LMFDB - Embedded newform 1521.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{2} +1.00000 q^{4} -9.00000 q^{5} -2.00000 q^{7} -21.0000 q^{8} +O(q^{10})$

$q+3.00000 q^{2} +1.00000 q^{4} -9.00000 q^{5} -2.00000 q^{7} -21.0000 q^{8} -27.0000 q^{10} +30.0000 q^{11} -6.00000 q^{14} -71.0000 q^{16} +111.000 q^{17} +46.0000 q^{19} -9.00000 q^{20} +90.0000 q^{22} +6.00000 q^{23} -44.0000 q^{25} -2.00000 q^{28} +105.000 q^{29} +100.000 q^{31} -45.0000 q^{32} +333.000 q^{34} +18.0000 q^{35} -17.0000 q^{37} +138.000 q^{38} +189.000 q^{40} -231.000 q^{41} -514.000 q^{43} +30.0000 q^{44} +18.0000 q^{46} -162.000 q^{47} -339.000 q^{49} -132.000 q^{50} -639.000 q^{53} -270.000 q^{55} +42.0000 q^{56} +315.000 q^{58} +600.000 q^{59} +233.000 q^{61} +300.000 q^{62} +433.000 q^{64} -926.000 q^{67} +111.000 q^{68} +54.0000 q^{70} -930.000 q^{71} +253.000 q^{73} -51.0000 q^{74} +46.0000 q^{76} -60.0000 q^{77} -1324.00 q^{79} +639.000 q^{80} -693.000 q^{82} +810.000 q^{83} -999.000 q^{85} -1542.00 q^{86} -630.000 q^{88} +498.000 q^{89} +6.00000 q^{92} -486.000 q^{94} -414.000 q^{95} -1358.00 q^{97} -1017.00 q^{98} +O(q^{100})$

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 $

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	−9.00000	−0.804984	−0.402492	−	0.915423i	$-0.631856\pi$
$5$	−9.00000	−0.804984	−0.402492	+	0.915423i	$0.631856\pi$
$6$	0	0
$7$	−2.00000	−0.107990	−0.0539949	−	0.998541i	$-0.517195\pi$
$7$	−2.00000	−0.107990	−0.0539949	+	0.998541i	$0.517195\pi$
$8$	−21.0000	−0.928078
$9$	0	0
$10$	−27.0000	−0.853815
$11$	30.0000	0.822304	0.411152	−	0.911567i	$-0.365127\pi$
$11$	30.0000	0.822304	0.411152	+	0.911567i	$0.365127\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−6.00000	−0.114541
$15$	0	0
$16$	−71.0000	−1.10938
$17$	111.000	1.58361	0.791807	−	0.610771i	$-0.209140\pi$
$17$	111.000	1.58361	0.791807	+	0.610771i	$0.209140\pi$
$18$	0	0
$19$	46.0000	0.555428	0.277714	−	0.960664i	$-0.410423\pi$
$19$	46.0000	0.555428	0.277714	+	0.960664i	$0.410423\pi$
$20$	−9.00000	−0.100623
$21$	0	0
$22$	90.0000	0.872185
$23$	6.00000	0.0543951	0.0271975	−	0.999630i	$-0.491342\pi$
$23$	6.00000	0.0543951	0.0271975	+	0.999630i	$0.491342\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	0	0
$27$	0	0
$28$	−2.00000	−0.0134987
$29$	105.000	0.672345	0.336173	−	0.941800i	$-0.390867\pi$
$29$	105.000	0.672345	0.336173	+	0.941800i	$0.390867\pi$
$30$	0	0
$31$	100.000	0.579372	0.289686	−	0.957122i	$-0.406449\pi$
$31$	100.000	0.579372	0.289686	+	0.957122i	$0.406449\pi$
$32$	−45.0000	−0.248592
$33$	0	0
$34$	333.000	1.67968
$35$	18.0000	0.0869302
$36$	0	0
$37$	−17.0000	−0.0755347	−0.0377673	−	0.999287i	$-0.512025\pi$
$37$	−17.0000	−0.0755347	−0.0377673	+	0.999287i	$0.512025\pi$
$38$	138.000	0.589120
$39$	0	0
$40$	189.000	0.747088
$41$	−231.000	−0.879906	−0.439953	−	0.898021i	$-0.645005\pi$
$41$	−231.000	−0.879906	−0.439953	+	0.898021i	$0.645005\pi$
$42$	0	0
$43$	−514.000	−1.82289	−0.911445	−	0.411422i	$-0.865032\pi$
$43$	−514.000	−1.82289	−0.911445	+	0.411422i	$0.865032\pi$
$44$	30.0000	0.102788
$45$	0	0
$46$	18.0000	0.0576947
$47$	−162.000	−0.502769	−0.251384	−	0.967887i	$-0.580886\pi$
$47$	−162.000	−0.502769	−0.251384	+	0.967887i	$0.580886\pi$
$48$	0	0
$49$	−339.000	−0.988338
$50$	−132.000	−0.373352
$51$	0	0
$52$	0	0
$53$	−639.000	−1.65610	−0.828051	−	0.560653i	$-0.810550\pi$
$53$	−639.000	−1.65610	−0.828051	+	0.560653i	$0.810550\pi$
$54$	0	0
$55$	−270.000	−0.661942
$56$	42.0000	0.100223
$57$	0	0
$58$	315.000	0.713130
$59$	600.000	1.32396	0.661978	−	0.749524i	$-0.269717\pi$
$59$	600.000	1.32396	0.661978	+	0.749524i	$0.269717\pi$
$60$	0	0
$61$	233.000	0.489059	0.244529	−	0.969642i	$-0.421367\pi$
$61$	233.000	0.489059	0.244529	+	0.969642i	$0.421367\pi$
$62$	300.000	0.614517
$63$	0	0
$64$	433.000	0.845703
$65$	0	0
$66$	0	0
$67$	−926.000	−1.68849	−0.844246	−	0.535957i	$-0.819951\pi$
$67$	−926.000	−1.68849	−0.844246	+	0.535957i	$0.819951\pi$
$68$	111.000	0.197952
$69$	0	0
$70$	54.0000	0.0922033
$71$	−930.000	−1.55452	−0.777258	−	0.629182i	$-0.783390\pi$
$71$	−930.000	−1.55452	−0.777258	+	0.629182i	$0.783390\pi$
$72$	0	0
$73$	253.000	0.405636	0.202818	−	0.979216i	$-0.434990\pi$
$73$	253.000	0.405636	0.202818	+	0.979216i	$0.434990\pi$
$74$	−51.0000	−0.0801166
$75$	0	0
$76$	46.0000	0.0694284
$77$	−60.0000	−0.0888004
$78$	0	0
$79$	−1324.00	−1.88559	−0.942795	−	0.333373i	$-0.891813\pi$
$79$	−1324.00	−1.88559	−0.942795	+	0.333373i	$0.891813\pi$
$80$	639.000	0.893030
$81$	0	0
$82$	−693.000	−0.933281
$83$	810.000	1.07119	0.535597	−	0.844474i	$-0.320087\pi$
$83$	810.000	1.07119	0.535597	+	0.844474i	$0.320087\pi$
