LMFDB - Embedded newform 8470.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +1.00000 q^{10} +1.61803 q^{12} -5.23607 q^{13} -1.00000 q^{14} -1.61803 q^{15} +1.00000 q^{16} +2.61803 q^{17} +0.381966 q^{18} +7.85410 q^{19} -1.00000 q^{20} +1.61803 q^{21} -0.472136 q^{23} -1.61803 q^{24} +1.00000 q^{25} +5.23607 q^{26} -5.47214 q^{27} +1.00000 q^{28} +4.47214 q^{29} +1.61803 q^{30} +2.47214 q^{31} -1.00000 q^{32} -2.61803 q^{34} -1.00000 q^{35} -0.381966 q^{36} -11.2361 q^{37} -7.85410 q^{38} -8.47214 q^{39} +1.00000 q^{40} +2.85410 q^{41} -1.61803 q^{42} -2.09017 q^{43} +0.381966 q^{45} +0.472136 q^{46} +4.76393 q^{47} +1.61803 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.23607 q^{51} -5.23607 q^{52} -2.76393 q^{53} +5.47214 q^{54} -1.00000 q^{56} +12.7082 q^{57} -4.47214 q^{58} +4.14590 q^{59} -1.61803 q^{60} +13.7082 q^{61} -2.47214 q^{62} -0.381966 q^{63} +1.00000 q^{64} +5.23607 q^{65} -3.38197 q^{67} +2.61803 q^{68} -0.763932 q^{69} +1.00000 q^{70} +10.9443 q^{71} +0.381966 q^{72} +2.85410 q^{73} +11.2361 q^{74} +1.61803 q^{75} +7.85410 q^{76} +8.47214 q^{78} +4.00000 q^{79} -1.00000 q^{80} -7.70820 q^{81} -2.85410 q^{82} -8.85410 q^{83} +1.61803 q^{84} -2.61803 q^{85} +2.09017 q^{86} +7.23607 q^{87} -14.8541 q^{89} -0.381966 q^{90} -5.23607 q^{91} -0.472136 q^{92} +4.00000 q^{93} -4.76393 q^{94} -7.85410 q^{95} -1.61803 q^{96} -8.38197 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + 2 q^{10} + q^{12} - 6 q^{13} - 2 q^{14} - q^{15} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 9 q^{19} - 2 q^{20} + q^{21} + 8 q^{23} - q^{24} + 2 q^{25} + 6 q^{26} - 2 q^{27} + 2 q^{28} + q^{30} - 4 q^{31} - 2 q^{32} - 3 q^{34} - 2 q^{35} - 3 q^{36} - 18 q^{37} - 9 q^{38} - 8 q^{39} + 2 q^{40} - q^{41} - q^{42} + 7 q^{43} + 3 q^{45} - 8 q^{46} + 14 q^{47} + q^{48} + 2 q^{49} - 2 q^{50} + 4 q^{51} - 6 q^{52} - 10 q^{53} + 2 q^{54} - 2 q^{56} + 12 q^{57} + 15 q^{59} - q^{60} + 14 q^{61} + 4 q^{62} - 3 q^{63} + 2 q^{64} + 6 q^{65} - 9 q^{67} + 3 q^{68} - 6 q^{69} + 2 q^{70} + 4 q^{71} + 3 q^{72} - q^{73} + 18 q^{74} + q^{75} + 9 q^{76} + 8 q^{78} + 8 q^{79} - 2 q^{80} - 2 q^{81} + q^{82} - 11 q^{83} + q^{84} - 3 q^{85} - 7 q^{86} + 10 q^{87} - 23 q^{89} - 3 q^{90} - 6 q^{91} + 8 q^{92} + 8 q^{93} - 14 q^{94} - 9 q^{95} - q^{96} - 19 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 + 2 * q^10 + q^12 - 6 * q^13 - 2 * q^14 - q^15 + 2 * q^16 + 3 * q^17 + 3 * q^18 + 9 * q^19 - 2 * q^20 + q^21 + 8 * q^23 - q^24 + 2 * q^25 + 6 * q^26 - 2 * q^27 + 2 * q^28 + q^30 - 4 * q^31 - 2 * q^32 - 3 * q^34 - 2 * q^35 - 3 * q^36 - 18 * q^37 - 9 * q^38 - 8 * q^39 + 2 * q^40 - q^41 - q^42 + 7 * q^43 + 3 * q^45 - 8 * q^46 + 14 * q^47 + q^48 + 2 * q^49 - 2 * q^50 + 4 * q^51 - 6 * q^52 - 10 * q^53 + 2 * q^54 - 2 * q^56 + 12 * q^57 + 15 * q^59 - q^60 + 14 * q^61 + 4 * q^62 - 3 * q^63 + 2 * q^64 + 6 * q^65 - 9 * q^67 + 3 * q^68 - 6 * q^69 + 2 * q^70 + 4 * q^71 + 3 * q^72 - q^73 + 18 * q^74 + q^75 + 9 * q^76 + 8 * q^78 + 8 * q^79 - 2 * q^80 - 2 * q^81 + q^82 - 11 * q^83 + q^84 - 3 * q^85 - 7 * q^86 + 10 * q^87 - 23 * q^89 - 3 * q^90 - 6 * q^91 + 8 * q^92 + 8 * q^93 - 14 * q^94 - 9 * q^95 - q^96 - 19 * q^97 - 2 * 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
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.61803	−0.660560
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−0.381966	−0.127322
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	1.61803	0.467086
$13$	−5.23607	−1.45222	−0.726112	−	0.687576i	$-0.758675\pi$
$13$	−5.23607	−1.45222	−0.726112	+	0.687576i	$0.758675\pi$
$14$	−1.00000	−0.267261
$15$	−1.61803	−0.417775
$16$	1.00000	0.250000
$17$	2.61803	0.634967	0.317483	−	0.948264i	$-0.397162\pi$
$17$	2.61803	0.634967	0.317483	+	0.948264i	$0.397162\pi$
$18$	0.381966	0.0900303
$19$	7.85410	1.80185	0.900927	−	0.433970i	$-0.142888\pi$
$19$	7.85410	1.80185	0.900927	+	0.433970i	$0.142888\pi$
$20$	−1.00000	−0.223607
$21$	1.61803	0.353084
$22$	0	0
$23$	−0.472136	−0.0984472	−0.0492236	−	0.998788i	$-0.515675\pi$
$23$	−0.472136	−0.0984472	−0.0492236	+	0.998788i	$0.515675\pi$
$24$	−1.61803	−0.330280
$25$	1.00000	0.200000
$26$	5.23607	1.02688
$27$	−5.47214	−1.05311
$28$	1.00000	0.188982
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	1.61803	0.295411
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.61803	−0.448989
$35$	−1.00000	−0.169031
$36$	−0.381966	−0.0636610
$37$	−11.2361	−1.84720	−0.923599	−	0.383360i	$-0.874767\pi$
$37$	−11.2361	−1.84720	−0.923599	+	0.383360i	$0.874767\pi$
$38$	−7.85410	−1.27410
$39$	−8.47214	−1.35663
$40$	1.00000	0.158114
$41$	2.85410	0.445736	0.222868	−	0.974849i	$-0.428458\pi$
$41$	2.85410	0.445736	0.222868	+	0.974849i	$0.428458\pi$
$42$	−1.61803	−0.249668
$43$	−2.09017	−0.318748	−0.159374	−	0.987218i	$-0.550948\pi$
$43$	−2.09017	−0.318748	−0.159374	+	0.987218i	$0.550948\pi$
$44$	0	0
$45$	0.381966	0.0569401
$46$	0.472136	0.0696126
$47$	4.76393	0.694891	0.347445	−	0.937700i	$-0.387049\pi$
$47$	4.76393	0.694891	0.347445	+	0.937700i	$0.387049\pi$
$48$	1.61803	0.233543
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	4.23607	0.593168
$52$	−5.23607	−0.726112
$53$	−2.76393	−0.379655	−0.189828	−	0.981817i	$-0.560793\pi$
$53$	−2.76393	−0.379655	−0.189828	+	0.981817i	$0.560793\pi$
$54$	5.47214	0.744663
$55$	0	0
$56$	−1.00000	−0.133631
$57$	12.7082	1.68324
$58$	−4.47214	−0.587220
$59$	4.14590	0.539750	0.269875	−	0.962895i	$-0.413018\pi$
$59$	4.14590	0.539750	0.269875	+	0.962895i	$0.413018\pi$
$60$	−1.61803	−0.208887
$61$	13.7082	1.75516	0.877578	−	0.479434i	$-0.159158\pi$
$61$	13.7082	1.75516	0.877578	+	0.479434i	$0.159158\pi$
