LMFDB - Embedded newform 1800.4.a.bw.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(1,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$106.203438010$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{6}, \sqrt{46})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 26x^{2} + 100 $ x^4 - 26*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.61591$ of defining polynomial
Character	$\chi$	$=$	1800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+28.2616 q^{7} +O(q^{10})$	$q+28.2616 q^{7} +47.8575 q^{11} +10.9302 q^{13} +28.6132 q^{17} +86.4530 q^{19} +49.2265 q^{23} +165.433 q^{29} -247.359 q^{31} +375.674 q^{37} +504.180 q^{41} -207.976 q^{43} -70.1325 q^{47} +455.718 q^{49} -286.425 q^{53} -92.3636 q^{59} -697.624 q^{61} -840.360 q^{67} -303.691 q^{71} -251.697 q^{73} +1352.53 q^{77} +745.171 q^{79} +1327.89 q^{83} -415.953 q^{89} +308.906 q^{91} +656.418 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 48 q^{17} + 80 q^{19} + 64 q^{23} - 192 q^{31} + 384 q^{47} + 228 q^{49} - 16 q^{53} - 664 q^{61} + 2752 q^{77} + 1120 q^{79} + 1856 q^{83} + 704 q^{91}+O(q^{100}) $ 4 * q + 48 * q^17 + 80 * q^19 + 64 * q^23 - 192 * q^31 + 384 * q^47 + 228 * q^49 - 16 * q^53 - 664 * q^61 + 2752 * q^77 + 1120 * q^79 + 1856 * q^83 + 704 * q^91

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	28.2616	1.52598	0.762991	−	0.646409i	$-0.223730\pi$
$7$	28.2616	1.52598	0.762991	+	0.646409i	$0.223730\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	47.8575	1.31178	0.655890	−	0.754856i	$-0.272293\pi$
$11$	47.8575	1.31178	0.655890	+	0.754856i	$0.272293\pi$
$12$	0	0
$13$	10.9302	0.233192	0.116596	−	0.993179i	$-0.462802\pi$
$13$	10.9302	0.233192	0.116596	+	0.993179i	$0.462802\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	28.6132	0.408220	0.204110	−	0.978948i	$-0.434570\pi$
$17$	28.6132	0.408220	0.204110	+	0.978948i	$0.434570\pi$
$18$	0	0
$19$	86.4530	1.04388	0.521939	−	0.852983i	$-0.325209\pi$
$19$	86.4530	1.04388	0.521939	+	0.852983i	$0.325209\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	49.2265	0.446280	0.223140	−	0.974786i	$-0.428369\pi$
$23$	49.2265	0.446280	0.223140	+	0.974786i	$0.428369\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	165.433	1.05932	0.529658	−	0.848212i	$-0.322320\pi$
$29$	165.433	1.05932	0.529658	+	0.848212i	$0.322320\pi$
$30$	0	0
$31$	−247.359	−1.43313	−0.716564	−	0.697521i	$-0.754286\pi$
$31$	−247.359	−1.43313	−0.716564	+	0.697521i	$0.754286\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	375.674	1.66920	0.834600	−	0.550856i	$-0.185699\pi$
$37$	375.674	1.66920	0.834600	+	0.550856i	$0.185699\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	504.180	1.92048	0.960239	−	0.279178i	$-0.0900619\pi$
$41$	504.180	1.92048	0.960239	+	0.279178i	$0.0900619\pi$
$42$	0	0
$43$	−207.976	−0.737584	−0.368792	−	0.929512i	$-0.620228\pi$
$43$	−207.976	−0.737584	−0.368792	+	0.929512i	$0.620228\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−70.1325	−0.217657	−0.108828	−	0.994061i	$-0.534710\pi$
$47$	−70.1325	−0.217657	−0.108828	+	0.994061i	$0.534710\pi$
$48$	0	0
$49$	455.718	1.32862
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−286.425	−0.742331	−0.371165	−	0.928567i	$-0.621042\pi$
$53$	−286.425	−0.742331	−0.371165	+	0.928567i	$0.621042\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−92.3636	−0.203809	−0.101904	−	0.994794i	$-0.532494\pi$
$59$	−92.3636	−0.203809	−0.101904	+	0.994794i	$0.532494\pi$
$60$	0	0
$61$	−697.624	−1.46429	−0.732144	−	0.681150i	$-0.761480\pi$
$61$	−697.624	−1.46429	−0.732144	+	0.681150i	$0.761480\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−840.360	−1.53233	−0.766166	−	0.642642i	$-0.777838\pi$
$67$	−840.360	−1.53233	−0.766166	+	0.642642i	$0.777838\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−303.691	−0.507627	−0.253814	−	0.967253i	$-0.581685\pi$
$71$	−303.691	−0.507627	−0.253814	+	0.967253i	$0.581685\pi$
$72$	0	0
$73$	−251.697	−0.403547	−0.201774	−	0.979432i	$-0.564671\pi$
$73$	−251.697	−0.403547	−0.201774	+	0.979432i	$0.564671\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1352.53	2.00175
$78$	0	0
$79$	745.171	1.06124	0.530622	−	0.847609i	$-0.321958\pi$
