LMFDB - Embedded newform 8470.2.a.dh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} + 69x^{4} - 10x^{3} - 70x^{2} + 10x + 5 $ x^8 - 16x^6 + 69x^4 - 10x^3 - 70x^2 + 10*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.12455$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.12455 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.73538 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.12455 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.73538 q^{9} -1.00000 q^{10} -1.12455 q^{12} +7.06945 q^{13} -1.00000 q^{14} +1.12455 q^{15} +1.00000 q^{16} +3.44477 q^{17} -1.73538 q^{18} +3.43616 q^{19} -1.00000 q^{20} +1.12455 q^{21} +3.63913 q^{23} -1.12455 q^{24} +1.00000 q^{25} +7.06945 q^{26} +5.32519 q^{27} -1.00000 q^{28} -8.69298 q^{29} +1.12455 q^{30} -4.52145 q^{31} +1.00000 q^{32} +3.44477 q^{34} +1.00000 q^{35} -1.73538 q^{36} -0.780576 q^{37} +3.43616 q^{38} -7.94999 q^{39} -1.00000 q^{40} -5.01886 q^{41} +1.12455 q^{42} -5.39500 q^{43} +1.73538 q^{45} +3.63913 q^{46} +9.82775 q^{47} -1.12455 q^{48} +1.00000 q^{49} +1.00000 q^{50} -3.87383 q^{51} +7.06945 q^{52} +11.3718 q^{53} +5.32519 q^{54} -1.00000 q^{56} -3.86415 q^{57} -8.69298 q^{58} -10.5247 q^{59} +1.12455 q^{60} +0.456235 q^{61} -4.52145 q^{62} +1.73538 q^{63} +1.00000 q^{64} -7.06945 q^{65} +10.4937 q^{67} +3.44477 q^{68} -4.09241 q^{69} +1.00000 q^{70} -6.81376 q^{71} -1.73538 q^{72} -15.1151 q^{73} -0.780576 q^{74} -1.12455 q^{75} +3.43616 q^{76} -7.94999 q^{78} -16.2368 q^{79} -1.00000 q^{80} -0.782335 q^{81} -5.01886 q^{82} +7.40344 q^{83} +1.12455 q^{84} -3.44477 q^{85} -5.39500 q^{86} +9.77573 q^{87} -4.97143 q^{89} +1.73538 q^{90} -7.06945 q^{91} +3.63913 q^{92} +5.08461 q^{93} +9.82775 q^{94} -3.43616 q^{95} -1.12455 q^{96} +2.88993 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^5 - 8 * q^7 + 8 * q^8 + 8 * q^9	$ 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - q^{13} - 8 q^{14} + 8 q^{16} + 6 q^{17} + 8 q^{18} + 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} - 8 q^{28} + 3 q^{29} - 8 q^{31} + 8 q^{32} + 6 q^{34} + 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} + 35 q^{39} - 8 q^{40} + 11 q^{41} - 5 q^{43} - 8 q^{45} + 10 q^{46} - 15 q^{47} + 8 q^{49} + 8 q^{50} - 6 q^{51} - q^{52} - 16 q^{53} - 8 q^{56} + 38 q^{57} + 3 q^{58} - 9 q^{59} + 32 q^{61} - 8 q^{62} - 8 q^{63} + 8 q^{64} + q^{65} + 33 q^{67} + 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} + 8 q^{72} - 34 q^{73} - 6 q^{74} + 5 q^{76} + 35 q^{78} + 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} + 50 q^{83} - 6 q^{85} - 5 q^{86} - 12 q^{87} + q^{89} - 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} - 15 q^{94} - 5 q^{95} - 4 q^{97} + 8 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^5 - 8 * q^7 + 8 * q^8 + 8 * q^9 - 8 * q^10 - q^13 - 8 * q^14 + 8 * q^16 + 6 * q^17 + 8 * q^18 + 5 * q^19 - 8 * q^20 + 10 * q^23 + 8 * q^25 - q^26 - 8 * q^28 + 3 * q^29 - 8 * q^31 + 8 * q^32 + 6 * q^34 + 8 * q^35 + 8 * q^36 - 6 * q^37 + 5 * q^38 + 35 * q^39 - 8 * q^40 + 11 * q^41 - 5 * q^43 - 8 * q^45 + 10 * q^46 - 15 * q^47 + 8 * q^49 + 8 * q^50 - 6 * q^51 - q^52 - 16 * q^53 - 8 * q^56 + 38 * q^57 + 3 * q^58 - 9 * q^59 + 32 * q^61 - 8 * q^62 - 8 * q^63 + 8 * q^64 + q^65 + 33 * q^67 + 6 * q^68 - 22 * q^69 + 8 * q^70 + 11 * q^71 + 8 * q^72 - 34 * q^73 - 6 * q^74 + 5 * q^76 + 35 * q^78 + 31 * q^79 - 8 * q^80 + 20 * q^81 + 11 * q^82 + 50 * q^83 - 6 * q^85 - 5 * q^86 - 12 * q^87 + q^89 - 8 * q^90 + q^91 + 10 * q^92 + 26 * q^93 - 15 * q^94 - 5 * q^95 - 4 * q^97 + 8 * 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
$3$	−1.12455	−0.649262	−0.324631	−	0.945841i	$-0.605240\pi$
$3$	−1.12455	−0.649262	−0.324631	+	0.945841i	$0.605240\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.12455	−0.459097
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−1.73538	−0.578459
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−1.12455	−0.324631
$13$	7.06945	1.96071	0.980357	−	0.197232i	$-0.0631953\pi$
$13$	7.06945	1.96071	0.980357	+	0.197232i	$0.0631953\pi$
$14$	−1.00000	−0.267261
$15$	1.12455	0.290359
$16$	1.00000	0.250000
$17$	3.44477	0.835479	0.417740	−	0.908567i	$-0.362822\pi$
$17$	3.44477	0.835479	0.417740	+	0.908567i	$0.362822\pi$
$18$	−1.73538	−0.409032
$19$	3.43616	0.788308	0.394154	−	0.919044i	$-0.371038\pi$
$19$	3.43616	0.788308	0.394154	+	0.919044i	$0.371038\pi$
$20$	−1.00000	−0.223607
$21$	1.12455	0.245398
$22$	0	0
$23$	3.63913	0.758812	0.379406	−	0.925230i	$-0.376128\pi$
$23$	3.63913	0.758812	0.379406	+	0.925230i	$0.376128\pi$
$24$	−1.12455	−0.229549
$25$	1.00000	0.200000
$26$	7.06945	1.38643
$27$	5.32519	1.02483
$28$	−1.00000	−0.188982
$29$	−8.69298	−1.61425	−0.807123	−	0.590383i	$-0.798977\pi$
$29$	−8.69298	−1.61425	−0.807123	+	0.590383i	$0.798977\pi$
$30$	1.12455	0.205315
$31$	−4.52145	−0.812076	−0.406038	−	0.913856i	$-0.633090\pi$
$31$	−4.52145	−0.812076	−0.406038	+	0.913856i	$0.633090\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.44477	0.590773
$35$	1.00000	0.169031
$36$	−1.73538	−0.289230
$37$	−0.780576	−0.128326	−0.0641630	−	0.997939i	$-0.520438\pi$
$37$	−0.780576	−0.128326	−0.0641630	+	0.997939i	$0.520438\pi$
$38$	3.43616	0.557418
$39$	−7.94999	−1.27302
$40$	−1.00000	−0.158114
$41$	−5.01886	−0.783814	−0.391907	−	0.920005i	$-0.628185\pi$
$41$	−5.01886	−0.783814	−0.391907	+	0.920005i	$0.628185\pi$
$42$	1.12455	0.173523
$43$	−5.39500	−0.822729	−0.411365	−	0.911471i	$-0.634948\pi$
$43$	−5.39500	−0.822729	−0.411365	+	0.911471i	$0.634948\pi$
$44$	0	0
$45$	1.73538	0.258695
$46$	3.63913	0.536561
$47$	9.82775	1.43352	0.716762	−	0.697318i	$-0.245623\pi$
$47$	9.82775	1.43352	0.716762	+	0.697318i	$0.245623\pi$
$48$	−1.12455	−0.162315
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−3.87383	−0.542445
