LMFDB - Embedded newform 8280.2.p.a.1241.3

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			1241.3
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.46

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -4.29385i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -4.29385i q^{7} +0.813503 q^{11} +1.00050 q^{13} +7.09518 q^{17} +3.82420i q^{19} +(-1.12608 + 4.66175i) q^{23} +1.00000 q^{25} +0.328907i q^{29} -4.61002 q^{31} +4.29385i q^{35} +10.4272i q^{37} -3.42126i q^{41} +7.16889i q^{43} -0.556882i q^{47} -11.4371 q^{49} +3.55889 q^{53} -0.813503 q^{55} +12.4450i q^{59} -9.65842i q^{61} -1.00050 q^{65} +1.86503i q^{67} -0.163405i q^{71} -8.45188 q^{73} -3.49306i q^{77} +15.2342i q^{79} +5.16406 q^{83} -7.09518 q^{85} +3.52407 q^{89} -4.29599i q^{91} -3.82420i q^{95} +7.94278i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	4.29385i		−	1.62292i	−0.584406	−	0.811461i	$-0.698672\pi$
$7$		−	4.29385i		−	1.62292i	0.584406	−	0.811461i	$-0.301328\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.813503			0.245280			0.122640	−	0.992451i	$-0.460864\pi$
$11$	0.813503			0.245280			0.122640	+	0.992451i	$0.460864\pi$
$12$		0			0
$13$	1.00050			0.277488			0.138744	−	0.990328i	$-0.455693\pi$
$13$	1.00050			0.277488			0.138744	+	0.990328i	$0.455693\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.09518			1.72083			0.860417	−	0.509590i	$-0.170203\pi$
$17$	7.09518			1.72083			0.860417	+	0.509590i	$0.170203\pi$
$18$		0			0
$19$			3.82420i			0.877331i	0.898650	+	0.438666i	$0.144549\pi$
$19$			3.82420i			0.877331i	−0.898650	+	0.438666i	$0.855451\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.12608	+	4.66175i	−0.234805	+	0.972043i
$23$	−1.12608	+	4.66175i	−0.234805	+	0.972043i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			0.328907i			0.0610765i	0.999534	+	0.0305383i	$0.00972214\pi$
$29$			0.328907i			0.0610765i	−0.999534	+	0.0305383i	$0.990278\pi$
$30$		0			0
$31$	−4.61002			−0.827983			−0.413992	−	0.910281i	$-0.635866\pi$
$31$	−4.61002			−0.827983			−0.413992	+	0.910281i	$0.635866\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			4.29385i			0.725793i
$36$		0			0
$37$			10.4272i			1.71422i	0.515133	+	0.857110i	$0.327743\pi$
$37$			10.4272i			1.71422i	−0.515133	+	0.857110i	$0.672257\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3.42126i		−	0.534311i	−0.963653	−	0.267155i	$-0.913916\pi$
$41$		−	3.42126i		−	0.534311i	0.963653	−	0.267155i	$-0.0860837\pi$
$42$		0			0
$43$			7.16889i			1.09325i	0.837379	+	0.546623i	$0.184087\pi$
$43$			7.16889i			1.09325i	−0.837379	+	0.546623i	$0.815913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	0.556882i		−	0.0812295i	−0.999175	−	0.0406148i	$-0.987068\pi$
$47$		−	0.556882i		−	0.0812295i	0.999175	−	0.0406148i	$-0.0129316\pi$
$48$		0			0
$49$	−11.4371			−1.63388
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.55889			0.488852			0.244426	−	0.969668i	$-0.421401\pi$
$53$	3.55889			0.488852			0.244426	+	0.969668i	$0.421401\pi$
$54$		0			0
$55$	−0.813503			−0.109693
$56$		0			0
$57$		0			0
$58$		0			0
$59$			12.4450i			1.62021i	0.586287	+	0.810104i	$0.300589\pi$
$59$			12.4450i			1.62021i	−0.586287	+	0.810104i	$0.699411\pi$
$60$		0			0
$61$		−	9.65842i		−	1.23663i	−0.785929	−	0.618317i	$-0.787815\pi$
$61$		−	9.65842i		−	1.23663i	0.785929	−	0.618317i	$-0.212185\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.00050			−0.124097
$66$		0			0
$67$			1.86503i			0.227849i	0.993489	+	0.113925i	$0.0363423\pi$
$67$			1.86503i			0.227849i	−0.993489	+	0.113925i	$0.963658\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.163405i		−	0.0193927i	−0.999953	−	0.00969633i	$-0.996914\pi$
$71$		−	0.163405i		−	0.0193927i	0.999953	−	0.00969633i	$-0.00308649\pi$
$72$		0			0
$73$	−8.45188			−0.989217			−0.494609	−	0.869116i	$-0.664689\pi$
$73$	−8.45188			−0.989217			−0.494609	+	0.869116i	$0.664689\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.49306i		−	0.398071i
$78$		0			0
$79$			15.2342i			1.71398i	0.515336	+	0.856988i	$0.327667\pi$
$79$			15.2342i			1.71398i	−0.515336	+	0.856988i	$0.672333\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.16406			0.566829			0.283414	−	0.958998i	$-0.408533\pi$
$83$	5.16406			0.566829			0.283414	+	0.958998i	$0.408533\pi$
$84$		0			0
$85$	−7.09518			−0.769581
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.52407			0.373550			0.186775	−	0.982403i	$-0.440196\pi$
$89$	3.52407			0.373550			0.186775	+	0.982403i	$0.440196\pi$
$90$		0			0
$91$		−	4.29599i		−	0.450342i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	3.82420i		−	0.392355i
$96$		0			0
$97$			7.94278i			0.806467i	0.915097	+	0.403233i	$0.132114\pi$
