LMFDB - Embedded newform 8470.2.a.by.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:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.23607 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.23607 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{9} -1.00000 q^{10} -2.23607 q^{12} -1.00000 q^{13} -1.00000 q^{14} +2.23607 q^{15} +1.00000 q^{16} +1.23607 q^{17} +2.00000 q^{18} +2.23607 q^{19} -1.00000 q^{20} +2.23607 q^{21} -3.76393 q^{23} -2.23607 q^{24} +1.00000 q^{25} -1.00000 q^{26} +2.23607 q^{27} -1.00000 q^{28} -4.47214 q^{29} +2.23607 q^{30} -5.23607 q^{31} +1.00000 q^{32} +1.23607 q^{34} +1.00000 q^{35} +2.00000 q^{36} +8.94427 q^{37} +2.23607 q^{38} +2.23607 q^{39} -1.00000 q^{40} +7.23607 q^{41} +2.23607 q^{42} -3.23607 q^{43} -2.00000 q^{45} -3.76393 q^{46} +7.70820 q^{47} -2.23607 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.76393 q^{51} -1.00000 q^{52} -5.70820 q^{53} +2.23607 q^{54} -1.00000 q^{56} -5.00000 q^{57} -4.47214 q^{58} +4.70820 q^{59} +2.23607 q^{60} +3.52786 q^{61} -5.23607 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -7.23607 q^{67} +1.23607 q^{68} +8.41641 q^{69} +1.00000 q^{70} -1.52786 q^{71} +2.00000 q^{72} +7.23607 q^{73} +8.94427 q^{74} -2.23607 q^{75} +2.23607 q^{76} +2.23607 q^{78} -10.2361 q^{79} -1.00000 q^{80} -11.0000 q^{81} +7.23607 q^{82} +0.236068 q^{83} +2.23607 q^{84} -1.23607 q^{85} -3.23607 q^{86} +10.0000 q^{87} -17.2361 q^{89} -2.00000 q^{90} +1.00000 q^{91} -3.76393 q^{92} +11.7082 q^{93} +7.70820 q^{94} -2.23607 q^{95} -2.23607 q^{96} +3.52786 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 + 4 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + 4 q^{9} - 2 q^{10} - 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 4 q^{18} - 2 q^{20} - 12 q^{23} + 2 q^{25} - 2 q^{26} - 2 q^{28} - 6 q^{31} + 2 q^{32} - 2 q^{34} + 2 q^{35} + 4 q^{36} - 2 q^{40} + 10 q^{41} - 2 q^{43} - 4 q^{45} - 12 q^{46} + 2 q^{47} + 2 q^{49} + 2 q^{50} - 10 q^{51} - 2 q^{52} + 2 q^{53} - 2 q^{56} - 10 q^{57} - 4 q^{59} + 16 q^{61} - 6 q^{62} - 4 q^{63} + 2 q^{64} + 2 q^{65} - 10 q^{67} - 2 q^{68} - 10 q^{69} + 2 q^{70} - 12 q^{71} + 4 q^{72} + 10 q^{73} - 16 q^{79} - 2 q^{80} - 22 q^{81} + 10 q^{82} - 4 q^{83} + 2 q^{85} - 2 q^{86} + 20 q^{87} - 30 q^{89} - 4 q^{90} + 2 q^{91} - 12 q^{92} + 10 q^{93} + 2 q^{94} + 16 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 + 4 * q^9 - 2 * q^10 - 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 + 4 * q^18 - 2 * q^20 - 12 * q^23 + 2 * q^25 - 2 * q^26 - 2 * q^28 - 6 * q^31 + 2 * q^32 - 2 * q^34 + 2 * q^35 + 4 * q^36 - 2 * q^40 + 10 * q^41 - 2 * q^43 - 4 * q^45 - 12 * q^46 + 2 * q^47 + 2 * q^49 + 2 * q^50 - 10 * q^51 - 2 * q^52 + 2 * q^53 - 2 * q^56 - 10 * q^57 - 4 * q^59 + 16 * q^61 - 6 * q^62 - 4 * q^63 + 2 * q^64 + 2 * q^65 - 10 * q^67 - 2 * q^68 - 10 * q^69 + 2 * q^70 - 12 * q^71 + 4 * q^72 + 10 * q^73 - 16 * q^79 - 2 * q^80 - 22 * q^81 + 10 * q^82 - 4 * q^83 + 2 * q^85 - 2 * q^86 + 20 * q^87 - 30 * q^89 - 4 * q^90 + 2 * q^91 - 12 * q^92 + 10 * q^93 + 2 * q^94 + 16 * 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
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.23607	−0.912871
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	2.00000	0.666667
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−2.23607	−0.645497
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−1.00000	−0.267261
$15$	2.23607	0.577350
$16$	1.00000	0.250000
$17$	1.23607	0.299791	0.149895	−	0.988702i	$-0.452106\pi$
$17$	1.23607	0.299791	0.149895	+	0.988702i	$0.452106\pi$
$18$	2.00000	0.471405
$19$	2.23607	0.512989	0.256495	−	0.966546i	$-0.417432\pi$
$19$	2.23607	0.512989	0.256495	+	0.966546i	$0.417432\pi$
$20$	−1.00000	−0.223607
$21$	2.23607	0.487950
$22$	0	0
$23$	−3.76393	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$23$	−3.76393	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$24$	−2.23607	−0.456435
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	2.23607	0.430331
$28$	−1.00000	−0.188982
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	2.23607	0.408248
$31$	−5.23607	−0.940426	−0.470213	−	0.882553i	$-0.655823\pi$
$31$	−5.23607	−0.940426	−0.470213	+	0.882553i	$0.655823\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.23607	0.211984
$35$	1.00000	0.169031
$36$	2.00000	0.333333
$37$	8.94427	1.47043	0.735215	−	0.677834i	$-0.237081\pi$
$37$	8.94427	1.47043	0.735215	+	0.677834i	$0.237081\pi$
$38$	2.23607	0.362738
$39$	2.23607	0.358057
$40$	−1.00000	−0.158114
$41$	7.23607	1.13008	0.565042	−	0.825062i	$-0.308860\pi$
$41$	7.23607	1.13008	0.565042	+	0.825062i	$0.308860\pi$
$42$	2.23607	0.345033
$43$	−3.23607	−0.493496	−0.246748	−	0.969080i	$-0.579362\pi$
$43$	−3.23607	−0.493496	−0.246748	+	0.969080i	$0.579362\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−3.76393	−0.554962
$47$	7.70820	1.12436	0.562179	−	0.827016i	$-0.309963\pi$
$47$	7.70820	1.12436	0.562179	+	0.827016i	$0.309963\pi$
$48$	−2.23607	−0.322749
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−2.76393	−0.387028
$52$	−1.00000	−0.138675
$53$	−5.70820	−0.784082	−0.392041	−	0.919948i	$-0.628231\pi$
$53$	−5.70820	−0.784082	−0.392041	+	0.919948i	$0.628231\pi$
$54$	2.23607	0.304290
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−5.00000	−0.662266
$58$	−4.47214	−0.587220
$59$	4.70820	0.612956	0.306478	−	0.951878i	$-0.400849\pi$
$59$	4.70820	0.612956	0.306478	+	0.951878i	$0.400849\pi$
$60$	2.23607	0.288675
$61$	3.52786	0.451697	0.225848	−	0.974162i	$-0.427485\pi$