$62$	−2.47214	−0.313962
$63$	−0.381966	−0.0481232
$64$	1.00000	0.125000
$65$	5.23607	0.649454
$66$	0	0
$67$	−3.38197	−0.413173	−0.206586	−	0.978428i	$-0.566235\pi$
$67$	−3.38197	−0.413173	−0.206586	+	0.978428i	$0.566235\pi$
$68$	2.61803	0.317483
$69$	−0.763932	−0.0919666
$70$	1.00000	0.119523
$71$	10.9443	1.29885	0.649423	−	0.760427i	$-0.275010\pi$
$71$	10.9443	1.29885	0.649423	+	0.760427i	$0.275010\pi$
$72$	0.381966	0.0450151
$73$	2.85410	0.334047	0.167024	−	0.985953i	$-0.446584\pi$
$73$	2.85410	0.334047	0.167024	+	0.985953i	$0.446584\pi$
$74$	11.2361	1.30617
$75$	1.61803	0.186834
$76$	7.85410	0.900927
$77$	0	0
$78$	8.47214	0.959280
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	−7.70820	−0.856467
$82$	−2.85410	−0.315183
$83$	−8.85410	−0.971864	−0.485932	−	0.873997i	$-0.661520\pi$
$83$	−8.85410	−0.971864	−0.485932	+	0.873997i	$0.661520\pi$
$84$	1.61803	0.176542
$85$	−2.61803	−0.283966
$86$	2.09017	0.225389
$87$	7.23607	0.775788
$88$	0	0
$89$	−14.8541	−1.57453	−0.787266	−	0.616614i	$-0.788504\pi$
$89$	−14.8541	−1.57453	−0.787266	+	0.616614i	$0.788504\pi$
$90$	−0.381966	−0.0402628
$91$	−5.23607	−0.548889
$92$	−0.472136	−0.0492236
$93$	4.00000	0.414781
$94$	−4.76393	−0.491362
$95$	−7.85410	−0.805814
$96$	−1.61803	−0.165140
$97$	−8.38197	−0.851060	−0.425530	−	0.904944i	$-0.639912\pi$
$97$	−8.38197	−0.851060	−0.425530	+	0.904944i	$0.639912\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−14.9443	−1.48701	−0.743505	−	0.668730i	$-0.766838\pi$
$101$	−14.9443	−1.48701	−0.743505	+	0.668730i	$0.766838\pi$
$102$	−4.23607	−0.419433
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	5.23607	0.513439
$105$	−1.61803	−0.157904
$106$	2.76393	0.268457
$107$	2.85410	0.275916	0.137958	−	0.990438i	$-0.455946\pi$
$107$	2.85410	0.275916	0.137958	+	0.990438i	$0.455946\pi$
$108$	−5.47214	−0.526557
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	−18.1803	−1.72560
$112$	1.00000	0.0944911
$113$	4.32624	0.406978	0.203489	−	0.979077i	$-0.434772\pi$
$113$	4.32624	0.406978	0.203489	+	0.979077i	$0.434772\pi$
$114$	−12.7082	−1.19023
$115$	0.472136	0.0440269
$116$	4.47214	0.415227
$117$	2.00000	0.184900
$118$	−4.14590	−0.381661
$119$	2.61803	0.239995
$120$	1.61803	0.147706
$121$	0	0
$122$	−13.7082	−1.24108
$123$	4.61803	0.416394
$124$	2.47214	0.222004
$125$	−1.00000	−0.0894427
$126$	0.381966	0.0340282
$127$	12.4721	1.10672	0.553362	−	0.832941i	$-0.313345\pi$
$127$	12.4721	1.10672	0.553362	+	0.832941i	$0.313345\pi$
$128$	−1.00000	−0.0883883
$129$	−3.38197	−0.297766
$130$	−5.23607	−0.459234
$131$	5.09017	0.444730	0.222365	−	0.974963i	$-0.428622\pi$
$131$	5.09017	0.444730	0.222365	+	0.974963i	$0.428622\pi$
$132$	0	0
$133$	7.85410	0.681037
$134$	3.38197	0.292157
$135$	5.47214	0.470966
$136$	−2.61803	−0.224495
$137$	4.85410	0.414714	0.207357	−	0.978265i	$-0.433514\pi$
$137$	4.85410	0.414714	0.207357	+	0.978265i	$0.433514\pi$
$138$	0.763932	0.0650302
$139$	−4.94427	−0.419368	−0.209684	−	0.977769i	$-0.567243\pi$
$139$	−4.94427	−0.419368	−0.209684	+	0.977769i	$0.567243\pi$
$140$	−1.00000	−0.0845154
$141$	7.70820	0.649148
$142$	−10.9443	−0.918423
$143$	0	0
$144$	−0.381966	−0.0318305
$145$	−4.47214	−0.371391
$146$	−2.85410	−0.236207
$147$	1.61803	0.133453
$148$	−11.2361	−0.923599
$149$	11.7082	0.959173	0.479587	−	0.877494i	$-0.340787\pi$
$149$	11.7082	0.959173	0.479587	+	0.877494i	$0.340787\pi$
$150$	−1.61803	−0.132112
$151$	18.4721	1.50324	0.751621	−	0.659596i	$-0.229272\pi$
$151$	18.4721	1.50324	0.751621	+	0.659596i	$0.229272\pi$
$152$	−7.85410	−0.637052
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−2.47214	−0.198567
$156$	−8.47214	−0.678314
$157$	−14.9443	−1.19268	−0.596341	−	0.802731i	$-0.703380\pi$
$157$	−14.9443	−1.19268	−0.596341	+	0.802731i	$0.703380\pi$
$158$	−4.00000	−0.318223
$159$	−4.47214	−0.354663
$160$	1.00000	0.0790569
$161$	−0.472136	−0.0372095
$162$	7.70820	0.605614
$163$	22.7984	1.78571	0.892853	−	0.450348i	$-0.148700\pi$
$163$	22.7984	1.78571	0.892853	+	0.450348i	$0.148700\pi$
$164$	2.85410	0.222868
$165$	0	0
$166$	8.85410	0.687212
$167$	8.29180	0.641638	0.320819	−	0.947140i	$-0.396042\pi$
$167$	8.29180	0.641638	0.320819	+	0.947140i	$0.396042\pi$
$168$	−1.61803	−0.124834
$169$	14.4164	1.10895
$170$	2.61803	0.200794
$171$	−3.00000	−0.229416
$172$	−2.09017	−0.159374
$173$	19.8885	1.51210	0.756049	−	0.654515i	$-0.227127\pi$
$173$	19.8885	1.51210	0.756049	+	0.654515i	$0.227127\pi$
$174$	−7.23607	−0.548565
$175$	1.00000	0.0755929
$176$	0	0
$177$	6.70820	0.504219
$178$	14.8541	1.11336
$179$	2.14590	0.160392	0.0801960	−	0.996779i	$-0.474445\pi$
$179$	2.14590	0.160392	0.0801960	+	0.996779i	$0.474445\pi$
$180$	0.381966	0.0284701
$181$	20.9443	1.55678	0.778388	−	0.627784i	$-0.216038\pi$
$181$	20.9443	1.55678	0.778388	+	0.627784i	$0.216038\pi$
$182$	5.23607	0.388123
$183$	22.1803	1.63962
$184$	0.472136	0.0348063
$185$	11.2361	0.826092
$186$	−4.00000	−0.293294
$187$	0	0
$188$	4.76393	0.347445
$189$	−5.47214	−0.398039
$190$	7.85410	0.569796
$191$	−1.23607	−0.0894387	−0.0447194	−	0.999000i	$-0.514239\pi$
$191$	−1.23607	−0.0894387	−0.0447194	+	0.999000i	$0.514239\pi$
$192$	1.61803	0.116772
$193$	−21.4164	−1.54159	−0.770793	−	0.637085i	$-0.780140\pi$
$193$	−21.4164	−1.54159	−0.770793	+	0.637085i	$0.780140\pi$
$194$	8.38197	0.601790
$195$	8.47214	0.606702
$196$	1.00000	0.0714286
$197$	11.5279	0.821326	0.410663	−	0.911787i	$-0.365297\pi$
$197$	11.5279	0.821326	0.410663	+	0.911787i	$0.365297\pi$
$198$	0	0
$199$	5.05573	0.358391	0.179196	−	0.983813i	$-0.442651\pi$
$199$	5.05573	0.358391	0.179196	+	0.983813i	$0.442651\pi$
$200$	−1.00000	−0.0707107
$201$	−5.47214	−0.385975