$97$			7.94278i			0.806467i	−0.915097	+	0.403233i	$0.867886\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	7.35867i		−	0.732215i	−0.930573	−	0.366107i	$-0.880690\pi$
$101$		−	7.35867i		−	0.732215i	0.930573	−	0.366107i	$-0.119310\pi$
$102$		0			0
$103$			5.03295i			0.495912i	0.968771	+	0.247956i	$0.0797588\pi$
$103$			5.03295i			0.495912i	−0.968771	+	0.247956i	$0.920241\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.80673			−0.561357			−0.280679	−	0.959802i	$-0.590560\pi$
$107$	−5.80673			−0.561357			−0.280679	+	0.959802i	$0.590560\pi$
$108$		0			0
$109$			2.93260i			0.280892i	0.990088	+	0.140446i	$0.0448537\pi$
$109$			2.93260i			0.280892i	−0.990088	+	0.140446i	$0.955146\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.1479			−0.954636			−0.477318	−	0.878731i	$-0.658391\pi$
$113$	−10.1479			−0.954636			−0.477318	+	0.878731i	$0.658391\pi$
$114$		0			0
$115$	1.12608	−	4.66175i	0.105008	−	0.434711i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	30.4657i		−	2.79278i
$120$		0			0
$121$	−10.3382			−0.939838
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−4.47190			−0.396817			−0.198409	−	0.980119i	$-0.563577\pi$
$127$	−4.47190			−0.396817			−0.198409	+	0.980119i	$0.563577\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			9.97112i			0.871181i	0.900145	+	0.435590i	$0.143460\pi$
$131$			9.97112i			0.871181i	−0.900145	+	0.435590i	$0.856540\pi$
$132$		0			0
$133$	16.4205			1.42384
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.36208			−0.543549			−0.271775	−	0.962361i	$-0.587611\pi$
$137$	−6.36208			−0.543549			−0.271775	+	0.962361i	$0.587611\pi$
$138$		0			0
$139$	13.2054			1.12007			0.560034	−	0.828470i	$-0.310788\pi$
$139$	13.2054			1.12007			0.560034	+	0.828470i	$0.310788\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.813908			0.0680624
$144$		0			0
$145$		−	0.328907i		−	0.0273142i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.6514			1.11836			0.559182	−	0.829045i	$-0.311115\pi$
$149$	13.6514			1.11836			0.559182	+	0.829045i	$0.311115\pi$
$150$		0			0
$151$	2.28358			0.185835			0.0929177	−	0.995674i	$-0.470381\pi$
$151$	2.28358			0.185835			0.0929177	+	0.995674i	$0.470381\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.61002			0.370285
$156$		0			0
$157$			23.1675i			1.84897i	0.381219	+	0.924485i	$0.375504\pi$
$157$			23.1675i			1.84897i	−0.381219	+	0.924485i	$0.624496\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	20.0169	+	4.83523i	1.57755	+	0.381070i
$162$		0			0
$163$	13.2846			1.04053			0.520265	−	0.854005i	$-0.325833\pi$
$163$	13.2846			1.04053			0.520265	+	0.854005i	$0.325833\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.37549i		−	0.338585i	−0.985566	−	0.169293i	$-0.945852\pi$
$167$		−	4.37549i		−	0.338585i	0.985566	−	0.169293i	$-0.0541483\pi$
$168$		0			0
$169$	−11.9990			−0.923000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	25.6230i		−	1.94808i	−0.226380	−	0.974039i	$-0.572689\pi$
$173$		−	25.6230i		−	1.94808i	0.226380	−	0.974039i	$-0.427311\pi$
$174$		0			0
$175$		−	4.29385i		−	0.324585i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			13.7392i			1.02692i	0.858115	+	0.513458i	$0.171636\pi$
$179$			13.7392i			1.02692i	−0.858115	+	0.513458i	$0.828364\pi$
$180$		0			0
$181$		−	22.6528i		−	1.68377i	−0.539658	−	0.841884i	$-0.681446\pi$
$181$		−	22.6528i		−	1.68377i	0.539658	−	0.841884i	$-0.318554\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	10.4272i		−	0.766623i
$186$		0			0
$187$	5.77195			0.422087
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.9372			1.51496			0.757482	−	0.652857i	$-0.226430\pi$
$191$	20.9372			1.51496			0.757482	+	0.652857i	$0.226430\pi$
$192$		0			0
$193$	−11.0000			−0.791800			−0.395900	−	0.918294i	$-0.629567\pi$
$193$	−11.0000			−0.791800			−0.395900	+	0.918294i	$0.629567\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			11.7476i			0.836982i	0.908221	+	0.418491i	$0.137441\pi$
$197$			11.7476i			0.836982i	−0.908221	+	0.418491i	$0.862559\pi$
$198$		0			0
$199$			25.4484i			1.80399i	0.431747	+	0.901995i	$0.357897\pi$
$199$			25.4484i			1.80399i	−0.431747	+	0.901995i	$0.642103\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.41228			0.0991225
$204$		0			0
$205$			3.42126i			0.238951i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3.11100i			0.215192i
$210$		0			0
$211$	4.31144			0.296812			0.148406	−	0.988927i	$-0.452586\pi$
$211$	4.31144			0.296812			0.148406	+	0.988927i	$0.452586\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	7.16889i		−	0.488914i
$216$		0			0
$217$			19.7947i			1.34375i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.09872			0.477511
$222$		0			0
$223$	19.6974			1.31904			0.659519	−	0.751688i	$-0.270760\pi$
$223$	19.6974			1.31904			0.659519	+	0.751688i	$0.270760\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.2783			−0.881312			−0.440656	−	0.897676i	$-0.645254\pi$
