LMFDB - Embedded newform 1900.4.c.b.1749.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,4,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$112.103629011$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.3
Root			$2.37228i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.4.c.b.1749.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.372281i q^{3} +3.51087i q^{7} +26.8614 q^{9} +O(q^{10})$	$q+0.372281i q^{3} +3.51087i q^{7} +26.8614 q^{9} -21.1386 q^{11} +14.0951i q^{13} +17.2337i q^{17} +19.0000 q^{19} -1.30703 q^{21} -171.965i q^{23} +20.0516i q^{27} -264.198 q^{29} +185.783 q^{31} -7.86950i q^{33} -212.978i q^{37} -5.24734 q^{39} -157.168 q^{41} -258.557i q^{43} +293.562i q^{47} +330.674 q^{49} -6.41578 q^{51} +215.791i q^{53} +7.07335i q^{57} +537.030 q^{59} -280.149 q^{61} +94.3070i q^{63} -147.291i q^{67} +64.0192 q^{69} -913.446 q^{71} -678.826i q^{73} -74.2150i q^{77} +608.277 q^{79} +717.793 q^{81} +282.440i q^{83} -98.3561i q^{87} +214.217 q^{89} -49.4861 q^{91} +69.1634i q^{93} -1670.50i q^{97} -567.812 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 50 q^{9}+O(q^{10}) $ 4 * q + 50 * q^9	$ 4 q + 50 q^{9} - 142 q^{11} + 76 q^{19} + 282 q^{21} - 310 q^{29} - 176 q^{31} - 538 q^{39} - 284 q^{41} - 56 q^{49} - 198 q^{51} + 1746 q^{59} + 890 q^{61} + 1922 q^{69} - 3424 q^{71} + 2548 q^{79} - 116 q^{81} + 1776 q^{89} + 2502 q^{91} - 950 q^{99}+O(q^{100}) $ 4 * q + 50 * q^9 - 142 * q^11 + 76 * q^19 + 282 * q^21 - 310 * q^29 - 176 * q^31 - 538 * q^39 - 284 * q^41 - 56 * q^49 - 198 * q^51 + 1746 * q^59 + 890 * q^61 + 1922 * q^69 - 3424 * q^71 + 2548 * q^79 - 116 * q^81 + 1776 * q^89 + 2502 * q^91 - 950 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.372281i	0.0716456i	0.999358	+	0.0358228i	$0.0114052\pi$
$3$	0.372281i	0.0716456i	−0.999358	+	0.0358228i	$0.988595\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.51087i	0.189569i	0.995498	+	0.0947847i	$0.0302163\pi$
$7$	3.51087i	0.189569i	−0.995498	+	0.0947847i	$0.969784\pi$
$8$	0	0
$9$	26.8614	0.994867
$10$	0	0
$11$	−21.1386	−0.579411	−0.289706	−	0.957116i	$-0.593557\pi$
$11$	−21.1386	−0.579411	−0.289706	+	0.957116i	$0.593557\pi$
$12$	0	0
$13$	14.0951i	0.300714i	0.988632	+	0.150357i	$0.0480422\pi$
$13$	14.0951i	0.300714i	−0.988632	+	0.150357i	$0.951958\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	17.2337i	0.245870i	0.992415	+	0.122935i	$0.0392306\pi$
$17$	17.2337i	0.245870i	−0.992415	+	0.122935i	$0.960769\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−1.30703	−0.0135818
$22$	0	0
$23$	− 171.965i	− 1.55900i	−0.626400	−	0.779502i	$-0.715472\pi$
$23$	− 171.965i	− 1.55900i	0.626400	−	0.779502i	$-0.284528\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	20.0516i	0.142923i
$28$	0	0
$29$	−264.198	−1.69174	−0.845869	−	0.533391i	$-0.820917\pi$
$29$	−264.198	−1.69174	−0.845869	+	0.533391i	$0.820917\pi$
$30$	0	0
$31$	185.783	1.07637	0.538186	−	0.842826i	$-0.319110\pi$
$31$	185.783	1.07637	0.538186	+	0.842826i	$0.319110\pi$
$32$	0	0
$33$	− 7.86950i	− 0.0415123i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 212.978i	− 0.946308i	−0.880980	−	0.473154i	$-0.843115\pi$
$37$	− 212.978i	− 0.946308i	0.880980	−	0.473154i	$-0.156885\pi$
$38$	0	0
$39$	−5.24734	−0.0215448
$40$	0	0
$41$	−157.168	−0.598673	−0.299336	−	0.954148i	$-0.596765\pi$
$41$	−157.168	−0.598673	−0.299336	+	0.954148i	$0.596765\pi$
$42$	0	0
$43$	− 258.557i	− 0.916967i	−0.888703	−	0.458483i	$-0.848393\pi$
$43$	− 258.557i	− 0.916967i	0.888703	−	0.458483i	$-0.151607\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	293.562i	0.911074i	0.890217	+	0.455537i	$0.150553\pi$
$47$	293.562i	0.911074i	−0.890217	+	0.455537i	$0.849447\pi$
$48$	0	0
$49$	330.674	0.964063
$50$	0	0
$51$	−6.41578	−0.0176155
$52$	0	0
$53$	215.791i	0.559266i	0.960107	+	0.279633i	$0.0902129\pi$
$53$	215.791i	0.559266i	−0.960107	+	0.279633i	$0.909787\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.07335i	0.0164366i
$58$	0	0
$59$	537.030	1.18501	0.592503	−	0.805568i	$-0.298140\pi$
$59$	537.030	1.18501	0.592503	+	0.805568i	$0.298140\pi$
$60$	0	0
$61$	−280.149	−0.588024	−0.294012	−	0.955802i	$-0.594990\pi$
$61$	−280.149	−0.588024	−0.294012	+	0.955802i	$0.594990\pi$
$62$	0	0
$63$	94.3070i	0.188596i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 147.291i	− 0.268574i	−0.990942	−	0.134287i	$-0.957126\pi$
$67$	− 147.291i	− 0.268574i	0.990942	−	0.134287i	$-0.0428744\pi$
$68$	0	0
$69$	64.0192	0.111696
$70$	0	0
$71$	−913.446	−1.52685	−0.763423	−	0.645899i	$-0.776483\pi$
$71$	−913.446	−1.52685	−0.763423	+	0.645899i	$0.776483\pi$
$72$	0	0
$73$	− 678.826i	− 1.08836i	−0.838967	−	0.544182i	$-0.816840\pi$
$73$	− 678.826i	− 1.08836i	0.838967	−	0.544182i	$-0.183160\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 74.2150i	− 0.109839i
