LMFDB - Embedded newform 8470.2.a.bl.1.1

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			1.1
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.73205 q^{12} +1.73205 q^{13} +1.00000 q^{14} +1.73205 q^{15} +1.00000 q^{16} +1.26795 q^{17} -1.73205 q^{19} -1.00000 q^{20} +1.73205 q^{21} +2.46410 q^{23} +1.73205 q^{24} +1.00000 q^{25} -1.73205 q^{26} +5.19615 q^{27} -1.00000 q^{28} +2.53590 q^{29} -1.73205 q^{30} -3.26795 q^{31} -1.00000 q^{32} -1.26795 q^{34} +1.00000 q^{35} -7.46410 q^{37} +1.73205 q^{38} -3.00000 q^{39} +1.00000 q^{40} -6.73205 q^{41} -1.73205 q^{42} -7.66025 q^{43} -2.46410 q^{46} -2.73205 q^{47} -1.73205 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.19615 q^{51} +1.73205 q^{52} -1.26795 q^{53} -5.19615 q^{54} +1.00000 q^{56} +3.00000 q^{57} -2.53590 q^{58} -1.00000 q^{59} +1.73205 q^{60} +4.00000 q^{61} +3.26795 q^{62} +1.00000 q^{64} -1.73205 q^{65} +14.1962 q^{67} +1.26795 q^{68} -4.26795 q^{69} -1.00000 q^{70} +2.92820 q^{71} +9.66025 q^{73} +7.46410 q^{74} -1.73205 q^{75} -1.73205 q^{76} +3.00000 q^{78} -0.0717968 q^{79} -1.00000 q^{80} -9.00000 q^{81} +6.73205 q^{82} +11.9282 q^{83} +1.73205 q^{84} -1.26795 q^{85} +7.66025 q^{86} -4.39230 q^{87} -4.73205 q^{89} -1.73205 q^{91} +2.46410 q^{92} +5.66025 q^{93} +2.73205 q^{94} +1.73205 q^{95} +1.73205 q^{96} +1.46410 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.666667\pi$
$3$	−1.73205	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$4$	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
$17$	1.26795	0.307523	0.153761	−	0.988108i	$-0.450861\pi$
$17$	1.26795	0.307523	0.153761	+	0.988108i	$0.450861\pi$
$18$	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
$23$	2.46410	0.513801	0.256900	−	0.966438i	$-0.417299\pi$
$23$	2.46410	0.513801	0.256900	+	0.966438i	$0.417299\pi$
$24$	1.73205	0.353553
$25$	1.00000	0.200000
$26$	−1.73205	−0.339683
$27$	5.19615	1.00000
$28$	−1.00000	−0.188982
$29$	2.53590	0.470905	0.235452	−	0.971886i	$-0.424343\pi$
$29$	2.53590	0.470905	0.235452	+	0.971886i	$0.424343\pi$
$30$	−1.73205	−0.316228
$31$	−3.26795	−0.586941	−0.293471	−	0.955968i	$-0.594810\pi$
$31$	−3.26795	−0.586941	−0.293471	+	0.955968i	$0.594810\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.26795	−0.217451
$35$	1.00000	0.169031
$36$	0	0
$37$	−7.46410	−1.22709	−0.613545	−	0.789659i	$-0.710257\pi$
$37$	−7.46410	−1.22709	−0.613545	+	0.789659i	$0.710257\pi$
$38$	1.73205	0.280976
$39$	−3.00000	−0.480384
$40$	1.00000	0.158114
$41$	−6.73205	−1.05137	−0.525685	−	0.850679i	$-0.676191\pi$
$41$	−6.73205	−1.05137	−0.525685	+	0.850679i	$0.676191\pi$
$42$	−1.73205	−0.267261
$43$	−7.66025	−1.16818	−0.584089	−	0.811690i	$-0.698548\pi$
$43$	−7.66025	−1.16818	−0.584089	+	0.811690i	$0.698548\pi$
$44$	0	0
$45$	0	0
$46$	−2.46410	−0.363312
$47$	−2.73205	−0.398511	−0.199255	−	0.979948i	$-0.563852\pi$
$47$	−2.73205	−0.398511	−0.199255	+	0.979948i	$0.563852\pi$
$48$	−1.73205	−0.250000
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.19615	−0.307523
$52$	1.73205	0.240192
$53$	−1.26795	−0.174166	−0.0870831	−	0.996201i	$-0.527755\pi$
$53$	−1.26795	−0.174166	−0.0870831	+	0.996201i	$0.527755\pi$
$54$	−5.19615	−0.707107
$55$	0	0
$56$	1.00000	0.133631
$57$	3.00000	0.397360
$58$	−2.53590	−0.332980
$59$	−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	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	3.26795	0.415030
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.73205	−0.214834
$66$	0	0
$67$	14.1962	1.73434	0.867168	−	0.498016i	$-0.165938\pi$
$67$	14.1962	1.73434	0.867168	+	0.498016i	$0.165938\pi$
$68$	1.26795	0.153761
$69$	−4.26795	−0.513801
$70$	−1.00000	−0.119523
$71$	2.92820	0.347514	0.173757	−	0.984789i	$-0.444409\pi$
$71$	2.92820	0.347514	0.173757	+	0.984789i	$0.444409\pi$
$72$	0	0
$73$	9.66025	1.13065	0.565324	−	0.824869i	$-0.308751\pi$
$73$	9.66025	1.13065	0.565324	+	0.824869i	$0.308751\pi$
$74$	7.46410	0.867684
$75$	−1.73205	−0.200000
$76$	−1.73205	−0.198680
$77$	0	0
$78$	3.00000	0.339683
$79$	−0.0717968	−0.00807777	−0.00403888	−	0.999992i	$-0.501286\pi$
$79$	−0.0717968	−0.00807777	−0.00403888	+	0.999992i	$0.501286\pi$
$80$	−1.00000	−0.111803
$81$	−9.00000	−1.00000
$82$	6.73205	0.743431
$83$	11.9282	1.30929	0.654645	−	0.755936i	$-0.272818\pi$
$83$	11.9282	1.30929	0.654645	+	0.755936i	$0.272818\pi$
$84$	1.73205	0.188982
$85$	−1.26795	−0.137528
$86$	7.66025	0.826026
$87$	−4.39230	−0.470905
$88$	0	0
$89$	−4.73205	−0.501596	−0.250798	−	0.968039i	$-0.580693\pi$
$89$	−4.73205	−0.501596	−0.250798	+	0.968039i	$0.580693\pi$
$90$	0	0
$91$	−1.73205	−0.181568
$92$	2.46410	0.256900
$93$	5.66025	0.586941
$94$	2.73205	0.281790
$95$	1.73205	0.177705
$96$	1.73205	0.176777
$97$	1.46410	0.148657	0.0743285	−	0.997234i	$-0.476319\pi$
$97$	1.46410	0.148657	0.0743285	+	0.997234i	$0.476319\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	7.92820	0.788886	0.394443	−	0.918920i	$-0.370938\pi$