$61$	3.52786	0.451697	0.225848	+	0.974162i	$0.427485\pi$
$62$	−5.23607	−0.664981
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	−7.23607	−0.884026	−0.442013	−	0.897009i	$-0.645736\pi$
$67$	−7.23607	−0.884026	−0.442013	+	0.897009i	$0.645736\pi$
$68$	1.23607	0.149895
$69$	8.41641	1.01322
$70$	1.00000	0.119523
$71$	−1.52786	−0.181324	−0.0906621	−	0.995882i	$-0.528898\pi$
$71$	−1.52786	−0.181324	−0.0906621	+	0.995882i	$0.528898\pi$
$72$	2.00000	0.235702
$73$	7.23607	0.846918	0.423459	−	0.905915i	$-0.360816\pi$
$73$	7.23607	0.846918	0.423459	+	0.905915i	$0.360816\pi$
$74$	8.94427	1.03975
$75$	−2.23607	−0.258199
$76$	2.23607	0.256495
$77$	0	0
$78$	2.23607	0.253185
$79$	−10.2361	−1.15165	−0.575824	−	0.817574i	$-0.695319\pi$
$79$	−10.2361	−1.15165	−0.575824	+	0.817574i	$0.695319\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	7.23607	0.799090
$83$	0.236068	0.0259118	0.0129559	−	0.999916i	$-0.495876\pi$
$83$	0.236068	0.0259118	0.0129559	+	0.999916i	$0.495876\pi$
$84$	2.23607	0.243975
$85$	−1.23607	−0.134070
$86$	−3.23607	−0.348954
$87$	10.0000	1.07211
$88$	0	0
$89$	−17.2361	−1.82702	−0.913510	−	0.406817i	$-0.866639\pi$
$89$	−17.2361	−1.82702	−0.913510	+	0.406817i	$0.866639\pi$
$90$	−2.00000	−0.210819
$91$	1.00000	0.104828
$92$	−3.76393	−0.392417
$93$	11.7082	1.21408
$94$	7.70820	0.795041
$95$	−2.23607	−0.229416
$96$	−2.23607	−0.228218
$97$	3.52786	0.358200	0.179100	−	0.983831i	$-0.442681\pi$
$97$	3.52786	0.358200	0.179100	+	0.983831i	$0.442681\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	1.00000	0.0995037	0.0497519	−	0.998762i	$-0.484157\pi$
$101$	1.00000	0.0995037	0.0497519	+	0.998762i	$0.484157\pi$
$102$	−2.76393	−0.273670
$103$	14.1803	1.39723	0.698615	−	0.715498i	$-0.253800\pi$
$103$	14.1803	1.39723	0.698615	+	0.715498i	$0.253800\pi$
$104$	−1.00000	−0.0980581
$105$	−2.23607	−0.218218
$106$	−5.70820	−0.554430
$107$	2.47214	0.238990	0.119495	−	0.992835i	$-0.461872\pi$
$107$	2.47214	0.238990	0.119495	+	0.992835i	$0.461872\pi$
$108$	2.23607	0.215166
$109$	3.70820	0.355182	0.177591	−	0.984104i	$-0.443170\pi$
$109$	3.70820	0.355182	0.177591	+	0.984104i	$0.443170\pi$
$110$	0	0
$111$	−20.0000	−1.89832
$112$	−1.00000	−0.0944911
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	−5.00000	−0.468293
$115$	3.76393	0.350988
$116$	−4.47214	−0.415227
$117$	−2.00000	−0.184900
$118$	4.70820	0.433425
$119$	−1.23607	−0.113310
$120$	2.23607	0.204124
$121$	0	0
$122$	3.52786	0.319398
$123$	−16.1803	−1.45893
$124$	−5.23607	−0.470213
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	−6.23607	−0.553362	−0.276681	−	0.960962i	$-0.589234\pi$
$127$	−6.23607	−0.553362	−0.276681	+	0.960962i	$0.589234\pi$
$128$	1.00000	0.0883883
$129$	7.23607	0.637100
$130$	1.00000	0.0877058
$131$	3.18034	0.277868	0.138934	−	0.990302i	$-0.455632\pi$
$131$	3.18034	0.277868	0.138934	+	0.990302i	$0.455632\pi$
$132$	0	0
$133$	−2.23607	−0.193892
$134$	−7.23607	−0.625101
$135$	−2.23607	−0.192450
$136$	1.23607	0.105992
$137$	13.4721	1.15100	0.575501	−	0.817801i	$-0.304807\pi$
$137$	13.4721	1.15100	0.575501	+	0.817801i	$0.304807\pi$
$138$	8.41641	0.716452
$139$	−4.23607	−0.359299	−0.179649	−	0.983731i	$-0.557496\pi$
$139$	−4.23607	−0.359299	−0.179649	+	0.983731i	$0.557496\pi$
$140$	1.00000	0.0845154
$141$	−17.2361	−1.45154
$142$	−1.52786	−0.128216
$143$	0	0
$144$	2.00000	0.166667
$145$	4.47214	0.371391
$146$	7.23607	0.598861
$147$	−2.23607	−0.184428
$148$	8.94427	0.735215
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	−2.23607	−0.182574
$151$	−12.2361	−0.995757	−0.497879	−	0.867247i	$-0.665887\pi$
$151$	−12.2361	−0.995757	−0.497879	+	0.867247i	$0.665887\pi$
$152$	2.23607	0.181369
$153$	2.47214	0.199860
$154$	0	0
$155$	5.23607	0.420571
$156$	2.23607	0.179029
$157$	8.52786	0.680598	0.340299	−	0.940317i	$-0.389472\pi$
$157$	8.52786	0.680598	0.340299	+	0.940317i	$0.389472\pi$
$158$	−10.2361	−0.814338
$159$	12.7639	1.01225
$160$	−1.00000	−0.0790569
$161$	3.76393	0.296639
$162$	−11.0000	−0.864242
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	7.23607	0.565042
$165$	0	0
$166$	0.236068	0.0183224
$167$	−1.70820	−0.132185	−0.0660924	−	0.997814i	$-0.521053\pi$
$167$	−1.70820	−0.132185	−0.0660924	+	0.997814i	$0.521053\pi$
$168$	2.23607	0.172516
$169$	−12.0000	−0.923077
$170$	−1.23607	−0.0948021
$171$	4.47214	0.341993
$172$	−3.23607	−0.246748
$173$	−19.8885	−1.51210	−0.756049	−	0.654515i	$-0.772873\pi$
$173$	−19.8885	−1.51210	−0.756049	+	0.654515i	$0.772873\pi$
$174$	10.0000	0.758098
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−10.5279	−0.791323
$178$	−17.2361	−1.29190
$179$	−11.2361	−0.839823	−0.419912	−	0.907565i	$-0.637939\pi$
$179$	−11.2361	−0.839823	−0.419912	+	0.907565i	$0.637939\pi$
$180$	−2.00000	−0.149071
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	1.00000	0.0741249
$183$	−7.88854	−0.583138
$184$	−3.76393	−0.277481
$185$	−8.94427	−0.657596
$186$	11.7082	0.858487
$187$	0	0
$188$	7.70820	0.562179
$189$	−2.23607	−0.162650
$190$	−2.23607	−0.162221
$191$	25.1803	1.82199	0.910993	−	0.412422i	$-0.135317\pi$
$191$	25.1803	1.82199	0.910993	+	0.412422i	$0.135317\pi$
$192$	−2.23607	−0.161374
$193$	−7.94427	−0.571841	−0.285921	−	0.958253i	$-0.592299\pi$
$193$	−7.94427	−0.571841	−0.285921	+	0.958253i	$0.592299\pi$
$194$	3.52786	0.253286
$195$	−2.23607	−0.160128
$196$	1.00000	0.0714286
$197$	7.52786	0.536338	0.268169	−	0.963372i	$-0.413581\pi$
$197$	7.52786	0.536338	0.268169	+	0.963372i	$0.413581\pi$
$198$	0	0
$199$	−4.47214	−0.317021	−0.158511	−	0.987357i	$-0.550669\pi$
$199$	−4.47214	−0.317021	−0.158511	+	0.987357i	$0.550669\pi$
