LMFDB - Embedded newform 832.2.bu.d.63.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,2,Mod(63,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 0, 7]))

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

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

chi := DirichletCharacter("832.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.bu (of order $12$, degree $4$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.64355344817$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			63.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	832.63
Dual form			832.2.bu.d.383.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.633975 - 0.633975i) q^{5} +(-1.50000 + 2.59808i) q^{9} +(-3.59808 - 0.232051i) q^{13} +(-6.86603 - 3.96410i) q^{17} -4.19615i q^{25} +(3.33013 + 5.76795i) q^{29} +(-11.6962 - 3.13397i) q^{37} +(-2.66987 + 9.96410i) q^{41} +(2.59808 - 0.696152i) q^{45} +(-6.06218 + 3.50000i) q^{49} -10.4641 q^{53} +(2.69615 - 4.66987i) q^{61} +(2.13397 + 2.42820i) q^{65} +(9.83013 - 9.83013i) q^{73} +(-4.50000 - 7.79423i) q^{81} +(1.83975 + 6.86603i) q^{85} +(-4.09808 - 1.09808i) q^{89} +(-6.83013 + 1.83013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} - 6 q^{9} - 4 q^{13} - 24 q^{17} - 4 q^{29} - 26 q^{37} - 28 q^{41} - 28 q^{53} - 10 q^{61} + 12 q^{65} + 22 q^{73} - 18 q^{81} + 42 q^{85} - 6 q^{89} - 10 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 - 6 * q^9 - 4 * q^13 - 24 * q^17 - 4 * q^29 - 26 * q^37 - 28 * q^41 - 28 * q^53 - 10 * q^61 + 12 * q^65 + 22 * q^73 - 18 * q^81 + 42 * q^85 - 6 * q^89 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$1$	$-1$	$e\left(\frac{7}{12}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$3$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$4$		0			0
$5$	−0.633975	−	0.633975i	−0.283522	−	0.283522i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−0.633975	−	0.633975i	−0.283522	−	0.283522i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$7$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$8$		0			0
$9$	−1.50000	+	2.59808i	−0.500000	+	0.866025i
$10$		0			0
$11$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$11$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$12$		0			0
$13$	−3.59808	−	0.232051i	−0.997927	−	0.0643593i
$13$	−3.59808	−	0.232051i	−0.997927	−	0.0643593i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.86603	−	3.96410i	−1.66526	−	0.961436i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−6.86603	−	3.96410i	−1.66526	−	0.961436i	−0.695113	−	0.718900i	$-0.744646\pi$
$18$		0			0
$19$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$19$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$		−	4.19615i		−	0.839230i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.33013	+	5.76795i	0.618389	+	1.07108i	0.989780	+	0.142605i	$0.0455477\pi$
$29$	3.33013	+	5.76795i	0.618389	+	1.07108i	−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−11.6962	−	3.13397i	−1.92284	−	0.515222i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−11.6962	−	3.13397i	−1.92284	−	0.515222i	−0.936442	−	0.350823i	$-0.885902\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.66987	+	9.96410i	−0.416964	+	1.55613i	0.363905	+	0.931436i	$0.381443\pi$
$41$	−2.66987	+	9.96410i	−0.416964	+	1.55613i	−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$43$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$44$		0			0
$45$	2.59808	−	0.696152i	0.387298	−	0.103776i
$46$		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
$49$	−6.06218	+	3.50000i	−0.866025	+	0.500000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.4641			−1.43735			−0.718677	−	0.695344i	$-0.755252\pi$