$101$	7.92820	0.788886	0.394443	+	0.918920i	$0.370938\pi$
$102$	2.19615	0.217451
$103$	1.80385	0.177738	0.0888692	−	0.996043i	$-0.471675\pi$
$103$	1.80385	0.177738	0.0888692	+	0.996043i	$0.471675\pi$
$104$	−1.73205	−0.169842
$105$	−1.73205	−0.169031
$106$	1.26795	0.123154
$107$	0.535898	0.0518073	0.0259036	−	0.999664i	$-0.491754\pi$
$107$	0.535898	0.0518073	0.0259036	+	0.999664i	$0.491754\pi$
$108$	5.19615	0.500000
$109$	0.196152	0.0187880	0.00939400	−	0.999956i	$-0.497010\pi$
$109$	0.196152	0.0187880	0.00939400	+	0.999956i	$0.497010\pi$
$110$	0	0
$111$	12.9282	1.22709
$112$	−1.00000	−0.0944911
$113$	−4.66025	−0.438400	−0.219200	−	0.975680i	$-0.570345\pi$
$113$	−4.66025	−0.438400	−0.219200	+	0.975680i	$0.570345\pi$
$114$	−3.00000	−0.280976
$115$	−2.46410	−0.229779
$116$	2.53590	0.235452
$117$	0	0
$118$	1.00000	0.0920575
$119$	−1.26795	−0.116233
$120$	−1.73205	−0.158114
$121$	0	0
$122$	−4.00000	−0.362143
$123$	11.6603	1.05137
$124$	−3.26795	−0.293471
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.1962	1.17097	0.585485	−	0.810684i	$-0.300904\pi$
$127$	13.1962	1.17097	0.585485	+	0.810684i	$0.300904\pi$
$128$	−1.00000	−0.0883883
$129$	13.2679	1.16818
$130$	1.73205	0.151911
$131$	4.26795	0.372892	0.186446	−	0.982465i	$-0.440303\pi$
$131$	4.26795	0.372892	0.186446	+	0.982465i	$0.440303\pi$
$132$	0	0
$133$	1.73205	0.150188
$134$	−14.1962	−1.22636
$135$	−5.19615	−0.447214
$136$	−1.26795	−0.108726
$137$	12.1244	1.03585	0.517927	−	0.855425i	$-0.326704\pi$
$137$	12.1244	1.03585	0.517927	+	0.855425i	$0.326704\pi$
$138$	4.26795	0.363312
$139$	1.33975	0.113636	0.0568179	−	0.998385i	$-0.481905\pi$
$139$	1.33975	0.113636	0.0568179	+	0.998385i	$0.481905\pi$
$140$	1.00000	0.0845154
$141$	4.73205	0.398511
$142$	−2.92820	−0.245729
$143$	0	0
$144$	0	0
$145$	−2.53590	−0.210595
$146$	−9.66025	−0.799488
$147$	−1.73205	−0.142857
$148$	−7.46410	−0.613545
$149$	0.928203	0.0760414	0.0380207	−	0.999277i	$-0.487895\pi$
$149$	0.928203	0.0760414	0.0380207	+	0.999277i	$0.487895\pi$
$150$	1.73205	0.141421
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	1.73205	0.140488
$153$	0	0
$154$	0	0
$155$	3.26795	0.262488
$156$	−3.00000	−0.240192
$157$	−10.4641	−0.835126	−0.417563	−	0.908648i	$-0.637116\pi$
$157$	−10.4641	−0.835126	−0.417563	+	0.908648i	$0.637116\pi$
$158$	0.0717968	0.00571184
$159$	2.19615	0.174166
$160$	1.00000	0.0790569
$161$	−2.46410	−0.194198
$162$	9.00000	0.707107
$163$	8.92820	0.699311	0.349655	−	0.936878i	$-0.386299\pi$
$163$	8.92820	0.699311	0.349655	+	0.936878i	$0.386299\pi$
$164$	−6.73205	−0.525685
$165$	0	0
$166$	−11.9282	−0.925808
$167$	6.73205	0.520942	0.260471	−	0.965482i	$-0.416122\pi$
$167$	6.73205	0.520942	0.260471	+	0.965482i	$0.416122\pi$
$168$	−1.73205	−0.133631
$169$	−10.0000	−0.769231
$170$	1.26795	0.0972473
$171$	0	0
$172$	−7.66025	−0.584089
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	4.39230	0.332980
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	1.73205	0.130189
$178$	4.73205	0.354682
$179$	−9.12436	−0.681986	−0.340993	−	0.940066i	$-0.610763\pi$
$179$	−9.12436	−0.681986	−0.340993	+	0.940066i	$0.610763\pi$
$180$	0	0
$181$	−4.12436	−0.306561	−0.153280	−	0.988183i	$-0.548984\pi$
$181$	−4.12436	−0.306561	−0.153280	+	0.988183i	$0.548984\pi$
$182$	1.73205	0.128388
$183$	−6.92820	−0.512148
$184$	−2.46410	−0.181656
$185$	7.46410	0.548772
$186$	−5.66025	−0.415030
$187$	0	0
$188$	−2.73205	−0.199255
$189$	−5.19615	−0.377964
$190$	−1.73205	−0.125656
$191$	−18.2679	−1.32182	−0.660911	−	0.750464i	$-0.729830\pi$
$191$	−18.2679	−1.32182	−0.660911	+	0.750464i	$0.729830\pi$
$192$	−1.73205	−0.125000
$193$	21.7846	1.56809	0.784045	−	0.620704i	$-0.213153\pi$
$193$	21.7846	1.56809	0.784045	+	0.620704i	$0.213153\pi$
$194$	−1.46410	−0.105116
$195$	3.00000	0.214834
$196$	1.00000	0.0714286
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	8.39230	0.594915	0.297457	−	0.954735i	$-0.403861\pi$
$199$	8.39230	0.594915	0.297457	+	0.954735i	$0.403861\pi$
$200$	−1.00000	−0.0707107
$201$	−24.5885	−1.73434
$202$	−7.92820	−0.557826
$203$	−2.53590	−0.177985
$204$	−2.19615	−0.153761
$205$	6.73205	0.470187
$206$	−1.80385	−0.125680
$207$	0	0
$208$	1.73205	0.120096
$209$	0	0
$210$	1.73205	0.119523
$211$	1.46410	0.100793	0.0503965	−	0.998729i	$-0.483952\pi$
$211$	1.46410	0.100793	0.0503965	+	0.998729i	$0.483952\pi$
$212$	−1.26795	−0.0870831
$213$	−5.07180	−0.347514
$214$	−0.535898	−0.0366333
$215$	7.66025	0.522425
$216$	−5.19615	−0.353553
$217$	3.26795	0.221843
$218$	−0.196152	−0.0132851
$219$	−16.7321	−1.13065
$220$	0	0
$221$	2.19615	0.147729
$222$	−12.9282	−0.867684
$223$	11.1244	0.744942	0.372471	−	0.928044i	$-0.378511\pi$
$223$	11.1244	0.744942	0.372471	+	0.928044i	$0.378511\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	4.66025	0.309995
$227$	7.46410	0.495410	0.247705	−	0.968836i	$-0.420324\pi$
$227$	7.46410	0.495410	0.247705	+	0.968836i	$0.420324\pi$
$228$	3.00000	0.198680
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	2.46410	0.162478