$78$	0	0
$79$	608.277	0.866285	0.433143	−	0.901325i	$-0.357405\pi$
$79$	608.277	0.866285	0.433143	+	0.901325i	$0.357405\pi$
$80$	0	0
$81$	717.793	0.984627
$82$	0	0
$83$	282.440i	0.373516i	0.982406	+	0.186758i	$0.0597980\pi$
$83$	282.440i	0.373516i	−0.982406	+	0.186758i	$0.940202\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 98.3561i	− 0.121206i
$88$	0	0
$89$	214.217	0.255135	0.127567	−	0.991830i	$-0.459283\pi$
$89$	214.217	0.255135	0.127567	+	0.991830i	$0.459283\pi$
$90$	0	0
$91$	−49.4861	−0.0570061
$92$	0	0
$93$	69.1634i	0.0771173i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1670.50i	− 1.74860i	−0.485387	−	0.874299i	$-0.661321\pi$
$97$	− 1670.50i	− 1.74860i	0.485387	−	0.874299i	$-0.338679\pi$
$98$	0	0
$99$	−567.812	−0.576437
$100$	0	0
$101$	166.316	0.163852	0.0819258	−	0.996638i	$-0.473893\pi$
$101$	166.316	0.163852	0.0819258	+	0.996638i	$0.473893\pi$
$102$	0	0
$103$	1219.78i	1.16688i	0.812156	+	0.583440i	$0.198294\pi$
$103$	1219.78i	1.16688i	−0.812156	+	0.583440i	$0.801706\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1736.73i	1.56912i	0.620052	+	0.784561i	$0.287112\pi$
$107$	1736.73i	1.56912i	−0.620052	+	0.784561i	$0.712888\pi$
$108$	0	0
$109$	18.2367	0.0160253	0.00801266	−	0.999968i	$-0.497449\pi$
$109$	18.2367	0.0160253	0.00801266	+	0.999968i	$0.497449\pi$
$110$	0	0
$111$	79.2878	0.0677988
$112$	0	0
$113$	− 1586.59i	− 1.32083i	−0.750902	−	0.660414i	$-0.770381\pi$
$113$	− 1586.59i	− 1.32083i	0.750902	−	0.660414i	$-0.229619\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	378.614i	0.299170i
$118$	0	0
$119$	−60.5053	−0.0466094
$120$	0	0
$121$	−884.160	−0.664282
$122$	0	0
$123$	− 58.5109i	− 0.0428923i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2327.85i	− 1.62648i	−0.581928	−	0.813240i	$-0.697701\pi$
$127$	− 2327.85i	− 1.62648i	0.581928	−	0.813240i	$-0.302299\pi$
$128$	0	0
$129$	96.2559	0.0656966
$130$	0	0
$131$	631.753	0.421347	0.210674	−	0.977556i	$-0.432434\pi$
$131$	631.753	0.421347	0.210674	+	0.977556i	$0.432434\pi$
$132$	0	0
$133$	66.7066i	0.0434902i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1098.25i	− 0.684892i	−0.939537	−	0.342446i	$-0.888745\pi$
$137$	− 1098.25i	− 0.684892i	0.939537	−	0.342446i	$-0.111255\pi$
$138$	0	0
$139$	−1523.38	−0.929577	−0.464788	−	0.885422i	$-0.653870\pi$
$139$	−1523.38	−0.929577	−0.464788	+	0.885422i	$0.653870\pi$
$140$	0	0
$141$	−109.288	−0.0652744
$142$	0	0
$143$	− 297.950i	− 0.174237i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	123.104i	0.0690709i
$148$	0	0
$149$	3160.49	1.73770	0.868849	−	0.495077i	$-0.164860\pi$
$149$	3160.49	1.73770	0.868849	+	0.495077i	$0.164860\pi$
$150$	0	0
$151$	−1930.72	−1.04053	−0.520263	−	0.854006i	$-0.674166\pi$
$151$	−1930.72	−1.04053	−0.520263	+	0.854006i	$0.674166\pi$
$152$	0	0
$153$	462.921i	0.244608i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1818.29i	− 0.924303i	−0.886801	−	0.462152i	$-0.847078\pi$
$157$	− 1818.29i	− 0.924303i	0.886801	−	0.462152i	$-0.152922\pi$
$158$	0	0
$159$	−80.3348	−0.0400690
$160$	0	0
$161$	603.746	0.295540
$162$	0	0
$163$	− 1259.17i	− 0.605068i	−0.953139	−	0.302534i	$-0.902167\pi$
$163$	− 1259.17i	− 0.605068i	0.953139	−	0.302534i	$-0.0978327\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1315.17i	0.609405i	0.952447	+	0.304703i	$0.0985571\pi$
$167$	1315.17i	0.609405i	−0.952447	+	0.304703i	$0.901443\pi$
$168$	0	0
$169$	1998.33	0.909571
$170$	0	0
$171$	510.367	0.228238
$172$	0	0
$173$	2407.00i	1.05781i	0.848682	+	0.528904i	$0.177397\pi$
$173$	2407.00i	1.05781i	−0.848682	+	0.528904i	$0.822603\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	199.926i	0.0849004i
$178$	0	0
$179$	−1082.32	−0.451935	−0.225968	−	0.974135i	$-0.572554\pi$
$179$	−1082.32	−0.451935	−0.225968	+	0.974135i	$0.572554\pi$
$180$	0	0
$181$	4317.41	1.77299	0.886494	−	0.462741i	$-0.153134\pi$
$181$	4317.41	1.77299	0.886494	+	0.462741i	$0.153134\pi$
$182$	0	0
$183$	− 104.294i	− 0.0421293i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 364.296i	− 0.142460i
$188$	0	0
$189$	−70.3986	−0.0270939
$190$	0	0
$191$	3208.27	1.21540	0.607702	−	0.794165i	$-0.292091\pi$
$191$	3208.27	1.21540	0.607702	+	0.794165i	$0.292091\pi$
$192$	0	0
$193$	− 4206.70i	− 1.56894i	−0.620168	−	0.784469i	$-0.712936\pi$
$193$	− 4206.70i	− 1.56894i	0.620168	−	0.784469i	$-0.287064\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4789.05i	− 1.73201i	−0.500037	−	0.866004i	$-0.666680\pi$
$197$	− 4789.05i	− 1.73201i	0.500037	−	0.866004i	$-0.333320\pi$
$198$	0	0
$199$	3504.05	1.24822	0.624110	−	0.781337i	$-0.285462\pi$
$199$	3504.05	1.24822	0.624110	+	0.781337i	$0.285462\pi$
$200$	0	0
$201$	54.8336	0.0192421
$202$	0	0
