LMFDB - Embedded newform 2240.4.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2240 = 2^{6} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2240.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:	$132.164278413$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+7.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +22.0000 q^{9} +9.00000 q^{11} -23.0000 q^{13} -35.0000 q^{15} +41.0000 q^{17} +34.0000 q^{19} -49.0000 q^{21} +6.00000 q^{23} +25.0000 q^{25} -35.0000 q^{27} -131.000 q^{29} -4.00000 q^{31} +63.0000 q^{33} +35.0000 q^{35} -26.0000 q^{37} -161.000 q^{39} -260.000 q^{41} -190.000 q^{43} -110.000 q^{45} -167.000 q^{47} +49.0000 q^{49} +287.000 q^{51} +368.000 q^{53} -45.0000 q^{55} +238.000 q^{57} +324.000 q^{59} +164.000 q^{61} -154.000 q^{63} +115.000 q^{65} +200.000 q^{67} +42.0000 q^{69} -784.000 q^{71} -410.000 q^{73} +175.000 q^{75} -63.0000 q^{77} -1211.00 q^{79} -839.000 q^{81} -1132.00 q^{83} -205.000 q^{85} -917.000 q^{87} -72.0000 q^{89} +161.000 q^{91} -28.0000 q^{93} -170.000 q^{95} -707.000 q^{97} +198.000 q^{99} +O(q^{100})

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	7.00000	1.34715	0.673575	−	0.739119i	$-0.264758\pi$
$3$	7.00000	1.34715	0.673575	+	0.739119i	$0.264758\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	22.0000	0.814815
$10$	0	0
$11$	9.00000	0.246691	0.123346	−	0.992364i	$-0.460638\pi$
$11$	9.00000	0.246691	0.123346	+	0.992364i	$0.460638\pi$
$12$	0	0
$13$	−23.0000	−0.490696	−0.245348	−	0.969435i	$-0.578902\pi$
$13$	−23.0000	−0.490696	−0.245348	+	0.969435i	$0.578902\pi$
$14$	0	0
$15$	−35.0000	−0.602464
$16$	0	0
$17$	41.0000	0.584939	0.292469	−	0.956275i	$-0.405523\pi$
$17$	41.0000	0.584939	0.292469	+	0.956275i	$0.405523\pi$
$18$	0	0
$19$	34.0000	0.410533	0.205267	−	0.978706i	$-0.434194\pi$
$19$	34.0000	0.410533	0.205267	+	0.978706i	$0.434194\pi$
$20$	0	0
$21$	−49.0000	−0.509175
$22$	0	0
$23$	6.00000	0.0543951	0.0271975	−	0.999630i	$-0.491342\pi$
$23$	6.00000	0.0543951	0.0271975	+	0.999630i	$0.491342\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−35.0000	−0.249472
$28$	0	0
$29$	−131.000	−0.838831	−0.419415	−	0.907794i	$-0.637765\pi$
$29$	−131.000	−0.838831	−0.419415	+	0.907794i	$0.637765\pi$
$30$	0	0
$31$	−4.00000	−0.0231749	−0.0115874	−	0.999933i	$-0.503688\pi$
$31$	−4.00000	−0.0231749	−0.0115874	+	0.999933i	$0.503688\pi$
$32$	0	0
$33$	63.0000	0.332330
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−26.0000	−0.115524	−0.0577618	−	0.998330i	$-0.518396\pi$
$37$	−26.0000	−0.115524	−0.0577618	+	0.998330i	$0.518396\pi$
$38$	0	0
$39$	−161.000	−0.661042
$40$	0	0
$41$	−260.000	−0.990370	−0.495185	−	0.868787i	$-0.664900\pi$
$41$	−260.000	−0.990370	−0.495185	+	0.868787i	$0.664900\pi$
$42$	0	0
$43$	−190.000	−0.673831	−0.336915	−	0.941535i	$-0.609384\pi$
$43$	−190.000	−0.673831	−0.336915	+	0.941535i	$0.609384\pi$
$44$	0	0
$45$	−110.000	−0.364396
$46$	0	0
$47$	−167.000	−0.518286	−0.259143	−	0.965839i	$-0.583440\pi$
$47$	−167.000	−0.518286	−0.259143	+	0.965839i	$0.583440\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	287.000	0.788001
$52$	0	0
$53$	368.000	0.953749	0.476874	−	0.878971i	$-0.341770\pi$
$53$	368.000	0.953749	0.476874	+	0.878971i	$0.341770\pi$
$54$	0	0
$55$	−45.0000	−0.110324
$56$	0	0
$57$	238.000	0.553050
$58$	0	0
$59$	324.000	0.714936	0.357468	−	0.933925i	$-0.383640\pi$
$59$	324.000	0.714936	0.357468	+	0.933925i	$0.383640\pi$
$60$	0	0
$61$	164.000	0.344230	0.172115	−	0.985077i	$-0.444940\pi$
$61$	164.000	0.344230	0.172115	+	0.985077i	$0.444940\pi$
$62$	0	0
$63$	−154.000	−0.307971
$64$	0	0
$65$	115.000	0.219446
$66$	0	0
$67$	200.000	0.364685	0.182342	−	0.983235i	$-0.441632\pi$
$67$	200.000	0.364685	0.182342	+	0.983235i	$0.441632\pi$
$68$	0	0
$69$	42.0000	0.0732783
$70$	0	0
$71$	−784.000	−1.31047	−0.655237	−	0.755423i	$-0.727431\pi$
$71$	−784.000	−1.31047	−0.655237	+	0.755423i	$0.727431\pi$
$72$	0	0
$73$	−410.000	−0.657354	−0.328677	−	0.944442i	$-0.606603\pi$
$73$	−410.000	−0.657354	−0.328677	+	0.944442i	$0.606603\pi$
$74$	0	0
$75$	175.000	0.269430
$76$	0	0
$77$	−63.0000	−0.0932405
$78$	0	0
$79$	−1211.00	−1.72466	−0.862330	−	0.506347i	$-0.830996\pi$
$79$	−1211.00	−1.72466	−0.862330	+	0.506347i	$0.830996\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	0	0
$83$	−1132.00	−1.49703	−0.748513	−	0.663120i	$-0.769232\pi$
$83$	−1132.00	−1.49703	−0.748513	+	0.663120i	$0.769232\pi$
$84$	0	0
