LMFDB - Embedded newform 5280.2.w.d.2641.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5280,2,Mod(2641,5280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5280.2641");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5280 = 2^{5} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5280.w (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:	$42.1610122672$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + x^{16} + 16x^{10} - 16x^{9} + 32x^{8} + 128x^{2} + 512 $ x^18 + x^16 + 16x^10 - 16x^9 + 32x^8 + 128x^2 + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 1320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2641.4
Root			$0.0805765 + 1.41192i$ of defining polynomial
Character	$\chi$	$=$	5280.2641
Dual form			5280.2.w.d.2641.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} -1.10929 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} -1.10929 q^{7} -1.00000 q^{9} -1.00000i q^{11} +3.85979i q^{13} -1.00000 q^{15} +1.10929 q^{17} -4.79786i q^{19} +1.10929i q^{21} -0.533975 q^{23} -1.00000 q^{25} +1.00000i q^{27} +2.68235i q^{29} +7.40654 q^{31} -1.00000 q^{33} +1.10929i q^{35} -2.36212i q^{37} +3.85979 q^{39} -3.93403 q^{41} -11.2135i q^{43} +1.00000i q^{45} +4.89887 q^{47} -5.76947 q^{49} -1.10929i q^{51} -1.13531i q^{53} -1.00000 q^{55} -4.79786 q^{57} -12.0787i q^{59} -1.68834i q^{61} +1.10929 q^{63} +3.85979 q^{65} +6.44653i q^{67} +0.533975i q^{69} -13.0060 q^{71} -11.9142 q^{73} +1.00000i q^{75} +1.10929i q^{77} -3.44379 q^{79} +1.00000 q^{81} -2.71306i q^{83} -1.10929i q^{85} +2.68235 q^{87} +13.9410 q^{89} -4.28163i q^{91} -7.40654i q^{93} -4.79786 q^{95} -19.1149 q^{97} +1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 18 q^{9}+O(q^{10}) $ 18 * q - 18 * q^9	$ 18 q - 18 q^{9} - 18 q^{15} - 18 q^{25} + 36 q^{31} - 18 q^{33} + 10 q^{49} - 18 q^{55} + 28 q^{57} - 8 q^{71} + 28 q^{73} - 8 q^{79} + 18 q^{81} - 28 q^{89} + 28 q^{95} - 52 q^{97}+O(q^{100}) $ 18 * q - 18 * q^9 - 18 * q^15 - 18 * q^25 + 36 * q^31 - 18 * q^33 + 10 * q^49 - 18 * q^55 + 28 * q^57 - 8 * q^71 + 28 * q^73 - 8 * q^79 + 18 * q^81 - 28 * q^89 + 28 * q^95 - 52 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5280\mathbb{Z}\right)^\times$.

$n$	$991$	$1057$	$3301$	$3521$	$3841$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	−1.10929	−0.419273	−0.209636	−	0.977779i	$-0.567228\pi$
$7$	−1.10929	−0.419273	−0.209636	+	0.977779i	$0.567228\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	3.85979i	1.07051i	0.844690	+	0.535256i	$0.179785\pi$
$13$	3.85979i	1.07051i	−0.844690	+	0.535256i	$0.820215\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.10929	0.269043	0.134521	−	0.990911i	$-0.457050\pi$
$17$	1.10929	0.269043	0.134521	+	0.990911i	$0.457050\pi$
$18$	0	0
$19$	− 4.79786i	− 1.10071i	−0.834932	−	0.550353i	$-0.814493\pi$
$19$	− 4.79786i	− 1.10071i	0.834932	−	0.550353i	$-0.185507\pi$
$20$	0	0
$21$	1.10929i	0.242067i
$22$	0	0
$23$	−0.533975	−0.111342	−0.0556708	−	0.998449i	$-0.517730\pi$
$23$	−0.533975	−0.111342	−0.0556708	+	0.998449i	$0.517730\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.68235i	0.498100i	0.968491	+	0.249050i	$0.0801184\pi$
$29$	2.68235i	0.498100i	−0.968491	+	0.249050i	$0.919882\pi$
$30$	0	0
$31$	7.40654	1.33025	0.665127	−	0.746731i	$-0.268378\pi$
$31$	7.40654	1.33025	0.665127	+	0.746731i	$0.268378\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	1.10929i	0.187504i
$36$	0	0
$37$	− 2.36212i	− 0.388331i	−0.980969	−	0.194165i	$-0.937800\pi$
$37$	− 2.36212i	− 0.388331i	0.980969	−	0.194165i	$-0.0621998\pi$
$38$	0	0
$39$	3.85979	0.618061
$40$	0	0
$41$	−3.93403	−0.614393	−0.307196	−	0.951646i	$-0.599391\pi$
$41$	−3.93403	−0.614393	−0.307196	+	0.951646i	$0.599391\pi$
$42$	0	0
$43$	− 11.2135i	− 1.71005i	−0.518588	−	0.855024i	$-0.673542\pi$
$43$	− 11.2135i	− 1.71005i	0.518588	−	0.855024i	$-0.326458\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	0	0
$47$	4.89887	0.714574	0.357287	−	0.933995i	$-0.383702\pi$
$47$	4.89887	0.714574	0.357287	+	0.933995i	$0.383702\pi$
$48$	0	0
$49$	−5.76947	−0.824210
$50$	0	0
$51$	− 1.10929i	− 0.155332i
$52$	0	0
$53$	− 1.13531i	− 0.155947i	−0.996955	−	0.0779736i	$-0.975155\pi$
$53$	− 1.13531i	− 0.155947i	0.996955	−	0.0779736i	$-0.0248450\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−4.79786	−0.635492
$58$	0	0
$59$	− 12.0787i	− 1.57251i	−0.617901	−	0.786256i	$-0.712017\pi$
$59$	− 12.0787i	− 1.57251i	0.617901	−	0.786256i	$-0.287983\pi$
$60$	0	0
$61$	− 1.68834i	− 0.216170i	−0.994142	−	0.108085i	$-0.965528\pi$
$61$	− 1.68834i	− 0.216170i	0.994142	−	0.108085i	$-0.0344718\pi$
$62$	0	0
$63$	1.10929	0.139758
$64$	0	0
$65$	3.85979	0.478748
$66$	0	0
$67$	6.44653i	0.787569i	0.919203	+	0.393785i	$0.128834\pi$
$67$	6.44653i	0.787569i	−0.919203	+	0.393785i	$0.871166\pi$
