LMFDB - Embedded newform 676.2.l.e.319.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [676,2,Mod(19,676)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(676, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("676.19"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	676.l (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,2,0,0,-6,0,0,8,-6,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$5.39788717664$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			319.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	676.319
Dual form			676.2.l.e.587.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(-0.633975 - 0.633975i) q^{5} +(2.00000 - 2.00000i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(-1.09808 - 0.633975i) q^{10} +(2.00000 - 3.46410i) q^{16} +(6.86603 - 3.96410i) q^{17} +(-3.00000 - 3.00000i) q^{18} +(-1.73205 - 0.464102i) q^{20} -4.19615i q^{25} +(-3.33013 + 5.76795i) q^{29} +(1.46410 - 5.46410i) q^{32} +(7.92820 - 7.92820i) q^{34} +(-5.19615 - 3.00000i) q^{36} +(3.13397 + 11.6962i) q^{37} -2.53590 q^{40} +(-9.96410 + 2.66987i) q^{41} +(-0.696152 + 2.59808i) q^{45} +(6.06218 + 3.50000i) q^{49} +(-1.53590 - 5.73205i) q^{50} +10.4641 q^{53} +(-2.43782 + 9.09808i) q^{58} +(-2.69615 - 4.66987i) q^{61} -8.00000i q^{64} +(7.92820 - 13.7321i) q^{68} +(-8.19615 - 2.19615i) q^{72} +(-9.83013 + 9.83013i) q^{73} +(8.56218 + 14.8301i) q^{74} +(-3.46410 + 0.928203i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(-12.6340 + 7.29423i) q^{82} +(-6.86603 - 1.83975i) q^{85} +(-1.09808 - 4.09808i) q^{89} +3.80385i q^{90} +(-1.83013 + 6.83013i) q^{97} +(9.56218 + 2.56218i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 6 q^{5} + 8 q^{8} - 6 q^{9} + 6 q^{10} + 8 q^{16} + 24 q^{17} - 12 q^{18} + 4 q^{29} - 8 q^{32} + 4 q^{34} + 16 q^{37} - 24 q^{40} - 26 q^{41} + 18 q^{45} - 20 q^{50} + 28 q^{53} - 34 q^{58}+ \cdots + 14 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 6 * q^5 + 8 * q^8 - 6 * q^9 + 6 * q^10 + 8 * q^16 + 24 * q^17 - 12 * q^18 + 4 * q^29 - 8 * q^32 + 4 * q^34 + 16 * q^37 - 24 * q^40 - 26 * q^41 + 18 * q^45 - 20 * q^50 + 28 * q^53 - 34 * q^58 + 10 * q^61 + 4 * q^68 - 12 * q^72 - 22 * q^73 + 10 * q^74 - 18 * q^81 - 54 * q^82 - 24 * q^85 + 6 * q^89 + 10 * q^97 + 14 * q^98

Character values

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

$n$	$339$	$509$
$\chi(n)$	$-1$	$e\left(\frac{11}{12}\right)$

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 $

$2$	1.36603	−	0.366025i	0.965926	−	0.258819i
$2$	1.36603	−	0.366025i	0.965926	−	0.258819i
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	1.73205	−	1.00000i	0.866025	−	0.500000i
$5$	−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		−0.965926	−	0.258819i	$-0.916667\pi$
$7$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$	−1.09808	−	0.633975i	−0.347242	−	0.200480i
$11$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$11$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	6.86603	−	3.96410i	1.66526	−	0.961436i	0.695113	−	0.718900i	$-0.255354\pi$
$17$	6.86603	−	3.96410i	1.66526	−	0.961436i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	−3.00000	−	3.00000i	−0.707107	−	0.707107i
$19$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$19$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$20$	−1.73205	−	0.464102i	−0.387298	−	0.103776i
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		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	0.371391	+	0.928477i	$0.378881\pi$
$29$	−3.33013	+	5.76795i	−0.618389	+	1.07108i	−0.989780	+	0.142605i	$0.954452\pi$
$30$		0			0
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$	1.46410	−	5.46410i	0.258819	−	0.965926i
$33$		0			0
$34$	7.92820	−	7.92820i	1.35968	−	1.35968i
$35$		0			0
$36$	−5.19615	−	3.00000i	−0.866025	−	0.500000i
$37$	3.13397	+	11.6962i	0.515222	+	1.92284i	0.350823	+	0.936442i	$0.385902\pi$
$37$	3.13397	+	11.6962i	0.515222	+	1.92284i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$	−2.53590			−0.400961
$41$	−9.96410	+	2.66987i	−1.55613	+	0.416964i	−0.931436	−	0.363905i	$-0.881443\pi$
