LMFDB - Embedded newform 400.4.a.r.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
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)

$f(q)$

$=$

$q+7.00000 q^{3} -34.0000 q^{7} +22.0000 q^{9} +O(q^{10})$

$q+7.00000 q^{3} -34.0000 q^{7} +22.0000 q^{9} -27.0000 q^{11} +28.0000 q^{13} -21.0000 q^{17} -35.0000 q^{19} -238.000 q^{21} -78.0000 q^{23} -35.0000 q^{27} -120.000 q^{29} -182.000 q^{31} -189.000 q^{33} -146.000 q^{37} +196.000 q^{39} +357.000 q^{41} -148.000 q^{43} -84.0000 q^{47} +813.000 q^{49} -147.000 q^{51} -702.000 q^{53} -245.000 q^{57} +840.000 q^{59} -238.000 q^{61} -748.000 q^{63} +461.000 q^{67} -546.000 q^{69} +708.000 q^{71} +133.000 q^{73} +918.000 q^{77} -650.000 q^{79} -839.000 q^{81} -903.000 q^{83} -840.000 q^{87} +735.000 q^{89} -952.000 q^{91} -1274.00 q^{93} -1106.00 q^{97} -594.000 q^{99} +O(q^{100})$

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
$3$	7.00000	1.34715	0.673575	−	0.739119i	$-0.264758\pi$
$3$	7.00000	1.34715	0.673575	+	0.739119i	$0.264758\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−34.0000	−1.83583	−0.917914	−	0.396780i	$-0.870128\pi$
$7$	−34.0000	−1.83583	−0.917914	+	0.396780i	$0.870128\pi$
$8$	0	0
$9$	22.0000	0.814815
$10$	0	0
$11$	−27.0000	−0.740073	−0.370037	−	0.929017i	$-0.620655\pi$
$11$	−27.0000	−0.740073	−0.370037	+	0.929017i	$0.620655\pi$
$12$	0	0
$13$	28.0000	0.597369	0.298685	−	0.954352i	$-0.403452\pi$
$13$	28.0000	0.597369	0.298685	+	0.954352i	$0.403452\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−21.0000	−0.299603	−0.149801	−	0.988716i	$-0.547863\pi$
$17$	−21.0000	−0.299603	−0.149801	+	0.988716i	$0.547863\pi$
$18$	0	0
$19$	−35.0000	−0.422608	−0.211304	−	0.977420i	$-0.567771\pi$
$19$	−35.0000	−0.422608	−0.211304	+	0.977420i	$0.567771\pi$
$20$	0	0
$21$	−238.000	−2.47314
$22$	0	0
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−35.0000	−0.249472
$28$	0	0
$29$	−120.000	−0.768395	−0.384197	−	0.923251i	$-0.625522\pi$
$29$	−120.000	−0.768395	−0.384197	+	0.923251i	$0.625522\pi$
$30$	0	0
$31$	−182.000	−1.05446	−0.527228	−	0.849724i	$-0.676769\pi$
$31$	−182.000	−1.05446	−0.527228	+	0.849724i	$0.676769\pi$
$32$	0	0
$33$	−189.000	−0.996990
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	0	0
$39$	196.000	0.804747
$40$	0	0
$41$	357.000	1.35985	0.679927	−	0.733280i	$-0.262011\pi$
$41$	357.000	1.35985	0.679927	+	0.733280i	$0.262011\pi$
$42$	0	0
$43$	−148.000	−0.524879	−0.262439	−	0.964948i	$-0.584527\pi$
$43$	−148.000	−0.524879	−0.262439	+	0.964948i	$0.584527\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−84.0000	−0.260695	−0.130347	−	0.991468i	$-0.541609\pi$
$47$	−84.0000	−0.260695	−0.130347	+	0.991468i	$0.541609\pi$
$48$	0	0
$49$	813.000	2.37026
$50$	0	0
$51$	−147.000	−0.403610
$52$	0	0
$53$	−702.000	−1.81938	−0.909690	−	0.415288i	$-0.863681\pi$
$53$	−702.000	−1.81938	−0.909690	+	0.415288i	$0.863681\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−245.000	−0.569317
$58$	0	0
$59$	840.000	1.85354	0.926769	−	0.375633i	$-0.122575\pi$
$59$	840.000	1.85354	0.926769	+	0.375633i	$0.122575\pi$
$60$	0	0
$61$	−238.000	−0.499554	−0.249777	−	0.968303i	$-0.580357\pi$
$61$	−238.000	−0.499554	−0.249777	+	0.968303i	$0.580357\pi$
$62$	0	0
$63$	−748.000	−1.49586
$64$	0	0
$65$	0	0
$66$	0	0
$67$	461.000	0.840599	0.420299	−	0.907386i	$-0.361925\pi$
$67$	461.000	0.840599	0.420299	+	0.907386i	$0.361925\pi$
$68$	0	0
$69$	−546.000	−0.952618
$70$	0	0
$71$	708.000	1.18344	0.591719	−	0.806144i	$-0.298449\pi$
$71$	708.000	1.18344	0.591719	+	0.806144i	$0.298449\pi$
$72$	0	0
$73$	133.000	0.213239	0.106620	−	0.994300i	$-0.465997\pi$
$73$	133.000	0.213239	0.106620	+	0.994300i	$0.465997\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	918.000	1.35865
$78$	0	0
$79$	−650.000	−0.925705	−0.462853	−	0.886435i	$-0.653174\pi$
$79$	−650.000	−0.925705	−0.462853	+	0.886435i	$0.653174\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	0	0
$83$	−903.000	−1.19418	−0.597091	−	0.802173i	$-0.703677\pi$
$83$	−903.000	−1.19418	−0.597091	+	0.802173i	$0.703677\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−840.000	−1.03514
$88$	0	0
$89$	735.000	0.875392	0.437696	−	0.899123i	$-0.355795\pi$
$89$	735.000	0.875392	0.437696	+	0.899123i	$0.355795\pi$
$90$	0	0
$91$	−952.000	−1.09667
$92$	0	0
$93$	−1274.00	−1.42051
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1106.00	−1.15770	−0.578852	−	0.815433i	$-0.696499\pi$
