LMFDB - Embedded newform 68.2.i.a.27.1

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			27.1
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	68.27
Dual form			68.2.i.a.63.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.541196 - 1.30656i) q^{2} +(-1.41421 - 1.41421i) q^{4} +(0.548594 - 0.109122i) q^{5} +(-2.61313 + 1.08239i) q^{8} +(1.14805 + 2.77164i) q^{9} +O(q^{10})$	\(q+(0.541196 - 1.30656i) q^{2} +(-1.41421 - 1.41421i) q^{4} +(0.548594 - 0.109122i) q^{5} +(-2.61313 + 1.08239i) q^{8} +(1.14805 + 2.77164i) q^{9} +(0.154322 - 0.775829i) q^{10} +(-2.83730 + 2.83730i) q^{13} +4.00000i q^{16} +(2.12132 - 3.53553i) q^{17} +4.24264 q^{18} +(-0.930151 - 0.621507i) q^{20} +(-4.33035 + 1.79369i) q^{25} +(2.17157 + 5.24264i) q^{26} +(-2.06566 - 10.3848i) q^{29} +(5.22625 + 2.16478i) q^{32} +(-3.47135 - 4.68506i) q^{34} +(2.29610 - 5.54328i) q^{36} +(-9.46149 - 6.32197i) q^{37} +(-1.31543 + 0.878944i) q^{40} +(9.60894 + 1.91134i) q^{41} +(0.932261 + 1.39523i) q^{45} +(6.46716 + 2.67878i) q^{49} +6.62861i q^{50} +8.02509 q^{52} +(-0.636039 + 1.53553i) q^{53} +(-14.6863 - 2.92128i) q^{58} +(-3.03040 + 15.2349i) q^{61} +(5.65685 - 5.65685i) q^{64} +(-1.24691 + 1.86614i) q^{65} +(-8.00000 + 2.00000i) q^{68} +(-6.00000 - 6.00000i) q^{72} +(8.83311 - 1.75701i) q^{73} +(-13.3806 + 8.94061i) q^{74} +(0.436489 + 2.19438i) q^{80} +(-6.36396 + 6.36396i) q^{81} +(7.69760 - 11.5203i) q^{82} +(0.777939 - 2.17106i) q^{85} +(10.8624 + 10.8624i) q^{89} +(2.32749 - 0.462966i) q^{90} +(-3.24865 - 16.3321i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$-1$	$e\left(\frac{3}{16}\right)$

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		−0.831470	−	0.555570i	$-0.812500\pi$
$3$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$4$	−1.41421	−	1.41421i	−0.707107	−	0.707107i
$5$	0.548594	−	0.109122i	0.245339	−	0.0488009i	−0.0708890	−	0.997484i	$-0.522584\pi$
$5$	0.548594	−	0.109122i	0.245339	−	0.0488009i	0.316228	+	0.948683i	$0.397584\pi$
$6$		0			0
$7$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$7$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$8$	−2.61313	+	1.08239i	−0.923880	+	0.382683i
$9$	1.14805	+	2.77164i	0.382683	+	0.923880i
$10$	0.154322	−	0.775829i	0.0488009	−	0.245339i
$11$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$11$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$12$		0			0
$13$	−2.83730	+	2.83730i	−0.786925	+	0.786925i	−0.980989	−	0.194064i	$-0.937833\pi$
$13$	−2.83730	+	2.83730i	−0.786925	+	0.786925i	0.194064	+	0.980989i	$0.437833\pi$
$14$		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	$0.625000\pi$
$20$	−0.930151	−	0.621507i	−0.207988	−	0.138973i
$21$		0			0
$22$		0			0
$23$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$23$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$24$		0			0
$25$	−4.33035	+	1.79369i	−0.866070	+	0.358738i
$26$	2.17157	+	5.24264i	0.425880	+	1.02817i
$27$		0			0
$28$		0			0
$29$	−2.06566	−	10.3848i	−0.383583	−	1.92840i	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−2.06566	−	10.3848i	−0.383583	−	1.92840i	−0.0121924	−	0.999926i	$-0.503881\pi$
$30$		0			0
$31$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$31$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$32$	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
$37$	−9.46149	−	6.32197i	−1.55546	−	1.03933i	−0.974222	−	0.225592i	$-0.927568\pi$
$37$	−9.46149	−	6.32197i	−1.55546	−	1.03933i	−0.581238	−	0.813733i	$-0.697432\pi$
$38$		0			0
$39$		0			0
$40$	−1.31543	+	0.878944i	−0.207988	+	0.138973i
$41$	9.60894	+	1.91134i	1.50066	+	0.298501i	0.875969	−	0.482368i	$-0.160223\pi$
$41$	9.60894	+	1.91134i	1.50066	+	0.298501i	0.624695	+	0.780869i	$0.285223\pi$
$42$		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
$45$	0.932261	+	1.39523i	0.138973	+	0.207988i
$46$		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
$50$			6.62861i			0.937427i
$51$		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
$58$	−14.6863	−	2.92128i	−1.92840	−	0.383583i
$59$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$59$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$60$		0			0
$61$	−3.03040	+	15.2349i	−0.388003	+	1.95063i	−0.0905357	+	0.995893i	$0.528858\pi$
$61$	−3.03040	+	15.2349i	−0.388003	+	1.95063i	−0.297468	+	0.954732i	$0.596142\pi$
$62$		0			0
$63$		0			0