$84$	0	0
$85$	−999.000	−1.27479
$86$	−1542.00	−1.93347
$87$	0	0
$88$	−630.000	−0.763162
$89$	498.000	0.593122	0.296561	−	0.955014i	$-0.404160\pi$
$89$	498.000	0.593122	0.296561	+	0.955014i	$0.404160\pi$
$90$	0	0
$91$	0	0
$92$	6.00000	0.00679938
$93$	0	0
$94$	−486.000	−0.533267
$95$	−414.000	−0.447111
$96$	0	0
$97$	−1358.00	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−1358.00	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−1017.00	−1.04829
$99$	0	0
$100$	−44.0000	−0.0440000
$101$	357.000	0.351711	0.175856	−	0.984416i	$-0.443731\pi$
$101$	357.000	0.351711	0.175856	+	0.984416i	$0.443731\pi$
$102$	0	0
$103$	1118.00	1.06951	0.534756	−	0.845006i	$-0.320403\pi$
$103$	1118.00	1.06951	0.534756	+	0.845006i	$0.320403\pi$
$104$	0	0
$105$	0	0
$106$	−1917.00	−1.75656
$107$	−714.000	−0.645093	−0.322547	−	0.946554i	$-0.604539\pi$
$107$	−714.000	−0.645093	−0.322547	+	0.946554i	$0.604539\pi$
$108$	0	0
$109$	−2006.00	−1.76275	−0.881376	−	0.472416i	$-0.843382\pi$
$109$	−2006.00	−1.76275	−0.881376	+	0.472416i	$0.843382\pi$
$110$	−810.000	−0.702095
$111$	0	0
$112$	142.000	0.119801
$113$	1119.00	0.931563	0.465782	−	0.884900i	$-0.345773\pi$
$113$	1119.00	0.931563	0.465782	+	0.884900i	$0.345773\pi$
$114$	0	0
$115$	−54.0000	−0.0437872
$116$	105.000	0.0840431
$117$	0	0
$118$	1800.00	1.40427
$119$	−222.000	−0.171014
$120$	0	0
$121$	−431.000	−0.323817
$122$	699.000	0.518725
$123$	0	0
$124$	100.000	0.0724215
$125$	1521.00	1.08834
$126$	0	0
$127$	−604.000	−0.422018	−0.211009	−	0.977484i	$-0.567675\pi$
$127$	−604.000	−0.422018	−0.211009	+	0.977484i	$0.567675\pi$
$128$	1659.00	1.14560
$129$	0	0
$130$	0	0
$131$	1584.00	1.05645	0.528224	−	0.849105i	$-0.322858\pi$
$131$	1584.00	1.05645	0.528224	+	0.849105i	$0.322858\pi$
$132$	0	0
$133$	−92.0000	−0.0599805
$134$	−2778.00	−1.79092
$135$	0	0
$136$	−2331.00	−1.46972
$137$	717.000	0.447135	0.223567	−	0.974688i	$-0.428230\pi$
$137$	717.000	0.447135	0.223567	+	0.974688i	$0.428230\pi$
$138$	0	0
$139$	−820.000	−0.500370	−0.250185	−	0.968198i	$-0.580492\pi$
$139$	−820.000	−0.500370	−0.250185	+	0.968198i	$0.580492\pi$
$140$	18.0000	0.0108663
$141$	0	0
$142$	−2790.00	−1.64881
$143$	0	0
$144$	0	0
$145$	−945.000	−0.541227
$146$	759.000	0.430242
$147$	0	0
$148$	−17.0000	−0.00944183
$149$	−1749.00	−0.961635	−0.480818	−	0.876821i	$-0.659660\pi$
$149$	−1749.00	−0.961635	−0.480818	+	0.876821i	$0.659660\pi$
$150$	0	0
$151$	370.000	0.199405	0.0997026	−	0.995017i	$-0.468211\pi$
$151$	370.000	0.199405	0.0997026	+	0.995017i	$0.468211\pi$
$152$	−966.000	−0.515480
$153$	0	0
$154$	−180.000	−0.0941871
$155$	−900.000	−0.466385
$156$	0	0
$157$	−2611.00	−1.32726	−0.663632	−	0.748059i	$-0.730986\pi$
$157$	−2611.00	−1.32726	−0.663632	+	0.748059i	$0.730986\pi$
$158$	−3972.00	−1.99997
$159$	0	0
$160$	405.000	0.200113
$161$	−12.0000	−0.00587411
$162$	0	0
$163$	1636.00	0.786144	0.393072	−	0.919508i	$-0.371412\pi$
$163$	1636.00	0.786144	0.393072	+	0.919508i	$0.371412\pi$
$164$	−231.000	−0.109988
$165$	0	0
$166$	2430.00	1.13617
$167$	264.000	0.122329	0.0611645	−	0.998128i	$-0.480519\pi$
$167$	264.000	0.122329	0.0611645	+	0.998128i	$0.480519\pi$
$168$	0	0
$169$	0	0
$170$	−2997.00	−1.35211
$171$	0	0
$172$	−514.000	−0.227861
$173$	−1410.00	−0.619655	−0.309827	−	0.950793i	$-0.600271\pi$
$173$	−1410.00	−0.619655	−0.309827	+	0.950793i	$0.600271\pi$
$174$	0	0
$175$	88.0000	0.0380124
$176$	−2130.00	−0.912243
$177$	0	0
$178$	1494.00	0.629101
$179$	474.000	0.197924	0.0989621	−	0.995091i	$-0.468448\pi$
$179$	474.000	0.197924	0.0989621	+	0.995091i	$0.468448\pi$
$180$	0	0
$181$	2249.00	0.923574	0.461787	−	0.886991i	$-0.347208\pi$
$181$	2249.00	0.923574	0.461787	+	0.886991i	$0.347208\pi$
$182$	0	0
$183$	0	0
$184$	−126.000	−0.0504828
$185$	153.000	0.0608042
$186$	0	0
$187$	3330.00	1.30221
$188$	−162.000	−0.0628461
$189$	0	0
$190$	−1242.00	−0.474232
$191$	−3444.00	−1.30471	−0.652354	−	0.757915i	$-0.726218\pi$
$191$	−3444.00	−1.30471	−0.652354	+	0.757915i	$0.726218\pi$
$192$	0	0
$193$	4273.00	1.59366	0.796832	−	0.604201i	$-0.206507\pi$
$193$	4273.00	1.59366	0.796832	+	0.604201i	$0.206507\pi$
$194$	−4074.00	−1.50771
$195$	0	0
$196$	−339.000	−0.123542
$197$	−1986.00	−0.718257	−0.359129	−	0.933288i	$-0.616926\pi$
$197$	−1986.00	−0.718257	−0.359129	+	0.933288i	$0.616926\pi$
$198$	0	0
$199$	−2386.00	−0.849945	−0.424973	−	0.905206i	$-0.639716\pi$
$199$	−2386.00	−0.849945	−0.424973	+	0.905206i	$0.639716\pi$
$200$	924.000	0.326683
$201$	0	0
$202$	1071.00	0.373046
$203$	−210.000	−0.0726065
$204$	0	0
$205$	2079.00	0.708311
$206$	3354.00	1.13439
$207$	0	0
$208$	0	0
