LMFDB - Embedded newform 46.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [46,4,Mod(1,46)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(46, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("46.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 46 = 2 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	46.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:	$2.71408786026$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	46.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} -20.0000 q^{5} -18.0000 q^{6} +2.00000 q^{7} +8.00000 q^{8} +54.0000 q^{9} -40.0000 q^{10} -52.0000 q^{11} -36.0000 q^{12} +43.0000 q^{13} +4.00000 q^{14} +180.000 q^{15} +16.0000 q^{16} -50.0000 q^{17} +108.000 q^{18} -74.0000 q^{19} -80.0000 q^{20} -18.0000 q^{21} -104.000 q^{22} -23.0000 q^{23} -72.0000 q^{24} +275.000 q^{25} +86.0000 q^{26} -243.000 q^{27} +8.00000 q^{28} -7.00000 q^{29} +360.000 q^{30} -273.000 q^{31} +32.0000 q^{32} +468.000 q^{33} -100.000 q^{34} -40.0000 q^{35} +216.000 q^{36} -4.00000 q^{37} -148.000 q^{38} -387.000 q^{39} -160.000 q^{40} +123.000 q^{41} -36.0000 q^{42} -152.000 q^{43} -208.000 q^{44} -1080.00 q^{45} -46.0000 q^{46} +75.0000 q^{47} -144.000 q^{48} -339.000 q^{49} +550.000 q^{50} +450.000 q^{51} +172.000 q^{52} +86.0000 q^{53} -486.000 q^{54} +1040.00 q^{55} +16.0000 q^{56} +666.000 q^{57} -14.0000 q^{58} -444.000 q^{59} +720.000 q^{60} +262.000 q^{61} -546.000 q^{62} +108.000 q^{63} +64.0000 q^{64} -860.000 q^{65} +936.000 q^{66} +764.000 q^{67} -200.000 q^{68} +207.000 q^{69} -80.0000 q^{70} -21.0000 q^{71} +432.000 q^{72} +681.000 q^{73} -8.00000 q^{74} -2475.00 q^{75} -296.000 q^{76} -104.000 q^{77} -774.000 q^{78} +426.000 q^{79} -320.000 q^{80} +729.000 q^{81} +246.000 q^{82} +902.000 q^{83} -72.0000 q^{84} +1000.00 q^{85} -304.000 q^{86} +63.0000 q^{87} -416.000 q^{88} -1272.00 q^{89} -2160.00 q^{90} +86.0000 q^{91} -92.0000 q^{92} +2457.00 q^{93} +150.000 q^{94} +1480.00 q^{95} -288.000 q^{96} -342.000 q^{97} -678.000 q^{98} -2808.00 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−9.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−9.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	4.00000	0.500000
$5$	−20.0000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−20.0000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−18.0000	−1.22474
$7$	2.00000	0.107990	0.0539949	−	0.998541i	$-0.482805\pi$
$7$	2.00000	0.107990	0.0539949	+	0.998541i	$0.482805\pi$
$8$	8.00000	0.353553
$9$	54.0000	2.00000
$10$	−40.0000	−1.26491
$11$	−52.0000	−1.42533	−0.712663	−	0.701506i	$-0.752511\pi$
$11$	−52.0000	−1.42533	−0.712663	+	0.701506i	$0.752511\pi$
$12$	−36.0000	−0.866025
$13$	43.0000	0.917389	0.458694	−	0.888594i	$-0.348317\pi$
$13$	43.0000	0.917389	0.458694	+	0.888594i	$0.348317\pi$
$14$	4.00000	0.0763604
$15$	180.000	3.09839
$16$	16.0000	0.250000
$17$	−50.0000	−0.713340	−0.356670	−	0.934230i	$-0.616088\pi$
$17$	−50.0000	−0.713340	−0.356670	+	0.934230i	$0.616088\pi$
$18$	108.000	1.41421
$19$	−74.0000	−0.893514	−0.446757	−	0.894655i	$-0.647421\pi$
$19$	−74.0000	−0.893514	−0.446757	+	0.894655i	$0.647421\pi$
$20$	−80.0000	−0.894427
$21$	−18.0000	−0.187044
$22$	−104.000	−1.00786
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−72.0000	−0.612372
$25$	275.000	2.20000
$26$	86.0000	0.648692
$27$	−243.000	−1.73205
$28$	8.00000	0.0539949
$29$	−7.00000	−0.0448230	−0.0224115	−	0.999749i	$-0.507134\pi$
$29$	−7.00000	−0.0448230	−0.0224115	+	0.999749i	$0.507134\pi$
$30$	360.000	2.19089
$31$	−273.000	−1.58169	−0.790843	−	0.612019i	$-0.790357\pi$
$31$	−273.000	−1.58169	−0.790843	+	0.612019i	$0.790357\pi$
$32$	32.0000	0.176777
$33$	468.000	2.46874
$34$	−100.000	−0.504408
$35$	−40.0000	−0.193178
$36$	216.000	1.00000
$37$	−4.00000	−0.0177729	−0.00888643	−	0.999961i	$-0.502829\pi$
$37$	−4.00000	−0.0177729	−0.00888643	+	0.999961i	$0.502829\pi$
$38$	−148.000	−0.631810
$39$	−387.000	−1.58896
$40$	−160.000	−0.632456
$41$	123.000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	123.000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	−36.0000	−0.132260
$43$	−152.000	−0.539065	−0.269532	−	0.962991i	$-0.586869\pi$
$43$	−152.000	−0.539065	−0.269532	+	0.962991i	$0.586869\pi$
$44$	−208.000	−0.712663
$45$	−1080.00	−3.57771
$46$	−46.0000	−0.147442
$47$	75.0000	0.232763	0.116382	−	0.993205i	$-0.462870\pi$
$47$	75.0000	0.232763	0.116382	+	0.993205i	$0.462870\pi$
$48$	−144.000	−0.433013
$49$	−339.000	−0.988338
$50$	550.000	1.55563
$51$	450.000	1.23554
$52$	172.000	0.458694
$53$	86.0000	0.222887	0.111443	−	0.993771i	$-0.464453\pi$
$53$	86.0000	0.222887	0.111443	+	0.993771i	$0.464453\pi$
$54$	−486.000	−1.22474
$55$	1040.00	2.54970
$56$	16.0000	0.0381802
$57$	666.000	1.54761
$58$	−14.0000	−0.0316947
$59$	−444.000	−0.979727	−0.489863	−	0.871799i	$-0.662953\pi$
$59$	−444.000	−0.979727	−0.489863	+	0.871799i	$0.662953\pi$
$60$	720.000	1.54919
$61$	262.000	0.549929	0.274964	−	0.961454i	$-0.411334\pi$
$61$	262.000	0.549929	0.274964	+	0.961454i	$0.411334\pi$
$62$	−546.000	−1.11842
$63$	108.000	0.215980
$64$	64.0000	0.125000
$65$	−860.000	−1.64107
$66$	936.000	1.74566
$67$	764.000	1.39310	0.696548	−	0.717510i	$-0.254718\pi$
$67$	764.000	1.39310	0.696548	+	0.717510i	$0.254718\pi$
$68$	−200.000	−0.356670
$69$	207.000	0.361158
$70$	−80.0000	−0.136598
$71$	−21.0000	−0.0351020	−0.0175510	−	0.999846i	$-0.505587\pi$
