LMFDB - Embedded newform 1936.4.a.bs.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1936,4,Mod(1,1936)] 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("1936.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1936.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3,0,-12,0,8] 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:	$114.227697771$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 82x^{4} + 161x^{3} + 1730x^{2} - 2271x - 5931 $ x^6 - 3x^5 - 82x^4 + 161x^3 + 1730x^2 - 2271*x - 5931
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 11 $
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.46299$ of defining polynomial
Character	$\chi$	$=$	1936.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.46299 q^{3} -0.0877400 q^{5} -24.4358 q^{7} +14.7703 q^{9} +33.0082 q^{13} +0.567063 q^{15} -48.6615 q^{17} -107.554 q^{19} +157.929 q^{21} -141.967 q^{23} -124.992 q^{25} +79.0405 q^{27} +122.803 q^{29} +124.724 q^{31} +2.14400 q^{35} -194.324 q^{37} -213.332 q^{39} +158.867 q^{41} -451.693 q^{43} -1.29595 q^{45} -609.110 q^{47} +254.110 q^{49} +314.499 q^{51} -348.442 q^{53} +695.118 q^{57} +93.3402 q^{59} -458.897 q^{61} -360.924 q^{63} -2.89614 q^{65} +768.836 q^{67} +917.529 q^{69} -1112.73 q^{71} -1057.88 q^{73} +807.824 q^{75} -49.0348 q^{79} -909.636 q^{81} -1001.34 q^{83} +4.26956 q^{85} -793.677 q^{87} +948.854 q^{89} -806.582 q^{91} -806.093 q^{93} +9.43675 q^{95} -409.188 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 12 q^{5} + 8 q^{7} + 11 q^{9} + 80 q^{13} + 98 q^{15} + 113 q^{17} - 53 q^{19} + 152 q^{21} + 194 q^{23} + 476 q^{25} - 72 q^{27} + 374 q^{29} - 16 q^{31} - 1044 q^{35} - 456 q^{37} - 592 q^{39}+ \cdots + 683 q^{97}+O(q^{100}) $ 6 * q - 3 * q^3 - 12 * q^5 + 8 * q^7 + 11 * q^9 + 80 * q^13 + 98 * q^15 + 113 * q^17 - 53 * q^19 + 152 * q^21 + 194 * q^23 + 476 * q^25 - 72 * q^27 + 374 * q^29 - 16 * q^31 - 1044 * q^35 - 456 * q^37 - 592 * q^39 + 825 * q^41 - 1165 * q^43 - 670 * q^45 - 1248 * q^47 - 4 * q^49 - 988 * q^51 - 122 * q^53 + 2282 * q^57 + 767 * q^59 + 1010 * q^61 - 254 * q^63 + 1946 * q^65 + 2197 * q^67 + 2648 * q^69 - 598 * q^71 + 1523 * q^73 - 609 * q^75 - 2116 * q^79 - 2106 * q^81 - 1213 * q^83 + 1066 * q^85 - 1518 * q^87 - 477 * q^89 - 2420 * q^91 - 714 * q^93 - 404 * q^95 + 683 * 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
$3$	−6.46299	−1.24380	−0.621902	−	0.783095i	$-0.713640\pi$
$3$	−6.46299	−1.24380	−0.621902	+	0.783095i	$0.713640\pi$
$4$	0	0
$5$	−0.0877400	−0.00784770	−0.00392385	−	0.999992i	$-0.501249\pi$
$5$	−0.0877400	−0.00784770	−0.00392385	+	0.999992i	$0.501249\pi$
$6$	0	0
$7$	−24.4358	−1.31941	−0.659706	−	0.751524i	$-0.729319\pi$
$7$	−24.4358	−1.31941	−0.659706	+	0.751524i	$0.729319\pi$
$8$	0	0
$9$	14.7703	0.547048
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	33.0082	0.704217	0.352108	−	0.935959i	$-0.385465\pi$
$13$	33.0082	0.704217	0.352108	+	0.935959i	$0.385465\pi$
$14$	0	0
$15$	0.567063	0.00976100
$16$	0	0
$17$	−48.6615	−0.694244	−0.347122	−	0.937820i	$-0.612841\pi$
$17$	−48.6615	−0.694244	−0.347122	+	0.937820i	$0.612841\pi$
$18$	0	0
$19$	−107.554	−1.29866	−0.649329	−	0.760508i	$-0.724950\pi$
$19$	−107.554	−1.29866	−0.649329	+	0.760508i	$0.724950\pi$
$20$	0	0
$21$	157.929	1.64109
$22$	0	0
$23$	−141.967	−1.28705	−0.643523	−	0.765427i	$-0.722528\pi$
$23$	−141.967	−1.28705	−0.643523	+	0.765427i	$0.722528\pi$
$24$	0	0
$25$	−124.992	−0.999938
$26$	0	0
$27$	79.0405	0.563384
$28$	0	0
$29$	122.803	0.786345	0.393172	−	0.919465i	$-0.371378\pi$
$29$	122.803	0.786345	0.393172	+	0.919465i	$0.371378\pi$
$30$	0	0
$31$	124.724	0.722618	0.361309	−	0.932446i	$-0.382330\pi$
$31$	124.724	0.722618	0.361309	+	0.932446i	$0.382330\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.14400	0.0103544
$36$	0	0
$37$	−194.324	−0.863422	−0.431711	−	0.902012i	$-0.642090\pi$
$37$	−194.324	−0.863422	−0.431711	+	0.902012i	$0.642090\pi$
$38$	0	0
$39$	−213.332	−0.875908
$40$	0	0
$41$	158.867	0.605141	0.302571	−	0.953127i	$-0.402155\pi$
$41$	158.867	0.605141	0.302571	+	0.953127i	$0.402155\pi$
$42$	0	0
$43$	−451.693	−1.60192	−0.800960	−	0.598718i	$-0.795677\pi$
$43$	−451.693	−1.60192	−0.800960	+	0.598718i	$0.795677\pi$
$44$	0	0
$45$	−1.29595	−0.00429307
$46$	0	0
$47$	−609.110	−1.89038	−0.945189	−	0.326522i	$-0.894123\pi$
$47$	−609.110	−1.89038	−0.945189	+	0.326522i	$0.894123\pi$
$48$	0	0
$49$	254.110	0.740846
$50$	0	0
$51$	314.499	0.863503
$52$	0	0
$53$	−348.442	−0.903059	−0.451530	−	0.892256i	$-0.649121\pi$
$53$	−348.442	−0.903059	−0.451530	+	0.892256i	$0.649121\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	695.118	1.61527
$58$	0	0
$59$	93.3402	0.205964	0.102982	−	0.994683i	$-0.467162\pi$
$59$	93.3402	0.205964	0.102982	+	0.994683i	$0.467162\pi$
