LMFDB - Embedded newform 368.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,4,Mod(1,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("368.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	368.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:	$21.7127028821$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} -20.0000 q^{5} -2.00000 q^{7} +54.0000 q^{9} +52.0000 q^{11} +43.0000 q^{13} -180.000 q^{15} -50.0000 q^{17} +74.0000 q^{19} -18.0000 q^{21} +23.0000 q^{23} +275.000 q^{25} +243.000 q^{27} -7.00000 q^{29} +273.000 q^{31} +468.000 q^{33} +40.0000 q^{35} -4.00000 q^{37} +387.000 q^{39} +123.000 q^{41} +152.000 q^{43} -1080.00 q^{45} -75.0000 q^{47} -339.000 q^{49} -450.000 q^{51} +86.0000 q^{53} -1040.00 q^{55} +666.000 q^{57} +444.000 q^{59} +262.000 q^{61} -108.000 q^{63} -860.000 q^{65} -764.000 q^{67} +207.000 q^{69} +21.0000 q^{71} +681.000 q^{73} +2475.00 q^{75} -104.000 q^{77} -426.000 q^{79} +729.000 q^{81} -902.000 q^{83} +1000.00 q^{85} -63.0000 q^{87} -1272.00 q^{89} -86.0000 q^{91} +2457.00 q^{93} -1480.00 q^{95} -342.000 q^{97} +2808.00 q^{99} +O(q^{100})$

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
$3$	9.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	9.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	−20.0000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−20.0000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	−2.00000	−0.107990	−0.0539949	−	0.998541i	$-0.517195\pi$
$7$	−2.00000	−0.107990	−0.0539949	+	0.998541i	$0.517195\pi$
$8$	0	0
$9$	54.0000	2.00000
$10$	0	0
$11$	52.0000	1.42533	0.712663	−	0.701506i	$-0.247489\pi$
$11$	52.0000	1.42533	0.712663	+	0.701506i	$0.247489\pi$
$12$	0	0
$13$	43.0000	0.917389	0.458694	−	0.888594i	$-0.348317\pi$
$13$	43.0000	0.917389	0.458694	+	0.888594i	$0.348317\pi$
$14$	0	0
$15$	−180.000	−3.09839
$16$	0	0
$17$	−50.0000	−0.713340	−0.356670	−	0.934230i	$-0.616088\pi$
$17$	−50.0000	−0.713340	−0.356670	+	0.934230i	$0.616088\pi$
$18$	0	0
$19$	74.0000	0.893514	0.446757	−	0.894655i	$-0.352579\pi$
$19$	74.0000	0.893514	0.446757	+	0.894655i	$0.352579\pi$
$20$	0	0
$21$	−18.0000	−0.187044
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	275.000	2.20000
$26$	0	0
$27$	243.000	1.73205
$28$	0	0
$29$	−7.00000	−0.0448230	−0.0224115	−	0.999749i	$-0.507134\pi$
$29$	−7.00000	−0.0448230	−0.0224115	+	0.999749i	$0.507134\pi$
$30$	0	0
$31$	273.000	1.58169	0.790843	−	0.612019i	$-0.209643\pi$
$31$	273.000	1.58169	0.790843	+	0.612019i	$0.209643\pi$
$32$	0	0
$33$	468.000	2.46874
$34$	0	0
$35$	40.0000	0.193178
$36$	0	0
$37$	−4.00000	−0.0177729	−0.00888643	−	0.999961i	$-0.502829\pi$
$37$	−4.00000	−0.0177729	−0.00888643	+	0.999961i	$0.502829\pi$
$38$	0	0
$39$	387.000	1.58896
$40$	0	0
$41$	123.000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	123.000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	152.000	0.539065	0.269532	−	0.962991i	$-0.413131\pi$
$43$	152.000	0.539065	0.269532	+	0.962991i	$0.413131\pi$
$44$	0	0
$45$	−1080.00	−3.57771
$46$	0	0
$47$	−75.0000	−0.232763	−0.116382	−	0.993205i	$-0.537130\pi$
$47$	−75.0000	−0.232763	−0.116382	+	0.993205i	$0.537130\pi$
$48$	0	0
$49$	−339.000	−0.988338
$50$	0	0
$51$	−450.000	−1.23554
$52$	0	0
$53$	86.0000	0.222887	0.111443	−	0.993771i	$-0.464453\pi$
$53$	86.0000	0.222887	0.111443	+	0.993771i	$0.464453\pi$
$54$	0	0
$55$	−1040.00	−2.54970
$56$	0	0
$57$	666.000	1.54761
$58$	0	0
$59$	444.000	0.979727	0.489863	−	0.871799i	$-0.337047\pi$
$59$	444.000	0.979727	0.489863	+	0.871799i	$0.337047\pi$
$60$	0	0
$61$	262.000	0.549929	0.274964	−	0.961454i	$-0.411334\pi$
$61$	262.000	0.549929	0.274964	+	0.961454i	$0.411334\pi$
$62$	0	0
$63$	−108.000	−0.215980
$64$	0	0
$65$	−860.000	−1.64107
$66$	0	0
$67$	−764.000	−1.39310	−0.696548	−	0.717510i	$-0.745282\pi$
$67$	−764.000	−1.39310	−0.696548	+	0.717510i	$0.745282\pi$
$68$	0	0
$69$	207.000	0.361158
$70$	0	0
$71$	21.0000	0.0351020	0.0175510	−	0.999846i	$-0.494413\pi$
$71$	21.0000	0.0351020	0.0175510	+	0.999846i	$0.494413\pi$
$72$	0	0
$73$	681.000	1.09185	0.545925	−	0.837834i	$-0.316178\pi$
$73$	681.000	1.09185	0.545925	+	0.837834i	$0.316178\pi$
$74$	0	0
$75$	2475.00	3.81051
$76$	0	0
$77$	−104.000	−0.153921
$78$	0	0
$79$	−426.000	−0.606693	−0.303346	−	0.952880i	$-0.598104\pi$
$79$	−426.000	−0.606693	−0.303346	+	0.952880i	$0.598104\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	−902.000	−1.19286	−0.596430	−	0.802665i	$-0.703415\pi$
