LMFDB - Embedded newform 8470.2.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.a (trivial)

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:	yes
Analytic conductor:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.73205 q^{12} +1.73205 q^{13} +1.00000 q^{14} -1.73205 q^{15} +1.00000 q^{16} -4.73205 q^{17} -1.73205 q^{19} -1.00000 q^{20} +1.73205 q^{21} -4.46410 q^{23} +1.73205 q^{24} +1.00000 q^{25} +1.73205 q^{26} -5.19615 q^{27} +1.00000 q^{28} -9.46410 q^{29} -1.73205 q^{30} -6.73205 q^{31} +1.00000 q^{32} -4.73205 q^{34} -1.00000 q^{35} -0.535898 q^{37} -1.73205 q^{38} +3.00000 q^{39} -1.00000 q^{40} +3.26795 q^{41} +1.73205 q^{42} -9.66025 q^{43} -4.46410 q^{46} +0.732051 q^{47} +1.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -8.19615 q^{51} +1.73205 q^{52} -4.73205 q^{53} -5.19615 q^{54} +1.00000 q^{56} -3.00000 q^{57} -9.46410 q^{58} -1.00000 q^{59} -1.73205 q^{60} -4.00000 q^{61} -6.73205 q^{62} +1.00000 q^{64} -1.73205 q^{65} +3.80385 q^{67} -4.73205 q^{68} -7.73205 q^{69} -1.00000 q^{70} -10.9282 q^{71} +7.66025 q^{73} -0.535898 q^{74} +1.73205 q^{75} -1.73205 q^{76} +3.00000 q^{78} +13.9282 q^{79} -1.00000 q^{80} -9.00000 q^{81} +3.26795 q^{82} +1.92820 q^{83} +1.73205 q^{84} +4.73205 q^{85} -9.66025 q^{86} -16.3923 q^{87} -1.26795 q^{89} +1.73205 q^{91} -4.46410 q^{92} -11.6603 q^{93} +0.732051 q^{94} +1.73205 q^{95} +1.73205 q^{96} -5.46410 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{14} + 2 q^{16} - 6 q^{17} - 2 q^{20} - 2 q^{23} + 2 q^{25} + 2 q^{28} - 12 q^{29} - 10 q^{31} + 2 q^{32} - 6 q^{34} - 2 q^{35} - 8 q^{37} + 6 q^{39} - 2 q^{40} + 10 q^{41} - 2 q^{43} - 2 q^{46} - 2 q^{47} + 2 q^{49} + 2 q^{50} - 6 q^{51} - 6 q^{53} + 2 q^{56} - 6 q^{57} - 12 q^{58} - 2 q^{59} - 8 q^{61} - 10 q^{62} + 2 q^{64} + 18 q^{67} - 6 q^{68} - 12 q^{69} - 2 q^{70} - 8 q^{71} - 2 q^{73} - 8 q^{74} + 6 q^{78} + 14 q^{79} - 2 q^{80} - 18 q^{81} + 10 q^{82} - 10 q^{83} + 6 q^{85} - 2 q^{86} - 12 q^{87} - 6 q^{89} - 2 q^{92} - 6 q^{93} - 2 q^{94} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 + 2 * q^14 + 2 * q^16 - 6 * q^17 - 2 * q^20 - 2 * q^23 + 2 * q^25 + 2 * q^28 - 12 * q^29 - 10 * q^31 + 2 * q^32 - 6 * q^34 - 2 * q^35 - 8 * q^37 + 6 * q^39 - 2 * q^40 + 10 * q^41 - 2 * q^43 - 2 * q^46 - 2 * q^47 + 2 * q^49 + 2 * q^50 - 6 * q^51 - 6 * q^53 + 2 * q^56 - 6 * q^57 - 12 * q^58 - 2 * q^59 - 8 * q^61 - 10 * q^62 + 2 * q^64 + 18 * q^67 - 6 * q^68 - 12 * q^69 - 2 * q^70 - 8 * q^71 - 2 * q^73 - 8 * q^74 + 6 * q^78 + 14 * q^79 - 2 * q^80 - 18 * q^81 + 10 * q^82 - 10 * q^83 + 6 * q^85 - 2 * q^86 - 12 * q^87 - 6 * q^89 - 2 * q^92 - 6 * q^93 - 2 * q^94 - 4 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.73205	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$3$	1.73205	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.73205	0.707107
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.73205	0.500000
$13$	1.73205	0.480384	0.240192	−	0.970725i	$-0.422790\pi$
$13$	1.73205	0.480384	0.240192	+	0.970725i	$0.422790\pi$
$14$	1.00000	0.267261
$15$	−1.73205	−0.447214
$16$	1.00000	0.250000
$17$	−4.73205	−1.14769	−0.573845	−	0.818964i	$-0.694549\pi$
$17$	−4.73205	−1.14769	−0.573845	+	0.818964i	$0.694549\pi$
$18$	0	0
$19$	−1.73205	−0.397360	−0.198680	−	0.980064i	$-0.563665\pi$
$19$	−1.73205	−0.397360	−0.198680	+	0.980064i	$0.563665\pi$
$20$	−1.00000	−0.223607
$21$	1.73205	0.377964
$22$	0	0
$23$	−4.46410	−0.930830	−0.465415	−	0.885093i	$-0.654095\pi$
$23$	−4.46410	−0.930830	−0.465415	+	0.885093i	$0.654095\pi$
$24$	1.73205	0.353553
$25$	1.00000	0.200000
$26$	1.73205	0.339683
$27$	−5.19615	−1.00000
$28$	1.00000	0.188982
$29$	−9.46410	−1.75744	−0.878720	−	0.477338i	$-0.841602\pi$
$29$	−9.46410	−1.75744	−0.878720	+	0.477338i	$0.841602\pi$
$30$	−1.73205	−0.316228
$31$	−6.73205	−1.20911	−0.604556	−	0.796563i	$-0.706649\pi$
$31$	−6.73205	−1.20911	−0.604556	+	0.796563i	$0.706649\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.73205	−0.811540
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−0.535898	−0.0881012	−0.0440506	−	0.999029i	$-0.514026\pi$
$37$	−0.535898	−0.0881012	−0.0440506	+	0.999029i	$0.514026\pi$
$38$	−1.73205	−0.280976
$39$	3.00000	0.480384
$40$	−1.00000	−0.158114
$41$	3.26795	0.510368	0.255184	−	0.966893i	$-0.417864\pi$
$41$	3.26795	0.510368	0.255184	+	0.966893i	$0.417864\pi$
$42$	1.73205	0.267261
$43$	−9.66025	−1.47317	−0.736587	−	0.676342i	$-0.763564\pi$
$43$	−9.66025	−1.47317	−0.736587	+	0.676342i	$0.763564\pi$
$44$	0	0
$45$	0	0
$46$	−4.46410	−0.658196
$47$	0.732051	0.106781	0.0533903	−	0.998574i	$-0.482997\pi$
$47$	0.732051	0.106781	0.0533903	+	0.998574i	$0.482997\pi$
$48$	1.73205	0.250000
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−8.19615	−1.14769
$52$	1.73205	0.240192
$53$	−4.73205	−0.649997	−0.324999	−	0.945715i	$-0.605364\pi$
$53$	−4.73205	−0.649997	−0.324999	+	0.945715i	$0.605364\pi$
$54$	−5.19615	−0.707107
$55$	0	0
$56$	1.00000	0.133631
$57$	−3.00000	−0.397360
$58$	−9.46410	−1.24270
$59$	−1.00000	−0.130189	−0.0650945	−	0.997879i	$-0.520735\pi$
$59$	−1.00000	−0.130189	−0.0650945	+	0.997879i	$0.520735\pi$
$60$	−1.73205	−0.223607
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−6.73205	−0.854971
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.73205	−0.214834
$66$	0	0
$67$	3.80385	0.464714	0.232357	−	0.972631i	$-0.425356\pi$
