LMFDB - Embedded newform 2320.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,4,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, 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("2320.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2320.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:	$136.884431213$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{3} -5.00000 q^{5} +14.0000 q^{7} +37.0000 q^{9} +O(q^{10})$

$q+8.00000 q^{3} -5.00000 q^{5} +14.0000 q^{7} +37.0000 q^{9} -62.0000 q^{11} +42.0000 q^{13} -40.0000 q^{15} -114.000 q^{17} +70.0000 q^{19} +112.000 q^{21} -62.0000 q^{23} +25.0000 q^{25} +80.0000 q^{27} -29.0000 q^{29} -142.000 q^{31} -496.000 q^{33} -70.0000 q^{35} +146.000 q^{37} +336.000 q^{39} +162.000 q^{41} -352.000 q^{43} -185.000 q^{45} +444.000 q^{47} -147.000 q^{49} -912.000 q^{51} -238.000 q^{53} +310.000 q^{55} +560.000 q^{57} -840.000 q^{59} +2.00000 q^{61} +518.000 q^{63} -210.000 q^{65} +154.000 q^{67} -496.000 q^{69} -892.000 q^{71} -38.0000 q^{73} +200.000 q^{75} -868.000 q^{77} -1050.00 q^{79} -359.000 q^{81} +778.000 q^{83} +570.000 q^{85} -232.000 q^{87} +1410.00 q^{89} +588.000 q^{91} -1136.00 q^{93} -350.000 q^{95} +466.000 q^{97} -2294.00 q^{99} +O(q^{100})$

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
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	14.0000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	14.0000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	−62.0000	−1.69943	−0.849714	−	0.527244i	$-0.823225\pi$
$11$	−62.0000	−1.69943	−0.849714	+	0.527244i	$0.823225\pi$
$12$	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
$15$	−40.0000	−0.688530
$16$	0	0
$17$	−114.000	−1.62642	−0.813208	−	0.581974i	$-0.802281\pi$
$17$	−114.000	−1.62642	−0.813208	+	0.581974i	$0.802281\pi$
$18$	0	0
$19$	70.0000	0.845216	0.422608	−	0.906313i	$-0.361115\pi$
$19$	70.0000	0.845216	0.422608	+	0.906313i	$0.361115\pi$
$20$	0	0
$21$	112.000	1.16383
$22$	0	0
$23$	−62.0000	−0.562082	−0.281041	−	0.959696i	$-0.590680\pi$
$23$	−62.0000	−0.562082	−0.281041	+	0.959696i	$0.590680\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	80.0000	0.570222
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−142.000	−0.822708	−0.411354	−	0.911476i	$-0.634944\pi$
$31$	−142.000	−0.822708	−0.411354	+	0.911476i	$0.634944\pi$
$32$	0	0
$33$	−496.000	−2.61644
$34$	0	0
$35$	−70.0000	−0.338062
$36$	0	0
$37$	146.000	0.648710	0.324355	−	0.945936i	$-0.394853\pi$
$37$	146.000	0.648710	0.324355	+	0.945936i	$0.394853\pi$
$38$	0	0
$39$	336.000	1.37957
$40$	0	0
$41$	162.000	0.617077	0.308538	−	0.951212i	$-0.400160\pi$
$41$	162.000	0.617077	0.308538	+	0.951212i	$0.400160\pi$
$42$	0	0
$43$	−352.000	−1.24836	−0.624180	−	0.781280i	$-0.714567\pi$
$43$	−352.000	−1.24836	−0.624180	+	0.781280i	$0.714567\pi$
$44$	0	0
$45$	−185.000	−0.612848
$46$	0	0
$47$	444.000	1.37796	0.688979	−	0.724781i	$-0.258059\pi$
$47$	444.000	1.37796	0.688979	+	0.724781i	$0.258059\pi$
$48$	0	0
$49$	−147.000	−0.428571
$50$	0	0
$51$	−912.000	−2.50403
$52$	0	0
$53$	−238.000	−0.616827	−0.308413	−	0.951252i	$-0.599798\pi$
$53$	−238.000	−0.616827	−0.308413	+	0.951252i	$0.599798\pi$
$54$	0	0
$55$	310.000	0.760007
$56$	0	0
$57$	560.000	1.30129
$58$	0	0
$59$	−840.000	−1.85354	−0.926769	−	0.375633i	$-0.877425\pi$
$59$	−840.000	−1.85354	−0.926769	+	0.375633i	$0.877425\pi$
$60$	0	0
$61$	2.00000	0.00419793	0.00209897	−	0.999998i	$-0.499332\pi$
$61$	2.00000	0.00419793	0.00209897	+	0.999998i	$0.499332\pi$
$62$	0	0
$63$	518.000	1.03590
$64$	0	0
$65$	−210.000	−0.400728
$66$	0	0
$67$	154.000	0.280807	0.140404	−	0.990094i	$-0.455160\pi$
$67$	154.000	0.280807	0.140404	+	0.990094i	$0.455160\pi$
$68$	0	0
$69$	−496.000	−0.865382
$70$	0	0
$71$	−892.000	−1.49100	−0.745499	−	0.666506i	$-0.767789\pi$
$71$	−892.000	−1.49100	−0.745499	+	0.666506i	$0.767789\pi$
$72$	0	0
$73$	−38.0000	−0.0609255	−0.0304628	−	0.999536i	$-0.509698\pi$
$73$	−38.0000	−0.0609255	−0.0304628	+	0.999536i	$0.509698\pi$
$74$	0	0
$75$	200.000	0.307920
$76$	0	0
$77$	−868.000	−1.28465
$78$	0	0
$79$	−1050.00	−1.49537	−0.747685	−	0.664054i	$-0.768835\pi$
$79$	−1050.00	−1.49537	−0.747685	+	0.664054i	$0.768835\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	778.000	1.02887	0.514437	−	0.857528i	$-0.328001\pi$
$83$	778.000	1.02887	0.514437	+	0.857528i	$0.328001\pi$
