LMFDB - Embedded newform 207.8.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,8,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("207.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	207.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:	$64.6637002752$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 1070 x^{10} + 4076 x^{9} + 403334 x^{8} - 1518684 x^{7} - 64710184 x^{6} + \cdots + 90709421512 $ x^12 - 4x^11 - 1070x^10 + 4076x^9 + 403334x^8 - 1518684x^7 - 64710184x^6 + 248484524x^5 + 4244585785x^4 - 14689737568x^3 - 89785869106x^2 + 190549532712*x + 90709421512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{7} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$8.40410$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.40410 q^{2} -39.5629 q^{4} +482.766 q^{5} +1331.25 q^{7} +1575.78 q^{8} +O(q^{10})$	\(q-9.40410 q^{2} -39.5629 q^{4} +482.766 q^{5} +1331.25 q^{7} +1575.78 q^{8} -4539.98 q^{10} +1518.96 q^{11} -14202.3 q^{13} -12519.2 q^{14} -9754.72 q^{16} -23664.6 q^{17} -41188.3 q^{19} -19099.7 q^{20} -14284.5 q^{22} +12167.0 q^{23} +154938. q^{25} +133560. q^{26} -52668.3 q^{28} -100862. q^{29} -249210. q^{31} -109965. q^{32} +222544. q^{34} +642685. q^{35} +184460. q^{37} +387339. q^{38} +760733. q^{40} -492710. q^{41} +32293.1 q^{43} -60094.5 q^{44} -114420. q^{46} -1.13531e6 q^{47} +948695. q^{49} -1.45706e6 q^{50} +561884. q^{52} -1.39324e6 q^{53} +733303. q^{55} +2.09776e6 q^{56} +948515. q^{58} -1.10433e6 q^{59} +1.35995e6 q^{61} +2.34360e6 q^{62} +2.28273e6 q^{64} -6.85639e6 q^{65} -4.46001e6 q^{67} +936242. q^{68} -6.04387e6 q^{70} +1.63027e6 q^{71} +2.89135e6 q^{73} -1.73468e6 q^{74} +1.62953e6 q^{76} +2.02212e6 q^{77} +2.40253e6 q^{79} -4.70925e6 q^{80} +4.63349e6 q^{82} +4.40005e6 q^{83} -1.14245e7 q^{85} -303688. q^{86} +2.39354e6 q^{88} +914433. q^{89} -1.89069e7 q^{91} -481362. q^{92} +1.06766e7 q^{94} -1.98843e7 q^{95} +3.86999e6 q^{97} -8.92162e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{2} + 640 q^{4} - 500 q^{5} - 228 q^{7} - 3072 q^{8}+O(q^{10}) $ 12 * q - 16 * q^2 + 640 * q^4 - 500 * q^5 - 228 * q^7 - 3072 * q^8	\( 12 q - 16 q^{2} + 640 q^{4} - 500 q^{5} - 228 q^{7} - 3072 q^{8} + 10270 q^{10} + 460 q^{11} - 21060 q^{13} - 4268 q^{14} + 56676 q^{16} - 73124 q^{17} + 8508 q^{19} - 170538 q^{20} + 124754 q^{22} + 146004 q^{23} + 194064 q^{25} - 206080 q^{26} - 390416 q^{28} - 268640 q^{29} - 191880 q^{31} - 1180172 q^{32} - 221436 q^{34} + 487244 q^{35} + 650332 q^{37} - 1432950 q^{38} + 1775722 q^{40} - 980088 q^{41} - 861276 q^{43} - 800666 q^{44} - 194672 q^{46} - 403868 q^{47} + 1699160 q^{49} - 2919092 q^{50} - 2369520 q^{52} + 201948 q^{53} - 1553512 q^{55} + 4848116 q^{56} + 3720672 q^{58} - 1302676 q^{59} + 2141364 q^{61} - 2160944 q^{62} + 9702136 q^{64} - 9099536 q^{65} - 6159260 q^{67} - 18442208 q^{68} - 10891632 q^{70} - 12584184 q^{71} + 7435872 q^{73} - 22491442 q^{74} + 5721386 q^{76} - 16450568 q^{77} + 3658028 q^{79} - 49905778 q^{80} - 5516316 q^{82} - 26137900 q^{83} + 5169556 q^{85} - 30678550 q^{86} + 14753046 q^{88} - 27235908 q^{89} - 7657216 q^{91} + 7786880 q^{92} - 23519352 q^{94} - 63623628 q^{95} + 22454720 q^{97} - 94951532 q^{98}+O(q^{100}) \) 12 * q - 16 * q^2 + 640 * q^4 - 500 * q^5 - 228 * q^7 - 3072 * q^8 + 10270 * q^10 + 460 * q^11 - 21060 * q^13 - 4268 * q^14 + 56676 * q^16 - 73124 * q^17 + 8508 * q^19 - 170538 * q^20 + 124754 * q^22 + 146004 * q^23 + 194064 * q^25 - 206080 * q^26 - 390416 * q^28 - 268640 * q^29 - 191880 * q^31 - 1180172 * q^32 - 221436 * q^34 + 487244 * q^35 + 650332 * q^37 - 1432950 * q^38 + 1775722 * q^40 - 980088 * q^41 - 861276 * q^43 - 800666 * q^44 - 194672 * q^46 - 403868 * q^47 + 1699160 * q^49 - 2919092 * q^50 - 2369520 * q^52 + 201948 * q^53 - 1553512 * q^55 + 4848116 * q^56 + 3720672 * q^58 - 1302676 * q^59 + 2141364 * q^61 - 2160944 * q^62 + 9702136 * q^64 - 9099536 * q^65 - 6159260 * q^67 - 18442208 * q^68 - 10891632 * q^70 - 12584184 * q^71 + 7435872 * q^73 - 22491442 * q^74 + 5721386 * q^76 - 16450568 * q^77 + 3658028 * q^79 - 49905778 * q^80 - 5516316 * q^82 - 26137900 * q^83 + 5169556 * q^85 - 30678550 * q^86 + 14753046 * q^88 - 27235908 * q^89 - 7657216 * q^91 + 7786880 * q^92 - 23519352 * q^94 - 63623628 * q^95 + 22454720 * q^97 - 94951532 * 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 $

$2$	−9.40410	−0.831213	−0.415606	−	0.909545i	$-0.636431\pi$
$2$	−9.40410	−0.831213	−0.415606	+	0.909545i	$0.636431\pi$
$3$	0	0
$3$	0	0
$4$	−39.5629	−0.309086
$5$	482.766	1.72720	0.863599	−	0.504180i	$-0.168205\pi$
$5$	482.766	1.72720	0.863599	+	0.504180i	$0.168205\pi$
$6$	0	0
$7$	1331.25	1.46696	0.733479	−	0.679712i	$-0.237895\pi$
$7$	1331.25	1.46696	0.733479	+	0.679712i	$0.237895\pi$
$8$	1575.78	1.08813
$9$	0	0
$10$	−4539.98	−1.43567
$11$	1518.96	0.344090	0.172045	−	0.985089i	$-0.444963\pi$
$11$	1518.96	0.344090	0.172045	+	0.985089i	$0.444963\pi$
$12$	0	0
$13$	−14202.3	−1.79290	−0.896451	−	0.443142i	$-0.853864\pi$
$13$	−14202.3	−1.79290	−0.896451	+	0.443142i	$0.853864\pi$
$14$	−12519.2	−1.21935
$15$	0	0
$16$	−9754.72	−0.595381
$17$	−23664.6	−1.16823	−0.584115	−	0.811671i	$-0.698558\pi$
$17$	−23664.6	−1.16823	−0.584115	+	0.811671i	$0.698558\pi$
$18$	0	0
$19$	−41188.3	−1.37764	−0.688821	−	0.724932i	$-0.741871\pi$
$19$	−41188.3	−1.37764	−0.688821	+	0.724932i	$0.741871\pi$
$20$	−19099.7	−0.533852
$21$	0	0
$22$	−14284.5	−0.286012
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	0	0
$25$	154938.	1.98321
$26$	133560.	1.49028
$27$	0	0
$28$	−52668.3	−0.453416
$29$	−100862.	−0.767952	−0.383976	−	0.923343i	$-0.625445\pi$
$29$	−100862.	−0.767952	−0.383976	+	0.923343i	$0.625445\pi$