$67$	3.80385	0.464714	0.232357	+	0.972631i	$0.425356\pi$
$68$	−4.73205	−0.573845
$69$	−7.73205	−0.930830
$70$	−1.00000	−0.119523
$71$	−10.9282	−1.29694	−0.648470	−	0.761241i	$-0.724591\pi$
$71$	−10.9282	−1.29694	−0.648470	+	0.761241i	$0.724591\pi$
$72$	0	0
$73$	7.66025	0.896565	0.448282	−	0.893892i	$-0.352036\pi$
$73$	7.66025	0.896565	0.448282	+	0.893892i	$0.352036\pi$
$74$	−0.535898	−0.0622969
$75$	1.73205	0.200000
$76$	−1.73205	−0.198680
$77$	0	0
$78$	3.00000	0.339683
$79$	13.9282	1.56705	0.783523	−	0.621363i	$-0.213421\pi$
$79$	13.9282	1.56705	0.783523	+	0.621363i	$0.213421\pi$
$80$	−1.00000	−0.111803
$81$	−9.00000	−1.00000
$82$	3.26795	0.360885
$83$	1.92820	0.211648	0.105824	−	0.994385i	$-0.466252\pi$
$83$	1.92820	0.211648	0.105824	+	0.994385i	$0.466252\pi$
$84$	1.73205	0.188982
$85$	4.73205	0.513263
$86$	−9.66025	−1.04169
$87$	−16.3923	−1.75744
$88$	0	0
$89$	−1.26795	−0.134402	−0.0672012	−	0.997739i	$-0.521407\pi$
$89$	−1.26795	−0.134402	−0.0672012	+	0.997739i	$0.521407\pi$
$90$	0	0
$91$	1.73205	0.181568
$92$	−4.46410	−0.465415
$93$	−11.6603	−1.20911
$94$	0.732051	0.0755053
$95$	1.73205	0.177705
$96$	1.73205	0.176777
$97$	−5.46410	−0.554795	−0.277398	−	0.960755i	$-0.589472\pi$
$97$	−5.46410	−0.554795	−0.277398	+	0.960755i	$0.589472\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	5.92820	0.589878	0.294939	−	0.955516i	$-0.404701\pi$
$101$	5.92820	0.589878	0.294939	+	0.955516i	$0.404701\pi$
$102$	−8.19615	−0.811540
$103$	12.1962	1.20172	0.600861	−	0.799353i	$-0.294824\pi$
$103$	12.1962	1.20172	0.600861	+	0.799353i	$0.294824\pi$
$104$	1.73205	0.169842
$105$	−1.73205	−0.169031
$106$	−4.73205	−0.459617
$107$	−7.46410	−0.721582	−0.360791	−	0.932647i	$-0.617493\pi$
$107$	−7.46410	−0.721582	−0.360791	+	0.932647i	$0.617493\pi$
$108$	−5.19615	−0.500000
$109$	10.1962	0.976614	0.488307	−	0.872672i	$-0.337615\pi$
$109$	10.1962	0.976614	0.488307	+	0.872672i	$0.337615\pi$
$110$	0	0
$111$	−0.928203	−0.0881012
$112$	1.00000	0.0944911
$113$	12.6603	1.19098	0.595488	−	0.803364i	$-0.296959\pi$
$113$	12.6603	1.19098	0.595488	+	0.803364i	$0.296959\pi$
$114$	−3.00000	−0.280976
$115$	4.46410	0.416280
$116$	−9.46410	−0.878720
$117$	0	0
$118$	−1.00000	−0.0920575
$119$	−4.73205	−0.433786
$120$	−1.73205	−0.158114
$121$	0	0
$122$	−4.00000	−0.362143
$123$	5.66025	0.510368
$124$	−6.73205	−0.604556
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.80385	−0.248801	−0.124401	−	0.992232i	$-0.539701\pi$
$127$	−2.80385	−0.248801	−0.124401	+	0.992232i	$0.539701\pi$
$128$	1.00000	0.0883883
$129$	−16.7321	−1.47317
$130$	−1.73205	−0.151911
$131$	−7.73205	−0.675552	−0.337776	−	0.941226i	$-0.609675\pi$
$131$	−7.73205	−0.675552	−0.337776	+	0.941226i	$0.609675\pi$
$132$	0	0
$133$	−1.73205	−0.150188
$134$	3.80385	0.328602
$135$	5.19615	0.447214
$136$	−4.73205	−0.405770
$137$	−12.1244	−1.03585	−0.517927	−	0.855425i	$-0.673296\pi$
$137$	−12.1244	−1.03585	−0.517927	+	0.855425i	$0.673296\pi$
$138$	−7.73205	−0.658196
$139$	−18.6603	−1.58274	−0.791371	−	0.611336i	$-0.790632\pi$
$139$	−18.6603	−1.58274	−0.791371	+	0.611336i	$0.790632\pi$
$140$	−1.00000	−0.0845154
$141$	1.26795	0.106781
$142$	−10.9282	−0.917074
$143$	0	0
$144$	0	0
$145$	9.46410	0.785951
$146$	7.66025	0.633967
$147$	1.73205	0.142857
$148$	−0.535898	−0.0440506
$149$	12.9282	1.05912	0.529560	−	0.848273i	$-0.322357\pi$
$149$	12.9282	1.05912	0.529560	+	0.848273i	$0.322357\pi$
$150$	1.73205	0.141421
$151$	13.0000	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$152$	−1.73205	−0.140488
$153$	0	0
$154$	0	0
$155$	6.73205	0.540731
$156$	3.00000	0.240192
$157$	−3.53590	−0.282195	−0.141098	−	0.989996i	$-0.545063\pi$
$157$	−3.53590	−0.282195	−0.141098	+	0.989996i	$0.545063\pi$
$158$	13.9282	1.10807
$159$	−8.19615	−0.649997
$160$	−1.00000	−0.0790569
$161$	−4.46410	−0.351820
$162$	−9.00000	−0.707107
$163$	−4.92820	−0.386007	−0.193003	−	0.981198i	$-0.561823\pi$
$163$	−4.92820	−0.386007	−0.193003	+	0.981198i	$0.561823\pi$
$164$	3.26795	0.255184
$165$	0	0
$166$	1.92820	0.149658
$167$	−3.26795	−0.252882	−0.126441	−	0.991974i	$-0.540355\pi$
$167$	−3.26795	−0.252882	−0.126441	+	0.991974i	$0.540355\pi$
$168$	1.73205	0.133631
$169$	−10.0000	−0.769231
$170$	4.73205	0.362932
$171$	0	0
$172$	−9.66025	−0.736587
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	−16.3923	−1.24270
$175$	1.00000	0.0755929
$176$	0	0
$177$	−1.73205	−0.130189
$178$	−1.26795	−0.0950368
$179$	15.1244	1.13045	0.565224	−	0.824938i	$-0.308790\pi$
$179$	15.1244	1.13045	0.565224	+	0.824938i	$0.308790\pi$
$180$	0	0
$181$	20.1244	1.49583	0.747916	−	0.663794i	$-0.231055\pi$
$181$	20.1244	1.49583	0.747916	+	0.663794i	$0.231055\pi$
$182$	1.73205	0.128388
$183$	−6.92820	−0.512148
$184$	−4.46410	−0.329098
$185$	0.535898	0.0394000
$186$	−11.6603	−0.854971
$187$	0	0
$188$	0.732051	0.0533903
$189$	−5.19615	−0.377964
$190$	1.73205	0.125656
$191$	−21.7321	−1.57248	−0.786238	−	0.617924i	$-0.787974\pi$
$191$	−21.7321	−1.57248	−0.786238	+	0.617924i	$0.787974\pi$
$192$	1.73205	0.125000
$193$	19.7846	1.42413	0.712064	−	0.702115i	$-0.247761\pi$
$193$	19.7846	1.42413	0.712064	+	0.702115i	$0.247761\pi$
$194$	−5.46410	−0.392300
$195$	−3.00000	−0.214834
$196$	1.00000	0.0714286
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	−12.3923	−0.878467	−0.439234	−	0.898373i	$-0.644750\pi$
