LMFDB - Embedded newform 2912.2.c.a.1457.1

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			1457.1
Character	$\chi$	$=$	2912.1457
Dual form			2912.2.c.a.1457.34

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16783i q^{3} -1.11795i q^{5} +1.00000 q^{7} -7.03512 q^{9} +4.51562i q^{11} -1.00000i q^{13} -3.54146 q^{15} -4.44816 q^{17} +6.89036i q^{19} -3.16783i q^{21} +2.66314 q^{23} +3.75020 q^{25} +12.7825i q^{27} +6.01139i q^{29} -10.4802 q^{31} +14.3047 q^{33} -1.11795i q^{35} -2.80922i q^{37} -3.16783 q^{39} -4.93559 q^{41} +4.88533i q^{43} +7.86489i q^{45} +6.35415 q^{47} +1.00000 q^{49} +14.0910i q^{51} +6.02877i q^{53} +5.04822 q^{55} +21.8275 q^{57} +15.1683i q^{59} -6.44954i q^{61} -7.03512 q^{63} -1.11795 q^{65} +5.98352i q^{67} -8.43637i q^{69} -0.527201 q^{71} -7.57660 q^{73} -11.8800i q^{75} +4.51562i q^{77} -4.85576 q^{79} +19.3875 q^{81} +2.98647i q^{83} +4.97281i q^{85} +19.0430 q^{87} +5.07956 q^{89} -1.00000i q^{91} +33.1995i q^{93} +7.70306 q^{95} -17.4191 q^{97} -31.7679i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 3.16783i	− 1.82894i	−0.404648	−	0.914472i	$-0.632606\pi$
$3$	− 3.16783i	− 1.82894i	0.404648	−	0.914472i	$-0.367394\pi$
$4$	0	0
$5$	− 1.11795i	− 0.499961i	−0.968251	−	0.249980i	$-0.919576\pi$
$5$	− 1.11795i	− 0.499961i	0.968251	−	0.249980i	$-0.0804242\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−7.03512	−2.34504
$10$	0	0
$11$	4.51562i	1.36151i	0.732511	+	0.680756i	$0.238348\pi$
$11$	4.51562i	1.36151i	−0.732511	+	0.680756i	$0.761652\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−3.54146	−0.914401
$16$	0	0
$17$	−4.44816	−1.07884	−0.539419	−	0.842038i	$-0.681356\pi$
$17$	−4.44816	−1.07884	−0.539419	+	0.842038i	$0.681356\pi$
$18$	0	0
$19$	6.89036i	1.58076i	0.612619	+	0.790379i	$0.290116\pi$
$19$	6.89036i	1.58076i	−0.612619	+	0.790379i	$0.709884\pi$
$20$	0	0
$21$	− 3.16783i	− 0.691276i
$22$	0	0
$23$	2.66314	0.555303	0.277652	−	0.960682i	$-0.410444\pi$
$23$	2.66314	0.555303	0.277652	+	0.960682i	$0.410444\pi$
$24$	0	0
$25$	3.75020	0.750039
$26$	0	0
$27$	12.7825i	2.46000i
$28$	0	0
$29$	6.01139i	1.11629i	0.829745	+	0.558143i	$0.188486\pi$
$29$	6.01139i	1.11629i	−0.829745	+	0.558143i	$0.811514\pi$
$30$	0	0
$31$	−10.4802	−1.88230	−0.941152	−	0.337983i	$-0.890255\pi$
$31$	−10.4802	−1.88230	−0.941152	+	0.337983i	$0.890255\pi$
$32$	0	0
$33$	14.3047	2.49013
$34$	0	0
$35$	− 1.11795i	− 0.188967i
$36$	0	0
$37$	− 2.80922i	− 0.461833i	−0.972974	−	0.230916i	$-0.925828\pi$
$37$	− 2.80922i	− 0.461833i	0.972974	−	0.230916i	$-0.0741724\pi$
$38$	0	0
$39$	−3.16783	−0.507258
$40$	0	0
$41$	−4.93559	−0.770810	−0.385405	−	0.922748i	$-0.625938\pi$
$41$	−4.93559	−0.770810	−0.385405	+	0.922748i	$0.625938\pi$
$42$	0	0
$43$	4.88533i	0.745006i	0.928031	+	0.372503i	$0.121500\pi$
$43$	4.88533i	0.745006i	−0.928031	+	0.372503i	$0.878500\pi$
$44$	0	0
$45$	7.86489i	1.17243i
$46$	0	0
$47$	6.35415	0.926849	0.463424	−	0.886136i	$-0.346621\pi$
$47$	6.35415	0.926849	0.463424	+	0.886136i	$0.346621\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	14.0910i	1.97314i
$52$	0	0
$53$	6.02877i	0.828116i	0.910251	+	0.414058i	$0.135889\pi$
$53$	6.02877i	0.828116i	−0.910251	+	0.414058i	$0.864111\pi$
$54$	0	0
$55$	5.04822	0.680702
$56$	0	0
$57$	21.8275	2.89112
$58$	0	0
$59$	15.1683i	1.97475i	0.158400	+	0.987375i	$0.449367\pi$
$59$	15.1683i	1.97475i	−0.158400	+	0.987375i	$0.550633\pi$
$60$	0	0
$61$	− 6.44954i	− 0.825779i	−0.910781	−	0.412890i	$-0.864520\pi$
$61$	− 6.44954i	− 0.825779i	0.910781	−	0.412890i	$-0.135480\pi$
$62$	0	0
$63$	−7.03512	−0.886341
$64$	0	0
$65$	−1.11795	−0.138664
$66$	0	0
$67$	5.98352i	0.731004i	0.930811	+	0.365502i	$0.119103\pi$
$67$	5.98352i	0.731004i	−0.930811	+	0.365502i	$0.880897\pi$
$68$	0	0
$69$	− 8.43637i	− 1.01562i
$70$	0	0
$71$	−0.527201	−0.0625672	−0.0312836	−	0.999511i	$-0.509960\pi$
$71$	−0.527201	−0.0625672	−0.0312836	+	0.999511i	$0.509960\pi$
$72$	0	0
$73$	−7.57660	−0.886774	−0.443387	−	0.896330i	$-0.646223\pi$
$73$	−7.57660	−0.886774	−0.443387	+	0.896330i	$0.646223\pi$
$74$	0	0
$75$	− 11.8800i	− 1.37178i
$76$	0	0
$77$	4.51562i	0.514603i
$78$	0	0
$79$	−4.85576	−0.546316	−0.273158	−	0.961969i	$-0.588068\pi$
$79$	−4.85576	−0.546316	−0.273158	+	0.961969i	$0.588068\pi$
$80$	0	0
$81$	19.3875	2.15417
$82$	0	0
$83$	2.98647i	0.327808i	0.986476	+	0.163904i	$0.0524087\pi$
$83$	2.98647i	0.327808i	−0.986476	+	0.163904i	$0.947591\pi$
$84$	0	0
$85$	4.97281i	0.539377i
$86$	0	0
$87$	19.0430	2.04163
$88$	0	0
$89$	5.07956	0.538432	0.269216	−	0.963080i	$-0.413235\pi$
