LMFDB - Embedded newform 68.2.i.a.7.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,2,Mod(3,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 1]))

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

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

chi := DirichletCharacter("68.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 68 = 2^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	68.i (of order $16$, degree $8$, 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:	$0.542982733745$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			7.1
Root			$-0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	68.7
Dual form			68.2.i.a.39.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.541196 + 1.30656i) q^{2} +(-1.41421 - 1.41421i) q^{4} +(0.865619 + 4.35176i) q^{5} +(2.61313 - 1.08239i) q^{8} +(-1.14805 - 2.77164i) q^{9} +O(q^{10})$	\(q+(-0.541196 + 1.30656i) q^{2} +(-1.41421 - 1.41421i) q^{4} +(0.865619 + 4.35176i) q^{5} +(2.61313 - 1.08239i) q^{8} +(-1.14805 - 2.77164i) q^{9} +(-6.15432 - 1.22417i) q^{10} +(2.83730 - 2.83730i) q^{13} +4.00000i q^{16} +(2.12132 - 3.53553i) q^{17} +4.24264 q^{18} +(4.93015 - 7.37849i) q^{20} +(-13.5691 + 5.62052i) q^{25} +(2.17157 + 5.24264i) q^{26} +(-1.93434 + 0.384765i) q^{29} +(-5.22625 - 2.16478i) q^{32} +(3.47135 + 4.68506i) q^{34} +(-2.29610 + 5.54328i) q^{36} +(2.39043 - 3.57753i) q^{37} +(6.97229 + 10.4348i) q^{40} +(-1.60894 + 8.08866i) q^{41} +(11.0677 - 7.39523i) q^{45} +(-6.46716 - 2.67878i) q^{49} -20.7707i q^{50} -8.02509 q^{52} +(-0.636039 + 1.53553i) q^{53} +(0.544139 - 2.73557i) q^{58} +(1.61619 + 0.321480i) q^{61} +(5.65685 - 5.65685i) q^{64} +(14.8033 + 9.89122i) q^{65} +(-8.00000 + 2.00000i) q^{68} +(-6.00000 - 6.00000i) q^{72} +(-2.83311 - 14.2430i) q^{73} +(3.38057 + 5.05939i) q^{74} +(-17.4071 + 3.46248i) q^{80} +(-6.36396 + 6.36396i) q^{81} +(-9.69760 - 6.47973i) q^{82} +(17.2221 + 6.17106i) q^{85} +(-10.8624 - 10.8624i) q^{89} +(3.67251 + 18.4630i) q^{90} +(10.3197 - 2.05272i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{10} + 16 q^{20} - 32 q^{25} + 40 q^{26} - 16 q^{29} + 32 q^{41} + 48 q^{45} - 56 q^{53} - 8 q^{65} - 64 q^{68} - 48 q^{72} + 24 q^{73} - 40 q^{74} - 8 q^{82} + 72 q^{85} + 24 q^{90} + 56 q^{98}+O(q^{100}) $ 8 * q - 24 * q^10 + 16 * q^20 - 32 * q^25 + 40 * q^26 - 16 * q^29 + 32 * q^41 + 48 * q^45 - 56 * q^53 - 8 * q^65 - 64 * q^68 - 48 * q^72 + 24 * q^73 - 40 * q^74 - 8 * q^82 + 72 * q^85 + 24 * q^90 + 56 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$-1$	$e\left(\frac{11}{16}\right)$

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 $

$2$	−0.541196	+	1.30656i	−0.382683	+	0.923880i
$2$	−0.541196	+	1.30656i	−0.382683	+	0.923880i
$3$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$3$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$4$	−1.41421	−	1.41421i	−0.707107	−	0.707107i
$5$	0.865619	+	4.35176i	0.387117	+	1.94617i	0.316228	+	0.948683i	$0.397584\pi$
$5$	0.865619	+	4.35176i	0.387117	+	1.94617i	0.0708890	+	0.997484i	$0.477416\pi$
$6$		0			0
$7$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$7$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$8$	2.61313	−	1.08239i	0.923880	−	0.382683i
$9$	−1.14805	−	2.77164i	−0.382683	−	0.923880i
$10$	−6.15432	−	1.22417i	−1.94617	−	0.387117i
$11$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$11$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$12$		0			0
$13$	2.83730	−	2.83730i	0.786925	−	0.786925i	−0.194064	−	0.980989i	$-0.562167\pi$
$13$	2.83730	−	2.83730i	0.786925	−	0.786925i	0.980989	+	0.194064i	$0.0621670\pi$
$14$		0			0
$15$		0			0
$16$			4.00000i			1.00000i
$17$	2.12132	−	3.53553i	0.514496	−	0.857493i
$17$	2.12132	−	3.53553i	0.514496	−	0.857493i
$18$	4.24264			1.00000
$19$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$19$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$20$	4.93015	−	7.37849i	1.10242	−	1.64988i
$21$		0			0
$22$		0			0
$23$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$23$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$24$		0			0
$25$	−13.5691	+	5.62052i	−2.71383	+	1.12410i
$26$	2.17157	+	5.24264i	0.425880	+	1.02817i
$27$		0			0
$28$		0			0
$29$	−1.93434	+	0.384765i	−0.359198	+	0.0714490i	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−1.93434	+	0.384765i	−0.359198	+	0.0714490i	0.0121924	+	0.999926i	$0.496119\pi$
$30$		0			0
$31$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$31$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$32$	−5.22625	−	2.16478i	−0.923880	−	0.382683i
$33$		0			0
$34$	3.47135	+	4.68506i	0.595331	+	0.803480i
$35$		0			0
$36$	−2.29610	+	5.54328i	−0.382683	+	0.923880i
$37$	2.39043	−	3.57753i	0.392984	−	0.588142i	−0.581238	−	0.813733i	$-0.697432\pi$
$37$	2.39043	−	3.57753i	0.392984	−	0.588142i	0.974222	+	0.225592i	$0.0724315\pi$
$38$		0			0
$39$		0			0
$40$	6.97229	+	10.4348i	1.10242	+	1.64988i
$41$	−1.60894	+	8.08866i	−0.251273	+	1.26324i	0.624695	+	0.780869i	$0.285223\pi$
$41$	−1.60894	+	8.08866i	−0.251273	+	1.26324i	−0.875969	+	0.482368i	$0.839777\pi$
$42$		0			0
$43$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$43$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$44$		0			0
$45$	11.0677	−	7.39523i	1.64988	−	1.10242i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$	−6.46716	−	2.67878i	−0.923880	−	0.382683i
$50$		−	20.7707i		−	2.93743i
$51$		0			0
$52$	−8.02509			−1.11288