$231$	0	0
$232$	−2.53590	−0.166490
$233$	22.8564	1.49737	0.748686	−	0.662924i	$-0.230685\pi$
$233$	22.8564	1.49737	0.748686	+	0.662924i	$0.230685\pi$
$234$	0	0
$235$	2.73205	0.178219
$236$	−1.00000	−0.0650945
$237$	0.124356	0.00807777
$238$	1.26795	0.0821889
$239$	2.46410	0.159389	0.0796947	−	0.996819i	$-0.474605\pi$
$239$	2.46410	0.159389	0.0796947	+	0.996819i	$0.474605\pi$
$240$	1.73205	0.111803
$241$	10.9282	0.703947	0.351974	−	0.936010i	$-0.385511\pi$
$241$	10.9282	0.703947	0.351974	+	0.936010i	$0.385511\pi$
$242$	0	0
$243$	0	0
$244$	4.00000	0.256074
$245$	−1.00000	−0.0638877
$246$	−11.6603	−0.743431
$247$	−3.00000	−0.190885
$248$	3.26795	0.207515
$249$	−20.6603	−1.30929
$250$	1.00000	0.0632456
$251$	−10.3923	−0.655956	−0.327978	−	0.944685i	$-0.606367\pi$
$251$	−10.3923	−0.655956	−0.327978	+	0.944685i	$0.606367\pi$
$252$	0	0
$253$	0	0
$254$	−13.1962	−0.828000
$255$	2.19615	0.137528
$256$	1.00000	0.0625000
$257$	−19.2679	−1.20190	−0.600951	−	0.799286i	$-0.705211\pi$
$257$	−19.2679	−1.20190	−0.600951	+	0.799286i	$0.705211\pi$
$258$	−13.2679	−0.826026
$259$	7.46410	0.463797
$260$	−1.73205	−0.107417
$261$	0	0
$262$	−4.26795	−0.263675
$263$	17.7321	1.09341	0.546703	−	0.837327i	$-0.315883\pi$
$263$	17.7321	1.09341	0.546703	+	0.837327i	$0.315883\pi$
$264$	0	0
$265$	1.26795	0.0778895
$266$	−1.73205	−0.106199
$267$	8.19615	0.501596
$268$	14.1962	0.867168
$269$	16.5167	1.00704	0.503519	−	0.863984i	$-0.332038\pi$
$269$	16.5167	1.00704	0.503519	+	0.863984i	$0.332038\pi$
$270$	5.19615	0.316228
$271$	32.3923	1.96769	0.983846	−	0.179016i	$-0.0572913\pi$
$271$	32.3923	1.96769	0.983846	+	0.179016i	$0.0572913\pi$
$272$	1.26795	0.0768807
$273$	3.00000	0.181568
$274$	−12.1244	−0.732459
$275$	0	0
$276$	−4.26795	−0.256900
$277$	−4.58846	−0.275694	−0.137847	−	0.990454i	$-0.544018\pi$
$277$	−4.58846	−0.275694	−0.137847	+	0.990454i	$0.544018\pi$
$278$	−1.33975	−0.0803526
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−7.73205	−0.461255	−0.230628	−	0.973042i	$-0.574078\pi$
$281$	−7.73205	−0.461255	−0.230628	+	0.973042i	$0.574078\pi$
$282$	−4.73205	−0.281790
$283$	17.7846	1.05719	0.528593	−	0.848876i	$-0.322720\pi$
$283$	17.7846	1.05719	0.528593	+	0.848876i	$0.322720\pi$
$284$	2.92820	0.173757
$285$	−3.00000	−0.177705
$286$	0	0
$287$	6.73205	0.397380
$288$	0	0
$289$	−15.3923	−0.905430
$290$	2.53590	0.148913
$291$	−2.53590	−0.148657
$292$	9.66025	0.565324
$293$	5.33975	0.311951	0.155976	−	0.987761i	$-0.450148\pi$
$293$	5.33975	0.311951	0.155976	+	0.987761i	$0.450148\pi$
$294$	1.73205	0.101015
$295$	1.00000	0.0582223
$296$	7.46410	0.433842
$297$	0	0
$298$	−0.928203	−0.0537694
$299$	4.26795	0.246822
$300$	−1.73205	−0.100000
$301$	7.66025	0.441530
$302$	13.0000	0.748066
$303$	−13.7321	−0.788886
$304$	−1.73205	−0.0993399
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−3.07180	−0.175317	−0.0876584	−	0.996151i	$-0.527938\pi$
$307$	−3.07180	−0.175317	−0.0876584	+	0.996151i	$0.527938\pi$
$308$	0	0
$309$	−3.12436	−0.177738
$310$	−3.26795	−0.185607
$311$	−32.1962	−1.82568	−0.912838	−	0.408322i	$-0.866114\pi$
$311$	−32.1962	−1.82568	−0.912838	+	0.408322i	$0.866114\pi$
$312$	3.00000	0.169842
$313$	−16.5359	−0.934664	−0.467332	−	0.884082i	$-0.654785\pi$
$313$	−16.5359	−0.934664	−0.467332	+	0.884082i	$0.654785\pi$
$314$	10.4641	0.590523
$315$	0	0
$316$	−0.0717968	−0.00403888
$317$	21.6603	1.21656	0.608281	−	0.793722i	$-0.291860\pi$
$317$	21.6603	1.21656	0.608281	+	0.793722i	$0.291860\pi$
$318$	−2.19615	−0.123154
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−0.928203	−0.0518073
$322$	2.46410	0.137319
$323$	−2.19615	−0.122197
$324$	−9.00000	−0.500000
$325$	1.73205	0.0960769
$326$	−8.92820	−0.494487
$327$	−0.339746	−0.0187880
$328$	6.73205	0.371715
$329$	2.73205	0.150623
$330$	0	0
$331$	−1.80385	−0.0991484	−0.0495742	−	0.998770i	$-0.515786\pi$
$331$	−1.80385	−0.0991484	−0.0495742	+	0.998770i	$0.515786\pi$
$332$	11.9282	0.654645
$333$	0	0
$334$	−6.73205	−0.368361
$335$	−14.1962	−0.775619
$336$	1.73205	0.0944911
$337$	−15.7846	−0.859842	−0.429921	−	0.902866i	$-0.641459\pi$
$337$	−15.7846	−0.859842	−0.429921	+	0.902866i	$0.641459\pi$
$338$	10.0000	0.543928
$339$	8.07180	0.438400
$340$	−1.26795	−0.0687642
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.66025	0.413013
$345$	4.26795	0.229779
$346$	10.0000	0.537603
$347$	−8.53590	−0.458231	−0.229116	−	0.973399i	$-0.573583\pi$
$347$	−8.53590	−0.458231	−0.229116	+	0.973399i	$0.573583\pi$
$348$	−4.39230	−0.235452
$349$	−3.00000	−0.160586	−0.0802932	−	0.996771i	$-0.525586\pi$
$349$	−3.00000	−0.160586	−0.0802932	+	0.996771i	$0.525586\pi$
$350$	1.00000	0.0534522
$351$	9.00000	0.480384
$352$	0	0
$353$	7.66025	0.407714	0.203857	−	0.979001i	$-0.434652\pi$
$353$	7.66025	0.407714	0.203857	+	0.979001i	$0.434652\pi$
$354$	−1.73205	−0.0920575
$355$	−2.92820	−0.155413
$356$	−4.73205	−0.250798
$357$	2.19615	0.116233
$358$	9.12436	0.482237