$53$	−10.4641			−1.43735			−0.718677	+	0.695344i	$0.755252\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$59$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$60$		0			0
$61$	2.69615	−	4.66987i	0.345207	−	0.597916i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	2.69615	−	4.66987i	0.345207	−	0.597916i	0.985391	+	0.170305i	$0.0544754\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.13397	+	2.42820i	0.264687	+	0.301182i
$66$		0			0
$67$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$67$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$71$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$72$		0			0
$73$	9.83013	−	9.83013i	1.15053	−	1.15053i	0.164083	−	0.986447i	$-0.447534\pi$
$73$	9.83013	−	9.83013i	1.15053	−	1.15053i	0.986447	−	0.164083i	$-0.0524664\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	1.83975	+	6.86603i	0.199548	+	0.744725i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.09808	−	1.09808i	−0.434395	−	0.116396i	0.0349934	−	0.999388i	$-0.488859\pi$
$89$	−4.09808	−	1.09808i	−0.434395	−	0.116396i	−0.469389	+	0.882992i	$0.655526\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.83013	+	1.83013i	−0.693494	+	0.185821i	−0.588315	−	0.808632i	$-0.700208\pi$
$97$	−6.83013	+	1.83013i	−0.693494	+	0.185821i	−0.105180	+	0.994453i	$0.533542\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.1603	−	5.86603i	1.01098	−	0.583691i	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	10.1603	−	5.86603i	1.01098	−	0.583691i	0.911479	+	0.411346i	$0.134941\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	7.00000	+	7.00000i	0.670478	+	0.670478i	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	7.00000	+	7.00000i	0.670478	+	0.670478i	−0.287348	+	0.957826i	$0.592774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.06218	−	3.57180i	0.193993	−	0.336006i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	2.06218	−	3.57180i	0.193993	−	0.336006i	0.946570	+	0.322498i	$0.104523\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.00000	−	9.00000i	0.554700	−	0.832050i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.52628	+	5.50000i	0.866025	+	0.500000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.83013	+	5.83013i	−0.521462	+	0.521462i
$126$		0			0
$127$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$127$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.03590	+	22.5263i	0.515682	+	1.92455i	0.341743	+	0.939793i	$0.388983\pi$
$137$	6.03590	+	22.5263i	0.515682	+	1.92455i	0.173939	+	0.984757i	$0.444351\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	1.54552	−	5.76795i	0.128348	−	0.479002i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.1603	+	4.06218i	−1.24198	+	0.332787i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−15.1603	+	4.06218i	−1.24198	+	0.332787i	−0.422744	+	0.906249i	$0.638933\pi$
$150$		0			0
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$	20.5981	−	11.8923i	1.66526	−	0.961436i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	25.0526			1.99941			0.999706	−	0.0242497i	$-0.00771967\pi$
$157$	25.0526			1.99941			0.999706	+	0.0242497i	$0.00771967\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$163$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$167$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$168$		0			0
$169$	12.8923	+	1.66987i	0.991716	+	0.128452i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.46410	+	2.00000i	0.263371	+	0.152057i	0.625871	−	0.779926i	$-0.284744\pi$
$173$	3.46410	+	2.00000i	0.263371	+	0.152057i	−0.362500	+	0.931984i	$0.618077\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$		−	26.3205i		−	1.95639i	−0.207693	−	0.978194i	$-0.566596\pi$
