LMFDB - Embedded newform 1620.2.r.g.109.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(109,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("1620.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1620 = 2^{2} \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1620.r (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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.3317760000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 25x^{4} + 625$ x^8 - 25*x^4 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			109.3
Root			$-0.578737 + 2.15988i$ of defining polynomial
Character	$\chi$	$=$	1620.109
Dual form			1620.2.r.g.1189.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.578737 + 2.15988i) q^{5} +(-0.866025 + 0.500000i) q^{7} +(-1.58114 - 2.73861i) q^{11} +(-2.59808 - 1.50000i) q^{13} -6.32456i q^{17} -3.00000 q^{19} +(2.73861 + 1.58114i) q^{23} +(-4.33013 + 2.50000i) q^{25} +(-4.74342 - 8.21584i) q^{29} +(1.00000 - 1.73205i) q^{31} +(-1.58114 - 1.58114i) q^{35} -1.00000i q^{37} +(-1.58114 + 2.73861i) q^{41} +(8.66025 - 5.00000i) q^{43} +(5.47723 - 3.16228i) q^{47} +(-3.00000 + 5.19615i) q^{49} -9.48683i q^{53} +(5.00000 - 5.00000i) q^{55} +(3.16228 - 5.47723i) q^{59} +(0.500000 + 0.866025i) q^{61} +(1.73621 - 6.47963i) q^{65} +(9.52628 + 5.50000i) q^{67} -9.48683 q^{71} +13.0000i q^{73} +(2.73861 + 1.58114i) q^{77} +(-1.50000 - 2.59808i) q^{79} +(13.6931 - 7.90569i) q^{83} +(13.6603 - 3.66025i) q^{85} -12.6491 q^{89} +3.00000 q^{91} +(-1.73621 - 6.47963i) q^{95} +(0.866025 - 0.500000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 24 q^{19} + 8 q^{31} - 24 q^{49} + 40 q^{55} + 4 q^{61} - 12 q^{79} + 40 q^{85} + 24 q^{91}+O(q^{100})$ 8 * q - 24 * q^19 + 8 * q^31 - 24 * q^49 + 40 * q^55 + 4 * q^61 - 12 * q^79 + 40 * q^85 + 24 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$ .

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.578737	+	2.15988i	0.258819	+	0.965926i
$5$	0.578737	+	2.15988i	0.258819	+	0.965926i
$6$		0			0
$7$	−0.866025	+	0.500000i	−0.327327	+	0.188982i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−0.866025	+	0.500000i	−0.327327	+	0.188982i	0.327327	+	0.944911i	$0.393852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.58114	−	2.73861i	−0.476731	−	0.825723i	0.522913	−	0.852386i	$-0.324845\pi$
$11$	−1.58114	−	2.73861i	−0.476731	−	0.825723i	−0.999644	+	0.0266631i	$0.991512\pi$
$12$		0			0
$13$	−2.59808	−	1.50000i	−0.720577	−	0.416025i	0.0943882	−	0.995535i	$-0.469911\pi$
$13$	−2.59808	−	1.50000i	−0.720577	−	0.416025i	−0.814965	+	0.579510i	$0.803244\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.32456i		−	1.53393i	−0.641689	−	0.766965i	$-0.721766\pi$
$17$		−	6.32456i		−	1.53393i	0.641689	−	0.766965i	$-0.278234\pi$
$18$		0			0
$19$	−3.00000			−0.688247			−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000			−0.688247			−0.344124	+	0.938924i	$0.611824\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.73861	+	1.58114i	0.571040	+	0.329690i	0.757565	−	0.652760i	$-0.226389\pi$
$23$	2.73861	+	1.58114i	0.571040	+	0.329690i	−0.186524	+	0.982450i	$0.559722\pi$
$24$		0			0
$25$	−4.33013	+	2.50000i	−0.866025	+	0.500000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.74342	−	8.21584i	−0.880830	−	1.52564i	−0.850419	−	0.526105i	$-0.823652\pi$
$29$	−4.74342	−	8.21584i	−0.880830	−	1.52564i	−0.0304110	−	0.999537i	$-0.509682\pi$
$30$		0			0
$31$	1.00000	−	1.73205i	0.179605	−	0.311086i	−0.762140	−	0.647412i	$-0.775851\pi$
$31$	1.00000	−	1.73205i	0.179605	−	0.311086i	0.941745	+	0.336327i	$0.109185\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.58114	−	1.58114i	−0.267261	−	0.267261i
$36$		0			0
$37$		−	1.00000i		−	0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$		−	1.00000i		−	0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.58114	+	2.73861i	−0.246932	+	0.427699i	−0.962673	−	0.270667i	$-0.912756\pi$
$41$	−1.58114	+	2.73861i	−0.246932	+	0.427699i	0.715741	+	0.698366i	$0.246089\pi$
$42$		0			0
$43$	8.66025	−	5.00000i	1.32068	−	0.762493i	0.336840	−	0.941562i	$-0.390642\pi$
$43$	8.66025	−	5.00000i	1.32068	−	0.762493i	0.983836	+	0.179069i	$0.0573086\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.47723	−	3.16228i	0.798935	−	0.461266i	−0.0441633	−	0.999024i	$-0.514062\pi$
$47$	5.47723	−	3.16228i	0.798935	−	0.461266i	0.843099	+	0.537759i	$0.180729\pi$
$48$		0			0
$49$	−3.00000	+	5.19615i	−0.428571	+	0.742307i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.48683i		−	1.30312i	−0.758599	−	0.651558i	$-0.774116\pi$
$53$		−	9.48683i		−	1.30312i	0.758599	−	0.651558i	$-0.225884\pi$
$54$		0			0
$55$	5.00000	−	5.00000i	0.674200	−	0.674200i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.16228	−	5.47723i	0.411693	−	0.713074i	−0.583382	−	0.812198i	$-0.698271\pi$
$59$	3.16228	−	5.47723i	0.411693	−	0.713074i	0.995075	+	0.0991242i	$0.0316041\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.73621	−	6.47963i	0.215350	−	0.803699i
$66$		0			0
$67$	9.52628	+	5.50000i	1.16382	+	0.671932i	0.952217	−	0.305424i	$-0.0987981\pi$
$67$	9.52628	+	5.50000i	1.16382	+	0.671932i	0.211604	+	0.977356i	$0.432131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.48683			−1.12588			−0.562940	−	0.826498i	$-0.690330\pi$
$71$	−9.48683			−1.12588			−0.562940	+	0.826498i	$0.690330\pi$
$72$		0			0
$73$			13.0000i			1.52153i	0.649025	+	0.760767i	$0.275177\pi$
$73$			13.0000i			1.52153i	−0.649025	+	0.760767i	$0.724823\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.73861	+	1.58114i	0.312094	+	0.180187i