$79$	745.171	1.06124	0.530622	+	0.847609i	$0.321958\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1327.89	1.75608	0.878041	−	0.478586i	$-0.158851\pi$
$83$	1327.89	1.75608	0.878041	+	0.478586i	$0.158851\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−415.953	−0.495403	−0.247702	−	0.968836i	$-0.579675\pi$
$89$	−415.953	−0.495403	−0.247702	+	0.968836i	$0.579675\pi$
$90$	0	0
$91$	308.906	0.355848
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	656.418	0.687105	0.343552	−	0.939134i	$-0.388370\pi$
$97$	656.418	0.687105	0.343552	+	0.939134i	$0.388370\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1228.14	−1.20995	−0.604974	−	0.796246i	$-0.706816\pi$
$101$	−1228.14	−1.20995	−0.604974	+	0.796246i	$0.706816\pi$
$102$	0	0
$103$	−598.808	−0.572838	−0.286419	−	0.958105i	$-0.592465\pi$
$103$	−598.808	−0.572838	−0.286419	+	0.958105i	$0.592465\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−462.607	−0.417962	−0.208981	−	0.977920i	$-0.567015\pi$
$107$	−462.607	−0.417962	−0.208981	+	0.977920i	$0.567015\pi$
$108$	0	0
$109$	−1427.44	−1.25434	−0.627172	−	0.778881i	$-0.715788\pi$
$109$	−1427.44	−1.25434	−0.627172	+	0.778881i	$0.715788\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−654.955	−0.545248	−0.272624	−	0.962121i	$-0.587891\pi$
$113$	−654.955	−0.545248	−0.272624	+	0.962121i	$0.587891\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	808.656	0.622936
$120$	0	0
$121$	959.342	0.720768
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	75.1229	0.0524888	0.0262444	−	0.999656i	$-0.491645\pi$
$127$	75.1229	0.0524888	0.0262444	+	0.999656i	$0.491645\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1888.12	−1.25928	−0.629641	−	0.776886i	$-0.716798\pi$
$131$	−1888.12	−1.25928	−0.629641	+	0.776886i	$0.716798\pi$
$132$	0	0
$133$	2443.30	1.59294
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3027.28	1.88787	0.943933	−	0.330137i	$-0.107095\pi$
$137$	3027.28	1.88787	0.943933	+	0.330137i	$0.107095\pi$
$138$	0	0
$139$	−911.513	−0.556212	−0.278106	−	0.960550i	$-0.589707\pi$
$139$	−911.513	−0.556212	−0.278106	+	0.960550i	$0.589707\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	523.094	0.305897
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3103.85	−1.70656	−0.853281	−	0.521452i	$-0.825391\pi$
$149$	−3103.85	−1.70656	−0.853281	+	0.521452i	$0.825391\pi$
$150$	0	0
$151$	1082.98	0.583655	0.291827	−	0.956471i	$-0.405737\pi$
$151$	1082.98	0.583655	0.291827	+	0.956471i	$0.405737\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−215.766	−0.109682	−0.0548408	−	0.998495i	$-0.517465\pi$
$157$	−215.766	−0.109682	−0.0548408	+	0.998495i	$0.517465\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1391.22	0.681015
$162$	0	0
$163$	1869.61	0.898401	0.449201	−	0.893431i	$-0.351709\pi$
$163$	1869.61	0.898401	0.449201	+	0.893431i	$0.351709\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3799.30	1.76047	0.880236	−	0.474536i	$-0.157384\pi$
$167$	3799.30	1.76047	0.880236	+	0.474536i	$0.157384\pi$
$168$	0	0
$169$	−2077.53	−0.945621
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1005.25	0.441781	0.220890	−	0.975299i	$-0.429104\pi$
$173$	1005.25	0.441781	0.220890	+	0.975299i	$0.429104\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2393.09	0.999261	0.499630	−	0.866239i	$-0.333469\pi$
$179$	2393.09	0.999261	0.499630	+	0.866239i	$0.333469\pi$
$180$	0	0
$181$	2378.31	0.976675	0.488338	−	0.872655i	$-0.337603\pi$
$181$	2378.31	0.976675	0.488338	+	0.872655i	$0.337603\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1369.36	0.535494
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3152.32	1.19421	0.597103	−	0.802164i	$-0.296318\pi$
$191$	3152.32	1.19421	0.597103	+	0.802164i	$0.296318\pi$
$192$	0	0
$193$	−4313.28	−1.60869	−0.804343	−	0.594165i	$-0.797482\pi$
$193$	−4313.28	−1.60869	−0.804343	+	0.594165i	$0.797482\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2892.08	1.04595	0.522976	−	0.852348i	$-0.324822\pi$
$197$	2892.08	1.04595	0.522976	+	0.852348i	$0.324822\pi$
$198$	0	0
$199$	−952.949	−0.339461	−0.169731	−	0.985491i	$-0.554290\pi$
$199$	−952.949	−0.339461	−0.169731	+	0.985491i	$0.554290\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4675.40	1.61650
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4137.43	1.36934
$210$	0	0