$52$	7.06945	0.980357
$53$	11.3718	1.56204	0.781020	−	0.624507i	$-0.214700\pi$
$53$	11.3718	1.56204	0.781020	+	0.624507i	$0.214700\pi$
$54$	5.32519	0.724667
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−3.86415	−0.511819
$58$	−8.69298	−1.14144
$59$	−10.5247	−1.37020	−0.685099	−	0.728450i	$-0.740241\pi$
$59$	−10.5247	−1.37020	−0.685099	+	0.728450i	$0.740241\pi$
$60$	1.12455	0.145179
$61$	0.456235	0.0584149	0.0292074	−	0.999573i	$-0.490702\pi$
$61$	0.456235	0.0584149	0.0292074	+	0.999573i	$0.490702\pi$
$62$	−4.52145	−0.574224
$63$	1.73538	0.218637
$64$	1.00000	0.125000
$65$	−7.06945	−0.876858
$66$	0	0
$67$	10.4937	1.28201	0.641006	−	0.767536i	$-0.278517\pi$
$67$	10.4937	1.28201	0.641006	+	0.767536i	$0.278517\pi$
$68$	3.44477	0.417740
$69$	−4.09241	−0.492668
$70$	1.00000	0.119523
$71$	−6.81376	−0.808645	−0.404322	−	0.914617i	$-0.632493\pi$
$71$	−6.81376	−0.808645	−0.404322	+	0.914617i	$0.632493\pi$
$72$	−1.73538	−0.204516
$73$	−15.1151	−1.76909	−0.884547	−	0.466452i	$-0.845532\pi$
$73$	−15.1151	−1.76909	−0.884547	+	0.466452i	$0.845532\pi$
$74$	−0.780576	−0.0907401
$75$	−1.12455	−0.129852
$76$	3.43616	0.394154
$77$	0	0
$78$	−7.94999	−0.900159
$79$	−16.2368	−1.82678	−0.913390	−	0.407087i	$-0.866545\pi$
$79$	−16.2368	−1.82678	−0.913390	+	0.407087i	$0.866545\pi$
$80$	−1.00000	−0.111803
$81$	−0.782335	−0.0869261
$82$	−5.01886	−0.554240
$83$	7.40344	0.812633	0.406317	−	0.913732i	$-0.366813\pi$
$83$	7.40344	0.812633	0.406317	+	0.913732i	$0.366813\pi$
$84$	1.12455	0.122699
$85$	−3.44477	−0.373638
$86$	−5.39500	−0.581757
$87$	9.77573	1.04807
$88$	0	0
$89$	−4.97143	−0.526970	−0.263485	−	0.964663i	$-0.584872\pi$
$89$	−4.97143	−0.526970	−0.263485	+	0.964663i	$0.584872\pi$
$90$	1.73538	0.182925
$91$	−7.06945	−0.741080
$92$	3.63913	0.379406
$93$	5.08461	0.527250
$94$	9.82775	1.01365
$95$	−3.43616	−0.352542
$96$	−1.12455	−0.114774
$97$	2.88993	0.293427	0.146714	−	0.989179i	$-0.453130\pi$
$97$	2.88993	0.293427	0.146714	+	0.989179i	$0.453130\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	12.2266	1.21659	0.608295	−	0.793711i	$-0.291854\pi$
$101$	12.2266	1.21659	0.608295	+	0.793711i	$0.291854\pi$
$102$	−3.87383	−0.383566
$103$	3.70633	0.365196	0.182598	−	0.983188i	$-0.441549\pi$
$103$	3.70633	0.365196	0.182598	+	0.983188i	$0.441549\pi$
$104$	7.06945	0.693217
$105$	−1.12455	−0.109745
$106$	11.3718	1.10453
$107$	−2.85107	−0.275623	−0.137812	−	0.990458i	$-0.544007\pi$
$107$	−2.85107	−0.275623	−0.137812	+	0.990458i	$0.544007\pi$
$108$	5.32519	0.512417
$109$	9.80121	0.938786	0.469393	−	0.882989i	$-0.344473\pi$
$109$	9.80121	0.938786	0.469393	+	0.882989i	$0.344473\pi$
$110$	0	0
$111$	0.877800	0.0833171
$112$	−1.00000	−0.0944911
$113$	10.8032	1.01628	0.508140	−	0.861275i	$-0.330333\pi$
$113$	10.8032	1.01628	0.508140	+	0.861275i	$0.330333\pi$
$114$	−3.86415	−0.361910
$115$	−3.63913	−0.339351
$116$	−8.69298	−0.807123
$117$	−12.2682	−1.13419
$118$	−10.5247	−0.968877
$119$	−3.44477	−0.315781
$120$	1.12455	0.102657
$121$	0	0
$122$	0.456235	0.0413055
$123$	5.64398	0.508901
$124$	−4.52145	−0.406038
$125$	−1.00000	−0.0894427
$126$	1.73538	0.154600
$127$	21.2610	1.88661	0.943303	−	0.331934i	$-0.107701\pi$
$127$	21.2610	1.88661	0.943303	+	0.331934i	$0.107701\pi$
$128$	1.00000	0.0883883
$129$	6.06697	0.534167
$130$	−7.06945	−0.620032
$131$	10.2637	0.896747	0.448373	−	0.893846i	$-0.352003\pi$
$131$	10.2637	0.896747	0.448373	+	0.893846i	$0.352003\pi$
$132$	0	0
$133$	−3.43616	−0.297953
$134$	10.4937	0.906519
$135$	−5.32519	−0.458319
$136$	3.44477	0.295386
$137$	−0.760911	−0.0650090	−0.0325045	−	0.999472i	$-0.510348\pi$
$137$	−0.760911	−0.0650090	−0.0325045	+	0.999472i	$0.510348\pi$
$138$	−4.09241	−0.348369
$139$	17.6377	1.49601	0.748007	−	0.663691i	$-0.231011\pi$
$139$	17.6377	1.49601	0.748007	+	0.663691i	$0.231011\pi$
$140$	1.00000	0.0845154
$141$	−11.0518	−0.930733
$142$	−6.81376	−0.571798
$143$	0	0
$144$	−1.73538	−0.144615
$145$	8.69298	0.721913
$146$	−15.1151	−1.25094
$147$	−1.12455	−0.0927517
$148$	−0.780576	−0.0641630
$149$	0.778180	0.0637510	0.0318755	−	0.999492i	$-0.489852\pi$
$149$	0.778180	0.0637510	0.0318755	+	0.999492i	$0.489852\pi$
$150$	−1.12455	−0.0918195
$151$	13.4154	1.09173	0.545864	−	0.837874i	$-0.316202\pi$
$151$	13.4154	1.09173	0.545864	+	0.837874i	$0.316202\pi$
$152$	3.43616	0.278709
$153$	−5.97797	−0.483290
$154$	0	0
$155$	4.52145	0.363171
$156$	−7.94999	−0.636508
$157$	−24.0361	−1.91829	−0.959146	−	0.282912i	$-0.908700\pi$
$157$	−24.0361	−1.91829	−0.959146	+	0.282912i	$0.908700\pi$
$158$	−16.2368	−1.29173
$159$	−12.7882	−1.01417
$160$	−1.00000	−0.0790569
$161$	−3.63913	−0.286804
$162$	−0.782335	−0.0614660
$163$	0.905391	0.0709157	0.0354579	−	0.999371i	$-0.488711\pi$
$163$	0.905391	0.0709157	0.0354579	+	0.999371i	$0.488711\pi$
$164$	−5.01886	−0.391907
$165$	0	0
$166$	7.40344	0.574618
$167$	−3.64082	−0.281735	−0.140867	−	0.990028i	$-0.544989\pi$
$167$	−3.64082	−0.281735	−0.140867	+	0.990028i	$0.544989\pi$
$168$	1.12455	0.0867613
$169$	36.9772	2.84440
$170$	−3.44477	−0.264202
$171$	−5.96303	−0.456004
$172$	−5.39500	−0.411365
$173$	−3.17934	−0.241721	−0.120860	−	0.992670i	$-0.538565\pi$
$173$	−3.17934	−0.241721	−0.120860	+	0.992670i	$0.538565\pi$
$174$	9.77573	0.741096
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	11.8356	0.889618
$178$	−4.97143	−0.372624
$179$	4.48137	0.334953	0.167477	−	0.985876i	$-0.446438\pi$
$179$	4.48137	0.334953	0.167477	+	0.985876i	$0.446438\pi$
$180$	1.73538	0.129347
$181$	−9.02937	−0.671148	−0.335574	−	0.942014i	$-0.608930\pi$
$181$	−9.02937	−0.671148	−0.335574	+	0.942014i	$0.608930\pi$
$182$	−7.06945	−0.524023
$183$	−0.513061	−0.0379265
$184$	3.63913	0.268281
$185$	0.780576	0.0573891
$186$	5.08461	0.372822
$187$	0	0
$188$	9.82775	0.716762
$189$	−5.32519	−0.387351
