LMFDB - Embedded newform 8280.2.p.a.1241.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1241,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8280.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.p (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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.11
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.78247i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.78247i q^{7} +1.03414 q^{11} -1.75504 q^{13} -4.53952 q^{17} +3.02274i q^{19} +(4.78190 + 0.365306i) q^{23} +1.00000 q^{25} -3.14710i q^{29} -2.28775 q^{31} +2.78247i q^{35} -10.4205i q^{37} +8.99716i q^{41} +3.28852i q^{43} -4.19754i q^{47} -0.742152 q^{49} +6.63128 q^{53} -1.03414 q^{55} -9.75064i q^{59} -12.2497i q^{61} +1.75504 q^{65} +5.18196i q^{67} +11.4297i q^{71} -15.1422 q^{73} -2.87745i q^{77} -0.759277i q^{79} -14.6261 q^{83} +4.53952 q^{85} -9.89856 q^{89} +4.88335i q^{91} -3.02274i q^{95} +0.520693i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{5}+O(q^{10}) $ 48 * q - 48 * q^5	$ 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) $ 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times$.

$n$	$1657$	$2071$	$3961$	$4141$	$4601$
$\chi(n)$	$1$	$1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	2.78247i		−	1.05168i	−0.850585	−	0.525838i	$-0.823752\pi$
$7$		−	2.78247i		−	1.05168i	0.850585	−	0.525838i	$-0.176248\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.03414			0.311804			0.155902	−	0.987773i	$-0.450172\pi$
$11$	1.03414			0.311804			0.155902	+	0.987773i	$0.450172\pi$
$12$		0			0
$13$	−1.75504			−0.486761			−0.243380	−	0.969931i	$-0.578256\pi$
$13$	−1.75504			−0.486761			−0.243380	+	0.969931i	$0.578256\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.53952			−1.10100			−0.550498	−	0.834837i	$-0.685562\pi$
$17$	−4.53952			−1.10100			−0.550498	+	0.834837i	$0.685562\pi$
$18$		0			0
$19$			3.02274i			0.693464i	0.937964	+	0.346732i	$0.112709\pi$
$19$			3.02274i			0.693464i	−0.937964	+	0.346732i	$0.887291\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.78190	+	0.365306i	0.997095	+	0.0761717i
$23$	4.78190	+	0.365306i	0.997095	+	0.0761717i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.14710i		−	0.584402i	−0.956357	−	0.292201i	$-0.905612\pi$
$29$		−	3.14710i		−	0.584402i	0.956357	−	0.292201i	$-0.0943876\pi$
$30$		0			0
$31$	−2.28775			−0.410892			−0.205446	−	0.978669i	$-0.565864\pi$
$31$	−2.28775			−0.410892			−0.205446	+	0.978669i	$0.565864\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.78247i			0.470324i
$36$		0			0
$37$		−	10.4205i		−	1.71311i	−0.516052	−	0.856557i	$-0.672599\pi$
$37$		−	10.4205i		−	1.71311i	0.516052	−	0.856557i	$-0.327401\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			8.99716i			1.40512i	0.711624	+	0.702560i	$0.247960\pi$
$41$			8.99716i			1.40512i	−0.711624	+	0.702560i	$0.752040\pi$
$42$		0			0
$43$			3.28852i			0.501494i	0.968053	+	0.250747i	$0.0806763\pi$
$43$			3.28852i			0.501494i	−0.968053	+	0.250747i	$0.919324\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	4.19754i		−	0.612274i	−0.951987	−	0.306137i	$-0.900963\pi$
$47$		−	4.19754i		−	0.612274i	0.951987	−	0.306137i	$-0.0990366\pi$
$48$		0			0
$49$	−0.742152			−0.106022
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.63128			0.910876			0.455438	−	0.890268i	$-0.349483\pi$
$53$	6.63128			0.910876			0.455438	+	0.890268i	$0.349483\pi$
$54$		0			0
$55$	−1.03414			−0.139443
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	9.75064i		−	1.26943i	−0.772748	−	0.634713i	$-0.781118\pi$
$59$		−	9.75064i		−	1.26943i	0.772748	−	0.634713i	$-0.218882\pi$
$60$		0			0
$61$		−	12.2497i		−	1.56841i	−0.620499	−	0.784207i	$-0.713070\pi$
$61$		−	12.2497i		−	1.56841i	0.620499	−	0.784207i	$-0.286930\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.75504			0.217686
$66$		0			0
$67$			5.18196i			0.633077i	0.948580	+	0.316539i	$0.102521\pi$
$67$			5.18196i			0.633077i	−0.948580	+	0.316539i	$0.897479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.4297i			1.35646i	0.734849	+	0.678230i	$0.237253\pi$
$71$			11.4297i			1.35646i	−0.734849	+	0.678230i	$0.762747\pi$
$72$		0			0
$73$	−15.1422			−1.77226			−0.886128	−	0.463440i	$-0.846615\pi$
$73$	−15.1422			−1.77226			−0.886128	+	0.463440i	$0.846615\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.87745i		−	0.327916i
$78$		0			0
$79$		−	0.759277i		−	0.0854253i	−0.999087	−	0.0427127i	$-0.986400\pi$
$79$		−	0.759277i		−	0.0854253i	0.999087	−	0.0427127i	$-0.0136000\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−14.6261			−1.60542			−0.802709	−	0.596371i	$-0.796609\pi$
$83$	−14.6261			−1.60542			−0.802709	+	0.596371i	$0.796609\pi$
$84$		0			0
$85$	4.53952			0.492380
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.89856			−1.04925			−0.524623	−	0.851335i	$-0.675794\pi$
