LMFDB - Embedded newform 1280.4.d.n.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 20)
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.n.641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{3} -5.00000i q^{5} +16.0000 q^{7} +11.0000 q^{9} +60.0000i q^{11} +86.0000i q^{13} +20.0000 q^{15} +18.0000 q^{17} +44.0000i q^{19} +64.0000i q^{21} -48.0000 q^{23} -25.0000 q^{25} +152.000i q^{27} -186.000i q^{29} +176.000 q^{31} -240.000 q^{33} -80.0000i q^{35} -254.000i q^{37} -344.000 q^{39} -186.000 q^{41} +100.000i q^{43} -55.0000i q^{45} +168.000 q^{47} -87.0000 q^{49} +72.0000i q^{51} +498.000i q^{53} +300.000 q^{55} -176.000 q^{57} +252.000i q^{59} -58.0000i q^{61} +176.000 q^{63} +430.000 q^{65} -1036.00i q^{67} -192.000i q^{69} -168.000 q^{71} -506.000 q^{73} -100.000i q^{75} +960.000i q^{77} +272.000 q^{79} -311.000 q^{81} +948.000i q^{83} -90.0000i q^{85} +744.000 q^{87} +1014.00 q^{89} +1376.00i q^{91} +704.000i q^{93} +220.000 q^{95} -766.000 q^{97} +660.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{7} + 22 q^{9} + 40 q^{15} + 36 q^{17} - 96 q^{23} - 50 q^{25} + 352 q^{31} - 480 q^{33} - 688 q^{39} - 372 q^{41} + 336 q^{47} - 174 q^{49} + 600 q^{55} - 352 q^{57} + 352 q^{63} + 860 q^{65}+ \cdots - 1532 q^{97}+O(q^{100}) $ 2 * q + 32 * q^7 + 22 * q^9 + 40 * q^15 + 36 * q^17 - 96 * q^23 - 50 * q^25 + 352 * q^31 - 480 * q^33 - 688 * q^39 - 372 * q^41 + 336 * q^47 - 174 * q^49 + 600 * q^55 - 352 * q^57 + 352 * q^63 + 860 * q^65 - 336 * q^71 - 1012 * q^73 + 544 * q^79 - 622 * q^81 + 1488 * q^87 + 2028 * q^89 + 440 * q^95 - 1532 * 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
$3$	4.00000i	0.769800i	0.922958	+	0.384900i	$0.125764\pi$
$3$	4.00000i	0.769800i	−0.922958	+	0.384900i	$0.874236\pi$
$4$	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
$11$	60.0000i	1.64461i	0.569049	+	0.822304i	$0.307311\pi$
$11$	60.0000i	1.64461i	−0.569049	+	0.822304i	$0.692689\pi$
$12$	0	0
$13$	86.0000i	1.83478i	0.397992	+	0.917389i	$0.369707\pi$
$13$	86.0000i	1.83478i	−0.397992	+	0.917389i	$0.630293\pi$
$14$	0	0
$15$	20.0000	0.344265
$16$	0	0
$17$	18.0000	0.256802	0.128401	−	0.991722i	$-0.459015\pi$
$17$	18.0000	0.256802	0.128401	+	0.991722i	$0.459015\pi$
$18$	0	0
$19$	44.0000i	0.531279i	0.964072	+	0.265639i	$0.0855830\pi$
$19$	44.0000i	0.531279i	−0.964072	+	0.265639i	$0.914417\pi$
$20$	0	0
$21$	64.0000i	0.665045i
$22$	0	0
$23$	−48.0000	−0.435161	−0.217580	−	0.976042i	$-0.569816\pi$
$23$	−48.0000	−0.435161	−0.217580	+	0.976042i	$0.569816\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	152.000i	1.08342i
$28$	0	0
$29$	− 186.000i	− 1.19101i	−0.803351	−	0.595506i	$-0.796952\pi$
$29$	− 186.000i	− 1.19101i	0.803351	−	0.595506i	$-0.203048\pi$
$30$	0	0
$31$	176.000	1.01969	0.509847	−	0.860265i	$-0.329702\pi$
$31$	176.000	1.01969	0.509847	+	0.860265i	$0.329702\pi$
$32$	0	0
$33$	−240.000	−1.26602
$34$	0	0
$35$	− 80.0000i	− 0.386356i
$36$	0	0
$37$	− 254.000i	− 1.12858i	−0.825578	−	0.564288i	$-0.809151\pi$
$37$	− 254.000i	− 1.12858i	0.825578	−	0.564288i	$-0.190849\pi$
$38$	0	0
$39$	−344.000	−1.41241
$40$	0	0
$41$	−186.000	−0.708496	−0.354248	−	0.935152i	$-0.615263\pi$
$41$	−186.000	−0.708496	−0.354248	+	0.935152i	$0.615263\pi$
$42$	0	0
$43$	100.000i	0.354648i	0.984153	+	0.177324i	$0.0567440\pi$
$43$	100.000i	0.354648i	−0.984153	+	0.177324i	$0.943256\pi$
$44$	0	0
$45$	− 55.0000i	− 0.182198i
$46$	0	0
$47$	168.000	0.521390	0.260695	−	0.965421i	$-0.416048\pi$
$47$	168.000	0.521390	0.260695	+	0.965421i	$0.416048\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	72.0000i	0.197687i
$52$	0	0
$53$	498.000i	1.29067i	0.763899	+	0.645335i	$0.223282\pi$
$53$	498.000i	1.29067i	−0.763899	+	0.645335i	$0.776718\pi$
$54$	0	0
$55$	300.000	0.735491
$56$	0	0
$57$	−176.000	−0.408978
$58$	0	0
$59$	252.000i	0.556061i	0.960572	+	0.278031i	$0.0896817\pi$
$59$	252.000i	0.556061i	−0.960572	+	0.278031i	$0.910318\pi$
$60$	0	0
$61$	− 58.0000i	− 0.121740i	−0.998146	−	0.0608700i	$-0.980612\pi$
$61$	− 58.0000i	− 0.121740i	0.998146	−	0.0608700i	$-0.0193875\pi$
$62$	0	0
$63$	176.000	0.351967
$64$	0	0
$65$	430.000	0.820537
$66$	0	0
$67$	− 1036.00i	− 1.88907i	−0.328414	−	0.944534i	$-0.606514\pi$
$67$	− 1036.00i	− 1.88907i	0.328414	−	0.944534i	$-0.393486\pi$
$68$	0	0
$69$	− 192.000i	− 0.334987i
$70$	0	0
$71$	−168.000	−0.280816	−0.140408	−	0.990094i	$-0.544841\pi$
$71$	−168.000	−0.280816	−0.140408	+	0.990094i	$0.544841\pi$
$72$	0	0
$73$	−506.000	−0.811272	−0.405636	−	0.914035i	$-0.632950\pi$
$73$	−506.000	−0.811272	−0.405636	+	0.914035i	$0.632950\pi$
$74$	0	0
$75$	− 100.000i	− 0.153960i
$76$	0	0
$77$	960.000i	1.42081i
$78$	0	0
$79$	272.000	0.387372	0.193686	−	0.981064i	$-0.437956\pi$
$79$	272.000	0.387372	0.193686	+	0.981064i	$0.437956\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	948.000i	1.25369i	0.779143	+	0.626846i	$0.215655\pi$
$83$	948.000i	1.25369i	−0.779143	+	0.626846i	$0.784345\pi$