$190$	−3.43616	−0.249285
$191$	2.90870	0.210466	0.105233	−	0.994448i	$-0.466441\pi$
$191$	2.90870	0.210466	0.105233	+	0.994448i	$0.466441\pi$
$192$	−1.12455	−0.0811577
$193$	−7.77320	−0.559527	−0.279764	−	0.960069i	$-0.590256\pi$
$193$	−7.77320	−0.559527	−0.279764	+	0.960069i	$0.590256\pi$
$194$	2.88993	0.207485
$195$	7.94999	0.569310
$196$	1.00000	0.0714286
$197$	26.2257	1.86851	0.934254	−	0.356609i	$-0.116067\pi$
$197$	26.2257	1.86851	0.934254	+	0.356609i	$0.116067\pi$
$198$	0	0
$199$	5.64963	0.400491	0.200246	−	0.979746i	$-0.435826\pi$
$199$	5.64963	0.400491	0.200246	+	0.979746i	$0.435826\pi$
$200$	1.00000	0.0707107
$201$	−11.8008	−0.832361
$202$	12.2266	0.860260
$203$	8.69298	0.610128
$204$	−3.87383	−0.271222
$205$	5.01886	0.350532
$206$	3.70633	0.258233
$207$	−6.31527	−0.438942
$208$	7.06945	0.490178
$209$	0	0
$210$	−1.12455	−0.0776016
$211$	7.42973	0.511484	0.255742	−	0.966745i	$-0.417680\pi$
$211$	7.42973	0.511484	0.255742	+	0.966745i	$0.417680\pi$
$212$	11.3718	0.781020
$213$	7.66245	0.525022
$214$	−2.85107	−0.194895
$215$	5.39500	0.367936
$216$	5.32519	0.362333
$217$	4.52145	0.306936
$218$	9.80121	0.663822
$219$	16.9978	1.14860
$220$	0	0
$221$	24.3526	1.63814
$222$	0.877800	0.0589141
$223$	1.51167	0.101229	0.0506145	−	0.998718i	$-0.483882\pi$
$223$	1.51167	0.101229	0.0506145	+	0.998718i	$0.483882\pi$
$224$	−1.00000	−0.0668153
$225$	−1.73538	−0.115692
$226$	10.8032	0.718618
$227$	13.8558	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$227$	13.8558	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$228$	−3.86415	−0.255909
$229$	20.5463	1.35774	0.678870	−	0.734258i	$-0.262470\pi$
$229$	20.5463	1.35774	0.678870	+	0.734258i	$0.262470\pi$
$230$	−3.63913	−0.239957
$231$	0	0
$232$	−8.69298	−0.570722
$233$	16.2345	1.06356	0.531780	−	0.846883i	$-0.321523\pi$
$233$	16.2345	1.06356	0.531780	+	0.846883i	$0.321523\pi$
$234$	−12.2682	−0.801995
$235$	−9.82775	−0.641092
$236$	−10.5247	−0.685099
$237$	18.2591	1.18606
$238$	−3.44477	−0.223291
$239$	−0.392921	−0.0254160	−0.0127080	−	0.999919i	$-0.504045\pi$
$239$	−0.392921	−0.0254160	−0.0127080	+	0.999919i	$0.504045\pi$
$240$	1.12455	0.0725897
$241$	−3.32681	−0.214298	−0.107149	−	0.994243i	$-0.534172\pi$
$241$	−3.32681	−0.214298	−0.107149	+	0.994243i	$0.534172\pi$
$242$	0	0
$243$	−15.0958	−0.968395
$244$	0.456235	0.0292074
$245$	−1.00000	−0.0638877
$246$	5.64398	0.359847
$247$	24.2918	1.54565
$248$	−4.52145	−0.287112
$249$	−8.32557	−0.527612
$250$	−1.00000	−0.0632456
$251$	17.2755	1.09042	0.545211	−	0.838299i	$-0.316450\pi$
$251$	17.2755	1.09042	0.545211	+	0.838299i	$0.316450\pi$
$252$	1.73538	0.109318
$253$	0	0
$254$	21.2610	1.33403
$255$	3.87383	0.242589
$256$	1.00000	0.0625000
$257$	−0.448545	−0.0279795	−0.0139897	−	0.999902i	$-0.504453\pi$
$257$	−0.448545	−0.0279795	−0.0139897	+	0.999902i	$0.504453\pi$
$258$	6.06697	0.377713
$259$	0.780576	0.0485026
$260$	−7.06945	−0.438429
$261$	15.0856	0.933775
$262$	10.2637	0.634096
$263$	26.1440	1.61211	0.806053	−	0.591843i	$-0.201600\pi$
$263$	26.1440	1.61211	0.806053	+	0.591843i	$0.201600\pi$
$264$	0	0
$265$	−11.3718	−0.698565
$266$	−3.43616	−0.210684
$267$	5.59064	0.342142
$268$	10.4937	0.641006
$269$	14.1617	0.863453	0.431727	−	0.902004i	$-0.357904\pi$
$269$	14.1617	0.863453	0.431727	+	0.902004i	$0.357904\pi$
$270$	−5.32519	−0.324081
$271$	−0.934211	−0.0567493	−0.0283746	−	0.999597i	$-0.509033\pi$
$271$	−0.934211	−0.0567493	−0.0283746	+	0.999597i	$0.509033\pi$
$272$	3.44477	0.208870
$273$	7.94999	0.481155
$274$	−0.760911	−0.0459683
$275$	0	0
$276$	−4.09241	−0.246334
$277$	−25.3249	−1.52162	−0.760812	−	0.648973i	$-0.775199\pi$
$277$	−25.3249	−1.52162	−0.760812	+	0.648973i	$0.775199\pi$
$278$	17.6377	1.05784
$279$	7.84641	0.469752
$280$	1.00000	0.0597614
$281$	27.7779	1.65709	0.828545	−	0.559922i	$-0.189169\pi$
$281$	27.7779	1.65709	0.828545	+	0.559922i	$0.189169\pi$
$282$	−11.0518	−0.658127
$283$	−19.5153	−1.16006	−0.580032	−	0.814594i	$-0.696960\pi$
$283$	−19.5153	−1.16006	−0.580032	+	0.814594i	$0.696960\pi$
$284$	−6.81376	−0.404322
$285$	3.86415	0.228892
$286$	0	0
$287$	5.01886	0.296254
$288$	−1.73538	−0.102258
$289$	−5.13357	−0.301975
$290$	8.69298	0.510469
$291$	−3.24988	−0.190511
$292$	−15.1151	−0.884547
$293$	−13.3505	−0.779945	−0.389972	−	0.920827i	$-0.627515\pi$
$293$	−13.3505	−0.779945	−0.389972	+	0.920827i	$0.627515\pi$
$294$	−1.12455	−0.0655854
$295$	10.5247	0.612771
$296$	−0.780576	−0.0453701
$297$	0	0
$298$	0.778180	0.0450788
$299$	25.7267	1.48781
$300$	−1.12455	−0.0649262
$301$	5.39500	0.310962
$302$	13.4154	0.771968
$303$	−13.7495	−0.789886
$304$	3.43616	0.197077
$305$	−0.456235	−0.0261239
$306$	−5.97797	−0.341738
$307$	25.7848	1.47162	0.735809	−	0.677189i	$-0.236802\pi$
$307$	25.7848	1.47162	0.735809	+	0.677189i	$0.236802\pi$
$308$	0	0
$309$	−4.16798	−0.237108
$310$	4.52145	0.256801
$311$	5.11540	0.290068	0.145034	−	0.989427i	$-0.453671\pi$
$311$	5.11540	0.290068	0.145034	+	0.989427i	$0.453671\pi$
$312$	−7.94999	−0.450079
$313$	16.5571	0.935861	0.467930	−	0.883765i	$-0.345000\pi$
$313$	16.5571	0.935861	0.467930	+	0.883765i	$0.345000\pi$
$314$	−24.0361	−1.35644
$315$	−1.73538	−0.0977774
$316$	−16.2368	−0.913390
$317$	15.3500	0.862143	0.431072	−	0.902318i	$-0.358136\pi$
$317$	15.3500	0.862143	0.431072	+	0.902318i	$0.358136\pi$
$318$	−12.7882	−0.717128
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	3.20618	0.178952
$322$	−3.63913	−0.202801
$323$	11.8368	0.658615
$324$	−0.782335	−0.0434630
$325$	7.06945	0.392143
$326$	0.905391	0.0501450
$327$	−11.0220	−0.609518
$328$	−5.01886	−0.277120
$329$	−9.82775	−0.541821
$330$	0	0
$331$	−6.59151	−0.362302	−0.181151	−	0.983455i	$-0.557982\pi$
$331$	−6.59151	−0.362302	−0.181151	+	0.983455i	$0.557982\pi$
$332$	7.40344	0.406317
$333$	1.35459	0.0742313
