LMFDB - Embedded newform 2475.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+3.00000 q^{2} +1.00000 q^{4} -7.00000 q^{7} -21.0000 q^{8} -11.0000 q^{11} -16.0000 q^{13} -21.0000 q^{14} -71.0000 q^{16} -21.0000 q^{17} +125.000 q^{19} -33.0000 q^{22} +81.0000 q^{23} -48.0000 q^{26} -7.00000 q^{28} -186.000 q^{29} -58.0000 q^{31} -45.0000 q^{32} -63.0000 q^{34} -253.000 q^{37} +375.000 q^{38} -63.0000 q^{41} -100.000 q^{43} -11.0000 q^{44} +243.000 q^{46} +219.000 q^{47} -294.000 q^{49} -16.0000 q^{52} +192.000 q^{53} +147.000 q^{56} -558.000 q^{58} -249.000 q^{59} -64.0000 q^{61} -174.000 q^{62} +433.000 q^{64} +272.000 q^{67} -21.0000 q^{68} +645.000 q^{71} -112.000 q^{73} -759.000 q^{74} +125.000 q^{76} +77.0000 q^{77} +509.000 q^{79} -189.000 q^{82} +1254.00 q^{83} -300.000 q^{86} +231.000 q^{88} -756.000 q^{89} +112.000 q^{91} +81.0000 q^{92} +657.000 q^{94} +839.000 q^{97} -882.000 q^{98} +O(q^{100})$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−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
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−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	$-0.547863\pi$
$17$	−21.0000	−0.299603	−0.149801	+	0.988716i	$0.547863\pi$
$18$	0	0
$19$	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
$21$	0	0
$22$	−33.0000	−0.319801
$23$	81.0000	0.734333	0.367167	−	0.930155i	$-0.380328\pi$
$23$	81.0000	0.734333	0.367167	+	0.930155i	$0.380328\pi$
$24$	0	0
$25$	0	0
$26$	−48.0000	−0.362061
$27$	0	0
$28$	−7.00000	−0.0472456
$29$	−186.000	−1.19101	−0.595506	−	0.803351i	$-0.703048\pi$
$29$	−186.000	−1.19101	−0.595506	+	0.803351i	$0.703048\pi$
$30$	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
$33$	0	0
$34$	−63.0000	−0.317777
$35$	0	0
$36$	0	0
$37$	−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
$39$	0	0
$40$	0	0
$41$	−63.0000	−0.239974	−0.119987	−	0.992775i	$-0.538285\pi$
$41$	−63.0000	−0.239974	−0.119987	+	0.992775i	$0.538285\pi$
$42$	0	0
$43$	−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	$-0.389629\pi$
$47$	219.000	0.679669	0.339834	+	0.940485i	$0.389629\pi$
$48$	0	0
$49$	−294.000	−0.857143
$50$	0	0
$51$	0	0
$52$	−16.0000	−0.0426692
$53$	192.000	0.497608	0.248804	−	0.968554i	$-0.419962\pi$
$53$	192.000	0.497608	0.248804	+	0.968554i	$0.419962\pi$
$54$	0	0
$55$	0	0
$56$	147.000	0.350780
$57$	0	0
$58$	−558.000	−1.26326
$59$	−249.000	−0.549441	−0.274721	−	0.961524i	$-0.588585\pi$
$59$	−249.000	−0.549441	−0.274721	+	0.961524i	$0.588585\pi$
$60$	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
$63$	0	0
$64$	433.000	0.845703
$65$	0	0
$66$	0	0
$67$	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
$69$	0	0
$70$	0	0
$71$	645.000	1.07813	0.539066	−	0.842263i	$-0.318777\pi$
$71$	645.000	1.07813	0.539066	+	0.842263i	$0.318777\pi$
$72$	0	0
$73$	−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
$78$	0	0
$79$	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
$81$	0	0
$82$	−189.000	−0.254531
$83$	1254.00	1.65837	0.829183	−	0.558977i	$-0.188806\pi$
$83$	1254.00	1.65837	0.829183	+	0.558977i	$0.188806\pi$
$84$	0	0
$85$	0	0
$86$	−300.000	−0.376161
$87$	0	0
$88$	231.000	0.279826
$89$	−756.000	−0.900403	−0.450201	−	0.892927i	$-0.648648\pi$
