LMFDB - Embedded newform 8470.2.a.dh.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:	$8$
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			1.2
Root			$-2.53932$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.53932 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.53932 q^{6} -1.00000 q^{7} +1.00000 q^{8} +3.44815 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.53932 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.53932 q^{6} -1.00000 q^{7} +1.00000 q^{8} +3.44815 q^{9} -1.00000 q^{10} -2.53932 q^{12} -4.11459 q^{13} -1.00000 q^{14} +2.53932 q^{15} +1.00000 q^{16} -6.84829 q^{17} +3.44815 q^{18} -6.62155 q^{19} -1.00000 q^{20} +2.53932 q^{21} -3.13877 q^{23} -2.53932 q^{24} +1.00000 q^{25} -4.11459 q^{26} -1.13800 q^{27} -1.00000 q^{28} +5.49561 q^{29} +2.53932 q^{30} -5.57409 q^{31} +1.00000 q^{32} -6.84829 q^{34} +1.00000 q^{35} +3.44815 q^{36} -7.78523 q^{37} -6.62155 q^{38} +10.4483 q^{39} -1.00000 q^{40} +2.90755 q^{41} +2.53932 q^{42} +3.23479 q^{43} -3.44815 q^{45} -3.13877 q^{46} -1.49988 q^{47} -2.53932 q^{48} +1.00000 q^{49} +1.00000 q^{50} +17.3900 q^{51} -4.11459 q^{52} -11.1313 q^{53} -1.13800 q^{54} -1.00000 q^{56} +16.8142 q^{57} +5.49561 q^{58} -7.42428 q^{59} +2.53932 q^{60} -2.28166 q^{61} -5.57409 q^{62} -3.44815 q^{63} +1.00000 q^{64} +4.11459 q^{65} +14.1969 q^{67} -6.84829 q^{68} +7.97035 q^{69} +1.00000 q^{70} +0.0608812 q^{71} +3.44815 q^{72} -3.91284 q^{73} -7.78523 q^{74} -2.53932 q^{75} -6.62155 q^{76} +10.4483 q^{78} -4.23983 q^{79} -1.00000 q^{80} -7.45471 q^{81} +2.90755 q^{82} +10.2468 q^{83} +2.53932 q^{84} +6.84829 q^{85} +3.23479 q^{86} -13.9551 q^{87} -6.54394 q^{89} -3.44815 q^{90} +4.11459 q^{91} -3.13877 q^{92} +14.1544 q^{93} -1.49988 q^{94} +6.62155 q^{95} -2.53932 q^{96} -0.691019 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.53932	−1.46608	−0.733039	−	0.680187i	$-0.761899\pi$
$3$	−2.53932	−1.46608	−0.733039	+	0.680187i	$0.761899\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.53932	−1.03667
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	3.44815	1.14938
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−2.53932	−0.733039
$13$	−4.11459	−1.14118	−0.570591	−	0.821234i	$-0.693286\pi$
$13$	−4.11459	−1.14118	−0.570591	+	0.821234i	$0.693286\pi$
$14$	−1.00000	−0.267261
$15$	2.53932	0.655650
$16$	1.00000	0.250000
$17$	−6.84829	−1.66095	−0.830477	−	0.557052i	$-0.811932\pi$
$17$	−6.84829	−1.66095	−0.830477	+	0.557052i	$0.811932\pi$
$18$	3.44815	0.812737
$19$	−6.62155	−1.51909	−0.759544	−	0.650456i	$-0.774578\pi$
$19$	−6.62155	−1.51909	−0.759544	+	0.650456i	$0.774578\pi$
$20$	−1.00000	−0.223607
$21$	2.53932	0.554125
$22$	0	0
$23$	−3.13877	−0.654479	−0.327240	−	0.944941i	$-0.606118\pi$
$23$	−3.13877	−0.654479	−0.327240	+	0.944941i	$0.606118\pi$
$24$	−2.53932	−0.518337
$25$	1.00000	0.200000
$26$	−4.11459	−0.806938
$27$	−1.13800	−0.219007
$28$	−1.00000	−0.188982
$29$	5.49561	1.02051	0.510255	−	0.860023i	$-0.329551\pi$
$29$	5.49561	1.02051	0.510255	+	0.860023i	$0.329551\pi$
$30$	2.53932	0.463614
$31$	−5.57409	−1.00114	−0.500568	−	0.865697i	$-0.666875\pi$
$31$	−5.57409	−1.00114	−0.500568	+	0.865697i	$0.666875\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.84829	−1.17447
$35$	1.00000	0.169031
$36$	3.44815	0.574692
$37$	−7.78523	−1.27988	−0.639942	−	0.768424i	$-0.721041\pi$
$37$	−7.78523	−1.27988	−0.639942	+	0.768424i	$0.721041\pi$
$38$	−6.62155	−1.07416
$39$	10.4483	1.67306
$40$	−1.00000	−0.158114
$41$	2.90755	0.454084	0.227042	−	0.973885i	$-0.427095\pi$
$41$	2.90755	0.454084	0.227042	+	0.973885i	$0.427095\pi$
$42$	2.53932	0.391826
$43$	3.23479	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$43$	3.23479	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$44$	0	0
$45$	−3.44815	−0.514020
$46$	−3.13877	−0.462787
$47$	−1.49988	−0.218780	−0.109390	−	0.993999i	$-0.534890\pi$
$47$	−1.49988	−0.218780	−0.109390	+	0.993999i	$0.534890\pi$
$48$	−2.53932	−0.366519
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	17.3900	2.43509
$52$	−4.11459	−0.570591
$53$	−11.1313	−1.52900	−0.764499	−	0.644625i	$-0.777013\pi$
$53$	−11.1313	−1.52900	−0.764499	+	0.644625i	$0.777013\pi$
$54$	−1.13800	−0.154862
$55$	0	0
$56$	−1.00000	−0.133631
$57$	16.8142	2.22710
$58$	5.49561	0.721610
$59$	−7.42428	−0.966560	−0.483280	−	0.875466i	$-0.660555\pi$
$59$	−7.42428	−0.966560	−0.483280	+	0.875466i	$0.660555\pi$
$60$	2.53932	0.327825
$61$	−2.28166	−0.292137	−0.146069	−	0.989274i	$-0.546662\pi$
$61$	−2.28166	−0.292137	−0.146069	+	0.989274i	$0.546662\pi$
$62$	−5.57409	−0.707910
$63$	−3.44815	−0.434426
$64$	1.00000	0.125000
$65$	4.11459	0.510352
$66$	0	0
$67$	14.1969	1.73442	0.867212	−	0.497939i	$-0.165910\pi$
$67$	14.1969	1.73442	0.867212	+	0.497939i	$0.165910\pi$
$68$	−6.84829	−0.830477
$69$	7.97035	0.959518
$70$	1.00000	0.119523
$71$	0.0608812	0.00722526	0.00361263	−	0.999993i	$-0.498850\pi$
$71$	0.0608812	0.00722526	0.00361263	+	0.999993i	$0.498850\pi$
$72$	3.44815	0.406368
$73$	−3.91284	−0.457963	−0.228982	−	0.973431i	$-0.573540\pi$
$73$	−3.91284	−0.457963	−0.228982	+	0.973431i	$0.573540\pi$
$74$	−7.78523	−0.905014
$75$	−2.53932	−0.293215
$76$	−6.62155	−0.759544
$77$	0	0
$78$	10.4483	1.18303
$79$	−4.23983	−0.477018	−0.238509	−	0.971140i	$-0.576659\pi$
$79$	−4.23983	−0.477018	−0.238509	+	0.971140i	$0.576659\pi$