$64$	5.65685	−	5.65685i	0.707107	−	0.707107i
$65$	−1.24691	+	1.86614i	−0.154660	+	0.231466i
$66$		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		−0.831470	−	0.555570i	$-0.812500\pi$
$71$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$72$	−6.00000	−	6.00000i	−0.707107	−	0.707107i
$73$	8.83311	−	1.75701i	1.03384	−	0.205643i	0.351123	−	0.936329i	$-0.385800\pi$
$73$	8.83311	−	1.75701i	1.03384	−	0.205643i	0.682713	+	0.730686i	$0.260800\pi$
$74$	−13.3806	+	8.94061i	−1.55546	+	1.03933i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$79$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$80$	0.436489	+	2.19438i	0.0488009	+	0.245339i
$81$	−6.36396	+	6.36396i	−0.707107	+	0.707107i
$82$	7.69760	−	11.5203i	0.850058	−	1.27220i
$83$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$83$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$84$		0			0
$85$	0.777939	−	2.17106i	0.0843793	−	0.235484i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.8624	+	10.8624i	1.15141	+	1.15141i	0.986270	+	0.165140i	$0.0528077\pi$
$89$	10.8624	+	10.8624i	1.15141	+	1.15141i	0.165140	+	0.986270i	$0.447192\pi$
$90$	2.32749	−	0.462966i	0.245339	−	0.0488009i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.24865	−	16.3321i	−0.329850	−	1.65827i	−0.688829	−	0.724924i	$-0.741875\pi$
$97$	−3.24865	−	16.3321i	−0.329850	−	1.65827i	0.358979	−	0.933346i	$-0.383125\pi$
$98$	7.00000	−	7.00000i	0.707107	−	0.707107i
$99$		0			0
$100$	8.66070	+	3.58738i	0.866070	+	0.358738i
$101$		−	17.7122i		−	1.76243i	−0.472714	−	0.881216i	$-0.656726\pi$
$101$		−	17.7122i		−	1.76243i	0.472714	−	0.881216i	$-0.343274\pi$
$102$		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		0.980785	−	0.195090i	$-0.0625000\pi$
$107$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$108$		0			0
$109$	13.0405	+	2.59392i	1.24905	+	0.248452i	0.774953	−	0.632019i	$-0.217774\pi$
$109$	13.0405	+	2.59392i	1.24905	+	0.248452i	0.474100	+	0.880471i	$0.342774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.20091	−	10.7769i	−0.677405	−	1.01381i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	−7.20091	−	10.7769i	−0.677405	−	1.01381i	0.320380	−	0.947289i	$-0.396189\pi$
$114$		0			0
$115$		0			0
$116$	−11.7650	+	17.6076i	−1.09235	+	1.63482i
$117$	−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
$122$	18.2653	+	12.2045i	1.65366	+	1.10494i
$123$		0			0
$124$		0			0
$125$	−4.50525	+	3.01031i	−0.402962	+	0.269251i
$126$		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
$130$	1.76340	+	2.63912i	0.154660	+	0.231466i
$131$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$131$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$132$		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	$-0.986017\pi$
$137$	−23.3868			−1.99807			−0.999035	+	0.0439140i	$0.986017\pi$
$138$		0			0
$139$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$139$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−11.0866	+	4.59220i	−0.923880	+	0.382683i
$145$	−2.26642	−	5.47161i	−0.188216	−	0.454393i
$146$	2.48479	−	12.4919i	0.205643	−	1.03384i
$147$		0			0
$148$	4.43996	+	22.3212i	0.364962	+	1.83479i
$149$	−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
$160$	3.10332	+	0.617288i	0.245339	+	0.0488009i
$161$		0			0
$162$	4.87076	+	11.7591i	0.382683	+	0.923880i
$163$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$163$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$164$	−10.8860	−	16.2921i	−0.850058	−	1.27220i
$165$		0			0
$166$		0			0
$167$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$167$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$168$		0			0
$169$		−	3.10051i		−	0.238500i
$170$	−2.41560	−	2.19139i	−0.185268	−	0.168072i
$171$		0			0
$172$		0			0
$173$	14.7758	+	9.87287i	1.12338	+	0.750621i	0.971326	−	0.237751i	$-0.0764102\pi$
$173$	14.7758	+	9.87287i	1.12338	+	0.750621i	0.152057	+	0.988372i	$0.451410\pi$
$174$		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	$0.375000\pi$
$180$	0.654733	−	3.29156i	0.0488009	−	0.245339i
$181$	7.64038	+	11.4346i	0.567905	+	0.849930i	0.998618	−	0.0525588i	$-0.0167377\pi$
$181$	7.64038	+	11.4346i	0.567905	+	0.849930i	−0.430713	+	0.902489i	$0.641738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.88039	−	2.43574i	−0.432335	−	0.179079i
$186$		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