$68$	0	0
$69$	0.533975i	0.0642831i
$70$	0	0
$71$	−13.0060	−1.54353	−0.771763	−	0.635911i	$-0.780625\pi$
$71$	−13.0060	−1.54353	−0.771763	+	0.635911i	$0.780625\pi$
$72$	0	0
$73$	−11.9142	−1.39445	−0.697224	−	0.716853i	$-0.745582\pi$
$73$	−11.9142	−1.39445	−0.697224	+	0.716853i	$0.745582\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	1.10929i	0.126416i
$78$	0	0
$79$	−3.44379	−0.387457	−0.193728	−	0.981055i	$-0.562058\pi$
$79$	−3.44379	−0.387457	−0.193728	+	0.981055i	$0.562058\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 2.71306i	− 0.297797i	−0.988852	−	0.148899i	$-0.952427\pi$
$83$	− 2.71306i	− 0.297797i	0.988852	−	0.148899i	$-0.0475728\pi$
$84$	0	0
$85$	− 1.10929i	− 0.120320i
$86$	0	0
$87$	2.68235	0.287578
$88$	0	0
$89$	13.9410	1.47775	0.738873	−	0.673844i	$-0.235358\pi$
$89$	13.9410	1.47775	0.738873	+	0.673844i	$0.235358\pi$
$90$	0	0
$91$	− 4.28163i	− 0.448837i
$92$	0	0
$93$	− 7.40654i	− 0.768022i
$94$	0	0
$95$	−4.79786	−0.492250
$96$	0	0
$97$	−19.1149	−1.94082	−0.970412	−	0.241456i	$-0.922375\pi$
$97$	−19.1149	−1.94082	−0.970412	+	0.241456i	$0.922375\pi$
$98$	0	0
$99$	1.00000i	0.100504i
$100$	0	0
$101$	− 1.75360i	− 0.174490i	−0.996187	−	0.0872448i	$-0.972194\pi$
$101$	− 1.75360i	− 0.174490i	0.996187	−	0.0872448i	$-0.0278062\pi$
$102$	0	0
$103$	−7.34041	−0.723272	−0.361636	−	0.932319i	$-0.617782\pi$
$103$	−7.34041	−0.723272	−0.361636	+	0.932319i	$0.617782\pi$
$104$	0	0
$105$	1.10929	0.108256
$106$	0	0
$107$	4.82102i	0.466066i	0.972469	+	0.233033i	$0.0748650\pi$
$107$	4.82102i	0.466066i	−0.972469	+	0.233033i	$0.925135\pi$
$108$	0	0
$109$	6.40843i	0.613816i	0.951739	+	0.306908i	$0.0992945\pi$
$109$	6.40843i	0.613816i	−0.951739	+	0.306908i	$0.900706\pi$
$110$	0	0
$111$	−2.36212	−0.224203
$112$	0	0
$113$	−6.18727	−0.582050	−0.291025	−	0.956715i	$-0.593996\pi$
$113$	−6.18727	−0.582050	−0.291025	+	0.956715i	$0.593996\pi$
$114$	0	0
$115$	0.533975i	0.0497935i
$116$	0	0
$117$	− 3.85979i	− 0.356838i
$118$	0	0
$119$	−1.23053	−0.112802
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	3.93403i	0.354720i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	0.205570	0.0182414	0.00912072	−	0.999958i	$-0.497097\pi$
$127$	0.205570	0.0182414	0.00912072	+	0.999958i	$0.497097\pi$
$128$	0	0
$129$	−11.2135	−0.987297
$130$	0	0
$131$	− 0.296752i	− 0.0259273i	−0.999916	−	0.0129637i	$-0.995873\pi$
$131$	− 0.296752i	− 0.0259273i	0.999916	−	0.0129637i	$-0.00412657\pi$
$132$	0	0
$133$	5.32223i	0.461496i
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	9.96720	0.851555	0.425778	−	0.904828i	$-0.360001\pi$
$137$	9.96720	0.851555	0.425778	+	0.904828i	$0.360001\pi$
$138$	0	0
$139$	− 10.1021i	− 0.856846i	−0.903578	−	0.428423i	$-0.859069\pi$
$139$	− 10.1021i	− 0.856846i	0.903578	−	0.428423i	$-0.140931\pi$
$140$	0	0
$141$	− 4.89887i	− 0.412559i
$142$	0	0
$143$	3.85979	0.322772
$144$	0	0
$145$	2.68235	0.222757
$146$	0	0
$147$	5.76947i	0.475858i
$148$	0	0
$149$	− 5.65265i	− 0.463083i	−0.972825	−	0.231542i	$-0.925623\pi$
$149$	− 5.65265i	− 0.463083i	0.972825	−	0.231542i	$-0.0743769\pi$
$150$	0	0
$151$	−10.4793	−0.852792	−0.426396	−	0.904536i	$-0.640217\pi$
$151$	−10.4793	−0.852792	−0.426396	+	0.904536i	$0.640217\pi$
$152$	0	0
$153$	−1.10929	−0.0896809
$154$	0	0
$155$	− 7.40654i	− 0.594907i
$156$	0	0
$157$	− 2.14258i	− 0.170997i	−0.996338	−	0.0854983i	$-0.972752\pi$
$157$	− 2.14258i	− 0.170997i	0.996338	−	0.0854983i	$-0.0272482\pi$
$158$	0	0
$159$	−1.13531	−0.0900362
$160$	0	0
$161$	0.592334	0.0466825
$162$	0	0
$163$	− 14.7976i	− 1.15904i	−0.814959	−	0.579519i	$-0.803240\pi$
$163$	− 14.7976i	− 1.15904i	0.814959	−	0.579519i	$-0.196760\pi$
$164$	0	0
$165$	1.00000i	0.0778499i
$166$	0	0
$167$	−19.6194	−1.51819	−0.759096	−	0.650978i	$-0.774359\pi$
$167$	−19.6194	−1.51819	−0.759096	+	0.650978i	$0.774359\pi$
$168$	0	0
$169$	−1.89797	−0.145997
$170$	0	0
$171$	4.79786i	0.366902i
$172$	0	0
$173$	10.7400i	0.816545i	0.912860	+	0.408272i	$0.133869\pi$
$173$	10.7400i	0.816545i	−0.912860	+	0.408272i	$0.866131\pi$
$174$	0	0
$175$	1.10929	0.0838546
$176$	0	0
$177$	−12.0787	−0.907890
$178$	0	0
$179$	2.41684i	0.180643i	0.995913	+	0.0903214i	$0.0287894\pi$
$179$	2.41684i	0.180643i	−0.995913	+	0.0903214i	$0.971211\pi$
$180$	0	0
$181$	− 5.82247i	− 0.432780i	−0.976307	−	0.216390i	$-0.930572\pi$
$181$	− 5.82247i	− 0.432780i	0.976307	−	0.216390i	$-0.0694283\pi$
$182$	0	0
$183$	−1.68834	−0.124806
$184$	0	0
$185$	−2.36212	−0.173667
$186$	0	0
$187$	− 1.10929i	− 0.0811194i
$188$	0	0
$189$	− 1.10929i	− 0.0806891i
$190$	0	0
$191$	−4.15396	−0.300570	−0.150285	−	0.988643i	$-0.548019\pi$
$191$	−4.15396	−0.300570	−0.150285	+	0.988643i	$0.548019\pi$
$192$	0	0
$193$	−12.2730	−0.883431	−0.441715	−	0.897155i	$-0.645630\pi$
$193$	−12.2730	−0.883431	−0.441715	+	0.897155i	$0.645630\pi$
$194$	0	0
