LMFDB - Embedded newform 2240.4.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.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:	$132.164278413$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} -26.0000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} -26.0000 q^{9} -39.0000 q^{11} +17.0000 q^{13} +5.00000 q^{15} -15.0000 q^{17} +74.0000 q^{19} +7.00000 q^{21} +14.0000 q^{23} +25.0000 q^{25} +53.0000 q^{27} +237.000 q^{29} +180.000 q^{31} +39.0000 q^{33} +35.0000 q^{35} +318.000 q^{37} -17.0000 q^{39} -348.000 q^{41} -22.0000 q^{43} +130.000 q^{45} +193.000 q^{47} +49.0000 q^{49} +15.0000 q^{51} +208.000 q^{53} +195.000 q^{55} -74.0000 q^{57} +452.000 q^{59} -340.000 q^{61} +182.000 q^{63} -85.0000 q^{65} -408.000 q^{67} -14.0000 q^{69} -528.000 q^{71} -554.000 q^{73} -25.0000 q^{75} +273.000 q^{77} -539.000 q^{79} +649.000 q^{81} +164.000 q^{83} +75.0000 q^{85} -237.000 q^{87} -576.000 q^{89} -119.000 q^{91} -180.000 q^{93} -370.000 q^{95} -827.000 q^{97} +1014.00 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.192450	−0.0962250	−	0.995360i	$-0.530677\pi$
$3$	−1.00000	−0.192450	−0.0962250	+	0.995360i	$0.530677\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−26.0000	−0.962963
$10$	0	0
$11$	−39.0000	−1.06899	−0.534497	−	0.845170i	$-0.679499\pi$
$11$	−39.0000	−1.06899	−0.534497	+	0.845170i	$0.679499\pi$
$12$	0	0
$13$	17.0000	0.362689	0.181344	−	0.983420i	$-0.441955\pi$
$13$	17.0000	0.362689	0.181344	+	0.983420i	$0.441955\pi$
$14$	0	0
$15$	5.00000	0.0860663
$16$	0	0
$17$	−15.0000	−0.214002	−0.107001	−	0.994259i	$-0.534125\pi$
$17$	−15.0000	−0.214002	−0.107001	+	0.994259i	$0.534125\pi$
$18$	0	0
$19$	74.0000	0.893514	0.446757	−	0.894655i	$-0.352579\pi$
$19$	74.0000	0.893514	0.446757	+	0.894655i	$0.352579\pi$
$20$	0	0
$21$	7.00000	0.0727393
$22$	0	0
$23$	14.0000	0.126922	0.0634609	−	0.997984i	$-0.479786\pi$
$23$	14.0000	0.126922	0.0634609	+	0.997984i	$0.479786\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	53.0000	0.377772
$28$	0	0
$29$	237.000	1.51758	0.758790	−	0.651336i	$-0.225791\pi$
$29$	237.000	1.51758	0.758790	+	0.651336i	$0.225791\pi$
$30$	0	0
$31$	180.000	1.04287	0.521435	−	0.853291i	$-0.325397\pi$
$31$	180.000	1.04287	0.521435	+	0.853291i	$0.325397\pi$
$32$	0	0
$33$	39.0000	0.205728
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	318.000	1.41294	0.706471	−	0.707742i	$-0.250286\pi$
$37$	318.000	1.41294	0.706471	+	0.707742i	$0.250286\pi$
$38$	0	0
$39$	−17.0000	−0.0697995
$40$	0	0
$41$	−348.000	−1.32557	−0.662786	−	0.748809i	$-0.730626\pi$
$41$	−348.000	−1.32557	−0.662786	+	0.748809i	$0.730626\pi$
$42$	0	0
$43$	−22.0000	−0.0780225	−0.0390113	−	0.999239i	$-0.512421\pi$
$43$	−22.0000	−0.0780225	−0.0390113	+	0.999239i	$0.512421\pi$
$44$	0	0
$45$	130.000	0.430650
$46$	0	0
$47$	193.000	0.598978	0.299489	−	0.954100i	$-0.403184\pi$
$47$	193.000	0.598978	0.299489	+	0.954100i	$0.403184\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	15.0000	0.0411847
$52$	0	0
$53$	208.000	0.539075	0.269538	−	0.962990i	$-0.413129\pi$
$53$	208.000	0.539075	0.269538	+	0.962990i	$0.413129\pi$
$54$	0	0
$55$	195.000	0.478069
$56$	0	0
$57$	−74.0000	−0.171957
$58$	0	0
$59$	452.000	0.997379	0.498690	−	0.866781i	$-0.333815\pi$
$59$	452.000	0.997379	0.498690	+	0.866781i	$0.333815\pi$
$60$	0	0
$61$	−340.000	−0.713648	−0.356824	−	0.934172i	$-0.616140\pi$
$61$	−340.000	−0.713648	−0.356824	+	0.934172i	$0.616140\pi$
$62$	0	0
$63$	182.000	0.363966
$64$	0	0
$65$	−85.0000	−0.162199
$66$	0	0
$67$	−408.000	−0.743957	−0.371979	−	0.928241i	$-0.621321\pi$
$67$	−408.000	−0.743957	−0.371979	+	0.928241i	$0.621321\pi$
$68$	0	0
$69$	−14.0000	−0.0244261
$70$	0	0
$71$	−528.000	−0.882564	−0.441282	−	0.897368i	$-0.645476\pi$
$71$	−528.000	−0.882564	−0.441282	+	0.897368i	$0.645476\pi$
$72$	0	0
$73$	−554.000	−0.888230	−0.444115	−	0.895970i	$-0.646482\pi$
$73$	−554.000	−0.888230	−0.444115	+	0.895970i	$0.646482\pi$
$74$	0	0
$75$	−25.0000	−0.0384900
$76$	0	0
$77$	273.000	0.404042
$78$	0	0
$79$	−539.000	−0.767623	−0.383812	−	0.923411i	$-0.625389\pi$
$79$	−539.000	−0.767623	−0.383812	+	0.923411i	$0.625389\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	0	0
$83$	164.000	0.216884	0.108442	−	0.994103i	$-0.465414\pi$
$83$	164.000	0.216884	0.108442	+	0.994103i	$0.465414\pi$
$84$	0	0
$85$	75.0000	0.0957046
$86$	0	0
$87$	−237.000	−0.292058
$88$	0	0