$71$	−21.0000	−0.0351020	−0.0175510	+	0.999846i	$0.505587\pi$
$72$	432.000	0.707107
$73$	681.000	1.09185	0.545925	−	0.837834i	$-0.316178\pi$
$73$	681.000	1.09185	0.545925	+	0.837834i	$0.316178\pi$
$74$	−8.00000	−0.0125673
$75$	−2475.00	−3.81051
$76$	−296.000	−0.446757
$77$	−104.000	−0.153921
$78$	−774.000	−1.12357
$79$	426.000	0.606693	0.303346	−	0.952880i	$-0.401896\pi$
$79$	426.000	0.606693	0.303346	+	0.952880i	$0.401896\pi$
$80$	−320.000	−0.447214
$81$	729.000	1.00000
$82$	246.000	0.331295
$83$	902.000	1.19286	0.596430	−	0.802665i	$-0.296585\pi$
$83$	902.000	1.19286	0.596430	+	0.802665i	$0.296585\pi$
$84$	−72.0000	−0.0935220
$85$	1000.00	1.27606
$86$	−304.000	−0.381176
$87$	63.0000	0.0776357
$88$	−416.000	−0.503929
$89$	−1272.00	−1.51496	−0.757482	−	0.652856i	$-0.773570\pi$
$89$	−1272.00	−1.51496	−0.757482	+	0.652856i	$0.773570\pi$
$90$	−2160.00	−2.52982
$91$	86.0000	0.0990687
$92$	−92.0000	−0.104257
$93$	2457.00	2.73956
$94$	150.000	0.164588
$95$	1480.00	1.59837
$96$	−288.000	−0.306186
$97$	−342.000	−0.357988	−0.178994	−	0.983850i	$-0.557284\pi$
$97$	−342.000	−0.357988	−0.178994	+	0.983850i	$0.557284\pi$
$98$	−678.000	−0.698861
$99$	−2808.00	−2.85065
$100$	1100.00	1.10000
$101$	−1426.00	−1.40487	−0.702437	−	0.711746i	$-0.747905\pi$
$101$	−1426.00	−1.40487	−0.702437	+	0.711746i	$0.747905\pi$
$102$	900.000	0.873660
$103$	−1190.00	−1.13839	−0.569195	−	0.822203i	$-0.692745\pi$
$103$	−1190.00	−1.13839	−0.569195	+	0.822203i	$0.692745\pi$
$104$	344.000	0.324346
$105$	360.000	0.334594
$106$	172.000	0.157605
$107$	−1210.00	−1.09323	−0.546613	−	0.837386i	$-0.684083\pi$
$107$	−1210.00	−1.09323	−0.546613	+	0.837386i	$0.684083\pi$
$108$	−972.000	−0.866025
$109$	−1680.00	−1.47628	−0.738141	−	0.674646i	$-0.764296\pi$
$109$	−1680.00	−1.47628	−0.738141	+	0.674646i	$0.764296\pi$
$110$	2080.00	1.80291
$111$	36.0000	0.0307835
$112$	32.0000	0.0269975
$113$	1030.00	0.857471	0.428736	−	0.903430i	$-0.358959\pi$
$113$	1030.00	0.857471	0.428736	+	0.903430i	$0.358959\pi$
$114$	1332.00	1.09433
$115$	460.000	0.373002
$116$	−28.0000	−0.0224115
$117$	2322.00	1.83478
$118$	−888.000	−0.692771
$119$	−100.000	−0.0770335
$120$	1440.00	1.09545
$121$	1373.00	1.03156
$122$	524.000	0.388858
$123$	−1107.00	−0.811503
$124$	−1092.00	−0.790843
$125$	−3000.00	−2.14663
$126$	216.000	0.152721
$127$	−2279.00	−1.59235	−0.796175	−	0.605066i	$-0.793147\pi$
$127$	−2279.00	−1.59235	−0.796175	+	0.605066i	$0.793147\pi$
$128$	128.000	0.0883883
$129$	1368.00	0.933687
$130$	−1720.00	−1.16042
$131$	987.000	0.658279	0.329140	−	0.944281i	$-0.393241\pi$
$131$	987.000	0.658279	0.329140	+	0.944281i	$0.393241\pi$
$132$	1872.00	1.23437
$133$	−148.000	−0.0964904
$134$	1528.00	0.985068
$135$	4860.00	3.09839
$136$	−400.000	−0.252204
$137$	−1644.00	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−1644.00	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	414.000	0.255377
$139$	2189.00	1.33575	0.667873	−	0.744276i	$-0.267205\pi$
$139$	2189.00	1.33575	0.667873	+	0.744276i	$0.267205\pi$
$140$	−160.000	−0.0965891
$141$	−675.000	−0.403158
$142$	−42.0000	−0.0248209
$143$	−2236.00	−1.30758
$144$	864.000	0.500000
$145$	140.000	0.0801818
$146$	1362.00	0.772054
$147$	3051.00	1.71185
$148$	−16.0000	−0.00888643
$149$	−946.000	−0.520130	−0.260065	−	0.965591i	$-0.583744\pi$
$149$	−946.000	−0.520130	−0.260065	+	0.965591i	$0.583744\pi$
$150$	−4950.00	−2.69444
$151$	−365.000	−0.196710	−0.0983552	−	0.995151i	$-0.531358\pi$
$151$	−365.000	−0.196710	−0.0983552	+	0.995151i	$0.531358\pi$
$152$	−592.000	−0.315905
$153$	−2700.00	−1.42668
$154$	−208.000	−0.108838
$155$	5460.00	2.82940
$156$	−1548.00	−0.794482
$157$	−108.000	−0.0549002	−0.0274501	−	0.999623i	$-0.508739\pi$
$157$	−108.000	−0.0549002	−0.0274501	+	0.999623i	$0.508739\pi$
$158$	852.000	0.428997
$159$	−774.000	−0.386052
$160$	−640.000	−0.316228
$161$	−46.0000	−0.0225174
$162$	1458.00	0.707107
$163$	−1415.00	−0.679947	−0.339973	−	0.940435i	$-0.610418\pi$
$163$	−1415.00	−0.679947	−0.339973	+	0.940435i	$0.610418\pi$
$164$	492.000	0.234261
$165$	−9360.00	−4.41621
$166$	1804.00	0.843479
$167$	1756.00	0.813673	0.406836	−	0.913501i	$-0.366632\pi$
$167$	1756.00	0.813673	0.406836	+	0.913501i	$0.366632\pi$
$168$	−144.000	−0.0661300
$169$	−348.000	−0.158398
$170$	2000.00	0.902312
$171$	−3996.00	−1.78703
$172$	−608.000	−0.269532
$173$	2358.00	1.03627	0.518137	−	0.855298i	$-0.326626\pi$
$173$	2358.00	1.03627	0.518137	+	0.855298i	$0.326626\pi$
$174$	126.000	0.0548968
$175$	550.000	0.237578
$176$	−832.000	−0.356332
$177$	3996.00	1.69694
$178$	−2544.00	−1.07124
$179$	1073.00	0.448043	0.224022	−	0.974584i	$-0.428081\pi$
$179$	1073.00	0.448043	0.224022	+	0.974584i	$0.428081\pi$
$180$	−4320.00	−1.78885
$181$	2868.00	1.17777	0.588886	−	0.808216i	$-0.299567\pi$
$181$	2868.00	1.17777	0.588886	+	0.808216i	$0.299567\pi$
$182$	172.000	0.0700521
$183$	−2358.00	−0.952505
$184$	−184.000	−0.0737210
$185$	80.0000	0.0317931
$186$	4914.00	1.93716
$187$	2600.00	1.01674
$188$	300.000	0.116382
$189$	−486.000	−0.187044
$190$	2960.00	1.13022
$191$	332.000	0.125773	0.0628866	−	0.998021i	$-0.479969\pi$
$191$	332.000	0.125773	0.0628866	+	0.998021i	$0.479969\pi$
$192$	−576.000	−0.216506
$193$	−2143.00	−0.799257	−0.399628	−	0.916677i	$-0.630861\pi$
$193$	−2143.00	−0.799257	−0.399628	+	0.916677i	$0.630861\pi$
$194$	−684.000	−0.253136
$195$	7740.00	2.84243
$196$	−1356.00	−0.494169