$195$	− 3.85979i	− 0.276405i
$196$	0	0
$197$	7.93158i	0.565102i	0.959252	+	0.282551i	$0.0911806\pi$
$197$	7.93158i	0.565102i	−0.959252	+	0.282551i	$0.908819\pi$
$198$	0	0
$199$	−0.922025	−0.0653606	−0.0326803	−	0.999466i	$-0.510404\pi$
$199$	−0.922025	−0.0653606	−0.0326803	+	0.999466i	$0.510404\pi$
$200$	0	0
$201$	6.44653	0.454703
$202$	0	0
$203$	− 2.97551i	− 0.208840i
$204$	0	0
$205$	3.93403i	0.274765i
$206$	0	0
$207$	0.533975	0.0371139
$208$	0	0
$209$	−4.79786	−0.331875
$210$	0	0
$211$	− 12.2179i	− 0.841115i	−0.907266	−	0.420558i	$-0.861834\pi$
$211$	− 12.2179i	− 0.841115i	0.907266	−	0.420558i	$-0.138166\pi$
$212$	0	0
$213$	13.0060i	0.891155i
$214$	0	0
$215$	−11.2135	−0.764757
$216$	0	0
$217$	−8.21601	−0.557739
$218$	0	0
$219$	11.9142i	0.805085i
$220$	0	0
$221$	4.28163i	0.288014i
$222$	0	0
$223$	10.4962	0.702879	0.351440	−	0.936211i	$-0.385692\pi$
$223$	10.4962	0.702879	0.351440	+	0.936211i	$0.385692\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	19.9398i	1.32345i	0.749746	+	0.661726i	$0.230176\pi$
$227$	19.9398i	1.32345i	−0.749746	+	0.661726i	$0.769824\pi$
$228$	0	0
$229$	22.1880i	1.46622i	0.680109	+	0.733111i	$0.261932\pi$
$229$	22.1880i	1.46622i	−0.680109	+	0.733111i	$0.738068\pi$
$230$	0	0
$231$	1.10929	0.0729860
$232$	0	0
$233$	−3.60397	−0.236104	−0.118052	−	0.993007i	$-0.537665\pi$
$233$	−3.60397	−0.236104	−0.118052	+	0.993007i	$0.537665\pi$
$234$	0	0
$235$	− 4.89887i	− 0.319567i
$236$	0	0
$237$	3.44379i	0.223698i
$238$	0	0
$239$	−24.8117	−1.60494	−0.802469	−	0.596694i	$-0.796481\pi$
$239$	−24.8117	−1.60494	−0.802469	+	0.596694i	$0.796481\pi$
$240$	0	0
$241$	11.7834	0.759034	0.379517	−	0.925185i	$-0.376090\pi$
$241$	11.7834	0.759034	0.379517	+	0.925185i	$0.376090\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	5.76947i	0.368598i
$246$	0	0
$247$	18.5187	1.17832
$248$	0	0
$249$	−2.71306	−0.171933
$250$	0	0
$251$	28.3697i	1.79068i	0.445385	+	0.895339i	$0.353067\pi$
$251$	28.3697i	1.79068i	−0.445385	+	0.895339i	$0.646933\pi$
$252$	0	0
$253$	0.533975i	0.0335707i
$254$	0	0
$255$	−1.10929	−0.0694665
$256$	0	0
$257$	−17.2750	−1.07759	−0.538793	−	0.842438i	$-0.681120\pi$
$257$	−17.2750	−1.07759	−0.538793	+	0.842438i	$0.681120\pi$
$258$	0	0
$259$	2.62028i	0.162816i
$260$	0	0
$261$	− 2.68235i	− 0.166033i
$262$	0	0
$263$	3.34321	0.206151	0.103076	−	0.994674i	$-0.467132\pi$
$263$	3.34321	0.206151	0.103076	+	0.994674i	$0.467132\pi$
$264$	0	0
$265$	−1.13531	−0.0697417
$266$	0	0
$267$	− 13.9410i	− 0.853177i
$268$	0	0
$269$	− 21.6071i	− 1.31741i	−0.752403	−	0.658703i	$-0.771106\pi$
$269$	− 21.6071i	− 1.31741i	0.752403	−	0.658703i	$-0.228894\pi$
$270$	0	0
$271$	−13.3858	−0.813132	−0.406566	−	0.913621i	$-0.633274\pi$
$271$	−13.3858	−0.813132	−0.406566	+	0.913621i	$0.633274\pi$
$272$	0	0
$273$	−4.28163	−0.259136
$274$	0	0
$275$	1.00000i	0.0603023i
$276$	0	0
$277$	− 1.55071i	− 0.0931733i	−0.998914	−	0.0465866i	$-0.985166\pi$
$277$	− 1.55071i	− 0.0931733i	0.998914	−	0.0465866i	$-0.0148344\pi$
$278$	0	0
$279$	−7.40654	−0.443418
$280$	0	0
$281$	−0.400588	−0.0238971	−0.0119485	−	0.999929i	$-0.503803\pi$
$281$	−0.400588	−0.0238971	−0.0119485	+	0.999929i	$0.503803\pi$
$282$	0	0
$283$	− 25.8493i	− 1.53658i	−0.640102	−	0.768290i	$-0.721108\pi$
$283$	− 25.8493i	− 1.53658i	0.640102	−	0.768290i	$-0.278892\pi$
$284$	0	0
$285$	4.79786i	0.284201i
$286$	0	0
$287$	4.36399	0.257598
$288$	0	0
$289$	−15.7695	−0.927616
$290$	0	0
$291$	19.1149i	1.12054i
$292$	0	0
$293$	7.62504i	0.445459i	0.974880	+	0.222730i	$0.0714967\pi$
$293$	7.62504i	0.445459i	−0.974880	+	0.222730i	$0.928503\pi$
$294$	0	0
$295$	−12.0787	−0.703249
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	− 2.06103i	− 0.119193i
$300$	0	0
$301$	12.4391i	0.716977i
$302$	0	0
$303$	−1.75360	−0.100742
$304$	0	0
$305$	−1.68834	−0.0966741
$306$	0	0
$307$	6.71091i	0.383012i	0.981491	+	0.191506i	$0.0613371\pi$
$307$	6.71091i	0.383012i	−0.981491	+	0.191506i	$0.938663\pi$
$308$	0	0
$309$	7.34041i	0.417581i
$310$	0	0
$311$	−15.1336	−0.858148	−0.429074	−	0.903269i	$-0.641160\pi$
$311$	−15.1336	−0.858148	−0.429074	+	0.903269i	$0.641160\pi$
$312$	0	0
$313$	15.2220	0.860398	0.430199	−	0.902734i	$-0.358443\pi$
$313$	15.2220	0.860398	0.430199	+	0.902734i	$0.358443\pi$
$314$	0	0
$315$	− 1.10929i	− 0.0625015i
$316$	0	0
$317$	33.4802i	1.88043i	0.340578	+	0.940216i	$0.389377\pi$
$317$	33.4802i	1.88043i	−0.340578	+	0.940216i	$0.610623\pi$
$318$	0	0
$319$	2.68235	0.150183
$320$	0	0
$321$	4.82102	0.269083
$322$	0	0
$323$	− 5.32223i	− 0.296137i
$324$	0	0
$325$	− 3.85979i	− 0.214103i
$326$	0	0
$327$	6.40843	0.354387
$328$	0	0
$329$	−5.43428	−0.299601
$330$	0	0
$331$	− 32.9618i	− 1.81174i	−0.423553	−	0.905872i	$-0.639217\pi$
$331$	− 32.9618i	− 1.81174i	0.423553	−	0.905872i	$-0.360783\pi$
$332$	0	0
$333$	2.36212i	0.129444i
