LMFDB - Embedded newform 480.4.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,4,Mod(1,480)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("480.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,-5,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$28.3209168028$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	480.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -5.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} -20.0000 q^{11} -58.0000 q^{13} -15.0000 q^{15} -70.0000 q^{17} -92.0000 q^{19} +36.0000 q^{21} +112.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +66.0000 q^{29} -108.000 q^{31} -60.0000 q^{33} -60.0000 q^{35} -58.0000 q^{37} -174.000 q^{39} +66.0000 q^{41} -388.000 q^{43} -45.0000 q^{45} -408.000 q^{47} -199.000 q^{49} -210.000 q^{51} +474.000 q^{53} +100.000 q^{55} -276.000 q^{57} -540.000 q^{59} +14.0000 q^{61} +108.000 q^{63} +290.000 q^{65} -276.000 q^{67} +336.000 q^{69} -96.0000 q^{71} -790.000 q^{73} +75.0000 q^{75} -240.000 q^{77} +308.000 q^{79} +81.0000 q^{81} -1036.00 q^{83} +350.000 q^{85} +198.000 q^{87} +1210.00 q^{89} -696.000 q^{91} -324.000 q^{93} +460.000 q^{95} +1426.00 q^{97} -180.000 q^{99} +O(q^{100})$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	12.0000	0.647939	0.323970	−	0.946068i	$-0.394982\pi$
$7$	12.0000	0.647939	0.323970	+	0.946068i	$0.394982\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−20.0000	−0.548202	−0.274101	−	0.961701i	$-0.588380\pi$
$11$	−20.0000	−0.548202	−0.274101	+	0.961701i	$0.588380\pi$
$12$	0	0
$13$	−58.0000	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
$13$	−58.0000	−1.23741	−0.618704	+	0.785624i	$0.712342\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−70.0000	−0.998676	−0.499338	−	0.866407i	$-0.666423\pi$
$17$	−70.0000	−0.998676	−0.499338	+	0.866407i	$0.666423\pi$
$18$	0	0
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	0	0
$21$	36.0000	0.374088
$22$	0	0
$23$	112.000	1.01537	0.507687	−	0.861541i	$-0.330501\pi$
$23$	112.000	1.01537	0.507687	+	0.861541i	$0.330501\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	66.0000	0.422617	0.211308	−	0.977419i	$-0.432228\pi$
$29$	66.0000	0.422617	0.211308	+	0.977419i	$0.432228\pi$
$30$	0	0
$31$	−108.000	−0.625722	−0.312861	−	0.949799i	$-0.601287\pi$
$31$	−108.000	−0.625722	−0.312861	+	0.949799i	$0.601287\pi$
$32$	0	0
$33$	−60.0000	−0.316505
$34$	0	0
$35$	−60.0000	−0.289767
$36$	0	0
$37$	−58.0000	−0.257707	−0.128853	−	0.991664i	$-0.541130\pi$
$37$	−58.0000	−0.257707	−0.128853	+	0.991664i	$0.541130\pi$
$38$	0	0
$39$	−174.000	−0.714418
$40$	0	0
$41$	66.0000	0.251402	0.125701	−	0.992068i	$-0.459882\pi$
$41$	66.0000	0.251402	0.125701	+	0.992068i	$0.459882\pi$
$42$	0	0
$43$	−388.000	−1.37603	−0.688017	−	0.725695i	$-0.741518\pi$
$43$	−388.000	−1.37603	−0.688017	+	0.725695i	$0.741518\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−408.000	−1.26623	−0.633116	−	0.774057i	$-0.718224\pi$
$47$	−408.000	−1.26623	−0.633116	+	0.774057i	$0.718224\pi$
$48$	0	0
$49$	−199.000	−0.580175
$50$	0	0
$51$	−210.000	−0.576586
$52$	0	0
$53$	474.000	1.22847	0.614235	−	0.789123i	$-0.289465\pi$
$53$	474.000	1.22847	0.614235	+	0.789123i	$0.289465\pi$
$54$	0	0
$55$	100.000	0.245164
$56$	0	0
$57$	−276.000	−0.641353
$58$	0	0
$59$	−540.000	−1.19156	−0.595780	−	0.803148i	$-0.703157\pi$
$59$	−540.000	−1.19156	−0.595780	+	0.803148i	$0.703157\pi$
$60$	0	0
$61$	14.0000	0.0293855	0.0146928	−	0.999892i	$-0.495323\pi$
$61$	14.0000	0.0293855	0.0146928	+	0.999892i	$0.495323\pi$
$62$	0	0
$63$	108.000	0.215980
$64$	0	0
$65$	290.000	0.553386
$66$	0	0
$67$	−276.000	−0.503265	−0.251633	−	0.967823i	$-0.580967\pi$
$67$	−276.000	−0.503265	−0.251633	+	0.967823i	$0.580967\pi$
$68$	0	0
$69$	336.000	0.586227
$70$	0	0
$71$	−96.0000	−0.160466	−0.0802331	−	0.996776i	$-0.525566\pi$
$71$	−96.0000	−0.160466	−0.0802331	+	0.996776i	$0.525566\pi$
$72$	0	0
$73$	−790.000	−1.26661	−0.633305	−	0.773902i	$-0.718302\pi$
$73$	−790.000	−1.26661	−0.633305	+	0.773902i	$0.718302\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−240.000	−0.355202
$78$	0	0
$79$	308.000	0.438642	0.219321	−	0.975653i	$-0.429616\pi$
$79$	308.000	0.438642	0.219321	+	0.975653i	$0.429616\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1036.00	−1.37007	−0.685035	−	0.728510i	$-0.740213\pi$
$83$	−1036.00	−1.37007	−0.685035	+	0.728510i	$0.740213\pi$