$97$	−1106.00	−1.15770	−0.578852	+	0.815433i	$0.696499\pi$
$98$	0	0
$99$	−594.000	−0.603023
$100$	0	0
$101$	462.000	0.455156	0.227578	−	0.973760i	$-0.426919\pi$
$101$	462.000	0.455156	0.227578	+	0.973760i	$0.426919\pi$
$102$	0	0
$103$	812.000	0.776784	0.388392	−	0.921494i	$-0.373031\pi$
$103$	812.000	0.776784	0.388392	+	0.921494i	$0.373031\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−789.000	−0.712855	−0.356428	−	0.934323i	$-0.616005\pi$
$107$	−789.000	−0.712855	−0.356428	+	0.934323i	$0.616005\pi$
$108$	0	0
$109$	230.000	0.202110	0.101055	−	0.994881i	$-0.467778\pi$
$109$	230.000	0.202110	0.101055	+	0.994881i	$0.467778\pi$
$110$	0	0
$111$	−1022.00	−0.873909
$112$	0	0
$113$	2073.00	1.72576	0.862882	−	0.505405i	$-0.168657\pi$
$113$	2073.00	1.72576	0.862882	+	0.505405i	$0.168657\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	616.000	0.486745
$118$	0	0
$119$	714.000	0.550019
$120$	0	0
$121$	−602.000	−0.452292
$122$	0	0
$123$	2499.00	1.83193
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1114.00	−0.778358	−0.389179	−	0.921162i	$-0.627241\pi$
$127$	−1114.00	−0.778358	−0.389179	+	0.921162i	$0.627241\pi$
$128$	0	0
$129$	−1036.00	−0.707091
$130$	0	0
$131$	−252.000	−0.168071	−0.0840357	−	0.996463i	$-0.526781\pi$
$131$	−252.000	−0.168071	−0.0840357	+	0.996463i	$0.526781\pi$
$132$	0	0
$133$	1190.00	0.775835
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1941.00	−1.21044	−0.605222	−	0.796057i	$-0.706916\pi$
$137$	−1941.00	−1.21044	−0.605222	+	0.796057i	$0.706916\pi$
$138$	0	0
$139$	1645.00	1.00379	0.501896	−	0.864928i	$-0.332636\pi$
$139$	1645.00	1.00379	0.501896	+	0.864928i	$0.332636\pi$
$140$	0	0
$141$	−588.000	−0.351195
$142$	0	0
$143$	−756.000	−0.442097
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5691.00	3.19310
$148$	0	0
$149$	−1800.00	−0.989676	−0.494838	−	0.868985i	$-0.664773\pi$
$149$	−1800.00	−0.989676	−0.494838	+	0.868985i	$0.664773\pi$
$150$	0	0
$151$	3298.00	1.77740	0.888700	−	0.458489i	$-0.151609\pi$
$151$	3298.00	1.77740	0.888700	+	0.458489i	$0.151609\pi$
$152$	0	0
$153$	−462.000	−0.244121
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−266.000	−0.135217	−0.0676086	−	0.997712i	$-0.521537\pi$
$157$	−266.000	−0.135217	−0.0676086	+	0.997712i	$0.521537\pi$
$158$	0	0
$159$	−4914.00	−2.45098
$160$	0	0
$161$	2652.00	1.29818
$162$	0	0
$163$	1157.00	0.555971	0.277985	−	0.960585i	$-0.410333\pi$
$163$	1157.00	0.555971	0.277985	+	0.960585i	$0.410333\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1764.00	−0.817380	−0.408690	−	0.912673i	$-0.634014\pi$
$167$	−1764.00	−0.817380	−0.408690	+	0.912673i	$0.634014\pi$
$168$	0	0
$169$	−1413.00	−0.643150
$170$	0	0
$171$	−770.000	−0.344347
$172$	0	0
$173$	1848.00	0.812144	0.406072	−	0.913841i	$-0.366898\pi$
$173$	1848.00	0.812144	0.406072	+	0.913841i	$0.366898\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5880.00	2.49699
$178$	0	0
$179$	−135.000	−0.0563708	−0.0281854	−	0.999603i	$-0.508973\pi$
$179$	−135.000	−0.0563708	−0.0281854	+	0.999603i	$0.508973\pi$
$180$	0	0
$181$	2282.00	0.937126	0.468563	−	0.883430i	$-0.344772\pi$
$181$	2282.00	0.937126	0.468563	+	0.883430i	$0.344772\pi$
$182$	0	0
$183$	−1666.00	−0.672974
$184$	0	0
$185$	0	0
$186$	0	0
$187$	567.000	0.221728
$188$	0	0
$189$	1190.00	0.457988
$190$	0	0
$191$	1398.00	0.529611	0.264806	−	0.964302i	$-0.414692\pi$
$191$	1398.00	0.529611	0.264806	+	0.964302i	$0.414692\pi$
$192$	0	0
$193$	−3317.00	−1.23711	−0.618557	−	0.785740i	$-0.712282\pi$
$193$	−3317.00	−1.23711	−0.618557	+	0.785740i	$0.712282\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1686.00	−0.609759	−0.304880	−	0.952391i	$-0.598616\pi$
$197$	−1686.00	−0.609759	−0.304880	+	0.952391i	$0.598616\pi$
$198$	0	0
$199$	1540.00	0.548581	0.274291	−	0.961647i	$-0.411557\pi$
$199$	1540.00	0.548581	0.274291	+	0.961647i	$0.411557\pi$
$200$	0	0
$201$	3227.00	1.13241
$202$	0	0
$203$	4080.00	1.41064
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1716.00	−0.576185
$208$	0	0
$209$	945.000	0.312761
$210$	0	0
$211$	3043.00	0.992838	0.496419	−	0.868083i	$-0.334648\pi$
$211$	3043.00	0.992838	0.496419	+	0.868083i	$0.334648\pi$
$212$	0	0
$213$	4956.00	1.59427
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6188.00	1.93580
$218$	0	0
$219$	931.000	0.287266
$220$	0	0
$221$	−588.000	−0.178974
$222$	0	0
$223$	3332.00	1.00057	0.500285	−	0.865861i	$-0.333228\pi$
