LMFDB - Embedded newform 676.2.f.e.99.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,2,Mod(99,676)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(676, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("676.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	676.f (of order $4$, degree $2$, 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:	$5.39788717664$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			99.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	676.99
Dual form			676.2.f.e.239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(0.633975 + 0.633975i) q^{5} +(-2.00000 + 2.00000i) q^{8} +3.00000 q^{9} +1.26795i q^{10} -4.00000 q^{16} +7.92820i q^{17} +(3.00000 + 3.00000i) q^{18} +(-1.26795 + 1.26795i) q^{20} -4.19615i q^{25} +6.66025 q^{29} +(-4.00000 - 4.00000i) q^{32} +(-7.92820 + 7.92820i) q^{34} +6.00000i q^{36} +(-8.56218 + 8.56218i) q^{37} -2.53590 q^{40} +(-7.29423 - 7.29423i) q^{41} +(1.90192 + 1.90192i) q^{45} -7.00000i q^{49} +(4.19615 - 4.19615i) q^{50} +10.4641 q^{53} +(6.66025 + 6.66025i) q^{58} +5.39230 q^{61} -8.00000i q^{64} -15.8564 q^{68} +(-6.00000 + 6.00000i) q^{72} +(9.83013 - 9.83013i) q^{73} -17.1244 q^{74} +(-2.53590 - 2.53590i) q^{80} +9.00000 q^{81} -14.5885i q^{82} +(-5.02628 + 5.02628i) q^{85} +(3.00000 - 3.00000i) q^{89} +3.80385i q^{90} +(5.00000 + 5.00000i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 6 q^{5} - 8 q^{8} + 12 q^{9} - 16 q^{16} + 12 q^{18} - 12 q^{20} - 8 q^{29} - 16 q^{32} - 4 q^{34} - 10 q^{37} - 24 q^{40} + 2 q^{41} + 18 q^{45} - 4 q^{50} + 28 q^{53} - 8 q^{58} - 20 q^{61}+ \cdots + 28 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 6 * q^5 - 8 * q^8 + 12 * q^9 - 16 * q^16 + 12 * q^18 - 12 * q^20 - 8 * q^29 - 16 * q^32 - 4 * q^34 - 10 * q^37 - 24 * q^40 + 2 * q^41 + 18 * q^45 - 4 * q^50 + 28 * q^53 - 8 * q^58 - 20 * q^61 - 8 * q^68 - 24 * q^72 + 22 * q^73 - 20 * q^74 - 24 * q^80 + 36 * q^81 + 18 * q^85 + 12 * q^89 + 20 * q^97 + 28 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$.

$n$	$339$	$509$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$			2.00000i			1.00000i
$5$	0.633975	+	0.633975i	0.283522	+	0.283522i	0.834512	−	0.550990i	$-0.185750\pi$
$5$	0.633975	+	0.633975i	0.283522	+	0.283522i	−0.550990	+	0.834512i	$0.685750\pi$
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$	3.00000			1.00000
$10$			1.26795i			0.400961i
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$			7.92820i			1.92287i	0.275029	+	0.961436i	$0.411312\pi$
$17$			7.92820i			1.92287i	−0.275029	+	0.961436i	$0.588688\pi$
$18$	3.00000	+	3.00000i	0.707107	+	0.707107i
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$	−1.26795	+	1.26795i	−0.283522	+	0.283522i
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$		−	4.19615i		−	0.839230i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.66025			1.23678			0.618389	−	0.785872i	$-0.287786\pi$
$29$	6.66025			1.23678			0.618389	+	0.785872i	$0.287786\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$	−4.00000	−	4.00000i	−0.707107	−	0.707107i
$33$		0			0
$34$	−7.92820	+	7.92820i	−1.35968	+	1.35968i
$35$		0			0
$36$			6.00000i			1.00000i
$37$	−8.56218	+	8.56218i	−1.40761	+	1.40761i	−0.635571	+	0.772043i	$0.719235\pi$
$37$	−8.56218	+	8.56218i	−1.40761	+	1.40761i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$		0			0
$40$	−2.53590			−0.400961
$41$	−7.29423	−	7.29423i	−1.13917	−	1.13917i	−0.988600	−	0.150567i	$-0.951890\pi$
$41$	−7.29423	−	7.29423i	−1.13917	−	1.13917i	−0.150567	−	0.988600i	$-0.548110\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	1.90192	+	1.90192i	0.283522	+	0.283522i
$46$		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
$49$		−	7.00000i		−	1.00000i
$50$	4.19615	−	4.19615i	0.593426	−	0.593426i
$51$		0			0
$52$		0			0
$53$	10.4641			1.43735			0.718677	−	0.695344i	$-0.244748\pi$
$53$	10.4641			1.43735			0.718677	+	0.695344i	$0.244748\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	6.66025	+	6.66025i	0.874534	+	0.874534i
