LMFDB - Embedded newform 735.2.p.b.509.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(374,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("735.374");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$735 = 3 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	735.p (of order $6$ , degree $2$ , 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:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.31116960000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81$ x^8 + x^6 - 8x^4 + 9x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			509.4
Root			$-0.306808 + 1.70466i$ of defining polynomial
Character	$\chi$	$=$	735.509
Dual form			735.2.p.b.374.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.62968 + 0.586627i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-1.93649 - 1.11803i) q^{5} +(2.31174 + 1.91203i) q^{9} +(5.12348 - 2.95804i) q^{11} +(0.613616 + 3.40932i) q^{12} +2.64575 q^{13} +(-2.50000 - 2.95804i) q^{15} +(-2.00000 + 3.46410i) q^{16} +(-1.93649 + 1.11803i) q^{17} -4.47214i q^{20} +(2.50000 + 4.33013i) q^{25} +(2.64575 + 4.47214i) q^{27} +5.91608i q^{29} +(10.0849 - 1.81510i) q^{33} +(-1.00000 + 5.91608i) q^{36} +(4.31174 + 1.55207i) q^{39} +(10.2470 + 5.91608i) q^{44} +(-2.33894 - 6.28724i) q^{45} +(-9.68246 - 5.59017i) q^{47} +(-5.29150 + 4.47214i) q^{48} +(-3.81174 + 0.686044i) q^{51} +(2.64575 + 4.58258i) q^{52} -13.2288 q^{55} +(2.62348 - 7.28817i) q^{60} -8.00000 q^{64} +(-5.12348 - 2.95804i) q^{65} +(-3.87298 - 2.23607i) q^{68} -11.8322i q^{71} +(-5.29150 - 9.16515i) q^{73} +(1.53404 + 8.52330i) q^{75} +(0.500000 - 0.866025i) q^{79} +(7.74597 - 4.47214i) q^{80} +(1.68826 + 8.84024i) q^{81} -8.94427i q^{83} +5.00000 q^{85} +(-3.47053 + 9.64134i) q^{87} +18.5203 q^{97} +(17.5000 + 2.95804i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 8 q^{4} - 2 q^{9} - 20 q^{15} - 16 q^{16} + 20 q^{25} - 8 q^{36} + 14 q^{39} - 10 q^{51} - 20 q^{60} - 64 q^{64} + 4 q^{79} + 34 q^{81} + 40 q^{85} + 140 q^{99}+O(q^{100})$ 8 * q + 8 * q^4 - 2 * q^9 - 20 * q^15 - 16 * q^16 + 20 * q^25 - 8 * q^36 + 14 * q^39 - 10 * q^51 - 20 * q^60 - 64 * q^64 + 4 * q^79 + 34 * q^81 + 40 * q^85 + 140 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$ .

$n$	$346$	$442$	$491$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$-1$

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		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$	1.62968	+	0.586627i	0.940898	+	0.338689i
$3$	1.62968	+	0.586627i	0.940898	+	0.338689i
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$	−1.93649	−	1.11803i	−0.866025	−	0.500000i
$5$	−1.93649	−	1.11803i	−0.866025	−	0.500000i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.31174	+	1.91203i	0.770579	+	0.637344i
$10$		0			0
$11$	5.12348	−	2.95804i	1.54479	−	0.891883i	0.546259	−	0.837616i	$-0.316051\pi$
$11$	5.12348	−	2.95804i	1.54479	−	0.891883i	0.998526	−	0.0542666i	$-0.0172821\pi$
$12$	0.613616	+	3.40932i	0.177136	+	0.984186i
$13$	2.64575			0.733799			0.366900	−	0.930261i	$-0.380419\pi$
$13$	2.64575			0.733799			0.366900	+	0.930261i	$0.380419\pi$
$14$		0			0
$15$	−2.50000	−	2.95804i	−0.645497	−	0.763763i
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	−1.93649	+	1.11803i	−0.469668	+	0.271163i	−0.716101	−	0.697997i	$-0.754075\pi$
$17$	−1.93649	+	1.11803i	−0.469668	+	0.271163i	0.246433	+	0.969160i	$0.420742\pi$
$18$		0			0
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		−	4.47214i		−	1.00000i
$21$		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
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$	2.64575	+	4.47214i	0.509175	+	0.860663i
$28$		0			0
$29$			5.91608i			1.09859i	0.835629	+	0.549294i	$0.185103\pi$
$29$			5.91608i			1.09859i	−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$	10.0849	−	1.81510i	1.75556	−	0.315968i
$34$		0			0
$35$		0			0
$36$	−1.00000	+	5.91608i	−0.166667	+	0.986013i
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	4.31174	+	1.55207i	0.690431	+	0.248530i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	10.2470	+	5.91608i	1.54479	+	0.891883i
$45$	−2.33894	−	6.28724i	−0.348669	−	0.937246i
$46$		0			0
$47$	−9.68246	−	5.59017i	−1.41233	−	0.815410i	−0.416724	−	0.909033i	$-0.636822\pi$
$47$	−9.68246	−	5.59017i	−1.41233	−	0.815410i	−0.995608	+	0.0936230i	$0.970155\pi$
$48$	−5.29150	+	4.47214i	−0.763763	+	0.645497i
