LMFDB - Embedded newform 960.4.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,4,Mod(1,960)] 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("960.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,5,0,12,0,9,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	960.1

$q$-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	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	0	0
$13$	58.0000	1.23741	0.618704	−	0.785624i	$-0.287658\pi$
$13$	58.0000	1.23741	0.618704	+	0.785624i	$0.287658\pi$
$14$	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	$-0.312555\pi$
$19$	92.0000	1.11086	0.555428	+	0.831565i	$0.312555\pi$
$20$	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	$-0.567772\pi$
$29$	−66.0000	−0.422617	−0.211308	+	0.977419i	$0.567772\pi$
$30$	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	$-0.458870\pi$
$37$	58.0000	0.257707	0.128853	+	0.991664i	$0.458870\pi$
$38$	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	$-0.258482\pi$
$43$	388.000	1.37603	0.688017	+	0.725695i	$0.258482\pi$
$44$	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	$-0.710535\pi$
$53$	−474.000	−1.22847	−0.614235	+	0.789123i	$0.710535\pi$
$54$	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	$-0.296843\pi$
$59$	540.000	1.19156	0.595780	+	0.803148i	$0.296843\pi$
$60$	0	0
$61$	−14.0000	−0.0293855	−0.0146928	−	0.999892i	$-0.504677\pi$
$61$	−14.0000	−0.0293855	−0.0146928	+	0.999892i	$0.504677\pi$
$62$	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	$-0.419033\pi$
$67$	276.000	0.503265	0.251633	+	0.967823i	$0.419033\pi$
$68$	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	$-0.259787\pi$
$83$	1036.00	1.37007	0.685035	+	0.728510i	$0.259787\pi$
$84$	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	$-0.511606\pi$
$101$	−74.0000	−0.0729037	−0.0364519	+	0.999335i	$0.511606\pi$
$102$	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	$-0.512082\pi$
$107$	−84.0000	−0.0758933	−0.0379467	+	0.999280i	$0.512082\pi$
$108$	0	0
$109$	250.000	0.219685	0.109842	−	0.993949i	$-0.464965\pi$
$109$	250.000	0.219685	0.109842	+	0.993949i	$0.464965\pi$
$110$	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	$-0.410068\pi$
$131$	836.000	0.557570	0.278785	+	0.960354i	$0.410068\pi$
$132$	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	$-0.234450\pi$
$139$	2428.00	1.48158	0.740792	+	0.671734i	$0.234450\pi$
$140$	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	$-0.659361\pi$
$149$	−1746.00	−0.959986	−0.479993	+	0.877272i	$0.659361\pi$
$150$	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	$-0.314814\pi$
$157$	2162.00	1.09902	0.549511	+	0.835487i	$0.314814\pi$
$158$	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	$-0.571887\pi$
$163$	−932.000	−0.447852	−0.223926	+	0.974606i	$0.571887\pi$
$164$	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	$-0.667195\pi$
$173$	−2282.00	−1.00287	−0.501437	+	0.865194i	$0.667195\pi$
$174$	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	$-0.362592\pi$
$179$	2004.00	0.836793	0.418397	+	0.908264i	$0.362592\pi$
$180$	0	0
$181$	4226.00	1.73545	0.867724	−	0.497046i	$-0.165582\pi$
$181$	4226.00	1.73545	0.867724	+	0.497046i	$0.165582\pi$
$182$	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	$-0.719180\pi$
$197$	−3514.00	−1.27087	−0.635437	+	0.772153i	$0.719180\pi$
$198$	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	$-0.696480\pi$
$211$	−3548.00	−1.15760	−0.578802	+	0.815468i	$0.696480\pi$
$212$	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	$-0.796711\pi$
$227$	−5492.00	−1.60580	−0.802901	+	0.596113i	$0.796711\pi$
$228$	0	0
$229$	−798.000	−0.230277	−0.115138	−	0.993349i	$-0.536731\pi$
$229$	−798.000	−0.230277	−0.115138	+	0.993349i	$0.536731\pi$
$230$	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	$-0.380506\pi$
$251$	2916.00	0.733292	0.366646	+	0.930361i	$0.380506\pi$
$252$	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	$-0.446138\pi$
$269$	1486.00	0.336814	0.168407	+	0.985718i	$0.446138\pi$
$270$	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	$-0.398229\pi$
