LMFDB - Embedded newform 1520.4.a.g.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
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)

$f(q)$

$=$

$q+4.00000 q^{3} +5.00000 q^{5} +20.0000 q^{7} -11.0000 q^{9} +O(q^{10})$

$q+4.00000 q^{3} +5.00000 q^{5} +20.0000 q^{7} -11.0000 q^{9} +44.0000 q^{11} +42.0000 q^{13} +20.0000 q^{15} -86.0000 q^{17} -19.0000 q^{19} +80.0000 q^{21} +164.000 q^{23} +25.0000 q^{25} -152.000 q^{27} -162.000 q^{29} +312.000 q^{31} +176.000 q^{33} +100.000 q^{35} +226.000 q^{37} +168.000 q^{39} +34.0000 q^{41} +432.000 q^{43} -55.0000 q^{45} -580.000 q^{47} +57.0000 q^{49} -344.000 q^{51} +506.000 q^{53} +220.000 q^{55} -76.0000 q^{57} -364.000 q^{59} +518.000 q^{61} -220.000 q^{63} +210.000 q^{65} -924.000 q^{67} +656.000 q^{69} -320.000 q^{71} -542.000 q^{73} +100.000 q^{75} +880.000 q^{77} +1208.00 q^{79} -311.000 q^{81} +1120.00 q^{83} -430.000 q^{85} -648.000 q^{87} -1022.00 q^{89} +840.000 q^{91} +1248.00 q^{93} -95.0000 q^{95} +1166.00 q^{97} -484.000 q^{99} +O(q^{100})$

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	$-0.374236\pi$
$3$	4.00000	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	20.0000	1.07990	0.539949	−	0.841698i	$-0.318443\pi$
$7$	20.0000	1.07990	0.539949	+	0.841698i	$0.318443\pi$
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	44.0000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	44.0000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	42.0000	0.896054	0.448027	−	0.894020i	$-0.352127\pi$
$13$	42.0000	0.896054	0.448027	+	0.894020i	$0.352127\pi$
$14$	0	0
$15$	20.0000	0.344265
$16$	0	0
$17$	−86.0000	−1.22694	−0.613472	−	0.789716i	$-0.710228\pi$
$17$	−86.0000	−1.22694	−0.613472	+	0.789716i	$0.710228\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	80.0000	0.831306
$22$	0	0
$23$	164.000	1.48680	0.743399	−	0.668848i	$-0.233212\pi$
$23$	164.000	1.48680	0.743399	+	0.668848i	$0.233212\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−152.000	−1.08342
$28$	0	0
$29$	−162.000	−1.03733	−0.518666	−	0.854977i	$-0.673571\pi$
$29$	−162.000	−1.03733	−0.518666	+	0.854977i	$0.673571\pi$
$30$	0	0
$31$	312.000	1.80764	0.903820	−	0.427912i	$-0.140751\pi$
$31$	312.000	1.80764	0.903820	+	0.427912i	$0.140751\pi$
$32$	0	0
$33$	176.000	0.928414
$34$	0	0
$35$	100.000	0.482945
$36$	0	0
$37$	226.000	1.00417	0.502083	−	0.864819i	$-0.332567\pi$
$37$	226.000	1.00417	0.502083	+	0.864819i	$0.332567\pi$
$38$	0	0
$39$	168.000	0.689783
$40$	0	0
$41$	34.0000	0.129510	0.0647550	−	0.997901i	$-0.479373\pi$
$41$	34.0000	0.129510	0.0647550	+	0.997901i	$0.479373\pi$
$42$	0	0
$43$	432.000	1.53208	0.766039	−	0.642794i	$-0.222225\pi$
$43$	432.000	1.53208	0.766039	+	0.642794i	$0.222225\pi$
$44$	0	0
$45$	−55.0000	−0.182198
$46$	0	0
$47$	−580.000	−1.80004	−0.900018	−	0.435853i	$-0.856447\pi$
$47$	−580.000	−1.80004	−0.900018	+	0.435853i	$0.856447\pi$
$48$	0	0
$49$	57.0000	0.166181
$50$	0	0
$51$	−344.000	−0.944503
$52$	0	0
$53$	506.000	1.31140	0.655702	−	0.755020i	$-0.272373\pi$
$53$	506.000	1.31140	0.655702	+	0.755020i	$0.272373\pi$
$54$	0	0
$55$	220.000	0.539360
$56$	0	0
$57$	−76.0000	−0.176604
$58$	0	0
$59$	−364.000	−0.803199	−0.401600	−	0.915815i	$-0.631546\pi$
$59$	−364.000	−0.803199	−0.401600	+	0.915815i	$0.631546\pi$
$60$	0	0
$61$	518.000	1.08726	0.543632	−	0.839324i	$-0.317049\pi$
$61$	518.000	1.08726	0.543632	+	0.839324i	$0.317049\pi$
$62$	0	0
$63$	−220.000	−0.439959
$64$	0	0
$65$	210.000	0.400728
$66$	0	0
$67$	−924.000	−1.68484	−0.842422	−	0.538818i	$-0.818871\pi$
$67$	−924.000	−1.68484	−0.842422	+	0.538818i	$0.818871\pi$
$68$	0	0
$69$	656.000	1.14454
$70$	0	0
$71$	−320.000	−0.534888	−0.267444	−	0.963573i	$-0.586179\pi$
$71$	−320.000	−0.534888	−0.267444	+	0.963573i	$0.586179\pi$
$72$	0	0
$73$	−542.000	−0.868990	−0.434495	−	0.900674i	$-0.643073\pi$
$73$	−542.000	−0.868990	−0.434495	+	0.900674i	$0.643073\pi$
$74$	0	0
$75$	100.000	0.153960
$76$	0	0
$77$	880.000	1.30241
$78$	0	0
$79$	1208.00	1.72039	0.860194	−	0.509967i	$-0.170343\pi$
$79$	1208.00	1.72039	0.860194	+	0.509967i	$0.170343\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	1120.00	1.48116	0.740578	−	0.671970i	$-0.234552\pi$
$83$	1120.00	1.48116	0.740578	+	0.671970i	$0.234552\pi$
$84$	0	0
$85$	−430.000	−0.548706
$86$	0	0
$87$	−648.000	−0.798539
$88$	0	0
