LMFDB - Embedded newform 270.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,4,Mod(1,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, 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("270.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	270.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:	$15.9305157015$
Analytic rank:	$0$
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	$\chi$	$=$	270.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -4.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -4.00000 q^{7} +8.00000 q^{8} -10.0000 q^{10} +42.0000 q^{11} +20.0000 q^{13} -8.00000 q^{14} +16.0000 q^{16} +93.0000 q^{17} +59.0000 q^{19} -20.0000 q^{20} +84.0000 q^{22} +9.00000 q^{23} +25.0000 q^{25} +40.0000 q^{26} -16.0000 q^{28} +120.000 q^{29} +47.0000 q^{31} +32.0000 q^{32} +186.000 q^{34} +20.0000 q^{35} -262.000 q^{37} +118.000 q^{38} -40.0000 q^{40} +126.000 q^{41} -178.000 q^{43} +168.000 q^{44} +18.0000 q^{46} +144.000 q^{47} -327.000 q^{49} +50.0000 q^{50} +80.0000 q^{52} +741.000 q^{53} -210.000 q^{55} -32.0000 q^{56} +240.000 q^{58} -444.000 q^{59} +221.000 q^{61} +94.0000 q^{62} +64.0000 q^{64} -100.000 q^{65} -538.000 q^{67} +372.000 q^{68} +40.0000 q^{70} +690.000 q^{71} -1126.00 q^{73} -524.000 q^{74} +236.000 q^{76} -168.000 q^{77} +665.000 q^{79} -80.0000 q^{80} +252.000 q^{82} +75.0000 q^{83} -465.000 q^{85} -356.000 q^{86} +336.000 q^{88} -1086.00 q^{89} -80.0000 q^{91} +36.0000 q^{92} +288.000 q^{94} -295.000 q^{95} +1544.00 q^{97} -654.000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	42.0000	1.15123	0.575613	−	0.817723i	$-0.304764\pi$
$11$	42.0000	1.15123	0.575613	+	0.817723i	$0.304764\pi$
$12$	0	0
$13$	20.0000	0.426692	0.213346	−	0.976977i	$-0.431564\pi$
$13$	20.0000	0.426692	0.213346	+	0.976977i	$0.431564\pi$
$14$	−8.00000	−0.152721
$15$	0	0
$16$	16.0000	0.250000
$17$	93.0000	1.32681	0.663406	−	0.748259i	$-0.269110\pi$
$17$	93.0000	1.32681	0.663406	+	0.748259i	$0.269110\pi$
$18$	0	0
$19$	59.0000	0.712396	0.356198	−	0.934410i	$-0.384073\pi$
$19$	59.0000	0.712396	0.356198	+	0.934410i	$0.384073\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	84.0000	0.814039
$23$	9.00000	0.0815926	0.0407963	−	0.999167i	$-0.487011\pi$
$23$	9.00000	0.0815926	0.0407963	+	0.999167i	$0.487011\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	40.0000	0.301717
$27$	0	0
$28$	−16.0000	−0.107990
$29$	120.000	0.768395	0.384197	−	0.923251i	$-0.374478\pi$
$29$	120.000	0.768395	0.384197	+	0.923251i	$0.374478\pi$
$30$	0	0
$31$	47.0000	0.272305	0.136152	−	0.990688i	$-0.456526\pi$
$31$	47.0000	0.272305	0.136152	+	0.990688i	$0.456526\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	186.000	0.938198
$35$	20.0000	0.0965891
$36$	0	0
$37$	−262.000	−1.16412	−0.582061	−	0.813145i	$-0.697754\pi$
$37$	−262.000	−1.16412	−0.582061	+	0.813145i	$0.697754\pi$
$38$	118.000	0.503740
$39$	0	0
$40$	−40.0000	−0.158114
$41$	126.000	0.479949	0.239974	−	0.970779i	$-0.422861\pi$
$41$	126.000	0.479949	0.239974	+	0.970779i	$0.422861\pi$
$42$	0	0
$43$	−178.000	−0.631273	−0.315637	−	0.948880i	$-0.602218\pi$
$43$	−178.000	−0.631273	−0.315637	+	0.948880i	$0.602218\pi$
$44$	168.000	0.575613
$45$	0	0
$46$	18.0000	0.0576947
$47$	144.000	0.446906	0.223453	−	0.974715i	$-0.428267\pi$
$47$	144.000	0.446906	0.223453	+	0.974715i	$0.428267\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	50.0000	0.141421
$51$	0	0
$52$	80.0000	0.213346
$53$	741.000	1.92046	0.960228	−	0.279217i	$-0.0900748\pi$
$53$	741.000	1.92046	0.960228	+	0.279217i	$0.0900748\pi$
$54$	0	0
$55$	−210.000	−0.514844
$56$	−32.0000	−0.0763604
$57$	0	0
$58$	240.000	0.543337
$59$	−444.000	−0.979727	−0.489863	−	0.871799i	$-0.662953\pi$
$59$	−444.000	−0.979727	−0.489863	+	0.871799i	$0.662953\pi$
$60$	0	0
$61$	221.000	0.463871	0.231936	−	0.972731i	$-0.425494\pi$
$61$	221.000	0.463871	0.231936	+	0.972731i	$0.425494\pi$
$62$	94.0000	0.192549
$63$	0	0
$64$	64.0000	0.125000
$65$	−100.000	−0.190823
$66$	0	0
$67$	−538.000	−0.981002	−0.490501	−	0.871441i	$-0.663186\pi$
$67$	−538.000	−0.981002	−0.490501	+	0.871441i	$0.663186\pi$
$68$	372.000	0.663406
$69$	0	0
$70$	40.0000	0.0682988
$71$	690.000	1.15335	0.576676	−	0.816973i	$-0.304350\pi$
$71$	690.000	1.15335	0.576676	+	0.816973i	$0.304350\pi$
$72$	0	0
$73$	−1126.00	−1.80532	−0.902660	−	0.430355i	$-0.858388\pi$
$73$	−1126.00	−1.80532	−0.902660	+	0.430355i	$0.858388\pi$
$74$	−524.000	−0.823159
$75$	0	0
$76$	236.000	0.356198
$77$	−168.000	−0.248641
$78$	0	0
$79$	665.000	0.947068	0.473534	−	0.880776i	$-0.342978\pi$
$79$	665.000	0.947068	0.473534	+	0.880776i	$0.342978\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	252.000	0.339375
$83$	75.0000	0.0991846	0.0495923	−	0.998770i	$-0.484208\pi$
