LMFDB - Embedded newform 8470.2.a.cv.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:	$0$
Dimension:	$6$
Coefficient field:	6.6.4642000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} - 9x - 1 $ x^6 - x^5 - 8x^4 + 5x^3 + 14x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			$0.803425$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.803425 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.803425 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.35451 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.803425 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.803425 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.35451 q^{9} -1.00000 q^{10} -0.803425 q^{12} +2.64314 q^{13} -1.00000 q^{14} -0.803425 q^{15} +1.00000 q^{16} -1.60450 q^{17} +2.35451 q^{18} -1.90940 q^{19} +1.00000 q^{20} -0.803425 q^{21} -4.22916 q^{23} +0.803425 q^{24} +1.00000 q^{25} -2.64314 q^{26} +4.30195 q^{27} +1.00000 q^{28} -8.97936 q^{29} +0.803425 q^{30} +2.47600 q^{31} -1.00000 q^{32} +1.60450 q^{34} +1.00000 q^{35} -2.35451 q^{36} -5.60877 q^{37} +1.90940 q^{38} -2.12356 q^{39} -1.00000 q^{40} +9.69579 q^{41} +0.803425 q^{42} +4.00169 q^{43} -2.35451 q^{45} +4.22916 q^{46} -6.54502 q^{47} -0.803425 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.28909 q^{51} +2.64314 q^{52} +10.0478 q^{53} -4.30195 q^{54} -1.00000 q^{56} +1.53406 q^{57} +8.97936 q^{58} +4.16663 q^{59} -0.803425 q^{60} -13.2629 q^{61} -2.47600 q^{62} -2.35451 q^{63} +1.00000 q^{64} +2.64314 q^{65} -10.5029 q^{67} -1.60450 q^{68} +3.39781 q^{69} -1.00000 q^{70} +4.58489 q^{71} +2.35451 q^{72} -3.69293 q^{73} +5.60877 q^{74} -0.803425 q^{75} -1.90940 q^{76} +2.12356 q^{78} +13.9477 q^{79} +1.00000 q^{80} +3.60723 q^{81} -9.69579 q^{82} +13.5603 q^{83} -0.803425 q^{84} -1.60450 q^{85} -4.00169 q^{86} +7.21424 q^{87} -3.41009 q^{89} +2.35451 q^{90} +2.64314 q^{91} -4.22916 q^{92} -1.98928 q^{93} +6.54502 q^{94} -1.90940 q^{95} +0.803425 q^{96} -17.4713 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} - q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 - q^9	$ 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} - q^{9} - 6 q^{10} - q^{12} + 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 21 q^{17} + q^{18} - 3 q^{19} + 6 q^{20} - q^{21} - 10 q^{23} + q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} + 6 q^{28} + 10 q^{29} + q^{30} - 4 q^{31} - 6 q^{32} - 21 q^{34} + 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} + 26 q^{39} - 6 q^{40} + 7 q^{41} + q^{42} + 19 q^{43} - q^{45} + 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} - 6 q^{50} - 4 q^{51} + 6 q^{52} - 16 q^{53} + 4 q^{54} - 6 q^{56} + 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} - 8 q^{61} + 4 q^{62} - q^{63} + 6 q^{64} + 6 q^{65} - 27 q^{67} + 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} + q^{72} + 13 q^{73} + 2 q^{74} - q^{75} - 3 q^{76} - 26 q^{78} + 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} + 51 q^{83} - q^{84} + 21 q^{85} - 19 q^{86} + 8 q^{87} + q^{89} + q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} - 10 q^{94} - 3 q^{95} + q^{96} + 7 q^{97} - 6 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 - q^9 - 6 * q^10 - q^12 + 6 * q^13 - 6 * q^14 - q^15 + 6 * q^16 + 21 * q^17 + q^18 - 3 * q^19 + 6 * q^20 - q^21 - 10 * q^23 + q^24 + 6 * q^25 - 6 * q^26 - 4 * q^27 + 6 * q^28 + 10 * q^29 + q^30 - 4 * q^31 - 6 * q^32 - 21 * q^34 + 6 * q^35 - q^36 - 2 * q^37 + 3 * q^38 + 26 * q^39 - 6 * q^40 + 7 * q^41 + q^42 + 19 * q^43 - q^45 + 10 * q^46 + 10 * q^47 - q^48 + 6 * q^49 - 6 * q^50 - 4 * q^51 + 6 * q^52 - 16 * q^53 + 4 * q^54 - 6 * q^56 + 16 * q^57 - 10 * q^58 - 3 * q^59 - q^60 - 8 * q^61 + 4 * q^62 - q^63 + 6 * q^64 + 6 * q^65 - 27 * q^67 + 21 * q^68 + 4 * q^69 - 6 * q^70 + 4 * q^71 + q^72 + 13 * q^73 + 2 * q^74 - q^75 - 3 * q^76 - 26 * q^78 + 14 * q^79 + 6 * q^80 - 14 * q^81 - 7 * q^82 + 51 * q^83 - q^84 + 21 * q^85 - 19 * q^86 + 8 * q^87 + q^89 + q^90 + 6 * q^91 - 10 * q^92 + 4 * q^93 - 10 * q^94 - 3 * q^95 + q^96 + 7 * 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
$3$	−0.803425	−0.463858	−0.231929	−	0.972733i	$-0.574504\pi$
$3$	−0.803425	−0.463858	−0.231929	+	0.972733i	$0.574504\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.803425	0.327997
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−2.35451	−0.784836
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−0.803425	−0.231929
$13$	2.64314	0.733075	0.366537	−	0.930403i	$-0.380543\pi$
$13$	2.64314	0.733075	0.366537	+	0.930403i	$0.380543\pi$
$14$	−1.00000	−0.267261
$15$	−0.803425	−0.207443
$16$	1.00000	0.250000
$17$	−1.60450	−0.389147	−0.194574	−	0.980888i	$-0.562332\pi$
$17$	−1.60450	−0.389147	−0.194574	+	0.980888i	$0.562332\pi$
$18$	2.35451	0.554963
$19$	−1.90940	−0.438046	−0.219023	−	0.975720i	$-0.570287\pi$
$19$	−1.90940	−0.438046	−0.219023	+	0.975720i	$0.570287\pi$
$20$	1.00000	0.223607
$21$	−0.803425	−0.175322
$22$	0	0
$23$	−4.22916	−0.881840	−0.440920	−	0.897546i	$-0.645348\pi$
$23$	−4.22916	−0.881840	−0.440920	+	0.897546i	$0.645348\pi$
$24$	0.803425	0.163998
$25$	1.00000	0.200000
$26$	−2.64314	−0.518362
$27$	4.30195	0.827910
$28$	1.00000	0.188982
$29$	−8.97936	−1.66742	−0.833712	−	0.552199i	$-0.813789\pi$
$29$	−8.97936	−1.66742	−0.833712	+	0.552199i	$0.813789\pi$
$30$	0.803425	0.146685
$31$	2.47600	0.444702	0.222351	−	0.974967i	$-0.428627\pi$
$31$	2.47600	0.444702	0.222351	+	0.974967i	$0.428627\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.60450	0.275169
$35$	1.00000	0.169031
$36$	−2.35451	−0.392418
$37$	−5.60877	−0.922076	−0.461038	−	0.887380i	$-0.652523\pi$
$37$	−5.60877	−0.922076	−0.461038	+	0.887380i	$0.652523\pi$
$38$	1.90940	0.309746
$39$	−2.12356	−0.340042
$40$	−1.00000	−0.158114
$41$	9.69579	1.51423	0.757114	−	0.653283i	$-0.226609\pi$
$41$	9.69579	1.51423	0.757114	+	0.653283i	$0.226609\pi$