$89$	−1022.00	−1.21721	−0.608606	−	0.793473i	$-0.708271\pi$
$89$	−1022.00	−1.21721	−0.608606	+	0.793473i	$0.708271\pi$
$90$	0	0
$91$	840.000	0.967648
$92$	0	0
$93$	1248.00	1.39152
$94$	0	0
$95$	−95.0000	−0.102598
$96$	0	0
$97$	1166.00	1.22051	0.610254	−	0.792205i	$-0.291067\pi$
$97$	1166.00	1.22051	0.610254	+	0.792205i	$0.291067\pi$
$98$	0	0
$99$	−484.000	−0.491352
$100$	0	0
$101$	−898.000	−0.884696	−0.442348	−	0.896843i	$-0.645854\pi$
$101$	−898.000	−0.884696	−0.442348	+	0.896843i	$0.645854\pi$
$102$	0	0
$103$	−1152.00	−1.10204	−0.551019	−	0.834493i	$-0.685761\pi$
$103$	−1152.00	−1.10204	−0.551019	+	0.834493i	$0.685761\pi$
$104$	0	0
$105$	400.000	0.371771
$106$	0	0
$107$	1156.00	1.04444	0.522218	−	0.852812i	$-0.325105\pi$
$107$	1156.00	1.04444	0.522218	+	0.852812i	$0.325105\pi$
$108$	0	0
$109$	1046.00	0.919162	0.459581	−	0.888136i	$-0.348000\pi$
$109$	1046.00	0.919162	0.459581	+	0.888136i	$0.348000\pi$
$110$	0	0
$111$	904.000	0.773008
$112$	0	0
$113$	−530.000	−0.441223	−0.220612	−	0.975362i	$-0.570805\pi$
$113$	−530.000	−0.441223	−0.220612	+	0.975362i	$0.570805\pi$
$114$	0	0
$115$	820.000	0.664916
$116$	0	0
$117$	−462.000	−0.365059
$118$	0	0
$119$	−1720.00	−1.32498
$120$	0	0
$121$	605.000	0.454545
$122$	0	0
$123$	136.000	0.0996968
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−568.000	−0.396865	−0.198432	−	0.980115i	$-0.563585\pi$
$127$	−568.000	−0.396865	−0.198432	+	0.980115i	$0.563585\pi$
$128$	0	0
$129$	1728.00	1.17939
$130$	0	0
$131$	924.000	0.616261	0.308131	−	0.951344i	$-0.400297\pi$
$131$	924.000	0.616261	0.308131	+	0.951344i	$0.400297\pi$
$132$	0	0
$133$	−380.000	−0.247746
$134$	0	0
$135$	−760.000	−0.484521
$136$	0	0
$137$	−702.000	−0.437780	−0.218890	−	0.975750i	$-0.570244\pi$
$137$	−702.000	−0.437780	−0.218890	+	0.975750i	$0.570244\pi$
$138$	0	0
$139$	2116.00	1.29120	0.645600	−	0.763676i	$-0.276607\pi$
$139$	2116.00	1.29120	0.645600	+	0.763676i	$0.276607\pi$
$140$	0	0
$141$	−2320.00	−1.38567
$142$	0	0
$143$	1848.00	1.08068
$144$	0	0
$145$	−810.000	−0.463909
$146$	0	0
$147$	228.000	0.127926
$148$	0	0
$149$	1454.00	0.799438	0.399719	−	0.916638i	$-0.369108\pi$
$149$	1454.00	0.799438	0.399719	+	0.916638i	$0.369108\pi$
$150$	0	0
$151$	680.000	0.366474	0.183237	−	0.983069i	$-0.441342\pi$
$151$	680.000	0.366474	0.183237	+	0.983069i	$0.441342\pi$
$152$	0	0
$153$	946.000	0.499866
$154$	0	0
$155$	1560.00	0.808401
$156$	0	0
$157$	462.000	0.234851	0.117426	−	0.993082i	$-0.462536\pi$
$157$	462.000	0.234851	0.117426	+	0.993082i	$0.462536\pi$
$158$	0	0
$159$	2024.00	1.00952
$160$	0	0
$161$	3280.00	1.60559
$162$	0	0
$163$	−4136.00	−1.98746	−0.993732	−	0.111792i	$-0.964341\pi$
$163$	−4136.00	−1.98746	−0.993732	+	0.111792i	$0.964341\pi$
$164$	0	0
$165$	880.000	0.415199
$166$	0	0
$167$	1464.00	0.678370	0.339185	−	0.940720i	$-0.389849\pi$
$167$	1464.00	0.678370	0.339185	+	0.940720i	$0.389849\pi$
$168$	0	0
$169$	−433.000	−0.197087
$170$	0	0
$171$	209.000	0.0934657
$172$	0	0
$173$	−1422.00	−0.624929	−0.312464	−	0.949929i	$-0.601154\pi$
$173$	−1422.00	−0.624929	−0.312464	+	0.949929i	$0.601154\pi$
$174$	0	0
$175$	500.000	0.215980
$176$	0	0
$177$	−1456.00	−0.618303
$178$	0	0
$179$	3212.00	1.34121	0.670604	−	0.741816i	$-0.266035\pi$
$179$	3212.00	1.34121	0.670604	+	0.741816i	$0.266035\pi$
$180$	0	0
$181$	−3234.00	−1.32807	−0.664037	−	0.747700i	$-0.731158\pi$
$181$	−3234.00	−1.32807	−0.664037	+	0.747700i	$0.731158\pi$
$182$	0	0
$183$	2072.00	0.836976
$184$	0	0
$185$	1130.00	0.449077
$186$	0	0
$187$	−3784.00	−1.47975
$188$	0	0
$189$	−3040.00	−1.16999
$190$	0	0
$191$	296.000	0.112135	0.0560676	−	0.998427i	$-0.482144\pi$
$191$	296.000	0.112135	0.0560676	+	0.998427i	$0.482144\pi$
$192$	0	0
$193$	−2770.00	−1.03310	−0.516552	−	0.856256i	$-0.672785\pi$
$193$	−2770.00	−1.03310	−0.516552	+	0.856256i	$0.672785\pi$
$194$	0	0
$195$	840.000	0.308480
$196$	0	0
$197$	−1154.00	−0.417356	−0.208678	−	0.977984i	$-0.566916\pi$
$197$	−1154.00	−0.417356	−0.208678	+	0.977984i	$0.566916\pi$
$198$	0	0
$199$	2680.00	0.954674	0.477337	−	0.878720i	$-0.341602\pi$
$199$	2680.00	0.954674	0.477337	+	0.878720i	$0.341602\pi$
$200$	0	0
$201$	−3696.00	−1.29699
$202$	0	0
$203$	−3240.00	−1.12021
$204$	0	0
$205$	170.000	0.0579186
$206$	0	0
$207$	−1804.00	−0.605733
$208$	0	0
$209$	−836.000	−0.276686
$210$	0	0
$211$	2180.00	0.711267	0.355634	−	0.934625i	$-0.384265\pi$