$203$	− 927.567i	− 0.320702i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4619.21i	− 1.55100i
$208$	0	0
$209$	−401.633	−0.132926
$210$	0	0
$211$	−3066.90	−1.00063	−0.500317	−	0.865842i	$-0.666783\pi$
$211$	−3066.90	−1.00063	−0.500317	+	0.865842i	$0.666783\pi$
$212$	0	0
$213$	− 340.059i	− 0.109392i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	652.259i	0.204047i
$218$	0	0
$219$	252.714	0.0779765
$220$	0	0
$221$	−242.910	−0.0739363
$222$	0	0
$223$	− 2117.06i	− 0.635733i	−0.948135	−	0.317867i	$-0.897034\pi$
$223$	− 2117.06i	− 0.635733i	0.948135	−	0.317867i	$-0.102966\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6476.34i	− 1.89361i	−0.321804	−	0.946806i	$-0.604289\pi$
$227$	− 6476.34i	− 1.89361i	0.321804	−	0.946806i	$-0.395711\pi$
$228$	0	0
$229$	−3686.92	−1.06392	−0.531962	−	0.846768i	$-0.678545\pi$
$229$	−3686.92	−1.06392	−0.531962	+	0.846768i	$0.678545\pi$
$230$	0	0
$231$	27.6288	0.00786946
$232$	0	0
$233$	1119.42i	0.314744i	0.987539	+	0.157372i	$0.0503022\pi$
$233$	1119.42i	0.314744i	−0.987539	+	0.157372i	$0.949698\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	226.450i	0.0620655i
$238$	0	0
$239$	6621.48	1.79208	0.896041	−	0.443971i	$-0.146431\pi$
$239$	6621.48	1.79208	0.896041	+	0.443971i	$0.146431\pi$
$240$	0	0
$241$	2784.49	0.744253	0.372127	−	0.928182i	$-0.378629\pi$
$241$	2784.49	0.744253	0.372127	+	0.928182i	$0.378629\pi$
$242$	0	0
$243$	808.614i	0.213468i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	267.807i	0.0689884i
$248$	0	0
$249$	−105.147	−0.0267608
$250$	0	0
$251$	−4670.19	−1.17442	−0.587210	−	0.809434i	$-0.699774\pi$
$251$	−4670.19	−1.17442	−0.587210	+	0.809434i	$0.699774\pi$
$252$	0	0
$253$	3635.09i	0.903305i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5167.86i	− 1.25433i	−0.778887	−	0.627164i	$-0.784216\pi$
$257$	− 5167.86i	− 1.25433i	0.778887	−	0.627164i	$-0.215784\pi$
$258$	0	0
$259$	747.740	0.179391
$260$	0	0
$261$	−7096.74	−1.68305
$262$	0	0
$263$	− 3254.16i	− 0.762966i	−0.924376	−	0.381483i	$-0.875413\pi$
$263$	− 3254.16i	− 0.762966i	0.924376	−	0.381483i	$-0.124587\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	79.7492i	0.0182793i
$268$	0	0
$269$	1547.77	0.350814	0.175407	−	0.984496i	$-0.443876\pi$
$269$	1547.77	0.350814	0.175407	+	0.984496i	$0.443876\pi$
$270$	0	0
$271$	−4969.53	−1.11394	−0.556969	−	0.830533i	$-0.688036\pi$
$271$	−4969.53	−1.11394	−0.556969	+	0.830533i	$0.688036\pi$
$272$	0	0
$273$	− 18.4228i	− 0.00408423i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 279.862i	− 0.0607049i	−0.999539	−	0.0303525i	$-0.990337\pi$
$277$	− 279.862i	− 0.0607049i	0.999539	−	0.0303525i	$-0.00966298\pi$
$278$	0	0
$279$	4990.38	1.07085
$280$	0	0
$281$	2092.54	0.444236	0.222118	−	0.975020i	$-0.428703\pi$
$281$	2092.54	0.444236	0.222118	+	0.975020i	$0.428703\pi$
$282$	0	0
$283$	− 3939.01i	− 0.827384i	−0.910417	−	0.413692i	$-0.864239\pi$
$283$	− 3939.01i	− 0.827384i	0.910417	−	0.413692i	$-0.135761\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 551.799i	− 0.113490i
$288$	0	0
$289$	4616.00	0.939548
$290$	0	0
$291$	621.898	0.125279
$292$	0	0
$293$	965.927i	0.192594i	0.995353	+	0.0962970i	$0.0306999\pi$
$293$	965.927i	0.192594i	−0.995353	+	0.0962970i	$0.969300\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 423.863i	− 0.0828114i
$298$	0	0
$299$	2423.86	0.468814
$300$	0	0
$301$	907.761	0.173829
$302$	0	0
$303$	61.9162i	0.0117392i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9718.56i	1.80673i	0.428867	+	0.903367i	$0.358913\pi$
$307$	9718.56i	1.80673i	−0.428867	+	0.903367i	$0.641087\pi$
$308$	0	0
$309$	−454.102	−0.0836019
$310$	0	0
$311$	3449.04	0.628865	0.314433	−	0.949280i	$-0.398186\pi$
$311$	3449.04	0.628865	0.314433	+	0.949280i	$0.398186\pi$
$312$	0	0
$313$	23.3728i	0.00422079i	0.999998	+	0.00211039i	$0.000671760\pi$
$313$	23.3728i	0.00422079i	−0.999998	+	0.00211039i	$0.999328\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 567.966i	− 0.100631i	−0.998733	−	0.0503157i	$-0.983977\pi$
$317$	− 567.966i	− 0.100631i	0.998733	−	0.0503157i	$-0.0160227\pi$
$318$	0	0
$319$	5584.78	0.980212
$320$	0	0
$321$	−646.552	−0.112421
$322$	0	0
$323$	327.440i	0.0564064i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.78918i	0.00114814i
$328$	0	0
$329$	−1030.66	−0.172712
$330$	0	0
$331$	−553.324	−0.0918835	−0.0459418	−	0.998944i	$-0.514629\pi$
$331$	−553.324	−0.0918835	−0.0459418	+	0.998944i	$0.514629\pi$
$332$	0	0
$333$	− 5720.90i	− 0.941451i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7328.60i	− 1.18461i	−0.805713	−	0.592306i	$-0.798218\pi$
$337$	− 7328.60i	− 1.18461i	0.805713	−	0.592306i	$-0.201782\pi$
$338$	0	0
$339$	590.657	0.0946315
$340$	0	0
$341$	−3927.18	−0.623662
$342$	0	0
