LMFDB - Embedded newform 936.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,4,Mod(1,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("936.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	936.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$55.2257877654$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{43}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 43 $ x^2 - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.55744$ of defining polynomial
Character	$\chi$	$=$	936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.1149 q^{5} +35.1149 q^{7} -26.0000 q^{11} -13.0000 q^{13} +36.2298 q^{17} -95.5744 q^{19} +161.379 q^{23} +240.379 q^{25} +91.3785 q^{29} -266.723 q^{31} -671.217 q^{35} -149.608 q^{37} +77.8041 q^{41} +183.608 q^{43} +60.6893 q^{47} +890.055 q^{49} -281.540 q^{53} +496.987 q^{55} +542.527 q^{59} +65.0810 q^{61} +248.493 q^{65} -1033.94 q^{67} -1041.81 q^{71} +483.311 q^{73} -912.987 q^{77} -1337.05 q^{79} -812.825 q^{83} -692.527 q^{85} -936.885 q^{89} -456.493 q^{91} +1826.89 q^{95} -954.068 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{5} + 44 q^{7} - 52 q^{11} - 26 q^{13} + 20 q^{17} - 60 q^{19} + 8 q^{23} + 166 q^{25} - 132 q^{29} - 140 q^{31} - 608 q^{35} + 68 q^{37} - 28 q^{41} - 36 q^{47} + 626 q^{49} - 668 q^{53} + 312 q^{55}+ \cdots - 1436 q^{97}+O(q^{100}) $ 2 * q - 12 * q^5 + 44 * q^7 - 52 * q^11 - 26 * q^13 + 20 * q^17 - 60 * q^19 + 8 * q^23 + 166 * q^25 - 132 * q^29 - 140 * q^31 - 608 * q^35 + 68 * q^37 - 28 * q^41 - 36 * q^47 + 626 * q^49 - 668 * q^53 + 312 * q^55 + 508 * q^59 + 340 * q^61 + 156 * q^65 - 940 * q^67 - 300 * q^71 + 1124 * q^73 - 1144 * q^77 - 1520 * q^79 - 524 * q^83 - 808 * q^85 - 1900 * q^89 - 572 * q^91 + 2080 * q^95 - 1436 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−19.1149	−1.70969	−0.854843	−	0.518886i	$-0.826347\pi$
$5$	−19.1149	−1.70969	−0.854843	+	0.518886i	$0.826347\pi$
$6$	0	0
$7$	35.1149	1.89603	0.948013	−	0.318233i	$-0.103089\pi$
$7$	35.1149	1.89603	0.948013	+	0.318233i	$0.103089\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−26.0000	−0.712663	−0.356332	−	0.934360i	$-0.615973\pi$
$11$	−26.0000	−0.712663	−0.356332	+	0.934360i	$0.615973\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	36.2298	0.516883	0.258441	−	0.966027i	$-0.416791\pi$
$17$	36.2298	0.516883	0.258441	+	0.966027i	$0.416791\pi$
$18$	0	0
$19$	−95.5744	−1.15401	−0.577007	−	0.816739i	$-0.695779\pi$
$19$	−95.5744	−1.15401	−0.577007	+	0.816739i	$0.695779\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	161.379	1.46303	0.731516	−	0.681824i	$-0.238813\pi$
$23$	161.379	1.46303	0.731516	+	0.681824i	$0.238813\pi$
$24$	0	0
$25$	240.379	1.92303
$26$	0	0
$27$	0	0
$28$	0	0
$29$	91.3785	0.585123	0.292561	−	0.956247i	$-0.405492\pi$
$29$	91.3785	0.585123	0.292561	+	0.956247i	$0.405492\pi$
$30$	0	0
$31$	−266.723	−1.54532	−0.772660	−	0.634821i	$-0.781074\pi$
$31$	−266.723	−1.54532	−0.772660	+	0.634821i	$0.781074\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−671.217	−3.24161
$36$	0	0
$37$	−149.608	−0.664742	−0.332371	−	0.943149i	$-0.607849\pi$
$37$	−149.608	−0.664742	−0.332371	+	0.943149i	$0.607849\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	77.8041	0.296365	0.148183	−	0.988960i	$-0.452658\pi$
$41$	77.8041	0.296365	0.148183	+	0.988960i	$0.452658\pi$
$42$	0	0
$43$	183.608	0.651163	0.325581	−	0.945514i	$-0.394440\pi$
$43$	183.608	0.651163	0.325581	+	0.945514i	$0.394440\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	60.6893	0.188350	0.0941749	−	0.995556i	$-0.469979\pi$
$47$	60.6893	0.188350	0.0941749	+	0.995556i	$0.469979\pi$
$48$	0	0
$49$	890.055	2.59491
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−281.540	−0.729671	−0.364835	−	0.931072i	$-0.618875\pi$
$53$	−281.540	−0.729671	−0.364835	+	0.931072i	$0.618875\pi$
$54$	0	0
$55$	496.987	1.21843
$56$	0	0
$57$	0	0
$58$	0	0
$59$	542.527	1.19714	0.598568	−	0.801072i	$-0.295737\pi$
$59$	542.527	1.19714	0.598568	+	0.801072i	$0.295737\pi$
$60$	0	0
$61$	65.0810	0.136603	0.0683014	−	0.997665i	$-0.478242\pi$
$61$	65.0810	0.136603	0.0683014	+	0.997665i	$0.478242\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	248.493	0.474182
$66$	0	0
$67$	−1033.94	−1.88531	−0.942656	−	0.333767i	$-0.891680\pi$
$67$	−1033.94	−1.88531	−0.942656	+	0.333767i	$0.891680\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1041.81	−1.74141	−0.870706	−	0.491803i	$-0.836338\pi$
$71$	−1041.81	−1.74141	−0.870706	+	0.491803i	$0.836338\pi$
$72$	0	0
$73$	483.311	0.774894	0.387447	−	0.921892i	$-0.373357\pi$