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	0	0
$361$	−16.0000	−0.842105
$362$	4.12436	0.216771
$363$	0	0
$364$	−1.73205	−0.0907841
$365$	−9.66025	−0.505641
$366$	6.92820	0.362143
$367$	−4.33975	−0.226533	−0.113266	−	0.993565i	$-0.536131\pi$
$367$	−4.33975	−0.226533	−0.113266	+	0.993565i	$0.536131\pi$
$368$	2.46410	0.128450
$369$	0	0
$370$	−7.46410	−0.388040
$371$	1.26795	0.0658286
$372$	5.66025	0.293471
$373$	−12.9282	−0.669397	−0.334698	−	0.942325i	$-0.608634\pi$
$373$	−12.9282	−0.669397	−0.334698	+	0.942325i	$0.608634\pi$
$374$	0	0
$375$	1.73205	0.0894427
$376$	2.73205	0.140895
$377$	4.39230	0.226215
$378$	5.19615	0.267261
$379$	−6.39230	−0.328351	−0.164175	−	0.986431i	$-0.552496\pi$
$379$	−6.39230	−0.328351	−0.164175	+	0.986431i	$0.552496\pi$
$380$	1.73205	0.0888523
$381$	−22.8564	−1.17097
$382$	18.2679	0.934670
$383$	−26.5885	−1.35861	−0.679303	−	0.733858i	$-0.737718\pi$
$383$	−26.5885	−1.35861	−0.679303	+	0.733858i	$0.737718\pi$
$384$	1.73205	0.0883883
$385$	0	0
$386$	−21.7846	−1.10881
$387$	0	0
$388$	1.46410	0.0743285
$389$	−10.8756	−0.551417	−0.275709	−	0.961241i	$-0.588912\pi$
$389$	−10.8756	−0.551417	−0.275709	+	0.961241i	$0.588912\pi$
$390$	−3.00000	−0.151911
$391$	3.12436	0.158005
$392$	−1.00000	−0.0505076
$393$	−7.39230	−0.372892
$394$	24.0000	1.20910
$395$	0.0717968	0.00361249
$396$	0	0
$397$	−9.85641	−0.494679	−0.247339	−	0.968929i	$-0.579556\pi$
$397$	−9.85641	−0.494679	−0.247339	+	0.968929i	$0.579556\pi$
$398$	−8.39230	−0.420668
$399$	−3.00000	−0.150188
$400$	1.00000	0.0500000
$401$	−30.2487	−1.51055	−0.755274	−	0.655409i	$-0.772496\pi$
$401$	−30.2487	−1.51055	−0.755274	+	0.655409i	$0.772496\pi$
$402$	24.5885	1.22636
$403$	−5.66025	−0.281957
$404$	7.92820	0.394443
$405$	9.00000	0.447214
$406$	2.53590	0.125855
$407$	0	0
$408$	2.19615	0.108726
$409$	25.1769	1.24492	0.622459	−	0.782652i	$-0.286134\pi$
$409$	25.1769	1.24492	0.622459	+	0.782652i	$0.286134\pi$
$410$	−6.73205	−0.332472
$411$	−21.0000	−1.03585
$412$	1.80385	0.0888692
$413$	1.00000	0.0492068
$414$	0	0
$415$	−11.9282	−0.585532
$416$	−1.73205	−0.0849208
$417$	−2.32051	−0.113636
$418$	0	0
$419$	−9.39230	−0.458844	−0.229422	−	0.973327i	$-0.573684\pi$
$419$	−9.39230	−0.458844	−0.229422	+	0.973327i	$0.573684\pi$
$420$	−1.73205	−0.0845154
$421$	−10.2487	−0.499492	−0.249746	−	0.968311i	$-0.580347\pi$
$421$	−10.2487	−0.499492	−0.249746	+	0.968311i	$0.580347\pi$
$422$	−1.46410	−0.0712714
$423$	0	0
$424$	1.26795	0.0615771
$425$	1.26795	0.0615046
$426$	5.07180	0.245729
$427$	−4.00000	−0.193574
$428$	0.535898	0.0259036
$429$	0	0
$430$	−7.66025	−0.369410
$431$	−18.4641	−0.889384	−0.444692	−	0.895683i	$-0.646687\pi$
$431$	−18.4641	−0.889384	−0.444692	+	0.895683i	$0.646687\pi$
$432$	5.19615	0.250000
$433$	12.1962	0.586110	0.293055	−	0.956096i	$-0.405328\pi$
$433$	12.1962	0.586110	0.293055	+	0.956096i	$0.405328\pi$
$434$	−3.26795	−0.156867
$435$	4.39230	0.210595
$436$	0.196152	0.00939400
$437$	−4.26795	−0.204164
$438$	16.7321	0.799488
$439$	−13.8038	−0.658822	−0.329411	−	0.944187i	$-0.606850\pi$
$439$	−13.8038	−0.658822	−0.329411	+	0.944187i	$0.606850\pi$
$440$	0	0
$441$	0	0
$442$	−2.19615	−0.104460
$443$	−18.9282	−0.899306	−0.449653	−	0.893203i	$-0.648452\pi$
$443$	−18.9282	−0.899306	−0.449653	+	0.893203i	$0.648452\pi$
$444$	12.9282	0.613545
$445$	4.73205	0.224321
$446$	−11.1244	−0.526754
$447$	−1.60770	−0.0760414
$448$	−1.00000	−0.0472456
$449$	−27.5359	−1.29950	−0.649750	−	0.760148i	$-0.725126\pi$
$449$	−27.5359	−1.29950	−0.649750	+	0.760148i	$0.725126\pi$
$450$	0	0
$451$	0	0
$452$	−4.66025	−0.219200
$453$	22.5167	1.05792
$454$	−7.46410	−0.350308
$455$	1.73205	0.0811998
$456$	−3.00000	−0.140488
$457$	−19.9282	−0.932202	−0.466101	−	0.884732i	$-0.654342\pi$
$457$	−19.9282	−0.932202	−0.466101	+	0.884732i	$0.654342\pi$
$458$	0	0
$459$	6.58846	0.307523
$460$	−2.46410	−0.114889
$461$	38.2487	1.78142	0.890710	−	0.454572i	$-0.150208\pi$
$461$	38.2487	1.78142	0.890710	+	0.454572i	$0.150208\pi$
$462$	0	0
$463$	35.0000	1.62659	0.813294	−	0.581853i	$-0.197672\pi$
$463$	35.0000	1.62659	0.813294	+	0.581853i	$0.197672\pi$
$464$	2.53590	0.117726
$465$	−5.66025	−0.262488
$466$	−22.8564	−1.05880
$467$	−5.87564	−0.271892	−0.135946	−	0.990716i	$-0.543407\pi$
$467$	−5.87564	−0.271892	−0.135946	+	0.990716i	$0.543407\pi$
$468$	0	0
$469$	−14.1962	−0.655517
$470$	−2.73205	−0.126020
$471$	18.1244	0.835126
$472$	1.00000	0.0460287
$473$	0	0
$474$	−0.124356	−0.00571184
$475$	−1.73205	−0.0794719
$476$	−1.26795	−0.0581164
$477$	0	0
$478$	−2.46410	−0.112705
$479$	−15.2679	−0.697610	−0.348805	−	0.937195i	$-0.613413\pi$
$479$	−15.2679	−0.697610	−0.348805	+	0.937195i	$0.613413\pi$
$480$	−1.73205	−0.0790569
$481$	−12.9282	−0.589475
$482$	−10.9282	−0.497766
$483$	4.26795	0.194198
$484$	0	0
$485$	−1.46410	−0.0664814
$486$	0	0
$487$	22.7128	1.02922	0.514608	−	0.857426i	$-0.327938\pi$
$487$	22.7128	1.02922	0.514608	+	0.857426i	$0.327938\pi$
$488$	−4.00000	−0.181071