$30$	0	0
$31$	−249210.	−1.50245	−0.751225	−	0.660046i	$-0.770537\pi$
$31$	−249210.	−1.50245	−0.751225	+	0.660046i	$0.770537\pi$
$32$	−109965.	−0.593241
$33$	0	0
$34$	222544.	0.971047
$35$	642685.	2.53373
$36$	0	0
$37$	184460.	0.598683	0.299342	−	0.954146i	$-0.403233\pi$
$37$	184460.	0.598683	0.299342	+	0.954146i	$0.403233\pi$
$38$	387339.	1.14511
$39$	0	0
$40$	760733.	1.87941
$41$	−492710.	−1.11647	−0.558236	−	0.829682i	$-0.688522\pi$
$41$	−492710.	−1.11647	−0.558236	+	0.829682i	$0.688522\pi$
$42$	0	0
$43$	32293.1	0.0619399	0.0309699	−	0.999520i	$-0.490140\pi$
$43$	32293.1	0.0619399	0.0309699	+	0.999520i	$0.490140\pi$
$44$	−60094.5	−0.106353
$45$	0	0
$46$	−114420.	−0.173320
$47$	−1.13531e6	−1.59504	−0.797521	−	0.603291i	$-0.793856\pi$
$47$	−1.13531e6	−1.59504	−0.797521	+	0.603291i	$0.793856\pi$
$48$	0	0
$49$	948695.	1.15197
$50$	−1.45706e6	−1.64847
$51$	0	0
$52$	561884.	0.554160
$53$	−1.39324e6	−1.28546	−0.642732	−	0.766091i	$-0.722199\pi$
$53$	−1.39324e6	−1.28546	−0.642732	+	0.766091i	$0.722199\pi$
$54$	0	0
$55$	733303.	0.594311
$56$	2.09776e6	1.59624
$57$	0	0
$58$	948515.	0.638331
$59$	−1.10433e6	−0.700031	−0.350016	−	0.936744i	$-0.613824\pi$
$59$	−1.10433e6	−0.700031	−0.350016	+	0.936744i	$0.613824\pi$
$60$	0	0
$61$	1.35995e6	0.767129	0.383565	−	0.923514i	$-0.374696\pi$
$61$	1.35995e6	0.767129	0.383565	+	0.923514i	$0.374696\pi$
$62$	2.34360e6	1.24886
$63$	0	0
$64$	2.28273e6	1.08849
$65$	−6.85639e6	−3.09670
$66$	0	0
$67$	−4.46001e6	−1.81165	−0.905825	−	0.423653i	$-0.860748\pi$
$67$	−4.46001e6	−1.81165	−0.905825	+	0.423653i	$0.860748\pi$
$68$	936242.	0.361083
$69$	0	0
$70$	−6.04387e6	−2.10607
$71$	1.63027e6	0.540574	0.270287	−	0.962780i	$-0.412881\pi$
$71$	1.63027e6	0.540574	0.270287	+	0.962780i	$0.412881\pi$
$72$	0	0
$73$	2.89135e6	0.869904	0.434952	−	0.900454i	$-0.356765\pi$
$73$	2.89135e6	0.869904	0.434952	+	0.900454i	$0.356765\pi$
$74$	−1.73468e6	−0.497633
$75$	0	0
$76$	1.62953e6	0.425809
$77$	2.02212e6	0.504766
$78$	0	0
$79$	2.40253e6	0.548245	0.274122	−	0.961695i	$-0.411613\pi$
$79$	2.40253e6	0.548245	0.274122	+	0.961695i	$0.411613\pi$
$80$	−4.70925e6	−1.02834
$81$	0	0
$82$	4.63349e6	0.928026
$83$	4.40005e6	0.844666	0.422333	−	0.906441i	$-0.361211\pi$
$83$	4.40005e6	0.844666	0.422333	+	0.906441i	$0.361211\pi$
$84$	0	0
$85$	−1.14245e7	−2.01776
$86$	−303688.	−0.0514852
$87$	0	0
$88$	2.39354e6	0.374414
$89$	914433.	0.137495	0.0687475	−	0.997634i	$-0.478100\pi$
$89$	914433.	0.137495	0.0687475	+	0.997634i	$0.478100\pi$
$90$	0	0
$91$	−1.89069e7	−2.63011
$92$	−481362.	−0.0644488
$93$	0	0
$94$	1.06766e7	1.32582
$95$	−1.98843e7	−2.37946
$96$	0	0
$97$	3.86999e6	0.430535	0.215268	−	0.976555i	$-0.430938\pi$
$97$	3.86999e6	0.430535	0.215268	+	0.976555i	$0.430938\pi$
$98$	−8.92162e6	−0.957531
$99$	0	0
$100$	−6.12982e6	−0.612982
$101$	−4.56409e6	−0.440787	−0.220394	−	0.975411i	$-0.570734\pi$
$101$	−4.56409e6	−0.440787	−0.220394	+	0.975411i	$0.570734\pi$
$102$	0	0
$103$	5.84620e6	0.527161	0.263581	−	0.964637i	$-0.415096\pi$
$103$	5.84620e6	0.527161	0.263581	+	0.964637i	$0.415096\pi$
$104$	−2.23797e7	−1.95091
$105$	0	0
$106$	1.31021e7	1.06849
$107$	9.52417e6	0.751595	0.375798	−	0.926702i	$-0.377369\pi$
$107$	9.52417e6	0.751595	0.375798	+	0.926702i	$0.377369\pi$
$108$	0	0
$109$	−1.65625e6	−0.122499	−0.0612495	−	0.998122i	$-0.519509\pi$
$109$	−1.65625e6	−0.122499	−0.0612495	+	0.998122i	$0.519509\pi$
$110$	−6.89605e6	−0.493999
$111$	0	0
$112$	−1.29860e7	−0.873399
$113$	−9.09617e6	−0.593040	−0.296520	−	0.955027i	$-0.595826\pi$
$113$	−9.09617e6	−0.593040	−0.296520	+	0.955027i	$0.595826\pi$
$114$	0	0
$115$	5.87382e6	0.360146
$116$	3.99039e6	0.237363
$117$	0	0
$118$	1.03852e7	0.581875
$119$	−3.15036e7	−1.71374
$120$	0	0
$121$	−1.71799e7	−0.881602
$122$	−1.27891e7	−0.637648
$123$	0	0
$124$	9.85950e6	0.464386
$125$	3.70829e7	1.69820
$126$	0	0
$127$	−1.58044e7	−0.684644	−0.342322	−	0.939583i	$-0.611213\pi$
$127$	−1.58044e7	−0.684644	−0.342322	+	0.939583i	$0.611213\pi$
$128$	−7.39144e6	−0.311526
$129$	0	0
$130$	6.44781e7	2.57401
$131$	−3.09769e7	−1.20390	−0.601948	−	0.798536i	$-0.705608\pi$
$131$	−3.09769e7	−1.20390	−0.601948	+	0.798536i	$0.705608\pi$
$132$	0	0
$133$	−5.48321e7	−2.02094
$134$	4.19424e7	1.50587
$135$	0	0
$136$	−3.72902e7	−1.27118
$137$	2.56464e7	0.852126	0.426063	−	0.904694i	$-0.359900\pi$
$137$	2.56464e7	0.852126	0.426063	+	0.904694i	$0.359900\pi$
$138$	0	0
$139$	7.56936e6	0.239060	0.119530	−	0.992831i	$-0.461861\pi$
$139$	7.56936e6	0.239060	0.119530	+	0.992831i	$0.461861\pi$
$140$	−2.54265e7	−0.783138
$141$	0	0
$142$	−1.53312e7	−0.449332
$143$	−2.15727e7	−0.616920
$144$	0	0
$145$	−4.86927e7	−1.32640
$146$	−2.71906e7	−0.723075
$147$	0	0
$148$	−7.29780e6	−0.185044
$149$	6.31552e7	1.56407	0.782037	−	0.623232i	$-0.214181\pi$
$149$	6.31552e7	1.56407	0.782037	+	0.623232i	$0.214181\pi$
$150$	0	0
$151$	−5.11440e7	−1.20886	−0.604429	−	0.796659i	$-0.706599\pi$
$151$	−5.11440e7	−1.20886	−0.604429	+	0.796659i	$0.706599\pi$
$152$	−6.49036e7	−1.49905
$153$	0	0
$154$	−1.90162e7	−0.419568
$155$	−1.20310e8	−2.59503
$156$	0	0
$157$	7.59131e7	1.56555	0.782777	−	0.622302i	$-0.213802\pi$
$157$	7.59131e7	1.56555	0.782777	+	0.622302i	$0.213802\pi$
$158$	−2.25936e7	−0.455708
$159$	0	0
$160$	−5.30876e7	−1.02464
$161$	1.61974e7	0.305882
$162$	0	0
$163$	−4.53090e7	−0.819459	−0.409730	−	0.912207i	$-0.634377\pi$
$163$	−4.53090e7	−0.819459	−0.409730	+	0.912207i	$0.634377\pi$
$164$	1.94931e7	0.345085
$165$	0	0
$166$	−4.13785e7	−0.702097
$167$	3.23087e7	0.536800	0.268400	−	0.963308i	$-0.413505\pi$
$167$	3.23087e7	0.536800	0.268400	+	0.963308i	$0.413505\pi$
$168$	0	0
$169$	1.38957e8	2.21450
$170$	1.07437e8	1.67719