$200$	1.00000	0.0707107
$201$	16.1803	1.14127
$202$	1.00000	0.0703598
$203$	4.47214	0.313882
$204$	−2.76393	−0.193514
$205$	−7.23607	−0.505389
$206$	14.1803	0.987991
$207$	−7.52786	−0.523223
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	−2.23607	−0.154303
$211$	−3.52786	−0.242868	−0.121434	−	0.992599i	$-0.538749\pi$
$211$	−3.52786	−0.242868	−0.121434	+	0.992599i	$0.538749\pi$
$212$	−5.70820	−0.392041
$213$	3.41641	0.234088
$214$	2.47214	0.168992
$215$	3.23607	0.220698
$216$	2.23607	0.152145
$217$	5.23607	0.355447
$218$	3.70820	0.251151
$219$	−16.1803	−1.09337
$220$	0	0
$221$	−1.23607	−0.0831469
$222$	−20.0000	−1.34231
$223$	−8.18034	−0.547796	−0.273898	−	0.961759i	$-0.588313\pi$
$223$	−8.18034	−0.547796	−0.273898	+	0.961759i	$0.588313\pi$
$224$	−1.00000	−0.0668153
$225$	2.00000	0.133333
$226$	−1.00000	−0.0665190
$227$	7.41641	0.492244	0.246122	−	0.969239i	$-0.420844\pi$
$227$	7.41641	0.492244	0.246122	+	0.969239i	$0.420844\pi$
$228$	−5.00000	−0.331133
$229$	6.94427	0.458890	0.229445	−	0.973322i	$-0.426309\pi$
$229$	6.94427	0.458890	0.229445	+	0.973322i	$0.426309\pi$
$230$	3.76393	0.248186
$231$	0	0
$232$	−4.47214	−0.293610
$233$	−10.4164	−0.682402	−0.341201	−	0.939990i	$-0.610834\pi$
$233$	−10.4164	−0.682402	−0.341201	+	0.939990i	$0.610834\pi$
$234$	−2.00000	−0.130744
$235$	−7.70820	−0.502828
$236$	4.70820	0.306478
$237$	22.8885	1.48677
$238$	−1.23607	−0.0801224
$239$	−10.7082	−0.692656	−0.346328	−	0.938113i	$-0.612572\pi$
$239$	−10.7082	−0.692656	−0.346328	+	0.938113i	$0.612572\pi$
$240$	2.23607	0.144338
$241$	22.3607	1.44038	0.720189	−	0.693778i	$-0.244055\pi$
$241$	22.3607	1.44038	0.720189	+	0.693778i	$0.244055\pi$
$242$	0	0
$243$	17.8885	1.14755
$244$	3.52786	0.225848
$245$	−1.00000	−0.0638877
$246$	−16.1803	−1.03162
$247$	−2.23607	−0.142278
$248$	−5.23607	−0.332491
$249$	−0.527864	−0.0334520
$250$	−1.00000	−0.0632456
$251$	−29.8885	−1.88655	−0.943274	−	0.332015i	$-0.892272\pi$
$251$	−29.8885	−1.88655	−0.943274	+	0.332015i	$0.892272\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	−6.23607	−0.391286
$255$	2.76393	0.173084
$256$	1.00000	0.0625000
$257$	−4.76393	−0.297166	−0.148583	−	0.988900i	$-0.547471\pi$
$257$	−4.76393	−0.297166	−0.148583	+	0.988900i	$0.547471\pi$
$258$	7.23607	0.450498
$259$	−8.94427	−0.555770
$260$	1.00000	0.0620174
$261$	−8.94427	−0.553637
$262$	3.18034	0.196482
$263$	−7.65248	−0.471872	−0.235936	−	0.971769i	$-0.575816\pi$
$263$	−7.65248	−0.471872	−0.235936	+	0.971769i	$0.575816\pi$
$264$	0	0
$265$	5.70820	0.350652
$266$	−2.23607	−0.137102
$267$	38.5410	2.35867
$268$	−7.23607	−0.442013
$269$	−27.8328	−1.69700	−0.848498	−	0.529198i	$-0.822493\pi$
$269$	−27.8328	−1.69700	−0.848498	+	0.529198i	$0.822493\pi$
$270$	−2.23607	−0.136083
$271$	−7.41641	−0.450515	−0.225257	−	0.974299i	$-0.572322\pi$
$271$	−7.41641	−0.450515	−0.225257	+	0.974299i	$0.572322\pi$
$272$	1.23607	0.0749476
$273$	−2.23607	−0.135333
$274$	13.4721	0.813881
$275$	0	0
$276$	8.41641	0.506608
$277$	−23.5967	−1.41779	−0.708896	−	0.705313i	$-0.750806\pi$
$277$	−23.5967	−1.41779	−0.708896	+	0.705313i	$0.750806\pi$
$278$	−4.23607	−0.254062
$279$	−10.4721	−0.626950
$280$	1.00000	0.0597614
$281$	−25.8328	−1.54106	−0.770528	−	0.637406i	$-0.780008\pi$
$281$	−25.8328	−1.54106	−0.770528	+	0.637406i	$0.780008\pi$
$282$	−17.2361	−1.02639
$283$	−26.2361	−1.55957	−0.779786	−	0.626046i	$-0.784672\pi$
$283$	−26.2361	−1.55957	−0.779786	+	0.626046i	$0.784672\pi$
$284$	−1.52786	−0.0906621
$285$	5.00000	0.296174
$286$	0	0
$287$	−7.23607	−0.427132
$288$	2.00000	0.117851
$289$	−15.4721	−0.910126
$290$	4.47214	0.262613
$291$	−7.88854	−0.462435
$292$	7.23607	0.423459
$293$	5.00000	0.292103	0.146052	−	0.989277i	$-0.453343\pi$
$293$	5.00000	0.292103	0.146052	+	0.989277i	$0.453343\pi$
$294$	−2.23607	−0.130410
$295$	−4.70820	−0.274122
$296$	8.94427	0.519875
$297$	0	0
$298$	−4.00000	−0.231714
$299$	3.76393	0.217674
$300$	−2.23607	−0.129099
$301$	3.23607	0.186524
$302$	−12.2361	−0.704107
$303$	−2.23607	−0.128459
$304$	2.23607	0.128247
$305$	−3.52786	−0.202005
$306$	2.47214	0.141323
$307$	−11.4164	−0.651569	−0.325784	−	0.945444i	$-0.605628\pi$
$307$	−11.4164	−0.651569	−0.325784	+	0.945444i	$0.605628\pi$
$308$	0	0
$309$	−31.7082	−1.80382
$310$	5.23607	0.297389
$311$	−27.1246	−1.53810	−0.769048	−	0.639191i	$-0.779269\pi$
$311$	−27.1246	−1.53810	−0.769048	+	0.639191i	$0.779269\pi$
$312$	2.23607	0.126592
$313$	19.5279	1.10378	0.551890	−	0.833917i	$-0.313907\pi$
$313$	19.5279	1.10378	0.551890	+	0.833917i	$0.313907\pi$
$314$	8.52786	0.481255
$315$	2.00000	0.112687
$316$	−10.2361	−0.575824
$317$	0.652476	0.0366467	0.0183233	−	0.999832i	$-0.494167\pi$
$317$	0.652476	0.0366467	0.0183233	+	0.999832i	$0.494167\pi$
$318$	12.7639	0.715766
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−5.52786	−0.308535
$322$	3.76393	0.209756
$323$	2.76393	0.153789
$324$	−11.0000	−0.611111
$325$	−1.00000	−0.0554700
$326$	−6.00000	−0.332309
$327$	−8.29180	−0.458537
$328$	7.23607	0.399545
$329$	−7.70820	−0.424967
$330$	0	0
$331$	−21.7082	−1.19319	−0.596595	−	0.802542i	$-0.703480\pi$
$331$	−21.7082	−1.19319	−0.596595	+	0.802542i	$0.703480\pi$
$332$	0.236068	0.0129559
$333$	17.8885	0.980286
$334$	−1.70820	−0.0934688
$335$	7.23607	0.395349
$336$	2.23607	0.121988
$337$	2.88854	0.157349	0.0786745	−	0.996900i	$-0.474931\pi$
$337$	2.88854	0.157349	0.0786745	+	0.996900i	$0.474931\pi$
$338$	−12.0000	−0.652714
$339$	2.23607	0.121447
$340$	−1.23607	−0.0670352
$341$	0	0
$342$	4.47214	0.241825
$343$	−1.00000	−0.0539949