$42$	0.803425	0.123971
$43$	4.00169	0.610252	0.305126	−	0.952312i	$-0.401301\pi$
$43$	4.00169	0.610252	0.305126	+	0.952312i	$0.401301\pi$
$44$	0	0
$45$	−2.35451	−0.350989
$46$	4.22916	0.623555
$47$	−6.54502	−0.954690	−0.477345	−	0.878716i	$-0.658401\pi$
$47$	−6.54502	−0.954690	−0.477345	+	0.878716i	$0.658401\pi$
$48$	−0.803425	−0.115964
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	1.28909	0.180509
$52$	2.64314	0.366537
$53$	10.0478	1.38017	0.690087	−	0.723727i	$-0.257572\pi$
$53$	10.0478	1.38017	0.690087	+	0.723727i	$0.257572\pi$
$54$	−4.30195	−0.585421
$55$	0	0
$56$	−1.00000	−0.133631
$57$	1.53406	0.203191
$58$	8.97936	1.17905
$59$	4.16663	0.542449	0.271225	−	0.962516i	$-0.412571\pi$
$59$	4.16663	0.542449	0.271225	+	0.962516i	$0.412571\pi$
$60$	−0.803425	−0.103722
$61$	−13.2629	−1.69814	−0.849068	−	0.528284i	$-0.822836\pi$
$61$	−13.2629	−1.69814	−0.849068	+	0.528284i	$0.822836\pi$
$62$	−2.47600	−0.314452
$63$	−2.35451	−0.296640
$64$	1.00000	0.125000
$65$	2.64314	0.327841
$66$	0	0
$67$	−10.5029	−1.28314	−0.641570	−	0.767065i	$-0.721716\pi$
$67$	−10.5029	−1.28314	−0.641570	+	0.767065i	$0.721716\pi$
$68$	−1.60450	−0.194574
$69$	3.39781	0.409048
$70$	−1.00000	−0.119523
$71$	4.58489	0.544126	0.272063	−	0.962279i	$-0.412294\pi$
$71$	4.58489	0.544126	0.272063	+	0.962279i	$0.412294\pi$
$72$	2.35451	0.277481
$73$	−3.69293	−0.432224	−0.216112	−	0.976369i	$-0.569338\pi$
$73$	−3.69293	−0.432224	−0.216112	+	0.976369i	$0.569338\pi$
$74$	5.60877	0.652006
$75$	−0.803425	−0.0927715
$76$	−1.90940	−0.219023
$77$	0	0
$78$	2.12356	0.240446
$79$	13.9477	1.56924	0.784620	−	0.619977i	$-0.212858\pi$
$79$	13.9477	1.56924	0.784620	+	0.619977i	$0.212858\pi$
$80$	1.00000	0.111803
$81$	3.60723	0.400804
$82$	−9.69579	−1.07072
$83$	13.5603	1.48843	0.744215	−	0.667940i	$-0.232824\pi$
$83$	13.5603	1.48843	0.744215	+	0.667940i	$0.232824\pi$
$84$	−0.803425	−0.0876609
$85$	−1.60450	−0.174032
$86$	−4.00169	−0.431513
$87$	7.21424	0.773448
$88$	0	0
$89$	−3.41009	−0.361469	−0.180735	−	0.983532i	$-0.557847\pi$
$89$	−3.41009	−0.361469	−0.180735	+	0.983532i	$0.557847\pi$
$90$	2.35451	0.248187
$91$	2.64314	0.277076
$92$	−4.22916	−0.440920
$93$	−1.98928	−0.206279
$94$	6.54502	0.675068
$95$	−1.90940	−0.195900
$96$	0.803425	0.0819992
$97$	−17.4713	−1.77394	−0.886968	−	0.461830i	$-0.847193\pi$
$97$	−17.4713	−1.77394	−0.886968	+	0.461830i	$0.847193\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	10.6494	1.05965	0.529827	−	0.848106i	$-0.322257\pi$
$101$	10.6494	1.05965	0.529827	+	0.848106i	$0.322257\pi$
$102$	−1.28909	−0.127639
$103$	−6.86666	−0.676592	−0.338296	−	0.941040i	$-0.609851\pi$
$103$	−6.86666	−0.676592	−0.338296	+	0.941040i	$0.609851\pi$
$104$	−2.64314	−0.259181
$105$	−0.803425	−0.0784063
$106$	−10.0478	−0.975930
$107$	2.88270	0.278681	0.139341	−	0.990245i	$-0.455502\pi$
$107$	2.88270	0.278681	0.139341	+	0.990245i	$0.455502\pi$
$108$	4.30195	0.413955
$109$	18.4938	1.77139	0.885693	−	0.464271i	$-0.153684\pi$
$109$	18.4938	1.77139	0.885693	+	0.464271i	$0.153684\pi$
$110$	0	0
$111$	4.50623	0.427712
$112$	1.00000	0.0944911
$113$	11.9566	1.12478	0.562391	−	0.826871i	$-0.309882\pi$
$113$	11.9566	1.12478	0.562391	+	0.826871i	$0.309882\pi$
$114$	−1.53406	−0.143678
$115$	−4.22916	−0.394371
$116$	−8.97936	−0.833712
$117$	−6.22329	−0.575343
$118$	−4.16663	−0.383570
$119$	−1.60450	−0.147084
$120$	0.803425	0.0733423
$121$	0	0
$122$	13.2629	1.20076
$123$	−7.78984	−0.702386
$124$	2.47600	0.222351
$125$	1.00000	0.0894427
$126$	2.35451	0.209756
$127$	13.4085	1.18982	0.594908	−	0.803794i	$-0.297189\pi$
$127$	13.4085	1.18982	0.594908	+	0.803794i	$0.297189\pi$
$128$	−1.00000	−0.0883883
$129$	−3.21506	−0.283070
$130$	−2.64314	−0.231819
$131$	8.49928	0.742585	0.371293	−	0.928516i	$-0.378915\pi$
$131$	8.49928	0.742585	0.371293	+	0.928516i	$0.378915\pi$
$132$	0	0
$133$	−1.90940	−0.165566
$134$	10.5029	0.907317
$135$	4.30195	0.370253
$136$	1.60450	0.137584
$137$	10.0747	0.860737	0.430369	−	0.902653i	$-0.358384\pi$
$137$	10.0747	0.860737	0.430369	+	0.902653i	$0.358384\pi$
$138$	−3.39781	−0.289241
$139$	−13.8200	−1.17220	−0.586100	−	0.810238i	$-0.699338\pi$
$139$	−13.8200	−1.17220	−0.586100	+	0.810238i	$0.699338\pi$
$140$	1.00000	0.0845154
$141$	5.25844	0.442840
$142$	−4.58489	−0.384755
$143$	0	0
$144$	−2.35451	−0.196209
$145$	−8.97936	−0.745695
$146$	3.69293	0.305629
$147$	−0.803425	−0.0662654
$148$	−5.60877	−0.461038
$149$	18.3638	1.50442	0.752211	−	0.658922i	$-0.228987\pi$
$149$	18.3638	1.50442	0.752211	+	0.658922i	$0.228987\pi$
$150$	0.803425	0.0655994
$151$	−20.5718	−1.67411	−0.837055	−	0.547118i	$-0.815725\pi$
$151$	−20.5718	−1.67411	−0.837055	+	0.547118i	$0.815725\pi$
$152$	1.90940	0.154873
$153$	3.77780	0.305417
$154$	0	0
$155$	2.47600	0.198877
$156$	−2.12356	−0.170021
$157$	−4.64886	−0.371019	−0.185510	−	0.982642i	$-0.559394\pi$
$157$	−4.64886	−0.371019	−0.185510	+	0.982642i	$0.559394\pi$
$158$	−13.9477	−1.10962
$159$	−8.07267	−0.640204
$160$	−1.00000	−0.0790569
$161$	−4.22916	−0.333304
$162$	−3.60723	−0.283411
$163$	4.99400	0.391160	0.195580	−	0.980688i	$-0.437341\pi$
$163$	4.99400	0.391160	0.195580	+	0.980688i	$0.437341\pi$
$164$	9.69579	0.757114
$165$	0	0
$166$	−13.5603	−1.05248
$167$	1.86847	0.144586	0.0722932	−	0.997383i	$-0.476968\pi$
$167$	1.86847	0.144586	0.0722932	+	0.997383i	$0.476968\pi$
$168$	0.803425	0.0619856
$169$	−6.01382	−0.462602
$170$	1.60450	0.123059
$171$	4.49570	0.343795
$172$	4.00169	0.305126
$173$	11.7799	0.895613	0.447806	−	0.894131i	$-0.352205\pi$
$173$	11.7799	0.895613	0.447806	+	0.894131i	$0.352205\pi$
$174$	−7.21424	−0.546910
$175$	1.00000	0.0755929
$176$	0	0
$177$	−3.34758	−0.251619
$178$	3.41009	0.255597
$179$	−10.3705	−0.775128	−0.387564	−	0.921843i	$-0.626683\pi$
$179$	−10.3705	−0.775128	−0.387564	+	0.921843i	$0.626683\pi$
$180$	−2.35451	−0.175495
$181$	−22.6738	−1.68533	−0.842664	−	0.538440i	$-0.819014\pi$