$89$	−756.000	−0.900403	−0.450201	+	0.892927i	$0.648648\pi$
$90$	0	0
$91$	112.000	0.129020
$92$	81.0000	0.0917917
$93$	0	0
$94$	657.000	0.720898
$95$	0	0
$96$	0	0
$97$	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
$99$	0	0
$100$	0	0
$101$	1413.00	1.39207	0.696033	−	0.718009i	$-0.254947\pi$
$101$	1413.00	1.39207	0.696033	+	0.718009i	$0.254947\pi$
$102$	0	0
$103$	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	$-0.606364\pi$
$107$	−726.000	−0.655935	−0.327968	+	0.944689i	$0.606364\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	497.000	0.419304
$113$	1128.00	0.939056	0.469528	−	0.882918i	$-0.344424\pi$
$113$	1128.00	0.939056	0.469528	+	0.882918i	$0.344424\pi$
$114$	0	0
$115$	0	0
$116$	−186.000	−0.148876
$117$	0	0
$118$	−747.000	−0.582771
$119$	147.000	0.113239
$120$	0	0
$121$	121.000	0.0909091
$122$	−192.000	−0.142482
$123$	0	0
$124$	−58.0000	−0.0420045
$125$	0	0
$126$	0	0
$127$	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
$129$	0	0
$130$	0	0
$131$	1122.00	0.748318	0.374159	−	0.927365i	$-0.377932\pi$
$131$	1122.00	0.748318	0.374159	+	0.927365i	$0.377932\pi$
$132$	0	0
$133$	−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	$-0.505360\pi$
$137$	−54.0000	−0.0336754	−0.0168377	+	0.999858i	$0.505360\pi$
$138$	0	0
$139$	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
$141$	0	0
$142$	1935.00	1.14353
$143$	176.000	0.102922
$144$	0	0
$145$	0	0
$146$	−336.000	−0.190463
$147$	0	0
$148$	−253.000	−0.140517
$149$	−1797.00	−0.988027	−0.494013	−	0.869454i	$-0.664471\pi$
$149$	−1797.00	−0.988027	−0.494013	+	0.869454i	$0.664471\pi$
$150$	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
$153$	0	0
$154$	231.000	0.120873
$155$	0	0
$156$	0	0
$157$	−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
$159$	0	0
$160$	0	0
$161$	−567.000	−0.277552
$162$	0	0
$163$	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	$-0.682386\pi$
$167$	−2340.00	−1.08428	−0.542140	+	0.840288i	$0.682386\pi$
$168$	0	0
$169$	−1941.00	−0.883477
$170$	0	0
$171$	0	0
$172$	−100.000	−0.0443310
$173$	−3747.00	−1.64670	−0.823350	−	0.567534i	$-0.807898\pi$
$173$	−3747.00	−1.64670	−0.823350	+	0.567534i	$0.807898\pi$
$174$	0	0
$175$	0	0
$176$	781.000	0.334489
$177$	0	0
$178$	−2268.00	−0.955021
$179$	267.000	0.111489	0.0557445	−	0.998445i	$-0.482247\pi$
$179$	267.000	0.111489	0.0557445	+	0.998445i	$0.482247\pi$
$180$	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
$183$	0	0
$184$	−1701.00	−0.681518
$185$	0	0
$186$	0	0
$187$	231.000	0.0903337
$188$	219.000	0.0849586
$189$	0	0
$190$	0	0
$191$	1797.00	0.680766	0.340383	−	0.940287i	$-0.389443\pi$
$191$	1797.00	0.680766	0.340383	+	0.940287i	$0.389443\pi$
$192$	0	0
$193$	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	$-0.705478\pi$
$197$	−3327.00	−1.20324	−0.601622	+	0.798781i	$0.705478\pi$
$198$	0	0
$199$	−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
$201$	0	0
$202$	4239.00	1.47651
$203$	1302.00	0.450160
$204$	0	0
$205$	0	0
$206$	4902.00	1.65795
$207$	0	0
$208$	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
$213$	0	0
$214$	−2178.00	−0.695724
$215$	0	0
$216$	0	0
$217$	406.000	0.127010
$218$	5136.00	1.59566
