LMFDB - Embedded newform 8470.2.a.da.1.3

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:	$6$
Coefficient field:	6.6.38498000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 12x^{4} + 5x^{3} + 38x^{2} - x - 19 $ x^6 - x^5 - 12x^4 + 5x^3 + 38*x^2 - x - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.824369$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.82437 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.82437 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.328323 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.82437 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.82437 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.328323 q^{9} -1.00000 q^{10} -1.82437 q^{12} -2.21432 q^{13} +1.00000 q^{14} +1.82437 q^{15} +1.00000 q^{16} -7.96638 q^{17} +0.328323 q^{18} -0.915653 q^{19} -1.00000 q^{20} -1.82437 q^{21} +6.07737 q^{23} -1.82437 q^{24} +1.00000 q^{25} -2.21432 q^{26} +4.87412 q^{27} +1.00000 q^{28} +5.06597 q^{29} +1.82437 q^{30} +9.33892 q^{31} +1.00000 q^{32} -7.96638 q^{34} -1.00000 q^{35} +0.328323 q^{36} +4.32937 q^{37} -0.915653 q^{38} +4.03973 q^{39} -1.00000 q^{40} -8.99953 q^{41} -1.82437 q^{42} +2.76947 q^{43} -0.328323 q^{45} +6.07737 q^{46} +4.22816 q^{47} -1.82437 q^{48} +1.00000 q^{49} +1.00000 q^{50} +14.5336 q^{51} -2.21432 q^{52} -9.33444 q^{53} +4.87412 q^{54} +1.00000 q^{56} +1.67049 q^{57} +5.06597 q^{58} +7.19487 q^{59} +1.82437 q^{60} -12.6895 q^{61} +9.33892 q^{62} +0.328323 q^{63} +1.00000 q^{64} +2.21432 q^{65} -1.29018 q^{67} -7.96638 q^{68} -11.0874 q^{69} -1.00000 q^{70} +0.0202370 q^{71} +0.328323 q^{72} +11.8463 q^{73} +4.32937 q^{74} -1.82437 q^{75} -0.915653 q^{76} +4.03973 q^{78} -9.39226 q^{79} -1.00000 q^{80} -9.87717 q^{81} -8.99953 q^{82} +0.747575 q^{83} -1.82437 q^{84} +7.96638 q^{85} +2.76947 q^{86} -9.24220 q^{87} -14.2639 q^{89} -0.328323 q^{90} -2.21432 q^{91} +6.07737 q^{92} -17.0376 q^{93} +4.22816 q^{94} +0.915653 q^{95} -1.82437 q^{96} -7.27431 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 5 q^{3} + 6 q^{4} - 6 q^{5} - 5 q^{6} + 6 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - 5 * q^3 + 6 * q^4 - 6 * q^5 - 5 * q^6 + 6 * q^7 + 6 * q^8 + 11 * q^9	$ 6 q + 6 q^{2} - 5 q^{3} + 6 q^{4} - 6 q^{5} - 5 q^{6} + 6 q^{7} + 6 q^{8} + 11 q^{9} - 6 q^{10} - 5 q^{12} + 6 q^{14} + 5 q^{15} + 6 q^{16} - 15 q^{17} + 11 q^{18} - 13 q^{19} - 6 q^{20} - 5 q^{21} - 2 q^{23} - 5 q^{24} + 6 q^{25} - 26 q^{27} + 6 q^{28} - 4 q^{29} + 5 q^{30} - 2 q^{31} + 6 q^{32} - 15 q^{34} - 6 q^{35} + 11 q^{36} - 13 q^{38} - 30 q^{39} - 6 q^{40} - 17 q^{41} - 5 q^{42} + 15 q^{43} - 11 q^{45} - 2 q^{46} - 18 q^{47} - 5 q^{48} + 6 q^{49} + 6 q^{50} + 6 q^{51} + 22 q^{53} - 26 q^{54} + 6 q^{56} - 2 q^{57} - 4 q^{58} - 7 q^{59} + 5 q^{60} - 20 q^{61} - 2 q^{62} + 11 q^{63} + 6 q^{64} - 29 q^{67} - 15 q^{68} + 12 q^{69} - 6 q^{70} - 2 q^{71} + 11 q^{72} + q^{73} - 5 q^{75} - 13 q^{76} - 30 q^{78} - 12 q^{79} - 6 q^{80} + 22 q^{81} - 17 q^{82} - 35 q^{83} - 5 q^{84} + 15 q^{85} + 15 q^{86} - 29 q^{89} - 11 q^{90} - 2 q^{92} + 8 q^{93} - 18 q^{94} + 13 q^{95} - 5 q^{96} - 19 q^{97} + 6 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - 5 * q^3 + 6 * q^4 - 6 * q^5 - 5 * q^6 + 6 * q^7 + 6 * q^8 + 11 * q^9 - 6 * q^10 - 5 * q^12 + 6 * q^14 + 5 * q^15 + 6 * q^16 - 15 * q^17 + 11 * q^18 - 13 * q^19 - 6 * q^20 - 5 * q^21 - 2 * q^23 - 5 * q^24 + 6 * q^25 - 26 * q^27 + 6 * q^28 - 4 * q^29 + 5 * q^30 - 2 * q^31 + 6 * q^32 - 15 * q^34 - 6 * q^35 + 11 * q^36 - 13 * q^38 - 30 * q^39 - 6 * q^40 - 17 * q^41 - 5 * q^42 + 15 * q^43 - 11 * q^45 - 2 * q^46 - 18 * q^47 - 5 * q^48 + 6 * q^49 + 6 * q^50 + 6 * q^51 + 22 * q^53 - 26 * q^54 + 6 * q^56 - 2 * q^57 - 4 * q^58 - 7 * q^59 + 5 * q^60 - 20 * q^61 - 2 * q^62 + 11 * q^63 + 6 * q^64 - 29 * q^67 - 15 * q^68 + 12 * q^69 - 6 * q^70 - 2 * q^71 + 11 * q^72 + q^73 - 5 * q^75 - 13 * q^76 - 30 * q^78 - 12 * q^79 - 6 * q^80 + 22 * q^81 - 17 * q^82 - 35 * q^83 - 5 * q^84 + 15 * q^85 + 15 * q^86 - 29 * q^89 - 11 * q^90 - 2 * q^92 + 8 * q^93 - 18 * q^94 + 13 * q^95 - 5 * q^96 - 19 * q^97 + 6 * 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
$3$	−1.82437	−1.05330	−0.526650	−	0.850082i	$-0.676552\pi$
$3$	−1.82437	−1.05330	−0.526650	+	0.850082i	$0.676552\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.82437	−0.744796
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0.328323	0.109441
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−1.82437	−0.526650
$13$	−2.21432	−0.614141	−0.307071	−	0.951687i	$-0.599349\pi$
$13$	−2.21432	−0.614141	−0.307071	+	0.951687i	$0.599349\pi$
$14$	1.00000	0.267261
$15$	1.82437	0.471050
$16$	1.00000	0.250000
$17$	−7.96638	−1.93213	−0.966066	−	0.258297i	$-0.916839\pi$
$17$	−7.96638	−1.93213	−0.966066	+	0.258297i	$0.916839\pi$
$18$	0.328323	0.0773866
$19$	−0.915653	−0.210065	−0.105033	−	0.994469i	$-0.533495\pi$
$19$	−0.915653	−0.210065	−0.105033	+	0.994469i	$0.533495\pi$
$20$	−1.00000	−0.223607
$21$	−1.82437	−0.398110
$22$	0	0
$23$	6.07737	1.26722	0.633610	−	0.773653i	$-0.281572\pi$
$23$	6.07737	1.26722	0.633610	+	0.773653i	$0.281572\pi$
$24$	−1.82437	−0.372398
$25$	1.00000	0.200000
$26$	−2.21432	−0.434263
$27$	4.87412	0.938026
$28$	1.00000	0.188982
$29$	5.06597	0.940727	0.470364	−	0.882473i	$-0.344123\pi$
$29$	5.06597	0.940727	0.470364	+	0.882473i	$0.344123\pi$
$30$	1.82437	0.333083
$31$	9.33892	1.67732	0.838660	−	0.544655i	$-0.183339\pi$
$31$	9.33892	1.67732	0.838660	+	0.544655i	$0.183339\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.96638	−1.36622
$35$	−1.00000	−0.169031
$36$	0.328323	0.0547206
$37$	4.32937	0.711744	0.355872	−	0.934535i	$-0.384184\pi$
$37$	4.32937	0.711744	0.355872	+	0.934535i	$0.384184\pi$
$38$	−0.915653	−0.148538
$39$	4.03973	0.646875
$40$	−1.00000	−0.158114
$41$	−8.99953	−1.40549	−0.702745	−	0.711441i	$-0.748043\pi$
$41$	−8.99953	−1.40549	−0.702745	+	0.711441i	$0.748043\pi$