$202$	14.9443	1.05148
$203$	4.47214	0.313882
$204$	4.23607	0.296584
$205$	−2.85410	−0.199339
$206$	6.00000	0.418040
$207$	0.180340	0.0125345
$208$	−5.23607	−0.363056
$209$	0	0
$210$	1.61803	0.111655
$211$	10.6738	0.734812	0.367406	−	0.930061i	$-0.380246\pi$
$211$	10.6738	0.734812	0.367406	+	0.930061i	$0.380246\pi$
$212$	−2.76393	−0.189828
$213$	17.7082	1.21335
$214$	−2.85410	−0.195102
$215$	2.09017	0.142548
$216$	5.47214	0.372332
$217$	2.47214	0.167820
$218$	−4.00000	−0.270914
$219$	4.61803	0.312058
$220$	0	0
$221$	−13.7082	−0.922114
$222$	18.1803	1.22018
$223$	13.1246	0.878889	0.439445	−	0.898270i	$-0.355175\pi$
$223$	13.1246	0.878889	0.439445	+	0.898270i	$0.355175\pi$
$224$	−1.00000	−0.0668153
$225$	−0.381966	−0.0254644
$226$	−4.32624	−0.287777
$227$	13.6738	0.907559	0.453780	−	0.891114i	$-0.350075\pi$
$227$	13.6738	0.907559	0.453780	+	0.891114i	$0.350075\pi$
$228$	12.7082	0.841621
$229$	22.3607	1.47764	0.738818	−	0.673905i	$-0.235384\pi$
$229$	22.3607	1.47764	0.738818	+	0.673905i	$0.235384\pi$
$230$	−0.472136	−0.0311317
$231$	0	0
$232$	−4.47214	−0.293610
$233$	0.673762	0.0441396	0.0220698	−	0.999756i	$-0.492974\pi$
$233$	0.673762	0.0441396	0.0220698	+	0.999756i	$0.492974\pi$
$234$	−2.00000	−0.130744
$235$	−4.76393	−0.310765
$236$	4.14590	0.269875
$237$	6.47214	0.420410
$238$	−2.61803	−0.169702
$239$	6.29180	0.406982	0.203491	−	0.979077i	$-0.434771\pi$
$239$	6.29180	0.406982	0.203491	+	0.979077i	$0.434771\pi$
$240$	−1.61803	−0.104444
$241$	−30.9787	−1.99551	−0.997757	−	0.0669372i	$-0.978677\pi$
$241$	−30.9787	−1.99551	−0.997757	+	0.0669372i	$0.978677\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	13.7082	0.877578
$245$	−1.00000	−0.0638877
$246$	−4.61803	−0.294435
$247$	−41.1246	−2.61670
$248$	−2.47214	−0.156981
$249$	−14.3262	−0.907888
$250$	1.00000	0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−0.381966	−0.0240616
$253$	0	0
$254$	−12.4721	−0.782571
$255$	−4.23607	−0.265273
$256$	1.00000	0.0625000
$257$	14.8541	0.926573	0.463287	−	0.886208i	$-0.346670\pi$
$257$	14.8541	0.926573	0.463287	+	0.886208i	$0.346670\pi$
$258$	3.38197	0.210552
$259$	−11.2361	−0.698175
$260$	5.23607	0.324727
$261$	−1.70820	−0.105735
$262$	−5.09017	−0.314472
$263$	26.3607	1.62547	0.812735	−	0.582634i	$-0.197978\pi$
$263$	26.3607	1.62547	0.812735	+	0.582634i	$0.197978\pi$
$264$	0	0
$265$	2.76393	0.169787
$266$	−7.85410	−0.481566
$267$	−24.0344	−1.47088
$268$	−3.38197	−0.206586
$269$	−18.6525	−1.13726	−0.568631	−	0.822593i	$-0.692527\pi$
$269$	−18.6525	−1.13726	−0.568631	+	0.822593i	$0.692527\pi$
$270$	−5.47214	−0.333024
$271$	−6.18034	−0.375429	−0.187714	−	0.982224i	$-0.560108\pi$
$271$	−6.18034	−0.375429	−0.187714	+	0.982224i	$0.560108\pi$
$272$	2.61803	0.158742
$273$	−8.47214	−0.512757
$274$	−4.85410	−0.293247
$275$	0	0
$276$	−0.763932	−0.0459833
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	4.94427	0.296538
$279$	−0.944272	−0.0565321
$280$	1.00000	0.0597614
$281$	11.3262	0.675667	0.337833	−	0.941206i	$-0.390306\pi$
$281$	11.3262	0.675667	0.337833	+	0.941206i	$0.390306\pi$
$282$	−7.70820	−0.459017
$283$	16.3607	0.972541	0.486271	−	0.873808i	$-0.338357\pi$
$283$	16.3607	0.972541	0.486271	+	0.873808i	$0.338357\pi$
$284$	10.9443	0.649423
$285$	−12.7082	−0.752769
$286$	0	0
$287$	2.85410	0.168472
$288$	0.381966	0.0225076
$289$	−10.1459	−0.596818
$290$	4.47214	0.262613
$291$	−13.5623	−0.795036
$292$	2.85410	0.167024
$293$	18.4721	1.07915	0.539577	−	0.841936i	$-0.318584\pi$
$293$	18.4721	1.07915	0.539577	+	0.841936i	$0.318584\pi$
$294$	−1.61803	−0.0943657
$295$	−4.14590	−0.241384
$296$	11.2361	0.653083
$297$	0	0
$298$	−11.7082	−0.678238
$299$	2.47214	0.142967
$300$	1.61803	0.0934172
$301$	−2.09017	−0.120475
$302$	−18.4721	−1.06295
$303$	−24.1803	−1.38912
$304$	7.85410	0.450464
$305$	−13.7082	−0.784929
$306$	1.00000	0.0571662
$307$	18.5623	1.05941	0.529703	−	0.848183i	$-0.322303\pi$
$307$	18.5623	1.05941	0.529703	+	0.848183i	$0.322303\pi$
$308$	0	0
$309$	−9.70820	−0.552280
$310$	2.47214	0.140408
$311$	−2.18034	−0.123636	−0.0618179	−	0.998087i	$-0.519690\pi$
$311$	−2.18034	−0.123636	−0.0618179	+	0.998087i	$0.519690\pi$
$312$	8.47214	0.479640
$313$	−31.2705	−1.76751	−0.883757	−	0.467946i	$-0.844994\pi$
$313$	−31.2705	−1.76751	−0.883757	+	0.467946i	$0.844994\pi$
$314$	14.9443	0.843354
$315$	0.381966	0.0215213
$316$	4.00000	0.225018
$317$	30.3607	1.70523	0.852613	−	0.522543i	$-0.175017\pi$
$317$	30.3607	1.70523	0.852613	+	0.522543i	$0.175017\pi$
$318$	4.47214	0.250785
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	4.61803	0.257754
$322$	0.472136	0.0263111
$323$	20.5623	1.14412
$324$	−7.70820	−0.428234
$325$	−5.23607	−0.290445
$326$	−22.7984	−1.26269
$327$	6.47214	0.357910
$328$	−2.85410	−0.157591
$329$	4.76393	0.262644
$330$	0	0
$331$	−0.965558	−0.0530719	−0.0265359	−	0.999648i	$-0.508448\pi$
$331$	−0.965558	−0.0530719	−0.0265359	+	0.999648i	$0.508448\pi$
$332$	−8.85410	−0.485932
$333$	4.29180	0.235189
$334$	−8.29180	−0.453707
$335$	3.38197	0.184777
$336$	1.61803	0.0882710
$337$	−1.20163	−0.0654567	−0.0327284	−	0.999464i	$-0.510420\pi$
$337$	−1.20163	−0.0654567	−0.0327284	+	0.999464i	$0.510420\pi$
$338$	−14.4164	−0.784149
$339$	7.00000	0.380188
$340$	−2.61803	−0.141983
$341$	0	0
$342$	3.00000	0.162221
$343$	1.00000	0.0539949
$344$	2.09017	0.112694
$345$	0.763932	0.0411287
$346$	−19.8885	−1.06921
$347$	15.0902	0.810083	0.405041	−	0.914298i	$-0.367257\pi$
$347$	15.0902	0.810083	0.405041	+	0.914298i	$0.367257\pi$
$348$	7.23607	0.387894
$349$	−19.4164	−1.03934	−0.519668	−	0.854368i	$-0.673944\pi$
$349$	−19.4164	−1.03934	−0.519668	+	0.854368i	$0.673944\pi$
