LMFDB - Embedded newform 352.2.s.a.79.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [352,2,Mod(79,352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(352, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 1]))

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

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

chi := DirichletCharacter("352.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 352 = 2^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	352.s (of order $10$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.81073415115$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.64000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			79.2
Root			$1.34500 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	352.79
Dual form			352.2.s.a.303.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.67625 - 1.94441i) q^{3} +(2.45454 - 7.55429i) q^{9} +(-3.25830 + 0.619233i) q^{11} +(0.998400 - 0.324400i) q^{17} +(4.72597 + 6.50475i) q^{19} +(-4.04508 + 2.93893i) q^{25} +(-5.05297 - 15.5514i) q^{27} +(-7.51600 + 7.99270i) q^{33} +(1.29913 + 1.78810i) q^{41} +6.88847i q^{43} +(-2.16312 - 6.65740i) q^{49} +(2.04120 - 2.80947i) q^{51} +(25.2958 + 8.21910i) q^{57} +(-2.79793 - 2.03282i) q^{59} +16.3138 q^{67} +(-4.20049 + 5.78148i) q^{73} +(-5.11118 + 15.7306i) q^{75} +(-24.4832 - 17.7881i) q^{81} +(-15.4499 + 5.01998i) q^{83} +0.182318 q^{89} +(-4.03261 + 12.4111i) q^{97} +(-3.31977 + 26.1341i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 2 q^{9} - 6 q^{11} + 10 q^{19} - 10 q^{25} - 38 q^{27} - 38 q^{33} + 14 q^{49} + 70 q^{51} + 70 q^{57} + 18 q^{59} + 28 q^{67} - 30 q^{75} - 8 q^{81} - 90 q^{83} - 36 q^{89} + 30 q^{97}+ \cdots - 6 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 - 2 * q^9 - 6 * q^11 + 10 * q^19 - 10 * q^25 - 38 * q^27 - 38 * q^33 + 14 * q^49 + 70 * q^51 + 70 * q^57 + 18 * q^59 + 28 * q^67 - 30 * q^75 - 8 * q^81 - 90 * q^83 - 36 * q^89 + 30 * q^97 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/352\mathbb{Z}\right)^\times$.

$n$	$133$	$287$	$321$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{10}\right)$

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
$3$	2.67625	−	1.94441i	1.54513	−	1.12261i	0.598123	−	0.801404i	$-0.295913\pi$
$3$	2.67625	−	1.94441i	1.54513	−	1.12261i	0.947011	−	0.321202i	$-0.104087\pi$
$4$		0			0
$5$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$5$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$6$		0			0
$7$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$7$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$8$		0			0
$9$	2.45454	−	7.55429i	0.818179	−	2.51810i
$10$		0			0
$11$	−3.25830	+	0.619233i	−0.982416	+	0.186706i
$11$	−3.25830	+	0.619233i	−0.982416	+	0.186706i
$12$		0			0
$13$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$13$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.998400	−	0.324400i	0.242148	−	0.0786785i	−0.185429	−	0.982658i	$-0.559367\pi$
$17$	0.998400	−	0.324400i	0.242148	−	0.0786785i	0.427576	+	0.903979i	$0.359367\pi$
$18$		0			0
$19$	4.72597	+	6.50475i	1.08421	+	1.49229i	0.854797	+	0.518962i	$0.173682\pi$
$19$	4.72597	+	6.50475i	1.08421	+	1.49229i	0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−4.04508	+	2.93893i	−0.809017	+	0.587785i
$26$		0			0
$27$	−5.05297	−	15.5514i	−0.972444	−	2.99288i
$28$		0			0
$29$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$29$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$		0			0
$33$	−7.51600	+	7.99270i	−1.30837	+	1.39135i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$37$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.29913	+	1.78810i	0.202890	+	0.279254i	0.898322	−	0.439338i	$-0.144787\pi$
$41$	1.29913	+	1.78810i	0.202890	+	0.279254i	−0.695432	+	0.718592i	$0.744787\pi$
$42$		0			0
$43$			6.88847i			1.05048i	0.850954	+	0.525241i	$0.176025\pi$
$43$			6.88847i			1.05048i	−0.850954	+	0.525241i	$0.823975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$47$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$48$		0			0
$49$	−2.16312	−	6.65740i	−0.309017	−	0.951057i
$50$		0			0
$51$	2.04120	−	2.80947i	0.285825	−	0.393405i
$52$		0			0
