LMFDB - Embedded newform 2400.4.a.b.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:	$1$
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} -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})

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	$-0.605018\pi$
$7$	−12.0000	−0.647939	−0.323970	+	0.946068i	$0.605018\pi$
$8$	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	$-0.287658\pi$
$13$	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	$-0.687445\pi$
$19$	−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	$-0.669499\pi$
$23$	−112.000	−1.01537	−0.507687	+	0.861541i	$0.669499\pi$
$24$	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	$-0.601287\pi$
$31$	−108.000	−0.625722	−0.312861	+	0.949799i	$0.601287\pi$
$32$	0	0
$33$	60.0000	0.316505
$34$	0	0
$35$	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	$-0.258482\pi$
$43$	388.000	1.37603	0.688017	+	0.725695i	$0.258482\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	408.000	1.26623	0.633116	−	0.774057i	$-0.281776\pi$
$47$	408.000	1.26623	0.633116	+	0.774057i	$0.281776\pi$
$48$	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	$-0.703157\pi$
$59$	−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
$65$	0	0
$66$	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	$-0.281698\pi$
$73$	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	$-0.429616\pi$
$79$	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
$85$	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	$-0.258988\pi$
$103$	1436.00	1.37372	0.686861	+	0.726789i	$0.258988\pi$
$104$	0	0
$105$	0	0
$106$	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	$-0.535035\pi$
$109$	−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	$-0.855368\pi$
$127$	−2572.00	−1.79707	−0.898536	+	0.438900i	$0.855368\pi$
$128$	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
$135$	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	$-0.765550\pi$
$139$	−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
$145$	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	$-0.309368\pi$
$151$	2092.00	1.12745	0.563724	+	0.825963i	$0.309368\pi$
$152$	0	0
$153$	630.000	0.332892
$154$	0	0
$155$	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	$-0.571887\pi$
$163$	−932.000	−0.447852	−0.223926	+	0.974606i	$0.571887\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3192.00	−1.47907	−0.739534	−	0.673119i	$-0.764954\pi$
$167$	−3192.00	−1.47907	−0.739534	+	0.673119i	$0.764954\pi$
$168$	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	$-0.637408\pi$
$179$	−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
$185$	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	$-0.332195\pi$
$191$	2656.00	1.00619	0.503093	+	0.864232i	$0.332195\pi$
$192$	0	0
$193$	−2162.00	−0.806343	−0.403171	−	0.915124i	$-0.632092\pi$
$193$	−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	$-0.556307\pi$
$199$	−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
$205$	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	$-0.303520\pi$
$211$	3548.00	1.15760	0.578802	+	0.815468i	$0.303520\pi$
$212$	0	0
$213$	288.000	0.0926452
$214$	0	0
$215$	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	$-0.535055\pi$
$223$	−732.000	−0.219813	−0.109907	+	0.993942i	$0.535055\pi$
$224$	0	0
$225$	0	0
$226$	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	$-0.463269\pi$
$229$	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	$-0.312992\pi$
$239$	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
$245$	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	$-0.619494\pi$
$251$	−2916.00	−0.733292	−0.366646	+	0.930361i	$0.619494\pi$
$252$	0	0
$253$	2240.00	0.556631
$254$	0	0
$255$	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	$-0.325046\pi$
$263$	4456.00	1.04475	0.522374	+	0.852716i	$0.325046\pi$
$264$	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	$-0.675588\pi$
$271$	−4676.00	−1.04814	−0.524072	+	0.851674i	$0.675588\pi$
$272$	0	0
$273$	2088.00	0.462899
$274$	0	0
$275$	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	$-0.307240\pi$
$283$	5420.00	1.13846	0.569232	+	0.822177i	$0.307240\pi$
$284$	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	$-0.689894\pi$
$307$	−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	$-0.482344\pi$
$313$	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	$-0.487628\pi$
$331$	468.000	0.0777148	0.0388574	+	0.999245i	$0.487628\pi$
$332$	0	0
$333$	522.000	0.0859022
$334$	0	0
$335$	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	$-0.725002\pi$
$347$	−8396.00	−1.29891	−0.649454	+	0.760401i	$0.725002\pi$
$348$	0	0
$349$	−11090.0	−1.70096	−0.850479	−	0.526010i	$-0.823688\pi$
$349$	−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	$-0.788880\pi$
$359$	−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
$365$	0	0
$366$	0	0
$367$	−4116.00	−0.585432	−0.292716	−	0.956199i	$-0.594559\pi$
$367$	−4116.00	−0.585432	−0.292716	+	0.956199i	$0.594559\pi$
$368$	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	$-0.819699\pi$
$379$	−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	$-0.189345\pi$
$383$	12416.0	1.65647	0.828235	+	0.560381i	$0.189345\pi$
$384$	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	$-0.726363\pi$
$419$	−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
$425$	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	$-0.436543\pi$
$431$	3544.00	0.396075	0.198038	+	0.980194i	$0.436543\pi$
$432$	0	0
$433$	−5810.00	−0.644829	−0.322414	−	0.946599i	$-0.604494\pi$
$433$	−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	$-0.675326\pi$
$439$	−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
$445$	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	$-0.324829\pi$