$60$	0	0
$61$	−458.897	−0.963208	−0.481604	−	0.876389i	$-0.659946\pi$
$61$	−458.897	−0.963208	−0.481604	+	0.876389i	$0.659946\pi$
$62$	0	0
$63$	−360.924	−0.721781
$64$	0	0
$65$	−2.89614	−0.00552649
$66$	0	0
$67$	768.836	1.40191	0.700957	−	0.713204i	$-0.252757\pi$
$67$	768.836	1.40191	0.700957	+	0.713204i	$0.252757\pi$
$68$	0	0
$69$	917.529	1.60083
$70$	0	0
$71$	−1112.73	−1.85996	−0.929979	−	0.367613i	$-0.880175\pi$
$71$	−1112.73	−1.85996	−0.929979	+	0.367613i	$0.880175\pi$
$72$	0	0
$73$	−1057.88	−1.69611	−0.848055	−	0.529909i	$-0.822226\pi$
$73$	−1057.88	−1.69611	−0.848055	+	0.529909i	$0.822226\pi$
$74$	0	0
$75$	807.824	1.24373
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−49.0348	−0.0698335	−0.0349168	−	0.999390i	$-0.511117\pi$
$79$	−49.0348	−0.0698335	−0.0349168	+	0.999390i	$0.511117\pi$
$80$	0	0
$81$	−909.636	−1.24779
$82$	0	0
$83$	−1001.34	−1.32424	−0.662118	−	0.749400i	$-0.730342\pi$
$83$	−1001.34	−1.32424	−0.662118	+	0.749400i	$0.730342\pi$
$84$	0	0
$85$	4.26956	0.00544822
$86$	0	0
$87$	−793.677	−0.978059
$88$	0	0
$89$	948.854	1.13009	0.565047	−	0.825059i	$-0.308858\pi$
$89$	948.854	1.13009	0.565047	+	0.825059i	$0.308858\pi$
$90$	0	0
$91$	−806.582	−0.929152
$92$	0	0
$93$	−806.093	−0.898795
$94$	0	0
$95$	9.43675	0.0101915
$96$	0	0
$97$	−409.188	−0.428317	−0.214159	−	0.976799i	$-0.568701\pi$
$97$	−409.188	−0.428317	−0.214159	+	0.976799i	$0.568701\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−295.630	−0.291250	−0.145625	−	0.989340i	$-0.546519\pi$
$101$	−295.630	−0.291250	−0.145625	+	0.989340i	$0.546519\pi$
$102$	0	0
$103$	−47.3100	−0.0452582	−0.0226291	−	0.999744i	$-0.507204\pi$
$103$	−47.3100	−0.0452582	−0.0226291	+	0.999744i	$0.507204\pi$
$104$	0	0
$105$	−13.8567	−0.0128788
$106$	0	0
$107$	132.367	0.119593	0.0597963	−	0.998211i	$-0.480955\pi$
$107$	132.367	0.119593	0.0597963	+	0.998211i	$0.480955\pi$
$108$	0	0
$109$	1326.34	1.16551	0.582756	−	0.812647i	$-0.301974\pi$
$109$	1326.34	1.16551	0.582756	+	0.812647i	$0.301974\pi$
$110$	0	0
$111$	1255.91	1.07393
$112$	0	0
$113$	−1315.89	−1.09547	−0.547736	−	0.836651i	$-0.684510\pi$
$113$	−1315.89	−1.09547	−0.547736	+	0.836651i	$0.684510\pi$
$114$	0	0
$115$	12.4561	0.0101004
$116$	0	0
$117$	487.540	0.385240
$118$	0	0
$119$	1189.08	0.915993
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−1026.75	−0.752677
$124$	0	0
$125$	21.9343	0.0156949
$126$	0	0
$127$	345.188	0.241185	0.120592	−	0.992702i	$-0.461521\pi$
$127$	345.188	0.241185	0.120592	+	0.992702i	$0.461521\pi$
$128$	0	0
$129$	2919.29	1.99247
$130$	0	0
$131$	31.6857	0.0211328	0.0105664	−	0.999944i	$-0.496637\pi$
$131$	31.6857	0.0211328	0.0105664	+	0.999944i	$0.496637\pi$
$132$	0	0
$133$	2628.16	1.71346
$134$	0	0
$135$	−6.93502	−0.00442127
$136$	0	0
$137$	954.964	0.595534	0.297767	−	0.954639i	$-0.403758\pi$
$137$	954.964	0.595534	0.297767	+	0.954639i	$0.403758\pi$
$138$	0	0
$139$	−1386.54	−0.846076	−0.423038	−	0.906112i	$-0.639036\pi$
$139$	−1386.54	−0.846076	−0.423038	+	0.906112i	$0.639036\pi$
$140$	0	0
$141$	3936.67	2.35126
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.7748	−0.00617100
$146$	0	0
$147$	−1642.31	−0.921467
$148$	0	0
$149$	−2651.61	−1.45791	−0.728955	−	0.684562i	$-0.759994\pi$
$149$	−2651.61	−1.45791	−0.728955	+	0.684562i	$0.759994\pi$
$150$	0	0
$151$	2019.53	1.08839	0.544196	−	0.838958i	$-0.316835\pi$
$151$	2019.53	1.08839	0.544196	+	0.838958i	$0.316835\pi$
$152$	0	0
$153$	−718.745	−0.379785
$154$	0	0
$155$	−10.9433	−0.00567090
$156$	0	0
$157$	−808.315	−0.410896	−0.205448	−	0.978668i	$-0.565865\pi$
$157$	−808.315	−0.410896	−0.205448	+	0.978668i	$0.565865\pi$
$158$	0	0
$159$	2251.98	1.12323
$160$	0	0
$161$	3469.07	1.69814
$162$	0	0
$163$	−92.1279	−0.0442700	−0.0221350	−	0.999755i	$-0.507046\pi$
$163$	−92.1279	−0.0442700	−0.0221350	+	0.999755i	$0.507046\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	634.393	0.293957	0.146978	−	0.989140i	$-0.453045\pi$
$167$	634.393	0.293957	0.146978	+	0.989140i	$0.453045\pi$
$168$	0	0
$169$	−1107.46	−0.504079
$170$	0	0
$171$	−1588.60	−0.710428
$172$	0	0
$173$	−2081.96	−0.914960	−0.457480	−	0.889220i	$-0.651248\pi$
$173$	−2081.96	−0.914960	−0.457480	+	0.889220i	$0.651248\pi$
$174$	0	0
$175$	3054.29	1.31933
$176$	0	0
$177$	−603.257	−0.256179
$178$	0	0
$179$	1046.15	0.436831	0.218415	−	0.975856i	$-0.429911\pi$
$179$	1046.15	0.436831	0.218415	+	0.975856i	$0.429911\pi$
$180$	0	0
$181$	−2555.26	−1.04934	−0.524671	−	0.851305i	$-0.675812\pi$
$181$	−2555.26	−1.04934	−0.524671	+	0.851305i	$0.675812\pi$
$182$	0	0
$183$	2965.85	1.19804
$184$	0	0
$185$	17.0500	0.00677588
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1931.42	−0.743335
$190$	0	0
$191$	478.156	0.181142	0.0905710	−	0.995890i	$-0.471131\pi$
