LMFDB - Embedded newform 1520.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,4,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1520.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1520.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:	$89.6829032087$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{3} -5.00000 q^{5} +22.0000 q^{7} -11.0000 q^{9} +O(q^{10})$

$q-4.00000 q^{3} -5.00000 q^{5} +22.0000 q^{7} -11.0000 q^{9} +12.0000 q^{11} +8.00000 q^{13} +20.0000 q^{15} -66.0000 q^{17} -19.0000 q^{19} -88.0000 q^{21} +30.0000 q^{23} +25.0000 q^{25} +152.000 q^{27} -6.00000 q^{29} +64.0000 q^{31} -48.0000 q^{33} -110.000 q^{35} -16.0000 q^{37} -32.0000 q^{39} +54.0000 q^{41} -182.000 q^{43} +55.0000 q^{45} -594.000 q^{47} +141.000 q^{49} +264.000 q^{51} +396.000 q^{53} -60.0000 q^{55} +76.0000 q^{57} +564.000 q^{59} -706.000 q^{61} -242.000 q^{63} -40.0000 q^{65} +628.000 q^{67} -120.000 q^{69} +984.000 q^{71} +14.0000 q^{73} -100.000 q^{75} +264.000 q^{77} +328.000 q^{79} -311.000 q^{81} +294.000 q^{83} +330.000 q^{85} +24.0000 q^{87} +918.000 q^{89} +176.000 q^{91} -256.000 q^{93} +95.0000 q^{95} -1564.00 q^{97} -132.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	22.0000	1.18789	0.593944	−	0.804506i	$-0.297570\pi$
$7$	22.0000	1.18789	0.593944	+	0.804506i	$0.297570\pi$
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	0	0
$13$	8.00000	0.170677	0.0853385	−	0.996352i	$-0.472803\pi$
$13$	8.00000	0.170677	0.0853385	+	0.996352i	$0.472803\pi$
$14$	0	0
$15$	20.0000	0.344265
$16$	0	0
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−88.0000	−0.914437
$22$	0	0
$23$	30.0000	0.271975	0.135988	−	0.990711i	$-0.456579\pi$
$23$	30.0000	0.271975	0.135988	+	0.990711i	$0.456579\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	152.000	1.08342
$28$	0	0
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	64.0000	0.370798	0.185399	−	0.982663i	$-0.440642\pi$
$31$	64.0000	0.370798	0.185399	+	0.982663i	$0.440642\pi$
$32$	0	0
$33$	−48.0000	−0.253204
$34$	0	0
$35$	−110.000	−0.531240
$36$	0	0
$37$	−16.0000	−0.0710915	−0.0355457	−	0.999368i	$-0.511317\pi$
$37$	−16.0000	−0.0710915	−0.0355457	+	0.999368i	$0.511317\pi$
$38$	0	0
$39$	−32.0000	−0.131387
$40$	0	0
$41$	54.0000	0.205692	0.102846	−	0.994697i	$-0.467205\pi$
$41$	54.0000	0.205692	0.102846	+	0.994697i	$0.467205\pi$
$42$	0	0
$43$	−182.000	−0.645459	−0.322730	−	0.946491i	$-0.604600\pi$
$43$	−182.000	−0.645459	−0.322730	+	0.946491i	$0.604600\pi$
$44$	0	0
$45$	55.0000	0.182198
$46$	0	0
$47$	−594.000	−1.84349	−0.921743	−	0.387802i	$-0.873234\pi$
$47$	−594.000	−1.84349	−0.921743	+	0.387802i	$0.873234\pi$
$48$	0	0
$49$	141.000	0.411079
$50$	0	0
$51$	264.000	0.724851
$52$	0	0
$53$	396.000	1.02632	0.513158	−	0.858294i	$-0.328475\pi$
$53$	396.000	1.02632	0.513158	+	0.858294i	$0.328475\pi$
$54$	0	0
$55$	−60.0000	−0.147098
$56$	0	0
$57$	76.0000	0.176604
$58$	0	0
$59$	564.000	1.24452	0.622259	−	0.782812i	$-0.286215\pi$
$59$	564.000	1.24452	0.622259	+	0.782812i	$0.286215\pi$
$60$	0	0
$61$	−706.000	−1.48187	−0.740935	−	0.671577i	$-0.765617\pi$
$61$	−706.000	−1.48187	−0.740935	+	0.671577i	$0.765617\pi$
$62$	0	0
$63$	−242.000	−0.483955
$64$	0	0
$65$	−40.0000	−0.0763291
$66$	0	0
$67$	628.000	1.14511	0.572555	−	0.819866i	$-0.305952\pi$
$67$	628.000	1.14511	0.572555	+	0.819866i	$0.305952\pi$
$68$	0	0
$69$	−120.000	−0.209367
$70$	0	0
$71$	984.000	1.64478	0.822390	−	0.568925i	$-0.192640\pi$
$71$	984.000	1.64478	0.822390	+	0.568925i	$0.192640\pi$
$72$	0	0
$73$	14.0000	0.0224462	0.0112231	−	0.999937i	$-0.496427\pi$
$73$	14.0000	0.0224462	0.0112231	+	0.999937i	$0.496427\pi$
$74$	0	0
$75$	−100.000	−0.153960
$76$	0	0
$77$	264.000	0.390722
$78$	0	0
$79$	328.000	0.467125	0.233563	−	0.972342i	$-0.424962\pi$
$79$	328.000	0.467125	0.233563	+	0.972342i	$0.424962\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	294.000	0.388804	0.194402	−	0.980922i	$-0.437723\pi$
$83$	294.000	0.388804	0.194402	+	0.980922i	$0.437723\pi$
$84$	0	0
$85$	330.000	0.421100
$86$	0	0
$87$	24.0000	0.0295755
$88$	0	0
$89$	918.000	1.09335	0.546673	−	0.837346i	$-0.315894\pi$
$89$	918.000	1.09335	0.546673	+	0.837346i	$0.315894\pi$
$90$	0	0
$91$	176.000	0.202745
$92$	0	0
$93$	−256.000	−0.285440
$94$	0	0
$95$	95.0000	0.102598
$96$	0	0
$97$	−1564.00	−1.63711	−0.818557	−	0.574425i	$-0.805226\pi$
