LMFDB - Embedded newform 400.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{3} +26.0000 q^{7} -23.0000 q^{9} +O(q^{10})$

$q+2.00000 q^{3} +26.0000 q^{7} -23.0000 q^{9} +28.0000 q^{11} -12.0000 q^{13} +64.0000 q^{17} +60.0000 q^{19} +52.0000 q^{21} -58.0000 q^{23} -100.000 q^{27} +90.0000 q^{29} +128.000 q^{31} +56.0000 q^{33} -236.000 q^{37} -24.0000 q^{39} +242.000 q^{41} +362.000 q^{43} +226.000 q^{47} +333.000 q^{49} +128.000 q^{51} +108.000 q^{53} +120.000 q^{57} +20.0000 q^{59} +542.000 q^{61} -598.000 q^{63} -434.000 q^{67} -116.000 q^{69} +1128.00 q^{71} -632.000 q^{73} +728.000 q^{77} +720.000 q^{79} +421.000 q^{81} -478.000 q^{83} +180.000 q^{87} -490.000 q^{89} -312.000 q^{91} +256.000 q^{93} -1456.00 q^{97} -644.000 q^{99} +O(q^{100})$

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
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	26.0000	1.40387	0.701934	−	0.712242i	$-0.252320\pi$
$7$	26.0000	1.40387	0.701934	+	0.712242i	$0.252320\pi$
$8$	0	0
$9$	−23.0000	−0.851852
$10$	0	0
$11$	28.0000	0.767483	0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000	0.767483	0.383742	+	0.923440i	$0.374635\pi$
$12$	0	0
$13$	−12.0000	−0.256015	−0.128008	−	0.991773i	$-0.540858\pi$
$13$	−12.0000	−0.256015	−0.128008	+	0.991773i	$0.540858\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	64.0000	0.913075	0.456538	−	0.889704i	$-0.349089\pi$
$17$	64.0000	0.913075	0.456538	+	0.889704i	$0.349089\pi$
$18$	0	0
$19$	60.0000	0.724471	0.362235	−	0.932087i	$-0.382014\pi$
$19$	60.0000	0.724471	0.362235	+	0.932087i	$0.382014\pi$
$20$	0	0
$21$	52.0000	0.540349
$22$	0	0
$23$	−58.0000	−0.525819	−0.262909	−	0.964821i	$-0.584682\pi$
$23$	−58.0000	−0.525819	−0.262909	+	0.964821i	$0.584682\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−100.000	−0.712778
$28$	0	0
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	128.000	0.741596	0.370798	−	0.928714i	$-0.379084\pi$
$31$	128.000	0.741596	0.370798	+	0.928714i	$0.379084\pi$
$32$	0	0
$33$	56.0000	0.295405
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−236.000	−1.04860	−0.524299	−	0.851534i	$-0.675673\pi$
$37$	−236.000	−1.04860	−0.524299	+	0.851534i	$0.675673\pi$
$38$	0	0
$39$	−24.0000	−0.0985404
$40$	0	0
$41$	242.000	0.921806	0.460903	−	0.887450i	$-0.347526\pi$
$41$	242.000	0.921806	0.460903	+	0.887450i	$0.347526\pi$
$42$	0	0
$43$	362.000	1.28383	0.641913	−	0.766778i	$-0.278141\pi$
$43$	362.000	1.28383	0.641913	+	0.766778i	$0.278141\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	226.000	0.701393	0.350697	−	0.936489i	$-0.385945\pi$
$47$	226.000	0.701393	0.350697	+	0.936489i	$0.385945\pi$
$48$	0	0
$49$	333.000	0.970845
$50$	0	0
$51$	128.000	0.351443
$52$	0	0
$53$	108.000	0.279905	0.139952	−	0.990158i	$-0.455305\pi$
$53$	108.000	0.279905	0.139952	+	0.990158i	$0.455305\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	120.000	0.278849
$58$	0	0
$59$	20.0000	0.0441318	0.0220659	−	0.999757i	$-0.492976\pi$
$59$	20.0000	0.0441318	0.0220659	+	0.999757i	$0.492976\pi$
$60$	0	0
$61$	542.000	1.13764	0.568820	−	0.822462i	$-0.307400\pi$
$61$	542.000	1.13764	0.568820	+	0.822462i	$0.307400\pi$
$62$	0	0
$63$	−598.000	−1.19589
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−434.000	−0.791366	−0.395683	−	0.918387i	$-0.629492\pi$
$67$	−434.000	−0.791366	−0.395683	+	0.918387i	$0.629492\pi$
$68$	0	0
$69$	−116.000	−0.202388
$70$	0	0
$71$	1128.00	1.88548	0.942739	−	0.333531i	$-0.108240\pi$
$71$	1128.00	1.88548	0.942739	+	0.333531i	$0.108240\pi$
$72$	0	0
$73$	−632.000	−1.01329	−0.506644	−	0.862155i	$-0.669114\pi$
$73$	−632.000	−1.01329	−0.506644	+	0.862155i	$0.669114\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	728.000	1.07745
$78$	0	0
$79$	720.000	1.02540	0.512698	−	0.858569i	$-0.328646\pi$
$79$	720.000	1.02540	0.512698	+	0.858569i	$0.328646\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	−478.000	−0.632136	−0.316068	−	0.948736i	$-0.602363\pi$
$83$	−478.000	−0.632136	−0.316068	+	0.948736i	$0.602363\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	180.000	0.221816
$88$	0	0
$89$	−490.000	−0.583594	−0.291797	−	0.956480i	$-0.594253\pi$
$89$	−490.000	−0.583594	−0.291797	+	0.956480i	$0.594253\pi$
$90$	0	0
$91$	−312.000	−0.359412
$92$	0	0
$93$	256.000	0.285440
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1456.00	−1.52407	−0.762033	−	0.647538i	$-0.775799\pi$
$97$	−1456.00	−1.52407	−0.762033	+	0.647538i	$0.775799\pi$
$98$	0	0
$99$	−644.000	−0.653782
$100$	0	0