$80$	−1.00000	−0.111803
$81$	−7.45471	−0.828301
$82$	2.90755	0.321086
$83$	10.2468	1.12473	0.562364	−	0.826890i	$-0.309892\pi$
$83$	10.2468	1.12473	0.562364	+	0.826890i	$0.309892\pi$
$84$	2.53932	0.277063
$85$	6.84829	0.742802
$86$	3.23479	0.348817
$87$	−13.9551	−1.49615
$88$	0	0
$89$	−6.54394	−0.693656	−0.346828	−	0.937929i	$-0.612741\pi$
$89$	−6.54394	−0.693656	−0.346828	+	0.937929i	$0.612741\pi$
$90$	−3.44815	−0.363467
$91$	4.11459	0.431326
$92$	−3.13877	−0.327240
$93$	14.1544	1.46774
$94$	−1.49988	−0.154701
$95$	6.62155	0.679357
$96$	−2.53932	−0.259168
$97$	−0.691019	−0.0701624	−0.0350812	−	0.999384i	$-0.511169\pi$
$97$	−0.691019	−0.0701624	−0.0350812	+	0.999384i	$0.511169\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	14.6190	1.45465	0.727324	−	0.686295i	$-0.240764\pi$
$101$	14.6190	1.45465	0.727324	+	0.686295i	$0.240764\pi$
$102$	17.3900	1.72187
$103$	1.07358	0.105783	0.0528914	−	0.998600i	$-0.483156\pi$
$103$	1.07358	0.105783	0.0528914	+	0.998600i	$0.483156\pi$
$104$	−4.11459	−0.403469
$105$	−2.53932	−0.247812
$106$	−11.1313	−1.08116
$107$	−18.8781	−1.82501	−0.912507	−	0.409062i	$-0.865856\pi$
$107$	−18.8781	−1.82501	−0.912507	+	0.409062i	$0.865856\pi$
$108$	−1.13800	−0.109504
$109$	4.22315	0.404504	0.202252	−	0.979333i	$-0.435174\pi$
$109$	4.22315	0.404504	0.202252	+	0.979333i	$0.435174\pi$
$110$	0	0
$111$	19.7692	1.87641
$112$	−1.00000	−0.0944911
$113$	7.36545	0.692883	0.346442	−	0.938072i	$-0.387390\pi$
$113$	7.36545	0.692883	0.346442	+	0.938072i	$0.387390\pi$
$114$	16.8142	1.57480
$115$	3.13877	0.292692
$116$	5.49561	0.510255
$117$	−14.1877	−1.31166
$118$	−7.42428	−0.683461
$119$	6.84829	0.627782
$120$	2.53932	0.231807
$121$	0	0
$122$	−2.28166	−0.206572
$123$	−7.38321	−0.665722
$124$	−5.57409	−0.500568
$125$	−1.00000	−0.0894427
$126$	−3.44815	−0.307186
$127$	−15.0031	−1.33131	−0.665656	−	0.746259i	$-0.731848\pi$
$127$	−15.0031	−1.33131	−0.665656	+	0.746259i	$0.731848\pi$
$128$	1.00000	0.0883883
$129$	−8.21418	−0.723218
$130$	4.11459	0.360874
$131$	−6.77296	−0.591757	−0.295878	−	0.955226i	$-0.595612\pi$
$131$	−6.77296	−0.591757	−0.295878	+	0.955226i	$0.595612\pi$
$132$	0	0
$133$	6.62155	0.574161
$134$	14.1969	1.22642
$135$	1.13800	0.0979431
$136$	−6.84829	−0.587236
$137$	−9.15100	−0.781822	−0.390911	−	0.920428i	$-0.627840\pi$
$137$	−9.15100	−0.781822	−0.390911	+	0.920428i	$0.627840\pi$
$138$	7.97035	0.678481
$139$	15.9131	1.34973	0.674865	−	0.737941i	$-0.264202\pi$
$139$	15.9131	1.34973	0.674865	+	0.737941i	$0.264202\pi$
$140$	1.00000	0.0845154
$141$	3.80868	0.320748
$142$	0.0608812	0.00510903
$143$	0	0
$144$	3.44815	0.287346
$145$	−5.49561	−0.456386
$146$	−3.91284	−0.323829
$147$	−2.53932	−0.209440
$148$	−7.78523	−0.639942
$149$	9.63068	0.788976	0.394488	−	0.918901i	$-0.370922\pi$
$149$	9.63068	0.788976	0.394488	+	0.918901i	$0.370922\pi$
$150$	−2.53932	−0.207335
$151$	20.3543	1.65641	0.828205	−	0.560425i	$-0.189362\pi$
$151$	20.3543	1.65641	0.828205	+	0.560425i	$0.189362\pi$
$152$	−6.62155	−0.537079
$153$	−23.6139	−1.90907
$154$	0	0
$155$	5.57409	0.447721
$156$	10.4483	0.836531
$157$	−6.01660	−0.480177	−0.240089	−	0.970751i	$-0.577177\pi$
$157$	−6.01660	−0.480177	−0.240089	+	0.970751i	$0.577177\pi$
$158$	−4.23983	−0.337303
$159$	28.2659	2.24163
$160$	−1.00000	−0.0790569
$161$	3.13877	0.247370
$162$	−7.45471	−0.585698
$163$	−8.61451	−0.674740	−0.337370	−	0.941372i	$-0.609537\pi$
$163$	−8.61451	−0.674740	−0.337370	+	0.941372i	$0.609537\pi$
$164$	2.90755	0.227042
$165$	0	0
$166$	10.2468	0.795303
$167$	−9.01590	−0.697671	−0.348836	−	0.937184i	$-0.613423\pi$
$167$	−9.01590	−0.697671	−0.348836	+	0.937184i	$0.613423\pi$
$168$	2.53932	0.195913
$169$	3.92987	0.302298
$170$	6.84829	0.525240
$171$	−22.8321	−1.74601
$172$	3.23479	0.246651
$173$	−12.9388	−0.983715	−0.491858	−	0.870676i	$-0.663682\pi$
$173$	−12.9388	−0.983715	−0.491858	+	0.870676i	$0.663682\pi$
$174$	−13.9551	−1.05794
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	18.8526	1.41705
$178$	−6.54394	−0.490489
$179$	15.7602	1.17797	0.588987	−	0.808142i	$-0.299527\pi$
$179$	15.7602	1.17797	0.588987	+	0.808142i	$0.299527\pi$
$180$	−3.44815	−0.257010
$181$	−14.6094	−1.08591	−0.542953	−	0.839763i	$-0.682694\pi$
$181$	−14.6094	−1.08591	−0.542953	+	0.839763i	$0.682694\pi$
$182$	4.11459	0.304994
$183$	5.79388	0.428296
$184$	−3.13877	−0.231393
$185$	7.78523	0.572381
$186$	14.1544	1.03785
$187$	0	0
$188$	−1.49988	−0.109390
$189$	1.13800	0.0827770
$190$	6.62155	0.480378
$191$	−17.9665	−1.30001	−0.650004	−	0.759931i	$-0.725233\pi$
$191$	−17.9665	−1.30001	−0.650004	+	0.759931i	$0.725233\pi$
$192$	−2.53932	−0.183260
$193$	23.8180	1.71446	0.857229	−	0.514935i	$-0.172184\pi$
$193$	23.8180	1.71446	0.857229	+	0.514935i	$0.172184\pi$
$194$	−0.691019	−0.0496123
$195$	−10.4483	−0.748216
$196$	1.00000	0.0714286
$197$	−1.85551	−0.132200	−0.0660998	−	0.997813i	$-0.521056\pi$
$197$	−1.85551	−0.132200	−0.0660998	+	0.997813i	$0.521056\pi$
$198$	0	0
$199$	21.8206	1.54682	0.773411	−	0.633905i	$-0.218549\pi$
$199$	21.8206	1.54682	0.773411	+	0.633905i	$0.218549\pi$
$200$	1.00000	0.0707107
$201$	−36.0504	−2.54280
$202$	14.6190	1.02859
$203$	−5.49561	−0.385717
$204$	17.3900	1.21754
$205$	−2.90755	−0.203072
$206$	1.07358	0.0747997
$207$	−10.8230	−0.752248
$208$	−4.11459	−0.285296
$209$	0	0
$210$	−2.53932	−0.175230
$211$	−26.5722	−1.82930	−0.914651	−	0.404244i	$-0.867535\pi$