$489$	−15.4641	−0.699311
$490$	1.00000	0.0451754
$491$	−17.5167	−0.790516	−0.395258	−	0.918570i	$-0.629345\pi$
$491$	−17.5167	−0.790516	−0.395258	+	0.918570i	$0.629345\pi$
$492$	11.6603	0.525685
$493$	3.21539	0.144814
$494$	3.00000	0.134976
$495$	0	0
$496$	−3.26795	−0.146735
$497$	−2.92820	−0.131348
$498$	20.6603	0.925808
$499$	1.60770	0.0719703	0.0359852	−	0.999352i	$-0.488543\pi$
$499$	1.60770	0.0719703	0.0359852	+	0.999352i	$0.488543\pi$
$500$	−1.00000	−0.0447214
$501$	−11.6603	−0.520942
$502$	10.3923	0.463831
$503$	−28.9808	−1.29219	−0.646094	−	0.763258i	$-0.723599\pi$
$503$	−28.9808	−1.29219	−0.646094	+	0.763258i	$0.723599\pi$
$504$	0	0
$505$	−7.92820	−0.352800
$506$	0	0
$507$	17.3205	0.769231
$508$	13.1962	0.585485
$509$	−35.1962	−1.56004	−0.780021	−	0.625753i	$-0.784792\pi$
$509$	−35.1962	−1.56004	−0.780021	+	0.625753i	$0.784792\pi$
$510$	−2.19615	−0.0972473
$511$	−9.66025	−0.427344
$512$	−1.00000	−0.0441942
$513$	−9.00000	−0.397360
$514$	19.2679	0.849873
$515$	−1.80385	−0.0794870
$516$	13.2679	0.584089
$517$	0	0
$518$	−7.46410	−0.327954
$519$	17.3205	0.760286
$520$	1.73205	0.0759555
$521$	−14.5359	−0.636829	−0.318415	−	0.947952i	$-0.603150\pi$
$521$	−14.5359	−0.636829	−0.318415	+	0.947952i	$0.603150\pi$
$522$	0	0
$523$	31.2487	1.36641	0.683205	−	0.730226i	$-0.260585\pi$
$523$	31.2487	1.36641	0.683205	+	0.730226i	$0.260585\pi$
$524$	4.26795	0.186446
$525$	1.73205	0.0755929
$526$	−17.7321	−0.773154
$527$	−4.14359	−0.180498
$528$	0	0
$529$	−16.9282	−0.736009
$530$	−1.26795	−0.0550762
$531$	0	0
$532$	1.73205	0.0750939
$533$	−11.6603	−0.505062
$534$	−8.19615	−0.354682
$535$	−0.535898	−0.0231689
$536$	−14.1962	−0.613180
$537$	15.8038	0.681986
$538$	−16.5167	−0.712084
$539$	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	$0.674540\pi$
$542$	−32.3923	−1.39137
$543$	7.14359	0.306561
$544$	−1.26795	−0.0543629
$545$	−0.196152	−0.00840225
$546$	−3.00000	−0.128388
$547$	26.5885	1.13684	0.568420	−	0.822738i	$-0.307555\pi$
$547$	26.5885	1.13684	0.568420	+	0.822738i	$0.307555\pi$
$548$	12.1244	0.517927
$549$	0	0
$550$	0	0
$551$	−4.39230	−0.187118
$552$	4.26795	0.181656
$553$	0.0717968	0.00305311
$554$	4.58846	0.194945
$555$	−12.9282	−0.548772
$556$	1.33975	0.0568179
$557$	−40.0526	−1.69708	−0.848541	−	0.529130i	$-0.822518\pi$
$557$	−40.0526	−1.69708	−0.848541	+	0.529130i	$0.822518\pi$
$558$	0	0
$559$	−13.2679	−0.561174
$560$	1.00000	0.0422577
$561$	0	0
$562$	7.73205	0.326157
$563$	42.4641	1.78965	0.894824	−	0.446419i	$-0.147301\pi$
$563$	42.4641	1.78965	0.894824	+	0.446419i	$0.147301\pi$
$564$	4.73205	0.199255
$565$	4.66025	0.196058
$566$	−17.7846	−0.747543
$567$	9.00000	0.377964
$568$	−2.92820	−0.122865
$569$	14.1244	0.592124	0.296062	−	0.955169i	$-0.404327\pi$
$569$	14.1244	0.592124	0.296062	+	0.955169i	$0.404327\pi$
$570$	3.00000	0.125656
$571$	0.679492	0.0284359	0.0142179	−	0.999899i	$-0.495474\pi$
$571$	0.679492	0.0284359	0.0142179	+	0.999899i	$0.495474\pi$
$572$	0	0
$573$	31.6410	1.32182
$574$	−6.73205	−0.280990
$575$	2.46410	0.102760
$576$	0	0
$577$	40.1962	1.67339	0.836694	−	0.547671i	$-0.184485\pi$
$577$	40.1962	1.67339	0.836694	+	0.547671i	$0.184485\pi$
$578$	15.3923	0.640235
$579$	−37.7321	−1.56809
$580$	−2.53590	−0.105297
$581$	−11.9282	−0.494865
$582$	2.53590	0.105116
$583$	0	0
$584$	−9.66025	−0.399744
$585$	0	0
$586$	−5.33975	−0.220583
$587$	−36.1244	−1.49101	−0.745506	−	0.666499i	$-0.767792\pi$
$587$	−36.1244	−1.49101	−0.745506	+	0.666499i	$0.767792\pi$
$588$	−1.73205	−0.0714286
$589$	5.66025	0.233227
$590$	−1.00000	−0.0411693
$591$	41.5692	1.70993
$592$	−7.46410	−0.306773
$593$	−2.92820	−0.120247	−0.0601234	−	0.998191i	$-0.519149\pi$
$593$	−2.92820	−0.120247	−0.0601234	+	0.998191i	$0.519149\pi$
$594$	0	0
$595$	1.26795	0.0519808
$596$	0.928203	0.0380207
$597$	−14.5359	−0.594915
$598$	−4.26795	−0.174529
$599$	2.66025	0.108695	0.0543475	−	0.998522i	$-0.482692\pi$
$599$	2.66025	0.108695	0.0543475	+	0.998522i	$0.482692\pi$
$600$	1.73205	0.0707107
$601$	5.41154	0.220741	0.110371	−	0.993890i	$-0.464796\pi$
$601$	5.41154	0.220741	0.110371	+	0.993890i	$0.464796\pi$
$602$	−7.66025	−0.312209
$603$	0	0
$604$	−13.0000	−0.528962
$605$	0	0
$606$	13.7321	0.557826
$607$	36.5359	1.48295	0.741473	−	0.670983i	$-0.234127\pi$
$607$	36.5359	1.48295	0.741473	+	0.670983i	$0.234127\pi$
$608$	1.73205	0.0702439
$609$	4.39230	0.177985
$610$	4.00000	0.161955
$611$	−4.73205	−0.191438
$612$	0	0
$613$	−16.9808	−0.685847	−0.342923	−	0.939363i	$-0.611417\pi$
$613$	−16.9808	−0.685847	−0.342923	+	0.939363i	$0.611417\pi$
$614$	3.07180	0.123968
$615$	−11.6603	−0.470187
$616$	0	0
$617$	2.39230	0.0963106	0.0481553	−	0.998840i	$-0.484666\pi$
$617$	2.39230	0.0963106	0.0481553	+	0.998840i	$0.484666\pi$
$618$	3.12436	0.125680
$619$	−0.607695	−0.0244253	−0.0122127	−	0.999925i	$-0.503888\pi$
$619$	−0.607695	−0.0244253	−0.0122127	+	0.999925i	$0.503888\pi$
$620$	3.26795	0.131244
$621$	12.8038	0.513801
$622$	32.1962	1.29095