$89$	−9.89856			−1.04925			−0.524623	+	0.851335i	$0.675794\pi$
$90$		0			0
$91$			4.88335i			0.511914i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	3.02274i		−	0.310127i
$96$		0			0
$97$			0.520693i			0.0528684i	0.999651	+	0.0264342i	$0.00841524\pi$
$97$			0.520693i			0.0528684i	−0.999651	+	0.0264342i	$0.991585\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			15.3462i			1.52700i	0.645806	+	0.763502i	$0.276522\pi$
$101$			15.3462i			1.52700i	−0.645806	+	0.763502i	$0.723478\pi$
$102$		0			0
$103$		−	7.07373i		−	0.696996i	−0.937310	−	0.348498i	$-0.886692\pi$
$103$		−	7.07373i		−	0.696996i	0.937310	−	0.348498i	$-0.113308\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.58106			−0.152847			−0.0764236	−	0.997075i	$-0.524350\pi$
$107$	−1.58106			−0.152847			−0.0764236	+	0.997075i	$0.524350\pi$
$108$		0			0
$109$		−	11.4628i		−	1.09794i	−0.835842	−	0.548970i	$-0.815020\pi$
$109$		−	11.4628i		−	1.09794i	0.835842	−	0.548970i	$-0.184980\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.83510			0.454848			0.227424	−	0.973796i	$-0.426970\pi$
$113$	4.83510			0.454848			0.227424	+	0.973796i	$0.426970\pi$
$114$		0			0
$115$	−4.78190	−	0.365306i	−0.445914	−	0.0340650i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			12.6311i			1.15789i
$120$		0			0
$121$	−9.93056			−0.902779
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−11.9837			−1.06338			−0.531690	−	0.846939i	$-0.678443\pi$
$127$	−11.9837			−1.06338			−0.531690	+	0.846939i	$0.678443\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			13.3972i			1.17052i	0.810846	+	0.585259i	$0.199007\pi$
$131$			13.3972i			1.17052i	−0.810846	+	0.585259i	$0.800993\pi$
$132$		0			0
$133$	8.41069			0.729299
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.8685			1.61204			0.806022	−	0.591886i	$-0.201616\pi$
$137$	18.8685			1.61204			0.806022	+	0.591886i	$0.201616\pi$
$138$		0			0
$139$	1.16167			0.0985319			0.0492660	−	0.998786i	$-0.484312\pi$
$139$	1.16167			0.0985319			0.0492660	+	0.998786i	$0.484312\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.81495			−0.151774
$144$		0			0
$145$			3.14710i			0.261353i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.02859			0.0842653			0.0421327	−	0.999112i	$-0.486585\pi$
$149$	1.02859			0.0842653			0.0421327	+	0.999112i	$0.486585\pi$
$150$		0			0
$151$	3.15652			0.256874			0.128437	−	0.991718i	$-0.459004\pi$
$151$	3.15652			0.256874			0.128437	+	0.991718i	$0.459004\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.28775			0.183756
$156$		0			0
$157$			9.11318i			0.727311i	0.931534	+	0.363656i	$0.118471\pi$
$157$			9.11318i			0.727311i	−0.931534	+	0.363656i	$0.881529\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.01646	−	13.3055i	0.0801079	−	1.04862i
$162$		0			0
$163$	−21.6335			−1.69447			−0.847233	−	0.531222i	$-0.821733\pi$
$163$	−21.6335			−1.69447			−0.847233	+	0.531222i	$0.821733\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	22.9282i		−	1.77424i	−0.461541	−	0.887119i	$-0.652703\pi$
$167$		−	22.9282i		−	1.77424i	0.461541	−	0.887119i	$-0.347297\pi$
$168$		0			0
$169$	−9.91983			−0.763064
$170$		0			0
$171$		0			0
$172$		0			0
$173$			10.6572i			0.810251i	0.914261	+	0.405125i	$0.132772\pi$
$173$			10.6572i			0.810251i	−0.914261	+	0.405125i	$0.867228\pi$
$174$		0			0
$175$		−	2.78247i		−	0.210335i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			8.11053i			0.606209i	0.952957	+	0.303105i	$0.0980232\pi$
$179$			8.11053i			0.606209i	−0.952957	+	0.303105i	$0.901977\pi$
$180$		0			0
$181$			19.1856i			1.42605i	0.701138	+	0.713026i	$0.252676\pi$
$181$			19.1856i			1.42605i	−0.701138	+	0.713026i	$0.747324\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			10.4205i			0.766128i
$186$		0			0
$187$	−4.69448			−0.343294
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.0615849			−0.00445613			−0.00222806	−	0.999998i	$-0.500709\pi$
$191$	−0.0615849			−0.00445613			−0.00222806	+	0.999998i	$0.500709\pi$
$192$		0			0
$193$	8.27305			0.595507			0.297754	−	0.954643i	$-0.403763\pi$
$193$	8.27305			0.595507			0.297754	+	0.954643i	$0.403763\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			13.4993i			0.961782i	0.876780	+	0.480891i	$0.159687\pi$
$197$			13.4993i			0.961782i	−0.876780	+	0.480891i	$0.840313\pi$
$198$		0			0
$199$			17.8657i			1.26647i	0.773960	+	0.633235i	$0.218273\pi$
$199$			17.8657i			1.26647i	−0.773960	+	0.633235i	$0.781727\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.75672			−0.614601
$204$		0			0
$205$		−	8.99716i		−	0.628389i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3.12592i			0.216225i
$210$		0			0
$211$	−11.3330			−0.780198			−0.390099	−	0.920773i	$-0.627559\pi$
$211$	−11.3330			−0.780198			−0.390099	+	0.920773i	$0.627559\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3.28852i		−	0.224275i