$53$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$53$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	25.2958	+	8.21910i	3.35051	+	1.08865i
$58$		0			0
$59$	−2.79793	−	2.03282i	−0.364260	−	0.264650i	0.390567	−	0.920575i	$-0.372279\pi$
$59$	−2.79793	−	2.03282i	−0.364260	−	0.264650i	−0.754827	+	0.655924i	$0.772279\pi$
$60$		0			0
$61$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$61$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	16.3138			1.99304			0.996522	−	0.0833352i	$-0.0265572\pi$
$67$	16.3138			1.99304			0.996522	+	0.0833352i	$0.0265572\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$71$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$72$		0			0
$73$	−4.20049	+	5.78148i	−0.491631	+	0.676671i	−0.980688	−	0.195580i	$-0.937341\pi$
$73$	−4.20049	+	5.78148i	−0.491631	+	0.676671i	0.489057	+	0.872252i	$0.337341\pi$
$74$		0			0
$75$	−5.11118	+	15.7306i	−0.590189	+	1.81641i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$79$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$80$		0			0
$81$	−24.4832	−	17.7881i	−2.72035	−	1.97645i
$82$		0			0
$83$	−15.4499	+	5.01998i	−1.69585	+	0.551014i	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−15.4499	+	5.01998i	−1.69585	+	0.551014i	−0.707968	+	0.706244i	$0.750388\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.182318			0.0193256			0.00966282	−	0.999953i	$-0.496924\pi$
$89$	0.182318			0.0193256			0.00966282	+	0.999953i	$0.496924\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.03261	+	12.4111i	−0.409449	+	1.26016i	0.507673	+	0.861550i	$0.330506\pi$
$97$	−4.03261	+	12.4111i	−0.409449	+	1.26016i	−0.917122	+	0.398606i	$0.869494\pi$
$98$		0			0
$99$	−3.31977	+	26.1341i	−0.333649	+	2.62658i
$100$		0			0
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$		0			0
$103$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$103$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.95030	−	9.56627i	−0.671911	−	0.924806i	0.327891	−	0.944716i	$-0.393662\pi$
$107$	−6.95030	−	9.56627i	−0.671911	−	0.924806i	−0.999802	+	0.0199092i	$0.993662\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.40116	+	4.65072i	−0.602171	+	0.437503i	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−6.40116	+	4.65072i	−0.602171	+	0.437503i	0.244478	+	0.969655i	$0.421383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.2331	−	4.03530i	0.930282	−	0.366845i
$122$		0			0
$123$	6.95359	+	2.25936i	0.626984	+	0.203720i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$127$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$128$		0			0
$129$	13.3940	+	18.4353i	1.17928	+	1.62314i
$130$		0			0
$131$		−	22.0214i		−	1.92402i	−0.273022	−	0.962008i	$-0.588023\pi$
$131$		−	22.0214i		−	1.92402i	0.273022	−	0.962008i	$-0.411977\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.07708	−	18.7033i	−0.519200	−	1.59793i	−0.775507	−	0.631338i	$-0.782506\pi$
$137$	−6.07708	−	18.7033i	−0.519200	−	1.59793i	0.256307	−	0.966595i	$-0.417494\pi$
$138$		0			0
$139$	−4.98752	+	6.86474i	−0.423036	+	0.582259i	−0.966337	−	0.257279i	$-0.917174\pi$
$139$	−4.98752	+	6.86474i	−0.423036	+	0.582259i	0.543301	+	0.839538i	$0.317174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−18.7338	−	13.6109i	−1.54513	−	1.12261i
$148$		0			0
$149$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$149$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$150$		0			0
$151$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$151$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$152$		0			0
$153$		−	8.33845i		−	0.674124i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$157$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.12369	−	12.6914i	0.322992	−	0.994068i	−0.649347	−	0.760493i	$-0.724958\pi$
$163$	4.12369	−	12.6914i	0.322992	−	0.994068i	0.972339	−	0.233575i	$-0.0750425\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$167$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$168$		0			0
$169$	10.5172	+	7.64121i	0.809017	+	0.587785i
$170$		0			0
$171$	60.7388	−	19.7352i	4.64481	−	1.50919i
$172$		0			0
$173$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$173$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−11.4406			−0.859929
$178$		0			0
