LMFDB - Embedded newform 1280.4.d.v.641.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$75.5224448073$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.1
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.4.d.v.641.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.47214i q^{3} -5.00000i q^{5} +31.3050 q^{7} +7.00000 q^{9} +O(q^{10})$	$q-4.47214i q^{3} -5.00000i q^{5} +31.3050 q^{7} +7.00000 q^{9} -8.94427i q^{11} +62.0000i q^{13} -22.3607 q^{15} -46.0000 q^{17} -107.331i q^{19} -140.000i q^{21} -192.302 q^{23} -25.0000 q^{25} -152.053i q^{27} +90.0000i q^{29} -152.053 q^{31} -40.0000 q^{33} -156.525i q^{35} -214.000i q^{37} +277.272 q^{39} +10.0000 q^{41} -67.0820i q^{43} -35.0000i q^{45} +398.020 q^{47} +637.000 q^{49} +205.718i q^{51} -678.000i q^{53} -44.7214 q^{55} -480.000 q^{57} -411.437i q^{59} -250.000i q^{61} +219.135 q^{63} +310.000 q^{65} -49.1935i q^{67} +860.000i q^{69} +366.715 q^{71} -522.000 q^{73} +111.803i q^{75} -280.000i q^{77} +876.539 q^{79} -491.000 q^{81} -380.132i q^{83} +230.000i q^{85} +402.492 q^{87} -970.000 q^{89} +1940.91i q^{91} +680.000i q^{93} -536.656 q^{95} -934.000 q^{97} -62.6099i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{9}+O(q^{10}) $ 4 * q + 28 * q^9	$ 4 q + 28 q^{9} - 184 q^{17} - 100 q^{25} - 160 q^{33} + 40 q^{41} + 2548 q^{49} - 1920 q^{57} + 1240 q^{65} - 2088 q^{73} - 1964 q^{81} - 3880 q^{89} - 3736 q^{97}+O(q^{100}) $ 4 * q + 28 * q^9 - 184 * q^17 - 100 * q^25 - 160 * q^33 + 40 * q^41 + 2548 * q^49 - 1920 * q^57 + 1240 * q^65 - 2088 * q^73 - 1964 * q^81 - 3880 * q^89 - 3736 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 4.47214i	− 0.860663i	−0.902671	−	0.430331i	$-0.858397\pi$
$3$	− 4.47214i	− 0.860663i	0.902671	−	0.430331i	$-0.141603\pi$
$4$	0	0
$5$	− 5.00000i	− 0.447214i
$5$	− 5.00000i	− 0.447214i
$6$	0	0
$7$	31.3050	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	31.3050	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	0	0
$9$	7.00000	0.259259
$10$	0	0
$11$	− 8.94427i	− 0.245164i	−0.992458	−	0.122582i	$-0.960883\pi$
$11$	− 8.94427i	− 0.245164i	0.992458	−	0.122582i	$-0.0391174\pi$
$12$	0	0
$13$	62.0000i	1.32275i	0.750057	+	0.661373i	$0.230026\pi$
$13$	62.0000i	1.32275i	−0.750057	+	0.661373i	$0.769974\pi$
$14$	0	0
$15$	−22.3607	−0.384900
$16$	0	0
$17$	−46.0000	−0.656273	−0.328136	−	0.944630i	$-0.606421\pi$
$17$	−46.0000	−0.656273	−0.328136	+	0.944630i	$0.606421\pi$
$18$	0	0
$19$	− 107.331i	− 1.29597i	−0.761652	−	0.647986i	$-0.775611\pi$
$19$	− 107.331i	− 1.29597i	0.761652	−	0.647986i	$-0.224389\pi$
$20$	0	0
$21$	− 140.000i	− 1.45479i
$22$	0	0
$23$	−192.302	−1.74338	−0.871689	−	0.490059i	$-0.836975\pi$
$23$	−192.302	−1.74338	−0.871689	+	0.490059i	$0.836975\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	− 152.053i	− 1.08380i
$28$	0	0
$29$	90.0000i	0.576296i	0.957586	+	0.288148i	$0.0930395\pi$
$29$	90.0000i	0.576296i	−0.957586	+	0.288148i	$0.906961\pi$
$30$	0	0
$31$	−152.053	−0.880950	−0.440475	−	0.897765i	$-0.645190\pi$
$31$	−152.053	−0.880950	−0.440475	+	0.897765i	$0.645190\pi$
$32$	0	0
$33$	−40.0000	−0.211003
$34$	0	0
$35$	− 156.525i	− 0.755929i
$36$	0	0
$37$	− 214.000i	− 0.950848i	−0.879757	−	0.475424i	$-0.842295\pi$
$37$	− 214.000i	− 0.950848i	0.879757	−	0.475424i	$-0.157705\pi$
$38$	0	0
$39$	277.272	1.13844
$40$	0	0
$41$	10.0000	0.0380912	0.0190456	−	0.999819i	$-0.493937\pi$
$41$	10.0000	0.0380912	0.0190456	+	0.999819i	$0.493937\pi$
$42$	0	0
$43$	− 67.0820i	− 0.237905i	−0.992900	−	0.118953i	$-0.962046\pi$
$43$	− 67.0820i	− 0.237905i	0.992900	−	0.118953i	$-0.0379536\pi$
$44$	0	0
$45$	− 35.0000i	− 0.115944i
$46$	0	0
$47$	398.020	1.23526	0.617630	−	0.786469i	$-0.288093\pi$
$47$	398.020	1.23526	0.617630	+	0.786469i	$0.288093\pi$
$48$	0	0
$49$	637.000	1.85714
$50$	0	0
$51$	205.718i	0.564830i
$52$	0	0
$53$	− 678.000i	− 1.75718i	−0.477578	−	0.878589i	$-0.658485\pi$
$53$	− 678.000i	− 1.75718i	0.477578	−	0.878589i	$-0.341515\pi$
$54$	0	0
$55$	−44.7214	−0.109640
$56$	0	0
$57$	−480.000	−1.11540
$58$	0	0
$59$	− 411.437i	− 0.907872i	−0.891034	−	0.453936i	$-0.850019\pi$
$59$	− 411.437i	− 0.907872i	0.891034	−	0.453936i	$-0.149981\pi$
$60$	0	0
$61$	− 250.000i	− 0.524741i	−0.964967	−	0.262371i	$-0.915496\pi$
$61$	− 250.000i	− 0.524741i	0.964967	−	0.262371i	$-0.0845043\pi$
$62$	0	0
$63$	219.135	0.438228
$64$	0	0
$65$	310.000	0.591550
$66$	0	0
$67$	− 49.1935i	− 0.0897006i	−0.998994	−	0.0448503i	$-0.985719\pi$
$67$	− 49.1935i	− 0.0897006i	0.998994	−	0.0448503i	$-0.0142811\pi$
$68$	0	0
$69$	860.000i	1.50046i
$70$	0	0
$71$	366.715	0.612973	0.306486	−	0.951875i	$-0.400847\pi$
$71$	366.715	0.612973	0.306486	+	0.951875i	$0.400847\pi$
$72$	0	0
$73$	−522.000	−0.836924	−0.418462	−	0.908234i	$-0.637431\pi$
$73$	−522.000	−0.836924	−0.418462	+	0.908234i	$0.637431\pi$
$74$	0	0
$75$	111.803i	0.172133i
$76$	0	0
