LMFDB - Embedded newform 1900.4.a.l.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1900,4,Mod(1,1900)] 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("1900.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1900.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$112.103629011$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 158x^{10} + 9117x^{8} - 255920x^{6} + 3734252x^{4} - 26837136x^{2} + 73273600 $ x^12 - 158x^10 + 9117x^8 - 255920x^6 + 3734252x^4 - 26837136*x^2 + 73273600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.73144$ of defining polynomial
Character	$\chi$	$=$	1900.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.73144 q^{3} +12.0773 q^{7} -13.0764 q^{9} +11.9264 q^{11} -55.5478 q^{13} +39.7521 q^{17} +19.0000 q^{19} -45.0655 q^{21} +42.6893 q^{23} +149.542 q^{27} +280.201 q^{29} -298.661 q^{31} -44.5027 q^{33} -444.568 q^{37} +207.273 q^{39} -53.2291 q^{41} +178.936 q^{43} +518.932 q^{47} -197.140 q^{49} -148.332 q^{51} +383.465 q^{53} -70.8973 q^{57} +243.246 q^{59} -129.042 q^{61} -157.927 q^{63} -624.571 q^{67} -159.292 q^{69} -60.2379 q^{71} -856.683 q^{73} +144.039 q^{77} -160.342 q^{79} -204.945 q^{81} +192.126 q^{83} -1045.55 q^{87} +1038.38 q^{89} -670.865 q^{91} +1114.43 q^{93} +1639.09 q^{97} -155.955 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9} - 110 q^{11} + 228 q^{19} - 8 q^{21} + 76 q^{29} + 80 q^{31} - 60 q^{39} - 648 q^{41} + 886 q^{49} - 328 q^{51} - 984 q^{59} - 1970 q^{61} - 364 q^{69} - 3060 q^{71} - 552 q^{79} - 3388 q^{81}+ \cdots - 2366 q^{99}+O(q^{100}) $ 12 * q - 8 * q^9 - 110 * q^11 + 228 * q^19 - 8 * q^21 + 76 * q^29 + 80 * q^31 - 60 * q^39 - 648 * q^41 + 886 * q^49 - 328 * q^51 - 984 * q^59 - 1970 * q^61 - 364 * q^69 - 3060 * q^71 - 552 * q^79 - 3388 * q^81 - 2428 * q^89 - 3468 * q^91 - 2366 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.73144	−0.718115	−0.359058	−	0.933315i	$-0.616902\pi$
$3$	−3.73144	−0.718115	−0.359058	+	0.933315i	$0.616902\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	12.0773	0.652111	0.326055	−	0.945351i	$-0.394280\pi$
$7$	12.0773	0.652111	0.326055	+	0.945351i	$0.394280\pi$
$8$	0	0
$9$	−13.0764	−0.484311
$10$	0	0
$11$	11.9264	0.326905	0.163453	−	0.986551i	$-0.447737\pi$
$11$	11.9264	0.326905	0.163453	+	0.986551i	$0.447737\pi$
$12$	0	0
$13$	−55.5478	−1.18509	−0.592545	−	0.805537i	$-0.701877\pi$
$13$	−55.5478	−1.18509	−0.592545	+	0.805537i	$0.701877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	39.7521	0.567135	0.283568	−	0.958952i	$-0.408482\pi$
$17$	39.7521	0.567135	0.283568	+	0.958952i	$0.408482\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−45.0655	−0.468290
$22$	0	0
$23$	42.6893	0.387015	0.193507	−	0.981099i	$-0.438014\pi$
$23$	42.6893	0.387015	0.193507	+	0.981099i	$0.438014\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	149.542	1.06591
$28$	0	0
$29$	280.201	1.79421	0.897105	−	0.441818i	$-0.145666\pi$
$29$	280.201	1.79421	0.897105	+	0.441818i	$0.145666\pi$
$30$	0	0
$31$	−298.661	−1.73036	−0.865179	−	0.501463i	$-0.832795\pi$
$31$	−298.661	−1.73036	−0.865179	+	0.501463i	$0.832795\pi$
$32$	0	0
$33$	−44.5027	−0.234756
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−444.568	−1.97531	−0.987655	−	0.156645i	$-0.949932\pi$
$37$	−444.568	−1.97531	−0.987655	+	0.156645i	$0.949932\pi$
$38$	0	0
$39$	207.273	0.851031
$40$	0	0
$41$	−53.2291	−0.202756	−0.101378	−	0.994848i	$-0.532325\pi$
$41$	−53.2291	−0.202756	−0.101378	+	0.994848i	$0.532325\pi$
$42$	0	0
$43$	178.936	0.634594	0.317297	−	0.948326i	$-0.397225\pi$
$43$	178.936	0.634594	0.317297	+	0.948326i	$0.397225\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	518.932	1.61051	0.805255	−	0.592929i	$-0.202028\pi$
$47$	518.932	1.61051	0.805255	+	0.592929i	$0.202028\pi$
$48$	0	0
$49$	−197.140	−0.574752
$50$	0	0
$51$	−148.332	−0.407269
$52$	0	0
$53$	383.465	0.993830	0.496915	−	0.867799i	$-0.334466\pi$
$53$	383.465	0.993830	0.496915	+	0.867799i	$0.334466\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−70.8973	−0.164747
$58$	0	0
$59$	243.246	0.536745	0.268372	−	0.963315i	$-0.413514\pi$
$59$	243.246	0.536745	0.268372	+	0.963315i	$0.413514\pi$
$60$	0	0
$61$	−129.042	−0.270854	−0.135427	−	0.990787i	$-0.543241\pi$
$61$	−129.042	−0.270854	−0.135427	+	0.990787i	$0.543241\pi$
$62$	0	0
$63$	−157.927	−0.315824
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−624.571	−1.13886	−0.569429	−	0.822041i	$-0.692836\pi$
$67$	−624.571	−1.13886	−0.569429	+	0.822041i	$0.692836\pi$
$68$	0	0
$69$	−159.292	−0.277921
$70$	0	0
$71$	−60.2379	−0.100689	−0.0503446	−	0.998732i	$-0.516032\pi$