$78$		0			0
$79$	−1.50000	−	2.59808i	−0.168763	−	0.292306i	0.769222	−	0.638982i	$-0.220644\pi$
$79$	−1.50000	−	2.59808i	−0.168763	−	0.292306i	−0.937985	+	0.346675i	$0.887311\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.6931	−	7.90569i	1.50301	−	0.867763i	0.503015	−	0.864278i	$-0.332224\pi$
$83$	13.6931	−	7.90569i	1.50301	−	0.867763i	0.999994	−	0.00348505i	$-0.00110933\pi$
$84$		0			0
$85$	13.6603	−	3.66025i	1.48166	−	0.397010i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.6491			−1.34080			−0.670402	−	0.741999i	$-0.733878\pi$
$89$	−12.6491			−1.34080			−0.670402	+	0.741999i	$0.733878\pi$
$90$		0			0
$91$	3.00000			0.314485
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.73621	−	6.47963i	−0.178131	−	0.664796i
$96$		0			0
$97$	0.866025	−	0.500000i	0.0879316	−	0.0507673i	−0.455389	−	0.890292i	$-0.650500\pi$
$97$	0.866025	−	0.500000i	0.0879316	−	0.0507673i	0.543321	+	0.839525i	$0.317167\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.16228	−	5.47723i	−0.314658	−	0.545004i	0.664706	−	0.747105i	$-0.268557\pi$
$101$	−3.16228	−	5.47723i	−0.314658	−	0.545004i	−0.979365	+	0.202100i	$0.935223\pi$
$102$		0			0
$103$	−14.7224	−	8.50000i	−1.45064	−	0.837530i	−0.452126	−	0.891954i	$-0.649334\pi$
$103$	−14.7224	−	8.50000i	−1.45064	−	0.837530i	−0.998518	+	0.0544240i	$0.982668\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			9.48683i			0.917127i	0.888662	+	0.458563i	$0.151636\pi$
$107$			9.48683i			0.917127i	−0.888662	+	0.458563i	$0.848364\pi$
$108$		0			0
$109$	−16.0000			−1.53252			−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000			−1.53252			−0.766261	+	0.642529i	$0.777885\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.21584	−	4.74342i	−0.772881	−	0.446223i	0.0610203	−	0.998137i	$-0.480565\pi$
$113$	−8.21584	−	4.74342i	−0.772881	−	0.446223i	−0.833901	+	0.551913i	$0.813898\pi$
$114$		0			0
$115$	−1.83013	+	6.83013i	−0.170660	+	0.636913i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.16228	+	5.47723i	0.289886	+	0.502096i
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.90569	−	7.90569i	−0.707107	−	0.707107i
$126$		0			0
$127$		−	10.0000i		−	0.887357i	−0.896186	−	0.443678i	$-0.853673\pi$
$127$		−	10.0000i		−	0.887357i	0.896186	−	0.443678i	$-0.146327\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.48683	−	16.4317i	0.828868	−	1.43564i	−0.0700581	−	0.997543i	$-0.522318\pi$
$131$	9.48683	−	16.4317i	0.828868	−	1.43564i	0.898927	−	0.438099i	$-0.144348\pi$
$132$		0			0
$133$	2.59808	−	1.50000i	0.225282	−	0.130066i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.21584	−	4.74342i	0.701926	−	0.405257i	−0.106138	−	0.994351i	$-0.533849\pi$
$137$	8.21584	−	4.74342i	0.701926	−	0.405257i	0.808065	+	0.589094i	$0.200515\pi$
$138$		0			0
$139$	−8.50000	+	14.7224i	−0.720961	+	1.24874i	0.239655	+	0.970858i	$0.422966\pi$
$139$	−8.50000	+	14.7224i	−0.720961	+	1.24874i	−0.960615	+	0.277882i	$0.910368\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			9.48683i			0.793329i
$144$		0			0
$145$	15.0000	−	15.0000i	1.24568	−	1.24568i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.32456	+	10.9545i	−0.518128	+	0.897424i	0.481650	+	0.876363i	$0.340037\pi$
$149$	−6.32456	+	10.9545i	−0.518128	+	0.897424i	−0.999778	+	0.0210602i	$0.993296\pi$
$150$		0			0
$151$	−0.500000	−	0.866025i	−0.0406894	−	0.0704761i	0.844963	−	0.534824i	$-0.179622\pi$
$151$	−0.500000	−	0.866025i	−0.0406894	−	0.0704761i	−0.885653	+	0.464348i	$0.846289\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.31975	+	1.15747i	0.346971	+	0.0929705i
$156$		0			0
$157$	−8.66025	−	5.00000i	−0.691164	−	0.399043i	0.112884	−	0.993608i	$-0.463991\pi$
$157$	−8.66025	−	5.00000i	−0.691164	−	0.399043i	−0.804048	+	0.594565i	$0.797324\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.16228			−0.249222
$162$		0			0
$163$			17.0000i			1.33154i	0.746156	+	0.665771i	$0.231897\pi$
$163$			17.0000i			1.33154i	−0.746156	+	0.665771i	$0.768103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−2.00000	−	3.46410i	−0.153846	−	0.266469i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.47723	+	3.16228i	−0.416426	+	0.240424i	−0.693547	−	0.720411i	$-0.743953\pi$
$173$	−5.47723	+	3.16228i	−0.416426	+	0.240424i	0.277121	+	0.960835i	$0.410620\pi$
$174$		0			0
$175$	2.50000	−	4.33013i	0.188982	−	0.327327i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.16228			0.236360			0.118180	−	0.992992i	$-0.462294\pi$
$179$	3.16228			0.236360			0.118180	+	0.992992i	$0.462294\pi$
$180$		0			0
$181$	11.0000			0.817624			0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000			0.817624			0.408812	+	0.912619i	$0.365943\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.15988	−	0.578737i	0.158797	−	0.0425496i
$186$		0			0
$187$	−17.3205	+	10.0000i	−1.26660	+	0.731272i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−6.06218	−	3.50000i	−0.436365	−	0.251936i	0.265689	−	0.964059i	$-0.414400\pi$
$193$	−6.06218	−	3.50000i	−0.436365	−	0.251936i	−0.702055	+	0.712123i	$0.747734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	12.6491i		−	0.901212i	−0.892723	−	0.450606i	$-0.851208\pi$
$197$		−	12.6491i		−	0.901212i	0.892723	−	0.450606i	$-0.148792\pi$
$198$		0			0
$199$	−27.0000			−1.91398			−0.956990	−	0.290122i	$-0.906304\pi$
$199$	−27.0000			−1.91398			−0.956990	+	0.290122i	$0.906304\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.21584	+	4.74342i	0.576639	+	0.332923i
$204$		0			0