$53$	−0.636039	+	1.53553i	−0.0873667	+	0.210922i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−0.636039	+	1.53553i	−0.0873667	+	0.210922i	0.874157	+	0.485643i	$0.161414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.544139	−	2.73557i	0.0714490	−	0.359198i
$59$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$59$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$60$		0			0
$61$	1.61619	+	0.321480i	0.206932	+	0.0411613i	0.297468	−	0.954732i	$-0.403858\pi$
$61$	1.61619	+	0.321480i	0.206932	+	0.0411613i	−0.0905357	+	0.995893i	$0.528858\pi$
$62$		0			0
$63$		0			0
$64$	5.65685	−	5.65685i	0.707107	−	0.707107i
$65$	14.8033	+	9.89122i	1.83612	+	1.22686i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−8.00000	+	2.00000i	−0.970143	+	0.242536i
$69$		0			0
$70$		0			0
$71$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$71$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$72$	−6.00000	−	6.00000i	−0.707107	−	0.707107i
$73$	−2.83311	−	14.2430i	−0.331590	−	1.66702i	−0.682713	−	0.730686i	$-0.739200\pi$
$73$	−2.83311	−	14.2430i	−0.331590	−	1.66702i	0.351123	−	0.936329i	$-0.385800\pi$
$74$	3.38057	+	5.05939i	0.392984	+	0.588142i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$79$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$80$	−17.4071	+	3.46248i	−1.94617	+	0.387117i
$81$	−6.36396	+	6.36396i	−0.707107	+	0.707107i
$82$	−9.69760	−	6.47973i	−1.07092	−	0.715566i
$83$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$83$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$84$		0			0
$85$	17.2221	+	6.17106i	1.86799	+	0.669345i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.8624	−	10.8624i	−1.15141	−	1.15141i	−0.986270	−	0.165140i	$-0.947192\pi$
$89$	−10.8624	−	10.8624i	−1.15141	−	1.15141i	−0.165140	−	0.986270i	$-0.552808\pi$
$90$	3.67251	+	18.4630i	0.387117	+	1.94617i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.3197	−	2.05272i	1.04781	−	0.208422i	0.358979	−	0.933346i	$-0.383125\pi$
$97$	10.3197	−	2.05272i	1.04781	−	0.208422i	0.688829	+	0.724924i	$0.258125\pi$
$98$	7.00000	−	7.00000i	0.707107	−	0.707107i
$99$		0			0
$100$	27.1383	+	11.2410i	2.71383	+	1.12410i
$101$			17.7122i			1.76243i	0.472714	+	0.881216i	$0.343274\pi$
$101$			17.7122i			1.76243i	−0.472714	+	0.881216i	$0.656726\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	4.34315	−	10.4853i	0.425880	−	1.02817i
$105$		0			0
$106$	−1.66205	−	1.66205i	−0.161433	−	0.161433i
$107$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$107$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$108$		0			0
$109$	−3.14100	+	15.7909i	−0.300853	+	1.51249i	0.474100	+	0.880471i	$0.342774\pi$
$109$	−3.14100	+	15.7909i	−0.300853	+	1.51249i	−0.774953	+	0.632019i	$0.782226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.0123	+	9.36271i	−1.31817	+	0.880770i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	−14.0123	+	9.36271i	−1.31817	+	0.880770i	−0.320380	+	0.947289i	$0.603811\pi$
$114$		0			0
$115$		0			0
$116$	3.27971	+	2.19143i	0.304514	+	0.203469i
$117$	−11.1213	−	4.60660i	−1.02817	−	0.425880i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	4.20952	−	10.1627i	0.382683	−	0.923880i
$122$	−1.29471	+	1.93767i	−0.117218	+	0.175428i
$123$		0			0
$124$		0			0
$125$	−23.8795	−	35.7382i	−2.13585	−	3.19652i
$126$		0			0
$127$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$127$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$128$	4.32957	+	10.4525i	0.382683	+	0.923880i
$129$		0			0
$130$	−20.9350	+	13.9883i	−1.83612	+	1.22686i
$131$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$131$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.71644	−	11.5349i	0.147184	−	0.989109i
$137$	23.3868			1.99807			0.999035	−	0.0439140i	$-0.0139827\pi$
$137$	23.3868			1.99807			0.999035	+	0.0439140i	$0.0139827\pi$
$138$		0			0
$139$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$139$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	11.0866	−	4.59220i	0.923880	−	0.382683i
$145$	−3.34881	−	8.08474i	−0.278103	−	0.671401i
$146$	20.1426	+	4.00662i	1.66702	+	0.331590i
$147$		0			0
$148$	−8.43996	+	1.67881i	−0.693760	+	0.137998i
$149$	−3.00000	+	3.00000i	−0.245770	+	0.245770i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−3.00000	+	3.00000i	−0.245770	+	0.245770i	0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$151$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$152$		0			0
$153$	−12.2346	−	1.82056i	−0.989109	−	0.147184i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.00000	+	5.00000i	0.399043	+	0.399043i	0.877896	−	0.478852i	$-0.158947\pi$
$157$	5.00000	+	5.00000i	0.399043	+	0.399043i	−0.478852	+	0.877896i	$0.658947\pi$
$158$		0			0
$159$		0			0
$160$	4.89668	−	24.6173i	0.387117	−	1.94617i
$161$		0			0
$162$	−4.87076	−	11.7591i	−0.382683	−	0.923880i
$163$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$163$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$164$	13.7145	−	9.16372i	1.07092	−	0.715566i
$165$		0			0
$166$		0			0
$167$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$167$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$168$		0			0
$169$		−	3.10051i		−	0.238500i
$170$	−17.3834	+	19.1620i	−1.33324	+	1.46965i
$171$		0			0
$172$		0			0
$173$	−10.7758	+	16.1271i	−0.819269	+	1.22612i	0.152057	+	0.988372i	$0.451410\pi$
$173$	−10.7758	+	16.1271i	−0.819269	+	1.22612i	−0.971326	+	0.237751i	$0.923590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	20.0711	−	8.31371i	1.50439	−	0.623139i
