LMFDB - Embedded newform 1280.4.d.m.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:	$1$
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 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{3} -5.00000i q^{5} +16.0000 q^{7} +11.0000 q^{9} +O(q^{10})$	$q-4.00000i q^{3} -5.00000i q^{5} +16.0000 q^{7} +11.0000 q^{9} +36.0000i q^{11} -42.0000i q^{13} -20.0000 q^{15} -110.000 q^{17} +116.000i q^{19} -64.0000i q^{21} +16.0000 q^{23} -25.0000 q^{25} -152.000i q^{27} +198.000i q^{29} -240.000 q^{31} +144.000 q^{33} -80.0000i q^{35} +258.000i q^{37} -168.000 q^{39} -442.000 q^{41} -292.000i q^{43} -55.0000i q^{45} -392.000 q^{47} -87.0000 q^{49} +440.000i q^{51} -142.000i q^{53} +180.000 q^{55} +464.000 q^{57} -348.000i q^{59} -570.000i q^{61} +176.000 q^{63} -210.000 q^{65} -692.000i q^{67} -64.0000i q^{69} +168.000 q^{71} +134.000 q^{73} +100.000i q^{75} +576.000i q^{77} -784.000 q^{79} -311.000 q^{81} -564.000i q^{83} +550.000i q^{85} +792.000 q^{87} -1034.00 q^{89} -672.000i q^{91} +960.000i q^{93} +580.000 q^{95} -382.000 q^{97} +396.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{7} + 22 q^{9}+O(q^{10}) $ 2 * q + 32 * q^7 + 22 * q^9	$ 2 q + 32 q^{7} + 22 q^{9} - 40 q^{15} - 220 q^{17} + 32 q^{23} - 50 q^{25} - 480 q^{31} + 288 q^{33} - 336 q^{39} - 884 q^{41} - 784 q^{47} - 174 q^{49} + 360 q^{55} + 928 q^{57} + 352 q^{63} - 420 q^{65} + 336 q^{71} + 268 q^{73} - 1568 q^{79} - 622 q^{81} + 1584 q^{87} - 2068 q^{89} + 1160 q^{95} - 764 q^{97}+O(q^{100}) $ 2 * q + 32 * q^7 + 22 * q^9 - 40 * q^15 - 220 * q^17 + 32 * q^23 - 50 * q^25 - 480 * q^31 + 288 * q^33 - 336 * q^39 - 884 * q^41 - 784 * q^47 - 174 * q^49 + 360 * q^55 + 928 * q^57 + 352 * q^63 - 420 * q^65 + 336 * q^71 + 268 * q^73 - 1568 * q^79 - 622 * q^81 + 1584 * q^87 - 2068 * q^89 + 1160 * q^95 - 764 * 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
$3$	− 4.00000i	− 0.769800i	−0.922958	−	0.384900i	$-0.874236\pi$
$3$	− 4.00000i	− 0.769800i	0.922958	−	0.384900i	$-0.125764\pi$
$4$	0	0
$5$	− 5.00000i	− 0.447214i
$5$	− 5.00000i	− 0.447214i
$6$	0	0
$7$	16.0000	0.863919	0.431959	−	0.901893i	$-0.357822\pi$
$7$	16.0000	0.863919	0.431959	+	0.901893i	$0.357822\pi$
$8$	0	0
$9$	11.0000	0.407407
$10$	0	0
$11$	36.0000i	0.986764i	0.869813	+	0.493382i	$0.164240\pi$
$11$	36.0000i	0.986764i	−0.869813	+	0.493382i	$0.835760\pi$
$12$	0	0
$13$	− 42.0000i	− 0.896054i	−0.894020	−	0.448027i	$-0.852127\pi$
$13$	− 42.0000i	− 0.896054i	0.894020	−	0.448027i	$-0.147873\pi$
$14$	0	0
$15$	−20.0000	−0.344265
$16$	0	0
$17$	−110.000	−1.56935	−0.784674	−	0.619909i	$-0.787170\pi$
$17$	−110.000	−1.56935	−0.784674	+	0.619909i	$0.787170\pi$
$18$	0	0
$19$	116.000i	1.40064i	0.713827	+	0.700322i	$0.246960\pi$
$19$	116.000i	1.40064i	−0.713827	+	0.700322i	$0.753040\pi$
$20$	0	0
$21$	− 64.0000i	− 0.665045i
$22$	0	0
$23$	16.0000	0.145054	0.0725268	−	0.997366i	$-0.476894\pi$
$23$	16.0000	0.145054	0.0725268	+	0.997366i	$0.476894\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	− 152.000i	− 1.08342i
$28$	0	0
$29$	198.000i	1.26785i	0.773394	+	0.633925i	$0.218557\pi$
$29$	198.000i	1.26785i	−0.773394	+	0.633925i	$0.781443\pi$
$30$	0	0
$31$	−240.000	−1.39049	−0.695246	−	0.718772i	$-0.744705\pi$
$31$	−240.000	−1.39049	−0.695246	+	0.718772i	$0.744705\pi$
$32$	0	0
$33$	144.000	0.759612
$34$	0	0
$35$	− 80.0000i	− 0.386356i
$36$	0	0
$37$	258.000i	1.14635i	0.819433	+	0.573175i	$0.194288\pi$
$37$	258.000i	1.14635i	−0.819433	+	0.573175i	$0.805712\pi$
$38$	0	0
$39$	−168.000	−0.689783
$40$	0	0
$41$	−442.000	−1.68363	−0.841815	−	0.539767i	$-0.818512\pi$
$41$	−442.000	−1.68363	−0.841815	+	0.539767i	$0.818512\pi$
$42$	0	0
$43$	− 292.000i	− 1.03557i	−0.855510	−	0.517786i	$-0.826756\pi$
$43$	− 292.000i	− 1.03557i	0.855510	−	0.517786i	$-0.173244\pi$
$44$	0	0
$45$	− 55.0000i	− 0.182198i
$46$	0	0
$47$	−392.000	−1.21658	−0.608288	−	0.793716i	$-0.708143\pi$
$47$	−392.000	−1.21658	−0.608288	+	0.793716i	$0.708143\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	440.000i	1.20808i
$52$	0	0
$53$	− 142.000i	− 0.368023i	−0.982924	−	0.184011i	$-0.941092\pi$
$53$	− 142.000i	− 0.368023i	0.982924	−	0.184011i	$-0.0589083\pi$
$54$	0	0
$55$	180.000	0.441294
$56$	0	0
$57$	464.000	1.07822
$58$	0	0
$59$	− 348.000i	− 0.767894i	−0.923355	−	0.383947i	$-0.874565\pi$
$59$	− 348.000i	− 0.767894i	0.923355	−	0.383947i	$-0.125435\pi$
$60$	0	0
$61$	− 570.000i	− 1.19641i	−0.801343	−	0.598205i	$-0.795881\pi$
$61$	− 570.000i	− 1.19641i	0.801343	−	0.598205i	$-0.204119\pi$
$62$	0	0
$63$	176.000	0.351967
$64$	0	0
$65$	−210.000	−0.400728
$66$	0	0
$67$	− 692.000i	− 1.26181i	−0.775860	−	0.630905i	$-0.782684\pi$
$67$	− 692.000i	− 1.26181i	0.775860	−	0.630905i	$-0.217316\pi$
$68$	0	0
$69$	− 64.0000i	− 0.111662i
$70$	0	0
$71$	168.000	0.280816	0.140408	−	0.990094i	$-0.455159\pi$
$71$	168.000	0.280816	0.140408	+	0.990094i	$0.455159\pi$
$72$	0	0
$73$	134.000	0.214843	0.107421	−	0.994214i	$-0.465741\pi$