$334$	0	0
$335$	6.44653	0.352212
$336$	0	0
$337$	12.0843	0.658274	0.329137	−	0.944282i	$-0.393242\pi$
$337$	12.0843	0.658274	0.329137	+	0.944282i	$0.393242\pi$
$338$	0	0
$339$	6.18727i	0.336047i
$340$	0	0
$341$	− 7.40654i	− 0.401086i
$342$	0	0
$343$	14.1651	0.764842
$344$	0	0
$345$	0.533975	0.0287483
$346$	0	0
$347$	12.9020i	0.692618i	0.938121	+	0.346309i	$0.112565\pi$
$347$	12.9020i	0.692618i	−0.938121	+	0.346309i	$0.887435\pi$
$348$	0	0
$349$	− 30.5861i	− 1.63723i	−0.574340	−	0.818617i	$-0.694741\pi$
$349$	− 30.5861i	− 1.63723i	0.574340	−	0.818617i	$-0.305259\pi$
$350$	0	0
$351$	−3.85979	−0.206020
$352$	0	0
$353$	31.7526	1.69002	0.845009	−	0.534751i	$-0.179595\pi$
$353$	31.7526	1.69002	0.845009	+	0.534751i	$0.179595\pi$
$354$	0	0
$355$	13.0060i	0.690286i
$356$	0	0
$357$	1.23053i	0.0651264i
$358$	0	0
$359$	−21.5000	−1.13473	−0.567363	−	0.823468i	$-0.692036\pi$
$359$	−21.5000	−1.13473	−0.567363	+	0.823468i	$0.692036\pi$
$360$	0	0
$361$	−4.01948	−0.211552
$362$	0	0
$363$	1.00000i	0.0524864i
$364$	0	0
$365$	11.9142i	0.623616i
$366$	0	0
$367$	−19.3760	−1.01142	−0.505710	−	0.862703i	$-0.668770\pi$
$367$	−19.3760	−1.01142	−0.505710	+	0.862703i	$0.668770\pi$
$368$	0	0
$369$	3.93403	0.204798
$370$	0	0
$371$	1.25939i	0.0653844i
$372$	0	0
$373$	− 25.4408i	− 1.31728i	−0.752460	−	0.658638i	$-0.771133\pi$
$373$	− 25.4408i	− 1.31728i	0.752460	−	0.658638i	$-0.228867\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−10.3533	−0.533222
$378$	0	0
$379$	− 1.23042i	− 0.0632022i	−0.999501	−	0.0316011i	$-0.989939\pi$
$379$	− 1.23042i	− 0.0632022i	0.999501	−	0.0316011i	$-0.0100606\pi$
$380$	0	0
$381$	− 0.205570i	− 0.0105317i
$382$	0	0
$383$	36.3893	1.85941	0.929704	−	0.368308i	$-0.120063\pi$
$383$	36.3893	1.85941	0.929704	+	0.368308i	$0.120063\pi$
$384$	0	0
$385$	1.10929	0.0565347
$386$	0	0
$387$	11.2135i	0.570016i
$388$	0	0
$389$	− 22.6608i	− 1.14895i	−0.818523	−	0.574474i	$-0.805207\pi$
$389$	− 22.6608i	− 1.14895i	0.818523	−	0.574474i	$-0.194793\pi$
$390$	0	0
$391$	−0.592334	−0.0299556
$392$	0	0
$393$	−0.296752	−0.0149691
$394$	0	0
$395$	3.44379i	0.173276i
$396$	0	0
$397$	− 4.71284i	− 0.236531i	−0.992982	−	0.118265i	$-0.962267\pi$
$397$	− 4.71284i	− 0.236531i	0.992982	−	0.118265i	$-0.0377333\pi$
$398$	0	0
$399$	5.32223	0.266445
$400$	0	0
$401$	6.44647	0.321921	0.160961	−	0.986961i	$-0.448541\pi$
$401$	6.44647	0.321921	0.160961	+	0.986961i	$0.448541\pi$
$402$	0	0
$403$	28.5877i	1.42405i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	−2.36212	−0.117086
$408$	0	0
$409$	−19.1426	−0.946541	−0.473271	−	0.880917i	$-0.656927\pi$
$409$	−19.1426	−0.946541	−0.473271	+	0.880917i	$0.656927\pi$
$410$	0	0
$411$	− 9.96720i	− 0.491646i
$412$	0	0
$413$	13.3988i	0.659311i
$414$	0	0
$415$	−2.71306	−0.133179
$416$	0	0
$417$	−10.1021	−0.494700
$418$	0	0
$419$	− 33.3655i	− 1.63001i	−0.579452	−	0.815006i	$-0.696733\pi$
$419$	− 33.3655i	− 1.63001i	0.579452	−	0.815006i	$-0.303267\pi$
$420$	0	0
$421$	− 27.6592i	− 1.34803i	−0.738719	−	0.674013i	$-0.764569\pi$
$421$	− 27.6592i	− 1.34803i	0.738719	−	0.674013i	$-0.235431\pi$
$422$	0	0
$423$	−4.89887	−0.238191
$424$	0	0
$425$	−1.10929	−0.0538085
$426$	0	0
$427$	1.87286i	0.0906341i
$428$	0	0
$429$	− 3.85979i	− 0.186352i
$430$	0	0
$431$	−28.8632	−1.39029	−0.695147	−	0.718868i	$-0.744661\pi$
$431$	−28.8632	−1.39029	−0.695147	+	0.718868i	$0.744661\pi$
$432$	0	0
$433$	22.0696	1.06060	0.530298	−	0.847811i	$-0.322080\pi$
$433$	22.0696	1.06060	0.530298	+	0.847811i	$0.322080\pi$
$434$	0	0
$435$	− 2.68235i	− 0.128609i
$436$	0	0
$437$	2.56194i	0.122554i
$438$	0	0
$439$	−21.8272	−1.04175	−0.520877	−	0.853631i	$-0.674395\pi$
$439$	−21.8272	−1.04175	−0.520877	+	0.853631i	$0.674395\pi$
$440$	0	0
$441$	5.76947	0.274737
$442$	0	0
$443$	− 9.51341i	− 0.451996i	−0.974128	−	0.225998i	$-0.927436\pi$
$443$	− 9.51341i	− 0.451996i	0.974128	−	0.225998i	$-0.0725642\pi$
$444$	0	0
$445$	− 13.9410i	− 0.660868i
$446$	0	0
$447$	−5.65265	−0.267361
$448$	0	0
$449$	−0.914410	−0.0431537	−0.0215768	−	0.999767i	$-0.506869\pi$
$449$	−0.914410	−0.0431537	−0.0215768	+	0.999767i	$0.506869\pi$
$450$	0	0
$451$	3.93403i	0.185246i
$452$	0	0
$453$	10.4793i	0.492360i
$454$	0	0
$455$	−4.28163	−0.200726
$456$	0	0
$457$	11.6844	0.546571	0.273286	−	0.961933i	$-0.411890\pi$
$457$	11.6844	0.546571	0.273286	+	0.961933i	$0.411890\pi$
$458$	0	0
$459$	1.10929i	0.0517773i
$460$	0	0
$461$	− 40.9715i	− 1.90823i	−0.299439	−	0.954115i	$-0.596800\pi$
$461$	− 40.9715i	− 1.90823i	0.299439	−	0.954115i	$-0.403200\pi$
$462$	0	0
$463$	−13.3735	−0.621521	−0.310760	−	0.950488i	$-0.600584\pi$
$463$	−13.3735	−0.621521	−0.310760	+	0.950488i	$0.600584\pi$
$464$	0	0
$465$	−7.40654	−0.343470
$466$	0	0
$467$	40.4093i	1.86992i	0.354752	+	0.934961i	$0.384565\pi$
$467$	40.4093i	1.86992i	−0.354752	+	0.934961i	$0.615435\pi$
