LMFDB - Embedded newform 1260.2.f.a.629.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,2,Mod(629,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1260.629");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1260.f (of order $2$, degree $1$, 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:	$10.0611506547$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.10070523904.11
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{4} + 81 $ x^8 - 10*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			629.3
Root			$-1.68014 + 0.420861i$ of defining polynomial
Character	$\chi$	$=$	1260.629
Dual form			1260.2.f.a.629.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.08495 - 1.95522i) q^{5} +(-0.595188 - 2.57794i) q^{7} -3.74166i q^{11} -3.36028 q^{13} -0.841723i q^{17} +5.59388i q^{19} -2.35425 q^{23} +(-2.64575 + 4.24264i) q^{25} +1.41421i q^{29} +8.66259i q^{31} +(-4.39467 + 3.96066i) q^{35} -5.15587i q^{37} -5.74103 q^{41} +3.32941i q^{43} -6.43560i q^{47} +(-6.29150 + 3.06871i) q^{49} +9.64575 q^{53} +(-7.31575 + 4.05952i) q^{55} -12.2508 q^{59} +(3.64575 + 6.57008i) q^{65} -1.82646i q^{67} -3.74166i q^{71} +0.979531 q^{73} +(-9.64575 + 2.22699i) q^{77} -6.58301 q^{79} -12.5730i q^{83} +(-1.64575 + 0.913230i) q^{85} +2.16991 q^{89} +(2.00000 + 8.66259i) q^{91} +(10.9373 - 6.06910i) q^{95} -12.0399 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{23} - 16 q^{35} - 8 q^{49} + 56 q^{53} + 8 q^{65} - 56 q^{77} + 32 q^{79} + 8 q^{85} + 16 q^{91} + 24 q^{95}+O(q^{100}) $ 8 * q - 40 * q^23 - 16 * q^35 - 8 * q^49 + 56 * q^53 + 8 * q^65 - 56 * q^77 + 32 * q^79 + 8 * q^85 + 16 * q^91 + 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$757$	$1081$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.08495	−	1.95522i	−0.485206	−	0.874400i
$5$	−1.08495	−	1.95522i	−0.485206	−	0.874400i
$6$		0			0
$7$	−0.595188	−	2.57794i	−0.224960	−	0.974368i
$7$	−0.595188	−	2.57794i	−0.224960	−	0.974368i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.74166i		−	1.12815i	−0.825723	−	0.564076i	$-0.809232\pi$
$11$		−	3.74166i		−	1.12815i	0.825723	−	0.564076i	$-0.190768\pi$
$12$		0			0
$13$	−3.36028			−0.931975			−0.465987	−	0.884791i	$-0.654301\pi$
$13$	−3.36028			−0.931975			−0.465987	+	0.884791i	$0.654301\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.841723i		−	0.204148i	−0.994777	−	0.102074i	$-0.967452\pi$
$17$		−	0.841723i		−	0.204148i	0.994777	−	0.102074i	$-0.0325478\pi$
$18$		0			0
$19$			5.59388i			1.28332i	0.766987	+	0.641662i	$0.221755\pi$
$19$			5.59388i			1.28332i	−0.766987	+	0.641662i	$0.778245\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.35425			−0.490895			−0.245447	−	0.969410i	$-0.578935\pi$
$23$	−2.35425			−0.490895			−0.245447	+	0.969410i	$0.578935\pi$
$24$		0			0
$25$	−2.64575	+	4.24264i	−0.529150	+	0.848528i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.41421i			0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$			1.41421i			0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$		0			0
$31$			8.66259i			1.55585i	0.628359	+	0.777924i	$0.283727\pi$
$31$			8.66259i			1.55585i	−0.628359	+	0.777924i	$0.716273\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.39467	+	3.96066i	−0.742835	+	0.669474i
$36$		0			0
$37$		−	5.15587i		−	0.847620i	−0.905751	−	0.423810i	$-0.860692\pi$
$37$		−	5.15587i		−	0.847620i	0.905751	−	0.423810i	$-0.139308\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.74103			−0.896599			−0.448300	−	0.893883i	$-0.647970\pi$
$41$	−5.74103			−0.896599			−0.448300	+	0.893883i	$0.647970\pi$
$42$		0			0
$43$			3.32941i			0.507730i	0.967240	+	0.253865i	$0.0817020\pi$
$43$			3.32941i			0.507730i	−0.967240	+	0.253865i	$0.918298\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.43560i		−	0.938729i	−0.883004	−	0.469365i	$-0.844483\pi$
$47$		−	6.43560i		−	0.938729i	0.883004	−	0.469365i	$-0.155517\pi$
$48$		0			0
$49$	−6.29150	+	3.06871i	−0.898786	+	0.438387i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.64575			1.32495			0.662473	−	0.749086i	$-0.269507\pi$
$53$	9.64575			1.32495			0.662473	+	0.749086i	$0.269507\pi$
$54$		0			0
$55$	−7.31575	+	4.05952i	−0.986456	+	0.547386i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.2508			−1.59491			−0.797456	−	0.603377i	$-0.793822\pi$
$59$	−12.2508			−1.59491			−0.797456	+	0.603377i	$0.793822\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.64575	+	6.57008i	0.452200	+	0.814919i
$66$		0			0
$67$		−	1.82646i		−	0.223138i	−0.993757	−	0.111569i	$-0.964412\pi$
$67$		−	1.82646i		−	0.223138i	0.993757	−	0.111569i	$-0.0355876\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.74166i		−	0.444053i	−0.975041	−	0.222027i	$-0.928733\pi$
$71$		−	3.74166i		−	0.444053i	0.975041	−	0.222027i	$-0.0712672\pi$
$72$		0			0
$73$	0.979531			0.114645			0.0573227	−	0.998356i	$-0.481744\pi$
$73$	0.979531			0.114645			0.0573227	+	0.998356i	$0.481744\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.64575	+	2.22699i	−1.09924	+	0.253789i
$78$		0			0
$79$	−6.58301			−0.740646			−0.370323	−	0.928903i	$-0.620753\pi$
$79$	−6.58301			−0.740646			−0.370323	+	0.928903i	$0.620753\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	12.5730i		−	1.38007i	−0.723776	−	0.690035i	$-0.757595\pi$
