LMFDB - Embedded newform 414.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,4,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("414.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	414.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:	$24.4267907424$
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	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} +4.00000 q^{4} +20.0000 q^{5} +2.00000 q^{7} -8.00000 q^{8} -40.0000 q^{10} +52.0000 q^{11} +43.0000 q^{13} -4.00000 q^{14} +16.0000 q^{16} +50.0000 q^{17} -74.0000 q^{19} +80.0000 q^{20} -104.000 q^{22} +23.0000 q^{23} +275.000 q^{25} -86.0000 q^{26} +8.00000 q^{28} +7.00000 q^{29} -273.000 q^{31} -32.0000 q^{32} -100.000 q^{34} +40.0000 q^{35} -4.00000 q^{37} +148.000 q^{38} -160.000 q^{40} -123.000 q^{41} -152.000 q^{43} +208.000 q^{44} -46.0000 q^{46} -75.0000 q^{47} -339.000 q^{49} -550.000 q^{50} +172.000 q^{52} -86.0000 q^{53} +1040.00 q^{55} -16.0000 q^{56} -14.0000 q^{58} +444.000 q^{59} +262.000 q^{61} +546.000 q^{62} +64.0000 q^{64} +860.000 q^{65} +764.000 q^{67} +200.000 q^{68} -80.0000 q^{70} +21.0000 q^{71} +681.000 q^{73} +8.00000 q^{74} -296.000 q^{76} +104.000 q^{77} +426.000 q^{79} +320.000 q^{80} +246.000 q^{82} -902.000 q^{83} +1000.00 q^{85} +304.000 q^{86} -416.000 q^{88} +1272.00 q^{89} +86.0000 q^{91} +92.0000 q^{92} +150.000 q^{94} -1480.00 q^{95} -342.000 q^{97} +678.000 q^{98} +O(q^{100})\)

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
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	20.0000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	20.0000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	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
$9$	0	0
$10$	−40.0000	−1.26491
$11$	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	$0.348317\pi$
$14$	−4.00000	−0.0763604
$15$	0	0
$16$	16.0000	0.250000
$17$	50.0000	0.713340	0.356670	−	0.934230i	$-0.383912\pi$
$17$	50.0000	0.713340	0.356670	+	0.934230i	$0.383912\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	−104.000	−1.00786
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	275.000	2.20000
$26$	−86.0000	−0.648692
$27$	0	0
$28$	8.00000	0.0539949
$29$	7.00000	0.0448230	0.0224115	−	0.999749i	$-0.492866\pi$
$29$	7.00000	0.0448230	0.0224115	+	0.999749i	$0.492866\pi$
$30$	0	0
$31$	−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
$33$	0	0
$34$	−100.000	−0.504408
$35$	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	$0.502829\pi$
$38$	148.000	0.631810
$39$	0	0
$40$	−160.000	−0.632456
$41$	−123.000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−123.000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	−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
$45$	0	0
$46$	−46.0000	−0.147442
$47$	−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
$50$	−550.000	−1.55563
$51$	0	0
$52$	172.000	0.458694
$53$	−86.0000	−0.222887	−0.111443	−	0.993771i	$-0.535547\pi$
$53$	−86.0000	−0.222887	−0.111443	+	0.993771i	$0.535547\pi$
$54$	0	0
$55$	1040.00	2.54970
$56$	−16.0000	−0.0381802
$57$	0	0
$58$	−14.0000	−0.0316947
$59$	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	$0.411334\pi$
$62$	546.000	1.11842
$63$	0	0
$64$	64.0000	0.125000
$65$	860.000	1.64107
$66$	0	0
$67$	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
$69$	0	0
$70$	−80.0000	−0.136598