$77$	− 280.000i	− 0.414402i
$78$	0	0
$79$	876.539	1.24833	0.624166	−	0.781291i	$-0.285439\pi$
$79$	876.539	1.24833	0.624166	+	0.781291i	$0.285439\pi$
$80$	0	0
$81$	−491.000	−0.673525
$82$	0	0
$83$	− 380.132i	− 0.502709i	−0.967895	−	0.251355i	$-0.919124\pi$
$83$	− 380.132i	− 0.502709i	0.967895	−	0.251355i	$-0.0808760\pi$
$84$	0	0
$85$	230.000i	0.293494i
$86$	0	0
$87$	402.492	0.495997
$88$	0	0
$89$	−970.000	−1.15528	−0.577639	−	0.816292i	$-0.696026\pi$
$89$	−970.000	−1.15528	−0.577639	+	0.816292i	$0.696026\pi$
$90$	0	0
$91$	1940.91i	2.23585i
$92$	0	0
$93$	680.000i	0.758201i
$94$	0	0
$95$	−536.656	−0.579577
$96$	0	0
$97$	−934.000	−0.977663	−0.488832	−	0.872378i	$-0.662577\pi$
$97$	−934.000	−0.977663	−0.488832	+	0.872378i	$0.662577\pi$
$98$	0	0
$99$	− 62.6099i	− 0.0635609i
$100$	0	0
$101$	− 602.000i	− 0.593082i	−0.955020	−	0.296541i	$-0.904167\pi$
$101$	− 602.000i	− 0.593082i	0.955020	−	0.296541i	$-0.0958331\pi$
$102$	0	0
$103$	1829.10	1.74978	0.874888	−	0.484325i	$-0.160935\pi$
$103$	1829.10	1.74978	0.874888	+	0.484325i	$0.160935\pi$
$104$	0	0
$105$	−700.000	−0.650600
$106$	0	0
$107$	− 1525.00i	− 1.37782i	−0.724845	−	0.688912i	$-0.758089\pi$
$107$	− 1525.00i	− 1.37782i	0.724845	−	0.688912i	$-0.241911\pi$
$108$	0	0
$109$	− 2154.00i	− 1.89281i	−0.322989	−	0.946403i	$-0.604688\pi$
$109$	− 2154.00i	− 1.89281i	0.322989	−	0.946403i	$-0.395312\pi$
$110$	0	0
$111$	−957.037	−0.818360
$112$	0	0
$113$	−2182.00	−1.81651	−0.908254	−	0.418420i	$-0.862584\pi$
$113$	−2182.00	−1.81651	−0.908254	+	0.418420i	$0.862584\pi$
$114$	0	0
$115$	961.509i	0.779663i
$116$	0	0
$117$	434.000i	0.342934i
$118$	0	0
$119$	−1440.03	−1.10930
$120$	0	0
$121$	1251.00	0.939895
$122$	0	0
$123$	− 44.7214i	− 0.0327837i
$124$	0	0
$125$	125.000i	0.0894427i
$126$	0	0
$127$	1310.34	0.915539	0.457770	−	0.889071i	$-0.348648\pi$
$127$	1310.34	0.915539	0.457770	+	0.889071i	$0.348648\pi$
$128$	0	0
$129$	−300.000	−0.204756
$130$	0	0
$131$	205.718i	0.137204i	0.997644	+	0.0686019i	$0.0218538\pi$
$131$	205.718i	0.137204i	−0.997644	+	0.0686019i	$0.978146\pi$
$132$	0	0
$133$	− 3360.00i	− 2.19059i
$134$	0	0
$135$	−760.263	−0.484689
$136$	0	0
$137$	2094.00	1.30586	0.652929	−	0.757419i	$-0.273540\pi$
$137$	2094.00	1.30586	0.652929	+	0.757419i	$0.273540\pi$
$138$	0	0
$139$	− 1377.42i	− 0.840511i	−0.907406	−	0.420256i	$-0.861940\pi$
$139$	− 1377.42i	− 0.840511i	0.907406	−	0.420256i	$-0.138060\pi$
$140$	0	0
$141$	− 1780.00i	− 1.06314i
$142$	0	0
$143$	554.545	0.324289
$144$	0	0
$145$	450.000	0.257727
$146$	0	0
$147$	− 2848.75i	− 1.59837i
$148$	0	0
$149$	− 334.000i	− 0.183640i	−0.995776	−	0.0918200i	$-0.970732\pi$
$149$	− 334.000i	− 0.183640i	0.995776	−	0.0918200i	$-0.0292684\pi$
$150$	0	0
$151$	−3139.44	−1.69195	−0.845973	−	0.533225i	$-0.820980\pi$
$151$	−3139.44	−1.69195	−0.845973	+	0.533225i	$0.820980\pi$
$152$	0	0
$153$	−322.000	−0.170145
$154$	0	0
$155$	760.263i	0.393973i
$156$	0	0
$157$	− 834.000i	− 0.423952i	−0.977275	−	0.211976i	$-0.932010\pi$
$157$	− 834.000i	− 0.423952i	0.977275	−	0.211976i	$-0.0679898\pi$
$158$	0	0
$159$	−3032.11	−1.51234
$160$	0	0
$161$	−6020.00	−2.94685
$162$	0	0
$163$	3090.25i	1.48495i	0.669874	+	0.742475i	$0.266348\pi$
$163$	3090.25i	1.48495i	−0.669874	+	0.742475i	$0.733652\pi$
$164$	0	0
$165$	200.000i	0.0943635i
$166$	0	0
$167$	−4.47214	−0.00207224	−0.00103612	−	0.999999i	$-0.500330\pi$
$167$	−4.47214	−0.00207224	−0.00103612	+	0.999999i	$0.500330\pi$
$168$	0	0
$169$	−1647.00	−0.749659
$170$	0	0
$171$	− 751.319i	− 0.335993i
$172$	0	0
$173$	1838.00i	0.807749i	0.914814	+	0.403874i	$0.132337\pi$
$173$	1838.00i	0.807749i	−0.914814	+	0.403874i	$0.867663\pi$
$174$	0	0
$175$	−782.624	−0.338062
$176$	0	0
$177$	−1840.00	−0.781372
$178$	0	0
$179$	− 1842.52i	− 0.769365i	−0.923049	−	0.384683i	$-0.874311\pi$
$179$	− 1842.52i	− 0.769365i	0.923049	−	0.384683i	$-0.125689\pi$
$180$	0	0
$181$	1862.00i	0.764648i	0.924028	+	0.382324i	$0.124876\pi$
$181$	1862.00i	0.764648i	−0.924028	+	0.382324i	$0.875124\pi$
$182$	0	0
$183$	−1118.03	−0.451625
$184$	0	0
$185$	−1070.00	−0.425232
$186$	0	0
$187$	411.437i	0.160894i
$188$	0	0
$189$	− 4760.00i	− 1.83195i
$190$	0	0
$191$	−2066.13	−0.782721	−0.391360	−	0.920237i	$-0.627995\pi$
$191$	−2066.13	−0.782721	−0.391360	+	0.920237i	$0.627995\pi$
$192$	0	0
$193$	3378.00	1.25986	0.629932	−	0.776650i	$-0.283083\pi$
$193$	3378.00	1.25986	0.629932	+	0.776650i	$0.283083\pi$
$194$	0	0
$195$	− 1386.36i	− 0.509125i
$196$	0	0
$197$	66.0000i	0.0238696i	0.999929	+	0.0119348i	$0.00379905\pi$
$197$	66.0000i	0.0238696i	−0.999929	+	0.0119348i	$0.996201\pi$
$198$	0	0
$199$	1216.42	0.433316	0.216658	−	0.976248i	$-0.430484\pi$
$199$	1216.42	0.433316	0.216658	+	0.976248i	$0.430484\pi$
$200$	0	0
$201$	−220.000	−0.0772020
$202$	0	0
$203$	2817.45i	0.974118i
$204$	0	0
$205$	− 50.0000i	− 0.0170349i
$206$	0	0
$207$	−1346.11	−0.451987
$208$	0	0