$223$	3332.00	1.00057	0.500285	+	0.865861i	$0.333228\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1596.00	0.466653	0.233327	−	0.972398i	$-0.425039\pi$
$227$	1596.00	0.466653	0.233327	+	0.972398i	$0.425039\pi$
$228$	0	0
$229$	4340.00	1.25238	0.626191	−	0.779670i	$-0.284613\pi$
$229$	4340.00	1.25238	0.626191	+	0.779670i	$0.284613\pi$
$230$	0	0
$231$	6426.00	1.83030
$232$	0	0
$233$	3018.00	0.848565	0.424283	−	0.905530i	$-0.360526\pi$
$233$	3018.00	0.848565	0.424283	+	0.905530i	$0.360526\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4550.00	−1.24706
$238$	0	0
$239$	−4440.00	−1.20167	−0.600836	−	0.799372i	$-0.705166\pi$
$239$	−4440.00	−1.20167	−0.600836	+	0.799372i	$0.705166\pi$
$240$	0	0
$241$	−3703.00	−0.989756	−0.494878	−	0.868962i	$-0.664787\pi$
$241$	−3703.00	−0.989756	−0.494878	+	0.868962i	$0.664787\pi$
$242$	0	0
$243$	−4928.00	−1.30095
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−980.000	−0.252453
$248$	0	0
$249$	−6321.00	−1.60874
$250$	0	0
$251$	−7077.00	−1.77967	−0.889833	−	0.456286i	$-0.849179\pi$
$251$	−7077.00	−1.77967	−0.889833	+	0.456286i	$0.849179\pi$
$252$	0	0
$253$	2106.00	0.523332
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6846.00	−1.66164	−0.830821	−	0.556540i	$-0.812128\pi$
$257$	−6846.00	−1.66164	−0.830821	+	0.556540i	$0.812128\pi$
$258$	0	0
$259$	4964.00	1.19092
$260$	0	0
$261$	−2640.00	−0.626099
$262$	0	0
$263$	−3438.00	−0.806069	−0.403035	−	0.915185i	$-0.632045\pi$
$263$	−3438.00	−0.806069	−0.403035	+	0.915185i	$0.632045\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5145.00	1.17928
$268$	0	0
$269$	−1680.00	−0.380786	−0.190393	−	0.981708i	$-0.560976\pi$
$269$	−1680.00	−0.380786	−0.190393	+	0.981708i	$0.560976\pi$
$270$	0	0
$271$	−5222.00	−1.17053	−0.585266	−	0.810842i	$-0.699010\pi$
$271$	−5222.00	−1.17053	−0.585266	+	0.810842i	$0.699010\pi$
$272$	0	0
$273$	−6664.00	−1.47738
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1384.00	0.300204	0.150102	−	0.988671i	$-0.452040\pi$
$277$	1384.00	0.300204	0.150102	+	0.988671i	$0.452040\pi$
$278$	0	0
$279$	−4004.00	−0.859187
$280$	0	0
$281$	−3858.00	−0.819036	−0.409518	−	0.912302i	$-0.634303\pi$
$281$	−3858.00	−0.819036	−0.409518	+	0.912302i	$0.634303\pi$
$282$	0	0
$283$	4277.00	0.898379	0.449190	−	0.893437i	$-0.351713\pi$
$283$	4277.00	0.898379	0.449190	+	0.893437i	$0.351713\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12138.0	−2.49646
$288$	0	0
$289$	−4472.00	−0.910238
$290$	0	0
$291$	−7742.00	−1.55960
$292$	0	0
$293$	−6342.00	−1.26452	−0.632259	−	0.774757i	$-0.717872\pi$
$293$	−6342.00	−1.26452	−0.632259	+	0.774757i	$0.717872\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	945.000	0.184628
$298$	0	0
$299$	−2184.00	−0.422421
$300$	0	0
$301$	5032.00	0.963587
$302$	0	0
$303$	3234.00	0.613163
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5831.00	1.08402	0.542008	−	0.840373i	$-0.317664\pi$
$307$	5831.00	1.08402	0.542008	+	0.840373i	$0.317664\pi$
$308$	0	0
$309$	5684.00	1.04644
$310$	0	0
$311$	2478.00	0.451815	0.225908	−	0.974149i	$-0.427465\pi$
$311$	2478.00	0.451815	0.225908	+	0.974149i	$0.427465\pi$
$312$	0	0
$313$	2758.00	0.498056	0.249028	−	0.968496i	$-0.419889\pi$
$313$	2758.00	0.498056	0.249028	+	0.968496i	$0.419889\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−636.000	−0.112686	−0.0563428	−	0.998411i	$-0.517944\pi$
$317$	−636.000	−0.112686	−0.0563428	+	0.998411i	$0.517944\pi$
$318$	0	0
$319$	3240.00	0.568668
$320$	0	0
$321$	−5523.00	−0.960323
$322$	0	0
$323$	735.000	0.126615
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1610.00	0.272273
$328$	0	0
$329$	2856.00	0.478591
$330$	0	0
$331$	−6887.00	−1.14364	−0.571818	−	0.820380i	$-0.693762\pi$
$331$	−6887.00	−1.14364	−0.571818	+	0.820380i	$0.693762\pi$
$332$	0	0
$333$	−3212.00	−0.528578
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7331.00	−1.18500	−0.592500	−	0.805570i	$-0.701859\pi$
$337$	−7331.00	−1.18500	−0.592500	+	0.805570i	$0.701859\pi$
$338$	0	0
$339$	14511.0	2.32487
$340$	0	0
$341$	4914.00	0.780375
$342$	0	0
$343$	−15980.0	−2.51557
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5349.00	−0.827520	−0.413760	−	0.910386i	$-0.635785\pi$
$347$	−5349.00	−0.827520	−0.413760	+	0.910386i	$0.635785\pi$
$348$	0	0
$349$	11270.0	1.72857	0.864283	−	0.503007i	$-0.167773\pi$
$349$	11270.0	1.72857	0.864283	+	0.503007i	$0.167773\pi$
$350$	0	0
