LMFDB - Embedded newform 207.8.a.g.1.8

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			1.8
Root			$-5.26584$ 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)

$f(q)$	$=$	$q+4.26584 q^{2} -109.803 q^{4} +101.908 q^{5} +827.161 q^{7} -1014.43 q^{8} +O(q^{10})$	\(q+4.26584 q^{2} -109.803 q^{4} +101.908 q^{5} +827.161 q^{7} -1014.43 q^{8} +434.722 q^{10} -4463.78 q^{11} +3664.05 q^{13} +3528.54 q^{14} +9727.36 q^{16} +15392.1 q^{17} -27461.5 q^{19} -11189.7 q^{20} -19041.8 q^{22} +12167.0 q^{23} -67739.8 q^{25} +15630.2 q^{26} -90824.5 q^{28} +75436.2 q^{29} +150445. q^{31} +171342. q^{32} +65660.0 q^{34} +84294.2 q^{35} -41357.1 q^{37} -117146. q^{38} -103378. q^{40} -86761.9 q^{41} +5654.17 q^{43} +490135. q^{44} +51902.4 q^{46} -1.14168e6 q^{47} -139347. q^{49} -288967. q^{50} -402322. q^{52} -1.45193e6 q^{53} -454894. q^{55} -839095. q^{56} +321799. q^{58} -1.45267e6 q^{59} +134473. q^{61} +641775. q^{62} -514185. q^{64} +373395. q^{65} +419718. q^{67} -1.69009e6 q^{68} +359585. q^{70} -1.27175e6 q^{71} -6.08795e6 q^{73} -176423. q^{74} +3.01535e6 q^{76} -3.69227e6 q^{77} -2.97501e6 q^{79} +991294. q^{80} -370112. q^{82} -1.22589e6 q^{83} +1.56857e6 q^{85} +24119.8 q^{86} +4.52818e6 q^{88} +1.60825e6 q^{89} +3.03076e6 q^{91} -1.33597e6 q^{92} -4.87021e6 q^{94} -2.79854e6 q^{95} -4.15029e6 q^{97} -594431. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.26584	0.377050	0.188525	−	0.982068i	$-0.439629\pi$
$2$	4.26584	0.377050	0.188525	+	0.982068i	$0.439629\pi$
$3$	0	0
$3$	0	0
$4$	−109.803	−0.857833
$5$	101.908	0.364597	0.182298	−	0.983243i	$-0.441646\pi$
$5$	101.908	0.364597	0.182298	+	0.983243i	$0.441646\pi$
$6$	0	0
$7$	827.161	0.911480	0.455740	−	0.890113i	$-0.349375\pi$
$7$	827.161	0.911480	0.455740	+	0.890113i	$0.349375\pi$
$8$	−1014.43	−0.700496
$9$	0	0
$10$	434.722	0.137471
$11$	−4463.78	−1.01118	−0.505590	−	0.862774i	$-0.668725\pi$
$11$	−4463.78	−1.01118	−0.505590	+	0.862774i	$0.668725\pi$
$12$	0	0
$13$	3664.05	0.462551	0.231276	−	0.972888i	$-0.425710\pi$
$13$	3664.05	0.462551	0.231276	+	0.972888i	$0.425710\pi$
$14$	3528.54	0.343674
$15$	0	0
$16$	9727.36	0.593711
$17$	15392.1	0.759846	0.379923	−	0.925018i	$-0.375951\pi$
$17$	15392.1	0.759846	0.379923	+	0.925018i	$0.375951\pi$
$18$	0	0
$19$	−27461.5	−0.918517	−0.459258	−	0.888303i	$-0.651885\pi$
$19$	−27461.5	−0.918517	−0.459258	+	0.888303i	$0.651885\pi$
$20$	−11189.7	−0.312763
$21$	0	0
$22$	−19041.8	−0.381266
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	0	0
$25$	−67739.8	−0.867069
$26$	15630.2	0.174405
$27$	0	0
$28$	−90824.5	−0.781898
$29$	75436.2	0.574364	0.287182	−	0.957876i	$-0.407282\pi$
$29$	75436.2	0.574364	0.287182	+	0.957876i	$0.407282\pi$
$30$	0	0
$31$	150445.	0.907011	0.453506	−	0.891253i	$-0.350173\pi$
$31$	150445.	0.907011	0.453506	+	0.891253i	$0.350173\pi$
$32$	171342.	0.924355
$33$	0	0
$34$	65660.0	0.286500
$35$	84294.2	0.332322
$36$	0	0
$37$	−41357.1	−0.134228	−0.0671142	−	0.997745i	$-0.521379\pi$
$37$	−41357.1	−0.134228	−0.0671142	+	0.997745i	$0.521379\pi$
$38$	−117146.	−0.346327
$39$	0	0
$40$	−103378.	−0.255399
$41$	−86761.9	−0.196601	−0.0983005	−	0.995157i	$-0.531341\pi$
$41$	−86761.9	−0.196601	−0.0983005	+	0.995157i	$0.531341\pi$
$42$	0	0
$43$	5654.17	0.0108450	0.00542249	−	0.999985i	$-0.498274\pi$
$43$	5654.17	0.0108450	0.00542249	+	0.999985i	$0.498274\pi$
$44$	490135.	0.867424
$45$	0	0
$46$	51902.4	0.0786204
$47$	−1.14168e6	−1.60399	−0.801994	−	0.597332i	$-0.796228\pi$
$47$	−1.14168e6	−1.60399	−0.801994	+	0.597332i	$0.796228\pi$
$48$	0	0
$49$	−139347.	−0.169204
$50$	−288967.	−0.326929
$51$	0	0
$52$	−402322.	−0.396792
$53$	−1.45193e6	−1.33962	−0.669808	−	0.742534i	$-0.733624\pi$
$53$	−1.45193e6	−1.33962	−0.669808	+	0.742534i	$0.733624\pi$
$54$	0	0
$55$	−454894.	−0.368673
$56$	−839095.	−0.638488
$57$	0	0
$58$	321799.	0.216564
$59$	−1.45267e6	−0.920840	−0.460420	−	0.887701i	$-0.652301\pi$
$59$	−1.45267e6	−0.920840	−0.460420	+	0.887701i	$0.652301\pi$
$60$	0	0
$61$	134473.	0.0758545	0.0379273	−	0.999281i	$-0.487924\pi$
$61$	134473.	0.0758545	0.0379273	+	0.999281i	$0.487924\pi$
$62$	641775.	0.341989
$63$	0	0
$64$	−514185.	−0.245183
$65$	373395.	0.168645
$66$	0	0
$67$	419718.	0.170489	0.0852444	−	0.996360i	$-0.472833\pi$
$67$	419718.	0.170489	0.0852444	+	0.996360i	$0.472833\pi$
$68$	−1.69009e6	−0.651821
$69$	0	0
$70$	359585.	0.125302
$71$	−1.27175e6	−0.421694	−0.210847	−	0.977519i	$-0.567622\pi$
$71$	−1.27175e6	−0.421694	−0.210847	+	0.977519i	$0.567622\pi$
$72$	0	0
$73$	−6.08795e6	−1.83164	−0.915822	−	0.401584i	$-0.868460\pi$
$73$	−6.08795e6	−1.83164	−0.915822	+	0.401584i	$0.868460\pi$
$74$	−176423.	−0.0506108
$75$	0	0
$76$	3.01535e6	0.787934
$77$	−3.69227e6	−0.921670
$78$	0	0
$79$	−2.97501e6	−0.678882	−0.339441	−	0.940627i	$-0.610238\pi$
$79$	−2.97501e6	−0.678882	−0.339441	+	0.940627i	$0.610238\pi$
$80$	991294.	0.216465
$81$	0	0
$82$	−370112.	−0.0741284
$83$	−1.22589e6	−0.235330	−0.117665	−	0.993053i	$-0.537541\pi$
$83$	−1.22589e6	−0.235330	−0.117665	+	0.993053i	$0.537541\pi$
$84$	0	0