$209$	−960.000	−0.317725
$210$	0	0
$211$	5286.06i	1.72468i	0.506329	+	0.862341i	$0.331002\pi$
$211$	5286.06i	1.72468i	−0.506329	+	0.862341i	$0.668998\pi$
$212$	0	0
$213$	− 1640.00i	− 0.527563i
$214$	0	0
$215$	−335.410	−0.106394
$216$	0	0
$217$	−4760.00	−1.48908
$218$	0	0
$219$	2334.45i	0.720310i
$220$	0	0
$221$	− 2852.00i	− 0.868083i
$222$	0	0
$223$	2965.03	0.890371	0.445186	−	0.895438i	$-0.353138\pi$
$223$	2965.03	0.890371	0.445186	+	0.895438i	$0.353138\pi$
$224$	0	0
$225$	−175.000	−0.0518519
$226$	0	0
$227$	4369.28i	1.27753i	0.769402	+	0.638765i	$0.220554\pi$
$227$	4369.28i	1.27753i	−0.769402	+	0.638765i	$0.779446\pi$
$228$	0	0
$229$	− 3250.00i	− 0.937843i	−0.883240	−	0.468921i	$-0.844643\pi$
$229$	− 3250.00i	− 0.937843i	0.883240	−	0.468921i	$-0.155357\pi$
$230$	0	0
$231$	−1252.20	−0.356661
$232$	0	0
$233$	−3298.00	−0.927293	−0.463646	−	0.886020i	$-0.653459\pi$
$233$	−3298.00	−0.927293	−0.463646	+	0.886020i	$0.653459\pi$
$234$	0	0
$235$	− 1990.10i	− 0.552425i
$236$	0	0
$237$	− 3920.00i	− 1.07439i
$238$	0	0
$239$	−554.545	−0.150086	−0.0750429	−	0.997180i	$-0.523909\pi$
$239$	−554.545	−0.150086	−0.0750429	+	0.997180i	$0.523909\pi$
$240$	0	0
$241$	5150.00	1.37652	0.688259	−	0.725465i	$-0.258375\pi$
$241$	5150.00	1.37652	0.688259	+	0.725465i	$0.258375\pi$
$242$	0	0
$243$	− 1909.60i	− 0.504119i
$244$	0	0
$245$	− 3185.00i	− 0.830540i
$246$	0	0
$247$	6654.54	1.71424
$248$	0	0
$249$	−1700.00	−0.432663
$250$	0	0
$251$	1386.36i	0.348631i	0.984690	+	0.174316i	$0.0557713\pi$
$251$	1386.36i	0.348631i	−0.984690	+	0.174316i	$0.944229\pi$
$252$	0	0
$253$	1720.00i	0.427413i
$254$	0	0
$255$	1028.59	0.252600
$256$	0	0
$257$	−4166.00	−1.01116	−0.505580	−	0.862780i	$-0.668721\pi$
$257$	−4166.00	−1.01116	−0.505580	+	0.862780i	$0.668721\pi$
$258$	0	0
$259$	− 6699.26i	− 1.60723i
$260$	0	0
$261$	630.000i	0.149410i
$262$	0	0
$263$	−961.509	−0.225434	−0.112717	−	0.993627i	$-0.535955\pi$
$263$	−961.509	−0.225434	−0.112717	+	0.993627i	$0.535955\pi$
$264$	0	0
$265$	−3390.00	−0.785834
$266$	0	0
$267$	4337.97i	0.994305i
$268$	0	0
$269$	1494.00i	0.338627i	0.985562	+	0.169314i	$0.0541551\pi$
$269$	1494.00i	0.338627i	−0.985562	+	0.169314i	$0.945845\pi$
$270$	0	0
$271$	5017.74	1.12474	0.562372	−	0.826884i	$-0.309889\pi$
$271$	5017.74	1.12474	0.562372	+	0.826884i	$0.309889\pi$
$272$	0	0
$273$	8680.00	1.92431
$274$	0	0
$275$	223.607i	0.0490327i
$276$	0	0
$277$	− 1006.00i	− 0.218212i	−0.994030	−	0.109106i	$-0.965201\pi$
$277$	− 1006.00i	− 0.218212i	0.994030	−	0.109106i	$-0.0347988\pi$
$278$	0	0
$279$	−1064.37	−0.228395
$280$	0	0
$281$	3210.00	0.681468	0.340734	−	0.940160i	$-0.389324\pi$
$281$	3210.00	0.681468	0.340734	+	0.940160i	$0.389324\pi$
$282$	0	0
$283$	3635.85i	0.763705i	0.924223	+	0.381853i	$0.124714\pi$
$283$	3635.85i	0.763705i	−0.924223	+	0.381853i	$0.875286\pi$
$284$	0	0
$285$	2400.00i	0.498820i
$286$	0	0
$287$	313.050	0.0643858
$288$	0	0
$289$	−2797.00	−0.569306
$290$	0	0
$291$	4176.97i	0.841439i
$292$	0	0
$293$	− 3622.00i	− 0.722183i	−0.932530	−	0.361091i	$-0.882404\pi$
$293$	− 3622.00i	− 0.722183i	0.932530	−	0.361091i	$-0.117596\pi$
$294$	0	0
$295$	−2057.18	−0.406013
$296$	0	0
$297$	−1360.00	−0.265708
$298$	0	0
$299$	− 11922.7i	− 2.30605i
$300$	0	0
$301$	− 2100.00i	− 0.402133i
$302$	0	0
$303$	−2692.23	−0.510443
$304$	0	0
$305$	−1250.00	−0.234671
$306$	0	0
$307$	− 2088.49i	− 0.388261i	−0.980976	−	0.194131i	$-0.937811\pi$
$307$	− 2088.49i	− 0.388261i	0.980976	−	0.194131i	$-0.0621886\pi$
$308$	0	0
$309$	− 8180.00i	− 1.50597i
$310$	0	0
$311$	8899.55	1.62266	0.811330	−	0.584589i	$-0.198744\pi$
$311$	8899.55	1.62266	0.811330	+	0.584589i	$0.198744\pi$
$312$	0	0
$313$	−8778.00	−1.58518	−0.792591	−	0.609754i	$-0.791268\pi$
$313$	−8778.00	−1.58518	−0.792591	+	0.609754i	$0.791268\pi$
$314$	0	0
$315$	− 1095.67i	− 0.195982i
$316$	0	0
$317$	5046.00i	0.894043i	0.894523	+	0.447021i	$0.147515\pi$
$317$	5046.00i	0.894043i	−0.894523	+	0.447021i	$0.852485\pi$
$318$	0	0
$319$	804.984	0.141287
$320$	0	0
$321$	−6820.00	−1.18584
$322$	0	0
$323$	4937.24i	0.850512i
$324$	0	0
$325$	− 1550.00i	− 0.264549i
$326$	0	0
$327$	−9632.98	−1.62907
$328$	0	0
$329$	12460.0	2.08797
$330$	0	0
$331$	− 313.050i	− 0.0519842i	−0.999662	−	0.0259921i	$-0.991726\pi$
$331$	− 313.050i	− 0.0519842i	0.999662	−	0.0259921i	$-0.00827447\pi$
$332$	0	0
$333$	− 1498.00i	− 0.246516i
$334$	0	0
$335$	−245.967	−0.0401153
$336$	0	0
$337$	−2574.00	−0.416067	−0.208034	−	0.978122i	$-0.566706\pi$
$337$	−2574.00	−0.416067	−0.208034	+	0.978122i	$0.566706\pi$
$338$	0	0
$339$	9758.20i	1.56340i
$340$	0	0
$341$	1360.00i	0.215977i
$342$	0	0
$343$	9203.66	1.44884
$344$	0	0
$345$	4300.00	0.671027
$346$	0	0
$347$	2643.03i	0.408892i	0.978878	+	0.204446i	$0.0655392\pi$
$347$	2643.03i	0.408892i	−0.978878	+	0.204446i	$0.934461\pi$
$348$	0	0
$349$	10170.0i	1.55985i	0.625873	+	0.779925i	$0.284743\pi$