$84$	0	0
$85$	350.000	0.446622
$86$	0	0
$87$	198.000	0.243998
$88$	0	0
$89$	1210.00	1.44112	0.720560	−	0.693392i	$-0.243885\pi$
$89$	1210.00	1.44112	0.720560	+	0.693392i	$0.243885\pi$
$90$	0	0
$91$	−696.000	−0.801765
$92$	0	0
$93$	−324.000	−0.361261
$94$	0	0
$95$	460.000	0.496790
$96$	0	0
$97$	1426.00	1.49266	0.746332	−	0.665574i	$-0.231813\pi$
$97$	1426.00	1.49266	0.746332	+	0.665574i	$0.231813\pi$
$98$	0	0
$99$	−180.000	−0.182734
$100$	0	0
$101$	74.0000	0.0729037	0.0364519	−	0.999335i	$-0.488394\pi$
$101$	74.0000	0.0729037	0.0364519	+	0.999335i	$0.488394\pi$
$102$	0	0
$103$	−1436.00	−1.37372	−0.686861	−	0.726789i	$-0.741012\pi$
$103$	−1436.00	−1.37372	−0.686861	+	0.726789i	$0.741012\pi$
$104$	0	0
$105$	−180.000	−0.167297
$106$	0	0
$107$	84.0000	0.0758933	0.0379467	−	0.999280i	$-0.487918\pi$
$107$	84.0000	0.0758933	0.0379467	+	0.999280i	$0.487918\pi$
$108$	0	0
$109$	−250.000	−0.219685	−0.109842	−	0.993949i	$-0.535035\pi$
$109$	−250.000	−0.219685	−0.109842	+	0.993949i	$0.535035\pi$
$110$	0	0
$111$	−174.000	−0.148787
$112$	0	0
$113$	−654.000	−0.544453	−0.272226	−	0.962233i	$-0.587760\pi$
$113$	−654.000	−0.544453	−0.272226	+	0.962233i	$0.587760\pi$
$114$	0	0
$115$	−560.000	−0.454089
$116$	0	0
$117$	−522.000	−0.412469
$118$	0	0
$119$	−840.000	−0.647081
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	198.000	0.145147
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2572.00	1.79707	0.898536	−	0.438900i	$-0.144632\pi$
$127$	2572.00	1.79707	0.898536	+	0.438900i	$0.144632\pi$
$128$	0	0
$129$	−1164.00	−0.794453
$130$	0	0
$131$	−836.000	−0.557570	−0.278785	−	0.960354i	$-0.589932\pi$
$131$	−836.000	−0.557570	−0.278785	+	0.960354i	$0.589932\pi$
$132$	0	0
$133$	−1104.00	−0.719766
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	1250.00	0.779523	0.389762	−	0.920916i	$-0.372557\pi$
$137$	1250.00	0.779523	0.389762	+	0.920916i	$0.372557\pi$
$138$	0	0
$139$	−2428.00	−1.48158	−0.740792	−	0.671734i	$-0.765550\pi$
$139$	−2428.00	−1.48158	−0.740792	+	0.671734i	$0.765550\pi$
$140$	0	0
$141$	−1224.00	−0.731060
$142$	0	0
$143$	1160.00	0.678350
$144$	0	0
$145$	−330.000	−0.189000
$146$	0	0
$147$	−597.000	−0.334964
$148$	0	0
$149$	1746.00	0.959986	0.479993	−	0.877272i	$-0.340639\pi$
$149$	1746.00	0.959986	0.479993	+	0.877272i	$0.340639\pi$
$150$	0	0
$151$	2092.00	1.12745	0.563724	−	0.825963i	$-0.309368\pi$
$151$	2092.00	1.12745	0.563724	+	0.825963i	$0.309368\pi$
$152$	0	0
$153$	−630.000	−0.332892
$154$	0	0
$155$	540.000	0.279831
$156$	0	0
$157$	−2162.00	−1.09902	−0.549511	−	0.835487i	$-0.685186\pi$
$157$	−2162.00	−1.09902	−0.549511	+	0.835487i	$0.685186\pi$
$158$	0	0
$159$	1422.00	0.709257
$160$	0	0
$161$	1344.00	0.657901
$162$	0	0
$163$	932.000	0.447852	0.223926	−	0.974606i	$-0.428113\pi$
$163$	932.000	0.447852	0.223926	+	0.974606i	$0.428113\pi$
$164$	0	0
$165$	300.000	0.141545
$166$	0	0
$167$	3192.00	1.47907	0.739534	−	0.673119i	$-0.235046\pi$
$167$	3192.00	1.47907	0.739534	+	0.673119i	$0.235046\pi$
$168$	0	0
$169$	1167.00	0.531179
$170$	0	0
$171$	−828.000	−0.370285
$172$	0	0
$173$	2282.00	1.00287	0.501437	−	0.865194i	$-0.332805\pi$
$173$	2282.00	1.00287	0.501437	+	0.865194i	$0.332805\pi$
$174$	0	0
$175$	300.000	0.129588
$176$	0	0
$177$	−1620.00	−0.687947
$178$	0	0
$179$	−2004.00	−0.836793	−0.418397	−	0.908264i	$-0.637408\pi$
$179$	−2004.00	−0.836793	−0.418397	+	0.908264i	$0.637408\pi$
$180$	0	0
$181$	−4226.00	−1.73545	−0.867724	−	0.497046i	$-0.834418\pi$
$181$	−4226.00	−1.73545	−0.867724	+	0.497046i	$0.834418\pi$
$182$	0	0
$183$	42.0000	0.0169657
$184$	0	0
$185$	290.000	0.115250
$186$	0	0
$187$	1400.00	0.547477
$188$	0	0
$189$	324.000	0.124696
$190$	0	0
$191$	2656.00	1.00619	0.503093	−	0.864232i	$-0.332195\pi$
$191$	2656.00	1.00619	0.503093	+	0.864232i	$0.332195\pi$
$192$	0	0
$193$	2162.00	0.806343	0.403171	−	0.915124i	$-0.367908\pi$
$193$	2162.00	0.806343	0.403171	+	0.915124i	$0.367908\pi$
$194$	0	0
$195$	870.000	0.319497
$196$	0	0
$197$	3514.00	1.27087	0.635437	−	0.772153i	$-0.280820\pi$
$197$	3514.00	1.27087	0.635437	+	0.772153i	$0.280820\pi$
$198$	0	0
$199$	−988.000	−0.351947	−0.175974	−	0.984395i	$-0.556307\pi$
$199$	−988.000	−0.351947	−0.175974	+	0.984395i	$0.556307\pi$
$200$	0	0
$201$	−828.000	−0.290560
$202$	0	0
$203$	792.000	0.273830
$204$	0	0
$205$	−330.000	−0.112430
$206$	0	0
$207$	1008.00	0.338458
$208$	0	0