$216$		0			0
$217$			6.36559i			0.432125i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.96704			0.535921
$222$		0			0
$223$	−11.4605			−0.767451			−0.383726	−	0.923447i	$-0.625359\pi$
$223$	−11.4605			−0.767451			−0.383726	+	0.923447i	$0.625359\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.4892			0.696192			0.348096	−	0.937459i	$-0.386828\pi$
$227$	10.4892			0.696192			0.348096	+	0.937459i	$0.386828\pi$
$228$		0			0
$229$			22.6608i			1.49747i	0.662871	+	0.748734i	$0.269338\pi$
$229$			22.6608i			1.49747i	−0.662871	+	0.748734i	$0.730662\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.10720i		−	0.138047i	−0.997615	−	0.0690236i	$-0.978012\pi$
$233$		−	2.10720i		−	0.138047i	0.997615	−	0.0690236i	$-0.0219884\pi$
$234$		0			0
$235$			4.19754i			0.273817i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.82702i		−	0.441603i	−0.975319	−	0.220802i	$-0.929133\pi$
$239$		−	6.82702i		−	0.441603i	0.975319	−	0.220802i	$-0.0708673\pi$
$240$		0			0
$241$		−	7.57730i		−	0.488097i	−0.969763	−	0.244048i	$-0.921524\pi$
$241$		−	7.57730i		−	0.488097i	0.969763	−	0.244048i	$-0.0784756\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.742152			0.0474143
$246$		0			0
$247$		−	5.30503i		−	0.337551i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.5124			−0.663538			−0.331769	−	0.943361i	$-0.607646\pi$
$251$	−10.5124			−0.663538			−0.331769	+	0.943361i	$0.607646\pi$
$252$		0			0
$253$	4.94513	+	0.377776i	0.310898	+	0.0237506i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	4.19357i		−	0.261588i	−0.991410	−	0.130794i	$-0.958247\pi$
$257$		−	4.19357i		−	0.261588i	0.991410	−	0.130794i	$-0.0417526\pi$
$258$		0			0
$259$	−28.9947			−1.80164
$260$		0			0
$261$		0			0
$262$		0			0
$263$	21.7688			1.34232			0.671162	−	0.741311i	$-0.265795\pi$
$263$	21.7688			1.34232			0.671162	+	0.741311i	$0.265795\pi$
$264$		0			0
$265$	−6.63128			−0.407356
$266$		0			0
$267$		0			0
$268$		0			0
$269$			28.0065i			1.70759i	0.520610	+	0.853795i	$0.325705\pi$
$269$			28.0065i			1.70759i	−0.520610	+	0.853795i	$0.674295\pi$
$270$		0			0
$271$	−10.7285			−0.651711			−0.325856	−	0.945420i	$-0.605652\pi$
$271$	−10.7285			−0.651711			−0.325856	+	0.945420i	$0.605652\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.03414			0.0623607
$276$		0			0
$277$	−17.4126			−1.04622			−0.523112	−	0.852264i	$-0.675229\pi$
$277$	−17.4126			−1.04622			−0.523112	+	0.852264i	$0.675229\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	20.8589			1.24434			0.622169	−	0.782883i	$-0.286252\pi$
$281$	20.8589			1.24434			0.622169	+	0.782883i	$0.286252\pi$
$282$		0			0
$283$			16.4546i			0.978123i	0.872249	+	0.489062i	$0.162661\pi$
$283$			16.4546i			0.978123i	−0.872249	+	0.489062i	$0.837339\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	25.0344			1.47773
$288$		0			0
$289$	3.60725			0.212191
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.02785			0.0600474			0.0300237	−	0.999549i	$-0.490442\pi$
$293$	1.02785			0.0600474			0.0300237	+	0.999549i	$0.490442\pi$
$294$		0			0
$295$			9.75064i			0.567704i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.39242	−	0.641128i	−0.485346	−	0.0370774i
$300$		0			0
$301$	9.15021			0.527409
$302$		0			0
$303$		0			0
$304$		0			0
$305$			12.2497i			0.701416i
$306$		0			0
$307$	29.0435			1.65760			0.828799	−	0.559546i	$-0.189025\pi$
$307$	29.0435			1.65760			0.828799	+	0.559546i	$0.189025\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			20.8118i			1.18013i	0.807356	+	0.590064i	$0.200898\pi$
$311$			20.8118i			1.18013i	−0.807356	+	0.590064i	$0.799102\pi$
$312$		0			0
$313$		−	2.43043i		−	0.137376i	−0.997638	−	0.0686881i	$-0.978119\pi$
$313$		−	2.43043i		−	0.137376i	0.997638	−	0.0686881i	$-0.0218813\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			31.9822i			1.79630i	0.439693	+	0.898148i	$0.355087\pi$
$317$			31.9822i			1.79630i	−0.439693	+	0.898148i	$0.644913\pi$
$318$		0			0
$319$		−	3.25453i		−	0.182219i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	13.7218i		−	0.763501i
$324$		0			0
$325$	−1.75504			−0.0973521
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.6795			−0.643914
$330$		0			0
$331$	13.0093			0.715057			0.357529	−	0.933902i	$-0.383619\pi$
$331$	13.0093			0.715057			0.357529	+	0.933902i	$0.383619\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	5.18196i		−	0.283121i
$336$		0			0
$337$		−	12.1940i		−	0.664247i	−0.943236	−	0.332123i	$-0.892235\pi$
$337$		−	12.1940i		−	0.664247i	0.943236	−	0.332123i	$-0.107765\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.36584			−0.128117
$342$		0			0
$343$		−	17.4123i		−	0.940175i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	11.9543i		−	0.641741i	−0.947123	−	0.320870i	$-0.896025\pi$