$84$	0	0
$85$	− 90.0000i	− 0.114846i
$86$	0	0
$87$	744.000	0.916841
$88$	0	0
$89$	1014.00	1.20768	0.603841	−	0.797104i	$-0.293636\pi$
$89$	1014.00	1.20768	0.603841	+	0.797104i	$0.293636\pi$
$90$	0	0
$91$	1376.00i	1.58510i
$92$	0	0
$93$	704.000i	0.784961i
$94$	0	0
$95$	220.000	0.237595
$96$	0	0
$97$	−766.000	−0.801809	−0.400905	−	0.916120i	$-0.631304\pi$
$97$	−766.000	−0.801809	−0.400905	+	0.916120i	$0.631304\pi$
$98$	0	0
$99$	660.000i	0.670025i
$100$	0	0
$101$	1314.00i	1.29453i	0.762264	+	0.647267i	$0.224088\pi$
$101$	1314.00i	1.29453i	−0.762264	+	0.647267i	$0.775912\pi$
$102$	0	0
$103$	448.000	0.428570	0.214285	−	0.976771i	$-0.431258\pi$
$103$	448.000	0.428570	0.214285	+	0.976771i	$0.431258\pi$
$104$	0	0
$105$	320.000	0.297417
$106$	0	0
$107$	− 1548.00i	− 1.39861i	−0.714826	−	0.699303i	$-0.753494\pi$
$107$	− 1548.00i	− 1.39861i	0.714826	−	0.699303i	$-0.246506\pi$
$108$	0	0
$109$	278.000i	0.244290i	0.992512	+	0.122145i	$0.0389772\pi$
$109$	278.000i	0.244290i	−0.992512	+	0.122145i	$0.961023\pi$
$110$	0	0
$111$	1016.00	0.868779
$112$	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
$115$	240.000i	0.194610i
$116$	0	0
$117$	946.000i	0.747502i
$118$	0	0
$119$	288.000	0.221856
$120$	0	0
$121$	−2269.00	−1.70473
$122$	0	0
$123$	− 744.000i	− 0.545400i
$124$	0	0
$125$	125.000i	0.0894427i
$126$	0	0
$127$	344.000	0.240355	0.120177	−	0.992752i	$-0.461654\pi$
$127$	344.000	0.240355	0.120177	+	0.992752i	$0.461654\pi$
$128$	0	0
$129$	−400.000	−0.273008
$130$	0	0
$131$	780.000i	0.520221i	0.965579	+	0.260110i	$0.0837590\pi$
$131$	780.000i	0.520221i	−0.965579	+	0.260110i	$0.916241\pi$
$132$	0	0
$133$	704.000i	0.458982i
$134$	0	0
$135$	760.000	0.484521
$136$	0	0
$137$	−666.000	−0.415330	−0.207665	−	0.978200i	$-0.566586\pi$
$137$	−666.000	−0.415330	−0.207665	+	0.978200i	$0.566586\pi$
$138$	0	0
$139$	− 884.000i	− 0.539424i	−0.962941	−	0.269712i	$-0.913072\pi$
$139$	− 884.000i	− 0.539424i	0.962941	−	0.269712i	$-0.0869285\pi$
$140$	0	0
$141$	672.000i	0.401366i
$142$	0	0
$143$	−5160.00	−3.01749
$144$	0	0
$145$	−930.000	−0.532637
$146$	0	0
$147$	− 348.000i	− 0.195255i
$148$	0	0
$149$	114.000i	0.0626795i	0.999509	+	0.0313397i	$0.00997738\pi$
$149$	114.000i	0.0626795i	−0.999509	+	0.0313397i	$0.990023\pi$
$150$	0	0
$151$	40.0000	0.0215573	0.0107787	−	0.999942i	$-0.496569\pi$
$151$	40.0000	0.0215573	0.0107787	+	0.999942i	$0.496569\pi$
$152$	0	0
$153$	198.000	0.104623
$154$	0	0
$155$	− 880.000i	− 0.456021i
$156$	0	0
$157$	− 154.000i	− 0.0782837i	−0.999234	−	0.0391418i	$-0.987538\pi$
$157$	− 154.000i	− 0.0782837i	0.999234	−	0.0391418i	$-0.0124624\pi$
$158$	0	0
$159$	−1992.00	−0.993559
$160$	0	0
$161$	−768.000	−0.375943
$162$	0	0
$163$	2180.00i	1.04755i	0.851856	+	0.523775i	$0.175477\pi$
$163$	2180.00i	1.04755i	−0.851856	+	0.523775i	$0.824523\pi$
$164$	0	0
$165$	1200.00i	0.566181i
$166$	0	0
$167$	−3696.00	−1.71261	−0.856303	−	0.516474i	$-0.827244\pi$
$167$	−3696.00	−1.71261	−0.856303	+	0.516474i	$0.827244\pi$
$168$	0	0
$169$	−5199.00	−2.36641
$170$	0	0
$171$	484.000i	0.216447i
$172$	0	0
$173$	1302.00i	0.572192i	0.958201	+	0.286096i	$0.0923576\pi$
$173$	1302.00i	0.572192i	−0.958201	+	0.286096i	$0.907642\pi$
$174$	0	0
$175$	−400.000	−0.172784
$176$	0	0
$177$	−1008.00	−0.428056
$178$	0	0
$179$	− 4308.00i	− 1.79885i	−0.437070	−	0.899427i	$-0.643984\pi$
$179$	− 4308.00i	− 1.79885i	0.437070	−	0.899427i	$-0.356016\pi$
$180$	0	0
$181$	− 1550.00i	− 0.636523i	−0.948003	−	0.318261i	$-0.896901\pi$
$181$	− 1550.00i	− 0.636523i	0.948003	−	0.318261i	$-0.103099\pi$
$182$	0	0
$183$	232.000	0.0937155
$184$	0	0
$185$	−1270.00	−0.504715
$186$	0	0
$187$	1080.00i	0.422339i
$188$	0	0
$189$	2432.00i	0.935989i
$190$	0	0
$191$	48.0000	0.0181841	0.00909204	−	0.999959i	$-0.497106\pi$
$191$	48.0000	0.0181841	0.00909204	+	0.999959i	$0.497106\pi$
$192$	0	0
$193$	1058.00	0.394593	0.197297	−	0.980344i	$-0.436784\pi$
$193$	1058.00	0.394593	0.197297	+	0.980344i	$0.436784\pi$
$194$	0	0
$195$	1720.00i	0.631650i
$196$	0	0
$197$	3714.00i	1.34321i	0.740911	+	0.671603i	$0.234394\pi$
$197$	3714.00i	1.34321i	−0.740911	+	0.671603i	$0.765606\pi$
$198$	0	0
$199$	1768.00	0.629800	0.314900	−	0.949125i	$-0.398029\pi$
$199$	1768.00	0.629800	0.314900	+	0.949125i	$0.398029\pi$
$200$	0	0
$201$	4144.00	1.45421
$202$	0	0
$203$	− 2976.00i	− 1.02894i
$204$	0	0
$205$	930.000i	0.316849i
$206$	0	0
$207$	−528.000	−0.177288
$208$	0	0
$209$	−2640.00	−0.873745
$210$	0	0
$211$	− 4036.00i	− 1.31682i	−0.752658	−	0.658412i	$-0.771229\pi$
$211$	− 4036.00i	− 1.31682i	0.752658	−	0.658412i	$-0.228771\pi$
$212$	0	0
$213$	− 672.000i	− 0.216172i
$214$	0	0
$215$	500.000	0.158603
$216$	0	0
$217$	2816.00	0.880933
$218$	0	0
$219$	− 2024.00i	− 0.624517i
$220$	0	0
$221$	1548.00i	0.471175i
$222$	0	0
$223$	680.000	0.204198	0.102099	−	0.994774i	$-0.467444\pi$
