LMFDB - Embedded newform 8470.2.a.ci.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.68585 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.68585 q^{9} -1.00000 q^{10} +1.14637 q^{12} -4.68585 q^{13} -1.00000 q^{14} +1.14637 q^{15} +1.00000 q^{16} +0.292731 q^{17} +1.68585 q^{18} -6.51806 q^{19} +1.00000 q^{20} +1.14637 q^{21} +2.85363 q^{23} -1.14637 q^{24} +1.00000 q^{25} +4.68585 q^{26} -5.37169 q^{27} +1.00000 q^{28} +1.43910 q^{29} -1.14637 q^{30} +0.978577 q^{31} -1.00000 q^{32} -0.292731 q^{34} +1.00000 q^{35} -1.68585 q^{36} +0.853635 q^{37} +6.51806 q^{38} -5.37169 q^{39} -1.00000 q^{40} +6.22533 q^{41} -1.14637 q^{42} +10.3503 q^{43} -1.68585 q^{45} -2.85363 q^{46} -9.95715 q^{47} +1.14637 q^{48} +1.00000 q^{49} -1.00000 q^{50} +0.335577 q^{51} -4.68585 q^{52} +5.43910 q^{53} +5.37169 q^{54} -1.00000 q^{56} -7.47208 q^{57} -1.43910 q^{58} -9.37169 q^{59} +1.14637 q^{60} +11.9572 q^{61} -0.978577 q^{62} -1.68585 q^{63} +1.00000 q^{64} -4.68585 q^{65} -0.585462 q^{67} +0.292731 q^{68} +3.27131 q^{69} -1.00000 q^{70} -0.335577 q^{71} +1.68585 q^{72} +3.70727 q^{73} -0.853635 q^{74} +1.14637 q^{75} -6.51806 q^{76} +5.37169 q^{78} +2.51806 q^{79} +1.00000 q^{80} -1.10038 q^{81} -6.22533 q^{82} +1.70727 q^{83} +1.14637 q^{84} +0.292731 q^{85} -10.3503 q^{86} +1.64973 q^{87} +13.0790 q^{89} +1.68585 q^{90} -4.68585 q^{91} +2.85363 q^{92} +1.12181 q^{93} +9.95715 q^{94} -6.51806 q^{95} -1.14637 q^{96} +9.10352 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 7 * q^9	$ 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} - 3 q^{10} + 2 q^{12} - 2 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} - 2 q^{17} - 7 q^{18} + 6 q^{19} + 3 q^{20} + 2 q^{21} + 10 q^{23} - 2 q^{24} + 3 q^{25} + 2 q^{26} + 8 q^{27} + 3 q^{28} - 2 q^{30} - 12 q^{31} - 3 q^{32} + 2 q^{34} + 3 q^{35} + 7 q^{36} + 4 q^{37} - 6 q^{38} + 8 q^{39} - 3 q^{40} - 4 q^{41} - 2 q^{42} - 8 q^{43} + 7 q^{45} - 10 q^{46} + 2 q^{48} + 3 q^{49} - 3 q^{50} + 28 q^{51} - 2 q^{52} + 12 q^{53} - 8 q^{54} - 3 q^{56} + 8 q^{57} - 4 q^{59} + 2 q^{60} + 6 q^{61} + 12 q^{62} + 7 q^{63} + 3 q^{64} - 2 q^{65} + 4 q^{67} - 2 q^{68} - 8 q^{69} - 3 q^{70} - 28 q^{71} - 7 q^{72} + 14 q^{73} - 4 q^{74} + 2 q^{75} + 6 q^{76} - 8 q^{78} - 18 q^{79} + 3 q^{80} + 3 q^{81} + 4 q^{82} + 8 q^{83} + 2 q^{84} - 2 q^{85} + 8 q^{86} + 44 q^{87} + 18 q^{89} - 7 q^{90} - 2 q^{91} + 10 q^{92} + 12 q^{93} + 6 q^{95} - 2 q^{96} - 4 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 7 * q^9 - 3 * q^10 + 2 * q^12 - 2 * q^13 - 3 * q^14 + 2 * q^15 + 3 * q^16 - 2 * q^17 - 7 * q^18 + 6 * q^19 + 3 * q^20 + 2 * q^21 + 10 * q^23 - 2 * q^24 + 3 * q^25 + 2 * q^26 + 8 * q^27 + 3 * q^28 - 2 * q^30 - 12 * q^31 - 3 * q^32 + 2 * q^34 + 3 * q^35 + 7 * q^36 + 4 * q^37 - 6 * q^38 + 8 * q^39 - 3 * q^40 - 4 * q^41 - 2 * q^42 - 8 * q^43 + 7 * q^45 - 10 * q^46 + 2 * q^48 + 3 * q^49 - 3 * q^50 + 28 * q^51 - 2 * q^52 + 12 * q^53 - 8 * q^54 - 3 * q^56 + 8 * q^57 - 4 * q^59 + 2 * q^60 + 6 * q^61 + 12 * q^62 + 7 * q^63 + 3 * q^64 - 2 * q^65 + 4 * q^67 - 2 * q^68 - 8 * q^69 - 3 * q^70 - 28 * q^71 - 7 * q^72 + 14 * q^73 - 4 * q^74 + 2 * q^75 + 6 * q^76 - 8 * q^78 - 18 * q^79 + 3 * q^80 + 3 * q^81 + 4 * q^82 + 8 * q^83 + 2 * q^84 - 2 * q^85 + 8 * q^86 + 44 * q^87 + 18 * q^89 - 7 * q^90 - 2 * q^91 + 10 * q^92 + 12 * q^93 + 6 * q^95 - 2 * q^96 - 4 * q^97 - 3 * 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
$3$	1.14637	0.661854	0.330927	−	0.943656i	$-0.392639\pi$
$3$	1.14637	0.661854	0.330927	+	0.943656i	$0.392639\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.14637	−0.468002
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−1.68585	−0.561949
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.14637	0.330927
$13$	−4.68585	−1.29962	−0.649810	−	0.760097i	$-0.725152\pi$
$13$	−4.68585	−1.29962	−0.649810	+	0.760097i	$0.725152\pi$
$14$	−1.00000	−0.267261
$15$	1.14637	0.295990
$16$	1.00000	0.250000
$17$	0.292731	0.0709977	0.0354988	−	0.999370i	$-0.488698\pi$
$17$	0.292731	0.0709977	0.0354988	+	0.999370i	$0.488698\pi$
$18$	1.68585	0.397358
$19$	−6.51806	−1.49535	−0.747673	−	0.664068i	$-0.768829\pi$
$19$	−6.51806	−1.49535	−0.747673	+	0.664068i	$0.768829\pi$
$20$	1.00000	0.223607
$21$	1.14637	0.250157
$22$	0	0
$23$	2.85363	0.595024	0.297512	−	0.954718i	$-0.403843\pi$
$23$	2.85363	0.595024	0.297512	+	0.954718i	$0.403843\pi$
$24$	−1.14637	−0.234001
$25$	1.00000	0.200000
$26$	4.68585	0.918970
$27$	−5.37169	−1.03378
$28$	1.00000	0.188982
$29$	1.43910	0.267234	0.133617	−	0.991033i	$-0.457341\pi$
$29$	1.43910	0.267234	0.133617	+	0.991033i	$0.457341\pi$
$30$	−1.14637	−0.209297
$31$	0.978577	0.175758	0.0878788	−	0.996131i	$-0.471991\pi$
$31$	0.978577	0.175758	0.0878788	+	0.996131i	$0.471991\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.292731	−0.0502029
$35$	1.00000	0.169031
$36$	−1.68585	−0.280974
$37$	0.853635	0.140337	0.0701683	−	0.997535i	$-0.477646\pi$
$37$	0.853635	0.140337	0.0701683	+	0.997535i	$0.477646\pi$
$38$	6.51806	1.05737
$39$	−5.37169	−0.860159
$40$	−1.00000	−0.158114
$41$	6.22533	0.972233	0.486116	−	0.873894i	$-0.338413\pi$
$41$	6.22533	0.972233	0.486116	+	0.873894i	$0.338413\pi$
$42$	−1.14637	−0.176888
$43$	10.3503	1.57840	0.789201	−	0.614135i	$-0.210495\pi$
$43$	10.3503	1.57840	0.789201	+	0.614135i	$0.210495\pi$
$44$	0	0
$45$	−1.68585	−0.251311
$46$	−2.85363	−0.420745
$47$	−9.95715	−1.45240	−0.726200	−	0.687483i	$-0.758715\pi$
$47$	−9.95715	−1.45240	−0.726200	+	0.687483i	$0.758715\pi$
$48$	1.14637	0.165464
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0.335577	0.0469901
$52$	−4.68585	−0.649810
$53$	5.43910	0.747117	0.373559	−	0.927607i	$-0.378137\pi$
$53$	5.43910	0.747117	0.373559	+	0.927607i	$0.378137\pi$
$54$	5.37169	0.730995