$227$	−13.2783			−0.881312			−0.440656	+	0.897676i	$0.645254\pi$
$228$		0			0
$229$		−	13.1695i		−	0.870263i	−0.900367	−	0.435132i	$-0.856702\pi$
$229$		−	13.1695i		−	0.870263i	0.900367	−	0.435132i	$-0.143298\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			7.20321i			0.471898i	0.971765	+	0.235949i	$0.0758198\pi$
$233$			7.20321i			0.471898i	−0.971765	+	0.235949i	$0.924180\pi$
$234$		0			0
$235$			0.556882i			0.0363270i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			14.0367i			0.907962i	0.891011	+	0.453981i	$0.149997\pi$
$239$			14.0367i			0.907962i	−0.891011	+	0.453981i	$0.850003\pi$
$240$		0			0
$241$		−	1.84347i		−	0.118748i	−0.998236	−	0.0593740i	$-0.981090\pi$
$241$		−	1.84347i		−	0.118748i	0.998236	−	0.0593740i	$-0.0189105\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.4371			0.730692
$246$		0			0
$247$			3.82610i			0.243449i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.77543			0.553900			0.276950	−	0.960884i	$-0.410676\pi$
$251$	8.77543			0.553900			0.276950	+	0.960884i	$0.410676\pi$
$252$		0			0
$253$	−0.916072	+	3.79235i	−0.0575929	+	0.238423i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	15.7886i		−	0.984867i	−0.870350	−	0.492433i	$-0.836108\pi$
$257$		−	15.7886i		−	0.984867i	0.870350	−	0.492433i	$-0.163892\pi$
$258$		0			0
$259$	44.7728			2.78205
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.6791			−0.781825			−0.390912	−	0.920428i	$-0.627840\pi$
$263$	−12.6791			−0.781825			−0.390912	+	0.920428i	$0.627840\pi$
$264$		0			0
$265$	−3.55889			−0.218621
$266$		0			0
$267$		0			0
$268$		0			0
$269$			3.58747i			0.218732i	0.994002	+	0.109366i	$0.0348821\pi$
$269$			3.58747i			0.218732i	−0.994002	+	0.109366i	$0.965118\pi$
$270$		0			0
$271$	17.6817			1.07408			0.537042	−	0.843555i	$-0.319542\pi$
$271$	17.6817			1.07408			0.537042	+	0.843555i	$0.319542\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.813503			0.0490561
$276$		0			0
$277$	15.3191			0.920435			0.460217	−	0.887806i	$-0.347771\pi$
$277$	15.3191			0.920435			0.460217	+	0.887806i	$0.347771\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.4562			1.33963			0.669813	−	0.742530i	$-0.266374\pi$
$281$	22.4562			1.33963			0.669813	+	0.742530i	$0.266374\pi$
$282$		0			0
$283$			24.0624i			1.43036i	0.698940	+	0.715181i	$0.253656\pi$
$283$			24.0624i			1.43036i	−0.698940	+	0.715181i	$0.746344\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−14.6904			−0.867145
$288$		0			0
$289$	33.3416			1.96127
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.0094			0.584756			0.292378	−	0.956303i	$-0.405553\pi$
$293$	10.0094			0.584756			0.292378	+	0.956303i	$0.405553\pi$
$294$		0			0
$295$		−	12.4450i		−	0.724579i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.12664	+	4.66408i	−0.0651555	+	0.269730i
$300$		0			0
$301$	30.7821			1.77425
$302$		0			0
$303$		0			0
$304$		0			0
$305$			9.65842i			0.553039i
$306$		0			0
$307$	3.95099			0.225495			0.112747	−	0.993624i	$-0.464035\pi$
$307$	3.95099			0.225495			0.112747	+	0.993624i	$0.464035\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	34.3304i		−	1.94670i	−0.229325	−	0.973350i	$-0.573652\pi$
$311$		−	34.3304i		−	1.94670i	0.229325	−	0.973350i	$-0.426348\pi$
$312$		0			0
$313$			21.1115i			1.19329i	0.802504	+	0.596647i	$0.203501\pi$
$313$			21.1115i			1.19329i	−0.802504	+	0.596647i	$0.796499\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	28.9788i		−	1.62761i	−0.581138	−	0.813805i	$-0.697392\pi$
$317$		−	28.9788i		−	1.62761i	0.581138	−	0.813805i	$-0.302608\pi$
$318$		0			0
$319$			0.267567i			0.0149809i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			27.1334i			1.50974i
$324$		0			0
$325$	1.00050			0.0554977
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.39117			−0.131829
$330$		0			0
$331$	−21.7836			−1.19734			−0.598668	−	0.800997i	$-0.704303\pi$
$331$	−21.7836			−1.19734			−0.598668	+	0.800997i	$0.704303\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1.86503i		−	0.101897i
$336$		0			0
$337$			2.82800i			0.154051i	0.997029	+	0.0770254i	$0.0245423\pi$
$337$			2.82800i			0.154051i	−0.997029	+	0.0770254i	$0.975458\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.75026			−0.203088
$342$		0			0
$343$			19.0524i			1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	8.04369i		−	0.431808i	−0.976415	−	0.215904i	$-0.930730\pi$
$347$		−	8.04369i		−	0.431808i	0.976415	−	0.215904i	$-0.0692698\pi$
$348$		0			0
$349$	−6.49701			−0.347777			−0.173889	−	0.984765i	$-0.555633\pi$
$349$	−6.49701			−0.347777			−0.173889	+	0.984765i	$0.555633\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			32.8289i			1.74731i	0.486548	+	0.873654i	$0.338256\pi$