$42$	−1.82437	−0.281506
$43$	2.76947	0.422340	0.211170	−	0.977449i	$-0.432273\pi$
$43$	2.76947	0.422340	0.211170	+	0.977449i	$0.432273\pi$
$44$	0	0
$45$	−0.328323	−0.0489436
$46$	6.07737	0.896060
$47$	4.22816	0.616740	0.308370	−	0.951266i	$-0.400216\pi$
$47$	4.22816	0.616740	0.308370	+	0.951266i	$0.400216\pi$
$48$	−1.82437	−0.263325
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	14.5336	2.03511
$52$	−2.21432	−0.307071
$53$	−9.33444	−1.28218	−0.641092	−	0.767464i	$-0.721518\pi$
$53$	−9.33444	−1.28218	−0.641092	+	0.767464i	$0.721518\pi$
$54$	4.87412	0.663284
$55$	0	0
$56$	1.00000	0.133631
$57$	1.67049	0.221262
$58$	5.06597	0.665195
$59$	7.19487	0.936693	0.468346	−	0.883545i	$-0.344850\pi$
$59$	7.19487	0.936693	0.468346	+	0.883545i	$0.344850\pi$
$60$	1.82437	0.235525
$61$	−12.6895	−1.62473	−0.812365	−	0.583150i	$-0.801820\pi$
$61$	−12.6895	−1.62473	−0.812365	+	0.583150i	$0.801820\pi$
$62$	9.33892	1.18604
$63$	0.328323	0.0413649
$64$	1.00000	0.125000
$65$	2.21432	0.274652
$66$	0	0
$67$	−1.29018	−0.157620	−0.0788101	−	0.996890i	$-0.525112\pi$
$67$	−1.29018	−0.157620	−0.0788101	+	0.996890i	$0.525112\pi$
$68$	−7.96638	−0.966066
$69$	−11.0874	−1.33476
$70$	−1.00000	−0.119523
$71$	0.0202370	0.00240169	0.00120084	−	0.999999i	$-0.499618\pi$
$71$	0.0202370	0.00240169	0.00120084	+	0.999999i	$0.499618\pi$
$72$	0.328323	0.0386933
$73$	11.8463	1.38650	0.693250	−	0.720698i	$-0.256178\pi$
$73$	11.8463	1.38650	0.693250	+	0.720698i	$0.256178\pi$
$74$	4.32937	0.503279
$75$	−1.82437	−0.210660
$76$	−0.915653	−0.105033
$77$	0	0
$78$	4.03973	0.457410
$79$	−9.39226	−1.05671	−0.528356	−	0.849023i	$-0.677191\pi$
$79$	−9.39226	−1.05671	−0.528356	+	0.849023i	$0.677191\pi$
$80$	−1.00000	−0.111803
$81$	−9.87717	−1.09746
$82$	−8.99953	−0.993832
$83$	0.747575	0.0820570	0.0410285	−	0.999158i	$-0.486937\pi$
$83$	0.747575	0.0820570	0.0410285	+	0.999158i	$0.486937\pi$
$84$	−1.82437	−0.199055
$85$	7.96638	0.864075
$86$	2.76947	0.298639
$87$	−9.24220	−0.990868
$88$	0	0
$89$	−14.2639	−1.51197	−0.755983	−	0.654591i	$-0.772841\pi$
$89$	−14.2639	−1.51197	−0.755983	+	0.654591i	$0.772841\pi$
$90$	−0.328323	−0.0346083
$91$	−2.21432	−0.232124
$92$	6.07737	0.633610
$93$	−17.0376	−1.76672
$94$	4.22816	0.436101
$95$	0.915653	0.0939440
$96$	−1.82437	−0.186199
$97$	−7.27431	−0.738594	−0.369297	−	0.929311i	$-0.620402\pi$
$97$	−7.27431	−0.738594	−0.369297	+	0.929311i	$0.620402\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−5.35283	−0.532627	−0.266313	−	0.963886i	$-0.585806\pi$
$101$	−5.35283	−0.532627	−0.266313	+	0.963886i	$0.585806\pi$
$102$	14.5336	1.43904
$103$	−14.3938	−1.41826	−0.709130	−	0.705078i	$-0.750912\pi$
$103$	−14.3938	−1.41826	−0.709130	+	0.705078i	$0.750912\pi$
$104$	−2.21432	−0.217132
$105$	1.82437	0.178040
$106$	−9.33444	−0.906641
$107$	−4.27627	−0.413402	−0.206701	−	0.978404i	$-0.566273\pi$
$107$	−4.27627	−0.413402	−0.206701	+	0.978404i	$0.566273\pi$
$108$	4.87412	0.469013
$109$	−5.04265	−0.482998	−0.241499	−	0.970401i	$-0.577639\pi$
$109$	−5.04265	−0.482998	−0.241499	+	0.970401i	$0.577639\pi$
$110$	0	0
$111$	−7.89837	−0.749680
$112$	1.00000	0.0944911
$113$	13.8169	1.29979	0.649894	−	0.760025i	$-0.274813\pi$
$113$	13.8169	1.29979	0.649894	+	0.760025i	$0.274813\pi$
$114$	1.67049	0.156456
$115$	−6.07737	−0.566718
$116$	5.06597	0.470364
$117$	−0.727012	−0.0672123
$118$	7.19487	0.662342
$119$	−7.96638	−0.730277
$120$	1.82437	0.166541
$121$	0	0
$122$	−12.6895	−1.14886
$123$	16.4185	1.48040
$124$	9.33892	0.838660
$125$	−1.00000	−0.0894427
$126$	0.328323	0.0292494
$127$	14.6820	1.30281	0.651407	−	0.758729i	$-0.274179\pi$
$127$	14.6820	1.30281	0.651407	+	0.758729i	$0.274179\pi$
$128$	1.00000	0.0883883
$129$	−5.05253	−0.444850
$130$	2.21432	0.194209
$131$	−7.62582	−0.666271	−0.333135	−	0.942879i	$-0.608107\pi$
$131$	−7.62582	−0.666271	−0.333135	+	0.942879i	$0.608107\pi$
$132$	0	0
$133$	−0.915653	−0.0793972
$134$	−1.29018	−0.111454
$135$	−4.87412	−0.419498
$136$	−7.96638	−0.683111
$137$	−4.93059	−0.421248	−0.210624	−	0.977567i	$-0.567550\pi$
$137$	−4.93059	−0.421248	−0.210624	+	0.977567i	$0.567550\pi$
$138$	−11.0874	−0.943820
$139$	−4.17718	−0.354304	−0.177152	−	0.984184i	$-0.556688\pi$
$139$	−4.17718	−0.354304	−0.177152	+	0.984184i	$0.556688\pi$
$140$	−1.00000	−0.0845154
$141$	−7.71373	−0.649613
$142$	0.0202370	0.00169825
$143$	0	0
$144$	0.328323	0.0273603
$145$	−5.06597	−0.420706
$146$	11.8463	0.980403
$147$	−1.82437	−0.150471
$148$	4.32937	0.355872
$149$	−13.4785	−1.10420	−0.552102	−	0.833776i	$-0.686174\pi$
$149$	−13.4785	−1.10420	−0.552102	+	0.833776i	$0.686174\pi$
$150$	−1.82437	−0.148959
$151$	17.6102	1.43309	0.716547	−	0.697538i	$-0.245721\pi$
$151$	17.6102	1.43309	0.716547	+	0.697538i	$0.245721\pi$
$152$	−0.915653	−0.0742692
$153$	−2.61555	−0.211455
$154$	0	0
$155$	−9.33892	−0.750121
$156$	4.03973	0.323438
$157$	11.4836	0.916491	0.458245	−	0.888826i	$-0.348478\pi$
$157$	11.4836	0.916491	0.458245	+	0.888826i	$0.348478\pi$
$158$	−9.39226	−0.747208
$159$	17.0295	1.35052
$160$	−1.00000	−0.0790569
$161$	6.07737	0.478964
$162$	−9.87717	−0.776024
$163$	−0.139250	−0.0109069	−0.00545347	−	0.999985i	$-0.501736\pi$
$163$	−0.139250	−0.0109069	−0.00545347	+	0.999985i	$0.501736\pi$
$164$	−8.99953	−0.702745
$165$	0	0
$166$	0.747575	0.0580230
$167$	7.04367	0.545056	0.272528	−	0.962148i	$-0.412140\pi$
$167$	7.04367	0.545056	0.272528	+	0.962148i	$0.412140\pi$
$168$	−1.82437	−0.140753
$169$	−8.09680	−0.622830
$170$	7.96638	0.610993
$171$	−0.300630	−0.0229898
$172$	2.76947	0.211170
$173$	−14.0443	−1.06777	−0.533883	−	0.845558i	$-0.679268\pi$
$173$	−14.0443	−1.06777	−0.533883	+	0.845558i	$0.679268\pi$
$174$	−9.24220	−0.700650
$175$	1.00000	0.0755929
$176$	0	0
$177$	−13.1261	−0.986619
$178$	−14.2639	−1.06912
$179$	12.8339	0.959254	0.479627	−	0.877473i	$-0.340772\pi$
$179$	12.8339	0.959254	0.479627	+	0.877473i	$0.340772\pi$
$180$	−0.328323	−0.0244718
$181$	19.2871	1.43360	0.716798	−	0.697281i	$-0.245607\pi$