$347$		−	11.9543i		−	0.641741i	0.947123	−	0.320870i	$-0.103975\pi$
$348$		0			0
$349$	−10.1866			−0.545275			−0.272638	−	0.962117i	$-0.587896\pi$
$349$	−10.1866			−0.545275			−0.272638	+	0.962117i	$0.587896\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			17.8461i			0.949850i	0.880026	+	0.474925i	$0.157525\pi$
$353$			17.8461i			0.949850i	−0.880026	+	0.474925i	$0.842475\pi$
$354$		0			0
$355$		−	11.4297i		−	0.606628i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.7096			−1.14579			−0.572894	−	0.819629i	$-0.694179\pi$
$359$	−21.7096			−1.14579			−0.572894	+	0.819629i	$0.694179\pi$
$360$		0			0
$361$	9.86304			0.519107
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.1422			0.792577
$366$		0			0
$367$		−	13.9623i		−	0.728827i	−0.931237	−	0.364413i	$-0.881270\pi$
$367$		−	13.9623i		−	0.728827i	0.931237	−	0.364413i	$-0.118730\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	18.4513i		−	0.957946i
$372$		0			0
$373$			14.1534i			0.732837i	0.930450	+	0.366418i	$0.119416\pi$
$373$			14.1534i			0.732837i	−0.930450	+	0.366418i	$0.880584\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			5.52329i			0.284464i
$378$		0			0
$379$			19.6415i			1.00891i	0.863437	+	0.504457i	$0.168307\pi$
$379$			19.6415i			1.00891i	−0.863437	+	0.504457i	$0.831693\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.02387			−0.358903			−0.179451	−	0.983767i	$-0.557432\pi$
$383$	−7.02387			−0.358903			−0.179451	+	0.983767i	$0.557432\pi$
$384$		0			0
$385$			2.87745i			0.146649i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−27.8758			−1.41336			−0.706680	−	0.707533i	$-0.749808\pi$
$389$	−27.8758			−1.41336			−0.706680	+	0.707533i	$0.749808\pi$
$390$		0			0
$391$	−21.7075	−	1.65832i	−1.09780	−	0.0838647i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			0.759277i			0.0382034i
$396$		0			0
$397$	−33.8757			−1.70017			−0.850086	−	0.526644i	$-0.823450\pi$
$397$	−33.8757			−1.70017			−0.850086	+	0.526644i	$0.823450\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−28.2213			−1.40930			−0.704652	−	0.709553i	$-0.748897\pi$
$401$	−28.2213			−1.40930			−0.704652	+	0.709553i	$0.748897\pi$
$402$		0			0
$403$	4.01509			0.200006
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	10.7762i		−	0.534155i
$408$		0			0
$409$	−7.08690			−0.350425			−0.175212	−	0.984531i	$-0.556061\pi$
$409$	−7.08690			−0.350425			−0.175212	+	0.984531i	$0.556061\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−27.1309			−1.33502
$414$		0			0
$415$	14.6261			0.717965
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.79099			0.234055			0.117028	−	0.993129i	$-0.462663\pi$
$419$	4.79099			0.234055			0.117028	+	0.993129i	$0.462663\pi$
$420$		0			0
$421$		−	10.4304i		−	0.508346i	−0.967159	−	0.254173i	$-0.918197\pi$
$421$		−	10.4304i		−	0.508346i	0.967159	−	0.254173i	$-0.0818033\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.53952			−0.220199
$426$		0			0
$427$	−34.0845			−1.64946
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.7476			0.662201			0.331100	−	0.943596i	$-0.392580\pi$
$431$	13.7476			0.662201			0.331100	+	0.943596i	$0.392580\pi$
$432$		0			0
$433$			16.9865i			0.816319i	0.912911	+	0.408160i	$0.133829\pi$
$433$			16.9865i			0.816319i	−0.912911	+	0.408160i	$0.866171\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.10423	+	14.4544i	−0.0528223	+	0.691449i
$438$		0			0
$439$	−39.0502			−1.86377			−0.931883	−	0.362760i	$-0.881835\pi$
$439$	−39.0502			−1.86377			−0.931883	+	0.362760i	$0.881835\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			26.3645i			1.25261i	0.779576	+	0.626307i	$0.215434\pi$
$443$			26.3645i			1.25261i	−0.779576	+	0.626307i	$0.784566\pi$
$444$		0			0
$445$	9.89856			0.469237
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	26.7249i		−	1.26123i	−0.776097	−	0.630614i	$-0.782803\pi$
$449$		−	26.7249i		−	1.26123i	0.776097	−	0.630614i	$-0.217197\pi$
$450$		0			0
$451$			9.30428i			0.438122i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	4.88335i		−	0.228935i
$456$		0			0
$457$		−	12.9051i		−	0.603677i	−0.953359	−	0.301839i	$-0.902400\pi$
$457$		−	12.9051i		−	0.603677i	0.953359	−	0.301839i	$-0.0976004\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			25.2417i			1.17562i	0.808997	+	0.587812i	$0.200011\pi$
$461$			25.2417i			1.17562i	−0.808997	+	0.587812i	$0.799989\pi$
$462$		0			0
$463$	−22.9882			−1.06835			−0.534176	−	0.845374i	$-0.679378\pi$
$463$	−22.9882			−1.06835			−0.534176	+	0.845374i	$0.679378\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−28.0022			−1.29579			−0.647893	−	0.761731i	$-0.724350\pi$
$467$	−28.0022			−1.29579			−0.647893	+	0.761731i	$0.724350\pi$
$468$		0			0
$469$	14.4187			0.665792
$470$		0			0
$471$		0			0
$472$		0			0
$473$			3.40077i			0.156368i
$474$		0			0