$209$	1840.00	0.608973
$210$	0	0
$211$	3548.00	1.15760	0.578802	−	0.815468i	$-0.303520\pi$
$211$	3548.00	1.15760	0.578802	+	0.815468i	$0.303520\pi$
$212$	0	0
$213$	−288.000	−0.0926452
$214$	0	0
$215$	1940.00	0.615381
$216$	0	0
$217$	−1296.00	−0.405430
$218$	0	0
$219$	−2370.00	−0.731277
$220$	0	0
$221$	4060.00	1.23577
$222$	0	0
$223$	732.000	0.219813	0.109907	−	0.993942i	$-0.464945\pi$
$223$	732.000	0.219813	0.109907	+	0.993942i	$0.464945\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	5492.00	1.60580	0.802901	−	0.596113i	$-0.203289\pi$
$227$	5492.00	1.60580	0.802901	+	0.596113i	$0.203289\pi$
$228$	0	0
$229$	798.000	0.230277	0.115138	−	0.993349i	$-0.463269\pi$
$229$	798.000	0.230277	0.115138	+	0.993349i	$0.463269\pi$
$230$	0	0
$231$	−720.000	−0.205076
$232$	0	0
$233$	−2886.00	−0.811451	−0.405726	−	0.913995i	$-0.632981\pi$
$233$	−2886.00	−0.811451	−0.405726	+	0.913995i	$0.632981\pi$
$234$	0	0
$235$	2040.00	0.566276
$236$	0	0
$237$	924.000	0.253250
$238$	0	0
$239$	4096.00	1.10857	0.554285	−	0.832327i	$-0.312992\pi$
$239$	4096.00	1.10857	0.554285	+	0.832327i	$0.312992\pi$
$240$	0	0
$241$	2354.00	0.629189	0.314594	−	0.949226i	$-0.398132\pi$
$241$	2354.00	0.629189	0.314594	+	0.949226i	$0.398132\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	995.000	0.259462
$246$	0	0
$247$	5336.00	1.37458
$248$	0	0
$249$	−3108.00	−0.791010
$250$	0	0
$251$	−2916.00	−0.733292	−0.366646	−	0.930361i	$-0.619494\pi$
$251$	−2916.00	−0.733292	−0.366646	+	0.930361i	$0.619494\pi$
$252$	0	0
$253$	−2240.00	−0.556631
$254$	0	0
$255$	1050.00	0.257857
$256$	0	0
$257$	882.000	0.214076	0.107038	−	0.994255i	$-0.465863\pi$
$257$	882.000	0.214076	0.107038	+	0.994255i	$0.465863\pi$
$258$	0	0
$259$	−696.000	−0.166978
$260$	0	0
$261$	594.000	0.140872
$262$	0	0
$263$	−4456.00	−1.04475	−0.522374	−	0.852716i	$-0.674954\pi$
$263$	−4456.00	−1.04475	−0.522374	+	0.852716i	$0.674954\pi$
$264$	0	0
$265$	−2370.00	−0.549388
$266$	0	0
$267$	3630.00	0.832031
$268$	0	0
$269$	−1486.00	−0.336814	−0.168407	−	0.985718i	$-0.553862\pi$
$269$	−1486.00	−0.336814	−0.168407	+	0.985718i	$0.553862\pi$
$270$	0	0
$271$	−4676.00	−1.04814	−0.524072	−	0.851674i	$-0.675588\pi$
$271$	−4676.00	−1.04814	−0.524072	+	0.851674i	$0.675588\pi$
$272$	0	0
$273$	−2088.00	−0.462899
$274$	0	0
$275$	−500.000	−0.109640
$276$	0	0
$277$	−2898.00	−0.628606	−0.314303	−	0.949323i	$-0.601771\pi$
$277$	−2898.00	−0.628606	−0.314303	+	0.949323i	$0.601771\pi$
$278$	0	0
$279$	−972.000	−0.208574
$280$	0	0
$281$	5194.00	1.10266	0.551331	−	0.834287i	$-0.314120\pi$
$281$	5194.00	1.10266	0.551331	+	0.834287i	$0.314120\pi$
$282$	0	0
$283$	−5420.00	−1.13846	−0.569232	−	0.822177i	$-0.692760\pi$
$283$	−5420.00	−1.13846	−0.569232	+	0.822177i	$0.692760\pi$
$284$	0	0
$285$	1380.00	0.286822
$286$	0	0
$287$	792.000	0.162893
$288$	0	0
$289$	−13.0000	−0.00264604
$290$	0	0
$291$	4278.00	0.861790
$292$	0	0
$293$	9130.00	1.82041	0.910205	−	0.414157i	$-0.135924\pi$
$293$	9130.00	1.82041	0.910205	+	0.414157i	$0.135924\pi$
$294$	0	0
$295$	2700.00	0.532882
$296$	0	0
$297$	−540.000	−0.105502
$298$	0	0
$299$	−6496.00	−1.25643
$300$	0	0
$301$	−4656.00	−0.891586
$302$	0	0
$303$	222.000	0.0420910
$304$	0	0
$305$	−70.0000	−0.0131416
$306$	0	0
$307$	6044.00	1.12361	0.561807	−	0.827269i	$-0.310106\pi$
$307$	6044.00	1.12361	0.561807	+	0.827269i	$0.310106\pi$
$308$	0	0
$309$	−4308.00	−0.793118
$310$	0	0
$311$	−6120.00	−1.11586	−0.557931	−	0.829887i	$-0.688405\pi$
$311$	−6120.00	−1.11586	−0.557931	+	0.829887i	$0.688405\pi$
$312$	0	0
$313$	−614.000	−0.110880	−0.0554398	−	0.998462i	$-0.517656\pi$
$313$	−614.000	−0.110880	−0.0554398	+	0.998462i	$0.517656\pi$
$314$	0	0
$315$	−540.000	−0.0965891
$316$	0	0
$317$	786.000	0.139262	0.0696312	−	0.997573i	$-0.477818\pi$
$317$	786.000	0.139262	0.0696312	+	0.997573i	$0.477818\pi$
$318$	0	0
$319$	−1320.00	−0.231680
$320$	0	0
$321$	252.000	0.0438170
$322$	0	0
$323$	6440.00	1.10938
$324$	0	0
$325$	−1450.00	−0.247482
$326$	0	0
$327$	−750.000	−0.126835
$328$	0	0
$329$	−4896.00	−0.820441
$330$	0	0
$331$	468.000	0.0777148	0.0388574	−	0.999245i	$-0.487628\pi$
$331$	468.000	0.0777148	0.0388574	+	0.999245i	$0.487628\pi$
$332$	0	0
$333$	−522.000	−0.0859022
$334$	0	0
$335$	1380.00	0.225067
$336$	0	0
$337$	1538.00	0.248606	0.124303	−	0.992244i	$-0.460331\pi$
$337$	1538.00	0.248606	0.124303	+	0.992244i	$0.460331\pi$
