LMFDB - Embedded newform 387.8.a.b.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,8,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, 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("387.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	387.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:	$120.893004862$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + \cdots + 238240894976 $ x^11 - 2x^10 - 977x^9 + 2592x^8 + 344686x^7 - 1160956x^6 - 53409536x^5 + 209758592x^4 + 3410917248x^3 - 14180732672x^2 - 60918607872x + 238240894976
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$19.7860$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+21.7860 q^{2} +346.631 q^{4} +409.924 q^{5} +476.571 q^{7} +4763.10 q^{8} +O(q^{10})$	\(q+21.7860 q^{2} +346.631 q^{4} +409.924 q^{5} +476.571 q^{7} +4763.10 q^{8} +8930.61 q^{10} -6867.50 q^{11} -8276.59 q^{13} +10382.6 q^{14} +59400.2 q^{16} +16111.6 q^{17} +27534.5 q^{19} +142092. q^{20} -149616. q^{22} +69123.4 q^{23} +89912.4 q^{25} -180314. q^{26} +165194. q^{28} +111490. q^{29} -169599. q^{31} +684418. q^{32} +351008. q^{34} +195358. q^{35} -60102.6 q^{37} +599867. q^{38} +1.95251e6 q^{40} +813919. q^{41} +79507.0 q^{43} -2.38049e6 q^{44} +1.50592e6 q^{46} +624677. q^{47} -596423. q^{49} +1.95883e6 q^{50} -2.86892e6 q^{52} -365470. q^{53} -2.81515e6 q^{55} +2.26995e6 q^{56} +2.42892e6 q^{58} -1.43500e6 q^{59} -153457. q^{61} -3.69489e6 q^{62} +7.30752e6 q^{64} -3.39277e6 q^{65} -1.27944e6 q^{67} +5.58478e6 q^{68} +4.25607e6 q^{70} -821210. q^{71} +4.08244e6 q^{73} -1.30940e6 q^{74} +9.54431e6 q^{76} -3.27285e6 q^{77} -6.74067e6 q^{79} +2.43496e7 q^{80} +1.77321e7 q^{82} -281235. q^{83} +6.60453e6 q^{85} +1.73214e6 q^{86} -3.27106e7 q^{88} -9.40951e6 q^{89} -3.94438e6 q^{91} +2.39603e7 q^{92} +1.36092e7 q^{94} +1.12870e7 q^{95} -6.78204e6 q^{97} -1.29937e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) $ 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8 - 1333 * q^10 - 1333 * q^11 - 17967 * q^13 + 22352 * q^14 - 34406 * q^16 + 63095 * q^17 - 54524 * q^19 + 280995 * q^20 - 289358 * q^22 + 138139 * q^23 + 3455 * q^25 + 132946 * q^26 - 12704 * q^28 + 308658 * q^29 - 209523 * q^31 + 644934 * q^32 + 762435 * q^34 + 578892 * q^35 - 298472 * q^37 + 369707 * q^38 + 2633173 * q^40 + 1346735 * q^41 + 874577 * q^43 - 3134292 * q^44 + 3588111 * q^46 - 499284 * q^47 + 2544563 * q^49 - 3049745 * q^50 + 983088 * q^52 + 2210495 * q^53 - 1855072 * q^55 + 469976 * q^56 + 4397067 * q^58 + 5824216 * q^59 - 4453034 * q^61 - 1002789 * q^62 + 4757538 * q^64 + 2162872 * q^65 - 6859513 * q^67 + 9397005 * q^68 + 845078 * q^70 + 10726554 * q^71 - 4456898 * q^73 - 1046637 * q^74 + 5861267 * q^76 + 17019816 * q^77 - 15541320 * q^79 + 15680911 * q^80 + 20233655 * q^82 + 11146767 * q^83 - 12471976 * q^85 + 1908168 * q^86 - 24463544 * q^88 + 13531356 * q^89 - 19746448 * q^91 + 26023161 * q^92 + 20288857 * q^94 + 12291624 * q^95 - 10999901 * q^97 - 29909168 * 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 $

$2$	21.7860	1.92563	0.962815	−	0.270160i	$-0.0870767\pi$
$2$	21.7860	1.92563	0.962815	+	0.270160i	$0.0870767\pi$
$3$	0	0
$3$	0	0
$4$	346.631	2.70805
$5$	409.924	1.46659	0.733294	−	0.679912i	$-0.237982\pi$
$5$	409.924	1.46659	0.733294	+	0.679912i	$0.237982\pi$
$6$	0	0
$7$	476.571	0.525151	0.262576	−	0.964911i	$-0.415428\pi$
$7$	476.571	0.525151	0.262576	+	0.964911i	$0.415428\pi$
$8$	4763.10	3.28908
$9$	0	0
$10$	8930.61	2.82411
$11$	−6867.50	−1.55569	−0.777847	−	0.628453i	$-0.783688\pi$
$11$	−6867.50	−1.55569	−0.777847	+	0.628453i	$0.783688\pi$
$12$	0	0
$13$	−8276.59	−1.04484	−0.522420	−	0.852688i	$-0.674971\pi$
$13$	−8276.59	−1.04484	−0.522420	+	0.852688i	$0.674971\pi$
$14$	10382.6	1.01125
$15$	0	0
$16$	59400.2	3.62550
$17$	16111.6	0.795367	0.397684	−	0.917523i	$-0.369814\pi$
$17$	16111.6	0.795367	0.397684	+	0.917523i	$0.369814\pi$
$18$	0	0
$19$	27534.5	0.920957	0.460479	−	0.887671i	$-0.347678\pi$
$19$	27534.5	0.920957	0.460479	+	0.887671i	$0.347678\pi$
$20$	142092.	3.97160
$21$	0	0
$22$	−149616.	−2.99569
$23$	69123.4	1.18462	0.592308	−	0.805711i	$-0.298217\pi$
$23$	69123.4	1.18462	0.592308	+	0.805711i	$0.298217\pi$
$24$	0	0
$25$	89912.4	1.15088
$26$	−180314.	−2.01198
$27$	0	0
$28$	165194.	1.42214
$29$	111490.	0.848871	0.424435	−	0.905458i	$-0.360473\pi$
$29$	111490.	0.848871	0.424435	+	0.905458i	$0.360473\pi$
$30$	0	0
$31$	−169599.	−1.02249	−0.511244	−	0.859436i	$-0.670815\pi$
$31$	−169599.	−1.02249	−0.511244	+	0.859436i	$0.670815\pi$
$32$	684418.	3.69230
$33$	0	0
$34$	351008.	1.53158
$35$	195358.	0.770180
$36$	0	0
$37$	−60102.6	−0.195068	−0.0975342	−	0.995232i	$-0.531096\pi$
$37$	−60102.6	−0.195068	−0.0975342	+	0.995232i	$0.531096\pi$