$83$	−902.000	−1.19286	−0.596430	+	0.802665i	$0.703415\pi$
$84$	0	0
$85$	1000.00	1.27606
$86$	0	0
$87$	−63.0000	−0.0776357
$88$	0	0
$89$	−1272.00	−1.51496	−0.757482	−	0.652856i	$-0.773570\pi$
$89$	−1272.00	−1.51496	−0.757482	+	0.652856i	$0.773570\pi$
$90$	0	0
$91$	−86.0000	−0.0990687
$92$	0	0
$93$	2457.00	2.73956
$94$	0	0
$95$	−1480.00	−1.59837
$96$	0	0
$97$	−342.000	−0.357988	−0.178994	−	0.983850i	$-0.557284\pi$
$97$	−342.000	−0.357988	−0.178994	+	0.983850i	$0.557284\pi$
$98$	0	0
$99$	2808.00	2.85065
$100$	0	0
$101$	−1426.00	−1.40487	−0.702437	−	0.711746i	$-0.747905\pi$
$101$	−1426.00	−1.40487	−0.702437	+	0.711746i	$0.747905\pi$
$102$	0	0
$103$	1190.00	1.13839	0.569195	−	0.822203i	$-0.307255\pi$
$103$	1190.00	1.13839	0.569195	+	0.822203i	$0.307255\pi$
$104$	0	0
$105$	360.000	0.334594
$106$	0	0
$107$	1210.00	1.09323	0.546613	−	0.837386i	$-0.315917\pi$
$107$	1210.00	1.09323	0.546613	+	0.837386i	$0.315917\pi$
$108$	0	0
$109$	−1680.00	−1.47628	−0.738141	−	0.674646i	$-0.764296\pi$
$109$	−1680.00	−1.47628	−0.738141	+	0.674646i	$0.764296\pi$
$110$	0	0
$111$	−36.0000	−0.0307835
$112$	0	0
$113$	1030.00	0.857471	0.428736	−	0.903430i	$-0.358959\pi$
$113$	1030.00	0.857471	0.428736	+	0.903430i	$0.358959\pi$
$114$	0	0
$115$	−460.000	−0.373002
$116$	0	0
$117$	2322.00	1.83478
$118$	0	0
$119$	100.000	0.0770335
$120$	0	0
$121$	1373.00	1.03156
$122$	0	0
$123$	1107.00	0.811503
$124$	0	0
$125$	−3000.00	−2.14663
$126$	0	0
$127$	2279.00	1.59235	0.796175	−	0.605066i	$-0.206853\pi$
$127$	2279.00	1.59235	0.796175	+	0.605066i	$0.206853\pi$
$128$	0	0
$129$	1368.00	0.933687
$130$	0	0
$131$	−987.000	−0.658279	−0.329140	−	0.944281i	$-0.606759\pi$
$131$	−987.000	−0.658279	−0.329140	+	0.944281i	$0.606759\pi$
$132$	0	0
$133$	−148.000	−0.0964904
$134$	0	0
$135$	−4860.00	−3.09839
$136$	0	0
$137$	−1644.00	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−1644.00	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−2189.00	−1.33575	−0.667873	−	0.744276i	$-0.732795\pi$
$139$	−2189.00	−1.33575	−0.667873	+	0.744276i	$0.732795\pi$
$140$	0	0
$141$	−675.000	−0.403158
$142$	0	0
$143$	2236.00	1.30758
$144$	0	0
$145$	140.000	0.0801818
$146$	0	0
$147$	−3051.00	−1.71185
$148$	0	0
$149$	−946.000	−0.520130	−0.260065	−	0.965591i	$-0.583744\pi$
$149$	−946.000	−0.520130	−0.260065	+	0.965591i	$0.583744\pi$
$150$	0	0
$151$	365.000	0.196710	0.0983552	−	0.995151i	$-0.468642\pi$
$151$	365.000	0.196710	0.0983552	+	0.995151i	$0.468642\pi$
$152$	0	0
$153$	−2700.00	−1.42668
$154$	0	0
$155$	−5460.00	−2.82940
$156$	0	0
$157$	−108.000	−0.0549002	−0.0274501	−	0.999623i	$-0.508739\pi$
$157$	−108.000	−0.0549002	−0.0274501	+	0.999623i	$0.508739\pi$
$158$	0	0
$159$	774.000	0.386052
$160$	0	0
$161$	−46.0000	−0.0225174
$162$	0	0
$163$	1415.00	0.679947	0.339973	−	0.940435i	$-0.389582\pi$
$163$	1415.00	0.679947	0.339973	+	0.940435i	$0.389582\pi$
$164$	0	0
$165$	−9360.00	−4.41621
$166$	0	0
$167$	−1756.00	−0.813673	−0.406836	−	0.913501i	$-0.633368\pi$
$167$	−1756.00	−0.813673	−0.406836	+	0.913501i	$0.633368\pi$
$168$	0	0
$169$	−348.000	−0.158398
$170$	0	0
$171$	3996.00	1.78703
$172$	0	0
$173$	2358.00	1.03627	0.518137	−	0.855298i	$-0.326626\pi$
$173$	2358.00	1.03627	0.518137	+	0.855298i	$0.326626\pi$
$174$	0	0
$175$	−550.000	−0.237578
$176$	0	0
$177$	3996.00	1.69694
$178$	0	0
$179$	−1073.00	−0.448043	−0.224022	−	0.974584i	$-0.571919\pi$
$179$	−1073.00	−0.448043	−0.224022	+	0.974584i	$0.571919\pi$
$180$	0	0
$181$	2868.00	1.17777	0.588886	−	0.808216i	$-0.299567\pi$
$181$	2868.00	1.17777	0.588886	+	0.808216i	$0.299567\pi$
$182$	0	0
$183$	2358.00	0.952505
$184$	0	0
$185$	80.0000	0.0317931
$186$	0	0
$187$	−2600.00	−1.01674
$188$	0	0
$189$	−486.000	−0.187044
$190$	0	0
$191$	−332.000	−0.125773	−0.0628866	−	0.998021i	$-0.520031\pi$
$191$	−332.000	−0.125773	−0.0628866	+	0.998021i	$0.520031\pi$
$192$	0	0
$193$	−2143.00	−0.799257	−0.399628	−	0.916677i	$-0.630861\pi$
$193$	−2143.00	−0.799257	−0.399628	+	0.916677i	$0.630861\pi$
$194$	0	0
$195$	−7740.00	−2.84243
$196$	0	0
$197$	−2739.00	−0.990587	−0.495294	−	0.868726i	$-0.664940\pi$
$197$	−2739.00	−0.990587	−0.495294	+	0.868726i	$0.664940\pi$
$198$	0	0
$199$	−752.000	−0.267879	−0.133939	−	0.990990i	$-0.542763\pi$
$199$	−752.000	−0.267879	−0.133939	+	0.990990i	$0.542763\pi$
$200$	0	0
$201$	−6876.00	−2.41291
$202$	0	0
$203$	14.0000	0.00484043
$204$	0	0