$209$	1380.00	0.456730
$210$	0	0
$211$	−1600.00	−0.522031	−0.261016	−	0.965335i	$-0.584057\pi$
$211$	−1600.00	−0.522031	−0.261016	+	0.965335i	$0.584057\pi$
$212$	−639.000	−0.207013
$213$	0	0
$214$	−2142.00	−0.684225
$215$	4626.00	1.46740
$216$	0	0
$217$	−200.000	−0.0625663
$218$	−6018.00	−1.86968
$219$	0	0
$220$	−270.000	−0.0827427
$221$	0	0
$222$	0	0
$223$	3832.00	1.15072	0.575358	−	0.817902i	$-0.304863\pi$
$223$	3832.00	1.15072	0.575358	+	0.817902i	$0.304863\pi$
$224$	90.0000	0.0268454
$225$	0	0
$226$	3357.00	0.988072
$227$	−1398.00	−0.408760	−0.204380	−	0.978892i	$-0.565518\pi$
$227$	−1398.00	−0.408760	−0.204380	+	0.978892i	$0.565518\pi$
$228$	0	0
$229$	−4466.00	−1.28874	−0.644370	−	0.764714i	$-0.722880\pi$
$229$	−4466.00	−1.28874	−0.644370	+	0.764714i	$0.722880\pi$
$230$	−162.000	−0.0464433
$231$	0	0
$232$	−2205.00	−0.623989
$233$	1638.00	0.460553	0.230277	−	0.973125i	$-0.426037\pi$
$233$	1638.00	0.460553	0.230277	+	0.973125i	$0.426037\pi$
$234$	0	0
$235$	1458.00	0.404721
$236$	600.000	0.165494
$237$	0	0
$238$	−666.000	−0.181388
$239$	−594.000	−0.160764	−0.0803821	−	0.996764i	$-0.525614\pi$
$239$	−594.000	−0.160764	−0.0803821	+	0.996764i	$0.525614\pi$
$240$	0	0
$241$	−2303.00	−0.615557	−0.307779	−	0.951458i	$-0.599586\pi$
$241$	−2303.00	−0.615557	−0.307779	+	0.951458i	$0.599586\pi$
$242$	−1293.00	−0.343459
$243$	0	0
$244$	233.000	0.0611324
$245$	3051.00	0.795597
$246$	0	0
$247$	0	0
$248$	−2100.00	−0.537702
$249$	0	0
$250$	4563.00	1.15436
$251$	−6324.00	−1.59031	−0.795154	−	0.606407i	$-0.792610\pi$
$251$	−6324.00	−1.59031	−0.795154	+	0.606407i	$0.792610\pi$
$252$	0	0
$253$	180.000	0.0447293
$254$	−1812.00	−0.447618
$255$	0	0
$256$	1513.00	0.369385
$257$	−7833.00	−1.90120	−0.950601	−	0.310414i	$-0.899532\pi$
$257$	−7833.00	−1.90120	−0.950601	+	0.310414i	$0.899532\pi$
$258$	0	0
$259$	34.0000	0.00815698
$260$	0	0
$261$	0	0
$262$	4752.00	1.12053
$263$	3030.00	0.710410	0.355205	−	0.934788i	$-0.384411\pi$
$263$	3030.00	0.710410	0.355205	+	0.934788i	$0.384411\pi$
$264$	0	0
$265$	5751.00	1.33314
$266$	−276.000	−0.0636190
$267$	0	0
$268$	−926.000	−0.211061
$269$	534.000	0.121036	0.0605178	−	0.998167i	$-0.480725\pi$
$269$	534.000	0.121036	0.0605178	+	0.998167i	$0.480725\pi$
$270$	0	0
$271$	3688.00	0.826679	0.413340	−	0.910577i	$-0.364362\pi$
$271$	3688.00	0.826679	0.413340	+	0.910577i	$0.364362\pi$
$272$	−7881.00	−1.75682
$273$	0	0
$274$	2151.00	0.474258
$275$	−1320.00	−0.289451
$276$	0	0
$277$	1865.00	0.404538	0.202269	−	0.979330i	$-0.435168\pi$
$277$	1865.00	0.404538	0.202269	+	0.979330i	$0.435168\pi$
$278$	−2460.00	−0.530723
$279$	0	0
$280$	−378.000	−0.0806779
$281$	2997.00	0.636249	0.318125	−	0.948049i	$-0.396947\pi$
$281$	2997.00	0.636249	0.318125	+	0.948049i	$0.396947\pi$
$282$	0	0
$283$	−4114.00	−0.864141	−0.432071	−	0.901840i	$-0.642217\pi$
$283$	−4114.00	−0.864141	−0.432071	+	0.901840i	$0.642217\pi$
$284$	−930.000	−0.194315
$285$	0	0
$286$	0	0
$287$	462.000	0.0950209
$288$	0	0
$289$	7408.00	1.50784
$290$	−2835.00	−0.574058
$291$	0	0
$292$	253.000	0.0507045
$293$	−4665.00	−0.930144	−0.465072	−	0.885273i	$-0.653972\pi$
$293$	−4665.00	−0.930144	−0.465072	+	0.885273i	$0.653972\pi$
$294$	0	0
$295$	−5400.00	−1.06576
$296$	357.000	0.0701020
$297$	0	0
$298$	−5247.00	−1.01997
$299$	0	0
$300$	0	0
$301$	1028.00	0.196854
$302$	1110.00	0.211501
$303$	0	0
$304$	−3266.00	−0.616177
$305$	−2097.00	−0.393685
$306$	0	0
$307$	−1502.00	−0.279230	−0.139615	−	0.990206i	$-0.544587\pi$
$307$	−1502.00	−0.279230	−0.139615	+	0.990206i	$0.544587\pi$
$308$	−60.0000	−0.0111001
$309$	0	0
$310$	−2700.00	−0.494676
$311$	−2106.00	−0.383988	−0.191994	−	0.981396i	$-0.561495\pi$
$311$	−2106.00	−0.383988	−0.191994	+	0.981396i	$0.561495\pi$
$312$	0	0
$313$	−3898.00	−0.703923	−0.351962	−	0.936014i	$-0.614485\pi$
$313$	−3898.00	−0.703923	−0.351962	+	0.936014i	$0.614485\pi$
$314$	−7833.00	−1.40778
$315$	0	0
$316$	−1324.00	−0.235699
$317$	9351.00	1.65680	0.828398	−	0.560140i	$-0.189253\pi$
$317$	9351.00	1.65680	0.828398	+	0.560140i	$0.189253\pi$
$318$	0	0
$319$	3150.00	0.552872
$320$	−3897.00	−0.680778
$321$	0	0
$322$	−36.0000	−0.00623044
$323$	5106.00	0.879583
$324$	0	0
$325$	0	0
$326$	4908.00	0.833831
$327$	0	0
$328$	4851.00	0.816621
$329$	324.000	0.0542939
$330$	0	0
$331$	9172.00	1.52308	0.761539	−	0.648119i	$-0.224444\pi$
$331$	9172.00	1.52308	0.761539	+	0.648119i	$0.224444\pi$
$332$	810.000	0.133899
$333$	0	0
$334$	792.000	0.129749
$335$	8334.00	1.35921
$336$	0	0
$337$	−11089.0	−1.79245	−0.896226	−	0.443598i	$-0.853702\pi$
$337$	−11089.0	−1.79245	−0.896226	+	0.443598i	$0.853702\pi$
$338$	0	0
$339$	0	0