$89$	−576.000	−0.686021	−0.343011	−	0.939332i	$-0.611447\pi$
$89$	−576.000	−0.686021	−0.343011	+	0.939332i	$0.611447\pi$
$90$	0	0
$91$	−119.000	−0.137083
$92$	0	0
$93$	−180.000	−0.200700
$94$	0	0
$95$	−370.000	−0.399592
$96$	0	0
$97$	−827.000	−0.865661	−0.432831	−	0.901475i	$-0.642485\pi$
$97$	−827.000	−0.865661	−0.432831	+	0.901475i	$0.642485\pi$
$98$	0	0
$99$	1014.00	1.02940
$100$	0	0
$101$	1114.00	1.09750	0.548748	−	0.835988i	$-0.315105\pi$
$101$	1114.00	1.09750	0.548748	+	0.835988i	$0.315105\pi$
$102$	0	0
$103$	−1643.00	−1.57174	−0.785872	−	0.618389i	$-0.787785\pi$
$103$	−1643.00	−1.57174	−0.785872	+	0.618389i	$0.787785\pi$
$104$	0	0
$105$	−35.0000	−0.0325300
$106$	0	0
$107$	710.000	0.641479	0.320740	−	0.947167i	$-0.396069\pi$
$107$	710.000	0.641479	0.320740	+	0.947167i	$0.396069\pi$
$108$	0	0
$109$	−1321.00	−1.16082	−0.580408	−	0.814326i	$-0.697107\pi$
$109$	−1321.00	−1.16082	−0.580408	+	0.814326i	$0.697107\pi$
$110$	0	0
$111$	−318.000	−0.271921
$112$	0	0
$113$	342.000	0.284714	0.142357	−	0.989815i	$-0.454532\pi$
$113$	342.000	0.284714	0.142357	+	0.989815i	$0.454532\pi$
$114$	0	0
$115$	−70.0000	−0.0567612
$116$	0	0
$117$	−442.000	−0.349256
$118$	0	0
$119$	105.000	0.0808852
$120$	0	0
$121$	190.000	0.142750
$122$	0	0
$123$	348.000	0.255107
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1348.00	0.941856	0.470928	−	0.882172i	$-0.343919\pi$
$127$	1348.00	0.941856	0.470928	+	0.882172i	$0.343919\pi$
$128$	0	0
$129$	22.0000	0.0150154
$130$	0	0
$131$	238.000	0.158734	0.0793670	−	0.996845i	$-0.474710\pi$
$131$	238.000	0.158734	0.0793670	+	0.996845i	$0.474710\pi$
$132$	0	0
$133$	−518.000	−0.337717
$134$	0	0
$135$	−265.000	−0.168945
$136$	0	0
$137$	−776.000	−0.483928	−0.241964	−	0.970285i	$-0.577792\pi$
$137$	−776.000	−0.483928	−0.241964	+	0.970285i	$0.577792\pi$
$138$	0	0
$139$	326.000	0.198928	0.0994639	−	0.995041i	$-0.468287\pi$
$139$	326.000	0.198928	0.0994639	+	0.995041i	$0.468287\pi$
$140$	0	0
$141$	−193.000	−0.115273
$142$	0	0
$143$	−663.000	−0.387712
$144$	0	0
$145$	−1185.00	−0.678682
$146$	0	0
$147$	−49.0000	−0.0274929
$148$	0	0
$149$	−3282.00	−1.80451	−0.902255	−	0.431203i	$-0.858089\pi$
$149$	−3282.00	−1.80451	−0.902255	+	0.431203i	$0.858089\pi$
$150$	0	0
$151$	−2627.00	−1.41578	−0.707888	−	0.706325i	$-0.750352\pi$
$151$	−2627.00	−1.41578	−0.707888	+	0.706325i	$0.750352\pi$
$152$	0	0
$153$	390.000	0.206076
$154$	0	0
$155$	−900.000	−0.466385
$156$	0	0
$157$	−258.000	−0.131151	−0.0655753	−	0.997848i	$-0.520888\pi$
$157$	−258.000	−0.131151	−0.0655753	+	0.997848i	$0.520888\pi$
$158$	0	0
$159$	−208.000	−0.103745
$160$	0	0
$161$	−98.0000	−0.0479719
$162$	0	0
$163$	134.000	0.0643907	0.0321954	−	0.999482i	$-0.489750\pi$
$163$	134.000	0.0643907	0.0321954	+	0.999482i	$0.489750\pi$
$164$	0	0
$165$	−195.000	−0.0920044
$166$	0	0
$167$	4181.00	1.93734	0.968669	−	0.248355i	$-0.0798899\pi$
$167$	4181.00	1.93734	0.968669	+	0.248355i	$0.0798899\pi$
$168$	0	0
$169$	−1908.00	−0.868457
$170$	0	0
$171$	−1924.00	−0.860421
$172$	0	0
$173$	−2901.00	−1.27491	−0.637454	−	0.770489i	$-0.720012\pi$
$173$	−2901.00	−1.27491	−0.637454	+	0.770489i	$0.720012\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−452.000	−0.191946
$178$	0	0
$179$	4404.00	1.83894	0.919470	−	0.393159i	$-0.128618\pi$
$179$	4404.00	1.83894	0.919470	+	0.393159i	$0.128618\pi$
$180$	0	0
$181$	−1752.00	−0.719476	−0.359738	−	0.933053i	$-0.617134\pi$
$181$	−1752.00	−0.719476	−0.359738	+	0.933053i	$0.617134\pi$
$182$	0	0
$183$	340.000	0.137342
$184$	0	0
$185$	−1590.00	−0.631887
$186$	0	0
$187$	585.000	0.228767
$188$	0	0
$189$	−371.000	−0.142785
$190$	0	0
$191$	3595.00	1.36191	0.680956	−	0.732325i	$-0.261565\pi$
$191$	3595.00	1.36191	0.680956	+	0.732325i	$0.261565\pi$
$192$	0	0
$193$	3448.00	1.28597	0.642986	−	0.765878i	$-0.277695\pi$
$193$	3448.00	1.28597	0.642986	+	0.765878i	$0.277695\pi$
$194$	0	0
$195$	85.0000	0.0312153
$196$	0	0
$197$	3576.00	1.29330	0.646648	−	0.762788i	$-0.276170\pi$
$197$	3576.00	1.29330	0.646648	+	0.762788i	$0.276170\pi$
$198$	0	0
$199$	−2868.00	−1.02164	−0.510822	−	0.859687i	$-0.670659\pi$
$199$	−2868.00	−1.02164	−0.510822	+	0.859687i	$0.670659\pi$
$200$	0	0
$201$	408.000	0.143175
$202$	0	0
$203$	−1659.00	−0.573591
$204$	0	0
$205$	1740.00	0.592814
$206$	0	0
$207$	−364.000	−0.122221
$208$	0	0
$209$	−2886.00	−0.955162
$210$	0	0
$211$	1359.00	0.443400	0.221700	−	0.975115i	$-0.428839\pi$
