LMFDB - Embedded newform 825.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.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:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +9.00000 q^{6} -7.00000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +9.00000 q^{6} -7.00000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +11.0000 q^{11} -3.00000 q^{12} -16.0000 q^{13} +21.0000 q^{14} -71.0000 q^{16} +21.0000 q^{17} -27.0000 q^{18} +125.000 q^{19} +21.0000 q^{21} -33.0000 q^{22} -81.0000 q^{23} -63.0000 q^{24} +48.0000 q^{26} -27.0000 q^{27} -7.00000 q^{28} +186.000 q^{29} -58.0000 q^{31} +45.0000 q^{32} -33.0000 q^{33} -63.0000 q^{34} +9.00000 q^{36} -253.000 q^{37} -375.000 q^{38} +48.0000 q^{39} +63.0000 q^{41} -63.0000 q^{42} -100.000 q^{43} +11.0000 q^{44} +243.000 q^{46} -219.000 q^{47} +213.000 q^{48} -294.000 q^{49} -63.0000 q^{51} -16.0000 q^{52} -192.000 q^{53} +81.0000 q^{54} -147.000 q^{56} -375.000 q^{57} -558.000 q^{58} +249.000 q^{59} -64.0000 q^{61} +174.000 q^{62} -63.0000 q^{63} +433.000 q^{64} +99.0000 q^{66} +272.000 q^{67} +21.0000 q^{68} +243.000 q^{69} -645.000 q^{71} +189.000 q^{72} -112.000 q^{73} +759.000 q^{74} +125.000 q^{76} -77.0000 q^{77} -144.000 q^{78} +509.000 q^{79} +81.0000 q^{81} -189.000 q^{82} -1254.00 q^{83} +21.0000 q^{84} +300.000 q^{86} -558.000 q^{87} +231.000 q^{88} +756.000 q^{89} +112.000 q^{91} -81.0000 q^{92} +174.000 q^{93} +657.000 q^{94} -135.000 q^{96} +839.000 q^{97} +882.000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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 $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	1.00000	0.125000
$5$	0	0
$5$	0	0
$6$	9.00000	0.612372
$7$	−7.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−7.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	21.0000	0.928078
$9$	9.00000	0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−3.00000	−0.0721688
$13$	−16.0000	−0.341354	−0.170677	−	0.985327i	$-0.554595\pi$
$13$	−16.0000	−0.341354	−0.170677	+	0.985327i	$0.554595\pi$
$14$	21.0000	0.400892
$15$	0	0
$16$	−71.0000	−1.10938
$17$	21.0000	0.299603	0.149801	−	0.988716i	$-0.452137\pi$
$17$	21.0000	0.299603	0.149801	+	0.988716i	$0.452137\pi$
$18$	−27.0000	−0.353553
$19$	125.000	1.50931	0.754657	−	0.656119i	$-0.227803\pi$
$19$	125.000	1.50931	0.754657	+	0.656119i	$0.227803\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	−33.0000	−0.319801
$23$	−81.0000	−0.734333	−0.367167	−	0.930155i	$-0.619672\pi$
$23$	−81.0000	−0.734333	−0.367167	+	0.930155i	$0.619672\pi$
$24$	−63.0000	−0.535826
$25$	0	0
$26$	48.0000	0.362061
$27$	−27.0000	−0.192450
$28$	−7.00000	−0.0472456
$29$	186.000	1.19101	0.595506	−	0.803351i	$-0.296952\pi$
$29$	186.000	1.19101	0.595506	+	0.803351i	$0.296952\pi$
$30$	0	0
$31$	−58.0000	−0.336036	−0.168018	−	0.985784i	$-0.553737\pi$
$31$	−58.0000	−0.336036	−0.168018	+	0.985784i	$0.553737\pi$
$32$	45.0000	0.248592
$33$	−33.0000	−0.174078
$34$	−63.0000	−0.317777
$35$	0	0
$36$	9.00000	0.0416667
$37$	−253.000	−1.12413	−0.562067	−	0.827092i	$-0.689994\pi$
$37$	−253.000	−1.12413	−0.562067	+	0.827092i	$0.689994\pi$
$38$	−375.000	−1.60087
$39$	48.0000	0.197081
$40$	0	0
$41$	63.0000	0.239974	0.119987	−	0.992775i	$-0.461715\pi$
$41$	63.0000	0.239974	0.119987	+	0.992775i	$0.461715\pi$
$42$	−63.0000	−0.231455
$43$	−100.000	−0.354648	−0.177324	−	0.984153i	$-0.556744\pi$
$43$	−100.000	−0.354648	−0.177324	+	0.984153i	$0.556744\pi$
$44$	11.0000	0.0376889
$45$	0	0
$46$	243.000	0.778878
$47$	−219.000	−0.679669	−0.339834	−	0.940485i	$-0.610371\pi$
$47$	−219.000	−0.679669	−0.339834	+	0.940485i	$0.610371\pi$
$48$	213.000	0.640498
$49$	−294.000	−0.857143
$50$	0	0
$51$	−63.0000	−0.172976
$52$	−16.0000	−0.0426692
$53$	−192.000	−0.497608	−0.248804	−	0.968554i	$-0.580038\pi$
$53$	−192.000	−0.497608	−0.248804	+	0.968554i	$0.580038\pi$
$54$	81.0000	0.204124
$55$	0	0
$56$	−147.000	−0.350780
$57$	−375.000	−0.871403
$58$	−558.000	−1.26326
$59$	249.000	0.549441	0.274721	−	0.961524i	$-0.411415\pi$
$59$	249.000	0.549441	0.274721	+	0.961524i	$0.411415\pi$
$60$	0	0
$61$	−64.0000	−0.134334	−0.0671669	−	0.997742i	$-0.521396\pi$
$61$	−64.0000	−0.134334	−0.0671669	+	0.997742i	$0.521396\pi$
$62$	174.000	0.356420
$63$	−63.0000	−0.125988
$64$	433.000	0.845703
$65$	0	0
$66$	99.0000	0.184637
$67$	272.000	0.495971	0.247986	−	0.968764i	$-0.420231\pi$
$67$	272.000	0.495971	0.247986	+	0.968764i	$0.420231\pi$
$68$	21.0000	0.0374504
$69$	243.000	0.423968
$70$	0	0
$71$	−645.000	−1.07813	−0.539066	−	0.842263i	$-0.681223\pi$
$71$	−645.000	−1.07813	−0.539066	+	0.842263i	$0.681223\pi$
$72$	189.000	0.309359
$73$	−112.000	−0.179570	−0.0897850	−	0.995961i	$-0.528618\pi$
$73$	−112.000	−0.179570	−0.0897850	+	0.995961i	$0.528618\pi$
$74$	759.000	1.19232
$75$	0	0
$76$	125.000	0.188664
$77$	−77.0000	−0.113961
$78$	−144.000	−0.209036
$79$	509.000	0.724898	0.362449	−	0.932004i	$-0.381941\pi$