$101$	−578.000	−0.569437	−0.284719	−	0.958611i	$-0.591900\pi$
$101$	−578.000	−0.569437	−0.284719	+	0.958611i	$0.591900\pi$
$102$	0	0
$103$	1462.00	1.39859	0.699297	−	0.714831i	$-0.253497\pi$
$103$	1462.00	1.39859	0.699297	+	0.714831i	$0.253497\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	966.000	0.872773	0.436387	−	0.899759i	$-0.356258\pi$
$107$	966.000	0.872773	0.436387	+	0.899759i	$0.356258\pi$
$108$	0	0
$109$	370.000	0.325134	0.162567	−	0.986698i	$-0.448023\pi$
$109$	370.000	0.325134	0.162567	+	0.986698i	$0.448023\pi$
$110$	0	0
$111$	−472.000	−0.403606
$112$	0	0
$113$	528.000	0.439558	0.219779	−	0.975550i	$-0.429466\pi$
$113$	528.000	0.439558	0.219779	+	0.975550i	$0.429466\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	276.000	0.218087
$118$	0	0
$119$	1664.00	1.28184
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	484.000	0.354803
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1534.00	−1.07181	−0.535907	−	0.844277i	$-0.680030\pi$
$127$	−1534.00	−1.07181	−0.535907	+	0.844277i	$0.680030\pi$
$128$	0	0
$129$	724.000	0.494145
$130$	0	0
$131$	−12.0000	−0.00800340	−0.00400170	−	0.999992i	$-0.501274\pi$
$131$	−12.0000	−0.00800340	−0.00400170	+	0.999992i	$0.501274\pi$
$132$	0	0
$133$	1560.00	1.01706
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1224.00	0.763309	0.381655	−	0.924305i	$-0.375354\pi$
$137$	1224.00	0.763309	0.381655	+	0.924305i	$0.375354\pi$
$138$	0	0
$139$	−3100.00	−1.89164	−0.945822	−	0.324685i	$-0.894742\pi$
$139$	−3100.00	−1.89164	−0.945822	+	0.324685i	$0.894742\pi$
$140$	0	0
$141$	452.000	0.269966
$142$	0	0
$143$	−336.000	−0.196488
$144$	0	0
$145$	0	0
$146$	0	0
$147$	666.000	0.373679
$148$	0	0
$149$	250.000	0.137455	0.0687275	−	0.997635i	$-0.478106\pi$
$149$	250.000	0.137455	0.0687275	+	0.997635i	$0.478106\pi$
$150$	0	0
$151$	−2152.00	−1.15978	−0.579892	−	0.814694i	$-0.696905\pi$
$151$	−2152.00	−1.15978	−0.579892	+	0.814694i	$0.696905\pi$
$152$	0	0
$153$	−1472.00	−0.777805
$154$	0	0
$155$	0	0
$156$	0	0
$157$	524.000	0.266368	0.133184	−	0.991091i	$-0.457480\pi$
$157$	524.000	0.266368	0.133184	+	0.991091i	$0.457480\pi$
$158$	0	0
$159$	216.000	0.107735
$160$	0	0
$161$	−1508.00	−0.738180
$162$	0	0
$163$	−3518.00	−1.69050	−0.845249	−	0.534373i	$-0.820548\pi$
$163$	−3518.00	−1.69050	−0.845249	+	0.534373i	$0.820548\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−534.000	−0.247438	−0.123719	−	0.992317i	$-0.539482\pi$
$167$	−534.000	−0.247438	−0.123719	+	0.992317i	$0.539482\pi$
$168$	0	0
$169$	−2053.00	−0.934456
$170$	0	0
$171$	−1380.00	−0.617142
$172$	0	0
$173$	−4252.00	−1.86863	−0.934317	−	0.356444i	$-0.883989\pi$
$173$	−4252.00	−1.86863	−0.934317	+	0.356444i	$0.883989\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	40.0000	0.0169864
$178$	0	0
$179$	−2500.00	−1.04390	−0.521952	−	0.852975i	$-0.674796\pi$
$179$	−2500.00	−1.04390	−0.521952	+	0.852975i	$0.674796\pi$
$180$	0	0
$181$	−2578.00	−1.05868	−0.529340	−	0.848410i	$-0.677561\pi$
$181$	−2578.00	−1.05868	−0.529340	+	0.848410i	$0.677561\pi$
$182$	0	0
$183$	1084.00	0.437878
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1792.00	0.700770
$188$	0	0
$189$	−2600.00	−1.00065
$190$	0	0
$191$	768.000	0.290945	0.145473	−	0.989362i	$-0.453530\pi$
$191$	768.000	0.290945	0.145473	+	0.989362i	$0.453530\pi$
$192$	0	0
$193$	2608.00	0.972684	0.486342	−	0.873769i	$-0.338331\pi$
$193$	2608.00	0.972684	0.486342	+	0.873769i	$0.338331\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5116.00	−1.85025	−0.925127	−	0.379659i	$-0.876041\pi$
$197$	−5116.00	−1.85025	−0.925127	+	0.379659i	$0.876041\pi$
$198$	0	0
$199$	3480.00	1.23965	0.619826	−	0.784739i	$-0.287203\pi$
$199$	3480.00	1.23965	0.619826	+	0.784739i	$0.287203\pi$
$200$	0	0
$201$	−868.000	−0.304597
$202$	0	0
$203$	2340.00	0.809043
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1334.00	0.447920
$208$	0	0
$209$	1680.00	0.556019
$210$	0	0
$211$	−3132.00	−1.02188	−0.510938	−	0.859618i	$-0.670702\pi$
$211$	−3132.00	−1.02188	−0.510938	+	0.859618i	$0.670702\pi$
$212$	0	0
$213$	2256.00	0.725721
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3328.00	1.04110
$218$	0	0
$219$	−1264.00	−0.390015
$220$	0	0
$221$	−768.000	−0.233761
$222$	0	0
$223$	62.0000	0.0186181	0.00930903	−	0.999957i	$-0.497037\pi$
$223$	62.0000	0.0186181	0.00930903	+	0.999957i	$0.497037\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5314.00	−1.55376	−0.776878	−	0.629651i	$-0.783198\pi$