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	5.39230			0.690414			0.345207	−	0.938527i	$-0.387809\pi$
$61$	5.39230			0.690414			0.345207	+	0.938527i	$0.387809\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	−15.8564			−1.92287
$69$		0			0
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$	−6.00000	+	6.00000i	−0.707107	+	0.707107i
$73$	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	$-0.0524664\pi$
$74$	−17.1244			−1.99067
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−2.53590	−	2.53590i	−0.283522	−	0.283522i
$81$	9.00000			1.00000
$82$		−	14.5885i		−	1.61103i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−5.02628	+	5.02628i	−0.545177	+	0.545177i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000	−	3.00000i	0.317999	−	0.317999i	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	3.00000	−	3.00000i	0.317999	−	0.317999i	0.847998	+	0.529999i	$0.177808\pi$
$90$			3.80385i			0.400961i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.00000	+	5.00000i	0.507673	+	0.507673i	0.913812	−	0.406138i	$-0.133125\pi$
$97$	5.00000	+	5.00000i	0.507673	+	0.507673i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	7.00000	−	7.00000i	0.707107	−	0.707107i
$99$		0			0
$100$	8.39230			0.839230
$101$		−	11.7321i		−	1.16738i	−0.811976	−	0.583691i	$-0.801608\pi$
$101$		−	11.7321i		−	1.16738i	0.811976	−	0.583691i	$-0.198392\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	10.4641	+	10.4641i	1.01636	+	1.01636i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	0.287348	−	0.957826i	$-0.407226\pi$
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.12436			−0.387987			−0.193993	−	0.981003i	$-0.562144\pi$
$113$	−4.12436			−0.387987			−0.193993	+	0.981003i	$0.562144\pi$
$114$		0			0
$115$		0			0
$116$			13.3205i			1.23678i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	11.0000i		−	1.00000i
$122$	5.39230	+	5.39230i	0.488196	+	0.488196i
$123$		0			0
$124$		0			0
$125$	5.83013	−	5.83013i	0.521462	−	0.521462i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	8.00000	−	8.00000i	0.707107	−	0.707107i
$129$		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
$136$	−15.8564	−	15.8564i	−1.35968	−	1.35968i
$137$	16.4904	−	16.4904i	1.40887	−	1.40887i	0.643013	−	0.765855i	$-0.277684\pi$
$137$	16.4904	−	16.4904i	1.40887	−	1.40887i	0.765855	−	0.643013i	$-0.222316\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$	4.22243	+	4.22243i	0.350654	+	0.350654i
$146$	19.6603			1.62709
$147$		0			0
$148$	−17.1244	−	17.1244i	−1.40761	−	1.40761i
$149$	−11.0981	−	11.0981i	−0.909190	−	0.909190i	0.0870170	−	0.996207i	$-0.472267\pi$
$149$	−11.0981	−	11.0981i	−0.909190	−	0.909190i	−0.996207	+	0.0870170i	$0.972267\pi$
$150$		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
$153$			23.7846i			1.92287i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−25.0526			−1.99941			−0.999706	−	0.0242497i	$-0.992280\pi$
$157$	−25.0526			−1.99941			−0.999706	+	0.0242497i	$0.992280\pi$
$158$		0			0
$159$		0			0
$160$		−	5.07180i		−	0.400961i
$161$		0			0
$162$	9.00000	+	9.00000i	0.707107	+	0.707107i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$	14.5885	−	14.5885i	1.13917	−	1.13917i
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$		0			0
$170$	−10.0526			−0.770996
$171$		0			0
$172$		0			0
$173$			4.00000i			0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$			4.00000i			0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	6.00000			0.449719
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−3.80385	+	3.80385i	−0.283522	+	0.283522i
$181$			26.3205i			1.95639i	0.207693	+	0.978194i	$0.433404\pi$
$181$			26.3205i			1.95639i	−0.207693	+	0.978194i	$0.566596\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.8564			−0.798179
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	13.9545	−	13.9545i	1.00447	−	1.00447i	0.00447566	−	0.999990i	$-0.498575\pi$
$193$	13.9545	−	13.9545i	1.00447	−	1.00447i	0.999990	−	0.00447566i	$-0.00142465\pi$
$194$			10.0000i			0.717958i
$195$		0			0
$196$	14.0000			1.00000
