LMFDB - Embedded newform 2400.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
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	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +12.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

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

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	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
$15$	0	0
$16$	0	0
$17$	70.0000	0.998676	0.499338	−	0.866407i	$-0.333577\pi$
$17$	70.0000	0.998676	0.499338	+	0.866407i	$0.333577\pi$
$18$	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
$25$	0	0
$26$	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	$-0.398713\pi$
$31$	108.000	0.625722	0.312861	+	0.949799i	$0.398713\pi$
$32$	0	0
$33$	60.0000	0.316505
$34$	0	0
$35$	0	0
$36$	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	$-0.741518\pi$
$43$	−388.000	−1.37603	−0.688017	+	0.725695i	$0.741518\pi$
$44$	0	0
$45$	0	0
$46$	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
$55$	0	0
$56$	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	$-0.495323\pi$
$61$	14.0000	0.0293855	0.0146928	+	0.999892i	$0.495323\pi$
$62$	0	0
$63$	108.000	0.215980
$64$	0	0
$65$	0	0
$66$	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	$-0.474434\pi$
$71$	96.0000	0.160466	0.0802331	+	0.996776i	$0.474434\pi$
$72$	0	0
$73$	790.000	1.26661	0.633305	−	0.773902i	$-0.281698\pi$
$73$	790.000	1.26661	0.633305	+	0.773902i	$0.281698\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	240.000	0.355202
$78$	0	0
$79$	−308.000	−0.438642	−0.219321	−	0.975653i	$-0.570384\pi$
$79$	−308.000	−0.438642	−0.219321	+	0.975653i	$0.570384\pi$
$80$	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
$85$	0	0
$86$	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
$95$	0	0
$96$	0	0
$97$	−1426.00	−1.49266	−0.746332	−	0.665574i	$-0.768187\pi$
$97$	−1426.00	−1.49266	−0.746332	+	0.665574i	$0.768187\pi$
$98$	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
$105$	0	0
$106$	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	$-0.412240\pi$
$113$	654.000	0.544453	0.272226	+	0.962233i	$0.412240\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	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
$135$	0	0
$136$	0	0
$137$	−1250.00	−0.779523	−0.389762	−	0.920916i	$-0.627443\pi$
$137$	−1250.00	−0.779523	−0.389762	+	0.920916i	$0.627443\pi$
$138$	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
$145$	0	0
$146$	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	$-0.690632\pi$
$151$	−2092.00	−1.12745	−0.563724	+	0.825963i	$0.690632\pi$
$152$	0	0
$153$	630.000	0.332892
$154$	0	0
$155$	0	0
$156$	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	$-0.428113\pi$
$163$	932.000	0.447852	0.223926	+	0.974606i	$0.428113\pi$
$164$	0	0
$165$	0	0
$166$	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
$175$	0	0
$176$	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	$-0.834418\pi$
$181$	−4226.00	−1.73545	−0.867724	+	0.497046i	$0.834418\pi$
$182$	0	0
$183$	42.0000	0.0169657
$184$	0	0
$185$	0	0
$186$	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	$-0.667805\pi$
$191$	−2656.00	−1.00619	−0.503093	+	0.864232i	$0.667805\pi$
$192$	0	0
$193$	−2162.00	−0.806343	−0.403171	−	0.915124i	$-0.632092\pi$
$193$	−2162.00	−0.806343	−0.403171	+	0.915124i	$0.632092\pi$
$194$	0	0
$195$	0	0
$196$	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	$-0.443693\pi$
$199$	988.000	0.351947	0.175974	+	0.984395i	$0.443693\pi$
$200$	0	0
$201$	−828.000	−0.290560
$202$	0	0
$203$	792.000	0.273830
$204$	0	0
$205$	0	0
$206$	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
$215$	0	0
$216$	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
$225$	0	0
$226$	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	$-0.367019\pi$
$233$	2886.00	0.811451	0.405726	+	0.913995i	$0.367019\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−924.000	−0.253250
$238$	0	0
$239$	−4096.00	−1.10857	−0.554285	−	0.832327i	$-0.687008\pi$
$239$	−4096.00	−1.10857	−0.554285	+	0.832327i	$0.687008\pi$
$240$	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
$245$	0	0
$246$	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
$255$	0	0
$256$	0	0
$257$	−882.000	−0.214076	−0.107038	−	0.994255i	$-0.534137\pi$
$257$	−882.000	−0.214076	−0.107038	+	0.994255i	$0.534137\pi$
$258$	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
$265$	0	0
$266$	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	$-0.324412\pi$
$271$	4676.00	1.04814	0.524072	+	0.851674i	$0.324412\pi$
$272$	0	0
$273$	2088.00	0.462899
$274$	0	0
$275$	0	0
$276$	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	$-0.692760\pi$
$283$	−5420.00	−1.13846	−0.569232	+	0.822177i	$0.692760\pi$
$284$	0	0
$285$	0	0