$89$	5.07956	0.538432	0.269216	+	0.963080i	$0.413235\pi$
$90$	0	0
$91$	− 1.00000i	− 0.104828i
$92$	0	0
$93$	33.1995i	3.44263i
$94$	0	0
$95$	7.70306	0.790317
$96$	0	0
$97$	−17.4191	−1.76864	−0.884320	−	0.466881i	$-0.845378\pi$
$97$	−17.4191	−1.76864	−0.884320	+	0.466881i	$0.845378\pi$
$98$	0	0
$99$	− 31.7679i	− 3.19280i
$100$	0	0
$101$	− 12.0937i	− 1.20337i	−0.798733	−	0.601685i	$-0.794496\pi$
$101$	− 12.0937i	− 1.20337i	0.798733	−	0.601685i	$-0.205504\pi$
$102$	0	0
$103$	10.2847	1.01338	0.506689	−	0.862129i	$-0.330869\pi$
$103$	10.2847	1.01338	0.506689	+	0.862129i	$0.330869\pi$
$104$	0	0
$105$	−3.54146	−0.345611
$106$	0	0
$107$	− 3.16408i	− 0.305883i	−0.988235	−	0.152942i	$-0.951125\pi$
$107$	− 3.16408i	− 0.305883i	0.988235	−	0.152942i	$-0.0488746\pi$
$108$	0	0
$109$	9.48432i	0.908433i	0.890891	+	0.454216i	$0.150081\pi$
$109$	9.48432i	0.908433i	−0.890891	+	0.454216i	$0.849919\pi$
$110$	0	0
$111$	−8.89912	−0.844667
$112$	0	0
$113$	−10.2704	−0.966163	−0.483081	−	0.875575i	$-0.660482\pi$
$113$	−10.2704	−0.966163	−0.483081	+	0.875575i	$0.660482\pi$
$114$	0	0
$115$	− 2.97725i	− 0.277630i
$116$	0	0
$117$	7.03512i	0.650397i
$118$	0	0
$119$	−4.44816	−0.407762
$120$	0	0
$121$	−9.39084	−0.853713
$122$	0	0
$123$	15.6351i	1.40977i
$124$	0	0
$125$	− 9.78225i	− 0.874951i
$126$	0	0
$127$	−21.8163	−1.93588	−0.967941	−	0.251179i	$-0.919182\pi$
$127$	−21.8163	−1.93588	−0.967941	+	0.251179i	$0.919182\pi$
$128$	0	0
$129$	15.4759	1.36258
$130$	0	0
$131$	− 15.7094i	− 1.37253i	−0.727350	−	0.686266i	$-0.759249\pi$
$131$	− 15.7094i	− 1.37253i	0.727350	−	0.686266i	$-0.240751\pi$
$132$	0	0
$133$	6.89036i	0.597470i
$134$	0	0
$135$	14.2902	1.22990
$136$	0	0
$137$	0.178216	0.0152260	0.00761299	−	0.999971i	$-0.497577\pi$
$137$	0.178216	0.0152260	0.00761299	+	0.999971i	$0.497577\pi$
$138$	0	0
$139$	− 12.0375i	− 1.02100i	−0.859877	−	0.510502i	$-0.829460\pi$
$139$	− 12.0375i	− 1.02100i	0.859877	−	0.510502i	$-0.170540\pi$
$140$	0	0
$141$	− 20.1289i	− 1.69516i
$142$	0	0
$143$	4.51562	0.377615
$144$	0	0
$145$	6.72041	0.558100
$146$	0	0
$147$	− 3.16783i	− 0.261278i
$148$	0	0
$149$	− 2.45649i	− 0.201243i	−0.994925	−	0.100622i	$-0.967917\pi$
$149$	− 2.45649i	− 0.201243i	0.994925	−	0.100622i	$-0.0320832\pi$
$150$	0	0
$151$	7.03048	0.572133	0.286066	−	0.958210i	$-0.407652\pi$
$151$	7.03048	0.572133	0.286066	+	0.958210i	$0.407652\pi$
$152$	0	0
$153$	31.2933	2.52992
$154$	0	0
$155$	11.7163i	0.941079i
$156$	0	0
$157$	− 3.93708i	− 0.314213i	−0.987582	−	0.157107i	$-0.949783\pi$
$157$	− 3.93708i	− 0.314213i	0.987582	−	0.157107i	$-0.0502166\pi$
$158$	0	0
$159$	19.0981	1.51458
$160$	0	0
$161$	2.66314	0.209885
$162$	0	0
$163$	13.7052i	1.07347i	0.843750	+	0.536736i	$0.180343\pi$
$163$	13.7052i	1.07347i	−0.843750	+	0.536736i	$0.819657\pi$
$164$	0	0
$165$	− 15.9919i	− 1.24497i
$166$	0	0
$167$	10.1840	0.788061	0.394031	−	0.919097i	$-0.371080\pi$
$167$	10.1840	0.788061	0.394031	+	0.919097i	$0.371080\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 48.4745i	− 3.70694i
$172$	0	0
$173$	6.19802i	0.471227i	0.971847	+	0.235614i	$0.0757099\pi$
$173$	6.19802i	0.471227i	−0.971847	+	0.235614i	$0.924290\pi$
$174$	0	0
$175$	3.75020	0.283488
$176$	0	0
$177$	48.0507	3.61171
$178$	0	0
$179$	11.4822i	0.858220i	0.903252	+	0.429110i	$0.141173\pi$
$179$	11.4822i	0.858220i	−0.903252	+	0.429110i	$0.858827\pi$
$180$	0	0
$181$	− 26.3811i	− 1.96089i	−0.196787	−	0.980446i	$-0.563051\pi$
$181$	− 26.3811i	− 1.96089i	0.196787	−	0.980446i	$-0.436949\pi$
$182$	0	0
$183$	−20.4310	−1.51030
$184$	0	0
$185$	−3.14056	−0.230898
$186$	0	0
$187$	− 20.0862i	− 1.46885i
$188$	0	0
$189$	12.7825i	0.929793i
$190$	0	0
$191$	−8.81111	−0.637549	−0.318775	−	0.947831i	$-0.603271\pi$
$191$	−8.81111	−0.637549	−0.318775	+	0.947831i	$0.603271\pi$
$192$	0	0
$193$	−0.353957	−0.0254784	−0.0127392	−	0.999919i	$-0.504055\pi$
$193$	−0.353957	−0.0254784	−0.0127392	+	0.999919i	$0.504055\pi$
$194$	0	0
$195$	3.54146i	0.253609i
$196$	0	0
$197$	5.33967i	0.380436i	0.981742	+	0.190218i	$0.0609195\pi$
$197$	5.33967i	0.380436i	−0.981742	+	0.190218i	$0.939081\pi$
$198$	0	0
$199$	−13.5233	−0.958642	−0.479321	−	0.877640i	$-0.659117\pi$
$199$	−13.5233	−0.958642	−0.479321	+	0.877640i	$0.659117\pi$
$200$	0	0
$201$	18.9548	1.33697
$202$	0	0
$203$	6.01139i	0.421917i
$204$	0	0
$205$	5.51773i	0.385375i
$206$	0	0
$207$	−18.7355	−1.30221
$208$	0	0
$209$	−31.1143	−2.15222
$210$	0	0
$211$	− 7.02940i	− 0.483924i	−0.970286	−	0.241962i	$-0.922209\pi$
$211$	− 7.02940i	− 0.483924i	0.970286	−	0.241962i	$-0.0777909\pi$
$212$	0	0
$213$	1.67008i	0.114432i
$214$	0	0
$215$	5.46154	0.372474