$85$	−205.000	−0.261593
$86$	0	0
$87$	−917.000	−1.13003
$88$	0	0
$89$	−72.0000	−0.0857526	−0.0428763	−	0.999080i	$-0.513652\pi$
$89$	−72.0000	−0.0857526	−0.0428763	+	0.999080i	$0.513652\pi$
$90$	0	0
$91$	161.000	0.185466
$92$	0	0
$93$	−28.0000	−0.0312201
$94$	0	0
$95$	−170.000	−0.183596
$96$	0	0
$97$	−707.000	−0.740051	−0.370026	−	0.929022i	$-0.620651\pi$
$97$	−707.000	−0.740051	−0.370026	+	0.929022i	$0.620651\pi$
$98$	0	0
$99$	198.000	0.201008
$100$	0	0
$101$	1410.00	1.38911	0.694556	−	0.719439i	$-0.255601\pi$
$101$	1410.00	1.38911	0.694556	+	0.719439i	$0.255601\pi$
$102$	0	0
$103$	813.000	0.777740	0.388870	−	0.921293i	$-0.372865\pi$
$103$	813.000	0.777740	0.388870	+	0.921293i	$0.372865\pi$
$104$	0	0
$105$	245.000	0.227710
$106$	0	0
$107$	1078.00	0.973964	0.486982	−	0.873412i	$-0.338098\pi$
$107$	1078.00	0.973964	0.486982	+	0.873412i	$0.338098\pi$
$108$	0	0
$109$	2135.00	1.87611	0.938055	−	0.346487i	$-0.112626\pi$
$109$	2135.00	1.87611	0.938055	+	0.346487i	$0.112626\pi$
$110$	0	0
$111$	−182.000	−0.155628
$112$	0	0
$113$	158.000	0.131534	0.0657672	−	0.997835i	$-0.479051\pi$
$113$	158.000	0.131534	0.0657672	+	0.997835i	$0.479051\pi$
$114$	0	0
$115$	−30.0000	−0.0243262
$116$	0	0
$117$	−506.000	−0.399827
$118$	0	0
$119$	−287.000	−0.221086
$120$	0	0
$121$	−1250.00	−0.939144
$122$	0	0
$123$	−1820.00	−1.33418
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−604.000	−0.422018	−0.211009	−	0.977484i	$-0.567675\pi$
$127$	−604.000	−0.422018	−0.211009	+	0.977484i	$0.567675\pi$
$128$	0	0
$129$	−1330.00	−0.907752
$130$	0	0
$131$	−1914.00	−1.27654	−0.638271	−	0.769812i	$-0.720350\pi$
$131$	−1914.00	−1.27654	−0.638271	+	0.769812i	$0.720350\pi$
$132$	0	0
$133$	−238.000	−0.155167
$134$	0	0
$135$	175.000	0.111567
$136$	0	0
$137$	−2144.00	−1.33704	−0.668519	−	0.743695i	$-0.733072\pi$
$137$	−2144.00	−1.33704	−0.668519	+	0.743695i	$0.733072\pi$
$138$	0	0
$139$	−2266.00	−1.38273	−0.691366	−	0.722505i	$-0.742991\pi$
$139$	−2266.00	−1.38273	−0.691366	+	0.722505i	$0.742991\pi$
$140$	0	0
$141$	−1169.00	−0.698210
$142$	0	0
$143$	−207.000	−0.121050
$144$	0	0
$145$	655.000	0.375136
$146$	0	0
$147$	343.000	0.192450
$148$	0	0
$149$	1854.00	1.01937	0.509683	−	0.860362i	$-0.329763\pi$
$149$	1854.00	1.01937	0.509683	+	0.860362i	$0.329763\pi$
$150$	0	0
$151$	1037.00	0.558873	0.279437	−	0.960164i	$-0.409852\pi$
$151$	1037.00	0.558873	0.279437	+	0.960164i	$0.409852\pi$
$152$	0	0
$153$	902.000	0.476617
$154$	0	0
$155$	20.0000	0.0103641
$156$	0	0
$157$	−1058.00	−0.537819	−0.268910	−	0.963165i	$-0.586663\pi$
$157$	−1058.00	−0.537819	−0.268910	+	0.963165i	$0.586663\pi$
$158$	0	0
$159$	2576.00	1.28484
$160$	0	0
$161$	−42.0000	−0.0205594
$162$	0	0
$163$	−2810.00	−1.35028	−0.675142	−	0.737688i	$-0.735918\pi$
$163$	−2810.00	−1.35028	−0.675142	+	0.737688i	$0.735918\pi$
$164$	0	0
$165$	−315.000	−0.148623
$166$	0	0
$167$	−1395.00	−0.646397	−0.323199	−	0.946331i	$-0.604758\pi$
$167$	−1395.00	−0.646397	−0.323199	+	0.946331i	$0.604758\pi$
$168$	0	0
$169$	−1668.00	−0.759217
$170$	0	0
$171$	748.000	0.334509
$172$	0	0
$173$	−4445.00	−1.95345	−0.976726	−	0.214492i	$-0.931190\pi$
$173$	−4445.00	−1.95345	−0.976726	+	0.214492i	$0.931190\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	2268.00	0.963126
$178$	0	0
$179$	−1740.00	−0.726557	−0.363279	−	0.931681i	$-0.618343\pi$
$179$	−1740.00	−0.726557	−0.363279	+	0.931681i	$0.618343\pi$
$180$	0	0
$181$	−2488.00	−1.02172	−0.510861	−	0.859663i	$-0.670673\pi$
$181$	−2488.00	−1.02172	−0.510861	+	0.859663i	$0.670673\pi$
$182$	0	0
$183$	1148.00	0.463730
$184$	0	0
$185$	130.000	0.0516637
$186$	0	0
$187$	369.000	0.144299
$188$	0	0
$189$	245.000	0.0942917
$190$	0	0
$191$	−2053.00	−0.777748	−0.388874	−	0.921291i	$-0.627136\pi$
$191$	−2053.00	−0.777748	−0.388874	+	0.921291i	$0.627136\pi$
$192$	0	0
$193$	448.000	0.167087	0.0835434	−	0.996504i	$-0.473376\pi$
$193$	448.000	0.167087	0.0835434	+	0.996504i	$0.473376\pi$
$194$	0	0
$195$	805.000	0.295627
$196$	0	0
$197$	−3312.00	−1.19782	−0.598909	−	0.800817i	$-0.704399\pi$
$197$	−3312.00	−1.19782	−0.598909	+	0.800817i	$0.704399\pi$
$198$	0	0
$199$	−1060.00	−0.377595	−0.188798	−	0.982016i	$-0.560459\pi$
$199$	−1060.00	−0.377595	−0.188798	+	0.982016i	$0.560459\pi$
$200$	0	0
$201$	1400.00	0.491286
$202$	0	0
$203$	917.000	0.317048
$204$	0	0
$205$	1300.00	0.442907
$206$	0	0
$207$	132.000	0.0443219
$208$	0	0
$209$	306.000	0.101275
$210$	0	0