$181$	19.2871	1.43360	0.716798	+	0.697281i	$0.245607\pi$
$182$	−2.21432	−0.164136
$183$	23.1504	1.71133
$184$	6.07737	0.448030
$185$	−4.32937	−0.318302
$186$	−17.0376	−1.24926
$187$	0	0
$188$	4.22816	0.308370
$189$	4.87412	0.354540
$190$	0.915653	0.0664284
$191$	−12.0109	−0.869076	−0.434538	−	0.900653i	$-0.643088\pi$
$191$	−12.0109	−0.869076	−0.434538	+	0.900653i	$0.643088\pi$
$192$	−1.82437	−0.131663
$193$	−17.2998	−1.24527	−0.622635	−	0.782512i	$-0.713938\pi$
$193$	−17.2998	−1.24527	−0.622635	+	0.782512i	$0.713938\pi$
$194$	−7.27431	−0.522265
$195$	−4.03973	−0.289291
$196$	1.00000	0.0714286
$197$	−13.1724	−0.938497	−0.469248	−	0.883066i	$-0.655475\pi$
$197$	−13.1724	−0.938497	−0.469248	+	0.883066i	$0.655475\pi$
$198$	0	0
$199$	−22.9779	−1.62886	−0.814431	−	0.580260i	$-0.802951\pi$
$199$	−22.9779	−1.62886	−0.814431	+	0.580260i	$0.802951\pi$
$200$	1.00000	0.0707107
$201$	2.35376	0.166021
$202$	−5.35283	−0.376624
$203$	5.06597	0.355561
$204$	14.5336	1.01756
$205$	8.99953	0.628555
$206$	−14.3938	−1.00286
$207$	1.99534	0.138686
$208$	−2.21432	−0.153535
$209$	0	0
$210$	1.82437	0.125893
$211$	−16.5609	−1.14010	−0.570050	−	0.821610i	$-0.693076\pi$
$211$	−16.5609	−1.14010	−0.570050	+	0.821610i	$0.693076\pi$
$212$	−9.33444	−0.641092
$213$	−0.0369197	−0.00252970
$214$	−4.27627	−0.292320
$215$	−2.76947	−0.188876
$216$	4.87412	0.331642
$217$	9.33892	0.633968
$218$	−5.04265	−0.341531
$219$	−21.6119	−1.46040
$220$	0	0
$221$	17.6401	1.18660
$222$	−7.89837	−0.530104
$223$	−19.1235	−1.28060	−0.640301	−	0.768124i	$-0.721190\pi$
$223$	−19.1235	−1.28060	−0.640301	+	0.768124i	$0.721190\pi$
$224$	1.00000	0.0668153
$225$	0.328323	0.0218882
$226$	13.8169	0.919088
$227$	19.6989	1.30746	0.653732	−	0.756726i	$-0.273202\pi$
$227$	19.6989	1.30746	0.653732	+	0.756726i	$0.273202\pi$
$228$	1.67049	0.110631
$229$	27.5813	1.82263	0.911313	−	0.411714i	$-0.135070\pi$
$229$	27.5813	1.82263	0.911313	+	0.411714i	$0.135070\pi$
$230$	−6.07737	−0.400730
$231$	0	0
$232$	5.06597	0.332597
$233$	−12.9463	−0.848141	−0.424071	−	0.905629i	$-0.639399\pi$
$233$	−12.9463	−0.848141	−0.424071	+	0.905629i	$0.639399\pi$
$234$	−0.727012	−0.0475263
$235$	−4.22816	−0.275815
$236$	7.19487	0.468346
$237$	17.1349	1.11303
$238$	−7.96638	−0.516384
$239$	−5.77626	−0.373635	−0.186818	−	0.982395i	$-0.559817\pi$
$239$	−5.77626	−0.373635	−0.186818	+	0.982395i	$0.559817\pi$
$240$	1.82437	0.117763
$241$	−25.0323	−1.61247	−0.806237	−	0.591593i	$-0.798499\pi$
$241$	−25.0323	−1.61247	−0.806237	+	0.591593i	$0.798499\pi$
$242$	0	0
$243$	3.39724	0.217933
$244$	−12.6895	−0.812365
$245$	−1.00000	−0.0638877
$246$	16.4185	1.04680
$247$	2.02755	0.129010
$248$	9.33892	0.593022
$249$	−1.36385	−0.0864306
$250$	−1.00000	−0.0632456
$251$	−6.66594	−0.420750	−0.210375	−	0.977621i	$-0.567469\pi$
$251$	−6.66594	−0.420750	−0.210375	+	0.977621i	$0.567469\pi$
$252$	0.328323	0.0206824
$253$	0	0
$254$	14.6820	0.921228
$255$	−14.5336	−0.910131
$256$	1.00000	0.0625000
$257$	−22.7153	−1.41694	−0.708472	−	0.705739i	$-0.750615\pi$
$257$	−22.7153	−1.41694	−0.708472	+	0.705739i	$0.750615\pi$
$258$	−5.05253	−0.314557
$259$	4.32937	0.269014
$260$	2.21432	0.137326
$261$	1.66328	0.102954
$262$	−7.62582	−0.471124
$263$	14.5015	0.894200	0.447100	−	0.894484i	$-0.352457\pi$
$263$	14.5015	0.894200	0.447100	+	0.894484i	$0.352457\pi$
$264$	0	0
$265$	9.33444	0.573410
$266$	−0.915653	−0.0561423
$267$	26.0225	1.59255
$268$	−1.29018	−0.0788101
$269$	−21.2753	−1.29718	−0.648590	−	0.761138i	$-0.724641\pi$
$269$	−21.2753	−1.29718	−0.648590	+	0.761138i	$0.724641\pi$
$270$	−4.87412	−0.296630
$271$	8.06611	0.489981	0.244991	−	0.969525i	$-0.421215\pi$
$271$	8.06611	0.489981	0.244991	+	0.969525i	$0.421215\pi$
$272$	−7.96638	−0.483033
$273$	4.03973	0.244496
$274$	−4.93059	−0.297868
$275$	0	0
$276$	−11.0874	−0.667382
$277$	17.4563	1.04885	0.524423	−	0.851458i	$-0.324281\pi$
$277$	17.4563	1.04885	0.524423	+	0.851458i	$0.324281\pi$
$278$	−4.17718	−0.250531
$279$	3.06619	0.183568
$280$	−1.00000	−0.0597614
$281$	−20.4688	−1.22106	−0.610532	−	0.791992i	$-0.709044\pi$
$281$	−20.4688	−1.22106	−0.610532	+	0.791992i	$0.709044\pi$
$282$	−7.71373	−0.459346
$283$	5.27282	0.313437	0.156718	−	0.987643i	$-0.449909\pi$
$283$	5.27282	0.313437	0.156718	+	0.987643i	$0.449909\pi$
$284$	0.0202370	0.00120084
$285$	−1.67049	−0.0989512
$286$	0	0
$287$	−8.99953	−0.531226
$288$	0.328323	0.0193466
$289$	46.4632	2.73313
$290$	−5.06597	−0.297484
$291$	13.2710	0.777961
$292$	11.8463	0.693250
$293$	6.40613	0.374250	0.187125	−	0.982336i	$-0.440083\pi$
$293$	6.40613	0.374250	0.187125	+	0.982336i	$0.440083\pi$
$294$	−1.82437	−0.106399
$295$	−7.19487	−0.418902
$296$	4.32937	0.251640
$297$	0	0
$298$	−13.4785	−0.780790
$299$	−13.4572	−0.778252
$300$	−1.82437	−0.105330
$301$	2.76947	0.159629
$302$	17.6102	1.01335
$303$	9.76555	0.561016
$304$	−0.915653	−0.0525163
$305$	12.6895	0.726601
$306$	−2.61555	−0.149521
$307$	5.35970	0.305894	0.152947	−	0.988234i	$-0.451124\pi$
$307$	5.35970	0.305894	0.152947	+	0.988234i	$0.451124\pi$
$308$	0	0
$309$	26.2595	1.49385
$310$	−9.33892	−0.530415
$311$	−26.9819	−1.53001	−0.765003	−	0.644027i	$-0.777262\pi$
$311$	−26.9819	−1.53001	−0.765003	+	0.644027i	$0.777262\pi$
$312$	4.03973	0.228705
$313$	−28.5633	−1.61449	−0.807245	−	0.590216i	$-0.799043\pi$
$313$	−28.5633	−1.61449	−0.807245	+	0.590216i	$0.799043\pi$
$314$	11.4836	0.648057
$315$	−0.328323	−0.0184989
$316$	−9.39226	−0.528356
$317$	21.0077	1.17991	0.589954	−	0.807437i	$-0.299146\pi$
$317$	21.0077	1.17991	0.589954	+	0.807437i	$0.299146\pi$
$318$	17.0295	0.954965
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	7.80149	0.435437
$322$	6.07737	0.338679
$323$	7.29444	0.405873
$324$	−9.87717	−0.548732
$325$	−2.21432	−0.122828
$326$	−0.139250	−0.00771236
$327$	9.19965	0.508742
$328$	−8.99953	−0.496916
$329$	4.22816	0.233106
$330$	0	0
$331$	11.6022	0.637717	0.318858	−	0.947802i	$-0.396701\pi$