$211$	1359.00	0.443400	0.221700	+	0.975115i	$0.428839\pi$
$212$	0	0
$213$	528.000	0.169850
$214$	0	0
$215$	110.000	0.0348927
$216$	0	0
$217$	−1260.00	−0.394168
$218$	0	0
$219$	554.000	0.170940
$220$	0	0
$221$	−255.000	−0.0776161
$222$	0	0
$223$	5693.00	1.70956	0.854779	−	0.518992i	$-0.173693\pi$
$223$	5693.00	1.70956	0.854779	+	0.518992i	$0.173693\pi$
$224$	0	0
$225$	−650.000	−0.192593
$226$	0	0
$227$	−6381.00	−1.86574	−0.932868	−	0.360220i	$-0.882702\pi$
$227$	−6381.00	−1.86574	−0.932868	+	0.360220i	$0.882702\pi$
$228$	0	0
$229$	−3556.00	−1.02614	−0.513072	−	0.858345i	$-0.671493\pi$
$229$	−3556.00	−1.02614	−0.513072	+	0.858345i	$0.671493\pi$
$230$	0	0
$231$	−273.000	−0.0777579
$232$	0	0
$233$	872.000	0.245179	0.122589	−	0.992457i	$-0.460880\pi$
$233$	872.000	0.245179	0.122589	+	0.992457i	$0.460880\pi$
$234$	0	0
$235$	−965.000	−0.267871
$236$	0	0
$237$	539.000	0.147729
$238$	0	0
$239$	2129.00	0.576207	0.288104	−	0.957599i	$-0.406975\pi$
$239$	2129.00	0.576207	0.288104	+	0.957599i	$0.406975\pi$
$240$	0	0
$241$	778.000	0.207948	0.103974	−	0.994580i	$-0.466844\pi$
$241$	778.000	0.207948	0.103974	+	0.994580i	$0.466844\pi$
$242$	0	0
$243$	−2080.00	−0.549103
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	1258.00	0.324067
$248$	0	0
$249$	−164.000	−0.0417393
$250$	0	0
$251$	−4990.00	−1.25484	−0.627422	−	0.778679i	$-0.715890\pi$
$251$	−4990.00	−1.25484	−0.627422	+	0.778679i	$0.715890\pi$
$252$	0	0
$253$	−546.000	−0.135679
$254$	0	0
$255$	−75.0000	−0.0184184
$256$	0	0
$257$	−3630.00	−0.881063	−0.440531	−	0.897737i	$-0.645210\pi$
$257$	−3630.00	−0.881063	−0.440531	+	0.897737i	$0.645210\pi$
$258$	0	0
$259$	−2226.00	−0.534042
$260$	0	0
$261$	−6162.00	−1.46137
$262$	0	0
$263$	290.000	0.0679930	0.0339965	−	0.999422i	$-0.489176\pi$
$263$	290.000	0.0679930	0.0339965	+	0.999422i	$0.489176\pi$
$264$	0	0
$265$	−1040.00	−0.241082
$266$	0	0
$267$	576.000	0.132025
$268$	0	0
$269$	1854.00	0.420224	0.210112	−	0.977677i	$-0.432617\pi$
$269$	1854.00	0.420224	0.210112	+	0.977677i	$0.432617\pi$
$270$	0	0
$271$	1836.00	0.411546	0.205773	−	0.978600i	$-0.434029\pi$
$271$	1836.00	0.411546	0.205773	+	0.978600i	$0.434029\pi$
$272$	0	0
$273$	119.000	0.0263817
$274$	0	0
$275$	−975.000	−0.213799
$276$	0	0
$277$	−6154.00	−1.33487	−0.667433	−	0.744670i	$-0.732607\pi$
$277$	−6154.00	−1.33487	−0.667433	+	0.744670i	$0.732607\pi$
$278$	0	0
$279$	−4680.00	−1.00424
$280$	0	0
$281$	−437.000	−0.0927731	−0.0463865	−	0.998924i	$-0.514771\pi$
$281$	−437.000	−0.0927731	−0.0463865	+	0.998924i	$0.514771\pi$
$282$	0	0
$283$	−347.000	−0.0728870	−0.0364435	−	0.999336i	$-0.511603\pi$
$283$	−347.000	−0.0728870	−0.0364435	+	0.999336i	$0.511603\pi$
$284$	0	0
$285$	370.000	0.0769014
$286$	0	0
$287$	2436.00	0.501019
$288$	0	0
$289$	−4688.00	−0.954203
$290$	0	0
$291$	827.000	0.166597
$292$	0	0
$293$	−1281.00	−0.255416	−0.127708	−	0.991812i	$-0.540762\pi$
$293$	−1281.00	−0.255416	−0.127708	+	0.991812i	$0.540762\pi$
$294$	0	0
$295$	−2260.00	−0.446042
$296$	0	0
$297$	−2067.00	−0.403837
$298$	0	0
$299$	238.000	0.0460331
$300$	0	0
$301$	154.000	0.0294897
$302$	0	0
$303$	−1114.00	−0.211213
$304$	0	0
$305$	1700.00	0.319153
$306$	0	0
$307$	−3611.00	−0.671305	−0.335653	−	0.941986i	$-0.608957\pi$
$307$	−3611.00	−0.671305	−0.335653	+	0.941986i	$0.608957\pi$
$308$	0	0
$309$	1643.00	0.302482
$310$	0	0
$311$	8206.00	1.49620	0.748102	−	0.663584i	$-0.230965\pi$
$311$	8206.00	1.49620	0.748102	+	0.663584i	$0.230965\pi$
$312$	0	0
$313$	4911.00	0.886857	0.443428	−	0.896310i	$-0.353762\pi$
$313$	4911.00	0.886857	0.443428	+	0.896310i	$0.353762\pi$
$314$	0	0
$315$	−910.000	−0.162770
$316$	0	0
$317$	−606.000	−0.107370	−0.0536851	−	0.998558i	$-0.517097\pi$
$317$	−606.000	−0.107370	−0.0536851	+	0.998558i	$0.517097\pi$
$318$	0	0
$319$	−9243.00	−1.62228
$320$	0	0
$321$	−710.000	−0.123453
$322$	0	0
$323$	−1110.00	−0.191214
$324$	0	0
$325$	425.000	0.0725377
$326$	0	0
$327$	1321.00	0.223399
$328$	0	0
$329$	−1351.00	−0.226392
$330$	0	0
$331$	5308.00	0.881432	0.440716	−	0.897647i	$-0.354725\pi$
$331$	5308.00	0.881432	0.440716	+	0.897647i	$0.354725\pi$
$332$	0	0
$333$	−8268.00	−1.36061
$334$	0	0
$335$	2040.00	0.332708
$336$	0	0
$337$	−10638.0	−1.71955	−0.859776	−	0.510672i	$-0.829397\pi$
$337$	−10638.0	−1.71955	−0.859776	+	0.510672i	$0.829397\pi$
$338$	0	0
$339$	−342.000	−0.0547932
$340$	0	0
