LMFDB - Embedded newform 2912.2.c.a.1457.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(1457,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.1457"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.9
Character	$\chi$	$=$	2912.1457
Dual form			2912.2.c.a.1457.26

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.60179i q^{3} -2.04118i q^{5} +1.00000 q^{7} +0.434268 q^{9} +3.30009i q^{11} +1.00000i q^{13} -3.26954 q^{15} -4.91083 q^{17} +4.94776i q^{19} -1.60179i q^{21} +9.20320 q^{23} +0.833579 q^{25} -5.50098i q^{27} -7.07766i q^{29} +7.44816 q^{31} +5.28605 q^{33} -2.04118i q^{35} -4.63185i q^{37} +1.60179 q^{39} +4.33522 q^{41} -7.48166i q^{43} -0.886420i q^{45} -0.246211 q^{47} +1.00000 q^{49} +7.86611i q^{51} -9.84928i q^{53} +6.73608 q^{55} +7.92528 q^{57} -4.04217i q^{59} -10.6561i q^{61} +0.434268 q^{63} +2.04118 q^{65} +10.2757i q^{67} -14.7416i q^{69} -11.1847 q^{71} +5.93073 q^{73} -1.33522i q^{75} +3.30009i q^{77} +6.84608 q^{79} -7.50861 q^{81} +8.53099i q^{83} +10.0239i q^{85} -11.3369 q^{87} -14.4730 q^{89} +1.00000i q^{91} -11.9304i q^{93} +10.0993 q^{95} -3.61044 q^{97} +1.43312i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) $ 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	− 1.60179i	− 0.924794i	−0.886673	−	0.462397i	$-0.846989\pi$
$3$	− 1.60179i	− 0.924794i	0.886673	−	0.462397i	$-0.153011\pi$
$4$	0	0
$5$	− 2.04118i	− 0.912844i	−0.889763	−	0.456422i	$-0.849131\pi$
$5$	− 2.04118i	− 0.912844i	0.889763	−	0.456422i	$-0.150869\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.434268	0.144756
$10$	0	0
$11$	3.30009i	0.995014i	0.867460	+	0.497507i	$0.165751\pi$
$11$	3.30009i	0.995014i	−0.867460	+	0.497507i	$0.834249\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−3.26954	−0.844193
$16$	0	0
$17$	−4.91083	−1.19105	−0.595525	−	0.803337i	$-0.703056\pi$
$17$	−4.91083	−1.19105	−0.595525	+	0.803337i	$0.703056\pi$
$18$	0	0
$19$	4.94776i	1.13509i	0.823341	+	0.567547i	$0.192108\pi$
$19$	4.94776i	1.13509i	−0.823341	+	0.567547i	$0.807892\pi$
$20$	0	0
$21$	− 1.60179i	− 0.349539i
$22$	0	0
$23$	9.20320	1.91900	0.959500	−	0.281709i	$-0.0909014\pi$
$23$	9.20320	1.91900	0.959500	+	0.281709i	$0.0909014\pi$
$24$	0	0
$25$	0.833579	0.166716
$26$	0	0
$27$	− 5.50098i	− 1.05866i
$28$	0	0
$29$	− 7.07766i	− 1.31429i	−0.753765	−	0.657144i	$-0.771764\pi$
$29$	− 7.07766i	− 1.31429i	0.753765	−	0.657144i	$-0.228236\pi$
$30$	0	0
$31$	7.44816	1.33773	0.668865	−	0.743384i	$-0.266780\pi$
$31$	7.44816	1.33773	0.668865	+	0.743384i	$0.266780\pi$
$32$	0	0
$33$	5.28605	0.920183
$34$	0	0
$35$	− 2.04118i	− 0.345023i
$36$	0	0
$37$	− 4.63185i	− 0.761471i	−0.924684	−	0.380735i	$-0.875671\pi$
$37$	− 4.63185i	− 0.761471i	0.924684	−	0.380735i	$-0.124329\pi$
$38$	0	0
$39$	1.60179	0.256492
$40$	0	0
$41$	4.33522	0.677047	0.338524	−	0.940958i	$-0.390072\pi$
$41$	4.33522	0.677047	0.338524	+	0.940958i	$0.390072\pi$
$42$	0	0
$43$	− 7.48166i	− 1.14094i	−0.821318	−	0.570471i	$-0.806761\pi$
$43$	− 7.48166i	− 1.14094i	0.821318	−	0.570471i	$-0.193239\pi$
$44$	0	0
$45$	− 0.886420i	− 0.132140i
$46$	0	0
$47$	−0.246211	−0.0359136	−0.0179568	−	0.999839i	$-0.505716\pi$
$47$	−0.246211	−0.0359136	−0.0179568	+	0.999839i	$0.505716\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.86611i	1.10148i
$52$	0	0
$53$	− 9.84928i	− 1.35290i	−0.736487	−	0.676451i	$-0.763517\pi$
$53$	− 9.84928i	− 1.35290i	0.736487	−	0.676451i	$-0.236483\pi$
$54$	0	0
$55$	6.73608	0.908293
$56$	0	0
$57$	7.92528	1.04973
$58$	0	0
$59$	− 4.04217i	− 0.526246i	−0.964762	−	0.263123i	$-0.915248\pi$
$59$	− 4.04217i	− 0.526246i	0.964762	−	0.263123i	$-0.0847525\pi$
$60$	0	0
$61$	− 10.6561i	− 1.36438i	−0.731175	−	0.682190i	$-0.761028\pi$
$61$	− 10.6561i	− 1.36438i	0.731175	−	0.682190i	$-0.238972\pi$
$62$	0	0
$63$	0.434268	0.0547126
$64$	0	0
$65$	2.04118	0.253177
$66$	0	0
$67$	10.2757i	1.25538i	0.778464	+	0.627689i	$0.215999\pi$
$67$	10.2757i	1.25538i	−0.778464	+	0.627689i	$0.784001\pi$
$68$	0	0
$69$	− 14.7416i	− 1.77468i
$70$	0	0
$71$	−11.1847	−1.32738	−0.663691	−	0.748007i	$-0.731011\pi$
$71$	−11.1847	−1.32738	−0.663691	+	0.748007i	$0.731011\pi$
$72$	0	0
$73$	5.93073	0.694140	0.347070	−	0.937839i	$-0.387177\pi$
$73$	5.93073	0.694140	0.347070	+	0.937839i	$0.387177\pi$
$74$	0	0
$75$	− 1.33522i	− 0.154178i
$76$	0	0
$77$	3.30009i	0.376080i
$78$	0	0
$79$	6.84608	0.770245	0.385122	−	0.922866i	$-0.374159\pi$
$79$	6.84608	0.770245	0.385122	+	0.922866i	$0.374159\pi$
$80$	0	0
$81$	−7.50861	−0.834290
$82$	0	0
$83$	8.53099i	0.936398i	0.883623	+	0.468199i	$0.155097\pi$
$83$	8.53099i	0.936398i	−0.883623	+	0.468199i	$0.844903\pi$