$83$	75.0000	0.0991846	0.0495923	+	0.998770i	$0.484208\pi$
$84$	0	0
$85$	−465.000	−0.593369
$86$	−356.000	−0.446378
$87$	0	0
$88$	336.000	0.407020
$89$	−1086.00	−1.29344	−0.646718	−	0.762729i	$-0.723859\pi$
$89$	−1086.00	−1.29344	−0.646718	+	0.762729i	$0.723859\pi$
$90$	0	0
$91$	−80.0000	−0.0921569
$92$	36.0000	0.0407963
$93$	0	0
$94$	288.000	0.316010
$95$	−295.000	−0.318593
$96$	0	0
$97$	1544.00	1.61618	0.808090	−	0.589059i	$-0.200501\pi$
$97$	1544.00	1.61618	0.808090	+	0.589059i	$0.200501\pi$
$98$	−654.000	−0.674122
$99$	0	0
$100$	100.000	0.100000
$101$	−132.000	−0.130044	−0.0650222	−	0.997884i	$-0.520712\pi$
$101$	−132.000	−0.130044	−0.0650222	+	0.997884i	$0.520712\pi$
$102$	0	0
$103$	−892.000	−0.853314	−0.426657	−	0.904413i	$-0.640309\pi$
$103$	−892.000	−0.853314	−0.426657	+	0.904413i	$0.640309\pi$
$104$	160.000	0.150859
$105$	0	0
$106$	1482.00	1.35797
$107$	−1140.00	−1.02998	−0.514990	−	0.857196i	$-0.672205\pi$
$107$	−1140.00	−1.02998	−0.514990	+	0.857196i	$0.672205\pi$
$108$	0	0
$109$	−1735.00	−1.52461	−0.762307	−	0.647216i	$-0.775933\pi$
$109$	−1735.00	−1.52461	−0.762307	+	0.647216i	$0.775933\pi$
$110$	−420.000	−0.364049
$111$	0	0
$112$	−64.0000	−0.0539949
$113$	−1434.00	−1.19380	−0.596900	−	0.802316i	$-0.703601\pi$
$113$	−1434.00	−1.19380	−0.596900	+	0.802316i	$0.703601\pi$
$114$	0	0
$115$	−45.0000	−0.0364893
$116$	480.000	0.384197
$117$	0	0
$118$	−888.000	−0.692771
$119$	−372.000	−0.286565
$120$	0	0
$121$	433.000	0.325319
$122$	442.000	0.328007
$123$	0	0
$124$	188.000	0.136152
$125$	−125.000	−0.0894427
$126$	0	0
$127$	686.000	0.479312	0.239656	−	0.970858i	$-0.422965\pi$
$127$	686.000	0.479312	0.239656	+	0.970858i	$0.422965\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−200.000	−0.134932
$131$	−114.000	−0.0760323	−0.0380161	−	0.999277i	$-0.512104\pi$
$131$	−114.000	−0.0760323	−0.0380161	+	0.999277i	$0.512104\pi$
$132$	0	0
$133$	−236.000	−0.153863
$134$	−1076.00	−0.693673
$135$	0	0
$136$	744.000	0.469099
$137$	159.000	0.0991554	0.0495777	−	0.998770i	$-0.484212\pi$
$137$	159.000	0.0991554	0.0495777	+	0.998770i	$0.484212\pi$
$138$	0	0
$139$	2276.00	1.38883	0.694417	−	0.719573i	$-0.255663\pi$
$139$	2276.00	1.38883	0.694417	+	0.719573i	$0.255663\pi$
$140$	80.0000	0.0482945
$141$	0	0
$142$	1380.00	0.815542
$143$	840.000	0.491219
$144$	0	0
$145$	−600.000	−0.343636
$146$	−2252.00	−1.27655
$147$	0	0
$148$	−1048.00	−0.582061
$149$	−1398.00	−0.768648	−0.384324	−	0.923198i	$-0.625566\pi$
$149$	−1398.00	−0.768648	−0.384324	+	0.923198i	$0.625566\pi$
$150$	0	0
$151$	2624.00	1.41416	0.707080	−	0.707134i	$-0.250012\pi$
$151$	2624.00	1.41416	0.707080	+	0.707134i	$0.250012\pi$
$152$	472.000	0.251870
$153$	0	0
$154$	−336.000	−0.175816
$155$	−235.000	−0.121778
$156$	0	0
$157$	−394.000	−0.200284	−0.100142	−	0.994973i	$-0.531930\pi$
$157$	−394.000	−0.200284	−0.100142	+	0.994973i	$0.531930\pi$
$158$	1330.00	0.669678
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−36.0000	−0.0176223
$162$	0	0
$163$	−3346.00	−1.60785	−0.803923	−	0.594733i	$-0.797258\pi$
$163$	−3346.00	−1.60785	−0.803923	+	0.594733i	$0.797258\pi$
$164$	504.000	0.239974
$165$	0	0
$166$	150.000	0.0701341
$167$	−1491.00	−0.690881	−0.345440	−	0.938441i	$-0.612270\pi$
$167$	−1491.00	−0.690881	−0.345440	+	0.938441i	$0.612270\pi$
$168$	0	0
$169$	−1797.00	−0.817934
$170$	−930.000	−0.419575
$171$	0	0
$172$	−712.000	−0.315637
$173$	2403.00	1.05605	0.528025	−	0.849229i	$-0.322933\pi$
$173$	2403.00	1.05605	0.528025	+	0.849229i	$0.322933\pi$
$174$	0	0
$175$	−100.000	−0.0431959
$176$	672.000	0.287806
$177$	0	0
$178$	−2172.00	−0.914597
$179$	−2640.00	−1.10236	−0.551181	−	0.834386i	$-0.685823\pi$
$179$	−2640.00	−1.10236	−0.551181	+	0.834386i	$0.685823\pi$
$180$	0	0
$181$	1073.00	0.440638	0.220319	−	0.975428i	$-0.429290\pi$
$181$	1073.00	0.440638	0.220319	+	0.975428i	$0.429290\pi$
$182$	−160.000	−0.0651648
$183$	0	0
$184$	72.0000	0.0288473
$185$	1310.00	0.520611
$186$	0	0
$187$	3906.00	1.52746
$188$	576.000	0.223453
$189$	0	0
$190$	−590.000	−0.225279
$191$	1470.00	0.556887	0.278444	−	0.960453i	$-0.410181\pi$
$191$	1470.00	0.556887	0.278444	+	0.960453i	$0.410181\pi$
$192$	0	0
$193$	−4720.00	−1.76038	−0.880189	−	0.474623i	$-0.842584\pi$
$193$	−4720.00	−1.76038	−0.880189	+	0.474623i	$0.842584\pi$
$194$	3088.00	1.14281
$195$	0	0
$196$	−1308.00	−0.476676
$197$	−765.000	−0.276670	−0.138335	−	0.990385i	$-0.544175\pi$
$197$	−765.000	−0.276670	−0.138335	+	0.990385i	$0.544175\pi$
$198$	0	0
$199$	668.000	0.237956	0.118978	−	0.992897i	$-0.462038\pi$
$199$	668.000	0.237956	0.118978	+	0.992897i	$0.462038\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	−264.000	−0.0919553
$203$	−480.000	−0.165958
$204$	0	0
$205$	−630.000	−0.214640
$206$	−1784.00	−0.603384