$211$	1711.00	0.558247	0.279123	−	0.960255i	$-0.409956\pi$
$211$	1711.00	0.558247	0.279123	+	0.960255i	$0.409956\pi$
$212$	0	0
$213$	−5488.00	−1.76541
$214$	0	0
$215$	950.000	0.301346
$216$	0	0
$217$	28.0000	0.00875928
$218$	0	0
$219$	−2870.00	−0.885555
$220$	0	0
$221$	−943.000	−0.287027
$222$	0	0
$223$	−795.000	−0.238732	−0.119366	−	0.992850i	$-0.538086\pi$
$223$	−795.000	−0.238732	−0.119366	+	0.992850i	$0.538086\pi$
$224$	0	0
$225$	550.000	0.162963
$226$	0	0
$227$	5899.00	1.72480	0.862402	−	0.506225i	$-0.168959\pi$
$227$	5899.00	1.72480	0.862402	+	0.506225i	$0.168959\pi$
$228$	0	0
$229$	2468.00	0.712184	0.356092	−	0.934451i	$-0.384109\pi$
$229$	2468.00	0.712184	0.356092	+	0.934451i	$0.384109\pi$
$230$	0	0
$231$	−441.000	−0.125609
$232$	0	0
$233$	−3960.00	−1.11343	−0.556713	−	0.830705i	$-0.687938\pi$
$233$	−3960.00	−1.11343	−0.556713	+	0.830705i	$0.687938\pi$
$234$	0	0
$235$	835.000	0.231785
$236$	0	0
$237$	−8477.00	−2.32338
$238$	0	0
$239$	−1183.00	−0.320175	−0.160088	−	0.987103i	$-0.551178\pi$
$239$	−1183.00	−0.320175	−0.160088	+	0.987103i	$0.551178\pi$
$240$	0	0
$241$	−2878.00	−0.769246	−0.384623	−	0.923074i	$-0.625669\pi$
$241$	−2878.00	−0.769246	−0.384623	+	0.923074i	$0.625669\pi$
$242$	0	0
$243$	−4928.00	−1.30095
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−782.000	−0.201447
$248$	0	0
$249$	−7924.00	−2.01672
$250$	0	0
$251$	5570.00	1.40070	0.700349	−	0.713800i	$-0.253028\pi$
$251$	5570.00	1.40070	0.700349	+	0.713800i	$0.253028\pi$
$252$	0	0
$253$	54.0000	0.0134188
$254$	0	0
$255$	−1435.00	−0.352405
$256$	0	0
$257$	2594.00	0.629608	0.314804	−	0.949157i	$-0.398061\pi$
$257$	2594.00	0.629608	0.314804	+	0.949157i	$0.398061\pi$
$258$	0	0
$259$	182.000	0.0436638
$260$	0	0
$261$	−2882.00	−0.683492
$262$	0	0
$263$	−638.000	−0.149585	−0.0747923	−	0.997199i	$-0.523829\pi$
$263$	−638.000	−0.149585	−0.0747923	+	0.997199i	$0.523829\pi$
$264$	0	0
$265$	−1840.00	−0.426529
$266$	0	0
$267$	−504.000	−0.115522
$268$	0	0
$269$	−1106.00	−0.250684	−0.125342	−	0.992114i	$-0.540003\pi$
$269$	−1106.00	−0.250684	−0.125342	+	0.992114i	$0.540003\pi$
$270$	0	0
$271$	8084.00	1.81206	0.906030	−	0.423214i	$-0.139098\pi$
$271$	8084.00	1.81206	0.906030	+	0.423214i	$0.139098\pi$
$272$	0	0
$273$	1127.00	0.249850
$274$	0	0
$275$	225.000	0.0493382
$276$	0	0
$277$	2854.00	0.619062	0.309531	−	0.950889i	$-0.399828\pi$
$277$	2854.00	0.619062	0.309531	+	0.950889i	$0.399828\pi$
$278$	0	0
$279$	−88.0000	−0.0188832
$280$	0	0
$281$	−165.000	−0.0350287	−0.0175144	−	0.999847i	$-0.505575\pi$
$281$	−165.000	−0.0350287	−0.0175144	+	0.999847i	$0.505575\pi$
$282$	0	0
$283$	4973.00	1.04457	0.522287	−	0.852770i	$-0.325079\pi$
$283$	4973.00	1.04457	0.522287	+	0.852770i	$0.325079\pi$
$284$	0	0
$285$	−1190.00	−0.247332
$286$	0	0
$287$	1820.00	0.374325
$288$	0	0
$289$	−3232.00	−0.657847
$290$	0	0
$291$	−4949.00	−0.996961
$292$	0	0
$293$	−7593.00	−1.51395	−0.756976	−	0.653443i	$-0.773324\pi$
$293$	−7593.00	−1.51395	−0.756976	+	0.653443i	$0.773324\pi$
$294$	0	0
$295$	−1620.00	−0.319729
$296$	0	0
$297$	−315.000	−0.0615426
$298$	0	0
$299$	−138.000	−0.0266915
$300$	0	0
$301$	1330.00	0.254684
$302$	0	0
$303$	9870.00	1.87134
$304$	0	0
$305$	−820.000	−0.153944
$306$	0	0
$307$	8205.00	1.52536	0.762678	−	0.646779i	$-0.223884\pi$
$307$	8205.00	1.52536	0.762678	+	0.646779i	$0.223884\pi$
$308$	0	0
$309$	5691.00	1.04773
$310$	0	0
$311$	1430.00	0.260733	0.130366	−	0.991466i	$-0.458385\pi$
$311$	1430.00	0.260733	0.130366	+	0.991466i	$0.458385\pi$
$312$	0	0
$313$	−1081.00	−0.195213	−0.0976066	−	0.995225i	$-0.531119\pi$
$313$	−1081.00	−0.195213	−0.0976066	+	0.995225i	$0.531119\pi$
$314$	0	0
$315$	770.000	0.137729
$316$	0	0
$317$	−5326.00	−0.943653	−0.471826	−	0.881691i	$-0.656405\pi$
$317$	−5326.00	−0.943653	−0.471826	+	0.881691i	$0.656405\pi$
$318$	0	0
$319$	−1179.00	−0.206932
$320$	0	0
$321$	7546.00	1.31208
$322$	0	0
$323$	1394.00	0.240137
$324$	0	0
$325$	−575.000	−0.0981393
$326$	0	0
$327$	14945.0	2.52740
$328$	0	0
$329$	1169.00	0.195894
$330$	0	0
$331$	−1732.00	−0.287611	−0.143806	−	0.989606i	$-0.545934\pi$
$331$	−1732.00	−0.287611	−0.143806	+	0.989606i	$0.545934\pi$
$332$	0	0
$333$	−572.000	−0.0941304
$334$	0	0
$335$	−1000.00	−0.163092
$336$	0	0
$337$	2002.00	0.323608	0.161804	−	0.986823i	$-0.448269\pi$
$337$	2002.00	0.323608	0.161804	+	0.986823i	$0.448269\pi$
$338$	0	0
$339$	1106.00	0.177197
$340$	0	0
$341$	−36.0000	−0.00571704
