LMFDB - Embedded newform 2312.2.b.l.577.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(577,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-24,0,0,0,12,0,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.419904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 9x^{2} + 1 $ x^6 + 6x^4 + 9x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.6
Root			$1.87939i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.l.577.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.22668i q^{3} +3.87939i q^{5} -0.467911i q^{7} -7.41147 q^{9} -1.30541i q^{11} -2.94356 q^{13} -12.5175 q^{15} +3.18479 q^{19} +1.50980 q^{21} +3.17024i q^{23} -10.0496 q^{25} -14.2344i q^{27} -8.80066i q^{29} -0.857097i q^{31} +4.21213 q^{33} +1.81521 q^{35} -3.49020i q^{37} -9.49794i q^{39} +8.12061i q^{41} -7.00774 q^{43} -28.7520i q^{45} -2.41147 q^{47} +6.78106 q^{49} -1.30541 q^{53} +5.06418 q^{55} +10.2763i q^{57} -9.96585 q^{59} +3.35504i q^{61} +3.46791i q^{63} -11.4192i q^{65} -12.7811 q^{67} -10.2294 q^{69} +12.6236i q^{71} -5.10607i q^{73} -32.4270i q^{75} -0.610815 q^{77} +4.06418i q^{79} +23.6955 q^{81} +8.51754 q^{83} +28.3969 q^{87} -4.31315 q^{89} +1.37733i q^{91} +2.76558 q^{93} +12.3550i q^{95} +6.93582i q^{97} +9.67499i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24 q^{9} + 12 q^{13} - 30 q^{15} + 12 q^{19} + 12 q^{21} - 6 q^{25} - 24 q^{33} + 18 q^{35} + 6 q^{43} + 6 q^{47} + 6 q^{49} - 12 q^{53} + 12 q^{55} - 18 q^{59} - 42 q^{67} - 30 q^{69} - 12 q^{77}+ \cdots + 78 q^{93}+O(q^{100}) $ 6 * q - 24 * q^9 + 12 * q^13 - 30 * q^15 + 12 * q^19 + 12 * q^21 - 6 * q^25 - 24 * q^33 + 18 * q^35 + 6 * q^43 + 6 * q^47 + 6 * q^49 - 12 * q^53 + 12 * q^55 - 18 * q^59 - 42 * q^67 - 30 * q^69 - 12 * q^77 + 6 * q^81 + 6 * q^83 + 114 * q^87 + 18 * q^89 + 78 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	3.22668i	1.86293i	0.363837	+	0.931463i	$0.381467\pi$
$3$	3.22668i	1.86293i	−0.363837	+	0.931463i	$0.618533\pi$
$4$	0	0
$5$	3.87939i	1.73491i	0.497512	+	0.867457i	$0.334247\pi$
$5$	3.87939i	1.73491i	−0.497512	+	0.867457i	$0.665753\pi$
$6$	0	0
$7$	− 0.467911i	− 0.176854i	−0.996083	−	0.0884269i	$-0.971816\pi$
$7$	− 0.467911i	− 0.176854i	0.996083	−	0.0884269i	$-0.0281840\pi$
$8$	0	0
$9$	−7.41147	−2.47049
$10$	0	0
$11$	− 1.30541i	− 0.393595i	−0.980444	−	0.196798i	$-0.936946\pi$
$11$	− 1.30541i	− 0.393595i	0.980444	−	0.196798i	$-0.0630542\pi$
$12$	0	0
$13$	−2.94356	−0.816397	−0.408199	−	0.912893i	$-0.633843\pi$
$13$	−2.94356	−0.816397	−0.408199	+	0.912893i	$0.633843\pi$
$14$	0	0
$15$	−12.5175	−3.23202
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.18479	0.730642	0.365321	−	0.930882i	$-0.380959\pi$
$19$	3.18479	0.730642	0.365321	+	0.930882i	$0.380959\pi$
$20$	0	0
$21$	1.50980	0.329465
$22$	0	0
$23$	3.17024i	0.661042i	0.943799	+	0.330521i	$0.107224\pi$
$23$	3.17024i	0.661042i	−0.943799	+	0.330521i	$0.892776\pi$
$24$	0	0
$25$	−10.0496	−2.00993
$26$	0	0
$27$	− 14.2344i	− 2.73942i
$28$	0	0
$29$	− 8.80066i	− 1.63424i	−0.576467	−	0.817121i	$-0.695569\pi$
$29$	− 8.80066i	− 1.63424i	0.576467	−	0.817121i	$-0.304431\pi$
$30$	0	0
$31$	− 0.857097i	− 0.153939i	−0.997033	−	0.0769695i	$-0.975476\pi$
$31$	− 0.857097i	− 0.153939i	0.997033	−	0.0769695i	$-0.0245244\pi$
$32$	0	0
$33$	4.21213	0.733238
$34$	0	0
$35$	1.81521	0.306826
$36$	0	0
$37$	− 3.49020i	− 0.573785i	−0.957963	−	0.286893i	$-0.907378\pi$
$37$	− 3.49020i	− 0.573785i	0.957963	−	0.286893i	$-0.0926223\pi$
$38$	0	0
$39$	− 9.49794i	− 1.52089i
$40$	0	0
$41$	8.12061i	1.26823i	0.773240	+	0.634113i	$0.218635\pi$
$41$	8.12061i	1.26823i	−0.773240	+	0.634113i	$0.781365\pi$
$42$	0	0
$43$	−7.00774	−1.06867	−0.534335	−	0.845273i	$-0.679438\pi$
$43$	−7.00774	−1.06867	−0.534335	+	0.845273i	$0.679438\pi$
$44$	0	0
$45$	− 28.7520i	− 4.28609i
$46$	0	0
$47$	−2.41147	−0.351750	−0.175875	−	0.984413i	$-0.556275\pi$
$47$	−2.41147	−0.351750	−0.175875	+	0.984413i	$0.556275\pi$
$48$	0	0
$49$	6.78106	0.968723
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.30541	−0.179311	−0.0896557	−	0.995973i	$-0.528577\pi$
$53$	−1.30541	−0.179311	−0.0896557	+	0.995973i	$0.528577\pi$
$54$	0	0
$55$	5.06418	0.682854
$56$	0	0
$57$	10.2763i	1.36113i
$58$	0	0
$59$	−9.96585	−1.29744	−0.648722	−	0.761026i	$-0.724696\pi$
$59$	−9.96585	−1.29744	−0.648722	+	0.761026i	$0.724696\pi$
$60$	0	0
$61$	3.35504i	0.429568i	0.976662	+	0.214784i	$0.0689048\pi$
$61$	3.35504i	0.429568i	−0.976662	+	0.214784i	$0.931095\pi$
$62$	0	0
$63$	3.46791i	0.436916i
$64$	0	0
$65$	− 11.4192i	− 1.41638i
$66$	0	0
$67$	−12.7811	−1.56145	−0.780727	−	0.624872i	$-0.785151\pi$
$67$	−12.7811	−1.56145	−0.780727	+	0.624872i	$0.785151\pi$
$68$	0	0
$69$	−10.2294	−1.23147
$70$	0	0
$71$	12.6236i	1.49815i	0.662487	+	0.749073i	$0.269501\pi$