$97$	−1564.00	−1.63711	−0.818557	+	0.574425i	$0.805226\pi$
$98$	0	0
$99$	−132.000	−0.134005
$100$	0	0
$101$	−294.000	−0.289644	−0.144822	−	0.989458i	$-0.546261\pi$
$101$	−294.000	−0.289644	−0.144822	+	0.989458i	$0.546261\pi$
$102$	0	0
$103$	−752.000	−0.719386	−0.359693	−	0.933071i	$-0.617119\pi$
$103$	−752.000	−0.719386	−0.359693	+	0.933071i	$0.617119\pi$
$104$	0	0
$105$	440.000	0.408949
$106$	0	0
$107$	216.000	0.195154	0.0975771	−	0.995228i	$-0.468891\pi$
$107$	216.000	0.195154	0.0975771	+	0.995228i	$0.468891\pi$
$108$	0	0
$109$	−754.000	−0.662570	−0.331285	−	0.943531i	$-0.607482\pi$
$109$	−754.000	−0.662570	−0.331285	+	0.943531i	$0.607482\pi$
$110$	0	0
$111$	64.0000	0.0547262
$112$	0	0
$113$	−12.0000	−0.00998996	−0.00499498	−	0.999988i	$-0.501590\pi$
$113$	−12.0000	−0.00998996	−0.00499498	+	0.999988i	$0.501590\pi$
$114$	0	0
$115$	−150.000	−0.121631
$116$	0	0
$117$	−88.0000	−0.0695351
$118$	0	0
$119$	−1452.00	−1.11853
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	−216.000	−0.158342
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−344.000	−0.240355	−0.120177	−	0.992752i	$-0.538346\pi$
$127$	−344.000	−0.240355	−0.120177	+	0.992752i	$0.538346\pi$
$128$	0	0
$129$	728.000	0.496875
$130$	0	0
$131$	−2520.00	−1.68071	−0.840357	−	0.542034i	$-0.817654\pi$
$131$	−2520.00	−1.68071	−0.840357	+	0.542034i	$0.817654\pi$
$132$	0	0
$133$	−418.000	−0.272520
$134$	0	0
$135$	−760.000	−0.484521
$136$	0	0
$137$	654.000	0.407847	0.203923	−	0.978987i	$-0.434631\pi$
$137$	654.000	0.407847	0.203923	+	0.978987i	$0.434631\pi$
$138$	0	0
$139$	2392.00	1.45962	0.729809	−	0.683652i	$-0.239609\pi$
$139$	2392.00	1.45962	0.729809	+	0.683652i	$0.239609\pi$
$140$	0	0
$141$	2376.00	1.41912
$142$	0	0
$143$	96.0000	0.0561393
$144$	0	0
$145$	30.0000	0.0171818
$146$	0	0
$147$	−564.000	−0.316449
$148$	0	0
$149$	−1266.00	−0.696072	−0.348036	−	0.937481i	$-0.613151\pi$
$149$	−1266.00	−0.696072	−0.348036	+	0.937481i	$0.613151\pi$
$150$	0	0
$151$	−3080.00	−1.65991	−0.829956	−	0.557828i	$-0.811635\pi$
$151$	−3080.00	−1.65991	−0.829956	+	0.557828i	$0.811635\pi$
$152$	0	0
$153$	726.000	0.383618
$154$	0	0
$155$	−320.000	−0.165826
$156$	0	0
$157$	1838.00	0.934321	0.467160	−	0.884173i	$-0.345277\pi$
$157$	1838.00	0.934321	0.467160	+	0.884173i	$0.345277\pi$
$158$	0	0
$159$	−1584.00	−0.790059
$160$	0	0
$161$	660.000	0.323076
$162$	0	0
$163$	850.000	0.408449	0.204224	−	0.978924i	$-0.434533\pi$
$163$	850.000	0.408449	0.204224	+	0.978924i	$0.434533\pi$
$164$	0	0
$165$	240.000	0.113236
$166$	0	0
$167$	−3804.00	−1.76265	−0.881324	−	0.472512i	$-0.843347\pi$
$167$	−3804.00	−1.76265	−0.881324	+	0.472512i	$0.843347\pi$
$168$	0	0
$169$	−2133.00	−0.970869
$170$	0	0
$171$	209.000	0.0934657
$172$	0	0
$173$	−564.000	−0.247862	−0.123931	−	0.992291i	$-0.539550\pi$
$173$	−564.000	−0.247862	−0.123931	+	0.992291i	$0.539550\pi$
$174$	0	0
$175$	550.000	0.237578
$176$	0	0
$177$	−2256.00	−0.958030
$178$	0	0
$179$	1812.00	0.756621	0.378311	−	0.925679i	$-0.376505\pi$
$179$	1812.00	0.756621	0.378311	+	0.925679i	$0.376505\pi$
$180$	0	0
$181$	−4498.00	−1.84715	−0.923574	−	0.383421i	$-0.874746\pi$
$181$	−4498.00	−1.84715	−0.923574	+	0.383421i	$0.874746\pi$
$182$	0	0
$183$	2824.00	1.14074
$184$	0	0
$185$	80.0000	0.0317931
$186$	0	0
$187$	−792.000	−0.309715
$188$	0	0
$189$	3344.00	1.28699
$190$	0	0
$191$	−3588.00	−1.35926	−0.679630	−	0.733555i	$-0.737860\pi$
$191$	−3588.00	−1.35926	−0.679630	+	0.733555i	$0.737860\pi$
$192$	0	0
$193$	−4492.00	−1.67534	−0.837672	−	0.546174i	$-0.816084\pi$
$193$	−4492.00	−1.67534	−0.837672	+	0.546174i	$0.816084\pi$
$194$	0	0
$195$	160.000	0.0587581
$196$	0	0
$197$	2466.00	0.891854	0.445927	−	0.895069i	$-0.352874\pi$
$197$	2466.00	0.891854	0.445927	+	0.895069i	$0.352874\pi$
$198$	0	0
$199$	−824.000	−0.293527	−0.146763	−	0.989172i	$-0.546886\pi$
$199$	−824.000	−0.293527	−0.146763	+	0.989172i	$0.546886\pi$
$200$	0	0
$201$	−2512.00	−0.881507
$202$	0	0
$203$	−132.000	−0.0456383
$204$	0	0
$205$	−270.000	−0.0919884
$206$	0	0
$207$	−330.000	−0.110805
$208$	0	0
$209$	−228.000	−0.0754598
$210$	0	0
$211$	−4244.00	−1.38469	−0.692344	−	0.721568i	$-0.743422\pi$
$211$	−4244.00	−1.38469	−0.692344	+	0.721568i	$0.743422\pi$
$212$	0	0
$213$	−3936.00	−1.26615
$214$	0	0
$215$	910.000	0.288658
$216$	0	0
