LMFDB - Embedded newform 68.2.i.a.23.1

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			23.1
Root			$0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	68.23
Dual form			68.2.i.a.3.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.30656 - 0.541196i) q^{2} +(1.41421 + 1.41421i) q^{4} +(1.52334 - 2.27983i) q^{5} +(-1.08239 - 2.61313i) q^{8} +(2.77164 - 1.14805i) q^{9} +O(q^{10})$	\(q+(-1.30656 - 0.541196i) q^{2} +(1.41421 + 1.41421i) q^{4} +(1.52334 - 2.27983i) q^{5} +(-1.08239 - 2.61313i) q^{8} +(2.77164 - 1.14805i) q^{9} +(-3.22417 + 2.15432i) q^{10} +(-4.23671 + 4.23671i) q^{13} +4.00000i q^{16} +(-2.12132 + 3.53553i) q^{17} -4.24264 q^{18} +(5.37849 - 1.06985i) q^{20} +(-0.963670 - 2.32650i) q^{25} +(7.82843 - 3.24264i) q^{26} +(-7.38476 - 4.93434i) q^{29} +(2.16478 - 5.22625i) q^{32} +(4.68506 - 3.47135i) q^{34} +(5.54328 + 2.29610i) q^{36} +(4.90775 - 0.976213i) q^{37} +(-7.60634 - 1.51299i) q^{40} +(7.08866 + 10.6089i) q^{41} +(1.60477 - 8.06774i) q^{45} +(-2.67878 + 6.46716i) q^{49} +3.56126i q^{50} -11.9832 q^{52} +(-13.3640 - 5.53553i) q^{53} +(6.97821 + 10.4436i) q^{58} +(8.16380 - 5.45488i) q^{61} +(-5.65685 + 5.65685i) q^{64} +(3.20506 + 16.1129i) q^{65} +(-8.00000 + 2.00000i) q^{68} +(-6.00000 - 6.00000i) q^{72} +(9.24299 - 13.8331i) q^{73} +(-6.94061 - 1.38057i) q^{74} +(9.11933 + 6.09334i) q^{80} +(6.36396 - 6.36396i) q^{81} +(-3.52027 - 17.6976i) q^{82} +(4.82894 + 10.2221i) q^{85} +(-7.74652 - 7.74652i) q^{89} +(-6.46297 + 9.67251i) q^{90} +(3.60414 + 2.40821i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$-1$	$e\left(\frac{15}{16}\right)$

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 $

$2$	−1.30656	−	0.541196i	−0.923880	−	0.382683i
$2$	−1.30656	−	0.541196i	−0.923880	−	0.382683i
$3$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$3$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$4$	1.41421	+	1.41421i	0.707107	+	0.707107i
$5$	1.52334	−	2.27983i	0.681256	−	1.01957i	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	1.52334	−	2.27983i	0.681256	−	1.01957i	0.997484	−	0.0708890i	$-0.0225836\pi$
$6$		0			0
$7$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$7$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$8$	−1.08239	−	2.61313i	−0.382683	−	0.923880i
$9$	2.77164	−	1.14805i	0.923880	−	0.382683i
$10$	−3.22417	+	2.15432i	−1.01957	+	0.681256i
$11$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$11$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$12$		0			0
$13$	−4.23671	+	4.23671i	−1.17505	+	1.17505i	−0.194064	+	0.980989i	$0.562167\pi$
$13$	−4.23671	+	4.23671i	−1.17505	+	1.17505i	−0.980989	+	0.194064i	$0.937833\pi$
$14$		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		−0.923880	−	0.382683i	$-0.875000\pi$
$19$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$20$	5.37849	−	1.06985i	1.20267	−	0.239225i
$21$		0			0
$22$		0			0
$23$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$23$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$24$		0			0
$25$	−0.963670	−	2.32650i	−0.192734	−	0.465301i
$26$	7.82843	−	3.24264i	1.53528	−	0.635934i
$27$		0			0
$28$		0			0
$29$	−7.38476	−	4.93434i	−1.37132	−	0.916284i	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−7.38476	−	4.93434i	−1.37132	−	0.916284i	−0.999926	+	0.0121924i	$0.996119\pi$
$30$		0			0
$31$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$31$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$32$	2.16478	−	5.22625i	0.382683	−	0.923880i
$33$		0			0
$34$	4.68506	−	3.47135i	0.803480	−	0.595331i
$35$		0			0
$36$	5.54328	+	2.29610i	0.923880	+	0.382683i
$37$	4.90775	−	0.976213i	0.806830	−	0.160488i	0.225592	−	0.974222i	$-0.427568\pi$
$37$	4.90775	−	0.976213i	0.806830	−	0.160488i	0.581238	+	0.813733i	$0.302568\pi$
$38$		0			0
$39$		0			0
$40$	−7.60634	−	1.51299i	−1.20267	−	0.239225i
$41$	7.08866	+	10.6089i	1.10706	+	1.65684i	0.624695	+	0.780869i	$0.285223\pi$
$41$	7.08866	+	10.6089i	1.10706	+	1.65684i	0.482368	+	0.875969i	$0.339777\pi$
$42$		0			0
$43$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$43$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$44$		0			0
$45$	1.60477	−	8.06774i	0.239225	−	1.20267i
$46$		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
$49$	−2.67878	+	6.46716i	−0.382683	+	0.923880i
$50$			3.56126i			0.503638i
$51$		0			0
$52$	−11.9832			−1.66178
$53$	−13.3640	−	5.53553i	−1.83568	−	0.760364i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−13.3640	−	5.53553i	−1.83568	−	0.760364i	−0.874157	−	0.485643i	$-0.838586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	6.97821	+	10.4436i	0.916284	+	1.37132i
$59$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$59$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$60$		0			0
$61$	8.16380	−	5.45488i	1.04527	−	0.698425i	0.0905357	−	0.995893i	$-0.471142\pi$
$61$	8.16380	−	5.45488i	1.04527	−	0.698425i	0.954732	+	0.297468i	$0.0961421\pi$
$62$		0			0
$63$		0			0
$64$	−5.65685	+	5.65685i	−0.707107	+	0.707107i
$65$	3.20506	+	16.1129i	0.397539	+	1.99856i
$66$		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		0.980785	−	0.195090i	$-0.0625000\pi$