$71$	12.6236i	1.49815i	−0.662487	+	0.749073i	$0.730499\pi$
$72$	0	0
$73$	− 5.10607i	− 0.597620i	−0.954313	−	0.298810i	$-0.903410\pi$
$73$	− 5.10607i	− 0.597620i	0.954313	−	0.298810i	$-0.0965897\pi$
$74$	0	0
$75$	− 32.4270i	− 3.74434i
$76$	0	0
$77$	−0.610815	−0.0696088
$78$	0	0
$79$	4.06418i	0.457256i	0.973514	+	0.228628i	$0.0734239\pi$
$79$	4.06418i	0.457256i	−0.973514	+	0.228628i	$0.926576\pi$
$80$	0	0
$81$	23.6955	2.63284
$82$	0	0
$83$	8.51754	0.934922	0.467461	−	0.884014i	$-0.345169\pi$
$83$	8.51754	0.934922	0.467461	+	0.884014i	$0.345169\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	28.3969	3.04447
$88$	0	0
$89$	−4.31315	−0.457193	−0.228596	−	0.973521i	$-0.573414\pi$
$89$	−4.31315	−0.457193	−0.228596	+	0.973521i	$0.573414\pi$
$90$	0	0
$91$	1.37733i	0.144383i
$92$	0	0
$93$	2.76558	0.286777
$94$	0	0
$95$	12.3550i	1.26760i
$96$	0	0
$97$	6.93582i	0.704226i	0.935957	+	0.352113i	$0.114537\pi$
$97$	6.93582i	0.704226i	−0.935957	+	0.352113i	$0.885463\pi$
$98$	0	0
$99$	9.67499i	0.972373i
$100$	0	0
$101$	13.9094	1.38404	0.692019	−	0.721879i	$-0.256721\pi$
$101$	13.9094	1.38404	0.692019	+	0.721879i	$0.256721\pi$
$102$	0	0
$103$	−13.7219	−1.35206	−0.676031	−	0.736873i	$-0.736302\pi$
$103$	−13.7219	−1.35206	−0.676031	+	0.736873i	$0.736302\pi$
$104$	0	0
$105$	5.85710i	0.571594i
$106$	0	0
$107$	15.9290i	1.53992i	0.638095	+	0.769958i	$0.279723\pi$
$107$	15.9290i	1.53992i	−0.638095	+	0.769958i	$0.720277\pi$
$108$	0	0
$109$	− 1.22163i	− 0.117011i	−0.998287	−	0.0585054i	$-0.981367\pi$
$109$	− 1.22163i	− 0.117011i	0.998287	−	0.0585054i	$-0.0186335\pi$
$110$	0	0
$111$	11.2618	1.06892
$112$	0	0
$113$	− 2.47565i	− 0.232890i	−0.993197	−	0.116445i	$-0.962850\pi$
$113$	− 2.47565i	− 0.232890i	0.993197	−	0.116445i	$-0.0371498\pi$
$114$	0	0
$115$	−12.2986	−1.14685
$116$	0	0
$117$	21.8161	2.01690
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.29591	0.845083
$122$	0	0
$123$	−26.2026	−2.36261
$124$	0	0
$125$	− 19.5895i	− 1.75213i
$126$	0	0
$127$	−2.36184	−0.209580	−0.104790	−	0.994494i	$-0.533417\pi$
$127$	−2.36184	−0.209580	−0.104790	+	0.994494i	$0.533417\pi$
$128$	0	0
$129$	− 22.6117i	− 1.99085i
$130$	0	0
$131$	1.07098i	0.0935724i	0.998905	+	0.0467862i	$0.0148979\pi$
$131$	1.07098i	0.0935724i	−0.998905	+	0.0467862i	$0.985102\pi$
$132$	0	0
$133$	− 1.49020i	− 0.129217i
$134$	0	0
$135$	55.2208	4.75265
$136$	0	0
$137$	19.5749	1.67240	0.836199	−	0.548426i	$-0.184773\pi$
$137$	19.5749	1.67240	0.836199	+	0.548426i	$0.184773\pi$
$138$	0	0
$139$	7.95811i	0.674998i	0.941326	+	0.337499i	$0.109581\pi$
$139$	7.95811i	0.674998i	−0.941326	+	0.337499i	$0.890419\pi$
$140$	0	0
$141$	− 7.78106i	− 0.655283i
$142$	0	0
$143$	3.84255i	0.321330i
$144$	0	0
$145$	34.1411	2.83527
$146$	0	0
$147$	21.8803i	1.80466i
$148$	0	0
$149$	1.65270	0.135395	0.0676974	−	0.997706i	$-0.478435\pi$
$149$	1.65270	0.135395	0.0676974	+	0.997706i	$0.478435\pi$
$150$	0	0
$151$	−15.8280	−1.28806	−0.644032	−	0.764998i	$-0.722740\pi$
$151$	−15.8280	−1.28806	−0.644032	+	0.764998i	$0.722740\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.32501	0.267071
$156$	0	0
$157$	−6.80840	−0.543370	−0.271685	−	0.962386i	$-0.587581\pi$
$157$	−6.80840	−0.543370	−0.271685	+	0.962386i	$0.587581\pi$
$158$	0	0
$159$	− 4.21213i	− 0.334044i
$160$	0	0
$161$	1.48339	0.116908
$162$	0	0
$163$	− 3.81521i	− 0.298830i	−0.988775	−	0.149415i	$-0.952261\pi$
$163$	− 3.81521i	− 0.298830i	0.988775	−	0.149415i	$-0.0477391\pi$
$164$	0	0
$165$	16.3405i	1.27211i
$166$	0	0
$167$	− 11.7023i	− 0.905554i	−0.891624	−	0.452777i	$-0.850433\pi$
$167$	− 11.7023i	− 0.905554i	0.891624	−	0.452777i	$-0.149567\pi$
$168$	0	0
$169$	−4.33544	−0.333495
$170$	0	0
$171$	−23.6040	−1.80504
$172$	0	0
$173$	− 15.1361i	− 1.15078i	−0.817881	−	0.575388i	$-0.804851\pi$
$173$	− 15.1361i	− 1.15078i	0.817881	−	0.575388i	$-0.195149\pi$
$174$	0	0
$175$	4.70233i	0.355463i
$176$	0	0
$177$	− 32.1566i	− 2.41704i
$178$	0	0
$179$	−6.10607	−0.456389	−0.228194	−	0.973616i	$-0.573282\pi$
$179$	−6.10607	−0.456389	−0.228194	+	0.973616i	$0.573282\pi$
$180$	0	0
$181$	7.28581i	0.541550i	0.962643	+	0.270775i	$0.0872799\pi$
$181$	7.28581i	0.541550i	−0.962643	+	0.270775i	$0.912720\pi$
$182$	0	0
$183$	−10.8256	−0.800254
$184$	0	0
$185$	13.5398	0.995468
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−6.66044	−0.484476
$190$	0	0
$191$	−16.5253	−1.19573	−0.597864	−	0.801598i	$-0.703984\pi$
$191$	−16.5253	−1.19573	−0.597864	+	0.801598i	$0.703984\pi$
$192$	0	0
$193$	− 5.65270i	− 0.406891i	−0.979086	−	0.203445i	$-0.934786\pi$
$193$	− 5.65270i	− 0.406891i	0.979086	−	0.203445i	$-0.0652139\pi$
$194$	0	0
$195$	36.8462	2.63861
$196$	0	0
$197$	0.834808i	0.0594776i	0.999558	+	0.0297388i	$0.00946755\pi$