$71$	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	$0.316178\pi$
$74$	8.00000	0.0125673
$75$	0	0
$76$	−296.000	−0.446757
$77$	104.000	0.153921
$78$	0	0
$79$	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
$81$	0	0
$82$	246.000	0.331295
$83$	−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
$86$	304.000	0.381176
$87$	0	0
$88$	−416.000	−0.503929
$89$	1272.00	1.51496	0.757482	−	0.652856i	$-0.226430\pi$
$89$	1272.00	1.51496	0.757482	+	0.652856i	$0.226430\pi$
$90$	0	0
$91$	86.0000	0.0990687
$92$	92.0000	0.104257
$93$	0	0
$94$	150.000	0.164588
$95$	−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	$0.557284\pi$
$98$	678.000	0.698861
$99$	0	0
$100$	1100.00	1.10000
$101$	1426.00	1.40487	0.702437	−	0.711746i	$-0.252095\pi$
$101$	1426.00	1.40487	0.702437	+	0.711746i	$0.252095\pi$
$102$	0	0
$103$	−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
$105$	0	0
$106$	172.000	0.157605
$107$	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	$0.764296\pi$
$110$	−2080.00	−1.80291
$111$	0	0
$112$	32.0000	0.0269975
$113$	−1030.00	−0.857471	−0.428736	−	0.903430i	$-0.641041\pi$
$113$	−1030.00	−0.857471	−0.428736	+	0.903430i	$0.641041\pi$
$114$	0	0
$115$	460.000	0.373002
$116$	28.0000	0.0224115
$117$	0	0
$118$	−888.000	−0.692771
$119$	100.000	0.0770335
$120$	0	0
$121$	1373.00	1.03156
$122$	−524.000	−0.388858
$123$	0	0
$124$	−1092.00	−0.790843
$125$	3000.00	2.14663
$126$	0	0
$127$	−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
$129$	0	0
$130$	−1720.00	−1.16042
$131$	−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
$134$	−1528.00	−0.985068
$135$	0	0
$136$	−400.000	−0.252204
$137$	1644.00	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	1644.00	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	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
$141$	0	0
$142$	−42.0000	−0.0248209
$143$	2236.00	1.30758
$144$	0	0
$145$	140.000	0.0801818
$146$	−1362.00	−0.772054
$147$	0	0
$148$	−16.0000	−0.00888643
$149$	946.000	0.520130	0.260065	−	0.965591i	$-0.416256\pi$
$149$	946.000	0.520130	0.260065	+	0.965591i	$0.416256\pi$
$150$	0	0
$151$	−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
$153$	0	0
$154$	−208.000	−0.108838
$155$	−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	$0.508739\pi$
$158$	−852.000	−0.428997
$159$	0	0
$160$	−640.000	−0.316228
$161$	46.0000	0.0225174
$162$	0	0
$163$	−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
$165$	0	0
$166$	1804.00	0.843479
$167$	−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
$170$	−2000.00	−0.902312
$171$	0	0
$172$	−608.000	−0.269532
$173$	−2358.00	−1.03627	−0.518137	−	0.855298i	$-0.673374\pi$
$173$	−2358.00	−1.03627	−0.518137	+	0.855298i	$0.673374\pi$
$174$	0	0
$175$	550.000	0.237578
$176$	832.000	0.356332
$177$	0	0
$178$	−2544.00	−1.07124
$179$	−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	$0.299567\pi$
$182$	−172.000	−0.0700521
$183$	0	0
$184$	−184.000	−0.0737210
$185$	−80.0000	−0.0317931
$186$	0	0
$187$	2600.00	1.01674
$188$	−300.000	−0.116382
$189$	0	0
$190$	2960.00	1.13022
$191$	−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	$0.630861\pi$
$194$	684.000	0.253136
$195$	0	0
$196$	−1356.00	−0.494169
$197$	2739.00	0.990587	0.495294	−	0.868726i	$-0.335060\pi$
$197$	2739.00	0.990587	0.495294	+	0.868726i	$0.335060\pi$
$198$	0	0
$199$	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