$71$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$72$	−6.00000	−	6.00000i	−0.707107	−	0.707107i
$73$	9.24299	−	13.8331i	1.08181	−	1.61904i	0.351123	−	0.936329i	$-0.385800\pi$
$73$	9.24299	−	13.8331i	1.08181	−	1.61904i	0.730686	−	0.682713i	$-0.239200\pi$
$74$	−6.94061	−	1.38057i	−0.806830	−	0.160488i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$79$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$80$	9.11933	+	6.09334i	1.01957	+	0.681256i
$81$	6.36396	−	6.36396i	0.707107	−	0.707107i
$82$	−3.52027	−	17.6976i	−0.388749	−	1.95437i
$83$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$83$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$84$		0			0
$85$	4.82894	+	10.2221i	0.523773	+	1.10874i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.74652	−	7.74652i	−0.821130	−	0.821130i	0.165140	−	0.986270i	$-0.447192\pi$
$89$	−7.74652	−	7.74652i	−0.821130	−	0.821130i	−0.986270	+	0.165140i	$0.947192\pi$
$90$	−6.46297	+	9.67251i	−0.681256	+	1.01957i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.60414	+	2.40821i	0.365944	+	0.244516i	0.724924	−	0.688829i	$-0.241875\pi$
$97$	3.60414	+	2.40821i	0.365944	+	0.244516i	−0.358979	+	0.933346i	$0.616875\pi$
$98$	7.00000	−	7.00000i	0.707107	−	0.707107i
$99$		0			0
$100$	1.92734	−	4.65301i	0.192734	−	0.465301i
$101$			9.50143i			0.945427i	0.881216	+	0.472714i	$0.156726\pi$
$101$			9.50143i			0.945427i	−0.881216	+	0.472714i	$0.843274\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	15.6569	+	6.48528i	1.53528	+	0.635934i
$105$		0			0
$106$	14.4650	+	14.4650i	1.40497	+	1.40497i
$107$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$107$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$108$		0			0
$109$	−11.5482	−	17.2831i	−1.10612	−	1.65542i	−0.632019	−	0.774953i	$-0.717774\pi$
$109$	−11.5482	−	17.2831i	−1.10612	−	1.65542i	−0.474100	−	0.880471i	$-0.657226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.536781	−	2.69858i	0.0504961	−	0.253861i	−0.947289	−	0.320380i	$-0.896189\pi$
$113$	0.536781	−	2.69858i	0.0504961	−	0.253861i	0.997785	+	0.0665190i	$0.0211893\pi$
$114$		0			0
$115$		0			0
$116$	−3.46542	−	17.4218i	−0.321756	−	1.61758i
$117$	−6.87868	+	16.6066i	−0.635934	+	1.53528i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.1627	−	4.20952i	−0.923880	−	0.382683i
$122$	−13.6187	+	2.70892i	−1.23298	+	0.245254i
$123$		0			0
$124$		0			0
$125$	6.67420	+	1.32758i	0.596958	+	0.118742i
$126$		0			0
$127$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$127$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$128$	10.4525	−	4.32957i	0.923880	−	0.382683i
$129$		0			0
$130$	4.53264	−	22.7871i	0.397539	−	1.99856i
$131$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$131$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	11.5349	+	1.71644i	0.989109	+	0.147184i
$137$	1.02800			0.0878279			0.0439140	−	0.999035i	$-0.486017\pi$
$137$	1.02800			0.0878279			0.0439140	+	0.999035i	$0.486017\pi$
$138$		0			0
$139$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$139$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	4.59220	+	11.0866i	0.382683	+	0.923880i
$145$	−22.4990	+	9.31937i	−1.86844	+	0.773932i
$146$	−19.5630	+	13.0716i	−1.61904	+	1.08181i
$147$		0			0
$148$	8.32119	+	5.56004i	0.683997	+	0.457032i
$149$	−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		0.382683	−	0.923880i	$-0.375000\pi$
$151$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$152$		0			0
$153$	−1.82056	+	12.2346i	−0.147184	+	0.989109i
$154$		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
$160$	−8.61729	−	12.8967i	−0.681256	−	1.01957i
$161$		0			0
$162$	−11.7591	+	4.87076i	−0.923880	+	0.382683i
$163$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$163$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$164$	−4.97842	+	25.0282i	−0.388749	+	1.95437i
$165$		0			0
$166$		0			0
$167$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$167$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$168$		0			0
$169$		−	22.8995i		−	1.76150i
$170$	−0.777180	−	15.9692i	−0.0596070	−	1.22478i
$171$		0			0
$172$		0			0
$173$	−1.12713	+	0.224199i	−0.0856938	+	0.0170456i	−0.237751	−	0.971326i	$-0.576410\pi$
$173$	−1.12713	+	0.224199i	−0.0856938	+	0.0170456i	0.152057	+	0.988372i	$0.451410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	5.92893	+	14.3137i	0.444392	+	1.07286i
$179$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$179$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$180$	13.6790	−	9.14001i	1.01957	−	0.681256i
$181$	−1.29328	+	6.50175i	−0.0961287	+	0.483271i	0.902489	+	0.430713i	$0.141738\pi$
$181$	−1.29328	+	6.50175i	−0.0961287	+	0.483271i	−0.998618	+	0.0525588i	$0.983262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.25056	−	12.6760i	0.386029	−	0.931955i
$186$		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
$193$	25.7653	+	5.12504i	1.85463	+	0.368909i	0.990851	−	0.134962i	$-0.0430913\pi$
$193$	25.7653	+	5.12504i	1.85463	+	0.368909i	0.863779	+	0.503871i	$0.168091\pi$
$194$	−3.40572	−	5.09702i	−0.244516	−	0.365944i
$195$		0			0
$196$	−12.9343	+	5.35757i	−0.923880	+	0.382683i