$331$	11.6022	0.637717	0.318858	+	0.947802i	$0.396701\pi$
$332$	0.747575	0.0410285
$333$	1.42143	0.0778941
$334$	7.04367	0.385413
$335$	1.29018	0.0704899
$336$	−1.82437	−0.0995275
$337$	2.34874	0.127944	0.0639720	−	0.997952i	$-0.479623\pi$
$337$	2.34874	0.127944	0.0639720	+	0.997952i	$0.479623\pi$
$338$	−8.09680	−0.440408
$339$	−25.2072	−1.36907
$340$	7.96638	0.432038
$341$	0	0
$342$	−0.300630	−0.0162562
$343$	1.00000	0.0539949
$344$	2.76947	0.149320
$345$	11.0874	0.596924
$346$	−14.0443	−0.755025
$347$	−31.4811	−1.68999	−0.844997	−	0.534771i	$-0.820398\pi$
$347$	−31.4811	−1.68999	−0.844997	+	0.534771i	$0.820398\pi$
$348$	−9.24220	−0.495434
$349$	2.60777	0.139591	0.0697955	−	0.997561i	$-0.477765\pi$
$349$	2.60777	0.139591	0.0697955	+	0.997561i	$0.477765\pi$
$350$	1.00000	0.0534522
$351$	−10.7929	−0.576080
$352$	0	0
$353$	−0.416007	−0.0221418	−0.0110709	−	0.999939i	$-0.503524\pi$
$353$	−0.416007	−0.0221418	−0.0110709	+	0.999939i	$0.503524\pi$
$354$	−13.1261	−0.697645
$355$	−0.0202370	−0.00107407
$356$	−14.2639	−0.755983
$357$	14.5336	0.769201
$358$	12.8339	0.678295
$359$	−11.6150	−0.613017	−0.306508	−	0.951868i	$-0.599161\pi$
$359$	−11.6150	−0.613017	−0.306508	+	0.951868i	$0.599161\pi$
$360$	−0.328323	−0.0173042
$361$	−18.1616	−0.955873
$362$	19.2871	1.01371
$363$	0	0
$364$	−2.21432	−0.116062
$365$	−11.8463	−0.620061
$366$	23.1504	1.21009
$367$	−8.00709	−0.417966	−0.208983	−	0.977919i	$-0.567015\pi$
$367$	−8.00709	−0.417966	−0.208983	+	0.977919i	$0.567015\pi$
$368$	6.07737	0.316805
$369$	−2.95476	−0.153819
$370$	−4.32937	−0.225073
$371$	−9.33444	−0.484620
$372$	−17.0376	−0.883361
$373$	−16.6329	−0.861217	−0.430609	−	0.902539i	$-0.641701\pi$
$373$	−16.6329	−0.861217	−0.430609	+	0.902539i	$0.641701\pi$
$374$	0	0
$375$	1.82437	0.0942100
$376$	4.22816	0.218051
$377$	−11.2177	−0.577739
$378$	4.87412	0.250698
$379$	−0.0831686	−0.00427208	−0.00213604	−	0.999998i	$-0.500680\pi$
$379$	−0.0831686	−0.00427208	−0.00213604	+	0.999998i	$0.500680\pi$
$380$	0.915653	0.0469720
$381$	−26.7853	−1.37225
$382$	−12.0109	−0.614530
$383$	6.28284	0.321038	0.160519	−	0.987033i	$-0.448683\pi$
$383$	6.28284	0.321038	0.160519	+	0.987033i	$0.448683\pi$
$384$	−1.82437	−0.0930995
$385$	0	0
$386$	−17.2998	−0.880539
$387$	0.909280	0.0462213
$388$	−7.27431	−0.369297
$389$	−3.67862	−0.186513	−0.0932567	−	0.995642i	$-0.529728\pi$
$389$	−3.67862	−0.186513	−0.0932567	+	0.995642i	$0.529728\pi$
$390$	−4.03973	−0.204560
$391$	−48.4147	−2.44844
$392$	1.00000	0.0505076
$393$	13.9123	0.701783
$394$	−13.1724	−0.663618
$395$	9.39226	0.472576
$396$	0	0
$397$	7.12152	0.357419	0.178709	−	0.983902i	$-0.442808\pi$
$397$	7.12152	0.357419	0.178709	+	0.983902i	$0.442808\pi$
$398$	−22.9779	−1.15178
$399$	1.67049	0.0836290
$400$	1.00000	0.0500000
$401$	5.70426	0.284857	0.142429	−	0.989805i	$-0.454509\pi$
$401$	5.70426	0.284857	0.142429	+	0.989805i	$0.454509\pi$
$402$	2.35376	0.117395
$403$	−20.6793	−1.03011
$404$	−5.35283	−0.266313
$405$	9.87717	0.490801
$406$	5.06597	0.251420
$407$	0	0
$408$	14.5336	0.719521
$409$	23.7324	1.17349	0.586745	−	0.809772i	$-0.300409\pi$
$409$	23.7324	1.17349	0.586745	+	0.809772i	$0.300409\pi$
$410$	8.99953	0.444455
$411$	8.99521	0.443701
$412$	−14.3938	−0.709130
$413$	7.19487	0.354037
$414$	1.99534	0.0980658
$415$	−0.747575	−0.0366970
$416$	−2.21432	−0.108566
$417$	7.62072	0.373188
$418$	0	0
$419$	26.3585	1.28770	0.643848	−	0.765154i	$-0.277337\pi$
$419$	26.3585	1.28770	0.643848	+	0.765154i	$0.277337\pi$
$420$	1.82437	0.0890201
$421$	−37.5160	−1.82842	−0.914209	−	0.405243i	$-0.867187\pi$
$421$	−37.5160	−1.82842	−0.914209	+	0.405243i	$0.867187\pi$
$422$	−16.5609	−0.806172
$423$	1.38820	0.0674968
$424$	−9.33444	−0.453320
$425$	−7.96638	−0.386426
$426$	−0.0369197	−0.00178877
$427$	−12.6895	−0.614090
$428$	−4.27627	−0.206701
$429$	0	0
$430$	−2.76947	−0.133556
$431$	−6.43804	−0.310110	−0.155055	−	0.987906i	$-0.549555\pi$
$431$	−6.43804	−0.310110	−0.155055	+	0.987906i	$0.549555\pi$
$432$	4.87412	0.234506
$433$	14.6478	0.703930	0.351965	−	0.936013i	$-0.385514\pi$
$433$	14.6478	0.703930	0.351965	+	0.936013i	$0.385514\pi$
$434$	9.33892	0.448283
$435$	9.24220	0.443130
$436$	−5.04265	−0.241499
$437$	−5.56476	−0.266199
$438$	−21.6119	−1.03266
$439$	−31.4583	−1.50143	−0.750713	−	0.660629i	$-0.770290\pi$
$439$	−31.4583	−1.50143	−0.750713	+	0.660629i	$0.770290\pi$
$440$	0	0
$441$	0.328323	0.0156344
$442$	17.6401	0.839054
$443$	−31.0007	−1.47289	−0.736444	−	0.676499i	$-0.763496\pi$
$443$	−31.0007	−1.47289	−0.736444	+	0.676499i	$0.763496\pi$
$444$	−7.89837	−0.374840
$445$	14.2639	0.676172
$446$	−19.1235	−0.905523
$447$	24.5898	1.16306
$448$	1.00000	0.0472456
$449$	17.5963	0.830419	0.415209	−	0.909726i	$-0.363708\pi$
$449$	17.5963	0.830419	0.415209	+	0.909726i	$0.363708\pi$
$450$	0.328323	0.0154773
$451$	0	0
$452$	13.8169	0.649894
$453$	−32.1274	−1.50948
$454$	19.6989	0.924517
$455$	2.21432	0.103809
$456$	1.67049	0.0782278
$457$	34.3749	1.60799	0.803996	−	0.594635i	$-0.202703\pi$
$457$	34.3749	1.60799	0.803996	+	0.594635i	$0.202703\pi$
$458$	27.5813	1.28879
$459$	−38.8291	−1.81239
$460$	−6.07737	−0.283359
$461$	−10.0757	−0.469271	−0.234635	−	0.972083i	$-0.575390\pi$
$461$	−10.0757	−0.469271	−0.234635	+	0.972083i	$0.575390\pi$
$462$	0	0
$463$	−30.0439	−1.39626	−0.698128	−	0.715973i	$-0.745983\pi$
$463$	−30.0439	−1.39626	−0.698128	+	0.715973i	$0.745983\pi$
$464$	5.06597	0.235182
$465$	17.0376	0.790102
$466$	−12.9463	−0.599726
$467$	−19.7362	−0.913280	−0.456640	−	0.889652i	$-0.650947\pi$
$467$	−19.7362	−0.913280	−0.456640	+	0.889652i	$0.650947\pi$
$468$	−0.727012	−0.0336062
$469$	−1.29018	−0.0595749
$470$	−4.22816	−0.195030
$471$	−20.9503	−0.965340
$472$	7.19487	0.331171
$473$	0	0
$474$	17.1349	0.787034
$475$	−0.915653	−0.0420130
$476$	−7.96638	−0.365138
$477$	−3.06471	−0.140324
$478$	−5.77626	−0.264200
$479$	5.39968	0.246718	0.123359	−	0.992362i	$-0.460633\pi$
$479$	5.39968	0.246718	0.123359	+	0.992362i	$0.460633\pi$