$197$	−2739.00	−0.990587	−0.495294	−	0.868726i	$-0.664940\pi$
$197$	−2739.00	−0.990587	−0.495294	+	0.868726i	$0.664940\pi$
$198$	−5616.00	−2.01572
$199$	752.000	0.267879	0.133939	−	0.990990i	$-0.457237\pi$
$199$	752.000	0.267879	0.133939	+	0.990990i	$0.457237\pi$
$200$	2200.00	0.777817
$201$	−6876.00	−2.41291
$202$	−2852.00	−0.993396
$203$	−14.0000	−0.00484043
$204$	1800.00	0.617771
$205$	−2460.00	−0.838116
$206$	−2380.00	−0.804963
$207$	−1242.00	−0.417029
$208$	688.000	0.229347
$209$	3848.00	1.27355
$210$	720.000	0.236594
$211$	−1016.00	−0.331490	−0.165745	−	0.986169i	$-0.553003\pi$
$211$	−1016.00	−0.331490	−0.165745	+	0.986169i	$0.553003\pi$
$212$	344.000	0.111443
$213$	189.000	0.0607984
$214$	−2420.00	−0.773027
$215$	3040.00	0.964308
$216$	−1944.00	−0.612372
$217$	−546.000	−0.170806
$218$	−3360.00	−1.04389
$219$	−6129.00	−1.89114
$220$	4160.00	1.27485
$221$	−2150.00	−0.654410
$222$	72.0000	0.0217672
$223$	−1120.00	−0.336326	−0.168163	−	0.985759i	$-0.553784\pi$
$223$	−1120.00	−0.336326	−0.168163	+	0.985759i	$0.553784\pi$
$224$	64.0000	0.0190901
$225$	14850.0	4.40000
$226$	2060.00	0.606324
$227$	2706.00	0.791205	0.395602	−	0.918422i	$-0.370536\pi$
$227$	2706.00	0.791205	0.395602	+	0.918422i	$0.370536\pi$
$228$	2664.00	0.773806
$229$	6140.00	1.77180	0.885901	−	0.463875i	$-0.153541\pi$
$229$	6140.00	1.77180	0.885901	+	0.463875i	$0.153541\pi$
$230$	920.000	0.263752
$231$	936.000	0.266599
$232$	−56.0000	−0.0158473
$233$	6567.00	1.84643	0.923216	−	0.384282i	$-0.125551\pi$
$233$	6567.00	1.84643	0.923216	+	0.384282i	$0.125551\pi$
$234$	4644.00	1.29738
$235$	−1500.00	−0.416380
$236$	−1776.00	−0.489863
$237$	−3834.00	−1.05082
$238$	−200.000	−0.0544709
$239$	−729.000	−0.197302	−0.0986508	−	0.995122i	$-0.531453\pi$
$239$	−729.000	−0.197302	−0.0986508	+	0.995122i	$0.531453\pi$
$240$	2880.00	0.774597
$241$	−2912.00	−0.778334	−0.389167	−	0.921167i	$-0.627237\pi$
$241$	−2912.00	−0.778334	−0.389167	+	0.921167i	$0.627237\pi$
$242$	2746.00	0.729420
$243$	0	0
$244$	1048.00	0.274964
$245$	6780.00	1.76799
$246$	−2214.00	−0.573819
$247$	−3182.00	−0.819700
$248$	−2184.00	−0.559210
$249$	−8118.00	−2.06609
$250$	−6000.00	−1.51789
$251$	398.000	0.100086	0.0500429	−	0.998747i	$-0.484064\pi$
$251$	398.000	0.100086	0.0500429	+	0.998747i	$0.484064\pi$
$252$	432.000	0.107990
$253$	1196.00	0.297201
$254$	−4558.00	−1.12596
$255$	−9000.00	−2.21020
$256$	256.000	0.0625000
$257$	8131.00	1.97353	0.986766	−	0.162149i	$-0.0518426\pi$
$257$	8131.00	1.97353	0.986766	+	0.162149i	$0.0518426\pi$
$258$	2736.00	0.660217
$259$	−8.00000	−0.00191929
$260$	−3440.00	−0.820537
$261$	−378.000	−0.0896460
$262$	1974.00	0.465474
$263$	−1978.00	−0.463759	−0.231880	−	0.972744i	$-0.574488\pi$
$263$	−1978.00	−0.463759	−0.231880	+	0.972744i	$0.574488\pi$
$264$	3744.00	0.872831
$265$	−1720.00	−0.398712
$266$	−296.000	−0.0682290
$267$	11448.0	2.62399
$268$	3056.00	0.696548
$269$	−8459.00	−1.91730	−0.958651	−	0.284584i	$-0.908145\pi$
$269$	−8459.00	−1.91730	−0.958651	+	0.284584i	$0.908145\pi$
$270$	9720.00	2.19089
$271$	−7240.00	−1.62287	−0.811437	−	0.584440i	$-0.801314\pi$
$271$	−7240.00	−1.62287	−0.811437	+	0.584440i	$0.801314\pi$
$272$	−800.000	−0.178335
$273$	−774.000	−0.171592
$274$	−3288.00	−0.724947
$275$	−14300.0	−3.13572
$276$	828.000	0.180579
$277$	−1319.00	−0.286105	−0.143052	−	0.989715i	$-0.545692\pi$
$277$	−1319.00	−0.286105	−0.143052	+	0.989715i	$0.545692\pi$
$278$	4378.00	0.944514
$279$	−14742.0	−3.16337
$280$	−320.000	−0.0682988
$281$	1770.00	0.375763	0.187881	−	0.982192i	$-0.439838\pi$
$281$	1770.00	0.375763	0.187881	+	0.982192i	$0.439838\pi$
$282$	−1350.00	−0.285076
$283$	4144.00	0.870443	0.435221	−	0.900324i	$-0.356670\pi$
$283$	4144.00	0.870443	0.435221	+	0.900324i	$0.356670\pi$
$284$	−84.0000	−0.0175510
$285$	−13320.0	−2.76845
$286$	−4472.00	−0.924598
$287$	246.000	0.0505955
$288$	1728.00	0.353553
$289$	−2413.00	−0.491146
$290$	280.000	0.0566971
$291$	3078.00	0.620053
$292$	2724.00	0.545925
$293$	−6812.00	−1.35823	−0.679115	−	0.734032i	$-0.737636\pi$
$293$	−6812.00	−1.35823	−0.679115	+	0.734032i	$0.737636\pi$
$294$	6102.00	1.21046
$295$	8880.00	1.75259
$296$	−32.0000	−0.00628366
$297$	12636.0	2.46874
$298$	−1892.00	−0.367787
$299$	−989.000	−0.191289
$300$	−9900.00	−1.90526
$301$	−304.000	−0.0582135
$302$	−730.000	−0.139095
$303$	12834.0	2.43331
$304$	−1184.00	−0.223378
$305$	−5240.00	−0.983743
$306$	−5400.00	−1.00882
$307$	−5692.00	−1.05817	−0.529087	−	0.848567i	$-0.677466\pi$
$307$	−5692.00	−1.05817	−0.529087	+	0.848567i	$0.677466\pi$
$308$	−416.000	−0.0769604
$309$	10710.0	1.97175
$310$	10920.0	2.00069
$311$	5267.00	0.960335	0.480167	−	0.877177i	$-0.340576\pi$
$311$	5267.00	0.960335	0.480167	+	0.877177i	$0.340576\pi$
$312$	−3096.00	−0.561784
$313$	6340.00	1.14491	0.572457	−	0.819935i	$-0.305990\pi$
$313$	6340.00	1.14491	0.572457	+	0.819935i	$0.305990\pi$
$314$	−216.000	−0.0388203
$315$	−2160.00	−0.386356
$316$	1704.00	0.303346
$317$	−8794.00	−1.55811	−0.779054	−	0.626957i	$-0.784300\pi$
$317$	−8794.00	−1.55811	−0.779054	+	0.626957i	$0.784300\pi$
$318$	−1548.00	−0.272980
$319$	364.000	0.0638874
$320$	−1280.00	−0.223607
$321$	10890.0	1.89352
$322$	−92.0000	−0.0159222
$323$	3700.00	0.637379
$324$	2916.00	0.500000
$325$	11825.0	2.01826
$326$	−2830.00	−0.480795
$327$	15120.0	2.55700
$328$	984.000	0.165647
$329$	150.000	0.0251361
$330$	−18720.0	−3.12273
