LMFDB - Embedded newform 8470.2.a.cq.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48718$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.919131 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.919131 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.15520 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.919131 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.919131 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.15520 q^{9} +1.00000 q^{10} -0.919131 q^{12} +3.72325 q^{13} +1.00000 q^{14} +0.919131 q^{15} +1.00000 q^{16} -1.31694 q^{17} +2.15520 q^{18} -2.70741 q^{19} -1.00000 q^{20} +0.919131 q^{21} -2.00000 q^{23} +0.919131 q^{24} +1.00000 q^{25} -3.72325 q^{26} +4.73830 q^{27} -1.00000 q^{28} -2.95002 q^{29} -0.919131 q^{30} +7.70820 q^{31} -1.00000 q^{32} +1.31694 q^{34} +1.00000 q^{35} -2.15520 q^{36} -8.04870 q^{37} +2.70741 q^{38} -3.42216 q^{39} +1.00000 q^{40} +3.24458 q^{41} -0.919131 q^{42} +6.48064 q^{43} +2.15520 q^{45} +2.00000 q^{46} -9.37090 q^{47} -0.919131 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.21044 q^{51} +3.72325 q^{52} +13.2848 q^{53} -4.73830 q^{54} +1.00000 q^{56} +2.48847 q^{57} +2.95002 q^{58} -2.83905 q^{59} +0.919131 q^{60} +1.44651 q^{61} -7.70820 q^{62} +2.15520 q^{63} +1.00000 q^{64} -3.72325 q^{65} -6.70741 q^{67} -1.31694 q^{68} +1.83826 q^{69} -1.00000 q^{70} +8.03365 q^{71} +2.15520 q^{72} -1.33071 q^{73} +8.04870 q^{74} -0.919131 q^{75} -2.70741 q^{76} +3.42216 q^{78} -7.72325 q^{79} -1.00000 q^{80} +2.11048 q^{81} -3.24458 q^{82} +0.285256 q^{83} +0.919131 q^{84} +1.31694 q^{85} -6.48064 q^{86} +2.71145 q^{87} -8.14015 q^{89} -2.15520 q^{90} -3.72325 q^{91} -2.00000 q^{92} -7.08485 q^{93} +9.37090 q^{94} +2.70741 q^{95} +0.919131 q^{96} +14.7984 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} + 4 q^{10} + 2 q^{12} - q^{13} + 4 q^{14} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 6 q^{18} + 3 q^{19} - 4 q^{20} - 2 q^{21} - 8 q^{23} - 2 q^{24} + 4 q^{25} + q^{26} + 14 q^{27} - 4 q^{28} - 15 q^{29} + 2 q^{30} + 4 q^{31} - 4 q^{32} + 2 q^{34} + 4 q^{35} + 6 q^{36} + 2 q^{37} - 3 q^{38} + q^{39} + 4 q^{40} - 11 q^{41} + 2 q^{42} - 7 q^{43} - 6 q^{45} + 8 q^{46} + 5 q^{47} + 2 q^{48} + 4 q^{49} - 4 q^{50} - 18 q^{51} - q^{52} + 10 q^{53} - 14 q^{54} + 4 q^{56} + 34 q^{57} + 15 q^{58} + 13 q^{59} - 2 q^{60} - 26 q^{61} - 4 q^{62} - 6 q^{63} + 4 q^{64} + q^{65} - 13 q^{67} - 2 q^{68} - 4 q^{69} - 4 q^{70} - 13 q^{71} - 6 q^{72} + 18 q^{73} - 2 q^{74} + 2 q^{75} + 3 q^{76} - q^{78} - 15 q^{79} - 4 q^{80} - 8 q^{81} + 11 q^{82} + 2 q^{83} - 2 q^{84} + 2 q^{85} + 7 q^{86} - 26 q^{87} - 7 q^{89} + 6 q^{90} + q^{91} - 8 q^{92} + 2 q^{93} - 5 q^{94} - 3 q^{95} - 2 q^{96} + 10 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 + 4 * q^10 + 2 * q^12 - q^13 + 4 * q^14 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 6 * q^18 + 3 * q^19 - 4 * q^20 - 2 * q^21 - 8 * q^23 - 2 * q^24 + 4 * q^25 + q^26 + 14 * q^27 - 4 * q^28 - 15 * q^29 + 2 * q^30 + 4 * q^31 - 4 * q^32 + 2 * q^34 + 4 * q^35 + 6 * q^36 + 2 * q^37 - 3 * q^38 + q^39 + 4 * q^40 - 11 * q^41 + 2 * q^42 - 7 * q^43 - 6 * q^45 + 8 * q^46 + 5 * q^47 + 2 * q^48 + 4 * q^49 - 4 * q^50 - 18 * q^51 - q^52 + 10 * q^53 - 14 * q^54 + 4 * q^56 + 34 * q^57 + 15 * q^58 + 13 * q^59 - 2 * q^60 - 26 * q^61 - 4 * q^62 - 6 * q^63 + 4 * q^64 + q^65 - 13 * q^67 - 2 * q^68 - 4 * q^69 - 4 * q^70 - 13 * q^71 - 6 * q^72 + 18 * q^73 - 2 * q^74 + 2 * q^75 + 3 * q^76 - q^78 - 15 * q^79 - 4 * q^80 - 8 * q^81 + 11 * q^82 + 2 * q^83 - 2 * q^84 + 2 * q^85 + 7 * q^86 - 26 * q^87 - 7 * q^89 + 6 * q^90 + q^91 - 8 * q^92 + 2 * q^93 - 5 * q^94 - 3 * q^95 - 2 * q^96 + 10 * q^97 - 4 * 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
$3$	−0.919131	−0.530660	−0.265330	−	0.964158i	$-0.585481\pi$
$3$	−0.919131	−0.530660	−0.265330	+	0.964158i	$0.585481\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.919131	0.375234
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−2.15520	−0.718400
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−0.919131	−0.265330
$13$	3.72325	1.03264	0.516322	−	0.856394i	$-0.327301\pi$
$13$	3.72325	1.03264	0.516322	+	0.856394i	$0.327301\pi$
$14$	1.00000	0.267261
$15$	0.919131	0.237319
$16$	1.00000	0.250000
$17$	−1.31694	−0.319404	−0.159702	−	0.987165i	$-0.551053\pi$
$17$	−1.31694	−0.319404	−0.159702	+	0.987165i	$0.551053\pi$
$18$	2.15520	0.507985
$19$	−2.70741	−0.621123	−0.310561	−	0.950553i	$-0.600517\pi$
$19$	−2.70741	−0.621123	−0.310561	+	0.950553i	$0.600517\pi$
$20$	−1.00000	−0.223607
$21$	0.919131	0.200571
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0.919131	0.187617
$25$	1.00000	0.200000
$26$	−3.72325	−0.730190
$27$	4.73830	0.911887
$28$	−1.00000	−0.188982
$29$	−2.95002	−0.547805	−0.273902	−	0.961757i	$-0.588315\pi$
$29$	−2.95002	−0.547805	−0.273902	+	0.961757i	$0.588315\pi$
$30$	−0.919131	−0.167810
$31$	7.70820	1.38443	0.692217	−	0.721689i	$-0.256634\pi$
$31$	7.70820	1.38443	0.692217	+	0.721689i	$0.256634\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.31694	0.225853
$35$	1.00000	0.169031
$36$	−2.15520	−0.359200
$37$	−8.04870	−1.32320	−0.661599	−	0.749858i	$-0.730122\pi$
$37$	−8.04870	−1.32320	−0.661599	+	0.749858i	$0.730122\pi$
$38$	2.70741	0.439200
$39$	−3.42216	−0.547984
$40$	1.00000	0.158114
$41$	3.24458	0.506718	0.253359	−	0.967372i	$-0.418465\pi$
$41$	3.24458	0.506718	0.253359	+	0.967372i	$0.418465\pi$
$42$	−0.919131	−0.141825
$43$	6.48064	0.988289	0.494145	−	0.869380i	$-0.335481\pi$
$43$	6.48064	0.988289	0.494145	+	0.869380i	$0.335481\pi$
$44$	0	0
$45$	2.15520	0.321278
$46$	2.00000	0.294884
$47$	−9.37090	−1.36689	−0.683443	−	0.730004i	$-0.739518\pi$
$47$	−9.37090	−1.36689	−0.683443	+	0.730004i	$0.739518\pi$
$48$	−0.919131	−0.132665
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	1.21044	0.169495