$227$	−5314.00	−1.55376	−0.776878	+	0.629651i	$0.783198\pi$
$228$	0	0
$229$	−190.000	−0.0548277	−0.0274139	−	0.999624i	$-0.508727\pi$
$229$	−190.000	−0.0548277	−0.0274139	+	0.999624i	$0.508727\pi$
$230$	0	0
$231$	1456.00	0.414709
$232$	0	0
$233$	2408.00	0.677053	0.338526	−	0.940957i	$-0.390072\pi$
$233$	2408.00	0.677053	0.338526	+	0.940957i	$0.390072\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1440.00	0.394675
$238$	0	0
$239$	5680.00	1.53727	0.768637	−	0.639685i	$-0.220935\pi$
$239$	5680.00	1.53727	0.768637	+	0.639685i	$0.220935\pi$
$240$	0	0
$241$	−278.000	−0.0743052	−0.0371526	−	0.999310i	$-0.511829\pi$
$241$	−278.000	−0.0743052	−0.0371526	+	0.999310i	$0.511829\pi$
$242$	0	0
$243$	3542.00	0.935059
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−720.000	−0.185476
$248$	0	0
$249$	−956.000	−0.243309
$250$	0	0
$251$	−3252.00	−0.817787	−0.408893	−	0.912582i	$-0.634085\pi$
$251$	−3252.00	−0.817787	−0.408893	+	0.912582i	$0.634085\pi$
$252$	0	0
$253$	−1624.00	−0.403557
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1536.00	−0.372813	−0.186407	−	0.982473i	$-0.559684\pi$
$257$	−1536.00	−0.372813	−0.186407	+	0.982473i	$0.559684\pi$
$258$	0	0
$259$	−6136.00	−1.47209
$260$	0	0
$261$	−2070.00	−0.490919
$262$	0	0
$263$	−4858.00	−1.13900	−0.569500	−	0.821991i	$-0.692863\pi$
$263$	−4858.00	−1.13900	−0.569500	+	0.821991i	$0.692863\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−980.000	−0.224626
$268$	0	0
$269$	2610.00	0.591578	0.295789	−	0.955253i	$-0.404417\pi$
$269$	2610.00	0.591578	0.295789	+	0.955253i	$0.404417\pi$
$270$	0	0
$271$	5168.00	1.15843	0.579213	−	0.815176i	$-0.303360\pi$
$271$	5168.00	1.15843	0.579213	+	0.815176i	$0.303360\pi$
$272$	0	0
$273$	−624.000	−0.138338
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1924.00	0.417336	0.208668	−	0.977987i	$-0.433087\pi$
$277$	1924.00	0.417336	0.208668	+	0.977987i	$0.433087\pi$
$278$	0	0
$279$	−2944.00	−0.631730
$280$	0	0
$281$	3042.00	0.645803	0.322901	−	0.946433i	$-0.395342\pi$
$281$	3042.00	0.645803	0.322901	+	0.946433i	$0.395342\pi$
$282$	0	0
$283$	−1718.00	−0.360864	−0.180432	−	0.983587i	$-0.557750\pi$
$283$	−1718.00	−0.360864	−0.180432	+	0.983587i	$0.557750\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6292.00	1.29409
$288$	0	0
$289$	−817.000	−0.166294
$290$	0	0
$291$	−2912.00	−0.586613
$292$	0	0
$293$	−2292.00	−0.456997	−0.228498	−	0.973544i	$-0.573382\pi$
$293$	−2292.00	−0.456997	−0.228498	+	0.973544i	$0.573382\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2800.00	−0.547045
$298$	0	0
$299$	696.000	0.134618
$300$	0	0
$301$	9412.00	1.80232
$302$	0	0
$303$	−1156.00	−0.219176
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5406.00	1.00501	0.502503	−	0.864576i	$-0.332413\pi$
$307$	5406.00	1.00501	0.502503	+	0.864576i	$0.332413\pi$
$308$	0	0
$309$	2924.00	0.538319
$310$	0	0
$311$	5688.00	1.03710	0.518548	−	0.855048i	$-0.326473\pi$
$311$	5688.00	1.03710	0.518548	+	0.855048i	$0.326473\pi$
$312$	0	0
$313$	−7352.00	−1.32767	−0.663833	−	0.747881i	$-0.731072\pi$
$313$	−7352.00	−1.32767	−0.663833	+	0.747881i	$0.731072\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3484.00	0.617290	0.308645	−	0.951177i	$-0.400124\pi$
$317$	3484.00	0.617290	0.308645	+	0.951177i	$0.400124\pi$
$318$	0	0
$319$	2520.00	0.442298
$320$	0	0
$321$	1932.00	0.335931
$322$	0	0
$323$	3840.00	0.661496
$324$	0	0
$325$	0	0
$326$	0	0
$327$	740.000	0.125144
$328$	0	0
$329$	5876.00	0.984664
$330$	0	0
$331$	7868.00	1.30654	0.653269	−	0.757125i	$-0.273397\pi$
$331$	7868.00	1.30654	0.653269	+	0.757125i	$0.273397\pi$
$332$	0	0
$333$	5428.00	0.893251
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−656.000	−0.106037	−0.0530187	−	0.998594i	$-0.516884\pi$
$337$	−656.000	−0.106037	−0.0530187	+	0.998594i	$0.516884\pi$
$338$	0	0
$339$	1056.00	0.169186
$340$	0	0
$341$	3584.00	0.569163
$342$	0	0
$343$	−260.000	−0.0409291
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5754.00	−0.890176	−0.445088	−	0.895487i	$-0.646828\pi$
$347$	−5754.00	−0.890176	−0.445088	+	0.895487i	$0.646828\pi$
$348$	0	0
$349$	−3110.00	−0.477004	−0.238502	−	0.971142i	$-0.576656\pi$
$349$	−3110.00	−0.477004	−0.238502	+	0.971142i	$0.576656\pi$
$350$	0	0
$351$	1200.00	0.182482
$352$	0	0
$353$	7808.00	1.17727	0.588637	−	0.808397i	$-0.299665\pi$
$353$	7808.00	1.17727	0.588637	+	0.808397i	$0.299665\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3328.00	0.493379
$358$	0	0
$359$	9240.00	1.35841	0.679204	−	0.733949i	$-0.262325\pi$