$341$	−7020.00	−1.11482
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	70.0000	0.0109237
$346$	0	0
$347$	−2194.00	−0.339424	−0.169712	−	0.985494i	$-0.554284\pi$
$347$	−2194.00	−0.339424	−0.169712	+	0.985494i	$0.554284\pi$
$348$	0	0
$349$	−938.000	−0.143868	−0.0719341	−	0.997409i	$-0.522917\pi$
$349$	−938.000	−0.143868	−0.0719341	+	0.997409i	$0.522917\pi$
$350$	0	0
$351$	901.000	0.137014
$352$	0	0
$353$	−4995.00	−0.753136	−0.376568	−	0.926389i	$-0.622896\pi$
$353$	−4995.00	−0.753136	−0.376568	+	0.926389i	$0.622896\pi$
$354$	0	0
$355$	2640.00	0.394695
$356$	0	0
$357$	−105.000	−0.0155664
$358$	0	0
$359$	752.000	0.110554	0.0552772	−	0.998471i	$-0.482396\pi$
$359$	752.000	0.110554	0.0552772	+	0.998471i	$0.482396\pi$
$360$	0	0
$361$	−1383.00	−0.201633
$362$	0	0
$363$	−190.000	−0.0274722
$364$	0	0
$365$	2770.00	0.397229
$366$	0	0
$367$	−1175.00	−0.167124	−0.0835620	−	0.996503i	$-0.526630\pi$
$367$	−1175.00	−0.167124	−0.0835620	+	0.996503i	$0.526630\pi$
$368$	0	0
$369$	9048.00	1.27648
$370$	0	0
$371$	−1456.00	−0.203751
$372$	0	0
$373$	−9520.00	−1.32152	−0.660760	−	0.750597i	$-0.729766\pi$
$373$	−9520.00	−1.32152	−0.660760	+	0.750597i	$0.729766\pi$
$374$	0	0
$375$	125.000	0.0172133
$376$	0	0
$377$	4029.00	0.550409
$378$	0	0
$379$	11620.0	1.57488	0.787440	−	0.616392i	$-0.211406\pi$
$379$	11620.0	1.57488	0.787440	+	0.616392i	$0.211406\pi$
$380$	0	0
$381$	−1348.00	−0.181260
$382$	0	0
$383$	1708.00	0.227871	0.113936	−	0.993488i	$-0.463654\pi$
$383$	1708.00	0.227871	0.113936	+	0.993488i	$0.463654\pi$
$384$	0	0
$385$	−1365.00	−0.180693
$386$	0	0
$387$	572.000	0.0751328
$388$	0	0
$389$	−125.000	−0.0162924	−0.00814621	−	0.999967i	$-0.502593\pi$
$389$	−125.000	−0.0162924	−0.00814621	+	0.999967i	$0.502593\pi$
$390$	0	0
$391$	−210.000	−0.0271615
$392$	0	0
$393$	−238.000	−0.0305484
$394$	0	0
$395$	2695.00	0.343292
$396$	0	0
$397$	−13077.0	−1.65319	−0.826594	−	0.562798i	$-0.809725\pi$
$397$	−13077.0	−1.65319	−0.826594	+	0.562798i	$0.809725\pi$
$398$	0	0
$399$	518.000	0.0649936
$400$	0	0
$401$	2161.00	0.269115	0.134558	−	0.990906i	$-0.457039\pi$
$401$	2161.00	0.269115	0.134558	+	0.990906i	$0.457039\pi$
$402$	0	0
$403$	3060.00	0.378237
$404$	0	0
$405$	−3245.00	−0.398137
$406$	0	0
$407$	−12402.0	−1.51043
$408$	0	0
$409$	−9614.00	−1.16230	−0.581151	−	0.813796i	$-0.697398\pi$
$409$	−9614.00	−1.16230	−0.581151	+	0.813796i	$0.697398\pi$
$410$	0	0
$411$	776.000	0.0931320
$412$	0	0
$413$	−3164.00	−0.376974
$414$	0	0
$415$	−820.000	−0.0969933
$416$	0	0
$417$	−326.000	−0.0382837
$418$	0	0
$419$	8808.00	1.02697	0.513483	−	0.858100i	$-0.328355\pi$
$419$	8808.00	1.02697	0.513483	+	0.858100i	$0.328355\pi$
$420$	0	0
$421$	13207.0	1.52891	0.764453	−	0.644679i	$-0.223009\pi$
$421$	13207.0	1.52891	0.764453	+	0.644679i	$0.223009\pi$
$422$	0	0
$423$	−5018.00	−0.576793
$424$	0	0
$425$	−375.000	−0.0428004
$426$	0	0
$427$	2380.00	0.269734
$428$	0	0
$429$	663.000	0.0746152
$430$	0	0
$431$	−13201.0	−1.47534	−0.737668	−	0.675164i	$-0.764073\pi$
$431$	−13201.0	−1.47534	−0.737668	+	0.675164i	$0.764073\pi$
$432$	0	0
$433$	−11890.0	−1.31962	−0.659812	−	0.751431i	$-0.729364\pi$
$433$	−11890.0	−1.31962	−0.659812	+	0.751431i	$0.729364\pi$
$434$	0	0
$435$	1185.00	0.130612
$436$	0	0
$437$	1036.00	0.113406
$438$	0	0
$439$	−4474.00	−0.486406	−0.243203	−	0.969975i	$-0.578198\pi$
$439$	−4474.00	−0.486406	−0.243203	+	0.969975i	$0.578198\pi$
$440$	0	0
$441$	−1274.00	−0.137566
$442$	0	0
$443$	−14302.0	−1.53388	−0.766940	−	0.641719i	$-0.778221\pi$
$443$	−14302.0	−1.53388	−0.766940	+	0.641719i	$0.778221\pi$
$444$	0	0
$445$	2880.00	0.306798
$446$	0	0
$447$	3282.00	0.347278
$448$	0	0
$449$	−9801.00	−1.03015	−0.515075	−	0.857145i	$-0.672236\pi$
$449$	−9801.00	−1.03015	−0.515075	+	0.857145i	$0.672236\pi$
$450$	0	0
$451$	13572.0	1.41703
$452$	0	0
$453$	2627.00	0.272466
$454$	0	0
$455$	595.000	0.0613056
$456$	0	0
$457$	4968.00	0.508519	0.254260	−	0.967136i	$-0.418168\pi$
$457$	4968.00	0.508519	0.254260	+	0.967136i	$0.418168\pi$
$458$	0	0
$459$	−795.000	−0.0808441
$460$	0	0
$461$	9408.00	0.950486	0.475243	−	0.879854i	$-0.342360\pi$
$461$	9408.00	0.950486	0.475243	+	0.879854i	$0.342360\pi$
$462$	0	0
$463$	1692.00	0.169836	0.0849178	−	0.996388i	$-0.472937\pi$
$463$	1692.00	0.169836	0.0849178	+	0.996388i	$0.472937\pi$
$464$	0	0
$465$	900.000	0.0897559
$466$	0	0
$467$	−4199.00	−0.416074	−0.208037	−	0.978121i	$-0.566707\pi$