$351$	−980.000	−0.149027
$352$	0	0
$353$	−1302.00	−0.196313	−0.0981565	−	0.995171i	$-0.531295\pi$
$353$	−1302.00	−0.196313	−0.0981565	+	0.995171i	$0.531295\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4998.00	0.740959
$358$	0	0
$359$	750.000	0.110260	0.0551302	−	0.998479i	$-0.482443\pi$
$359$	750.000	0.110260	0.0551302	+	0.998479i	$0.482443\pi$
$360$	0	0
$361$	−5634.00	−0.821403
$362$	0	0
$363$	−4214.00	−0.609305
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1204.00	−0.171249	−0.0856244	−	0.996327i	$-0.527288\pi$
$367$	−1204.00	−0.171249	−0.0856244	+	0.996327i	$0.527288\pi$
$368$	0	0
$369$	7854.00	1.10803
$370$	0	0
$371$	23868.0	3.34007
$372$	0	0
$373$	1198.00	0.166301	0.0831503	−	0.996537i	$-0.473502\pi$
$373$	1198.00	0.166301	0.0831503	+	0.996537i	$0.473502\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3360.00	−0.459015
$378$	0	0
$379$	−8105.00	−1.09849	−0.549243	−	0.835663i	$-0.685084\pi$
$379$	−8105.00	−1.09849	−0.549243	+	0.835663i	$0.685084\pi$
$380$	0	0
$381$	−7798.00	−1.04857
$382$	0	0
$383$	−3318.00	−0.442668	−0.221334	−	0.975198i	$-0.571041\pi$
$383$	−3318.00	−0.442668	−0.221334	+	0.975198i	$0.571041\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3256.00	−0.427679
$388$	0	0
$389$	−13770.0	−1.79477	−0.897387	−	0.441245i	$-0.854537\pi$
$389$	−13770.0	−1.79477	−0.897387	+	0.441245i	$0.854537\pi$
$390$	0	0
$391$	1638.00	0.211860
$392$	0	0
$393$	−1764.00	−0.226417
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3724.00	0.470786	0.235393	−	0.971900i	$-0.424362\pi$
$397$	3724.00	0.470786	0.235393	+	0.971900i	$0.424362\pi$
$398$	0	0
$399$	8330.00	1.04517
$400$	0	0
$401$	6117.00	0.761767	0.380883	−	0.924623i	$-0.375620\pi$
$401$	6117.00	0.761767	0.380883	+	0.924623i	$0.375620\pi$
$402$	0	0
$403$	−5096.00	−0.629900
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3942.00	0.480093
$408$	0	0
$409$	6125.00	0.740493	0.370247	−	0.928933i	$-0.379273\pi$
$409$	6125.00	0.740493	0.370247	+	0.928933i	$0.379273\pi$
$410$	0	0
$411$	−13587.0	−1.63065
$412$	0	0
$413$	−28560.0	−3.40277
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11515.0	1.35226
$418$	0	0
$419$	−1575.00	−0.183637	−0.0918184	−	0.995776i	$-0.529268\pi$
$419$	−1575.00	−0.183637	−0.0918184	+	0.995776i	$0.529268\pi$
$420$	0	0
$421$	−988.000	−0.114376	−0.0571879	−	0.998363i	$-0.518213\pi$
$421$	−988.000	−0.114376	−0.0571879	+	0.998363i	$0.518213\pi$
$422$	0	0
$423$	−1848.00	−0.212418
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8092.00	0.917094
$428$	0	0
$429$	−5292.00	−0.595571
$430$	0	0
$431$	558.000	0.0623617	0.0311809	−	0.999514i	$-0.490073\pi$
$431$	558.000	0.0623617	0.0311809	+	0.999514i	$0.490073\pi$
$432$	0	0
$433$	2443.00	0.271139	0.135569	−	0.990768i	$-0.456714\pi$
$433$	2443.00	0.271139	0.135569	+	0.990768i	$0.456714\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2730.00	0.298841
$438$	0	0
$439$	−12320.0	−1.33941	−0.669706	−	0.742627i	$-0.733580\pi$
$439$	−12320.0	−1.33941	−0.669706	+	0.742627i	$0.733580\pi$
$440$	0	0
$441$	17886.0	1.93132
$442$	0	0
$443$	−2343.00	−0.251285	−0.125643	−	0.992076i	$-0.540099\pi$
$443$	−2343.00	−0.251285	−0.125643	+	0.992076i	$0.540099\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12600.0	−1.33324
$448$	0	0
$449$	10905.0	1.14619	0.573094	−	0.819489i	$-0.305743\pi$
$449$	10905.0	1.14619	0.573094	+	0.819489i	$0.305743\pi$
$450$	0	0
$451$	−9639.00	−1.00639
$452$	0	0
$453$	23086.0	2.39443
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3319.00	0.339729	0.169865	−	0.985467i	$-0.445667\pi$
$457$	3319.00	0.339729	0.169865	+	0.985467i	$0.445667\pi$
$458$	0	0
$459$	735.000	0.0747426
$460$	0	0
$461$	−6468.00	−0.653459	−0.326730	−	0.945118i	$-0.605947\pi$
$461$	−6468.00	−0.653459	−0.326730	+	0.945118i	$0.605947\pi$
$462$	0	0
$463$	11972.0	1.20170	0.600849	−	0.799363i	$-0.294829\pi$
$463$	11972.0	1.20170	0.600849	+	0.799363i	$0.294829\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6636.00	0.657553	0.328777	−	0.944408i	$-0.393364\pi$
$467$	6636.00	0.657553	0.328777	+	0.944408i	$0.393364\pi$
$468$	0	0
$469$	−15674.0	−1.54319
$470$	0	0
$471$	−1862.00	−0.182158
$472$	0	0
$473$	3996.00	0.388449
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−15444.0	−1.48246
$478$	0	0
$479$	630.000	0.0600949	0.0300474	−	0.999548i	$-0.490434\pi$
$479$	630.000	0.0600949	0.0300474	+	0.999548i	$0.490434\pi$
$480$	0	0