$211$	2180.00	0.711267	0.355634	+	0.934625i	$0.384265\pi$
$212$	0	0
$213$	−1280.00	−0.411757
$214$	0	0
$215$	2160.00	0.685166
$216$	0	0
$217$	6240.00	1.95207
$218$	0	0
$219$	−2168.00	−0.668949
$220$	0	0
$221$	−3612.00	−1.09941
$222$	0	0
$223$	4296.00	1.29005	0.645026	−	0.764161i	$-0.276847\pi$
$223$	4296.00	1.29005	0.645026	+	0.764161i	$0.276847\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	−500.000	−0.146195	−0.0730973	−	0.997325i	$-0.523288\pi$
$227$	−500.000	−0.146195	−0.0730973	+	0.997325i	$0.523288\pi$
$228$	0	0
$229$	4366.00	1.25988	0.629942	−	0.776642i	$-0.283079\pi$
$229$	4366.00	1.25988	0.629942	+	0.776642i	$0.283079\pi$
$230$	0	0
$231$	3520.00	1.00259
$232$	0	0
$233$	2970.00	0.835069	0.417535	−	0.908661i	$-0.362894\pi$
$233$	2970.00	0.835069	0.417535	+	0.908661i	$0.362894\pi$
$234$	0	0
$235$	−2900.00	−0.805001
$236$	0	0
$237$	4832.00	1.32435
$238$	0	0
$239$	144.000	0.0389732	0.0194866	−	0.999810i	$-0.493797\pi$
$239$	144.000	0.0389732	0.0194866	+	0.999810i	$0.493797\pi$
$240$	0	0
$241$	1738.00	0.464541	0.232271	−	0.972651i	$-0.425385\pi$
$241$	1738.00	0.464541	0.232271	+	0.972651i	$0.425385\pi$
$242$	0	0
$243$	2860.00	0.755017
$244$	0	0
$245$	285.000	0.0743183
$246$	0	0
$247$	−798.000	−0.205569
$248$	0	0
$249$	4480.00	1.14019
$250$	0	0
$251$	3012.00	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	3012.00	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	7216.00	1.79315
$254$	0	0
$255$	−1720.00	−0.422394
$256$	0	0
$257$	1014.00	0.246115	0.123058	−	0.992400i	$-0.460730\pi$
$257$	1014.00	0.246115	0.123058	+	0.992400i	$0.460730\pi$
$258$	0	0
$259$	4520.00	1.08440
$260$	0	0
$261$	1782.00	0.422617
$262$	0	0
$263$	−4284.00	−1.00442	−0.502211	−	0.864745i	$-0.667480\pi$
$263$	−4284.00	−1.00442	−0.502211	+	0.864745i	$0.667480\pi$
$264$	0	0
$265$	2530.00	0.586478
$266$	0	0
$267$	−4088.00	−0.937010
$268$	0	0
$269$	38.0000	0.00861301	0.00430651	−	0.999991i	$-0.498629\pi$
$269$	38.0000	0.00861301	0.00430651	+	0.999991i	$0.498629\pi$
$270$	0	0
$271$	−5888.00	−1.31982	−0.659909	−	0.751346i	$-0.729405\pi$
$271$	−5888.00	−1.31982	−0.659909	+	0.751346i	$0.729405\pi$
$272$	0	0
$273$	3360.00	0.744895
$274$	0	0
$275$	1100.00	0.241209
$276$	0	0
$277$	5254.00	1.13965	0.569824	−	0.821767i	$-0.307012\pi$
$277$	5254.00	1.13965	0.569824	+	0.821767i	$0.307012\pi$
$278$	0	0
$279$	−3432.00	−0.736446
$280$	0	0
$281$	−2558.00	−0.543052	−0.271526	−	0.962431i	$-0.587528\pi$
$281$	−2558.00	−0.543052	−0.271526	+	0.962431i	$0.587528\pi$
$282$	0	0
$283$	6776.00	1.42329	0.711646	−	0.702539i	$-0.247950\pi$
$283$	6776.00	1.42329	0.711646	+	0.702539i	$0.247950\pi$
$284$	0	0
$285$	−380.000	−0.0789799
$286$	0	0
$287$	680.000	0.139858
$288$	0	0
$289$	2483.00	0.505394
$290$	0	0
$291$	4664.00	0.939548
$292$	0	0
$293$	−2086.00	−0.415923	−0.207961	−	0.978137i	$-0.566683\pi$
$293$	−2086.00	−0.415923	−0.207961	+	0.978137i	$0.566683\pi$
$294$	0	0
$295$	−1820.00	−0.359202
$296$	0	0
$297$	−6688.00	−1.30666
$298$	0	0
$299$	6888.00	1.33225
$300$	0	0
$301$	8640.00	1.65449
$302$	0	0
$303$	−3592.00	−0.681040
$304$	0	0
$305$	2590.00	0.486239
$306$	0	0
$307$	−7428.00	−1.38091	−0.690453	−	0.723377i	$-0.742589\pi$
$307$	−7428.00	−1.38091	−0.690453	+	0.723377i	$0.742589\pi$
$308$	0	0
$309$	−4608.00	−0.848349
$310$	0	0
$311$	4040.00	0.736615	0.368308	−	0.929704i	$-0.379937\pi$
$311$	4040.00	0.736615	0.368308	+	0.929704i	$0.379937\pi$
$312$	0	0
$313$	−3318.00	−0.599184	−0.299592	−	0.954067i	$-0.596850\pi$
$313$	−3318.00	−0.599184	−0.299592	+	0.954067i	$0.596850\pi$
$314$	0	0
$315$	−1100.00	−0.196755
$316$	0	0
$317$	−7302.00	−1.29376	−0.646879	−	0.762593i	$-0.723926\pi$
$317$	−7302.00	−1.29376	−0.646879	+	0.762593i	$0.723926\pi$
$318$	0	0
$319$	−7128.00	−1.25107
$320$	0	0
$321$	4624.00	0.804008
$322$	0	0
$323$	1634.00	0.281480
$324$	0	0
$325$	1050.00	0.179211
$326$	0	0
$327$	4184.00	0.707571
$328$	0	0
$329$	−11600.0	−1.94386
$330$	0	0
$331$	−1428.00	−0.237130	−0.118565	−	0.992946i	$-0.537829\pi$
$331$	−1428.00	−0.237130	−0.118565	+	0.992946i	$0.537829\pi$
$332$	0	0
$333$	−2486.00	−0.409105
$334$	0	0
$335$	−4620.00	−0.753485
$336$	0	0
$337$	6302.00	1.01867	0.509335	−	0.860568i	$-0.329891\pi$
$337$	6302.00	1.01867	0.509335	+	0.860568i	$0.329891\pi$
$338$	0	0
$339$	−2120.00	−0.339654
$340$	0	0
$341$	13728.0	2.18010