$84$	0	0
$85$	10.0239i	1.08724i
$86$	0	0
$87$	−11.3369	−1.21545
$88$	0	0
$89$	−14.4730	−1.53413	−0.767066	−	0.641568i	$-0.778284\pi$
$89$	−14.4730	−1.53413	−0.767066	+	0.641568i	$0.778284\pi$
$90$	0	0
$91$	1.00000i	0.104828i
$92$	0	0
$93$	− 11.9304i	− 1.23712i
$94$	0	0
$95$	10.0993	1.03616
$96$	0	0
$97$	−3.61044	−0.366584	−0.183292	−	0.983058i	$-0.558675\pi$
$97$	−3.61044	−0.366584	−0.183292	+	0.983058i	$0.558675\pi$
$98$	0	0
$99$	1.43312i	0.144034i
$100$	0	0
$101$	6.51050i	0.647819i	0.946088	+	0.323909i	$0.104997\pi$
$101$	6.51050i	0.647819i	−0.946088	+	0.323909i	$0.895003\pi$
$102$	0	0
$103$	−18.3047	−1.80362	−0.901809	−	0.432134i	$-0.857761\pi$
$103$	−18.3047	−1.80362	−0.901809	+	0.432134i	$0.857761\pi$
$104$	0	0
$105$	−3.26954	−0.319075
$106$	0	0
$107$	16.0316i	1.54983i	0.632064	+	0.774916i	$0.282208\pi$
$107$	16.0316i	1.54983i	−0.632064	+	0.774916i	$0.717792\pi$
$108$	0	0
$109$	10.7704i	1.03162i	0.856703	+	0.515810i	$0.172509\pi$
$109$	10.7704i	1.03162i	−0.856703	+	0.515810i	$0.827491\pi$
$110$	0	0
$111$	−7.41925	−0.704204
$112$	0	0
$113$	5.36213	0.504426	0.252213	−	0.967672i	$-0.418842\pi$
$113$	5.36213	0.504426	0.252213	+	0.967672i	$0.418842\pi$
$114$	0	0
$115$	− 18.7854i	− 1.75175i
$116$	0	0
$117$	0.434268i	0.0401481i
$118$	0	0
$119$	−4.91083	−0.450175
$120$	0	0
$121$	0.109419	0.00994719
$122$	0	0
$123$	− 6.94411i	− 0.626129i
$124$	0	0
$125$	− 11.9074i	− 1.06503i
$126$	0	0
$127$	18.4402	1.63630	0.818152	−	0.575002i	$-0.194999\pi$
$127$	18.4402	1.63630	0.818152	+	0.575002i	$0.194999\pi$
$128$	0	0
$129$	−11.9841	−1.05514
$130$	0	0
$131$	− 9.95924i	− 0.870143i	−0.900396	−	0.435072i	$-0.856723\pi$
$131$	− 9.95924i	− 0.870143i	0.900396	−	0.435072i	$-0.143277\pi$
$132$	0	0
$133$	4.94776i	0.429025i
$134$	0	0
$135$	−11.2285	−0.966395
$136$	0	0
$137$	11.7779	1.00626	0.503128	−	0.864212i	$-0.332182\pi$
$137$	11.7779	1.00626	0.503128	+	0.864212i	$0.332182\pi$
$138$	0	0
$139$	− 6.87146i	− 0.582829i	−0.956597	−	0.291415i	$-0.905874\pi$
$139$	− 6.87146i	− 0.582829i	0.956597	−	0.291415i	$-0.0941259\pi$
$140$	0	0
$141$	0.394379i	0.0332127i
$142$	0	0
$143$	−3.30009	−0.275967
$144$	0	0
$145$	−14.4468	−1.19974
$146$	0	0
$147$	− 1.60179i	− 0.132113i
$148$	0	0
$149$	− 15.7608i	− 1.29118i	−0.763686	−	0.645588i	$-0.776612\pi$
$149$	− 15.7608i	− 1.29118i	0.763686	−	0.645588i	$-0.223388\pi$
$150$	0	0
$151$	3.89237	0.316757	0.158378	−	0.987378i	$-0.449373\pi$
$151$	3.89237	0.316757	0.158378	+	0.987378i	$0.449373\pi$
$152$	0	0
$153$	−2.13261	−0.172412
$154$	0	0
$155$	− 15.2031i	− 1.22114i
$156$	0	0
$157$	10.5821i	0.844545i	0.906469	+	0.422273i	$0.138768\pi$
$157$	10.5821i	0.844545i	−0.906469	+	0.422273i	$0.861232\pi$
$158$	0	0
$159$	−15.7765	−1.25116
$160$	0	0
$161$	9.20320	0.725314
$162$	0	0
$163$	− 18.4249i	− 1.44315i	−0.692338	−	0.721573i	$-0.743419\pi$
$163$	− 18.4249i	− 1.44315i	0.692338	−	0.721573i	$-0.256581\pi$
$164$	0	0
$165$	− 10.7898i	− 0.839984i
$166$	0	0
$167$	13.1609	1.01842	0.509212	−	0.860641i	$-0.329937\pi$
$167$	13.1609	1.01842	0.509212	+	0.860641i	$0.329937\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	2.14866i	0.164312i
$172$	0	0
$173$	− 7.42763i	− 0.564713i	−0.959310	−	0.282356i	$-0.908884\pi$
$173$	− 7.42763i	− 0.564713i	0.959310	−	0.282356i	$-0.0911160\pi$
$174$	0	0
$175$	0.833579	0.0630126
$176$	0	0
$177$	−6.47471	−0.486669
$178$	0	0
$179$	− 18.5579i	− 1.38708i	−0.720419	−	0.693540i	$-0.756050\pi$
$179$	− 18.5579i	− 1.38708i	0.720419	−	0.693540i	$-0.243950\pi$
$180$	0	0
$181$	18.9137i	1.40584i	0.711267	+	0.702922i	$0.248122\pi$
$181$	18.9137i	1.40584i	−0.711267	+	0.702922i	$0.751878\pi$
$182$	0	0
$183$	−17.0689	−1.26177
$184$	0	0
$185$	−9.45444	−0.695104
$186$	0	0
$187$	− 16.2062i	− 1.18511i
$188$	0	0
$189$	− 5.50098i	− 0.400137i
$190$	0	0
$191$	−17.3720	−1.25699	−0.628496	−	0.777813i	$-0.716329\pi$
$191$	−17.3720	−1.25699	−0.628496	+	0.777813i	$0.716329\pi$
$192$	0	0
$193$	10.9562	0.788647	0.394324	−	0.918972i	$-0.370979\pi$
$193$	10.9562	0.788647	0.394324	+	0.918972i	$0.370979\pi$
$194$	0	0
$195$	− 3.26954i	− 0.234137i
$196$	0	0
$197$	4.36246i	0.310812i	0.987851	+	0.155406i	$0.0496687\pi$
$197$	4.36246i	0.310812i	−0.987851	+	0.155406i	$0.950331\pi$
$198$	0	0
$199$	−10.1073	−0.716484	−0.358242	−	0.933629i	$-0.616624\pi$
$199$	−10.1073	−0.716484	−0.358242	+	0.933629i	$0.616624\pi$
$200$	0	0
$201$	16.4595	1.16097
$202$	0	0
$203$	− 7.07766i	− 0.496754i
$204$	0	0
$205$	− 8.84897i	− 0.618039i
$206$	0	0
$207$	3.99665	0.277787
$208$	0	0
$209$	−16.3281	−1.12944
$210$	0	0
$211$	24.7203i	1.70181i	0.525317	+	0.850906i	$0.323947\pi$