$41$	−9.96410	+	2.66987i	−1.55613	+	0.416964i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$43$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$44$		0			0
$45$	−0.696152	+	2.59808i	−0.103776	+	0.387298i
$46$		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
$50$	−1.53590	−	5.73205i	−0.217209	−	0.810634i
$51$		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
$58$	−2.43782	+	9.09808i	−0.320102	+	1.19464i
$59$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$59$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$60$		0			0
$61$	−2.69615	−	4.66987i	−0.345207	−	0.597916i	0.640184	−	0.768221i	$-0.278858\pi$
$61$	−2.69615	−	4.66987i	−0.345207	−	0.597916i	−0.985391	+	0.170305i	$0.945525\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$67$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$68$	7.92820	−	13.7321i	0.961436	−	1.66526i
$69$		0			0
$70$		0			0
$71$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$71$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$72$	−8.19615	−	2.19615i	−0.965926	−	0.258819i
$73$	−9.83013	+	9.83013i	−1.15053	+	1.15053i	−0.164083	+	0.986447i	$0.552466\pi$
$73$	−9.83013	+	9.83013i	−1.15053	+	1.15053i	−0.986447	+	0.164083i	$0.947534\pi$
$74$	8.56218	+	14.8301i	0.995333	+	1.72397i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−3.46410	+	0.928203i	−0.387298	+	0.103776i
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$	−12.6340	+	7.29423i	−1.39519	+	0.805513i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−6.86603	−	1.83975i	−0.744725	−	0.199548i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.09808	−	4.09808i	−0.116396	−	0.434395i	0.882992	−	0.469389i	$-0.155526\pi$
$89$	−1.09808	−	4.09808i	−0.116396	−	0.434395i	−0.999388	+	0.0349934i	$0.988859\pi$
$90$			3.80385i			0.400961i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.83013	+	6.83013i	−0.185821	+	0.693494i	0.808632	+	0.588315i	$0.200208\pi$
$97$	−1.83013	+	6.83013i	−0.185821	+	0.693494i	−0.994453	+	0.105180i	$0.966458\pi$
$98$	9.56218	+	2.56218i	0.965926	+	0.258819i
$99$		0			0
$100$	−4.19615	−	7.26795i	−0.419615	−	0.726795i
$101$	10.1603	+	5.86603i	1.01098	+	0.583691i	0.911479	−	0.411346i	$-0.134941\pi$
$101$	10.1603	+	5.86603i	1.01098	+	0.583691i	0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	14.2942	−	3.83013i	1.38838	−	0.372015i
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		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	0.946570	−	0.322498i	$-0.104523\pi$
$113$	2.06218	+	3.57180i	0.193993	+	0.336006i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$		0			0
$116$			13.3205i			1.23678i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.52628	+	5.50000i	−0.866025	+	0.500000i
$122$	−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		−0.866025	−	0.500000i	$-0.833333\pi$
$127$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$128$	−2.92820	−	10.9282i	−0.258819	−	0.965926i
$129$		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
$136$	5.80385	−	21.6603i	0.497676	−	1.85735i
$137$	22.5263	+	6.03590i	1.92455	+	0.515682i	0.984757	+	0.173939i	$0.0556494\pi$
$137$	22.5263	+	6.03590i	1.92455	+	0.515682i	0.939793	+	0.341743i	$0.111017\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$	5.76795	−	1.54552i	0.479002	−	0.128348i
$146$	−9.83013	+	17.0263i	−0.813547	+	1.40910i
$147$		0			0
$148$	17.1244	+	17.1244i	1.40761	+	1.40761i
$149$	4.06218	−	15.1603i	0.332787	−	1.24198i	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	4.06218	−	15.1603i	0.332787	−	1.24198i	0.906249	−	0.422744i	$-0.138933\pi$
$150$		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	$-0.992280\pi$
$157$	−25.0526			−1.99941			−0.999706	+	0.0242497i	$0.992280\pi$
$158$		0			0
$159$		0			0
$160$	−4.39230	+	2.53590i	−0.347242	+	0.200480i
$161$		0			0
$162$	−3.29423	+	12.2942i	−0.258819	+	0.965926i
$163$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$163$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$164$	−14.5885	+	14.5885i	−1.13917	+	1.13917i
$165$		0			0
$166$		0			0
$167$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$167$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$168$		0			0
$169$		0			0
$170$	−10.0526			−0.770996