$338$	0	0
$339$	−1962.00	−0.314340
$340$	0	0
$341$	2160.00	0.343022
$342$	0	0
$343$	−6504.00	−1.02386
$344$	0	0
$345$	−1680.00	−0.262169
$346$	0	0
$347$	8396.00	1.29891	0.649454	−	0.760401i	$-0.274998\pi$
$347$	8396.00	1.29891	0.649454	+	0.760401i	$0.274998\pi$
$348$	0	0
$349$	−11090.0	−1.70096	−0.850479	−	0.526010i	$-0.823688\pi$
$349$	−11090.0	−1.70096	−0.850479	+	0.526010i	$0.823688\pi$
$350$	0	0
$351$	−1566.00	−0.238139
$352$	0	0
$353$	3298.00	0.497266	0.248633	−	0.968598i	$-0.420019\pi$
$353$	3298.00	0.497266	0.248633	+	0.968598i	$0.420019\pi$
$354$	0	0
$355$	480.000	0.0717627
$356$	0	0
$357$	−2520.00	−0.373593
$358$	0	0
$359$	−10720.0	−1.57599	−0.787994	−	0.615682i	$-0.788880\pi$
$359$	−10720.0	−1.57599	−0.787994	+	0.615682i	$0.788880\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	−2793.00	−0.403842
$364$	0	0
$365$	3950.00	0.566445
$366$	0	0
$367$	4116.00	0.585432	0.292716	−	0.956199i	$-0.405441\pi$
$367$	4116.00	0.585432	0.292716	+	0.956199i	$0.405441\pi$
$368$	0	0
$369$	594.000	0.0838006
$370$	0	0
$371$	5688.00	0.795974
$372$	0	0
$373$	3590.00	0.498346	0.249173	−	0.968459i	$-0.419841\pi$
$373$	3590.00	0.498346	0.249173	+	0.968459i	$0.419841\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−3828.00	−0.522950
$378$	0	0
$379$	−12452.0	−1.68764	−0.843821	−	0.536625i	$-0.819699\pi$
$379$	−12452.0	−1.68764	−0.843821	+	0.536625i	$0.819699\pi$
$380$	0	0
$381$	7716.00	1.03754
$382$	0	0
$383$	−12416.0	−1.65647	−0.828235	−	0.560381i	$-0.810655\pi$
$383$	−12416.0	−1.65647	−0.828235	+	0.560381i	$0.810655\pi$
$384$	0	0
$385$	1200.00	0.158851
$386$	0	0
$387$	−3492.00	−0.458678
$388$	0	0
$389$	2370.00	0.308904	0.154452	−	0.988000i	$-0.450639\pi$
$389$	2370.00	0.308904	0.154452	+	0.988000i	$0.450639\pi$
$390$	0	0
$391$	−7840.00	−1.01403
$392$	0	0
$393$	−2508.00	−0.321913
$394$	0	0
$395$	−1540.00	−0.196167
$396$	0	0
$397$	9486.00	1.19922	0.599608	−	0.800294i	$-0.295323\pi$
$397$	9486.00	1.19922	0.599608	+	0.800294i	$0.295323\pi$
$398$	0	0
$399$	−3312.00	−0.415557
$400$	0	0
$401$	−10630.0	−1.32378	−0.661891	−	0.749600i	$-0.730246\pi$
$401$	−10630.0	−1.32378	−0.661891	+	0.749600i	$0.730246\pi$
$402$	0	0
$403$	6264.00	0.774273
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	1160.00	0.141275
$408$	0	0
$409$	12890.0	1.55836	0.779180	−	0.626800i	$-0.215636\pi$
$409$	12890.0	1.55836	0.779180	+	0.626800i	$0.215636\pi$
$410$	0	0
$411$	3750.00	0.450058
$412$	0	0
$413$	−6480.00	−0.772058
$414$	0	0
$415$	5180.00	0.612714
$416$	0	0
$417$	−7284.00	−0.855393
$418$	0	0
$419$	−11196.0	−1.30539	−0.652697	−	0.757619i	$-0.726363\pi$
$419$	−11196.0	−1.30539	−0.652697	+	0.757619i	$0.726363\pi$
$420$	0	0
$421$	−8594.00	−0.994883	−0.497442	−	0.867497i	$-0.665727\pi$
$421$	−8594.00	−0.994883	−0.497442	+	0.867497i	$0.665727\pi$
$422$	0	0
$423$	−3672.00	−0.422077
$424$	0	0
$425$	−1750.00	−0.199735
$426$	0	0
$427$	168.000	0.0190400
$428$	0	0
$429$	3480.00	0.391646
$430$	0	0
$431$	3544.00	0.396075	0.198038	−	0.980194i	$-0.436543\pi$
$431$	3544.00	0.396075	0.198038	+	0.980194i	$0.436543\pi$
$432$	0	0
$433$	5810.00	0.644829	0.322414	−	0.946599i	$-0.395506\pi$
$433$	5810.00	0.644829	0.322414	+	0.946599i	$0.395506\pi$
$434$	0	0
$435$	−990.000	−0.109119
$436$	0	0
$437$	−10304.0	−1.12793
$438$	0	0
$439$	−9628.00	−1.04674	−0.523371	−	0.852105i	$-0.675326\pi$
$439$	−9628.00	−1.04674	−0.523371	+	0.852105i	$0.675326\pi$
$440$	0	0
$441$	−1791.00	−0.193392
$442$	0	0
$443$	−6100.00	−0.654221	−0.327110	−	0.944986i	$-0.606075\pi$
$443$	−6100.00	−0.654221	−0.327110	+	0.944986i	$0.606075\pi$
$444$	0	0
$445$	−6050.00	−0.644489
$446$	0	0
$447$	5238.00	0.554248
$448$	0	0
$449$	−7750.00	−0.814577	−0.407289	−	0.913300i	$-0.633526\pi$
$449$	−7750.00	−0.814577	−0.407289	+	0.913300i	$0.633526\pi$
$450$	0	0
$451$	−1320.00	−0.137819
$452$	0	0
$453$	6276.00	0.650932
$454$	0	0
$455$	3480.00	0.358560
$456$	0	0
$457$	18314.0	1.87460	0.937301	−	0.348522i	$-0.113316\pi$
$457$	18314.0	1.87460	0.937301	+	0.348522i	$0.113316\pi$
$458$	0	0
$459$	−1890.00	−0.192195
$460$	0	0
$461$	6122.00	0.618503	0.309252	−	0.950980i	$-0.399921\pi$
$461$	6122.00	0.618503	0.309252	+	0.950980i	$0.399921\pi$
$462$	0	0
$463$	−10420.0	−1.04591	−0.522957	−	0.852359i	$-0.675171\pi$
$463$	−10420.0	−1.04591	−0.522957	+	0.852359i	$0.675171\pi$
$464$	0	0
$465$	1620.00	0.161561
$466$	0	0