$179$	21.1961	−	15.3999i	1.58427	−	1.15104i	0.672692	−	0.739923i	$-0.265138\pi$
$179$	21.1961	−	15.3999i	1.58427	−	1.15104i	0.911582	−	0.411119i	$-0.134862\pi$
$180$		0			0
$181$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$181$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.05221	+	1.67524i	−0.223200	+	0.122505i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$191$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$192$		0			0
$193$	16.1400	−	5.24419i	1.16178	−	0.377485i	0.336211	−	0.941787i	$-0.390854\pi$
$193$	16.1400	−	5.24419i	1.16178	−	0.377485i	0.825569	+	0.564301i	$0.190854\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	43.6597	−	31.7206i	3.07952	−	2.23740i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−19.4266	−	18.2680i	−1.34377	−	1.26362i
$210$		0			0
$211$	−20.1444	−	6.54531i	−1.38680	−	0.450598i	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−20.1444	−	6.54531i	−1.38680	−	0.450598i	−0.904898	+	0.425628i	$0.860053\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$			23.6402i			1.59746i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$223$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$224$		0			0
$225$	12.2727	+	37.7714i	0.818179	+	2.51810i
$226$		0			0
$227$	−11.7097	+	16.1171i	−0.777203	+	1.06973i	0.218382	+	0.975863i	$0.429922\pi$
$227$	−11.7097	+	16.1171i	−0.777203	+	1.06973i	−0.995585	+	0.0938647i	$0.970078\pi$
$228$		0			0
$229$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$229$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−28.7978	−	9.35696i	−1.88660	−	0.612995i	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−28.7978	−	9.35696i	−1.88660	−	0.612995i	−0.903921	−	0.427698i	$-0.859325\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$239$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$240$		0			0
$241$		−	29.0119i		−	1.86882i	−0.356199	−	0.934410i	$-0.615928\pi$
$241$		−	29.0119i		−	1.86882i	0.356199	−	0.934410i	$-0.384072\pi$
$242$		0			0
$243$	−51.0552			−3.27520
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−31.5869	+	43.4757i	−2.00174	+	2.75516i
$250$		0			0
$251$	−1.85410	+	5.70634i	−0.117030	+	0.360181i	−0.992365	−	0.123336i	$-0.960641\pi$
$251$	−1.85410	+	5.70634i	−0.117030	+	0.360181i	0.875335	+	0.483517i	$0.160641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.2050	−	11.7736i	−1.01084	−	0.734419i	−0.0464552	−	0.998920i	$-0.514792\pi$
$257$	−16.2050	−	11.7736i	−1.01084	−	0.734419i	−0.964385	+	0.264502i	$0.914792\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.487928	−	0.354501i	0.0298607	−	0.0216951i
$268$		0			0
$269$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$269$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.3602	−	12.0808i	0.685048	−	0.728498i
$276$		0			0
$277$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$277$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	24.5937	−	7.99097i	1.46714	−	0.476702i	0.536895	−	0.843649i	$-0.319597\pi$
$281$	24.5937	−	7.99097i	1.46714	−	0.476702i	0.930242	+	0.366947i	$0.119597\pi$
$282$		0			0
$283$	14.9626	+	20.5942i	0.889432	+	1.22420i	0.973718	+	0.227757i	$0.0731392\pi$
$283$	14.9626	+	20.5942i	0.889432	+	1.22420i	−0.0842855	+	0.996442i	$0.526861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−12.8617	+	9.34459i	−0.756572	+	0.549682i
$290$		0			0
$291$	13.3400	+	41.0562i	0.782004	+	2.40676i
$292$		0			0
$293$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$293$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	26.0941	+	47.5424i	1.51413	+	2.75869i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	34.9580i		−	1.99516i	−0.0695319	−	0.997580i	$-0.522151\pi$
$307$		−	34.9580i		−	1.99516i	0.0695319	−	0.997580i	$-0.477849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$311$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$312$		0			0
$313$	−3.66492	−	11.2795i	−0.207153	−	0.637553i	−0.999618	−	0.0276348i	$-0.991202\pi$
$313$	−3.66492	−	11.2795i	−0.207153	−	0.637553i	0.792465	−	0.609918i	$-0.208798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$317$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−37.2015	−	12.0875i	−2.07639	−	0.674659i
$322$		0			0