$217$	1408.00	0.440467
$218$	0	0
$219$	−56.0000	−0.0172791
$220$	0	0
$221$	−528.000	−0.160711
$222$	0	0
$223$	−5480.00	−1.64560	−0.822798	−	0.568334i	$-0.807588\pi$
$223$	−5480.00	−1.64560	−0.822798	+	0.568334i	$0.807588\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	−5544.00	−1.62101	−0.810503	−	0.585735i	$-0.800806\pi$
$227$	−5544.00	−1.62101	−0.810503	+	0.585735i	$0.800806\pi$
$228$	0	0
$229$	2930.00	0.845502	0.422751	−	0.906246i	$-0.361065\pi$
$229$	2930.00	0.845502	0.422751	+	0.906246i	$0.361065\pi$
$230$	0	0
$231$	−1056.00	−0.300778
$232$	0	0
$233$	−4398.00	−1.23658	−0.618289	−	0.785951i	$-0.712174\pi$
$233$	−4398.00	−1.23658	−0.618289	+	0.785951i	$0.712174\pi$
$234$	0	0
$235$	2970.00	0.824432
$236$	0	0
$237$	−1312.00	−0.359593
$238$	0	0
$239$	6480.00	1.75379	0.876896	−	0.480680i	$-0.159610\pi$
$239$	6480.00	1.75379	0.876896	+	0.480680i	$0.159610\pi$
$240$	0	0
$241$	−5770.00	−1.54223	−0.771117	−	0.636694i	$-0.780302\pi$
$241$	−5770.00	−1.54223	−0.771117	+	0.636694i	$0.780302\pi$
$242$	0	0
$243$	−2860.00	−0.755017
$244$	0	0
$245$	−705.000	−0.183840
$246$	0	0
$247$	−152.000	−0.0391560
$248$	0	0
$249$	−1176.00	−0.299301
$250$	0	0
$251$	624.000	0.156918	0.0784592	−	0.996917i	$-0.475000\pi$
$251$	624.000	0.156918	0.0784592	+	0.996917i	$0.475000\pi$
$252$	0	0
$253$	360.000	0.0894585
$254$	0	0
$255$	−1320.00	−0.324163
$256$	0	0
$257$	120.000	0.0291260	0.0145630	−	0.999894i	$-0.495364\pi$
$257$	120.000	0.0291260	0.0145630	+	0.999894i	$0.495364\pi$
$258$	0	0
$259$	−352.000	−0.0844487
$260$	0	0
$261$	66.0000	0.0156525
$262$	0	0
$263$	−1842.00	−0.431873	−0.215936	−	0.976407i	$-0.569280\pi$
$263$	−1842.00	−0.431873	−0.215936	+	0.976407i	$0.569280\pi$
$264$	0	0
$265$	−1980.00	−0.458983
$266$	0	0
$267$	−3672.00	−0.841658
$268$	0	0
$269$	−3234.00	−0.733013	−0.366506	−	0.930416i	$-0.619446\pi$
$269$	−3234.00	−0.733013	−0.366506	+	0.930416i	$0.619446\pi$
$270$	0	0
$271$	−4664.00	−1.04545	−0.522727	−	0.852500i	$-0.675085\pi$
$271$	−4664.00	−1.04545	−0.522727	+	0.852500i	$0.675085\pi$
$272$	0	0
$273$	−704.000	−0.156073
$274$	0	0
$275$	300.000	0.0657843
$276$	0	0
$277$	5222.00	1.13271	0.566353	−	0.824163i	$-0.308354\pi$
$277$	5222.00	1.13271	0.566353	+	0.824163i	$0.308354\pi$
$278$	0	0
$279$	−704.000	−0.151066
$280$	0	0
$281$	7566.00	1.60623	0.803113	−	0.595826i	$-0.203175\pi$
$281$	7566.00	1.60623	0.803113	+	0.595826i	$0.203175\pi$
$282$	0	0
$283$	5614.00	1.17921	0.589607	−	0.807690i	$-0.299283\pi$
$283$	5614.00	1.17921	0.589607	+	0.807690i	$0.299283\pi$
$284$	0	0
$285$	−380.000	−0.0789799
$286$	0	0
$287$	1188.00	0.244339
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	6256.00	1.26025
$292$	0	0
$293$	924.000	0.184234	0.0921172	−	0.995748i	$-0.470637\pi$
$293$	924.000	0.184234	0.0921172	+	0.995748i	$0.470637\pi$
$294$	0	0
$295$	−2820.00	−0.556565
$296$	0	0
$297$	1824.00	0.356361
$298$	0	0
$299$	240.000	0.0464199
$300$	0	0
$301$	−4004.00	−0.766733
$302$	0	0
$303$	1176.00	0.222968
$304$	0	0
$305$	3530.00	0.662712
$306$	0	0
$307$	4480.00	0.832857	0.416429	−	0.909168i	$-0.363282\pi$
$307$	4480.00	0.832857	0.416429	+	0.909168i	$0.363282\pi$
$308$	0	0
$309$	3008.00	0.553784
$310$	0	0
$311$	−1272.00	−0.231924	−0.115962	−	0.993254i	$-0.536995\pi$
$311$	−1272.00	−0.231924	−0.115962	+	0.993254i	$0.536995\pi$
$312$	0	0
$313$	1370.00	0.247402	0.123701	−	0.992320i	$-0.460524\pi$
$313$	1370.00	0.247402	0.123701	+	0.992320i	$0.460524\pi$
$314$	0	0
$315$	1210.00	0.216431
$316$	0	0
$317$	3552.00	0.629338	0.314669	−	0.949201i	$-0.398106\pi$
$317$	3552.00	0.629338	0.314669	+	0.949201i	$0.398106\pi$
$318$	0	0
$319$	−72.0000	−0.0126371
$320$	0	0
$321$	−864.000	−0.150230
$322$	0	0
$323$	1254.00	0.216020
$324$	0	0
$325$	200.000	0.0341354
$326$	0	0
$327$	3016.00	0.510046
$328$	0	0
$329$	−13068.0	−2.18985
$330$	0	0
$331$	−4532.00	−0.752572	−0.376286	−	0.926504i	$-0.622799\pi$
$331$	−4532.00	−0.752572	−0.376286	+	0.926504i	$0.622799\pi$
$332$	0	0
$333$	176.000	0.0289632
$334$	0	0
$335$	−3140.00	−0.512109
$336$	0	0
$337$	−10036.0	−1.62224	−0.811121	−	0.584878i	$-0.801142\pi$
$337$	−10036.0	−1.62224	−0.811121	+	0.584878i	$0.801142\pi$
$338$	0	0
$339$	48.0000	0.00769027
$340$	0	0
$341$	768.000	0.121963
$342$	0	0
$343$	−4444.00	−0.699573
$344$	0	0
$345$	600.000	0.0936316
$346$	0	0
$347$	2178.00	0.336949	0.168474	−	0.985706i	$-0.446116\pi$
