LMFDB - Embedded newform 756.4.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(1,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.55552.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 18x^{2} - 12x + 46 $ x^4 - 18x^2 - 12x + 46
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.86258$ of defining polynomial
Character	$\chi$	$=$	756.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.27322 q^{5} +7.00000 q^{7} +54.3063 q^{11} +40.9117 q^{13} -25.0929 q^{17} +23.0000 q^{19} -139.979 q^{23} -85.6468 q^{25} -73.1260 q^{29} -57.7351 q^{31} -43.9125 q^{35} +277.382 q^{37} -346.810 q^{41} +364.382 q^{43} +599.707 q^{47} +49.0000 q^{49} +91.7610 q^{53} -340.675 q^{55} +771.052 q^{59} +373.823 q^{61} -256.648 q^{65} +718.029 q^{67} -248.591 q^{71} +376.735 q^{73} +380.144 q^{77} +305.795 q^{79} -210.767 q^{83} +157.413 q^{85} -133.890 q^{89} +286.382 q^{91} -144.284 q^{95} +1045.47 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{7} - 40 q^{13} + 92 q^{19} + 472 q^{25} + 380 q^{31} - 316 q^{37} + 32 q^{43} + 196 q^{49} + 1692 q^{55} + 1088 q^{61} + 632 q^{67} + 896 q^{73} + 3056 q^{79} + 3888 q^{85} - 280 q^{91} + 2960 q^{97}+O(q^{100}) $ 4 * q + 28 * q^7 - 40 * q^13 + 92 * q^19 + 472 * q^25 + 380 * q^31 - 316 * q^37 + 32 * q^43 + 196 * q^49 + 1692 * q^55 + 1088 * q^61 + 632 * q^67 + 896 * q^73 + 3056 * q^79 + 3888 * q^85 - 280 * q^91 + 2960 * 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
$5$	−6.27322	−0.561094	−0.280547	−	0.959840i	$-0.590516\pi$
$5$	−6.27322	−0.561094	−0.280547	+	0.959840i	$0.590516\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	54.3063	1.48854	0.744271	−	0.667877i	$-0.232797\pi$
$11$	54.3063	1.48854	0.744271	+	0.667877i	$0.232797\pi$
$12$	0	0
$13$	40.9117	0.872835	0.436418	−	0.899744i	$-0.356247\pi$
$13$	40.9117	0.872835	0.436418	+	0.899744i	$0.356247\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−25.0929	−0.357995	−0.178997	−	0.983850i	$-0.557285\pi$
$17$	−25.0929	−0.357995	−0.178997	+	0.983850i	$0.557285\pi$
$18$	0	0
$19$	23.0000	0.277714	0.138857	−	0.990312i	$-0.455657\pi$
$19$	23.0000	0.277714	0.138857	+	0.990312i	$0.455657\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−139.979	−1.26903	−0.634513	−	0.772912i	$-0.718799\pi$
$23$	−139.979	−1.26903	−0.634513	+	0.772912i	$0.718799\pi$
$24$	0	0
$25$	−85.6468	−0.685174
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−73.1260	−0.468247	−0.234123	−	0.972207i	$-0.575222\pi$
$29$	−73.1260	−0.468247	−0.234123	+	0.972207i	$0.575222\pi$
$30$	0	0
$31$	−57.7351	−0.334501	−0.167250	−	0.985914i	$-0.553489\pi$
$31$	−57.7351	−0.334501	−0.167250	+	0.985914i	$0.553489\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−43.9125	−0.212073
$36$	0	0
$37$	277.382	1.23247	0.616234	−	0.787563i	$-0.288658\pi$
$37$	277.382	1.23247	0.616234	+	0.787563i	$0.288658\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−346.810	−1.32104	−0.660520	−	0.750808i	$-0.729664\pi$
$41$	−346.810	−1.32104	−0.660520	+	0.750808i	$0.729664\pi$
$42$	0	0
$43$	364.382	1.29227	0.646136	−	0.763222i	$-0.276384\pi$
$43$	364.382	1.29227	0.646136	+	0.763222i	$0.276384\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	599.707	1.86120	0.930598	−	0.366042i	$-0.119287\pi$
$47$	599.707	1.86120	0.930598	+	0.366042i	$0.119287\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	91.7610	0.237818	0.118909	−	0.992905i	$-0.462060\pi$
$53$	91.7610	0.237818	0.118909	+	0.992905i	$0.462060\pi$
$54$	0	0
$55$	−340.675	−0.835212
$56$	0	0
$57$	0	0
$58$	0	0
$59$	771.052	1.70140	0.850698	−	0.525655i	$-0.176180\pi$
$59$	771.052	1.70140	0.850698	+	0.525655i	$0.176180\pi$
$60$	0	0
$61$	373.823	0.784642	0.392321	−	0.919828i	$-0.371672\pi$
$61$	373.823	0.784642	0.392321	+	0.919828i	$0.371672\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−256.648	−0.489742
$66$	0	0
$67$	718.029	1.30927	0.654635	−	0.755945i	$-0.272822\pi$
$67$	718.029	1.30927	0.654635	+	0.755945i	$0.272822\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−248.591	−0.415526	−0.207763	−	0.978179i	$-0.566618\pi$
$71$	−248.591	−0.415526	−0.207763	+	0.978179i	$0.566618\pi$
$72$	0	0
$73$	376.735	0.604021	0.302010	−	0.953305i	$-0.402342\pi$
$73$	376.735	0.604021	0.302010	+	0.953305i	$0.402342\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	380.144	0.562616
$78$	0	0
$79$	305.795	0.435501	0.217751	−	0.976004i	$-0.430128\pi$
$79$	305.795	0.435501	0.217751	+	0.976004i	$0.430128\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−210.767	−0.278732	−0.139366	−	0.990241i	$-0.544506\pi$