$211$	−26.5722	−1.82930	−0.914651	+	0.404244i	$0.867535\pi$
$212$	−11.1313	−0.764499
$213$	−0.154597	−0.0105928
$214$	−18.8781	−1.29048
$215$	−3.23479	−0.220611
$216$	−1.13800	−0.0774308
$217$	5.57409	0.378394
$218$	4.22315	0.286028
$219$	9.93595	0.671409
$220$	0	0
$221$	28.1779	1.89545
$222$	19.7692	1.32682
$223$	−9.89233	−0.662439	−0.331220	−	0.943554i	$-0.607460\pi$
$223$	−9.89233	−0.662439	−0.331220	+	0.943554i	$0.607460\pi$
$224$	−1.00000	−0.0668153
$225$	3.44815	0.229877
$226$	7.36545	0.489942
$227$	13.9299	0.924563	0.462282	−	0.886733i	$-0.347031\pi$
$227$	13.9299	0.924563	0.462282	+	0.886733i	$0.347031\pi$
$228$	16.8142	1.11355
$229$	−10.1930	−0.673570	−0.336785	−	0.941581i	$-0.609340\pi$
$229$	−10.1930	−0.673570	−0.336785	+	0.941581i	$0.609340\pi$
$230$	3.13877	0.206965
$231$	0	0
$232$	5.49561	0.360805
$233$	−13.2121	−0.865554	−0.432777	−	0.901501i	$-0.642466\pi$
$233$	−13.2121	−0.865554	−0.432777	+	0.901501i	$0.642466\pi$
$234$	−14.1877	−0.927481
$235$	1.49988	0.0978414
$236$	−7.42428	−0.483280
$237$	10.7663	0.699345
$238$	6.84829	0.443909
$239$	5.10128	0.329974	0.164987	−	0.986296i	$-0.447242\pi$
$239$	5.10128	0.329974	0.164987	+	0.986296i	$0.447242\pi$
$240$	2.53932	0.163912
$241$	24.0075	1.54646	0.773229	−	0.634127i	$-0.218640\pi$
$241$	24.0075	1.54646	0.773229	+	0.634127i	$0.218640\pi$
$242$	0	0
$243$	22.3439	1.43336
$244$	−2.28166	−0.146069
$245$	−1.00000	−0.0638877
$246$	−7.38321	−0.470736
$247$	27.2450	1.73356
$248$	−5.57409	−0.353955
$249$	−26.0198	−1.64894
$250$	−1.00000	−0.0632456
$251$	15.4967	0.978140	0.489070	−	0.872245i	$-0.337336\pi$
$251$	15.4967	0.978140	0.489070	+	0.872245i	$0.337336\pi$
$252$	−3.44815	−0.217213
$253$	0	0
$254$	−15.0031	−0.941380
$255$	−17.3900	−1.08900
$256$	1.00000	0.0625000
$257$	−4.12482	−0.257299	−0.128650	−	0.991690i	$-0.541064\pi$
$257$	−4.12482	−0.257299	−0.128650	+	0.991690i	$0.541064\pi$
$258$	−8.21418	−0.511392
$259$	7.78523	0.483750
$260$	4.11459	0.255176
$261$	18.9497	1.17296
$262$	−6.77296	−0.418435
$263$	0.308382	0.0190156	0.00950781	−	0.999955i	$-0.496974\pi$
$263$	0.308382	0.0190156	0.00950781	+	0.999955i	$0.496974\pi$
$264$	0	0
$265$	11.1313	0.683788
$266$	6.62155	0.405993
$267$	16.6172	1.01695
$268$	14.1969	0.867212
$269$	−17.1417	−1.04515	−0.522573	−	0.852594i	$-0.675028\pi$
$269$	−17.1417	−1.04515	−0.522573	+	0.852594i	$0.675028\pi$
$270$	1.13800	0.0692562
$271$	26.6258	1.61740	0.808701	−	0.588221i	$-0.200171\pi$
$271$	26.6258	1.61740	0.808701	+	0.588221i	$0.200171\pi$
$272$	−6.84829	−0.415239
$273$	−10.4483	−0.632358
$274$	−9.15100	−0.552832
$275$	0	0
$276$	7.97035	0.479759
$277$	−15.7129	−0.944098	−0.472049	−	0.881572i	$-0.656485\pi$
$277$	−15.7129	−0.944098	−0.472049	+	0.881572i	$0.656485\pi$
$278$	15.9131	0.954404
$279$	−19.2203	−1.15069
$280$	1.00000	0.0597614
$281$	−17.4342	−1.04004	−0.520020	−	0.854154i	$-0.674075\pi$
$281$	−17.4342	−1.04004	−0.520020	+	0.854154i	$0.674075\pi$
$282$	3.80868	0.226803
$283$	15.0195	0.892814	0.446407	−	0.894830i	$-0.352703\pi$
$283$	15.0195	0.892814	0.446407	+	0.894830i	$0.352703\pi$
$284$	0.0608812	0.00361263
$285$	−16.8142	−0.995990
$286$	0	0
$287$	−2.90755	−0.171627
$288$	3.44815	0.203184
$289$	29.8991	1.75877
$290$	−5.49561	−0.322714
$291$	1.75472	0.102863
$292$	−3.91284	−0.228982
$293$	7.50667	0.438544	0.219272	−	0.975664i	$-0.429632\pi$
$293$	7.50667	0.438544	0.219272	+	0.975664i	$0.429632\pi$
$294$	−2.53932	−0.148096
$295$	7.42428	0.432259
$296$	−7.78523	−0.452507
$297$	0	0
$298$	9.63068	0.557891
$299$	12.9148	0.746881
$300$	−2.53932	−0.146608
$301$	−3.23479	−0.186450
$302$	20.3543	1.17126
$303$	−37.1224	−2.13263
$304$	−6.62155	−0.379772
$305$	2.28166	0.130648
$306$	−23.6139	−1.34992
$307$	−5.08116	−0.289997	−0.144998	−	0.989432i	$-0.546318\pi$
$307$	−5.08116	−0.289997	−0.144998	+	0.989432i	$0.546318\pi$
$308$	0	0
$309$	−2.72616	−0.155086
$310$	5.57409	0.316587
$311$	10.8436	0.614884	0.307442	−	0.951567i	$-0.400527\pi$
$311$	10.8436	0.614884	0.307442	+	0.951567i	$0.400527\pi$
$312$	10.4483	0.591517
$313$	33.4044	1.88813	0.944064	−	0.329764i	$-0.106969\pi$
$313$	33.4044	1.88813	0.944064	+	0.329764i	$0.106969\pi$
$314$	−6.01660	−0.339537
$315$	3.44815	0.194281
$316$	−4.23983	−0.238509
$317$	−27.7007	−1.55583	−0.777914	−	0.628371i	$-0.783722\pi$
$317$	−27.7007	−1.55583	−0.777914	+	0.628371i	$0.783722\pi$
$318$	28.2659	1.58507
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	47.9375	2.67561
$322$	3.13877	0.174917
$323$	45.3463	2.52314
$324$	−7.45471	−0.414151
$325$	−4.11459	−0.228237
$326$	−8.61451	−0.477113
$327$	−10.7239	−0.593035
$328$	2.90755	0.160543
$329$	1.49988	0.0826911
$330$	0	0
$331$	−28.8317	−1.58473	−0.792367	−	0.610045i	$-0.791152\pi$
$331$	−28.8317	−1.58473	−0.792367	+	0.610045i	$0.791152\pi$
$332$	10.2468	0.562364
$333$	−26.8446	−1.47108
$334$	−9.01590	−0.493328
$335$	−14.1969	−0.775658
$336$	2.53932	0.138531
$337$	−16.9730	−0.924577	−0.462288	−	0.886730i	$-0.652972\pi$
$337$	−16.9730	−0.924577	−0.462288	+	0.886730i	$0.652972\pi$
$338$	3.92987	0.213757
$339$	−18.7032	−1.01582
$340$	6.84829	0.371401
$341$	0	0
$342$	−22.8321	−1.23462
$343$	−1.00000	−0.0539949
$344$	3.23479	0.174408
$345$	−7.97035	−0.429109
$346$	−12.9388	−0.695592
$347$	22.8463	1.22645	0.613227	−	0.789907i	$-0.289871\pi$
$347$	22.8463	1.22645	0.613227	+	0.789907i	$0.289871\pi$