$49$		0			0
$50$		0			0
$51$	−3.81174	+	0.686044i	−0.533750	+	0.0960653i
$52$	2.64575	+	4.58258i	0.366900	+	0.635489i
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$	−13.2288			−1.78377
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$	2.62348	−	7.28817i	0.338689	−	0.940898i
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	−5.12348	−	2.95804i	−0.635489	−	0.366900i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−3.87298	−	2.23607i	−0.469668	−	0.271163i
$69$		0			0
$70$		0			0
$71$		−	11.8322i		−	1.40422i	−0.712069	−	0.702109i	$-0.752242\pi$
$71$		−	11.8322i		−	1.40422i	0.712069	−	0.702109i	$-0.247758\pi$
$72$		0			0
$73$	−5.29150	−	9.16515i	−0.619324	−	1.07270i	−0.989609	−	0.143782i	$-0.954074\pi$
$73$	−5.29150	−	9.16515i	−0.619324	−	1.07270i	0.370286	−	0.928918i	$-0.379260\pi$
$74$		0			0
$75$	1.53404	+	8.52330i	0.177136	+	0.984186i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	−0.836527	−	0.547926i	$-0.815418\pi$
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	0.892781	+	0.450490i	$0.148751\pi$
$80$	7.74597	−	4.47214i	0.866025	−	0.500000i
$81$	1.68826	+	8.84024i	0.187585	+	0.982248i
$82$		0			0
$83$		−	8.94427i		−	0.981761i	−0.871227	−	0.490881i	$-0.836675\pi$
$83$		−	8.94427i		−	0.981761i	0.871227	−	0.490881i	$-0.163325\pi$
$84$		0			0
$85$	5.00000			0.542326
$86$		0			0
$87$	−3.47053	+	9.64134i	−0.372080	+	1.03366i
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	18.5203			1.88045			0.940224	−	0.340557i	$-0.110616\pi$
$97$	18.5203			1.88045			0.940224	+	0.340557i	$0.110616\pi$
$98$		0			0
$99$	17.5000	+	2.95804i	1.75882	+	0.297294i
$100$	−5.00000	+	8.66025i	−0.500000	+	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	−1.32288	+	2.29129i	−0.130347	+	0.225767i	−0.923810	−	0.382851i	$-0.874942\pi$
$103$	−1.32288	+	2.29129i	−0.130347	+	0.225767i	0.793463	+	0.608618i	$0.208276\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$	−5.10022	+	9.05471i	−0.490768	+	0.871290i
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	−0.999512	−	0.0312328i	$-0.990057\pi$
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	0.472708	−	0.881219i	$-0.343277\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	−10.2470	+	5.91608i	−0.951405	+	0.549294i
$117$	6.11628	+	5.05876i	0.565451	+	0.467683i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	12.0000	−	20.7846i	1.09091	−	1.88951i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$	13.2288	+	15.6525i	1.15142	+	1.36237i
$133$		0			0
$134$		0			0
$135$	−0.123475	−	11.6183i	−0.0106271	−	0.999944i
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−12.5000	−	14.7902i	−1.05269	−	1.24556i
$142$		0			0
$143$	13.5554	−	7.82624i	1.13356	−	0.654463i
$144$	−11.2470	+	4.18403i	−0.937246	+	0.348669i
$145$	6.61438	−	11.4564i	0.549294	−	0.951405i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−20.4939	−	11.8322i	−1.67893	−	0.969328i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−20.4939	−	11.8322i	−1.67893	−	0.969328i	−0.716578	−	0.697507i	$-0.754293\pi$
$150$		0			0
$151$	−8.50000	−	14.7224i	−0.691720	−	1.19809i	−0.971274	−	0.237964i	$-0.923520\pi$
$151$	−8.50000	−	14.7224i	−0.691720	−	1.19809i	0.279554	−	0.960130i	$-0.409814\pi$
$152$		0			0
$153$	−6.61438	−	1.11803i	−0.534741	−	0.0903877i
$154$		0			0
$155$		0			0
$156$	1.62348	+	9.02022i	0.129982	+	0.722195i
$157$	10.5830	+	18.3303i	0.844616	+	1.46292i	0.885954	+	0.463772i	$0.153504\pi$
$157$	10.5830	+	18.3303i	0.844616	+	1.46292i	−0.0413387	+	0.999145i	$0.513162\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$	−21.5587	−	7.76034i	−1.67834	−	0.604142i
$166$		0			0
$167$			24.5967i			1.90335i	0.307102	+	0.951677i	$0.400641\pi$
$167$			24.5967i			1.90335i	−0.307102	+	0.951677i	$0.599359\pi$
$168$		0			0
$169$	−6.00000			−0.461538
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.68246	−	5.59017i	−0.736144	−	0.425013i	0.0845218	−	0.996422i	$-0.473064\pi$
$173$	−9.68246	−	5.59017i	−0.736144	−	0.425013i	−0.820666	+	0.571409i	$0.806397\pi$
$174$		0			0
$175$		0			0
$176$			23.6643i			1.78377i
$177$		0			0
$178$		0			0
$179$	−10.2470	+	5.91608i	−0.765893	+	0.442189i	−0.831408	−	0.555663i	$-0.812464\pi$
$179$	−10.2470	+	5.91608i	−0.765893	+	0.442189i	0.0655145	+	0.997852i	$0.479131\pi$
$180$	8.55087	−	10.3384i	0.637344	−	0.770579i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.61438	+	11.4564i	−0.483691	+	0.837778i