$71$	−60.2379	−0.100689	−0.0503446	+	0.998732i	$0.516032\pi$
$72$	0	0
$73$	−856.683	−1.37352	−0.686761	−	0.726883i	$-0.740968\pi$
$73$	−856.683	−1.37352	−0.686761	+	0.726883i	$0.740968\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	144.039	0.213178
$78$	0	0
$79$	−160.342	−0.228352	−0.114176	−	0.993461i	$-0.536423\pi$
$79$	−160.342	−0.228352	−0.114176	+	0.993461i	$0.536423\pi$
$80$	0	0
$81$	−204.945	−0.281132
$82$	0	0
$83$	192.126	0.254079	0.127040	−	0.991898i	$-0.459452\pi$
$83$	192.126	0.254079	0.127040	+	0.991898i	$0.459452\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1045.55	−1.28845
$88$	0	0
$89$	1038.38	1.23671	0.618357	−	0.785897i	$-0.287798\pi$
$89$	1038.38	1.23671	0.618357	+	0.785897i	$0.287798\pi$
$90$	0	0
$91$	−670.865	−0.772810
$92$	0	0
$93$	1114.43	1.24260
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1639.09	1.71572	0.857859	−	0.513885i	$-0.171794\pi$
$97$	1639.09	1.71572	0.857859	+	0.513885i	$0.171794\pi$
$98$	0	0
$99$	−155.955	−0.158324
$100$	0	0
$101$	−1956.55	−1.92757	−0.963783	−	0.266687i	$-0.914071\pi$
$101$	−1956.55	−1.92757	−0.963783	+	0.266687i	$0.914071\pi$
$102$	0	0
$103$	373.058	0.356878	0.178439	−	0.983951i	$-0.442895\pi$
$103$	373.058	0.356878	0.178439	+	0.983951i	$0.442895\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2110.00	1.90637	0.953185	−	0.302388i	$-0.0977838\pi$
$107$	2110.00	1.90637	0.953185	+	0.302388i	$0.0977838\pi$
$108$	0	0
$109$	−1171.53	−1.02947	−0.514736	−	0.857349i	$-0.672110\pi$
$109$	−1171.53	−1.02947	−0.514736	+	0.857349i	$0.672110\pi$
$110$	0	0
$111$	1658.88	1.41850
$112$	0	0
$113$	499.266	0.415637	0.207818	−	0.978167i	$-0.433364\pi$
$113$	499.266	0.415637	0.207818	+	0.978167i	$0.433364\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	726.364	0.573952
$118$	0	0
$119$	480.097	0.369835
$120$	0	0
$121$	−1188.76	−0.893133
$122$	0	0
$123$	198.621	0.145602
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2435.04	1.70138	0.850690	−	0.525668i	$-0.176185\pi$
$127$	2435.04	1.70138	0.850690	+	0.525668i	$0.176185\pi$
$128$	0	0
$129$	−667.689	−0.455711
$130$	0	0
$131$	282.543	0.188442	0.0942210	−	0.995551i	$-0.469964\pi$
$131$	282.543	0.188442	0.0942210	+	0.995551i	$0.469964\pi$
$132$	0	0
$133$	229.468	0.149604
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2434.93	−1.51847	−0.759233	−	0.650819i	$-0.774426\pi$
$137$	−2434.93	−1.51847	−0.759233	+	0.650819i	$0.774426\pi$
$138$	0	0
$139$	−2250.93	−1.37353	−0.686767	−	0.726877i	$-0.740971\pi$
$139$	−2250.93	−1.37353	−0.686767	+	0.726877i	$0.740971\pi$
$140$	0	0
$141$	−1936.36	−1.15653
$142$	0	0
$143$	−662.487	−0.387412
$144$	0	0
$145$	0	0
$146$	0	0
$147$	735.614	0.412738
$148$	0	0
$149$	−648.317	−0.356458	−0.178229	−	0.983989i	$-0.557037\pi$
$149$	−648.317	−0.356458	−0.178229	+	0.983989i	$0.557037\pi$
$150$	0	0
$151$	2539.56	1.36865	0.684326	−	0.729176i	$-0.260097\pi$
$151$	2539.56	1.36865	0.684326	+	0.729176i	$0.260097\pi$
$152$	0	0
$153$	−519.814	−0.274670
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−132.602	−0.0674066	−0.0337033	−	0.999432i	$-0.510730\pi$
$157$	−132.602	−0.0674066	−0.0337033	+	0.999432i	$0.510730\pi$
$158$	0	0
$159$	−1430.88	−0.713685
$160$	0	0
$161$	515.570	0.252376
$162$	0	0
$163$	−2271.38	−1.09146	−0.545731	−	0.837960i	$-0.683748\pi$
$163$	−2271.38	−1.09146	−0.545731	+	0.837960i	$0.683748\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−280.518	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$167$	−280.518	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$168$	0	0
$169$	888.554	0.404440
$170$	0	0
$171$	−248.451	−0.111109
$172$	0	0
$173$	2716.18	1.19368	0.596841	−	0.802359i	$-0.296422\pi$
$173$	2716.18	1.19368	0.596841	+	0.802359i	$0.296422\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−907.657	−0.385444
$178$	0	0
$179$	−3512.90	−1.46685	−0.733426	−	0.679769i	$-0.762080\pi$
$179$	−3512.90	−1.46685	−0.733426	+	0.679769i	$0.762080\pi$
$180$	0	0
$181$	−1681.05	−0.690340	−0.345170	−	0.938540i	$-0.612179\pi$
$181$	−1681.05	−0.690340	−0.345170	+	0.938540i	$0.612179\pi$
$182$	0	0
$183$	481.511	0.194505
$184$	0	0
$185$	0	0
$186$	0	0
$187$	474.101	0.185400
$188$	0	0
$189$	1806.06	0.695089
$190$	0	0
$191$	−3346.47	−1.26776	−0.633880	−	0.773431i	$-0.718539\pi$
$191$	−3346.47	−1.26776	−0.633880	+	0.773431i	$0.718539\pi$
$192$	0	0
$193$	−1057.68	−0.394474	−0.197237	−	0.980356i	$-0.563197\pi$
$193$	−1057.68	−0.394474	−0.197237	+	0.980356i	$0.563197\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−339.367	−0.122736	−0.0613678	−	0.998115i	$-0.519546\pi$