$85$	1.56857e6	0.277037
$86$	24119.8	0.00408911
$87$	0	0
$88$	4.52818e6	0.708328
$89$	1.60825e6	0.241818	0.120909	−	0.992664i	$-0.461419\pi$
$89$	1.60825e6	0.241818	0.120909	+	0.992664i	$0.461419\pi$
$90$	0	0
$91$	3.03076e6	0.421606
$92$	−1.33597e6	−0.178871
$93$	0	0
$94$	−4.87021e6	−0.604784
$95$	−2.79854e6	−0.334888
$96$	0	0
$97$	−4.15029e6	−0.461719	−0.230859	−	0.972987i	$-0.574154\pi$
$97$	−4.15029e6	−0.461719	−0.230859	+	0.972987i	$0.574154\pi$
$98$	−594431.	−0.0637985
$99$	0	0
$100$	7.43801e6	0.743801
$101$	−1.49474e7	−1.44358	−0.721789	−	0.692113i	$-0.756680\pi$
$101$	−1.49474e7	−1.44358	−0.721789	+	0.692113i	$0.756680\pi$
$102$	0	0
$103$	1.27347e7	1.14831	0.574153	−	0.818748i	$-0.305332\pi$
$103$	1.27347e7	1.14831	0.574153	+	0.818748i	$0.305332\pi$
$104$	−3.71691e6	−0.324015
$105$	0	0
$106$	−6.19370e6	−0.505102
$107$	−1.03257e7	−0.814846	−0.407423	−	0.913240i	$-0.633573\pi$
$107$	−1.03257e7	−0.814846	−0.407423	+	0.913240i	$0.633573\pi$
$108$	0	0
$109$	5.98027e6	0.442311	0.221156	−	0.975239i	$-0.429017\pi$
$109$	5.98027e6	0.442311	0.221156	+	0.975239i	$0.429017\pi$
$110$	−1.94050e6	−0.139008
$111$	0	0
$112$	8.04610e6	0.541156
$113$	−2.33066e7	−1.51951	−0.759756	−	0.650208i	$-0.774682\pi$
$113$	−2.33066e7	−1.51951	−0.759756	+	0.650208i	$0.774682\pi$
$114$	0	0
$115$	1.23991e6	0.0760236
$116$	−8.28310e6	−0.492708
$117$	0	0
$118$	−6.19684e6	−0.347203
$119$	1.27317e7	0.692584
$120$	0	0
$121$	438174.	0.0224853
$122$	573641.	0.0286010
$123$	0	0
$124$	−1.65193e7	−0.778064
$125$	−1.48648e7	−0.680727
$126$	0	0
$127$	3.17582e7	1.37576	0.687880	−	0.725824i	$-0.258541\pi$
$127$	3.17582e7	1.37576	0.687880	+	0.725824i	$0.258541\pi$
$128$	−2.41252e7	−1.01680
$129$	0	0
$130$	1.59284e6	0.0635875
$131$	−1.37995e7	−0.536306	−0.268153	−	0.963376i	$-0.586413\pi$
$131$	−1.37995e7	−0.536306	−0.268153	+	0.963376i	$0.586413\pi$
$132$	0	0
$133$	−2.27151e7	−0.837210
$134$	1.79045e6	0.0642828
$135$	0	0
$136$	−1.56141e7	−0.532269
$137$	1.28991e7	0.428584	0.214292	−	0.976770i	$-0.431256\pi$
$137$	1.28991e7	0.428584	0.214292	+	0.976770i	$0.431256\pi$
$138$	0	0
$139$	−1.16204e7	−0.367003	−0.183502	−	0.983019i	$-0.558743\pi$
$139$	−1.16204e7	−0.367003	−0.183502	+	0.983019i	$0.558743\pi$
$140$	−9.25573e6	−0.285077
$141$	0	0
$142$	−5.42507e6	−0.159000
$143$	−1.63555e7	−0.467723
$144$	0	0
$145$	7.68754e6	0.209411
$146$	−2.59702e7	−0.690622
$147$	0	0
$148$	4.54112e6	0.115145
$149$	−4.65905e7	−1.15384	−0.576919	−	0.816801i	$-0.695745\pi$
$149$	−4.65905e7	−1.15384	−0.576919	+	0.816801i	$0.695745\pi$
$150$	0	0
$151$	4.14857e7	0.980572	0.490286	−	0.871562i	$-0.336892\pi$
$151$	4.14857e7	0.980572	0.490286	+	0.871562i	$0.336892\pi$
$152$	2.78577e7	0.643417
$153$	0	0
$154$	−1.57506e7	−0.347516
$155$	1.53315e7	0.330693
$156$	0	0
$157$	2.09733e7	0.432531	0.216265	−	0.976335i	$-0.430612\pi$
$157$	2.09733e7	0.432531	0.216265	+	0.976335i	$0.430612\pi$
$158$	−1.26909e7	−0.255972
$159$	0	0
$160$	1.74611e7	0.337017
$161$	1.00641e7	0.190057
$162$	0	0
$163$	2.50542e7	0.453130	0.226565	−	0.973996i	$-0.427250\pi$
$163$	2.50542e7	0.453130	0.226565	+	0.973996i	$0.427250\pi$
$164$	9.52669e6	0.168651
$165$	0	0
$166$	−5.22945e6	−0.0887314
$167$	−5.55811e7	−0.923464	−0.461732	−	0.887020i	$-0.652772\pi$
$167$	−5.55811e7	−0.923464	−0.461732	+	0.887020i	$0.652772\pi$
$168$	0	0
$169$	−4.93232e7	−0.786046
$170$	6.69127e6	0.104457
$171$	0	0
$172$	−620843.	−0.00930319
$173$	1.22298e8	1.79581	0.897903	−	0.440193i	$-0.145090\pi$
$173$	1.22298e8	1.79581	0.897903	+	0.440193i	$0.145090\pi$
$174$	0	0
$175$	−5.60317e7	−0.790316
$176$	−4.34208e7	−0.600349
$177$	0	0
$178$	6.86054e6	0.0911776
$179$	5.12543e7	0.667951	0.333975	−	0.942582i	$-0.391610\pi$
$179$	5.12543e7	0.667951	0.333975	+	0.942582i	$0.391610\pi$
$180$	0	0
$181$	−2.58764e7	−0.324361	−0.162181	−	0.986761i	$-0.551853\pi$
$181$	−2.58764e7	−0.324361	−0.162181	+	0.986761i	$0.551853\pi$
$182$	1.29287e7	0.158967
$183$	0	0
$184$	−1.23425e7	−0.146064
$185$	−4.21461e6	−0.0489392
$186$	0	0
$187$	−6.87068e7	−0.768341
$188$	1.25359e8	1.37595
$189$	0	0
$190$	−1.19381e7	−0.126270
$191$	−1.10362e8	−1.14605	−0.573023	−	0.819539i	$-0.694229\pi$
$191$	−1.10362e8	−1.14605	−0.573023	+	0.819539i	$0.694229\pi$
$192$	0	0
$193$	−3.88101e7	−0.388592	−0.194296	−	0.980943i	$-0.562242\pi$
$193$	−3.88101e7	−0.388592	−0.194296	+	0.980943i	$0.562242\pi$
$194$	−1.77045e7	−0.174091
$195$	0	0
$196$	1.53007e7	0.145149
$197$	8.02355e7	0.747713	0.373856	−	0.927487i	$-0.378035\pi$
$197$	8.02355e7	0.747713	0.373856	+	0.927487i	$0.378035\pi$
$198$	0	0
$199$	4.37503e7	0.393546	0.196773	−	0.980449i	$-0.436954\pi$
$199$	4.37503e7	0.393546	0.196773	+	0.980449i	$0.436954\pi$
$200$	6.87171e7	0.607379
$201$	0	0
$202$	−6.37631e7	−0.544301
$203$	6.23979e7	0.523521
$204$	0	0
$205$	−8.84172e6	−0.0716800
$206$	5.43240e7	0.432969
$207$	0	0
$208$	3.56415e7	0.274622
$209$	1.22582e8	0.928786
$210$	0	0
$211$	2.13936e8	1.56782	0.783908	−	0.620878i	$-0.213224\pi$
$211$	2.13936e8	1.56782	0.783908	+	0.620878i	$0.213224\pi$
$212$	1.59426e8	1.14917