$211$	2536.79	0.827678	0.413839	−	0.910350i	$-0.364188\pi$
$211$	2536.79	0.827678	0.413839	+	0.910350i	$0.364188\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6990.76	−2.18693
$218$	0	0
$219$	0	0
$220$	0	0
$221$	312.750	0.0951937
$222$	0	0
$223$	1956.33	0.587469	0.293735	−	0.955887i	$-0.405102\pi$
$223$	1956.33	0.587469	0.293735	+	0.955887i	$0.405102\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3664.11	1.07135	0.535673	−	0.844425i	$-0.320058\pi$
$227$	3664.11	1.07135	0.535673	+	0.844425i	$0.320058\pi$
$228$	0	0
$229$	−2296.19	−0.662604	−0.331302	−	0.943525i	$-0.607488\pi$
$229$	−2296.19	−0.662604	−0.331302	+	0.943525i	$0.607488\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2886.31	0.811540	0.405770	−	0.913975i	$-0.367003\pi$
$233$	2886.31	0.811540	0.405770	+	0.913975i	$0.367003\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4994.15	−1.35165	−0.675826	−	0.737061i	$-0.736213\pi$
$239$	−4994.15	−1.35165	−0.675826	+	0.737061i	$0.736213\pi$
$240$	0	0
$241$	−5735.37	−1.53298	−0.766489	−	0.642258i	$-0.777998\pi$
$241$	−5735.37	−1.53298	−0.766489	+	0.642258i	$0.777998\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	944.952	0.243424
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3503.44	0.881017	0.440509	−	0.897748i	$-0.354798\pi$
$251$	3503.44	0.881017	0.440509	+	0.897748i	$0.354798\pi$
$252$	0	0
$253$	2355.86	0.585421
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3512.51	0.852547	0.426274	−	0.904594i	$-0.359826\pi$
$257$	3512.51	0.852547	0.426274	+	0.904594i	$0.359826\pi$
$258$	0	0
$259$	10617.1	2.54717
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6722.79	−1.57622	−0.788108	−	0.615536i	$-0.788939\pi$
$263$	−6722.79	−1.57622	−0.788108	+	0.615536i	$0.788939\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3727.78	0.844933	0.422466	−	0.906379i	$-0.361164\pi$
$269$	3727.78	0.844933	0.422466	+	0.906379i	$0.361164\pi$
$270$	0	0
$271$	−14.0769	−0.00315539	−0.00157770	−	0.999999i	$-0.500502\pi$
$271$	−14.0769	−0.00315539	−0.00157770	+	0.999999i	$0.500502\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2055.06	0.445765	0.222882	−	0.974845i	$-0.428453\pi$
$277$	2055.06	0.445765	0.222882	+	0.974845i	$0.428453\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4287.07	0.910125	0.455062	−	0.890460i	$-0.349617\pi$
$281$	4287.07	0.910125	0.455062	+	0.890460i	$0.349617\pi$
$282$	0	0
$283$	−4888.77	−1.02688	−0.513441	−	0.858125i	$-0.671629\pi$
$283$	−4888.77	−1.02688	−0.513441	+	0.858125i	$0.671629\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14248.9	2.93062
$288$	0	0
$289$	−4094.28	−0.833357
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5724.92	1.14148	0.570740	−	0.821131i	$-0.306657\pi$
$293$	5724.92	1.14148	0.570740	+	0.821131i	$0.306657\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	538.057	0.104069
$300$	0	0
$301$	−5877.74	−1.12554
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6253.78	1.16261	0.581307	−	0.813685i	$-0.302542\pi$
$307$	6253.78	1.16261	0.581307	+	0.813685i	$0.302542\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4527.71	−0.825541	−0.412770	−	0.910835i	$-0.635439\pi$
$311$	−4527.71	−0.825541	−0.412770	+	0.910835i	$0.635439\pi$
$312$	0	0
$313$	−7737.03	−1.39720	−0.698599	−	0.715513i	$-0.746193\pi$
$313$	−7737.03	−1.39720	−0.698599	+	0.715513i	$0.746193\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1151.57	0.204033	0.102016	−	0.994783i	$-0.467471\pi$
$317$	1151.57	0.204033	0.102016	+	0.994783i	$0.467471\pi$
$318$	0	0
$319$	7917.21	1.38959
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2473.70	0.426131
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1982.06	−0.332141
$330$	0	0
$331$	−6596.81	−1.09545	−0.547724	−	0.836659i	$-0.684506\pi$
$331$	−6596.81	−1.09545	−0.547724	+	0.836659i	$0.684506\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4365.09	0.705583	0.352792	−	0.935702i	$-0.385232\pi$
$337$	4365.09	0.705583	0.352792	+	0.935702i	$0.385232\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11838.0	−1.87995
$342$	0	0
$343$	3185.59	0.501474
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1002.43	−0.155081	−0.0775405	−	0.996989i	$-0.524707\pi$
$347$	−1002.43	−0.155081	−0.0775405	+	0.996989i	$0.524707\pi$