$219$	0	0
$220$	0	0
$221$	336.000	0.102271
$222$	0	0
$223$	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	$-0.435053\pi$
$227$	1386.00	0.405251	0.202626	+	0.979256i	$0.435053\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	3906.00	1.10535
$233$	−975.000	−0.274139	−0.137069	−	0.990561i	$-0.543768\pi$
$233$	−975.000	−0.274139	−0.137069	+	0.990561i	$0.543768\pi$
$234$	0	0
$235$	0	0
$236$	−249.000	−0.0686802
$237$	0	0
$238$	441.000	0.120108
$239$	1524.00	0.412466	0.206233	−	0.978503i	$-0.433880\pi$
$239$	1524.00	0.412466	0.206233	+	0.978503i	$0.433880\pi$
$240$	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
$243$	0	0
$244$	−64.0000	−0.0167917
$245$	0	0
$246$	0	0
$247$	−2000.00	−0.515210
$248$	1218.00	0.311867
$249$	0	0
$250$	0	0
$251$	−3864.00	−0.971687	−0.485844	−	0.874046i	$-0.661488\pi$
$251$	−3864.00	−0.971687	−0.485844	+	0.874046i	$0.661488\pi$
$252$	0	0
$253$	−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	$-0.315276\pi$
$257$	4518.00	1.09660	0.548298	+	0.836283i	$0.315276\pi$
$258$	0	0
$259$	1771.00	0.424883
$260$	0	0
$261$	0	0
$262$	3366.00	0.793711
$263$	438.000	0.102693	0.0513465	−	0.998681i	$-0.483649\pi$
$263$	438.000	0.102693	0.0513465	+	0.998681i	$0.483649\pi$
$264$	0	0
$265$	0	0
$266$	−2625.00	−0.605072
$267$	0	0
$268$	272.000	0.0619964
$269$	−4902.00	−1.11108	−0.555539	−	0.831490i	$-0.687488\pi$
$269$	−4902.00	−1.11108	−0.555539	+	0.831490i	$0.687488\pi$
$270$	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
$273$	0	0
$274$	−162.000	−0.0357182
$275$	0	0
$276$	0	0
$277$	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
$279$	0	0
$280$	0	0
$281$	6093.00	1.29352	0.646758	−	0.762695i	$-0.276124\pi$
$281$	6093.00	1.29352	0.646758	+	0.762695i	$0.276124\pi$
$282$	0	0
$283$	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
$288$	0	0
$289$	−4472.00	−0.910238
$290$	0	0
$291$	0	0
$292$	−112.000	−0.0224462
$293$	4623.00	0.921770	0.460885	−	0.887460i	$-0.347532\pi$
$293$	4623.00	0.921770	0.460885	+	0.887460i	$0.347532\pi$
$294$	0	0
$295$	0	0
$296$	5313.00	1.04328
$297$	0	0
$298$	−5391.00	−1.04796
$299$	−1296.00	−0.250668
$300$	0	0
$301$	700.000	0.134044
$302$	3120.00	0.594489
$303$	0	0
$304$	−8875.00	−1.67440
$305$	0	0
$306$	0	0
$307$	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
$309$	0	0
$310$	0	0
$311$	−2616.00	−0.476977	−0.238488	−	0.971145i	$-0.576652\pi$
$311$	−2616.00	−0.476977	−0.238488	+	0.971145i	$0.576652\pi$
$312$	0	0
$313$	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	$-0.600469\pi$
$317$	−3504.00	−0.620834	−0.310417	+	0.950601i	$0.600469\pi$
$318$	0	0
$319$	2046.00	0.359103
$320$	0	0
$321$	0	0
$322$	−1701.00	−0.294388
$323$	−2625.00	−0.452195
$324$	0	0
$325$	0	0
$326$	7296.00	1.23953
$327$	0	0
$328$	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
$333$	0	0
$334$	−7020.00	−1.15005
$335$	0	0
$336$	0	0
$337$	−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
$339$	0	0
$340$	0	0
$341$	638.000	0.101319
$342$	0	0
$343$	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	$-0.409865\pi$
$347$	3612.00	0.558796	0.279398	+	0.960175i	$0.409865\pi$
$348$	0	0
$349$	−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