$213$	0	0
$214$	−4.40477e7	−0.307238
$215$	576204.	0.00395405
$216$	0	0
$217$	1.24443e8	0.826722
$218$	2.55108e7	0.166773
$219$	0	0
$220$	4.99486e7	0.316260
$221$	5.63973e7	0.351468
$222$	0	0
$223$	3.27936e7	0.198026	0.0990129	−	0.995086i	$-0.468432\pi$
$223$	3.27936e7	0.198026	0.0990129	+	0.995086i	$0.468432\pi$
$224$	1.41727e8	0.842531
$225$	0	0
$226$	−9.94221e7	−0.572932
$227$	−1.71273e7	−0.0971847	−0.0485923	−	0.998819i	$-0.515474\pi$
$227$	−1.71273e7	−0.0971847	−0.0485923	+	0.998819i	$0.515474\pi$
$228$	0	0
$229$	−1.55840e8	−0.857539	−0.428770	−	0.903414i	$-0.641053\pi$
$229$	−1.55840e8	−0.857539	−0.428770	+	0.903414i	$0.641053\pi$
$230$	5.28926e6	0.0286647
$231$	0	0
$232$	−7.65246e7	−0.402340
$233$	−9.29896e7	−0.481602	−0.240801	−	0.970574i	$-0.577410\pi$
$233$	−9.29896e7	−0.481602	−0.240801	+	0.970574i	$0.577410\pi$
$234$	0	0
$235$	−1.16346e8	−0.584809
$236$	1.59507e8	0.789927
$237$	0	0
$238$	5.43114e7	0.261139
$239$	−3.18551e8	−1.50934	−0.754668	−	0.656107i	$-0.772202\pi$
$239$	−3.18551e8	−1.50934	−0.754668	+	0.656107i	$0.772202\pi$
$240$	0	0
$241$	2.74935e8	1.26523	0.632617	−	0.774465i	$-0.281981\pi$
$241$	2.74935e8	1.26523	0.632617	+	0.774465i	$0.281981\pi$
$242$	1.86918e6	0.00847807
$243$	0	0
$244$	−1.47655e7	−0.0650705
$245$	−1.42005e7	−0.0616913
$246$	0	0
$247$	−1.00620e8	−0.424861
$248$	−1.52616e8	−0.635358
$249$	0	0
$250$	−6.34106e7	−0.256668
$251$	−1.03140e8	−0.411691	−0.205845	−	0.978585i	$-0.565994\pi$
$251$	−1.03140e8	−0.411691	−0.205845	+	0.978585i	$0.565994\pi$
$252$	0	0
$253$	−5.43108e7	−0.210846
$254$	1.35475e8	0.518731
$255$	0	0
$256$	−3.70985e7	−0.138202
$257$	3.98328e8	1.46378	0.731889	−	0.681424i	$-0.238639\pi$
$257$	3.98328e8	1.46378	0.731889	+	0.681424i	$0.238639\pi$
$258$	0	0
$259$	−3.42090e7	−0.122346
$260$	−4.09998e7	−0.144669
$261$	0	0
$262$	−5.88662e7	−0.202214
$263$	1.30894e8	0.443686	0.221843	−	0.975082i	$-0.428793\pi$
$263$	1.30894e8	0.443686	0.221843	+	0.975082i	$0.428793\pi$
$264$	0	0
$265$	−1.47963e8	−0.488419
$266$	−9.68989e7	−0.315670
$267$	0	0
$268$	−4.60861e7	−0.146251
$269$	−1.97336e8	−0.618122	−0.309061	−	0.951042i	$-0.600015\pi$
$269$	−1.97336e8	−0.618122	−0.309061	+	0.951042i	$0.600015\pi$
$270$	0	0
$271$	3.34396e8	1.02063	0.510316	−	0.859987i	$-0.329528\pi$
$271$	3.34396e8	1.02063	0.510316	+	0.859987i	$0.329528\pi$
$272$	1.49724e8	0.451129
$273$	0	0
$274$	5.50253e7	0.161598
$275$	3.02376e8	0.876763
$276$	0	0
$277$	4.33780e8	1.22628	0.613141	−	0.789974i	$-0.289906\pi$
$277$	4.33780e8	1.22628	0.613141	+	0.789974i	$0.289906\pi$
$278$	−4.95708e7	−0.138379
$279$	0	0
$280$	−8.55103e7	−0.232791
$281$	6.38437e8	1.71651	0.858254	−	0.513226i	$-0.171550\pi$
$281$	6.38437e8	1.71651	0.858254	+	0.513226i	$0.171550\pi$
$282$	0	0
$283$	1.70705e8	0.447707	0.223854	−	0.974623i	$-0.428136\pi$
$283$	1.70705e8	0.447707	0.223854	+	0.974623i	$0.428136\pi$
$284$	1.39641e8	0.361743
$285$	0	0
$286$	−6.97700e7	−0.176355
$287$	−7.17661e7	−0.179198
$288$	0	0
$289$	−1.73423e8	−0.422634
$290$	3.27938e7	0.0789585
$291$	0	0
$292$	6.68473e8	1.57125
$293$	6.77650e7	0.157387	0.0786934	−	0.996899i	$-0.474925\pi$
$293$	6.77650e7	0.157387	0.0786934	+	0.996899i	$0.474925\pi$
$294$	0	0
$295$	−1.48038e8	−0.335735
$296$	4.19538e7	0.0940264
$297$	0	0
$298$	−1.98747e8	−0.435055
$299$	4.45805e7	0.0964486
$300$	0	0
$301$	4.67691e6	0.00988499
$302$	1.76971e8	0.369725
$303$	0	0
$304$	−2.67128e8	−0.545333
$305$	1.37039e7	0.0276563
$306$	0	0
$307$	5.05649e8	0.997389	0.498694	−	0.866778i	$-0.333813\pi$
$307$	5.05649e8	0.997389	0.498694	+	0.866778i	$0.333813\pi$
$308$	4.05421e8	0.790640
$309$	0	0
$310$	6.54019e7	0.124688
$311$	−3.72486e8	−0.702180	−0.351090	−	0.936342i	$-0.614189\pi$
$311$	−3.72486e8	−0.702180	−0.351090	+	0.936342i	$0.614189\pi$
$312$	0	0
$313$	2.85640e8	0.526519	0.263260	−	0.964725i	$-0.415202\pi$
$313$	2.85640e8	0.526519	0.263260	+	0.964725i	$0.415202\pi$
$314$	8.94685e7	0.163086
$315$	0	0
$316$	3.26664e8	0.582367
$317$	−5.54461e8	−0.977605	−0.488802	−	0.872395i	$-0.662566\pi$
$317$	−5.54461e8	−0.977605	−0.488802	+	0.872395i	$0.662566\pi$
$318$	0	0
$319$	−3.36731e8	−0.580785
$320$	−5.23995e7	−0.0893927
$321$	0	0
$322$	4.29317e7	0.0716609
$323$	−4.22689e8	−0.697931
$324$	0	0
$325$	−2.48202e8	−0.401064
$326$	1.06877e8	0.170853
$327$	0	0
$328$	8.80136e7	0.137718
$329$	−9.44352e8	−1.46200
$330$	0	0
$331$	−6.21927e8	−0.942630	−0.471315	−	0.881965i	$-0.656220\pi$
$331$	−6.21927e8	−0.942630	−0.471315	+	0.881965i	$0.656220\pi$
$332$	1.34606e8	0.201874
$333$	0	0
$334$	−2.37100e8	−0.348192
$335$	4.27725e7	0.0621596
$336$	0	0
$337$	−5.54850e8	−0.789716	−0.394858	−	0.918742i	$-0.629206\pi$
$337$	−5.54850e8	−0.789716	−0.394858	+	0.918742i	$0.629206\pi$
$338$	−2.10405e8	−0.296379
$339$	0	0
$340$	−1.72233e8	−0.237652
$341$	−6.71555e8	−0.917152
$342$	0	0
$343$	−7.96465e8	−1.06571
$344$	−5.73574e6	−0.00759688
$345$	0	0
$346$	5.21705e8	0.677109
$347$	5.53293e8	0.710890	0.355445	−	0.934697i	$-0.384329\pi$
$347$	5.53293e8	0.710890	0.355445	+	0.934697i	$0.384329\pi$
$348$	0	0