$475$			3.02274i			0.138693i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	33.3105			1.52200			0.760998	−	0.648754i	$-0.224710\pi$
$479$	33.3105			1.52200			0.760998	+	0.648754i	$0.224710\pi$
$480$		0			0
$481$			18.2883i			0.833876i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	0.520693i		−	0.0236434i
$486$		0			0
$487$	−20.6958			−0.937818			−0.468909	−	0.883246i	$-0.655353\pi$
$487$	−20.6958			−0.937818			−0.468909	+	0.883246i	$0.655353\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7.90703i		−	0.356839i	−0.983954	−	0.178420i	$-0.942902\pi$
$491$		−	7.90703i		−	0.356839i	0.983954	−	0.178420i	$-0.0570984\pi$
$492$		0			0
$493$			14.2863i			0.643424i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	31.8029			1.42656
$498$		0			0
$499$	−28.7068			−1.28509			−0.642547	−	0.766246i	$-0.722122\pi$
$499$	−28.7068			−1.28509			−0.642547	+	0.766246i	$0.722122\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−29.6073			−1.32012			−0.660061	−	0.751212i	$-0.729470\pi$
$503$	−29.6073			−1.32012			−0.660061	+	0.751212i	$0.729470\pi$
$504$		0			0
$505$		−	15.3462i		−	0.682897i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	14.4750i		−	0.641595i	−0.947148	−	0.320798i	$-0.896049\pi$
$509$		−	14.4750i		−	0.641595i	0.947148	−	0.320798i	$-0.103951\pi$
$510$		0			0
$511$			42.1326i			1.86384i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			7.07373i			0.311706i
$516$		0			0
$517$		−	4.34083i		−	0.190909i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−42.3007			−1.85323			−0.926613	−	0.376016i	$-0.877294\pi$
$521$	−42.3007			−1.85323			−0.926613	+	0.376016i	$0.877294\pi$
$522$		0			0
$523$		−	4.15789i		−	0.181812i	−0.995859	−	0.0909059i	$-0.971024\pi$
$523$		−	4.15789i		−	0.181812i	0.995859	−	0.0909059i	$-0.0289763\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.3853			0.452390
$528$		0			0
$529$	22.7331	+	3.49372i	0.988396	+	0.151901i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	15.7904i		−	0.683957i
$534$		0			0
$535$	1.58106			0.0683553
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.767485			−0.0330579
$540$		0			0
$541$	31.9895			1.37534			0.687669	−	0.726024i	$-0.258634\pi$
$541$	31.9895			1.37534			0.687669	+	0.726024i	$0.258634\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			11.4628i			0.491013i
$546$		0			0
$547$	13.7213			0.586681			0.293340	−	0.956008i	$-0.405233\pi$
$547$	13.7213			0.586681			0.293340	+	0.956008i	$0.405233\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9.51287			0.405262
$552$		0			0
$553$	−2.11267			−0.0898397
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.9939			−0.762426			−0.381213	−	0.924487i	$-0.624493\pi$
$557$	−17.9939			−0.762426			−0.381213	+	0.924487i	$0.624493\pi$
$558$		0			0
$559$		−	5.77148i		−	0.244108i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8.10047			−0.341394			−0.170697	−	0.985324i	$-0.554602\pi$
$563$	−8.10047			−0.341394			−0.170697	+	0.985324i	$0.554602\pi$
$564$		0			0
$565$	−4.83510			−0.203414
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.7768			0.451786			0.225893	−	0.974152i	$-0.427470\pi$
$569$	10.7768			0.451786			0.225893	+	0.974152i	$0.427470\pi$
$570$		0			0
$571$		−	44.8404i		−	1.87651i	−0.345941	−	0.938256i	$-0.612440\pi$
$571$		−	44.8404i		−	1.87651i	0.345941	−	0.938256i	$-0.387560\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.78190	+	0.365306i	0.199419	+	0.0152343i
$576$		0			0
$577$	−41.1818			−1.71442			−0.857211	−	0.514966i	$-0.827805\pi$
$577$	−41.1818			−1.71442			−0.857211	+	0.514966i	$0.827805\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			40.6966i			1.68838i
$582$		0			0
$583$	6.85764			0.284014
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	3.00956i		−	0.124218i	−0.998069	−	0.0621090i	$-0.980217\pi$
$587$		−	3.00956i		−	0.124218i	0.998069	−	0.0621090i	$-0.0197826\pi$
$588$		0			0
$589$		−	6.91527i		−	0.284939i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			38.7575i			1.59158i	0.605574	+	0.795789i	$0.292944\pi$
$593$			38.7575i			1.59158i	−0.605574	+	0.795789i	$0.707056\pi$
$594$		0			0
$595$		−	12.6311i		−	0.517824i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			39.1760i			1.60069i	0.599540	+	0.800344i	$0.295350\pi$
$599$			39.1760i			1.60069i	−0.599540	+	0.800344i	$0.704650\pi$
$600$		0			0
$601$	6.29610			0.256823			0.128412	−	0.991721i	$-0.459012\pi$
$601$	6.29610			0.256823			0.128412	+	0.991721i	$0.459012\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.93056			0.403735
$606$		0			0
$607$	−25.4771			−1.03408			−0.517041	−	0.855960i	$-0.672967\pi$
$607$	−25.4771			−1.03408			−0.517041	+	0.855960i	$0.672967\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			7.36685i			0.298031i
$612$		0			0
$613$			31.5665i			1.27496i	0.770467	+	0.637480i	$0.220023\pi$