$349$	10170.0i	1.55985i	−0.625873	+	0.779925i	$0.715257\pi$
$350$	0	0
$351$	9427.26	1.43359
$352$	0	0
$353$	−318.000	−0.0479474	−0.0239737	−	0.999713i	$-0.507632\pi$
$353$	−318.000	−0.0479474	−0.0239737	+	0.999713i	$0.507632\pi$
$354$	0	0
$355$	− 1833.58i	− 0.274130i
$356$	0	0
$357$	6440.00i	0.954737i
$358$	0	0
$359$	−12378.9	−1.81987	−0.909933	−	0.414755i	$-0.863867\pi$
$359$	−12378.9	−1.81987	−0.909933	+	0.414755i	$0.863867\pi$
$360$	0	0
$361$	−4661.00	−0.679545
$362$	0	0
$363$	− 5594.64i	− 0.808933i
$364$	0	0
$365$	2610.00i	0.374284i
$366$	0	0
$367$	−3072.36	−0.436991	−0.218496	−	0.975838i	$-0.570115\pi$
$367$	−3072.36	−0.436991	−0.218496	+	0.975838i	$0.570115\pi$
$368$	0	0
$369$	70.0000	0.00987549
$370$	0	0
$371$	− 21224.8i	− 2.97017i
$372$	0	0
$373$	− 3278.00i	− 0.455036i	−0.973774	−	0.227518i	$-0.926939\pi$
$373$	− 3278.00i	− 0.455036i	0.973774	−	0.227518i	$-0.0730610\pi$
$374$	0	0
$375$	559.017	0.0769800
$376$	0	0
$377$	−5580.00	−0.762293
$378$	0	0
$379$	5116.12i	0.693397i	0.937977	+	0.346699i	$0.112697\pi$
$379$	5116.12i	0.693397i	−0.937977	+	0.346699i	$0.887303\pi$
$380$	0	0
$381$	− 5860.00i	− 0.787971i
$382$	0	0
$383$	−1149.34	−0.153338	−0.0766690	−	0.997057i	$-0.524428\pi$
$383$	−1149.34	−0.153338	−0.0766690	+	0.997057i	$0.524428\pi$
$384$	0	0
$385$	−1400.00	−0.185326
$386$	0	0
$387$	− 469.574i	− 0.0616791i
$388$	0	0
$389$	834.000i	0.108703i	0.998522	+	0.0543515i	$0.0173092\pi$
$389$	834.000i	0.108703i	−0.998522	+	0.0543515i	$0.982691\pi$
$390$	0	0
$391$	8845.88	1.14413
$392$	0	0
$393$	920.000	0.118086
$394$	0	0
$395$	− 4382.69i	− 0.558271i
$396$	0	0
$397$	8734.00i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	8734.00i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	−15026.4	−1.88536
$400$	0	0
$401$	242.000	0.0301369	0.0150685	−	0.999886i	$-0.495203\pi$
$401$	242.000	0.0301369	0.0150685	+	0.999886i	$0.495203\pi$
$402$	0	0
$403$	− 9427.26i	− 1.16527i
$404$	0	0
$405$	2455.00i	0.301210i
$406$	0	0
$407$	−1914.07	−0.233113
$408$	0	0
$409$	6514.00	0.787522	0.393761	−	0.919213i	$-0.371174\pi$
$409$	6514.00	0.787522	0.393761	+	0.919213i	$0.371174\pi$
$410$	0	0
$411$	− 9364.65i	− 1.12390i
$412$	0	0
$413$	− 12880.0i	− 1.53458i
$414$	0	0
$415$	−1900.66	−0.224818
$416$	0	0
$417$	−6160.00	−0.723397
$418$	0	0
$419$	− 16081.8i	− 1.87505i	−0.347913	−	0.937527i	$-0.613110\pi$
$419$	− 16081.8i	− 1.87505i	0.347913	−	0.937527i	$-0.386890\pi$
$420$	0	0
$421$	7250.00i	0.839295i	0.907687	+	0.419648i	$0.137846\pi$
$421$	7250.00i	0.839295i	−0.907687	+	0.419648i	$0.862154\pi$
$422$	0	0
$423$	2786.14	0.320252
$424$	0	0
$425$	1150.00	0.131255
$426$	0	0
$427$	− 7826.24i	− 0.886975i
$428$	0	0
$429$	− 2480.00i	− 0.279104i
$430$	0	0
$431$	4981.96	0.556781	0.278390	−	0.960468i	$-0.410199\pi$
$431$	4981.96	0.556781	0.278390	+	0.960468i	$0.410199\pi$
$432$	0	0
$433$	11482.0	1.27434	0.637171	−	0.770723i	$-0.280105\pi$
$433$	11482.0	1.27434	0.637171	+	0.770723i	$0.280105\pi$
$434$	0	0
$435$	− 2012.46i	− 0.221816i
$436$	0	0
$437$	20640.0i	2.25937i
$438$	0	0
$439$	−3792.37	−0.412301	−0.206150	−	0.978520i	$-0.566094\pi$
$439$	−3792.37	−0.412301	−0.206150	+	0.978520i	$0.566094\pi$
$440$	0	0
$441$	4459.00	0.481481
$442$	0	0
$443$	− 746.847i	− 0.0800988i	−0.999198	−	0.0400494i	$-0.987248\pi$
$443$	− 746.847i	− 0.0800988i	0.999198	−	0.0400494i	$-0.0127515\pi$
$444$	0	0
$445$	4850.00i	0.516656i
$446$	0	0
$447$	−1493.69	−0.158052
$448$	0	0
$449$	−1306.00	−0.137269	−0.0686347	−	0.997642i	$-0.521864\pi$
$449$	−1306.00	−0.137269	−0.0686347	+	0.997642i	$0.521864\pi$
$450$	0	0
$451$	− 89.4427i	− 0.00933857i
$452$	0	0
$453$	14040.0i	1.45620i
$454$	0	0
$455$	9704.54	0.999902
$456$	0	0
$457$	9526.00	0.975071	0.487536	−	0.873103i	$-0.337896\pi$
$457$	9526.00	0.975071	0.487536	+	0.873103i	$0.337896\pi$
$458$	0	0
$459$	6994.42i	0.711267i
$460$	0	0
$461$	− 1518.00i	− 0.153363i	−0.997056	−	0.0766815i	$-0.975568\pi$
$461$	− 1518.00i	− 0.153363i	0.997056	−	0.0766815i	$-0.0244325\pi$
$462$	0	0
$463$	17293.7	1.73587	0.867936	−	0.496676i	$-0.165446\pi$
$463$	17293.7	1.73587	0.867936	+	0.496676i	$0.165446\pi$
$464$	0	0
$465$	3400.00	0.339078
$466$	0	0
$467$	16980.7i	1.68260i	0.540570	+	0.841299i	$0.318208\pi$
$467$	16980.7i	1.68260i	−0.540570	+	0.841299i	$0.681792\pi$
$468$	0	0
$469$	− 1540.00i	− 0.151622i
$470$	0	0
$471$	−3729.76	−0.364880
$472$	0	0
$473$	−600.000	−0.0583256
$474$	0	0
$475$	2683.28i	0.259195i
$476$	0	0
$477$	− 4746.00i	− 0.455565i
$478$	0	0
$479$	3810.26	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$479$	3810.26	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$480$	0	0
$481$	13268.0	1.25773
$482$	0	0
$483$	26922.3i	2.53624i
$484$	0	0
$485$	4670.00i	0.437224i
$486$	0	0
$487$	−1310.34	−0.121924	−0.0609620	−	0.998140i	$-0.519417\pi$
$487$	−1310.34	−0.121924	−0.0609620	+	0.998140i	$0.519417\pi$
$488$	0	0
$489$	13820.0	1.27804