$353$			32.8289i			1.74731i	−0.486548	+	0.873654i	$0.661744\pi$
$354$		0			0
$355$			0.163405i			0.00867266i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−27.4672			−1.44966			−0.724832	−	0.688925i	$-0.758083\pi$
$359$	−27.4672			−1.44966			−0.724832	+	0.688925i	$0.758083\pi$
$360$		0			0
$361$	4.37550			0.230290
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.45188			0.442391
$366$		0			0
$367$			6.84036i			0.357064i	0.983934	+	0.178532i	$0.0571348\pi$
$367$			6.84036i			0.357064i	−0.983934	+	0.178532i	$0.942865\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	15.2814i		−	0.793368i
$372$		0			0
$373$		−	21.2371i		−	1.09962i	−0.835290	−	0.549809i	$-0.814701\pi$
$373$		−	21.2371i		−	1.09962i	0.835290	−	0.549809i	$-0.185299\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			0.329071i			0.0169480i
$378$		0			0
$379$			17.9444i			0.921742i	0.887467	+	0.460871i	$0.152463\pi$
$379$			17.9444i			0.921742i	−0.887467	+	0.460871i	$0.847537\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.3781			0.632489			0.316244	−	0.948678i	$-0.397578\pi$
$383$	12.3781			0.632489			0.316244	+	0.948678i	$0.397578\pi$
$384$		0			0
$385$			3.49306i			0.178023i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.9600			0.707801			0.353900	−	0.935283i	$-0.384855\pi$
$389$	13.9600			0.707801			0.353900	+	0.935283i	$0.384855\pi$
$390$		0			0
$391$	−7.98977	+	33.0760i	−0.404060	+	1.67272i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	15.2342i		−	0.766514i
$396$		0			0
$397$	−13.2916			−0.667087			−0.333543	−	0.942735i	$-0.608244\pi$
$397$	−13.2916			−0.667087			−0.333543	+	0.942735i	$0.608244\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	21.0805			1.05271			0.526355	−	0.850265i	$-0.323558\pi$
$401$	21.0805			1.05271			0.526355	+	0.850265i	$0.323558\pi$
$402$		0			0
$403$	−4.61231			−0.229756
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8.48255i			0.420465i
$408$		0			0
$409$	23.4381			1.15894			0.579469	−	0.814994i	$-0.303260\pi$
$409$	23.4381			1.15894			0.579469	+	0.814994i	$0.303260\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	53.4372			2.62947
$414$		0			0
$415$	−5.16406			−0.253494
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.681486			−0.0332928			−0.0166464	−	0.999861i	$-0.505299\pi$
$419$	−0.681486			−0.0332928			−0.0166464	+	0.999861i	$0.505299\pi$
$420$		0			0
$421$			12.4676i			0.607633i	0.952731	+	0.303816i	$0.0982610\pi$
$421$			12.4676i			0.607633i	−0.952731	+	0.303816i	$0.901739\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.09518			0.344167
$426$		0			0
$427$	−41.4718			−2.00696
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15.4770			0.745501			0.372751	−	0.927932i	$-0.378415\pi$
$431$	15.4770			0.745501			0.372751	+	0.927932i	$0.378415\pi$
$432$		0			0
$433$		−	21.3471i		−	1.02588i	−0.858426	−	0.512938i	$-0.828557\pi$
$433$		−	21.3471i		−	1.02588i	0.858426	−	0.512938i	$-0.171443\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−17.8275	−	4.30637i	−0.852803	−	0.206001i
$438$		0			0
$439$	−8.69629			−0.415051			−0.207526	−	0.978230i	$-0.566541\pi$
$439$	−8.69629			−0.415051			−0.207526	+	0.978230i	$0.566541\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	14.5706i		−	0.692272i	−0.938184	−	0.346136i	$-0.887494\pi$
$443$		−	14.5706i		−	0.692272i	0.938184	−	0.346136i	$-0.112506\pi$
$444$		0			0
$445$	−3.52407			−0.167057
$446$		0			0
$447$		0			0
$448$		0			0
$449$			32.6200i			1.53943i	0.638386	+	0.769717i	$0.279602\pi$
$449$			32.6200i			1.53943i	−0.638386	+	0.769717i	$0.720398\pi$
$450$		0			0
$451$		−	2.78320i		−	0.131056i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			4.29599i			0.201399i
$456$		0			0
$457$			11.9080i			0.557035i	0.960431	+	0.278517i	$0.0898430\pi$
$457$			11.9080i			0.557035i	−0.960431	+	0.278517i	$0.910157\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			23.5935i			1.09886i	0.835540	+	0.549429i	$0.185155\pi$
$461$			23.5935i			1.09886i	−0.835540	+	0.549429i	$0.814845\pi$
$462$		0			0
$463$	−37.7038			−1.75224			−0.876122	−	0.482090i	$-0.839878\pi$
$463$	−37.7038			−1.75224			−0.876122	+	0.482090i	$0.839878\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.80548			0.314920			0.157460	−	0.987525i	$-0.449669\pi$
$467$	6.80548			0.314920			0.157460	+	0.987525i	$0.449669\pi$
$468$		0			0
$469$	8.00815			0.369782
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5.83191i			0.268152i
$474$		0			0
$475$			3.82420i			0.175466i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.183570			0.00838754			0.00419377	−	0.999991i	$-0.498665\pi$
$479$	0.183570			0.00838754			0.00419377	+	0.999991i	$0.498665\pi$
$480$		0			0