$623$	4.73205	0.189586
$624$	−3.00000	−0.120096
$625$	1.00000	0.0400000
$626$	16.5359	0.660907
$627$	0	0
$628$	−10.4641	−0.417563
$629$	−9.46410	−0.377358
$630$	0	0
$631$	6.53590	0.260190	0.130095	−	0.991502i	$-0.458472\pi$
$631$	6.53590	0.260190	0.130095	+	0.991502i	$0.458472\pi$
$632$	0.0717968	0.00285592
$633$	−2.53590	−0.100793
$634$	−21.6603	−0.860239
$635$	−13.1962	−0.523673
$636$	2.19615	0.0870831
$637$	1.73205	0.0686264
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	−20.3205	−0.802612	−0.401306	−	0.915944i	$-0.631444\pi$
$641$	−20.3205	−0.802612	−0.401306	+	0.915944i	$0.631444\pi$
$642$	0.928203	0.0366333
$643$	−40.2487	−1.58725	−0.793627	−	0.608404i	$-0.791810\pi$
$643$	−40.2487	−1.58725	−0.793627	+	0.608404i	$0.791810\pi$
$644$	−2.46410	−0.0970992
$645$	−13.2679	−0.522425
$646$	2.19615	0.0864065
$647$	−28.4449	−1.11828	−0.559141	−	0.829072i	$-0.688869\pi$
$647$	−28.4449	−1.11828	−0.559141	+	0.829072i	$0.688869\pi$
$648$	9.00000	0.353553
$649$	0	0
$650$	−1.73205	−0.0679366
$651$	−5.66025	−0.221843
$652$	8.92820	0.349655
$653$	−4.05256	−0.158589	−0.0792944	−	0.996851i	$-0.525267\pi$
$653$	−4.05256	−0.158589	−0.0792944	+	0.996851i	$0.525267\pi$
$654$	0.339746	0.0132851
$655$	−4.26795	−0.166763
$656$	−6.73205	−0.262842
$657$	0	0
$658$	−2.73205	−0.106506
$659$	−14.8756	−0.579473	−0.289736	−	0.957106i	$-0.593568\pi$
$659$	−14.8756	−0.579473	−0.289736	+	0.957106i	$0.593568\pi$
$660$	0	0
$661$	−5.58846	−0.217366	−0.108683	−	0.994076i	$-0.534663\pi$
$661$	−5.58846	−0.217366	−0.108683	+	0.994076i	$0.534663\pi$
$662$	1.80385	0.0701085
$663$	−3.80385	−0.147729
$664$	−11.9282	−0.462904
$665$	−1.73205	−0.0671660
$666$	0	0
$667$	6.24871	0.241951
$668$	6.73205	0.260471
$669$	−19.2679	−0.744942
$670$	14.1962	0.548445
$671$	0	0
$672$	−1.73205	−0.0668153
$673$	33.7846	1.30230	0.651150	−	0.758949i	$-0.274287\pi$
$673$	33.7846	1.30230	0.651150	+	0.758949i	$0.274287\pi$
$674$	15.7846	0.608000
$675$	5.19615	0.200000
$676$	−10.0000	−0.384615
$677$	−25.7321	−0.988963	−0.494482	−	0.869188i	$-0.664642\pi$
$677$	−25.7321	−0.988963	−0.494482	+	0.869188i	$0.664642\pi$
$678$	−8.07180	−0.309995
$679$	−1.46410	−0.0561871
$680$	1.26795	0.0486236
$681$	−12.9282	−0.495410
$682$	0	0
$683$	4.39230	0.168067	0.0840334	−	0.996463i	$-0.473220\pi$
$683$	4.39230	0.168067	0.0840334	+	0.996463i	$0.473220\pi$
$684$	0	0
$685$	−12.1244	−0.463248
$686$	1.00000	0.0381802
$687$	0	0
$688$	−7.66025	−0.292044
$689$	−2.19615	−0.0836667
$690$	−4.26795	−0.162478
$691$	−18.6410	−0.709138	−0.354569	−	0.935030i	$-0.615372\pi$
$691$	−18.6410	−0.709138	−0.354569	+	0.935030i	$0.615372\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	8.53590	0.324018
$695$	−1.33975	−0.0508195
$696$	4.39230	0.166490
$697$	−8.53590	−0.323320
$698$	3.00000	0.113552
$699$	−39.5885	−1.49737
$700$	−1.00000	−0.0377964
$701$	−7.41154	−0.279930	−0.139965	−	0.990156i	$-0.544699\pi$
$701$	−7.41154	−0.279930	−0.139965	+	0.990156i	$0.544699\pi$
$702$	−9.00000	−0.339683
$703$	12.9282	0.487596
$704$	0	0
$705$	−4.73205	−0.178219
$706$	−7.66025	−0.288297
$707$	−7.92820	−0.298171
$708$	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	$0.439871\pi$
$710$	2.92820	0.109894
$711$	0	0
$712$	4.73205	0.177341
$713$	−8.05256	−0.301571
$714$	−2.19615	−0.0821889
$715$	0	0
$716$	−9.12436	−0.340993
$717$	−4.26795	−0.159389
$718$	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
$721$	−1.80385	−0.0671788
$722$	16.0000	0.595458
$723$	−18.9282	−0.703947
$724$	−4.12436	−0.153280
$725$	2.53590	0.0941809
$726$	0	0
$727$	51.7128	1.91792	0.958961	−	0.283538i	$-0.0915082\pi$
$727$	51.7128	1.91792	0.958961	+	0.283538i	$0.0915082\pi$
$728$	1.73205	0.0641941
$729$	27.0000	1.00000
$730$	9.66025	0.357542
$731$	−9.71281	−0.359241
$732$	−6.92820	−0.256074
$733$	−26.5167	−0.979415	−0.489708	−	0.871887i	$-0.662896\pi$
$733$	−26.5167	−0.979415	−0.489708	+	0.871887i	$0.662896\pi$
$734$	4.33975	0.160183
$735$	1.73205	0.0638877
$736$	−2.46410	−0.0908280
$737$	0	0
$738$	0	0
$739$	8.33975	0.306783	0.153391	−	0.988166i	$-0.450981\pi$
$739$	8.33975	0.306783	0.153391	+	0.988166i	$0.450981\pi$
$740$	7.46410	0.274386
$741$	5.19615	0.190885
$742$	−1.26795	−0.0465479
$743$	23.1769	0.850279	0.425139	−	0.905128i	$-0.360225\pi$
$743$	23.1769	0.850279	0.425139	+	0.905128i	$0.360225\pi$
$744$	−5.66025	−0.207515
$745$	−0.928203	−0.0340067
$746$	12.9282	0.473335
$747$	0	0
$748$	0	0
$749$	−0.535898	−0.0195813
$750$	−1.73205	−0.0632456
$751$	−30.6603	−1.11881	−0.559404	−	0.828895i	$-0.688970\pi$
$751$	−30.6603	−1.11881	−0.559404	+	0.828895i	$0.688970\pi$
$752$	−2.73205	−0.0996276
$753$	18.0000	0.655956
$754$	−4.39230	−0.159958
$755$	13.0000	0.473118
$756$	−5.19615	−0.188982
$757$	−26.9808	−0.980632	−0.490316	−	0.871545i	$-0.663119\pi$
$757$	−26.9808	−0.980632	−0.490316	+	0.871545i	$0.663119\pi$
$758$	6.39230	0.232179
$759$	0	0
$760$	−1.73205	−0.0628281
$761$	−21.5167	−0.779978	−0.389989	−	0.920819i	$-0.627521\pi$