$350$	−1.00000	−0.0534522
$351$	28.6525	1.52936
$352$	0	0
$353$	−11.5623	−0.615399	−0.307700	−	0.951484i	$-0.599559\pi$
$353$	−11.5623	−0.615399	−0.307700	+	0.951484i	$0.599559\pi$
$354$	−6.70820	−0.356537
$355$	−10.9443	−0.580862
$356$	−14.8541	−0.787266
$357$	4.23607	0.224196
$358$	−2.14590	−0.113414
$359$	29.5967	1.56206	0.781028	−	0.624496i	$-0.214695\pi$
$359$	29.5967	1.56206	0.781028	+	0.624496i	$0.214695\pi$
$360$	−0.381966	−0.0201314
$361$	42.6869	2.24668
$362$	−20.9443	−1.10081
$363$	0	0
$364$	−5.23607	−0.274445
$365$	−2.85410	−0.149391
$366$	−22.1803	−1.15938
$367$	15.8885	0.829375	0.414688	−	0.909964i	$-0.363891\pi$
$367$	15.8885	0.829375	0.414688	+	0.909964i	$0.363891\pi$
$368$	−0.472136	−0.0246118
$369$	−1.09017	−0.0567520
$370$	−11.2361	−0.584135
$371$	−2.76393	−0.143496
$372$	4.00000	0.207390
$373$	14.6525	0.758676	0.379338	−	0.925258i	$-0.376152\pi$
$373$	14.6525	0.758676	0.379338	+	0.925258i	$0.376152\pi$
$374$	0	0
$375$	−1.61803	−0.0835549
$376$	−4.76393	−0.245681
$377$	−23.4164	−1.20601
$378$	5.47214	0.281456
$379$	15.6180	0.802245	0.401122	−	0.916025i	$-0.368620\pi$
$379$	15.6180	0.802245	0.401122	+	0.916025i	$0.368620\pi$
$380$	−7.85410	−0.402907
$381$	20.1803	1.03387
$382$	1.23607	0.0632427
$383$	−20.1803	−1.03117	−0.515584	−	0.856839i	$-0.672425\pi$
$383$	−20.1803	−1.03117	−0.515584	+	0.856839i	$0.672425\pi$
$384$	−1.61803	−0.0825700
$385$	0	0
$386$	21.4164	1.09007
$387$	0.798374	0.0405836
$388$	−8.38197	−0.425530
$389$	25.7082	1.30346	0.651729	−	0.758452i	$-0.274044\pi$
$389$	25.7082	1.30346	0.651729	+	0.758452i	$0.274044\pi$
$390$	−8.47214	−0.429003
$391$	−1.23607	−0.0625106
$392$	−1.00000	−0.0505076
$393$	8.23607	0.415455
$394$	−11.5279	−0.580765
$395$	−4.00000	−0.201262
$396$	0	0
$397$	0.111456	0.00559383	0.00279691	−	0.999996i	$-0.499110\pi$
$397$	0.111456	0.00559383	0.00279691	+	0.999996i	$0.499110\pi$
$398$	−5.05573	−0.253421
$399$	12.7082	0.636206
$400$	1.00000	0.0500000
$401$	2.56231	0.127955	0.0639777	−	0.997951i	$-0.479621\pi$
$401$	2.56231	0.127955	0.0639777	+	0.997951i	$0.479621\pi$
$402$	5.47214	0.272925
$403$	−12.9443	−0.644800
$404$	−14.9443	−0.743505
$405$	7.70820	0.383024
$406$	−4.47214	−0.221948
$407$	0	0
$408$	−4.23607	−0.209717
$409$	−22.9443	−1.13452	−0.567261	−	0.823538i	$-0.691997\pi$
$409$	−22.9443	−1.13452	−0.567261	+	0.823538i	$0.691997\pi$
$410$	2.85410	0.140954
$411$	7.85410	0.387414
$412$	−6.00000	−0.295599
$413$	4.14590	0.204006
$414$	−0.180340	−0.00886322
$415$	8.85410	0.434631
$416$	5.23607	0.256719
$417$	−8.00000	−0.391762
$418$	0	0
$419$	22.5623	1.10224	0.551120	−	0.834426i	$-0.314200\pi$
$419$	22.5623	1.10224	0.551120	+	0.834426i	$0.314200\pi$
$420$	−1.61803	−0.0789520
$421$	−36.8328	−1.79512	−0.897561	−	0.440891i	$-0.854663\pi$
$421$	−36.8328	−1.79512	−0.897561	+	0.440891i	$0.854663\pi$
$422$	−10.6738	−0.519591
$423$	−1.81966	−0.0884749
$424$	2.76393	0.134228
$425$	2.61803	0.126993
$426$	−17.7082	−0.857965
$427$	13.7082	0.663386
$428$	2.85410	0.137958
$429$	0	0
$430$	−2.09017	−0.100797
$431$	−0.652476	−0.0314287	−0.0157143	−	0.999877i	$-0.505002\pi$
$431$	−0.652476	−0.0314287	−0.0157143	+	0.999877i	$0.505002\pi$
$432$	−5.47214	−0.263278
$433$	−8.79837	−0.422823	−0.211411	−	0.977397i	$-0.567806\pi$
$433$	−8.79837	−0.422823	−0.211411	+	0.977397i	$0.567806\pi$
$434$	−2.47214	−0.118666
$435$	−7.23607	−0.346943
$436$	4.00000	0.191565
$437$	−3.70820	−0.177387
$438$	−4.61803	−0.220658
$439$	30.5410	1.45764	0.728822	−	0.684704i	$-0.240068\pi$
$439$	30.5410	1.45764	0.728822	+	0.684704i	$0.240068\pi$
$440$	0	0
$441$	−0.381966	−0.0181889
$442$	13.7082	0.652033
$443$	−28.5623	−1.35704	−0.678518	−	0.734584i	$-0.737377\pi$
$443$	−28.5623	−1.35704	−0.678518	+	0.734584i	$0.737377\pi$
$444$	−18.1803	−0.862801
$445$	14.8541	0.704152
$446$	−13.1246	−0.621468
$447$	18.9443	0.896033
$448$	1.00000	0.0472456
$449$	−20.0344	−0.945484	−0.472742	−	0.881201i	$-0.656736\pi$
$449$	−20.0344	−0.945484	−0.472742	+	0.881201i	$0.656736\pi$
$450$	0.381966	0.0180061
$451$	0	0
$452$	4.32624	0.203489
$453$	29.8885	1.40429
$454$	−13.6738	−0.641741
$455$	5.23607	0.245471
$456$	−12.7082	−0.595116
$457$	−25.9787	−1.21523	−0.607616	−	0.794231i	$-0.707874\pi$
$457$	−25.9787	−1.21523	−0.607616	+	0.794231i	$0.707874\pi$
$458$	−22.3607	−1.04485
$459$	−14.3262	−0.668692
$460$	0.472136	0.0220135
$461$	23.0557	1.07381	0.536906	−	0.843642i	$-0.319593\pi$
$461$	23.0557	1.07381	0.536906	+	0.843642i	$0.319593\pi$
$462$	0	0
$463$	−10.9443	−0.508623	−0.254312	−	0.967122i	$-0.581849\pi$
$463$	−10.9443	−0.508623	−0.254312	+	0.967122i	$0.581849\pi$
$464$	4.47214	0.207614
$465$	−4.00000	−0.185496
$466$	−0.673762	−0.0312114
$467$	12.9443	0.598989	0.299495	−	0.954098i	$-0.403182\pi$
$467$	12.9443	0.598989	0.299495	+	0.954098i	$0.403182\pi$
$468$	2.00000	0.0924500
$469$	−3.38197	−0.156165
$470$	4.76393	0.219744
$471$	−24.1803	−1.11417
$472$	−4.14590	−0.190830
$473$	0	0
$474$	−6.47214	−0.297275
$475$	7.85410	0.360371
$476$	2.61803	0.119997
$477$	1.05573	0.0483385
$478$	−6.29180	−0.287780
$479$	22.2918	1.01854	0.509269	−	0.860607i	$-0.329916\pi$
$479$	22.2918	1.01854	0.509269	+	0.860607i	$0.329916\pi$
$480$	1.61803	0.0738528
$481$	58.8328	2.68255
$482$	30.9787	1.41104
$483$	−0.763932	−0.0347601
$484$	0	0
$485$	8.38197	0.380605
$486$	−3.94427	−0.178916
$487$	38.1803	1.73012	0.865058	−	0.501672i	$-0.167281\pi$
$487$	38.1803	1.73012	0.865058	+	0.501672i	$0.167281\pi$
$488$	−13.7082	−0.620541
$489$	36.8885	1.66816
$490$	1.00000	0.0451754
$491$	−19.2148	−0.867151	−0.433575	−	0.901117i	$-0.642748\pi$
$491$	−19.2148	−0.867151	−0.433575	+	0.901117i	$0.642748\pi$