$344$	−3.23607	−0.174477
$345$	−8.41641	−0.453124
$346$	−19.8885	−1.06921
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	10.0000	0.536056
$349$	−0.416408	−0.0222898	−0.0111449	−	0.999938i	$-0.503548\pi$
$349$	−0.416408	−0.0222898	−0.0111449	+	0.999938i	$0.503548\pi$
$350$	−1.00000	−0.0534522
$351$	−2.23607	−0.119352
$352$	0	0
$353$	−4.76393	−0.253559	−0.126779	−	0.991931i	$-0.540464\pi$
$353$	−4.76393	−0.253559	−0.126779	+	0.991931i	$0.540464\pi$
$354$	−10.5279	−0.559550
$355$	1.52786	0.0810906
$356$	−17.2361	−0.913510
$357$	2.76393	0.146283
$358$	−11.2361	−0.593845
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	−2.00000	−0.105409
$361$	−14.0000	−0.736842
$362$	7.00000	0.367912
$363$	0	0
$364$	1.00000	0.0524142
$365$	−7.23607	−0.378753
$366$	−7.88854	−0.412341
$367$	8.18034	0.427010	0.213505	−	0.976942i	$-0.431512\pi$
$367$	8.18034	0.427010	0.213505	+	0.976942i	$0.431512\pi$
$368$	−3.76393	−0.196209
$369$	14.4721	0.753389
$370$	−8.94427	−0.464991
$371$	5.70820	0.296355
$372$	11.7082	0.607042
$373$	5.52786	0.286222	0.143111	−	0.989707i	$-0.454289\pi$
$373$	5.52786	0.286222	0.143111	+	0.989707i	$0.454289\pi$
$374$	0	0
$375$	2.23607	0.115470
$376$	7.70820	0.397520
$377$	4.47214	0.230327
$378$	−2.23607	−0.115011
$379$	11.4164	0.586421	0.293211	−	0.956048i	$-0.405276\pi$
$379$	11.4164	0.586421	0.293211	+	0.956048i	$0.405276\pi$
$380$	−2.23607	−0.114708
$381$	13.9443	0.714387
$382$	25.1803	1.28834
$383$	−17.2361	−0.880722	−0.440361	−	0.897821i	$-0.645150\pi$
$383$	−17.2361	−0.880722	−0.440361	+	0.897821i	$0.645150\pi$
$384$	−2.23607	−0.114109
$385$	0	0
$386$	−7.94427	−0.404353
$387$	−6.47214	−0.328997
$388$	3.52786	0.179100
$389$	−18.2918	−0.927431	−0.463715	−	0.885984i	$-0.653484\pi$
$389$	−18.2918	−0.927431	−0.463715	+	0.885984i	$0.653484\pi$
$390$	−2.23607	−0.113228
$391$	−4.65248	−0.235286
$392$	1.00000	0.0505076
$393$	−7.11146	−0.358726
$394$	7.52786	0.379248
$395$	10.2361	0.515032
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	−4.47214	−0.224168
$399$	5.00000	0.250313
$400$	1.00000	0.0500000
$401$	−28.8328	−1.43984	−0.719921	−	0.694056i	$-0.755822\pi$
$401$	−28.8328	−1.43984	−0.719921	+	0.694056i	$0.755822\pi$
$402$	16.1803	0.807002
$403$	5.23607	0.260827
$404$	1.00000	0.0497519
$405$	11.0000	0.546594
$406$	4.47214	0.221948
$407$	0	0
$408$	−2.76393	−0.136835
$409$	−11.5279	−0.570016	−0.285008	−	0.958525i	$-0.591996\pi$
$409$	−11.5279	−0.570016	−0.285008	+	0.958525i	$0.591996\pi$
$410$	−7.23607	−0.357364
$411$	−30.1246	−1.48594
$412$	14.1803	0.698615
$413$	−4.70820	−0.231676
$414$	−7.52786	−0.369974
$415$	−0.236068	−0.0115881
$416$	−1.00000	−0.0490290
$417$	9.47214	0.463852
$418$	0	0
$419$	−28.2361	−1.37942	−0.689711	−	0.724085i	$-0.742262\pi$
$419$	−28.2361	−1.37942	−0.689711	+	0.724085i	$0.742262\pi$
$420$	−2.23607	−0.109109
$421$	−3.05573	−0.148927	−0.0744635	−	0.997224i	$-0.523724\pi$
$421$	−3.05573	−0.148927	−0.0744635	+	0.997224i	$0.523724\pi$
$422$	−3.52786	−0.171734
$423$	15.4164	0.749571
$424$	−5.70820	−0.277215
$425$	1.23607	0.0599581
$426$	3.41641	0.165526
$427$	−3.52786	−0.170725
$428$	2.47214	0.119495
$429$	0	0
$430$	3.23607	0.156057
$431$	33.6525	1.62098	0.810491	−	0.585751i	$-0.199200\pi$
$431$	33.6525	1.62098	0.810491	+	0.585751i	$0.199200\pi$
$432$	2.23607	0.107583
$433$	−12.6525	−0.608039	−0.304020	−	0.952666i	$-0.598329\pi$
$433$	−12.6525	−0.608039	−0.304020	+	0.952666i	$0.598329\pi$
$434$	5.23607	0.251339
$435$	−10.0000	−0.479463
$436$	3.70820	0.177591
$437$	−8.41641	−0.402611
$438$	−16.1803	−0.773127
$439$	−15.1246	−0.721858	−0.360929	−	0.932593i	$-0.617540\pi$
$439$	−15.1246	−0.721858	−0.360929	+	0.932593i	$0.617540\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−1.23607	−0.0587938
$443$	−24.3607	−1.15741	−0.578705	−	0.815537i	$-0.696442\pi$
$443$	−24.3607	−1.15741	−0.578705	+	0.815537i	$0.696442\pi$
$444$	−20.0000	−0.949158
$445$	17.2361	0.817068
$446$	−8.18034	−0.387350
$447$	8.94427	0.423050
$448$	−1.00000	−0.0472456
$449$	23.8328	1.12474	0.562370	−	0.826886i	$-0.309890\pi$
$449$	23.8328	1.12474	0.562370	+	0.826886i	$0.309890\pi$
$450$	2.00000	0.0942809
$451$	0	0
$452$	−1.00000	−0.0470360
$453$	27.3607	1.28552
$454$	7.41641	0.348069
$455$	−1.00000	−0.0468807
$456$	−5.00000	−0.234146
$457$	36.4164	1.70349	0.851744	−	0.523958i	$-0.175545\pi$
$457$	36.4164	1.70349	0.851744	+	0.523958i	$0.175545\pi$
$458$	6.94427	0.324485
$459$	2.76393	0.129009
$460$	3.76393	0.175494
$461$	23.8885	1.11260	0.556300	−	0.830981i	$-0.312220\pi$
$461$	23.8885	1.11260	0.556300	+	0.830981i	$0.312220\pi$
$462$	0	0
$463$	10.7082	0.497652	0.248826	−	0.968548i	$-0.419955\pi$
$463$	10.7082	0.497652	0.248826	+	0.968548i	$0.419955\pi$
$464$	−4.47214	−0.207614
$465$	−11.7082	−0.542955
$466$	−10.4164	−0.482531
$467$	−2.81966	−0.130478	−0.0652392	−	0.997870i	$-0.520781\pi$
$467$	−2.81966	−0.130478	−0.0652392	+	0.997870i	$0.520781\pi$
$468$	−2.00000	−0.0924500
$469$	7.23607	0.334131
$470$	−7.70820	−0.355553
$471$	−19.0689	−0.878648
$472$	4.70820	0.216713
$473$	0	0
$474$	22.8885	1.05131
$475$	2.23607	0.102598
$476$	−1.23607	−0.0566551
$477$	−11.4164	−0.522721
$478$	−10.7082	−0.489782
$479$	27.5967	1.26093	0.630464	−	0.776219i	$-0.282865\pi$
$479$	27.5967	1.26093	0.630464	+	0.776219i	$0.282865\pi$
$480$	2.23607	0.102062
$481$	−8.94427	−0.407824
$482$	22.3607	1.01850
$483$	−8.41641	−0.382960
$484$	0	0
$485$	−3.52786	−0.160192
$486$	17.8885	0.811441
$487$	2.23607	0.101326	0.0506630	−	0.998716i	$-0.483867\pi$
$487$	2.23607	0.101326	0.0506630	+	0.998716i	$0.483867\pi$