$193$	10.1250	−	6.76534i	0.728817	−	0.486980i	−0.134962	−	0.990851i	$-0.543091\pi$
$193$	10.1250	−	6.76534i	0.728817	−	0.486980i	0.863779	+	0.503871i	$0.168091\pi$
$194$	−23.0970	−	4.59428i	−1.65827	−	0.329850i
$195$		0			0
$196$	−5.35757	−	12.9343i	−0.382683	−	0.923880i
$197$	1.44837	−	7.28145i	0.103192	−	0.518782i	−0.894267	−	0.447535i	$-0.852302\pi$
$197$	1.44837	−	7.28145i	0.103192	−	0.518782i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$199$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$200$	9.37427	−	9.37427i	0.662861	−	0.662861i
$201$		0			0
$202$	−23.1421	−	9.58579i	−1.62827	−	0.674454i
$203$		0			0
$204$		0			0
$205$	5.47997			0.382738
$206$		0			0
$207$		0			0
$208$	−11.3492	−	11.3492i	−0.786925	−	0.786925i
$209$		0			0
$210$		0			0
$211$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$211$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$212$	3.07107	−	1.27208i	0.210922	−	0.0873667i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	10.4466	−	15.6344i	0.707531	−	1.05890i
$219$		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
$225$	−9.94292	−	9.94292i	−0.662861	−	0.662861i
$226$	−17.9778	+	3.57601i	−1.19587	+	0.237873i
$227$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$227$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$228$		0			0
$229$	16.9853	−	7.03555i	1.12242	−	0.464922i	0.257223	−	0.966352i	$-0.417192\pi$
$229$	16.9853	−	7.03555i	1.12242	−	0.464922i	0.865198	+	0.501430i	$0.167192\pi$
$230$		0			0
$231$		0			0
$232$	16.6382	+	24.9008i	1.09235	+	1.63482i
$233$	5.95169	+	29.9211i	0.389908	+	1.96020i	0.231621	+	0.972806i	$0.425597\pi$
$233$	5.95169	+	29.9211i	0.389908	+	1.96020i	0.158287	+	0.987393i	$0.449403\pi$
$234$	−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
$241$	−24.4358	−	16.3275i	−1.57405	−	1.05174i	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−24.4358	−	16.3275i	−1.57405	−	1.05174i	−0.607811	−	0.794081i	$-0.707952\pi$
$242$	11.0000	+	11.0000i	0.707107	+	0.707107i
$243$		0			0
$244$	25.8310	−	17.2597i	1.65366	−	1.10494i
$245$	3.84016	+	0.763855i	0.245339	+	0.0488009i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	1.49494	+	7.51556i	0.0945482	+	0.475326i
$251$		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	$0.355131\pi$
$257$	−8.11794	+	19.5984i	−0.506383	+	1.22252i	−0.945951	+	0.324308i	$0.894869\pi$
$258$		0			0
$259$		0			0
$260$	4.40252	−	0.875715i	0.273032	−	0.0543095i
$261$	26.4113	−	17.6475i	1.63482	−	1.09235i
$262$		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
$265$	−0.181366	+	0.911791i	−0.0111413	+	0.0560109i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.07986	+	12.0924i	−0.492638	+	0.737284i	−0.991600	−	0.129339i	$-0.958714\pi$
$269$	−8.07986	+	12.0924i	−0.492638	+	0.737284i	0.498963	+	0.866623i	$0.333714\pi$
$270$		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
$277$	−30.3785	+	6.04265i	−1.82527	+	0.363068i	−0.984087	−	0.177690i	$-0.943138\pi$
$277$	−30.3785	+	6.04265i	−1.82527	+	0.363068i	−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		−0.555570	−	0.831470i	$-0.687500\pi$
$283$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$284$		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
$290$	−8.37558			−0.491831
$291$		0			0
$292$	−14.9767	−	10.0071i	−0.876445	−	0.585622i
$293$	−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
$296$	31.5669	+	6.27905i	1.83479	+	0.364962i
$297$		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
$305$			8.68844i			0.497499i
$306$	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		0.980785	−	0.195090i	$-0.0625000\pi$
$311$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$312$		0			0
$313$	−29.6026	−	5.88833i	−1.67324	−	0.332828i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−29.6026	−	5.88833i	−1.67324	−	0.332828i	−0.938436	+	0.345452i	$0.887726\pi$
$314$	9.23880	−	3.82683i	0.521375	−	0.215961i
$315$		0			0
$316$		0			0
$317$	6.14383	+	9.19489i	0.345072	+	0.516437i	0.962893	−	0.269882i	$-0.0869846\pi$
$317$	6.14383	+	9.19489i	0.345072	+	0.516437i	−0.617822	+	0.786318i	$0.711985\pi$
$318$		0			0
$319$		0			0
$320$	2.48603	−	3.72061i	0.138973	−	0.207988i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	7.19726	−	17.3757i	0.399232	−	0.963831i
$326$		0			0
$327$		0			0
$328$	−27.1782	+	5.40607i	−1.50066	+	0.298501i
$329$		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