$492$	4.61803	0.208197
$493$	11.7082	0.527311
$494$	41.1246	1.85028
$495$	0	0
$496$	2.47214	0.111002
$497$	10.9443	0.490918
$498$	14.3262	0.641974
$499$	−19.9787	−0.894370	−0.447185	−	0.894441i	$-0.647573\pi$
$499$	−19.9787	−0.894370	−0.447185	+	0.894441i	$0.647573\pi$
$500$	−1.00000	−0.0447214
$501$	13.4164	0.599401
$502$	−20.0000	−0.892644
$503$	−20.6525	−0.920848	−0.460424	−	0.887699i	$-0.652303\pi$
$503$	−20.6525	−0.920848	−0.460424	+	0.887699i	$0.652303\pi$
$504$	0.381966	0.0170141
$505$	14.9443	0.665011
$506$	0	0
$507$	23.3262	1.03595
$508$	12.4721	0.553362
$509$	13.1246	0.581738	0.290869	−	0.956763i	$-0.406056\pi$
$509$	13.1246	0.581738	0.290869	+	0.956763i	$0.406056\pi$
$510$	4.23607	0.187576
$511$	2.85410	0.126258
$512$	−1.00000	−0.0441942
$513$	−42.9787	−1.89756
$514$	−14.8541	−0.655186
$515$	6.00000	0.264392
$516$	−3.38197	−0.148883
$517$	0	0
$518$	11.2361	0.493684
$519$	32.1803	1.41256
$520$	−5.23607	−0.229617
$521$	−9.67376	−0.423815	−0.211908	−	0.977290i	$-0.567968\pi$
$521$	−9.67376	−0.423815	−0.211908	+	0.977290i	$0.567968\pi$
$522$	1.70820	0.0747661
$523$	−20.7984	−0.909449	−0.454725	−	0.890632i	$-0.650262\pi$
$523$	−20.7984	−0.909449	−0.454725	+	0.890632i	$0.650262\pi$
$524$	5.09017	0.222365
$525$	1.61803	0.0706168
$526$	−26.3607	−1.14938
$527$	6.47214	0.281931
$528$	0	0
$529$	−22.7771	−0.990308
$530$	−2.76393	−0.120058
$531$	−1.58359	−0.0687220
$532$	7.85410	0.340519
$533$	−14.9443	−0.647308
$534$	24.0344	1.04007
$535$	−2.85410	−0.123394
$536$	3.38197	0.146079
$537$	3.47214	0.149834
$538$	18.6525	0.804165
$539$	0	0
$540$	5.47214	0.235483
$541$	−43.0132	−1.84928	−0.924640	−	0.380842i	$-0.875634\pi$
$541$	−43.0132	−1.84928	−0.924640	+	0.380842i	$0.875634\pi$
$542$	6.18034	0.265468
$543$	33.8885	1.45430
$544$	−2.61803	−0.112247
$545$	−4.00000	−0.171341
$546$	8.47214	0.362574
$547$	−22.1459	−0.946890	−0.473445	−	0.880823i	$-0.656990\pi$
$547$	−22.1459	−0.946890	−0.473445	+	0.880823i	$0.656990\pi$
$548$	4.85410	0.207357
$549$	−5.23607	−0.223470
$550$	0	0
$551$	35.1246	1.49636
$552$	0.763932	0.0325151
$553$	4.00000	0.170097
$554$	12.0000	0.509831
$555$	18.1803	0.771712
$556$	−4.94427	−0.209684
$557$	18.7639	0.795053	0.397527	−	0.917591i	$-0.369869\pi$
$557$	18.7639	0.795053	0.397527	+	0.917591i	$0.369869\pi$
$558$	0.944272	0.0399742
$559$	10.9443	0.462893
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−11.3262	−0.477769
$563$	29.6180	1.24825	0.624126	−	0.781324i	$-0.285455\pi$
$563$	29.6180	1.24825	0.624126	+	0.781324i	$0.285455\pi$
$564$	7.70820	0.324574
$565$	−4.32624	−0.182006
$566$	−16.3607	−0.687691
$567$	−7.70820	−0.323714
$568$	−10.9443	−0.459211
$569$	44.3262	1.85825	0.929126	−	0.369763i	$-0.120561\pi$
$569$	44.3262	1.85825	0.929126	+	0.369763i	$0.120561\pi$
$570$	12.7082	0.532288
$571$	−34.8328	−1.45771	−0.728854	−	0.684669i	$-0.759947\pi$
$571$	−34.8328	−1.45771	−0.728854	+	0.684669i	$0.759947\pi$
$572$	0	0
$573$	−2.00000	−0.0835512
$574$	−2.85410	−0.119128
$575$	−0.472136	−0.0196894
$576$	−0.381966	−0.0159153
$577$	−26.3951	−1.09884	−0.549422	−	0.835545i	$-0.685152\pi$
$577$	−26.3951	−1.09884	−0.549422	+	0.835545i	$0.685152\pi$
$578$	10.1459	0.422014
$579$	−34.6525	−1.44011
$580$	−4.47214	−0.185695
$581$	−8.85410	−0.367330
$582$	13.5623	0.562176
$583$	0	0
$584$	−2.85410	−0.118104
$585$	−2.00000	−0.0826898
$586$	−18.4721	−0.763077
$587$	−8.21478	−0.339060	−0.169530	−	0.985525i	$-0.554225\pi$
$587$	−8.21478	−0.339060	−0.169530	+	0.985525i	$0.554225\pi$
$588$	1.61803	0.0667266
$589$	19.4164	0.800039
$590$	4.14590	0.170684
$591$	18.6525	0.767260
$592$	−11.2361	−0.461800
$593$	−34.0344	−1.39763	−0.698814	−	0.715304i	$-0.746288\pi$
$593$	−34.0344	−1.39763	−0.698814	+	0.715304i	$0.746288\pi$
$594$	0	0
$595$	−2.61803	−0.107329
$596$	11.7082	0.479587
$597$	8.18034	0.334799
$598$	−2.47214	−0.101093
$599$	−36.0689	−1.47373	−0.736867	−	0.676037i	$-0.763696\pi$
$599$	−36.0689	−1.47373	−0.736867	+	0.676037i	$0.763696\pi$
$600$	−1.61803	−0.0660560
$601$	−33.3262	−1.35941	−0.679703	−	0.733488i	$-0.737891\pi$
$601$	−33.3262	−1.35941	−0.679703	+	0.733488i	$0.737891\pi$
$602$	2.09017	0.0851890
$603$	1.29180	0.0526060
$604$	18.4721	0.751621
$605$	0	0
$606$	24.1803	0.982259
$607$	−6.83282	−0.277335	−0.138668	−	0.990339i	$-0.544282\pi$
$607$	−6.83282	−0.277335	−0.138668	+	0.990339i	$0.544282\pi$
$608$	−7.85410	−0.318526
$609$	7.23607	0.293220
$610$	13.7082	0.555029
$611$	−24.9443	−1.00914
$612$	−1.00000	−0.0404226
$613$	−10.4721	−0.422966	−0.211483	−	0.977382i	$-0.567829\pi$
$613$	−10.4721	−0.422966	−0.211483	+	0.977382i	$0.567829\pi$
$614$	−18.5623	−0.749114
$615$	−4.61803	−0.186217
$616$	0	0
$617$	−5.56231	−0.223930	−0.111965	−	0.993712i	$-0.535714\pi$
$617$	−5.56231	−0.223930	−0.111965	+	0.993712i	$0.535714\pi$
$618$	9.70820	0.390521
$619$	43.2705	1.73919	0.869594	−	0.493767i	$-0.164380\pi$
$619$	43.2705	1.73919	0.869594	+	0.493767i	$0.164380\pi$
$620$	−2.47214	−0.0992834
$621$	2.58359	0.103676
$622$	2.18034	0.0874237
$623$	−14.8541	−0.595117
$624$	−8.47214	−0.339157
$625$	1.00000	0.0400000
$626$	31.2705	1.24982
$627$	0	0
$628$	−14.9443	−0.596341
$629$	−29.4164	−1.17291
$630$	−0.381966	−0.0152179
$631$	25.1246	1.00020	0.500098	−	0.865969i	$-0.333297\pi$
$631$	25.1246	1.00020	0.500098	+	0.865969i	$0.333297\pi$
$632$	−4.00000	−0.159111
$633$	17.2705	0.686441
$634$	−30.3607	−1.20578
$635$	−12.4721	−0.494942
$636$	−4.47214	−0.177332
$637$	−5.23607	−0.207461
$638$	0	0
$639$	−4.18034	−0.165372
$640$	1.00000	0.0395285
$641$	−40.8541	−1.61364	−0.806820	−	0.590797i	$-0.798813\pi$
$641$	−40.8541	−1.61364	−0.806820	+	0.590797i	$0.798813\pi$
$642$	−4.61803	−0.182259