$188$		−	22.3607i		−	1.63082i
$189$		0			0
$190$		0			0
$191$	−5.12348	−	2.95804i	−0.370722	−	0.214036i	0.303052	−	0.952974i	$-0.401994\pi$
$191$	−5.12348	−	2.95804i	−0.370722	−	0.214036i	−0.673774	+	0.738938i	$0.735328\pi$
$192$	−13.0375	−	4.69302i	−0.940898	−	0.338689i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$	−6.61438	−	7.82624i	−0.473665	−	0.560449i
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	−5.00000	−	5.91608i	−0.350070	−	0.414208i
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−5.29150	+	9.16515i	−0.366900	+	0.635489i
$209$		0			0
$210$		0			0
$211$	23.0000			1.58339			0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000			1.58339			0.791693	+	0.610920i	$0.209200\pi$
$212$		0			0
$213$	6.94106	−	19.2827i	0.475594	−	1.32123i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−3.24695	−	18.0404i	−0.219409	−	1.21906i
$220$	−13.2288	−	22.9129i	−0.891883	−	1.54479i
$221$	−5.12348	+	2.95804i	−0.344642	+	0.198979i
$222$		0			0
$223$	−29.1033			−1.94890			−0.974449	−	0.224607i	$-0.927890\pi$
$223$	−29.1033			−1.94890			−0.974449	+	0.224607i	$0.927890\pi$
$224$		0			0
$225$	−2.50000	+	14.7902i	−0.166667	+	0.986013i
$226$		0			0
$227$	−25.1744	+	14.5344i	−1.67088	+	0.964685i	−0.703738	+	0.710460i	$0.748487\pi$
$227$	−25.1744	+	14.5344i	−1.67088	+	0.964685i	−0.967145	+	0.254225i	$0.918180\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$	12.5000	+	21.6506i	0.815410	+	1.41233i
$236$		0			0
$237$	1.32288	−	1.11803i	0.0859300	−	0.0726241i
$238$		0			0
$239$		−	29.5804i		−	1.91340i	−0.291081	−	0.956698i	$-0.594015\pi$
$239$		−	29.5804i		−	1.91340i	0.291081	−	0.956698i	$-0.405985\pi$
$240$	15.2470	−	2.74417i	0.984186	−	0.177136i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		0			0
$243$	−2.43459	+	15.3972i	−0.156179	+	0.987729i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	5.24695	−	14.5763i	0.332512	−	0.923738i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	8.14842	+	2.93313i	0.510274	+	0.183680i
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−3.87298	−	2.23607i	−0.241590	−	0.139482i	0.374317	−	0.927301i	$-0.377877\pi$
$257$	−3.87298	−	2.23607i	−0.241590	−	0.139482i	−0.615907	+	0.787819i	$0.711210\pi$
$258$		0			0
$259$		0			0
$260$		−	11.8322i		−	0.733799i
$261$	−11.3117	+	13.6764i	−0.700179	+	0.846549i
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		−	8.94427i		−	0.542326i
$273$		0			0
$274$		0			0
$275$	25.6174	+	14.7902i	1.54479	+	0.891883i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5.91608i			0.352924i	0.984307	+	0.176462i	$0.0564652\pi$
$281$			5.91608i			0.352924i	−0.984307	+	0.176462i	$0.943535\pi$
$282$		0			0
$283$	−1.32288	−	2.29129i	−0.0786368	−	0.136203i	0.824025	−	0.566553i	$-0.191723\pi$
$283$	−1.32288	−	2.29129i	−0.0786368	−	0.136203i	−0.902662	+	0.430350i	$0.858390\pi$
$284$	20.4939	−	11.8322i	1.21609	−	0.702109i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−6.00000	+	10.3923i	−0.352941	+	0.611312i
$290$		0			0
$291$	30.1822	+	10.8645i	1.76931	+	0.636887i
$292$	10.5830	−	18.3303i	0.619324	−	1.07270i
$293$			24.5967i			1.43696i	0.695549	+	0.718479i	$0.255161\pi$
$293$			24.5967i			1.43696i	−0.695549	+	0.718479i	$0.744839\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	26.7842	+	15.0866i	1.55418	+	0.875416i
$298$		0			0
$299$		0			0
$300$	−13.2288	+	11.1803i	−0.763763	+	0.645497i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	34.3948			1.96301			0.981507	−	0.191429i	$-0.0613121\pi$
$307$	34.3948			1.96301			0.981507	+	0.191429i	$0.0613121\pi$
$308$		0			0
$309$	−3.50000	+	2.95804i	−0.199108	+	0.168277i
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	14.5516	−	25.2042i	0.822507	−	1.42462i	−0.0813030	−	0.996689i	$-0.525908\pi$
$313$	14.5516	−	25.2042i	0.822507	−	1.42462i	0.903810	−	0.427934i	$-0.140759\pi$
$314$		0			0
$315$		0			0
$316$	2.00000			0.112509
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$	17.5000	+	30.3109i	0.979812	+	1.69708i
$320$	15.4919	+	8.94427i	0.866025	+	0.500000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−13.6235	+	11.7644i	−0.756860	+	0.653577i
$325$	6.61438	+	11.4564i	0.366900	+	0.635489i
$326$		0			0
$327$	−3.37489	−	18.7513i	−0.186632	−	1.03695i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	−0.954765	−	0.297361i	$-0.903893\pi$