$205$	−6.83013	−	1.83013i	−0.477037	−	0.127822i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.74342	+	8.21584i	0.328109	+	0.568301i
$210$		0			0
$211$	10.5000	−	18.1865i	0.722850	−	1.25201i	−0.237003	−	0.971509i	$-0.576165\pi$
$211$	10.5000	−	18.1865i	0.722850	−	1.25201i	0.959853	−	0.280504i	$-0.0905015\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.8114	+	15.8114i	1.07833	+	1.07833i
$216$		0			0
$217$			2.00000i			0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.48683	+	16.4317i	−0.638153	+	1.10531i
$222$		0			0
$223$	8.66025	−	5.00000i	0.579934	−	0.334825i	−0.181173	−	0.983451i	$-0.557990\pi$
$223$	8.66025	−	5.00000i	0.579934	−	0.334825i	0.761107	+	0.648626i	$0.224656\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.6931	+	7.90569i	−0.908841	+	0.524719i	−0.880058	−	0.474866i	$-0.842496\pi$
$227$	−13.6931	+	7.90569i	−0.908841	+	0.524719i	−0.0287826	+	0.999586i	$0.509163\pi$
$228$		0			0
$229$	−8.00000	+	13.8564i	−0.528655	+	0.915657i	0.470787	+	0.882247i	$0.343970\pi$
$229$	−8.00000	+	13.8564i	−0.528655	+	0.915657i	−0.999442	+	0.0334101i	$0.989363\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.32456i		−	0.414335i	−0.978305	−	0.207168i	$-0.933575\pi$
$233$		−	6.32456i		−	0.414335i	0.978305	−	0.207168i	$-0.0664246\pi$
$234$		0			0
$235$	10.0000	+	10.0000i	0.652328	+	0.652328i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.58114	−	2.73861i	0.102275	−	0.177146i	−0.810346	−	0.585951i	$-0.800721\pi$
$239$	1.58114	−	2.73861i	0.102275	−	0.177146i	0.912622	+	0.408805i	$0.134054\pi$
$240$		0			0
$241$	5.50000	+	9.52628i	0.354286	+	0.613642i	0.986996	−	0.160748i	$-0.0513906\pi$
$241$	5.50000	+	9.52628i	0.354286	+	0.613642i	−0.632709	+	0.774389i	$0.718057\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−12.9593	−	3.47242i	−0.827936	−	0.221845i
$246$		0			0
$247$	7.79423	+	4.50000i	0.495935	+	0.286328i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	28.4605			1.79641			0.898205	−	0.439576i	$-0.144871\pi$
$251$	28.4605			1.79641			0.898205	+	0.439576i	$0.144871\pi$
$252$		0			0
$253$		−	10.0000i		−	0.628695i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.4317	−	9.48683i	−1.02498	−	0.591772i	−0.109437	−	0.993994i	$-0.534905\pi$
$257$	−16.4317	−	9.48683i	−1.02498	−	0.591772i	−0.915542	+	0.402221i	$0.868238\pi$
$258$		0			0
$259$	0.500000	+	0.866025i	0.0310685	+	0.0538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−27.3861	+	15.8114i	−1.68870	+	0.974972i	−0.733187	+	0.680028i	$0.761968\pi$
$263$	−27.3861	+	15.8114i	−1.68870	+	0.974972i	−0.955514	+	0.294944i	$0.904699\pi$
$264$		0			0
$265$	20.4904	−	5.49038i	1.25871	−	0.337271i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.9737			1.15684			0.578422	−	0.815737i	$-0.303669\pi$
$269$	18.9737			1.15684			0.578422	+	0.815737i	$0.303669\pi$
$270$		0			0
$271$	19.0000			1.15417			0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000			1.15417			0.577084	+	0.816685i	$0.304191\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	13.6931	+	7.90569i	0.825723	+	0.476731i
$276$		0			0
$277$	17.3205	−	10.0000i	1.04069	−	0.600842i	0.120660	−	0.992694i	$-0.461499\pi$
$277$	17.3205	−	10.0000i	1.04069	−	0.600842i	0.920028	+	0.391852i	$0.128166\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.48683	+	16.4317i	0.565937	+	0.980232i	0.996962	+	0.0778916i	$0.0248188\pi$
$281$	9.48683	+	16.4317i	0.565937	+	0.980232i	−0.431025	+	0.902340i	$0.641848\pi$
$282$		0			0
$283$	8.66025	+	5.00000i	0.514799	+	0.297219i	0.734804	−	0.678280i	$-0.237274\pi$
$283$	8.66025	+	5.00000i	0.514799	+	0.297219i	−0.220005	+	0.975499i	$0.570607\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	3.16228i		−	0.186663i
$288$		0			0
$289$	−23.0000			−1.35294
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.21584	+	4.74342i	0.479974	+	0.277113i	0.720406	−	0.693553i	$-0.243956\pi$
$293$	8.21584	+	4.74342i	0.479974	+	0.277113i	−0.240431	+	0.970666i	$0.577289\pi$
$294$		0			0
$295$	13.6603	+	3.66025i	0.795331	+	0.213108i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.74342	−	8.21584i	−0.274319	−	0.475134i
$300$		0			0
$301$	−5.00000	+	8.66025i	−0.288195	+	0.499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.58114	+	1.58114i	−0.0905357	+	0.0905357i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.48683	+	16.4317i	−0.537949	+	0.931755i	0.461065	+	0.887366i	$0.347467\pi$
$311$	−9.48683	+	16.4317i	−0.537949	+	0.931755i	−0.999014	+	0.0443887i	$0.985866\pi$
$312$		0			0
$313$	6.06218	−	3.50000i	0.342655	−	0.197832i	−0.318791	−	0.947825i	$-0.603277\pi$
$313$	6.06218	−	3.50000i	0.342655	−	0.197832i	0.661445	+	0.749993i	$0.269943\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−21.9089	+	12.6491i	−1.23053	+	0.710445i	−0.967140	−	0.254245i	$-0.918173\pi$
$317$	−21.9089	+	12.6491i	−1.23053	+	0.710445i	−0.263387	+	0.964690i	$0.584840\pi$
$318$		0			0
$319$	−15.0000	+	25.9808i	−0.839839	+	1.45464i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			18.9737i			1.05572i
$324$		0			0
$325$	15.0000			0.832050
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.16228	+	5.47723i	−0.174342	+	0.301969i
$330$		0			0
$331$	−9.50000	−	16.4545i	−0.522167	−	0.904420i	−0.999667	−	0.0257885i	$-0.991790\pi$
$331$	−9.50000	−	16.4545i	−0.522167	−	0.904420i	0.477500	−	0.878632i	$-0.341543\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.36611	+	23.7586i	−0.347818	+	1.29807i
$336$		0			0
$337$	−9.52628	−	5.50000i	−0.518930	−	0.299604i	0.217567	−	0.976045i	$-0.430188\pi$