$216$	0	0
$217$	−10.4802	−0.711444
$218$	0	0
$219$	24.0013i	1.62186i
$220$	0	0
$221$	4.44816i	0.299216i
$222$	0	0
$223$	15.1940	1.01747	0.508734	−	0.860924i	$-0.330114\pi$
$223$	15.1940	1.01747	0.508734	+	0.860924i	$0.330114\pi$
$224$	0	0
$225$	−26.3831	−1.75887
$226$	0	0
$227$	3.08660i	0.204865i	0.994740	+	0.102432i	$0.0326625\pi$
$227$	3.08660i	0.204865i	−0.994740	+	0.102432i	$0.967337\pi$
$228$	0	0
$229$	8.10820i	0.535805i	0.963446	+	0.267902i	$0.0863305\pi$
$229$	8.10820i	0.535805i	−0.963446	+	0.267902i	$0.913669\pi$
$230$	0	0
$231$	14.3047	0.941180
$232$	0	0
$233$	11.7007	0.766541	0.383270	−	0.923636i	$-0.374798\pi$
$233$	11.7007	0.766541	0.383270	+	0.923636i	$0.374798\pi$
$234$	0	0
$235$	− 7.10361i	− 0.463388i
$236$	0	0
$237$	15.3822i	0.999181i
$238$	0	0
$239$	26.1557	1.69187	0.845935	−	0.533286i	$-0.179043\pi$
$239$	26.1557	1.69187	0.845935	+	0.533286i	$0.179043\pi$
$240$	0	0
$241$	−13.3340	−0.858916	−0.429458	−	0.903087i	$-0.641295\pi$
$241$	−13.3340	−0.858916	−0.429458	+	0.903087i	$0.641295\pi$
$242$	0	0
$243$	− 23.0686i	− 1.47985i
$244$	0	0
$245$	− 1.11795i	− 0.0714230i
$246$	0	0
$247$	6.89036	0.438423
$248$	0	0
$249$	9.46061	0.599542
$250$	0	0
$251$	15.0492i	0.949896i	0.880014	+	0.474948i	$0.157533\pi$
$251$	15.0492i	0.949896i	−0.880014	+	0.474948i	$0.842467\pi$
$252$	0	0
$253$	12.0257i	0.756052i
$254$	0	0
$255$	15.7530	0.986490
$256$	0	0
$257$	1.36042	0.0848609	0.0424305	−	0.999099i	$-0.486490\pi$
$257$	1.36042	0.0848609	0.0424305	+	0.999099i	$0.486490\pi$
$258$	0	0
$259$	− 2.80922i	− 0.174556i
$260$	0	0
$261$	− 42.2908i	− 2.61774i
$262$	0	0
$263$	11.1149	0.685373	0.342687	−	0.939450i	$-0.388663\pi$
$263$	11.1149	0.685373	0.342687	+	0.939450i	$0.388663\pi$
$264$	0	0
$265$	6.73985	0.414026
$266$	0	0
$267$	− 16.0912i	− 0.984763i
$268$	0	0
$269$	7.18479i	0.438064i	0.975718	+	0.219032i	$0.0702900\pi$
$269$	7.18479i	0.438064i	−0.975718	+	0.219032i	$0.929710\pi$
$270$	0	0
$271$	−26.3646	−1.60154	−0.800769	−	0.598974i	$-0.795575\pi$
$271$	−26.3646	−1.60154	−0.800769	+	0.598974i	$0.795575\pi$
$272$	0	0
$273$	−3.16783	−0.191726
$274$	0	0
$275$	16.9345i	1.02119i
$276$	0	0
$277$	9.62372i	0.578233i	0.957294	+	0.289116i	$0.0933615\pi$
$277$	9.62372i	0.578233i	−0.957294	+	0.289116i	$0.906639\pi$
$278$	0	0
$279$	73.7296	4.41408
$280$	0	0
$281$	−13.8037	−0.823460	−0.411730	−	0.911306i	$-0.635075\pi$
$281$	−13.8037	−0.823460	−0.411730	+	0.911306i	$0.635075\pi$
$282$	0	0
$283$	− 7.76345i	− 0.461489i	−0.973014	−	0.230744i	$-0.925884\pi$
$283$	− 7.76345i	− 0.461489i	0.973014	−	0.230744i	$-0.0741161\pi$
$284$	0	0
$285$	− 24.4019i	− 1.44545i
$286$	0	0
$287$	−4.93559	−0.291339
$288$	0	0
$289$	2.78615	0.163891
$290$	0	0
$291$	55.1806i	3.23475i
$292$	0	0
$293$	23.4220i	1.36833i	0.729329	+	0.684163i	$0.239832\pi$
$293$	23.4220i	1.36833i	−0.729329	+	0.684163i	$0.760168\pi$
$294$	0	0
$295$	16.9574	0.987298
$296$	0	0
$297$	−57.7211	−3.34932
$298$	0	0
$299$	− 2.66314i	− 0.154013i
$300$	0	0
$301$	4.88533i	0.281586i
$302$	0	0
$303$	−38.3108	−2.20090
$304$	0	0
$305$	−7.21024	−0.412857
$306$	0	0
$307$	− 7.12411i	− 0.406594i	−0.979117	−	0.203297i	$-0.934834\pi$
$307$	− 7.12411i	− 0.406594i	0.979117	−	0.203297i	$-0.0651657\pi$
$308$	0	0
$309$	− 32.5800i	− 1.85341i
$310$	0	0
$311$	16.3917	0.929486	0.464743	−	0.885446i	$-0.346147\pi$
$311$	16.3917	0.929486	0.464743	+	0.885446i	$0.346147\pi$
$312$	0	0
$313$	14.5603	0.822998	0.411499	−	0.911410i	$-0.365005\pi$
$313$	14.5603	0.822998	0.411499	+	0.911410i	$0.365005\pi$
$314$	0	0
$315$	7.86489i	0.443136i
$316$	0	0
$317$	− 24.5206i	− 1.37721i	−0.725135	−	0.688607i	$-0.758223\pi$
$317$	− 24.5206i	− 1.37721i	0.725135	−	0.688607i	$-0.241777\pi$
$318$	0	0
$319$	−27.1452	−1.51984
$320$	0	0
$321$	−10.0233	−0.559443
$322$	0	0
$323$	− 30.6494i	− 1.70538i
$324$	0	0
$325$	− 3.75020i	− 0.208023i
$326$	0	0
$327$	30.0447	1.66147
$328$	0	0
$329$	6.35415	0.350316
$330$	0	0
$331$	1.15297i	0.0633727i	0.999498	+	0.0316864i	$0.0100878\pi$
$331$	1.15297i	0.0633727i	−0.999498	+	0.0316864i	$0.989912\pi$
$332$	0	0
$333$	19.7632i	1.08302i
$334$	0	0
$335$	6.68926	0.365473
$336$	0	0
$337$	34.0718	1.85601	0.928004	−	0.372569i	$-0.121523\pi$
$337$	34.0718	1.85601	0.928004	+	0.372569i	$0.121523\pi$
$338$	0	0
$339$	32.5350i	1.76706i
$340$	0	0
$341$	− 47.3247i	− 2.56278i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−9.43141	−0.507770
$346$	0	0
$347$	− 10.9924i	− 0.590103i	−0.955481	−	0.295051i	$-0.904663\pi$
$347$	− 10.9924i	− 0.590103i	0.955481	−	0.295051i	$-0.0953367\pi$
$348$	0	0
$349$	− 6.05336i	− 0.324029i	−0.986788	−	0.162014i	$-0.948201\pi$