$333$	6.65994	−	33.4818i	0.364962	−	1.83479i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	18.1920	−	27.2262i	0.990980	−	1.48311i	0.119405	−	0.992846i	$-0.461901\pi$
$337$	18.1920	−	27.2262i	0.990980	−	1.48311i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	−4.05101	−	1.67798i	−0.220346	−	0.0912701i
$339$		0			0
$340$	−4.17051	+	1.97017i	−0.226178	+	0.106847i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	20.8961	−	13.9624i	1.12338	−	0.750621i
$347$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$347$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$348$		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	$0.625000\pi$
$360$	−3.94630	−	2.63683i	−0.207988	−	0.138973i
$361$	−13.4350	−	13.4350i	−0.707107	−	0.707107i
$362$	19.0750	−	3.79426i	1.00256	−	0.199422i
$363$		0			0
$364$		0			0
$365$	4.65406	−	1.92778i	0.243605	−	0.100904i
$366$		0			0
$367$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$367$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$368$		0			0
$369$	5.73401	+	28.8268i	0.298501	+	1.50066i
$370$	−6.36489	+	6.36489i	−0.330895	+	0.330895i
$371$		0			0
$372$		0			0
$373$			26.7109i			1.38304i	0.722358	+	0.691519i	$0.243058\pi$
$373$			26.7109i			1.38304i	−0.722358	+	0.691519i	$0.756942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	35.3255	+	23.6038i	1.81936	+	1.21566i
$378$		0			0
$379$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$379$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$380$		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
$386$	−3.35971	−	16.8904i	−0.171005	−	0.859698i
$387$		0			0
$388$	−18.5027	+	27.6913i	−0.939334	+	1.40581i
$389$	−18.4776	−	7.65367i	−0.936851	−	0.388056i	−0.138578	−	0.990352i	$-0.544253\pi$
$389$	−18.4776	−	7.65367i	−0.936851	−	0.388056i	−0.798273	+	0.602295i	$0.794253\pi$
$390$		0			0
$391$		0			0
$392$	−19.7990			−1.00000
$393$		0			0
$394$	−8.72982	−	5.83308i	−0.439802	−	0.293866i
$395$		0			0
$396$		0			0
$397$	−29.0422	+	19.4054i	−1.45759	+	0.973928i	−0.461353	+	0.887217i	$0.652636\pi$
$397$	−29.0422	+	19.4054i	−1.45759	+	0.973928i	−0.996233	+	0.0867112i	$0.972364\pi$
$398$		0			0
$399$		0			0
$400$	−7.17476	−	17.3214i	−0.358738	−	0.866070i
$401$	−1.13973	+	5.72979i	−0.0569152	+	0.286132i	−0.998752	−	0.0499376i	$-0.984098\pi$
$401$	−1.13973	+	5.72979i	−0.0569152	+	0.286132i	0.941837	+	0.336070i	$0.109098\pi$
$402$		0			0
$403$		0			0
$404$	−25.0489	+	25.0489i	−1.24623	+	1.24623i
$405$	−2.79678	+	4.18568i	−0.138973	+	0.207988i
$406$		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	$0.202607\pi$
$410$	2.96574	−	7.15993i	0.146467	−	0.353604i
$411$		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		−0.555570	−	0.831470i	$-0.687500\pi$
$419$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$420$		0			0
$421$	10.1739	−	10.1739i	0.495847	−	0.495847i	−0.414295	−	0.910143i	$-0.635972\pi$
$421$	10.1739	−	10.1739i	0.495847	−	0.495847i	0.910143	+	0.414295i	$0.135972\pi$
$422$		0			0
$423$		0			0
$424$		−	4.70099i		−	0.228300i
$425$	−2.84441	+	19.1151i	−0.137974	+	0.927218i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$431$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$432$		0			0
$433$	−22.1731	+	9.18440i	−1.06557	+	0.441374i	−0.845426	−	0.534093i	$-0.820653\pi$
$433$	−22.1731	+	9.18440i	−1.06557	+	0.441374i	−0.220146	+	0.975467i	$0.570653\pi$
$434$		0			0
$435$		0			0
$436$	−14.7737	−	22.1104i	−0.707531	−	1.05890i
$437$		0			0
$438$		0			0
$439$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$439$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$440$		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
$445$	7.14437	+	4.77371i	0.338675	+	0.226296i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.2127	−	6.60642i	−1.56740	−	0.311776i	−0.666403	−	0.745592i	$-0.732167\pi$
$449$	−33.2127	−	6.60642i	−1.56740	−	0.311776i	−0.901002	+	0.433816i	$0.857167\pi$
$450$	−18.3721	+	7.60998i	−0.866070	+	0.358738i
$451$		0			0
$452$	−5.05725	+	25.4245i	−0.237873	+	1.19587i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	32.6641	+	13.5299i	1.52796	+	0.632902i	0.979167	−	0.203056i	$-0.0650872\pi$
$457$	32.6641	+	13.5299i	1.52796	+	0.632902i	0.548794	+	0.835958i	$0.315087\pi$
$458$		−	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	$0.250000\pi$
$464$	41.5391	−	8.26263i	1.92840	−	0.383583i
