LMFDB - Embedded newform 1280.4.d.i.641.2

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.4.d.i.641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000i q^{3} +5.00000i q^{5} +2.00000 q^{7} -37.0000 q^{9} -22.0000i q^{11} -10.0000i q^{13} -40.0000 q^{15} +10.0000 q^{17} +110.000i q^{19} +16.0000i q^{21} -154.000 q^{23} -25.0000 q^{25} -80.0000i q^{27} -222.000i q^{29} +92.0000 q^{31} +176.000 q^{33} +10.0000i q^{35} -34.0000i q^{37} +80.0000 q^{39} -398.000 q^{41} -268.000i q^{43} -185.000i q^{45} +10.0000 q^{47} -339.000 q^{49} +80.0000i q^{51} -582.000i q^{53} +110.000 q^{55} -880.000 q^{57} -746.000i q^{59} +226.000i q^{61} -74.0000 q^{63} +50.0000 q^{65} -172.000i q^{67} -1232.00i q^{69} +928.000 q^{71} -570.000 q^{73} -200.000i q^{75} -44.0000i q^{77} -64.0000 q^{79} -359.000 q^{81} -864.000i q^{83} +50.0000i q^{85} +1776.00 q^{87} +874.000 q^{89} -20.0000i q^{91} +736.000i q^{93} -550.000 q^{95} +306.000 q^{97} +814.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} - 74 q^{9} - 80 q^{15} + 20 q^{17} - 308 q^{23} - 50 q^{25} + 184 q^{31} + 352 q^{33} + 160 q^{39} - 796 q^{41} + 20 q^{47} - 678 q^{49} + 220 q^{55} - 1760 q^{57} - 148 q^{63} + 100 q^{65}+ \cdots + 612 q^{97}+O(q^{100}) $ 2 * q + 4 * q^7 - 74 * q^9 - 80 * q^15 + 20 * q^17 - 308 * q^23 - 50 * q^25 + 184 * q^31 + 352 * q^33 + 160 * q^39 - 796 * q^41 + 20 * q^47 - 678 * q^49 + 220 * q^55 - 1760 * q^57 - 148 * q^63 + 100 * q^65 + 1856 * q^71 - 1140 * q^73 - 128 * q^79 - 718 * q^81 + 3552 * q^87 + 1748 * q^89 - 1100 * q^95 + 612 * q^97

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
$3$	8.00000i	1.53960i	0.638285	+	0.769800i	$0.279644\pi$
$3$	8.00000i	1.53960i	−0.638285	+	0.769800i	$0.720356\pi$
$4$	0	0
$5$	5.00000i	0.447214i
$5$	5.00000i	0.447214i
$6$	0	0
$7$	2.00000	0.107990	0.0539949	−	0.998541i	$-0.482805\pi$
$7$	2.00000	0.107990	0.0539949	+	0.998541i	$0.482805\pi$
$8$	0	0
$9$	−37.0000	−1.37037
$10$	0	0
$11$	− 22.0000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 22.0000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	− 10.0000i	− 0.213346i	−0.994294	−	0.106673i	$-0.965980\pi$
$13$	− 10.0000i	− 0.213346i	0.994294	−	0.106673i	$-0.0340198\pi$
$14$	0	0
$15$	−40.0000	−0.688530
$16$	0	0
$17$	10.0000	0.142668	0.0713340	−	0.997452i	$-0.477274\pi$
$17$	10.0000	0.142668	0.0713340	+	0.997452i	$0.477274\pi$
$18$	0	0
$19$	110.000i	1.32820i	0.747645	+	0.664098i	$0.231184\pi$
$19$	110.000i	1.32820i	−0.747645	+	0.664098i	$0.768816\pi$
$20$	0	0
$21$	16.0000i	0.166261i
$22$	0	0
$23$	−154.000	−1.39614	−0.698070	−	0.716030i	$-0.745958\pi$
$23$	−154.000	−1.39614	−0.698070	+	0.716030i	$0.745958\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	− 80.0000i	− 0.570222i
$28$	0	0
$29$	− 222.000i	− 1.42153i	−0.703430	−	0.710765i	$-0.748349\pi$
$29$	− 222.000i	− 1.42153i	0.703430	−	0.710765i	$-0.251651\pi$
$30$	0	0
$31$	92.0000	0.533022	0.266511	−	0.963832i	$-0.414129\pi$
$31$	92.0000	0.533022	0.266511	+	0.963832i	$0.414129\pi$
$32$	0	0
$33$	176.000	0.928414
$34$	0	0
$35$	10.0000i	0.0482945i
$36$	0	0
$37$	− 34.0000i	− 0.151069i	−0.997143	−	0.0755347i	$-0.975934\pi$
$37$	− 34.0000i	− 0.151069i	0.997143	−	0.0755347i	$-0.0240664\pi$
$38$	0	0
$39$	80.0000	0.328468
$40$	0	0
$41$	−398.000	−1.51603	−0.758014	−	0.652238i	$-0.773830\pi$
$41$	−398.000	−1.51603	−0.758014	+	0.652238i	$0.773830\pi$
$42$	0	0
$43$	− 268.000i	− 0.950456i	−0.879863	−	0.475228i	$-0.842366\pi$
$43$	− 268.000i	− 0.950456i	0.879863	−	0.475228i	$-0.157634\pi$
$44$	0	0
$45$	− 185.000i	− 0.612848i
$46$	0	0
$47$	10.0000	0.0310351	0.0155176	−	0.999880i	$-0.495060\pi$
$47$	10.0000	0.0310351	0.0155176	+	0.999880i	$0.495060\pi$
$48$	0	0
$49$	−339.000	−0.988338
$50$	0	0
$51$	80.0000i	0.219652i
$52$	0	0
$53$	− 582.000i	− 1.50837i	−0.656659	−	0.754187i	$-0.728031\pi$
$53$	− 582.000i	− 1.50837i	0.656659	−	0.754187i	$-0.271969\pi$
$54$	0	0
$55$	110.000	0.269680
$56$	0	0
$57$	−880.000	−2.04489
$58$	0	0
$59$	− 746.000i	− 1.64612i	−0.567956	−	0.823059i	$-0.692266\pi$
$59$	− 746.000i	− 1.64612i	0.567956	−	0.823059i	$-0.307734\pi$
$60$	0	0
$61$	226.000i	0.474366i	0.971465	+	0.237183i	$0.0762241\pi$
$61$	226.000i	0.474366i	−0.971465	+	0.237183i	$0.923776\pi$
$62$	0	0
$63$	−74.0000	−0.147986
$64$	0	0
$65$	50.0000	0.0954113
$66$	0	0
$67$	− 172.000i	− 0.313629i	−0.987628	−	0.156815i	$-0.949878\pi$
$67$	− 172.000i	− 0.313629i	0.987628	−	0.156815i	$-0.0501225\pi$
$68$	0	0
$69$	− 1232.00i	− 2.14950i
$70$	0	0
$71$	928.000	1.55117	0.775587	−	0.631241i	$-0.217454\pi$
$71$	928.000	1.55117	0.775587	+	0.631241i	$0.217454\pi$
$72$	0	0
$73$	−570.000	−0.913883	−0.456941	−	0.889497i	$-0.651055\pi$
$73$	−570.000	−0.913883	−0.456941	+	0.889497i	$0.651055\pi$
$74$	0	0
$75$	− 200.000i	− 0.307920i