$79$	509.000	0.724898	0.362449	+	0.932004i	$0.381941\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−189.000	−0.254531
$83$	−1254.00	−1.65837	−0.829183	−	0.558977i	$-0.811194\pi$
$83$	−1254.00	−1.65837	−0.829183	+	0.558977i	$0.811194\pi$
$84$	21.0000	0.0272772
$85$	0	0
$86$	300.000	0.376161
$87$	−558.000	−0.687631
$88$	231.000	0.279826
$89$	756.000	0.900403	0.450201	−	0.892927i	$-0.351352\pi$
$89$	756.000	0.900403	0.450201	+	0.892927i	$0.351352\pi$
$90$	0	0
$91$	112.000	0.129020
$92$	−81.0000	−0.0917917
$93$	174.000	0.194010
$94$	657.000	0.720898
$95$	0	0
$96$	−135.000	−0.143525
$97$	839.000	0.878222	0.439111	−	0.898433i	$-0.355293\pi$
$97$	839.000	0.878222	0.439111	+	0.898433i	$0.355293\pi$
$98$	882.000	0.909137
$99$	99.0000	0.100504
$100$	0	0
$101$	−1413.00	−1.39207	−0.696033	−	0.718009i	$-0.745053\pi$
$101$	−1413.00	−1.39207	−0.696033	+	0.718009i	$0.745053\pi$
$102$	189.000	0.183469
$103$	1634.00	1.56313	0.781567	−	0.623821i	$-0.214421\pi$
$103$	1634.00	1.56313	0.781567	+	0.623821i	$0.214421\pi$
$104$	−336.000	−0.316803
$105$	0	0
$106$	576.000	0.527793
$107$	726.000	0.655935	0.327968	−	0.944689i	$-0.393636\pi$
$107$	726.000	0.655935	0.327968	+	0.944689i	$0.393636\pi$
$108$	−27.0000	−0.0240563
$109$	1712.00	1.50440	0.752201	−	0.658934i	$-0.228992\pi$
$109$	1712.00	1.50440	0.752201	+	0.658934i	$0.228992\pi$
$110$	0	0
$111$	759.000	0.649019
$112$	497.000	0.419304
$113$	−1128.00	−0.939056	−0.469528	−	0.882918i	$-0.655576\pi$
$113$	−1128.00	−0.939056	−0.469528	+	0.882918i	$0.655576\pi$
$114$	1125.00	0.924262
$115$	0	0
$116$	186.000	0.148876
$117$	−144.000	−0.113785
$118$	−747.000	−0.582771
$119$	−147.000	−0.113239
$120$	0	0
$121$	121.000	0.0909091
$122$	192.000	0.142482
$123$	−189.000	−0.138549
$124$	−58.0000	−0.0420045
$125$	0	0
$126$	189.000	0.133631
$127$	1127.00	0.787442	0.393721	−	0.919230i	$-0.371188\pi$
$127$	1127.00	0.787442	0.393721	+	0.919230i	$0.371188\pi$
$128$	−1659.00	−1.14560
$129$	300.000	0.204756
$130$	0	0
$131$	−1122.00	−0.748318	−0.374159	−	0.927365i	$-0.622068\pi$
$131$	−1122.00	−0.748318	−0.374159	+	0.927365i	$0.622068\pi$
$132$	−33.0000	−0.0217597
$133$	−875.000	−0.570467
$134$	−816.000	−0.526057
$135$	0	0
$136$	441.000	0.278055
$137$	54.0000	0.0336754	0.0168377	−	0.999858i	$-0.494640\pi$
$137$	54.0000	0.0336754	0.0168377	+	0.999858i	$0.494640\pi$
$138$	−729.000	−0.449686
$139$	1748.00	1.06664	0.533322	−	0.845913i	$-0.320944\pi$
$139$	1748.00	1.06664	0.533322	+	0.845913i	$0.320944\pi$
$140$	0	0
$141$	657.000	0.392407
$142$	1935.00	1.14353
$143$	−176.000	−0.102922
$144$	−639.000	−0.369792
$145$	0	0
$146$	336.000	0.190463
$147$	882.000	0.494872
$148$	−253.000	−0.140517
$149$	1797.00	0.988027	0.494013	−	0.869454i	$-0.335529\pi$
$149$	1797.00	0.988027	0.494013	+	0.869454i	$0.335529\pi$
$150$	0	0
$151$	1040.00	0.560490	0.280245	−	0.959928i	$-0.409584\pi$
$151$	1040.00	0.560490	0.280245	+	0.959928i	$0.409584\pi$
$152$	2625.00	1.40076
$153$	189.000	0.0998676
$154$	231.000	0.120873
$155$	0	0
$156$	48.0000	0.0246351
$157$	−562.000	−0.285685	−0.142842	−	0.989745i	$-0.545624\pi$
$157$	−562.000	−0.285685	−0.142842	+	0.989745i	$0.545624\pi$
$158$	−1527.00	−0.768871
$159$	576.000	0.287294
$160$	0	0
$161$	567.000	0.277552
$162$	−243.000	−0.117851
$163$	2432.00	1.16864	0.584322	−	0.811522i	$-0.301361\pi$
$163$	2432.00	1.16864	0.584322	+	0.811522i	$0.301361\pi$
$164$	63.0000	0.0299968
$165$	0	0
$166$	3762.00	1.75896
$167$	2340.00	1.08428	0.542140	−	0.840288i	$-0.317614\pi$
$167$	2340.00	1.08428	0.542140	+	0.840288i	$0.317614\pi$
$168$	441.000	0.202523
$169$	−1941.00	−0.883477
$170$	0	0
$171$	1125.00	0.503105
$172$	−100.000	−0.0443310
$173$	3747.00	1.64670	0.823350	−	0.567534i	$-0.192102\pi$
$173$	3747.00	1.64670	0.823350	+	0.567534i	$0.192102\pi$
$174$	1674.00	0.729343
$175$	0	0
$176$	−781.000	−0.334489
$177$	−747.000	−0.317220
$178$	−2268.00	−0.955021
$179$	−267.000	−0.111489	−0.0557445	−	0.998445i	$-0.517753\pi$
$179$	−267.000	−0.111489	−0.0557445	+	0.998445i	$0.517753\pi$
$180$	0	0
$181$	4277.00	1.75639	0.878196	−	0.478301i	$-0.158747\pi$
$181$	4277.00	1.75639	0.878196	+	0.478301i	$0.158747\pi$
$182$	−336.000	−0.136846
$183$	192.000	0.0775576
$184$	−1701.00	−0.681518
$185$	0	0
$186$	−522.000	−0.205779
$187$	231.000	0.0903337
$188$	−219.000	−0.0849586
$189$	189.000	0.0727393
$190$	0	0
$191$	−1797.00	−0.680766	−0.340383	−	0.940287i	$-0.610557\pi$
$191$	−1797.00	−0.680766	−0.340383	+	0.940287i	$0.610557\pi$
$192$	−1299.00	−0.488267
$193$	1988.00	0.741448	0.370724	−	0.928743i	$-0.379110\pi$
$193$	1988.00	0.741448	0.370724	+	0.928743i	$0.379110\pi$
$194$	−2517.00	−0.931495
$195$	0	0
$196$	−294.000	−0.107143
$197$	3327.00	1.20324	0.601622	−	0.798781i	$-0.294522\pi$
$197$	3327.00	1.20324	0.601622	+	0.798781i	$0.294522\pi$
$198$	−297.000	−0.106600
$199$	−1780.00	−0.634075	−0.317037	−	0.948413i	$-0.602688\pi$
$199$	−1780.00	−0.634075	−0.317037	+	0.948413i	$0.602688\pi$
$200$	0	0
$201$	−816.000	−0.286349