$205$	−2460.00	−0.838116
$206$	0	0
$207$	1242.00	0.417029
$208$	0	0
$209$	3848.00	1.27355
$210$	0	0
$211$	1016.00	0.331490	0.165745	−	0.986169i	$-0.446997\pi$
$211$	1016.00	0.331490	0.165745	+	0.986169i	$0.446997\pi$
$212$	0	0
$213$	189.000	0.0607984
$214$	0	0
$215$	−3040.00	−0.964308
$216$	0	0
$217$	−546.000	−0.170806
$218$	0	0
$219$	6129.00	1.89114
$220$	0	0
$221$	−2150.00	−0.654410
$222$	0	0
$223$	1120.00	0.336326	0.168163	−	0.985759i	$-0.446216\pi$
$223$	1120.00	0.336326	0.168163	+	0.985759i	$0.446216\pi$
$224$	0	0
$225$	14850.0	4.40000
$226$	0	0
$227$	−2706.00	−0.791205	−0.395602	−	0.918422i	$-0.629464\pi$
$227$	−2706.00	−0.791205	−0.395602	+	0.918422i	$0.629464\pi$
$228$	0	0
$229$	6140.00	1.77180	0.885901	−	0.463875i	$-0.153541\pi$
$229$	6140.00	1.77180	0.885901	+	0.463875i	$0.153541\pi$
$230$	0	0
$231$	−936.000	−0.266599
$232$	0	0
$233$	6567.00	1.84643	0.923216	−	0.384282i	$-0.125551\pi$
$233$	6567.00	1.84643	0.923216	+	0.384282i	$0.125551\pi$
$234$	0	0
$235$	1500.00	0.416380
$236$	0	0
$237$	−3834.00	−1.05082
$238$	0	0
$239$	729.000	0.197302	0.0986508	−	0.995122i	$-0.468547\pi$
$239$	729.000	0.197302	0.0986508	+	0.995122i	$0.468547\pi$
$240$	0	0
$241$	−2912.00	−0.778334	−0.389167	−	0.921167i	$-0.627237\pi$
$241$	−2912.00	−0.778334	−0.389167	+	0.921167i	$0.627237\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6780.00	1.76799
$246$	0	0
$247$	3182.00	0.819700
$248$	0	0
$249$	−8118.00	−2.06609
$250$	0	0
$251$	−398.000	−0.100086	−0.0500429	−	0.998747i	$-0.515936\pi$
$251$	−398.000	−0.100086	−0.0500429	+	0.998747i	$0.515936\pi$
$252$	0	0
$253$	1196.00	0.297201
$254$	0	0
$255$	9000.00	2.21020
$256$	0	0
$257$	8131.00	1.97353	0.986766	−	0.162149i	$-0.0518426\pi$
$257$	8131.00	1.97353	0.986766	+	0.162149i	$0.0518426\pi$
$258$	0	0
$259$	8.00000	0.00191929
$260$	0	0
$261$	−378.000	−0.0896460
$262$	0	0
$263$	1978.00	0.463759	0.231880	−	0.972744i	$-0.425512\pi$
$263$	1978.00	0.463759	0.231880	+	0.972744i	$0.425512\pi$
$264$	0	0
$265$	−1720.00	−0.398712
$266$	0	0
$267$	−11448.0	−2.62399
$268$	0	0
$269$	−8459.00	−1.91730	−0.958651	−	0.284584i	$-0.908145\pi$
$269$	−8459.00	−1.91730	−0.958651	+	0.284584i	$0.908145\pi$
$270$	0	0
$271$	7240.00	1.62287	0.811437	−	0.584440i	$-0.198686\pi$
$271$	7240.00	1.62287	0.811437	+	0.584440i	$0.198686\pi$
$272$	0	0
$273$	−774.000	−0.171592
$274$	0	0
$275$	14300.0	3.13572
$276$	0	0
$277$	−1319.00	−0.286105	−0.143052	−	0.989715i	$-0.545692\pi$
$277$	−1319.00	−0.286105	−0.143052	+	0.989715i	$0.545692\pi$
$278$	0	0
$279$	14742.0	3.16337
$280$	0	0
$281$	1770.00	0.375763	0.187881	−	0.982192i	$-0.439838\pi$
$281$	1770.00	0.375763	0.187881	+	0.982192i	$0.439838\pi$
$282$	0	0
$283$	−4144.00	−0.870443	−0.435221	−	0.900324i	$-0.643330\pi$
$283$	−4144.00	−0.870443	−0.435221	+	0.900324i	$0.643330\pi$
$284$	0	0
$285$	−13320.0	−2.76845
$286$	0	0
$287$	−246.000	−0.0505955
$288$	0	0
$289$	−2413.00	−0.491146
$290$	0	0
$291$	−3078.00	−0.620053
$292$	0	0
$293$	−6812.00	−1.35823	−0.679115	−	0.734032i	$-0.737636\pi$
$293$	−6812.00	−1.35823	−0.679115	+	0.734032i	$0.737636\pi$
$294$	0	0
$295$	−8880.00	−1.75259
$296$	0	0
$297$	12636.0	2.46874
$298$	0	0
$299$	989.000	0.191289
$300$	0	0
$301$	−304.000	−0.0582135
$302$	0	0
$303$	−12834.0	−2.43331
$304$	0	0
$305$	−5240.00	−0.983743
$306$	0	0
$307$	5692.00	1.05817	0.529087	−	0.848567i	$-0.322534\pi$
$307$	5692.00	1.05817	0.529087	+	0.848567i	$0.322534\pi$
$308$	0	0
$309$	10710.0	1.97175
$310$	0	0
$311$	−5267.00	−0.960335	−0.480167	−	0.877177i	$-0.659424\pi$
$311$	−5267.00	−0.960335	−0.480167	+	0.877177i	$0.659424\pi$
$312$	0	0
$313$	6340.00	1.14491	0.572457	−	0.819935i	$-0.305990\pi$
$313$	6340.00	1.14491	0.572457	+	0.819935i	$0.305990\pi$
$314$	0	0
$315$	2160.00	0.386356
$316$	0	0
$317$	−8794.00	−1.55811	−0.779054	−	0.626957i	$-0.784300\pi$
$317$	−8794.00	−1.55811	−0.779054	+	0.626957i	$0.784300\pi$
$318$	0	0
$319$	−364.000	−0.0638874
$320$	0	0
$321$	10890.0	1.89352
$322$	0	0
$323$	−3700.00	−0.637379
$324$	0	0
$325$	11825.0	2.01826
$326$	0	0
$327$	−15120.0	−2.55700
$328$	0	0
$329$	150.000	0.0251361
$330$	0	0
$331$	−8225.00	−1.36582	−0.682911	−	0.730502i	$-0.739286\pi$
$331$	−8225.00	−1.36582	−0.682911	+	0.730502i	$0.739286\pi$
$332$	0	0
$333$	−216.000	−0.0355457
$334$	0	0
$335$	15280.0	2.49205
$336$	0	0
$337$	−2576.00	−0.416391	−0.208195	−	0.978087i	$-0.566759\pi$
$337$	−2576.00	−0.416391	−0.208195	+	0.978087i	$0.566759\pi$