$181$		−	26.3205i		−	1.95639i	0.207693	−	0.978194i	$-0.433404\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.42820	+	9.40192i	0.399089	+	0.691243i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−19.0622	−	5.10770i	−1.37213	−	0.367660i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−19.0622	−	5.10770i	−1.37213	−	0.367660i	−0.868255	+	0.496119i	$0.834758\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.49038	+	20.4904i	−0.391173	+	1.45988i	0.437028	+	0.899448i	$0.356031\pi$
$197$	−5.49038	+	20.4904i	−0.391173	+	1.45988i	−0.828201	+	0.560431i	$0.810635\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	8.00962	−	4.62436i	0.559416	−	0.322979i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	23.7846	+	15.8564i	1.59993	+	1.06662i
$222$		0			0
$223$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$223$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$224$		0			0
$225$	10.9019	+	6.29423i	0.726795	+	0.419615i
$226$		0			0
$227$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$227$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$228$		0			0
$229$	17.0000	−	17.0000i	1.12339	−	1.12339i	0.132164	−	0.991228i	$-0.457808\pi$
$229$	17.0000	−	17.0000i	1.12339	−	1.12339i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−6.96410	−	25.9904i	−0.448597	−	1.67419i	−0.706260	−	0.707953i	$-0.749619\pi$
$241$	−6.96410	−	25.9904i	−0.448597	−	1.67419i	0.257663	−	0.966235i	$-0.417048\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.06218	+	1.62436i	0.387298	+	0.103776i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$251$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.3564	−	7.13397i	0.770771	−	0.445005i	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	12.3564	−	7.13397i	0.770771	−	0.445005i	0.833150	+	0.553047i	$0.186535\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−19.9808			−1.23678
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$	6.63397	+	6.63397i	0.407522	+	0.407522i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.0000	+	17.3205i	−0.609711	+	1.05605i	0.381577	+	0.924337i	$0.375381\pi$
$269$	−10.0000	+	17.3205i	−0.609711	+	1.05605i	−0.991288	+	0.131713i	$0.957952\pi$
$270$		0			0
$271$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$271$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.37564	−	0.794229i	−0.0826545	−	0.0477206i	0.458103	−	0.888899i	$-0.348529\pi$
$277$	−1.37564	−	0.794229i	−0.0826545	−	0.0477206i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.6865	+	12.6865i	−0.756815	+	0.756815i	−0.975741	−	0.218926i	$-0.929745\pi$
$281$	−12.6865	+	12.6865i	−0.756815	+	0.756815i	0.218926	+	0.975741i	$0.429745\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	22.9282	+	39.7128i	1.34872	+	2.33605i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.27757	−	4.76795i	−0.0746363	−	0.278547i	0.918514	−	0.395388i	$-0.129390\pi$
$293$	−1.27757	−	4.76795i	−0.0746363	−	0.278547i	−0.993151	+	0.116841i	$0.962723\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.66987	+	1.25129i	−0.267396	+	0.0716486i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−24.0000			−1.35656			−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000			−1.35656			−0.678280	+	0.734803i	$0.737274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−23.1506	−	23.1506i	−1.30027	−	1.30027i	−0.928208	−	0.372061i	$-0.878651\pi$
$317$	−23.1506	−	23.1506i	−1.30027	−	1.30027i	−0.372061	−	0.928208i	$-0.621349\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.973721	+	15.0981i	−0.0540123	+	0.837491i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$331$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$332$		0			0
$333$	25.6865	−	25.6865i	1.40761	−	1.40761i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	18.7128i		−	1.01935i	−0.860366	−	0.509676i	$-0.829765\pi$