$349$	9.19891e8	1.15837	0.579185	−	0.815196i	$-0.303371\pi$
$349$	9.19891e8	1.15837	0.579185	+	0.815196i	$0.303371\pi$
$350$	−2.39022e8	−0.297989
$351$	0	0
$352$	−7.64833e8	−0.934690
$353$	−1.10966e9	−1.34270	−0.671352	−	0.741139i	$-0.734286\pi$
$353$	−1.10966e9	−1.34270	−0.671352	+	0.741139i	$0.734286\pi$
$354$	0	0
$355$	−1.29601e8	−0.153748
$356$	−1.76590e8	−0.207440
$357$	0	0
$358$	2.18642e8	0.251851
$359$	−1.37121e9	−1.56413	−0.782065	−	0.623196i	$-0.785834\pi$
$359$	−1.37121e9	−1.56413	−0.782065	+	0.623196i	$0.785834\pi$
$360$	0	0
$361$	−1.39737e8	−0.156327
$362$	−1.10385e8	−0.122301
$363$	0	0
$364$	−3.32786e8	−0.361668
$365$	−6.20410e8	−0.667811
$366$	0	0
$367$	−1.00617e9	−1.06252	−0.531261	−	0.847208i	$-0.678282\pi$
$367$	−1.00617e9	−1.06252	−0.531261	+	0.847208i	$0.678282\pi$
$368$	1.18353e8	0.123797
$369$	0	0
$370$	−1.79788e7	−0.0184525
$371$	−1.20098e9	−1.22103
$372$	0	0
$373$	−1.30996e9	−1.30701	−0.653504	−	0.756923i	$-0.726702\pi$
$373$	−1.30996e9	−1.30701	−0.653504	+	0.756923i	$0.726702\pi$
$374$	−2.93092e8	−0.289703
$375$	0	0
$376$	1.15815e9	1.12359
$377$	2.76402e8	0.265673
$378$	0	0
$379$	1.97871e9	1.86700	0.933502	−	0.358573i	$-0.116737\pi$
$379$	1.97871e9	1.86700	0.933502	+	0.358573i	$0.116737\pi$
$380$	3.07288e8	0.287278
$381$	0	0
$382$	−4.70785e8	−0.432117
$383$	−1.50430e9	−1.36816	−0.684082	−	0.729405i	$-0.739797\pi$
$383$	−1.50430e9	−1.36816	−0.684082	+	0.729405i	$0.739797\pi$
$384$	0	0
$385$	−3.76271e8	−0.336038
$386$	−1.65557e8	−0.146519
$387$	0	0
$388$	4.55713e8	0.396078
$389$	−6.71657e8	−0.578528	−0.289264	−	0.957249i	$-0.593410\pi$
$389$	−6.71657e8	−0.578528	−0.289264	+	0.957249i	$0.593410\pi$
$390$	0	0
$391$	1.87275e8	0.158439
$392$	1.41357e8	0.118527
$393$	0	0
$394$	3.42272e8	0.281925
$395$	−3.03177e8	−0.247518
$396$	0	0
$397$	1.20878e9	0.969569	0.484785	−	0.874634i	$-0.338898\pi$
$397$	1.20878e9	0.969569	0.484785	+	0.874634i	$0.338898\pi$
$398$	1.86632e8	0.148386
$399$	0	0
$400$	−6.58929e8	−0.514789
$401$	1.05571e9	0.817598	0.408799	−	0.912624i	$-0.365948\pi$
$401$	1.05571e9	0.817598	0.408799	+	0.912624i	$0.365948\pi$
$402$	0	0
$403$	5.51239e8	0.419539
$404$	1.64126e9	1.23835
$405$	0	0
$406$	2.66179e8	0.197394
$407$	1.84609e8	0.135729
$408$	0	0
$409$	−1.26626e9	−0.915150	−0.457575	−	0.889171i	$-0.651282\pi$
$409$	−1.26626e9	−0.915150	−0.457575	+	0.889171i	$0.651282\pi$
$410$	−3.77173e7	−0.0270270
$411$	0	0
$412$	−1.39830e9	−0.985054
$413$	−1.20159e9	−0.839327
$414$	0	0
$415$	−1.24928e8	−0.0858007
$416$	6.27806e8	0.427562
$417$	0	0
$418$	5.22916e8	0.350199
$419$	−1.95145e9	−1.29601	−0.648006	−	0.761636i	$-0.724397\pi$
$419$	−1.95145e9	−1.29601	−0.648006	+	0.761636i	$0.724397\pi$
$420$	0	0
$421$	−9.17106e8	−0.599007	−0.299504	−	0.954095i	$-0.596821\pi$
$421$	−9.17106e8	−0.599007	−0.299504	+	0.954095i	$0.596821\pi$
$422$	9.12615e8	0.591145
$423$	0	0
$424$	1.47288e9	0.938396
$425$	−1.04265e9	−0.658839
$426$	0	0
$427$	1.11231e8	0.0691399
$428$	1.13379e9	0.699002
$429$	0	0
$430$	2.45799e6	0.00149087
$431$	−1.44942e9	−0.872015	−0.436007	−	0.899943i	$-0.643608\pi$
$431$	−1.44942e9	−0.872015	−0.436007	+	0.899943i	$0.643608\pi$
$432$	0	0
$433$	3.03820e8	0.179849	0.0899247	−	0.995949i	$-0.471337\pi$
$433$	3.03820e8	0.179849	0.0899247	+	0.995949i	$0.471337\pi$
$434$	5.30851e8	0.311716
$435$	0	0
$436$	−6.56649e8	−0.379429
$437$	−3.34124e8	−0.191524
$438$	0	0
$439$	4.49460e8	0.253551	0.126776	−	0.991931i	$-0.459537\pi$
$439$	4.49460e8	0.253551	0.126776	+	0.991931i	$0.459537\pi$
$440$	4.61457e8	0.258254
$441$	0	0
$442$	2.40582e8	0.132521
$443$	2.13044e9	1.16428	0.582139	−	0.813090i	$-0.302216\pi$
$443$	2.13044e9	1.16428	0.582139	+	0.813090i	$0.302216\pi$
$444$	0	0
$445$	1.63893e8	0.0881661
$446$	1.39892e8	0.0746656
$447$	0	0
$448$	−4.25314e8	−0.223479
$449$	−1.68653e9	−0.879291	−0.439645	−	0.898171i	$-0.644896\pi$
$449$	−1.68653e9	−0.879291	−0.439645	+	0.898171i	$0.644896\pi$
$450$	0	0
$451$	3.87286e8	0.198799
$452$	2.55913e9	1.30349
$453$	0	0
$454$	−7.30621e7	−0.0366435
$455$	3.08858e8	0.153716
$456$	0	0
$457$	−1.80858e8	−0.0886401	−0.0443201	−	0.999017i	$-0.514112\pi$
$457$	−1.80858e8	−0.0886401	−0.0443201	+	0.999017i	$0.514112\pi$
$458$	−6.64787e8	−0.323335
$459$	0	0
$460$	−1.36146e8	−0.0652156
$461$	−1.25708e9	−0.597600	−0.298800	−	0.954316i	$-0.596586\pi$
$461$	−1.25708e9	−0.597600	−0.298800	+	0.954316i	$0.596586\pi$
$462$	0	0
$463$	−1.05626e8	−0.0494579	−0.0247290	−	0.999694i	$-0.507872\pi$
$463$	−1.05626e8	−0.0494579	−0.0247290	+	0.999694i	$0.507872\pi$
$464$	7.33795e8	0.341006
$465$	0	0
$466$	−3.96678e8	−0.181588
$467$	2.42471e9	1.10167	0.550833	−	0.834616i	$-0.314310\pi$
$467$	2.42471e9	1.10167	0.550833	+	0.834616i	$0.314310\pi$
$468$	0	0
$469$	3.47174e8	0.155397
$470$	−4.96313e8	−0.220502
$471$	0	0
$472$	1.47363e9	0.645045
$473$	−2.52390e7	−0.0109662
$474$	0	0
$475$	1.86024e9	0.796418
$476$	−1.39798e9	−0.594122
$477$	0	0
$478$	−1.35888e9	−0.569095
$479$	−1.74881e8	−0.0727056	−0.0363528	−	0.999339i	$-0.511574\pi$
$479$	−1.74881e8	−0.0727056	−0.0363528	+	0.999339i	$0.511574\pi$