$338$	0	0
$339$	9270.00	1.48518
$340$	0	0
$341$	14196.0	2.25442
$342$	0	0
$343$	1364.00	0.214720
$344$	0	0
$345$	−4140.00	−0.646058
$346$	0	0
$347$	−596.000	−0.0922045	−0.0461022	−	0.998937i	$-0.514680\pi$
$347$	−596.000	−0.0922045	−0.0461022	+	0.998937i	$0.514680\pi$
$348$	0	0
$349$	−9271.00	−1.42196	−0.710982	−	0.703210i	$-0.751749\pi$
$349$	−9271.00	−1.42196	−0.710982	+	0.703210i	$0.751749\pi$
$350$	0	0
$351$	10449.0	1.58896
$352$	0	0
$353$	8141.00	1.22748	0.613742	−	0.789507i	$-0.289664\pi$
$353$	8141.00	1.22748	0.613742	+	0.789507i	$0.289664\pi$
$354$	0	0
$355$	−420.000	−0.0627924
$356$	0	0
$357$	900.000	0.133426
$358$	0	0
$359$	2130.00	0.313140	0.156570	−	0.987667i	$-0.449956\pi$
$359$	2130.00	0.313140	0.156570	+	0.987667i	$0.449956\pi$
$360$	0	0
$361$	−1383.00	−0.201633
$362$	0	0
$363$	12357.0	1.78671
$364$	0	0
$365$	−13620.0	−1.95316
$366$	0	0
$367$	2574.00	0.366108	0.183054	−	0.983103i	$-0.441402\pi$
$367$	2574.00	0.366108	0.183054	+	0.983103i	$0.441402\pi$
$368$	0	0
$369$	6642.00	0.937043
$370$	0	0
$371$	−172.000	−0.0240695
$372$	0	0
$373$	−4504.00	−0.625223	−0.312612	−	0.949881i	$-0.601204\pi$
$373$	−4504.00	−0.625223	−0.312612	+	0.949881i	$0.601204\pi$
$374$	0	0
$375$	−27000.0	−3.71806
$376$	0	0
$377$	−301.000	−0.0411201
$378$	0	0
$379$	−2740.00	−0.371357	−0.185679	−	0.982611i	$-0.559448\pi$
$379$	−2740.00	−0.371357	−0.185679	+	0.982611i	$0.559448\pi$
$380$	0	0
$381$	20511.0	2.75803
$382$	0	0
$383$	−6948.00	−0.926961	−0.463481	−	0.886107i	$-0.653400\pi$
$383$	−6948.00	−0.926961	−0.463481	+	0.886107i	$0.653400\pi$
$384$	0	0
$385$	2080.00	0.275342
$386$	0	0
$387$	8208.00	1.07813
$388$	0	0
$389$	−1404.00	−0.182996	−0.0914982	−	0.995805i	$-0.529166\pi$
$389$	−1404.00	−0.182996	−0.0914982	+	0.995805i	$0.529166\pi$
$390$	0	0
$391$	−1150.00	−0.148742
$392$	0	0
$393$	−8883.00	−1.14017
$394$	0	0
$395$	8520.00	1.08529
$396$	0	0
$397$	−8641.00	−1.09239	−0.546196	−	0.837658i	$-0.683924\pi$
$397$	−8641.00	−1.09239	−0.546196	+	0.837658i	$0.683924\pi$
$398$	0	0
$399$	−1332.00	−0.167126
$400$	0	0
$401$	−1140.00	−0.141967	−0.0709836	−	0.997477i	$-0.522614\pi$
$401$	−1140.00	−0.141967	−0.0709836	+	0.997477i	$0.522614\pi$
$402$	0	0
$403$	11739.0	1.45102
$404$	0	0
$405$	−14580.0	−1.78885
$406$	0	0
$407$	−208.000	−0.0253321
$408$	0	0
$409$	12529.0	1.51472	0.757358	−	0.652999i	$-0.226490\pi$
$409$	12529.0	1.51472	0.757358	+	0.652999i	$0.226490\pi$
$410$	0	0
$411$	−14796.0	−1.77575
$412$	0	0
$413$	−888.000	−0.105801
$414$	0	0
$415$	18040.0	2.13385
$416$	0	0
$417$	−19701.0	−2.31358
$418$	0	0
$419$	−3252.00	−0.379166	−0.189583	−	0.981865i	$-0.560714\pi$
$419$	−3252.00	−0.379166	−0.189583	+	0.981865i	$0.560714\pi$
$420$	0	0
$421$	2206.00	0.255377	0.127689	−	0.991814i	$-0.459244\pi$
$421$	2206.00	0.255377	0.127689	+	0.991814i	$0.459244\pi$
$422$	0	0
$423$	−4050.00	−0.465527
$424$	0	0
$425$	−13750.0	−1.56935
$426$	0	0
$427$	−524.000	−0.0593867
$428$	0	0
$429$	20124.0	2.26479
$430$	0	0
$431$	−14316.0	−1.59995	−0.799974	−	0.600035i	$-0.795153\pi$
$431$	−14316.0	−1.59995	−0.799974	+	0.600035i	$0.795153\pi$
$432$	0	0
$433$	7828.00	0.868798	0.434399	−	0.900720i	$-0.356961\pi$
$433$	7828.00	0.868798	0.434399	+	0.900720i	$0.356961\pi$
$434$	0	0
$435$	1260.00	0.138879
$436$	0	0
$437$	1702.00	0.186311
$438$	0	0
$439$	16039.0	1.74374	0.871868	−	0.489742i	$-0.162909\pi$
$439$	16039.0	1.74374	0.871868	+	0.489742i	$0.162909\pi$
$440$	0	0
$441$	−18306.0	−1.97668
$442$	0	0
$443$	−11747.0	−1.25986	−0.629929	−	0.776653i	$-0.716916\pi$
$443$	−11747.0	−1.25986	−0.629929	+	0.776653i	$0.716916\pi$
$444$	0	0
$445$	25440.0	2.71005
$446$	0	0
$447$	−8514.00	−0.900891
$448$	0	0
$449$	−2890.00	−0.303758	−0.151879	−	0.988399i	$-0.548532\pi$
$449$	−2890.00	−0.303758	−0.151879	+	0.988399i	$0.548532\pi$
$450$	0	0
$451$	6396.00	0.667796
$452$	0	0
$453$	3285.00	0.340713
$454$	0	0
$455$	1720.00	0.177219
$456$	0	0
$457$	−13126.0	−1.34356	−0.671782	−	0.740749i	$-0.734471\pi$
$457$	−13126.0	−1.34356	−0.671782	+	0.740749i	$0.734471\pi$
$458$	0	0
$459$	−12150.0	−1.23554
$460$	0	0
$461$	14481.0	1.46301	0.731505	−	0.681836i	$-0.238818\pi$
$461$	14481.0	1.46301	0.731505	+	0.681836i	$0.238818\pi$
$462$	0	0
$463$	−5272.00	−0.529181	−0.264590	−	0.964361i	$-0.585237\pi$
$463$	−5272.00	−0.529181	−0.264590	+	0.964361i	$0.585237\pi$
$464$	0	0
$465$	−49140.0	−4.90067
$466$	0	0
$467$	13466.0	1.33433	0.667165	−	0.744910i	$-0.267508\pi$