$83$	−210.767	−0.278732	−0.139366	+	0.990241i	$0.544506\pi$
$84$	0	0
$85$	157.413	0.200869
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−133.890	−0.159464	−0.0797322	−	0.996816i	$-0.525407\pi$
$89$	−133.890	−0.159464	−0.0797322	+	0.996816i	$0.525407\pi$
$90$	0	0
$91$	286.382	0.329901
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−144.284	−0.155823
$96$	0	0
$97$	1045.47	1.09434	0.547172	−	0.837020i	$-0.315704\pi$
$97$	1045.47	1.09434	0.547172	+	0.837020i	$0.315704\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	422.273	0.416018	0.208009	−	0.978127i	$-0.433302\pi$
$101$	422.273	0.416018	0.208009	+	0.978127i	$0.433302\pi$
$102$	0	0
$103$	1315.38	1.25833	0.629167	−	0.777270i	$-0.283396\pi$
$103$	1315.38	1.25833	0.629167	+	0.777270i	$0.283396\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1700.16	−1.53608	−0.768042	−	0.640400i	$-0.778769\pi$
$107$	−1700.16	−1.53608	−0.768042	+	0.640400i	$0.778769\pi$
$108$	0	0
$109$	−176.499	−0.155096	−0.0775482	−	0.996989i	$-0.524709\pi$
$109$	−176.499	−0.155096	−0.0775482	+	0.996989i	$0.524709\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−710.841	−0.591773	−0.295886	−	0.955223i	$-0.595615\pi$
$113$	−710.841	−0.591773	−0.295886	+	0.955223i	$0.595615\pi$
$114$	0	0
$115$	878.117	0.712042
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−175.650	−0.135309
$120$	0	0
$121$	1618.18	1.21576
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1321.43	0.945540
$126$	0	0
$127$	176.969	0.123649	0.0618246	−	0.998087i	$-0.480308\pi$
$127$	176.969	0.123649	0.0618246	+	0.998087i	$0.480308\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−227.619	−0.151810	−0.0759052	−	0.997115i	$-0.524185\pi$
$131$	−227.619	−0.151810	−0.0759052	+	0.997115i	$0.524185\pi$
$132$	0	0
$133$	161.000	0.104966
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1866.16	−1.16377	−0.581886	−	0.813271i	$-0.697685\pi$
$137$	−1866.16	−1.16377	−0.581886	+	0.813271i	$0.697685\pi$
$138$	0	0
$139$	1477.47	0.901564	0.450782	−	0.892634i	$-0.351145\pi$
$139$	1477.47	0.901564	0.450782	+	0.892634i	$0.351145\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2221.76	1.29925
$144$	0	0
$145$	458.735	0.262730
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3263.05	1.79409	0.897046	−	0.441937i	$-0.145708\pi$
$149$	3263.05	1.79409	0.897046	+	0.441937i	$0.145708\pi$
$150$	0	0
$151$	−219.117	−0.118089	−0.0590446	−	0.998255i	$-0.518805\pi$
$151$	−219.117	−0.118089	−0.0590446	+	0.998255i	$0.518805\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	362.185	0.187686
$156$	0	0
$157$	1589.62	0.808060	0.404030	−	0.914746i	$-0.367609\pi$
$157$	1589.62	0.808060	0.404030	+	0.914746i	$0.367609\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−979.851	−0.479646
$162$	0	0
$163$	−1769.68	−0.850378	−0.425189	−	0.905105i	$-0.639792\pi$
$163$	−1769.68	−0.850378	−0.425189	+	0.905105i	$0.639792\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4002.92	−1.85482	−0.927412	−	0.374042i	$-0.877972\pi$
$167$	−4002.92	−1.85482	−0.927412	+	0.374042i	$0.877972\pi$
$168$	0	0
$169$	−523.234	−0.238158
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−186.229	−0.0818422	−0.0409211	−	0.999162i	$-0.513029\pi$
$173$	−186.229	−0.0818422	−0.0409211	+	0.999162i	$0.513029\pi$
$174$	0	0
$175$	−599.527	−0.258971
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1700.16	0.709923	0.354961	−	0.934881i	$-0.384494\pi$
$179$	1700.16	0.709923	0.354961	+	0.934881i	$0.384494\pi$
$180$	0	0
$181$	1539.58	0.632245	0.316123	−	0.948718i	$-0.397619\pi$
$181$	1539.58	0.632245	0.316123	+	0.948718i	$0.397619\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1740.08	−0.691529
$186$	0	0
$187$	−1362.70	−0.532891
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3713.13	1.40666	0.703331	−	0.710863i	$-0.251695\pi$
$191$	3713.13	1.40666	0.703331	+	0.710863i	$0.251695\pi$
$192$	0	0
$193$	4442.88	1.65702	0.828511	−	0.559972i	$-0.189188\pi$
$193$	4442.88	1.65702	0.828511	+	0.559972i	$0.189188\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2589.85	−0.936646	−0.468323	−	0.883557i	$-0.655142\pi$
$197$	−2589.85	−0.936646	−0.468323	+	0.883557i	$0.655142\pi$
$198$	0	0
$199$	−4488.91	−1.59905	−0.799523	−	0.600635i	$-0.794915\pi$
$199$	−4488.91	−1.59905	−0.799523	+	0.600635i	$0.794915\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−511.882	−0.176981
$204$	0	0
$205$	2175.62	0.741227