$202$	4239.00	1.47651
$203$	−1302.00	−0.450160
$204$	−63.0000	−0.0216220
$205$	0	0
$206$	−4902.00	−1.65795
$207$	−729.000	−0.244778
$208$	1136.00	0.378690
$209$	1375.00	0.455075
$210$	0	0
$211$	2180.00	0.711267	0.355634	−	0.934625i	$-0.384265\pi$
$211$	2180.00	0.711267	0.355634	+	0.934625i	$0.384265\pi$
$212$	−192.000	−0.0622010
$213$	1935.00	0.622460
$214$	−2178.00	−0.695724
$215$	0	0
$216$	−567.000	−0.178609
$217$	406.000	0.127010
$218$	−5136.00	−1.59566
$219$	336.000	0.103675
$220$	0	0
$221$	−336.000	−0.102271
$222$	−2277.00	−0.688388
$223$	3848.00	1.15552	0.577760	−	0.816206i	$-0.303927\pi$
$223$	3848.00	1.15552	0.577760	+	0.816206i	$0.303927\pi$
$224$	−315.000	−0.0939590
$225$	0	0
$226$	3384.00	0.996019
$227$	−1386.00	−0.405251	−0.202626	−	0.979256i	$-0.564947\pi$
$227$	−1386.00	−0.405251	−0.202626	+	0.979256i	$0.564947\pi$
$228$	−375.000	−0.108925
$229$	−991.000	−0.285970	−0.142985	−	0.989725i	$-0.545670\pi$
$229$	−991.000	−0.285970	−0.142985	+	0.989725i	$0.545670\pi$
$230$	0	0
$231$	231.000	0.0657952
$232$	3906.00	1.10535
$233$	975.000	0.274139	0.137069	−	0.990561i	$-0.456232\pi$
$233$	975.000	0.274139	0.137069	+	0.990561i	$0.456232\pi$
$234$	432.000	0.120687
$235$	0	0
$236$	249.000	0.0686802
$237$	−1527.00	−0.418520
$238$	441.000	0.120108
$239$	−1524.00	−0.412466	−0.206233	−	0.978503i	$-0.566120\pi$
$239$	−1524.00	−0.412466	−0.206233	+	0.978503i	$0.566120\pi$
$240$	0	0
$241$	−2230.00	−0.596045	−0.298023	−	0.954559i	$-0.596327\pi$
$241$	−2230.00	−0.596045	−0.298023	+	0.954559i	$0.596327\pi$
$242$	−363.000	−0.0964237
$243$	−243.000	−0.0641500
$244$	−64.0000	−0.0167917
$245$	0	0
$246$	567.000	0.146954
$247$	−2000.00	−0.515210
$248$	−1218.00	−0.311867
$249$	3762.00	0.957458
$250$	0	0
$251$	3864.00	0.971687	0.485844	−	0.874046i	$-0.338512\pi$
$251$	3864.00	0.971687	0.485844	+	0.874046i	$0.338512\pi$
$252$	−63.0000	−0.0157485
$253$	−891.000	−0.221410
$254$	−3381.00	−0.835208
$255$	0	0
$256$	1513.00	0.369385
$257$	−4518.00	−1.09660	−0.548298	−	0.836283i	$-0.684724\pi$
$257$	−4518.00	−1.09660	−0.548298	+	0.836283i	$0.684724\pi$
$258$	−900.000	−0.217177
$259$	1771.00	0.424883
$260$	0	0
$261$	1674.00	0.397004
$262$	3366.00	0.793711
$263$	−438.000	−0.102693	−0.0513465	−	0.998681i	$-0.516351\pi$
$263$	−438.000	−0.102693	−0.0513465	+	0.998681i	$0.516351\pi$
$264$	−693.000	−0.161558
$265$	0	0
$266$	2625.00	0.605072
$267$	−2268.00	−0.519848
$268$	272.000	0.0619964
$269$	4902.00	1.11108	0.555539	−	0.831490i	$-0.312512\pi$
$269$	4902.00	1.11108	0.555539	+	0.831490i	$0.312512\pi$
$270$	0	0
$271$	−2455.00	−0.550298	−0.275149	−	0.961402i	$-0.588727\pi$
$271$	−2455.00	−0.550298	−0.275149	+	0.961402i	$0.588727\pi$
$272$	−1491.00	−0.332372
$273$	−336.000	−0.0744895
$274$	−162.000	−0.0357182
$275$	0	0
$276$	243.000	0.0529959
$277$	1868.00	0.405189	0.202594	−	0.979263i	$-0.435063\pi$
$277$	1868.00	0.405189	0.202594	+	0.979263i	$0.435063\pi$
$278$	−5244.00	−1.13135
$279$	−522.000	−0.112012
$280$	0	0
$281$	−6093.00	−1.29352	−0.646758	−	0.762695i	$-0.723876\pi$
$281$	−6093.00	−1.29352	−0.646758	+	0.762695i	$0.723876\pi$
$282$	−1971.00	−0.416210
$283$	2543.00	0.534154	0.267077	−	0.963675i	$-0.413942\pi$
$283$	2543.00	0.534154	0.267077	+	0.963675i	$0.413942\pi$
$284$	−645.000	−0.134767
$285$	0	0
$286$	528.000	0.109165
$287$	−441.000	−0.0907018
$288$	405.000	0.0828641
$289$	−4472.00	−0.910238
$290$	0	0
$291$	−2517.00	−0.507042
$292$	−112.000	−0.0224462
$293$	−4623.00	−0.921770	−0.460885	−	0.887460i	$-0.652468\pi$
$293$	−4623.00	−0.921770	−0.460885	+	0.887460i	$0.652468\pi$
$294$	−2646.00	−0.524891
$295$	0	0
$296$	−5313.00	−1.04328
$297$	−297.000	−0.0580259
$298$	−5391.00	−1.04796
$299$	1296.00	0.250668
$300$	0	0
$301$	700.000	0.134044
$302$	−3120.00	−0.594489
$303$	4239.00	0.803710
$304$	−8875.00	−1.67440
$305$	0	0
$306$	−567.000	−0.105926
$307$	644.000	0.119723	0.0598616	−	0.998207i	$-0.480934\pi$
$307$	644.000	0.119723	0.0598616	+	0.998207i	$0.480934\pi$
$308$	−77.0000	−0.0142451
$309$	−4902.00	−0.902476
$310$	0	0
$311$	2616.00	0.476977	0.238488	−	0.971145i	$-0.423348\pi$
$311$	2616.00	0.476977	0.238488	+	0.971145i	$0.423348\pi$
$312$	1008.00	0.182906
$313$	4079.00	0.736609	0.368305	−	0.929705i	$-0.379938\pi$
$313$	4079.00	0.736609	0.368305	+	0.929705i	$0.379938\pi$
$314$	1686.00	0.303014
$315$	0	0
$316$	509.000	0.0906123
$317$	3504.00	0.620834	0.310417	−	0.950601i	$-0.399531\pi$
$317$	3504.00	0.620834	0.310417	+	0.950601i	$0.399531\pi$
$318$	−1728.00	−0.304721
$319$	2046.00	0.359103
$320$	0	0
$321$	−2178.00	−0.378704
$322$	−1701.00	−0.294388
$323$	2625.00	0.452195
$324$	81.0000	0.0138889
$325$	0	0
$326$	−7296.00	−1.23953
$327$	−5136.00	−0.868567
$328$	1323.00	0.222715
$329$	1533.00	0.256891
$330$	0	0
$331$	4100.00	0.680835	0.340417	−	0.940274i	$-0.389432\pi$
$331$	4100.00	0.680835	0.340417	+	0.940274i	$0.389432\pi$
$332$	−1254.00	−0.207296
$333$	−2277.00	−0.374711
$334$	−7020.00	−1.15005