$171$	0	0
$172$	−1.27761e6	−0.0191447
$173$	−9.20786e7	−1.35206	−0.676032	−	0.736872i	$-0.736302\pi$
$173$	−9.20786e7	−1.35206	−0.676032	+	0.736872i	$0.736302\pi$
$174$	0	0
$175$	2.06262e8	2.90929
$176$	−1.48170e7	−0.204865
$177$	0	0
$178$	−8.59942e6	−0.114288
$179$	−7.37536e7	−0.961165	−0.480582	−	0.876950i	$-0.659575\pi$
$179$	−7.37536e7	−0.961165	−0.480582	+	0.876950i	$0.659575\pi$
$180$	0	0
$181$	1.08441e8	1.35931	0.679654	−	0.733533i	$-0.262130\pi$
$181$	1.08441e8	1.35931	0.679654	+	0.733533i	$0.262130\pi$
$182$	1.77802e8	2.18618
$183$	0	0
$184$	1.91725e7	0.226890
$185$	8.90513e7	1.03404
$186$	0	0
$187$	−3.59456e7	−0.401976
$188$	4.49162e7	0.493004
$189$	0	0
$190$	1.86994e8	1.97784
$191$	4.58396e7	0.476019	0.238010	−	0.971263i	$-0.423505\pi$
$191$	4.58396e7	0.476019	0.238010	+	0.971263i	$0.423505\pi$
$192$	0	0
$193$	9.72581e7	0.973813	0.486906	−	0.873454i	$-0.338125\pi$
$193$	9.72581e7	0.973813	0.486906	+	0.873454i	$0.338125\pi$
$194$	−3.63937e7	−0.357866
$195$	0	0
$196$	−3.75332e7	−0.356057
$197$	4.66575e7	0.434800	0.217400	−	0.976083i	$-0.430242\pi$
$197$	4.66575e7	0.434800	0.217400	+	0.976083i	$0.430242\pi$
$198$	0	0
$199$	3.28428e7	0.295430	0.147715	−	0.989030i	$-0.452808\pi$
$199$	3.28428e7	0.295430	0.147715	+	0.989030i	$0.452808\pi$
$200$	2.44148e8	2.15799
$201$	0	0
$202$	4.29211e7	0.366388
$203$	−1.34273e8	−1.12655
$204$	0	0
$205$	−2.37864e8	−1.92837
$206$	−5.49783e7	−0.438183
$207$	0	0
$208$	1.38539e8	1.06746
$209$	−6.25634e7	−0.474033
$210$	0	0
$211$	6.89921e7	0.505604	0.252802	−	0.967518i	$-0.418648\pi$
$211$	6.89921e7	0.505604	0.252802	+	0.967518i	$0.418648\pi$
$212$	5.51206e7	0.397318
$213$	0	0
$214$	−8.95662e7	−0.624735
$215$	1.55900e7	0.106982
$216$	0	0
$217$	−3.31763e8	−2.20403
$218$	1.55755e7	0.101823
$219$	0	0
$220$	−2.90116e7	−0.183693
$221$	3.36092e8	2.09452
$222$	0	0
$223$	2.83191e8	1.71006	0.855032	−	0.518574i	$-0.173537\pi$
$223$	2.83191e8	1.71006	0.855032	+	0.518574i	$0.173537\pi$
$224$	−1.46392e8	−0.870259
$225$	0	0
$226$	8.55413e7	0.492942
$227$	2.67834e8	1.51976	0.759879	−	0.650064i	$-0.225258\pi$
$227$	2.67834e8	1.51976	0.759879	+	0.650064i	$0.225258\pi$
$228$	0	0
$229$	−1.65795e8	−0.912318	−0.456159	−	0.889898i	$-0.650775\pi$
$229$	−1.65795e8	−0.912318	−0.456159	+	0.889898i	$0.650775\pi$
$230$	−5.52380e7	−0.299358
$231$	0	0
$232$	−1.58936e8	−0.835630
$233$	−5.85915e6	−0.0303451	−0.0151726	−	0.999885i	$-0.504830\pi$
$233$	−5.85915e6	−0.0303451	−0.0151726	+	0.999885i	$0.504830\pi$
$234$	0	0
$235$	−5.48090e8	−2.75495
$236$	4.36906e7	0.216370
$237$	0	0
$238$	2.96263e8	1.42449
$239$	5.24280e6	0.0248411	0.0124206	−	0.999923i	$-0.496046\pi$
$239$	5.24280e6	0.0248411	0.0124206	+	0.999923i	$0.496046\pi$
$240$	0	0
$241$	−3.18258e8	−1.46460	−0.732300	−	0.680982i	$-0.761553\pi$
$241$	−3.18258e8	−1.46460	−0.732300	+	0.680982i	$0.761553\pi$
$242$	1.61562e8	0.732799
$243$	0	0
$244$	−5.38036e7	−0.237109
$245$	4.57998e8	1.98968
$246$	0	0
$247$	5.84968e8	2.46998
$248$	−3.92700e8	−1.63486
$249$	0	0
$250$	−3.48731e8	−1.41156
$251$	−1.18488e8	−0.472950	−0.236475	−	0.971638i	$-0.575992\pi$
$251$	−1.18488e8	−0.472950	−0.236475	+	0.971638i	$0.575992\pi$
$252$	0	0
$253$	1.84812e7	0.0717477
$254$	1.48626e8	0.569084
$255$	0	0
$256$	−2.22679e8	−0.829545
$257$	−3.80441e8	−1.39805	−0.699023	−	0.715099i	$-0.746382\pi$
$257$	−3.80441e8	−1.39805	−0.699023	+	0.715099i	$0.746382\pi$
$258$	0	0
$259$	2.45564e8	0.878244
$260$	2.71259e8	0.957144
$261$	0	0
$262$	2.91310e8	1.00069
$263$	4.62036e7	0.156614	0.0783070	−	0.996929i	$-0.475049\pi$
$263$	4.62036e7	0.156614	0.0783070	+	0.996929i	$0.475049\pi$
$264$	0	0
$265$	−6.72608e8	−2.22025
$266$	5.15647e8	1.67983
$267$	0	0
$268$	1.76451e8	0.559955
$269$	−1.48180e8	−0.464147	−0.232074	−	0.972698i	$-0.574551\pi$
$269$	−1.48180e8	−0.464147	−0.232074	+	0.972698i	$0.574551\pi$
$270$	0	0
$271$	−4.54874e8	−1.38835	−0.694175	−	0.719807i	$-0.744230\pi$
$271$	−4.54874e8	−1.38835	−0.694175	+	0.719807i	$0.744230\pi$
$272$	2.30842e8	0.695541
$273$	0	0
$274$	−2.41181e8	−0.708298
$275$	2.35345e8	0.682403
$276$	0	0
$277$	−2.16604e8	−0.612333	−0.306166	−	0.951978i	$-0.599046\pi$
$277$	−2.16604e8	−0.612333	−0.306166	+	0.951978i	$0.599046\pi$
$278$	−7.11830e7	−0.198710
$279$	0	0
$280$	1.01273e9	2.75702
$281$	−7.09101e8	−1.90650	−0.953249	−	0.302187i	$-0.902283\pi$
$281$	−7.09101e8	−1.90650	−0.953249	+	0.302187i	$0.902283\pi$
$282$	0	0
$283$	2.05296e8	0.538427	0.269214	−	0.963080i	$-0.413236\pi$
$283$	2.05296e8	0.538427	0.269214	+	0.963080i	$0.413236\pi$
$284$	−6.44983e7	−0.167084
$285$	0	0
$286$	2.02872e8	0.512792
$287$	−6.55923e8	−1.63782
$288$	0	0
$289$	1.49675e8	0.364761
$290$	4.57911e8	1.10252
$291$	0	0
$292$	−1.14390e8	−0.268875
$293$	−4.72329e8	−1.09700	−0.548502	−	0.836149i	$-0.684802\pi$
$293$	−4.72329e8	−1.09700	−0.548502	+	0.836149i	$0.684802\pi$
$294$	0	0
$295$	−5.33134e8	−1.20909
$296$	2.90669e8	0.651444
$297$	0	0
$298$	−5.93918e8	−1.30008
$299$	−1.72799e8	−0.373846
$300$	0	0
$301$	4.29903e7	0.0908632
$302$	4.80963e8	1.00482
$303$	0	0
$304$	4.01780e8	0.820221
$305$	6.56538e8	1.32498
$306$	0	0
$307$	6.72011e8	1.32554	0.662769	−	0.748824i	$-0.269381\pi$
$307$	6.72011e8	1.32554	0.662769	+	0.748824i	$0.269381\pi$
$308$	−8.00011e7	−0.156016
$309$	0	0
$310$	1.13141e9	2.15702
$311$	6.83526e8	1.28853	0.644264	−	0.764804i	$-0.277164\pi$
$311$	6.83526e8	1.28853	0.644264	+	0.764804i	$0.277164\pi$
$312$	0	0
$313$	3.07159e8	0.566184	0.283092	−	0.959093i	$-0.408640\pi$
$313$	3.07159e8	0.566184	0.283092	+	0.959093i	$0.408640\pi$
$314$	−7.13895e8	−1.30131
$315$	0	0
$316$	−9.50512e7	−0.169454
$317$	1.02815e9	1.81280	0.906402	−	0.422416i	$-0.138818\pi$
$317$	1.02815e9	1.81280	0.906402	+	0.422416i	$0.138818\pi$