$347$	2178.00	0.336949	0.168474	+	0.985706i	$0.446116\pi$
$348$	0	0
$349$	−10042.0	−1.54022	−0.770109	−	0.637913i	$-0.779798\pi$
$349$	−10042.0	−1.54022	−0.770109	+	0.637913i	$0.779798\pi$
$350$	0	0
$351$	1216.00	0.184915
$352$	0	0
$353$	6102.00	0.920047	0.460024	−	0.887907i	$-0.347841\pi$
$353$	6102.00	0.920047	0.460024	+	0.887907i	$0.347841\pi$
$354$	0	0
$355$	−4920.00	−0.735568
$356$	0	0
$357$	5808.00	0.861042
$358$	0	0
$359$	1140.00	0.167596	0.0837979	−	0.996483i	$-0.473295\pi$
$359$	1140.00	0.167596	0.0837979	+	0.996483i	$0.473295\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	4748.00	0.686516
$364$	0	0
$365$	−70.0000	−0.0100383
$366$	0	0
$367$	5614.00	0.798497	0.399249	−	0.916843i	$-0.369271\pi$
$367$	5614.00	0.798497	0.399249	+	0.916843i	$0.369271\pi$
$368$	0	0
$369$	−594.000	−0.0838006
$370$	0	0
$371$	8712.00	1.21915
$372$	0	0
$373$	−3652.00	−0.506953	−0.253476	−	0.967342i	$-0.581574\pi$
$373$	−3652.00	−0.506953	−0.253476	+	0.967342i	$0.581574\pi$
$374$	0	0
$375$	500.000	0.0688530
$376$	0	0
$377$	−48.0000	−0.00655736
$378$	0	0
$379$	316.000	0.0428280	0.0214140	−	0.999771i	$-0.493183\pi$
$379$	316.000	0.0428280	0.0214140	+	0.999771i	$0.493183\pi$
$380$	0	0
$381$	1376.00	0.185025
$382$	0	0
$383$	8844.00	1.17991	0.589957	−	0.807434i	$-0.299145\pi$
$383$	8844.00	1.17991	0.589957	+	0.807434i	$0.299145\pi$
$384$	0	0
$385$	−1320.00	−0.174736
$386$	0	0
$387$	2002.00	0.262965
$388$	0	0
$389$	−7230.00	−0.942354	−0.471177	−	0.882039i	$-0.656171\pi$
$389$	−7230.00	−0.942354	−0.471177	+	0.882039i	$0.656171\pi$
$390$	0	0
$391$	−1980.00	−0.256094
$392$	0	0
$393$	10080.0	1.29381
$394$	0	0
$395$	−1640.00	−0.208905
$396$	0	0
$397$	3446.00	0.435642	0.217821	−	0.975989i	$-0.430105\pi$
$397$	3446.00	0.435642	0.217821	+	0.975989i	$0.430105\pi$
$398$	0	0
$399$	1672.00	0.209786
$400$	0	0
$401$	−14478.0	−1.80298	−0.901492	−	0.432795i	$-0.857527\pi$
$401$	−14478.0	−1.80298	−0.901492	+	0.432795i	$0.857527\pi$
$402$	0	0
$403$	512.000	0.0632867
$404$	0	0
$405$	1555.00	0.190787
$406$	0	0
$407$	−192.000	−0.0233835
$408$	0	0
$409$	9074.00	1.09702	0.548509	−	0.836145i	$-0.315196\pi$
$409$	9074.00	1.09702	0.548509	+	0.836145i	$0.315196\pi$
$410$	0	0
$411$	−2616.00	−0.313960
$412$	0	0
$413$	12408.0	1.47835
$414$	0	0
$415$	−1470.00	−0.173878
$416$	0	0
$417$	−9568.00	−1.12361
$418$	0	0
$419$	7068.00	0.824092	0.412046	−	0.911163i	$-0.364814\pi$
$419$	7068.00	0.824092	0.412046	+	0.911163i	$0.364814\pi$
$420$	0	0
$421$	−7342.00	−0.849946	−0.424973	−	0.905206i	$-0.639716\pi$
$421$	−7342.00	−0.849946	−0.424973	+	0.905206i	$0.639716\pi$
$422$	0	0
$423$	6534.00	0.751050
$424$	0	0
$425$	−1650.00	−0.188322
$426$	0	0
$427$	−15532.0	−1.76030
$428$	0	0
$429$	−384.000	−0.0432161
$430$	0	0
$431$	−2976.00	−0.332596	−0.166298	−	0.986076i	$-0.553181\pi$
$431$	−2976.00	−0.332596	−0.166298	+	0.986076i	$0.553181\pi$
$432$	0	0
$433$	3476.00	0.385787	0.192894	−	0.981220i	$-0.438213\pi$
$433$	3476.00	0.385787	0.192894	+	0.981220i	$0.438213\pi$
$434$	0	0
$435$	−120.000	−0.0132266
$436$	0	0
$437$	−570.000	−0.0623954
$438$	0	0
$439$	−6200.00	−0.674054	−0.337027	−	0.941495i	$-0.609421\pi$
$439$	−6200.00	−0.674054	−0.337027	+	0.941495i	$0.609421\pi$
$440$	0	0
$441$	−1551.00	−0.167477
$442$	0	0
$443$	−16026.0	−1.71878	−0.859389	−	0.511323i	$-0.829156\pi$
$443$	−16026.0	−1.71878	−0.859389	+	0.511323i	$0.829156\pi$
$444$	0	0
$445$	−4590.00	−0.488959
$446$	0	0
$447$	5064.00	0.535837
$448$	0	0
$449$	−1830.00	−0.192345	−0.0961726	−	0.995365i	$-0.530660\pi$
$449$	−1830.00	−0.192345	−0.0961726	+	0.995365i	$0.530660\pi$
$450$	0	0
$451$	648.000	0.0676566
$452$	0	0
$453$	12320.0	1.27780
$454$	0	0
$455$	−880.000	−0.0906704
$456$	0	0
$457$	12986.0	1.32923	0.664616	−	0.747185i	$-0.268595\pi$
$457$	12986.0	1.32923	0.664616	+	0.747185i	$0.268595\pi$
$458$	0	0
$459$	−10032.0	−1.02016
$460$	0	0
$461$	−10506.0	−1.06142	−0.530708	−	0.847554i	$-0.678074\pi$
$461$	−10506.0	−1.06142	−0.530708	+	0.847554i	$0.678074\pi$
$462$	0	0
$463$	−1562.00	−0.156787	−0.0783934	−	0.996923i	$-0.524979\pi$
$463$	−1562.00	−0.156787	−0.0783934	+	0.996923i	$0.524979\pi$
$464$	0	0
$465$	1280.00	0.127653
$466$	0	0
$467$	−6.00000	−0.000594533 0	−0.000297266	−	1.00000i	$-0.500095\pi$
$467$	−6.00000	−0.000594533 0	−0.000297266		1.00000i	$0.500095\pi$