$73$	483.311	0.774894	0.387447	+	0.921892i	$0.373357\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−912.987	−1.35123
$78$	0	0
$79$	−1337.05	−1.90418	−0.952091	−	0.305815i	$-0.901071\pi$
$79$	−1337.05	−1.90418	−0.952091	+	0.305815i	$0.901071\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−812.825	−1.07493	−0.537465	−	0.843286i	$-0.680618\pi$
$83$	−812.825	−1.07493	−0.537465	+	0.843286i	$0.680618\pi$
$84$	0	0
$85$	−692.527	−0.883707
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−936.885	−1.11584	−0.557919	−	0.829895i	$-0.688400\pi$
$89$	−936.885	−1.11584	−0.557919	+	0.829895i	$0.688400\pi$
$90$	0	0
$91$	−456.493	−0.525863
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1826.89	1.97300
$96$	0	0
$97$	−954.068	−0.998669	−0.499335	−	0.866409i	$-0.666422\pi$
$97$	−954.068	−0.998669	−0.499335	+	0.866409i	$0.666422\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1401.15	−1.38039	−0.690196	−	0.723623i	$-0.742476\pi$
$101$	−1401.15	−1.38039	−0.690196	+	0.723623i	$0.742476\pi$
$102$	0	0
$103$	954.960	0.913544	0.456772	−	0.889584i	$-0.349005\pi$
$103$	954.960	0.913544	0.456772	+	0.889584i	$0.349005\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	924.595	0.835364	0.417682	−	0.908593i	$-0.362843\pi$
$107$	924.595	0.835364	0.417682	+	0.908593i	$0.362843\pi$
$108$	0	0
$109$	172.893	0.151928	0.0759638	−	0.997111i	$-0.475797\pi$
$109$	172.893	0.151928	0.0759638	+	0.997111i	$0.475797\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1241.81	−1.03380	−0.516902	−	0.856045i	$-0.672915\pi$
$113$	−1241.81	−1.03380	−0.516902	+	0.856045i	$0.672915\pi$
$114$	0	0
$115$	−3084.73	−2.50133
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1272.20	0.980023
$120$	0	0
$121$	−655.000	−0.492111
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2205.45	−1.57809
$126$	0	0
$127$	−1079.93	−0.754555	−0.377278	−	0.926100i	$-0.623140\pi$
$127$	−1079.93	−0.754555	−0.377278	+	0.926100i	$0.623140\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	918.026	0.612277	0.306139	−	0.951987i	$-0.400963\pi$
$131$	918.026	0.612277	0.306139	+	0.951987i	$0.400963\pi$
$132$	0	0
$133$	−3356.08	−2.18804
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1311.06	−0.817603	−0.408801	−	0.912623i	$-0.634053\pi$
$137$	−1311.06	−0.817603	−0.408801	+	0.912623i	$0.634053\pi$
$138$	0	0
$139$	−227.593	−0.138879	−0.0694396	−	0.997586i	$-0.522121\pi$
$139$	−227.593	−0.138879	−0.0694396	+	0.997586i	$0.522121\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	338.000	0.197657
$144$	0	0
$145$	−1746.69	−1.00038
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1325.48	−0.728776	−0.364388	−	0.931247i	$-0.618722\pi$
$149$	−1325.48	−0.728776	−0.364388	+	0.931247i	$0.618722\pi$
$150$	0	0
$151$	252.426	0.136040	0.0680202	−	0.997684i	$-0.478332\pi$
$151$	252.426	0.136040	0.0680202	+	0.997684i	$0.478332\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5098.38	2.64201
$156$	0	0
$157$	−2268.62	−1.15322	−0.576611	−	0.817019i	$-0.695625\pi$
$157$	−2268.62	−1.15322	−0.576611	+	0.817019i	$0.695625\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5666.79	2.77395
$162$	0	0
$163$	−1083.45	−0.520630	−0.260315	−	0.965524i	$-0.583826\pi$
$163$	−1083.45	−0.520630	−0.260315	+	0.965524i	$0.583826\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	796.041	0.368859	0.184430	−	0.982846i	$-0.440956\pi$
$167$	796.041	0.368859	0.184430	+	0.982846i	$0.440956\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.09598	0.000481654 0	0.000240827	−	1.00000i	$-0.499923\pi$
$173$	1.09598	0.000481654 0	0.000240827		1.00000i	$0.499923\pi$
$174$	0	0
$175$	8440.86	3.64611
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3250.06	−1.35710	−0.678549	−	0.734555i	$-0.737391\pi$
$179$	−3250.06	−1.35710	−0.678549	+	0.734555i	$0.737391\pi$
$180$	0	0
$181$	2980.84	1.22411	0.612055	−	0.790815i	$-0.290343\pi$
$181$	2980.84	1.22411	0.612055	+	0.790815i	$0.290343\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2859.74	1.13650
$186$	0	0
$187$	−941.974	−0.368363
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−612.595	−0.232072	−0.116036	−	0.993245i	$-0.537019\pi$
$191$	−612.595	−0.232072	−0.116036	+	0.993245i	$0.537019\pi$
$192$	0	0
$193$	−4185.81	−1.56115	−0.780573	−	0.625064i	$-0.785073\pi$
$193$	−4185.81	−1.56115	−0.780573	+	0.625064i	$0.785073\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.06029	−0.000745124 0	−0.000372562	−	1.00000i	$-0.500119\pi$
$197$	−2.06029	−0.000745124 0	−0.000372562		1.00000i	$0.500119\pi$
$198$	0	0
$199$	1492.39	0.531622	0.265811	−	0.964025i	$-0.414360\pi$