$467$	13466.0	1.33433	0.667165	+	0.744910i	$0.267508\pi$
$468$	0	0
$469$	1528.00	0.150440
$470$	0	0
$471$	−972.000	−0.0950900
$472$	0	0
$473$	7904.00	0.768343
$474$	0	0
$475$	20350.0	1.96573
$476$	0	0
$477$	4644.00	0.445774
$478$	0	0
$479$	4526.00	0.431729	0.215865	−	0.976423i	$-0.430743\pi$
$479$	4526.00	0.431729	0.215865	+	0.976423i	$0.430743\pi$
$480$	0	0
$481$	−172.000	−0.0163046
$482$	0	0
$483$	−414.000	−0.0390014
$484$	0	0
$485$	6840.00	0.640388
$486$	0	0
$487$	8795.00	0.818356	0.409178	−	0.912455i	$-0.365815\pi$
$487$	8795.00	0.818356	0.409178	+	0.912455i	$0.365815\pi$
$488$	0	0
$489$	12735.0	1.17770
$490$	0	0
$491$	1275.00	0.117189	0.0585946	−	0.998282i	$-0.481338\pi$
$491$	1275.00	0.117189	0.0585946	+	0.998282i	$0.481338\pi$
$492$	0	0
$493$	350.000	0.0319741
$494$	0	0
$495$	−56160.0	−5.09940
$496$	0	0
$497$	−42.0000	−0.00379066
$498$	0	0
$499$	9533.00	0.855222	0.427611	−	0.903963i	$-0.359355\pi$
$499$	9533.00	0.855222	0.427611	+	0.903963i	$0.359355\pi$
$500$	0	0
$501$	−15804.0	−1.40932
$502$	0	0
$503$	13398.0	1.18765	0.593824	−	0.804595i	$-0.297617\pi$
$503$	13398.0	1.18765	0.593824	+	0.804595i	$0.297617\pi$
$504$	0	0
$505$	28520.0	2.51312
$506$	0	0
$507$	−3132.00	−0.274353
$508$	0	0
$509$	−8031.00	−0.699347	−0.349674	−	0.936872i	$-0.613708\pi$
$509$	−8031.00	−0.699347	−0.349674	+	0.936872i	$0.613708\pi$
$510$	0	0
$511$	−1362.00	−0.117909
$512$	0	0
$513$	17982.0	1.54761
$514$	0	0
$515$	−23800.0	−2.03641
$516$	0	0
$517$	−3900.00	−0.331764
$518$	0	0
$519$	21222.0	1.79488
$520$	0	0
$521$	−21184.0	−1.78136	−0.890679	−	0.454632i	$-0.849771\pi$
$521$	−21184.0	−1.78136	−0.890679	+	0.454632i	$0.849771\pi$
$522$	0	0
$523$	21706.0	1.81479	0.907397	−	0.420275i	$-0.138066\pi$
$523$	21706.0	1.81479	0.907397	+	0.420275i	$0.138066\pi$
$524$	0	0
$525$	−4950.00	−0.411497
$526$	0	0
$527$	−13650.0	−1.12828
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	23976.0	1.95945
$532$	0	0
$533$	5289.00	0.429816
$534$	0	0
$535$	−24200.0	−1.95562
$536$	0	0
$537$	−9657.00	−0.776034
$538$	0	0
$539$	−17628.0	−1.40870
$540$	0	0
$541$	5781.00	0.459417	0.229709	−	0.973259i	$-0.426223\pi$
$541$	5781.00	0.459417	0.229709	+	0.973259i	$0.426223\pi$
$542$	0	0
$543$	25812.0	2.03996
$544$	0	0
$545$	33600.0	2.64085
$546$	0	0
$547$	−7809.00	−0.610400	−0.305200	−	0.952288i	$-0.598723\pi$
$547$	−7809.00	−0.610400	−0.305200	+	0.952288i	$0.598723\pi$
$548$	0	0
$549$	14148.0	1.09986
$550$	0	0
$551$	−518.000	−0.0400500
$552$	0	0
$553$	852.000	0.0655167
$554$	0	0
$555$	720.000	0.0550672
$556$	0	0
$557$	−20240.0	−1.53967	−0.769835	−	0.638243i	$-0.779662\pi$
$557$	−20240.0	−1.53967	−0.769835	+	0.638243i	$0.779662\pi$
$558$	0	0
$559$	6536.00	0.494532
$560$	0	0
$561$	−23400.0	−1.76105
$562$	0	0
$563$	−7612.00	−0.569818	−0.284909	−	0.958555i	$-0.591963\pi$
$563$	−7612.00	−0.569818	−0.284909	+	0.958555i	$0.591963\pi$
$564$	0	0
$565$	−20600.0	−1.53389
$566$	0	0
$567$	−1458.00	−0.107990
$568$	0	0
$569$	19484.0	1.43552	0.717761	−	0.696290i	$-0.245167\pi$
$569$	19484.0	1.43552	0.717761	+	0.696290i	$0.245167\pi$
$570$	0	0
$571$	−6614.00	−0.484741	−0.242371	−	0.970184i	$-0.577925\pi$
$571$	−6614.00	−0.484741	−0.242371	+	0.970184i	$0.577925\pi$
$572$	0	0
$573$	−2988.00	−0.217846
$574$	0	0
$575$	6325.00	0.458732
$576$	0	0
$577$	−639.000	−0.0461038	−0.0230519	−	0.999734i	$-0.507338\pi$
$577$	−639.000	−0.0461038	−0.0230519	+	0.999734i	$0.507338\pi$
$578$	0	0
$579$	−19287.0	−1.38435
$580$	0	0
$581$	1804.00	0.128817
$582$	0	0
$583$	4472.00	0.317687
$584$	0	0
$585$	−46440.0	−3.28215
$586$	0	0
$587$	829.000	0.0582904	0.0291452	−	0.999575i	$-0.490721\pi$
$587$	829.000	0.0582904	0.0291452	+	0.999575i	$0.490721\pi$
$588$	0	0
$589$	20202.0	1.41326
$590$	0	0
$591$	−24651.0	−1.71575
$592$	0	0
$593$	−20610.0	−1.42724	−0.713618	−	0.700535i	$-0.752945\pi$
$593$	−20610.0	−1.42724	−0.713618	+	0.700535i	$0.752945\pi$
$594$	0	0
$595$	−2000.00	−0.137802
$596$	0	0
$597$	−6768.00	−0.463980
$598$	0	0
$599$	17240.0	1.17597	0.587986	−	0.808871i	$-0.299921\pi$
$599$	17240.0	1.17597	0.587986	+	0.808871i	$0.299921\pi$
$600$	0	0
$601$	−8459.00	−0.574126	−0.287063	−	0.957912i	$-0.592679\pi$
$601$	−8459.00	−0.574126	−0.287063	+	0.957912i	$0.592679\pi$
$602$	0	0
$603$	−41256.0	−2.78619
$604$	0	0
$605$	−27460.0	−1.84530
$606$	0	0
$607$	−17840.0	−1.19292	−0.596461	−	0.802642i	$-0.703427\pi$
$607$	−17840.0	−1.19292	−0.596461	+	0.802642i	$0.703427\pi$