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−7.47208	−0.989701
$58$	−1.43910	−0.188963
$59$	−9.37169	−1.22009	−0.610045	−	0.792367i	$-0.708849\pi$
$59$	−9.37169	−1.22009	−0.610045	+	0.792367i	$0.708849\pi$
$60$	1.14637	0.147995
$61$	11.9572	1.53096	0.765478	−	0.643462i	$-0.222502\pi$
$61$	11.9572	1.53096	0.765478	+	0.643462i	$0.222502\pi$
$62$	−0.978577	−0.124279
$63$	−1.68585	−0.212397
$64$	1.00000	0.125000
$65$	−4.68585	−0.581208
$66$	0	0
$67$	−0.585462	−0.0715256	−0.0357628	−	0.999360i	$-0.511386\pi$
$67$	−0.585462	−0.0715256	−0.0357628	+	0.999360i	$0.511386\pi$
$68$	0.292731	0.0354988
$69$	3.27131	0.393819
$70$	−1.00000	−0.119523
$71$	−0.335577	−0.0398256	−0.0199128	−	0.999802i	$-0.506339\pi$
$71$	−0.335577	−0.0398256	−0.0199128	+	0.999802i	$0.506339\pi$
$72$	1.68585	0.198679
$73$	3.70727	0.433903	0.216952	−	0.976182i	$-0.430389\pi$
$73$	3.70727	0.433903	0.216952	+	0.976182i	$0.430389\pi$
$74$	−0.853635	−0.0992330
$75$	1.14637	0.132371
$76$	−6.51806	−0.747673
$77$	0	0
$78$	5.37169	0.608224
$79$	2.51806	0.283304	0.141652	−	0.989917i	$-0.454759\pi$
$79$	2.51806	0.283304	0.141652	+	0.989917i	$0.454759\pi$
$80$	1.00000	0.111803
$81$	−1.10038	−0.122265
$82$	−6.22533	−0.687472
$83$	1.70727	0.187397	0.0936986	−	0.995601i	$-0.470131\pi$
$83$	1.70727	0.187397	0.0936986	+	0.995601i	$0.470131\pi$
$84$	1.14637	0.125079
$85$	0.292731	0.0317511
$86$	−10.3503	−1.11610
$87$	1.64973	0.176870
$88$	0	0
$89$	13.0790	1.38637	0.693184	−	0.720761i	$-0.256208\pi$
$89$	13.0790	1.38637	0.693184	+	0.720761i	$0.256208\pi$
$90$	1.68585	0.177704
$91$	−4.68585	−0.491210
$92$	2.85363	0.297512
$93$	1.12181	0.116326
$94$	9.95715	1.02700
$95$	−6.51806	−0.668739
$96$	−1.14637	−0.117000
$97$	9.10352	0.924322	0.462161	−	0.886796i	$-0.347074\pi$
$97$	9.10352	0.924322	0.462161	+	0.886796i	$0.347074\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	10.7862	1.07327	0.536635	−	0.843814i	$-0.319695\pi$
$101$	10.7862	1.07327	0.536635	+	0.843814i	$0.319695\pi$
$102$	−0.335577	−0.0332270
$103$	7.66442	0.755198	0.377599	−	0.925969i	$-0.376750\pi$
$103$	7.66442	0.755198	0.377599	+	0.925969i	$0.376750\pi$
$104$	4.68585	0.459485
$105$	1.14637	0.111874
$106$	−5.43910	−0.528292
$107$	−9.56404	−0.924591	−0.462295	−	0.886726i	$-0.652974\pi$
$107$	−9.56404	−0.924591	−0.462295	+	0.886726i	$0.652974\pi$
$108$	−5.37169	−0.516891
$109$	−3.14637	−0.301367	−0.150684	−	0.988582i	$-0.548147\pi$
$109$	−3.14637	−0.301367	−0.150684	+	0.988582i	$0.548147\pi$
$110$	0	0
$111$	0.978577	0.0928824
$112$	1.00000	0.0944911
$113$	−2.58546	−0.243220	−0.121610	−	0.992578i	$-0.538806\pi$
$113$	−2.58546	−0.243220	−0.121610	+	0.992578i	$0.538806\pi$
$114$	7.47208	0.699824
$115$	2.85363	0.266103
$116$	1.43910	0.133617
$117$	7.89962	0.730320
$118$	9.37169	0.862734
$119$	0.292731	0.0268346
$120$	−1.14637	−0.104648
$121$	0	0
$122$	−11.9572	−1.08255
$123$	7.13650	0.643477
$124$	0.978577	0.0878788
$125$	1.00000	0.0894427
$126$	1.68585	0.150187
$127$	13.3717	1.18655	0.593273	−	0.805001i	$-0.297836\pi$
$127$	13.3717	1.18655	0.593273	+	0.805001i	$0.297836\pi$
$128$	−1.00000	−0.0883883
$129$	11.8652	1.04467
$130$	4.68585	0.410976
$131$	11.4391	0.999438	0.499719	−	0.866187i	$-0.333437\pi$
$131$	11.4391	0.999438	0.499719	+	0.866187i	$0.333437\pi$
$132$	0	0
$133$	−6.51806	−0.565187
$134$	0.585462	0.0505762
$135$	−5.37169	−0.462322
$136$	−0.292731	−0.0251015
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−3.27131	−0.278472
$139$	14.1825	1.20294	0.601471	−	0.798895i	$-0.294581\pi$
$139$	14.1825	1.20294	0.601471	+	0.798895i	$0.294581\pi$
$140$	1.00000	0.0845154
$141$	−11.4145	−0.961278
$142$	0.335577	0.0281610
$143$	0	0
$144$	−1.68585	−0.140487
$145$	1.43910	0.119510
$146$	−3.70727	−0.306816
$147$	1.14637	0.0945506
$148$	0.853635	0.0701683
$149$	−11.5970	−0.950065	−0.475032	−	0.879968i	$-0.657564\pi$
$149$	−11.5970	−0.950065	−0.475032	+	0.879968i	$0.657564\pi$
$150$	−1.14637	−0.0936004
$151$	−1.73183	−0.140934	−0.0704671	−	0.997514i	$-0.522449\pi$
$151$	−1.73183	−0.140934	−0.0704671	+	0.997514i	$0.522449\pi$
$152$	6.51806	0.528684
$153$	−0.493499	−0.0398971
$154$	0	0
$155$	0.978577	0.0786012
$156$	−5.37169	−0.430080
$157$	−12.2927	−0.981067	−0.490533	−	0.871422i	$-0.663198\pi$
$157$	−12.2927	−0.981067	−0.490533	+	0.871422i	$0.663198\pi$
$158$	−2.51806	−0.200326
$159$	6.23519	0.494483
$160$	−1.00000	−0.0790569
$161$	2.85363	0.224898
$162$	1.10038	0.0864543
$163$	23.9143	1.87311	0.936557	−	0.350516i	$-0.113994\pi$
$163$	23.9143	1.87311	0.936557	+	0.350516i	$0.113994\pi$
$164$	6.22533	0.486116
$165$	0	0
$166$	−1.70727	−0.132510
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−1.14637	−0.0884440
$169$	8.95715	0.689012
$170$	−0.292731	−0.0224514
$171$	10.9884	0.840307
$172$	10.3503	0.789201
$173$	7.37169	0.560459	0.280230	−	0.959933i	$-0.409589\pi$
$173$	7.37169	0.560459	0.280230	+	0.959933i	$0.409589\pi$
$174$	−1.64973	−0.125066
$175$	1.00000	0.0755929
$176$	0	0
$177$	−10.7434	−0.807522
$178$	−13.0790	−0.980310
$179$	1.56404	0.116902	0.0584509	−	0.998290i	$-0.481384\pi$
$179$	1.56404	0.116902	0.0584509	+	0.998290i	$0.481384\pi$
$180$	−1.68585	−0.125656
$181$	14.7862	1.09905	0.549526	−	0.835477i	$-0.314808\pi$
$181$	14.7862	1.09905	0.549526	+	0.835477i	$0.314808\pi$
$182$	4.68585	0.347338
$183$	13.7073	1.01327
$184$	−2.85363	−0.210373
$185$	0.853635	0.0627605
$186$	−1.12181	−0.0822549
$187$	0	0
$188$	−9.95715	−0.726200
$189$	−5.37169	−0.390733
$190$	6.51806	0.472870
$191$	−17.9572	−1.29933	−0.649667	−	0.760219i	$-0.725092\pi$
$191$	−17.9572	−1.29933	−0.649667	+	0.760219i	$0.725092\pi$
$192$	1.14637	0.0827318
$193$	−18.6430	−1.34195	−0.670976	−	0.741479i	$-0.734125\pi$
$193$	−18.6430	−1.34195	−0.670976	+	0.741479i	$0.734125\pi$
$194$	−9.10352	−0.653595