$481$	−4088.00	−0.387519
$482$	0	0
$483$	18564.0	1.74884
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10646.0	0.990588	0.495294	−	0.868725i	$-0.335060\pi$
$487$	10646.0	0.990588	0.495294	+	0.868725i	$0.335060\pi$
$488$	0	0
$489$	8099.00	0.748976
$490$	0	0
$491$	2388.00	0.219489	0.109744	−	0.993960i	$-0.464997\pi$
$491$	2388.00	0.219489	0.109744	+	0.993960i	$0.464997\pi$
$492$	0	0
$493$	2520.00	0.230213
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24072.0	−2.17259
$498$	0	0
$499$	10540.0	0.945562	0.472781	−	0.881180i	$-0.343250\pi$
$499$	10540.0	0.945562	0.472781	+	0.881180i	$0.343250\pi$
$500$	0	0
$501$	−12348.0	−1.10113
$502$	0	0
$503$	6972.00	0.618024	0.309012	−	0.951058i	$-0.400002\pi$
$503$	6972.00	0.618024	0.309012	+	0.951058i	$0.400002\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9891.00	−0.866420
$508$	0	0
$509$	−9030.00	−0.786341	−0.393171	−	0.919466i	$-0.628622\pi$
$509$	−9030.00	−0.786341	−0.393171	+	0.919466i	$0.628622\pi$
$510$	0	0
$511$	−4522.00	−0.391471
$512$	0	0
$513$	1225.00	0.105429
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2268.00	0.192933
$518$	0	0
$519$	12936.0	1.09408
$520$	0	0
$521$	10437.0	0.877645	0.438823	−	0.898574i	$-0.355396\pi$
$521$	10437.0	0.877645	0.438823	+	0.898574i	$0.355396\pi$
$522$	0	0
$523$	−8113.00	−0.678311	−0.339156	−	0.940730i	$-0.610141\pi$
$523$	−8113.00	−0.678311	−0.339156	+	0.940730i	$0.610141\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3822.00	0.315918
$528$	0	0
$529$	−6083.00	−0.499959
$530$	0	0
$531$	18480.0	1.51029
$532$	0	0
$533$	9996.00	0.812336
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−945.000	−0.0759400
$538$	0	0
$539$	−21951.0	−1.75417
$540$	0	0
$541$	−14848.0	−1.17997	−0.589986	−	0.807413i	$-0.700867\pi$
$541$	−14848.0	−1.17997	−0.589986	+	0.807413i	$0.700867\pi$
$542$	0	0
$543$	15974.0	1.26245
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20329.0	−1.58904	−0.794520	−	0.607238i	$-0.792278\pi$
$547$	−20329.0	−1.58904	−0.794520	+	0.607238i	$0.792278\pi$
$548$	0	0
$549$	−5236.00	−0.407044
$550$	0	0
$551$	4200.00	0.324730
$552$	0	0
$553$	22100.0	1.69944
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15324.0	1.16571	0.582853	−	0.812577i	$-0.301936\pi$
$557$	15324.0	1.16571	0.582853	+	0.812577i	$0.301936\pi$
$558$	0	0
$559$	−4144.00	−0.313547
$560$	0	0
$561$	3969.00	0.298701
$562$	0	0
$563$	−9408.00	−0.704263	−0.352131	−	0.935951i	$-0.614543\pi$
$563$	−9408.00	−0.704263	−0.352131	+	0.935951i	$0.614543\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	28526.0	2.11284
$568$	0	0
$569$	−24375.0	−1.79588	−0.897938	−	0.440122i	$-0.854935\pi$
$569$	−24375.0	−1.79588	−0.897938	+	0.440122i	$0.854935\pi$
$570$	0	0
$571$	988.000	0.0724107	0.0362054	−	0.999344i	$-0.488473\pi$
$571$	988.000	0.0724107	0.0362054	+	0.999344i	$0.488473\pi$
$572$	0	0
$573$	9786.00	0.713466
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16429.0	1.18535	0.592676	−	0.805441i	$-0.298071\pi$
$577$	16429.0	1.18535	0.592676	+	0.805441i	$0.298071\pi$
$578$	0	0
$579$	−23219.0	−1.66658
$580$	0	0
$581$	30702.0	2.19231
$582$	0	0
$583$	18954.0	1.34647
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13839.0	−0.973078	−0.486539	−	0.873659i	$-0.661741\pi$
$587$	−13839.0	−0.973078	−0.486539	+	0.873659i	$0.661741\pi$
$588$	0	0
$589$	6370.00	0.445622
$590$	0	0
$591$	−11802.0	−0.821437
$592$	0	0
$593$	−14007.0	−0.969981	−0.484990	−	0.874520i	$-0.661177\pi$
$593$	−14007.0	−0.969981	−0.484990	+	0.874520i	$0.661177\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10780.0	0.739022
$598$	0	0
$599$	−21090.0	−1.43859	−0.719294	−	0.694706i	$-0.755535\pi$
$599$	−21090.0	−1.43859	−0.719294	+	0.694706i	$0.755535\pi$
$600$	0	0
$601$	5747.00	0.390058	0.195029	−	0.980797i	$-0.437520\pi$
$601$	5747.00	0.390058	0.195029	+	0.980797i	$0.437520\pi$
$602$	0	0
$603$	10142.0	0.684932
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17696.0	1.18329	0.591646	−	0.806198i	$-0.298478\pi$
$607$	17696.0	1.18329	0.591646	+	0.806198i	$0.298478\pi$
$608$	0	0
$609$	28560.0	1.90034
$610$	0	0
$611$	−2352.00	−0.155731
$612$	0	0
$613$	−26102.0	−1.71982	−0.859910	−	0.510445i	$-0.829481\pi$
$613$	−26102.0	−1.71982	−0.859910	+	0.510445i	$0.829481\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4194.00	0.273653	0.136827	−	0.990595i	$-0.456310\pi$
$617$	4194.00	0.273653	0.136827	+	0.990595i	$0.456310\pi$