$337$		−	18.7128i		−	1.01935i	0.860366	−	0.509676i	$-0.170235\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−31.4186	−	8.41858i	−1.68180	−	0.450636i	−0.713545	−	0.700609i	$-0.752912\pi$
$349$	−31.4186	−	8.41858i	−1.68180	−	0.450636i	−0.968253	+	0.249973i	$0.919578\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.42820	+	27.7224i	−0.395363	+	1.47552i	0.425797	+	0.904819i	$0.359994\pi$
$353$	−7.42820	+	27.7224i	−0.395363	+	1.47552i	−0.821160	+	0.570697i	$0.806673\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$	16.4545	−	9.50000i	0.866025	−	0.500000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.4641			−0.652401
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$	−21.8827	−	21.8827i	−1.13917	−	1.13917i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−15.0622	+	26.0885i	−0.779890	+	1.35081i	0.152115	+	0.988363i	$0.451392\pi$
$373$	−15.0622	+	26.0885i	−0.779890	+	1.35081i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10.6436	−	21.5263i	−0.548173	−	1.10866i
$378$		0			0
$379$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$379$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$383$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	0.320508i		−	0.0162504i	−0.999967	−	0.00812520i	$-0.997414\pi$
$389$		−	0.320508i		−	0.0162504i	0.999967	−	0.00812520i	$-0.00258636\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.15064	+	34.1506i	0.459257	+	1.71397i	0.675261	+	0.737579i	$0.264031\pi$
$397$	9.15064	+	34.1506i	0.459257	+	1.71397i	−0.216004	+	0.976392i	$0.569302\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−36.8205	−	9.86603i	−1.83873	−	0.492686i	−0.839976	−	0.542623i	$-0.817431\pi$
$401$	−36.8205	−	9.86603i	−1.83873	−	0.492686i	−0.998752	+	0.0499376i	$0.984098\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−2.08846	+	7.79423i	−0.103776	+	0.387298i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−15.5981	+	4.17949i	−0.771275	+	0.206663i	−0.622935	−	0.782274i	$-0.714060\pi$
$409$	−15.5981	+	4.17949i	−0.771275	+	0.206663i	−0.148340	+	0.988936i	$0.547393\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	−13.6340	−	13.6340i	−0.664479	−	0.664479i	0.291953	−	0.956433i	$-0.405695\pi$
$421$	−13.6340	−	13.6340i	−0.664479	−	0.664479i	−0.956433	+	0.291953i	$0.905695\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.6340	+	28.8109i	−0.806866	+	1.39753i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$431$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$432$		0			0
$433$	−15.1077	−	8.72243i	−0.726029	−	0.419173i	0.0909384	−	0.995857i	$-0.471013\pi$
$433$	−15.1077	−	8.72243i	−0.726029	−	0.419173i	−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		−	21.0000i		−	1.00000i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	1.90192	+	3.29423i	0.0901598	+	0.156161i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.88269	+	36.8827i	0.466393	+	1.74060i	0.652230	+	0.758021i	$0.273834\pi$
$449$	9.88269	+	36.8827i	0.466393	+	1.74060i	−0.185837	+	0.982581i	$0.559500\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.813467	−	3.03590i	0.0380524	−	0.142013i	−0.944286	−	0.329125i	$-0.893246\pi$
$457$	0.813467	−	3.03590i	0.0380524	−	0.142013i	0.982339	+	0.187112i	$0.0599128\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−28.1603	+	7.54552i	−1.31155	+	0.351430i	−0.845807	−	0.533488i	$-0.820881\pi$
$461$	−28.1603	+	7.54552i	−1.31155	+	0.351430i	−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	15.6962	−	27.1865i	0.718677	−	1.24479i
$478$		0			0
$479$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$479$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$480$		0			0
$481$	41.3564	+	13.9904i	1.88569	+	0.637906i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.49038	+	3.16987i	0.249305	+	0.143937i
$486$		0			0