$84$	0	0
$85$	570.000	0.727355
$86$	0	0
$87$	−232.000	−0.285897
$88$	0	0
$89$	1410.00	1.67932	0.839661	−	0.543110i	$-0.182754\pi$
$89$	1410.00	1.67932	0.839661	+	0.543110i	$0.182754\pi$
$90$	0	0
$91$	588.000	0.677353
$92$	0	0
$93$	−1136.00	−1.26664
$94$	0	0
$95$	−350.000	−0.377992
$96$	0	0
$97$	466.000	0.487785	0.243892	−	0.969802i	$-0.421576\pi$
$97$	466.000	0.487785	0.243892	+	0.969802i	$0.421576\pi$
$98$	0	0
$99$	−2294.00	−2.32885
$100$	0	0
$101$	−878.000	−0.864993	−0.432496	−	0.901636i	$-0.642367\pi$
$101$	−878.000	−0.864993	−0.432496	+	0.901636i	$0.642367\pi$
$102$	0	0
$103$	−1062.00	−1.01594	−0.507971	−	0.861374i	$-0.669604\pi$
$103$	−1062.00	−1.01594	−0.507971	+	0.861374i	$0.669604\pi$
$104$	0	0
$105$	−560.000	−0.520480
$106$	0	0
$107$	−1826.00	−1.64978	−0.824888	−	0.565296i	$-0.808762\pi$
$107$	−1826.00	−1.64978	−0.824888	+	0.565296i	$0.808762\pi$
$108$	0	0
$109$	−1270.00	−1.11600	−0.558000	−	0.829841i	$-0.688431\pi$
$109$	−1270.00	−1.11600	−0.558000	+	0.829841i	$0.688431\pi$
$110$	0	0
$111$	1168.00	0.998754
$112$	0	0
$113$	82.0000	0.0682647	0.0341324	−	0.999417i	$-0.489133\pi$
$113$	82.0000	0.0682647	0.0341324	+	0.999417i	$0.489133\pi$
$114$	0	0
$115$	310.000	0.251371
$116$	0	0
$117$	1554.00	1.22793
$118$	0	0
$119$	−1596.00	−1.22945
$120$	0	0
$121$	2513.00	1.88805
$122$	0	0
$123$	1296.00	0.950052
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−296.000	−0.206817	−0.103408	−	0.994639i	$-0.532975\pi$
$127$	−296.000	−0.206817	−0.103408	+	0.994639i	$0.532975\pi$
$128$	0	0
$129$	−2816.00	−1.92198
$130$	0	0
$131$	198.000	0.132056	0.0660280	−	0.997818i	$-0.478967\pi$
$131$	198.000	0.132056	0.0660280	+	0.997818i	$0.478967\pi$
$132$	0	0
$133$	980.000	0.638923
$134$	0	0
$135$	−400.000	−0.255011
$136$	0	0
$137$	166.000	0.103521	0.0517604	−	0.998660i	$-0.483517\pi$
$137$	166.000	0.103521	0.0517604	+	0.998660i	$0.483517\pi$
$138$	0	0
$139$	80.0000	0.0488166	0.0244083	−	0.999702i	$-0.492230\pi$
$139$	80.0000	0.0488166	0.0244083	+	0.999702i	$0.492230\pi$
$140$	0	0
$141$	3552.00	2.12151
$142$	0	0
$143$	−2604.00	−1.52278
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	−1176.00	−0.659829
$148$	0	0
$149$	3150.00	1.73193	0.865967	−	0.500102i	$-0.166704\pi$
$149$	3150.00	1.73193	0.865967	+	0.500102i	$0.166704\pi$
$150$	0	0
$151$	1868.00	1.00673	0.503363	−	0.864075i	$-0.332096\pi$
$151$	1868.00	1.00673	0.503363	+	0.864075i	$0.332096\pi$
$152$	0	0
$153$	−4218.00	−2.22879
$154$	0	0
$155$	710.000	0.367926
$156$	0	0
$157$	−354.000	−0.179951	−0.0899754	−	0.995944i	$-0.528679\pi$
$157$	−354.000	−0.179951	−0.0899754	+	0.995944i	$0.528679\pi$
$158$	0	0
$159$	−1904.00	−0.949667
$160$	0	0
$161$	−868.000	−0.424894
$162$	0	0
$163$	2268.00	1.08984	0.544919	−	0.838489i	$-0.316561\pi$
$163$	2268.00	1.08984	0.544919	+	0.838489i	$0.316561\pi$
$164$	0	0
$165$	2480.00	1.17011
$166$	0	0
$167$	−2046.00	−0.948049	−0.474025	−	0.880512i	$-0.657199\pi$
$167$	−2046.00	−0.948049	−0.474025	+	0.880512i	$0.657199\pi$
$168$	0	0
$169$	−433.000	−0.197087
$170$	0	0
$171$	2590.00	1.15826
$172$	0	0
$173$	−2118.00	−0.930801	−0.465400	−	0.885100i	$-0.654090\pi$
$173$	−2118.00	−0.930801	−0.465400	+	0.885100i	$0.654090\pi$
$174$	0	0
$175$	350.000	0.151186
$176$	0	0
$177$	−6720.00	−2.85371
$178$	0	0
$179$	−1260.00	−0.526127	−0.263064	−	0.964778i	$-0.584733\pi$
$179$	−1260.00	−0.526127	−0.263064	+	0.964778i	$0.584733\pi$
$180$	0	0
$181$	1342.00	0.551105	0.275553	−	0.961286i	$-0.411139\pi$
$181$	1342.00	0.551105	0.275553	+	0.961286i	$0.411139\pi$
$182$	0	0
$183$	16.0000	0.00646314
$184$	0	0
$185$	−730.000	−0.290112
$186$	0	0
$187$	7068.00	2.76398
$188$	0	0
$189$	1120.00	0.431048
$190$	0	0
$191$	−1622.00	−0.614470	−0.307235	−	0.951634i	$-0.599404\pi$
$191$	−1622.00	−0.614470	−0.307235	+	0.951634i	$0.599404\pi$
$192$	0	0
$193$	−4058.00	−1.51348	−0.756739	−	0.653717i	$-0.773209\pi$
$193$	−4058.00	−1.51348	−0.756739	+	0.653717i	$0.773209\pi$
$194$	0	0
$195$	−1680.00	−0.616961
$196$	0	0
$197$	266.000	0.0962016	0.0481008	−	0.998842i	$-0.484683\pi$
$197$	266.000	0.0962016	0.0481008	+	0.998842i	$0.484683\pi$
$198$	0	0
$199$	2220.00	0.790812	0.395406	−	0.918506i	$-0.370604\pi$
$199$	2220.00	0.790812	0.395406	+	0.918506i	$0.370604\pi$
$200$	0	0
$201$	1232.00	0.432331
$202$	0	0
$203$	−406.000	−0.140372
$204$	0	0