$179$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$179$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$180$	−26.1106	−	5.19372i	−1.94617	−	0.387117i
$181$	19.2297	−	12.8489i	1.42933	−	0.955048i	0.430713	−	0.902489i	$-0.358262\pi$
$181$	19.2297	−	12.8489i	1.42933	−	0.955048i	0.998618	−	0.0525588i	$-0.0167377\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	17.6377	+	7.30579i	1.29675	+	0.537133i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$191$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$192$		0			0
$193$	13.8750	+	20.7653i	0.998741	+	1.49472i	0.863779	+	0.503871i	$0.168091\pi$
$193$	13.8750	+	20.7653i	0.998741	+	1.49472i	0.134962	+	0.990851i	$0.456909\pi$
$194$	−2.90298	+	14.5943i	−0.208422	+	1.04781i
$195$		0			0
$196$	5.35757	+	12.9343i	0.382683	+	0.923880i
$197$	26.5516	+	5.28145i	1.89173	+	0.376288i	0.997459	−	0.0712470i	$-0.0226979\pi$
$197$	26.5516	+	5.28145i	1.89173	+	0.376288i	0.894267	+	0.447535i	$0.147698\pi$
$198$		0			0
$199$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$199$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$200$	−29.3743	+	29.3743i	−2.07707	+	2.07707i
$201$		0			0
$202$	−23.1421	−	9.58579i	−1.62827	−	0.674454i
$203$		0			0
$204$		0			0
$205$	−36.5927			−2.55574
$206$		0			0
$207$		0			0
$208$	11.3492	+	11.3492i	0.786925	+	0.786925i
$209$		0			0
$210$		0			0
$211$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$211$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$212$	3.07107	−	1.27208i	0.210922	−	0.0873667i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−18.9319	−	12.6499i	−1.28223	−	0.856757i
$219$		0			0
$220$		0			0
$221$	−4.01254	−	16.0502i	−0.269913	−	1.07965i
$222$		0			0
$223$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$223$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$224$		0			0
$225$	31.1561	+	31.1561i	2.07707	+	2.07707i
$226$	−4.64958	−	23.3750i	−0.309285	−	1.55488i
$227$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$227$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$228$		0			0
$229$	−16.9853	+	7.03555i	−1.12242	+	0.464922i	−0.865198	−	0.501430i	$-0.832808\pi$
$229$	−16.9853	+	7.03555i	−1.12242	+	0.464922i	−0.257223	+	0.966352i	$0.582808\pi$
$230$		0			0
$231$		0			0
$232$	−4.63821	+	3.09915i	−0.304514	+	0.203469i
$233$	1.11938	−	0.222659i	0.0733332	−	0.0145869i	−0.158287	−	0.987393i	$-0.550597\pi$
$233$	1.11938	−	0.222659i	0.0733332	−	0.0145869i	0.231621	+	0.972806i	$0.425597\pi$
$234$	12.0376	−	12.0376i	0.786925	−	0.786925i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−5.56423	+	8.32746i	−0.358424	+	0.536419i	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−5.56423	+	8.32746i	−0.358424	+	0.536419i	0.607811	+	0.794081i	$0.292048\pi$
$242$	11.0000	+	11.0000i	0.707107	+	0.707107i
$243$		0			0
$244$	−1.83100	−	2.74028i	−0.117218	−	0.175428i
$245$	6.05934	−	30.4623i	0.387117	−	1.94617i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	59.6178	−	11.8587i	3.77056	−	0.750011i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−16.0000			−1.00000
$257$	8.11794	−	19.5984i	0.506383	−	1.22252i	−0.439568	−	0.898209i	$-0.644869\pi$
$257$	8.11794	−	19.5984i	0.506383	−	1.22252i	0.945951	−	0.324308i	$-0.105131\pi$
$258$		0			0
$259$		0			0
$260$	−6.94667	−	34.9233i	−0.430814	−	2.16585i
$261$	3.28715	+	4.91957i	0.203469	+	0.304514i
$262$		0			0
$263$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$263$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$264$		0			0
$265$	−7.23285	−	1.43870i	−0.444310	−	0.0883788i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−24.4471	−	16.3350i	−1.49056	−	0.995963i	−0.991600	−	0.129339i	$-0.958714\pi$
$269$	−24.4471	−	16.3350i	−1.49056	−	0.995963i	−0.498963	−	0.866623i	$-0.666286\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	14.1421	+	8.48528i	0.857493	+	0.514496i
$273$		0			0
$274$	−12.6569	+	30.5563i	−0.764629	+	1.84598i
$275$		0			0
$276$		0			0
$277$	2.37846	+	11.9573i	0.142908	+	0.718447i	0.984087	+	0.177690i	$0.0568624\pi$
$277$	2.37846	+	11.9573i	0.142908	+	0.718447i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.84924	−	23.7782i	−0.587557	−	1.41849i	−0.885832	−	0.464007i	$-0.846411\pi$
$281$	−9.84924	−	23.7782i	−0.587557	−	1.41849i	0.298275	−	0.954480i	$-0.403589\pi$
$282$		0			0
$283$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$283$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$			16.9706i			1.00000i
$289$	−8.00000	−	15.0000i	−0.470588	−	0.882353i
$290$	12.3756			0.726719
$291$		0			0
$292$	−16.1360	+	24.1492i	−0.944288	+	1.41323i
$293$	−2.82843	−	2.82843i	−0.165238	−	0.165238i	0.619644	−	0.784883i	$-0.287277\pi$
$293$	−2.82843	−	2.82843i	−0.165238	−	0.165238i	−0.784883	+	0.619644i	$0.787277\pi$
$294$		0			0
$295$		0			0
$296$	2.37420	−	11.9359i	0.137998	−	0.693760i
$297$		0			0
$298$	−2.29610	−	5.54328i	−0.133010	−	0.321113i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$			7.31156i			0.418659i
$306$	9.00000	−	15.0000i	0.514496	−	0.857493i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$311$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$312$		0			0
$313$	3.60264	−	18.1117i	0.203633	−	1.02373i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	3.60264	−	18.1117i	0.203633	−	1.02373i	0.938436	−	0.345452i	$-0.112274\pi$
$314$	−9.23880	+	3.82683i	−0.521375	+	0.215961i
$315$		0			0
$316$		0			0