$318$	0	0
$319$	−1.53205e8	−0.264245
$320$	1.10202e9	1.88004
$321$	0	0
$322$	−1.52322e8	−0.254253
$323$	9.74705e8	1.60940
$324$	0	0
$325$	−2.20048e9	−3.55570
$326$	4.26090e8	0.681145
$327$	0	0
$328$	−7.76402e8	−1.21487
$329$	−1.51139e9	−2.33986
$330$	0	0
$331$	−5.00653e8	−0.758820	−0.379410	−	0.925229i	$-0.623873\pi$
$331$	−5.00653e8	−0.758820	−0.379410	+	0.925229i	$0.623873\pi$
$332$	−1.74079e8	−0.261074
$333$	0	0
$334$	−3.03834e8	−0.446195
$335$	−2.15314e9	−3.12908
$336$	0	0
$337$	5.53100e8	0.787225	0.393613	−	0.919276i	$-0.371225\pi$
$337$	5.53100e8	0.787225	0.393613	+	0.919276i	$0.371225\pi$
$338$	−1.30676e9	−1.84072
$339$	0	0
$340$	4.51986e8	0.623661
$341$	−3.78541e8	−0.516978
$342$	0	0
$343$	1.66610e8	0.222931
$344$	5.08868e7	0.0673985
$345$	0	0
$346$	8.65916e8	1.12385
$347$	−5.42774e8	−0.697374	−0.348687	−	0.937239i	$-0.613372\pi$
$347$	−5.42774e8	−0.697374	−0.348687	+	0.937239i	$0.613372\pi$
$348$	0	0
$349$	8.98846e8	1.13187	0.565934	−	0.824450i	$-0.308516\pi$
$349$	8.98846e8	1.13187	0.565934	+	0.824450i	$0.308516\pi$
$350$	−1.93971e9	−2.41824
$351$	0	0
$352$	−1.67033e8	−0.204128
$353$	−1.05660e9	−1.27849	−0.639247	−	0.769001i	$-0.720754\pi$
$353$	−1.05660e9	−1.27849	−0.639247	+	0.769001i	$0.720754\pi$
$354$	0	0
$355$	7.87040e8	0.933678
$356$	−3.61777e7	−0.0424977
$357$	0	0
$358$	6.93587e8	0.798932
$359$	−8.63780e8	−0.985309	−0.492654	−	0.870225i	$-0.663973\pi$
$359$	−8.63780e8	−0.985309	−0.492654	+	0.870225i	$0.663973\pi$
$360$	0	0
$361$	8.02605e8	0.897897
$362$	−1.01979e9	−1.12987
$363$	0	0
$364$	7.48011e8	0.812930
$365$	1.39585e9	1.50250
$366$	0	0
$367$	−6.10681e8	−0.644886	−0.322443	−	0.946589i	$-0.604504\pi$
$367$	−6.10681e8	−0.644886	−0.322443	+	0.946589i	$0.604504\pi$
$368$	−1.18686e8	−0.124145
$369$	0	0
$370$	−8.37447e8	−0.859510
$371$	−1.85475e9	−1.88572
$372$	0	0
$373$	−1.22094e9	−1.21819	−0.609093	−	0.793099i	$-0.708466\pi$
$373$	−1.22094e9	−1.21819	−0.609093	+	0.793099i	$0.708466\pi$
$374$	3.38036e8	0.334128
$375$	0	0
$376$	−1.78900e9	−1.73561
$377$	1.43247e9	1.37686
$378$	0	0
$379$	1.69636e9	1.60060	0.800298	−	0.599602i	$-0.204675\pi$
$379$	1.69636e9	1.60060	0.800298	+	0.599602i	$0.204675\pi$
$380$	7.86683e8	0.735456
$381$	0	0
$382$	−4.31080e8	−0.395673
$383$	1.33564e9	1.21477	0.607386	−	0.794407i	$-0.292218\pi$
$383$	1.33564e9	1.21477	0.607386	+	0.794407i	$0.292218\pi$
$384$	0	0
$385$	9.76213e8	0.871830
$386$	−9.14625e8	−0.809445
$387$	0	0
$388$	−1.53108e8	−0.133072
$389$	−6.72397e8	−0.579165	−0.289583	−	0.957153i	$-0.593516\pi$
$389$	−6.72397e8	−0.579165	−0.289583	+	0.957153i	$0.593516\pi$
$390$	0	0
$391$	−2.87927e8	−0.243593
$392$	1.49493e9	1.25349
$393$	0	0
$394$	−4.38771e8	−0.361411
$395$	1.15986e9	0.946927
$396$	0	0
$397$	−8.73918e8	−0.700977	−0.350488	−	0.936567i	$-0.613984\pi$
$397$	−8.73918e8	−0.700977	−0.350488	+	0.936567i	$0.613984\pi$
$398$	−3.08857e8	−0.245565
$399$	0	0
$400$	−1.51138e9	−1.18077
$401$	−2.42002e9	−1.87419	−0.937097	−	0.349070i	$-0.886497\pi$
$401$	−2.42002e9	−1.87419	−0.937097	+	0.349070i	$0.886497\pi$
$402$	0	0
$403$	3.53936e9	2.69375
$404$	1.80569e8	0.136241
$405$	0	0
$406$	1.26271e9	0.936406
$407$	2.80188e8	0.206001
$408$	0	0
$409$	−1.27236e9	−0.919559	−0.459780	−	0.888033i	$-0.652072\pi$
$409$	−1.27236e9	−0.919559	−0.459780	+	0.888033i	$0.652072\pi$
$410$	2.23690e9	1.60288
$411$	0	0
$412$	−2.31293e8	−0.162938
$413$	−1.47015e9	−1.02692
$414$	0	0
$415$	2.12420e9	1.45890
$416$	1.56176e9	1.06362
$417$	0	0
$418$	5.88352e8	0.394022
$419$	2.60489e7	0.0172998	0.00864990	−	0.999963i	$-0.497247\pi$
$419$	2.60489e7	0.0172998	0.00864990	+	0.999963i	$0.497247\pi$
$420$	0	0
$421$	2.20507e9	1.44024	0.720121	−	0.693849i	$-0.244086\pi$
$421$	2.20507e9	1.44024	0.720121	+	0.693849i	$0.244086\pi$
$422$	−6.48808e8	−0.420264
$423$	0	0
$424$	−2.19543e9	−1.39875
$425$	−3.66656e9	−2.31685
$426$	0	0
$427$	1.81044e9	1.12535
$428$	−3.76804e8	−0.232307
$429$	0	0
$430$	−1.46610e8	−0.0889251
$431$	−2.84579e9	−1.71211	−0.856056	−	0.516882i	$-0.827092\pi$
$431$	−2.84579e9	−1.71211	−0.856056	+	0.516882i	$0.827092\pi$
$432$	0	0
$433$	−1.25825e9	−0.744832	−0.372416	−	0.928066i	$-0.621470\pi$
$433$	−1.25825e9	−0.744832	−0.372416	+	0.928066i	$0.621470\pi$
$434$	3.11993e9	1.83202
$435$	0	0
$436$	6.55260e7	0.0378627
$437$	−5.01138e8	−0.287258
$438$	0	0
$439$	1.88747e9	1.06477	0.532383	−	0.846503i	$-0.321297\pi$
$439$	1.88747e9	1.06477	0.532383	+	0.846503i	$0.321297\pi$
$440$	1.15552e9	0.646687
$441$	0	0
$442$	−3.16064e9	−1.74099
$443$	2.00114e9	1.09361	0.546806	−	0.837259i	$-0.315843\pi$
$443$	2.00114e9	1.09361	0.546806	+	0.837259i	$0.315843\pi$
$444$	0	0
$445$	4.41457e8	0.237481
$446$	−2.66316e9	−1.42143
$447$	0	0
$448$	3.03889e9	1.59677
$449$	1.29242e9	0.673818	0.336909	−	0.941537i	$-0.390618\pi$
$449$	1.29242e9	0.673818	0.336909	+	0.941537i	$0.390618\pi$
$450$	0	0
$451$	−7.48407e8	−0.384167
$452$	3.59871e8	0.183300
$453$	0	0
$454$	−2.51873e9	−1.26324
$455$	−9.12760e9	−4.54273
$456$	0	0
$457$	3.42750e9	1.67985	0.839925	−	0.542703i	$-0.182599\pi$
$457$	3.42750e9	1.67985	0.839925	+	0.542703i	$0.182599\pi$
$458$	1.55915e9	0.758330
$459$	0	0
$460$	−2.32386e8	−0.111316
$461$	5.62123e8	0.267225	0.133613	−	0.991034i	$-0.457342\pi$
$461$	5.62123e8	0.267225	0.133613	+	0.991034i	$0.457342\pi$
$462$	0	0
$463$	6.30958e8	0.295438	0.147719	−	0.989029i	$-0.452807\pi$
$463$	6.30958e8	0.295438	0.147719	+	0.989029i	$0.452807\pi$
$464$	9.83879e8	0.457224
$465$	0	0
$466$	5.51000e7	0.0252232
$467$	−3.03752e8	−0.138010	−0.0690049	−	0.997616i	$-0.521982\pi$
$467$	−3.03752e8	−0.138010	−0.0690049	+	0.997616i	$0.521982\pi$
$468$	0	0
$469$	−5.93741e9	−2.65762
$470$	5.15429e9	2.28995