$618$	0	0
$619$	7420.00	0.481801	0.240901	−	0.970550i	$-0.422557\pi$
$619$	7420.00	0.481801	0.240901	+	0.970550i	$0.422557\pi$
$620$	0	0
$621$	2730.00	0.176411
$622$	0	0
$623$	−24990.0	−1.60707
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6615.00	0.421336
$628$	0	0
$629$	3066.00	0.194355
$630$	0	0
$631$	5818.00	0.367054	0.183527	−	0.983015i	$-0.441249\pi$
$631$	5818.00	0.367054	0.183527	+	0.983015i	$0.441249\pi$
$632$	0	0
$633$	21301.0	1.33750
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22764.0	1.41592
$638$	0	0
$639$	15576.0	0.964283
$640$	0	0
$641$	−29478.0	−1.81640	−0.908199	−	0.418539i	$-0.862542\pi$
$641$	−29478.0	−1.81640	−0.908199	+	0.418539i	$0.862542\pi$
$642$	0	0
$643$	5852.00	0.358912	0.179456	−	0.983766i	$-0.442566\pi$
$643$	5852.00	0.358912	0.179456	+	0.983766i	$0.442566\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29484.0	−1.79155	−0.895777	−	0.444503i	$-0.853380\pi$
$647$	−29484.0	−1.79155	−0.895777	+	0.444503i	$0.853380\pi$
$648$	0	0
$649$	−22680.0	−1.37175
$650$	0	0
$651$	43316.0	2.60782
$652$	0	0
$653$	3498.00	0.209628	0.104814	−	0.994492i	$-0.466575\pi$
$653$	3498.00	0.209628	0.104814	+	0.994492i	$0.466575\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2926.00	0.173751
$658$	0	0
$659$	−7905.00	−0.467276	−0.233638	−	0.972324i	$-0.575063\pi$
$659$	−7905.00	−0.467276	−0.233638	+	0.972324i	$0.575063\pi$
$660$	0	0
$661$	27272.0	1.60478	0.802389	−	0.596802i	$-0.203562\pi$
$661$	27272.0	1.60478	0.802389	+	0.596802i	$0.203562\pi$
$662$	0	0
$663$	−4116.00	−0.241104
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9360.00	0.543359
$668$	0	0
$669$	23324.0	1.34792
$670$	0	0
$671$	6426.00	0.369706
$672$	0	0
$673$	−12602.0	−0.721800	−0.360900	−	0.932605i	$-0.617530\pi$
$673$	−12602.0	−0.721800	−0.360900	+	0.932605i	$0.617530\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25536.0	−1.44967	−0.724836	−	0.688921i	$-0.758085\pi$
$677$	−25536.0	−1.44967	−0.724836	+	0.688921i	$0.758085\pi$
$678$	0	0
$679$	37604.0	2.12534
$680$	0	0
$681$	11172.0	0.628652
$682$	0	0
$683$	2127.00	0.119162	0.0595808	−	0.998223i	$-0.481024\pi$
$683$	2127.00	0.119162	0.0595808	+	0.998223i	$0.481024\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	30380.0	1.68715
$688$	0	0
$689$	−19656.0	−1.08684
$690$	0	0
$691$	8953.00	0.492892	0.246446	−	0.969157i	$-0.420737\pi$
$691$	8953.00	0.492892	0.246446	+	0.969157i	$0.420737\pi$
$692$	0	0
$693$	20196.0	1.10705
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−7497.00	−0.407416
$698$	0	0
$699$	21126.0	1.14315
$700$	0	0
$701$	23172.0	1.24849	0.624247	−	0.781227i	$-0.285406\pi$
$701$	23172.0	1.24849	0.624247	+	0.781227i	$0.285406\pi$
$702$	0	0
$703$	5110.00	0.274150
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15708.0	−0.835587
$708$	0	0
$709$	9620.00	0.509572	0.254786	−	0.966997i	$-0.417995\pi$
$709$	9620.00	0.509572	0.254786	+	0.966997i	$0.417995\pi$
$710$	0	0
$711$	−14300.0	−0.754278
$712$	0	0
$713$	14196.0	0.745644
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−31080.0	−1.61883
$718$	0	0
$719$	−4830.00	−0.250527	−0.125263	−	0.992124i	$-0.539978\pi$
$719$	−4830.00	−0.250527	−0.125263	+	0.992124i	$0.539978\pi$
$720$	0	0
$721$	−27608.0	−1.42604
$722$	0	0
$723$	−25921.0	−1.33335
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11816.0	0.602794	0.301397	−	0.953499i	$-0.402547\pi$
$727$	11816.0	0.602794	0.301397	+	0.953499i	$0.402547\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	0	0
$731$	3108.00	0.157255
$732$	0	0
$733$	−23492.0	−1.18376	−0.591881	−	0.806026i	$-0.701614\pi$
$733$	−23492.0	−1.18376	−0.591881	+	0.806026i	$0.701614\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12447.0	−0.622105
$738$	0	0
$739$	−35300.0	−1.75715	−0.878573	−	0.477607i	$-0.841504\pi$
$739$	−35300.0	−1.75715	−0.878573	+	0.477607i	$0.841504\pi$
$740$	0	0
$741$	−6860.00	−0.340092
$742$	0	0
$743$	16242.0	0.801967	0.400983	−	0.916085i	$-0.368669\pi$
$743$	16242.0	0.801967	0.400983	+	0.916085i	$0.368669\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−19866.0	−0.973037
$748$	0	0
$749$	26826.0	1.30868
$750$	0	0
$751$	−10712.0	−0.520488	−0.260244	−	0.965543i	$-0.583803\pi$
$751$	−10712.0	−0.520488	−0.260244	+	0.965543i	$0.583803\pi$
$752$	0	0
$753$	−49539.0	−2.39748
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13504.0	0.648364	0.324182	−	0.945995i	$-0.394911\pi$