$171$		0			0
$172$		0			0
$173$	3.46410	−	2.00000i	0.263371	−	0.152057i	−0.362500	−	0.931984i	$-0.618077\pi$
$173$	3.46410	−	2.00000i	0.263371	−	0.152057i	0.625871	+	0.779926i	$0.284744\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−3.00000	−	5.19615i	−0.224860	−	0.389468i
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$	1.39230	+	5.19615i	0.103776	+	0.387298i
$181$			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
$185$	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	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−5.10770	−	19.0622i	−0.367660	−	1.37213i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−5.10770	−	19.0622i	−0.367660	−	1.37213i	0.496119	−	0.868255i	$-0.334758\pi$
$194$			10.0000i			0.717958i
$195$		0			0
$196$	14.0000			1.00000
$197$	20.4904	−	5.49038i	1.45988	−	0.391173i	0.560431	−	0.828201i	$-0.310635\pi$
$197$	20.4904	−	5.49038i	1.45988	−	0.391173i	0.899448	+	0.437028i	$0.143969\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−8.39230	−	8.39230i	−0.593426	−	0.593426i
$201$		0			0
$202$	16.0263	+	4.29423i	1.12761	+	0.302141i
$203$		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	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$	18.1244	−	10.4641i	1.24479	−	0.718677i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	12.1244	+	7.00000i	0.821165	+	0.474100i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$223$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$224$		0			0
$225$	−10.9019	+	6.29423i	−0.726795	+	0.419615i
$226$	4.12436	+	4.12436i	0.274348	+	0.274348i
$227$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$227$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$228$		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
$232$	4.87564	+	18.1962i	0.320102	+	1.19464i
$233$		−	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
$241$	−25.9904	−	6.96410i	−1.67419	−	0.448597i	−0.707953	−	0.706260i	$-0.750381\pi$
$241$	−25.9904	−	6.96410i	−1.67419	−	0.448597i	−0.966235	+	0.257663i	$0.917048\pi$
$242$	−11.0000	+	11.0000i	−0.707107	+	0.707107i
$243$		0			0
$244$	−9.33975	−	5.39230i	−0.597916	−	0.345207i
$245$	−1.62436	−	6.06218i	−0.103776	−	0.387298i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−5.83013	+	10.0981i	−0.368730	+	0.638658i
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−12.3564	−	7.13397i	−0.770771	−	0.445005i	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	−12.3564	−	7.13397i	−0.770771	−	0.445005i	−0.833150	+	0.553047i	$0.813465\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	19.9808			1.23678
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		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	0.991288	+	0.131713i	$0.0420477\pi$
$269$	10.0000	+	17.3205i	0.609711	+	1.05605i	−0.381577	+	0.924337i	$0.624619\pi$
$270$		0			0
$271$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$271$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$272$		−	31.7128i		−	1.92287i
$273$		0			0
$274$	32.9808			1.99244
$275$		0			0
$276$		0			0
$277$	−1.37564	+	0.794229i	−0.0826545	+	0.0477206i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−1.37564	+	0.794229i	−0.0826545	+	0.0477206i	0.458103	+	0.888899i	$0.348529\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.6865	−	12.6865i	0.756815	−	0.756815i	−0.218926	−	0.975741i	$-0.570255\pi$
$281$	12.6865	−	12.6865i	0.756815	−	0.756815i	0.975741	+	0.218926i	$0.0702554\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−16.3923	+	4.39230i	−0.965926	+	0.258819i
$289$	22.9282	−	39.7128i	1.34872	−	2.33605i
$290$	7.31347	−	4.22243i	0.429462	−	0.247950i
$291$		0			0
$292$	−7.19615	+	26.8564i	−0.421123	+	1.57165i
$293$	4.76795	+	1.27757i	0.278547	+	0.0746363i	0.395388	−	0.918514i	$-0.370610\pi$
$293$	4.76795	+	1.27757i	0.278547	+	0.0746363i	−0.116841	+	0.993151i	$0.537277\pi$
$294$		0			0
$295$		0			0
$296$	29.6603	+	17.1244i	1.72397	+	0.995333i
$297$		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
$305$	−1.25129	+	4.66987i	−0.0716486	+	0.267396i
$306$	−32.4904	−	8.70577i	−1.85735	−	0.497676i
$307$		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	$0.737274\pi$