$331$	8225.00	1.36582	0.682911	−	0.730502i	$-0.260714\pi$
$331$	8225.00	1.36582	0.682911	+	0.730502i	$0.260714\pi$
$332$	3608.00	0.596430
$333$	−216.000	−0.0355457
$334$	3512.00	0.575354
$335$	−15280.0	−2.49205
$336$	−288.000	−0.0467610
$337$	−2576.00	−0.416391	−0.208195	−	0.978087i	$-0.566759\pi$
$337$	−2576.00	−0.416391	−0.208195	+	0.978087i	$0.566759\pi$
$338$	−696.000	−0.112004
$339$	−9270.00	−1.48518
$340$	4000.00	0.638031
$341$	14196.0	2.25442
$342$	−7992.00	−1.26362
$343$	−1364.00	−0.214720
$344$	−1216.00	−0.190588
$345$	−4140.00	−0.646058
$346$	4716.00	0.732756
$347$	596.000	0.0922045	0.0461022	−	0.998937i	$-0.485320\pi$
$347$	596.000	0.0922045	0.0461022	+	0.998937i	$0.485320\pi$
$348$	252.000	0.0388179
$349$	−9271.00	−1.42196	−0.710982	−	0.703210i	$-0.751749\pi$
$349$	−9271.00	−1.42196	−0.710982	+	0.703210i	$0.751749\pi$
$350$	1100.00	0.167993
$351$	−10449.0	−1.58896
$352$	−1664.00	−0.251964
$353$	8141.00	1.22748	0.613742	−	0.789507i	$-0.289664\pi$
$353$	8141.00	1.22748	0.613742	+	0.789507i	$0.289664\pi$
$354$	7992.00	1.19992
$355$	420.000	0.0627924
$356$	−5088.00	−0.757482
$357$	900.000	0.133426
$358$	2146.00	0.316815
$359$	−2130.00	−0.313140	−0.156570	−	0.987667i	$-0.550044\pi$
$359$	−2130.00	−0.313140	−0.156570	+	0.987667i	$0.550044\pi$
$360$	−8640.00	−1.26491
$361$	−1383.00	−0.201633
$362$	5736.00	0.832811
$363$	−12357.0	−1.78671
$364$	344.000	0.0495343
$365$	−13620.0	−1.95316
$366$	−4716.00	−0.673523
$367$	−2574.00	−0.366108	−0.183054	−	0.983103i	$-0.558598\pi$
$367$	−2574.00	−0.366108	−0.183054	+	0.983103i	$0.558598\pi$
$368$	−368.000	−0.0521286
$369$	6642.00	0.937043
$370$	160.000	0.0224811
$371$	172.000	0.0240695
$372$	9828.00	1.36978
$373$	−4504.00	−0.625223	−0.312612	−	0.949881i	$-0.601204\pi$
$373$	−4504.00	−0.625223	−0.312612	+	0.949881i	$0.601204\pi$
$374$	5200.00	0.718945
$375$	27000.0	3.71806
$376$	600.000	0.0822942
$377$	−301.000	−0.0411201
$378$	−972.000	−0.132260
$379$	2740.00	0.371357	0.185679	−	0.982611i	$-0.440552\pi$
$379$	2740.00	0.371357	0.185679	+	0.982611i	$0.440552\pi$
$380$	5920.00	0.799183
$381$	20511.0	2.75803
$382$	664.000	0.0889351
$383$	6948.00	0.926961	0.463481	−	0.886107i	$-0.346600\pi$
$383$	6948.00	0.926961	0.463481	+	0.886107i	$0.346600\pi$
$384$	−1152.00	−0.153093
$385$	2080.00	0.275342
$386$	−4286.00	−0.565160
$387$	−8208.00	−1.07813
$388$	−1368.00	−0.178994
$389$	−1404.00	−0.182996	−0.0914982	−	0.995805i	$-0.529166\pi$
$389$	−1404.00	−0.182996	−0.0914982	+	0.995805i	$0.529166\pi$
$390$	15480.0	2.00990
$391$	1150.00	0.148742
$392$	−2712.00	−0.349430
$393$	−8883.00	−1.14017
$394$	−5478.00	−0.700451
$395$	−8520.00	−1.08529
$396$	−11232.0	−1.42533
$397$	−8641.00	−1.09239	−0.546196	−	0.837658i	$-0.683924\pi$
$397$	−8641.00	−1.09239	−0.546196	+	0.837658i	$0.683924\pi$
$398$	1504.00	0.189419
$399$	1332.00	0.167126
$400$	4400.00	0.550000
$401$	−1140.00	−0.141967	−0.0709836	−	0.997477i	$-0.522614\pi$
$401$	−1140.00	−0.141967	−0.0709836	+	0.997477i	$0.522614\pi$
$402$	−13752.0	−1.70619
$403$	−11739.0	−1.45102
$404$	−5704.00	−0.702437
$405$	−14580.0	−1.78885
$406$	−28.0000	−0.00342270
$407$	208.000	0.0253321
$408$	3600.00	0.436830
$409$	12529.0	1.51472	0.757358	−	0.652999i	$-0.226490\pi$
$409$	12529.0	1.51472	0.757358	+	0.652999i	$0.226490\pi$
$410$	−4920.00	−0.592638
$411$	14796.0	1.77575
$412$	−4760.00	−0.569195
$413$	−888.000	−0.105801
$414$	−2484.00	−0.294884
$415$	−18040.0	−2.13385
$416$	1376.00	0.162173
$417$	−19701.0	−2.31358
$418$	7696.00	0.900535
$419$	3252.00	0.379166	0.189583	−	0.981865i	$-0.439286\pi$
$419$	3252.00	0.379166	0.189583	+	0.981865i	$0.439286\pi$
$420$	1440.00	0.167297
$421$	2206.00	0.255377	0.127689	−	0.991814i	$-0.459244\pi$
$421$	2206.00	0.255377	0.127689	+	0.991814i	$0.459244\pi$
$422$	−2032.00	−0.234399
$423$	4050.00	0.465527
$424$	688.000	0.0788024
$425$	−13750.0	−1.56935
$426$	378.000	0.0429910
$427$	524.000	0.0593867
$428$	−4840.00	−0.546613
$429$	20124.0	2.26479
$430$	6080.00	0.681869
$431$	14316.0	1.59995	0.799974	−	0.600035i	$-0.204847\pi$
$431$	14316.0	1.59995	0.799974	+	0.600035i	$0.204847\pi$
$432$	−3888.00	−0.433013
$433$	7828.00	0.868798	0.434399	−	0.900720i	$-0.356961\pi$
$433$	7828.00	0.868798	0.434399	+	0.900720i	$0.356961\pi$
$434$	−1092.00	−0.120778
$435$	−1260.00	−0.138879
$436$	−6720.00	−0.738141
$437$	1702.00	0.186311
$438$	−12258.0	−1.33724
$439$	−16039.0	−1.74374	−0.871868	−	0.489742i	$-0.837091\pi$
$439$	−16039.0	−1.74374	−0.871868	+	0.489742i	$0.837091\pi$
$440$	8320.00	0.901456
$441$	−18306.0	−1.97668
$442$	−4300.00	−0.462738
$443$	11747.0	1.25986	0.629929	−	0.776653i	$-0.283084\pi$
$443$	11747.0	1.25986	0.629929	+	0.776653i	$0.283084\pi$
$444$	144.000	0.0153918
$445$	25440.0	2.71005
$446$	−2240.00	−0.237819
$447$	8514.00	0.900891
$448$	128.000	0.0134987
$449$	−2890.00	−0.303758	−0.151879	−	0.988399i	$-0.548532\pi$
$449$	−2890.00	−0.303758	−0.151879	+	0.988399i	$0.548532\pi$
$450$	29700.0	3.11127
$451$	−6396.00	−0.667796
$452$	4120.00	0.428736
$453$	3285.00	0.340713
$454$	5412.00	0.559466
$455$	−1720.00	−0.177219
$456$	5328.00	0.547163
$457$	−13126.0	−1.34356	−0.671782	−	0.740749i	$-0.734471\pi$
$457$	−13126.0	−1.34356	−0.671782	+	0.740749i	$0.734471\pi$
$458$	12280.0	1.25285
$459$	12150.0	1.23554
$460$	1840.00	0.186501
$461$	14481.0	1.46301	0.731505	−	0.681836i	$-0.238818\pi$
$461$	14481.0	1.46301	0.731505	+	0.681836i	$0.238818\pi$
$462$	1872.00	0.188514