$487$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$487$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		−	52.8038i		−	2.37817i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$	−10.1603	−	2.72243i	−0.452125	−	0.121147i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.3301	+	38.5526i	−0.457875	+	1.70881i	0.221621	+	0.975133i	$0.428865\pi$
$509$	−10.3301	+	38.5526i	−0.457875	+	1.70881i	−0.679496	+	0.733679i	$0.737801\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.947441			0.0415081			0.0207541	−	0.999785i	$-0.493393\pi$
$521$	0.947441			0.0415081			0.0207541	+	0.999785i	$0.493393\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	11.9186	−	35.2321i	0.516251	−	1.52607i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−32.3468	+	32.3468i	−1.39070	+	1.39070i	−0.566933	+	0.823764i	$0.691870\pi$
$541$	−32.3468	+	32.3468i	−1.39070	+	1.39070i	−0.823764	+	0.566933i	$0.808130\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	8.87564i		−	0.380191i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	8.08846	+	14.0096i	0.345207	+	0.597916i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−42.4545	−	11.3756i	−1.79885	−	0.482002i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−42.4545	−	11.3756i	−1.79885	−	0.482002i	−0.993798	+	0.111198i	$0.964531\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$	−3.57180	+	0.957060i	−0.150267	+	0.0402638i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−34.6410	+	20.0000i	−1.45223	+	0.838444i	−0.998608	−	0.0527519i	$-0.983201\pi$
$569$	−34.6410	+	20.0000i	−1.45223	+	0.838444i	−0.453619	+	0.891196i	$0.649867\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	10.1506	+	10.1506i	0.422576	+	0.422576i	0.886090	−	0.463513i	$-0.153411\pi$
$577$	10.1506	+	10.1506i	0.422576	+	0.422576i	−0.463513	+	0.886090i	$0.653411\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−9.50962	+	1.90192i	−0.393174	+	0.0786349i
$586$		0			0
$587$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$587$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	19.3468	−	19.3468i	0.794477	−	0.794477i	−0.187741	−	0.982219i	$-0.560117\pi$
$593$	19.3468	−	19.3468i	0.794477	−	0.794477i	0.982219	+	0.187741i	$0.0601166\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−16.3301	−	28.2846i	−0.666120	−	1.15375i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−16.3301	−	28.2846i	−0.666120	−	1.15375i	0.312861	−	0.949799i	$-0.398713\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.55256	−	9.52628i	−0.103776	−	0.387298i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	10.9115	−	40.7224i	0.440713	−	1.64476i	−0.286300	−	0.958140i	$-0.592425\pi$
$613$	10.9115	−	40.7224i	0.440713	−	1.64476i	0.727013	−	0.686624i	$-0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.3564	+	5.45448i	−0.819518	+	0.219589i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	−20.3564	+	5.45448i	−0.819518	+	0.219589i	−0.175382	+	0.984500i	$0.556116\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−13.5885			−0.543538
$626$		0			0
$627$		0			0
$628$		0			0
$629$	67.8827	+	67.8827i	2.70666	+	2.70666i
$630$		0			0
$631$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$631$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	22.6244	−	11.1865i	0.896410	−	0.443227i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.6506	+	9.03590i	0.618163	+	0.356897i	0.776153	−	0.630544i	$-0.217168\pi$
$641$	15.6506	+	9.03590i	0.618163	+	0.356897i	−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$643$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.0000	+	38.1051i	0.860927	+	1.49117i	0.871036	+	0.491220i	$0.163449\pi$
$653$	22.0000	+	38.1051i	0.860927	+	1.49117i	−0.0101092	+	0.999949i	$0.503218\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	10.7942	+	40.2846i	0.421123	+	1.57165i