$199$	−12.3923	−0.878467	−0.439234	+	0.898373i	$0.644750\pi$
$200$	1.00000	0.0707107
$201$	6.58846	0.464714
$202$	5.92820	0.417107
$203$	−9.46410	−0.664250
$204$	−8.19615	−0.573845
$205$	−3.26795	−0.228243
$206$	12.1962	0.849746
$207$	0	0
$208$	1.73205	0.120096
$209$	0	0
$210$	−1.73205	−0.119523
$211$	5.46410	0.376164	0.188082	−	0.982153i	$-0.439773\pi$
$211$	5.46410	0.376164	0.188082	+	0.982153i	$0.439773\pi$
$212$	−4.73205	−0.324999
$213$	−18.9282	−1.29694
$214$	−7.46410	−0.510235
$215$	9.66025	0.658824
$216$	−5.19615	−0.353553
$217$	−6.73205	−0.457001
$218$	10.1962	0.690571
$219$	13.2679	0.896565
$220$	0	0
$221$	−8.19615	−0.551333
$222$	−0.928203	−0.0622969
$223$	−13.1244	−0.878872	−0.439436	−	0.898274i	$-0.644822\pi$
$223$	−13.1244	−0.878872	−0.439436	+	0.898274i	$0.644822\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	12.6603	0.842148
$227$	−0.535898	−0.0355688	−0.0177844	−	0.999842i	$-0.505661\pi$
$227$	−0.535898	−0.0355688	−0.0177844	+	0.999842i	$0.505661\pi$
$228$	−3.00000	−0.198680
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	4.46410	0.294354
$231$	0	0
$232$	−9.46410	−0.621349
$233$	4.85641	0.318154	0.159077	−	0.987266i	$-0.449148\pi$
$233$	4.85641	0.318154	0.159077	+	0.987266i	$0.449148\pi$
$234$	0	0
$235$	−0.732051	−0.0477537
$236$	−1.00000	−0.0650945
$237$	24.1244	1.56705
$238$	−4.73205	−0.306733
$239$	4.46410	0.288759	0.144379	−	0.989522i	$-0.453881\pi$
$239$	4.46410	0.288759	0.144379	+	0.989522i	$0.453881\pi$
$240$	−1.73205	−0.111803
$241$	2.92820	0.188622	0.0943111	−	0.995543i	$-0.469935\pi$
$241$	2.92820	0.188622	0.0943111	+	0.995543i	$0.469935\pi$
$242$	0	0
$243$	0	0
$244$	−4.00000	−0.256074
$245$	−1.00000	−0.0638877
$246$	5.66025	0.360885
$247$	−3.00000	−0.190885
$248$	−6.73205	−0.427486
$249$	3.33975	0.211648
$250$	−1.00000	−0.0632456
$251$	10.3923	0.655956	0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923	0.655956	0.327978	+	0.944685i	$0.393633\pi$
$252$	0	0
$253$	0	0
$254$	−2.80385	−0.175929
$255$	8.19615	0.513263
$256$	1.00000	0.0625000
$257$	−22.7321	−1.41799	−0.708993	−	0.705215i	$-0.750850\pi$
$257$	−22.7321	−1.41799	−0.708993	+	0.705215i	$0.750850\pi$
$258$	−16.7321	−1.04169
$259$	−0.535898	−0.0332991
$260$	−1.73205	−0.107417
$261$	0	0
$262$	−7.73205	−0.477688
$263$	−14.2679	−0.879799	−0.439900	−	0.898047i	$-0.644986\pi$
$263$	−14.2679	−0.879799	−0.439900	+	0.898047i	$0.644986\pi$
$264$	0	0
$265$	4.73205	0.290688
$266$	−1.73205	−0.106199
$267$	−2.19615	−0.134402
$268$	3.80385	0.232357
$269$	−28.5167	−1.73869	−0.869346	−	0.494204i	$-0.835459\pi$
$269$	−28.5167	−1.73869	−0.869346	+	0.494204i	$0.835459\pi$
$270$	5.19615	0.316228
$271$	−11.6077	−0.705117	−0.352559	−	0.935790i	$-0.614688\pi$
$271$	−11.6077	−0.705117	−0.352559	+	0.935790i	$0.614688\pi$
$272$	−4.73205	−0.286923
$273$	3.00000	0.181568
$274$	−12.1244	−0.732459
$275$	0	0
$276$	−7.73205	−0.465415
$277$	−26.5885	−1.59755	−0.798773	−	0.601633i	$-0.794517\pi$
$277$	−26.5885	−1.59755	−0.798773	+	0.601633i	$0.794517\pi$
$278$	−18.6603	−1.11917
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	4.26795	0.254605	0.127302	−	0.991864i	$-0.459368\pi$
$281$	4.26795	0.254605	0.127302	+	0.991864i	$0.459368\pi$
$282$	1.26795	0.0755053
$283$	23.7846	1.41385	0.706924	−	0.707289i	$-0.250082\pi$
$283$	23.7846	1.41385	0.706924	+	0.707289i	$0.250082\pi$
$284$	−10.9282	−0.648470
$285$	3.00000	0.177705
$286$	0	0
$287$	3.26795	0.192901
$288$	0	0
$289$	5.39230	0.317194
$290$	9.46410	0.555751
$291$	−9.46410	−0.554795
$292$	7.66025	0.448282
$293$	−22.6603	−1.32383	−0.661913	−	0.749581i	$-0.730255\pi$
$293$	−22.6603	−1.32383	−0.661913	+	0.749581i	$0.730255\pi$
$294$	1.73205	0.101015
$295$	1.00000	0.0582223
$296$	−0.535898	−0.0311485
$297$	0	0
$298$	12.9282	0.748911
$299$	−7.73205	−0.447156
$300$	1.73205	0.100000
$301$	−9.66025	−0.556808
$302$	13.0000	0.748066
$303$	10.2679	0.589878
$304$	−1.73205	−0.0993399
$305$	4.00000	0.229039
$306$	0	0
$307$	16.9282	0.966144	0.483072	−	0.875581i	$-0.339521\pi$
$307$	16.9282	0.966144	0.483072	+	0.875581i	$0.339521\pi$
$308$	0	0
$309$	21.1244	1.20172
$310$	6.73205	0.382355
$311$	−21.8038	−1.23638	−0.618191	−	0.786028i	$-0.712134\pi$
$311$	−21.8038	−1.23638	−0.618191	+	0.786028i	$0.712134\pi$
$312$	3.00000	0.169842
$313$	−23.4641	−1.32627	−0.663135	−	0.748500i	$-0.730774\pi$
$313$	−23.4641	−1.32627	−0.663135	+	0.748500i	$0.730774\pi$
$314$	−3.53590	−0.199542
$315$	0	0
$316$	13.9282	0.783523
$317$	4.33975	0.243744	0.121872	−	0.992546i	$-0.461110\pi$
$317$	4.33975	0.243744	0.121872	+	0.992546i	$0.461110\pi$
$318$	−8.19615	−0.459617
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−12.9282	−0.721582
$322$	−4.46410	−0.248775
$323$	8.19615	0.456046
$324$	−9.00000	−0.500000
$325$	1.73205	0.0960769
$326$	−4.92820	−0.272948
$327$	17.6603	0.976614
$328$	3.26795	0.180442
$329$	0.732051	0.0403593
$330$	0	0
$331$	−12.1962	−0.670361	−0.335181	−	0.942154i	$-0.608797\pi$
$331$	−12.1962	−0.670361	−0.335181	+	0.942154i	$0.608797\pi$
$332$	1.92820	0.105824
$333$	0	0
$334$	−3.26795	−0.178814
$335$	−3.80385	−0.207826
$336$	1.73205	0.0944911
$337$	−25.7846	−1.40458	−0.702289	−	0.711892i	$-0.747838\pi$
$337$	−25.7846	−1.40458	−0.702289	+	0.711892i	$0.747838\pi$
$338$	−10.0000	−0.543928
$339$	21.9282	1.19098
$340$	4.73205	0.256631
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.66025	−0.520846