$343$	2365.18i	0.372326i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4828.56i	− 0.747005i	−0.927629	−	0.373502i	$-0.878157\pi$
$347$	− 4828.56i	− 0.747005i	0.927629	−	0.373502i	$-0.121843\pi$
$348$	0	0
$349$	10256.6	1.57313	0.786564	−	0.617508i	$-0.211858\pi$
$349$	10256.6	1.57313	0.786564	+	0.617508i	$0.211858\pi$
$350$	0	0
$351$	−282.629	−0.0429790
$352$	0	0
$353$	2003.83i	0.302133i	0.988524	+	0.151067i	$0.0482708\pi$
$353$	2003.83i	0.302133i	−0.988524	+	0.151067i	$0.951729\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 22.5250i	− 0.00333935i
$358$	0	0
$359$	−4841.62	−0.711786	−0.355893	−	0.934527i	$-0.615823\pi$
$359$	−4841.62	−0.711786	−0.355893	+	0.934527i	$0.615823\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	− 329.156i	− 0.0475929i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7004.47i	− 0.996268i	−0.867100	−	0.498134i	$-0.834019\pi$
$367$	− 7004.47i	− 0.996268i	0.867100	−	0.498134i	$-0.165981\pi$
$368$	0	0
$369$	−4221.77	−0.595600
$370$	0	0
$371$	−757.614	−0.106020
$372$	0	0
$373$	− 11718.1i	− 1.62665i	−0.581812	−	0.813324i	$-0.697656\pi$
$373$	− 11718.1i	− 1.62665i	0.581812	−	0.813324i	$-0.302344\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3723.90i	− 0.508728i
$378$	0	0
$379$	4807.97	0.651633	0.325817	−	0.945433i	$-0.394361\pi$
$379$	4807.97	0.651633	0.325817	+	0.945433i	$0.394361\pi$
$380$	0	0
$381$	866.614	0.116530
$382$	0	0
$383$	5998.09i	0.800230i	0.916465	+	0.400115i	$0.131030\pi$
$383$	5998.09i	0.800230i	−0.916465	+	0.400115i	$0.868970\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 6945.20i	− 0.912260i
$388$	0	0
$389$	−7480.25	−0.974971	−0.487486	−	0.873131i	$-0.662086\pi$
$389$	−7480.25	−0.974971	−0.487486	+	0.873131i	$0.662086\pi$
$390$	0	0
$391$	2963.58	0.383312
$392$	0	0
$393$	235.190i	0.0301877i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10366.4i	1.31052i	0.755406	+	0.655258i	$0.227440\pi$
$397$	10366.4i	1.31052i	−0.755406	+	0.655258i	$0.772560\pi$
$398$	0	0
$399$	−24.8336	−0.00311588
$400$	0	0
$401$	801.467	0.0998088	0.0499044	−	0.998754i	$-0.484108\pi$
$401$	801.467	0.0998088	0.0499044	+	0.998754i	$0.484108\pi$
$402$	0	0
$403$	2618.62i	0.323680i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4502.06i	0.548302i
$408$	0	0
$409$	9396.88	1.13605	0.568027	−	0.823010i	$-0.307707\pi$
$409$	9396.88	1.13605	0.568027	+	0.823010i	$0.307707\pi$
$410$	0	0
$411$	408.860	0.0490695
$412$	0	0
$413$	1885.44i	0.224641i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 567.125i	− 0.0666001i
$418$	0	0
$419$	5721.15	0.667056	0.333528	−	0.942740i	$-0.391761\pi$
$419$	5721.15	0.667056	0.333528	+	0.942740i	$0.391761\pi$
$420$	0	0
$421$	678.987	0.0786029	0.0393014	−	0.999227i	$-0.487487\pi$
$421$	678.987	0.0786029	0.0393014	+	0.999227i	$0.487487\pi$
$422$	0	0
$423$	7885.50i	0.906398i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 983.569i	− 0.111471i
$428$	0	0
$429$	110.921	0.0124833
$430$	0	0
$431$	−16238.9	−1.81485	−0.907423	−	0.420219i	$-0.861953\pi$
$431$	−16238.9	−1.81485	−0.907423	+	0.420219i	$0.861953\pi$
$432$	0	0
$433$	− 12712.1i	− 1.41087i	−0.708777	−	0.705433i	$-0.750753\pi$
$433$	− 12712.1i	− 1.41087i	0.708777	−	0.705433i	$-0.249247\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3267.33i	− 0.357660i
$438$	0	0
$439$	−15433.5	−1.67791	−0.838955	−	0.544200i	$-0.816833\pi$
$439$	−15433.5	−1.67791	−0.838955	+	0.544200i	$0.816833\pi$
$440$	0	0
$441$	8882.36	0.959115
$442$	0	0
$443$	− 14185.4i	− 1.52137i	−0.649122	−	0.760685i	$-0.724863\pi$
$443$	− 14185.4i	− 1.52137i	0.649122	−	0.760685i	$-0.275137\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1176.59i	0.124498i
$448$	0	0
$449$	3674.17	0.386180	0.193090	−	0.981181i	$-0.438149\pi$
$449$	3674.17	0.386180	0.193090	+	0.981181i	$0.438149\pi$
$450$	0	0
$451$	3322.32	0.346878
$452$	0	0
$453$	− 718.770i	− 0.0745491i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2247.26i	0.230027i	0.993364	+	0.115013i	$0.0366911\pi$
$457$	2247.26i	0.230027i	−0.993364	+	0.115013i	$0.963309\pi$
$458$	0	0
$459$	−345.563	−0.0351405
$460$	0	0
$461$	6336.44	0.640168	0.320084	−	0.947389i	$-0.396289\pi$
$461$	6336.44	0.640168	0.320084	+	0.947389i	$0.396289\pi$
$462$	0	0
$463$	− 14249.5i	− 1.43030i	−0.698969	−	0.715152i	$-0.746357\pi$
$463$	− 14249.5i	− 1.43030i	0.698969	−	0.715152i	$-0.253643\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7043.41i	− 0.697923i	−0.937137	−	0.348961i	$-0.886534\pi$
$467$	− 7043.41i	− 0.697923i	0.937137	−	0.348961i	$-0.113466\pi$
$468$	0	0
$469$	517.120	0.0509134
$470$	0	0
$471$	676.917	0.0662222
$472$	0	0
$473$	5465.53i	0.531301i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5796.44i	0.556396i
$478$	0	0
$479$	13038.5	1.24372	0.621860	−	0.783128i	$-0.286377\pi$