$199$	1492.39	0.531622	0.265811	+	0.964025i	$0.414360\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3208.75	1.10941
$204$	0	0
$205$	−1487.22	−0.506691
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2484.93	0.822423
$210$	0	0
$211$	2748.27	0.896677	0.448338	−	0.893864i	$-0.352016\pi$
$211$	2748.27	0.896677	0.448338	+	0.893864i	$0.352016\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3509.65	−1.11328
$216$	0	0
$217$	−9365.95	−2.92996
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−470.987	−0.143357
$222$	0	0
$223$	186.655	0.0560510	0.0280255	−	0.999607i	$-0.491078\pi$
$223$	186.655	0.0560510	0.0280255	+	0.999607i	$0.491078\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3932.57	−1.14984	−0.574920	−	0.818210i	$-0.694967\pi$
$227$	−3932.57	−1.14984	−0.574920	+	0.818210i	$0.694967\pi$
$228$	0	0
$229$	5951.00	1.71726	0.858632	−	0.512593i	$-0.171315\pi$
$229$	5951.00	1.71726	0.858632	+	0.512593i	$0.171315\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6548.94	−1.84135	−0.920676	−	0.390327i	$-0.872362\pi$
$233$	−6548.94	−1.84135	−0.920676	+	0.390327i	$0.872362\pi$
$234$	0	0
$235$	−1160.07	−0.322019
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3462.35	0.937076	0.468538	−	0.883443i	$-0.344781\pi$
$239$	3462.35	0.937076	0.468538	+	0.883443i	$0.344781\pi$
$240$	0	0
$241$	4581.96	1.22469	0.612345	−	0.790591i	$-0.290226\pi$
$241$	4581.96	1.22469	0.612345	+	0.790591i	$0.290226\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−17013.3	−4.43649
$246$	0	0
$247$	1242.47	0.320066
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4746.76	−1.19368	−0.596838	−	0.802362i	$-0.703576\pi$
$251$	−4746.76	−1.19368	−0.596838	+	0.802362i	$0.703576\pi$
$252$	0	0
$253$	−4195.84	−1.04265
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3373.64	0.818840	0.409420	−	0.912346i	$-0.365731\pi$
$257$	3373.64	0.818840	0.409420	+	0.912346i	$0.365731\pi$
$258$	0	0
$259$	−5253.48	−1.26037
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3527.19	−0.826981	−0.413491	−	0.910508i	$-0.635691\pi$
$263$	−3527.19	−0.826981	−0.413491	+	0.910508i	$0.635691\pi$
$264$	0	0
$265$	5381.61	1.24751
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1144.30	−0.259365	−0.129682	−	0.991556i	$-0.541396\pi$
$269$	−1144.30	−0.259365	−0.129682	+	0.991556i	$0.541396\pi$
$270$	0	0
$271$	2254.25	0.505298	0.252649	−	0.967558i	$-0.418698\pi$
$271$	2254.25	0.505298	0.252649	+	0.967558i	$0.418698\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6249.84	−1.37047
$276$	0	0
$277$	7101.62	1.54042	0.770208	−	0.637793i	$-0.220152\pi$
$277$	7101.62	1.54042	0.770208	+	0.637793i	$0.220152\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3831.81	0.813475	0.406738	−	0.913545i	$-0.366666\pi$
$281$	3831.81	0.813475	0.406738	+	0.913545i	$0.366666\pi$
$282$	0	0
$283$	−3032.56	−0.636985	−0.318493	−	0.947925i	$-0.603177\pi$
$283$	−3032.56	−0.636985	−0.318493	+	0.947925i	$0.603177\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2732.08	0.561915
$288$	0	0
$289$	−3600.40	−0.732832
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2296.44	0.457882	0.228941	−	0.973440i	$-0.426474\pi$
$293$	2296.44	0.457882	0.228941	+	0.973440i	$0.426474\pi$
$294$	0	0
$295$	−10370.3	−2.04673
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2097.92	−0.405772
$300$	0	0
$301$	6447.38	1.23462
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1244.01	−0.233548
$306$	0	0
$307$	5174.26	0.961925	0.480962	−	0.876741i	$-0.340287\pi$
$307$	5174.26	0.961925	0.480962	+	0.876741i	$0.340287\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7481.54	1.36411	0.682057	−	0.731299i	$-0.261086\pi$
$311$	7481.54	1.36411	0.682057	+	0.731299i	$0.261086\pi$
$312$	0	0
$313$	−4158.97	−0.751051	−0.375526	−	0.926812i	$-0.622538\pi$
$313$	−4158.97	−0.751051	−0.375526	+	0.926812i	$0.622538\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	370.535	0.0656508	0.0328254	−	0.999461i	$-0.489549\pi$
$317$	370.535	0.0656508	0.0328254	+	0.999461i	$0.489549\pi$
$318$	0	0
$319$	−2375.84	−0.416996
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3462.64	−0.596490
$324$	0	0
$325$	−3124.92	−0.533352
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2131.10	0.357116
$330$	0	0
$331$	1820.84	0.302364	0.151182	−	0.988506i	$-0.451692\pi$
$331$	1820.84	0.302364	0.151182	+	0.988506i	$0.451692\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19763.6	3.22329
$336$	0	0
$337$	−11838.2	−1.91356	−0.956781	−	0.290809i	$-0.906076\pi$
$337$	−11838.2	−1.91356	−0.956781	+	0.290809i	$0.906076\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6934.80	1.10129