$201$	0	0
$202$	−2852.00	−0.993396
$203$	14.0000	0.00484043
$204$	0	0
$205$	−2460.00	−0.838116
$206$	2380.00	0.804963
$207$	0	0
$208$	688.000	0.229347
$209$	−3848.00	−1.27355
$210$	0	0
$211$	−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
$213$	0	0
$214$	−2420.00	−0.773027
$215$	−3040.00	−0.964308
$216$	0	0
$217$	−546.000	−0.170806
$218$	3360.00	1.04389
$219$	0	0
$220$	4160.00	1.27485
$221$	2150.00	0.654410
$222$	0	0
$223$	−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
$225$	0	0
$226$	2060.00	0.606324
$227$	−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	$0.153541\pi$
$230$	−920.000	−0.263752
$231$	0	0
$232$	−56.0000	−0.0158473
$233$	−6567.00	−1.84643	−0.923216	−	0.384282i	$-0.874449\pi$
$233$	−6567.00	−1.84643	−0.923216	+	0.384282i	$0.874449\pi$
$234$	0	0
$235$	−1500.00	−0.416380
$236$	1776.00	0.489863
$237$	0	0
$238$	−200.000	−0.0544709
$239$	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	$0.627237\pi$
$242$	−2746.00	−0.729420
$243$	0	0
$244$	1048.00	0.274964
$245$	−6780.00	−1.76799
$246$	0	0
$247$	−3182.00	−0.819700
$248$	2184.00	0.559210
$249$	0	0
$250$	−6000.00	−1.51789
$251$	−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
$254$	4558.00	1.12596
$255$	0	0
$256$	256.000	0.0625000
$257$	−8131.00	−1.97353	−0.986766	−	0.162149i	$-0.948157\pi$
$257$	−8131.00	−1.97353	−0.986766	+	0.162149i	$0.948157\pi$
$258$	0	0
$259$	−8.00000	−0.00191929
$260$	3440.00	0.820537
$261$	0	0
$262$	1974.00	0.465474
$263$	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
$266$	296.000	0.0682290
$267$	0	0
$268$	3056.00	0.696548
$269$	8459.00	1.91730	0.958651	−	0.284584i	$-0.0918554\pi$
$269$	8459.00	1.91730	0.958651	+	0.284584i	$0.0918554\pi$
$270$	0	0
$271$	−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
$273$	0	0
$274$	−3288.00	−0.724947
$275$	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	$0.545692\pi$
$278$	−4378.00	−0.944514
$279$	0	0
$280$	−320.000	−0.0682988
$281$	−1770.00	−0.375763	−0.187881	−	0.982192i	$-0.560162\pi$
$281$	−1770.00	−0.375763	−0.187881	+	0.982192i	$0.560162\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	−4472.00	−0.924598
$287$	−246.000	−0.0505955
$288$	0	0
$289$	−2413.00	−0.491146
$290$	−280.000	−0.0566971
$291$	0	0
$292$	2724.00	0.545925
$293$	6812.00	1.35823	0.679115	−	0.734032i	$-0.262364\pi$
$293$	6812.00	1.35823	0.679115	+	0.734032i	$0.262364\pi$
$294$	0	0
$295$	8880.00	1.75259
$296$	32.0000	0.00628366
$297$	0	0
$298$	−1892.00	−0.367787
$299$	989.000	0.191289
$300$	0	0
$301$	−304.000	−0.0582135
$302$	730.000	0.139095
$303$	0	0
$304$	−1184.00	−0.223378
$305$	5240.00	0.983743
$306$	0	0
$307$	−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
$309$	0	0
$310$	10920.0	2.00069
$311$	−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	$0.305990\pi$
$314$	216.000	0.0388203
$315$	0	0
$316$	1704.00	0.303346
$317$	8794.00	1.55811	0.779054	−	0.626957i	$-0.215700\pi$
$317$	8794.00	1.55811	0.779054	+	0.626957i	$0.215700\pi$
$318$	0	0
$319$	364.000	0.0638874
$320$	1280.00	0.223607
$321$	0	0
$322$	−92.0000	−0.0159222
$323$	−3700.00	−0.637379
$324$	0	0
$325$	11825.0	2.01826
$326$	2830.00	0.480795
$327$	0	0
$328$	984.000	0.165647
$329$	−150.000	−0.0251361
$330$	0	0
$331$	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
$333$	0	0
$334$	3512.00	0.575354
$335$	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	$0.566759\pi$