$334$	−3.64082	−0.199217
$335$	−10.4937	−0.573333
$336$	1.12455	0.0613495
$337$	−19.1547	−1.04342	−0.521711	−	0.853123i	$-0.674706\pi$
$337$	−19.1547	−1.04342	−0.521711	+	0.853123i	$0.674706\pi$
$338$	36.9772	2.01129
$339$	−12.1488	−0.659832
$340$	−3.44477	−0.186819
$341$	0	0
$342$	−5.96303	−0.322444
$343$	−1.00000	−0.0539949
$344$	−5.39500	−0.290879
$345$	4.09241	0.220328
$346$	−3.17934	−0.170922
$347$	−9.00265	−0.483288	−0.241644	−	0.970365i	$-0.577687\pi$
$347$	−9.00265	−0.483288	−0.241644	+	0.970365i	$0.577687\pi$
$348$	9.77573	0.524034
$349$	−6.72881	−0.360185	−0.180092	−	0.983650i	$-0.557640\pi$
$349$	−6.72881	−0.360185	−0.180092	+	0.983650i	$0.557640\pi$
$350$	−1.00000	−0.0534522
$351$	37.6462	2.00940
$352$	0	0
$353$	−29.9182	−1.59238	−0.796192	−	0.605044i	$-0.793156\pi$
$353$	−29.9182	−1.59238	−0.796192	+	0.605044i	$0.793156\pi$
$354$	11.8356	0.629055
$355$	6.81376	0.361637
$356$	−4.97143	−0.263485
$357$	3.87383	0.205025
$358$	4.48137	0.236848
$359$	−23.5622	−1.24356	−0.621782	−	0.783190i	$-0.713591\pi$
$359$	−23.5622	−1.24356	−0.621782	+	0.783190i	$0.713591\pi$
$360$	1.73538	0.0914624
$361$	−7.19282	−0.378570
$362$	−9.02937	−0.474573
$363$	0	0
$364$	−7.06945	−0.370540
$365$	15.1151	0.791163
$366$	−0.513061	−0.0268181
$367$	−10.1699	−0.530862	−0.265431	−	0.964130i	$-0.585514\pi$
$367$	−10.1699	−0.530862	−0.265431	+	0.964130i	$0.585514\pi$
$368$	3.63913	0.189703
$369$	8.70961	0.453404
$370$	0.780576	0.0405802
$371$	−11.3718	−0.590395
$372$	5.08461	0.263625
$373$	10.7767	0.557996	0.278998	−	0.960292i	$-0.409998\pi$
$373$	10.7767	0.557996	0.278998	+	0.960292i	$0.409998\pi$
$374$	0	0
$375$	1.12455	0.0580717
$376$	9.82775	0.506827
$377$	−61.4546	−3.16507
$378$	−5.32519	−0.273898
$379$	−5.39945	−0.277351	−0.138676	−	0.990338i	$-0.544285\pi$
$379$	−5.39945	−0.277351	−0.138676	+	0.990338i	$0.544285\pi$
$380$	−3.43616	−0.176271
$381$	−23.9091	−1.22490
$382$	2.90870	0.148822
$383$	7.17505	0.366628	0.183314	−	0.983054i	$-0.441318\pi$
$383$	7.17505	0.366628	0.183314	+	0.983054i	$0.441318\pi$
$384$	−1.12455	−0.0573872
$385$	0	0
$386$	−7.77320	−0.395646
$387$	9.36235	0.475915
$388$	2.88993	0.146714
$389$	16.5388	0.838549	0.419274	−	0.907860i	$-0.362285\pi$
$389$	16.5388	0.838549	0.419274	+	0.907860i	$0.362285\pi$
$390$	7.94999	0.402563
$391$	12.5360	0.633972
$392$	1.00000	0.0505076
$393$	−11.5421	−0.582224
$394$	26.2257	1.32123
$395$	16.2368	0.816960
$396$	0	0
$397$	16.7413	0.840221	0.420111	−	0.907473i	$-0.361991\pi$
$397$	16.7413	0.840221	0.420111	+	0.907473i	$0.361991\pi$
$398$	5.64963	0.283190
$399$	3.86415	0.193449
$400$	1.00000	0.0500000
$401$	−20.3412	−1.01579	−0.507896	−	0.861419i	$-0.669576\pi$
$401$	−20.3412	−1.01579	−0.507896	+	0.861419i	$0.669576\pi$
$402$	−11.8008	−0.588568
$403$	−31.9641	−1.59225
$404$	12.2266	0.608295
$405$	0.782335	0.0388745
$406$	8.69298	0.431425
$407$	0	0
$408$	−3.87383	−0.191783
$409$	−0.443589	−0.0219341	−0.0109670	−	0.999940i	$-0.503491\pi$
$409$	−0.443589	−0.0219341	−0.0109670	+	0.999940i	$0.503491\pi$
$410$	5.01886	0.247864
$411$	0.855686	0.0422079
$412$	3.70633	0.182598
$413$	10.5247	0.517886
$414$	−6.31527	−0.310379
$415$	−7.40344	−0.363421
$416$	7.06945	0.346608
$417$	−19.8346	−0.971305
$418$	0	0
$419$	−18.6545	−0.911330	−0.455665	−	0.890151i	$-0.650599\pi$
$419$	−18.6545	−0.911330	−0.455665	+	0.890151i	$0.650599\pi$
$420$	−1.12455	−0.0548726
$421$	12.2716	0.598081	0.299040	−	0.954240i	$-0.403333\pi$
$421$	12.2716	0.598081	0.299040	+	0.954240i	$0.403333\pi$
$422$	7.42973	0.361674
$423$	−17.0549	−0.829235
$424$	11.3718	0.552264
$425$	3.44477	0.167096
$426$	7.66245	0.371247
$427$	−0.456235	−0.0220787
$428$	−2.85107	−0.137812
$429$	0	0
$430$	5.39500	0.260170
$431$	21.9795	1.05872	0.529358	−	0.848398i	$-0.322433\pi$
$431$	21.9795	1.05872	0.529358	+	0.848398i	$0.322433\pi$
$432$	5.32519	0.256208
$433$	22.2345	1.06852	0.534262	−	0.845319i	$-0.320590\pi$
$433$	22.2345	1.06852	0.534262	+	0.845319i	$0.320590\pi$
$434$	4.52145	0.217036
$435$	−9.77573	−0.468710
$436$	9.80121	0.469393
$437$	12.5046	0.598178
$438$	16.9978	0.812186
$439$	−11.0302	−0.526444	−0.263222	−	0.964735i	$-0.584785\pi$
$439$	−11.0302	−0.526444	−0.263222	+	0.964735i	$0.584785\pi$
$440$	0	0
$441$	−1.73538	−0.0826370
$442$	24.3526	1.15834
$443$	30.8702	1.46669	0.733345	−	0.679857i	$-0.237958\pi$
$443$	30.8702	1.46669	0.733345	+	0.679857i	$0.237958\pi$
$444$	0.877800	0.0416586
$445$	4.97143	0.235668
$446$	1.51167	0.0715797
$447$	−0.875106	−0.0413911
$448$	−1.00000	−0.0472456
$449$	−13.9952	−0.660475	−0.330238	−	0.943898i	$-0.607129\pi$
$449$	−13.9952	−0.660475	−0.330238	+	0.943898i	$0.607129\pi$
$450$	−1.73538	−0.0818065
$451$	0	0
$452$	10.8032	0.508140
$453$	−15.0863	−0.708817
$454$	13.8558	0.650284
$455$	7.06945	0.331421
$456$	−3.86415	−0.180955
$457$	−7.06533	−0.330502	−0.165251	−	0.986252i	$-0.552843\pi$
$457$	−7.06533	−0.330502	−0.165251	+	0.986252i	$0.552843\pi$
$458$	20.5463	0.960067
$459$	18.3440	0.856227
$460$	−3.63913	−0.169676
$461$	10.6011	0.493744	0.246872	−	0.969048i	$-0.420597\pi$
$461$	10.6011	0.493744	0.246872	+	0.969048i	$0.420597\pi$
$462$	0	0
$463$	4.59137	0.213379	0.106690	−	0.994292i	$-0.465975\pi$
$463$	4.59137	0.213379	0.106690	+	0.994292i	$0.465975\pi$
$464$	−8.69298	−0.403561
$465$	−5.08461	−0.235793
$466$	16.2345	0.752050
$467$	−30.9287	−1.43121	−0.715606	−	0.698504i	$-0.753849\pi$
$467$	−30.9287	−1.43121	−0.715606	+	0.698504i	$0.753849\pi$
$468$	−12.2682	−0.567096
$469$	−10.4937	−0.484555
$470$	−9.82775	−0.453320
$471$	27.0299	1.24547
$472$	−10.5247	−0.484438
$473$	0	0
$474$	18.2591	0.838670
$475$	3.43616	0.157662
$476$	−3.44477	−0.157891
$477$	−19.7344	−0.903576
$478$	−0.392921	−0.0179718
$479$	3.01362	0.137696	0.0688480	−	0.997627i	$-0.478068\pi$