$359$	9240.00	1.35841	0.679204	+	0.733949i	$0.262325\pi$
$360$	0	0
$361$	−3259.00	−0.475142
$362$	0	0
$363$	−1094.00	−0.158182
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3214.00	−0.457137	−0.228569	−	0.973528i	$-0.573405\pi$
$367$	−3214.00	−0.457137	−0.228569	+	0.973528i	$0.573405\pi$
$368$	0	0
$369$	−5566.00	−0.785242
$370$	0	0
$371$	2808.00	0.392949
$372$	0	0
$373$	348.000	0.0483077	0.0241538	−	0.999708i	$-0.492311\pi$
$373$	348.000	0.0483077	0.0241538	+	0.999708i	$0.492311\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1080.00	−0.147541
$378$	0	0
$379$	−4940.00	−0.669527	−0.334764	−	0.942302i	$-0.608656\pi$
$379$	−4940.00	−0.669527	−0.334764	+	0.942302i	$0.608656\pi$
$380$	0	0
$381$	−3068.00	−0.412542
$382$	0	0
$383$	6142.00	0.819430	0.409715	−	0.912214i	$-0.365628\pi$
$383$	6142.00	0.819430	0.409715	+	0.912214i	$0.365628\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8326.00	−1.09363
$388$	0	0
$389$	3050.00	0.397535	0.198768	−	0.980047i	$-0.436306\pi$
$389$	3050.00	0.397535	0.198768	+	0.980047i	$0.436306\pi$
$390$	0	0
$391$	−3712.00	−0.480112
$392$	0	0
$393$	−24.0000	−0.00308051
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5396.00	−0.682160	−0.341080	−	0.940034i	$-0.610793\pi$
$397$	−5396.00	−0.682160	−0.341080	+	0.940034i	$0.610793\pi$
$398$	0	0
$399$	3120.00	0.391467
$400$	0	0
$401$	14482.0	1.80348	0.901741	−	0.432276i	$-0.142289\pi$
$401$	14482.0	1.80348	0.901741	+	0.432276i	$0.142289\pi$
$402$	0	0
$403$	−1536.00	−0.189860
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6608.00	−0.804782
$408$	0	0
$409$	−1090.00	−0.131778	−0.0658888	−	0.997827i	$-0.520988\pi$
$409$	−1090.00	−0.131778	−0.0658888	+	0.997827i	$0.520988\pi$
$410$	0	0
$411$	2448.00	0.293798
$412$	0	0
$413$	520.000	0.0619553
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6200.00	−0.728094
$418$	0	0
$419$	7180.00	0.837150	0.418575	−	0.908182i	$-0.362530\pi$
$419$	7180.00	0.837150	0.418575	+	0.908182i	$0.362530\pi$
$420$	0	0
$421$	−8138.00	−0.942095	−0.471047	−	0.882108i	$-0.656124\pi$
$421$	−8138.00	−0.942095	−0.471047	+	0.882108i	$0.656124\pi$
$422$	0	0
$423$	−5198.00	−0.597483
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14092.0	1.59710
$428$	0	0
$429$	−672.000	−0.0756281
$430$	0	0
$431$	208.000	0.0232460	0.0116230	−	0.999932i	$-0.496300\pi$
$431$	208.000	0.0232460	0.0116230	+	0.999932i	$0.496300\pi$
$432$	0	0
$433$	−12992.0	−1.44193	−0.720965	−	0.692971i	$-0.756301\pi$
$433$	−12992.0	−1.44193	−0.720965	+	0.692971i	$0.756301\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3480.00	−0.380940
$438$	0	0
$439$	−1080.00	−0.117416	−0.0587080	−	0.998275i	$-0.518698\pi$
$439$	−1080.00	−0.117416	−0.0587080	+	0.998275i	$0.518698\pi$
$440$	0	0
$441$	−7659.00	−0.827017
$442$	0	0
$443$	−9078.00	−0.973609	−0.486805	−	0.873511i	$-0.661838\pi$
$443$	−9078.00	−0.973609	−0.486805	+	0.873511i	$0.661838\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	500.000	0.0529065
$448$	0	0
$449$	14310.0	1.50408	0.752039	−	0.659119i	$-0.229071\pi$
$449$	14310.0	1.50408	0.752039	+	0.659119i	$0.229071\pi$
$450$	0	0
$451$	6776.00	0.707471
$452$	0	0
$453$	−4304.00	−0.446401
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2344.00	0.239929	0.119965	−	0.992778i	$-0.461722\pi$
$457$	2344.00	0.239929	0.119965	+	0.992778i	$0.461722\pi$
$458$	0	0
$459$	−6400.00	−0.650820
$460$	0	0
$461$	11382.0	1.14992	0.574959	−	0.818182i	$-0.305018\pi$
$461$	11382.0	1.14992	0.574959	+	0.818182i	$0.305018\pi$
$462$	0	0
$463$	16062.0	1.61223	0.806117	−	0.591756i	$-0.201565\pi$
$463$	16062.0	1.61223	0.806117	+	0.591756i	$0.201565\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17166.0	1.70096	0.850479	−	0.526008i	$-0.176312\pi$
$467$	17166.0	1.70096	0.850479	+	0.526008i	$0.176312\pi$
$468$	0	0
$469$	−11284.0	−1.11097
$470$	0	0
$471$	1048.00	0.102525
$472$	0	0
$473$	10136.0	0.985315
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2484.00	−0.238437
$478$	0	0
$479$	−7520.00	−0.717323	−0.358661	−	0.933468i	$-0.616767\pi$
$479$	−7520.00	−0.717323	−0.358661	+	0.933468i	$0.616767\pi$
$480$	0	0
$481$	2832.00	0.268458
$482$	0	0
$483$	−3016.00	−0.284126
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11814.0	−1.09927	−0.549634	−	0.835406i	$-0.685233\pi$
$487$	−11814.0	−1.09927	−0.549634	+	0.835406i	$0.685233\pi$
$488$	0	0
$489$	−7036.00	−0.650673
$490$	0	0
$491$	−14052.0	−1.29156	−0.645782	−	0.763522i	$-0.723468\pi$
$491$	−14052.0	−1.29156	−0.645782	+	0.763522i	$0.723468\pi$