$613$			31.5665i			1.27496i	−0.770467	+	0.637480i	$0.779977\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.6701			−0.429563			−0.214782	−	0.976662i	$-0.568904\pi$
$617$	−10.6701			−0.429563			−0.214782	+	0.976662i	$0.568904\pi$
$618$		0			0
$619$			35.5335i			1.42821i	0.700038	+	0.714105i	$0.253166\pi$
$619$			35.5335i			1.42821i	−0.700038	+	0.714105i	$0.746834\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			27.5425i			1.10347i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			47.3039i			1.88613i
$630$		0			0
$631$		−	23.6745i		−	0.942467i	−0.882008	−	0.471234i	$-0.843809\pi$
$631$		−	23.6745i		−	0.942467i	0.882008	−	0.471234i	$-0.156191\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.9837			0.475558
$636$		0			0
$637$	1.30251			0.0516072
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.94897			−0.234970			−0.117485	−	0.993075i	$-0.537483\pi$
$641$	−5.94897			−0.234970			−0.117485	+	0.993075i	$0.537483\pi$
$642$		0			0
$643$		−	40.6981i		−	1.60498i	−0.596668	−	0.802488i	$-0.703509\pi$
$643$		−	40.6981i		−	1.60498i	0.596668	−	0.802488i	$-0.296491\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	48.0432i		−	1.88877i	−0.328839	−	0.944386i	$-0.606657\pi$
$647$		−	48.0432i		−	1.88877i	0.328839	−	0.944386i	$-0.393343\pi$
$648$		0			0
$649$		−	10.0835i		−	0.395811i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	26.4361i		−	1.03452i	−0.855827	−	0.517262i	$-0.826951\pi$
$653$		−	26.4361i		−	1.03452i	0.855827	−	0.517262i	$-0.173049\pi$
$654$		0			0
$655$		−	13.3972i		−	0.523472i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.5539			−0.644848			−0.322424	−	0.946595i	$-0.604498\pi$
$659$	−16.5539			−0.644848			−0.322424	+	0.946595i	$0.604498\pi$
$660$		0			0
$661$			10.7988i			0.420024i	0.977699	+	0.210012i	$0.0673503\pi$
$661$			10.7988i			0.420024i	−0.977699	+	0.210012i	$0.932650\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.41069			−0.326153
$666$		0			0
$667$	1.14966	−	15.0491i	0.0445149	−	0.582704i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	12.6679i		−	0.489037i
$672$		0			0
$673$	−8.02067			−0.309174			−0.154587	−	0.987979i	$-0.549405\pi$
$673$	−8.02067			−0.309174			−0.154587	+	0.987979i	$0.549405\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.9741			0.844535			0.422268	−	0.906471i	$-0.361234\pi$
$677$	21.9741			0.844535			0.422268	+	0.906471i	$0.361234\pi$
$678$		0			0
$679$	1.44881			0.0556004
$680$		0			0
$681$		0			0
$682$		0			0
$683$			0.546314i			0.0209041i	0.999945	+	0.0104521i	$0.00332705\pi$
$683$			0.546314i			0.0209041i	−0.999945	+	0.0104521i	$0.996673\pi$
$684$		0			0
$685$	−18.8685			−0.720928
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11.6382			−0.443378
$690$		0			0
$691$	−3.24022			−0.123264			−0.0616318	−	0.998099i	$-0.519630\pi$
$691$	−3.24022			−0.123264			−0.0616318	+	0.998099i	$0.519630\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.16167			−0.0440648
$696$		0			0
$697$		−	40.8428i		−	1.54703i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	40.7146			1.53777			0.768886	−	0.639386i	$-0.220812\pi$
$701$	40.7146			1.53777			0.768886	+	0.639386i	$0.220812\pi$
$702$		0			0
$703$	31.4984			1.18798
$704$		0			0
$705$		0			0
$706$		0			0
$707$	42.7004			1.60591
$708$		0			0
$709$		−	26.5198i		−	0.995971i	−0.867185	−	0.497986i	$-0.834073\pi$
$709$		−	26.5198i		−	0.995971i	0.867185	−	0.497986i	$-0.165927\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−10.9398	−	0.835729i	−0.409698	−	0.0312983i
$714$		0			0
$715$	1.81495			0.0678753
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	32.5891i		−	1.21537i	−0.794179	−	0.607684i	$-0.792099\pi$
$719$		−	32.5891i		−	1.21537i	0.794179	−	0.607684i	$-0.207901\pi$
$720$		0			0
$721$	−19.6825			−0.733013
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	3.14710i		−	0.116880i
$726$		0			0
$727$		−	18.9548i		−	0.702996i	−0.936189	−	0.351498i	$-0.885672\pi$
$727$		−	18.9548i		−	0.702996i	0.936189	−	0.351498i	$-0.114328\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	14.9283i		−	0.552143i
$732$		0			0
$733$		−	22.9103i		−	0.846212i	−0.906080	−	0.423106i	$-0.860940\pi$
$733$		−	22.9103i		−	0.846212i	0.906080	−	0.423106i	$-0.139060\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			5.35885i			0.197396i
$738$		0			0
$739$	−30.7808			−1.13229			−0.566144	−	0.824306i	$-0.691566\pi$
$739$	−30.7808			−1.13229			−0.566144	+	0.824306i	$0.691566\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	32.8990			1.20695			0.603473	−	0.797383i	$-0.293783\pi$
$743$	32.8990			1.20695			0.603473	+	0.797383i	$0.293783\pi$
$744$		0			0
$745$	−1.02859			−0.0376846
$746$		0			0
$747$		0			0
$748$		0			0
$749$			4.39927i			0.160746i
$750$		0			0
$751$		−	21.5983i		−	0.788132i	−0.919082	−	0.394066i	$-0.871068\pi$
$751$		−	21.5983i		−	0.788132i	0.919082	−	0.394066i	$-0.128932\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.15652			−0.114877