$348$	0	0
$349$	4042.17	0.619978	0.309989	−	0.950740i	$-0.399675\pi$
$349$	4042.17	0.619978	0.309989	+	0.950740i	$0.399675\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10488.1	1.58137	0.790687	−	0.612220i	$-0.209723\pi$
$353$	10488.1	1.58137	0.790687	+	0.612220i	$0.209723\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6852.95	−1.00748	−0.503740	−	0.863856i	$-0.668043\pi$
$359$	−6852.95	−1.00748	−0.503740	+	0.863856i	$0.668043\pi$
$360$	0	0
$361$	615.120	0.0896807
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2137.98	0.304091	0.152046	−	0.988373i	$-0.451414\pi$
$367$	2137.98	0.304091	0.152046	+	0.988373i	$0.451414\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8094.83	−1.13278
$372$	0	0
$373$	13254.9	1.83998	0.919988	−	0.391946i	$-0.128198\pi$
$373$	13254.9	1.83998	0.919988	+	0.391946i	$0.128198\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1808.22	0.247024
$378$	0	0
$379$	2309.84	0.313056	0.156528	−	0.987673i	$-0.449970\pi$
$379$	2309.84	0.313056	0.156528	+	0.987673i	$0.449970\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3123.01	−0.416654	−0.208327	−	0.978059i	$-0.566802\pi$
$383$	−3123.01	−0.416654	−0.208327	+	0.978059i	$0.566802\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4031.11	0.525413	0.262706	−	0.964876i	$-0.415385\pi$
$389$	4031.11	0.525413	0.262706	+	0.964876i	$0.415385\pi$
$390$	0	0
$391$	1408.53	0.182180
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7703.16	0.973830	0.486915	−	0.873449i	$-0.338122\pi$
$397$	7703.16	0.973830	0.486915	+	0.873449i	$0.338122\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6871.01	−0.855666	−0.427833	−	0.903858i	$-0.640723\pi$
$401$	−6871.01	−0.855666	−0.427833	+	0.903858i	$0.640723\pi$
$402$	0	0
$403$	−2703.69	−0.334195
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17978.8	2.18962
$408$	0	0
$409$	−12212.6	−1.47647	−0.738236	−	0.674543i	$-0.764341\pi$
$409$	−12212.6	−1.47647	−0.738236	+	0.674543i	$0.764341\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2610.34	−0.311009
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3634.55	−0.423769	−0.211884	−	0.977295i	$-0.567960\pi$
$419$	−3634.55	−0.423769	−0.211884	+	0.977295i	$0.567960\pi$
$420$	0	0
$421$	−5104.50	−0.590921	−0.295461	−	0.955355i	$-0.595473\pi$
$421$	−5104.50	−0.590921	−0.295461	+	0.955355i	$0.595473\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−19716.0	−2.23448
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14460.5	1.61610	0.808049	−	0.589115i	$-0.200523\pi$
$431$	14460.5	1.61610	0.808049	+	0.589115i	$0.200523\pi$
$432$	0	0
$433$	−8668.46	−0.962078	−0.481039	−	0.876699i	$-0.659740\pi$
$433$	−8668.46	−0.962078	−0.481039	+	0.876699i	$0.659740\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4255.78	0.465861
$438$	0	0
$439$	11041.5	1.20041	0.600206	−	0.799845i	$-0.295085\pi$
$439$	11041.5	1.20041	0.600206	+	0.799845i	$0.295085\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5711.44	0.612548	0.306274	−	0.951943i	$-0.400918\pi$
$443$	5711.44	0.612548	0.306274	+	0.951943i	$0.400918\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9454.89	−0.993773	−0.496886	−	0.867816i	$-0.665523\pi$
$449$	−9454.89	−0.993773	−0.496886	+	0.867816i	$0.665523\pi$
$450$	0	0
$451$	24128.8	2.51925
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1649.44	0.168835	0.0844174	−	0.996430i	$-0.473097\pi$
$457$	1649.44	0.168835	0.0844174	+	0.996430i	$0.473097\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11194.3	1.13096	0.565480	−	0.824762i	$-0.308691\pi$
$461$	11194.3	1.13096	0.565480	+	0.824762i	$0.308691\pi$
$462$	0	0
$463$	−9702.91	−0.973936	−0.486968	−	0.873420i	$-0.661897\pi$
$463$	−9702.91	−0.973936	−0.486968	+	0.873420i	$0.661897\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10614.4	−1.05176	−0.525882	−	0.850557i	$-0.676265\pi$
$467$	−10614.4	−1.05176	−0.525882	+	0.850557i	$0.676265\pi$
$468$	0	0
$469$	−23749.9	−2.33831
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9953.23	−0.967548
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18297.3	1.74535	0.872676	−	0.488299i	$-0.162383\pi$
$479$	18297.3	1.74535	0.872676	+	0.488299i	$0.162383\pi$
$480$	0	0
$481$	4106.21	0.389245