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	1.30385	+	0.349365i	0.0507138	+	0.0135887i	0.284087	−	0.958799i	$-0.408310\pi$
$661$	1.30385	+	0.349365i	0.0507138	+	0.0135887i	−0.233373	+	0.972387i	$0.574976\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.91858	+	1.10770i	−0.0739560	+	0.0426985i	−0.536522	−	0.843886i	$-0.680262\pi$
$673$	−1.91858	+	1.10770i	−0.0739560	+	0.0426985i	0.462566	+	0.886585i	$0.346929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.00000			−0.0768662			−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000			−0.0768662			−0.0384331	+	0.999261i	$0.512237\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$683$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$684$		0			0
$685$	10.4545	−	18.1077i	0.399445	−	0.691859i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	37.6506	+	2.42820i	1.43437	+	0.0925072i
$690$		0			0
$691$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$691$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	57.8301	−	57.8301i	2.19047	−	2.19047i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	10.0000i		−	0.377695i	−0.982006	−	0.188847i	$-0.939525\pi$
$701$		−	10.0000i		−	0.377695i	0.982006	−	0.188847i	$-0.0604752\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	10.4474	+	38.9904i	0.392362	+	1.46431i	0.826227	+	0.563337i	$0.190483\pi$
$709$	10.4474	+	38.9904i	0.392362	+	1.46431i	−0.433865	+	0.900978i	$0.642851\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	24.2032	−	13.9737i	0.898884	−	0.518971i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−36.1506	−	36.1506i	−1.33525	−	1.33525i	−0.900595	−	0.434659i	$-0.856869\pi$
$733$	−36.1506	−	36.1506i	−1.33525	−	1.33525i	−0.434659	−	0.900595i	$-0.643131\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$739$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$743$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$744$		0			0
$745$	12.1865	+	7.03590i	0.446480	+	0.257775i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	9.00000	+	15.5885i	0.327111	+	0.566572i	0.981937	−	0.189207i	$-0.0605917\pi$
$757$	9.00000	+	15.5885i	0.327111	+	0.566572i	−0.654827	+	0.755779i	$0.727258\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.366025	+	1.36603i	0.0132684	+	0.0495184i	0.972243	−	0.233975i	$-0.0751733\pi$
$761$	0.366025	+	1.36603i	0.0132684	+	0.0495184i	−0.958974	+	0.283493i	$0.908507\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−20.5981	−	5.51924i	−0.744725	−	0.199548i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−13.5429	+	50.5429i	−0.488371	+	1.82263i	0.0760054	+	0.997107i	$0.475783\pi$
$769$	−13.5429	+	50.5429i	−0.488371	+	1.82263i	−0.564376	+	0.825518i	$0.690883\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.83013	−	1.83013i	0.245663	−	0.0658251i	−0.133887	−	0.990997i	$-0.542746\pi$
$773$	6.83013	−	1.83013i	0.245663	−	0.0658251i	0.379549	+	0.925172i	$0.376079\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.8827	−	15.8827i	−0.566877	−	0.566877i
$786$		0			0
$787$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$787$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−10.7846	+	16.1769i	−0.382973	+	0.574459i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.0526	−	11.0000i	−0.674876	−	0.389640i	0.123045	−	0.992401i	$-0.460734\pi$
$797$	−19.0526	−	11.0000i	−0.674876	−	0.389640i	−0.797922	+	0.602761i	$0.794067\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	9.00000	−	9.00000i	0.317999	−	0.317999i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.66987	+	16.7487i	0.339975	+	0.588853i	0.984428	−	0.175791i	$-0.0562482\pi$
$809$	9.66987	+	16.7487i	0.339975	+	0.588853i	−0.644453	+	0.764644i	$0.722915\pi$