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−210.000	−0.0327711
$346$	0	0
$347$	6054.00	0.936587	0.468294	−	0.883573i	$-0.344869\pi$
$347$	6054.00	0.936587	0.468294	+	0.883573i	$0.344869\pi$
$348$	0	0
$349$	−10162.0	−1.55862	−0.779311	−	0.626637i	$-0.784431\pi$
$349$	−10162.0	−1.55862	−0.779311	+	0.626637i	$0.784431\pi$
$350$	0	0
$351$	805.000	0.122415
$352$	0	0
$353$	5893.00	0.888535	0.444267	−	0.895894i	$-0.353464\pi$
$353$	5893.00	0.888535	0.444267	+	0.895894i	$0.353464\pi$
$354$	0	0
$355$	3920.00	0.586062
$356$	0	0
$357$	−2009.00	−0.297836
$358$	0	0
$359$	−6896.00	−1.01381	−0.506904	−	0.862003i	$-0.669210\pi$
$359$	−6896.00	−1.01381	−0.506904	+	0.862003i	$0.669210\pi$
$360$	0	0
$361$	−5703.00	−0.831462
$362$	0	0
$363$	−8750.00	−1.26517
$364$	0	0
$365$	2050.00	0.293978
$366$	0	0
$367$	5729.00	0.814854	0.407427	−	0.913238i	$-0.366426\pi$
$367$	5729.00	0.814854	0.407427	+	0.913238i	$0.366426\pi$
$368$	0	0
$369$	−5720.00	−0.806968
$370$	0	0
$371$	−2576.00	−0.360483
$372$	0	0
$373$	11800.0	1.63802	0.819009	−	0.573780i	$-0.194524\pi$
$373$	11800.0	1.63802	0.819009	+	0.573780i	$0.194524\pi$
$374$	0	0
$375$	−875.000	−0.120493
$376$	0	0
$377$	3013.00	0.411611
$378$	0	0
$379$	8548.00	1.15853	0.579263	−	0.815141i	$-0.303340\pi$
$379$	8548.00	1.15853	0.579263	+	0.815141i	$0.303340\pi$
$380$	0	0
$381$	−4228.00	−0.568522
$382$	0	0
$383$	−1668.00	−0.222535	−0.111267	−	0.993791i	$-0.535491\pi$
$383$	−1668.00	−0.222535	−0.111267	+	0.993791i	$0.535491\pi$
$384$	0	0
$385$	315.000	0.0416984
$386$	0	0
$387$	−4180.00	−0.549047
$388$	0	0
$389$	739.000	0.0963208	0.0481604	−	0.998840i	$-0.484664\pi$
$389$	739.000	0.0963208	0.0481604	+	0.998840i	$0.484664\pi$
$390$	0	0
$391$	246.000	0.0318178
$392$	0	0
$393$	−13398.0	−1.71969
$394$	0	0
$395$	6055.00	0.771291
$396$	0	0
$397$	1235.00	0.156128	0.0780641	−	0.996948i	$-0.475126\pi$
$397$	1235.00	0.156128	0.0780641	+	0.996948i	$0.475126\pi$
$398$	0	0
$399$	−1666.00	−0.209033
$400$	0	0
$401$	11313.0	1.40884	0.704419	−	0.709784i	$-0.251208\pi$
$401$	11313.0	1.40884	0.704419	+	0.709784i	$0.251208\pi$
$402$	0	0
$403$	92.0000	0.0113718
$404$	0	0
$405$	4195.00	0.514694
$406$	0	0
$407$	−234.000	−0.0284986
$408$	0	0
$409$	−6942.00	−0.839266	−0.419633	−	0.907694i	$-0.637841\pi$
$409$	−6942.00	−0.839266	−0.419633	+	0.907694i	$0.637841\pi$
$410$	0	0
$411$	−15008.0	−1.80119
$412$	0	0
$413$	−2268.00	−0.270220
$414$	0	0
$415$	5660.00	0.669490
$416$	0	0
$417$	−15862.0	−1.86275
$418$	0	0
$419$	−120.000	−0.0139914	−0.00699568	−	0.999976i	$-0.502227\pi$
$419$	−120.000	−0.0139914	−0.00699568	+	0.999976i	$0.502227\pi$
$420$	0	0
$421$	9479.00	1.09734	0.548668	−	0.836041i	$-0.315135\pi$
$421$	9479.00	1.09734	0.548668	+	0.836041i	$0.315135\pi$
$422$	0	0
$423$	−3674.00	−0.422307
$424$	0	0
$425$	1025.00	0.116988
$426$	0	0
$427$	−1148.00	−0.130107
$428$	0	0
$429$	−1449.00	−0.163073
$430$	0	0
$431$	16239.0	1.81486	0.907431	−	0.420202i	$-0.138041\pi$
$431$	16239.0	1.81486	0.907431	+	0.420202i	$0.138041\pi$
$432$	0	0
$433$	−7058.00	−0.783339	−0.391670	−	0.920106i	$-0.628102\pi$
$433$	−7058.00	−0.783339	−0.391670	+	0.920106i	$0.628102\pi$
$434$	0	0
$435$	4585.00	0.505365
$436$	0	0
$437$	204.000	0.0223310
$438$	0	0
$439$	−10810.0	−1.17525	−0.587623	−	0.809135i	$-0.699936\pi$
$439$	−10810.0	−1.17525	−0.587623	+	0.809135i	$0.699936\pi$
$440$	0	0
$441$	1078.00	0.116402
$442$	0	0
$443$	−17830.0	−1.91225	−0.956127	−	0.292951i	$-0.905363\pi$
$443$	−17830.0	−1.91225	−0.956127	+	0.292951i	$0.905363\pi$
$444$	0	0
$445$	360.000	0.0383497
$446$	0	0
$447$	12978.0	1.37324
$448$	0	0
$449$	4743.00	0.498521	0.249261	−	0.968436i	$-0.419812\pi$
$449$	4743.00	0.498521	0.249261	+	0.968436i	$0.419812\pi$
$450$	0	0
$451$	−2340.00	−0.244316
$452$	0	0
$453$	7259.00	0.752886
$454$	0	0
$455$	−805.000	−0.0829428
$456$	0	0
$457$	−2976.00	−0.304620	−0.152310	−	0.988333i	$-0.548671\pi$
$457$	−2976.00	−0.304620	−0.152310	+	0.988333i	$0.548671\pi$
$458$	0	0
$459$	−1435.00	−0.145926
$460$	0	0
$461$	−7872.00	−0.795305	−0.397652	−	0.917536i	$-0.630175\pi$
$461$	−7872.00	−0.795305	−0.397652	+	0.917536i	$0.630175\pi$
$462$	0	0
$463$	−3148.00	−0.315983	−0.157991	−	0.987441i	$-0.550502\pi$
$463$	−3148.00	−0.315983	−0.157991	+	0.987441i	$0.550502\pi$
$464$	0	0
$465$	140.000	0.0139620
$466$	0	0
$467$	8321.00	0.824518	0.412259	−	0.911067i	$-0.364740\pi$
$467$	8321.00	0.824518	0.412259	+	0.911067i	$0.364740\pi$