$197$	0.834808i	0.0594776i	−0.999558	+	0.0297388i	$0.990532\pi$
$198$	0	0
$199$	23.0719i	1.63552i	0.575556	+	0.817762i	$0.304786\pi$
$199$	23.0719i	1.63552i	−0.575556	+	0.817762i	$0.695214\pi$
$200$	0	0
$201$	− 41.2404i	− 2.90887i
$202$	0	0
$203$	−4.11793	−0.289022
$204$	0	0
$205$	−31.5030	−2.20026
$206$	0	0
$207$	− 23.4962i	− 1.63310i
$208$	0	0
$209$	− 4.15745i	− 0.287577i
$210$	0	0
$211$	17.8530i	1.22905i	0.788897	+	0.614525i	$0.210652\pi$
$211$	17.8530i	1.22905i	−0.788897	+	0.614525i	$0.789348\pi$
$212$	0	0
$213$	−40.7324	−2.79094
$214$	0	0
$215$	− 27.1857i	− 1.85405i
$216$	0	0
$217$	−0.401045	−0.0272247
$218$	0	0
$219$	16.4757	1.11332
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.6750	−0.982710	−0.491355	−	0.870959i	$-0.663498\pi$
$223$	−14.6750	−0.982710	−0.491355	+	0.870959i	$0.663498\pi$
$224$	0	0
$225$	74.4826	4.96550
$226$	0	0
$227$	− 18.6013i	− 1.23461i	−0.786723	−	0.617306i	$-0.788224\pi$
$227$	− 18.6013i	− 1.23461i	0.786723	−	0.617306i	$-0.211776\pi$
$228$	0	0
$229$	−7.78880	−0.514698	−0.257349	−	0.966318i	$-0.582849\pi$
$229$	−7.78880	−0.514698	−0.257349	+	0.966318i	$0.582849\pi$
$230$	0	0
$231$	− 1.97090i	− 0.129676i
$232$	0	0
$233$	15.4466i	1.01194i	0.862552	+	0.505969i	$0.168865\pi$
$233$	15.4466i	1.01194i	−0.862552	+	0.505969i	$0.831135\pi$
$234$	0	0
$235$	− 9.35504i	− 0.610255i
$236$	0	0
$237$	−13.1138	−0.851833
$238$	0	0
$239$	26.9368	1.74239	0.871197	−	0.490934i	$-0.163344\pi$
$239$	26.9368	1.74239	0.871197	+	0.490934i	$0.163344\pi$
$240$	0	0
$241$	19.7246i	1.27057i	0.772276	+	0.635287i	$0.219118\pi$
$241$	19.7246i	1.27057i	−0.772276	+	0.635287i	$0.780882\pi$
$242$	0	0
$243$	33.7547i	2.16536i
$244$	0	0
$245$	26.3063i	1.68065i
$246$	0	0
$247$	−9.37464	−0.596494
$248$	0	0
$249$	27.4834i	1.74169i
$250$	0	0
$251$	26.9864	1.70337	0.851683	−	0.524058i	$-0.175582\pi$
$251$	26.9864	1.70337	0.851683	+	0.524058i	$0.175582\pi$
$252$	0	0
$253$	4.13846	0.260183
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.2422	−1.76170	−0.880849	−	0.473398i	$-0.843027\pi$
$257$	−28.2422	−1.76170	−0.880849	+	0.473398i	$0.843027\pi$
$258$	0	0
$259$	−1.63310	−0.101476
$260$	0	0
$261$	65.2259i	4.03738i
$262$	0	0
$263$	−5.96585	−0.367870	−0.183935	−	0.982938i	$-0.558884\pi$
$263$	−5.96585	−0.367870	−0.183935	+	0.982938i	$0.558884\pi$
$264$	0	0
$265$	− 5.06418i	− 0.311090i
$266$	0	0
$267$	− 13.9172i	− 0.851716i
$268$	0	0
$269$	− 23.7392i	− 1.44740i	−0.690113	−	0.723701i	$-0.742439\pi$
$269$	− 23.7392i	− 1.44740i	0.690113	−	0.723701i	$-0.257561\pi$
$270$	0	0
$271$	−6.56212	−0.398620	−0.199310	−	0.979936i	$-0.563870\pi$
$271$	−6.56212	−0.398620	−0.199310	+	0.979936i	$0.563870\pi$
$272$	0	0
$273$	−4.44419	−0.268975
$274$	0	0
$275$	13.1189i	0.791097i
$276$	0	0
$277$	− 13.7638i	− 0.826988i	−0.910507	−	0.413494i	$-0.864308\pi$
$277$	− 13.7638i	− 0.826988i	0.910507	−	0.413494i	$-0.135692\pi$
$278$	0	0
$279$	6.35235i	0.380305i
$280$	0	0
$281$	23.0446	1.37472	0.687362	−	0.726315i	$-0.258769\pi$
$281$	23.0446	1.37472	0.687362	+	0.726315i	$0.258769\pi$
$282$	0	0
$283$	− 24.0077i	− 1.42711i	−0.700598	−	0.713556i	$-0.747083\pi$
$283$	− 24.0077i	− 1.42711i	0.700598	−	0.713556i	$-0.252917\pi$
$284$	0	0
$285$	−39.8658	−2.36144
$286$	0	0
$287$	3.79973	0.224291
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−22.3797	−1.31192
$292$	0	0
$293$	−21.2472	−1.24128	−0.620638	−	0.784097i	$-0.713126\pi$
$293$	−21.2472	−1.24128	−0.620638	+	0.784097i	$0.713126\pi$
$294$	0	0
$295$	− 38.6614i	− 2.25095i
$296$	0	0
$297$	−18.5817	−1.07822
$298$	0	0
$299$	− 9.33181i	− 0.539673i
$300$	0	0
$301$	3.27900i	0.188998i
$302$	0	0
$303$	44.8813i	2.57836i
$304$	0	0
$305$	−13.0155	−0.745264
$306$	0	0
$307$	13.3277	0.760652	0.380326	−	0.924853i	$-0.375812\pi$
$307$	13.3277	0.760652	0.380326	+	0.924853i	$0.375812\pi$
$308$	0	0
$309$	− 44.2763i	− 2.51879i
$310$	0	0
$311$	− 10.5895i	− 0.600473i	−0.953865	−	0.300237i	$-0.902934\pi$
$311$	− 10.5895i	− 0.600473i	0.953865	−	0.300237i	$-0.0970656\pi$
$312$	0	0
$313$	− 4.38919i	− 0.248091i	−0.992277	−	0.124046i	$-0.960413\pi$
$313$	− 4.38919i	− 0.248091i	0.992277	−	0.124046i	$-0.0395869\pi$
$314$	0	0
$315$	−13.4534	−0.758011
$316$	0	0
$317$	− 1.36184i	− 0.0764888i	−0.999268	−	0.0382444i	$-0.987823\pi$
$317$	− 1.36184i	− 0.0764888i	0.999268	−	0.0382444i	$-0.0121765\pi$
$318$	0	0
$319$	−11.4884	−0.643229
$320$	0	0
$321$	−51.3979	−2.86875
$322$	0	0
$323$	0	0
$324$	0	0
$325$	29.5817	1.64090
$326$	0	0
$327$	3.94181	0.217982
$328$	0	0
$329$	1.12836i	0.0622083i
$330$	0	0
$331$	−17.9418	−0.986171	−0.493085	−	0.869981i	$-0.664131\pi$
$331$	−17.9418	−0.986171	−0.493085	+	0.869981i	$0.664131\pi$
$332$	0	0
$333$	25.8675i	1.41753i
$334$	0	0
$335$	− 49.5827i	− 2.70899i
$336$	0	0