$197$	−339.367	−0.122736	−0.0613678	+	0.998115i	$0.519546\pi$
$198$	0	0
$199$	−4295.06	−1.53000	−0.764998	−	0.644033i	$-0.777260\pi$
$199$	−4295.06	−1.53000	−0.764998	+	0.644033i	$0.777260\pi$
$200$	0	0
$201$	2330.54	0.817830
$202$	0	0
$203$	3384.06	1.17002
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−558.222	−0.187435
$208$	0	0
$209$	226.602	0.0749972
$210$	0	0
$211$	491.959	0.160511	0.0802556	−	0.996774i	$-0.474426\pi$
$211$	491.959	0.160511	0.0802556	+	0.996774i	$0.474426\pi$
$212$	0	0
$213$	224.774	0.0723064
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3607.01	−1.12838
$218$	0	0
$219$	3196.66	0.986347
$220$	0	0
$221$	−2208.14	−0.672107
$222$	0	0
$223$	−2751.65	−0.826296	−0.413148	−	0.910664i	$-0.635571\pi$
$223$	−2751.65	−0.826296	−0.413148	+	0.910664i	$0.635571\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2313.98	−0.676583	−0.338291	−	0.941041i	$-0.609849\pi$
$227$	−2313.98	−0.676583	−0.338291	+	0.941041i	$0.609849\pi$
$228$	0	0
$229$	3756.02	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$229$	3756.02	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$230$	0	0
$231$	−537.471	−0.153087
$232$	0	0
$233$	−6113.21	−1.71884	−0.859420	−	0.511271i	$-0.829175\pi$
$233$	−6113.21	−1.71884	−0.859420	+	0.511271i	$0.829175\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	598.304	0.163983
$238$	0	0
$239$	−4138.87	−1.12017	−0.560086	−	0.828434i	$-0.689232\pi$
$239$	−4138.87	−1.12017	−0.560086	+	0.828434i	$0.689232\pi$
$240$	0	0
$241$	−525.293	−0.140403	−0.0702015	−	0.997533i	$-0.522364\pi$
$241$	−525.293	−0.140403	−0.0702015	+	0.997533i	$0.522364\pi$
$242$	0	0
$243$	−3272.91	−0.864021
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1055.41	−0.271878
$248$	0	0
$249$	−716.907	−0.182458
$250$	0	0
$251$	−4652.73	−1.17003	−0.585015	−	0.811023i	$-0.698911\pi$
$251$	−4652.73	−1.17003	−0.585015	+	0.811023i	$0.698911\pi$
$252$	0	0
$253$	509.132	0.126517
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−733.325	−0.177990	−0.0889952	−	0.996032i	$-0.528366\pi$
$257$	−733.325	−0.177990	−0.0889952	+	0.996032i	$0.528366\pi$
$258$	0	0
$259$	−5369.16	−1.28812
$260$	0	0
$261$	−3664.02	−0.868955
$262$	0	0
$263$	−4221.63	−0.989799	−0.494899	−	0.868950i	$-0.664795\pi$
$263$	−4221.63	−0.989799	−0.494899	+	0.868950i	$0.664795\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3874.63	−0.888103
$268$	0	0
$269$	130.149	0.0294993	0.0147496	−	0.999891i	$-0.495305\pi$
$269$	130.149	0.0294993	0.0147496	+	0.999891i	$0.495305\pi$
$270$	0	0
$271$	−2120.68	−0.475359	−0.237680	−	0.971344i	$-0.576387\pi$
$271$	−2120.68	−0.475359	−0.237680	+	0.971344i	$0.576387\pi$
$272$	0	0
$273$	2503.29	0.554967
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6551.71	1.42113	0.710567	−	0.703630i	$-0.248439\pi$
$277$	6551.71	1.42113	0.710567	+	0.703630i	$0.248439\pi$
$278$	0	0
$279$	3905.41	0.838031
$280$	0	0
$281$	−6787.79	−1.44102	−0.720508	−	0.693446i	$-0.756091\pi$
$281$	−6787.79	−1.44102	−0.720508	+	0.693446i	$0.756091\pi$
$282$	0	0
$283$	−6066.00	−1.27416	−0.637078	−	0.770799i	$-0.719857\pi$
$283$	−6066.00	−1.27416	−0.637078	+	0.770799i	$0.719857\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−642.862	−0.132219
$288$	0	0
$289$	−3332.77	−0.678357
$290$	0	0
$291$	−6116.17	−1.23208
$292$	0	0
$293$	956.350	0.190684	0.0953422	−	0.995445i	$-0.469605\pi$
$293$	956.350	0.190684	0.0953422	+	0.995445i	$0.469605\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1783.51	0.348450
$298$	0	0
$299$	−2371.30	−0.458647
$300$	0	0
$301$	2161.06	0.413825
$302$	0	0
$303$	7300.75	1.38421
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7123.36	−1.32427	−0.662136	−	0.749384i	$-0.730350\pi$
$307$	−7123.36	−1.32427	−0.662136	+	0.749384i	$0.730350\pi$
$308$	0	0
$309$	−1392.04	−0.256280
$310$	0	0
$311$	1335.62	0.243524	0.121762	−	0.992559i	$-0.461146\pi$
$311$	1335.62	0.243524	0.121762	+	0.992559i	$0.461146\pi$
$312$	0	0
$313$	6997.22	1.26360	0.631799	−	0.775132i	$-0.282317\pi$
$313$	6997.22	1.26360	0.631799	+	0.775132i	$0.282317\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1498.13	−0.265437	−0.132718	−	0.991154i	$-0.542371\pi$
$317$	−1498.13	−0.265437	−0.132718	+	0.991154i	$0.542371\pi$
$318$	0	0
$319$	3341.80	0.586536
$320$	0	0
$321$	−7873.33	−1.36899
$322$	0	0
$323$	755.290	0.130110
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4371.50	0.739280
$328$	0	0
$329$	6267.27	1.05023
$330$	0	0
$331$	−4427.94	−0.735292	−0.367646	−	0.929966i	$-0.619836\pi$
$331$	−4427.94	−0.735292	−0.367646	+	0.929966i	$0.619836\pi$
$332$	0	0
$333$	5813.34	0.956664