$197$	20.2814	−	13.5516i	1.44499	−	0.965514i	0.447535	−	0.894267i	$-0.352302\pi$
$197$	20.2814	−	13.5516i	1.44499	−	0.965514i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$199$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$200$	−5.03638	+	5.03638i	−0.356126	+	0.356126i
$201$		0			0
$202$	5.14214	−	12.4142i	0.361799	−	0.873461i
$203$		0			0
$204$		0			0
$205$	34.9850			2.44346
$206$		0			0
$207$		0			0
$208$	−16.9469	−	16.9469i	−1.17505	−	1.17505i
$209$		0			0
$210$		0			0
$211$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$211$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$212$	−11.0711	−	26.7279i	−0.760364	−	1.83568i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	5.73491	+	28.8314i	0.388417	+	1.95271i
$219$		0			0
$220$		0			0
$221$	−5.99162	−	23.9665i	−0.403040	−	1.61216i
$222$		0			0
$223$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$223$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$224$		0			0
$225$	−5.34189	−	5.34189i	−0.356126	−	0.356126i
$226$	−2.16180	+	3.23536i	−0.143801	+	0.215213i
$227$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$227$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$228$		0			0
$229$	7.03555	+	16.9853i	0.464922	+	1.12242i	0.966352	+	0.257223i	$0.0828075\pi$
$229$	7.03555	+	16.9853i	0.464922	+	1.12242i	−0.501430	+	0.865198i	$0.667192\pi$
$230$		0			0
$231$		0			0
$232$	−4.90085	+	24.6382i	−0.321756	+	1.61758i
$233$	−18.6074	−	12.4331i	−1.21901	−	0.814519i	−0.231621	−	0.972806i	$-0.574403\pi$
$233$	−18.6074	−	12.4331i	−1.21901	−	0.814519i	−0.987393	+	0.158287i	$0.949403\pi$
$234$	17.9749	−	17.9749i	1.17505	−	1.17505i
$235$		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
$241$	−27.3275	+	5.43577i	−1.76032	+	0.350149i	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−27.3275	+	5.43577i	−1.76032	+	0.350149i	−0.794081	+	0.607811i	$0.792048\pi$
$242$	11.0000	+	11.0000i	0.707107	+	0.707107i
$243$		0			0
$244$	19.2597	+	3.83100i	1.23298	+	0.245254i
$245$	10.6634	+	15.9588i	0.681256	+	1.01957i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−8.00178	−	5.34662i	−0.506077	−	0.338150i
$251$		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
$257$	19.5984	+	8.11794i	1.22252	+	0.506383i	0.898209	−	0.439568i	$-0.144869\pi$
$257$	19.5984	+	8.11794i	1.22252	+	0.506383i	0.324308	+	0.945951i	$0.394869\pi$
$258$		0			0
$259$		0			0
$260$	−18.2545	+	27.3198i	−1.13210	+	1.69430i
$261$	−26.1328	−	5.19813i	−1.61758	−	0.321756i
$262$		0			0
$263$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$263$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$264$		0			0
$265$	−32.9779	+	22.0351i	−2.02582	+	1.35361i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.04978	+	10.3049i	0.124977	+	0.628302i	0.991600	+	0.129339i	$0.0412856\pi$
$269$	2.04978	+	10.3049i	0.124977	+	0.628302i	−0.866623	+	0.498963i	$0.833714\pi$
$270$		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
$274$	−1.34315	−	0.556349i	−0.0811424	−	0.0336103i
$275$		0			0
$276$		0			0
$277$	−16.9573	+	25.3785i	−1.01887	+	1.52484i	−0.177690	+	0.984087i	$0.556862\pi$
$277$	−16.9573	+	25.3785i	−1.01887	+	1.52484i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	19.8492	−	8.22183i	1.18411	−	0.490473i	0.298275	−	0.954480i	$-0.403589\pi$
$281$	19.8492	−	8.22183i	1.18411	−	0.490473i	0.885832	+	0.464007i	$0.153589\pi$
$282$		0			0
$283$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$283$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$284$		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
$290$	34.4399			2.02238
$291$		0			0
$292$	32.6345	−	6.49141i	1.90979	−	0.379881i
$293$	2.82843	+	2.82843i	0.165238	+	0.165238i	0.784883	−	0.619644i	$-0.212723\pi$
$293$	2.82843	+	2.82843i	0.165238	+	0.165238i	−0.619644	+	0.784883i	$0.712723\pi$
$294$		0			0
$295$		0			0
$296$	−7.86308	−	11.7679i	−0.457032	−	0.683997i
$297$		0			0
$298$	5.54328	−	2.29610i	0.321113	−	0.133010i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	26.9217i		−	1.54153i
$306$	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		0.555570	−	0.831470i	$-0.312500\pi$
$311$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$312$		0			0
$313$	−19.1117	−	28.6026i	−1.08026	−	1.61672i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−19.1117	−	28.6026i	−1.08026	−	1.61672i	−0.345452	−	0.938436i	$-0.612274\pi$
$314$	−3.82683	−	9.23880i	−0.215961	−	0.521375i
$315$		0			0
$316$		0			0
$317$	−6.19489	+	31.1438i	−0.347940	+	1.74921i	0.269882	+	0.962893i	$0.413015\pi$
$317$	−6.19489	+	31.1438i	−0.347940	+	1.74921i	−0.617822	+	0.786318i	$0.711985\pi$
$318$		0			0
$319$		0			0
$320$	4.27939	+	21.5140i	0.239225	+	1.20267i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	13.9395	+	5.77394i	0.773226	+	0.320281i
$326$		0			0
$327$		0			0
$328$	20.0498	−	30.0066i	1.10706	−	1.65684i
$329$		0			0
$330$		0			0
$331$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$331$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$332$		0			0
$333$	12.4818	−	8.34006i	0.683997	−	0.457032i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.22622	−	11.1920i	−0.121270	−	0.609666i	−0.992846	−	0.119405i	$-0.961901\pi$