$340$	−999.000	−0.159348
$341$	3000.00	0.476420
$342$	0	0
$343$	1364.00	0.214720
$344$	10794.0	1.69178
$345$	0	0
$346$	−4230.00	−0.657243
$347$	−9762.00	−1.51024	−0.755118	−	0.655589i	$-0.772420\pi$
$347$	−9762.00	−1.51024	−0.755118	+	0.655589i	$0.772420\pi$
$348$	0	0
$349$	8290.00	1.27150	0.635750	−	0.771895i	$-0.280691\pi$
$349$	8290.00	1.27150	0.635750	+	0.771895i	$0.280691\pi$
$350$	264.000	0.0403183
$351$	0	0
$352$	−1350.00	−0.204418
$353$	12405.0	1.87040	0.935200	−	0.354119i	$-0.115219\pi$
$353$	12405.0	1.87040	0.935200	+	0.354119i	$0.115219\pi$
$354$	0	0
$355$	8370.00	1.25136
$356$	498.000	0.0741403
$357$	0	0
$358$	1422.00	0.209930
$359$	−1098.00	−0.161421	−0.0807106	−	0.996738i	$-0.525719\pi$
$359$	−1098.00	−0.161421	−0.0807106	+	0.996738i	$0.525719\pi$
$360$	0	0
$361$	−4743.00	−0.691500
$362$	6747.00	0.979598
$363$	0	0
$364$	0	0
$365$	−2277.00	−0.326530
$366$	0	0
$367$	−5734.00	−0.815565	−0.407783	−	0.913079i	$-0.633698\pi$
$367$	−5734.00	−0.815565	−0.407783	+	0.913079i	$0.633698\pi$
$368$	−426.000	−0.0603445
$369$	0	0
$370$	459.000	0.0644926
$371$	1278.00	0.178842
$372$	0	0
$373$	−8971.00	−1.24531	−0.622655	−	0.782496i	$-0.713946\pi$
$373$	−8971.00	−1.24531	−0.622655	+	0.782496i	$0.713946\pi$
$374$	9990.00	1.38120
$375$	0	0
$376$	3402.00	0.466608
$377$	0	0
$378$	0	0
$379$	−7244.00	−0.981792	−0.490896	−	0.871218i	$-0.663331\pi$
$379$	−7244.00	−0.981792	−0.490896	+	0.871218i	$0.663331\pi$
$380$	−414.000	−0.0558888
$381$	0	0
$382$	−10332.0	−1.38385
$383$	−6312.00	−0.842110	−0.421055	−	0.907035i	$-0.638340\pi$
$383$	−6312.00	−0.842110	−0.421055	+	0.907035i	$0.638340\pi$
$384$	0	0
$385$	540.000	0.0714830
$386$	12819.0	1.69034
$387$	0	0
$388$	−1358.00	−0.177686
$389$	−3627.00	−0.472741	−0.236370	−	0.971663i	$-0.575958\pi$
$389$	−3627.00	−0.472741	−0.236370	+	0.971663i	$0.575958\pi$
$390$	0	0
$391$	666.000	0.0861408
$392$	7119.00	0.917255
$393$	0	0
$394$	−5958.00	−0.761827
$395$	11916.0	1.51787
$396$	0	0
$397$	3898.00	0.492783	0.246392	−	0.969170i	$-0.420755\pi$
$397$	3898.00	0.492783	0.246392	+	0.969170i	$0.420755\pi$
$398$	−7158.00	−0.901503
$399$	0	0
$400$	3124.00	0.390500
$401$	−5703.00	−0.710210	−0.355105	−	0.934826i	$-0.615555\pi$
$401$	−5703.00	−0.710210	−0.355105	+	0.934826i	$0.615555\pi$
$402$	0	0
$403$	0	0
$404$	357.000	0.0439639
$405$	0	0
$406$	−630.000	−0.0770108
$407$	−510.000	−0.0621124
$408$	0	0
$409$	−6311.00	−0.762980	−0.381490	−	0.924373i	$-0.624589\pi$
$409$	−6311.00	−0.762980	−0.381490	+	0.924373i	$0.624589\pi$
$410$	6237.00	0.751277
$411$	0	0
$412$	1118.00	0.133689
$413$	−1200.00	−0.142974
$414$	0	0
$415$	−7290.00	−0.862294
$416$	0	0
$417$	0	0
$418$	4140.00	0.484435
$419$	2328.00	0.271433	0.135716	−	0.990748i	$-0.456666\pi$
$419$	2328.00	0.271433	0.135716	+	0.990748i	$0.456666\pi$
$420$	0	0
$421$	−2045.00	−0.236739	−0.118370	−	0.992970i	$-0.537767\pi$
$421$	−2045.00	−0.236739	−0.118370	+	0.992970i	$0.537767\pi$
$422$	−4800.00	−0.553697
$423$	0	0
$424$	13419.0	1.53699
$425$	−4884.00	−0.557432
$426$	0	0
$427$	−466.000	−0.0528134
$428$	−714.000	−0.0806367
$429$	0	0
$430$	13878.0	1.55641
$431$	5034.00	0.562597	0.281298	−	0.959620i	$-0.409235\pi$
$431$	5034.00	0.562597	0.281298	+	0.959620i	$0.409235\pi$
$432$	0	0
$433$	4283.00	0.475353	0.237676	−	0.971344i	$-0.423614\pi$
$433$	4283.00	0.475353	0.237676	+	0.971344i	$0.423614\pi$
$434$	−600.000	−0.0663616
$435$	0	0
$436$	−2006.00	−0.220344
$437$	276.000	0.0302125
$438$	0	0
$439$	−1306.00	−0.141986	−0.0709931	−	0.997477i	$-0.522617\pi$
$439$	−1306.00	−0.141986	−0.0709931	+	0.997477i	$0.522617\pi$
$440$	5670.00	0.614333
$441$	0	0
$442$	0	0
$443$	5796.00	0.621617	0.310808	−	0.950473i	$-0.399400\pi$
$443$	5796.00	0.621617	0.310808	+	0.950473i	$0.399400\pi$
$444$	0	0
$445$	−4482.00	−0.477454
$446$	11496.0	1.22052
$447$	0	0
$448$	−866.000	−0.0913274
$449$	2706.00	0.284419	0.142209	−	0.989837i	$-0.454579\pi$
$449$	2706.00	0.284419	0.142209	+	0.989837i	$0.454579\pi$
$450$	0	0
$451$	−6930.00	−0.723550
$452$	1119.00	0.116445
$453$	0	0
$454$	−4194.00	−0.433555
$455$	0	0
$456$	0	0
$457$	829.000	0.0848555	0.0424278	−	0.999100i	$-0.486491\pi$
$457$	829.000	0.0848555	0.0424278	+	0.999100i	$0.486491\pi$
$458$	−13398.0	−1.36692
$459$	0	0
$460$	−54.0000	−0.00547340
$461$	−5493.00	−0.554956	−0.277478	−	0.960732i	$-0.589498\pi$
$461$	−5493.00	−0.554956	−0.277478	+	0.960732i	$0.589498\pi$
$462$	0	0
$463$	15346.0	1.54037	0.770183	−	0.637823i	$-0.220165\pi$
$463$	15346.0	1.54037	0.770183	+	0.637823i	$0.220165\pi$
$464$	−7455.00	−0.745883
$465$	0	0
$466$	4914.00	0.488491
$467$	9594.00	0.950658	0.475329	−	0.879808i	$-0.342329\pi$