$76$	0	0
$77$	− 44.0000i	− 0.0651203i
$78$	0	0
$79$	−64.0000	−0.0911464	−0.0455732	−	0.998961i	$-0.514511\pi$
$79$	−64.0000	−0.0911464	−0.0455732	+	0.998961i	$0.514511\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	− 864.000i	− 1.14261i	−0.820739	−	0.571303i	$-0.806438\pi$
$83$	− 864.000i	− 1.14261i	0.820739	−	0.571303i	$-0.193562\pi$
$84$	0	0
$85$	50.0000i	0.0638031i
$86$	0	0
$87$	1776.00	2.18859
$88$	0	0
$89$	874.000	1.04094	0.520471	−	0.853879i	$-0.325756\pi$
$89$	874.000	1.04094	0.520471	+	0.853879i	$0.325756\pi$
$90$	0	0
$91$	− 20.0000i	− 0.0230392i
$92$	0	0
$93$	736.000i	0.820641i
$94$	0	0
$95$	−550.000	−0.593987
$96$	0	0
$97$	306.000	0.320305	0.160153	−	0.987092i	$-0.448801\pi$
$97$	306.000	0.320305	0.160153	+	0.987092i	$0.448801\pi$
$98$	0	0
$99$	814.000i	0.826364i
$100$	0	0
$101$	− 930.000i	− 0.916222i	−0.888895	−	0.458111i	$-0.848526\pi$
$101$	− 930.000i	− 0.916222i	0.888895	−	0.458111i	$-0.151474\pi$
$102$	0	0
$103$	462.000	0.441963	0.220982	−	0.975278i	$-0.429074\pi$
$103$	462.000	0.441963	0.220982	+	0.975278i	$0.429074\pi$
$104$	0	0
$105$	−80.0000	−0.0743543
$106$	0	0
$107$	1560.00i	1.40945i	0.709482	+	0.704724i	$0.248929\pi$
$107$	1560.00i	1.40945i	−0.709482	+	0.704724i	$0.751071\pi$
$108$	0	0
$109$	1186.00i	1.04219i	0.853500	+	0.521093i	$0.174475\pi$
$109$	1186.00i	1.04219i	−0.853500	+	0.521093i	$0.825525\pi$
$110$	0	0
$111$	272.000	0.232586
$112$	0	0
$113$	−1382.00	−1.15051	−0.575255	−	0.817974i	$-0.695097\pi$
$113$	−1382.00	−1.15051	−0.575255	+	0.817974i	$0.695097\pi$
$114$	0	0
$115$	− 770.000i	− 0.624373i
$116$	0	0
$117$	370.000i	0.292363i
$118$	0	0
$119$	20.0000	0.0154067
$120$	0	0
$121$	847.000	0.636364
$122$	0	0
$123$	− 3184.00i	− 2.33408i
$124$	0	0
$125$	− 125.000i	− 0.0894427i
$126$	0	0
$127$	−1198.00	−0.837050	−0.418525	−	0.908205i	$-0.637453\pi$
$127$	−1198.00	−0.837050	−0.418525	+	0.908205i	$0.637453\pi$
$128$	0	0
$129$	2144.00	1.46332
$130$	0	0
$131$	2314.00i	1.54332i	0.636034	+	0.771661i	$0.280574\pi$
$131$	2314.00i	1.54332i	−0.636034	+	0.771661i	$0.719426\pi$
$132$	0	0
$133$	220.000i	0.143432i
$134$	0	0
$135$	400.000	0.255011
$136$	0	0
$137$	−2394.00	−1.49294	−0.746472	−	0.665417i	$-0.768254\pi$
$137$	−2394.00	−1.49294	−0.746472	+	0.665417i	$0.768254\pi$
$138$	0	0
$139$	− 826.000i	− 0.504032i	−0.967723	−	0.252016i	$-0.918906\pi$
$139$	− 826.000i	− 0.504032i	0.967723	−	0.252016i	$-0.0810935\pi$
$140$	0	0
$141$	80.0000i	0.0477817i
$142$	0	0
$143$	−220.000	−0.128653
$144$	0	0
$145$	1110.00	0.635727
$146$	0	0
$147$	− 2712.00i	− 1.52165i
$148$	0	0
$149$	− 2298.00i	− 1.26349i	−0.775178	−	0.631743i	$-0.782340\pi$
$149$	− 2298.00i	− 1.26349i	0.775178	−	0.631743i	$-0.217660\pi$
$150$	0	0
$151$	2120.00	1.14254	0.571269	−	0.820763i	$-0.306451\pi$
$151$	2120.00	1.14254	0.571269	+	0.820763i	$0.306451\pi$
$152$	0	0
$153$	−370.000	−0.195508
$154$	0	0
$155$	460.000i	0.238375i
$156$	0	0
$157$	− 838.000i	− 0.425985i	−0.977054	−	0.212993i	$-0.931679\pi$
$157$	− 838.000i	− 0.425985i	0.977054	−	0.212993i	$-0.0683210\pi$
$158$	0	0
$159$	4656.00	2.32229
$160$	0	0
$161$	−308.000	−0.150769
$162$	0	0
$163$	1440.00i	0.691960i	0.938242	+	0.345980i	$0.112454\pi$
$163$	1440.00i	0.691960i	−0.938242	+	0.345980i	$0.887546\pi$
$164$	0	0
$165$	880.000i	0.415199i
$166$	0	0
$167$	2122.00	0.983265	0.491633	−	0.870803i	$-0.336400\pi$
$167$	2122.00	0.983265	0.491633	+	0.870803i	$0.336400\pi$
$168$	0	0
$169$	2097.00	0.954483
$170$	0	0
$171$	− 4070.00i	− 1.82012i
$172$	0	0
$173$	3306.00i	1.45289i	0.687223	+	0.726447i	$0.258830\pi$
$173$	3306.00i	1.45289i	−0.687223	+	0.726447i	$0.741170\pi$
$174$	0	0
$175$	−50.0000	−0.0215980
$176$	0	0
$177$	5968.00	2.53436
$178$	0	0
$179$	− 2746.00i	− 1.14662i	−0.819337	−	0.573312i	$-0.805658\pi$
$179$	− 2746.00i	− 1.14662i	0.819337	−	0.573312i	$-0.194342\pi$
$180$	0	0
$181$	4062.00i	1.66810i	0.551689	+	0.834050i	$0.313984\pi$
$181$	4062.00i	1.66810i	−0.551689	+	0.834050i	$0.686016\pi$
$182$	0	0
$183$	−1808.00	−0.730334
$184$	0	0
$185$	170.000	0.0675603
$186$	0	0
$187$	− 220.000i	− 0.0860320i
$188$	0	0
$189$	− 160.000i	− 0.0615782i
$190$	0	0
$191$	1908.00	0.722817	0.361408	−	0.932408i	$-0.382296\pi$
$191$	1908.00	0.722817	0.361408	+	0.932408i	$0.382296\pi$
$192$	0	0
$193$	−2758.00	−1.02863	−0.514314	−	0.857602i	$-0.671953\pi$
$193$	−2758.00	−1.02863	−0.514314	+	0.857602i	$0.671953\pi$
$194$	0	0
$195$	400.000i	0.146895i
$196$	0	0
$197$	4322.00i	1.56310i	0.623846	+	0.781548i	$0.285569\pi$
$197$	4322.00i	1.56310i	−0.623846	+	0.781548i	$0.714431\pi$
$198$	0	0
$199$	−2852.00	−1.01594	−0.507972	−	0.861373i	$-0.669605\pi$
$199$	−2852.00	−1.01594	−0.507972	+	0.861373i	$0.669605\pi$
$200$	0	0
$201$	1376.00	0.482863
$202$	0	0
$203$	− 444.000i	− 0.153511i
$204$	0	0
$205$	− 1990.00i	− 0.677988i