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2787.35	0.450553	0.225277	−	0.974295i	$-0.427671\pi$
$337$	2787.35	0.450553	0.225277	+	0.974295i	$0.427671\pi$
$338$	0	0
$339$	−1862.98	−0.298475
$340$	0	0
$341$	−3561.96	−0.565663
$342$	0	0
$343$	−6523.41	−1.02691
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6454.76	0.998587	0.499293	−	0.866433i	$-0.333593\pi$
$347$	6454.76	0.998587	0.499293	+	0.866433i	$0.333593\pi$
$348$	0	0
$349$	9480.34	1.45407	0.727036	−	0.686600i	$-0.240898\pi$
$349$	9480.34	1.45407	0.727036	+	0.686600i	$0.240898\pi$
$350$	0	0
$351$	−8306.75	−1.26319
$352$	0	0
$353$	4159.06	0.627095	0.313547	−	0.949573i	$-0.398483\pi$
$353$	4159.06	0.627095	0.313547	+	0.949573i	$0.398483\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1791.45	−0.265584
$358$	0	0
$359$	925.090	0.136001	0.0680005	−	0.997685i	$-0.478338\pi$
$359$	925.090	0.136001	0.0680005	+	0.997685i	$0.478338\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	4435.78	0.641372
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7110.72	−1.01138	−0.505690	−	0.862715i	$-0.668762\pi$
$367$	−7110.72	−1.01138	−0.505690	+	0.862715i	$0.668762\pi$
$368$	0	0
$369$	696.045	0.0981969
$370$	0	0
$371$	4631.21	0.648087
$372$	0	0
$373$	−941.543	−0.130700	−0.0653502	−	0.997862i	$-0.520816\pi$
$373$	−941.543	−0.130700	−0.0653502	+	0.997862i	$0.520816\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15564.6	−2.12630
$378$	0	0
$379$	−2765.66	−0.374835	−0.187417	−	0.982280i	$-0.560012\pi$
$379$	−2765.66	−0.374835	−0.187417	+	0.982280i	$0.560012\pi$
$380$	0	0
$381$	−9086.21	−1.22179
$382$	0	0
$383$	−8948.71	−1.19388	−0.596942	−	0.802284i	$-0.703618\pi$
$383$	−8948.71	−1.19388	−0.596942	+	0.802284i	$0.703618\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2339.84	−0.307341
$388$	0	0
$389$	−1300.33	−0.169484	−0.0847420	−	0.996403i	$-0.527007\pi$
$389$	−1300.33	−0.169484	−0.0847420	+	0.996403i	$0.527007\pi$
$390$	0	0
$391$	1696.99	0.219490
$392$	0	0
$393$	−1054.29	−0.135323
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1637.53	−0.207016	−0.103508	−	0.994629i	$-0.533007\pi$
$397$	−1637.53	−0.207016	−0.103508	+	0.994629i	$0.533007\pi$
$398$	0	0
$399$	−856.245	−0.107433
$400$	0	0
$401$	−9798.46	−1.22023	−0.610114	−	0.792313i	$-0.708877\pi$
$401$	−9798.46	−1.22023	−0.610114	+	0.792313i	$0.708877\pi$
$402$	0	0
$403$	16589.9	2.05063
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5302.11	−0.645739
$408$	0	0
$409$	−11052.1	−1.33616	−0.668080	−	0.744090i	$-0.732884\pi$
$409$	−11052.1	−1.33616	−0.668080	+	0.744090i	$0.732884\pi$
$410$	0	0
$411$	9085.77	1.09043
$412$	0	0
$413$	2937.74	0.350017
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8399.20	0.986356
$418$	0	0
$419$	−4192.79	−0.488857	−0.244428	−	0.969667i	$-0.578600\pi$
$419$	−4192.79	−0.488857	−0.244428	+	0.969667i	$0.578600\pi$
$420$	0	0
$421$	−8223.34	−0.951974	−0.475987	−	0.879452i	$-0.657909\pi$
$421$	−8223.34	−0.951974	−0.475987	+	0.879452i	$0.657909\pi$
$422$	0	0
$423$	−6785.75	−0.779987
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1558.47	−0.176627
$428$	0	0
$429$	2472.03	0.278207
$430$	0	0
$431$	−10799.6	−1.20696	−0.603478	−	0.797380i	$-0.706219\pi$
$431$	−10799.6	−1.20696	−0.603478	+	0.797380i	$0.706219\pi$
$432$	0	0
$433$	3032.54	0.336570	0.168285	−	0.985738i	$-0.446177\pi$
$433$	3032.54	0.336570	0.168285	+	0.985738i	$0.446177\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	811.097	0.0887873
$438$	0	0
$439$	−6014.92	−0.653932	−0.326966	−	0.945036i	$-0.606026\pi$
$439$	−6014.92	−0.653932	−0.326966	+	0.945036i	$0.606026\pi$
$440$	0	0
$441$	2577.88	0.278358
$442$	0	0
$443$	999.949	0.107244	0.0536219	−	0.998561i	$-0.482923\pi$
$443$	999.949	0.107244	0.0536219	+	0.998561i	$0.482923\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2419.15	0.255978
$448$	0	0
$449$	5439.68	0.571747	0.285874	−	0.958267i	$-0.407716\pi$
$449$	5439.68	0.571747	0.285874	+	0.958267i	$0.407716\pi$
$450$	0	0
$451$	−634.834	−0.0662820
$452$	0	0
$453$	−9476.20	−0.982850
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13597.2	1.39179	0.695897	−	0.718141i	$-0.255007\pi$
$457$	13597.2	1.39179	0.695897	+	0.718141i	$0.255007\pi$
$458$	0	0
$459$	5944.63	0.604513
$460$	0	0
$461$	10712.0	1.08222	0.541112	−	0.840950i	$-0.318003\pi$
$461$	10712.0	1.08222	0.541112	+	0.840950i	$0.318003\pi$
$462$	0	0
$463$	−18450.5	−1.85198	−0.925988	−	0.377552i	$-0.876766\pi$
$463$	−18450.5	−1.85198	−0.925988	+	0.377552i	$0.876766\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7508.05	0.743964	0.371982	−	0.928240i	$-0.378678\pi$