$83$		−	12.5730i		−	1.38007i	0.723776	−	0.690035i	$-0.242405\pi$
$84$		0			0
$85$	−1.64575	+	0.913230i	−0.178507	+	0.0990537i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.16991			0.230010			0.115005	−	0.993365i	$-0.463312\pi$
$89$	2.16991			0.230010			0.115005	+	0.993365i	$0.463312\pi$
$90$		0			0
$91$	2.00000	+	8.66259i	0.209657	+	0.908086i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.9373	−	6.06910i	1.12214	−	0.622677i
$96$		0			0
$97$	−12.0399			−1.22247			−0.611234	−	0.791450i	$-0.709327\pi$
$97$	−12.0399			−1.22247			−0.611234	+	0.791450i	$0.709327\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.74103			0.571254			0.285627	−	0.958341i	$-0.407798\pi$
$101$	5.74103			0.571254			0.285627	+	0.958341i	$0.407798\pi$
$102$		0			0
$103$	−13.4411			−1.32439			−0.662197	−	0.749330i	$-0.730376\pi$
$103$	−13.4411			−1.32439			−0.662197	+	0.749330i	$0.730376\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.9373			−1.63739			−0.818693	−	0.574231i	$-0.805301\pi$
$107$	−16.9373			−1.63739			−0.818693	+	0.574231i	$0.805301\pi$
$108$		0			0
$109$	17.8745			1.71207			0.856034	−	0.516920i	$-0.172922\pi$
$109$	17.8745			1.71207			0.856034	+	0.516920i	$0.172922\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.93725			0.464458			0.232229	−	0.972661i	$-0.425398\pi$
$113$	4.93725			0.464458			0.232229	+	0.972661i	$0.425398\pi$
$114$		0			0
$115$	2.55425	+	4.60307i	0.238185	+	0.429238i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.16991	+	0.500983i	−0.198915	+	0.0459251i
$120$		0			0
$121$	−3.00000			−0.272727
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.1658	+	0.569951i	0.998700	+	0.0509780i
$126$		0			0
$127$		−	10.3117i		−	0.915019i	−0.889205	−	0.457510i	$-0.848742\pi$
$127$		−	10.3117i		−	0.915019i	0.889205	−	0.457510i	$-0.151258\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.91094			−0.691182			−0.345591	−	0.938385i	$-0.612322\pi$
$131$	−7.91094			−0.691182			−0.345591	+	0.938385i	$0.612322\pi$
$132$		0			0
$133$	14.4207	−	3.32941i	1.25043	−	0.288696i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.2288			−1.04477			−0.522387	−	0.852709i	$-0.674958\pi$
$137$	−12.2288			−1.04477			−0.522387	+	0.852709i	$0.674958\pi$
$138$		0			0
$139$		−	3.61226i		−	0.306388i	−0.988196	−	0.153194i	$-0.951044\pi$
$139$		−	3.61226i		−	0.306388i	0.988196	−	0.153194i	$-0.0489559\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			12.5730i			1.05141i
$144$		0			0
$145$	2.76510	−	1.53436i	0.229629	−	0.127421i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	17.3828i		−	1.42406i	−0.702151	−	0.712028i	$-0.747777\pi$
$149$		−	17.3828i		−	1.42406i	0.702151	−	0.712028i	$-0.252223\pi$
$150$		0			0
$151$	−5.29150			−0.430616			−0.215308	−	0.976546i	$-0.569076\pi$
$151$	−5.29150			−0.430616			−0.215308	+	0.976546i	$0.569076\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.9373	−	9.39851i	1.36043	−	0.754907i
$156$		0			0
$157$	20.3725			1.62591			0.812953	−	0.582329i	$-0.197859\pi$
$157$	20.3725			1.62591			0.812953	+	0.582329i	$0.197859\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.40122	+	6.06910i	0.110432	+	0.478312i
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	20.6921i		−	1.60120i	−0.599198	−	0.800601i	$-0.704514\pi$
$167$		−	20.6921i		−	1.60120i	0.599198	−	0.800601i	$-0.295486\pi$
$168$		0			0
$169$	−1.70850			−0.131423
$170$		0			0
$171$		0			0
$172$		0			0
$173$			18.4651i			1.40387i	0.712239	+	0.701937i	$0.247681\pi$
$173$			18.4651i			1.40387i	−0.712239	+	0.701937i	$0.752319\pi$
$174$		0			0
$175$	12.5120	+	4.29541i	0.945816	+	0.324702i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			8.39655i			0.627587i	0.949491	+	0.313794i	$0.101600\pi$
$179$			8.39655i			0.627587i	−0.949491	+	0.313794i	$0.898400\pi$
$180$		0			0
$181$		−	26.5313i		−	1.97206i	−0.166575	−	0.986029i	$-0.553271\pi$
$181$		−	26.5313i		−	1.97206i	0.166575	−	0.986029i	$-0.446729\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.0808	+	5.59388i	−0.741159	+	0.411270i
$186$		0			0
$187$	−3.14944			−0.230310
$188$		0			0
$189$		0			0
$190$		0			0
$191$			16.8818i			1.22153i	0.791813	+	0.610763i	$0.209137\pi$
$191$			16.8818i			1.22153i	−0.791813	+	0.610763i	$0.790863\pi$
$192$		0			0
$193$			15.4676i			1.11338i	0.830719	+	0.556692i	$0.187929\pi$
$193$			15.4676i			1.11338i	−0.830719	+	0.556692i	$0.812071\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−26.8118			−1.91026			−0.955129	−	0.296189i	$-0.904284\pi$
$197$	−26.8118			−1.91026			−0.955129	+	0.296189i	$0.904284\pi$
$198$		0			0
$199$			6.68097i			0.473601i	0.971558	+	0.236801i	$0.0760988\pi$
$199$			6.68097i			0.473601i	−0.971558	+	0.236801i	$0.923901\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.64575	−	0.841723i	0.255882	−	0.0590774i
$204$		0			0
$205$	6.22876	+	11.2250i	0.435035	+	0.783986i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	20.9304			1.44779
$210$		0			0
$211$	−5.29150			−0.364282			−0.182141	−	0.983272i	$-0.558303\pi$
$211$	−5.29150			−0.364282			−0.182141	+	0.983272i	$0.558303\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.50972	−	3.61226i	0.443959	−	0.246354i
$216$		0			0
$217$	22.3316	−	5.15587i	1.51597	−	0.350003i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			2.82843i			0.190261i