$480$	0	0
$481$	−1.51535e8	−0.0620875
$482$	1.17283e9	0.477056
$483$	0	0
$484$	−4.81127e7	−0.0192886
$485$	−4.22947e8	−0.168341
$486$	0	0
$487$	2.75662e9	1.08150	0.540749	−	0.841184i	$-0.318141\pi$
$487$	2.75662e9	1.08150	0.540749	+	0.841184i	$0.318141\pi$
$488$	−1.36413e8	−0.0531358
$489$	0	0
$490$	−6.05772e7	−0.0232607
$491$	−2.54281e9	−0.969458	−0.484729	−	0.874664i	$-0.661082\pi$
$491$	−2.54281e9	−0.969458	−0.484729	+	0.874664i	$0.661082\pi$
$492$	0	0
$493$	1.16112e9	0.436428
$494$	−4.29230e8	−0.160194
$495$	0	0
$496$	1.46344e9	0.538502
$497$	−1.05194e9	−0.384365
$498$	0	0
$499$	−2.20288e9	−0.793669	−0.396835	−	0.917890i	$-0.629891\pi$
$499$	−2.20288e9	−0.793669	−0.396835	+	0.917890i	$0.629891\pi$
$500$	1.63219e9	0.583950
$501$	0	0
$502$	−4.39980e8	−0.155228
$503$	3.31501e9	1.16144	0.580721	−	0.814103i	$-0.302771\pi$
$503$	3.31501e9	1.16144	0.580721	+	0.814103i	$0.302771\pi$
$504$	0	0
$505$	−1.52325e9	−0.526323
$506$	−2.31681e8	−0.0794994
$507$	0	0
$508$	−3.48713e9	−1.18017
$509$	4.50690e9	1.51484	0.757418	−	0.652930i	$-0.226461\pi$
$509$	4.50690e9	1.51484	0.757418	+	0.652930i	$0.226461\pi$
$510$	0	0
$511$	−5.03572e9	−1.66951
$512$	2.92977e9	0.964692
$513$	0	0
$514$	1.69920e9	0.551918
$515$	1.29776e9	0.418668
$516$	0	0
$517$	5.09620e9	1.62192
$518$	−1.45930e8	−0.0461307
$519$	0	0
$520$	−3.78782e8	−0.118135
$521$	7.29463e8	0.225981	0.112990	−	0.993596i	$-0.463957\pi$
$521$	7.29463e8	0.225981	0.112990	+	0.993596i	$0.463957\pi$
$522$	0	0
$523$	−1.91935e9	−0.586676	−0.293338	−	0.956009i	$-0.594766\pi$
$523$	−1.91935e9	−0.586676	−0.293338	+	0.956009i	$0.594766\pi$
$524$	1.51522e9	0.460061
$525$	0	0
$526$	5.58374e8	0.167292
$527$	2.31566e9	0.689189
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	−6.31186e8	−0.184159
$531$	0	0
$532$	2.49418e9	0.718186
$533$	−3.17900e8	−0.0909380
$534$	0	0
$535$	−1.05227e9	−0.297090
$536$	−4.25773e8	−0.119427
$537$	0	0
$538$	−8.41805e8	−0.233063
$539$	6.22014e8	0.171096
$540$	0	0
$541$	2.30258e9	0.625207	0.312604	−	0.949884i	$-0.398799\pi$
$541$	2.30258e9	0.625207	0.312604	+	0.949884i	$0.398799\pi$
$542$	1.42648e9	0.384829
$543$	0	0
$544$	2.63731e9	0.702367
$545$	6.09436e8	0.161265
$546$	0	0
$547$	1.64323e9	0.429281	0.214641	−	0.976693i	$-0.431142\pi$
$547$	1.64323e9	0.429281	0.214641	+	0.976693i	$0.431142\pi$
$548$	−1.41635e9	−0.367654
$549$	0	0
$550$	1.28988e9	0.330584
$551$	−2.07159e9	−0.527563
$552$	0	0
$553$	−2.46082e9	−0.618787
$554$	1.85043e9	0.462370
$555$	0	0
$556$	1.27595e9	0.314828
$557$	−2.83543e9	−0.695224	−0.347612	−	0.937638i	$-0.613007\pi$
$557$	−2.83543e9	−0.695224	−0.347612	+	0.937638i	$0.613007\pi$
$558$	0	0
$559$	2.07172e7	0.00501636
$560$	8.19960e8	0.197303
$561$	0	0
$562$	2.72347e9	0.647209
$563$	2.73965e9	0.647017	0.323509	−	0.946225i	$-0.395138\pi$
$563$	2.73965e9	0.647017	0.323509	+	0.946225i	$0.395138\pi$
$564$	0	0
$565$	−2.37512e9	−0.554009
$566$	7.28200e8	0.168808
$567$	0	0
$568$	1.29010e9	0.295395
$569$	−2.41033e9	−0.548509	−0.274255	−	0.961657i	$-0.588431\pi$
$569$	−2.41033e9	−0.548509	−0.274255	+	0.961657i	$0.588431\pi$
$570$	0	0
$571$	−5.29039e9	−1.18922	−0.594609	−	0.804015i	$-0.702693\pi$
$571$	−5.29039e9	−1.18922	−0.594609	+	0.804015i	$0.702693\pi$
$572$	1.79588e9	0.401228
$573$	0	0
$574$	−3.06142e8	−0.0675666
$575$	−8.24190e8	−0.180796
$576$	0	0
$577$	−7.66014e9	−1.66005	−0.830026	−	0.557725i	$-0.811674\pi$
$577$	−7.66014e9	−1.66005	−0.830026	+	0.557725i	$0.811674\pi$
$578$	−7.39795e8	−0.159354
$579$	0	0
$580$	−8.44112e8	−0.179640
$581$	−1.01401e9	−0.214499
$582$	0	0
$583$	6.48110e9	1.35459
$584$	6.17578e9	1.28306
$585$	0	0
$586$	2.89074e8	0.0593427
$587$	6.94466e9	1.41716	0.708578	−	0.705632i	$-0.249337\pi$
$587$	6.94466e9	1.41716	0.708578	+	0.705632i	$0.249337\pi$
$588$	0	0
$589$	−4.13146e9	−0.833105
$590$	−6.31507e8	−0.126589
$591$	0	0
$592$	−4.02295e8	−0.0796928
$593$	−1.25889e9	−0.247912	−0.123956	−	0.992288i	$-0.539558\pi$
$593$	−1.25889e9	−0.247912	−0.123956	+	0.992288i	$0.539558\pi$
$594$	0	0
$595$	1.29746e9	0.252514
$596$	5.11576e9	0.989801
$597$	0	0
$598$	1.90173e8	0.0363660
$599$	5.20134e9	0.988830	0.494415	−	0.869226i	$-0.335382\pi$
$599$	5.20134e9	0.988830	0.494415	+	0.869226i	$0.335382\pi$
$600$	0	0
$601$	2.97472e9	0.558966	0.279483	−	0.960151i	$-0.409837\pi$
$601$	2.97472e9	0.558966	0.279483	+	0.960151i	$0.409837\pi$
$602$	1.99509e7	0.00372714
$603$	0	0
$604$	−4.55524e9	−0.841167
$605$	4.46534e7	0.00819805
$606$	0	0
$607$	−2.57539e9	−0.467393	−0.233697	−	0.972310i	$-0.575082\pi$
$607$	−2.57539e9	−0.467393	−0.233697	+	0.972310i	$0.575082\pi$
$608$	−4.70531e9	−0.849036
$609$	0	0
$610$	5.84585e7	0.0104278
$611$	−4.18317e9	−0.741927
$612$	0	0
$613$	−9.83357e9	−1.72425	−0.862124	−	0.506698i	$-0.830866\pi$
$613$	−9.83357e9	−1.72425	−0.862124	+	0.506698i	$0.830866\pi$
$614$	2.15701e9	0.376066
$615$	0	0
$616$	3.74554e9	0.645627
$617$	4.64605e9	0.796317	0.398158	−	0.917317i	$-0.369649\pi$
$617$	4.64605e9	0.796317	0.398158	+	0.917317i	$0.369649\pi$
$618$	0	0
$619$	5.82142e9	0.986533	0.493266	−	0.869878i	$-0.335803\pi$