$337$	−9.52628	−	5.50000i	−0.518930	−	0.299604i	−0.736497	+	0.676441i	$0.763521\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.32456			−0.342494
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	24.6475	+	14.2302i	1.32315	+	0.763920i	0.984230	−	0.176896i	$-0.0566056\pi$
$347$	24.6475	+	14.2302i	1.32315	+	0.763920i	0.338918	+	0.940816i	$0.389939\pi$
$348$		0			0
$349$	−8.50000	−	14.7224i	−0.454995	−	0.788074i	0.543693	−	0.839284i	$-0.317025\pi$
$349$	−8.50000	−	14.7224i	−0.454995	−	0.788074i	−0.998688	+	0.0512103i	$0.983692\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.9545	−	6.32456i	0.583047	−	0.336622i	−0.179297	−	0.983795i	$-0.557382\pi$
$353$	10.9545	−	6.32456i	0.583047	−	0.336622i	0.762343	+	0.647173i	$0.224049\pi$
$354$		0			0
$355$	−5.49038	−	20.4904i	−0.291399	−	1.08752i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.4605			1.50209			0.751044	−	0.660252i	$-0.229551\pi$
$359$	28.4605			1.50209			0.751044	+	0.660252i	$0.229551\pi$
$360$		0			0
$361$	−10.0000			−0.526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−28.0784	+	7.52358i	−1.46969	+	0.393802i
$366$		0			0
$367$	−16.4545	+	9.50000i	−0.858917	+	0.495896i	−0.863649	−	0.504093i	$-0.831827\pi$
$367$	−16.4545	+	9.50000i	−0.858917	+	0.495896i	0.00473247	+	0.999989i	$0.498494\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.74342	+	8.21584i	0.246266	+	0.426545i
$372$		0			0
$373$	23.3827	+	13.5000i	1.21071	+	0.699004i	0.962914	−	0.269809i	$-0.0869605\pi$
$373$	23.3827	+	13.5000i	1.21071	+	0.699004i	0.247796	+	0.968812i	$0.420294\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			28.4605i			1.46579i
$378$		0			0
$379$	−23.0000			−1.18143			−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000			−1.18143			−0.590715	+	0.806880i	$0.701154\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.9545	−	6.32456i	−0.559746	−	0.323170i	0.193297	−	0.981140i	$-0.438082\pi$
$383$	−10.9545	−	6.32456i	−0.559746	−	0.323170i	−0.753044	+	0.657971i	$0.771415\pi$
$384$		0			0
$385$	−1.83013	+	6.83013i	−0.0932719	+	0.348096i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.32456	+	10.9545i	0.320668	+	0.555413i	0.980626	−	0.195890i	$-0.0627594\pi$
$389$	6.32456	+	10.9545i	0.320668	+	0.555413i	−0.659958	+	0.751302i	$0.729426\pi$
$390$		0			0
$391$	10.0000	−	17.3205i	0.505722	−	0.875936i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.74342	−	4.74342i	0.238667	−	0.238667i
$396$		0			0
$397$			20.0000i			1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$			20.0000i			1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	11.0680	−	19.1703i	0.552708	−	0.957319i	−0.445370	−	0.895347i	$-0.646928\pi$
$401$	11.0680	−	19.1703i	0.552708	−	0.957319i	0.998078	−	0.0619718i	$-0.0197389\pi$
$402$		0			0
$403$	−5.19615	+	3.00000i	−0.258839	+	0.149441i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.73861	+	1.58114i	−0.135748	+	0.0783741i
$408$		0			0
$409$	−3.50000	+	6.06218i	−0.173064	+	0.299755i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−3.50000	+	6.06218i	−0.173064	+	0.299755i	0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			6.32456i			0.311211i
$414$		0			0
$415$	25.0000	+	25.0000i	1.22720	+	1.22720i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.32456	−	10.9545i	0.308975	−	0.535160i	−0.669164	−	0.743115i	$-0.733347\pi$
$419$	6.32456	−	10.9545i	0.308975	−	0.535160i	0.978138	+	0.207955i	$0.0666808\pi$
$420$		0			0
$421$	14.5000	+	25.1147i	0.706687	+	1.22402i	0.966079	+	0.258245i	$0.0831443\pi$
$421$	14.5000	+	25.1147i	0.706687	+	1.22402i	−0.259393	+	0.965772i	$0.583522\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	15.8114	+	27.3861i	0.766965	+	1.32842i
$426$		0			0
$427$	−0.866025	−	0.500000i	−0.0419099	−	0.0241967i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	22.1359			1.06625			0.533125	−	0.846036i	$-0.321017\pi$
$431$	22.1359			1.06625			0.533125	+	0.846036i	$0.321017\pi$
$432$		0			0
$433$		−	20.0000i		−	0.961139i	−0.876957	−	0.480569i	$-0.840430\pi$
$433$		−	20.0000i		−	0.961139i	0.876957	−	0.480569i	$-0.159570\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.21584	−	4.74342i	−0.393017	−	0.226908i
$438$		0			0
$439$	3.00000	+	5.19615i	0.143182	+	0.247999i	0.928693	−	0.370849i	$-0.120933\pi$
$439$	3.00000	+	5.19615i	0.143182	+	0.247999i	−0.785511	+	0.618848i	$0.787600\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.4317	+	9.48683i	−0.780693	+	0.450733i	−0.836676	−	0.547699i	$-0.815504\pi$
$443$	−16.4317	+	9.48683i	−0.780693	+	0.450733i	0.0559831	+	0.998432i	$0.482171\pi$
$444$		0			0
$445$	−7.32051	−	27.3205i	−0.347025	−	1.29512i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−37.9473			−1.79085			−0.895423	−	0.445217i	$-0.853127\pi$
$449$	−37.9473			−1.79085			−0.895423	+	0.445217i	$0.853127\pi$
$450$		0			0
$451$	10.0000			0.470882
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.73621	+	6.47963i	0.0813948	+	0.303770i
$456$		0			0
$457$	17.3205	−	10.0000i	0.810219	−	0.467780i	−0.0368128	−	0.999322i	$-0.511721\pi$
$457$	17.3205	−	10.0000i	0.810219	−	0.467780i	0.847032	+	0.531542i	$0.178387\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$	−2.59808	−	1.50000i	−0.120743	−	0.0697109i	0.438412	−	0.898774i	$-0.355541\pi$
$463$	−2.59808	−	1.50000i	−0.120743	−	0.0697109i	−0.559155	+	0.829063i	$0.688874\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			37.9473i			1.75599i	0.478667	+	0.877997i	$0.341120\pi$
$467$			37.9473i			1.75599i	−0.478667	+	0.877997i	$0.658880\pi$
$468$		0			0
$469$	−11.0000			−0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−27.3861	−	15.8114i	−1.25922	−	0.727008i