$222$		0			0
$223$	−14.2098			−0.951560			−0.475780	−	0.879564i	$-0.657834\pi$
$223$	−14.2098			−0.951560			−0.475780	+	0.879564i	$0.657834\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			17.0270i			1.13012i	0.825049	+	0.565061i	$0.191147\pi$
$227$			17.0270i			1.13012i	−0.825049	+	0.565061i	$0.808853\pi$
$228$		0			0
$229$		−	25.4442i		−	1.68140i	−0.541499	−	0.840701i	$-0.682143\pi$
$229$		−	25.4442i		−	1.68140i	0.541499	−	0.840701i	$-0.317857\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21.6458			1.41806			0.709030	−	0.705178i	$-0.249133\pi$
$233$	21.6458			1.41806			0.709030	+	0.705178i	$0.249133\pi$
$234$		0			0
$235$	−12.5830	+	6.98233i	−0.820825	+	0.455477i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			15.0554i			0.973851i	0.873444	+	0.486925i	$0.161882\pi$
$239$			15.0554i			0.973851i	−0.873444	+	0.486925i	$0.838118\pi$
$240$		0			0
$241$		−	8.11905i		−	0.522994i	−0.965204	−	0.261497i	$-0.915784\pi$
$241$		−	8.11905i		−	0.522994i	0.965204	−	0.261497i	$-0.0842161\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.8260	+	8.97185i	0.819422	+	0.573190i
$246$		0			0
$247$		−	18.7970i		−	1.19603i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.5906			1.04719			0.523594	−	0.851968i	$-0.324591\pi$
$251$	16.5906			1.04719			0.523594	+	0.851968i	$0.324591\pi$
$252$		0			0
$253$			8.80879i			0.553804i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	5.89206i		−	0.367537i	−0.982970	−	0.183768i	$-0.941170\pi$
$257$		−	5.89206i		−	0.367537i	0.982970	−	0.183768i	$-0.0588296\pi$
$258$		0			0
$259$	−13.2915	+	3.06871i	−0.825894	+	0.190681i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.64575			−0.594783			−0.297391	−	0.954756i	$-0.596117\pi$
$263$	−9.64575			−0.594783			−0.297391	+	0.954756i	$0.596117\pi$
$264$		0			0
$265$	−10.4652	−	18.8595i	−0.642872	−	1.15853i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	25.9027			1.57932			0.789659	−	0.613546i	$-0.210258\pi$
$269$	25.9027			1.57932			0.789659	+	0.613546i	$0.210258\pi$
$270$		0			0
$271$		−	14.8000i		−	0.899037i	−0.893271	−	0.449519i	$-0.851596\pi$
$271$		−	14.8000i		−	0.899037i	0.893271	−	0.449519i	$-0.148404\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.8745	+	9.89949i	0.957269	+	0.596962i
$276$		0			0
$277$		−	25.4558i		−	1.52949i	−0.644331	−	0.764747i	$-0.722864\pi$
$277$		−	25.4558i		−	1.52949i	0.644331	−	0.764747i	$-0.277136\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5.06713i			0.302280i	0.988512	+	0.151140i	$0.0482944\pi$
$281$			5.06713i			0.302280i	−0.988512	+	0.151140i	$0.951706\pi$
$282$		0			0
$283$	27.6510			1.64368			0.821839	−	0.569719i	$-0.192948\pi$
$283$	27.6510			1.64368			0.821839	+	0.569719i	$0.192948\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.41699	+	14.8000i	0.201699	+	0.873617i
$288$		0			0
$289$	16.2915			0.958324
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	19.2540i		−	1.12483i	−0.826855	−	0.562415i	$-0.809872\pi$
$293$		−	19.2540i		−	1.12483i	0.826855	−	0.562415i	$-0.190128\pi$
$294$		0			0
$295$	13.2915	+	23.9529i	0.773861	+	1.39459i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.91094			0.457502
$300$		0			0
$301$	8.58301	−	1.98162i	0.494716	−	0.114219i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−18.5496			−1.05868			−0.529342	−	0.848409i	$-0.677561\pi$
$307$	−18.5496			−1.05868			−0.529342	+	0.848409i	$0.677561\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.9304			−1.18685			−0.593427	−	0.804888i	$-0.702225\pi$
$311$	−20.9304			−1.18685			−0.593427	+	0.804888i	$0.702225\pi$
$312$		0			0
$313$	4.55066			0.257218			0.128609	−	0.991695i	$-0.458949\pi$
$313$	4.55066			0.257218			0.128609	+	0.991695i	$0.458949\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.8118			−0.831911			−0.415956	−	0.909385i	$-0.636553\pi$
$317$	−14.8118			−0.831911			−0.415956	+	0.909385i	$0.636553\pi$
$318$		0			0
$319$	5.29150			0.296267
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.70850			0.261988
$324$		0			0
$325$	8.89047	−	14.2565i	0.493155	−	0.790807i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−16.5906	+	3.83039i	−0.914668	+	0.211176i
$330$		0			0
$331$	11.8745			0.652682			0.326341	−	0.945252i	$-0.394184\pi$
$331$	11.8745			0.652682			0.326341	+	0.945252i	$0.394184\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.57113	+	1.98162i	−0.195112	+	0.108268i
$336$		0			0
$337$		−	23.9529i		−	1.30480i	−0.757876	−	0.652399i	$-0.773763\pi$
$337$		−	23.9529i		−	1.30480i	0.757876	−	0.652399i	$-0.226237\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	32.4125			1.75523
$342$		0			0
$343$	11.6556	+	14.3926i	0.629342	+	0.777129i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−28.9373			−1.55343			−0.776717	−	0.629850i	$-0.783116\pi$
$347$	−28.9373			−1.55343			−0.776717	+	0.629850i	$0.783116\pi$
$348$		0			0
$349$		−	29.6000i		−	1.58445i	−0.610227	−	0.792227i	$-0.708922\pi$
$349$		−	29.6000i		−	1.58445i	0.610227	−	0.792227i	$-0.291078\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.4651i			0.982797i	0.870935	+	0.491399i	$0.163514\pi$
$353$			18.4651i			0.982797i	−0.870935	+	0.491399i	$0.836486\pi$
$354$		0			0