$342$	0	0
$343$	19209.8	3.02399
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10603.5	−1.64043	−0.820214	−	0.572057i	$-0.806145\pi$
$347$	−10603.5	−1.64043	−0.820214	+	0.572057i	$0.806145\pi$
$348$	0	0
$349$	−6398.72	−0.981421	−0.490710	−	0.871323i	$-0.663263\pi$
$349$	−6398.72	−0.981421	−0.490710	+	0.871323i	$0.663263\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7031.27	1.06016	0.530080	−	0.847948i	$-0.322162\pi$
$353$	7031.27	1.06016	0.530080	+	0.847948i	$0.322162\pi$
$354$	0	0
$355$	19914.1	2.97727
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5053.75	−0.742972	−0.371486	−	0.928439i	$-0.621152\pi$
$359$	−5053.75	−0.742972	−0.371486	+	0.928439i	$0.621152\pi$
$360$	0	0
$361$	2275.46	0.331749
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9238.43	−1.32483
$366$	0	0
$367$	−81.1902	−0.0115479	−0.00577397	−	0.999983i	$-0.501838\pi$
$367$	−81.1902	−0.0115479	−0.00577397	+	0.999983i	$0.501838\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9886.26	−1.38347
$372$	0	0
$373$	8295.79	1.15158	0.575790	−	0.817597i	$-0.304695\pi$
$373$	8295.79	1.15158	0.575790	+	0.817597i	$0.304695\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1187.92	−0.162284
$378$	0	0
$379$	−6592.90	−0.893548	−0.446774	−	0.894647i	$-0.647427\pi$
$379$	−6592.90	−0.893548	−0.446774	+	0.894647i	$0.647427\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7472.96	0.996999	0.498499	−	0.866890i	$-0.333885\pi$
$383$	7472.96	0.996999	0.498499	+	0.866890i	$0.333885\pi$
$384$	0	0
$385$	17451.6	2.31018
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3175.88	0.413943	0.206971	−	0.978347i	$-0.433639\pi$
$389$	3175.88	0.413943	0.206971	+	0.978347i	$0.433639\pi$
$390$	0	0
$391$	5846.70	0.756216
$392$	0	0
$393$	0	0
$394$	0	0
$395$	25557.6	3.25555
$396$	0	0
$397$	11394.6	1.44050	0.720251	−	0.693713i	$-0.244027\pi$
$397$	11394.6	1.44050	0.720251	+	0.693713i	$0.244027\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13046.7	−1.62474	−0.812370	−	0.583142i	$-0.801823\pi$
$401$	−13046.7	−1.62474	−0.812370	+	0.583142i	$0.801823\pi$
$402$	0	0
$403$	3467.40	0.428594
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3889.82	0.473737
$408$	0	0
$409$	−2888.77	−0.349243	−0.174622	−	0.984636i	$-0.555870\pi$
$409$	−2888.77	−0.349243	−0.174622	+	0.984636i	$0.555870\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	19050.8	2.26980
$414$	0	0
$415$	15537.0	1.83779
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14260.8	1.66273	0.831367	−	0.555724i	$-0.187559\pi$
$419$	14260.8	1.66273	0.831367	+	0.555724i	$0.187559\pi$
$420$	0	0
$421$	890.723	0.103114	0.0515572	−	0.998670i	$-0.483582\pi$
$421$	890.723	0.103114	0.0515572	+	0.998670i	$0.483582\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8708.85	0.993980
$426$	0	0
$427$	2285.31	0.259002
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1323.12	0.147871	0.0739353	−	0.997263i	$-0.476444\pi$
$431$	1323.12	0.147871	0.0739353	+	0.997263i	$0.476444\pi$
$432$	0	0
$433$	−9399.47	−1.04321	−0.521605	−	0.853187i	$-0.674666\pi$
$433$	−9399.47	−1.04321	−0.521605	+	0.853187i	$0.674666\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15423.7	−1.68836
$438$	0	0
$439$	−6115.04	−0.664817	−0.332409	−	0.943135i	$-0.607861\pi$
$439$	−6115.04	−0.664817	−0.332409	+	0.943135i	$0.607861\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12931.1	1.38685	0.693425	−	0.720529i	$-0.256101\pi$
$443$	12931.1	1.38685	0.693425	+	0.720529i	$0.256101\pi$
$444$	0	0
$445$	17908.4	1.90773
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2050.73	0.215546	0.107773	−	0.994176i	$-0.465628\pi$
$449$	2050.73	0.215546	0.107773	+	0.994176i	$0.465628\pi$
$450$	0	0
$451$	−2022.91	−0.211208
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8725.82	0.899061
$456$	0	0
$457$	−11923.7	−1.22049	−0.610247	−	0.792211i	$-0.708930\pi$
$457$	−11923.7	−1.22049	−0.610247	+	0.792211i	$0.708930\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1324.14	−0.133777	−0.0668886	−	0.997760i	$-0.521307\pi$
$461$	−1324.14	−0.133777	−0.0668886	+	0.997760i	$0.521307\pi$
$462$	0	0
$463$	−6840.65	−0.686635	−0.343318	−	0.939219i	$-0.611551\pi$
$463$	−6840.65	−0.686635	−0.343318	+	0.939219i	$0.611551\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9041.73	0.895935	0.447967	−	0.894050i	$-0.352148\pi$
$467$	9041.73	0.895935	0.447967	+	0.894050i	$0.352148\pi$
$468$	0	0
$469$	−36306.7	−3.57460
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4773.82	−0.464060
$474$	0	0