$468$	0	0
$469$	− 7.15108i	− 0.330206i
$470$	0	0
$471$	−2.14258	−0.0987249
$472$	0	0
$473$	−11.2135	−0.515599
$474$	0	0
$475$	4.79786i	0.220141i
$476$	0	0
$477$	1.13531i	0.0519824i
$478$	0	0
$479$	−22.5139	−1.02868	−0.514342	−	0.857585i	$-0.671964\pi$
$479$	−22.5139	−1.02868	−0.514342	+	0.857585i	$0.671964\pi$
$480$	0	0
$481$	9.11730	0.415713
$482$	0	0
$483$	− 0.592334i	− 0.0269522i
$484$	0	0
$485$	19.1149i	0.867963i
$486$	0	0
$487$	14.3851	0.651849	0.325924	−	0.945396i	$-0.394324\pi$
$487$	14.3851	0.651849	0.325924	+	0.945396i	$0.394324\pi$
$488$	0	0
$489$	−14.7976	−0.669171
$490$	0	0
$491$	2.28511i	0.103126i	0.998670	+	0.0515628i	$0.0164202\pi$
$491$	2.28511i	0.103126i	−0.998670	+	0.0515628i	$0.983580\pi$
$492$	0	0
$493$	2.97551i	0.134010i
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	14.4274	0.647158
$498$	0	0
$499$	− 20.1669i	− 0.902795i	−0.892323	−	0.451398i	$-0.850926\pi$
$499$	− 20.1669i	− 0.902795i	0.892323	−	0.451398i	$-0.149074\pi$
$500$	0	0
$501$	19.6194i	0.876529i
$502$	0	0
$503$	−9.88520	−0.440759	−0.220380	−	0.975414i	$-0.570730\pi$
$503$	−9.88520	−0.440759	−0.220380	+	0.975414i	$0.570730\pi$
$504$	0	0
$505$	−1.75360	−0.0780341
$506$	0	0
$507$	1.89797i	0.0842916i
$508$	0	0
$509$	18.4339i	0.817070i	0.912743	+	0.408535i	$0.133960\pi$
$509$	18.4339i	0.817070i	−0.912743	+	0.408535i	$0.866040\pi$
$510$	0	0
$511$	13.2163	0.584654
$512$	0	0
$513$	4.79786	0.211831
$514$	0	0
$515$	7.34041i	0.323457i
$516$	0	0
$517$	− 4.89887i	− 0.215452i
$518$	0	0
$519$	10.7400	0.471432
$520$	0	0
$521$	−6.65531	−0.291574	−0.145787	−	0.989316i	$-0.546571\pi$
$521$	−6.65531	−0.291574	−0.145787	+	0.989316i	$0.546571\pi$
$522$	0	0
$523$	20.6236i	0.901806i	0.892573	+	0.450903i	$0.148898\pi$
$523$	20.6236i	0.901806i	−0.892573	+	0.450903i	$0.851102\pi$
$524$	0	0
$525$	− 1.10929i	− 0.0484135i
$526$	0	0
$527$	8.21601	0.357895
$528$	0	0
$529$	−22.7149	−0.987603
$530$	0	0
$531$	12.0787i	0.524171i
$532$	0	0
$533$	− 15.1845i	− 0.657715i
$534$	0	0
$535$	4.82102	0.208431
$536$	0	0
$537$	2.41684	0.104294
$538$	0	0
$539$	5.76947i	0.248509i
$540$	0	0
$541$	11.5706i	0.497460i	0.968573	+	0.248730i	$0.0800132\pi$
$541$	11.5706i	0.497460i	−0.968573	+	0.248730i	$0.919987\pi$
$542$	0	0
$543$	−5.82247	−0.249866
$544$	0	0
$545$	6.40843	0.274507
$546$	0	0
$547$	17.0495i	0.728984i	0.931207	+	0.364492i	$0.118757\pi$
$547$	17.0495i	0.728984i	−0.931207	+	0.364492i	$0.881243\pi$
$548$	0	0
$549$	1.68834i	0.0720566i
$550$	0	0
$551$	12.8696	0.548261
$552$	0	0
$553$	3.82017	0.162450
$554$	0	0
$555$	2.36212i	0.100267i
$556$	0	0
$557$	7.00000i	0.296600i	0.988942	+	0.148300i	$0.0473801\pi$
$557$	7.00000i	0.296600i	−0.988942	+	0.148300i	$0.952620\pi$
$558$	0	0
$559$	43.2819	1.83063
$560$	0	0
$561$	−1.10929	−0.0468343
$562$	0	0
$563$	− 17.1369i	− 0.722234i	−0.932521	−	0.361117i	$-0.882396\pi$
$563$	− 17.1369i	− 0.722234i	0.932521	−	0.361117i	$-0.117604\pi$
$564$	0	0
$565$	6.18727i	0.260301i
$566$	0	0
$567$	−1.10929	−0.0465859
$568$	0	0
$569$	17.0090	0.713055	0.356528	−	0.934285i	$-0.383961\pi$
$569$	17.0090	0.713055	0.356528	+	0.934285i	$0.383961\pi$
$570$	0	0
$571$	34.6935i	1.45188i	0.687758	+	0.725940i	$0.258595\pi$
$571$	34.6935i	1.45188i	−0.687758	+	0.725940i	$0.741405\pi$
$572$	0	0
$573$	4.15396i	0.173534i
$574$	0	0
$575$	0.533975	0.0222683
$576$	0	0
$577$	−22.9523	−0.955519	−0.477759	−	0.878491i	$-0.658551\pi$
$577$	−22.9523	−0.955519	−0.477759	+	0.878491i	$0.658551\pi$
$578$	0	0
$579$	12.2730i	0.510049i
$580$	0	0
$581$	3.00958i	0.124858i
$582$	0	0
$583$	−1.13531	−0.0470199
$584$	0	0
$585$	−3.85979	−0.159583
$586$	0	0
$587$	1.75446i	0.0724142i	0.999344	+	0.0362071i	$0.0115276\pi$
$587$	1.75446i	0.0724142i	−0.999344	+	0.0362071i	$0.988472\pi$
$588$	0	0
$589$	− 35.5355i	− 1.46422i
$590$	0	0
$591$	7.93158	0.326262
$592$	0	0
$593$	−18.4126	−0.756115	−0.378057	−	0.925782i	$-0.623408\pi$
$593$	−18.4126	−0.756115	−0.378057	+	0.925782i	$0.623408\pi$
$594$	0	0
$595$	1.23053i	0.0504467i
$596$	0	0
$597$	0.922025i	0.0377360i
$598$	0	0
$599$	−18.7239	−0.765038	−0.382519	−	0.923948i	$-0.624943\pi$
$599$	−18.7239	−0.765038	−0.382519	+	0.923948i	$0.624943\pi$
$600$	0	0
$601$	42.3253	1.72648	0.863242	−	0.504791i	$-0.168430\pi$
$601$	42.3253	1.72648	0.863242	+	0.504791i	$0.168430\pi$
$602$	0	0
$603$	− 6.44653i	− 0.262523i
$604$	0	0
$605$	1.00000i	0.0406558i
$606$	0	0
$607$	−32.9914	−1.33908	−0.669539	−	0.742777i	$-0.733508\pi$
$607$	−32.9914	−1.33908	−0.669539	+	0.742777i	$0.733508\pi$
$608$	0	0
$609$	−2.97551	−0.120574
$610$	0	0
$611$	18.9086i	0.764961i
$612$	0	0
$613$	− 35.9397i	− 1.45159i	−0.687911	−	0.725795i	$-0.741472\pi$
$613$	− 35.9397i	− 1.45159i	0.687911	−	0.725795i	$-0.258528\pi$
$614$	0	0
$615$	3.93403	0.158636
$616$	0	0
$617$	−5.13014	−0.206532	−0.103266	−	0.994654i	$-0.532929\pi$