$351$	0	0
$352$	495.000	0.0749534
$353$	816.000	0.123035	0.0615174	−	0.998106i	$-0.480406\pi$
$353$	816.000	0.123035	0.0615174	+	0.998106i	$0.480406\pi$
$354$	0	0
$355$	0	0
$356$	−756.000	−0.112550
$357$	0	0
$358$	801.000	0.118252
$359$	−4818.00	−0.708313	−0.354156	−	0.935186i	$-0.615232\pi$
$359$	−4818.00	−0.708313	−0.354156	+	0.935186i	$0.615232\pi$
$360$	0	0
$361$	8766.00	1.27803
$362$	12831.0	1.86293
$363$	0	0
$364$	112.000	0.0161275
$365$	0	0
$366$	0	0
$367$	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
$369$	0	0
$370$	0	0
$371$	−1344.00	−0.188078
$372$	0	0
$373$	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
$378$	0	0
$379$	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
$381$	0	0
$382$	5391.00	0.722062
$383$	−11472.0	−1.53053	−0.765263	−	0.643717i	$-0.777391\pi$
$383$	−11472.0	−1.53053	−0.765263	+	0.643717i	$0.777391\pi$
$384$	0	0
$385$	0	0
$386$	5964.00	0.786424
$387$	0	0
$388$	839.000	0.109778
$389$	3462.00	0.451235	0.225617	−	0.974216i	$-0.427560\pi$
$389$	3462.00	0.451235	0.225617	+	0.974216i	$0.427560\pi$
$390$	0	0
$391$	−1701.00	−0.220008
$392$	6174.00	0.795495
$393$	0	0
$394$	−9981.00	−1.27623
$395$	0	0
$396$	0	0
$397$	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
$399$	0	0
$400$	0	0
$401$	9024.00	1.12378	0.561892	−	0.827211i	$-0.310074\pi$
$401$	9024.00	1.12378	0.561892	+	0.827211i	$0.310074\pi$
$402$	0	0
$403$	928.000	0.114707
$404$	1413.00	0.174008
$405$	0	0
$406$	3906.00	0.477467
$407$	2783.00	0.338939
$408$	0	0
$409$	−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
$411$	0	0
$412$	1634.00	0.195392
$413$	1743.00	0.207669
$414$	0	0
$415$	0	0
$416$	720.000	0.0848579
$417$	0	0
$418$	−4125.00	−0.482680
$419$	3201.00	0.373220	0.186610	−	0.982434i	$-0.440250\pi$
$419$	3201.00	0.373220	0.186610	+	0.982434i	$0.440250\pi$
$420$	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
$423$	0	0
$424$	−4032.00	−0.461819
$425$	0	0
$426$	0	0
$427$	448.000	0.0507734
$428$	−726.000	−0.0819919
$429$	0	0
$430$	0	0
$431$	354.000	0.0395628	0.0197814	−	0.999804i	$-0.493703\pi$
$431$	354.000	0.0395628	0.0197814	+	0.999804i	$0.493703\pi$
$432$	0	0
$433$	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
$438$	0	0
$439$	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
$441$	0	0
$442$	1008.00	0.108474
$443$	1551.00	0.166344	0.0831718	−	0.996535i	$-0.473495\pi$
$443$	1551.00	0.166344	0.0831718	+	0.996535i	$0.473495\pi$
$444$	0	0
$445$	0	0
$446$	11544.0	1.22561
$447$	0	0
$448$	−3031.00	−0.319646
$449$	1092.00	0.114777	0.0573883	−	0.998352i	$-0.481723\pi$
$449$	1092.00	0.114777	0.0573883	+	0.998352i	$0.481723\pi$
$450$	0	0
$451$	693.000	0.0723550
$452$	1128.00	0.117382
$453$	0	0
$454$	4158.00	0.429834
$455$	0	0
$456$	0	0
$457$	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
$459$	0	0
$460$	0	0
$461$	−15510.0	−1.56697	−0.783485	−	0.621411i	$-0.786560\pi$
$461$	−15510.0	−1.56697	−0.783485	+	0.621411i	$0.786560\pi$
$462$	0	0
$463$	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	$-0.201208\pi$
$467$	16284.0	1.61356	0.806781	+	0.590850i	$0.201208\pi$
$468$	0	0
$469$	−1904.00	−0.187460
$470$	0	0
$471$	0	0
$472$	5229.00	0.509924
$473$	1100.00	0.106930