$337$	−2.22622	−	11.1920i	−0.121270	−	0.609666i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	−12.3931	+	29.9196i	−0.674097	+	1.62741i
$339$		0			0
$340$	−7.62702	+	21.2853i	−0.413633	+	1.15436i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.59400	+	0.317066i	0.0856938	+	0.0170456i
$347$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$347$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$348$		0			0
$349$	−4.19239	+	1.73654i	−0.224413	+	0.0929551i	−0.492057	−	0.870563i	$-0.663755\pi$
$349$	−4.19239	+	1.73654i	−0.224413	+	0.0929551i	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	−0.940887	−	0.338719i	$-0.890006\pi$
$353$	−11.3137	+	11.3137i	−0.602168	+	0.602168i	0.338719	+	0.940887i	$0.390006\pi$
$354$		0			0
$355$		0			0
$356$		−	21.9105i		−	1.16125i
$357$		0			0
$358$		0			0
$359$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$359$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$360$	−22.8190	+	4.53898i	−1.20267	+	0.239225i
$361$	13.4350	+	13.4350i	0.707107	+	0.707107i
$362$	5.20847	−	7.79503i	0.273751	−	0.409698i
$363$		0			0
$364$		0			0
$365$	−17.4570	−	42.1449i	−0.913741	−	2.20597i
$366$		0			0
$367$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$367$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$368$		0			0
$369$	31.8268	+	21.2660i	1.65684	+	1.10706i
$370$	−13.7204	+	13.7204i	−0.713288	+	0.713288i
$371$		0			0
$372$		0			0
$373$			27.9021i			1.44472i	0.691519	+	0.722358i	$0.256942\pi$
$373$			27.9021i			1.44472i	−0.691519	+	0.722358i	$0.743058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	52.1925	−	10.3817i	2.68805	−	0.534687i
$378$		0			0
$379$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$379$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$383$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$384$		0			0
$385$		0			0
$386$	−30.8904	−	20.6403i	−1.57228	−	1.05056i
$387$		0			0
$388$	1.69130	+	8.50273i	0.0858627	+	0.431661i
$389$	7.65367	−	18.4776i	0.388056	−	0.936851i	−0.602295	−	0.798273i	$-0.705747\pi$
$389$	7.65367	−	18.4776i	0.388056	−	0.936851i	0.990352	−	0.138578i	$-0.0442530\pi$
$390$		0			0
$391$		0			0
$392$	19.7990			1.00000
$393$		0			0
$394$	−33.8331	+	6.72982i	−1.70449	+	0.339043i
$395$		0			0
$396$		0			0
$397$	10.9201	+	2.17214i	0.548064	+	0.109017i	0.461353	−	0.887217i	$-0.347364\pi$
$397$	10.9201	+	2.17214i	0.548064	+	0.109017i	0.0867112	+	0.996233i	$0.472364\pi$
$398$		0			0
$399$		0			0
$400$	9.30602	−	3.85468i	0.465301	−	0.192734i
$401$	−26.7298	+	17.8603i	−1.33482	+	0.891900i	−0.998752	−	0.0499376i	$-0.984098\pi$
$401$	−26.7298	+	17.8603i	−1.33482	+	0.891900i	−0.336070	+	0.941837i	$0.609098\pi$
$402$		0			0
$403$		0			0
$404$	−13.4370	+	13.4370i	−0.668518	+	0.668518i
$405$	−4.81432	−	24.2032i	−0.239225	−	1.20267i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−32.5269			−1.60835			−0.804176	−	0.594391i	$-0.797393\pi$
$409$	−32.5269			−1.60835			−0.804176	+	0.594391i	$0.797393\pi$
$410$	−45.7101	−	18.9338i	−2.25746	−	0.935071i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	12.9706	+	31.3137i	0.635934	+	1.53528i
$417$		0			0
$418$		0			0
$419$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$419$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$420$		0			0
$421$	27.1752	−	27.1752i	1.32444	−	1.32444i	0.414295	−	0.910143i	$-0.364028\pi$
$421$	27.1752	−	27.1752i	1.32444	−	1.32444i	0.910143	−	0.414295i	$-0.135972\pi$
$422$		0			0
$423$		0			0
$424$			40.9133i			1.98693i
$425$	10.2697	+	1.52817i	0.498153	+	0.0741273i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$431$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$432$		0			0
$433$	9.18440	+	22.1731i	0.441374	+	1.06557i	0.975467	+	0.220146i	$0.0706533\pi$
$433$	9.18440	+	22.1731i	0.441374	+	1.06557i	−0.534093	+	0.845426i	$0.679347\pi$
$434$		0			0
$435$		0			0
$436$	8.11039	−	40.7737i	0.388417	−	1.95271i
$437$		0			0
$438$		0			0
$439$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$439$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$	−5.14214	+	34.5563i	−0.244586	+	1.64368i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−29.4613	+	5.86022i	−1.39660	+	0.277801i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.29308	+	4.92844i	0.155410	+	0.232587i	0.901002	−	0.433816i	$-0.142833\pi$
$449$	3.29308	+	4.92844i	0.155410	+	0.232587i	−0.745592	+	0.666403i	$0.767833\pi$
$450$	4.08850	+	9.87052i	0.192734	+	0.465301i
$451$		0			0
$452$	4.57550	−	3.05725i	0.215213	−	0.143801i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	13.5299	−	32.6641i	0.632902	−	1.52796i	−0.203056	−	0.979167i	$-0.565087\pi$
$457$	13.5299	−	32.6641i	0.632902	−	1.52796i	0.835958	−	0.548794i	$-0.184913\pi$
$458$		−	26.0000i		−	1.21490i
$459$		0			0
$460$		0			0
$461$	3.63604	+	1.50610i	0.169347	+	0.0701459i	0.465746	−	0.884918i	$-0.345786\pi$
$461$	3.63604	+	1.50610i	0.169347	+	0.0701459i	−0.296399	+	0.955064i	$0.595786\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$	19.7374	−	29.5391i	0.916284	−	1.37132i
$465$		0			0
$466$	17.5830	+	26.3149i	0.814519	+	1.21901i