$337$	32.6091i	1.77633i	0.459526	+	0.888164i	$0.348019\pi$
$337$	32.6091i	1.77633i	−0.459526	+	0.888164i	$0.651981\pi$
$338$	0	0
$339$	7.98814	0.433856
$340$	0	0
$341$	−1.11886	−0.0605897
$342$	0	0
$343$	− 6.44831i	− 0.348176i
$344$	0	0
$345$	− 39.6837i	− 2.13650i
$346$	0	0
$347$	11.3696i	0.610351i	0.952296	+	0.305176i	$0.0987152\pi$
$347$	11.3696i	0.610351i	−0.952296	+	0.305176i	$0.901285\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	41.8999i	2.23645i
$352$	0	0
$353$	−11.1506	−0.593489	−0.296744	−	0.954957i	$-0.595901\pi$
$353$	−11.1506	−0.593489	−0.296744	+	0.954957i	$0.595901\pi$
$354$	0	0
$355$	−48.9718	−2.59916
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.4442	1.02623	0.513113	−	0.858321i	$-0.328492\pi$
$359$	19.4442	1.02623	0.513113	+	0.858321i	$0.328492\pi$
$360$	0	0
$361$	−8.85710	−0.466163
$362$	0	0
$363$	29.9949i	1.57433i
$364$	0	0
$365$	19.8084	1.03682
$366$	0	0
$367$	− 29.4662i	− 1.53812i	−0.639176	−	0.769060i	$-0.720724\pi$
$367$	− 29.4662i	− 1.53812i	0.639176	−	0.769060i	$-0.279276\pi$
$368$	0	0
$369$	− 60.1857i	− 3.13314i
$370$	0	0
$371$	0.610815i	0.0317119i
$372$	0	0
$373$	−31.2891	−1.62009	−0.810044	−	0.586369i	$-0.800557\pi$
$373$	−31.2891	−1.62009	−0.810044	+	0.586369i	$0.800557\pi$
$374$	0	0
$375$	63.2089	3.26410
$376$	0	0
$377$	25.9053i	1.33419i
$378$	0	0
$379$	31.9932i	1.64338i	0.569935	+	0.821690i	$0.306968\pi$
$379$	31.9932i	1.64338i	−0.569935	+	0.821690i	$0.693032\pi$
$380$	0	0
$381$	− 7.62092i	− 0.390432i
$382$	0	0
$383$	4.00505	0.204649	0.102324	−	0.994751i	$-0.467372\pi$
$383$	4.00505	0.204649	0.102324	+	0.994751i	$0.467372\pi$
$384$	0	0
$385$	− 2.36959i	− 0.120765i
$386$	0	0
$387$	51.9377	2.64014
$388$	0	0
$389$	−13.0155	−0.659911	−0.329956	−	0.943996i	$-0.607034\pi$
$389$	−13.0155	−0.659911	−0.329956	+	0.943996i	$0.607034\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3.45573	−0.174318
$394$	0	0
$395$	−15.7665	−0.793299
$396$	0	0
$397$	− 37.0523i	− 1.85960i	−0.368062	−	0.929801i	$-0.619979\pi$
$397$	− 37.0523i	− 1.85960i	0.368062	−	0.929801i	$-0.380021\pi$
$398$	0	0
$399$	4.80840	0.240721
$400$	0	0
$401$	8.84255i	0.441576i	0.975322	+	0.220788i	$0.0708629\pi$
$401$	8.84255i	0.441576i	−0.975322	+	0.220788i	$0.929137\pi$
$402$	0	0
$403$	2.52292i	0.125675i
$404$	0	0
$405$	91.9241i	4.56774i
$406$	0	0
$407$	−4.55613	−0.225839
$408$	0	0
$409$	10.0942	0.499126	0.249563	−	0.968359i	$-0.419713\pi$
$409$	10.0942	0.499126	0.249563	+	0.968359i	$0.419713\pi$
$410$	0	0
$411$	63.1620i	3.11555i
$412$	0	0
$413$	4.66313i	0.229458i
$414$	0	0
$415$	33.0428i	1.62201i
$416$	0	0
$417$	−25.6783	−1.25747
$418$	0	0
$419$	− 19.2226i	− 0.939084i	−0.882910	−	0.469542i	$-0.844419\pi$
$419$	− 19.2226i	− 0.939084i	0.882910	−	0.469542i	$-0.155581\pi$
$420$	0	0
$421$	13.1925	0.642965	0.321482	−	0.946916i	$-0.395819\pi$
$421$	13.1925	0.642965	0.321482	+	0.946916i	$0.395819\pi$
$422$	0	0
$423$	17.8726	0.868994
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.56986	0.0759708
$428$	0	0
$429$	−12.3987	−0.598614
$430$	0	0
$431$	18.8307i	0.907042i	0.891245	+	0.453521i	$0.149832\pi$
$431$	18.8307i	0.907042i	−0.891245	+	0.453521i	$0.850168\pi$
$432$	0	0
$433$	4.19665	0.201678	0.100839	−	0.994903i	$-0.467847\pi$
$433$	4.19665	0.201678	0.100839	+	0.994903i	$0.467847\pi$
$434$	0	0
$435$	110.163i	5.28189i
$436$	0	0
$437$	10.0966i	0.482985i
$438$	0	0
$439$	32.8708i	1.56884i	0.620231	+	0.784419i	$0.287039\pi$
$439$	32.8708i	1.56884i	−0.620231	+	0.784419i	$0.712961\pi$
$440$	0	0
$441$	−50.2576	−2.39322
$442$	0	0
$443$	−13.3979	−0.636552	−0.318276	−	0.947998i	$-0.603104\pi$
$443$	−13.3979	−0.636552	−0.318276	+	0.947998i	$0.603104\pi$
$444$	0	0
$445$	− 16.7324i	− 0.793190i
$446$	0	0
$447$	5.33275i	0.252230i
$448$	0	0
$449$	− 10.3446i	− 0.488192i	−0.969751	−	0.244096i	$-0.921509\pi$
$449$	− 10.3446i	− 0.488192i	0.969751	−	0.244096i	$-0.0784912\pi$
$450$	0	0
$451$	10.6007	0.499168
$452$	0	0
$453$	− 51.0719i	− 2.39957i
$454$	0	0
$455$	−5.34318	−0.250492
$456$	0	0
$457$	20.4929	0.958617	0.479308	−	0.877647i	$-0.340888\pi$
$457$	20.4929	0.958617	0.479308	+	0.877647i	$0.340888\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.9017	−0.600891	−0.300445	−	0.953799i	$-0.597135\pi$
$461$	−12.9017	−0.600891	−0.300445	+	0.953799i	$0.597135\pi$
$462$	0	0
$463$	9.54488	0.443588	0.221794	−	0.975094i	$-0.428809\pi$
$463$	9.54488	0.443588	0.221794	+	0.975094i	$0.428809\pi$
$464$	0	0
$465$	10.7287i	0.497533i
$466$	0	0
$467$	−16.5790	−0.767186	−0.383593	−	0.923502i	$-0.625313\pi$
$467$	−16.5790	−0.767186	−0.383593	+	0.923502i	$0.625313\pi$
$468$	0	0
$469$	5.98040i	0.276149i
$470$	0	0
$471$	− 21.9685i	− 1.01226i
$472$	0	0
$473$	9.14796i	0.420623i
$474$	0	0
$475$	−32.0060	−1.46854
$476$	0	0