$207$	0	0
$208$	320.000	0.106673
$209$	2478.00	0.820128
$210$	0	0
$211$	4601.00	1.50117	0.750583	−	0.660777i	$-0.229773\pi$
$211$	4601.00	1.50117	0.750583	+	0.660777i	$0.229773\pi$
$212$	2964.00	0.960228
$213$	0	0
$214$	−2280.00	−0.728307
$215$	890.000	0.282314
$216$	0	0
$217$	−188.000	−0.0588123
$218$	−3470.00	−1.07806
$219$	0	0
$220$	−840.000	−0.257422
$221$	1860.00	0.566141
$222$	0	0
$223$	−2158.00	−0.648029	−0.324014	−	0.946052i	$-0.605033\pi$
$223$	−2158.00	−0.648029	−0.324014	+	0.946052i	$0.605033\pi$
$224$	−128.000	−0.0381802
$225$	0	0
$226$	−2868.00	−0.844144
$227$	3123.00	0.913131	0.456566	−	0.889690i	$-0.349079\pi$
$227$	3123.00	0.913131	0.456566	+	0.889690i	$0.349079\pi$
$228$	0	0
$229$	2027.00	0.584925	0.292463	−	0.956277i	$-0.405525\pi$
$229$	2027.00	0.584925	0.292463	+	0.956277i	$0.405525\pi$
$230$	−90.0000	−0.0258018
$231$	0	0
$232$	960.000	0.271668
$233$	−438.000	−0.123152	−0.0615758	−	0.998102i	$-0.519613\pi$
$233$	−438.000	−0.123152	−0.0615758	+	0.998102i	$0.519613\pi$
$234$	0	0
$235$	−720.000	−0.199862
$236$	−1776.00	−0.489863
$237$	0	0
$238$	−744.000	−0.202632
$239$	6414.00	1.73593	0.867965	−	0.496626i	$-0.165428\pi$
$239$	6414.00	1.73593	0.867965	+	0.496626i	$0.165428\pi$
$240$	0	0
$241$	3431.00	0.917055	0.458527	−	0.888680i	$-0.348377\pi$
$241$	3431.00	0.917055	0.458527	+	0.888680i	$0.348377\pi$
$242$	866.000	0.230035
$243$	0	0
$244$	884.000	0.231936
$245$	1635.00	0.426352
$246$	0	0
$247$	1180.00	0.303974
$248$	376.000	0.0962743
$249$	0	0
$250$	−250.000	−0.0632456
$251$	7308.00	1.83776	0.918878	−	0.394541i	$-0.129096\pi$
$251$	7308.00	1.83776	0.918878	+	0.394541i	$0.129096\pi$
$252$	0	0
$253$	378.000	0.0939314
$254$	1372.00	0.338925
$255$	0	0
$256$	256.000	0.0625000
$257$	−3729.00	−0.905092	−0.452546	−	0.891741i	$-0.649484\pi$
$257$	−3729.00	−0.905092	−0.452546	+	0.891741i	$0.649484\pi$
$258$	0	0
$259$	1048.00	0.251427
$260$	−400.000	−0.0954113
$261$	0	0
$262$	−228.000	−0.0537629
$263$	−1956.00	−0.458601	−0.229301	−	0.973356i	$-0.573644\pi$
$263$	−1956.00	−0.458601	−0.229301	+	0.973356i	$0.573644\pi$
$264$	0	0
$265$	−3705.00	−0.858854
$266$	−472.000	−0.108798
$267$	0	0
$268$	−2152.00	−0.490501
$269$	990.000	0.224392	0.112196	−	0.993686i	$-0.464212\pi$
$269$	990.000	0.224392	0.112196	+	0.993686i	$0.464212\pi$
$270$	0	0
$271$	8495.00	1.90419	0.952093	−	0.305808i	$-0.0989266\pi$
$271$	8495.00	1.90419	0.952093	+	0.305808i	$0.0989266\pi$
$272$	1488.00	0.331703
$273$	0	0
$274$	318.000	0.0701134
$275$	1050.00	0.230245
$276$	0	0
$277$	−1366.00	−0.296300	−0.148150	−	0.988965i	$-0.547332\pi$
$277$	−1366.00	−0.296300	−0.148150	+	0.988965i	$0.547332\pi$
$278$	4552.00	0.982053
$279$	0	0
$280$	160.000	0.0341494
$281$	5520.00	1.17187	0.585935	−	0.810358i	$-0.300727\pi$
$281$	5520.00	1.17187	0.585935	+	0.810358i	$0.300727\pi$
$282$	0	0
$283$	5438.00	1.14225	0.571123	−	0.820865i	$-0.306508\pi$
$283$	5438.00	1.14225	0.571123	+	0.820865i	$0.306508\pi$
$284$	2760.00	0.576676
$285$	0	0
$286$	1680.00	0.347344
$287$	−504.000	−0.103659
$288$	0	0
$289$	3736.00	0.760432
$290$	−1200.00	−0.242988
$291$	0	0
$292$	−4504.00	−0.902660
$293$	−8253.00	−1.64555	−0.822774	−	0.568369i	$-0.807575\pi$
$293$	−8253.00	−1.64555	−0.822774	+	0.568369i	$0.807575\pi$
$294$	0	0
$295$	2220.00	0.438147
$296$	−2096.00	−0.411579
$297$	0	0
$298$	−2796.00	−0.543517
$299$	180.000	0.0348149
$300$	0	0
$301$	712.000	0.136342
$302$	5248.00	0.999962
$303$	0	0
$304$	944.000	0.178099
$305$	−1105.00	−0.207450
$306$	0	0
$307$	9290.00	1.72706	0.863531	−	0.504295i	$-0.168248\pi$
$307$	9290.00	1.72706	0.863531	+	0.504295i	$0.168248\pi$
$308$	−672.000	−0.124321
$309$	0	0
$310$	−470.000	−0.0861103
$311$	−8112.00	−1.47907	−0.739533	−	0.673121i	$-0.764953\pi$
$311$	−8112.00	−1.47907	−0.739533	+	0.673121i	$0.764953\pi$
$312$	0	0
$313$	−7900.00	−1.42663	−0.713314	−	0.700845i	$-0.752807\pi$
$313$	−7900.00	−1.42663	−0.713314	+	0.700845i	$0.752807\pi$
$314$	−788.000	−0.141622
$315$	0	0
$316$	2660.00	0.473534
$317$	−4419.00	−0.782952	−0.391476	−	0.920188i	$-0.628035\pi$
$317$	−4419.00	−0.782952	−0.391476	+	0.920188i	$0.628035\pi$
$318$	0	0
$319$	5040.00	0.884595
$320$	−320.000	−0.0559017
$321$	0	0
$322$	−72.0000	−0.0124609
$323$	5487.00	0.945216
$324$	0	0
$325$	500.000	0.0853385
$326$	−6692.00	−1.13692
$327$	0	0
$328$	1008.00	0.169687
$329$	−576.000	−0.0965225
$330$	0	0
$331$	−8200.00	−1.36167	−0.680835	−	0.732437i	$-0.738383\pi$
$331$	−8200.00	−1.36167	−0.680835	+	0.732437i	$0.738383\pi$
$332$	300.000	0.0495923
$333$	0	0
$334$	−2982.00	−0.488526
$335$	2690.00	0.438718
$336$	0	0
$337$	−9556.00	−1.54465	−0.772327	−	0.635225i	$-0.780907\pi$
$337$	−9556.00	−1.54465	−0.772327	+	0.635225i	$0.780907\pi$
$338$	−3594.00	−0.578366
$339$	0	0
$340$	−1860.00	−0.296684