$345$	7.73205	0.416280
$346$	10.0000	0.537603
$347$	15.4641	0.830156	0.415078	−	0.909786i	$-0.363754\pi$
$347$	15.4641	0.830156	0.415078	+	0.909786i	$0.363754\pi$
$348$	−16.3923	−0.878720
$349$	3.00000	0.160586	0.0802932	−	0.996771i	$-0.474414\pi$
$349$	3.00000	0.160586	0.0802932	+	0.996771i	$0.474414\pi$
$350$	1.00000	0.0534522
$351$	−9.00000	−0.480384
$352$	0	0
$353$	−9.66025	−0.514163	−0.257082	−	0.966390i	$-0.582761\pi$
$353$	−9.66025	−0.514163	−0.257082	+	0.966390i	$0.582761\pi$
$354$	−1.73205	−0.0920575
$355$	10.9282	0.580009
$356$	−1.26795	−0.0672012
$357$	−8.19615	−0.433786
$358$	15.1244	0.799347
$359$	14.0000	0.738892	0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000	0.738892	0.369446	+	0.929252i	$0.379548\pi$
$360$	0	0
$361$	−16.0000	−0.842105
$362$	20.1244	1.05771
$363$	0	0
$364$	1.73205	0.0907841
$365$	−7.66025	−0.400956
$366$	−6.92820	−0.362143
$367$	−21.6603	−1.13066	−0.565328	−	0.824866i	$-0.691250\pi$
$367$	−21.6603	−1.13066	−0.565328	+	0.824866i	$0.691250\pi$
$368$	−4.46410	−0.232707
$369$	0	0
$370$	0.535898	0.0278600
$371$	−4.73205	−0.245676
$372$	−11.6603	−0.604556
$373$	−0.928203	−0.0480605	−0.0240303	−	0.999711i	$-0.507650\pi$
$373$	−0.928203	−0.0480605	−0.0240303	+	0.999711i	$0.507650\pi$
$374$	0	0
$375$	−1.73205	−0.0894427
$376$	0.732051	0.0377526
$377$	−16.3923	−0.844247
$378$	−5.19615	−0.267261
$379$	14.3923	0.739283	0.369642	−	0.929174i	$-0.379480\pi$
$379$	14.3923	0.739283	0.369642	+	0.929174i	$0.379480\pi$
$380$	1.73205	0.0888523
$381$	−4.85641	−0.248801
$382$	−21.7321	−1.11191
$383$	4.58846	0.234459	0.117230	−	0.993105i	$-0.462599\pi$
$383$	4.58846	0.234459	0.117230	+	0.993105i	$0.462599\pi$
$384$	1.73205	0.0883883
$385$	0	0
$386$	19.7846	1.00701
$387$	0	0
$388$	−5.46410	−0.277398
$389$	−35.1244	−1.78088	−0.890438	−	0.455105i	$-0.849602\pi$
$389$	−35.1244	−1.78088	−0.890438	+	0.455105i	$0.849602\pi$
$390$	−3.00000	−0.151911
$391$	21.1244	1.06830
$392$	1.00000	0.0505076
$393$	−13.3923	−0.675552
$394$	24.0000	1.20910
$395$	−13.9282	−0.700804
$396$	0	0
$397$	17.8564	0.896187	0.448094	−	0.893987i	$-0.352103\pi$
$397$	17.8564	0.896187	0.448094	+	0.893987i	$0.352103\pi$
$398$	−12.3923	−0.621170
$399$	−3.00000	−0.150188
$400$	1.00000	0.0500000
$401$	18.2487	0.911297	0.455649	−	0.890160i	$-0.349407\pi$
$401$	18.2487	0.911297	0.455649	+	0.890160i	$0.349407\pi$
$402$	6.58846	0.328602
$403$	−11.6603	−0.580839
$404$	5.92820	0.294939
$405$	9.00000	0.447214
$406$	−9.46410	−0.469695
$407$	0	0
$408$	−8.19615	−0.405770
$409$	37.1769	1.83828	0.919140	−	0.393931i	$-0.128885\pi$
$409$	37.1769	1.83828	0.919140	+	0.393931i	$0.128885\pi$
$410$	−3.26795	−0.161393
$411$	−21.0000	−1.03585
$412$	12.1962	0.600861
$413$	−1.00000	−0.0492068
$414$	0	0
$415$	−1.92820	−0.0946518
$416$	1.73205	0.0849208
$417$	−32.3205	−1.58274
$418$	0	0
$419$	11.3923	0.556551	0.278275	−	0.960501i	$-0.410237\pi$
$419$	11.3923	0.556551	0.278275	+	0.960501i	$0.410237\pi$
$420$	−1.73205	−0.0845154
$421$	38.2487	1.86413	0.932064	−	0.362294i	$-0.118006\pi$
$421$	38.2487	1.86413	0.932064	+	0.362294i	$0.118006\pi$
$422$	5.46410	0.265988
$423$	0	0
$424$	−4.73205	−0.229809
$425$	−4.73205	−0.229538
$426$	−18.9282	−0.917074
$427$	−4.00000	−0.193574
$428$	−7.46410	−0.360791
$429$	0	0
$430$	9.66025	0.465859
$431$	11.5359	0.555665	0.277832	−	0.960630i	$-0.410384\pi$
$431$	11.5359	0.555665	0.277832	+	0.960630i	$0.410384\pi$
$432$	−5.19615	−0.250000
$433$	1.80385	0.0866874	0.0433437	−	0.999060i	$-0.486199\pi$
$433$	1.80385	0.0866874	0.0433437	+	0.999060i	$0.486199\pi$
$434$	−6.73205	−0.323149
$435$	16.3923	0.785951
$436$	10.1962	0.488307
$437$	7.73205	0.369874
$438$	13.2679	0.633967
$439$	24.1962	1.15482	0.577410	−	0.816455i	$-0.304064\pi$
$439$	24.1962	1.15482	0.577410	+	0.816455i	$0.304064\pi$
$440$	0	0
$441$	0	0
$442$	−8.19615	−0.389851
$443$	−5.07180	−0.240968	−0.120484	−	0.992715i	$-0.538445\pi$
$443$	−5.07180	−0.240968	−0.120484	+	0.992715i	$0.538445\pi$
$444$	−0.928203	−0.0440506
$445$	1.26795	0.0601066
$446$	−13.1244	−0.621456
$447$	22.3923	1.05912
$448$	1.00000	0.0472456
$449$	−34.4641	−1.62646	−0.813231	−	0.581941i	$-0.802293\pi$
$449$	−34.4641	−1.62646	−0.813231	+	0.581941i	$0.802293\pi$
$450$	0	0
$451$	0	0
$452$	12.6603	0.595488
$453$	22.5167	1.05792
$454$	−0.535898	−0.0251510
$455$	−1.73205	−0.0811998
$456$	−3.00000	−0.140488
$457$	6.07180	0.284027	0.142013	−	0.989865i	$-0.454642\pi$
$457$	6.07180	0.284027	0.142013	+	0.989865i	$0.454642\pi$
$458$	0	0
$459$	24.5885	1.14769
$460$	4.46410	0.208140
$461$	10.2487	0.477330	0.238665	−	0.971102i	$-0.423290\pi$
$461$	10.2487	0.477330	0.238665	+	0.971102i	$0.423290\pi$
$462$	0	0
$463$	35.0000	1.62659	0.813294	−	0.581853i	$-0.197672\pi$
$463$	35.0000	1.62659	0.813294	+	0.581853i	$0.197672\pi$
$464$	−9.46410	−0.439360
$465$	11.6603	0.540731
$466$	4.85641	0.224969
$467$	−30.1244	−1.39399	−0.696994	−	0.717077i	$-0.745480\pi$
$467$	−30.1244	−1.39399	−0.696994	+	0.717077i	$0.745480\pi$
$468$	0	0
$469$	3.80385	0.175645
$470$	−0.732051	−0.0337670
$471$	−6.12436	−0.282195
$472$	−1.00000	−0.0460287
$473$	0	0
$474$	24.1244	1.10807
$475$	−1.73205	−0.0794719
$476$	−4.73205	−0.216893
$477$	0	0
$478$	4.46410	0.204183
$479$	18.7321	0.855889	0.427945	−	0.903805i	$-0.359238\pi$
$479$	18.7321	0.855889	0.427945	+	0.903805i	$0.359238\pi$
$480$	−1.73205	−0.0790569