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	0.734905	+	0.678170i	$0.237227\pi$
$332$	15.4919	−	8.94427i	0.850230	−	0.490881i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	5.00000	+	8.66025i	0.271163	+	0.469668i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$	−20.1698	+	3.63020i	−1.08122	+	0.194599i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	7.00000	+	11.8322i	0.373632	+	0.631554i
$352$		0			0
$353$	−25.1744	+	14.5344i	−1.33990	+	0.773590i	−0.986792	−	0.161993i	$-0.948208\pi$
$353$	−25.1744	+	14.5344i	−1.33990	+	0.773590i	−0.353106	+	0.935583i	$0.614874\pi$
$354$		0			0
$355$	−13.2288	+	22.9129i	−0.702109	+	1.21609i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.2470	+	5.91608i	0.540813	+	0.312239i	0.745409	−	0.666608i	$-0.232254\pi$
$359$	10.2470	+	5.91608i	0.540813	+	0.312239i	−0.204595	+	0.978847i	$0.565588\pi$
$360$		0			0
$361$	−9.50000	−	16.4545i	−0.500000	−	0.866025i
$362$		0			0
$363$	31.7490	−	26.8328i	1.66639	−	1.40836i
$364$		0			0
$365$			23.6643i			1.23865i
$366$		0			0
$367$	−9.26013	−	16.0390i	−0.483375	−	0.837230i	0.516443	−	0.856322i	$-0.327256\pi$
$367$	−9.26013	−	16.0390i	−0.483375	−	0.837230i	−0.999818	+	0.0190919i	$0.993923\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	6.55869	−	18.2204i	0.338689	−	0.940898i
$376$		0			0
$377$			15.6525i			0.806144i
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	30.9839	+	17.8885i	1.58320	+	0.914062i	0.994388	+	0.105793i	$0.0337381\pi$
$383$	30.9839	+	17.8885i	1.58320	+	0.914062i	0.588813	+	0.808269i	$0.299595\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	18.5203	+	32.0780i	0.940224	+	1.62852i
$389$	5.12348	−	2.95804i	0.259771	−	0.149979i	−0.364459	−	0.931219i	$-0.618746\pi$
$389$	5.12348	−	2.95804i	0.259771	−	0.149979i	0.624230	+	0.781241i	$0.285413\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.93649	+	1.11803i	−0.0974355	+	0.0562544i
$396$	12.3765	+	33.2689i	0.621944	+	1.67183i
$397$	−17.1974	+	29.7867i	−0.863112	+	1.49495i	0.00579782	+	0.999983i	$0.498154\pi$
$397$	−17.1974	+	29.7867i	−0.863112	+	1.49495i	−0.868910	+	0.494971i	$0.835179\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$	25.6174	+	14.7902i	1.27927	+	0.738587i	0.976714	−	0.214544i	$-0.0688266\pi$
$401$	25.6174	+	14.7902i	1.27927	+	0.738587i	0.302556	+	0.953131i	$0.402160\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	6.61438	−	19.0066i	0.328671	−	0.944444i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$	−5.29150			−0.260694
$413$		0			0
$414$		0			0
$415$	−10.0000	+	17.3205i	−0.490881	+	0.850230i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−37.0000			−1.80327			−0.901635	−	0.432498i	$-0.857632\pi$
$421$	−37.0000			−1.80327			−0.901635	+	0.432498i	$0.857632\pi$
$422$		0			0
$423$	−11.6947	−	31.4362i	−0.568617	−	1.52848i
$424$		0			0
$425$	−9.68246	−	5.59017i	−0.469668	−	0.271163i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	26.6822	−	4.80230i	1.28823	−	0.231857i
$430$		0			0
$431$	35.8643	−	20.7063i	1.72752	−	0.997386i	0.827636	−	0.561266i	$-0.189685\pi$
$431$	35.8643	−	20.7063i	1.72752	−	0.997386i	0.899888	−	0.436121i	$-0.143648\pi$
$432$	−20.7834	+	0.220879i	−0.999944	+	0.0106271i
$433$	10.5830			0.508587			0.254293	−	0.967127i	$-0.418157\pi$
$433$	10.5830			0.508587			0.254293	+	0.967127i	$0.418157\pi$
$434$		0			0
$435$	17.5000	−	14.7902i	0.839061	−	0.709136i
$436$	11.0000	−	19.0526i	0.526804	−	0.912452i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−26.4575	−	31.3050i	−1.25140	−	1.48067i
$448$		0			0
$449$			41.4126i			1.95438i	0.212368	+	0.977190i	$0.431882\pi$
$449$			41.4126i			1.95438i	−0.212368	+	0.977190i	$0.568118\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−5.21574	−	28.9792i	−0.245057	−	1.36156i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$	−10.1235	−	5.70221i	−0.472523	−	0.266157i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−20.4939	−	11.8322i	−0.951405	−	0.549294i
$465$		0			0
$466$		0			0
$467$	36.7933	+	21.2426i	1.70259	+	0.982992i	0.943119	+	0.332454i	$0.107877\pi$
$467$	36.7933	+	21.2426i	1.70259	+	0.982992i	0.759473	+	0.650538i	$0.225457\pi$
$468$	−2.64575	+	15.6525i	−0.122300	+	0.723536i
$469$		0			0
$470$		0			0
$471$	6.49390	+	36.0809i	0.299223	+	1.66252i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	48.0000			2.18182