$341$	1974.00	0.313484
$342$	0	0
$343$	2680.00	0.421885
$344$	−1424.00	−0.223189
$345$	0	0
$346$	4806.00	0.746740
$347$	−10116.0	−1.56500	−0.782500	−	0.622650i	$-0.786056\pi$
$347$	−10116.0	−1.56500	−0.782500	+	0.622650i	$0.786056\pi$
$348$	0	0
$349$	−6751.00	−1.03545	−0.517726	−	0.855546i	$-0.673221\pi$
$349$	−6751.00	−1.03545	−0.517726	+	0.855546i	$0.673221\pi$
$350$	−200.000	−0.0305441
$351$	0	0
$352$	1344.00	0.203510
$353$	−4062.00	−0.612460	−0.306230	−	0.951958i	$-0.599068\pi$
$353$	−4062.00	−0.612460	−0.306230	+	0.951958i	$0.599068\pi$
$354$	0	0
$355$	−3450.00	−0.515794
$356$	−4344.00	−0.646718
$357$	0	0
$358$	−5280.00	−0.779488
$359$	−8778.00	−1.29049	−0.645244	−	0.763977i	$-0.723244\pi$
$359$	−8778.00	−1.29049	−0.645244	+	0.763977i	$0.723244\pi$
$360$	0	0
$361$	−3378.00	−0.492492
$362$	2146.00	0.311578
$363$	0	0
$364$	−320.000	−0.0460785
$365$	5630.00	0.807363
$366$	0	0
$367$	956.000	0.135975	0.0679875	−	0.997686i	$-0.478342\pi$
$367$	956.000	0.135975	0.0679875	+	0.997686i	$0.478342\pi$
$368$	144.000	0.0203981
$369$	0	0
$370$	2620.00	0.368128
$371$	−2964.00	−0.414780
$372$	0	0
$373$	2300.00	0.319275	0.159637	−	0.987176i	$-0.448968\pi$
$373$	2300.00	0.319275	0.159637	+	0.987176i	$0.448968\pi$
$374$	7812.00	1.08008
$375$	0	0
$376$	1152.00	0.158005
$377$	2400.00	0.327868
$378$	0	0
$379$	29.0000	0.00393042	0.00196521	−	0.999998i	$-0.499374\pi$
$379$	29.0000	0.00393042	0.00196521	+	0.999998i	$0.499374\pi$
$380$	−1180.00	−0.159297
$381$	0	0
$382$	2940.00	0.393779
$383$	−8127.00	−1.08426	−0.542128	−	0.840296i	$-0.682381\pi$
$383$	−8127.00	−1.08426	−0.542128	+	0.840296i	$0.682381\pi$
$384$	0	0
$385$	840.000	0.111196
$386$	−9440.00	−1.24478
$387$	0	0
$388$	6176.00	0.808090
$389$	7938.00	1.03463	0.517317	−	0.855794i	$-0.326931\pi$
$389$	7938.00	1.03463	0.517317	+	0.855794i	$0.326931\pi$
$390$	0	0
$391$	837.000	0.108258
$392$	−2616.00	−0.337061
$393$	0	0
$394$	−1530.00	−0.195635
$395$	−3325.00	−0.423542
$396$	0	0
$397$	272.000	0.0343861	0.0171931	−	0.999852i	$-0.494527\pi$
$397$	272.000	0.0343861	0.0171931	+	0.999852i	$0.494527\pi$
$398$	1336.00	0.168260
$399$	0	0
$400$	400.000	0.0500000
$401$	−4554.00	−0.567122	−0.283561	−	0.958954i	$-0.591516\pi$
$401$	−4554.00	−0.567122	−0.283561	+	0.958954i	$0.591516\pi$
$402$	0	0
$403$	940.000	0.116190
$404$	−528.000	−0.0650222
$405$	0	0
$406$	−960.000	−0.117350
$407$	−11004.0	−1.34017
$408$	0	0
$409$	1001.00	0.121018	0.0605089	−	0.998168i	$-0.480728\pi$
$409$	1001.00	0.121018	0.0605089	+	0.998168i	$0.480728\pi$
$410$	−1260.00	−0.151773
$411$	0	0
$412$	−3568.00	−0.426657
$413$	1776.00	0.211601
$414$	0	0
$415$	−375.000	−0.0443567
$416$	640.000	0.0754293
$417$	0	0
$418$	4956.00	0.579918
$419$	1794.00	0.209171	0.104585	−	0.994516i	$-0.466648\pi$
$419$	1794.00	0.209171	0.104585	+	0.994516i	$0.466648\pi$
$420$	0	0
$421$	−16129.0	−1.86717	−0.933586	−	0.358354i	$-0.883338\pi$
$421$	−16129.0	−1.86717	−0.933586	+	0.358354i	$0.883338\pi$
$422$	9202.00	1.06148
$423$	0	0
$424$	5928.00	0.678984
$425$	2325.00	0.265363
$426$	0	0
$427$	−884.000	−0.100187
$428$	−4560.00	−0.514990
$429$	0	0
$430$	1780.00	0.199626
$431$	13356.0	1.49266	0.746329	−	0.665577i	$-0.231814\pi$
$431$	13356.0	1.49266	0.746329	+	0.665577i	$0.231814\pi$
$432$	0	0
$433$	−11500.0	−1.27634	−0.638169	−	0.769896i	$-0.720308\pi$
$433$	−11500.0	−1.27634	−0.638169	+	0.769896i	$0.720308\pi$
$434$	−376.000	−0.0415866
$435$	0	0
$436$	−6940.00	−0.762307
$437$	531.000	0.0581263
$438$	0	0
$439$	−11149.0	−1.21210	−0.606051	−	0.795426i	$-0.707247\pi$
$439$	−11149.0	−1.21210	−0.606051	+	0.795426i	$0.707247\pi$
$440$	−1680.00	−0.182025
$441$	0	0
$442$	3720.00	0.400322
$443$	−3849.00	−0.412803	−0.206401	−	0.978467i	$-0.566175\pi$
$443$	−3849.00	−0.412803	−0.206401	+	0.978467i	$0.566175\pi$
$444$	0	0
$445$	5430.00	0.578442
$446$	−4316.00	−0.458225
$447$	0	0
$448$	−256.000	−0.0269975
$449$	−18048.0	−1.89697	−0.948483	−	0.316828i	$-0.897382\pi$
$449$	−18048.0	−1.89697	−0.948483	+	0.316828i	$0.897382\pi$
$450$	0	0
$451$	5292.00	0.552529
$452$	−5736.00	−0.596900
$453$	0	0
$454$	6246.00	0.645681
$455$	400.000	0.0412138
$456$	0	0
$457$	−4264.00	−0.436458	−0.218229	−	0.975898i	$-0.570028\pi$
$457$	−4264.00	−0.436458	−0.218229	+	0.975898i	$0.570028\pi$
$458$	4054.00	0.413605
$459$	0	0
$460$	−180.000	−0.0182447
$461$	10242.0	1.03475	0.517373	−	0.855760i	$-0.326910\pi$
$461$	10242.0	1.03475	0.517373	+	0.855760i	$0.326910\pi$
$462$	0	0
$463$	3302.00	0.331441	0.165720	−	0.986173i	$-0.447005\pi$
$463$	3302.00	0.331441	0.165720	+	0.986173i	$0.447005\pi$
$464$	1920.00	0.192099
$465$	0	0
$466$	−876.000	−0.0870814
$467$	1923.00	0.190548	0.0952739	−	0.995451i	$-0.469627\pi$
$467$	1923.00	0.190548	0.0952739	+	0.995451i	$0.469627\pi$
$468$	0	0