$617$	−5.13014	−0.206532	−0.103266	+	0.994654i	$0.532929\pi$
$618$	0	0
$619$	− 30.6167i	− 1.23059i	−0.788297	−	0.615295i	$-0.789037\pi$
$619$	− 30.6167i	− 1.23059i	0.788297	−	0.615295i	$-0.210963\pi$
$620$	0	0
$621$	− 0.533975i	− 0.0214277i
$622$	0	0
$623$	−15.4647	−0.619579
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.79786i	0.191608i
$628$	0	0
$629$	− 2.62028i	− 0.104478i
$630$	0	0
$631$	−18.7668	−0.747096	−0.373548	−	0.927611i	$-0.621859\pi$
$631$	−18.7668	−0.747096	−0.373548	+	0.927611i	$0.621859\pi$
$632$	0	0
$633$	−12.2179	−0.485618
$634$	0	0
$635$	− 0.205570i	− 0.00815782i
$636$	0	0
$637$	− 22.2689i	− 0.882328i
$638$	0	0
$639$	13.0060	0.514509
$640$	0	0
$641$	−7.01696	−0.277153	−0.138577	−	0.990352i	$-0.544253\pi$
$641$	−7.01696	−0.277153	−0.138577	+	0.990352i	$0.544253\pi$
$642$	0	0
$643$	− 14.4766i	− 0.570901i	−0.958393	−	0.285450i	$-0.907857\pi$
$643$	− 14.4766i	− 0.570901i	0.958393	−	0.285450i	$-0.0921432\pi$
$644$	0	0
$645$	11.2135i	0.441533i
$646$	0	0
$647$	18.5693	0.730033	0.365017	−	0.931001i	$-0.381063\pi$
$647$	18.5693	0.730033	0.365017	+	0.931001i	$0.381063\pi$
$648$	0	0
$649$	−12.0787	−0.474130
$650$	0	0
$651$	8.21601i	0.322011i
$652$	0	0
$653$	9.60926i	0.376039i	0.982165	+	0.188020i	$0.0602069\pi$
$653$	9.60926i	0.376039i	−0.982165	+	0.188020i	$0.939793\pi$
$654$	0	0
$655$	−0.296752	−0.0115950
$656$	0	0
$657$	11.9142	0.464816
$658$	0	0
$659$	15.1640i	0.590707i	0.955388	+	0.295354i	$0.0954375\pi$
$659$	15.1640i	0.590707i	−0.955388	+	0.295354i	$0.904563\pi$
$660$	0	0
$661$	− 15.3449i	− 0.596846i	−0.954434	−	0.298423i	$-0.903539\pi$
$661$	− 15.3449i	− 0.596846i	0.954434	−	0.298423i	$-0.0964607\pi$
$662$	0	0
$663$	4.28163	0.166285
$664$	0	0
$665$	5.32223	0.206387
$666$	0	0
$667$	− 1.43231i	− 0.0554593i
$668$	0	0
$669$	− 10.4962i	− 0.405807i
$670$	0	0
$671$	−1.68834	−0.0651777
$672$	0	0
$673$	−26.2662	−1.01249	−0.506244	−	0.862390i	$-0.668966\pi$
$673$	−26.2662	−1.01249	−0.506244	+	0.862390i	$0.668966\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	23.0202i	0.884740i	0.896833	+	0.442370i	$0.145862\pi$
$677$	23.0202i	0.884740i	−0.896833	+	0.442370i	$0.854138\pi$
$678$	0	0
$679$	21.2040	0.813735
$680$	0	0
$681$	19.9398	0.764095
$682$	0	0
$683$	34.2805i	1.31171i	0.754888	+	0.655854i	$0.227691\pi$
$683$	34.2805i	1.31171i	−0.754888	+	0.655854i	$0.772309\pi$
$684$	0	0
$685$	− 9.96720i	− 0.380827i
$686$	0	0
$687$	22.1880	0.846524
$688$	0	0
$689$	4.38207	0.166943
$690$	0	0
$691$	− 41.1950i	− 1.56713i	−0.621309	−	0.783566i	$-0.713399\pi$
$691$	− 41.1950i	− 1.56713i	0.621309	−	0.783566i	$-0.286601\pi$
$692$	0	0
$693$	− 1.10929i	− 0.0421385i
$694$	0	0
$695$	−10.1021	−0.383193
$696$	0	0
$697$	−4.36399	−0.165298
$698$	0	0
$699$	3.60397i	0.136315i
$700$	0	0
$701$	27.4610i	1.03719i	0.855021	+	0.518594i	$0.173544\pi$
$701$	27.4610i	1.03719i	−0.855021	+	0.518594i	$0.826456\pi$
$702$	0	0
$703$	−11.3331	−0.427438
$704$	0	0
$705$	−4.89887	−0.184502
$706$	0	0
$707$	1.94525i	0.0731587i
$708$	0	0
$709$	− 31.3070i	− 1.17576i	−0.808948	−	0.587880i	$-0.799963\pi$
$709$	− 31.3070i	− 1.17576i	0.808948	−	0.587880i	$-0.200037\pi$
$710$	0	0
$711$	3.44379	0.129152
$712$	0	0
$713$	−3.95491	−0.148112
$714$	0	0
$715$	− 3.85979i	− 0.144348i
$716$	0	0
$717$	24.8117i	0.926612i
$718$	0	0
$719$	32.3409	1.20611	0.603057	−	0.797698i	$-0.293949\pi$
$719$	32.3409	1.20611	0.603057	+	0.797698i	$0.293949\pi$
$720$	0	0
$721$	8.14265	0.303248
$722$	0	0
$723$	− 11.7834i	− 0.438228i
$724$	0	0
$725$	− 2.68235i	− 0.0996200i
$726$	0	0
$727$	48.5576	1.80090	0.900451	−	0.434957i	$-0.143236\pi$
$727$	48.5576	1.80090	0.900451	+	0.434957i	$0.143236\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 12.4391i	− 0.460076i
$732$	0	0
$733$	− 26.8562i	− 0.991955i	−0.868335	−	0.495978i	$-0.834810\pi$
$733$	− 26.8562i	− 0.991955i	0.868335	−	0.495978i	$-0.165190\pi$
$734$	0	0
$735$	5.76947	0.212810
$736$	0	0
$737$	6.44653	0.237461
$738$	0	0
$739$	− 5.69031i	− 0.209321i	−0.994508	−	0.104661i	$-0.966624\pi$
$739$	− 5.69031i	− 0.209321i	0.994508	−	0.104661i	$-0.0333756\pi$
$740$	0	0
$741$	− 18.5187i	− 0.680303i
$742$	0	0
$743$	21.3939	0.784866	0.392433	−	0.919781i	$-0.371634\pi$
$743$	21.3939	0.784866	0.392433	+	0.919781i	$0.371634\pi$
$744$	0	0
$745$	−5.65265	−0.207097
$746$	0	0
$747$	2.71306i	0.0992657i
$748$	0	0
$749$	− 5.34792i	− 0.195409i
$750$	0	0
$751$	−0.0564198	−0.00205879	−0.00102939	−	0.999999i	$-0.500328\pi$
$751$	−0.0564198	−0.00205879	−0.00102939	+	0.999999i	$0.500328\pi$
$752$	0	0
$753$	28.3697	1.03385
$754$	0	0
$755$	10.4793i	0.381380i
$756$	0	0
$757$	18.4756i	0.671508i	0.941950	+	0.335754i	$0.108991\pi$
$757$	18.4756i	0.671508i	−0.941950	+	0.335754i	$0.891009\pi$
$758$	0	0
$759$	0.533975	0.0193821
$760$	0	0
$761$	41.0714	1.48884	0.744418	−	0.667714i	$-0.232727\pi$