$38$	599867.	1.77342
$39$	0	0
$40$	1.95251e6	4.82372
$41$	813919.	1.84433	0.922163	−	0.386802i	$-0.126420\pi$
$41$	813919.	1.84433	0.922163	+	0.386802i	$0.126420\pi$
$42$	0	0
$43$	79507.0	0.152499
$43$	79507.0	0.152499
$44$	−2.38049e6	−4.21290
$45$	0	0
$46$	1.50592e6	2.28113
$47$	624677.	0.877633	0.438817	−	0.898577i	$-0.355398\pi$
$47$	624677.	0.877633	0.438817	+	0.898577i	$0.355398\pi$
$48$	0	0
$49$	−596423.	−0.724216
$50$	1.95883e6	2.21617
$51$	0	0
$52$	−2.86892e6	−2.82948
$53$	−365470.	−0.337199	−0.168600	−	0.985685i	$-0.553924\pi$
$53$	−365470.	−0.337199	−0.168600	+	0.985685i	$0.553924\pi$
$54$	0	0
$55$	−2.81515e6	−2.28156
$56$	2.26995e6	1.72726
$57$	0	0
$58$	2.42892e6	1.63461
$59$	−1.43500e6	−0.909643	−0.454822	−	0.890583i	$-0.650297\pi$
$59$	−1.43500e6	−0.909643	−0.454822	+	0.890583i	$0.650297\pi$
$60$	0	0
$61$	−153457.	−0.0865630	−0.0432815	−	0.999063i	$-0.513781\pi$
$61$	−153457.	−0.0865630	−0.0432815	+	0.999063i	$0.513781\pi$
$62$	−3.69489e6	−1.96893
$63$	0	0
$64$	7.30752e6	3.48450
$65$	−3.39277e6	−1.53235
$66$	0	0
$67$	−1.27944e6	−0.519708	−0.259854	−	0.965648i	$-0.583674\pi$
$67$	−1.27944e6	−0.519708	−0.259854	+	0.965648i	$0.583674\pi$
$68$	5.58478e6	2.15390
$69$	0	0
$70$	4.25607e6	1.48308
$71$	−821210.	−0.272302	−0.136151	−	0.990688i	$-0.543473\pi$
$71$	−821210.	−0.272302	−0.136151	+	0.990688i	$0.543473\pi$
$72$	0	0
$73$	4.08244e6	1.22826	0.614130	−	0.789205i	$-0.289507\pi$
$73$	4.08244e6	1.22826	0.614130	+	0.789205i	$0.289507\pi$
$74$	−1.30940e6	−0.375630
$75$	0	0
$76$	9.54431e6	2.49400
$77$	−3.27285e6	−0.816975
$78$	0	0
$79$	−6.74067e6	−1.53818	−0.769091	−	0.639139i	$-0.779291\pi$
$79$	−6.74067e6	−1.53818	−0.769091	+	0.639139i	$0.779291\pi$
$80$	2.43496e7	5.31712
$81$	0	0
$82$	1.77321e7	3.55149
$83$	−281235.	−0.0539878	−0.0269939	−	0.999636i	$-0.508593\pi$
$83$	−281235.	−0.0539878	−0.0269939	+	0.999636i	$0.508593\pi$
$84$	0	0
$85$	6.60453e6	1.16648
$86$	1.73214e6	0.293656
$87$	0	0
$88$	−3.27106e7	−5.11681
$89$	−9.40951e6	−1.41482	−0.707411	−	0.706802i	$-0.750137\pi$
$89$	−9.40951e6	−1.41482	−0.707411	+	0.706802i	$0.750137\pi$
$90$	0	0
$91$	−3.94438e6	−0.548699
$92$	2.39603e7	3.20801
$93$	0	0
$94$	1.36092e7	1.69000
$95$	1.12870e7	1.35066
$96$	0	0
$97$	−6.78204e6	−0.754500	−0.377250	−	0.926112i	$-0.623130\pi$
$97$	−6.78204e6	−0.754500	−0.377250	+	0.926112i	$0.623130\pi$
$98$	−1.29937e7	−1.39457