$481$			10.4324i			0.475676i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	7.94278i		−	0.360663i
$486$		0			0
$487$	−16.8484			−0.763476			−0.381738	−	0.924271i	$-0.624674\pi$
$487$	−16.8484			−0.763476			−0.381738	+	0.924271i	$0.624674\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			30.5373i			1.37813i	0.724700	+	0.689065i	$0.241979\pi$
$491$			30.5373i			1.37813i	−0.724700	+	0.689065i	$0.758021\pi$
$492$		0			0
$493$			2.33366i			0.105103i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.701639			−0.0314728
$498$		0			0
$499$	−30.5806			−1.36898			−0.684488	−	0.729024i	$-0.739974\pi$
$499$	−30.5806			−1.36898			−0.684488	+	0.729024i	$0.739974\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	0.671716			0.0299503			0.0149752	−	0.999888i	$-0.495233\pi$
$503$	0.671716			0.0299503			0.0149752	+	0.999888i	$0.495233\pi$
$504$		0			0
$505$			7.35867i			0.327456i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	34.8846i		−	1.54623i	−0.634265	−	0.773116i	$-0.718697\pi$
$509$		−	34.8846i		−	1.54623i	0.634265	−	0.773116i	$-0.281303\pi$
$510$		0			0
$511$			36.2911i			1.60542i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	5.03295i		−	0.221778i
$516$		0			0
$517$		−	0.453025i		−	0.0199240i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−14.5506			−0.637473			−0.318736	−	0.947843i	$-0.603258\pi$
$521$	−14.5506			−0.637473			−0.318736	+	0.947843i	$0.603258\pi$
$522$		0			0
$523$		−	22.7614i		−	0.995289i	−0.867381	−	0.497644i	$-0.834198\pi$
$523$		−	22.7614i		−	0.995289i	0.867381	−	0.497644i	$-0.165802\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−32.7089			−1.42482
$528$		0			0
$529$	−20.4639	−	10.4990i	−0.889734	−	0.456480i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	3.42296i		−	0.148265i
$534$		0			0
$535$	5.80673			0.251047
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.30415			−0.400758
$540$		0			0
$541$	−14.1629			−0.608909			−0.304455	−	0.952527i	$-0.598474\pi$
$541$	−14.1629			−0.608909			−0.304455	+	0.952527i	$0.598474\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	2.93260i		−	0.125619i
$546$		0			0
$547$	22.9864			0.982827			0.491413	−	0.870926i	$-0.336480\pi$
$547$	22.9864			0.982827			0.491413	+	0.870926i	$0.336480\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.25781			−0.0535843
$552$		0			0
$553$	65.4132			2.78165
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.43773			−0.188032			−0.0940162	−	0.995571i	$-0.529971\pi$
$557$	−4.43773			−0.188032			−0.0940162	+	0.995571i	$0.529971\pi$
$558$		0			0
$559$			7.17246i			0.303363i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.5077			0.864295			0.432148	−	0.901803i	$-0.357756\pi$
$563$	20.5077			0.864295			0.432148	+	0.901803i	$0.357756\pi$
$564$		0			0
$565$	10.1479			0.426926
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11.6778			0.489559			0.244779	−	0.969579i	$-0.421284\pi$
$569$	11.6778			0.489559			0.244779	+	0.969579i	$0.421284\pi$
$570$		0			0
$571$			29.3337i			1.22758i	0.789471	+	0.613788i	$0.210355\pi$
$571$			29.3337i			1.22758i	−0.789471	+	0.613788i	$0.789645\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.12608	+	4.66175i	−0.0469609	+	0.194409i
$576$		0			0
$577$	17.6579			0.735109			0.367555	−	0.930002i	$-0.380195\pi$
$577$	17.6579			0.735109			0.367555	+	0.930002i	$0.380195\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	22.1737i		−	0.919919i
$582$		0			0
$583$	2.89517			0.119906
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	12.2214i		−	0.504430i	−0.967671	−	0.252215i	$-0.918841\pi$
$587$		−	12.2214i		−	0.504430i	0.967671	−	0.252215i	$-0.0811590\pi$
$588$		0			0
$589$		−	17.6296i		−	0.726416i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	10.6127i		−	0.435812i	−0.975970	−	0.217906i	$-0.930077\pi$
$593$		−	10.6127i		−	0.435812i	0.975970	−	0.217906i	$-0.0699226\pi$
$594$		0			0
$595$			30.4657i			1.24897i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			2.07408i			0.0847445i	0.999102	+	0.0423722i	$0.0134915\pi$
$599$			2.07408i			0.0847445i	−0.999102	+	0.0423722i	$0.986508\pi$
$600$		0			0
$601$	−40.5425			−1.65376			−0.826882	−	0.562375i	$-0.809888\pi$
$601$	−40.5425			−1.65376			−0.826882	+	0.562375i	$0.809888\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.3382			0.420308
$606$		0			0
$607$	−35.2667			−1.43143			−0.715716	−	0.698391i	$-0.753900\pi$
$607$	−35.2667			−1.43143			−0.715716	+	0.698391i	$0.753900\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	0.557159i		−	0.0225402i
$612$		0			0
$613$		−	12.7368i		−	0.514433i	−0.966354	−	0.257216i	$-0.917195\pi$
$613$		−	12.7368i		−	0.514433i	0.966354	−	0.257216i	$-0.0828053\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	38.9494			1.56804			0.784022	−	0.620733i	$-0.213165\pi$