$223$	680.000	0.204198	0.102099	+	0.994774i	$0.467444\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	2388.00i	0.698225i	0.937081	+	0.349113i	$0.113517\pi$
$227$	2388.00i	0.698225i	−0.937081	+	0.349113i	$0.886483\pi$
$228$	0	0
$229$	3874.00i	1.11791i	0.829198	+	0.558954i	$0.188797\pi$
$229$	3874.00i	1.11791i	−0.829198	+	0.558954i	$0.811203\pi$
$230$	0	0
$231$	−3840.00	−1.09374
$232$	0	0
$233$	−3162.00	−0.889054	−0.444527	−	0.895766i	$-0.646628\pi$
$233$	−3162.00	−0.889054	−0.444527	+	0.895766i	$0.646628\pi$
$234$	0	0
$235$	− 840.000i	− 0.233173i
$236$	0	0
$237$	1088.00i	0.298199i
$238$	0	0
$239$	5424.00	1.46799	0.733995	−	0.679155i	$-0.237654\pi$
$239$	5424.00	1.46799	0.733995	+	0.679155i	$0.237654\pi$
$240$	0	0
$241$	−3886.00	−1.03867	−0.519335	−	0.854571i	$-0.673820\pi$
$241$	−3886.00	−1.03867	−0.519335	+	0.854571i	$0.673820\pi$
$242$	0	0
$243$	2860.00i	0.755017i
$244$	0	0
$245$	435.000i	0.113433i
$246$	0	0
$247$	−3784.00	−0.974778
$248$	0	0
$249$	−3792.00	−0.965093
$250$	0	0
$251$	5100.00i	1.28251i	0.767329	+	0.641253i	$0.221585\pi$
$251$	5100.00i	1.28251i	−0.767329	+	0.641253i	$0.778415\pi$
$252$	0	0
$253$	− 2880.00i	− 0.715668i
$254$	0	0
$255$	360.000	0.0884081
$256$	0	0
$257$	2178.00	0.528638	0.264319	−	0.964435i	$-0.414853\pi$
$257$	2178.00	0.528638	0.264319	+	0.964435i	$0.414853\pi$
$258$	0	0
$259$	− 4064.00i	− 0.974999i
$260$	0	0
$261$	− 2046.00i	− 0.485227i
$262$	0	0
$263$	6144.00	1.44051	0.720257	−	0.693707i	$-0.244024\pi$
$263$	6144.00	1.44051	0.720257	+	0.693707i	$0.244024\pi$
$264$	0	0
$265$	2490.00	0.577206
$266$	0	0
$267$	4056.00i	0.929675i
$268$	0	0
$269$	822.000i	0.186313i	0.995651	+	0.0931566i	$0.0296957\pi$
$269$	822.000i	0.186313i	−0.995651	+	0.0931566i	$0.970304\pi$
$270$	0	0
$271$	8480.00	1.90082	0.950412	−	0.310994i	$-0.100662\pi$
$271$	8480.00	1.90082	0.950412	+	0.310994i	$0.100662\pi$
$272$	0	0
$273$	−5504.00	−1.22021
$274$	0	0
$275$	− 1500.00i	− 0.328921i
$276$	0	0
$277$	1138.00i	0.246844i	0.992354	+	0.123422i	$0.0393869\pi$
$277$	1138.00i	0.246844i	−0.992354	+	0.123422i	$0.960613\pi$
$278$	0	0
$279$	1936.00	0.415431
$280$	0	0
$281$	−5706.00	−1.21136	−0.605679	−	0.795709i	$-0.707098\pi$
$281$	−5706.00	−1.21136	−0.605679	+	0.795709i	$0.707098\pi$
$282$	0	0
$283$	3028.00i	0.636028i	0.948086	+	0.318014i	$0.103016\pi$
$283$	3028.00i	0.636028i	−0.948086	+	0.318014i	$0.896984\pi$
$284$	0	0
$285$	880.000i	0.182901i
$286$	0	0
$287$	−2976.00	−0.612083
$288$	0	0
$289$	−4589.00	−0.934053
$290$	0	0
$291$	− 3064.00i	− 0.617233i
$292$	0	0
$293$	− 3390.00i	− 0.675925i	−0.941160	−	0.337962i	$-0.890262\pi$
$293$	− 3390.00i	− 0.675925i	0.941160	−	0.337962i	$-0.109738\pi$
$294$	0	0
$295$	1260.00	0.248678
$296$	0	0
$297$	−9120.00	−1.78180
$298$	0	0
$299$	− 4128.00i	− 0.798423i
$300$	0	0
$301$	1600.00i	0.306387i
$302$	0	0
$303$	−5256.00	−0.996532
$304$	0	0
$305$	−290.000	−0.0544438
$306$	0	0
$307$	− 4156.00i	− 0.772624i	−0.922368	−	0.386312i	$-0.873749\pi$
$307$	− 4156.00i	− 0.772624i	0.922368	−	0.386312i	$-0.126251\pi$
$308$	0	0
$309$	1792.00i	0.329914i
$310$	0	0
$311$	−6552.00	−1.19463	−0.597315	−	0.802007i	$-0.703766\pi$
$311$	−6552.00	−1.19463	−0.597315	+	0.802007i	$0.703766\pi$
$312$	0	0
$313$	1366.00	0.246680	0.123340	−	0.992364i	$-0.460639\pi$
$313$	1366.00	0.246680	0.123340	+	0.992364i	$0.460639\pi$
$314$	0	0
$315$	− 880.000i	− 0.157404i
$316$	0	0
$317$	2598.00i	0.460310i	0.973154	+	0.230155i	$0.0739233\pi$
$317$	2598.00i	0.460310i	−0.973154	+	0.230155i	$0.926077\pi$
$318$	0	0
$319$	11160.0	1.95875
$320$	0	0
$321$	6192.00	1.07665
$322$	0	0
$323$	792.000i	0.136434i
$324$	0	0
$325$	− 2150.00i	− 0.366956i
$326$	0	0
$327$	−1112.00	−0.188054
$328$	0	0
$329$	2688.00	0.450438
$330$	0	0
$331$	3292.00i	0.546661i	0.961920	+	0.273330i	$0.0881252\pi$
$331$	3292.00i	0.546661i	−0.961920	+	0.273330i	$0.911875\pi$
$332$	0	0
$333$	− 2794.00i	− 0.459791i
$334$	0	0
$335$	−5180.00	−0.844817
$336$	0	0
$337$	6194.00	1.00121	0.500606	−	0.865675i	$-0.333110\pi$
$337$	6194.00	1.00121	0.500606	+	0.865675i	$0.333110\pi$
$338$	0	0
$339$	− 2232.00i	− 0.357598i
$340$	0	0
$341$	10560.0i	1.67700i
$342$	0	0
$343$	−6880.00	−1.08305
$344$	0	0
$345$	−960.000	−0.149811
$346$	0	0
$347$	10020.0i	1.55015i	0.631870	+	0.775075i	$0.282288\pi$
$347$	10020.0i	1.55015i	−0.631870	+	0.775075i	$0.717712\pi$
$348$	0	0
$349$	− 3130.00i	− 0.480072i	−0.970764	−	0.240036i	$-0.922841\pi$
$349$	− 3130.00i	− 0.480072i	0.970764	−	0.240036i	$-0.0771592\pi$
$350$	0	0
$351$	−13072.0	−1.98784
$352$	0	0
$353$	4194.00	0.632363	0.316181	−	0.948699i	$-0.397599\pi$
$353$	4194.00	0.632363	0.316181	+	0.948699i	$0.397599\pi$
$354$	0	0
$355$	840.000i	0.125585i
$356$	0	0
$357$	1152.00i	0.170785i
$358$	0	0
$359$	4104.00	0.603345	0.301672	−	0.953412i	$-0.402455\pi$
$359$	4104.00	0.603345	0.301672	+	0.953412i	$0.402455\pi$
$360$	0	0
$361$	4923.00	0.717743
$362$	0	0
$363$	− 9076.00i	− 1.31230i
$364$	0	0
$365$	2530.00i	0.362812i
$366$	0	0