$467$	12612.0	1.24971	0.624854	−	0.780742i	$-0.285158\pi$
$467$	12612.0	1.24971	0.624854	+	0.780742i	$0.285158\pi$
$468$	0	0
$469$	−3312.00	−0.326085
$470$	0	0
$471$	−6486.00	−0.634520
$472$	0	0
$473$	7760.00	0.754345
$474$	0	0
$475$	−2300.00	−0.222171
$476$	0	0
$477$	4266.00	0.409490
$478$	0	0
$479$	−3352.00	−0.319743	−0.159871	−	0.987138i	$-0.551108\pi$
$479$	−3352.00	−0.319743	−0.159871	+	0.987138i	$0.551108\pi$
$480$	0	0
$481$	3364.00	0.318888
$482$	0	0
$483$	4032.00	0.379839
$484$	0	0
$485$	−7130.00	−0.667539
$486$	0	0
$487$	−17108.0	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$487$	−17108.0	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$488$	0	0
$489$	2796.00	0.258567
$490$	0	0
$491$	−11388.0	−1.04671	−0.523354	−	0.852116i	$-0.675319\pi$
$491$	−11388.0	−1.04671	−0.523354	+	0.852116i	$0.675319\pi$
$492$	0	0
$493$	−4620.00	−0.422057
$494$	0	0
$495$	900.000	0.0817212
$496$	0	0
$497$	−1152.00	−0.103972
$498$	0	0
$499$	−8996.00	−0.807047	−0.403523	−	0.914969i	$-0.632215\pi$
$499$	−8996.00	−0.807047	−0.403523	+	0.914969i	$0.632215\pi$
$500$	0	0
$501$	9576.00	0.853940
$502$	0	0
$503$	−19504.0	−1.72891	−0.864454	−	0.502713i	$-0.832335\pi$
$503$	−19504.0	−1.72891	−0.864454	+	0.502713i	$0.832335\pi$
$504$	0	0
$505$	−370.000	−0.0326035
$506$	0	0
$507$	3501.00	0.306676
$508$	0	0
$509$	8306.00	0.723295	0.361647	−	0.932315i	$-0.382214\pi$
$509$	8306.00	0.723295	0.361647	+	0.932315i	$0.382214\pi$
$510$	0	0
$511$	−9480.00	−0.820686
$512$	0	0
$513$	−2484.00	−0.213784
$514$	0	0
$515$	7180.00	0.614347
$516$	0	0
$517$	8160.00	0.694152
$518$	0	0
$519$	6846.00	0.579010
$520$	0	0
$521$	14850.0	1.24873	0.624367	−	0.781131i	$-0.285357\pi$
$521$	14850.0	1.24873	0.624367	+	0.781131i	$0.285357\pi$
$522$	0	0
$523$	20044.0	1.67584	0.837919	−	0.545795i	$-0.183772\pi$
$523$	20044.0	1.67584	0.837919	+	0.545795i	$0.183772\pi$
$524$	0	0
$525$	900.000	0.0748176
$526$	0	0
$527$	7560.00	0.624893
$528$	0	0
$529$	377.000	0.0309855
$530$	0	0
$531$	−4860.00	−0.397187
$532$	0	0
$533$	−3828.00	−0.311086
$534$	0	0
$535$	−420.000	−0.0339405
$536$	0	0
$537$	−6012.00	−0.483123
$538$	0	0
$539$	3980.00	0.318053
$540$	0	0
$541$	−21930.0	−1.74278	−0.871390	−	0.490590i	$-0.836781\pi$
$541$	−21930.0	−1.74278	−0.871390	+	0.490590i	$0.836781\pi$
$542$	0	0
$543$	−12678.0	−1.00196
$544$	0	0
$545$	1250.00	0.0982461
$546$	0	0
$547$	19988.0	1.56239	0.781193	−	0.624290i	$-0.214611\pi$
$547$	19988.0	1.56239	0.781193	+	0.624290i	$0.214611\pi$
$548$	0	0
$549$	126.000	0.00979517
$550$	0	0
$551$	−6072.00	−0.469466
$552$	0	0
$553$	3696.00	0.284213
$554$	0	0
$555$	870.000	0.0665395
$556$	0	0
$557$	−11718.0	−0.891396	−0.445698	−	0.895183i	$-0.647044\pi$
$557$	−11718.0	−0.891396	−0.445698	+	0.895183i	$0.647044\pi$
$558$	0	0
$559$	22504.0	1.70272
$560$	0	0
$561$	4200.00	0.316086
$562$	0	0
$563$	12756.0	0.954887	0.477443	−	0.878662i	$-0.341564\pi$
$563$	12756.0	0.954887	0.477443	+	0.878662i	$0.341564\pi$
$564$	0	0
$565$	3270.00	0.243487
$566$	0	0
$567$	972.000	0.0719932
$568$	0	0
$569$	−15790.0	−1.16336	−0.581679	−	0.813418i	$-0.697604\pi$
$569$	−15790.0	−1.16336	−0.581679	+	0.813418i	$0.697604\pi$
$570$	0	0
$571$	2500.00	0.183225	0.0916127	−	0.995795i	$-0.470798\pi$
$571$	2500.00	0.183225	0.0916127	+	0.995795i	$0.470798\pi$
$572$	0	0
$573$	7968.00	0.580921
$574$	0	0
$575$	2800.00	0.203075
$576$	0	0
$577$	13778.0	0.994083	0.497041	−	0.867727i	$-0.334420\pi$
$577$	13778.0	0.994083	0.497041	+	0.867727i	$0.334420\pi$
$578$	0	0
$579$	6486.00	0.465542
$580$	0	0
$581$	−12432.0	−0.887722
$582$	0	0
$583$	−9480.00	−0.673450
$584$	0	0
$585$	2610.00	0.184462
$586$	0	0
$587$	−11724.0	−0.824363	−0.412182	−	0.911102i	$-0.635233\pi$
$587$	−11724.0	−0.824363	−0.412182	+	0.911102i	$0.635233\pi$
$588$	0	0
$589$	9936.00	0.695086
$590$	0	0
$591$	10542.0	0.733739
$592$	0	0
$593$	−27054.0	−1.87348	−0.936741	−	0.350024i	$-0.886174\pi$
$593$	−27054.0	−1.87348	−0.936741	+	0.350024i	$0.886174\pi$
$594$	0	0
$595$	4200.00	0.289384
$596$	0	0
$597$	−2964.00	−0.203197
$598$	0	0
$599$	15328.0	1.04555	0.522776	−	0.852470i	$-0.324897\pi$
$599$	15328.0	1.04555	0.522776	+	0.852470i	$0.324897\pi$
$600$	0	0
$601$	−25286.0	−1.71620	−0.858101	−	0.513481i	$-0.828356\pi$
$601$	−25286.0	−1.71620	−0.858101	+	0.513481i	$0.828356\pi$
$602$	0	0
$603$	−2484.00	−0.167755
$604$	0	0
$605$	4655.00	0.312814
$606$	0	0