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1249.05	0.413389
$210$	0	0
$211$	−1317.65	−0.429908	−0.214954	−	0.976624i	$-0.568960\pi$
$211$	−1317.65	−0.429908	−0.214954	+	0.976624i	$0.568960\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2285.85	−0.725086
$216$	0	0
$217$	−404.145	−0.126429
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1026.59	−0.312471
$222$	0	0
$223$	2546.79	0.764779	0.382389	−	0.924001i	$-0.375101\pi$
$223$	2546.79	0.764779	0.382389	+	0.924001i	$0.375101\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3213.61	−0.939624	−0.469812	−	0.882767i	$-0.655678\pi$
$227$	−3213.61	−0.939624	−0.469812	+	0.882767i	$0.655678\pi$
$228$	0	0
$229$	4056.41	1.17055	0.585273	−	0.810836i	$-0.300987\pi$
$229$	4056.41	1.17055	0.585273	+	0.810836i	$0.300987\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3030.08	0.851963	0.425982	−	0.904732i	$-0.359929\pi$
$233$	3030.08	0.851963	0.425982	+	0.904732i	$0.359929\pi$
$234$	0	0
$235$	−3762.09	−1.04431
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5276.14	1.42797	0.713985	−	0.700160i	$-0.246888\pi$
$239$	5276.14	1.42797	0.713985	+	0.700160i	$0.246888\pi$
$240$	0	0
$241$	1332.85	0.356251	0.178125	−	0.984008i	$-0.442997\pi$
$241$	1332.85	0.356251	0.178125	+	0.984008i	$0.442997\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−307.388	−0.0801562
$246$	0	0
$247$	940.969	0.242398
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7220.10	1.81565	0.907826	−	0.419348i	$-0.137741\pi$
$251$	7220.10	1.81565	0.907826	+	0.419348i	$0.137741\pi$
$252$	0	0
$253$	−7601.73	−1.88900
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5500.07	1.33496	0.667480	−	0.744627i	$-0.267373\pi$
$257$	5500.07	1.33496	0.667480	+	0.744627i	$0.267373\pi$
$258$	0	0
$259$	1941.67	0.465829
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1094.25	−0.256556	−0.128278	−	0.991738i	$-0.540945\pi$
$263$	−1094.25	−0.256556	−0.128278	+	0.991738i	$0.540945\pi$
$264$	0	0
$265$	−575.636	−0.133438
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1038.28	−0.235334	−0.117667	−	0.993053i	$-0.537542\pi$
$269$	−1038.28	−0.235334	−0.117667	+	0.993053i	$0.537542\pi$
$270$	0	0
$271$	−4774.18	−1.07015	−0.535075	−	0.844805i	$-0.679717\pi$
$271$	−4774.18	−1.07015	−0.535075	+	0.844805i	$0.679717\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4651.16	−1.01991
$276$	0	0
$277$	−4393.79	−0.953058	−0.476529	−	0.879159i	$-0.658105\pi$
$277$	−4393.79	−0.953058	−0.476529	+	0.879159i	$0.658105\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3527.57	−0.748888	−0.374444	−	0.927250i	$-0.622166\pi$
$281$	−3527.57	−0.748888	−0.374444	+	0.927250i	$0.622166\pi$
$282$	0	0
$283$	−4939.35	−1.03751	−0.518753	−	0.854924i	$-0.673603\pi$
$283$	−4939.35	−1.03751	−0.518753	+	0.854924i	$0.673603\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2427.67	−0.499306
$288$	0	0
$289$	−4283.35	−0.871840
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1296.22	−0.258450	−0.129225	−	0.991615i	$-0.541249\pi$
$293$	−1296.22	−0.258450	−0.129225	+	0.991615i	$0.541249\pi$
$294$	0	0
$295$	−4836.97	−0.954642
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5726.77	−1.10765
$300$	0	0
$301$	2550.67	0.488433
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2345.08	−0.440258
$306$	0	0
$307$	−5738.77	−1.06687	−0.533434	−	0.845841i	$-0.679099\pi$
$307$	−5738.77	−1.06687	−0.533434	+	0.845841i	$0.679099\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4188.97	0.763777	0.381888	−	0.924208i	$-0.375274\pi$
$311$	4188.97	0.763777	0.381888	+	0.924208i	$0.375274\pi$
$312$	0	0
$313$	5361.14	0.968146	0.484073	−	0.875028i	$-0.339157\pi$
$313$	5361.14	0.968146	0.484073	+	0.875028i	$0.339157\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2837.28	0.502705	0.251352	−	0.967896i	$-0.419125\pi$
$317$	2837.28	0.502705	0.251352	+	0.967896i	$0.419125\pi$
$318$	0	0
$319$	−3971.20	−0.697005
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−577.136	−0.0994201
$324$	0	0
$325$	−3503.95	−0.598044
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4197.95	0.703466
$330$	0	0
$331$	−4514.05	−0.749591	−0.374796	−	0.927107i	$-0.622287\pi$
$331$	−4514.05	−0.749591	−0.374796	+	0.927107i	$0.622287\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4504.35	−0.734623
$336$	0	0
$337$	2632.46	0.425517	0.212758	−	0.977105i	$-0.431755\pi$
$337$	2632.46	0.425517	0.212758	+	0.977105i	$0.431755\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3135.38	−0.497919