$206$	0	0
$207$	5698.00	1.91323
$208$	0	0
$209$	2420.00	0.800933
$210$	0	0
$211$	− 4310.00i	− 1.40622i	−0.711081	−	0.703111i	$-0.751794\pi$
$211$	− 4310.00i	− 1.40622i	0.711081	−	0.703111i	$-0.248206\pi$
$212$	0	0
$213$	7424.00i	2.38819i
$214$	0	0
$215$	1340.00	0.425057
$216$	0	0
$217$	184.000	0.0575610
$218$	0	0
$219$	− 4560.00i	− 1.40701i
$220$	0	0
$221$	− 100.000i	− 0.0304377i
$222$	0	0
$223$	2742.00	0.823399	0.411699	−	0.911320i	$-0.364935\pi$
$223$	2742.00	0.823399	0.411699	+	0.911320i	$0.364935\pi$
$224$	0	0
$225$	925.000	0.274074
$226$	0	0
$227$	− 1932.00i	− 0.564896i	−0.959283	−	0.282448i	$-0.908854\pi$
$227$	− 1932.00i	− 0.564896i	0.959283	−	0.282448i	$-0.0911464\pi$
$228$	0	0
$229$	− 6114.00i	− 1.76430i	−0.470969	−	0.882150i	$-0.656096\pi$
$229$	− 6114.00i	− 1.76430i	0.470969	−	0.882150i	$-0.343904\pi$
$230$	0	0
$231$	352.000	0.100259
$232$	0	0
$233$	−690.000	−0.194006	−0.0970030	−	0.995284i	$-0.530926\pi$
$233$	−690.000	−0.194006	−0.0970030	+	0.995284i	$0.530926\pi$
$234$	0	0
$235$	50.0000i	0.0138793i
$236$	0	0
$237$	− 512.000i	− 0.140329i
$238$	0	0
$239$	1264.00	0.342098	0.171049	−	0.985263i	$-0.445284\pi$
$239$	1264.00	0.342098	0.171049	+	0.985263i	$0.445284\pi$
$240$	0	0
$241$	−3246.00	−0.867607	−0.433803	−	0.901008i	$-0.642829\pi$
$241$	−3246.00	−0.867607	−0.433803	+	0.901008i	$0.642829\pi$
$242$	0	0
$243$	− 5032.00i	− 1.32841i
$244$	0	0
$245$	− 1695.00i	− 0.441998i
$246$	0	0
$247$	1100.00	0.283366
$248$	0	0
$249$	6912.00	1.75916
$250$	0	0
$251$	− 2422.00i	− 0.609065i	−0.952502	−	0.304532i	$-0.901500\pi$
$251$	− 2422.00i	− 0.609065i	0.952502	−	0.304532i	$-0.0985002\pi$
$252$	0	0
$253$	3388.00i	0.841904i
$254$	0	0
$255$	−400.000	−0.0982313
$256$	0	0
$257$	−1158.00	−0.281066	−0.140533	−	0.990076i	$-0.544882\pi$
$257$	−1158.00	−0.281066	−0.140533	+	0.990076i	$0.544882\pi$
$258$	0	0
$259$	− 68.0000i	− 0.0163140i
$260$	0	0
$261$	8214.00i	1.94802i
$262$	0	0
$263$	3038.00	0.712286	0.356143	−	0.934432i	$-0.384092\pi$
$263$	3038.00	0.712286	0.356143	+	0.934432i	$0.384092\pi$
$264$	0	0
$265$	2910.00	0.674566
$266$	0	0
$267$	6992.00i	1.60263i
$268$	0	0
$269$	− 3846.00i	− 0.871728i	−0.900013	−	0.435864i	$-0.856443\pi$
$269$	− 3846.00i	− 0.871728i	0.900013	−	0.435864i	$-0.143557\pi$
$270$	0	0
$271$	−6612.00	−1.48210	−0.741052	−	0.671447i	$-0.765673\pi$
$271$	−6612.00	−1.48210	−0.741052	+	0.671447i	$0.765673\pi$
$272$	0	0
$273$	160.000	0.0354712
$274$	0	0
$275$	550.000i	0.120605i
$276$	0	0
$277$	− 7314.00i	− 1.58648i	−0.608907	−	0.793241i	$-0.708392\pi$
$277$	− 7314.00i	− 1.58648i	0.608907	−	0.793241i	$-0.291608\pi$
$278$	0	0
$279$	−3404.00	−0.730438
$280$	0	0
$281$	2102.00	0.446245	0.223122	−	0.974790i	$-0.428375\pi$
$281$	2102.00	0.446245	0.223122	+	0.974790i	$0.428375\pi$
$282$	0	0
$283$	4620.00i	0.970426i	0.874396	+	0.485213i	$0.161258\pi$
$283$	4620.00i	0.970426i	−0.874396	+	0.485213i	$0.838742\pi$
$284$	0	0
$285$	− 4400.00i	− 0.914504i
$286$	0	0
$287$	−796.000	−0.163716
$288$	0	0
$289$	−4813.00	−0.979646
$290$	0	0
$291$	2448.00i	0.493142i
$292$	0	0
$293$	− 4018.00i	− 0.801140i	−0.916266	−	0.400570i	$-0.868812\pi$
$293$	− 4018.00i	− 0.801140i	0.916266	−	0.400570i	$-0.131188\pi$
$294$	0	0
$295$	3730.00	0.736166
$296$	0	0
$297$	−1760.00	−0.343857
$298$	0	0
$299$	1540.00i	0.297861i
$300$	0	0
$301$	− 536.000i	− 0.102640i
$302$	0	0
$303$	7440.00	1.41062
$304$	0	0
$305$	−1130.00	−0.212143
$306$	0	0
$307$	− 8596.00i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 8596.00i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	3696.00i	0.680447i
$310$	0	0
$311$	1312.00	0.239218	0.119609	−	0.992821i	$-0.461836\pi$
$311$	1312.00	0.239218	0.119609	+	0.992821i	$0.461836\pi$
$312$	0	0
$313$	−1154.00	−0.208396	−0.104198	−	0.994557i	$-0.533228\pi$
$313$	−1154.00	−0.208396	−0.104198	+	0.994557i	$0.533228\pi$
$314$	0	0
$315$	− 370.000i	− 0.0661814i
$316$	0	0
$317$	7262.00i	1.28667i	0.765585	+	0.643335i	$0.222450\pi$
$317$	7262.00i	1.28667i	−0.765585	+	0.643335i	$0.777550\pi$
$318$	0	0
$319$	−4884.00	−0.857215
$320$	0	0
$321$	−12480.0	−2.16999
$322$	0	0
$323$	1100.00i	0.189491i
$324$	0	0
$325$	250.000i	0.0426692i
$326$	0	0
$327$	−9488.00	−1.60455
$328$	0	0
$329$	20.0000	0.00335148
$330$	0	0
$331$	3098.00i	0.514446i	0.966352	+	0.257223i	$0.0828074\pi$
$331$	3098.00i	0.514446i	−0.966352	+	0.257223i	$0.917193\pi$
$332$	0	0
$333$	1258.00i	0.207021i
$334$	0	0
$335$	860.000	0.140259
$336$	0	0
$337$	−7062.00	−1.14152	−0.570759	−	0.821118i	$-0.693351\pi$
$337$	−7062.00	−1.14152	−0.570759	+	0.821118i	$0.693351\pi$
$338$	0	0
$339$	− 11056.0i	− 1.77133i
$340$	0	0
$341$	− 2024.00i	− 0.321424i
$342$	0	0
$343$	−1364.00	−0.214720
$344$	0	0
$345$	6160.00	0.961285
$346$	0	0
$347$	2240.00i	0.346540i	0.984874	+	0.173270i	$0.0554334\pi$