$323$	6.82855	+	4.96123i	0.379951	+	0.276050i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−16.1755			−0.889086			−0.444543	−	0.895757i	$-0.646634\pi$
$331$	−16.1755			−0.889086			−0.444543	+	0.895757i	$0.646634\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	11.3031	−	15.5574i	0.615718	−	0.847463i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	11.3031	−	15.5574i	0.615718	−	0.847463i	0.997032	+	0.0769821i	$0.0245284\pi$
$338$		0			0
$339$	−8.08822	+	24.8930i	−0.439292	+	1.35200i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−24.9372	+	8.10257i	−1.33870	+	0.434969i	−0.888875	−	0.458149i	$-0.848512\pi$
$347$	−24.9372	+	8.10257i	−1.33870	+	0.434969i	−0.449822	+	0.893118i	$0.648512\pi$
$348$		0			0
$349$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$349$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	37.5706			1.99968			0.999840	−	0.0178943i	$-0.00569624\pi$
$353$	37.5706			1.99968			0.999840	+	0.0178943i	$0.00569624\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$359$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$360$		0			0
$361$	−14.1056	+	43.4124i	−0.742398	+	2.28487i
$362$		0			0
$363$	19.5401	−	30.6968i	1.02559	−	1.61117i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$367$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$368$		0			0
$369$	16.6966	−	5.42505i	0.869189	−	0.282417i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.13492	+	3.49291i	0.0582967	+	0.179419i	0.975964	−	0.217930i	$-0.0699304\pi$
$379$	1.13492	+	3.49291i	0.0582967	+	0.179419i	−0.917668	+	0.397349i	$0.869930\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$383$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	52.0375	+	16.9080i	2.64521	+	0.859482i
$388$		0			0
$389$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$389$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−42.8186	−	58.9347i	−2.15991	−	2.97286i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.2105	+	37.5801i	0.609763	+	1.87666i	0.459951	+	0.887945i	$0.347867\pi$
$401$	12.2105	+	37.5801i	0.609763	+	1.87666i	0.149813	+	0.988714i	$0.452133\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	32.2799	+	10.4884i	1.59614	+	0.518617i	0.966149	−	0.257985i	$-0.0830586\pi$
$409$	32.2799	+	10.4884i	1.59614	+	0.518617i	0.629991	+	0.776603i	$0.283059\pi$
$410$		0			0
$411$	−52.6307	−	38.2385i	−2.59608	−	1.88617i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			28.0695i			1.37457i
$418$		0			0
$419$	29.4076			1.43666			0.718328	−	0.695705i	$-0.244908\pi$
$419$	29.4076			1.43666			0.718328	+	0.695705i	$0.244908\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.08523	+	4.24645i	−0.149655	+	0.205983i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$431$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$432$		0			0
$433$	16.8013	+	12.2069i	0.807421	+	0.586626i	0.913082	−	0.407777i	$-0.133696\pi$
$433$	16.8013	+	12.2069i	0.807421	+	0.586626i	−0.105661	+	0.994402i	$0.533696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−55.6013			−2.64768
$442$		0			0
$443$	−8.32375	+	6.04756i	−0.395473	+	0.287328i	−0.767695	−	0.640816i	$-0.778596\pi$
$443$	−8.32375	+	6.04756i	−0.395473	+	0.287328i	0.372221	+	0.928144i	$0.378596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.47251	+	29.1534i	−0.447036	+	1.37583i	0.433200	+	0.901298i	$0.357384\pi$
$449$	−9.47251	+	29.1534i	−0.447036	+	1.37583i	−0.880236	+	0.474536i	$0.842616\pi$
$450$		0			0
$451$	−5.34021	−	5.02171i	−0.251461	−	0.236463i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−33.4923	+	10.8823i	−1.56670	+	0.509052i	−0.958588	−	0.284797i	$-0.908074\pi$
$457$	−33.4923	+	10.8823i	−1.56670	+	0.509052i	−0.608114	+	0.793849i	$0.708074\pi$
$458$		0			0
$459$	−10.0898	−	13.8874i	−0.470950	−	0.648207i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.27051	−	28.5317i	−0.428988	−	1.32029i	−0.899123	−	0.437695i	$-0.855795\pi$
$467$	−9.27051	−	28.5317i	−0.428988	−	1.32029i	0.470135	−	0.882594i	$-0.344205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.26557	−	22.4447i	−0.196131	−	1.03201i
$474$		0			0