$349$	− 6.05336i	− 0.324029i	0.986788	−	0.162014i	$-0.0517991\pi$
$350$	0	0
$351$	12.7825	0.682282
$352$	0	0
$353$	6.11657	0.325552	0.162776	−	0.986663i	$-0.447955\pi$
$353$	6.11657	0.325552	0.162776	+	0.986663i	$0.447955\pi$
$354$	0	0
$355$	0.589382i	0.0312812i
$356$	0	0
$357$	14.0910i	0.745775i
$358$	0	0
$359$	−1.19941	−0.0633027	−0.0316513	−	0.999499i	$-0.510077\pi$
$359$	−1.19941	−0.0633027	−0.0316513	+	0.999499i	$0.510077\pi$
$360$	0	0
$361$	−28.4771	−1.49879
$362$	0	0
$363$	29.7485i	1.56139i
$364$	0	0
$365$	8.47023i	0.443352i
$366$	0	0
$367$	−6.89769	−0.360056	−0.180028	−	0.983661i	$-0.557619\pi$
$367$	−6.89769	−0.360056	−0.180028	+	0.983661i	$0.557619\pi$
$368$	0	0
$369$	34.7225	1.80758
$370$	0	0
$371$	6.02877i	0.312998i
$372$	0	0
$373$	11.8425i	0.613180i	0.951842	+	0.306590i	$0.0991881\pi$
$373$	11.8425i	0.613180i	−0.951842	+	0.306590i	$0.900812\pi$
$374$	0	0
$375$	−30.9885	−1.60024
$376$	0	0
$377$	6.01139	0.309602
$378$	0	0
$379$	25.1816i	1.29349i	0.762705	+	0.646746i	$0.223871\pi$
$379$	25.1816i	1.29349i	−0.762705	+	0.646746i	$0.776129\pi$
$380$	0	0
$381$	69.1101i	3.54062i
$382$	0	0
$383$	5.10219	0.260710	0.130355	−	0.991467i	$-0.458388\pi$
$383$	5.10219	0.260710	0.130355	+	0.991467i	$0.458388\pi$
$384$	0	0
$385$	5.04822	0.257281
$386$	0	0
$387$	− 34.3689i	− 1.74707i
$388$	0	0
$389$	10.1680i	0.515536i	0.966207	+	0.257768i	$0.0829871\pi$
$389$	10.1680i	0.515536i	−0.966207	+	0.257768i	$0.917013\pi$
$390$	0	0
$391$	−11.8461	−0.599082
$392$	0	0
$393$	−49.7645	−2.51029
$394$	0	0
$395$	5.42848i	0.273137i
$396$	0	0
$397$	18.3066i	0.918779i	0.888235	+	0.459390i	$0.151932\pi$
$397$	18.3066i	0.918779i	−0.888235	+	0.459390i	$0.848068\pi$
$398$	0	0
$399$	21.8275	1.09274
$400$	0	0
$401$	17.2034	0.859099	0.429549	−	0.903043i	$-0.358672\pi$
$401$	17.2034	0.859099	0.429549	+	0.903043i	$0.358672\pi$
$402$	0	0
$403$	10.4802i	0.522057i
$404$	0	0
$405$	− 21.6742i	− 1.07700i
$406$	0	0
$407$	12.6854	0.628791
$408$	0	0
$409$	8.95836	0.442962	0.221481	−	0.975165i	$-0.428911\pi$
$409$	8.95836	0.442962	0.221481	+	0.975165i	$0.428911\pi$
$410$	0	0
$411$	− 0.564556i	− 0.0278475i
$412$	0	0
$413$	15.1683i	0.746385i
$414$	0	0
$415$	3.33871	0.163891
$416$	0	0
$417$	−38.1325	−1.86736
$418$	0	0
$419$	0.476426i	0.0232749i	0.999932	+	0.0116375i	$0.00370440\pi$
$419$	0.476426i	0.0232749i	−0.999932	+	0.0116375i	$0.996296\pi$
$420$	0	0
$421$	4.92425i	0.239993i	0.992774	+	0.119997i	$0.0382884\pi$
$421$	4.92425i	0.239993i	−0.992774	+	0.119997i	$0.961712\pi$
$422$	0	0
$423$	−44.7022	−2.17350
$424$	0	0
$425$	−16.6815	−0.809171
$426$	0	0
$427$	− 6.44954i	− 0.312115i
$428$	0	0
$429$	− 14.3047i	− 0.690638i
$430$	0	0
$431$	−17.7040	−0.852770	−0.426385	−	0.904542i	$-0.640213\pi$
$431$	−17.7040	−0.852770	−0.426385	+	0.904542i	$0.640213\pi$
$432$	0	0
$433$	−13.1961	−0.634165	−0.317083	−	0.948398i	$-0.602703\pi$
$433$	−13.1961	−0.634165	−0.317083	+	0.948398i	$0.602703\pi$
$434$	0	0
$435$	− 21.2891i	− 1.02073i
$436$	0	0
$437$	18.3500i	0.877800i
$438$	0	0
$439$	18.7265	0.893767	0.446884	−	0.894592i	$-0.352534\pi$
$439$	18.7265	0.893767	0.446884	+	0.894592i	$0.352534\pi$
$440$	0	0
$441$	−7.03512	−0.335006
$442$	0	0
$443$	− 20.1309i	− 0.956450i	−0.878237	−	0.478225i	$-0.841280\pi$
$443$	− 20.1309i	− 0.956450i	0.878237	−	0.478225i	$-0.158720\pi$
$444$	0	0
$445$	− 5.67868i	− 0.269195i
$446$	0	0
$447$	−7.78173	−0.368063
$448$	0	0
$449$	−10.7517	−0.507403	−0.253702	−	0.967283i	$-0.581648\pi$
$449$	−10.7517	−0.507403	−0.253702	+	0.967283i	$0.581648\pi$
$450$	0	0
$451$	− 22.2873i	− 1.04947i
$452$	0	0
$453$	− 22.2713i	− 1.04640i
$454$	0	0
$455$	−1.11795	−0.0524101
$456$	0	0
$457$	3.24345	0.151722	0.0758610	−	0.997118i	$-0.475829\pi$
$457$	3.24345	0.151722	0.0758610	+	0.997118i	$0.475829\pi$
$458$	0	0
$459$	− 56.8588i	− 2.65394i
$460$	0	0
$461$	27.3854i	1.27546i	0.770258	+	0.637732i	$0.220127\pi$
$461$	27.3854i	1.27546i	−0.770258	+	0.637732i	$0.779873\pi$
$462$	0	0
$463$	−41.5405	−1.93055	−0.965276	−	0.261233i	$-0.915871\pi$
$463$	−41.5405	−1.93055	−0.965276	+	0.261233i	$0.915871\pi$
$464$	0	0
$465$	37.1153	1.72118
$466$	0	0
$467$	22.2758i	1.03080i	0.856949	+	0.515400i	$0.172357\pi$
$467$	22.2758i	1.03080i	−0.856949	+	0.515400i	$0.827643\pi$
$468$	0	0
$469$	5.98352i	0.276293i
$470$	0	0
$471$	−12.4720	−0.574678
$472$	0	0
$473$	−22.0603	−1.01433
$474$	0	0
$475$	25.8402i	1.18563i
$476$	0	0
$477$	− 42.4131i	− 1.94196i
$478$	0	0
$479$	−33.6848	−1.53910	−0.769549	−	0.638588i	$-0.779519\pi$
$479$	−33.6848	−1.53910	−0.769549	+	0.638588i	$0.779519\pi$
$480$	0	0