$469$	2152.00	0.211877
$470$	−1440.00	−0.141324
$471$	0	0
$472$	−3552.00	−0.346386
$473$	−7476.00	−0.726738
$474$	0	0
$475$	1475.00	0.142479
$476$	−1488.00	−0.143282
$477$	0	0
$478$	12828.0	1.22749
$479$	−15246.0	−1.45430	−0.727148	−	0.686481i	$-0.759154\pi$
$479$	−15246.0	−1.45430	−0.727148	+	0.686481i	$0.759154\pi$
$480$	0	0
$481$	−5240.00	−0.496722
$482$	6862.00	0.648455
$483$	0	0
$484$	1732.00	0.162660
$485$	−7720.00	−0.722778
$486$	0	0
$487$	−8206.00	−0.763551	−0.381776	−	0.924255i	$-0.624687\pi$
$487$	−8206.00	−0.763551	−0.381776	+	0.924255i	$0.624687\pi$
$488$	1768.00	0.164003
$489$	0	0
$490$	3270.00	0.301477
$491$	16806.0	1.54469	0.772346	−	0.635202i	$-0.219083\pi$
$491$	16806.0	1.54469	0.772346	+	0.635202i	$0.219083\pi$
$492$	0	0
$493$	11160.0	1.01952
$494$	2360.00	0.214942
$495$	0	0
$496$	752.000	0.0680762
$497$	−2760.00	−0.249100
$498$	0	0
$499$	−5425.00	−0.486686	−0.243343	−	0.969940i	$-0.578244\pi$
$499$	−5425.00	−0.486686	−0.243343	+	0.969940i	$0.578244\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	14616.0	1.29949
$503$	19665.0	1.74318	0.871589	−	0.490236i	$-0.163090\pi$
$503$	19665.0	1.74318	0.871589	+	0.490236i	$0.163090\pi$
$504$	0	0
$505$	660.000	0.0581577
$506$	756.000	0.0664196
$507$	0	0
$508$	2744.00	0.239656
$509$	14724.0	1.28218	0.641090	−	0.767466i	$-0.278482\pi$
$509$	14724.0	1.28218	0.641090	+	0.767466i	$0.278482\pi$
$510$	0	0
$511$	4504.00	0.389912
$512$	512.000	0.0441942
$513$	0	0
$514$	−7458.00	−0.639997
$515$	4460.00	0.381614
$516$	0	0
$517$	6048.00	0.514489
$518$	2096.00	0.177786
$519$	0	0
$520$	−800.000	−0.0674660
$521$	−2058.00	−0.173057	−0.0865284	−	0.996249i	$-0.527577\pi$
$521$	−2058.00	−0.173057	−0.0865284	+	0.996249i	$0.527577\pi$
$522$	0	0
$523$	11912.0	0.995938	0.497969	−	0.867195i	$-0.334079\pi$
$523$	11912.0	0.995938	0.497969	+	0.867195i	$0.334079\pi$
$524$	−456.000	−0.0380161
$525$	0	0
$526$	−3912.00	−0.324280
$527$	4371.00	0.361297
$528$	0	0
$529$	−12086.0	−0.993343
$530$	−7410.00	−0.607302
$531$	0	0
$532$	−944.000	−0.0769316
$533$	2520.00	0.204790
$534$	0	0
$535$	5700.00	0.460621
$536$	−4304.00	−0.346837
$537$	0	0
$538$	1980.00	0.158669
$539$	−13734.0	−1.09752
$540$	0	0
$541$	−5170.00	−0.410861	−0.205430	−	0.978672i	$-0.565859\pi$
$541$	−5170.00	−0.410861	−0.205430	+	0.978672i	$0.565859\pi$
$542$	16990.0	1.34646
$543$	0	0
$544$	2976.00	0.234550
$545$	8675.00	0.681828
$546$	0	0
$547$	−4186.00	−0.327204	−0.163602	−	0.986526i	$-0.552311\pi$
$547$	−4186.00	−0.327204	−0.163602	+	0.986526i	$0.552311\pi$
$548$	636.000	0.0495777
$549$	0	0
$550$	2100.00	0.162808
$551$	7080.00	0.547401
$552$	0	0
$553$	−2660.00	−0.204547
$554$	−2732.00	−0.209515
$555$	0	0
$556$	9104.00	0.694417
$557$	−13026.0	−0.990896	−0.495448	−	0.868637i	$-0.664996\pi$
$557$	−13026.0	−0.990896	−0.495448	+	0.868637i	$0.664996\pi$
$558$	0	0
$559$	−3560.00	−0.269359
$560$	320.000	0.0241473
$561$	0	0
$562$	11040.0	0.828638
$563$	10668.0	0.798584	0.399292	−	0.916824i	$-0.369256\pi$
$563$	10668.0	0.798584	0.399292	+	0.916824i	$0.369256\pi$
$564$	0	0
$565$	7170.00	0.533883
$566$	10876.0	0.807690
$567$	0	0
$568$	5520.00	0.407771
$569$	15372.0	1.13256	0.566281	−	0.824212i	$-0.308382\pi$
$569$	15372.0	1.13256	0.566281	+	0.824212i	$0.308382\pi$
$570$	0	0
$571$	−14989.0	−1.09855	−0.549273	−	0.835643i	$-0.685095\pi$
$571$	−14989.0	−1.09855	−0.549273	+	0.835643i	$0.685095\pi$
$572$	3360.00	0.245610
$573$	0	0
$574$	−1008.00	−0.0732981
$575$	225.000	0.0163185
$576$	0	0
$577$	−1066.00	−0.0769119	−0.0384559	−	0.999260i	$-0.512244\pi$
$577$	−1066.00	−0.0769119	−0.0384559	+	0.999260i	$0.512244\pi$
$578$	7472.00	0.537706
$579$	0	0
$580$	−2400.00	−0.171818
$581$	−300.000	−0.0214219
$582$	0	0
$583$	31122.0	2.21088
$584$	−9008.00	−0.638277
$585$	0	0
$586$	−16506.0	−1.16358
$587$	621.000	0.0436651	0.0218325	−	0.999762i	$-0.493050\pi$
$587$	621.000	0.0436651	0.0218325	+	0.999762i	$0.493050\pi$
$588$	0	0
$589$	2773.00	0.193989
$590$	4440.00	0.309817
$591$	0	0
$592$	−4192.00	−0.291031
$593$	20187.0	1.39794	0.698972	−	0.715149i	$-0.253641\pi$
$593$	20187.0	1.39794	0.698972	+	0.715149i	$0.253641\pi$
$594$	0	0
$595$	1860.00	0.128156
$596$	−5592.00	−0.384324
$597$	0	0
$598$	360.000	0.0246179
$599$	18228.0	1.24337	0.621683	−	0.783269i	$-0.286449\pi$
$599$	18228.0	1.24337	0.621683	+	0.783269i	$0.286449\pi$
$600$	0	0
$601$	−11743.0	−0.797017	−0.398508	−	0.917165i	$-0.630472\pi$
$601$	−11743.0	−0.797017	−0.398508	+	0.917165i	$0.630472\pi$
$602$	1424.00	0.0964085
$603$	0	0
$604$	10496.0	0.707080
$605$	−2165.00	−0.145487
$606$	0	0
$607$	−24418.0	−1.63278	−0.816389	−	0.577503i	$-0.804027\pi$
$607$	−24418.0	−1.63278	−0.816389	+	0.577503i	$0.804027\pi$
$608$	1888.00	0.125935
$609$	0	0
$610$	−2210.00	−0.146689
$611$	2880.00	0.190691