$467$	−4199.00	−0.416074	−0.208037	+	0.978121i	$0.566707\pi$
$468$	0	0
$469$	2856.00	0.281189
$470$	0	0
$471$	258.000	0.0252399
$472$	0	0
$473$	858.000	0.0834057
$474$	0	0
$475$	1850.00	0.178703
$476$	0	0
$477$	−5408.00	−0.519110
$478$	0	0
$479$	−398.000	−0.0379647	−0.0189823	−	0.999820i	$-0.506043\pi$
$479$	−398.000	−0.0379647	−0.0189823	+	0.999820i	$0.506043\pi$
$480$	0	0
$481$	5406.00	0.512458
$482$	0	0
$483$	98.0000	0.00923220
$484$	0	0
$485$	4135.00	0.387135
$486$	0	0
$487$	−13786.0	−1.28276	−0.641379	−	0.767224i	$-0.721637\pi$
$487$	−13786.0	−1.28276	−0.641379	+	0.767224i	$0.721637\pi$
$488$	0	0
$489$	−134.000	−0.0123920
$490$	0	0
$491$	1891.00	0.173808	0.0869039	−	0.996217i	$-0.472303\pi$
$491$	1891.00	0.173808	0.0869039	+	0.996217i	$0.472303\pi$
$492$	0	0
$493$	−3555.00	−0.324765
$494$	0	0
$495$	−5070.00	−0.460363
$496$	0	0
$497$	3696.00	0.333578
$498$	0	0
$499$	−3171.00	−0.284476	−0.142238	−	0.989832i	$-0.545430\pi$
$499$	−3171.00	−0.284476	−0.142238	+	0.989832i	$0.545430\pi$
$500$	0	0
$501$	−4181.00	−0.372841
$502$	0	0
$503$	21803.0	1.93270	0.966350	−	0.257232i	$-0.0828105\pi$
$503$	21803.0	1.93270	0.966350	+	0.257232i	$0.0828105\pi$
$504$	0	0
$505$	−5570.00	−0.490815
$506$	0	0
$507$	1908.00	0.167135
$508$	0	0
$509$	19174.0	1.66969	0.834845	−	0.550484i	$-0.185557\pi$
$509$	19174.0	1.66969	0.834845	+	0.550484i	$0.185557\pi$
$510$	0	0
$511$	3878.00	0.335719
$512$	0	0
$513$	3922.00	0.337545
$514$	0	0
$515$	8215.00	0.702905
$516$	0	0
$517$	−7527.00	−0.640304
$518$	0	0
$519$	2901.00	0.245356
$520$	0	0
$521$	−3210.00	−0.269928	−0.134964	−	0.990850i	$-0.543092\pi$
$521$	−3210.00	−0.269928	−0.134964	+	0.990850i	$0.543092\pi$
$522$	0	0
$523$	−3220.00	−0.269218	−0.134609	−	0.990899i	$-0.542978\pi$
$523$	−3220.00	−0.269218	−0.134609	+	0.990899i	$0.542978\pi$
$524$	0	0
$525$	175.000	0.0145479
$526$	0	0
$527$	−2700.00	−0.223176
$528$	0	0
$529$	−11971.0	−0.983891
$530$	0	0
$531$	−11752.0	−0.960439
$532$	0	0
$533$	−5916.00	−0.480770
$534$	0	0
$535$	−3550.00	−0.286878
$536$	0	0
$537$	−4404.00	−0.353904
$538$	0	0
$539$	−1911.00	−0.152714
$540$	0	0
$541$	−5443.00	−0.432556	−0.216278	−	0.976332i	$-0.569392\pi$
$541$	−5443.00	−0.432556	−0.216278	+	0.976332i	$0.569392\pi$
$542$	0	0
$543$	1752.00	0.138463
$544$	0	0
$545$	6605.00	0.519132
$546$	0	0
$547$	14988.0	1.17156	0.585778	−	0.810472i	$-0.300789\pi$
$547$	14988.0	1.17156	0.585778	+	0.810472i	$0.300789\pi$
$548$	0	0
$549$	8840.00	0.687217
$550$	0	0
$551$	17538.0	1.35598
$552$	0	0
$553$	3773.00	0.290134
$554$	0	0
$555$	1590.00	0.121607
$556$	0	0
$557$	3024.00	0.230038	0.115019	−	0.993363i	$-0.463307\pi$
$557$	3024.00	0.230038	0.115019	+	0.993363i	$0.463307\pi$
$558$	0	0
$559$	−374.000	−0.0282979
$560$	0	0
$561$	−585.000	−0.0440262
$562$	0	0
$563$	−8964.00	−0.671026	−0.335513	−	0.942036i	$-0.608910\pi$
$563$	−8964.00	−0.671026	−0.335513	+	0.942036i	$0.608910\pi$
$564$	0	0
$565$	−1710.00	−0.127328
$566$	0	0
$567$	−4543.00	−0.336487
$568$	0	0
$569$	−14022.0	−1.03310	−0.516549	−	0.856258i	$-0.672784\pi$
$569$	−14022.0	−1.03310	−0.516549	+	0.856258i	$0.672784\pi$
$570$	0	0
$571$	−20020.0	−1.46727	−0.733635	−	0.679544i	$-0.762178\pi$
$571$	−20020.0	−1.46727	−0.733635	+	0.679544i	$0.762178\pi$
$572$	0	0
$573$	−3595.00	−0.262100
$574$	0	0
$575$	350.000	0.0253844
$576$	0	0
$577$	−18349.0	−1.32388	−0.661940	−	0.749557i	$-0.730267\pi$
$577$	−18349.0	−1.32388	−0.661940	+	0.749557i	$0.730267\pi$
$578$	0	0
$579$	−3448.00	−0.247485
$580$	0	0
$581$	−1148.00	−0.0819743
$582$	0	0
$583$	−8112.00	−0.576269
$584$	0	0
$585$	2210.00	0.156192
$586$	0	0
$587$	−5616.00	−0.394884	−0.197442	−	0.980315i	$-0.563263\pi$
$587$	−5616.00	−0.394884	−0.197442	+	0.980315i	$0.563263\pi$
$588$	0	0
$589$	13320.0	0.931818
$590$	0	0
$591$	−3576.00	−0.248895
$592$	0	0
$593$	8243.00	0.570825	0.285413	−	0.958405i	$-0.407869\pi$
$593$	8243.00	0.570825	0.285413	+	0.958405i	$0.407869\pi$
$594$	0	0
$595$	−525.000	−0.0361729
$596$	0	0
$597$	2868.00	0.196615
$598$	0	0
$599$	−4385.00	−0.299109	−0.149554	−	0.988753i	$-0.547784\pi$
$599$	−4385.00	−0.299109	−0.149554	+	0.988753i	$0.547784\pi$
$600$	0	0
$601$	14378.0	0.975858	0.487929	−	0.872883i	$-0.337752\pi$
$601$	14378.0	0.975858	0.487929	+	0.872883i	$0.337752\pi$
$602$	0	0
$603$	10608.0	0.716403
$604$	0	0
$605$	−950.000	−0.0638397
$606$	0	0
$607$	7235.00	0.483788	0.241894	−	0.970303i	$-0.422231\pi$