$468$	0	0
$469$	13816.0	1.36026
$470$	0	0
$471$	−7352.00	−0.719241
$472$	0	0
$473$	−2184.00	−0.212305
$474$	0	0
$475$	−475.000	−0.0458831
$476$	0	0
$477$	−4356.00	−0.418129
$478$	0	0
$479$	−3132.00	−0.298757	−0.149379	−	0.988780i	$-0.547727\pi$
$479$	−3132.00	−0.298757	−0.149379	+	0.988780i	$0.547727\pi$
$480$	0	0
$481$	−128.000	−0.0121337
$482$	0	0
$483$	−2640.00	−0.248704
$484$	0	0
$485$	7820.00	0.732140
$486$	0	0
$487$	12436.0	1.15714	0.578572	−	0.815631i	$-0.303610\pi$
$487$	12436.0	1.15714	0.578572	+	0.815631i	$0.303610\pi$
$488$	0	0
$489$	−3400.00	−0.314424
$490$	0	0
$491$	7848.00	0.721335	0.360667	−	0.932695i	$-0.382549\pi$
$491$	7848.00	0.721335	0.360667	+	0.932695i	$0.382549\pi$
$492$	0	0
$493$	396.000	0.0361764
$494$	0	0
$495$	660.000	0.0599289
$496$	0	0
$497$	21648.0	1.95381
$498$	0	0
$499$	−17720.0	−1.58969	−0.794846	−	0.606811i	$-0.792448\pi$
$499$	−17720.0	−1.58969	−0.794846	+	0.606811i	$0.792448\pi$
$500$	0	0
$501$	15216.0	1.35689
$502$	0	0
$503$	5094.00	0.451551	0.225776	−	0.974179i	$-0.427508\pi$
$503$	5094.00	0.451551	0.225776	+	0.974179i	$0.427508\pi$
$504$	0	0
$505$	1470.00	0.129533
$506$	0	0
$507$	8532.00	0.747376
$508$	0	0
$509$	−5670.00	−0.493749	−0.246875	−	0.969047i	$-0.579404\pi$
$509$	−5670.00	−0.493749	−0.246875	+	0.969047i	$0.579404\pi$
$510$	0	0
$511$	308.000	0.0266636
$512$	0	0
$513$	−2888.00	−0.248554
$514$	0	0
$515$	3760.00	0.321719
$516$	0	0
$517$	−7128.00	−0.606362
$518$	0	0
$519$	2256.00	0.190804
$520$	0	0
$521$	−20670.0	−1.73814	−0.869068	−	0.494692i	$-0.835281\pi$
$521$	−20670.0	−1.73814	−0.869068	+	0.494692i	$0.835281\pi$
$522$	0	0
$523$	16816.0	1.40595	0.702975	−	0.711214i	$-0.251854\pi$
$523$	16816.0	1.40595	0.702975	+	0.711214i	$0.251854\pi$
$524$	0	0
$525$	−2200.00	−0.182887
$526$	0	0
$527$	−4224.00	−0.349147
$528$	0	0
$529$	−11267.0	−0.926029
$530$	0	0
$531$	−6204.00	−0.507026
$532$	0	0
$533$	432.000	0.0351069
$534$	0	0
$535$	−1080.00	−0.0872756
$536$	0	0
$537$	−7248.00	−0.582447
$538$	0	0
$539$	1692.00	0.135213
$540$	0	0
$541$	9530.00	0.757351	0.378675	−	0.925530i	$-0.376380\pi$
$541$	9530.00	0.757351	0.378675	+	0.925530i	$0.376380\pi$
$542$	0	0
$543$	17992.0	1.42193
$544$	0	0
$545$	3770.00	0.296310
$546$	0	0
$547$	−5264.00	−0.411467	−0.205733	−	0.978608i	$-0.565958\pi$
$547$	−5264.00	−0.411467	−0.205733	+	0.978608i	$0.565958\pi$
$548$	0	0
$549$	7766.00	0.603725
$550$	0	0
$551$	114.000	0.00881409
$552$	0	0
$553$	7216.00	0.554892
$554$	0	0
$555$	−320.000	−0.0244743
$556$	0	0
$557$	16542.0	1.25836	0.629180	−	0.777259i	$-0.283391\pi$
$557$	16542.0	1.25836	0.629180	+	0.777259i	$0.283391\pi$
$558$	0	0
$559$	−1456.00	−0.110165
$560$	0	0
$561$	3168.00	0.238419
$562$	0	0
$563$	5232.00	0.391656	0.195828	−	0.980638i	$-0.437261\pi$
$563$	5232.00	0.391656	0.195828	+	0.980638i	$0.437261\pi$
$564$	0	0
$565$	60.0000	0.00446764
$566$	0	0
$567$	−6842.00	−0.506767
$568$	0	0
$569$	15114.0	1.11355	0.556777	−	0.830662i	$-0.312038\pi$
$569$	15114.0	1.11355	0.556777	+	0.830662i	$0.312038\pi$
$570$	0	0
$571$	11764.0	0.862186	0.431093	−	0.902308i	$-0.358128\pi$
$571$	11764.0	0.862186	0.431093	+	0.902308i	$0.358128\pi$
$572$	0	0
$573$	14352.0	1.04636
$574$	0	0
$575$	750.000	0.0543951
$576$	0	0
$577$	−25198.0	−1.81804	−0.909018	−	0.416757i	$-0.863166\pi$
$577$	−25198.0	−1.81804	−0.909018	+	0.416757i	$0.863166\pi$
$578$	0	0
$579$	17968.0	1.28968
$580$	0	0
$581$	6468.00	0.461855
$582$	0	0
$583$	4752.00	0.337578
$584$	0	0
$585$	440.000	0.0310970
$586$	0	0
$587$	−20190.0	−1.41964	−0.709822	−	0.704382i	$-0.751224\pi$
$587$	−20190.0	−1.41964	−0.709822	+	0.704382i	$0.751224\pi$
$588$	0	0
$589$	−1216.00	−0.0850669
$590$	0	0
$591$	−9864.00	−0.686549
$592$	0	0
$593$	−2886.00	−0.199855	−0.0999273	−	0.994995i	$-0.531861\pi$
$593$	−2886.00	−0.199855	−0.0999273	+	0.994995i	$0.531861\pi$
$594$	0	0
$595$	7260.00	0.500220
$596$	0	0
$597$	3296.00	0.225957
$598$	0	0
$599$	4464.00	0.304498	0.152249	−	0.988342i	$-0.451348\pi$
$599$	4464.00	0.304498	0.152249	+	0.988342i	$0.451348\pi$
$600$	0	0
$601$	−15874.0	−1.07739	−0.538697	−	0.842499i	$-0.681083\pi$
$601$	−15874.0	−1.07739	−0.538697	+	0.842499i	$0.681083\pi$
$602$	0	0
$603$	−6908.00	−0.466527
$604$	0	0
$605$	5935.00	0.398830
$606$	0	0
$607$	18916.0	1.26487	0.632436	−	0.774613i	$-0.282055\pi$
$607$	18916.0	1.26487	0.632436	+	0.774613i	$0.282055\pi$