$471$	0	0
$472$	−1.74018e9	−0.761724
$473$	4.90520e7	0.0213129
$474$	0	0
$475$	−6.38165e9	−2.73215
$476$	1.24638e9	0.529694
$477$	0	0
$478$	−4.93038e7	−0.0206482
$479$	−2.16410e9	−0.899710	−0.449855	−	0.893102i	$-0.648524\pi$
$479$	−2.16410e9	−0.899710	−0.449855	+	0.893102i	$0.648524\pi$
$480$	0	0
$481$	−2.61976e9	−1.07338
$482$	2.99292e9	1.21739
$483$	0	0
$484$	6.79689e8	0.272490
$485$	1.86830e9	0.743619
$486$	0	0
$487$	−4.69077e9	−1.84032	−0.920160	−	0.391543i	$-0.871941\pi$
$487$	−4.69077e9	−1.84032	−0.920160	+	0.391543i	$0.871941\pi$
$488$	2.14298e9	0.834735
$489$	0	0
$490$	−4.30706e9	−1.65384
$491$	2.35568e9	0.898114	0.449057	−	0.893503i	$-0.351760\pi$
$491$	2.35568e9	0.898114	0.449057	+	0.893503i	$0.351760\pi$
$492$	0	0
$493$	2.38686e9	0.897144
$494$	−5.50110e9	−2.05308
$495$	0	0
$496$	2.43098e9	0.894530
$497$	2.17030e9	0.793000
$498$	0	0
$499$	2.87478e9	1.03575	0.517873	−	0.855457i	$-0.326724\pi$
$499$	2.87478e9	1.03575	0.517873	+	0.855457i	$0.326724\pi$
$500$	−1.46711e9	−0.524889
$501$	0	0
$502$	1.11427e9	0.393122
$503$	2.89255e9	1.01343	0.506715	−	0.862114i	$-0.330860\pi$
$503$	2.89255e9	1.01343	0.506715	+	0.862114i	$0.330860\pi$
$504$	0	0
$505$	−2.20339e9	−0.761327
$506$	−1.73799e8	−0.0596376
$507$	0	0
$508$	6.25268e8	0.211613
$509$	2.34162e8	0.0787052	0.0393526	−	0.999225i	$-0.487470\pi$
$509$	2.34162e8	0.0787052	0.0393526	+	0.999225i	$0.487470\pi$
$510$	0	0
$511$	3.84913e9	1.27611
$512$	3.04020e9	1.00105
$513$	0	0
$514$	3.57771e9	1.16207
$515$	2.82235e9	0.910512
$516$	0	0
$517$	−1.72449e9	−0.548838
$518$	−2.30930e9	−0.730007
$519$	0	0
$520$	−1.08041e10	−3.36960
$521$	−4.62358e9	−1.43234	−0.716170	−	0.697926i	$-0.754107\pi$
$521$	−4.62358e9	−1.43234	−0.716170	+	0.697926i	$0.754107\pi$
$522$	0	0
$523$	3.21781e8	0.0983567	0.0491783	−	0.998790i	$-0.484340\pi$
$523$	3.21781e8	0.0983567	0.0491783	+	0.998790i	$0.484340\pi$
$524$	1.22554e9	0.372107
$525$	0	0
$526$	−4.34503e8	−0.130179
$527$	5.89747e9	1.75521
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	6.32527e9	1.84550
$531$	0	0
$532$	2.16932e9	0.624644
$533$	6.99761e9	2.00173
$534$	0	0
$535$	4.59795e9	1.29815
$536$	−7.02799e9	−1.97131
$537$	0	0
$538$	1.39350e9	0.385805
$539$	1.44103e9	0.396381
$540$	0	0
$541$	2.74569e9	0.745523	0.372762	−	0.927927i	$-0.378411\pi$
$541$	2.74569e9	0.745523	0.372762	+	0.927927i	$0.378411\pi$
$542$	4.27768e9	1.15401
$543$	0	0
$544$	2.60229e9	0.693041
$545$	−7.99581e8	−0.211580
$546$	0	0
$547$	5.80387e8	0.151622	0.0758110	−	0.997122i	$-0.475845\pi$
$547$	5.80387e8	0.151622	0.0758110	+	0.997122i	$0.475845\pi$
$548$	−1.01465e9	−0.263380
$549$	0	0
$550$	−2.21321e9	−0.567222
$551$	4.15433e9	1.05796
$552$	0	0
$553$	3.19838e9	0.804252
$554$	2.03697e9	0.508979
$555$	0	0
$556$	−2.99466e8	−0.0738900
$557$	−5.01593e9	−1.22987	−0.614934	−	0.788578i	$-0.710817\pi$
$557$	−5.01593e9	−1.22987	−0.614934	+	0.788578i	$0.710817\pi$
$558$	0	0
$559$	−4.58636e8	−0.111052
$560$	−6.26921e9	−1.50853
$561$	0	0
$562$	6.66846e9	1.58470
$563$	7.13595e8	0.168528	0.0842640	−	0.996443i	$-0.473146\pi$
$563$	7.13595e8	0.168528	0.0842640	+	0.996443i	$0.473146\pi$
$564$	0	0
$565$	−4.39132e9	−1.02430
$566$	−1.93062e9	−0.447548
$567$	0	0
$568$	2.56894e9	0.588214
$569$	−3.55449e9	−0.808881	−0.404440	−	0.914564i	$-0.632534\pi$
$569$	−3.55449e9	−0.808881	−0.404440	+	0.914564i	$0.632534\pi$
$570$	0	0
$571$	−1.49825e9	−0.336789	−0.168395	−	0.985720i	$-0.553858\pi$
$571$	−1.49825e9	−0.336789	−0.168395	+	0.985720i	$0.553858\pi$
$572$	8.53480e8	0.190681
$573$	0	0
$574$	6.16836e9	1.36138
$575$	1.88513e9	0.413528
$576$	0	0
$577$	5.55203e9	1.20320	0.601598	−	0.798799i	$-0.294531\pi$
$577$	5.55203e9	1.20320	0.601598	+	0.798799i	$0.294531\pi$
$578$	−1.40756e9	−0.303194
$579$	0	0
$580$	1.92643e9	0.409972
$581$	5.85759e9	1.23909
$582$	0	0
$583$	−2.11627e9	−0.442315
$584$	4.55613e9	0.946567
$585$	0	0
$586$	4.44183e9	0.911843
$587$	−1.53213e8	−0.0312652	−0.0156326	−	0.999878i	$-0.504976\pi$
$587$	−1.53213e8	−0.0312652	−0.0156326	+	0.999878i	$0.504976\pi$
$588$	0	0
$589$	1.02646e10	2.06984
$590$	5.01365e9	1.00501
$591$	0	0
$592$	−1.79936e9	−0.356444
$593$	−8.12848e9	−1.60073	−0.800364	−	0.599514i	$-0.795361\pi$
$593$	−8.12848e9	−1.60073	−0.800364	+	0.599514i	$0.795361\pi$
$594$	0	0
$595$	−1.52089e10	−2.95998
$596$	−2.49861e9	−0.483433
$597$	0	0
$598$	1.62502e9	0.310746
$599$	8.03048e8	0.152668	0.0763340	−	0.997082i	$-0.475678\pi$
$599$	8.03048e8	0.152668	0.0763340	+	0.997082i	$0.475678\pi$
$600$	0	0
$601$	7.88379e9	1.48141	0.740703	−	0.671833i	$-0.234493\pi$
$601$	7.88379e9	1.48141	0.740703	+	0.671833i	$0.234493\pi$
$602$	−4.04285e8	−0.0755267
$603$	0	0
$604$	2.02341e9	0.373641
$605$	−8.29389e9	−1.52270
$606$	0	0
$607$	4.56811e9	0.829041	0.414520	−	0.910040i	$-0.363949\pi$
$607$	4.56811e9	0.829041	0.414520	+	0.910040i	$0.363949\pi$
$608$	4.52928e9	0.817273
$609$	0	0
$610$	−6.17415e9	−1.10134
$611$	1.61240e10	2.85975
$612$	0	0
$613$	−6.03167e9	−1.05761	−0.528805	−	0.848743i	$-0.677360\pi$
$613$	−6.03167e9	−1.05761	−0.528805	+	0.848743i	$0.677360\pi$
$614$	−6.31966e9	−1.10180
$615$	0	0
$616$	3.18642e9	0.549250
$617$	4.81058e9	0.824516	0.412258	−	0.911067i	$-0.364740\pi$
$617$	4.81058e9	0.824516	0.412258	+	0.911067i	$0.364740\pi$
$618$	0	0
$619$	6.35197e9	1.07644	0.538221	−	0.842804i	$-0.319096\pi$
$619$	6.35197e9	1.07644	0.538221	+	0.842804i	$0.319096\pi$
$620$	4.75983e9	0.802086
$621$	0	0
$622$	−6.42794e9	−1.07104
$623$	1.21734e9	0.201699
$624$	0	0
$625$	5.79781e9	0.949914
$626$	−2.88855e9	−0.470619
$627$	0	0
$628$	−3.00335e9	−0.483890
$629$	−4.36518e9	−0.699399
$630$	0	0
$631$	4.66561e9	0.739274	0.369637	−	0.929176i	$-0.379482\pi$