$181$	−22.6738	−1.68533	−0.842664	+	0.538440i	$0.819014\pi$
$182$	−2.64314	−0.195922
$183$	10.6557	0.787693
$184$	4.22916	0.311778
$185$	−5.60877	−0.412365
$186$	1.98928	0.145861
$187$	0	0
$188$	−6.54502	−0.477345
$189$	4.30195	0.312921
$190$	1.90940	0.138522
$191$	−8.48643	−0.614056	−0.307028	−	0.951700i	$-0.599335\pi$
$191$	−8.48643	−0.614056	−0.307028	+	0.951700i	$0.599335\pi$
$192$	−0.803425	−0.0579822
$193$	25.1578	1.81090	0.905451	−	0.424452i	$-0.139533\pi$
$193$	25.1578	1.81090	0.905451	+	0.424452i	$0.139533\pi$
$194$	17.4713	1.25436
$195$	−2.12356	−0.152072
$196$	1.00000	0.0714286
$197$	13.2603	0.944757	0.472379	−	0.881396i	$-0.343396\pi$
$197$	13.2603	0.944757	0.472379	+	0.881396i	$0.343396\pi$
$198$	0	0
$199$	3.59813	0.255065	0.127532	−	0.991834i	$-0.459294\pi$
$199$	3.59813	0.255065	0.127532	+	0.991834i	$0.459294\pi$
$200$	−1.00000	−0.0707107
$201$	8.43833	0.595194
$202$	−10.6494	−0.749288
$203$	−8.97936	−0.630227
$204$	1.28909	0.0902545
$205$	9.69579	0.677183
$206$	6.86666	0.478423
$207$	9.95758	0.692100
$208$	2.64314	0.183269
$209$	0	0
$210$	0.803425	0.0554416
$211$	10.0058	0.688828	0.344414	−	0.938818i	$-0.388078\pi$
$211$	10.0058	0.688828	0.344414	+	0.938818i	$0.388078\pi$
$212$	10.0478	0.690087
$213$	−3.68361	−0.252397
$214$	−2.88270	−0.197057
$215$	4.00169	0.272913
$216$	−4.30195	−0.292710
$217$	2.47600	0.168082
$218$	−18.4938	−1.25256
$219$	2.96699	0.200491
$220$	0	0
$221$	−4.24090	−0.285274
$222$	−4.50623	−0.302438
$223$	25.0397	1.67678	0.838391	−	0.545069i	$-0.183497\pi$
$223$	25.0397	1.67678	0.838391	+	0.545069i	$0.183497\pi$
$224$	−1.00000	−0.0668153
$225$	−2.35451	−0.156967
$226$	−11.9566	−0.795341
$227$	0.182924	0.0121411	0.00607054	−	0.999982i	$-0.498068\pi$
$227$	0.182924	0.0121411	0.00607054	+	0.999982i	$0.498068\pi$
$228$	1.53406	0.101596
$229$	−0.134371	−0.00887950	−0.00443975	−	0.999990i	$-0.501413\pi$
$229$	−0.134371	−0.00887950	−0.00443975	+	0.999990i	$0.501413\pi$
$230$	4.22916	0.278862
$231$	0	0
$232$	8.97936	0.589524
$233$	2.93010	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$233$	2.93010	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$234$	6.22329	0.406829
$235$	−6.54502	−0.426950
$236$	4.16663	0.271225
$237$	−11.2059	−0.727904
$238$	1.60450	0.104004
$239$	−8.27410	−0.535207	−0.267604	−	0.963529i	$-0.586232\pi$
$239$	−8.27410	−0.535207	−0.267604	+	0.963529i	$0.586232\pi$
$240$	−0.803425	−0.0518609
$241$	−16.5842	−1.06828	−0.534142	−	0.845395i	$-0.679365\pi$
$241$	−16.5842	−1.06828	−0.534142	+	0.845395i	$0.679365\pi$
$242$	0	0
$243$	−15.8040	−1.01383
$244$	−13.2629	−0.849068
$245$	1.00000	0.0638877
$246$	7.78984	0.496662
$247$	−5.04681	−0.321121
$248$	−2.47600	−0.157226
$249$	−10.8946	−0.690420
$250$	−1.00000	−0.0632456
$251$	−13.5486	−0.855180	−0.427590	−	0.903973i	$-0.640637\pi$
$251$	−13.5486	−0.855180	−0.427590	+	0.903973i	$0.640637\pi$
$252$	−2.35451	−0.148320
$253$	0	0
$254$	−13.4085	−0.841327
$255$	1.28909	0.0807261
$256$	1.00000	0.0625000
$257$	18.4226	1.14917	0.574584	−	0.818446i	$-0.305164\pi$
$257$	18.4226	1.14917	0.574584	+	0.818446i	$0.305164\pi$
$258$	3.21506	0.200161
$259$	−5.60877	−0.348512
$260$	2.64314	0.163920
$261$	21.1420	1.30865
$262$	−8.49928	−0.525087
$263$	19.1658	1.18181	0.590907	−	0.806740i	$-0.298770\pi$
$263$	19.1658	1.18181	0.590907	+	0.806740i	$0.298770\pi$
$264$	0	0
$265$	10.0478	0.617232
$266$	1.90940	0.117073
$267$	2.73975	0.167670
$268$	−10.5029	−0.641570
$269$	9.08666	0.554023	0.277012	−	0.960867i	$-0.410656\pi$
$269$	9.08666	0.554023	0.277012	+	0.960867i	$0.410656\pi$
$270$	−4.30195	−0.261808
$271$	−19.1739	−1.16473	−0.582366	−	0.812927i	$-0.697873\pi$
$271$	−19.1739	−1.16473	−0.582366	+	0.812927i	$0.697873\pi$
$272$	−1.60450	−0.0972869
$273$	−2.12356	−0.128524
$274$	−10.0747	−0.608633
$275$	0	0
$276$	3.39781	0.204524
$277$	20.6691	1.24189	0.620944	−	0.783855i	$-0.286749\pi$
$277$	20.6691	1.24189	0.620944	+	0.783855i	$0.286749\pi$
$278$	13.8200	0.828871
$279$	−5.82976	−0.349018
$280$	−1.00000	−0.0597614
$281$	1.37153	0.0818184	0.0409092	−	0.999163i	$-0.486975\pi$
$281$	1.37153	0.0818184	0.0409092	+	0.999163i	$0.486975\pi$
$282$	−5.25844	−0.313135
$283$	6.43513	0.382528	0.191264	−	0.981539i	$-0.438741\pi$
$283$	6.43513	0.382528	0.191264	+	0.981539i	$0.438741\pi$
$284$	4.58489	0.272063
$285$	1.53406	0.0908699
$286$	0	0
$287$	9.69579	0.572324
$288$	2.35451	0.138741
$289$	−14.4256	−0.848564
$290$	8.97936	0.527286
$291$	14.0368	0.822854
$292$	−3.69293	−0.216112
$293$	−4.59731	−0.268578	−0.134289	−	0.990942i	$-0.542875\pi$
$293$	−4.59731	−0.268578	−0.134289	+	0.990942i	$0.542875\pi$
$294$	0.803425	0.0468567
$295$	4.16663	0.242591
$296$	5.60877	0.326003
$297$	0	0
$298$	−18.3638	−1.06379
$299$	−11.1782	−0.646454
$300$	−0.803425	−0.0463858
$301$	4.00169	0.230654
$302$	20.5718	1.18377
$303$	−8.55598	−0.491528
$304$	−1.90940	−0.109512
$305$	−13.2629	−0.759429
$306$	−3.77780	−0.215962
$307$	24.8042	1.41565	0.707825	−	0.706388i	$-0.249677\pi$
$307$	24.8042	1.41565	0.707825	+	0.706388i	$0.249677\pi$
$308$	0	0
$309$	5.51685	0.313843
$310$	−2.47600	−0.140627
$311$	25.8023	1.46311	0.731556	−	0.681782i	$-0.238795\pi$
$311$	25.8023	1.46311	0.731556	+	0.681782i	$0.238795\pi$
$312$	2.12356	0.120223
$313$	16.5081	0.933093	0.466547	−	0.884497i	$-0.345498\pi$
$313$	16.5081	0.933093	0.466547	+	0.884497i	$0.345498\pi$
$314$	4.64886	0.262350
$315$	−2.35451	−0.132662
$316$	13.9477	0.784620
$317$	2.33084	0.130913	0.0654566	−	0.997855i	$-0.479150\pi$
$317$	2.33084	0.130913	0.0654566	+	0.997855i	$0.479150\pi$
$318$	8.07267	0.452693
$319$	0	0
$320$	1.00000	0.0559017
$321$	−2.31603	−0.129268
$322$	4.22916	0.235682
$323$	3.06363	0.170465
$324$	3.60723	0.200402
$325$	2.64314	0.146615
$326$	−4.99400	−0.276592
$327$	−14.8584	−0.821671
$328$	−9.69579	−0.535361
$329$	−6.54502	−0.360839
$330$	0	0
$331$	6.54411	0.359697	0.179848	−	0.983694i	$-0.442439\pi$