$485$	−35.8643	−	20.7063i	−1.62852	−	0.940224i
$486$		0			0
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	29.5804i		−	1.33494i	−0.744635	−	0.667472i	$-0.767376\pi$
$491$		−	29.5804i		−	1.33494i	0.744635	−	0.667472i	$-0.232624\pi$
$492$		0			0
$493$	−6.61438	−	11.4564i	−0.297897	−	0.515972i
$494$		0			0
$495$	−30.5814	−	25.2938i	−1.37453	−	1.13687i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−20.5000	+	35.5070i	−0.917706	+	1.58951i	−0.114816	+	0.993387i	$0.536628\pi$
$499$	−20.5000	+	35.5070i	−0.917706	+	1.58951i	−0.802890	+	0.596127i	$0.796706\pi$
$500$	19.3649	−	11.1803i	0.866025	−	0.500000i
$501$	−14.4291	+	40.0849i	−0.644645	+	1.79086i
$502$		0			0
$503$			38.0132i			1.69492i	0.530857	+	0.847461i	$0.321870\pi$
$503$			38.0132i			1.69492i	−0.530857	+	0.847461i	$0.678130\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−9.77810	−	3.51976i	−0.434261	−	0.156318i
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.12348	−	2.95804i	0.225767	−	0.130347i
$516$		0			0
$517$	−66.1438			−2.90900
$518$		0			0
$519$	−12.5000	−	14.7902i	−0.548689	−	0.649218i
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	18.5203	−	32.0780i	0.809834	−	1.40267i	−0.103144	−	0.994666i	$-0.532890\pi$
$523$	18.5203	−	32.0780i	0.809834	−	1.40267i	0.912978	−	0.408008i	$-0.133776\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−13.8821	+	38.5654i	−0.604142	+	1.67834i
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−20.1698	+	3.63020i	−0.870392	+	0.156655i
$538$		0			0
$539$		0			0
$540$	20.0000	−	11.8322i	0.860663	−	0.509175i
$541$	21.5000	−	37.2391i	0.924357	−	1.60103i	0.131765	−	0.991281i	$-0.457935\pi$
$541$	21.5000	−	37.2391i	0.924357	−	1.60103i	0.792592	−	0.609753i	$-0.208731\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			24.5967i			1.05361i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−17.5000	+	14.7902i	−0.738851	+	0.624443i
$562$		0			0
$563$	38.7298	−	22.3607i	1.63227	−	0.942390i	0.648876	−	0.760894i	$-0.275239\pi$
$563$	38.7298	−	22.3607i	1.63227	−	0.942390i	0.983392	−	0.181496i	$-0.0580941\pi$
$564$	13.1174	−	36.4408i	0.552341	−	1.53444i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	40.9878	+	23.6643i	1.71830	+	0.992060i	0.922032	+	0.387113i	$0.126528\pi$
$569$	40.9878	+	23.6643i	1.71830	+	0.992060i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	−0.978022	−	0.208502i	$-0.933141\pi$
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	0.308443	−	0.951243i	$-0.400192\pi$
$572$	27.1109	+	15.6525i	1.13356	+	0.654463i
$573$	−6.61438	−	7.82624i	−0.276320	−	0.326946i
$574$		0			0
$575$		0			0
$576$	−18.4939	−	15.2963i	−0.770579	−	0.637344i
$577$	−17.1974	−	29.7867i	−0.715936	−	1.24004i	−0.962597	−	0.270936i	$-0.912667\pi$
$577$	−17.1974	−	29.7867i	−0.715936	−	1.24004i	0.246661	−	0.969102i	$-0.420667\pi$
$578$		0			0
$579$		0			0
$580$	26.4575			1.09859
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−6.18826	−	16.6345i	−0.255853	−	0.687750i
$586$		0			0
$587$		−	8.94427i		−	0.369170i	−0.982817	−	0.184585i	$-0.940906\pi$
$587$		−	8.94427i		−	0.369170i	0.982817	−	0.184585i	$-0.0590940\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	36.7933	+	21.2426i	1.51092	+	0.872331i	0.999919	+	0.0127518i	$0.00405915\pi$
$593$	36.7933	+	21.2426i	1.51092	+	0.872331i	0.511003	+	0.859579i	$0.329274\pi$
$594$		0			0
$595$		0			0
$596$		−	47.3286i		−	1.93866i
$597$		0			0
$598$		0			0
$599$	−25.6174	+	14.7902i	−1.04670	+	0.604311i	−0.921723	−	0.387849i	$-0.873218\pi$
$599$	−25.6174	+	14.7902i	−1.04670	+	0.604311i	−0.124975	+	0.992160i	$0.539885\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	17.0000	−	29.4449i	0.691720	−	1.19809i
$605$	−46.4758	+	26.8328i	−1.88951	+	1.09091i
$606$		0			0
$607$	22.4889	−	38.9519i	0.912796	−	1.58101i	0.102699	−	0.994712i	$-0.467252\pi$
$607$	22.4889	−	38.9519i	0.912796	−	1.58101i	0.810097	−	0.586296i	$-0.199414\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−25.6174	−	14.7902i	−1.03637	−	0.598347i
$612$	−4.67789	−	12.5745i	−0.189092	−	0.508293i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−14.0000	+	11.8322i	−0.560449	+	0.473665i
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−21.1660	+	36.6606i	−0.844616	+	1.46292i
$629$		0			0
$630$		0			0
$631$	47.0000			1.87104			0.935520	−	0.353273i	$-0.114931\pi$
$631$	47.0000			1.87104			0.935520	+	0.353273i	$0.114931\pi$