$52$	3.72325	0.516322
$53$	13.2848	1.82480	0.912402	−	0.409296i	$-0.134226\pi$
$53$	13.2848	1.82480	0.912402	+	0.409296i	$0.134226\pi$
$54$	−4.73830	−0.644801
$55$	0	0
$56$	1.00000	0.133631
$57$	2.48847	0.329605
$58$	2.95002	0.387357
$59$	−2.83905	−0.369613	−0.184807	−	0.982775i	$-0.559166\pi$
$59$	−2.83905	−0.369613	−0.184807	+	0.982775i	$0.559166\pi$
$60$	0.919131	0.118659
$61$	1.44651	0.185206	0.0926030	−	0.995703i	$-0.470481\pi$
$61$	1.44651	0.185206	0.0926030	+	0.995703i	$0.470481\pi$
$62$	−7.70820	−0.978943
$63$	2.15520	0.271530
$64$	1.00000	0.125000
$65$	−3.72325	−0.461813
$66$	0	0
$67$	−6.70741	−0.819441	−0.409720	−	0.912211i	$-0.634374\pi$
$67$	−6.70741	−0.819441	−0.409720	+	0.912211i	$0.634374\pi$
$68$	−1.31694	−0.159702
$69$	1.83826	0.221301
$70$	−1.00000	−0.119523
$71$	8.03365	0.953419	0.476709	−	0.879061i	$-0.341829\pi$
$71$	8.03365	0.953419	0.476709	+	0.879061i	$0.341829\pi$
$72$	2.15520	0.253993
$73$	−1.33071	−0.155747	−0.0778736	−	0.996963i	$-0.524813\pi$
$73$	−1.33071	−0.155747	−0.0778736	+	0.996963i	$0.524813\pi$
$74$	8.04870	0.935642
$75$	−0.919131	−0.106132
$76$	−2.70741	−0.310561
$77$	0	0
$78$	3.42216	0.387483
$79$	−7.72325	−0.868934	−0.434467	−	0.900688i	$-0.643063\pi$
$79$	−7.72325	−0.868934	−0.434467	+	0.900688i	$0.643063\pi$
$80$	−1.00000	−0.111803
$81$	2.11048	0.234498
$82$	−3.24458	−0.358304
$83$	0.285256	0.0313109	0.0156555	−	0.999877i	$-0.495017\pi$
$83$	0.285256	0.0313109	0.0156555	+	0.999877i	$0.495017\pi$
$84$	0.919131	0.100285
$85$	1.31694	0.142842
$86$	−6.48064	−0.698826
$87$	2.71145	0.290698
$88$	0	0
$89$	−8.14015	−0.862854	−0.431427	−	0.902148i	$-0.641990\pi$
$89$	−8.14015	−0.862854	−0.431427	+	0.902148i	$0.641990\pi$
$90$	−2.15520	−0.227178
$91$	−3.72325	−0.390303
$92$	−2.00000	−0.208514
$93$	−7.08485	−0.734664
$94$	9.37090	0.966534
$95$	2.70741	0.277775
$96$	0.919131	0.0938084
$97$	14.7984	1.50255	0.751274	−	0.659991i	$-0.229440\pi$
$97$	14.7984	1.50255	0.751274	+	0.659991i	$0.229440\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−14.1000	−1.40300	−0.701499	−	0.712670i	$-0.747486\pi$
$101$	−14.1000	−1.40300	−0.701499	+	0.712670i	$0.747486\pi$
$102$	−1.21044	−0.119851
$103$	11.7627	1.15901	0.579504	−	0.814969i	$-0.303246\pi$
$103$	11.7627	1.15901	0.579504	+	0.814969i	$0.303246\pi$
$104$	−3.72325	−0.365095
$105$	−0.919131	−0.0896980
$106$	−13.2848	−1.29033
$107$	19.8272	1.91677	0.958383	−	0.285484i	$-0.0921544\pi$
$107$	19.8272	1.91677	0.958383	+	0.285484i	$0.0921544\pi$
$108$	4.73830	0.455943
$109$	−0.995533	−0.0953548	−0.0476774	−	0.998863i	$-0.515182\pi$
$109$	−0.995533	−0.0953548	−0.0476774	+	0.998863i	$0.515182\pi$
$110$	0	0
$111$	7.39781	0.702169
$112$	−1.00000	−0.0944911
$113$	11.0691	1.04129	0.520645	−	0.853773i	$-0.325691\pi$
$113$	11.0691	1.04129	0.520645	+	0.853773i	$0.325691\pi$
$114$	−2.48847	−0.233066
$115$	2.00000	0.186501
$116$	−2.95002	−0.273902
$117$	−8.02435	−0.741851
$118$	2.83905	0.261356
$119$	1.31694	0.120723
$120$	−0.919131	−0.0839048
$121$	0	0
$122$	−1.44651	−0.130960
$123$	−2.98219	−0.268895
$124$	7.70820	0.692217
$125$	−1.00000	−0.0894427
$126$	−2.15520	−0.192000
$127$	6.50223	0.576980	0.288490	−	0.957483i	$-0.406847\pi$
$127$	6.50223	0.576980	0.288490	+	0.957483i	$0.406847\pi$
$128$	−1.00000	−0.0883883
$129$	−5.95656	−0.524446
$130$	3.72325	0.326551
$131$	−21.7622	−1.90137	−0.950684	−	0.310160i	$-0.899617\pi$
$131$	−21.7622	−1.90137	−0.950684	+	0.310160i	$0.899617\pi$
$132$	0	0
$133$	2.70741	0.234762
$134$	6.70741	0.579432
$135$	−4.73830	−0.407808
$136$	1.31694	0.112926
$137$	−2.90408	−0.248112	−0.124056	−	0.992275i	$-0.539590\pi$
$137$	−2.90408	−0.248112	−0.124056	+	0.992275i	$0.539590\pi$
$138$	−1.83826	−0.156483
$139$	−1.26775	−0.107529	−0.0537645	−	0.998554i	$-0.517122\pi$
$139$	−1.26775	−0.107529	−0.0537645	+	0.998554i	$0.517122\pi$
$140$	1.00000	0.0845154
$141$	8.61308	0.725352
$142$	−8.03365	−0.674169
$143$	0	0
$144$	−2.15520	−0.179600
$145$	2.95002	0.244986
$146$	1.33071	0.110130
$147$	−0.919131	−0.0758086
$148$	−8.04870	−0.661599
$149$	5.85489	0.479652	0.239826	−	0.970816i	$-0.422910\pi$
$149$	5.85489	0.479652	0.239826	+	0.970816i	$0.422910\pi$
$150$	0.919131	0.0750467
$151$	10.2243	0.832039	0.416020	−	0.909356i	$-0.363425\pi$
$151$	10.2243	0.832039	0.416020	+	0.909356i	$0.363425\pi$
$152$	2.70741	0.219600
$153$	2.83826	0.229460
$154$	0	0
$155$	−7.70820	−0.619138
$156$	−3.42216	−0.273992
$157$	1.18004	0.0941772	0.0470886	−	0.998891i	$-0.485006\pi$
$157$	1.18004	0.0941772	0.0470886	+	0.998891i	$0.485006\pi$
$158$	7.72325	0.614429
$159$	−12.2104	−0.968351
$160$	1.00000	0.0790569
$161$	2.00000	0.157622
$162$	−2.11048	−0.165815
$163$	13.5818	1.06381	0.531905	−	0.846804i	$-0.321476\pi$
$163$	13.5818	1.06381	0.531905	+	0.846804i	$0.321476\pi$
$164$	3.24458	0.253359
$165$	0	0
$166$	−0.285256	−0.0221402
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−0.919131	−0.0709125
$169$	0.862611	0.0663547
$170$	−1.31694	−0.101004
$171$	5.83501	0.446214
$172$	6.48064	0.494145
$173$	−12.4741	−0.948389	−0.474194	−	0.880420i	$-0.657261\pi$
$173$	−12.4741	−0.948389	−0.474194	+	0.880420i	$0.657261\pi$
$174$	−2.71145	−0.205555
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	2.60946	0.196139
$178$	8.14015	0.610130
$179$	12.2295	0.914078	0.457039	−	0.889447i	$-0.348910\pi$
$179$	12.2295	0.914078	0.457039	+	0.889447i	$0.348910\pi$
$180$	2.15520	0.160639
$181$	20.9718	1.55882	0.779411	−	0.626513i	$-0.215518\pi$
$181$	20.9718	1.55882	0.779411	+	0.626513i	$0.215518\pi$
$182$	3.72325	0.275986
$183$	−1.32953	−0.0982815
$184$	2.00000	0.147442
$185$	8.04870	0.591752
$186$	7.08485	0.519486
$187$	0	0
$188$	−9.37090	−0.683443
$189$	−4.73830	−0.344661
$190$	−2.70741	−0.196416
$191$	6.74632	0.488147	0.244073	−	0.969757i	$-0.421516\pi$
$191$	6.74632	0.488147	0.244073	+	0.969757i	$0.421516\pi$