$355$	−7.31575	+	4.05952i	−0.388280	+	0.215457i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	29.1975i		−	1.54099i	−0.637449	−	0.770493i	$-0.720010\pi$
$359$		−	29.1975i		−	1.54099i	0.637449	−	0.770493i	$-0.279990\pi$
$360$		0			0
$361$	−12.2915			−0.646921
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.06275	−	1.91520i	−0.0556267	−	0.100246i
$366$		0			0
$367$	27.6510			1.44337			0.721684	−	0.692223i	$-0.243368\pi$
$367$	27.6510			1.44337			0.721684	+	0.692223i	$0.243368\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.74103	−	24.8661i	−0.298060	−	1.29098i
$372$		0			0
$373$			30.9352i			1.60177i	0.598821	+	0.800883i	$0.295636\pi$
$373$			30.9352i			1.60177i	−0.598821	+	0.800883i	$0.704364\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	4.75216i		−	0.244749i
$378$		0			0
$379$	−2.70850			−0.139126			−0.0695631	−	0.997578i	$-0.522161\pi$
$379$	−2.70850			−0.139126			−0.0695631	+	0.997578i	$0.522161\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	20.6921i		−	1.05732i	−0.848835	−	0.528658i	$-0.822695\pi$
$383$		−	20.6921i		−	1.05732i	0.848835	−	0.528658i	$-0.177305\pi$
$384$		0			0
$385$	14.8194	+	16.4434i	0.755269	+	0.838031i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			5.06713i			0.256914i	0.991715	+	0.128457i	$0.0410024\pi$
$389$			5.06713i			0.256914i	−0.991715	+	0.128457i	$0.958998\pi$
$390$		0			0
$391$			1.98162i			0.100215i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.14226	+	12.8712i	0.359366	+	0.647621i
$396$		0			0
$397$	16.0327			0.804660			0.402330	−	0.915495i	$-0.368201\pi$
$397$	16.0327			0.804660			0.402330	+	0.915495i	$0.368201\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	12.5504i		−	0.626740i	−0.949631	−	0.313370i	$-0.898542\pi$
$401$		−	12.5504i		−	0.626740i	0.949631	−	0.313370i	$-0.101458\pi$
$402$		0			0
$403$		−	29.1088i		−	1.45001i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−19.2915			−0.956244
$408$		0			0
$409$			14.2565i			0.704937i	0.935824	+	0.352469i	$0.114658\pi$
$409$			14.2565i			0.704937i	−0.935824	+	0.352469i	$0.885342\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.29150	+	31.5817i	0.358791	+	1.55403i
$414$		0			0
$415$	−24.5830	+	13.6412i	−1.20673	+	0.669618i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.14226			−0.348922			−0.174461	−	0.984664i	$-0.555818\pi$
$419$	−7.14226			−0.348922			−0.174461	+	0.984664i	$0.555818\pi$
$420$		0			0
$421$	23.8745			1.16357			0.581786	−	0.813342i	$-0.302354\pi$
$421$	23.8745			1.16357			0.581786	+	0.813342i	$0.302354\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.57113	+	2.22699i	0.173225	+	0.108025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			8.39655i			0.404447i	0.979339	+	0.202224i	$0.0648168\pi$
$431$			8.39655i			0.404447i	−0.979339	+	0.202224i	$0.935183\pi$
$432$		0			0
$433$	−35.7727			−1.71913			−0.859564	−	0.511028i	$-0.829265\pi$
$433$	−35.7727			−1.71913			−0.859564	+	0.511028i	$0.829265\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	13.1694i		−	0.629977i
$438$		0			0
$439$			4.50679i			0.215098i	0.994200	+	0.107549i	$0.0343002\pi$
$439$			4.50679i			0.215098i	−0.994200	+	0.107549i	$0.965700\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.64575			−0.458283			−0.229142	−	0.973393i	$-0.573592\pi$
$443$	−9.64575			−0.458283			−0.229142	+	0.973393i	$0.573592\pi$
$444$		0			0
$445$	−2.35425	−	4.24264i	−0.111602	−	0.201120i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			28.6965i			1.35427i	0.735858	+	0.677136i	$0.236779\pi$
$449$			28.6965i			1.35427i	−0.735858	+	0.677136i	$0.763221\pi$
$450$		0			0
$451$			21.4810i			1.01150i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	14.7673	−	13.3089i	0.692304	−	0.623933i
$456$		0			0
$457$			9.98823i			0.467230i	0.972329	+	0.233615i	$0.0750555\pi$
$457$			9.98823i			0.467230i	−0.972329	+	0.233615i	$0.924945\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	31.0112			1.44434			0.722169	−	0.691717i	$-0.243145\pi$
$461$	31.0112			1.44434			0.722169	+	0.691717i	$0.243145\pi$
$462$		0			0
$463$		−	16.9706i		−	0.788689i	−0.918963	−	0.394344i	$-0.870972\pi$
$463$		−	16.9706i		−	0.788689i	0.918963	−	0.394344i	$-0.129028\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	18.7105i		−	0.865816i	−0.901438	−	0.432908i	$-0.857487\pi$
$467$		−	18.7105i		−	0.865816i	0.901438	−	0.432908i	$-0.142513\pi$
$468$		0			0
$469$	−4.70850	+	1.08709i	−0.217418	+	0.0501970i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.4575			0.572797
$474$		0			0
$475$	−23.7328	−	14.8000i	−1.08894	−	0.679071i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−20.9304			−0.956334			−0.478167	−	0.878269i	$-0.658699\pi$
$479$	−20.9304			−0.956334			−0.478167	+	0.878269i	$0.658699\pi$
$480$		0			0
$481$			17.3252i			0.789960i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	13.0627	+	23.5406i	0.593149	+	1.06893i
$486$		0			0
$487$		−	6.98233i		−	0.316400i	−0.987407	−	0.158200i	$-0.949431\pi$
$487$		−	6.98233i		−	0.316400i	0.987407	−	0.158200i	$-0.0505690\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	26.1916i		−	1.18201i	−0.806668	−	0.591005i	$-0.798731\pi$
$491$		−	26.1916i		−	1.18201i	0.806668	−	0.591005i	$-0.201269\pi$
$492$		0			0
$493$	1.19038			0.0536118
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.64575	+	2.22699i	−0.432671	+	0.0998941i