$465$		0			0
$466$	42.3149	+	8.41695i	1.96020	+	0.389908i
$467$		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		−0.831470	−	0.555570i	$-0.812500\pi$
$479$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$480$		0			0
$481$	44.7824	−	8.90777i	2.04190	−	0.406159i
$482$	−34.5574	+	23.0905i	−1.57405	+	1.05174i
$483$		0			0
$484$	20.3253	−	8.41904i	0.923880	−	0.382683i
$485$	−3.56438	−	8.60517i	−0.161850	−	0.390741i
$486$		0			0
$487$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$487$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$488$	−8.57128	−	43.0907i	−0.388003	−	1.95063i
$489$		0			0
$490$	3.07630	−	4.60401i	0.138973	−	0.207988i
$491$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$491$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$492$		0			0
$493$	−41.0976	−	14.7262i	−1.85094	−	0.663235i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$499$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$500$	10.6286	+	2.11416i	0.475326	+	0.0945482i
$501$		0			0
$502$		0			0
$503$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$503$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$504$		0			0
$505$	−1.93280	−	9.71682i	−0.0860083	−	0.432393i
$506$		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
$520$	1.23845	−	6.22610i	0.0543095	−	0.273032i
$521$	−25.2681	−	37.8163i	−1.10701	−	1.65676i	−0.625096	−	0.780548i	$-0.714940\pi$
$521$	−25.2681	−	37.8163i	−1.10701	−	1.65676i	−0.481919	−	0.876216i	$-0.660060\pi$
$522$	−8.76385	−	44.0588i	−0.383583	−	1.92840i
$523$		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
$530$	1.09316	+	0.730424i	0.0474837	+	0.0317276i
$531$		0			0
$532$		0			0
$533$	−32.6864	+	21.8404i	−1.41581	+	0.946012i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	11.4266	+	17.1012i	0.492638	+	0.737284i
$539$		0			0
$540$		0			0
$541$	−15.4501	+	23.1228i	−0.664253	+	0.994125i	0.334410	+	0.942428i	$0.391463\pi$
$541$	−15.4501	+	23.1228i	−0.664253	+	0.994125i	−0.998663	+	0.0516971i	$0.983537\pi$
$542$		0			0
$543$		0			0
$544$	18.7402	−	13.8854i	0.803480	−	0.595331i
$545$	7.43699			0.318566
$546$		0			0
$547$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$547$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$548$	33.0740	+	33.0740i	1.41285	+	1.41285i
$549$	−45.7046	+	9.09121i	−1.95063	+	0.388003i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−8.54560	+	42.9616i	−0.363068	+	1.82527i
$555$		0			0
$556$		0			0
$557$	28.5746	−	28.5746i	1.21074	−	1.21074i	0.239963	−	0.970782i	$-0.422865\pi$
$557$	28.5746	−	28.5746i	1.21074	−	1.21074i	0.970782	−	0.239963i	$-0.0771353\pi$
$558$		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
$565$	−5.12638	−	5.12638i	−0.215668	−	0.215668i
$566$		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		0.195090	−	0.980785i	$-0.437500\pi$
$571$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$572$		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	$0.388255\pi$
$577$			45.1116i			1.87802i	−0.343890	+	0.939010i	$0.611745\pi$
$578$	−23.9280	+	2.33456i	−0.995274	+	0.0971050i
$579$		0			0
$580$	−4.53283	+	10.9432i	−0.188216	+	0.454393i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−21.1802	+	14.1522i	−0.876445	+	0.585622i
$585$	−6.60377	−	1.31357i	−0.273032	−	0.0543095i
$586$	−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
$592$	25.2879	−	37.8460i	1.03933	−	1.55546i
$593$	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
$601$	−29.0865	+	19.4350i	−1.18646	+	0.792770i	−0.982510	−	0.186210i	$-0.940379\pi$
$601$	−29.0865	+	19.4350i	−1.18646	+	0.792770i	−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.20034	+	6.03454i	−0.0488009	+	0.245339i
$606$		0			0
$607$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$607$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$608$		0			0
$609$		0			0
$610$	11.3520	+	4.70215i	0.459629	+	0.190385i
$611$		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
$617$	46.8018	−	9.30946i	1.88417	−	0.374785i	0.887823	−	0.460186i	$-0.152217\pi$
$617$	46.8018	−	9.30946i	1.88417	−	0.374785i	0.996347	+	0.0854011i	$0.0272172\pi$
$618$		0			0
$619$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$619$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	14.4285	−	14.4285i	0.577139	−	0.577139i
$626$	−23.7143	+	35.4910i	−0.947814	+	1.41850i
$627$		0			0