$492$	0	0
$493$	5760.00	0.526202
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29328.0	2.64696
$498$	0	0
$499$	−7620.00	−0.683603	−0.341802	−	0.939772i	$-0.611037\pi$
$499$	−7620.00	−0.683603	−0.341802	+	0.939772i	$0.611037\pi$
$500$	0	0
$501$	−1068.00	−0.0952390
$502$	0	0
$503$	−1818.00	−0.161154	−0.0805772	−	0.996748i	$-0.525676\pi$
$503$	−1818.00	−0.161154	−0.0805772	+	0.996748i	$0.525676\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4106.00	−0.359672
$508$	0	0
$509$	17850.0	1.55440	0.777198	−	0.629256i	$-0.216640\pi$
$509$	17850.0	1.55440	0.777198	+	0.629256i	$0.216640\pi$
$510$	0	0
$511$	−16432.0	−1.42252
$512$	0	0
$513$	−6000.00	−0.516387
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6328.00	0.538308
$518$	0	0
$519$	−8504.00	−0.719237
$520$	0	0
$521$	−19238.0	−1.61772	−0.808860	−	0.588001i	$-0.799915\pi$
$521$	−19238.0	−1.61772	−0.808860	+	0.588001i	$0.799915\pi$
$522$	0	0
$523$	−6278.00	−0.524891	−0.262445	−	0.964947i	$-0.584529\pi$
$523$	−6278.00	−0.524891	−0.262445	+	0.964947i	$0.584529\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8192.00	0.677133
$528$	0	0
$529$	−8803.00	−0.723514
$530$	0	0
$531$	−460.000	−0.0375938
$532$	0	0
$533$	−2904.00	−0.235997
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5000.00	−0.401799
$538$	0	0
$539$	9324.00	0.745108
$540$	0	0
$541$	−9818.00	−0.780238	−0.390119	−	0.920764i	$-0.627566\pi$
$541$	−9818.00	−0.780238	−0.390119	+	0.920764i	$0.627566\pi$
$542$	0	0
$543$	−5156.00	−0.407486
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12514.0	−0.978172	−0.489086	−	0.872236i	$-0.662670\pi$
$547$	−12514.0	−0.978172	−0.489086	+	0.872236i	$0.662670\pi$
$548$	0	0
$549$	−12466.0	−0.969100
$550$	0	0
$551$	5400.00	0.417509
$552$	0	0
$553$	18720.0	1.43952
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10596.0	−0.806045	−0.403022	−	0.915190i	$-0.632040\pi$
$557$	−10596.0	−0.806045	−0.403022	+	0.915190i	$0.632040\pi$
$558$	0	0
$559$	−4344.00	−0.328679
$560$	0	0
$561$	3584.00	0.269727
$562$	0	0
$563$	14002.0	1.04816	0.524080	−	0.851669i	$-0.324409\pi$
$563$	14002.0	1.04816	0.524080	+	0.851669i	$0.324409\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10946.0	0.810739
$568$	0	0
$569$	−7330.00	−0.540052	−0.270026	−	0.962853i	$-0.587032\pi$
$569$	−7330.00	−0.540052	−0.270026	+	0.962853i	$0.587032\pi$
$570$	0	0
$571$	−5812.00	−0.425963	−0.212981	−	0.977056i	$-0.568317\pi$
$571$	−5812.00	−0.425963	−0.212981	+	0.977056i	$0.568317\pi$
$572$	0	0
$573$	1536.00	0.111985
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−16736.0	−1.20750	−0.603751	−	0.797173i	$-0.706328\pi$
$577$	−16736.0	−1.20750	−0.603751	+	0.797173i	$0.706328\pi$
$578$	0	0
$579$	5216.00	0.374386
$580$	0	0
$581$	−12428.0	−0.887436
$582$	0	0
$583$	3024.00	0.214822
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7434.00	−0.522716	−0.261358	−	0.965242i	$-0.584170\pi$
$587$	−7434.00	−0.522716	−0.261358	+	0.965242i	$0.584170\pi$
$588$	0	0
$589$	7680.00	0.537265
$590$	0	0
$591$	−10232.0	−0.712163
$592$	0	0
$593$	−25872.0	−1.79163	−0.895814	−	0.444429i	$-0.853407\pi$
$593$	−25872.0	−1.79163	−0.895814	+	0.444429i	$0.853407\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6960.00	0.477142
$598$	0	0
$599$	3720.00	0.253748	0.126874	−	0.991919i	$-0.459506\pi$
$599$	3720.00	0.253748	0.126874	+	0.991919i	$0.459506\pi$
$600$	0	0
$601$	−12958.0	−0.879481	−0.439740	−	0.898125i	$-0.644930\pi$
$601$	−12958.0	−0.879481	−0.439740	+	0.898125i	$0.644930\pi$
$602$	0	0
$603$	9982.00	0.674127
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7214.00	−0.482384	−0.241192	−	0.970477i	$-0.577538\pi$
$607$	−7214.00	−0.482384	−0.241192	+	0.970477i	$0.577538\pi$
$608$	0	0
$609$	4680.00	0.311401
$610$	0	0
$611$	−2712.00	−0.179568
$612$	0	0
$613$	4828.00	0.318109	0.159055	−	0.987270i	$-0.449155\pi$
$613$	4828.00	0.318109	0.159055	+	0.987270i	$0.449155\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27656.0	−1.80452	−0.902260	−	0.431193i	$-0.858093\pi$
$617$	−27656.0	−1.80452	−0.902260	+	0.431193i	$0.858093\pi$
$618$	0	0
$619$	21220.0	1.37787	0.688937	−	0.724821i	$-0.258078\pi$
$619$	21220.0	1.37787	0.688937	+	0.724821i	$0.258078\pi$
$620$	0	0
$621$	5800.00	0.374792
$622$	0	0
$623$	−12740.0	−0.819289
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3360.00	0.214012
$628$	0	0
$629$	−15104.0	−0.957450
$630$	0	0
$631$	−17672.0	−1.11491	−0.557457	−	0.830206i	$-0.688223\pi$