$488$	3.52786	0.159699
$489$	13.4164	0.606711
$490$	−1.00000	−0.0451754
$491$	−31.2361	−1.40966	−0.704832	−	0.709374i	$-0.748978\pi$
$491$	−31.2361	−1.40966	−0.704832	+	0.709374i	$0.748978\pi$
$492$	−16.1803	−0.729466
$493$	−5.52786	−0.248962
$494$	−2.23607	−0.100605
$495$	0	0
$496$	−5.23607	−0.235106
$497$	1.52786	0.0685341
$498$	−0.527864	−0.0236542
$499$	26.8328	1.20120	0.600601	−	0.799549i	$-0.294928\pi$
$499$	26.8328	1.20120	0.600601	+	0.799549i	$0.294928\pi$
$500$	−1.00000	−0.0447214
$501$	3.81966	0.170650
$502$	−29.8885	−1.33399
$503$	−25.5967	−1.14130	−0.570651	−	0.821192i	$-0.693309\pi$
$503$	−25.5967	−1.14130	−0.570651	+	0.821192i	$0.693309\pi$
$504$	−2.00000	−0.0890871
$505$	−1.00000	−0.0444994
$506$	0	0
$507$	26.8328	1.19169
$508$	−6.23607	−0.276681
$509$	28.8885	1.28046	0.640231	−	0.768182i	$-0.278839\pi$
$509$	28.8885	1.28046	0.640231	+	0.768182i	$0.278839\pi$
$510$	2.76393	0.122389
$511$	−7.23607	−0.320105
$512$	1.00000	0.0441942
$513$	5.00000	0.220755
$514$	−4.76393	−0.210128
$515$	−14.1803	−0.624860
$516$	7.23607	0.318550
$517$	0	0
$518$	−8.94427	−0.392989
$519$	44.4721	1.95211
$520$	1.00000	0.0438529
$521$	−1.05573	−0.0462523	−0.0231261	−	0.999733i	$-0.507362\pi$
$521$	−1.05573	−0.0462523	−0.0231261	+	0.999733i	$0.507362\pi$
$522$	−8.94427	−0.391480
$523$	35.6525	1.55897	0.779487	−	0.626418i	$-0.215480\pi$
$523$	35.6525	1.55897	0.779487	+	0.626418i	$0.215480\pi$
$524$	3.18034	0.138934
$525$	2.23607	0.0975900
$526$	−7.65248	−0.333664
$527$	−6.47214	−0.281931
$528$	0	0
$529$	−8.83282	−0.384035
$530$	5.70820	0.247949
$531$	9.41641	0.408637
$532$	−2.23607	−0.0969458
$533$	−7.23607	−0.313429
$534$	38.5410	1.66783
$535$	−2.47214	−0.106880
$536$	−7.23607	−0.312551
$537$	25.1246	1.08421
$538$	−27.8328	−1.19996
$539$	0	0
$540$	−2.23607	−0.0962250
$541$	35.3050	1.51788	0.758939	−	0.651161i	$-0.225718\pi$
$541$	35.3050	1.51788	0.758939	+	0.651161i	$0.225718\pi$
$542$	−7.41641	−0.318562
$543$	−15.6525	−0.671712
$544$	1.23607	0.0529960
$545$	−3.70820	−0.158842
$546$	−2.23607	−0.0956949
$547$	−6.65248	−0.284439	−0.142220	−	0.989835i	$-0.545424\pi$
$547$	−6.65248	−0.284439	−0.142220	+	0.989835i	$0.545424\pi$
$548$	13.4721	0.575501
$549$	7.05573	0.301131
$550$	0	0
$551$	−10.0000	−0.426014
$552$	8.41641	0.358226
$553$	10.2361	0.435282
$554$	−23.5967	−1.00253
$555$	20.0000	0.848953
$556$	−4.23607	−0.179649
$557$	31.1246	1.31879	0.659396	−	0.751796i	$-0.270812\pi$
$557$	31.1246	1.31879	0.659396	+	0.751796i	$0.270812\pi$
$558$	−10.4721	−0.443321
$559$	3.23607	0.136871
$560$	1.00000	0.0422577
$561$	0	0
$562$	−25.8328	−1.08969
$563$	36.5967	1.54237	0.771185	−	0.636612i	$-0.219665\pi$
$563$	36.5967	1.54237	0.771185	+	0.636612i	$0.219665\pi$
$564$	−17.2361	−0.725769
$565$	1.00000	0.0420703
$566$	−26.2361	−1.10278
$567$	11.0000	0.461957
$568$	−1.52786	−0.0641078
$569$	−7.47214	−0.313248	−0.156624	−	0.987658i	$-0.550061\pi$
$569$	−7.47214	−0.313248	−0.156624	+	0.987658i	$0.550061\pi$
$570$	5.00000	0.209427
$571$	−36.3607	−1.52165	−0.760824	−	0.648959i	$-0.775205\pi$
$571$	−36.3607	−1.52165	−0.760824	+	0.648959i	$0.775205\pi$
$572$	0	0
$573$	−56.3050	−2.35217
$574$	−7.23607	−0.302028
$575$	−3.76393	−0.156967
$576$	2.00000	0.0833333
$577$	−35.0132	−1.45762	−0.728808	−	0.684718i	$-0.759925\pi$
$577$	−35.0132	−1.45762	−0.728808	+	0.684718i	$0.759925\pi$
$578$	−15.4721	−0.643556
$579$	17.7639	0.738244
$580$	4.47214	0.185695
$581$	−0.236068	−0.00979375
$582$	−7.88854	−0.326991
$583$	0	0
$584$	7.23607	0.299431
$585$	2.00000	0.0826898
$586$	5.00000	0.206548
$587$	−23.0689	−0.952155	−0.476077	−	0.879403i	$-0.657942\pi$
$587$	−23.0689	−0.952155	−0.476077	+	0.879403i	$0.657942\pi$
$588$	−2.23607	−0.0922139
$589$	−11.7082	−0.482428
$590$	−4.70820	−0.193834
$591$	−16.8328	−0.692410
$592$	8.94427	0.367607
$593$	−36.9443	−1.51712	−0.758560	−	0.651604i	$-0.774097\pi$
$593$	−36.9443	−1.51712	−0.758560	+	0.651604i	$0.774097\pi$
$594$	0	0
$595$	1.23607	0.0506738
$596$	−4.00000	−0.163846
$597$	10.0000	0.409273
$598$	3.76393	0.153919
$599$	27.0689	1.10600	0.553002	−	0.833180i	$-0.313482\pi$
$599$	27.0689	1.10600	0.553002	+	0.833180i	$0.313482\pi$
$600$	−2.23607	−0.0912871
$601$	−26.6525	−1.08718	−0.543589	−	0.839352i	$-0.682935\pi$
$601$	−26.6525	−1.08718	−0.543589	+	0.839352i	$0.682935\pi$
$602$	3.23607	0.131892
$603$	−14.4721	−0.589351
$604$	−12.2361	−0.497879
$605$	0	0
$606$	−2.23607	−0.0908341
$607$	−21.0557	−0.854626	−0.427313	−	0.904104i	$-0.640540\pi$
$607$	−21.0557	−0.854626	−0.427313	+	0.904104i	$0.640540\pi$
$608$	2.23607	0.0906845
$609$	−10.0000	−0.405220
$610$	−3.52786	−0.142839
$611$	−7.70820	−0.311841
$612$	2.47214	0.0999302
$613$	−0.875388	−0.0353566	−0.0176783	−	0.999844i	$-0.505627\pi$
$613$	−0.875388	−0.0353566	−0.0176783	+	0.999844i	$0.505627\pi$
$614$	−11.4164	−0.460729
$615$	16.1803	0.652454
$616$	0	0
$617$	−43.8885	−1.76689	−0.883443	−	0.468538i	$-0.844781\pi$
$617$	−43.8885	−1.76689	−0.883443	+	0.468538i	$0.844781\pi$
$618$	−31.7082	−1.27549
$619$	−47.5410	−1.91083	−0.955417	−	0.295258i	$-0.904594\pi$
$619$	−47.5410	−1.91083	−0.955417	+	0.295258i	$0.904594\pi$
$620$	5.23607	0.210286
$621$	−8.41641	−0.337739
$622$	−27.1246	−1.08760
$623$	17.2361	0.690548
$624$	2.23607	0.0895144
$625$	1.00000	0.0400000
$626$	19.5279	0.780490
$627$	0	0
$628$	8.52786	0.340299
$629$	11.0557	0.440821
$630$	2.00000	0.0796819
$631$	−16.3607	−0.651308	−0.325654	−	0.945489i	$-0.605585\pi$
$631$	−16.3607	−0.651308	−0.325654	+	0.945489i	$0.605585\pi$
$632$	−10.2361	−0.407169
$633$	7.88854	0.313541