$191$	478.156	0.181142	0.0905710	+	0.995890i	$0.471131\pi$
$192$	0	0
$193$	1769.57	0.659981	0.329990	−	0.943984i	$-0.392954\pi$
$193$	1769.57	0.659981	0.329990	+	0.943984i	$0.392954\pi$
$194$	0	0
$195$	18.7177	0.00687386
$196$	0	0
$197$	524.670	0.189752	0.0948761	−	0.995489i	$-0.469755\pi$
$197$	524.670	0.189752	0.0948761	+	0.995489i	$0.469755\pi$
$198$	0	0
$199$	1061.88	0.378264	0.189132	−	0.981952i	$-0.439433\pi$
$199$	1061.88	0.378264	0.189132	+	0.981952i	$0.439433\pi$
$200$	0	0
$201$	−4968.98	−1.74371
$202$	0	0
$203$	−3000.80	−1.03751
$204$	0	0
$205$	−13.9390	−0.00474897
$206$	0	0
$207$	−2096.89	−0.704076
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5374.01	1.75338	0.876688	−	0.481059i	$-0.159748\pi$
$211$	5374.01	1.75338	0.876688	+	0.481059i	$0.159748\pi$
$212$	0	0
$213$	7191.58	2.31342
$214$	0	0
$215$	39.6315	0.0125714
$216$	0	0
$217$	−3047.75	−0.953431
$218$	0	0
$219$	6837.10	2.10963
$220$	0	0
$221$	−1606.23	−0.488898
$222$	0	0
$223$	−1153.64	−0.346429	−0.173215	−	0.984884i	$-0.555415\pi$
$223$	−1153.64	−0.346429	−0.173215	+	0.984884i	$0.555415\pi$
$224$	0	0
$225$	−1846.17	−0.547014
$226$	0	0
$227$	6073.94	1.77595	0.887977	−	0.459887i	$-0.152110\pi$
$227$	6073.94	1.77595	0.887977	+	0.459887i	$0.152110\pi$
$228$	0	0
$229$	−2752.78	−0.794363	−0.397182	−	0.917740i	$-0.630012\pi$
$229$	−2752.78	−0.794363	−0.397182	+	0.917740i	$0.630012\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2953.86	0.830531	0.415265	−	0.909700i	$-0.363689\pi$
$233$	2953.86	0.830531	0.415265	+	0.909700i	$0.363689\pi$
$234$	0	0
$235$	53.4433	0.0148351
$236$	0	0
$237$	316.912	0.0868592
$238$	0	0
$239$	5493.73	1.48686	0.743430	−	0.668814i	$-0.233198\pi$
$239$	5493.73	1.48686	0.743430	+	0.668814i	$0.233198\pi$
$240$	0	0
$241$	−4038.21	−1.07935	−0.539676	−	0.841873i	$-0.681453\pi$
$241$	−4038.21	−1.07935	−0.539676	+	0.841873i	$0.681453\pi$
$242$	0	0
$243$	3744.88	0.988618
$244$	0	0
$245$	−22.2956	−0.00581394
$246$	0	0
$247$	−3550.15	−0.914536
$248$	0	0
$249$	6471.67	1.64709
$250$	0	0
$251$	−4469.34	−1.12391	−0.561957	−	0.827167i	$-0.689951\pi$
$251$	−4469.34	−1.12391	−0.561957	+	0.827167i	$0.689951\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−27.5941	−0.00677652
$256$	0	0
$257$	−4738.19	−1.15004	−0.575020	−	0.818139i	$-0.695006\pi$
$257$	−4738.19	−1.15004	−0.575020	+	0.818139i	$0.695006\pi$
$258$	0	0
$259$	4748.46	1.13921
$260$	0	0
$261$	1813.84	0.430168
$262$	0	0
$263$	3942.64	0.924386	0.462193	−	0.886779i	$-0.347063\pi$
$263$	3942.64	0.924386	0.462193	+	0.886779i	$0.347063\pi$
$264$	0	0
$265$	30.5723	0.00708694
$266$	0	0
$267$	−6132.44	−1.40561
$268$	0	0
$269$	−3402.46	−0.771196	−0.385598	−	0.922667i	$-0.626005\pi$
$269$	−3402.46	−0.771196	−0.385598	+	0.922667i	$0.626005\pi$
$270$	0	0
$271$	3984.02	0.893033	0.446516	−	0.894775i	$-0.352664\pi$
$271$	3984.02	0.893033	0.446516	+	0.894775i	$0.352664\pi$
$272$	0	0
$273$	5212.94	1.15568
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3537.07	−0.767227	−0.383614	−	0.923494i	$-0.625321\pi$
$277$	−3537.07	−0.767227	−0.383614	+	0.923494i	$0.625321\pi$
$278$	0	0
$279$	1842.22	0.395307
$280$	0	0
$281$	4377.68	0.929362	0.464681	−	0.885478i	$-0.346169\pi$
$281$	4377.68	0.929362	0.464681	+	0.885478i	$0.346169\pi$
$282$	0	0
$283$	2481.81	0.521302	0.260651	−	0.965433i	$-0.416063\pi$
$283$	2481.81	0.521302	0.260651	+	0.965433i	$0.416063\pi$
$284$	0	0
$285$	−60.9897	−0.0126762
$286$	0	0
$287$	−3882.04	−0.798431
$288$	0	0
$289$	−2545.06	−0.518025
$290$	0	0
$291$	2644.58	0.532743
$292$	0	0
$293$	4468.30	0.890924	0.445462	−	0.895301i	$-0.353039\pi$
$293$	4468.30	0.890924	0.445462	+	0.895301i	$0.353039\pi$
$294$	0	0
$295$	−8.18967	−0.00161634
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4686.05	−0.906360
$300$	0	0
$301$	11037.5	2.11359
$302$	0	0
$303$	1910.66	0.362258
$304$	0	0
$305$	40.2636	0.00755897
$306$	0	0
$307$	−3630.47	−0.674925	−0.337462	−	0.941339i	$-0.609569\pi$
$307$	−3630.47	−0.674925	−0.337462	+	0.941339i	$0.609569\pi$
$308$	0	0
$309$	305.764	0.0562923
$310$	0	0
$311$	8775.39	1.60002	0.800011	−	0.599986i	$-0.204827\pi$
$311$	8775.39	1.60002	0.800011	+	0.599986i	$0.204827\pi$
$312$	0	0
$313$	3371.15	0.608782	0.304391	−	0.952547i	$-0.401547\pi$
$313$	3371.15	0.608782	0.304391	+	0.952547i	$0.401547\pi$
$314$	0	0
$315$	31.6675	0.00566433
$316$	0	0
$317$	−4514.05	−0.799794	−0.399897	−	0.916560i	$-0.630954\pi$
$317$	−4514.05	−0.799794	−0.399897	+	0.916560i	$0.630954\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−855.488	−0.148750
$322$	0	0
$323$	5233.72	0.901585
$324$	0	0
$325$	−4125.77	−0.704173
$326$	0	0
$327$	−8572.16	−1.44967
$328$	0	0
$329$	14884.1	2.49419
$330$	0	0
$331$	−575.331	−0.0955379	−0.0477690	−	0.998858i	$-0.515211\pi$
$331$	−575.331	−0.0955379	−0.0477690	+	0.998858i	$0.515211\pi$