$468$	0	0
$469$	−1400.00	−0.137838
$470$	0	0
$471$	−7406.00	−0.724523
$472$	0	0
$473$	−1710.00	−0.166228
$474$	0	0
$475$	850.000	0.0821067
$476$	0	0
$477$	8096.00	0.777129
$478$	0	0
$479$	−8518.00	−0.812521	−0.406260	−	0.913757i	$-0.633167\pi$
$479$	−8518.00	−0.812521	−0.406260	+	0.913757i	$0.633167\pi$
$480$	0	0
$481$	598.000	0.0566870
$482$	0	0
$483$	−294.000	−0.0276966
$484$	0	0
$485$	3535.00	0.330961
$486$	0	0
$487$	934.000	0.0869067	0.0434534	−	0.999055i	$-0.486164\pi$
$487$	934.000	0.0869067	0.0434534	+	0.999055i	$0.486164\pi$
$488$	0	0
$489$	−19670.0	−1.81904
$490$	0	0
$491$	10515.0	0.966467	0.483234	−	0.875492i	$-0.339462\pi$
$491$	10515.0	0.966467	0.483234	+	0.875492i	$0.339462\pi$
$492$	0	0
$493$	−5371.00	−0.490665
$494$	0	0
$495$	−990.000	−0.0898933
$496$	0	0
$497$	5488.00	0.495313
$498$	0	0
$499$	10157.0	0.911202	0.455601	−	0.890184i	$-0.349424\pi$
$499$	10157.0	0.911202	0.455601	+	0.890184i	$0.349424\pi$
$500$	0	0
$501$	−9765.00	−0.870794
$502$	0	0
$503$	−3197.00	−0.283394	−0.141697	−	0.989910i	$-0.545256\pi$
$503$	−3197.00	−0.283394	−0.141697	+	0.989910i	$0.545256\pi$
$504$	0	0
$505$	−7050.00	−0.621229
$506$	0	0
$507$	−11676.0	−1.02278
$508$	0	0
$509$	−8250.00	−0.718418	−0.359209	−	0.933257i	$-0.616953\pi$
$509$	−8250.00	−0.718418	−0.359209	+	0.933257i	$0.616953\pi$
$510$	0	0
$511$	2870.00	0.248457
$512$	0	0
$513$	−1190.00	−0.102417
$514$	0	0
$515$	−4065.00	−0.347816
$516$	0	0
$517$	−1503.00	−0.127857
$518$	0	0
$519$	−31115.0	−2.63159
$520$	0	0
$521$	−8818.00	−0.741504	−0.370752	−	0.928732i	$-0.620900\pi$
$521$	−8818.00	−0.741504	−0.370752	+	0.928732i	$0.620900\pi$
$522$	0	0
$523$	16844.0	1.40829	0.704146	−	0.710055i	$-0.251330\pi$
$523$	16844.0	1.40829	0.704146	+	0.710055i	$0.251330\pi$
$524$	0	0
$525$	−1225.00	−0.101835
$526$	0	0
$527$	−164.000	−0.0135559
$528$	0	0
$529$	−12131.0	−0.997041
$530$	0	0
$531$	7128.00	0.582540
$532$	0	0
$533$	5980.00	0.485971
$534$	0	0
$535$	−5390.00	−0.435570
$536$	0	0
$537$	−12180.0	−0.978782
$538$	0	0
$539$	441.000	0.0352416
$540$	0	0
$541$	−12659.0	−1.00601	−0.503006	−	0.864283i	$-0.667773\pi$
$541$	−12659.0	−1.00601	−0.503006	+	0.864283i	$0.667773\pi$
$542$	0	0
$543$	−17416.0	−1.37641
$544$	0	0
$545$	−10675.0	−0.839022
$546$	0	0
$547$	21092.0	1.64868	0.824341	−	0.566094i	$-0.191546\pi$
$547$	21092.0	1.64868	0.824341	+	0.566094i	$0.191546\pi$
$548$	0	0
$549$	3608.00	0.280484
$550$	0	0
$551$	−4454.00	−0.344368
$552$	0	0
$553$	8477.00	0.651860
$554$	0	0
$555$	910.000	0.0695988
$556$	0	0
$557$	−1232.00	−0.0937191	−0.0468595	−	0.998901i	$-0.514921\pi$
$557$	−1232.00	−0.0937191	−0.0468595	+	0.998901i	$0.514921\pi$
$558$	0	0
$559$	4370.00	0.330646
$560$	0	0
$561$	2583.00	0.194393
$562$	0	0
$563$	9148.00	0.684800	0.342400	−	0.939554i	$-0.388760\pi$
$563$	9148.00	0.684800	0.342400	+	0.939554i	$0.388760\pi$
$564$	0	0
$565$	−790.000	−0.0588240
$566$	0	0
$567$	5873.00	0.434996
$568$	0	0
$569$	19066.0	1.40472	0.702362	−	0.711820i	$-0.252129\pi$
$569$	19066.0	1.40472	0.702362	+	0.711820i	$0.252129\pi$
$570$	0	0
$571$	−18884.0	−1.38401	−0.692006	−	0.721892i	$-0.743273\pi$
$571$	−18884.0	−1.38401	−0.692006	+	0.721892i	$0.743273\pi$
$572$	0	0
$573$	−14371.0	−1.04774
$574$	0	0
$575$	150.000	0.0108790
$576$	0	0
$577$	−14197.0	−1.02431	−0.512157	−	0.858892i	$-0.671153\pi$
$577$	−14197.0	−1.02431	−0.512157	+	0.858892i	$0.671153\pi$
$578$	0	0
$579$	3136.00	0.225091
$580$	0	0
$581$	7924.00	0.565823
$582$	0	0
$583$	3312.00	0.235281
$584$	0	0
$585$	2530.00	0.178808
$586$	0	0
$587$	−2080.00	−0.146253	−0.0731267	−	0.997323i	$-0.523298\pi$
$587$	−2080.00	−0.146253	−0.0731267	+	0.997323i	$0.523298\pi$
$588$	0	0
$589$	−136.000	−0.00951406
$590$	0	0
$591$	−23184.0	−1.61364
$592$	0	0
$593$	22763.0	1.57633	0.788166	−	0.615463i	$-0.211031\pi$
$593$	22763.0	1.57633	0.788166	+	0.615463i	$0.211031\pi$
$594$	0	0
$595$	1435.00	0.0988727
$596$	0	0
$597$	−7420.00	−0.508677
$598$	0	0
$599$	−6113.00	−0.416979	−0.208489	−	0.978025i	$-0.566855\pi$
$599$	−6113.00	−0.416979	−0.208489	+	0.978025i	$0.566855\pi$
$600$	0	0
$601$	−10230.0	−0.694327	−0.347163	−	0.937805i	$-0.612855\pi$
$601$	−10230.0	−0.694327	−0.347163	+	0.937805i	$0.612855\pi$
$602$	0	0
$603$	4400.00	0.297151
$604$	0	0
$605$	6250.00	0.419998
$606$	0	0
$607$	−6597.00	−0.441127	−0.220563	−	0.975373i	$-0.570790\pi$
$607$	−6597.00	−0.441127	−0.220563	+	0.975373i	$0.570790\pi$
$608$	0	0