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18775.1	−1.74699	−0.873493	−	0.486837i	$-0.838151\pi$
$487$	−18775.1	−1.74699	−0.873493	+	0.486837i	$0.838151\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5582.90	−0.513142	−0.256571	−	0.966525i	$-0.582593\pi$
$491$	−5582.90	−0.513142	−0.256571	+	0.966525i	$0.582593\pi$
$492$	0	0
$493$	4733.58	0.432433
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8582.80	−0.774630
$498$	0	0
$499$	2517.00	0.225804	0.112902	−	0.993606i	$-0.463985\pi$
$499$	2517.00	0.225804	0.112902	+	0.993606i	$0.463985\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9246.22	−0.819619	−0.409810	−	0.912171i	$-0.634405\pi$
$503$	−9246.22	−0.819619	−0.409810	+	0.912171i	$0.634405\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10051.4	−0.875284	−0.437642	−	0.899149i	$-0.644186\pi$
$509$	−10051.4	−0.875284	−0.437642	+	0.899149i	$0.644186\pi$
$510$	0	0
$511$	−7113.37	−0.615806
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3356.37	−0.285518
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4805.38	−0.404083	−0.202042	−	0.979377i	$-0.564758\pi$
$521$	−4805.38	−0.404083	−0.202042	+	0.979377i	$0.564758\pi$
$522$	0	0
$523$	18434.3	1.54125	0.770626	−	0.637287i	$-0.219943\pi$
$523$	18434.3	1.54125	0.770626	+	0.637287i	$0.219943\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7077.74	−0.585031
$528$	0	0
$529$	−9743.75	−0.800834
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5510.80	0.447841
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	21809.5	1.74286
$540$	0	0
$541$	6437.04	0.511553	0.255776	−	0.966736i	$-0.417669\pi$
$541$	6437.04	0.511553	0.255776	+	0.966736i	$0.417669\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10635.8	0.831359	0.415679	−	0.909511i	$-0.363544\pi$
$547$	10635.8	0.831359	0.415679	+	0.909511i	$0.363544\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14302.2	1.10580
$552$	0	0
$553$	21059.7	1.61944
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3571.98	0.271723	0.135861	−	0.990728i	$-0.456620\pi$
$557$	3571.98	0.271723	0.135861	+	0.990728i	$0.456620\pi$
$558$	0	0
$559$	−2273.23	−0.171999
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26445.3	1.97964	0.989821	−	0.142321i	$-0.0454566\pi$
$563$	26445.3	1.97964	0.989821	+	0.142321i	$0.0454566\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11021.0	0.811993	0.405997	−	0.913875i	$-0.366924\pi$
$569$	11021.0	0.811993	0.405997	+	0.913875i	$0.366924\pi$
$570$	0	0
$571$	18553.4	1.35978	0.679891	−	0.733313i	$-0.262027\pi$
$571$	18553.4	1.35978	0.679891	+	0.733313i	$0.262027\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10253.4	−0.739783	−0.369891	−	0.929075i	$-0.620605\pi$
$577$	−10253.4	−0.739783	−0.369891	+	0.929075i	$0.620605\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	37528.3	2.67975
$582$	0	0
$583$	−13707.6	−0.973775
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20577.3	−1.44688	−0.723439	−	0.690388i	$-0.757440\pi$
$587$	−20577.3	−1.44688	−0.723439	+	0.690388i	$0.757440\pi$
$588$	0	0
$589$	−21384.9	−1.49601
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12428.4	−0.860662	−0.430331	−	0.902671i	$-0.641603\pi$
$593$	−12428.4	−0.860662	−0.430331	+	0.902671i	$0.641603\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22657.2	−1.54549	−0.772746	−	0.634716i	$-0.781117\pi$
$599$	−22657.2	−1.54549	−0.772746	+	0.634716i	$0.781117\pi$
$600$	0	0
$601$	−11771.5	−0.798948	−0.399474	−	0.916745i	$-0.630807\pi$
$601$	−11771.5	−0.798948	−0.399474	+	0.916745i	$0.630807\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24031.3	1.60692	0.803460	−	0.595359i	$-0.202990\pi$
$607$	24031.3	1.60692	0.803460	+	0.595359i	$0.202990\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−766.565	−0.0507560
$612$	0	0
$613$	−21456.8	−1.41375	−0.706877	−	0.707337i	$-0.749896\pi$
$613$	−21456.8	−1.41375	−0.706877	+	0.707337i	$0.749896\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21534.5	1.40510	0.702550	−	0.711634i	$-0.252045\pi$
$617$	21534.5	1.40510	0.702550	+	0.711634i	$0.252045\pi$
$618$	0	0
$619$	24043.5	1.56121	0.780607	−	0.625023i	$-0.214910\pi$
$619$	24043.5	1.56121	0.780607	+	0.625023i	$0.214910\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11755.5	−0.755977