$467$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$467$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$468$	−33.2132	+	13.7574i	−1.53528	+	0.635934i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−43.3951			−1.98693
$478$		0			0
$479$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$479$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$480$		0			0
$481$	−16.6568	+	24.9287i	−0.759486	+	1.13665i
$482$	38.6469	+	7.68734i	1.76032	+	0.350149i
$483$		0			0
$484$	−8.41904	−	20.3253i	−0.382683	−	0.923880i
$485$	10.9806	−	4.54832i	0.498604	−	0.206529i
$486$		0			0
$487$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$487$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$488$	−23.0907	−	15.4287i	−1.04527	−	0.698425i
$489$		0			0
$490$	−5.29548	−	26.6222i	−0.239225	−	1.20267i
$491$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$491$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$492$		0			0
$493$	33.1110	−	15.6418i	1.49124	−	0.704470i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$499$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$500$	7.56126	+	11.3162i	0.338150	+	0.506077i
$501$		0			0
$502$		0			0
$503$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$503$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$504$		0			0
$505$	21.6617	+	14.4739i	0.963932	+	0.644079i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			38.1838i			1.69247i	0.532813	+	0.846233i	$0.321135\pi$
$509$			38.1838i			1.69247i	−0.532813	+	0.846233i	$0.678865\pi$
$510$		0			0
$511$		0			0
$512$	20.9050	+	8.65914i	0.923880	+	0.382683i
$513$		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
$520$	38.6360	−	25.8158i	1.69430	−	1.13210i
$521$	6.81634	−	34.2681i	0.298629	−	1.50131i	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	6.81634	−	34.2681i	0.298629	−	1.50131i	0.780548	−	0.625096i	$-0.214940\pi$
$522$	31.3309	+	20.9346i	1.37132	+	0.916284i
$523$		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
$529$	21.2492	+	8.80172i	0.923880	+	0.382683i
$530$	55.0130	−	10.9428i	2.38961	−	0.475324i
$531$		0			0
$532$		0			0
$533$	−74.9797	−	14.9144i	−3.24773	−	0.646014i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	2.89882	−	14.5734i	0.124977	−	0.628302i
$539$		0			0
$540$		0			0
$541$	−8.98062	−	45.1486i	−0.386107	−	1.94109i	−0.334410	−	0.942428i	$-0.608537\pi$
$541$	−8.98062	−	45.1486i	−0.386107	−	1.94109i	−0.0516971	−	0.998663i	$-0.516463\pi$
$542$		0			0
$543$		0			0
$544$	13.8854	+	18.7402i	0.595331	+	0.803480i
$545$	−56.9945			−2.44138
$546$		0			0
$547$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$547$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$548$	1.45381	+	1.45381i	0.0621037	+	0.0621037i
$549$	16.3646	−	24.4914i	0.698425	−	1.04527i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	35.8906	−	23.9813i	1.52484	−	1.01887i
$555$		0			0
$556$		0			0
$557$	−17.2480	+	17.2480i	−0.730819	+	0.730819i	−0.970782	−	0.239963i	$-0.922865\pi$
$557$	−17.2480	+	17.2480i	−0.730819	+	0.730819i	0.239963	+	0.970782i	$0.422865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−30.3839			−1.28167
$563$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$563$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$564$		0			0
$565$	−5.33462	−	5.33462i	−0.224429	−	0.224429i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.9497	+	43.3345i	0.752493	+	1.81668i	0.544988	+	0.838444i	$0.316534\pi$
$569$	17.9497	+	43.3345i	0.752493	+	1.81668i	0.207504	+	0.978234i	$0.433466\pi$
$570$		0			0
$571$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$571$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−9.18440	+	22.1731i	−0.382683	+	0.923880i
$577$		−	16.5210i		−	0.687780i	−0.939010	−	0.343890i	$-0.888255\pi$
$577$		−	16.5210i		−	0.687780i	0.939010	−	0.343890i	$-0.111745\pi$
$578$	2.33456	+	23.9280i	0.0971050	+	0.995274i
$579$		0			0
$580$	−44.9979	−	18.6387i	−1.86844	−	0.773932i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−46.1522	−	9.18024i	−1.90979	−	0.379881i
$585$	27.3817	+	40.9797i	1.13210	+	1.69430i
$586$	−2.16478	−	5.22625i	−0.0894264	−	0.215894i
$587$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$587$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	3.90485	+	19.6310i	0.160488	+	0.806830i
$593$	−13.9203	+	33.6066i	−0.571639	+	1.38006i	0.328521	+	0.944497i	$0.393450\pi$
$593$	−13.9203	+	33.6066i	−0.571639	+	1.38006i	−0.900159	+	0.435561i	$0.856550\pi$
$594$		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
$601$	−0.434995	−	0.0865259i	−0.0177438	−	0.00352947i	0.186210	−	0.982510i	$-0.440379\pi$
$601$	−0.434995	−	0.0865259i	−0.0177438	−	0.00352947i	−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−25.0782	+	16.7567i	−1.01957	+	0.681256i
$606$		0			0
$607$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$607$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$608$		0			0
$609$		0			0
$610$	−14.5699	+	35.1749i	−0.589919	+	1.42419i
$611$		0			0
$612$	−19.8770	+	14.7277i	−0.803480	+	0.595331i
$613$	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
$617$	−13.3180	+	19.9317i	−0.536161	+	0.802422i	−0.996347	−	0.0854011i	$-0.972783\pi$