$467$	7508.05	0.743964	0.371982	+	0.928240i	$0.378678\pi$
$468$	0	0
$469$	−7543.10	−0.742661
$470$	0	0
$471$	494.798	0.0484057
$472$	0	0
$473$	2134.07	0.207452
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5014.34	−0.481323
$478$	0	0
$479$	5986.88	0.571081	0.285540	−	0.958367i	$-0.407827\pi$
$479$	5986.88	0.571081	0.285540	+	0.958367i	$0.407827\pi$
$480$	0	0
$481$	24694.7	2.34092
$482$	0	0
$483$	−1923.82	−0.181235
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17195.1	1.59997	0.799984	−	0.600021i	$-0.204841\pi$
$487$	17195.1	1.59997	0.799984	+	0.600021i	$0.204841\pi$
$488$	0	0
$489$	8475.52	0.783796
$490$	0	0
$491$	−7217.97	−0.663426	−0.331713	−	0.943380i	$-0.607627\pi$
$491$	−7217.97	−0.663426	−0.331713	+	0.943380i	$0.607627\pi$
$492$	0	0
$493$	11138.6	1.01756
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−727.509	−0.0656604
$498$	0	0
$499$	−15943.5	−1.43032	−0.715161	−	0.698960i	$-0.753647\pi$
$499$	−15943.5	−1.43032	−0.715161	+	0.698960i	$0.753647\pi$
$500$	0	0
$501$	1046.73	0.0933425
$502$	0	0
$503$	−13780.1	−1.22152	−0.610760	−	0.791815i	$-0.709136\pi$
$503$	−13780.1	−1.22152	−0.610760	+	0.791815i	$0.709136\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3315.58	−0.290434
$508$	0	0
$509$	18733.2	1.63131	0.815654	−	0.578540i	$-0.196377\pi$
$509$	18733.2	1.63131	0.815654	+	0.578540i	$0.196377\pi$
$510$	0	0
$511$	−10346.4	−0.895688
$512$	0	0
$513$	2841.31	0.244536
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6189.01	0.526484
$518$	0	0
$519$	−10135.2	−0.857201
$520$	0	0
$521$	15939.0	1.34031	0.670154	−	0.742222i	$-0.266228\pi$
$521$	15939.0	1.34031	0.670154	+	0.742222i	$0.266228\pi$
$522$	0	0
$523$	2414.07	0.201836	0.100918	−	0.994895i	$-0.467822\pi$
$523$	2414.07	0.201836	0.100918	+	0.994895i	$0.467822\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11872.4	−0.981347
$528$	0	0
$529$	−10344.6	−0.850220
$530$	0	0
$531$	−3180.78	−0.259951
$532$	0	0
$533$	2956.76	0.240284
$534$	0	0
$535$	0	0
$536$	0	0
$537$	13108.2	1.05337
$538$	0	0
$539$	−2351.18	−0.187889
$540$	0	0
$541$	17831.4	1.41706	0.708530	−	0.705680i	$-0.249359\pi$
$541$	17831.4	1.41706	0.708530	+	0.705680i	$0.249359\pi$
$542$	0	0
$543$	6272.73	0.495744
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5416.08	0.423354	0.211677	−	0.977340i	$-0.432107\pi$
$547$	5416.08	0.423354	0.211677	+	0.977340i	$0.432107\pi$
$548$	0	0
$549$	1687.40	0.131178
$550$	0	0
$551$	5323.82	0.411620
$552$	0	0
$553$	−1936.49	−0.148911
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9994.53	0.760290	0.380145	−	0.924927i	$-0.375874\pi$
$557$	9994.53	0.760290	0.380145	+	0.924927i	$0.375874\pi$
$558$	0	0
$559$	−9939.51	−0.752051
$560$	0	0
$561$	−1769.08	−0.133138
$562$	0	0
$563$	9425.83	0.705597	0.352799	−	0.935699i	$-0.385230\pi$
$563$	9425.83	0.705597	0.352799	+	0.935699i	$0.385230\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2475.18	−0.183329
$568$	0	0
$569$	899.764	0.0662919	0.0331459	−	0.999451i	$-0.489447\pi$
$569$	899.764	0.0662919	0.0331459	+	0.999451i	$0.489447\pi$
$570$	0	0
$571$	12647.0	0.926901	0.463451	−	0.886123i	$-0.346611\pi$
$571$	12647.0	0.926901	0.463451	+	0.886123i	$0.346611\pi$
$572$	0	0
$573$	12487.1	0.910397
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−9330.00	−0.673160	−0.336580	−	0.941655i	$-0.609270\pi$
$577$	−9330.00	−0.673160	−0.336580	+	0.941655i	$0.609270\pi$
$578$	0	0
$579$	3946.67	0.283278
$580$	0	0
$581$	2320.36	0.165688
$582$	0	0
$583$	4573.38	0.324888
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14668.1	1.03137	0.515687	−	0.856777i	$-0.327537\pi$
$587$	14668.1	1.03137	0.515687	+	0.856777i	$0.327537\pi$
$588$	0	0
$589$	−5674.56	−0.396971
$590$	0	0
$591$	1266.33	0.0881383
$592$	0	0
$593$	−21581.4	−1.49450	−0.747252	−	0.664540i	$-0.768627\pi$
$593$	−21581.4	−1.49450	−0.747252	+	0.664540i	$0.768627\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16026.8	1.09871
$598$	0	0
$599$	−12710.6	−0.867017	−0.433508	−	0.901150i	$-0.642725\pi$
$599$	−12710.6	−0.867017	−0.433508	+	0.901150i	$0.642725\pi$
$600$	0	0
$601$	7671.87	0.520702	0.260351	−	0.965514i	$-0.416162\pi$
$601$	7671.87	0.520702	0.260351	+	0.965514i	$0.416162\pi$
$602$	0	0
$603$	8167.13	0.551561
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4570.51	−0.305620	−0.152810	−	0.988256i	$-0.548832\pi$
$607$	−4570.51	−0.305620	−0.152810	+	0.988256i	$0.548832\pi$
$608$	0	0
$609$	−12627.4	−0.840211
$610$	0	0
$611$	−28825.5	−1.90860
$612$	0	0
$613$	−21692.6	−1.42929	−0.714647	−	0.699485i	$-0.753413\pi$
$613$	−21692.6	−1.42929	−0.714647	+	0.699485i	$0.753413\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23216.9	−1.51487	−0.757437	−	0.652908i	$-0.773549\pi$