$463$	5272.00	0.529181	0.264590	−	0.964361i	$-0.414763\pi$
$463$	5272.00	0.529181	0.264590	+	0.964361i	$0.414763\pi$
$464$	−112.000	−0.0112058
$465$	−49140.0	−4.90067
$466$	13134.0	1.30562
$467$	−13466.0	−1.33433	−0.667165	−	0.744910i	$-0.732492\pi$
$467$	−13466.0	−1.33433	−0.667165	+	0.744910i	$0.732492\pi$
$468$	9288.00	0.917389
$469$	1528.00	0.150440
$470$	−3000.00	−0.294425
$471$	972.000	0.0950900
$472$	−3552.00	−0.346386
$473$	7904.00	0.768343
$474$	−7668.00	−0.743044
$475$	−20350.0	−1.96573
$476$	−400.000	−0.0385167
$477$	4644.00	0.445774
$478$	−1458.00	−0.139513
$479$	−4526.00	−0.431729	−0.215865	−	0.976423i	$-0.569257\pi$
$479$	−4526.00	−0.431729	−0.215865	+	0.976423i	$0.569257\pi$
$480$	5760.00	0.547723
$481$	−172.000	−0.0163046
$482$	−5824.00	−0.550365
$483$	414.000	0.0390014
$484$	5492.00	0.515778
$485$	6840.00	0.640388
$486$	0	0
$487$	−8795.00	−0.818356	−0.409178	−	0.912455i	$-0.634185\pi$
$487$	−8795.00	−0.818356	−0.409178	+	0.912455i	$0.634185\pi$
$488$	2096.00	0.194429
$489$	12735.0	1.17770
$490$	13560.0	1.25016
$491$	−1275.00	−0.117189	−0.0585946	−	0.998282i	$-0.518662\pi$
$491$	−1275.00	−0.117189	−0.0585946	+	0.998282i	$0.518662\pi$
$492$	−4428.00	−0.405751
$493$	350.000	0.0319741
$494$	−6364.00	−0.579615
$495$	56160.0	5.09940
$496$	−4368.00	−0.395421
$497$	−42.0000	−0.00379066
$498$	−16236.0	−1.46095
$499$	−9533.00	−0.855222	−0.427611	−	0.903963i	$-0.640645\pi$
$499$	−9533.00	−0.855222	−0.427611	+	0.903963i	$0.640645\pi$
$500$	−12000.0	−1.07331
$501$	−15804.0	−1.40932
$502$	796.000	0.0707714
$503$	−13398.0	−1.18765	−0.593824	−	0.804595i	$-0.702383\pi$
$503$	−13398.0	−1.18765	−0.593824	+	0.804595i	$0.702383\pi$
$504$	864.000	0.0763604
$505$	28520.0	2.51312
$506$	2392.00	0.210153
$507$	3132.00	0.274353
$508$	−9116.00	−0.796175
$509$	−8031.00	−0.699347	−0.349674	−	0.936872i	$-0.613708\pi$
$509$	−8031.00	−0.699347	−0.349674	+	0.936872i	$0.613708\pi$
$510$	−18000.0	−1.56285
$511$	1362.00	0.117909
$512$	512.000	0.0441942
$513$	17982.0	1.54761
$514$	16262.0	1.39550
$515$	23800.0	2.03641
$516$	5472.00	0.466844
$517$	−3900.00	−0.331764
$518$	−16.0000	−0.00135714
$519$	−21222.0	−1.79488
$520$	−6880.00	−0.580208
$521$	−21184.0	−1.78136	−0.890679	−	0.454632i	$-0.849771\pi$
$521$	−21184.0	−1.78136	−0.890679	+	0.454632i	$0.849771\pi$
$522$	−756.000	−0.0633893
$523$	−21706.0	−1.81479	−0.907397	−	0.420275i	$-0.861934\pi$
$523$	−21706.0	−1.81479	−0.907397	+	0.420275i	$0.861934\pi$
$524$	3948.00	0.329140
$525$	−4950.00	−0.411497
$526$	−3956.00	−0.327927
$527$	13650.0	1.12828
$528$	7488.00	0.617184
$529$	529.000	0.0434783
$530$	−3440.00	−0.281932
$531$	−23976.0	−1.95945
$532$	−592.000	−0.0482452
$533$	5289.00	0.429816
$534$	22896.0	1.85544
$535$	24200.0	1.95562
$536$	6112.00	0.492534
$537$	−9657.00	−0.776034
$538$	−16918.0	−1.35574
$539$	17628.0	1.40870
$540$	19440.0	1.54919
$541$	5781.00	0.459417	0.229709	−	0.973259i	$-0.426223\pi$
$541$	5781.00	0.459417	0.229709	+	0.973259i	$0.426223\pi$
$542$	−14480.0	−1.14754
$543$	−25812.0	−2.03996
$544$	−1600.00	−0.126102
$545$	33600.0	2.64085
$546$	−1548.00	−0.121334
$547$	7809.00	0.610400	0.305200	−	0.952288i	$-0.401277\pi$
$547$	7809.00	0.610400	0.305200	+	0.952288i	$0.401277\pi$
$548$	−6576.00	−0.512615
$549$	14148.0	1.09986
$550$	−28600.0	−2.21729
$551$	518.000	0.0400500
$552$	1656.00	0.127688
$553$	852.000	0.0655167
$554$	−2638.00	−0.202307
$555$	−720.000	−0.0550672
$556$	8756.00	0.667873
$557$	−20240.0	−1.53967	−0.769835	−	0.638243i	$-0.779662\pi$
$557$	−20240.0	−1.53967	−0.769835	+	0.638243i	$0.779662\pi$
$558$	−29484.0	−2.23684
$559$	−6536.00	−0.494532
$560$	−640.000	−0.0482945
$561$	−23400.0	−1.76105
$562$	3540.00	0.265704
$563$	7612.00	0.569818	0.284909	−	0.958555i	$-0.408037\pi$
$563$	7612.00	0.569818	0.284909	+	0.958555i	$0.408037\pi$
$564$	−2700.00	−0.201579
$565$	−20600.0	−1.53389
$566$	8288.00	0.615496
$567$	1458.00	0.107990
$568$	−168.000	−0.0124104
$569$	19484.0	1.43552	0.717761	−	0.696290i	$-0.245167\pi$
$569$	19484.0	1.43552	0.717761	+	0.696290i	$0.245167\pi$
$570$	−26640.0	−1.95759
$571$	6614.00	0.484741	0.242371	−	0.970184i	$-0.422075\pi$
$571$	6614.00	0.484741	0.242371	+	0.970184i	$0.422075\pi$
$572$	−8944.00	−0.653789
$573$	−2988.00	−0.217846
$574$	492.000	0.0357765
$575$	−6325.00	−0.458732
$576$	3456.00	0.250000
$577$	−639.000	−0.0461038	−0.0230519	−	0.999734i	$-0.507338\pi$
$577$	−639.000	−0.0461038	−0.0230519	+	0.999734i	$0.507338\pi$
$578$	−4826.00	−0.347293
$579$	19287.0	1.38435
$580$	560.000	0.0400909
$581$	1804.00	0.128817
$582$	6156.00	0.438444
$583$	−4472.00	−0.317687
$584$	5448.00	0.386027
$585$	−46440.0	−3.28215
$586$	−13624.0	−0.960413
$587$	−829.000	−0.0582904	−0.0291452	−	0.999575i	$-0.509279\pi$
$587$	−829.000	−0.0582904	−0.0291452	+	0.999575i	$0.509279\pi$
$588$	12204.0	0.855926
$589$	20202.0	1.41326
$590$	17760.0	1.23927
$591$	24651.0	1.71575
$592$	−64.0000	−0.00444322
$593$	−20610.0	−1.42724	−0.713618	−	0.700535i	$-0.752945\pi$
$593$	−20610.0	−1.42724	−0.713618	+	0.700535i	$0.752945\pi$
$594$	25272.0	1.74566
$595$	2000.00	0.137802
$596$	−3784.00	−0.260065
$597$	−6768.00	−0.463980
$598$	−1978.00	−0.135262
$599$	−17240.0	−1.17597	−0.587986	−	0.808871i	$-0.700079\pi$
$599$	−17240.0	−1.17597	−0.587986	+	0.808871i	$0.700079\pi$
$600$	−19800.0	−1.34722
$601$	−8459.00	−0.574126	−0.287063	−	0.957912i	$-0.592679\pi$