$211$	24.7203i	1.70181i	−0.525317	+	0.850906i	$0.676053\pi$
$212$	0	0
$213$	17.9156i	1.22756i
$214$	0	0
$215$	−15.2714	−1.04150
$216$	0	0
$217$	7.44816	0.505614
$218$	0	0
$219$	− 9.49979i	− 0.641936i
$220$	0	0
$221$	− 4.91083i	− 0.330338i
$222$	0	0
$223$	9.86265	0.660452	0.330226	−	0.943902i	$-0.392875\pi$
$223$	9.86265	0.660452	0.330226	+	0.943902i	$0.392875\pi$
$224$	0	0
$225$	0.361997	0.0241331
$226$	0	0
$227$	− 3.27410i	− 0.217310i	−0.994080	−	0.108655i	$-0.965346\pi$
$227$	− 3.27410i	− 0.217310i	0.994080	−	0.108655i	$-0.0346543\pi$
$228$	0	0
$229$	− 0.592074i	− 0.0391254i	−0.999809	−	0.0195627i	$-0.993773\pi$
$229$	− 0.592074i	− 0.0391254i	0.999809	−	0.0195627i	$-0.00622739\pi$
$230$	0	0
$231$	5.28605	0.347796
$232$	0	0
$233$	15.3194	1.00361	0.501804	−	0.864981i	$-0.332670\pi$
$233$	15.3194	1.00361	0.501804	+	0.864981i	$0.332670\pi$
$234$	0	0
$235$	0.502562i	0.0327835i
$236$	0	0
$237$	− 10.9660i	− 0.712318i
$238$	0	0
$239$	8.41048	0.544028	0.272014	−	0.962293i	$-0.412310\pi$
$239$	8.41048	0.544028	0.272014	+	0.962293i	$0.412310\pi$
$240$	0	0
$241$	−21.7077	−1.39831	−0.699157	−	0.714968i	$-0.746441\pi$
$241$	−21.7077	−1.39831	−0.699157	+	0.714968i	$0.746441\pi$
$242$	0	0
$243$	− 4.47572i	− 0.287117i
$244$	0	0
$245$	− 2.04118i	− 0.130406i
$246$	0	0
$247$	−4.94776	−0.314819
$248$	0	0
$249$	13.6649	0.865975
$250$	0	0
$251$	− 14.6293i	− 0.923394i	−0.887038	−	0.461697i	$-0.847241\pi$
$251$	− 14.6293i	− 0.923394i	0.887038	−	0.461697i	$-0.152759\pi$
$252$	0	0
$253$	30.3714i	1.90943i
$254$	0	0
$255$	16.0562	1.00548
$256$	0	0
$257$	−16.6321	−1.03748	−0.518740	−	0.854932i	$-0.673599\pi$
$257$	−16.6321	−1.03748	−0.518740	+	0.854932i	$0.673599\pi$
$258$	0	0
$259$	− 4.63185i	− 0.287809i
$260$	0	0
$261$	− 3.07360i	− 0.190251i
$262$	0	0
$263$	15.6456	0.964750	0.482375	−	0.875965i	$-0.339774\pi$
$263$	15.6456	0.964750	0.482375	+	0.875965i	$0.339774\pi$
$264$	0	0
$265$	−20.1042	−1.23499
$266$	0	0
$267$	23.1827i	1.41876i
$268$	0	0
$269$	11.9287i	0.727305i	0.931535	+	0.363652i	$0.118470\pi$
$269$	11.9287i	0.727305i	−0.931535	+	0.363652i	$0.881530\pi$
$270$	0	0
$271$	5.99184	0.363978	0.181989	−	0.983301i	$-0.441746\pi$
$271$	5.99184	0.363978	0.181989	+	0.983301i	$0.441746\pi$
$272$	0	0
$273$	1.60179	0.0969448
$274$	0	0
$275$	2.75088i	0.165884i
$276$	0	0
$277$	− 12.9046i	− 0.775363i	−0.921793	−	0.387682i	$-0.873276\pi$
$277$	− 12.9046i	− 0.775363i	0.921793	−	0.387682i	$-0.126724\pi$
$278$	0	0
$279$	3.23450	0.193644
$280$	0	0
$281$	−17.9330	−1.06979	−0.534896	−	0.844918i	$-0.679649\pi$
$281$	−17.9330	−1.06979	−0.534896	+	0.844918i	$0.679649\pi$
$282$	0	0
$283$	12.8586i	0.764364i	0.924087	+	0.382182i	$0.124827\pi$
$283$	12.8586i	0.764364i	−0.924087	+	0.382182i	$0.875173\pi$
$284$	0	0
$285$	− 16.1769i	− 0.958239i
$286$	0	0
$287$	4.33522	0.255900
$288$	0	0
$289$	7.11620	0.418600
$290$	0	0
$291$	5.78316i	0.339015i
$292$	0	0
$293$	26.7615i	1.56342i	0.623641	+	0.781711i	$0.285653\pi$
$293$	26.7615i	1.56342i	−0.623641	+	0.781711i	$0.714347\pi$
$294$	0	0
$295$	−8.25080	−0.480380
$296$	0	0
$297$	18.1537	1.05338
$298$	0	0
$299$	9.20320i	0.532235i
$300$	0	0
$301$	− 7.48166i	− 0.431236i
$302$	0	0
$303$	10.4285	0.599099
$304$	0	0
$305$	−21.7511	−1.24547
$306$	0	0
$307$	− 10.2784i	− 0.586622i	−0.956017	−	0.293311i	$-0.905243\pi$
$307$	− 10.2784i	− 0.586622i	0.956017	−	0.293311i	$-0.0947571\pi$
$308$	0	0
$309$	29.3203i	1.66798i
$310$	0	0
$311$	4.51151	0.255824	0.127912	−	0.991786i	$-0.459172\pi$
$311$	4.51151	0.255824	0.127912	+	0.991786i	$0.459172\pi$
$312$	0	0
$313$	−2.90958	−0.164459	−0.0822296	−	0.996613i	$-0.526204\pi$
$313$	−2.90958	−0.164459	−0.0822296	+	0.996613i	$0.526204\pi$
$314$	0	0
$315$	− 0.886420i	− 0.0499441i
$316$	0	0
$317$	− 8.78517i	− 0.493424i	−0.969089	−	0.246712i	$-0.920650\pi$
$317$	− 8.78517i	− 0.493424i	0.969089	−	0.246712i	$-0.0793502\pi$
$318$	0	0
$319$	23.3569	1.30773
$320$	0	0
$321$	25.6793	1.43328
$322$	0	0
$323$	− 24.2976i	− 1.35195i
$324$	0	0
$325$	0.833579i	0.0462386i
$326$	0	0
$327$	17.2520	0.954036
$328$	0	0
$329$	−0.246211	−0.0135741
$330$	0	0
$331$	11.5276i	0.633614i	0.948490	+	0.316807i	$0.102611\pi$
$331$	11.5276i	0.633614i	−0.948490	+	0.316807i	$0.897389\pi$
$332$	0	0
$333$	− 2.01146i	− 0.110227i
$334$	0	0
$335$	20.9746	1.14597
$336$	0	0
$337$	17.7814	0.968615	0.484307	−	0.874898i	$-0.339072\pi$
$337$	17.7814	0.968615	0.484307	+	0.874898i	$0.339072\pi$
$338$	0	0
$339$	− 8.58900i	− 0.466490i
$340$	0	0
$341$	24.5796i	1.33106i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−30.0903	−1.62001
$346$	0	0