$195$	−5.37169	−0.384675
$196$	1.00000	0.0714286
$197$	15.0361	1.07128	0.535639	−	0.844447i	$-0.320071\pi$
$197$	15.0361	1.07128	0.535639	+	0.844447i	$0.320071\pi$
$198$	0	0
$199$	15.3288	1.08663	0.543317	−	0.839528i	$-0.317168\pi$
$199$	15.3288	1.08663	0.543317	+	0.839528i	$0.317168\pi$
$200$	−1.00000	−0.0707107
$201$	−0.671153	−0.0473395
$202$	−10.7862	−0.758917
$203$	1.43910	0.101005
$204$	0.335577	0.0234951
$205$	6.22533	0.434796
$206$	−7.66442	−0.534006
$207$	−4.81079	−0.334373
$208$	−4.68585	−0.324905
$209$	0	0
$210$	−1.14637	−0.0791067
$211$	−2.04285	−0.140635	−0.0703176	−	0.997525i	$-0.522401\pi$
$211$	−2.04285	−0.140635	−0.0703176	+	0.997525i	$0.522401\pi$
$212$	5.43910	0.373559
$213$	−0.384694	−0.0263588
$214$	9.56404	0.653784
$215$	10.3503	0.705883
$216$	5.37169	0.365497
$217$	0.978577	0.0664301
$218$	3.14637	0.213099
$219$	4.24989	0.287181
$220$	0	0
$221$	−1.37169	−0.0922700
$222$	−0.978577	−0.0656778
$223$	11.0790	0.741902	0.370951	−	0.928652i	$-0.379032\pi$
$223$	11.0790	0.741902	0.370951	+	0.928652i	$0.379032\pi$
$224$	−1.00000	−0.0668153
$225$	−1.68585	−0.112390
$226$	2.58546	0.171982
$227$	18.5426	1.23072	0.615358	−	0.788248i	$-0.289011\pi$
$227$	18.5426	1.23072	0.615358	+	0.788248i	$0.289011\pi$
$228$	−7.47208	−0.494850
$229$	−8.01469	−0.529626	−0.264813	−	0.964300i	$-0.585310\pi$
$229$	−8.01469	−0.529626	−0.264813	+	0.964300i	$0.585310\pi$
$230$	−2.85363	−0.188163
$231$	0	0
$232$	−1.43910	−0.0944813
$233$	−27.9572	−1.83153	−0.915767	−	0.401710i	$-0.868416\pi$
$233$	−27.9572	−1.83153	−0.915767	+	0.401710i	$0.868416\pi$
$234$	−7.89962	−0.516414
$235$	−9.95715	−0.649533
$236$	−9.37169	−0.610045
$237$	2.88661	0.187506
$238$	−0.292731	−0.0189749
$239$	28.8108	1.86362	0.931808	−	0.362953i	$-0.118231\pi$
$239$	28.8108	1.86362	0.931808	+	0.362953i	$0.118231\pi$
$240$	1.14637	0.0739976
$241$	8.51806	0.548696	0.274348	−	0.961630i	$-0.411538\pi$
$241$	8.51806	0.548696	0.274348	+	0.961630i	$0.411538\pi$
$242$	0	0
$243$	14.8536	0.952861
$244$	11.9572	0.765478
$245$	1.00000	0.0638877
$246$	−7.13650	−0.455007
$247$	30.5426	1.94338
$248$	−0.978577	−0.0621397
$249$	1.95715	0.124030
$250$	−1.00000	−0.0632456
$251$	−23.1281	−1.45983	−0.729916	−	0.683537i	$-0.760441\pi$
$251$	−23.1281	−1.45983	−0.729916	+	0.683537i	$0.760441\pi$
$252$	−1.68585	−0.106198
$253$	0	0
$254$	−13.3717	−0.839015
$255$	0.335577	0.0210146
$256$	1.00000	0.0625000
$257$	14.8108	0.923872	0.461936	−	0.886913i	$-0.347155\pi$
$257$	14.8108	0.923872	0.461936	+	0.886913i	$0.347155\pi$
$258$	−11.8652	−0.738695
$259$	0.853635	0.0530423
$260$	−4.68585	−0.290604
$261$	−2.42610	−0.150172
$262$	−11.4391	−0.706710
$263$	−18.7434	−1.15577	−0.577883	−	0.816119i	$-0.696121\pi$
$263$	−18.7434	−1.15577	−0.577883	+	0.816119i	$0.696121\pi$
$264$	0	0
$265$	5.43910	0.334121
$266$	6.51806	0.399648
$267$	14.9933	0.917573
$268$	−0.585462	−0.0357628
$269$	12.6858	0.773470	0.386735	−	0.922191i	$-0.373603\pi$
$269$	12.6858	0.773470	0.386735	+	0.922191i	$0.373603\pi$
$270$	5.37169	0.326911
$271$	−1.12181	−0.0681449	−0.0340725	−	0.999419i	$-0.510848\pi$
$271$	−1.12181	−0.0681449	−0.0340725	+	0.999419i	$0.510848\pi$
$272$	0.292731	0.0177494
$273$	−5.37169	−0.325110
$274$	−10.0000	−0.604122
$275$	0	0
$276$	3.27131	0.196910
$277$	−30.1151	−1.80944	−0.904720	−	0.426007i	$-0.859920\pi$
$277$	−30.1151	−1.80944	−0.904720	+	0.426007i	$0.859920\pi$
$278$	−14.1825	−0.850609
$279$	−1.64973	−0.0987668
$280$	−1.00000	−0.0597614
$281$	14.3356	0.855189	0.427594	−	0.903971i	$-0.359361\pi$
$281$	14.3356	0.855189	0.427594	+	0.903971i	$0.359361\pi$
$282$	11.4145	0.679726
$283$	−3.21377	−0.191039	−0.0955194	−	0.995428i	$-0.530451\pi$
$283$	−3.21377	−0.191039	−0.0955194	+	0.995428i	$0.530451\pi$
$284$	−0.335577	−0.0199128
$285$	−7.47208	−0.442608
$286$	0	0
$287$	6.22533	0.367469
$288$	1.68585	0.0993394
$289$	−16.9143	−0.994959
$290$	−1.43910	−0.0845067
$291$	10.4360	0.611767
$292$	3.70727	0.216952
$293$	16.6858	0.974798	0.487399	−	0.873179i	$-0.337946\pi$
$293$	16.6858	0.974798	0.487399	+	0.873179i	$0.337946\pi$
$294$	−1.14637	−0.0668574
$295$	−9.37169	−0.545641
$296$	−0.853635	−0.0496165
$297$	0	0
$298$	11.5970	0.671797
$299$	−13.3717	−0.773305
$300$	1.14637	0.0661854
$301$	10.3503	0.596580
$302$	1.73183	0.0996555
$303$	12.3650	0.710349
$304$	−6.51806	−0.373836
$305$	11.9572	0.684665
$306$	0.493499	0.0282115
$307$	6.29273	0.359145	0.179573	−	0.983745i	$-0.442529\pi$
$307$	6.29273	0.359145	0.179573	+	0.983745i	$0.442529\pi$
$308$	0	0
$309$	8.78623	0.499831
$310$	−0.978577	−0.0555794
$311$	−20.3931	−1.15639	−0.578194	−	0.815900i	$-0.696242\pi$
$311$	−20.3931	−1.15639	−0.578194	+	0.815900i	$0.696242\pi$
$312$	5.37169	0.304112
$313$	12.0674	0.682090	0.341045	−	0.940047i	$-0.389219\pi$
$313$	12.0674	0.682090	0.341045	+	0.940047i	$0.389219\pi$
$314$	12.2927	0.693719
$315$	−1.68585	−0.0949867
$316$	2.51806	0.141652
$317$	28.7679	1.61577	0.807884	−	0.589341i	$-0.200613\pi$
$317$	28.7679	1.61577	0.807884	+	0.589341i	$0.200613\pi$
$318$	−6.23519	−0.349652
$319$	0	0
$320$	1.00000	0.0559017
$321$	−10.9639	−0.611944
$322$	−2.85363	−0.159027
$323$	−1.90804	−0.106166
$324$	−1.10038	−0.0611325
$325$	−4.68585	−0.259924
$326$	−23.9143	−1.32449
$327$	−3.60688	−0.199461
$328$	−6.22533	−0.343736
$329$	−9.95715	−0.548956
$330$	0	0
$331$	−3.80765	−0.209288	−0.104644	−	0.994510i	$-0.533370\pi$
$331$	−3.80765	−0.209288	−0.104644	+	0.994510i	$0.533370\pi$
$332$	1.70727	0.0936986
$333$	−1.43910	−0.0788620
$334$	−8.00000	−0.437741
$335$	−0.585462	−0.0319872
$336$	1.14637	0.0625394
$337$	−15.3717	−0.837349	−0.418675	−	0.908136i	$-0.637505\pi$
$337$	−15.3717	−0.837349	−0.418675	+	0.908136i	$0.637505\pi$
$338$	−8.95715	−0.487205
$339$	−2.96388	−0.160976
$340$	0.292731	0.0158756