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7132.46	−1.10343	−0.551715	−	0.834033i	$-0.686026\pi$
$347$	−7132.46	−1.10343	−0.551715	+	0.834033i	$0.686026\pi$
$348$	0	0
$349$	−2406.64	−0.369124	−0.184562	−	0.982821i	$-0.559087\pi$
$349$	−2406.64	−0.369124	−0.184562	+	0.982821i	$0.559087\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4385.16	0.661185	0.330593	−	0.943774i	$-0.392751\pi$
$353$	4385.16	0.661185	0.330593	+	0.943774i	$0.392751\pi$
$354$	0	0
$355$	1559.47	0.233149
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5618.09	−0.825937	−0.412969	−	0.910745i	$-0.635508\pi$
$359$	−5618.09	−0.825937	−0.412969	+	0.910745i	$0.635508\pi$
$360$	0	0
$361$	−6330.00	−0.922875
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2363.34	−0.338912
$366$	0	0
$367$	−6934.25	−0.986281	−0.493141	−	0.869950i	$-0.664151\pi$
$367$	−6934.25	−0.986281	−0.493141	+	0.869950i	$0.664151\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	642.327	0.0898866
$372$	0	0
$373$	9272.79	1.28720	0.643602	−	0.765361i	$-0.277439\pi$
$373$	9272.79	1.28720	0.643602	+	0.765361i	$0.277439\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2991.71	−0.408702
$378$	0	0
$379$	−13842.6	−1.87611	−0.938053	−	0.346491i	$-0.887373\pi$
$379$	−13842.6	−1.87611	−0.938053	+	0.346491i	$0.887373\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7076.43	0.944096	0.472048	−	0.881573i	$-0.343515\pi$
$383$	7076.43	0.944096	0.472048	+	0.881573i	$0.343515\pi$
$384$	0	0
$385$	−2384.73	−0.315680
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10637.8	1.38652	0.693260	−	0.720687i	$-0.256174\pi$
$389$	10637.8	1.38652	0.693260	+	0.720687i	$0.256174\pi$
$390$	0	0
$391$	3512.47	0.454305
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1918.32	−0.244357
$396$	0	0
$397$	−2436.82	−0.308061	−0.154031	−	0.988066i	$-0.549225\pi$
$397$	−2436.82	−0.308061	−0.154031	+	0.988066i	$0.549225\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11945.4	−1.48759	−0.743794	−	0.668409i	$-0.766976\pi$
$401$	−11945.4	−1.48759	−0.743794	+	0.668409i	$0.766976\pi$
$402$	0	0
$403$	−2362.04	−0.291964
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15063.6	1.83458
$408$	0	0
$409$	577.213	0.0697832	0.0348916	−	0.999391i	$-0.488891\pi$
$409$	577.213	0.0697832	0.0348916	+	0.999391i	$0.488891\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5397.36	0.643067
$414$	0	0
$415$	1322.19	0.156395
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−659.178	−0.0768567	−0.0384284	−	0.999261i	$-0.512235\pi$
$419$	−659.178	−0.0768567	−0.0384284	+	0.999261i	$0.512235\pi$
$420$	0	0
$421$	−8707.27	−1.00800	−0.503998	−	0.863705i	$-0.668138\pi$
$421$	−8707.27	−1.00800	−0.503998	+	0.863705i	$0.668138\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2149.12	0.245289
$426$	0	0
$427$	2616.76	0.296567
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9391.86	1.04963	0.524814	−	0.851217i	$-0.324135\pi$
$431$	9391.86	1.04963	0.524814	+	0.851217i	$0.324135\pi$
$432$	0	0
$433$	−322.964	−0.0358444	−0.0179222	−	0.999839i	$-0.505705\pi$
$433$	−322.964	−0.0358444	−0.0179222	+	0.999839i	$0.505705\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3219.51	−0.352426
$438$	0	0
$439$	9837.40	1.06951	0.534754	−	0.845008i	$-0.320404\pi$
$439$	9837.40	1.06951	0.534754	+	0.845008i	$0.320404\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3000.01	−0.321749	−0.160874	−	0.986975i	$-0.551431\pi$
$443$	−3000.01	−0.321749	−0.160874	+	0.986975i	$0.551431\pi$
$444$	0	0
$445$	839.922	0.0894744
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1257.84	−0.132207	−0.0661037	−	0.997813i	$-0.521057\pi$
$449$	−1257.84	−0.132207	−0.0661037	+	0.997813i	$0.521057\pi$
$450$	0	0
$451$	−18834.0	−1.96643
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1796.54	−0.185105
$456$	0	0
$457$	13042.3	1.33499	0.667497	−	0.744613i	$-0.267366\pi$
$457$	13042.3	1.33499	0.667497	+	0.744613i	$0.267366\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6558.71	0.662624	0.331312	−	0.943521i	$-0.392509\pi$
$461$	6558.71	0.662624	0.331312	+	0.943521i	$0.392509\pi$
$462$	0	0
$463$	−16745.9	−1.68088	−0.840442	−	0.541902i	$-0.817704\pi$
$463$	−16745.9	−1.68088	−0.840442	+	0.541902i	$0.817704\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9711.12	0.962263	0.481132	−	0.876648i	$-0.340226\pi$
$467$	9711.12	0.962263	0.481132	+	0.876648i	$0.340226\pi$
$468$	0	0
$469$	5026.20	0.494858
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19788.2	1.92360