$338$	696.000	0.112004
$339$	0	0
$340$	4000.00	0.638031
$341$	−14196.0	−2.25442
$342$	0	0
$343$	−1364.00	−0.214720
$344$	1216.00	0.190588
$345$	0	0
$346$	4716.00	0.732756
$347$	−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	$0.751749\pi$
$350$	−1100.00	−0.167993
$351$	0	0
$352$	−1664.00	−0.251964
$353$	−8141.00	−1.22748	−0.613742	−	0.789507i	$-0.710336\pi$
$353$	−8141.00	−1.22748	−0.613742	+	0.789507i	$0.710336\pi$
$354$	0	0
$355$	420.000	0.0627924
$356$	5088.00	0.757482
$357$	0	0
$358$	2146.00	0.316815
$359$	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
$362$	−5736.00	−0.832811
$363$	0	0
$364$	344.000	0.0495343
$365$	13620.0	1.95316
$366$	0	0
$367$	−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
$369$	0	0
$370$	160.000	0.0224811
$371$	−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	$0.601204\pi$
$374$	−5200.00	−0.718945
$375$	0	0
$376$	600.000	0.0822942
$377$	301.000	0.0411201
$378$	0	0
$379$	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
$381$	0	0
$382$	664.000	0.0889351
$383$	−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
$386$	4286.00	0.565160
$387$	0	0
$388$	−1368.00	−0.178994
$389$	1404.00	0.182996	0.0914982	−	0.995805i	$-0.470834\pi$
$389$	1404.00	0.182996	0.0914982	+	0.995805i	$0.470834\pi$
$390$	0	0
$391$	1150.00	0.148742
$392$	2712.00	0.349430
$393$	0	0
$394$	−5478.00	−0.700451
$395$	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	$0.683924\pi$
$398$	−1504.00	−0.189419
$399$	0	0
$400$	4400.00	0.550000
$401$	1140.00	0.141967	0.0709836	−	0.997477i	$-0.477386\pi$
$401$	1140.00	0.141967	0.0709836	+	0.997477i	$0.477386\pi$
$402$	0	0
$403$	−11739.0	−1.45102
$404$	5704.00	0.702437
$405$	0	0
$406$	−28.0000	−0.00342270
$407$	−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	$0.226490\pi$
$410$	4920.00	0.592638
$411$	0	0
$412$	−4760.00	−0.569195
$413$	888.000	0.105801
$414$	0	0
$415$	−18040.0	−2.13385
$416$	−1376.00	−0.162173
$417$	0	0
$418$	7696.00	0.900535
$419$	−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	$0.459244\pi$
$422$	2032.00	0.234399
$423$	0	0
$424$	688.000	0.0788024
$425$	13750.0	1.56935
$426$	0	0
$427$	524.000	0.0593867
$428$	4840.00	0.546613
$429$	0	0
$430$	6080.00	0.681869
$431$	−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	$0.356961\pi$
$434$	1092.00	0.120778
$435$	0	0
$436$	−6720.00	−0.738141
$437$	−1702.00	−0.186311
$438$	0	0
$439$	−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
$441$	0	0
$442$	−4300.00	−0.462738
$443$	−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
$446$	2240.00	0.237819
$447$	0	0
$448$	128.000	0.0134987
$449$	2890.00	0.303758	0.151879	−	0.988399i	$-0.451468\pi$
$449$	2890.00	0.303758	0.151879	+	0.988399i	$0.451468\pi$
$450$	0	0
$451$	−6396.00	−0.667796
$452$	−4120.00	−0.428736
$453$	0	0
$454$	5412.00	0.559466
$455$	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	$0.734471\pi$
$458$	−12280.0	−1.25285
$459$	0	0
$460$	1840.00	0.186501
$461$	−14481.0	−1.46301	−0.731505	−	0.681836i	$-0.761182\pi$
$461$	−14481.0	−1.46301	−0.731505	+	0.681836i	$0.761182\pi$
$462$	0	0
$463$	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
$465$	0	0
$466$	13134.0	1.30562
$467$	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
$470$	3000.00	0.294425
$471$	0	0
$472$	−3552.00	−0.346386
$473$	−7904.00	−0.768343
$474$	0	0
$475$	−20350.0	−1.96573