$314$	−34.2224	+	9.16987i	−1.93128	+	0.517486i
$315$		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
$320$	−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
$328$	−14.5885	+	25.2679i	−0.805513	+	1.39519i
$329$		0			0
$330$		0			0
$331$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$331$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$332$		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
$340$	−13.7321	+	3.67949i	−0.744725	+	0.199548i
$341$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	8.41858	+	31.4186i	0.450636	+	1.68180i	0.700609	+	0.713545i	$0.252912\pi$
$349$	8.41858	+	31.4186i	0.450636	+	1.68180i	−0.249973	+	0.968253i	$0.580422\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−27.7224	+	7.42820i	−1.47552	+	0.395363i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−27.7224	+	7.42820i	−1.47552	+	0.395363i	−0.570697	+	0.821160i	$0.693327\pi$
$354$		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	$0.750000\pi$
$360$	3.80385	+	6.58846i	0.200480	+	0.347242i
$361$	−16.4545	−	9.50000i	−0.866025	−	0.500000i
$362$	9.63397	+	35.9545i	0.506350	+	1.88973i
$363$		0			0
$364$		0			0
$365$	12.4641			0.652401
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$	21.8827	+	21.8827i	1.13917	+	1.13917i
$370$	3.97372	−	14.8301i	0.206584	−	0.770982i
$371$		0			0
$372$		0			0
$373$	15.0622	+	26.0885i	0.779890	+	1.35081i	0.932005	+	0.362446i	$0.118058\pi$
$373$	15.0622	+	26.0885i	0.779890	+	1.35081i	−0.152115	+	0.988363i	$0.548608\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$379$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$383$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$384$		0			0
$385$		0			0
$386$	−13.9545	−	24.1699i	−0.710264	−	1.23021i
$387$		0			0
$388$	3.66025	+	13.6603i	0.185821	+	0.693494i
$389$			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
$392$	19.1244	−	5.12436i	0.965926	−	0.258819i
$393$		0			0
$394$	25.9808	−	15.0000i	1.30889	−	0.755689i
$395$		0			0
$396$		0			0
$397$	−34.1506	−	9.15064i	−1.71397	−	0.459257i	−0.737579	−	0.675261i	$-0.764031\pi$
$397$	−34.1506	−	9.15064i	−1.71397	−	0.459257i	−0.976392	+	0.216004i	$0.930698\pi$
$398$		0			0
$399$		0			0
$400$	−14.5359	−	8.39230i	−0.726795	−	0.419615i
$401$	−9.86603	−	36.8205i	−0.492686	−	1.83873i	−0.542623	−	0.839976i	$-0.682569\pi$
$401$	−9.86603	−	36.8205i	−0.492686	−	1.83873i	0.0499376	−	0.998752i	$-0.484098\pi$
$402$		0			0
$403$		0			0
$404$	23.4641			1.16738
$405$	7.79423	−	2.08846i	0.387298	−	0.103776i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−4.17949	+	15.5981i	−0.206663	+	0.771275i	0.782274	+	0.622935i	$0.214060\pi$
$409$	−4.17949	+	15.5981i	−0.206663	+	0.771275i	−0.988936	+	0.148340i	$0.952607\pi$
$410$	12.6340	+	3.38526i	0.623948	+	0.167186i
$411$		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	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		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
$424$	20.9282	−	20.9282i	1.01636	−	1.01636i
$425$	−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		0.965926	−	0.258819i	$-0.0833333\pi$
$431$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$432$		0			0
$433$	15.1077	−	8.72243i	0.726029	−	0.419173i	−0.0909384	−	0.995857i	$-0.528987\pi$
$433$	15.1077	−	8.72243i	0.726029	−	0.419173i	0.816968	+	0.576683i	$0.195653\pi$
$434$		0			0
$435$		0			0
$436$	19.1244	+	5.12436i	0.915891	+	0.245412i
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		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
$449$	36.8827	+	9.88269i	1.74060	+	0.466393i	0.982581	−	0.185837i	$-0.0594997\pi$
$449$	36.8827	+	9.88269i	1.74060	+	0.466393i	0.758021	+	0.652230i	$0.226166\pi$
$450$	−12.5885	+	12.5885i	−0.593426	+	0.593426i
$451$		0			0
$452$	7.14359	+	4.12436i	0.336006	+	0.193993i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.03590	−	0.813467i	0.142013	−	0.0380524i	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	3.03590	−	0.813467i	0.142013	−	0.0380524i	0.329125	+	0.944286i	$0.393246\pi$
$458$	17.0000	−	29.4449i	0.794358	−	1.37587i
$459$		0			0
$460$		0			0
$461$	7.54552	−	28.1603i	0.351430	−	1.31155i	−0.533488	−	0.845807i	$-0.679119\pi$