$73$	134.000	0.214843	0.107421	+	0.994214i	$0.465741\pi$
$74$	0	0
$75$	100.000i	0.153960i
$76$	0	0
$77$	576.000i	0.852484i
$78$	0	0
$79$	−784.000	−1.11654	−0.558271	−	0.829658i	$-0.688535\pi$
$79$	−784.000	−1.11654	−0.558271	+	0.829658i	$0.688535\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	− 564.000i	− 0.745868i	−0.927858	−	0.372934i	$-0.878352\pi$
$83$	− 564.000i	− 0.745868i	0.927858	−	0.372934i	$-0.121648\pi$
$84$	0	0
$85$	550.000i	0.701834i
$86$	0	0
$87$	792.000	0.975992
$88$	0	0
$89$	−1034.00	−1.23150	−0.615752	−	0.787940i	$-0.711148\pi$
$89$	−1034.00	−1.23150	−0.615752	+	0.787940i	$0.711148\pi$
$90$	0	0
$91$	− 672.000i	− 0.774118i
$92$	0	0
$93$	960.000i	1.07040i
$94$	0	0
$95$	580.000	0.626387
$96$	0	0
$97$	−382.000	−0.399858	−0.199929	−	0.979810i	$-0.564071\pi$
$97$	−382.000	−0.399858	−0.199929	+	0.979810i	$0.564071\pi$
$98$	0	0
$99$	396.000i	0.402015i
$100$	0	0
$101$	674.000i	0.664015i	0.943277	+	0.332007i	$0.107726\pi$
$101$	674.000i	0.664015i	−0.943277	+	0.332007i	$0.892274\pi$
$102$	0	0
$103$	−992.000	−0.948977	−0.474489	−	0.880262i	$-0.657367\pi$
$103$	−992.000	−0.948977	−0.474489	+	0.880262i	$0.657367\pi$
$104$	0	0
$105$	−320.000	−0.297417
$106$	0	0
$107$	− 500.000i	− 0.451746i	−0.974157	−	0.225873i	$-0.927477\pi$
$107$	− 500.000i	− 0.451746i	0.974157	−	0.225873i	$-0.0725234\pi$
$108$	0	0
$109$	1046.00i	0.919162i	0.888136	+	0.459581i	$0.152000\pi$
$109$	1046.00i	0.919162i	−0.888136	+	0.459581i	$0.848000\pi$
$110$	0	0
$111$	1032.00	0.882460
$112$	0	0
$113$	−558.000	−0.464533	−0.232266	−	0.972652i	$-0.574614\pi$
$113$	−558.000	−0.464533	−0.232266	+	0.972652i	$0.574614\pi$
$114$	0	0
$115$	− 80.0000i	− 0.0648699i
$116$	0	0
$117$	− 462.000i	− 0.365059i
$118$	0	0
$119$	−1760.00	−1.35579
$120$	0	0
$121$	35.0000	0.0262960
$122$	0	0
$123$	1768.00i	1.29606i
$124$	0	0
$125$	125.000i	0.0894427i
$126$	0	0
$127$	328.000	0.229176	0.114588	−	0.993413i	$-0.463445\pi$
$127$	328.000	0.229176	0.114588	+	0.993413i	$0.463445\pi$
$128$	0	0
$129$	−1168.00	−0.797183
$130$	0	0
$131$	212.000i	0.141393i	0.997498	+	0.0706967i	$0.0225222\pi$
$131$	212.000i	0.141393i	−0.997498	+	0.0706967i	$0.977478\pi$
$132$	0	0
$133$	1856.00i	1.21004i
$134$	0	0
$135$	−760.000	−0.484521
$136$	0	0
$137$	−1434.00	−0.894269	−0.447135	−	0.894467i	$-0.647556\pi$
$137$	−1434.00	−0.894269	−0.447135	+	0.894467i	$0.647556\pi$
$138$	0	0
$139$	2196.00i	1.34002i	0.742354	+	0.670008i	$0.233709\pi$
$139$	2196.00i	1.34002i	−0.742354	+	0.670008i	$0.766291\pi$
$140$	0	0
$141$	1568.00i	0.936521i
$142$	0	0
$143$	1512.00	0.884194
$144$	0	0
$145$	990.000	0.567000
$146$	0	0
$147$	348.000i	0.195255i
$148$	0	0
$149$	2418.00i	1.32946i	0.747081	+	0.664732i	$0.231454\pi$
$149$	2418.00i	1.32946i	−0.747081	+	0.664732i	$0.768546\pi$
$150$	0	0
$151$	3672.00	1.97896	0.989481	−	0.144666i	$-0.0462108\pi$
$151$	3672.00	1.97896	0.989481	+	0.144666i	$0.0462108\pi$
$152$	0	0
$153$	−1210.00	−0.639364
$154$	0	0
$155$	1200.00i	0.621847i
$156$	0	0
$157$	358.000i	0.181984i	0.995852	+	0.0909921i	$0.0290038\pi$
$157$	358.000i	0.181984i	−0.995852	+	0.0909921i	$0.970996\pi$
$158$	0	0
$159$	−568.000	−0.283304
$160$	0	0
$161$	256.000	0.125314
$162$	0	0
$163$	− 2564.00i	− 1.23207i	−0.787717	−	0.616037i	$-0.788737\pi$
$163$	− 2564.00i	− 1.23207i	0.787717	−	0.616037i	$-0.211263\pi$
$164$	0	0
$165$	− 720.000i	− 0.339709i
$166$	0	0
$167$	−3056.00	−1.41605	−0.708025	−	0.706187i	$-0.750414\pi$
$167$	−3056.00	−1.41605	−0.708025	+	0.706187i	$0.750414\pi$
$168$	0	0
$169$	433.000	0.197087
$170$	0	0
$171$	1276.00i	0.570633i
$172$	0	0
$173$	− 234.000i	− 0.102836i	−0.998677	−	0.0514182i	$-0.983626\pi$
$173$	− 234.000i	− 0.102836i	0.998677	−	0.0514182i	$-0.0163741\pi$
$174$	0	0
$175$	−400.000	−0.172784
$176$	0	0
$177$	−1392.00	−0.591125
$178$	0	0
$179$	− 524.000i	− 0.218802i	−0.993998	−	0.109401i	$-0.965107\pi$
$179$	− 524.000i	− 0.218802i	0.993998	−	0.109401i	$-0.0348933\pi$
$180$	0	0
$181$	1138.00i	0.467331i	0.972317	+	0.233665i	$0.0750720\pi$
$181$	1138.00i	0.467331i	−0.972317	+	0.233665i	$0.924928\pi$
$182$	0	0
$183$	−2280.00	−0.920997
$184$	0	0
$185$	1290.00	0.512663
$186$	0	0
$187$	− 3960.00i	− 1.54858i
$188$	0	0
$189$	− 2432.00i	− 0.935989i
$190$	0	0
$191$	−1520.00	−0.575829	−0.287915	−	0.957656i	$-0.592962\pi$
$191$	−1520.00	−0.575829	−0.287915	+	0.957656i	$0.592962\pi$
$192$	0	0
$193$	−2142.00	−0.798884	−0.399442	−	0.916759i	$-0.630796\pi$
$193$	−2142.00	−0.798884	−0.399442	+	0.916759i	$0.630796\pi$
$194$	0	0
$195$	840.000i	0.308480i
$196$	0	0
$197$	2306.00i	0.833988i	0.908909	+	0.416994i	$0.136916\pi$
$197$	2306.00i	0.833988i	−0.908909	+	0.416994i	$0.863084\pi$
$198$	0	0
$199$	3288.00	1.17126	0.585628	−	0.810580i	$-0.300848\pi$
$199$	3288.00	1.17126	0.585628	+	0.810580i	$0.300848\pi$
$200$	0	0
$201$	−2768.00	−0.971342
$202$	0	0
$203$	3168.00i	1.09532i
$204$	0	0
$205$	2210.00i	0.752942i
$206$	0	0
$207$	176.000	0.0590959
$208$	0	0
$209$	−4176.00	−1.38211