$608$	0	0
$609$	126.000	0.00838387
$610$	0	0
$611$	−3225.00	−0.213534
$612$	0	0
$613$	2534.00	0.166961	0.0834807	−	0.996509i	$-0.473396\pi$
$613$	2534.00	0.166961	0.0834807	+	0.996509i	$0.473396\pi$
$614$	0	0
$615$	−22140.0	−1.45166
$616$	0	0
$617$	−5610.00	−0.366046	−0.183023	−	0.983109i	$-0.558588\pi$
$617$	−5610.00	−0.366046	−0.183023	+	0.983109i	$0.558588\pi$
$618$	0	0
$619$	11948.0	0.775817	0.387908	−	0.921698i	$-0.373198\pi$
$619$	11948.0	0.775817	0.387908	+	0.921698i	$0.373198\pi$
$620$	0	0
$621$	5589.00	0.361158
$622$	0	0
$623$	2544.00	0.163601
$624$	0	0
$625$	25625.0	1.64000
$626$	0	0
$627$	34632.0	2.20585
$628$	0	0
$629$	200.000	0.0126781
$630$	0	0
$631$	7840.00	0.494620	0.247310	−	0.968936i	$-0.420453\pi$
$631$	7840.00	0.494620	0.247310	+	0.968936i	$0.420453\pi$
$632$	0	0
$633$	9144.00	0.574157
$634$	0	0
$635$	−45580.0	−2.84848
$636$	0	0
$637$	−14577.0	−0.906690
$638$	0	0
$639$	1134.00	0.0702040
$640$	0	0
$641$	−2320.00	−0.142956	−0.0714778	−	0.997442i	$-0.522771\pi$
$641$	−2320.00	−0.142956	−0.0714778	+	0.997442i	$0.522771\pi$
$642$	0	0
$643$	−1864.00	−0.114322	−0.0571610	−	0.998365i	$-0.518205\pi$
$643$	−1864.00	−0.114322	−0.0571610	+	0.998365i	$0.518205\pi$
$644$	0	0
$645$	−27360.0	−1.67023
$646$	0	0
$647$	−11939.0	−0.725457	−0.362728	−	0.931895i	$-0.618155\pi$
$647$	−11939.0	−0.725457	−0.362728	+	0.931895i	$0.618155\pi$
$648$	0	0
$649$	23088.0	1.39643
$650$	0	0
$651$	−4914.00	−0.295845
$652$	0	0
$653$	10503.0	0.629424	0.314712	−	0.949187i	$-0.398092\pi$
$653$	10503.0	0.629424	0.314712	+	0.949187i	$0.398092\pi$
$654$	0	0
$655$	19740.0	1.17757
$656$	0	0
$657$	36774.0	2.18370
$658$	0	0
$659$	−10950.0	−0.647271	−0.323635	−	0.946182i	$-0.604905\pi$
$659$	−10950.0	−0.647271	−0.323635	+	0.946182i	$0.604905\pi$
$660$	0	0
$661$	−3210.00	−0.188887	−0.0944437	−	0.995530i	$-0.530107\pi$
$661$	−3210.00	−0.188887	−0.0944437	+	0.995530i	$0.530107\pi$
$662$	0	0
$663$	−19350.0	−1.13347
$664$	0	0
$665$	2960.00	0.172607
$666$	0	0
$667$	−161.000	−0.00934624
$668$	0	0
$669$	10080.0	0.582534
$670$	0	0
$671$	13624.0	0.783828
$672$	0	0
$673$	−13517.0	−0.774208	−0.387104	−	0.922036i	$-0.626525\pi$
$673$	−13517.0	−0.774208	−0.387104	+	0.922036i	$0.626525\pi$
$674$	0	0
$675$	66825.0	3.81051
$676$	0	0
$677$	−7494.00	−0.425433	−0.212716	−	0.977114i	$-0.568231\pi$
$677$	−7494.00	−0.425433	−0.212716	+	0.977114i	$0.568231\pi$
$678$	0	0
$679$	684.000	0.0386591
$680$	0	0
$681$	−24354.0	−1.37041
$682$	0	0
$683$	−17865.0	−1.00086	−0.500428	−	0.865778i	$-0.666824\pi$
$683$	−17865.0	−1.00086	−0.500428	+	0.865778i	$0.666824\pi$
$684$	0	0
$685$	32880.0	1.83399
$686$	0	0
$687$	55260.0	3.06885
$688$	0	0
$689$	3698.00	0.204474
$690$	0	0
$691$	−22364.0	−1.23121	−0.615605	−	0.788055i	$-0.711088\pi$
$691$	−22364.0	−1.23121	−0.615605	+	0.788055i	$0.711088\pi$
$692$	0	0
$693$	−5616.00	−0.307842
$694$	0	0
$695$	43780.0	2.38945
$696$	0	0
$697$	−6150.00	−0.334215
$698$	0	0
$699$	59103.0	3.19811
$700$	0	0
$701$	7842.00	0.422522	0.211261	−	0.977430i	$-0.432243\pi$
$701$	7842.00	0.422522	0.211261	+	0.977430i	$0.432243\pi$
$702$	0	0
$703$	−296.000	−0.0158803
$704$	0	0
$705$	13500.0	0.721191
$706$	0	0
$707$	2852.00	0.151712
$708$	0	0
$709$	11234.0	0.595066	0.297533	−	0.954712i	$-0.403836\pi$
$709$	11234.0	0.595066	0.297533	+	0.954712i	$0.403836\pi$
$710$	0	0
$711$	−23004.0	−1.21339
$712$	0	0
$713$	6279.00	0.329804
$714$	0	0
$715$	−44720.0	−2.33907
$716$	0	0
$717$	6561.00	0.341736
$718$	0	0
$719$	17568.0	0.911232	0.455616	−	0.890176i	$-0.349419\pi$
$719$	17568.0	0.911232	0.455616	+	0.890176i	$0.349419\pi$
$720$	0	0
$721$	−2380.00	−0.122935
$722$	0	0
$723$	−26208.0	−1.34811
$724$	0	0
$725$	−1925.00	−0.0986106
$726$	0	0
$727$	35664.0	1.81940	0.909701	−	0.415265i	$-0.136311\pi$
$727$	35664.0	1.81940	0.909701	+	0.415265i	$0.136311\pi$
$728$	0	0
$729$	−19683.0	−1.00000
$730$	0	0
$731$	−7600.00	−0.384536
$732$	0	0
$733$	−27914.0	−1.40659	−0.703293	−	0.710900i	$-0.748288\pi$
$733$	−27914.0	−1.40659	−0.703293	+	0.710900i	$0.748288\pi$
$734$	0	0
$735$	61020.0	3.06225
$736$	0	0
$737$	−39728.0	−1.98562
$738$	0	0
$739$	−39529.0	−1.96766	−0.983828	−	0.179116i	$-0.942676\pi$
$739$	−39529.0	−1.96766	−0.983828	+	0.179116i	$0.942676\pi$
$740$	0	0
$741$	28638.0	1.41976
$742$	0	0
$743$	−10062.0	−0.496822	−0.248411	−	0.968655i	$-0.579908\pi$
$743$	−10062.0	−0.496822	−0.248411	+	0.968655i	$0.579908\pi$