$99$	0	0
$100$	3.11664e7	3.11664
$101$	−1.37928e7	−1.33208	−0.666038	−	0.745918i	$-0.732011\pi$
$101$	−1.37928e7	−1.33208	−0.666038	+	0.745918i	$0.732011\pi$
$102$	0	0
$103$	−1.39555e7	−1.25839	−0.629197	−	0.777246i	$-0.716616\pi$
$103$	−1.39555e7	−1.25839	−0.629197	+	0.777246i	$0.716616\pi$
$104$	−3.94222e7	−3.43656
$105$	0	0
$106$	−7.96214e6	−0.649321
$107$	−4.53042e6	−0.357516	−0.178758	−	0.983893i	$-0.557208\pi$
$107$	−4.53042e6	−0.357516	−0.178758	+	0.983893i	$0.557208\pi$
$108$	0	0
$109$	1.43535e7	1.06161	0.530806	−	0.847493i	$-0.321889\pi$
$109$	1.43535e7	1.06161	0.530806	+	0.847493i	$0.321889\pi$
$110$	−6.13310e7	−4.39345
$111$	0	0
$112$	2.83084e7	1.90394
$113$	1.27510e7	0.831322	0.415661	−	0.909520i	$-0.363550\pi$
$113$	1.27510e7	0.831322	0.415661	+	0.909520i	$0.363550\pi$
$114$	0	0
$115$	2.83353e7	1.73734
$116$	3.86458e7	2.29879
$117$	0	0
$118$	−3.12630e7	−1.75164
$119$	7.67832e6	0.417688
$120$	0	0
$121$	2.76754e7	1.42019
$122$	−3.34322e6	−0.166688
$123$	0	0
$124$	−5.87884e7	−2.76895
$125$	4.83193e6	0.221277
$126$	0	0
$127$	−1.14084e7	−0.494210	−0.247105	−	0.968989i	$-0.579479\pi$
$127$	−1.14084e7	−0.494210	−0.247105	+	0.968989i	$0.579479\pi$
$128$	7.15964e7	3.01756
$129$	0	0
$130$	−7.39150e7	−2.95074
$131$	−3.48381e6	−0.135396	−0.0676978	−	0.997706i	$-0.521565\pi$
$131$	−3.48381e6	−0.135396	−0.0676978	+	0.997706i	$0.521565\pi$
$132$	0	0
$133$	1.31221e7	0.483642
$134$	−2.78740e7	−1.00077
$135$	0	0
$136$	7.67412e7	2.61603
$137$	−2.52123e7	−0.837703	−0.418852	−	0.908055i	$-0.637567\pi$
$137$	−2.52123e7	−0.837703	−0.418852	+	0.908055i	$0.637567\pi$
$138$	0	0
$139$	2.46536e7	0.778625	0.389313	−	0.921106i	$-0.372713\pi$
$139$	2.46536e7	0.778625	0.389313	+	0.921106i	$0.372713\pi$
$140$	6.77170e7	2.08569
$141$	0	0
$142$	−1.78909e7	−0.524352
$143$	5.68395e7	1.62545
$144$	0	0
$145$	4.57023e7	1.24494
$146$	8.89402e7	2.36518
$147$	0	0
$148$	−2.08334e7	−0.528256
$149$	4.27780e7	1.05942	0.529710	−	0.848179i	$-0.322301\pi$
$149$	4.27780e7	1.05942	0.529710	+	0.848179i	$0.322301\pi$
$150$	0	0
$151$	6.03840e7	1.42726	0.713629	−	0.700524i	$-0.247050\pi$
$151$	6.03840e7	1.42726	0.713629	+	0.700524i	$0.247050\pi$
$152$	1.31149e8	3.02910
$153$	0	0
$154$	−7.13024e7	−1.57319
$155$	−6.95228e7	−1.49957
$156$	0	0
$157$	−5.70927e7	−1.17742	−0.588711	−	0.808344i	$-0.700364\pi$
$157$	−5.70927e7	−1.17742	−0.588711	+	0.808344i	$0.700364\pi$