$761$	−21.5167	−0.779978	−0.389989	+	0.920819i	$0.627521\pi$
$762$	22.8564	0.828000
$763$	−0.196152	−0.00710119
$764$	−18.2679	−0.660911
$765$	0	0
$766$	26.5885	0.960680
$767$	−1.73205	−0.0625407
$768$	−1.73205	−0.0625000
$769$	−30.3923	−1.09597	−0.547987	−	0.836487i	$-0.684606\pi$
$769$	−30.3923	−1.09597	−0.547987	+	0.836487i	$0.684606\pi$
$770$	0	0
$771$	33.3731	1.20190
$772$	21.7846	0.784045
$773$	−6.60770	−0.237662	−0.118831	−	0.992914i	$-0.537915\pi$
$773$	−6.60770	−0.237662	−0.118831	+	0.992914i	$0.537915\pi$
$774$	0	0
$775$	−3.26795	−0.117388
$776$	−1.46410	−0.0525582
$777$	−12.9282	−0.463797
$778$	10.8756	0.389911
$779$	11.6603	0.417772
$780$	3.00000	0.107417
$781$	0	0
$782$	−3.12436	−0.111727
$783$	13.1769	0.470905
$784$	1.00000	0.0357143
$785$	10.4641	0.373480
$786$	7.39230	0.263675
$787$	−25.7128	−0.916563	−0.458281	−	0.888807i	$-0.651535\pi$
$787$	−25.7128	−0.916563	−0.458281	+	0.888807i	$0.651535\pi$
$788$	−24.0000	−0.854965
$789$	−30.7128	−1.09341
$790$	−0.0717968	−0.00255441
$791$	4.66025	0.165700
$792$	0	0
$793$	6.92820	0.246028
$794$	9.85641	0.349791
$795$	−2.19615	−0.0778895
$796$	8.39230	0.297457
$797$	−11.6795	−0.413709	−0.206854	−	0.978372i	$-0.566323\pi$
$797$	−11.6795	−0.413709	−0.206854	+	0.978372i	$0.566323\pi$
$798$	3.00000	0.106199
$799$	−3.46410	−0.122551
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	30.2487	1.06812
$803$	0	0
$804$	−24.5885	−0.867168
$805$	2.46410	0.0868482
$806$	5.66025	0.199374
$807$	−28.6077	−1.00704
$808$	−7.92820	−0.278913
$809$	3.85641	0.135584	0.0677920	−	0.997699i	$-0.478405\pi$
$809$	3.85641	0.135584	0.0677920	+	0.997699i	$0.478405\pi$
$810$	−9.00000	−0.316228
$811$	7.32051	0.257058	0.128529	−	0.991706i	$-0.458974\pi$
$811$	7.32051	0.257058	0.128529	+	0.991706i	$0.458974\pi$
$812$	−2.53590	−0.0889926
$813$	−56.1051	−1.96769
$814$	0	0
$815$	−8.92820	−0.312741
$816$	−2.19615	−0.0768807
$817$	13.2679	0.464187
$818$	−25.1769	−0.880290
$819$	0	0
$820$	6.73205	0.235093
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	21.0000	0.732459
$823$	−49.7128	−1.73288	−0.866440	−	0.499281i	$-0.833597\pi$
$823$	−49.7128	−1.73288	−0.866440	+	0.499281i	$0.833597\pi$
$824$	−1.80385	−0.0628400
$825$	0	0
$826$	−1.00000	−0.0347945
$827$	−36.9808	−1.28595	−0.642974	−	0.765888i	$-0.722299\pi$
$827$	−36.9808	−1.28595	−0.642974	+	0.765888i	$0.722299\pi$
$828$	0	0
$829$	38.3731	1.33275	0.666376	−	0.745616i	$-0.267845\pi$
$829$	38.3731	1.33275	0.666376	+	0.745616i	$0.267845\pi$
$830$	11.9282	0.414034
$831$	7.94744	0.275694
$832$	1.73205	0.0600481
$833$	1.26795	0.0439318
$834$	2.32051	0.0803526
$835$	−6.73205	−0.232972
$836$	0	0
$837$	−16.9808	−0.586941
$838$	9.39230	0.324452
$839$	5.66025	0.195414	0.0977068	−	0.995215i	$-0.468849\pi$
$839$	5.66025	0.195414	0.0977068	+	0.995215i	$0.468849\pi$
$840$	1.73205	0.0597614
$841$	−22.5692	−0.778249
$842$	10.2487	0.353194
$843$	13.3923	0.461255
$844$	1.46410	0.0503965
$845$	10.0000	0.344010
$846$	0	0
$847$	0	0
$848$	−1.26795	−0.0435416
$849$	−30.8038	−1.05719
$850$	−1.26795	−0.0434903
$851$	−18.3923	−0.630480
$852$	−5.07180	−0.173757
$853$	−45.7321	−1.56584	−0.782918	−	0.622125i	$-0.786269\pi$
$853$	−45.7321	−1.56584	−0.782918	+	0.622125i	$0.786269\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	−0.535898	−0.0183166
$857$	−28.3923	−0.969863	−0.484931	−	0.874552i	$-0.661155\pi$
$857$	−28.3923	−0.969863	−0.484931	+	0.874552i	$0.661155\pi$
$858$	0	0
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	7.66025	0.261212
$861$	−11.6603	−0.397380
$862$	18.4641	0.628890
$863$	−38.6410	−1.31536	−0.657678	−	0.753299i	$-0.728461\pi$
$863$	−38.6410	−1.31536	−0.657678	+	0.753299i	$0.728461\pi$
$864$	−5.19615	−0.176777
$865$	10.0000	0.340010
$866$	−12.1962	−0.414442
$867$	26.6603	0.905430
$868$	3.26795	0.110921
$869$	0	0
$870$	−4.39230	−0.148913
$871$	24.5885	0.833148
$872$	−0.196152	−0.00664256
$873$	0	0
$874$	4.26795	0.144366
$875$	1.00000	0.0338062
$876$	−16.7321	−0.565324
$877$	−19.1244	−0.645784	−0.322892	−	0.946436i	$-0.604655\pi$
$877$	−19.1244	−0.645784	−0.322892	+	0.946436i	$0.604655\pi$
$878$	13.8038	0.465857
$879$	−9.24871	−0.311951
$880$	0	0
$881$	−33.3205	−1.12260	−0.561298	−	0.827614i	$-0.689698\pi$
$881$	−33.3205	−1.12260	−0.561298	+	0.827614i	$0.689698\pi$
$882$	0	0
$883$	34.3923	1.15739	0.578697	−	0.815543i	$-0.303562\pi$
$883$	34.3923	1.15739	0.578697	+	0.815543i	$0.303562\pi$
$884$	2.19615	0.0738646
$885$	−1.73205	−0.0582223
$886$	18.9282	0.635905
$887$	12.9808	0.435851	0.217926	−	0.975965i	$-0.430071\pi$
$887$	12.9808	0.435851	0.217926	+	0.975965i	$0.430071\pi$
$888$	−12.9282	−0.433842
$889$	−13.1962	−0.442585
$890$	−4.73205	−0.158619
$891$	0	0
$892$	11.1244	0.372471
$893$	4.73205	0.158352
$894$	1.60770	0.0537694
$895$	9.12436	0.304994
$896$	1.00000	0.0334077
$897$	−7.39230	−0.246822
$898$	27.5359	0.918885
$899$	−8.28719	−0.276393
$900$	0	0
$901$	−1.60770	−0.0535601
$902$	0	0
$903$	−13.2679	−0.441530