$210$	0	0
$211$	3876.00i	1.26462i	0.774715	+	0.632310i	$0.217893\pi$
$211$	3876.00i	1.26462i	−0.774715	+	0.632310i	$0.782107\pi$
$212$	0	0
$213$	− 672.000i	− 0.216172i
$214$	0	0
$215$	−1460.00	−0.463122
$216$	0	0
$217$	−3840.00	−1.20127
$218$	0	0
$219$	− 536.000i	− 0.165386i
$220$	0	0
$221$	4620.00i	1.40622i
$222$	0	0
$223$	5688.00	1.70806	0.854028	−	0.520226i	$-0.174152\pi$
$223$	5688.00	1.70806	0.854028	+	0.520226i	$0.174152\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	2796.00i	0.817520i	0.912642	+	0.408760i	$0.134039\pi$
$227$	2796.00i	0.817520i	−0.912642	+	0.408760i	$0.865961\pi$
$228$	0	0
$229$	− 4446.00i	− 1.28297i	−0.767136	−	0.641485i	$-0.778319\pi$
$229$	− 4446.00i	− 1.28297i	0.767136	−	0.641485i	$-0.221681\pi$
$230$	0	0
$231$	2304.00	0.656243
$232$	0	0
$233$	−2522.00	−0.709106	−0.354553	−	0.935036i	$-0.615367\pi$
$233$	−2522.00	−0.709106	−0.354553	+	0.935036i	$0.615367\pi$
$234$	0	0
$235$	1960.00i	0.544069i
$236$	0	0
$237$	3136.00i	0.859515i
$238$	0	0
$239$	−816.000	−0.220848	−0.110424	−	0.993885i	$-0.535221\pi$
$239$	−816.000	−0.220848	−0.110424	+	0.993885i	$0.535221\pi$
$240$	0	0
$241$	−5422.00	−1.44922	−0.724609	−	0.689160i	$-0.757980\pi$
$241$	−5422.00	−1.44922	−0.724609	+	0.689160i	$0.757980\pi$
$242$	0	0
$243$	− 2860.00i	− 0.755017i
$244$	0	0
$245$	435.000i	0.113433i
$246$	0	0
$247$	4872.00	1.25505
$248$	0	0
$249$	−2256.00	−0.574169
$250$	0	0
$251$	− 5900.00i	− 1.48368i	−0.670575	−	0.741842i	$-0.733952\pi$
$251$	− 5900.00i	− 1.48368i	0.670575	−	0.741842i	$-0.266048\pi$
$252$	0	0
$253$	576.000i	0.143134i
$254$	0	0
$255$	2200.00	0.540272
$256$	0	0
$257$	5250.00	1.27426	0.637132	−	0.770754i	$-0.280120\pi$
$257$	5250.00	1.27426	0.637132	+	0.770754i	$0.280120\pi$
$258$	0	0
$259$	4128.00i	0.990353i
$260$	0	0
$261$	2178.00i	0.516532i
$262$	0	0
$263$	6240.00	1.46302	0.731511	−	0.681829i	$-0.238815\pi$
$263$	6240.00	1.46302	0.731511	+	0.681829i	$0.238815\pi$
$264$	0	0
$265$	−710.000	−0.164585
$266$	0	0
$267$	4136.00i	0.948012i
$268$	0	0
$269$	− 714.000i	− 0.161834i	−0.996721	−	0.0809170i	$-0.974215\pi$
$269$	− 714.000i	− 0.161834i	0.996721	−	0.0809170i	$-0.0257849\pi$
$270$	0	0
$271$	−2144.00	−0.480586	−0.240293	−	0.970700i	$-0.577243\pi$
$271$	−2144.00	−0.480586	−0.240293	+	0.970700i	$0.577243\pi$
$272$	0	0
$273$	−2688.00	−0.595916
$274$	0	0
$275$	− 900.000i	− 0.197353i
$276$	0	0
$277$	4466.00i	0.968722i	0.874868	+	0.484361i	$0.160948\pi$
$277$	4466.00i	0.968722i	−0.874868	+	0.484361i	$0.839052\pi$
$278$	0	0
$279$	−2640.00	−0.566497
$280$	0	0
$281$	5302.00	1.12559	0.562795	−	0.826596i	$-0.309726\pi$
$281$	5302.00	1.12559	0.562795	+	0.826596i	$0.309726\pi$
$282$	0	0
$283$	− 6932.00i	− 1.45606i	−0.685546	−	0.728029i	$-0.740436\pi$
$283$	− 6932.00i	− 1.45606i	0.685546	−	0.728029i	$-0.259564\pi$
$284$	0	0
$285$	− 2320.00i	− 0.482193i
$286$	0	0
$287$	−7072.00	−1.45452
$288$	0	0
$289$	7187.00	1.46285
$290$	0	0
$291$	1528.00i	0.307811i
$292$	0	0
$293$	4034.00i	0.804330i	0.915567	+	0.402165i	$0.131742\pi$
$293$	4034.00i	0.804330i	−0.915567	+	0.402165i	$0.868258\pi$
$294$	0	0
$295$	−1740.00	−0.343413
$296$	0	0
$297$	5472.00	1.06908
$298$	0	0
$299$	− 672.000i	− 0.129976i
$300$	0	0
$301$	− 4672.00i	− 0.894650i
$302$	0	0
$303$	2696.00	0.511159
$304$	0	0
$305$	−2850.00	−0.535051
$306$	0	0
$307$	3836.00i	0.713134i	0.934270	+	0.356567i	$0.116053\pi$
$307$	3836.00i	0.713134i	−0.934270	+	0.356567i	$0.883947\pi$
$308$	0	0
$309$	3968.00i	0.730523i
$310$	0	0
$311$	664.000	0.121067	0.0605337	−	0.998166i	$-0.480720\pi$
$311$	664.000	0.121067	0.0605337	+	0.998166i	$0.480720\pi$
$312$	0	0
$313$	−2986.00	−0.539229	−0.269615	−	0.962968i	$-0.586896\pi$
$313$	−2986.00	−0.539229	−0.269615	+	0.962968i	$0.586896\pi$
$314$	0	0
$315$	− 880.000i	− 0.157404i
$316$	0	0
$317$	2726.00i	0.482989i	0.970402	+	0.241494i	$0.0776375\pi$
$317$	2726.00i	0.482989i	−0.970402	+	0.241494i	$0.922362\pi$
$318$	0	0
$319$	−7128.00	−1.25107
$320$	0	0
$321$	−2000.00	−0.347754
$322$	0	0
$323$	− 12760.0i	− 2.19810i
$324$	0	0
$325$	1050.00i	0.179211i
$326$	0	0
$327$	4184.00	0.707571
$328$	0	0
$329$	−6272.00	−1.05102
$330$	0	0
$331$	− 9212.00i	− 1.52972i	−0.644197	−	0.764860i	$-0.722808\pi$
$331$	− 9212.00i	− 1.52972i	0.644197	−	0.764860i	$-0.277192\pi$
$332$	0	0
$333$	2838.00i	0.467031i
$334$	0	0
$335$	−3460.00	−0.564298
$336$	0	0
$337$	−3278.00	−0.529864	−0.264932	−	0.964267i	$-0.585349\pi$
$337$	−3278.00	−0.529864	−0.264932	+	0.964267i	$0.585349\pi$
$338$	0	0
$339$	2232.00i	0.357598i
$340$	0	0
$341$	− 8640.00i	− 1.37209i
$342$	0	0
$343$	−6880.00	−1.08305
$344$	0	0
$345$	−320.000	−0.0499369
$346$	0	0
$347$	4956.00i	0.766721i	0.923599	+	0.383360i	$0.125233\pi$
$347$	4956.00i	0.766721i	−0.923599	+	0.383360i	$0.874767\pi$
$348$	0	0
$349$	4678.00i	0.717500i	0.933434	+	0.358750i	$0.116797\pi$
$349$	4678.00i	0.717500i	−0.933434	+	0.358750i	$0.883203\pi$
$350$	0	0
$351$	−6384.00	−0.970805
$352$	0	0