$158$	−1.46852e8	−2.96197
$159$	0	0
$160$	2.80559e8	5.41508
$161$	3.29422e7	0.622103
$162$	0	0
$163$	−2.03054e7	−0.367245	−0.183622	−	0.982997i	$-0.558782\pi$
$163$	−2.03054e7	−0.367245	−0.183622	+	0.982997i	$0.558782\pi$
$164$	2.82129e8	4.99453
$165$	0	0
$166$	−6.12699e6	−0.103961
$167$	−3.18949e7	−0.529925	−0.264962	−	0.964259i	$-0.585360\pi$
$167$	−3.18949e7	−0.529925	−0.264962	+	0.964259i	$0.585360\pi$
$168$	0	0
$169$	5.75340e6	0.0916898
$170$	1.43886e8	2.24620
$171$	0	0
$172$	2.75596e7	0.412974
$173$	2.81474e7	0.413311	0.206656	−	0.978414i	$-0.433742\pi$
$173$	2.81474e7	0.413311	0.206656	+	0.978414i	$0.433742\pi$
$174$	0	0
$175$	4.28496e7	0.604385
$176$	−4.07931e8	−5.64017
$177$	0	0
$178$	−2.04996e8	−2.72443
$179$	3.87229e6	0.0504641	0.0252321	−	0.999682i	$-0.491968\pi$
$179$	3.87229e6	0.0504641	0.0252321	+	0.999682i	$0.491968\pi$
$180$	0	0
$181$	−5.34400e7	−0.669871	−0.334935	−	0.942241i	$-0.608715\pi$
$181$	−5.34400e7	−0.669871	−0.334935	+	0.942241i	$0.608715\pi$
$182$	−8.59324e7	−1.05659
$183$	0	0
$184$	3.29242e8	3.89630
$185$	−2.46375e7	−0.286085
$186$	0	0
$187$	−1.10647e8	−1.23735
$188$	2.16532e8	2.37668
$189$	0	0
$190$	2.45900e8	2.60088
$191$	1.82945e7	0.189978	0.0949891	−	0.995478i	$-0.469718\pi$
$191$	1.82945e7	0.189978	0.0949891	+	0.995478i	$0.469718\pi$
$192$	0	0
$193$	8.24942e7	0.825986	0.412993	−	0.910734i	$-0.364483\pi$
$193$	8.24942e7	0.825986	0.412993	+	0.910734i	$0.364483\pi$
$194$	−1.47754e8	−1.45289
$195$	0	0
$196$	−2.06739e8	−1.96122
$197$	−7.08196e7	−0.659966	−0.329983	−	0.943987i	$-0.607043\pi$
$197$	−7.08196e7	−0.659966	−0.329983	+	0.943987i	$0.607043\pi$
$198$	0	0
$199$	1.61028e8	1.44849	0.724247	−	0.689540i	$-0.242187\pi$
$199$	1.61028e8	1.44849	0.724247	+	0.689540i	$0.242187\pi$
$200$	4.28262e8	3.78533
$201$	0	0
$202$	−3.00491e8	−2.56509
$203$	5.31327e7	0.445785
$204$	0	0
$205$	3.33645e8	2.70486
$206$	−3.04036e8	−2.42320
$207$	0	0
$208$	−4.91631e8	−3.78807
$209$	−1.89093e8	−1.43273
$210$	0	0
$211$	−2.05501e8	−1.50600	−0.753002	−	0.658018i	$-0.771395\pi$
$211$	−2.05501e8	−1.50600	−0.753002	+	0.658018i	$0.771395\pi$
$212$	−1.26683e8	−0.913153
$213$	0	0
$214$	−9.86999e7	−0.688444
$215$	3.25918e7	0.223652
$216$	0	0
$217$	−8.08261e7	−0.536961
$218$	3.12706e8	2.04427
$219$	0	0
$220$	−9.75818e8	−6.17859
$221$	−1.33349e8	−0.831031
$222$	0	0