$607$	22060.0	1.47510	0.737552	−	0.675291i	$-0.235982\pi$
$607$	22060.0	1.47510	0.737552	+	0.675291i	$0.235982\pi$
$608$	0	0
$609$	2376.00	0.158096
$610$	0	0
$611$	23664.0	1.56685
$612$	0	0
$613$	−8810.00	−0.580477	−0.290239	−	0.956954i	$-0.593735\pi$
$613$	−8810.00	−0.580477	−0.290239	+	0.956954i	$0.593735\pi$
$614$	0	0
$615$	−990.000	−0.0649116
$616$	0	0
$617$	−11766.0	−0.767717	−0.383858	−	0.923392i	$-0.625405\pi$
$617$	−11766.0	−0.767717	−0.383858	+	0.923392i	$0.625405\pi$
$618$	0	0
$619$	−28316.0	−1.83864	−0.919318	−	0.393515i	$-0.871259\pi$
$619$	−28316.0	−1.83864	−0.919318	+	0.393515i	$0.871259\pi$
$620$	0	0
$621$	3024.00	0.195409
$622$	0	0
$623$	14520.0	0.933758
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	5520.00	0.351591
$628$	0	0
$629$	4060.00	0.257365
$630$	0	0
$631$	1388.00	0.0875680	0.0437840	−	0.999041i	$-0.486059\pi$
$631$	1388.00	0.0875680	0.0437840	+	0.999041i	$0.486059\pi$
$632$	0	0
$633$	10644.0	0.668343
$634$	0	0
$635$	−12860.0	−0.803675
$636$	0	0
$637$	11542.0	0.717913
$638$	0	0
$639$	−864.000	−0.0534888
$640$	0	0
$641$	−15974.0	−0.984298	−0.492149	−	0.870511i	$-0.663788\pi$
$641$	−15974.0	−0.984298	−0.492149	+	0.870511i	$0.663788\pi$
$642$	0	0
$643$	13044.0	0.800008	0.400004	−	0.916513i	$-0.369009\pi$
$643$	13044.0	0.800008	0.400004	+	0.916513i	$0.369009\pi$
$644$	0	0
$645$	5820.00	0.355290
$646$	0	0
$647$	−18016.0	−1.09472	−0.547359	−	0.836898i	$-0.684367\pi$
$647$	−18016.0	−1.09472	−0.547359	+	0.836898i	$0.684367\pi$
$648$	0	0
$649$	10800.0	0.653216
$650$	0	0
$651$	−3888.00	−0.234075
$652$	0	0
$653$	−17830.0	−1.06852	−0.534259	−	0.845321i	$-0.679409\pi$
$653$	−17830.0	−1.06852	−0.534259	+	0.845321i	$0.679409\pi$
$654$	0	0
$655$	4180.00	0.249353
$656$	0	0
$657$	−7110.00	−0.422203
$658$	0	0
$659$	−20740.0	−1.22597	−0.612986	−	0.790094i	$-0.710032\pi$
$659$	−20740.0	−1.22597	−0.612986	+	0.790094i	$0.710032\pi$
$660$	0	0
$661$	12070.0	0.710240	0.355120	−	0.934821i	$-0.384440\pi$
$661$	12070.0	0.710240	0.355120	+	0.934821i	$0.384440\pi$
$662$	0	0
$663$	12180.0	0.713472
$664$	0	0
$665$	5520.00	0.321889
$666$	0	0
$667$	7392.00	0.429115
$668$	0	0
$669$	2196.00	0.126909
$670$	0	0
$671$	−280.000	−0.0161092
$672$	0	0
$673$	20514.0	1.17497	0.587486	−	0.809234i	$-0.300118\pi$
$673$	20514.0	1.17497	0.587486	+	0.809234i	$0.300118\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−13326.0	−0.756514	−0.378257	−	0.925701i	$-0.623476\pi$
$677$	−13326.0	−0.756514	−0.378257	+	0.925701i	$0.623476\pi$
$678$	0	0
$679$	17112.0	0.967155
$680$	0	0
$681$	16476.0	0.927110
$682$	0	0
$683$	2988.00	0.167398	0.0836989	−	0.996491i	$-0.473327\pi$
$683$	2988.00	0.167398	0.0836989	+	0.996491i	$0.473327\pi$
$684$	0	0
$685$	−6250.00	−0.348613
$686$	0	0
$687$	2394.00	0.132950
$688$	0	0
$689$	−27492.0	−1.52012
$690$	0	0
$691$	16628.0	0.915425	0.457713	−	0.889100i	$-0.348669\pi$
$691$	16628.0	0.915425	0.457713	+	0.889100i	$0.348669\pi$
$692$	0	0
$693$	−2160.00	−0.118401
$694$	0	0
$695$	12140.0	0.662585
$696$	0	0
$697$	−4620.00	−0.251069
$698$	0	0
$699$	−8658.00	−0.468492
$700$	0	0
$701$	28082.0	1.51304	0.756521	−	0.653969i	$-0.226897\pi$
$701$	28082.0	1.51304	0.756521	+	0.653969i	$0.226897\pi$
$702$	0	0
$703$	5336.00	0.286275
$704$	0	0
$705$	6120.00	0.326940
$706$	0	0
$707$	888.000	0.0472372
$708$	0	0
$709$	20158.0	1.06777	0.533885	−	0.845557i	$-0.320731\pi$
$709$	20158.0	1.06777	0.533885	+	0.845557i	$0.320731\pi$
$710$	0	0
$711$	2772.00	0.146214
$712$	0	0
$713$	−12096.0	−0.635342
$714$	0	0
$715$	−5800.00	−0.303367
$716$	0	0
$717$	12288.0	0.640033
$718$	0	0
$719$	30536.0	1.58387	0.791934	−	0.610607i	$-0.209075\pi$
$719$	30536.0	1.58387	0.791934	+	0.610607i	$0.209075\pi$
$720$	0	0
$721$	−17232.0	−0.890088
$722$	0	0
$723$	7062.00	0.363262
$724$	0	0
$725$	1650.00	0.0845234
$726$	0	0
$727$	−7204.00	−0.367512	−0.183756	−	0.982972i	$-0.558826\pi$
$727$	−7204.00	−0.367512	−0.183756	+	0.982972i	$0.558826\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	27160.0	1.37421
$732$	0	0
$733$	5398.00	0.272005	0.136003	−	0.990708i	$-0.456574\pi$
$733$	5398.00	0.272005	0.136003	+	0.990708i	$0.456574\pi$
$734$	0	0
$735$	2985.00	0.149801
$736$	0	0
$737$	5520.00	0.275891
$738$	0	0
$739$	5660.00	0.281741	0.140870	−	0.990028i	$-0.455010\pi$
$739$	5660.00	0.281741	0.140870	+	0.990028i	$0.455010\pi$
$740$	0	0
$741$	16008.0	0.793615
$742$	0	0