$474$	0	0
$475$	−1969.88	−0.190282
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7967.97	−0.760054	−0.380027	−	0.924975i	$-0.624085\pi$
$479$	−7967.97	−0.760054	−0.380027	+	0.924975i	$0.624085\pi$
$480$	0	0
$481$	11348.2	1.07574
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6558.46	−0.614030
$486$	0	0
$487$	−7386.38	−0.687287	−0.343643	−	0.939100i	$-0.611661\pi$
$487$	−7386.38	−0.687287	−0.343643	+	0.939100i	$0.611661\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11918.5	−1.09547	−0.547736	−	0.836651i	$-0.684510\pi$
$491$	−11918.5	−1.09547	−0.547736	+	0.836651i	$0.684510\pi$
$492$	0	0
$493$	1834.94	0.167630
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1740.14	−0.157054
$498$	0	0
$499$	−1973.67	−0.177061	−0.0885307	−	0.996073i	$-0.528217\pi$
$499$	−1973.67	−0.177061	−0.0885307	+	0.996073i	$0.528217\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10451.8	−0.926490	−0.463245	−	0.886230i	$-0.653315\pi$
$503$	−10451.8	−0.926490	−0.463245	+	0.886230i	$0.653315\pi$
$504$	0	0
$505$	−2649.01	−0.233425
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12655.2	−1.10202	−0.551012	−	0.834497i	$-0.685758\pi$
$509$	−12655.2	−1.10202	−0.551012	+	0.834497i	$0.685758\pi$
$510$	0	0
$511$	2637.15	0.228298
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8251.68	−0.706043
$516$	0	0
$517$	32567.9	2.77047
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1754.84	−0.147564	−0.0737820	−	0.997274i	$-0.523507\pi$
$521$	−1754.84	−0.147564	−0.0737820	+	0.997274i	$0.523507\pi$
$522$	0	0
$523$	955.291	0.0798699	0.0399349	−	0.999202i	$-0.487285\pi$
$523$	955.291	0.0798699	0.0399349	+	0.999202i	$0.487285\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1448.74	0.119750
$528$	0	0
$529$	7427.04	0.610425
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14188.6	−1.15305
$534$	0	0
$535$	10665.5	0.861887
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2661.01	0.212649
$540$	0	0
$541$	−2203.95	−0.175149	−0.0875743	−	0.996158i	$-0.527912\pi$
$541$	−2203.95	−0.175149	−0.0875743	+	0.996158i	$0.527912\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1107.21	0.0870236
$546$	0	0
$547$	−18506.4	−1.44658	−0.723288	−	0.690547i	$-0.757370\pi$
$547$	−18506.4	−1.44658	−0.723288	+	0.690547i	$0.757370\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1681.90	−0.130039
$552$	0	0
$553$	2140.56	0.164604
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9152.49	−0.696236	−0.348118	−	0.937451i	$-0.613179\pi$
$557$	−9152.49	−0.696236	−0.348118	+	0.937451i	$0.613179\pi$
$558$	0	0
$559$	14907.5	1.12794
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10809.7	0.809188	0.404594	−	0.914496i	$-0.367413\pi$
$563$	10809.7	0.809188	0.404594	+	0.914496i	$0.367413\pi$
$564$	0	0
$565$	4459.26	0.332040
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5125.71	0.377647	0.188823	−	0.982011i	$-0.439533\pi$
$569$	5125.71	0.377647	0.188823	+	0.982011i	$0.439533\pi$
$570$	0	0
$571$	−11485.9	−0.841802	−0.420901	−	0.907107i	$-0.638286\pi$
$571$	−11485.9	−0.841802	−0.420901	+	0.907107i	$0.638286\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11988.7	0.869503
$576$	0	0
$577$	−15827.6	−1.14196	−0.570980	−	0.820964i	$-0.693436\pi$
$577$	−15827.6	−1.14196	−0.570980	+	0.820964i	$0.693436\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1475.37	−0.105351
$582$	0	0
$583$	4983.20	0.354002
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11595.0	0.815292	0.407646	−	0.913140i	$-0.366350\pi$
$587$	11595.0	0.815292	0.407646	+	0.913140i	$0.366350\pi$
$588$	0	0
$589$	−1327.91	−0.0928955
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1904.90	0.131914	0.0659569	−	0.997822i	$-0.478990\pi$
$593$	1904.90	0.131914	0.0659569	+	0.997822i	$0.478990\pi$
$594$	0	0
$595$	1101.89	0.0759212
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11629.3	0.793256	0.396628	−	0.917979i	$-0.370180\pi$
$599$	11629.3	0.793256	0.396628	+	0.917979i	$0.370180\pi$
$600$	0	0
$601$	18945.8	1.28588	0.642942	−	0.765915i	$-0.277713\pi$
$601$	18945.8	1.28588	0.642942	+	0.765915i	$0.277713\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10151.2	−0.682155
$606$	0	0
$607$	562.301	0.0375998	0.0187999	−	0.999823i	$-0.494015\pi$
$607$	562.301	0.0375998	0.0187999	+	0.999823i	$0.494015\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24535.0	1.62452
$612$	0	0
$613$	22628.2	1.49094	0.745470	−	0.666539i	$-0.232225\pi$