$223$	−8.77217e7	−0.529712	−0.264856	−	0.964288i	$-0.585324\pi$
$223$	−8.77217e7	−0.529712	−0.264856	+	0.964288i	$0.585324\pi$
$224$	3.26174e8	1.93901
$225$	0	0
$226$	2.77793e8	1.60082
$227$	1.96811e8	1.11676	0.558378	−	0.829586i	$-0.311424\pi$
$227$	1.96811e8	1.11676	0.558378	+	0.829586i	$0.311424\pi$
$228$	0	0
$229$	−1.91603e8	−1.05434	−0.527168	−	0.849761i	$-0.676746\pi$
$229$	−1.91603e8	−1.05434	−0.527168	+	0.849761i	$0.676746\pi$
$230$	6.17314e8	3.34548
$231$	0	0
$232$	5.31036e8	2.79201
$233$	7.48220e7	0.387511	0.193755	−	0.981050i	$-0.437933\pi$
$233$	7.48220e7	0.387511	0.193755	+	0.981050i	$0.437933\pi$
$234$	0	0
$235$	2.56070e8	1.28713
$236$	−4.97417e8	−2.46336
$237$	0	0
$238$	1.67280e8	0.804313
$239$	−3.09828e8	−1.46801	−0.734004	−	0.679145i	$-0.762351\pi$
$239$	−3.09828e8	−1.46801	−0.734004	+	0.679145i	$0.762351\pi$
$240$	0	0
$241$	−2.29329e8	−1.05536	−0.527678	−	0.849445i	$-0.676937\pi$
$241$	−2.29329e8	−1.05536	−0.527678	+	0.849445i	$0.676937\pi$
$242$	6.02937e8	2.73475
$243$	0	0
$244$	−5.31929e7	−0.234417
$245$	−2.44488e8	−1.06213
$246$	0	0
$247$	−2.27892e8	−0.962253
$248$	−8.07818e8	−3.36305
$249$	0	0
$250$	1.05269e8	0.426097
$251$	−7.43216e7	−0.296659	−0.148329	−	0.988938i	$-0.547390\pi$
$251$	−7.43216e7	−0.296659	−0.148329	+	0.988938i	$0.547390\pi$
$252$	0	0
$253$	−4.74705e8	−1.84290
$254$	−2.48544e8	−0.951666
$255$	0	0
$256$	6.24437e8	2.32621
$257$	−8.21908e7	−0.302035	−0.151017	−	0.988531i	$-0.548255\pi$
$257$	−8.21908e7	−0.302035	−0.151017	+	0.988531i	$0.548255\pi$
$258$	0	0
$259$	−2.86431e7	−0.102440
$260$	−1.17604e9	−4.14968
$261$	0	0
$262$	−7.58983e7	−0.260722
$263$	−4.73912e8	−1.60640	−0.803198	−	0.595713i	$-0.796870\pi$
$263$	−4.73912e8	−1.60640	−0.803198	+	0.595713i	$0.796870\pi$
$264$	0	0
$265$	−1.49815e8	−0.494532
$266$	2.85879e8	0.931315
$267$	0	0
$268$	−4.43495e8	−1.40740
$269$	1.61866e8	0.507016	0.253508	−	0.967333i	$-0.418416\pi$
$269$	1.61866e8	0.507016	0.253508	+	0.967333i	$0.418416\pi$
$270$	0	0
$271$	9.56080e7	0.291811	0.145905	−	0.989299i	$-0.453390\pi$
$271$	9.56080e7	0.291811	0.145905	+	0.989299i	$0.453390\pi$
$272$	9.57033e8	2.88361
$273$	0	0
$274$	−5.49275e8	−1.61311
$275$	−6.17473e8	−1.79042
$276$	0	0
$277$	5.77704e8	1.63315	0.816575	−	0.577240i	$-0.195870\pi$
$277$	5.77704e8	1.63315	0.816575	+	0.577240i	$0.195870\pi$
$278$	5.37104e8	1.49934
$279$	0	0
$280$	9.30507e8	2.53318