$335$	0	0
$336$	−1491.00	−0.242085
$337$	−10924.0	−1.76578	−0.882891	−	0.469579i	$-0.844406\pi$
$337$	−10924.0	−1.76578	−0.882891	+	0.469579i	$0.844406\pi$
$338$	5823.00	0.937069
$339$	3384.00	0.542164
$340$	0	0
$341$	−638.000	−0.101319
$342$	−3375.00	−0.533623
$343$	4459.00	0.701934
$344$	−2100.00	−0.329141
$345$	0	0
$346$	−11241.0	−1.74659
$347$	−3612.00	−0.558796	−0.279398	−	0.960175i	$-0.590135\pi$
$347$	−3612.00	−0.558796	−0.279398	+	0.960175i	$0.590135\pi$
$348$	−558.000	−0.0859538
$349$	−406.000	−0.0622713	−0.0311356	−	0.999515i	$-0.509912\pi$
$349$	−406.000	−0.0622713	−0.0311356	+	0.999515i	$0.509912\pi$
$350$	0	0
$351$	432.000	0.0656936
$352$	495.000	0.0749534
$353$	−816.000	−0.123035	−0.0615174	−	0.998106i	$-0.519594\pi$
$353$	−816.000	−0.123035	−0.0615174	+	0.998106i	$0.519594\pi$
$354$	2241.00	0.336463
$355$	0	0
$356$	756.000	0.112550
$357$	441.000	0.0653787
$358$	801.000	0.118252
$359$	4818.00	0.708313	0.354156	−	0.935186i	$-0.384768\pi$
$359$	4818.00	0.708313	0.354156	+	0.935186i	$0.384768\pi$
$360$	0	0
$361$	8766.00	1.27803
$362$	−12831.0	−1.86293
$363$	−363.000	−0.0524864
$364$	112.000	0.0161275
$365$	0	0
$366$	−576.000	−0.0822623
$367$	2306.00	0.327990	0.163995	−	0.986461i	$-0.447562\pi$
$367$	2306.00	0.327990	0.163995	+	0.986461i	$0.447562\pi$
$368$	5751.00	0.814651
$369$	567.000	0.0799914
$370$	0	0
$371$	1344.00	0.188078
$372$	174.000	0.0242513
$373$	3134.00	0.435047	0.217523	−	0.976055i	$-0.430202\pi$
$373$	3134.00	0.435047	0.217523	+	0.976055i	$0.430202\pi$
$374$	−693.000	−0.0958133
$375$	0	0
$376$	−4599.00	−0.630785
$377$	−2976.00	−0.406556
$378$	−567.000	−0.0771517
$379$	7202.00	0.976100	0.488050	−	0.872816i	$-0.337708\pi$
$379$	7202.00	0.976100	0.488050	+	0.872816i	$0.337708\pi$
$380$	0	0
$381$	−3381.00	−0.454630
$382$	5391.00	0.722062
$383$	11472.0	1.53053	0.765263	−	0.643717i	$-0.222609\pi$
$383$	11472.0	1.53053	0.765263	+	0.643717i	$0.222609\pi$
$384$	4977.00	0.661410
$385$	0	0
$386$	−5964.00	−0.786424
$387$	−900.000	−0.118216
$388$	839.000	0.109778
$389$	−3462.00	−0.451235	−0.225617	−	0.974216i	$-0.572440\pi$
$389$	−3462.00	−0.451235	−0.225617	+	0.974216i	$0.572440\pi$
$390$	0	0
$391$	−1701.00	−0.220008
$392$	−6174.00	−0.795495
$393$	3366.00	0.432041
$394$	−9981.00	−1.27623
$395$	0	0
$396$	99.0000	0.0125630
$397$	2486.00	0.314279	0.157140	−	0.987576i	$-0.449773\pi$
$397$	2486.00	0.314279	0.157140	+	0.987576i	$0.449773\pi$
$398$	5340.00	0.672538
$399$	2625.00	0.329359
$400$	0	0
$401$	−9024.00	−1.12378	−0.561892	−	0.827211i	$-0.689926\pi$
$401$	−9024.00	−1.12378	−0.561892	+	0.827211i	$0.689926\pi$
$402$	2448.00	0.303719
$403$	928.000	0.114707
$404$	−1413.00	−0.174008
$405$	0	0
$406$	3906.00	0.477467
$407$	−2783.00	−0.338939
$408$	−1323.00	−0.160535
$409$	−14488.0	−1.75155	−0.875777	−	0.482716i	$-0.839650\pi$
$409$	−14488.0	−1.75155	−0.875777	+	0.482716i	$0.839650\pi$
$410$	0	0
$411$	−162.000	−0.0194425
$412$	1634.00	0.195392
$413$	−1743.00	−0.207669
$414$	2187.00	0.259626
$415$	0	0
$416$	−720.000	−0.0848579
$417$	−5244.00	−0.615827
$418$	−4125.00	−0.482680
$419$	−3201.00	−0.373220	−0.186610	−	0.982434i	$-0.559750\pi$
$419$	−3201.00	−0.373220	−0.186610	+	0.982434i	$0.559750\pi$
$420$	0	0
$421$	−6721.00	−0.778056	−0.389028	−	0.921226i	$-0.627189\pi$
$421$	−6721.00	−0.778056	−0.389028	+	0.921226i	$0.627189\pi$
$422$	−6540.00	−0.754413
$423$	−1971.00	−0.226556
$424$	−4032.00	−0.461819
$425$	0	0
$426$	−5805.00	−0.660219
$427$	448.000	0.0507734
$428$	726.000	0.0819919
$429$	528.000	0.0594221
$430$	0	0
$431$	−354.000	−0.0395628	−0.0197814	−	0.999804i	$-0.506297\pi$
$431$	−354.000	−0.0395628	−0.0197814	+	0.999804i	$0.506297\pi$
$432$	1917.00	0.213499
$433$	7682.00	0.852594	0.426297	−	0.904583i	$-0.359818\pi$
$433$	7682.00	0.852594	0.426297	+	0.904583i	$0.359818\pi$
$434$	−1218.00	−0.134714
$435$	0	0
$436$	1712.00	0.188050
$437$	−10125.0	−1.10834
$438$	−1008.00	−0.109964
$439$	5465.00	0.594146	0.297073	−	0.954855i	$-0.403989\pi$
$439$	5465.00	0.594146	0.297073	+	0.954855i	$0.403989\pi$
$440$	0	0
$441$	−2646.00	−0.285714
$442$	1008.00	0.108474
$443$	−1551.00	−0.166344	−0.0831718	−	0.996535i	$-0.526505\pi$
$443$	−1551.00	−0.166344	−0.0831718	+	0.996535i	$0.526505\pi$
$444$	759.000	0.0811274
$445$	0	0
$446$	−11544.0	−1.22561
$447$	−5391.00	−0.570437
$448$	−3031.00	−0.319646
$449$	−1092.00	−0.114777	−0.0573883	−	0.998352i	$-0.518277\pi$
$449$	−1092.00	−0.114777	−0.0573883	+	0.998352i	$0.518277\pi$
$450$	0	0
$451$	693.000	0.0723550
$452$	−1128.00	−0.117382
$453$	−3120.00	−0.323599
$454$	4158.00	0.429834
$455$	0	0
$456$	−7875.00	−0.808730
$457$	10046.0	1.02830	0.514149	−	0.857701i	$-0.328108\pi$
$457$	10046.0	1.02830	0.514149	+	0.857701i	$0.328108\pi$
$458$	2973.00	0.303317
$459$	−567.000	−0.0576586
$460$	0	0
$461$	15510.0	1.56697	0.783485	−	0.621411i	$-0.213440\pi$
$461$	15510.0	1.56697	0.783485	+	0.621411i	$0.213440\pi$
$462$	−693.000	−0.0697863