$341$	0	0
$342$	−10.9884	−0.594187
$343$	1.00000	0.0539949
$344$	−10.3503	−0.558049
$345$	3.27131	0.176121
$346$	−7.37169	−0.396305
$347$	−3.02142	−0.162198	−0.0810992	−	0.996706i	$-0.525843\pi$
$347$	−3.02142	−0.162198	−0.0810992	+	0.996706i	$0.525843\pi$
$348$	1.64973	0.0884348
$349$	−11.0361	−0.590750	−0.295375	−	0.955381i	$-0.595445\pi$
$349$	−11.0361	−0.590750	−0.295375	+	0.955381i	$0.595445\pi$
$350$	−1.00000	−0.0534522
$351$	25.1709	1.34352
$352$	0	0
$353$	23.9817	1.27642	0.638209	−	0.769863i	$-0.279676\pi$
$353$	23.9817	1.27642	0.638209	+	0.769863i	$0.279676\pi$
$354$	10.7434	0.571004
$355$	−0.335577	−0.0178106
$356$	13.0790	0.693184
$357$	0.335577	0.0177606
$358$	−1.56404	−0.0826620
$359$	10.5181	0.555122	0.277561	−	0.960708i	$-0.410474\pi$
$359$	10.5181	0.555122	0.277561	+	0.960708i	$0.410474\pi$
$360$	1.68585	0.0888519
$361$	23.4851	1.23606
$362$	−14.7862	−0.777147
$363$	0	0
$364$	−4.68585	−0.245605
$365$	3.70727	0.194047
$366$	−13.7073	−0.716490
$367$	−35.5787	−1.85719	−0.928597	−	0.371089i	$-0.878985\pi$
$367$	−35.5787	−1.85719	−0.928597	+	0.371089i	$0.878985\pi$
$368$	2.85363	0.148756
$369$	−10.4949	−0.546345
$370$	−0.853635	−0.0443783
$371$	5.43910	0.282384
$372$	1.12181	0.0581630
$373$	−7.50650	−0.388672	−0.194336	−	0.980935i	$-0.562255\pi$
$373$	−7.50650	−0.388672	−0.194336	+	0.980935i	$0.562255\pi$
$374$	0	0
$375$	1.14637	0.0591981
$376$	9.95715	0.513501
$377$	−6.74338	−0.347302
$378$	5.37169	0.276290
$379$	33.4868	1.72010	0.860050	−	0.510210i	$-0.170432\pi$
$379$	33.4868	1.72010	0.860050	+	0.510210i	$0.170432\pi$
$380$	−6.51806	−0.334369
$381$	15.3288	0.785321
$382$	17.9572	0.918768
$383$	15.6644	0.800415	0.400207	−	0.916425i	$-0.368938\pi$
$383$	15.6644	0.800415	0.400207	+	0.916425i	$0.368938\pi$
$384$	−1.14637	−0.0585002
$385$	0	0
$386$	18.6430	0.948904
$387$	−17.4490	−0.886981
$388$	9.10352	0.462161
$389$	−19.6216	−0.994853	−0.497427	−	0.867506i	$-0.665722\pi$
$389$	−19.6216	−0.994853	−0.497427	+	0.867506i	$0.665722\pi$
$390$	5.37169	0.272006
$391$	0.835347	0.0422453
$392$	−1.00000	−0.0505076
$393$	13.1134	0.661483
$394$	−15.0361	−0.757509
$395$	2.51806	0.126697
$396$	0	0
$397$	10.2499	0.514427	0.257213	−	0.966355i	$-0.417196\pi$
$397$	10.2499	0.514427	0.257213	+	0.966355i	$0.417196\pi$
$398$	−15.3288	−0.768366
$399$	−7.47208	−0.374072
$400$	1.00000	0.0500000
$401$	−16.8866	−0.843277	−0.421639	−	0.906764i	$-0.638545\pi$
$401$	−16.8866	−0.843277	−0.421639	+	0.906764i	$0.638545\pi$
$402$	0.671153	0.0334741
$403$	−4.58546	−0.228418
$404$	10.7862	0.536635
$405$	−1.10038	−0.0546785
$406$	−1.43910	−0.0714212
$407$	0	0
$408$	−0.335577	−0.0166135
$409$	−18.2744	−0.903613	−0.451807	−	0.892116i	$-0.649220\pi$
$409$	−18.2744	−0.903613	−0.451807	+	0.892116i	$0.649220\pi$
$410$	−6.22533	−0.307447
$411$	11.4637	0.565460
$412$	7.66442	0.377599
$413$	−9.37169	−0.461151
$414$	4.81079	0.236437
$415$	1.70727	0.0838065
$416$	4.68585	0.229743
$417$	16.2583	0.796173
$418$	0	0
$419$	−37.2860	−1.82154	−0.910770	−	0.412914i	$-0.864511\pi$
$419$	−37.2860	−1.82154	−0.910770	+	0.412914i	$0.864511\pi$
$420$	1.14637	0.0559369
$421$	−2.78623	−0.135793	−0.0678963	−	0.997692i	$-0.521629\pi$
$421$	−2.78623	−0.135793	−0.0678963	+	0.997692i	$0.521629\pi$
$422$	2.04285	0.0994442
$423$	16.7862	0.816174
$424$	−5.43910	−0.264146
$425$	0.292731	0.0141995
$426$	0.384694	0.0186385
$427$	11.9572	0.578647
$428$	−9.56404	−0.462295
$429$	0	0
$430$	−10.3503	−0.499134
$431$	−38.0477	−1.83269	−0.916346	−	0.400387i	$-0.868876\pi$
$431$	−38.0477	−1.83269	−0.916346	+	0.400387i	$0.868876\pi$
$432$	−5.37169	−0.258446
$433$	−10.8108	−0.519533	−0.259767	−	0.965671i	$-0.583646\pi$
$433$	−10.8108	−0.519533	−0.259767	+	0.965671i	$0.583646\pi$
$434$	−0.978577	−0.0469732
$435$	1.64973	0.0790985
$436$	−3.14637	−0.150684
$437$	−18.6002	−0.889766
$438$	−4.24989	−0.203067
$439$	−30.9933	−1.47923	−0.739614	−	0.673031i	$-0.764992\pi$
$439$	−30.9933	−1.47923	−0.739614	+	0.673031i	$0.764992\pi$
$440$	0	0
$441$	−1.68585	−0.0802784
$442$	1.37169	0.0652448
$443$	25.3717	1.20545	0.602723	−	0.797951i	$-0.294083\pi$
$443$	25.3717	1.20545	0.602723	+	0.797951i	$0.294083\pi$
$444$	0.978577	0.0464412
$445$	13.0790	0.620002
$446$	−11.0790	−0.524604
$447$	−13.2944	−0.628805
$448$	1.00000	0.0472456
$449$	−0.886615	−0.0418419	−0.0209210	−	0.999781i	$-0.506660\pi$
$449$	−0.886615	−0.0418419	−0.0209210	+	0.999781i	$0.506660\pi$
$450$	1.68585	0.0794716
$451$	0	0
$452$	−2.58546	−0.121610
$453$	−1.98531	−0.0932779
$454$	−18.5426	−0.870248
$455$	−4.68585	−0.219676
$456$	7.47208	0.349912
$457$	−29.9143	−1.39933	−0.699666	−	0.714470i	$-0.746668\pi$
$457$	−29.9143	−1.39933	−0.699666	+	0.714470i	$0.746668\pi$
$458$	8.01469	0.374502
$459$	−1.57246	−0.0733962
$460$	2.85363	0.133051
$461$	2.33558	0.108779	0.0543893	−	0.998520i	$-0.482679\pi$
$461$	2.33558	0.108779	0.0543893	+	0.998520i	$0.482679\pi$
$462$	0	0
$463$	0.110250	0.00512374	0.00256187	−	0.999997i	$-0.499185\pi$
$463$	0.110250	0.00512374	0.00256187	+	0.999997i	$0.499185\pi$
$464$	1.43910	0.0668084
$465$	1.12181	0.0520226
$466$	27.9572	1.29509
$467$	36.6760	1.69716	0.848581	−	0.529066i	$-0.177457\pi$
$467$	36.6760	1.69716	0.848581	+	0.529066i	$0.177457\pi$
$468$	7.89962	0.365160
$469$	−0.585462	−0.0270341
$470$	9.95715	0.459289
$471$	−14.0920	−0.649323
$472$	9.37169	0.431367
$473$	0	0
$474$	−2.88661	−0.132587
$475$	−6.51806	−0.299069
$476$	0.292731	0.0134173
$477$	−9.16948	−0.419842
$478$	−28.8108	−1.31777
$479$	42.7434	1.95300	0.976498	−	0.215529i	$-0.0691474\pi$
$479$	42.7434	1.95300	0.976498	+	0.215529i	$0.0691474\pi$
$480$	−1.14637	−0.0523242
$481$	−4.00000	−0.182384
$482$	−8.51806	−0.387987
$483$	3.27131	0.148850
$484$	0	0
$485$	9.10352	0.413370
$486$	−14.8536	−0.673775