$348$	−13.9551	−0.748073
$349$	−15.1421	−0.810540	−0.405270	−	0.914197i	$-0.632822\pi$
$349$	−15.1421	−0.810540	−0.405270	+	0.914197i	$0.632822\pi$
$350$	−1.00000	−0.0534522
$351$	4.68239	0.249927
$352$	0	0
$353$	−9.01014	−0.479561	−0.239781	−	0.970827i	$-0.577076\pi$
$353$	−9.01014	−0.479561	−0.239781	+	0.970827i	$0.577076\pi$
$354$	18.8526	1.00201
$355$	−0.0608812	−0.00323124
$356$	−6.54394	−0.346828
$357$	−17.3900	−0.920377
$358$	15.7602	0.832954
$359$	36.8929	1.94713	0.973566	−	0.228408i	$-0.0733519\pi$
$359$	36.8929	1.94713	0.973566	+	0.228408i	$0.0733519\pi$
$360$	−3.44815	−0.181733
$361$	24.8450	1.30763
$362$	−14.6094	−0.767852
$363$	0	0
$364$	4.11459	0.215663
$365$	3.91284	0.204807
$366$	5.79388	0.302851
$367$	34.5081	1.80131	0.900654	−	0.434537i	$-0.143088\pi$
$367$	34.5081	1.80131	0.900654	+	0.434537i	$0.143088\pi$
$368$	−3.13877	−0.163620
$369$	10.0257	0.521916
$370$	7.78523	0.404735
$371$	11.1313	0.577907
$372$	14.1544	0.733871
$373$	2.31601	0.119918	0.0599592	−	0.998201i	$-0.480903\pi$
$373$	2.31601	0.119918	0.0599592	+	0.998201i	$0.480903\pi$
$374$	0	0
$375$	2.53932	0.131130
$376$	−1.49988	−0.0773504
$377$	−22.6122	−1.16459
$378$	1.13800	0.0585322
$379$	−7.92393	−0.407025	−0.203513	−	0.979072i	$-0.565236\pi$
$379$	−7.92393	−0.407025	−0.203513	+	0.979072i	$0.565236\pi$
$380$	6.62155	0.339678
$381$	38.0977	1.95181
$382$	−17.9665	−0.919244
$383$	19.2045	0.981304	0.490652	−	0.871356i	$-0.336759\pi$
$383$	19.2045	0.981304	0.490652	+	0.871356i	$0.336759\pi$
$384$	−2.53932	−0.129584
$385$	0	0
$386$	23.8180	1.21231
$387$	11.1540	0.566992
$388$	−0.691019	−0.0350812
$389$	−7.65932	−0.388343	−0.194172	−	0.980968i	$-0.562202\pi$
$389$	−7.65932	−0.388343	−0.194172	+	0.980968i	$0.562202\pi$
$390$	−10.4483	−0.529069
$391$	21.4952	1.08706
$392$	1.00000	0.0505076
$393$	17.1987	0.867561
$394$	−1.85551	−0.0934793
$395$	4.23983	0.213329
$396$	0	0
$397$	−20.0239	−1.00497	−0.502486	−	0.864585i	$-0.667581\pi$
$397$	−20.0239	−1.00497	−0.502486	+	0.864585i	$0.667581\pi$
$398$	21.8206	1.09377
$399$	−16.8142	−0.841765
$400$	1.00000	0.0500000
$401$	20.7562	1.03651	0.518257	−	0.855225i	$-0.326581\pi$
$401$	20.7562	1.03651	0.518257	+	0.855225i	$0.326581\pi$
$402$	−36.0504	−1.79803
$403$	22.9351	1.14248
$404$	14.6190	0.727324
$405$	7.45471	0.370428
$406$	−5.49561	−0.272743
$407$	0	0
$408$	17.3900	0.860934
$409$	32.8553	1.62459	0.812295	−	0.583247i	$-0.198218\pi$
$409$	32.8553	1.62459	0.812295	+	0.583247i	$0.198218\pi$
$410$	−2.90755	−0.143594
$411$	23.2373	1.14621
$412$	1.07358	0.0528914
$413$	7.42428	0.365325
$414$	−10.8230	−0.531919
$415$	−10.2468	−0.502994
$416$	−4.11459	−0.201734
$417$	−40.4084	−1.97881
$418$	0	0
$419$	−16.3625	−0.799361	−0.399680	−	0.916655i	$-0.630879\pi$
$419$	−16.3625	−0.799361	−0.399680	+	0.916655i	$0.630879\pi$
$420$	−2.53932	−0.123906
$421$	26.0472	1.26946	0.634731	−	0.772733i	$-0.281111\pi$
$421$	26.0472	1.26946	0.634731	+	0.772733i	$0.281111\pi$
$422$	−26.5722	−1.29351
$423$	−5.17181	−0.251462
$424$	−11.1313	−0.540582
$425$	−6.84829	−0.332191
$426$	−0.154597	−0.00749024
$427$	2.28166	0.110417
$428$	−18.8781	−0.912507
$429$	0	0
$430$	−3.23479	−0.155996
$431$	18.2575	0.879433	0.439716	−	0.898137i	$-0.355079\pi$
$431$	18.2575	0.879433	0.439716	+	0.898137i	$0.355079\pi$
$432$	−1.13800	−0.0547518
$433$	32.6096	1.56712	0.783560	−	0.621316i	$-0.213402\pi$
$433$	32.6096	1.56712	0.783560	+	0.621316i	$0.213402\pi$
$434$	5.57409	0.267565
$435$	13.9551	0.669097
$436$	4.22315	0.202252
$437$	20.7836	0.994212
$438$	9.93595	0.474758
$439$	−29.2540	−1.39622	−0.698108	−	0.715993i	$-0.745974\pi$
$439$	−29.2540	−1.39622	−0.698108	+	0.715993i	$0.745974\pi$
$440$	0	0
$441$	3.44815	0.164198
$442$	28.1779	1.34029
$443$	11.4091	0.542063	0.271032	−	0.962570i	$-0.412635\pi$
$443$	11.4091	0.542063	0.271032	+	0.962570i	$0.412635\pi$
$444$	19.7692	0.938204
$445$	6.54394	0.310212
$446$	−9.89233	−0.468415
$447$	−24.4554	−1.15670
$448$	−1.00000	−0.0472456
$449$	13.6315	0.643312	0.321656	−	0.946857i	$-0.395760\pi$
$449$	13.6315	0.643312	0.321656	+	0.946857i	$0.395760\pi$
$450$	3.44815	0.162547
$451$	0	0
$452$	7.36545	0.346442
$453$	−51.6861	−2.42843
$454$	13.9299	0.653765
$455$	−4.11459	−0.192895
$456$	16.8142	0.787399
$457$	−6.66357	−0.311708	−0.155854	−	0.987780i	$-0.549813\pi$
$457$	−6.66357	−0.311708	−0.155854	+	0.987780i	$0.549813\pi$
$458$	−10.1930	−0.476286
$459$	7.79333	0.363761
$460$	3.13877	0.146346
$461$	18.1977	0.847553	0.423776	−	0.905767i	$-0.360704\pi$
$461$	18.1977	0.847553	0.423776	+	0.905767i	$0.360704\pi$
$462$	0	0
$463$	11.3971	0.529666	0.264833	−	0.964294i	$-0.414683\pi$
$463$	11.3971	0.529666	0.264833	+	0.964294i	$0.414683\pi$
$464$	5.49561	0.255128
$465$	−14.1544	−0.656394
$466$	−13.2121	−0.612039
$467$	22.9793	1.06336	0.531679	−	0.846946i	$-0.321561\pi$
$467$	22.9793	1.06336	0.531679	+	0.846946i	$0.321561\pi$
$468$	−14.1877	−0.655828
$469$	−14.1969	−0.655551
$470$	1.49988	0.0691843
$471$	15.2781	0.703977
$472$	−7.42428	−0.341730
$473$	0	0
$474$	10.7663	0.494512
$475$	−6.62155	−0.303818
$476$	6.84829	0.313891
$477$	−38.3823	−1.75740
$478$	5.10128	0.233327
$479$	1.36694	0.0624573	0.0312286	−	0.999512i	$-0.490058\pi$
$479$	1.36694	0.0624573	0.0312286	+	0.999512i	$0.490058\pi$
$480$	2.53932	0.115904
$481$	32.0330	1.46058
$482$	24.0075	1.09351
$483$	−7.97035	−0.362664
$484$	0	0