$197$	15.0000	+	15.0000i	1.06871	+	1.06871i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	15.0000	+	15.0000i	1.06871	+	1.06871i	0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	8.39230	+	8.39230i	0.593426	+	0.593426i
$201$		0			0
$202$	11.7321	−	11.7321i	0.825464	−	0.825464i
$203$		0			0
$204$		0			0
$205$		−	9.24871i		−	0.645958i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			20.9282i			1.43735i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		−	14.0000i		−	0.948200i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		−	12.5885i		−	0.839230i
$226$	−4.12436	−	4.12436i	−0.274348	−	0.274348i
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$	−17.0000	+	17.0000i	−1.12339	+	1.12339i	−0.132164	+	0.991228i	$0.542192\pi$
$229$	−17.0000	+	17.0000i	−1.12339	+	1.12339i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$		0			0
$232$	−13.3205	+	13.3205i	−0.874534	+	0.874534i
$233$		−	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
$241$	−19.0263	+	19.0263i	−1.22559	+	1.22559i	−0.259975	+	0.965615i	$0.583714\pi$
$241$	−19.0263	+	19.0263i	−1.22559	+	1.22559i	−0.965615	+	0.259975i	$0.916286\pi$
$242$	11.0000	−	11.0000i	0.707107	−	0.707107i
$243$		0			0
$244$			10.7846i			0.690414i
$245$	4.43782	−	4.43782i	0.283522	−	0.283522i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	11.6603			0.737459
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$			14.2679i			0.890010i	0.895528	+	0.445005i	$0.146798\pi$
$257$			14.2679i			0.890010i	−0.895528	+	0.445005i	$0.853202\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	19.9808			1.23678
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	6.63397	+	6.63397i	0.407522	+	0.407522i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$271$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$272$		−	31.7128i		−	1.92287i
$273$		0			0
$274$	32.9808			1.99244
$275$		0			0
$276$		0			0
$277$		−	1.58846i		−	0.0954411i	−0.998861	−	0.0477206i	$-0.984804\pi$
$277$		−	1.58846i		−	0.0954411i	0.998861	−	0.0477206i	$-0.0151957\pi$
$278$		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			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−12.0000	−	12.0000i	−0.707107	−	0.707107i
$289$	−45.8564			−2.69744
$290$			8.44486i			0.495899i
$291$		0			0
$292$	19.6603	+	19.6603i	1.15053	+	1.15053i
$293$	3.49038	−	3.49038i	0.203910	−	0.203910i	−0.597763	−	0.801673i	$-0.703944\pi$
$293$	3.49038	−	3.49038i	0.203910	−	0.203910i	0.801673	+	0.597763i	$0.203944\pi$
$294$		0			0
$295$		0			0
$296$		−	34.2487i		−	1.99067i
$297$		0			0
$298$		−	22.1962i		−	1.28579i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.41858	+	3.41858i	0.195748	+	0.195748i
$306$	−23.7846	+	23.7846i	−1.35968	+	1.35968i
$307$		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	$0.737274\pi$
$314$	−25.0526	−	25.0526i	−1.41380	−	1.41380i
$315$		0			0
$316$		0			0
$317$	23.1506	+	23.1506i	1.30027	+	1.30027i	0.928208	+	0.372061i	$0.121349\pi$
$317$	23.1506	+	23.1506i	1.30027	+	1.30027i	0.372061	+	0.928208i	$0.378651\pi$
$318$		0			0
$319$		0			0
$320$	5.07180	−	5.07180i	0.283522	−	0.283522i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	29.1769			1.61103
$329$		0			0
$330$		0			0
$331$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$331$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$332$		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
$340$	−10.0526	−	10.0526i	−0.545177	−	0.545177i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−4.00000	+	4.00000i	−0.215041	+	0.215041i
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	−23.0000	+	23.0000i	−1.23116	+	1.23116i	−0.267644	+	0.963518i	$0.586245\pi$
$349$	−23.0000	+	23.0000i	−1.23116	+	1.23116i	−0.963518	+	0.267644i	$0.913755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−20.2942	−	20.2942i	−1.08015	−	1.08015i	−0.996495	−	0.0836583i	$-0.973340\pi$
$353$	−20.2942	−	20.2942i	−1.08015	−	1.08015i	−0.0836583	−	0.996495i	$-0.526660\pi$
$354$		0			0
$355$		0			0
$356$	6.00000	+	6.00000i	0.317999	+	0.317999i