$756$		0			0
$757$			23.5642i			0.856456i	0.903671	+	0.428228i	$0.140862\pi$
$757$			23.5642i			0.856456i	−0.903671	+	0.428228i	$0.859138\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	22.9185i		−	0.830797i	−0.909640	−	0.415398i	$-0.863642\pi$
$761$		−	22.9185i		−	0.830797i	0.909640	−	0.415398i	$-0.136358\pi$
$762$		0			0
$763$	−31.8950			−1.15468
$764$		0			0
$765$		0			0
$766$		0			0
$767$			17.1128i			0.617906i
$768$		0			0
$769$		−	16.9885i		−	0.612623i	−0.951931	−	0.306311i	$-0.900905\pi$
$769$		−	16.9885i		−	0.612623i	0.951931	−	0.306311i	$-0.0990948\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.98783			−0.0714972			−0.0357486	−	0.999361i	$-0.511382\pi$
$773$	−1.98783			−0.0714972			−0.0357486	+	0.999361i	$0.511382\pi$
$774$		0			0
$775$	−2.28775			−0.0821783
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.1961			−0.974401
$780$		0			0
$781$			11.8199i			0.422949i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	9.11318i		−	0.325263i
$786$		0			0
$787$		−	34.8230i		−	1.24131i	−0.784086	−	0.620653i	$-0.786868\pi$
$787$		−	34.8230i		−	1.24131i	0.784086	−	0.620653i	$-0.213132\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	13.4535i		−	0.478353i
$792$		0			0
$793$			21.4987i			0.763442i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	44.3485			1.57090			0.785452	−	0.618923i	$-0.212431\pi$
$797$	44.3485			1.57090			0.785452	+	0.618923i	$0.212431\pi$
$798$		0			0
$799$			19.0548i			0.674111i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−15.6590			−0.552596
$804$		0			0
$805$	−1.01646	+	13.3055i	−0.0358253	+	0.468957i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			4.82977i			0.169806i	0.996389	+	0.0849029i	$0.0270580\pi$
$809$			4.82977i			0.169806i	−0.996389	+	0.0849029i	$0.972942\pi$
$810$		0			0
$811$	13.2347			0.464733			0.232366	−	0.972628i	$-0.425353\pi$
$811$	13.2347			0.464733			0.232366	+	0.972628i	$0.425353\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21.6335			0.757788
$816$		0			0
$817$	−9.94033			−0.347768
$818$		0			0
$819$		0			0
$820$		0			0
$821$			26.5938i			0.928128i	0.885802	+	0.464064i	$0.153609\pi$
$821$			26.5938i			0.928128i	−0.885802	+	0.464064i	$0.846391\pi$
$822$		0			0
$823$	−5.83941			−0.203549			−0.101775	−	0.994807i	$-0.532452\pi$
$823$	−5.83941			−0.203549			−0.101775	+	0.994807i	$0.532452\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.20576			0.320116			0.160058	−	0.987108i	$-0.448832\pi$
$827$	9.20576			0.320116			0.160058	+	0.987108i	$0.448832\pi$
$828$		0			0
$829$	42.0210			1.45945			0.729724	−	0.683742i	$-0.239649\pi$
$829$	42.0210			1.45945			0.729724	+	0.683742i	$0.239649\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.36901			0.116729
$834$		0			0
$835$			22.9282i			0.793463i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−32.0408			−1.10617			−0.553086	−	0.833124i	$-0.686550\pi$
$839$	−32.0408			−1.10617			−0.553086	+	0.833124i	$0.686550\pi$
$840$		0			0
$841$	19.0958			0.658474
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.91983			0.341253
$846$		0			0
$847$			27.6315i			0.949430i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.80666	−	49.8296i	0.130491	−	1.70814i
$852$		0			0
$853$	43.6257			1.49372			0.746858	−	0.664984i	$-0.231562\pi$
$853$	43.6257			1.49372			0.746858	+	0.664984i	$0.231562\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			26.8396i			0.916822i	0.888740	+	0.458411i	$0.151581\pi$
$857$			26.8396i			0.916822i	−0.888740	+	0.458411i	$0.848419\pi$
$858$		0			0
$859$	13.0488			0.445220			0.222610	−	0.974908i	$-0.428542\pi$
$859$	13.0488			0.445220			0.222610	+	0.974908i	$0.428542\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	45.4883i		−	1.54844i	−0.632917	−	0.774219i	$-0.718143\pi$
$863$		−	45.4883i		−	1.54844i	0.632917	−	0.774219i	$-0.281857\pi$
$864$		0			0
$865$		−	10.6572i		−	0.362355i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	0.785195i		−	0.0266359i
$870$		0			0
$871$		−	9.09455i		−	0.308157i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			2.78247i			0.0940647i
$876$		0			0
$877$	45.9402			1.55129			0.775645	−	0.631169i	$-0.217425\pi$
$877$	45.9402			1.55129			0.775645	+	0.631169i	$0.217425\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.0346			−0.472836			−0.236418	−	0.971651i	$-0.575974\pi$
$881$	−14.0346			−0.472836			−0.236418	+	0.971651i	$0.575974\pi$
$882$		0			0
$883$	−25.6379			−0.862785			−0.431392	−	0.902164i	$-0.641978\pi$
$883$	−25.6379			−0.862785			−0.431392	+	0.902164i	$0.641978\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			31.2808i			1.05031i	0.851007	+	0.525154i	$0.175992\pi$
$887$			31.2808i			1.05031i	−0.851007	+	0.525154i	$0.824008\pi$
$888$		0			0