$467$	9594.00	0.950658	0.475329	+	0.879808i	$0.342329\pi$
$468$	0	0
$469$	1852.00	0.182340
$470$	4374.00	0.429271
$471$	0	0
$472$	−12600.0	−1.22873
$473$	−15420.0	−1.49897
$474$	0	0
$475$	−2024.00	−0.195511
$476$	−222.000	−0.0213768
$477$	0	0
$478$	−1782.00	−0.170516
$479$	−12840.0	−1.22479	−0.612395	−	0.790552i	$-0.709794\pi$
$479$	−12840.0	−1.22479	−0.612395	+	0.790552i	$0.709794\pi$
$480$	0	0
$481$	0	0
$482$	−6909.00	−0.652897
$483$	0	0
$484$	−431.000	−0.0404771
$485$	12222.0	1.14427
$486$	0	0
$487$	14086.0	1.31067	0.655336	−	0.755337i	$-0.272527\pi$
$487$	14086.0	1.31067	0.655336	+	0.755337i	$0.272527\pi$
$488$	−4893.00	−0.453885
$489$	0	0
$490$	9153.00	0.843858
$491$	−11694.0	−1.07483	−0.537416	−	0.843317i	$-0.680600\pi$
$491$	−11694.0	−1.07483	−0.537416	+	0.843317i	$0.680600\pi$
$492$	0	0
$493$	11655.0	1.06474
$494$	0	0
$495$	0	0
$496$	−7100.00	−0.642741
$497$	1860.00	0.167872
$498$	0	0
$499$	3688.00	0.330857	0.165428	−	0.986222i	$-0.447099\pi$
$499$	3688.00	0.330857	0.165428	+	0.986222i	$0.447099\pi$
$500$	1521.00	0.136042
$501$	0	0
$502$	−18972.0	−1.68678
$503$	4746.00	0.420703	0.210352	−	0.977626i	$-0.432539\pi$
$503$	4746.00	0.420703	0.210352	+	0.977626i	$0.432539\pi$
$504$	0	0
$505$	−3213.00	−0.283122
$506$	540.000	0.0474425
$507$	0	0
$508$	−604.000	−0.0527523
$509$	−14505.0	−1.26311	−0.631555	−	0.775331i	$-0.717583\pi$
$509$	−14505.0	−1.26311	−0.631555	+	0.775331i	$0.717583\pi$
$510$	0	0
$511$	−506.000	−0.0438045
$512$	−8733.00	−0.753804
$513$	0	0
$514$	−23499.0	−2.01653
$515$	−10062.0	−0.860941
$516$	0	0
$517$	−4860.00	−0.413429
$518$	102.000	0.00865178
$519$	0	0
$520$	0	0
$521$	−5085.00	−0.427597	−0.213798	−	0.976878i	$-0.568584\pi$
$521$	−5085.00	−0.427597	−0.213798	+	0.976878i	$0.568584\pi$
$522$	0	0
$523$	−10882.0	−0.909821	−0.454911	−	0.890537i	$-0.650329\pi$
$523$	−10882.0	−0.909821	−0.454911	+	0.890537i	$0.650329\pi$
$524$	1584.00	0.132056
$525$	0	0
$526$	9090.00	0.753503
$527$	11100.0	0.917502
$528$	0	0
$529$	−12131.0	−0.997041
$530$	17253.0	1.41400
$531$	0	0
$532$	−92.0000	−0.00749757
$533$	0	0
$534$	0	0
$535$	6426.00	0.519290
$536$	19446.0	1.56705
$537$	0	0
$538$	1602.00	0.128378
$539$	−10170.0	−0.812714
$540$	0	0
$541$	4699.00	0.373430	0.186715	−	0.982414i	$-0.440216\pi$
$541$	4699.00	0.373430	0.186715	+	0.982414i	$0.440216\pi$
$542$	11064.0	0.876826
$543$	0	0
$544$	−4995.00	−0.393674
$545$	18054.0	1.41899
$546$	0	0
$547$	8270.00	0.646434	0.323217	−	0.946325i	$-0.395236\pi$
$547$	8270.00	0.646434	0.323217	+	0.946325i	$0.395236\pi$
$548$	717.000	0.0558918
$549$	0	0
$550$	−3960.00	−0.307009
$551$	4830.00	0.373439
$552$	0	0
$553$	2648.00	0.203625
$554$	5595.00	0.429077
$555$	0	0
$556$	−820.000	−0.0625463
$557$	−22785.0	−1.73327	−0.866635	−	0.498943i	$-0.833722\pi$
$557$	−22785.0	−1.73327	−0.866635	+	0.498943i	$0.833722\pi$
$558$	0	0
$559$	0	0
$560$	−1278.00	−0.0964381
$561$	0	0
$562$	8991.00	0.674844
$563$	11928.0	0.892905	0.446452	−	0.894807i	$-0.352687\pi$
$563$	11928.0	0.892905	0.446452	+	0.894807i	$0.352687\pi$
$564$	0	0
$565$	−10071.0	−0.749894
$566$	−12342.0	−0.916560
$567$	0	0
$568$	19530.0	1.44271
$569$	7962.00	0.586616	0.293308	−	0.956018i	$-0.405244\pi$
$569$	7962.00	0.586616	0.293308	+	0.956018i	$0.405244\pi$
$570$	0	0
$571$	20618.0	1.51110	0.755549	−	0.655093i	$-0.227370\pi$
$571$	20618.0	1.51110	0.755549	+	0.655093i	$0.227370\pi$
$572$	0	0
$573$	0	0
$574$	1386.00	0.100785
$575$	−264.000	−0.0191471
$576$	0	0
$577$	3493.00	0.252020	0.126010	−	0.992029i	$-0.459783\pi$
$577$	3493.00	0.252020	0.126010	+	0.992029i	$0.459783\pi$
$578$	22224.0	1.59930
$579$	0	0
$580$	−945.000	−0.0676534
$581$	−1620.00	−0.115678
$582$	0	0
$583$	−19170.0	−1.36182
$584$	−5313.00	−0.376461
$585$	0	0
$586$	−13995.0	−0.986567
$587$	10416.0	0.732392	0.366196	−	0.930538i	$-0.380660\pi$
$587$	10416.0	0.732392	0.366196	+	0.930538i	$0.380660\pi$
$588$	0	0
$589$	4600.00	0.321799
$590$	−16200.0	−1.13041
$591$	0	0
$592$	1207.00	0.0837963
$593$	2061.00	0.142724	0.0713618	−	0.997450i	$-0.477266\pi$
$593$	2061.00	0.142724	0.0713618	+	0.997450i	$0.477266\pi$
$594$	0	0
$595$	1998.00	0.137664
$596$	−1749.00	−0.120204
$597$	0	0
$598$	0	0
$599$	−12456.0	−0.849647	−0.424823	−	0.905276i	$-0.639664\pi$
$599$	−12456.0	−0.849647	−0.424823	+	0.905276i	$0.639664\pi$
$600$	0	0
$601$	−781.000	−0.0530077	−0.0265039	−	0.999649i	$-0.508437\pi$
$601$	−781.000	−0.0530077	−0.0265039	+	0.999649i	$0.508437\pi$
$602$	3084.00	0.208795
$603$	0	0
$604$	370.000	0.0249256
$605$	3879.00	0.260667
$606$	0	0
$607$	19304.0	1.29082	0.645408	−	0.763838i	$-0.276687\pi$