$608$	0	0
$609$	528.000	0.0351324
$610$	0	0
$611$	−4752.00	−0.314640
$612$	0	0
$613$	−3058.00	−0.201487	−0.100743	−	0.994912i	$-0.532122\pi$
$613$	−3058.00	−0.201487	−0.100743	+	0.994912i	$0.532122\pi$
$614$	0	0
$615$	1080.00	0.0708127
$616$	0	0
$617$	4158.00	0.271304	0.135652	−	0.990757i	$-0.456687\pi$
$617$	4158.00	0.271304	0.135652	+	0.990757i	$0.456687\pi$
$618$	0	0
$619$	22864.0	1.48462	0.742312	−	0.670055i	$-0.233729\pi$
$619$	22864.0	1.48462	0.742312	+	0.670055i	$0.233729\pi$
$620$	0	0
$621$	4560.00	0.294664
$622$	0	0
$623$	20196.0	1.29877
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	912.000	0.0580890
$628$	0	0
$629$	1056.00	0.0669403
$630$	0	0
$631$	−18536.0	−1.16942	−0.584712	−	0.811241i	$-0.698792\pi$
$631$	−18536.0	−1.16942	−0.584712	+	0.811241i	$0.698792\pi$
$632$	0	0
$633$	16976.0	1.06593
$634$	0	0
$635$	1720.00	0.107490
$636$	0	0
$637$	1128.00	0.0701617
$638$	0	0
$639$	−10824.0	−0.670095
$640$	0	0
$641$	15630.0	0.963101	0.481551	−	0.876418i	$-0.340074\pi$
$641$	15630.0	0.963101	0.481551	+	0.876418i	$0.340074\pi$
$642$	0	0
$643$	27574.0	1.69115	0.845577	−	0.533853i	$-0.179256\pi$
$643$	27574.0	1.69115	0.845577	+	0.533853i	$0.179256\pi$
$644$	0	0
$645$	−3640.00	−0.222209
$646$	0	0
$647$	8826.00	0.536300	0.268150	−	0.963377i	$-0.413588\pi$
$647$	8826.00	0.536300	0.268150	+	0.963377i	$0.413588\pi$
$648$	0	0
$649$	6768.00	0.409349
$650$	0	0
$651$	−5632.00	−0.339071
$652$	0	0
$653$	18678.0	1.11934	0.559668	−	0.828717i	$-0.310929\pi$
$653$	18678.0	1.11934	0.559668	+	0.828717i	$0.310929\pi$
$654$	0	0
$655$	12600.0	0.751638
$656$	0	0
$657$	−154.000	−0.00914477
$658$	0	0
$659$	−16980.0	−1.00371	−0.501857	−	0.864951i	$-0.667349\pi$
$659$	−16980.0	−1.00371	−0.501857	+	0.864951i	$0.667349\pi$
$660$	0	0
$661$	−9358.00	−0.550657	−0.275328	−	0.961350i	$-0.588787\pi$
$661$	−9358.00	−0.550657	−0.275328	+	0.961350i	$0.588787\pi$
$662$	0	0
$663$	2112.00	0.123715
$664$	0	0
$665$	2090.00	0.121875
$666$	0	0
$667$	−180.000	−0.0104492
$668$	0	0
$669$	21920.0	1.26678
$670$	0	0
$671$	−8472.00	−0.487419
$672$	0	0
$673$	−16120.0	−0.923299	−0.461650	−	0.887062i	$-0.652742\pi$
$673$	−16120.0	−0.923299	−0.461650	+	0.887062i	$0.652742\pi$
$674$	0	0
$675$	3800.00	0.216685
$676$	0	0
$677$	27876.0	1.58251	0.791257	−	0.611484i	$-0.209427\pi$
$677$	27876.0	1.58251	0.791257	+	0.611484i	$0.209427\pi$
$678$	0	0
$679$	−34408.0	−1.94471
$680$	0	0
$681$	22176.0	1.24785
$682$	0	0
$683$	4872.00	0.272946	0.136473	−	0.990644i	$-0.456423\pi$
$683$	4872.00	0.272946	0.136473	+	0.990644i	$0.456423\pi$
$684$	0	0
$685$	−3270.00	−0.182395
$686$	0	0
$687$	−11720.0	−0.650867
$688$	0	0
$689$	3168.00	0.175169
$690$	0	0
$691$	−13412.0	−0.738374	−0.369187	−	0.929355i	$-0.620364\pi$
$691$	−13412.0	−0.738374	−0.369187	+	0.929355i	$0.620364\pi$
$692$	0	0
$693$	−2904.00	−0.159183
$694$	0	0
$695$	−11960.0	−0.652761
$696$	0	0
$697$	−3564.00	−0.193682
$698$	0	0
$699$	17592.0	0.951918
$700$	0	0
$701$	1926.00	0.103772	0.0518859	−	0.998653i	$-0.483477\pi$
$701$	1926.00	0.103772	0.0518859	+	0.998653i	$0.483477\pi$
$702$	0	0
$703$	304.000	0.0163095
$704$	0	0
$705$	−11880.0	−0.634648
$706$	0	0
$707$	−6468.00	−0.344065
$708$	0	0
$709$	17534.0	0.928777	0.464389	−	0.885631i	$-0.346274\pi$
$709$	17534.0	0.928777	0.464389	+	0.885631i	$0.346274\pi$
$710$	0	0
$711$	−3608.00	−0.190310
$712$	0	0
$713$	1920.00	0.100848
$714$	0	0
$715$	−480.000	−0.0251063
$716$	0	0
$717$	−25920.0	−1.35007
$718$	0	0
$719$	11220.0	0.581969	0.290984	−	0.956728i	$-0.406017\pi$
$719$	11220.0	0.581969	0.290984	+	0.956728i	$0.406017\pi$
$720$	0	0
$721$	−16544.0	−0.854550
$722$	0	0
$723$	23080.0	1.18721
$724$	0	0
$725$	−150.000	−0.00768395
$726$	0	0
$727$	25078.0	1.27936	0.639678	−	0.768643i	$-0.279068\pi$
$727$	25078.0	1.27936	0.639678	+	0.768643i	$0.279068\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	12012.0	0.607770
$732$	0	0
$733$	434.000	0.0218692	0.0109346	−	0.999940i	$-0.496519\pi$
$733$	434.000	0.0218692	0.0109346	+	0.999940i	$0.496519\pi$
$734$	0	0
$735$	2820.00	0.141520
$736$	0	0
$737$	7536.00	0.376651
$738$	0	0
$739$	−2564.00	−0.127630	−0.0638148	−	0.997962i	$-0.520327\pi$
$739$	−2564.00	−0.127630	−0.0638148	+	0.997962i	$0.520327\pi$
$740$	0	0
$741$	608.000	0.0301423
$742$	0	0
$743$	−21948.0	−1.08371	−0.541853	−	0.840473i	$-0.682277\pi$