$743$	3624.00	0.178939	0.0894695	−	0.995990i	$-0.471483\pi$
$743$	3624.00	0.178939	0.0894695	+	0.995990i	$0.471483\pi$
$744$	0	0
$745$	−8730.00	−0.429319
$746$	0	0
$747$	−9324.00	−0.456690
$748$	0	0
$749$	1008.00	0.0491743
$750$	0	0
$751$	12532.0	0.608920	0.304460	−	0.952525i	$-0.401524\pi$
$751$	12532.0	0.608920	0.304460	+	0.952525i	$0.401524\pi$
$752$	0	0
$753$	−8748.00	−0.423366
$754$	0	0
$755$	−10460.0	−0.504210
$756$	0	0
$757$	−9026.00	−0.433363	−0.216681	−	0.976242i	$-0.569523\pi$
$757$	−9026.00	−0.433363	−0.216681	+	0.976242i	$0.569523\pi$
$758$	0	0
$759$	−6720.00	−0.321371
$760$	0	0
$761$	4674.00	0.222644	0.111322	−	0.993784i	$-0.464491\pi$
$761$	4674.00	0.222644	0.111322	+	0.993784i	$0.464491\pi$
$762$	0	0
$763$	−3000.00	−0.142342
$764$	0	0
$765$	3150.00	0.148874
$766$	0	0
$767$	31320.0	1.47445
$768$	0	0
$769$	38386.0	1.80004	0.900022	−	0.435843i	$-0.143550\pi$
$769$	38386.0	1.80004	0.900022	+	0.435843i	$0.143550\pi$
$770$	0	0
$771$	2646.00	0.123597
$772$	0	0
$773$	−16774.0	−0.780490	−0.390245	−	0.920711i	$-0.627610\pi$
$773$	−16774.0	−0.780490	−0.390245	+	0.920711i	$0.627610\pi$
$774$	0	0
$775$	−2700.00	−0.125144
$776$	0	0
$777$	−2088.00	−0.0964049
$778$	0	0
$779$	−6072.00	−0.279271
$780$	0	0
$781$	1920.00	0.0879680
$782$	0	0
$783$	1782.00	0.0813327
$784$	0	0
$785$	10810.0	0.491497
$786$	0	0
$787$	−27116.0	−1.22818	−0.614092	−	0.789234i	$-0.710478\pi$
$787$	−27116.0	−1.22818	−0.614092	+	0.789234i	$0.710478\pi$
$788$	0	0
$789$	−13368.0	−0.603186
$790$	0	0
$791$	−7848.00	−0.352772
$792$	0	0
$793$	−812.000	−0.0363619
$794$	0	0
$795$	−7110.00	−0.317190
$796$	0	0
$797$	−17494.0	−0.777502	−0.388751	−	0.921343i	$-0.627093\pi$
$797$	−17494.0	−0.777502	−0.388751	+	0.921343i	$0.627093\pi$
$798$	0	0
$799$	28560.0	1.26456
$800$	0	0
$801$	10890.0	0.480374
$802$	0	0
$803$	15800.0	0.694359
$804$	0	0
$805$	−6720.00	−0.294222
$806$	0	0
$807$	−4458.00	−0.194460
$808$	0	0
$809$	9298.00	0.404079	0.202040	−	0.979377i	$-0.435243\pi$
$809$	9298.00	0.404079	0.202040	+	0.979377i	$0.435243\pi$
$810$	0	0
$811$	−21252.0	−0.920171	−0.460085	−	0.887875i	$-0.652181\pi$
$811$	−21252.0	−0.920171	−0.460085	+	0.887875i	$0.652181\pi$
$812$	0	0
$813$	−14028.0	−0.605146
$814$	0	0
$815$	−4660.00	−0.200285
$816$	0	0
$817$	35696.0	1.52857
$818$	0	0
$819$	−6264.00	−0.267255
$820$	0	0
$821$	1578.00	0.0670799	0.0335399	−	0.999437i	$-0.489322\pi$
$821$	1578.00	0.0670799	0.0335399	+	0.999437i	$0.489322\pi$
$822$	0	0
$823$	−9652.00	−0.408806	−0.204403	−	0.978887i	$-0.565525\pi$
$823$	−9652.00	−0.408806	−0.204403	+	0.978887i	$0.565525\pi$
$824$	0	0
$825$	−1500.00	−0.0633010
$826$	0	0
$827$	−15612.0	−0.656448	−0.328224	−	0.944600i	$-0.606450\pi$
$827$	−15612.0	−0.656448	−0.328224	+	0.944600i	$0.606450\pi$
$828$	0	0
$829$	−13194.0	−0.552770	−0.276385	−	0.961047i	$-0.589137\pi$
$829$	−13194.0	−0.552770	−0.276385	+	0.961047i	$0.589137\pi$
$830$	0	0
$831$	−8694.00	−0.362926
$832$	0	0
$833$	13930.0	0.579407
$834$	0	0
$835$	−15960.0	−0.661459
$836$	0	0
$837$	−2916.00	−0.120420
$838$	0	0
$839$	22512.0	0.926342	0.463171	−	0.886269i	$-0.346712\pi$
$839$	22512.0	0.926342	0.463171	+	0.886269i	$0.346712\pi$
$840$	0	0
$841$	−20033.0	−0.821395
$842$	0	0
$843$	15582.0	0.636622
$844$	0	0
$845$	−5835.00	−0.237550
$846$	0	0
$847$	−11172.0	−0.453217
$848$	0	0
$849$	−16260.0	−0.657293
$850$	0	0
$851$	−6496.00	−0.261669
$852$	0	0
$853$	−3114.00	−0.124996	−0.0624978	−	0.998045i	$-0.519907\pi$
$853$	−3114.00	−0.124996	−0.0624978	+	0.998045i	$0.519907\pi$
$854$	0	0
$855$	4140.00	0.165597
$856$	0	0
$857$	−36254.0	−1.44506	−0.722528	−	0.691342i	$-0.757020\pi$
$857$	−36254.0	−1.44506	−0.722528	+	0.691342i	$0.757020\pi$
$858$	0	0
$859$	27644.0	1.09802	0.549011	−	0.835815i	$-0.315004\pi$
$859$	27644.0	1.09802	0.549011	+	0.835815i	$0.315004\pi$
$860$	0	0
$861$	2376.00	0.0940463
$862$	0	0
$863$	8608.00	0.339536	0.169768	−	0.985484i	$-0.445698\pi$
$863$	8608.00	0.339536	0.169768	+	0.985484i	$0.445698\pi$
$864$	0	0
$865$	−11410.0	−0.448499
$866$	0	0
$867$	−39.0000	−0.00152769
$868$	0	0
$869$	−6160.00	−0.240465
$870$	0	0
$871$	16008.0	0.622744
$872$	0	0
$873$	12834.0	0.497555
$874$	0	0
$875$	−1500.00	−0.0579534
$876$	0	0
$877$	37294.0	1.43595	0.717975	−	0.696068i	$-0.245069\pi$
$877$	37294.0	1.43595	0.717975	+	0.696068i	$0.245069\pi$
$878$	0	0
$879$	27390.0	1.05101
$880$	0	0
$881$	−5742.00	−0.219583	−0.109792	−	0.993955i	$-0.535018\pi$