$475$	−38.2339	−	12.4230i	−1.75429	−	0.570004i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$479$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$487$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$488$		0			0
$489$	−13.6413	−	41.9835i	−0.616880	−	1.89856i
$490$		0			0
$491$	−26.0474	+	35.8512i	−1.17550	+	1.61794i	−0.579149	+	0.815222i	$0.696615\pi$
$491$	−26.0474	+	35.8512i	−1.17550	+	1.61794i	−0.596355	+	0.802721i	$0.703385\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	29.1438	+	21.1742i	1.30465	+	0.947887i	0.999990	−	0.00457310i	$-0.00145567\pi$
$499$	29.1438	+	21.1742i	1.30465	+	0.947887i	0.304664	+	0.952460i	$0.401456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$503$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	43.0044			1.90989
$508$		0			0
$509$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$509$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	77.2779	−	106.364i	3.41190	−	4.69608i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	17.5929	+	12.7820i	0.770759	+	0.559989i	0.902191	−	0.431336i	$-0.141958\pi$
$521$	17.5929	+	12.7820i	0.770759	+	0.559989i	−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	−1.65639	+	0.538193i	−0.0724288	+	0.0235335i	−0.345007	−	0.938600i	$-0.612124\pi$
$523$	−1.65639	+	0.538193i	−0.0724288	+	0.0235335i	0.272578	+	0.962134i	$0.412124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	−22.2241	+	16.1468i	−0.964445	+	0.700710i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	26.7825	−	82.4279i	1.15575	−	3.55703i
$538$		0			0
$539$	11.1706	+	20.3523i	0.481151	+	0.876638i
$540$		0			0
$541$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$541$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	19.9276	+	27.4280i	0.852043	+	1.17274i	0.983409	+	0.181402i	$0.0580636\pi$
$547$	19.9276	+	27.4280i	0.852043	+	1.17274i	−0.131366	+	0.991334i	$0.541936\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$557$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−4.91114	+	10.4181i	−0.207349	+	0.439853i
$562$		0			0
$563$	37.9416	+	12.3280i	1.59905	+	0.519562i	0.966871	−	0.255264i	$-0.0821625\pi$
$563$	37.9416	+	12.3280i	1.59905	+	0.519562i	0.632175	+	0.774826i	$0.282163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	25.2706	+	34.7820i	1.05940	+	1.45814i	0.880366	+	0.474295i	$0.157297\pi$
$569$	25.2706	+	34.7820i	1.05940	+	1.45814i	0.179034	+	0.983843i	$0.442703\pi$
$570$		0			0
$571$		−	42.4264i		−	1.77549i	−0.460336	−	0.887745i	$-0.652271\pi$
$571$		−	42.4264i		−	1.77549i	0.460336	−	0.887745i	$-0.347729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−14.6649	−	45.1340i	−0.610509	−	1.87895i	−0.453218	−	0.891400i	$-0.649724\pi$
$577$	−14.6649	−	45.1340i	−0.610509	−	1.87895i	−0.157290	−	0.987552i	$-0.550276\pi$
$578$		0			0
$579$	32.9977	−	45.4175i	1.37134	−	1.88749i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	26.7920	+	19.4655i	1.10582	+	0.803428i	0.982001	−	0.188876i	$-0.0604844\pi$
$587$	26.7920	+	19.4655i	1.10582	+	0.803428i	0.123823	+	0.992304i	$0.460484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			47.1921i			1.93795i	0.247167	+	0.968973i	$0.420500\pi$
$593$			47.1921i			1.93795i	−0.247167	+	0.968973i	$0.579500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$599$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$600$		0			0
$601$	−28.7972	+	39.6360i	−1.17466	+	1.61679i	−0.555566	+	0.831472i	$0.687498\pi$
$601$	−28.7972	+	39.6360i	−1.17466	+	1.61679i	−0.619098	+	0.785314i	$0.712502\pi$
$602$		0			0
$603$	40.0427	−	123.239i	1.63067	−	5.01867i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$607$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$613$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.995401			0.0400733			0.0200367	−	0.999799i	$-0.493622\pi$
$617$	0.995401			0.0400733			0.0200367	+	0.999799i	$0.493622\pi$
$618$		0			0
$619$	−37.1922	+	27.0217i	−1.49488	+	1.08609i	−0.522514	+	0.852631i	$0.675006\pi$
$619$	−37.1922	+	27.0217i	−1.49488	+	1.08609i	−0.972366	+	0.233463i	$0.924994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.72542	−	23.7764i	0.309017	−	0.951057i
$626$		0			0
$627$	−87.5109	−	11.1163i	−3.49485	−	0.443944i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$631$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$632$		0			0