$479$	13038.5	1.24372	0.621860	+	0.783128i	$0.286377\pi$
$480$	0	0
$481$	3001.95	0.284568
$482$	0	0
$483$	224.763i	0.0211741i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 18977.6i	− 1.76583i	−0.469536	−	0.882913i	$-0.655579\pi$
$487$	− 18977.6i	− 1.76583i	0.469536	−	0.882913i	$-0.344421\pi$
$488$	0	0
$489$	468.767	0.0433505
$490$	0	0
$491$	3177.78	0.292080	0.146040	−	0.989279i	$-0.453347\pi$
$491$	3177.78	0.292080	0.146040	+	0.989279i	$0.453347\pi$
$492$	0	0
$493$	− 4553.11i	− 0.415947i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3206.99i	− 0.289443i
$498$	0	0
$499$	9180.49	0.823597	0.411799	−	0.911275i	$-0.364901\pi$
$499$	9180.49	0.823597	0.411799	+	0.911275i	$0.364901\pi$
$500$	0	0
$501$	−489.612	−0.0436612
$502$	0	0
$503$	10409.3i	0.922723i	0.887212	+	0.461362i	$0.152639\pi$
$503$	10409.3i	0.922723i	−0.887212	+	0.461362i	$0.847361\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	743.940i	0.0651668i
$508$	0	0
$509$	6173.36	0.537582	0.268791	−	0.963198i	$-0.413376\pi$
$509$	6173.36	0.537582	0.268791	+	0.963198i	$0.413376\pi$
$510$	0	0
$511$	2383.27	0.206321
$512$	0	0
$513$	380.980i	0.0327889i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6205.50i	− 0.527887i
$518$	0	0
$519$	−896.081	−0.0757873
$520$	0	0
$521$	660.989	0.0555825	0.0277912	−	0.999614i	$-0.491153\pi$
$521$	660.989	0.0555825	0.0277912	+	0.999614i	$0.491153\pi$
$522$	0	0
$523$	19526.4i	1.63257i	0.577652	+	0.816283i	$0.303969\pi$
$523$	19526.4i	1.63257i	−0.577652	+	0.816283i	$0.696031\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3201.72i	0.264647i
$528$	0	0
$529$	−17404.8	−1.43049
$530$	0	0
$531$	14425.4	1.17892
$532$	0	0
$533$	− 2215.30i	− 0.180029i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 402.927i	− 0.0323791i
$538$	0	0
$539$	−6989.98	−0.558589
$540$	0	0
$541$	−9259.01	−0.735815	−0.367907	−	0.929862i	$-0.619926\pi$
$541$	−9259.01	−0.735815	−0.367907	+	0.929862i	$0.619926\pi$
$542$	0	0
$543$	1607.29i	0.127027i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 11947.9i	− 0.933924i	−0.884277	−	0.466962i	$-0.845348\pi$
$547$	− 11947.9i	− 0.933924i	0.884277	−	0.466962i	$-0.154652\pi$
$548$	0	0
$549$	−7525.20	−0.585005
$550$	0	0
$551$	−5019.77	−0.388111
$552$	0	0
$553$	2135.58i	0.164221i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3317.62i	0.252373i	0.992007	+	0.126187i	$0.0402738\pi$
$557$	3317.62i	0.252373i	−0.992007	+	0.126187i	$0.959726\pi$
$558$	0	0
$559$	3644.38	0.275744
$560$	0	0
$561$	135.621	0.0102066
$562$	0	0
$563$	21580.3i	1.61546i	0.589555	+	0.807729i	$0.299303\pi$
$563$	21580.3i	1.61546i	−0.589555	+	0.807729i	$0.700697\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2520.08i	0.186655i
$568$	0	0
$569$	−11040.1	−0.813401	−0.406700	−	0.913562i	$-0.633321\pi$
$569$	−11040.1	−0.813401	−0.406700	+	0.913562i	$0.633321\pi$
$570$	0	0
$571$	−19559.0	−1.43348	−0.716742	−	0.697338i	$-0.754368\pi$
$571$	−19559.0	−1.43348	−0.716742	+	0.697338i	$0.754368\pi$
$572$	0	0
$573$	1194.38i	0.0870784i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4148.41i	0.299308i	0.988738	+	0.149654i	$0.0478160\pi$
$577$	4148.41i	0.299308i	−0.988738	+	0.149654i	$0.952184\pi$
$578$	0	0
$579$	1566.08	0.112407
$580$	0	0
$581$	−991.612	−0.0708072
$582$	0	0
$583$	− 4561.51i	− 0.324045i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23530.2i	1.65451i	0.561829	+	0.827253i	$0.310098\pi$
$587$	23530.2i	1.65451i	−0.561829	+	0.827253i	$0.689902\pi$
$588$	0	0
$589$	3529.87	0.246937
$590$	0	0
$591$	1782.87	0.124091
$592$	0	0
$593$	− 3378.46i	− 0.233957i	−0.993134	−	0.116979i	$-0.962679\pi$
$593$	− 3378.46i	− 0.233957i	0.993134	−	0.116979i	$-0.0373209\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1304.49i	0.0894294i
$598$	0	0
$599$	−15748.7	−1.07424	−0.537122	−	0.843504i	$-0.680489\pi$
$599$	−15748.7	−1.07424	−0.537122	+	0.843504i	$0.680489\pi$
$600$	0	0
$601$	17980.4	1.22036	0.610180	−	0.792263i	$-0.291097\pi$
$601$	17980.4	1.22036	0.610180	+	0.792263i	$0.291097\pi$
$602$	0	0
$603$	− 3956.44i	− 0.267195i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4314.26i	0.288485i	0.989542	+	0.144243i	$0.0460746\pi$
$607$	4314.26i	0.288485i	−0.989542	+	0.144243i	$0.953925\pi$
$608$	0	0
$609$	345.316	0.0229769
$610$	0	0
$611$	−4137.79	−0.273972
$612$	0	0
$613$	− 3891.03i	− 0.256374i	−0.991750	−	0.128187i	$-0.959084\pi$
$613$	− 3891.03i	− 0.256374i	0.991750	−	0.128187i	$-0.0409158\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6001.52i	− 0.391592i	−0.980645	−	0.195796i	$-0.937271\pi$
$617$	− 6001.52i	− 0.391592i	0.980645	−	0.195796i	$-0.0627290\pi$
$618$	0	0
$619$	−16195.8	−1.05164	−0.525820	−	0.850596i	$-0.676241\pi$
$619$	−16195.8	−1.05164	−0.525820	+	0.850596i	$0.676241\pi$
$620$	0	0