$490$	0	0
$491$	2960.55i	0.272114i	0.990701	+	0.136057i	$0.0434430\pi$
$491$	2960.55i	0.272114i	−0.990701	+	0.136057i	$0.956557\pi$
$492$	0	0
$493$	− 4140.00i	− 0.378207i
$494$	0	0
$495$	−313.050	−0.0284253
$496$	0	0
$497$	11480.0	1.03611
$498$	0	0
$499$	19319.6i	1.73320i	0.499006	+	0.866598i	$0.333699\pi$
$499$	19319.6i	1.73320i	−0.499006	+	0.866598i	$0.666301\pi$
$500$	0	0
$501$	20.0000i	0.00178350i
$502$	0	0
$503$	−3072.36	−0.272345	−0.136173	−	0.990685i	$-0.543480\pi$
$503$	−3072.36	−0.272345	−0.136173	+	0.990685i	$0.543480\pi$
$504$	0	0
$505$	−3010.00	−0.265234
$506$	0	0
$507$	7365.61i	0.645203i
$508$	0	0
$509$	− 18550.0i	− 1.61535i	−0.589626	−	0.807676i	$-0.700725\pi$
$509$	− 18550.0i	− 1.61535i	0.589626	−	0.807676i	$-0.299275\pi$
$510$	0	0
$511$	−16341.2	−1.41466
$512$	0	0
$513$	−16320.0	−1.40457
$514$	0	0
$515$	− 9145.52i	− 0.782524i
$516$	0	0
$517$	− 3560.00i	− 0.302841i
$518$	0	0
$519$	8219.79	0.695200
$520$	0	0
$521$	2102.00	0.176757	0.0883784	−	0.996087i	$-0.471832\pi$
$521$	2102.00	0.176757	0.0883784	+	0.996087i	$0.471832\pi$
$522$	0	0
$523$	17696.2i	1.47955i	0.672856	+	0.739773i	$0.265067\pi$
$523$	17696.2i	1.47955i	−0.672856	+	0.739773i	$0.734933\pi$
$524$	0	0
$525$	3500.00i	0.290957i
$526$	0	0
$527$	6994.42	0.578144
$528$	0	0
$529$	24813.0	2.03937
$530$	0	0
$531$	− 2880.06i	− 0.235374i
$532$	0	0
$533$	620.000i	0.0503850i
$534$	0	0
$535$	−7624.99	−0.616182
$536$	0	0
$537$	−8240.00	−0.662164
$538$	0	0
$539$	− 5697.50i	− 0.455304i
$540$	0	0
$541$	9922.00i	0.788503i	0.919003	+	0.394251i	$0.128996\pi$
$541$	9922.00i	0.788503i	−0.919003	+	0.394251i	$0.871004\pi$
$542$	0	0
$543$	8327.12	0.658105
$544$	0	0
$545$	−10770.0	−0.846488
$546$	0	0
$547$	− 3716.34i	− 0.290493i	−0.989396	−	0.145246i	$-0.953603\pi$
$547$	− 3716.34i	− 0.290493i	0.989396	−	0.145246i	$-0.0463975\pi$
$548$	0	0
$549$	− 1750.00i	− 0.136044i
$550$	0	0
$551$	9659.81	0.746864
$552$	0	0
$553$	27440.0	2.11007
$554$	0	0
$555$	4785.19i	0.365982i
$556$	0	0
$557$	15094.0i	1.14821i	0.818781	+	0.574105i	$0.194650\pi$
$557$	15094.0i	1.14821i	−0.818781	+	0.574105i	$0.805350\pi$
$558$	0	0
$559$	4159.09	0.314688
$560$	0	0
$561$	1840.00	0.138476
$562$	0	0
$563$	− 5657.25i	− 0.423490i	−0.977325	−	0.211745i	$-0.932085\pi$
$563$	− 5657.25i	− 0.423490i	0.977325	−	0.211745i	$-0.0679146\pi$
$564$	0	0
$565$	10910.0i	0.812367i
$566$	0	0
$567$	−15370.7	−1.13847
$568$	0	0
$569$	5906.00	0.435136	0.217568	−	0.976045i	$-0.430188\pi$
$569$	5906.00	0.435136	0.217568	+	0.976045i	$0.430188\pi$
$570$	0	0
$571$	− 4892.52i	− 0.358573i	−0.983797	−	0.179287i	$-0.942621\pi$
$571$	− 4892.52i	− 0.358573i	0.983797	−	0.179287i	$-0.0573790\pi$
$572$	0	0
$573$	9240.00i	0.673659i
$574$	0	0
$575$	4807.55	0.348676
$576$	0	0
$577$	−13286.0	−0.958585	−0.479292	−	0.877655i	$-0.659107\pi$
$577$	−13286.0	−0.958585	−0.479292	+	0.877655i	$0.659107\pi$
$578$	0	0
$579$	− 15106.9i	− 1.08432i
$580$	0	0
$581$	− 11900.0i	− 0.849734i
$582$	0	0
$583$	−6064.22	−0.430796
$584$	0	0
$585$	2170.00	0.153365
$586$	0	0
$587$	9029.24i	0.634884i	0.948278	+	0.317442i	$0.102824\pi$
$587$	9029.24i	0.634884i	−0.948278	+	0.317442i	$0.897176\pi$
$588$	0	0
$589$	16320.0i	1.14169i
$590$	0	0
$591$	295.161	0.0205437
$592$	0	0
$593$	11442.0	0.792355	0.396178	−	0.918174i	$-0.370336\pi$
$593$	11442.0	0.792355	0.396178	+	0.918174i	$0.370336\pi$
$594$	0	0
$595$	7200.14i	0.496096i
$596$	0	0
$597$	− 5440.00i	− 0.372939i
$598$	0	0
$599$	14149.8	0.965187	0.482593	−	0.875845i	$-0.339695\pi$
$599$	14149.8	0.965187	0.482593	+	0.875845i	$0.339695\pi$
$600$	0	0
$601$	−3110.00	−0.211081	−0.105540	−	0.994415i	$-0.533657\pi$
$601$	−3110.00	−0.211081	−0.105540	+	0.994415i	$0.533657\pi$
$602$	0	0
$603$	− 344.354i	− 0.0232557i
$604$	0	0
$605$	− 6255.00i	− 0.420334i
$606$	0	0
$607$	−11193.8	−0.748502	−0.374251	−	0.927327i	$-0.622100\pi$
$607$	−11193.8	−0.748502	−0.374251	+	0.927327i	$0.622100\pi$
$608$	0	0
$609$	12600.0	0.838387
$610$	0	0
$611$	24677.2i	1.63394i
$612$	0	0
$613$	− 5342.00i	− 0.351976i	−0.984392	−	0.175988i	$-0.943688\pi$
$613$	− 5342.00i	− 0.351976i	0.984392	−	0.175988i	$-0.0563120\pi$
$614$	0	0
$615$	−223.607	−0.0146613
$616$	0	0
$617$	−19714.0	−1.28631	−0.643157	−	0.765734i	$-0.722376\pi$
$617$	−19714.0	−1.28631	−0.643157	+	0.765734i	$0.722376\pi$
$618$	0	0
$619$	− 13166.0i	− 0.854903i	−0.904038	−	0.427451i	$-0.859411\pi$
$619$	− 13166.0i	− 0.854903i	0.904038	−	0.427451i	$-0.140589\pi$
$620$	0	0
$621$	29240.0i	1.88947i
$622$	0	0
$623$	−30365.8	−1.95278
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4293.25i	0.273454i
$628$	0	0
$629$	9844.00i	0.624016i
$630$	0	0
$631$	−12262.6	−0.773639	−0.386820	−	0.922155i	$-0.626426\pi$
$631$	−12262.6	−0.773639	−0.386820	+	0.922155i	$0.626426\pi$
$632$	0	0
$633$	23640.0	1.48437
$634$	0	0
$635$	− 6551.68i	− 0.409442i
$636$	0	0
$637$	39494.0i	2.45653i