$498$		0			0
$499$	18.7085			0.837507			0.418754	−	0.908100i	$-0.362467\pi$
$499$	18.7085			0.837507			0.418754	+	0.908100i	$0.362467\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			7.82087i			0.348715i	0.984682	+	0.174358i	$0.0557849\pi$
$503$			7.82087i			0.348715i	−0.984682	+	0.174358i	$0.944215\pi$
$504$		0			0
$505$	−6.22876	−	11.2250i	−0.277176	−	0.499505i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−30.2425			−1.34048			−0.670239	−	0.742146i	$-0.733808\pi$
$509$	−30.2425			−1.34048			−0.670239	+	0.742146i	$0.733808\pi$
$510$		0			0
$511$	−0.583005	−	2.52517i	−0.0257906	−	0.111707i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	14.5830	+	26.2803i	0.642604	+	1.15805i
$516$		0			0
$517$	−24.0798			−1.05903
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.2231			−0.754558			−0.377279	−	0.926100i	$-0.623140\pi$
$521$	−17.2231			−0.754558			−0.377279	+	0.926100i	$0.623140\pi$
$522$		0			0
$523$	−13.4411			−0.587740			−0.293870	−	0.955845i	$-0.594943\pi$
$523$	−13.4411			−0.587740			−0.293870	+	0.955845i	$0.594943\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.29150			0.317623
$528$		0			0
$529$	−17.4575			−0.759022
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.2915			0.835608
$534$		0			0
$535$	18.3761	+	33.1160i	0.794470	+	1.43173i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.4821	+	23.5406i	0.494568	+	1.01397i
$540$		0			0
$541$	−19.8745			−0.854472			−0.427236	−	0.904140i	$-0.640513\pi$
$541$	−19.8745			−0.854472			−0.427236	+	0.904140i	$0.640513\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19.3930	−	34.9486i	−0.830705	−	1.49703i
$546$		0			0
$547$			25.4558i			1.08841i	0.838951	+	0.544207i	$0.183169\pi$
$547$			25.4558i			1.08841i	−0.838951	+	0.544207i	$0.816831\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.91094			−0.337017
$552$		0			0
$553$	3.91813	+	16.9706i	0.166616	+	0.721662i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.5203			−0.827100			−0.413550	−	0.910481i	$-0.635711\pi$
$557$	−19.5203			−0.827100			−0.413550	+	0.910481i	$0.635711\pi$
$558$		0			0
$559$		−	11.1878i		−	0.473192i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	2.27980i		−	0.0960823i	−0.998845	−	0.0480411i	$-0.984702\pi$
$563$		−	2.27980i		−	0.0960823i	0.998845	−	0.0480411i	$-0.0152979\pi$
$564$		0			0
$565$	−5.35669	−	9.65341i	−0.225358	−	0.406122i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	22.2152i		−	0.931308i	−0.884967	−	0.465654i	$-0.845819\pi$
$569$		−	22.2152i		−	0.931308i	0.884967	−	0.465654i	$-0.154181\pi$
$570$		0			0
$571$	22.5830			0.945069			0.472535	−	0.881312i	$-0.343339\pi$
$571$	22.5830			0.945069			0.472535	+	0.881312i	$0.343339\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.22876	−	9.98823i	0.259757	−	0.416538i
$576$		0			0
$577$	−27.0931			−1.12790			−0.563950	−	0.825809i	$-0.690719\pi$
$577$	−27.0931			−1.12790			−0.563950	+	0.825809i	$0.690719\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−32.4125	+	7.48331i	−1.34470	+	0.310460i
$582$		0			0
$583$		−	36.0911i		−	1.49474i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	32.9669i		−	1.36069i	−0.732892	−	0.680345i	$-0.761830\pi$
$587$		−	32.9669i		−	1.36069i	0.732892	−	0.680345i	$-0.238170\pi$
$588$		0			0
$589$	−48.4575			−1.99666
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	27.3730i		−	1.12408i	−0.827111	−	0.562038i	$-0.810017\pi$
$593$		−	27.3730i		−	1.12408i	0.827111	−	0.562038i	$-0.189983\pi$
$594$		0			0
$595$	3.33378	+	3.69910i	0.136672	+	0.151648i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	26.1916i		−	1.07016i	−0.844801	−	0.535080i	$-0.820281\pi$
$599$		−	26.1916i		−	1.07016i	0.844801	−	0.535080i	$-0.179719\pi$
$600$		0			0
$601$			27.4259i			1.11872i	0.828923	+	0.559362i	$0.188954\pi$
$601$			27.4259i			1.11872i	−0.828923	+	0.559362i	$0.811046\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.25486	+	5.86565i	0.132329	+	0.238473i
$606$		0			0
$607$	−34.3715			−1.39510			−0.697548	−	0.716538i	$-0.745726\pi$
$607$	−34.3715			−1.39510			−0.697548	+	0.716538i	$0.745726\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			21.6255i			0.874872i
$612$		0			0
$613$			27.2823i			1.10192i	0.834531	+	0.550961i	$0.185739\pi$
$613$			27.2823i			1.10192i	−0.834531	+	0.550961i	$0.814261\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.52026			0.302754			0.151377	−	0.988476i	$-0.451629\pi$
$617$	7.52026			0.302754			0.151377	+	0.988476i	$0.451629\pi$
$618$		0			0
$619$			20.9374i			0.841547i	0.907166	+	0.420773i	$0.138241\pi$
$619$			20.9374i			0.841547i	−0.907166	+	0.420773i	$0.861759\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.29150	−	5.59388i	−0.0517430	−	0.224114i
$624$		0			0
$625$	−11.0000	−	22.4499i	−0.440000	−	0.897998i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.33981			−0.173040
$630$		0			0
$631$	−21.1660			−0.842606			−0.421303	−	0.906920i	$-0.638427\pi$
$631$	−21.1660			−0.842606			−0.421303	+	0.906920i	$0.638427\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−20.1617	+	11.1878i	−0.800093	+	0.443973i
$636$		0			0
$637$	21.1412	−	10.3117i	0.837646	−	0.408566i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	2.23871i		−	0.0884236i	−0.999022	−	0.0442118i	$-0.985922\pi$