$474$	0	0
$475$	0	0
$476$	147.000	0.0141549
$477$	0	0
$478$	4572.00	0.437486
$479$	−6732.00	−0.642156	−0.321078	−	0.947053i	$-0.604045\pi$
$479$	−6732.00	−0.642156	−0.321078	+	0.947053i	$0.604045\pi$
$480$	0	0
$481$	4048.00	0.383727
$482$	−6690.00	−0.632202
$483$	0	0
$484$	121.000	0.0113636
$485$	0	0
$486$	0	0
$487$	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
$489$	0	0
$490$	0	0
$491$	−2454.00	−0.225555	−0.112777	−	0.993620i	$-0.535975\pi$
$491$	−2454.00	−0.225555	−0.112777	+	0.993620i	$0.535975\pi$
$492$	0	0
$493$	3906.00	0.356830
$494$	−6000.00	−0.546463
$495$	0	0
$496$	4118.00	0.372790
$497$	−4515.00	−0.407496
$498$	0	0
$499$	−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
$501$	0	0
$502$	−11592.0	−1.03063
$503$	1956.00	0.173387	0.0866936	−	0.996235i	$-0.472370\pi$
$503$	1956.00	0.173387	0.0866936	+	0.996235i	$0.472370\pi$
$504$	0	0
$505$	0	0
$506$	−2673.00	−0.234841
$507$	0	0
$508$	1127.00	0.0984302
$509$	−18240.0	−1.58836	−0.794179	−	0.607684i	$-0.792099\pi$
$509$	−18240.0	−1.58836	−0.794179	+	0.607684i	$0.792099\pi$
$510$	0	0
$511$	784.000	0.0678711
$512$	−8733.00	−0.753804
$513$	0	0
$514$	13554.0	1.16312
$515$	0	0
$516$	0	0
$517$	−2409.00	−0.204928
$518$	5313.00	0.450656
$519$	0	0
$520$	0	0
$521$	−20790.0	−1.74823	−0.874114	−	0.485721i	$-0.838557\pi$
$521$	−20790.0	−1.74823	−0.874114	+	0.485721i	$0.838557\pi$
$522$	0	0
$523$	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
$528$	0	0
$529$	−5606.00	−0.460754
$530$	0	0
$531$	0	0
$532$	−875.000	−0.0713084
$533$	1008.00	0.0819162
$534$	0	0
$535$	0	0
$536$	−5712.00	−0.460300
$537$	0	0
$538$	−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
$543$	0	0
$544$	945.000	0.0744789
$545$	0	0
$546$	0	0
$547$	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
$549$	0	0
$550$	0	0
$551$	−23250.0	−1.79761
$552$	0	0
$553$	−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	$-0.114178\pi$
$557$	24618.0	1.87271	0.936354	+	0.351058i	$0.114178\pi$
$558$	0	0
$559$	1600.00	0.121060
$560$	0	0
$561$	0	0
$562$	18279.0	1.37198
$563$	−6438.00	−0.481935	−0.240967	−	0.970533i	$-0.577465\pi$
$563$	−6438.00	−0.481935	−0.240967	+	0.970533i	$0.577465\pi$
$564$	0	0
$565$	0	0
$566$	7629.00	0.566556
$567$	0	0
$568$	−13545.0	−1.00059
$569$	18183.0	1.33967	0.669834	−	0.742511i	$-0.266365\pi$
$569$	18183.0	1.33967	0.669834	+	0.742511i	$0.266365\pi$
$570$	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
$573$	0	0
$574$	1323.00	0.0962038
$575$	0	0
$576$	0	0
$577$	−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
$579$	0	0
$580$	0	0
$581$	−8778.00	−0.626803
$582$	0	0
$583$	−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	$-0.833064\pi$
$587$	−24621.0	−1.73121	−0.865603	+	0.500732i	$0.833064\pi$
$588$	0	0
$589$	−7250.00	−0.507183
$590$	0	0
$591$	0	0
$592$	17963.0	1.24709
$593$	9066.00	0.627818	0.313909	−	0.949453i	$-0.398361\pi$
$593$	9066.00	0.627818	0.313909	+	0.949453i	$0.398361\pi$
$594$	0	0
$595$	0	0
$596$	−1797.00	−0.123503
$597$	0	0
$598$	−3888.00	−0.265873
$599$	−22353.0	−1.52474	−0.762370	−	0.647142i	$-0.775964\pi$
$599$	−22353.0	−1.52474	−0.762370	+	0.647142i	$0.775964\pi$
$600$	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