$347$	2240.00i	0.346540i	−0.984874	+	0.173270i	$0.944567\pi$
$348$	0	0
$349$	− 12166.0i	− 1.86599i	−0.359887	−	0.932996i	$-0.617185\pi$
$349$	− 12166.0i	− 1.86599i	0.359887	−	0.932996i	$-0.382815\pi$
$350$	0	0
$351$	−800.000	−0.121655
$352$	0	0
$353$	−9318.00	−1.40495	−0.702475	−	0.711709i	$-0.747922\pi$
$353$	−9318.00	−1.40495	−0.702475	+	0.711709i	$0.747922\pi$
$354$	0	0
$355$	4640.00i	0.693706i
$356$	0	0
$357$	160.000i	0.0237202i
$358$	0	0
$359$	5196.00	0.763884	0.381942	−	0.924186i	$-0.375255\pi$
$359$	5196.00	0.763884	0.381942	+	0.924186i	$0.375255\pi$
$360$	0	0
$361$	−5241.00	−0.764106
$362$	0	0
$363$	6776.00i	0.979746i
$364$	0	0
$365$	− 2850.00i	− 0.408701i
$366$	0	0
$367$	12970.0	1.84476	0.922382	−	0.386279i	$-0.126240\pi$
$367$	12970.0	1.84476	0.922382	+	0.386279i	$0.126240\pi$
$368$	0	0
$369$	14726.0	2.07752
$370$	0	0
$371$	− 1164.00i	− 0.162889i
$372$	0	0
$373$	702.000i	0.0974482i	0.998812	+	0.0487241i	$0.0155155\pi$
$373$	702.000i	0.0974482i	−0.998812	+	0.0487241i	$0.984484\pi$
$374$	0	0
$375$	1000.00	0.137706
$376$	0	0
$377$	−2220.00	−0.303278
$378$	0	0
$379$	6630.00i	0.898576i	0.893387	+	0.449288i	$0.148322\pi$
$379$	6630.00i	0.898576i	−0.893387	+	0.449288i	$0.851678\pi$
$380$	0	0
$381$	− 9584.00i	− 1.28872i
$382$	0	0
$383$	46.0000	0.00613705	0.00306853	−	0.999995i	$-0.499023\pi$
$383$	46.0000	0.00613705	0.00306853	+	0.999995i	$0.499023\pi$
$384$	0	0
$385$	220.000	0.0291227
$386$	0	0
$387$	9916.00i	1.30248i
$388$	0	0
$389$	− 11914.0i	− 1.55286i	−0.630202	−	0.776432i	$-0.717028\pi$
$389$	− 11914.0i	− 1.55286i	0.630202	−	0.776432i	$-0.282972\pi$
$390$	0	0
$391$	−1540.00	−0.199185
$392$	0	0
$393$	−18512.0	−2.37610
$394$	0	0
$395$	− 320.000i	− 0.0407619i
$396$	0	0
$397$	− 2994.00i	− 0.378500i	−0.981929	−	0.189250i	$-0.939394\pi$
$397$	− 2994.00i	− 0.378500i	0.981929	−	0.189250i	$-0.0606057\pi$
$398$	0	0
$399$	−1760.00	−0.220828
$400$	0	0
$401$	−6402.00	−0.797258	−0.398629	−	0.917112i	$-0.630514\pi$
$401$	−6402.00	−0.797258	−0.398629	+	0.917112i	$0.630514\pi$
$402$	0	0
$403$	− 920.000i	− 0.113718i
$404$	0	0
$405$	− 1795.00i	− 0.220233i
$406$	0	0
$407$	−748.000	−0.0910982
$408$	0	0
$409$	−6.00000	−0.000725381 0	−0.000362691	−	1.00000i	$-0.500115\pi$
$409$	−6.00000	−0.000725381 0	−0.000362691		1.00000i	$0.500115\pi$
$410$	0	0
$411$	− 19152.0i	− 2.29854i
$412$	0	0
$413$	− 1492.00i	− 0.177764i
$414$	0	0
$415$	4320.00	0.510989
$416$	0	0
$417$	6608.00	0.776008
$418$	0	0
$419$	− 1770.00i	− 0.206373i	−0.994662	−	0.103186i	$-0.967096\pi$
$419$	− 1770.00i	− 0.206373i	0.994662	−	0.103186i	$-0.0329038\pi$
$420$	0	0
$421$	3638.00i	0.421153i	0.977577	+	0.210576i	$0.0675340\pi$
$421$	3638.00i	0.421153i	−0.977577	+	0.210576i	$0.932466\pi$
$422$	0	0
$423$	−370.000	−0.0425296
$424$	0	0
$425$	−250.000	−0.0285336
$426$	0	0
$427$	452.000i	0.0512267i
$428$	0	0
$429$	− 1760.00i	− 0.198074i
$430$	0	0
$431$	−13492.0	−1.50786	−0.753929	−	0.656956i	$-0.771844\pi$
$431$	−13492.0	−1.50786	−0.753929	+	0.656956i	$0.771844\pi$
$432$	0	0
$433$	−4478.00	−0.496995	−0.248498	−	0.968633i	$-0.579937\pi$
$433$	−4478.00	−0.496995	−0.248498	+	0.968633i	$0.579937\pi$
$434$	0	0
$435$	8880.00i	0.978766i
$436$	0	0
$437$	− 16940.0i	− 1.85435i
$438$	0	0
$439$	−6796.00	−0.738851	−0.369425	−	0.929260i	$-0.620445\pi$
$439$	−6796.00	−0.738851	−0.369425	+	0.929260i	$0.620445\pi$
$440$	0	0
$441$	12543.0	1.35439
$442$	0	0
$443$	− 7692.00i	− 0.824962i	−0.910966	−	0.412481i	$-0.864662\pi$
$443$	− 7692.00i	− 0.824962i	0.910966	−	0.412481i	$-0.135338\pi$
$444$	0	0
$445$	4370.00i	0.465523i
$446$	0	0
$447$	18384.0	1.94526
$448$	0	0
$449$	18818.0	1.97790	0.988949	−	0.148255i	$-0.0473657\pi$
$449$	18818.0	1.97790	0.988949	+	0.148255i	$0.0473657\pi$
$450$	0	0
$451$	8756.00i	0.914199i
$452$	0	0
$453$	16960.0i	1.75905i
$454$	0	0
$455$	100.000	0.0103035
$456$	0	0
$457$	−11722.0	−1.19985	−0.599926	−	0.800056i	$-0.704803\pi$
$457$	−11722.0	−1.19985	−0.599926	+	0.800056i	$0.704803\pi$
$458$	0	0
$459$	− 800.000i	− 0.0813525i
$460$	0	0
$461$	− 6846.00i	− 0.691649i	−0.938299	−	0.345824i	$-0.887599\pi$
$461$	− 6846.00i	− 0.691649i	0.938299	−	0.345824i	$-0.112401\pi$
$462$	0	0
$463$	−13802.0	−1.38539	−0.692693	−	0.721233i	$-0.743576\pi$
$463$	−13802.0	−1.38539	−0.692693	+	0.721233i	$0.743576\pi$
$464$	0	0
$465$	−3680.00	−0.367002
$466$	0	0
$467$	15396.0i	1.52557i	0.646651	+	0.762786i	$0.276169\pi$
$467$	15396.0i	1.52557i	−0.646651	+	0.762786i	$0.723831\pi$
$468$	0	0
$469$	− 344.000i	− 0.0338688i
$470$	0	0
$471$	6704.00	0.655847
$472$	0	0
$473$	−5896.00	−0.573147
$474$	0	0
$475$	− 2750.00i	− 0.265639i
$476$	0	0
$477$	21534.0i	2.06703i
$478$	0	0
$479$	−14584.0	−1.39115	−0.695574	−	0.718454i	$-0.744850\pi$
$479$	−14584.0	−1.39115	−0.695574	+	0.718454i	$0.744850\pi$