$192$	−0.919131	−0.0663325
$193$	−19.9673	−1.43728	−0.718640	−	0.695382i	$-0.755235\pi$
$193$	−19.9673	−1.43728	−0.718640	+	0.695382i	$0.755235\pi$
$194$	−14.7984	−1.06246
$195$	3.42216	0.245066
$196$	1.00000	0.0714286
$197$	−21.1717	−1.50842	−0.754212	−	0.656631i	$-0.771981\pi$
$197$	−21.1717	−1.50842	−0.754212	+	0.656631i	$0.771981\pi$
$198$	0	0
$199$	8.27872	0.586863	0.293431	−	0.955980i	$-0.405203\pi$
$199$	8.27872	0.586863	0.293431	+	0.955980i	$0.405203\pi$
$200$	−1.00000	−0.0707107
$201$	6.16499	0.434845
$202$	14.1000	0.992070
$203$	2.95002	0.207051
$204$	1.21044	0.0847476
$205$	−3.24458	−0.226611
$206$	−11.7627	−0.819543
$207$	4.31040	0.299593
$208$	3.72325	0.258161
$209$	0	0
$210$	0.919131	0.0634260
$211$	−15.5912	−1.07334	−0.536671	−	0.843792i	$-0.680318\pi$
$211$	−15.5912	−1.07334	−0.536671	+	0.843792i	$0.680318\pi$
$212$	13.2848	0.912402
$213$	−7.38397	−0.505942
$214$	−19.8272	−1.35536
$215$	−6.48064	−0.441976
$216$	−4.73830	−0.322401
$217$	−7.70820	−0.523267
$218$	0.995533	0.0674260
$219$	1.22309	0.0826489
$220$	0	0
$221$	−4.90329	−0.329831
$222$	−7.39781	−0.496508
$223$	−16.6313	−1.11372	−0.556858	−	0.830608i	$-0.687993\pi$
$223$	−16.6313	−1.11372	−0.556858	+	0.830608i	$0.687993\pi$
$224$	1.00000	0.0668153
$225$	−2.15520	−0.143680
$226$	−11.0691	−0.736304
$227$	−1.19716	−0.0794582	−0.0397291	−	0.999210i	$-0.512649\pi$
$227$	−1.19716	−0.0794582	−0.0397291	+	0.999210i	$0.512649\pi$
$228$	2.48847	0.164803
$229$	14.8613	0.982064	0.491032	−	0.871141i	$-0.336620\pi$
$229$	14.8613	0.982064	0.491032	+	0.871141i	$0.336620\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	2.95002	0.193678
$233$	6.33199	0.414822	0.207411	−	0.978254i	$-0.433496\pi$
$233$	6.33199	0.414822	0.207411	+	0.978254i	$0.433496\pi$
$234$	8.02435	0.524568
$235$	9.37090	0.611290
$236$	−2.83905	−0.184807
$237$	7.09868	0.461109
$238$	−1.31694	−0.0853644
$239$	11.6906	0.756202	0.378101	−	0.925764i	$-0.376577\pi$
$239$	11.6906	0.756202	0.378101	+	0.925764i	$0.376577\pi$
$240$	0.919131	0.0593296
$241$	−10.3945	−0.669566	−0.334783	−	0.942295i	$-0.608663\pi$
$241$	−10.3945	−0.669566	−0.334783	+	0.942295i	$0.608663\pi$
$242$	0	0
$243$	−16.1547	−1.03633
$244$	1.44651	0.0926030
$245$	−1.00000	−0.0638877
$246$	2.98219	0.190138
$247$	−10.0804	−0.641399
$248$	−7.70820	−0.489471
$249$	−0.262188	−0.0166155
$250$	1.00000	0.0632456
$251$	−22.3395	−1.41006	−0.705029	−	0.709179i	$-0.749066\pi$
$251$	−22.3395	−1.41006	−0.705029	+	0.709179i	$0.749066\pi$
$252$	2.15520	0.135765
$253$	0	0
$254$	−6.50223	−0.407986
$255$	−1.21044	−0.0758005
$256$	1.00000	0.0625000
$257$	−5.28201	−0.329482	−0.164741	−	0.986337i	$-0.552679\pi$
$257$	−5.28201	−0.329482	−0.164741	+	0.986337i	$0.552679\pi$
$258$	5.95656	0.370839
$259$	8.04870	0.500122
$260$	−3.72325	−0.230906
$261$	6.35788	0.393543
$262$	21.7622	1.34447
$263$	−25.3847	−1.56529	−0.782645	−	0.622469i	$-0.786130\pi$
$263$	−25.3847	−1.56529	−0.782645	+	0.622469i	$0.786130\pi$
$264$	0	0
$265$	−13.2848	−0.816077
$266$	−2.70741	−0.166002
$267$	7.48186	0.457883
$268$	−6.70741	−0.409720
$269$	−15.3782	−0.937627	−0.468814	−	0.883297i	$-0.655318\pi$
$269$	−15.3782	−0.937627	−0.468814	+	0.883297i	$0.655318\pi$
$270$	4.73830	0.288364
$271$	−10.3274	−0.627346	−0.313673	−	0.949531i	$-0.601560\pi$
$271$	−10.3274	−0.627346	−0.313673	+	0.949531i	$0.601560\pi$
$272$	−1.31694	−0.0798510
$273$	3.42216	0.207118
$274$	2.90408	0.175442
$275$	0	0
$276$	1.83826	0.110650
$277$	22.7614	1.36760	0.683799	−	0.729670i	$-0.260326\pi$
$277$	22.7614	1.36760	0.683799	+	0.729670i	$0.260326\pi$
$278$	1.26775	0.0760345
$279$	−16.6127	−0.994577
$280$	−1.00000	−0.0597614
$281$	3.07890	0.183672	0.0918359	−	0.995774i	$-0.470726\pi$
$281$	3.07890	0.183672	0.0918359	+	0.995774i	$0.470726\pi$
$282$	−8.61308	−0.512901
$283$	10.9791	0.652642	0.326321	−	0.945259i	$-0.394191\pi$
$283$	10.9791	0.652642	0.326321	+	0.945259i	$0.394191\pi$
$284$	8.03365	0.476709
$285$	−2.48847	−0.147404
$286$	0	0
$287$	−3.24458	−0.191521
$288$	2.15520	0.126996
$289$	−15.2657	−0.897981
$290$	−2.95002	−0.173231
$291$	−13.6016	−0.797342
$292$	−1.33071	−0.0778736
$293$	−8.05126	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$293$	−8.05126	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$294$	0.919131	0.0536048
$295$	2.83905	0.165296
$296$	8.04870	0.467821
$297$	0	0
$298$	−5.85489	−0.339165
$299$	−7.44651	−0.430643
$300$	−0.919131	−0.0530660
$301$	−6.48064	−0.373538
$302$	−10.2243	−0.588341
$303$	12.9597	0.744516
$304$	−2.70741	−0.155281
$305$	−1.44651	−0.0828267
$306$	−2.83826	−0.162253
$307$	27.5963	1.57501	0.787503	−	0.616311i	$-0.211373\pi$
$307$	27.5963	1.57501	0.787503	+	0.616311i	$0.211373\pi$
$308$	0	0
$309$	−10.8114	−0.615040
$310$	7.70820	0.437797
$311$	29.7870	1.68907	0.844533	−	0.535504i	$-0.179878\pi$
$311$	29.7870	1.68907	0.844533	+	0.535504i	$0.179878\pi$
$312$	3.42216	0.193741
$313$	30.9153	1.74744	0.873718	−	0.486433i	$-0.161702\pi$
$313$	30.9153	1.74744	0.873718	+	0.486433i	$0.161702\pi$
$314$	−1.18004	−0.0665934
$315$	−2.15520	−0.121432
$316$	−7.72325	−0.434467
$317$	−2.32189	−0.130411	−0.0652053	−	0.997872i	$-0.520770\pi$
$317$	−2.32189	−0.130411	−0.0652053	+	0.997872i	$0.520770\pi$
$318$	12.2104	0.684727
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−18.2238	−1.01715
$322$	−2.00000	−0.111456
$323$	3.56549	0.198389
$324$	2.11048	0.117249
$325$	3.72325	0.206529
$326$	−13.5818	−0.752228
$327$	0.915025	0.0506010
$328$	−3.24458	−0.179152
$329$	9.37090	0.516634
$330$	0	0
$331$	−9.95686	−0.547279	−0.273639	−	0.961832i	$-0.588227\pi$
$331$	−9.95686	−0.547279	−0.273639	+	0.961832i	$0.588227\pi$
$332$	0.285256	0.0156555
$333$	17.3465	0.950585
$334$	12.0000	0.656611
$335$	6.70741	0.366465
$336$	0.919131	0.0501427
$337$	−20.7682	−1.13132	−0.565658	−	0.824640i	$-0.691378\pi$