$481$	−0.928203	−0.0423224
$482$	2.92820	0.133376
$483$	−7.73205	−0.351820
$484$	0	0
$485$	5.46410	0.248112
$486$	0	0
$487$	−32.7128	−1.48236	−0.741180	−	0.671307i	$-0.765733\pi$
$487$	−32.7128	−1.48236	−0.741180	+	0.671307i	$0.765733\pi$
$488$	−4.00000	−0.181071
$489$	−8.53590	−0.386007
$490$	−1.00000	−0.0451754
$491$	−27.5167	−1.24181	−0.620905	−	0.783886i	$-0.713235\pi$
$491$	−27.5167	−1.24181	−0.620905	+	0.783886i	$0.713235\pi$
$492$	5.66025	0.255184
$493$	44.7846	2.01700
$494$	−3.00000	−0.134976
$495$	0	0
$496$	−6.73205	−0.302278
$497$	−10.9282	−0.490197
$498$	3.33975	0.149658
$499$	22.3923	1.00242	0.501209	−	0.865326i	$-0.332889\pi$
$499$	22.3923	1.00242	0.501209	+	0.865326i	$0.332889\pi$
$500$	−1.00000	−0.0447214
$501$	−5.66025	−0.252882
$502$	10.3923	0.463831
$503$	−22.9808	−1.02466	−0.512331	−	0.858788i	$-0.671218\pi$
$503$	−22.9808	−1.02466	−0.512331	+	0.858788i	$0.671218\pi$
$504$	0	0
$505$	−5.92820	−0.263802
$506$	0	0
$507$	−17.3205	−0.769231
$508$	−2.80385	−0.124401
$509$	−24.8038	−1.09941	−0.549706	−	0.835358i	$-0.685260\pi$
$509$	−24.8038	−1.09941	−0.549706	+	0.835358i	$0.685260\pi$
$510$	8.19615	0.362932
$511$	7.66025	0.338870
$512$	1.00000	0.0441942
$513$	9.00000	0.397360
$514$	−22.7321	−1.00267
$515$	−12.1962	−0.537427
$516$	−16.7321	−0.736587
$517$	0	0
$518$	−0.535898	−0.0235460
$519$	17.3205	0.760286
$520$	−1.73205	−0.0759555
$521$	−21.4641	−0.940359	−0.470180	−	0.882571i	$-0.655811\pi$
$521$	−21.4641	−0.940359	−0.470180	+	0.882571i	$0.655811\pi$
$522$	0	0
$523$	17.2487	0.754233	0.377117	−	0.926166i	$-0.376916\pi$
$523$	17.2487	0.754233	0.377117	+	0.926166i	$0.376916\pi$
$524$	−7.73205	−0.337776
$525$	1.73205	0.0755929
$526$	−14.2679	−0.622112
$527$	31.8564	1.38769
$528$	0	0
$529$	−3.07180	−0.133556
$530$	4.73205	0.205547
$531$	0	0
$532$	−1.73205	−0.0750939
$533$	5.66025	0.245173
$534$	−2.19615	−0.0950368
$535$	7.46410	0.322701
$536$	3.80385	0.164301
$537$	26.1962	1.13045
$538$	−28.5167	−1.22944
$539$	0	0
$540$	5.19615	0.223607
$541$	−24.2487	−1.04253	−0.521267	−	0.853394i	$-0.674540\pi$
$541$	−24.2487	−1.04253	−0.521267	+	0.853394i	$0.674540\pi$
$542$	−11.6077	−0.498593
$543$	34.8564	1.49583
$544$	−4.73205	−0.202885
$545$	−10.1962	−0.436755
$546$	3.00000	0.128388
$547$	4.58846	0.196188	0.0980941	−	0.995177i	$-0.468725\pi$
$547$	4.58846	0.196188	0.0980941	+	0.995177i	$0.468725\pi$
$548$	−12.1244	−0.517927
$549$	0	0
$550$	0	0
$551$	16.3923	0.698336
$552$	−7.73205	−0.329098
$553$	13.9282	0.592287
$554$	−26.5885	−1.12964
$555$	0.928203	0.0394000
$556$	−18.6603	−0.791371
$557$	1.94744	0.0825157	0.0412579	−	0.999149i	$-0.486863\pi$
$557$	1.94744	0.0825157	0.0412579	+	0.999149i	$0.486863\pi$
$558$	0	0
$559$	−16.7321	−0.707690
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	4.26795	0.180033
$563$	−35.5359	−1.49766	−0.748830	−	0.662762i	$-0.769384\pi$
$563$	−35.5359	−1.49766	−0.748830	+	0.662762i	$0.769384\pi$
$564$	1.26795	0.0533903
$565$	−12.6603	−0.532621
$566$	23.7846	0.999742
$567$	−9.00000	−0.377964
$568$	−10.9282	−0.458537
$569$	10.1244	0.424435	0.212218	−	0.977222i	$-0.431931\pi$
$569$	10.1244	0.424435	0.212218	+	0.977222i	$0.431931\pi$
$570$	3.00000	0.125656
$571$	−35.3205	−1.47812	−0.739059	−	0.673641i	$-0.764729\pi$
$571$	−35.3205	−1.47812	−0.739059	+	0.673641i	$0.764729\pi$
$572$	0	0
$573$	−37.6410	−1.57248
$574$	3.26795	0.136402
$575$	−4.46410	−0.186166
$576$	0	0
$577$	29.8038	1.24075	0.620375	−	0.784305i	$-0.286980\pi$
$577$	29.8038	1.24075	0.620375	+	0.784305i	$0.286980\pi$
$578$	5.39230	0.224290
$579$	34.2679	1.42413
$580$	9.46410	0.392975
$581$	1.92820	0.0799953
$582$	−9.46410	−0.392300
$583$	0	0
$584$	7.66025	0.316984
$585$	0	0
$586$	−22.6603	−0.936086
$587$	−11.8756	−0.490160	−0.245080	−	0.969503i	$-0.578814\pi$
$587$	−11.8756	−0.490160	−0.245080	+	0.969503i	$0.578814\pi$
$588$	1.73205	0.0714286
$589$	11.6603	0.480452
$590$	1.00000	0.0411693
$591$	41.5692	1.70993
$592$	−0.535898	−0.0220253
$593$	−10.9282	−0.448768	−0.224384	−	0.974501i	$-0.572037\pi$
$593$	−10.9282	−0.448768	−0.224384	+	0.974501i	$0.572037\pi$
$594$	0	0
$595$	4.73205	0.193995
$596$	12.9282	0.529560
$597$	−21.4641	−0.878467
$598$	−7.73205	−0.316187
$599$	−14.6603	−0.599002	−0.299501	−	0.954096i	$-0.596820\pi$
$599$	−14.6603	−0.599002	−0.299501	+	0.954096i	$0.596820\pi$
$600$	1.73205	0.0707107
$601$	−36.5885	−1.49247	−0.746237	−	0.665680i	$-0.768142\pi$
$601$	−36.5885	−1.49247	−0.746237	+	0.665680i	$0.768142\pi$
$602$	−9.66025	−0.393723
$603$	0	0
$604$	13.0000	0.528962
$605$	0	0
$606$	10.2679	0.417107
$607$	−43.4641	−1.76415	−0.882077	−	0.471106i	$-0.843855\pi$
$607$	−43.4641	−1.76415	−0.882077	+	0.471106i	$0.843855\pi$
$608$	−1.73205	−0.0702439
$609$	−16.3923	−0.664250
$610$	4.00000	0.161955
$611$	1.26795	0.0512957
$612$	0	0
$613$	−34.9808	−1.41286	−0.706430	−	0.707783i	$-0.749695\pi$
$613$	−34.9808	−1.41286	−0.706430	+	0.707783i	$0.749695\pi$
$614$	16.9282	0.683167
$615$	−5.66025	−0.228243
$616$	0	0
$617$	−18.3923	−0.740446	−0.370223	−	0.928943i	$-0.620719\pi$
$617$	−18.3923	−0.740446	−0.370223	+	0.928943i	$0.620719\pi$
$618$	21.1244	0.849746
$619$	−21.3923	−0.859829	−0.429915	−	0.902870i	$-0.641456\pi$
$619$	−21.3923	−0.859829	−0.429915	+	0.902870i	$0.641456\pi$
$620$	6.73205	0.270366
$621$	23.1962	0.930830
$622$	−21.8038	−0.874255
$623$	−1.26795	−0.0507993
$624$	3.00000	0.120096