$633$	−66.6383	+	21.6521i	−2.64863	+	0.860593i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	32.2625	−	23.4401i	1.27429	−	0.925828i	0.274928	−	0.961465i	$-0.411346\pi$
$641$	32.2625	−	23.4401i	1.27429	−	0.925828i	0.999365	+	0.0356372i	$0.0113461\pi$
$642$		0			0
$643$	−7.26834	−	22.3696i	−0.286635	−	0.882173i	−0.985904	−	0.167313i	$-0.946491\pi$
$643$	−7.26834	−	22.3696i	−0.286635	−	0.882173i	0.699269	−	0.714859i	$-0.253509\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$647$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$648$		0			0
$649$	10.3753	+	4.89097i	0.407267	+	0.191987i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$653$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	33.3647	+	45.9226i	1.30168	+	1.79161i
$658$		0			0
$659$			28.3200i			1.10319i	0.834111	+	0.551596i	$0.185981\pi$
$659$			28.3200i			1.10319i	−0.834111	+	0.551596i	$0.814019\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−33.5823	−	10.9116i	−1.29450	−	0.420610i	−0.420838	−	0.907136i	$-0.638264\pi$
$673$	−33.5823	−	10.9116i	−1.29450	−	0.420610i	−0.873666	+	0.486526i	$0.838264\pi$
$674$		0			0
$675$	66.1442	+	48.0566i	2.54589	+	1.84970i
$676$		0			0
$677$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$677$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			65.9019i			2.52537i
$682$		0			0
$683$	42.0000			1.60709			0.803543	−	0.595247i	$-0.202946\pi$
$683$	42.0000			1.60709			0.803543	+	0.595247i	$0.202946\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−6.87631	+	21.1631i	−0.261587	+	0.805083i	0.730873	+	0.682514i	$0.239113\pi$
$691$	−6.87631	+	21.1631i	−0.261587	+	0.805083i	−0.992460	+	0.122569i	$0.960887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.87711	+	1.36380i	0.0711006	+	0.0516576i
$698$		0			0
$699$	−95.2638	+	30.9531i	−3.60321	+	1.17075i
$700$		0			0
$701$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$701$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$709$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−56.4110	−	77.6431i	−2.09795	−	2.88758i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−63.1871	+	45.9081i	−2.34026	+	1.70030i
$730$		0			0
$731$	2.23462	+	6.87745i	0.0826503	+	0.254372i
$732$		0			0
$733$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$733$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−53.1552	+	10.1020i	−1.95800	+	0.372113i
$738$		0			0
$739$	−13.6372	−	4.43099i	−0.501652	−	0.162997i	0.0472504	−	0.998883i	$-0.484954\pi$
$739$	−13.6372	−	4.43099i	−0.501652	−	0.162997i	−0.548902	+	0.835886i	$0.684954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$743$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			129.035i			4.72113i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$751$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$752$		0			0
$753$	6.13342	+	18.8767i	0.223514	+	0.687906i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$757$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	38.8919	+	12.6368i	1.40983	+	0.458082i	0.912356	−	0.409397i	$-0.134261\pi$
$761$	38.8919	+	12.6368i	1.40983	+	0.458082i	0.497475	+	0.867479i	$0.334261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	50.9117i		−	1.83592i	−0.396670	−	0.917961i	$-0.629834\pi$
$769$		−	50.9117i		−	1.83592i	0.396670	−	0.917961i	$-0.370166\pi$
$770$		0			0
$771$	−66.2614			−2.38635
$772$		0			0
$773$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$773$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.49147	+	16.9010i	−0.196752	+	0.605542i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	47.5371	−	15.4457i	1.69451	−	0.550581i	0.706877	−	0.707336i	$-0.250103\pi$
$787$	47.5371	−	15.4457i	1.69451	−	0.550581i	0.987638	+	0.156755i	$0.0501033\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$797$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	0.447506	−	1.37728i	0.0158118	−	0.0486638i
$802$		0			0
$803$	10.1064	−	21.4389i	0.356647	−	0.756563i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−46.8791	+	15.2319i	−1.64818	+	0.535527i	−0.978345	−	0.206981i	$-0.933636\pi$
$809$	−46.8791	+	15.2319i	−1.64818	+	0.535527i	−0.669837	+	0.742508i	$0.733636\pi$
$810$		0			0