$631$	4.66561e9	0.739274	0.369637	+	0.929176i	$0.379482\pi$
$632$	3.78586e9	0.596561
$633$	0	0
$634$	−9.66886e9	−1.50683
$635$	−7.62983e9	−1.18251
$636$	0	0
$637$	−1.34736e10	−2.06537
$638$	1.44076e9	0.219643
$639$	0	0
$640$	−3.56834e9	−0.538067
$641$	5.88172e9	0.882066	0.441033	−	0.897491i	$-0.354612\pi$
$641$	5.88172e9	0.882066	0.441033	+	0.897491i	$0.354612\pi$
$642$	0	0
$643$	−3.88885e9	−0.576876	−0.288438	−	0.957499i	$-0.593136\pi$
$643$	−3.88885e9	−0.576876	−0.288438	+	0.957499i	$0.593136\pi$
$644$	−6.40816e8	−0.0945437
$645$	0	0
$646$	−9.16622e9	−1.33776
$647$	−1.34673e10	−1.95486	−0.977429	−	0.211266i	$-0.932241\pi$
$647$	−1.34673e10	−1.95486	−0.977429	+	0.211266i	$0.932241\pi$
$648$	0	0
$649$	−1.67744e9	−0.240874
$650$	2.06935e10	2.95555
$651$	0	0
$652$	1.79256e9	0.253283
$653$	5.54037e9	0.778650	0.389325	−	0.921100i	$-0.372708\pi$
$653$	5.54037e9	0.778650	0.389325	+	0.921100i	$0.372708\pi$
$654$	0	0
$655$	−1.49546e10	−2.07936
$656$	4.80625e9	0.664726
$657$	0	0
$658$	1.42132e10	1.94492
$659$	−4.27349e8	−0.0581679	−0.0290840	−	0.999577i	$-0.509259\pi$
$659$	−4.27349e8	−0.0581679	−0.0290840	+	0.999577i	$0.509259\pi$
$660$	0	0
$661$	−1.07403e10	−1.44648	−0.723238	−	0.690599i	$-0.757347\pi$
$661$	−1.07403e10	−1.44648	−0.723238	+	0.690599i	$0.757347\pi$
$662$	4.70819e9	0.630741
$663$	0	0
$664$	6.93351e9	0.919105
$665$	−2.64711e10	−3.49057
$666$	0	0
$667$	−1.22719e9	−0.160129
$668$	−1.27823e9	−0.165917
$669$	0	0
$670$	2.02484e10	2.60093
$671$	2.06571e9	0.263962
$672$	0	0
$673$	−1.05916e10	−1.33939	−0.669696	−	0.742635i	$-0.733576\pi$
$673$	−1.05916e10	−1.33939	−0.669696	+	0.742635i	$0.733576\pi$
$674$	−5.20141e9	−0.654351
$675$	0	0
$676$	−5.49753e9	−0.684470
$677$	−1.39606e10	−1.72919	−0.864595	−	0.502469i	$-0.832425\pi$
$677$	−1.39606e10	−1.72919	−0.864595	+	0.502469i	$0.832425\pi$
$678$	0	0
$679$	5.15194e9	0.631577
$680$	−1.80024e10	−2.19559
$681$	0	0
$682$	3.55983e9	0.429719
$683$	2.38059e9	0.285898	0.142949	−	0.989730i	$-0.454341\pi$
$683$	2.38059e9	0.285898	0.142949	+	0.989730i	$0.454341\pi$
$684$	0	0
$685$	1.23812e10	1.47179
$686$	−1.56681e9	−0.185303
$687$	0	0
$688$	−3.15010e8	−0.0368778
$689$	1.97872e10	2.30471
$690$	0	0
$691$	−7.07114e9	−0.815298	−0.407649	−	0.913139i	$-0.633651\pi$
$691$	−7.07114e9	−0.815298	−0.407649	+	0.913139i	$0.633651\pi$
$692$	3.64290e9	0.417904
$693$	0	0
$694$	5.10430e9	0.579666
$695$	3.65423e9	0.412904
$696$	0	0
$697$	1.16598e10	1.30430
$698$	−8.45283e9	−0.940824
$699$	0	0
$700$	−8.16035e9	−0.899219
$701$	−1.27425e10	−1.39714	−0.698570	−	0.715542i	$-0.746180\pi$
$701$	−1.27425e10	−1.39714	−0.698570	+	0.715542i	$0.746180\pi$
$702$	0	0
$703$	−7.59761e9	−0.824771
$704$	3.46737e9	0.374538
$705$	0	0
$706$	9.93636e9	1.06270
$707$	−6.07596e9	−0.646617
$708$	0	0
$709$	9.74714e8	0.102711	0.0513553	−	0.998680i	$-0.483646\pi$
$709$	9.74714e8	0.102711	0.0513553	+	0.998680i	$0.483646\pi$
$710$	−7.40140e9	−0.776085
$711$	0	0
$712$	1.44094e9	0.149612
$713$	−3.03214e9	−0.313283
$714$	0	0
$715$	−1.04146e10	−1.06554
$716$	2.91791e9	0.297082
$717$	0	0
$718$	8.12307e9	0.819001
$719$	5.28906e8	0.0530673	0.0265337	−	0.999648i	$-0.491553\pi$
$719$	5.28906e8	0.0530673	0.0265337	+	0.999648i	$0.491553\pi$
$720$	0	0
$721$	7.78278e9	0.773324
$722$	−7.54777e9	−0.746343
$723$	0	0
$724$	−4.29024e9	−0.420142
$725$	−1.56274e10	−1.52301
$726$	0	0
$727$	1.48628e8	0.0143460	0.00717298	−	0.999974i	$-0.497717\pi$
$727$	1.48628e8	0.0143460	0.00717298	+	0.999974i	$0.497717\pi$
$728$	−2.97930e10	−2.86190
$729$	0	0
$730$	−1.31267e10	−1.24889
$731$	−7.64204e8	−0.0723600
$732$	0	0
$733$	−7.75155e9	−0.726984	−0.363492	−	0.931597i	$-0.618416\pi$
$733$	−7.75155e9	−0.726984	−0.363492	+	0.931597i	$0.618416\pi$
$734$	5.74290e9	0.536038
$735$	0	0
$736$	−1.33795e9	−0.123699
$737$	−6.77458e9	−0.623370
$738$	0	0
$739$	−4.55376e9	−0.415063	−0.207532	−	0.978228i	$-0.566543\pi$
$739$	−4.55376e9	−0.415063	−0.207532	+	0.978228i	$0.566543\pi$
$740$	−3.52313e9	−0.319608
$741$	0	0
$742$	1.74423e10	1.56744
$743$	5.02367e9	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$743$	5.02367e9	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$744$	0	0
$745$	3.04892e10	2.70146
$746$	1.14818e10	1.01257
$747$	0	0
$748$	1.42211e9	0.124245
$749$	1.26791e10	1.10256
$750$	0	0
$751$	5.97002e9	0.514323	0.257161	−	0.966368i	$-0.417213\pi$
$751$	5.97002e9	0.514323	0.257161	+	0.966368i	$0.417213\pi$
$752$	1.10746e10	0.949657
$753$	0	0
$754$	−1.34711e10	−1.14447
$755$	−2.46906e10	−2.08794
$756$	0	0
$757$	1.40526e10	1.17739	0.588696	−	0.808354i	$-0.299641\pi$
$757$	1.40526e10	1.17739	0.588696	+	0.808354i	$0.299641\pi$
$758$	−1.59528e10	−1.33044
$759$	0	0
$760$	−3.13333e10	−2.58916
$761$	2.55883e9	0.210473	0.105236	−	0.994447i	$-0.466440\pi$
$761$	2.55883e9	0.210473	0.105236	+	0.994447i	$0.466440\pi$
$762$	0	0
$763$	−2.20489e9	−0.179701
$764$	−1.81355e9	−0.147131
$765$	0	0
$766$	−1.25605e10	−1.00973
$767$	1.56840e10	1.25509
$768$	0	0
$769$	2.06379e9	0.163653	0.0818264	−	0.996647i	$-0.473925\pi$
$769$	2.06379e9	0.163653	0.0818264	+	0.996647i	$0.473925\pi$
$770$	−9.18040e9	−0.724676
$771$	0	0
$772$	−3.84782e9	−0.300991
$773$	−1.69063e9	−0.131650	−0.0658248	−	0.997831i	$-0.520968\pi$
$773$	−1.69063e9	−0.131650	−0.0658248	+	0.997831i	$0.520968\pi$
$774$	0	0
$775$	−3.86122e10	−2.97968
$776$	6.09824e9	0.468478
$777$	0	0
$778$	6.32329e9	0.481409
$779$	2.02939e10	1.53810
$780$	0	0
$781$	2.47632e9	0.186006
$782$	2.70770e9	0.202477
$783$	0	0
$784$	−9.25425e9	−0.685860
$785$	3.66483e10	2.70402
$786$	0	0
$787$	−1.21027e10	−0.885056	−0.442528	−	0.896755i	$-0.645918\pi$
$787$	−1.21027e10	−0.885056	−0.442528	+	0.896755i	$0.645918\pi$
$788$	−1.84591e9	−0.134390
$789$	0	0