$638$	0	0
$639$	2567.01	0.158919
$640$	0	0
$641$	−2690.00	−0.165754	−0.0828772	−	0.996560i	$-0.526411\pi$
$641$	−2690.00	−0.165754	−0.0828772	+	0.996560i	$0.526411\pi$
$642$	0	0
$643$	− 12240.2i	− 0.750712i	−0.926881	−	0.375356i	$-0.877520\pi$
$643$	− 12240.2i	− 0.750712i	0.926881	−	0.375356i	$-0.122480\pi$
$644$	0	0
$645$	1500.00i	0.0915697i
$646$	0	0
$647$	17973.5	1.09214	0.546068	−	0.837741i	$-0.316124\pi$
$647$	17973.5	1.09214	0.546068	+	0.837741i	$0.316124\pi$
$648$	0	0
$649$	−3680.00	−0.222577
$650$	0	0
$651$	21287.4i	1.28159i
$652$	0	0
$653$	3478.00i	0.208430i	0.994555	+	0.104215i	$0.0332330\pi$
$653$	3478.00i	0.208430i	−0.994555	+	0.104215i	$0.966767\pi$
$654$	0	0
$655$	1028.59	0.0613594
$656$	0	0
$657$	−3654.00	−0.216980
$658$	0	0
$659$	10572.1i	0.624934i	0.949929	+	0.312467i	$0.101155\pi$
$659$	10572.1i	0.624934i	−0.949929	+	0.312467i	$0.898845\pi$
$660$	0	0
$661$	− 110.000i	− 0.00647277i	−0.999995	−	0.00323639i	$-0.998970\pi$
$661$	− 110.000i	− 0.00647277i	0.999995	−	0.00323639i	$-0.00103018\pi$
$662$	0	0
$663$	−12754.5	−0.747127
$664$	0	0
$665$	−16800.0	−0.979663
$666$	0	0
$667$	− 17307.2i	− 1.00470i
$668$	0	0
$669$	− 13260.0i	− 0.766310i
$670$	0	0
$671$	−2236.07	−0.128647
$672$	0	0
$673$	−14278.0	−0.817796	−0.408898	−	0.912580i	$-0.634087\pi$
$673$	−14278.0	−0.817796	−0.408898	+	0.912580i	$0.634087\pi$
$674$	0	0
$675$	3801.32i	0.216760i
$676$	0	0
$677$	18386.0i	1.04377i	0.853016	+	0.521884i	$0.174771\pi$
$677$	18386.0i	1.04377i	−0.853016	+	0.521884i	$0.825229\pi$
$678$	0	0
$679$	−29238.8	−1.65255
$680$	0	0
$681$	19540.0	1.09952
$682$	0	0
$683$	15317.1i	0.858113i	0.903278	+	0.429057i	$0.141154\pi$
$683$	15317.1i	0.858113i	−0.903278	+	0.429057i	$0.858846\pi$
$684$	0	0
$685$	− 10470.0i	− 0.583997i
$686$	0	0
$687$	−14534.4	−0.807167
$688$	0	0
$689$	42036.0	2.32430
$690$	0	0
$691$	− 9507.76i	− 0.523433i	−0.965145	−	0.261717i	$-0.915711\pi$
$691$	− 9507.76i	− 0.523433i	0.965145	−	0.261717i	$-0.0842886\pi$
$692$	0	0
$693$	− 1960.00i	− 0.107438i
$694$	0	0
$695$	−6887.09	−0.375888
$696$	0	0
$697$	−460.000	−0.0249982
$698$	0	0
$699$	14749.1i	0.798086i
$700$	0	0
$701$	15830.0i	0.852911i	0.904508	+	0.426456i	$0.140238\pi$
$701$	15830.0i	0.852911i	−0.904508	+	0.426456i	$0.859762\pi$
$702$	0	0
$703$	−22968.9	−1.23227
$704$	0	0
$705$	−8900.00	−0.475452
$706$	0	0
$707$	− 18845.6i	− 1.00249i
$708$	0	0
$709$	− 20050.0i	− 1.06205i	−0.847356	−	0.531025i	$-0.821807\pi$
$709$	− 20050.0i	− 1.06205i	0.847356	−	0.531025i	$-0.178193\pi$
$710$	0	0
$711$	6135.77	0.323642
$712$	0	0
$713$	29240.0	1.53583
$714$	0	0
$715$	− 2772.72i	− 0.145027i
$716$	0	0
$717$	2480.00i	0.129173i
$718$	0	0
$719$	21126.4	1.09580	0.547900	−	0.836544i	$-0.315427\pi$
$719$	21126.4	1.09580	0.547900	+	0.836544i	$0.315427\pi$
$720$	0	0
$721$	57260.0	2.95766
$722$	0	0
$723$	− 23031.5i	− 1.18472i
$724$	0	0
$725$	− 2250.00i	− 0.115259i
$726$	0	0
$727$	11336.9	0.578351	0.289175	−	0.957276i	$-0.406619\pi$
$727$	11336.9	0.578351	0.289175	+	0.957276i	$0.406619\pi$
$728$	0	0
$729$	−21797.0	−1.10740
$730$	0	0
$731$	3085.77i	0.156131i
$732$	0	0
$733$	17198.0i	0.866607i	0.901248	+	0.433303i	$0.142652\pi$
$733$	17198.0i	0.866607i	−0.901248	+	0.433303i	$0.857348\pi$
$734$	0	0
$735$	−14243.8	−0.714815
$736$	0	0
$737$	−440.000	−0.0219913
$738$	0	0
$739$	− 4597.36i	− 0.228845i	−0.993432	−	0.114423i	$-0.963498\pi$
$739$	− 4597.36i	− 0.228845i	0.993432	−	0.114423i	$-0.0365018\pi$
$740$	0	0
$741$	− 29760.0i	− 1.47539i
$742$	0	0
$743$	2419.43	0.119462	0.0597309	−	0.998215i	$-0.480976\pi$
$743$	2419.43	0.119462	0.0597309	+	0.998215i	$0.480976\pi$
$744$	0	0
$745$	−1670.00	−0.0821263
$746$	0	0
$747$	− 2660.92i	− 0.130332i
$748$	0	0
$749$	− 47740.0i	− 2.32895i
$750$	0	0
$751$	7432.69	0.361149	0.180574	−	0.983561i	$-0.442204\pi$
$751$	7432.69	0.361149	0.180574	+	0.983561i	$0.442204\pi$
$752$	0	0
$753$	6200.00	0.300054
$754$	0	0
$755$	15697.2i	0.756662i
$756$	0	0
$757$	11474.0i	0.550898i	0.961316	+	0.275449i	$0.0888265\pi$
$757$	11474.0i	0.550898i	−0.961316	+	0.275449i	$0.911174\pi$
$758$	0	0
$759$	7692.07	0.367858
$760$	0	0
$761$	−31802.0	−1.51488	−0.757439	−	0.652906i	$-0.773550\pi$
$761$	−31802.0	−1.51488	−0.757439	+	0.652906i	$0.773550\pi$
$762$	0	0
$763$	− 67430.9i	− 3.19942i
$764$	0	0
$765$	1610.00i	0.0760911i
$766$	0	0
$767$	25509.1	1.20089
$768$	0	0
$769$	−5310.00	−0.249003	−0.124502	−	0.992219i	$-0.539733\pi$
$769$	−5310.00	−0.249003	−0.124502	+	0.992219i	$0.539733\pi$
$770$	0	0
$771$	18630.9i	0.870267i
$772$	0	0
$773$	37938.0i	1.76525i	0.470082	+	0.882623i	$0.344224\pi$
$773$	37938.0i	1.76525i	−0.470082	+	0.882623i	$0.655776\pi$
$774$	0	0
$775$	3801.32	0.176190
$776$	0	0
$777$	−29960.0	−1.38328
$778$	0	0
$779$	− 1073.31i	− 0.0493651i
$780$	0	0
$781$	− 3280.00i	− 0.150279i
$782$	0	0
$783$	13684.7	0.624588
$784$	0	0
$785$	−4170.00	−0.189597
$786$	0	0