$607$	7235.00	0.483788	0.241894	+	0.970303i	$0.422231\pi$
$608$	0	0
$609$	1659.00	0.110388
$610$	0	0
$611$	3281.00	0.217242
$612$	0	0
$613$	25350.0	1.67027	0.835136	−	0.550043i	$-0.185389\pi$
$613$	25350.0	1.67027	0.835136	+	0.550043i	$0.185389\pi$
$614$	0	0
$615$	−1740.00	−0.114087
$616$	0	0
$617$	−4634.00	−0.302363	−0.151181	−	0.988506i	$-0.548308\pi$
$617$	−4634.00	−0.302363	−0.151181	+	0.988506i	$0.548308\pi$
$618$	0	0
$619$	12446.0	0.808153	0.404077	−	0.914725i	$-0.367593\pi$
$619$	12446.0	0.808153	0.404077	+	0.914725i	$0.367593\pi$
$620$	0	0
$621$	742.000	0.0479476
$622$	0	0
$623$	4032.00	0.259292
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	2886.00	0.183821
$628$	0	0
$629$	−4770.00	−0.302373
$630$	0	0
$631$	−7145.00	−0.450773	−0.225387	−	0.974269i	$-0.572365\pi$
$631$	−7145.00	−0.450773	−0.225387	+	0.974269i	$0.572365\pi$
$632$	0	0
$633$	−1359.00	−0.0853324
$634$	0	0
$635$	−6740.00	−0.421211
$636$	0	0
$637$	833.000	0.0518127
$638$	0	0
$639$	13728.0	0.849877
$640$	0	0
$641$	−20846.0	−1.28450	−0.642252	−	0.766493i	$-0.722000\pi$
$641$	−20846.0	−1.28450	−0.642252	+	0.766493i	$0.722000\pi$
$642$	0	0
$643$	1751.00	0.107391	0.0536957	−	0.998557i	$-0.482900\pi$
$643$	1751.00	0.107391	0.0536957	+	0.998557i	$0.482900\pi$
$644$	0	0
$645$	−110.000	−0.00671511
$646$	0	0
$647$	−15016.0	−0.912426	−0.456213	−	0.889871i	$-0.650795\pi$
$647$	−15016.0	−0.912426	−0.456213	+	0.889871i	$0.650795\pi$
$648$	0	0
$649$	−17628.0	−1.06619
$650$	0	0
$651$	1260.00	0.0758576
$652$	0	0
$653$	−7902.00	−0.473552	−0.236776	−	0.971564i	$-0.576091\pi$
$653$	−7902.00	−0.473552	−0.236776	+	0.971564i	$0.576091\pi$
$654$	0	0
$655$	−1190.00	−0.0709880
$656$	0	0
$657$	14404.0	0.855333
$658$	0	0
$659$	−10019.0	−0.592238	−0.296119	−	0.955151i	$-0.595692\pi$
$659$	−10019.0	−0.592238	−0.296119	+	0.955151i	$0.595692\pi$
$660$	0	0
$661$	11456.0	0.674110	0.337055	−	0.941485i	$-0.390569\pi$
$661$	11456.0	0.674110	0.337055	+	0.941485i	$0.390569\pi$
$662$	0	0
$663$	255.000	0.0149372
$664$	0	0
$665$	2590.00	0.151031
$666$	0	0
$667$	3318.00	0.192614
$668$	0	0
$669$	−5693.00	−0.329005
$670$	0	0
$671$	13260.0	0.762886
$672$	0	0
$673$	−30912.0	−1.77053	−0.885267	−	0.465082i	$-0.846025\pi$
$673$	−30912.0	−1.77053	−0.885267	+	0.465082i	$0.846025\pi$
$674$	0	0
$675$	1325.00	0.0755545
$676$	0	0
$677$	−27839.0	−1.58041	−0.790207	−	0.612841i	$-0.790027\pi$
$677$	−27839.0	−1.58041	−0.790207	+	0.612841i	$0.790027\pi$
$678$	0	0
$679$	5789.00	0.327189
$680$	0	0
$681$	6381.00	0.359061
$682$	0	0
$683$	−19996.0	−1.12024	−0.560121	−	0.828411i	$-0.689245\pi$
$683$	−19996.0	−1.12024	−0.560121	+	0.828411i	$0.689245\pi$
$684$	0	0
$685$	3880.00	0.216419
$686$	0	0
$687$	3556.00	0.197482
$688$	0	0
$689$	3536.00	0.195517
$690$	0	0
$691$	4356.00	0.239812	0.119906	−	0.992785i	$-0.461741\pi$
$691$	4356.00	0.239812	0.119906	+	0.992785i	$0.461741\pi$
$692$	0	0
$693$	−7098.00	−0.389078
$694$	0	0
$695$	−1630.00	−0.0889632
$696$	0	0
$697$	5220.00	0.283675
$698$	0	0
$699$	−872.000	−0.0471846
$700$	0	0
$701$	−19327.0	−1.04133	−0.520664	−	0.853762i	$-0.674316\pi$
$701$	−19327.0	−1.04133	−0.520664	+	0.853762i	$0.674316\pi$
$702$	0	0
$703$	23532.0	1.26248
$704$	0	0
$705$	965.000	0.0515518
$706$	0	0
$707$	−7798.00	−0.414815
$708$	0	0
$709$	6917.00	0.366394	0.183197	−	0.983076i	$-0.441355\pi$
$709$	6917.00	0.366394	0.183197	+	0.983076i	$0.441355\pi$
$710$	0	0
$711$	14014.0	0.739193
$712$	0	0
$713$	2520.00	0.132363
$714$	0	0
$715$	3315.00	0.173390
$716$	0	0
$717$	−2129.00	−0.110891
$718$	0	0
$719$	27106.0	1.40596	0.702979	−	0.711211i	$-0.251853\pi$
$719$	27106.0	1.40596	0.702979	+	0.711211i	$0.251853\pi$
$720$	0	0
$721$	11501.0	0.594063
$722$	0	0
$723$	−778.000	−0.0400195
$724$	0	0
$725$	5925.00	0.303516
$726$	0	0
$727$	−18024.0	−0.919495	−0.459748	−	0.888050i	$-0.652060\pi$
$727$	−18024.0	−0.919495	−0.459748	+	0.888050i	$0.652060\pi$
$728$	0	0
$729$	−15443.0	−0.784586
$730$	0	0
$731$	330.000	0.0166970
$732$	0	0
$733$	−5347.00	−0.269435	−0.134718	−	0.990884i	$-0.543013\pi$
$733$	−5347.00	−0.269435	−0.134718	+	0.990884i	$0.543013\pi$
$734$	0	0
$735$	245.000	0.0122952
$736$	0	0
$737$	15912.0	0.795286
$738$	0	0
$739$	3833.00	0.190797	0.0953986	−	0.995439i	$-0.469587\pi$
$739$	3833.00	0.190797	0.0953986	+	0.995439i	$0.469587\pi$
$740$	0	0
$741$	−1258.00	−0.0623668
$742$	0	0
$743$	−32120.0	−1.58596	−0.792980	−	0.609247i	$-0.791472\pi$