$481$	−2.80922	−0.128089
$482$	0	0
$483$	− 8.43637i	− 0.383868i
$484$	0	0
$485$	19.4736i	0.884251i
$486$	0	0
$487$	−8.82614	−0.399951	−0.199975	−	0.979801i	$-0.564086\pi$
$487$	−8.82614	−0.399951	−0.199975	+	0.979801i	$0.564086\pi$
$488$	0	0
$489$	43.4156	1.96332
$490$	0	0
$491$	− 8.33126i	− 0.375984i	−0.982171	−	0.187992i	$-0.939802\pi$
$491$	− 8.33126i	− 0.375984i	0.982171	−	0.187992i	$-0.0601980\pi$
$492$	0	0
$493$	− 26.7396i	− 1.20429i
$494$	0	0
$495$	−35.5149	−1.59627
$496$	0	0
$497$	−0.527201	−0.0236482
$498$	0	0
$499$	24.1391i	1.08061i	0.841468	+	0.540306i	$0.181692\pi$
$499$	24.1391i	1.08061i	−0.841468	+	0.540306i	$0.818308\pi$
$500$	0	0
$501$	− 32.2611i	− 1.44132i
$502$	0	0
$503$	−23.5917	−1.05190	−0.525952	−	0.850514i	$-0.676291\pi$
$503$	−23.5917	−1.05190	−0.525952	+	0.850514i	$0.676291\pi$
$504$	0	0
$505$	−13.5201	−0.601638
$506$	0	0
$507$	3.16783i	0.140688i
$508$	0	0
$509$	− 28.5768i	− 1.26664i	−0.773889	−	0.633322i	$-0.781691\pi$
$509$	− 28.5768i	− 1.26664i	0.773889	−	0.633322i	$-0.218309\pi$
$510$	0	0
$511$	−7.57660	−0.335169
$512$	0	0
$513$	−88.0764	−3.88867
$514$	0	0
$515$	− 11.4977i	− 0.506650i
$516$	0	0
$517$	28.6930i	1.26192i
$518$	0	0
$519$	19.6343	0.861848
$520$	0	0
$521$	−13.2632	−0.581070	−0.290535	−	0.956864i	$-0.593833\pi$
$521$	−13.2632	−0.581070	−0.290535	+	0.956864i	$0.593833\pi$
$522$	0	0
$523$	21.7999i	0.953243i	0.879109	+	0.476621i	$0.158139\pi$
$523$	21.7999i	0.953243i	−0.879109	+	0.476621i	$0.841861\pi$
$524$	0	0
$525$	− 11.8800i	− 0.518484i
$526$	0	0
$527$	46.6178	2.03070
$528$	0	0
$529$	−15.9077	−0.691638
$530$	0	0
$531$	− 106.711i	− 4.63087i
$532$	0	0
$533$	4.93559i	0.213784i
$534$	0	0
$535$	−3.53727	−0.152930
$536$	0	0
$537$	36.3736	1.56964
$538$	0	0
$539$	4.51562i	0.194502i
$540$	0	0
$541$	35.2967i	1.51753i	0.651367	+	0.758763i	$0.274196\pi$
$541$	35.2967i	1.51753i	−0.651367	+	0.758763i	$0.725804\pi$
$542$	0	0
$543$	−83.5707	−3.58636
$544$	0	0
$545$	10.6030	0.454181
$546$	0	0
$547$	25.6660i	1.09740i	0.836020	+	0.548699i	$0.184877\pi$
$547$	25.6660i	1.09740i	−0.836020	+	0.548699i	$0.815123\pi$
$548$	0	0
$549$	45.3733i	1.93648i
$550$	0	0
$551$	−41.4206	−1.76458
$552$	0	0
$553$	−4.85576	−0.206488
$554$	0	0
$555$	9.94874i	0.422301i
$556$	0	0
$557$	− 23.9095i	− 1.01308i	−0.862218	−	0.506538i	$-0.830925\pi$
$557$	− 23.9095i	− 1.01308i	0.862218	−	0.506538i	$-0.169075\pi$
$558$	0	0
$559$	4.88533	0.206628
$560$	0	0
$561$	−63.6296	−2.68645
$562$	0	0
$563$	4.23760i	0.178594i	0.996005	+	0.0892969i	$0.0284620\pi$
$563$	4.23760i	0.178594i	−0.996005	+	0.0892969i	$0.971538\pi$
$564$	0	0
$565$	11.4818i	0.483044i
$566$	0	0
$567$	19.3875	0.814199
$568$	0	0
$569$	14.1681	0.593958	0.296979	−	0.954884i	$-0.404021\pi$
$569$	14.1681	0.593958	0.296979	+	0.954884i	$0.404021\pi$
$570$	0	0
$571$	− 27.7699i	− 1.16213i	−0.813856	−	0.581066i	$-0.802636\pi$
$571$	− 27.7699i	− 1.16213i	0.813856	−	0.581066i	$-0.197364\pi$
$572$	0	0
$573$	27.9120i	1.16604i
$574$	0	0
$575$	9.98730	0.416499
$576$	0	0
$577$	10.7261	0.446534	0.223267	−	0.974757i	$-0.428328\pi$
$577$	10.7261	0.446534	0.223267	+	0.974757i	$0.428328\pi$
$578$	0	0
$579$	1.12128i	0.0465986i
$580$	0	0
$581$	2.98647i	0.123900i
$582$	0	0
$583$	−27.2237	−1.12749
$584$	0	0
$585$	7.86489	0.325173
$586$	0	0
$587$	38.0092i	1.56881i	0.620251	+	0.784403i	$0.287031\pi$
$587$	38.0092i	1.56881i	−0.620251	+	0.784403i	$0.712969\pi$
$588$	0	0
$589$	− 72.2125i	− 2.97547i
$590$	0	0
$591$	16.9151	0.695796
$592$	0	0
$593$	16.4663	0.676190	0.338095	−	0.941112i	$-0.390217\pi$
$593$	16.4663	0.676190	0.338095	+	0.941112i	$0.390217\pi$
$594$	0	0
$595$	4.97281i	0.203865i
$596$	0	0
$597$	42.8395i	1.75330i
$598$	0	0
$599$	−28.8517	−1.17885	−0.589425	−	0.807823i	$-0.700646\pi$
$599$	−28.8517	−1.17885	−0.589425	+	0.807823i	$0.700646\pi$
$600$	0	0
$601$	6.29534	0.256792	0.128396	−	0.991723i	$-0.459017\pi$
$601$	6.29534	0.256792	0.128396	+	0.991723i	$0.459017\pi$
$602$	0	0
$603$	− 42.0948i	− 1.71423i
$604$	0	0
$605$	10.4985i	0.426823i
$606$	0	0
$607$	−15.7914	−0.640954	−0.320477	−	0.947256i	$-0.603843\pi$
$607$	−15.7914	−0.640954	−0.320477	+	0.947256i	$0.603843\pi$
$608$	0	0
$609$	19.0430	0.771662
$610$	0	0
$611$	− 6.35415i	− 0.257062i
$612$	0	0
$613$	44.8229i	1.81038i	0.425008	+	0.905189i	$0.360271\pi$
$613$	44.8229i	1.81038i	−0.425008	+	0.905189i	$0.639729\pi$
$614$	0	0
$615$	17.4792	0.704829
$616$	0	0
$617$	41.5256	1.67176	0.835879	−	0.548913i	$-0.184958\pi$
$617$	41.5256	1.67176	0.835879	+	0.548913i	$0.184958\pi$
$618$	0	0
$619$	40.6374i	1.63335i	0.577095	+	0.816677i	$0.304186\pi$