$631$	−17672.0	−1.11491	−0.557457	+	0.830206i	$0.688223\pi$
$632$	0	0
$633$	−6264.00	−0.393320
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3996.00	−0.248551
$638$	0	0
$639$	−25944.0	−1.60615
$640$	0	0
$641$	7322.00	0.451173	0.225586	−	0.974223i	$-0.427570\pi$
$641$	7322.00	0.451173	0.225586	+	0.974223i	$0.427570\pi$
$642$	0	0
$643$	−8238.00	−0.505249	−0.252624	−	0.967564i	$-0.581294\pi$
$643$	−8238.00	−0.505249	−0.252624	+	0.967564i	$0.581294\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6426.00	0.390467	0.195233	−	0.980757i	$-0.437454\pi$
$647$	6426.00	0.390467	0.195233	+	0.980757i	$0.437454\pi$
$648$	0	0
$649$	560.000	0.0338705
$650$	0	0
$651$	6656.00	0.400721
$652$	0	0
$653$	5908.00	0.354055	0.177027	−	0.984206i	$-0.443352\pi$
$653$	5908.00	0.354055	0.177027	+	0.984206i	$0.443352\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14536.0	0.863171
$658$	0	0
$659$	26780.0	1.58301	0.791503	−	0.611166i	$-0.209299\pi$
$659$	26780.0	1.58301	0.791503	+	0.611166i	$0.209299\pi$
$660$	0	0
$661$	−24538.0	−1.44390	−0.721950	−	0.691945i	$-0.756754\pi$
$661$	−24538.0	−1.44390	−0.721950	+	0.691945i	$0.756754\pi$
$662$	0	0
$663$	−1536.00	−0.0899748
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5220.00	−0.303027
$668$	0	0
$669$	124.000	0.00716609
$670$	0	0
$671$	15176.0	0.873119
$672$	0	0
$673$	28848.0	1.65232	0.826158	−	0.563439i	$-0.190522\pi$
$673$	28848.0	1.65232	0.826158	+	0.563439i	$0.190522\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26884.0	1.52620	0.763099	−	0.646282i	$-0.223677\pi$
$677$	26884.0	1.52620	0.763099	+	0.646282i	$0.223677\pi$
$678$	0	0
$679$	−37856.0	−2.13959
$680$	0	0
$681$	−10628.0	−0.598041
$682$	0	0
$683$	14282.0	0.800125	0.400063	−	0.916488i	$-0.368988\pi$
$683$	14282.0	0.800125	0.400063	+	0.916488i	$0.368988\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−380.000	−0.0211032
$688$	0	0
$689$	−1296.00	−0.0716599
$690$	0	0
$691$	3428.00	0.188723	0.0943613	−	0.995538i	$-0.469919\pi$
$691$	3428.00	0.188723	0.0943613	+	0.995538i	$0.469919\pi$
$692$	0	0
$693$	−16744.0	−0.917824
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15488.0	0.841678
$698$	0	0
$699$	4816.00	0.260598
$700$	0	0
$701$	26942.0	1.45162	0.725810	−	0.687895i	$-0.241465\pi$
$701$	26942.0	1.45162	0.725810	+	0.687895i	$0.241465\pi$
$702$	0	0
$703$	−14160.0	−0.759679
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15028.0	−0.799415
$708$	0	0
$709$	−1950.00	−0.103292	−0.0516458	−	0.998665i	$-0.516447\pi$
$709$	−1950.00	−0.103292	−0.0516458	+	0.998665i	$0.516447\pi$
$710$	0	0
$711$	−16560.0	−0.873486
$712$	0	0
$713$	−7424.00	−0.389945
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11360.0	0.591697
$718$	0	0
$719$	−12080.0	−0.626576	−0.313288	−	0.949658i	$-0.601430\pi$
$719$	−12080.0	−0.626576	−0.313288	+	0.949658i	$0.601430\pi$
$720$	0	0
$721$	38012.0	1.96344
$722$	0	0
$723$	−556.000	−0.0286001
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17226.0	0.878785	0.439393	−	0.898295i	$-0.355194\pi$
$727$	17226.0	0.878785	0.439393	+	0.898295i	$0.355194\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	0	0
$731$	23168.0	1.17223
$732$	0	0
$733$	788.000	0.0397073	0.0198536	−	0.999803i	$-0.493680\pi$
$733$	788.000	0.0397073	0.0198536	+	0.999803i	$0.493680\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12152.0	−0.607360
$738$	0	0
$739$	2060.00	0.102542	0.0512709	−	0.998685i	$-0.483673\pi$
$739$	2060.00	0.102542	0.0512709	+	0.998685i	$0.483673\pi$
$740$	0	0
$741$	−1440.00	−0.0713896
$742$	0	0
$743$	−3258.00	−0.160867	−0.0804337	−	0.996760i	$-0.525631\pi$
$743$	−3258.00	−0.160867	−0.0804337	+	0.996760i	$0.525631\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10994.0	0.538487
$748$	0	0
$749$	25116.0	1.22526
$750$	0	0
$751$	4528.00	0.220012	0.110006	−	0.993931i	$-0.464913\pi$
$751$	4528.00	0.220012	0.110006	+	0.993931i	$0.464913\pi$
$752$	0	0
$753$	−6504.00	−0.314766
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−18236.0	−0.875560	−0.437780	−	0.899082i	$-0.644235\pi$
$757$	−18236.0	−0.875560	−0.437780	+	0.899082i	$0.644235\pi$
$758$	0	0
$759$	−3248.00	−0.155329
$760$	0	0
$761$	−18678.0	−0.889720	−0.444860	−	0.895600i	$-0.646747\pi$
$761$	−18678.0	−0.889720	−0.444860	+	0.895600i	$0.646747\pi$
$762$	0	0
$763$	9620.00	0.456445
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−240.000	−0.0112984
$768$	0	0
$769$	27390.0	1.28441	0.642203	−	0.766534i	$-0.278020\pi$
$769$	27390.0	1.28441	0.642203	+	0.766534i	$0.278020\pi$