$609$	6419.00	0.427112
$610$	0	0
$611$	3841.00	0.254321
$612$	0	0
$613$	−20762.0	−1.36798	−0.683988	−	0.729493i	$-0.739756\pi$
$613$	−20762.0	−1.36798	−0.683988	+	0.729493i	$0.739756\pi$
$614$	0	0
$615$	9100.00	0.596662
$616$	0	0
$617$	−1994.00	−0.130106	−0.0650530	−	0.997882i	$-0.520722\pi$
$617$	−1994.00	−0.130106	−0.0650530	+	0.997882i	$0.520722\pi$
$618$	0	0
$619$	−14362.0	−0.932565	−0.466282	−	0.884636i	$-0.654407\pi$
$619$	−14362.0	−0.932565	−0.466282	+	0.884636i	$0.654407\pi$
$620$	0	0
$621$	−210.000	−0.0135701
$622$	0	0
$623$	504.000	0.0324115
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	2142.00	0.136433
$628$	0	0
$629$	−1066.00	−0.0675743
$630$	0	0
$631$	−13337.0	−0.841422	−0.420711	−	0.907195i	$-0.638219\pi$
$631$	−13337.0	−0.841422	−0.420711	+	0.907195i	$0.638219\pi$
$632$	0	0
$633$	11977.0	0.752043
$634$	0	0
$635$	3020.00	0.188732
$636$	0	0
$637$	−1127.00	−0.0700995
$638$	0	0
$639$	−17248.0	−1.06779
$640$	0	0
$641$	12882.0	0.793773	0.396886	−	0.917868i	$-0.370091\pi$
$641$	12882.0	0.793773	0.396886	+	0.917868i	$0.370091\pi$
$642$	0	0
$643$	−2465.00	−0.151182	−0.0755911	−	0.997139i	$-0.524084\pi$
$643$	−2465.00	−0.151182	−0.0755911	+	0.997139i	$0.524084\pi$
$644$	0	0
$645$	6650.00	0.405959
$646$	0	0
$647$	18696.0	1.13604	0.568018	−	0.823016i	$-0.307710\pi$
$647$	18696.0	1.13604	0.568018	+	0.823016i	$0.307710\pi$
$648$	0	0
$649$	2916.00	0.176368
$650$	0	0
$651$	196.000	0.0118001
$652$	0	0
$653$	−1542.00	−0.0924091	−0.0462045	−	0.998932i	$-0.514713\pi$
$653$	−1542.00	−0.0924091	−0.0462045	+	0.998932i	$0.514713\pi$
$654$	0	0
$655$	9570.00	0.570887
$656$	0	0
$657$	−9020.00	−0.535622
$658$	0	0
$659$	−2819.00	−0.166635	−0.0833176	−	0.996523i	$-0.526552\pi$
$659$	−2819.00	−0.166635	−0.0833176	+	0.996523i	$0.526552\pi$
$660$	0	0
$661$	30128.0	1.77283	0.886417	−	0.462887i	$-0.153187\pi$
$661$	30128.0	1.77283	0.886417	+	0.462887i	$0.153187\pi$
$662$	0	0
$663$	−6601.00	−0.386669
$664$	0	0
$665$	1190.00	0.0693928
$666$	0	0
$667$	−786.000	−0.0456282
$668$	0	0
$669$	−5565.00	−0.321607
$670$	0	0
$671$	1476.00	0.0849186
$672$	0	0
$673$	12520.0	0.717103	0.358552	−	0.933510i	$-0.383271\pi$
$673$	12520.0	0.717103	0.358552	+	0.933510i	$0.383271\pi$
$674$	0	0
$675$	−875.000	−0.0498945
$676$	0	0
$677$	8521.00	0.483735	0.241868	−	0.970309i	$-0.422240\pi$
$677$	8521.00	0.483735	0.241868	+	0.970309i	$0.422240\pi$
$678$	0	0
$679$	4949.00	0.279713
$680$	0	0
$681$	41293.0	2.32357
$682$	0	0
$683$	−16788.0	−0.940520	−0.470260	−	0.882528i	$-0.655840\pi$
$683$	−16788.0	−0.940520	−0.470260	+	0.882528i	$0.655840\pi$
$684$	0	0
$685$	10720.0	0.597942
$686$	0	0
$687$	17276.0	0.959419
$688$	0	0
$689$	−8464.00	−0.468001
$690$	0	0
$691$	−14508.0	−0.798712	−0.399356	−	0.916796i	$-0.630766\pi$
$691$	−14508.0	−0.798712	−0.399356	+	0.916796i	$0.630766\pi$
$692$	0	0
$693$	−1386.00	−0.0759737
$694$	0	0
$695$	11330.0	0.618376
$696$	0	0
$697$	−10660.0	−0.579306
$698$	0	0
$699$	−27720.0	−1.49995
$700$	0	0
$701$	17681.0	0.952642	0.476321	−	0.879271i	$-0.341970\pi$
$701$	17681.0	0.952642	0.476321	+	0.879271i	$0.341970\pi$
$702$	0	0
$703$	−884.000	−0.0474263
$704$	0	0
$705$	5845.00	0.312249
$706$	0	0
$707$	−9870.00	−0.525035
$708$	0	0
$709$	1093.00	0.0578963	0.0289481	−	0.999581i	$-0.490784\pi$
$709$	1093.00	0.0578963	0.0289481	+	0.999581i	$0.490784\pi$
$710$	0	0
$711$	−26642.0	−1.40528
$712$	0	0
$713$	−24.0000	−0.00126060
$714$	0	0
$715$	1035.00	0.0541354
$716$	0	0
$717$	−8281.00	−0.431324
$718$	0	0
$719$	−8870.00	−0.460077	−0.230038	−	0.973182i	$-0.573885\pi$
$719$	−8870.00	−0.460077	−0.230038	+	0.973182i	$0.573885\pi$
$720$	0	0
$721$	−5691.00	−0.293958
$722$	0	0
$723$	−20146.0	−1.03629
$724$	0	0
$725$	−3275.00	−0.167766
$726$	0	0
$727$	23368.0	1.19212	0.596060	−	0.802940i	$-0.296732\pi$
$727$	23368.0	1.19212	0.596060	+	0.802940i	$0.296732\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	0	0
$731$	−7790.00	−0.394150
$732$	0	0
$733$	−6667.00	−0.335950	−0.167975	−	0.985791i	$-0.553723\pi$
$733$	−6667.00	−0.335950	−0.167975	+	0.985791i	$0.553723\pi$
$734$	0	0
$735$	−1715.00	−0.0860663
$736$	0	0
$737$	1800.00	0.0899645
$738$	0	0
$739$	−5015.00	−0.249634	−0.124817	−	0.992180i	$-0.539834\pi$
$739$	−5015.00	−0.249634	−0.124817	+	0.992180i	$0.539834\pi$
$740$	0	0
$741$	−5474.00	−0.271380
$742$	0	0
$743$	15896.0	0.784882	0.392441	−	0.919777i	$-0.371631\pi$
$743$	15896.0	0.784882	0.392441	+	0.919777i	$0.371631\pi$