$743$	−21948.0	−1.08371	−0.541853	+	0.840473i	$0.682277\pi$
$744$	0	0
$745$	6330.00	0.311293
$746$	0	0
$747$	−3234.00	−0.158401
$748$	0	0
$749$	4752.00	0.231821
$750$	0	0
$751$	7648.00	0.371610	0.185805	−	0.982587i	$-0.440511\pi$
$751$	7648.00	0.371610	0.185805	+	0.982587i	$0.440511\pi$
$752$	0	0
$753$	−2496.00	−0.120796
$754$	0	0
$755$	15400.0	0.742336
$756$	0	0
$757$	−30190.0	−1.44950	−0.724752	−	0.689010i	$-0.758046\pi$
$757$	−30190.0	−1.44950	−0.724752	+	0.689010i	$0.758046\pi$
$758$	0	0
$759$	−1440.00	−0.0688652
$760$	0	0
$761$	−1242.00	−0.0591622	−0.0295811	−	0.999562i	$-0.509417\pi$
$761$	−1242.00	−0.0591622	−0.0295811	+	0.999562i	$0.509417\pi$
$762$	0	0
$763$	−16588.0	−0.787059
$764$	0	0
$765$	−3630.00	−0.171559
$766$	0	0
$767$	4512.00	0.212411
$768$	0	0
$769$	−28738.0	−1.34762	−0.673809	−	0.738905i	$-0.735343\pi$
$769$	−28738.0	−1.34762	−0.673809	+	0.738905i	$0.735343\pi$
$770$	0	0
$771$	−480.000	−0.0224212
$772$	0	0
$773$	−40128.0	−1.86715	−0.933573	−	0.358387i	$-0.883327\pi$
$773$	−40128.0	−1.86715	−0.933573	+	0.358387i	$0.883327\pi$
$774$	0	0
$775$	1600.00	0.0741596
$776$	0	0
$777$	1408.00	0.0650086
$778$	0	0
$779$	−1026.00	−0.0471890
$780$	0	0
$781$	11808.0	0.541003
$782$	0	0
$783$	−912.000	−0.0416248
$784$	0	0
$785$	−9190.00	−0.417841
$786$	0	0
$787$	15448.0	0.699697	0.349849	−	0.936806i	$-0.386233\pi$
$787$	15448.0	0.699697	0.349849	+	0.936806i	$0.386233\pi$
$788$	0	0
$789$	7368.00	0.332456
$790$	0	0
$791$	−264.000	−0.0118670
$792$	0	0
$793$	−5648.00	−0.252921
$794$	0	0
$795$	7920.00	0.353325
$796$	0	0
$797$	−27324.0	−1.21439	−0.607193	−	0.794554i	$-0.707705\pi$
$797$	−27324.0	−1.21439	−0.607193	+	0.794554i	$0.707705\pi$
$798$	0	0
$799$	39204.0	1.73584
$800$	0	0
$801$	−10098.0	−0.445437
$802$	0	0
$803$	168.000	0.00738305
$804$	0	0
$805$	−3300.00	−0.144484
$806$	0	0
$807$	12936.0	0.564274
$808$	0	0
$809$	−17766.0	−0.772088	−0.386044	−	0.922480i	$-0.626159\pi$
$809$	−17766.0	−0.772088	−0.386044	+	0.922480i	$0.626159\pi$
$810$	0	0
$811$	7396.00	0.320233	0.160116	−	0.987098i	$-0.448813\pi$
$811$	7396.00	0.320233	0.160116	+	0.987098i	$0.448813\pi$
$812$	0	0
$813$	18656.0	0.804790
$814$	0	0
$815$	−4250.00	−0.182664
$816$	0	0
$817$	3458.00	0.148078
$818$	0	0
$819$	−1936.00	−0.0825999
$820$	0	0
$821$	26898.0	1.14342	0.571709	−	0.820456i	$-0.306281\pi$
$821$	26898.0	1.14342	0.571709	+	0.820456i	$0.306281\pi$
$822$	0	0
$823$	24442.0	1.03523	0.517615	−	0.855614i	$-0.326820\pi$
$823$	24442.0	1.03523	0.517615	+	0.855614i	$0.326820\pi$
$824$	0	0
$825$	−1200.00	−0.0506408
$826$	0	0
$827$	−13524.0	−0.568652	−0.284326	−	0.958728i	$-0.591770\pi$
$827$	−13524.0	−0.568652	−0.284326	+	0.958728i	$0.591770\pi$
$828$	0	0
$829$	−7714.00	−0.323183	−0.161591	−	0.986858i	$-0.551663\pi$
$829$	−7714.00	−0.323183	−0.161591	+	0.986858i	$0.551663\pi$
$830$	0	0
$831$	−20888.0	−0.871958
$832$	0	0
$833$	−9306.00	−0.387075
$834$	0	0
$835$	19020.0	0.788281
$836$	0	0
$837$	9728.00	0.401731
$838$	0	0
$839$	16248.0	0.668586	0.334293	−	0.942469i	$-0.391503\pi$
$839$	16248.0	0.668586	0.334293	+	0.942469i	$0.391503\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	−30264.0	−1.23647
$844$	0	0
$845$	10665.0	0.434186
$846$	0	0
$847$	−26114.0	−1.05937
$848$	0	0
$849$	−22456.0	−0.907760
$850$	0	0
$851$	−480.000	−0.0193351
$852$	0	0
$853$	35498.0	1.42489	0.712443	−	0.701730i	$-0.247589\pi$
$853$	35498.0	1.42489	0.712443	+	0.701730i	$0.247589\pi$
$854$	0	0
$855$	−1045.00	−0.0417991
$856$	0	0
$857$	−40344.0	−1.60808	−0.804040	−	0.594575i	$-0.797320\pi$
$857$	−40344.0	−1.60808	−0.804040	+	0.594575i	$0.797320\pi$
$858$	0	0
$859$	−31484.0	−1.25055	−0.625274	−	0.780406i	$-0.715013\pi$
$859$	−31484.0	−1.25055	−0.625274	+	0.780406i	$0.715013\pi$
$860$	0	0
$861$	−4752.00	−0.188093
$862$	0	0
$863$	28836.0	1.13741	0.568707	−	0.822540i	$-0.307444\pi$
$863$	28836.0	1.13741	0.568707	+	0.822540i	$0.307444\pi$
$864$	0	0
$865$	2820.00	0.110847
$866$	0	0
$867$	2228.00	0.0872743
$868$	0	0
$869$	3936.00	0.153647
$870$	0	0
$871$	5024.00	0.195444
$872$	0	0
$873$	17204.0	0.666973
$874$	0	0
$875$	−2750.00	−0.106248
$876$	0	0
$877$	22796.0	0.877727	0.438863	−	0.898554i	$-0.355381\pi$
$877$	22796.0	0.877727	0.438863	+	0.898554i	$0.355381\pi$
$878$	0	0
$879$	−3696.00	−0.141824
$880$	0	0