$643$	−14.2016	−0.560057	−0.280029	−	0.959992i	$-0.590344\pi$
$643$	−14.2016	−0.560057	−0.280029	+	0.959992i	$0.590344\pi$
$644$	−0.472136	−0.0186048
$645$	3.38197	0.133165
$646$	−20.5623	−0.809013
$647$	37.7082	1.48246	0.741231	−	0.671250i	$-0.234242\pi$
$647$	37.7082	1.48246	0.741231	+	0.671250i	$0.234242\pi$
$648$	7.70820	0.302807
$649$	0	0
$650$	5.23607	0.205375
$651$	4.00000	0.156772
$652$	22.7984	0.892853
$653$	44.6525	1.74739	0.873693	−	0.486477i	$-0.161718\pi$
$653$	44.6525	1.74739	0.873693	+	0.486477i	$0.161718\pi$
$654$	−6.47214	−0.253081
$655$	−5.09017	−0.198889
$656$	2.85410	0.111434
$657$	−1.09017	−0.0425316
$658$	−4.76393	−0.185717
$659$	−20.9787	−0.817215	−0.408607	−	0.912710i	$-0.633985\pi$
$659$	−20.9787	−0.817215	−0.408607	+	0.912710i	$0.633985\pi$
$660$	0	0
$661$	1.81966	0.0707766	0.0353883	−	0.999374i	$-0.488733\pi$
$661$	1.81966	0.0707766	0.0353883	+	0.999374i	$0.488733\pi$
$662$	0.965558	0.0375275
$663$	−22.1803	−0.861413
$664$	8.85410	0.343606
$665$	−7.85410	−0.304569
$666$	−4.29180	−0.166304
$667$	−2.11146	−0.0817559
$668$	8.29180	0.320819
$669$	21.2361	0.821034
$670$	−3.38197	−0.130657
$671$	0	0
$672$	−1.61803	−0.0624170
$673$	15.4508	0.595586	0.297793	−	0.954630i	$-0.403749\pi$
$673$	15.4508	0.595586	0.297793	+	0.954630i	$0.403749\pi$
$674$	1.20163	0.0462849
$675$	−5.47214	−0.210623
$676$	14.4164	0.554477
$677$	11.4164	0.438768	0.219384	−	0.975639i	$-0.429595\pi$
$677$	11.4164	0.438768	0.219384	+	0.975639i	$0.429595\pi$
$678$	−7.00000	−0.268833
$679$	−8.38197	−0.321670
$680$	2.61803	0.100397
$681$	22.1246	0.847817
$682$	0	0
$683$	21.5279	0.823741	0.411870	−	0.911242i	$-0.364876\pi$
$683$	21.5279	0.823741	0.411870	+	0.911242i	$0.364876\pi$
$684$	−3.00000	−0.114708
$685$	−4.85410	−0.185466
$686$	−1.00000	−0.0381802
$687$	36.1803	1.38037
$688$	−2.09017	−0.0796870
$689$	14.4721	0.551344
$690$	−0.763932	−0.0290824
$691$	−7.56231	−0.287684	−0.143842	−	0.989601i	$-0.545946\pi$
$691$	−7.56231	−0.287684	−0.143842	+	0.989601i	$0.545946\pi$
$692$	19.8885	0.756049
$693$	0	0
$694$	−15.0902	−0.572815
$695$	4.94427	0.187547
$696$	−7.23607	−0.274282
$697$	7.47214	0.283027
$698$	19.4164	0.734922
$699$	1.09017	0.0412340
$700$	1.00000	0.0377964
$701$	−34.8328	−1.31562	−0.657809	−	0.753185i	$-0.728516\pi$
$701$	−34.8328	−1.31562	−0.657809	+	0.753185i	$0.728516\pi$
$702$	−28.6525	−1.08142
$703$	−88.2492	−3.32838
$704$	0	0
$705$	−7.70820	−0.290308
$706$	11.5623	0.435153
$707$	−14.9443	−0.562037
$708$	6.70820	0.252110
$709$	33.7771	1.26853	0.634263	−	0.773118i	$-0.281304\pi$
$709$	33.7771	1.26853	0.634263	+	0.773118i	$0.281304\pi$
$710$	10.9443	0.410731
$711$	−1.52786	−0.0572994
$712$	14.8541	0.556681
$713$	−1.16718	−0.0437114
$714$	−4.23607	−0.158531
$715$	0	0
$716$	2.14590	0.0801960
$717$	10.1803	0.380192
$718$	−29.5967	−1.10454
$719$	24.0689	0.897618	0.448809	−	0.893628i	$-0.351848\pi$
$719$	24.0689	0.897618	0.448809	+	0.893628i	$0.351848\pi$
$720$	0.381966	0.0142350
$721$	−6.00000	−0.223452
$722$	−42.6869	−1.58864
$723$	−50.1246	−1.86415
$724$	20.9443	0.778388
$725$	4.47214	0.166091
$726$	0	0
$727$	50.2492	1.86364	0.931820	−	0.362920i	$-0.118220\pi$
$727$	50.2492	1.86364	0.931820	+	0.362920i	$0.118220\pi$
$728$	5.23607	0.194062
$729$	29.5066	1.09284
$730$	2.85410	0.105635
$731$	−5.47214	−0.202394
$732$	22.1803	0.819809
$733$	−35.5967	−1.31480	−0.657398	−	0.753544i	$-0.728343\pi$
$733$	−35.5967	−1.31480	−0.657398	+	0.753544i	$0.728343\pi$
$734$	−15.8885	−0.586457
$735$	−1.61803	−0.0596821
$736$	0.472136	0.0174032
$737$	0	0
$738$	1.09017	0.0401297
$739$	10.1459	0.373223	0.186611	−	0.982434i	$-0.440249\pi$
$739$	10.1459	0.373223	0.186611	+	0.982434i	$0.440249\pi$
$740$	11.2361	0.413046
$741$	−66.5410	−2.44445
$742$	2.76393	0.101467
$743$	−40.2492	−1.47660	−0.738300	−	0.674472i	$-0.764371\pi$
$743$	−40.2492	−1.47660	−0.738300	+	0.674472i	$0.764371\pi$
$744$	−4.00000	−0.146647
$745$	−11.7082	−0.428955
$746$	−14.6525	−0.536465
$747$	3.38197	0.123740
$748$	0	0
$749$	2.85410	0.104287
$750$	1.61803	0.0590822
$751$	−45.3050	−1.65320	−0.826601	−	0.562789i	$-0.809728\pi$
$751$	−45.3050	−1.65320	−0.826601	+	0.562789i	$0.809728\pi$
$752$	4.76393	0.173723
$753$	32.3607	1.17929
$754$	23.4164	0.852775
$755$	−18.4721	−0.672270
$756$	−5.47214	−0.199020
$757$	44.6525	1.62292	0.811461	−	0.584407i	$-0.198673\pi$
$757$	44.6525	1.62292	0.811461	+	0.584407i	$0.198673\pi$
$758$	−15.6180	−0.567273
$759$	0	0
$760$	7.85410	0.284898
$761$	−49.8115	−1.80567	−0.902833	−	0.429991i	$-0.858517\pi$
$761$	−49.8115	−1.80567	−0.902833	+	0.429991i	$0.858517\pi$
$762$	−20.1803	−0.731057
$763$	4.00000	0.144810
$764$	−1.23607	−0.0447194
$765$	1.00000	0.0361551
$766$	20.1803	0.729145
$767$	−21.7082	−0.783838
$768$	1.61803	0.0583858
$769$	31.3050	1.12889	0.564443	−	0.825472i	$-0.309091\pi$
$769$	31.3050	1.12889	0.564443	+	0.825472i	$0.309091\pi$
$770$	0	0
$771$	24.0344	0.865579
$772$	−21.4164	−0.770793
$773$	34.7639	1.25037	0.625186	−	0.780476i	$-0.285023\pi$
$773$	34.7639	1.25037	0.625186	+	0.780476i	$0.285023\pi$
$774$	−0.798374	−0.0286970
$775$	2.47214	0.0888017
$776$	8.38197	0.300895
$777$	−18.1803	−0.652216
$778$	−25.7082	−0.921684
$779$	22.4164	0.803151
$780$	8.47214	0.303351
$781$	0	0
$782$	1.23607	0.0442017
$783$	−24.4721	−0.874563
$784$	1.00000	0.0357143
$785$	14.9443	0.533384
$786$	−8.23607	−0.293771
$787$	43.2705	1.54243	0.771214	−	0.636577i	$-0.219650\pi$
$787$	43.2705	1.54243	0.771214	+	0.636577i	$0.219650\pi$
$788$	11.5279	0.410663
$789$	42.6525	1.51847
$790$	4.00000	0.142314
$791$	4.32624	0.153823
$792$	0	0
$793$	−71.7771	−2.54888
$794$	−0.111456	−0.00395543
$795$	4.47214	0.158610