$601$	−8459.00	−0.574126	−0.287063	+	0.957912i	$0.592679\pi$
$602$	−608.000	−0.0411632
$603$	41256.0	2.78619
$604$	−1460.00	−0.0983552
$605$	−27460.0	−1.84530
$606$	25668.0	1.72061
$607$	17840.0	1.19292	0.596461	−	0.802642i	$-0.296573\pi$
$607$	17840.0	1.19292	0.596461	+	0.802642i	$0.296573\pi$
$608$	−2368.00	−0.157952
$609$	126.000	0.00838387
$610$	−10480.0	−0.695611
$611$	3225.00	0.213534
$612$	−10800.0	−0.713340
$613$	2534.00	0.166961	0.0834807	−	0.996509i	$-0.473396\pi$
$613$	2534.00	0.166961	0.0834807	+	0.996509i	$0.473396\pi$
$614$	−11384.0	−0.748242
$615$	22140.0	1.45166
$616$	−832.000	−0.0544192
$617$	−5610.00	−0.366046	−0.183023	−	0.983109i	$-0.558588\pi$
$617$	−5610.00	−0.366046	−0.183023	+	0.983109i	$0.558588\pi$
$618$	21420.0	1.39424
$619$	−11948.0	−0.775817	−0.387908	−	0.921698i	$-0.626802\pi$
$619$	−11948.0	−0.775817	−0.387908	+	0.921698i	$0.626802\pi$
$620$	21840.0	1.41470
$621$	5589.00	0.361158
$622$	10534.0	0.679059
$623$	−2544.00	−0.163601
$624$	−6192.00	−0.397241
$625$	25625.0	1.64000
$626$	12680.0	0.809576
$627$	−34632.0	−2.20585
$628$	−432.000	−0.0274501
$629$	200.000	0.0126781
$630$	−4320.00	−0.273195
$631$	−7840.00	−0.494620	−0.247310	−	0.968936i	$-0.579547\pi$
$631$	−7840.00	−0.494620	−0.247310	+	0.968936i	$0.579547\pi$
$632$	3408.00	0.214498
$633$	9144.00	0.574157
$634$	−17588.0	−1.10175
$635$	45580.0	2.84848
$636$	−3096.00	−0.193026
$637$	−14577.0	−0.906690
$638$	728.000	0.0451752
$639$	−1134.00	−0.0702040
$640$	−2560.00	−0.158114
$641$	−2320.00	−0.142956	−0.0714778	−	0.997442i	$-0.522771\pi$
$641$	−2320.00	−0.142956	−0.0714778	+	0.997442i	$0.522771\pi$
$642$	21780.0	1.33892
$643$	1864.00	0.114322	0.0571610	−	0.998365i	$-0.481795\pi$
$643$	1864.00	0.114322	0.0571610	+	0.998365i	$0.481795\pi$
$644$	−184.000	−0.0112587
$645$	−27360.0	−1.67023
$646$	7400.00	0.450695
$647$	11939.0	0.725457	0.362728	−	0.931895i	$-0.381845\pi$
$647$	11939.0	0.725457	0.362728	+	0.931895i	$0.381845\pi$
$648$	5832.00	0.353553
$649$	23088.0	1.39643
$650$	23650.0	1.42712
$651$	4914.00	0.295845
$652$	−5660.00	−0.339973
$653$	10503.0	0.629424	0.314712	−	0.949187i	$-0.398092\pi$
$653$	10503.0	0.629424	0.314712	+	0.949187i	$0.398092\pi$
$654$	30240.0	1.80807
$655$	−19740.0	−1.17757
$656$	1968.00	0.117130
$657$	36774.0	2.18370
$658$	300.000	0.0177739
$659$	10950.0	0.647271	0.323635	−	0.946182i	$-0.395095\pi$
$659$	10950.0	0.647271	0.323635	+	0.946182i	$0.395095\pi$
$660$	−37440.0	−2.20811
$661$	−3210.00	−0.188887	−0.0944437	−	0.995530i	$-0.530107\pi$
$661$	−3210.00	−0.188887	−0.0944437	+	0.995530i	$0.530107\pi$
$662$	16450.0	0.965782
$663$	19350.0	1.13347
$664$	7216.00	0.421740
$665$	2960.00	0.172607
$666$	−432.000	−0.0251346
$667$	161.000	0.00934624
$668$	7024.00	0.406836
$669$	10080.0	0.582534
$670$	−30560.0	−1.76214
$671$	−13624.0	−0.783828
$672$	−576.000	−0.0330650
$673$	−13517.0	−0.774208	−0.387104	−	0.922036i	$-0.626525\pi$
$673$	−13517.0	−0.774208	−0.387104	+	0.922036i	$0.626525\pi$
$674$	−5152.00	−0.294433
$675$	−66825.0	−3.81051
$676$	−1392.00	−0.0791989
$677$	−7494.00	−0.425433	−0.212716	−	0.977114i	$-0.568231\pi$
$677$	−7494.00	−0.425433	−0.212716	+	0.977114i	$0.568231\pi$
$678$	−18540.0	−1.05018
$679$	−684.000	−0.0386591
$680$	8000.00	0.451156
$681$	−24354.0	−1.37041
$682$	28392.0	1.59411
$683$	17865.0	1.00086	0.500428	−	0.865778i	$-0.333176\pi$
$683$	17865.0	1.00086	0.500428	+	0.865778i	$0.333176\pi$
$684$	−15984.0	−0.893514
$685$	32880.0	1.83399
$686$	−2728.00	−0.151830
$687$	−55260.0	−3.06885
$688$	−2432.00	−0.134766
$689$	3698.00	0.204474
$690$	−8280.00	−0.456832
$691$	22364.0	1.23121	0.615605	−	0.788055i	$-0.288912\pi$
$691$	22364.0	1.23121	0.615605	+	0.788055i	$0.288912\pi$
$692$	9432.00	0.518137
$693$	−5616.00	−0.307842
$694$	1192.00	0.0651984
$695$	−43780.0	−2.38945
$696$	504.000	0.0274484
$697$	−6150.00	−0.334215
$698$	−18542.0	−1.00548
$699$	−59103.0	−3.19811
$700$	2200.00	0.118789
$701$	7842.00	0.422522	0.211261	−	0.977430i	$-0.432243\pi$
$701$	7842.00	0.422522	0.211261	+	0.977430i	$0.432243\pi$
$702$	−20898.0	−1.12357
$703$	296.000	0.0158803
$704$	−3328.00	−0.178166
$705$	13500.0	0.721191
$706$	16282.0	0.867962
$707$	−2852.00	−0.151712
$708$	15984.0	0.848468
$709$	11234.0	0.595066	0.297533	−	0.954712i	$-0.403836\pi$
$709$	11234.0	0.595066	0.297533	+	0.954712i	$0.403836\pi$
$710$	840.000	0.0444009
$711$	23004.0	1.21339
$712$	−10176.0	−0.535620
$713$	6279.00	0.329804
$714$	1800.00	0.0943464
$715$	44720.0	2.33907
$716$	4292.00	0.224022
$717$	6561.00	0.341736
$718$	−4260.00	−0.221423
$719$	−17568.0	−0.911232	−0.455616	−	0.890176i	$-0.650581\pi$
$719$	−17568.0	−0.911232	−0.455616	+	0.890176i	$0.650581\pi$
$720$	−17280.0	−0.894427
$721$	−2380.00	−0.122935
$722$	−2766.00	−0.142576
$723$	26208.0	1.34811
$724$	11472.0	0.588886
$725$	−1925.00	−0.0986106
$726$	−24714.0	−1.26339
$727$	−35664.0	−1.81940	−0.909701	−	0.415265i	$-0.863689\pi$
$727$	−35664.0	−1.81940	−0.909701	+	0.415265i	$0.863689\pi$
$728$	688.000	0.0350261
$729$	−19683.0	−1.00000
$730$	−27240.0	−1.38109
$731$	7600.00	0.384536
$732$	−9432.00	−0.476252
$733$	−27914.0	−1.40659	−0.703293	−	0.710900i	$-0.748288\pi$
$733$	−27914.0	−1.40659	−0.703293	+	0.710900i	$0.748288\pi$
$734$	−5148.00	−0.258878
$735$	−61020.0	−3.06225
$736$	−736.000	−0.0368605
$737$	−39728.0	−1.98562
$738$	13284.0	0.662589
$739$	39529.0	1.96766	0.983828	−	0.179116i	$-0.0573237\pi$
$739$	39529.0	1.96766	0.983828	+	0.179116i	$0.0573237\pi$