$480$	1.82437	0.0832707
$481$	−9.58660	−0.437111
$482$	−25.0323	−1.14019
$483$	−11.0874	−0.504493
$484$	0	0
$485$	7.27431	0.330309
$486$	3.39724	0.154102
$487$	−12.6656	−0.573933	−0.286967	−	0.957941i	$-0.592647\pi$
$487$	−12.6656	−0.573933	−0.286967	+	0.957941i	$0.592647\pi$
$488$	−12.6895	−0.574428
$489$	0.254044	0.0114883
$490$	−1.00000	−0.0451754
$491$	−28.7791	−1.29878	−0.649392	−	0.760454i	$-0.724977\pi$
$491$	−28.7791	−1.29878	−0.649392	+	0.760454i	$0.724977\pi$
$492$	16.4185	0.740202
$493$	−40.3575	−1.81761
$494$	2.02755	0.0912236
$495$	0	0
$496$	9.33892	0.419330
$497$	0.0202370	0.000907752 0
$498$	−1.36385	−0.0611157
$499$	−19.3858	−0.867826	−0.433913	−	0.900955i	$-0.642867\pi$
$499$	−19.3858	−0.867826	−0.433913	+	0.900955i	$0.642867\pi$
$500$	−1.00000	−0.0447214
$501$	−12.8503	−0.574107
$502$	−6.66594	−0.297516
$503$	1.14878	0.0512218	0.0256109	−	0.999672i	$-0.491847\pi$
$503$	1.14878	0.0512218	0.0256109	+	0.999672i	$0.491847\pi$
$504$	0.328323	0.0146247
$505$	5.35283	0.238198
$506$	0	0
$507$	14.7715	0.656027
$508$	14.6820	0.651407
$509$	−1.32413	−0.0586909	−0.0293454	−	0.999569i	$-0.509342\pi$
$509$	−1.32413	−0.0586909	−0.0293454	+	0.999569i	$0.509342\pi$
$510$	−14.5336	−0.643559
$511$	11.8463	0.524047
$512$	1.00000	0.0441942
$513$	−4.46301	−0.197047
$514$	−22.7153	−1.00193
$515$	14.3938	0.634265
$516$	−5.05253	−0.222425
$517$	0	0
$518$	4.32937	0.190222
$519$	25.6220	1.12468
$520$	2.21432	0.0971043
$521$	−27.2341	−1.19315	−0.596573	−	0.802559i	$-0.703471\pi$
$521$	−27.2341	−1.19315	−0.596573	+	0.802559i	$0.703471\pi$
$522$	1.66328	0.0727997
$523$	23.8982	1.04499	0.522497	−	0.852641i	$-0.325001\pi$
$523$	23.8982	1.04499	0.522497	+	0.852641i	$0.325001\pi$
$524$	−7.62582	−0.333135
$525$	−1.82437	−0.0796220
$526$	14.5015	0.632295
$527$	−74.3974	−3.24080
$528$	0	0
$529$	13.9345	0.605847
$530$	9.33444	0.405462
$531$	2.36225	0.102513
$532$	−0.915653	−0.0396986
$533$	19.9278	0.863170
$534$	26.0225	1.12611
$535$	4.27627	0.184879
$536$	−1.29018	−0.0557272
$537$	−23.4138	−1.01038
$538$	−21.2753	−0.917244
$539$	0	0
$540$	−4.87412	−0.209749
$541$	−33.8027	−1.45329	−0.726645	−	0.687013i	$-0.758921\pi$
$541$	−33.8027	−1.45329	−0.726645	+	0.687013i	$0.758921\pi$
$542$	8.06611	0.346469
$543$	−35.1867	−1.51001
$544$	−7.96638	−0.341556
$545$	5.04265	0.216003
$546$	4.03973	0.172885
$547$	−23.6009	−1.00910	−0.504550	−	0.863382i	$-0.668342\pi$
$547$	−23.6009	−1.00910	−0.504550	+	0.863382i	$0.668342\pi$
$548$	−4.93059	−0.210624
$549$	−4.16627	−0.177812
$550$	0	0
$551$	−4.63867	−0.197614
$552$	−11.0874	−0.471910
$553$	−9.39226	−0.399399
$554$	17.4563	0.741646
$555$	7.89837	0.335267
$556$	−4.17718	−0.177152
$557$	6.01368	0.254808	0.127404	−	0.991851i	$-0.459336\pi$
$557$	6.01368	0.254808	0.127404	+	0.991851i	$0.459336\pi$
$558$	3.06619	0.129802
$559$	−6.13248	−0.259376
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−20.4688	−0.863423
$563$	6.87116	0.289585	0.144793	−	0.989462i	$-0.453749\pi$
$563$	6.87116	0.289585	0.144793	+	0.989462i	$0.453749\pi$
$564$	−7.71373	−0.324806
$565$	−13.8169	−0.581283
$566$	5.27282	0.221633
$567$	−9.87717	−0.414802
$568$	0.0202370	0.000849124 0
$569$	18.9976	0.796420	0.398210	−	0.917294i	$-0.369632\pi$
$569$	18.9976	0.796420	0.398210	+	0.917294i	$0.369632\pi$
$570$	−1.67049	−0.0699691
$571$	0.393382	0.0164625	0.00823127	−	0.999966i	$-0.497380\pi$
$571$	0.393382	0.0164625	0.00823127	+	0.999966i	$0.497380\pi$
$572$	0	0
$573$	21.9123	0.915398
$574$	−8.99953	−0.375633
$575$	6.07737	0.253444
$576$	0.328323	0.0136801
$577$	8.43804	0.351280	0.175640	−	0.984454i	$-0.443800\pi$
$577$	8.43804	0.351280	0.175640	+	0.984454i	$0.443800\pi$
$578$	46.4632	1.93261
$579$	31.5613	1.31164
$580$	−5.06597	−0.210353
$581$	0.747575	0.0310146
$582$	13.2710	0.550102
$583$	0	0
$584$	11.8463	0.490201
$585$	0.727012	0.0300583
$586$	6.40613	0.264635
$587$	46.3445	1.91284	0.956422	−	0.291988i	$-0.0943166\pi$
$587$	46.3445	1.91284	0.956422	+	0.291988i	$0.0943166\pi$
$588$	−1.82437	−0.0752357
$589$	−8.55121	−0.352347
$590$	−7.19487	−0.296208
$591$	24.0314	0.988519
$592$	4.32937	0.177936
$593$	−25.9525	−1.06574	−0.532871	−	0.846197i	$-0.678887\pi$
$593$	−25.9525	−1.06574	−0.532871	+	0.846197i	$0.678887\pi$
$594$	0	0
$595$	7.96638	0.326590
$596$	−13.4785	−0.552102
$597$	41.9202	1.71568
$598$	−13.4572	−0.550307
$599$	4.80045	0.196141	0.0980705	−	0.995179i	$-0.468733\pi$
$599$	4.80045	0.196141	0.0980705	+	0.995179i	$0.468733\pi$
$600$	−1.82437	−0.0744796
$601$	−30.5112	−1.24458	−0.622289	−	0.782787i	$-0.713797\pi$
$601$	−30.5112	−1.24458	−0.622289	+	0.782787i	$0.713797\pi$
$602$	2.76947	0.112875
$603$	−0.423595	−0.0172501
$604$	17.6102	0.716547
$605$	0	0
$606$	9.76555	0.396698
$607$	16.5488	0.671697	0.335848	−	0.941916i	$-0.390977\pi$
$607$	16.5488	0.671697	0.335848	+	0.941916i	$0.390977\pi$
$608$	−0.915653	−0.0371346
$609$	−9.24220	−0.374513
$610$	12.6895	0.513784
$611$	−9.36249	−0.378766
$612$	−2.61555	−0.105727
$613$	−42.0921	−1.70008	−0.850041	−	0.526716i	$-0.823423\pi$
$613$	−42.0921	−1.70008	−0.850041	+	0.526716i	$0.823423\pi$
$614$	5.35970	0.216300
$615$	−16.4185	−0.662057
$616$	0	0
$617$	7.32983	0.295088	0.147544	−	0.989056i	$-0.452863\pi$
$617$	7.32983	0.295088	0.147544	+	0.989056i	$0.452863\pi$
$618$	26.2595	1.05631
$619$	−11.9509	−0.480347	−0.240173	−	0.970730i	$-0.577204\pi$
$619$	−11.9509	−0.480347	−0.240173	+	0.970730i	$0.577204\pi$
$620$	−9.33892	−0.375060
$621$	29.6219	1.18869
$622$	−26.9819	−1.08188
$623$	−14.2639	−0.571469
$624$	4.03973	0.161719
$625$	1.00000	0.0400000
$626$	−28.5633	−1.14162
$627$	0	0
$628$	11.4836	0.458245
$629$	−34.4894	−1.37518
$630$	−0.328323	−0.0130807
$631$	43.7266	1.74073	0.870363	−	0.492410i	$-0.163884\pi$
$631$	43.7266	1.74073	0.870363	+	0.492410i	$0.163884\pi$
$632$	−9.39226	−0.373604
$633$	30.2132	1.20087
$634$	21.0077	0.834321
$635$	−14.6820	−0.582636
$636$	17.0295	0.675262
$637$	−2.21432	−0.0877345