$475$	−22974.0	−2.21920
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4650.71	−0.443625	−0.221812	−	0.975089i	$-0.571197\pi$
$479$	−4650.71	−0.443625	−0.221812	+	0.975089i	$0.571197\pi$
$480$	0	0
$481$	1944.91	0.184366
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18236.9	1.70741
$486$	0	0
$487$	−19632.4	−1.82675	−0.913375	−	0.407120i	$-0.866533\pi$
$487$	−19632.4	−1.82675	−0.913375	+	0.407120i	$0.866533\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14472.0	1.33017	0.665084	−	0.746769i	$-0.268396\pi$
$491$	14472.0	1.33017	0.665084	+	0.746769i	$0.268396\pi$
$492$	0	0
$493$	3310.62	0.302440
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−36583.1	−3.30176
$498$	0	0
$499$	−7065.91	−0.633895	−0.316947	−	0.948443i	$-0.602658\pi$
$499$	−7065.91	−0.633895	−0.316947	+	0.948443i	$0.602658\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11955.1	−1.05974	−0.529871	−	0.848078i	$-0.677760\pi$
$503$	−11955.1	−1.05974	−0.529871	+	0.848078i	$0.677760\pi$
$504$	0	0
$505$	26782.8	2.36004
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8321.15	0.724614	0.362307	−	0.932059i	$-0.381989\pi$
$509$	8321.15	0.724614	0.362307	+	0.932059i	$0.381989\pi$
$510$	0	0
$511$	16971.4	1.46922
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−18254.0	−1.56187
$516$	0	0
$517$	−1577.92	−0.134230
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20130.2	1.69274	0.846372	−	0.532592i	$-0.178782\pi$
$521$	20130.2	1.69274	0.846372	+	0.532592i	$0.178782\pi$
$522$	0	0
$523$	8810.52	0.736630	0.368315	−	0.929701i	$-0.379935\pi$
$523$	8810.52	0.736630	0.368315	+	0.929701i	$0.379935\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9663.31	−0.798749
$528$	0	0
$529$	13876.0	1.14046
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1011.45	−0.0821969
$534$	0	0
$535$	−17673.5	−1.42821
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−23141.4	−1.84930
$540$	0	0
$541$	7765.62	0.617135	0.308567	−	0.951202i	$-0.400150\pi$
$541$	7765.62	0.617135	0.308567	+	0.951202i	$0.400150\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3304.82	−0.259749
$546$	0	0
$547$	19782.4	1.54631	0.773156	−	0.634216i	$-0.218677\pi$
$547$	19782.4	1.54631	0.773156	+	0.634216i	$0.218677\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8733.45	−0.675240
$552$	0	0
$553$	−46950.5	−3.61038
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10153.9	0.772413	0.386206	−	0.922412i	$-0.373785\pi$
$557$	10153.9	0.772413	0.386206	+	0.922412i	$0.373785\pi$
$558$	0	0
$559$	−2386.91	−0.180600
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18388.2	−1.37650	−0.688252	−	0.725472i	$-0.741622\pi$
$563$	−18388.2	−1.37650	−0.688252	+	0.725472i	$0.741622\pi$
$564$	0	0
$565$	23737.1	1.76748
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16365.0	−1.20573	−0.602863	−	0.797845i	$-0.705973\pi$
$569$	−16365.0	−1.20573	−0.602863	+	0.797845i	$0.705973\pi$
$570$	0	0
$571$	544.241	0.0398875	0.0199437	−	0.999801i	$-0.493651\pi$
$571$	544.241	0.0398875	0.0199437	+	0.999801i	$0.493651\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	38791.9	2.81345
$576$	0	0
$577$	−798.738	−0.0576289	−0.0288145	−	0.999585i	$-0.509173\pi$
$577$	−798.738	−0.0576289	−0.0288145	+	0.999585i	$0.509173\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−28542.2	−2.03809
$582$	0	0
$583$	7320.05	0.520010
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9187.53	−0.646013	−0.323007	−	0.946397i	$-0.604694\pi$
$587$	−9187.53	−0.646013	−0.323007	+	0.946397i	$0.604694\pi$
$588$	0	0
$589$	25491.9	1.78332
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4371.80	0.302746	0.151373	−	0.988477i	$-0.451631\pi$
$593$	4371.80	0.302746	0.151373	+	0.988477i	$0.451631\pi$
$594$	0	0
$595$	−24318.0	−1.67553
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6780.78	0.462530	0.231265	−	0.972891i	$-0.425714\pi$
$599$	6780.78	0.462530	0.231265	+	0.972891i	$0.425714\pi$
$600$	0	0
$601$	−21348.5	−1.44895	−0.724477	−	0.689299i	$-0.757919\pi$
$601$	−21348.5	−1.44895	−0.724477	+	0.689299i	$0.757919\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12520.2	0.841356
$606$	0	0
$607$	11161.8	0.746362	0.373181	−	0.927758i	$-0.378267\pi$
$607$	11161.8	0.746362	0.373181	+	0.927758i	$0.378267\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−788.960	−0.0522388
$612$	0	0
$613$	12407.8	0.817528	0.408764	−	0.912640i	$-0.365960\pi$
$613$	12407.8	0.817528	0.408764	+	0.912640i	$0.365960\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5035.01	−0.328528	−0.164264	−	0.986416i	$-0.552525\pi$
$617$	−5035.01	−0.328528	−0.164264	+	0.986416i	$0.552525\pi$