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10749.3	0.681400
$630$	0	0
$631$	7000.28	0.441643	0.220822	−	0.975314i	$-0.429126\pi$
$631$	7000.28	0.441643	0.220822	+	0.975314i	$0.429126\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4981.11	0.309825
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19557.4	−1.20511	−0.602553	−	0.798079i	$-0.705850\pi$
$641$	−19557.4	−1.20511	−0.602553	+	0.798079i	$0.705850\pi$
$642$	0	0
$643$	5640.24	0.345925	0.172962	−	0.984928i	$-0.444666\pi$
$643$	5640.24	0.345925	0.172962	+	0.984928i	$0.444666\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23410.0	−1.42247	−0.711237	−	0.702953i	$-0.751865\pi$
$647$	−23410.0	−1.42247	−0.711237	+	0.702953i	$0.751865\pi$
$648$	0	0
$649$	−4420.29	−0.267352
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24659.1	−1.47777	−0.738887	−	0.673830i	$-0.764648\pi$
$653$	−24659.1	−1.47777	−0.738887	+	0.673830i	$0.764648\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−21423.4	−1.26637	−0.633185	−	0.774001i	$-0.718253\pi$
$659$	−21423.4	−1.26637	−0.633185	+	0.774001i	$0.718253\pi$
$660$	0	0
$661$	−7120.12	−0.418972	−0.209486	−	0.977812i	$-0.567179\pi$
$661$	−7120.12	−0.418972	−0.209486	+	0.977812i	$0.567179\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8143.69	0.472751
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33386.5	−1.92082
$672$	0	0
$673$	−6092.55	−0.348961	−0.174480	−	0.984661i	$-0.555825\pi$
$673$	−6092.55	−0.348961	−0.174480	+	0.984661i	$0.555825\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24575.9	−1.39517	−0.697583	−	0.716504i	$-0.745741\pi$
$677$	−24575.9	−1.39517	−0.697583	+	0.716504i	$0.745741\pi$
$678$	0	0
$679$	18551.4	1.04851
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6977.95	−0.390928	−0.195464	−	0.980711i	$-0.562621\pi$
$683$	−6977.95	−0.390928	−0.195464	+	0.980711i	$0.562621\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3130.70	−0.173106
$690$	0	0
$691$	−11730.7	−0.645815	−0.322907	−	0.946431i	$-0.604660\pi$
$691$	−11730.7	−0.645815	−0.322907	+	0.946431i	$0.604660\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	14426.2	0.783977
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6659.02	−0.358784	−0.179392	−	0.983778i	$-0.557413\pi$
$701$	−6659.02	−0.358784	−0.179392	+	0.983778i	$0.557413\pi$
$702$	0	0
$703$	32478.1	1.74244
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−34709.2	−1.84636
$708$	0	0
$709$	15052.8	0.797350	0.398675	−	0.917092i	$-0.369470\pi$
$709$	15052.8	0.797350	0.398675	+	0.917092i	$0.369470\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12176.6	−0.639576
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	634.195	0.0328949	0.0164475	−	0.999865i	$-0.494764\pi$
$719$	634.195	0.0328949	0.0164475	+	0.999865i	$0.494764\pi$
$720$	0	0
$721$	−16923.3	−0.874141
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−9672.48	−0.493442	−0.246721	−	0.969087i	$-0.579353\pi$
$727$	−9672.48	−0.493442	−0.246721	+	0.969087i	$0.579353\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5950.88	−0.301096
$732$	0	0
$733$	−5566.87	−0.280514	−0.140257	−	0.990115i	$-0.544793\pi$
$733$	−5566.87	−0.280514	−0.140257	+	0.990115i	$0.544793\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−40217.5	−2.01008
$738$	0	0
$739$	22861.0	1.13797	0.568983	−	0.822349i	$-0.307337\pi$
$739$	22861.0	1.13797	0.568983	+	0.822349i	$0.307337\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15838.3	0.782032	0.391016	−	0.920384i	$-0.372124\pi$
$743$	15838.3	0.782032	0.391016	+	0.920384i	$0.372124\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13074.0	−0.637802
$750$	0	0
$751$	−26296.1	−1.27771	−0.638855	−	0.769327i	$-0.720592\pi$
$751$	−26296.1	−1.27771	−0.638855	+	0.769327i	$0.720592\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33171.7	−1.59266	−0.796331	−	0.604861i	$-0.793229\pi$
$757$	−33171.7	−1.59266	−0.796331	+	0.604861i	$0.793229\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−343.309	−0.0163534	−0.00817670	−	0.999967i	$-0.502603\pi$
$761$	−343.309	−0.0163534	−0.00817670	+	0.999967i	$0.502603\pi$
$762$	0	0
$763$	−40341.6	−1.91411
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1009.56	−0.0475267
$768$	0	0
$769$	4648.82	0.217998	0.108999	−	0.994042i	$-0.465235\pi$