$476$	400.000	0.0385167
$477$	0	0
$478$	−1458.00	−0.139513
$479$	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
$482$	5824.00	0.550365
$483$	0	0
$484$	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
$489$	0	0
$490$	13560.0	1.25016
$491$	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
$494$	6364.00	0.579615
$495$	0	0
$496$	−4368.00	−0.395421
$497$	42.0000	0.00379066
$498$	0	0
$499$	−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
$501$	0	0
$502$	796.000	0.0707714
$503$	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
$506$	−2392.00	−0.210153
$507$	0	0
$508$	−9116.00	−0.796175
$509$	8031.00	0.699347	0.349674	−	0.936872i	$-0.386292\pi$
$509$	8031.00	0.699347	0.349674	+	0.936872i	$0.386292\pi$
$510$	0	0
$511$	1362.00	0.117909
$512$	−512.000	−0.0441942
$513$	0	0
$514$	16262.0	1.39550
$515$	−23800.0	−2.03641
$516$	0	0
$517$	−3900.00	−0.331764
$518$	16.0000	0.00135714
$519$	0	0
$520$	−6880.00	−0.580208
$521$	21184.0	1.78136	0.890679	−	0.454632i	$-0.150229\pi$
$521$	21184.0	1.78136	0.890679	+	0.454632i	$0.150229\pi$
$522$	0	0
$523$	−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
$525$	0	0
$526$	−3956.00	−0.327927
$527$	−13650.0	−1.12828
$528$	0	0
$529$	529.000	0.0434783
$530$	3440.00	0.281932
$531$	0	0
$532$	−592.000	−0.0482452
$533$	−5289.00	−0.429816
$534$	0	0
$535$	24200.0	1.95562
$536$	−6112.00	−0.492534
$537$	0	0
$538$	−16918.0	−1.35574
$539$	−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	$0.426223\pi$
$542$	14480.0	1.14754
$543$	0	0
$544$	−1600.00	−0.126102
$545$	−33600.0	−2.64085
$546$	0	0
$547$	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
$549$	0	0
$550$	−28600.0	−2.21729
$551$	−518.000	−0.0400500
$552$	0	0
$553$	852.000	0.0655167
$554$	2638.00	0.202307
$555$	0	0
$556$	8756.00	0.667873
$557$	20240.0	1.53967	0.769835	−	0.638243i	$-0.220338\pi$
$557$	20240.0	1.53967	0.769835	+	0.638243i	$0.220338\pi$
$558$	0	0
$559$	−6536.00	−0.494532
$560$	640.000	0.0482945
$561$	0	0
$562$	3540.00	0.265704
$563$	−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
$566$	−8288.00	−0.615496
$567$	0	0
$568$	−168.000	−0.0124104
$569$	−19484.0	−1.43552	−0.717761	−	0.696290i	$-0.754833\pi$
$569$	−19484.0	−1.43552	−0.717761	+	0.696290i	$0.754833\pi$
$570$	0	0
$571$	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
$573$	0	0
$574$	492.000	0.0357765
$575$	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	$0.507338\pi$
$578$	4826.00	0.347293
$579$	0	0
$580$	560.000	0.0400909
$581$	−1804.00	−0.128817
$582$	0	0
$583$	−4472.00	−0.317687
$584$	−5448.00	−0.386027
$585$	0	0
$586$	−13624.0	−0.960413
$587$	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
$590$	−17760.0	−1.23927
$591$	0	0
$592$	−64.0000	−0.00444322
$593$	20610.0	1.42724	0.713618	−	0.700535i	$-0.247055\pi$
$593$	20610.0	1.42724	0.713618	+	0.700535i	$0.247055\pi$
$594$	0	0
$595$	2000.00	0.137802
$596$	3784.00	0.260065
$597$	0	0
$598$	−1978.00	−0.135262
$599$	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	$0.592679\pi$
$602$	608.000	0.0411632
$603$	0	0
$604$	−1460.00	−0.0983552
$605$	27460.0	1.84530
$606$	0	0
$607$	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
$609$	0	0
$610$	−10480.0	−0.695611
$611$	−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	$0.473396\pi$
$614$	11384.0	0.748242
$615$	0	0
$616$	−832.000	−0.0544192
$617$	5610.00	0.366046	0.183023	−	0.983109i	$-0.441412\pi$