$480$	0	0
$481$	−340.000	−0.0322301
$482$	0	0
$483$	− 2464.00i	− 0.232124i
$484$	0	0
$485$	1530.00i	0.143245i
$486$	0	0
$487$	−6910.00	−0.642961	−0.321480	−	0.946916i	$-0.604180\pi$
$487$	−6910.00	−0.642961	−0.321480	+	0.946916i	$0.604180\pi$
$488$	0	0
$489$	−11520.0	−1.06534
$490$	0	0
$491$	− 2710.00i	− 0.249085i	−0.992214	−	0.124542i	$-0.960254\pi$
$491$	− 2710.00i	− 0.249085i	0.992214	−	0.124542i	$-0.0397463\pi$
$492$	0	0
$493$	− 2220.00i	− 0.202807i
$494$	0	0
$495$	−4070.00	−0.369561
$496$	0	0
$497$	1856.00	0.167511
$498$	0	0
$499$	− 5522.00i	− 0.495388i	−0.968838	−	0.247694i	$-0.920327\pi$
$499$	− 5522.00i	− 0.495388i	0.968838	−	0.247694i	$-0.0796728\pi$
$500$	0	0
$501$	16976.0i	1.51384i
$502$	0	0
$503$	−3450.00	−0.305821	−0.152910	−	0.988240i	$-0.548865\pi$
$503$	−3450.00	−0.305821	−0.152910	+	0.988240i	$0.548865\pi$
$504$	0	0
$505$	4650.00	0.409747
$506$	0	0
$507$	16776.0i	1.46952i
$508$	0	0
$509$	− 4590.00i	− 0.399702i	−0.979826	−	0.199851i	$-0.935954\pi$
$509$	− 4590.00i	− 0.399702i	0.979826	−	0.199851i	$-0.0640458\pi$
$510$	0	0
$511$	−1140.00	−0.0986901
$512$	0	0
$513$	8800.00	0.757367
$514$	0	0
$515$	2310.00i	0.197652i
$516$	0	0
$517$	− 220.000i	− 0.0187149i
$518$	0	0
$519$	−26448.0	−2.23688
$520$	0	0
$521$	−7242.00	−0.608978	−0.304489	−	0.952516i	$-0.598486\pi$
$521$	−7242.00	−0.608978	−0.304489	+	0.952516i	$0.598486\pi$
$522$	0	0
$523$	− 2732.00i	− 0.228417i	−0.993457	−	0.114208i	$-0.963567\pi$
$523$	− 2732.00i	− 0.228417i	0.993457	−	0.114208i	$-0.0364332\pi$
$524$	0	0
$525$	− 400.000i	− 0.0332522i
$526$	0	0
$527$	920.000	0.0760452
$528$	0	0
$529$	11549.0	0.949207
$530$	0	0
$531$	27602.0i	2.25579i
$532$	0	0
$533$	3980.00i	0.323439i
$534$	0	0
$535$	−7800.00	−0.630324
$536$	0	0
$537$	21968.0	1.76534
$538$	0	0
$539$	7458.00i	0.595990i
$540$	0	0
$541$	− 2486.00i	− 0.197563i	−0.995109	−	0.0987814i	$-0.968506\pi$
$541$	− 2486.00i	− 0.197563i	0.995109	−	0.0987814i	$-0.0314945\pi$
$542$	0	0
$543$	−32496.0	−2.56821
$544$	0	0
$545$	−5930.00	−0.466079
$546$	0	0
$547$	11860.0i	0.927051i	0.886084	+	0.463526i	$0.153416\pi$
$547$	11860.0i	0.927051i	−0.886084	+	0.463526i	$0.846584\pi$
$548$	0	0
$549$	− 8362.00i	− 0.650057i
$550$	0	0
$551$	24420.0	1.88807
$552$	0	0
$553$	−128.000	−0.00984288
$554$	0	0
$555$	1360.00i	0.104016i
$556$	0	0
$557$	23546.0i	1.79116i	0.444901	+	0.895580i	$0.353239\pi$
$557$	23546.0i	1.79116i	−0.444901	+	0.895580i	$0.646761\pi$
$558$	0	0
$559$	−2680.00	−0.202776
$560$	0	0
$561$	1760.00	0.132455
$562$	0	0
$563$	11792.0i	0.882724i	0.897329	+	0.441362i	$0.145505\pi$
$563$	11792.0i	0.882724i	−0.897329	+	0.441362i	$0.854495\pi$
$564$	0	0
$565$	− 6910.00i	− 0.514524i
$566$	0	0
$567$	−718.000	−0.0531802
$568$	0	0
$569$	−3702.00	−0.272752	−0.136376	−	0.990657i	$-0.543546\pi$
$569$	−3702.00	−0.272752	−0.136376	+	0.990657i	$0.543546\pi$
$570$	0	0
$571$	12170.0i	0.891942i	0.895048	+	0.445971i	$0.147141\pi$
$571$	12170.0i	0.891942i	−0.895048	+	0.445971i	$0.852859\pi$
$572$	0	0
$573$	15264.0i	1.11285i
$574$	0	0
$575$	3850.00	0.279228
$576$	0	0
$577$	−6526.00	−0.470851	−0.235425	−	0.971892i	$-0.575648\pi$
$577$	−6526.00	−0.470851	−0.235425	+	0.971892i	$0.575648\pi$
$578$	0	0
$579$	− 22064.0i	− 1.58368i
$580$	0	0
$581$	− 1728.00i	− 0.123390i
$582$	0	0
$583$	−12804.0	−0.909584
$584$	0	0
$585$	−1850.00	−0.130749
$586$	0	0
$587$	− 2624.00i	− 0.184504i	−0.995736	−	0.0922522i	$-0.970593\pi$
$587$	− 2624.00i	− 0.184504i	0.995736	−	0.0922522i	$-0.0294066\pi$
$588$	0	0
$589$	10120.0i	0.707958i
$590$	0	0
$591$	−34576.0	−2.40654
$592$	0	0
$593$	17218.0	1.19234	0.596171	−	0.802858i	$-0.296688\pi$
$593$	17218.0	1.19234	0.596171	+	0.802858i	$0.296688\pi$
$594$	0	0
$595$	100.000i	0.00689008i
$596$	0	0
$597$	− 22816.0i	− 1.56415i
$598$	0	0
$599$	14236.0	0.971064	0.485532	−	0.874219i	$-0.338626\pi$
$599$	14236.0	0.971064	0.485532	+	0.874219i	$0.338626\pi$
$600$	0	0
$601$	19030.0	1.29160	0.645799	−	0.763508i	$-0.276525\pi$
$601$	19030.0	1.29160	0.645799	+	0.763508i	$0.276525\pi$
$602$	0	0
$603$	6364.00i	0.429788i
$604$	0	0
$605$	4235.00i	0.284590i
$606$	0	0
$607$	−1966.00	−0.131462	−0.0657310	−	0.997837i	$-0.520938\pi$
$607$	−1966.00	−0.131462	−0.0657310	+	0.997837i	$0.520938\pi$
$608$	0	0
$609$	3552.00	0.236345
$610$	0	0
$611$	− 100.000i	− 0.00662122i
$612$	0	0
$613$	− 16518.0i	− 1.08835i	−0.838973	−	0.544173i	$-0.816844\pi$
$613$	− 16518.0i	− 1.08835i	0.838973	−	0.544173i	$-0.183156\pi$
$614$	0	0
$615$	15920.0	1.04383
$616$	0	0
$617$	−17954.0	−1.17148	−0.585738	−	0.810500i	$-0.699195\pi$
$617$	−17954.0	−1.17148	−0.585738	+	0.810500i	$0.699195\pi$
$618$	0	0
$619$	2494.00i	0.161942i	0.996716	+	0.0809712i	$0.0258022\pi$
$619$	2494.00i	0.161942i	−0.996716	+	0.0809712i	$0.974198\pi$
$620$	0	0