$337$	−20.7682	−1.13132	−0.565658	+	0.824640i	$0.691378\pi$
$338$	−0.862611	−0.0469198
$339$	−10.1739	−0.552572
$340$	1.31694	0.0714210
$341$	0	0
$342$	−5.83501	−0.315521
$343$	−1.00000	−0.0539949
$344$	−6.48064	−0.349413
$345$	−1.83826	−0.0989687
$346$	12.4741	0.670612
$347$	−15.7606	−0.846072	−0.423036	−	0.906113i	$-0.639036\pi$
$347$	−15.7606	−0.846072	−0.423036	+	0.906113i	$0.639036\pi$
$348$	2.71145	0.145349
$349$	−23.2953	−1.24697	−0.623484	−	0.781836i	$-0.714283\pi$
$349$	−23.2953	−1.24697	−0.623484	+	0.781836i	$0.714283\pi$
$350$	1.00000	0.0534522
$351$	17.6419	0.941655
$352$	0	0
$353$	27.1303	1.44400	0.721999	−	0.691894i	$-0.243224\pi$
$353$	27.1303	1.44400	0.721999	+	0.691894i	$0.243224\pi$
$354$	−2.60946	−0.138691
$355$	−8.03365	−0.426382
$356$	−8.14015	−0.431427
$357$	−1.21044	−0.0640631
$358$	−12.2295	−0.646351
$359$	−32.0637	−1.69226	−0.846130	−	0.532977i	$-0.821073\pi$
$359$	−32.0637	−1.69226	−0.846130	+	0.532977i	$0.821073\pi$
$360$	−2.15520	−0.113589
$361$	−11.6699	−0.614206
$362$	−20.9718	−1.10225
$363$	0	0
$364$	−3.72325	−0.195151
$365$	1.33071	0.0696523
$366$	1.32953	0.0694955
$367$	−11.6037	−0.605709	−0.302854	−	0.953037i	$-0.597940\pi$
$367$	−11.6037	−0.605709	−0.302854	+	0.953037i	$0.597940\pi$
$368$	−2.00000	−0.104257
$369$	−6.99271	−0.364026
$370$	−8.04870	−0.418432
$371$	−13.2848	−0.689711
$372$	−7.08485	−0.367332
$373$	17.7252	0.917777	0.458889	−	0.888494i	$-0.348248\pi$
$373$	17.7252	0.917777	0.458889	+	0.888494i	$0.348248\pi$
$374$	0	0
$375$	0.919131	0.0474637
$376$	9.37090	0.483267
$377$	−10.9837	−0.565688
$378$	4.73830	0.243712
$379$	−12.0540	−0.619174	−0.309587	−	0.950871i	$-0.600191\pi$
$379$	−12.0540	−0.619174	−0.309587	+	0.950871i	$0.600191\pi$
$380$	2.70741	0.138887
$381$	−5.97640	−0.306180
$382$	−6.74632	−0.345172
$383$	6.75538	0.345184	0.172592	−	0.984993i	$-0.444786\pi$
$383$	6.75538	0.345184	0.172592	+	0.984993i	$0.444786\pi$
$384$	0.919131	0.0469042
$385$	0	0
$386$	19.9673	1.01631
$387$	−13.9671	−0.709986
$388$	14.7984	0.751274
$389$	−24.1441	−1.22416	−0.612078	−	0.790797i	$-0.709666\pi$
$389$	−24.1441	−1.22416	−0.612078	+	0.790797i	$0.709666\pi$
$390$	−3.42216	−0.173288
$391$	2.63387	0.133201
$392$	−1.00000	−0.0505076
$393$	20.0023	1.00898
$394$	21.1717	1.06662
$395$	7.72325	0.388599
$396$	0	0
$397$	10.5163	0.527798	0.263899	−	0.964550i	$-0.414991\pi$
$397$	10.5163	0.527798	0.263899	+	0.964550i	$0.414991\pi$
$398$	−8.27872	−0.414975
$399$	−2.48847	−0.124579
$400$	1.00000	0.0500000
$401$	−37.1997	−1.85766	−0.928831	−	0.370503i	$-0.879185\pi$
$401$	−37.1997	−1.85766	−0.928831	+	0.370503i	$0.879185\pi$
$402$	−6.16499	−0.307482
$403$	28.6996	1.42963
$404$	−14.1000	−0.701499
$405$	−2.11048	−0.104870
$406$	−2.95002	−0.146407
$407$	0	0
$408$	−1.21044	−0.0599256
$409$	−26.3908	−1.30494	−0.652470	−	0.757815i	$-0.726267\pi$
$409$	−26.3908	−1.30494	−0.652470	+	0.757815i	$0.726267\pi$
$410$	3.24458	0.160238
$411$	2.66923	0.131663
$412$	11.7627	0.579504
$413$	2.83905	0.139701
$414$	−4.31040	−0.211844
$415$	−0.285256	−0.0140027
$416$	−3.72325	−0.182547
$417$	1.16523	0.0570614
$418$	0	0
$419$	13.0447	0.637273	0.318637	−	0.947877i	$-0.396775\pi$
$419$	13.0447	0.637273	0.318637	+	0.947877i	$0.396775\pi$
$420$	−0.919131	−0.0448490
$421$	−13.4510	−0.655563	−0.327782	−	0.944753i	$-0.606301\pi$
$421$	−13.4510	−0.655563	−0.327782	+	0.944753i	$0.606301\pi$
$422$	15.5912	0.758967
$423$	20.1961	0.981970
$424$	−13.2848	−0.645165
$425$	−1.31694	−0.0638808
$426$	7.38397	0.357755
$427$	−1.44651	−0.0700013
$428$	19.8272	0.958383
$429$	0	0
$430$	6.48064	0.312524
$431$	−32.7451	−1.57728	−0.788638	−	0.614858i	$-0.789213\pi$
$431$	−32.7451	−1.57728	−0.788638	+	0.614858i	$0.789213\pi$
$432$	4.73830	0.227972
$433$	9.00434	0.432721	0.216361	−	0.976314i	$-0.430581\pi$
$433$	9.00434	0.432721	0.216361	+	0.976314i	$0.430581\pi$
$434$	7.70820	0.370006
$435$	−2.71145	−0.130004
$436$	−0.995533	−0.0476774
$437$	5.41482	0.259026
$438$	−1.22309	−0.0584416
$439$	−3.88952	−0.185637	−0.0928184	−	0.995683i	$-0.529588\pi$
$439$	−3.88952	−0.185637	−0.0928184	+	0.995683i	$0.529588\pi$
$440$	0	0
$441$	−2.15520	−0.102629
$442$	4.90329	0.233226
$443$	5.63377	0.267669	0.133834	−	0.991004i	$-0.457271\pi$
$443$	5.63377	0.267669	0.133834	+	0.991004i	$0.457271\pi$
$444$	7.39781	0.351084
$445$	8.14015	0.385880
$446$	16.6313	0.787515
$447$	−5.38141	−0.254532
$448$	−1.00000	−0.0472456
$449$	−5.62855	−0.265628	−0.132814	−	0.991141i	$-0.542401\pi$
$449$	−5.62855	−0.265628	−0.132814	+	0.991141i	$0.542401\pi$
$450$	2.15520	0.101597
$451$	0	0
$452$	11.0691	0.520645
$453$	−9.39744	−0.441530
$454$	1.19716	0.0561855
$455$	3.72325	0.174549
$456$	−2.48847	−0.116533
$457$	−1.15770	−0.0541548	−0.0270774	−	0.999633i	$-0.508620\pi$
$457$	−1.15770	−0.0541548	−0.0270774	+	0.999633i	$0.508620\pi$
$458$	−14.8613	−0.694424
$459$	−6.24005	−0.291260
$460$	2.00000	0.0932505
$461$	−10.3088	−0.480129	−0.240065	−	0.970757i	$-0.577169\pi$
$461$	−10.3088	−0.480129	−0.240065	+	0.970757i	$0.577169\pi$
$462$	0	0
$463$	−29.4740	−1.36977	−0.684887	−	0.728649i	$-0.740149\pi$
$463$	−29.4740	−1.36977	−0.684887	+	0.728649i	$0.740149\pi$
$464$	−2.95002	−0.136951
$465$	7.08485	0.328552
$466$	−6.33199	−0.293324
$467$	20.5977	0.953146	0.476573	−	0.879135i	$-0.341879\pi$
$467$	20.5977	0.953146	0.476573	+	0.879135i	$0.341879\pi$
$468$	−8.02435	−0.370926
$469$	6.70741	0.309720
$470$	−9.37090	−0.432247
$471$	−1.08461	−0.0499761
$472$	2.83905	0.130678
$473$	0	0
$474$	−7.09868	−0.326053
$475$	−2.70741	−0.124225
$476$	1.31694	0.0603617
$477$	−28.6313	−1.31094
$478$	−11.6906	−0.534715
$479$	4.18526	0.191229	0.0956146	−	0.995418i	$-0.469518\pi$
$479$	4.18526	0.191229	0.0956146	+	0.995418i	$0.469518\pi$
$480$	−0.919131	−0.0419524
$481$	−29.9673	−1.36639
$482$	10.3945	0.473454
$483$	−1.83826	−0.0836438
$484$	0	0
$485$	−14.7984	−0.671960