$286$	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
$295$	0	0
$296$	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
$305$	0	0
$306$	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	$-0.311595\pi$
$311$	6120.00	1.11586	0.557931	+	0.829887i	$0.311595\pi$
$312$	0	0
$313$	614.000	0.110880	0.0554398	−	0.998462i	$-0.482344\pi$
$313$	614.000	0.110880	0.0554398	+	0.998462i	$0.482344\pi$
$314$	0	0
$315$	0	0
$316$	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
$325$	0	0
$326$	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
$335$	0	0
$336$	0	0
$337$	−1538.00	−0.248606	−0.124303	−	0.992244i	$-0.539669\pi$
$337$	−1538.00	−0.248606	−0.124303	+	0.992244i	$0.539669\pi$
$338$	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
$345$	0	0
$346$	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	$-0.579981\pi$
$353$	−3298.00	−0.497266	−0.248633	+	0.968598i	$0.579981\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2520.00	0.373593
$358$	0	0
$359$	10720.0	1.57599	0.787994	−	0.615682i	$-0.211120\pi$
$359$	10720.0	1.57599	0.787994	+	0.615682i	$0.211120\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	−2793.00	−0.403842
$364$	0	0
$365$	0	0
$366$	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
$375$	0	0
$376$	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
$385$	0	0
$386$	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
$395$	0	0
$396$	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
$405$	0	0
$406$	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
$415$	0	0
$416$	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	$-0.665727\pi$
$421$	−8594.00	−0.994883	−0.497442	+	0.867497i	$0.665727\pi$
$422$	0	0
$423$	−3672.00	−0.422077
$424$	0	0
$425$	0	0
$426$	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	$-0.563457\pi$
$431$	−3544.00	−0.396075	−0.198038	+	0.980194i	$0.563457\pi$
$432$	0	0
$433$	−5810.00	−0.644829	−0.322414	−	0.946599i	$-0.604494\pi$
$433$	−5810.00	−0.644829	−0.322414	+	0.946599i	$0.604494\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10304.0	1.12793
$438$	0	0
$439$	9628.00	1.04674	0.523371	−	0.852105i	$-0.324674\pi$
$439$	9628.00	1.04674	0.523371	+	0.852105i	$0.324674\pi$
$440$	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
$445$	0	0
$446$	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
$455$	0	0
$456$	0	0
$457$	−18314.0	−1.87460	−0.937301	−	0.348522i	$-0.886684\pi$
$457$	−18314.0	−1.87460	−0.937301	+	0.348522i	$0.886684\pi$
$458$	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
$465$	0	0
$466$	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
$475$	0	0
$476$	0	0
$477$	−4266.00	−0.409490
$478$	0	0
$479$	3352.00	0.319743	0.159871	−	0.987138i	$-0.448892\pi$
$479$	3352.00	0.319743	0.159871	+	0.987138i	$0.448892\pi$
$480$	0	0
$481$	3364.00	0.318888
$482$	0	0
$483$	4032.00	0.379839
$484$	0	0
$485$	0	0
$486$	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
$495$	0	0
$496$	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
$505$	0	0
$506$	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
$515$	0	0
$516$	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
$525$	0	0
$526$	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
$535$	0	0
$536$	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
$545$	0	0
$546$	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
$555$	0	0
$556$	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	$-0.341564\pi$
$563$	12756.0	0.954887	0.477443	+	0.878662i	$0.341564\pi$
$564$	0	0
$565$	0	0
$566$	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
$575$	0	0
$576$	0	0
$577$	−13778.0	−0.994083	−0.497041	−	0.867727i	$-0.665580\pi$
$577$	−13778.0	−0.994083	−0.497041	+	0.867727i	$0.665580\pi$
$578$	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
$585$	0	0
$586$	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	$-0.113826\pi$
$593$	27054.0	1.87348	0.936741	+	0.350024i	$0.113826\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2964.00	0.203197
$598$	0	0
$599$	−15328.0	−1.04555	−0.522776	−	0.852470i	$-0.675103\pi$
$599$	−15328.0	−1.04555	−0.522776	+	0.852470i	$0.675103\pi$
$600$	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
$605$	0	0
$606$	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
$615$	0	0
$616$	0	0
$617$	11766.0	0.767717	0.383858	−	0.923392i	$-0.374595\pi$
$617$	11766.0	0.767717	0.383858	+	0.923392i	$0.374595\pi$
$618$	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
$625$	0	0
$626$	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	$-0.513941\pi$
$631$	−1388.00	−0.0875680	−0.0437840	+	0.999041i	$0.513941\pi$
$632$	0	0
$633$	−10644.0	−0.668343
$634$	0	0
$635$	0	0
$636$	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
$645$	0	0
$646$	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