$463$	6878.00	0.690384	0.345192	−	0.938532i	$-0.387814\pi$
$463$	6878.00	0.690384	0.345192	+	0.938532i	$0.387814\pi$
$464$	−13206.0	−1.32128
$465$	0	0
$466$	−2925.00	−0.290768
$467$	−16284.0	−1.61356	−0.806781	−	0.590850i	$-0.798792\pi$
$467$	−16284.0	−1.61356	−0.806781	+	0.590850i	$0.798792\pi$
$468$	−144.000	−0.0142231
$469$	−1904.00	−0.187460
$470$	0	0
$471$	1686.00	0.164940
$472$	5229.00	0.509924
$473$	−1100.00	−0.106930
$474$	4581.00	0.443908
$475$	0	0
$476$	−147.000	−0.0141549
$477$	−1728.00	−0.165869
$478$	4572.00	0.437486
$479$	6732.00	0.642156	0.321078	−	0.947053i	$-0.395955\pi$
$479$	6732.00	0.642156	0.321078	+	0.947053i	$0.395955\pi$
$480$	0	0
$481$	4048.00	0.383727
$482$	6690.00	0.632202
$483$	−1701.00	−0.160245
$484$	121.000	0.0113636
$485$	0	0
$486$	729.000	0.0680414
$487$	17498.0	1.62815	0.814076	−	0.580758i	$-0.197244\pi$
$487$	17498.0	1.62815	0.814076	+	0.580758i	$0.197244\pi$
$488$	−1344.00	−0.124672
$489$	−7296.00	−0.674717
$490$	0	0
$491$	2454.00	0.225555	0.112777	−	0.993620i	$-0.464025\pi$
$491$	2454.00	0.225555	0.112777	+	0.993620i	$0.464025\pi$
$492$	−189.000	−0.0173187
$493$	3906.00	0.356830
$494$	6000.00	0.546463
$495$	0	0
$496$	4118.00	0.372790
$497$	4515.00	0.407496
$498$	−11286.0	−1.01554
$499$	−20716.0	−1.85847	−0.929234	−	0.369492i	$-0.879532\pi$
$499$	−20716.0	−1.85847	−0.929234	+	0.369492i	$0.879532\pi$
$500$	0	0
$501$	−7020.00	−0.626009
$502$	−11592.0	−1.03063
$503$	−1956.00	−0.173387	−0.0866936	−	0.996235i	$-0.527630\pi$
$503$	−1956.00	−0.173387	−0.0866936	+	0.996235i	$0.527630\pi$
$504$	−1323.00	−0.116927
$505$	0	0
$506$	2673.00	0.234841
$507$	5823.00	0.510076
$508$	1127.00	0.0984302
$509$	18240.0	1.58836	0.794179	−	0.607684i	$-0.207901\pi$
$509$	18240.0	1.58836	0.794179	+	0.607684i	$0.207901\pi$
$510$	0	0
$511$	784.000	0.0678711
$512$	8733.00	0.753804
$513$	−3375.00	−0.290468
$514$	13554.0	1.16312
$515$	0	0
$516$	300.000	0.0255945
$517$	−2409.00	−0.204928
$518$	−5313.00	−0.450656
$519$	−11241.0	−0.950723
$520$	0	0
$521$	20790.0	1.74823	0.874114	−	0.485721i	$-0.161443\pi$
$521$	20790.0	1.74823	0.874114	+	0.485721i	$0.161443\pi$
$522$	−5022.00	−0.421086
$523$	11897.0	0.994684	0.497342	−	0.867555i	$-0.334310\pi$
$523$	11897.0	0.994684	0.497342	+	0.867555i	$0.334310\pi$
$524$	−1122.00	−0.0935397
$525$	0	0
$526$	1314.00	0.108922
$527$	−1218.00	−0.100677
$528$	2343.00	0.193117
$529$	−5606.00	−0.460754
$530$	0	0
$531$	2241.00	0.183147
$532$	−875.000	−0.0713084
$533$	−1008.00	−0.0819162
$534$	6804.00	0.551382
$535$	0	0
$536$	5712.00	0.460300
$537$	801.000	0.0643682
$538$	−14706.0	−1.17848
$539$	−3234.00	−0.258438
$540$	0	0
$541$	20336.0	1.61611	0.808053	−	0.589110i	$-0.200522\pi$
$541$	20336.0	1.61611	0.808053	+	0.589110i	$0.200522\pi$
$542$	7365.00	0.583679
$543$	−12831.0	−1.01405
$544$	945.000	0.0744789
$545$	0	0
$546$	1008.00	0.0790081
$547$	16481.0	1.28826	0.644129	−	0.764917i	$-0.277220\pi$
$547$	16481.0	1.28826	0.644129	+	0.764917i	$0.277220\pi$
$548$	54.0000	0.00420943
$549$	−576.000	−0.0447779
$550$	0	0
$551$	23250.0	1.79761
$552$	5103.00	0.393475
$553$	−3563.00	−0.273986
$554$	−5604.00	−0.429767
$555$	0	0
$556$	1748.00	0.133330
$557$	−24618.0	−1.87271	−0.936354	−	0.351058i	$-0.885822\pi$
$557$	−24618.0	−1.87271	−0.936354	+	0.351058i	$0.885822\pi$
$558$	1566.00	0.118807
$559$	1600.00	0.121060
$560$	0	0
$561$	−693.000	−0.0521542
$562$	18279.0	1.37198
$563$	6438.00	0.481935	0.240967	−	0.970533i	$-0.422535\pi$
$563$	6438.00	0.481935	0.240967	+	0.970533i	$0.422535\pi$
$564$	657.000	0.0490509
$565$	0	0
$566$	−7629.00	−0.566556
$567$	−567.000	−0.0419961
$568$	−13545.0	−1.00059
$569$	−18183.0	−1.33967	−0.669834	−	0.742511i	$-0.733635\pi$
$569$	−18183.0	−1.33967	−0.669834	+	0.742511i	$0.733635\pi$
$570$	0	0
$571$	−17656.0	−1.29401	−0.647006	−	0.762485i	$-0.723979\pi$
$571$	−17656.0	−1.29401	−0.647006	+	0.762485i	$0.723979\pi$
$572$	−176.000	−0.0128653
$573$	5391.00	0.393041
$574$	1323.00	0.0962038
$575$	0	0
$576$	3897.00	0.281901
$577$	−17155.0	−1.23773	−0.618867	−	0.785496i	$-0.712408\pi$
$577$	−17155.0	−1.23773	−0.618867	+	0.785496i	$0.712408\pi$
$578$	13416.0	0.965453
$579$	−5964.00	−0.428075
$580$	0	0
$581$	8778.00	0.626803
$582$	7551.00	0.537799
$583$	−2112.00	−0.150034
$584$	−2352.00	−0.166655
$585$	0	0
$586$	13869.0	0.977684
$587$	24621.0	1.73121	0.865603	−	0.500732i	$-0.166936\pi$
$587$	24621.0	1.73121	0.865603	+	0.500732i	$0.166936\pi$
$588$	882.000	0.0618590
$589$	−7250.00	−0.507183
$590$	0	0
$591$	−9981.00	−0.694693
$592$	17963.0	1.24709
$593$	−9066.00	−0.627818	−0.313909	−	0.949453i	$-0.601639\pi$
$593$	−9066.00	−0.627818	−0.313909	+	0.949453i	$0.601639\pi$
$594$	891.000	0.0615457
$595$	0	0
$596$	1797.00	0.123503
$597$	5340.00	0.366083
$598$	−3888.00	−0.265873
$599$	22353.0	1.52474	0.762370	−	0.647142i	$-0.224036\pi$
$599$	22353.0	1.52474	0.762370	+	0.647142i	$0.224036\pi$
$600$	0	0
$601$	−6304.00	−0.427863	−0.213931	−	0.976849i	$-0.568627\pi$