$463$	10420.0	1.04591	0.522957	+	0.852359i	$0.324829\pi$
$464$	0	0
$465$	0	0
$466$	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
$475$	0	0
$476$	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
$485$	0	0
$486$	0	0
$487$	17108.0	1.59186	0.795932	−	0.605386i	$-0.206981\pi$
$487$	17108.0	1.59186	0.795932	+	0.605386i	$0.206981\pi$
$488$	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
$495$	0	0
$496$	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	$-0.167665\pi$
$503$	19504.0	1.72891	0.864454	+	0.502713i	$0.167665\pi$
$504$	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	$-0.816228\pi$
$523$	−20044.0	−1.67584	−0.837919	+	0.545795i	$0.816228\pi$
$524$	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	$-0.785389\pi$
$547$	−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
$555$	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	$-0.658436\pi$
$563$	−12756.0	−0.954887	−0.477443	+	0.878662i	$0.658436\pi$
$564$	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	$-0.470798\pi$
$571$	2500.00	0.183225	0.0916127	+	0.995795i	$0.470798\pi$
$572$	0	0
$573$	−7968.00	−0.580921
$574$	0	0
$575$	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	$-0.364767\pi$
$587$	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	$-0.113826\pi$
$593$	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	$-0.324897\pi$
$599$	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
$605$	0	0
$606$	0	0
$607$	−22060.0	−1.47510	−0.737552	−	0.675291i	$-0.764018\pi$
$607$	−22060.0	−1.47510	−0.737552	+	0.675291i	$0.764018\pi$
$608$	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	$-0.871259\pi$
$619$	−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
$625$	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	$-0.486059\pi$
$631$	1388.00	0.0875680	0.0437840	+	0.999041i	$0.486059\pi$
$632$	0	0
$633$	−10644.0	−0.668343
$634$	0	0
$635$	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	$-0.630991\pi$
$643$	−13044.0	−0.800008	−0.400004	+	0.916513i	$0.630991\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18016.0	1.09472	0.547359	−	0.836898i	$-0.315633\pi$
$647$	18016.0	1.09472	0.547359	+	0.836898i	$0.315633\pi$
$648$	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	$-0.710032\pi$
$659$	−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
$665$	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	$-0.526673\pi$
$683$	−2988.00	−0.167398	−0.0836989	+	0.996491i	$0.526673\pi$
$684$	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	$-0.348669\pi$
$691$	16628.0	0.915425	0.457713	+	0.889100i	$0.348669\pi$
$692$	0	0
$693$	2160.00	0.118401
$694$	0	0
$695$	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	$-0.209075\pi$
$719$	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
$725$	0	0
$726$	0	0
$727$	7204.00	0.367512	0.183756	−	0.982972i	$-0.441174\pi$
$727$	7204.00	0.367512	0.183756	+	0.982972i	$0.441174\pi$
$728$	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	$-0.455010\pi$
$739$	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	$-0.528517\pi$
$743$	−3624.00	−0.178939	−0.0894695	+	0.995990i	$0.528517\pi$
$744$	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	$-0.401524\pi$
$751$	12532.0	0.608920	0.304460	+	0.952525i	$0.401524\pi$
$752$	0	0
$753$	8748.00	0.423366
$754$	0	0
$755$	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	$-0.289522\pi$
$787$	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
$795$	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	$-0.652181\pi$
$811$	−21252.0	−0.920171	−0.460085	+	0.887875i	$0.652181\pi$
$812$	0	0
$813$	14028.0	0.605146
$814$	0	0
$815$	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	$-0.434475\pi$
$823$	9652.00	0.408806	0.204403	+	0.978887i	$0.434475\pi$
$824$	0	0
$825$	0	0
$826$	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	$-0.589137\pi$
$829$	−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	$-0.346712\pi$
$839$	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
$845$	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	$-0.315004\pi$
$859$	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	$-0.554302\pi$
$863$	−8608.00	−0.339536	−0.169768	+	0.985484i	$0.554302\pi$
$864$	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	$-0.159472\pi$
$883$	46028.0	1.75421	0.877104	+	0.480301i	$0.159472\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10136.0	0.383691	0.191845	−	0.981425i	$-0.438553\pi$
$887$	10136.0	0.383691	0.191845	+	0.981425i	$0.438553\pi$
$888$	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	$-0.308641\pi$
$907$	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
$915$	0	0
$916$	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
$925$	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	$-0.290712\pi$
$947$	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	$-0.407121\pi$
$953$	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	$-0.347783\pi$
$967$	27676.0	0.920372	0.460186	+	0.887822i	$0.347783\pi$
$968$	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
$975$	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	$-0.320075\pi$
$983$	33016.0	1.07126	0.535629	+	0.844453i	$0.320075\pi$
$984$	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	$-0.488386\pi$
$991$	2276.00	0.0729561	0.0364781	+	0.999334i	$0.488386\pi$
$992$	0	0
$993$	−1404.00	−0.0448687
$994$	0	0
$995$	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

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

)

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2400.4.a.b.1.1

Level	$2400$
Weight	$4$
Character	2400.1
Self dual	yes
Analytic conductor	$141.605$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$