$317$	−28.1438	+	18.8051i	−1.58072	+	1.05620i	−0.617822	+	0.786318i	$0.711985\pi$
$317$	−28.1438	+	18.8051i	−1.58072	+	1.05620i	−0.962893	+	0.269882i	$0.913015\pi$
$318$		0			0
$319$		0			0
$320$	29.5140	+	19.7206i	1.64988	+	1.10242i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	−22.5526	+	54.4468i	−1.25099	+	3.02016i
$326$		0			0
$327$		0			0
$328$	4.55076	+	22.8782i	0.251273	+	1.26324i
$329$		0			0
$330$		0			0
$331$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$331$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$332$		0			0
$333$	−12.6599	−	2.51822i	−0.693760	−	0.137998i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	13.8080	+	9.22622i	0.752171	+	0.502584i	0.871576	−	0.490261i	$-0.163099\pi$
$337$	13.8080	+	9.22622i	0.752171	+	0.502584i	−0.119405	+	0.992846i	$0.538099\pi$
$338$	4.05101	+	1.67798i	0.220346	+	0.0912701i
$339$		0			0
$340$	−15.6285	−	33.0829i	−0.847573	−	1.79417i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−15.2393	−	22.8072i	−0.819269	−	1.22612i
$347$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$347$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$348$		0			0
$349$	14.1924	+	34.2635i	0.759701	+	1.83408i	0.492057	+	0.870563i	$0.336245\pi$
$349$	14.1924	+	34.2635i	0.759701	+	1.83408i	0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.3137	−	11.3137i	0.602168	−	0.602168i	−0.338719	−	0.940887i	$-0.609994\pi$
$353$	11.3137	−	11.3137i	0.602168	−	0.602168i	0.940887	+	0.338719i	$0.109994\pi$
$354$		0			0
$355$		0			0
$356$			30.7235i			1.62834i
$357$		0			0
$358$		0			0
$359$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$359$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$360$	20.9169	−	31.3043i	1.10242	−	1.64988i
$361$	−13.4350	−	13.4350i	−0.707107	−	0.707107i
$362$	6.38082	+	32.0785i	0.335368	+	1.68601i
$363$		0			0
$364$		0			0
$365$	59.5297	−	24.6580i	3.11593	−	1.29066i
$366$		0			0
$367$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$367$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$368$		0			0
$369$	24.2660	−	4.82681i	1.26324	−	0.251273i
$370$	−19.0910	+	19.0910i	−0.992492	+	0.992492i
$371$		0			0
$372$		0			0
$373$		−	26.7109i		−	1.38304i	−0.722358	−	0.691519i	$-0.756942\pi$
$373$		−	26.7109i		−	1.38304i	0.722358	−	0.691519i	$-0.243058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.39661	+	6.57999i	−0.226437	+	0.338887i
$378$		0			0
$379$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$379$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$383$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$384$		0			0
$385$		0			0
$386$	−34.6403	+	6.89038i	−1.76314	+	0.350711i
$387$		0			0
$388$	−17.4973	−	11.6913i	−0.888289	−	0.593536i
$389$	18.4776	+	7.65367i	0.936851	+	0.388056i	0.798273	−	0.602295i	$-0.205747\pi$
$389$	18.4776	+	7.65367i	0.936851	+	0.388056i	0.138578	+	0.990352i	$0.455747\pi$
$390$		0			0
$391$		0			0
$392$	−19.7990			−1.00000
$393$		0			0
$394$	−21.2702	+	31.8331i	−1.07158	+	1.60373i
$395$		0			0
$396$		0			0
$397$	10.6574	+	15.9500i	0.534881	+	0.800506i	0.996233	−	0.0867112i	$-0.0276357\pi$
$397$	10.6574	+	15.9500i	0.534881	+	0.800506i	−0.461353	+	0.887217i	$0.652636\pi$
$398$		0			0
$399$		0			0
$400$	−22.4821	−	54.2766i	−1.12410	−	2.71383i
$401$	−38.8603	−	7.72979i	−1.94059	−	0.386007i	−0.998752	−	0.0499376i	$-0.984098\pi$
$401$	−38.8603	−	7.72979i	−1.94059	−	0.386007i	−0.941837	−	0.336070i	$-0.890902\pi$
$402$		0			0
$403$		0			0
$404$	25.0489	−	25.0489i	1.24623	−	1.24623i
$405$	−33.2032	−	22.1857i	−1.64988	−	1.10242i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	32.5269			1.60835			0.804176	−	0.594391i	$-0.202607\pi$
$409$	32.5269			1.60835			0.804176	+	0.594391i	$0.202607\pi$
$410$	19.8038	−	47.8106i	0.978041	−	2.36120i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−20.9706	+	8.68629i	−1.02817	+	0.425880i
$417$		0			0
$418$		0			0
$419$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$419$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$420$		0			0
$421$	−10.1739	+	10.1739i	−0.495847	+	0.495847i	−0.910143	−	0.414295i	$-0.864028\pi$
$421$	−10.1739	+	10.1739i	−0.495847	+	0.495847i	0.414295	+	0.910143i	$0.364028\pi$
$422$		0			0
$423$		0			0
$424$			4.70099i			0.228300i
$425$	−8.91295	+	59.8971i	−0.432342	+	2.90544i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$431$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$432$		0			0
$433$	22.1731	−	9.18440i	1.06557	−	0.441374i	0.220146	−	0.975467i	$-0.429347\pi$
$433$	22.1731	−	9.18440i	1.06557	−	0.441374i	0.845426	+	0.534093i	$0.179347\pi$
$434$		0			0
$435$		0			0
$436$	26.7737	−	17.8896i	1.28223	−	0.856757i
$437$		0			0
$438$		0			0
$439$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$439$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$	23.1421	+	3.44365i	1.10076	+	0.163798i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	37.8678	−	56.6732i	1.79511	−	2.68657i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4.97106	+	24.9912i	−0.234599	+	1.17941i	0.666403	+	0.745592i	$0.267833\pi$
$449$	−4.97106	+	24.9912i	−0.234599	+	1.17941i	−0.901002	+	0.433816i	$0.857167\pi$
$450$	−57.5690	+	23.8459i	−2.71383	+	1.12410i
$451$		0			0
$452$	33.0572	+	6.57550i	1.55488	+	0.309285i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−32.6641	−	13.5299i	−1.52796	−	0.632902i	−0.548794	−	0.835958i	$-0.684913\pi$
$457$	−32.6641	−	13.5299i	−1.52796	−	0.632902i	−0.979167	+	0.203056i	$0.934913\pi$