$744$	0	0
$745$	−9270.00	−0.455875
$746$	0	0
$747$	−24904.0	−1.21980
$748$	0	0
$749$	−7546.00	−0.368124
$750$	0	0
$751$	16533.0	0.803326	0.401663	−	0.915788i	$-0.368432\pi$
$751$	16533.0	0.803326	0.401663	+	0.915788i	$0.368432\pi$
$752$	0	0
$753$	38990.0	1.88695
$754$	0	0
$755$	−5185.00	−0.249936
$756$	0	0
$757$	16376.0	0.786256	0.393128	−	0.919484i	$-0.371393\pi$
$757$	16376.0	0.786256	0.393128	+	0.919484i	$0.371393\pi$
$758$	0	0
$759$	378.000	0.0180771
$760$	0	0
$761$	29482.0	1.40436	0.702182	−	0.711997i	$-0.252209\pi$
$761$	29482.0	1.40436	0.702182	+	0.711997i	$0.252209\pi$
$762$	0	0
$763$	−14945.0	−0.709103
$764$	0	0
$765$	−4510.00	−0.213150
$766$	0	0
$767$	−7452.00	−0.350816
$768$	0	0
$769$	−12942.0	−0.606893	−0.303446	−	0.952849i	$-0.598137\pi$
$769$	−12942.0	−0.606893	−0.303446	+	0.952849i	$0.598137\pi$
$770$	0	0
$771$	18158.0	0.848177
$772$	0	0
$773$	−11599.0	−0.539699	−0.269849	−	0.962903i	$-0.586974\pi$
$773$	−11599.0	−0.539699	−0.269849	+	0.962903i	$0.586974\pi$
$774$	0	0
$775$	−100.000	−0.00463498
$776$	0	0
$777$	1274.00	0.0588217
$778$	0	0
$779$	−8840.00	−0.406580
$780$	0	0
$781$	−7056.00	−0.323282
$782$	0	0
$783$	4585.00	0.209265
$784$	0	0
$785$	5290.00	0.240520
$786$	0	0
$787$	11987.0	0.542936	0.271468	−	0.962448i	$-0.412491\pi$
$787$	11987.0	0.542936	0.271468	+	0.962448i	$0.412491\pi$
$788$	0	0
$789$	−4466.00	−0.201513
$790$	0	0
$791$	−1106.00	−0.0497153
$792$	0	0
$793$	−3772.00	−0.168913
$794$	0	0
$795$	−12880.0	−0.574599
$796$	0	0
$797$	−40989.0	−1.82171	−0.910856	−	0.412724i	$-0.864577\pi$
$797$	−40989.0	−1.82171	−0.910856	+	0.412724i	$0.864577\pi$
$798$	0	0
$799$	−6847.00	−0.303166
$800$	0	0
$801$	−1584.00	−0.0698725
$802$	0	0
$803$	−3690.00	−0.162163
$804$	0	0
$805$	210.000	0.00919444
$806$	0	0
$807$	−7742.00	−0.337709
$808$	0	0
$809$	−8239.00	−0.358057	−0.179028	−	0.983844i	$-0.557295\pi$
$809$	−8239.00	−0.358057	−0.179028	+	0.983844i	$0.557295\pi$
$810$	0	0
$811$	18018.0	0.780145	0.390072	−	0.920784i	$-0.372450\pi$
$811$	18018.0	0.780145	0.390072	+	0.920784i	$0.372450\pi$
$812$	0	0
$813$	56588.0	2.44112
$814$	0	0
$815$	14050.0	0.603865
$816$	0	0
$817$	−6460.00	−0.276630
$818$	0	0
$819$	3542.00	0.151120
$820$	0	0
$821$	31155.0	1.32438	0.662191	−	0.749335i	$-0.269627\pi$
$821$	31155.0	1.32438	0.662191	+	0.749335i	$0.269627\pi$
$822$	0	0
$823$	−32860.0	−1.39177	−0.695886	−	0.718153i	$-0.744988\pi$
$823$	−32860.0	−1.39177	−0.695886	+	0.718153i	$0.744988\pi$
$824$	0	0
$825$	1575.00	0.0664660
$826$	0	0
$827$	−9486.00	−0.398864	−0.199432	−	0.979912i	$-0.563910\pi$
$827$	−9486.00	−0.398864	−0.199432	+	0.979912i	$0.563910\pi$
$828$	0	0
$829$	−5860.00	−0.245508	−0.122754	−	0.992437i	$-0.539173\pi$
$829$	−5860.00	−0.245508	−0.122754	+	0.992437i	$0.539173\pi$
$830$	0	0
$831$	19978.0	0.833970
$832$	0	0
$833$	2009.00	0.0835627
$834$	0	0
$835$	6975.00	0.289078
$836$	0	0
$837$	140.000	0.00578149
$838$	0	0
$839$	32454.0	1.33544	0.667721	−	0.744411i	$-0.267270\pi$
$839$	32454.0	1.33544	0.667721	+	0.744411i	$0.267270\pi$
$840$	0	0
$841$	−7228.00	−0.296363
$842$	0	0
$843$	−1155.00	−0.0471890
$844$	0	0
$845$	8340.00	0.339532
$846$	0	0
$847$	8750.00	0.354963
$848$	0	0
$849$	34811.0	1.40720
$850$	0	0
$851$	−156.000	−0.00628391
$852$	0	0
$853$	−45938.0	−1.84395	−0.921974	−	0.387252i	$-0.873424\pi$
$853$	−45938.0	−1.84395	−0.921974	+	0.387252i	$0.873424\pi$
$854$	0	0
$855$	−3740.00	−0.149597
$856$	0	0
$857$	−9930.00	−0.395802	−0.197901	−	0.980222i	$-0.563412\pi$
$857$	−9930.00	−0.395802	−0.197901	+	0.980222i	$0.563412\pi$
$858$	0	0
$859$	4628.00	0.183825	0.0919123	−	0.995767i	$-0.470702\pi$
$859$	4628.00	0.183825	0.0919123	+	0.995767i	$0.470702\pi$
$860$	0	0
$861$	12740.0	0.504272
$862$	0	0
$863$	46008.0	1.81475	0.907376	−	0.420320i	$-0.138082\pi$
$863$	46008.0	1.81475	0.907376	+	0.420320i	$0.138082\pi$
$864$	0	0
$865$	22225.0	0.873610
$866$	0	0
$867$	−22624.0	−0.886218
$868$	0	0
$869$	−10899.0	−0.425458
$870$	0	0
$871$	−4600.00	−0.178950
$872$	0	0
$873$	−15554.0	−0.603005
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	40074.0	1.54299	0.771495	−	0.636235i	$-0.219509\pi$
$877$	40074.0	1.54299	0.771495	+	0.636235i	$0.219509\pi$
$878$	0	0
$879$	−53151.0	−2.03952
$880$	0	0
$881$	−11744.0	−0.449109	−0.224555	−	0.974461i	$-0.572093\pi$
$881$	−11744.0	−0.449109	−0.224555	+	0.974461i	$0.572093\pi$
$882$	0	0