$790$	−1.09075e10	−0.787097
$791$	−1.21093e10	−0.869965
$792$	0	0
$793$	−1.93144e10	−1.37539
$794$	8.21841e9	0.582661
$795$	0	0
$796$	−1.29936e9	−0.0913131
$797$	−7.66224e9	−0.536107	−0.268054	−	0.963404i	$-0.586380\pi$
$797$	−7.66224e9	−0.536107	−0.268054	+	0.963404i	$0.586380\pi$
$798$	0	0
$799$	2.68667e10	1.86337
$800$	−1.70378e10	−1.17652
$801$	0	0
$802$	2.27581e10	1.55785
$803$	4.39185e9	0.299325
$804$	0	0
$805$	7.81955e9	0.528319
$806$	−3.32845e10	−2.23908
$807$	0	0
$808$	−7.19199e9	−0.479633
$809$	−2.52937e9	−0.167955	−0.0839774	−	0.996468i	$-0.526762\pi$
$809$	−2.52937e9	−0.167955	−0.0839774	+	0.996468i	$0.526762\pi$
$810$	0	0
$811$	−6.75898e9	−0.444947	−0.222473	−	0.974939i	$-0.571413\pi$
$811$	−6.75898e9	−0.444947	−0.222473	+	0.974939i	$0.571413\pi$
$812$	5.31223e9	0.348202
$813$	0	0
$814$	−2.63492e9	−0.171231
$815$	−2.18736e10	−1.41537
$816$	0	0
$817$	−1.33010e9	−0.0853309
$818$	1.19654e10	0.764349
$819$	0	0
$820$	9.41060e9	0.596031
$821$	1.62913e10	1.02743	0.513717	−	0.857959i	$-0.328268\pi$
$821$	1.62913e10	1.02743	0.513717	+	0.857959i	$0.328268\pi$
$822$	0	0
$823$	−5.49597e9	−0.343672	−0.171836	−	0.985126i	$-0.554970\pi$
$823$	−5.49597e9	−0.343672	−0.171836	+	0.985126i	$0.554970\pi$
$824$	9.21232e9	0.573619
$825$	0	0
$826$	1.38254e10	0.853586
$827$	−1.11151e10	−0.683351	−0.341675	−	0.939818i	$-0.610994\pi$
$827$	−1.11151e10	−0.683351	−0.341675	+	0.939818i	$0.610994\pi$
$828$	0	0
$829$	1.71246e10	1.04395	0.521976	−	0.852960i	$-0.325195\pi$
$829$	1.71246e10	1.04395	0.521976	+	0.852960i	$0.325195\pi$
$830$	−1.99762e10	−1.21266
$831$	0	0
$832$	−3.24200e10	−1.95156
$833$	−2.24505e10	−1.34576
$834$	0	0
$835$	1.55976e10	0.927159
$836$	2.47519e9	0.146517
$837$	0	0
$838$	−2.44967e8	−0.0143798
$839$	6.82571e9	0.399008	0.199504	−	0.979897i	$-0.436067\pi$
$839$	6.82571e9	0.399008	0.199504	+	0.979897i	$0.436067\pi$
$840$	0	0
$841$	−7.07676e9	−0.410250
$842$	−2.07367e10	−1.19715
$843$	0	0
$844$	−2.72953e9	−0.156275
$845$	6.70835e10	3.82488
$846$	0	0
$847$	−2.28709e10	−1.29327
$848$	1.35906e10	0.765340
$849$	0	0
$850$	3.44807e10	1.92579
$851$	2.24433e9	0.124834
$852$	0	0
$853$	−3.24228e9	−0.178867	−0.0894333	−	0.995993i	$-0.528506\pi$
$853$	−3.24228e9	−0.178867	−0.0894333	+	0.995993i	$0.528506\pi$
$854$	−1.70256e10	−0.935403
$855$	0	0
$856$	1.50080e10	0.817832
$857$	−1.38532e10	−0.751827	−0.375914	−	0.926655i	$-0.622671\pi$
$857$	−1.38532e10	−0.751827	−0.375914	+	0.926655i	$0.622671\pi$
$858$	0	0
$859$	2.44691e10	1.31717	0.658585	−	0.752507i	$-0.271155\pi$
$859$	2.44691e10	1.31717	0.658585	+	0.752507i	$0.271155\pi$
$860$	−6.16787e8	−0.0330667
$861$	0	0
$862$	2.67621e10	1.42313
$863$	1.63426e10	0.865530	0.432765	−	0.901507i	$-0.357538\pi$
$863$	1.63426e10	0.865530	0.432765	+	0.901507i	$0.357538\pi$
$864$	0	0
$865$	−4.44525e10	−2.33528
$866$	1.18327e10	0.619114
$867$	0	0
$868$	1.31255e10	0.681235
$869$	3.64935e9	0.188645
$870$	0	0
$871$	6.33424e10	3.24811
$872$	−2.60988e9	−0.133295
$873$	0	0
$874$	4.71275e9	0.238773
$875$	4.93668e10	2.49119
$876$	0	0
$877$	−7.74167e9	−0.387557	−0.193779	−	0.981045i	$-0.562074\pi$
$877$	−7.74167e9	−0.387557	−0.193779	+	0.981045i	$0.562074\pi$
$878$	−1.77500e10	−0.885047
$879$	0	0
$880$	−7.15316e9	−0.353841
$881$	−8.53252e8	−0.0420399	−0.0210200	−	0.999779i	$-0.506691\pi$
$881$	−8.53252e8	−0.0420399	−0.0210200	+	0.999779i	$0.506691\pi$
$882$	0	0
$883$	1.99397e10	0.974665	0.487332	−	0.873216i	$-0.337970\pi$
$883$	1.99397e10	0.974665	0.487332	+	0.873216i	$0.337970\pi$
$884$	−1.32968e10	−0.647386
$885$	0	0
$886$	−1.88189e10	−0.909024
$887$	−2.92670e10	−1.40814	−0.704070	−	0.710130i	$-0.748636\pi$
$887$	−2.92670e10	−1.40814	−0.704070	+	0.710130i	$0.748636\pi$
$888$	0	0
$889$	−2.10397e10	−1.00434
$890$	−4.15151e9	−0.197397
$891$	0	0
$892$	−1.12039e10	−0.528556
$893$	4.67615e10	2.19740
$894$	0	0
$895$	−3.56058e10	−1.66012
$896$	−9.83989e9	−0.456996
$897$	0	0
$898$	−1.21541e10	−0.560086
$899$	2.51358e10	1.15381
$900$	0	0
$901$	3.29704e10	1.50172
$902$	7.03809e9	0.319324
$903$	0	0
$904$	−1.43335e10	−0.645304
$905$	5.23516e10	2.34779
$906$	0	0
$907$	−2.22820e10	−0.991584	−0.495792	−	0.868441i	$-0.665122\pi$
$907$	−2.22820e10	−0.991584	−0.495792	+	0.868441i	$0.665122\pi$
$908$	−1.05963e10	−0.469735
$909$	0	0
$910$	8.58368e10	3.77597
$911$	−2.17481e10	−0.953030	−0.476515	−	0.879166i	$-0.658100\pi$
$911$	−2.17481e10	−0.953030	−0.476515	+	0.879166i	$0.658100\pi$
$912$	0	0
$913$	6.68351e9	0.290641
$914$	−3.22325e10	−1.39631
$915$	0	0
$916$	6.55932e9	0.281984
$917$	−4.12382e10	−1.76606
$918$	0	0
$919$	2.66114e10	1.13100	0.565501	−	0.824747i	$-0.308683\pi$
$919$	2.66114e10	1.13100	0.565501	+	0.824747i	$0.308683\pi$
$920$	9.25583e9	0.391885
$921$	0	0
$922$	−5.28626e9	−0.222121
$923$	−2.31536e10	−0.969197
$924$	0	0
$925$	2.85800e10	1.18731
$926$	−5.93359e9	−0.245572
$927$	0	0
$928$	1.10913e10	0.455580
$929$	−2.16598e9	−0.0886339	−0.0443169	−	0.999018i	$-0.514111\pi$
$929$	−2.16598e9	−0.0886339	−0.0443169	+	0.999018i	$0.514111\pi$
$930$	0	0
$931$	−3.90752e10	−1.58700
$932$	2.31805e8	0.00937923
$933$	0	0
$934$	2.85651e9	0.114715
$935$	−1.73533e10	−0.694292
$936$	0	0
$937$	1.23177e10	0.489147	0.244574	−	0.969631i	$-0.421352\pi$
$937$	1.23177e10	0.489147	0.244574	+	0.969631i	$0.421352\pi$
$938$	5.58360e10	2.20904
$939$	0	0
$940$	2.16840e10	0.851516
$941$	1.73653e10	0.679389	0.339695	−	0.940536i	$-0.389676\pi$
$941$	1.73653e10	0.679389	0.339695	+	0.940536i	$0.389676\pi$
$942$	0	0
$943$	−5.99480e9	−0.232801
$944$	1.07724e10	0.416785
$945$	0	0
$946$	−4.61289e8	−0.0177155
$947$	−3.93511e10	−1.50568	−0.752839	−	0.658205i	$-0.771316\pi$
$947$	−3.93511e10	−1.50568	−0.752839	+	0.658205i	$0.771316\pi$