$621$	12320.0i	0.796110i
$622$	0	0
$623$	1748.00	0.112411
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	19360.0i	1.23312i
$628$	0	0
$629$	− 340.000i	− 0.0215528i
$630$	0	0
$631$	−10600.0	−0.668747	−0.334373	−	0.942441i	$-0.608525\pi$
$631$	−10600.0	−0.668747	−0.334373	+	0.942441i	$0.608525\pi$
$632$	0	0
$633$	34480.0	2.16502
$634$	0	0
$635$	− 5990.00i	− 0.374340i
$636$	0	0
$637$	3390.00i	0.210858i
$638$	0	0
$639$	−34336.0	−2.12568
$640$	0	0
$641$	4386.00	0.270260	0.135130	−	0.990828i	$-0.456855\pi$
$641$	4386.00	0.270260	0.135130	+	0.990828i	$0.456855\pi$
$642$	0	0
$643$	− 22128.0i	− 1.35714i	−0.734534	−	0.678572i	$-0.762599\pi$
$643$	− 22128.0i	− 1.35714i	0.734534	−	0.678572i	$-0.237401\pi$
$644$	0	0
$645$	10720.0i	0.654418i
$646$	0	0
$647$	5650.00	0.343314	0.171657	−	0.985157i	$-0.445088\pi$
$647$	5650.00	0.343314	0.171657	+	0.985157i	$0.445088\pi$
$648$	0	0
$649$	−16412.0	−0.992646
$650$	0	0
$651$	1472.00i	0.0886209i
$652$	0	0
$653$	15238.0i	0.913184i	0.889676	+	0.456592i	$0.150930\pi$
$653$	15238.0i	0.913184i	−0.889676	+	0.456592i	$0.849070\pi$
$654$	0	0
$655$	−11570.0	−0.690194
$656$	0	0
$657$	21090.0	1.25236
$658$	0	0
$659$	13318.0i	0.787247i	0.919272	+	0.393623i	$0.128779\pi$
$659$	13318.0i	0.787247i	−0.919272	+	0.393623i	$0.871221\pi$
$660$	0	0
$661$	5838.00i	0.343528i	0.985138	+	0.171764i	$0.0549466\pi$
$661$	5838.00i	0.343528i	−0.985138	+	0.171764i	$0.945053\pi$
$662$	0	0
$663$	800.000	0.0468619
$664$	0	0
$665$	−1100.00	−0.0641446
$666$	0	0
$667$	34188.0i	1.98465i
$668$	0	0
$669$	21936.0i	1.26771i
$670$	0	0
$671$	4972.00	0.286054
$672$	0	0
$673$	74.0000	0.00423847	0.00211924	−	0.999998i	$-0.499325\pi$
$673$	74.0000	0.00423847	0.00211924	+	0.999998i	$0.499325\pi$
$674$	0	0
$675$	2000.00i	0.114044i
$676$	0	0
$677$	6194.00i	0.351632i	0.984423	+	0.175816i	$0.0562564\pi$
$677$	6194.00i	0.351632i	−0.984423	+	0.175816i	$0.943744\pi$
$678$	0	0
$679$	612.000	0.0345897
$680$	0	0
$681$	15456.0	0.869714
$682$	0	0
$683$	− 23532.0i	− 1.31834i	−0.751993	−	0.659171i	$-0.770908\pi$
$683$	− 23532.0i	− 1.31834i	0.751993	−	0.659171i	$-0.229092\pi$
$684$	0	0
$685$	− 11970.0i	− 0.667665i
$686$	0	0
$687$	48912.0	2.71632
$688$	0	0
$689$	−5820.00	−0.321806
$690$	0	0
$691$	18530.0i	1.02014i	0.860134	+	0.510068i	$0.170380\pi$
$691$	18530.0i	1.02014i	−0.860134	+	0.510068i	$0.829620\pi$
$692$	0	0
$693$	1628.00i	0.0892390i
$694$	0	0
$695$	4130.00	0.225410
$696$	0	0
$697$	−3980.00	−0.216289
$698$	0	0
$699$	− 5520.00i	− 0.298692i
$700$	0	0
$701$	− 5142.00i	− 0.277048i	−0.990359	−	0.138524i	$-0.955764\pi$
$701$	− 5142.00i	− 0.277048i	0.990359	−	0.138524i	$-0.0442358\pi$
$702$	0	0
$703$	3740.00	0.200650
$704$	0	0
$705$	−400.000	−0.0213686
$706$	0	0
$707$	− 1860.00i	− 0.0989427i
$708$	0	0
$709$	21438.0i	1.13557i	0.823176	+	0.567786i	$0.192200\pi$
$709$	21438.0i	1.13557i	−0.823176	+	0.567786i	$0.807800\pi$
$710$	0	0
$711$	2368.00	0.124904
$712$	0	0
$713$	−14168.0	−0.744174
$714$	0	0
$715$	− 1100.00i	− 0.0575352i
$716$	0	0
$717$	10112.0i	0.526694i
$718$	0	0
$719$	2624.00	0.136104	0.0680519	−	0.997682i	$-0.478322\pi$
$719$	2624.00	0.136104	0.0680519	+	0.997682i	$0.478322\pi$
$720$	0	0
$721$	924.000	0.0477275
$722$	0	0
$723$	− 25968.0i	− 1.33577i
$724$	0	0
$725$	5550.00i	0.284306i
$726$	0	0
$727$	9410.00	0.480052	0.240026	−	0.970766i	$-0.422844\pi$
$727$	9410.00	0.480052	0.240026	+	0.970766i	$0.422844\pi$
$728$	0	0
$729$	30563.0	1.55276
$730$	0	0
$731$	− 2680.00i	− 0.135600i
$732$	0	0
$733$	− 19142.0i	− 0.964565i	−0.876016	−	0.482282i	$-0.839808\pi$
$733$	− 19142.0i	− 0.964565i	0.876016	−	0.482282i	$-0.160192\pi$
$734$	0	0
$735$	13560.0	0.680501
$736$	0	0
$737$	−3784.00	−0.189125
$738$	0	0
$739$	− 2930.00i	− 0.145848i	−0.997337	−	0.0729241i	$-0.976767\pi$
$739$	− 2930.00i	− 0.145848i	0.997337	−	0.0729241i	$-0.0232331\pi$
$740$	0	0
$741$	8800.00i	0.436270i
$742$	0	0
$743$	−20250.0	−0.999866	−0.499933	−	0.866064i	$-0.666642\pi$
$743$	−20250.0	−0.999866	−0.499933	+	0.866064i	$0.666642\pi$
$744$	0	0
$745$	11490.0	0.565048
$746$	0	0
$747$	31968.0i	1.56579i
$748$	0	0
$749$	3120.00i	0.152206i
$750$	0	0
$751$	−1916.00	−0.0930970	−0.0465485	−	0.998916i	$-0.514822\pi$
$751$	−1916.00	−0.0930970	−0.0465485	+	0.998916i	$0.514822\pi$
$752$	0	0
$753$	19376.0	0.937717
$754$	0	0
$755$	10600.0i	0.510958i
$756$	0	0
$757$	20494.0i	0.983972i	0.870603	+	0.491986i	$0.163729\pi$
$757$	20494.0i	0.983972i	−0.870603	+	0.491986i	$0.836271\pi$
$758$	0	0
$759$	−27104.0	−1.29620
$760$	0	0
$761$	−14826.0	−0.706231	−0.353116	−	0.935580i	$-0.614878\pi$
$761$	−14826.0	−0.706231	−0.353116	+	0.935580i	$0.614878\pi$
$762$	0	0
$763$	2372.00i	0.112545i
$764$	0	0
$765$	− 1850.00i	− 0.0874338i
$766$	0	0
$767$	−7460.00	−0.351193
$768$	0	0
$769$	−39194.0	−1.83793	−0.918967	−	0.394334i	$-0.870975\pi$