$811$	28.9488	+	39.8446i	1.01653	+	1.39913i	0.914609	+	0.404340i	$0.132499\pi$
$811$	28.9488	+	39.8446i	1.01653	+	1.39913i	0.101921	+	0.994792i	$0.467501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−44.8077	+	32.5547i	−1.56762	+	1.13895i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$821$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$822$		0			0
$823$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$823$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$824$		0			0
$825$	6.91289	−	54.4201i	0.240676	−	1.89467i
$826$		0			0
$827$	−14.9532	−	4.85858i	−0.519972	−	0.168949i	0.0372604	−	0.999306i	$-0.488137\pi$
$827$	−14.9532	−	4.85858i	−0.519972	−	0.168949i	−0.557233	+	0.830356i	$0.688137\pi$
$828$		0			0
$829$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$829$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.31932	−	5.94503i	−0.149655	−	0.205983i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$839$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$840$		0			0
$841$	−8.96149	−	27.5806i	−0.309017	−	0.951057i
$842$		0			0
$843$	50.2812	−	69.2061i	1.73178	−	2.38358i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	80.0872	+	26.0219i	2.74858	+	0.893069i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$853$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	58.3493i		−	1.99317i	−0.0825467	−	0.996587i	$-0.526305\pi$
$857$		−	58.3493i		−	1.99317i	0.0825467	−	0.996587i	$-0.473695\pi$
$858$		0			0
$859$	51.9105			1.77116			0.885582	−	0.464483i	$-0.153760\pi$
$859$	51.9105			1.77116			0.885582	+	0.464483i	$0.153760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$863$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−16.2515	+	50.0169i	−0.551929	+	1.69866i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	83.8588	+	60.9270i	2.83819	+	2.06207i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$877$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	59.3622			1.99996			0.999981	−	0.00609171i	$-0.00193906\pi$
$881$	59.3622			1.99996			0.999981	+	0.00609171i	$0.00193906\pi$
$882$		0			0
$883$	−45.2013	+	32.8406i	−1.52114	+	1.10518i	−0.560227	+	0.828339i	$0.689286\pi$
$883$	−45.2013	+	32.8406i	−1.52114	+	1.10518i	−0.960917	+	0.276836i	$0.910714\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$887$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	90.7887	+	42.7982i	3.04153	+	1.43379i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−13.2886	−	40.8981i	−0.441241	−	1.35800i	−0.886554	−	0.462625i	$-0.846908\pi$
$907$	−13.2886	−	40.8981i	−0.441241	−	1.35800i	0.445313	−	0.895375i	$-0.353092\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$911$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$912$		0			0
$913$	47.2319	−	25.9237i	1.56315	−	0.857949i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$919$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$920$		0			0
$921$	−67.9727	−	93.5564i	−2.23978	−	3.08279i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.6374	+	57.3601i	0.611474	+	1.88192i	0.443935	+	0.896059i	$0.353582\pi$
$929$	18.6374	+	57.3601i	0.611474	+	1.88192i	0.167539	+	0.985865i	$0.446418\pi$
$930$		0			0
$931$	33.0818	−	45.5332i	1.08421	−	1.49229i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−58.1791	−	18.9035i	−1.90063	−	0.617552i	−0.962522	−	0.271204i	$-0.912578\pi$
$937$	−58.1791	−	18.9035i	−1.90063	−	0.617552i	−0.938106	−	0.346348i	$-0.887422\pi$
$938$		0			0
$939$	−31.7401	−	23.0606i	−1.03580	−	0.752553i
$940$		0			0
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−55.8582			−1.81515			−0.907573	−	0.419894i	$-0.862067\pi$
$947$	−55.8582			−1.81515			−0.907573	+	0.419894i	$0.862067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	31.6986	−	43.6294i	1.02682	−	1.41329i	0.119508	−	0.992833i	$-0.461868\pi$
$953$	31.6986	−	43.6294i	1.02682	−	1.41329i	0.907311	−	0.420461i	$-0.138132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−25.0795	−	18.2213i	−0.809017	−	0.587785i
$962$		0			0
$963$	−89.3262	+	29.0238i	−2.87850	+	0.935280i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$	27.9216			0.896970
$970$		0			0
$971$	43.6869	−	31.7404i	1.40198	−	1.01860i	0.407552	−	0.913182i	$-0.366383\pi$