$486$	16.1547	0.732792
$487$	−8.06730	−0.365564	−0.182782	−	0.983153i	$-0.558510\pi$
$487$	−8.06730	−0.365564	−0.182782	+	0.983153i	$0.558510\pi$
$488$	−1.44651	−0.0654802
$489$	−12.4835	−0.564522
$490$	1.00000	0.0451754
$491$	23.5882	1.06452	0.532261	−	0.846580i	$-0.321342\pi$
$491$	23.5882	1.06452	0.532261	+	0.846580i	$0.321342\pi$
$492$	−2.98219	−0.134448
$493$	3.88499	0.174971
$494$	10.0804	0.453538
$495$	0	0
$496$	7.70820	0.346109
$497$	−8.03365	−0.360358
$498$	0.262188	0.0117489
$499$	7.22181	0.323293	0.161646	−	0.986849i	$-0.448320\pi$
$499$	7.22181	0.323293	0.161646	+	0.986849i	$0.448320\pi$
$500$	−1.00000	−0.0447214
$501$	11.0296	0.492765
$502$	22.3395	0.997061
$503$	11.3828	0.507532	0.253766	−	0.967266i	$-0.418331\pi$
$503$	11.3828	0.507532	0.253766	+	0.967266i	$0.418331\pi$
$504$	−2.15520	−0.0960002
$505$	14.1000	0.627440
$506$	0	0
$507$	−0.792852	−0.0352118
$508$	6.50223	0.288490
$509$	12.1422	0.538192	0.269096	−	0.963113i	$-0.413275\pi$
$509$	12.1422	0.538192	0.269096	+	0.963113i	$0.413275\pi$
$510$	1.21044	0.0535991
$511$	1.33071	0.0588669
$512$	−1.00000	−0.0441942
$513$	−12.8285	−0.566394
$514$	5.28201	0.232979
$515$	−11.7627	−0.518324
$516$	−5.95656	−0.262223
$517$	0	0
$518$	−8.04870	−0.353640
$519$	11.4653	0.503272
$520$	3.72325	0.163275
$521$	12.1565	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$521$	12.1565	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$522$	−6.35788	−0.278277
$523$	−29.3199	−1.28207	−0.641035	−	0.767512i	$-0.721495\pi$
$523$	−29.3199	−1.28207	−0.641035	+	0.767512i	$0.721495\pi$
$524$	−21.7622	−0.950684
$525$	0.919131	0.0401142
$526$	25.3847	1.10683
$527$	−10.1512	−0.442194
$528$	0	0
$529$	−19.0000	−0.826087
$530$	13.2848	0.577053
$531$	6.11872	0.265530
$532$	2.70741	0.117381
$533$	12.0804	0.523259
$534$	−7.48186	−0.323772
$535$	−19.8272	−0.857204
$536$	6.70741	0.289716
$537$	−11.2405	−0.485065
$538$	15.3782	0.663002
$539$	0	0
$540$	−4.73830	−0.203904
$541$	2.54640	0.109478	0.0547392	−	0.998501i	$-0.482567\pi$
$541$	2.54640	0.109478	0.0547392	+	0.998501i	$0.482567\pi$
$542$	10.3274	0.443600
$543$	−19.2758	−0.827205
$544$	1.31694	0.0564632
$545$	0.995533	0.0426439
$546$	−3.42216	−0.146455
$547$	32.1359	1.37403	0.687016	−	0.726642i	$-0.258920\pi$
$547$	32.1359	1.37403	0.687016	+	0.726642i	$0.258920\pi$
$548$	−2.90408	−0.124056
$549$	−3.11751	−0.133052
$550$	0	0
$551$	7.98692	0.340254
$552$	−1.83826	−0.0782416
$553$	7.72325	0.328426
$554$	−22.7614	−0.967038
$555$	−7.39781	−0.314019
$556$	−1.26775	−0.0537645
$557$	5.98042	0.253399	0.126699	−	0.991941i	$-0.459562\pi$
$557$	5.98042	0.253399	0.126699	+	0.991941i	$0.459562\pi$
$558$	16.6127	0.703272
$559$	24.1291	1.02055
$560$	1.00000	0.0422577
$561$	0	0
$562$	−3.07890	−0.129876
$563$	−1.80284	−0.0759807	−0.0379903	−	0.999278i	$-0.512096\pi$
$563$	−1.80284	−0.0759807	−0.0379903	+	0.999278i	$0.512096\pi$
$564$	8.61308	0.362676
$565$	−11.0691	−0.465679
$566$	−10.9791	−0.461488
$567$	−2.11048	−0.0886317
$568$	−8.03365	−0.337084
$569$	−30.1377	−1.26344	−0.631718	−	0.775198i	$-0.717650\pi$
$569$	−30.1377	−1.26344	−0.631718	+	0.775198i	$0.717650\pi$
$570$	2.48847	0.104230
$571$	−30.3322	−1.26936	−0.634681	−	0.772774i	$-0.718869\pi$
$571$	−30.3322	−1.26936	−0.634681	+	0.772774i	$0.718869\pi$
$572$	0	0
$573$	−6.20075	−0.259040
$574$	3.24458	0.135426
$575$	−2.00000	−0.0834058
$576$	−2.15520	−0.0897999
$577$	−15.7849	−0.657135	−0.328568	−	0.944480i	$-0.606566\pi$
$577$	−15.7849	−0.657135	−0.328568	+	0.944480i	$0.606566\pi$
$578$	15.2657	0.634968
$579$	18.3526	0.762708
$580$	2.95002	0.122493
$581$	−0.285256	−0.0118344
$582$	13.6016	0.563806
$583$	0	0
$584$	1.33071	0.0550650
$585$	8.02435	0.331766
$586$	8.05126	0.332595
$587$	25.9219	1.06991	0.534956	−	0.844880i	$-0.320328\pi$
$587$	25.9219	1.06991	0.534956	+	0.844880i	$0.320328\pi$
$588$	−0.919131	−0.0379043
$589$	−20.8693	−0.859904
$590$	−2.83905	−0.116882
$591$	19.4596	0.800460
$592$	−8.04870	−0.330799
$593$	2.58469	0.106140	0.0530702	−	0.998591i	$-0.483099\pi$
$593$	2.58469	0.106140	0.0530702	+	0.998591i	$0.483099\pi$
$594$	0	0
$595$	−1.31694	−0.0539892
$596$	5.85489	0.239826
$597$	−7.60922	−0.311425
$598$	7.44651	0.304510
$599$	8.01529	0.327496	0.163748	−	0.986502i	$-0.447642\pi$
$599$	8.01529	0.327496	0.163748	+	0.986502i	$0.447642\pi$
$600$	0.919131	0.0375234
$601$	−1.07488	−0.0438454	−0.0219227	−	0.999760i	$-0.506979\pi$
$601$	−1.07488	−0.0438454	−0.0219227	+	0.999760i	$0.506979\pi$
$602$	6.48064	0.264131
$603$	14.4558	0.588686
$604$	10.2243	0.416020
$605$	0	0
$606$	−12.9597	−0.526452
$607$	−12.6752	−0.514472	−0.257236	−	0.966349i	$-0.582812\pi$
$607$	−12.6752	−0.514472	−0.257236	+	0.966349i	$0.582812\pi$
$608$	2.70741	0.109800
$609$	−2.71145	−0.109874
$610$	1.44651	0.0585673
$611$	−34.8902	−1.41151
$612$	2.83826	0.114730
$613$	21.0929	0.851935	0.425968	−	0.904738i	$-0.359934\pi$
$613$	21.0929	0.851935	0.425968	+	0.904738i	$0.359934\pi$
$614$	−27.5963	−1.11370
$615$	2.98219	0.120254
$616$	0	0
$617$	−15.4070	−0.620263	−0.310131	−	0.950694i	$-0.600373\pi$
$617$	−15.4070	−0.620263	−0.310131	+	0.950694i	$0.600373\pi$
$618$	10.8114	0.434899
$619$	18.6431	0.749328	0.374664	−	0.927161i	$-0.377758\pi$
$619$	18.6431	0.749328	0.374664	+	0.927161i	$0.377758\pi$
$620$	−7.70820	−0.309569
$621$	−9.47660	−0.380283
$622$	−29.7870	−1.19435
$623$	8.14015	0.326128
$624$	−3.42216	−0.136996
$625$	1.00000	0.0400000
$626$	−30.9153	−1.23562
$627$	0	0
$628$	1.18004	0.0470886
$629$	10.5996	0.422635
$630$	2.15520	0.0858652
$631$	27.4709	1.09360	0.546799	−	0.837264i	$-0.315846\pi$
$631$	27.4709	1.09360	0.546799	+	0.837264i	$0.315846\pi$
$632$	7.72325	0.307214
$633$	14.3303	0.569580
$634$	2.32189	0.0922142
$635$	−6.50223	−0.258033
$636$	−12.2104	−0.484175
$637$	3.72325	0.147521
$638$	0	0
$639$	−17.3141	−0.684936
$640$	1.00000	0.0395285