$205$	−810.000	−0.275965
$206$	0	0
$207$	−2294.00	−0.770261
$208$	0	0
$209$	−4340.00	−1.43638
$210$	0	0
$211$	3478.00	1.13476	0.567382	−	0.823454i	$-0.307956\pi$
$211$	3478.00	1.13476	0.567382	+	0.823454i	$0.307956\pi$
$212$	0	0
$213$	−7136.00	−2.29554
$214$	0	0
$215$	1760.00	0.558284
$216$	0	0
$217$	−1988.00	−0.621909
$218$	0	0
$219$	−304.000	−0.0938010
$220$	0	0
$221$	−4788.00	−1.45736
$222$	0	0
$223$	−6622.00	−1.98853	−0.994264	−	0.106950i	$-0.965891\pi$
$223$	−6622.00	−1.98853	−0.994264	+	0.106950i	$0.965891\pi$
$224$	0	0
$225$	925.000	0.274074
$226$	0	0
$227$	−3666.00	−1.07190	−0.535949	−	0.844250i	$-0.680046\pi$
$227$	−3666.00	−1.07190	−0.535949	+	0.844250i	$0.680046\pi$
$228$	0	0
$229$	−350.000	−0.100998	−0.0504992	−	0.998724i	$-0.516081\pi$
$229$	−350.000	−0.100998	−0.0504992	+	0.998724i	$0.516081\pi$
$230$	0	0
$231$	−6944.00	−1.97784
$232$	0	0
$233$	2.00000	0.000562336 0	0.000281168	−	1.00000i	$-0.499911\pi$
$233$	2.00000	0.000562336 0	0.000281168		1.00000i	$0.499911\pi$
$234$	0	0
$235$	−2220.00	−0.616242
$236$	0	0
$237$	−8400.00	−2.30227
$238$	0	0
$239$	−700.000	−0.189453	−0.0947264	−	0.995503i	$-0.530198\pi$
$239$	−700.000	−0.189453	−0.0947264	+	0.995503i	$0.530198\pi$
$240$	0	0
$241$	−1018.00	−0.272096	−0.136048	−	0.990702i	$-0.543440\pi$
$241$	−1018.00	−0.272096	−0.136048	+	0.990702i	$0.543440\pi$
$242$	0	0
$243$	−5032.00	−1.32841
$244$	0	0
$245$	735.000	0.191663
$246$	0	0
$247$	2940.00	0.757359
$248$	0	0
$249$	6224.00	1.58406
$250$	0	0
$251$	−7102.00	−1.78595	−0.892977	−	0.450103i	$-0.851387\pi$
$251$	−7102.00	−1.78595	−0.892977	+	0.450103i	$0.851387\pi$
$252$	0	0
$253$	3844.00	0.955218
$254$	0	0
$255$	4560.00	1.11984
$256$	0	0
$257$	5906.00	1.43349	0.716743	−	0.697337i	$-0.245632\pi$
$257$	5906.00	1.43349	0.716743	+	0.697337i	$0.245632\pi$
$258$	0	0
$259$	2044.00	0.490378
$260$	0	0
$261$	−1073.00	−0.254471
$262$	0	0
$263$	−6252.00	−1.46584	−0.732918	−	0.680317i	$-0.761842\pi$
$263$	−6252.00	−1.46584	−0.732918	+	0.680317i	$0.761842\pi$
$264$	0	0
$265$	1190.00	0.275853
$266$	0	0
$267$	11280.0	2.58549
$268$	0	0
$269$	−7230.00	−1.63874	−0.819370	−	0.573266i	$-0.805676\pi$
$269$	−7230.00	−1.63874	−0.819370	+	0.573266i	$0.805676\pi$
$270$	0	0
$271$	7838.00	1.75692	0.878459	−	0.477818i	$-0.158572\pi$
$271$	7838.00	1.75692	0.878459	+	0.477818i	$0.158572\pi$
$272$	0	0
$273$	4704.00	1.04285
$274$	0	0
$275$	−1550.00	−0.339886
$276$	0	0
$277$	5386.00	1.16828	0.584140	−	0.811653i	$-0.301432\pi$
$277$	5386.00	1.16828	0.584140	+	0.811653i	$0.301432\pi$
$278$	0	0
$279$	−5254.00	−1.12741
$280$	0	0
$281$	6502.00	1.38034	0.690172	−	0.723645i	$-0.257535\pi$
$281$	6502.00	1.38034	0.690172	+	0.723645i	$0.257535\pi$
$282$	0	0
$283$	6478.00	1.36070	0.680348	−	0.732889i	$-0.261829\pi$
$283$	6478.00	1.36070	0.680348	+	0.732889i	$0.261829\pi$
$284$	0	0
$285$	−2800.00	−0.581957
$286$	0	0
$287$	2268.00	0.466466
$288$	0	0
$289$	8083.00	1.64523
$290$	0	0
$291$	3728.00	0.750994
$292$	0	0
$293$	−6858.00	−1.36740	−0.683701	−	0.729762i	$-0.739631\pi$
$293$	−6858.00	−1.36740	−0.683701	+	0.729762i	$0.739631\pi$
$294$	0	0
$295$	4200.00	0.828927
$296$	0	0
$297$	−4960.00	−0.969052
$298$	0	0
$299$	−2604.00	−0.503656
$300$	0	0
$301$	−4928.00	−0.943672
$302$	0	0
$303$	−7024.00	−1.33174
$304$	0	0
$305$	−10.0000	−0.00187737
$306$	0	0
$307$	3684.00	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	3684.00	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−8496.00	−1.56414
$310$	0	0
$311$	138.000	0.0251616	0.0125808	−	0.999921i	$-0.495995\pi$
$311$	138.000	0.0251616	0.0125808	+	0.999921i	$0.495995\pi$
$312$	0	0
$313$	8802.00	1.58952	0.794758	−	0.606927i	$-0.207598\pi$
$313$	8802.00	1.58952	0.794758	+	0.606927i	$0.207598\pi$
$314$	0	0
$315$	−2590.00	−0.463270
$316$	0	0
$317$	5546.00	0.982632	0.491316	−	0.870981i	$-0.336516\pi$
$317$	5546.00	0.982632	0.491316	+	0.870981i	$0.336516\pi$
$318$	0	0
$319$	1798.00	0.315576
$320$	0	0
$321$	−14608.0	−2.54000
$322$	0	0
$323$	−7980.00	−1.37467
$324$	0	0
$325$	1050.00	0.179211
$326$	0	0
$327$	−10160.0	−1.71819
$328$	0	0
$329$	6216.00	1.04164
$330$	0	0
$331$	−3082.00	−0.511789	−0.255894	−	0.966705i	$-0.582370\pi$
$331$	−3082.00	−0.511789	−0.255894	+	0.966705i	$0.582370\pi$
$332$	0	0
$333$	5402.00	0.888972
$334$	0	0
$335$	−770.000	−0.125581
$336$	0	0