$342$	0	0
$343$	−5720.00	−0.900440
$344$	0	0
$345$	3280.00	0.511853
$346$	0	0
$347$	−2608.00	−0.403472	−0.201736	−	0.979440i	$-0.564658\pi$
$347$	−2608.00	−0.403472	−0.201736	+	0.979440i	$0.564658\pi$
$348$	0	0
$349$	−8234.00	−1.26291	−0.631455	−	0.775412i	$-0.717542\pi$
$349$	−8234.00	−1.26291	−0.631455	+	0.775412i	$0.717542\pi$
$350$	0	0
$351$	−6384.00	−0.970805
$352$	0	0
$353$	−7254.00	−1.09374	−0.546872	−	0.837216i	$-0.684181\pi$
$353$	−7254.00	−1.09374	−0.546872	+	0.837216i	$0.684181\pi$
$354$	0	0
$355$	−1600.00	−0.239209
$356$	0	0
$357$	−6880.00	−1.01997
$358$	0	0
$359$	−6000.00	−0.882083	−0.441042	−	0.897487i	$-0.645391\pi$
$359$	−6000.00	−0.882083	−0.441042	+	0.897487i	$0.645391\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	2420.00	0.349909
$364$	0	0
$365$	−2710.00	−0.388624
$366$	0	0
$367$	10204.0	1.45135	0.725674	−	0.688039i	$-0.241528\pi$
$367$	10204.0	1.45135	0.725674	+	0.688039i	$0.241528\pi$
$368$	0	0
$369$	−374.000	−0.0527633
$370$	0	0
$371$	10120.0	1.41618
$372$	0	0
$373$	−3094.00	−0.429494	−0.214747	−	0.976670i	$-0.568893\pi$
$373$	−3094.00	−0.429494	−0.214747	+	0.976670i	$0.568893\pi$
$374$	0	0
$375$	500.000	0.0688530
$376$	0	0
$377$	−6804.00	−0.929506
$378$	0	0
$379$	−2708.00	−0.367020	−0.183510	−	0.983018i	$-0.558746\pi$
$379$	−2708.00	−0.367020	−0.183510	+	0.983018i	$0.558746\pi$
$380$	0	0
$381$	−2272.00	−0.305507
$382$	0	0
$383$	8272.00	1.10360	0.551801	−	0.833976i	$-0.313941\pi$
$383$	8272.00	1.10360	0.551801	+	0.833976i	$0.313941\pi$
$384$	0	0
$385$	4400.00	0.582454
$386$	0	0
$387$	−4752.00	−0.624180
$388$	0	0
$389$	−4066.00	−0.529960	−0.264980	−	0.964254i	$-0.585365\pi$
$389$	−4066.00	−0.529960	−0.264980	+	0.964254i	$0.585365\pi$
$390$	0	0
$391$	−14104.0	−1.82422
$392$	0	0
$393$	3696.00	0.474398
$394$	0	0
$395$	6040.00	0.769381
$396$	0	0
$397$	15438.0	1.95167	0.975833	−	0.218520i	$-0.0701228\pi$
$397$	15438.0	1.95167	0.975833	+	0.218520i	$0.0701228\pi$
$398$	0	0
$399$	−1520.00	−0.190715
$400$	0	0
$401$	14514.0	1.80747	0.903734	−	0.428095i	$-0.140815\pi$
$401$	14514.0	1.80747	0.903734	+	0.428095i	$0.140815\pi$
$402$	0	0
$403$	13104.0	1.61974
$404$	0	0
$405$	−1555.00	−0.190787
$406$	0	0
$407$	9944.00	1.21107
$408$	0	0
$409$	7818.00	0.945172	0.472586	−	0.881285i	$-0.343321\pi$
$409$	7818.00	0.945172	0.472586	+	0.881285i	$0.343321\pi$
$410$	0	0
$411$	−2808.00	−0.337003
$412$	0	0
$413$	−7280.00	−0.867374
$414$	0	0
$415$	5600.00	0.662393
$416$	0	0
$417$	8464.00	0.993966
$418$	0	0
$419$	3684.00	0.429535	0.214768	−	0.976665i	$-0.431101\pi$
$419$	3684.00	0.429535	0.214768	+	0.976665i	$0.431101\pi$
$420$	0	0
$421$	−5834.00	−0.675372	−0.337686	−	0.941259i	$-0.609644\pi$
$421$	−5834.00	−0.675372	−0.337686	+	0.941259i	$0.609644\pi$
$422$	0	0
$423$	6380.00	0.733348
$424$	0	0
$425$	−2150.00	−0.245389
$426$	0	0
$427$	10360.0	1.17413
$428$	0	0
$429$	7392.00	0.831909
$430$	0	0
$431$	−6944.00	−0.776057	−0.388029	−	0.921647i	$-0.626844\pi$
$431$	−6944.00	−0.776057	−0.388029	+	0.921647i	$0.626844\pi$
$432$	0	0
$433$	−738.000	−0.0819077	−0.0409538	−	0.999161i	$-0.513040\pi$
$433$	−738.000	−0.0819077	−0.0409538	+	0.999161i	$0.513040\pi$
$434$	0	0
$435$	−3240.00	−0.357117
$436$	0	0
$437$	−3116.00	−0.341095
$438$	0	0
$439$	−544.000	−0.0591428	−0.0295714	−	0.999563i	$-0.509414\pi$
$439$	−544.000	−0.0591428	−0.0295714	+	0.999563i	$0.509414\pi$
$440$	0	0
$441$	−627.000	−0.0677033
$442$	0	0
$443$	792.000	0.0849414	0.0424707	−	0.999098i	$-0.486477\pi$
$443$	792.000	0.0849414	0.0424707	+	0.999098i	$0.486477\pi$
$444$	0	0
$445$	−5110.00	−0.544353
$446$	0	0
$447$	5816.00	0.615408
$448$	0	0
$449$	7362.00	0.773796	0.386898	−	0.922123i	$-0.373547\pi$
$449$	7362.00	0.773796	0.386898	+	0.922123i	$0.373547\pi$
$450$	0	0
$451$	1496.00	0.156195
$452$	0	0
$453$	2720.00	0.282112
$454$	0	0
$455$	4200.00	0.432745
$456$	0	0
$457$	−14278.0	−1.46148	−0.730740	−	0.682656i	$-0.760825\pi$
$457$	−14278.0	−1.46148	−0.730740	+	0.682656i	$0.760825\pi$
$458$	0	0
$459$	13072.0	1.32930
$460$	0	0
$461$	−1002.00	−0.101232	−0.0506158	−	0.998718i	$-0.516118\pi$
$461$	−1002.00	−0.101232	−0.0506158	+	0.998718i	$0.516118\pi$
$462$	0	0
$463$	−8916.00	−0.894950	−0.447475	−	0.894297i	$-0.647677\pi$
$463$	−8916.00	−0.894950	−0.447475	+	0.894297i	$0.647677\pi$
$464$	0	0
$465$	6240.00	0.622308
$466$	0	0