$655$	0	0
$656$	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	$-0.384440\pi$
$661$	12070.0	0.710240	0.355120	+	0.934821i	$0.384440\pi$
$662$	0	0
$663$	12180.0	0.713472
$664$	0	0
$665$	0	0
$666$	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	$-0.699882\pi$
$673$	−20514.0	−1.17497	−0.587486	+	0.809234i	$0.699882\pi$
$674$	0	0
$675$	0	0
$676$	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	$-0.473327\pi$
$683$	2988.00	0.167398	0.0836989	+	0.996491i	$0.473327\pi$
$684$	0	0
$685$	0	0
$686$	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
$695$	0	0
$696$	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
$705$	0	0
$706$	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
$715$	0	0
$716$	0	0
$717$	−12288.0	−0.640033
$718$	0	0
$719$	−30536.0	−1.58387	−0.791934	−	0.610607i	$-0.790925\pi$
$719$	−30536.0	−1.58387	−0.791934	+	0.610607i	$0.790925\pi$
$720$	0	0
$721$	−17232.0	−0.890088
$722$	0	0
$723$	7062.00	0.363262
$724$	0	0
$725$	0	0
$726$	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
$735$	0	0
$736$	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
$745$	0	0
$746$	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	$-0.598476\pi$
$751$	−12532.0	−0.608920	−0.304460	+	0.952525i	$0.598476\pi$
$752$	0	0
$753$	8748.00	0.423366
$754$	0	0
$755$	0	0
$756$	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
$765$	0	0
$766$	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
$775$	0	0
$776$	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
$785$	0	0
$786$	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
$795$	0	0
$796$	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
$805$	0	0
$806$	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
$815$	0	0
$816$	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
$825$	0	0
$826$	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
$835$	0	0
$836$	0	0
$837$	2916.00	0.120420
$838$	0	0
$839$	−22512.0	−0.926342	−0.463171	−	0.886269i	$-0.653288\pi$
$839$	−22512.0	−0.926342	−0.463171	+	0.886269i	$0.653288\pi$
$840$	0	0
$841$	−20033.0	−0.821395
$842$	0	0
$843$	15582.0	0.636622
$844$	0	0
$845$	0	0
$846$	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
$855$	0	0
$856$	0	0
$857$	36254.0	1.44506	0.722528	−	0.691342i	$-0.242980\pi$
$857$	36254.0	1.44506	0.722528	+	0.691342i	$0.242980\pi$
$858$	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
$865$	0	0
$866$	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
$875$	0	0
$876$	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	$-0.840528\pi$
$883$	−46028.0	−1.75421	−0.877104	+	0.480301i	$0.840528\pi$
$884$	0	0
$885$	0	0
$886$	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
$895$	0	0
$896$	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
$905$	0	0
$906$	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	$-0.193389\pi$
$911$	45152.0	1.64210	0.821050	+	0.570857i	$0.193389\pi$
$912$	0	0
$913$	−20720.0	−0.751075
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10032.0	0.361271
$918$	0	0
$919$	−30044.0	−1.07841	−0.539206	−	0.842174i	$-0.681275\pi$
$919$	−30044.0	−1.07841	−0.539206	+	0.842174i	$0.681275\pi$
$920$	0	0
$921$	18132.0	0.648718
$922$	0	0
$923$	5568.00	0.198562
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	34758.0	1.21184	0.605920	−	0.795525i	$-0.292805\pi$
$937$	34758.0	1.21184	0.605920	+	0.795525i	$0.292805\pi$
$938$	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
$945$	0	0
$946$	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	$-0.407121\pi$
$953$	16926.0	0.575327	0.287664	+	0.957731i	$0.407121\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	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
$975$	0	0
$976$	0	0
$977$	−27594.0	−0.903593	−0.451796	−	0.892121i	$-0.649217\pi$
$977$	−27594.0	−0.903593	−0.451796	+	0.892121i	$0.649217\pi$
$978$	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
$985$	0	0
$986$	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	$-0.511614\pi$
$991$	−2276.00	−0.0729561	−0.0364781	+	0.999334i	$0.511614\pi$
$992$	0	0
$993$	−1404.00	−0.0448687
$994$	0	0
$995$	0	0
$996$	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	2400.4.a.u.1.1		1
4.3	odd	2		2400.4.a.b.1.1		1
5.4	even	2		480.4.a.a.1.1	✓	1
15.14	odd	2		1440.4.a.l.1.1		1
20.19	odd	2		480.4.a.h.1.1	yes	1
40.19	odd	2		960.4.a.p.1.1		1
40.29	even	2		960.4.a.be.1.1		1
60.59	even	2		1440.4.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.a.1.1	✓	1	5.4	even	2
480.4.a.h.1.1	yes	1	20.19	odd	2
960.4.a.p.1.1		1	40.19	odd	2
960.4.a.be.1.1		1	40.29	even	2
1440.4.a.l.1.1		1	15.14	odd	2
1440.4.a.q.1.1		1	60.59	even	2
2400.4.a.b.1.1		1	4.3	odd	2
2400.4.a.u.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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