0	0
$865$	−12.7383	−0.433116
$866$	0	0
$867$	0	0
$868$	0	0
$869$	33.2674i	1.12852i
$870$	0	0
$871$	− 3.50348i	− 0.118711i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.41421	−0.0478091
$876$	0	0
$877$	−20.2456	−0.683644	−0.341822	−	0.939765i	$-0.611044\pi$
$877$	−20.2456	−0.683644	−0.341822	+	0.939765i	$0.611044\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 8.09628i	− 0.272771i	−0.990656	−	0.136385i	$-0.956451\pi$
$881$	− 8.09628i	− 0.272771i	0.990656	−	0.136385i	$-0.0435485\pi$
$882$	0	0
$883$	40.6923i	1.36940i	0.728823	+	0.684702i	$0.240068\pi$
$883$	40.6923i	1.36940i	−0.728823	+	0.684702i	$0.759932\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.2265	0.779870	0.389935	−	0.920842i	$-0.372498\pi$
$887$	23.2265	0.779870	0.389935	+	0.920842i	$0.372498\pi$
$888$	0	0
$889$	1.13610	0.0381037
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 35.0562i	− 1.17311i
$894$	0	0
$895$	13.1452i	0.439395i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41.3993	−1.38075
$900$	0	0
$901$	−7.89968	−0.263176
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 7.70701i	− 0.256190i
$906$	0	0
$907$	14.1083i	0.468457i	0.972182	+	0.234229i	$0.0752564\pi$
$907$	14.1083i	0.468457i	−0.972182	+	0.234229i	$0.924744\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.3111	1.13678	0.568389	−	0.822760i	$-0.307567\pi$
$911$	34.3111	1.13678	0.568389	+	0.822760i	$0.307567\pi$
$912$	0	0
$913$	48.0204	1.58924
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16.7236i	− 0.552263i
$918$	0	0
$919$	− 52.0278i	− 1.71624i	−0.513450	−	0.858120i	$-0.671633\pi$
$919$	− 52.0278i	− 1.71624i	0.513450	−	0.858120i	$-0.328367\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.19864	−0.0723692
$924$	0	0
$925$	1.01773	0.0334629
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15.1836i	0.498156i	0.968483	+	0.249078i	$0.0801276\pi$
$929$	15.1836i	0.498156i	−0.968483	+	0.249078i	$0.919872\pi$
$930$	0	0
$931$	26.3019i	0.862009i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17.0460	0.557463
$936$	0	0
$937$	−27.4057	−0.895305	−0.447652	−	0.894208i	$-0.647740\pi$
$937$	−27.4057	−0.895305	−0.447652	+	0.894208i	$0.647740\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 11.1214i	− 0.362548i	−0.983433	−	0.181274i	$-0.941978\pi$
$941$	− 11.1214i	− 0.362548i	0.983433	−	0.181274i	$-0.0580220\pi$
$942$	0	0
$943$	− 7.62550i	− 0.248321i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.6863	0.412249	0.206125	−	0.978526i	$-0.433915\pi$
$947$	12.6863	0.412249	0.206125	+	0.978526i	$0.433915\pi$
$948$	0	0
$949$	3.94349	0.128011
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 34.7543i	− 1.12580i	−0.826524	−	0.562901i	$-0.809685\pi$
$953$	− 34.7543i	− 1.12580i	0.826524	−	0.562901i	$-0.190315\pi$
$954$	0	0
$955$	11.1863i	0.361979i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	23.9246	0.772565
$960$	0	0
$961$	5.92649	0.191177
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.8952i	0.447302i
$966$	0	0
$967$	− 8.70080i	− 0.279799i	−0.990166	−	0.139899i	$-0.955322\pi$
$967$	− 8.70080i	− 0.279799i	0.990166	−	0.139899i	$-0.0446779\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.82379	0.0585281	0.0292640	−	0.999572i	$-0.490684\pi$
$971$	1.82379	0.0585281	0.0292640	+	0.999572i	$0.490684\pi$
$972$	0	0
$973$	−24.2530	−0.777516
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.7099i	0.758548i	0.925284	+	0.379274i	$0.123826\pi$
$977$	23.7099i	0.758548i	−0.925284	+	0.379274i	$0.876174\pi$
$978$	0	0
$979$	− 12.2948i	− 0.392945i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.06988	0.257389	0.128695	−	0.991684i	$-0.458921\pi$
$983$	8.06988	0.257389	0.128695	+	0.991684i	$0.458921\pi$
$984$	0	0
$985$	−14.5354	−0.463138
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.41414i	0.0449669i
$990$	0	0
$991$	− 53.5636i	− 1.70150i	−0.525568	−	0.850751i	$-0.676147\pi$
$991$	− 53.5636i	− 1.70150i	0.525568	−	0.850751i	$-0.323853\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−18.7144	−0.593285
$996$	0	0
$997$	43.9985	1.39345	0.696723	−	0.717340i	$-0.254641\pi$
$997$	43.9985	1.39345	0.696723	+	0.717340i	$0.254641\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5760.2.h.c.1151.6	yes	8
3.2	odd	2		5760.2.h.h.1151.1	yes	8
4.3	odd	2		5760.2.h.h.1151.7	yes	8
8.3	odd	2		5760.2.h.b.1151.4	✓	8
8.5	even	2		5760.2.h.e.1151.1	yes	8
12.11	even	2	inner	5760.2.h.c.1151.4	yes	8
24.5	odd	2		5760.2.h.b.1151.6	yes	8
24.11	even	2		5760.2.h.e.1151.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5760.2.h.b.1151.4	✓	8	8.3	odd	2
5760.2.h.b.1151.6	yes	8	24.5	odd	2
5760.2.h.c.1151.4	yes	8	12.11	even	2	inner
5760.2.h.c.1151.6	yes	8	1.1	even	1	trivial
5760.2.h.e.1151.1	yes	8	8.5	even	2
5760.2.h.e.1151.7	yes	8	24.11	even	2
5760.2.h.h.1151.1	yes	8	3.2	odd	2
5760.2.h.h.1151.7	yes	8	4.3	odd	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:	\( N \)	\(=\)	\( 5760 = 2^{7} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5760.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -1.41421i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} -1.41421i q^{7} +3.84616 q^{11} +0.406868 q^{13} -4.43195i q^{17} +5.26038i q^{19} -0.992654 q^{23} -1.00000 q^{25} +8.26772i q^{29} +5.00735i q^{31} +1.41421 q^{35} -1.01773 q^{37} +7.68193i q^{41} -1.42460i q^{43} -6.66420 q^{47} +5.00000 q^{49} -1.78244i q^{53} +3.84616i q^{55} -8.45703 q^{59} +10.8431 q^{61} +0.406868i q^{65} -8.61087i q^{67} -5.40383 q^{71} +9.69232 q^{73} -5.43929i q^{77} +8.64951i q^{79} +12.4853 q^{83} +4.43195 q^{85} -3.19665i q^{89} -0.575398i q^{91} -5.26038 q^{95} +0.575398 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{11} + 8 q^{13} - 24 q^{23} - 8 q^{25} + 8 q^{37} + 8 q^{47} + 40 q^{49} - 8 q^{59} + 32 q^{61} - 32 q^{71} + 32 q^{83} + 8 q^{85} + 8 q^{95} + 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^11 + 8 * q^13 - 24 * q^23 - 8 * q^25 + 8 * q^37 + 8 * q^47 + 40 * q^49 - 8 * q^59 + 32 * q^61 - 32 * q^71 + 32 * q^83 + 8 * q^85 + 8 * q^95 + 16 * q^97

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