$641$	−32.4224	−1.28061	−0.640304	−	0.768121i	$-0.721192\pi$
$641$	−32.4224	−1.28061	−0.640304	+	0.768121i	$0.721192\pi$
$642$	18.2238	0.719235
$643$	1.62945	0.0642591	0.0321296	−	0.999484i	$-0.489771\pi$
$643$	1.62945	0.0642591	0.0321296	+	0.999484i	$0.489771\pi$
$644$	2.00000	0.0788110
$645$	5.95656	0.234539
$646$	−3.56549	−0.140282
$647$	36.2914	1.42676	0.713382	−	0.700775i	$-0.247163\pi$
$647$	36.2914	1.42676	0.713382	+	0.700775i	$0.247163\pi$
$648$	−2.11048	−0.0829074
$649$	0	0
$650$	−3.72325	−0.146038
$651$	7.08485	0.277677
$652$	13.5818	0.531905
$653$	−32.1275	−1.25725	−0.628623	−	0.777710i	$-0.716381\pi$
$653$	−32.1275	−1.25725	−0.628623	+	0.777710i	$0.716381\pi$
$654$	−0.915025	−0.0357803
$655$	21.7622	0.850318
$656$	3.24458	0.126679
$657$	2.86793	0.111889
$658$	−9.37090	−0.365316
$659$	24.7215	0.963012	0.481506	−	0.876443i	$-0.340090\pi$
$659$	24.7215	0.963012	0.481506	+	0.876443i	$0.340090\pi$
$660$	0	0
$661$	−24.3370	−0.946598	−0.473299	−	0.880902i	$-0.656937\pi$
$661$	−24.3370	−0.946598	−0.473299	+	0.880902i	$0.656937\pi$
$662$	9.95686	0.386984
$663$	4.50676	0.175028
$664$	−0.285256	−0.0110701
$665$	−2.70741	−0.104989
$666$	−17.3465	−0.672165
$667$	5.90004	0.228450
$668$	−12.0000	−0.464294
$669$	15.2864	0.591004
$670$	−6.70741	−0.259130
$671$	0	0
$672$	−0.919131	−0.0354562
$673$	−43.7873	−1.68787	−0.843937	−	0.536442i	$-0.819768\pi$
$673$	−43.7873	−1.68787	−0.843937	+	0.536442i	$0.819768\pi$
$674$	20.7682	0.799962
$675$	4.73830	0.182377
$676$	0.862611	0.0331773
$677$	20.9199	0.804018	0.402009	−	0.915636i	$-0.368312\pi$
$677$	20.9199	0.804018	0.402009	+	0.915636i	$0.368312\pi$
$678$	10.1739	0.390727
$679$	−14.7984	−0.567909
$680$	−1.31694	−0.0505022
$681$	1.10035	0.0421653
$682$	0	0
$683$	−6.79955	−0.260178	−0.130089	−	0.991502i	$-0.541526\pi$
$683$	−6.79955	−0.260178	−0.130089	+	0.991502i	$0.541526\pi$
$684$	5.83501	0.223107
$685$	2.90408	0.110959
$686$	1.00000	0.0381802
$687$	−13.6595	−0.521143
$688$	6.48064	0.247072
$689$	49.4625	1.88437
$690$	1.83826	0.0699814
$691$	33.0127	1.25586	0.627931	−	0.778269i	$-0.283902\pi$
$691$	33.0127	1.25586	0.627931	+	0.778269i	$0.283902\pi$
$692$	−12.4741	−0.474194
$693$	0	0
$694$	15.7606	0.598263
$695$	1.26775	0.0480885
$696$	−2.71145	−0.102777
$697$	−4.27290	−0.161848
$698$	23.2953	0.881740
$699$	−5.81992	−0.220130
$700$	−1.00000	−0.0377964
$701$	−34.2118	−1.29216	−0.646081	−	0.763269i	$-0.723593\pi$
$701$	−34.2118	−1.29216	−0.646081	+	0.763269i	$0.723593\pi$
$702$	−17.6419	−0.665850
$703$	21.7911	0.821869
$704$	0	0
$705$	−8.61308	−0.324387
$706$	−27.1303	−1.02106
$707$	14.1000	0.530284
$708$	2.60946	0.0980695
$709$	−10.7815	−0.404909	−0.202455	−	0.979292i	$-0.564892\pi$
$709$	−10.7815	−0.404909	−0.202455	+	0.979292i	$0.564892\pi$
$710$	8.03365	0.301498
$711$	16.6451	0.624242
$712$	8.14015	0.305065
$713$	−15.4164	−0.577349
$714$	1.21044	0.0452995
$715$	0	0
$716$	12.2295	0.457039
$717$	−10.7452	−0.401286
$718$	32.0637	1.19661
$719$	−48.7392	−1.81767	−0.908833	−	0.417160i	$-0.863026\pi$
$719$	−48.7392	−1.81767	−0.908833	+	0.417160i	$0.863026\pi$
$720$	2.15520	0.0803195
$721$	−11.7627	−0.438064
$722$	11.6699	0.434309
$723$	9.55386	0.355312
$724$	20.9718	0.779411
$725$	−2.95002	−0.109561
$726$	0	0
$727$	11.9416	0.442891	0.221446	−	0.975173i	$-0.428922\pi$
$727$	11.9416	0.442891	0.221446	+	0.975173i	$0.428922\pi$
$728$	3.72325	0.137993
$729$	8.51686	0.315439
$730$	−1.33071	−0.0492516
$731$	−8.53460	−0.315664
$732$	−1.32953	−0.0491408
$733$	−38.9332	−1.43803	−0.719015	−	0.694995i	$-0.755407\pi$
$733$	−38.9332	−1.43803	−0.719015	+	0.694995i	$0.755407\pi$
$734$	11.6037	0.428301
$735$	0.919131	0.0339026
$736$	2.00000	0.0737210
$737$	0	0
$738$	6.99271	0.257405
$739$	−29.3583	−1.07996	−0.539981	−	0.841677i	$-0.681569\pi$
$739$	−29.3583	−1.07996	−0.539981	+	0.841677i	$0.681569\pi$
$740$	8.04870	0.295876
$741$	9.26519	0.340365
$742$	13.2848	0.487699
$743$	−27.1351	−0.995491	−0.497746	−	0.867323i	$-0.665839\pi$
$743$	−27.1351	−0.995491	−0.497746	+	0.867323i	$0.665839\pi$
$744$	7.08485	0.259743
$745$	−5.85489	−0.214507
$746$	−17.7252	−0.648966
$747$	−0.614784	−0.0224938
$748$	0	0
$749$	−19.8272	−0.724470
$750$	−0.919131	−0.0335619
$751$	−36.8507	−1.34470	−0.672350	−	0.740233i	$-0.734715\pi$
$751$	−36.8507	−1.34470	−0.672350	+	0.740233i	$0.734715\pi$
$752$	−9.37090	−0.341721
$753$	20.5329	0.748262
$754$	10.9837	0.400002
$755$	−10.2243	−0.372099
$756$	−4.73830	−0.172330
$757$	1.04667	0.0380417	0.0190209	−	0.999819i	$-0.493945\pi$
$757$	1.04667	0.0380417	0.0190209	+	0.999819i	$0.493945\pi$
$758$	12.0540	0.437822
$759$	0	0
$760$	−2.70741	−0.0982082
$761$	−2.62959	−0.0953227	−0.0476614	−	0.998864i	$-0.515177\pi$
$761$	−2.62959	−0.0953227	−0.0476614	+	0.998864i	$0.515177\pi$
$762$	5.97640	0.216502
$763$	0.995533	0.0360407
$764$	6.74632	0.244073
$765$	−2.83826	−0.102618
$766$	−6.75538	−0.244082
$767$	−10.5705	−0.381679
$768$	−0.919131	−0.0331663
$769$	−31.0270	−1.11886	−0.559431	−	0.828877i	$-0.688980\pi$
$769$	−31.0270	−1.11886	−0.559431	+	0.828877i	$0.688980\pi$
$770$	0	0
$771$	4.85485	0.174843
$772$	−19.9673	−0.718640
$773$	−8.84786	−0.318236	−0.159118	−	0.987260i	$-0.550865\pi$
$773$	−8.84786	−0.318236	−0.159118	+	0.987260i	$0.550865\pi$
$774$	13.9671	0.502036
$775$	7.70820	0.276887
$776$	−14.7984	−0.531231
$777$	−7.39781	−0.265395
$778$	24.1441	0.865609
$779$	−8.78441	−0.314734
$780$	3.42216	0.122533
$781$	0	0
$782$	−2.63387	−0.0941872
$783$	−13.9781	−0.499536
$784$	1.00000	0.0357143
$785$	−1.18004	−0.0421173
$786$	−20.0023	−0.713457
$787$	−6.15195	−0.219293	−0.109647	−	0.993971i	$-0.534972\pi$
$787$	−6.15195	−0.219293	−0.109647	+	0.993971i	$0.534972\pi$
$788$	−21.1717	−0.754212
$789$	23.3319	0.830637
$790$	−7.72325	−0.274781
$791$	−11.0691	−0.393571
$792$	0	0
$793$	5.38571	0.191252
$794$	−10.5163	−0.373210
$795$	12.2104	0.433060