$769$	4648.82	0.217998	0.108999	+	0.994042i	$0.465235\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5421.42	−0.252257	−0.126129	−	0.992014i	$-0.540255\pi$
$773$	−5421.42	−0.252257	−0.126129	+	0.992014i	$0.540255\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	43587.8	2.00475
$780$	0	0
$781$	−14533.9	−0.665895
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18357.5	0.831478	0.415739	−	0.909484i	$-0.363523\pi$
$787$	18357.5	0.831478	0.415739	+	0.909484i	$0.363523\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−18510.1	−0.832039
$792$	0	0
$793$	−7625.20	−0.341461
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5677.15	0.252315	0.126157	−	0.992010i	$-0.459736\pi$
$797$	5677.15	0.252315	0.126157	+	0.992010i	$0.459736\pi$
$798$	0	0
$799$	−2006.72	−0.0888518
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−12045.6	−0.529365
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24875.0	−1.08104	−0.540518	−	0.841333i	$-0.681772\pi$
$809$	−24875.0	−1.08104	−0.540518	+	0.841333i	$0.681772\pi$
$810$	0	0
$811$	6323.63	0.273801	0.136901	−	0.990585i	$-0.456286\pi$
$811$	6323.63	0.273801	0.136901	+	0.990585i	$0.456286\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−17980.2	−0.769947
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31744.6	−1.34944	−0.674722	−	0.738072i	$-0.735736\pi$
$821$	−31744.6	−1.34944	−0.674722	+	0.738072i	$0.735736\pi$
$822$	0	0
$823$	−6599.93	−0.279537	−0.139769	−	0.990184i	$-0.544636\pi$
$823$	−6599.93	−0.279537	−0.139769	+	0.990184i	$0.544636\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1552.94	−0.0652975	−0.0326487	−	0.999467i	$-0.510394\pi$
$827$	−1552.94	−0.0652975	−0.0326487	+	0.999467i	$0.510394\pi$
$828$	0	0
$829$	−18949.1	−0.793884	−0.396942	−	0.917844i	$-0.629929\pi$
$829$	−18949.1	−0.793884	−0.396942	+	0.917844i	$0.629929\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13039.6	0.542370
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−33786.4	−1.39027	−0.695134	−	0.718880i	$-0.744655\pi$
$839$	−33786.4	−1.39027	−0.695134	+	0.718880i	$0.744655\pi$
$840$	0	0
$841$	2979.09	0.122149
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	27112.5	1.09988
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18493.1	0.744930
$852$	0	0
$853$	−27376.2	−1.09888	−0.549440	−	0.835533i	$-0.685159\pi$
$853$	−27376.2	−1.09888	−0.549440	+	0.835533i	$0.685159\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23105.5	−0.920969	−0.460485	−	0.887668i	$-0.652324\pi$
$857$	−23105.5	−0.920969	−0.460485	+	0.887668i	$0.652324\pi$
$858$	0	0
$859$	22947.0	0.911457	0.455728	−	0.890119i	$-0.349379\pi$
$859$	22947.0	0.911457	0.455728	+	0.890119i	$0.349379\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30590.2	−1.20661	−0.603304	−	0.797511i	$-0.706150\pi$
$863$	−30590.2	−1.20661	−0.603304	+	0.797511i	$0.706150\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	35662.0	1.39212
$870$	0	0
$871$	−9185.33	−0.357329
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13685.2	−0.526928	−0.263464	−	0.964669i	$-0.584865\pi$
$877$	−13685.2	−0.526928	−0.263464	+	0.964669i	$0.584865\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33737.2	1.29017	0.645083	−	0.764113i	$-0.276823\pi$
$881$	33737.2	1.29017	0.645083	+	0.764113i	$0.276823\pi$
$882$	0	0
$883$	3738.74	0.142490	0.0712450	−	0.997459i	$-0.477303\pi$
$883$	3738.74	0.142490	0.0712450	+	0.997459i	$0.477303\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37168.3	−1.40698	−0.703489	−	0.710706i	$-0.748375\pi$
$887$	−37168.3	−1.40698	−0.703489	+	0.710706i	$0.748375\pi$
$888$	0	0
$889$	2123.09	0.0800971
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6063.16	−0.227207
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−40921.3	−1.51813
$900$	0	0
$901$	−8195.56	−0.303034
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−21277.6	−0.778954	−0.389477	−	0.921036i	$-0.627344\pi$
$907$	−21277.6	−0.778954	−0.389477	+	0.921036i	$0.627344\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	41401.1	1.50568	0.752842	−	0.658201i	$-0.228682\pi$
$911$	41401.1	1.50568	0.752842	+	0.658201i	$0.228682\pi$
$912$	0	0
$913$	63549.5	2.30359
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−53361.4	−1.92164