$603$	0	0
$604$	1040.00	0.0700613
$605$	0	0
$606$	0	0
$607$	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
$609$	0	0
$610$	0	0
$611$	−3504.00	−0.232008
$612$	0	0
$613$	−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	$-0.487847\pi$
$617$	1170.00	0.0763410	0.0381705	+	0.999271i	$0.487847\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	−7848.00	−0.505910
$623$	5292.00	0.340320
$624$	0	0
$625$	0	0
$626$	12237.0	0.781292
$627$	0	0
$628$	−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
$633$	0	0
$634$	−10512.0	−0.658493
$635$	0	0
$636$	0	0
$637$	4704.00	0.292589
$638$	6138.00	0.380887
$639$	0	0
$640$	0	0
$641$	−28308.0	−1.74430	−0.872152	−	0.489235i	$-0.837276\pi$
$641$	−28308.0	−1.74430	−0.872152	+	0.489235i	$0.837276\pi$
$642$	0	0
$643$	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	$-0.751192\pi$
$647$	−23361.0	−1.41950	−0.709749	+	0.704454i	$0.751192\pi$
$648$	0	0
$649$	2739.00	0.165663
$650$	0	0
$651$	0	0
$652$	2432.00	0.146080
$653$	−12294.0	−0.736756	−0.368378	−	0.929676i	$-0.620087\pi$
$653$	−12294.0	−0.736756	−0.368378	+	0.929676i	$0.620087\pi$
$654$	0	0
$655$	0	0
$656$	4473.00	0.266222
$657$	0	0
$658$	−4599.00	−0.272474
$659$	−31896.0	−1.88542	−0.942710	−	0.333613i	$-0.891732\pi$
$659$	−31896.0	−1.88542	−0.942710	+	0.333613i	$0.891732\pi$
$660$	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
$663$	0	0
$664$	−26334.0	−1.53909
$665$	0	0
$666$	0	0
$667$	−15066.0	−0.874599
$668$	−2340.00	−0.135535
$669$	0	0
$670$	0	0
$671$	704.000	0.0405032
$672$	0	0
$673$	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	$-0.466381\pi$
$677$	3714.00	0.210843	0.105421	+	0.994428i	$0.466381\pi$
$678$	0	0
$679$	−5873.00	−0.331937
$680$	0	0
$681$	0	0
$682$	1914.00	0.107465
$683$	13065.0	0.731945	0.365972	−	0.930626i	$-0.380736\pi$
$683$	13065.0	0.731945	0.365972	+	0.930626i	$0.380736\pi$
$684$	0	0
$685$	0	0
$686$	13377.0	0.744513
$687$	0	0
$688$	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
$693$	0	0
$694$	10836.0	0.592693
$695$	0	0
$696$	0	0
$697$	1323.00	0.0718970
$698$	−1218.00	−0.0660487
$699$	0	0
$700$	0	0
$701$	−33405.0	−1.79984	−0.899921	−	0.436053i	$-0.856376\pi$
$701$	−33405.0	−1.79984	−0.899921	+	0.436053i	$0.856376\pi$
$702$	0	0
$703$	−31625.0	−1.69667
$704$	−4763.00	−0.254989
$705$	0	0
$706$	2448.00	0.130498
$707$	−9891.00	−0.526152
$708$	0	0
$709$	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
$711$	0	0
$712$	15876.0	0.835644
$713$	−4698.00	−0.246762
$714$	0	0
$715$	0	0
$716$	267.000	0.0139361
$717$	0	0
$718$	−14454.0	−0.751279
$719$	−12336.0	−0.639854	−0.319927	−	0.947442i	$-0.603658\pi$
$719$	−12336.0	−0.639854	−0.319927	+	0.947442i	$0.603658\pi$
$720$	0	0
$721$	−11438.0	−0.590809
$722$	26298.0	1.35555
$723$	0	0
$724$	4277.00	0.219549
$725$	0	0
$726$	0	0
$727$	−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
$729$	0	0
$730$	0	0
$731$	2100.00	0.106253
$732$	0	0
$733$	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
$738$	0	0
$739$	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
$741$	0	0
$742$	−4032.00	−0.199487
$743$	6000.00	0.296257	0.148128	−	0.988968i	$-0.452675\pi$
$743$	6000.00	0.296257	0.148128	+	0.988968i	$0.452675\pi$