$618$	0	0
$619$	−1623.67	−0.105430	−0.0527148	−	0.998610i	$-0.516787\pi$
$619$	−1623.67	−0.105430	−0.0527148	+	0.998610i	$0.516787\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−32898.6	−2.11566
$624$	0	0
$625$	12109.5	0.775009
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5420.27	−0.343594
$630$	0	0
$631$	24809.0	1.56519	0.782593	−	0.622534i	$-0.213897\pi$
$631$	24809.0	1.56519	0.782593	+	0.622534i	$0.213897\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	20642.8	1.29005
$636$	0	0
$637$	−11570.7	−0.719699
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18076.6	−1.11386	−0.556928	−	0.830561i	$-0.688020\pi$
$641$	−18076.6	−1.11386	−0.556928	+	0.830561i	$0.688020\pi$
$642$	0	0
$643$	19954.2	1.22382	0.611910	−	0.790928i	$-0.290402\pi$
$643$	19954.2	1.22382	0.611910	+	0.790928i	$0.290402\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11206.1	0.680921	0.340461	−	0.940259i	$-0.389417\pi$
$647$	11206.1	0.680921	0.340461	+	0.940259i	$0.389417\pi$
$648$	0	0
$649$	−14105.7	−0.853155
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2456.10	−0.147189	−0.0735947	−	0.997288i	$-0.523447\pi$
$653$	−2456.10	−0.147189	−0.0735947	+	0.997288i	$0.523447\pi$
$654$	0	0
$655$	−17548.0	−1.04680
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23280.8	1.37616	0.688081	−	0.725633i	$-0.258453\pi$
$659$	23280.8	1.37616	0.688081	+	0.725633i	$0.258453\pi$
$660$	0	0
$661$	−6999.49	−0.411874	−0.205937	−	0.978565i	$-0.566024\pi$
$661$	−6999.49	−0.411874	−0.205937	+	0.978565i	$0.566024\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	64151.1	3.74086
$666$	0	0
$667$	14746.5	0.856054
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1692.11	−0.0973517
$672$	0	0
$673$	33166.1	1.89964	0.949820	−	0.312798i	$-0.101266\pi$
$673$	33166.1	1.89964	0.949820	+	0.312798i	$0.101266\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19998.6	−1.13532	−0.567658	−	0.823265i	$-0.692150\pi$
$677$	−19998.6	−1.13532	−0.567658	+	0.823265i	$0.692150\pi$
$678$	0	0
$679$	−33502.0	−1.89350
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−22857.9	−1.28058	−0.640289	−	0.768134i	$-0.721185\pi$
$683$	−22857.9	−1.28058	−0.640289	+	0.768134i	$0.721185\pi$
$684$	0	0
$685$	25060.8	1.39784
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3660.03	0.202374
$690$	0	0
$691$	−2866.56	−0.157813	−0.0789066	−	0.996882i	$-0.525143\pi$
$691$	−2866.56	−0.157813	−0.0789066	+	0.996882i	$0.525143\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4350.42	0.237440
$696$	0	0
$697$	2818.82	0.153186
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30788.1	−1.65885	−0.829424	−	0.558620i	$-0.811331\pi$
$701$	−30788.1	−1.65885	−0.829424	+	0.558620i	$0.811331\pi$
$702$	0	0
$703$	14298.7	0.767121
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−49201.2	−2.61726
$708$	0	0
$709$	−4389.07	−0.232489	−0.116245	−	0.993221i	$-0.537086\pi$
$709$	−4389.07	−0.232489	−0.116245	+	0.993221i	$0.537086\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−43043.4	−2.26085
$714$	0	0
$715$	−6460.83	−0.337932
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8729.09	0.452768	0.226384	−	0.974038i	$-0.427310\pi$
$719$	8729.09	0.452768	0.226384	+	0.974038i	$0.427310\pi$
$720$	0	0
$721$	33533.3	1.73210
$722$	0	0
$723$	0	0
$724$	0	0
$725$	21965.4	1.12521
$726$	0	0
$727$	1471.81	0.0750845	0.0375423	−	0.999295i	$-0.488047\pi$
$727$	1471.81	0.0750845	0.0375423	+	0.999295i	$0.488047\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6652.08	0.336575
$732$	0	0
$733$	−14082.8	−0.709632	−0.354816	−	0.934936i	$-0.615457\pi$
$733$	−14082.8	−0.709632	−0.354816	+	0.934936i	$0.615457\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26882.4	1.34359
$738$	0	0
$739$	12529.5	0.623687	0.311843	−	0.950133i	$-0.399054\pi$
$739$	12529.5	0.623687	0.311843	+	0.950133i	$0.399054\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	19044.9	0.940363	0.470182	−	0.882570i	$-0.344188\pi$
$743$	19044.9	0.940363	0.470182	+	0.882570i	$0.344188\pi$
$744$	0	0
$745$	25336.4	1.24598
$746$	0	0
$747$	0	0
$748$	0	0
$749$	32467.0	1.58387
$750$	0	0
$751$	−22009.9	−1.06944	−0.534722	−	0.845028i	$-0.679584\pi$
$751$	−22009.9	−1.06944	−0.534722	+	0.845028i	$0.679584\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4825.08	−0.232587
$756$	0	0
$757$	−8198.54	−0.393634	−0.196817	−	0.980440i	$-0.563061\pi$
$757$	−8198.54	−0.393634	−0.196817	+	0.980440i	$0.563061\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31929.6	1.52096	0.760478	−	0.649363i	$-0.224964\pi$