$461$	7.54552	−	28.1603i	0.351430	−	1.31155i	0.884918	−	0.465746i	$-0.154214\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	13.3205	+	23.0718i	0.618389	+	1.07108i
$465$		0			0
$466$	−5.85641	−	21.8564i	−0.271293	−	1.01248i
$467$		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		−0.258819	−	0.965926i	$-0.583333\pi$
$479$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$480$		0			0
$481$		0			0
$482$	−38.0526			−1.73325
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$	5.49038	−	3.16987i	0.249305	−	0.143937i
$486$		0			0
$487$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$487$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$488$	−14.7321	−	3.94744i	−0.666889	−	0.178692i
$489$		0			0
$490$	−4.43782	−	7.68653i	−0.200480	−	0.347242i
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		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	$0.250000\pi$
$500$	−4.26795	+	15.9282i	−0.190868	+	0.712331i
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$	−2.72243	−	10.1603i	−0.121147	−	0.452125i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	38.5526	−	10.3301i	1.70881	−	0.457875i	0.733679	−	0.679496i	$-0.237801\pi$
$509$	38.5526	−	10.3301i	1.70881	−	0.457875i	0.975133	+	0.221621i	$0.0711348\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$	−19.4904	−	5.22243i	−0.859684	−	0.230352i
$515$		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	$0.493393\pi$
$522$	27.2942	−	7.31347i	1.19464	−	0.320102i
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$	−11.4904	−	6.63397i	−0.499110	−	0.288161i
$531$		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	$0.691870\pi$
$541$	−32.3468	+	32.3468i	−1.39070	+	1.39070i	−0.823764	+	0.566933i	$0.808130\pi$
$542$		0			0
$543$		0			0
$544$	−11.6077	−	43.3205i	−0.497676	−	1.85735i
$545$		−	8.87564i		−	0.380191i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	45.0526	−	12.0718i	1.92455	−	0.515682i
$549$	−8.08846	+	14.0096i	−0.345207	+	0.597916i
$550$		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
$557$	11.3756	+	42.4545i	0.482002	+	1.79885i	0.593199	+	0.805056i	$0.297865\pi$
$557$	11.3756	+	42.4545i	0.482002	+	1.79885i	−0.111198	+	0.993798i	$0.535469\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	12.6865	−	21.9737i	0.535149	−	0.926905i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	0.957060	−	3.57180i	0.0402638	−	0.150267i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	34.6410	+	20.0000i	1.45223	+	0.838444i	0.998608	−	0.0527519i	$-0.0167993\pi$
$569$	34.6410	+	20.0000i	1.45223	+	0.838444i	0.453619	+	0.891196i	$0.350133\pi$
$570$		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
$576$	−20.7846	+	12.0000i	−0.866025	+	0.500000i
$577$	−10.1506	−	10.1506i	−0.422576	−	0.422576i	0.463513	−	0.886090i	$-0.346589\pi$
$577$	−10.1506	−	10.1506i	−0.422576	−	0.422576i	−0.886090	+	0.463513i	$0.846589\pi$
$578$	16.7846	−	62.6410i	0.698148	−	2.60552i
$579$		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		0.965926	−	0.258819i	$-0.0833333\pi$
$587$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	46.7846	+	12.5359i	1.92284	+	0.515222i
$593$	−19.3468	+	19.3468i	−0.794477	+	0.794477i	−0.982219	−	0.187741i	$-0.939883\pi$
$593$	−19.3468	+	19.3468i	−0.794477	+	0.794477i	0.187741	+	0.982219i	$0.439883\pi$
$594$		0			0
$595$		0			0
$596$	−8.12436	−	30.3205i	−0.332787	−	1.24198i
$597$		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	0.312861	+	0.949799i	$0.398713\pi$
$601$	−16.3301	+	28.2846i	−0.666120	+	1.15375i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.52628	+	2.55256i	0.387298	+	0.103776i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$			6.83717i			0.276829i
$611$		0			0
$612$	−47.5692			−1.92287
$613$	−40.7224	+	10.9115i	−1.64476	+	0.440713i	−0.958140	−	0.286300i	$-0.907575\pi$
$613$	−40.7224	+	10.9115i	−1.64476	+	0.440713i	−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.45448	+	20.3564i	−0.219589	+	0.819518i	0.764911	+	0.644136i	$0.222783\pi$
$617$	−5.45448	+	20.3564i	−0.219589	+	0.819518i	−0.984500	+	0.175382i	$0.943884\pi$
$618$		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