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$	−7.60770			−0.400961
$361$			19.0000i			1.00000i
$362$	−26.3205	+	26.3205i	−1.38338	+	1.38338i
$363$		0			0
$364$		0			0
$365$	12.4641			0.652401
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−21.8827	−	21.8827i	−1.13917	−	1.13917i
$370$	−10.8564	−	10.8564i	−0.564398	−	0.564398i
$371$		0			0
$372$		0			0
$373$	−30.1244			−1.55978			−0.779890	−	0.625917i	$-0.784725\pi$
$373$	−30.1244			−1.55978			−0.779890	+	0.625917i	$0.784725\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$379$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$	27.9090			1.42053
$387$		0			0
$388$	−10.0000	+	10.0000i	−0.507673	+	0.507673i
$389$			0.320508i			0.0162504i	0.999967	+	0.00812520i	$0.00258636\pi$
$389$			0.320508i			0.0162504i	−0.999967	+	0.00812520i	$0.997414\pi$
$390$		0			0
$391$		0			0
$392$	14.0000	+	14.0000i	0.707107	+	0.707107i
$393$		0			0
$394$			30.0000i			1.51138i
$395$		0			0
$396$		0			0
$397$	−25.0000	+	25.0000i	−1.25471	+	1.25471i	−0.301131	+	0.953583i	$0.597364\pi$
$397$	−25.0000	+	25.0000i	−1.25471	+	1.25471i	−0.953583	+	0.301131i	$0.902636\pi$
$398$		0			0
$399$		0			0
$400$			16.7846i			0.839230i
$401$	26.9545	−	26.9545i	1.34604	−	1.34604i	0.456129	−	0.889914i	$-0.349236\pi$
$401$	26.9545	−	26.9545i	1.34604	−	1.34604i	0.889914	−	0.456129i	$-0.150764\pi$
$402$		0			0
$403$		0			0
$404$	23.4641			1.16738
$405$	5.70577	+	5.70577i	0.283522	+	0.283522i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	11.4186	+	11.4186i	0.564613	+	0.564613i	0.930614	−	0.366002i	$-0.119274\pi$
$409$	11.4186	+	11.4186i	0.564613	+	0.564613i	−0.366002	+	0.930614i	$0.619274\pi$
$410$	9.24871	−	9.24871i	0.456761	−	0.456761i
$411$		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		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	13.6340	+	13.6340i	0.664479	+	0.664479i	0.956433	−	0.291953i	$-0.0943052\pi$
$421$	13.6340	+	13.6340i	0.664479	+	0.664479i	−0.291953	+	0.956433i	$0.594305\pi$
$422$		0			0
$423$		0			0
$424$	−20.9282	+	20.9282i	−1.01636	+	1.01636i
$425$	33.2679			1.61373
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$		0			0
$433$			17.4449i			0.838347i	0.907906	+	0.419173i	$0.137680\pi$
$433$			17.4449i			0.838347i	−0.907906	+	0.419173i	$0.862320\pi$
$434$		0			0
$435$		0			0
$436$	14.0000	−	14.0000i	0.670478	−	0.670478i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		−	21.0000i		−	1.00000i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	3.80385			0.180320
$446$		0			0
$447$		0			0
$448$		0			0
$449$	27.0000	−	27.0000i	1.27421	−	1.27421i	0.330350	−	0.943858i	$-0.392833\pi$
$449$	27.0000	−	27.0000i	1.27421	−	1.27421i	0.943858	−	0.330350i	$-0.107167\pi$
$450$	12.5885	−	12.5885i	0.593426	−	0.593426i
$451$		0			0
$452$		−	8.24871i		−	0.387987i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.22243	+	2.22243i	0.103961	+	0.103961i	0.757174	−	0.653213i	$-0.226579\pi$
$457$	2.22243	+	2.22243i	0.103961	+	0.103961i	−0.653213	+	0.757174i	$0.726579\pi$
$458$	−34.0000			−1.58872
$459$		0			0
$460$		0			0
$461$	−20.6147	−	20.6147i	−0.960124	−	0.960124i	0.0391109	−	0.999235i	$-0.487547\pi$
$461$	−20.6147	−	20.6147i	−0.960124	−	0.960124i	−0.999235	+	0.0391109i	$0.987547\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	−26.6410			−1.23678
$465$		0			0
$466$	16.0000	−	16.0000i	0.741186	−	0.741186i
$467$		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
$477$	31.3923			1.43735
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$		0			0
$482$	−38.0526			−1.73325
$483$		0			0
$484$	22.0000			1.00000
$485$			6.33975i			0.287873i
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	−10.7846	+	10.7846i	−0.488196	+	0.488196i
$489$		0			0
$490$	8.87564			0.400961
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		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	$0.250000\pi$
$500$	11.6603	+	11.6603i	0.521462	+	0.521462i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	7.43782	−	7.43782i	0.330979	−	0.330979i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	28.2224	+	28.2224i	1.25094	+	1.25094i	0.955300	+	0.295637i	$0.0955319\pi$