$770$	0	0
$771$	−3072.00	−0.143496
$772$	0	0
$773$	−9252.00	−0.430493	−0.215247	−	0.976560i	$-0.569056\pi$
$773$	−9252.00	−0.430493	−0.215247	+	0.976560i	$0.569056\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12272.0	−0.566609
$778$	0	0
$779$	14520.0	0.667822
$780$	0	0
$781$	31584.0	1.44707
$782$	0	0
$783$	−9000.00	−0.410771
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5726.00	0.259352	0.129676	−	0.991556i	$-0.458606\pi$
$787$	5726.00	0.259352	0.129676	+	0.991556i	$0.458606\pi$
$788$	0	0
$789$	−9716.00	−0.438401
$790$	0	0
$791$	13728.0	0.617082
$792$	0	0
$793$	−6504.00	−0.291253
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27236.0	−1.21048	−0.605238	−	0.796045i	$-0.706922\pi$
$797$	−27236.0	−1.21048	−0.605238	+	0.796045i	$0.706922\pi$
$798$	0	0
$799$	14464.0	0.640425
$800$	0	0
$801$	11270.0	0.497136
$802$	0	0
$803$	−17696.0	−0.777682
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5220.00	0.227699
$808$	0	0
$809$	10950.0	0.475873	0.237937	−	0.971281i	$-0.423529\pi$
$809$	10950.0	0.475873	0.237937	+	0.971281i	$0.423529\pi$
$810$	0	0
$811$	8828.00	0.382236	0.191118	−	0.981567i	$-0.438789\pi$
$811$	8828.00	0.382236	0.191118	+	0.981567i	$0.438789\pi$
$812$	0	0
$813$	10336.0	0.445879
$814$	0	0
$815$	0	0
$816$	0	0
$817$	21720.0	0.930094
$818$	0	0
$819$	7176.00	0.306166
$820$	0	0
$821$	−16058.0	−0.682616	−0.341308	−	0.939951i	$-0.610870\pi$
$821$	−16058.0	−0.682616	−0.341308	+	0.939951i	$0.610870\pi$
$822$	0	0
$823$	41862.0	1.77305	0.886523	−	0.462684i	$-0.153113\pi$
$823$	41862.0	1.77305	0.886523	+	0.462684i	$0.153113\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12154.0	−0.511047	−0.255524	−	0.966803i	$-0.582248\pi$
$827$	−12154.0	−0.511047	−0.255524	+	0.966803i	$0.582248\pi$
$828$	0	0
$829$	−15390.0	−0.644773	−0.322386	−	0.946608i	$-0.604485\pi$
$829$	−15390.0	−0.644773	−0.322386	+	0.946608i	$0.604485\pi$
$830$	0	0
$831$	3848.00	0.160633
$832$	0	0
$833$	21312.0	0.886455
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−12800.0	−0.528593
$838$	0	0
$839$	4280.00	0.176117	0.0880584	−	0.996115i	$-0.471934\pi$
$839$	4280.00	0.176117	0.0880584	+	0.996115i	$0.471934\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	0	0
$843$	6084.00	0.248570
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−14222.0	−0.576947
$848$	0	0
$849$	−3436.00	−0.138897
$850$	0	0
$851$	13688.0	0.551373
$852$	0	0
$853$	−14452.0	−0.580102	−0.290051	−	0.957011i	$-0.593672\pi$
$853$	−14452.0	−0.580102	−0.290051	+	0.957011i	$0.593672\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22584.0	0.900181	0.450090	−	0.892983i	$-0.351392\pi$
$857$	22584.0	0.900181	0.450090	+	0.892983i	$0.351392\pi$
$858$	0	0
$859$	26740.0	1.06212	0.531058	−	0.847336i	$-0.321795\pi$
$859$	26740.0	1.06212	0.531058	+	0.847336i	$0.321795\pi$
$860$	0	0
$861$	12584.0	0.498097
$862$	0	0
$863$	−498.000	−0.0196432	−0.00982162	−	0.999952i	$-0.503126\pi$
$863$	−498.000	−0.0196432	−0.00982162	+	0.999952i	$0.503126\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1634.00	−0.0640064
$868$	0	0
$869$	20160.0	0.786975
$870$	0	0
$871$	5208.00	0.202602
$872$	0	0
$873$	33488.0	1.29828
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13244.0	0.509941	0.254970	−	0.966949i	$-0.417934\pi$
$877$	13244.0	0.509941	0.254970	+	0.966949i	$0.417934\pi$
$878$	0	0
$879$	−4584.00	−0.175898
$880$	0	0
$881$	40842.0	1.56186	0.780932	−	0.624616i	$-0.214745\pi$
$881$	40842.0	1.56186	0.780932	+	0.624616i	$0.214745\pi$
$882$	0	0
$883$	−12078.0	−0.460314	−0.230157	−	0.973154i	$-0.573924\pi$
$883$	−12078.0	−0.460314	−0.230157	+	0.973154i	$0.573924\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−18294.0	−0.692506	−0.346253	−	0.938141i	$-0.612546\pi$
$887$	−18294.0	−0.692506	−0.346253	+	0.938141i	$0.612546\pi$
$888$	0	0
$889$	−39884.0	−1.50469
$890$	0	0
$891$	11788.0	0.443224
$892$	0	0
$893$	13560.0	0.508139
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1392.00	0.0518144
$898$	0	0
$899$	11520.0	0.427379
$900$	0	0
$901$	6912.00	0.255574
$902$	0	0
$903$	18824.0	0.693714
$904$	0	0
$905$	0	0
$906$	0	0
$907$	22566.0	0.826121	0.413060	−	0.910704i	$-0.364460\pi$
$907$	22566.0	0.826121	0.413060	+	0.910704i	$0.364460\pi$
$908$	0	0
$909$	13294.0	0.485076
$910$	0	0
$911$	6768.00	0.246140	0.123070	−	0.992398i	$-0.460726\pi$
$911$	6768.00	0.246140	0.123070	+	0.992398i	$0.460726\pi$
$912$	0	0
$913$	−13384.0	−0.485154
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−312.000	−0.0112357