$607$	19304.0	1.29082	0.645408	+	0.763838i	$0.276687\pi$
$608$	−2070.00	−0.138075
$609$	0	0
$610$	−6291.00	−0.417566
$611$	0	0
$612$	0	0
$613$	−12041.0	−0.793363	−0.396681	−	0.917956i	$-0.629838\pi$
$613$	−12041.0	−0.793363	−0.396681	+	0.917956i	$0.629838\pi$
$614$	−4506.00	−0.296168
$615$	0	0
$616$	1260.00	0.0824137
$617$	9717.00	0.634022	0.317011	−	0.948422i	$-0.397321\pi$
$617$	9717.00	0.634022	0.317011	+	0.948422i	$0.397321\pi$
$618$	0	0
$619$	21040.0	1.36619	0.683093	−	0.730332i	$-0.260634\pi$
$619$	21040.0	1.36619	0.683093	+	0.730332i	$0.260634\pi$
$620$	−900.000	−0.0582982
$621$	0	0
$622$	−6318.00	−0.407281
$623$	−996.000	−0.0640512
$624$	0	0
$625$	−8189.00	−0.524096
$626$	−11694.0	−0.746623
$627$	0	0
$628$	−2611.00	−0.165908
$629$	−1887.00	−0.119618
$630$	0	0
$631$	5068.00	0.319737	0.159868	−	0.987138i	$-0.448893\pi$
$631$	5068.00	0.319737	0.159868	+	0.987138i	$0.448893\pi$
$632$	27804.0	1.74997
$633$	0	0
$634$	28053.0	1.75730
$635$	5436.00	0.339718
$636$	0	0
$637$	0	0
$638$	9450.00	0.586409
$639$	0	0
$640$	−14931.0	−0.922187
$641$	−10185.0	−0.627587	−0.313794	−	0.949491i	$-0.601600\pi$
$641$	−10185.0	−0.627587	−0.313794	+	0.949491i	$0.601600\pi$
$642$	0	0
$643$	−25928.0	−1.59020	−0.795101	−	0.606476i	$-0.792582\pi$
$643$	−25928.0	−1.59020	−0.795101	+	0.606476i	$0.792582\pi$
$644$	−12.0000	−0.000734264 0
$645$	0	0
$646$	15318.0	0.932939
$647$	−23160.0	−1.40729	−0.703643	−	0.710554i	$-0.748444\pi$
$647$	−23160.0	−1.40729	−0.703643	+	0.710554i	$0.748444\pi$
$648$	0	0
$649$	18000.0	1.08869
$650$	0	0
$651$	0	0
$652$	1636.00	0.0982680
$653$	−16626.0	−0.996364	−0.498182	−	0.867073i	$-0.665999\pi$
$653$	−16626.0	−0.996364	−0.498182	+	0.867073i	$0.665999\pi$
$654$	0	0
$655$	−14256.0	−0.850424
$656$	16401.0	0.976146
$657$	0	0
$658$	972.000	0.0575874
$659$	14808.0	0.875323	0.437661	−	0.899140i	$-0.355807\pi$
$659$	14808.0	0.875323	0.437661	+	0.899140i	$0.355807\pi$
$660$	0	0
$661$	−4853.00	−0.285567	−0.142784	−	0.989754i	$-0.545605\pi$
$661$	−4853.00	−0.285567	−0.142784	+	0.989754i	$0.545605\pi$
$662$	27516.0	1.61547
$663$	0	0
$664$	−17010.0	−0.994151
$665$	828.000	0.0482834
$666$	0	0
$667$	630.000	0.0365723
$668$	264.000	0.0152911
$669$	0	0
$670$	25002.0	1.44166
$671$	6990.00	0.402155
$672$	0	0
$673$	−16165.0	−0.925877	−0.462938	−	0.886391i	$-0.653205\pi$
$673$	−16165.0	−0.925877	−0.462938	+	0.886391i	$0.653205\pi$
$674$	−33267.0	−1.90118
$675$	0	0
$676$	0	0
$677$	25686.0	1.45819	0.729094	−	0.684414i	$-0.239942\pi$
$677$	25686.0	1.45819	0.729094	+	0.684414i	$0.239942\pi$
$678$	0	0
$679$	2716.00	0.153506
$680$	20979.0	1.18310
$681$	0	0
$682$	9000.00	0.505319
$683$	−19056.0	−1.06758	−0.533790	−	0.845617i	$-0.679233\pi$
$683$	−19056.0	−1.06758	−0.533790	+	0.845617i	$0.679233\pi$
$684$	0	0
$685$	−6453.00	−0.359936
$686$	4092.00	0.227745
$687$	0	0
$688$	36494.0	2.02227
$689$	0	0
$690$	0	0
$691$	16390.0	0.902323	0.451161	−	0.892442i	$-0.351010\pi$
$691$	16390.0	0.902323	0.451161	+	0.892442i	$0.351010\pi$
$692$	−1410.00	−0.0774569
$693$	0	0
$694$	−29286.0	−1.60185
$695$	7380.00	0.402790
$696$	0	0
$697$	−25641.0	−1.39343
$698$	24870.0	1.34863
$699$	0	0
$700$	88.0000	0.00475155
$701$	27846.0	1.50033	0.750163	−	0.661253i	$-0.229975\pi$
$701$	27846.0	1.50033	0.750163	+	0.661253i	$0.229975\pi$
$702$	0	0
$703$	−782.000	−0.0419540
$704$	12990.0	0.695425
$705$	0	0
$706$	37215.0	1.98386
$707$	−714.000	−0.0379812
$708$	0	0
$709$	12283.0	0.650632	0.325316	−	0.945605i	$-0.394529\pi$
$709$	12283.0	0.650632	0.325316	+	0.945605i	$0.394529\pi$
$710$	25110.0	1.32727
$711$	0	0
$712$	−10458.0	−0.550464
$713$	600.000	0.0315150
$714$	0	0
$715$	0	0
$716$	474.000	0.0247405
$717$	0	0
$718$	−3294.00	−0.171213
$719$	−25512.0	−1.32328	−0.661639	−	0.749822i	$-0.730139\pi$
$719$	−25512.0	−1.32328	−0.661639	+	0.749822i	$0.730139\pi$
$720$	0	0
$721$	−2236.00	−0.115497
$722$	−14229.0	−0.733447
$723$	0	0
$724$	2249.00	0.115447
$725$	−4620.00	−0.236666
$726$	0	0
$727$	6110.00	0.311702	0.155851	−	0.987781i	$-0.450188\pi$
$727$	6110.00	0.311702	0.155851	+	0.987781i	$0.450188\pi$
$728$	0	0
$729$	0	0
$730$	−6831.00	−0.346338
$731$	−57054.0	−2.88676
$732$	0	0
$733$	27127.0	1.36693	0.683464	−	0.729984i	$-0.260473\pi$
$733$	27127.0	1.36693	0.683464	+	0.729984i	$0.260473\pi$
$734$	−17202.0	−0.865037
$735$	0	0
$736$	−270.000	−0.0135222
$737$	−27780.0	−1.38845
$738$	0	0
$739$	880.000	0.0438042	0.0219021	−	0.999760i	$-0.493028\pi$
$739$	880.000	0.0438042	0.0219021	+	0.999760i	$0.493028\pi$
$740$	153.000	0.00760053
$741$	0	0
$742$	3834.00	0.189691
$743$	−21876.0	−1.08015	−0.540076	−	0.841616i	$-0.681604\pi$
$743$	−21876.0	−1.08015	−0.540076	+	0.841616i	$0.681604\pi$