$796$	8.27872	0.293431
$797$	−47.3736	−1.67806	−0.839030	−	0.544085i	$-0.816877\pi$
$797$	−47.3736	−1.67806	−0.839030	+	0.544085i	$0.816877\pi$
$798$	2.48847	0.0880907
$799$	12.3409	0.436589
$800$	−1.00000	−0.0353553
$801$	17.5436	0.619874
$802$	37.1997	1.31357
$803$	0	0
$804$	6.16499	0.217422
$805$	−2.00000	−0.0704907
$806$	−28.6996	−1.01090
$807$	14.1346	0.497562
$808$	14.1000	0.496035
$809$	−9.79837	−0.344492	−0.172246	−	0.985054i	$-0.555102\pi$
$809$	−9.79837	−0.344492	−0.172246	+	0.985054i	$0.555102\pi$
$810$	2.11048	0.0741546
$811$	−20.9999	−0.737406	−0.368703	−	0.929547i	$-0.620198\pi$
$811$	−20.9999	−0.737406	−0.368703	+	0.929547i	$0.620198\pi$
$812$	2.95002	0.103525
$813$	9.49224	0.332908
$814$	0	0
$815$	−13.5818	−0.475750
$816$	1.21044	0.0423738
$817$	−17.5458	−0.613849
$818$	26.3908	0.922732
$819$	8.02435	0.280393
$820$	−3.24458	−0.113306
$821$	−48.1483	−1.68039	−0.840194	−	0.542286i	$-0.817559\pi$
$821$	−48.1483	−1.68039	−0.840194	+	0.542286i	$0.817559\pi$
$822$	−2.66923	−0.0931001
$823$	−35.5348	−1.23867	−0.619333	−	0.785128i	$-0.712597\pi$
$823$	−35.5348	−1.23867	−0.619333	+	0.785128i	$0.712597\pi$
$824$	−11.7627	−0.409771
$825$	0	0
$826$	−2.83905	−0.0987833
$827$	−44.0977	−1.53343	−0.766714	−	0.641988i	$-0.778110\pi$
$827$	−44.0977	−1.53343	−0.766714	+	0.641988i	$0.778110\pi$
$828$	4.31040	0.149797
$829$	−6.77860	−0.235430	−0.117715	−	0.993047i	$-0.537557\pi$
$829$	−6.77860	−0.235430	−0.117715	+	0.993047i	$0.537557\pi$
$830$	0.285256	0.00990139
$831$	−20.9207	−0.725730
$832$	3.72325	0.129081
$833$	−1.31694	−0.0456292
$834$	−1.16523	−0.0403485
$835$	12.0000	0.415277
$836$	0	0
$837$	36.5238	1.26245
$838$	−13.0447	−0.450620
$839$	−53.6528	−1.85230	−0.926150	−	0.377155i	$-0.876902\pi$
$839$	−53.6528	−1.85230	−0.926150	+	0.377155i	$0.876902\pi$
$840$	0.919131	0.0317130
$841$	−20.2974	−0.699910
$842$	13.4510	0.463553
$843$	−2.82991	−0.0974673
$844$	−15.5912	−0.536671
$845$	−0.862611	−0.0296747
$846$	−20.1961	−0.694358
$847$	0	0
$848$	13.2848	0.456201
$849$	−10.0913	−0.346331
$850$	1.31694	0.0451706
$851$	16.0974	0.551812
$852$	−7.38397	−0.252971
$853$	−8.65701	−0.296410	−0.148205	−	0.988957i	$-0.547350\pi$
$853$	−8.65701	−0.296410	−0.148205	+	0.988957i	$0.547350\pi$
$854$	1.44651	0.0494984
$855$	−5.83501	−0.199553
$856$	−19.8272	−0.677679
$857$	−46.7535	−1.59707	−0.798534	−	0.601950i	$-0.794391\pi$
$857$	−46.7535	−1.59707	−0.798534	+	0.601950i	$0.794391\pi$
$858$	0	0
$859$	−35.5377	−1.21253	−0.606265	−	0.795263i	$-0.707333\pi$
$859$	−35.5377	−1.21253	−0.606265	+	0.795263i	$0.707333\pi$
$860$	−6.48064	−0.220988
$861$	2.98219	0.101633
$862$	32.7451	1.11530
$863$	40.0753	1.36418	0.682089	−	0.731269i	$-0.261072\pi$
$863$	40.0753	1.36418	0.682089	+	0.731269i	$0.261072\pi$
$864$	−4.73830	−0.161200
$865$	12.4741	0.424132
$866$	−9.00434	−0.305980
$867$	14.0312	0.476523
$868$	−7.70820	−0.261633
$869$	0	0
$870$	2.71145	0.0919269
$871$	−24.9734	−0.846191
$872$	0.995533	0.0337130
$873$	−31.8934	−1.07943
$874$	−5.41482	−0.183159
$875$	1.00000	0.0338062
$876$	1.22309	0.0413244
$877$	37.0839	1.25223	0.626117	−	0.779729i	$-0.284643\pi$
$877$	37.0839	1.25223	0.626117	+	0.779729i	$0.284643\pi$
$878$	3.88952	0.131265
$879$	7.40016	0.249601
$880$	0	0
$881$	−24.7764	−0.834737	−0.417369	−	0.908737i	$-0.637048\pi$
$881$	−24.7764	−0.834737	−0.417369	+	0.908737i	$0.637048\pi$
$882$	2.15520	0.0725693
$883$	−5.49721	−0.184996	−0.0924980	−	0.995713i	$-0.529485\pi$
$883$	−5.49721	−0.184996	−0.0924980	+	0.995713i	$0.529485\pi$
$884$	−4.90329	−0.164915
$885$	−2.60946	−0.0877161
$886$	−5.63377	−0.189270
$887$	22.5044	0.755625	0.377813	−	0.925882i	$-0.376676\pi$
$887$	22.5044	0.755625	0.377813	+	0.925882i	$0.376676\pi$
$888$	−7.39781	−0.248254
$889$	−6.50223	−0.218078
$890$	−8.14015	−0.272858
$891$	0	0
$892$	−16.6313	−0.556858
$893$	25.3709	0.849004
$894$	5.38141	0.179981
$895$	−12.2295	−0.408788
$896$	1.00000	0.0334077
$897$	6.84431	0.228525
$898$	5.62855	0.187827
$899$	−22.7394	−0.758400
$900$	−2.15520	−0.0718400
$901$	−17.4952	−0.582850
$902$	0	0
$903$	5.95656	0.198222
$904$	−11.0691	−0.368152
$905$	−20.9718	−0.697126
$906$	9.39744	0.312209
$907$	−10.0800	−0.334702	−0.167351	−	0.985897i	$-0.553521\pi$
$907$	−10.0800	−0.334702	−0.167351	+	0.985897i	$0.553521\pi$
$908$	−1.19716	−0.0397291
$909$	30.3882	1.00791
$910$	−3.72325	−0.123425
$911$	16.3242	0.540846	0.270423	−	0.962742i	$-0.412836\pi$
$911$	16.3242	0.540846	0.270423	+	0.962742i	$0.412836\pi$
$912$	2.48847	0.0824013
$913$	0	0
$914$	1.15770	0.0382932
$915$	1.32953	0.0439528
$916$	14.8613	0.491032
$917$	21.7622	0.718650
$918$	6.24005	0.205952
$919$	18.9580	0.625366	0.312683	−	0.949858i	$-0.398772\pi$
$919$	18.9580	0.625366	0.312683	+	0.949858i	$0.398772\pi$
$920$	−2.00000	−0.0659380
$921$	−25.3646	−0.835793
$922$	10.3088	0.339503
$923$	29.9113	0.984543
$924$	0	0
$925$	−8.04870	−0.264640
$926$	29.4740	0.968577
$927$	−25.3509	−0.832631
$928$	2.95002	0.0968392
$929$	−15.5873	−0.511404	−0.255702	−	0.966756i	$-0.582307\pi$
$929$	−15.5873	−0.511404	−0.255702	+	0.966756i	$0.582307\pi$
$930$	−7.08485	−0.232321
$931$	−2.70741	−0.0887319
$932$	6.33199	0.207411
$933$	−27.3781	−0.896320
$934$	−20.5977	−0.673976
$935$	0	0
$936$	8.02435	0.262284
$937$	32.7215	1.06896	0.534482	−	0.845180i	$-0.320507\pi$
$937$	32.7215	1.06896	0.534482	+	0.845180i	$0.320507\pi$
$938$	−6.70741	−0.219005
$939$	−28.4152	−0.927295
$940$	9.37090	0.305645
$941$	−14.5193	−0.473314	−0.236657	−	0.971593i	$-0.576052\pi$
$941$	−14.5193	−0.473314	−0.236657	+	0.971593i	$0.576052\pi$
$942$	1.08461	0.0353385
$943$	−6.48915	−0.211316
$944$	−2.83905	−0.0924033
$945$	4.73830	0.154137
$946$	0	0
$947$	16.5854	0.538954	0.269477	−	0.963007i	$-0.413149\pi$
$947$	16.5854	0.538954	0.269477	+	0.963007i	$0.413149\pi$
$948$	7.09868	0.230554
$949$	−4.95455	−0.160832
$950$	2.70741	0.0878401