$619$	5.82142e9	0.986533	0.493266	+	0.869878i	$0.335803\pi$
$620$	−1.68344e9	−0.283679
$621$	0	0
$622$	−1.58896e9	−0.264757
$623$	1.33028e9	0.220413
$624$	0	0
$625$	3.77734e9	0.618879
$626$	1.21849e9	0.198524
$627$	0	0
$628$	−2.30292e9	−0.371039
$629$	−6.36571e8	−0.101993
$630$	0	0
$631$	−2.42986e9	−0.385015	−0.192508	−	0.981295i	$-0.561662\pi$
$631$	−2.42986e9	−0.385015	−0.192508	+	0.981295i	$0.561662\pi$
$632$	3.01793e9	0.475554
$633$	0	0
$634$	−2.36524e9	−0.368606
$635$	3.23641e9	0.501598
$636$	0	0
$637$	−5.10574e8	−0.0782656
$638$	−1.43644e9	−0.218985
$639$	0	0
$640$	−2.45855e9	−0.370722
$641$	−4.78517e9	−0.717620	−0.358810	−	0.933411i	$-0.616817\pi$
$641$	−4.78517e9	−0.717620	−0.358810	+	0.933411i	$0.616817\pi$
$642$	0	0
$643$	9.81453e9	1.45590	0.727949	−	0.685631i	$-0.240474\pi$
$643$	9.81453e9	1.45590	0.727949	+	0.685631i	$0.240474\pi$
$644$	−1.10506e9	−0.163037
$645$	0	0
$646$	−1.80312e9	−0.263155
$647$	−4.49992e9	−0.653190	−0.326595	−	0.945164i	$-0.605901\pi$
$647$	−4.49992e9	−0.653190	−0.326595	+	0.945164i	$0.605901\pi$
$648$	0	0
$649$	6.48439e9	0.931135
$650$	−1.05879e9	−0.151221
$651$	0	0
$652$	−2.75101e9	−0.388710
$653$	−6.11060e9	−0.858792	−0.429396	−	0.903116i	$-0.641274\pi$
$653$	−6.11060e9	−0.858792	−0.429396	+	0.903116i	$0.641274\pi$
$654$	0	0
$655$	−1.40627e9	−0.195535
$656$	−8.43964e8	−0.116724
$657$	0	0
$658$	−4.02845e9	−0.551249
$659$	5.95108e9	0.810021	0.405011	−	0.914312i	$-0.367268\pi$
$659$	5.95108e9	0.810021	0.405011	+	0.914312i	$0.367268\pi$
$660$	0	0
$661$	1.54958e9	0.208694	0.104347	−	0.994541i	$-0.466725\pi$
$661$	1.54958e9	0.208694	0.104347	+	0.994541i	$0.466725\pi$
$662$	−2.65304e9	−0.355419
$663$	0	0
$664$	1.24358e9	0.164848
$665$	−2.31485e9	−0.305244
$666$	0	0
$667$	9.17833e8	0.119763
$668$	6.10296e9	0.792178
$669$	0	0
$670$	1.82461e8	0.0234373
$671$	−6.00259e8	−0.0767026
$672$	0	0
$673$	7.44125e9	0.941008	0.470504	−	0.882398i	$-0.344072\pi$
$673$	7.44125e9	0.941008	0.470504	+	0.882398i	$0.344072\pi$
$674$	−2.36690e9	−0.297762
$675$	0	0
$676$	5.41582e9	0.674297
$677$	1.02760e10	1.27281	0.636404	−	0.771356i	$-0.280421\pi$
$677$	1.02760e10	1.27281	0.636404	+	0.771356i	$0.280421\pi$
$678$	0	0
$679$	−3.43296e9	−0.420847
$680$	−1.59120e9	−0.194063
$681$	0	0
$682$	−2.86474e9	−0.345812
$683$	−8.42180e9	−1.01142	−0.505711	−	0.862703i	$-0.668770\pi$
$683$	−8.42180e9	−1.01142	−0.505711	+	0.862703i	$0.668770\pi$
$684$	0	0
$685$	1.31452e9	0.156260
$686$	−3.39759e9	−0.401825
$687$	0	0
$688$	5.50001e7	0.00643879
$689$	−5.31995e9	−0.619641
$690$	0	0
$691$	−6.02799e9	−0.695023	−0.347511	−	0.937676i	$-0.612973\pi$
$691$	−6.02799e9	−0.695023	−0.347511	+	0.937676i	$0.612973\pi$
$692$	−1.34287e10	−1.54050
$693$	0	0
$694$	2.36026e9	0.268041
$695$	−1.18421e9	−0.133808
$696$	0	0
$697$	−1.33544e9	−0.149386
$698$	3.92411e9	0.436764
$699$	0	0
$700$	6.15243e9	0.677960
$701$	−1.59740e10	−1.75146	−0.875732	−	0.482798i	$-0.839621\pi$
$701$	−1.59740e10	−1.75146	−0.875732	+	0.482798i	$0.839621\pi$
$702$	0	0
$703$	1.13573e9	0.123291
$704$	2.29521e9	0.247924
$705$	0	0
$706$	−4.73365e9	−0.506267
$707$	−1.23639e10	−1.31579
$708$	0	0
$709$	−1.64889e10	−1.73752	−0.868759	−	0.495236i	$-0.835082\pi$
$709$	−1.64889e10	−1.73752	−0.868759	+	0.495236i	$0.835082\pi$
$710$	−5.52857e8	−0.0579707
$711$	0	0
$712$	−1.63145e9	−0.169393
$713$	1.83047e9	0.189125
$714$	0	0
$715$	−1.66676e9	−0.170530
$716$	−5.62785e9	−0.572990
$717$	0	0
$718$	−5.84935e9	−0.589756
$719$	−1.48508e10	−1.49004	−0.745020	−	0.667043i	$-0.767560\pi$
$719$	−1.48508e10	−1.49004	−0.745020	+	0.667043i	$0.767560\pi$
$720$	0	0
$721$	1.05336e10	1.04666
$722$	−5.96093e8	−0.0589432
$723$	0	0
$724$	2.84130e9	0.278248
$725$	−5.11003e9	−0.498013
$726$	0	0
$727$	6.61355e8	0.0638358	0.0319179	−	0.999490i	$-0.489838\pi$
$727$	6.61355e8	0.0638358	0.0319179	+	0.999490i	$0.489838\pi$
$728$	−3.07449e9	−0.295334
$729$	0	0
$730$	−2.64657e9	−0.251798
$731$	8.70293e7	0.00824052
$732$	0	0
$733$	1.84444e10	1.72982	0.864909	−	0.501929i	$-0.167376\pi$
$733$	1.84444e10	1.72982	0.864909	+	0.501929i	$0.167376\pi$
$734$	−4.29214e9	−0.400624
$735$	0	0
$736$	2.08472e9	0.192741
$737$	−1.87353e9	−0.172395
$738$	0	0
$739$	1.27825e10	1.16509	0.582546	−	0.812798i	$-0.302057\pi$
$739$	1.27825e10	1.16509	0.582546	+	0.812798i	$0.302057\pi$
$740$	4.62776e8	0.0419816
$741$	0	0
$742$	−5.12319e9	−0.460391
$743$	−6.16352e8	−0.0551275	−0.0275637	−	0.999620i	$-0.508775\pi$
$743$	−6.16352e8	−0.0551275	−0.0275637	+	0.999620i	$0.508775\pi$
$744$	0	0
$745$	−4.74793e9	−0.420686
$746$	−5.58809e9	−0.492807
$747$	0	0
$748$	7.54419e9	0.659108
$749$	−8.54101e9	−0.742716
$750$	0	0
$751$	−1.02940e10	−0.886839	−0.443419	−	0.896314i	$-0.646235\pi$
$751$	−1.02940e10	−0.886839	−0.443419	+	0.896314i	$0.646235\pi$
$752$	−1.11055e10	−0.952305
$753$	0	0
$754$	1.17909e9	0.100172
$755$	4.22772e9	0.357513
$756$	0	0
$757$	1.05762e10	0.886127	0.443064	−	0.896490i	$-0.353892\pi$
$757$	1.05762e10	0.886127	0.443064	+	0.896490i	$0.353892\pi$
$758$	8.44086e9	0.703954