$458$		−	26.0000i		−	1.21490i
$459$		0			0
$460$		0			0
$461$	16.3640	−	39.5061i	0.762146	−	1.83998i	0.296399	−	0.955064i	$-0.404214\pi$
$461$	16.3640	−	39.5061i	0.762146	−	1.83998i	0.465746	−	0.884918i	$-0.345786\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$	−1.53906	−	7.73737i	−0.0714490	−	0.359198i
$465$		0			0
$466$	−0.314887	+	1.58305i	−0.0145869	+	0.0733332i
$467$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$467$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$468$	9.21320	+	22.2426i	0.425880	+	1.02817i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.98615			0.228300
$478$		0			0
$479$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$479$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$480$		0			0
$481$	−3.36815	−	16.9329i	−0.153575	−	0.772072i
$482$	−7.86901	−	11.7768i	−0.358424	−	0.536419i
$483$		0			0
$484$	−20.3253	+	8.41904i	−0.923880	+	0.382683i
$485$	17.8659	+	43.1321i	0.811248	+	1.95853i
$486$		0			0
$487$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$487$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$488$	4.57128	−	0.909283i	0.206932	−	0.0411613i
$489$		0			0
$490$	36.5217	+	24.4030i	1.64988	+	1.10242i
$491$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$491$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$492$		0			0
$493$	−2.74301	+	7.65514i	−0.123539	+	0.344770i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$499$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$500$	−16.7707	+	84.3122i	−0.750011	+	3.77056i
$501$		0			0
$502$		0			0
$503$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$503$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$504$		0			0
$505$	−77.0794	+	15.3320i	−3.42999	+	0.682267i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	38.1838i		−	1.69247i	−0.532813	−	0.846233i	$-0.678865\pi$
$509$		−	38.1838i		−	1.69247i	0.532813	−	0.846233i	$-0.321135\pi$
$510$		0			0
$511$		0			0
$512$	8.65914	−	20.9050i	0.382683	−	0.923880i
$513$		0			0
$514$	21.2132	+	21.2132i	0.935674	+	0.935674i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	49.3890	+	9.82408i	2.16585	+	0.430814i
$521$	3.26807	−	2.18366i	0.143177	−	0.0956677i	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	3.26807	−	2.18366i	0.143177	−	0.0956677i	0.625096	+	0.780548i	$0.285060\pi$
$522$	−8.20672	+	1.63242i	−0.359198	+	0.0714490i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−8.80172	+	21.2492i	−0.382683	+	0.923880i
$530$	5.79414	−	8.67155i	0.251682	−	0.376668i
$531$		0			0
$532$		0			0
$533$	18.3849	+	27.5150i	0.796339	+	1.19181i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	34.5734	−	23.1012i	1.49056	−	0.995963i
$539$		0			0
$540$		0			0
$541$	31.0065	+	20.7179i	1.33307	+	0.890731i	0.998663	−	0.0516971i	$-0.0164630\pi$
$541$	31.0065	+	20.7179i	1.33307	+	0.890731i	0.334410	+	0.942428i	$0.391463\pi$
$542$		0			0
$543$		0			0
$544$	−18.7402	+	13.8854i	−0.803480	+	0.595331i
$545$	−71.4370			−3.06002
$546$		0			0
$547$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$547$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$548$	−33.0740	−	33.0740i	−1.41285	−	1.41285i
$549$	−0.964441	−	4.84857i	−0.0411613	−	0.206932i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−16.9102	−	3.36366i	−0.718447	−	0.142908i
$555$		0			0
$556$		0			0
$557$	−28.5746	+	28.5746i	−1.21074	+	1.21074i	−0.239963	+	0.970782i	$0.577135\pi$
$557$	−28.5746	+	28.5746i	−1.21074	+	1.21074i	−0.970782	+	0.239963i	$0.922865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	36.3981			1.53536
$563$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$563$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$564$		0			0
$565$	−52.8736	−	52.8736i	−2.22441	−	2.22441i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.05025	−	3.33452i	0.337484	−	0.139791i	−0.207504	−	0.978234i	$-0.566534\pi$
$569$	8.05025	−	3.33452i	0.337484	−	0.139791i	0.544988	+	0.838444i	$0.316534\pi$
$570$		0			0
$571$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$571$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−22.1731	−	9.18440i	−0.923880	−	0.382683i
$577$		−	45.1116i		−	1.87802i	−0.343890	−	0.939010i	$-0.611745\pi$
$577$		−	45.1116i		−	1.87802i	0.343890	−	0.939010i	$-0.388255\pi$
$578$	23.9280	−	2.33456i	0.995274	−	0.0971050i
$579$		0			0
$580$	−6.69762	+	16.1695i	−0.278103	+	0.671401i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−22.8198	−	34.1522i	−0.944288	−	1.41323i
$585$	10.4200	−	52.3849i	0.430814	−	2.16585i
$586$	5.22625	−	2.16478i	0.215894	−	0.0894264i
$587$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$587$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	14.3101	+	9.56171i	0.588142	+	0.392984i
$593$	29.9203	+	12.3934i	1.22868	+	0.508936i	0.900159	−	0.435561i	$-0.143450\pi$
$593$	29.9203	+	12.3934i	1.22868	+	0.508936i	0.328521	+	0.944497i	$0.393450\pi$
$594$		0			0
$595$		0			0
$596$	8.48528			0.347571
$597$		0			0
$598$		0			0
$599$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$599$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$600$		0			0
$601$	19.0865	+	28.5650i	0.778556	+	1.16519i	0.982510	+	0.186210i	$0.0596206\pi$
$601$	19.0865	+	28.5650i	0.778556	+	1.16519i	−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	47.8694	+	9.52181i	1.94617	+	0.387117i
$606$		0			0
$607$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$607$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$608$		0			0