$612$	0	0
$613$	2672.00	0.176054	0.0880270	−	0.996118i	$-0.471944\pi$
$613$	2672.00	0.176054	0.0880270	+	0.996118i	$0.471944\pi$
$614$	18580.0	1.22122
$615$	0	0
$616$	−1344.00	−0.0879080
$617$	8601.00	0.561205	0.280602	−	0.959824i	$-0.409466\pi$
$617$	8601.00	0.561205	0.280602	+	0.959824i	$0.409466\pi$
$618$	0	0
$619$	21308.0	1.38359	0.691794	−	0.722095i	$-0.256821\pi$
$619$	21308.0	1.38359	0.691794	+	0.722095i	$0.256821\pi$
$620$	−940.000	−0.0608892
$621$	0	0
$622$	−16224.0	−1.04586
$623$	4344.00	0.279356
$624$	0	0
$625$	625.000	0.0400000
$626$	−15800.0	−1.00878
$627$	0	0
$628$	−1576.00	−0.100142
$629$	−24366.0	−1.54457
$630$	0	0
$631$	−19015.0	−1.19964	−0.599822	−	0.800134i	$-0.704762\pi$
$631$	−19015.0	−1.19964	−0.599822	+	0.800134i	$0.704762\pi$
$632$	5320.00	0.334839
$633$	0	0
$634$	−8838.00	−0.553631
$635$	−3430.00	−0.214355
$636$	0	0
$637$	−6540.00	−0.406788
$638$	10080.0	0.625503
$639$	0	0
$640$	−640.000	−0.0395285
$641$	4416.00	0.272108	0.136054	−	0.990701i	$-0.456558\pi$
$641$	4416.00	0.272108	0.136054	+	0.990701i	$0.456558\pi$
$642$	0	0
$643$	7580.00	0.464893	0.232446	−	0.972609i	$-0.425327\pi$
$643$	7580.00	0.464893	0.232446	+	0.972609i	$0.425327\pi$
$644$	−144.000	−0.00881117
$645$	0	0
$646$	10974.0	0.668369
$647$	14901.0	0.905439	0.452719	−	0.891653i	$-0.350454\pi$
$647$	14901.0	0.905439	0.452719	+	0.891653i	$0.350454\pi$
$648$	0	0
$649$	−18648.0	−1.12789
$650$	1000.00	0.0603434
$651$	0	0
$652$	−13384.0	−0.803923
$653$	−12915.0	−0.773971	−0.386985	−	0.922086i	$-0.626484\pi$
$653$	−12915.0	−0.773971	−0.386985	+	0.922086i	$0.626484\pi$
$654$	0	0
$655$	570.000	0.0340027
$656$	2016.00	0.119987
$657$	0	0
$658$	−1152.00	−0.0682517
$659$	−28128.0	−1.66269	−0.831344	−	0.555758i	$-0.812428\pi$
$659$	−28128.0	−1.66269	−0.831344	+	0.555758i	$0.812428\pi$
$660$	0	0
$661$	−8362.00	−0.492049	−0.246024	−	0.969264i	$-0.579124\pi$
$661$	−8362.00	−0.492049	−0.246024	+	0.969264i	$0.579124\pi$
$662$	−16400.0	−0.962846
$663$	0	0
$664$	600.000	0.0350670
$665$	1180.00	0.0688097
$666$	0	0
$667$	1080.00	0.0626953
$668$	−5964.00	−0.345440
$669$	0	0
$670$	5380.00	0.310220
$671$	9282.00	0.534020
$672$	0	0
$673$	29708.0	1.70157	0.850787	−	0.525511i	$-0.176126\pi$
$673$	29708.0	1.70157	0.850787	+	0.525511i	$0.176126\pi$
$674$	−19112.0	−1.09224
$675$	0	0
$676$	−7188.00	−0.408967
$677$	−6762.00	−0.383877	−0.191939	−	0.981407i	$-0.561477\pi$
$677$	−6762.00	−0.383877	−0.191939	+	0.981407i	$0.561477\pi$
$678$	0	0
$679$	−6176.00	−0.349062
$680$	−3720.00	−0.209787
$681$	0	0
$682$	3948.00	0.221667
$683$	19155.0	1.07313	0.536563	−	0.843860i	$-0.319722\pi$
$683$	19155.0	1.07313	0.536563	+	0.843860i	$0.319722\pi$
$684$	0	0
$685$	−795.000	−0.0443436
$686$	5360.00	0.298317
$687$	0	0
$688$	−2848.00	−0.157818
$689$	14820.0	0.819444
$690$	0	0
$691$	−22975.0	−1.26485	−0.632424	−	0.774622i	$-0.717940\pi$
$691$	−22975.0	−1.26485	−0.632424	+	0.774622i	$0.717940\pi$
$692$	9612.00	0.528025
$693$	0	0
$694$	−20232.0	−1.10662
$695$	−11380.0	−0.621105
$696$	0	0
$697$	11718.0	0.636802
$698$	−13502.0	−0.732175
$699$	0	0
$700$	−400.000	−0.0215980
$701$	6450.00	0.347522	0.173761	−	0.984788i	$-0.444408\pi$
$701$	6450.00	0.347522	0.173761	+	0.984788i	$0.444408\pi$
$702$	0	0
$703$	−15458.0	−0.829317
$704$	2688.00	0.143903
$705$	0	0
$706$	−8124.00	−0.433075
$707$	528.000	0.0280870
$708$	0	0
$709$	34538.0	1.82948	0.914740	−	0.404042i	$-0.132395\pi$
$709$	34538.0	1.82948	0.914740	+	0.404042i	$0.132395\pi$
$710$	−6900.00	−0.364722
$711$	0	0
$712$	−8688.00	−0.457299
$713$	423.000	0.0222181
$714$	0	0
$715$	−4200.00	−0.219680
$716$	−10560.0	−0.551181
$717$	0	0
$718$	−17556.0	−0.912513
$719$	−27114.0	−1.40637	−0.703186	−	0.711006i	$-0.748240\pi$
$719$	−27114.0	−1.40637	−0.703186	+	0.711006i	$0.748240\pi$
$720$	0	0
$721$	3568.00	0.184299
$722$	−6756.00	−0.348244
$723$	0	0
$724$	4292.00	0.220319
$725$	3000.00	0.153679
$726$	0	0
$727$	236.000	0.0120396	0.00601978	−	0.999982i	$-0.498084\pi$
$727$	236.000	0.0120396	0.00601978	+	0.999982i	$0.498084\pi$
$728$	−640.000	−0.0325824
$729$	0	0
$730$	11260.0	0.570892
$731$	−16554.0	−0.837581
$732$	0	0
$733$	27128.0	1.36698	0.683489	−	0.729960i	$-0.260462\pi$
$733$	27128.0	1.36698	0.683489	+	0.729960i	$0.260462\pi$
$734$	1912.00	0.0961488
$735$	0	0
$736$	288.000	0.0144237
$737$	−22596.0	−1.12935
$738$	0	0
$739$	5249.00	0.261282	0.130641	−	0.991430i	$-0.458296\pi$
$739$	5249.00	0.261282	0.130641	+	0.991430i	$0.458296\pi$
$740$	5240.00	0.260306
$741$	0	0
$742$	−5928.00	−0.293293
$743$	−13896.0	−0.686130	−0.343065	−	0.939312i	$-0.611465\pi$
$743$	−13896.0	−0.686130	−0.343065	+	0.939312i	$0.611465\pi$
$744$	0	0
$745$	6990.00	0.343750
$746$	4600.00	0.225761
$747$	0	0
$748$	15624.0	0.763730
$749$	4560.00	0.222455