$744$	0	0
$745$	18920.0	0.930436
$746$	0	0
$747$	−48708.0	−2.38572
$748$	0	0
$749$	−2420.00	−0.118057
$750$	0	0
$751$	−25644.0	−1.24602	−0.623011	−	0.782213i	$-0.714091\pi$
$751$	−25644.0	−1.24602	−0.623011	+	0.782213i	$0.714091\pi$
$752$	0	0
$753$	−3582.00	−0.173354
$754$	0	0
$755$	−7300.00	−0.351886
$756$	0	0
$757$	37368.0	1.79414	0.897069	−	0.441890i	$-0.145692\pi$
$757$	37368.0	1.79414	0.897069	+	0.441890i	$0.145692\pi$
$758$	0	0
$759$	10764.0	0.514767
$760$	0	0
$761$	105.000	0.00500164	0.00250082	−	0.999997i	$-0.499204\pi$
$761$	105.000	0.00500164	0.00250082	+	0.999997i	$0.499204\pi$
$762$	0	0
$763$	3360.00	0.159424
$764$	0	0
$765$	54000.0	2.55212
$766$	0	0
$767$	19092.0	0.898790
$768$	0	0
$769$	−15464.0	−0.725157	−0.362579	−	0.931953i	$-0.618104\pi$
$769$	−15464.0	−0.725157	−0.362579	+	0.931953i	$0.618104\pi$
$770$	0	0
$771$	73179.0	3.41826
$772$	0	0
$773$	35168.0	1.63636	0.818179	−	0.574963i	$-0.194984\pi$
$773$	35168.0	1.63636	0.818179	+	0.574963i	$0.194984\pi$
$774$	0	0
$775$	75075.0	3.47971
$776$	0	0
$777$	72.0000	0.00332431
$778$	0	0
$779$	9102.00	0.418630
$780$	0	0
$781$	1092.00	0.0500318
$782$	0	0
$783$	−1701.00	−0.0776357
$784$	0	0
$785$	2160.00	0.0982085
$786$	0	0
$787$	21216.0	0.960951	0.480476	−	0.877008i	$-0.340464\pi$
$787$	21216.0	0.960951	0.480476	+	0.877008i	$0.340464\pi$
$788$	0	0
$789$	17802.0	0.803255
$790$	0	0
$791$	−2060.00	−0.0925982
$792$	0	0
$793$	11266.0	0.504499
$794$	0	0
$795$	−15480.0	−0.690590
$796$	0	0
$797$	9506.00	0.422484	0.211242	−	0.977434i	$-0.432249\pi$
$797$	9506.00	0.422484	0.211242	+	0.977434i	$0.432249\pi$
$798$	0	0
$799$	3750.00	0.166039
$800$	0	0
$801$	−68688.0	−3.02993
$802$	0	0
$803$	35412.0	1.55624
$804$	0	0
$805$	920.000	0.0402804
$806$	0	0
$807$	−76131.0	−3.32087
$808$	0	0
$809$	−20550.0	−0.893077	−0.446539	−	0.894764i	$-0.647343\pi$
$809$	−20550.0	−0.893077	−0.446539	+	0.894764i	$0.647343\pi$
$810$	0	0
$811$	5161.00	0.223461	0.111731	−	0.993739i	$-0.464361\pi$
$811$	5161.00	0.223461	0.111731	+	0.993739i	$0.464361\pi$
$812$	0	0
$813$	65160.0	2.81090
$814$	0	0
$815$	−28300.0	−1.21633
$816$	0	0
$817$	11248.0	0.481662
$818$	0	0
$819$	−4644.00	−0.198137
$820$	0	0
$821$	7866.00	0.334379	0.167190	−	0.985925i	$-0.446531\pi$
$821$	7866.00	0.334379	0.167190	+	0.985925i	$0.446531\pi$
$822$	0	0
$823$	22317.0	0.945227	0.472613	−	0.881270i	$-0.343311\pi$
$823$	22317.0	0.945227	0.472613	+	0.881270i	$0.343311\pi$
$824$	0	0
$825$	128700.	5.43122
$826$	0	0
$827$	26196.0	1.10148	0.550740	−	0.834677i	$-0.314346\pi$
$827$	26196.0	1.10148	0.550740	+	0.834677i	$0.314346\pi$
$828$	0	0
$829$	5886.00	0.246597	0.123299	−	0.992370i	$-0.460653\pi$
$829$	5886.00	0.246597	0.123299	+	0.992370i	$0.460653\pi$
$830$	0	0
$831$	−11871.0	−0.495548
$832$	0	0
$833$	16950.0	0.705021
$834$	0	0
$835$	35120.0	1.45554
$836$	0	0
$837$	66339.0	2.73956
$838$	0	0
$839$	−32394.0	−1.33297	−0.666487	−	0.745517i	$-0.732203\pi$
$839$	−32394.0	−1.33297	−0.666487	+	0.745517i	$0.732203\pi$
$840$	0	0
$841$	−24340.0	−0.997991
$842$	0	0
$843$	15930.0	0.650840
$844$	0	0
$845$	6960.00	0.283351
$846$	0	0
$847$	−2746.00	−0.111397
$848$	0	0
$849$	−37296.0	−1.50765
$850$	0	0
$851$	−92.0000	−0.00370590
$852$	0	0
$853$	−31286.0	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$853$	−31286.0	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$854$	0	0
$855$	−79920.0	−3.19673
$856$	0	0
$857$	−2913.00	−0.116110	−0.0580550	−	0.998313i	$-0.518490\pi$
$857$	−2913.00	−0.116110	−0.0580550	+	0.998313i	$0.518490\pi$
$858$	0	0
$859$	−15451.0	−0.613715	−0.306858	−	0.951755i	$-0.599278\pi$
$859$	−15451.0	−0.613715	−0.306858	+	0.951755i	$0.599278\pi$
$860$	0	0
$861$	−2214.00	−0.0876341
$862$	0	0
$863$	20627.0	0.813617	0.406808	−	0.913514i	$-0.366642\pi$
$863$	20627.0	0.813617	0.406808	+	0.913514i	$0.366642\pi$
$864$	0	0
$865$	−47160.0	−1.85374
$866$	0	0
$867$	−21717.0	−0.850690
$868$	0	0
$869$	−22152.0	−0.864735
$870$	0	0
$871$	−32852.0	−1.27801
$872$	0	0
$873$	−18468.0	−0.715976
$874$	0	0
$875$	6000.00	0.231814
$876$	0	0
$877$	−6966.00	−0.268216	−0.134108	−	0.990967i	$-0.542817\pi$
$877$	−6966.00	−0.268216	−0.134108	+	0.990967i	$0.542817\pi$
$878$	0	0
$879$	−61308.0	−2.35252
$880$	0	0
$881$	−37590.0	−1.43750	−0.718751	−	0.695268i	$-0.755286\pi$
$881$	−37590.0	−1.43750	−0.718751	+	0.695268i	$0.755286\pi$
$882$	0	0
$883$	−27876.0	−1.06240	−0.531202	−	0.847245i	$-0.678259\pi$