$331$	6.54411	0.359697	0.179848	+	0.983694i	$0.442439\pi$
$332$	13.5603	0.744215
$333$	13.2059	0.723678
$334$	−1.86847	−0.102238
$335$	−10.5029	−0.573837
$336$	−0.803425	−0.0438304
$337$	−16.6761	−0.908407	−0.454204	−	0.890898i	$-0.650076\pi$
$337$	−16.6761	−0.908407	−0.454204	+	0.890898i	$0.650076\pi$
$338$	6.01382	0.327109
$339$	−9.60623	−0.521739
$340$	−1.60450	−0.0870160
$341$	0	0
$342$	−4.49570	−0.243099
$343$	1.00000	0.0539949
$344$	−4.00169	−0.215757
$345$	3.39781	0.182932
$346$	−11.7799	−0.633294
$347$	−29.2816	−1.57192	−0.785959	−	0.618279i	$-0.787830\pi$
$347$	−29.2816	−1.57192	−0.785959	+	0.618279i	$0.787830\pi$
$348$	7.21424	0.386724
$349$	16.7836	0.898406	0.449203	−	0.893430i	$-0.351708\pi$
$349$	16.7836	0.898406	0.449203	+	0.893430i	$0.351708\pi$
$350$	−1.00000	−0.0534522
$351$	11.3706	0.606920
$352$	0	0
$353$	22.9110	1.21943	0.609715	−	0.792621i	$-0.291284\pi$
$353$	22.9110	1.21943	0.609715	+	0.792621i	$0.291284\pi$
$354$	3.34758	0.177922
$355$	4.58489	0.243340
$356$	−3.41009	−0.180735
$357$	1.28909	0.0682260
$358$	10.3705	0.548098
$359$	−8.36518	−0.441497	−0.220749	−	0.975331i	$-0.570850\pi$
$359$	−8.36518	−0.441497	−0.220749	+	0.975331i	$0.570850\pi$
$360$	2.35451	0.124093
$361$	−15.3542	−0.808115
$362$	22.6738	1.19171
$363$	0	0
$364$	2.64314	0.138538
$365$	−3.69293	−0.193297
$366$	−10.6557	−0.556983
$367$	−1.80322	−0.0941273	−0.0470637	−	0.998892i	$-0.514986\pi$
$367$	−1.80322	−0.0941273	−0.0470637	+	0.998892i	$0.514986\pi$
$368$	−4.22916	−0.220460
$369$	−22.8288	−1.18842
$370$	5.60877	0.291586
$371$	10.0478	0.521657
$372$	−1.98928	−0.103139
$373$	−9.60451	−0.497303	−0.248651	−	0.968593i	$-0.579987\pi$
$373$	−9.60451	−0.497303	−0.248651	+	0.968593i	$0.579987\pi$
$374$	0	0
$375$	−0.803425	−0.0414887
$376$	6.54502	0.337534
$377$	−23.7337	−1.22235
$378$	−4.30195	−0.221268
$379$	−5.94046	−0.305141	−0.152570	−	0.988293i	$-0.548755\pi$
$379$	−5.94046	−0.305141	−0.152570	+	0.988293i	$0.548755\pi$
$380$	−1.90940	−0.0979502
$381$	−10.7728	−0.551905
$382$	8.48643	0.434203
$383$	−29.4258	−1.50359	−0.751794	−	0.659399i	$-0.770811\pi$
$383$	−29.4258	−1.50359	−0.751794	+	0.659399i	$0.770811\pi$
$384$	0.803425	0.0409996
$385$	0	0
$386$	−25.1578	−1.28050
$387$	−9.42201	−0.478948
$388$	−17.4713	−0.886968
$389$	−18.8617	−0.956324	−0.478162	−	0.878272i	$-0.658697\pi$
$389$	−18.8617	−0.956324	−0.478162	+	0.878272i	$0.658697\pi$
$390$	2.12356	0.107531
$391$	6.78566	0.343166
$392$	−1.00000	−0.0505076
$393$	−6.82853	−0.344454
$394$	−13.2603	−0.668044
$395$	13.9477	0.701785
$396$	0	0
$397$	25.4542	1.27751	0.638756	−	0.769409i	$-0.279449\pi$
$397$	25.4542	1.27751	0.638756	+	0.769409i	$0.279449\pi$
$398$	−3.59813	−0.180358
$399$	1.53406	0.0767991
$400$	1.00000	0.0500000
$401$	14.9867	0.748399	0.374199	−	0.927348i	$-0.377918\pi$
$401$	14.9867	0.748399	0.374199	+	0.927348i	$0.377918\pi$
$402$	−8.43833	−0.420866
$403$	6.54440	0.326000
$404$	10.6494	0.529827
$405$	3.60723	0.179245
$406$	8.97936	0.445638
$407$	0	0
$408$	−1.28909	−0.0638196
$409$	−4.85775	−0.240200	−0.120100	−	0.992762i	$-0.538322\pi$
$409$	−4.85775	−0.240200	−0.120100	+	0.992762i	$0.538322\pi$
$410$	−9.69579	−0.478841
$411$	−8.09424	−0.399260
$412$	−6.86666	−0.338296
$413$	4.16663	0.205027
$414$	−9.95758	−0.489388
$415$	13.5603	0.665647
$416$	−2.64314	−0.129590
$417$	11.1034	0.543734
$418$	0	0
$419$	−1.77493	−0.0867109	−0.0433554	−	0.999060i	$-0.513805\pi$
$419$	−1.77493	−0.0867109	−0.0433554	+	0.999060i	$0.513805\pi$
$420$	−0.803425	−0.0392031
$421$	−16.9245	−0.824851	−0.412426	−	0.910991i	$-0.635318\pi$
$421$	−16.9245	−0.824851	−0.412426	+	0.910991i	$0.635318\pi$
$422$	−10.0058	−0.487075
$423$	15.4103	0.749275
$424$	−10.0478	−0.487965
$425$	−1.60450	−0.0778295
$426$	3.68361	0.178472
$427$	−13.2629	−0.641835
$428$	2.88270	0.139341
$429$	0	0
$430$	−4.00169	−0.192979
$431$	−4.15621	−0.200198	−0.100099	−	0.994977i	$-0.531916\pi$
$431$	−4.15621	−0.200198	−0.100099	+	0.994977i	$0.531916\pi$
$432$	4.30195	0.206977
$433$	22.2410	1.06883	0.534417	−	0.845221i	$-0.320531\pi$
$433$	22.2410	1.06883	0.534417	+	0.845221i	$0.320531\pi$
$434$	−2.47600	−0.118852
$435$	7.21424	0.345896
$436$	18.4938	0.885693
$437$	8.07515	0.386287
$438$	−2.96699	−0.141768
$439$	21.6172	1.03173	0.515866	−	0.856669i	$-0.327470\pi$
$439$	21.6172	1.03173	0.515866	+	0.856669i	$0.327470\pi$
$440$	0	0
$441$	−2.35451	−0.112119
$442$	4.24090	0.201719
$443$	−12.9462	−0.615092	−0.307546	−	0.951533i	$-0.599508\pi$
$443$	−12.9462	−0.615092	−0.307546	+	0.951533i	$0.599508\pi$
$444$	4.50623	0.213856
$445$	−3.41009	−0.161654
$446$	−25.0397	−1.18566
$447$	−14.7540	−0.697838
$448$	1.00000	0.0472456
$449$	13.8561	0.653910	0.326955	−	0.945040i	$-0.393977\pi$
$449$	13.8561	0.653910	0.326955	+	0.945040i	$0.393977\pi$
$450$	2.35451	0.110993
$451$	0	0
$452$	11.9566	0.562391
$453$	16.5279	0.776549
$454$	−0.182924	−0.00858503
$455$	2.64314	0.123912
$456$	−1.53406	−0.0718389
$457$	9.98796	0.467217	0.233608	−	0.972331i	$-0.424947\pi$
$457$	9.98796	0.467217	0.233608	+	0.972331i	$0.424947\pi$
$458$	0.134371	0.00627876
$459$	−6.90246	−0.322179
$460$	−4.22916	−0.197185
$461$	−27.8198	−1.29570	−0.647850	−	0.761768i	$-0.724332\pi$
$461$	−27.8198	−1.29570	−0.647850	+	0.761768i	$0.724332\pi$
$462$	0	0
$463$	−10.2727	−0.477413	−0.238707	−	0.971092i	$-0.576723\pi$
$463$	−10.2727	−0.477413	−0.238707	+	0.971092i	$0.576723\pi$
$464$	−8.97936	−0.416856
$465$	−1.98928	−0.0922506
$466$	−2.93010	−0.135734
$467$	9.03790	0.418224	0.209112	−	0.977892i	$-0.432943\pi$
$467$	9.03790	0.418224	0.209112	+	0.977892i	$0.432943\pi$
$468$	−6.22329	−0.287672
$469$	−10.5029	−0.484981
$470$	6.54502	0.301899
$471$	3.73501	0.172100
$472$	−4.16663	−0.191785
$473$	0	0
$474$	11.2059	0.514706
$475$	−1.90940	−0.0876093
$476$	−1.60450	−0.0735420
$477$	−23.6577	−1.08321
$478$	8.27410	0.378449
$479$	39.4300	1.80160	0.900801	−	0.434233i	$-0.142980\pi$