$634$	0.652476	0.0259131
$635$	6.23607	0.247471
$636$	12.7639	0.506123
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−3.05573	−0.120883
$640$	−1.00000	−0.0395285
$641$	−23.8328	−0.941340	−0.470670	−	0.882309i	$-0.655988\pi$
$641$	−23.8328	−0.941340	−0.470670	+	0.882309i	$0.655988\pi$
$642$	−5.52786	−0.218167
$643$	6.11146	0.241012	0.120506	−	0.992713i	$-0.461548\pi$
$643$	6.11146	0.241012	0.120506	+	0.992713i	$0.461548\pi$
$644$	3.76393	0.148320
$645$	−7.23607	−0.284920
$646$	2.76393	0.108745
$647$	−9.23607	−0.363107	−0.181554	−	0.983381i	$-0.558113\pi$
$647$	−9.23607	−0.363107	−0.181554	+	0.983381i	$0.558113\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	−1.00000	−0.0392232
$651$	−11.7082	−0.458881
$652$	−6.00000	−0.234978
$653$	26.2918	1.02888	0.514439	−	0.857527i	$-0.328000\pi$
$653$	26.2918	1.02888	0.514439	+	0.857527i	$0.328000\pi$
$654$	−8.29180	−0.324235
$655$	−3.18034	−0.124266
$656$	7.23607	0.282521
$657$	14.4721	0.564612
$658$	−7.70820	−0.300497
$659$	20.7639	0.808848	0.404424	−	0.914572i	$-0.367472\pi$
$659$	20.7639	0.808848	0.404424	+	0.914572i	$0.367472\pi$
$660$	0	0
$661$	4.41641	0.171778	0.0858892	−	0.996305i	$-0.472627\pi$
$661$	4.41641	0.171778	0.0858892	+	0.996305i	$0.472627\pi$
$662$	−21.7082	−0.843713
$663$	2.76393	0.107342
$664$	0.236068	0.00916121
$665$	2.23607	0.0867110
$666$	17.8885	0.693167
$667$	16.8328	0.651769
$668$	−1.70820	−0.0660924
$669$	18.2918	0.707202
$670$	7.23607	0.279554
$671$	0	0
$672$	2.23607	0.0862582
$673$	10.0557	0.387620	0.193810	−	0.981039i	$-0.437915\pi$
$673$	10.0557	0.387620	0.193810	+	0.981039i	$0.437915\pi$
$674$	2.88854	0.111263
$675$	2.23607	0.0860663
$676$	−12.0000	−0.461538
$677$	45.8328	1.76150	0.880749	−	0.473583i	$-0.157040\pi$
$677$	45.8328	1.76150	0.880749	+	0.473583i	$0.157040\pi$
$678$	2.23607	0.0858757
$679$	−3.52786	−0.135387
$680$	−1.23607	−0.0474010
$681$	−16.5836	−0.635485
$682$	0	0
$683$	−6.36068	−0.243385	−0.121692	−	0.992568i	$-0.538832\pi$
$683$	−6.36068	−0.243385	−0.121692	+	0.992568i	$0.538832\pi$
$684$	4.47214	0.170996
$685$	−13.4721	−0.514744
$686$	−1.00000	−0.0381802
$687$	−15.5279	−0.592425
$688$	−3.23607	−0.123374
$689$	5.70820	0.217465
$690$	−8.41641	−0.320407
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	−19.8885	−0.756049
$693$	0	0
$694$	6.00000	0.227757
$695$	4.23607	0.160683
$696$	10.0000	0.379049
$697$	8.94427	0.338788
$698$	−0.416408	−0.0157613
$699$	23.2918	0.880977
$700$	−1.00000	−0.0377964
$701$	17.5967	0.664620	0.332310	−	0.943170i	$-0.392172\pi$
$701$	17.5967	0.664620	0.332310	+	0.943170i	$0.392172\pi$
$702$	−2.23607	−0.0843949
$703$	20.0000	0.754314
$704$	0	0
$705$	17.2361	0.649148
$706$	−4.76393	−0.179293
$707$	−1.00000	−0.0376089
$708$	−10.5279	−0.395661
$709$	19.0557	0.715653	0.357826	−	0.933788i	$-0.383518\pi$
$709$	19.0557	0.715653	0.357826	+	0.933788i	$0.383518\pi$
$710$	1.52786	0.0573397
$711$	−20.4721	−0.767765
$712$	−17.2361	−0.645949
$713$	19.7082	0.738078
$714$	2.76393	0.103438
$715$	0	0
$716$	−11.2361	−0.419912
$717$	23.9443	0.894215
$718$	−16.0000	−0.597115
$719$	6.83282	0.254821	0.127411	−	0.991850i	$-0.459333\pi$
$719$	6.83282	0.254821	0.127411	+	0.991850i	$0.459333\pi$
$720$	−2.00000	−0.0745356
$721$	−14.1803	−0.528103
$722$	−14.0000	−0.521026
$723$	−50.0000	−1.85952
$724$	7.00000	0.260153
$725$	−4.47214	−0.166091
$726$	0	0
$727$	−39.3050	−1.45774	−0.728870	−	0.684652i	$-0.759954\pi$
$727$	−39.3050	−1.45774	−0.728870	+	0.684652i	$0.759954\pi$
$728$	1.00000	0.0370625
$729$	−7.00000	−0.259259
$730$	−7.23607	−0.267819
$731$	−4.00000	−0.147945
$732$	−7.88854	−0.291569
$733$	48.4164	1.78830	0.894150	−	0.447767i	$-0.147780\pi$
$733$	48.4164	1.78830	0.894150	+	0.447767i	$0.147780\pi$
$734$	8.18034	0.301942
$735$	2.23607	0.0824786
$736$	−3.76393	−0.138740
$737$	0	0
$738$	14.4721	0.532727
$739$	−18.0689	−0.664675	−0.332337	−	0.943161i	$-0.607837\pi$
$739$	−18.0689	−0.664675	−0.332337	+	0.943161i	$0.607837\pi$
$740$	−8.94427	−0.328798
$741$	5.00000	0.183680
$742$	5.70820	0.209555
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	11.7082	0.429244
$745$	4.00000	0.146549
$746$	5.52786	0.202389
$747$	0.472136	0.0172746
$748$	0	0
$749$	−2.47214	−0.0903299
$750$	2.23607	0.0816497
$751$	−11.1803	−0.407976	−0.203988	−	0.978973i	$-0.565390\pi$
$751$	−11.1803	−0.407976	−0.203988	+	0.978973i	$0.565390\pi$
$752$	7.70820	0.281089
$753$	66.8328	2.43552
$754$	4.47214	0.162866
$755$	12.2361	0.445316
$756$	−2.23607	−0.0813250
$757$	−42.6525	−1.55023	−0.775115	−	0.631820i	$-0.782308\pi$
$757$	−42.6525	−1.55023	−0.775115	+	0.631820i	$0.782308\pi$
$758$	11.4164	0.414663
$759$	0	0
$760$	−2.23607	−0.0811107
$761$	30.5410	1.10711	0.553556	−	0.832812i	$-0.313271\pi$
$761$	30.5410	1.10711	0.553556	+	0.832812i	$0.313271\pi$
$762$	13.9443	0.505148
$763$	−3.70820	−0.134246
$764$	25.1803	0.910993
$765$	−2.47214	−0.0893803
$766$	−17.2361	−0.622764
$767$	−4.70820	−0.170003
$768$	−2.23607	−0.0806872
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	10.6525	0.383640
$772$	−7.94427	−0.285921
$773$	−27.9443	−1.00509	−0.502543	−	0.864552i	$-0.667602\pi$
$773$	−27.9443	−1.00509	−0.502543	+	0.864552i	$0.667602\pi$
$774$	−6.47214	−0.232636
$775$	−5.23607	−0.188085
$776$	3.52786	0.126643
$777$	20.0000	0.717496
$778$	−18.2918	−0.655793
$779$	16.1803	0.579721
$780$	−2.23607	−0.0800641
$781$	0	0
$782$	−4.65248	−0.166372
$783$	−10.0000	−0.357371
$784$	1.00000	0.0357143
$785$	−8.52786	−0.304373
$786$	−7.11146	−0.253657
$787$	31.7771	1.13273	0.566365	−	0.824154i	$-0.308349\pi$