$759$	0	0
$760$	2.83892e9	0.234588
$761$	3.63789e9	0.299229	0.149615	−	0.988744i	$-0.452197\pi$
$761$	3.63789e9	0.299229	0.149615	+	0.988744i	$0.452197\pi$
$762$	0	0
$763$	4.94665e9	0.403158
$764$	1.21180e10	0.983116
$765$	0	0
$766$	−6.41709e9	−0.515867
$767$	−5.32265e9	−0.425936
$768$	0	0
$769$	9.81178e9	0.778047	0.389023	−	0.921228i	$-0.372813\pi$
$769$	9.81178e9	0.778047	0.389023	+	0.921228i	$0.372813\pi$
$770$	−1.60511e9	−0.126703
$771$	0	0
$772$	4.26145e9	0.333347
$773$	9.03802e9	0.703793	0.351896	−	0.936039i	$-0.385537\pi$
$773$	9.03802e9	0.703793	0.351896	+	0.936039i	$0.385537\pi$
$774$	0	0
$775$	−1.01911e10	−0.786442
$776$	4.21017e9	0.323432
$777$	0	0
$778$	−2.86518e9	−0.218134
$779$	2.38261e9	0.180581
$780$	0	0
$781$	5.67681e9	0.426408
$782$	7.98885e8	0.0597394
$783$	0	0
$784$	−1.35548e9	−0.100458
$785$	2.13734e9	0.157699
$786$	0	0
$787$	1.18573e8	0.00867113	0.00433556	−	0.999991i	$-0.498620\pi$
$787$	1.18573e8	0.00867113	0.00433556	+	0.999991i	$0.498620\pi$
$788$	−8.81007e9	−0.641413
$789$	0	0
$790$	−1.29330e9	−0.0933267
$791$	−1.92783e10	−1.38501
$792$	0	0
$793$	4.92717e8	0.0350866
$794$	5.15644e9	0.365576
$795$	0	0
$796$	−4.80390e9	−0.337596
$797$	1.15934e10	0.811159	0.405579	−	0.914060i	$-0.367070\pi$
$797$	1.15934e10	0.811159	0.405579	+	0.914060i	$0.367070\pi$
$798$	0	0
$799$	−1.75728e10	−1.21878
$800$	−1.16067e10	−0.801480
$801$	0	0
$802$	4.50349e9	0.308276
$803$	2.71753e10	1.85212
$804$	0	0
$805$	1.02561e9	0.0692940
$806$	2.35150e9	0.158187
$807$	0	0
$808$	1.51630e10	1.01122
$809$	8.22895e9	0.546418	0.273209	−	0.961955i	$-0.411915\pi$
$809$	8.22895e9	0.546418	0.273209	+	0.961955i	$0.411915\pi$
$810$	0	0
$811$	2.84646e9	0.187384	0.0936918	−	0.995601i	$-0.470133\pi$
$811$	2.84646e9	0.187384	0.0936918	+	0.995601i	$0.470133\pi$
$812$	−6.85146e9	−0.449094
$813$	0	0
$814$	7.87512e8	0.0511766
$815$	2.55321e9	0.165210
$816$	0	0
$817$	−1.55272e8	−0.00996130
$818$	−5.40167e9	−0.345057
$819$	0	0
$820$	9.70844e8	0.0614895
$821$	1.95817e10	1.23495	0.617476	−	0.786590i	$-0.288155\pi$
$821$	1.95817e10	1.23495	0.617476	+	0.786590i	$0.288155\pi$
$822$	0	0
$823$	5.44395e9	0.340420	0.170210	−	0.985408i	$-0.445555\pi$
$823$	5.44395e9	0.340420	0.170210	+	0.985408i	$0.445555\pi$
$824$	−1.29184e10	−0.804384
$825$	0	0
$826$	−5.12579e9	−0.316469
$827$	7.65476e9	0.470611	0.235306	−	0.971921i	$-0.424391\pi$
$827$	7.65476e9	0.470611	0.235306	+	0.971921i	$0.424391\pi$
$828$	0	0
$829$	−1.31924e10	−0.804235	−0.402117	−	0.915588i	$-0.631726\pi$
$829$	−1.31924e10	−0.804235	−0.402117	+	0.915588i	$0.631726\pi$
$830$	−5.32921e8	−0.0323512
$831$	0	0
$832$	−1.88400e9	−0.113409
$833$	−2.14484e9	−0.128569
$834$	0	0
$835$	−5.66415e9	−0.336692
$836$	−1.34599e10	−0.796743
$837$	0	0
$838$	−8.32457e9	−0.488661
$839$	3.14397e10	1.83786	0.918929	−	0.394423i	$-0.129056\pi$
$839$	3.14397e10	1.83786	0.918929	+	0.394423i	$0.129056\pi$
$840$	0	0
$841$	−1.15593e10	−0.670106
$842$	−3.91222e9	−0.225856
$843$	0	0
$844$	−2.34907e10	−1.34492
$845$	−5.02643e9	−0.286590
$846$	0	0
$847$	3.62441e8	0.0204949
$848$	−1.41234e10	−0.795344
$849$	0	0
$850$	−4.44779e9	−0.248415
$851$	−5.03192e8	−0.0279885
$852$	0	0
$853$	−3.63919e9	−0.200763	−0.100382	−	0.994949i	$-0.532006\pi$
$853$	−3.63919e9	−0.200763	−0.100382	+	0.994949i	$0.532006\pi$
$854$	4.74494e8	0.0260692
$855$	0	0
$856$	1.04747e10	0.570797
$857$	−1.52755e9	−0.0829016	−0.0414508	−	0.999141i	$-0.513198\pi$
$857$	−1.52755e9	−0.0829016	−0.0414508	+	0.999141i	$0.513198\pi$
$858$	0	0
$859$	−1.47653e10	−0.794813	−0.397407	−	0.917643i	$-0.630090\pi$
$859$	−1.47653e10	−0.794813	−0.397407	+	0.917643i	$0.630090\pi$
$860$	−6.32687e7	−0.00339191
$861$	0	0
$862$	−6.18299e9	−0.328793
$863$	−2.09885e9	−0.111159	−0.0555795	−	0.998454i	$-0.517701\pi$
$863$	−2.09885e9	−0.111159	−0.0555795	+	0.998454i	$0.517701\pi$
$864$	0	0
$865$	1.24632e10	0.654745
$866$	1.29605e9	0.0678122
$867$	0	0
$868$	−1.36641e10	−0.709190
$869$	1.32798e10	0.686472
$870$	0	0
$871$	1.53787e9	0.0788598
$872$	−6.06654e9	−0.309837
$873$	0	0
$874$	−1.42532e9	−0.0722141
$875$	−1.22956e10	−0.620469
$876$	0	0
$877$	1.04225e10	0.521762	0.260881	−	0.965371i	$-0.415987\pi$
$877$	1.04225e10	0.521762	0.260881	+	0.965371i	$0.415987\pi$
$878$	1.91732e9	0.0956015
$879$	0	0
$880$	−4.42492e9	−0.218885
$881$	−6.15622e9	−0.303318	−0.151659	−	0.988433i	$-0.548462\pi$
$881$	−6.15622e9	−0.303318	−0.151659	+	0.988433i	$0.548462\pi$
$882$	0	0
$883$	4.92709e9	0.240840	0.120420	−	0.992723i	$-0.461576\pi$
$883$	4.92709e9	0.240840	0.120420	+	0.992723i	$0.461576\pi$
$884$	−6.19257e9	−0.301501
$885$	0	0
$886$	9.08811e9	0.438991
$887$	−4.08057e9	−0.196331	−0.0981655	−	0.995170i	$-0.531297\pi$
$887$	−4.08057e9	−0.196331	−0.0981655	+	0.995170i	$0.531297\pi$
$888$	0	0
$889$	2.62692e10	1.25398
$890$	6.99143e8	0.0332430
$891$	0	0
$892$	−3.60082e9	−0.169873
$893$	3.13522e10	1.47329
$894$	0	0
$895$	5.22321e9	0.243533
$896$	−1.99554e10	−0.926794
$897$	0	0
$898$	−7.19447e9	−0.331537
$899$	1.13490e10	0.520954
$900$	0	0
$901$	−2.23482e10	−1.01790
$902$	1.65210e9	0.0749572