$951$	2.13412	0.0692037
$952$	−1.31694	−0.0426822
$953$	29.5354	0.956745	0.478373	−	0.878157i	$-0.341227\pi$
$953$	29.5354	0.956745	0.478373	+	0.878157i	$0.341227\pi$
$954$	28.6313	0.926973
$955$	−6.74632	−0.218306
$956$	11.6906	0.378101
$957$	0	0
$958$	−4.18526	−0.135220
$959$	2.90408	0.0937777
$960$	0.919131	0.0296648
$961$	28.4164	0.916658
$962$	29.9673	0.966186
$963$	−42.7315	−1.37700
$964$	−10.3945	−0.334783
$965$	19.9673	0.642771
$966$	1.83826	0.0591451
$967$	36.7434	1.18159	0.590794	−	0.806823i	$-0.298815\pi$
$967$	36.7434	1.18159	0.590794	+	0.806823i	$0.298815\pi$
$968$	0	0
$969$	−3.27715	−0.105277
$970$	14.7984	0.475147
$971$	−22.2877	−0.715245	−0.357622	−	0.933866i	$-0.616412\pi$
$971$	−22.2877	−0.715245	−0.357622	+	0.933866i	$0.616412\pi$
$972$	−16.1547	−0.518163
$973$	1.26775	0.0406422
$974$	8.06730	0.258493
$975$	−3.42216	−0.109597
$976$	1.44651	0.0463015
$977$	−40.5549	−1.29747	−0.648733	−	0.761016i	$-0.724701\pi$
$977$	−40.5549	−1.29747	−0.648733	+	0.761016i	$0.724701\pi$
$978$	12.4835	0.399177
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	2.14557	0.0685028
$982$	−23.5882	−0.752731
$983$	−39.8706	−1.27167	−0.635837	−	0.771824i	$-0.719345\pi$
$983$	−39.8706	−1.27167	−0.635837	+	0.771824i	$0.719345\pi$
$984$	2.98219	0.0950688
$985$	21.1717	0.674587
$986$	−3.88499	−0.123723
$987$	−8.61308	−0.274157
$988$	−10.0804	−0.320700
$989$	−12.9613	−0.412145
$990$	0	0
$991$	35.5410	1.12900	0.564498	−	0.825435i	$-0.309070\pi$
$991$	35.5410	1.12900	0.564498	+	0.825435i	$0.309070\pi$
$992$	−7.70820	−0.244736
$993$	9.15166	0.290419
$994$	8.03365	0.254812
$995$	−8.27872	−0.262453
$996$	−0.262188	−0.00830774
$997$	47.4408	1.50246	0.751232	−	0.660038i	$-0.229460\pi$
$997$	47.4408	1.50246	0.751232	+	0.660038i	$0.229460\pi$
$998$	−7.22181	−0.228602
$999$	−38.1372	−1.20661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cq.1.2		4
11.3	even	5		770.2.n.e.141.2	yes	8
11.4	even	5		770.2.n.e.71.2	✓	8
11.10	odd	2		8470.2.a.ct.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.e.71.2	✓	8	11.4	even	5
770.2.n.e.141.2	yes	8	11.3	even	5
8470.2.a.cq.1.2		4	1.1	even	1	trivial
8470.2.a.ct.1.2		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.919131 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.919131 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.15520 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.919131 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.919131 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.15520 q^{9} +1.00000 q^{10} -0.919131 q^{12} +3.72325 q^{13} +1.00000 q^{14} +0.919131 q^{15} +1.00000 q^{16} -1.31694 q^{17} +2.15520 q^{18} -2.70741 q^{19} -1.00000 q^{20} +0.919131 q^{21} -2.00000 q^{23} +0.919131 q^{24} +1.00000 q^{25} -3.72325 q^{26} +4.73830 q^{27} -1.00000 q^{28} -2.95002 q^{29} -0.919131 q^{30} +7.70820 q^{31} -1.00000 q^{32} +1.31694 q^{34} +1.00000 q^{35} -2.15520 q^{36} -8.04870 q^{37} +2.70741 q^{38} -3.42216 q^{39} +1.00000 q^{40} +3.24458 q^{41} -0.919131 q^{42} +6.48064 q^{43} +2.15520 q^{45} +2.00000 q^{46} -9.37090 q^{47} -0.919131 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.21044 q^{51} +3.72325 q^{52} +13.2848 q^{53} -4.73830 q^{54} +1.00000 q^{56} +2.48847 q^{57} +2.95002 q^{58} -2.83905 q^{59} +0.919131 q^{60} +1.44651 q^{61} -7.70820 q^{62} +2.15520 q^{63} +1.00000 q^{64} -3.72325 q^{65} -6.70741 q^{67} -1.31694 q^{68} +1.83826 q^{69} -1.00000 q^{70} +8.03365 q^{71} +2.15520 q^{72} -1.33071 q^{73} +8.04870 q^{74} -0.919131 q^{75} -2.70741 q^{76} +3.42216 q^{78} -7.72325 q^{79} -1.00000 q^{80} +2.11048 q^{81} -3.24458 q^{82} +0.285256 q^{83} +0.919131 q^{84} +1.31694 q^{85} -6.48064 q^{86} +2.71145 q^{87} -8.14015 q^{89} -2.15520 q^{90} -3.72325 q^{91} -2.00000 q^{92} -7.08485 q^{93} +9.37090 q^{94} +2.70741 q^{95} +0.919131 q^{96} +14.7984 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} + 4 q^{10} + 2 q^{12} - q^{13} + 4 q^{14} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 6 q^{18} + 3 q^{19} - 4 q^{20} - 2 q^{21} - 8 q^{23} - 2 q^{24} + 4 q^{25} + q^{26} + 14 q^{27} - 4 q^{28} - 15 q^{29} + 2 q^{30} + 4 q^{31} - 4 q^{32} + 2 q^{34} + 4 q^{35} + 6 q^{36} + 2 q^{37} - 3 q^{38} + q^{39} + 4 q^{40} - 11 q^{41} + 2 q^{42} - 7 q^{43} - 6 q^{45} + 8 q^{46} + 5 q^{47} + 2 q^{48} + 4 q^{49} - 4 q^{50} - 18 q^{51} - q^{52} + 10 q^{53} - 14 q^{54} + 4 q^{56} + 34 q^{57} + 15 q^{58} + 13 q^{59} - 2 q^{60} - 26 q^{61} - 4 q^{62} - 6 q^{63} + 4 q^{64} + q^{65} - 13 q^{67} - 2 q^{68} - 4 q^{69} - 4 q^{70} - 13 q^{71} - 6 q^{72} + 18 q^{73} - 2 q^{74} + 2 q^{75} + 3 q^{76} - q^{78} - 15 q^{79} - 4 q^{80} - 8 q^{81} + 11 q^{82} + 2 q^{83} - 2 q^{84} + 2 q^{85} + 7 q^{86} - 26 q^{87} - 7 q^{89} + 6 q^{90} + q^{91} - 8 q^{92} + 2 q^{93} - 5 q^{94} - 3 q^{95} - 2 q^{96} + 10 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 + 4 * q^10 + 2 * q^12 - q^13 + 4 * q^14 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 6 * q^18 + 3 * q^19 - 4 * q^20 - 2 * q^21 - 8 * q^23 - 2 * q^24 + 4 * q^25 + q^26 + 14 * q^27 - 4 * q^28 - 15 * q^29 + 2 * q^30 + 4 * q^31 - 4 * q^32 + 2 * q^34 + 4 * q^35 + 6 * q^36 + 2 * q^37 - 3 * q^38 + q^39 + 4 * q^40 - 11 * q^41 + 2 * q^42 - 7 * q^43 - 6 * q^45 + 8 * q^46 + 5 * q^47 + 2 * q^48 + 4 * q^49 - 4 * q^50 - 18 * q^51 - q^52 + 10 * q^53 - 14 * q^54 + 4 * q^56 + 34 * q^57 + 15 * q^58 + 13 * q^59 - 2 * q^60 - 26 * q^61 - 4 * q^62 - 6 * q^63 + 4 * q^64 + q^65 - 13 * q^67 - 2 * q^68 - 4 * q^69 - 4 * q^70 - 13 * q^71 - 6 * q^72 + 18 * q^73 - 2 * q^74 + 2 * q^75 + 3 * q^76 - q^78 - 15 * q^79 - 4 * q^80 - 8 * q^81 + 11 * q^82 + 2 * q^83 - 2 * q^84 + 2 * q^85 + 7 * q^86 - 26 * q^87 - 7 * q^89 + 6 * q^90 + q^91 - 8 * q^92 + 2 * q^93 - 5 * q^94 - 3 * q^95 - 2 * q^96 + 10 * q^97 - 4 * q^98

Introduction

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