$750$	0	0
$751$	27665.0	1.34422	0.672111	−	0.740451i	$-0.265388\pi$
$751$	27665.0	1.34422	0.672111	+	0.740451i	$0.265388\pi$
$752$	2304.00	0.111726
$753$	0	0
$754$	4800.00	0.231838
$755$	−13120.0	−0.632431
$756$	0	0
$757$	−8122.00	−0.389959	−0.194980	−	0.980807i	$-0.562464\pi$
$757$	−8122.00	−0.389959	−0.194980	+	0.980807i	$0.562464\pi$
$758$	58.0000	0.00277923
$759$	0	0
$760$	−2360.00	−0.112640
$761$	−10584.0	−0.504165	−0.252083	−	0.967706i	$-0.581115\pi$
$761$	−10584.0	−0.504165	−0.252083	+	0.967706i	$0.581115\pi$
$762$	0	0
$763$	6940.00	0.329286
$764$	5880.00	0.278444
$765$	0	0
$766$	−16254.0	−0.766685
$767$	−8880.00	−0.418042
$768$	0	0
$769$	−18619.0	−0.873106	−0.436553	−	0.899679i	$-0.643801\pi$
$769$	−18619.0	−0.873106	−0.436553	+	0.899679i	$0.643801\pi$
$770$	1680.00	0.0786273
$771$	0	0
$772$	−18880.0	−0.880189
$773$	−22251.0	−1.03533	−0.517667	−	0.855582i	$-0.673199\pi$
$773$	−22251.0	−1.03533	−0.517667	+	0.855582i	$0.673199\pi$
$774$	0	0
$775$	1175.00	0.0544610
$776$	12352.0	0.571406
$777$	0	0
$778$	15876.0	0.731597
$779$	7434.00	0.341914
$780$	0	0
$781$	28980.0	1.32777
$782$	1674.00	0.0765500
$783$	0	0
$784$	−5232.00	−0.238338
$785$	1970.00	0.0895698
$786$	0	0
$787$	24854.0	1.12573	0.562865	−	0.826549i	$-0.309699\pi$
$787$	24854.0	1.12573	0.562865	+	0.826549i	$0.309699\pi$
$788$	−3060.00	−0.138335
$789$	0	0
$790$	−6650.00	−0.299489
$791$	5736.00	0.257837
$792$	0	0
$793$	4420.00	0.197930
$794$	544.000	0.0243147
$795$	0	0
$796$	2672.00	0.118978
$797$	−3681.00	−0.163598	−0.0817991	−	0.996649i	$-0.526067\pi$
$797$	−3681.00	−0.163598	−0.0817991	+	0.996649i	$0.526067\pi$
$798$	0	0
$799$	13392.0	0.592960
$800$	800.000	0.0353553
$801$	0	0
$802$	−9108.00	−0.401016
$803$	−47292.0	−2.07833
$804$	0	0
$805$	180.000	0.00788095
$806$	1880.00	0.0821590
$807$	0	0
$808$	−1056.00	−0.0459777
$809$	5142.00	0.223465	0.111732	−	0.993738i	$-0.464360\pi$
$809$	5142.00	0.223465	0.111732	+	0.993738i	$0.464360\pi$
$810$	0	0
$811$	−18484.0	−0.800322	−0.400161	−	0.916445i	$-0.631046\pi$
$811$	−18484.0	−0.800322	−0.400161	+	0.916445i	$0.631046\pi$
$812$	−1920.00	−0.0829788
$813$	0	0
$814$	−22008.0	−0.947641
$815$	16730.0	0.719051
$816$	0	0
$817$	−10502.0	−0.449717
$818$	2002.00	0.0855725
$819$	0	0
$820$	−2520.00	−0.107320
$821$	−25014.0	−1.06333	−0.531665	−	0.846954i	$-0.678434\pi$
$821$	−25014.0	−1.06333	−0.531665	+	0.846954i	$0.678434\pi$
$822$	0	0
$823$	−32146.0	−1.36153	−0.680765	−	0.732502i	$-0.738352\pi$
$823$	−32146.0	−1.36153	−0.680765	+	0.732502i	$0.738352\pi$
$824$	−7136.00	−0.301692
$825$	0	0
$826$	3552.00	0.149625
$827$	10977.0	0.461557	0.230779	−	0.973006i	$-0.425873\pi$
$827$	10977.0	0.461557	0.230779	+	0.973006i	$0.425873\pi$
$828$	0	0
$829$	36602.0	1.53346	0.766731	−	0.641969i	$-0.221882\pi$
$829$	36602.0	1.53346	0.766731	+	0.641969i	$0.221882\pi$
$830$	−750.000	−0.0313649
$831$	0	0
$832$	1280.00	0.0533366
$833$	−30411.0	−1.26492
$834$	0	0
$835$	7455.00	0.308971
$836$	9912.00	0.410064
$837$	0	0
$838$	3588.00	0.147906
$839$	−11076.0	−0.455764	−0.227882	−	0.973689i	$-0.573180\pi$
$839$	−11076.0	−0.455764	−0.227882	+	0.973689i	$0.573180\pi$
$840$	0	0
$841$	−9989.00	−0.409570
$842$	−32258.0	−1.32029
$843$	0	0
$844$	18404.0	0.750583
$845$	8985.00	0.365791
$846$	0	0
$847$	−1732.00	−0.0702624
$848$	11856.0	0.480114
$849$	0	0
$850$	4650.00	0.187640
$851$	−2358.00	−0.0949838
$852$	0	0
$853$	36848.0	1.47908	0.739538	−	0.673115i	$-0.235044\pi$
$853$	36848.0	1.47908	0.739538	+	0.673115i	$0.235044\pi$
$854$	−1768.00	−0.0708428
$855$	0	0
$856$	−9120.00	−0.364153
$857$	26961.0	1.07464	0.537322	−	0.843377i	$-0.319436\pi$
$857$	26961.0	1.07464	0.537322	+	0.843377i	$0.319436\pi$
$858$	0	0
$859$	−415.000	−0.0164838	−0.00824192	−	0.999966i	$-0.502624\pi$
$859$	−415.000	−0.0164838	−0.00824192	+	0.999966i	$0.502624\pi$
$860$	3560.00	0.141157
$861$	0	0
$862$	26712.0	1.05547
$863$	45501.0	1.79475	0.897377	−	0.441265i	$-0.145470\pi$
$863$	45501.0	1.79475	0.897377	+	0.441265i	$0.145470\pi$
$864$	0	0
$865$	−12015.0	−0.472280
$866$	−23000.0	−0.902508
$867$	0	0
$868$	−752.000	−0.0294062
$869$	27930.0	1.09029
$870$	0	0
$871$	−10760.0	−0.418586
$872$	−13880.0	−0.539032
$873$	0	0
$874$	1062.00	0.0411015
$875$	500.000	0.0193178
$876$	0	0
$877$	35042.0	1.34924	0.674620	−	0.738165i	$-0.264307\pi$
$877$	35042.0	1.34924	0.674620	+	0.738165i	$0.264307\pi$
$878$	−22298.0	−0.857085
$879$	0	0
$880$	−3360.00	−0.128711
$881$	−1080.00	−0.0413009	−0.0206505	−	0.999787i	$-0.506574\pi$
$881$	−1080.00	−0.0413009	−0.0206505	+	0.999787i	$0.506574\pi$
$882$	0	0
$883$	−20164.0	−0.768485	−0.384243	−	0.923232i	$-0.625537\pi$
$883$	−20164.0	−0.768485	−0.384243	+	0.923232i	$0.625537\pi$
$884$	7440.00	0.283070
$885$	0	0
$886$	−7698.00	−0.291895