$810$		0			0
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.02628	−	15.0263i	0.140518	−	0.524421i	−0.859396	−	0.511311i	$-0.829160\pi$
$821$	4.02628	−	15.0263i	0.140518	−	0.524421i	0.999914	−	0.0131101i	$-0.00417319\pi$
$822$		0			0
$823$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$823$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	49.1603	−	28.3827i	1.70741	−	0.985771i	0.769657	−	0.638457i	$-0.220427\pi$
$829$	49.1603	−	28.3827i	1.70741	−	0.985771i	0.937749	−	0.347314i	$-0.112906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	55.4974			1.92287
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$839$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$840$		0			0
$841$	−7.67949	+	13.3013i	−0.264810	+	0.458664i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.11474	−	9.23205i	−0.244754	−	0.317592i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	16.1699	−	16.1699i	0.553646	−	0.553646i	−0.373845	−	0.927491i	$-0.621961\pi$
$853$	16.1699	−	16.1699i	0.553646	−	0.553646i	0.927491	+	0.373845i	$0.121961\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	54.2295i		−	1.85244i	−0.376979	−	0.926222i	$-0.623037\pi$
$857$		−	54.2295i		−	1.85244i	0.376979	−	0.926222i	$-0.376963\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	−0.928203	−	3.46410i	−0.0315599	−	0.117783i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	5.49038	−	20.4904i	0.185821	−	0.693494i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	51.1147	−	13.6962i	1.72602	−	0.462486i	0.746762	−	0.665092i	$-0.231608\pi$
$877$	51.1147	−	13.6962i	1.72602	−	0.462486i	0.979260	+	0.202606i	$0.0649409\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	51.3564	−	29.6506i	1.73024	−	0.998955i	0.842271	−	0.539054i	$-0.181218\pi$
$881$	51.3564	−	29.6506i	1.73024	−	0.998955i	0.887970	−	0.459902i	$-0.152115\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	71.8468	+	41.4808i	2.39356	+	1.38192i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16.6865	+	16.6865i	−0.554679	+	0.554679i
$906$		0			0
$907$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$907$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$908$		0			0
$909$			35.1962i			1.16738i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−13.1506	+	49.0788i	−0.432390	+	1.61370i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	32.9186	−	8.82051i	1.08002	−	0.289391i	0.325418	−	0.945570i	$-0.394495\pi$
$929$	32.9186	−	8.82051i	1.08002	−	0.289391i	0.754606	+	0.656179i	$0.227828\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−60.5692			−1.97871			−0.989355	−	0.145522i	$-0.953514\pi$
$937$	−60.5692			−1.97871			−0.989355	+	0.145522i	$0.953514\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−19.0000	−	19.0000i	−0.619382	−	0.619382i	0.325991	−	0.945373i	$-0.394302\pi$
$941$	−19.0000	−	19.0000i	−0.619382	−	0.619382i	−0.945373	+	0.325991i	$0.894302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$947$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$948$		0			0
$949$	−37.6506	+	33.0885i	−1.22219	+	1.07410i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−48.4974	−	28.0000i	−1.57099	−	0.907009i	−0.996048	−	0.0888114i	$-0.971693\pi$
$953$	−48.4974	−	28.0000i	−1.57099	−	0.907009i	−0.574937	−	0.818198i	$-0.694974\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$			31.0000i			1.00000i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.84679	+	15.3231i	0.284788	+	0.493268i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.15321	+	22.9641i	−0.196859	+	0.734687i	0.794919	+	0.606715i	$0.207513\pi$