$625$	1.00000	0.0400000
$626$	−23.4641	−0.937814
$627$	0	0
$628$	−3.53590	−0.141098
$629$	2.53590	0.101113
$630$	0	0
$631$	13.4641	0.535997	0.267999	−	0.963419i	$-0.413638\pi$
$631$	13.4641	0.535997	0.267999	+	0.963419i	$0.413638\pi$
$632$	13.9282	0.554034
$633$	9.46410	0.376164
$634$	4.33975	0.172353
$635$	2.80385	0.111267
$636$	−8.19615	−0.324999
$637$	1.73205	0.0686264
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	14.3205	0.565626	0.282813	−	0.959175i	$-0.408732\pi$
$641$	14.3205	0.565626	0.282813	+	0.959175i	$0.408732\pi$
$642$	−12.9282	−0.510235
$643$	8.24871	0.325297	0.162649	−	0.986684i	$-0.447996\pi$
$643$	8.24871	0.325297	0.162649	+	0.986684i	$0.447996\pi$
$644$	−4.46410	−0.175910
$645$	16.7321	0.658824
$646$	8.19615	0.322473
$647$	30.4449	1.19691	0.598456	−	0.801156i	$-0.295781\pi$
$647$	30.4449	1.19691	0.598456	+	0.801156i	$0.295781\pi$
$648$	−9.00000	−0.353553
$649$	0	0
$650$	1.73205	0.0679366
$651$	−11.6603	−0.457001
$652$	−4.92820	−0.193003
$653$	34.0526	1.33258	0.666290	−	0.745693i	$-0.267881\pi$
$653$	34.0526	1.33258	0.666290	+	0.745693i	$0.267881\pi$
$654$	17.6603	0.690571
$655$	7.73205	0.302116
$656$	3.26795	0.127592
$657$	0	0
$658$	0.732051	0.0285383
$659$	39.1244	1.52407	0.762034	−	0.647537i	$-0.224201\pi$
$659$	39.1244	1.52407	0.762034	+	0.647537i	$0.224201\pi$
$660$	0	0
$661$	25.5885	0.995276	0.497638	−	0.867385i	$-0.334201\pi$
$661$	25.5885	0.995276	0.497638	+	0.867385i	$0.334201\pi$
$662$	−12.1962	−0.474017
$663$	−14.1962	−0.551333
$664$	1.92820	0.0748288
$665$	1.73205	0.0671660
$666$	0	0
$667$	42.2487	1.63588
$668$	−3.26795	−0.126441
$669$	−22.7321	−0.878872
$670$	−3.80385	−0.146955
$671$	0	0
$672$	1.73205	0.0668153
$673$	7.78461	0.300075	0.150037	−	0.988680i	$-0.452061\pi$
$673$	7.78461	0.300075	0.150037	+	0.988680i	$0.452061\pi$
$674$	−25.7846	−0.993186
$675$	−5.19615	−0.200000
$676$	−10.0000	−0.384615
$677$	22.2679	0.855827	0.427913	−	0.903820i	$-0.359249\pi$
$677$	22.2679	0.855827	0.427913	+	0.903820i	$0.359249\pi$
$678$	21.9282	0.842148
$679$	−5.46410	−0.209693
$680$	4.73205	0.181466
$681$	−0.928203	−0.0355688
$682$	0	0
$683$	−16.3923	−0.627234	−0.313617	−	0.949550i	$-0.601541\pi$
$683$	−16.3923	−0.627234	−0.313617	+	0.949550i	$0.601541\pi$
$684$	0	0
$685$	12.1244	0.463248
$686$	1.00000	0.0381802
$687$	0	0
$688$	−9.66025	−0.368294
$689$	−8.19615	−0.312249
$690$	7.73205	0.294354
$691$	50.6410	1.92648	0.963238	−	0.268651i	$-0.0865779\pi$
$691$	50.6410	1.92648	0.963238	+	0.268651i	$0.0865779\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	15.4641	0.587009
$695$	18.6603	0.707824
$696$	−16.3923	−0.621349
$697$	−15.4641	−0.585745
$698$	3.00000	0.113552
$699$	8.41154	0.318154
$700$	1.00000	0.0377964
$701$	38.5885	1.45747	0.728733	−	0.684798i	$-0.240110\pi$
$701$	38.5885	1.45747	0.728733	+	0.684798i	$0.240110\pi$
$702$	−9.00000	−0.339683
$703$	0.928203	0.0350078
$704$	0	0
$705$	−1.26795	−0.0477537
$706$	−9.66025	−0.363568
$707$	5.92820	0.222953
$708$	−1.73205	−0.0650945
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	10.9282	0.410128
$711$	0	0
$712$	−1.26795	−0.0475184
$713$	30.0526	1.12548
$714$	−8.19615	−0.306733
$715$	0	0
$716$	15.1244	0.565224
$717$	7.73205	0.288759
$718$	14.0000	0.522475
$719$	32.0000	1.19340	0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	0	0
$721$	12.1962	0.454208
$722$	−16.0000	−0.595458
$723$	5.07180	0.188622
$724$	20.1244	0.747916
$725$	−9.46410	−0.351488
$726$	0	0
$727$	−3.71281	−0.137701	−0.0688503	−	0.997627i	$-0.521933\pi$
$727$	−3.71281	−0.137701	−0.0688503	+	0.997627i	$0.521933\pi$
$728$	1.73205	0.0641941
$729$	27.0000	1.00000
$730$	−7.66025	−0.283519
$731$	45.7128	1.69075
$732$	−6.92820	−0.256074
$733$	−18.5167	−0.683928	−0.341964	−	0.939713i	$-0.611092\pi$
$733$	−18.5167	−0.683928	−0.341964	+	0.939713i	$0.611092\pi$
$734$	−21.6603	−0.799495
$735$	−1.73205	−0.0638877
$736$	−4.46410	−0.164549
$737$	0	0
$738$	0	0
$739$	−25.6603	−0.943928	−0.471964	−	0.881618i	$-0.656455\pi$
$739$	−25.6603	−0.943928	−0.471964	+	0.881618i	$0.656455\pi$
$740$	0.535898	0.0197000
$741$	−5.19615	−0.190885
$742$	−4.73205	−0.173719
$743$	39.1769	1.43726	0.718631	−	0.695392i	$-0.244769\pi$
$743$	39.1769	1.43726	0.718631	+	0.695392i	$0.244769\pi$
$744$	−11.6603	−0.427486
$745$	−12.9282	−0.473653
$746$	−0.928203	−0.0339839
$747$	0	0
$748$	0	0
$749$	−7.46410	−0.272732
$750$	−1.73205	−0.0632456
$751$	−13.3397	−0.486774	−0.243387	−	0.969929i	$-0.578259\pi$
$751$	−13.3397	−0.486774	−0.243387	+	0.969929i	$0.578259\pi$
$752$	0.732051	0.0266951
$753$	18.0000	0.655956
$754$	−16.3923	−0.596973
$755$	−13.0000	−0.473118
$756$	−5.19615	−0.188982
$757$	24.9808	0.907941	0.453971	−	0.891017i	$-0.350007\pi$
$757$	24.9808	0.907941	0.453971	+	0.891017i	$0.350007\pi$
$758$	14.3923	0.522752
$759$	0	0
$760$	1.73205	0.0628281
$761$	−23.5167	−0.852478	−0.426239	−	0.904611i	$-0.640162\pi$
$761$	−23.5167	−0.852478	−0.426239	+	0.904611i	$0.640162\pi$
$762$	−4.85641	−0.175929
$763$	10.1962	0.369126
$764$	−21.7321	−0.786238
$765$	0	0
$766$	4.58846	0.165788
$767$	−1.73205	−0.0625407
$768$	1.73205	0.0625000
$769$	9.60770	0.346462	0.173231	−	0.984881i	$-0.444579\pi$
$769$	9.60770	0.346462	0.173231	+	0.984881i	$0.444579\pi$
$770$	0	0
$771$	−39.3731	−1.41799
$772$	19.7846	0.712064
$773$	−27.3923	−0.985233	−0.492616	−	0.870247i	$-0.663959\pi$