$485$	0.691019	0.0313776
$486$	22.3439	1.01354
$487$	−12.2098	−0.553277	−0.276638	−	0.960974i	$-0.589220\pi$
$487$	−12.2098	−0.553277	−0.276638	+	0.960974i	$0.589220\pi$
$488$	−2.28166	−0.103286
$489$	21.8750	0.989221
$490$	−1.00000	−0.0451754
$491$	39.8546	1.79861	0.899307	−	0.437318i	$-0.144072\pi$
$491$	39.8546	1.79861	0.899307	+	0.437318i	$0.144072\pi$
$492$	−7.38321	−0.332861
$493$	−37.6356	−1.69502
$494$	27.2450	1.22581
$495$	0	0
$496$	−5.57409	−0.250284
$497$	−0.0608812	−0.00273089
$498$	−26.0198	−1.16598
$499$	36.7505	1.64518	0.822589	−	0.568636i	$-0.192529\pi$
$499$	36.7505	1.64518	0.822589	+	0.568636i	$0.192529\pi$
$500$	−1.00000	−0.0447214
$501$	22.8943	1.02284
$502$	15.4967	0.691650
$503$	11.9782	0.534081	0.267041	−	0.963685i	$-0.413954\pi$
$503$	11.9782	0.534081	0.267041	+	0.963685i	$0.413954\pi$
$504$	−3.44815	−0.153593
$505$	−14.6190	−0.650538
$506$	0	0
$507$	−9.97920	−0.443192
$508$	−15.0031	−0.665656
$509$	−23.6608	−1.04875	−0.524374	−	0.851488i	$-0.675701\pi$
$509$	−23.6608	−1.04875	−0.524374	+	0.851488i	$0.675701\pi$
$510$	−17.3900	−0.770043
$511$	3.91284	0.173094
$512$	1.00000	0.0441942
$513$	7.53530	0.332691
$514$	−4.12482	−0.181938
$515$	−1.07358	−0.0473075
$516$	−8.21418	−0.361609
$517$	0	0
$518$	7.78523	0.342063
$519$	32.8556	1.44220
$520$	4.11459	0.180437
$521$	−9.40537	−0.412057	−0.206028	−	0.978546i	$-0.566054\pi$
$521$	−9.40537	−0.412057	−0.206028	+	0.978546i	$0.566054\pi$
$522$	18.9497	0.829406
$523$	−28.4936	−1.24594	−0.622968	−	0.782247i	$-0.714074\pi$
$523$	−28.4936	−1.24594	−0.622968	+	0.782247i	$0.714074\pi$
$524$	−6.77296	−0.295878
$525$	2.53932	0.110825
$526$	0.308382	0.0134461
$527$	38.1730	1.66284
$528$	0	0
$529$	−13.1481	−0.571657
$530$	11.1313	0.483511
$531$	−25.6000	−1.11095
$532$	6.62155	0.287081
$533$	−11.9634	−0.518192
$534$	16.6172	0.719095
$535$	18.8781	0.816171
$536$	14.1969	0.613211
$537$	−40.0203	−1.72700
$538$	−17.1417	−0.739030
$539$	0	0
$540$	1.13800	0.0489715
$541$	−2.39709	−0.103059	−0.0515294	−	0.998671i	$-0.516410\pi$
$541$	−2.39709	−0.103059	−0.0515294	+	0.998671i	$0.516410\pi$
$542$	26.6258	1.14368
$543$	37.0979	1.59202
$544$	−6.84829	−0.293618
$545$	−4.22315	−0.180900
$546$	−10.4483	−0.447145
$547$	13.0332	0.557260	0.278630	−	0.960399i	$-0.410120\pi$
$547$	13.0332	0.557260	0.278630	+	0.960399i	$0.410120\pi$
$548$	−9.15100	−0.390911
$549$	−7.86752	−0.335777
$550$	0	0
$551$	−36.3895	−1.55024
$552$	7.97035	0.339241
$553$	4.23983	0.180296
$554$	−15.7129	−0.667578
$555$	−19.7692	−0.839155
$556$	15.9131	0.674865
$557$	20.1348	0.853138	0.426569	−	0.904455i	$-0.359722\pi$
$557$	20.1348	0.853138	0.426569	+	0.904455i	$0.359722\pi$
$558$	−19.2203	−0.813660
$559$	−13.3099	−0.562947
$560$	1.00000	0.0422577
$561$	0	0
$562$	−17.4342	−0.735419
$563$	10.8907	0.458988	0.229494	−	0.973310i	$-0.426293\pi$
$563$	10.8907	0.458988	0.229494	+	0.973310i	$0.426293\pi$
$564$	3.80868	0.160374
$565$	−7.36545	−0.309867
$566$	15.0195	0.631315
$567$	7.45471	0.313069
$568$	0.0608812	0.00255452
$569$	−22.5186	−0.944027	−0.472014	−	0.881591i	$-0.656473\pi$
$569$	−22.5186	−0.944027	−0.472014	+	0.881591i	$0.656473\pi$
$570$	−16.8142	−0.704271
$571$	14.1198	0.590893	0.295447	−	0.955359i	$-0.404532\pi$
$571$	14.1198	0.590893	0.295447	+	0.955359i	$0.404532\pi$
$572$	0	0
$573$	45.6226	1.90591
$574$	−2.90755	−0.121359
$575$	−3.13877	−0.130896
$576$	3.44815	0.143673
$577$	−12.5952	−0.524344	−0.262172	−	0.965021i	$-0.584439\pi$
$577$	−12.5952	−0.524344	−0.262172	+	0.965021i	$0.584439\pi$
$578$	29.8991	1.24364
$579$	−60.4816	−2.51353
$580$	−5.49561	−0.228193
$581$	−10.2468	−0.425107
$582$	1.75472	0.0727355
$583$	0	0
$584$	−3.91284	−0.161914
$585$	14.1877	0.586590
$586$	7.50667	0.310098
$587$	−15.4018	−0.635701	−0.317851	−	0.948141i	$-0.602961\pi$
$587$	−15.4018	−0.635701	−0.317851	+	0.948141i	$0.602961\pi$
$588$	−2.53932	−0.104720
$589$	36.9091	1.52081
$590$	7.42428	0.305653
$591$	4.71174	0.193815
$592$	−7.78523	−0.319971
$593$	27.0005	1.10878	0.554389	−	0.832258i	$-0.312952\pi$
$593$	27.0005	1.10878	0.554389	+	0.832258i	$0.312952\pi$
$594$	0	0
$595$	−6.84829	−0.280753
$596$	9.63068	0.394488
$597$	−55.4095	−2.26776
$598$	12.9148	0.528124
$599$	38.5299	1.57429	0.787145	−	0.616768i	$-0.211558\pi$
$599$	38.5299	1.57429	0.787145	+	0.616768i	$0.211558\pi$
$600$	−2.53932	−0.103667
$601$	15.9195	0.649371	0.324686	−	0.945822i	$-0.394742\pi$
$601$	15.9195	0.649371	0.324686	+	0.945822i	$0.394742\pi$
$602$	−3.23479	−0.131840
$603$	48.9529	1.99352
$604$	20.3543	0.828205
$605$	0	0
$606$	−37.1224	−1.50799
$607$	20.2644	0.822509	0.411254	−	0.911521i	$-0.365091\pi$
$607$	20.2644	0.822509	0.411254	+	0.911521i	$0.365091\pi$
$608$	−6.62155	−0.268539
$609$	13.9551	0.565490
$610$	2.28166	0.0923819
$611$	6.17139	0.249668
$612$	−23.6139	−0.954537
$613$	−28.0631	−1.13346	−0.566728	−	0.823905i	$-0.691791\pi$
$613$	−28.0631	−1.13346	−0.566728	+	0.823905i	$0.691791\pi$
$614$	−5.08116	−0.205059
$615$	7.38321	0.297720
$616$	0	0
$617$	38.0285	1.53097	0.765484	−	0.643455i	$-0.222500\pi$
$617$	38.0285	1.53097	0.765484	+	0.643455i	$0.222500\pi$
$618$	−2.72616	−0.109662
$619$	−34.7958	−1.39856	−0.699281	−	0.714847i	$-0.746496\pi$
$619$	−34.7958	−1.39856	−0.699281	+	0.714847i	$0.746496\pi$
$620$	5.57409	0.223861
$621$	3.57191	0.143336
$622$	10.8436	0.434788
$623$	6.54394	0.262177
$624$	10.4483	0.418266