$337$	−6414.00	−1.03677	−0.518387	−	0.855146i	$-0.673467\pi$
$337$	−6414.00	−1.03677	−0.518387	+	0.855146i	$0.673467\pi$
$338$	0	0
$339$	656.000	0.105100
$340$	0	0
$341$	8804.00	1.39813
$342$	0	0
$343$	−6860.00	−1.07990
$344$	0	0
$345$	2480.00	0.387011
$346$	0	0
$347$	−1486.00	−0.229892	−0.114946	−	0.993372i	$-0.536670\pi$
$347$	−1486.00	−0.229892	−0.114946	+	0.993372i	$0.536670\pi$
$348$	0	0
$349$	1370.00	0.210127	0.105064	−	0.994466i	$-0.466495\pi$
$349$	1370.00	0.210127	0.105064	+	0.994466i	$0.466495\pi$
$350$	0	0
$351$	3360.00	0.510950
$352$	0	0
$353$	1122.00	0.169173	0.0845865	−	0.996416i	$-0.473043\pi$
$353$	1122.00	0.169173	0.0845865	+	0.996416i	$0.473043\pi$
$354$	0	0
$355$	4460.00	0.666795
$356$	0	0
$357$	−12768.0	−1.89287
$358$	0	0
$359$	4230.00	0.621869	0.310934	−	0.950431i	$-0.399358\pi$
$359$	4230.00	0.621869	0.310934	+	0.950431i	$0.399358\pi$
$360$	0	0
$361$	−1959.00	−0.285610
$362$	0	0
$363$	20104.0	2.90685
$364$	0	0
$365$	190.000	0.0272467
$366$	0	0
$367$	−4016.00	−0.571208	−0.285604	−	0.958348i	$-0.592194\pi$
$367$	−4016.00	−0.571208	−0.285604	+	0.958348i	$0.592194\pi$
$368$	0	0
$369$	5994.00	0.845624
$370$	0	0
$371$	−3332.00	−0.466277
$372$	0	0
$373$	3802.00	0.527775	0.263888	−	0.964553i	$-0.414995\pi$
$373$	3802.00	0.527775	0.263888	+	0.964553i	$0.414995\pi$
$374$	0	0
$375$	−1000.00	−0.137706
$376$	0	0
$377$	−1218.00	−0.166393
$378$	0	0
$379$	4430.00	0.600406	0.300203	−	0.953875i	$-0.402946\pi$
$379$	4430.00	0.600406	0.300203	+	0.953875i	$0.402946\pi$
$380$	0	0
$381$	−2368.00	−0.318416
$382$	0	0
$383$	−8642.00	−1.15296	−0.576482	−	0.817110i	$-0.695575\pi$
$383$	−8642.00	−1.15296	−0.576482	+	0.817110i	$0.695575\pi$
$384$	0	0
$385$	4340.00	0.574511
$386$	0	0
$387$	−13024.0	−1.71072
$388$	0	0
$389$	10.0000	0.00130339	0.000651697	−	1.00000i	$-0.499793\pi$
$389$	10.0000	0.00130339	0.000651697		1.00000i	$0.499793\pi$
$390$	0	0
$391$	7068.00	0.914179
$392$	0	0
$393$	1584.00	0.203314
$394$	0	0
$395$	5250.00	0.668750
$396$	0	0
$397$	−7654.00	−0.967615	−0.483808	−	0.875174i	$-0.660747\pi$
$397$	−7654.00	−0.967615	−0.483808	+	0.875174i	$0.660747\pi$
$398$	0	0
$399$	7840.00	0.983687
$400$	0	0
$401$	8402.00	1.04632	0.523162	−	0.852233i	$-0.324752\pi$
$401$	8402.00	1.04632	0.523162	+	0.852233i	$0.324752\pi$
$402$	0	0
$403$	−5964.00	−0.737191
$404$	0	0
$405$	1795.00	0.220233
$406$	0	0
$407$	−9052.00	−1.10243
$408$	0	0
$409$	−790.000	−0.0955085	−0.0477543	−	0.998859i	$-0.515206\pi$
$409$	−790.000	−0.0955085	−0.0477543	+	0.998859i	$0.515206\pi$
$410$	0	0
$411$	1328.00	0.159381
$412$	0	0
$413$	−11760.0	−1.40114
$414$	0	0
$415$	−3890.00	−0.460127
$416$	0	0
$417$	640.000	0.0751581
$418$	0	0
$419$	15220.0	1.77457	0.887286	−	0.461220i	$-0.152588\pi$
$419$	15220.0	1.77457	0.887286	+	0.461220i	$0.152588\pi$
$420$	0	0
$421$	4122.00	0.477183	0.238591	−	0.971120i	$-0.423314\pi$
$421$	4122.00	0.477183	0.238591	+	0.971120i	$0.423314\pi$
$422$	0	0
$423$	16428.0	1.88831
$424$	0	0
$425$	−2850.00	−0.325283
$426$	0	0
$427$	28.0000	0.00317334
$428$	0	0
$429$	−20832.0	−2.34447
$430$	0	0
$431$	12168.0	1.35989	0.679944	−	0.733264i	$-0.262004\pi$
$431$	12168.0	1.35989	0.679944	+	0.733264i	$0.262004\pi$
$432$	0	0
$433$	11822.0	1.31208	0.656038	−	0.754728i	$-0.272231\pi$
$433$	11822.0	1.31208	0.656038	+	0.754728i	$0.272231\pi$
$434$	0	0
$435$	1160.00	0.127857
$436$	0	0
$437$	−4340.00	−0.475081
$438$	0	0
$439$	6480.00	0.704496	0.352248	−	0.935907i	$-0.385417\pi$
$439$	6480.00	0.704496	0.352248	+	0.935907i	$0.385417\pi$
$440$	0	0
$441$	−5439.00	−0.587302
$442$	0	0
$443$	5308.00	0.569279	0.284640	−	0.958635i	$-0.408126\pi$
$443$	5308.00	0.569279	0.284640	+	0.958635i	$0.408126\pi$
$444$	0	0
$445$	−7050.00	−0.751016
$446$	0	0
$447$	25200.0	2.66649
$448$	0	0
$449$	5210.00	0.547606	0.273803	−	0.961786i	$-0.411718\pi$
$449$	5210.00	0.547606	0.273803	+	0.961786i	$0.411718\pi$
$450$	0	0
$451$	−10044.0	−1.04868
$452$	0	0
$453$	14944.0	1.54996
$454$	0	0
$455$	−2940.00	−0.302922
$456$	0	0
$457$	5626.00	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$457$	5626.00	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$458$	0	0
$459$	−9120.00	−0.927419
$460$	0	0
$461$	−4278.00	−0.432205	−0.216102	−	0.976371i	$-0.569334\pi$
$461$	−4278.00	−0.432205	−0.216102	+	0.976371i	$0.569334\pi$
$462$	0	0
$463$	−9642.00	−0.967822	−0.483911	−	0.875117i	$-0.660784\pi$