$796$	5.05573	0.179196
$797$	−8.18034	−0.289763	−0.144881	−	0.989449i	$-0.546280\pi$
$797$	−8.18034	−0.289763	−0.144881	+	0.989449i	$0.546280\pi$
$798$	−12.7082	−0.449866
$799$	12.4721	0.441232
$800$	−1.00000	−0.0353553
$801$	5.67376	0.200473
$802$	−2.56231	−0.0904782
$803$	0	0
$804$	−5.47214	−0.192987
$805$	0.472136	0.0166406
$806$	12.9443	0.455943
$807$	−30.1803	−1.06240
$808$	14.9443	0.525738
$809$	27.1591	0.954861	0.477431	−	0.878669i	$-0.341568\pi$
$809$	27.1591	0.954861	0.477431	+	0.878669i	$0.341568\pi$
$810$	−7.70820	−0.270839
$811$	35.6869	1.25314	0.626569	−	0.779366i	$-0.284459\pi$
$811$	35.6869	1.25314	0.626569	+	0.779366i	$0.284459\pi$
$812$	4.47214	0.156941
$813$	−10.0000	−0.350715
$814$	0	0
$815$	−22.7984	−0.798592
$816$	4.23607	0.148292
$817$	−16.4164	−0.574337
$818$	22.9443	0.802228
$819$	2.00000	0.0698857
$820$	−2.85410	−0.0996696
$821$	−0.111456	−0.00388985	−0.00194492	−	0.999998i	$-0.500619\pi$
$821$	−0.111456	−0.00388985	−0.00194492	+	0.999998i	$0.500619\pi$
$822$	−7.85410	−0.273943
$823$	−3.81966	−0.133145	−0.0665725	−	0.997782i	$-0.521206\pi$
$823$	−3.81966	−0.133145	−0.0665725	+	0.997782i	$0.521206\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	−4.14590	−0.144254
$827$	52.3262	1.81956	0.909781	−	0.415089i	$-0.136250\pi$
$827$	52.3262	1.81956	0.909781	+	0.415089i	$0.136250\pi$
$828$	0.180340	0.00626724
$829$	33.5279	1.16447	0.582235	−	0.813020i	$-0.302178\pi$
$829$	33.5279	1.16447	0.582235	+	0.813020i	$0.302178\pi$
$830$	−8.85410	−0.307330
$831$	−19.4164	−0.673548
$832$	−5.23607	−0.181528
$833$	2.61803	0.0907095
$834$	8.00000	0.277017
$835$	−8.29180	−0.286949
$836$	0	0
$837$	−13.5279	−0.467591
$838$	−22.5623	−0.779402
$839$	36.2918	1.25293	0.626466	−	0.779449i	$-0.284501\pi$
$839$	36.2918	1.25293	0.626466	+	0.779449i	$0.284501\pi$
$840$	1.61803	0.0558275
$841$	−9.00000	−0.310345
$842$	36.8328	1.26934
$843$	18.3262	0.631189
$844$	10.6738	0.367406
$845$	−14.4164	−0.495940
$846$	1.81966	0.0625612
$847$	0	0
$848$	−2.76393	−0.0949138
$849$	26.4721	0.908521
$850$	−2.61803	−0.0897978
$851$	5.30495	0.181851
$852$	17.7082	0.606673
$853$	−13.4164	−0.459369	−0.229685	−	0.973265i	$-0.573769\pi$
$853$	−13.4164	−0.459369	−0.229685	+	0.973265i	$0.573769\pi$
$854$	−13.7082	−0.469085
$855$	3.00000	0.102598
$856$	−2.85410	−0.0975512
$857$	24.5623	0.839032	0.419516	−	0.907748i	$-0.362200\pi$
$857$	24.5623	0.839032	0.419516	+	0.907748i	$0.362200\pi$
$858$	0	0
$859$	14.3262	0.488805	0.244402	−	0.969674i	$-0.421408\pi$
$859$	14.3262	0.488805	0.244402	+	0.969674i	$0.421408\pi$
$860$	2.09017	0.0712742
$861$	4.61803	0.157382
$862$	0.652476	0.0222234
$863$	−14.1115	−0.480360	−0.240180	−	0.970728i	$-0.577206\pi$
$863$	−14.1115	−0.480360	−0.240180	+	0.970728i	$0.577206\pi$
$864$	5.47214	0.186166
$865$	−19.8885	−0.676231
$866$	8.79837	0.298981
$867$	−16.4164	−0.557530
$868$	2.47214	0.0839098
$869$	0	0
$870$	7.23607	0.245326
$871$	17.7082	0.600020
$872$	−4.00000	−0.135457
$873$	3.20163	0.108359
$874$	3.70820	0.125432
$875$	−1.00000	−0.0338062
$876$	4.61803	0.156029
$877$	−2.76393	−0.0933314	−0.0466657	−	0.998911i	$-0.514860\pi$
$877$	−2.76393	−0.0933314	−0.0466657	+	0.998911i	$0.514860\pi$
$878$	−30.5410	−1.03071
$879$	29.8885	1.00812
$880$	0	0
$881$	14.6738	0.494372	0.247186	−	0.968968i	$-0.420494\pi$
$881$	14.6738	0.494372	0.247186	+	0.968968i	$0.420494\pi$
$882$	0.381966	0.0128615
$883$	−43.2148	−1.45429	−0.727147	−	0.686482i	$-0.759154\pi$
$883$	−43.2148	−1.45429	−0.727147	+	0.686482i	$0.759154\pi$
$884$	−13.7082	−0.461057
$885$	−6.70820	−0.225494
$886$	28.5623	0.959569
$887$	26.5410	0.891160	0.445580	−	0.895242i	$-0.352997\pi$
$887$	26.5410	0.891160	0.445580	+	0.895242i	$0.352997\pi$
$888$	18.1803	0.610092
$889$	12.4721	0.418302
$890$	−14.8541	−0.497911
$891$	0	0
$892$	13.1246	0.439445
$893$	37.4164	1.25209
$894$	−18.9443	−0.633591
$895$	−2.14590	−0.0717295
$896$	−1.00000	−0.0334077
$897$	4.00000	0.133556
$898$	20.0344	0.668558
$899$	11.0557	0.368729
$900$	−0.381966	−0.0127322
$901$	−7.23607	−0.241068
$902$	0	0
$903$	−3.38197	−0.112545
$904$	−4.32624	−0.143889
$905$	−20.9443	−0.696211
$906$	−29.8885	−0.992980
$907$	25.8541	0.858471	0.429236	−	0.903193i	$-0.358783\pi$
$907$	25.8541	0.858471	0.429236	+	0.903193i	$0.358783\pi$
$908$	13.6738	0.453780
$909$	5.70820	0.189329
$910$	−5.23607	−0.173574
$911$	9.52786	0.315672	0.157836	−	0.987465i	$-0.449548\pi$
$911$	9.52786	0.315672	0.157836	+	0.987465i	$0.449548\pi$
$912$	12.7082	0.420811
$913$	0	0
$914$	25.9787	0.859299
$915$	−22.1803	−0.733259
$916$	22.3607	0.738818
$917$	5.09017	0.168092
$918$	14.3262	0.472836
$919$	7.05573	0.232747	0.116373	−	0.993206i	$-0.462873\pi$
$919$	7.05573	0.232747	0.116373	+	0.993206i	$0.462873\pi$
$920$	−0.472136	−0.0155659
$921$	30.0344	0.989669
$922$	−23.0557	−0.759300
$923$	−57.3050	−1.88622
$924$	0	0
$925$	−11.2361	−0.369440
$926$	10.9443	0.359651
$927$	2.29180	0.0752725
$928$	−4.47214	−0.146805
$929$	10.9098	0.357940	0.178970	−	0.983855i	$-0.442724\pi$
$929$	10.9098	0.357940	0.178970	+	0.983855i	$0.442724\pi$
$930$	4.00000	0.131165
$931$	7.85410	0.257408
$932$	0.673762	0.0220698
$933$	−3.52786	−0.115497
$934$	−12.9443	−0.423550
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−28.5066	−0.931269	−0.465635	−	0.884977i	$-0.654174\pi$
$937$	−28.5066	−0.931269	−0.465635	+	0.884977i	$0.654174\pi$
$938$	3.38197	0.110425
$939$	−50.5967	−1.65116
$940$	−4.76393	−0.155382
$941$	−24.3607	−0.794135	−0.397068	−	0.917789i	$-0.629972\pi$
$941$	−24.3607	−0.794135	−0.397068	+	0.917789i	$0.629972\pi$
$942$	24.1803	0.787838
$943$	−1.34752	−0.0438814
$944$	4.14590	0.134937
$945$	5.47214	0.178009