$628$		−	14.1421i		−	0.564333i
$629$	−42.4224	+	20.0405i	−1.69149	+	0.799068i
$630$		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
$634$	15.3387	−	3.05106i	0.609179	−	0.121173i
$635$		0			0
$636$		0			0
$637$	−25.9497	+	10.7487i	−1.02817	+	0.425880i
$638$		0			0
$639$		0			0
$640$	−3.51578	−	5.26173i	−0.138973	−	0.207988i
$641$	4.12163	+	20.7208i	0.162794	+	0.818423i	0.972737	+	0.231913i	$0.0744985\pi$
$641$	4.12163	+	20.7208i	0.162794	+	0.818423i	−0.809942	+	0.586510i	$0.800502\pi$
$642$		0			0
$643$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$643$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$644$		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
$650$	−18.8073	−	18.8073i	−0.737685	−	0.737685i
$651$		0			0
$652$		0			0
$653$	−16.5914	−	3.30023i	−0.649272	−	0.129148i	−0.140542	−	0.990075i	$-0.544885\pi$
$653$	−16.5914	−	3.30023i	−0.649272	−	0.129148i	−0.508729	+	0.860927i	$0.669885\pi$
$654$		0			0
$655$		0			0
$656$	−7.64534	+	38.4357i	−0.298501	+	1.50066i
$657$	15.0107	+	22.4650i	0.585622	+	0.876445i
$658$		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
$666$	−40.1417	−	26.8218i	−1.55546	−	1.03933i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−4.86046	+	24.4352i	−0.187357	+	0.941908i	0.766637	+	0.642081i	$0.221929\pi$
$673$	−4.86046	+	24.4352i	−0.187357	+	0.941908i	−0.953994	+	0.299827i	$0.903071\pi$
$674$	−25.7273	−	38.5037i	−0.990980	−	1.48311i
$675$		0			0
$676$	−4.38478	+	4.38478i	−0.168645	+	0.168645i
$677$	23.4036	−	35.0259i	0.899472	−	1.34615i	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	23.4036	−	35.0259i	0.899472	−	1.34615i	0.937905	−	0.346893i	$-0.112763\pi$
$678$		0			0
$679$		0			0
$680$	0.317082	+	6.51528i	0.0121595	+	0.249850i
$681$		0			0
$682$		0			0
$683$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$683$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$684$		0			0
$685$	−12.8299	+	2.55202i	−0.490204	+	0.0975077i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.55213	−	6.16140i	−0.0972286	−	0.234731i
$690$		0			0
$691$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$691$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$692$	−6.93378	−	34.8585i	−0.263583	−	1.32512i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	27.1412	−	29.9182i	1.02805	−	1.13323i
$698$	52.4482			1.98519
$699$		0			0
$700$		0			0
$701$	−7.53828	−	7.53828i	−0.284717	−	0.284717i	0.550270	−	0.834987i	$-0.314525\pi$
$701$	−7.53828	−	7.53828i	−0.284717	−	0.284717i	−0.834987	+	0.550270i	$0.814525\pi$
$702$		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
$709$	−7.27723	−	36.5851i	−0.273302	−	1.37398i	−0.836639	−	0.547755i	$-0.815483\pi$
$709$	−7.27723	−	36.5851i	−0.273302	−	1.37398i	0.563337	−	0.826227i	$-0.309517\pi$
$710$		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		0.980785	−	0.195090i	$-0.0625000\pi$
$719$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$720$	−5.58091	+	3.72904i	−0.207988	+	0.138973i
$721$		0			0
$722$	−24.8247	+	10.2827i	−0.923880	+	0.382683i
$723$		0			0
$724$	5.36589	−	26.9762i	0.199422	−	1.00256i
$725$	27.5721	+	41.2645i	1.02400	+	1.53253i
$726$		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
$730$		−	7.12413i		−	0.263676i
$731$		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
$738$	40.7673	+	8.10911i	1.50066	+	0.298501i
$739$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$739$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$740$	4.87147	+	11.7608i	0.179079	+	0.432335i
$741$		0			0
$742$		0			0
$743$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$743$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$744$		0			0
$745$	−1.31842	+	1.97315i	−0.0483030	+	0.0722906i
$746$	34.8995	+	14.4558i	1.27776	+	0.529266i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$751$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$752$		0			0
$753$		0			0
$754$	49.9579	−	33.3808i	1.81936	−	1.21566i
$755$		0			0
$756$		0			0
$757$	18.9419	+	45.7297i	0.688454	+	1.66207i	0.747873	+	0.663841i	$0.231075\pi$
$757$	18.9419	+	45.7297i	0.688454	+	1.66207i	−0.0594198	+	0.998233i	$0.518925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.8485	+	15.8485i	−0.574509	+	0.574509i	−0.933385	−	0.358876i	$-0.883160\pi$