$883$	−14128.0	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−14128.0	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	−11340.0	−0.430723
$886$	0	0
$887$	−16336.0	−0.618387	−0.309193	−	0.950999i	$-0.600059\pi$
$887$	−16336.0	−0.618387	−0.309193	+	0.950999i	$0.600059\pi$
$888$	0	0
$889$	4228.00	0.159508
$890$	0	0
$891$	−7551.00	−0.283915
$892$	0	0
$893$	−5678.00	−0.212774
$894$	0	0
$895$	8700.00	0.324926
$896$	0	0
$897$	−966.000	−0.0359574
$898$	0	0
$899$	524.000	0.0194398
$900$	0	0
$901$	15088.0	0.557885
$902$	0	0
$903$	9310.00	0.343098
$904$	0	0
$905$	12440.0	0.456928
$906$	0	0
$907$	45202.0	1.65480	0.827402	−	0.561610i	$-0.189818\pi$
$907$	45202.0	1.65480	0.827402	+	0.561610i	$0.189818\pi$
$908$	0	0
$909$	31020.0	1.13187
$910$	0	0
$911$	35352.0	1.28569	0.642845	−	0.765996i	$-0.277754\pi$
$911$	35352.0	1.28569	0.642845	+	0.765996i	$0.277754\pi$
$912$	0	0
$913$	−10188.0	−0.369303
$914$	0	0
$915$	−5740.00	−0.207386
$916$	0	0
$917$	13398.0	0.482487
$918$	0	0
$919$	−7527.00	−0.270177	−0.135089	−	0.990834i	$-0.543132\pi$
$919$	−7527.00	−0.270177	−0.135089	+	0.990834i	$0.543132\pi$
$920$	0	0
$921$	57435.0	2.05488
$922$	0	0
$923$	18032.0	0.643045
$924$	0	0
$925$	−650.000	−0.0231047
$926$	0	0
$927$	17886.0	0.633714
$928$	0	0
$929$	28896.0	1.02050	0.510251	−	0.860025i	$-0.329552\pi$
$929$	28896.0	1.02050	0.510251	+	0.860025i	$0.329552\pi$
$930$	0	0
$931$	1666.00	0.0586476
$932$	0	0
$933$	10010.0	0.351246
$934$	0	0
$935$	−1845.00	−0.0645326
$936$	0	0
$937$	3013.00	0.105048	0.0525242	−	0.998620i	$-0.483273\pi$
$937$	3013.00	0.105048	0.0525242	+	0.998620i	$0.483273\pi$
$938$	0	0
$939$	−7567.00	−0.262982
$940$	0	0
$941$	51788.0	1.79409	0.897046	−	0.441937i	$-0.145709\pi$
$941$	51788.0	1.79409	0.897046	+	0.441937i	$0.145709\pi$
$942$	0	0
$943$	−1560.00	−0.0538713
$944$	0	0
$945$	−1225.00	−0.0421685
$946$	0	0
$947$	35600.0	1.22159	0.610794	−	0.791789i	$-0.290850\pi$
$947$	35600.0	1.22159	0.610794	+	0.791789i	$0.290850\pi$
$948$	0	0
$949$	9430.00	0.322561
$950$	0	0
$951$	−37282.0	−1.27124
$952$	0	0
$953$	−17800.0	−0.605035	−0.302518	−	0.953144i	$-0.597827\pi$
$953$	−17800.0	−0.605035	−0.302518	+	0.953144i	$0.597827\pi$
$954$	0	0
$955$	10265.0	0.347819
$956$	0	0
$957$	−8253.00	−0.278769
$958$	0	0
$959$	15008.0	0.505353
$960$	0	0
$961$	−29775.0	−0.999463
$962$	0	0
$963$	23716.0	0.793601
$964$	0	0
$965$	−2240.00	−0.0747235
$966$	0	0
$967$	20406.0	0.678607	0.339303	−	0.940677i	$-0.389809\pi$
$967$	20406.0	0.678607	0.339303	+	0.940677i	$0.389809\pi$
$968$	0	0
$969$	9758.00	0.323501
$970$	0	0
$971$	−25128.0	−0.830480	−0.415240	−	0.909712i	$-0.636302\pi$
$971$	−25128.0	−0.830480	−0.415240	+	0.909712i	$0.636302\pi$
$972$	0	0
$973$	15862.0	0.522623
$974$	0	0
$975$	−4025.00	−0.132208
$976$	0	0
$977$	−37470.0	−1.22699	−0.613496	−	0.789698i	$-0.710237\pi$
$977$	−37470.0	−1.22699	−0.613496	+	0.789698i	$0.710237\pi$
$978$	0	0
$979$	−648.000	−0.0211544
$980$	0	0
$981$	46970.0	1.52868
$982$	0	0
$983$	27507.0	0.892510	0.446255	−	0.894906i	$-0.352757\pi$
$983$	27507.0	0.892510	0.446255	+	0.894906i	$0.352757\pi$
$984$	0	0
$985$	16560.0	0.535681
$986$	0	0
$987$	8183.00	0.263898
$988$	0	0
$989$	−1140.00	−0.0366531
$990$	0	0
$991$	2960.00	0.0948814	0.0474407	−	0.998874i	$-0.484893\pi$
$991$	2960.00	0.0948814	0.0474407	+	0.998874i	$0.484893\pi$
$992$	0	0
$993$	−12124.0	−0.387456
$994$	0	0
$995$	5300.00	0.168866
$996$	0	0
$997$	−27769.0	−0.882099	−0.441050	−	0.897483i	$-0.645394\pi$
$997$	−27769.0	−0.882099	−0.441050	+	0.897483i	$0.645394\pi$
$998$	0	0
$999$	910.000	0.0288199

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.bg.1.1		1
4.3	odd	2		2240.4.a.e.1.1		1
8.3	odd	2		280.4.a.d.1.1	✓	1
8.5	even	2		560.4.a.d.1.1		1
40.3	even	4		1400.4.g.a.449.2		2
40.19	odd	2		1400.4.a.a.1.1		1
40.27	even	4		1400.4.g.a.449.1		2
56.27	even	2		1960.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.d.1.1	✓	1	8.3	odd	2
560.4.a.d.1.1		1	8.5	even	2
1400.4.a.a.1.1		1	40.19	odd	2
1400.4.g.a.449.1		2	40.27	even	4
1400.4.g.a.449.2		2	40.3	even	4
1960.4.a.b.1.1		1	56.27	even	2
2240.4.a.e.1.1		1	4.3	odd	2
2240.4.a.bg.1.1		1	1.1	even	1	trivial

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Hilbert	Bianchi

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

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2240.4.a.bg.1.1

Level	$2240$
Weight	$4$
Character	2240.1
Self dual	yes
Analytic conductor	$132.164$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$