$761$	41.0714	1.48884	0.744418	+	0.667714i	$0.232727\pi$
$762$	0	0
$763$	− 7.10882i	− 0.257357i
$764$	0	0
$765$	1.10929i	0.0401065i
$766$	0	0
$767$	46.6212	1.68339
$768$	0	0
$769$	−40.1555	−1.44804	−0.724021	−	0.689778i	$-0.757708\pi$
$769$	−40.1555	−1.44804	−0.724021	+	0.689778i	$0.757708\pi$
$770$	0	0
$771$	17.2750i	0.622145i
$772$	0	0
$773$	− 31.6221i	− 1.13737i	−0.822556	−	0.568684i	$-0.807453\pi$
$773$	− 31.6221i	− 1.13737i	0.822556	−	0.568684i	$-0.192547\pi$
$774$	0	0
$775$	−7.40654	−0.266051
$776$	0	0
$777$	2.62028	0.0940021
$778$	0	0
$779$	18.8749i	0.676265i
$780$	0	0
$781$	13.0060i	0.465390i
$782$	0	0
$783$	−2.68235	−0.0958594
$784$	0	0
$785$	−2.14258	−0.0764720
$786$	0	0
$787$	39.8478i	1.42042i	0.703989	+	0.710210i	$0.251400\pi$
$787$	39.8478i	1.42042i	−0.703989	+	0.710210i	$0.748600\pi$
$788$	0	0
$789$	− 3.34321i	− 0.119021i
$790$	0	0
$791$	6.86349	0.244038
$792$	0	0
$793$	6.51664	0.231413
$794$	0	0
$795$	1.13531i	0.0402654i
$796$	0	0
$797$	− 30.6821i	− 1.08681i	−0.839469	−	0.543407i	$-0.817134\pi$
$797$	− 30.6821i	− 1.08681i	0.839469	−	0.543407i	$-0.182866\pi$
$798$	0	0
$799$	5.43428	0.192251
$800$	0	0
$801$	−13.9410	−0.492582
$802$	0	0
$803$	11.9142i	0.420442i
$804$	0	0
$805$	− 0.592334i	− 0.0208770i
$806$	0	0
$807$	−21.6071	−0.760604
$808$	0	0
$809$	−28.0006	−0.984449	−0.492224	−	0.870468i	$-0.663816\pi$
$809$	−28.0006	−0.984449	−0.492224	+	0.870468i	$0.663816\pi$
$810$	0	0
$811$	− 31.9277i	− 1.12113i	−0.828109	−	0.560567i	$-0.810583\pi$
$811$	− 31.9277i	− 1.12113i	0.828109	−	0.560567i	$-0.189417\pi$
$812$	0	0
$813$	13.3858i	0.469462i
$814$	0	0
$815$	−14.7976	−0.518338
$816$	0	0
$817$	−53.8010	−1.88226
$818$	0	0
$819$	4.28163i	0.149612i
$820$	0	0
$821$	− 30.2311i	− 1.05507i	−0.849532	−	0.527536i	$-0.823116\pi$
$821$	− 30.2311i	− 1.05507i	0.849532	−	0.527536i	$-0.176884\pi$
$822$	0	0
$823$	1.53246	0.0534184	0.0267092	−	0.999643i	$-0.491497\pi$
$823$	1.53246	0.0534184	0.0267092	+	0.999643i	$0.491497\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	− 29.2197i	− 1.01607i	−0.861337	−	0.508035i	$-0.830372\pi$
$827$	− 29.2197i	− 1.01607i	0.861337	−	0.508035i	$-0.169628\pi$
$828$	0	0
$829$	− 21.7095i	− 0.754002i	−0.926213	−	0.377001i	$-0.876955\pi$
$829$	− 21.7095i	− 0.754002i	0.926213	−	0.377001i	$-0.123045\pi$
$830$	0	0
$831$	−1.55071	−0.0537936
$832$	0	0
$833$	−6.40003	−0.221748
$834$	0	0
$835$	19.6194i	0.678956i
$836$	0	0
$837$	7.40654i	0.256007i
$838$	0	0
$839$	44.7521	1.54501	0.772507	−	0.635006i	$-0.219002\pi$
$839$	44.7521	1.54501	0.772507	+	0.635006i	$0.219002\pi$
$840$	0	0
$841$	21.8050	0.751896
$842$	0	0
$843$	0.400588i	0.0137970i
$844$	0	0
$845$	1.89797i	0.0652920i
$846$	0	0
$847$	1.10929	0.0381157
$848$	0	0
$849$	−25.8493	−0.887145
$850$	0	0
$851$	1.26132i	0.0432374i
$852$	0	0
$853$	− 11.4901i	− 0.393413i	−0.980462	−	0.196706i	$-0.936975\pi$
$853$	− 11.4901i	− 0.393413i	0.980462	−	0.196706i	$-0.0630246\pi$
$854$	0	0
$855$	4.79786	0.164083
$856$	0	0
$857$	−35.9738	−1.22884	−0.614421	−	0.788978i	$-0.710610\pi$
$857$	−35.9738	−1.22884	−0.614421	+	0.788978i	$0.710610\pi$
$858$	0	0
$859$	− 42.4735i	− 1.44918i	−0.689183	−	0.724588i	$-0.742030\pi$
$859$	− 42.4735i	− 1.44918i	0.689183	−	0.724588i	$-0.257970\pi$
$860$	0	0
$861$	− 4.36399i	− 0.148724i
$862$	0	0
$863$	−23.1283	−0.787297	−0.393648	−	0.919261i	$-0.628787\pi$
$863$	−23.1283	−0.787297	−0.393648	+	0.919261i	$0.628787\pi$
$864$	0	0
$865$	10.7400	0.365170
$866$	0	0
$867$	15.7695i	0.535559i
$868$	0	0
$869$	3.44379i	0.116823i
$870$	0	0
$871$	−24.8822	−0.843103
$872$	0	0
$873$	19.1149	0.646941
$874$	0	0
$875$	− 1.10929i	− 0.0375009i
$876$	0	0
$877$	− 27.7216i	− 0.936090i	−0.883705	−	0.468045i	$-0.844958\pi$
$877$	− 27.7216i	− 0.936090i	0.883705	−	0.468045i	$-0.155042\pi$
$878$	0	0
$879$	7.62504	0.257186
$880$	0	0
$881$	−4.00917	−0.135072	−0.0675361	−	0.997717i	$-0.521514\pi$
$881$	−4.00917	−0.135072	−0.0675361	+	0.997717i	$0.521514\pi$
$882$	0	0
$883$	8.72477i	0.293612i	0.989165	+	0.146806i	$0.0468993\pi$
$883$	8.72477i	0.293612i	−0.989165	+	0.146806i	$0.953101\pi$
$884$	0	0
$885$	12.0787i	0.406021i
$886$	0	0
$887$	20.3233	0.682391	0.341195	−	0.939992i	$-0.389168\pi$
$887$	20.3233	0.682391	0.341195	+	0.939992i	$0.389168\pi$
$888$	0	0
$889$	−0.228038	−0.00764814
$890$	0	0
$891$	− 1.00000i	− 0.0335013i
$892$	0	0
$893$	− 23.5041i	− 0.786535i
$894$	0	0
$895$	2.41684	0.0807859
$896$	0	0
$897$	−2.06103	−0.0688159
$898$	0	0
$899$	19.8669i	0.662599i
$900$	0	0
$901$	− 1.25939i	− 0.0419565i
$902$	0	0
$903$	12.4391	0.413947
$904$	0	0
$905$	−5.82247	−0.193545
$906$	0	0
$907$	− 28.5645i	− 0.948468i	−0.880399	−	0.474234i	$-0.842725\pi$
$907$	− 28.5645i	− 0.948468i	0.880399	−	0.474234i	$-0.157275\pi$
$908$	0	0
$909$	1.75360i	0.0581632i
$910$	0	0