$761$	−15.8485	+	15.8485i	−0.574509	+	0.574509i	0.358876	+	0.933385i	$0.383160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	6.91050	−	0.336316i	0.249850	−	0.0121595i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−2.14885	−	2.14885i	−0.0774896	−	0.0774896i	0.667300	−	0.744789i	$-0.267450\pi$
$769$	−2.14885	−	2.14885i	−0.0774896	−	0.0774896i	−0.744789	+	0.667300i	$0.767450\pi$
$770$		0			0
$771$		0			0
$772$	−23.8866	−	4.75134i	−0.859698	−	0.171005i
$773$	40.6507	−	16.8381i	1.46210	−	0.605623i	0.497061	−	0.867715i	$-0.334412\pi$
$773$	40.6507	−	16.8381i	1.46210	−	0.605623i	0.965043	+	0.262092i	$0.0844124\pi$
$774$		0			0
$775$		0			0
$776$	26.1668	+	39.1614i	0.939334	+	1.40581i
$777$		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
$785$	3.28858	+	2.19736i	0.117375	+	0.0784271i
$786$		0			0
$787$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$787$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$788$	−12.3458	+	8.24922i	−0.439802	+	0.293866i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−34.6277	−	51.8240i	−1.22967	−	1.84032i
$794$	9.63682	+	48.4476i	0.341998	+	1.71934i
$795$		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
$800$	−26.5145			−0.937427
$801$	−17.6360	+	42.5772i	−0.623139	+	1.50439i
$802$	6.86952	+	4.59006i	0.242571	+	0.162081i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	19.1716	+	46.2843i	0.674454	+	1.62827i
$809$	7.99999	−	40.2187i	0.281265	−	1.41401i	−0.539133	−	0.842220i	$-0.681248\pi$
$809$	7.99999	−	40.2187i	0.281265	−	1.41401i	0.820398	−	0.571793i	$-0.193752\pi$
$810$	3.95525	+	5.91945i	0.138973	+	0.207988i
$811$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$811$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$812$		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
$820$	−7.74985	−	7.74985i	−0.270637	−	0.270637i
$821$	−17.3672	+	3.45456i	−0.606120	+	0.120565i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−17.3672	+	3.45456i	−0.606120	+	0.120565i	−0.117517	+	0.993071i	$0.537493\pi$
$822$		0			0
$823$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$823$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$827$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$828$		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		0.831470	−	0.555570i	$-0.187500\pi$
$839$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$840$		0			0
$841$	−76.7839	+	31.8049i	−2.64772	+	1.09672i
$842$	−7.78680	−	18.7990i	−0.268351	−	0.647856i
$843$		0			0
$844$		0			0
$845$	−0.338334	−	1.70092i	−0.0116390	−	0.0585134i
$846$		0			0
$847$		0			0
$848$	−6.14214	−	2.54416i	−0.210922	−	0.0873667i
$849$		0			0
$850$	23.4357	+	14.0614i	0.803837	+	0.482302i
$851$		0			0
$852$		0			0
$853$	−31.6731	−	21.1633i	−1.08447	−	0.724616i	−0.121054	−	0.992646i	$-0.538628\pi$
$853$	−31.6731	−	21.1633i	−1.08447	−	0.724616i	−0.963411	+	0.268029i	$0.913628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.40230	−	0.676760i	−0.116220	−	0.0231177i	0.136637	−	0.990621i	$-0.456370\pi$
$857$	−3.40230	−	0.676760i	−0.116220	−	0.0231177i	−0.252858	+	0.967503i	$0.581370\pi$
$858$		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
$865$	9.18327	+	3.80383i	0.312240	+	0.129334i
$866$			33.9411i			1.15337i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−36.8841	+	7.33670i	−1.24905	+	0.248452i
$873$	41.5369	−	27.7541i	1.40581	−	0.939334i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	4.33987	−	21.8180i	0.146547	−	0.736741i	−0.835705	−	0.549178i	$-0.814941\pi$
$877$	4.33987	−	21.8180i	0.146547	−	0.736741i	0.982252	−	0.187564i	$-0.0600591\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−13.2199	+	19.7850i	−0.445391	+	0.666574i	−0.984444	−	0.175697i	$-0.943782\pi$
$881$	−13.2199	+	19.7850i	−0.445391	+	0.666574i	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		−0.831470	−	0.555570i	$-0.812500\pi$
$887$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$888$		0			0
$889$		0			0
$890$	10.1037	−	6.75105i	0.338675	−	0.226296i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−26.6063	+	39.8191i	−0.887863	+	1.32878i
$899$		0			0
$900$			28.1228i			0.937427i
$901$	4.07969	+	5.50610i	0.135914	+	0.183435i
$902$		0			0
$903$		0			0
$904$	30.4818	+	20.3673i	1.01381	+	0.677405i
$905$	5.43924	+	5.43924i	0.180807	+	0.180807i
$906$		0			0
$907$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$907$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$908$		0			0