$479$	3.01362	0.137696	0.0688480	+	0.997627i	$0.478068\pi$
$480$	1.12455	0.0513287
$481$	−5.51825	−0.251610
$482$	−3.32681	−0.151532
$483$	4.09241	0.186211
$484$	0	0
$485$	−2.88993	−0.131225
$486$	−15.0958	−0.684759
$487$	3.00551	0.136193	0.0680963	−	0.997679i	$-0.478307\pi$
$487$	3.00551	0.136193	0.0680963	+	0.997679i	$0.478307\pi$
$488$	0.456235	0.0206528
$489$	−1.01816	−0.0460429
$490$	−1.00000	−0.0451754
$491$	34.6723	1.56474	0.782369	−	0.622815i	$-0.214011\pi$
$491$	34.6723	1.56474	0.782369	+	0.622815i	$0.214011\pi$
$492$	5.64398	0.254450
$493$	−29.9453	−1.34867
$494$	24.2918	1.09294
$495$	0	0
$496$	−4.52145	−0.203019
$497$	6.81376	0.305639
$498$	−8.32557	−0.373078
$499$	−5.25216	−0.235119	−0.117559	−	0.993066i	$-0.537507\pi$
$499$	−5.25216	−0.235119	−0.117559	+	0.993066i	$0.537507\pi$
$500$	−1.00000	−0.0447214
$501$	4.09430	0.182920
$502$	17.2755	0.771044
$503$	33.6525	1.50049	0.750245	−	0.661160i	$-0.229936\pi$
$503$	33.6525	1.50049	0.750245	+	0.661160i	$0.229936\pi$
$504$	1.73538	0.0772998
$505$	−12.2266	−0.544076
$506$	0	0
$507$	−41.5828	−1.84676
$508$	21.2610	0.943303
$509$	−20.1633	−0.893724	−0.446862	−	0.894603i	$-0.647458\pi$
$509$	−20.1633	−0.893724	−0.446862	+	0.894603i	$0.647458\pi$
$510$	3.87383	0.171536
$511$	15.1151	0.668654
$512$	1.00000	0.0441942
$513$	18.2982	0.807885
$514$	−0.448545	−0.0197845
$515$	−3.70633	−0.163321
$516$	6.06697	0.267083
$517$	0	0
$518$	0.780576	0.0342965
$519$	3.57534	0.156940
$520$	−7.06945	−0.310016
$521$	−34.1002	−1.49396	−0.746979	−	0.664848i	$-0.768496\pi$
$521$	−34.1002	−1.49396	−0.746979	+	0.664848i	$0.768496\pi$
$522$	15.0856	0.660279
$523$	−37.5694	−1.64280	−0.821398	−	0.570355i	$-0.806806\pi$
$523$	−37.5694	−1.64280	−0.821398	+	0.570355i	$0.806806\pi$
$524$	10.2637	0.448373
$525$	1.12455	0.0490796
$526$	26.1440	1.13993
$527$	−15.5753	−0.678472
$528$	0	0
$529$	−9.75670	−0.424204
$530$	−11.3718	−0.493960
$531$	18.2643	0.792604
$532$	−3.43616	−0.148976
$533$	−35.4806	−1.53683
$534$	5.59064	0.241931
$535$	2.85107	0.123263
$536$	10.4937	0.453260
$537$	−5.03954	−0.217472
$538$	14.1617	0.610554
$539$	0	0
$540$	−5.32519	−0.229160
$541$	−22.6441	−0.973547	−0.486773	−	0.873528i	$-0.661826\pi$
$541$	−22.6441	−0.973547	−0.486773	+	0.873528i	$0.661826\pi$
$542$	−0.934211	−0.0401278
$543$	10.1540	0.435751
$544$	3.44477	0.147693
$545$	−9.80121	−0.419838
$546$	7.94999	0.340228
$547$	24.3655	1.04179	0.520896	−	0.853620i	$-0.325598\pi$
$547$	24.3655	1.04179	0.520896	+	0.853620i	$0.325598\pi$
$548$	−0.760911	−0.0325045
$549$	−0.791739	−0.0337906
$550$	0	0
$551$	−29.8704	−1.27252
$552$	−4.09241	−0.174184
$553$	16.2368	0.690458
$554$	−25.3249	−1.07595
$555$	−0.877800	−0.0372605
$556$	17.6377	0.748007
$557$	25.1506	1.06567	0.532833	−	0.846220i	$-0.321127\pi$
$557$	25.1506	1.06567	0.532833	+	0.846220i	$0.321127\pi$
$558$	7.84641	0.332165
$559$	−38.1397	−1.61314
$560$	1.00000	0.0422577
$561$	0	0
$562$	27.7779	1.17174
$563$	−20.1972	−0.851212	−0.425606	−	0.904909i	$-0.639939\pi$
$563$	−20.1972	−0.851212	−0.425606	+	0.904909i	$0.639939\pi$
$564$	−11.0518	−0.465366
$565$	−10.8032	−0.454494
$566$	−19.5153	−0.820289
$567$	0.782335	0.0328550
$568$	−6.81376	−0.285899
$569$	25.8942	1.08554	0.542771	−	0.839881i	$-0.317375\pi$
$569$	25.8942	1.08554	0.542771	+	0.839881i	$0.317375\pi$
$570$	3.86415	0.161851
$571$	22.4564	0.939771	0.469885	−	0.882727i	$-0.344295\pi$
$571$	22.4564	0.939771	0.469885	+	0.882727i	$0.344295\pi$
$572$	0	0
$573$	−3.27099	−0.136648
$574$	5.01886	0.209483
$575$	3.63913	0.151762
$576$	−1.73538	−0.0723074
$577$	−33.8577	−1.40952	−0.704758	−	0.709448i	$-0.748945\pi$
$577$	−33.8577	−1.40952	−0.704758	+	0.709448i	$0.748945\pi$
$578$	−5.13357	−0.213528
$579$	8.74139	0.363280
$580$	8.69298	0.360956
$581$	−7.40344	−0.307146
$582$	−3.24988	−0.134712
$583$	0	0
$584$	−15.1151	−0.625469
$585$	12.2682	0.507226
$586$	−13.3505	−0.551504
$587$	28.5545	1.17857	0.589285	−	0.807925i	$-0.299409\pi$
$587$	28.5545	1.17857	0.589285	+	0.807925i	$0.299409\pi$
$588$	−1.12455	−0.0463758
$589$	−15.5364	−0.640166
$590$	10.5247	0.433295
$591$	−29.4923	−1.21315
$592$	−0.780576	−0.0320815
$593$	34.9417	1.43488	0.717442	−	0.696618i	$-0.245313\pi$
$593$	34.9417	1.43488	0.717442	+	0.696618i	$0.245313\pi$
$594$	0	0
$595$	3.44477	0.141222
$596$	0.778180	0.0318755
$597$	−6.35331	−0.260024
$598$	25.7267	1.05204
$599$	−31.0639	−1.26924	−0.634618	−	0.772826i	$-0.718843\pi$
$599$	−31.0639	−1.26924	−0.634618	+	0.772826i	$0.718843\pi$
$600$	−1.12455	−0.0459097
$601$	−43.2602	−1.76462	−0.882310	−	0.470668i	$-0.844013\pi$
$601$	−43.2602	−1.76462	−0.882310	+	0.470668i	$0.844013\pi$
$602$	5.39500	0.219884
$603$	−18.2106	−0.741591
$604$	13.4154	0.545864
$605$	0	0
$606$	−13.7495	−0.558534
$607$	−33.1013	−1.34354	−0.671771	−	0.740759i	$-0.734466\pi$
$607$	−33.1013	−1.34354	−0.671771	+	0.740759i	$0.734466\pi$
$608$	3.43616	0.139355
$609$	−9.77573	−0.396133
$610$	−0.456235	−0.0184724
$611$	69.4768	2.81073
$612$	−5.97797	−0.241645
$613$	−17.0269	−0.687708	−0.343854	−	0.939023i	$-0.611733\pi$
$613$	−17.0269	−0.687708	−0.343854	+	0.939023i	$0.611733\pi$
$614$	25.7848	1.04059
$615$	−5.64398	−0.227587
$616$	0	0
$617$	3.82168	0.153855	0.0769276	−	0.997037i	$-0.475489\pi$
$617$	3.82168	0.153855	0.0769276	+	0.997037i	$0.475489\pi$
$618$	−4.16798	−0.167661
$619$	−28.4586	−1.14385	−0.571924	−	0.820307i	$-0.693803\pi$
$619$	−28.4586	−1.14385	−0.571924	+	0.820307i	$0.693803\pi$
$620$	4.52145	0.181586
$621$	19.3791	0.777656
$622$	5.11540	0.205109
$623$	4.97143	0.199176
$624$	−7.94999	−0.318254
$625$	1.00000	0.0400000
$626$	16.5571	0.661753
$627$	0	0
$628$	−24.0361	−0.959146
$629$	−2.68890	−0.107214
$630$	−1.73538	−0.0691391
$631$	8.90784	0.354615	0.177308	−	0.984155i	$-0.443261\pi$
$631$	8.90784	0.354615	0.177308	+	0.984155i	$0.443261\pi$