$641$		−	2.23871i		−	0.0884236i	0.999022	−	0.0442118i	$-0.0140776\pi$
$642$		0			0
$643$	−30.0317			−1.18433			−0.592167	−	0.805815i	$-0.701728\pi$
$643$	−30.0317			−1.18433			−0.592167	+	0.805815i	$0.701728\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			25.1461i			0.988593i	0.869293	+	0.494297i	$0.164574\pi$
$647$			25.1461i			0.988593i	−0.869293	+	0.494297i	$0.835426\pi$
$648$		0			0
$649$			45.8381i			1.79930i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.1033			0.864968			0.432484	−	0.901642i	$-0.357637\pi$
$653$	22.1033			0.864968			0.432484	+	0.901642i	$0.357637\pi$
$654$		0			0
$655$	8.58301	+	15.4676i	0.335366	+	0.604370i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1.09070i			0.0424878i	0.999774	+	0.0212439i	$0.00676264\pi$
$659$			1.09070i			0.0424878i	−0.999774	+	0.0212439i	$0.993237\pi$
$660$		0			0
$661$		−	6.13742i		−	0.238718i	−0.992851	−	0.119359i	$-0.961916\pi$
$661$		−	6.13742i		−	0.238718i	0.992851	−	0.119359i	$-0.0380840\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−22.1555	−	24.5833i	−0.859152	−	0.953299i
$666$		0			0
$667$		−	3.32941i		−	0.128915i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			8.16177i			0.314613i	0.987550	+	0.157307i	$0.0502811\pi$
$673$			8.16177i			0.314613i	−0.987550	+	0.157307i	$0.949719\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			33.8086i			1.29937i	0.760203	+	0.649686i	$0.225100\pi$
$677$			33.8086i			1.29937i	−0.760203	+	0.649686i	$0.774900\pi$
$678$		0			0
$679$	7.16601	+	31.0381i	0.275006	+	1.19113i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.5203			−0.746922			−0.373461	−	0.927646i	$-0.621829\pi$
$683$	−19.5203			−0.746922			−0.373461	+	0.927646i	$0.621829\pi$
$684$		0			0
$685$	13.2676	+	23.9099i	0.506930	+	0.913550i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−32.4125			−1.23482
$690$		0			0
$691$			2.52517i			0.0960619i	0.998846	+	0.0480310i	$0.0152946\pi$
$691$			2.52517i			0.0960619i	−0.998846	+	0.0480310i	$0.984705\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.06275	+	3.91913i	−0.267905	+	0.148661i
$696$		0			0
$697$			4.83236i			0.183039i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	13.7299i		−	0.518571i	−0.965801	−	0.259285i	$-0.916513\pi$
$701$		−	13.7299i		−	0.518571i	0.965801	−	0.259285i	$-0.0834870\pi$
$702$		0			0
$703$	28.8413			1.08777
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.41699	−	14.8000i	−0.128509	−	0.556612i
$708$		0			0
$709$	49.1660			1.84647			0.923234	−	0.384238i	$-0.125536\pi$
$709$	49.1660			1.84647			0.923234	+	0.384238i	$0.125536\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	20.3939i		−	0.763757i
$714$		0			0
$715$	24.5830	−	13.6412i	0.919352	−	0.510150i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	38.2896			1.42796			0.713981	−	0.700165i	$-0.246890\pi$
$719$	38.2896			1.42796			0.713981	+	0.700165i	$0.246890\pi$
$720$		0			0
$721$	8.00000	+	34.6504i	0.297936	+	1.29045i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.00000	−	3.74166i	−0.222834	−	0.138962i
$726$		0			0
$727$	3.91813			0.145315			0.0726576	−	0.997357i	$-0.476852\pi$
$727$	3.91813			0.145315			0.0726576	+	0.997357i	$0.476852\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.80244			0.103652
$732$		0			0
$733$	13.9990			0.517064			0.258532	−	0.966003i	$-0.416761\pi$
$733$	13.9990			0.517064			0.258532	+	0.966003i	$0.416761\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.83399			−0.251733
$738$		0			0
$739$	−9.16601			−0.337177			−0.168589	−	0.985687i	$-0.553921\pi$
$739$	−9.16601			−0.337177			−0.168589	+	0.985687i	$0.553921\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−28.9373			−1.06160			−0.530802	−	0.847496i	$-0.678109\pi$
$743$	−28.9373			−1.06160			−0.530802	+	0.847496i	$0.678109\pi$
$744$		0			0
$745$	−33.9872	+	18.8595i	−1.24519	+	0.690960i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.0808	+	43.6631i	0.368346	+	1.59542i
$750$		0			0
$751$	−29.2915			−1.06886			−0.534431	−	0.845212i	$-0.679474\pi$
$751$	−29.2915			−1.06886			−0.534431	+	0.845212i	$0.679474\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.74103	+	10.3460i	0.208938	+	0.376531i
$756$		0			0
$757$		−	13.6412i		−	0.495796i	−0.968786	−	0.247898i	$-0.920260\pi$
$757$		−	13.6412i		−	0.495796i	0.968786	−	0.247898i	$-0.0797398\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−51.1729			−1.85502			−0.927509	−	0.373802i	$-0.878054\pi$
$761$	−51.1729			−1.85502			−0.927509	+	0.373802i	$0.878054\pi$
$762$		0			0
$763$	−10.6387	−	46.0793i	−0.385146	−	1.66818i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	41.1660			1.48642
$768$		0			0
$769$			27.4259i			0.989002i	0.869177	+	0.494501i	$0.164649\pi$
$769$			27.4259i			0.989002i	−0.869177	+	0.494501i	$0.835351\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	11.1349i		−	0.400496i	−0.979745	−	0.200248i	$-0.935825\pi$
$773$		−	11.1349i		−	0.400496i	0.979745	−	0.200248i	$-0.0641748\pi$
$774$		0			0
$775$	−36.7523	−	22.9191i	−1.32018	−	0.823277i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	32.1147i		−	1.15063i
$780$		0			0
$781$	−14.0000			−0.500959
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−22.1033	−	39.8328i	−0.788899	−	1.42169i
$786$		0			0