$332$	0	0
$333$	−2870.22	−0.472333
$334$	0	0
$335$	−67.4576	−0.0110018
$336$	0	0
$337$	6858.18	1.10857	0.554286	−	0.832326i	$-0.312991\pi$
$337$	6858.18	1.10857	0.554286	+	0.832326i	$0.312991\pi$
$338$	0	0
$339$	8504.58	1.36255
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2172.10	0.341930
$344$	0	0
$345$	−80.5040	−0.0125629
$346$	0	0
$347$	−7229.04	−1.11837	−0.559186	−	0.829042i	$-0.688886\pi$
$347$	−7229.04	−1.11837	−0.559186	+	0.829042i	$0.688886\pi$
$348$	0	0
$349$	12831.1	1.96801	0.984003	−	0.178151i	$-0.0570116\pi$
$349$	12831.1	1.96801	0.984003	+	0.178151i	$0.0570116\pi$
$350$	0	0
$351$	2608.98	0.396744
$352$	0	0
$353$	9780.78	1.47473	0.737363	−	0.675497i	$-0.236071\pi$
$353$	9780.78	1.47473	0.737363	+	0.675497i	$0.236071\pi$
$354$	0	0
$355$	97.6311	0.0145964
$356$	0	0
$357$	−7685.05	−1.13932
$358$	0	0
$359$	551.839	0.0811279	0.0405640	−	0.999177i	$-0.487085\pi$
$359$	551.839	0.0811279	0.0405640	+	0.999177i	$0.487085\pi$
$360$	0	0
$361$	4708.77	0.686510
$362$	0	0
$363$	0	0
$364$	0	0
$365$	92.8188	0.0133106
$366$	0	0
$367$	4462.72	0.634746	0.317373	−	0.948301i	$-0.397199\pi$
$367$	4462.72	0.634746	0.317373	+	0.948301i	$0.397199\pi$
$368$	0	0
$369$	2346.51	0.331041
$370$	0	0
$371$	8514.46	1.19151
$372$	0	0
$373$	10086.8	1.40020	0.700102	−	0.714043i	$-0.253138\pi$
$373$	10086.8	1.40020	0.700102	+	0.714043i	$0.253138\pi$
$374$	0	0
$375$	−141.761	−0.0195214
$376$	0	0
$377$	4053.51	0.553757
$378$	0	0
$379$	478.104	0.0647983	0.0323991	−	0.999475i	$-0.489685\pi$
$379$	478.104	0.0647983	0.0323991	+	0.999475i	$0.489685\pi$
$380$	0	0
$381$	−2230.95	−0.299986
$382$	0	0
$383$	17.4676	0.00233043	0.00116521	−	0.999999i	$-0.499629\pi$
$383$	17.4676	0.00233043	0.00116521	+	0.999999i	$0.499629\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6671.64	−0.876326
$388$	0	0
$389$	−11552.5	−1.50575	−0.752873	−	0.658165i	$-0.771333\pi$
$389$	−11552.5	−1.50575	−0.752873	+	0.658165i	$0.771333\pi$
$390$	0	0
$391$	6908.30	0.893524
$392$	0	0
$393$	−204.785	−0.0262850
$394$	0	0
$395$	4.30231	0.000548033 0
$396$	0	0
$397$	−1357.56	−0.171622	−0.0858109	−	0.996311i	$-0.527348\pi$
$397$	−1357.56	−0.171622	−0.0858109	+	0.996311i	$0.527348\pi$
$398$	0	0
$399$	−16985.8	−2.13121
$400$	0	0
$401$	8849.49	1.10205	0.551026	−	0.834488i	$-0.314237\pi$
$401$	8849.49	1.10205	0.551026	+	0.834488i	$0.314237\pi$
$402$	0	0
$403$	4116.92	0.508880
$404$	0	0
$405$	79.8115	0.00979226
$406$	0	0
$407$	0	0
$408$	0	0
$409$	8506.04	1.02835	0.514177	−	0.857684i	$-0.328098\pi$
$409$	8506.04	1.02835	0.514177	+	0.857684i	$0.328098\pi$
$410$	0	0
$411$	−6171.93	−0.740727
$412$	0	0
$413$	−2280.85	−0.271751
$414$	0	0
$415$	87.8577	0.0103922
$416$	0	0
$417$	8961.18	1.05235
$418$	0	0
$419$	8675.96	1.01157	0.505786	−	0.862659i	$-0.331203\pi$
$419$	8675.96	1.01157	0.505786	+	0.862659i	$0.331203\pi$
$420$	0	0
$421$	9073.31	1.05037	0.525185	−	0.850988i	$-0.323996\pi$
$421$	9073.31	1.05037	0.525185	+	0.850988i	$0.323996\pi$
$422$	0	0
$423$	−8996.73	−1.03413
$424$	0	0
$425$	6082.31	0.694201
$426$	0	0
$427$	11213.5	1.27087
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15056.4	−1.68269	−0.841346	−	0.540496i	$-0.818237\pi$
$431$	−15056.4	−1.68269	−0.841346	+	0.540496i	$0.818237\pi$
$432$	0	0
$433$	−4529.27	−0.502685	−0.251343	−	0.967898i	$-0.580872\pi$
$433$	−4529.27	−0.502685	−0.251343	+	0.967898i	$0.580872\pi$
$434$	0	0
$435$	69.6372	0.00767551
$436$	0	0
$437$	15269.0	1.67143
$438$	0	0
$439$	1061.24	0.115376	0.0576881	−	0.998335i	$-0.481627\pi$
$439$	1061.24	0.115376	0.0576881	+	0.998335i	$0.481627\pi$
$440$	0	0
$441$	3753.28	0.405278
$442$	0	0
$443$	−5691.66	−0.610427	−0.305213	−	0.952284i	$-0.598728\pi$
$443$	−5691.66	−0.610427	−0.305213	+	0.952284i	$0.598728\pi$
$444$	0	0
$445$	−83.2524	−0.00886864
$446$	0	0
$447$	17137.4	1.81335
$448$	0	0
$449$	−8065.89	−0.847779	−0.423889	−	0.905714i	$-0.639335\pi$
$449$	−8065.89	−0.847779	−0.423889	+	0.905714i	$0.639335\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−13052.2	−1.35375
$454$	0	0
$455$	70.7695	0.00729171
$456$	0	0
$457$	3229.97	0.330617	0.165308	−	0.986242i	$-0.447138\pi$
$457$	3229.97	0.330617	0.165308	+	0.986242i	$0.447138\pi$
$458$	0	0
$459$	−3846.23	−0.391126
$460$	0	0
$461$	−14797.9	−1.49503	−0.747515	−	0.664245i	$-0.768753\pi$
$461$	−14797.9	−1.49503	−0.747515	+	0.664245i	$0.768753\pi$
$462$	0	0
$463$	11153.9	1.11958	0.559792	−	0.828633i	$-0.310881\pi$
$463$	11153.9	1.11958	0.559792	+	0.828633i	$0.310881\pi$
$464$	0	0
$465$	70.7266	0.00705348
$466$	0	0
$467$	−19020.7	−1.88474	−0.942368	−	0.334579i	$-0.891406\pi$
$467$	−19020.7	−1.88474	−0.942368	+	0.334579i	$0.891406\pi$
$468$	0	0
$469$	−18787.1	−1.84970
$470$	0	0
$471$	5224.14	0.511073
$472$	0	0
$473$	0	0
$474$	0	0