$609$		0			0
$610$	−9.55301	−	3.95699i	−0.386790	−	0.160214i
$611$		0			0
$612$	14.7277	+	19.8770i	0.595331	+	0.803480i
$613$	36.0000			1.45403			0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000			1.45403			0.727013	+	0.686624i	$0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.69568	+	13.5521i	0.108524	+	0.545587i	0.996347	+	0.0854011i	$0.0272172\pi$
$617$	2.69568	+	13.5521i	0.108524	+	0.545587i	−0.887823	+	0.460186i	$0.847783\pi$
$618$		0			0
$619$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$619$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	82.9269	−	82.9269i	3.31707	−	3.31707i
$626$	21.7143	+	14.5090i	0.867878	+	0.579898i
$627$		0			0
$628$		−	14.1421i		−	0.564333i
$629$	−7.57761	−	16.0405i	−0.302139	−	0.639577i
$630$		0			0
$631$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$631$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$632$		0			0
$633$		0			0
$634$	−9.33872	−	46.9489i	−0.370888	−	1.86458i
$635$		0			0
$636$		0			0
$637$	−25.9497	+	10.7487i	−1.02817	+	0.425880i
$638$		0			0
$639$		0			0
$640$	−41.7391	+	27.8891i	−1.64988	+	1.10242i
$641$	−45.1338	+	8.97767i	−1.78268	+	0.354597i	−0.972737	−	0.231913i	$-0.925502\pi$
$641$	−45.1338	+	8.97767i	−1.78268	+	0.354597i	−0.809942	+	0.586510i	$0.800502\pi$
$642$		0			0
$643$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$643$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−9.74153	+	23.5181i	−0.382683	+	0.923880i
$649$		0			0
$650$	−58.9328	−	58.9328i	−2.31153	−	2.31153i
$651$		0			0
$652$		0			0
$653$	−9.40860	+	47.3002i	−0.368187	+	1.85100i	0.140542	+	0.990075i	$0.455115\pi$
$653$	−9.40860	+	47.3002i	−0.368187	+	1.85100i	−0.508729	+	0.860927i	$0.669885\pi$
$654$		0			0
$655$		0			0
$656$	−32.3547	−	6.43574i	−1.26324	−	0.251273i
$657$	−36.2239	+	24.2040i	−1.41323	+	0.944288i
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	46.9203	+	19.4350i	1.82499	+	0.755935i	0.972387	+	0.233373i	$0.0749763\pi$
$661$	46.9203	+	19.4350i	1.82499	+	0.755935i	0.852601	+	0.522562i	$0.175024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	10.1417	−	15.1782i	0.392984	−	0.588142i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−44.6370	−	8.87885i	−1.72063	−	0.342255i	−0.766637	−	0.642081i	$-0.778071\pi$
$673$	−44.6370	−	8.87885i	−1.72063	−	0.342255i	−0.953994	+	0.299827i	$0.903071\pi$
$674$	−19.5275	+	13.0479i	−0.752171	+	0.502584i
$675$		0			0
$676$	−4.38478	+	4.38478i	−0.168645	+	0.168645i
$677$	−25.4036	−	16.9741i	−0.976338	−	0.652368i	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−25.4036	−	16.9741i	−0.976338	−	0.652368i	−0.937905	+	0.346893i	$0.887237\pi$
$678$		0			0
$679$		0			0
$680$	51.6829	−	2.51528i	1.98195	−	0.0964565i
$681$		0			0
$682$		0			0
$683$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$683$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$684$		0			0
$685$	20.2441	+	101.774i	0.773487	+	3.88858i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.55213	+	6.16140i	0.0972286	+	0.234731i
$690$		0			0
$691$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$691$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$692$	38.0465	−	7.56792i	1.44631	−	0.287689i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	25.1847	+	22.8471i	0.953938	+	0.865395i
$698$	−52.4482			−1.98519
$699$		0			0
$700$		0			0
$701$	7.53828	+	7.53828i	0.284717	+	0.284717i	0.834987	−	0.550270i	$-0.185475\pi$
$701$	7.53828	+	7.53828i	0.284717	+	0.284717i	−0.550270	+	0.834987i	$0.685475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	8.65914	+	20.9050i	0.325891	+	0.786770i
$707$		0			0
$708$		0			0
$709$	37.2772	−	7.41490i	1.39998	−	0.278472i	0.563337	−	0.826227i	$-0.309517\pi$
$709$	37.2772	−	7.41490i	1.39998	−	0.278472i	0.836639	+	0.547755i	$0.184517\pi$
$710$		0			0
$711$		0			0
$712$	−40.1421	−	16.6274i	−1.50439	−	0.623139i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$719$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$720$	29.5809	+	44.2710i	1.10242	+	1.64988i
$721$		0			0
$722$	24.8247	−	10.2827i	0.923880	−	0.382683i
$723$		0			0
$724$	−45.3659	−	9.02384i	−1.68601	−	0.335368i
$725$	24.0848	−	16.0929i	0.894487	−	0.597677i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	24.9447	+	10.3325i	0.923880	+	0.382683i
$730$			91.1241i			3.37266i
$731$		0			0
$732$		0			0
$733$	9.32233	−	22.5061i	0.344328	−	0.831282i	−0.652940	−	0.757410i	$-0.726464\pi$
$733$	9.32233	−	22.5061i	0.344328	−	0.831282i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−6.82613	+	34.3173i	−0.251273	+	1.26324i
$739$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$739$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$740$	−14.6116	−	35.2755i	−0.537133	−	1.29675i
$741$		0			0
$742$		0			0
$743$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$743$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$744$		0			0
$745$	−15.6521	−	10.4584i	−0.573450	−	0.383167i
$746$	34.8995	+	14.4558i	1.27776	+	0.529266i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$751$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$752$		0			0
$753$		0			0
$754$	−6.21775	−	9.30552i	−0.226437	−	0.338887i
$755$		0			0
$756$		0			0
$757$	−18.9419	−	45.7297i	−0.688454	−	1.66207i	−0.747873	−	0.663841i	$-0.768925\pi$
$757$	−18.9419	−	45.7297i	−0.688454	−	1.66207i	0.0594198	−	0.998233i	$-0.481075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.8485	−	15.8485i	0.574509	−	0.574509i	−0.358876	−	0.933385i	$-0.616840\pi$