$477$	9.67499	0.442987
$478$	0	0
$479$	− 1.13247i	− 0.0517441i	−0.999665	−	0.0258720i	$-0.991764\pi$
$479$	− 1.13247i	− 0.0517441i	0.999665	−	0.0258720i	$-0.00823624\pi$
$480$	0	0
$481$	10.2736i	0.468437i
$482$	0	0
$483$	4.78644i	0.217790i
$484$	0	0
$485$	−26.9067	−1.22177
$486$	0	0
$487$	26.2267i	1.18844i	0.804301	+	0.594222i	$0.202540\pi$
$487$	26.2267i	1.18844i	−0.804301	+	0.594222i	$0.797460\pi$
$488$	0	0
$489$	12.3105	0.556698
$490$	0	0
$491$	−3.40104	−0.153487	−0.0767435	−	0.997051i	$-0.524452\pi$
$491$	−3.40104	−0.153487	−0.0767435	+	0.997051i	$0.524452\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−37.5330	−1.68698
$496$	0	0
$497$	5.90673	0.264953
$498$	0	0
$499$	29.6587i	1.32771i	0.747863	+	0.663853i	$0.231080\pi$
$499$	29.6587i	1.32771i	−0.747863	+	0.663853i	$0.768920\pi$
$500$	0	0
$501$	37.7597	1.68698
$502$	0	0
$503$	12.7861i	0.570105i	0.958512	+	0.285052i	$0.0920110\pi$
$503$	12.7861i	0.570105i	−0.958512	+	0.285052i	$0.907989\pi$
$504$	0	0
$505$	53.9600i	2.40119i
$506$	0	0
$507$	− 13.9891i	− 0.621277i
$508$	0	0
$509$	−29.3337	−1.30019	−0.650096	−	0.759852i	$-0.725271\pi$
$509$	−29.3337	−1.30019	−0.650096	+	0.759852i	$0.725271\pi$
$510$	0	0
$511$	−2.38919	−0.105691
$512$	0	0
$513$	− 45.3337i	− 2.00153i
$514$	0	0
$515$	− 53.2327i	− 2.34571i
$516$	0	0
$517$	3.14796i	0.138447i
$518$	0	0
$519$	48.8394	2.14381
$520$	0	0
$521$	22.8075i	0.999213i	0.866252	+	0.499607i	$0.166522\pi$
$521$	22.8075i	0.999213i	−0.866252	+	0.499607i	$0.833478\pi$
$522$	0	0
$523$	−14.5422	−0.635886	−0.317943	−	0.948110i	$-0.602992\pi$
$523$	−14.5422	−0.635886	−0.317943	+	0.948110i	$0.602992\pi$
$524$	0	0
$525$	−15.1729	−0.662201
$526$	0	0
$527$	0	0
$528$	0	0
$529$	12.9495	0.563024
$530$	0	0
$531$	73.8617	3.20532
$532$	0	0
$533$	− 23.9035i	− 1.03538i
$534$	0	0
$535$	−61.7948	−2.67162
$536$	0	0
$537$	− 19.7023i	− 0.850218i
$538$	0	0
$539$	− 8.85204i	− 0.381285i
$540$	0	0
$541$	− 20.0651i	− 0.862667i	−0.902193	−	0.431333i	$-0.858043\pi$
$541$	− 20.0651i	− 0.862667i	0.902193	−	0.431333i	$-0.141957\pi$
$542$	0	0
$543$	−23.5090	−1.00887
$544$	0	0
$545$	4.73917	0.203004
$546$	0	0
$547$	− 14.1019i	− 0.602956i	−0.953473	−	0.301478i	$-0.902520\pi$
$547$	− 14.1019i	− 0.602956i	0.953473	−	0.301478i	$-0.0974799\pi$
$548$	0	0
$549$	− 24.8658i	− 1.06125i
$550$	0	0
$551$	− 28.0283i	− 1.19404i
$552$	0	0
$553$	1.90167	0.0808674
$554$	0	0
$555$	43.6887i	1.85448i
$556$	0	0
$557$	−27.2344	−1.15396	−0.576980	−	0.816758i	$-0.695769\pi$
$557$	−27.2344	−1.15396	−0.576980	+	0.816758i	$0.695769\pi$
$558$	0	0
$559$	20.6277	0.872460
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.6869	0.492542	0.246271	−	0.969201i	$-0.420795\pi$
$563$	11.6869	0.492542	0.246271	+	0.969201i	$0.420795\pi$
$564$	0	0
$565$	9.60401	0.404044
$566$	0	0
$567$	− 11.0874i	− 0.465627i
$568$	0	0
$569$	10.7297	0.449811	0.224906	−	0.974381i	$-0.427793\pi$
$569$	10.7297	0.449811	0.224906	+	0.974381i	$0.427793\pi$
$570$	0	0
$571$	− 8.86484i	− 0.370982i	−0.982646	−	0.185491i	$-0.940612\pi$
$571$	− 8.86484i	− 0.370982i	0.982646	−	0.185491i	$-0.0593875\pi$
$572$	0	0
$573$	− 53.3218i	− 2.22755i
$574$	0	0
$575$	− 31.8598i	− 1.32864i
$576$	0	0
$577$	28.8307	1.20024	0.600119	−	0.799911i	$-0.295120\pi$
$577$	28.8307	1.20024	0.600119	+	0.799911i	$0.295120\pi$
$578$	0	0
$579$	18.2395	0.758007
$580$	0	0
$581$	− 3.98545i	− 0.165344i
$582$	0	0
$583$	1.70409i	0.0705761i
$584$	0	0
$585$	84.6332i	3.49915i
$586$	0	0
$587$	4.58584	0.189278	0.0946389	−	0.995512i	$-0.469830\pi$
$587$	4.58584	0.189278	0.0946389	+	0.995512i	$0.469830\pi$
$588$	0	0
$589$	− 2.72967i	− 0.112474i
$590$	0	0
$591$	−2.69366	−0.110802
$592$	0	0
$593$	21.8108	0.895661	0.447830	−	0.894119i	$-0.352197\pi$
$593$	21.8108	0.895661	0.447830	+	0.894119i	$0.352197\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−74.4457	−3.04686
$598$	0	0
$599$	−5.33544	−0.218000	−0.109000	−	0.994042i	$-0.534765\pi$
$599$	−5.33544	−0.218000	−0.109000	+	0.994042i	$0.534765\pi$
$600$	0	0
$601$	16.7273i	0.682321i	0.940005	+	0.341161i	$0.110820\pi$
$601$	16.7273i	0.682321i	−0.940005	+	0.341161i	$0.889180\pi$
$602$	0	0
$603$	94.7265	3.85756
$604$	0	0
$605$	36.0624i	1.46615i
$606$	0	0
$607$	− 30.8794i	− 1.25336i	−0.779278	−	0.626678i	$-0.784414\pi$
$607$	− 30.8794i	− 1.25336i	0.779278	−	0.626678i	$-0.215586\pi$
$608$	0	0
$609$	− 13.2872i	− 0.538426i
$610$	0	0
$611$	7.09833	0.287168
$612$	0	0
$613$	−46.6432	−1.88390	−0.941951	−	0.335751i	$-0.891010\pi$
$613$	−46.6432	−1.88390	−0.941951	+	0.335751i	$0.891010\pi$
$614$	0	0
$615$	− 101.650i	− 4.09893i
$616$	0	0
$617$	− 2.08109i	− 0.0837815i	−0.999122	−	0.0418908i	$-0.986662\pi$
$617$	− 2.08109i	− 0.0837815i	0.999122	−	0.0418908i	$-0.0133381\pi$
$618$	0	0