$621$	3448.16	0.222818
$622$	0	0
$623$	752.091i	0.0483658i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 149.521i	− 0.00952357i
$628$	0	0
$629$	3670.40	0.232668
$630$	0	0
$631$	−29691.7	−1.87323	−0.936615	−	0.350361i	$-0.886059\pi$
$631$	−29691.7	−1.87323	−0.936615	+	0.350361i	$0.886059\pi$
$632$	0	0
$633$	− 1141.75i	− 0.0716910i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4660.88i	0.289907i
$638$	0	0
$639$	−24536.4	−1.51901
$640$	0	0
$641$	19562.7	1.20543	0.602716	−	0.797956i	$-0.294085\pi$
$641$	19562.7	1.20543	0.602716	+	0.797956i	$0.294085\pi$
$642$	0	0
$643$	20169.7i	1.23704i	0.785770	+	0.618518i	$0.212267\pi$
$643$	20169.7i	1.23704i	−0.785770	+	0.618518i	$0.787733\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19276.8i	1.17133i	0.810555	+	0.585663i	$0.199166\pi$
$647$	19276.8i	1.17133i	−0.810555	+	0.585663i	$0.800834\pi$
$648$	0	0
$649$	−11352.1	−0.686606
$650$	0	0
$651$	−242.824	−0.0146191
$652$	0	0
$653$	− 17256.6i	− 1.03416i	−0.855938	−	0.517078i	$-0.827020\pi$
$653$	− 17256.6i	− 1.03416i	0.855938	−	0.517078i	$-0.172980\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 18234.2i	− 1.08278i
$658$	0	0
$659$	15812.2	0.934681	0.467340	−	0.884078i	$-0.345212\pi$
$659$	15812.2	0.934681	0.467340	+	0.884078i	$0.345212\pi$
$660$	0	0
$661$	−21264.8	−1.25130	−0.625648	−	0.780106i	$-0.715165\pi$
$661$	−21264.8	−1.25130	−0.625648	+	0.780106i	$0.715165\pi$
$662$	0	0
$663$	− 90.4310i	− 0.00529721i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45432.8i	2.63743i
$668$	0	0
$669$	788.140	0.0455475
$670$	0	0
$671$	5921.96	0.340708
$672$	0	0
$673$	10290.0i	0.589374i	0.955594	+	0.294687i	$0.0952154\pi$
$673$	10290.0i	0.589374i	−0.955594	+	0.294687i	$0.904785\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3204.13i	0.181897i	0.995856	+	0.0909487i	$0.0289899\pi$
$677$	3204.13i	0.181897i	−0.995856	+	0.0909487i	$0.971010\pi$
$678$	0	0
$679$	5864.93	0.331481
$680$	0	0
$681$	2411.02	0.135669
$682$	0	0
$683$	22703.2i	1.27191i	0.771727	+	0.635954i	$0.219393\pi$
$683$	22703.2i	1.27191i	−0.771727	+	0.635954i	$0.780607\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1372.57i	− 0.0762254i
$688$	0	0
$689$	−3041.59	−0.168179
$690$	0	0
$691$	14141.0	0.778509	0.389255	−	0.921130i	$-0.372733\pi$
$691$	14141.0	0.778509	0.389255	+	0.921130i	$0.372733\pi$
$692$	0	0
$693$	− 1993.52i	− 0.109275i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2708.59i	− 0.147195i
$698$	0	0
$699$	−416.738	−0.0225500
$700$	0	0
$701$	−8699.08	−0.468701	−0.234351	−	0.972152i	$-0.575296\pi$
$701$	−8699.08	−0.468701	−0.234351	+	0.972152i	$0.575296\pi$
$702$	0	0
$703$	− 4046.59i	− 0.217098i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	583.913i	0.0310613i
$708$	0	0
$709$	19611.9	1.03884	0.519422	−	0.854518i	$-0.326147\pi$
$709$	19611.9	1.03884	0.519422	+	0.854518i	$0.326147\pi$
$710$	0	0
$711$	16339.2	0.861838
$712$	0	0
$713$	− 31948.0i	− 1.67807i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2465.05i	0.128395i
$718$	0	0
$719$	−3735.43	−0.193752	−0.0968762	−	0.995296i	$-0.530885\pi$
$719$	−3735.43	−0.193752	−0.0968762	+	0.995296i	$0.530885\pi$
$720$	0	0
$721$	−4282.50	−0.221205
$722$	0	0
$723$	1036.62i	0.0533225i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4091.05i	0.208705i	0.994540	+	0.104353i	$0.0332770\pi$
$727$	4091.05i	0.208705i	−0.994540	+	0.104353i	$0.966723\pi$
$728$	0	0
$729$	19079.4	0.969333
$730$	0	0
$731$	4455.89	0.225454
$732$	0	0
$733$	21206.4i	1.06859i	0.845299	+	0.534294i	$0.179423\pi$
$733$	21206.4i	1.06859i	−0.845299	+	0.534294i	$0.820577\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3113.52i	0.155615i
$738$	0	0
$739$	−8939.54	−0.444988	−0.222494	−	0.974934i	$-0.571420\pi$
$739$	−8939.54	−0.444988	−0.222494	+	0.974934i	$0.571420\pi$
$740$	0	0
$741$	−99.6995	−0.00494271
$742$	0	0
$743$	26137.1i	1.29055i	0.763952	+	0.645273i	$0.223257\pi$
$743$	26137.1i	1.29055i	−0.763952	+	0.645273i	$0.776743\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7586.74i	0.371599i
$748$	0	0
$749$	−6097.44	−0.297458
$750$	0	0
$751$	−11466.6	−0.557155	−0.278578	−	0.960414i	$-0.589863\pi$
$751$	−11466.6	−0.557155	−0.278578	+	0.960414i	$0.589863\pi$
$752$	0	0
$753$	− 1738.62i	− 0.0841421i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8936.66i	0.429073i	0.976716	+	0.214537i	$0.0688241\pi$
$757$	8936.66i	0.429073i	−0.976716	+	0.214537i	$0.931176\pi$
$758$	0	0
$759$	−1353.28	−0.0647178
$760$	0	0
$761$	−9971.09	−0.474969	−0.237485	−	0.971391i	$-0.576323\pi$
$761$	−9971.09	−0.474969	−0.237485	+	0.971391i	$0.576323\pi$
$762$	0	0
$763$	64.0268i	0.00303791i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7569.49i	0.356347i
$768$	0	0
$769$	−25560.0	−1.19859	−0.599296	−	0.800527i	$-0.704553\pi$