$509$	28.2224	+	28.2224i	1.25094	+	1.25094i	0.295637	+	0.955300i	$0.404468\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$		0			0
$514$	−14.2679	+	14.2679i	−0.629332	+	0.629332i
$515$		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	$0.493393\pi$
$522$	19.9808	+	19.9808i	0.874534	+	0.874534i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$			13.2679i			0.576323i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−20.0000	−	20.0000i	−0.862261	−	0.862261i
$539$		0			0
$540$		0			0
$541$	32.3468	−	32.3468i	1.39070	−	1.39070i	0.566933	−	0.823764i	$-0.308130\pi$
$541$	32.3468	−	32.3468i	1.39070	−	1.39070i	0.823764	−	0.566933i	$-0.191870\pi$
$542$		0			0
$543$		0			0
$544$	31.7128	−	31.7128i	1.35968	−	1.35968i
$545$		−	8.87564i		−	0.380191i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	32.9808	+	32.9808i	1.40887	+	1.40887i
$549$	16.1769			0.690414
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.58846	−	1.58846i	0.0674871	−	0.0674871i
$555$		0			0
$556$		0			0
$557$	−31.0788	+	31.0788i	−1.31685	+	1.31685i	−0.400599	+	0.916253i	$0.631198\pi$
$557$	−31.0788	+	31.0788i	−1.31685	+	1.31685i	−0.916253	+	0.400599i	$0.868802\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−25.3731			−1.07030
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$	−2.61474	−	2.61474i	−0.110003	−	0.110003i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	40.0000i		−	1.67689i	−0.544988	−	0.838444i	$-0.683466\pi$
$569$		−	40.0000i		−	1.67689i	0.544988	−	0.838444i	$-0.316534\pi$
$570$		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
$576$		−	24.0000i		−	1.00000i
$577$	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	$0.653411\pi$
$578$	−45.8564	−	45.8564i	−1.90738	−	1.90738i
$579$		0			0
$580$	−8.44486	+	8.44486i	−0.350654	+	0.350654i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			39.3205i			1.62709i
$585$		0			0
$586$	6.98076			0.288373
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	34.2487	−	34.2487i	1.40761	−	1.40761i
$593$	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
$596$	22.1962	−	22.1962i	0.909190	−	0.909190i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	32.6603			1.33224			0.666120	−	0.745845i	$-0.267954\pi$
$601$	32.6603			1.33224			0.666120	+	0.745845i	$0.267954\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.97372	−	6.97372i	0.283522	−	0.283522i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$			6.83717i			0.276829i
$611$		0			0
$612$	−47.5692			−1.92287
$613$	−29.8109	−	29.8109i	−1.20405	−	1.20405i	−0.972924	−	0.231127i	$-0.925759\pi$
$613$	−29.8109	−	29.8109i	−1.20405	−	1.20405i	−0.231127	−	0.972924i	$-0.574241\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.9019	+	14.9019i	0.599929	+	0.599929i	0.940294	−	0.340365i	$-0.110551\pi$
$617$	14.9019	+	14.9019i	0.599929	+	0.599929i	−0.340365	+	0.940294i	$0.610551\pi$
$618$		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
$626$	−24.0000	−	24.0000i	−0.959233	−	0.959233i
$627$		0			0
$628$		−	50.1051i		−	1.99941i
$629$	−67.8827	−	67.8827i	−2.70666	−	2.70666i
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$		0			0
$633$		0			0
$634$			46.3013i			1.83886i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	10.1436			0.400961
$641$		−	18.0718i		−	0.713793i	−0.934144	−	0.356897i	$-0.883835\pi$
$641$		−	18.0718i		−	0.713793i	0.934144	−	0.356897i	$-0.116165\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−18.0000	+	18.0000i	−0.707107	+	0.707107i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	44.0000			1.72185			0.860927	−	0.508729i	$-0.169885\pi$
$653$	44.0000			1.72185			0.860927	+	0.508729i	$0.169885\pi$
$654$		0			0
$655$		0			0
$656$	29.1769	+	29.1769i	1.13917	+	1.13917i
$657$	29.4904	−	29.4904i	1.15053	−	1.15053i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	0.954483	−	0.954483i	0.0371251	−	0.0371251i	−0.688301	−	0.725426i	$-0.741643\pi$