$787$	31.7771	1.13273	0.566365	+	0.824154i	$0.308349\pi$
$788$	7.52786	0.268169
$789$	17.1115	0.609184
$790$	10.2361	0.364183
$791$	1.00000	0.0355559
$792$	0	0
$793$	−3.52786	−0.125278
$794$	6.00000	0.212932
$795$	−12.7639	−0.452690
$796$	−4.47214	−0.158511
$797$	28.8885	1.02328	0.511642	−	0.859199i	$-0.329037\pi$
$797$	28.8885	1.02328	0.511642	+	0.859199i	$0.329037\pi$
$798$	5.00000	0.176998
$799$	9.52786	0.337072
$800$	1.00000	0.0353553
$801$	−34.4721	−1.21801
$802$	−28.8328	−1.01812
$803$	0	0
$804$	16.1803	0.570637
$805$	−3.76393	−0.132661
$806$	5.23607	0.184433
$807$	62.2361	2.19081
$808$	1.00000	0.0351799
$809$	1.63932	0.0576354	0.0288177	−	0.999585i	$-0.490826\pi$
$809$	1.63932	0.0576354	0.0288177	+	0.999585i	$0.490826\pi$
$810$	11.0000	0.386501
$811$	−2.47214	−0.0868084	−0.0434042	−	0.999058i	$-0.513820\pi$
$811$	−2.47214	−0.0868084	−0.0434042	+	0.999058i	$0.513820\pi$
$812$	4.47214	0.156941
$813$	16.5836	0.581612
$814$	0	0
$815$	6.00000	0.210171
$816$	−2.76393	−0.0967570
$817$	−7.23607	−0.253158
$818$	−11.5279	−0.403062
$819$	2.00000	0.0698857
$820$	−7.23607	−0.252694
$821$	−13.0557	−0.455648	−0.227824	−	0.973702i	$-0.573161\pi$
$821$	−13.0557	−0.455648	−0.227824	+	0.973702i	$0.573161\pi$
$822$	−30.1246	−1.05072
$823$	16.3607	0.570297	0.285149	−	0.958483i	$-0.407957\pi$
$823$	16.3607	0.570297	0.285149	+	0.958483i	$0.407957\pi$
$824$	14.1803	0.493996
$825$	0	0
$826$	−4.70820	−0.163819
$827$	−12.1803	−0.423552	−0.211776	−	0.977318i	$-0.567925\pi$
$827$	−12.1803	−0.423552	−0.211776	+	0.977318i	$0.567925\pi$
$828$	−7.52786	−0.261611
$829$	−11.9443	−0.414842	−0.207421	−	0.978252i	$-0.566507\pi$
$829$	−11.9443	−0.414842	−0.207421	+	0.978252i	$0.566507\pi$
$830$	−0.236068	−0.00819404
$831$	52.7639	1.83036
$832$	−1.00000	−0.0346688
$833$	1.23607	0.0428272
$834$	9.47214	0.327993
$835$	1.70820	0.0591148
$836$	0	0
$837$	−11.7082	−0.404695
$838$	−28.2361	−0.975399
$839$	−13.5967	−0.469412	−0.234706	−	0.972066i	$-0.575413\pi$
$839$	−13.5967	−0.469412	−0.234706	+	0.972066i	$0.575413\pi$
$840$	−2.23607	−0.0771517
$841$	−9.00000	−0.310345
$842$	−3.05573	−0.105307
$843$	57.7639	1.98950
$844$	−3.52786	−0.121434
$845$	12.0000	0.412813
$846$	15.4164	0.530027
$847$	0	0
$848$	−5.70820	−0.196021
$849$	58.6656	2.01340
$850$	1.23607	0.0423968
$851$	−33.6656	−1.15404
$852$	3.41641	0.117044
$853$	−3.47214	−0.118884	−0.0594418	−	0.998232i	$-0.518932\pi$
$853$	−3.47214	−0.118884	−0.0594418	+	0.998232i	$0.518932\pi$
$854$	−3.52786	−0.120721
$855$	−4.47214	−0.152944
$856$	2.47214	0.0844959
$857$	−10.8328	−0.370042	−0.185021	−	0.982735i	$-0.559235\pi$
$857$	−10.8328	−0.370042	−0.185021	+	0.982735i	$0.559235\pi$
$858$	0	0
$859$	0.944272	0.0322181	0.0161091	−	0.999870i	$-0.494872\pi$
$859$	0.944272	0.0322181	0.0161091	+	0.999870i	$0.494872\pi$
$860$	3.23607	0.110349
$861$	16.1803	0.551425
$862$	33.6525	1.14621
$863$	−56.7214	−1.93082	−0.965409	−	0.260741i	$-0.916033\pi$
$863$	−56.7214	−1.93082	−0.965409	+	0.260741i	$0.916033\pi$
$864$	2.23607	0.0760726
$865$	19.8885	0.676231
$866$	−12.6525	−0.429949
$867$	34.5967	1.17497
$868$	5.23607	0.177724
$869$	0	0
$870$	−10.0000	−0.339032
$871$	7.23607	0.245185
$872$	3.70820	0.125576
$873$	7.05573	0.238800
$874$	−8.41641	−0.284689
$875$	1.00000	0.0338062
$876$	−16.1803	−0.546683
$877$	51.9574	1.75448	0.877239	−	0.480054i	$-0.159383\pi$
$877$	51.9574	1.75448	0.877239	+	0.480054i	$0.159383\pi$
$878$	−15.1246	−0.510431
$879$	−11.1803	−0.377104
$880$	0	0
$881$	−4.94427	−0.166577	−0.0832884	−	0.996525i	$-0.526542\pi$
$881$	−4.94427	−0.166577	−0.0832884	+	0.996525i	$0.526542\pi$
$882$	2.00000	0.0673435
$883$	−22.4721	−0.756248	−0.378124	−	0.925755i	$-0.623431\pi$
$883$	−22.4721	−0.756248	−0.378124	+	0.925755i	$0.623431\pi$
$884$	−1.23607	−0.0415735
$885$	10.5279	0.353890
$886$	−24.3607	−0.818413
$887$	−5.23607	−0.175810	−0.0879050	−	0.996129i	$-0.528017\pi$
$887$	−5.23607	−0.175810	−0.0879050	+	0.996129i	$0.528017\pi$
$888$	−20.0000	−0.671156
$889$	6.23607	0.209151
$890$	17.2361	0.577754
$891$	0	0
$892$	−8.18034	−0.273898
$893$	17.2361	0.576783
$894$	8.94427	0.299141
$895$	11.2361	0.375580
$896$	−1.00000	−0.0334077
$897$	−8.41641	−0.281016
$898$	23.8328	0.795311
$899$	23.4164	0.780981
$900$	2.00000	0.0666667
$901$	−7.05573	−0.235060
$902$	0	0
$903$	−7.23607	−0.240801
$904$	−1.00000	−0.0332595
$905$	−7.00000	−0.232688
$906$	27.3607	0.908998
$907$	40.3607	1.34015	0.670077	−	0.742291i	$-0.266261\pi$
$907$	40.3607	1.34015	0.670077	+	0.742291i	$0.266261\pi$
$908$	7.41641	0.246122
$909$	2.00000	0.0663358
$910$	−1.00000	−0.0331497
$911$	−43.6525	−1.44627	−0.723136	−	0.690706i	$-0.757300\pi$
$911$	−43.6525	−1.44627	−0.723136	+	0.690706i	$0.757300\pi$
$912$	−5.00000	−0.165567
$913$	0	0
$914$	36.4164	1.20455
$915$	7.88854	0.260787
$916$	6.94427	0.229445
$917$	−3.18034	−0.105024
$918$	2.76393	0.0912234
$919$	−12.9443	−0.426992	−0.213496	−	0.976944i	$-0.568485\pi$
$919$	−12.9443	−0.426992	−0.213496	+	0.976944i	$0.568485\pi$
$920$	3.76393	0.124093
$921$	25.5279	0.841172
$922$	23.8885	0.786727
$923$	1.52786	0.0502903
$924$	0	0
$925$	8.94427	0.294086
$926$	10.7082	0.351893
$927$	28.3607	0.931487
$928$	−4.47214	−0.146805
$929$	−42.8328	−1.40530	−0.702650	−	0.711536i	$-0.748000\pi$
$929$	−42.8328	−1.40530	−0.702650	+	0.711536i	$0.748000\pi$
$930$	−11.7082	−0.383927
$931$	2.23607	0.0732842
$932$	−10.4164	−0.341201
$933$	60.6525	1.98567
$934$	−2.81966	−0.0922621
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−20.4721	−0.668796	−0.334398	−	0.942432i	$-0.608533\pi$
$937$	−20.4721	−0.668796	−0.334398	+	0.942432i	$0.608533\pi$