$353$	1890.00	0.284970	0.142485	−	0.989797i	$-0.454491\pi$
$353$	1890.00	0.284970	0.142485	+	0.989797i	$0.454491\pi$
$354$	0	0
$355$	− 840.000i	− 0.125585i
$356$	0	0
$357$	7040.00i	1.04369i
$358$	0	0
$359$	−6472.00	−0.951474	−0.475737	−	0.879588i	$-0.657819\pi$
$359$	−6472.00	−0.951474	−0.475737	+	0.879588i	$0.657819\pi$
$360$	0	0
$361$	−6597.00	−0.961802
$362$	0	0
$363$	− 140.000i	− 0.0202427i
$364$	0	0
$365$	− 670.000i	− 0.0960806i
$366$	0	0
$367$	−1960.00	−0.278777	−0.139389	−	0.990238i	$-0.544514\pi$
$367$	−1960.00	−0.278777	−0.139389	+	0.990238i	$0.544514\pi$
$368$	0	0
$369$	−4862.00	−0.685923
$370$	0	0
$371$	− 2272.00i	− 0.317942i
$372$	0	0
$373$	− 8750.00i	− 1.21463i	−0.794460	−	0.607316i	$-0.792246\pi$
$373$	− 8750.00i	− 1.21463i	0.794460	−	0.607316i	$-0.207754\pi$
$374$	0	0
$375$	500.000	0.0688530
$376$	0	0
$377$	8316.00	1.13606
$378$	0	0
$379$	− 380.000i	− 0.0515021i	−0.999668	−	0.0257510i	$-0.991802\pi$
$379$	− 380.000i	− 0.0515021i	0.999668	−	0.0257510i	$-0.00819772\pi$
$380$	0	0
$381$	− 1312.00i	− 0.176419i
$382$	0	0
$383$	9688.00	1.29252	0.646258	−	0.763119i	$-0.276333\pi$
$383$	9688.00	1.29252	0.646258	+	0.763119i	$0.276333\pi$
$384$	0	0
$385$	2880.00	0.381243
$386$	0	0
$387$	− 3212.00i	− 0.421900i
$388$	0	0
$389$	− 3870.00i	− 0.504413i	−0.967673	−	0.252207i	$-0.918844\pi$
$389$	− 3870.00i	− 0.504413i	0.967673	−	0.252207i	$-0.0811563\pi$
$390$	0	0
$391$	−1760.00	−0.227639
$392$	0	0
$393$	848.000	0.108845
$394$	0	0
$395$	3920.00i	0.499333i
$396$	0	0
$397$	1622.00i	0.205053i	0.994730	+	0.102526i	$0.0326926\pi$
$397$	1622.00i	0.205053i	−0.994730	+	0.102526i	$0.967307\pi$
$398$	0	0
$399$	7424.00	0.931491
$400$	0	0
$401$	9906.00	1.23362	0.616811	−	0.787112i	$-0.288424\pi$
$401$	9906.00	1.23362	0.616811	+	0.787112i	$0.288424\pi$
$402$	0	0
$403$	10080.0i	1.24596i
$404$	0	0
$405$	1555.00i	0.190787i
$406$	0	0
$407$	−9288.00	−1.13118
$408$	0	0
$409$	4214.00	0.509459	0.254730	−	0.967012i	$-0.418014\pi$
$409$	4214.00	0.509459	0.254730	+	0.967012i	$0.418014\pi$
$410$	0	0
$411$	5736.00i	0.688409i
$412$	0	0
$413$	− 5568.00i	− 0.663398i
$414$	0	0
$415$	−2820.00	−0.333562
$416$	0	0
$417$	8784.00	1.03155
$418$	0	0
$419$	7012.00i	0.817562i	0.912632	+	0.408781i	$0.134046\pi$
$419$	7012.00i	0.817562i	−0.912632	+	0.408781i	$0.865954\pi$
$420$	0	0
$421$	1602.00i	0.185455i	0.995692	+	0.0927277i	$0.0295586\pi$
$421$	1602.00i	0.185455i	−0.995692	+	0.0927277i	$0.970441\pi$
$422$	0	0
$423$	−4312.00	−0.495642
$424$	0	0
$425$	2750.00	0.313870
$426$	0	0
$427$	− 9120.00i	− 1.03360i
$428$	0	0
$429$	− 6048.00i	− 0.680653i
$430$	0	0
$431$	3584.00	0.400546	0.200273	−	0.979740i	$-0.435817\pi$
$431$	3584.00	0.400546	0.200273	+	0.979740i	$0.435817\pi$
$432$	0	0
$433$	−3470.00	−0.385121	−0.192561	−	0.981285i	$-0.561679\pi$
$433$	−3470.00	−0.385121	−0.192561	+	0.981285i	$0.561679\pi$
$434$	0	0
$435$	− 3960.00i	− 0.436477i
$436$	0	0
$437$	1856.00i	0.203168i
$438$	0	0
$439$	−3416.00	−0.371382	−0.185691	−	0.982608i	$-0.559452\pi$
$439$	−3416.00	−0.371382	−0.185691	+	0.982608i	$0.559452\pi$
$440$	0	0
$441$	−957.000	−0.103337
$442$	0	0
$443$	9708.00i	1.04118i	0.853808	+	0.520588i	$0.174287\pi$
$443$	9708.00i	1.04118i	−0.853808	+	0.520588i	$0.825713\pi$
$444$	0	0
$445$	5170.00i	0.550745i
$446$	0	0
$447$	9672.00	1.02342
$448$	0	0
$449$	−10366.0	−1.08954	−0.544768	−	0.838587i	$-0.683382\pi$
$449$	−10366.0	−1.08954	−0.544768	+	0.838587i	$0.683382\pi$
$450$	0	0
$451$	− 15912.0i	− 1.66135i
$452$	0	0
$453$	− 14688.0i	− 1.52340i
$454$	0	0
$455$	−3360.00	−0.346196
$456$	0	0
$457$	16742.0	1.71369	0.856847	−	0.515572i	$-0.172420\pi$
$457$	16742.0	1.71369	0.856847	+	0.515572i	$0.172420\pi$
$458$	0	0
$459$	16720.0i	1.70027i
$460$	0	0
$461$	− 1258.00i	− 0.127095i	−0.997979	−	0.0635476i	$-0.979759\pi$
$461$	− 1258.00i	− 0.127095i	0.997979	−	0.0635476i	$-0.0202415\pi$
$462$	0	0
$463$	−13528.0	−1.35788	−0.678941	−	0.734193i	$-0.737561\pi$
$463$	−13528.0	−1.35788	−0.678941	+	0.734193i	$0.737561\pi$
$464$	0	0
$465$	4800.00	0.478698
$466$	0	0
$467$	− 6916.00i	− 0.685298i	−0.939463	−	0.342649i	$-0.888676\pi$
$467$	− 6916.00i	− 0.685298i	0.939463	−	0.342649i	$-0.111324\pi$
$468$	0	0
$469$	− 11072.0i	− 1.09010i
$470$	0	0
$471$	1432.00	0.140091
$472$	0	0
$473$	10512.0	1.02187
$474$	0	0
$475$	− 2900.00i	− 0.280129i
$476$	0	0
$477$	− 1562.00i	− 0.149935i
$478$	0	0
$479$	−1728.00	−0.164832	−0.0824158	−	0.996598i	$-0.526264\pi$
$479$	−1728.00	−0.164832	−0.0824158	+	0.996598i	$0.526264\pi$
$480$	0	0
$481$	10836.0	1.02719
$482$	0	0
$483$	− 1024.00i	− 0.0964671i
$484$	0	0
$485$	1910.00i	0.178822i
$486$	0	0
$487$	16656.0	1.54981	0.774903	−	0.632080i	$-0.217799\pi$
$487$	16656.0	1.54981	0.774903	+	0.632080i	$0.217799\pi$
$488$	0	0
$489$	−10256.0	−0.948451
$490$	0	0
$491$	− 1084.00i	− 0.0996339i	−0.998758	−	0.0498169i	$-0.984136\pi$
$491$	− 1084.00i	− 0.0996339i	0.998758	−	0.0498169i	$-0.0158638\pi$
$492$	0	0
$493$	− 21780.0i	− 1.98970i
$494$	0	0