$467$	−6344.00	−0.628620	−0.314310	−	0.949320i	$-0.601773\pi$
$467$	−6344.00	−0.628620	−0.314310	+	0.949320i	$0.601773\pi$
$468$	0	0
$469$	−18480.0	−1.81946
$470$	0	0
$471$	1848.00	0.180788
$472$	0	0
$473$	19008.0	1.84776
$474$	0	0
$475$	−475.000	−0.0458831
$476$	0	0
$477$	−5566.00	−0.534276
$478$	0	0
$479$	1800.00	0.171700	0.0858498	−	0.996308i	$-0.472639\pi$
$479$	1800.00	0.171700	0.0858498	+	0.996308i	$0.472639\pi$
$480$	0	0
$481$	9492.00	0.899788
$482$	0	0
$483$	13120.0	1.23598
$484$	0	0
$485$	5830.00	0.545828
$486$	0	0
$487$	−3704.00	−0.344649	−0.172325	−	0.985040i	$-0.555128\pi$
$487$	−3704.00	−0.344649	−0.172325	+	0.985040i	$0.555128\pi$
$488$	0	0
$489$	−16544.0	−1.52995
$490$	0	0
$491$	−3468.00	−0.318755	−0.159377	−	0.987218i	$-0.550949\pi$
$491$	−3468.00	−0.318755	−0.159377	+	0.987218i	$0.550949\pi$
$492$	0	0
$493$	13932.0	1.27275
$494$	0	0
$495$	−2420.00	−0.219739
$496$	0	0
$497$	−6400.00	−0.577624
$498$	0	0
$499$	−5348.00	−0.479778	−0.239889	−	0.970800i	$-0.577111\pi$
$499$	−5348.00	−0.479778	−0.239889	+	0.970800i	$0.577111\pi$
$500$	0	0
$501$	5856.00	0.522209
$502$	0	0
$503$	2228.00	0.197498	0.0987491	−	0.995112i	$-0.468516\pi$
$503$	2228.00	0.197498	0.0987491	+	0.995112i	$0.468516\pi$
$504$	0	0
$505$	−4490.00	−0.395648
$506$	0	0
$507$	−1732.00	−0.151718
$508$	0	0
$509$	−18082.0	−1.57460	−0.787299	−	0.616571i	$-0.788521\pi$
$509$	−18082.0	−1.57460	−0.787299	+	0.616571i	$0.788521\pi$
$510$	0	0
$511$	−10840.0	−0.938421
$512$	0	0
$513$	2888.00	0.248554
$514$	0	0
$515$	−5760.00	−0.492846
$516$	0	0
$517$	−25520.0	−2.17093
$518$	0	0
$519$	−5688.00	−0.481070
$520$	0	0
$521$	−10166.0	−0.854857	−0.427429	−	0.904049i	$-0.640580\pi$
$521$	−10166.0	−0.854857	−0.427429	+	0.904049i	$0.640580\pi$
$522$	0	0
$523$	−6268.00	−0.524054	−0.262027	−	0.965060i	$-0.584391\pi$
$523$	−6268.00	−0.524054	−0.262027	+	0.965060i	$0.584391\pi$
$524$	0	0
$525$	2000.00	0.166261
$526$	0	0
$527$	−26832.0	−2.21788
$528$	0	0
$529$	14729.0	1.21057
$530$	0	0
$531$	4004.00	0.327229
$532$	0	0
$533$	1428.00	0.116048
$534$	0	0
$535$	5780.00	0.467086
$536$	0	0
$537$	12848.0	1.03246
$538$	0	0
$539$	2508.00	0.200422
$540$	0	0
$541$	−1594.00	−0.126675	−0.0633377	−	0.997992i	$-0.520175\pi$
$541$	−1594.00	−0.126675	−0.0633377	+	0.997992i	$0.520175\pi$
$542$	0	0
$543$	−12936.0	−1.02235
$544$	0	0
$545$	5230.00	0.411062
$546$	0	0
$547$	−7940.00	−0.620640	−0.310320	−	0.950632i	$-0.600436\pi$
$547$	−7940.00	−0.620640	−0.310320	+	0.950632i	$0.600436\pi$
$548$	0	0
$549$	−5698.00	−0.442959
$550$	0	0
$551$	3078.00	0.237980
$552$	0	0
$553$	24160.0	1.85784
$554$	0	0
$555$	4520.00	0.345700
$556$	0	0
$557$	−12626.0	−0.960468	−0.480234	−	0.877140i	$-0.659448\pi$
$557$	−12626.0	−0.960468	−0.480234	+	0.877140i	$0.659448\pi$
$558$	0	0
$559$	18144.0	1.37283
$560$	0	0
$561$	−15136.0	−1.13911
$562$	0	0
$563$	21660.0	1.62142	0.810711	−	0.585447i	$-0.199081\pi$
$563$	21660.0	1.62142	0.810711	+	0.585447i	$0.199081\pi$
$564$	0	0
$565$	−2650.00	−0.197321
$566$	0	0
$567$	−6220.00	−0.460697
$568$	0	0
$569$	14346.0	1.05697	0.528485	−	0.848943i	$-0.322760\pi$
$569$	14346.0	1.05697	0.528485	+	0.848943i	$0.322760\pi$
$570$	0	0
$571$	5404.00	0.396060	0.198030	−	0.980196i	$-0.436546\pi$
$571$	5404.00	0.396060	0.198030	+	0.980196i	$0.436546\pi$
$572$	0	0
$573$	1184.00	0.0863217
$574$	0	0
$575$	4100.00	0.297360
$576$	0	0
$577$	1426.00	0.102886	0.0514429	−	0.998676i	$-0.483618\pi$
$577$	1426.00	0.102886	0.0514429	+	0.998676i	$0.483618\pi$
$578$	0	0
$579$	−11080.0	−0.795283
$580$	0	0
$581$	22400.0	1.59950
$582$	0	0
$583$	22264.0	1.58161
$584$	0	0
$585$	−2310.00	−0.163259
$586$	0	0
$587$	18704.0	1.31516	0.657578	−	0.753386i	$-0.271581\pi$
$587$	18704.0	1.31516	0.657578	+	0.753386i	$0.271581\pi$
$588$	0	0
$589$	−5928.00	−0.414701
$590$	0	0
$591$	−4616.00	−0.321281
$592$	0	0
$593$	−14270.0	−0.988193	−0.494097	−	0.869407i	$-0.664501\pi$
$593$	−14270.0	−0.988193	−0.494097	+	0.869407i	$0.664501\pi$
$594$	0	0
$595$	−8600.00	−0.592547
$596$	0	0
$597$	10720.0	0.734909
$598$	0	0
$599$	−15296.0	−1.04337	−0.521684	−	0.853139i	$-0.674696\pi$
$599$	−15296.0	−1.04337	−0.521684	+	0.853139i	$0.674696\pi$
$600$	0	0
$601$	10002.0	0.678852	0.339426	−	0.940633i	$-0.389767\pi$
$601$	10002.0	0.678852	0.339426	+	0.940633i	$0.389767\pi$
$602$	0	0
$603$	10164.0	0.686418
$604$	0	0
$605$	3025.00	0.203279
$606$	0	0