$619$	35.6168i	1.43156i	0.698326	+	0.715780i	$0.253929\pi$
$619$	35.6168i	1.43156i	−0.698326	+	0.715780i	$0.746071\pi$
$620$	0	0
$621$	45.1266	1.81087
$622$	0	0
$623$	2.01817i	0.0808563i
$624$	0	0
$625$	25.7469	1.02988
$626$	0	0
$627$	13.4148	0.535734
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−23.7939	−0.947218	−0.473609	−	0.880735i	$-0.657049\pi$
$631$	−23.7939	−0.947218	−0.473609	+	0.880735i	$0.657049\pi$
$632$	0	0
$633$	−57.6059	−2.28963
$634$	0	0
$635$	− 9.16250i	− 0.363603i
$636$	0	0
$637$	−19.9605	−0.790863
$638$	0	0
$639$	− 93.5595i	− 3.70116i
$640$	0	0
$641$	9.01455i	0.356053i	0.984026	+	0.178027i	$0.0569713\pi$
$641$	9.01455i	0.356053i	−0.984026	+	0.178027i	$0.943029\pi$
$642$	0	0
$643$	3.71419i	0.146473i	0.997315	+	0.0732367i	$0.0233329\pi$
$643$	3.71419i	0.146473i	−0.997315	+	0.0732367i	$0.976667\pi$
$644$	0	0
$645$	87.7197	3.45396
$646$	0	0
$647$	4.28405	0.168423	0.0842117	−	0.996448i	$-0.473163\pi$
$647$	4.28405	0.168423	0.0842117	+	0.996448i	$0.473163\pi$
$648$	0	0
$649$	13.0095i	0.510667i
$650$	0	0
$651$	− 1.29404i	− 0.0507176i
$652$	0	0
$653$	− 4.64765i	− 0.181877i	−0.995857	−	0.0909383i	$-0.971013\pi$
$653$	− 4.64765i	− 0.181877i	0.995857	−	0.0909383i	$-0.0289866\pi$
$654$	0	0
$655$	−4.15476	−0.162340
$656$	0	0
$657$	37.8435i	1.47641i
$658$	0	0
$659$	−45.0259	−1.75396	−0.876980	−	0.480526i	$-0.840446\pi$
$659$	−45.0259	−1.75396	−0.876980	+	0.480526i	$0.840446\pi$
$660$	0	0
$661$	17.7537	0.690540	0.345270	−	0.938503i	$-0.387787\pi$
$661$	17.7537	0.690540	0.345270	+	0.938503i	$0.387787\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.78106	0.224180
$666$	0	0
$667$	27.9002	1.08030
$668$	0	0
$669$	− 47.3515i	− 1.83072i
$670$	0	0
$671$	4.37969	0.169076
$672$	0	0
$673$	34.5357i	1.33125i	0.746285	+	0.665627i	$0.231836\pi$
$673$	34.5357i	1.33125i	−0.746285	+	0.665627i	$0.768164\pi$
$674$	0	0
$675$	143.051i	5.50602i
$676$	0	0
$677$	− 29.1138i	− 1.11893i	−0.828852	−	0.559467i	$-0.811006\pi$
$677$	− 29.1138i	− 1.11893i	0.828852	−	0.559467i	$-0.188994\pi$
$678$	0	0
$679$	3.24535	0.124545
$680$	0	0
$681$	60.0205	2.29999
$682$	0	0
$683$	5.03508i	0.192662i	0.995349	+	0.0963310i	$0.0307107\pi$
$683$	5.03508i	0.192662i	−0.995349	+	0.0963310i	$0.969289\pi$
$684$	0	0
$685$	75.9386i	2.90147i
$686$	0	0
$687$	− 25.1320i	− 0.958845i
$688$	0	0
$689$	3.84255	0.146389
$690$	0	0
$691$	11.4270i	0.434702i	0.976094	+	0.217351i	$0.0697416\pi$
$691$	11.4270i	0.434702i	−0.976094	+	0.217351i	$0.930258\pi$
$692$	0	0
$693$	4.52704	0.171968
$694$	0	0
$695$	−30.8726	−1.17106
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−49.8411	−1.88516
$700$	0	0
$701$	38.0806	1.43828	0.719142	−	0.694863i	$-0.244535\pi$
$701$	38.0806	1.43828	0.719142	+	0.694863i	$0.244535\pi$
$702$	0	0
$703$	− 11.1156i	− 0.419231i
$704$	0	0
$705$	30.1857	1.13686
$706$	0	0
$707$	− 6.50837i	− 0.244772i
$708$	0	0
$709$	13.7382i	0.515950i	0.966152	+	0.257975i	$0.0830553\pi$
$709$	13.7382i	0.515950i	−0.966152	+	0.257975i	$0.916945\pi$
$710$	0	0
$711$	− 30.1215i	− 1.12965i
$712$	0	0
$713$	2.71721	0.101760
$714$	0	0
$715$	−14.9067	−0.557480
$716$	0	0
$717$	86.9163i	3.24595i
$718$	0	0
$719$	− 8.36722i	− 0.312045i	−0.987754	−	0.156022i	$-0.950133\pi$
$719$	− 8.36722i	− 0.312045i	0.987754	−	0.156022i	$-0.0498672\pi$
$720$	0	0
$721$	6.42065i	0.239117i
$722$	0	0
$723$	−63.6451	−2.36699
$724$	0	0
$725$	88.4434i	3.28470i
$726$	0	0
$727$	−2.78518	−0.103297	−0.0516483	−	0.998665i	$-0.516447\pi$
$727$	−2.78518	−0.103297	−0.0516483	+	0.998665i	$0.516447\pi$
$728$	0	0
$729$	−37.8289	−1.40107
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−5.34730	−0.197507	−0.0987534	−	0.995112i	$-0.531486\pi$
$733$	−5.34730	−0.197507	−0.0987534	+	0.995112i	$0.531486\pi$
$734$	0	0
$735$	−84.8822	−3.13093
$736$	0	0
$737$	16.6845i	0.614581i
$738$	0	0
$739$	21.7374	0.799624	0.399812	−	0.916597i	$-0.369075\pi$
$739$	21.7374	0.799624	0.399812	+	0.916597i	$0.369075\pi$
$740$	0	0
$741$	− 30.2490i	− 1.11122i
$742$	0	0
$743$	31.2371i	1.14598i	0.819563	+	0.572989i	$0.194216\pi$
$743$	31.2371i	1.14598i	−0.819563	+	0.572989i	$0.805784\pi$
$744$	0	0
$745$	6.41147i	0.234898i
$746$	0	0
$747$	−63.1275	−2.30972
$748$	0	0
$749$	7.45336	0.272340
$750$	0	0
$751$	22.9608i	0.837851i	0.908021	+	0.418926i	$0.137593\pi$
$751$	22.9608i	0.837851i	−0.908021	+	0.418926i	$0.862407\pi$
$752$	0	0
$753$	87.0765i	3.17324i
$754$	0	0
$755$	− 61.4029i	− 2.23468i
$756$	0	0
$757$	32.1756	1.16944	0.584721	−	0.811234i	$-0.301204\pi$
$757$	32.1756	1.16944	0.584721	+	0.811234i	$0.301204\pi$
$758$	0	0
$759$	13.3535i	0.484701i
$760$	0	0
$761$	−24.8348	−0.900261	−0.450131	−	0.892963i	$-0.648623\pi$
$761$	−24.8348	−0.900261	−0.450131	+	0.892963i	$0.648623\pi$
$762$	0	0
$763$	−0.571614	−0.0206938
$764$	0	0