$638$	0	0
$639$	0.00664427	0.000262843 0
$640$	−1.00000	−0.0395285
$641$	37.6723	1.48796	0.743982	−	0.668199i	$-0.232935\pi$
$641$	37.6723	1.48796	0.743982	+	0.668199i	$0.232935\pi$
$642$	7.80149	0.307900
$643$	24.7735	0.976973	0.488486	−	0.872572i	$-0.337549\pi$
$643$	24.7735	0.976973	0.488486	+	0.872572i	$0.337549\pi$
$644$	6.07737	0.239482
$645$	5.05253	0.198943
$646$	7.29444	0.286996
$647$	36.2066	1.42343	0.711714	−	0.702470i	$-0.247919\pi$
$647$	36.2066	1.42343	0.711714	+	0.702470i	$0.247919\pi$
$648$	−9.87717	−0.388012
$649$	0	0
$650$	−2.21432	−0.0868527
$651$	−17.0376	−0.667758
$652$	−0.139250	−0.00545347
$653$	−16.7136	−0.654055	−0.327027	−	0.945015i	$-0.606047\pi$
$653$	−16.7136	−0.654055	−0.327027	+	0.945015i	$0.606047\pi$
$654$	9.19965	0.359735
$655$	7.62582	0.297965
$656$	−8.99953	−0.351373
$657$	3.88940	0.151740
$658$	4.22816	0.164831
$659$	34.0997	1.32834	0.664168	−	0.747584i	$-0.268786\pi$
$659$	34.0997	1.32834	0.664168	+	0.747584i	$0.268786\pi$
$660$	0	0
$661$	−29.1326	−1.13313	−0.566563	−	0.824018i	$-0.691727\pi$
$661$	−29.1326	−1.13313	−0.566563	+	0.824018i	$0.691727\pi$
$662$	11.6022	0.450934
$663$	−32.1821	−1.24985
$664$	0.747575	0.0290115
$665$	0.915653	0.0355075
$666$	1.42143	0.0550794
$667$	30.7878	1.19211
$668$	7.04367	0.272528
$669$	34.8883	1.34886
$670$	1.29018	0.0498439
$671$	0	0
$672$	−1.82437	−0.0703766
$673$	27.1266	1.04565	0.522826	−	0.852439i	$-0.324878\pi$
$673$	27.1266	1.04565	0.522826	+	0.852439i	$0.324878\pi$
$674$	2.34874	0.0904700
$675$	4.87412	0.187605
$676$	−8.09680	−0.311415
$677$	−45.5790	−1.75174	−0.875872	−	0.482543i	$-0.839713\pi$
$677$	−45.5790	−1.75174	−0.875872	+	0.482543i	$0.839713\pi$
$678$	−25.2072	−0.968076
$679$	−7.27431	−0.279162
$680$	7.96638	0.305497
$681$	−35.9381	−1.37715
$682$	0	0
$683$	−6.68497	−0.255793	−0.127897	−	0.991787i	$-0.540823\pi$
$683$	−6.68497	−0.255793	−0.127897	+	0.991787i	$0.540823\pi$
$684$	−0.300630	−0.0114949
$685$	4.93059	0.188388
$686$	1.00000	0.0381802
$687$	−50.3185	−1.91977
$688$	2.76947	0.105585
$689$	20.6694	0.787442
$690$	11.0874	0.422089
$691$	2.37244	0.0902518	0.0451259	−	0.998981i	$-0.485631\pi$
$691$	2.37244	0.0902518	0.0451259	+	0.998981i	$0.485631\pi$
$692$	−14.0443	−0.533883
$693$	0	0
$694$	−31.4811	−1.19501
$695$	4.17718	0.158450
$696$	−9.24220	−0.350325
$697$	71.6937	2.71559
$698$	2.60777	0.0987057
$699$	23.6189	0.893347
$700$	1.00000	0.0377964
$701$	−30.5560	−1.15408	−0.577042	−	0.816714i	$-0.695793\pi$
$701$	−30.5560	−1.15408	−0.577042	+	0.816714i	$0.695793\pi$
$702$	−10.7929	−0.407350
$703$	−3.96420	−0.149513
$704$	0	0
$705$	7.71373	0.290516
$706$	−0.416007	−0.0156566
$707$	−5.35283	−0.201314
$708$	−13.1261	−0.493309
$709$	10.5970	0.397980	0.198990	−	0.980002i	$-0.436234\pi$
$709$	10.5970	0.397980	0.198990	+	0.980002i	$0.436234\pi$
$710$	−0.0202370	−0.000759480 0
$711$	−3.08370	−0.115648
$712$	−14.2639	−0.534561
$713$	56.7561	2.12553
$714$	14.5336	0.543907
$715$	0	0
$716$	12.8339	0.479627
$717$	10.5380	0.393550
$718$	−11.6150	−0.433468
$719$	11.8939	0.443568	0.221784	−	0.975096i	$-0.428812\pi$
$719$	11.8939	0.443568	0.221784	+	0.975096i	$0.428812\pi$
$720$	−0.328323	−0.0122359
$721$	−14.3938	−0.536052
$722$	−18.1616	−0.675904
$723$	45.6682	1.69842
$724$	19.2871	0.716798
$725$	5.06597	0.188145
$726$	0	0
$727$	−23.4180	−0.868527	−0.434264	−	0.900786i	$-0.642991\pi$
$727$	−23.4180	−0.868527	−0.434264	+	0.900786i	$0.642991\pi$
$728$	−2.21432	−0.0820681
$729$	23.4337	0.867915
$730$	−11.8463	−0.438450
$731$	−22.0626	−0.816016
$732$	23.1504	0.855664
$733$	12.0816	0.446242	0.223121	−	0.974791i	$-0.428375\pi$
$733$	12.0816	0.446242	0.223121	+	0.974791i	$0.428375\pi$
$734$	−8.00709	−0.295547
$735$	1.82437	0.0672929
$736$	6.07737	0.224015
$737$	0	0
$738$	−2.95476	−0.108766
$739$	35.0133	1.28798	0.643992	−	0.765032i	$-0.277277\pi$
$739$	35.0133	1.28798	0.643992	+	0.765032i	$0.277277\pi$
$740$	−4.32937	−0.159151
$741$	−3.69899	−0.135886
$742$	−9.33444	−0.342678
$743$	−48.5288	−1.78035	−0.890174	−	0.455620i	$-0.849417\pi$
$743$	−48.5288	−1.78035	−0.890174	+	0.455620i	$0.849417\pi$
$744$	−17.0376	−0.624630
$745$	13.4785	0.493815
$746$	−16.6329	−0.608973
$747$	0.245446	0.00898041
$748$	0	0
$749$	−4.27627	−0.156251
$750$	1.82437	0.0666165
$751$	−0.0449682	−0.00164091	−0.000820457	−	1.00000i	$-0.500261\pi$
$751$	−0.0449682	−0.00164091	−0.000820457		1.00000i	$0.500261\pi$
$752$	4.22816	0.154185
$753$	12.1611	0.443177
$754$	−11.2177	−0.408523
$755$	−17.6102	−0.640899
$756$	4.87412	0.177270
$757$	28.8381	1.04814	0.524069	−	0.851676i	$-0.324413\pi$
$757$	28.8381	1.04814	0.524069	+	0.851676i	$0.324413\pi$
$758$	−0.0831686	−0.00302082
$759$	0	0
$760$	0.915653	0.0332142
$761$	−18.3746	−0.666078	−0.333039	−	0.942913i	$-0.608074\pi$
$761$	−18.3746	−0.666078	−0.333039	+	0.942913i	$0.608074\pi$
$762$	−26.7853	−0.970330
$763$	−5.04265	−0.182556
$764$	−12.0109	−0.434538
$765$	2.61555	0.0945654
$766$	6.28284	0.227008
$767$	−15.9317	−0.575262
$768$	−1.82437	−0.0658313
$769$	20.2436	0.730003	0.365002	−	0.931007i	$-0.381068\pi$
$769$	20.2436	0.730003	0.365002	+	0.931007i	$0.381068\pi$
$770$	0	0
$771$	41.4412	1.49247
$772$	−17.2998	−0.622635
$773$	−27.2469	−0.980003	−0.490001	−	0.871722i	$-0.663004\pi$
$773$	−27.2469	−0.980003	−0.490001	+	0.871722i	$0.663004\pi$
$774$	0.909280	0.0326834
$775$	9.33892	0.335464
$776$	−7.27431	−0.261132
$777$	−7.89837	−0.283352
$778$	−3.67862	−0.131885
$779$	8.24045	0.295245
$780$	−4.03973	−0.144646
$781$	0	0
$782$	−48.4147	−1.73131
$783$	24.6922	0.882426
$784$	1.00000	0.0357143
$785$	−11.4836	−0.409867
$786$	13.9123	0.496235
$787$	−26.8288	−0.956342	−0.478171	−	0.878267i	$-0.658700\pi$
$787$	−26.8288	−0.956342	−0.478171	+	0.878267i	$0.658700\pi$
$788$	−13.1724	−0.469248
$789$	−26.4561	−0.941861
$790$	9.39226	0.334161
$791$	13.8169	0.491273
$792$	0	0
$793$	28.0987	0.997813
$794$	7.12152	0.252733
$795$	−17.0295	−0.603973
$796$	−22.9779	−0.814431