$887$	−20067.0	−0.759621	−0.379811	−	0.925064i	$-0.624011\pi$
$887$	−20067.0	−0.759621	−0.379811	+	0.925064i	$0.624011\pi$
$888$	0	0
$889$	−2744.00	−0.103522
$890$	10860.0	0.409020
$891$	0	0
$892$	−8632.00	−0.324014
$893$	8496.00	0.318374
$894$	0	0
$895$	13200.0	0.492991
$896$	−512.000	−0.0190901
$897$	0	0
$898$	−36096.0	−1.34136
$899$	5640.00	0.209238
$900$	0	0
$901$	68913.0	2.54809
$902$	10584.0	0.390697
$903$	0	0
$904$	−11472.0	−0.422072
$905$	−5365.00	−0.197059
$906$	0	0
$907$	−26524.0	−0.971020	−0.485510	−	0.874231i	$-0.661366\pi$
$907$	−26524.0	−0.971020	−0.485510	+	0.874231i	$0.661366\pi$
$908$	12492.0	0.456566
$909$	0	0
$910$	800.000	0.0291426
$911$	−35568.0	−1.29355	−0.646773	−	0.762683i	$-0.723882\pi$
$911$	−35568.0	−1.29355	−0.646773	+	0.762683i	$0.723882\pi$
$912$	0	0
$913$	3150.00	0.114184
$914$	−8528.00	−0.308623
$915$	0	0
$916$	8108.00	0.292463
$917$	456.000	0.0164214
$918$	0	0
$919$	−23704.0	−0.850841	−0.425420	−	0.904996i	$-0.639874\pi$
$919$	−23704.0	−0.850841	−0.425420	+	0.904996i	$0.639874\pi$
$920$	−360.000	−0.0129009
$921$	0	0
$922$	20484.0	0.731675
$923$	13800.0	0.492126
$924$	0	0
$925$	−6550.00	−0.232825
$926$	6604.00	0.234364
$927$	0	0
$928$	3840.00	0.135834
$929$	40590.0	1.43349	0.716746	−	0.697334i	$-0.245631\pi$
$929$	40590.0	1.43349	0.716746	+	0.697334i	$0.245631\pi$
$930$	0	0
$931$	−19293.0	−0.679165
$932$	−1752.00	−0.0615758
$933$	0	0
$934$	3846.00	0.134738
$935$	−19530.0	−0.683101
$936$	0	0
$937$	−12964.0	−0.451991	−0.225995	−	0.974128i	$-0.572563\pi$
$937$	−12964.0	−0.451991	−0.225995	+	0.974128i	$0.572563\pi$
$938$	4304.00	0.149819
$939$	0	0
$940$	−2880.00	−0.0999311
$941$	−29922.0	−1.03659	−0.518294	−	0.855203i	$-0.673433\pi$
$941$	−29922.0	−1.03659	−0.518294	+	0.855203i	$0.673433\pi$
$942$	0	0
$943$	1134.00	0.0391603
$944$	−7104.00	−0.244932
$945$	0	0
$946$	−14952.0	−0.513881
$947$	−5241.00	−0.179841	−0.0899206	−	0.995949i	$-0.528661\pi$
$947$	−5241.00	−0.179841	−0.0899206	+	0.995949i	$0.528661\pi$
$948$	0	0
$949$	−22520.0	−0.770316
$950$	2950.00	0.100748
$951$	0	0
$952$	−2976.00	−0.101316
$953$	−26214.0	−0.891033	−0.445517	−	0.895274i	$-0.646980\pi$
$953$	−26214.0	−0.891033	−0.445517	+	0.895274i	$0.646980\pi$
$954$	0	0
$955$	−7350.00	−0.249048
$956$	25656.0	0.867965
$957$	0	0
$958$	−30492.0	−1.02834
$959$	−636.000	−0.0214155
$960$	0	0
$961$	−27582.0	−0.925850
$962$	−10480.0	−0.351236
$963$	0	0
$964$	13724.0	0.458527
$965$	23600.0	0.787265
$966$	0	0
$967$	18278.0	0.607840	0.303920	−	0.952698i	$-0.401705\pi$
$967$	18278.0	0.607840	0.303920	+	0.952698i	$0.401705\pi$
$968$	3464.00	0.115018
$969$	0	0
$970$	−15440.0	−0.511081
$971$	24942.0	0.824333	0.412166	−	0.911109i	$-0.364772\pi$
$971$	24942.0	0.824333	0.412166	+	0.911109i	$0.364772\pi$
$972$	0	0
$973$	−9104.00	−0.299960
$974$	−16412.0	−0.539912
$975$	0	0
$976$	3536.00	0.115968
$977$	11226.0	0.367607	0.183803	−	0.982963i	$-0.441159\pi$
$977$	11226.0	0.367607	0.183803	+	0.982963i	$0.441159\pi$
$978$	0	0
$979$	−45612.0	−1.48904
$980$	6540.00	0.213176
$981$	0	0
$982$	33612.0	1.09226
$983$	23073.0	0.748641	0.374321	−	0.927299i	$-0.377876\pi$
$983$	23073.0	0.748641	0.374321	+	0.927299i	$0.377876\pi$
$984$	0	0
$985$	3825.00	0.123731
$986$	22320.0	0.720906
$987$	0	0
$988$	4720.00	0.151987
$989$	−1602.00	−0.0515072
$990$	0	0
$991$	22037.0	0.706386	0.353193	−	0.935551i	$-0.385096\pi$
$991$	22037.0	0.706386	0.353193	+	0.935551i	$0.385096\pi$
$992$	1504.00	0.0481371
$993$	0	0
$994$	−5520.00	−0.176141
$995$	−3340.00	−0.106417
$996$	0	0
$997$	19082.0	0.606151	0.303076	−	0.952966i	$-0.401986\pi$
$997$	19082.0	0.606151	0.303076	+	0.952966i	$0.401986\pi$
$998$	−10850.0	−0.344139
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.4.a.i.1.1	yes	1
3.2	odd	2		270.4.a.e.1.1	✓	1
4.3	odd	2		2160.4.a.e.1.1		1
5.2	odd	4		1350.4.c.q.649.2		2
5.3	odd	4		1350.4.c.q.649.1		2
5.4	even	2		1350.4.a.g.1.1		1
9.2	odd	6		810.4.e.q.271.1		2
9.4	even	3		810.4.e.h.541.1		2
9.5	odd	6		810.4.e.q.541.1		2
9.7	even	3		810.4.e.h.271.1		2
12.11	even	2		2160.4.a.o.1.1		1
15.2	even	4		1350.4.c.d.649.1		2
15.8	even	4		1350.4.c.d.649.2		2
15.14	odd	2		1350.4.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.e.1.1	✓	1	3.2	odd	2
270.4.a.i.1.1	yes	1	1.1	even	1	trivial
810.4.e.h.271.1		2	9.7	even	3
810.4.e.h.541.1		2	9.4	even	3
810.4.e.q.271.1		2	9.2	odd	6
810.4.e.q.541.1		2	9.5	odd	6
1350.4.a.g.1.1		1	5.4	even	2
1350.4.a.u.1.1		1	15.14	odd	2
1350.4.c.d.649.1		2	15.2	even	4
1350.4.c.d.649.2		2	15.8	even	4
1350.4.c.q.649.1		2	5.3	odd	4
1350.4.c.q.649.2		2	5.2	odd	4
2160.4.a.e.1.1		1	4.3	odd	2
2160.4.a.o.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	270.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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