$625$	1.00000	0.0400000
$626$	33.4044	1.33511
$627$	0	0
$628$	−6.01660	−0.240089
$629$	53.3155	2.12583
$630$	3.44815	0.137378
$631$	−29.7828	−1.18564	−0.592818	−	0.805337i	$-0.701985\pi$
$631$	−29.7828	−1.18564	−0.592818	+	0.805337i	$0.701985\pi$
$632$	−4.23983	−0.168651
$633$	67.4752	2.68190
$634$	−27.7007	−1.10014
$635$	15.0031	0.595381
$636$	28.2659	1.12081
$637$	−4.11459	−0.163026
$638$	0	0
$639$	0.209927	0.00830460
$640$	−1.00000	−0.0395285
$641$	0.579841	0.0229023	0.0114512	−	0.999934i	$-0.496355\pi$
$641$	0.579841	0.0229023	0.0114512	+	0.999934i	$0.496355\pi$
$642$	47.9375	1.89194
$643$	47.4766	1.87229	0.936147	−	0.351609i	$-0.114365\pi$
$643$	47.4766	1.87229	0.936147	+	0.351609i	$0.114365\pi$
$644$	3.13877	0.123685
$645$	8.21418	0.323433
$646$	45.3463	1.78413
$647$	−41.5395	−1.63309	−0.816543	−	0.577284i	$-0.804112\pi$
$647$	−41.5395	−1.63309	−0.816543	+	0.577284i	$0.804112\pi$
$648$	−7.45471	−0.292849
$649$	0	0
$650$	−4.11459	−0.161388
$651$	−14.1544	−0.554755
$652$	−8.61451	−0.337370
$653$	−26.4931	−1.03676	−0.518378	−	0.855152i	$-0.673464\pi$
$653$	−26.4931	−1.03676	−0.518378	+	0.855152i	$0.673464\pi$
$654$	−10.7239	−0.419339
$655$	6.77296	0.264642
$656$	2.90755	0.113521
$657$	−13.4921	−0.526375
$658$	1.49988	0.0584714
$659$	−46.8092	−1.82343	−0.911713	−	0.410827i	$-0.865240\pi$
$659$	−46.8092	−1.82343	−0.911713	+	0.410827i	$0.865240\pi$
$660$	0	0
$661$	−35.7060	−1.38880	−0.694401	−	0.719588i	$-0.744331\pi$
$661$	−35.7060	−1.38880	−0.694401	+	0.719588i	$0.744331\pi$
$662$	−28.8317	−1.12058
$663$	−71.5528	−2.77888
$664$	10.2468	0.397652
$665$	−6.62155	−0.256773
$666$	−26.8446	−1.04021
$667$	−17.2495	−0.667903
$668$	−9.01590	−0.348836
$669$	25.1198	0.971188
$670$	−14.1969	−0.548473
$671$	0	0
$672$	2.53932	0.0979564
$673$	15.6243	0.602272	0.301136	−	0.953581i	$-0.402634\pi$
$673$	15.6243	0.602272	0.301136	+	0.953581i	$0.402634\pi$
$674$	−16.9730	−0.653775
$675$	−1.13800	−0.0438015
$676$	3.92987	0.151149
$677$	11.3333	0.435574	0.217787	−	0.975996i	$-0.430116\pi$
$677$	11.3333	0.435574	0.217787	+	0.975996i	$0.430116\pi$
$678$	−18.7032	−0.718294
$679$	0.691019	0.0265189
$680$	6.84829	0.262620
$681$	−35.3726	−1.35548
$682$	0	0
$683$	−12.5315	−0.479504	−0.239752	−	0.970834i	$-0.577066\pi$
$683$	−12.5315	−0.479504	−0.239752	+	0.970834i	$0.577066\pi$
$684$	−22.8321	−0.873007
$685$	9.15100	0.349642
$686$	−1.00000	−0.0381802
$687$	25.8832	0.987506
$688$	3.23479	0.123325
$689$	45.8006	1.74487
$690$	−7.97035	−0.303426
$691$	3.41506	0.129915	0.0649574	−	0.997888i	$-0.479309\pi$
$691$	3.41506	0.129915	0.0649574	+	0.997888i	$0.479309\pi$
$692$	−12.9388	−0.491858
$693$	0	0
$694$	22.8463	0.867234
$695$	−15.9131	−0.603618
$696$	−13.9551	−0.528968
$697$	−19.9118	−0.754212
$698$	−15.1421	−0.573138
$699$	33.5498	1.26897
$700$	−1.00000	−0.0377964
$701$	35.1062	1.32594	0.662971	−	0.748645i	$-0.269295\pi$
$701$	35.1062	1.32594	0.662971	+	0.748645i	$0.269295\pi$
$702$	4.68239	0.176725
$703$	51.5503	1.94426
$704$	0	0
$705$	−3.80868	−0.143443
$706$	−9.01014	−0.339101
$707$	−14.6190	−0.549805
$708$	18.8526	0.708526
$709$	−28.9326	−1.08659	−0.543293	−	0.839543i	$-0.682823\pi$
$709$	−28.9326	−1.08659	−0.543293	+	0.839543i	$0.682823\pi$
$710$	−0.0608812	−0.00228483
$711$	−14.6196	−0.548276
$712$	−6.54394	−0.245244
$713$	17.4958	0.655223
$714$	−17.3900	−0.650805
$715$	0	0
$716$	15.7602	0.588987
$717$	−12.9538	−0.483768
$718$	36.8929	1.37683
$719$	−7.06973	−0.263656	−0.131828	−	0.991273i	$-0.542085\pi$
$719$	−7.06973	−0.263656	−0.131828	+	0.991273i	$0.542085\pi$
$720$	−3.44815	−0.128505
$721$	−1.07358	−0.0399821
$722$	24.8450	0.924634
$723$	−60.9627	−2.26723
$724$	−14.6094	−0.542953
$725$	5.49561	0.204102
$726$	0	0
$727$	−28.5131	−1.05749	−0.528747	−	0.848779i	$-0.677338\pi$
$727$	−28.5131	−1.05749	−0.528747	+	0.848779i	$0.677338\pi$
$728$	4.11459	0.152497
$729$	−34.3742	−1.27312
$730$	3.91284	0.144821
$731$	−22.1528	−0.819351
$732$	5.79388	0.214148
$733$	40.1218	1.48193	0.740967	−	0.671542i	$-0.234368\pi$
$733$	40.1218	1.48193	0.740967	+	0.671542i	$0.234368\pi$
$734$	34.5081	1.27372
$735$	2.53932	0.0936643
$736$	−3.13877	−0.115697
$737$	0	0
$738$	10.0257	0.369050
$739$	−36.5692	−1.34522	−0.672610	−	0.739998i	$-0.734827\pi$
$739$	−36.5692	−1.34522	−0.672610	+	0.739998i	$0.734827\pi$
$740$	7.78523	0.286191
$741$	−69.1838	−2.54153
$742$	11.1313	0.408642
$743$	−0.0491486	−0.00180309	−0.000901543	−	1.00000i	$-0.500287\pi$
$743$	−0.0491486	−0.00180309	−0.000901543		1.00000i	$0.500287\pi$
$744$	14.1544	0.518925
$745$	−9.63068	−0.352841
$746$	2.31601	0.0847951
$747$	35.3324	1.29274
$748$	0	0
$749$	18.8781	0.689790
$750$	2.53932	0.0927229
$751$	6.39915	0.233508	0.116754	−	0.993161i	$-0.462751\pi$
$751$	6.39915	0.233508	0.116754	+	0.993161i	$0.462751\pi$
$752$	−1.49988	−0.0546950
$753$	−39.3510	−1.43403
$754$	−22.6122	−0.823488
$755$	−20.3543	−0.740769
$756$	1.13800	0.0413885
$757$	−19.2059	−0.698050	−0.349025	−	0.937114i	$-0.613487\pi$
$757$	−19.2059	−0.698050	−0.349025	+	0.937114i	$0.613487\pi$
$758$	−7.92393	−0.287810
$759$	0	0
$760$	6.62155	0.240189
$761$	−2.87463	−0.104205	−0.0521026	−	0.998642i	$-0.516592\pi$
$761$	−2.87463	−0.104205	−0.0521026	+	0.998642i	$0.516592\pi$
$762$	38.0977	1.38014
$763$	−4.22315	−0.152888
$764$	−17.9665	−0.650004
$765$	23.6139	0.853764
$766$	19.2045	0.693887
$767$	30.5479	1.10302