$601$	−6304.00	−0.427863	−0.213931	+	0.976849i	$0.568627\pi$
$602$	−2100.00	−0.142175
$603$	2448.00	0.165324
$604$	1040.00	0.0700613
$605$	0	0
$606$	−12717.0	−0.852463
$607$	23672.0	1.58289	0.791447	−	0.611238i	$-0.209328\pi$
$607$	23672.0	1.58289	0.791447	+	0.611238i	$0.209328\pi$
$608$	5625.00	0.375204
$609$	3906.00	0.259900
$610$	0	0
$611$	3504.00	0.232008
$612$	189.000	0.0124835
$613$	−3028.00	−0.199510	−0.0997551	−	0.995012i	$-0.531806\pi$
$613$	−3028.00	−0.199510	−0.0997551	+	0.995012i	$0.531806\pi$
$614$	−1932.00	−0.126986
$615$	0	0
$616$	−1617.00	−0.105764
$617$	−1170.00	−0.0763410	−0.0381705	−	0.999271i	$-0.512153\pi$
$617$	−1170.00	−0.0763410	−0.0381705	+	0.999271i	$0.512153\pi$
$618$	14706.0	0.957220
$619$	9626.00	0.625043	0.312521	−	0.949911i	$-0.398826\pi$
$619$	9626.00	0.625043	0.312521	+	0.949911i	$0.398826\pi$
$620$	0	0
$621$	2187.00	0.141323
$622$	−7848.00	−0.505910
$623$	−5292.00	−0.340320
$624$	−3408.00	−0.218637
$625$	0	0
$626$	−12237.0	−0.781292
$627$	−4125.00	−0.262738
$628$	−562.000	−0.0357106
$629$	−5313.00	−0.336794
$630$	0	0
$631$	−5794.00	−0.365540	−0.182770	−	0.983156i	$-0.558506\pi$
$631$	−5794.00	−0.365540	−0.182770	+	0.983156i	$0.558506\pi$
$632$	10689.0	0.672762
$633$	−6540.00	−0.410650
$634$	−10512.0	−0.658493
$635$	0	0
$636$	576.000	0.0359118
$637$	4704.00	0.292589
$638$	−6138.00	−0.380887
$639$	−5805.00	−0.359378
$640$	0	0
$641$	28308.0	1.74430	0.872152	−	0.489235i	$-0.162724\pi$
$641$	28308.0	1.74430	0.872152	+	0.489235i	$0.162724\pi$
$642$	6534.00	0.401677
$643$	2270.00	0.139222	0.0696112	−	0.997574i	$-0.477824\pi$
$643$	2270.00	0.139222	0.0696112	+	0.997574i	$0.477824\pi$
$644$	567.000	0.0346940
$645$	0	0
$646$	−7875.00	−0.479625
$647$	23361.0	1.41950	0.709749	−	0.704454i	$-0.248808\pi$
$647$	23361.0	1.41950	0.709749	+	0.704454i	$0.248808\pi$
$648$	1701.00	0.103120
$649$	2739.00	0.165663
$650$	0	0
$651$	−1218.00	−0.0733290
$652$	2432.00	0.146080
$653$	12294.0	0.736756	0.368378	−	0.929676i	$-0.379913\pi$
$653$	12294.0	0.736756	0.368378	+	0.929676i	$0.379913\pi$
$654$	15408.0	0.921255
$655$	0	0
$656$	−4473.00	−0.266222
$657$	−1008.00	−0.0598567
$658$	−4599.00	−0.272474
$659$	31896.0	1.88542	0.942710	−	0.333613i	$-0.108268\pi$
$659$	31896.0	1.88542	0.942710	+	0.333613i	$0.108268\pi$
$660$	0	0
$661$	13469.0	0.792562	0.396281	−	0.918129i	$-0.370301\pi$
$661$	13469.0	0.792562	0.396281	+	0.918129i	$0.370301\pi$
$662$	−12300.0	−0.722135
$663$	1008.00	0.0590460
$664$	−26334.0	−1.53909
$665$	0	0
$666$	6831.00	0.397441
$667$	−15066.0	−0.874599
$668$	2340.00	0.135535
$669$	−11544.0	−0.667140
$670$	0	0
$671$	−704.000	−0.0405032
$672$	945.000	0.0542473
$673$	13712.0	0.785377	0.392689	−	0.919672i	$-0.371545\pi$
$673$	13712.0	0.785377	0.392689	+	0.919672i	$0.371545\pi$
$674$	32772.0	1.87289
$675$	0	0
$676$	−1941.00	−0.110435
$677$	−3714.00	−0.210843	−0.105421	−	0.994428i	$-0.533619\pi$
$677$	−3714.00	−0.210843	−0.105421	+	0.994428i	$0.533619\pi$
$678$	−10152.0	−0.575052
$679$	−5873.00	−0.331937
$680$	0	0
$681$	4158.00	0.233972
$682$	1914.00	0.107465
$683$	−13065.0	−0.731945	−0.365972	−	0.930626i	$-0.619264\pi$
$683$	−13065.0	−0.731945	−0.365972	+	0.930626i	$0.619264\pi$
$684$	1125.00	0.0628881
$685$	0	0
$686$	−13377.0	−0.744513
$687$	2973.00	0.165105
$688$	7100.00	0.393437
$689$	3072.00	0.169860
$690$	0	0
$691$	12512.0	0.688826	0.344413	−	0.938818i	$-0.388078\pi$
$691$	12512.0	0.688826	0.344413	+	0.938818i	$0.388078\pi$
$692$	3747.00	0.205838
$693$	−693.000	−0.0379869
$694$	10836.0	0.592693
$695$	0	0
$696$	−11718.0	−0.638175
$697$	1323.00	0.0718970
$698$	1218.00	0.0660487
$699$	−2925.00	−0.158274
$700$	0	0
$701$	33405.0	1.79984	0.899921	−	0.436053i	$-0.143624\pi$
$701$	33405.0	1.79984	0.899921	+	0.436053i	$0.143624\pi$
$702$	−1296.00	−0.0696786
$703$	−31625.0	−1.69667
$704$	4763.00	0.254989
$705$	0	0
$706$	2448.00	0.130498
$707$	9891.00	0.526152
$708$	−747.000	−0.0396525
$709$	22217.0	1.17684	0.588418	−	0.808557i	$-0.299751\pi$
$709$	22217.0	1.17684	0.588418	+	0.808557i	$0.299751\pi$
$710$	0	0
$711$	4581.00	0.241633
$712$	15876.0	0.835644
$713$	4698.00	0.246762
$714$	−1323.00	−0.0693446
$715$	0	0
$716$	−267.000	−0.0139361
$717$	4572.00	0.238137
$718$	−14454.0	−0.751279
$719$	12336.0	0.639854	0.319927	−	0.947442i	$-0.396342\pi$
$719$	12336.0	0.639854	0.319927	+	0.947442i	$0.396342\pi$
$720$	0	0
$721$	−11438.0	−0.590809
$722$	−26298.0	−1.35555
$723$	6690.00	0.344127
$724$	4277.00	0.219549
$725$	0	0
$726$	1089.00	0.0556702
$727$	−10720.0	−0.546881	−0.273441	−	0.961889i	$-0.588162\pi$
$727$	−10720.0	−0.546881	−0.273441	+	0.961889i	$0.588162\pi$
$728$	2352.00	0.119740
$729$	729.000	0.0370370
$730$	0	0
$731$	−2100.00	−0.106253
$732$	192.000	0.00969471
$733$	10820.0	0.545219	0.272610	−	0.962125i	$-0.412113\pi$
$733$	10820.0	0.545219	0.272610	+	0.962125i	$0.412113\pi$
$734$	−6918.00	−0.347886
$735$	0	0
$736$	−3645.00	−0.182550
$737$	2992.00	0.149541
$738$	−1701.00	−0.0848437