$475$	13443.4	1.29858
$476$	0	0
$477$	−5146.58	−0.494017
$478$	0	0
$479$	−10637.4	−1.01469	−0.507345	−	0.861743i	$-0.669373\pi$
$479$	−10637.4	−1.01469	−0.507345	+	0.861743i	$0.669373\pi$
$480$	0	0
$481$	−6414.27	−0.608037
$482$	0	0
$483$	−22420.6	−2.11216
$484$	0	0
$485$	35.9022	0.00336131
$486$	0	0
$487$	−7747.98	−0.720933	−0.360467	−	0.932772i	$-0.617383\pi$
$487$	−7747.98	−0.720933	−0.360467	+	0.932772i	$0.617383\pi$
$488$	0	0
$489$	595.422	0.0550632
$490$	0	0
$491$	−10054.0	−0.924093	−0.462047	−	0.886856i	$-0.652885\pi$
$491$	−10054.0	−0.924093	−0.462047	+	0.886856i	$0.652885\pi$
$492$	0	0
$493$	−5975.79	−0.545915
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27190.5	2.45405
$498$	0	0
$499$	−4382.69	−0.393178	−0.196589	−	0.980486i	$-0.562987\pi$
$499$	−4382.69	−0.393178	−0.196589	+	0.980486i	$0.562987\pi$
$500$	0	0
$501$	−4100.08	−0.365624
$502$	0	0
$503$	5236.71	0.464201	0.232101	−	0.972692i	$-0.425440\pi$
$503$	5236.71	0.464201	0.232101	+	0.972692i	$0.425440\pi$
$504$	0	0
$505$	25.9386	0.00228565
$506$	0	0
$507$	7157.51	0.626975
$508$	0	0
$509$	−3776.96	−0.328901	−0.164451	−	0.986385i	$-0.552585\pi$
$509$	−3776.96	−0.328901	−0.164451	+	0.986385i	$0.552585\pi$
$510$	0	0
$511$	25850.3	2.23787
$512$	0	0
$513$	−8501.09	−0.731642
$514$	0	0
$515$	4.15098	0.000355173 0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	13455.7	1.13803
$520$	0	0
$521$	−81.4864	−0.00685218	−0.00342609	−	0.999994i	$-0.501091\pi$
$521$	−81.4864	−0.00685218	−0.00342609	+	0.999994i	$0.501091\pi$
$522$	0	0
$523$	−15891.3	−1.32864	−0.664320	−	0.747448i	$-0.731279\pi$
$523$	−15891.3	−1.32864	−0.664320	+	0.747448i	$0.731279\pi$
$524$	0	0
$525$	−19739.9	−1.64099
$526$	0	0
$527$	−6069.28	−0.501673
$528$	0	0
$529$	7987.49	0.656488
$530$	0	0
$531$	1378.66	0.112672
$532$	0	0
$533$	5243.90	0.426151
$534$	0	0
$535$	−11.6139	−0.000938528 0
$536$	0	0
$537$	−6761.25	−0.543332
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7267.61	−0.577558	−0.288779	−	0.957396i	$-0.593249\pi$
$541$	−7267.61	−0.577558	−0.288779	+	0.957396i	$0.593249\pi$
$542$	0	0
$543$	16514.6	1.30518
$544$	0	0
$545$	−116.374	−0.00914659
$546$	0	0
$547$	−17435.1	−1.36283	−0.681417	−	0.731895i	$-0.738636\pi$
$547$	−17435.1	−1.36283	−0.681417	+	0.731895i	$0.738636\pi$
$548$	0	0
$549$	−6778.04	−0.526921
$550$	0	0
$551$	−13207.9	−1.02119
$552$	0	0
$553$	1198.21	0.0921391
$554$	0	0
$555$	−110.194	−0.00842787
$556$	0	0
$557$	12535.6	0.953594	0.476797	−	0.879013i	$-0.341798\pi$
$557$	12535.6	0.953594	0.476797	+	0.879013i	$0.341798\pi$
$558$	0	0
$559$	−14909.6	−1.12810
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4448.14	0.332979	0.166489	−	0.986043i	$-0.446757\pi$
$563$	4448.14	0.332979	0.166489	+	0.986043i	$0.446757\pi$
$564$	0	0
$565$	115.456	0.00859694
$566$	0	0
$567$	22227.7	1.64634
$568$	0	0
$569$	6577.85	0.484636	0.242318	−	0.970197i	$-0.422092\pi$
$569$	6577.85	0.484636	0.242318	+	0.970197i	$0.422092\pi$
$570$	0	0
$571$	17109.0	1.25392	0.626962	−	0.779050i	$-0.284298\pi$
$571$	17109.0	1.25392	0.626962	+	0.779050i	$0.284298\pi$
$572$	0	0
$573$	−3090.32	−0.225305
$574$	0	0
$575$	17744.7	1.28697
$576$	0	0
$577$	−10528.4	−0.759623	−0.379811	−	0.925064i	$-0.624011\pi$
$577$	−10528.4	−0.759623	−0.379811	+	0.925064i	$0.624011\pi$
$578$	0	0
$579$	−11436.7	−0.820887
$580$	0	0
$581$	24468.6	1.74721
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−42.7768	−0.00302325
$586$	0	0
$587$	−14551.9	−1.02321	−0.511603	−	0.859222i	$-0.670948\pi$
$587$	−14551.9	−1.02321	−0.511603	+	0.859222i	$0.670948\pi$
$588$	0	0
$589$	−13414.6	−0.938433
$590$	0	0
$591$	−3390.94	−0.236014
$592$	0	0
$593$	−4585.83	−0.317568	−0.158784	−	0.987313i	$-0.550757\pi$
$593$	−4585.83	−0.317568	−0.158784	+	0.987313i	$0.550757\pi$
$594$	0	0
$595$	−104.330	−0.00718845
$596$	0	0
$597$	−6862.91	−0.470486
$598$	0	0
$599$	−23496.7	−1.60276	−0.801378	−	0.598158i	$-0.795899\pi$
$599$	−23496.7	−1.60276	−0.801378	+	0.598158i	$0.795899\pi$
$600$	0	0
$601$	3758.44	0.255092	0.127546	−	0.991833i	$-0.459290\pi$
$601$	3758.44	0.255092	0.127546	+	0.991833i	$0.459290\pi$
$602$	0	0
$603$	11355.9	0.766914
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15705.2	−1.05018	−0.525088	−	0.851048i	$-0.675967\pi$
$607$	−15705.2	−1.05018	−0.525088	+	0.851048i	$0.675967\pi$
$608$	0	0
$609$	19394.2	1.29046
$610$	0	0
$611$	−20105.6	−1.33124
$612$	0	0
$613$	−2731.67	−0.179985	−0.0899926	−	0.995942i	$-0.528684\pi$
$613$	−2731.67	−0.179985	−0.0899926	+	0.995942i	$0.528684\pi$
$614$	0	0
$615$	90.0874	0.00590679
$616$	0	0
$617$	3140.17	0.204892	0.102446	−	0.994739i	$-0.467333\pi$
$617$	3140.17	0.204892	0.102446	+	0.994739i	$0.467333\pi$
$618$	0	0
$619$	−5204.95	−0.337972	−0.168986	−	0.985618i	$-0.554049\pi$