$661$	0.954483	−	0.954483i	0.0371251	−	0.0371251i	0.725426	+	0.688301i	$0.241643\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−51.3731			−1.99067
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	2.21539i		−	0.0853970i	−0.999088	−	0.0426985i	$-0.986405\pi$
$673$		−	2.21539i		−	0.0853970i	0.999088	−	0.0426985i	$-0.0135955\pi$
$674$	18.7128	−	18.7128i	0.720791	−	0.720791i
$675$		0			0
$676$		0			0
$677$	2.00000			0.0768662			0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000			0.0768662			0.0384331	+	0.999261i	$0.487763\pi$
$678$		0			0
$679$		0			0
$680$		−	20.1051i		−	0.770996i
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	20.9090			0.798891
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$691$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$692$	−8.00000			−0.304114
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	57.8301	−	57.8301i	2.19047	−	2.19047i
$698$	−46.0000			−1.74113
$699$		0			0
$700$		0			0
$701$			10.0000i			0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$			10.0000i			0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		−	40.5885i		−	1.52757i
$707$		0			0
$708$		0			0
$709$	−28.5429	+	28.5429i	−1.07195	+	1.07195i	−0.0747503	+	0.997202i	$0.523816\pi$
$709$	−28.5429	+	28.5429i	−1.07195	+	1.07195i	−0.997202	+	0.0747503i	$0.976184\pi$
$710$		0			0
$711$		0			0
$712$			12.0000i			0.449719i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−7.60770	−	7.60770i	−0.283522	−	0.283522i
$721$		0			0
$722$	−19.0000	+	19.0000i	−0.707107	+	0.707107i
$723$		0			0
$724$	−52.6410			−1.95639
$725$		−	27.9474i		−	1.03794i
$726$		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
$730$	12.4641	+	12.4641i	0.461317	+	0.461317i
$731$		0			0
$732$		0			0
$733$	36.1506	+	36.1506i	1.33525	+	1.33525i	0.900595	+	0.434659i	$0.143131\pi$
$733$	36.1506	+	36.1506i	1.33525	+	1.33525i	0.434659	+	0.900595i	$0.356869\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		−	43.7654i		−	1.61103i
$739$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$739$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$740$		−	21.7128i		−	0.798179i
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		−	14.0718i		−	0.515551i
$746$	−30.1244	−	30.1244i	−1.10293	−	1.10293i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	18.0000			0.654221			0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000			0.654221			0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.00000	−	1.00000i	0.0362500	−	0.0362500i	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	1.00000	−	1.00000i	0.0362500	−	0.0362500i	0.724999	+	0.688749i	$0.241840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−15.0788	+	15.0788i	−0.545177	+	0.545177i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−37.0000	−	37.0000i	−1.33425	−	1.33425i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−37.0000	−	37.0000i	−1.33425	−	1.33425i	−0.432731	−	0.901523i	$-0.642450\pi$
$770$		0			0
$771$		0			0
$772$	27.9090	+	27.9090i	1.00447	+	1.00447i
$773$	5.00000	+	5.00000i	0.179838	+	0.179838i	0.791285	−	0.611448i	$-0.209412\pi$
$773$	5.00000	+	5.00000i	0.179838	+	0.179838i	−0.611448	+	0.791285i	$0.709412\pi$
$774$		0			0
$775$		0			0
$776$	−20.0000			−0.717958
$777$		0			0
$778$	−0.320508	+	0.320508i	−0.0114908	+	0.0114908i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			28.0000i			1.00000i
$785$	−15.8827	−	15.8827i	−0.566877	−	0.566877i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−30.0000	+	30.0000i	−1.06871	+	1.06871i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−50.0000			−1.77443
$795$		0			0
$796$		0			0
$797$		−	22.0000i		−	0.779280i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$		−	22.0000i		−	0.779280i	0.920967	−	0.389640i	$-0.127401\pi$
$798$		0			0
$799$		0			0
$800$	−16.7846	+	16.7846i	−0.593426	+	0.593426i
$801$	9.00000	−	9.00000i	0.317999	−	0.317999i
$802$	53.9090			1.90359
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	23.4641	+	23.4641i	0.825464	+	0.825464i
$809$	−19.3397			−0.679949			−0.339975	−	0.940435i	$-0.610418\pi$