$463$	−9642.00	−0.967822	−0.483911	+	0.875117i	$0.660784\pi$
$464$	0	0
$465$	5680.00	0.566460
$466$	0	0
$467$	2204.00	0.218392	0.109196	−	0.994020i	$-0.465172\pi$
$467$	2204.00	0.218392	0.109196	+	0.994020i	$0.465172\pi$
$468$	0	0
$469$	2156.00	0.212270
$470$	0	0
$471$	−2832.00	−0.277052
$472$	0	0
$473$	21824.0	2.12150
$474$	0	0
$475$	1750.00	0.169043
$476$	0	0
$477$	−8806.00	−0.845281
$478$	0	0
$479$	−11870.0	−1.13226	−0.566132	−	0.824315i	$-0.691561\pi$
$479$	−11870.0	−1.13226	−0.566132	+	0.824315i	$0.691561\pi$
$480$	0	0
$481$	6132.00	0.581279
$482$	0	0
$483$	−6944.00	−0.654168
$484$	0	0
$485$	−2330.00	−0.218144
$486$	0	0
$487$	−12626.0	−1.17482	−0.587411	−	0.809289i	$-0.699853\pi$
$487$	−12626.0	−1.17482	−0.587411	+	0.809289i	$0.699853\pi$
$488$	0	0
$489$	18144.0	1.67791
$490$	0	0
$491$	−922.000	−0.0847439	−0.0423720	−	0.999102i	$-0.513491\pi$
$491$	−922.000	−0.0847439	−0.0423720	+	0.999102i	$0.513491\pi$
$492$	0	0
$493$	3306.00	0.302018
$494$	0	0
$495$	11470.0	1.04149
$496$	0	0
$497$	−12488.0	−1.12709
$498$	0	0
$499$	−18060.0	−1.62019	−0.810097	−	0.586296i	$-0.800586\pi$
$499$	−18060.0	−1.62019	−0.810097	+	0.586296i	$0.800586\pi$
$500$	0	0
$501$	−16368.0	−1.45962
$502$	0	0
$503$	−13272.0	−1.17648	−0.588240	−	0.808687i	$-0.700179\pi$
$503$	−13272.0	−1.17648	−0.588240	+	0.808687i	$0.700179\pi$
$504$	0	0
$505$	4390.00	0.386837
$506$	0	0
$507$	−3464.00	−0.303435
$508$	0	0
$509$	2410.00	0.209865	0.104933	−	0.994479i	$-0.466537\pi$
$509$	2410.00	0.209865	0.104933	+	0.994479i	$0.466537\pi$
$510$	0	0
$511$	−532.000	−0.0460554
$512$	0	0
$513$	5600.00	0.481961
$514$	0	0
$515$	5310.00	0.454343
$516$	0	0
$517$	−27528.0	−2.34174
$518$	0	0
$519$	−16944.0	−1.43306
$520$	0	0
$521$	−4018.00	−0.337873	−0.168936	−	0.985627i	$-0.554033\pi$
$521$	−4018.00	−0.337873	−0.168936	+	0.985627i	$0.554033\pi$
$522$	0	0
$523$	3618.00	0.302493	0.151247	−	0.988496i	$-0.451671\pi$
$523$	3618.00	0.302493	0.151247	+	0.988496i	$0.451671\pi$
$524$	0	0
$525$	2800.00	0.232766
$526$	0	0
$527$	16188.0	1.33807
$528$	0	0
$529$	−8323.00	−0.684063
$530$	0	0
$531$	−31080.0	−2.54003
$532$	0	0
$533$	6804.00	0.552934
$534$	0	0
$535$	9130.00	0.737802
$536$	0	0
$537$	−10080.0	−0.810026
$538$	0	0
$539$	9114.00	0.728326
$540$	0	0
$541$	20522.0	1.63089	0.815443	−	0.578837i	$-0.196493\pi$
$541$	20522.0	1.63089	0.815443	+	0.578837i	$0.196493\pi$
$542$	0	0
$543$	10736.0	0.848482
$544$	0	0
$545$	6350.00	0.499090
$546$	0	0
$547$	−20026.0	−1.56536	−0.782678	−	0.622427i	$-0.786147\pi$
$547$	−20026.0	−1.56536	−0.782678	+	0.622427i	$0.786147\pi$
$548$	0	0
$549$	74.0000	0.00575272
$550$	0	0
$551$	−2030.00	−0.156953
$552$	0	0
$553$	−14700.0	−1.13039
$554$	0	0
$555$	−5840.00	−0.446656
$556$	0	0
$557$	9186.00	0.698785	0.349393	−	0.936976i	$-0.386388\pi$
$557$	9186.00	0.698785	0.349393	+	0.936976i	$0.386388\pi$
$558$	0	0
$559$	−14784.0	−1.11860
$560$	0	0
$561$	56544.0	4.25542
$562$	0	0
$563$	17068.0	1.27767	0.638837	−	0.769342i	$-0.279416\pi$
$563$	17068.0	1.27767	0.638837	+	0.769342i	$0.279416\pi$
$564$	0	0
$565$	−410.000	−0.0305289
$566$	0	0
$567$	−5026.00	−0.372261
$568$	0	0
$569$	−15270.0	−1.12505	−0.562523	−	0.826781i	$-0.690169\pi$
$569$	−15270.0	−1.12505	−0.562523	+	0.826781i	$0.690169\pi$
$570$	0	0
$571$	−15492.0	−1.13541	−0.567706	−	0.823232i	$-0.692169\pi$
$571$	−15492.0	−1.13541	−0.567706	+	0.823232i	$0.692169\pi$
$572$	0	0
$573$	−12976.0	−0.946039
$574$	0	0
$575$	−1550.00	−0.112416
$576$	0	0
$577$	−14554.0	−1.05007	−0.525035	−	0.851080i	$-0.675948\pi$
$577$	−14554.0	−1.05007	−0.525035	+	0.851080i	$0.675948\pi$
$578$	0	0
$579$	−32464.0	−2.33015
$580$	0	0
$581$	10892.0	0.777756
$582$	0	0
$583$	14756.0	1.04825
$584$	0	0
$585$	−7770.00	−0.549145
$586$	0	0
$587$	−13226.0	−0.929975	−0.464988	−	0.885317i	$-0.653941\pi$
$587$	−13226.0	−0.929975	−0.464988	+	0.885317i	$0.653941\pi$
$588$	0	0
$589$	−9940.00	−0.695366
$590$	0	0
$591$	2128.00	0.148112
$592$	0	0
$593$	4642.00	0.321457	0.160729	−	0.986999i	$-0.448616\pi$
$593$	4642.00	0.321457	0.160729	+	0.986999i	$0.448616\pi$
$594$	0	0
$595$	7980.00	0.549829
$596$	0	0
$597$	17760.0	1.21754
$598$	0	0
$599$	−16530.0	−1.12754	−0.563771	−	0.825931i	$-0.690650\pi$
$599$	−16530.0	−1.12754	−0.563771	+	0.825931i	$0.690650\pi$
$600$	0	0
$601$	−23198.0	−1.57449	−0.787243	−	0.616643i	$-0.788492\pi$