$487$	−36.1396	−1.63764	−0.818822	−	0.574048i	$-0.805372\pi$
$487$	−36.1396	−1.63764	−0.818822	+	0.574048i	$0.805372\pi$
$488$	−11.9572	−0.541275
$489$	27.4145	1.23973
$490$	−1.00000	−0.0451754
$491$	−2.54262	−0.114747	−0.0573733	−	0.998353i	$-0.518273\pi$
$491$	−2.54262	−0.114747	−0.0573733	+	0.998353i	$0.518273\pi$
$492$	7.13650	0.321738
$493$	0.421268	0.0189730
$494$	−30.5426	−1.37418
$495$	0	0
$496$	0.978577	0.0439394
$497$	−0.335577	−0.0150527
$498$	−1.95715	−0.0877022
$499$	3.13650	0.140409	0.0702045	−	0.997533i	$-0.477635\pi$
$499$	3.13650	0.140409	0.0702045	+	0.997533i	$0.477635\pi$
$500$	1.00000	0.0447214
$501$	9.17092	0.409727
$502$	23.1281	1.03226
$503$	28.5855	1.27456	0.637281	−	0.770631i	$-0.280059\pi$
$503$	28.5855	1.27456	0.637281	+	0.770631i	$0.280059\pi$
$504$	1.68585	0.0750936
$505$	10.7862	0.479981
$506$	0	0
$507$	10.2682	0.456026
$508$	13.3717	0.593273
$509$	2.20077	0.0975473	0.0487737	−	0.998810i	$-0.484469\pi$
$509$	2.20077	0.0975473	0.0487737	+	0.998810i	$0.484469\pi$
$510$	−0.335577	−0.0148596
$511$	3.70727	0.164000
$512$	−1.00000	−0.0441942
$513$	35.0130	1.54586
$514$	−14.8108	−0.653276
$515$	7.66442	0.337735
$516$	11.8652	0.522336
$517$	0	0
$518$	−0.853635	−0.0375065
$519$	8.45065	0.370943
$520$	4.68585	0.205488
$521$	37.1940	1.62950	0.814750	−	0.579812i	$-0.196874\pi$
$521$	37.1940	1.62950	0.814750	+	0.579812i	$0.196874\pi$
$522$	2.42610	0.106187
$523$	−30.4078	−1.32964	−0.664820	−	0.747003i	$-0.731492\pi$
$523$	−30.4078	−1.32964	−0.664820	+	0.747003i	$0.731492\pi$
$524$	11.4391	0.499719
$525$	1.14637	0.0500315
$526$	18.7434	0.817250
$527$	0.286460	0.0124784
$528$	0	0
$529$	−14.8568	−0.645947
$530$	−5.43910	−0.236259
$531$	15.7992	0.685628
$532$	−6.51806	−0.282594
$533$	−29.1709	−1.26353
$534$	−14.9933	−0.648822
$535$	−9.56404	−0.413489
$536$	0.585462	0.0252881
$537$	1.79296	0.0773720
$538$	−12.6858	−0.546926
$539$	0	0
$540$	−5.37169	−0.231161
$541$	−39.0607	−1.67935	−0.839675	−	0.543090i	$-0.817254\pi$
$541$	−39.0607	−1.67935	−0.839675	+	0.543090i	$0.817254\pi$
$542$	1.12181	0.0481857
$543$	16.9504	0.727412
$544$	−0.292731	−0.0125507
$545$	−3.14637	−0.134775
$546$	5.37169	0.229887
$547$	0.585462	0.0250325	0.0125163	−	0.999922i	$-0.496016\pi$
$547$	0.585462	0.0250325	0.0125163	+	0.999922i	$0.496016\pi$
$548$	10.0000	0.427179
$549$	−20.1579	−0.860319
$550$	0	0
$551$	−9.38011	−0.399606
$552$	−3.27131	−0.139236
$553$	2.51806	0.107079
$554$	30.1151	1.27947
$555$	0.978577	0.0415383
$556$	14.1825	0.601471
$557$	−17.4637	−0.739959	−0.369979	−	0.929040i	$-0.620635\pi$
$557$	−17.4637	−0.739959	−0.369979	+	0.929040i	$0.620635\pi$
$558$	1.64973	0.0698387
$559$	−48.4998	−2.05132
$560$	1.00000	0.0422577
$561$	0	0
$562$	−14.3356	−0.604710
$563$	18.1579	0.765265	0.382633	−	0.923901i	$-0.375018\pi$
$563$	18.1579	0.765265	0.382633	+	0.923901i	$0.375018\pi$
$564$	−11.4145	−0.480639
$565$	−2.58546	−0.108771
$566$	3.21377	0.135085
$567$	−1.10038	−0.0462118
$568$	0.335577	0.0140805
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	7.47208	0.312971
$571$	9.03612	0.378150	0.189075	−	0.981963i	$-0.439451\pi$
$571$	9.03612	0.378150	0.189075	+	0.981963i	$0.439451\pi$
$572$	0	0
$573$	−20.5855	−0.859970
$574$	−6.22533	−0.259840
$575$	2.85363	0.119005
$576$	−1.68585	−0.0702436
$577$	19.3963	0.807476	0.403738	−	0.914875i	$-0.367711\pi$
$577$	19.3963	0.807476	0.403738	+	0.914875i	$0.367711\pi$
$578$	16.9143	0.703542
$579$	−21.3717	−0.888177
$580$	1.43910	0.0597552
$581$	1.70727	0.0708295
$582$	−10.4360	−0.432584
$583$	0	0
$584$	−3.70727	−0.153408
$585$	7.89962	0.326609
$586$	−16.6858	−0.689286
$587$	17.2614	0.712456	0.356228	−	0.934399i	$-0.384063\pi$
$587$	17.2614	0.712456	0.356228	+	0.934399i	$0.384063\pi$
$588$	1.14637	0.0472753
$589$	−6.37842	−0.262818
$590$	9.37169	0.385826
$591$	17.2369	0.709031
$592$	0.853635	0.0350842
$593$	11.0361	0.453199	0.226599	−	0.973988i	$-0.427239\pi$
$593$	11.0361	0.453199	0.226599	+	0.973988i	$0.427239\pi$
$594$	0	0
$595$	0.292731	0.0120008
$596$	−11.5970	−0.475032
$597$	17.5725	0.719193
$598$	13.3717	0.546809
$599$	−41.7367	−1.70531	−0.852657	−	0.522472i	$-0.825010\pi$
$599$	−41.7367	−1.70531	−0.852657	+	0.522472i	$0.825010\pi$
$600$	−1.14637	−0.0468002
$601$	−39.3106	−1.60351	−0.801756	−	0.597652i	$-0.796100\pi$
$601$	−39.3106	−1.60351	−0.801756	+	0.597652i	$0.796100\pi$
$602$	−10.3503	−0.421845
$603$	0.986999	0.0401937
$604$	−1.73183	−0.0704671
$605$	0	0
$606$	−12.3650	−0.502292
$607$	6.35027	0.257749	0.128875	−	0.991661i	$-0.458863\pi$
$607$	6.35027	0.257749	0.128875	+	0.991661i	$0.458863\pi$
$608$	6.51806	0.264342
$609$	1.64973	0.0668505
$610$	−11.9572	−0.484131
$611$	46.6577	1.88757
$612$	−0.493499	−0.0199485
$613$	42.4507	1.71457	0.857283	−	0.514846i	$-0.172151\pi$
$613$	42.4507	1.71457	0.857283	+	0.514846i	$0.172151\pi$
$614$	−6.29273	−0.253954
$615$	7.13650	0.287771
$616$	0	0
$617$	−0.677425	−0.0272721	−0.0136360	−	0.999907i	$-0.504341\pi$
$617$	−0.677425	−0.0272721	−0.0136360	+	0.999907i	$0.504341\pi$
$618$	−8.78623	−0.353434
$619$	−18.5426	−0.745291	−0.372645	−	0.927974i	$-0.621549\pi$
$619$	−18.5426	−0.745291	−0.372645	+	0.927974i	$0.621549\pi$
$620$	0.978577	0.0393006
$621$	−15.3288	−0.615125
$622$	20.3931	0.817689
$623$	13.0790	0.523998
$624$	−5.37169	−0.215040
$625$	1.00000	0.0400000
$626$	−12.0674	−0.482310
$627$	0	0
$628$	−12.2927	−0.490533
$629$	0.249885	0.00996358
$630$	1.68585	0.0671657
$631$	21.3717	0.850794	0.425397	−	0.905007i	$-0.360135\pi$
$631$	21.3717	0.850794	0.425397	+	0.905007i	$0.360135\pi$
$632$	−2.51806	−0.100163
$633$	−2.34185	−0.0930801
$634$	−28.7679	−1.14252
$635$	13.3717	0.530639
$636$	6.23519	0.247241
$637$	−4.68585	−0.185660
$638$	0	0
$639$	0.565731	0.0223800
$640$	−1.00000	−0.0395285
$641$	23.8715	0.942866	0.471433	−	0.881902i	$-0.343737\pi$