$474$		0			0
$475$	12.9904	−	7.50000i	0.596040	−	0.344124i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.0680	+	19.1703i	0.505709	+	0.875913i	0.999978	+	0.00660425i	$0.00210221\pi$
$479$	11.0680	+	19.1703i	0.505709	+	0.875913i	−0.494270	+	0.869309i	$0.664564\pi$
$480$		0			0
$481$	−1.50000	+	2.59808i	−0.0683941	+	0.118462i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.58114	+	1.58114i	0.0717958	+	0.0717958i
$486$		0			0
$487$		−	21.0000i		−	0.951601i	−0.879553	−	0.475800i	$-0.842158\pi$
$487$		−	21.0000i		−	0.951601i	0.879553	−	0.475800i	$-0.157842\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.74342	−	8.21584i	0.214067	−	0.370776i	−0.738916	−	0.673797i	$-0.764662\pi$
$491$	4.74342	−	8.21584i	0.214067	−	0.370776i	0.952984	+	0.303022i	$0.0979955\pi$
$492$		0			0
$493$	−51.9615	+	30.0000i	−2.34023	+	1.35113i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.21584	−	4.74342i	0.368531	−	0.212771i
$498$		0			0
$499$	−12.0000	+	20.7846i	−0.537194	+	0.930447i	0.461860	+	0.886953i	$0.347182\pi$
$499$	−12.0000	+	20.7846i	−0.537194	+	0.930447i	−0.999054	+	0.0434940i	$0.986151\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.48683i		−	0.422997i	−0.977378	−	0.211498i	$-0.932166\pi$
$503$		−	9.48683i		−	0.422997i	0.977378	−	0.211498i	$-0.0678343\pi$
$504$		0			0
$505$	10.0000	−	10.0000i	0.444994	−	0.444994i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.2302	−	24.6475i	0.630745	−	1.09248i	−0.356655	−	0.934236i	$-0.616083\pi$
$509$	14.2302	−	24.6475i	0.630745	−	1.09248i	0.987400	−	0.158246i	$-0.0505838\pi$
$510$		0			0
$511$	−6.50000	−	11.2583i	−0.287543	−	0.498039i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9.83853	−	36.7179i	0.433537	−	1.61798i
$516$		0			0
$517$	−17.3205	−	10.0000i	−0.761755	−	0.439799i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−22.1359			−0.969793			−0.484897	−	0.874571i	$-0.661143\pi$
$521$	−22.1359			−0.969793			−0.484897	+	0.874571i	$0.661143\pi$
$522$		0			0
$523$		−	7.00000i		−	0.306089i	−0.988219	−	0.153044i	$-0.951092\pi$
$523$		−	7.00000i		−	0.306089i	0.988219	−	0.153044i	$-0.0489077\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.9545	−	6.32456i	−0.477183	−	0.275502i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	8.21584	−	4.74342i	0.355867	−	0.205460i
$534$		0			0
$535$	−20.4904	+	5.49038i	−0.885876	+	0.237370i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.9737			0.817254
$540$		0			0
$541$	19.0000			0.816874			0.408437	−	0.912787i	$-0.366074\pi$
$541$	19.0000			0.816874			0.408437	+	0.912787i	$0.366074\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.25979	−	34.5580i	−0.396646	−	1.48030i
$546$		0			0
$547$	9.52628	−	5.50000i	0.407314	−	0.235163i	−0.282321	−	0.959320i	$-0.591104\pi$
$547$	9.52628	−	5.50000i	0.407314	−	0.235163i	0.689635	+	0.724157i	$0.257771\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.2302	+	24.6475i	0.606229	+	1.05002i
$552$		0			0
$553$	2.59808	+	1.50000i	0.110481	+	0.0637865i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	18.9737i		−	0.803940i	−0.915653	−	0.401970i	$-0.868326\pi$
$557$		−	18.9737i		−	0.803940i	0.915653	−	0.401970i	$-0.131674\pi$
$558$		0			0
$559$	−30.0000			−1.26886
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.9089	+	12.6491i	0.923350	+	0.533096i	0.884702	−	0.466156i	$-0.154362\pi$
$563$	21.9089	+	12.6491i	0.923350	+	0.533096i	0.0386478	+	0.999253i	$0.487695\pi$
$564$		0			0
$565$	5.49038	−	20.4904i	0.230982	−	0.862037i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.16228	−	5.47723i	−0.132570	−	0.229617i	0.792097	−	0.610395i	$-0.208989\pi$
$569$	−3.16228	−	5.47723i	−0.132570	−	0.229617i	−0.924666	+	0.380778i	$0.875656\pi$
$570$		0			0
$571$	5.50000	−	9.52628i	0.230168	−	0.398662i	−0.727690	−	0.685907i	$-0.759406\pi$
$571$	5.50000	−	9.52628i	0.230168	−	0.398662i	0.957857	+	0.287244i	$0.0927391\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−15.8114			−0.659380
$576$		0			0
$577$			41.0000i			1.70685i	0.521214	+	0.853426i	$0.325479\pi$
$577$			41.0000i			1.70685i	−0.521214	+	0.853426i	$0.674521\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.90569	+	13.6931i	−0.327983	+	0.568084i
$582$		0			0
$583$	−25.9808	+	15.0000i	−1.07601	+	0.621237i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.21584	−	4.74342i	0.339104	−	0.195782i	−0.320772	−	0.947157i	$-0.603942\pi$
$587$	8.21584	−	4.74342i	0.339104	−	0.195782i	0.659876	+	0.751375i	$0.270609\pi$
$588$		0			0
$589$	−3.00000	+	5.19615i	−0.123613	+	0.214104i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			28.4605i			1.16873i	0.811490	+	0.584366i	$0.198657\pi$
$593$			28.4605i			1.16873i	−0.811490	+	0.584366i	$0.801343\pi$
$594$		0			0
$595$	−10.0000	+	10.0000i	−0.409960	+	0.409960i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−14.2302	+	24.6475i	−0.581432	+	1.00707i	0.413878	+	0.910333i	$0.364174\pi$
$599$	−14.2302	+	24.6475i	−0.581432	+	1.00707i	−0.995310	+	0.0967377i	$0.969159\pi$
$600$		0			0
$601$	−4.00000	−	6.92820i	−0.163163	−	0.282607i	0.772838	−	0.634603i	$-0.218836\pi$
$601$	−4.00000	−	6.92820i	−0.163163	−	0.282607i	−0.936002	+	0.351996i	$0.885503\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.15988	+	0.578737i	0.0878114	+	0.0235290i
$606$		0			0
$607$	9.52628	+	5.50000i	0.386660	+	0.223238i	0.680712	−	0.732551i	$-0.261671\pi$
$607$	9.52628	+	5.50000i	0.386660	+	0.223238i	−0.294052	+	0.955789i	$0.595004\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.9737			−0.767592