$281$	−1.72244e8	−0.463096	−0.231548	−	0.972823i	$-0.574379\pi$
$281$	−1.72244e8	−0.463096	−0.231548	+	0.972823i	$0.574379\pi$
$282$	0	0
$283$	−3.33472e8	−0.874595	−0.437298	−	0.899317i	$-0.644064\pi$
$283$	−3.33472e8	−0.874595	−0.437298	+	0.899317i	$0.644064\pi$
$284$	−2.84657e8	−0.737407
$285$	0	0
$286$	1.23831e9	3.13002
$287$	3.87890e8	0.968549
$288$	0	0
$289$	−1.50755e8	−0.367391
$290$	9.95670e8	2.39730
$291$	0	0
$292$	1.41510e9	3.32619
$293$	2.96954e8	0.689688	0.344844	−	0.938660i	$-0.387932\pi$
$293$	2.96954e8	0.689688	0.344844	+	0.938660i	$0.387932\pi$
$294$	0	0
$295$	−5.88242e8	−1.33407
$296$	−2.86274e8	−0.641596
$297$	0	0
$298$	9.31962e8	2.04005
$299$	−5.72106e8	−1.23773
$300$	0	0
$301$	3.78907e7	0.0800848
$302$	1.31553e9	2.74837
$303$	0	0
$304$	1.63555e9	3.33893
$305$	−6.29056e7	−0.126952
$306$	0	0
$307$	3.16973e8	0.625228	0.312614	−	0.949880i	$-0.398795\pi$
$307$	3.16973e8	0.625228	0.312614	+	0.949880i	$0.398795\pi$
$308$	−1.13447e9	−2.21241
$309$	0	0
$310$	−1.51462e9	−2.88761
$311$	8.80151e8	1.65919	0.829594	−	0.558367i	$-0.188572\pi$
$311$	8.80151e8	1.65919	0.829594	+	0.558367i	$0.188572\pi$
$312$	0	0
$313$	−8.29846e8	−1.52965	−0.764825	−	0.644238i	$-0.777175\pi$
$313$	−8.29846e8	−1.52965	−0.764825	+	0.644238i	$0.777175\pi$
$314$	−1.24382e9	−2.26728
$315$	0	0
$316$	−2.33652e9	−4.16548
$317$	2.04074e8	0.359817	0.179908	−	0.983683i	$-0.442420\pi$
$317$	2.04074e8	0.359817	0.179908	+	0.983683i	$0.442420\pi$
$318$	0	0
$319$	−7.65655e8	−1.32058
$320$	2.99553e9	5.11032
$321$	0	0
$322$	7.17680e8	1.19794
$323$	4.43625e8	0.732499
$324$	0	0
$325$	−7.44168e8	−1.20248
$326$	−4.42375e8	−0.707178
$327$	0	0
$328$	3.87677e9	6.06614
$329$	2.97703e8	0.460890
$330$	0	0
$331$	−4.43600e8	−0.672347	−0.336174	−	0.941800i	$-0.609133\pi$
$331$	−4.43600e8	−0.672347	−0.336174	+	0.941800i	$0.609133\pi$
$332$	−9.74847e7	−0.146202
$333$	0	0
$334$	−6.94864e8	−1.02044
$335$	−5.24475e8	−0.762198
$336$	0	0
$337$	9.56240e8	1.36101	0.680506	−	0.732742i	$-0.261760\pi$
$337$	9.56240e8	1.36101	0.680506	+	0.732742i	$0.261760\pi$
$338$	1.25344e8	0.176561
$339$	0	0
$340$	2.28933e9	3.15888
$341$	1.16472e9	1.59068
$342$	0	0
$343$	−6.76714e8	−0.905474
$344$	3.78700e8	0.501580
$345$	0	0
$346$	6.13220e8	0.795885
$347$	−7.21806e8	−0.927401	−0.463700	−	0.885992i	$-0.653479\pi$
$347$	−7.21806e8	−0.927401	−0.463700	+	0.885992i	$0.653479\pi$