$881$	−18822.0	−0.719784	−0.359892	−	0.932994i	$-0.617186\pi$
$881$	−18822.0	−0.719784	−0.359892	+	0.932994i	$0.617186\pi$
$882$	0	0
$883$	−7526.00	−0.286829	−0.143415	−	0.989663i	$-0.545808\pi$
$883$	−7526.00	−0.286829	−0.143415	+	0.989663i	$0.545808\pi$
$884$	0	0
$885$	11280.0	0.428444
$886$	0	0
$887$	−33816.0	−1.28008	−0.640040	−	0.768342i	$-0.721082\pi$
$887$	−33816.0	−1.28008	−0.640040	+	0.768342i	$0.721082\pi$
$888$	0	0
$889$	−7568.00	−0.285515
$890$	0	0
$891$	−3732.00	−0.140322
$892$	0	0
$893$	11286.0	0.422925
$894$	0	0
$895$	−9060.00	−0.338371
$896$	0	0
$897$	−960.000	−0.0357341
$898$	0	0
$899$	−384.000	−0.0142460
$900$	0	0
$901$	−26136.0	−0.966389
$902$	0	0
$903$	16016.0	0.590232
$904$	0	0
$905$	22490.0	0.826069
$906$	0	0
$907$	33784.0	1.23680	0.618401	−	0.785863i	$-0.287781\pi$
$907$	33784.0	1.23680	0.618401	+	0.785863i	$0.287781\pi$
$908$	0	0
$909$	3234.00	0.118003
$910$	0	0
$911$	15216.0	0.553379	0.276690	−	0.960959i	$-0.410763\pi$
$911$	15216.0	0.553379	0.276690	+	0.960959i	$0.410763\pi$
$912$	0	0
$913$	3528.00	0.127886
$914$	0	0
$915$	−14120.0	−0.510156
$916$	0	0
$917$	−55440.0	−1.99650
$918$	0	0
$919$	−19760.0	−0.709273	−0.354637	−	0.935004i	$-0.615395\pi$
$919$	−19760.0	−0.709273	−0.354637	+	0.935004i	$0.615395\pi$
$920$	0	0
$921$	−17920.0	−0.641134
$922$	0	0
$923$	7872.00	0.280726
$924$	0	0
$925$	−400.000	−0.0142183
$926$	0	0
$927$	8272.00	0.293083
$928$	0	0
$929$	16278.0	0.574880	0.287440	−	0.957799i	$-0.407196\pi$
$929$	16278.0	0.574880	0.287440	+	0.957799i	$0.407196\pi$
$930$	0	0
$931$	−2679.00	−0.0943079
$932$	0	0
$933$	5088.00	0.178536
$934$	0	0
$935$	3960.00	0.138509
$936$	0	0
$937$	−6994.00	−0.243846	−0.121923	−	0.992540i	$-0.538906\pi$
$937$	−6994.00	−0.243846	−0.121923	+	0.992540i	$0.538906\pi$
$938$	0	0
$939$	−5480.00	−0.190451
$940$	0	0
$941$	32502.0	1.12597	0.562983	−	0.826468i	$-0.309653\pi$
$941$	32502.0	1.12597	0.562983	+	0.826468i	$0.309653\pi$
$942$	0	0
$943$	1620.00	0.0559432
$944$	0	0
$945$	−16720.0	−0.575557
$946$	0	0
$947$	−50358.0	−1.72800	−0.864000	−	0.503493i	$-0.832048\pi$
$947$	−50358.0	−1.72800	−0.864000	+	0.503493i	$0.832048\pi$
$948$	0	0
$949$	112.000	0.00383106
$950$	0	0
$951$	−14208.0	−0.484465
$952$	0	0
$953$	39816.0	1.35338	0.676688	−	0.736270i	$-0.263415\pi$
$953$	39816.0	1.35338	0.676688	+	0.736270i	$0.263415\pi$
$954$	0	0
$955$	17940.0	0.607879
$956$	0	0
$957$	288.000	0.00972802
$958$	0	0
$959$	14388.0	0.484476
$960$	0	0
$961$	−25695.0	−0.862509
$962$	0	0
$963$	−2376.00	−0.0795073
$964$	0	0
$965$	22460.0	0.749236
$966$	0	0
$967$	−590.000	−0.0196206	−0.00981030	−	0.999952i	$-0.503123\pi$
$967$	−590.000	−0.0196206	−0.00981030	+	0.999952i	$0.503123\pi$
$968$	0	0
$969$	−5016.00	−0.166292
$970$	0	0
$971$	−26820.0	−0.886400	−0.443200	−	0.896423i	$-0.646157\pi$
$971$	−26820.0	−0.886400	−0.443200	+	0.896423i	$0.646157\pi$
$972$	0	0
$973$	52624.0	1.73386
$974$	0	0
$975$	−800.000	−0.0262774
$976$	0	0
$977$	−33312.0	−1.09083	−0.545417	−	0.838165i	$-0.683629\pi$
$977$	−33312.0	−1.09083	−0.545417	+	0.838165i	$0.683629\pi$
$978$	0	0
$979$	11016.0	0.359625
$980$	0	0
$981$	8294.00	0.269936
$982$	0	0
$983$	−612.000	−0.0198573	−0.00992867	−	0.999951i	$-0.503160\pi$
$983$	−612.000	−0.0198573	−0.00992867	+	0.999951i	$0.503160\pi$
$984$	0	0
$985$	−12330.0	−0.398849
$986$	0	0
$987$	52272.0	1.68575
$988$	0	0
$989$	−5460.00	−0.175549
$990$	0	0
$991$	−39416.0	−1.26346	−0.631731	−	0.775188i	$-0.717655\pi$
$991$	−39416.0	−1.26346	−0.631731	+	0.775188i	$0.717655\pi$
$992$	0	0
$993$	18128.0	0.579330
$994$	0	0
$995$	4120.00	0.131269
$996$	0	0
$997$	36614.0	1.16307	0.581533	−	0.813523i	$-0.302453\pi$
$997$	36614.0	1.16307	0.581533	+	0.813523i	$0.302453\pi$
$998$	0	0
$999$	−2432.00	−0.0770221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.4.a.b.1.1		1
4.3	odd	2		95.4.a.a.1.1	✓	1
12.11	even	2		855.4.a.e.1.1		1
20.3	even	4		475.4.b.e.324.2		2
20.7	even	4		475.4.b.e.324.1		2
20.19	odd	2		475.4.a.d.1.1		1
76.75	even	2		1805.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.4.a.a.1.1	✓	1	4.3	odd	2
475.4.a.d.1.1		1	20.19	odd	2
475.4.b.e.324.1		2	20.7	even	4
475.4.b.e.324.2		2	20.3	even	4
855.4.a.e.1.1		1	12.11	even	2
1520.4.a.b.1.1		1	1.1	even	1	trivial
1805.4.a.f.1.1		1	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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