$765$	0	0
$766$	0	0
$767$	29.3351	1.05923
$768$	0	0
$769$	−2.52671	−0.0911156	−0.0455578	−	0.998962i	$-0.514507\pi$
$769$	−2.52671	−0.0911156	−0.0455578	+	0.998962i	$0.514507\pi$
$770$	0	0
$771$	− 91.1285i	− 3.28191i
$772$	0	0
$773$	35.9968	1.29472	0.647358	−	0.762186i	$-0.275874\pi$
$773$	35.9968	1.29472	0.647358	+	0.762186i	$0.275874\pi$
$774$	0	0
$775$	8.61350i	0.309406i
$776$	0	0
$777$	− 5.26950i	− 0.189042i
$778$	0	0
$779$	25.8625i	0.926619i
$780$	0	0
$781$	16.4789	0.589663
$782$	0	0
$783$	−125.272	−4.47687
$784$	0	0
$785$	− 26.4124i	− 0.942699i
$786$	0	0
$787$	24.4338i	0.870970i	0.900196	+	0.435485i	$0.143423\pi$
$787$	24.4338i	0.870970i	−0.900196	+	0.435485i	$0.856577\pi$
$788$	0	0
$789$	− 19.2499i	− 0.685315i
$790$	0	0
$791$	−1.15839	−0.0411874
$792$	0	0
$793$	− 9.87576i	− 0.350699i
$794$	0	0
$795$	16.3405	0.579537
$796$	0	0
$797$	−6.45161	−0.228528	−0.114264	−	0.993450i	$-0.536451\pi$
$797$	−6.45161	−0.228528	−0.114264	+	0.993450i	$0.536451\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	31.9668	1.12949
$802$	0	0
$803$	−6.66550	−0.235220
$804$	0	0
$805$	5.75465i	0.202825i
$806$	0	0
$807$	76.5987	2.69640
$808$	0	0
$809$	22.6777i	0.797305i	0.917102	+	0.398652i	$0.130522\pi$
$809$	22.6777i	0.797305i	−0.917102	+	0.398652i	$0.869478\pi$
$810$	0	0
$811$	20.7989i	0.730348i	0.930939	+	0.365174i	$0.118991\pi$
$811$	20.7989i	0.730348i	−0.930939	+	0.365174i	$0.881009\pi$
$812$	0	0
$813$	− 21.1739i	− 0.742600i
$814$	0	0
$815$	14.8007	0.518444
$816$	0	0
$817$	−22.3182	−0.780815
$818$	0	0
$819$	− 10.2080i	− 0.356697i
$820$	0	0
$821$	42.5381i	1.48459i	0.670074	+	0.742295i	$0.266262\pi$
$821$	42.5381i	1.48459i	−0.670074	+	0.742295i	$0.733738\pi$
$822$	0	0
$823$	49.3474i	1.72014i	0.510174	+	0.860071i	$0.329581\pi$
$823$	49.3474i	1.72014i	−0.510174	+	0.860071i	$0.670419\pi$
$824$	0	0
$825$	−42.3304	−1.47375
$826$	0	0
$827$	− 20.3577i	− 0.707907i	−0.935263	−	0.353954i	$-0.884837\pi$
$827$	− 20.3577i	− 0.707907i	0.935263	−	0.353954i	$-0.115163\pi$
$828$	0	0
$829$	24.0820	0.836403	0.418202	−	0.908354i	$-0.362661\pi$
$829$	24.0820	0.836403	0.418202	+	0.908354i	$0.362661\pi$
$830$	0	0
$831$	44.4115	1.54062
$832$	0	0
$833$	0	0
$834$	0	0
$835$	45.3979	1.57106
$836$	0	0
$837$	−12.2003	−0.421703
$838$	0	0
$839$	51.1513i	1.76594i	0.469432	+	0.882969i	$0.344459\pi$
$839$	51.1513i	1.76594i	−0.469432	+	0.882969i	$0.655541\pi$
$840$	0	0
$841$	−48.4516	−1.67075
$842$	0	0
$843$	74.3575i	2.56101i
$844$	0	0
$845$	− 16.8188i	− 0.578585i
$846$	0	0
$847$	− 4.34966i	− 0.149456i
$848$	0	0
$849$	77.4653	2.65860
$850$	0	0
$851$	11.0648	0.379296
$852$	0	0
$853$	− 0.737415i	− 0.0252486i	−0.999920	−	0.0126243i	$-0.995981\pi$
$853$	− 0.737415i	− 0.0252486i	0.999920	−	0.0126243i	$-0.00401855\pi$
$854$	0	0
$855$	− 91.5690i	− 3.13160i
$856$	0	0
$857$	− 18.2932i	− 0.624885i	−0.949937	−	0.312442i	$-0.898853\pi$
$857$	− 18.2932i	− 0.624885i	0.949937	−	0.312442i	$-0.101147\pi$
$858$	0	0
$859$	37.9077	1.29339	0.646696	−	0.762748i	$-0.276150\pi$
$859$	37.9077	1.29339	0.646696	+	0.762748i	$0.276150\pi$
$860$	0	0
$861$	12.2605i	0.417837i
$862$	0	0
$863$	−38.2719	−1.30279	−0.651395	−	0.758739i	$-0.725816\pi$
$863$	−38.2719	−1.30279	−0.651395	+	0.758739i	$0.725816\pi$
$864$	0	0
$865$	58.7187	1.99650
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.30541	0.179974
$870$	0	0
$871$	37.6219	1.27477
$872$	0	0
$873$	− 51.4047i	− 1.73978i
$874$	0	0
$875$	−9.16613	−0.309872
$876$	0	0
$877$	− 38.2249i	− 1.29076i	−0.763860	−	0.645382i	$-0.776698\pi$
$877$	− 38.2249i	− 1.29076i	0.763860	−	0.645382i	$-0.223302\pi$
$878$	0	0
$879$	− 68.5580i	− 2.31240i
$880$	0	0
$881$	6.38144i	0.214996i	0.994205	+	0.107498i	$0.0342840\pi$
$881$	6.38144i	0.214996i	−0.994205	+	0.107498i	$0.965716\pi$
$882$	0	0
$883$	26.8266	0.902786	0.451393	−	0.892325i	$-0.350927\pi$
$883$	26.8266	0.902786	0.451393	+	0.892325i	$0.350927\pi$
$884$	0	0
$885$	124.748	4.19336
$886$	0	0
$887$	− 8.87433i	− 0.297971i	−0.988839	−	0.148985i	$-0.952399\pi$
$887$	− 8.87433i	− 0.297971i	0.988839	−	0.148985i	$-0.0476008\pi$
$888$	0	0
$889$	1.10513i	0.0370650i
$890$	0	0
$891$	− 30.9323i	− 1.03627i
$892$	0	0
$893$	−7.68004	−0.257003
$894$	0	0
$895$	− 23.6878i	− 0.791795i
$896$	0	0
$897$	30.1108	1.00537
$898$	0	0
$899$	−7.54301	−0.251574
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−10.5803	−0.352090
$904$	0	0
$905$	−28.2645	−0.939542
$906$	0	0
$907$	37.5827i	1.24791i	0.781460	+	0.623956i	$0.214475\pi$
$907$	37.5827i	1.24791i	−0.781460	+	0.623956i	$0.785525\pi$
$908$	0	0
$909$	−103.089	−3.41926
$910$	0	0
$911$	− 36.5289i	− 1.21026i	−0.796128	−	0.605128i	$-0.793122\pi$
$911$	− 36.5289i	− 1.21026i	0.796128	−	0.605128i	$-0.206878\pi$
$912$	0	0
$913$	− 11.1189i	− 0.367981i