$617$	38.9494			1.56804			0.784022	+	0.620733i	$0.213165\pi$
$618$		0			0
$619$			23.4023i			0.940618i	0.882502	+	0.470309i	$0.155857\pi$
$619$			23.4023i			0.940618i	−0.882502	+	0.470309i	$0.844143\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	15.1318i		−	0.606243i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			73.9829i			2.94989i
$630$		0			0
$631$		−	13.2157i		−	0.526109i	−0.964781	−	0.263054i	$-0.915270\pi$
$631$		−	13.2157i		−	0.526109i	0.964781	−	0.263054i	$-0.0847298\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.47190			0.177462
$636$		0			0
$637$	−11.4428			−0.453382
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20.4461			0.807572			0.403786	−	0.914854i	$-0.367694\pi$
$641$	20.4461			0.807572			0.403786	+	0.914854i	$0.367694\pi$
$642$		0			0
$643$			26.0848i			1.02869i	0.857585	+	0.514343i	$0.171964\pi$
$643$			26.0848i			1.02869i	−0.857585	+	0.514343i	$0.828036\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	37.0363i		−	1.45605i	−0.685551	−	0.728025i	$-0.740439\pi$
$647$		−	37.0363i		−	1.45605i	0.685551	−	0.728025i	$-0.259561\pi$
$648$		0			0
$649$			10.1241i			0.397405i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			6.48658i			0.253839i	0.991913	+	0.126920i	$0.0405091\pi$
$653$			6.48658i			0.253839i	−0.991913	+	0.126920i	$0.959491\pi$
$654$		0			0
$655$		−	9.97112i		−	0.389604i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	28.7980			1.12181			0.560905	−	0.827880i	$-0.310453\pi$
$659$	28.7980			1.12181			0.560905	+	0.827880i	$0.310453\pi$
$660$		0			0
$661$			23.6161i			0.918561i	0.888291	+	0.459280i	$0.151893\pi$
$661$			23.6161i			0.918561i	−0.888291	+	0.459280i	$0.848107\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−16.4205			−0.636761
$666$		0			0
$667$	−1.53328	−	0.370377i	−0.0593690	−	0.0143410i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	7.85715i		−	0.303322i
$672$		0			0
$673$	−31.8499			−1.22772			−0.613861	−	0.789414i	$-0.710385\pi$
$673$	−31.8499			−1.22772			−0.613861	+	0.789414i	$0.710385\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	35.4831			1.36373			0.681864	−	0.731479i	$-0.261170\pi$
$677$	35.4831			1.36373			0.681864	+	0.731479i	$0.261170\pi$
$678$		0			0
$679$	34.1051			1.30883
$680$		0			0
$681$		0			0
$682$		0			0
$683$			47.5707i			1.82024i	0.414344	+	0.910120i	$0.364011\pi$
$683$			47.5707i			1.82024i	−0.414344	+	0.910120i	$0.635989\pi$
$684$		0			0
$685$	6.36208			0.243083
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.56067			0.135651
$690$		0			0
$691$	34.0730			1.29620			0.648098	−	0.761557i	$-0.275565\pi$
$691$	34.0730			1.29620			0.648098	+	0.761557i	$0.275565\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.2054			−0.500910
$696$		0			0
$697$		−	24.2745i		−	0.919460i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.7857			−0.407372			−0.203686	−	0.979036i	$-0.565292\pi$
$701$	−10.7857			−0.407372			−0.203686	+	0.979036i	$0.565292\pi$
$702$		0			0
$703$	−39.8757			−1.50394
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−31.5970			−1.18833
$708$		0			0
$709$		−	8.57056i		−	0.321874i	−0.986965	−	0.160937i	$-0.948548\pi$
$709$		−	8.57056i		−	0.321874i	0.986965	−	0.160937i	$-0.0514516\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.19126	−	21.4908i	0.194414	−	0.804835i
$714$		0			0
$715$	−0.813908			−0.0304384
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	0.199893i		−	0.00745476i	−0.999993	−	0.00372738i	$-0.998814\pi$
$719$		−	0.199893i		−	0.00745476i	0.999993	−	0.00372738i	$-0.00118646\pi$
$720$		0			0
$721$	21.6107			0.804826
$722$		0			0
$723$		0			0
$724$		0			0
$725$			0.328907i			0.0122153i
$726$		0			0
$727$		−	16.7842i		−	0.622492i	−0.950329	−	0.311246i	$-0.899254\pi$
$727$		−	16.7842i		−	0.622492i	0.950329	−	0.311246i	$-0.100746\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			50.8646i			1.88129i
$732$		0			0
$733$			20.7005i			0.764592i	0.924040	+	0.382296i	$0.124866\pi$
$733$			20.7005i			0.764592i	−0.924040	+	0.382296i	$0.875134\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.51721i			0.0558870i
$738$		0			0
$739$	29.0959			1.07031			0.535155	−	0.844754i	$-0.320253\pi$
$739$	29.0959			1.07031			0.535155	+	0.844754i	$0.320253\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−36.6338			−1.34396			−0.671982	−	0.740567i	$-0.734557\pi$
$743$	−36.6338			−1.34396			−0.671982	+	0.740567i	$0.734557\pi$
$744$		0			0
$745$	−13.6514			−0.500148
$746$		0			0
$747$		0			0
$748$		0			0
$749$			24.9332i			0.911040i
$750$		0			0
$751$			6.74192i			0.246016i	0.992406	+	0.123008i	$0.0392541\pi$
$751$			6.74192i			0.246016i	−0.992406	+	0.123008i	$0.960746\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.28358			−0.0831082
$756$		0			0