$632$		0			0
$633$	37.4827	+	13.4924i	1.48980	+	0.536276i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	22.6235	−	27.3528i	0.894971	−	1.08206i
$640$		0			0
$641$	−40.9878	+	23.6643i	−1.61892	+	0.934684i	−0.631721	+	0.775196i	$0.717651\pi$
$641$	−40.9878	+	23.6643i	−1.61892	+	0.934684i	−0.987200	+	0.159489i	$0.949015\pi$
$642$		0			0
$643$	50.2693			1.98243			0.991213	−	0.132273i	$-0.0422275\pi$
$643$	50.2693			1.98243			0.991213	+	0.132273i	$0.0422275\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	15.4919	−	8.94427i	0.609051	−	0.351636i	−0.163543	−	0.986536i	$-0.552292\pi$
$647$	15.4919	−	8.94427i	0.609051	−	0.351636i	0.772594	+	0.634901i	$0.218959\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	5.29150	−	31.3050i	0.206441	−	1.22132i
$658$		0			0
$659$			5.91608i			0.230458i	0.993339	+	0.115229i	$0.0367601\pi$
$659$			5.91608i			0.230458i	−0.993339	+	0.115229i	$0.963240\pi$
$660$	−8.11738	−	45.1011i	−0.315968	−	1.75556i
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$	−10.0849	+	1.81510i	−0.391665	+	0.0704927i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−42.6028	+	24.5967i	−1.64835	+	0.951677i
$669$	−47.4291	−	17.0728i	−1.83372	−	0.660071i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−12.7505	+	22.6368i	−0.490768	+	0.871290i
$676$	−6.00000	−	10.3923i	−0.230769	−	0.399704i
$677$	−44.5393	−	25.7148i	−1.71178	−	0.988299i	−0.932156	−	0.362058i	$-0.882074\pi$
$677$	−44.5393	−	25.7148i	−1.71178	−	0.988299i	−0.779629	−	0.626242i	$-0.784592\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−49.5526	+	8.91857i	−1.89886	+	0.341760i
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		−	22.3607i		−	0.850026i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			41.4126i			1.56413i	0.623196	+	0.782065i	$0.285834\pi$
$701$			41.4126i			1.56413i	−0.623196	+	0.782065i	$0.714166\pi$
$702$		0			0
$703$		0			0
$704$	−40.9878	+	23.6643i	−1.54479	+	0.891883i
$705$	7.67020	+	42.6165i	0.288876	+	1.60503i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	−0.856484	−	0.516174i	$-0.827356\pi$
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	0.875262	+	0.483650i	$0.160689\pi$
$710$		0			0
$711$	2.81174	−	1.04601i	0.105448	−	0.0392283i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−35.0000			−1.30893
$716$	−20.4939	−	11.8322i	−0.765893	−	0.442189i
$717$	17.3527	−	48.2067i	0.648047	−	1.80031i
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$	26.4575	+	4.47214i	0.986013	+	0.166667i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−25.6174	+	14.7902i	−0.951405	+	0.549294i
$726$		0			0
$727$	−5.29150			−0.196251			−0.0981255	−	0.995174i	$-0.531285\pi$
$727$	−5.29150			−0.196251			−0.0981255	+	0.995174i	$0.531285\pi$
$728$		0			0
$729$	−13.0000	+	23.6643i	−0.481481	+	0.876456i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−25.1346	+	43.5345i	−0.928369	+	1.60798i	−0.142318	+	0.989821i	$0.545455\pi$
$733$	−25.1346	+	43.5345i	−0.928369	+	1.60798i	−0.786051	+	0.618161i	$0.787878\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−5.50000	−	9.52628i	−0.202321	−	0.350430i	0.746955	−	0.664875i	$-0.231515\pi$
$739$	−5.50000	−	9.52628i	−0.202321	−	0.350430i	−0.949276	+	0.314445i	$0.898182\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	26.4575	+	45.8258i	0.969328	+	1.67893i
$746$		0			0
$747$	17.1017	−	20.6768i	0.625720	−	0.756525i
$748$	−26.4575			−0.967382
$749$		0			0
$750$		0			0
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	−0.722718	−	0.691143i	$-0.757107\pi$
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	0.959906	+	0.280321i	$0.0904408\pi$
$752$	38.7298	−	22.3607i	1.41233	−	0.815410i
$753$		0			0
$754$		0			0
$755$			38.0132i			1.38344i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$		−	11.8322i		−	0.428073i
$765$	11.5587	+	9.56016i	0.417905	+	0.345648i
$766$		0			0
$767$		0			0
$768$	−4.90893	−	27.2746i	−0.177136	−	0.984186i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−5.00000	−	5.91608i	−0.180071	−	0.213062i
$772$		0			0
$773$	−1.93649	+	1.11803i	−0.0696508	+	0.0402129i	−0.534421	−	0.845218i	$-0.679470\pi$
$773$	−1.93649	+	1.11803i	−0.0696508	+	0.0402129i	0.464770	+	0.885431i	$0.346137\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	6.94106	−	19.2827i	0.248530	−	0.690431i
$781$	−35.0000	−	60.6218i	−1.25240	−	2.16922i
$782$		0			0