$347$	14.2957i	0.767435i	0.923451	+	0.383717i	$0.125356\pi$
$347$	14.2957i	0.767435i	−0.923451	+	0.383717i	$0.874644\pi$
$348$	0	0
$349$	− 27.0196i	− 1.44632i	−0.690678	−	0.723162i	$-0.742688\pi$
$349$	− 27.0196i	− 1.44632i	0.690678	−	0.723162i	$-0.257312\pi$
$350$	0	0
$351$	5.50098	0.293620
$352$	0	0
$353$	21.9915	1.17049	0.585244	−	0.810857i	$-0.300999\pi$
$353$	21.9915	1.17049	0.585244	+	0.810857i	$0.300999\pi$
$354$	0	0
$355$	22.8300i	1.21169i
$356$	0	0
$357$	7.86611i	0.416319i
$358$	0	0
$359$	−3.86395	−0.203931	−0.101966	−	0.994788i	$-0.532513\pi$
$359$	−3.86395	−0.203931	−0.101966	+	0.994788i	$0.532513\pi$
$360$	0	0
$361$	−5.48036	−0.288440
$362$	0	0
$363$	− 0.175266i	− 0.00919910i
$364$	0	0
$365$	− 12.1057i	− 0.633641i
$366$	0	0
$367$	−29.1363	−1.52090	−0.760450	−	0.649396i	$-0.775022\pi$
$367$	−29.1363	−1.52090	−0.760450	+	0.649396i	$0.775022\pi$
$368$	0	0
$369$	1.88265	0.0980067
$370$	0	0
$371$	− 9.84928i	− 0.511349i
$372$	0	0
$373$	− 16.0903i	− 0.833123i	−0.909107	−	0.416562i	$-0.863235\pi$
$373$	− 16.0903i	− 0.833123i	0.909107	−	0.416562i	$-0.136765\pi$
$374$	0	0
$375$	−19.0731	−0.984933
$376$	0	0
$377$	7.07766	0.364518
$378$	0	0
$379$	23.9974i	1.23266i	0.787487	+	0.616331i	$0.211382\pi$
$379$	23.9974i	1.23266i	−0.787487	+	0.616331i	$0.788618\pi$
$380$	0	0
$381$	− 29.5374i	− 1.51324i
$382$	0	0
$383$	−18.3411	−0.937186	−0.468593	−	0.883414i	$-0.655239\pi$
$383$	−18.3411	−0.937186	−0.468593	+	0.883414i	$0.655239\pi$
$384$	0	0
$385$	6.73608	0.343302
$386$	0	0
$387$	− 3.24905i	− 0.165158i
$388$	0	0
$389$	− 0.628023i	− 0.0318420i	−0.999873	−	0.0159210i	$-0.994932\pi$
$389$	− 0.628023i	− 0.0318420i	0.999873	−	0.0159210i	$-0.00506803\pi$
$390$	0	0
$391$	−45.1953	−2.28562
$392$	0	0
$393$	−15.9526	−0.804703
$394$	0	0
$395$	− 13.9741i	− 0.703113i
$396$	0	0
$397$	− 5.60043i	− 0.281078i	−0.990075	−	0.140539i	$-0.955117\pi$
$397$	− 5.60043i	− 0.281078i	0.990075	−	0.140539i	$-0.0448835\pi$
$398$	0	0
$399$	7.92528	0.396760
$400$	0	0
$401$	16.1431	0.806147	0.403073	−	0.915168i	$-0.367942\pi$
$401$	16.1431	0.806147	0.403073	+	0.915168i	$0.367942\pi$
$402$	0	0
$403$	7.44816i	0.371019i
$404$	0	0
$405$	15.3264i	0.761576i
$406$	0	0
$407$	15.2855	0.757674
$408$	0	0
$409$	35.6500	1.76278	0.881390	−	0.472389i	$-0.156608\pi$
$409$	35.6500	1.76278	0.881390	+	0.472389i	$0.156608\pi$
$410$	0	0
$411$	− 18.8658i	− 0.930580i
$412$	0	0
$413$	− 4.04217i	− 0.198902i
$414$	0	0
$415$	17.4133	0.854785
$416$	0	0
$417$	−11.0066	−0.538997
$418$	0	0
$419$	− 31.0570i	− 1.51724i	−0.651536	−	0.758618i	$-0.725875\pi$
$419$	− 31.0570i	− 1.51724i	0.651536	−	0.758618i	$-0.274125\pi$
$420$	0	0
$421$	1.20004i	0.0584864i	0.999572	+	0.0292432i	$0.00930972\pi$
$421$	1.20004i	0.0584864i	−0.999572	+	0.0292432i	$0.990690\pi$
$422$	0	0
$423$	−0.106922	−0.00519871
$424$	0	0
$425$	−4.09356	−0.198567
$426$	0	0
$427$	− 10.6561i	− 0.515687i
$428$	0	0
$429$	5.28605i	0.255213i
$430$	0	0
$431$	−22.6281	−1.08996	−0.544978	−	0.838450i	$-0.683462\pi$
$431$	−22.6281	−1.08996	−0.544978	+	0.838450i	$0.683462\pi$
$432$	0	0
$433$	−7.81312	−0.375475	−0.187737	−	0.982219i	$-0.560115\pi$
$433$	−7.81312	−0.375475	−0.187737	+	0.982219i	$0.560115\pi$
$434$	0	0
$435$	23.1407i	1.10951i
$436$	0	0
$437$	45.5352i	2.17825i
$438$	0	0
$439$	6.45379	0.308023	0.154011	−	0.988069i	$-0.450781\pi$
$439$	6.45379	0.308023	0.154011	+	0.988069i	$0.450781\pi$
$440$	0	0
$441$	0.434268	0.0206794
$442$	0	0
$443$	6.15949i	0.292646i	0.989237	+	0.146323i	$0.0467439\pi$
$443$	6.15949i	0.292646i	−0.989237	+	0.146323i	$0.953256\pi$
$444$	0	0
$445$	29.5420i	1.40042i
$446$	0	0
$447$	−25.2455	−1.19407
$448$	0	0
$449$	−3.77506	−0.178156	−0.0890781	−	0.996025i	$-0.528392\pi$
$449$	−3.77506	−0.178156	−0.0890781	+	0.996025i	$0.528392\pi$
$450$	0	0
$451$	14.3066i	0.673672i
$452$	0	0
$453$	− 6.23477i	− 0.292935i
$454$	0	0
$455$	2.04118	0.0956921
$456$	0	0
$457$	16.8871	0.789945	0.394972	−	0.918693i	$-0.370754\pi$
$457$	16.8871	0.789945	0.394972	+	0.918693i	$0.370754\pi$
$458$	0	0
$459$	27.0143i	1.26092i
$460$	0	0
$461$	− 22.1909i	− 1.03353i	−0.856127	−	0.516766i	$-0.827136\pi$
$461$	− 22.1909i	− 1.03353i	0.856127	−	0.516766i	$-0.172864\pi$
$462$	0	0
$463$	0.870232	0.0404431	0.0202216	−	0.999796i	$-0.493563\pi$
$463$	0.870232	0.0404431	0.0202216	+	0.999796i	$0.493563\pi$
$464$	0	0
$465$	−24.3521	−1.12930
$466$	0	0
$467$	8.29553i	0.383871i	0.981408	+	0.191936i	$0.0614765\pi$
$467$	8.29553i	0.383871i	−0.981408	+	0.191936i	$0.938523\pi$
$468$	0	0
$469$	10.2757i	0.474489i
$470$	0	0
$471$	16.9503	0.781030
$472$	0	0
$473$	24.6901	1.13525
$474$	0	0
$475$	4.12435i	0.189238i
$476$	0	0