$626$	−32.7846	+	8.78461i	−1.31034	+	0.351104i
$627$		0			0
$628$	−43.3923	+	25.0526i	−1.73154	+	0.999706i
$629$	67.8827	+	67.8827i	2.70666	+	2.70666i
$630$		0			0
$631$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$631$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$632$		0			0
$633$		0			0
$634$	−40.0981	−	23.1506i	−1.59250	−	0.919429i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−5.07180	+	8.78461i	−0.200480	+	0.347242i
$641$	−15.6506	+	9.03590i	−0.618163	+	0.356897i	−0.776153	−	0.630544i	$-0.782832\pi$
$641$	−15.6506	+	9.03590i	−0.618163	+	0.356897i	0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$643$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	6.58846	+	24.5885i	0.258819	+	0.965926i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.0000	+	38.1051i	−0.860927	+	1.49117i	0.0101092	+	0.999949i	$0.496782\pi$
$653$	−22.0000	+	38.1051i	−0.860927	+	1.49117i	−0.871036	+	0.491220i	$0.836551\pi$
$654$		0			0
$655$		0			0
$656$	−10.6795	+	39.8564i	−0.416964	+	1.55613i
$657$	40.2846	+	10.7942i	1.57165	+	0.421123i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	−0.349365	−	1.30385i	−0.0135887	−	0.0507138i	0.958799	−	0.284087i	$-0.0916904\pi$
$661$	−0.349365	−	1.30385i	−0.0135887	−	0.0507138i	−0.972387	+	0.233373i	$0.925024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	25.6865	−	44.4904i	0.995333	−	1.72397i
$667$		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	$-0.319738\pi$
$673$	1.91858	+	1.10770i	0.0739560	+	0.0426985i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	−6.84936	−	25.5622i	−0.263828	−	0.984618i
$675$		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
$680$	−17.4115	+	10.0526i	−0.667702	+	0.385498i
$681$		0			0
$682$		0			0
$683$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$683$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$684$		0			0
$685$	−10.4545	−	18.1077i	−0.399445	−	0.691859i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$691$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$692$	4.00000	−	6.92820i	0.152057	−	0.263371i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−57.8301	+	57.8301i	−2.19047	+	2.19047i
$698$	23.0000	+	39.8372i	0.870563	+	1.50786i
$699$		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
$706$	−35.1506	+	20.2942i	−1.32291	+	0.763783i
$707$		0			0
$708$		0			0
$709$	−38.9904	−	10.4474i	−1.46431	−	0.392362i	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−38.9904	−	10.4474i	−1.46431	−	0.392362i	−0.900978	+	0.433865i	$0.857149\pi$
$710$		0			0
$711$		0			0
$712$	−10.3923	−	6.00000i	−0.389468	−	0.224860i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$	7.60770	+	7.60770i	0.283522	+	0.283522i
$721$		0			0
$722$	−25.9545	−	6.95448i	−0.965926	−	0.258819i
$723$		0			0
$724$	26.3205	+	45.5885i	0.978194	+	1.69428i
$725$	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
$730$	17.0263	−	4.56218i	0.630171	−	0.168854i
$731$		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
$738$	37.9019	+	21.8827i	1.39519	+	0.805513i
$739$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$739$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$740$		−	21.7128i		−	0.798179i
$741$		0			0
$742$		0			0
$743$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$743$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$744$		0			0
$745$	−12.1865	+	7.03590i	−0.446480	+	0.257775i
$746$	30.1244	+	30.1244i	1.10293	+	1.10293i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		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	$-0.939408\pi$
$757$	−9.00000	+	15.5885i	−0.327111	+	0.566572i	0.654827	+	0.755779i	$0.272742\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.36603	+	0.366025i	0.0495184	+	0.0132684i	0.283493	−	0.958974i	$-0.408507\pi$
$761$	1.36603	+	0.366025i	0.0495184	+	0.0132684i	−0.233975	+	0.972243i	$0.575173\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	5.51924	+	20.5981i	0.199548	+	0.744725i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−50.5429	+	13.5429i	−1.82263	+	0.488371i	−0.997107	−	0.0760054i	$-0.975783\pi$
$769$	−50.5429	+	13.5429i	−1.82263	+	0.488371i	−0.825518	+	0.564376i	$0.809117\pi$