$787$	24.0798			0.858353			0.429177	−	0.903221i	$-0.358804\pi$
$787$	24.0798			0.858353			0.429177	+	0.903221i	$0.358804\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.93859	−	12.7279i	−0.104484	−	0.452553i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	26.2860i		−	0.931096i	−0.885023	−	0.465548i	$-0.845857\pi$
$797$		−	26.2860i		−	0.931096i	0.885023	−	0.465548i	$-0.154143\pi$
$798$		0			0
$799$	−5.41699			−0.191639
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	3.66507i		−	0.129338i
$804$		0			0
$805$	10.3462	−	9.32438i	0.364654	−	0.328641i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	50.1445i		−	1.76299i	−0.472197	−	0.881493i	$-0.656539\pi$
$809$		−	50.1445i		−	1.76299i	0.472197	−	0.881493i	$-0.343461\pi$
$810$		0			0
$811$		−	14.8000i		−	0.519699i	−0.965649	−	0.259849i	$-0.916327\pi$
$811$		−	14.8000i		−	0.519699i	0.965649	−	0.259849i	$-0.0836729\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−18.6243			−0.651583
$818$		0			0
$819$		0			0
$820$		0			0
$821$			9.89949i			0.345495i	0.984966	+	0.172747i	$0.0552644\pi$
$821$			9.89949i			0.345495i	−0.984966	+	0.172747i	$0.944736\pi$
$822$		0			0
$823$			27.2823i			0.951001i	0.879715	+	0.475501i	$0.157733\pi$
$823$			27.2823i			0.951001i	−0.879715	+	0.475501i	$0.842267\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	52.9373			1.84081			0.920404	−	0.390968i	$-0.127860\pi$
$827$	52.9373			1.84081			0.920404	+	0.390968i	$0.127860\pi$
$828$		0			0
$829$		−	15.3436i		−	0.532904i	−0.963848	−	0.266452i	$-0.914149\pi$
$829$		−	15.3436i		−	0.532904i	0.963848	−	0.266452i	$-0.0858514\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.58301	+	5.29570i	0.0894958	+	0.183485i
$834$		0			0
$835$	−40.4575	+	22.4499i	−1.40009	+	0.776912i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.33981			0.149827			0.0749135	−	0.997190i	$-0.476132\pi$
$839$	4.33981			0.149827			0.0749135	+	0.997190i	$0.476132\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.85364	+	3.34048i	0.0637672	+	0.114916i
$846$		0			0
$847$	1.78556	+	7.73381i	0.0613527	+	0.265737i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			12.1382i			0.416092i
$852$		0			0
$853$	0.210845			0.00721918			0.00360959	−	0.999993i	$-0.498851\pi$
$853$	0.210845			0.00721918			0.00360959	+	0.999993i	$0.498851\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	16.1853i		−	0.552879i	−0.961031	−	0.276439i	$-0.910845\pi$
$857$		−	16.1853i		−	0.552879i	0.961031	−	0.276439i	$-0.0891545\pi$
$858$		0			0
$859$		−	39.1572i		−	1.33603i	−0.744150	−	0.668013i	$-0.767145\pi$
$859$		−	39.1572i		−	1.33603i	0.744150	−	0.668013i	$-0.232855\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−48.6863			−1.65730			−0.828650	−	0.559767i	$-0.810891\pi$
$863$	−48.6863			−1.65730			−0.828650	+	0.559767i	$0.810891\pi$
$864$		0			0
$865$	36.1033	−	20.0338i	1.22755	−	0.681168i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			24.6314i			0.835561i
$870$		0			0
$871$			6.13742i			0.207959i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.17645	−	29.1239i	−0.174996	−	0.984569i
$876$		0			0
$877$		−	16.9706i		−	0.573055i	−0.958072	−	0.286528i	$-0.907499\pi$
$877$		−	16.9706i		−	0.573055i	0.958072	−	0.286528i	$-0.0925010\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15.1894			−0.511742			−0.255871	−	0.966711i	$-0.582362\pi$
$881$	−15.1894			−0.511742			−0.255871	+	0.966711i	$0.582362\pi$
$882$		0			0
$883$			1.50295i			0.0505783i	0.999680	+	0.0252891i	$0.00805064\pi$
$883$			1.50295i			0.0505783i	−0.999680	+	0.0252891i	$0.991949\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			3.66507i			0.123061i	0.998105	+	0.0615305i	$0.0195982\pi$
$887$			3.66507i			0.123061i	−0.998105	+	0.0615305i	$0.980402\pi$
$888$		0			0
$889$	−26.5830	+	6.13742i	−0.891565	+	0.205843i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	36.0000			1.20469
$894$		0			0
$895$	16.4171	−	9.10986i	0.548762	−	0.304509i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.2508			−0.408586
$900$		0			0
$901$		−	8.11905i		−	0.270485i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−51.8745	+	28.7853i	−1.72437	+	0.956854i
$906$		0			0
$907$			29.4323i			0.977283i	0.872485	+	0.488641i	$0.162507\pi$
$907$			29.4323i			0.977283i	−0.872485	+	0.488641i	$0.837493\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			49.6435i			1.64476i	0.568936	+	0.822381i	$0.307355\pi$
$911$			49.6435i			1.64476i	−0.568936	+	0.822381i	$0.692645\pi$
$912$		0			0
$913$	−47.0440			−1.55693
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.70850	+	20.3939i	0.155488	+	0.673466i
$918$		0			0
$919$	47.8745			1.57923			0.789617	−	0.613600i	$-0.210279\pi$
$919$	47.8745			1.57923			0.789617	+	0.613600i	$0.210279\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			12.5730i			0.413846i
$924$		0			0
$925$	21.8745	+	13.6412i	0.719229	+	0.448518i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.70728			−0.121632			−0.0608160	−	0.998149i	$-0.519370\pi$
$929$	−3.70728			−0.121632			−0.0608160	+	0.998149i	$0.519370\pi$
$930$		0			0
$931$	−17.1660	−	35.1939i	−0.562593	−	1.15343i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.41699	+	6.15784i	0.111748	+	0.201383i
$936$		0			0
$937$	−22.7533			−0.743318			−0.371659	−	0.928369i	$-0.621211\pi$