$617$	−13.3180	+	19.9317i	−0.536161	+	0.802422i	0.460186	+	0.887823i	$0.347783\pi$
$618$		0			0
$619$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$619$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	22.0969	−	22.0969i	0.883874	−	0.883874i
$626$	9.49097	+	47.7143i	0.379335	+	1.90705i
$627$		0			0
$628$			14.1421i			0.564333i
$629$	−6.95949	+	19.4224i	−0.277493	+	0.774422i
$630$		0			0
$631$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$631$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$632$		0			0
$633$		0			0
$634$	24.9489	−	37.3387i	0.990849	−	1.48291i
$635$		0			0
$636$		0			0
$637$	−16.0503	−	38.7487i	−0.635934	−	1.53528i
$638$		0			0
$639$		0			0
$640$	6.05198	−	30.4253i	0.239225	−	1.20267i
$641$	14.6345	+	9.77848i	0.578029	+	0.386227i	0.809942	−	0.586510i	$-0.199498\pi$
$641$	14.6345	+	9.77848i	0.578029	+	0.386227i	−0.231913	+	0.972737i	$0.574498\pi$
$642$		0			0
$643$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$643$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−23.5181	−	9.74153i	−0.923880	−	0.382683i
$649$		0			0
$650$	−15.0880	−	15.0880i	−0.591801	−	0.591801i
$651$		0			0
$652$		0			0
$653$	12.3002	+	18.4086i	0.481345	+	0.720384i	0.990075	−	0.140542i	$-0.0448846\pi$
$653$	12.3002	+	18.4086i	0.481345	+	0.720384i	−0.508729	+	0.860927i	$0.669885\pi$
$654$		0			0
$655$		0			0
$656$	−42.4357	+	28.3547i	−1.65684	+	1.10706i
$657$	9.73712	−	48.9518i	0.379881	−	1.90979i
$658$		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
$661$	3.07969	−	7.43503i	0.119786	−	0.289189i	−0.852601	−	0.522562i	$-0.824976\pi$
$661$	3.07969	−	7.43503i	0.119786	−	0.289189i	0.972387	+	0.233373i	$0.0749763\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−20.8218	+	4.14172i	−0.806830	+	0.160488i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	41.4058	−	27.6664i	1.59607	−	1.06646i	0.642081	−	0.766637i	$-0.278071\pi$
$673$	41.4058	−	27.6664i	1.59607	−	1.06646i	0.953994	−	0.299827i	$-0.0969288\pi$
$674$	−3.14836	+	15.8279i	−0.121270	+	0.609666i
$675$		0			0
$676$	32.3848	−	32.3848i	1.24557	−	1.24557i
$677$	−10.0259	−	50.4036i	−0.385326	−	1.93717i	−0.346893	−	0.937905i	$-0.612763\pi$
$677$	−10.0259	−	50.4036i	−0.385326	−	1.93717i	−0.0384331	−	0.999261i	$-0.512237\pi$
$678$		0			0
$679$		0			0
$680$	21.4847	−	23.6829i	0.823901	−	0.908198i
$681$		0			0
$682$		0			0
$683$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$683$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$684$		0			0
$685$	1.56599	−	2.34367i	0.0598333	−	0.0895469i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	80.0718	−	33.1668i	3.05049	−	1.26355i
$690$		0			0
$691$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$691$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$692$	−1.91106	−	1.27693i	−0.0726477	−	0.0485416i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−52.5456	+	2.55726i	−1.99031	+	0.0968632i
$698$	6.41743			0.242903
$699$		0			0
$700$		0			0
$701$	36.6766	+	36.6766i	1.38526	+	1.38526i	0.834987	+	0.550270i	$0.185475\pi$
$701$	36.6766	+	36.6766i	1.38526	+	1.38526i	0.550270	+	0.834987i	$0.314525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	20.9050	−	8.65914i	0.786770	−	0.325891i
$707$		0			0
$708$		0			0
$709$	0.414902	+	0.277228i	0.0155820	+	0.0104115i	0.563337	−	0.826227i	$-0.309517\pi$
$709$	0.414902	+	0.277228i	0.0155820	+	0.0104115i	−0.547755	+	0.836639i	$0.684517\pi$
$710$		0			0
$711$		0			0
$712$	−11.8579	+	28.6274i	−0.444392	+	1.07286i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$719$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$720$	32.2710	+	6.41909i	1.20267	+	0.239225i
$721$		0			0
$722$	−10.2827	−	24.8247i	−0.382683	−	0.923880i
$723$		0			0
$724$	−11.0238	+	7.36589i	−0.409698	+	0.273751i
$725$	−4.36329	+	21.9358i	−0.162049	+	0.814674i
$726$		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
$729$	10.3325	−	24.9447i	0.382683	−	0.923880i
$730$			64.5127i			2.38772i
$731$		0			0
$732$		0			0
$733$	44.6777	+	18.5061i	1.65021	+	0.683538i	0.997268	−	0.0738717i	$-0.0235355\pi$
$733$	44.6777	+	18.5061i	1.65021	+	0.683538i	0.652940	+	0.757410i	$0.273536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−30.0747	−	45.0099i	−1.10706	−	1.65684i
$739$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$739$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$740$	25.3519	−	10.5011i	0.931955	−	0.386029i
$741$		0			0
$742$		0			0
$743$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$743$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$744$		0			0
$745$	2.26949	+	11.4095i	0.0831477	+	0.418012i
$746$	15.1005	−	36.4558i	0.552869	−	1.33474i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$751$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$752$		0			0
$753$		0			0
$754$	−73.8114	−	14.6820i	−2.68805	−	0.534687i
$755$		0			0
$756$		0			0
$757$	−45.7297	+	18.9419i	−1.66207	+	0.688454i	−0.998233	−	0.0594198i	$-0.981075\pi$
$757$	−45.7297	+	18.9419i	−1.66207	+	0.688454i	−0.663841	+	0.747873i	$0.731075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−35.6486	+	35.6486i	−1.29226	+	1.29226i	−0.358876	+	0.933385i	$0.616840\pi$