$770$		0			0
$771$		0			0
$772$	−27.9090	−	27.9090i	−1.00447	−	1.00447i
$773$	−1.83013	+	6.83013i	−0.0658251	+	0.245663i	−0.990997	−	0.133887i	$-0.957254\pi$
$773$	−1.83013	+	6.83013i	−0.0658251	+	0.245663i	0.925172	+	0.379549i	$0.123921\pi$
$774$		0			0
$775$		0			0
$776$	10.0000	+	17.3205i	0.358979	+	0.621770i
$777$		0			0
$778$	0.117314	+	0.437822i	0.00420591	+	0.0156967i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	24.2487	−	14.0000i	0.866025	−	0.500000i
$785$	15.8827	+	15.8827i	0.566877	+	0.566877i
$786$		0			0
$787$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$787$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$788$	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
$797$	−19.0526	+	11.0000i	−0.674876	+	0.389640i	−0.797922	−	0.602761i	$-0.794067\pi$
$797$	−19.0526	+	11.0000i	−0.674876	+	0.389640i	0.123045	+	0.992401i	$0.460734\pi$
$798$		0			0
$799$		0			0
$800$	−22.9282	−	6.14359i	−0.810634	−	0.217209i
$801$	−9.00000	+	9.00000i	−0.317999	+	0.317999i
$802$	−26.9545	−	46.6865i	−0.951796	−	1.64856i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	32.0526	−	8.58846i	1.12761	−	0.302141i
$809$	9.66987	−	16.7487i	0.339975	−	0.588853i	−0.644453	−	0.764644i	$-0.722915\pi$
$809$	9.66987	−	16.7487i	0.339975	−	0.588853i	0.984428	+	0.175791i	$0.0562482\pi$
$810$	9.88269	−	5.70577i	0.347242	−	0.200480i
$811$		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
$821$	−15.0263	+	4.02628i	−0.524421	+	0.140518i	−0.511311	−	0.859396i	$-0.670840\pi$
$821$	−15.0263	+	4.02628i	−0.524421	+	0.140518i	−0.0131101	+	0.999914i	$0.504173\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		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	0.937749	+	0.347314i	$0.112906\pi$
$829$	49.1603	+	28.3827i	1.70741	+	0.985771i	0.769657	+	0.638457i	$0.220427\pi$
$830$		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		−0.965926	−	0.258819i	$-0.916667\pi$
$839$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$840$		0			0
$841$	−7.67949	−	13.3013i	−0.264810	−	0.458664i
$842$	−23.6147	−	13.6340i	−0.813818	−	0.469858i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	20.9282	−	36.2487i	0.718677	−	1.24479i
$849$		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	−0.373845	−	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
$865$	−3.46410	−	0.928203i	−0.117783	−	0.0315599i
$866$	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
$873$	20.4904	−	5.49038i	0.693494	−	0.185821i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−13.6962	+	51.1147i	−0.462486	+	1.72602i	0.202606	+	0.979260i	$0.435059\pi$
$877$	−13.6962	+	51.1147i	−0.462486	+	1.72602i	−0.665092	+	0.746762i	$0.731608\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−51.3564	−	29.6506i	−1.73024	−	0.998955i	−0.887970	−	0.459902i	$-0.847885\pi$
$881$	−51.3564	−	29.6506i	−1.73024	−	0.998955i	−0.842271	−	0.539054i	$-0.818782\pi$
$882$	−7.68653	−	28.6865i	−0.258819	−	0.965926i
$883$		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	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$	−1.39230	+	5.19615i	−0.0466702	+	0.174175i
$891$		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
$900$	−12.5885	+	21.8038i	−0.419615	+	0.726795i
$901$	71.8468	−	41.4808i	2.39356	−	1.38192i
$902$		0			0
$903$		0			0
$904$	11.2679	+	3.01924i	0.374766	+	0.100418i
$905$	16.6865	−	16.6865i	0.554679	−	0.554679i
$906$		0			0
$907$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$907$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$908$		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
$914$	3.84936	−	2.22243i	0.127326	−	0.0735115i
$915$		0			0
$916$	12.4449	−	46.4449i	0.411190	−	1.53458i
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$		−	41.2295i		−	1.35782i
$923$		0			0
$924$		0			0
$925$	49.0788	−	13.1506i	1.61370	−	0.432390i
$926$		0			0
$927$		0			0
$928$	26.6410	+	26.6410i	0.874534	+	0.874534i
$929$	8.82051	−	32.9186i	0.289391	−	1.08002i	−0.656179	−	0.754606i	$-0.727828\pi$
$929$	8.82051	−	32.9186i	0.289391	−	1.08002i	0.945570	−	0.325418i	$-0.105505\pi$
$930$		0			0
$931$		0			0
$932$	−16.0000	−	27.7128i	−0.524097	−	0.907763i