$881$	−5742.00	−0.219583	−0.109792	+	0.993955i	$0.535018\pi$
$882$	0	0
$883$	−46028.0	−1.75421	−0.877104	−	0.480301i	$-0.840528\pi$
$883$	−46028.0	−1.75421	−0.877104	+	0.480301i	$0.840528\pi$
$884$	0	0
$885$	8100.00	0.307659
$886$	0	0
$887$	−10136.0	−0.383691	−0.191845	−	0.981425i	$-0.561447\pi$
$887$	−10136.0	−0.383691	−0.191845	+	0.981425i	$0.561447\pi$
$888$	0	0
$889$	30864.0	1.16439
$890$	0	0
$891$	−1620.00	−0.0609114
$892$	0	0
$893$	37536.0	1.40660
$894$	0	0
$895$	10020.0	0.374225
$896$	0	0
$897$	−19488.0	−0.725402
$898$	0	0
$899$	−7128.00	−0.264441
$900$	0	0
$901$	−33180.0	−1.22684
$902$	0	0
$903$	−13968.0	−0.514757
$904$	0	0
$905$	21130.0	0.776116
$906$	0	0
$907$	−30900.0	−1.13122	−0.565611	−	0.824672i	$-0.691359\pi$
$907$	−30900.0	−1.13122	−0.565611	+	0.824672i	$0.691359\pi$
$908$	0	0
$909$	666.000	0.0243012
$910$	0	0
$911$	−45152.0	−1.64210	−0.821050	−	0.570857i	$-0.806611\pi$
$911$	−45152.0	−1.64210	−0.821050	+	0.570857i	$0.806611\pi$
$912$	0	0
$913$	20720.0	0.751075
$914$	0	0
$915$	−210.000	−0.00758731
$916$	0	0
$917$	−10032.0	−0.361271
$918$	0	0
$919$	30044.0	1.07841	0.539206	−	0.842174i	$-0.318725\pi$
$919$	30044.0	1.07841	0.539206	+	0.842174i	$0.318725\pi$
$920$	0	0
$921$	18132.0	0.648718
$922$	0	0
$923$	5568.00	0.198562
$924$	0	0
$925$	−1450.00	−0.0515413
$926$	0	0
$927$	−12924.0	−0.457907
$928$	0	0
$929$	−11382.0	−0.401971	−0.200986	−	0.979594i	$-0.564414\pi$
$929$	−11382.0	−0.401971	−0.200986	+	0.979594i	$0.564414\pi$
$930$	0	0
$931$	18308.0	0.644490
$932$	0	0
$933$	−18360.0	−0.644244
$934$	0	0
$935$	−7000.00	−0.244839
$936$	0	0
$937$	−34758.0	−1.21184	−0.605920	−	0.795525i	$-0.707195\pi$
$937$	−34758.0	−1.21184	−0.605920	+	0.795525i	$0.707195\pi$
$938$	0	0
$939$	−1842.00	−0.0640164
$940$	0	0
$941$	−13270.0	−0.459713	−0.229856	−	0.973225i	$-0.573826\pi$
$941$	−13270.0	−0.459713	−0.229856	+	0.973225i	$0.573826\pi$
$942$	0	0
$943$	7392.00	0.255267
$944$	0	0
$945$	−1620.00	−0.0557657
$946$	0	0
$947$	−35620.0	−1.22228	−0.611138	−	0.791524i	$-0.709288\pi$
$947$	−35620.0	−1.22228	−0.611138	+	0.791524i	$0.709288\pi$
$948$	0	0
$949$	45820.0	1.56731
$950$	0	0
$951$	2358.00	0.0804031
$952$	0	0
$953$	−16926.0	−0.575327	−0.287664	−	0.957731i	$-0.592879\pi$
$953$	−16926.0	−0.575327	−0.287664	+	0.957731i	$0.592879\pi$
$954$	0	0
$955$	−13280.0	−0.449980
$956$	0	0
$957$	−3960.00	−0.133760
$958$	0	0
$959$	15000.0	0.505084
$960$	0	0
$961$	−18127.0	−0.608472
$962$	0	0
$963$	756.000	0.0252978
$964$	0	0
$965$	−10810.0	−0.360607
$966$	0	0
$967$	−27676.0	−0.920372	−0.460186	−	0.887822i	$-0.652217\pi$
$967$	−27676.0	−0.920372	−0.460186	+	0.887822i	$0.652217\pi$
$968$	0	0
$969$	19320.0	0.640503
$970$	0	0
$971$	−46916.0	−1.55057	−0.775286	−	0.631610i	$-0.782394\pi$
$971$	−46916.0	−1.55057	−0.775286	+	0.631610i	$0.782394\pi$
$972$	0	0
$973$	−29136.0	−0.959977
$974$	0	0
$975$	−4350.00	−0.142884
$976$	0	0
$977$	27594.0	0.903593	0.451796	−	0.892121i	$-0.350783\pi$
$977$	27594.0	0.903593	0.451796	+	0.892121i	$0.350783\pi$
$978$	0	0
$979$	−24200.0	−0.790026
$980$	0	0
$981$	−2250.00	−0.0732283
$982$	0	0
$983$	−33016.0	−1.07126	−0.535629	−	0.844453i	$-0.679925\pi$
$983$	−33016.0	−1.07126	−0.535629	+	0.844453i	$0.679925\pi$
$984$	0	0
$985$	−17570.0	−0.568352
$986$	0	0
$987$	−14688.0	−0.473682
$988$	0	0
$989$	−43456.0	−1.39719
$990$	0	0
$991$	2276.00	0.0729561	0.0364781	−	0.999334i	$-0.488386\pi$
$991$	2276.00	0.0729561	0.0364781	+	0.999334i	$0.488386\pi$
$992$	0	0
$993$	1404.00	0.0448687
$994$	0	0
$995$	4940.00	0.157396
$996$	0	0
$997$	57654.0	1.83141	0.915707	−	0.401846i	$-0.131631\pi$
$997$	57654.0	1.83141	0.915707	+	0.401846i	$0.131631\pi$
$998$	0	0
$999$	−1566.00	−0.0495956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.a.h.1.1	yes	1
3.2	odd	2		1440.4.a.q.1.1		1
4.3	odd	2		480.4.a.a.1.1	✓	1
5.4	even	2		2400.4.a.b.1.1		1
8.3	odd	2		960.4.a.be.1.1		1
8.5	even	2		960.4.a.p.1.1		1
12.11	even	2		1440.4.a.l.1.1		1
20.19	odd	2		2400.4.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.a.1.1	✓	1	4.3	odd	2
480.4.a.h.1.1	yes	1	1.1	even	1	trivial
960.4.a.p.1.1		1	8.5	even	2
960.4.a.be.1.1		1	8.3	odd	2
1440.4.a.l.1.1		1	12.11	even	2
1440.4.a.q.1.1		1	3.2	odd	2
2400.4.a.b.1.1		1	5.4	even	2
2400.4.a.u.1.1		1	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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