$479$	39.4300	1.80160	0.900801	+	0.434233i	$0.142980\pi$
$480$	0.803425	0.0366712
$481$	−14.8247	−0.675950
$482$	16.5842	0.755391
$483$	3.39781	0.154606
$484$	0	0
$485$	−17.4713	−0.793329
$486$	15.8040	0.716883
$487$	−18.9186	−0.857282	−0.428641	−	0.903475i	$-0.641007\pi$
$487$	−18.9186	−0.857282	−0.428641	+	0.903475i	$0.641007\pi$
$488$	13.2629	0.600381
$489$	−4.01230	−0.181443
$490$	−1.00000	−0.0451754
$491$	1.51922	0.0685615	0.0342808	−	0.999412i	$-0.489086\pi$
$491$	1.51922	0.0685615	0.0342808	+	0.999412i	$0.489086\pi$
$492$	−7.78984	−0.351193
$493$	14.4073	0.648874
$494$	5.04681	0.227067
$495$	0	0
$496$	2.47600	0.111176
$497$	4.58489	0.205660
$498$	10.8946	0.488201
$499$	11.8172	0.529012	0.264506	−	0.964384i	$-0.414791\pi$
$499$	11.8172	0.529012	0.264506	+	0.964384i	$0.414791\pi$
$500$	1.00000	0.0447214
$501$	−1.50117	−0.0670675
$502$	13.5486	0.604703
$503$	16.0491	0.715592	0.357796	−	0.933800i	$-0.383528\pi$
$503$	16.0491	0.715592	0.357796	+	0.933800i	$0.383528\pi$
$504$	2.35451	0.104878
$505$	10.6494	0.473891
$506$	0	0
$507$	4.83166	0.214581
$508$	13.4085	0.594908
$509$	40.2765	1.78522	0.892612	−	0.450826i	$-0.148870\pi$
$509$	40.2765	1.78522	0.892612	+	0.450826i	$0.148870\pi$
$510$	−1.28909	−0.0570820
$511$	−3.69293	−0.163365
$512$	−1.00000	−0.0441942
$513$	−8.21414	−0.362663
$514$	−18.4226	−0.812584
$515$	−6.86666	−0.302581
$516$	−3.21506	−0.141535
$517$	0	0
$518$	5.60877	0.246435
$519$	−9.46431	−0.415437
$520$	−2.64314	−0.115909
$521$	−27.4901	−1.20436	−0.602182	−	0.798359i	$-0.705702\pi$
$521$	−27.4901	−1.20436	−0.602182	+	0.798359i	$0.705702\pi$
$522$	−21.1420	−0.925359
$523$	−20.2364	−0.884875	−0.442438	−	0.896799i	$-0.645886\pi$
$523$	−20.2364	−0.884875	−0.442438	+	0.896799i	$0.645886\pi$
$524$	8.49928	0.371293
$525$	−0.803425	−0.0350643
$526$	−19.1658	−0.835668
$527$	−3.97273	−0.173055
$528$	0	0
$529$	−5.11424	−0.222358
$530$	−10.0478	−0.436449
$531$	−9.81037	−0.425734
$532$	−1.90940	−0.0827830
$533$	25.6273	1.11004
$534$	−2.73975	−0.118561
$535$	2.88270	0.124630
$536$	10.5029	0.453658
$537$	8.33193	0.359549
$538$	−9.08666	−0.391754
$539$	0	0
$540$	4.30195	0.185126
$541$	−26.5254	−1.14042	−0.570209	−	0.821500i	$-0.693138\pi$
$541$	−26.5254	−1.14042	−0.570209	+	0.821500i	$0.693138\pi$
$542$	19.1739	0.823590
$543$	18.2167	0.781752
$544$	1.60450	0.0687922
$545$	18.4938	0.792188
$546$	2.12356	0.0908801
$547$	−3.17578	−0.135787	−0.0678933	−	0.997693i	$-0.521628\pi$
$547$	−3.17578	−0.135787	−0.0678933	+	0.997693i	$0.521628\pi$
$548$	10.0747	0.430369
$549$	31.2275	1.33276
$550$	0	0
$551$	17.1452	0.730409
$552$	−3.39781	−0.144620
$553$	13.9477	0.593117
$554$	−20.6691	−0.878148
$555$	4.50623	0.191279
$556$	−13.8200	−0.586100
$557$	29.2227	1.23821	0.619103	−	0.785310i	$-0.287497\pi$
$557$	29.2227	1.23821	0.619103	+	0.785310i	$0.287497\pi$
$558$	5.82976	0.246793
$559$	10.5770	0.447360
$560$	1.00000	0.0422577
$561$	0	0
$562$	−1.37153	−0.0578543
$563$	6.15480	0.259394	0.129697	−	0.991554i	$-0.458600\pi$
$563$	6.15480	0.259394	0.129697	+	0.991554i	$0.458600\pi$
$564$	5.25844	0.221420
$565$	11.9566	0.503018
$566$	−6.43513	−0.270488
$567$	3.60723	0.151490
$568$	−4.58489	−0.192378
$569$	3.59047	0.150520	0.0752601	−	0.997164i	$-0.476021\pi$
$569$	3.59047	0.150520	0.0752601	+	0.997164i	$0.476021\pi$
$570$	−1.53406	−0.0642547
$571$	28.1128	1.17649	0.588243	−	0.808684i	$-0.299820\pi$
$571$	28.1128	1.17649	0.588243	+	0.808684i	$0.299820\pi$
$572$	0	0
$573$	6.81821	0.284835
$574$	−9.69579	−0.404695
$575$	−4.22916	−0.176368
$576$	−2.35451	−0.0981045
$577$	28.5378	1.18804	0.594022	−	0.804448i	$-0.297539\pi$
$577$	28.5378	1.18804	0.594022	+	0.804448i	$0.297539\pi$
$578$	14.4256	0.600026
$579$	−20.2124	−0.840000
$580$	−8.97936	−0.372847
$581$	13.5603	0.562574
$582$	−14.0368	−0.581846
$583$	0	0
$584$	3.69293	0.152814
$585$	−6.22329	−0.257301
$586$	4.59731	0.189913
$587$	−19.5107	−0.805291	−0.402646	−	0.915356i	$-0.631909\pi$
$587$	−19.5107	−0.805291	−0.402646	+	0.915356i	$0.631909\pi$
$588$	−0.803425	−0.0331327
$589$	−4.72767	−0.194800
$590$	−4.16663	−0.171538
$591$	−10.6537	−0.438233
$592$	−5.60877	−0.230519
$593$	−4.84298	−0.198877	−0.0994386	−	0.995044i	$-0.531705\pi$
$593$	−4.84298	−0.198877	−0.0994386	+	0.995044i	$0.531705\pi$
$594$	0	0
$595$	−1.60450	−0.0657779
$596$	18.3638	0.752211
$597$	−2.89083	−0.118314
$598$	11.1782	0.457112
$599$	−3.67149	−0.150013	−0.0750064	−	0.997183i	$-0.523898\pi$
$599$	−3.67149	−0.150013	−0.0750064	+	0.997183i	$0.523898\pi$
$600$	0.803425	0.0327997
$601$	1.73373	0.0707203	0.0353601	−	0.999375i	$-0.488742\pi$
$601$	1.73373	0.0707203	0.0353601	+	0.999375i	$0.488742\pi$
$602$	−4.00169	−0.163097
$603$	24.7293	1.00705
$604$	−20.5718	−0.837055
$605$	0	0
$606$	8.55598	0.347563
$607$	28.4589	1.15511	0.577555	−	0.816351i	$-0.304007\pi$
$607$	28.4589	1.15511	0.577555	+	0.816351i	$0.304007\pi$
$608$	1.90940	0.0774364
$609$	7.21424	0.292336
$610$	13.2629	0.536997
$611$	−17.2994	−0.699859
$612$	3.77780	0.152708
$613$	8.07904	0.326309	0.163155	−	0.986601i	$-0.447833\pi$
$613$	8.07904	0.326309	0.163155	+	0.986601i	$0.447833\pi$
$614$	−24.8042	−1.00102
$615$	−7.78984	−0.314117
$616$	0	0
$617$	2.53152	0.101915	0.0509577	−	0.998701i	$-0.483773\pi$
$617$	2.53152	0.101915	0.0509577	+	0.998701i	$0.483773\pi$
$618$	−5.51685	−0.221920
$619$	−27.6305	−1.11056	−0.555281	−	0.831663i	$-0.687389\pi$
$619$	−27.6305	−1.11056	−0.555281	+	0.831663i	$0.687389\pi$
$620$	2.47600	0.0994385
$621$	−18.1936	−0.730084
$622$	−25.8023	−1.03458
$623$	−3.41009	−0.136622
$624$	−2.12356	−0.0850106
$625$	1.00000	0.0400000
$626$	−16.5081	−0.659797
$627$	0	0
$628$	−4.64886	−0.185510
$629$	8.99925	0.358823
$630$	2.35451	0.0938058
$631$	−32.2983	−1.28577	−0.642887	−	0.765961i	$-0.722263\pi$
$631$	−32.2983	−1.28577	−0.642887	+	0.765961i	$0.722263\pi$
$632$	−13.9477	−0.554810
$633$	−8.03891	−0.319518
$634$	−2.33084	−0.0925696
$635$	13.4085	0.532102
$636$	−8.07267	−0.320102