$641$	23.8715	0.942866	0.471433	+	0.881902i	$0.343737\pi$
$642$	10.9639	0.432710
$643$	0.311018	0.0122654	0.00613268	−	0.999981i	$-0.498048\pi$
$643$	0.311018	0.0122654	0.00613268	+	0.999981i	$0.498048\pi$
$644$	2.85363	0.112449
$645$	11.8652	0.467191
$646$	1.90804	0.0750707
$647$	−9.62158	−0.378263	−0.189132	−	0.981952i	$-0.560567\pi$
$647$	−9.62158	−0.378263	−0.189132	+	0.981952i	$0.560567\pi$
$648$	1.10038	0.0432272
$649$	0	0
$650$	4.68585	0.183794
$651$	1.12181	0.0439671
$652$	23.9143	0.936557
$653$	−27.0607	−1.05897	−0.529483	−	0.848321i	$-0.677614\pi$
$653$	−27.0607	−1.05897	−0.529483	+	0.848321i	$0.677614\pi$
$654$	3.60688	0.141040
$655$	11.4391	0.446962
$656$	6.22533	0.243058
$657$	−6.24989	−0.243831
$658$	9.95715	0.388170
$659$	26.4935	1.03204	0.516020	−	0.856576i	$-0.327413\pi$
$659$	26.4935	1.03204	0.516020	+	0.856576i	$0.327413\pi$
$660$	0	0
$661$	35.0852	1.36466	0.682329	−	0.731046i	$-0.260967\pi$
$661$	35.0852	1.36466	0.682329	+	0.731046i	$0.260967\pi$
$662$	3.80765	0.147989
$663$	−1.57246	−0.0610693
$664$	−1.70727	−0.0662549
$665$	−6.51806	−0.252759
$666$	1.43910	0.0557639
$667$	4.10666	0.159010
$668$	8.00000	0.309529
$669$	12.7005	0.491031
$670$	0.585462	0.0226184
$671$	0	0
$672$	−1.14637	−0.0442220
$673$	−32.6577	−1.25886	−0.629431	−	0.777057i	$-0.716712\pi$
$673$	−32.6577	−1.25886	−0.629431	+	0.777057i	$0.716712\pi$
$674$	15.3717	0.592095
$675$	−5.37169	−0.206757
$676$	8.95715	0.344506
$677$	−9.12808	−0.350821	−0.175410	−	0.984495i	$-0.556125\pi$
$677$	−9.12808	−0.350821	−0.175410	+	0.984495i	$0.556125\pi$
$678$	2.96388	0.113827
$679$	9.10352	0.349361
$680$	−0.292731	−0.0112257
$681$	21.2566	0.814555
$682$	0	0
$683$	10.4935	0.401523	0.200761	−	0.979640i	$-0.435658\pi$
$683$	10.4935	0.401523	0.200761	+	0.979640i	$0.435658\pi$
$684$	10.9884	0.420154
$685$	10.0000	0.382080
$686$	−1.00000	−0.0381802
$687$	−9.18777	−0.350535
$688$	10.3503	0.394600
$689$	−25.4868	−0.970969
$690$	−3.27131	−0.124537
$691$	1.70727	0.0649476	0.0324738	−	0.999473i	$-0.489661\pi$
$691$	1.70727	0.0649476	0.0324738	+	0.999473i	$0.489661\pi$
$692$	7.37169	0.280230
$693$	0	0
$694$	3.02142	0.114692
$695$	14.1825	0.537972
$696$	−1.64973	−0.0625329
$697$	1.82235	0.0690263
$698$	11.0361	0.417723
$699$	−32.0491	−1.21221
$700$	1.00000	0.0377964
$701$	37.3534	1.41082	0.705409	−	0.708800i	$-0.250763\pi$
$701$	37.3534	1.41082	0.705409	+	0.708800i	$0.250763\pi$
$702$	−25.1709	−0.950015
$703$	−5.56404	−0.209852
$704$	0	0
$705$	−11.4145	−0.429896
$706$	−23.9817	−0.902564
$707$	10.7862	0.405658
$708$	−10.7434	−0.403761
$709$	−8.20704	−0.308222	−0.154111	−	0.988054i	$-0.549251\pi$
$709$	−8.20704	−0.308222	−0.154111	+	0.988054i	$0.549251\pi$
$710$	0.335577	0.0125940
$711$	−4.24506	−0.159202
$712$	−13.0790	−0.490155
$713$	2.79250	0.104580
$714$	−0.335577	−0.0125586
$715$	0	0
$716$	1.56404	0.0584509
$717$	33.0277	1.23344
$718$	−10.5181	−0.392530
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−1.68585	−0.0628278
$721$	7.66442	0.285438
$722$	−23.4851	−0.874024
$723$	9.76481	0.363157
$724$	14.7862	0.549526
$725$	1.43910	0.0534467
$726$	0	0
$727$	−46.9442	−1.74106	−0.870531	−	0.492113i	$-0.836225\pi$
$727$	−46.9442	−1.74106	−0.870531	+	0.492113i	$0.836225\pi$
$728$	4.68585	0.173669
$729$	20.3288	0.752920
$730$	−3.70727	−0.137212
$731$	3.02984	0.112063
$732$	13.7073	0.506635
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	35.5787	1.31323
$735$	1.14637	0.0422843
$736$	−2.85363	−0.105186
$737$	0	0
$738$	10.4949	0.386324
$739$	0.871922	0.0320742	0.0160371	−	0.999871i	$-0.494895\pi$
$739$	0.871922	0.0320742	0.0160371	+	0.999871i	$0.494895\pi$
$740$	0.853635	0.0313802
$741$	35.0130	1.28623
$742$	−5.43910	−0.199676
$743$	24.6712	0.905097	0.452548	−	0.891740i	$-0.350515\pi$
$743$	24.6712	0.905097	0.452548	+	0.891740i	$0.350515\pi$
$744$	−1.12181	−0.0411274
$745$	−11.5970	−0.424882
$746$	7.50650	0.274833
$747$	−2.87819	−0.105308
$748$	0	0
$749$	−9.56404	−0.349462
$750$	−1.14637	−0.0418593
$751$	−3.75011	−0.136844	−0.0684218	−	0.997656i	$-0.521796\pi$
$751$	−3.75011	−0.136844	−0.0684218	+	0.997656i	$0.521796\pi$
$752$	−9.95715	−0.363100
$753$	−26.5132	−0.966196
$754$	6.74338	0.245580
$755$	−1.73183	−0.0630277
$756$	−5.37169	−0.195367
$757$	15.7318	0.571783	0.285891	−	0.958262i	$-0.407710\pi$
$757$	15.7318	0.571783	0.285891	+	0.958262i	$0.407710\pi$
$758$	−33.4868	−1.21629
$759$	0	0
$760$	6.51806	0.236435
$761$	−15.2614	−0.553227	−0.276613	−	0.960981i	$-0.589212\pi$
$761$	−15.2614	−0.553227	−0.276613	+	0.960981i	$0.589212\pi$
$762$	−15.3288	−0.555306
$763$	−3.14637	−0.113906
$764$	−17.9572	−0.649667
$765$	−0.493499	−0.0178425
$766$	−15.6644	−0.565979
$767$	43.9143	1.58565
$768$	1.14637	0.0413659
$769$	9.63986	0.347622	0.173811	−	0.984779i	$-0.444392\pi$
$769$	9.63986	0.347622	0.173811	+	0.984779i	$0.444392\pi$
$770$	0	0
$771$	16.9786	0.611469
$772$	−18.6430	−0.670976
$773$	12.8291	0.461430	0.230715	−	0.973021i	$-0.425894\pi$
$773$	12.8291	0.461430	0.230715	+	0.973021i	$0.425894\pi$
$774$	17.4490	0.627190
$775$	0.978577	0.0351515
$776$	−9.10352	−0.326797
$777$	0.978577	0.0351063
$778$	19.6216	0.703468
$779$	−40.5770	−1.45382
$780$	−5.37169	−0.192337
$781$	0	0
$782$	−0.835347	−0.0298720
$783$	−7.73038	−0.276261
$784$	1.00000	0.0357143
$785$	−12.2927	−0.438746
$786$	−13.1134	−0.467739
$787$	−10.8291	−0.386015	−0.193007	−	0.981197i	$-0.561824\pi$
$787$	−10.8291	−0.386015	−0.193007	+	0.981197i	$0.561824\pi$
$788$	15.0361	0.535639
$789$	−21.4868	−0.764949
$790$	−2.51806	−0.0895885
$791$	−2.58546	−0.0919284
$792$	0	0
$793$	−56.0294	−1.98966
$794$	−10.2499	−0.363755
$795$	6.23519	0.221139
$796$	15.3288	0.543317
$797$	−21.0790	−0.746655	−0.373328	−	0.927700i	$-0.621783\pi$