$619$	40.6374i	1.63335i	−0.577095	+	0.816677i	$0.695814\pi$
$620$	0	0
$621$	34.0417i	1.36605i
$622$	0	0
$623$	5.07956	0.203508
$624$	0	0
$625$	7.81494	0.312598
$626$	0	0
$627$	98.5646i	3.93629i
$628$	0	0
$629$	12.4959i	0.498243i
$630$	0	0
$631$	−19.3579	−0.770624	−0.385312	−	0.922786i	$-0.625906\pi$
$631$	−19.3579	−0.770624	−0.385312	+	0.922786i	$0.625906\pi$
$632$	0	0
$633$	−22.2679	−0.885069
$634$	0	0
$635$	24.3894i	0.967865i
$636$	0	0
$637$	− 1.00000i	− 0.0396214i
$638$	0	0
$639$	3.70892	0.146723
$640$	0	0
$641$	23.1842	0.915719	0.457860	−	0.889024i	$-0.348616\pi$
$641$	23.1842	0.915719	0.457860	+	0.889024i	$0.348616\pi$
$642$	0	0
$643$	− 36.1071i	− 1.42393i	−0.702217	−	0.711963i	$-0.747806\pi$
$643$	− 36.1071i	− 1.42393i	0.702217	−	0.711963i	$-0.252194\pi$
$644$	0	0
$645$	− 17.3012i	− 0.681235i
$646$	0	0
$647$	−23.0515	−0.906248	−0.453124	−	0.891447i	$-0.649691\pi$
$647$	−23.0515	−0.906248	−0.453124	+	0.891447i	$0.649691\pi$
$648$	0	0
$649$	−68.4945	−2.68864
$650$	0	0
$651$	33.1995i	1.30119i
$652$	0	0
$653$	10.6277i	0.415893i	0.978140	+	0.207946i	$0.0666780\pi$
$653$	10.6277i	0.415893i	−0.978140	+	0.207946i	$0.933322\pi$
$654$	0	0
$655$	−17.5622	−0.686213
$656$	0	0
$657$	53.3023	2.07952
$658$	0	0
$659$	31.8733i	1.24161i	0.783965	+	0.620804i	$0.213194\pi$
$659$	31.8733i	1.24161i	−0.783965	+	0.620804i	$0.786806\pi$
$660$	0	0
$661$	− 32.4525i	− 1.26225i	−0.775679	−	0.631127i	$-0.782593\pi$
$661$	− 32.4525i	− 1.26225i	0.775679	−	0.631127i	$-0.217407\pi$
$662$	0	0
$663$	14.0910	0.547249
$664$	0	0
$665$	7.70306	0.298712
$666$	0	0
$667$	16.0092i	0.619878i
$668$	0	0
$669$	− 48.1321i	− 1.86089i
$670$	0	0
$671$	29.1237	1.12431
$672$	0	0
$673$	−26.9401	−1.03846	−0.519232	−	0.854633i	$-0.673782\pi$
$673$	−26.9401	−1.03846	−0.519232	+	0.854633i	$0.673782\pi$
$674$	0	0
$675$	47.9370i	1.84510i
$676$	0	0
$677$	27.8214i	1.06926i	0.845085	+	0.534632i	$0.179550\pi$
$677$	27.8214i	1.06926i	−0.845085	+	0.534632i	$0.820450\pi$
$678$	0	0
$679$	−17.4191	−0.668483
$680$	0	0
$681$	9.77780	0.374686
$682$	0	0
$683$	− 36.9656i	− 1.41445i	−0.706990	−	0.707224i	$-0.749947\pi$
$683$	− 36.9656i	− 1.41445i	0.706990	−	0.707224i	$-0.250053\pi$
$684$	0	0
$685$	− 0.199235i	− 0.00761240i
$686$	0	0
$687$	25.6854	0.979958
$688$	0	0
$689$	6.02877	0.229678
$690$	0	0
$691$	23.6432i	0.899428i	0.893173	+	0.449714i	$0.148474\pi$
$691$	23.6432i	0.899428i	−0.893173	+	0.449714i	$0.851526\pi$
$692$	0	0
$693$	− 31.7679i	− 1.20676i
$694$	0	0
$695$	−13.4572	−0.510462
$696$	0	0
$697$	21.9543	0.831579
$698$	0	0
$699$	− 37.0659i	− 1.40196i
$700$	0	0
$701$	− 7.24572i	− 0.273667i	−0.990594	−	0.136833i	$-0.956307\pi$
$701$	− 7.24572i	− 0.273667i	0.990594	−	0.136833i	$-0.0436925\pi$
$702$	0	0
$703$	19.3565	0.730046
$704$	0	0
$705$	−22.5030	−0.847511
$706$	0	0
$707$	− 12.0937i	− 0.454831i
$708$	0	0
$709$	− 6.77224i	− 0.254337i	−0.991881	−	0.127168i	$-0.959411\pi$
$709$	− 6.77224i	− 0.254337i	0.991881	−	0.127168i	$-0.0405889\pi$
$710$	0	0
$711$	34.1608	1.28113
$712$	0	0
$713$	−27.9103	−1.04525
$714$	0	0
$715$	− 5.04822i	− 0.188793i
$716$	0	0
$717$	− 82.8566i	− 3.09434i
$718$	0	0
$719$	49.7201	1.85425	0.927124	−	0.374756i	$-0.122273\pi$
$719$	49.7201	1.85425	0.927124	+	0.374756i	$0.122273\pi$
$720$	0	0
$721$	10.2847	0.383021
$722$	0	0
$723$	42.2397i	1.57091i
$724$	0	0
$725$	22.5439i	0.837259i
$726$	0	0
$727$	12.0287	0.446121	0.223060	−	0.974805i	$-0.428395\pi$
$727$	12.0287	0.446121	0.223060	+	0.974805i	$0.428395\pi$
$728$	0	0
$729$	−14.9149	−0.552402
$730$	0	0
$731$	− 21.7308i	− 0.803741i
$732$	0	0
$733$	− 31.1191i	− 1.14941i	−0.818360	−	0.574706i	$-0.805117\pi$
$733$	− 31.1191i	− 1.14941i	0.818360	−	0.574706i	$-0.194883\pi$
$734$	0	0
$735$	−3.54146	−0.130629
$736$	0	0
$737$	−27.0193	−0.995270
$738$	0	0
$739$	− 44.6644i	− 1.64301i	−0.570204	−	0.821503i	$-0.693136\pi$
$739$	− 44.6644i	− 1.64301i	0.570204	−	0.821503i	$-0.306864\pi$
$740$	0	0
$741$	− 21.8275i	− 0.801852i
$742$	0	0
$743$	17.5023	0.642099	0.321049	−	0.947062i	$-0.395964\pi$
$743$	17.5023	0.642099	0.321049	+	0.947062i	$0.395964\pi$
$744$	0	0
$745$	−2.74622	−0.100614
$746$	0	0
$747$	− 21.0102i	− 0.768722i
$748$	0	0
$749$	− 3.16408i	− 0.115613i
$750$	0	0
$751$	26.6034	0.970774	0.485387	−	0.874300i	$-0.338679\pi$
$751$	26.6034	0.970774	0.485387	+	0.874300i	$0.338679\pi$
$752$	0	0
$753$	47.6732	1.73731
$754$	0	0
$755$	− 7.85971i	− 0.286044i
$756$	0	0
$757$	28.6275i	1.04048i	0.854019	+	0.520242i	$0.174158\pi$
$757$	28.6275i	1.04048i	−0.854019	+	0.520242i	$0.825842\pi$
$758$	0	0
$759$	38.0954	1.38278
$760$	0	0