$637$	2.64314	0.104725
$638$	0	0
$639$	−10.7952	−0.427050
$640$	−1.00000	−0.0395285
$641$	31.3973	1.24012	0.620060	−	0.784554i	$-0.287108\pi$
$641$	31.3973	1.24012	0.620060	+	0.784554i	$0.287108\pi$
$642$	2.31603	0.0914066
$643$	4.12543	0.162691	0.0813456	−	0.996686i	$-0.474078\pi$
$643$	4.12543	0.162691	0.0813456	+	0.996686i	$0.474078\pi$
$644$	−4.22916	−0.166652
$645$	−3.21506	−0.126593
$646$	−3.06363	−0.120537
$647$	20.9288	0.822794	0.411397	−	0.911456i	$-0.365041\pi$
$647$	20.9288	0.822794	0.411397	+	0.911456i	$0.365041\pi$
$648$	−3.60723	−0.141705
$649$	0	0
$650$	−2.64314	−0.103672
$651$	−1.98928	−0.0779660
$652$	4.99400	0.195580
$653$	−2.00850	−0.0785988	−0.0392994	−	0.999227i	$-0.512513\pi$
$653$	−2.00850	−0.0785988	−0.0392994	+	0.999227i	$0.512513\pi$
$654$	14.8584	0.581009
$655$	8.49928	0.332094
$656$	9.69579	0.378557
$657$	8.69502	0.339225
$658$	6.54502	0.255152
$659$	−11.7132	−0.456282	−0.228141	−	0.973628i	$-0.573265\pi$
$659$	−11.7132	−0.456282	−0.228141	+	0.973628i	$0.573265\pi$
$660$	0	0
$661$	35.8654	1.39500	0.697502	−	0.716583i	$-0.254295\pi$
$661$	35.8654	1.39500	0.697502	+	0.716583i	$0.254295\pi$
$662$	−6.54411	−0.254344
$663$	3.40725	0.132327
$664$	−13.5603	−0.526240
$665$	−1.90940	−0.0740434
$666$	−13.2059	−0.511718
$667$	37.9751	1.47040
$668$	1.86847	0.0722932
$669$	−20.1175	−0.777788
$670$	10.5029	0.405764
$671$	0	0
$672$	0.803425	0.0309928
$673$	33.3617	1.28600	0.642999	−	0.765867i	$-0.277690\pi$
$673$	33.3617	1.28600	0.642999	+	0.765867i	$0.277690\pi$
$674$	16.6761	0.642341
$675$	4.30195	0.165582
$676$	−6.01382	−0.231301
$677$	13.2733	0.510134	0.255067	−	0.966923i	$-0.417903\pi$
$677$	13.2733	0.510134	0.255067	+	0.966923i	$0.417903\pi$
$678$	9.60623	0.368925
$679$	−17.4713	−0.670485
$680$	1.60450	0.0615296
$681$	−0.146965	−0.00563173
$682$	0	0
$683$	−15.3702	−0.588123	−0.294061	−	0.955787i	$-0.595007\pi$
$683$	−15.3702	−0.588123	−0.294061	+	0.955787i	$0.595007\pi$
$684$	4.49570	0.171897
$685$	10.0747	0.384933
$686$	−1.00000	−0.0381802
$687$	0.107957	0.00411883
$688$	4.00169	0.152563
$689$	26.5578	1.01177
$690$	−3.39781	−0.129352
$691$	20.7798	0.790501	0.395251	−	0.918573i	$-0.370658\pi$
$691$	20.7798	0.790501	0.395251	+	0.918573i	$0.370658\pi$
$692$	11.7799	0.447806
$693$	0	0
$694$	29.2816	1.11151
$695$	−13.8200	−0.524224
$696$	−7.21424	−0.273455
$697$	−15.5569	−0.589258
$698$	−16.7836	−0.635269
$699$	−2.35411	−0.0890407
$700$	1.00000	0.0377964
$701$	4.18595	0.158101	0.0790506	−	0.996871i	$-0.474811\pi$
$701$	4.18595	0.158101	0.0790506	+	0.996871i	$0.474811\pi$
$702$	−11.3706	−0.429157
$703$	10.7094	0.403912
$704$	0	0
$705$	5.25844	0.198044
$706$	−22.9110	−0.862267
$707$	10.6494	0.400511
$708$	−3.34758	−0.125810
$709$	5.48528	0.206004	0.103002	−	0.994681i	$-0.467155\pi$
$709$	5.48528	0.206004	0.103002	+	0.994681i	$0.467155\pi$
$710$	−4.58489	−0.172068
$711$	−32.8400	−1.23160
$712$	3.41009	0.127799
$713$	−10.4714	−0.392156
$714$	−1.28909	−0.0482431
$715$	0	0
$716$	−10.3705	−0.387564
$717$	6.64762	0.248260
$718$	8.36518	0.312186
$719$	−16.6013	−0.619122	−0.309561	−	0.950880i	$-0.600182\pi$
$719$	−16.6013	−0.619122	−0.309561	+	0.950880i	$0.600182\pi$
$720$	−2.35451	−0.0877473
$721$	−6.86666	−0.255728
$722$	15.3542	0.571424
$723$	13.3242	0.495532
$724$	−22.6738	−0.842664
$725$	−8.97936	−0.333485
$726$	0	0
$727$	−22.9499	−0.851164	−0.425582	−	0.904920i	$-0.639931\pi$
$727$	−22.9499	−0.851164	−0.425582	+	0.904920i	$0.639931\pi$
$728$	−2.64314	−0.0979612
$729$	1.87562	0.0694673
$730$	3.69293	0.136681
$731$	−6.42070	−0.237478
$732$	10.6557	0.393847
$733$	15.9482	0.589060	0.294530	−	0.955642i	$-0.404837\pi$
$733$	15.9482	0.589060	0.294530	+	0.955642i	$0.404837\pi$
$734$	1.80322	0.0665581
$735$	−0.803425	−0.0296348
$736$	4.22916	0.155889
$737$	0	0
$738$	22.8288	0.840340
$739$	41.6561	1.53234	0.766171	−	0.642636i	$-0.222159\pi$
$739$	41.6561	1.53234	0.766171	+	0.642636i	$0.222159\pi$
$740$	−5.60877	−0.206182
$741$	4.05473	0.148954
$742$	−10.0478	−0.368867
$743$	41.9839	1.54024	0.770120	−	0.637899i	$-0.220196\pi$
$743$	41.9839	1.54024	0.770120	+	0.637899i	$0.220196\pi$
$744$	1.98928	0.0729305
$745$	18.3638	0.672798
$746$	9.60451	0.351646
$747$	−31.9277	−1.16817
$748$	0	0
$749$	2.88270	0.105332
$750$	0.803425	0.0293369
$751$	30.9563	1.12961	0.564806	−	0.825223i	$-0.308951\pi$
$751$	30.9563	1.12961	0.564806	+	0.825223i	$0.308951\pi$
$752$	−6.54502	−0.238672
$753$	10.8853	0.396682
$754$	23.7337	0.864330
$755$	−20.5718	−0.748685
$756$	4.30195	0.156460
$757$	19.9821	0.726261	0.363130	−	0.931738i	$-0.381708\pi$
$757$	19.9821	0.726261	0.363130	+	0.931738i	$0.381708\pi$
$758$	5.94046	0.215767
$759$	0	0
$760$	1.90940	0.0692612
$761$	−8.32474	−0.301771	−0.150886	−	0.988551i	$-0.548213\pi$
$761$	−8.32474	−0.301771	−0.150886	+	0.988551i	$0.548213\pi$
$762$	10.7728	0.390256
$763$	18.4938	0.669521
$764$	−8.48643	−0.307028
$765$	3.77780	0.136587
$766$	29.4258	1.06320
$767$	11.0130	0.397656
$768$	−0.803425	−0.0289911
$769$	38.3934	1.38450	0.692251	−	0.721657i	$-0.256619\pi$
$769$	38.3934	1.38450	0.692251	+	0.721657i	$0.256619\pi$
$770$	0	0
$771$	−14.8011	−0.533050
$772$	25.1578	0.905451
$773$	28.1213	1.01145	0.505726	−	0.862694i	$-0.331225\pi$
$773$	28.1213	1.01145	0.505726	+	0.862694i	$0.331225\pi$
$774$	9.42201	0.338667
$775$	2.47600	0.0889405
$776$	17.4713	0.627181
$777$	4.50623	0.161660
$778$	18.8617	0.676223
$779$	−18.5131	−0.663302
$780$	−2.12356	−0.0760358
$781$	0	0
$782$	−6.78566	−0.242655
$783$	−38.6287	−1.38048
$784$	1.00000	0.0357143
$785$	−4.64886	−0.165925
$786$	6.82853	0.243566
$787$	30.1966	1.07639	0.538197	−	0.842819i	$-0.319106\pi$
$787$	30.1966	1.07639	0.538197	+	0.842819i	$0.319106\pi$
$788$	13.2603	0.472379
$789$	−15.3983	−0.548193
$790$	−13.9477	−0.496237
$791$	11.9566	0.425128
$792$	0	0
$793$	−35.0556	−1.24486
$794$	−25.4542	−0.903337
$795$	−8.07267	−0.286308
$796$	3.59813	0.127532