$761$	−35.6486	+	35.6486i	−1.29226	+	1.29226i	−0.933385	+	0.358876i	$0.883160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	25.1195	+	22.7880i	0.908198	+	0.823901i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−39.1584	−	39.1584i	−1.41209	−	1.41209i	−0.744789	−	0.667300i	$-0.767450\pi$
$769$	−39.1584	−	39.1584i	−1.41209	−	1.41209i	−0.667300	−	0.744789i	$-0.732550\pi$
$770$		0			0
$771$		0			0
$772$	29.1898	+	43.6856i	1.05056	+	1.57228i
$773$	−16.8381	−	40.6507i	−0.605623	−	1.46210i	−0.867715	−	0.497061i	$-0.834412\pi$
$773$	−16.8381	−	40.6507i	−0.605623	−	1.46210i	0.262092	−	0.965043i	$-0.415588\pi$
$774$		0			0
$775$		0			0
$776$	2.39186	−	12.0247i	0.0858627	−	0.431661i
$777$		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
$784$	−25.8686	−	10.7151i	−0.923880	−	0.382683i
$785$	19.0158	−	3.78249i	0.678705	−	0.135003i
$786$		0			0
$787$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$787$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$788$	47.8472	+	9.51740i	1.70449	+	0.339043i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−11.4769	+	57.6984i	−0.407558	+	2.04893i
$794$	−13.0922	−	8.74795i	−0.464626	−	0.310453i
$795$		0			0
$796$		0			0
$797$	0.162951	−	0.393398i	0.00577202	−	0.0139349i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	0.162951	−	0.393398i	0.00577202	−	0.0139349i	0.926739	+	0.375705i	$0.122599\pi$
$798$		0			0
$799$		0			0
$800$	−14.2450			−0.503638
$801$	−30.3640	−	12.5772i	−1.07286	−	0.444392i
$802$	44.5901	−	8.86952i	1.57453	−	0.313193i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	24.8284	−	10.2843i	0.873461	−	0.361799i
$809$	−47.2897	+	31.5980i	−1.66262	+	1.11093i	−0.820398	+	0.571793i	$0.806248\pi$
$809$	−47.2897	+	31.5980i	−1.66262	+	1.11093i	−0.842220	+	0.539133i	$0.818752\pi$
$810$	−6.80848	+	34.2285i	−0.239225	+	1.20267i
$811$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$811$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	42.4985	+	17.6034i	1.48592	+	0.615490i
$819$		0			0
$820$	49.4763	+	49.4763i	1.72779	+	1.72779i
$821$	14.4546	−	21.6328i	0.504467	−	0.754989i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	14.4546	−	21.6328i	0.504467	−	0.754989i	0.993071	+	0.117517i	$0.0374935\pi$
$822$		0			0
$823$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$823$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$827$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$828$		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
$832$		−	47.9329i		−	1.66178i
$833$	−17.1823	−	23.1898i	−0.595331	−	0.803480i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$839$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$840$		0			0
$841$	19.0892	+	46.0854i	0.658248	+	1.58915i
$842$	−50.2132	+	20.7990i	−1.73046	+	0.716781i
$843$		0			0
$844$		0			0
$845$	−52.2070	−	34.8836i	−1.79598	−	1.20003i
$846$		0			0
$847$		0			0
$848$	22.1421	−	53.4558i	0.760364	−	1.83568i
$849$		0			0
$850$	−12.5910	−	7.55457i	−0.431866	−	0.259120i
$851$		0			0
$852$		0			0
$853$	−4.29258	+	0.853847i	−0.146975	+	0.0292352i	−0.268029	−	0.963411i	$-0.586372\pi$
$853$	−4.29258	+	0.853847i	−0.146975	+	0.0292352i	0.121054	+	0.992646i	$0.461372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.3232	−	36.4023i	−0.830866	−	1.24348i	−0.967503	−	0.252858i	$-0.918630\pi$
$857$	−24.3232	−	36.4023i	−0.830866	−	1.24348i	0.136637	−	0.990621i	$-0.456370\pi$
$858$		0			0
$859$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$859$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$860$		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
$865$	−1.20585	+	2.91119i	−0.0410003	+	0.0989834i
$866$		−	33.9411i		−	1.15337i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−32.6633	+	48.8841i	−1.10612	+	1.65542i
$873$	12.7541	+	2.53695i	0.431661	+	0.0858627i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	19.1942	−	12.8252i	0.648142	−	0.433075i	−0.187564	−	0.982252i	$-0.560059\pi$
$877$	19.1942	−	12.8252i	0.648142	−	0.433075i	0.835705	+	0.549178i	$0.185059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	10.7850	+	54.2199i	0.363356	+	1.82672i	0.539054	+	0.842271i	$0.318782\pi$
$881$	10.7850	+	54.2199i	0.363356	+	1.82672i	−0.175697	+	0.984444i	$0.556218\pi$
$882$	11.3651	−	27.4378i	0.382683	−	0.923880i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$	25.4203	−	42.3671i	0.854977	−	1.42496i
$885$		0			0
$886$		0			0
$887$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$887$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$888$		0			0
$889$		0			0
$890$	41.6646	+	8.28761i	1.39660	+	0.277801i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.63536	−	8.22151i	−0.0545727	−	0.274355i
$899$		0			0
$900$		−	15.1091i		−	0.503638i
$901$	47.9203	−	35.5061i	1.59646	−	1.18288i
$902$		0			0
$903$		0			0
$904$	−7.63274	+	1.51825i	−0.253861	+	0.0504961i
$905$	12.8528	+	12.8528i	0.427242	+	0.427242i
$906$		0			0
$907$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$907$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$908$		0			0
$909$	10.9081	+	26.3345i	0.361799	+	0.873461i