$761$	23.5126	0.852332	0.426166	−	0.904645i	$-0.359864\pi$
$761$	23.5126	0.852332	0.426166	+	0.904645i	$0.359864\pi$
$762$	0	0
$763$	9.48432i	0.343355i
$764$	0	0
$765$	− 34.9843i	− 1.26486i
$766$	0	0
$767$	15.1683	0.547697
$768$	0	0
$769$	1.86478	0.0672456	0.0336228	−	0.999435i	$-0.489296\pi$
$769$	1.86478	0.0672456	0.0336228	+	0.999435i	$0.489296\pi$
$770$	0	0
$771$	− 4.30959i	− 0.155206i
$772$	0	0
$773$	30.6421i	1.10212i	0.834466	+	0.551060i	$0.185776\pi$
$773$	30.6421i	1.10212i	−0.834466	+	0.551060i	$0.814224\pi$
$774$	0	0
$775$	−39.3029	−1.41180
$776$	0	0
$777$	−8.89912	−0.319254
$778$	0	0
$779$	− 34.0080i	− 1.21846i
$780$	0	0
$781$	− 2.38064i	− 0.0851860i
$782$	0	0
$783$	−76.8408	−2.74607
$784$	0	0
$785$	−4.40144	−0.157094
$786$	0	0
$787$	− 52.8326i	− 1.88328i	−0.336622	−	0.941640i	$-0.609285\pi$
$787$	− 52.8326i	− 1.88328i	0.336622	−	0.941640i	$-0.390715\pi$
$788$	0	0
$789$	− 35.2100i	− 1.25351i
$790$	0	0
$791$	−10.2704	−0.365175
$792$	0	0
$793$	−6.44954	−0.229030
$794$	0	0
$795$	− 21.3507i	− 0.757230i
$796$	0	0
$797$	1.45717i	0.0516158i	0.999667	+	0.0258079i	$0.00821582\pi$
$797$	1.45717i	0.0516158i	−0.999667	+	0.0258079i	$0.991784\pi$
$798$	0	0
$799$	−28.2643	−0.999920
$800$	0	0
$801$	−35.7353	−1.26264
$802$	0	0
$803$	− 34.2131i	− 1.20735i
$804$	0	0
$805$	− 2.97725i	− 0.104934i
$806$	0	0
$807$	22.7602	0.801196
$808$	0	0
$809$	12.8081	0.450308	0.225154	−	0.974323i	$-0.427712\pi$
$809$	12.8081	0.450308	0.225154	+	0.974323i	$0.427712\pi$
$810$	0	0
$811$	− 25.1200i	− 0.882083i	−0.897487	−	0.441042i	$-0.854609\pi$
$811$	− 25.1200i	− 0.882083i	0.897487	−	0.441042i	$-0.145391\pi$
$812$	0	0
$813$	83.5185i	2.92912i
$814$	0	0
$815$	15.3217	0.536694
$816$	0	0
$817$	−33.6617	−1.17767
$818$	0	0
$819$	7.03512i	0.245827i
$820$	0	0
$821$	− 38.6333i	− 1.34831i	−0.738590	−	0.674155i	$-0.764508\pi$
$821$	− 38.6333i	− 1.34831i	0.738590	−	0.674155i	$-0.235492\pi$
$822$	0	0
$823$	−23.7386	−0.827476	−0.413738	−	0.910396i	$-0.635777\pi$
$823$	−23.7386	−0.827476	−0.413738	+	0.910396i	$0.635777\pi$
$824$	0	0
$825$	53.6454	1.86769
$826$	0	0
$827$	− 7.45758i	− 0.259325i	−0.991558	−	0.129663i	$-0.958611\pi$
$827$	− 7.45758i	− 0.259325i	0.991558	−	0.129663i	$-0.0413895\pi$
$828$	0	0
$829$	− 13.2732i	− 0.460998i	−0.973073	−	0.230499i	$-0.925964\pi$
$829$	− 13.2732i	− 0.460998i	0.973073	−	0.230499i	$-0.0740358\pi$
$830$	0	0
$831$	30.4862	1.05756
$832$	0	0
$833$	−4.44816	−0.154120
$834$	0	0
$835$	− 11.3852i	− 0.394000i
$836$	0	0
$837$	− 133.964i	− 4.63047i
$838$	0	0
$839$	33.6190	1.16066	0.580328	−	0.814383i	$-0.302924\pi$
$839$	33.6190	1.16066	0.580328	+	0.814383i	$0.302924\pi$
$840$	0	0
$841$	−7.13678	−0.246096
$842$	0	0
$843$	43.7277i	1.50606i
$844$	0	0
$845$	1.11795i	0.0384585i
$846$	0	0
$847$	−9.39084	−0.322673
$848$	0	0
$849$	−24.5932	−0.844038
$850$	0	0
$851$	− 7.48135i	− 0.256457i
$852$	0	0
$853$	5.30251i	0.181554i	0.995871	+	0.0907772i	$0.0289351\pi$
$853$	5.30251i	0.181554i	−0.995871	+	0.0907772i	$0.971065\pi$
$854$	0	0
$855$	−54.1919	−1.85332
$856$	0	0
$857$	−44.8103	−1.53069	−0.765346	−	0.643619i	$-0.777432\pi$
$857$	−44.8103	−1.53069	−0.765346	+	0.643619i	$0.777432\pi$
$858$	0	0
$859$	− 25.3400i	− 0.864590i	−0.901732	−	0.432295i	$-0.857704\pi$
$859$	− 25.3400i	− 0.864590i	0.901732	−	0.432295i	$-0.142296\pi$
$860$	0	0
$861$	15.6351i	0.532843i
$862$	0	0
$863$	22.2518	0.757462	0.378731	−	0.925507i	$-0.376361\pi$
$863$	22.2518	0.757462	0.378731	+	0.925507i	$0.376361\pi$
$864$	0	0
$865$	6.92906	0.235595
$866$	0	0
$867$	− 8.82604i	− 0.299748i
$868$	0	0
$869$	− 21.9268i	− 0.743815i
$870$	0	0
$871$	5.98352	0.202744
$872$	0	0
$873$	122.545	4.14753
$874$	0	0
$875$	− 9.78225i	− 0.330700i
$876$	0	0
$877$	16.4120i	0.554194i	0.960842	+	0.277097i	$0.0893723\pi$
$877$	16.4120i	0.554194i	−0.960842	+	0.277097i	$0.910628\pi$
$878$	0	0
$879$	74.1967	2.50259
$880$	0	0
$881$	−13.3626	−0.450197	−0.225098	−	0.974336i	$-0.572270\pi$
$881$	−13.3626	−0.450197	−0.225098	+	0.974336i	$0.572270\pi$
$882$	0	0
$883$	49.7871i	1.67547i	0.546077	+	0.837735i	$0.316120\pi$
$883$	49.7871i	1.67547i	−0.546077	+	0.837735i	$0.683880\pi$
$884$	0	0
$885$	− 53.7181i	− 1.80571i
$886$	0	0
$887$	4.91098	0.164895	0.0824473	−	0.996595i	$-0.473726\pi$
$887$	4.91098	0.164895	0.0824473	+	0.996595i	$0.473726\pi$
$888$	0	0
$889$	−21.8163	−0.731694
$890$	0	0
$891$	87.5467i	2.93293i
$892$	0	0
$893$	43.7824i	1.46512i
$894$	0	0
$895$	12.8365	0.429076
$896$	0	0
$897$	−8.43637	−0.281682
$898$	0	0
$899$	− 63.0007i	− 2.10119i
$900$	0	0
$901$	− 26.8170i	− 0.893403i
$902$	0	0
$903$	15.4759	0.515005
$904$	0	0