$495$	1980.00	0.179787
$496$	0	0
$497$	2688.00	0.242602
$498$	0	0
$499$	− 5804.00i	− 0.520687i	−0.965516	−	0.260343i	$-0.916164\pi$
$499$	− 5804.00i	− 0.520687i	0.965516	−	0.260343i	$-0.0838358\pi$
$500$	0	0
$501$	12224.0i	1.09008i
$502$	0	0
$503$	10512.0	0.931823	0.465911	−	0.884831i	$-0.345727\pi$
$503$	10512.0	0.931823	0.465911	+	0.884831i	$0.345727\pi$
$504$	0	0
$505$	3370.00	0.296956
$506$	0	0
$507$	− 1732.00i	− 0.151718i
$508$	0	0
$509$	− 4314.00i	− 0.375667i	−0.982201	−	0.187834i	$-0.939853\pi$
$509$	− 4314.00i	− 0.375667i	0.982201	−	0.187834i	$-0.0601466\pi$
$510$	0	0
$511$	2144.00	0.185607
$512$	0	0
$513$	17632.0	1.51749
$514$	0	0
$515$	4960.00i	0.424396i
$516$	0	0
$517$	− 14112.0i	− 1.20047i
$518$	0	0
$519$	−936.000	−0.0791635
$520$	0	0
$521$	1190.00	0.100067	0.0500334	−	0.998748i	$-0.484067\pi$
$521$	1190.00	0.100067	0.0500334	+	0.998748i	$0.484067\pi$
$522$	0	0
$523$	− 3780.00i	− 0.316038i	−0.987436	−	0.158019i	$-0.949489\pi$
$523$	− 3780.00i	− 0.316038i	0.987436	−	0.158019i	$-0.0505107\pi$
$524$	0	0
$525$	1600.00i	0.133009i
$526$	0	0
$527$	26400.0	2.18217
$528$	0	0
$529$	−11911.0	−0.978959
$530$	0	0
$531$	− 3828.00i	− 0.312846i
$532$	0	0
$533$	18564.0i	1.50862i
$534$	0	0
$535$	−2500.00	−0.202027
$536$	0	0
$537$	−2096.00	−0.168434
$538$	0	0
$539$	− 3132.00i	− 0.250287i
$540$	0	0
$541$	− 11002.0i	− 0.874331i	−0.899381	−	0.437165i	$-0.855982\pi$
$541$	− 11002.0i	− 0.874331i	0.899381	−	0.437165i	$-0.144018\pi$
$542$	0	0
$543$	4552.00	0.359751
$544$	0	0
$545$	5230.00	0.411062
$546$	0	0
$547$	− 5908.00i	− 0.461806i	−0.972977	−	0.230903i	$-0.925832\pi$
$547$	− 5908.00i	− 0.461806i	0.972977	−	0.230903i	$-0.0741680\pi$
$548$	0	0
$549$	− 6270.00i	− 0.487426i
$550$	0	0
$551$	−22968.0	−1.77581
$552$	0	0
$553$	−12544.0	−0.964602
$554$	0	0
$555$	− 5160.00i	− 0.394648i
$556$	0	0
$557$	14806.0i	1.12630i	0.826354	+	0.563151i	$0.190411\pi$
$557$	14806.0i	1.12630i	−0.826354	+	0.563151i	$0.809589\pi$
$558$	0	0
$559$	−12264.0	−0.927928
$560$	0	0
$561$	−15840.0	−1.19210
$562$	0	0
$563$	684.000i	0.0512028i	0.999672	+	0.0256014i	$0.00815007\pi$
$563$	684.000i	0.0512028i	−0.999672	+	0.0256014i	$0.991850\pi$
$564$	0	0
$565$	2790.00i	0.207745i
$566$	0	0
$567$	−4976.00	−0.368558
$568$	0	0
$569$	2582.00	0.190234	0.0951169	−	0.995466i	$-0.469677\pi$
$569$	2582.00	0.190234	0.0951169	+	0.995466i	$0.469677\pi$
$570$	0	0
$571$	− 2540.00i	− 0.186157i	−0.995659	−	0.0930785i	$-0.970329\pi$
$571$	− 2540.00i	− 0.186157i	0.995659	−	0.0930785i	$-0.0296708\pi$
$572$	0	0
$573$	6080.00i	0.443273i
$574$	0	0
$575$	−400.000	−0.0290107
$576$	0	0
$577$	22786.0	1.64401	0.822005	−	0.569480i	$-0.192856\pi$
$577$	22786.0	1.64401	0.822005	+	0.569480i	$0.192856\pi$
$578$	0	0
$579$	8568.00i	0.614981i
$580$	0	0
$581$	− 9024.00i	− 0.644369i
$582$	0	0
$583$	5112.00	0.363152
$584$	0	0
$585$	−2310.00	−0.163259
$586$	0	0
$587$	7884.00i	0.554357i	0.960818	+	0.277178i	$0.0893993\pi$
$587$	7884.00i	0.554357i	−0.960818	+	0.277178i	$0.910601\pi$
$588$	0	0
$589$	− 27840.0i	− 1.94758i
$590$	0	0
$591$	9224.00	0.642005
$592$	0	0
$593$	−21902.0	−1.51671	−0.758354	−	0.651843i	$-0.773996\pi$
$593$	−21902.0	−1.51671	−0.758354	+	0.651843i	$0.773996\pi$
$594$	0	0
$595$	8800.00i	0.606327i
$596$	0	0
$597$	− 13152.0i	− 0.901634i
$598$	0	0
$599$	15080.0	1.02863	0.514317	−	0.857600i	$-0.328045\pi$
$599$	15080.0	1.02863	0.514317	+	0.857600i	$0.328045\pi$
$600$	0	0
$601$	19702.0	1.33721	0.668603	−	0.743619i	$-0.266892\pi$
$601$	19702.0	1.33721	0.668603	+	0.743619i	$0.266892\pi$
$602$	0	0
$603$	− 7612.00i	− 0.514071i
$604$	0	0
$605$	− 175.000i	− 0.0117599i
$606$	0	0
$607$	−7320.00	−0.489472	−0.244736	−	0.969590i	$-0.578701\pi$
$607$	−7320.00	−0.489472	−0.244736	+	0.969590i	$0.578701\pi$
$608$	0	0
$609$	12672.0	0.843178
$610$	0	0
$611$	16464.0i	1.09012i
$612$	0	0
$613$	− 24350.0i	− 1.60438i	−0.597066	−	0.802192i	$-0.703667\pi$
$613$	− 24350.0i	− 1.60438i	0.597066	−	0.802192i	$-0.296333\pi$
$614$	0	0
$615$	8840.00	0.579615
$616$	0	0
$617$	−19546.0	−1.27535	−0.637676	−	0.770305i	$-0.720104\pi$
$617$	−19546.0	−1.27535	−0.637676	+	0.770305i	$0.720104\pi$
$618$	0	0
$619$	3476.00i	0.225706i	0.993612	+	0.112853i	$0.0359990\pi$
$619$	3476.00i	0.225706i	−0.993612	+	0.112853i	$0.964001\pi$
$620$	0	0
$621$	− 2432.00i	− 0.157154i
$622$	0	0
$623$	−16544.0	−1.06392
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	16704.0i	1.06394i
$628$	0	0
$629$	− 28380.0i	− 1.79902i
$630$	0	0
$631$	21880.0	1.38039	0.690197	−	0.723621i	$-0.257524\pi$
$631$	21880.0	1.38039	0.690197	+	0.723621i	$0.257524\pi$
$632$	0	0
$633$	15504.0	0.973505
$634$	0	0
$635$	− 1640.00i	− 0.102490i
$636$	0	0
$637$	3654.00i	0.227279i
$638$	0	0
$639$	1848.00	0.114406
$640$	0	0
$641$	20994.0	1.29362	0.646812	−	0.762649i	$-0.276102\pi$
$641$	20994.0	1.29362	0.646812	+	0.762649i	$0.276102\pi$
$642$	0	0
$643$	18204.0i	1.11648i	0.829680	+	0.558239i	$0.188523\pi$
$643$	18204.0i	1.11648i	−0.829680	+	0.558239i	$0.811477\pi$