$613$	22628.2	1.49094	0.745470	+	0.666539i	$0.232225\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11749.9	0.766667	0.383333	−	0.923610i	$-0.374776\pi$
$617$	11749.9	0.766667	0.383333	+	0.923610i	$0.374776\pi$
$618$	0	0
$619$	15391.3	0.999398	0.499699	−	0.866199i	$-0.333444\pi$
$619$	15391.3	0.999398	0.499699	+	0.866199i	$0.333444\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−937.231	−0.0602719
$624$	0	0
$625$	2416.21	0.154637
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6960.30	−0.441217
$630$	0	0
$631$	−1288.97	−0.0813205	−0.0406603	−	0.999173i	$-0.512946\pi$
$631$	−1288.97	−0.0813205	−0.0406603	+	0.999173i	$0.512946\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1110.16	−0.0693788
$636$	0	0
$637$	2004.67	0.124691
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−28695.1	−1.76816	−0.884080	−	0.467336i	$-0.845214\pi$
$641$	−28695.1	−1.76816	−0.884080	+	0.467336i	$0.845214\pi$
$642$	0	0
$643$	3141.24	0.192657	0.0963286	−	0.995350i	$-0.469290\pi$
$643$	3141.24	0.192657	0.0963286	+	0.995350i	$0.469290\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26078.3	−1.58461	−0.792306	−	0.610124i	$-0.791119\pi$
$647$	−26078.3	−1.58461	−0.792306	+	0.610124i	$0.791119\pi$
$648$	0	0
$649$	41873.0	2.53260
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11132.9	−0.667175	−0.333587	−	0.942719i	$-0.608259\pi$
$653$	−11132.9	−0.667175	−0.333587	+	0.942719i	$0.608259\pi$
$654$	0	0
$655$	1427.90	0.0851799
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17907.1	1.05852	0.529259	−	0.848461i	$-0.322470\pi$
$659$	17907.1	1.05852	0.529259	+	0.848461i	$0.322470\pi$
$660$	0	0
$661$	−7229.66	−0.425418	−0.212709	−	0.977116i	$-0.568229\pi$
$661$	−7229.66	−0.425418	−0.212709	+	0.977116i	$0.568229\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1009.99	−0.0588957
$666$	0	0
$667$	10236.1	0.594217
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20301.0	1.16797
$672$	0	0
$673$	−29668.5	−1.69931	−0.849656	−	0.527338i	$-0.823190\pi$
$673$	−29668.5	−1.69931	−0.849656	+	0.527338i	$0.823190\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9401.82	−0.533739	−0.266870	−	0.963733i	$-0.585989\pi$
$677$	−9401.82	−0.533739	−0.266870	+	0.963733i	$0.585989\pi$
$678$	0	0
$679$	7318.29	0.413623
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28270.7	−1.58382	−0.791910	−	0.610638i	$-0.790913\pi$
$683$	−28270.7	−1.58382	−0.791910	+	0.610638i	$0.790913\pi$
$684$	0	0
$685$	11706.8	0.652984
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3754.10	0.207576
$690$	0	0
$691$	22644.2	1.24664	0.623319	−	0.781968i	$-0.285784\pi$
$691$	22644.2	1.24664	0.623319	+	0.781968i	$0.285784\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9268.49	−0.505862
$696$	0	0
$697$	8702.46	0.472926
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33108.1	1.78385	0.891923	−	0.452188i	$-0.149356\pi$
$701$	33108.1	1.78385	0.891923	+	0.452188i	$0.149356\pi$
$702$	0	0
$703$	6379.78	0.342273
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2955.91	0.157240
$708$	0	0
$709$	28543.9	1.51197	0.755986	−	0.654588i	$-0.227158\pi$
$709$	28543.9	1.51197	0.755986	+	0.654588i	$0.227158\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8081.68	0.424490
$714$	0	0
$715$	−13937.6	−0.729002
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6684.10	−0.346697	−0.173348	−	0.984861i	$-0.555459\pi$
$719$	−6684.10	−0.346697	−0.173348	+	0.984861i	$0.555459\pi$
$720$	0	0
$721$	9207.67	0.475606
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6263.00	0.320830
$726$	0	0
$727$	12114.4	0.618018	0.309009	−	0.951059i	$-0.400003\pi$
$727$	12114.4	0.618018	0.309009	+	0.951059i	$0.400003\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9143.38	−0.462627
$732$	0	0
$733$	22864.6	1.15214	0.576072	−	0.817399i	$-0.304585\pi$
$733$	22864.6	1.15214	0.576072	+	0.817399i	$0.304585\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38993.5	1.94891
$738$	0	0
$739$	−5748.45	−0.286144	−0.143072	−	0.989712i	$-0.545698\pi$
$739$	−5748.45	−0.286144	−0.143072	+	0.989712i	$0.545698\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38887.5	−1.92011	−0.960057	−	0.279805i	$-0.909730\pi$
$743$	−38887.5	−1.92011	−0.960057	+	0.279805i	$0.909730\pi$
$744$	0	0
$745$	−20469.8	−1.00665
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−11901.1	−0.580585
$750$	0	0
$751$	11613.5	0.564290	0.282145	−	0.959372i	$-0.408954\pi$
$751$	11613.5	0.564290	0.282145	+	0.959372i	$0.408954\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1374.57	0.0662591