$761$	15.8485	−	15.8485i	0.574509	−	0.574509i	0.933385	+	0.358876i	$0.116840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2.66786	−	54.8180i	−0.0964565	−	1.98195i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.14885	+	2.14885i	0.0774896	+	0.0774896i	0.744789	−	0.667300i	$-0.232550\pi$
$769$	2.14885	+	2.14885i	0.0774896	+	0.0774896i	−0.667300	+	0.744789i	$0.732550\pi$
$770$		0			0
$771$		0			0
$772$	9.74447	−	48.9888i	0.350711	−	1.76314i
$773$	−40.6507	+	16.8381i	−1.46210	+	0.605623i	−0.965043	−	0.262092i	$-0.915588\pi$
$773$	−40.6507	+	16.8381i	−1.46210	+	0.605623i	−0.497061	+	0.867715i	$0.665588\pi$
$774$		0			0
$775$		0			0
$776$	24.7449	−	16.5340i	0.888289	−	0.593536i
$777$		0			0
$778$	−20.0000	+	20.0000i	−0.717035	+	0.717035i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	10.7151	−	25.8686i	0.382683	−	0.923880i
$785$	−17.4307	+	26.0869i	−0.622129	+	0.931082i
$786$		0			0
$787$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$787$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$788$	−30.0806	−	45.0188i	−1.07158	−	1.60373i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	5.49775	−	3.67348i	0.195231	−	0.130449i
$794$	−26.6074	+	5.29254i	−0.944261	+	0.187825i
$795$		0			0
$796$		0			0
$797$	−52.1630	−	21.6066i	−1.84771	−	0.765345i	−0.926739	−	0.375705i	$-0.877401\pi$
$797$	−52.1630	−	21.6066i	−1.84771	−	0.765345i	−0.920967	−	0.389640i	$-0.872599\pi$
$798$		0			0
$799$		0			0
$800$	83.0830			2.93743
$801$	−17.6360	+	42.5772i	−0.623139	+	1.50439i
$802$	31.1305	−	46.5901i	1.09926	−	1.64515i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	19.1716	+	46.2843i	0.674454	+	1.62827i
$809$	38.6691	+	7.69175i	1.35953	+	0.270428i	0.820398	−	0.571793i	$-0.193752\pi$
$809$	38.6691	+	7.69175i	1.35953	+	0.270428i	0.539133	+	0.842220i	$0.318752\pi$
$810$	46.9564	−	31.3753i	1.64988	−	1.10242i
$811$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$811$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−17.6034	+	42.4985i	−0.615490	+	1.48592i
$819$		0			0
$820$	51.7499	+	51.7499i	1.80718	+	1.80718i
$821$	−10.6328	−	53.4546i	−0.371086	−	1.86558i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−10.6328	−	53.4546i	−0.371086	−	1.86558i	0.117517	−	0.993071i	$-0.462507\pi$
$822$		0			0
$823$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$823$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$827$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$828$		0			0
$829$	−37.0000	+	37.0000i	−1.28506	+	1.28506i	−0.347314	+	0.937749i	$0.612906\pi$
$829$	−37.0000	+	37.0000i	−1.28506	+	1.28506i	−0.937749	+	0.347314i	$0.887094\pi$
$830$		0			0
$831$		0			0
$832$		−	32.1003i		−	1.11288i
$833$	−23.1898	+	17.1823i	−0.803480	+	0.595331i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$839$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$840$		0			0
$841$	−23.1989	+	9.60929i	−0.799961	+	0.331355i
$842$	−7.78680	−	18.7990i	−0.268351	−	0.647856i
$843$		0			0
$844$		0			0
$845$	13.4927	−	2.68386i	0.464162	−	0.0923275i
$846$		0			0
$847$		0			0
$848$	−6.14214	−	2.54416i	−0.210922	−	0.0873667i
$849$		0			0
$850$	−73.4357	−	44.0614i	−2.51882	−	1.51129i
$851$		0			0
$852$		0			0
$853$	24.6020	−	36.8195i	0.842356	−	1.26068i	−0.121054	−	0.992646i	$-0.538628\pi$
$853$	24.6020	−	36.8195i	0.842356	−	1.26068i	0.963411	−	0.268029i	$-0.0863725\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11.4023	−	57.3232i	0.389495	−	1.95812i	0.136637	−	0.990621i	$-0.456370\pi$
$857$	11.4023	−	57.3232i	0.389495	−	1.95812i	0.252858	−	0.967503i	$-0.418630\pi$
$858$		0			0
$859$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$859$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$	−79.5092	−	32.9338i	−2.70339	−	1.11978i
$866$			33.9411i			1.15337i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	8.88408	+	44.6633i	0.300853	+	1.51249i
$873$	−17.5369	−	26.2459i	−0.593536	−	0.888289i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−53.8373	−	10.7089i	−1.81796	−	0.361614i	−0.835705	−	0.549178i	$-0.814941\pi$
$877$	−53.8373	−	10.7089i	−1.81796	−	0.361614i	−0.982252	+	0.187564i	$0.939941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	45.2199	+	30.2150i	1.52350	+	1.01797i	0.984444	+	0.175697i	$0.0562180\pi$
$881$	45.2199	+	30.2150i	1.52350	+	1.01797i	0.539054	+	0.842271i	$0.318782\pi$
$882$	−27.4378	−	11.3651i	−0.923880	−	0.382683i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$	−17.0238	+	28.3730i	−0.572572	+	0.954286i
$885$		0			0
$886$		0			0
$887$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$887$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$888$		0			0
$889$		0			0
$890$	53.5532	+	80.1480i	1.79511	+	2.68657i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−29.9623	−	20.0201i	−0.999853	−	0.668081i
$899$		0			0
$900$		−	88.1228i		−	2.93743i
$901$	4.07969	+	5.50610i	0.135914	+	0.183435i
$902$		0			0
$903$		0			0
$904$	−26.4818	+	39.6327i	−0.880770	+	1.31817i
$905$	72.5608	+	72.5608i	2.41200	+	2.41200i
$906$		0			0
$907$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$907$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$908$		0			0
$909$	49.0919	−	20.3345i	1.62827	−	0.674454i
$910$		0			0
$911$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$911$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$912$		0			0
$913$		0			0
$914$	35.3553	−	35.3553i	1.16945	−	1.16945i