$918$	0	0
$919$	−2049.62	−0.0735697	−0.0367849	−	0.999323i	$-0.511712\pi$
$919$	−2049.62	−0.0735697	−0.0367849	+	0.999323i	$0.511712\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3319.42	−0.118375
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14404.0	0.508699	0.254350	−	0.967112i	$-0.418139\pi$
$929$	14404.0	0.508699	0.254350	+	0.967112i	$0.418139\pi$
$930$	0	0
$931$	39398.2	1.38692
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	36602.2	1.27614	0.638068	−	0.769980i	$-0.279734\pi$
$937$	36602.2	1.27614	0.638068	+	0.769980i	$0.279734\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5472.52	−0.189584	−0.0947922	−	0.995497i	$-0.530219\pi$
$941$	−5472.52	−0.189584	−0.0947922	+	0.995497i	$0.530219\pi$
$942$	0	0
$943$	24819.0	0.857071
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22467.1	0.770944	0.385472	−	0.922720i	$-0.374039\pi$
$947$	22467.1	0.770944	0.385472	+	0.922720i	$0.374039\pi$
$948$	0	0
$949$	−2751.11	−0.0941042
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13989.7	−0.475520	−0.237760	−	0.971324i	$-0.576413\pi$
$953$	−13989.7	−0.475520	−0.237760	+	0.971324i	$0.576413\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	85555.6	2.88085
$960$	0	0
$961$	31395.5	1.05386
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−37809.4	−1.25736	−0.628681	−	0.777663i	$-0.716405\pi$
$967$	−37809.4	−1.25736	−0.628681	+	0.777663i	$0.716405\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8527.36	0.281829	0.140915	−	0.990022i	$-0.454996\pi$
$971$	8527.36	0.281829	0.140915	+	0.990022i	$0.454996\pi$
$972$	0	0
$973$	−25760.8	−0.848770
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19536.5	−0.639744	−0.319872	−	0.947461i	$-0.603640\pi$
$977$	−19536.5	−0.639744	−0.319872	+	0.947461i	$0.603640\pi$
$978$	0	0
$979$	−19906.5	−0.649860
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17524.1	−0.568599	−0.284300	−	0.958735i	$-0.591761\pi$
$983$	−17524.1	−0.568599	−0.284300	+	0.958735i	$0.591761\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10237.9	−0.329169
$990$	0	0
$991$	2581.67	0.0827541	0.0413771	−	0.999144i	$-0.486826\pi$
$991$	2581.67	0.0827541	0.0413771	+	0.999144i	$0.486826\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42014.1	1.33460	0.667301	−	0.744788i	$-0.267449\pi$
$997$	42014.1	1.33460	0.667301	+	0.744788i	$0.267449\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.a.bw.1.4		4
3.2	odd	2		1800.4.a.bv.1.4		4
5.2	odd	4		360.4.f.f.289.4	yes	8
5.3	odd	4		360.4.f.f.289.3	✓	8
5.4	even	2		1800.4.a.bv.1.1		4
15.2	even	4		360.4.f.f.289.5	yes	8
15.8	even	4		360.4.f.f.289.6	yes	8
15.14	odd	2	inner	1800.4.a.bw.1.1		4
20.3	even	4		720.4.f.n.289.3		8
20.7	even	4		720.4.f.n.289.4		8
60.23	odd	4		720.4.f.n.289.6		8
60.47	odd	4		720.4.f.n.289.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.f.f.289.3	✓	8	5.3	odd	4
360.4.f.f.289.4	yes	8	5.2	odd	4
360.4.f.f.289.5	yes	8	15.2	even	4
360.4.f.f.289.6	yes	8	15.8	even	4
720.4.f.n.289.3		8	20.3	even	4
720.4.f.n.289.4		8	20.7	even	4
720.4.f.n.289.5		8	60.47	odd	4
720.4.f.n.289.6		8	60.23	odd	4
1800.4.a.bv.1.1		4	5.4	even	2
1800.4.a.bv.1.4		4	3.2	odd	2
1800.4.a.bw.1.1		4	15.14	odd	2	inner
1800.4.a.bw.1.4		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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)

\(f(q)\)	\(=\)	\(q+28.2616 q^{7} +O(q^{10})\)	\(q+28.2616 q^{7} +47.8575 q^{11} +10.9302 q^{13} +28.6132 q^{17} +86.4530 q^{19} +49.2265 q^{23} +165.433 q^{29} -247.359 q^{31} +375.674 q^{37} +504.180 q^{41} -207.976 q^{43} -70.1325 q^{47} +455.718 q^{49} -286.425 q^{53} -92.3636 q^{59} -697.624 q^{61} -840.360 q^{67} -303.691 q^{71} -251.697 q^{73} +1352.53 q^{77} +745.171 q^{79} +1327.89 q^{83} -415.953 q^{89} +308.906 q^{91} +656.418 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 48 q^{17} + 80 q^{19} + 64 q^{23} - 192 q^{31} + 384 q^{47} + 228 q^{49} - 16 q^{53} - 664 q^{61} + 2752 q^{77} + 1120 q^{79} + 1856 q^{83} + 704 q^{91}+O(q^{100}) \) 4 * q + 48 * q^17 + 80 * q^19 + 64 * q^23 - 192 * q^31 + 384 * q^47 + 228 * q^49 - 16 * q^53 - 664 * q^61 + 2752 * q^77 + 1120 * q^79 + 1856 * q^83 + 704 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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