$617$	−23216.9	−1.51487	−0.757437	+	0.652908i	$0.773549\pi$
$618$	0	0
$619$	−2105.96	−0.136746	−0.0683729	−	0.997660i	$-0.521781\pi$
$619$	−2105.96	−0.136746	−0.0683729	+	0.997660i	$0.521781\pi$
$620$	0	0
$621$	6383.87	0.412521
$622$	0	0
$623$	12540.7	0.806475
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−845.552	−0.0538566
$628$	0	0
$629$	−17672.5	−1.12027
$630$	0	0
$631$	−1392.01	−0.0878211	−0.0439105	−	0.999035i	$-0.513982\pi$
$631$	−1392.01	−0.0878211	−0.0439105	+	0.999035i	$0.513982\pi$
$632$	0	0
$633$	−1835.71	−0.115265
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10950.7	0.681133
$638$	0	0
$639$	787.695	0.0487648
$640$	0	0
$641$	12336.5	0.760158	0.380079	−	0.924954i	$-0.375897\pi$
$641$	12336.5	0.760158	0.380079	+	0.924954i	$0.375897\pi$
$642$	0	0
$643$	7028.47	0.431067	0.215533	−	0.976496i	$-0.430851\pi$
$643$	7028.47	0.431067	0.215533	+	0.976496i	$0.430851\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	322.899	0.0196205	0.00981025	−	0.999952i	$-0.496877\pi$
$647$	322.899	0.0196205	0.00981025	+	0.999952i	$0.496877\pi$
$648$	0	0
$649$	2901.06	0.175465
$650$	0	0
$651$	13459.3	0.810310
$652$	0	0
$653$	−5602.85	−0.335768	−0.167884	−	0.985807i	$-0.553693\pi$
$653$	−5602.85	−0.335768	−0.167884	+	0.985807i	$0.553693\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11202.3	0.665212
$658$	0	0
$659$	22171.0	1.31056	0.655281	−	0.755385i	$-0.272550\pi$
$659$	22171.0	1.31056	0.655281	+	0.755385i	$0.272550\pi$
$660$	0	0
$661$	−183.870	−0.0108196	−0.00540978	−	0.999985i	$-0.501722\pi$
$661$	−183.870	−0.0108196	−0.00540978	+	0.999985i	$0.501722\pi$
$662$	0	0
$663$	8239.53	0.482650
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11961.6	0.694385
$668$	0	0
$669$	10267.6	0.593376
$670$	0	0
$671$	−1539.01	−0.0885437
$672$	0	0
$673$	889.554	0.0509506	0.0254753	−	0.999675i	$-0.491890\pi$
$673$	889.554	0.0509506	0.0254753	+	0.999675i	$0.491890\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14387.8	−0.816793	−0.408396	−	0.912805i	$-0.633912\pi$
$677$	−14387.8	−0.816793	−0.408396	+	0.912805i	$0.633912\pi$
$678$	0	0
$679$	19795.7	1.11884
$680$	0	0
$681$	8634.47	0.485864
$682$	0	0
$683$	−18351.4	−1.02811	−0.514053	−	0.857758i	$-0.671857\pi$
$683$	−18351.4	−1.02811	−0.514053	+	0.857758i	$0.671857\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−14015.3	−0.778338
$688$	0	0
$689$	−21300.6	−1.17778
$690$	0	0
$691$	9153.77	0.503945	0.251972	−	0.967734i	$-0.418921\pi$
$691$	9153.77	0.503945	0.251972	+	0.967734i	$0.418921\pi$
$692$	0	0
$693$	−1883.51	−0.103245
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2115.97	−0.114990
$698$	0	0
$699$	22811.0	1.23432
$700$	0	0
$701$	−14795.6	−0.797181	−0.398590	−	0.917129i	$-0.630500\pi$
$701$	−14795.6	−0.797181	−0.398590	+	0.917129i	$0.630500\pi$
$702$	0	0
$703$	−8446.79	−0.453167
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−23629.8	−1.25699
$708$	0	0
$709$	20247.5	1.07251	0.536256	−	0.844056i	$-0.319838\pi$
$709$	20247.5	1.07251	0.536256	+	0.844056i	$0.319838\pi$
$710$	0	0
$711$	2096.69	0.110594
$712$	0	0
$713$	−12749.6	−0.669674
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15443.9	0.804412
$718$	0	0
$719$	−8824.67	−0.457725	−0.228863	−	0.973459i	$-0.573501\pi$
$719$	−8824.67	−0.457725	−0.228863	+	0.973459i	$0.573501\pi$
$720$	0	0
$721$	4505.51	0.232724
$722$	0	0
$723$	1960.10	0.100825
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22085.6	−1.12670	−0.563350	−	0.826218i	$-0.690488\pi$
$727$	−22085.6	−1.12670	−0.563350	+	0.826218i	$0.690488\pi$
$728$	0	0
$729$	17746.2	0.901598
$730$	0	0
$731$	7113.10	0.359901
$732$	0	0
$733$	−5044.63	−0.254199	−0.127099	−	0.991890i	$-0.540567\pi$
$733$	−5044.63	−0.254199	−0.127099	+	0.991890i	$0.540567\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7448.90	−0.372298
$738$	0	0
$739$	−1581.66	−0.0787313	−0.0393657	−	0.999225i	$-0.512534\pi$
$739$	−1581.66	−0.0787313	−0.0393657	+	0.999225i	$0.512534\pi$
$740$	0	0
$741$	3938.18	0.195240
$742$	0	0
$743$	19670.3	0.971243	0.485622	−	0.874169i	$-0.338593\pi$
$743$	19670.3	0.971243	0.485622	+	0.874169i	$0.338593\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2512.32	−0.123053
$748$	0	0
$749$	25483.0	1.24316
$750$	0	0
$751$	24790.3	1.20454	0.602271	−	0.798292i	$-0.294263\pi$
$751$	24790.3	1.20454	0.602271	+	0.798292i	$0.294263\pi$
$752$	0	0
$753$	17361.3	0.840216
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2952.53	0.141759	0.0708794	−	0.997485i	$-0.477419\pi$
$757$	2952.53	0.141759	0.0708794	+	0.997485i	$0.477419\pi$