$773$	−27.3923	−0.985233	−0.492616	+	0.870247i	$0.663959\pi$
$774$	0	0
$775$	−6.73205	−0.241822
$776$	−5.46410	−0.196150
$777$	−0.928203	−0.0332991
$778$	−35.1244	−1.25927
$779$	−5.66025	−0.202800
$780$	−3.00000	−0.107417
$781$	0	0
$782$	21.1244	0.755405
$783$	49.1769	1.75744
$784$	1.00000	0.0357143
$785$	3.53590	0.126202
$786$	−13.3923	−0.477688
$787$	−29.7128	−1.05915	−0.529574	−	0.848264i	$-0.677648\pi$
$787$	−29.7128	−1.05915	−0.529574	+	0.848264i	$0.677648\pi$
$788$	24.0000	0.854965
$789$	−24.7128	−0.879799
$790$	−13.9282	−0.495543
$791$	12.6603	0.450147
$792$	0	0
$793$	−6.92820	−0.246028
$794$	17.8564	0.633700
$795$	8.19615	0.290688
$796$	−12.3923	−0.439234
$797$	−46.3205	−1.64076	−0.820378	−	0.571821i	$-0.806237\pi$
$797$	−46.3205	−1.64076	−0.820378	+	0.571821i	$0.806237\pi$
$798$	−3.00000	−0.106199
$799$	−3.46410	−0.122551
$800$	1.00000	0.0353553
$801$	0	0
$802$	18.2487	0.644384
$803$	0	0
$804$	6.58846	0.232357
$805$	4.46410	0.157339
$806$	−11.6603	−0.410715
$807$	−49.3923	−1.73869
$808$	5.92820	0.208553
$809$	23.8564	0.838747	0.419373	−	0.907814i	$-0.362250\pi$
$809$	23.8564	0.838747	0.419373	+	0.907814i	$0.362250\pi$
$810$	9.00000	0.316228
$811$	27.3205	0.959353	0.479676	−	0.877445i	$-0.340754\pi$
$811$	27.3205	0.959353	0.479676	+	0.877445i	$0.340754\pi$
$812$	−9.46410	−0.332125
$813$	−20.1051	−0.705117
$814$	0	0
$815$	4.92820	0.172627
$816$	−8.19615	−0.286923
$817$	16.7321	0.585380
$818$	37.1769	1.29986
$819$	0	0
$820$	−3.26795	−0.114122
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	−21.0000	−0.732459
$823$	5.71281	0.199136	0.0995681	−	0.995031i	$-0.468254\pi$
$823$	5.71281	0.199136	0.0995681	+	0.995031i	$0.468254\pi$
$824$	12.1962	0.424873
$825$	0	0
$826$	−1.00000	−0.0347945
$827$	−14.9808	−0.520932	−0.260466	−	0.965483i	$-0.583876\pi$
$827$	−14.9808	−0.520932	−0.260466	+	0.965483i	$0.583876\pi$
$828$	0	0
$829$	−34.3731	−1.19383	−0.596913	−	0.802306i	$-0.703606\pi$
$829$	−34.3731	−1.19383	−0.596913	+	0.802306i	$0.703606\pi$
$830$	−1.92820	−0.0669289
$831$	−46.0526	−1.59755
$832$	1.73205	0.0600481
$833$	−4.73205	−0.163956
$834$	−32.3205	−1.11917
$835$	3.26795	0.113092
$836$	0	0
$837$	34.9808	1.20911
$838$	11.3923	0.393541
$839$	−11.6603	−0.402557	−0.201278	−	0.979534i	$-0.564510\pi$
$839$	−11.6603	−0.402557	−0.201278	+	0.979534i	$0.564510\pi$
$840$	−1.73205	−0.0597614
$841$	60.5692	2.08859
$842$	38.2487	1.31814
$843$	7.39230	0.254605
$844$	5.46410	0.188082
$845$	10.0000	0.344010
$846$	0	0
$847$	0	0
$848$	−4.73205	−0.162499
$849$	41.1962	1.41385
$850$	−4.73205	−0.162308
$851$	2.39230	0.0820072
$852$	−18.9282	−0.648470
$853$	42.2679	1.44723	0.723614	−	0.690205i	$-0.242480\pi$
$853$	42.2679	1.44723	0.723614	+	0.690205i	$0.242480\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−7.46410	−0.255118
$857$	7.60770	0.259874	0.129937	−	0.991522i	$-0.458523\pi$
$857$	7.60770	0.259874	0.129937	+	0.991522i	$0.458523\pi$
$858$	0	0
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	9.66025	0.329412
$861$	5.66025	0.192901
$862$	11.5359	0.392914
$863$	30.6410	1.04303	0.521516	−	0.853241i	$-0.325367\pi$
$863$	30.6410	1.04303	0.521516	+	0.853241i	$0.325367\pi$
$864$	−5.19615	−0.176777
$865$	−10.0000	−0.340010
$866$	1.80385	0.0612972
$867$	9.33975	0.317194
$868$	−6.73205	−0.228501
$869$	0	0
$870$	16.3923	0.555751
$871$	6.58846	0.223241
$872$	10.1962	0.345285
$873$	0	0
$874$	7.73205	0.261541
$875$	−1.00000	−0.0338062
$876$	13.2679	0.448282
$877$	−5.12436	−0.173037	−0.0865186	−	0.996250i	$-0.527574\pi$
$877$	−5.12436	−0.173037	−0.0865186	+	0.996250i	$0.527574\pi$
$878$	24.1962	0.816581
$879$	−39.2487	−1.32383
$880$	0	0
$881$	1.32051	0.0444890	0.0222445	−	0.999753i	$-0.492919\pi$
$881$	1.32051	0.0444890	0.0222445	+	0.999753i	$0.492919\pi$
$882$	0	0
$883$	13.6077	0.457935	0.228968	−	0.973434i	$-0.426465\pi$
$883$	13.6077	0.457935	0.228968	+	0.973434i	$0.426465\pi$
$884$	−8.19615	−0.275666
$885$	1.73205	0.0582223
$886$	−5.07180	−0.170390
$887$	38.9808	1.30885	0.654423	−	0.756129i	$-0.272912\pi$
$887$	38.9808	1.30885	0.654423	+	0.756129i	$0.272912\pi$
$888$	−0.928203	−0.0311485
$889$	−2.80385	−0.0940380
$890$	1.26795	0.0425018
$891$	0	0
$892$	−13.1244	−0.439436
$893$	−1.26795	−0.0424303
$894$	22.3923	0.748911
$895$	−15.1244	−0.505551
$896$	1.00000	0.0334077
$897$	−13.3923	−0.447156
$898$	−34.4641	−1.15008
$899$	63.7128	2.12494
$900$	0	0
$901$	22.3923	0.745996
$902$	0	0
$903$	−16.7321	−0.556808
$904$	12.6603	0.421074
$905$	−20.1244	−0.668956
$906$	22.5167	0.748066
$907$	−1.07180	−0.0355884	−0.0177942	−	0.999842i	$-0.505664\pi$
$907$	−1.07180	−0.0355884	−0.0177942	+	0.999842i	$0.505664\pi$
$908$	−0.535898	−0.0177844
$909$	0	0
$910$	−1.73205	−0.0574169
$911$	−46.2679	−1.53293	−0.766463	−	0.642289i	$-0.777985\pi$
$911$	−46.2679	−1.53293	−0.766463	+	0.642289i	$0.777985\pi$
$912$	−3.00000	−0.0993399
$913$	0	0
$914$	6.07180	0.200837
$915$	6.92820	0.229039
$916$	0	0
$917$	−7.73205	−0.255335
$918$	24.5885	0.811540
$919$	49.7128	1.63987	0.819937	−	0.572453i	$-0.194008\pi$
$919$	49.7128	1.63987	0.819937	+	0.572453i	$0.194008\pi$
$920$	4.46410	0.147177
$921$	29.3205	0.966144
$922$	10.2487	0.337523
$923$	−18.9282	−0.623029
$924$	0	0
$925$	−0.535898	−0.0176202
$926$	35.0000	1.15017
$927$	0	0
$928$	−9.46410	−0.310674
$929$	−3.32051	−0.108942	−0.0544712	−	0.998515i	$-0.517347\pi$