$367$	7496.00	1.06618	0.533090	−	0.846059i	$-0.321031\pi$
$367$	7496.00	1.06618	0.533090	+	0.846059i	$0.321031\pi$
$368$	0	0
$369$	−2046.00	−0.288646
$370$	0	0
$371$	7968.00i	1.11503i
$372$	0	0
$373$	5842.00i	0.810958i	0.914104	+	0.405479i	$0.132895\pi$
$373$	5842.00i	0.810958i	−0.914104	+	0.405479i	$0.867105\pi$
$374$	0	0
$375$	−500.000	−0.0688530
$376$	0	0
$377$	15996.0	2.18524
$378$	0	0
$379$	412.000i	0.0558391i	0.999610	+	0.0279195i	$0.00888822\pi$
$379$	412.000i	0.0558391i	−0.999610	+	0.0279195i	$0.991112\pi$
$380$	0	0
$381$	1376.00i	0.185025i
$382$	0	0
$383$	2568.00	0.342607	0.171304	−	0.985218i	$-0.445202\pi$
$383$	2568.00	0.342607	0.171304	+	0.985218i	$0.445202\pi$
$384$	0	0
$385$	4800.00	0.635404
$386$	0	0
$387$	1100.00i	0.144486i
$388$	0	0
$389$	− 13086.0i	− 1.70562i	−0.522221	−	0.852810i	$-0.674896\pi$
$389$	− 13086.0i	− 1.70562i	0.522221	−	0.852810i	$-0.325104\pi$
$390$	0	0
$391$	−864.000	−0.111750
$392$	0	0
$393$	−3120.00	−0.400466
$394$	0	0
$395$	− 1360.00i	− 0.173238i
$396$	0	0
$397$	10454.0i	1.32159i	0.750566	+	0.660795i	$0.229781\pi$
$397$	10454.0i	1.32159i	−0.750566	+	0.660795i	$0.770219\pi$
$398$	0	0
$399$	−2816.00	−0.353324
$400$	0	0
$401$	−10830.0	−1.34869	−0.674345	−	0.738417i	$-0.735574\pi$
$401$	−10830.0	−1.34869	−0.674345	+	0.738417i	$0.735574\pi$
$402$	0	0
$403$	15136.0i	1.87091i
$404$	0	0
$405$	1555.00i	0.190787i
$406$	0	0
$407$	15240.0	1.85607
$408$	0	0
$409$	8566.00	1.03560	0.517801	−	0.855501i	$-0.326751\pi$
$409$	8566.00	1.03560	0.517801	+	0.855501i	$0.326751\pi$
$410$	0	0
$411$	− 2664.00i	− 0.319721i
$412$	0	0
$413$	4032.00i	0.480392i
$414$	0	0
$415$	4740.00	0.560669
$416$	0	0
$417$	3536.00	0.415249
$418$	0	0
$419$	13884.0i	1.61880i	0.587257	+	0.809401i	$0.300208\pi$
$419$	13884.0i	1.61880i	−0.587257	+	0.809401i	$0.699792\pi$
$420$	0	0
$421$	− 4286.00i	− 0.496168i	−0.968738	−	0.248084i	$-0.920199\pi$
$421$	− 4286.00i	− 0.496168i	0.968738	−	0.248084i	$-0.0798010\pi$
$422$	0	0
$423$	1848.00	0.212418
$424$	0	0
$425$	−450.000	−0.0513605
$426$	0	0
$427$	− 928.000i	− 0.105173i
$428$	0	0
$429$	− 20640.0i	− 2.32286i
$430$	0	0
$431$	6336.00	0.708108	0.354054	−	0.935225i	$-0.384803\pi$
$431$	6336.00	0.708108	0.354054	+	0.935225i	$0.384803\pi$
$432$	0	0
$433$	−8974.00	−0.995988	−0.497994	−	0.867180i	$-0.665930\pi$
$433$	−8974.00	−0.995988	−0.497994	+	0.867180i	$0.665930\pi$
$434$	0	0
$435$	− 3720.00i	− 0.410024i
$436$	0	0
$437$	− 2112.00i	− 0.231191i
$438$	0	0
$439$	2968.00	0.322676	0.161338	−	0.986899i	$-0.448419\pi$
$439$	2968.00	0.322676	0.161338	+	0.986899i	$0.448419\pi$
$440$	0	0
$441$	−957.000	−0.103337
$442$	0	0
$443$	12372.0i	1.32689i	0.748226	+	0.663444i	$0.230906\pi$
$443$	12372.0i	1.32689i	−0.748226	+	0.663444i	$0.769094\pi$
$444$	0	0
$445$	− 5070.00i	− 0.540092i
$446$	0	0
$447$	−456.000	−0.0482507
$448$	0	0
$449$	11394.0	1.19759	0.598793	−	0.800904i	$-0.295647\pi$
$449$	11394.0	1.19759	0.598793	+	0.800904i	$0.295647\pi$
$450$	0	0
$451$	− 11160.0i	− 1.16520i
$452$	0	0
$453$	160.000i	0.0165948i
$454$	0	0
$455$	6880.00	0.708878
$456$	0	0
$457$	358.000	0.0366445	0.0183222	−	0.999832i	$-0.494168\pi$
$457$	358.000	0.0366445	0.0183222	+	0.999832i	$0.494168\pi$
$458$	0	0
$459$	2736.00i	0.278226i
$460$	0	0
$461$	− 7530.00i	− 0.760753i	−0.924832	−	0.380376i	$-0.875794\pi$
$461$	− 7530.00i	− 0.760753i	0.924832	−	0.380376i	$-0.124206\pi$
$462$	0	0
$463$	−13768.0	−1.38197	−0.690986	−	0.722868i	$-0.742823\pi$
$463$	−13768.0	−1.38197	−0.690986	+	0.722868i	$0.742823\pi$
$464$	0	0
$465$	3520.00	0.351045
$466$	0	0
$467$	13380.0i	1.32581i	0.748704	+	0.662904i	$0.230676\pi$
$467$	13380.0i	1.32581i	−0.748704	+	0.662904i	$0.769324\pi$
$468$	0	0
$469$	− 16576.0i	− 1.63200i
$470$	0	0
$471$	616.000	0.0602628
$472$	0	0
$473$	−6000.00	−0.583256
$474$	0	0
$475$	− 1100.00i	− 0.106256i
$476$	0	0
$477$	5478.00i	0.525829i
$478$	0	0
$479$	−6336.00	−0.604383	−0.302191	−	0.953247i	$-0.597718\pi$
$479$	−6336.00	−0.604383	−0.302191	+	0.953247i	$0.597718\pi$
$480$	0	0
$481$	21844.0	2.07069
$482$	0	0
$483$	− 3072.00i	− 0.289401i
$484$	0	0
$485$	3830.00i	0.358580i
$486$	0	0
$487$	5008.00	0.465984	0.232992	−	0.972479i	$-0.425148\pi$
$487$	5008.00	0.465984	0.232992	+	0.972479i	$0.425148\pi$
$488$	0	0
$489$	−8720.00	−0.806405
$490$	0	0
$491$	− 12900.0i	− 1.18568i	−0.805320	−	0.592840i	$-0.798007\pi$
$491$	− 12900.0i	− 1.18568i	0.805320	−	0.592840i	$-0.201993\pi$
$492$	0	0
$493$	− 3348.00i	− 0.305855i
$494$	0	0
$495$	3300.00	0.299644
$496$	0	0
$497$	−2688.00	−0.242602
$498$	0	0
$499$	− 8116.00i	− 0.728100i	−0.931379	−	0.364050i	$-0.881394\pi$
$499$	− 8116.00i	− 0.728100i	0.931379	−	0.364050i	$-0.118606\pi$
$500$	0	0
$501$	− 14784.0i	− 1.31836i
$502$	0	0
$503$	4944.00	0.438255	0.219127	−	0.975696i	$-0.429679\pi$
$503$	4944.00	0.438255	0.219127	+	0.975696i	$0.429679\pi$
$504$	0	0
$505$	6570.00	0.578933
$506$	0	0
$507$	− 20796.0i	− 1.82166i
$508$	0	0
$509$	− 5466.00i	− 0.475985i	−0.971267	−	0.237992i	$-0.923511\pi$