$769$	−39194.0	−1.83793	−0.918967	+	0.394334i	$0.870975\pi$
$770$	0	0
$771$	− 9264.00i	− 0.432730i
$772$	0	0
$773$	− 35862.0i	− 1.66865i	−0.551273	−	0.834325i	$-0.685858\pi$
$773$	− 35862.0i	− 1.66865i	0.551273	−	0.834325i	$-0.314142\pi$
$774$	0	0
$775$	−2300.00	−0.106604
$776$	0	0
$777$	544.000	0.0251170
$778$	0	0
$779$	− 43780.0i	− 2.01358i
$780$	0	0
$781$	− 20416.0i	− 0.935393i
$782$	0	0
$783$	−17760.0	−0.810588
$784$	0	0
$785$	4190.00	0.190506
$786$	0	0
$787$	23060.0i	1.04447i	0.852801	+	0.522236i	$0.174902\pi$
$787$	23060.0i	1.04447i	−0.852801	+	0.522236i	$0.825098\pi$
$788$	0	0
$789$	24304.0i	1.09664i
$790$	0	0
$791$	−2764.00	−0.124243
$792$	0	0
$793$	2260.00	0.101204
$794$	0	0
$795$	23280.0i	1.03856i
$796$	0	0
$797$	− 40466.0i	− 1.79847i	−0.437468	−	0.899234i	$-0.644125\pi$
$797$	− 40466.0i	− 1.79847i	0.437468	−	0.899234i	$-0.355875\pi$
$798$	0	0
$799$	100.000	0.00442772
$800$	0	0
$801$	−32338.0	−1.42648
$802$	0	0
$803$	12540.0i	0.551092i
$804$	0	0
$805$	− 1540.00i	− 0.0674259i
$806$	0	0
$807$	30768.0	1.34211
$808$	0	0
$809$	−36090.0	−1.56843	−0.784213	−	0.620492i	$-0.786933\pi$
$809$	−36090.0	−1.56843	−0.784213	+	0.620492i	$0.786933\pi$
$810$	0	0
$811$	− 12022.0i	− 0.520530i	−0.965537	−	0.260265i	$-0.916190\pi$
$811$	− 12022.0i	− 0.520530i	0.965537	−	0.260265i	$-0.0838099\pi$
$812$	0	0
$813$	− 52896.0i	− 2.28185i
$814$	0	0
$815$	−7200.00	−0.309454
$816$	0	0
$817$	29480.0	1.26239
$818$	0	0
$819$	740.000i	0.0315723i
$820$	0	0
$821$	− 15250.0i	− 0.648269i	−0.946011	−	0.324134i	$-0.894927\pi$
$821$	− 15250.0i	− 0.648269i	0.946011	−	0.324134i	$-0.105073\pi$
$822$	0	0
$823$	−45970.0	−1.94704	−0.973520	−	0.228603i	$-0.926584\pi$
$823$	−45970.0	−1.94704	−0.973520	+	0.228603i	$0.926584\pi$
$824$	0	0
$825$	−4400.00	−0.185683
$826$	0	0
$827$	8160.00i	0.343109i	0.985175	+	0.171554i	$0.0548789\pi$
$827$	8160.00i	0.343109i	−0.985175	+	0.171554i	$0.945121\pi$
$828$	0	0
$829$	82.0000i	0.00343544i	0.999999	+	0.00171772i	$0.000546767\pi$
$829$	82.0000i	0.00343544i	−0.999999	+	0.00171772i	$0.999453\pi$
$830$	0	0
$831$	58512.0	2.44255
$832$	0	0
$833$	−3390.00	−0.141004
$834$	0	0
$835$	10610.0i	0.439730i
$836$	0	0
$837$	− 7360.00i	− 0.303941i
$838$	0	0
$839$	−46796.0	−1.92560	−0.962799	−	0.270217i	$-0.912904\pi$
$839$	−46796.0	−1.92560	−0.962799	+	0.270217i	$0.912904\pi$
$840$	0	0
$841$	−24895.0	−1.02075
$842$	0	0
$843$	16816.0i	0.687039i
$844$	0	0
$845$	10485.0i	0.426858i
$846$	0	0
$847$	1694.00	0.0687208
$848$	0	0
$849$	−36960.0	−1.49407
$850$	0	0
$851$	5236.00i	0.210914i
$852$	0	0
$853$	− 18566.0i	− 0.745238i	−0.927984	−	0.372619i	$-0.878460\pi$
$853$	− 18566.0i	− 0.745238i	0.927984	−	0.372619i	$-0.121540\pi$
$854$	0	0
$855$	20350.0	0.813983
$856$	0	0
$857$	−21266.0	−0.847646	−0.423823	−	0.905745i	$-0.639312\pi$
$857$	−21266.0	−0.847646	−0.423823	+	0.905745i	$0.639312\pi$
$858$	0	0
$859$	− 22106.0i	− 0.878052i	−0.898474	−	0.439026i	$-0.855324\pi$
$859$	− 22106.0i	− 0.878052i	0.898474	−	0.439026i	$-0.144676\pi$
$860$	0	0
$861$	− 6368.00i	− 0.252057i
$862$	0	0
$863$	6150.00	0.242582	0.121291	−	0.992617i	$-0.461297\pi$
$863$	6150.00	0.242582	0.121291	+	0.992617i	$0.461297\pi$
$864$	0	0
$865$	−16530.0	−0.649754
$866$	0	0
$867$	− 38504.0i	− 1.50826i
$868$	0	0
$869$	1408.00i	0.0549633i
$870$	0	0
$871$	−1720.00	−0.0669116
$872$	0	0
$873$	−11322.0	−0.438937
$874$	0	0
$875$	− 250.000i	− 0.00965891i
$876$	0	0
$877$	− 48038.0i	− 1.84963i	−0.380414	−	0.924816i	$-0.624218\pi$
$877$	− 48038.0i	− 1.84963i	0.380414	−	0.924816i	$-0.375782\pi$
$878$	0	0
$879$	32144.0	1.23344
$880$	0	0
$881$	34750.0	1.32890	0.664448	−	0.747335i	$-0.268667\pi$
$881$	34750.0	1.32890	0.664448	+	0.747335i	$0.268667\pi$
$882$	0	0
$883$	46608.0i	1.77631i	0.459541	+	0.888156i	$0.348014\pi$
$883$	46608.0i	1.77631i	−0.459541	+	0.888156i	$0.651986\pi$
$884$	0	0
$885$	29840.0i	1.13340i
$886$	0	0
$887$	33906.0	1.28349	0.641743	−	0.766920i	$-0.278212\pi$
$887$	33906.0	1.28349	0.641743	+	0.766920i	$0.278212\pi$
$888$	0	0
$889$	−2396.00	−0.0903929
$890$	0	0
$891$	7898.00i	0.296962i
$892$	0	0
$893$	1100.00i	0.0412207i
$894$	0	0
$895$	13730.0	0.512786
$896$	0	0
$897$	−12320.0	−0.458587
$898$	0	0
$899$	− 20424.0i	− 0.757707i
$900$	0	0
$901$	− 5820.00i	− 0.215197i
$902$	0	0
$903$	4288.00	0.158024
$904$	0	0
$905$	−20310.0	−0.745997
$906$	0	0
$907$	46256.0i	1.69339i	0.532078	+	0.846695i	$0.321411\pi$
$907$	46256.0i	1.69339i	−0.532078	+	0.846695i	$0.678589\pi$
$908$	0	0
$909$	34410.0i	1.25556i
$910$	0	0
$911$	−52092.0	−1.89450	−0.947248	−	0.320503i	$-0.896148\pi$
$911$	−52092.0	−1.89450	−0.947248	+	0.320503i	$0.896148\pi$
$912$	0	0
$913$	−19008.0	−0.689018
$914$	0	0
$915$	− 9040.00i	− 0.326615i
$916$	0	0
$917$	4628.00i	0.166663i
$918$	0	0