$914$	0	0
$915$	− 41.9968i	− 1.38837i
$916$	0	0
$917$	0.501126	0.0165486
$918$	0	0
$919$	−37.8108	−1.24726	−0.623631	−	0.781719i	$-0.714343\pi$
$919$	−37.8108	−1.24726	−0.623631	+	0.781719i	$0.714343\pi$
$920$	0	0
$921$	43.0042i	1.41704i
$922$	0	0
$923$	− 37.1584i	− 1.22308i
$924$	0	0
$925$	35.0752i	1.15327i
$926$	0	0
$927$	101.700	3.34026
$928$	0	0
$929$	10.5645i	0.346609i	0.984868	+	0.173305i	$0.0554445\pi$
$929$	10.5645i	0.346609i	−0.984868	+	0.173305i	$0.944555\pi$
$930$	0	0
$931$	21.5963	0.707789
$932$	0	0
$933$	34.1688	1.11864
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.0378	−0.915954	−0.457977	−	0.888964i	$-0.651426\pi$
$937$	−28.0378	−0.915954	−0.457977	+	0.888964i	$0.651426\pi$
$938$	0	0
$939$	14.1625	0.462176
$940$	0	0
$941$	26.5871i	0.866715i	0.901222	+	0.433357i	$0.142671\pi$
$941$	26.5871i	0.866715i	−0.901222	+	0.433357i	$0.857329\pi$
$942$	0	0
$943$	−25.7443	−0.838351
$944$	0	0
$945$	− 25.8384i	− 0.840524i
$946$	0	0
$947$	17.7888i	0.578058i	0.957320	+	0.289029i	$0.0933325\pi$
$947$	17.7888i	0.578058i	−0.957320	+	0.289029i	$0.906668\pi$
$948$	0	0
$949$	15.0300i	0.487895i
$950$	0	0
$951$	4.39424	0.142493
$952$	0	0
$953$	−20.1061	−0.651299	−0.325650	−	0.945490i	$-0.605583\pi$
$953$	−20.1061	−0.651299	−0.325650	+	0.945490i	$0.605583\pi$
$954$	0	0
$955$	− 64.1079i	− 2.07448i
$956$	0	0
$957$	− 37.0696i	− 1.19829i
$958$	0	0
$959$	− 9.15932i	− 0.295770i
$960$	0	0
$961$	30.2654	0.976303
$962$	0	0
$963$	− 118.057i	− 3.80435i
$964$	0	0
$965$	21.9290	0.705920
$966$	0	0
$967$	−37.0068	−1.19006	−0.595029	−	0.803704i	$-0.702860\pi$
$967$	−37.0068	−1.19006	−0.595029	+	0.803704i	$0.702860\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−13.3618	−0.428802	−0.214401	−	0.976746i	$-0.568780\pi$
$971$	−13.3618	−0.428802	−0.214401	+	0.976746i	$0.568780\pi$
$972$	0	0
$973$	3.72369	0.119376
$974$	0	0
$975$	95.4508i	3.05687i
$976$	0	0
$977$	20.6304	0.660025	0.330013	−	0.943976i	$-0.392947\pi$
$977$	20.6304	0.660025	0.330013	+	0.943976i	$0.392947\pi$
$978$	0	0
$979$	5.63041i	0.179949i
$980$	0	0
$981$	9.05407i	0.289074i
$982$	0	0
$983$	16.8452i	0.537280i	0.963241	+	0.268640i	$0.0865741\pi$
$983$	16.8452i	0.537280i	−0.963241	+	0.268640i	$0.913426\pi$
$984$	0	0
$985$	−3.23854	−0.103189
$986$	0	0
$987$	−3.64084	−0.115889
$988$	0	0
$989$	− 22.2163i	− 0.706436i
$990$	0	0
$991$	− 42.3432i	− 1.34508i	−0.740063	−	0.672538i	$-0.765204\pi$
$991$	− 42.3432i	− 1.34508i	0.740063	−	0.672538i	$-0.234796\pi$
$992$	0	0
$993$	− 57.8925i	− 1.83716i
$994$	0	0
$995$	−89.5049	−2.83749
$996$	0	0
$997$	− 24.4097i	− 0.773064i	−0.922276	−	0.386532i	$-0.873673\pi$
$997$	− 24.4097i	− 0.773064i	0.922276	−	0.386532i	$-0.126327\pi$
$998$	0	0
$999$	−49.6810	−1.57184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.b.l.577.6		6
17.4	even	4		2312.2.a.r.1.3	yes	3
17.13	even	4		2312.2.a.o.1.1	✓	3
17.16	even	2	inner	2312.2.b.l.577.1		6
68.47	odd	4		4624.2.a.bi.1.3		3
68.55	odd	4		4624.2.a.bb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.o.1.1	✓	3	17.13	even	4
2312.2.a.r.1.3	yes	3	17.4	even	4
2312.2.b.l.577.1		6	17.16	even	2	inner
2312.2.b.l.577.6		6	1.1	even	1	trivial
4624.2.a.bb.1.1		3	68.55	odd	4
4624.2.a.bi.1.3		3	68.47	odd	4

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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.22668i q^{3} +3.87939i q^{5} -0.467911i q^{7} -7.41147 q^{9} -1.30541i q^{11} -2.94356 q^{13} -12.5175 q^{15} +3.18479 q^{19} +1.50980 q^{21} +3.17024i q^{23} -10.0496 q^{25} -14.2344i q^{27} -8.80066i q^{29} -0.857097i q^{31} +4.21213 q^{33} +1.81521 q^{35} -3.49020i q^{37} -9.49794i q^{39} +8.12061i q^{41} -7.00774 q^{43} -28.7520i q^{45} -2.41147 q^{47} +6.78106 q^{49} -1.30541 q^{53} +5.06418 q^{55} +10.2763i q^{57} -9.96585 q^{59} +3.35504i q^{61} +3.46791i q^{63} -11.4192i q^{65} -12.7811 q^{67} -10.2294 q^{69} +12.6236i q^{71} -5.10607i q^{73} -32.4270i q^{75} -0.610815 q^{77} +4.06418i q^{79} +23.6955 q^{81} +8.51754 q^{83} +28.3969 q^{87} -4.31315 q^{89} +1.37733i q^{91} +2.76558 q^{93} +12.3550i q^{95} +6.93582i q^{97} +9.67499i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24 q^{9} + 12 q^{13} - 30 q^{15} + 12 q^{19} + 12 q^{21} - 6 q^{25} - 24 q^{33} + 18 q^{35} + 6 q^{43} + 6 q^{47} + 6 q^{49} - 12 q^{53} + 12 q^{55} - 18 q^{59} - 42 q^{67} - 30 q^{69} - 12 q^{77}+ \cdots + 78 q^{93}+O(q^{100}) \) 6 * q - 24 * q^9 + 12 * q^13 - 30 * q^15 + 12 * q^19 + 12 * q^21 - 6 * q^25 - 24 * q^33 + 18 * q^35 + 6 * q^43 + 6 * q^47 + 6 * q^49 - 12 * q^53 + 12 * q^55 - 18 * q^59 - 42 * q^67 - 30 * q^69 - 12 * q^77 + 6 * q^81 + 6 * q^83 + 114 * q^87 + 18 * q^89 + 78 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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