$744$	0	0
$745$	15741.0	0.774102
$746$	−26913.0	−1.32085
$747$	0	0
$748$	3330.00	0.162777
$749$	1428.00	0.0696635
$750$	0	0
$751$	11798.0	0.573256	0.286628	−	0.958042i	$-0.407466\pi$
$751$	11798.0	0.573256	0.286628	+	0.958042i	$0.407466\pi$
$752$	11502.0	0.557759
$753$	0	0
$754$	0	0
$755$	−3330.00	−0.160518
$756$	0	0
$757$	−8074.00	−0.387655	−0.193827	−	0.981036i	$-0.562090\pi$
$757$	−8074.00	−0.387655	−0.193827	+	0.981036i	$0.562090\pi$
$758$	−21732.0	−1.04135
$759$	0	0
$760$	8694.00	0.414953
$761$	19554.0	0.931448	0.465724	−	0.884930i	$-0.345794\pi$
$761$	19554.0	0.931448	0.465724	+	0.884930i	$0.345794\pi$
$762$	0	0
$763$	4012.00	0.190359
$764$	−3444.00	−0.163088
$765$	0	0
$766$	−18936.0	−0.893193
$767$	0	0
$768$	0	0
$769$	−14030.0	−0.657913	−0.328956	−	0.944345i	$-0.606697\pi$
$769$	−14030.0	−0.657913	−0.328956	+	0.944345i	$0.606697\pi$
$770$	1620.00	0.0758192
$771$	0	0
$772$	4273.00	0.199208
$773$	36042.0	1.67703	0.838513	−	0.544882i	$-0.183426\pi$
$773$	36042.0	1.67703	0.838513	+	0.544882i	$0.183426\pi$
$774$	0	0
$775$	−4400.00	−0.203939
$776$	28518.0	1.31925
$777$	0	0
$778$	−10881.0	−0.501417
$779$	−10626.0	−0.488724
$780$	0	0
$781$	−27900.0	−1.27828
$782$	1998.00	0.0913662
$783$	0	0
$784$	24069.0	1.09644
$785$	23499.0	1.06843
$786$	0	0
$787$	−28628.0	−1.29667	−0.648334	−	0.761356i	$-0.724534\pi$
$787$	−28628.0	−1.29667	−0.648334	+	0.761356i	$0.724534\pi$
$788$	−1986.00	−0.0897821
$789$	0	0
$790$	35748.0	1.60995
$791$	−2238.00	−0.100599
$792$	0	0
$793$	0	0
$794$	11694.0	0.522676
$795$	0	0
$796$	−2386.00	−0.106243
$797$	37434.0	1.66371	0.831857	−	0.554990i	$-0.187278\pi$
$797$	37434.0	1.66371	0.831857	+	0.554990i	$0.187278\pi$
$798$	0	0
$799$	−17982.0	−0.796192
$800$	1980.00	0.0875045
$801$	0	0
$802$	−17109.0	−0.753292
$803$	7590.00	0.333556
$804$	0	0
$805$	108.000	0.00472857
$806$	0	0
$807$	0	0
$808$	−7497.00	−0.326415
$809$	−37569.0	−1.63270	−0.816351	−	0.577556i	$-0.804006\pi$
$809$	−37569.0	−1.63270	−0.816351	+	0.577556i	$0.804006\pi$
$810$	0	0
$811$	−5516.00	−0.238832	−0.119416	−	0.992844i	$-0.538102\pi$
$811$	−5516.00	−0.238832	−0.119416	+	0.992844i	$0.538102\pi$
$812$	−210.000	−0.00907581
$813$	0	0
$814$	−1530.00	−0.0658802
$815$	−14724.0	−0.632833
$816$	0	0
$817$	−23644.0	−1.01248
$818$	−18933.0	−0.809263
$819$	0	0
$820$	2079.00	0.0885388
$821$	8778.00	0.373148	0.186574	−	0.982441i	$-0.440262\pi$
$821$	8778.00	0.373148	0.186574	+	0.982441i	$0.440262\pi$
$822$	0	0
$823$	−3088.00	−0.130791	−0.0653955	−	0.997859i	$-0.520831\pi$
$823$	−3088.00	−0.130791	−0.0653955	+	0.997859i	$0.520831\pi$
$824$	−23478.0	−0.992591
$825$	0	0
$826$	−3600.00	−0.151647
$827$	13176.0	0.554020	0.277010	−	0.960867i	$-0.410657\pi$
$827$	13176.0	0.554020	0.277010	+	0.960867i	$0.410657\pi$
$828$	0	0
$829$	−2359.00	−0.0988317	−0.0494158	−	0.998778i	$-0.515736\pi$
$829$	−2359.00	−0.0988317	−0.0494158	+	0.998778i	$0.515736\pi$
$830$	−21870.0	−0.914601
$831$	0	0
$832$	0	0
$833$	−37629.0	−1.56515
$834$	0	0
$835$	−2376.00	−0.0984729
$836$	1380.00	0.0570913
$837$	0	0
$838$	6984.00	0.287898
$839$	−2676.00	−0.110114	−0.0550571	−	0.998483i	$-0.517534\pi$
$839$	−2676.00	−0.110114	−0.0550571	+	0.998483i	$0.517534\pi$
$840$	0	0
$841$	−13364.0	−0.547952
$842$	−6135.00	−0.251100
$843$	0	0
$844$	−1600.00	−0.0652539
$845$	0	0
$846$	0	0
$847$	862.000	0.0349689
$848$	45369.0	1.83724
$849$	0	0
$850$	−14652.0	−0.591246
$851$	−102.000	−0.00410871
$852$	0	0
$853$	−2477.00	−0.0994266	−0.0497133	−	0.998764i	$-0.515831\pi$
$853$	−2477.00	−0.0994266	−0.0497133	+	0.998764i	$0.515831\pi$
$854$	−1398.00	−0.0560171
$855$	0	0
$856$	14994.0	0.598697
$857$	17199.0	0.685539	0.342769	−	0.939420i	$-0.388635\pi$
$857$	17199.0	0.685539	0.342769	+	0.939420i	$0.388635\pi$
$858$	0	0
$859$	24338.0	0.966708	0.483354	−	0.875425i	$-0.339418\pi$
$859$	24338.0	0.966708	0.483354	+	0.875425i	$0.339418\pi$
$860$	4626.00	0.183425
$861$	0	0
$862$	15102.0	0.596724
$863$	25146.0	0.991865	0.495933	−	0.868361i	$-0.334826\pi$
$863$	25146.0	0.991865	0.495933	+	0.868361i	$0.334826\pi$
$864$	0	0
$865$	12690.0	0.498813
$866$	12849.0	0.504188
$867$	0	0
$868$	−200.000	−0.00782079
$869$	−39720.0	−1.55053
$870$	0	0
$871$	0	0
$872$	42126.0	1.63597
$873$	0	0
$874$	828.000	0.0320452
$875$	−3042.00	−0.117530
$876$	0	0
$877$	−18089.0	−0.696490	−0.348245	−	0.937403i	$-0.613222\pi$
$877$	−18089.0	−0.696490	−0.348245	+	0.937403i	$0.613222\pi$
$878$	−3918.00	−0.150599
$879$	0	0
$880$	19170.0	0.734342
$881$	15099.0	0.577410	0.288705	−	0.957418i	$-0.406775\pi$
$881$	15099.0	0.577410	0.288705	+	0.957418i	$0.406775\pi$
$882$	0	0