$758$	0	0
$759$	−1899.79	−0.0908539
$760$	0	0
$761$	10719.9	0.510637	0.255318	−	0.966857i	$-0.417820\pi$
$761$	10719.9	0.510637	0.255318	+	0.966857i	$0.417820\pi$
$762$	0	0
$763$	−14148.9	−0.671330
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13511.8	−0.636091
$768$	0	0
$769$	−892.638	−0.0418587	−0.0209294	−	0.999781i	$-0.506663\pi$
$769$	−892.638	−0.0418587	−0.0209294	+	0.999781i	$0.506663\pi$
$770$	0	0
$771$	2736.35	0.127818
$772$	0	0
$773$	18992.0	0.883692	0.441846	−	0.897091i	$-0.354324\pi$
$773$	18992.0	0.883692	0.441846	+	0.897091i	$0.354324\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20034.7	0.925019
$778$	0	0
$779$	−1011.35	−0.0465154
$780$	0	0
$781$	−718.424	−0.0329158
$782$	0	0
$783$	41902.0	1.91246
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31759.1	1.43849	0.719244	−	0.694758i	$-0.244488\pi$
$787$	31759.1	1.43849	0.719244	+	0.694758i	$0.244488\pi$
$788$	0	0
$789$	15752.7	0.710789
$790$	0	0
$791$	6029.76	0.271041
$792$	0	0
$793$	7167.98	0.320987
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14311.6	−0.636066	−0.318033	−	0.948080i	$-0.603022\pi$
$797$	−14311.6	−0.636066	−0.318033	+	0.948080i	$0.603022\pi$
$798$	0	0
$799$	20628.6	0.913377
$800$	0	0
$801$	−13578.2	−0.598954
$802$	0	0
$803$	−10217.2	−0.449012
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−485.641	−0.0211839
$808$	0	0
$809$	−9987.11	−0.434027	−0.217014	−	0.976169i	$-0.569632\pi$
$809$	−9987.11	−0.434027	−0.217014	+	0.976169i	$0.569632\pi$
$810$	0	0
$811$	−19713.1	−0.853538	−0.426769	−	0.904361i	$-0.640348\pi$
$811$	−19713.1	−0.853538	−0.426769	+	0.904361i	$0.640348\pi$
$812$	0	0
$813$	7913.20	0.341363
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3399.79	0.145586
$818$	0	0
$819$	8772.49	0.374280
$820$	0	0
$821$	−27603.3	−1.17340	−0.586701	−	0.809804i	$-0.699574\pi$
$821$	−27603.3	−1.17340	−0.586701	+	0.809804i	$0.699574\pi$
$822$	0	0
$823$	−42308.6	−1.79196	−0.895981	−	0.444092i	$-0.853526\pi$
$823$	−42308.6	−1.79196	−0.895981	+	0.444092i	$0.853526\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2569.96	0.108061	0.0540305	−	0.998539i	$-0.482793\pi$
$827$	2569.96	0.108061	0.0540305	+	0.998539i	$0.482793\pi$
$828$	0	0
$829$	−13102.3	−0.548929	−0.274464	−	0.961597i	$-0.588501\pi$
$829$	−13102.3	−0.548929	−0.274464	+	0.961597i	$0.588501\pi$
$830$	0	0
$831$	−24447.3	−1.02054
$832$	0	0
$833$	−7836.72	−0.325962
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−44662.5	−1.84440
$838$	0	0
$839$	32633.7	1.34284	0.671419	−	0.741078i	$-0.265685\pi$
$839$	32633.7	1.34284	0.671419	+	0.741078i	$0.265685\pi$
$840$	0	0
$841$	54123.7	2.21919
$842$	0	0
$843$	25328.2	1.03482
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−14357.0	−0.582422
$848$	0	0
$849$	22634.9	0.914991
$850$	0	0
$851$	−18978.3	−0.764474
$852$	0	0
$853$	11689.0	0.469197	0.234599	−	0.972092i	$-0.424622\pi$
$853$	11689.0	0.469197	0.234599	+	0.972092i	$0.424622\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27684.8	1.10350	0.551748	−	0.834011i	$-0.313961\pi$
$857$	27684.8	1.10350	0.551748	+	0.834011i	$0.313961\pi$
$858$	0	0
$859$	44308.5	1.75994	0.879970	−	0.475030i	$-0.157563\pi$
$859$	44308.5	1.75994	0.879970	+	0.475030i	$0.157563\pi$
$860$	0	0
$861$	2398.80	0.0949487
$862$	0	0
$863$	−9691.73	−0.382283	−0.191141	−	0.981562i	$-0.561219\pi$
$863$	−9691.73	−0.382283	−0.191141	+	0.981562i	$0.561219\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12436.0	0.487139
$868$	0	0
$869$	−1912.30	−0.0746496
$870$	0	0
$871$	34693.5	1.34965
$872$	0	0
$873$	−21433.4	−0.830941
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27750.6	−1.06850	−0.534248	−	0.845328i	$-0.679405\pi$
$877$	−27750.6	−1.06850	−0.534248	+	0.845328i	$0.679405\pi$
$878$	0	0
$879$	−3568.56	−0.136933
$880$	0	0
$881$	8716.52	0.333334	0.166667	−	0.986013i	$-0.446700\pi$
$881$	8716.52	0.333334	0.166667	+	0.986013i	$0.446700\pi$
$882$	0	0
$883$	8880.61	0.338456	0.169228	−	0.985577i	$-0.445873\pi$
$883$	8880.61	0.338456	0.169228	+	0.985577i	$0.445873\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42224.2	1.59837	0.799183	−	0.601088i	$-0.205266\pi$
$887$	42224.2	1.59837	0.799183	+	0.601088i	$0.205266\pi$
$888$	0	0
$889$	29408.7	1.10949
$890$	0	0
$891$	−2444.27	−0.0919036
$892$	0	0
$893$	9859.70	0.369476
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8848.34	0.329362
$898$	0	0
$899$	−83685.2	−3.10462
$900$	0	0
$901$	15243.6	0.563637
$902$	0	0
$903$	−8063.86	−0.297174
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−38157.5	−1.39691	−0.698456	−	0.715653i	$-0.746129\pi$