$783$	−26.4575	+	15.6525i	−0.945514	+	0.559374i
$784$		0			0
$785$		−	47.3286i		−	1.68923i
$786$		0			0
$787$	22.4889	+	38.9519i	0.801642	+	1.38849i	0.918535	+	0.395340i	$0.129373\pi$
$787$	22.4889	+	38.9519i	0.801642	+	1.38849i	−0.116892	+	0.993145i	$0.537293\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	55.9017i		−	1.98014i	−0.140576	−	0.990070i	$-0.544895\pi$
$797$		−	55.9017i		−	1.98014i	0.140576	−	0.990070i	$-0.455105\pi$
$798$		0			0
$799$	25.0000			0.884436
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−54.2218	−	31.3050i	−1.91344	−	1.10473i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	35.8643	−	20.7063i	1.26092	−	0.727994i	0.287670	−	0.957730i	$-0.407120\pi$
$809$	35.8643	−	20.7063i	1.26092	−	0.727994i	0.973253	+	0.229736i	$0.0737862\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	5.24695	−	14.5763i	0.183680	−	0.510274i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.12348	−	2.95804i	−0.178811	−	0.103236i	0.407923	−	0.913016i	$-0.366253\pi$
$821$	−5.12348	−	2.95804i	−0.178811	−	0.103236i	−0.586734	+	0.809780i	$0.699586\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	33.0719	+	39.1312i	1.15142	+	1.36237i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$	−21.1660			−0.733799
$833$		0			0
$834$		0			0
$835$	27.5000	−	47.6314i	0.951677	−	1.64835i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−6.00000			−0.206897
$842$		0			0
$843$	−3.47053	+	9.64134i	−0.119531	+	0.332065i
$844$	23.0000	+	39.8372i	0.791693	+	1.37125i
$845$	11.6190	+	6.70820i	0.399704	+	0.230769i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−0.811738	−	4.51011i	−0.0278588	−	0.154787i
$850$		0			0
$851$		0			0
$852$	40.3396	−	7.26040i	1.38201	−	0.248737i
$853$	42.3320			1.44942			0.724710	−	0.689054i	$-0.241974\pi$
$853$	42.3320			1.44942			0.724710	+	0.689054i	$0.241974\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−42.6028	+	24.5967i	−1.45528	+	0.840209i	−0.998774	−	0.0495090i	$-0.984234\pi$
$857$	−42.6028	+	24.5967i	−1.45528	+	0.840209i	−0.456511	+	0.889718i	$0.650901\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$	12.5000	+	21.6506i	0.425013	+	0.736144i
$866$		0			0
$867$	−15.8745	+	13.4164i	−0.539127	+	0.455645i
$868$		0			0
$869$		−	5.91608i		−	0.200689i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	42.8140	+	35.4113i	1.44903	+	1.19849i
$874$		0			0
$875$		0			0
$876$	28.0000	−	23.6643i	0.946032	−	0.799543i
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$	−14.4291	+	40.0849i	−0.486682	+	1.35203i
$880$	26.4575	−	45.8258i	0.891883	−	1.54479i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−10.2470	−	5.91608i	−0.344642	−	0.198979i
$885$		0			0
$886$		0			0
$887$	30.9839	+	17.8885i	1.04034	+	0.600639i	0.919929	−	0.392086i	$-0.128246\pi$
$887$	30.9839	+	17.8885i	1.04034	+	0.600639i	0.120408	+	0.992725i	$0.461580\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	34.7995	+	40.2988i	1.16583	+	1.35006i
$892$	−29.1033	−	50.4083i	−0.974449	−	1.68780i
$893$		0			0
$894$		0			0
$895$	26.4575			0.884377
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−28.1174	+	10.4601i	−0.937246	+	0.348669i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	−50.3488	−	29.0689i	−1.67088	−	0.964685i
$909$		0			0
$910$		0			0
$911$			59.1608i			1.96008i	0.198789	+	0.980042i	$0.436299\pi$
$911$			59.1608i			1.96008i	−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$	−26.4575	−	45.8258i	−0.875616	−	1.51661i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−14.5000	+	25.1147i	−0.478311	+	0.828459i	−0.999691	−	0.0248659i	$-0.992084\pi$
$919$	−14.5000	+	25.1147i	−0.478311	+	0.828459i	0.521380	+	0.853325i	$0.325417\pi$
$920$		0			0
$921$	56.0526	+	20.1769i	1.84700	+	0.664851i
$922$		0			0
$923$		−	31.3050i		−	1.03042i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−7.43916	+	2.76748i	−0.244334	+	0.0908958i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	25.6174	−	14.7902i	0.837778	−	0.483691i
$936$		0			0
$937$	−60.8523			−1.98796			−0.993979	−	0.109574i	$-0.965051\pi$
$937$	−60.8523			−1.98796			−0.993979	+	0.109574i	$0.965051\pi$
$938$		0			0
$939$	38.5000	−	32.5384i	1.25640	−	1.06185i
$940$	−25.0000	+	43.3013i	−0.815410	+	1.41233i
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$	3.25937	+	1.17325i	0.105859	+	0.0381055i