$619$	−5204.95	−0.337972	−0.168986	+	0.985618i	$0.554049\pi$
$620$	0	0
$621$	−11221.1	−0.725101
$622$	0	0
$623$	−23186.0	−1.49106
$624$	0	0
$625$	15622.1	0.999815
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9456.09	0.599426
$630$	0	0
$631$	−21694.2	−1.36867	−0.684337	−	0.729166i	$-0.739908\pi$
$631$	−21694.2	−1.36867	−0.684337	+	0.729166i	$0.739908\pi$
$632$	0	0
$633$	−34732.2	−2.18086
$634$	0	0
$635$	−30.2868	−0.00189275
$636$	0	0
$637$	8387.71	0.521716
$638$	0	0
$639$	−16435.4	−1.01749
$640$	0	0
$641$	−29091.2	−1.79256	−0.896282	−	0.443485i	$-0.853742\pi$
$641$	−29091.2	−1.79256	−0.896282	+	0.443485i	$0.853742\pi$
$642$	0	0
$643$	24207.4	1.48467	0.742337	−	0.670026i	$-0.233717\pi$
$643$	24207.4	1.48467	0.742337	+	0.670026i	$0.233717\pi$
$644$	0	0
$645$	−256.138	−0.0156363
$646$	0	0
$647$	−13203.6	−0.802300	−0.401150	−	0.916012i	$-0.631389\pi$
$647$	−13203.6	−0.802300	−0.401150	+	0.916012i	$0.631389\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	19697.6	1.18588
$652$	0	0
$653$	−19289.5	−1.15598	−0.577990	−	0.816044i	$-0.696163\pi$
$653$	−19289.5	−1.15598	−0.577990	+	0.816044i	$0.696163\pi$
$654$	0	0
$655$	−2.78010	−0.000165844 0
$656$	0	0
$657$	−15625.3	−0.927853
$658$	0	0
$659$	−7086.80	−0.418911	−0.209456	−	0.977818i	$-0.567169\pi$
$659$	−7086.80	−0.418911	−0.209456	+	0.977818i	$0.567169\pi$
$660$	0	0
$661$	−23077.2	−1.35794	−0.678969	−	0.734167i	$-0.737573\pi$
$661$	−23077.2	−1.35794	−0.678969	+	0.734167i	$0.737573\pi$
$662$	0	0
$663$	10381.0	0.608094
$664$	0	0
$665$	−230.595	−0.0134468
$666$	0	0
$667$	−17434.0	−1.01206
$668$	0	0
$669$	7455.99	0.430890
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−23030.6	−1.31911	−0.659556	−	0.751655i	$-0.729256\pi$
$673$	−23030.6	−1.31911	−0.659556	+	0.751655i	$0.729256\pi$
$674$	0	0
$675$	−9879.46	−0.563349
$676$	0	0
$677$	28434.0	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	28434.0	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	9998.86	0.565127
$680$	0	0
$681$	−39255.9	−2.20894
$682$	0	0
$683$	−11600.3	−0.649887	−0.324943	−	0.945733i	$-0.605345\pi$
$683$	−11600.3	−0.649887	−0.324943	+	0.945733i	$0.605345\pi$
$684$	0	0
$685$	−83.7886	−0.00467357
$686$	0	0
$687$	17791.2	0.988032
$688$	0	0
$689$	−11501.4	−0.635949
$690$	0	0
$691$	−17422.2	−0.959149	−0.479574	−	0.877501i	$-0.659209\pi$
$691$	−17422.2	−0.959149	−0.479574	+	0.877501i	$0.659209\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	121.655	0.00663975
$696$	0	0
$697$	−7730.69	−0.420116
$698$	0	0
$699$	−19090.8	−1.03302
$700$	0	0
$701$	−1287.92	−0.0693926	−0.0346963	−	0.999398i	$-0.511046\pi$
$701$	−1287.92	−0.0693926	−0.0346963	+	0.999398i	$0.511046\pi$
$702$	0	0
$703$	20900.2	1.12129
$704$	0	0
$705$	−345.404	−0.0184520
$706$	0	0
$707$	7223.97	0.384279
$708$	0	0
$709$	3087.56	0.163548	0.0817741	−	0.996651i	$-0.473941\pi$
$709$	3087.56	0.163548	0.0817741	+	0.996651i	$0.473941\pi$
$710$	0	0
$711$	−724.258	−0.0382023
$712$	0	0
$713$	−17706.7	−0.930043
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−35505.9	−1.84936
$718$	0	0
$719$	17909.4	0.928940	0.464470	−	0.885589i	$-0.346245\pi$
$719$	17909.4	0.928940	0.464470	+	0.885589i	$0.346245\pi$
$720$	0	0
$721$	1156.06	0.0597141
$722$	0	0
$723$	26098.9	1.34250
$724$	0	0
$725$	−15349.5	−0.786296
$726$	0	0
$727$	−5694.09	−0.290484	−0.145242	−	0.989396i	$-0.546396\pi$
$727$	−5694.09	−0.290484	−0.145242	+	0.989396i	$0.546396\pi$
$728$	0	0
$729$	357.045	0.0181397
$730$	0	0
$731$	21980.1	1.11212
$732$	0	0
$733$	−10055.1	−0.506678	−0.253339	−	0.967378i	$-0.581529\pi$
$733$	−10055.1	−0.506678	−0.253339	+	0.967378i	$0.581529\pi$
$734$	0	0
$735$	144.097	0.00723140
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−8697.71	−0.432951	−0.216475	−	0.976288i	$-0.569456\pi$
$739$	−8697.71	−0.432951	−0.216475	+	0.976288i	$0.569456\pi$
$740$	0	0
$741$	22944.6	1.13750
$742$	0	0
$743$	26970.3	1.33169	0.665844	−	0.746091i	$-0.268072\pi$
$743$	26970.3	1.33169	0.665844	+	0.746091i	$0.268072\pi$
$744$	0	0
$745$	232.653	0.0114412
$746$	0	0
$747$	−14790.1	−0.724420
$748$	0	0
$749$	−3234.50	−0.157792
$750$	0	0
$751$	38692.8	1.88005	0.940027	−	0.341101i	$-0.110800\pi$
$751$	38692.8	1.88005	0.940027	+	0.341101i	$0.110800\pi$
$752$	0	0
$753$	28885.3	1.39793
$754$	0	0
$755$	−177.194	−0.00854138
$756$	0	0
$757$	−12815.1	−0.615285	−0.307643	−	0.951502i	$-0.599540\pi$
$757$	−12815.1	−0.615285	−0.307643	+	0.951502i	$0.599540\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16670.0	0.794070	0.397035	−	0.917804i	$-0.370039\pi$
$761$	16670.0	0.794070	0.397035	+	0.917804i	$0.370039\pi$
$762$	0	0
$763$	−32410.4	−1.53779
$764$	0	0
$765$	63.0627	0.00298044
$766$	0	0
$767$	3080.99	0.145043
$768$	0	0
$769$	−3634.44	−0.170431	−0.0852154	−	0.996363i	$-0.527158\pi$