$739$	10127.0	0.504097	0.252049	−	0.967715i	$-0.418896\pi$
$739$	10127.0	0.504097	0.252049	+	0.967715i	$0.418896\pi$
$740$	0	0
$741$	6000.00	0.297457
$742$	−4032.00	−0.199487
$743$	−6000.00	−0.296257	−0.148128	−	0.988968i	$-0.547325\pi$
$743$	−6000.00	−0.296257	−0.148128	+	0.988968i	$0.547325\pi$
$744$	3654.00	0.180057
$745$	0	0
$746$	−9402.00	−0.461437
$747$	−11286.0	−0.552789
$748$	231.000	0.0112917
$749$	−5082.00	−0.247920
$750$	0	0
$751$	26132.0	1.26973	0.634867	−	0.772621i	$-0.281055\pi$
$751$	26132.0	1.26973	0.634867	+	0.772621i	$0.281055\pi$
$752$	15549.0	0.754008
$753$	−11592.0	−0.561004
$754$	8928.00	0.431218
$755$	0	0
$756$	189.000	0.00909241
$757$	25850.0	1.24113	0.620564	−	0.784156i	$-0.286904\pi$
$757$	25850.0	1.24113	0.620564	+	0.784156i	$0.286904\pi$
$758$	−21606.0	−1.03531
$759$	2673.00	0.127831
$760$	0	0
$761$	−3654.00	−0.174057	−0.0870285	−	0.996206i	$-0.527737\pi$
$761$	−3654.00	−0.174057	−0.0870285	+	0.996206i	$0.527737\pi$
$762$	10143.0	0.482208
$763$	−11984.0	−0.568611
$764$	−1797.00	−0.0850958
$765$	0	0
$766$	−34416.0	−1.62337
$767$	−3984.00	−0.187554
$768$	−4539.00	−0.213264
$769$	6248.00	0.292989	0.146495	−	0.989211i	$-0.453201\pi$
$769$	6248.00	0.292989	0.146495	+	0.989211i	$0.453201\pi$
$770$	0	0
$771$	13554.0	0.633120
$772$	1988.00	0.0926809
$773$	−23952.0	−1.11448	−0.557240	−	0.830351i	$-0.688140\pi$
$773$	−23952.0	−1.11448	−0.557240	+	0.830351i	$0.688140\pi$
$774$	2700.00	0.125387
$775$	0	0
$776$	17619.0	0.815058
$777$	−5313.00	−0.245306
$778$	10386.0	0.478607
$779$	7875.00	0.362197
$780$	0	0
$781$	−7095.00	−0.325069
$782$	5103.00	0.233354
$783$	−5022.00	−0.229210
$784$	20874.0	0.950893
$785$	0	0
$786$	−10098.0	−0.458249
$787$	−39631.0	−1.79504	−0.897518	−	0.440979i	$-0.854631\pi$
$787$	−39631.0	−1.79504	−0.897518	+	0.440979i	$0.854631\pi$
$788$	3327.00	0.150405
$789$	1314.00	0.0592898
$790$	0	0
$791$	7896.00	0.354930
$792$	2079.00	0.0932753
$793$	1024.00	0.0458554
$794$	−7458.00	−0.333343
$795$	0	0
$796$	−1780.00	−0.0792593
$797$	−7530.00	−0.334663	−0.167331	−	0.985901i	$-0.553515\pi$
$797$	−7530.00	−0.334663	−0.167331	+	0.985901i	$0.553515\pi$
$798$	−7875.00	−0.349338
$799$	−4599.00	−0.203631
$800$	0	0
$801$	6804.00	0.300134
$802$	27072.0	1.19195
$803$	−1232.00	−0.0541424
$804$	−816.000	−0.0357937
$805$	0	0
$806$	−2784.00	−0.121665
$807$	−14706.0	−0.641482
$808$	−29673.0	−1.29195
$809$	−7239.00	−0.314598	−0.157299	−	0.987551i	$-0.550279\pi$
$809$	−7239.00	−0.314598	−0.157299	+	0.987551i	$0.550279\pi$
$810$	0	0
$811$	−35611.0	−1.54189	−0.770944	−	0.636903i	$-0.780215\pi$
$811$	−35611.0	−1.54189	−0.770944	+	0.636903i	$0.780215\pi$
$812$	−1302.00	−0.0562700
$813$	7365.00	0.317714
$814$	8349.00	0.359499
$815$	0	0
$816$	4473.00	0.191895
$817$	−12500.0	−0.535275
$818$	43464.0	1.85780
$819$	1008.00	0.0430066
$820$	0	0
$821$	−42054.0	−1.78769	−0.893846	−	0.448375i	$-0.852003\pi$
$821$	−42054.0	−1.78769	−0.893846	+	0.448375i	$0.852003\pi$
$822$	486.000	0.0206219
$823$	10172.0	0.430831	0.215415	−	0.976523i	$-0.430889\pi$
$823$	10172.0	0.430831	0.215415	+	0.976523i	$0.430889\pi$
$824$	34314.0	1.45071
$825$	0	0
$826$	5229.00	0.220267
$827$	4560.00	0.191737	0.0958686	−	0.995394i	$-0.469437\pi$
$827$	4560.00	0.191737	0.0958686	+	0.995394i	$0.469437\pi$
$828$	−729.000	−0.0305972
$829$	13202.0	0.553105	0.276553	−	0.960999i	$-0.410808\pi$
$829$	13202.0	0.553105	0.276553	+	0.960999i	$0.410808\pi$
$830$	0	0
$831$	−5604.00	−0.233936
$832$	−6928.00	−0.288684
$833$	−6174.00	−0.256802
$834$	15732.0	0.653183
$835$	0	0
$836$	1375.00	0.0568844
$837$	1566.00	0.0646701
$838$	9603.00	0.395859
$839$	6216.00	0.255781	0.127890	−	0.991788i	$-0.459179\pi$
$839$	6216.00	0.255781	0.127890	+	0.991788i	$0.459179\pi$
$840$	0	0
$841$	10207.0	0.418508
$842$	20163.0	0.825253
$843$	18279.0	0.746812
$844$	2180.00	0.0889084
$845$	0	0
$846$	5913.00	0.240299
$847$	−847.000	−0.0343604
$848$	13632.0	0.552034
$849$	−7629.00	−0.308394
$850$	0	0
$851$	20493.0	0.825489
$852$	1935.00	0.0778075
$853$	22718.0	0.911899	0.455949	−	0.890006i	$-0.349300\pi$
$853$	22718.0	0.911899	0.455949	+	0.890006i	$0.349300\pi$
$854$	−1344.00	−0.0538533
$855$	0	0
$856$	15246.0	0.608759
$857$	−27435.0	−1.09354	−0.546769	−	0.837284i	$-0.684142\pi$
$857$	−27435.0	−1.09354	−0.546769	+	0.837284i	$0.684142\pi$
$858$	−1584.00	−0.0630267
$859$	37556.0	1.49173	0.745864	−	0.666098i	$-0.232037\pi$
$859$	37556.0	1.49173	0.745864	+	0.666098i	$0.232037\pi$
$860$	0	0
$861$	1323.00	0.0523667
$862$	1062.00	0.0419627
$863$	−14976.0	−0.590717	−0.295359	−	0.955386i	$-0.595439\pi$
$863$	−14976.0	−0.590717	−0.295359	+	0.955386i	$0.595439\pi$
$864$	−1215.00	−0.0478416
$865$	0	0
$866$	−23046.0	−0.904313
$867$	13416.0	0.525526
$868$	406.000	0.0158762
$869$	5599.00	0.218565
$870$	0	0
$871$	−4352.00	−0.169302
$872$	35952.0	1.39620
$873$	7551.00	0.292741
$874$	30375.0	1.17557
$875$	0	0
$876$	336.000	0.0129593