$757$		−	17.6311i		−	0.640812i	−0.947280	−	0.320406i	$-0.896181\pi$
$757$		−	17.6311i		−	0.640812i	0.947280	−	0.320406i	$-0.103819\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	28.4539i		−	1.03145i	−0.856753	−	0.515726i	$-0.827522\pi$
$761$		−	28.4539i		−	1.03145i	0.856753	−	0.515726i	$-0.172478\pi$
$762$		0			0
$763$	12.5922			0.455867
$764$		0			0
$765$		0			0
$766$		0			0
$767$			12.4512i			0.449588i
$768$		0			0
$769$		−	51.5377i		−	1.85850i	−0.369454	−	0.929249i	$-0.620455\pi$
$769$		−	51.5377i		−	1.85850i	0.369454	−	0.929249i	$-0.379545\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	38.8535			1.39746			0.698732	−	0.715384i	$-0.253748\pi$
$773$	38.8535			1.39746			0.698732	+	0.715384i	$0.253748\pi$
$774$		0			0
$775$	−4.61002			−0.165597
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.0836			0.468768
$780$		0			0
$781$		−	0.132931i		−	0.00475664i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	23.1675i		−	0.826884i
$786$		0			0
$787$			20.4094i			0.727517i	0.931493	+	0.363759i	$0.118507\pi$
$787$			20.4094i			0.727517i	−0.931493	+	0.363759i	$0.881493\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			43.5737i			1.54930i
$792$		0			0
$793$		−	9.66323i		−	0.343151i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	20.8255			0.737675			0.368838	−	0.929494i	$-0.379756\pi$
$797$	20.8255			0.737675			0.368838	+	0.929494i	$0.379756\pi$
$798$		0			0
$799$		−	3.95118i		−	0.139783i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.87562			−0.242636
$804$		0			0
$805$	−20.0169	−	4.83523i	−0.705502	−	0.170420i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			6.85576i			0.241036i	0.992711	+	0.120518i	$0.0384555\pi$
$809$			6.85576i			0.241036i	−0.992711	+	0.120518i	$0.961545\pi$
$810$		0			0
$811$	−7.87797			−0.276633			−0.138317	−	0.990388i	$-0.544169\pi$
$811$	−7.87797			−0.276633			−0.138317	+	0.990388i	$0.544169\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.2846			−0.465339
$816$		0			0
$817$	−27.4153			−0.959139
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	15.2914i		−	0.533673i	−0.963742	−	0.266836i	$-0.914022\pi$
$821$		−	15.2914i		−	0.533673i	0.963742	−	0.266836i	$-0.0859784\pi$
$822$		0			0
$823$	5.65471			0.197111			0.0985555	−	0.995132i	$-0.468578\pi$
$823$	5.65471			0.197111			0.0985555	+	0.995132i	$0.468578\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−21.9187			−0.762190			−0.381095	−	0.924536i	$-0.624453\pi$
$827$	−21.9187			−0.762190			−0.381095	+	0.924536i	$0.624453\pi$
$828$		0			0
$829$	−36.2401			−1.25867			−0.629335	−	0.777134i	$-0.716672\pi$
$829$	−36.2401			−1.25867			−0.629335	+	0.777134i	$0.716672\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−81.1486			−2.81163
$834$		0			0
$835$			4.37549i			0.151420i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	26.2613			0.906641			0.453320	−	0.891348i	$-0.350239\pi$
$839$	26.2613			0.906641			0.453320	+	0.891348i	$0.350239\pi$
$840$		0			0
$841$	28.8918			0.996270
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.9990			0.412778
$846$		0			0
$847$			44.3907i			1.52528i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−48.6090	−	11.7419i	−1.66630	−	0.402507i
$852$		0			0
$853$	2.65148			0.0907848			0.0453924	−	0.998969i	$-0.485546\pi$
$853$	2.65148			0.0907848			0.0453924	+	0.998969i	$0.485546\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	29.3275i		−	1.00181i	−0.865503	−	0.500904i	$-0.833001\pi$
$857$		−	29.3275i		−	1.00181i	0.865503	−	0.500904i	$-0.166999\pi$
$858$		0			0
$859$	3.66128			0.124921			0.0624607	−	0.998047i	$-0.480105\pi$
$859$	3.66128			0.124921			0.0624607	+	0.998047i	$0.480105\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	32.2061i		−	1.09631i	−0.836378	−	0.548154i	$-0.815331\pi$
$863$		−	32.2061i		−	1.09631i	0.836378	−	0.548154i	$-0.184669\pi$
$864$		0			0
$865$			25.6230i			0.871207i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			12.3930i			0.420405i
$870$		0			0
$871$			1.86596i			0.0632256i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			4.29385i			0.145159i
$876$		0			0
$877$	−41.4007			−1.39800			−0.699001	−	0.715121i	$-0.746372\pi$
$877$	−41.4007			−1.39800			−0.699001	+	0.715121i	$0.746372\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	24.6009			0.828826			0.414413	−	0.910089i	$-0.363987\pi$
$881$	24.6009			0.828826			0.414413	+	0.910089i	$0.363987\pi$
$882$		0			0
$883$	18.6407			0.627308			0.313654	−	0.949537i	$-0.398447\pi$
$883$	18.6407			0.627308			0.313654	+	0.949537i	$0.398447\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			16.7054i			0.560911i	0.959867	+	0.280456i	$0.0904856\pi$
$887$			16.7054i			0.560911i	−0.959867	+	0.280456i	$0.909514\pi$
$888$		0			0