$744$	0	0
$745$	0	0
$746$	9402.00	0.461437
$747$	0	0
$748$	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
$753$	0	0
$754$	8928.00	0.431218
$755$	0	0
$756$	0	0
$757$	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
$759$	0	0
$760$	0	0
$761$	3654.00	0.174057	0.0870285	−	0.996206i	$-0.472263\pi$
$761$	3654.00	0.174057	0.0870285	+	0.996206i	$0.472263\pi$
$762$	0	0
$763$	−11984.0	−0.568611
$764$	1797.00	0.0850958
$765$	0	0
$766$	−34416.0	−1.62337
$767$	3984.00	0.187554
$768$	0	0
$769$	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
$771$	0	0
$772$	1988.00	0.0926809
$773$	23952.0	1.11448	0.557240	−	0.830351i	$-0.311860\pi$
$773$	23952.0	1.11448	0.557240	+	0.830351i	$0.311860\pi$
$774$	0	0
$775$	0	0
$776$	−17619.0	−0.815058
$777$	0	0
$778$	10386.0	0.478607
$779$	−7875.00	−0.362197
$780$	0	0
$781$	−7095.00	−0.325069
$782$	−5103.00	−0.233354
$783$	0	0
$784$	20874.0	0.950893
$785$	0	0
$786$	0	0
$787$	−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
$789$	0	0
$790$	0	0
$791$	−7896.00	−0.354930
$792$	0	0
$793$	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	$-0.446485\pi$
$797$	7530.00	0.334663	0.167331	+	0.985901i	$0.446485\pi$
$798$	0	0
$799$	−4599.00	−0.203631
$800$	0	0
$801$	0	0
$802$	27072.0	1.19195
$803$	1232.00	0.0541424
$804$	0	0
$805$	0	0
$806$	2784.00	0.121665
$807$	0	0
$808$	−29673.0	−1.29195
$809$	7239.00	0.314598	0.157299	−	0.987551i	$-0.449721\pi$
$809$	7239.00	0.314598	0.157299	+	0.987551i	$0.449721\pi$
$810$	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
$813$	0	0
$814$	8349.00	0.359499
$815$	0	0
$816$	0	0
$817$	−12500.0	−0.535275
$818$	−43464.0	−1.85780
$819$	0	0
$820$	0	0
$821$	42054.0	1.78769	0.893846	−	0.448375i	$-0.147997\pi$
$821$	42054.0	1.78769	0.893846	+	0.448375i	$0.147997\pi$
$822$	0	0
$823$	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	$-0.530563\pi$
$827$	−4560.00	−0.191737	−0.0958686	+	0.995394i	$0.530563\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	−6928.00	−0.288684
$833$	6174.00	0.256802
$834$	0	0
$835$	0	0
$836$	−1375.00	−0.0568844
$837$	0	0
$838$	9603.00	0.395859
$839$	−6216.00	−0.255781	−0.127890	−	0.991788i	$-0.540821\pi$
$839$	−6216.00	−0.255781	−0.127890	+	0.991788i	$0.540821\pi$
$840$	0	0
$841$	10207.0	0.418508
$842$	−20163.0	−0.825253
$843$	0	0
$844$	2180.00	0.0889084
$845$	0	0
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−13632.0	−0.552034
$849$	0	0
$850$	0	0
$851$	−20493.0	−0.825489
$852$	0	0
$853$	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	$-0.315858\pi$
$857$	27435.0	1.09354	0.546769	+	0.837284i	$0.315858\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	1062.00	0.0419627
$863$	14976.0	0.590717	0.295359	−	0.955386i	$-0.404561\pi$
$863$	14976.0	0.590717	0.295359	+	0.955386i	$0.404561\pi$
$864$	0	0
$865$	0	0
$866$	23046.0	0.904313
$867$	0	0
$868$	406.000	0.0158762
$869$	−5599.00	−0.218565
$870$	0	0
$871$	−4352.00	−0.169302
$872$	−35952.0	−1.39620
$873$	0	0
$874$	30375.0	1.17557
$875$	0	0
$876$	0	0
$877$	−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
$879$	0	0
$880$	0	0
$881$	−2916.00	−0.111513	−0.0557563	−	0.998444i	$-0.517757\pi$