$809$	−19.3397			−0.679949			−0.339975	+	0.940435i	$0.610418\pi$
$810$			11.4115i			0.400961i
$811$		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
$818$			22.8372i			0.798483i
$819$		0			0
$820$	18.4974			0.645958
$821$	−11.0000	−	11.0000i	−0.383903	−	0.383903i	0.488603	−	0.872506i	$-0.337507\pi$
$821$	−11.0000	−	11.0000i	−0.383903	−	0.383903i	−0.872506	+	0.488603i	$0.837507\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		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
$829$		−	56.7654i		−	1.97154i	−0.168091	−	0.985771i	$-0.553760\pi$
$829$		−	56.7654i		−	1.97154i	0.168091	−	0.985771i	$-0.446240\pi$
$830$		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		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	15.3590			0.529620
$842$			27.2679i			0.939716i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−41.8564			−1.43735
$849$		0			0
$850$	33.2679	+	33.2679i	1.14108	+	1.14108i
$851$		0			0
$852$		0			0
$853$	−16.1699	+	16.1699i	−0.553646	+	0.553646i	−0.927491	−	0.373845i	$-0.878039\pi$
$853$	−16.1699	+	16.1699i	−0.553646	+	0.553646i	0.373845	+	0.927491i	$0.378039\pi$
$854$		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
$865$	−2.53590	+	2.53590i	−0.0862231	+	0.0862231i
$866$	−17.4449	+	17.4449i	−0.592801	+	0.592801i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	28.0000			0.948200
$873$	15.0000	+	15.0000i	0.507673	+	0.507673i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	37.4186	+	37.4186i	1.26354	+	1.26354i	0.949367	+	0.314169i	$0.101726\pi$
$877$	37.4186	+	37.4186i	1.26354	+	1.26354i	0.314169	+	0.949367i	$0.398274\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			59.3013i			1.99791i	0.0456985	+	0.998955i	$0.485449\pi$
$881$			59.3013i			1.99791i	−0.0456985	+	0.998955i	$0.514551\pi$
$882$	21.0000	−	21.0000i	0.707107	−	0.707107i
$883$		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		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$	3.80385	+	3.80385i	0.127505	+	0.127505i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	54.0000			1.80200
$899$		0			0
$900$	25.1769			0.839230
$901$			82.9615i			2.76385i
$902$		0			0
$903$		0			0
$904$	8.24871	−	8.24871i	0.274348	−	0.274348i
$905$	−16.6865	+	16.6865i	−0.554679	+	0.554679i
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		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
$914$			4.44486i			0.147023i
$915$		0			0
$916$	−34.0000	−	34.0000i	−1.12339	−	1.12339i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		−	41.2295i		−	1.35782i
$923$		0			0
$924$		0			0
$925$	35.9282	+	35.9282i	1.18131	+	1.18131i
$926$		0			0
$927$		0			0
$928$	−26.6410	−	26.6410i	−0.874534	−	0.874534i
$929$	−24.0981	−	24.0981i	−0.790632	−	0.790632i	0.190965	−	0.981597i	$-0.438838\pi$
$929$	−24.0981	−	24.0981i	−0.790632	−	0.790632i	−0.981597	+	0.190965i	$0.938838\pi$
$930$		0			0
$931$		0			0
$932$	32.0000			1.04819
$933$		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	0.945373	−	0.325991i	$-0.105698\pi$
$941$	19.0000	+	19.0000i	0.619382	+	0.619382i	−0.325991	+	0.945373i	$0.605698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			56.0000i			1.81402i	0.421111	+	0.907009i	$0.361640\pi$
$953$			56.0000i			1.81402i	−0.421111	+	0.907009i	$0.638360\pi$
$954$	31.3923	+	31.3923i	1.01636	+	1.01636i
$955$		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
$964$	−38.0526	−	38.0526i	−1.22559	−	1.22559i
$965$	17.6936			0.569576
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	22.0000	+	22.0000i	0.707107	+	0.707107i
$969$		0			0
$970$	−6.33975	+	6.33975i	−0.203557	+	0.203557i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−21.5692			−0.690414
$977$	−16.8109	−	16.8109i	−0.537828	−	0.537828i	0.385063	−	0.922890i	$-0.374180\pi$
$977$	−16.8109	−	16.8109i	−0.537828	−	0.537828i	−0.922890	+	0.385063i	$0.874180\pi$
$978$		0			0
$979$		0			0
$980$	8.87564	+	8.87564i	0.283522	+	0.283522i
$981$	−21.0000	−	21.0000i	−0.670478	−	0.670478i
$982$		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
$985$			19.0192i			0.606003i