$644$	0	0
$645$	5840.00i	0.356511i
$646$	0	0
$647$	−2064.00	−0.125416	−0.0627080	−	0.998032i	$-0.519974\pi$
$647$	−2064.00	−0.125416	−0.0627080	+	0.998032i	$0.519974\pi$
$648$	0	0
$649$	12528.0	0.757730
$650$	0	0
$651$	15360.0i	0.924740i
$652$	0	0
$653$	9942.00i	0.595805i	0.954596	+	0.297902i	$0.0962870\pi$
$653$	9942.00i	0.595805i	−0.954596	+	0.297902i	$0.903713\pi$
$654$	0	0
$655$	1060.00	0.0632330
$656$	0	0
$657$	1474.00	0.0875285
$658$	0	0
$659$	− 24236.0i	− 1.43263i	−0.697779	−	0.716313i	$-0.745828\pi$
$659$	− 24236.0i	− 1.43263i	0.697779	−	0.716313i	$-0.254172\pi$
$660$	0	0
$661$	− 17614.0i	− 1.03647i	−0.855239	−	0.518234i	$-0.826590\pi$
$661$	− 17614.0i	− 1.03647i	0.855239	−	0.518234i	$-0.173410\pi$
$662$	0	0
$663$	18480.0	1.08251
$664$	0	0
$665$	9280.00	0.541147
$666$	0	0
$667$	3168.00i	0.183906i
$668$	0	0
$669$	− 22752.0i	− 1.31486i
$670$	0	0
$671$	20520.0	1.18057
$672$	0	0
$673$	13058.0	0.747918	0.373959	−	0.927445i	$-0.378000\pi$
$673$	13058.0	0.747918	0.373959	+	0.927445i	$0.378000\pi$
$674$	0	0
$675$	3800.00i	0.216685i
$676$	0	0
$677$	33186.0i	1.88396i	0.335668	+	0.941980i	$0.391038\pi$
$677$	33186.0i	1.88396i	−0.335668	+	0.941980i	$0.608962\pi$
$678$	0	0
$679$	−6112.00	−0.345445
$680$	0	0
$681$	11184.0	0.629327
$682$	0	0
$683$	− 31716.0i	− 1.77684i	−0.459035	−	0.888418i	$-0.651805\pi$
$683$	− 31716.0i	− 1.77684i	0.459035	−	0.888418i	$-0.348195\pi$
$684$	0	0
$685$	7170.00i	0.399929i
$686$	0	0
$687$	−17784.0	−0.987630
$688$	0	0
$689$	−5964.00	−0.329768
$690$	0	0
$691$	2084.00i	0.114731i	0.998353	+	0.0573655i	$0.0182700\pi$
$691$	2084.00i	0.114731i	−0.998353	+	0.0573655i	$0.981730\pi$
$692$	0	0
$693$	6336.00i	0.347308i
$694$	0	0
$695$	10980.0	0.599274
$696$	0	0
$697$	48620.0	2.64220
$698$	0	0
$699$	10088.0i	0.545870i
$700$	0	0
$701$	− 7418.00i	− 0.399678i	−0.979829	−	0.199839i	$-0.935958\pi$
$701$	− 7418.00i	− 0.399678i	0.979829	−	0.199839i	$-0.0640418\pi$
$702$	0	0
$703$	−29928.0	−1.60563
$704$	0	0
$705$	7840.00	0.418825
$706$	0	0
$707$	10784.0i	0.573655i
$708$	0	0
$709$	18242.0i	0.966280i	0.875543	+	0.483140i	$0.160504\pi$
$709$	18242.0i	0.966280i	−0.875543	+	0.483140i	$0.839496\pi$
$710$	0	0
$711$	−8624.00	−0.454888
$712$	0	0
$713$	−3840.00	−0.201696
$714$	0	0
$715$	− 7560.00i	− 0.395424i
$716$	0	0
$717$	3264.00i	0.170009i
$718$	0	0
$719$	−3024.00	−0.156851	−0.0784257	−	0.996920i	$-0.524989\pi$
$719$	−3024.00	−0.156851	−0.0784257	+	0.996920i	$0.524989\pi$
$720$	0	0
$721$	−15872.0	−0.819839
$722$	0	0
$723$	21688.0i	1.11561i
$724$	0	0
$725$	− 4950.00i	− 0.253570i
$726$	0	0
$727$	−26176.0	−1.33537	−0.667685	−	0.744444i	$-0.732715\pi$
$727$	−26176.0	−1.33537	−0.667685	+	0.744444i	$0.732715\pi$
$728$	0	0
$729$	−19837.0	−1.00782
$730$	0	0
$731$	32120.0i	1.62517i
$732$	0	0
$733$	− 17818.0i	− 0.897848i	−0.893570	−	0.448924i	$-0.851807\pi$
$733$	− 17818.0i	− 0.897848i	0.893570	−	0.448924i	$-0.148193\pi$
$734$	0	0
$735$	1740.00	0.0873209
$736$	0	0
$737$	24912.0	1.24511
$738$	0	0
$739$	22052.0i	1.09769i	0.835923	+	0.548847i	$0.184933\pi$
$739$	22052.0i	1.09769i	−0.835923	+	0.548847i	$0.815067\pi$
$740$	0	0
$741$	− 19488.0i	− 0.966140i
$742$	0	0
$743$	15840.0	0.782117	0.391059	−	0.920366i	$-0.372109\pi$
$743$	15840.0	0.782117	0.391059	+	0.920366i	$0.372109\pi$
$744$	0	0
$745$	12090.0	0.594555
$746$	0	0
$747$	− 6204.00i	− 0.303872i
$748$	0	0
$749$	− 8000.00i	− 0.390272i
$750$	0	0
$751$	−21024.0	−1.02154	−0.510770	−	0.859717i	$-0.670640\pi$
$751$	−21024.0	−1.02154	−0.510770	+	0.859717i	$0.670640\pi$
$752$	0	0
$753$	−23600.0	−1.14214
$754$	0	0
$755$	− 18360.0i	− 0.885018i
$756$	0	0
$757$	38034.0i	1.82612i	0.407831	+	0.913058i	$0.366285\pi$
$757$	38034.0i	1.82612i	−0.407831	+	0.913058i	$0.633715\pi$
$758$	0	0
$759$	2304.00	0.110184
$760$	0	0
$761$	−37802.0	−1.80069	−0.900343	−	0.435182i	$-0.856684\pi$
$761$	−37802.0	−1.80069	−0.900343	+	0.435182i	$0.856684\pi$
$762$	0	0
$763$	16736.0i	0.794081i
$764$	0	0
$765$	6050.00i	0.285932i
$766$	0	0
$767$	−14616.0	−0.688075
$768$	0	0
$769$	15042.0	0.705369	0.352684	−	0.935742i	$-0.385269\pi$
$769$	15042.0	0.705369	0.352684	+	0.935742i	$0.385269\pi$
$770$	0	0
$771$	− 21000.0i	− 0.980929i
$772$	0	0
$773$	− 5950.00i	− 0.276852i	−0.990373	−	0.138426i	$-0.955796\pi$
$773$	− 5950.00i	− 0.276852i	0.990373	−	0.138426i	$-0.0442043\pi$
$774$	0	0
$775$	6000.00	0.278099
$776$	0	0
$777$	16512.0	0.762374
$778$	0	0
$779$	− 51272.0i	− 2.35816i
$780$	0	0
$781$	6048.00i	0.277099i
$782$	0	0
$783$	30096.0	1.37362
$784$	0	0
$785$	1790.00	0.0813858
$786$	0	0
$787$	− 23364.0i	− 1.05824i	−0.848546	−	0.529121i	$-0.822522\pi$
$787$	− 23364.0i	− 1.05824i	0.848546	−	0.529121i	$-0.177478\pi$
$788$	0	0
$789$	− 24960.0i	− 1.12624i
$790$	0	0
$791$	−8928.00	−0.401319
$792$	0	0
$793$	−23940.0	−1.07205
$794$	0	0
$795$	2840.00i	0.126697i
$796$	0	0
$797$	19846.0i	0.882034i	0.897499	+	0.441017i	$0.145382\pi$
$797$	19846.0i	0.882034i	−0.897499	+	0.441017i	$0.854618\pi$