$768$	−2.53932	−0.0916298
$769$	−35.4758	−1.27929	−0.639646	−	0.768670i	$-0.720919\pi$
$769$	−35.4758	−1.27929	−0.639646	+	0.768670i	$0.720919\pi$
$770$	0	0
$771$	10.4742	0.377220
$772$	23.8180	0.857229
$773$	−31.2613	−1.12439	−0.562195	−	0.827005i	$-0.690043\pi$
$773$	−31.2613	−1.12439	−0.562195	+	0.827005i	$0.690043\pi$
$774$	11.1540	0.400924
$775$	−5.57409	−0.200227
$776$	−0.691019	−0.0248062
$777$	−19.7692	−0.709216
$778$	−7.65932	−0.274600
$779$	−19.2525	−0.689793
$780$	−10.4483	−0.374108
$781$	0	0
$782$	21.4952	0.768668
$783$	−6.25398	−0.223499
$784$	1.00000	0.0357143
$785$	6.01660	0.214742
$786$	17.1987	0.613458
$787$	−12.9176	−0.460462	−0.230231	−	0.973136i	$-0.573948\pi$
$787$	−12.9176	−0.460462	−0.230231	+	0.973136i	$0.573948\pi$
$788$	−1.85551	−0.0660998
$789$	−0.783080	−0.0278784
$790$	4.23983	0.150846
$791$	−7.36545	−0.261885
$792$	0	0
$793$	9.38812	0.333382
$794$	−20.0239	−0.710622
$795$	−28.2659	−1.00249
$796$	21.8206	0.773411
$797$	10.1101	0.358119	0.179060	−	0.983838i	$-0.442694\pi$
$797$	10.1101	0.358119	0.179060	+	0.983838i	$0.442694\pi$
$798$	−16.8142	−0.595218
$799$	10.2716	0.363384
$800$	1.00000	0.0353553
$801$	−22.5645	−0.797276
$802$	20.7562	0.732926
$803$	0	0
$804$	−36.0504	−1.27140
$805$	−3.13877	−0.110627
$806$	22.9351	0.807854
$807$	43.5282	1.53227
$808$	14.6190	0.514296
$809$	5.06439	0.178054	0.0890272	−	0.996029i	$-0.471624\pi$
$809$	5.06439	0.178054	0.0890272	+	0.996029i	$0.471624\pi$
$810$	7.45471	0.261932
$811$	−9.52150	−0.334345	−0.167173	−	0.985928i	$-0.553464\pi$
$811$	−9.52150	−0.334345	−0.167173	+	0.985928i	$0.553464\pi$
$812$	−5.49561	−0.192858
$813$	−67.6114	−2.37124
$814$	0	0
$815$	8.61451	0.301753
$816$	17.3900	0.608772
$817$	−21.4193	−0.749368
$818$	32.8553	1.14876
$819$	14.1877	0.495759
$820$	−2.90755	−0.101536
$821$	17.4006	0.607286	0.303643	−	0.952786i	$-0.401797\pi$
$821$	17.4006	0.607286	0.303643	+	0.952786i	$0.401797\pi$
$822$	23.2373	0.810494
$823$	18.4859	0.644379	0.322189	−	0.946675i	$-0.395581\pi$
$823$	18.4859	0.644379	0.322189	+	0.946675i	$0.395581\pi$
$824$	1.07358	0.0373998
$825$	0	0
$826$	7.42428	0.258324
$827$	−55.2703	−1.92194	−0.960968	−	0.276660i	$-0.910772\pi$
$827$	−55.2703	−1.92194	−0.960968	+	0.276660i	$0.910772\pi$
$828$	−10.8230	−0.376124
$829$	−20.0696	−0.697046	−0.348523	−	0.937300i	$-0.613317\pi$
$829$	−20.0696	−0.697046	−0.348523	+	0.937300i	$0.613317\pi$
$830$	−10.2468	−0.355670
$831$	39.9002	1.38412
$832$	−4.11459	−0.142648
$833$	−6.84829	−0.237279
$834$	−40.4084	−1.39923
$835$	9.01590	0.312008
$836$	0	0
$837$	6.34329	0.219256
$838$	−16.3625	−0.565233
$839$	−10.7623	−0.371556	−0.185778	−	0.982592i	$-0.559481\pi$
$839$	−10.7623	−0.371556	−0.185778	+	0.982592i	$0.559481\pi$
$840$	−2.53932	−0.0876149
$841$	1.20178	0.0414407
$842$	26.0472	0.897645
$843$	44.2711	1.52478
$844$	−26.5722	−0.914651
$845$	−3.92987	−0.135192
$846$	−5.17181	−0.177811
$847$	0	0
$848$	−11.1313	−0.382249
$849$	−38.1392	−1.30893
$850$	−6.84829	−0.234894
$851$	24.4361	0.837657
$852$	−0.154597	−0.00529640
$853$	−14.6593	−0.501923	−0.250962	−	0.967997i	$-0.580747\pi$
$853$	−14.6593	−0.501923	−0.250962	+	0.967997i	$0.580747\pi$
$854$	2.28166	0.0780769
$855$	22.8321	0.780841
$856$	−18.8781	−0.645240
$857$	34.6532	1.18373	0.591865	−	0.806037i	$-0.298392\pi$
$857$	34.6532	1.18373	0.591865	+	0.806037i	$0.298392\pi$
$858$	0	0
$859$	9.13441	0.311662	0.155831	−	0.987784i	$-0.450194\pi$
$859$	9.13441	0.311662	0.155831	+	0.987784i	$0.450194\pi$
$860$	−3.23479	−0.110306
$861$	7.38321	0.251619
$862$	18.2575	0.621853
$863$	23.4381	0.797841	0.398921	−	0.916986i	$-0.369385\pi$
$863$	23.4381	0.797841	0.398921	+	0.916986i	$0.369385\pi$
$864$	−1.13800	−0.0387154
$865$	12.9388	0.439931
$866$	32.6096	1.10812
$867$	−75.9234	−2.57849
$868$	5.57409	0.189197
$869$	0	0
$870$	13.9551	0.473123
$871$	−58.4143	−1.97929
$872$	4.22315	0.143014
$873$	−2.38274	−0.0806435
$874$	20.7836	0.703014
$875$	1.00000	0.0338062
$876$	9.93595	0.335705
$877$	−50.9962	−1.72202	−0.861010	−	0.508589i	$-0.830167\pi$
$877$	−50.9962	−1.72202	−0.861010	+	0.508589i	$0.830167\pi$
$878$	−29.2540	−0.987274
$879$	−19.0618	−0.642940
$880$	0	0
$881$	−1.75183	−0.0590207	−0.0295103	−	0.999564i	$-0.509395\pi$
$881$	−1.75183	−0.0590207	−0.0295103	+	0.999564i	$0.509395\pi$
$882$	3.44815	0.116105
$883$	−40.8658	−1.37524	−0.687621	−	0.726070i	$-0.741345\pi$
$883$	−40.8658	−1.37524	−0.687621	+	0.726070i	$0.741345\pi$
$884$	28.1779	0.947726
$885$	−18.8526	−0.633725
$886$	11.4091	0.383297
$887$	−23.5552	−0.790906	−0.395453	−	0.918486i	$-0.629412\pi$
$887$	−23.5552	−0.790906	−0.395453	+	0.918486i	$0.629412\pi$
$888$	19.7692	0.663410
$889$	15.0031	0.503189
$890$	6.54394	0.219353
$891$	0	0
$892$	−9.89233	−0.331220
$893$	9.93153	0.332346
$894$	−24.4554	−0.817911
$895$	−15.7602	−0.526806
$896$	−1.00000	−0.0334077
$897$	−32.7947	−1.09498
$898$	13.6315	0.454891
$899$	−30.6330	−1.02167
$900$	3.44815	0.114938
$901$	76.2302	2.53960
$902$	0	0
$903$	8.21418	0.273351
$904$	7.36545	0.244971
$905$	14.6094	0.485632
$906$	−51.6861	−1.71716
$907$	−41.4267	−1.37555	−0.687776	−	0.725923i	$-0.741413\pi$
$907$	−41.4267	−1.37555	−0.687776	+	0.725923i	$0.741413\pi$
$908$	13.9299	0.462282
$909$	50.4086	1.67195
$910$	−4.11459	−0.136397
$911$	45.1138	1.49469	0.747344	−	0.664437i	$-0.231329\pi$
$911$	45.1138	1.49469	0.747344	+	0.664437i	$0.231329\pi$