$986$	−52.8038	+	52.8038i	−1.68162	+	1.68162i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−20.6077			−0.652652			−0.326326	−	0.945257i	$-0.605811\pi$
$997$	−20.6077			−0.652652			−0.326326	+	0.945257i	$0.605811\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.2.f.e.99.1		4
4.3	odd	2	CM	676.2.f.e.99.1		4
13.2	odd	12		52.2.l.a.19.1	yes	4
13.3	even	3		676.2.l.c.319.1		4
13.4	even	6		676.2.l.d.427.1		4
13.5	odd	4	inner	676.2.f.e.239.1		4
13.6	odd	12		676.2.l.c.587.1		4
13.7	odd	12		676.2.l.e.587.1		4
13.8	odd	4		676.2.f.d.239.2		4
13.9	even	3		52.2.l.a.11.1	✓	4
13.10	even	6		676.2.l.e.319.1		4
13.11	odd	12		676.2.l.d.19.1		4
13.12	even	2		676.2.f.d.99.2		4
39.2	even	12		468.2.cb.d.19.1		4
39.35	odd	6		468.2.cb.d.271.1		4
52.3	odd	6		676.2.l.c.319.1		4
52.7	even	12		676.2.l.e.587.1		4
52.11	even	12		676.2.l.d.19.1		4
52.15	even	12		52.2.l.a.19.1	yes	4
52.19	even	12		676.2.l.c.587.1		4
52.23	odd	6		676.2.l.e.319.1		4
52.31	even	4	inner	676.2.f.e.239.1		4
52.35	odd	6		52.2.l.a.11.1	✓	4
52.43	odd	6		676.2.l.d.427.1		4
52.47	even	4		676.2.f.d.239.2		4
52.51	odd	2		676.2.f.d.99.2		4
104.35	odd	6		832.2.bu.d.63.1		4
104.61	even	6		832.2.bu.d.63.1		4
104.67	even	12		832.2.bu.d.383.1		4
104.93	odd	12		832.2.bu.d.383.1		4
156.35	even	6		468.2.cb.d.271.1		4
156.119	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	13.9	even	3
52.2.l.a.11.1	✓	4	52.35	odd	6
52.2.l.a.19.1	yes	4	13.2	odd	12
52.2.l.a.19.1	yes	4	52.15	even	12
468.2.cb.d.19.1		4	39.2	even	12
468.2.cb.d.19.1		4	156.119	odd	12
468.2.cb.d.271.1		4	39.35	odd	6
468.2.cb.d.271.1		4	156.35	even	6
676.2.f.d.99.2		4	13.12	even	2
676.2.f.d.99.2		4	52.51	odd	2
676.2.f.d.239.2		4	13.8	odd	4
676.2.f.d.239.2		4	52.47	even	4
676.2.f.e.99.1		4	1.1	even	1	trivial
676.2.f.e.99.1		4	4.3	odd	2	CM
676.2.f.e.239.1		4	13.5	odd	4	inner
676.2.f.e.239.1		4	52.31	even	4	inner
676.2.l.c.319.1		4	13.3	even	3
676.2.l.c.319.1		4	52.3	odd	6
676.2.l.c.587.1		4	13.6	odd	12
676.2.l.c.587.1		4	52.19	even	12
676.2.l.d.19.1		4	13.11	odd	12
676.2.l.d.19.1		4	52.11	even	12
676.2.l.d.427.1		4	13.4	even	6
676.2.l.d.427.1		4	52.43	odd	6
676.2.l.e.319.1		4	13.10	even	6
676.2.l.e.319.1		4	52.23	odd	6
676.2.l.e.587.1		4	13.7	odd	12
676.2.l.e.587.1		4	52.7	even	12
832.2.bu.d.63.1		4	104.35	odd	6
832.2.bu.d.63.1		4	104.61	even	6
832.2.bu.d.383.1		4	104.67	even	12
832.2.bu.d.383.1		4	104.93	odd	12

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:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	676.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(0.633975 + 0.633975i) q^{5} +(-2.00000 + 2.00000i) q^{8} +3.00000 q^{9} +1.26795i q^{10} -4.00000 q^{16} +7.92820i q^{17} +(3.00000 + 3.00000i) q^{18} +(-1.26795 + 1.26795i) q^{20} -4.19615i q^{25} +6.66025 q^{29} +(-4.00000 - 4.00000i) q^{32} +(-7.92820 + 7.92820i) q^{34} +6.00000i q^{36} +(-8.56218 + 8.56218i) q^{37} -2.53590 q^{40} +(-7.29423 - 7.29423i) q^{41} +(1.90192 + 1.90192i) q^{45} -7.00000i q^{49} +(4.19615 - 4.19615i) q^{50} +10.4641 q^{53} +(6.66025 + 6.66025i) q^{58} +5.39230 q^{61} -8.00000i q^{64} -15.8564 q^{68} +(-6.00000 + 6.00000i) q^{72} +(9.83013 - 9.83013i) q^{73} -17.1244 q^{74} +(-2.53590 - 2.53590i) q^{80} +9.00000 q^{81} -14.5885i q^{82} +(-5.02628 + 5.02628i) q^{85} +(3.00000 - 3.00000i) q^{89} +3.80385i q^{90} +(5.00000 + 5.00000i) q^{97} +(7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 6 q^{5} - 8 q^{8} + 12 q^{9} - 16 q^{16} + 12 q^{18} - 12 q^{20} - 8 q^{29} - 16 q^{32} - 4 q^{34} - 10 q^{37} - 24 q^{40} + 2 q^{41} + 18 q^{45} - 4 q^{50} + 28 q^{53} - 8 q^{58} - 20 q^{61}+ \cdots + 28 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 6 * q^5 - 8 * q^8 + 12 * q^9 - 16 * q^16 + 12 * q^18 - 12 * q^20 - 8 * q^29 - 16 * q^32 - 4 * q^34 - 10 * q^37 - 24 * q^40 + 2 * q^41 + 18 * q^45 - 4 * q^50 + 28 * q^53 - 8 * q^58 - 20 * q^61 - 8 * q^68 - 24 * q^72 + 22 * q^73 - 20 * q^74 - 24 * q^80 + 36 * q^81 + 18 * q^85 + 12 * q^89 + 20 * q^97 + 28 * q^98

Introduction

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