$919$	−19988.0	−0.717457	−0.358729	−	0.933442i	$-0.616790\pi$
$919$	−19988.0	−0.717457	−0.358729	+	0.933442i	$0.616790\pi$
$920$	0	0
$921$	68768.0	2.46035
$922$	0	0
$923$	− 9280.00i	− 0.330937i
$924$	0	0
$925$	850.000i	0.0302139i
$926$	0	0
$927$	−17094.0	−0.605653
$928$	0	0
$929$	−31950.0	−1.12836	−0.564179	−	0.825652i	$-0.690807\pi$
$929$	−31950.0	−1.12836	−0.564179	+	0.825652i	$0.690807\pi$
$930$	0	0
$931$	− 37290.0i	− 1.31271i
$932$	0	0
$933$	10496.0i	0.368300i
$934$	0	0
$935$	1100.00	0.0384747
$936$	0	0
$937$	−18874.0	−0.658043	−0.329022	−	0.944322i	$-0.606719\pi$
$937$	−18874.0	−0.658043	−0.329022	+	0.944322i	$0.606719\pi$
$938$	0	0
$939$	− 9232.00i	− 0.320847i
$940$	0	0
$941$	− 24038.0i	− 0.832749i	−0.909193	−	0.416374i	$-0.863301\pi$
$941$	− 24038.0i	− 0.832749i	0.909193	−	0.416374i	$-0.136699\pi$
$942$	0	0
$943$	61292.0	2.11659
$944$	0	0
$945$	800.000	0.0275386
$946$	0	0
$947$	− 11316.0i	− 0.388301i	−0.980972	−	0.194150i	$-0.937805\pi$
$947$	− 11316.0i	− 0.388301i	0.980972	−	0.194150i	$-0.0621949\pi$
$948$	0	0
$949$	5700.00i	0.194973i
$950$	0	0
$951$	−58096.0	−1.98096
$952$	0	0
$953$	48390.0	1.64481	0.822406	−	0.568901i	$-0.192631\pi$
$953$	48390.0	1.64481	0.822406	+	0.568901i	$0.192631\pi$
$954$	0	0
$955$	9540.00i	0.323254i
$956$	0	0
$957$	− 39072.0i	− 1.31977i
$958$	0	0
$959$	−4788.00	−0.161223
$960$	0	0
$961$	−21327.0	−0.715887
$962$	0	0
$963$	− 57720.0i	− 1.93147i
$964$	0	0
$965$	− 13790.0i	− 0.460016i
$966$	0	0
$967$	−21126.0	−0.702551	−0.351275	−	0.936272i	$-0.614252\pi$
$967$	−21126.0	−0.702551	−0.351275	+	0.936272i	$0.614252\pi$
$968$	0	0
$969$	−8800.00	−0.291741
$970$	0	0
$971$	− 42894.0i	− 1.41765i	−0.705387	−	0.708823i	$-0.749227\pi$
$971$	− 42894.0i	− 1.41765i	0.705387	−	0.708823i	$-0.250773\pi$
$972$	0	0
$973$	− 1652.00i	− 0.0544303i
$974$	0	0
$975$	−2000.00	−0.0656936
$976$	0	0
$977$	9594.00	0.314165	0.157083	−	0.987585i	$-0.449791\pi$
$977$	9594.00	0.314165	0.157083	+	0.987585i	$0.449791\pi$
$978$	0	0
$979$	− 19228.0i	− 0.627711i
$980$	0	0
$981$	− 43882.0i	− 1.42818i
$982$	0	0
$983$	55742.0	1.80864	0.904320	−	0.426855i	$-0.140378\pi$
$983$	55742.0	1.80864	0.904320	+	0.426855i	$0.140378\pi$
$984$	0	0
$985$	−21610.0	−0.699037
$986$	0	0
$987$	160.000i	0.00515994i
$988$	0	0
$989$	41272.0i	1.32697i
$990$	0	0
$991$	11788.0	0.377859	0.188929	−	0.981991i	$-0.439498\pi$
$991$	11788.0	0.377859	0.188929	+	0.981991i	$0.439498\pi$
$992$	0	0
$993$	−24784.0	−0.792041
$994$	0	0
$995$	− 14260.0i	− 0.454344i
$996$	0	0
$997$	5074.00i	0.161179i	0.996747	+	0.0805894i	$0.0256802\pi$
$997$	5074.00i	0.161179i	−0.996747	+	0.0805894i	$0.974320\pi$
$998$	0	0
$999$	−2720.00	−0.0861431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.d.i.641.2		2
4.3	odd	2		1280.4.d.h.641.1		2
8.3	odd	2		1280.4.d.h.641.2		2
8.5	even	2	inner	1280.4.d.i.641.1		2
16.3	odd	4		640.4.a.d.1.1	yes	1
16.5	even	4		640.4.a.c.1.1	yes	1
16.11	odd	4		640.4.a.a.1.1	✓	1
16.13	even	4		640.4.a.b.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.4.a.a.1.1	✓	1	16.11	odd	4
640.4.a.b.1.1	yes	1	16.13	even	4
640.4.a.c.1.1	yes	1	16.5	even	4
640.4.a.d.1.1	yes	1	16.3	odd	4
1280.4.d.h.641.1		2	4.3	odd	2
1280.4.d.h.641.2		2	8.3	odd	2
1280.4.d.i.641.1		2	8.5	even	2	inner
1280.4.d.i.641.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+8.00000i q^{3} +5.00000i q^{5} +2.00000 q^{7} -37.0000 q^{9} -22.0000i q^{11} -10.0000i q^{13} -40.0000 q^{15} +10.0000 q^{17} +110.000i q^{19} +16.0000i q^{21} -154.000 q^{23} -25.0000 q^{25} -80.0000i q^{27} -222.000i q^{29} +92.0000 q^{31} +176.000 q^{33} +10.0000i q^{35} -34.0000i q^{37} +80.0000 q^{39} -398.000 q^{41} -268.000i q^{43} -185.000i q^{45} +10.0000 q^{47} -339.000 q^{49} +80.0000i q^{51} -582.000i q^{53} +110.000 q^{55} -880.000 q^{57} -746.000i q^{59} +226.000i q^{61} -74.0000 q^{63} +50.0000 q^{65} -172.000i q^{67} -1232.00i q^{69} +928.000 q^{71} -570.000 q^{73} -200.000i q^{75} -44.0000i q^{77} -64.0000 q^{79} -359.000 q^{81} -864.000i q^{83} +50.0000i q^{85} +1776.00 q^{87} +874.000 q^{89} -20.0000i q^{91} +736.000i q^{93} -550.000 q^{95} +306.000 q^{97} +814.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} - 74 q^{9} - 80 q^{15} + 20 q^{17} - 308 q^{23} - 50 q^{25} + 184 q^{31} + 352 q^{33} + 160 q^{39} - 796 q^{41} + 20 q^{47} - 678 q^{49} + 220 q^{55} - 1760 q^{57} - 148 q^{63} + 100 q^{65}+ \cdots + 612 q^{97}+O(q^{100}) \) 2 * q + 4 * q^7 - 74 * q^9 - 80 * q^15 + 20 * q^17 - 308 * q^23 - 50 * q^25 + 184 * q^31 + 352 * q^33 + 160 * q^39 - 796 * q^41 + 20 * q^47 - 678 * q^49 + 220 * q^55 - 1760 * q^57 - 148 * q^63 + 100 * q^65 + 1856 * q^71 - 1140 * q^73 - 128 * q^79 - 718 * q^81 + 3552 * q^87 + 1748 * q^89 - 1100 * q^95 + 612 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more