$477$	− 4.27723i	− 0.195841i
$478$	0	0
$479$	−14.4135	−0.658572	−0.329286	−	0.944230i	$-0.606808\pi$
$479$	−14.4135	−0.658572	−0.329286	+	0.944230i	$0.606808\pi$
$480$	0	0
$481$	4.63185	0.211194
$482$	0	0
$483$	− 14.7416i	− 0.670766i
$484$	0	0
$485$	7.36955i	0.334634i
$486$	0	0
$487$	13.8036	0.625503	0.312751	−	0.949835i	$-0.398749\pi$
$487$	13.8036	0.625503	0.312751	+	0.949835i	$0.398749\pi$
$488$	0	0
$489$	−29.5128	−1.33461
$490$	0	0
$491$	25.7469i	1.16194i	0.813925	+	0.580970i	$0.197327\pi$
$491$	25.7469i	1.16194i	−0.813925	+	0.580970i	$0.802673\pi$
$492$	0	0
$493$	34.7571i	1.56538i
$494$	0	0
$495$	2.92526	0.131481
$496$	0	0
$497$	−11.1847	−0.501703
$498$	0	0
$499$	12.7680i	0.571575i	0.958293	+	0.285788i	$0.0922552\pi$
$499$	12.7680i	0.571575i	−0.958293	+	0.285788i	$0.907745\pi$
$500$	0	0
$501$	− 21.0811i	− 0.941832i
$502$	0	0
$503$	−11.8527	−0.528486	−0.264243	−	0.964456i	$-0.585122\pi$
$503$	−11.8527	−0.528486	−0.264243	+	0.964456i	$0.585122\pi$
$504$	0	0
$505$	13.2891	0.591358
$506$	0	0
$507$	1.60179i	0.0711380i
$508$	0	0
$509$	20.3465i	0.901841i	0.892564	+	0.450920i	$0.148904\pi$
$509$	20.3465i	0.901841i	−0.892564	+	0.450920i	$0.851096\pi$
$510$	0	0
$511$	5.93073	0.262360
$512$	0	0
$513$	27.2175	1.20168
$514$	0	0
$515$	37.3633i	1.64642i
$516$	0	0
$517$	− 0.812519i	− 0.0357346i
$518$	0	0
$519$	−11.8975	−0.522243
$520$	0	0
$521$	38.3974	1.68222	0.841111	−	0.540863i	$-0.181902\pi$
$521$	38.3974	1.68222	0.841111	+	0.540863i	$0.181902\pi$
$522$	0	0
$523$	− 23.6884i	− 1.03582i	−0.855435	−	0.517911i	$-0.826710\pi$
$523$	− 23.6884i	− 1.03582i	0.855435	−	0.517911i	$-0.173290\pi$
$524$	0	0
$525$	− 1.33522i	− 0.0582737i
$526$	0	0
$527$	−36.5766	−1.59330
$528$	0	0
$529$	61.6989	2.68256
$530$	0	0
$531$	− 1.75539i	− 0.0761772i
$532$	0	0
$533$	4.33522i	0.187779i
$534$	0	0
$535$	32.7234	1.41476
$536$	0	0
$537$	−29.7258	−1.28276
$538$	0	0
$539$	3.30009i	0.142145i
$540$	0	0
$541$	− 3.94405i	− 0.169568i	−0.996399	−	0.0847839i	$-0.972980\pi$
$541$	− 3.94405i	− 0.169568i	0.996399	−	0.0847839i	$-0.0270200\pi$
$542$	0	0
$543$	30.2958	1.30012
$544$	0	0
$545$	21.9844	0.941708
$546$	0	0
$547$	18.7051i	0.799771i	0.916565	+	0.399886i	$0.130950\pi$
$547$	18.7051i	0.799771i	−0.916565	+	0.399886i	$0.869050\pi$
$548$	0	0
$549$	− 4.62762i	− 0.197502i
$550$	0	0
$551$	35.0186	1.49184
$552$	0	0
$553$	6.84608	0.291125
$554$	0	0
$555$	15.1440i	0.642828i
$556$	0	0
$557$	− 17.9717i	− 0.761487i	−0.924681	−	0.380744i	$-0.875668\pi$
$557$	− 17.9717i	− 0.761487i	0.924681	−	0.380744i	$-0.124332\pi$
$558$	0	0
$559$	7.48166	0.316441
$560$	0	0
$561$	−25.9589	−1.09598
$562$	0	0
$563$	3.54863i	0.149557i	0.997200	+	0.0747784i	$0.0238249\pi$
$563$	3.54863i	0.149557i	−0.997200	+	0.0747784i	$0.976175\pi$
$564$	0	0
$565$	− 10.9451i	− 0.460463i
$566$	0	0
$567$	−7.50861	−0.315332
$568$	0	0
$569$	−26.3743	−1.10567	−0.552835	−	0.833291i	$-0.686454\pi$
$569$	−26.3743	−1.10567	−0.552835	+	0.833291i	$0.686454\pi$
$570$	0	0
$571$	28.8831i	1.20872i	0.796711	+	0.604361i	$0.206571\pi$
$571$	28.8831i	1.20872i	−0.796711	+	0.604361i	$0.793429\pi$
$572$	0	0
$573$	27.8263i	1.16246i
$574$	0	0
$575$	7.67159	0.319927
$576$	0	0
$577$	−2.90293	−0.120851	−0.0604253	−	0.998173i	$-0.519246\pi$
$577$	−2.90293	−0.120851	−0.0604253	+	0.998173i	$0.519246\pi$
$578$	0	0
$579$	− 17.5496i	− 0.729336i
$580$	0	0
$581$	8.53099i	0.353925i
$582$	0	0
$583$	32.5035	1.34616
$584$	0	0
$585$	0.886420	0.0366489
$586$	0	0
$587$	− 37.8683i	− 1.56299i	−0.623909	−	0.781497i	$-0.714456\pi$
$587$	− 37.8683i	− 1.56299i	0.623909	−	0.781497i	$-0.285544\pi$
$588$	0	0
$589$	36.8517i	1.51845i
$590$	0	0
$591$	6.98775	0.287438
$592$	0	0
$593$	−6.49295	−0.266634	−0.133317	−	0.991073i	$-0.542563\pi$
$593$	−6.49295	−0.266634	−0.133317	+	0.991073i	$0.542563\pi$
$594$	0	0
$595$	10.0239i	0.410939i
$596$	0	0
$597$	16.1897i	0.662600i
$598$	0	0
$599$	−25.2803	−1.03293	−0.516463	−	0.856310i	$-0.672752\pi$
$599$	−25.2803	−1.03293	−0.516463	+	0.856310i	$0.672752\pi$
$600$	0	0
$601$	−16.8805	−0.688571	−0.344285	−	0.938865i	$-0.611879\pi$
$601$	−16.8805	−0.688571	−0.344285	+	0.938865i	$0.611879\pi$
$602$	0	0
$603$	4.46242i	0.181724i
$604$	0	0
$605$	− 0.223344i	− 0.00908023i
$606$	0	0
$607$	7.10773	0.288494	0.144247	−	0.989542i	$-0.453924\pi$
$607$	7.10773	0.288494	0.144247	+	0.989542i	$0.453924\pi$
$608$	0	0
$609$	−11.3369	−0.459395
$610$	0	0
$611$	− 0.246211i	− 0.00996065i
$612$	0	0
$613$	44.5259i	1.79838i	0.437554	+	0.899192i	$0.355845\pi$
$613$	44.5259i	1.79838i	−0.437554	+	0.899192i	$0.644155\pi$
$614$	0	0
$615$	−14.1742	−0.571558
$616$	0	0