$601$	−23198.0	−1.57449	−0.787243	+	0.616643i	$0.788492\pi$
$602$	0	0
$603$	5698.00	0.384810
$604$	0	0
$605$	−12565.0	−0.844363
$606$	0	0
$607$	−7196.00	−0.481181	−0.240590	−	0.970627i	$-0.577341\pi$
$607$	−7196.00	−0.481181	−0.240590	+	0.970627i	$0.577341\pi$
$608$	0	0
$609$	−3248.00	−0.216118
$610$	0	0
$611$	18648.0	1.23473
$612$	0	0
$613$	9282.00	0.611577	0.305788	−	0.952100i	$-0.401080\pi$
$613$	9282.00	0.611577	0.305788	+	0.952100i	$0.401080\pi$
$614$	0	0
$615$	−6480.00	−0.424876
$616$	0	0
$617$	−7974.00	−0.520294	−0.260147	−	0.965569i	$-0.583771\pi$
$617$	−7974.00	−0.520294	−0.260147	+	0.965569i	$0.583771\pi$
$618$	0	0
$619$	15890.0	1.03178	0.515891	−	0.856654i	$-0.327461\pi$
$619$	15890.0	1.03178	0.515891	+	0.856654i	$0.327461\pi$
$620$	0	0
$621$	−4960.00	−0.320512
$622$	0	0
$623$	19740.0	1.26945
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−34720.0	−2.21146
$628$	0	0
$629$	−16644.0	−1.05507
$630$	0	0
$631$	19788.0	1.24841	0.624206	−	0.781260i	$-0.285423\pi$
$631$	19788.0	1.24841	0.624206	+	0.781260i	$0.285423\pi$
$632$	0	0
$633$	27824.0	1.74708
$634$	0	0
$635$	1480.00	0.0924914
$636$	0	0
$637$	−6174.00	−0.384023
$638$	0	0
$639$	−33004.0	−2.04322
$640$	0	0
$641$	−9878.00	−0.608670	−0.304335	−	0.952565i	$-0.598434\pi$
$641$	−9878.00	−0.608670	−0.304335	+	0.952565i	$0.598434\pi$
$642$	0	0
$643$	23918.0	1.46693	0.733463	−	0.679729i	$-0.237903\pi$
$643$	23918.0	1.46693	0.733463	+	0.679729i	$0.237903\pi$
$644$	0	0
$645$	14080.0	0.859534
$646$	0	0
$647$	−28646.0	−1.74063	−0.870317	−	0.492492i	$-0.836086\pi$
$647$	−28646.0	−1.74063	−0.870317	+	0.492492i	$0.836086\pi$
$648$	0	0
$649$	52080.0	3.14995
$650$	0	0
$651$	−15904.0	−0.957491
$652$	0	0
$653$	−23578.0	−1.41298	−0.706492	−	0.707721i	$-0.749723\pi$
$653$	−23578.0	−1.41298	−0.706492	+	0.707721i	$0.749723\pi$
$654$	0	0
$655$	−990.000	−0.0590573
$656$	0	0
$657$	−1406.00	−0.0834905
$658$	0	0
$659$	−22770.0	−1.34597	−0.672984	−	0.739657i	$-0.734988\pi$
$659$	−22770.0	−1.34597	−0.672984	+	0.739657i	$0.734988\pi$
$660$	0	0
$661$	−1298.00	−0.0763787	−0.0381894	−	0.999271i	$-0.512159\pi$
$661$	−1298.00	−0.0763787	−0.0381894	+	0.999271i	$0.512159\pi$
$662$	0	0
$663$	−38304.0	−2.24375
$664$	0	0
$665$	−4900.00	−0.285735
$666$	0	0
$667$	1798.00	0.104376
$668$	0	0
$669$	−52976.0	−3.06154
$670$	0	0
$671$	−124.000	−0.00713408
$672$	0	0
$673$	1362.00	0.0780108	0.0390054	−	0.999239i	$-0.487581\pi$
$673$	1362.00	0.0780108	0.0390054	+	0.999239i	$0.487581\pi$
$674$	0	0
$675$	2000.00	0.114044
$676$	0	0
$677$	12326.0	0.699744	0.349872	−	0.936798i	$-0.386225\pi$
$677$	12326.0	0.699744	0.349872	+	0.936798i	$0.386225\pi$
$678$	0	0
$679$	6524.00	0.368731
$680$	0	0
$681$	−29328.0	−1.65030
$682$	0	0
$683$	22538.0	1.26265	0.631327	−	0.775517i	$-0.282511\pi$
$683$	22538.0	1.26265	0.631327	+	0.775517i	$0.282511\pi$
$684$	0	0
$685$	−830.000	−0.0462959
$686$	0	0
$687$	−2800.00	−0.155497
$688$	0	0
$689$	−9996.00	−0.552710
$690$	0	0
$691$	21148.0	1.16427	0.582133	−	0.813094i	$-0.302218\pi$
$691$	21148.0	1.16427	0.582133	+	0.813094i	$0.302218\pi$
$692$	0	0
$693$	−32116.0	−1.76044
$694$	0	0
$695$	−400.000	−0.0218315
$696$	0	0
$697$	−18468.0	−1.00362
$698$	0	0
$699$	16.0000	0.000865773 0
$700$	0	0
$701$	−24498.0	−1.31994	−0.659969	−	0.751293i	$-0.729431\pi$
$701$	−24498.0	−1.31994	−0.659969	+	0.751293i	$0.729431\pi$
$702$	0	0
$703$	10220.0	0.548300
$704$	0	0
$705$	−17760.0	−0.948766
$706$	0	0
$707$	−12292.0	−0.653873
$708$	0	0
$709$	17890.0	0.947635	0.473817	−	0.880623i	$-0.342876\pi$
$709$	17890.0	0.947635	0.473817	+	0.880623i	$0.342876\pi$
$710$	0	0
$711$	−38850.0	−2.04921
$712$	0	0
$713$	8804.00	0.462430
$714$	0	0
$715$	13020.0	0.681008
$716$	0	0
$717$	−5600.00	−0.291682
$718$	0	0
$719$	25140.0	1.30398	0.651992	−	0.758226i	$-0.273934\pi$
$719$	25140.0	1.30398	0.651992	+	0.758226i	$0.273934\pi$
$720$	0	0
$721$	−14868.0	−0.767980
$722$	0	0
$723$	−8144.00	−0.418919
$724$	0	0
$725$	−725.000	−0.0371391
$726$	0	0
$727$	9944.00	0.507294	0.253647	−	0.967297i	$-0.418370\pi$
$727$	9944.00	0.507294	0.253647	+	0.967297i	$0.418370\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	40128.0	2.03035
$732$	0	0
$733$	−25318.0	−1.27577	−0.637887	−	0.770130i	$-0.720191\pi$
$733$	−25318.0	−1.27577	−0.637887	+	0.770130i	$0.720191\pi$
$734$	0	0
$735$	5880.00	0.295084