$937$	−22.7533			−0.743318			−0.371659	+	0.928369i	$0.621211\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.50972			0.212211			0.106105	−	0.994355i	$-0.466162\pi$
$941$	6.50972			0.212211			0.106105	+	0.994355i	$0.466162\pi$
$942$		0			0
$943$	13.5158			0.440136
$944$		0			0
$945$		0			0
$946$		0			0
$947$	28.9373			0.940334			0.470167	−	0.882577i	$-0.344194\pi$
$947$	28.9373			0.940334			0.470167	+	0.882577i	$0.344194\pi$
$948$		0			0
$949$	−3.29150			−0.106847
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.81176			0.0910819			0.0455409	−	0.998962i	$-0.485499\pi$
$953$	2.81176			0.0910819			0.0455409	+	0.998962i	$0.485499\pi$
$954$		0			0
$955$	33.0076	−	18.3160i	1.06810	−	0.592692i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.27841	+	31.5249i	0.235032	+	1.01799i
$960$		0			0
$961$	−44.0405			−1.42066
$962$		0			0
$963$		0			0
$964$		0			0
$965$	30.2425	−	16.7816i	0.973542	−	0.540220i
$966$		0			0
$967$			5.47938i			0.176205i	0.996111	+	0.0881025i	$0.0280803\pi$
$967$			5.47938i			0.176205i	−0.996111	+	0.0881025i	$0.971920\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	41.8608			1.34338			0.671688	−	0.740834i	$-0.265570\pi$
$971$	41.8608			1.34338			0.671688	+	0.740834i	$0.265570\pi$
$972$		0			0
$973$	−9.31216	+	2.14997i	−0.298534	+	0.0689249i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	60.6863			1.94153			0.970763	−	0.240040i	$-0.0771607\pi$
$977$	60.6863			1.94153			0.970763	+	0.240040i	$0.0771607\pi$
$978$		0			0
$979$		−	8.11905i		−	0.259486i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			11.9767i			0.381996i	0.981590	+	0.190998i	$0.0611725\pi$
$983$			11.9767i			0.381996i	−0.981590	+	0.190998i	$0.938828\pi$
$984$		0			0
$985$	29.0895	+	52.4228i	0.926869	+	1.67033i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	7.83826i		−	0.249242i
$990$		0			0
$991$	14.4575			0.459258			0.229629	−	0.973278i	$-0.426249\pi$
$991$	14.4575			0.459258			0.229629	+	0.973278i	$0.426249\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	13.0627	−	7.24854i	0.414117	−	0.229794i
$996$		0			0
$997$	−8.46878			−0.268209			−0.134105	−	0.990967i	$-0.542816\pi$
$997$	−8.46878			−0.268209			−0.134105	+	0.990967i	$0.542816\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1260.2.f.a.629.3	✓	8
3.2	odd	2		1260.2.f.b.629.6	yes	8
4.3	odd	2		5040.2.k.f.1889.3		8
5.2	odd	4		6300.2.d.f.3401.13		16
5.3	odd	4		6300.2.d.f.3401.3		16
5.4	even	2		1260.2.f.b.629.4	yes	8
7.6	odd	2	inner	1260.2.f.a.629.6	yes	8
12.11	even	2		5040.2.k.e.1889.6		8
15.2	even	4		6300.2.d.f.3401.14		16
15.8	even	4		6300.2.d.f.3401.4		16
15.14	odd	2	inner	1260.2.f.a.629.5	yes	8
20.19	odd	2		5040.2.k.e.1889.4		8
21.20	even	2		1260.2.f.b.629.3	yes	8
28.27	even	2		5040.2.k.f.1889.6		8
35.13	even	4		6300.2.d.f.3401.1		16
35.27	even	4		6300.2.d.f.3401.15		16
35.34	odd	2		1260.2.f.b.629.5	yes	8
60.59	even	2		5040.2.k.f.1889.5		8
84.83	odd	2		5040.2.k.e.1889.3		8
105.62	odd	4		6300.2.d.f.3401.16		16
105.83	odd	4		6300.2.d.f.3401.2		16
105.104	even	2	inner	1260.2.f.a.629.4	yes	8
140.139	even	2		5040.2.k.e.1889.5		8
420.419	odd	2		5040.2.k.f.1889.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.f.a.629.3	✓	8	1.1	even	1	trivial
1260.2.f.a.629.4	yes	8	105.104	even	2	inner
1260.2.f.a.629.5	yes	8	15.14	odd	2	inner
1260.2.f.a.629.6	yes	8	7.6	odd	2	inner
1260.2.f.b.629.3	yes	8	21.20	even	2
1260.2.f.b.629.4	yes	8	5.4	even	2
1260.2.f.b.629.5	yes	8	35.34	odd	2
1260.2.f.b.629.6	yes	8	3.2	odd	2
5040.2.k.e.1889.3		8	84.83	odd	2
5040.2.k.e.1889.4		8	20.19	odd	2
5040.2.k.e.1889.5		8	140.139	even	2
5040.2.k.e.1889.6		8	12.11	even	2
5040.2.k.f.1889.3		8	4.3	odd	2
5040.2.k.f.1889.4		8	420.419	odd	2
5040.2.k.f.1889.5		8	60.59	even	2
5040.2.k.f.1889.6		8	28.27	even	2
6300.2.d.f.3401.1		16	35.13	even	4
6300.2.d.f.3401.2		16	105.83	odd	4
6300.2.d.f.3401.3		16	5.3	odd	4
6300.2.d.f.3401.4		16	15.8	even	4
6300.2.d.f.3401.13		16	5.2	odd	4
6300.2.d.f.3401.14		16	15.2	even	4
6300.2.d.f.3401.15		16	35.27	even	4
6300.2.d.f.3401.16		16	105.62	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:	\( N \)	\(=\)	\( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1260.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.08495 - 1.95522i) q^{5} +(-0.595188 - 2.57794i) q^{7} -3.74166i q^{11} -3.36028 q^{13} -0.841723i q^{17} +5.59388i q^{19} -2.35425 q^{23} +(-2.64575 + 4.24264i) q^{25} +1.41421i q^{29} +8.66259i q^{31} +(-4.39467 + 3.96066i) q^{35} -5.15587i q^{37} -5.74103 q^{41} +3.32941i q^{43} -6.43560i q^{47} +(-6.29150 + 3.06871i) q^{49} +9.64575 q^{53} +(-7.31575 + 4.05952i) q^{55} -12.2508 q^{59} +(3.64575 + 6.57008i) q^{65} -1.82646i q^{67} -3.74166i q^{71} +0.979531 q^{73} +(-9.64575 + 2.22699i) q^{77} -6.58301 q^{79} -12.5730i q^{83} +(-1.64575 + 0.913230i) q^{85} +2.16991 q^{89} +(2.00000 + 8.66259i) q^{91} +(10.9373 - 6.06910i) q^{95} -12.0399 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{23} - 16 q^{35} - 8 q^{49} + 56 q^{53} + 8 q^{65} - 56 q^{77} + 32 q^{79} + 8 q^{85} + 16 q^{91} + 24 q^{95}+O(q^{100}) \) 8 * q - 40 * q^23 - 16 * q^35 - 8 * q^49 + 56 * q^53 + 8 * q^65 - 56 * q^77 + 32 * q^79 + 8 * q^85 + 16 * q^91 + 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more