$769$	−3634.44	−0.170431	−0.0852154	+	0.996363i	$0.527158\pi$
$770$	0	0
$771$	30622.9	1.43042
$772$	0	0
$773$	24666.2	1.14771	0.573857	−	0.818956i	$-0.305447\pi$
$773$	24666.2	1.14771	0.573857	+	0.818956i	$0.305447\pi$
$774$	0	0
$775$	−15589.6	−0.722574
$776$	0	0
$777$	−30689.3	−1.41695
$778$	0	0
$779$	−17086.7	−0.785871
$780$	0	0
$781$	0	0
$782$	0	0
$783$	9706.44	0.443014
$784$	0	0
$785$	70.9216	0.00322459
$786$	0	0
$787$	16770.2	0.759585	0.379793	−	0.925072i	$-0.375995\pi$
$787$	16770.2	0.759585	0.379793	+	0.925072i	$0.375995\pi$
$788$	0	0
$789$	−25481.2	−1.14975
$790$	0	0
$791$	32154.8	1.44538
$792$	0	0
$793$	−15147.3	−0.678307
$794$	0	0
$795$	−197.588	−0.00881476
$796$	0	0
$797$	9571.10	0.425377	0.212689	−	0.977120i	$-0.431778\pi$
$797$	9571.10	0.425377	0.212689	+	0.977120i	$0.431778\pi$
$798$	0	0
$799$	29640.2	1.31238
$800$	0	0
$801$	14014.8	0.618215
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−304.376	−0.0133265
$806$	0	0
$807$	21990.1	0.959216
$808$	0	0
$809$	43981.3	1.91137	0.955685	−	0.294391i	$-0.0951168\pi$
$809$	43981.3	1.91137	0.955685	+	0.294391i	$0.0951168\pi$
$810$	0	0
$811$	−29970.8	−1.29768	−0.648840	−	0.760925i	$-0.724745\pi$
$811$	−29970.8	−1.29768	−0.648840	+	0.760925i	$0.724745\pi$
$812$	0	0
$813$	−25748.7	−1.11076
$814$	0	0
$815$	8.08330	0.000347418 0
$816$	0	0
$817$	48581.2	2.08034
$818$	0	0
$819$	−11913.5	−0.508290
$820$	0	0
$821$	1215.26	0.0516600	0.0258300	−	0.999666i	$-0.491777\pi$
$821$	1215.26	0.0516600	0.0258300	+	0.999666i	$0.491777\pi$
$822$	0	0
$823$	9499.78	0.402359	0.201180	−	0.979554i	$-0.435523\pi$
$823$	9499.78	0.402359	0.201180	+	0.979554i	$0.435523\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28746.2	−1.20871	−0.604355	−	0.796715i	$-0.706569\pi$
$827$	−28746.2	−1.20871	−0.604355	+	0.796715i	$0.706569\pi$
$828$	0	0
$829$	12365.1	0.518043	0.259022	−	0.965872i	$-0.416600\pi$
$829$	12365.1	0.518043	0.259022	+	0.965872i	$0.416600\pi$
$830$	0	0
$831$	22860.1	0.954280
$832$	0	0
$833$	−12365.4	−0.514328
$834$	0	0
$835$	−55.6616	−0.00230689
$836$	0	0
$837$	9858.29	0.407111
$838$	0	0
$839$	3194.11	0.131434	0.0657169	−	0.997838i	$-0.479067\pi$
$839$	3194.11	0.131434	0.0657169	+	0.997838i	$0.479067\pi$
$840$	0	0
$841$	−9308.35	−0.381662
$842$	0	0
$843$	−28292.9	−1.15594
$844$	0	0
$845$	97.1686	0.00395586
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−16039.9	−0.648397
$850$	0	0
$851$	27587.5	1.11126
$852$	0	0
$853$	−5962.48	−0.239333	−0.119667	−	0.992814i	$-0.538183\pi$
$853$	−5962.48	−0.239333	−0.119667	+	0.992814i	$0.538183\pi$
$854$	0	0
$855$	139.384	0.00557523
$856$	0	0
$857$	17358.2	0.691886	0.345943	−	0.938256i	$-0.387559\pi$
$857$	17358.2	0.691886	0.345943	+	0.938256i	$0.387559\pi$
$858$	0	0
$859$	22138.3	0.879337	0.439669	−	0.898160i	$-0.355096\pi$
$859$	22138.3	0.879337	0.439669	+	0.898160i	$0.355096\pi$
$860$	0	0
$861$	25089.6	0.993091
$862$	0	0
$863$	11181.4	0.441043	0.220522	−	0.975382i	$-0.429224\pi$
$863$	11181.4	0.441043	0.220522	+	0.975382i	$0.429224\pi$
$864$	0	0
$865$	182.671	0.00718034
$866$	0	0
$867$	16448.7	0.644322
$868$	0	0
$869$	0	0
$870$	0	0
$871$	25377.9	0.987251
$872$	0	0
$873$	−6043.83	−0.234310
$874$	0	0
$875$	−535.984	−0.0207081
$876$	0	0
$877$	11145.6	0.429147	0.214573	−	0.976708i	$-0.431164\pi$
$877$	11145.6	0.429147	0.214573	+	0.976708i	$0.431164\pi$
$878$	0	0
$879$	−28878.6	−1.10813
$880$	0	0
$881$	−35280.1	−1.34917	−0.674584	−	0.738198i	$-0.735677\pi$
$881$	−35280.1	−1.34917	−0.674584	+	0.738198i	$0.735677\pi$
$882$	0	0
$883$	−737.156	−0.0280943	−0.0140471	−	0.999901i	$-0.504471\pi$
$883$	−737.156	−0.0280943	−0.0140471	+	0.999901i	$0.504471\pi$
$884$	0	0
$885$	52.9298	0.00201041
$886$	0	0
$887$	−18421.0	−0.697312	−0.348656	−	0.937251i	$-0.613362\pi$
$887$	−18421.0	−0.697312	−0.348656	+	0.937251i	$0.613362\pi$
$888$	0	0
$889$	−8434.95	−0.318222
$890$	0	0
$891$	0	0
$892$	0	0
$893$	65511.9	2.45495
$894$	0	0
$895$	−91.7890	−0.00342812
$896$	0	0
$897$	30285.9	1.12733
$898$	0	0
$899$	15316.6	0.568227
$900$	0	0
$901$	16955.7	0.626943
$902$	0	0
$903$	−71335.3	−2.62889
$904$	0	0
$905$	224.199	0.00823493
$906$	0	0
$907$	−9341.10	−0.341969	−0.170985	−	0.985274i	$-0.554695\pi$
$907$	−9341.10	−0.341969	−0.170985	+	0.985274i	$0.554695\pi$
$908$	0	0
$909$	−4366.54	−0.159328
$910$	0	0
$911$	−30360.6	−1.10416	−0.552081	−	0.833791i	$-0.686166\pi$
$911$	−30360.6	−1.10416	−0.552081	+	0.833791i	$0.686166\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−260.223	−0.00940188
$916$	0	0
$917$	−774.267	−0.0278828
$918$	0	0
$919$	28842.0	1.03527	0.517633	−	0.855603i	$-0.326813\pi$
$919$	28842.0	1.03527	0.517633	+	0.855603i	$0.326813\pi$
$920$	0	0
$921$	23463.7	0.839474
$922$	0	0
$923$	−36729.2	−1.30981
$924$	0	0
$925$	24289.0	0.863369
$926$	0	0
$927$	−698.782	−0.0247584