$797$	46.4224	1.64436	0.822182	−	0.569224i	$-0.192756\pi$
$797$	46.4224	1.64436	0.822182	+	0.569224i	$0.192756\pi$
$798$	−1.53406	−0.0543051
$799$	10.5015	0.371515
$800$	−1.00000	−0.0353553
$801$	8.02909	0.283694
$802$	−14.9867	−0.529198
$803$	0	0
$804$	8.43833	0.297597
$805$	−4.22916	−0.149058
$806$	−6.54440	−0.230517
$807$	−7.30045	−0.256988
$808$	−10.6494	−0.374644
$809$	12.0678	0.424282	0.212141	−	0.977239i	$-0.431956\pi$
$809$	12.0678	0.424282	0.212141	+	0.977239i	$0.431956\pi$
$810$	−3.60723	−0.126745
$811$	−14.9538	−0.525098	−0.262549	−	0.964919i	$-0.584563\pi$
$811$	−14.9538	−0.525098	−0.262549	+	0.964919i	$0.584563\pi$
$812$	−8.97936	−0.315114
$813$	15.4048	0.540270
$814$	0	0
$815$	4.99400	0.174932
$816$	1.28909	0.0451273
$817$	−7.64083	−0.267319
$818$	4.85775	0.169847
$819$	−6.22329	−0.217459
$820$	9.69579	0.338592
$821$	−23.8592	−0.832693	−0.416347	−	0.909206i	$-0.636690\pi$
$821$	−23.8592	−0.832693	−0.416347	+	0.909206i	$0.636690\pi$
$822$	8.09424	0.282319
$823$	−42.3899	−1.47762	−0.738809	−	0.673914i	$-0.764612\pi$
$823$	−42.3899	−1.47762	−0.738809	+	0.673914i	$0.764612\pi$
$824$	6.86666	0.239212
$825$	0	0
$826$	−4.16663	−0.144976
$827$	52.9595	1.84158	0.920790	−	0.390058i	$-0.127545\pi$
$827$	52.9595	1.84158	0.920790	+	0.390058i	$0.127545\pi$
$828$	9.95758	0.346050
$829$	44.8068	1.55620	0.778102	−	0.628137i	$-0.216182\pi$
$829$	44.8068	1.55620	0.778102	+	0.628137i	$0.216182\pi$
$830$	−13.5603	−0.470683
$831$	−16.6061	−0.576060
$832$	2.64314	0.0916343
$833$	−1.60450	−0.0555925
$834$	−11.1034	−0.384478
$835$	1.86847	0.0646610
$836$	0	0
$837$	10.6516	0.368174
$838$	1.77493	0.0613138
$839$	−40.6109	−1.40204	−0.701022	−	0.713140i	$-0.747272\pi$
$839$	−40.6109	−1.40204	−0.701022	+	0.713140i	$0.747272\pi$
$840$	0.803425	0.0277208
$841$	51.6288	1.78030
$842$	16.9245	0.583258
$843$	−1.10192	−0.0379521
$844$	10.0058	0.344414
$845$	−6.01382	−0.206882
$846$	−15.4103	−0.529817
$847$	0	0
$848$	10.0478	0.345043
$849$	−5.17014	−0.177439
$850$	1.60450	0.0550338
$851$	23.7204	0.813123
$852$	−3.68361	−0.126198
$853$	0.116843	0.00400061	0.00200031	−	0.999998i	$-0.499363\pi$
$853$	0.116843	0.00400061	0.00200031	+	0.999998i	$0.499363\pi$
$854$	13.2629	0.453846
$855$	4.49570	0.153750
$856$	−2.88270	−0.0985287
$857$	−22.2623	−0.760465	−0.380233	−	0.924891i	$-0.624156\pi$
$857$	−22.2623	−0.760465	−0.380233	+	0.924891i	$0.624156\pi$
$858$	0	0
$859$	37.0142	1.26291	0.631454	−	0.775413i	$-0.282458\pi$
$859$	37.0142	1.26291	0.631454	+	0.775413i	$0.282458\pi$
$860$	4.00169	0.136457
$861$	−7.78984	−0.265477
$862$	4.15621	0.141561
$863$	44.0257	1.49865	0.749326	−	0.662201i	$-0.230377\pi$
$863$	44.0257	1.49865	0.749326	+	0.662201i	$0.230377\pi$
$864$	−4.30195	−0.146355
$865$	11.7799	0.400530
$866$	−22.2410	−0.755780
$867$	11.5899	0.393613
$868$	2.47600	0.0840408
$869$	0	0
$870$	−7.21424	−0.244586
$871$	−27.7607	−0.940637
$872$	−18.4938	−0.626280
$873$	41.1362	1.39225
$874$	−8.07515	−0.273146
$875$	1.00000	0.0338062
$876$	2.96699	0.100245
$877$	29.4099	0.993100	0.496550	−	0.868008i	$-0.334600\pi$
$877$	29.4099	0.993100	0.496550	+	0.868008i	$0.334600\pi$
$878$	−21.6172	−0.729545
$879$	3.69359	0.124582
$880$	0	0
$881$	34.0756	1.14804	0.574019	−	0.818842i	$-0.305384\pi$
$881$	34.0756	1.14804	0.574019	+	0.818842i	$0.305384\pi$
$882$	2.35451	0.0792804
$883$	−18.4353	−0.620397	−0.310199	−	0.950672i	$-0.600396\pi$
$883$	−18.4353	−0.620397	−0.310199	+	0.950672i	$0.600396\pi$
$884$	−4.24090	−0.142637
$885$	−3.34758	−0.112528
$886$	12.9462	0.434936
$887$	−2.19623	−0.0737421	−0.0368711	−	0.999320i	$-0.511739\pi$
$887$	−2.19623	−0.0737421	−0.0368711	+	0.999320i	$0.511739\pi$
$888$	−4.50623	−0.151219
$889$	13.4085	0.449708
$890$	3.41009	0.114307
$891$	0	0
$892$	25.0397	0.838391
$893$	12.4971	0.418198
$894$	14.7540	0.493446
$895$	−10.3705	−0.346648
$896$	−1.00000	−0.0334077
$897$	8.98088	0.299863
$898$	−13.8561	−0.462384
$899$	−22.2329	−0.741508
$900$	−2.35451	−0.0784836
$901$	−16.1217	−0.537091
$902$	0	0
$903$	−3.21506	−0.106990
$904$	−11.9566	−0.397671
$905$	−22.6738	−0.753702
$906$	−16.5279	−0.549103
$907$	4.84606	0.160911	0.0804554	−	0.996758i	$-0.474363\pi$
$907$	4.84606	0.160911	0.0804554	+	0.996758i	$0.474363\pi$
$908$	0.182924	0.00607054
$909$	−25.0741	−0.831654
$910$	−2.64314	−0.0876192
$911$	59.6573	1.97653	0.988267	−	0.152736i	$-0.0488085\pi$
$911$	59.6573	1.97653	0.988267	+	0.152736i	$0.0488085\pi$
$912$	1.53406	0.0507978
$913$	0	0
$914$	−9.98796	−0.330372
$915$	10.6557	0.352267
$916$	−0.134371	−0.00443975
$917$	8.49928	0.280671
$918$	6.90246	0.227815
$919$	16.9163	0.558018	0.279009	−	0.960288i	$-0.409994\pi$
$919$	16.9163	0.558018	0.279009	+	0.960288i	$0.409994\pi$
$920$	4.22916	0.139431
$921$	−19.9283	−0.656660
$922$	27.8198	0.916198
$923$	12.1185	0.398885
$924$	0	0
$925$	−5.60877	−0.184415
$926$	10.2727	0.337582
$927$	16.1676	0.531014
$928$	8.97936	0.294762
$929$	23.9924	0.787164	0.393582	−	0.919290i	$-0.371236\pi$
$929$	23.9924	0.787164	0.393582	+	0.919290i	$0.371236\pi$
$930$	1.98928	0.0652310
$931$	−1.90940	−0.0625781
$932$	2.93010	0.0959785
$933$	−20.7302	−0.678676
$934$	−9.03790	−0.295729
$935$	0	0
$936$	6.22329	0.203415
$937$	−30.5167	−0.996937	−0.498468	−	0.866908i	$-0.666104\pi$
$937$	−30.5167	−0.996937	−0.498468	+	0.866908i	$0.666104\pi$
$938$	10.5029	0.342933
$939$	−13.2630	−0.432822
$940$	−6.54502	−0.213475
$941$	−17.6656	−0.575882	−0.287941	−	0.957648i	$-0.592971\pi$
$941$	−17.6656	−0.575882	−0.287941	+	0.957648i	$0.592971\pi$
$942$	−3.73501	−0.121693
$943$	−41.0050	−1.33531
$944$	4.16663	0.135612
$945$	4.30195	0.139942
$946$	0	0
$947$	4.02455	0.130780	0.0653902	−	0.997860i	$-0.479171\pi$
$947$	4.02455	0.130780	0.0653902	+	0.997860i	$0.479171\pi$
$948$	−11.2059	−0.363952
$949$	−9.76091	−0.316853
$950$	1.90940	0.0619491
$951$	−1.87266	−0.0607251
$952$	1.60450	0.0520020
$953$	51.5532	1.66997	0.834985	−	0.550272i	$-0.185476\pi$