$769$	−25560.0	−1.19859	−0.599296	+	0.800527i	$0.704553\pi$
$770$	0	0
$771$	1923.90	0.0898670
$772$	0	0
$773$	− 15918.8i	− 0.740698i	−0.928893	−	0.370349i	$-0.879238\pi$
$773$	− 15918.8i	− 0.740698i	0.928893	−	0.370349i	$-0.120762\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	278.370i	0.0128526i
$778$	0	0
$779$	−2986.20	−0.137345
$780$	0	0
$781$	19309.0	0.884672
$782$	0	0
$783$	− 5297.60i	− 0.241789i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	11278.0i	0.510822i	0.966833	+	0.255411i	$0.0822108\pi$
$787$	11278.0i	0.510822i	−0.966833	+	0.255411i	$0.917789\pi$
$788$	0	0
$789$	1211.46	0.0546631
$790$	0	0
$791$	5570.31	0.250389
$792$	0	0
$793$	− 3948.73i	− 0.176827i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23845.2i	1.05978i	0.848068	+	0.529888i	$0.177766\pi$
$797$	23845.2i	1.05978i	−0.848068	+	0.529888i	$0.822234\pi$
$798$	0	0
$799$	−5059.16	−0.224005
$800$	0	0
$801$	5754.18	0.253825
$802$	0	0
$803$	14349.4i	0.630611i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	576.204i	0.0251343i
$808$	0	0
$809$	−12729.7	−0.553216	−0.276608	−	0.960983i	$-0.589210\pi$
$809$	−12729.7	−0.553216	−0.276608	+	0.960983i	$0.589210\pi$
$810$	0	0
$811$	−5859.98	−0.253726	−0.126863	−	0.991920i	$-0.540491\pi$
$811$	−5859.98	−0.253726	−0.126863	+	0.991920i	$0.540491\pi$
$812$	0	0
$813$	− 1850.06i	− 0.0798088i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4912.58i	− 0.210367i
$818$	0	0
$819$	−1329.27	−0.0567135
$820$	0	0
$821$	12778.2	0.543195	0.271598	−	0.962411i	$-0.412448\pi$
$821$	12778.2	0.543195	0.271598	+	0.962411i	$0.412448\pi$
$822$	0	0
$823$	− 15511.0i	− 0.656963i	−0.944510	−	0.328482i	$-0.893463\pi$
$823$	− 15511.0i	− 0.656963i	0.944510	−	0.328482i	$-0.106537\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12501.0i	0.525639i	0.964845	+	0.262819i	$0.0846523\pi$
$827$	12501.0i	0.525639i	−0.964845	+	0.262819i	$0.915348\pi$
$828$	0	0
$829$	2013.80	0.0843694	0.0421847	−	0.999110i	$-0.486568\pi$
$829$	2013.80	0.0843694	0.0421847	+	0.999110i	$0.486568\pi$
$830$	0	0
$831$	104.187	0.00434924
$832$	0	0
$833$	5698.73i	0.237034i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3725.24i	0.153839i
$838$	0	0
$839$	−16805.2	−0.691513	−0.345757	−	0.938324i	$-0.612378\pi$
$839$	−16805.2	−0.691513	−0.345757	+	0.938324i	$0.612378\pi$
$840$	0	0
$841$	45411.7	1.86198
$842$	0	0
$843$	779.013i	0.0318275i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3104.17i	− 0.125928i
$848$	0	0
$849$	1466.42	0.0592784
$850$	0	0
$851$	−36624.7	−1.47530
$852$	0	0
$853$	− 16648.7i	− 0.668279i	−0.942524	−	0.334139i	$-0.891554\pi$
$853$	− 16648.7i	− 0.668279i	0.942524	−	0.334139i	$-0.108446\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 12094.5i	− 0.482078i	−0.970515	−	0.241039i	$-0.922512\pi$
$857$	− 12094.5i	− 0.482078i	0.970515	−	0.241039i	$-0.0774881\pi$
$858$	0	0
$859$	−10296.1	−0.408962	−0.204481	−	0.978871i	$-0.565551\pi$
$859$	−10296.1	−0.408962	−0.204481	+	0.978871i	$0.565551\pi$
$860$	0	0
$861$	205.424	0.00813106
$862$	0	0
$863$	27367.0i	1.07947i	0.841834	+	0.539736i	$0.181476\pi$
$863$	27367.0i	1.07947i	−0.841834	+	0.539736i	$0.818524\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1718.45i	0.0673145i
$868$	0	0
$869$	−12858.1	−0.501936
$870$	0	0
$871$	2076.08	0.0807638
$872$	0	0
$873$	− 44872.1i	− 1.73962i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29880.2i	1.15049i	0.817979	+	0.575247i	$0.195094\pi$
$877$	29880.2i	1.15049i	−0.817979	+	0.575247i	$0.804906\pi$
$878$	0	0
$879$	−359.597	−0.0137985
$880$	0	0
$881$	−28639.8	−1.09523	−0.547617	−	0.836729i	$-0.684465\pi$
$881$	−28639.8	−1.09523	−0.547617	+	0.836729i	$0.684465\pi$
$882$	0	0
$883$	− 16394.7i	− 0.624832i	−0.949945	−	0.312416i	$-0.898862\pi$
$883$	− 16394.7i	− 0.624832i	0.949945	−	0.312416i	$-0.101138\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16041.2i	0.607226i	0.952795	+	0.303613i	$0.0981930\pi$
$887$	16041.2i	0.607226i	−0.952795	+	0.303613i	$0.901807\pi$
$888$	0	0
$889$	8172.78	0.308331
$890$	0	0
$891$	−15173.1	−0.570504
$892$	0	0
$893$	5577.69i	0.209015i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	902.357i	0.0335884i
$898$	0	0
$899$	−49083.4	−1.82094
$900$	0	0
$901$	−3718.87	−0.137507
$902$	0	0
$903$	337.942i	0.0124541i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10953.4i	0.400994i	0.979694	+	0.200497i	$0.0642556\pi$
$907$	10953.4i	0.400994i	−0.979694	+	0.200497i	$0.935744\pi$
$908$	0	0
$909$	4467.47	0.163011
$910$	0	0
$911$	12739.1	0.463299	0.231649	−	0.972799i	$-0.425588\pi$
$911$	12739.1	0.463299	0.231649	+	0.972799i	$0.425588\pi$
$912$	0	0
$913$	− 5970.39i	− 0.216419i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2218.00i	0.0798745i
$918$	0	0