$607$	2992.00	0.200068	0.100034	−	0.994984i	$-0.468105\pi$
$607$	2992.00	0.200068	0.100034	+	0.994984i	$0.468105\pi$
$608$	0	0
$609$	−12960.0	−0.862341
$610$	0	0
$611$	−24360.0	−1.61293
$612$	0	0
$613$	−13930.0	−0.917826	−0.458913	−	0.888481i	$-0.651761\pi$
$613$	−13930.0	−0.917826	−0.458913	+	0.888481i	$0.651761\pi$
$614$	0	0
$615$	680.000	0.0445858
$616$	0	0
$617$	−23022.0	−1.50216	−0.751078	−	0.660213i	$-0.770466\pi$
$617$	−23022.0	−1.50216	−0.751078	+	0.660213i	$0.770466\pi$
$618$	0	0
$619$	−26284.0	−1.70669	−0.853347	−	0.521344i	$-0.825431\pi$
$619$	−26284.0	−1.70669	−0.853347	+	0.521344i	$0.825431\pi$
$620$	0	0
$621$	−24928.0	−1.61083
$622$	0	0
$623$	−20440.0	−1.31446
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−3344.00	−0.212993
$628$	0	0
$629$	−19436.0	−1.23206
$630$	0	0
$631$	−8408.00	−0.530455	−0.265228	−	0.964186i	$-0.585447\pi$
$631$	−8408.00	−0.530455	−0.265228	+	0.964186i	$0.585447\pi$
$632$	0	0
$633$	8720.00	0.547534
$634$	0	0
$635$	−2840.00	−0.177483
$636$	0	0
$637$	2394.00	0.148907
$638$	0	0
$639$	3520.00	0.217917
$640$	0	0
$641$	−16262.0	−1.00204	−0.501022	−	0.865434i	$-0.667042\pi$
$641$	−16262.0	−1.00204	−0.501022	+	0.865434i	$0.667042\pi$
$642$	0	0
$643$	−24448.0	−1.49943	−0.749716	−	0.661760i	$-0.769810\pi$
$643$	−24448.0	−1.49943	−0.749716	+	0.661760i	$0.769810\pi$
$644$	0	0
$645$	8640.00	0.527441
$646$	0	0
$647$	−10548.0	−0.640935	−0.320467	−	0.947260i	$-0.603840\pi$
$647$	−10548.0	−0.640935	−0.320467	+	0.947260i	$0.603840\pi$
$648$	0	0
$649$	−16016.0	−0.968695
$650$	0	0
$651$	24960.0	1.50270
$652$	0	0
$653$	7662.00	0.459169	0.229584	−	0.973289i	$-0.426263\pi$
$653$	7662.00	0.459169	0.229584	+	0.973289i	$0.426263\pi$
$654$	0	0
$655$	4620.00	0.275601
$656$	0	0
$657$	5962.00	0.354033
$658$	0	0
$659$	−32452.0	−1.91829	−0.959143	−	0.282922i	$-0.908696\pi$
$659$	−32452.0	−1.91829	−0.959143	+	0.282922i	$0.908696\pi$
$660$	0	0
$661$	−17690.0	−1.04094	−0.520470	−	0.853880i	$-0.674243\pi$
$661$	−17690.0	−1.04094	−0.520470	+	0.853880i	$0.674243\pi$
$662$	0	0
$663$	−14448.0	−0.846326
$664$	0	0
$665$	−1900.00	−0.110795
$666$	0	0
$667$	−26568.0	−1.54230
$668$	0	0
$669$	17184.0	0.993082
$670$	0	0
$671$	22792.0	1.31129
$672$	0	0
$673$	−15802.0	−0.905085	−0.452543	−	0.891743i	$-0.649483\pi$
$673$	−15802.0	−0.905085	−0.452543	+	0.891743i	$0.649483\pi$
$674$	0	0
$675$	−3800.00	−0.216685
$676$	0	0
$677$	−10518.0	−0.597104	−0.298552	−	0.954393i	$-0.596504\pi$
$677$	−10518.0	−0.597104	−0.298552	+	0.954393i	$0.596504\pi$
$678$	0	0
$679$	23320.0	1.31803
$680$	0	0
$681$	−2000.00	−0.112541
$682$	0	0
$683$	7092.00	0.397317	0.198659	−	0.980069i	$-0.436341\pi$
$683$	7092.00	0.397317	0.198659	+	0.980069i	$0.436341\pi$
$684$	0	0
$685$	−3510.00	−0.195781
$686$	0	0
$687$	17464.0	0.969859
$688$	0	0
$689$	21252.0	1.17509
$690$	0	0
$691$	−508.000	−0.0279670	−0.0139835	−	0.999902i	$-0.504451\pi$
$691$	−508.000	−0.0279670	−0.0139835	+	0.999902i	$0.504451\pi$
$692$	0	0
$693$	−9680.00	−0.530610
$694$	0	0
$695$	10580.0	0.577442
$696$	0	0
$697$	−2924.00	−0.158902
$698$	0	0
$699$	11880.0	0.642837
$700$	0	0
$701$	−31466.0	−1.69537	−0.847685	−	0.530500i	$-0.822004\pi$
$701$	−31466.0	−1.69537	−0.847685	+	0.530500i	$0.822004\pi$
$702$	0	0
$703$	−4294.00	−0.230372
$704$	0	0
$705$	−11600.0	−0.619690
$706$	0	0
$707$	−17960.0	−0.955382
$708$	0	0
$709$	−31282.0	−1.65701	−0.828505	−	0.559982i	$-0.810808\pi$
$709$	−31282.0	−1.65701	−0.828505	+	0.559982i	$0.810808\pi$
$710$	0	0
$711$	−13288.0	−0.700899
$712$	0	0
$713$	51168.0	2.68760
$714$	0	0
$715$	9240.00	0.483296
$716$	0	0
$717$	576.000	0.0300016
$718$	0	0
$719$	−18232.0	−0.945673	−0.472836	−	0.881150i	$-0.656770\pi$
$719$	−18232.0	−0.945673	−0.472836	+	0.881150i	$0.656770\pi$
$720$	0	0
$721$	−23040.0	−1.19009
$722$	0	0
$723$	6952.00	0.357604
$724$	0	0
$725$	−4050.00	−0.207467
$726$	0	0
$727$	8884.00	0.453218	0.226609	−	0.973986i	$-0.427236\pi$
$727$	8884.00	0.453218	0.226609	+	0.973986i	$0.427236\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	−37152.0	−1.87978
$732$	0	0
$733$	6838.00	0.344567	0.172283	−	0.985047i	$-0.444886\pi$
$733$	6838.00	0.344567	0.172283	+	0.985047i	$0.444886\pi$
$734$	0	0
$735$	1140.00	0.0572102
$736$	0	0
$737$	−40656.0	−2.03200
$738$	0	0
$739$	−32380.0	−1.61180	−0.805898	−	0.592054i	$-0.798317\pi$