$918$	0	0
$919$	−22200.0	−0.796856	−0.398428	−	0.917200i	$-0.630444\pi$
$919$	−22200.0	−0.796856	−0.398428	+	0.917200i	$0.630444\pi$
$920$	0	0
$921$	10812.0	0.386827
$922$	0	0
$923$	−13536.0	−0.482712
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−33626.0	−1.19139
$928$	0	0
$929$	−6330.00	−0.223553	−0.111776	−	0.993733i	$-0.535654\pi$
$929$	−6330.00	−0.223553	−0.111776	+	0.993733i	$0.535654\pi$
$930$	0	0
$931$	19980.0	0.703349
$932$	0	0
$933$	11376.0	0.399178
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19544.0	0.681403	0.340702	−	0.940172i	$-0.389335\pi$
$937$	19544.0	0.681403	0.340702	+	0.940172i	$0.389335\pi$
$938$	0	0
$939$	−14704.0	−0.511019
$940$	0	0
$941$	−9898.00	−0.342896	−0.171448	−	0.985193i	$-0.554845\pi$
$941$	−9898.00	−0.342896	−0.171448	+	0.985193i	$0.554845\pi$
$942$	0	0
$943$	−14036.0	−0.484703
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41406.0	1.42082	0.710409	−	0.703789i	$-0.248510\pi$
$947$	41406.0	1.42082	0.710409	+	0.703789i	$0.248510\pi$
$948$	0	0
$949$	7584.00	0.259417
$950$	0	0
$951$	6968.00	0.237595
$952$	0	0
$953$	−25432.0	−0.864453	−0.432226	−	0.901765i	$-0.642272\pi$
$953$	−25432.0	−0.864453	−0.432226	+	0.901765i	$0.642272\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5040.00	0.170240
$958$	0	0
$959$	31824.0	1.07159
$960$	0	0
$961$	−13407.0	−0.450035
$962$	0	0
$963$	−22218.0	−0.743474
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12106.0	0.402588	0.201294	−	0.979531i	$-0.435485\pi$
$967$	12106.0	0.402588	0.201294	+	0.979531i	$0.435485\pi$
$968$	0	0
$969$	7680.00	0.254610
$970$	0	0
$971$	−7812.00	−0.258186	−0.129093	−	0.991632i	$-0.541207\pi$
$971$	−7812.00	−0.258186	−0.129093	+	0.991632i	$0.541207\pi$
$972$	0	0
$973$	−80600.0	−2.65562
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12576.0	−0.411814	−0.205907	−	0.978572i	$-0.566014\pi$
$977$	−12576.0	−0.411814	−0.205907	+	0.978572i	$0.566014\pi$
$978$	0	0
$979$	−13720.0	−0.447899
$980$	0	0
$981$	−8510.00	−0.276966
$982$	0	0
$983$	4342.00	0.140883	0.0704417	−	0.997516i	$-0.477559\pi$
$983$	4342.00	0.140883	0.0704417	+	0.997516i	$0.477559\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11752.0	0.378997
$988$	0	0
$989$	−20996.0	−0.675060
$990$	0	0
$991$	−26272.0	−0.842137	−0.421068	−	0.907029i	$-0.638345\pi$
$991$	−26272.0	−0.842137	−0.421068	+	0.907029i	$0.638345\pi$
$992$	0	0
$993$	15736.0	0.502887
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−44796.0	−1.42297	−0.711486	−	0.702700i	$-0.751978\pi$
$997$	−44796.0	−1.42297	−0.711486	+	0.702700i	$0.751978\pi$
$998$	0	0
$999$	23600.0	0.747418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.a.n.1.1		1
4.3	odd	2		50.4.a.b.1.1		1
5.2	odd	4		80.4.c.a.49.1		2
5.3	odd	4		80.4.c.a.49.2		2
5.4	even	2		400.4.a.h.1.1		1
8.3	odd	2		1600.4.a.bh.1.1		1
8.5	even	2		1600.4.a.t.1.1		1
12.11	even	2		450.4.a.k.1.1		1
15.2	even	4		720.4.f.f.289.1		2
15.8	even	4		720.4.f.f.289.2		2
20.3	even	4		10.4.b.a.9.2	yes	2
20.7	even	4		10.4.b.a.9.1	✓	2
20.19	odd	2		50.4.a.d.1.1		1
28.27	even	2		2450.4.a.o.1.1		1
40.3	even	4		320.4.c.d.129.2		2
40.13	odd	4		320.4.c.c.129.1		2
40.19	odd	2		1600.4.a.u.1.1		1
40.27	even	4		320.4.c.d.129.1		2
40.29	even	2		1600.4.a.bg.1.1		1
40.37	odd	4		320.4.c.c.129.2		2
60.23	odd	4		90.4.c.b.19.1		2
60.47	odd	4		90.4.c.b.19.2		2
60.59	even	2		450.4.a.j.1.1		1
140.27	odd	4		490.4.c.b.99.1		2
140.83	odd	4		490.4.c.b.99.2		2
140.139	even	2		2450.4.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.b.a.9.1	✓	2	20.7	even	4
10.4.b.a.9.2	yes	2	20.3	even	4
50.4.a.b.1.1		1	4.3	odd	2
50.4.a.d.1.1		1	20.19	odd	2
80.4.c.a.49.1		2	5.2	odd	4
80.4.c.a.49.2		2	5.3	odd	4
90.4.c.b.19.1		2	60.23	odd	4
90.4.c.b.19.2		2	60.47	odd	4
320.4.c.c.129.1		2	40.13	odd	4
320.4.c.c.129.2		2	40.37	odd	4
320.4.c.d.129.1		2	40.27	even	4
320.4.c.d.129.2		2	40.3	even	4
400.4.a.h.1.1		1	5.4	even	2
400.4.a.n.1.1		1	1.1	even	1	trivial
450.4.a.j.1.1		1	60.59	even	2
450.4.a.k.1.1		1	12.11	even	2
490.4.c.b.99.1		2	140.27	odd	4
490.4.c.b.99.2		2	140.83	odd	4
720.4.f.f.289.1		2	15.2	even	4
720.4.f.f.289.2		2	15.8	even	4
1600.4.a.t.1.1		1	8.5	even	2
1600.4.a.u.1.1		1	40.19	odd	2
1600.4.a.bg.1.1		1	40.29	even	2
1600.4.a.bh.1.1		1	8.3	odd	2
2450.4.a.o.1.1		1	28.27	even	2
2450.4.a.bb.1.1		1	140.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

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L-functions

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