$632$	−16.2368	−0.645864
$633$	−8.35514	−0.332087
$634$	15.3500	0.609627
$635$	−21.2610	−0.843716
$636$	−12.7882	−0.507086
$637$	7.06945	0.280102
$638$	0	0
$639$	11.8244	0.467768
$640$	−1.00000	−0.0395285
$641$	28.2250	1.11482	0.557410	−	0.830237i	$-0.311795\pi$
$641$	28.2250	1.11482	0.557410	+	0.830237i	$0.311795\pi$
$642$	3.20618	0.126538
$643$	27.7470	1.09423	0.547117	−	0.837056i	$-0.315725\pi$
$643$	27.7470	1.09423	0.547117	+	0.837056i	$0.315725\pi$
$644$	−3.63913	−0.143402
$645$	−6.06697	−0.238887
$646$	11.8368	0.465711
$647$	26.0679	1.02484	0.512418	−	0.858736i	$-0.328750\pi$
$647$	26.0679	1.02484	0.512418	+	0.858736i	$0.328750\pi$
$648$	−0.782335	−0.0307330
$649$	0	0
$650$	7.06945	0.277287
$651$	−5.08461	−0.199282
$652$	0.905391	0.0354579
$653$	10.8574	0.424884	0.212442	−	0.977174i	$-0.431858\pi$
$653$	10.8574	0.424884	0.212442	+	0.977174i	$0.431858\pi$
$654$	−11.0220	−0.430994
$655$	−10.2637	−0.401037
$656$	−5.01886	−0.195954
$657$	26.2305	1.02335
$658$	−9.82775	−0.383126
$659$	−8.76914	−0.341597	−0.170799	−	0.985306i	$-0.554635\pi$
$659$	−8.76914	−0.341597	−0.170799	+	0.985306i	$0.554635\pi$
$660$	0	0
$661$	−18.7665	−0.729934	−0.364967	−	0.931020i	$-0.618920\pi$
$661$	−18.7665	−0.729934	−0.364967	+	0.931020i	$0.618920\pi$
$662$	−6.59151	−0.256186
$663$	−27.3859	−1.06358
$664$	7.40344	0.287309
$665$	3.43616	0.133248
$666$	1.35459	0.0524894
$667$	−31.6349	−1.22491
$668$	−3.64082	−0.140867
$669$	−1.69996	−0.0657241
$670$	−10.4937	−0.405408
$671$	0	0
$672$	1.12455	0.0433806
$673$	13.0246	0.502061	0.251031	−	0.967979i	$-0.419230\pi$
$673$	13.0246	0.502061	0.251031	+	0.967979i	$0.419230\pi$
$674$	−19.1547	−0.737810
$675$	5.32519	0.204967
$676$	36.9772	1.42220
$677$	−10.0804	−0.387422	−0.193711	−	0.981059i	$-0.562052\pi$
$677$	−10.0804	−0.387422	−0.193711	+	0.981059i	$0.562052\pi$
$678$	−12.1488	−0.466571
$679$	−2.88993	−0.110905
$680$	−3.44477	−0.132101
$681$	−15.5816	−0.597087
$682$	0	0
$683$	−8.98900	−0.343955	−0.171977	−	0.985101i	$-0.555016\pi$
$683$	−8.98900	−0.343955	−0.171977	+	0.985101i	$0.555016\pi$
$684$	−5.96303	−0.228002
$685$	0.760911	0.0290729
$686$	−1.00000	−0.0381802
$687$	−23.1055	−0.881529
$688$	−5.39500	−0.205682
$689$	80.3925	3.06271
$690$	4.09241	0.155795
$691$	45.5777	1.73386	0.866929	−	0.498432i	$-0.166090\pi$
$691$	45.5777	1.73386	0.866929	+	0.498432i	$0.166090\pi$
$692$	−3.17934	−0.120860
$693$	0	0
$694$	−9.00265	−0.341736
$695$	−17.6377	−0.669038
$696$	9.77573	0.370548
$697$	−17.2888	−0.654860
$698$	−6.72881	−0.254689
$699$	−18.2566	−0.690529
$700$	−1.00000	−0.0377964
$701$	11.8996	0.449442	0.224721	−	0.974423i	$-0.427853\pi$
$701$	11.8996	0.449442	0.224721	+	0.974423i	$0.427853\pi$
$702$	37.6462	1.42086
$703$	−2.68218	−0.101160
$704$	0	0
$705$	11.0518	0.416236
$706$	−29.9182	−1.12599
$707$	−12.2266	−0.459828
$708$	11.8356	0.444809
$709$	−51.2048	−1.92304	−0.961518	−	0.274740i	$-0.911408\pi$
$709$	−51.2048	−1.92304	−0.961518	+	0.274740i	$0.911408\pi$
$710$	6.81376	0.255716
$711$	28.1769	1.05672
$712$	−4.97143	−0.186312
$713$	−16.4541	−0.616213
$714$	3.87383	0.144974
$715$	0	0
$716$	4.48137	0.167477
$717$	0.441861	0.0165016
$718$	−23.5622	−0.879333
$719$	−27.8658	−1.03922	−0.519610	−	0.854403i	$-0.673923\pi$
$719$	−27.8658	−1.03922	−0.519610	+	0.854403i	$0.673923\pi$
$720$	1.73538	0.0646737
$721$	−3.70633	−0.138031
$722$	−7.19282	−0.267689
$723$	3.74118	0.139136
$724$	−9.02937	−0.335574
$725$	−8.69298	−0.322849
$726$	0	0
$727$	41.3933	1.53519	0.767597	−	0.640933i	$-0.221453\pi$
$727$	41.3933	1.53519	0.767597	+	0.640933i	$0.221453\pi$
$728$	−7.06945	−0.262011
$729$	19.3230	0.715668
$730$	15.1151	0.559436
$731$	−18.5845	−0.687373
$732$	−0.513061	−0.0189633
$733$	−5.84935	−0.216051	−0.108025	−	0.994148i	$-0.534453\pi$
$733$	−5.84935	−0.216051	−0.108025	+	0.994148i	$0.534453\pi$
$734$	−10.1699	−0.375376
$735$	1.12455	0.0414798
$736$	3.63913	0.134140
$737$	0	0
$738$	8.70961	0.320605
$739$	48.5521	1.78602	0.893009	−	0.450039i	$-0.148590\pi$
$739$	48.5521	1.78602	0.893009	+	0.450039i	$0.148590\pi$
$740$	0.780576	0.0286945
$741$	−27.3174	−1.00353
$742$	−11.3718	−0.417473
$743$	33.8585	1.24215	0.621073	−	0.783752i	$-0.286697\pi$
$743$	33.8585	1.24215	0.621073	+	0.783752i	$0.286697\pi$
$744$	5.08461	0.186411
$745$	−0.778180	−0.0285103
$746$	10.7767	0.394562
$747$	−12.8478	−0.470075
$748$	0	0
$749$	2.85107	0.104176
$750$	1.12455	0.0410629
$751$	34.9061	1.27374	0.636870	−	0.770971i	$-0.280229\pi$
$751$	34.9061	1.27374	0.636870	+	0.770971i	$0.280229\pi$
$752$	9.82775	0.358381
$753$	−19.4273	−0.707969
$754$	−61.4546	−2.23804
$755$	−13.4154	−0.488235
$756$	−5.32519	−0.193675
$757$	−18.7354	−0.680948	−0.340474	−	0.940254i	$-0.610588\pi$
$757$	−18.7354	−0.680948	−0.340474	+	0.940254i	$0.610588\pi$
$758$	−5.39945	−0.196117
$759$	0	0
$760$	−3.43616	−0.124643
$761$	−26.6630	−0.966531	−0.483266	−	0.875474i	$-0.660549\pi$
$761$	−26.6630	−0.966531	−0.483266	+	0.875474i	$0.660549\pi$
$762$	−23.9091	−0.866136
$763$	−9.80121	−0.354828
$764$	2.90870	0.105233
$765$	5.97797	0.216134
$766$	7.17505	0.259245
$767$	−74.4038	−2.68657
$768$	−1.12455	−0.0405789
$769$	−38.2777	−1.38033	−0.690165	−	0.723652i	$-0.742462\pi$
$769$	−38.2777	−1.38033	−0.690165	+	0.723652i	$0.742462\pi$
$770$	0	0
$771$	0.504413	0.0181660
$772$	−7.77320	−0.279764
$773$	7.52232	0.270559	0.135280	−	0.990807i	$-0.456807\pi$
$773$	7.52232	0.270559	0.135280	+	0.990807i	$0.456807\pi$
$774$	9.36235	0.336523
$775$	−4.52145	−0.162415
$776$	2.88993	0.103742
$777$	−0.877800	−0.0314909
$778$	16.5388	0.592943
$779$	−17.2456	−0.617887
$780$	7.94999	0.284655
$781$	0	0
$782$	12.5360	0.448286
$783$	−46.2918	−1.65433
$784$	1.00000	0.0357143
$785$	24.0361	0.857886
$786$	−11.5421	−0.411694
$787$	0.215236	0.00767235	0.00383618	−	0.999993i	$-0.498779\pi$