$907$	−38157.5	−1.39691	−0.698456	+	0.715653i	$0.746129\pi$
$908$	0	0
$909$	25584.6	0.933541
$910$	0	0
$911$	35165.8	1.27892	0.639460	−	0.768824i	$-0.279158\pi$
$911$	35165.8	1.27892	0.639460	+	0.768824i	$0.279158\pi$
$912$	0	0
$913$	2291.38	0.0830599
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3412.34	0.122885
$918$	0	0
$919$	−28297.8	−1.01573	−0.507866	−	0.861436i	$-0.669565\pi$
$919$	−28297.8	−1.01573	−0.507866	+	0.861436i	$0.669565\pi$
$920$	0	0
$921$	26580.3	0.950979
$922$	0	0
$923$	3346.08	0.119326
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4878.25	−0.172840
$928$	0	0
$929$	35859.6	1.26643	0.633215	−	0.773976i	$-0.281735\pi$
$929$	35859.6	1.26643	0.633215	+	0.773976i	$0.281735\pi$
$930$	0	0
$931$	−3745.66	−0.131857
$932$	0	0
$933$	−4983.77	−0.174878
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−33337.3	−1.16231	−0.581154	−	0.813793i	$-0.697399\pi$
$937$	−33337.3	−1.16231	−0.581154	+	0.813793i	$0.697399\pi$
$938$	0	0
$939$	−26109.7	−0.907409
$940$	0	0
$941$	−1590.15	−0.0550874	−0.0275437	−	0.999621i	$-0.508769\pi$
$941$	−1590.15	−0.0550874	−0.0275437	+	0.999621i	$0.508769\pi$
$942$	0	0
$943$	−2272.32	−0.0784695
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48614.0	1.66815	0.834077	−	0.551648i	$-0.186001\pi$
$947$	48614.0	1.66815	0.834077	+	0.551648i	$0.186001\pi$
$948$	0	0
$949$	47586.8	1.62775
$950$	0	0
$951$	5590.18	0.190614
$952$	0	0
$953$	33010.3	1.12204	0.561022	−	0.827801i	$-0.310408\pi$
$953$	33010.3	1.12204	0.561022	+	0.827801i	$0.310408\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12469.7	−0.421201
$958$	0	0
$959$	−29407.2	−0.990208
$960$	0	0
$961$	59407.4	1.99414
$962$	0	0
$963$	−27591.2	−0.923276
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−59137.1	−1.96662	−0.983309	−	0.181944i	$-0.941761\pi$
$967$	−59137.1	−1.96662	−0.983309	+	0.181944i	$0.941761\pi$
$968$	0	0
$969$	−2818.32	−0.0934338
$970$	0	0
$971$	24624.8	0.813849	0.406925	−	0.913462i	$-0.366601\pi$
$971$	24624.8	0.813849	0.406925	+	0.913462i	$0.366601\pi$
$972$	0	0
$973$	−27185.1	−0.895697
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21010.2	0.688001	0.344000	−	0.938970i	$-0.388218\pi$
$977$	21010.2	0.688001	0.344000	+	0.938970i	$0.388218\pi$
$978$	0	0
$979$	12384.1	0.404289
$980$	0	0
$981$	15319.4	0.498585
$982$	0	0
$983$	−11609.2	−0.376679	−0.188340	−	0.982104i	$-0.560311\pi$
$983$	−11609.2	−0.376679	−0.188340	+	0.982104i	$0.560311\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−23385.9	−0.754186
$988$	0	0
$989$	7638.67	0.245597
$990$	0	0
$991$	−30991.4	−0.993416	−0.496708	−	0.867918i	$-0.665458\pi$
$991$	−30991.4	−0.993416	−0.496708	+	0.867918i	$0.665458\pi$
$992$	0	0
$993$	16522.6	0.528024
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22116.9	−0.702557	−0.351279	−	0.936271i	$-0.614253\pi$
$997$	−22116.9	−0.702557	−0.351279	+	0.936271i	$0.614253\pi$
$998$	0	0
$999$	−66481.7	−2.10549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.4.a.l.1.5		12
5.2	odd	4		380.4.c.a.229.8	yes	12
5.3	odd	4		380.4.c.a.229.5	✓	12
5.4	even	2	inner	1900.4.a.l.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.4.c.a.229.5	✓	12	5.3	odd	4
380.4.c.a.229.8	yes	12	5.2	odd	4
1900.4.a.l.1.5		12	1.1	even	1	trivial
1900.4.a.l.1.8		12	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1900.a (trivial)

\(f(q)\)	\(=\)	\(q-3.73144 q^{3} +12.0773 q^{7} -13.0764 q^{9} +11.9264 q^{11} -55.5478 q^{13} +39.7521 q^{17} +19.0000 q^{19} -45.0655 q^{21} +42.6893 q^{23} +149.542 q^{27} +280.201 q^{29} -298.661 q^{31} -44.5027 q^{33} -444.568 q^{37} +207.273 q^{39} -53.2291 q^{41} +178.936 q^{43} +518.932 q^{47} -197.140 q^{49} -148.332 q^{51} +383.465 q^{53} -70.8973 q^{57} +243.246 q^{59} -129.042 q^{61} -157.927 q^{63} -624.571 q^{67} -159.292 q^{69} -60.2379 q^{71} -856.683 q^{73} +144.039 q^{77} -160.342 q^{79} -204.945 q^{81} +192.126 q^{83} -1045.55 q^{87} +1038.38 q^{89} -670.865 q^{91} +1114.43 q^{93} +1639.09 q^{97} -155.955 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9} - 110 q^{11} + 228 q^{19} - 8 q^{21} + 76 q^{29} + 80 q^{31} - 60 q^{39} - 648 q^{41} + 886 q^{49} - 328 q^{51} - 984 q^{59} - 1970 q^{61} - 364 q^{69} - 3060 q^{71} - 552 q^{79} - 3388 q^{81}+ \cdots - 2366 q^{99}+O(q^{100}) \) 12 * q - 8 * q^9 - 110 * q^11 + 228 * q^19 - 8 * q^21 + 76 * q^29 + 80 * q^31 - 60 * q^39 - 648 * q^41 + 886 * q^49 - 328 * q^51 - 984 * q^59 - 1970 * q^61 - 364 * q^69 - 3060 * q^71 - 552 * q^79 - 3388 * q^81 - 2428 * q^89 - 3468 * q^91 - 2366 * q^99

Introduction

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