$509$	− 5466.00i	− 0.475985i	0.971267	−	0.237992i	$-0.0764893\pi$
$510$	0	0
$511$	−8096.00	−0.700873
$512$	0	0
$513$	−6688.00	−0.575599
$514$	0	0
$515$	− 2240.00i	− 0.191663i
$516$	0	0
$517$	10080.0i	0.857481i
$518$	0	0
$519$	−5208.00	−0.440474
$520$	0	0
$521$	−10074.0	−0.847121	−0.423560	−	0.905868i	$-0.639220\pi$
$521$	−10074.0	−0.847121	−0.423560	+	0.905868i	$0.639220\pi$
$522$	0	0
$523$	13828.0i	1.15613i	0.815991	+	0.578065i	$0.196192\pi$
$523$	13828.0i	1.15613i	−0.815991	+	0.578065i	$0.803808\pi$
$524$	0	0
$525$	− 1600.00i	− 0.133009i
$526$	0	0
$527$	3168.00	0.261860
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	2772.00i	0.226543i
$532$	0	0
$533$	− 15996.0i	− 1.29993i
$534$	0	0
$535$	−7740.00	−0.625475
$536$	0	0
$537$	17232.0	1.38476
$538$	0	0
$539$	− 5220.00i	− 0.417145i
$540$	0	0
$541$	− 15226.0i	− 1.21001i	−0.796221	−	0.605006i	$-0.793171\pi$
$541$	− 15226.0i	− 1.21001i	0.796221	−	0.605006i	$-0.206829\pi$
$542$	0	0
$543$	6200.00	0.489995
$544$	0	0
$545$	1390.00	0.109250
$546$	0	0
$547$	− 13228.0i	− 1.03398i	−0.855991	−	0.516991i	$-0.827052\pi$
$547$	− 13228.0i	− 1.03398i	0.855991	−	0.516991i	$-0.172948\pi$
$548$	0	0
$549$	− 638.000i	− 0.0495978i
$550$	0	0
$551$	8184.00	0.632759
$552$	0	0
$553$	4352.00	0.334658
$554$	0	0
$555$	− 5080.00i	− 0.388530i
$556$	0	0
$557$	− 8490.00i	− 0.645840i	−0.946426	−	0.322920i	$-0.895336\pi$
$557$	− 8490.00i	− 0.645840i	0.946426	−	0.322920i	$-0.104664\pi$
$558$	0	0
$559$	−8600.00	−0.650700
$560$	0	0
$561$	−4320.00	−0.325117
$562$	0	0
$563$	− 10284.0i	− 0.769838i	−0.922950	−	0.384919i	$-0.874229\pi$
$563$	− 10284.0i	− 0.769838i	0.922950	−	0.384919i	$-0.125771\pi$
$564$	0	0
$565$	2790.00i	0.207745i
$566$	0	0
$567$	−4976.00	−0.368558
$568$	0	0
$569$	−1770.00	−0.130408	−0.0652041	−	0.997872i	$-0.520770\pi$
$569$	−1770.00	−0.130408	−0.0652041	+	0.997872i	$0.520770\pi$
$570$	0	0
$571$	− 6068.00i	− 0.444725i	−0.974964	−	0.222362i	$-0.928623\pi$
$571$	− 6068.00i	− 0.444725i	0.974964	−	0.222362i	$-0.0713768\pi$
$572$	0	0
$573$	192.000i	0.0139981i
$574$	0	0
$575$	1200.00	0.0870321
$576$	0	0
$577$	21506.0	1.55166	0.775829	−	0.630943i	$-0.217332\pi$
$577$	21506.0	1.55166	0.775829	+	0.630943i	$0.217332\pi$
$578$	0	0
$579$	4232.00i	0.303758i
$580$	0	0
$581$	15168.0i	1.08309i
$582$	0	0
$583$	−29880.0	−2.12265
$584$	0	0
$585$	4730.00	0.334293
$586$	0	0
$587$	− 12108.0i	− 0.851364i	−0.904873	−	0.425682i	$-0.860034\pi$
$587$	− 12108.0i	− 0.851364i	0.904873	−	0.425682i	$-0.139966\pi$
$588$	0	0
$589$	7744.00i	0.541742i
$590$	0	0
$591$	−14856.0	−1.03400
$592$	0	0
$593$	15474.0	1.07157	0.535785	−	0.844354i	$-0.320016\pi$
$593$	15474.0	1.07157	0.535785	+	0.844354i	$0.320016\pi$
$594$	0	0
$595$	− 1440.00i	− 0.0992172i
$596$	0	0
$597$	7072.00i	0.484820i
$598$	0	0
$599$	2520.00	0.171894	0.0859469	−	0.996300i	$-0.472608\pi$
$599$	2520.00	0.171894	0.0859469	+	0.996300i	$0.472608\pi$
$600$	0	0
$601$	12790.0	0.868078	0.434039	−	0.900894i	$-0.357088\pi$
$601$	12790.0	0.868078	0.434039	+	0.900894i	$0.357088\pi$
$602$	0	0
$603$	− 11396.0i	− 0.769620i
$604$	0	0
$605$	11345.0i	0.762380i
$606$	0	0
$607$	11576.0	0.774062	0.387031	−	0.922067i	$-0.373501\pi$
$607$	11576.0	0.774062	0.387031	+	0.922067i	$0.373501\pi$
$608$	0	0
$609$	11904.0	0.792076
$610$	0	0
$611$	14448.0i	0.956634i
$612$	0	0
$613$	− 20126.0i	− 1.32607i	−0.748588	−	0.663035i	$-0.769268\pi$
$613$	− 20126.0i	− 1.32607i	0.748588	−	0.663035i	$-0.230732\pi$
$614$	0	0
$615$	−3720.00	−0.243910
$616$	0	0
$617$	27942.0	1.82318	0.911590	−	0.411100i	$-0.134855\pi$
$617$	27942.0	1.82318	0.911590	+	0.411100i	$0.134855\pi$
$618$	0	0
$619$	22540.0i	1.46358i	0.681528	+	0.731792i	$0.261316\pi$
$619$	22540.0i	1.46358i	−0.681528	+	0.731792i	$0.738684\pi$
$620$	0	0
$621$	− 7296.00i	− 0.471463i
$622$	0	0
$623$	16224.0	1.04334
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	− 10560.0i	− 0.672609i
$628$	0	0
$629$	− 4572.00i	− 0.289821i
$630$	0	0
$631$	5128.00	0.323522	0.161761	−	0.986830i	$-0.448283\pi$
$631$	5128.00	0.323522	0.161761	+	0.986830i	$0.448283\pi$
$632$	0	0
$633$	16144.0	1.01369
$634$	0	0
$635$	− 1720.00i	− 0.107490i
$636$	0	0
$637$	− 7482.00i	− 0.465381i
$638$	0	0
$639$	−1848.00	−0.114406
$640$	0	0
$641$	−12798.0	−0.788597	−0.394298	−	0.918982i	$-0.629012\pi$
$641$	−12798.0	−0.788597	−0.394298	+	0.918982i	$0.629012\pi$
$642$	0	0
$643$	− 21148.0i	− 1.29704i	−0.761198	−	0.648519i	$-0.775389\pi$
$643$	− 21148.0i	− 1.29704i	0.761198	−	0.648519i	$-0.224611\pi$
$644$	0	0
$645$	2000.00i	0.122093i
$646$	0	0
$647$	−16464.0	−1.00041	−0.500206	−	0.865906i	$-0.666742\pi$
$647$	−16464.0	−1.00041	−0.500206	+	0.865906i	$0.666742\pi$
$648$	0	0
$649$	−15120.0	−0.914502
$650$	0	0
$651$	11264.0i	0.678143i
$652$	0	0
$653$	− 24234.0i	− 1.45230i	−0.687538	−	0.726148i	$-0.741309\pi$
$653$	− 24234.0i	− 1.45230i	0.687538	−	0.726148i	$-0.258691\pi$
$654$	0	0