$740$	320.000	0.0158965
$741$	28638.0	1.41976
$742$	344.000	0.0170197
$743$	10062.0	0.496822	0.248411	−	0.968655i	$-0.420092\pi$
$743$	10062.0	0.496822	0.248411	+	0.968655i	$0.420092\pi$
$744$	19656.0	0.968581
$745$	18920.0	0.930436
$746$	−9008.00	−0.442100
$747$	48708.0	2.38572
$748$	10400.0	0.508371
$749$	−2420.00	−0.118057
$750$	54000.0	2.62907
$751$	25644.0	1.24602	0.623011	−	0.782213i	$-0.285909\pi$
$751$	25644.0	1.24602	0.623011	+	0.782213i	$0.285909\pi$
$752$	1200.00	0.0581908
$753$	−3582.00	−0.173354
$754$	−602.000	−0.0290763
$755$	7300.00	0.351886
$756$	−1944.00	−0.0935220
$757$	37368.0	1.79414	0.897069	−	0.441890i	$-0.145692\pi$
$757$	37368.0	1.79414	0.897069	+	0.441890i	$0.145692\pi$
$758$	5480.00	0.262589
$759$	−10764.0	−0.514767
$760$	11840.0	0.565108
$761$	105.000	0.00500164	0.00250082	−	0.999997i	$-0.499204\pi$
$761$	105.000	0.00500164	0.00250082	+	0.999997i	$0.499204\pi$
$762$	41022.0	1.95022
$763$	−3360.00	−0.159424
$764$	1328.00	0.0628866
$765$	54000.0	2.55212
$766$	13896.0	0.655461
$767$	−19092.0	−0.898790
$768$	−2304.00	−0.108253
$769$	−15464.0	−0.725157	−0.362579	−	0.931953i	$-0.618104\pi$
$769$	−15464.0	−0.725157	−0.362579	+	0.931953i	$0.618104\pi$
$770$	4160.00	0.194696
$771$	−73179.0	−3.41826
$772$	−8572.00	−0.399628
$773$	35168.0	1.63636	0.818179	−	0.574963i	$-0.194984\pi$
$773$	35168.0	1.63636	0.818179	+	0.574963i	$0.194984\pi$
$774$	−16416.0	−0.762353
$775$	−75075.0	−3.47971
$776$	−2736.00	−0.126568
$777$	72.0000	0.00332431
$778$	−2808.00	−0.129398
$779$	−9102.00	−0.418630
$780$	30960.0	1.42121
$781$	1092.00	0.0500318
$782$	2300.00	0.105176
$783$	1701.00	0.0776357
$784$	−5424.00	−0.247085
$785$	2160.00	0.0982085
$786$	−17766.0	−0.806224
$787$	−21216.0	−0.960951	−0.480476	−	0.877008i	$-0.659536\pi$
$787$	−21216.0	−0.960951	−0.480476	+	0.877008i	$0.659536\pi$
$788$	−10956.0	−0.495294
$789$	17802.0	0.803255
$790$	−17040.0	−0.767413
$791$	2060.00	0.0925982
$792$	−22464.0	−1.00786
$793$	11266.0	0.504499
$794$	−17282.0	−0.772437
$795$	15480.0	0.690590
$796$	3008.00	0.133939
$797$	9506.00	0.422484	0.211242	−	0.977434i	$-0.432249\pi$
$797$	9506.00	0.422484	0.211242	+	0.977434i	$0.432249\pi$
$798$	2664.00	0.118176
$799$	−3750.00	−0.166039
$800$	8800.00	0.388909
$801$	−68688.0	−3.02993
$802$	−2280.00	−0.100386
$803$	−35412.0	−1.55624
$804$	−27504.0	−1.20646
$805$	920.000	0.0402804
$806$	−23478.0	−1.02603
$807$	76131.0	3.32087
$808$	−11408.0	−0.496698
$809$	−20550.0	−0.893077	−0.446539	−	0.894764i	$-0.647343\pi$
$809$	−20550.0	−0.893077	−0.446539	+	0.894764i	$0.647343\pi$
$810$	−29160.0	−1.26491
$811$	−5161.00	−0.223461	−0.111731	−	0.993739i	$-0.535639\pi$
$811$	−5161.00	−0.223461	−0.111731	+	0.993739i	$0.535639\pi$
$812$	−56.0000	−0.00242022
$813$	65160.0	2.81090
$814$	416.000	0.0179125
$815$	28300.0	1.21633
$816$	7200.00	0.308885
$817$	11248.0	0.481662
$818$	25058.0	1.07107
$819$	4644.00	0.198137
$820$	−9840.00	−0.419058
$821$	7866.00	0.334379	0.167190	−	0.985925i	$-0.446531\pi$
$821$	7866.00	0.334379	0.167190	+	0.985925i	$0.446531\pi$
$822$	29592.0	1.25564
$823$	−22317.0	−0.945227	−0.472613	−	0.881270i	$-0.656689\pi$
$823$	−22317.0	−0.945227	−0.472613	+	0.881270i	$0.656689\pi$
$824$	−9520.00	−0.402482
$825$	128700.	5.43122
$826$	−1776.00	−0.0748123
$827$	−26196.0	−1.10148	−0.550740	−	0.834677i	$-0.685654\pi$
$827$	−26196.0	−1.10148	−0.550740	+	0.834677i	$0.685654\pi$
$828$	−4968.00	−0.208514
$829$	5886.00	0.246597	0.123299	−	0.992370i	$-0.460653\pi$
$829$	5886.00	0.246597	0.123299	+	0.992370i	$0.460653\pi$
$830$	−36080.0	−1.50886
$831$	11871.0	0.495548
$832$	2752.00	0.114674
$833$	16950.0	0.705021
$834$	−39402.0	−1.63595
$835$	−35120.0	−1.45554
$836$	15392.0	0.636774
$837$	66339.0	2.73956
$838$	6504.00	0.268111
$839$	32394.0	1.33297	0.666487	−	0.745517i	$-0.267797\pi$
$839$	32394.0	1.33297	0.666487	+	0.745517i	$0.267797\pi$
$840$	2880.00	0.118297
$841$	−24340.0	−0.997991
$842$	4412.00	0.180579
$843$	−15930.0	−0.650840
$844$	−4064.00	−0.165745
$845$	6960.00	0.283351
$846$	8100.00	0.329177
$847$	2746.00	0.111397
$848$	1376.00	0.0557217
$849$	−37296.0	−1.50765
$850$	−27500.0	−1.10970
$851$	92.0000	0.00370590
$852$	756.000	0.0303992
$853$	−31286.0	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$853$	−31286.0	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$854$	1048.00	0.0419928
$855$	79920.0	3.19673
$856$	−9680.00	−0.386514
$857$	−2913.00	−0.116110	−0.0580550	−	0.998313i	$-0.518490\pi$
$857$	−2913.00	−0.116110	−0.0580550	+	0.998313i	$0.518490\pi$
$858$	40248.0	1.60145
$859$	15451.0	0.613715	0.306858	−	0.951755i	$-0.400722\pi$
$859$	15451.0	0.613715	0.306858	+	0.951755i	$0.400722\pi$
$860$	12160.0	0.482154
$861$	−2214.00	−0.0876341
$862$	28632.0	1.13133
$863$	−20627.0	−0.813617	−0.406808	−	0.913514i	$-0.633358\pi$
$863$	−20627.0	−0.813617	−0.406808	+	0.913514i	$0.633358\pi$
$864$	−7776.00	−0.306186
$865$	−47160.0	−1.85374
$866$	15656.0	0.614333
$867$	21717.0	0.850690
$868$	−2184.00	−0.0854030
$869$	−22152.0	−0.864735
$870$	−2520.00	−0.0982023
$871$	32852.0	1.27801
$872$	−13440.0	−0.521945
$873$	−18468.0	−0.715976
$874$	3404.00	0.131741
$875$	−6000.00	−0.231814
$876$	−24516.0	−0.945569
$877$	−6966.00	−0.268216	−0.134108	−	0.990967i	$-0.542817\pi$
$877$	−6966.00	−0.268216	−0.134108	+	0.990967i	$0.542817\pi$
$878$	−32078.0	−1.23301
$879$	61308.0	2.35252
$880$	16640.0	0.637425