$798$	0	0
$799$	43120.0	1.90923
$800$	0	0
$801$	−11374.0	−0.501724
$802$	0	0
$803$	4824.00i	0.211999i
$804$	0	0
$805$	− 1280.00i	− 0.0560423i
$806$	0	0
$807$	−2856.00	−0.124580
$808$	0	0
$809$	−24762.0	−1.07613	−0.538063	−	0.842905i	$-0.680844\pi$
$809$	−24762.0	−1.07613	−0.538063	+	0.842905i	$0.680844\pi$
$810$	0	0
$811$	16644.0i	0.720653i	0.932826	+	0.360327i	$0.117335\pi$
$811$	16644.0i	0.720653i	−0.932826	+	0.360327i	$0.882665\pi$
$812$	0	0
$813$	8576.00i	0.369955i
$814$	0	0
$815$	−12820.0	−0.551000
$816$	0	0
$817$	33872.0	1.45047
$818$	0	0
$819$	− 7392.00i	− 0.315381i
$820$	0	0
$821$	− 3182.00i	− 0.135265i	−0.997710	−	0.0676325i	$-0.978455\pi$
$821$	− 3182.00i	− 0.135265i	0.997710	−	0.0676325i	$-0.0215445\pi$
$822$	0	0
$823$	−7504.00	−0.317829	−0.158914	−	0.987292i	$-0.550799\pi$
$823$	−7504.00	−0.317829	−0.158914	+	0.987292i	$0.550799\pi$
$824$	0	0
$825$	−3600.00	−0.151922
$826$	0	0
$827$	12604.0i	0.529969i	0.964253	+	0.264984i	$0.0853668\pi$
$827$	12604.0i	0.529969i	−0.964253	+	0.264984i	$0.914633\pi$
$828$	0	0
$829$	12230.0i	0.512383i	0.966626	+	0.256191i	$0.0824678\pi$
$829$	12230.0i	0.512383i	−0.966626	+	0.256191i	$0.917532\pi$
$830$	0	0
$831$	17864.0	0.745722
$832$	0	0
$833$	9570.00	0.398056
$834$	0	0
$835$	15280.0i	0.633277i
$836$	0	0
$837$	36480.0i	1.50649i
$838$	0	0
$839$	9656.00	0.397333	0.198666	−	0.980067i	$-0.436339\pi$
$839$	9656.00	0.397333	0.198666	+	0.980067i	$0.436339\pi$
$840$	0	0
$841$	−14815.0	−0.607446
$842$	0	0
$843$	− 21208.0i	− 0.866480i
$844$	0	0
$845$	− 2165.00i	− 0.0881400i
$846$	0	0
$847$	560.000	0.0227176
$848$	0	0
$849$	−27728.0	−1.12087
$850$	0	0
$851$	4128.00i	0.166282i
$852$	0	0
$853$	− 5806.00i	− 0.233052i	−0.993188	−	0.116526i	$-0.962824\pi$
$853$	− 5806.00i	− 0.233052i	0.993188	−	0.116526i	$-0.0371759\pi$
$854$	0	0
$855$	6380.00	0.255195
$856$	0	0
$857$	39094.0	1.55826	0.779128	−	0.626865i	$-0.215662\pi$
$857$	39094.0	1.55826	0.779128	+	0.626865i	$0.215662\pi$
$858$	0	0
$859$	− 18876.0i	− 0.749756i	−0.927074	−	0.374878i	$-0.877685\pi$
$859$	− 18876.0i	− 0.749756i	0.927074	−	0.374878i	$-0.122315\pi$
$860$	0	0
$861$	28288.0i	1.11969i
$862$	0	0
$863$	−32296.0	−1.27389	−0.636946	−	0.770909i	$-0.719803\pi$
$863$	−32296.0	−1.27389	−0.636946	+	0.770909i	$0.719803\pi$
$864$	0	0
$865$	−1170.00	−0.0459898
$866$	0	0
$867$	− 28748.0i	− 1.12611i
$868$	0	0
$869$	− 28224.0i	− 1.10176i
$870$	0	0
$871$	−29064.0	−1.13065
$872$	0	0
$873$	−4202.00	−0.162905
$874$	0	0
$875$	2000.00i	0.0772712i
$876$	0	0
$877$	− 9578.00i	− 0.368787i	−0.982853	−	0.184393i	$-0.940968\pi$
$877$	− 9578.00i	− 0.368787i	0.982853	−	0.184393i	$-0.0590321\pi$
$878$	0	0
$879$	16136.0	0.619174
$880$	0	0
$881$	−41710.0	−1.59506	−0.797529	−	0.603281i	$-0.793860\pi$
$881$	−41710.0	−1.59506	−0.797529	+	0.603281i	$0.793860\pi$
$882$	0	0
$883$	− 2260.00i	− 0.0861326i	−0.999072	−	0.0430663i	$-0.986287\pi$
$883$	− 2260.00i	− 0.0861326i	0.999072	−	0.0430663i	$-0.0137127\pi$
$884$	0	0
$885$	6960.00i	0.264359i
$886$	0	0
$887$	−33696.0	−1.27554	−0.637768	−	0.770228i	$-0.720142\pi$
$887$	−33696.0	−1.27554	−0.637768	+	0.770228i	$0.720142\pi$
$888$	0	0
$889$	5248.00	0.197989
$890$	0	0
$891$	− 11196.0i	− 0.420965i
$892$	0	0
$893$	− 45472.0i	− 1.70399i
$894$	0	0
$895$	−2620.00	−0.0978513
$896$	0	0
$897$	−2688.00	−0.100055
$898$	0	0
$899$	− 47520.0i	− 1.76294i
$900$	0	0
$901$	15620.0i	0.577556i
$902$	0	0
$903$	−18688.0	−0.688702
$904$	0	0
$905$	5690.00	0.208997
$906$	0	0
$907$	7756.00i	0.283940i	0.989871	+	0.141970i	$0.0453437\pi$
$907$	7756.00i	0.283940i	−0.989871	+	0.141970i	$0.954656\pi$
$908$	0	0
$909$	7414.00i	0.270525i
$910$	0	0
$911$	5312.00	0.193188	0.0965941	−	0.995324i	$-0.469205\pi$
$911$	5312.00	0.193188	0.0965941	+	0.995324i	$0.469205\pi$
$912$	0	0
$913$	20304.0	0.735996
$914$	0	0
$915$	11400.0i	0.411882i
$916$	0	0
$917$	3392.00i	0.122152i
$918$	0	0
$919$	−23576.0	−0.846246	−0.423123	−	0.906072i	$-0.639066\pi$
$919$	−23576.0	−0.846246	−0.423123	+	0.906072i	$0.639066\pi$
$920$	0	0
$921$	15344.0	0.548971
$922$	0	0
$923$	− 7056.00i	− 0.251626i
$924$	0	0
$925$	− 6450.00i	− 0.229270i
$926$	0	0
$927$	−10912.0	−0.386620
$928$	0	0
$929$	−19038.0	−0.672354	−0.336177	−	0.941799i	$-0.609134\pi$
$929$	−19038.0	−0.672354	−0.336177	+	0.941799i	$0.609134\pi$
$930$	0	0
$931$	− 10092.0i	− 0.355265i
$932$	0	0
$933$	− 2656.00i	− 0.0931978i
$934$	0	0
$935$	−19800.0	−0.692545
$936$	0	0
$937$	−20570.0	−0.717175	−0.358587	−	0.933496i	$-0.616741\pi$
$937$	−20570.0	−0.717175	−0.358587	+	0.933496i	$0.616741\pi$
$938$	0	0
$939$	11944.0i	0.415099i
$940$	0	0
$941$	− 21386.0i	− 0.740875i	−0.928857	−	0.370438i	$-0.879208\pi$
$941$	− 21386.0i	− 0.740875i	0.928857	−	0.370438i	$-0.120792\pi$
$942$	0	0
$943$	−7072.00	−0.244216
$944$	0	0
$945$	−12160.0	−0.418587
$946$	0	0
$947$	− 38020.0i	− 1.30463i	−0.757948	−	0.652315i	$-0.773798\pi$
$947$	− 38020.0i	− 1.30463i	0.757948	−	0.652315i	$-0.226202\pi$