$787$	0.215236	0.00767235	0.00383618	+	0.999993i	$0.498779\pi$
$788$	26.2257	0.934254
$789$	−29.4003	−1.04668
$790$	16.2368	0.577678
$791$	−10.8032	−0.384118
$792$	0	0
$793$	3.22533	0.114535
$794$	16.7413	0.594126
$795$	12.7882	0.453552
$796$	5.64963	0.200246
$797$	28.2639	1.00116	0.500580	−	0.865690i	$-0.333120\pi$
$797$	28.2639	1.00116	0.500580	+	0.865690i	$0.333120\pi$
$798$	3.86415	0.136789
$799$	33.8543	1.19768
$800$	1.00000	0.0353553
$801$	8.62730	0.304831
$802$	−20.3412	−0.718273
$803$	0	0
$804$	−11.8008	−0.416181
$805$	3.63913	0.128263
$806$	−31.9641	−1.12589
$807$	−15.9256	−0.560607
$808$	12.2266	0.430130
$809$	−4.57788	−0.160950	−0.0804748	−	0.996757i	$-0.525644\pi$
$809$	−4.57788	−0.160950	−0.0804748	+	0.996757i	$0.525644\pi$
$810$	0.782335	0.0274884
$811$	44.4278	1.56007	0.780035	−	0.625736i	$-0.215201\pi$
$811$	44.4278	1.56007	0.780035	+	0.625736i	$0.215201\pi$
$812$	8.69298	0.305064
$813$	1.05057	0.0368451
$814$	0	0
$815$	−0.905391	−0.0317145
$816$	−3.87383	−0.135611
$817$	−18.5381	−0.648564
$818$	−0.443589	−0.0155097
$819$	12.2682	0.428684
$820$	5.01886	0.175266
$821$	51.5288	1.79837	0.899184	−	0.437571i	$-0.144161\pi$
$821$	51.5288	1.79837	0.899184	+	0.437571i	$0.144161\pi$
$822$	0.855686	0.0298455
$823$	3.01093	0.104954	0.0524772	−	0.998622i	$-0.483288\pi$
$823$	3.01093	0.104954	0.0524772	+	0.998622i	$0.483288\pi$
$824$	3.70633	0.129116
$825$	0	0
$826$	10.5247	0.366201
$827$	−17.5581	−0.610555	−0.305277	−	0.952263i	$-0.598749\pi$
$827$	−17.5581	−0.610555	−0.305277	+	0.952263i	$0.598749\pi$
$828$	−6.31527	−0.219471
$829$	−54.9773	−1.90944	−0.954720	−	0.297507i	$-0.903845\pi$
$829$	−54.9773	−1.90944	−0.954720	+	0.297507i	$0.903845\pi$
$830$	−7.40344	−0.256977
$831$	28.4792	0.987932
$832$	7.06945	0.245089
$833$	3.44477	0.119354
$834$	−19.8346	−0.686816
$835$	3.64082	0.125996
$836$	0	0
$837$	−24.0776	−0.832242
$838$	−18.6545	−0.644408
$839$	8.14300	0.281127	0.140564	−	0.990072i	$-0.455109\pi$
$839$	8.14300	0.281127	0.140564	+	0.990072i	$0.455109\pi$
$840$	−1.12455	−0.0388008
$841$	46.5679	1.60579
$842$	12.2716	0.422907
$843$	−31.2378	−1.07589
$844$	7.42973	0.255742
$845$	−36.9772	−1.27205
$846$	−17.0549	−0.586358
$847$	0	0
$848$	11.3718	0.390510
$849$	21.9460	0.753185
$850$	3.44477	0.118155
$851$	−2.84062	−0.0973753
$852$	7.66245	0.262511
$853$	−18.4464	−0.631593	−0.315796	−	0.948827i	$-0.602272\pi$
$853$	−18.4464	−0.631593	−0.315796	+	0.948827i	$0.602272\pi$
$854$	−0.456235	−0.0156120
$855$	5.96303	0.203931
$856$	−2.85107	−0.0974476
$857$	36.0108	1.23011	0.615054	−	0.788485i	$-0.289134\pi$
$857$	36.0108	1.23011	0.615054	+	0.788485i	$0.289134\pi$
$858$	0	0
$859$	25.6553	0.875349	0.437674	−	0.899134i	$-0.355802\pi$
$859$	25.6553	0.875349	0.437674	+	0.899134i	$0.355802\pi$
$860$	5.39500	0.183968
$861$	−5.64398	−0.192346
$862$	21.9795	0.748626
$863$	15.4519	0.525990	0.262995	−	0.964797i	$-0.415290\pi$
$863$	15.4519	0.525990	0.262995	+	0.964797i	$0.415290\pi$
$864$	5.32519	0.181167
$865$	3.17934	0.108101
$866$	22.2345	0.755561
$867$	5.77298	0.196061
$868$	4.52145	0.153468
$869$	0	0
$870$	−9.77573	−0.331428
$871$	74.1848	2.51366
$872$	9.80121	0.331911
$873$	−5.01511	−0.169736
$874$	12.5046	0.422976
$875$	1.00000	0.0338062
$876$	16.9978	0.574302
$877$	27.9195	0.942774	0.471387	−	0.881926i	$-0.343753\pi$
$877$	27.9195	0.942774	0.471387	+	0.881926i	$0.343753\pi$
$878$	−11.0302	−0.372252
$879$	15.0134	0.506388
$880$	0	0
$881$	23.1798	0.780946	0.390473	−	0.920614i	$-0.372311\pi$
$881$	23.1798	0.780946	0.390473	+	0.920614i	$0.372311\pi$
$882$	−1.73538	−0.0584332
$883$	24.4908	0.824181	0.412090	−	0.911143i	$-0.364799\pi$
$883$	24.4908	0.824181	0.412090	+	0.911143i	$0.364799\pi$
$884$	24.3526	0.819068
$885$	−11.8356	−0.397849
$886$	30.8702	1.03711
$887$	−22.6184	−0.759453	−0.379726	−	0.925099i	$-0.623982\pi$
$887$	−22.6184	−0.759453	−0.379726	+	0.925099i	$0.623982\pi$
$888$	0.877800	0.0294570
$889$	−21.2610	−0.713070
$890$	4.97143	0.166643
$891$	0	0
$892$	1.51167	0.0506145
$893$	33.7697	1.13006
$894$	−0.875106	−0.0292679
$895$	−4.48137	−0.149796
$896$	−1.00000	−0.0334077
$897$	−28.9311	−0.965980
$898$	−13.9952	−0.467027
$899$	39.3048	1.31089
$900$	−1.73538	−0.0578459
$901$	39.1733	1.30505
$902$	0	0
$903$	−6.06697	−0.201896
$904$	10.8032	0.359309
$905$	9.02937	0.300146
$906$	−15.0863	−0.501209
$907$	29.1264	0.967127	0.483563	−	0.875309i	$-0.339342\pi$
$907$	29.1264	0.967127	0.483563	+	0.875309i	$0.339342\pi$
$908$	13.8558	0.459820
$909$	−21.2177	−0.703748
$910$	7.06945	0.234350
$911$	−0.781467	−0.0258912	−0.0129456	−	0.999916i	$-0.504121\pi$
$911$	−0.781467	−0.0258912	−0.0129456	+	0.999916i	$0.504121\pi$
$912$	−3.86415	−0.127955
$913$	0	0
$914$	−7.06533	−0.233700
$915$	0.513061	0.0169613
$916$	20.5463	0.678870
$917$	−10.2637	−0.338938
$918$	18.3440	0.605444
$919$	0.902346	0.0297656	0.0148828	−	0.999889i	$-0.495262\pi$
$919$	0.902346	0.0297656	0.0148828	+	0.999889i	$0.495262\pi$
$920$	−3.63913	−0.119979
$921$	−28.9965	−0.955466
$922$	10.6011	0.349130
$923$	−48.1696	−1.58552
$924$	0	0
$925$	−0.780576	−0.0256652
$926$	4.59137	0.150882
$927$	−6.43189	−0.211251
$928$	−8.69298	−0.285361
$929$	56.9745	1.86927	0.934637	−	0.355603i	$-0.115724\pi$
$929$	56.9745	1.86927	0.934637	+	0.355603i	$0.115724\pi$
$930$	−5.08461	−0.166731
$931$	3.43616	0.112615
$932$	16.2345	0.531780
$933$	−5.75254	−0.188330
$934$	−30.9287	−1.01202
$935$	0	0
$936$	−12.2682	−0.400998
$937$	35.7088	1.16656	0.583278	−	0.812273i	$-0.301770\pi$
$937$	35.7088	1.16656	0.583278	+	0.812273i	$0.301770\pi$
$938$	−10.4937	−0.342632
$939$	−18.6193	−0.607619
$940$	−9.82775	−0.320546
$941$	9.23591	0.301082	0.150541	−	0.988604i	$-0.451898\pi$
$941$	9.23591	0.301082	0.150541	+	0.988604i	$0.451898\pi$
$942$	27.0299	0.880683
$943$	−18.2643	−0.594768