$617$	5610.00	0.366046	0.183023	+	0.983109i	$0.441412\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	10534.0	0.679059
$623$	2544.00	0.163601
$624$	0	0
$625$	25625.0	1.64000
$626$	−12680.0	−0.809576
$627$	0	0
$628$	−432.000	−0.0274501
$629$	−200.000	−0.0126781
$630$	0	0
$631$	−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
$633$	0	0
$634$	−17588.0	−1.10175
$635$	−45580.0	−2.84848
$636$	0	0
$637$	−14577.0	−0.906690
$638$	−728.000	−0.0451752
$639$	0	0
$640$	−2560.00	−0.158114
$641$	2320.00	0.142956	0.0714778	−	0.997442i	$-0.477229\pi$
$641$	2320.00	0.142956	0.0714778	+	0.997442i	$0.477229\pi$
$642$	0	0
$643$	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
$645$	0	0
$646$	7400.00	0.450695
$647$	−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
$650$	−23650.0	−1.42712
$651$	0	0
$652$	−5660.00	−0.339973
$653$	−10503.0	−0.629424	−0.314712	−	0.949187i	$-0.601908\pi$
$653$	−10503.0	−0.629424	−0.314712	+	0.949187i	$0.601908\pi$
$654$	0	0
$655$	−19740.0	−1.17757
$656$	−1968.00	−0.117130
$657$	0	0
$658$	300.000	0.0177739
$659$	−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	$0.530107\pi$
$662$	−16450.0	−0.965782
$663$	0	0
$664$	7216.00	0.421740
$665$	−2960.00	−0.172607
$666$	0	0
$667$	161.000	0.00934624
$668$	−7024.00	−0.406836
$669$	0	0
$670$	−30560.0	−1.76214
$671$	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	$0.626525\pi$
$674$	5152.00	0.294433
$675$	0	0
$676$	−1392.00	−0.0791989
$677$	7494.00	0.425433	0.212716	−	0.977114i	$-0.431769\pi$
$677$	7494.00	0.425433	0.212716	+	0.977114i	$0.431769\pi$
$678$	0	0
$679$	−684.000	−0.0386591
$680$	−8000.00	−0.451156
$681$	0	0
$682$	28392.0	1.59411
$683$	−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
$686$	2728.00	0.151830
$687$	0	0
$688$	−2432.00	−0.134766
$689$	−3698.00	−0.204474
$690$	0	0
$691$	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
$693$	0	0
$694$	1192.00	0.0651984
$695$	43780.0	2.38945
$696$	0	0
$697$	−6150.00	−0.334215
$698$	18542.0	1.00548
$699$	0	0
$700$	2200.00	0.118789
$701$	−7842.00	−0.422522	−0.211261	−	0.977430i	$-0.567757\pi$
$701$	−7842.00	−0.422522	−0.211261	+	0.977430i	$0.567757\pi$
$702$	0	0
$703$	296.000	0.0158803
$704$	3328.00	0.178166
$705$	0	0
$706$	16282.0	0.867962
$707$	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	$0.403836\pi$
$710$	−840.000	−0.0444009
$711$	0	0
$712$	−10176.0	−0.535620
$713$	−6279.00	−0.329804
$714$	0	0
$715$	44720.0	2.33907
$716$	−4292.00	−0.224022
$717$	0	0
$718$	−4260.00	−0.221423
$719$	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
$722$	2766.00	0.142576
$723$	0	0
$724$	11472.0	0.588886
$725$	1925.00	0.0986106
$726$	0	0
$727$	−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
$729$	0	0
$730$	−27240.0	−1.38109
$731$	−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	$0.748288\pi$
$734$	5148.00	0.258878
$735$	0	0
$736$	−736.000	−0.0368605
$737$	39728.0	1.98562
$738$	0	0
$739$	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
$741$	0	0
$742$	344.000	0.0170197
$743$	−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
$746$	9008.00	0.442100
$747$	0	0
$748$	10400.0	0.508371
$749$	2420.00	0.118057
$750$	0	0
$751$	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
$753$	0	0
$754$	−602.000	−0.0290763
$755$	−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	$0.145692\pi$