$948$	0	0
$949$	−4.10639e10	−1.55965
$950$	6.00136e10	2.27100
$951$	0	0
$952$	−4.96427e10	−1.86477
$953$	−1.84054e10	−0.688842	−0.344421	−	0.938815i	$-0.611925\pi$
$953$	−1.84054e10	−0.688842	−0.344421	+	0.938815i	$0.611925\pi$
$954$	0	0
$955$	2.21298e10	0.822179
$956$	−2.07421e8	−0.00767803
$957$	0	0
$958$	2.03514e10	0.747850
$959$	3.41418e10	1.25003
$960$	0	0
$961$	3.45932e10	1.25736
$962$	2.46365e10	0.892207
$963$	0	0
$964$	1.25912e10	0.452687
$965$	4.69530e10	1.68197
$966$	0	0
$967$	−1.75195e10	−0.623058	−0.311529	−	0.950237i	$-0.600841\pi$
$967$	−1.75195e10	−0.623058	−0.311529	+	0.950237i	$0.600841\pi$
$968$	−2.70718e10	−0.959296
$969$	0	0
$970$	−1.75697e10	−0.618106
$971$	1.54785e10	0.542578	0.271289	−	0.962498i	$-0.412550\pi$
$971$	1.54785e10	0.542578	0.271289	+	0.962498i	$0.412550\pi$
$972$	0	0
$973$	1.00767e10	0.350691
$974$	4.41125e10	1.52970
$975$	0	0
$976$	−1.32659e10	−0.456734
$977$	4.59508e10	1.57638	0.788192	−	0.615429i	$-0.211017\pi$
$977$	4.59508e10	1.57638	0.788192	+	0.615429i	$0.211017\pi$
$978$	0	0
$979$	1.38899e9	0.0473106
$980$	−1.81198e10	−0.614980
$981$	0	0
$982$	−2.21531e10	−0.746524
$983$	2.28656e10	0.767796	0.383898	−	0.923376i	$-0.374581\pi$
$983$	2.28656e10	0.767796	0.383898	+	0.923376i	$0.374581\pi$
$984$	0	0
$985$	2.25246e10	0.750985
$986$	−2.24462e10	−0.745718
$987$	0	0
$988$	−2.31431e10	−0.763434
$989$	3.92910e8	0.0129154
$990$	0	0
$991$	−3.74255e10	−1.22155	−0.610774	−	0.791805i	$-0.709141\pi$
$991$	−3.74255e10	−1.22155	−0.610774	+	0.791805i	$0.709141\pi$
$992$	2.74045e10	0.891315
$993$	0	0
$994$	−2.04098e10	−0.659152
$995$	1.58554e10	0.510266
$996$	0	0
$997$	−4.91574e10	−1.57093	−0.785463	−	0.618908i	$-0.787575\pi$
$997$	−4.91574e10	−1.57093	−0.785463	+	0.618908i	$0.787575\pi$
$998$	−2.70348e10	−0.860925
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.8.a.g.1.5	✓	12
3.2	odd	2		207.8.a.h.1.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.8.a.g.1.5	✓	12	1.1	even	1	trivial
207.8.a.h.1.8	yes	12	3.2	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-9.40410 q^{2} -39.5629 q^{4} +482.766 q^{5} +1331.25 q^{7} +1575.78 q^{8} +O(q^{10})\)	\(q-9.40410 q^{2} -39.5629 q^{4} +482.766 q^{5} +1331.25 q^{7} +1575.78 q^{8} -4539.98 q^{10} +1518.96 q^{11} -14202.3 q^{13} -12519.2 q^{14} -9754.72 q^{16} -23664.6 q^{17} -41188.3 q^{19} -19099.7 q^{20} -14284.5 q^{22} +12167.0 q^{23} +154938. q^{25} +133560. q^{26} -52668.3 q^{28} -100862. q^{29} -249210. q^{31} -109965. q^{32} +222544. q^{34} +642685. q^{35} +184460. q^{37} +387339. q^{38} +760733. q^{40} -492710. q^{41} +32293.1 q^{43} -60094.5 q^{44} -114420. q^{46} -1.13531e6 q^{47} +948695. q^{49} -1.45706e6 q^{50} +561884. q^{52} -1.39324e6 q^{53} +733303. q^{55} +2.09776e6 q^{56} +948515. q^{58} -1.10433e6 q^{59} +1.35995e6 q^{61} +2.34360e6 q^{62} +2.28273e6 q^{64} -6.85639e6 q^{65} -4.46001e6 q^{67} +936242. q^{68} -6.04387e6 q^{70} +1.63027e6 q^{71} +2.89135e6 q^{73} -1.73468e6 q^{74} +1.62953e6 q^{76} +2.02212e6 q^{77} +2.40253e6 q^{79} -4.70925e6 q^{80} +4.63349e6 q^{82} +4.40005e6 q^{83} -1.14245e7 q^{85} -303688. q^{86} +2.39354e6 q^{88} +914433. q^{89} -1.89069e7 q^{91} -481362. q^{92} +1.06766e7 q^{94} -1.98843e7 q^{95} +3.86999e6 q^{97} -8.92162e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{2} + 640 q^{4} - 500 q^{5} - 228 q^{7} - 3072 q^{8}+O(q^{10}) \) 12 * q - 16 * q^2 + 640 * q^4 - 500 * q^5 - 228 * q^7 - 3072 * q^8	\( 12 q - 16 q^{2} + 640 q^{4} - 500 q^{5} - 228 q^{7} - 3072 q^{8} + 10270 q^{10} + 460 q^{11} - 21060 q^{13} - 4268 q^{14} + 56676 q^{16} - 73124 q^{17} + 8508 q^{19} - 170538 q^{20} + 124754 q^{22} + 146004 q^{23} + 194064 q^{25} - 206080 q^{26} - 390416 q^{28} - 268640 q^{29} - 191880 q^{31} - 1180172 q^{32} - 221436 q^{34} + 487244 q^{35} + 650332 q^{37} - 1432950 q^{38} + 1775722 q^{40} - 980088 q^{41} - 861276 q^{43} - 800666 q^{44} - 194672 q^{46} - 403868 q^{47} + 1699160 q^{49} - 2919092 q^{50} - 2369520 q^{52} + 201948 q^{53} - 1553512 q^{55} + 4848116 q^{56} + 3720672 q^{58} - 1302676 q^{59} + 2141364 q^{61} - 2160944 q^{62} + 9702136 q^{64} - 9099536 q^{65} - 6159260 q^{67} - 18442208 q^{68} - 10891632 q^{70} - 12584184 q^{71} + 7435872 q^{73} - 22491442 q^{74} + 5721386 q^{76} - 16450568 q^{77} + 3658028 q^{79} - 49905778 q^{80} - 5516316 q^{82} - 26137900 q^{83} + 5169556 q^{85} - 30678550 q^{86} + 14753046 q^{88} - 27235908 q^{89} - 7657216 q^{91} + 7786880 q^{92} - 23519352 q^{94} - 63623628 q^{95} + 22454720 q^{97} - 94951532 q^{98}+O(q^{100}) \) 12 * q - 16 * q^2 + 640 * q^4 - 500 * q^5 - 228 * q^7 - 3072 * q^8 + 10270 * q^10 + 460 * q^11 - 21060 * q^13 - 4268 * q^14 + 56676 * q^16 - 73124 * q^17 + 8508 * q^19 - 170538 * q^20 + 124754 * q^22 + 146004 * q^23 + 194064 * q^25 - 206080 * q^26 - 390416 * q^28 - 268640 * q^29 - 191880 * q^31 - 1180172 * q^32 - 221436 * q^34 + 487244 * q^35 + 650332 * q^37 - 1432950 * q^38 + 1775722 * q^40 - 980088 * q^41 - 861276 * q^43 - 800666 * q^44 - 194672 * q^46 - 403868 * q^47 + 1699160 * q^49 - 2919092 * q^50 - 2369520 * q^52 + 201948 * q^53 - 1553512 * q^55 + 4848116 * q^56 + 3720672 * q^58 - 1302676 * q^59 + 2141364 * q^61 - 2160944 * q^62 + 9702136 * q^64 - 9099536 * q^65 - 6159260 * q^67 - 18442208 * q^68 - 10891632 * q^70 - 12584184 * q^71 + 7435872 * q^73 - 22491442 * q^74 + 5721386 * q^76 - 16450568 * q^77 + 3658028 * q^79 - 49905778 * q^80 - 5516316 * q^82 - 26137900 * q^83 + 5169556 * q^85 - 30678550 * q^86 + 14753046 * q^88 - 27235908 * q^89 - 7657216 * q^91 + 7786880 * q^92 - 23519352 * q^94 - 63623628 * q^95 + 22454720 * q^97 - 94951532 * q^98

Introduction

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