$277$	2898.00	0.628606	0.314303	+	0.949323i	$0.398229\pi$
$278$	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	$-0.307240\pi$
$283$	5420.00	1.13846	0.569232	+	0.822177i	$0.307240\pi$
$284$	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	$-0.864076\pi$
$293$	−9130.00	−1.82041	−0.910205	+	0.414157i	$0.864076\pi$
$294$	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	$-0.689894\pi$
$307$	−6044.00	−1.12361	−0.561807	+	0.827269i	$0.689894\pi$
$308$	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	$-0.522182\pi$
$317$	−786.000	−0.139262	−0.0696312	+	0.997573i	$0.522182\pi$
$318$	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	$-0.512372\pi$
$331$	−468.000	−0.0777148	−0.0388574	+	0.999245i	$0.512372\pi$
$332$	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	$-0.725002\pi$
$347$	−8396.00	−1.29891	−0.649454	+	0.760401i	$0.725002\pi$
$348$	0	0
$349$	11090.0	1.70096	0.850479	−	0.526010i	$-0.176312\pi$
$349$	11090.0	1.70096	0.850479	+	0.526010i	$0.176312\pi$
$350$	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	$-0.580159\pi$
$373$	−3590.00	−0.498346	−0.249173	+	0.968459i	$0.580159\pi$
$374$	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	$-0.180301\pi$
$379$	12452.0	1.68764	0.843821	+	0.536625i	$0.180301\pi$
$380$	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	$-0.549361\pi$
$389$	−2370.00	−0.308904	−0.154452	+	0.988000i	$0.549361\pi$
$390$	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	$-0.704677\pi$
$397$	−9486.00	−1.19922	−0.599608	+	0.800294i	$0.704677\pi$
$398$	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	$-0.273637\pi$
$419$	11196.0	1.30539	0.652697	+	0.757619i	$0.273637\pi$
$420$	0	0
$421$	8594.00	0.994883	0.497442	−	0.867497i	$-0.334273\pi$
$421$	8594.00	0.994883	0.497442	+	0.867497i	$0.334273\pi$
$422$	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	$-0.393925\pi$
$443$	6100.00	0.654221	0.327110	+	0.944986i	$0.393925\pi$
$444$	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	$-0.600079\pi$
$461$	−6122.00	−0.618503	−0.309252	+	0.950980i	$0.600079\pi$
$462$	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	$-0.714842\pi$
$467$	−12612.0	−1.24971	−0.624854	+	0.780742i	$0.714842\pi$
$468$	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	$-0.324681\pi$
$491$	11388.0	1.04671	0.523354	+	0.852116i	$0.324681\pi$
$492$	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	$-0.367785\pi$
$499$	8996.00	0.807047	0.403523	+	0.914969i	$0.367785\pi$
$500$	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	$-0.617786\pi$
$509$	−8306.00	−0.723295	−0.361647	+	0.932315i	$0.617786\pi$
$510$	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	$-0.816228\pi$
$523$	−20044.0	−1.67584	−0.837919	+	0.545795i	$0.816228\pi$
$524$	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	$-0.163219\pi$
$541$	21930.0	1.74278	0.871390	+	0.490590i	$0.163219\pi$
$542$	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	$-0.785389\pi$
$547$	−19988.0	−1.56239	−0.781193	+	0.624290i	$0.785389\pi$
$548$	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	$-0.352956\pi$
$557$	11718.0	0.891396	0.445698	+	0.895183i	$0.352956\pi$
$558$	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	$-0.658436\pi$
$563$	−12756.0	−0.954887	−0.477443	+	0.878662i	$0.658436\pi$
$564$	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	$-0.529202\pi$
$571$	−2500.00	−0.183225	−0.0916127	+	0.995795i	$0.529202\pi$
$572$	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	$-0.364767\pi$
$587$	11724.0	0.824363	0.412182	+	0.911102i	$0.364767\pi$
$588$	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	$-0.406265\pi$
$613$	8810.00	0.580477	0.290239	+	0.956954i	$0.406265\pi$
$614$	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	$-0.128741\pi$
$619$	28316.0	1.83864	0.919318	+	0.393515i	$0.128741\pi$
$620$	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	$-0.630991\pi$
$643$	−13044.0	−0.800008	−0.400004	+	0.916513i	$0.630991\pi$