$739$	−32380.0	−1.61180	−0.805898	+	0.592054i	$0.798317\pi$
$740$	0	0
$741$	−3192.00	−0.158247
$742$	0	0
$743$	4584.00	0.226340	0.113170	−	0.993576i	$-0.463900\pi$
$743$	4584.00	0.226340	0.113170	+	0.993576i	$0.463900\pi$
$744$	0	0
$745$	7270.00	0.357520
$746$	0	0
$747$	−12320.0	−0.603434
$748$	0	0
$749$	23120.0	1.12789
$750$	0	0
$751$	−10384.0	−0.504551	−0.252275	−	0.967655i	$-0.581179\pi$
$751$	−10384.0	−0.504551	−0.252275	+	0.967655i	$0.581179\pi$
$752$	0	0
$753$	12048.0	0.583072
$754$	0	0
$755$	3400.00	0.163892
$756$	0	0
$757$	−2002.00	−0.0961214	−0.0480607	−	0.998844i	$-0.515304\pi$
$757$	−2002.00	−0.0961214	−0.0480607	+	0.998844i	$0.515304\pi$
$758$	0	0
$759$	28864.0	1.38036
$760$	0	0
$761$	−2854.00	−0.135949	−0.0679747	−	0.997687i	$-0.521654\pi$
$761$	−2854.00	−0.135949	−0.0679747	+	0.997687i	$0.521654\pi$
$762$	0	0
$763$	20920.0	0.992601
$764$	0	0
$765$	4730.00	0.223547
$766$	0	0
$767$	−15288.0	−0.719710
$768$	0	0
$769$	2322.00	0.108886	0.0544431	−	0.998517i	$-0.482662\pi$
$769$	2322.00	0.108886	0.0544431	+	0.998517i	$0.482662\pi$
$770$	0	0
$771$	4056.00	0.189459
$772$	0	0
$773$	−27678.0	−1.28785	−0.643925	−	0.765088i	$-0.722695\pi$
$773$	−27678.0	−1.28785	−0.643925	+	0.765088i	$0.722695\pi$
$774$	0	0
$775$	7800.00	0.361528
$776$	0	0
$777$	18080.0	0.834770
$778$	0	0
$779$	−646.000	−0.0297116
$780$	0	0
$781$	−14080.0	−0.645099
$782$	0	0
$783$	24624.0	1.12387
$784$	0	0
$785$	2310.00	0.105029
$786$	0	0
$787$	−12532.0	−0.567621	−0.283810	−	0.958880i	$-0.591599\pi$
$787$	−12532.0	−0.567621	−0.283810	+	0.958880i	$0.591599\pi$
$788$	0	0
$789$	−17136.0	−0.773204
$790$	0	0
$791$	−10600.0	−0.476476
$792$	0	0
$793$	21756.0	0.974247
$794$	0	0
$795$	10120.0	0.451471
$796$	0	0
$797$	4562.00	0.202753	0.101377	−	0.994848i	$-0.467675\pi$
$797$	4562.00	0.202753	0.101377	+	0.994848i	$0.467675\pi$
$798$	0	0
$799$	49880.0	2.20855
$800$	0	0
$801$	11242.0	0.495901
$802$	0	0
$803$	−23848.0	−1.04804
$804$	0	0
$805$	16400.0	0.718042
$806$	0	0
$807$	152.000	0.00663030
$808$	0	0
$809$	−1350.00	−0.0586693	−0.0293347	−	0.999570i	$-0.509339\pi$
$809$	−1350.00	−0.0586693	−0.0293347	+	0.999570i	$0.509339\pi$
$810$	0	0
$811$	−12892.0	−0.558199	−0.279099	−	0.960262i	$-0.590036\pi$
$811$	−12892.0	−0.558199	−0.279099	+	0.960262i	$0.590036\pi$
$812$	0	0
$813$	−23552.0	−1.01600
$814$	0	0
$815$	−20680.0	−0.888821
$816$	0	0
$817$	−8208.00	−0.351483
$818$	0	0
$819$	−9240.00	−0.394227
$820$	0	0
$821$	−32802.0	−1.39439	−0.697197	−	0.716879i	$-0.745570\pi$
$821$	−32802.0	−1.39439	−0.697197	+	0.716879i	$0.745570\pi$
$822$	0	0
$823$	−24916.0	−1.05531	−0.527653	−	0.849460i	$-0.676928\pi$
$823$	−24916.0	−1.05531	−0.527653	+	0.849460i	$0.676928\pi$
$824$	0	0
$825$	4400.00	0.185683
$826$	0	0
$827$	1596.00	0.0671081	0.0335540	−	0.999437i	$-0.489317\pi$
$827$	1596.00	0.0671081	0.0335540	+	0.999437i	$0.489317\pi$
$828$	0	0
$829$	25910.0	1.08551	0.542757	−	0.839890i	$-0.317380\pi$
$829$	25910.0	1.08551	0.542757	+	0.839890i	$0.317380\pi$
$830$	0	0
$831$	21016.0	0.877301
$832$	0	0
$833$	−4902.00	−0.203895
$834$	0	0
$835$	7320.00	0.303376
$836$	0	0
$837$	−47424.0	−1.95844
$838$	0	0
$839$	−27504.0	−1.13176	−0.565878	−	0.824489i	$-0.691463\pi$
$839$	−27504.0	−1.13176	−0.565878	+	0.824489i	$0.691463\pi$
$840$	0	0
$841$	1855.00	0.0760589
$842$	0	0
$843$	−10232.0	−0.418041
$844$	0	0
$845$	−2165.00	−0.0881400
$846$	0	0
$847$	12100.0	0.490863
$848$	0	0
$849$	27104.0	1.09565
$850$	0	0
$851$	37064.0	1.49299
$852$	0	0
$853$	−14514.0	−0.582591	−0.291295	−	0.956633i	$-0.594086\pi$
$853$	−14514.0	−0.582591	−0.291295	+	0.956633i	$0.594086\pi$
$854$	0	0
$855$	1045.00	0.0417991
$856$	0	0
$857$	41790.0	1.66572	0.832858	−	0.553486i	$-0.186703\pi$
$857$	41790.0	1.66572	0.832858	+	0.553486i	$0.186703\pi$
$858$	0	0
$859$	6332.00	0.251508	0.125754	−	0.992061i	$-0.459865\pi$
$859$	6332.00	0.251508	0.125754	+	0.992061i	$0.459865\pi$
$860$	0	0
$861$	2720.00	0.107662
$862$	0	0
$863$	−3488.00	−0.137582	−0.0687908	−	0.997631i	$-0.521914\pi$
$863$	−3488.00	−0.137582	−0.0687908	+	0.997631i	$0.521914\pi$
$864$	0	0
$865$	−7110.00	−0.279477
$866$	0	0
$867$	9932.00	0.389052
$868$	0	0
$869$	53152.0	2.07487
$870$	0	0
$871$	−38808.0	−1.50971
$872$	0	0
$873$	−12826.0	−0.497244
$874$	0	0
$875$	2500.00	0.0965891
$876$	0	0
$877$	23426.0	0.901984	0.450992	−	0.892528i	$-0.351070\pi$