$617$	−38.8785	−1.56519	−0.782596	−	0.622530i	$-0.786105\pi$
$617$	−38.8785	−1.56519	−0.782596	+	0.622530i	$0.786105\pi$
$618$	0	0
$619$	12.3262i	0.495431i	0.968833	+	0.247716i	$0.0796798\pi$
$619$	12.3262i	0.495431i	−0.968833	+	0.247716i	$0.920320\pi$
$620$	0	0
$621$	− 50.6266i	− 2.03157i
$622$	0	0
$623$	−14.4730	−0.579847
$624$	0	0
$625$	−20.1373	−0.805490
$626$	0	0
$627$	26.1541i	1.04449i
$628$	0	0
$629$	22.7462i	0.906950i
$630$	0	0
$631$	−20.7693	−0.826815	−0.413407	−	0.910546i	$-0.635661\pi$
$631$	−20.7693	−0.826815	−0.413407	+	0.910546i	$0.635661\pi$
$632$	0	0
$633$	39.5967	1.57383
$634$	0	0
$635$	− 37.6398i	− 1.49369i
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	0	0
$639$	−4.85717	−0.192147
$640$	0	0
$641$	−19.0642	−0.752989	−0.376495	−	0.926419i	$-0.622871\pi$
$641$	−19.0642	−0.752989	−0.376495	+	0.926419i	$0.622871\pi$
$642$	0	0
$643$	34.7530i	1.37052i	0.728297	+	0.685262i	$0.240312\pi$
$643$	34.7530i	1.37052i	−0.728297	+	0.685262i	$0.759688\pi$
$644$	0	0
$645$	24.4616i	0.963176i
$646$	0	0
$647$	15.8459	0.622967	0.311483	−	0.950252i	$-0.399174\pi$
$647$	15.8459	0.622967	0.311483	+	0.950252i	$0.399174\pi$
$648$	0	0
$649$	13.3395	0.523622
$650$	0	0
$651$	− 11.9304i	− 0.467589i
$652$	0	0
$653$	− 7.69423i	− 0.301098i	−0.988603	−	0.150549i	$-0.951896\pi$
$653$	− 7.69423i	− 0.301098i	0.988603	−	0.150549i	$-0.0481042\pi$
$654$	0	0
$655$	−20.3286	−0.794305
$656$	0	0
$657$	2.57553	0.100481
$658$	0	0
$659$	− 40.5243i	− 1.57860i	−0.614006	−	0.789301i	$-0.710443\pi$
$659$	− 40.5243i	− 1.57860i	0.614006	−	0.789301i	$-0.289557\pi$
$660$	0	0
$661$	− 22.9742i	− 0.893594i	−0.894635	−	0.446797i	$-0.852565\pi$
$661$	− 22.9742i	− 0.893594i	0.894635	−	0.446797i	$-0.147435\pi$
$662$	0	0
$663$	−7.86611	−0.305494
$664$	0	0
$665$	10.0993	0.391633
$666$	0	0
$667$	− 65.1371i	− 2.52212i
$668$	0	0
$669$	− 15.7979i	− 0.610782i
$670$	0	0
$671$	35.1662	1.35758
$672$	0	0
$673$	23.7759	0.916494	0.458247	−	0.888825i	$-0.348477\pi$
$673$	23.7759	0.916494	0.458247	+	0.888825i	$0.348477\pi$
$674$	0	0
$675$	− 4.58550i	− 0.176496i
$676$	0	0
$677$	− 28.8401i	− 1.10842i	−0.832378	−	0.554208i	$-0.813021\pi$
$677$	− 28.8401i	− 1.10842i	0.832378	−	0.554208i	$-0.186979\pi$
$678$	0	0
$679$	−3.61044	−0.138556
$680$	0	0
$681$	−5.24442	−0.200967
$682$	0	0
$683$	14.1254i	0.540492i	0.962791	+	0.270246i	$0.0871050\pi$
$683$	14.1254i	0.540492i	−0.962791	+	0.270246i	$0.912895\pi$
$684$	0	0
$685$	− 24.0409i	− 0.918555i
$686$	0	0
$687$	−0.948379	−0.0361829
$688$	0	0
$689$	9.84928	0.375228
$690$	0	0
$691$	21.5228i	0.818766i	0.912363	+	0.409383i	$0.134256\pi$
$691$	21.5228i	0.818766i	−0.912363	+	0.409383i	$0.865744\pi$
$692$	0	0
$693$	1.43312i	0.0544398i
$694$	0	0
$695$	−14.0259	−0.532032
$696$	0	0
$697$	−21.2895	−0.806397
$698$	0	0
$699$	− 24.5385i	− 0.928131i
$700$	0	0
$701$	28.0267i	1.05855i	0.848449	+	0.529277i	$0.177537\pi$
$701$	28.0267i	1.05855i	−0.848449	+	0.529277i	$0.822463\pi$
$702$	0	0
$703$	22.9173	0.864342
$704$	0	0
$705$	0.804999	0.0303180
$706$	0	0
$707$	6.51050i	0.244853i
$708$	0	0
$709$	28.2833i	1.06220i	0.847308	+	0.531101i	$0.178222\pi$
$709$	28.2833i	1.06220i	−0.847308	+	0.531101i	$0.821778\pi$
$710$	0	0
$711$	2.97303	0.111498
$712$	0	0
$713$	68.5469	2.56710
$714$	0	0
$715$	6.73608i	0.251915i
$716$	0	0
$717$	− 13.4718i	− 0.503114i
$718$	0	0
$719$	−13.6899	−0.510545	−0.255273	−	0.966869i	$-0.582165\pi$
$719$	−13.6899	−0.510545	−0.255273	+	0.966869i	$0.582165\pi$
$720$	0	0
$721$	−18.3047	−0.681704
$722$	0	0
$723$	34.7712i	1.29315i
$724$	0	0
$725$	− 5.89978i	− 0.219112i
$726$	0	0
$727$	−13.1391	−0.487303	−0.243651	−	0.969863i	$-0.578345\pi$
$727$	−13.1391	−0.487303	−0.243651	+	0.969863i	$0.578345\pi$
$728$	0	0
$729$	−29.6950	−1.09981
$730$	0	0
$731$	36.7411i	1.35892i
$732$	0	0
$733$	4.10881i	0.151762i	0.997117	+	0.0758812i	$0.0241770\pi$
$733$	4.10881i	0.151762i	−0.997117	+	0.0758812i	$0.975823\pi$
$734$	0	0
$735$	−3.26954	−0.120599
$736$	0	0
$737$	−33.9108	−1.24912
$738$	0	0
$739$	− 5.60947i	− 0.206348i	−0.994663	−	0.103174i	$-0.967100\pi$
$739$	− 5.60947i	− 0.206348i	0.994663	−	0.103174i	$-0.0328998\pi$
$740$	0	0
$741$	7.92528i	0.291142i
$742$	0	0
$743$	15.2173	0.558268	0.279134	−	0.960252i	$-0.409953\pi$
$743$	15.2173	0.558268	0.279134	+	0.960252i	$0.409953\pi$
$744$	0	0
$745$	−32.1707	−1.17864
$746$	0	0
$747$	3.70474i	0.135549i
$748$	0	0
$749$	16.0316i	0.585782i
$750$	0	0
$751$	30.0338	1.09595	0.547974	−	0.836495i	$-0.315399\pi$
$751$	30.0338	1.09595	0.547974	+	0.836495i	$0.315399\pi$
$752$	0	0
$753$	−23.4331	−0.853949
$754$	0	0
$755$	− 7.94504i	− 0.289150i
$756$	0	0