$919$	10544.1	0.378475	0.189238	−	0.981931i	$-0.439398\pi$
$919$	10544.1	0.378475	0.189238	+	0.981931i	$0.439398\pi$
$920$	0	0
$921$	−3618.04	−0.129445
$922$	0	0
$923$	− 12875.1i	− 0.459143i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	32765.1i	1.16089i
$928$	0	0
$929$	28682.8	1.01297	0.506487	−	0.862247i	$-0.330944\pi$
$929$	28682.8	1.01297	0.506487	+	0.862247i	$0.330944\pi$
$930$	0	0
$931$	6282.80	0.221171
$932$	0	0
$933$	1284.01i	0.0450554i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39763.3i	1.38635i	0.720769	+	0.693175i	$0.243789\pi$
$937$	39763.3i	1.38635i	−0.720769	+	0.693175i	$0.756211\pi$
$938$	0	0
$939$	−8.70124	−0.000302401 0
$940$	0	0
$941$	27875.7	0.965696	0.482848	−	0.875704i	$-0.339602\pi$
$941$	27875.7	0.965696	0.482848	+	0.875704i	$0.339602\pi$
$942$	0	0
$943$	27027.4i	0.933333i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 54572.7i	− 1.87262i	−0.351169	−	0.936312i	$-0.614216\pi$
$947$	− 54572.7i	− 1.87262i	0.351169	−	0.936312i	$-0.385784\pi$
$948$	0	0
$949$	9568.12	0.327286
$950$	0	0
$951$	211.443	0.00720979
$952$	0	0
$953$	− 8065.62i	− 0.274157i	−0.990560	−	0.137078i	$-0.956229\pi$
$953$	− 8065.62i	− 0.274157i	0.990560	−	0.137078i	$-0.0437712\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2079.11i	0.0702279i
$958$	0	0
$959$	3855.83	0.129835
$960$	0	0
$961$	4724.14	0.158576
$962$	0	0
$963$	46651.0i	1.56107i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 14289.8i	− 0.475210i	−0.971362	−	0.237605i	$-0.923638\pi$
$967$	− 14289.8i	− 0.475210i	0.971362	−	0.237605i	$-0.0763624\pi$
$968$	0	0
$969$	−121.900	−0.00404127
$970$	0	0
$971$	−14397.3	−0.475832	−0.237916	−	0.971286i	$-0.576464\pi$
$971$	−14397.3	−0.475832	−0.237916	+	0.971286i	$0.576464\pi$
$972$	0	0
$973$	− 5348.39i	− 0.176219i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41896.2i	1.37193i	0.727634	+	0.685966i	$0.240620\pi$
$977$	41896.2i	1.37193i	−0.727634	+	0.685966i	$0.759380\pi$
$978$	0	0
$979$	−4528.26	−0.147828
$980$	0	0
$981$	489.863	0.0159431
$982$	0	0
$983$	19373.7i	0.628612i	0.949322	+	0.314306i	$0.101772\pi$
$983$	19373.7i	0.628612i	−0.949322	+	0.314306i	$0.898228\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 383.696i	− 0.0123740i
$988$	0	0
$989$	−44462.6	−1.42955
$990$	0	0
$991$	−13250.6	−0.424741	−0.212371	−	0.977189i	$-0.568118\pi$
$991$	−13250.6	−0.424741	−0.212371	+	0.977189i	$0.568118\pi$
$992$	0	0
$993$	− 205.992i	− 0.00658305i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4129.59i	0.131179i	0.997847	+	0.0655895i	$0.0208928\pi$
$997$	4129.59i	0.131179i	−0.997847	+	0.0655895i	$0.979107\pi$
$998$	0	0
$999$	4270.55	0.135250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.4.c.b.1749.3		4
5.2	odd	4		76.4.a.a.1.2	✓	2
5.3	odd	4		1900.4.a.b.1.1		2
5.4	even	2	inner	1900.4.c.b.1749.2		4
15.2	even	4		684.4.a.g.1.2		2
20.7	even	4		304.4.a.f.1.1		2
40.27	even	4		1216.4.a.h.1.2		2
40.37	odd	4		1216.4.a.o.1.1		2
95.37	even	4		1444.4.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.a.a.1.2	✓	2	5.2	odd	4
304.4.a.f.1.1		2	20.7	even	4
684.4.a.g.1.2		2	15.2	even	4
1216.4.a.h.1.2		2	40.27	even	4
1216.4.a.o.1.1		2	40.37	odd	4
1444.4.a.d.1.1		2	95.37	even	4
1900.4.a.b.1.1		2	5.3	odd	4
1900.4.c.b.1749.2		4	5.4	even	2	inner
1900.4.c.b.1749.3		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.372281i q^{3} +3.51087i q^{7} +26.8614 q^{9} +O(q^{10})\)	\(q+0.372281i q^{3} +3.51087i q^{7} +26.8614 q^{9} -21.1386 q^{11} +14.0951i q^{13} +17.2337i q^{17} +19.0000 q^{19} -1.30703 q^{21} -171.965i q^{23} +20.0516i q^{27} -264.198 q^{29} +185.783 q^{31} -7.86950i q^{33} -212.978i q^{37} -5.24734 q^{39} -157.168 q^{41} -258.557i q^{43} +293.562i q^{47} +330.674 q^{49} -6.41578 q^{51} +215.791i q^{53} +7.07335i q^{57} +537.030 q^{59} -280.149 q^{61} +94.3070i q^{63} -147.291i q^{67} +64.0192 q^{69} -913.446 q^{71} -678.826i q^{73} -74.2150i q^{77} +608.277 q^{79} +717.793 q^{81} +282.440i q^{83} -98.3561i q^{87} +214.217 q^{89} -49.4861 q^{91} +69.1634i q^{93} -1670.50i q^{97} -567.812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 50 q^{9}+O(q^{10}) \) 4 * q + 50 * q^9	\( 4 q + 50 q^{9} - 142 q^{11} + 76 q^{19} + 282 q^{21} - 310 q^{29} - 176 q^{31} - 538 q^{39} - 284 q^{41} - 56 q^{49} - 198 q^{51} + 1746 q^{59} + 890 q^{61} + 1922 q^{69} - 3424 q^{71} + 2548 q^{79} - 116 q^{81} + 1776 q^{89} + 2502 q^{91} - 950 q^{99}+O(q^{100}) \) 4 * q + 50 * q^9 - 142 * q^11 + 76 * q^19 + 282 * q^21 - 310 * q^29 - 176 * q^31 - 538 * q^39 - 284 * q^41 - 56 * q^49 - 198 * q^51 + 1746 * q^59 + 890 * q^61 + 1922 * q^69 - 3424 * q^71 + 2548 * q^79 - 116 * q^81 + 1776 * q^89 + 2502 * q^91 - 950 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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