$761$	31929.6	1.52096	0.760478	+	0.649363i	$0.224964\pi$
$762$	0	0
$763$	6071.10	0.288059
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7052.85	−0.332026
$768$	0	0
$769$	−26725.8	−1.25326	−0.626630	−	0.779317i	$-0.715566\pi$
$769$	−26725.8	−1.25326	−0.626630	+	0.779317i	$0.715566\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−23760.8	−1.10558	−0.552792	−	0.833320i	$-0.686437\pi$
$773$	−23760.8	−1.10558	−0.552792	+	0.833320i	$0.686437\pi$
$774$	0	0
$775$	−64114.5	−2.97169
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7436.08	−0.342009
$780$	0	0
$781$	27087.1	1.24104
$782$	0	0
$783$	0	0
$784$	0	0
$785$	43364.4	1.97165
$786$	0	0
$787$	−23647.2	−1.07107	−0.535536	−	0.844513i	$-0.679890\pi$
$787$	−23647.2	−1.07107	−0.535536	+	0.844513i	$0.679890\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−43606.1	−1.96012
$792$	0	0
$793$	−846.053	−0.0378868
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21067.5	−0.936324	−0.468162	−	0.883643i	$-0.655084\pi$
$797$	−21067.5	−0.936324	−0.468162	+	0.883643i	$0.655084\pi$
$798$	0	0
$799$	2198.76	0.0973547
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−12566.1	−0.552238
$804$	0	0
$805$	−108320.	−4.74258
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1877.67	−0.0816011	−0.0408005	−	0.999167i	$-0.512991\pi$
$809$	−1877.67	−0.0816011	−0.0408005	+	0.999167i	$0.512991\pi$
$810$	0	0
$811$	2178.85	0.0943401	0.0471700	−	0.998887i	$-0.484980\pi$
$811$	2178.85	0.0943401	0.0471700	+	0.998887i	$0.484980\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20710.1	0.890114
$816$	0	0
$817$	−17548.2	−0.751451
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11731.4	−0.498696	−0.249348	−	0.968414i	$-0.580216\pi$
$821$	−11731.4	−0.498696	−0.249348	+	0.968414i	$0.580216\pi$
$822$	0	0
$823$	−6925.69	−0.293334	−0.146667	−	0.989186i	$-0.546855\pi$
$823$	−6925.69	−0.293334	−0.146667	+	0.989186i	$0.546855\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35912.7	1.51004	0.755022	−	0.655700i	$-0.227626\pi$
$827$	35912.7	1.51004	0.755022	+	0.655700i	$0.227626\pi$
$828$	0	0
$829$	−7107.67	−0.297780	−0.148890	−	0.988854i	$-0.547570\pi$
$829$	−7107.67	−0.297780	−0.148890	+	0.988854i	$0.547570\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	32246.5	1.34126
$834$	0	0
$835$	−15216.2	−0.630634
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37844.0	−1.55724	−0.778618	−	0.627498i	$-0.784079\pi$
$839$	−37844.0	−1.55724	−0.778618	+	0.627498i	$0.784079\pi$
$840$	0	0
$841$	−16039.0	−0.657631
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3230.41	−0.131514
$846$	0	0
$847$	−23000.2	−0.933055
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24143.6	−0.972539
$852$	0	0
$853$	23280.9	0.934492	0.467246	−	0.884127i	$-0.345246\pi$
$853$	23280.9	0.934492	0.467246	+	0.884127i	$0.345246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10395.1	−0.414342	−0.207171	−	0.978305i	$-0.566426\pi$
$857$	−10395.1	−0.414342	−0.207171	+	0.978305i	$0.566426\pi$
$858$	0	0
$859$	−6698.89	−0.266080	−0.133040	−	0.991111i	$-0.542474\pi$
$859$	−6698.89	−0.266080	−0.133040	+	0.991111i	$0.542474\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31844.8	−1.25609	−0.628047	−	0.778175i	$-0.716146\pi$
$863$	−31844.8	−1.25609	−0.628047	+	0.778175i	$0.716146\pi$
$864$	0	0
$865$	−20.9496	−0.000823477 0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	34763.4	1.35704
$870$	0	0
$871$	13441.2	0.522891
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−77444.0	−2.99210
$876$	0	0
$877$	−20114.1	−0.774465	−0.387233	−	0.921982i	$-0.626569\pi$
$877$	−20114.1	−0.774465	−0.387233	+	0.921982i	$0.626569\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2596.61	0.0992987	0.0496493	−	0.998767i	$-0.484190\pi$
$881$	2596.61	0.0992987	0.0496493	+	0.998767i	$0.484190\pi$
$882$	0	0
$883$	−46374.7	−1.76742	−0.883710	−	0.468035i	$-0.844962\pi$
$883$	−46374.7	−1.76742	−0.883710	+	0.468035i	$0.844962\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31863.1	1.20615	0.603077	−	0.797683i	$-0.293941\pi$
$887$	31863.1	1.20615	0.603077	+	0.797683i	$0.293941\pi$
$888$	0	0
$889$	−37921.7	−1.43066
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5800.34	−0.217358
$894$	0	0
$895$	62124.4	2.32021
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−24372.8	−0.904202
$900$	0	0
$901$	−10200.1	−0.377154
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−56978.4	−2.09285
$906$	0	0
$907$	3004.29	0.109984	0.0549922	−	0.998487i	$-0.482487\pi$
$907$	3004.29	0.109984	0.0549922	+	0.998487i	$0.482487\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9985.05	0.363139	0.181569	−	0.983378i	$-0.441882\pi$