$756$	0	0
$757$	−19081.3	−0.916146	−0.458073	−	0.888915i	$-0.651460\pi$
$757$	−19081.3	−0.916146	−0.458073	+	0.888915i	$0.651460\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24926.4	−1.18736	−0.593681	−	0.804701i	$-0.702326\pi$
$761$	−24926.4	−1.18736	−0.593681	+	0.804701i	$0.702326\pi$
$762$	0	0
$763$	−1235.49	−0.0586209
$764$	0	0
$765$	0	0
$766$	0	0
$767$	31545.0	1.48504
$768$	0	0
$769$	3138.69	0.147183	0.0735917	−	0.997288i	$-0.476554\pi$
$769$	3138.69	0.147183	0.0735917	+	0.997288i	$0.476554\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24628.5	1.14596	0.572979	−	0.819570i	$-0.305787\pi$
$773$	24628.5	1.14596	0.572979	+	0.819570i	$0.305787\pi$
$774$	0	0
$775$	4944.82	0.229191
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7976.63	−0.366871
$780$	0	0
$781$	−13500.1	−0.618529
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9972.02	−0.453397
$786$	0	0
$787$	−41304.8	−1.87085	−0.935424	−	0.353529i	$-0.884982\pi$
$787$	−41304.8	−1.87085	−0.935424	+	0.353529i	$0.884982\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4975.89	−0.223669
$792$	0	0
$793$	15293.7	0.684864
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10843.8	0.481941	0.240970	−	0.970532i	$-0.422534\pi$
$797$	10843.8	0.481941	0.240970	+	0.970532i	$0.422534\pi$
$798$	0	0
$799$	−15048.4	−0.666299
$800$	0	0
$801$	0	0
$802$	0	0
$803$	20459.1	0.899111
$804$	0	0
$805$	6146.82	0.269127
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11240.7	0.488506	0.244253	−	0.969712i	$-0.421457\pi$
$809$	11240.7	0.488506	0.244253	+	0.969712i	$0.421457\pi$
$810$	0	0
$811$	38327.8	1.65952	0.829761	−	0.558119i	$-0.188477\pi$
$811$	38327.8	1.65952	0.829761	+	0.558119i	$0.188477\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11101.6	0.477142
$816$	0	0
$817$	8380.78	0.358882
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38112.9	1.62016	0.810079	−	0.586321i	$-0.199424\pi$
$821$	38112.9	1.62016	0.810079	+	0.586321i	$0.199424\pi$
$822$	0	0
$823$	−28622.6	−1.21230	−0.606148	−	0.795352i	$-0.707286\pi$
$823$	−28622.6	−1.21230	−0.606148	+	0.795352i	$0.707286\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28694.4	−1.20653	−0.603265	−	0.797541i	$-0.706134\pi$
$827$	−28694.4	−1.20653	−0.603265	+	0.797541i	$0.706134\pi$
$828$	0	0
$829$	414.299	0.0173573	0.00867864	−	0.999962i	$-0.497237\pi$
$829$	414.299	0.0173573	0.00867864	+	0.999962i	$0.497237\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1229.55	−0.0511421
$834$	0	0
$835$	25111.2	1.04073
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12623.5	−0.519441	−0.259721	−	0.965684i	$-0.583630\pi$
$839$	−12623.5	−0.519441	−0.259721	+	0.965684i	$0.583630\pi$
$840$	0	0
$841$	−19041.6	−0.780745
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3282.36	0.133629
$846$	0	0
$847$	11327.2	0.459514
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−38827.6	−1.56403
$852$	0	0
$853$	−18263.6	−0.733099	−0.366549	−	0.930399i	$-0.619461\pi$
$853$	−18263.6	−0.733099	−0.366549	+	0.930399i	$0.619461\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28587.5	1.13947	0.569737	−	0.821827i	$-0.307045\pi$
$857$	28587.5	1.13947	0.569737	+	0.821827i	$0.307045\pi$
$858$	0	0
$859$	−18946.9	−0.752571	−0.376286	−	0.926504i	$-0.622799\pi$
$859$	−18946.9	−0.752571	−0.376286	+	0.926504i	$0.622799\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22689.6	−0.894974	−0.447487	−	0.894290i	$-0.647681\pi$
$863$	−22689.6	−0.894974	−0.447487	+	0.894290i	$0.647681\pi$
$864$	0	0
$865$	1168.25	0.0459211
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16606.6	0.648262
$870$	0	0
$871$	29375.8	1.14278
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9250.03	0.357381
$876$	0	0
$877$	556.805	0.0214390	0.0107195	−	0.999943i	$-0.496588\pi$
$877$	556.805	0.0214390	0.0107195	+	0.999943i	$0.496588\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42746.8	−1.63471	−0.817354	−	0.576136i	$-0.804560\pi$
$881$	−42746.8	−1.63471	−0.817354	+	0.576136i	$0.804560\pi$
$882$	0	0
$883$	3022.30	0.115185	0.0575926	−	0.998340i	$-0.481658\pi$
$883$	3022.30	0.115185	0.0575926	+	0.998340i	$0.481658\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41135.8	1.55717	0.778583	−	0.627542i	$-0.215939\pi$
$887$	41135.8	1.55717	0.778583	+	0.627542i	$0.215939\pi$
$888$	0	0
$889$	1238.78	0.0467350
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13793.3	0.516880
$894$	0	0
$895$	−10665.5	−0.398333
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4221.93	0.156629