$953$	51.5532	1.66997	0.834985	+	0.550272i	$0.185476\pi$
$954$	23.6577	0.765945
$955$	−8.48643	−0.274614
$956$	−8.27410	−0.267604
$957$	0	0
$958$	−39.4300	−1.27392
$959$	10.0747	0.325328
$960$	−0.803425	−0.0259304
$961$	−24.8694	−0.802240
$962$	14.8247	0.477969
$963$	−6.78734	−0.218719
$964$	−16.5842	−0.534142
$965$	25.1578	0.809860
$966$	−3.39781	−0.109323
$967$	−25.3238	−0.814359	−0.407179	−	0.913348i	$-0.633488\pi$
$967$	−25.3238	−0.814359	−0.407179	+	0.913348i	$0.633488\pi$
$968$	0	0
$969$	−2.46139	−0.0790713
$970$	17.4713	0.560968
$971$	−41.2627	−1.32418	−0.662091	−	0.749423i	$-0.730331\pi$
$971$	−41.2627	−1.32418	−0.662091	+	0.749423i	$0.730331\pi$
$972$	−15.8040	−0.506913
$973$	−13.8200	−0.443050
$974$	18.9186	0.606190
$975$	−2.12356	−0.0680085
$976$	−13.2629	−0.424534
$977$	14.0714	0.450184	0.225092	−	0.974337i	$-0.427732\pi$
$977$	14.0714	0.450184	0.225092	+	0.974337i	$0.427732\pi$
$978$	4.01230	0.128299
$979$	0	0
$980$	1.00000	0.0319438
$981$	−43.5438	−1.39025
$982$	−1.51922	−0.0484803
$983$	51.4025	1.63949	0.819743	−	0.572732i	$-0.194117\pi$
$983$	51.4025	1.63949	0.819743	+	0.572732i	$0.194117\pi$
$984$	7.78984	0.248331
$985$	13.2603	0.422508
$986$	−14.4073	−0.458823
$987$	5.25844	0.167378
$988$	−5.04681	−0.160560
$989$	−16.9238	−0.538145
$990$	0	0
$991$	45.9460	1.45952	0.729761	−	0.683702i	$-0.239631\pi$
$991$	45.9460	1.45952	0.729761	+	0.683702i	$0.239631\pi$
$992$	−2.47600	−0.0786130
$993$	−5.25770	−0.166848
$994$	−4.58489	−0.145424
$995$	3.59813	0.114068
$996$	−10.8946	−0.345210
$997$	−12.2089	−0.386659	−0.193329	−	0.981134i	$-0.561929\pi$
$997$	−12.2089	−0.386659	−0.193329	+	0.981134i	$0.561929\pi$
$998$	−11.8172	−0.374068
$999$	−24.1286	−0.763396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cv.1.3		6
11.2	odd	10		770.2.n.g.631.2	yes	12
11.6	odd	10		770.2.n.g.421.2	✓	12
11.10	odd	2		8470.2.a.db.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.g.421.2	✓	12	11.6	odd	10
770.2.n.g.631.2	yes	12	11.2	odd	10
8470.2.a.cv.1.3		6	1.1	even	1	trivial
8470.2.a.db.1.3		6	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.803425 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.803425 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.35451 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.803425 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.803425 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.35451 q^{9} -1.00000 q^{10} -0.803425 q^{12} +2.64314 q^{13} -1.00000 q^{14} -0.803425 q^{15} +1.00000 q^{16} -1.60450 q^{17} +2.35451 q^{18} -1.90940 q^{19} +1.00000 q^{20} -0.803425 q^{21} -4.22916 q^{23} +0.803425 q^{24} +1.00000 q^{25} -2.64314 q^{26} +4.30195 q^{27} +1.00000 q^{28} -8.97936 q^{29} +0.803425 q^{30} +2.47600 q^{31} -1.00000 q^{32} +1.60450 q^{34} +1.00000 q^{35} -2.35451 q^{36} -5.60877 q^{37} +1.90940 q^{38} -2.12356 q^{39} -1.00000 q^{40} +9.69579 q^{41} +0.803425 q^{42} +4.00169 q^{43} -2.35451 q^{45} +4.22916 q^{46} -6.54502 q^{47} -0.803425 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.28909 q^{51} +2.64314 q^{52} +10.0478 q^{53} -4.30195 q^{54} -1.00000 q^{56} +1.53406 q^{57} +8.97936 q^{58} +4.16663 q^{59} -0.803425 q^{60} -13.2629 q^{61} -2.47600 q^{62} -2.35451 q^{63} +1.00000 q^{64} +2.64314 q^{65} -10.5029 q^{67} -1.60450 q^{68} +3.39781 q^{69} -1.00000 q^{70} +4.58489 q^{71} +2.35451 q^{72} -3.69293 q^{73} +5.60877 q^{74} -0.803425 q^{75} -1.90940 q^{76} +2.12356 q^{78} +13.9477 q^{79} +1.00000 q^{80} +3.60723 q^{81} -9.69579 q^{82} +13.5603 q^{83} -0.803425 q^{84} -1.60450 q^{85} -4.00169 q^{86} +7.21424 q^{87} -3.41009 q^{89} +2.35451 q^{90} +2.64314 q^{91} -4.22916 q^{92} -1.98928 q^{93} +6.54502 q^{94} -1.90940 q^{95} +0.803425 q^{96} -17.4713 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} - q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 - q^9	\( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} - q^{9} - 6 q^{10} - q^{12} + 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 21 q^{17} + q^{18} - 3 q^{19} + 6 q^{20} - q^{21} - 10 q^{23} + q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} + 6 q^{28} + 10 q^{29} + q^{30} - 4 q^{31} - 6 q^{32} - 21 q^{34} + 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} + 26 q^{39} - 6 q^{40} + 7 q^{41} + q^{42} + 19 q^{43} - q^{45} + 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} - 6 q^{50} - 4 q^{51} + 6 q^{52} - 16 q^{53} + 4 q^{54} - 6 q^{56} + 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} - 8 q^{61} + 4 q^{62} - q^{63} + 6 q^{64} + 6 q^{65} - 27 q^{67} + 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} + q^{72} + 13 q^{73} + 2 q^{74} - q^{75} - 3 q^{76} - 26 q^{78} + 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} + 51 q^{83} - q^{84} + 21 q^{85} - 19 q^{86} + 8 q^{87} + q^{89} + q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} - 10 q^{94} - 3 q^{95} + q^{96} + 7 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 - q^9 - 6 * q^10 - q^12 + 6 * q^13 - 6 * q^14 - q^15 + 6 * q^16 + 21 * q^17 + q^18 - 3 * q^19 + 6 * q^20 - q^21 - 10 * q^23 + q^24 + 6 * q^25 - 6 * q^26 - 4 * q^27 + 6 * q^28 + 10 * q^29 + q^30 - 4 * q^31 - 6 * q^32 - 21 * q^34 + 6 * q^35 - q^36 - 2 * q^37 + 3 * q^38 + 26 * q^39 - 6 * q^40 + 7 * q^41 + q^42 + 19 * q^43 - q^45 + 10 * q^46 + 10 * q^47 - q^48 + 6 * q^49 - 6 * q^50 - 4 * q^51 + 6 * q^52 - 16 * q^53 + 4 * q^54 - 6 * q^56 + 16 * q^57 - 10 * q^58 - 3 * q^59 - q^60 - 8 * q^61 + 4 * q^62 - q^63 + 6 * q^64 + 6 * q^65 - 27 * q^67 + 21 * q^68 + 4 * q^69 - 6 * q^70 + 4 * q^71 + q^72 + 13 * q^73 + 2 * q^74 - q^75 - 3 * q^76 - 26 * q^78 + 14 * q^79 + 6 * q^80 - 14 * q^81 - 7 * q^82 + 51 * q^83 - q^84 + 21 * q^85 - 19 * q^86 + 8 * q^87 + q^89 + q^90 + 6 * q^91 - 10 * q^92 + 4 * q^93 - 10 * q^94 - 3 * q^95 + q^96 + 7 * q^97 - 6 * q^98

Introduction

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