$944$	−10.5247	−0.342550
$945$	5.32519	0.173228
$946$	0	0
$947$	−26.7339	−0.868736	−0.434368	−	0.900735i	$-0.643028\pi$
$947$	−26.7339	−0.868736	−0.434368	+	0.900735i	$0.643028\pi$
$948$	18.2591	0.593029
$949$	−106.856	−3.46869
$950$	3.43616	0.111484
$951$	−17.2619	−0.559757
$952$	−3.44477	−0.111646
$953$	−41.1619	−1.33337	−0.666683	−	0.745341i	$-0.732287\pi$
$953$	−41.1619	−1.33337	−0.666683	+	0.745341i	$0.732287\pi$
$954$	−19.7344	−0.638924
$955$	−2.90870	−0.0941234
$956$	−0.392921	−0.0127080
$957$	0	0
$958$	3.01362	0.0973658
$959$	0.760911	0.0245711
$960$	1.12455	0.0362948
$961$	−10.5565	−0.340533
$962$	−5.51825	−0.177915
$963$	4.94768	0.159437
$964$	−3.32681	−0.107149
$965$	7.77320	0.250228
$966$	4.09241	0.131671
$967$	31.2558	1.00512	0.502559	−	0.864543i	$-0.332392\pi$
$967$	31.2558	1.00512	0.502559	+	0.864543i	$0.332392\pi$
$968$	0	0
$969$	−13.3111	−0.427614
$970$	−2.88993	−0.0927899
$971$	−10.2903	−0.330232	−0.165116	−	0.986274i	$-0.552800\pi$
$971$	−10.2903	−0.330232	−0.165116	+	0.986274i	$0.552800\pi$
$972$	−15.0958	−0.484198
$973$	−17.6377	−0.565440
$974$	3.00551	0.0963028
$975$	−7.94999	−0.254603
$976$	0.456235	0.0146037
$977$	−44.9080	−1.43673	−0.718367	−	0.695665i	$-0.755110\pi$
$977$	−44.9080	−1.43673	−0.718367	+	0.695665i	$0.755110\pi$
$978$	−1.01816	−0.0325572
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−17.0088	−0.543049
$982$	34.6723	1.10644
$983$	11.9639	0.381588	0.190794	−	0.981630i	$-0.438894\pi$
$983$	11.9639	0.381588	0.190794	+	0.981630i	$0.438894\pi$
$984$	5.64398	0.179924
$985$	−26.2257	−0.835622
$986$	−29.9453	−0.953653
$987$	11.0518	0.351784
$988$	24.2918	0.772824
$989$	−19.6331	−0.624297
$990$	0	0
$991$	25.7714	0.818656	0.409328	−	0.912387i	$-0.365763\pi$
$991$	25.7714	0.818656	0.409328	+	0.912387i	$0.365763\pi$
$992$	−4.52145	−0.143556
$993$	7.41252	0.235229
$994$	6.81376	0.216119
$995$	−5.64963	−0.179105
$996$	−8.32557	−0.263806
$997$	9.16694	0.290320	0.145160	−	0.989408i	$-0.453630\pi$
$997$	9.16694	0.290320	0.145160	+	0.989408i	$0.453630\pi$
$998$	−5.25216	−0.166254
$999$	−4.15672	−0.131513

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.dh.1.3		8
11.5	even	5		770.2.n.k.421.2	✓	16
11.9	even	5		770.2.n.k.631.2	yes	16
11.10	odd	2		8470.2.a.dg.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.k.421.2	✓	16	11.5	even	5
770.2.n.k.631.2	yes	16	11.9	even	5
8470.2.a.dg.1.3		8	11.10	odd	2
8470.2.a.dh.1.3		8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.12455 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.73538 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.12455 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.73538 q^{9} -1.00000 q^{10} -1.12455 q^{12} +7.06945 q^{13} -1.00000 q^{14} +1.12455 q^{15} +1.00000 q^{16} +3.44477 q^{17} -1.73538 q^{18} +3.43616 q^{19} -1.00000 q^{20} +1.12455 q^{21} +3.63913 q^{23} -1.12455 q^{24} +1.00000 q^{25} +7.06945 q^{26} +5.32519 q^{27} -1.00000 q^{28} -8.69298 q^{29} +1.12455 q^{30} -4.52145 q^{31} +1.00000 q^{32} +3.44477 q^{34} +1.00000 q^{35} -1.73538 q^{36} -0.780576 q^{37} +3.43616 q^{38} -7.94999 q^{39} -1.00000 q^{40} -5.01886 q^{41} +1.12455 q^{42} -5.39500 q^{43} +1.73538 q^{45} +3.63913 q^{46} +9.82775 q^{47} -1.12455 q^{48} +1.00000 q^{49} +1.00000 q^{50} -3.87383 q^{51} +7.06945 q^{52} +11.3718 q^{53} +5.32519 q^{54} -1.00000 q^{56} -3.86415 q^{57} -8.69298 q^{58} -10.5247 q^{59} +1.12455 q^{60} +0.456235 q^{61} -4.52145 q^{62} +1.73538 q^{63} +1.00000 q^{64} -7.06945 q^{65} +10.4937 q^{67} +3.44477 q^{68} -4.09241 q^{69} +1.00000 q^{70} -6.81376 q^{71} -1.73538 q^{72} -15.1151 q^{73} -0.780576 q^{74} -1.12455 q^{75} +3.43616 q^{76} -7.94999 q^{78} -16.2368 q^{79} -1.00000 q^{80} -0.782335 q^{81} -5.01886 q^{82} +7.40344 q^{83} +1.12455 q^{84} -3.44477 q^{85} -5.39500 q^{86} +9.77573 q^{87} -4.97143 q^{89} +1.73538 q^{90} -7.06945 q^{91} +3.63913 q^{92} +5.08461 q^{93} +9.82775 q^{94} -3.43616 q^{95} -1.12455 q^{96} +2.88993 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^5 - 8 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - q^{13} - 8 q^{14} + 8 q^{16} + 6 q^{17} + 8 q^{18} + 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} - 8 q^{28} + 3 q^{29} - 8 q^{31} + 8 q^{32} + 6 q^{34} + 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} + 35 q^{39} - 8 q^{40} + 11 q^{41} - 5 q^{43} - 8 q^{45} + 10 q^{46} - 15 q^{47} + 8 q^{49} + 8 q^{50} - 6 q^{51} - q^{52} - 16 q^{53} - 8 q^{56} + 38 q^{57} + 3 q^{58} - 9 q^{59} + 32 q^{61} - 8 q^{62} - 8 q^{63} + 8 q^{64} + q^{65} + 33 q^{67} + 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} + 8 q^{72} - 34 q^{73} - 6 q^{74} + 5 q^{76} + 35 q^{78} + 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} + 50 q^{83} - 6 q^{85} - 5 q^{86} - 12 q^{87} + q^{89} - 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} - 15 q^{94} - 5 q^{95} - 4 q^{97} + 8 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^5 - 8 * q^7 + 8 * q^8 + 8 * q^9 - 8 * q^10 - q^13 - 8 * q^14 + 8 * q^16 + 6 * q^17 + 8 * q^18 + 5 * q^19 - 8 * q^20 + 10 * q^23 + 8 * q^25 - q^26 - 8 * q^28 + 3 * q^29 - 8 * q^31 + 8 * q^32 + 6 * q^34 + 8 * q^35 + 8 * q^36 - 6 * q^37 + 5 * q^38 + 35 * q^39 - 8 * q^40 + 11 * q^41 - 5 * q^43 - 8 * q^45 + 10 * q^46 - 15 * q^47 + 8 * q^49 + 8 * q^50 - 6 * q^51 - q^52 - 16 * q^53 - 8 * q^56 + 38 * q^57 + 3 * q^58 - 9 * q^59 + 32 * q^61 - 8 * q^62 - 8 * q^63 + 8 * q^64 + q^65 + 33 * q^67 + 6 * q^68 - 22 * q^69 + 8 * q^70 + 11 * q^71 + 8 * q^72 - 34 * q^73 - 6 * q^74 + 5 * q^76 + 35 * q^78 + 31 * q^79 - 8 * q^80 + 20 * q^81 + 11 * q^82 + 50 * q^83 - 6 * q^85 - 5 * q^86 - 12 * q^87 + q^89 - 8 * q^90 + q^91 + 10 * q^92 + 26 * q^93 - 15 * q^94 - 5 * q^95 - 4 * q^97 + 8 * q^98

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