$900$	0	0
$901$	−2302.55	−0.0851375
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9658.15	−0.354749
$906$	0	0
$907$	−52198.9	−1.91095	−0.955477	−	0.295067i	$-0.904658\pi$
$907$	−52198.9	−1.91095	−0.955477	+	0.295067i	$0.904658\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27366.7	−0.995279	−0.497640	−	0.867384i	$-0.665800\pi$
$911$	−27366.7	−0.995279	−0.497640	+	0.867384i	$0.665800\pi$
$912$	0	0
$913$	−11446.0	−0.414904
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1593.33	−0.0573790
$918$	0	0
$919$	12079.8	0.433599	0.216799	−	0.976216i	$-0.430438\pi$
$919$	12079.8	0.433599	0.216799	+	0.976216i	$0.430438\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10170.3	−0.362686
$924$	0	0
$925$	−23756.9	−0.844455
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22201.4	−0.784073	−0.392036	−	0.919950i	$-0.628229\pi$
$929$	−22201.4	−0.784073	−0.392036	+	0.919950i	$0.628229\pi$
$930$	0	0
$931$	1127.00	0.0396734
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8548.52	0.299002
$936$	0	0
$937$	5170.30	0.180263	0.0901315	−	0.995930i	$-0.471271\pi$
$937$	5170.30	0.180263	0.0901315	+	0.995930i	$0.471271\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30559.9	−1.05869	−0.529343	−	0.848408i	$-0.677562\pi$
$941$	−30559.9	−1.05869	−0.529343	+	0.848408i	$0.677562\pi$
$942$	0	0
$943$	48546.1	1.67643
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17034.0	−0.584511	−0.292256	−	0.956340i	$-0.594406\pi$
$947$	−17034.0	−0.584511	−0.292256	+	0.956340i	$0.594406\pi$
$948$	0	0
$949$	15412.9	0.527211
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19824.1	−0.673837	−0.336919	−	0.941534i	$-0.609385\pi$
$953$	−19824.1	−0.673837	−0.336919	+	0.941534i	$0.609385\pi$
$954$	0	0
$955$	−23293.2	−0.789269
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13063.1	−0.439864
$960$	0	0
$961$	−26457.7	−0.888109
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−27871.1	−0.929745
$966$	0	0
$967$	−49000.7	−1.62953	−0.814765	−	0.579791i	$-0.803134\pi$
$967$	−49000.7	−1.62953	−0.814765	+	0.579791i	$0.803134\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25383.3	0.838918	0.419459	−	0.907774i	$-0.362220\pi$
$971$	25383.3	0.838918	0.419459	+	0.907774i	$0.362220\pi$
$972$	0	0
$973$	10342.3	0.340759
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17385.8	0.569314	0.284657	−	0.958629i	$-0.408120\pi$
$977$	17385.8	0.569314	0.284657	+	0.958629i	$0.408120\pi$
$978$	0	0
$979$	−7271.08	−0.237370
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10951.6	−0.355343	−0.177671	−	0.984090i	$-0.556856\pi$
$983$	−10951.6	−0.355343	−0.177671	+	0.984090i	$0.556856\pi$
$984$	0	0
$985$	16246.7	0.525546
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−51005.7	−1.63993
$990$	0	0
$991$	48190.0	1.54471	0.772354	−	0.635193i	$-0.219079\pi$
$991$	48190.0	1.54471	0.772354	+	0.635193i	$0.219079\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	28159.9	0.897215
$996$	0	0
$997$	−21310.0	−0.676925	−0.338463	−	0.940980i	$-0.609907\pi$
$997$	−21310.0	−0.676925	−0.338463	+	0.940980i	$0.609907\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.a.l.1.2	✓	4
3.2	odd	2	inner	756.4.a.l.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.a.l.1.2	✓	4	1.1	even	1	trivial
756.4.a.l.1.3	yes	4	3.2	odd	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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.a (trivial)

\(f(q)\)	\(=\)	\(q-6.27322 q^{5} +7.00000 q^{7} +54.3063 q^{11} +40.9117 q^{13} -25.0929 q^{17} +23.0000 q^{19} -139.979 q^{23} -85.6468 q^{25} -73.1260 q^{29} -57.7351 q^{31} -43.9125 q^{35} +277.382 q^{37} -346.810 q^{41} +364.382 q^{43} +599.707 q^{47} +49.0000 q^{49} +91.7610 q^{53} -340.675 q^{55} +771.052 q^{59} +373.823 q^{61} -256.648 q^{65} +718.029 q^{67} -248.591 q^{71} +376.735 q^{73} +380.144 q^{77} +305.795 q^{79} -210.767 q^{83} +157.413 q^{85} -133.890 q^{89} +286.382 q^{91} -144.284 q^{95} +1045.47 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{7} - 40 q^{13} + 92 q^{19} + 472 q^{25} + 380 q^{31} - 316 q^{37} + 32 q^{43} + 196 q^{49} + 1692 q^{55} + 1088 q^{61} + 632 q^{67} + 896 q^{73} + 3056 q^{79} + 3888 q^{85} - 280 q^{91} + 2960 q^{97}+O(q^{100}) \) 4 * q + 28 * q^7 - 40 * q^13 + 92 * q^19 + 472 * q^25 + 380 * q^31 - 316 * q^37 + 32 * q^43 + 196 * q^49 + 1692 * q^55 + 1088 * q^61 + 632 * q^67 + 896 * q^73 + 3056 * q^79 + 3888 * q^85 - 280 * q^91 + 2960 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more