$883$	33488.0	1.27629	0.638143	−	0.769918i	$-0.279703\pi$
$883$	33488.0	1.27629	0.638143	+	0.769918i	$0.279703\pi$
$884$	0	0
$885$	0	0
$886$	17388.0	0.659324
$887$	39768.0	1.50539	0.752694	−	0.658371i	$-0.228754\pi$
$887$	39768.0	1.50539	0.752694	+	0.658371i	$0.228754\pi$
$888$	0	0
$889$	1208.00	0.0455737
$890$	−13446.0	−0.506417
$891$	0	0
$892$	3832.00	0.143840
$893$	−7452.00	−0.279252
$894$	0	0
$895$	−4266.00	−0.159326
$896$	−3318.00	−0.123713
$897$	0	0
$898$	8118.00	0.301672
$899$	10500.0	0.389538
$900$	0	0
$901$	−70929.0	−2.62263
$902$	−20790.0	−0.767440
$903$	0	0
$904$	−23499.0	−0.864563
$905$	−20241.0	−0.743463
$906$	0	0
$907$	32156.0	1.17720	0.588601	−	0.808424i	$-0.299679\pi$
$907$	32156.0	1.17720	0.588601	+	0.808424i	$0.299679\pi$
$908$	−1398.00	−0.0510950
$909$	0	0
$910$	0	0
$911$	−11520.0	−0.418962	−0.209481	−	0.977813i	$-0.567177\pi$
$911$	−11520.0	−0.418962	−0.209481	+	0.977813i	$0.567177\pi$
$912$	0	0
$913$	24300.0	0.880846
$914$	2487.00	0.0900029
$915$	0	0
$916$	−4466.00	−0.161093
$917$	−3168.00	−0.114086
$918$	0	0
$919$	4952.00	0.177749	0.0888745	−	0.996043i	$-0.471673\pi$
$919$	4952.00	0.177749	0.0888745	+	0.996043i	$0.471673\pi$
$920$	1134.00	0.0406379
$921$	0	0
$922$	−16479.0	−0.588619
$923$	0	0
$924$	0	0
$925$	748.000	0.0265882
$926$	46038.0	1.63380
$927$	0	0
$928$	−4725.00	−0.167140
$929$	8781.00	0.310113	0.155057	−	0.987906i	$-0.450444\pi$
$929$	8781.00	0.310113	0.155057	+	0.987906i	$0.450444\pi$
$930$	0	0
$931$	−15594.0	−0.548950
$932$	1638.00	0.0575692
$933$	0	0
$934$	28782.0	1.00833
$935$	−29970.0	−1.04826
$936$	0	0
$937$	50039.0	1.74461	0.872307	−	0.488959i	$-0.162623\pi$
$937$	50039.0	1.74461	0.872307	+	0.488959i	$0.162623\pi$
$938$	5556.00	0.193401
$939$	0	0
$940$	1458.00	0.0505901
$941$	−50670.0	−1.75536	−0.877681	−	0.479246i	$-0.840910\pi$
$941$	−50670.0	−1.75536	−0.877681	+	0.479246i	$0.840910\pi$
$942$	0	0
$943$	−1386.00	−0.0478625
$944$	−42600.0	−1.46876
$945$	0	0
$946$	−46260.0	−1.58990
$947$	42384.0	1.45438	0.727188	−	0.686438i	$-0.240827\pi$
$947$	42384.0	1.45438	0.727188	+	0.686438i	$0.240827\pi$
$948$	0	0
$949$	0	0
$950$	−6072.00	−0.207370
$951$	0	0
$952$	4662.00	0.158715
$953$	50538.0	1.71782	0.858912	−	0.512123i	$-0.171141\pi$
$953$	50538.0	1.71782	0.858912	+	0.512123i	$0.171141\pi$
$954$	0	0
$955$	30996.0	1.05027
$956$	−594.000	−0.0200955
$957$	0	0
$958$	−38520.0	−1.29909
$959$	−1434.00	−0.0482860
$960$	0	0
$961$	−19791.0	−0.664328
$962$	0	0
$963$	0	0
$964$	−2303.00	−0.0769446
$965$	−38457.0	−1.28288
$966$	0	0
$967$	6886.00	0.228996	0.114498	−	0.993423i	$-0.463474\pi$
$967$	6886.00	0.228996	0.114498	+	0.993423i	$0.463474\pi$
$968$	9051.00	0.300527
$969$	0	0
$970$	36666.0	1.21368
$971$	−9060.00	−0.299433	−0.149716	−	0.988729i	$-0.547836\pi$
$971$	−9060.00	−0.299433	−0.149716	+	0.988729i	$0.547836\pi$
$972$	0	0
$973$	1640.00	0.0540349
$974$	42258.0	1.39018
$975$	0	0
$976$	−16543.0	−0.542550
$977$	−28311.0	−0.927072	−0.463536	−	0.886078i	$-0.653419\pi$
$977$	−28311.0	−0.927072	−0.463536	+	0.886078i	$0.653419\pi$
$978$	0	0
$979$	14940.0	0.487727
$980$	3051.00	0.0994496
$981$	0	0
$982$	−35082.0	−1.14003
$983$	4284.00	0.139001	0.0695007	−	0.997582i	$-0.477859\pi$
$983$	4284.00	0.139001	0.0695007	+	0.997582i	$0.477859\pi$
$984$	0	0
$985$	17874.0	0.578186
$986$	34965.0	1.12932
$987$	0	0
$988$	0	0
$989$	−3084.00	−0.0991562
$990$	0	0
$991$	−2458.00	−0.0787901	−0.0393950	−	0.999224i	$-0.512543\pi$
$991$	−2458.00	−0.0787901	−0.0393950	+	0.999224i	$0.512543\pi$
$992$	−4500.00	−0.144027
$993$	0	0
$994$	5580.00	0.178055
$995$	21474.0	0.684193
$996$	0	0
$997$	24101.0	0.765583	0.382792	−	0.923835i	$-0.374963\pi$
$997$	24101.0	0.765583	0.382792	+	0.923835i	$0.374963\pi$
$998$	11064.0	0.350927
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.j.1.1		1
3.2	odd	2		507.4.a.a.1.1		1
13.4	even	6		117.4.g.b.55.1		2
13.10	even	6		117.4.g.b.100.1		2
13.12	even	2		1521.4.a.c.1.1		1
39.5	even	4		507.4.b.c.337.2		2
39.8	even	4		507.4.b.c.337.1		2
39.17	odd	6		39.4.e.a.16.1	✓	2
39.23	odd	6		39.4.e.a.22.1	yes	2
39.38	odd	2		507.4.a.e.1.1		1
156.23	even	6		624.4.q.b.529.1		2
156.95	even	6		624.4.q.b.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.a.16.1	✓	2	39.17	odd	6
39.4.e.a.22.1	yes	2	39.23	odd	6
117.4.g.b.55.1		2	13.4	even	6
117.4.g.b.100.1		2	13.10	even	6
507.4.a.a.1.1		1	3.2	odd	2
507.4.a.e.1.1		1	39.38	odd	2
507.4.b.c.337.1		2	39.8	even	4
507.4.b.c.337.2		2	39.5	even	4
624.4.q.b.289.1		2	156.95	even	6
624.4.q.b.529.1		2	156.23	even	6
1521.4.a.c.1.1		1	13.12	even	2
1521.4.a.j.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more