$949$	−14.0000	−	24.2487i	−0.454459	−	0.787146i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	6.61438	+	11.4564i	0.214036	+	0.370722i
$956$	51.2348	−	29.5804i	1.65705	−	0.956698i
$957$	10.7383	+	59.6631i	0.347119	+	1.92864i
$958$		0			0
$959$		0			0
$960$	20.0000	+	23.6643i	0.645497	+	0.763763i
$961$	−15.5000	+	26.8468i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$	−29.1033	+	11.1803i	−0.933488	+	0.358610i
$973$		0			0
$974$		0			0
$975$	4.05869	+	22.5505i	0.129982	+	0.722195i
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	5.50000	−	32.5384i	0.175601	−	1.03887i
$982$		0			0
$983$	−25.1744	+	14.5344i	−0.802938	+	0.463577i	−0.844498	−	0.535559i	$-0.820101\pi$
$983$	−25.1744	+	14.5344i	−0.802938	+	0.463577i	0.0415592	+	0.999136i	$0.486767\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	26.0000	+	45.0333i	0.825917	+	1.43053i	0.901216	+	0.433370i	$0.142676\pi$
$991$	26.0000	+	45.0333i	0.825917	+	1.43053i	−0.0752991	+	0.997161i	$0.523991\pi$
$992$		0			0
$993$	−10.5830	+	8.94427i	−0.335842	+	0.283838i
$994$		0			0
$995$		0			0
$996$	30.4939	−	5.48835i	0.966236	−	0.173905i
$997$	−9.26013	−	16.0390i	−0.293271	−	0.507961i	0.681310	−	0.731995i	$-0.261411\pi$
$997$	−9.26013	−	16.0390i	−0.293271	−	0.507961i	−0.974581	+	0.224034i	$0.928077\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.p.b.509.4		8
3.2	odd	2	inner	735.2.p.b.509.2		8
5.4	even	2	inner	735.2.p.b.509.1		8
7.2	even	3		105.2.g.b.104.2	yes	4
7.3	odd	6	inner	735.2.p.b.374.3		8
7.4	even	3	inner	735.2.p.b.374.2		8
7.5	odd	6		105.2.g.b.104.3	yes	4
7.6	odd	2	inner	735.2.p.b.509.1		8
15.14	odd	2	inner	735.2.p.b.509.3		8
21.2	odd	6		105.2.g.b.104.1	✓	4
21.5	even	6		105.2.g.b.104.4	yes	4
21.11	odd	6	inner	735.2.p.b.374.4		8
21.17	even	6	inner	735.2.p.b.374.1		8
21.20	even	2	inner	735.2.p.b.509.3		8
28.19	even	6		1680.2.k.b.209.2		4
28.23	odd	6		1680.2.k.b.209.3		4
35.2	odd	12		525.2.b.f.251.4		4
35.4	even	6	inner	735.2.p.b.374.3		8
35.9	even	6		105.2.g.b.104.3	yes	4
35.12	even	12		525.2.b.f.251.1		4
35.19	odd	6		105.2.g.b.104.2	yes	4
35.23	odd	12		525.2.b.f.251.1		4
35.24	odd	6	inner	735.2.p.b.374.2		8
35.33	even	12		525.2.b.f.251.4		4
35.34	odd	2	CM	735.2.p.b.509.4		8
84.23	even	6		1680.2.k.b.209.4		4
84.47	odd	6		1680.2.k.b.209.1		4
105.2	even	12		525.2.b.f.251.2		4
105.23	even	12		525.2.b.f.251.3		4
105.44	odd	6		105.2.g.b.104.4	yes	4
105.47	odd	12		525.2.b.f.251.3		4
105.59	even	6	inner	735.2.p.b.374.4		8
105.68	odd	12		525.2.b.f.251.2		4
105.74	odd	6	inner	735.2.p.b.374.1		8
105.89	even	6		105.2.g.b.104.1	✓	4
105.104	even	2	inner	735.2.p.b.509.2		8
140.19	even	6		1680.2.k.b.209.3		4
140.79	odd	6		1680.2.k.b.209.2		4
420.299	odd	6		1680.2.k.b.209.4		4
420.359	even	6		1680.2.k.b.209.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.g.b.104.1	✓	4	21.2	odd	6
105.2.g.b.104.1	✓	4	105.89	even	6
105.2.g.b.104.2	yes	4	7.2	even	3
105.2.g.b.104.2	yes	4	35.19	odd	6
105.2.g.b.104.3	yes	4	7.5	odd	6
105.2.g.b.104.3	yes	4	35.9	even	6
105.2.g.b.104.4	yes	4	21.5	even	6
105.2.g.b.104.4	yes	4	105.44	odd	6
525.2.b.f.251.1		4	35.12	even	12
525.2.b.f.251.1		4	35.23	odd	12
525.2.b.f.251.2		4	105.2	even	12
525.2.b.f.251.2		4	105.68	odd	12
525.2.b.f.251.3		4	105.23	even	12
525.2.b.f.251.3		4	105.47	odd	12
525.2.b.f.251.4		4	35.2	odd	12
525.2.b.f.251.4		4	35.33	even	12
735.2.p.b.374.1		8	21.17	even	6	inner
735.2.p.b.374.1		8	105.74	odd	6	inner
735.2.p.b.374.2		8	7.4	even	3	inner
735.2.p.b.374.2		8	35.24	odd	6	inner
735.2.p.b.374.3		8	7.3	odd	6	inner
735.2.p.b.374.3		8	35.4	even	6	inner
735.2.p.b.374.4		8	21.11	odd	6	inner
735.2.p.b.374.4		8	105.59	even	6	inner
735.2.p.b.509.1		8	5.4	even	2	inner
735.2.p.b.509.1		8	7.6	odd	2	inner
735.2.p.b.509.2		8	3.2	odd	2	inner
735.2.p.b.509.2		8	105.104	even	2	inner
735.2.p.b.509.3		8	15.14	odd	2	inner
735.2.p.b.509.3		8	21.20	even	2	inner
735.2.p.b.509.4		8	1.1	even	1	trivial
735.2.p.b.509.4		8	35.34	odd	2	CM
1680.2.k.b.209.1		4	84.47	odd	6
1680.2.k.b.209.1		4	420.359	even	6
1680.2.k.b.209.2		4	28.19	even	6
1680.2.k.b.209.2		4	140.79	odd	6
1680.2.k.b.209.3		4	28.23	odd	6
1680.2.k.b.209.3		4	140.19	even	6
1680.2.k.b.209.4		4	84.23	even	6
1680.2.k.b.209.4		4	420.299	odd	6

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	735.2.p.b.509.4

Level	$735$
Weight	$2$
Character	735.509
Analytic conductor	$5.869$
Analytic rank	$0$
Dimension	$8$
CM discriminant	-35
Inner twists	$16$