$757$	− 24.4007i	− 0.886858i	−0.896310	−	0.443429i	$-0.853762\pi$
$757$	− 24.4007i	− 0.886858i	0.896310	−	0.443429i	$-0.146238\pi$
$758$	0	0
$759$	48.6486	1.76583
$760$	0	0
$761$	−50.2706	−1.82231	−0.911153	−	0.412068i	$-0.864807\pi$
$761$	−50.2706	−1.82231	−0.911153	+	0.412068i	$0.864807\pi$
$762$	0	0
$763$	10.7704i	0.389916i
$764$	0	0
$765$	4.35305i	0.157385i
$766$	0	0
$767$	4.04217	0.145954
$768$	0	0
$769$	−20.7181	−0.747115	−0.373557	−	0.927607i	$-0.621862\pi$
$769$	−20.7181	−0.747115	−0.373557	+	0.927607i	$0.621862\pi$
$770$	0	0
$771$	26.6411i	0.959454i
$772$	0	0
$773$	46.0405i	1.65596i	0.560756	+	0.827981i	$0.310511\pi$
$773$	46.0405i	1.65596i	−0.560756	+	0.827981i	$0.689489\pi$
$774$	0	0
$775$	6.20863	0.223021
$776$	0	0
$777$	−7.41925	−0.266164
$778$	0	0
$779$	21.4496i	0.768513i
$780$	0	0
$781$	− 36.9106i	− 1.32076i
$782$	0	0
$783$	−38.9340	−1.39139
$784$	0	0
$785$	21.6000	0.770938
$786$	0	0
$787$	42.3137i	1.50832i	0.656691	+	0.754160i	$0.271956\pi$
$787$	42.3137i	1.50832i	−0.656691	+	0.754160i	$0.728044\pi$
$788$	0	0
$789$	− 25.0610i	− 0.892195i
$790$	0	0
$791$	5.36213	0.190655
$792$	0	0
$793$	10.6561	0.378411
$794$	0	0
$795$	32.2027i	1.14211i
$796$	0	0
$797$	17.9127i	0.634502i	0.948342	+	0.317251i	$0.102760\pi$
$797$	17.9127i	0.634502i	−0.948342	+	0.317251i	$0.897240\pi$
$798$	0	0
$799$	1.20910	0.0427749
$800$	0	0
$801$	−6.28515	−0.222075
$802$	0	0
$803$	19.5719i	0.690679i
$804$	0	0
$805$	− 18.7854i	− 0.662098i
$806$	0	0
$807$	19.1073	0.672607
$808$	0	0
$809$	−4.46270	−0.156900	−0.0784500	−	0.996918i	$-0.524997\pi$
$809$	−4.46270	−0.156900	−0.0784500	+	0.996918i	$0.524997\pi$
$810$	0	0
$811$	30.1488i	1.05867i	0.848413	+	0.529334i	$0.177558\pi$
$811$	30.1488i	1.05867i	−0.848413	+	0.529334i	$0.822442\pi$
$812$	0	0
$813$	− 9.59767i	− 0.336605i
$814$	0	0
$815$	−37.6085	−1.31737
$816$	0	0
$817$	37.0175	1.29508
$818$	0	0
$819$	0.434268i	0.0151746i
$820$	0	0
$821$	16.5840i	0.578786i	0.957210	+	0.289393i	$0.0934535\pi$
$821$	16.5840i	0.578786i	−0.957210	+	0.289393i	$0.906547\pi$
$822$	0	0
$823$	−33.0671	−1.15265	−0.576324	−	0.817221i	$-0.695513\pi$
$823$	−33.0671	−1.15265	−0.576324	+	0.817221i	$0.695513\pi$
$824$	0	0
$825$	4.40634	0.153409
$826$	0	0
$827$	24.9129i	0.866308i	0.901320	+	0.433154i	$0.142599\pi$
$827$	24.9129i	0.866308i	−0.901320	+	0.433154i	$0.857401\pi$
$828$	0	0
$829$	42.3263i	1.47005i	0.678038	+	0.735027i	$0.262831\pi$
$829$	42.3263i	1.47005i	−0.678038	+	0.735027i	$0.737169\pi$
$830$	0	0
$831$	−20.6705	−0.717051
$832$	0	0
$833$	−4.91083	−0.170150
$834$	0	0
$835$	− 26.8639i	− 0.929662i
$836$	0	0
$837$	− 40.9722i	− 1.41621i
$838$	0	0
$839$	22.8739	0.789695	0.394847	−	0.918747i	$-0.370797\pi$
$839$	22.8739	0.789695	0.394847	+	0.918747i	$0.370797\pi$
$840$	0	0
$841$	−21.0932	−0.727352
$842$	0	0
$843$	28.7249i	0.989338i
$844$	0	0
$845$	2.04118i	0.0702188i
$846$	0	0
$847$	0.109419	0.00375968
$848$	0	0
$849$	20.5968	0.706879
$850$	0	0
$851$	− 42.6278i	− 1.46126i
$852$	0	0
$853$	8.85133i	0.303064i	0.988452	+	0.151532i	$0.0484206\pi$
$853$	8.85133i	0.303064i	−0.988452	+	0.151532i	$0.951579\pi$
$854$	0	0
$855$	4.38579	0.149991
$856$	0	0
$857$	−8.20832	−0.280391	−0.140195	−	0.990124i	$-0.544773\pi$
$857$	−8.20832	−0.280391	−0.140195	+	0.990124i	$0.544773\pi$
$858$	0	0
$859$	21.9662i	0.749476i	0.927131	+	0.374738i	$0.122267\pi$
$859$	21.9662i	0.749476i	−0.927131	+	0.374738i	$0.877733\pi$
$860$	0	0
$861$	− 6.94411i	− 0.236655i
$862$	0	0
$863$	33.8494	1.15225	0.576123	−	0.817363i	$-0.304565\pi$
$863$	33.8494	1.15225	0.576123	+	0.817363i	$0.304565\pi$
$864$	0	0
$865$	−15.1611	−0.515495
$866$	0	0
$867$	− 11.3987i	− 0.387119i
$868$	0	0
$869$	22.5927i	0.766404i
$870$	0	0
$871$	−10.2757	−0.348179
$872$	0	0
$873$	−1.56790	−0.0530653
$874$	0	0
$875$	− 11.9074i	− 0.402543i
$876$	0	0
$877$	− 26.3189i	− 0.888728i	−0.895846	−	0.444364i	$-0.853430\pi$
$877$	− 26.3189i	− 0.888728i	0.895846	−	0.444364i	$-0.146570\pi$
$878$	0	0
$879$	42.8663	1.44584
$880$	0	0
$881$	36.5036	1.22984	0.614919	−	0.788590i	$-0.289188\pi$
$881$	36.5036	1.22984	0.614919	+	0.788590i	$0.289188\pi$
$882$	0	0
$883$	35.3164i	1.18849i	0.804283	+	0.594246i	$0.202549\pi$
$883$	35.3164i	1.18849i	−0.804283	+	0.594246i	$0.797451\pi$
$884$	0	0
$885$	13.2161i	0.444253i
$886$	0	0
$887$	−23.3302	−0.783351	−0.391676	−	0.920103i	$-0.628104\pi$
$887$	−23.3302	−0.783351	−0.391676	+	0.920103i	$0.628104\pi$
$888$	0	0
$889$	18.4402	0.618465
$890$	0	0
$891$	− 24.7791i	− 0.830130i
$892$	0	0
$893$	− 1.21820i	− 0.0407654i
$894$	0	0
$895$	−37.8799	−1.26619
$896$	0	0
$897$	14.7416	0.492207
$898$	0	0