$889$			33.3443i			1.11833i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	12.6881			0.424590
$894$		0			0
$895$		−	8.11053i		−	0.271105i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			7.19977i			0.240126i
$900$		0			0
$901$	−30.1028			−1.00287
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	19.1856i		−	0.637749i
$906$		0			0
$907$		−	8.89199i		−	0.295254i	−0.989043	−	0.147627i	$-0.952837\pi$
$907$		−	8.89199i		−	0.295254i	0.989043	−	0.147627i	$-0.0471635\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.72773			0.123505			0.0617527	−	0.998091i	$-0.480331\pi$
$911$	3.72773			0.123505			0.0617527	+	0.998091i	$0.480331\pi$
$912$		0			0
$913$	−15.1253			−0.500575
$914$		0			0
$915$		0			0
$916$		0			0
$917$	37.2773			1.23101
$918$		0			0
$919$			14.7073i			0.485150i	0.970133	+	0.242575i	$0.0779921\pi$
$919$			14.7073i			0.485150i	−0.970133	+	0.242575i	$0.922008\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	20.0597i		−	0.660272i
$924$		0			0
$925$		−	10.4205i		−	0.342623i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1.44246i			0.0473255i	0.999720	+	0.0236628i	$0.00753279\pi$
$929$			1.44246i			0.0473255i	−0.999720	+	0.0236628i	$0.992467\pi$
$930$		0			0
$931$		−	2.24333i		−	0.0735222i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.69448			0.153526
$936$		0			0
$937$		−	8.93792i		−	0.291989i	−0.989285	−	0.145995i	$-0.953362\pi$
$937$		−	8.93792i		−	0.291989i	0.989285	−	0.145995i	$-0.0466382\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.9247			0.388735			0.194367	−	0.980929i	$-0.437735\pi$
$941$	11.9247			0.388735			0.194367	+	0.980929i	$0.437735\pi$
$942$		0			0
$943$	−3.28672	+	43.0235i	−0.107030	+	1.40104i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			22.9620i			0.746165i	0.927798	+	0.373083i	$0.121699\pi$
$947$			22.9620i			0.746165i	−0.927798	+	0.373083i	$0.878301\pi$
$948$		0			0
$949$	26.5751			0.862664
$950$		0			0
$951$		0			0
$952$		0			0
$953$	57.1630			1.85169			0.925845	−	0.377902i	$-0.123355\pi$
$953$	57.1630			1.85169			0.925845	+	0.377902i	$0.123355\pi$
$954$		0			0
$955$	0.0615849			0.00199284
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	52.5010i		−	1.69535i
$960$		0			0
$961$	−25.7662			−0.831168
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.27305			−0.266319
$966$		0			0
$967$	42.4177			1.36406			0.682031	−	0.731324i	$-0.261097\pi$
$967$	42.4177			1.36406			0.682031	+	0.731324i	$0.261097\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−42.6671			−1.36925			−0.684627	−	0.728894i	$-0.740035\pi$
$971$	−42.6671			−1.36925			−0.684627	+	0.728894i	$0.740035\pi$
$972$		0			0
$973$		−	3.23233i		−	0.103624i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	23.6368			0.756209			0.378105	−	0.925763i	$-0.376576\pi$
$977$	23.6368			0.756209			0.378105	+	0.925763i	$0.376576\pi$
$978$		0			0
$979$	−10.2365			−0.327158
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−52.3095			−1.66841			−0.834207	−	0.551452i	$-0.814074\pi$
$983$	−52.3095			−1.66841			−0.834207	+	0.551452i	$0.814074\pi$
$984$		0			0
$985$		−	13.4993i		−	0.430122i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.20132	+	15.7254i	−0.0381997	+	0.500037i
$990$		0			0
$991$	−32.5613			−1.03434			−0.517172	−	0.855881i	$-0.673015\pi$
$991$	−32.5613			−1.03434			−0.517172	+	0.855881i	$0.673015\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	17.8657i		−	0.566382i
$996$		0			0
$997$	25.9215			0.820942			0.410471	−	0.911874i	$-0.365364\pi$
$997$	25.9215			0.820942			0.410471	+	0.911874i	$0.365364\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.p.a.1241.11	✓	48
3.2	odd	2		8280.2.p.b.1241.11	yes	48
23.22	odd	2		8280.2.p.b.1241.38	yes	48
69.68	even	2	inner	8280.2.p.a.1241.38	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.11	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.38	yes	48	69.68	even	2	inner
8280.2.p.b.1241.11	yes	48	3.2	odd	2
8280.2.p.b.1241.38	yes	48	23.22	odd	2

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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.78247i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.78247i q^{7} +1.03414 q^{11} -1.75504 q^{13} -4.53952 q^{17} +3.02274i q^{19} +(4.78190 + 0.365306i) q^{23} +1.00000 q^{25} -3.14710i q^{29} -2.28775 q^{31} +2.78247i q^{35} -10.4205i q^{37} +8.99716i q^{41} +3.28852i q^{43} -4.19754i q^{47} -0.742152 q^{49} +6.63128 q^{53} -1.03414 q^{55} -9.75064i q^{59} -12.2497i q^{61} +1.75504 q^{65} +5.18196i q^{67} +11.4297i q^{71} -15.1422 q^{73} -2.87745i q^{77} -0.759277i q^{79} -14.6261 q^{83} +4.53952 q^{85} -9.89856 q^{89} +4.88335i q^{91} -3.02274i q^{95} +0.520693i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 48 * q - 48 * q^5	\( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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