$889$			19.2017i			0.644003i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.12963			0.0712652
$894$		0			0
$895$		−	13.7392i		−	0.459251i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1.51627i		−	0.0505703i
$900$		0			0
$901$	25.2510			0.841233
$902$		0			0
$903$		0			0
$904$		0			0
$905$			22.6528i			0.753004i
$906$		0			0
$907$		−	18.5243i		−	0.615088i	−0.951534	−	0.307544i	$-0.900493\pi$
$907$		−	18.5243i		−	0.615088i	0.951534	−	0.307544i	$-0.0995071\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−57.9245			−1.91913			−0.959563	−	0.281495i	$-0.909170\pi$
$911$	−57.9245			−1.91913			−0.959563	+	0.281495i	$0.909170\pi$
$912$		0			0
$913$	4.20097			0.139032
$914$		0			0
$915$		0			0
$916$		0			0
$917$	42.8145			1.41386
$918$		0			0
$919$		−	12.3244i		−	0.406545i	−0.979122	−	0.203272i	$-0.934842\pi$
$919$		−	12.3244i		−	0.406545i	0.979122	−	0.203272i	$-0.0651577\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	0.163487i		−	0.00538124i
$924$		0			0
$925$			10.4272i			0.342844i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	10.4788i		−	0.343799i	−0.985114	−	0.171900i	$-0.945010\pi$
$929$		−	10.4788i		−	0.343799i	0.985114	−	0.171900i	$-0.0549904\pi$
$930$		0			0
$931$		−	43.7379i		−	1.43345i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.77195			−0.188763
$936$		0			0
$937$			20.8551i			0.681307i	0.940189	+	0.340654i	$0.110648\pi$
$937$			20.8551i			0.681307i	−0.940189	+	0.340654i	$0.889352\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	7.47330			0.243623			0.121811	−	0.992553i	$-0.461130\pi$
$941$	7.47330			0.243623			0.121811	+	0.992553i	$0.461130\pi$
$942$		0			0
$943$	15.9491	+	3.85262i	0.519373	+	0.125459i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	58.1883i		−	1.89087i	−0.325815	−	0.945433i	$-0.605639\pi$
$947$		−	58.1883i		−	1.89087i	0.325815	−	0.945433i	$-0.394361\pi$
$948$		0			0
$949$	−8.45609			−0.274496
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.55884			−0.309641			−0.154821	−	0.987943i	$-0.549480\pi$
$953$	−9.55884			−0.309641			−0.154821	+	0.987943i	$0.549480\pi$
$954$		0			0
$955$	−20.9372			−0.677512
$956$		0			0
$957$		0			0
$958$		0			0
$959$			27.3178i			0.882138i
$960$		0			0
$961$	−9.74775			−0.314444
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.0000			0.354104
$966$		0			0
$967$	35.7262			1.14888			0.574438	−	0.818548i	$-0.305221\pi$
$967$	35.7262			1.14888			0.574438	+	0.818548i	$0.305221\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.90868			0.189619			0.0948093	−	0.995495i	$-0.469776\pi$
$971$	5.90868			0.189619			0.0948093	+	0.995495i	$0.469776\pi$
$972$		0			0
$973$		−	56.7020i		−	1.81778i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.19574			−0.166227			−0.0831133	−	0.996540i	$-0.526486\pi$
$977$	−5.19574			−0.166227			−0.0831133	+	0.996540i	$0.526486\pi$
$978$		0			0
$979$	2.86684			0.0916245
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.21814			−0.0388527			−0.0194264	−	0.999811i	$-0.506184\pi$
$983$	−1.21814			−0.0388527			−0.0194264	+	0.999811i	$0.506184\pi$
$984$		0			0
$985$		−	11.7476i		−	0.374310i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−33.4196	−	8.07277i	−1.06268	−	0.256699i
$990$		0			0
$991$	−16.4780			−0.523440			−0.261720	−	0.965144i	$-0.584290\pi$
$991$	−16.4780			−0.523440			−0.261720	+	0.965144i	$0.584290\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	25.4484i		−	0.806768i
$996$		0			0
$997$	−44.8877			−1.42161			−0.710804	−	0.703390i	$-0.751669\pi$
$997$	−44.8877			−1.42161			−0.710804	+	0.703390i	$0.751669\pi$
$998$		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.1241.3	✓	48
3.2	odd	2		8280.2.p.b.1241.3	yes	48
23.22	odd	2		8280.2.p.b.1241.46	yes	48
69.68	even	2	inner	8280.2.p.a.1241.46	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.3	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.46	yes	48	69.68	even	2	inner
8280.2.p.b.1241.3	yes	48	3.2	odd	2
8280.2.p.b.1241.46	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -4.29385i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -4.29385i q^{7} +0.813503 q^{11} +1.00050 q^{13} +7.09518 q^{17} +3.82420i q^{19} +(-1.12608 + 4.66175i) q^{23} +1.00000 q^{25} +0.328907i q^{29} -4.61002 q^{31} +4.29385i q^{35} +10.4272i q^{37} -3.42126i q^{41} +7.16889i q^{43} -0.556882i q^{47} -11.4371 q^{49} +3.55889 q^{53} -0.813503 q^{55} +12.4450i q^{59} -9.65842i q^{61} -1.00050 q^{65} +1.86503i q^{67} -0.163405i q^{71} -8.45188 q^{73} -3.49306i q^{77} +15.2342i q^{79} +5.16406 q^{83} -7.09518 q^{85} +3.52407 q^{89} -4.29599i q^{91} -3.82420i q^{95} +7.94278i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 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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