$655$	3900.00	0.232650
$656$	0	0
$657$	−5566.00	−0.330518
$658$	0	0
$659$	− 22836.0i	− 1.34987i	−0.737877	−	0.674935i	$-0.764172\pi$
$659$	− 22836.0i	− 1.34987i	0.737877	−	0.674935i	$-0.235828\pi$
$660$	0	0
$661$	− 26318.0i	− 1.54864i	−0.632794	−	0.774320i	$-0.718092\pi$
$661$	− 26318.0i	− 1.54864i	0.632794	−	0.774320i	$-0.281908\pi$
$662$	0	0
$663$	−6192.00	−0.362711
$664$	0	0
$665$	3520.00	0.205263
$666$	0	0
$667$	8928.00i	0.518281i
$668$	0	0
$669$	2720.00i	0.157192i
$670$	0	0
$671$	3480.00	0.200214
$672$	0	0
$673$	28802.0	1.64968	0.824841	−	0.565365i	$-0.191265\pi$
$673$	28802.0	1.64968	0.824841	+	0.565365i	$0.191265\pi$
$674$	0	0
$675$	− 3800.00i	− 0.216685i
$676$	0	0
$677$	− 2526.00i	− 0.143400i	−0.997426	−	0.0717002i	$-0.977158\pi$
$677$	− 2526.00i	− 0.143400i	0.997426	−	0.0717002i	$-0.0228425\pi$
$678$	0	0
$679$	−12256.0	−0.692698
$680$	0	0
$681$	−9552.00	−0.537494
$682$	0	0
$683$	23076.0i	1.29279i	0.763001	+	0.646397i	$0.223725\pi$
$683$	23076.0i	1.29279i	−0.763001	+	0.646397i	$0.776275\pi$
$684$	0	0
$685$	3330.00i	0.185741i
$686$	0	0
$687$	−15496.0	−0.860567
$688$	0	0
$689$	−42828.0	−2.36809
$690$	0	0
$691$	7868.00i	0.433159i	0.976265	+	0.216579i	$0.0694900\pi$
$691$	7868.00i	0.433159i	−0.976265	+	0.216579i	$0.930510\pi$
$692$	0	0
$693$	10560.0i	0.578847i
$694$	0	0
$695$	−4420.00	−0.241238
$696$	0	0
$697$	−3348.00	−0.181943
$698$	0	0
$699$	− 12648.0i	− 0.684394i
$700$	0	0
$701$	21510.0i	1.15895i	0.814991	+	0.579473i	$0.196742\pi$
$701$	21510.0i	1.15895i	−0.814991	+	0.579473i	$0.803258\pi$
$702$	0	0
$703$	11176.0	0.599589
$704$	0	0
$705$	3360.00	0.179496
$706$	0	0
$707$	21024.0i	1.11837i
$708$	0	0
$709$	− 30014.0i	− 1.58984i	−0.606712	−	0.794922i	$-0.707512\pi$
$709$	− 30014.0i	− 1.58984i	0.606712	−	0.794922i	$-0.292488\pi$
$710$	0	0
$711$	2992.00	0.157818
$712$	0	0
$713$	−8448.00	−0.443731
$714$	0	0
$715$	25800.0i	1.34946i
$716$	0	0
$717$	21696.0i	1.13006i
$718$	0	0
$719$	−816.000	−0.0423250	−0.0211625	−	0.999776i	$-0.506737\pi$
$719$	−816.000	−0.0423250	−0.0211625	+	0.999776i	$0.506737\pi$
$720$	0	0
$721$	7168.00	0.370250
$722$	0	0
$723$	− 15544.0i	− 0.799568i
$724$	0	0
$725$	4650.00i	0.238202i
$726$	0	0
$727$	9952.00	0.507702	0.253851	−	0.967243i	$-0.418303\pi$
$727$	9952.00	0.507702	0.253851	+	0.967243i	$0.418303\pi$
$728$	0	0
$729$	−19837.0	−1.00782
$730$	0	0
$731$	1800.00i	0.0910744i
$732$	0	0
$733$	− 33946.0i	− 1.71054i	−0.518185	−	0.855269i	$-0.673392\pi$
$733$	− 33946.0i	− 1.71054i	0.518185	−	0.855269i	$-0.326608\pi$
$734$	0	0
$735$	−1740.00	−0.0873209
$736$	0	0
$737$	62160.0	3.10677
$738$	0	0
$739$	23420.0i	1.16579i	0.812548	+	0.582895i	$0.198080\pi$
$739$	23420.0i	1.16579i	−0.812548	+	0.582895i	$0.801920\pi$
$740$	0	0
$741$	− 15136.0i	− 0.750384i
$742$	0	0
$743$	14592.0	0.720496	0.360248	−	0.932857i	$-0.382692\pi$
$743$	14592.0	0.720496	0.360248	+	0.932857i	$0.382692\pi$
$744$	0	0
$745$	570.000	0.0280311
$746$	0	0
$747$	10428.0i	0.510764i
$748$	0	0
$749$	− 24768.0i	− 1.20828i
$750$	0	0
$751$	9056.00	0.440024	0.220012	−	0.975497i	$-0.429390\pi$
$751$	9056.00	0.440024	0.220012	+	0.975497i	$0.429390\pi$
$752$	0	0
$753$	−20400.0	−0.987274
$754$	0	0
$755$	− 200.000i	− 0.00964072i
$756$	0	0
$757$	17554.0i	0.842815i	0.906871	+	0.421408i	$0.138464\pi$
$757$	17554.0i	0.842815i	−0.906871	+	0.421408i	$0.861536\pi$
$758$	0	0
$759$	11520.0	0.550922
$760$	0	0
$761$	36438.0	1.73571	0.867856	−	0.496816i	$-0.165498\pi$
$761$	36438.0	1.73571	0.867856	+	0.496816i	$0.165498\pi$
$762$	0	0
$763$	4448.00i	0.211046i
$764$	0	0
$765$	− 990.000i	− 0.0467889i
$766$	0	0
$767$	−21672.0	−1.02025
$768$	0	0
$769$	−9022.00	−0.423071	−0.211536	−	0.977370i	$-0.567846\pi$
$769$	−9022.00	−0.423071	−0.211536	+	0.977370i	$0.567846\pi$
$770$	0	0
$771$	8712.00i	0.406946i
$772$	0	0
$773$	− 1470.00i	− 0.0683987i	−0.999415	−	0.0341994i	$-0.989112\pi$
$773$	− 1470.00i	− 0.0683987i	0.999415	−	0.0341994i	$-0.0108881\pi$
$774$	0	0
$775$	−4400.00	−0.203939
$776$	0	0
$777$	16256.0	0.750554
$778$	0	0
$779$	− 8184.00i	− 0.376409i
$780$	0	0
$781$	− 10080.0i	− 0.461832i
$782$	0	0
$783$	28272.0	1.29037
$784$	0	0
$785$	−770.000	−0.0350095
$786$	0	0
$787$	5252.00i	0.237883i	0.992901	+	0.118941i	$0.0379500\pi$
$787$	5252.00i	0.237883i	−0.992901	+	0.118941i	$0.962050\pi$
$788$	0	0
$789$	24576.0i	1.10891i
$790$	0	0
$791$	−8928.00	−0.401319
$792$	0	0
$793$	4988.00	0.223366
$794$	0	0
$795$	9960.00i	0.444333i
$796$	0	0
$797$	12294.0i	0.546394i	0.961958	+	0.273197i	$0.0880810\pi$
$797$	12294.0i	0.546394i	−0.961958	+	0.273197i	$0.911919\pi$
$798$	0	0
$799$	3024.00	0.133894
$800$	0	0
$801$	11154.0	0.492019
$802$	0	0
$803$	− 30360.0i	− 1.33422i
$804$	0	0
$805$	3840.00i	0.168127i
$806$	0	0
$807$	−3288.00	−0.143424
$808$	0	0
$809$	−15546.0	−0.675610	−0.337805	−	0.941216i	$-0.609684\pi$
$809$	−15546.0	−0.675610	−0.337805	+	0.941216i	$0.609684\pi$
$810$	0	0
$811$	− 19364.0i	− 0.838424i	−0.907888	−	0.419212i	$-0.862306\pi$