$977$	−6.15321	+	22.9641i	−0.196859	+	0.734687i	−0.991778	+	0.127971i	$0.959153\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−28.6865	+	7.68653i	−0.915891	+	0.245412i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	16.4711	−	9.50962i	0.524814	−	0.303002i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−10.3038	+	17.8468i	−0.326326	+	0.565213i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−10.3038	+	17.8468i	−0.326326	+	0.565213i	0.655454	+	0.755235i	$0.272477\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.bu.d.63.1		4
4.3	odd	2	CM	832.2.bu.d.63.1		4
8.3	odd	2		52.2.l.a.11.1	✓	4
8.5	even	2		52.2.l.a.11.1	✓	4
13.6	odd	12	inner	832.2.bu.d.383.1		4
24.5	odd	2		468.2.cb.d.271.1		4
24.11	even	2		468.2.cb.d.271.1		4
52.19	even	12	inner	832.2.bu.d.383.1		4
104.3	odd	6		676.2.f.e.99.1		4
104.5	odd	4		676.2.l.c.587.1		4
104.11	even	12		676.2.f.d.239.2		4
104.19	even	12		52.2.l.a.19.1	yes	4
104.21	odd	4		676.2.l.e.587.1		4
104.29	even	6		676.2.f.e.99.1		4
104.35	odd	6		676.2.l.c.319.1		4
104.37	odd	12		676.2.f.d.239.2		4
104.43	odd	6		676.2.l.e.319.1		4
104.45	odd	12		52.2.l.a.19.1	yes	4
104.51	odd	2		676.2.l.d.427.1		4
104.59	even	12		676.2.l.d.19.1		4
104.61	even	6		676.2.l.c.319.1		4
104.67	even	12		676.2.f.e.239.1		4
104.69	even	6		676.2.l.e.319.1		4
104.75	odd	6		676.2.f.d.99.2		4
104.77	even	2		676.2.l.d.427.1		4
104.83	even	4		676.2.l.c.587.1		4
104.85	odd	12		676.2.l.d.19.1		4
104.93	odd	12		676.2.f.e.239.1		4
104.99	even	4		676.2.l.e.587.1		4
104.101	even	6		676.2.f.d.99.2		4
312.149	even	12		468.2.cb.d.19.1		4
312.227	odd	12		468.2.cb.d.19.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.l.a.11.1	✓	4	8.3	odd	2
52.2.l.a.11.1	✓	4	8.5	even	2
52.2.l.a.19.1	yes	4	104.19	even	12
52.2.l.a.19.1	yes	4	104.45	odd	12
468.2.cb.d.19.1		4	312.149	even	12
468.2.cb.d.19.1		4	312.227	odd	12
468.2.cb.d.271.1		4	24.5	odd	2
468.2.cb.d.271.1		4	24.11	even	2
676.2.f.d.99.2		4	104.75	odd	6
676.2.f.d.99.2		4	104.101	even	6
676.2.f.d.239.2		4	104.11	even	12
676.2.f.d.239.2		4	104.37	odd	12
676.2.f.e.99.1		4	104.3	odd	6
676.2.f.e.99.1		4	104.29	even	6
676.2.f.e.239.1		4	104.67	even	12
676.2.f.e.239.1		4	104.93	odd	12
676.2.l.c.319.1		4	104.35	odd	6
676.2.l.c.319.1		4	104.61	even	6
676.2.l.c.587.1		4	104.5	odd	4
676.2.l.c.587.1		4	104.83	even	4
676.2.l.d.19.1		4	104.59	even	12
676.2.l.d.19.1		4	104.85	odd	12
676.2.l.d.427.1		4	104.51	odd	2
676.2.l.d.427.1		4	104.77	even	2
676.2.l.e.319.1		4	104.43	odd	6
676.2.l.e.319.1		4	104.69	even	6
676.2.l.e.587.1		4	104.21	odd	4
676.2.l.e.587.1		4	104.99	even	4
832.2.bu.d.63.1		4	1.1	even	1	trivial
832.2.bu.d.63.1		4	4.3	odd	2	CM
832.2.bu.d.383.1		4	13.6	odd	12	inner
832.2.bu.d.383.1		4	52.19	even	12	inner

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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.bu (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.633975 - 0.633975i) q^{5} +(-1.50000 + 2.59808i) q^{9} +(-3.59808 - 0.232051i) q^{13} +(-6.86603 - 3.96410i) q^{17} -4.19615i q^{25} +(3.33013 + 5.76795i) q^{29} +(-11.6962 - 3.13397i) q^{37} +(-2.66987 + 9.96410i) q^{41} +(2.59808 - 0.696152i) q^{45} +(-6.06218 + 3.50000i) q^{49} -10.4641 q^{53} +(2.69615 - 4.66987i) q^{61} +(2.13397 + 2.42820i) q^{65} +(9.83013 - 9.83013i) q^{73} +(-4.50000 - 7.79423i) q^{81} +(1.83975 + 6.86603i) q^{85} +(-4.09808 - 1.09808i) q^{89} +(-6.83013 + 1.83013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} - 6 q^{9} - 4 q^{13} - 24 q^{17} - 4 q^{29} - 26 q^{37} - 28 q^{41} - 28 q^{53} - 10 q^{61} + 12 q^{65} + 22 q^{73} - 18 q^{81} + 42 q^{85} - 6 q^{89} - 10 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 - 6 * q^9 - 4 * q^13 - 24 * q^17 - 4 * q^29 - 26 * q^37 - 28 * q^41 - 28 * q^53 - 10 * q^61 + 12 * q^65 + 22 * q^73 - 18 * q^81 + 42 * q^85 - 6 * q^89 - 10 * q^97

Introduction

L-functions

Modular forms

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