$736$	0	0
$737$	−9548.00	−0.477212
$738$	0	0
$739$	7830.00	0.389758	0.194879	−	0.980827i	$-0.437569\pi$
$739$	7830.00	0.389758	0.194879	+	0.980827i	$0.437569\pi$
$740$	0	0
$741$	23520.0	1.16603
$742$	0	0
$743$	30008.0	1.48168	0.740839	−	0.671683i	$-0.234428\pi$
$743$	30008.0	1.48168	0.740839	+	0.671683i	$0.234428\pi$
$744$	0	0
$745$	−15750.0	−0.774544
$746$	0	0
$747$	28786.0	1.40994
$748$	0	0
$749$	−25564.0	−1.24711
$750$	0	0
$751$	−15562.0	−0.756146	−0.378073	−	0.925776i	$-0.623413\pi$
$751$	−15562.0	−0.756146	−0.378073	+	0.925776i	$0.623413\pi$
$752$	0	0
$753$	−56816.0	−2.74965
$754$	0	0
$755$	−9340.00	−0.450222
$756$	0	0
$757$	−6914.00	−0.331960	−0.165980	−	0.986129i	$-0.553079\pi$
$757$	−6914.00	−0.331960	−0.165980	+	0.986129i	$0.553079\pi$
$758$	0	0
$759$	30752.0	1.47065
$760$	0	0
$761$	−13018.0	−0.620108	−0.310054	−	0.950719i	$-0.600347\pi$
$761$	−13018.0	−0.620108	−0.310054	+	0.950719i	$0.600347\pi$
$762$	0	0
$763$	−17780.0	−0.843616
$764$	0	0
$765$	21090.0	0.996746
$766$	0	0
$767$	−35280.0	−1.66087
$768$	0	0
$769$	18450.0	0.865181	0.432590	−	0.901591i	$-0.357600\pi$
$769$	18450.0	0.865181	0.432590	+	0.901591i	$0.357600\pi$
$770$	0	0
$771$	47248.0	2.20700
$772$	0	0
$773$	−1838.00	−0.0855217	−0.0427608	−	0.999085i	$-0.513615\pi$
$773$	−1838.00	−0.0855217	−0.0427608	+	0.999085i	$0.513615\pi$
$774$	0	0
$775$	−3550.00	−0.164542
$776$	0	0
$777$	16352.0	0.754987
$778$	0	0
$779$	11340.0	0.521563
$780$	0	0
$781$	55304.0	2.53384
$782$	0	0
$783$	−2320.00	−0.105888
$784$	0	0
$785$	1770.00	0.0804764
$786$	0	0
$787$	5274.00	0.238879	0.119440	−	0.992841i	$-0.461890\pi$
$787$	5274.00	0.238879	0.119440	+	0.992841i	$0.461890\pi$
$788$	0	0
$789$	−50016.0	−2.25680
$790$	0	0
$791$	1148.00	0.0516033
$792$	0	0
$793$	84.0000	0.00376157
$794$	0	0
$795$	9520.00	0.424704
$796$	0	0
$797$	21926.0	0.974478	0.487239	−	0.873269i	$-0.338004\pi$
$797$	21926.0	0.974478	0.487239	+	0.873269i	$0.338004\pi$
$798$	0	0
$799$	−50616.0	−2.24113
$800$	0	0
$801$	52170.0	2.30129
$802$	0	0
$803$	2356.00	0.103539
$804$	0	0
$805$	4340.00	0.190019
$806$	0	0
$807$	−57840.0	−2.52300
$808$	0	0
$809$	35010.0	1.52149	0.760745	−	0.649050i	$-0.224834\pi$
$809$	35010.0	1.52149	0.760745	+	0.649050i	$0.224834\pi$
$810$	0	0
$811$	2888.00	0.125045	0.0625224	−	0.998044i	$-0.480086\pi$
$811$	2888.00	0.125045	0.0625224	+	0.998044i	$0.480086\pi$
$812$	0	0
$813$	62704.0	2.70495
$814$	0	0
$815$	−11340.0	−0.487390
$816$	0	0
$817$	−24640.0	−1.05513
$818$	0	0
$819$	21756.0	0.928225
$820$	0	0
$821$	3542.00	0.150568	0.0752842	−	0.997162i	$-0.476014\pi$
$821$	3542.00	0.150568	0.0752842	+	0.997162i	$0.476014\pi$
$822$	0	0
$823$	11468.0	0.485722	0.242861	−	0.970061i	$-0.421914\pi$
$823$	11468.0	0.485722	0.242861	+	0.970061i	$0.421914\pi$
$824$	0	0
$825$	−12400.0	−0.523288
$826$	0	0
$827$	−16616.0	−0.698664	−0.349332	−	0.936999i	$-0.613591\pi$
$827$	−16616.0	−0.698664	−0.349332	+	0.936999i	$0.613591\pi$
$828$	0	0
$829$	33970.0	1.42319	0.711596	−	0.702588i	$-0.247972\pi$
$829$	33970.0	1.42319	0.711596	+	0.702588i	$0.247972\pi$
$830$	0	0
$831$	43088.0	1.79868
$832$	0	0
$833$	16758.0	0.697035
$834$	0	0
$835$	10230.0	0.423981
$836$	0	0
$837$	−11360.0	−0.469127
$838$	0	0
$839$	43410.0	1.78627	0.893134	−	0.449790i	$-0.148501\pi$
$839$	43410.0	1.78627	0.893134	+	0.449790i	$0.148501\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	52016.0	2.12518
$844$	0	0
$845$	2165.00	0.0881400
$846$	0	0
$847$	35182.0	1.42723
$848$	0	0
$849$	51824.0	2.09493
$850$	0	0
$851$	−9052.00	−0.364628
$852$	0	0
$853$	82.0000	0.00329147	0.00164574	−	0.999999i	$-0.499476\pi$
$853$	82.0000	0.00329147	0.00164574	+	0.999999i	$0.499476\pi$
$854$	0	0
$855$	−12950.0	−0.517989
$856$	0	0
$857$	29306.0	1.16811	0.584057	−	0.811713i	$-0.301464\pi$
$857$	29306.0	1.16811	0.584057	+	0.811713i	$0.301464\pi$
$858$	0	0
$859$	490.000	0.0194628	0.00973142	−	0.999953i	$-0.496902\pi$
$859$	490.000	0.0194628	0.00973142	+	0.999953i	$0.496902\pi$
$860$	0	0
$861$	18144.0	0.718172
$862$	0	0
$863$	−31642.0	−1.24810	−0.624048	−	0.781386i	$-0.714513\pi$
$863$	−31642.0	−1.24810	−0.624048	+	0.781386i	$0.714513\pi$
$864$	0	0
$865$	10590.0	0.416267
$866$	0	0
$867$	64664.0	2.53299
$868$	0	0
$869$	65100.0	2.54127
$870$	0	0
$871$	6468.00	0.251619
$872$	0	0
$873$	17242.0	0.668446
$874$	0	0
$875$	−1750.00	−0.0676123