$948$	0	0
$949$	− 5628.00i	− 0.192511i
$950$	0	0
$951$	10904.0	0.371805
$952$	0	0
$953$	−20202.0	−0.686681	−0.343340	−	0.939211i	$-0.611558\pi$
$953$	−20202.0	−0.686681	−0.343340	+	0.939211i	$0.611558\pi$
$954$	0	0
$955$	7600.00i	0.257519i
$956$	0	0
$957$	28512.0i	0.963074i
$958$	0	0
$959$	−22944.0	−0.772576
$960$	0	0
$961$	27809.0	0.933470
$962$	0	0
$963$	− 5500.00i	− 0.184045i
$964$	0	0
$965$	10710.0i	0.357272i
$966$	0	0
$967$	29840.0	0.992337	0.496168	−	0.868226i	$-0.334740\pi$
$967$	29840.0	0.992337	0.496168	+	0.868226i	$0.334740\pi$
$968$	0	0
$969$	−51040.0	−1.69210
$970$	0	0
$971$	− 12476.0i	− 0.412332i	−0.978517	−	0.206166i	$-0.933901\pi$
$971$	− 12476.0i	− 0.412332i	0.978517	−	0.206166i	$-0.0660986\pi$
$972$	0	0
$973$	35136.0i	1.15767i
$974$	0	0
$975$	4200.00	0.137957
$976$	0	0
$977$	−36974.0	−1.21075	−0.605375	−	0.795940i	$-0.706977\pi$
$977$	−36974.0	−1.21075	−0.605375	+	0.795940i	$0.706977\pi$
$978$	0	0
$979$	− 37224.0i	− 1.21520i
$980$	0	0
$981$	11506.0i	0.374473i
$982$	0	0
$983$	−16368.0	−0.531087	−0.265543	−	0.964099i	$-0.585551\pi$
$983$	−16368.0	−0.531087	−0.265543	+	0.964099i	$0.585551\pi$
$984$	0	0
$985$	11530.0	0.372971
$986$	0	0
$987$	25088.0i	0.809078i
$988$	0	0
$989$	− 4672.00i	− 0.150213i
$990$	0	0
$991$	49552.0	1.58837	0.794183	−	0.607678i	$-0.207899\pi$
$991$	49552.0	1.58837	0.794183	+	0.607678i	$0.207899\pi$
$992$	0	0
$993$	−36848.0	−1.17758
$994$	0	0
$995$	− 16440.0i	− 0.523802i
$996$	0	0
$997$	− 24414.0i	− 0.775526i	−0.921759	−	0.387763i	$-0.873248\pi$
$997$	− 24414.0i	− 0.775526i	0.921759	−	0.387763i	$-0.126752\pi$
$998$	0	0
$999$	39216.0	1.24198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.d.m.641.1		2
4.3	odd	2		1280.4.d.d.641.2		2
8.3	odd	2		1280.4.d.d.641.1		2
8.5	even	2	inner	1280.4.d.m.641.2		2
16.3	odd	4		320.4.a.e.1.1		1
16.5	even	4		80.4.a.b.1.1		1
16.11	odd	4		40.4.a.b.1.1	✓	1
16.13	even	4		320.4.a.j.1.1		1
48.5	odd	4		720.4.a.d.1.1		1
48.11	even	4		360.4.a.f.1.1		1
80.19	odd	4		1600.4.a.bk.1.1		1
80.27	even	4		200.4.c.f.49.1		2
80.29	even	4		1600.4.a.q.1.1		1
80.37	odd	4		400.4.c.h.49.2		2
80.43	even	4		200.4.c.f.49.2		2
80.53	odd	4		400.4.c.h.49.1		2
80.59	odd	4		200.4.a.d.1.1		1
80.69	even	4		400.4.a.p.1.1		1
112.27	even	4		1960.4.a.e.1.1		1
240.59	even	4		1800.4.a.h.1.1		1
240.107	odd	4		1800.4.f.d.649.2		2
240.203	odd	4		1800.4.f.d.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.b.1.1	✓	1	16.11	odd	4
80.4.a.b.1.1		1	16.5	even	4
200.4.a.d.1.1		1	80.59	odd	4
200.4.c.f.49.1		2	80.27	even	4
200.4.c.f.49.2		2	80.43	even	4
320.4.a.e.1.1		1	16.3	odd	4
320.4.a.j.1.1		1	16.13	even	4
360.4.a.f.1.1		1	48.11	even	4
400.4.a.p.1.1		1	80.69	even	4
400.4.c.h.49.1		2	80.53	odd	4
400.4.c.h.49.2		2	80.37	odd	4
720.4.a.d.1.1		1	48.5	odd	4
1280.4.d.d.641.1		2	8.3	odd	2
1280.4.d.d.641.2		2	4.3	odd	2
1280.4.d.m.641.1		2	1.1	even	1	trivial
1280.4.d.m.641.2		2	8.5	even	2	inner
1600.4.a.q.1.1		1	80.29	even	4
1600.4.a.bk.1.1		1	80.19	odd	4
1800.4.a.h.1.1		1	240.59	even	4
1800.4.f.d.649.1		2	240.203	odd	4
1800.4.f.d.649.2		2	240.107	odd	4
1960.4.a.e.1.1		1	112.27	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)

\(f(q)\)	\(=\)	\(q-4.00000i q^{3} -5.00000i q^{5} +16.0000 q^{7} +11.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{3} -5.00000i q^{5} +16.0000 q^{7} +11.0000 q^{9} +36.0000i q^{11} -42.0000i q^{13} -20.0000 q^{15} -110.000 q^{17} +116.000i q^{19} -64.0000i q^{21} +16.0000 q^{23} -25.0000 q^{25} -152.000i q^{27} +198.000i q^{29} -240.000 q^{31} +144.000 q^{33} -80.0000i q^{35} +258.000i q^{37} -168.000 q^{39} -442.000 q^{41} -292.000i q^{43} -55.0000i q^{45} -392.000 q^{47} -87.0000 q^{49} +440.000i q^{51} -142.000i q^{53} +180.000 q^{55} +464.000 q^{57} -348.000i q^{59} -570.000i q^{61} +176.000 q^{63} -210.000 q^{65} -692.000i q^{67} -64.0000i q^{69} +168.000 q^{71} +134.000 q^{73} +100.000i q^{75} +576.000i q^{77} -784.000 q^{79} -311.000 q^{81} -564.000i q^{83} +550.000i q^{85} +792.000 q^{87} -1034.00 q^{89} -672.000i q^{91} +960.000i q^{93} +580.000 q^{95} -382.000 q^{97} +396.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{7} + 22 q^{9}+O(q^{10}) \) 2 * q + 32 * q^7 + 22 * q^9	\( 2 q + 32 q^{7} + 22 q^{9} - 40 q^{15} - 220 q^{17} + 32 q^{23} - 50 q^{25} - 480 q^{31} + 288 q^{33} - 336 q^{39} - 884 q^{41} - 784 q^{47} - 174 q^{49} + 360 q^{55} + 928 q^{57} + 352 q^{63} - 420 q^{65} + 336 q^{71} + 268 q^{73} - 1568 q^{79} - 622 q^{81} + 1584 q^{87} - 2068 q^{89} + 1160 q^{95} - 764 q^{97}+O(q^{100}) \) 2 * q + 32 * q^7 + 22 * q^9 - 40 * q^15 - 220 * q^17 + 32 * q^23 - 50 * q^25 - 480 * q^31 + 288 * q^33 - 336 * q^39 - 884 * q^41 - 784 * q^47 - 174 * q^49 + 360 * q^55 + 928 * q^57 + 352 * q^63 - 420 * q^65 + 336 * q^71 + 268 * q^73 - 1568 * q^79 - 622 * q^81 + 1584 * q^87 - 2068 * q^89 + 1160 * q^95 - 764 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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