$910$		0			0
$911$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$911$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$912$		0			0
$913$		0			0
$914$	−35.3553	+	35.3553i	−1.16945	+	1.16945i
$915$		0			0
$916$	−14.0711	+	33.9706i	−0.464922	+	1.12242i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	−3.93562	−	3.93562i	−0.129613	−	0.129613i
$923$		0			0
$924$		0			0
$925$	−7.00062	−	10.4772i	−0.230179	−	0.344487i
$926$		0			0
$927$		0			0
$928$	−41.7745	+	27.9129i	−1.37132	+	0.916284i
$929$	9.40444	−	47.2793i	0.308550	−	1.55118i	−0.446056	−	0.895005i	$-0.647172\pi$
$929$	9.40444	−	47.2793i	0.308550	−	1.55118i	0.754606	−	0.656179i	$-0.227828\pi$
$930$		0			0
$931$		0			0
$932$	−8.73184	−	43.8979i	−0.286021	−	1.43792i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	50.8406			1.66178
$937$	−54.4056	−	22.5355i	−1.77735	−	0.736204i	−0.993307	−	0.115501i	$-0.963153\pi$
$937$	−54.4056	−	22.5355i	−1.77735	−	0.736204i	−0.784046	−	0.620703i	$-0.786847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	51.9657	+	10.3366i	1.69403	+	0.336964i	0.945373	−	0.325991i	$-0.105698\pi$
$941$	51.9657	+	10.3366i	1.69403	+	0.336964i	0.748660	+	0.662955i	$0.230698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$947$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$948$		0			0
$949$	19.4470	+	97.7668i	0.631277	+	3.17364i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	45.4511			1.47231			0.736153	−	0.676815i	$-0.236640\pi$
$953$	45.4511			1.47231			0.736153	+	0.676815i	$0.236640\pi$
$954$	56.6985	+	23.4853i	1.83568	+	0.760364i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−28.6403	+	11.8632i	−0.923880	+	0.382683i
$962$	35.2545	−	23.5563i	1.13665	−	0.759486i
$963$		0			0
$964$	−46.3342	−	30.9595i	−1.49232	−	0.997139i
$965$	50.9335	−	50.9335i	1.63961	−	1.63961i
$966$		0			0
$967$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$967$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$968$			31.1127i			1.00000i
$969$		0			0
$970$	−16.8084			−0.539685
$971$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$971$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	21.8195	+	32.6552i	0.698425	+	1.04527i
$977$	−23.0919	−	55.7487i	−0.738775	−	1.78356i	−0.610803	−	0.791782i	$-0.709153\pi$
$977$	−23.0919	−	55.7487i	−0.738775	−	1.78356i	−0.127971	−	0.991778i	$-0.540847\pi$
$978$		0			0
$979$		0			0
$980$	−7.48894	+	37.6494i	−0.239225	+	1.20267i
$981$	−51.8494	−	34.6447i	−1.65542	−	1.10612i
$982$		0			0
$983$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$983$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$984$		0			0
$985$		−	66.8820i		−	2.13104i
$986$	−51.7268	+	2.51742i	−1.64732	+	0.0801709i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$991$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−11.4065	+	57.3442i	−0.361246	+	1.81611i	0.190022	+	0.981780i	$0.439144\pi$
$997$	−11.4065	+	57.3442i	−0.361246	+	1.81611i	−0.551268	+	0.834328i	$0.685856\pi$
$998$		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.23.1	yes	8
3.2	odd	2		612.2.bd.a.91.1		8
4.3	odd	2	CM	68.2.i.a.23.1	yes	8
12.11	even	2		612.2.bd.a.91.1		8
17.3	odd	16	inner	68.2.i.a.3.1	✓	8
51.20	even	16		612.2.bd.a.343.1		8
68.3	even	16	inner	68.2.i.a.3.1	✓	8
204.71	odd	16		612.2.bd.a.343.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.i.a.3.1	✓	8	17.3	odd	16	inner
68.2.i.a.3.1	✓	8	68.3	even	16	inner
68.2.i.a.23.1	yes	8	1.1	even	1	trivial
68.2.i.a.23.1	yes	8	4.3	odd	2	CM
612.2.bd.a.91.1		8	3.2	odd	2
612.2.bd.a.91.1		8	12.11	even	2
612.2.bd.a.343.1		8	51.20	even	16
612.2.bd.a.343.1		8	204.71	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)

\(f(q)\)	\(=\)	\(q+(-1.30656 - 0.541196i) q^{2} +(1.41421 + 1.41421i) q^{4} +(1.52334 - 2.27983i) q^{5} +(-1.08239 - 2.61313i) q^{8} +(2.77164 - 1.14805i) q^{9} +O(q^{10})\)	\(q+(-1.30656 - 0.541196i) q^{2} +(1.41421 + 1.41421i) q^{4} +(1.52334 - 2.27983i) q^{5} +(-1.08239 - 2.61313i) q^{8} +(2.77164 - 1.14805i) q^{9} +(-3.22417 + 2.15432i) q^{10} +(-4.23671 + 4.23671i) q^{13} +4.00000i q^{16} +(-2.12132 + 3.53553i) q^{17} -4.24264 q^{18} +(5.37849 - 1.06985i) q^{20} +(-0.963670 - 2.32650i) q^{25} +(7.82843 - 3.24264i) q^{26} +(-7.38476 - 4.93434i) q^{29} +(2.16478 - 5.22625i) q^{32} +(4.68506 - 3.47135i) q^{34} +(5.54328 + 2.29610i) q^{36} +(4.90775 - 0.976213i) q^{37} +(-7.60634 - 1.51299i) q^{40} +(7.08866 + 10.6089i) q^{41} +(1.60477 - 8.06774i) q^{45} +(-2.67878 + 6.46716i) q^{49} +3.56126i q^{50} -11.9832 q^{52} +(-13.3640 - 5.53553i) q^{53} +(6.97821 + 10.4436i) q^{58} +(8.16380 - 5.45488i) q^{61} +(-5.65685 + 5.65685i) q^{64} +(3.20506 + 16.1129i) q^{65} +(-8.00000 + 2.00000i) q^{68} +(-6.00000 - 6.00000i) q^{72} +(9.24299 - 13.8331i) q^{73} +(-6.94061 - 1.38057i) q^{74} +(9.11933 + 6.09334i) q^{80} +(6.36396 - 6.36396i) q^{81} +(-3.52027 - 17.6976i) q^{82} +(4.82894 + 10.2221i) q^{85} +(-7.74652 - 7.74652i) q^{89} +(-6.46297 + 9.67251i) q^{90} +(3.60414 + 2.40821i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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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