$612$		0			0
$613$		−	23.0000i		−	0.928961i	−0.885583	−	0.464481i	$-0.846241\pi$
$613$		−	23.0000i		−	0.928961i	0.885583	−	0.464481i	$-0.153759\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.73861	−	1.58114i	−0.110252	−	0.0636543i	0.443860	−	0.896096i	$-0.353609\pi$
$617$	−2.73861	−	1.58114i	−0.110252	−	0.0636543i	−0.554112	+	0.832442i	$0.686942\pi$
$618$		0			0
$619$	8.50000	+	14.7224i	0.341644	+	0.591744i	0.984738	−	0.174042i	$-0.0556830\pi$
$619$	8.50000	+	14.7224i	0.341644	+	0.591744i	−0.643094	+	0.765787i	$0.722350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.9545	−	6.32456i	0.438881	−	0.253388i
$624$		0			0
$625$	12.5000	−	21.6506i	0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.32456			−0.252177
$630$		0			0
$631$	−1.00000			−0.0398094			−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000			−0.0398094			−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	21.5988	−	5.78737i	0.857121	−	0.229665i
$636$		0			0
$637$	15.5885	−	9.00000i	0.617637	−	0.356593i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.32456	−	10.9545i	−0.249805	−	0.432675i	0.713667	−	0.700485i	$-0.247033\pi$
$641$	−6.32456	−	10.9545i	−0.249805	−	0.432675i	−0.963472	+	0.267811i	$0.913700\pi$
$642$		0			0
$643$	25.9808	+	15.0000i	1.02458	+	0.591542i	0.915428	−	0.402483i	$-0.131853\pi$
$643$	25.9808	+	15.0000i	1.02458	+	0.591542i	0.109154	+	0.994025i	$0.465186\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	28.4605i		−	1.11890i	−0.828865	−	0.559449i	$-0.811013\pi$
$647$		−	28.4605i		−	1.11890i	0.828865	−	0.559449i	$-0.188987\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.9545	+	6.32456i	0.428681	+	0.247499i	0.698784	−	0.715332i	$-0.253725\pi$
$653$	10.9545	+	6.32456i	0.428681	+	0.247499i	−0.270104	+	0.962831i	$0.587058\pi$
$654$		0			0
$655$	40.9808	+	10.9808i	1.60125	+	0.429054i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.32456	−	10.9545i	−0.246370	−	0.426725i	0.716146	−	0.697950i	$-0.245904\pi$
$659$	−6.32456	−	10.9545i	−0.246370	−	0.426725i	−0.962516	+	0.271226i	$0.912571\pi$
$660$		0			0
$661$	−5.50000	+	9.52628i	−0.213925	+	0.370529i	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−5.50000	+	9.52628i	−0.213925	+	0.370529i	0.739014	+	0.673690i	$0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.74342	+	4.74342i	0.183942	+	0.183942i
$666$		0			0
$667$		−	30.0000i		−	1.16160i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.58114	−	2.73861i	0.0610392	−	0.105723i
$672$		0			0
$673$	2.59808	−	1.50000i	0.100148	−	0.0578208i	−0.449089	−	0.893487i	$-0.648251\pi$
$673$	2.59808	−	1.50000i	0.100148	−	0.0578208i	0.549238	+	0.835666i	$0.314918\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	27.3861	−	15.8114i	1.05253	−	0.607681i	0.129177	−	0.991622i	$-0.458767\pi$
$677$	27.3861	−	15.8114i	1.05253	−	0.607681i	0.923358	+	0.383941i	$0.125433\pi$
$678$		0			0
$679$	−0.500000	+	0.866025i	−0.0191882	+	0.0332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	22.1359i		−	0.847008i	−0.905894	−	0.423504i	$-0.860800\pi$
$683$		−	22.1359i		−	0.847008i	0.905894	−	0.423504i	$-0.139200\pi$
$684$		0			0
$685$	15.0000	+	15.0000i	0.573121	+	0.573121i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.2302	+	24.6475i	−0.542129	+	0.938996i
$690$		0			0
$691$	1.00000	+	1.73205i	0.0380418	+	0.0658903i	0.884419	−	0.466693i	$-0.154555\pi$
$691$	1.00000	+	1.73205i	0.0380418	+	0.0658903i	−0.846378	+	0.532583i	$0.821221\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−36.7179	−	9.83853i	−1.39279	−	0.373197i
$696$		0			0
$697$	17.3205	+	10.0000i	0.656061	+	0.378777i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.16228			0.119438			0.0597188	−	0.998215i	$-0.480980\pi$
$701$	3.16228			0.119438			0.0597188	+	0.998215i	$0.480980\pi$
$702$		0			0
$703$			3.00000i			0.113147i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.47723	+	3.16228i	0.205992	+	0.118930i
$708$		0			0
$709$	1.50000	+	2.59808i	0.0563337	+	0.0975728i	0.892817	−	0.450420i	$-0.148726\pi$
$709$	1.50000	+	2.59808i	0.0563337	+	0.0975728i	−0.836483	+	0.547992i	$0.815392\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.47723	−	3.16228i	0.205124	−	0.118428i
$714$		0			0
$715$	−20.4904	+	5.49038i	−0.766297	+	0.205329i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−41.1096			−1.53313			−0.766565	−	0.642167i	$-0.778036\pi$
$719$	−41.1096			−1.53313			−0.766565	+	0.642167i	$0.778036\pi$
$720$		0			0
$721$	17.0000			0.633113
$722$		0			0
$723$		0			0
$724$		0			0
$725$	41.0792	+	23.7171i	1.52564	+	0.880830i
$726$		0			0
$727$	25.9808	−	15.0000i	0.963573	−	0.556319i	0.0663022	−	0.997800i	$-0.478880\pi$
$727$	25.9808	−	15.0000i	0.963573	−	0.556319i	0.897271	+	0.441480i	$0.145547\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−31.6228	−	54.7723i	−1.16961	−	2.02583i
$732$		0			0
$733$	34.6410	+	20.0000i	1.27950	+	0.738717i	0.976756	−	0.214356i	$-0.0687654\pi$
$733$	34.6410	+	20.0000i	1.27950	+	0.738717i	0.302740	+	0.953073i	$0.402099\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	34.7851i		−	1.28132i
$738$		0			0
$739$	26.0000			0.956425			0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000			0.956425			0.478213	+	0.878244i	$0.341285\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.4317	−	9.48683i	−0.602820	−	0.348038i	0.167330	−	0.985901i	$-0.446485\pi$
$743$	−16.4317	−	9.48683i	−0.602820	−	0.348038i	−0.770150	+	0.637863i	$0.779819\pi$
$744$		0			0
$745$	−27.3205	−	7.32051i	−1.00095	−	0.268203i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.74342	−	8.21584i	−0.173321	−	0.300200i