$876$	0	0
$877$	−7774.00	−0.299326	−0.149663	−	0.988737i	$-0.547819\pi$
$877$	−7774.00	−0.299326	−0.149663	+	0.988737i	$0.547819\pi$
$878$	0	0
$879$	−54864.0	−2.10525
$880$	0	0
$881$	14682.0	0.561463	0.280732	−	0.959786i	$-0.409423\pi$
$881$	14682.0	0.561463	0.280732	+	0.959786i	$0.409423\pi$
$882$	0	0
$883$	−922.000	−0.0351390	−0.0175695	−	0.999846i	$-0.505593\pi$
$883$	−922.000	−0.0351390	−0.0175695	+	0.999846i	$0.505593\pi$
$884$	0	0
$885$	33600.0	1.27622
$886$	0	0
$887$	9664.00	0.365823	0.182912	−	0.983129i	$-0.441448\pi$
$887$	9664.00	0.365823	0.182912	+	0.983129i	$0.441448\pi$
$888$	0	0
$889$	−4144.00	−0.156339
$890$	0	0
$891$	22258.0	0.836892
$892$	0	0
$893$	31080.0	1.16467
$894$	0	0
$895$	6300.00	0.235291
$896$	0	0
$897$	−20832.0	−0.775429
$898$	0	0
$899$	4118.00	0.152773
$900$	0	0
$901$	27132.0	1.00322
$902$	0	0
$903$	−39424.0	−1.45288
$904$	0	0
$905$	−6710.00	−0.246462
$906$	0	0
$907$	31904.0	1.16798	0.583988	−	0.811762i	$-0.301491\pi$
$907$	31904.0	1.16798	0.583988	+	0.811762i	$0.301491\pi$
$908$	0	0
$909$	−32486.0	−1.18536
$910$	0	0
$911$	27518.0	1.00078	0.500391	−	0.865800i	$-0.333190\pi$
$911$	27518.0	1.00078	0.500391	+	0.865800i	$0.333190\pi$
$912$	0	0
$913$	−48236.0	−1.74850
$914$	0	0
$915$	−80.0000	−0.00289040
$916$	0	0
$917$	2772.00	0.0998250
$918$	0	0
$919$	19420.0	0.697069	0.348535	−	0.937296i	$-0.386679\pi$
$919$	19420.0	0.697069	0.348535	+	0.937296i	$0.386679\pi$
$920$	0	0
$921$	29472.0	1.05444
$922$	0	0
$923$	−37464.0	−1.33602
$924$	0	0
$925$	3650.00	0.129742
$926$	0	0
$927$	−39294.0	−1.39222
$928$	0	0
$929$	39010.0	1.37769	0.688846	−	0.724907i	$-0.258117\pi$
$929$	39010.0	1.37769	0.688846	+	0.724907i	$0.258117\pi$
$930$	0	0
$931$	−10290.0	−0.362235
$932$	0	0
$933$	1104.00	0.0387388
$934$	0	0
$935$	−35340.0	−1.23609
$936$	0	0
$937$	−18734.0	−0.653162	−0.326581	−	0.945169i	$-0.605897\pi$
$937$	−18734.0	−0.653162	−0.326581	+	0.945169i	$0.605897\pi$
$938$	0	0
$939$	70416.0	2.44722
$940$	0	0
$941$	−29298.0	−1.01497	−0.507485	−	0.861660i	$-0.669425\pi$
$941$	−29298.0	−1.01497	−0.507485	+	0.861660i	$0.669425\pi$
$942$	0	0
$943$	−10044.0	−0.346848
$944$	0	0
$945$	−5600.00	−0.192770
$946$	0	0
$947$	45044.0	1.54565	0.772826	−	0.634617i	$-0.218842\pi$
$947$	45044.0	1.54565	0.772826	+	0.634617i	$0.218842\pi$
$948$	0	0
$949$	−1596.00	−0.0545926
$950$	0	0
$951$	44368.0	1.51286
$952$	0	0
$953$	30282.0	1.02931	0.514654	−	0.857398i	$-0.327920\pi$
$953$	30282.0	1.02931	0.514654	+	0.857398i	$0.327920\pi$
$954$	0	0
$955$	8110.00	0.274799
$956$	0	0
$957$	14384.0	0.485861
$958$	0	0
$959$	2324.00	0.0782543
$960$	0	0
$961$	−9627.00	−0.323151
$962$	0	0
$963$	−67562.0	−2.26080
$964$	0	0
$965$	20290.0	0.676848
$966$	0	0
$967$	15364.0	0.510934	0.255467	−	0.966818i	$-0.417771\pi$
$967$	15364.0	0.510934	0.255467	+	0.966818i	$0.417771\pi$
$968$	0	0
$969$	−63840.0	−2.11645
$970$	0	0
$971$	−44462.0	−1.46947	−0.734734	−	0.678355i	$-0.762693\pi$
$971$	−44462.0	−1.46947	−0.734734	+	0.678355i	$0.762693\pi$
$972$	0	0
$973$	1120.00	0.0369019
$974$	0	0
$975$	8400.00	0.275913
$976$	0	0
$977$	−18534.0	−0.606914	−0.303457	−	0.952845i	$-0.598141\pi$
$977$	−18534.0	−0.606914	−0.303457	+	0.952845i	$0.598141\pi$
$978$	0	0
$979$	−87420.0	−2.85389
$980$	0	0
$981$	−46990.0	−1.52933
$982$	0	0
$983$	−14272.0	−0.463078	−0.231539	−	0.972826i	$-0.574376\pi$
$983$	−14272.0	−0.463078	−0.231539	+	0.972826i	$0.574376\pi$
$984$	0	0
$985$	−1330.00	−0.0430227
$986$	0	0
$987$	49728.0	1.60371
$988$	0	0
$989$	21824.0	0.701681
$990$	0	0
$991$	6508.00	0.208611	0.104305	−	0.994545i	$-0.466738\pi$
$991$	6508.00	0.208611	0.104305	+	0.994545i	$0.466738\pi$
$992$	0	0
$993$	−24656.0	−0.787950
$994$	0	0
$995$	−11100.0	−0.353662
$996$	0	0
$997$	−39874.0	−1.26662	−0.633311	−	0.773897i	$-0.718305\pi$
$997$	−39874.0	−1.26662	−0.633311	+	0.773897i	$0.718305\pi$
$998$	0	0
$999$	11680.0	0.369909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.4.a.f.1.1		1
4.3	odd	2		145.4.a.a.1.1	✓	1
12.11	even	2		1305.4.a.b.1.1		1
20.19	odd	2		725.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.4.a.a.1.1	✓	1	4.3	odd	2
725.4.a.a.1.1		1	20.19	odd	2
1305.4.a.b.1.1		1	12.11	even	2
2320.4.a.f.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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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