$881$	−37590.0	−1.43750	−0.718751	−	0.695268i	$-0.755286\pi$
$881$	−37590.0	−1.43750	−0.718751	+	0.695268i	$0.755286\pi$
$882$	−36612.0	−1.39772
$883$	27876.0	1.06240	0.531202	−	0.847245i	$-0.321741\pi$
$883$	27876.0	1.06240	0.531202	+	0.847245i	$0.321741\pi$
$884$	−8600.00	−0.327205
$885$	−79920.0	−3.03557
$886$	23494.0	0.890854
$887$	9471.00	0.358518	0.179259	−	0.983802i	$-0.442630\pi$
$887$	9471.00	0.358518	0.179259	+	0.983802i	$0.442630\pi$
$888$	288.000	0.0108836
$889$	−4558.00	−0.171958
$890$	50880.0	1.91629
$891$	−37908.0	−1.42533
$892$	−4480.00	−0.168163
$893$	−5550.00	−0.207977
$894$	17028.0	0.637026
$895$	−21460.0	−0.801485
$896$	256.000	0.00954504
$897$	8901.00	0.331322
$898$	−5780.00	−0.214790
$899$	1911.00	0.0708959
$900$	59400.0	2.20000
$901$	−4300.00	−0.158994
$902$	−12792.0	−0.472203
$903$	2736.00	0.100829
$904$	8240.00	0.303162
$905$	−57360.0	−2.10686
$906$	6570.00	0.240920
$907$	28366.0	1.03845	0.519227	−	0.854636i	$-0.326220\pi$
$907$	28366.0	1.03845	0.519227	+	0.854636i	$0.326220\pi$
$908$	10824.0	0.395602
$909$	−77004.0	−2.80975
$910$	−3440.00	−0.125313
$911$	−7210.00	−0.262215	−0.131108	−	0.991368i	$-0.541853\pi$
$911$	−7210.00	−0.262215	−0.131108	+	0.991368i	$0.541853\pi$
$912$	10656.0	0.386903
$913$	−46904.0	−1.70021
$914$	−26252.0	−0.950043
$915$	47160.0	1.70389
$916$	24560.0	0.885901
$917$	1974.00	0.0710875
$918$	24300.0	0.873660
$919$	17198.0	0.617312	0.308656	−	0.951174i	$-0.400121\pi$
$919$	17198.0	0.617312	0.308656	+	0.951174i	$0.400121\pi$
$920$	3680.00	0.131876
$921$	51228.0	1.83281
$922$	28962.0	1.03450
$923$	−903.000	−0.0322022
$924$	3744.00	0.133299
$925$	−1100.00	−0.0391003
$926$	10544.0	0.374187
$927$	−64260.0	−2.27678
$928$	−224.000	−0.00792366
$929$	−51033.0	−1.80230	−0.901151	−	0.433505i	$-0.857276\pi$
$929$	−51033.0	−1.80230	−0.901151	+	0.433505i	$0.857276\pi$
$930$	−98280.0	−3.46530
$931$	25086.0	0.883094
$932$	26268.0	0.923216
$933$	−47403.0	−1.66335
$934$	−26932.0	−0.943514
$935$	−52000.0	−1.81880
$936$	18576.0	0.648692
$937$	33328.0	1.16198	0.580992	−	0.813910i	$-0.302665\pi$
$937$	33328.0	1.16198	0.580992	+	0.813910i	$0.302665\pi$
$938$	3056.00	0.106377
$939$	−57060.0	−1.98305
$940$	−6000.00	−0.208190
$941$	−20166.0	−0.698611	−0.349305	−	0.937009i	$-0.613582\pi$
$941$	−20166.0	−0.698611	−0.349305	+	0.937009i	$0.613582\pi$
$942$	1944.00	0.0672388
$943$	−2829.00	−0.0976934
$944$	−7104.00	−0.244932
$945$	9720.00	0.334594
$946$	15808.0	0.543301
$947$	−28629.0	−0.982384	−0.491192	−	0.871051i	$-0.663439\pi$
$947$	−28629.0	−0.982384	−0.491192	+	0.871051i	$0.663439\pi$
$948$	−15336.0	−0.525412
$949$	29283.0	1.00165
$950$	−40700.0	−1.38998
$951$	79146.0	2.69872
$952$	−800.000	−0.0272355
$953$	38146.0	1.29661	0.648305	−	0.761380i	$-0.275478\pi$
$953$	38146.0	1.29661	0.648305	+	0.761380i	$0.275478\pi$
$954$	9288.00	0.315210
$955$	−6640.00	−0.224990
$956$	−2916.00	−0.0986508
$957$	−3276.00	−0.110656
$958$	−9052.00	−0.305279
$959$	−3288.00	−0.110714
$960$	11520.0	0.387298
$961$	44738.0	1.50173
$962$	−344.000	−0.0115291
$963$	−65340.0	−2.18645
$964$	−11648.0	−0.389167
$965$	42860.0	1.42975
$966$	828.000	0.0275781
$967$	44621.0	1.48388	0.741941	−	0.670465i	$-0.233905\pi$
$967$	44621.0	1.48388	0.741941	+	0.670465i	$0.233905\pi$
$968$	10984.0	0.364710
$969$	−33300.0	−1.10397
$970$	13680.0	0.452823
$971$	5950.00	0.196647	0.0983237	−	0.995154i	$-0.468652\pi$
$971$	5950.00	0.196647	0.0983237	+	0.995154i	$0.468652\pi$
$972$	0	0
$973$	4378.00	0.144247
$974$	−17590.0	−0.578665
$975$	−106425.	−3.49572
$976$	4192.00	0.137482
$977$	40836.0	1.33722	0.668608	−	0.743615i	$-0.266891\pi$
$977$	40836.0	1.33722	0.668608	+	0.743615i	$0.266891\pi$
$978$	25470.0	0.832762
$979$	66144.0	2.15932
$980$	27120.0	0.883997
$981$	−90720.0	−2.95257
$982$	−2550.00	−0.0828653
$983$	26874.0	0.871971	0.435985	−	0.899954i	$-0.356400\pi$
$983$	26874.0	0.871971	0.435985	+	0.899954i	$0.356400\pi$
$984$	−8856.00	−0.286910
$985$	54780.0	1.77202
$986$	700.000	0.0226091
$987$	−1350.00	−0.0435370
$988$	−12728.0	−0.409850
$989$	3496.00	0.112403
$990$	112320.	3.60582
$991$	21472.0	0.688275	0.344138	−	0.938919i	$-0.388171\pi$
$991$	21472.0	0.688275	0.344138	+	0.938919i	$0.388171\pi$
$992$	−8736.00	−0.279605
$993$	−74025.0	−2.36567
$994$	−84.0000	−0.00268040
$995$	−15040.0	−0.479196
$996$	−32472.0	−1.03305
$997$	6286.00	0.199679	0.0998393	−	0.995004i	$-0.468167\pi$
$997$	6286.00	0.199679	0.0998393	+	0.995004i	$0.468167\pi$
$998$	−19066.0	−0.604733
$999$	972.000	0.0307835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	46.4.a.b.1.1	✓	1
3.2	odd	2		414.4.a.b.1.1		1
4.3	odd	2		368.4.a.e.1.1		1
5.2	odd	4		1150.4.b.a.599.2		2
5.3	odd	4		1150.4.b.a.599.1		2
5.4	even	2		1150.4.a.d.1.1		1
7.6	odd	2		2254.4.a.b.1.1		1
8.3	odd	2		1472.4.a.a.1.1		1
8.5	even	2		1472.4.a.j.1.1		1
23.22	odd	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	1.1	even	1	trivial
368.4.a.e.1.1		1	4.3	odd	2
414.4.a.b.1.1		1	3.2	odd	2
1058.4.a.b.1.1		1	23.22	odd	2
1150.4.a.d.1.1		1	5.4	even	2
1150.4.b.a.599.1		2	5.3	odd	4
1150.4.b.a.599.2		2	5.2	odd	4
1472.4.a.a.1.1		1	8.3	odd	2
1472.4.a.j.1.1		1	8.5	even	2
2254.4.a.b.1.1		1	7.6	odd	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:	\( N \)	\(=\)	\( 46 = 2 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	46.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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