$905$	−29.4927	−0.980370
$906$	0	0
$907$	− 14.1867i	− 0.471062i	−0.971867	−	0.235531i	$-0.924317\pi$
$907$	− 14.1867i	− 0.471062i	0.971867	−	0.235531i	$-0.0756829\pi$
$908$	0	0
$909$	85.0808i	2.82195i
$910$	0	0
$911$	5.49425	0.182033	0.0910163	−	0.995849i	$-0.470988\pi$
$911$	5.49425	0.182033	0.0910163	+	0.995849i	$0.470988\pi$
$912$	0	0
$913$	−13.4858	−0.446314
$914$	0	0
$915$	22.8408i	0.755093i
$916$	0	0
$917$	− 15.7094i	− 0.518768i
$918$	0	0
$919$	−53.9477	−1.77957	−0.889785	−	0.456379i	$-0.849146\pi$
$919$	−53.9477	−1.77957	−0.889785	+	0.456379i	$0.849146\pi$
$920$	0	0
$921$	−22.5679	−0.743638
$922$	0	0
$923$	0.527201i	0.0173530i
$924$	0	0
$925$	− 10.5351i	− 0.346393i
$926$	0	0
$927$	−72.3539	−2.37641
$928$	0	0
$929$	−4.12817	−0.135441	−0.0677204	−	0.997704i	$-0.521573\pi$
$929$	−4.12817	−0.135441	−0.0677204	+	0.997704i	$0.521573\pi$
$930$	0	0
$931$	6.89036i	0.225822i
$932$	0	0
$933$	− 51.9259i	− 1.69998i
$934$	0	0
$935$	−22.4553	−0.734368
$936$	0	0
$937$	6.95266	0.227134	0.113567	−	0.993530i	$-0.463772\pi$
$937$	6.95266	0.227134	0.113567	+	0.993530i	$0.463772\pi$
$938$	0	0
$939$	− 46.1245i	− 1.50522i
$940$	0	0
$941$	20.5079i	0.668539i	0.942478	+	0.334270i	$0.108490\pi$
$941$	20.5079i	0.668539i	−0.942478	+	0.334270i	$0.891510\pi$
$942$	0	0
$943$	−13.1442	−0.428033
$944$	0	0
$945$	14.2902	0.464860
$946$	0	0
$947$	20.6521i	0.671102i	0.942022	+	0.335551i	$0.108923\pi$
$947$	20.6521i	0.671102i	−0.942022	+	0.335551i	$0.891077\pi$
$948$	0	0
$949$	7.57660i	0.245947i
$950$	0	0
$951$	−77.6769	−2.51885
$952$	0	0
$953$	61.2746	1.98488	0.992440	−	0.122730i	$-0.0391649\pi$
$953$	61.2746	1.98488	0.992440	+	0.122730i	$0.0391649\pi$
$954$	0	0
$955$	9.85035i	0.318750i
$956$	0	0
$957$	85.9911i	2.77970i
$958$	0	0
$959$	0.178216	0.00575488
$960$	0	0
$961$	78.8351	2.54307
$962$	0	0
$963$	22.2597i	0.717308i
$964$	0	0
$965$	0.395705i	0.0127382i
$966$	0	0
$967$	−19.5810	−0.629683	−0.314842	−	0.949144i	$-0.601951\pi$
$967$	−19.5810	−0.629683	−0.314842	+	0.949144i	$0.601951\pi$
$968$	0	0
$969$	−97.0921	−3.11905
$970$	0	0
$971$	38.5437i	1.23693i	0.785814	+	0.618463i	$0.212244\pi$
$971$	38.5437i	1.23693i	−0.785814	+	0.618463i	$0.787756\pi$
$972$	0	0
$973$	− 12.0375i	− 0.385903i
$974$	0	0
$975$	−11.8800	−0.380463
$976$	0	0
$977$	27.8078	0.889651	0.444825	−	0.895617i	$-0.353266\pi$
$977$	27.8078	0.889651	0.444825	+	0.895617i	$0.353266\pi$
$978$	0	0
$979$	22.9374i	0.733082i
$980$	0	0
$981$	− 66.7233i	− 2.13031i
$982$	0	0
$983$	−3.36489	−0.107323	−0.0536616	−	0.998559i	$-0.517089\pi$
$983$	−3.36489	−0.107323	−0.0536616	+	0.998559i	$0.517089\pi$
$984$	0	0
$985$	5.96947	0.190203
$986$	0	0
$987$	− 20.1289i	− 0.640708i
$988$	0	0
$989$	13.0103i	0.413705i
$990$	0	0
$991$	−22.7862	−0.723826	−0.361913	−	0.932212i	$-0.617876\pi$
$991$	−22.7862	−0.723826	−0.361913	+	0.932212i	$0.617876\pi$
$992$	0	0
$993$	3.65239	0.115905
$994$	0	0
$995$	15.1183i	0.479284i
$996$	0	0
$997$	− 41.0678i	− 1.30063i	−0.759664	−	0.650315i	$-0.774637\pi$
$997$	− 41.0678i	− 1.30063i	0.759664	−	0.650315i	$-0.225363\pi$
$998$	0	0
$999$	35.9090	1.13611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.c.a.1457.1		34
4.3	odd	2		728.2.c.a.365.11	✓	34
8.3	odd	2		728.2.c.a.365.12	yes	34
8.5	even	2	inner	2912.2.c.a.1457.34		34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.c.a.365.11	✓	34	4.3	odd	2
728.2.c.a.365.12	yes	34	8.3	odd	2
2912.2.c.a.1457.1		34	1.1	even	1	trivial
2912.2.c.a.1457.34		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)

\(f(q)\)	\(=\)	\(q-3.16783i q^{3} -1.11795i q^{5} +1.00000 q^{7} -7.03512 q^{9} +4.51562i q^{11} -1.00000i q^{13} -3.54146 q^{15} -4.44816 q^{17} +6.89036i q^{19} -3.16783i q^{21} +2.66314 q^{23} +3.75020 q^{25} +12.7825i q^{27} +6.01139i q^{29} -10.4802 q^{31} +14.3047 q^{33} -1.11795i q^{35} -2.80922i q^{37} -3.16783 q^{39} -4.93559 q^{41} +4.88533i q^{43} +7.86489i q^{45} +6.35415 q^{47} +1.00000 q^{49} +14.0910i q^{51} +6.02877i q^{53} +5.04822 q^{55} +21.8275 q^{57} +15.1683i q^{59} -6.44954i q^{61} -7.03512 q^{63} -1.11795 q^{65} +5.98352i q^{67} -8.43637i q^{69} -0.527201 q^{71} -7.57660 q^{73} -11.8800i q^{75} +4.51562i q^{77} -4.85576 q^{79} +19.3875 q^{81} +2.98647i q^{83} +4.97281i q^{85} +19.0430 q^{87} +5.07956 q^{89} -1.00000i q^{91} +33.1995i q^{93} +7.70306 q^{95} -17.4191 q^{97} -31.7679i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) \) 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

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Groups

Database

Properties

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