$811$	− 19364.0i	− 0.838424i	0.907888	−	0.419212i	$-0.137694\pi$
$812$	0	0
$813$	33920.0i	1.46326i
$814$	0	0
$815$	10900.0	0.468479
$816$	0	0
$817$	−4400.00	−0.188417
$818$	0	0
$819$	15136.0i	0.645781i
$820$	0	0
$821$	7314.00i	0.310914i	0.987843	+	0.155457i	$0.0496850\pi$
$821$	7314.00i	0.310914i	−0.987843	+	0.155457i	$0.950315\pi$
$822$	0	0
$823$	−11984.0	−0.507577	−0.253789	−	0.967260i	$-0.581677\pi$
$823$	−11984.0	−0.507577	−0.253789	+	0.967260i	$0.581677\pi$
$824$	0	0
$825$	6000.00	0.253204
$826$	0	0
$827$	− 13500.0i	− 0.567643i	−0.958877	−	0.283822i	$-0.908398\pi$
$827$	− 13500.0i	− 0.567643i	0.958877	−	0.283822i	$-0.0916024\pi$
$828$	0	0
$829$	− 44602.0i	− 1.86863i	−0.356453	−	0.934313i	$-0.616014\pi$
$829$	− 44602.0i	− 1.86863i	0.356453	−	0.934313i	$-0.383986\pi$
$830$	0	0
$831$	−4552.00	−0.190021
$832$	0	0
$833$	−1566.00	−0.0651365
$834$	0	0
$835$	18480.0i	0.765900i
$836$	0	0
$837$	26752.0i	1.10476i
$838$	0	0
$839$	−35448.0	−1.45864	−0.729321	−	0.684172i	$-0.760164\pi$
$839$	−35448.0	−1.45864	−0.729321	+	0.684172i	$0.760164\pi$
$840$	0	0
$841$	−10207.0	−0.418508
$842$	0	0
$843$	− 22824.0i	− 0.932503i
$844$	0	0
$845$	25995.0i	1.05829i
$846$	0	0
$847$	−36304.0	−1.47275
$848$	0	0
$849$	−12112.0	−0.489615
$850$	0	0
$851$	12192.0i	0.491112i
$852$	0	0
$853$	− 12590.0i	− 0.505362i	−0.967550	−	0.252681i	$-0.918688\pi$
$853$	− 12590.0i	− 0.505362i	0.967550	−	0.252681i	$-0.0813122\pi$
$854$	0	0
$855$	2420.00	0.0967980
$856$	0	0
$857$	−24906.0	−0.992734	−0.496367	−	0.868113i	$-0.665333\pi$
$857$	−24906.0	−0.992734	−0.496367	+	0.868113i	$0.665333\pi$
$858$	0	0
$859$	− 23204.0i	− 0.921665i	−0.887487	−	0.460833i	$-0.847551\pi$
$859$	− 23204.0i	− 0.921665i	0.887487	−	0.460833i	$-0.152449\pi$
$860$	0	0
$861$	− 11904.0i	− 0.471181i
$862$	0	0
$863$	19848.0	0.782890	0.391445	−	0.920202i	$-0.371975\pi$
$863$	19848.0	0.782890	0.391445	+	0.920202i	$0.371975\pi$
$864$	0	0
$865$	6510.00	0.255892
$866$	0	0
$867$	− 18356.0i	− 0.719034i
$868$	0	0
$869$	16320.0i	0.637075i
$870$	0	0
$871$	89096.0	3.46602
$872$	0	0
$873$	−8426.00	−0.326663
$874$	0	0
$875$	2000.00i	0.0772712i
$876$	0	0
$877$	27542.0i	1.06046i	0.847852	+	0.530232i	$0.177895\pi$
$877$	27542.0i	1.06046i	−0.847852	+	0.530232i	$0.822105\pi$
$878$	0	0
$879$	13560.0	0.520327
$880$	0	0
$881$	−20718.0	−0.792290	−0.396145	−	0.918188i	$-0.629652\pi$
$881$	−20718.0	−0.792290	−0.396145	+	0.918188i	$0.629652\pi$
$882$	0	0
$883$	25172.0i	0.959349i	0.877446	+	0.479675i	$0.159245\pi$
$883$	25172.0i	0.959349i	−0.877446	+	0.479675i	$0.840755\pi$
$884$	0	0
$885$	5040.00i	0.191432i
$886$	0	0
$887$	12864.0	0.486957	0.243478	−	0.969906i	$-0.421711\pi$
$887$	12864.0	0.486957	0.243478	+	0.969906i	$0.421711\pi$
$888$	0	0
$889$	5504.00	0.207647
$890$	0	0
$891$	− 18660.0i	− 0.701609i
$892$	0	0
$893$	7392.00i	0.277003i
$894$	0	0
$895$	−21540.0	−0.804472
$896$	0	0
$897$	16512.0	0.614626
$898$	0	0
$899$	− 32736.0i	− 1.21447i
$900$	0	0
$901$	8964.00i	0.331447i
$902$	0	0
$903$	−6400.00	−0.235857
$904$	0	0
$905$	−7750.00	−0.284662
$906$	0	0
$907$	23092.0i	0.845377i	0.906275	+	0.422689i	$0.138914\pi$
$907$	23092.0i	0.845377i	−0.906275	+	0.422689i	$0.861086\pi$
$908$	0	0
$909$	14454.0i	0.527403i
$910$	0	0
$911$	−14208.0	−0.516720	−0.258360	−	0.966049i	$-0.583182\pi$
$911$	−14208.0	−0.516720	−0.258360	+	0.966049i	$0.583182\pi$
$912$	0	0
$913$	−56880.0	−2.06183
$914$	0	0
$915$	− 1160.00i	− 0.0419108i
$916$	0	0
$917$	12480.0i	0.449428i
$918$	0	0
$919$	26584.0	0.954217	0.477108	−	0.878844i	$-0.341685\pi$
$919$	26584.0	0.954217	0.477108	+	0.878844i	$0.341685\pi$
$920$	0	0
$921$	16624.0	0.594766
$922$	0	0
$923$	− 14448.0i	− 0.515235i
$924$	0	0
$925$	6350.00i	0.225715i
$926$	0	0
$927$	4928.00	0.174603
$928$	0	0
$929$	162.000	0.00572126	0.00286063	−	0.999996i	$-0.499089\pi$
$929$	162.000	0.00572126	0.00286063	+	0.999996i	$0.499089\pi$
$930$	0	0
$931$	− 3828.00i	− 0.134756i
$932$	0	0
$933$	− 26208.0i	− 0.919626i
$934$	0	0
$935$	5400.00	0.188876
$936$	0	0
$937$	29734.0	1.03668	0.518339	−	0.855175i	$-0.326551\pi$
$937$	29734.0	1.03668	0.518339	+	0.855175i	$0.326551\pi$
$938$	0	0
$939$	5464.00i	0.189894i
$940$	0	0
$941$	17142.0i	0.593850i	0.954901	+	0.296925i	$0.0959612\pi$
$941$	17142.0i	0.593850i	−0.954901	+	0.296925i	$0.904039\pi$
$942$	0	0
$943$	8928.00	0.308309
$944$	0	0
$945$	12160.0	0.418587
$946$	0	0
$947$	26436.0i	0.907133i	0.891223	+	0.453566i	$0.149848\pi$
$947$	26436.0i	0.907133i	−0.891223	+	0.453566i	$0.850152\pi$
$948$	0	0
$949$	− 43516.0i	− 1.48850i
$950$	0	0
$951$	−10392.0	−0.354347
$952$	0	0
$953$	−27882.0	−0.947730	−0.473865	−	0.880598i	$-0.657142\pi$
$953$	−27882.0	−0.947730	−0.473865	+	0.880598i	$0.657142\pi$
$954$	0	0
$955$	− 240.000i	− 0.00813217i
$956$	0	0
$957$	44640.0i	1.50784i
$958$	0	0
$959$	−10656.0	−0.358811
$960$	0	0
$961$	1185.00	0.0397771
$962$	0	0
$963$	− 17028.0i	− 0.569802i
$964$	0	0