$946$	0	0
$947$	−53.6312	−1.74278	−0.871390	−	0.490591i	$-0.836781\pi$
$947$	−53.6312	−1.74278	−0.871390	+	0.490591i	$0.836781\pi$
$948$	6.47214	0.210205
$949$	−14.9443	−0.485112
$950$	−7.85410	−0.254821
$951$	49.1246	1.59297
$952$	−2.61803	−0.0848510
$953$	0.854102	0.0276671	0.0138335	−	0.999904i	$-0.495597\pi$
$953$	0.854102	0.0276671	0.0138335	+	0.999904i	$0.495597\pi$
$954$	−1.05573	−0.0341805
$955$	1.23607	0.0399982
$956$	6.29180	0.203491
$957$	0	0
$958$	−22.2918	−0.720215
$959$	4.85410	0.156747
$960$	−1.61803	−0.0522218
$961$	−24.8885	−0.802856
$962$	−58.8328	−1.89685
$963$	−1.09017	−0.0351302
$964$	−30.9787	−0.997757
$965$	21.4164	0.689419
$966$	0.763932	0.0245791
$967$	17.3475	0.557859	0.278929	−	0.960312i	$-0.410020\pi$
$967$	17.3475	0.557859	0.278929	+	0.960312i	$0.410020\pi$
$968$	0	0
$969$	33.2705	1.06880
$970$	−8.38197	−0.269129
$971$	−34.8328	−1.11784	−0.558919	−	0.829222i	$-0.688784\pi$
$971$	−34.8328	−1.11784	−0.558919	+	0.829222i	$0.688784\pi$
$972$	3.94427	0.126513
$973$	−4.94427	−0.158506
$974$	−38.1803	−1.22338
$975$	−8.47214	−0.271325
$976$	13.7082	0.438789
$977$	−37.1935	−1.18993	−0.594963	−	0.803753i	$-0.702833\pi$
$977$	−37.1935	−1.18993	−0.594963	+	0.803753i	$0.702833\pi$
$978$	−36.8885	−1.17957
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−1.52786	−0.0487809
$982$	19.2148	0.613168
$983$	−8.94427	−0.285278	−0.142639	−	0.989775i	$-0.545559\pi$
$983$	−8.94427	−0.285278	−0.142639	+	0.989775i	$0.545559\pi$
$984$	−4.61803	−0.147218
$985$	−11.5279	−0.367308
$986$	−11.7082	−0.372865
$987$	7.70820	0.245355
$988$	−41.1246	−1.30835
$989$	0.986844	0.0313798
$990$	0	0
$991$	34.3607	1.09150	0.545751	−	0.837947i	$-0.316244\pi$
$991$	34.3607	1.09150	0.545751	+	0.837947i	$0.316244\pi$
$992$	−2.47214	−0.0784904
$993$	−1.56231	−0.0495783
$994$	−10.9443	−0.347131
$995$	−5.05573	−0.160277
$996$	−14.3262	−0.453944
$997$	−48.8328	−1.54655	−0.773275	−	0.634070i	$-0.781383\pi$
$997$	−48.8328	−1.54655	−0.773275	+	0.634070i	$0.781383\pi$
$998$	19.9787	0.632415
$999$	61.4853	1.94531

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bp.1.2		2
11.5	even	5		770.2.n.d.421.1	✓	4
11.9	even	5		770.2.n.d.631.1	yes	4
11.10	odd	2		8470.2.a.cb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.d.421.1	✓	4	11.5	even	5
770.2.n.d.631.1	yes	4	11.9	even	5
8470.2.a.bp.1.2		2	1.1	even	1	trivial
8470.2.a.cb.1.2		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +1.00000 q^{10} +1.61803 q^{12} -5.23607 q^{13} -1.00000 q^{14} -1.61803 q^{15} +1.00000 q^{16} +2.61803 q^{17} +0.381966 q^{18} +7.85410 q^{19} -1.00000 q^{20} +1.61803 q^{21} -0.472136 q^{23} -1.61803 q^{24} +1.00000 q^{25} +5.23607 q^{26} -5.47214 q^{27} +1.00000 q^{28} +4.47214 q^{29} +1.61803 q^{30} +2.47214 q^{31} -1.00000 q^{32} -2.61803 q^{34} -1.00000 q^{35} -0.381966 q^{36} -11.2361 q^{37} -7.85410 q^{38} -8.47214 q^{39} +1.00000 q^{40} +2.85410 q^{41} -1.61803 q^{42} -2.09017 q^{43} +0.381966 q^{45} +0.472136 q^{46} +4.76393 q^{47} +1.61803 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.23607 q^{51} -5.23607 q^{52} -2.76393 q^{53} +5.47214 q^{54} -1.00000 q^{56} +12.7082 q^{57} -4.47214 q^{58} +4.14590 q^{59} -1.61803 q^{60} +13.7082 q^{61} -2.47214 q^{62} -0.381966 q^{63} +1.00000 q^{64} +5.23607 q^{65} -3.38197 q^{67} +2.61803 q^{68} -0.763932 q^{69} +1.00000 q^{70} +10.9443 q^{71} +0.381966 q^{72} +2.85410 q^{73} +11.2361 q^{74} +1.61803 q^{75} +7.85410 q^{76} +8.47214 q^{78} +4.00000 q^{79} -1.00000 q^{80} -7.70820 q^{81} -2.85410 q^{82} -8.85410 q^{83} +1.61803 q^{84} -2.61803 q^{85} +2.09017 q^{86} +7.23607 q^{87} -14.8541 q^{89} -0.381966 q^{90} -5.23607 q^{91} -0.472136 q^{92} +4.00000 q^{93} -4.76393 q^{94} -7.85410 q^{95} -1.61803 q^{96} -8.38197 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + 2 q^{10} + q^{12} - 6 q^{13} - 2 q^{14} - q^{15} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 9 q^{19} - 2 q^{20} + q^{21} + 8 q^{23} - q^{24} + 2 q^{25} + 6 q^{26} - 2 q^{27} + 2 q^{28} + q^{30} - 4 q^{31} - 2 q^{32} - 3 q^{34} - 2 q^{35} - 3 q^{36} - 18 q^{37} - 9 q^{38} - 8 q^{39} + 2 q^{40} - q^{41} - q^{42} + 7 q^{43} + 3 q^{45} - 8 q^{46} + 14 q^{47} + q^{48} + 2 q^{49} - 2 q^{50} + 4 q^{51} - 6 q^{52} - 10 q^{53} + 2 q^{54} - 2 q^{56} + 12 q^{57} + 15 q^{59} - q^{60} + 14 q^{61} + 4 q^{62} - 3 q^{63} + 2 q^{64} + 6 q^{65} - 9 q^{67} + 3 q^{68} - 6 q^{69} + 2 q^{70} + 4 q^{71} + 3 q^{72} - q^{73} + 18 q^{74} + q^{75} + 9 q^{76} + 8 q^{78} + 8 q^{79} - 2 q^{80} - 2 q^{81} + q^{82} - 11 q^{83} + q^{84} - 3 q^{85} - 7 q^{86} + 10 q^{87} - 23 q^{89} - 3 q^{90} - 6 q^{91} + 8 q^{92} + 8 q^{93} - 14 q^{94} - 9 q^{95} - q^{96} - 19 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 + 2 * q^10 + q^12 - 6 * q^13 - 2 * q^14 - q^15 + 2 * q^16 + 3 * q^17 + 3 * q^18 + 9 * q^19 - 2 * q^20 + q^21 + 8 * q^23 - q^24 + 2 * q^25 + 6 * q^26 - 2 * q^27 + 2 * q^28 + q^30 - 4 * q^31 - 2 * q^32 - 3 * q^34 - 2 * q^35 - 3 * q^36 - 18 * q^37 - 9 * q^38 - 8 * q^39 + 2 * q^40 - q^41 - q^42 + 7 * q^43 + 3 * q^45 - 8 * q^46 + 14 * q^47 + q^48 + 2 * q^49 - 2 * q^50 + 4 * q^51 - 6 * q^52 - 10 * q^53 + 2 * q^54 - 2 * q^56 + 12 * q^57 + 15 * q^59 - q^60 + 14 * q^61 + 4 * q^62 - 3 * q^63 + 2 * q^64 + 6 * q^65 - 9 * q^67 + 3 * q^68 - 6 * q^69 + 2 * q^70 + 4 * q^71 + 3 * q^72 - q^73 + 18 * q^74 + q^75 + 9 * q^76 + 8 * q^78 + 8 * q^79 - 2 * q^80 - 2 * q^81 + q^82 - 11 * q^83 + q^84 - 3 * q^85 - 7 * q^86 + 10 * q^87 - 23 * q^89 - 3 * q^90 - 6 * q^91 + 8 * q^92 + 8 * q^93 - 14 * q^94 - 9 * q^95 - q^96 - 19 * q^97 - 2 * q^98

Introduction

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