$904$	4.66025	0.154998
$905$	4.12436	0.137098
$906$	−22.5167	−0.748066
$907$	−14.9282	−0.495683	−0.247841	−	0.968801i	$-0.579721\pi$
$907$	−14.9282	−0.495683	−0.247841	+	0.968801i	$0.579721\pi$
$908$	7.46410	0.247705
$909$	0	0
$910$	−1.73205	−0.0574169
$911$	−49.7321	−1.64770	−0.823848	−	0.566811i	$-0.808177\pi$
$911$	−49.7321	−1.64770	−0.823848	+	0.566811i	$0.808177\pi$
$912$	3.00000	0.0993399
$913$	0	0
$914$	19.9282	0.659166
$915$	6.92820	0.229039
$916$	0	0
$917$	−4.26795	−0.140940
$918$	−6.58846	−0.217451
$919$	5.71281	0.188448	0.0942242	−	0.995551i	$-0.469963\pi$
$919$	5.71281	0.188448	0.0942242	+	0.995551i	$0.469963\pi$
$920$	2.46410	0.0812390
$921$	5.32051	0.175317
$922$	−38.2487	−1.25965
$923$	5.07180	0.166940
$924$	0	0
$925$	−7.46410	−0.245418
$926$	−35.0000	−1.15017
$927$	0	0
$928$	−2.53590	−0.0832449
$929$	31.3205	1.02759	0.513796	−	0.857912i	$-0.328239\pi$
$929$	31.3205	1.02759	0.513796	+	0.857912i	$0.328239\pi$
$930$	5.66025	0.185607
$931$	−1.73205	−0.0567657
$932$	22.8564	0.748686
$933$	55.7654	1.82568
$934$	5.87564	0.192257
$935$	0	0
$936$	0	0
$937$	−50.6410	−1.65437	−0.827185	−	0.561930i	$-0.810059\pi$
$937$	−50.6410	−1.65437	−0.827185	+	0.561930i	$0.810059\pi$
$938$	14.1962	0.463521
$939$	28.6410	0.934664
$940$	2.73205	0.0891097
$941$	2.28719	0.0745602	0.0372801	−	0.999305i	$-0.488131\pi$
$941$	2.28719	0.0745602	0.0372801	+	0.999305i	$0.488131\pi$
$942$	−18.1244	−0.590523
$943$	−16.5885	−0.540194
$944$	−1.00000	−0.0325472
$945$	5.19615	0.169031
$946$	0	0
$947$	3.26795	0.106194	0.0530970	−	0.998589i	$-0.483091\pi$
$947$	3.26795	0.106194	0.0530970	+	0.998589i	$0.483091\pi$
$948$	0.124356	0.00403888
$949$	16.7321	0.543145
$950$	1.73205	0.0561951
$951$	−37.5167	−1.21656
$952$	1.26795	0.0410945
$953$	−6.07180	−0.196685	−0.0983424	−	0.995153i	$-0.531354\pi$
$953$	−6.07180	−0.196685	−0.0983424	+	0.995153i	$0.531354\pi$
$954$	0	0
$955$	18.2679	0.591137
$956$	2.46410	0.0796947
$957$	0	0
$958$	15.2679	0.493285
$959$	−12.1244	−0.391516
$960$	1.73205	0.0559017
$961$	−20.3205	−0.655500
$962$	12.9282	0.416822
$963$	0	0
$964$	10.9282	0.351974
$965$	−21.7846	−0.701271
$966$	−4.26795	−0.137319
$967$	−52.4974	−1.68820	−0.844102	−	0.536183i	$-0.819866\pi$
$967$	−52.4974	−1.68820	−0.844102	+	0.536183i	$0.819866\pi$
$968$	0	0
$969$	3.80385	0.122197
$970$	1.46410	0.0470095
$971$	−6.60770	−0.212051	−0.106026	−	0.994363i	$-0.533813\pi$
$971$	−6.60770	−0.212051	−0.106026	+	0.994363i	$0.533813\pi$
$972$	0	0
$973$	−1.33975	−0.0429503
$974$	−22.7128	−0.727765
$975$	−3.00000	−0.0960769
$976$	4.00000	0.128037
$977$	−40.5167	−1.29624	−0.648121	−	0.761537i	$-0.724445\pi$
$977$	−40.5167	−1.29624	−0.648121	+	0.761537i	$0.724445\pi$
$978$	15.4641	0.494487
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	17.5167	0.558979
$983$	23.0718	0.735876	0.367938	−	0.929850i	$-0.380064\pi$
$983$	23.0718	0.735876	0.367938	+	0.929850i	$0.380064\pi$
$984$	−11.6603	−0.371715
$985$	24.0000	0.764704
$986$	−3.21539	−0.102399
$987$	−4.73205	−0.150623
$988$	−3.00000	−0.0954427
$989$	−18.8756	−0.600211
$990$	0	0
$991$	24.5167	0.778797	0.389399	−	0.921069i	$-0.372683\pi$
$991$	24.5167	0.778797	0.389399	+	0.921069i	$0.372683\pi$
$992$	3.26795	0.103757
$993$	3.12436	0.0991484
$994$	2.92820	0.0928770
$995$	−8.39230	−0.266054
$996$	−20.6603	−0.654645
$997$	3.19615	0.101223	0.0506116	−	0.998718i	$-0.483883\pi$
$997$	3.19615	0.101223	0.0506116	+	0.998718i	$0.483883\pi$
$998$	−1.60770	−0.0508907
$999$	−38.7846	−1.22709

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.73205 q^{12} +1.73205 q^{13} +1.00000 q^{14} +1.73205 q^{15} +1.00000 q^{16} +1.26795 q^{17} -1.73205 q^{19} -1.00000 q^{20} +1.73205 q^{21} +2.46410 q^{23} +1.73205 q^{24} +1.00000 q^{25} -1.73205 q^{26} +5.19615 q^{27} -1.00000 q^{28} +2.53590 q^{29} -1.73205 q^{30} -3.26795 q^{31} -1.00000 q^{32} -1.26795 q^{34} +1.00000 q^{35} -7.46410 q^{37} +1.73205 q^{38} -3.00000 q^{39} +1.00000 q^{40} -6.73205 q^{41} -1.73205 q^{42} -7.66025 q^{43} -2.46410 q^{46} -2.73205 q^{47} -1.73205 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.19615 q^{51} +1.73205 q^{52} -1.26795 q^{53} -5.19615 q^{54} +1.00000 q^{56} +3.00000 q^{57} -2.53590 q^{58} -1.00000 q^{59} +1.73205 q^{60} +4.00000 q^{61} +3.26795 q^{62} +1.00000 q^{64} -1.73205 q^{65} +14.1962 q^{67} +1.26795 q^{68} -4.26795 q^{69} -1.00000 q^{70} +2.92820 q^{71} +9.66025 q^{73} +7.46410 q^{74} -1.73205 q^{75} -1.73205 q^{76} +3.00000 q^{78} -0.0717968 q^{79} -1.00000 q^{80} -9.00000 q^{81} +6.73205 q^{82} +11.9282 q^{83} +1.73205 q^{84} -1.26795 q^{85} +7.66025 q^{86} -4.39230 q^{87} -4.73205 q^{89} -1.73205 q^{91} +2.46410 q^{92} +5.66025 q^{93} +2.73205 q^{94} +1.73205 q^{95} +1.73205 q^{96} +1.46410 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 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

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Database

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