$929$	−3.32051	−0.108942	−0.0544712	+	0.998515i	$0.517347\pi$
$930$	11.6603	0.382355
$931$	−1.73205	−0.0567657
$932$	4.85641	0.159077
$933$	−37.7654	−1.23638
$934$	−30.1244	−0.985699
$935$	0	0
$936$	0	0
$937$	−18.6410	−0.608975	−0.304488	−	0.952516i	$-0.598485\pi$
$937$	−18.6410	−0.608975	−0.304488	+	0.952516i	$0.598485\pi$
$938$	3.80385	0.124200
$939$	−40.6410	−1.32627
$940$	−0.732051	−0.0238769
$941$	−57.7128	−1.88138	−0.940692	−	0.339262i	$-0.889823\pi$
$941$	−57.7128	−1.88138	−0.940692	+	0.339262i	$0.889823\pi$
$942$	−6.12436	−0.199542
$943$	−14.5885	−0.475066
$944$	−1.00000	−0.0325472
$945$	5.19615	0.169031
$946$	0	0
$947$	6.73205	0.218762	0.109381	−	0.994000i	$-0.465113\pi$
$947$	6.73205	0.218762	0.109381	+	0.994000i	$0.465113\pi$
$948$	24.1244	0.783523
$949$	13.2679	0.430696
$950$	−1.73205	−0.0561951
$951$	7.51666	0.243744
$952$	−4.73205	−0.153367
$953$	19.9282	0.645538	0.322769	−	0.946478i	$-0.395386\pi$
$953$	19.9282	0.645538	0.322769	+	0.946478i	$0.395386\pi$
$954$	0	0
$955$	21.7321	0.703233
$956$	4.46410	0.144379
$957$	0	0
$958$	18.7321	0.605205
$959$	−12.1244	−0.391516
$960$	−1.73205	−0.0559017
$961$	14.3205	0.461952
$962$	−0.928203	−0.0299265
$963$	0	0
$964$	2.92820	0.0943111
$965$	−19.7846	−0.636889
$966$	−7.73205	−0.248775
$967$	−44.4974	−1.43094	−0.715470	−	0.698643i	$-0.753788\pi$
$967$	−44.4974	−1.43094	−0.715470	+	0.698643i	$0.753788\pi$
$968$	0	0
$969$	14.1962	0.456046
$970$	5.46410	0.175442
$971$	−27.3923	−0.879061	−0.439530	−	0.898228i	$-0.644855\pi$
$971$	−27.3923	−0.879061	−0.439530	+	0.898228i	$0.644855\pi$
$972$	0	0
$973$	−18.6603	−0.598220
$974$	−32.7128	−1.04819
$975$	3.00000	0.0960769
$976$	−4.00000	−0.128037
$977$	4.51666	0.144501	0.0722504	−	0.997387i	$-0.476982\pi$
$977$	4.51666	0.144501	0.0722504	+	0.997387i	$0.476982\pi$
$978$	−8.53590	−0.272948
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−27.5167	−0.878092
$983$	36.9282	1.17783	0.588913	−	0.808196i	$-0.299556\pi$
$983$	36.9282	1.17783	0.588913	+	0.808196i	$0.299556\pi$
$984$	5.66025	0.180442
$985$	−24.0000	−0.764704
$986$	44.7846	1.42623
$987$	1.26795	0.0403593
$988$	−3.00000	−0.0954427
$989$	43.1244	1.37127
$990$	0	0
$991$	−20.5167	−0.651733	−0.325867	−	0.945416i	$-0.605656\pi$
$991$	−20.5167	−0.651733	−0.325867	+	0.945416i	$0.605656\pi$
$992$	−6.73205	−0.213743
$993$	−21.1244	−0.670361
$994$	−10.9282	−0.346622
$995$	12.3923	0.392862
$996$	3.33975	0.105824
$997$	7.19615	0.227904	0.113952	−	0.993486i	$-0.463649\pi$
$997$	7.19615	0.227904	0.113952	+	0.993486i	$0.463649\pi$
$998$	22.3923	0.708816
$999$	2.78461	0.0881012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.ca.1.2	yes	2
11.10	odd	2		8470.2.a.bl.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.bl.1.2	✓	2	11.10	odd	2
8470.2.a.ca.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.73205 q^{12} +1.73205 q^{13} +1.00000 q^{14} -1.73205 q^{15} +1.00000 q^{16} -4.73205 q^{17} -1.73205 q^{19} -1.00000 q^{20} +1.73205 q^{21} -4.46410 q^{23} +1.73205 q^{24} +1.00000 q^{25} +1.73205 q^{26} -5.19615 q^{27} +1.00000 q^{28} -9.46410 q^{29} -1.73205 q^{30} -6.73205 q^{31} +1.00000 q^{32} -4.73205 q^{34} -1.00000 q^{35} -0.535898 q^{37} -1.73205 q^{38} +3.00000 q^{39} -1.00000 q^{40} +3.26795 q^{41} +1.73205 q^{42} -9.66025 q^{43} -4.46410 q^{46} +0.732051 q^{47} +1.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -8.19615 q^{51} +1.73205 q^{52} -4.73205 q^{53} -5.19615 q^{54} +1.00000 q^{56} -3.00000 q^{57} -9.46410 q^{58} -1.00000 q^{59} -1.73205 q^{60} -4.00000 q^{61} -6.73205 q^{62} +1.00000 q^{64} -1.73205 q^{65} +3.80385 q^{67} -4.73205 q^{68} -7.73205 q^{69} -1.00000 q^{70} -10.9282 q^{71} +7.66025 q^{73} -0.535898 q^{74} +1.73205 q^{75} -1.73205 q^{76} +3.00000 q^{78} +13.9282 q^{79} -1.00000 q^{80} -9.00000 q^{81} +3.26795 q^{82} +1.92820 q^{83} +1.73205 q^{84} +4.73205 q^{85} -9.66025 q^{86} -16.3923 q^{87} -1.26795 q^{89} +1.73205 q^{91} -4.46410 q^{92} -11.6603 q^{93} +0.732051 q^{94} +1.73205 q^{95} +1.73205 q^{96} -5.46410 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{14} + 2 q^{16} - 6 q^{17} - 2 q^{20} - 2 q^{23} + 2 q^{25} + 2 q^{28} - 12 q^{29} - 10 q^{31} + 2 q^{32} - 6 q^{34} - 2 q^{35} - 8 q^{37} + 6 q^{39} - 2 q^{40} + 10 q^{41} - 2 q^{43} - 2 q^{46} - 2 q^{47} + 2 q^{49} + 2 q^{50} - 6 q^{51} - 6 q^{53} + 2 q^{56} - 6 q^{57} - 12 q^{58} - 2 q^{59} - 8 q^{61} - 10 q^{62} + 2 q^{64} + 18 q^{67} - 6 q^{68} - 12 q^{69} - 2 q^{70} - 8 q^{71} - 2 q^{73} - 8 q^{74} + 6 q^{78} + 14 q^{79} - 2 q^{80} - 18 q^{81} + 10 q^{82} - 10 q^{83} + 6 q^{85} - 2 q^{86} - 12 q^{87} - 6 q^{89} - 2 q^{92} - 6 q^{93} - 2 q^{94} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 + 2 * q^14 + 2 * q^16 - 6 * q^17 - 2 * q^20 - 2 * q^23 + 2 * q^25 + 2 * q^28 - 12 * q^29 - 10 * q^31 + 2 * q^32 - 6 * q^34 - 2 * q^35 - 8 * q^37 + 6 * q^39 - 2 * q^40 + 10 * q^41 - 2 * q^43 - 2 * q^46 - 2 * q^47 + 2 * q^49 + 2 * q^50 - 6 * q^51 - 6 * q^53 + 2 * q^56 - 6 * q^57 - 12 * q^58 - 2 * q^59 - 8 * q^61 - 10 * q^62 + 2 * q^64 + 18 * q^67 - 6 * q^68 - 12 * q^69 - 2 * q^70 - 8 * q^71 - 2 * q^73 - 8 * q^74 + 6 * q^78 + 14 * q^79 - 2 * q^80 - 18 * q^81 + 10 * q^82 - 10 * q^83 + 6 * q^85 - 2 * q^86 - 12 * q^87 - 6 * q^89 - 2 * q^92 - 6 * q^93 - 2 * q^94 - 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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