$883$	−27876.0	−1.06240	−0.531202	+	0.847245i	$0.678259\pi$
$884$	0	0
$885$	−79920.0	−3.03557
$886$	0	0
$887$	−9471.00	−0.358518	−0.179259	−	0.983802i	$-0.557370\pi$
$887$	−9471.00	−0.358518	−0.179259	+	0.983802i	$0.557370\pi$
$888$	0	0
$889$	−4558.00	−0.171958
$890$	0	0
$891$	37908.0	1.42533
$892$	0	0
$893$	−5550.00	−0.207977
$894$	0	0
$895$	21460.0	0.801485
$896$	0	0
$897$	8901.00	0.331322
$898$	0	0
$899$	−1911.00	−0.0708959
$900$	0	0
$901$	−4300.00	−0.158994
$902$	0	0
$903$	−2736.00	−0.100829
$904$	0	0
$905$	−57360.0	−2.10686
$906$	0	0
$907$	−28366.0	−1.03845	−0.519227	−	0.854636i	$-0.673780\pi$
$907$	−28366.0	−1.03845	−0.519227	+	0.854636i	$0.673780\pi$
$908$	0	0
$909$	−77004.0	−2.80975
$910$	0	0
$911$	7210.00	0.262215	0.131108	−	0.991368i	$-0.458147\pi$
$911$	7210.00	0.262215	0.131108	+	0.991368i	$0.458147\pi$
$912$	0	0
$913$	−46904.0	−1.70021
$914$	0	0
$915$	−47160.0	−1.70389
$916$	0	0
$917$	1974.00	0.0710875
$918$	0	0
$919$	−17198.0	−0.617312	−0.308656	−	0.951174i	$-0.599879\pi$
$919$	−17198.0	−0.617312	−0.308656	+	0.951174i	$0.599879\pi$
$920$	0	0
$921$	51228.0	1.83281
$922$	0	0
$923$	903.000	0.0322022
$924$	0	0
$925$	−1100.00	−0.0391003
$926$	0	0
$927$	64260.0	2.27678
$928$	0	0
$929$	−51033.0	−1.80230	−0.901151	−	0.433505i	$-0.857276\pi$
$929$	−51033.0	−1.80230	−0.901151	+	0.433505i	$0.857276\pi$
$930$	0	0
$931$	−25086.0	−0.883094
$932$	0	0
$933$	−47403.0	−1.66335
$934$	0	0
$935$	52000.0	1.81880
$936$	0	0
$937$	33328.0	1.16198	0.580992	−	0.813910i	$-0.302665\pi$
$937$	33328.0	1.16198	0.580992	+	0.813910i	$0.302665\pi$
$938$	0	0
$939$	57060.0	1.98305
$940$	0	0
$941$	−20166.0	−0.698611	−0.349305	−	0.937009i	$-0.613582\pi$
$941$	−20166.0	−0.698611	−0.349305	+	0.937009i	$0.613582\pi$
$942$	0	0
$943$	2829.00	0.0976934
$944$	0	0
$945$	9720.00	0.334594
$946$	0	0
$947$	28629.0	0.982384	0.491192	−	0.871051i	$-0.336561\pi$
$947$	28629.0	0.982384	0.491192	+	0.871051i	$0.336561\pi$
$948$	0	0
$949$	29283.0	1.00165
$950$	0	0
$951$	−79146.0	−2.69872
$952$	0	0
$953$	38146.0	1.29661	0.648305	−	0.761380i	$-0.275478\pi$
$953$	38146.0	1.29661	0.648305	+	0.761380i	$0.275478\pi$
$954$	0	0
$955$	6640.00	0.224990
$956$	0	0
$957$	−3276.00	−0.110656
$958$	0	0
$959$	3288.00	0.110714
$960$	0	0
$961$	44738.0	1.50173
$962$	0	0
$963$	65340.0	2.18645
$964$	0	0
$965$	42860.0	1.42975
$966$	0	0
$967$	−44621.0	−1.48388	−0.741941	−	0.670465i	$-0.766095\pi$
$967$	−44621.0	−1.48388	−0.741941	+	0.670465i	$0.766095\pi$
$968$	0	0
$969$	−33300.0	−1.10397
$970$	0	0
$971$	−5950.00	−0.196647	−0.0983237	−	0.995154i	$-0.531348\pi$
$971$	−5950.00	−0.196647	−0.0983237	+	0.995154i	$0.531348\pi$
$972$	0	0
$973$	4378.00	0.144247
$974$	0	0
$975$	106425.	3.49572
$976$	0	0
$977$	40836.0	1.33722	0.668608	−	0.743615i	$-0.266891\pi$
$977$	40836.0	1.33722	0.668608	+	0.743615i	$0.266891\pi$
$978$	0	0
$979$	−66144.0	−2.15932
$980$	0	0
$981$	−90720.0	−2.95257
$982$	0	0
$983$	−26874.0	−0.871971	−0.435985	−	0.899954i	$-0.643600\pi$
$983$	−26874.0	−0.871971	−0.435985	+	0.899954i	$0.643600\pi$
$984$	0	0
$985$	54780.0	1.77202
$986$	0	0
$987$	1350.00	0.0435370
$988$	0	0
$989$	3496.00	0.112403
$990$	0	0
$991$	−21472.0	−0.688275	−0.344138	−	0.938919i	$-0.611829\pi$
$991$	−21472.0	−0.688275	−0.344138	+	0.938919i	$0.611829\pi$
$992$	0	0
$993$	−74025.0	−2.36567
$994$	0	0
$995$	15040.0	0.479196
$996$	0	0
$997$	6286.00	0.199679	0.0998393	−	0.995004i	$-0.468167\pi$
$997$	6286.00	0.199679	0.0998393	+	0.995004i	$0.468167\pi$
$998$	0	0
$999$	−972.000	−0.0307835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.4.a.e.1.1		1
4.3	odd	2		46.4.a.b.1.1	✓	1
8.3	odd	2		1472.4.a.j.1.1		1
8.5	even	2		1472.4.a.a.1.1		1
12.11	even	2		414.4.a.b.1.1		1
20.3	even	4		1150.4.b.a.599.1		2
20.7	even	4		1150.4.b.a.599.2		2
20.19	odd	2		1150.4.a.d.1.1		1
28.27	even	2		2254.4.a.b.1.1		1
92.91	even	2		1058.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.4.a.b.1.1	✓	1	4.3	odd	2
368.4.a.e.1.1		1	1.1	even	1	trivial
414.4.a.b.1.1		1	12.11	even	2
1058.4.a.b.1.1		1	92.91	even	2
1150.4.a.d.1.1		1	20.19	odd	2
1150.4.b.a.599.1		2	20.3	even	4
1150.4.b.a.599.2		2	20.7	even	4
1472.4.a.a.1.1		1	8.5	even	2
1472.4.a.j.1.1		1	8.3	odd	2
2254.4.a.b.1.1		1	28.27	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:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	368.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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