$909$	49.0919	−	20.3345i	1.62827	−	0.674454i
$910$		0			0
$911$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$911$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$912$		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
$925$	52.3112	+	10.4054i	1.71998	+	0.342126i
$926$		0			0
$927$		0			0
$928$	11.6851	−	58.7451i	0.383583	−	1.92840i
$929$	−4.27931	−	6.40444i	−0.140400	−	0.210123i	0.754606	−	0.656179i	$-0.227828\pi$
$929$	−4.27931	−	6.40444i	−0.140400	−	0.210123i	−0.895005	+	0.446056i	$0.852828\pi$
$930$		0			0
$931$		0			0
$932$	33.8979	−	50.7318i	1.11036	−	1.66178i
$933$		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
$941$	49.3366	−	32.9657i	1.60833	−	1.07465i	0.662955	−	0.748660i	$-0.269302\pi$
$941$	49.3366	−	32.9657i	1.60833	−	1.07465i	0.945373	−	0.325991i	$-0.105698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$947$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$948$		0			0
$949$	−20.0770	+	30.0473i	−0.651726	+	0.975377i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−41.7875			−1.35363			−0.676815	−	0.736153i	$-0.736640\pi$
$953$	−41.7875			−1.35363			−0.676815	+	0.736153i	$0.736640\pi$
$954$	−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
$962$	12.5975	−	63.3318i	0.406159	−	2.04190i
$963$		0			0
$964$	11.4669	+	57.6479i	0.369323	+	1.85671i
$965$	4.81629	−	4.81629i	0.155042	−	0.155042i
$966$		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
$970$	−13.1722			−0.422935
$971$		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
$976$	−60.9395	−	12.1216i	−1.95063	−	0.388003i
$977$	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
$980$	−4.35055	−	6.51106i	−0.138973	−	0.207988i
$981$	7.78175	+	39.1215i	0.248452	+	1.24905i
$982$		0			0
$983$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$983$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$984$		0			0
$985$		−	4.15261i		−	0.132313i
$986$	−41.4826	+	45.7268i	−1.32107	+	1.45624i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$991$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	32.3442	+	48.4065i	1.02435	+	1.53305i	0.834328	+	0.551268i	$0.185856\pi$
$997$	32.3442	+	48.4065i	1.02435	+	1.53305i	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.27.1	✓	8
3.2	odd	2		612.2.bd.a.163.1		8
4.3	odd	2	CM	68.2.i.a.27.1	✓	8
12.11	even	2		612.2.bd.a.163.1		8
17.12	odd	16	inner	68.2.i.a.63.1	yes	8
51.29	even	16		612.2.bd.a.199.1		8
68.63	even	16	inner	68.2.i.a.63.1	yes	8
204.131	odd	16		612.2.bd.a.199.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.i.a.27.1	✓	8	1.1	even	1	trivial
68.2.i.a.27.1	✓	8	4.3	odd	2	CM
68.2.i.a.63.1	yes	8	17.12	odd	16	inner
68.2.i.a.63.1	yes	8	68.63	even	16	inner
612.2.bd.a.163.1		8	3.2	odd	2
612.2.bd.a.163.1		8	12.11	even	2
612.2.bd.a.199.1		8	51.29	even	16
612.2.bd.a.199.1		8	204.131	odd	16

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)

\(f(q)\)	\(=\)	\(q+(0.541196 - 1.30656i) q^{2} +(-1.41421 - 1.41421i) q^{4} +(0.548594 - 0.109122i) q^{5} +(-2.61313 + 1.08239i) q^{8} +(1.14805 + 2.77164i) q^{9} +O(q^{10})\)	\(q+(0.541196 - 1.30656i) q^{2} +(-1.41421 - 1.41421i) q^{4} +(0.548594 - 0.109122i) q^{5} +(-2.61313 + 1.08239i) q^{8} +(1.14805 + 2.77164i) q^{9} +(0.154322 - 0.775829i) q^{10} +(-2.83730 + 2.83730i) q^{13} +4.00000i q^{16} +(2.12132 - 3.53553i) q^{17} +4.24264 q^{18} +(-0.930151 - 0.621507i) q^{20} +(-4.33035 + 1.79369i) q^{25} +(2.17157 + 5.24264i) q^{26} +(-2.06566 - 10.3848i) q^{29} +(5.22625 + 2.16478i) q^{32} +(-3.47135 - 4.68506i) q^{34} +(2.29610 - 5.54328i) q^{36} +(-9.46149 - 6.32197i) q^{37} +(-1.31543 + 0.878944i) q^{40} +(9.60894 + 1.91134i) q^{41} +(0.932261 + 1.39523i) q^{45} +(6.46716 + 2.67878i) q^{49} +6.62861i q^{50} +8.02509 q^{52} +(-0.636039 + 1.53553i) q^{53} +(-14.6863 - 2.92128i) q^{58} +(-3.03040 + 15.2349i) q^{61} +(5.65685 - 5.65685i) q^{64} +(-1.24691 + 1.86614i) q^{65} +(-8.00000 + 2.00000i) q^{68} +(-6.00000 - 6.00000i) q^{72} +(8.83311 - 1.75701i) q^{73} +(-13.3806 + 8.94061i) q^{74} +(0.436489 + 2.19438i) q^{80} +(-6.36396 + 6.36396i) q^{81} +(7.69760 - 11.5203i) q^{82} +(0.777939 - 2.17106i) q^{85} +(10.8624 + 10.8624i) q^{89} +(2.32749 - 0.462966i) q^{90} +(-3.24865 - 16.3321i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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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