$758$	−5480.00	−0.262589
$759$	0	0
$760$	11840.0	0.565108
$761$	−105.000	−0.00500164	−0.00250082	−	0.999997i	$-0.500796\pi$
$761$	−105.000	−0.00500164	−0.00250082	+	0.999997i	$0.500796\pi$
$762$	0	0
$763$	−3360.00	−0.159424
$764$	−1328.00	−0.0628866
$765$	0	0
$766$	13896.0	0.655461
$767$	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	$0.618104\pi$
$770$	−4160.00	−0.194696
$771$	0	0
$772$	−8572.00	−0.399628
$773$	−35168.0	−1.63636	−0.818179	−	0.574963i	$-0.805016\pi$
$773$	−35168.0	−1.63636	−0.818179	+	0.574963i	$0.805016\pi$
$774$	0	0
$775$	−75075.0	−3.47971
$776$	2736.00	0.126568
$777$	0	0
$778$	−2808.00	−0.129398
$779$	9102.00	0.418630
$780$	0	0
$781$	1092.00	0.0500318
$782$	−2300.00	−0.105176
$783$	0	0
$784$	−5424.00	−0.247085
$785$	−2160.00	−0.0982085
$786$	0	0
$787$	−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
$789$	0	0
$790$	−17040.0	−0.767413
$791$	−2060.00	−0.0925982
$792$	0	0
$793$	11266.0	0.504499
$794$	17282.0	0.772437
$795$	0	0
$796$	3008.00	0.133939
$797$	−9506.00	−0.422484	−0.211242	−	0.977434i	$-0.567751\pi$
$797$	−9506.00	−0.422484	−0.211242	+	0.977434i	$0.567751\pi$
$798$	0	0
$799$	−3750.00	−0.166039
$800$	−8800.00	−0.388909
$801$	0	0
$802$	−2280.00	−0.100386
$803$	35412.0	1.55624
$804$	0	0
$805$	920.000	0.0402804
$806$	23478.0	1.02603
$807$	0	0
$808$	−11408.0	−0.496698
$809$	20550.0	0.893077	0.446539	−	0.894764i	$-0.352657\pi$
$809$	20550.0	0.893077	0.446539	+	0.894764i	$0.352657\pi$
$810$	0	0
$811$	−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
$813$	0	0
$814$	416.000	0.0179125
$815$	−28300.0	−1.21633
$816$	0	0
$817$	11248.0	0.481662
$818$	−25058.0	−1.07107
$819$	0	0
$820$	−9840.00	−0.419058
$821$	−7866.00	−0.334379	−0.167190	−	0.985925i	$-0.553469\pi$
$821$	−7866.00	−0.334379	−0.167190	+	0.985925i	$0.553469\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	−1776.00	−0.0748123
$827$	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	$0.460653\pi$
$830$	36080.0	1.50886
$831$	0	0
$832$	2752.00	0.114674
$833$	−16950.0	−0.705021
$834$	0	0
$835$	−35120.0	−1.45554
$836$	−15392.0	−0.636774
$837$	0	0
$838$	6504.00	0.268111
$839$	−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
$842$	−4412.00	−0.180579
$843$	0	0
$844$	−4064.00	−0.165745
$845$	−6960.00	−0.283351
$846$	0	0
$847$	2746.00	0.111397
$848$	−1376.00	−0.0557217
$849$	0	0
$850$	−27500.0	−1.10970
$851$	−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	$0.716089\pi$
$854$	−1048.00	−0.0419928
$855$	0	0
$856$	−9680.00	−0.386514
$857$	2913.00	0.116110	0.0580550	−	0.998313i	$-0.481510\pi$
$857$	2913.00	0.116110	0.0580550	+	0.998313i	$0.481510\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	28632.0	1.13133
$863$	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
$866$	−15656.0	−0.614333
$867$	0	0
$868$	−2184.00	−0.0854030
$869$	22152.0	0.864735
$870$	0	0
$871$	32852.0	1.27801
$872$	13440.0	0.521945
$873$	0	0
$874$	3404.00	0.131741
$875$	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	$0.542817\pi$
$878$	32078.0	1.23301
$879$	0	0
$880$	16640.0	0.637425
$881$	37590.0	1.43750	0.718751	−	0.695268i	$-0.244714\pi$
$881$	37590.0	1.43750	0.718751	+	0.695268i	$0.244714\pi$
$882$	0	0
$883$	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
$885$	0	0
$886$	23494.0	0.890854
$887$	−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
$890$	−50880.0	−1.91629
$891$	0	0