$743$	−32120.0	−1.58596	−0.792980	+	0.609247i	$0.791472\pi$
$744$	0	0
$745$	16410.0	0.807001
$746$	0	0
$747$	−4264.00	−0.208851
$748$	0	0
$749$	−4970.00	−0.242456
$750$	0	0
$751$	−18859.0	−0.916344	−0.458172	−	0.888863i	$-0.651496\pi$
$751$	−18859.0	−0.916344	−0.458172	+	0.888863i	$0.651496\pi$
$752$	0	0
$753$	4990.00	0.241495
$754$	0	0
$755$	13135.0	0.633154
$756$	0	0
$757$	−26776.0	−1.28559	−0.642794	−	0.766039i	$-0.722225\pi$
$757$	−26776.0	−1.28559	−0.642794	+	0.766039i	$0.722225\pi$
$758$	0	0
$759$	546.000	0.0261114
$760$	0	0
$761$	14722.0	0.701277	0.350639	−	0.936511i	$-0.385965\pi$
$761$	14722.0	0.701277	0.350639	+	0.936511i	$0.385965\pi$
$762$	0	0
$763$	9247.00	0.438747
$764$	0	0
$765$	−1950.00	−0.0921600
$766$	0	0
$767$	7684.00	0.361738
$768$	0	0
$769$	41042.0	1.92459	0.962297	−	0.272002i	$-0.0876857\pi$
$769$	41042.0	1.92459	0.962297	+	0.272002i	$0.0876857\pi$
$770$	0	0
$771$	3630.00	0.169561
$772$	0	0
$773$	−17735.0	−0.825205	−0.412603	−	0.910911i	$-0.635380\pi$
$773$	−17735.0	−0.825205	−0.412603	+	0.910911i	$0.635380\pi$
$774$	0	0
$775$	4500.00	0.208574
$776$	0	0
$777$	2226.00	0.102776
$778$	0	0
$779$	−25752.0	−1.18442
$780$	0	0
$781$	20592.0	0.943457
$782$	0	0
$783$	12561.0	0.573300
$784$	0	0
$785$	1290.00	0.0586523
$786$	0	0
$787$	43979.0	1.99197	0.995986	−	0.0895086i	$-0.0285297\pi$
$787$	43979.0	1.99197	0.995986	+	0.0895086i	$0.0285297\pi$
$788$	0	0
$789$	−290.000	−0.0130853
$790$	0	0
$791$	−2394.00	−0.107612
$792$	0	0
$793$	−5780.00	−0.258832
$794$	0	0
$795$	1040.00	0.0463962
$796$	0	0
$797$	−5301.00	−0.235597	−0.117799	−	0.993038i	$-0.537584\pi$
$797$	−5301.00	−0.235597	−0.117799	+	0.993038i	$0.537584\pi$
$798$	0	0
$799$	−2895.00	−0.128182
$800$	0	0
$801$	14976.0	0.660613
$802$	0	0
$803$	21606.0	0.949513
$804$	0	0
$805$	490.000	0.0214537
$806$	0	0
$807$	−1854.00	−0.0808722
$808$	0	0
$809$	6737.00	0.292782	0.146391	−	0.989227i	$-0.453234\pi$
$809$	6737.00	0.292782	0.146391	+	0.989227i	$0.453234\pi$
$810$	0	0
$811$	−4366.00	−0.189039	−0.0945197	−	0.995523i	$-0.530132\pi$
$811$	−4366.00	−0.189039	−0.0945197	+	0.995523i	$0.530132\pi$
$812$	0	0
$813$	−1836.00	−0.0792021
$814$	0	0
$815$	−670.000	−0.0287964
$816$	0	0
$817$	−1628.00	−0.0697142
$818$	0	0
$819$	3094.00	0.132006
$820$	0	0
$821$	36963.0	1.57128	0.785638	−	0.618686i	$-0.212335\pi$
$821$	36963.0	1.57128	0.785638	+	0.618686i	$0.212335\pi$
$822$	0	0
$823$	30924.0	1.30977	0.654886	−	0.755727i	$-0.272716\pi$
$823$	30924.0	1.30977	0.654886	+	0.755727i	$0.272716\pi$
$824$	0	0
$825$	975.000	0.0411456
$826$	0	0
$827$	−1806.00	−0.0759381	−0.0379690	−	0.999279i	$-0.512089\pi$
$827$	−1806.00	−0.0759381	−0.0379690	+	0.999279i	$0.512089\pi$
$828$	0	0
$829$	−12780.0	−0.535426	−0.267713	−	0.963499i	$-0.586268\pi$
$829$	−12780.0	−0.535426	−0.267713	+	0.963499i	$0.586268\pi$
$830$	0	0
$831$	6154.00	0.256895
$832$	0	0
$833$	−735.000	−0.0305717
$834$	0	0
$835$	−20905.0	−0.866404
$836$	0	0
$837$	9540.00	0.393967
$838$	0	0
$839$	−20706.0	−0.852027	−0.426013	−	0.904717i	$-0.640082\pi$
$839$	−20706.0	−0.852027	−0.426013	+	0.904717i	$0.640082\pi$
$840$	0	0
$841$	31780.0	1.30305
$842$	0	0
$843$	437.000	0.0178542
$844$	0	0
$845$	9540.00	0.388386
$846$	0	0
$847$	−1330.00	−0.0539544
$848$	0	0
$849$	347.000	0.0140271
$850$	0	0
$851$	4452.00	0.179333
$852$	0	0
$853$	20494.0	0.822627	0.411314	−	0.911494i	$-0.365070\pi$
$853$	20494.0	0.822627	0.411314	+	0.911494i	$0.365070\pi$
$854$	0	0
$855$	9620.00	0.384792
$856$	0	0
$857$	36246.0	1.44474	0.722369	−	0.691508i	$-0.243053\pi$
$857$	36246.0	1.44474	0.722369	+	0.691508i	$0.243053\pi$
$858$	0	0
$859$	−39596.0	−1.57276	−0.786378	−	0.617745i	$-0.788046\pi$
$859$	−39596.0	−1.57276	−0.786378	+	0.617745i	$0.788046\pi$
$860$	0	0
$861$	−2436.00	−0.0964212
$862$	0	0
$863$	16224.0	0.639944	0.319972	−	0.947427i	$-0.396327\pi$
$863$	16224.0	0.639944	0.319972	+	0.947427i	$0.396327\pi$
$864$	0	0
$865$	14505.0	0.570156
$866$	0	0
$867$	4688.00	0.183636
$868$	0	0
$869$	21021.0	0.820585
$870$	0	0
$871$	−6936.00	−0.269825
$872$	0	0
$873$	21502.0	0.833600
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−11022.0	−0.424386	−0.212193	−	0.977228i	$-0.568061\pi$
$877$	−11022.0	−0.424386	−0.212193	+	0.977228i	$0.568061\pi$
$878$	0	0
$879$	1281.00	0.0491548
$880$	0	0
$881$	33600.0	1.28492	0.642459	−	0.766320i	$-0.277914\pi$