$757$	13504.0	0.648364	0.324182	+	0.945995i	$0.394911\pi$
$758$	0	0
$759$	14742.0	0.705008
$760$	0	0
$761$	−18123.0	−0.863283	−0.431641	−	0.902045i	$-0.642065\pi$
$761$	−18123.0	−0.863283	−0.431641	+	0.902045i	$0.642065\pi$
$762$	0	0
$763$	−7820.00	−0.371039
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23520.0	1.10725
$768$	0	0
$769$	9485.00	0.444783	0.222391	−	0.974957i	$-0.428614\pi$
$769$	9485.00	0.444783	0.222391	+	0.974957i	$0.428614\pi$
$770$	0	0
$771$	−47922.0	−2.23848
$772$	0	0
$773$	31248.0	1.45396	0.726981	−	0.686658i	$-0.240923\pi$
$773$	31248.0	1.45396	0.726981	+	0.686658i	$0.240923\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	34748.0	1.60435
$778$	0	0
$779$	−12495.0	−0.574685
$780$	0	0
$781$	−19116.0	−0.875831
$782$	0	0
$783$	4200.00	0.191693
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−15484.0	−0.701328	−0.350664	−	0.936501i	$-0.614044\pi$
$787$	−15484.0	−0.701328	−0.350664	+	0.936501i	$0.614044\pi$
$788$	0	0
$789$	−24066.0	−1.08590
$790$	0	0
$791$	−70482.0	−3.16821
$792$	0	0
$793$	−6664.00	−0.298418
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−34146.0	−1.51758	−0.758791	−	0.651334i	$-0.774210\pi$
$797$	−34146.0	−1.51758	−0.758791	+	0.651334i	$0.774210\pi$
$798$	0	0
$799$	1764.00	0.0781049
$800$	0	0
$801$	16170.0	0.713282
$802$	0	0
$803$	−3591.00	−0.157813
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11760.0	−0.512976
$808$	0	0
$809$	−36030.0	−1.56582	−0.782909	−	0.622136i	$-0.786265\pi$
$809$	−36030.0	−1.56582	−0.782909	+	0.622136i	$0.786265\pi$
$810$	0	0
$811$	6748.00	0.292175	0.146088	−	0.989272i	$-0.453332\pi$
$811$	6748.00	0.292175	0.146088	+	0.989272i	$0.453332\pi$
$812$	0	0
$813$	−36554.0	−1.57688
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5180.00	0.221818
$818$	0	0
$819$	−20944.0	−0.893581
$820$	0	0
$821$	−5598.00	−0.237968	−0.118984	−	0.992896i	$-0.537964\pi$
$821$	−5598.00	−0.237968	−0.118984	+	0.992896i	$0.537964\pi$
$822$	0	0
$823$	5732.00	0.242776	0.121388	−	0.992605i	$-0.461265\pi$
$823$	5732.00	0.242776	0.121388	+	0.992605i	$0.461265\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39999.0	−1.68186	−0.840932	−	0.541141i	$-0.817993\pi$
$827$	−39999.0	−1.68186	−0.840932	+	0.541141i	$0.817993\pi$
$828$	0	0
$829$	16940.0	0.709711	0.354856	−	0.934921i	$-0.384530\pi$
$829$	16940.0	0.709711	0.354856	+	0.934921i	$0.384530\pi$
$830$	0	0
$831$	9688.00	0.404420
$832$	0	0
$833$	−17073.0	−0.710137
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6370.00	0.263058
$838$	0	0
$839$	45360.0	1.86651	0.933255	−	0.359216i	$-0.116956\pi$
$839$	45360.0	1.86651	0.933255	+	0.359216i	$0.116956\pi$
$840$	0	0
$841$	−9989.00	−0.409570
$842$	0	0
$843$	−27006.0	−1.10336
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20468.0	0.830329
$848$	0	0
$849$	29939.0	1.21025
$850$	0	0
$851$	11388.0	0.458726
$852$	0	0
$853$	43918.0	1.76286	0.881432	−	0.472310i	$-0.156580\pi$
$853$	43918.0	1.76286	0.881432	+	0.472310i	$0.156580\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3339.00	0.133090	0.0665450	−	0.997783i	$-0.478802\pi$
$857$	3339.00	0.133090	0.0665450	+	0.997783i	$0.478802\pi$
$858$	0	0
$859$	36925.0	1.46666	0.733332	−	0.679870i	$-0.237964\pi$
$859$	36925.0	1.46666	0.733332	+	0.679870i	$0.237964\pi$
$860$	0	0
$861$	−84966.0	−3.36311
$862$	0	0
$863$	−40608.0	−1.60175	−0.800876	−	0.598830i	$-0.795633\pi$
$863$	−40608.0	−1.60175	−0.800876	+	0.598830i	$0.795633\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−31304.0	−1.22623
$868$	0	0
$869$	17550.0	0.685090
$870$	0	0
$871$	12908.0	0.502148
$872$	0	0
$873$	−24332.0	−0.943314
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35156.0	−1.35363	−0.676815	−	0.736153i	$-0.736640\pi$
$877$	−35156.0	−1.35363	−0.676815	+	0.736153i	$0.736640\pi$
$878$	0	0
$879$	−44394.0	−1.70350
$880$	0	0
$881$	2142.00	0.0819135	0.0409568	−	0.999161i	$-0.486959\pi$
$881$	2142.00	0.0819135	0.0409568	+	0.999161i	$0.486959\pi$
$882$	0	0
$883$	−19153.0	−0.729954	−0.364977	−	0.931016i	$-0.618923\pi$
$883$	−19153.0	−0.729954	−0.364977	+	0.931016i	$0.618923\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24444.0	−0.925309	−0.462655	−	0.886539i	$-0.653103\pi$
$887$	−24444.0	−0.925309	−0.462655	+	0.886539i	$0.653103\pi$
$888$	0	0
$889$	37876.0	1.42893
$890$	0	0
$891$	22653.0	0.851744
$892$	0	0
$893$	2940.00	0.110172
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−15288.0	−0.569065