$644$	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	$-0.320591\pi$
$653$	17830.0	1.06852	0.534259	+	0.845321i	$0.320591\pi$
$654$	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	$-0.289968\pi$
$659$	20740.0	1.22597	0.612986	+	0.790094i	$0.289968\pi$
$660$	0	0
$661$	−12070.0	−0.710240	−0.355120	−	0.934821i	$-0.615560\pi$
$661$	−12070.0	−0.710240	−0.355120	+	0.934821i	$0.615560\pi$
$662$	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	$-0.376524\pi$
$677$	13326.0	0.756514	0.378257	+	0.925701i	$0.376524\pi$
$678$	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	$-0.526673\pi$
$683$	−2988.00	−0.167398	−0.0836989	+	0.996491i	$0.526673\pi$
$684$	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	$-0.651331\pi$
$691$	−16628.0	−0.915425	−0.457713	+	0.889100i	$0.651331\pi$
$692$	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	$-0.773103\pi$
$701$	−28082.0	−1.51304	−0.756521	+	0.653969i	$0.773103\pi$
$702$	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	$-0.679269\pi$
$709$	−20158.0	−1.06777	−0.533885	+	0.845557i	$0.679269\pi$
$710$	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	$-0.543426\pi$
$733$	−5398.00	−0.272005	−0.136003	+	0.990708i	$0.543426\pi$
$734$	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	$-0.544990\pi$
$739$	−5660.00	−0.281741	−0.140870	+	0.990028i	$0.544990\pi$
$740$	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	$-0.430477\pi$
$757$	9026.00	0.433363	0.216681	+	0.976242i	$0.430477\pi$
$758$	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	$-0.372390\pi$
$773$	16774.0	0.780490	0.390245	+	0.920711i	$0.372390\pi$
$774$	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	$-0.289522\pi$
$787$	27116.0	1.22818	0.614092	+	0.789234i	$0.289522\pi$
$788$	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	$-0.372907\pi$
$797$	17494.0	0.777502	0.388751	+	0.921343i	$0.372907\pi$
$798$	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	$-0.347819\pi$
$811$	21252.0	0.920171	0.460085	+	0.887875i	$0.347819\pi$
$812$	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	$-0.510678\pi$
$821$	−1578.00	−0.0670799	−0.0335399	+	0.999437i	$0.510678\pi$
$822$	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	$-0.393550\pi$
$827$	15612.0	0.656448	0.328224	+	0.944600i	$0.393550\pi$
$828$	0	0
$829$	13194.0	0.552770	0.276385	−	0.961047i	$-0.410863\pi$
$829$	13194.0	0.552770	0.276385	+	0.961047i	$0.410863\pi$
$830$	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	$-0.480093\pi$
$853$	3114.00	0.124996	0.0624978	+	0.998045i	$0.480093\pi$
$854$	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	$-0.684996\pi$
$859$	−27644.0	−1.09802	−0.549011	+	0.835815i	$0.684996\pi$
$860$	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	$-0.754931\pi$
$877$	−37294.0	−1.43595	−0.717975	+	0.696068i	$0.754931\pi$
$878$	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	$-0.159472\pi$
$883$	46028.0	1.75421	0.877104	+	0.480301i	$0.159472\pi$
$884$	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	$-0.308641\pi$
$907$	30900.0	1.13122	0.565611	+	0.824672i	$0.308641\pi$
$908$	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	$-0.426174\pi$
$941$	13270.0	0.459713	0.229856	+	0.973225i	$0.426174\pi$
$942$	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	$-0.290712\pi$
$947$	35620.0	1.22228	0.611138	+	0.791524i	$0.290712\pi$
$948$	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	$-0.217606\pi$
$971$	46916.0	1.55057	0.775286	+	0.631610i	$0.217606\pi$
$972$	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	$-0.868369\pi$
$997$	−57654.0	−1.83141	−0.915707	+	0.401846i	$0.868369\pi$
$998$	0	0
$999$	−1566.00	−0.0495956

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.a.1.1	✓	1	8.3	odd	2
480.4.a.h.1.1	yes	1	8.5	even	2
960.4.a.p.1.1		1	1.1	even	1	trivial
960.4.a.be.1.1		1	4.3	odd	2
1440.4.a.l.1.1		1	24.11	even	2
1440.4.a.q.1.1		1	24.5	odd	2
2400.4.a.b.1.1		1	40.29	even	2
2400.4.a.u.1.1		1	40.19	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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