$750$		0			0
$751$	24.5000	−	42.4352i	0.894018	−	1.54848i	0.0590021	−	0.998258i	$-0.481208\pi$
$751$	24.5000	−	42.4352i	0.894018	−	1.54848i	0.835016	−	0.550226i	$-0.185459\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.58114	−	1.58114i	0.0575435	−	0.0575435i
$756$		0			0
$757$			29.0000i			1.05402i	0.849858	+	0.527011i	$0.176688\pi$
$757$			29.0000i			1.05402i	−0.849858	+	0.527011i	$0.823312\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.74342	−	8.21584i	0.171949	−	0.297824i	−0.767152	−	0.641465i	$-0.778327\pi$
$761$	4.74342	−	8.21584i	0.171949	−	0.297824i	0.939101	+	0.343641i	$0.111660\pi$
$762$		0			0
$763$	13.8564	−	8.00000i	0.501636	−	0.289619i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−16.4317	+	9.48683i	−0.593313	+	0.342550i
$768$		0			0
$769$	−1.50000	+	2.59808i	−0.0540914	+	0.0936890i	−0.891803	−	0.452423i	$-0.850560\pi$
$769$	−1.50000	+	2.59808i	−0.0540914	+	0.0936890i	0.837712	+	0.546113i	$0.183893\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	47.4342i		−	1.70609i	−0.521839	−	0.853044i	$-0.674754\pi$
$773$		−	47.4342i		−	1.70609i	0.521839	−	0.853044i	$-0.325246\pi$
$774$		0			0
$775$			10.0000i			0.359211i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.74342	−	8.21584i	0.169951	−	0.294363i
$780$		0			0
$781$	15.0000	+	25.9808i	0.536742	+	0.929665i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.78737	−	21.5988i	0.206560	−	0.770893i
$786$		0			0
$787$	0.866025	+	0.500000i	0.0308705	+	0.0178231i	0.515356	−	0.856976i	$-0.327660\pi$
$787$	0.866025	+	0.500000i	0.0308705	+	0.0178231i	−0.484485	+	0.874799i	$0.660993\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.48683			0.337313
$792$		0			0
$793$		−	3.00000i		−	0.106533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.47723	+	3.16228i	0.194013	+	0.112014i	0.593860	−	0.804568i	$-0.297603\pi$
$797$	5.47723	+	3.16228i	0.194013	+	0.112014i	−0.399847	+	0.916582i	$0.630937\pi$
$798$		0			0
$799$	−20.0000	−	34.6410i	−0.707549	−	1.22551i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	35.6020	−	20.5548i	1.25637	−	0.725363i
$804$		0			0
$805$	−1.83013	−	6.83013i	−0.0645035	−	0.240730i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.9737			0.667079			0.333539	−	0.942736i	$-0.391757\pi$
$809$	18.9737			0.667079			0.333539	+	0.942736i	$0.391757\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−36.7179	+	9.83853i	−1.28617	+	0.344629i
$816$		0			0
$817$	−25.9808	+	15.0000i	−0.908952	+	0.524784i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.1359	−	38.3406i	−0.772550	−	1.33810i	−0.936161	−	0.351571i	$-0.885648\pi$
$821$	−22.1359	−	38.3406i	−0.772550	−	1.33810i	0.163612	−	0.986525i	$-0.447686\pi$
$822$		0			0
$823$	2.59808	+	1.50000i	0.0905632	+	0.0522867i	0.544598	−	0.838697i	$-0.316682\pi$
$823$	2.59808	+	1.50000i	0.0905632	+	0.0522867i	−0.454034	+	0.890984i	$0.650016\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	44.2719i		−	1.53948i	−0.638355	−	0.769742i	$-0.720385\pi$
$827$		−	44.2719i		−	1.53948i	0.638355	−	0.769742i	$-0.279615\pi$
$828$		0			0
$829$	−3.00000			−0.104194			−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000			−0.104194			−0.0520972	+	0.998642i	$0.516591\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	32.8634	+	18.9737i	1.13865	+	0.657399i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.8114	−	27.3861i	−0.545870	−	0.945474i	−0.998552	−	0.0538020i	$-0.982866\pi$
$839$	−15.8114	−	27.3861i	−0.545870	−	0.945474i	0.452682	−	0.891672i	$-0.350467\pi$
$840$		0			0
$841$	−30.5000	+	52.8275i	−1.05172	+	1.82164i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.32456	−	6.32456i	0.217571	−	0.217571i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.58114	−	2.73861i	0.0542007	−	0.0938784i
$852$		0			0
$853$	−6.06218	+	3.50000i	−0.207565	+	0.119838i	−0.600179	−	0.799866i	$-0.704904\pi$
$853$	−6.06218	+	3.50000i	−0.207565	+	0.119838i	0.392614	+	0.919703i	$0.371571\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.21584	+	4.74342i	−0.280648	+	0.162032i	−0.633717	−	0.773565i	$-0.718471\pi$
$857$	−8.21584	+	4.74342i	−0.280648	+	0.162032i	0.353069	+	0.935597i	$0.385138\pi$
$858$		0			0
$859$	−6.50000	+	11.2583i	−0.221777	+	0.384129i	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−6.50000	+	11.2583i	−0.221777	+	0.384129i	0.733571	+	0.679613i	$0.237852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−10.0000	−	10.0000i	−0.340010	−	0.340010i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.74342	+	8.21584i	−0.160909	+	0.278703i
$870$		0			0
$871$	−16.5000	−	28.5788i	−0.559081	−	0.968357i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.7994	+	2.89368i	0.365086	+	0.0978244i
$876$		0			0
$877$	−42.4352	−	24.5000i	−1.43294	−	0.827306i	−0.435593	−	0.900144i	$-0.643461\pi$
$877$	−42.4352	−	24.5000i	−1.43294	−	0.827306i	−0.997344	+	0.0728377i	$0.976794\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.48683			−0.319620			−0.159810	−	0.987148i	$-0.551088\pi$
$881$	−9.48683			−0.319620			−0.159810	+	0.987148i	$0.551088\pi$
$882$		0			0
$883$		−	27.0000i		−	0.908622i	−0.890843	−	0.454311i	$-0.849885\pi$
$883$		−	27.0000i		−	0.908622i	0.890843	−	0.454311i	$-0.150115\pi$
$884$		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
$889$	5.00000	+	8.66025i	0.167695	+	0.290456i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.4317	+	9.48683i	−0.549865	+	0.317465i
$894$		0			0