$912$	16.8142	0.556775
$913$	0	0
$914$	−6.66357	−0.220411
$915$	−5.79388	−0.191540
$916$	−10.1930	−0.336785
$917$	6.77296	0.223663
$918$	7.79333	0.257218
$919$	5.91882	0.195244	0.0976219	−	0.995224i	$-0.468876\pi$
$919$	5.91882	0.195244	0.0976219	+	0.995224i	$0.468876\pi$
$920$	3.13877	0.103482
$921$	12.9027	0.425158
$922$	18.1977	0.599310
$923$	−0.250501	−0.00824535
$924$	0	0
$925$	−7.78523	−0.255977
$926$	11.3971	0.374531
$927$	3.70186	0.121585
$928$	5.49561	0.180402
$929$	−55.2709	−1.81338	−0.906690	−	0.421798i	$-0.861399\pi$
$929$	−55.2709	−1.81338	−0.906690	+	0.421798i	$0.861399\pi$
$930$	−14.1544	−0.464141
$931$	−6.62155	−0.217013
$932$	−13.2121	−0.432777
$933$	−27.5354	−0.901467
$934$	22.9793	0.751907
$935$	0	0
$936$	−14.1877	−0.463740
$937$	−16.0968	−0.525859	−0.262930	−	0.964815i	$-0.584689\pi$
$937$	−16.0968	−0.525859	−0.262930	+	0.964815i	$0.584689\pi$
$938$	−14.1969	−0.463544
$939$	−84.8244	−2.76814
$940$	1.49988	0.0489207
$941$	−49.8464	−1.62495	−0.812474	−	0.582998i	$-0.801880\pi$
$941$	−49.8464	−1.62495	−0.812474	+	0.582998i	$0.801880\pi$
$942$	15.2781	0.497787
$943$	−9.12615	−0.297188
$944$	−7.42428	−0.241640
$945$	−1.13800	−0.0370190
$946$	0	0
$947$	−31.1447	−1.01207	−0.506033	−	0.862514i	$-0.668889\pi$
$947$	−31.1447	−1.01207	−0.506033	+	0.862514i	$0.668889\pi$
$948$	10.7663	0.349673
$949$	16.0997	0.522619
$950$	−6.62155	−0.214832
$951$	70.3410	2.28096
$952$	6.84829	0.221954
$953$	−38.8286	−1.25778	−0.628891	−	0.777494i	$-0.716491\pi$
$953$	−38.8286	−1.25778	−0.628891	+	0.777494i	$0.716491\pi$
$954$	−38.3823	−1.24267
$955$	17.9665	0.581381
$956$	5.10128	0.164987
$957$	0	0
$958$	1.36694	0.0441640
$959$	9.15100	0.295501
$960$	2.53932	0.0819562
$961$	0.0704520	0.00227265
$962$	32.0330	1.03279
$963$	−65.0944	−2.09764
$964$	24.0075	0.773229
$965$	−23.8180	−0.766729
$966$	−7.97035	−0.256442
$967$	18.4412	0.593030	0.296515	−	0.955028i	$-0.404176\pi$
$967$	18.4412	0.593030	0.296515	+	0.955028i	$0.404176\pi$
$968$	0	0
$969$	−115.149	−3.69911
$970$	0.691019	0.0221873
$971$	6.48256	0.208035	0.104018	−	0.994575i	$-0.466830\pi$
$971$	6.48256	0.208035	0.104018	+	0.994575i	$0.466830\pi$
$972$	22.3439	0.716681
$973$	−15.9131	−0.510150
$974$	−12.2098	−0.391226
$975$	10.4483	0.334612
$976$	−2.28166	−0.0730343
$977$	−27.1448	−0.868438	−0.434219	−	0.900807i	$-0.642976\pi$
$977$	−27.1448	−0.868438	−0.434219	+	0.900807i	$0.642976\pi$
$978$	21.8750	0.699485
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	14.5621	0.464930
$982$	39.8546	1.27181
$983$	−9.91081	−0.316106	−0.158053	−	0.987431i	$-0.550522\pi$
$983$	−9.91081	−0.316106	−0.158053	+	0.987431i	$0.550522\pi$
$984$	−7.38321	−0.235368
$985$	1.85551	0.0591215
$986$	−37.6356	−1.19856
$987$	−3.80868	−0.121231
$988$	27.2450	0.866779
$989$	−10.1533	−0.322856
$990$	0	0
$991$	23.8739	0.758380	0.379190	−	0.925319i	$-0.376203\pi$
$991$	23.8739	0.758380	0.379190	+	0.925319i	$0.376203\pi$
$992$	−5.57409	−0.176977
$993$	73.2130	2.32334
$994$	−0.0608812	−0.00193103
$995$	−21.8206	−0.691760
$996$	−26.0198	−0.824470
$997$	−17.9566	−0.568691	−0.284345	−	0.958722i	$-0.591776\pi$
$997$	−17.9566	−0.568691	−0.284345	+	0.958722i	$0.591776\pi$
$998$	36.7505	1.16332
$999$	8.85955	0.280304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.dh.1.2		8
11.3	even	5		770.2.n.k.141.4	yes	16
11.4	even	5		770.2.n.k.71.4	✓	16
11.10	odd	2		8470.2.a.dg.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.k.71.4	✓	16	11.4	even	5
770.2.n.k.141.4	yes	16	11.3	even	5
8470.2.a.dg.1.2		8	11.10	odd	2
8470.2.a.dh.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.53932 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.53932 q^{6} -1.00000 q^{7} +1.00000 q^{8} +3.44815 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.53932 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.53932 q^{6} -1.00000 q^{7} +1.00000 q^{8} +3.44815 q^{9} -1.00000 q^{10} -2.53932 q^{12} -4.11459 q^{13} -1.00000 q^{14} +2.53932 q^{15} +1.00000 q^{16} -6.84829 q^{17} +3.44815 q^{18} -6.62155 q^{19} -1.00000 q^{20} +2.53932 q^{21} -3.13877 q^{23} -2.53932 q^{24} +1.00000 q^{25} -4.11459 q^{26} -1.13800 q^{27} -1.00000 q^{28} +5.49561 q^{29} +2.53932 q^{30} -5.57409 q^{31} +1.00000 q^{32} -6.84829 q^{34} +1.00000 q^{35} +3.44815 q^{36} -7.78523 q^{37} -6.62155 q^{38} +10.4483 q^{39} -1.00000 q^{40} +2.90755 q^{41} +2.53932 q^{42} +3.23479 q^{43} -3.44815 q^{45} -3.13877 q^{46} -1.49988 q^{47} -2.53932 q^{48} +1.00000 q^{49} +1.00000 q^{50} +17.3900 q^{51} -4.11459 q^{52} -11.1313 q^{53} -1.13800 q^{54} -1.00000 q^{56} +16.8142 q^{57} +5.49561 q^{58} -7.42428 q^{59} +2.53932 q^{60} -2.28166 q^{61} -5.57409 q^{62} -3.44815 q^{63} +1.00000 q^{64} +4.11459 q^{65} +14.1969 q^{67} -6.84829 q^{68} +7.97035 q^{69} +1.00000 q^{70} +0.0608812 q^{71} +3.44815 q^{72} -3.91284 q^{73} -7.78523 q^{74} -2.53932 q^{75} -6.62155 q^{76} +10.4483 q^{78} -4.23983 q^{79} -1.00000 q^{80} -7.45471 q^{81} +2.90755 q^{82} +10.2468 q^{83} +2.53932 q^{84} +6.84829 q^{85} +3.23479 q^{86} -13.9551 q^{87} -6.54394 q^{89} -3.44815 q^{90} +4.11459 q^{91} -3.13877 q^{92} +14.1544 q^{93} -1.49988 q^{94} +6.62155 q^{95} -2.53932 q^{96} -0.691019 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 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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