$881$	33600.0	1.28492	0.642459	+	0.766320i	$0.277914\pi$
$882$	0	0
$883$	43240.0	1.64795	0.823976	−	0.566625i	$-0.191751\pi$
$883$	43240.0	1.64795	0.823976	+	0.566625i	$0.191751\pi$
$884$	0	0
$885$	2260.00	0.0858408
$886$	0	0
$887$	−3024.00	−0.114471	−0.0572356	−	0.998361i	$-0.518229\pi$
$887$	−3024.00	−0.114471	−0.0572356	+	0.998361i	$0.518229\pi$
$888$	0	0
$889$	−9436.00	−0.355988
$890$	0	0
$891$	−25311.0	−0.951684
$892$	0	0
$893$	14282.0	0.535195
$894$	0	0
$895$	−22020.0	−0.822399
$896$	0	0
$897$	−238.000	−0.00885907
$898$	0	0
$899$	42660.0	1.58264
$900$	0	0
$901$	−3120.00	−0.115363
$902$	0	0
$903$	−154.000	−0.00567530
$904$	0	0
$905$	8760.00	0.321759
$906$	0	0
$907$	−40694.0	−1.48977	−0.744885	−	0.667192i	$-0.767496\pi$
$907$	−40694.0	−1.48977	−0.744885	+	0.667192i	$0.767496\pi$
$908$	0	0
$909$	−28964.0	−1.05685
$910$	0	0
$911$	−28728.0	−1.04479	−0.522394	−	0.852704i	$-0.674961\pi$
$911$	−28728.0	−1.04479	−0.522394	+	0.852704i	$0.674961\pi$
$912$	0	0
$913$	−6396.00	−0.231847
$914$	0	0
$915$	−1700.00	−0.0614211
$916$	0	0
$917$	−1666.00	−0.0599958
$918$	0	0
$919$	−9239.00	−0.331628	−0.165814	−	0.986157i	$-0.553025\pi$
$919$	−9239.00	−0.331628	−0.165814	+	0.986157i	$0.553025\pi$
$920$	0	0
$921$	3611.00	0.129193
$922$	0	0
$923$	−8976.00	−0.320096
$924$	0	0
$925$	7950.00	0.282589
$926$	0	0
$927$	42718.0	1.51353
$928$	0	0
$929$	30192.0	1.06627	0.533136	−	0.846029i	$-0.321013\pi$
$929$	30192.0	1.06627	0.533136	+	0.846029i	$0.321013\pi$
$930$	0	0
$931$	3626.00	0.127645
$932$	0	0
$933$	−8206.00	−0.287945
$934$	0	0
$935$	−2925.00	−0.102308
$936$	0	0
$937$	−36515.0	−1.27310	−0.636549	−	0.771236i	$-0.719639\pi$
$937$	−36515.0	−1.27310	−0.636549	+	0.771236i	$0.719639\pi$
$938$	0	0
$939$	−4911.00	−0.170676
$940$	0	0
$941$	46356.0	1.60591	0.802956	−	0.596039i	$-0.203260\pi$
$941$	46356.0	1.60591	0.802956	+	0.596039i	$0.203260\pi$
$942$	0	0
$943$	−4872.00	−0.168244
$944$	0	0
$945$	1855.00	0.0638552
$946$	0	0
$947$	−23800.0	−0.816680	−0.408340	−	0.912830i	$-0.633892\pi$
$947$	−23800.0	−0.816680	−0.408340	+	0.912830i	$0.633892\pi$
$948$	0	0
$949$	−9418.00	−0.322151
$950$	0	0
$951$	606.000	0.0206634
$952$	0	0
$953$	44752.0	1.52115	0.760577	−	0.649248i	$-0.224916\pi$
$953$	44752.0	1.52115	0.760577	+	0.649248i	$0.224916\pi$
$954$	0	0
$955$	−17975.0	−0.609065
$956$	0	0
$957$	9243.00	0.312209
$958$	0	0
$959$	5432.00	0.182908
$960$	0	0
$961$	2609.00	0.0875768
$962$	0	0
$963$	−18460.0	−0.617721
$964$	0	0
$965$	−17240.0	−0.575104
$966$	0	0
$967$	8870.00	0.294974	0.147487	−	0.989064i	$-0.452882\pi$
$967$	8870.00	0.294974	0.147487	+	0.989064i	$0.452882\pi$
$968$	0	0
$969$	1110.00	0.0367991
$970$	0	0
$971$	−35616.0	−1.17711	−0.588554	−	0.808458i	$-0.700303\pi$
$971$	−35616.0	−1.17711	−0.588554	+	0.808458i	$0.700303\pi$
$972$	0	0
$973$	−2282.00	−0.0751876
$974$	0	0
$975$	−425.000	−0.0139599
$976$	0	0
$977$	−21894.0	−0.716941	−0.358470	−	0.933541i	$-0.616702\pi$
$977$	−21894.0	−0.716941	−0.358470	+	0.933541i	$0.616702\pi$
$978$	0	0
$979$	22464.0	0.733353
$980$	0	0
$981$	34346.0	1.11782
$982$	0	0
$983$	−56037.0	−1.81821	−0.909106	−	0.416564i	$-0.863234\pi$
$983$	−56037.0	−1.81821	−0.909106	+	0.416564i	$0.863234\pi$
$984$	0	0
$985$	−17880.0	−0.578380
$986$	0	0
$987$	1351.00	0.0435692
$988$	0	0
$989$	−308.000	−0.00990276
$990$	0	0
$991$	−13168.0	−0.422094	−0.211047	−	0.977476i	$-0.567687\pi$
$991$	−13168.0	−0.422094	−0.211047	+	0.977476i	$0.567687\pi$
$992$	0	0
$993$	−5308.00	−0.169632
$994$	0	0
$995$	14340.0	0.456893
$996$	0	0
$997$	24911.0	0.791313	0.395657	−	0.918399i	$-0.370517\pi$
$997$	24911.0	0.791313	0.395657	+	0.918399i	$0.370517\pi$
$998$	0	0
$999$	16854.0	0.533771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.q.1.1		1
4.3	odd	2		2240.4.a.t.1.1		1
8.3	odd	2		280.4.a.b.1.1	✓	1
8.5	even	2		560.4.a.j.1.1		1
40.3	even	4		1400.4.g.g.449.1		2
40.19	odd	2		1400.4.a.e.1.1		1
40.27	even	4		1400.4.g.g.449.2		2
56.27	even	2		1960.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.b.1.1	✓	1	8.3	odd	2
560.4.a.j.1.1		1	8.5	even	2
1400.4.a.e.1.1		1	40.19	odd	2
1400.4.g.g.449.1		2	40.3	even	4
1400.4.g.g.449.2		2	40.27	even	4
1960.4.a.f.1.1		1	56.27	even	2
2240.4.a.q.1.1		1	1.1	even	1	trivial
2240.4.a.t.1.1		1	4.3	odd	2

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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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