$797$	36.4937	1.29267	0.646337	−	0.763052i	$-0.276300\pi$
$797$	36.4937	1.29267	0.646337	+	0.763052i	$0.276300\pi$
$798$	1.67049	0.0591347
$799$	−33.6831	−1.19162
$800$	1.00000	0.0353553
$801$	−4.68316	−0.165471
$802$	5.70426	0.201424
$803$	0	0
$804$	2.35376	0.0830107
$805$	−6.07737	−0.214199
$806$	−20.6793	−0.728399
$807$	38.8141	1.36632
$808$	−5.35283	−0.188312
$809$	−8.52614	−0.299763	−0.149882	−	0.988704i	$-0.547889\pi$
$809$	−8.52614	−0.299763	−0.149882	+	0.988704i	$0.547889\pi$
$810$	9.87717	0.347049
$811$	43.5605	1.52962	0.764808	−	0.644258i	$-0.222834\pi$
$811$	43.5605	1.52962	0.764808	+	0.644258i	$0.222834\pi$
$812$	5.06597	0.177781
$813$	−14.7156	−0.516097
$814$	0	0
$815$	0.139250	0.00487773
$816$	14.5336	0.508778
$817$	−2.53587	−0.0887188
$818$	23.7324	0.829783
$819$	−0.727012	−0.0254039
$820$	8.99953	0.314277
$821$	−22.0579	−0.769825	−0.384913	−	0.922953i	$-0.625768\pi$
$821$	−22.0579	−0.769825	−0.384913	+	0.922953i	$0.625768\pi$
$822$	8.99521	0.313744
$823$	3.37693	0.117712	0.0588562	−	0.998266i	$-0.481255\pi$
$823$	3.37693	0.117712	0.0588562	+	0.998266i	$0.481255\pi$
$824$	−14.3938	−0.501431
$825$	0	0
$826$	7.19487	0.250342
$827$	17.2810	0.600919	0.300460	−	0.953795i	$-0.402860\pi$
$827$	17.2810	0.600919	0.300460	+	0.953795i	$0.402860\pi$
$828$	1.99534	0.0693430
$829$	−23.8021	−0.826682	−0.413341	−	0.910576i	$-0.635638\pi$
$829$	−23.8021	−0.826682	−0.413341	+	0.910576i	$0.635638\pi$
$830$	−0.747575	−0.0259487
$831$	−31.8467	−1.10475
$832$	−2.21432	−0.0767677
$833$	−7.96638	−0.276019
$834$	7.62072	0.263884
$835$	−7.04367	−0.243756
$836$	0	0
$837$	45.5191	1.57337
$838$	26.3585	0.910538
$839$	2.14765	0.0741451	0.0370726	−	0.999313i	$-0.488197\pi$
$839$	2.14765	0.0741451	0.0370726	+	0.999313i	$0.488197\pi$
$840$	1.82437	0.0629467
$841$	−3.33593	−0.115032
$842$	−37.5160	−1.29289
$843$	37.3426	1.28615
$844$	−16.5609	−0.570050
$845$	8.09680	0.278538
$846$	1.38820	0.0477274
$847$	0	0
$848$	−9.33444	−0.320546
$849$	−9.61957	−0.330143
$850$	−7.96638	−0.273245
$851$	26.3112	0.901936
$852$	−0.0369197	−0.00126485
$853$	10.0469	0.343998	0.171999	−	0.985097i	$-0.444977\pi$
$853$	10.0469	0.343998	0.171999	+	0.985097i	$0.444977\pi$
$854$	−12.6895	−0.434227
$855$	0.300630	0.0102813
$856$	−4.27627	−0.146160
$857$	−27.9578	−0.955019	−0.477510	−	0.878626i	$-0.658460\pi$
$857$	−27.9578	−0.955019	−0.477510	+	0.878626i	$0.658460\pi$
$858$	0	0
$859$	−20.7449	−0.707805	−0.353903	−	0.935282i	$-0.615146\pi$
$859$	−20.7449	−0.707805	−0.353903	+	0.935282i	$0.615146\pi$
$860$	−2.76947	−0.0944380
$861$	16.4185	0.559540
$862$	−6.43804	−0.219281
$863$	2.03460	0.0692587	0.0346293	−	0.999400i	$-0.488975\pi$
$863$	2.03460	0.0692587	0.0346293	+	0.999400i	$0.488975\pi$
$864$	4.87412	0.165821
$865$	14.0443	0.477520
$866$	14.6478	0.497754
$867$	−84.7661	−2.87881
$868$	9.33892	0.316984
$869$	0	0
$870$	9.24220	0.313340
$871$	2.85686	0.0968011
$872$	−5.04265	−0.170766
$873$	−2.38833	−0.0808326
$874$	−5.56476	−0.188231
$875$	−1.00000	−0.0338062
$876$	−21.6119	−0.730200
$877$	32.2740	1.08982	0.544908	−	0.838496i	$-0.316565\pi$
$877$	32.2740	1.08982	0.544908	+	0.838496i	$0.316565\pi$
$878$	−31.4583	−1.06167
$879$	−11.6872	−0.394198
$880$	0	0
$881$	−3.99069	−0.134450	−0.0672248	−	0.997738i	$-0.521414\pi$
$881$	−3.99069	−0.134450	−0.0672248	+	0.997738i	$0.521414\pi$
$882$	0.328323	0.0110552
$883$	16.2835	0.547982	0.273991	−	0.961732i	$-0.411656\pi$
$883$	16.2835	0.547982	0.273991	+	0.961732i	$0.411656\pi$
$884$	17.6401	0.593301
$885$	13.1261	0.441229
$886$	−31.0007	−1.04149
$887$	−37.4984	−1.25907	−0.629536	−	0.776971i	$-0.716755\pi$
$887$	−37.4984	−1.25907	−0.629536	+	0.776971i	$0.716755\pi$
$888$	−7.89837	−0.265052
$889$	14.6820	0.492417
$890$	14.2639	0.478126
$891$	0	0
$892$	−19.1235	−0.640301
$893$	−3.87153	−0.129556
$894$	24.5898	0.822407
$895$	−12.8339	−0.428991
$896$	1.00000	0.0334077
$897$	24.5510	0.819733
$898$	17.5963	0.587195
$899$	47.3107	1.57790
$900$	0.328323	0.0109441
$901$	74.3617	2.47735
$902$	0	0
$903$	−5.05253	−0.168138
$904$	13.8169	0.459544
$905$	−19.2871	−0.641124
$906$	−32.1274	−1.06736
$907$	4.16415	0.138268	0.0691341	−	0.997607i	$-0.477976\pi$
$907$	4.16415	0.138268	0.0691341	+	0.997607i	$0.477976\pi$
$908$	19.6989	0.653732
$909$	−1.75746	−0.0582913
$910$	2.21432	0.0734039
$911$	−45.0657	−1.49309	−0.746546	−	0.665333i	$-0.768289\pi$
$911$	−45.0657	−1.49309	−0.746546	+	0.665333i	$0.768289\pi$
$912$	1.67049	0.0553154
$913$	0	0
$914$	34.3749	1.13702
$915$	−23.1504	−0.765329
$916$	27.5813	0.911313
$917$	−7.62582	−0.251827
$918$	−38.8291	−1.28155
$919$	32.0466	1.05712	0.528561	−	0.848896i	$-0.322732\pi$
$919$	32.0466	1.05712	0.528561	+	0.848896i	$0.322732\pi$
$920$	−6.07737	−0.200365
$921$	−9.77808	−0.322199
$922$	−10.0757	−0.331824
$923$	−0.0448111	−0.00147497
$924$	0	0
$925$	4.32937	0.142349
$926$	−30.0439	−0.987302
$927$	−4.72581	−0.155216
$928$	5.06597	0.166299
$929$	51.0738	1.67568	0.837839	−	0.545918i	$-0.183819\pi$
$929$	51.0738	1.67568	0.837839	+	0.545918i	$0.183819\pi$
$930$	17.0376	0.558686
$931$	−0.915653	−0.0300093
$932$	−12.9463	−0.424071
$933$	49.2250	1.61156
$934$	−19.7362	−0.645787
$935$	0	0
$936$	−0.727012	−0.0237631
$937$	−5.41242	−0.176816	−0.0884080	−	0.996084i	$-0.528178\pi$
$937$	−5.41242	−0.176816	−0.0884080	+	0.996084i	$0.528178\pi$
$938$	−1.29018	−0.0421258
$939$	52.1099	1.70054
$940$	−4.22816	−0.137907
$941$	4.74009	0.154523	0.0772613	−	0.997011i	$-0.475382\pi$
$941$	4.74009	0.154523	0.0772613	+	0.997011i	$0.475382\pi$
$942$	−20.9503	−0.682598
$943$	−54.6935	−1.78107
$944$	7.19487	0.234173
$945$	−4.87412	−0.158555
$946$	0	0
$947$	16.4424	0.534306	0.267153	−	0.963654i	$-0.413917\pi$
$947$	16.4424	0.534306	0.267153	+	0.963654i	$0.413917\pi$
$948$	17.1349	0.556517
$949$	−26.2314	−0.851506
$950$	−0.915653	−0.0297077
$951$	−38.3258	−1.24280
$952$	−7.96638	−0.258192
$953$	45.6007	1.47715	0.738575	−	0.674171i	$-0.235499\pi$