$797$	−21.0790	−0.746655	−0.373328	+	0.927700i	$0.621783\pi$
$798$	7.47208	0.264509
$799$	−2.91477	−0.103117
$800$	−1.00000	−0.0353553
$801$	−22.0491	−0.779067
$802$	16.8866	0.596287
$803$	0	0
$804$	−0.671153	−0.0236698
$805$	2.85363	0.100577
$806$	4.58546	0.161516
$807$	14.5426	0.511924
$808$	−10.7862	−0.379458
$809$	−3.62158	−0.127328	−0.0636639	−	0.997971i	$-0.520279\pi$
$809$	−3.62158	−0.127328	−0.0636639	+	0.997971i	$0.520279\pi$
$810$	1.10038	0.0386636
$811$	−34.7188	−1.21914	−0.609571	−	0.792731i	$-0.708658\pi$
$811$	−34.7188	−1.21914	−0.609571	+	0.792731i	$0.708658\pi$
$812$	1.43910	0.0505024
$813$	−1.28600	−0.0451020
$814$	0	0
$815$	23.9143	0.837682
$816$	0.335577	0.0117475
$817$	−67.4637	−2.36025
$818$	18.2744	0.638951
$819$	7.89962	0.276035
$820$	6.22533	0.217398
$821$	37.7549	1.31766	0.658828	−	0.752293i	$-0.271052\pi$
$821$	37.7549	1.31766	0.658828	+	0.752293i	$0.271052\pi$
$822$	−11.4637	−0.399841
$823$	11.1892	0.390031	0.195016	−	0.980800i	$-0.437524\pi$
$823$	11.1892	0.390031	0.195016	+	0.980800i	$0.437524\pi$
$824$	−7.66442	−0.267003
$825$	0	0
$826$	9.37169	0.326083
$827$	7.91431	0.275207	0.137604	−	0.990487i	$-0.456060\pi$
$827$	7.91431	0.275207	0.137604	+	0.990487i	$0.456060\pi$
$828$	−4.81079	−0.167186
$829$	−44.7152	−1.55302	−0.776512	−	0.630102i	$-0.783013\pi$
$829$	−44.7152	−1.55302	−0.776512	+	0.630102i	$0.783013\pi$
$830$	−1.70727	−0.0592602
$831$	−34.5229	−1.19759
$832$	−4.68585	−0.162452
$833$	0.292731	0.0101425
$834$	−16.2583	−0.562979
$835$	8.00000	0.276851
$836$	0	0
$837$	−5.25662	−0.181695
$838$	37.2860	1.28802
$839$	21.6791	0.748446	0.374223	−	0.927339i	$-0.377909\pi$
$839$	21.6791	0.748446	0.374223	+	0.927339i	$0.377909\pi$
$840$	−1.14637	−0.0395534
$841$	−26.9290	−0.928586
$842$	2.78623	0.0960198
$843$	16.4338	0.566010
$844$	−2.04285	−0.0703176
$845$	8.95715	0.308135
$846$	−16.7862	−0.577122
$847$	0	0
$848$	5.43910	0.186779
$849$	−3.68415	−0.126440
$850$	−0.292731	−0.0100406
$851$	2.43596	0.0835037
$852$	−0.384694	−0.0131794
$853$	−9.41454	−0.322348	−0.161174	−	0.986926i	$-0.551528\pi$
$853$	−9.41454	−0.322348	−0.161174	+	0.986926i	$0.551528\pi$
$854$	−11.9572	−0.409165
$855$	10.9884	0.375797
$856$	9.56404	0.326892
$857$	25.8652	0.883538	0.441769	−	0.897129i	$-0.354351\pi$
$857$	25.8652	0.883538	0.441769	+	0.897129i	$0.354351\pi$
$858$	0	0
$859$	−29.2860	−0.999225	−0.499613	−	0.866249i	$-0.666524\pi$
$859$	−29.2860	−0.999225	−0.499613	+	0.866249i	$0.666524\pi$
$860$	10.3503	0.352941
$861$	7.13650	0.243211
$862$	38.0477	1.29591
$863$	−20.3110	−0.691395	−0.345698	−	0.938346i	$-0.612358\pi$
$863$	−20.3110	−0.691395	−0.345698	+	0.938346i	$0.612358\pi$
$864$	5.37169	0.182749
$865$	7.37169	0.250645
$866$	10.8108	0.367366
$867$	−19.3900	−0.658518
$868$	0.978577	0.0332151
$869$	0	0
$870$	−1.64973	−0.0559311
$871$	2.74338	0.0929560
$872$	3.14637	0.106549
$873$	−15.3471	−0.519422
$874$	18.6002	0.629160
$875$	1.00000	0.0338062
$876$	4.24989	0.143590
$877$	38.3650	1.29549	0.647746	−	0.761856i	$-0.275712\pi$
$877$	38.3650	1.29549	0.647746	+	0.761856i	$0.275712\pi$
$878$	30.9933	1.04597
$879$	19.1281	0.645174
$880$	0	0
$881$	40.5426	1.36592	0.682958	−	0.730458i	$-0.260693\pi$
$881$	40.5426	1.36592	0.682958	+	0.730458i	$0.260693\pi$
$882$	1.68585	0.0567654
$883$	28.0491	0.943928	0.471964	−	0.881618i	$-0.343545\pi$
$883$	28.0491	0.943928	0.471964	+	0.881618i	$0.343545\pi$
$884$	−1.37169	−0.0461350
$885$	−10.7434	−0.361135
$886$	−25.3717	−0.852379
$887$	6.65769	0.223543	0.111772	−	0.993734i	$-0.464347\pi$
$887$	6.65769	0.223543	0.111772	+	0.993734i	$0.464347\pi$
$888$	−0.978577	−0.0328389
$889$	13.3717	0.448472
$890$	−13.0790	−0.438408
$891$	0	0
$892$	11.0790	0.370951
$893$	64.9013	2.17184
$894$	13.2944	0.444632
$895$	1.56404	0.0522801
$896$	−1.00000	−0.0334077
$897$	−15.3288	−0.511815
$898$	0.886615	0.0295867
$899$	1.40827	0.0469683
$900$	−1.68585	−0.0561949
$901$	1.59219	0.0530436
$902$	0	0
$903$	11.8652	0.394849
$904$	2.58546	0.0859912
$905$	14.7862	0.491511
$906$	1.98531	0.0659574
$907$	39.5787	1.31419	0.657095	−	0.753808i	$-0.271785\pi$
$907$	39.5787	1.31419	0.657095	+	0.753808i	$0.271785\pi$
$908$	18.5426	0.615358
$909$	−18.1839	−0.603123
$910$	4.68585	0.155334
$911$	−24.3356	−0.806274	−0.403137	−	0.915140i	$-0.632080\pi$
$911$	−24.3356	−0.806274	−0.403137	+	0.915140i	$0.632080\pi$
$912$	−7.47208	−0.247425
$913$	0	0
$914$	29.9143	0.989477
$915$	13.7073	0.453148
$916$	−8.01469	−0.264813
$917$	11.4391	0.377752
$918$	1.57246	0.0518989
$919$	−50.6331	−1.67023	−0.835117	−	0.550073i	$-0.814600\pi$
$919$	−50.6331	−1.67023	−0.835117	+	0.550073i	$0.814600\pi$
$920$	−2.85363	−0.0940815
$921$	7.21377	0.237702
$922$	−2.33558	−0.0769181
$923$	1.57246	0.0517582
$924$	0	0
$925$	0.853635	0.0280673
$926$	−0.110250	−0.00362303
$927$	−12.9210	−0.424383
$928$	−1.43910	−0.0472407
$929$	47.1512	1.54698	0.773490	−	0.633808i	$-0.218509\pi$
$929$	47.1512	1.54698	0.773490	+	0.633808i	$0.218509\pi$
$930$	−1.12181	−0.0367855
$931$	−6.51806	−0.213621
$932$	−27.9572	−0.915767
$933$	−23.3780	−0.765360
$934$	−36.6760	−1.20007
$935$	0	0
$936$	−7.89962	−0.258207
$937$	42.0294	1.37304	0.686520	−	0.727111i	$-0.259137\pi$
$937$	42.0294	1.37304	0.686520	+	0.727111i	$0.259137\pi$
$938$	0.585462	0.0191160
$939$	13.8337	0.451444
$940$	−9.95715	−0.324767
$941$	36.7434	1.19780	0.598900	−	0.800824i	$-0.295605\pi$
$941$	36.7434	1.19780	0.598900	+	0.800824i	$0.295605\pi$
$942$	14.0920	0.459141
$943$	17.7648	0.578502
$944$	−9.37169	−0.305023
$945$	−5.37169	−0.174741
$946$	0	0
$947$	18.8782	0.613459	0.306729	−	0.951797i	$-0.400765\pi$
$947$	18.8782	0.613459	0.306729	+	0.951797i	$0.400765\pi$
$948$	2.88661	0.0937529
$949$	−17.3717	−0.563909
$950$	6.51806	0.211474
$951$	32.9786	1.06940
$952$	−0.292731	−0.00948747