$938$	7.23607	0.236266
$939$	−43.6656	−1.42497
$940$	−7.70820	−0.251414
$941$	55.8885	1.82191	0.910957	−	0.412501i	$-0.135345\pi$
$941$	55.8885	1.82191	0.910957	+	0.412501i	$0.135345\pi$
$942$	−19.0689	−0.621298
$943$	−27.2361	−0.886928
$944$	4.70820	0.153239
$945$	2.23607	0.0727393
$946$	0	0
$947$	10.0689	0.327195	0.163597	−	0.986527i	$-0.447690\pi$
$947$	10.0689	0.327195	0.163597	+	0.986527i	$0.447690\pi$
$948$	22.8885	0.743385
$949$	−7.23607	−0.234893
$950$	2.23607	0.0725476
$951$	−1.45898	−0.0473107
$952$	−1.23607	−0.0400612
$953$	−24.8885	−0.806219	−0.403110	−	0.915152i	$-0.632071\pi$
$953$	−24.8885	−0.806219	−0.403110	+	0.915152i	$0.632071\pi$
$954$	−11.4164	−0.369620
$955$	−25.1803	−0.814817
$956$	−10.7082	−0.346328
$957$	0	0
$958$	27.5967	0.891610
$959$	−13.4721	−0.435038
$960$	2.23607	0.0721688
$961$	−3.58359	−0.115600
$962$	−8.94427	−0.288375
$963$	4.94427	0.159327
$964$	22.3607	0.720189
$965$	7.94427	0.255735
$966$	−8.41641	−0.270793
$967$	−55.1935	−1.77490	−0.887452	−	0.460901i	$-0.847526\pi$
$967$	−55.1935	−1.77490	−0.887452	+	0.460901i	$0.847526\pi$
$968$	0	0
$969$	−6.18034	−0.198541
$970$	−3.52786	−0.113273
$971$	−16.1246	−0.517463	−0.258732	−	0.965949i	$-0.583305\pi$
$971$	−16.1246	−0.517463	−0.258732	+	0.965949i	$0.583305\pi$
$972$	17.8885	0.573775
$973$	4.23607	0.135802
$974$	2.23607	0.0716482
$975$	2.23607	0.0716115
$976$	3.52786	0.112924
$977$	−45.9443	−1.46989	−0.734944	−	0.678128i	$-0.762791\pi$
$977$	−45.9443	−1.46989	−0.734944	+	0.678128i	$0.762791\pi$
$978$	13.4164	0.429009
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	7.41641	0.236788
$982$	−31.2361	−0.996783
$983$	43.1935	1.37766	0.688829	−	0.724924i	$-0.258125\pi$
$983$	43.1935	1.37766	0.688829	+	0.724924i	$0.258125\pi$
$984$	−16.1803	−0.515810
$985$	−7.52786	−0.239858
$986$	−5.52786	−0.176043
$987$	17.2361	0.548630
$988$	−2.23607	−0.0711388
$989$	12.1803	0.387312
$990$	0	0
$991$	19.2918	0.612824	0.306412	−	0.951899i	$-0.400871\pi$
$991$	19.2918	0.612824	0.306412	+	0.951899i	$0.400871\pi$
$992$	−5.23607	−0.166245
$993$	48.5410	1.54040
$994$	1.52786	0.0484609
$995$	4.47214	0.141776
$996$	−0.527864	−0.0167260
$997$	59.8328	1.89492	0.947462	−	0.319868i	$-0.103639\pi$
$997$	59.8328	1.89492	0.947462	+	0.319868i	$0.103639\pi$
$998$	26.8328	0.849378
$999$	20.0000	0.632772

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.bn.1.1	✓	2	11.10	odd	2
8470.2.a.by.1.1	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.23607 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.23607 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{9} -1.00000 q^{10} -2.23607 q^{12} -1.00000 q^{13} -1.00000 q^{14} +2.23607 q^{15} +1.00000 q^{16} +1.23607 q^{17} +2.00000 q^{18} +2.23607 q^{19} -1.00000 q^{20} +2.23607 q^{21} -3.76393 q^{23} -2.23607 q^{24} +1.00000 q^{25} -1.00000 q^{26} +2.23607 q^{27} -1.00000 q^{28} -4.47214 q^{29} +2.23607 q^{30} -5.23607 q^{31} +1.00000 q^{32} +1.23607 q^{34} +1.00000 q^{35} +2.00000 q^{36} +8.94427 q^{37} +2.23607 q^{38} +2.23607 q^{39} -1.00000 q^{40} +7.23607 q^{41} +2.23607 q^{42} -3.23607 q^{43} -2.00000 q^{45} -3.76393 q^{46} +7.70820 q^{47} -2.23607 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.76393 q^{51} -1.00000 q^{52} -5.70820 q^{53} +2.23607 q^{54} -1.00000 q^{56} -5.00000 q^{57} -4.47214 q^{58} +4.70820 q^{59} +2.23607 q^{60} +3.52786 q^{61} -5.23607 q^{62} -2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -7.23607 q^{67} +1.23607 q^{68} +8.41641 q^{69} +1.00000 q^{70} -1.52786 q^{71} +2.00000 q^{72} +7.23607 q^{73} +8.94427 q^{74} -2.23607 q^{75} +2.23607 q^{76} +2.23607 q^{78} -10.2361 q^{79} -1.00000 q^{80} -11.0000 q^{81} +7.23607 q^{82} +0.236068 q^{83} +2.23607 q^{84} -1.23607 q^{85} -3.23607 q^{86} +10.0000 q^{87} -17.2361 q^{89} -2.00000 q^{90} +1.00000 q^{91} -3.76393 q^{92} +11.7082 q^{93} +7.70820 q^{94} -2.23607 q^{95} -2.23607 q^{96} +3.52786 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 + 4 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + 4 q^{9} - 2 q^{10} - 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 4 q^{18} - 2 q^{20} - 12 q^{23} + 2 q^{25} - 2 q^{26} - 2 q^{28} - 6 q^{31} + 2 q^{32} - 2 q^{34} + 2 q^{35} + 4 q^{36} - 2 q^{40} + 10 q^{41} - 2 q^{43} - 4 q^{45} - 12 q^{46} + 2 q^{47} + 2 q^{49} + 2 q^{50} - 10 q^{51} - 2 q^{52} + 2 q^{53} - 2 q^{56} - 10 q^{57} - 4 q^{59} + 16 q^{61} - 6 q^{62} - 4 q^{63} + 2 q^{64} + 2 q^{65} - 10 q^{67} - 2 q^{68} - 10 q^{69} + 2 q^{70} - 12 q^{71} + 4 q^{72} + 10 q^{73} - 16 q^{79} - 2 q^{80} - 22 q^{81} + 10 q^{82} - 4 q^{83} + 2 q^{85} - 2 q^{86} + 20 q^{87} - 30 q^{89} - 4 q^{90} + 2 q^{91} - 12 q^{92} + 10 q^{93} + 2 q^{94} + 16 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 + 4 * q^9 - 2 * q^10 - 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 + 4 * q^18 - 2 * q^20 - 12 * q^23 + 2 * q^25 - 2 * q^26 - 2 * q^28 - 6 * q^31 + 2 * q^32 - 2 * q^34 + 2 * q^35 + 4 * q^36 - 2 * q^40 + 10 * q^41 - 2 * q^43 - 4 * q^45 - 12 * q^46 + 2 * q^47 + 2 * q^49 + 2 * q^50 - 10 * q^51 - 2 * q^52 + 2 * q^53 - 2 * q^56 - 10 * q^57 - 4 * q^59 + 16 * q^61 - 6 * q^62 - 4 * q^63 + 2 * q^64 + 2 * q^65 - 10 * q^67 - 2 * q^68 - 10 * q^69 + 2 * q^70 - 12 * q^71 + 4 * q^72 + 10 * q^73 - 16 * q^79 - 2 * q^80 - 22 * q^81 + 10 * q^82 - 4 * q^83 + 2 * q^85 - 2 * q^86 + 20 * q^87 - 30 * q^89 - 4 * q^90 + 2 * q^91 - 12 * q^92 + 10 * q^93 + 2 * q^94 + 16 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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