$877$	−30718.0	−1.18275	−0.591376	−	0.806396i	$-0.701415\pi$
$877$	−30718.0	−1.18275	−0.591376	+	0.806396i	$0.701415\pi$
$878$	−16395.0	−0.630187
$879$	13869.0	0.532184
$880$	0	0
$881$	2916.00	0.111513	0.0557563	−	0.998444i	$-0.482243\pi$
$881$	2916.00	0.111513	0.0557563	+	0.998444i	$0.482243\pi$
$882$	7938.00	0.303046
$883$	−39670.0	−1.51189	−0.755947	−	0.654633i	$-0.772823\pi$
$883$	−39670.0	−1.51189	−0.755947	+	0.654633i	$0.772823\pi$
$884$	−336.000	−0.0127838
$885$	0	0
$886$	4653.00	0.176434
$887$	−23724.0	−0.898054	−0.449027	−	0.893518i	$-0.648229\pi$
$887$	−23724.0	−0.898054	−0.449027	+	0.893518i	$0.648229\pi$
$888$	15939.0	0.602340
$889$	−7889.00	−0.297625
$890$	0	0
$891$	891.000	0.0335013
$892$	3848.00	0.144440
$893$	−27375.0	−1.02583
$894$	16173.0	0.605040
$895$	0	0
$896$	11613.0	0.432995
$897$	−3888.00	−0.144723
$898$	3276.00	0.121739
$899$	−10788.0	−0.400222
$900$	0	0
$901$	−4032.00	−0.149085
$902$	−2079.00	−0.0767440
$903$	−2100.00	−0.0773905
$904$	−23688.0	−0.871517
$905$	0	0
$906$	9360.00	0.343229
$907$	−42166.0	−1.54366	−0.771830	−	0.635829i	$-0.780658\pi$
$907$	−42166.0	−1.54366	−0.771830	+	0.635829i	$0.780658\pi$
$908$	−1386.00	−0.0506564
$909$	−12717.0	−0.464022
$910$	0	0
$911$	32139.0	1.16884	0.584420	−	0.811452i	$-0.301322\pi$
$911$	32139.0	1.16884	0.584420	+	0.811452i	$0.301322\pi$
$912$	26625.0	0.966713
$913$	−13794.0	−0.500016
$914$	−30138.0	−1.09067
$915$	0	0
$916$	−991.000	−0.0357462
$917$	7854.00	0.282837
$918$	1701.00	0.0611562
$919$	44525.0	1.59820	0.799099	−	0.601199i	$-0.205310\pi$
$919$	44525.0	1.59820	0.799099	+	0.601199i	$0.205310\pi$
$920$	0	0
$921$	−1932.00	−0.0691222
$922$	−46530.0	−1.66202
$923$	10320.0	0.368025
$924$	231.000	0.00822440
$925$	0	0
$926$	−20634.0	−0.732263
$927$	14706.0	0.521045
$928$	8370.00	0.296076
$929$	−5964.00	−0.210627	−0.105314	−	0.994439i	$-0.533585\pi$
$929$	−5964.00	−0.210627	−0.105314	+	0.994439i	$0.533585\pi$
$930$	0	0
$931$	−36750.0	−1.29370
$932$	975.000	0.0342674
$933$	−7848.00	−0.275383
$934$	48852.0	1.71144
$935$	0	0
$936$	−3024.00	−0.105601
$937$	6662.00	0.232271	0.116136	−	0.993233i	$-0.462949\pi$
$937$	6662.00	0.232271	0.116136	+	0.993233i	$0.462949\pi$
$938$	5712.00	0.198831
$939$	−12237.0	−0.425282
$940$	0	0
$941$	−42129.0	−1.45948	−0.729738	−	0.683727i	$-0.760358\pi$
$941$	−42129.0	−1.45948	−0.729738	+	0.683727i	$0.760358\pi$
$942$	−5058.00	−0.174945
$943$	−5103.00	−0.176221
$944$	−17679.0	−0.609536
$945$	0	0
$946$	3300.00	0.113417
$947$	23049.0	0.790910	0.395455	−	0.918485i	$-0.370587\pi$
$947$	23049.0	0.790910	0.395455	+	0.918485i	$0.370587\pi$
$948$	−1527.00	−0.0523150
$949$	1792.00	0.0612969
$950$	0	0
$951$	−10512.0	−0.358438
$952$	−3087.00	−0.105095
$953$	10221.0	0.347419	0.173710	−	0.984797i	$-0.444425\pi$
$953$	10221.0	0.347419	0.173710	+	0.984797i	$0.444425\pi$
$954$	5184.00	0.175931
$955$	0	0
$956$	−1524.00	−0.0515582
$957$	−6138.00	−0.207328
$958$	−20196.0	−0.681110
$959$	−378.000	−0.0127281
$960$	0	0
$961$	−26427.0	−0.887080
$962$	−12144.0	−0.407004
$963$	6534.00	0.218645
$964$	−2230.00	−0.0745057
$965$	0	0
$966$	5103.00	0.169965
$967$	−1072.00	−0.0356496	−0.0178248	−	0.999841i	$-0.505674\pi$
$967$	−1072.00	−0.0356496	−0.0178248	+	0.999841i	$0.505674\pi$
$968$	2541.00	0.0843707
$969$	−7875.00	−0.261075
$970$	0	0
$971$	−50337.0	−1.66364	−0.831818	−	0.555048i	$-0.812700\pi$
$971$	−50337.0	−1.66364	−0.831818	+	0.555048i	$0.812700\pi$
$972$	−243.000	−0.00801875
$973$	−12236.0	−0.403153
$974$	−52494.0	−1.72692
$975$	0	0
$976$	4544.00	0.149027
$977$	49638.0	1.62545	0.812723	−	0.582651i	$-0.197984\pi$
$977$	49638.0	1.62545	0.812723	+	0.582651i	$0.197984\pi$
$978$	21888.0	0.715645
$979$	8316.00	0.271482
$980$	0	0
$981$	15408.0	0.501467
$982$	−7362.00	−0.239237
$983$	−1143.00	−0.0370865	−0.0185433	−	0.999828i	$-0.505903\pi$
$983$	−1143.00	−0.0370865	−0.0185433	+	0.999828i	$0.505903\pi$
$984$	−3969.00	−0.128584
$985$	0	0
$986$	−11718.0	−0.378476
$987$	−4599.00	−0.148316
$988$	−2000.00	−0.0644013
$989$	8100.00	0.260430
$990$	0	0
$991$	35060.0	1.12383	0.561916	−	0.827194i	$-0.310064\pi$
$991$	35060.0	1.12383	0.561916	+	0.827194i	$0.310064\pi$
$992$	−2610.00	−0.0835359
$993$	−12300.0	−0.393080
$994$	−13545.0	−0.432215
$995$	0	0
$996$	3762.00	0.119682
$997$	−55582.0	−1.76560	−0.882798	−	0.469752i	$-0.844343\pi$
$997$	−55582.0	−1.76560	−0.882798	+	0.469752i	$0.844343\pi$
$998$	62148.0	1.97120
$999$	6831.00	0.216340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.c.1.1	✓	1
3.2	odd	2		2475.4.a.i.1.1		1
5.2	odd	4		825.4.c.d.199.1		2
5.3	odd	4		825.4.c.d.199.2		2
5.4	even	2		825.4.a.g.1.1	yes	1
15.14	odd	2		2475.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.c.1.1	✓	1	1.1	even	1	trivial
825.4.a.g.1.1	yes	1	5.4	even	2
825.4.c.d.199.1		2	5.2	odd	4
825.4.c.d.199.2		2	5.3	odd	4
2475.4.a.d.1.1		1	15.14	odd	2
2475.4.a.i.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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