$953$	43.2285	1.40031	0.700154	−	0.713992i	$-0.253115\pi$
$953$	43.2285	1.40031	0.700154	+	0.713992i	$0.253115\pi$
$954$	9.16948	0.296873
$955$	−17.9572	−0.581080
$956$	28.8108	0.931808
$957$	0	0
$958$	−42.7434	−1.38098
$959$	10.0000	0.322917
$960$	1.14637	0.0369988
$961$	−30.0424	−0.969109
$962$	4.00000	0.128965
$963$	16.1235	0.519572
$964$	8.51806	0.274348
$965$	−18.6430	−0.600139
$966$	−3.27131	−0.105253
$967$	−48.7299	−1.56705	−0.783524	−	0.621361i	$-0.786580\pi$
$967$	−48.7299	−1.56705	−0.783524	+	0.621361i	$0.786580\pi$
$968$	0	0
$969$	−2.18731	−0.0702665
$970$	−9.10352	−0.292296
$971$	−25.9865	−0.833948	−0.416974	−	0.908918i	$-0.636909\pi$
$971$	−25.9865	−0.833948	−0.416974	+	0.908918i	$0.636909\pi$
$972$	14.8536	0.476431
$973$	14.1825	0.454669
$974$	36.1396	1.15799
$975$	−5.37169	−0.172032
$976$	11.9572	0.382739
$977$	2.67115	0.0854578	0.0427289	−	0.999087i	$-0.486395\pi$
$977$	2.67115	0.0854578	0.0427289	+	0.999087i	$0.486395\pi$
$978$	−27.4145	−0.876620
$979$	0	0
$980$	1.00000	0.0319438
$981$	5.30429	0.169353
$982$	2.54262	0.0811381
$983$	−8.33558	−0.265864	−0.132932	−	0.991125i	$-0.542439\pi$
$983$	−8.33558	−0.265864	−0.132932	+	0.991125i	$0.542439\pi$
$984$	−7.13650	−0.227503
$985$	15.0361	0.479090
$986$	−0.421268	−0.0134159
$987$	−11.4145	−0.363329
$988$	30.5426	0.971690
$989$	29.5359	0.939187
$990$	0	0
$991$	46.2730	1.46991	0.734955	−	0.678116i	$-0.237203\pi$
$991$	46.2730	1.46991	0.734955	+	0.678116i	$0.237203\pi$
$992$	−0.978577	−0.0310699
$993$	−4.36496	−0.138518
$994$	0.335577	0.0106438
$995$	15.3288	0.485957
$996$	1.95715	0.0620148
$997$	−35.8715	−1.13606	−0.568030	−	0.823008i	$-0.692294\pi$
$997$	−35.8715	−1.13606	−0.568030	+	0.823008i	$0.692294\pi$
$998$	−3.13650	−0.0992842
$999$	−4.58546	−0.145078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.ci.1.2		3
11.10	odd	2		770.2.a.m.1.2	✓	3
33.32	even	2		6930.2.a.ce.1.3		3
44.43	even	2		6160.2.a.bf.1.2		3
55.32	even	4		3850.2.c.ba.1849.5		6
55.43	even	4		3850.2.c.ba.1849.2		6
55.54	odd	2		3850.2.a.bt.1.2		3
77.76	even	2		5390.2.a.ca.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.m.1.2	✓	3	11.10	odd	2
3850.2.a.bt.1.2		3	55.54	odd	2
3850.2.c.ba.1849.2		6	55.43	even	4
3850.2.c.ba.1849.5		6	55.32	even	4
5390.2.a.ca.1.2		3	77.76	even	2
6160.2.a.bf.1.2		3	44.43	even	2
6930.2.a.ce.1.3		3	33.32	even	2
8470.2.a.ci.1.2		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.68585 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.68585 q^{9} -1.00000 q^{10} +1.14637 q^{12} -4.68585 q^{13} -1.00000 q^{14} +1.14637 q^{15} +1.00000 q^{16} +0.292731 q^{17} +1.68585 q^{18} -6.51806 q^{19} +1.00000 q^{20} +1.14637 q^{21} +2.85363 q^{23} -1.14637 q^{24} +1.00000 q^{25} +4.68585 q^{26} -5.37169 q^{27} +1.00000 q^{28} +1.43910 q^{29} -1.14637 q^{30} +0.978577 q^{31} -1.00000 q^{32} -0.292731 q^{34} +1.00000 q^{35} -1.68585 q^{36} +0.853635 q^{37} +6.51806 q^{38} -5.37169 q^{39} -1.00000 q^{40} +6.22533 q^{41} -1.14637 q^{42} +10.3503 q^{43} -1.68585 q^{45} -2.85363 q^{46} -9.95715 q^{47} +1.14637 q^{48} +1.00000 q^{49} -1.00000 q^{50} +0.335577 q^{51} -4.68585 q^{52} +5.43910 q^{53} +5.37169 q^{54} -1.00000 q^{56} -7.47208 q^{57} -1.43910 q^{58} -9.37169 q^{59} +1.14637 q^{60} +11.9572 q^{61} -0.978577 q^{62} -1.68585 q^{63} +1.00000 q^{64} -4.68585 q^{65} -0.585462 q^{67} +0.292731 q^{68} +3.27131 q^{69} -1.00000 q^{70} -0.335577 q^{71} +1.68585 q^{72} +3.70727 q^{73} -0.853635 q^{74} +1.14637 q^{75} -6.51806 q^{76} +5.37169 q^{78} +2.51806 q^{79} +1.00000 q^{80} -1.10038 q^{81} -6.22533 q^{82} +1.70727 q^{83} +1.14637 q^{84} +0.292731 q^{85} -10.3503 q^{86} +1.64973 q^{87} +13.0790 q^{89} +1.68585 q^{90} -4.68585 q^{91} +2.85363 q^{92} +1.12181 q^{93} +9.95715 q^{94} -6.51806 q^{95} -1.14637 q^{96} +9.10352 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 7 * q^9	\( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} - 3 q^{10} + 2 q^{12} - 2 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} - 2 q^{17} - 7 q^{18} + 6 q^{19} + 3 q^{20} + 2 q^{21} + 10 q^{23} - 2 q^{24} + 3 q^{25} + 2 q^{26} + 8 q^{27} + 3 q^{28} - 2 q^{30} - 12 q^{31} - 3 q^{32} + 2 q^{34} + 3 q^{35} + 7 q^{36} + 4 q^{37} - 6 q^{38} + 8 q^{39} - 3 q^{40} - 4 q^{41} - 2 q^{42} - 8 q^{43} + 7 q^{45} - 10 q^{46} + 2 q^{48} + 3 q^{49} - 3 q^{50} + 28 q^{51} - 2 q^{52} + 12 q^{53} - 8 q^{54} - 3 q^{56} + 8 q^{57} - 4 q^{59} + 2 q^{60} + 6 q^{61} + 12 q^{62} + 7 q^{63} + 3 q^{64} - 2 q^{65} + 4 q^{67} - 2 q^{68} - 8 q^{69} - 3 q^{70} - 28 q^{71} - 7 q^{72} + 14 q^{73} - 4 q^{74} + 2 q^{75} + 6 q^{76} - 8 q^{78} - 18 q^{79} + 3 q^{80} + 3 q^{81} + 4 q^{82} + 8 q^{83} + 2 q^{84} - 2 q^{85} + 8 q^{86} + 44 q^{87} + 18 q^{89} - 7 q^{90} - 2 q^{91} + 10 q^{92} + 12 q^{93} + 6 q^{95} - 2 q^{96} - 4 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 7 * q^9 - 3 * q^10 + 2 * q^12 - 2 * q^13 - 3 * q^14 + 2 * q^15 + 3 * q^16 - 2 * q^17 - 7 * q^18 + 6 * q^19 + 3 * q^20 + 2 * q^21 + 10 * q^23 - 2 * q^24 + 3 * q^25 + 2 * q^26 + 8 * q^27 + 3 * q^28 - 2 * q^30 - 12 * q^31 - 3 * q^32 + 2 * q^34 + 3 * q^35 + 7 * q^36 + 4 * q^37 - 6 * q^38 + 8 * q^39 - 3 * q^40 - 4 * q^41 - 2 * q^42 - 8 * q^43 + 7 * q^45 - 10 * q^46 + 2 * q^48 + 3 * q^49 - 3 * q^50 + 28 * q^51 - 2 * q^52 + 12 * q^53 - 8 * q^54 - 3 * q^56 + 8 * q^57 - 4 * q^59 + 2 * q^60 + 6 * q^61 + 12 * q^62 + 7 * q^63 + 3 * q^64 - 2 * q^65 + 4 * q^67 - 2 * q^68 - 8 * q^69 - 3 * q^70 - 28 * q^71 - 7 * q^72 + 14 * q^73 - 4 * q^74 + 2 * q^75 + 6 * q^76 - 8 * q^78 - 18 * q^79 + 3 * q^80 + 3 * q^81 + 4 * q^82 + 8 * q^83 + 2 * q^84 - 2 * q^85 + 8 * q^86 + 44 * q^87 + 18 * q^89 - 7 * q^90 - 2 * q^91 + 10 * q^92 + 12 * q^93 + 6 * q^95 - 2 * q^96 - 4 * q^97 - 3 * q^98

Introduction

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