$933$		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		−0.258819	−	0.965926i	$-0.583333\pi$
$947$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	48.4974	−	28.0000i	1.57099	−	0.907009i	0.574937	−	0.818198i	$-0.305026\pi$
$953$	48.4974	−	28.0000i	1.57099	−	0.907009i	0.996048	−	0.0888114i	$-0.0283068\pi$
$954$	−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
$964$	−51.9808	+	13.9282i	−1.67419	+	0.448597i
$965$	−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	$0.250000\pi$
$968$	−8.05256	+	30.0526i	−0.258819	+	0.965926i
$969$		0			0
$970$	6.33975	−	6.33975i	0.203557	−	0.203557i
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−21.5692			−0.690414
$977$	−22.9641	+	6.15321i	−0.734687	+	0.196859i	−0.606715	−	0.794919i	$-0.707513\pi$
$977$	−22.9641	+	6.15321i	−0.734687	+	0.196859i	−0.127971	+	0.991778i	$0.540847\pi$
$978$		0			0
$979$		0			0
$980$	−8.87564	−	8.87564i	−0.283522	−	0.283522i
$981$	7.68653	−	28.6865i	0.245412	−	0.915891i
$982$		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
$986$	19.3275	+	72.1314i	0.615515	+	2.29713i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		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	$-0.0608559\pi$
$997$	10.3038	+	17.8468i	0.326326	+	0.565213i	−0.655454	+	0.755235i	$0.727523\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.l.a.11.1	✓	4	13.10	even	6
52.2.l.a.11.1	✓	4	52.23	odd	6
52.2.l.a.19.1	yes	4	13.8	odd	4
52.2.l.a.19.1	yes	4	52.47	even	4
468.2.cb.d.19.1		4	39.8	even	4
468.2.cb.d.19.1		4	156.47	odd	4
468.2.cb.d.271.1		4	39.23	odd	6
468.2.cb.d.271.1		4	156.23	even	6
676.2.f.d.99.2		4	13.9	even	3
676.2.f.d.99.2		4	52.35	odd	6
676.2.f.d.239.2		4	13.6	odd	12
676.2.f.d.239.2		4	52.19	even	12
676.2.f.e.99.1		4	13.4	even	6
676.2.f.e.99.1		4	52.43	odd	6
676.2.f.e.239.1		4	13.7	odd	12
676.2.f.e.239.1		4	52.7	even	12
676.2.l.c.319.1		4	13.12	even	2
676.2.l.c.319.1		4	52.51	odd	2
676.2.l.c.587.1		4	13.11	odd	12
676.2.l.c.587.1		4	52.11	even	12
676.2.l.d.19.1		4	13.5	odd	4
676.2.l.d.19.1		4	52.31	even	4
676.2.l.d.427.1		4	13.3	even	3
676.2.l.d.427.1		4	52.3	odd	6
676.2.l.e.319.1		4	1.1	even	1	trivial
676.2.l.e.319.1		4	4.3	odd	2	CM
676.2.l.e.587.1		4	13.2	odd	12	inner
676.2.l.e.587.1		4	52.15	even	12	inner
832.2.bu.d.63.1		4	104.75	odd	6
832.2.bu.d.63.1		4	104.101	even	6
832.2.bu.d.383.1		4	104.21	odd	4
832.2.bu.d.383.1		4	104.99	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	676.l (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(-0.633975 - 0.633975i) q^{5} +(2.00000 - 2.00000i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(-1.09808 - 0.633975i) q^{10} +(2.00000 - 3.46410i) q^{16} +(6.86603 - 3.96410i) q^{17} +(-3.00000 - 3.00000i) q^{18} +(-1.73205 - 0.464102i) q^{20} -4.19615i q^{25} +(-3.33013 + 5.76795i) q^{29} +(1.46410 - 5.46410i) q^{32} +(7.92820 - 7.92820i) q^{34} +(-5.19615 - 3.00000i) q^{36} +(3.13397 + 11.6962i) q^{37} -2.53590 q^{40} +(-9.96410 + 2.66987i) q^{41} +(-0.696152 + 2.59808i) q^{45} +(6.06218 + 3.50000i) q^{49} +(-1.53590 - 5.73205i) q^{50} +10.4641 q^{53} +(-2.43782 + 9.09808i) q^{58} +(-2.69615 - 4.66987i) q^{61} -8.00000i q^{64} +(7.92820 - 13.7321i) q^{68} +(-8.19615 - 2.19615i) q^{72} +(-9.83013 + 9.83013i) q^{73} +(8.56218 + 14.8301i) q^{74} +(-3.46410 + 0.928203i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(-12.6340 + 7.29423i) q^{82} +(-6.86603 - 1.83975i) q^{85} +(-1.09808 - 4.09808i) q^{89} +3.80385i q^{90} +(-1.83013 + 6.83013i) q^{97} +(9.56218 + 2.56218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 6 q^{5} + 8 q^{8} - 6 q^{9} + 6 q^{10} + 8 q^{16} + 24 q^{17} - 12 q^{18} + 4 q^{29} - 8 q^{32} + 4 q^{34} + 16 q^{37} - 24 q^{40} - 26 q^{41} + 18 q^{45} - 20 q^{50} + 28 q^{53} - 34 q^{58}+ \cdots + 14 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 6 * q^5 + 8 * q^8 - 6 * q^9 + 6 * q^10 + 8 * q^16 + 24 * q^17 - 12 * q^18 + 4 * q^29 - 8 * q^32 + 4 * q^34 + 16 * q^37 - 24 * q^40 - 26 * q^41 + 18 * q^45 - 20 * q^50 + 28 * q^53 - 34 * q^58 + 10 * q^61 + 4 * q^68 - 12 * q^72 - 22 * q^73 + 10 * q^74 - 18 * q^81 - 54 * q^82 - 24 * q^85 + 6 * q^89 + 10 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more