$965$	− 5290.00i	− 0.176467i
$966$	0	0
$967$	−12656.0	−0.420879	−0.210439	−	0.977607i	$-0.567489\pi$
$967$	−12656.0	−0.420879	−0.210439	+	0.977607i	$0.567489\pi$
$968$	0	0
$969$	−3168.00	−0.105027
$970$	0	0
$971$	− 2916.00i	− 0.0963737i	−0.998838	−	0.0481869i	$-0.984656\pi$
$971$	− 2916.00i	− 0.0963737i	0.998838	−	0.0481869i	$-0.0153443\pi$
$972$	0	0
$973$	− 14144.0i	− 0.466018i
$974$	0	0
$975$	8600.00	0.282482
$976$	0	0
$977$	−6894.00	−0.225751	−0.112875	−	0.993609i	$-0.536006\pi$
$977$	−6894.00	−0.225751	−0.112875	+	0.993609i	$0.536006\pi$
$978$	0	0
$979$	60840.0i	1.98616i
$980$	0	0
$981$	3058.00i	0.0995254i
$982$	0	0
$983$	45264.0	1.46866	0.734332	−	0.678790i	$-0.237495\pi$
$983$	45264.0	1.46866	0.734332	+	0.678790i	$0.237495\pi$
$984$	0	0
$985$	18570.0	0.600700
$986$	0	0
$987$	10752.0i	0.346748i
$988$	0	0
$989$	− 4800.00i	− 0.154329i
$990$	0	0
$991$	52016.0	1.66735	0.833674	−	0.552256i	$-0.186233\pi$
$991$	52016.0	1.66735	0.833674	+	0.552256i	$0.186233\pi$
$992$	0	0
$993$	−13168.0	−0.420820
$994$	0	0
$995$	− 8840.00i	− 0.281655i
$996$	0	0
$997$	13858.0i	0.440208i	0.975476	+	0.220104i	$0.0706397\pi$
$997$	13858.0i	0.440208i	−0.975476	+	0.220104i	$0.929360\pi$
$998$	0	0
$999$	38608.0	1.22273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.d.n.641.2		2
4.3	odd	2		1280.4.d.c.641.1		2
8.3	odd	2		1280.4.d.c.641.2		2
8.5	even	2	inner	1280.4.d.n.641.1		2
16.3	odd	4		320.4.a.k.1.1		1
16.5	even	4		20.4.a.a.1.1	✓	1
16.11	odd	4		80.4.a.c.1.1		1
16.13	even	4		320.4.a.d.1.1		1
48.5	odd	4		180.4.a.a.1.1		1
48.11	even	4		720.4.a.k.1.1		1
80.19	odd	4		1600.4.a.p.1.1		1
80.27	even	4		400.4.c.j.49.2		2
80.29	even	4		1600.4.a.bl.1.1		1
80.37	odd	4		100.4.c.a.49.1		2
80.43	even	4		400.4.c.j.49.1		2
80.53	odd	4		100.4.c.a.49.2		2
80.59	odd	4		400.4.a.o.1.1		1
80.69	even	4		100.4.a.a.1.1		1
112.5	odd	12		980.4.i.n.361.1		2
112.37	even	12		980.4.i.e.361.1		2
112.53	even	12		980.4.i.e.961.1		2
112.69	odd	4		980.4.a.c.1.1		1
112.101	odd	12		980.4.i.n.961.1		2
144.5	odd	12		1620.4.i.j.541.1		2
144.85	even	12		1620.4.i.d.541.1		2
144.101	odd	12		1620.4.i.j.1081.1		2
144.133	even	12		1620.4.i.d.1081.1		2
176.21	odd	4		2420.4.a.d.1.1		1
240.53	even	4		900.4.d.k.649.2		2
240.149	odd	4		900.4.a.m.1.1		1
240.197	even	4		900.4.d.k.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.a.a.1.1	✓	1	16.5	even	4
80.4.a.c.1.1		1	16.11	odd	4
100.4.a.a.1.1		1	80.69	even	4
100.4.c.a.49.1		2	80.37	odd	4
100.4.c.a.49.2		2	80.53	odd	4
180.4.a.a.1.1		1	48.5	odd	4
320.4.a.d.1.1		1	16.13	even	4
320.4.a.k.1.1		1	16.3	odd	4
400.4.a.o.1.1		1	80.59	odd	4
400.4.c.j.49.1		2	80.43	even	4
400.4.c.j.49.2		2	80.27	even	4
720.4.a.k.1.1		1	48.11	even	4
900.4.a.m.1.1		1	240.149	odd	4
900.4.d.k.649.1		2	240.197	even	4
900.4.d.k.649.2		2	240.53	even	4
980.4.a.c.1.1		1	112.69	odd	4
980.4.i.e.361.1		2	112.37	even	12
980.4.i.e.961.1		2	112.53	even	12
980.4.i.n.361.1		2	112.5	odd	12
980.4.i.n.961.1		2	112.101	odd	12
1280.4.d.c.641.1		2	4.3	odd	2
1280.4.d.c.641.2		2	8.3	odd	2
1280.4.d.n.641.1		2	8.5	even	2	inner
1280.4.d.n.641.2		2	1.1	even	1	trivial
1600.4.a.p.1.1		1	80.19	odd	4
1600.4.a.bl.1.1		1	80.29	even	4
1620.4.i.d.541.1		2	144.85	even	12
1620.4.i.d.1081.1		2	144.133	even	12
1620.4.i.j.541.1		2	144.5	odd	12
1620.4.i.j.1081.1		2	144.101	odd	12
2420.4.a.d.1.1		1	176.21	odd	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)

\(f(q)\)	\(=\)	\(q+4.00000i q^{3} -5.00000i q^{5} +16.0000 q^{7} +11.0000 q^{9} +60.0000i q^{11} +86.0000i q^{13} +20.0000 q^{15} +18.0000 q^{17} +44.0000i q^{19} +64.0000i q^{21} -48.0000 q^{23} -25.0000 q^{25} +152.000i q^{27} -186.000i q^{29} +176.000 q^{31} -240.000 q^{33} -80.0000i q^{35} -254.000i q^{37} -344.000 q^{39} -186.000 q^{41} +100.000i q^{43} -55.0000i q^{45} +168.000 q^{47} -87.0000 q^{49} +72.0000i q^{51} +498.000i q^{53} +300.000 q^{55} -176.000 q^{57} +252.000i q^{59} -58.0000i q^{61} +176.000 q^{63} +430.000 q^{65} -1036.00i q^{67} -192.000i q^{69} -168.000 q^{71} -506.000 q^{73} -100.000i q^{75} +960.000i q^{77} +272.000 q^{79} -311.000 q^{81} +948.000i q^{83} -90.0000i q^{85} +744.000 q^{87} +1014.00 q^{89} +1376.00i q^{91} +704.000i q^{93} +220.000 q^{95} -766.000 q^{97} +660.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{7} + 22 q^{9} + 40 q^{15} + 36 q^{17} - 96 q^{23} - 50 q^{25} + 352 q^{31} - 480 q^{33} - 688 q^{39} - 372 q^{41} + 336 q^{47} - 174 q^{49} + 600 q^{55} - 352 q^{57} + 352 q^{63} + 860 q^{65}+ \cdots - 1532 q^{97}+O(q^{100}) \) 2 * q + 32 * q^7 + 22 * q^9 + 40 * q^15 + 36 * q^17 - 96 * q^23 - 50 * q^25 + 352 * q^31 - 480 * q^33 - 688 * q^39 - 372 * q^41 + 336 * q^47 - 174 * q^49 + 600 * q^55 - 352 * q^57 + 352 * q^63 + 860 * q^65 - 336 * q^71 - 1012 * q^73 + 544 * q^79 - 622 * q^81 + 1488 * q^87 + 2028 * q^89 + 440 * q^95 - 1532 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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