$787$	− 37633.0i	− 1.70454i	−0.523103	−	0.852270i	$-0.675226\pi$
$787$	− 37633.0i	− 1.70454i	0.523103	−	0.852270i	$-0.324774\pi$
$788$	0	0
$789$	4300.00i	0.194023i
$790$	0	0
$791$	−68307.4	−3.07046
$792$	0	0
$793$	15500.0	0.694100
$794$	0	0
$795$	15160.5i	0.676338i
$796$	0	0
$797$	17526.0i	0.778924i	0.921042	+	0.389462i	$0.127339\pi$
$797$	17526.0i	0.778924i	−0.921042	+	0.389462i	$0.872661\pi$
$798$	0	0
$799$	−18308.9	−0.810667
$800$	0	0
$801$	−6790.00	−0.299517
$802$	0	0
$803$	4668.91i	0.205183i
$804$	0	0
$805$	30100.0i	1.31787i
$806$	0	0
$807$	6681.37	0.291444
$808$	0	0
$809$	−8970.00	−0.389825	−0.194912	−	0.980821i	$-0.562442\pi$
$809$	−8970.00	−0.389825	−0.194912	+	0.980821i	$0.562442\pi$
$810$	0	0
$811$	− 3550.88i	− 0.153746i	−0.997041	−	0.0768731i	$-0.975506\pi$
$811$	− 3550.88i	− 0.153746i	0.997041	−	0.0768731i	$-0.0244936\pi$
$812$	0	0
$813$	− 22440.0i	− 0.968026i
$814$	0	0
$815$	15451.2	0.664090
$816$	0	0
$817$	−7200.00	−0.308318
$818$	0	0
$819$	13586.3i	0.579665i
$820$	0	0
$821$	− 15550.0i	− 0.661022i	−0.943802	−	0.330511i	$-0.892779\pi$
$821$	− 15550.0i	− 0.661022i	0.943802	−	0.330511i	$-0.107221\pi$
$822$	0	0
$823$	26712.1	1.13138	0.565689	−	0.824619i	$-0.308610\pi$
$823$	26712.1	1.13138	0.565689	+	0.824619i	$0.308610\pi$
$824$	0	0
$825$	1000.00	0.0422006
$826$	0	0
$827$	− 863.122i	− 0.0362923i	−0.999835	−	0.0181461i	$-0.994224\pi$
$827$	− 863.122i	− 0.0362923i	0.999835	−	0.0181461i	$-0.00577641\pi$
$828$	0	0
$829$	− 19066.0i	− 0.798781i	−0.916781	−	0.399391i	$-0.869222\pi$
$829$	− 19066.0i	− 0.798781i	0.916781	−	0.399391i	$-0.130778\pi$
$830$	0	0
$831$	−4498.97	−0.187807
$832$	0	0
$833$	−29302.0	−1.21879
$834$	0	0
$835$	22.3607i	0 0.000926734i
$836$	0	0
$837$	23120.0i	0.954772i
$838$	0	0
$839$	47744.5	1.96463	0.982315	−	0.187238i	$-0.0599534\pi$
$839$	47744.5	1.96463	0.982315	+	0.187238i	$0.0599534\pi$
$840$	0	0
$841$	16289.0	0.667883
$842$	0	0
$843$	− 14355.6i	− 0.586514i
$844$	0	0
$845$	8235.00i	0.335258i
$846$	0	0
$847$	39162.5	1.58871
$848$	0	0
$849$	16260.0	0.657293
$850$	0	0
$851$	41152.6i	1.65769i
$852$	0	0
$853$	− 14462.0i	− 0.580503i	−0.956950	−	0.290252i	$-0.906261\pi$
$853$	− 14462.0i	− 0.580503i	0.956950	−	0.290252i	$-0.0937390\pi$
$854$	0	0
$855$	−3756.59	−0.150261
$856$	0	0
$857$	−29346.0	−1.16971	−0.584854	−	0.811138i	$-0.698848\pi$
$857$	−29346.0	−1.16971	−0.584854	+	0.811138i	$0.698848\pi$
$858$	0	0
$859$	22807.9i	0.905932i	0.891528	+	0.452966i	$0.149634\pi$
$859$	22807.9i	0.905932i	−0.891528	+	0.452966i	$0.850366\pi$
$860$	0	0
$861$	− 1400.00i	− 0.0554145i
$862$	0	0
$863$	24753.3	0.976375	0.488187	−	0.872739i	$-0.337658\pi$
$863$	24753.3	0.976375	0.488187	+	0.872739i	$0.337658\pi$
$864$	0	0
$865$	9190.00	0.361236
$866$	0	0
$867$	12508.6i	0.489981i
$868$	0	0
$869$	− 7840.00i	− 0.306046i
$870$	0	0
$871$	3050.00	0.118651
$872$	0	0
$873$	−6538.00	−0.253468
$874$	0	0
$875$	3913.12i	0.151186i
$876$	0	0
$877$	32126.0i	1.23696i	0.785799	+	0.618482i	$0.212252\pi$
$877$	32126.0i	1.23696i	−0.785799	+	0.618482i	$0.787748\pi$
$878$	0	0
$879$	−16198.1	−0.621556
$880$	0	0
$881$	−33570.0	−1.28377	−0.641885	−	0.766801i	$-0.721848\pi$
$881$	−33570.0	−1.28377	−0.641885	+	0.766801i	$0.721848\pi$
$882$	0	0
$883$	6435.40i	0.245265i	0.992452	+	0.122632i	$0.0391336\pi$
$883$	6435.40i	0.245265i	−0.992452	+	0.122632i	$0.960866\pi$
$884$	0	0
$885$	9200.00i	0.349440i
$886$	0	0
$887$	46827.7	1.77263	0.886314	−	0.463084i	$-0.153257\pi$
$887$	46827.7	1.77263	0.886314	+	0.463084i	$0.153257\pi$
$888$	0	0
$889$	41020.0	1.54754
$890$	0	0
$891$	4391.64i	0.165124i
$892$	0	0
$893$	− 42720.0i	− 1.60086i
$894$	0	0
$895$	−9212.60	−0.344071
$896$	0	0
$897$	−53320.0	−1.98473
$898$	0	0
$899$	− 13684.7i	− 0.507688i
$900$	0	0
$901$	31188.0i	1.15319i
$902$	0	0
$903$	−9391.49	−0.346101
$904$	0	0
$905$	9310.00	0.341961
$906$	0	0
$907$	11980.9i	0.438608i	0.975657	+	0.219304i	$0.0703787\pi$
$907$	11980.9i	0.438608i	−0.975657	+	0.219304i	$0.929621\pi$
$908$	0	0
$909$	− 4214.00i	− 0.153762i
$910$	0	0
$911$	−24194.3	−0.879903	−0.439951	−	0.898022i	$-0.645004\pi$
$911$	−24194.3	−0.879903	−0.439951	+	0.898022i	$0.645004\pi$
$912$	0	0
$913$	−3400.00	−0.123246
$914$	0	0
$915$	5590.17i	0.201973i
$916$	0	0
$917$	6440.00i	0.231917i
$918$	0	0
$919$	37512.3	1.34648	0.673240	−	0.739424i	$-0.264902\pi$
$919$	37512.3	1.34648	0.673240	+	0.739424i	$0.264902\pi$
$920$	0	0
$921$	−9340.00	−0.334162
$922$	0	0
$923$	22736.3i	0.810808i
$924$	0	0
$925$	5350.00i	0.190170i
$926$	0	0
$927$	12803.7	0.453646
$928$	0	0
$929$	−21994.0	−0.776749	−0.388374	−	0.921502i	$-0.626963\pi$
$929$	−21994.0	−0.776749	−0.388374	+	0.921502i	$0.626963\pi$
$930$	0	0
$931$	− 68370.0i	− 2.40681i
$932$	0	0
$933$	− 39800.0i	− 1.39656i
$934$	0	0
$935$	2057.18	0.0719541
$936$	0	0
$937$	16286.0	0.567813	0.283906	−	0.958852i	$-0.408370\pi$
$937$	16286.0	0.567813	0.283906	+	0.958852i	$0.408370\pi$