$971$	43.6869	−	31.7404i	1.40198	−	1.01860i	0.994428	−	0.105416i	$-0.0336174\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.85410	−	5.70634i	0.0593180	−	0.182562i	−0.917007	−	0.398871i	$-0.869402\pi$
$977$	1.85410	−	5.70634i	0.0593180	−	0.182562i	0.976325	+	0.216309i	$0.0694020\pi$
$978$		0			0
$979$	−0.594047	+	0.112897i	−0.0189858	+	0.00360821i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$983$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−43.2897	+	31.4518i	−1.37376	+	0.998093i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$997$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	352.2.s.a.79.2		8
4.3	odd	2		88.2.k.a.35.2	✓	8
8.3	odd	2	CM	352.2.s.a.79.2		8
8.5	even	2		88.2.k.a.35.2	✓	8
11.4	even	5		3872.2.g.b.1935.1		8
11.6	odd	10	inner	352.2.s.a.303.2		8
11.7	odd	10		3872.2.g.b.1935.2		8
12.11	even	2		792.2.bp.a.739.1		8
24.5	odd	2		792.2.bp.a.739.1		8
44.3	odd	10		968.2.k.c.403.2		8
44.7	even	10		968.2.g.a.483.4		8
44.15	odd	10		968.2.g.a.483.8		8
44.19	even	10		968.2.k.d.403.1		8
44.27	odd	10		968.2.k.b.699.1		8
44.31	odd	10		968.2.k.d.723.1		8
44.35	even	10		968.2.k.c.723.2		8
44.39	even	10		88.2.k.a.83.2	yes	8
44.43	even	2		968.2.k.b.475.1		8
88.5	even	10		968.2.k.b.699.1		8
88.13	odd	10		968.2.k.c.723.2		8
88.21	odd	2		968.2.k.b.475.1		8
88.29	odd	10		968.2.g.a.483.4		8
88.37	even	10		968.2.g.a.483.8		8
88.51	even	10		3872.2.g.b.1935.2		8
88.53	even	10		968.2.k.d.723.1		8
88.59	odd	10		3872.2.g.b.1935.1		8
88.61	odd	10		88.2.k.a.83.2	yes	8
88.69	even	10		968.2.k.c.403.2		8
88.83	even	10	inner	352.2.s.a.303.2		8
88.85	odd	10		968.2.k.d.403.1		8
132.83	odd	10		792.2.bp.a.523.1		8
264.149	even	10		792.2.bp.a.523.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.k.a.35.2	✓	8	4.3	odd	2
88.2.k.a.35.2	✓	8	8.5	even	2
88.2.k.a.83.2	yes	8	44.39	even	10
88.2.k.a.83.2	yes	8	88.61	odd	10
352.2.s.a.79.2		8	1.1	even	1	trivial
352.2.s.a.79.2		8	8.3	odd	2	CM
352.2.s.a.303.2		8	11.6	odd	10	inner
352.2.s.a.303.2		8	88.83	even	10	inner
792.2.bp.a.523.1		8	132.83	odd	10
792.2.bp.a.523.1		8	264.149	even	10
792.2.bp.a.739.1		8	12.11	even	2
792.2.bp.a.739.1		8	24.5	odd	2
968.2.g.a.483.4		8	44.7	even	10
968.2.g.a.483.4		8	88.29	odd	10
968.2.g.a.483.8		8	44.15	odd	10
968.2.g.a.483.8		8	88.37	even	10
968.2.k.b.475.1		8	44.43	even	2
968.2.k.b.475.1		8	88.21	odd	2
968.2.k.b.699.1		8	44.27	odd	10
968.2.k.b.699.1		8	88.5	even	10
968.2.k.c.403.2		8	44.3	odd	10
968.2.k.c.403.2		8	88.69	even	10
968.2.k.c.723.2		8	44.35	even	10
968.2.k.c.723.2		8	88.13	odd	10
968.2.k.d.403.1		8	44.19	even	10
968.2.k.d.403.1		8	88.85	odd	10
968.2.k.d.723.1		8	44.31	odd	10
968.2.k.d.723.1		8	88.53	even	10
3872.2.g.b.1935.1		8	11.4	even	5
3872.2.g.b.1935.1		8	88.59	odd	10
3872.2.g.b.1935.2		8	11.7	odd	10
3872.2.g.b.1935.2		8	88.51	even	10

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:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.s (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.67625 - 1.94441i) q^{3} +(2.45454 - 7.55429i) q^{9} +(-3.25830 + 0.619233i) q^{11} +(0.998400 - 0.324400i) q^{17} +(4.72597 + 6.50475i) q^{19} +(-4.04508 + 2.93893i) q^{25} +(-5.05297 - 15.5514i) q^{27} +(-7.51600 + 7.99270i) q^{33} +(1.29913 + 1.78810i) q^{41} +6.88847i q^{43} +(-2.16312 - 6.65740i) q^{49} +(2.04120 - 2.80947i) q^{51} +(25.2958 + 8.21910i) q^{57} +(-2.79793 - 2.03282i) q^{59} +16.3138 q^{67} +(-4.20049 + 5.78148i) q^{73} +(-5.11118 + 15.7306i) q^{75} +(-24.4832 - 17.7881i) q^{81} +(-15.4499 + 5.01998i) q^{83} +0.182318 q^{89} +(-4.03261 + 12.4111i) q^{97} +(-3.31977 + 26.1341i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 2 q^{9} - 6 q^{11} + 10 q^{19} - 10 q^{25} - 38 q^{27} - 38 q^{33} + 14 q^{49} + 70 q^{51} + 70 q^{57} + 18 q^{59} + 28 q^{67} - 30 q^{75} - 8 q^{81} - 90 q^{83} - 36 q^{89} + 30 q^{97}+ \cdots - 6 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 2 * q^9 - 6 * q^11 + 10 * q^19 - 10 * q^25 - 38 * q^27 - 38 * q^33 + 14 * q^49 + 70 * q^51 + 70 * q^57 + 18 * q^59 + 28 * q^67 - 30 * q^75 - 8 * q^81 - 90 * q^83 - 36 * q^89 + 30 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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