$898$	0	0
$899$	21840.0	0.810239
$900$	0	0
$901$	14742.0	0.545091
$902$	0	0
$903$	35224.0	1.29810
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20036.0	0.733500	0.366750	−	0.930320i	$-0.380470\pi$
$907$	20036.0	0.733500	0.366750	+	0.930320i	$0.380470\pi$
$908$	0	0
$909$	10164.0	0.370868
$910$	0	0
$911$	33468.0	1.21717	0.608586	−	0.793488i	$-0.291737\pi$
$911$	33468.0	1.21717	0.608586	+	0.793488i	$0.291737\pi$
$912$	0	0
$913$	24381.0	0.883782
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8568.00	0.308550
$918$	0	0
$919$	−35090.0	−1.25953	−0.629767	−	0.776784i	$-0.716850\pi$
$919$	−35090.0	−1.25953	−0.629767	+	0.776784i	$0.716850\pi$
$920$	0	0
$921$	40817.0	1.46033
$922$	0	0
$923$	19824.0	0.706950
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17864.0	0.632935
$928$	0	0
$929$	−32130.0	−1.13472	−0.567358	−	0.823471i	$-0.692034\pi$
$929$	−32130.0	−1.13472	−0.567358	+	0.823471i	$0.692034\pi$
$930$	0	0
$931$	−28455.0	−1.00169
$932$	0	0
$933$	17346.0	0.608663
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20531.0	−0.715815	−0.357907	−	0.933757i	$-0.616510\pi$
$937$	−20531.0	−0.715815	−0.357907	+	0.933757i	$0.616510\pi$
$938$	0	0
$939$	19306.0	0.670956
$940$	0	0
$941$	7812.00	0.270631	0.135316	−	0.990803i	$-0.456795\pi$
$941$	7812.00	0.270631	0.135316	+	0.990803i	$0.456795\pi$
$942$	0	0
$943$	−27846.0	−0.961602
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23256.0	0.798013	0.399007	−	0.916948i	$-0.369355\pi$
$947$	23256.0	0.798013	0.399007	+	0.916948i	$0.369355\pi$
$948$	0	0
$949$	3724.00	0.127383
$950$	0	0
$951$	−4452.00	−0.151804
$952$	0	0
$953$	−14097.0	−0.479167	−0.239584	−	0.970876i	$-0.577011\pi$
$953$	−14097.0	−0.479167	−0.239584	+	0.970876i	$0.577011\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	22680.0	0.766082
$958$	0	0
$959$	65994.0	2.22217
$960$	0	0
$961$	3333.00	0.111879
$962$	0	0
$963$	−17358.0	−0.580845
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−34144.0	−1.13547	−0.567734	−	0.823212i	$-0.692180\pi$
$967$	−34144.0	−1.13547	−0.567734	+	0.823212i	$0.692180\pi$
$968$	0	0
$969$	5145.00	0.170569
$970$	0	0
$971$	32613.0	1.07786	0.538929	−	0.842351i	$-0.318829\pi$
$971$	32613.0	1.07786	0.538929	+	0.842351i	$0.318829\pi$
$972$	0	0
$973$	−55930.0	−1.84279
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43359.0	1.41983	0.709917	−	0.704286i	$-0.248733\pi$
$977$	43359.0	1.41983	0.709917	+	0.704286i	$0.248733\pi$
$978$	0	0
$979$	−19845.0	−0.647854
$980$	0	0
$981$	5060.00	0.164682
$982$	0	0
$983$	−28518.0	−0.925313	−0.462657	−	0.886538i	$-0.653104\pi$
$983$	−28518.0	−0.925313	−0.462657	+	0.886538i	$0.653104\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	19992.0	0.644734
$988$	0	0
$989$	11544.0	0.371161
$990$	0	0
$991$	−18122.0	−0.580892	−0.290446	−	0.956891i	$-0.593804\pi$
$991$	−18122.0	−0.580892	−0.290446	+	0.956891i	$0.593804\pi$
$992$	0	0
$993$	−48209.0	−1.54065
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28924.0	0.918789	0.459394	−	0.888232i	$-0.348066\pi$
$997$	28924.0	0.918789	0.459394	+	0.888232i	$0.348066\pi$
$998$	0	0
$999$	5110.00	0.161835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.a.r.1.1		1
4.3	odd	2		50.4.a.a.1.1	✓	1
5.2	odd	4		400.4.c.d.49.1		2
5.3	odd	4		400.4.c.d.49.2		2
5.4	even	2		400.4.a.d.1.1		1
8.3	odd	2		1600.4.a.bu.1.1		1
8.5	even	2		1600.4.a.g.1.1		1
12.11	even	2		450.4.a.t.1.1		1
20.3	even	4		50.4.b.b.49.2		2
20.7	even	4		50.4.b.b.49.1		2
20.19	odd	2		50.4.a.e.1.1	yes	1
28.27	even	2		2450.4.a.t.1.1		1
40.19	odd	2		1600.4.a.f.1.1		1
40.29	even	2		1600.4.a.bv.1.1		1
60.23	odd	4		450.4.c.c.199.1		2
60.47	odd	4		450.4.c.c.199.2		2
60.59	even	2		450.4.a.a.1.1		1
140.139	even	2		2450.4.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.4.a.a.1.1	✓	1	4.3	odd	2
50.4.a.e.1.1	yes	1	20.19	odd	2
50.4.b.b.49.1		2	20.7	even	4
50.4.b.b.49.2		2	20.3	even	4
400.4.a.d.1.1		1	5.4	even	2
400.4.a.r.1.1		1	1.1	even	1	trivial
400.4.c.d.49.1		2	5.2	odd	4
400.4.c.d.49.2		2	5.3	odd	4
450.4.a.a.1.1		1	60.59	even	2
450.4.a.t.1.1		1	12.11	even	2
450.4.c.c.199.1		2	60.23	odd	4
450.4.c.c.199.2		2	60.47	odd	4
1600.4.a.f.1.1		1	40.19	odd	2
1600.4.a.g.1.1		1	8.5	even	2
1600.4.a.bu.1.1		1	8.3	odd	2
1600.4.a.bv.1.1		1	40.29	even	2
2450.4.a.t.1.1		1	28.27	even	2
2450.4.a.y.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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