$899$	− 52.7155i	− 1.75816i
$900$	0	0
$901$	48.3681i	1.61137i
$902$	0	0
$903$	−11.9841	−0.398804
$904$	0	0
$905$	38.6063	1.28332
$906$	0	0
$907$	30.1817i	1.00217i	0.865399	+	0.501083i	$0.167065\pi$
$907$	30.1817i	1.00217i	−0.865399	+	0.501083i	$0.832935\pi$
$908$	0	0
$909$	2.82730i	0.0937757i
$910$	0	0
$911$	−7.93273	−0.262823	−0.131412	−	0.991328i	$-0.541951\pi$
$911$	−7.93273	−0.262823	−0.131412	+	0.991328i	$0.541951\pi$
$912$	0	0
$913$	−28.1530	−0.931729
$914$	0	0
$915$	34.8407i	1.15180i
$916$	0	0
$917$	− 9.95924i	− 0.328883i
$918$	0	0
$919$	−29.3071	−0.966750	−0.483375	−	0.875413i	$-0.660589\pi$
$919$	−29.3071	−0.966750	−0.483375	+	0.875413i	$0.660589\pi$
$920$	0	0
$921$	−16.4639	−0.542504
$922$	0	0
$923$	− 11.1847i	− 0.368150i
$924$	0	0
$925$	− 3.86101i	− 0.126949i
$926$	0	0
$927$	−7.94916	−0.261085
$928$	0	0
$929$	2.37267	0.0778447	0.0389224	−	0.999242i	$-0.487607\pi$
$929$	2.37267	0.0778447	0.0389224	+	0.999242i	$0.487607\pi$
$930$	0	0
$931$	4.94776i	0.162156i
$932$	0	0
$933$	− 7.22649i	− 0.236585i
$934$	0	0
$935$	−33.0797	−1.08182
$936$	0	0
$937$	32.6751	1.06745	0.533725	−	0.845658i	$-0.320792\pi$
$937$	32.6751	1.06745	0.533725	+	0.845658i	$0.320792\pi$
$938$	0	0
$939$	4.66054i	0.152091i
$940$	0	0
$941$	− 38.3641i	− 1.25063i	−0.780371	−	0.625316i	$-0.784970\pi$
$941$	− 38.3641i	− 1.25063i	0.780371	−	0.625316i	$-0.215030\pi$
$942$	0	0
$943$	39.8979	1.29925
$944$	0	0
$945$	−11.2285	−0.365263
$946$	0	0
$947$	− 9.22751i	− 0.299854i	−0.988697	−	0.149927i	$-0.952096\pi$
$947$	− 9.22751i	− 0.299854i	0.988697	−	0.149927i	$-0.0479039\pi$
$948$	0	0
$949$	5.93073i	0.192520i
$950$	0	0
$951$	−14.0720	−0.456316
$952$	0	0
$953$	20.3151	0.658072	0.329036	−	0.944317i	$-0.393276\pi$
$953$	20.3151	0.658072	0.329036	+	0.944317i	$0.393276\pi$
$954$	0	0
$955$	35.4593i	1.14744i
$956$	0	0
$957$	− 37.4128i	− 1.20939i
$958$	0	0
$959$	11.7779	0.380329
$960$	0	0
$961$	24.4751	0.789521
$962$	0	0
$963$	6.96201i	0.224348i
$964$	0	0
$965$	− 22.3637i	− 0.719912i
$966$	0	0
$967$	−13.9577	−0.448851	−0.224425	−	0.974491i	$-0.572050\pi$
$967$	−13.9577	−0.448851	−0.224425	+	0.974491i	$0.572050\pi$
$968$	0	0
$969$	−38.9197	−1.25028
$970$	0	0
$971$	− 11.4952i	− 0.368898i	−0.982842	−	0.184449i	$-0.940950\pi$
$971$	− 11.4952i	− 0.368898i	0.982842	−	0.184449i	$-0.0590501\pi$
$972$	0	0
$973$	− 6.87146i	− 0.220289i
$974$	0	0
$975$	1.33522	0.0427612
$976$	0	0
$977$	−7.39204	−0.236492	−0.118246	−	0.992984i	$-0.537727\pi$
$977$	−7.39204	−0.236492	−0.118246	+	0.992984i	$0.537727\pi$
$978$	0	0
$979$	− 47.7621i	− 1.52648i
$980$	0	0
$981$	4.67725i	0.149333i
$982$	0	0
$983$	34.2578	1.09265	0.546327	−	0.837572i	$-0.316026\pi$
$983$	34.2578	1.09265	0.546327	+	0.837572i	$0.316026\pi$
$984$	0	0
$985$	8.90457	0.283723
$986$	0	0
$987$	0.394379i	0.0125532i
$988$	0	0
$989$	− 68.8552i	− 2.18947i
$990$	0	0
$991$	−50.9734	−1.61922	−0.809612	−	0.586966i	$-0.800322\pi$
$991$	−50.9734	−1.61922	−0.809612	+	0.586966i	$0.800322\pi$
$992$	0	0
$993$	18.4648	0.585962
$994$	0	0
$995$	20.6307i	0.654038i
$996$	0	0
$997$	− 7.97874i	− 0.252689i	−0.991986	−	0.126345i	$-0.959675\pi$
$997$	− 7.97874i	− 0.252689i	0.991986	−	0.126345i	$-0.0403245\pi$
$998$	0	0
$999$	−25.4797	−0.806141

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.c.a.1457.9		34
4.3	odd	2		728.2.c.a.365.1	✓	34
8.3	odd	2		728.2.c.a.365.2	yes	34
8.5	even	2	inner	2912.2.c.a.1457.26		34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.c.a.365.1	✓	34	4.3	odd	2
728.2.c.a.365.2	yes	34	8.3	odd	2
2912.2.c.a.1457.9		34	1.1	even	1	trivial
2912.2.c.a.1457.26		34	8.5	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.60179i q^{3} -2.04118i q^{5} +1.00000 q^{7} +0.434268 q^{9} +3.30009i q^{11} +1.00000i q^{13} -3.26954 q^{15} -4.91083 q^{17} +4.94776i q^{19} -1.60179i q^{21} +9.20320 q^{23} +0.833579 q^{25} -5.50098i q^{27} -7.07766i q^{29} +7.44816 q^{31} +5.28605 q^{33} -2.04118i q^{35} -4.63185i q^{37} +1.60179 q^{39} +4.33522 q^{41} -7.48166i q^{43} -0.886420i q^{45} -0.246211 q^{47} +1.00000 q^{49} +7.86611i q^{51} -9.84928i q^{53} +6.73608 q^{55} +7.92528 q^{57} -4.04217i q^{59} -10.6561i q^{61} +0.434268 q^{63} +2.04118 q^{65} +10.2757i q^{67} -14.7416i q^{69} -11.1847 q^{71} +5.93073 q^{73} -1.33522i q^{75} +3.30009i q^{77} +6.84608 q^{79} -7.50861 q^{81} +8.53099i q^{83} +10.0239i q^{85} -11.3369 q^{87} -14.4730 q^{89} +1.00000i q^{91} -11.9304i q^{93} +10.0993 q^{95} -3.61044 q^{97} +1.43312i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) \) 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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