$903$	0	0
$904$	2.36428e10	1.06441
$905$	−2.63701e9	−0.118261
$906$	0	0
$907$	1.18006e10	0.525143	0.262571	−	0.964913i	$-0.415430\pi$
$907$	1.18006e10	0.525143	0.262571	+	0.964913i	$0.415430\pi$
$908$	1.88062e9	0.0833682
$909$	0	0
$910$	1.31754e9	0.0579587
$911$	2.35502e10	1.03200	0.516001	−	0.856588i	$-0.327420\pi$
$911$	2.35502e10	1.03200	0.516001	+	0.856588i	$0.327420\pi$
$912$	0	0
$913$	5.47211e9	0.237962
$914$	−7.71510e8	−0.0334218
$915$	0	0
$916$	1.71116e10	0.735626
$917$	−1.14144e10	−0.488832
$918$	0	0
$919$	−1.32888e10	−0.564785	−0.282392	−	0.959299i	$-0.591128\pi$
$919$	−1.32888e10	−0.564785	−0.282392	+	0.959299i	$0.591128\pi$
$920$	−1.25780e9	−0.0532543
$921$	0	0
$922$	−5.36251e9	−0.225325
$923$	−4.65975e9	−0.195055
$924$	0	0
$925$	2.80152e9	0.116385
$926$	−4.50582e8	−0.0186481
$927$	0	0
$928$	1.29254e10	0.530916
$929$	3.44441e10	1.40948	0.704741	−	0.709464i	$-0.251063\pi$
$929$	3.44441e10	1.40948	0.704741	+	0.709464i	$0.251063\pi$
$930$	0	0
$931$	3.82668e9	0.155417
$932$	1.02105e10	0.413134
$933$	0	0
$934$	1.03434e10	0.415383
$935$	−7.00176e9	−0.280134
$936$	0	0
$937$	5.57331e9	0.221322	0.110661	−	0.993858i	$-0.464703\pi$
$937$	5.57331e9	0.221322	0.110661	+	0.993858i	$0.464703\pi$
$938$	1.48099e9	0.0585925
$939$	0	0
$940$	1.27751e10	0.501668
$941$	−3.51408e10	−1.37483	−0.687414	−	0.726266i	$-0.741254\pi$
$941$	−3.51408e10	−1.37483	−0.687414	+	0.726266i	$0.741254\pi$
$942$	0	0
$943$	−1.05563e9	−0.0409941
$944$	−1.41306e10	−0.546713
$945$	0	0
$946$	−1.07665e8	−0.00413482
$947$	4.39421e10	1.68134	0.840671	−	0.541546i	$-0.182161\pi$
$947$	4.39421e10	1.68134	0.840671	+	0.541546i	$0.182161\pi$
$948$	0	0
$949$	−2.23066e10	−0.847229
$950$	7.93547e9	0.300289
$951$	0	0
$952$	−1.29154e10	−0.485153
$953$	3.10018e10	1.16028	0.580138	−	0.814518i	$-0.302998\pi$
$953$	3.10018e10	1.16028	0.580138	+	0.814518i	$0.302998\pi$
$954$	0	0
$955$	−1.12467e10	−0.417844
$956$	3.49777e10	1.29476
$957$	0	0
$958$	−7.46013e8	−0.0274137
$959$	1.06696e10	0.390646
$960$	0	0
$961$	−4.87884e9	−0.177331
$962$	−6.46422e8	−0.0234101
$963$	0	0
$964$	−3.01886e10	−1.08536
$965$	−3.95505e9	−0.141679
$966$	0	0
$967$	4.69745e9	0.167059	0.0835294	−	0.996505i	$-0.473381\pi$
$967$	4.69745e9	0.167059	0.0835294	+	0.996505i	$0.473381\pi$
$968$	−4.44496e8	−0.0157508
$969$	0	0
$970$	−1.80422e9	−0.0634730
$971$	3.00012e10	1.05165	0.525826	−	0.850592i	$-0.323756\pi$
$971$	3.00012e10	1.05165	0.525826	+	0.850592i	$0.323756\pi$
$972$	0	0
$973$	−9.61196e9	−0.334516
$974$	1.17593e10	0.407779
$975$	0	0
$976$	1.30807e9	0.0450357
$977$	8.33551e9	0.285957	0.142979	−	0.989726i	$-0.454332\pi$
$977$	8.33551e9	0.285957	0.142979	+	0.989726i	$0.454332\pi$
$978$	0	0
$979$	−7.17889e9	−0.244522
$980$	1.55926e9	0.0529208
$981$	0	0
$982$	−1.08472e10	−0.365534
$983$	4.04004e10	1.35659	0.678295	−	0.734790i	$-0.262719\pi$
$983$	4.04004e10	1.35659	0.678295	+	0.734790i	$0.262719\pi$
$984$	0	0
$985$	8.17663e9	0.272614
$986$	4.95314e9	0.164555
$987$	0	0
$988$	1.10484e10	0.364460
$989$	6.87943e7	0.00226134
$990$	0	0
$991$	−2.78434e10	−0.908793	−0.454396	−	0.890800i	$-0.650145\pi$
$991$	−2.78434e10	−0.908793	−0.454396	+	0.890800i	$0.650145\pi$
$992$	2.57776e10	0.838400
$993$	0	0
$994$	−4.48741e9	−0.144925
$995$	4.45850e9	0.143485
$996$	0	0
$997$	3.80309e10	1.21536	0.607679	−	0.794183i	$-0.292101\pi$
$997$	3.80309e10	1.21536	0.607679	+	0.794183i	$0.292101\pi$
$998$	−9.39714e9	−0.299253
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.8.a.g.1.8	✓	12	1.1	even	1	trivial
207.8.a.h.1.5	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)

\(f(q)\)	\(=\)	\(q+4.26584 q^{2} -109.803 q^{4} +101.908 q^{5} +827.161 q^{7} -1014.43 q^{8} +O(q^{10})\)	\(q+4.26584 q^{2} -109.803 q^{4} +101.908 q^{5} +827.161 q^{7} -1014.43 q^{8} +434.722 q^{10} -4463.78 q^{11} +3664.05 q^{13} +3528.54 q^{14} +9727.36 q^{16} +15392.1 q^{17} -27461.5 q^{19} -11189.7 q^{20} -19041.8 q^{22} +12167.0 q^{23} -67739.8 q^{25} +15630.2 q^{26} -90824.5 q^{28} +75436.2 q^{29} +150445. q^{31} +171342. q^{32} +65660.0 q^{34} +84294.2 q^{35} -41357.1 q^{37} -117146. q^{38} -103378. q^{40} -86761.9 q^{41} +5654.17 q^{43} +490135. q^{44} +51902.4 q^{46} -1.14168e6 q^{47} -139347. q^{49} -288967. q^{50} -402322. q^{52} -1.45193e6 q^{53} -454894. q^{55} -839095. q^{56} +321799. q^{58} -1.45267e6 q^{59} +134473. q^{61} +641775. q^{62} -514185. q^{64} +373395. q^{65} +419718. q^{67} -1.69009e6 q^{68} +359585. q^{70} -1.27175e6 q^{71} -6.08795e6 q^{73} -176423. q^{74} +3.01535e6 q^{76} -3.69227e6 q^{77} -2.97501e6 q^{79} +991294. q^{80} -370112. q^{82} -1.22589e6 q^{83} +1.56857e6 q^{85} +24119.8 q^{86} +4.52818e6 q^{88} +1.60825e6 q^{89} +3.03076e6 q^{91} -1.33597e6 q^{92} -4.87021e6 q^{94} -2.79854e6 q^{95} -4.15029e6 q^{97} -594431. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{2} + 640 q^{4} - 500 q^{5} - 228 q^{7} - 3072 q^{8}+O(q^{10}) \) 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

L-functions

Modular forms

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