$915$		0			0
$916$	33.9706	+	14.0711i	1.12242	+	0.464922i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	42.7611	+	42.7611i	1.40826	+	1.40826i
$923$		0			0
$924$		0			0
$925$	−12.3285	+	61.9794i	−0.405358	+	2.03787i
$926$		0			0
$927$		0			0
$928$	10.9423	+	2.17656i	0.359198	+	0.0714490i
$929$	50.2793	−	33.5956i	1.64961	−	1.10223i	0.754606	−	0.656179i	$-0.227828\pi$
$929$	50.2793	−	33.5956i	1.64961	−	1.10223i	0.895005	−	0.446056i	$-0.147172\pi$
$930$		0			0
$931$		0			0
$932$	−1.89793	−	1.26816i	−0.0621689	−	0.0415399i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−34.0476			−1.11288
$937$	6.40559	−	15.4645i	0.209262	−	0.505202i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	6.40559	−	15.4645i	0.209262	−	0.505202i	0.993307	+	0.115501i	$0.0368473\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.66339	+	12.9657i	0.282418	+	0.422669i	0.945373	−	0.325991i	$-0.105698\pi$
$941$	8.66339	+	12.9657i	0.282418	+	0.422669i	−0.662955	+	0.748660i	$0.730698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$947$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$948$		0			0
$949$	−48.4499	−	32.3732i	−1.57275	−	1.05088i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	41.7875			1.35363			0.676815	−	0.736153i	$-0.263360\pi$
$953$	41.7875			1.35363			0.676815	+	0.736153i	$0.263360\pi$
$954$	−2.69848	+	6.51472i	−0.0873667	+	0.210922i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11.8632	+	28.6403i	0.382683	+	0.923880i
$962$	23.9467	+	4.76329i	0.772072	+	0.153575i
$963$		0			0
$964$	19.6458	−	3.90780i	0.632749	−	0.125862i
$965$	−78.3554	+	78.3554i	−2.52235	+	2.52235i
$966$		0			0
$967$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$967$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$968$		−	31.1127i		−	1.00000i
$969$		0			0
$970$	−66.0237			−2.11989
$971$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$971$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.28592	+	6.46476i	−0.0411613	+	0.206932i
$977$	15.0919	−	6.25126i	0.482832	−	0.199996i	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	15.0919	−	6.25126i	0.482832	−	0.199996i	0.610803	+	0.791782i	$0.290847\pi$
$978$		0			0
$979$		0			0
$980$	−51.6494	+	34.5111i	−1.64988	+	1.10242i
$981$	47.3726	−	9.42299i	1.51249	−	0.300853i
$982$		0			0
$983$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$983$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$984$		0			0
$985$			120.118i			3.82728i
$986$	−8.51742	−	7.72685i	−0.271250	−	0.246073i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$991$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−20.3442	+	13.5935i	−0.644306	+	0.430512i	−0.834328	−	0.551268i	$-0.814144\pi$
$997$	−20.3442	+	13.5935i	−0.644306	+	0.430512i	0.190022	+	0.981780i	$0.439144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.2.i.a.7.1	✓	8
3.2	odd	2		612.2.bd.a.415.1		8
4.3	odd	2	CM	68.2.i.a.7.1	✓	8
12.11	even	2		612.2.bd.a.415.1		8
17.5	odd	16	inner	68.2.i.a.39.1	yes	8
51.5	even	16		612.2.bd.a.379.1		8
68.39	even	16	inner	68.2.i.a.39.1	yes	8
204.107	odd	16		612.2.bd.a.379.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.i.a.7.1	✓	8	1.1	even	1	trivial
68.2.i.a.7.1	✓	8	4.3	odd	2	CM
68.2.i.a.39.1	yes	8	17.5	odd	16	inner
68.2.i.a.39.1	yes	8	68.39	even	16	inner
612.2.bd.a.379.1		8	51.5	even	16
612.2.bd.a.379.1		8	204.107	odd	16
612.2.bd.a.415.1		8	3.2	odd	2
612.2.bd.a.415.1		8	12.11	even	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:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	68.i (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.541196 + 1.30656i) q^{2} +(-1.41421 - 1.41421i) q^{4} +(0.865619 + 4.35176i) q^{5} +(2.61313 - 1.08239i) q^{8} +(-1.14805 - 2.77164i) q^{9} +O(q^{10})\)	\(q+(-0.541196 + 1.30656i) q^{2} +(-1.41421 - 1.41421i) q^{4} +(0.865619 + 4.35176i) q^{5} +(2.61313 - 1.08239i) q^{8} +(-1.14805 - 2.77164i) q^{9} +(-6.15432 - 1.22417i) q^{10} +(2.83730 - 2.83730i) q^{13} +4.00000i q^{16} +(2.12132 - 3.53553i) q^{17} +4.24264 q^{18} +(4.93015 - 7.37849i) q^{20} +(-13.5691 + 5.62052i) q^{25} +(2.17157 + 5.24264i) q^{26} +(-1.93434 + 0.384765i) q^{29} +(-5.22625 - 2.16478i) q^{32} +(3.47135 + 4.68506i) q^{34} +(-2.29610 + 5.54328i) q^{36} +(2.39043 - 3.57753i) q^{37} +(6.97229 + 10.4348i) q^{40} +(-1.60894 + 8.08866i) q^{41} +(11.0677 - 7.39523i) q^{45} +(-6.46716 - 2.67878i) q^{49} -20.7707i q^{50} -8.02509 q^{52} +(-0.636039 + 1.53553i) q^{53} +(0.544139 - 2.73557i) q^{58} +(1.61619 + 0.321480i) q^{61} +(5.65685 - 5.65685i) q^{64} +(14.8033 + 9.89122i) q^{65} +(-8.00000 + 2.00000i) q^{68} +(-6.00000 - 6.00000i) q^{72} +(-2.83311 - 14.2430i) q^{73} +(3.38057 + 5.05939i) q^{74} +(-17.4071 + 3.46248i) q^{80} +(-6.36396 + 6.36396i) q^{81} +(-9.69760 - 6.47973i) q^{82} +(17.2221 + 6.17106i) q^{85} +(-10.8624 - 10.8624i) q^{89} +(3.67251 + 18.4630i) q^{90} +(10.3197 - 2.05272i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{10} + 16 q^{20} - 32 q^{25} + 40 q^{26} - 16 q^{29} + 32 q^{41} + 48 q^{45} - 56 q^{53} - 8 q^{65} - 64 q^{68} - 48 q^{72} + 24 q^{73} - 40 q^{74} - 8 q^{82} + 72 q^{85} + 24 q^{90} + 56 q^{98}+O(q^{100}) \) 8 * q - 24 * q^10 + 16 * q^20 - 32 * q^25 + 40 * q^26 - 16 * q^29 + 32 * q^41 + 48 * q^45 - 56 * q^53 - 8 * q^65 - 64 * q^68 - 48 * q^72 + 24 * q^73 - 40 * q^74 - 8 * q^82 + 72 * q^85 + 24 * q^90 + 56 * q^98

Introduction

L-functions

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