$911$	−26.2768	−0.870589	−0.435295	−	0.900288i	$-0.643356\pi$
$911$	−26.2768	−0.870589	−0.435295	+	0.900288i	$0.643356\pi$
$912$	0	0
$913$	−2.71306	−0.0897892
$914$	0	0
$915$	1.68834i	0.0558148i
$916$	0	0
$917$	0.329184i	0.0108706i
$918$	0	0
$919$	15.9484	0.526088	0.263044	−	0.964784i	$-0.415274\pi$
$919$	15.9484	0.526088	0.263044	+	0.964784i	$0.415274\pi$
$920$	0	0
$921$	6.71091	0.221132
$922$	0	0
$923$	− 50.2003i	− 1.65236i
$924$	0	0
$925$	2.36212i	0.0776661i
$926$	0	0
$927$	7.34041	0.241091
$928$	0	0
$929$	37.8391	1.24146	0.620731	−	0.784024i	$-0.286836\pi$
$929$	37.8391	1.24146	0.620731	+	0.784024i	$0.286836\pi$
$930$	0	0
$931$	27.6811i	0.907212i
$932$	0	0
$933$	15.1336i	0.495452i
$934$	0	0
$935$	−1.10929	−0.0362777
$936$	0	0
$937$	32.3622	1.05723	0.528613	−	0.848863i	$-0.322712\pi$
$937$	32.3622	1.05723	0.528613	+	0.848863i	$0.322712\pi$
$938$	0	0
$939$	− 15.2220i	− 0.496751i
$940$	0	0
$941$	45.6044i	1.48666i	0.668925	+	0.743330i	$0.266755\pi$
$941$	45.6044i	1.48666i	−0.668925	+	0.743330i	$0.733245\pi$
$942$	0	0
$943$	2.10068	0.0684075
$944$	0	0
$945$	−1.10929	−0.0360853
$946$	0	0
$947$	− 17.4628i	− 0.567463i	−0.958904	−	0.283732i	$-0.908428\pi$
$947$	− 17.4628i	− 0.567463i	0.958904	−	0.283732i	$-0.0915725\pi$
$948$	0	0
$949$	− 45.9862i	− 1.49277i
$950$	0	0
$951$	33.4802	1.08567
$952$	0	0
$953$	51.0904	1.65498	0.827490	−	0.561481i	$-0.189768\pi$
$953$	51.0904	1.65498	0.827490	+	0.561481i	$0.189768\pi$
$954$	0	0
$955$	4.15396i	0.134419i
$956$	0	0
$957$	− 2.68235i	− 0.0867081i
$958$	0	0
$959$	−11.0565	−0.357034
$960$	0	0
$961$	23.8568	0.769573
$962$	0	0
$963$	− 4.82102i	− 0.155355i
$964$	0	0
$965$	12.2730i	0.395082i
$966$	0	0
$967$	47.2549	1.51961	0.759807	−	0.650148i	$-0.225293\pi$
$967$	47.2549	1.51961	0.759807	+	0.650148i	$0.225293\pi$
$968$	0	0
$969$	−5.32223	−0.170975
$970$	0	0
$971$	− 26.5400i	− 0.851710i	−0.904791	−	0.425855i	$-0.859973\pi$
$971$	− 26.5400i	− 0.851710i	0.904791	−	0.425855i	$-0.140027\pi$
$972$	0	0
$973$	11.2061i	0.359252i
$974$	0	0
$975$	−3.85979	−0.123612
$976$	0	0
$977$	30.8413	0.986699	0.493350	−	0.869831i	$-0.335772\pi$
$977$	30.8413	0.986699	0.493350	+	0.869831i	$0.335772\pi$
$978$	0	0
$979$	− 13.9410i	− 0.445557i
$980$	0	0
$981$	− 6.40843i	− 0.204605i
$982$	0	0
$983$	25.8204	0.823544	0.411772	−	0.911287i	$-0.364910\pi$
$983$	25.8204	0.823544	0.411772	+	0.911287i	$0.364910\pi$
$984$	0	0
$985$	7.93158	0.252721
$986$	0	0
$987$	5.43428i	0.172975i
$988$	0	0
$989$	5.98775i	0.190399i
$990$	0	0
$991$	34.0696	1.08226	0.541128	−	0.840940i	$-0.317997\pi$
$991$	34.0696	1.08226	0.541128	+	0.840940i	$0.317997\pi$
$992$	0	0
$993$	−32.9618	−1.04601
$994$	0	0
$995$	0.922025i	0.0292302i
$996$	0	0
$997$	39.8148i	1.26095i	0.776210	+	0.630474i	$0.217140\pi$
$997$	39.8148i	1.26095i	−0.776210	+	0.630474i	$0.782860\pi$
$998$	0	0
$999$	2.36212	0.0747343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5280.2.w.d.2641.4		18
4.3	odd	2		1320.2.w.d.661.10	yes	18
8.3	odd	2		1320.2.w.d.661.9	✓	18
8.5	even	2	inner	5280.2.w.d.2641.13		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.w.d.661.9	✓	18	8.3	odd	2
1320.2.w.d.661.10	yes	18	4.3	odd	2
5280.2.w.d.2641.4		18	1.1	even	1	trivial
5280.2.w.d.2641.13		18	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 5280 = 2^{5} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5280.w (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} -1.10929 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} -1.10929 q^{7} -1.00000 q^{9} -1.00000i q^{11} +3.85979i q^{13} -1.00000 q^{15} +1.10929 q^{17} -4.79786i q^{19} +1.10929i q^{21} -0.533975 q^{23} -1.00000 q^{25} +1.00000i q^{27} +2.68235i q^{29} +7.40654 q^{31} -1.00000 q^{33} +1.10929i q^{35} -2.36212i q^{37} +3.85979 q^{39} -3.93403 q^{41} -11.2135i q^{43} +1.00000i q^{45} +4.89887 q^{47} -5.76947 q^{49} -1.10929i q^{51} -1.13531i q^{53} -1.00000 q^{55} -4.79786 q^{57} -12.0787i q^{59} -1.68834i q^{61} +1.10929 q^{63} +3.85979 q^{65} +6.44653i q^{67} +0.533975i q^{69} -13.0060 q^{71} -11.9142 q^{73} +1.00000i q^{75} +1.10929i q^{77} -3.44379 q^{79} +1.00000 q^{81} -2.71306i q^{83} -1.10929i q^{85} +2.68235 q^{87} +13.9410 q^{89} -4.28163i q^{91} -7.40654i q^{93} -4.79786 q^{95} -19.1149 q^{97} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 18 q^{9}+O(q^{10}) \) 18 * q - 18 * q^9	\( 18 q - 18 q^{9} - 18 q^{15} - 18 q^{25} + 36 q^{31} - 18 q^{33} + 10 q^{49} - 18 q^{55} + 28 q^{57} - 8 q^{71} + 28 q^{73} - 8 q^{79} + 18 q^{81} - 28 q^{89} + 28 q^{95} - 52 q^{97}+O(q^{100}) \) 18 * q - 18 * q^9 - 18 * q^15 - 18 * q^25 + 36 * q^31 - 18 * q^33 + 10 * q^49 - 18 * q^55 + 28 * q^57 - 8 * q^71 + 28 * q^73 - 8 * q^79 + 18 * q^81 - 28 * q^89 + 28 * q^95 - 52 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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