$877$	23426.0	0.901984	0.450992	+	0.892528i	$0.351070\pi$
$878$	0	0
$879$	−8344.00	−0.320178
$880$	0	0
$881$	−31230.0	−1.19429	−0.597143	−	0.802135i	$-0.703697\pi$
$881$	−31230.0	−1.19429	−0.597143	+	0.802135i	$0.703697\pi$
$882$	0	0
$883$	−24120.0	−0.919256	−0.459628	−	0.888112i	$-0.652017\pi$
$883$	−24120.0	−0.919256	−0.459628	+	0.888112i	$0.652017\pi$
$884$	0	0
$885$	−7280.00	−0.276514
$886$	0	0
$887$	33696.0	1.27554	0.637768	−	0.770228i	$-0.279858\pi$
$887$	33696.0	1.27554	0.637768	+	0.770228i	$0.279858\pi$
$888$	0	0
$889$	−11360.0	−0.428574
$890$	0	0
$891$	−13684.0	−0.514513
$892$	0	0
$893$	11020.0	0.412957
$894$	0	0
$895$	16060.0	0.599806
$896$	0	0
$897$	27552.0	1.02557
$898$	0	0
$899$	−50544.0	−1.87512
$900$	0	0
$901$	−43516.0	−1.60902
$902$	0	0
$903$	34560.0	1.27363
$904$	0	0
$905$	−16170.0	−0.593933
$906$	0	0
$907$	7412.00	0.271347	0.135673	−	0.990754i	$-0.456680\pi$
$907$	7412.00	0.271347	0.135673	+	0.990754i	$0.456680\pi$
$908$	0	0
$909$	9878.00	0.360432
$910$	0	0
$911$	−18288.0	−0.665103	−0.332551	−	0.943085i	$-0.607909\pi$
$911$	−18288.0	−0.665103	−0.332551	+	0.943085i	$0.607909\pi$
$912$	0	0
$913$	49280.0	1.78634
$914$	0	0
$915$	10360.0	0.374307
$916$	0	0
$917$	18480.0	0.665500
$918$	0	0
$919$	34040.0	1.22185	0.610923	−	0.791690i	$-0.290799\pi$
$919$	34040.0	1.22185	0.610923	+	0.791690i	$0.290799\pi$
$920$	0	0
$921$	−29712.0	−1.06302
$922$	0	0
$923$	−13440.0	−0.479288
$924$	0	0
$925$	5650.00	0.200833
$926$	0	0
$927$	12672.0	0.448979
$928$	0	0
$929$	41778.0	1.47545	0.737724	−	0.675102i	$-0.235900\pi$
$929$	41778.0	1.47545	0.737724	+	0.675102i	$0.235900\pi$
$930$	0	0
$931$	−1083.00	−0.0381245
$932$	0	0
$933$	16160.0	0.567047
$934$	0	0
$935$	−18920.0	−0.661765
$936$	0	0
$937$	14834.0	0.517189	0.258594	−	0.965986i	$-0.416741\pi$
$937$	14834.0	0.517189	0.258594	+	0.965986i	$0.416741\pi$
$938$	0	0
$939$	−13272.0	−0.461252
$940$	0	0
$941$	−18026.0	−0.624475	−0.312237	−	0.950004i	$-0.601078\pi$
$941$	−18026.0	−0.624475	−0.312237	+	0.950004i	$0.601078\pi$
$942$	0	0
$943$	5576.00	0.192555
$944$	0	0
$945$	−15200.0	−0.523234
$946$	0	0
$947$	−57112.0	−1.95976	−0.979879	−	0.199593i	$-0.936038\pi$
$947$	−57112.0	−1.95976	−0.979879	+	0.199593i	$0.936038\pi$
$948$	0	0
$949$	−22764.0	−0.778662
$950$	0	0
$951$	−29208.0	−0.995935
$952$	0	0
$953$	−7842.00	−0.266555	−0.133278	−	0.991079i	$-0.542550\pi$
$953$	−7842.00	−0.266555	−0.133278	+	0.991079i	$0.542550\pi$
$954$	0	0
$955$	1480.00	0.0501484
$956$	0	0
$957$	−28512.0	−0.963074
$958$	0	0
$959$	−14040.0	−0.472758
$960$	0	0
$961$	67553.0	2.26756
$962$	0	0
$963$	−12716.0	−0.425511
$964$	0	0
$965$	−13850.0	−0.462018
$966$	0	0
$967$	25068.0	0.833643	0.416821	−	0.908988i	$-0.363144\pi$
$967$	25068.0	0.833643	0.416821	+	0.908988i	$0.363144\pi$
$968$	0	0
$969$	6536.00	0.216684
$970$	0	0
$971$	−11628.0	−0.384305	−0.192153	−	0.981365i	$-0.561547\pi$
$971$	−11628.0	−0.384305	−0.192153	+	0.981365i	$0.561547\pi$
$972$	0	0
$973$	42320.0	1.39436
$974$	0	0
$975$	4200.00	0.137957
$976$	0	0
$977$	−22554.0	−0.738553	−0.369277	−	0.929320i	$-0.620394\pi$
$977$	−22554.0	−0.738553	−0.369277	+	0.929320i	$0.620394\pi$
$978$	0	0
$979$	−44968.0	−1.46801
$980$	0	0
$981$	−11506.0	−0.374473
$982$	0	0
$983$	38136.0	1.23739	0.618693	−	0.785633i	$-0.287663\pi$
$983$	38136.0	1.23739	0.618693	+	0.785633i	$0.287663\pi$
$984$	0	0
$985$	−5770.00	−0.186647
$986$	0	0
$987$	−46400.0	−1.49638
$988$	0	0
$989$	70848.0	2.27789
$990$	0	0
$991$	−22720.0	−0.728279	−0.364140	−	0.931344i	$-0.618637\pi$
$991$	−22720.0	−0.728279	−0.364140	+	0.931344i	$0.618637\pi$
$992$	0	0
$993$	−5712.00	−0.182543
$994$	0	0
$995$	13400.0	0.426943
$996$	0	0
$997$	18230.0	0.579087	0.289544	−	0.957165i	$-0.406496\pi$
$997$	18230.0	0.579087	0.289544	+	0.957165i	$0.406496\pi$
$998$	0	0
$999$	−34352.0	−1.08794

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.4.a.g.1.1		1
4.3	odd	2		190.4.a.c.1.1	✓	1
12.11	even	2		1710.4.a.b.1.1		1
20.3	even	4		950.4.b.a.799.1		2
20.7	even	4		950.4.b.a.799.2		2
20.19	odd	2		950.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.4.a.c.1.1	✓	1	4.3	odd	2
950.4.a.a.1.1		1	20.19	odd	2
950.4.b.a.799.1		2	20.3	even	4
950.4.b.a.799.2		2	20.7	even	4
1520.4.a.g.1.1		1	1.1	even	1	trivial
1710.4.a.b.1.1		1	12.11	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

Related objects

Downloads

Learn more