$928$	0	0
$929$	−295.881	−0.0104494	−0.00522472	−	0.999986i	$-0.501663\pi$
$929$	−295.881	−0.0104494	−0.00522472	+	0.999986i	$0.501663\pi$
$930$	0	0
$931$	−27330.5	−0.962105
$932$	0	0
$933$	−56715.3	−1.99011
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−49518.4	−1.72646	−0.863231	−	0.504810i	$-0.831563\pi$
$937$	−49518.4	−1.72646	−0.863231	+	0.504810i	$0.831563\pi$
$938$	0	0
$939$	−21787.7	−0.757206
$940$	0	0
$941$	21174.7	0.733554	0.366777	−	0.930309i	$-0.380461\pi$
$941$	21174.7	0.733554	0.366777	+	0.930309i	$0.380461\pi$
$942$	0	0
$943$	−22553.7	−0.778845
$944$	0	0
$945$	169.463	0.00583347
$946$	0	0
$947$	−26386.2	−0.905425	−0.452713	−	0.891656i	$-0.649544\pi$
$947$	−26386.2	−0.905425	−0.452713	+	0.891656i	$0.649544\pi$
$948$	0	0
$949$	−34918.8	−1.19443
$950$	0	0
$951$	29174.3	0.994786
$952$	0	0
$953$	−23955.8	−0.814277	−0.407138	−	0.913366i	$-0.633473\pi$
$953$	−23955.8	−0.814277	−0.407138	+	0.913366i	$0.633473\pi$
$954$	0	0
$955$	−41.9534	−0.00142155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−23335.4	−0.785754
$960$	0	0
$961$	−14234.8	−0.477823
$962$	0	0
$963$	1955.10	0.0654229
$964$	0	0
$965$	−155.262	−0.00517933
$966$	0	0
$967$	−18414.5	−0.612380	−0.306190	−	0.951970i	$-0.599054\pi$
$967$	−18414.5	−0.612380	−0.306190	+	0.951970i	$0.599054\pi$
$968$	0	0
$969$	−33825.5	−1.12139
$970$	0	0
$971$	33277.5	1.09982	0.549910	−	0.835224i	$-0.314662\pi$
$971$	33277.5	1.09982	0.549910	+	0.835224i	$0.314662\pi$
$972$	0	0
$973$	33881.2	1.11632
$974$	0	0
$975$	26664.8	0.875854
$976$	0	0
$977$	−8390.58	−0.274758	−0.137379	−	0.990519i	$-0.543868\pi$
$977$	−8390.58	−0.274758	−0.137379	+	0.990519i	$0.543868\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	19590.5	0.637591
$982$	0	0
$983$	54879.7	1.78066	0.890331	−	0.455314i	$-0.150473\pi$
$983$	54879.7	1.78066	0.890331	+	0.455314i	$0.150473\pi$
$984$	0	0
$985$	−46.0345	−0.00148912
$986$	0	0
$987$	−96195.9	−3.10228
$988$	0	0
$989$	64125.3	2.06174
$990$	0	0
$991$	−45505.7	−1.45866	−0.729332	−	0.684160i	$-0.760169\pi$
$991$	−45505.7	−1.45866	−0.729332	+	0.684160i	$0.760169\pi$
$992$	0	0
$993$	3718.36	0.118830
$994$	0	0
$995$	−93.1692	−0.00296850
$996$	0	0
$997$	−3271.56	−0.103923	−0.0519616	−	0.998649i	$-0.516547\pi$
$997$	−3271.56	−0.103923	−0.0519616	+	0.998649i	$0.516547\pi$
$998$	0	0
$999$	−15359.5	−0.486438

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.bs.1.2		6
4.3	odd	2		484.4.a.h.1.5		6
11.2	odd	10		176.4.m.d.81.1		12
11.6	odd	10		176.4.m.d.113.1		12
11.10	odd	2		1936.4.a.br.1.2		6
44.35	even	10		44.4.e.a.37.3	yes	12
44.39	even	10		44.4.e.a.25.3	✓	12
44.43	even	2		484.4.a.i.1.5		6
132.35	odd	10		396.4.j.d.37.2		12
132.83	odd	10		396.4.j.d.289.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.4.e.a.25.3	✓	12	44.39	even	10
44.4.e.a.37.3	yes	12	44.35	even	10
176.4.m.d.81.1		12	11.2	odd	10
176.4.m.d.113.1		12	11.6	odd	10
396.4.j.d.37.2		12	132.35	odd	10
396.4.j.d.289.2		12	132.83	odd	10
484.4.a.h.1.5		6	4.3	odd	2
484.4.a.i.1.5		6	44.43	even	2
1936.4.a.br.1.2		6	11.10	odd	2
1936.4.a.bs.1.2		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q-6.46299 q^{3} -0.0877400 q^{5} -24.4358 q^{7} +14.7703 q^{9} +33.0082 q^{13} +0.567063 q^{15} -48.6615 q^{17} -107.554 q^{19} +157.929 q^{21} -141.967 q^{23} -124.992 q^{25} +79.0405 q^{27} +122.803 q^{29} +124.724 q^{31} +2.14400 q^{35} -194.324 q^{37} -213.332 q^{39} +158.867 q^{41} -451.693 q^{43} -1.29595 q^{45} -609.110 q^{47} +254.110 q^{49} +314.499 q^{51} -348.442 q^{53} +695.118 q^{57} +93.3402 q^{59} -458.897 q^{61} -360.924 q^{63} -2.89614 q^{65} +768.836 q^{67} +917.529 q^{69} -1112.73 q^{71} -1057.88 q^{73} +807.824 q^{75} -49.0348 q^{79} -909.636 q^{81} -1001.34 q^{83} +4.26956 q^{85} -793.677 q^{87} +948.854 q^{89} -806.582 q^{91} -806.093 q^{93} +9.43675 q^{95} -409.188 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 12 q^{5} + 8 q^{7} + 11 q^{9} + 80 q^{13} + 98 q^{15} + 113 q^{17} - 53 q^{19} + 152 q^{21} + 194 q^{23} + 476 q^{25} - 72 q^{27} + 374 q^{29} - 16 q^{31} - 1044 q^{35} - 456 q^{37} - 592 q^{39}+ \cdots + 683 q^{97}+O(q^{100}) \) 6 * q - 3 * q^3 - 12 * q^5 + 8 * q^7 + 11 * q^9 + 80 * q^13 + 98 * q^15 + 113 * q^17 - 53 * q^19 + 152 * q^21 + 194 * q^23 + 476 * q^25 - 72 * q^27 + 374 * q^29 - 16 * q^31 - 1044 * q^35 - 456 * q^37 - 592 * q^39 + 825 * q^41 - 1165 * q^43 - 670 * q^45 - 1248 * q^47 - 4 * q^49 - 988 * q^51 - 122 * q^53 + 2282 * q^57 + 767 * q^59 + 1010 * q^61 - 254 * q^63 + 1946 * q^65 + 2197 * q^67 + 2648 * q^69 - 598 * q^71 + 1523 * q^73 - 609 * q^75 - 2116 * q^79 - 2106 * q^81 - 1213 * q^83 + 1066 * q^85 - 1518 * q^87 - 477 * q^89 - 2420 * q^91 - 714 * q^93 - 404 * q^95 + 683 * q^97

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