$881$	−2916.00	−0.111513	−0.0557563	+	0.998444i	$0.517757\pi$
$882$	0	0
$883$	−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	$-0.351771\pi$
$887$	23724.0	0.898054	0.449027	+	0.893518i	$0.351771\pi$
$888$	0	0
$889$	−7889.00	−0.297625
$890$	0	0
$891$	0	0
$892$	3848.00	0.144440
$893$	27375.0	1.02583
$894$	0	0
$895$	0	0
$896$	−11613.0	−0.432995
$897$	0	0
$898$	3276.00	0.121739
$899$	10788.0	0.400222
$900$	0	0
$901$	−4032.00	−0.149085
$902$	2079.00	0.0767440
$903$	0	0
$904$	−23688.0	−0.871517
$905$	0	0
$906$	0	0
$907$	−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
$909$	0	0
$910$	0	0
$911$	−32139.0	−1.16884	−0.584420	−	0.811452i	$-0.698678\pi$
$911$	−32139.0	−1.16884	−0.584420	+	0.811452i	$0.698678\pi$
$912$	0	0
$913$	−13794.0	−0.500016
$914$	30138.0	1.09067
$915$	0	0
$916$	−991.000	−0.0357462
$917$	−7854.00	−0.282837
$918$	0	0
$919$	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
$921$	0	0
$922$	−46530.0	−1.66202
$923$	−10320.0	−0.368025
$924$	0	0
$925$	0	0
$926$	20634.0	0.732263
$927$	0	0
$928$	8370.00	0.296076
$929$	5964.00	0.210627	0.105314	−	0.994439i	$-0.466415\pi$
$929$	5964.00	0.210627	0.105314	+	0.994439i	$0.466415\pi$
$930$	0	0
$931$	−36750.0	−1.29370
$932$	−975.000	−0.0342674
$933$	0	0
$934$	48852.0	1.71144
$935$	0	0
$936$	0	0
$937$	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
$939$	0	0
$940$	0	0
$941$	42129.0	1.45948	0.729738	−	0.683727i	$-0.239642\pi$
$941$	42129.0	1.45948	0.729738	+	0.683727i	$0.239642\pi$
$942$	0	0
$943$	−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	$-0.629413\pi$
$947$	−23049.0	−0.790910	−0.395455	+	0.918485i	$0.629413\pi$
$948$	0	0
$949$	1792.00	0.0612969
$950$	0	0
$951$	0	0
$952$	−3087.00	−0.105095
$953$	−10221.0	−0.347419	−0.173710	−	0.984797i	$-0.555575\pi$
$953$	−10221.0	−0.347419	−0.173710	+	0.984797i	$0.555575\pi$
$954$	0	0
$955$	0	0
$956$	1524.00	0.0515582
$957$	0	0
$958$	−20196.0	−0.681110
$959$	378.000	0.0127281
$960$	0	0
$961$	−26427.0	−0.887080
$962$	12144.0	0.407004
$963$	0	0
$964$	−2230.00	−0.0745057
$965$	0	0
$966$	0	0
$967$	−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
$969$	0	0
$970$	0	0
$971$	50337.0	1.66364	0.831818	−	0.555048i	$-0.187300\pi$
$971$	50337.0	1.66364	0.831818	+	0.555048i	$0.187300\pi$
$972$	0	0
$973$	−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	$-0.802016\pi$
$977$	−49638.0	−1.62545	−0.812723	+	0.582651i	$0.802016\pi$
$978$	0	0
$979$	8316.00	0.271482
$980$	0	0
$981$	0	0
$982$	−7362.00	−0.239237
$983$	1143.00	0.0370865	0.0185433	−	0.999828i	$-0.494097\pi$
$983$	1143.00	0.0370865	0.0185433	+	0.999828i	$0.494097\pi$
$984$	0	0
$985$	0	0
$986$	11718.0	0.378476
$987$	0	0
$988$	−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
$993$	0	0
$994$	−13545.0	−0.432215
$995$	0	0
$996$	0	0
$997$	−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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.c.1.1	✓	1	3.2	odd	2
825.4.a.g.1.1	yes	1	15.14	odd	2
825.4.c.d.199.1		2	15.2	even	4
825.4.c.d.199.2		2	15.8	even	4
2475.4.a.d.1.1		1	5.4	even	2
2475.4.a.i.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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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