$911$	9985.05	0.363139	0.181569	+	0.983378i	$0.441882\pi$
$912$	0	0
$913$	21133.4	0.766062
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32236.4	1.16089
$918$	0	0
$919$	−8216.29	−0.294919	−0.147459	−	0.989068i	$-0.547110\pi$
$919$	−8216.29	−0.294919	−0.147459	+	0.989068i	$0.547110\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13543.6	0.482981
$924$	0	0
$925$	−35962.6	−1.27832
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17832.8	0.629791	0.314896	−	0.949126i	$-0.398030\pi$
$929$	17832.8	0.629791	0.314896	+	0.949126i	$0.398030\pi$
$930$	0	0
$931$	−85066.4	−2.99456
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18005.7	0.629786
$936$	0	0
$937$	−444.289	−0.0154902	−0.00774509	−	0.999970i	$-0.502465\pi$
$937$	−444.289	−0.0154902	−0.00774509	+	0.999970i	$0.502465\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30762.5	−1.06571	−0.532853	−	0.846208i	$-0.678880\pi$
$941$	−30762.5	−1.06571	−0.532853	+	0.846208i	$0.678880\pi$
$942$	0	0
$943$	12555.9	0.433592
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8791.45	0.301672	0.150836	−	0.988559i	$-0.451803\pi$
$947$	8791.45	0.301672	0.150836	+	0.988559i	$0.451803\pi$
$948$	0	0
$949$	−6283.04	−0.214917
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1754.41	−0.0596336	−0.0298168	−	0.999555i	$-0.509492\pi$
$953$	−1754.41	−0.0596336	−0.0298168	+	0.999555i	$0.509492\pi$
$954$	0	0
$955$	11709.7	0.396771
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−46037.8	−1.55020
$960$	0	0
$961$	41350.2	1.38801
$962$	0	0
$963$	0	0
$964$	0	0
$965$	80011.3	2.66907
$966$	0	0
$967$	26655.9	0.886447	0.443224	−	0.896411i	$-0.353835\pi$
$967$	26655.9	0.886447	0.443224	+	0.896411i	$0.353835\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42151.7	1.39311	0.696556	−	0.717502i	$-0.254715\pi$
$971$	42151.7	1.39311	0.696556	+	0.717502i	$0.254715\pi$
$972$	0	0
$973$	−7991.91	−0.263318
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14594.7	−0.477919	−0.238959	−	0.971030i	$-0.576806\pi$
$977$	−14594.7	−0.477919	−0.238959	+	0.971030i	$0.576806\pi$
$978$	0	0
$979$	24359.0	0.795217
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57077.8	−1.85198	−0.925991	−	0.377546i	$-0.876768\pi$
$983$	−57077.8	−1.85198	−0.925991	+	0.377546i	$0.876768\pi$
$984$	0	0
$985$	39.3821	0.00127393
$986$	0	0
$987$	0	0
$988$	0	0
$989$	29630.4	0.952672
$990$	0	0
$991$	11986.6	0.384224	0.192112	−	0.981373i	$-0.438466\pi$
$991$	11986.6	0.384224	0.192112	+	0.981373i	$0.438466\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−28526.9	−0.908908
$996$	0	0
$997$	32210.1	1.02317	0.511587	−	0.859231i	$-0.329058\pi$
$997$	32210.1	1.02317	0.511587	+	0.859231i	$0.329058\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	936.4.a.c.1.1		2
3.2	odd	2		312.4.a.f.1.2	✓	2
4.3	odd	2		1872.4.a.v.1.1		2
12.11	even	2		624.4.a.l.1.2		2
24.5	odd	2		2496.4.a.u.1.1		2
24.11	even	2		2496.4.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.f.1.2	✓	2	3.2	odd	2
624.4.a.l.1.2		2	12.11	even	2
936.4.a.c.1.1		2	1.1	even	1	trivial
1872.4.a.v.1.1		2	4.3	odd	2
2496.4.a.u.1.1		2	24.5	odd	2
2496.4.a.bd.1.1		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	936.a (trivial)

\(f(q)\)	\(=\)	\(q-19.1149 q^{5} +35.1149 q^{7} -26.0000 q^{11} -13.0000 q^{13} +36.2298 q^{17} -95.5744 q^{19} +161.379 q^{23} +240.379 q^{25} +91.3785 q^{29} -266.723 q^{31} -671.217 q^{35} -149.608 q^{37} +77.8041 q^{41} +183.608 q^{43} +60.6893 q^{47} +890.055 q^{49} -281.540 q^{53} +496.987 q^{55} +542.527 q^{59} +65.0810 q^{61} +248.493 q^{65} -1033.94 q^{67} -1041.81 q^{71} +483.311 q^{73} -912.987 q^{77} -1337.05 q^{79} -812.825 q^{83} -692.527 q^{85} -936.885 q^{89} -456.493 q^{91} +1826.89 q^{95} -954.068 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{5} + 44 q^{7} - 52 q^{11} - 26 q^{13} + 20 q^{17} - 60 q^{19} + 8 q^{23} + 166 q^{25} - 132 q^{29} - 140 q^{31} - 608 q^{35} + 68 q^{37} - 28 q^{41} - 36 q^{47} + 626 q^{49} - 668 q^{53} + 312 q^{55}+ \cdots - 1436 q^{97}+O(q^{100}) \) 2 * q - 12 * q^5 + 44 * q^7 - 52 * q^11 - 26 * q^13 + 20 * q^17 - 60 * q^19 + 8 * q^23 + 166 * q^25 - 132 * q^29 - 140 * q^31 - 608 * q^35 + 68 * q^37 - 28 * q^41 - 36 * q^47 + 626 * q^49 - 668 * q^53 + 312 * q^55 + 508 * q^59 + 340 * q^61 + 156 * q^65 - 940 * q^67 - 300 * q^71 + 1124 * q^73 - 1144 * q^77 - 1520 * q^79 - 524 * q^83 - 808 * q^85 - 1900 * q^89 - 572 * q^91 + 2080 * q^95 - 1436 * q^97

Introduction

L-functions

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