$938$	0	0
$939$	39256.4i	1.36431i
$940$	0	0
$941$	− 24302.0i	− 0.841894i	−0.907085	−	0.420947i	$-0.861698\pi$
$941$	− 24302.0i	− 0.841894i	0.907085	−	0.420947i	$-0.138302\pi$
$942$	0	0
$943$	−1923.02	−0.0664073
$944$	0	0
$945$	−23800.0	−0.819274
$946$	0	0
$947$	− 19869.7i	− 0.681815i	−0.940097	−	0.340907i	$-0.889266\pi$
$947$	− 19869.7i	− 0.681815i	0.940097	−	0.340907i	$-0.110734\pi$
$948$	0	0
$949$	− 32364.0i	− 1.10704i
$950$	0	0
$951$	22566.4	0.769470
$952$	0	0
$953$	22422.0	0.762140	0.381070	−	0.924546i	$-0.375556\pi$
$953$	22422.0	0.762140	0.381070	+	0.924546i	$0.375556\pi$
$954$	0	0
$955$	10330.6i	0.350043i
$956$	0	0
$957$	− 3600.00i	− 0.121600i
$958$	0	0
$959$	65552.6	2.20730
$960$	0	0
$961$	−6671.00	−0.223927
$962$	0	0
$963$	− 10675.0i	− 0.357214i
$964$	0	0
$965$	− 16890.0i	− 0.563428i
$966$	0	0
$967$	−43777.7	−1.45584	−0.727920	−	0.685662i	$-0.759513\pi$
$967$	−43777.7	−1.45584	−0.727920	+	0.685662i	$0.759513\pi$
$968$	0	0
$969$	22080.0	0.732004
$970$	0	0
$971$	− 25714.8i	− 0.849873i	−0.905223	−	0.424936i	$-0.860296\pi$
$971$	− 25714.8i	− 0.849873i	0.905223	−	0.424936i	$-0.139704\pi$
$972$	0	0
$973$	− 43120.0i	− 1.42072i
$974$	0	0
$975$	−6931.81	−0.227688
$976$	0	0
$977$	28986.0	0.949175	0.474588	−	0.880208i	$-0.342597\pi$
$977$	28986.0	0.949175	0.474588	+	0.880208i	$0.342597\pi$
$978$	0	0
$979$	8675.94i	0.283232i
$980$	0	0
$981$	− 15078.0i	− 0.490727i
$982$	0	0
$983$	−32123.4	−1.04229	−0.521147	−	0.853467i	$-0.674496\pi$
$983$	−32123.4	−1.04229	−0.521147	+	0.853467i	$0.674496\pi$
$984$	0	0
$985$	330.000	0.0106748
$986$	0	0
$987$	− 55722.8i	− 1.79704i
$988$	0	0
$989$	12900.0i	0.414758i
$990$	0	0
$991$	−11994.3	−0.384471	−0.192235	−	0.981349i	$-0.561574\pi$
$991$	−11994.3	−0.384471	−0.192235	+	0.981349i	$0.561574\pi$
$992$	0	0
$993$	−1400.00	−0.0447408
$994$	0	0
$995$	− 6082.10i	− 0.193785i
$996$	0	0
$997$	− 406.000i	− 0.0128968i	−0.999979	−	0.00644842i	$-0.997947\pi$
$997$	− 406.000i	− 0.0128968i	0.999979	−	0.00644842i	$-0.00205261\pi$
$998$	0	0
$999$	−32539.3	−1.03053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.d.v.641.1		4
4.3	odd	2	inner	1280.4.d.v.641.3		4
8.3	odd	2	inner	1280.4.d.v.641.2		4
8.5	even	2	inner	1280.4.d.v.641.4		4
16.3	odd	4		160.4.a.d.1.1	✓	2
16.5	even	4		320.4.a.q.1.1		2
16.11	odd	4		320.4.a.q.1.2		2
16.13	even	4		160.4.a.d.1.2	yes	2
48.29	odd	4		1440.4.a.bb.1.1		2
48.35	even	4		1440.4.a.bb.1.2		2
80.3	even	4		800.4.c.l.449.2		4
80.13	odd	4		800.4.c.l.449.3		4
80.19	odd	4		800.4.a.o.1.2		2
80.29	even	4		800.4.a.o.1.1		2
80.59	odd	4		1600.4.a.cg.1.1		2
80.67	even	4		800.4.c.l.449.4		4
80.69	even	4		1600.4.a.cg.1.2		2
80.77	odd	4		800.4.c.l.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.d.1.1	✓	2	16.3	odd	4
160.4.a.d.1.2	yes	2	16.13	even	4
320.4.a.q.1.1		2	16.5	even	4
320.4.a.q.1.2		2	16.11	odd	4
800.4.a.o.1.1		2	80.29	even	4
800.4.a.o.1.2		2	80.19	odd	4
800.4.c.l.449.1		4	80.77	odd	4
800.4.c.l.449.2		4	80.3	even	4
800.4.c.l.449.3		4	80.13	odd	4
800.4.c.l.449.4		4	80.67	even	4
1280.4.d.v.641.1		4	1.1	even	1	trivial
1280.4.d.v.641.2		4	8.3	odd	2	inner
1280.4.d.v.641.3		4	4.3	odd	2	inner
1280.4.d.v.641.4		4	8.5	even	2	inner
1440.4.a.bb.1.1		2	48.29	odd	4
1440.4.a.bb.1.2		2	48.35	even	4
1600.4.a.cg.1.1		2	80.59	odd	4
1600.4.a.cg.1.2		2	80.69	even	4

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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.47214i q^{3} -5.00000i q^{5} +31.3050 q^{7} +7.00000 q^{9} +O(q^{10})\)	\(q-4.47214i q^{3} -5.00000i q^{5} +31.3050 q^{7} +7.00000 q^{9} -8.94427i q^{11} +62.0000i q^{13} -22.3607 q^{15} -46.0000 q^{17} -107.331i q^{19} -140.000i q^{21} -192.302 q^{23} -25.0000 q^{25} -152.053i q^{27} +90.0000i q^{29} -152.053 q^{31} -40.0000 q^{33} -156.525i q^{35} -214.000i q^{37} +277.272 q^{39} +10.0000 q^{41} -67.0820i q^{43} -35.0000i q^{45} +398.020 q^{47} +637.000 q^{49} +205.718i q^{51} -678.000i q^{53} -44.7214 q^{55} -480.000 q^{57} -411.437i q^{59} -250.000i q^{61} +219.135 q^{63} +310.000 q^{65} -49.1935i q^{67} +860.000i q^{69} +366.715 q^{71} -522.000 q^{73} +111.803i q^{75} -280.000i q^{77} +876.539 q^{79} -491.000 q^{81} -380.132i q^{83} +230.000i q^{85} +402.492 q^{87} -970.000 q^{89} +1940.91i q^{91} +680.000i q^{93} -536.656 q^{95} -934.000 q^{97} -62.6099i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{9}+O(q^{10}) \) 4 * q + 28 * q^9	\( 4 q + 28 q^{9} - 184 q^{17} - 100 q^{25} - 160 q^{33} + 40 q^{41} + 2548 q^{49} - 1920 q^{57} + 1240 q^{65} - 2088 q^{73} - 1964 q^{81} - 3880 q^{89} - 3736 q^{97}+O(q^{100}) \) 4 * q + 28 * q^9 - 184 * q^17 - 100 * q^25 - 160 * q^33 + 40 * q^41 + 2548 * q^49 - 1920 * q^57 + 1240 * q^65 - 2088 * q^73 - 1964 * q^81 - 3880 * q^89 - 3736 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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