$892$	−4480.00	−0.168163
$893$	5550.00	0.207977
$894$	0	0
$895$	−21460.0	−0.801485
$896$	−256.000	−0.00954504
$897$	0	0
$898$	−5780.00	−0.214790
$899$	−1911.00	−0.0708959
$900$	0	0
$901$	−4300.00	−0.158994
$902$	12792.0	0.472203
$903$	0	0
$904$	8240.00	0.303162
$905$	57360.0	2.10686
$906$	0	0
$907$	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
$909$	0	0
$910$	−3440.00	−0.125313
$911$	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
$914$	26252.0	0.950043
$915$	0	0
$916$	24560.0	0.885901
$917$	−1974.00	−0.0710875
$918$	0	0
$919$	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
$921$	0	0
$922$	28962.0	1.03450
$923$	903.000	0.0322022
$924$	0	0
$925$	−1100.00	−0.0391003
$926$	−10544.0	−0.374187
$927$	0	0
$928$	−224.000	−0.00792366
$929$	51033.0	1.80230	0.901151	−	0.433505i	$-0.142724\pi$
$929$	51033.0	1.80230	0.901151	+	0.433505i	$0.142724\pi$
$930$	0	0
$931$	25086.0	0.883094
$932$	−26268.0	−0.923216
$933$	0	0
$934$	−26932.0	−0.943514
$935$	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	$0.302665\pi$
$938$	−3056.00	−0.106377
$939$	0	0
$940$	−6000.00	−0.208190
$941$	20166.0	0.698611	0.349305	−	0.937009i	$-0.386418\pi$
$941$	20166.0	0.698611	0.349305	+	0.937009i	$0.386418\pi$
$942$	0	0
$943$	−2829.00	−0.0976934
$944$	7104.00	0.244932
$945$	0	0
$946$	15808.0	0.543301
$947$	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
$950$	40700.0	1.38998
$951$	0	0
$952$	−800.000	−0.0272355
$953$	−38146.0	−1.29661	−0.648305	−	0.761380i	$-0.724522\pi$
$953$	−38146.0	−1.29661	−0.648305	+	0.761380i	$0.724522\pi$
$954$	0	0
$955$	−6640.00	−0.224990
$956$	2916.00	0.0986508
$957$	0	0
$958$	−9052.00	−0.305279
$959$	3288.00	0.110714
$960$	0	0
$961$	44738.0	1.50173
$962$	344.000	0.0115291
$963$	0	0
$964$	−11648.0	−0.389167
$965$	−42860.0	−1.42975
$966$	0	0
$967$	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
$969$	0	0
$970$	13680.0	0.452823
$971$	−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
$974$	17590.0	0.578665
$975$	0	0
$976$	4192.00	0.137482
$977$	−40836.0	−1.33722	−0.668608	−	0.743615i	$-0.733109\pi$
$977$	−40836.0	−1.33722	−0.668608	+	0.743615i	$0.733109\pi$
$978$	0	0
$979$	66144.0	2.15932
$980$	−27120.0	−0.883997
$981$	0	0
$982$	−2550.00	−0.0828653
$983$	−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
$986$	−700.000	−0.0226091
$987$	0	0
$988$	−12728.0	−0.409850
$989$	−3496.00	−0.112403
$990$	0	0
$991$	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
$993$	0	0
$994$	−84.0000	−0.00268040
$995$	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	$0.468167\pi$
$998$	19066.0	0.604733
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.4.a.b.1.1		1
3.2	odd	2		46.4.a.b.1.1	✓	1
12.11	even	2		368.4.a.e.1.1		1
15.2	even	4		1150.4.b.a.599.2		2
15.8	even	4		1150.4.b.a.599.1		2
15.14	odd	2		1150.4.a.d.1.1		1
21.20	even	2		2254.4.a.b.1.1		1
24.5	odd	2		1472.4.a.j.1.1		1
24.11	even	2		1472.4.a.a.1.1		1
69.68	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	3.2	odd	2
368.4.a.e.1.1		1	12.11	even	2
414.4.a.b.1.1		1	1.1	even	1	trivial
1058.4.a.b.1.1		1	69.68	even	2
1150.4.a.d.1.1		1	15.14	odd	2
1150.4.b.a.599.1		2	15.8	even	4
1150.4.b.a.599.2		2	15.2	even	4
1472.4.a.a.1.1		1	24.11	even	2
1472.4.a.j.1.1		1	24.5	odd	2
2254.4.a.b.1.1		1	21.20	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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more