$953$	45.6007	1.47715	0.738575	+	0.674171i	$0.235499\pi$
$954$	−3.06471	−0.0992238
$955$	12.0109	0.388663
$956$	−5.77626	−0.186818
$957$	0	0
$958$	5.39968	0.174456
$959$	−4.93059	−0.159217
$960$	1.82437	0.0588813
$961$	56.2155	1.81340
$962$	−9.58660	−0.309084
$963$	−1.40400	−0.0452432
$964$	−25.0323	−0.806237
$965$	17.2998	0.556902
$966$	−11.0874	−0.356730
$967$	28.9513	0.931012	0.465506	−	0.885045i	$-0.345872\pi$
$967$	28.9513	0.931012	0.465506	+	0.885045i	$0.345872\pi$
$968$	0	0
$969$	−13.3077	−0.427506
$970$	7.27431	0.233564
$971$	7.09959	0.227837	0.113918	−	0.993490i	$-0.463660\pi$
$971$	7.09959	0.227837	0.113918	+	0.993490i	$0.463660\pi$
$972$	3.39724	0.108966
$973$	−4.17718	−0.133914
$974$	−12.6656	−0.405832
$975$	4.03973	0.129375
$976$	−12.6895	−0.406182
$977$	−0.0166861	−0.000533837 0	−0.000266918	−	1.00000i	$-0.500085\pi$
$977$	−0.0166861	−0.000533837 0	−0.000266918		1.00000i	$0.500085\pi$
$978$	0.254044	0.00812343
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−1.65562	−0.0528599
$982$	−28.7791	−0.918379
$983$	−30.3585	−0.968286	−0.484143	−	0.874989i	$-0.660869\pi$
$983$	−30.3585	−0.968286	−0.484143	+	0.874989i	$0.660869\pi$
$984$	16.4185	0.523402
$985$	13.1724	0.419709
$986$	−40.3575	−1.28524
$987$	−7.71373	−0.245531
$988$	2.02755	0.0645048
$989$	16.8311	0.535197
$990$	0	0
$991$	−55.9611	−1.77766	−0.888831	−	0.458234i	$-0.848482\pi$
$991$	−55.9611	−1.77766	−0.888831	+	0.458234i	$0.848482\pi$
$992$	9.33892	0.296511
$993$	−21.1668	−0.671707
$994$	0.0202370	0.000641878 0
$995$	22.9779	0.728449
$996$	−1.36385	−0.0432153
$997$	46.4336	1.47057	0.735284	−	0.677759i	$-0.237049\pi$
$997$	46.4336	1.47057	0.735284	+	0.677759i	$0.237049\pi$
$998$	−19.3858	−0.613645
$999$	21.1019	0.667634

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.da.1.3		6
11.5	even	5		770.2.n.h.421.2	✓	12
11.9	even	5		770.2.n.h.631.2	yes	12
11.10	odd	2		8470.2.a.cu.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.h.421.2	✓	12	11.5	even	5
770.2.n.h.631.2	yes	12	11.9	even	5
8470.2.a.cu.1.3		6	11.10	odd	2
8470.2.a.da.1.3		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.82437 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.82437 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.328323 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.82437 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.82437 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.328323 q^{9} -1.00000 q^{10} -1.82437 q^{12} -2.21432 q^{13} +1.00000 q^{14} +1.82437 q^{15} +1.00000 q^{16} -7.96638 q^{17} +0.328323 q^{18} -0.915653 q^{19} -1.00000 q^{20} -1.82437 q^{21} +6.07737 q^{23} -1.82437 q^{24} +1.00000 q^{25} -2.21432 q^{26} +4.87412 q^{27} +1.00000 q^{28} +5.06597 q^{29} +1.82437 q^{30} +9.33892 q^{31} +1.00000 q^{32} -7.96638 q^{34} -1.00000 q^{35} +0.328323 q^{36} +4.32937 q^{37} -0.915653 q^{38} +4.03973 q^{39} -1.00000 q^{40} -8.99953 q^{41} -1.82437 q^{42} +2.76947 q^{43} -0.328323 q^{45} +6.07737 q^{46} +4.22816 q^{47} -1.82437 q^{48} +1.00000 q^{49} +1.00000 q^{50} +14.5336 q^{51} -2.21432 q^{52} -9.33444 q^{53} +4.87412 q^{54} +1.00000 q^{56} +1.67049 q^{57} +5.06597 q^{58} +7.19487 q^{59} +1.82437 q^{60} -12.6895 q^{61} +9.33892 q^{62} +0.328323 q^{63} +1.00000 q^{64} +2.21432 q^{65} -1.29018 q^{67} -7.96638 q^{68} -11.0874 q^{69} -1.00000 q^{70} +0.0202370 q^{71} +0.328323 q^{72} +11.8463 q^{73} +4.32937 q^{74} -1.82437 q^{75} -0.915653 q^{76} +4.03973 q^{78} -9.39226 q^{79} -1.00000 q^{80} -9.87717 q^{81} -8.99953 q^{82} +0.747575 q^{83} -1.82437 q^{84} +7.96638 q^{85} +2.76947 q^{86} -9.24220 q^{87} -14.2639 q^{89} -0.328323 q^{90} -2.21432 q^{91} +6.07737 q^{92} -17.0376 q^{93} +4.22816 q^{94} +0.915653 q^{95} -1.82437 q^{96} -7.27431 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 5 q^{3} + 6 q^{4} - 6 q^{5} - 5 q^{6} + 6 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - 5 * q^3 + 6 * q^4 - 6 * q^5 - 5 * q^6 + 6 * q^7 + 6 * q^8 + 11 * q^9	\( 6 q + 6 q^{2} - 5 q^{3} + 6 q^{4} - 6 q^{5} - 5 q^{6} + 6 q^{7} + 6 q^{8} + 11 q^{9} - 6 q^{10} - 5 q^{12} + 6 q^{14} + 5 q^{15} + 6 q^{16} - 15 q^{17} + 11 q^{18} - 13 q^{19} - 6 q^{20} - 5 q^{21} - 2 q^{23} - 5 q^{24} + 6 q^{25} - 26 q^{27} + 6 q^{28} - 4 q^{29} + 5 q^{30} - 2 q^{31} + 6 q^{32} - 15 q^{34} - 6 q^{35} + 11 q^{36} - 13 q^{38} - 30 q^{39} - 6 q^{40} - 17 q^{41} - 5 q^{42} + 15 q^{43} - 11 q^{45} - 2 q^{46} - 18 q^{47} - 5 q^{48} + 6 q^{49} + 6 q^{50} + 6 q^{51} + 22 q^{53} - 26 q^{54} + 6 q^{56} - 2 q^{57} - 4 q^{58} - 7 q^{59} + 5 q^{60} - 20 q^{61} - 2 q^{62} + 11 q^{63} + 6 q^{64} - 29 q^{67} - 15 q^{68} + 12 q^{69} - 6 q^{70} - 2 q^{71} + 11 q^{72} + q^{73} - 5 q^{75} - 13 q^{76} - 30 q^{78} - 12 q^{79} - 6 q^{80} + 22 q^{81} - 17 q^{82} - 35 q^{83} - 5 q^{84} + 15 q^{85} + 15 q^{86} - 29 q^{89} - 11 q^{90} - 2 q^{92} + 8 q^{93} - 18 q^{94} + 13 q^{95} - 5 q^{96} - 19 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - 5 * q^3 + 6 * q^4 - 6 * q^5 - 5 * q^6 + 6 * q^7 + 6 * q^8 + 11 * q^9 - 6 * q^10 - 5 * q^12 + 6 * q^14 + 5 * q^15 + 6 * q^16 - 15 * q^17 + 11 * q^18 - 13 * q^19 - 6 * q^20 - 5 * q^21 - 2 * q^23 - 5 * q^24 + 6 * q^25 - 26 * q^27 + 6 * q^28 - 4 * q^29 + 5 * q^30 - 2 * q^31 + 6 * q^32 - 15 * q^34 - 6 * q^35 + 11 * q^36 - 13 * q^38 - 30 * q^39 - 6 * q^40 - 17 * q^41 - 5 * q^42 + 15 * q^43 - 11 * q^45 - 2 * q^46 - 18 * q^47 - 5 * q^48 + 6 * q^49 + 6 * q^50 + 6 * q^51 + 22 * q^53 - 26 * q^54 + 6 * q^56 - 2 * q^57 - 4 * q^58 - 7 * q^59 + 5 * q^60 - 20 * q^61 - 2 * q^62 + 11 * q^63 + 6 * q^64 - 29 * q^67 - 15 * q^68 + 12 * q^69 - 6 * q^70 - 2 * q^71 + 11 * q^72 + q^73 - 5 * q^75 - 13 * q^76 - 30 * q^78 - 12 * q^79 - 6 * q^80 + 22 * q^81 - 17 * q^82 - 35 * q^83 - 5 * q^84 + 15 * q^85 + 15 * q^86 - 29 * q^89 - 11 * q^90 - 2 * q^92 + 8 * q^93 - 18 * q^94 + 13 * q^95 - 5 * q^96 - 19 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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