$895$	1.83013	+	6.83013i	0.0611744	+	0.228306i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−18.9737			−0.632807
$900$		0			0
$901$	−60.0000			−1.99889
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.36611	+	23.7586i	0.211617	+	0.789764i
$906$		0			0
$907$	−18.1865	+	10.5000i	−0.603874	+	0.348647i	−0.770564	−	0.637363i	$-0.780025\pi$
$907$	−18.1865	+	10.5000i	−0.603874	+	0.348647i	0.166690	+	0.986009i	$0.446692\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.48683	+	16.4317i	0.314313	+	0.544406i	0.979291	−	0.202457i	$-0.0648926\pi$
$911$	9.48683	+	16.4317i	0.314313	+	0.544406i	−0.664978	+	0.746863i	$0.731559\pi$
$912$		0			0
$913$	−43.3013	−	25.0000i	−1.43306	−	0.827379i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			18.9737i			0.626566i
$918$		0			0
$919$	6.00000			0.197922			0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000			0.197922			0.0989609	+	0.995091i	$0.468448\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	24.6475	+	14.2302i	0.811283	+	0.468394i
$924$		0			0
$925$	2.50000	+	4.33013i	0.0821995	+	0.142374i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.74342	−	8.21584i	−0.155626	−	0.269553i	0.777661	−	0.628684i	$-0.216406\pi$
$929$	−4.74342	−	8.21584i	−0.155626	−	0.269553i	−0.933287	+	0.359132i	$0.883073\pi$
$930$		0			0
$931$	9.00000	−	15.5885i	0.294963	−	0.510891i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−31.6228	−	31.6228i	−1.03418	−	1.03418i
$936$		0			0
$937$		−	11.0000i		−	0.359354i	−0.983726	−	0.179677i	$-0.942495\pi$
$937$		−	11.0000i		−	0.359354i	0.983726	−	0.179677i	$-0.0575053\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	7.90569	−	13.6931i	0.257718	−	0.446381i	−0.707912	−	0.706301i	$-0.750363\pi$
$941$	7.90569	−	13.6931i	0.257718	−	0.446381i	0.965630	+	0.259919i	$0.0836960\pi$
$942$		0			0
$943$	−8.66025	+	5.00000i	−0.282017	+	0.162822i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	49.2950	−	28.4605i	1.60187	−	0.924842i	0.610761	−	0.791815i	$-0.290864\pi$
$947$	49.2950	−	28.4605i	1.60187	−	0.924842i	0.991112	−	0.133027i	$-0.0424696\pi$
$948$		0			0
$949$	19.5000	−	33.7750i	0.632997	−	1.09638i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	25.2982i		−	0.819490i	−0.912200	−	0.409745i	$-0.865618\pi$
$953$		−	25.2982i		−	0.819490i	0.912200	−	0.409745i	$-0.134382\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.74342	+	8.21584i	−0.153173	+	0.265303i
$960$		0			0
$961$	13.5000	+	23.3827i	0.435484	+	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.05116	−	15.1191i	0.130411	−	0.486702i
$966$		0			0
$967$	9.52628	+	5.50000i	0.306344	+	0.176868i	0.645290	−	0.763938i	$-0.276737\pi$
$967$	9.52628	+	5.50000i	0.306344	+	0.176868i	−0.338945	+	0.940806i	$0.610070\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.9737			−0.608894			−0.304447	−	0.952529i	$-0.598472\pi$
$971$	−18.9737			−0.608894			−0.304447	+	0.952529i	$0.598472\pi$
$972$		0			0
$973$		−	17.0000i		−	0.544995i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.6475	+	14.2302i	0.788544	+	0.455266i	0.839450	−	0.543437i	$-0.182877\pi$
$977$	24.6475	+	14.2302i	0.788544	+	0.455266i	−0.0509058	+	0.998703i	$0.516211\pi$
$978$		0			0
$979$	20.0000	+	34.6410i	0.639203	+	1.10713i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.73861	−	1.58114i	0.0873482	−	0.0504305i	−0.455690	−	0.890139i	$-0.650607\pi$
$983$	2.73861	−	1.58114i	0.0873482	−	0.0504305i	0.543038	+	0.839708i	$0.317274\pi$
$984$		0			0
$985$	27.3205	−	7.32051i	0.870504	−	0.233251i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	31.6228			1.00555
$990$		0			0
$991$	19.0000			0.603555			0.301777	−	0.953378i	$-0.402420\pi$
$991$	19.0000			0.603555			0.301777	+	0.953378i	$0.402420\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−15.6259	−	58.3166i	−0.495374	−	1.84876i
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		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	1620.2.r.g.109.3		8
3.2	odd	2	inner	1620.2.r.g.109.2		8
5.4	even	2	inner	1620.2.r.g.109.1		8
9.2	odd	6	inner	1620.2.r.g.1189.4		8
9.4	even	3		540.2.d.c.109.3	yes	4
9.5	odd	6		540.2.d.c.109.2	yes	4
9.7	even	3	inner	1620.2.r.g.1189.1		8
15.14	odd	2	inner	1620.2.r.g.109.4		8
36.23	even	6		2160.2.f.l.1729.2		4
36.31	odd	6		2160.2.f.l.1729.3		4
45.4	even	6		540.2.d.c.109.4	yes	4
45.13	odd	12		2700.2.a.v.1.2		2
45.14	odd	6		540.2.d.c.109.1	✓	4
45.22	odd	12		2700.2.a.w.1.2		2
45.23	even	12		2700.2.a.v.1.1		2
45.29	odd	6	inner	1620.2.r.g.1189.2		8
45.32	even	12		2700.2.a.w.1.1		2
45.34	even	6	inner	1620.2.r.g.1189.3		8
180.59	even	6		2160.2.f.l.1729.1		4
180.139	odd	6		2160.2.f.l.1729.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.2.d.c.109.1	✓	4	45.14	odd	6
540.2.d.c.109.2	yes	4	9.5	odd	6
540.2.d.c.109.3	yes	4	9.4	even	3
540.2.d.c.109.4	yes	4	45.4	even	6
1620.2.r.g.109.1		8	5.4	even	2	inner
1620.2.r.g.109.2		8	3.2	odd	2	inner
1620.2.r.g.109.3		8	1.1	even	1	trivial
1620.2.r.g.109.4		8	15.14	odd	2	inner
1620.2.r.g.1189.1		8	9.7	even	3	inner
1620.2.r.g.1189.2		8	45.29	odd	6	inner
1620.2.r.g.1189.3		8	45.34	even	6	inner
1620.2.r.g.1189.4		8	9.2	odd	6	inner
2160.2.f.l.1729.1		4	180.59	even	6
2160.2.f.l.1729.2		4	36.23	even	6
2160.2.f.l.1729.3		4	36.31	odd	6
2160.2.f.l.1729.4		4	180.139	odd	6
2700.2.a.v.1.1		2	45.23	even	12
2700.2.a.v.1.2		2	45.13	odd	12
2700.2.a.w.1.1		2	45.32	even	12
2700.2.a.w.1.2		2	45.22	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1620.2.r.g.109.3

Level	$1620$
Weight	$2$
Character	1620.109
Analytic conductor	$12.936$
Analytic rank	$0$
Dimension	$8$
Inner twists	$8$