$348$	0	0
$349$	4.87486e8	0.613865	0.306933	−	0.951731i	$-0.400697\pi$
$349$	4.87486e8	0.613865	0.306933	+	0.951731i	$0.400697\pi$
$350$	9.33523e8	1.16382
$351$	0	0
$352$	−4.70024e9	−5.74409
$353$	−2.31713e8	−0.280375	−0.140187	−	0.990125i	$-0.544771\pi$
$353$	−2.31713e8	−0.280375	−0.140187	+	0.990125i	$0.544771\pi$
$354$	0	0
$355$	−3.36633e8	−0.399354
$356$	−3.26163e9	−3.83142
$357$	0	0
$358$	8.43619e7	0.0971753
$359$	1.05857e8	0.120750	0.0603751	−	0.998176i	$-0.480770\pi$
$359$	1.05857e8	0.120750	0.0603751	+	0.998176i	$0.480770\pi$
$360$	0	0
$361$	−1.35724e8	−0.151838
$362$	−1.16424e9	−1.28992
$363$	0	0
$364$	−1.36724e9	−1.48591
$365$	1.67349e9	1.80135
$366$	0	0
$367$	2.03126e8	0.214504	0.107252	−	0.994232i	$-0.465795\pi$
$367$	2.03126e8	0.214504	0.107252	+	0.994232i	$0.465795\pi$
$368$	4.10595e9	4.29483
$369$	0	0
$370$	−5.36752e8	−0.550894
$371$	−1.74172e8	−0.177080
$372$	0	0
$373$	−7.70434e8	−0.768696	−0.384348	−	0.923188i	$-0.625574\pi$
$373$	−7.70434e8	−0.768696	−0.384348	+	0.923188i	$0.625574\pi$
$374$	−2.41055e9	−2.38268
$375$	0	0
$376$	2.97540e9	2.88661
$377$	−9.22754e8	−0.886934
$378$	0	0
$379$	9.79682e8	0.924375	0.462187	−	0.886782i	$-0.347065\pi$
$379$	9.79682e8	0.924375	0.462187	+	0.886782i	$0.347065\pi$
$380$	3.91244e9	3.65767
$381$	0	0
$382$	3.98564e8	0.365828
$383$	1.47413e9	1.34072	0.670362	−	0.742034i	$-0.266139\pi$
$383$	1.47413e9	1.34072	0.670362	+	0.742034i	$0.266139\pi$
$384$	0	0
$385$	−1.34162e9	−1.19816
$386$	1.79722e9	1.59054
$387$	0	0
$388$	−2.35086e9	−2.04323
$389$	−1.14843e9	−0.989194	−0.494597	−	0.869122i	$-0.664684\pi$
$389$	−1.14843e9	−0.989194	−0.494597	+	0.869122i	$0.664684\pi$
$390$	0	0
$391$	1.11369e9	0.942205
$392$	−2.84082e9	−2.38201
$393$	0	0
$394$	−1.54288e9	−1.27085
$395$	−2.76316e9	−2.25588
$396$	0	0
$397$	1.53079e9	1.22786	0.613928	−	0.789362i	$-0.289588\pi$
$397$	1.53079e9	1.22786	0.613928	+	0.789362i	$0.289588\pi$
$398$	3.50817e9	2.78927
$399$	0	0
$400$	5.34082e9	4.17251
$401$	5.05503e8	0.391488	0.195744	−	0.980655i	$-0.437288\pi$
$401$	5.05503e8	0.391488	0.195744	+	0.980655i	$0.437288\pi$
$402$	0	0
$403$	1.40370e9	1.06834
$404$	−4.78103e9	−3.60734
$405$	0	0
$406$	1.15755e9	0.858418
$407$	4.12754e8	0.303467
$408$	0	0
$409$	1.60782e9	1.16200	0.580998	−	0.813905i	$-0.302662\pi$
$409$	1.60782e9	1.16200	0.580998	+	0.813905i	$0.302662\pi$
$410$	7.26879e9	5.20857
$411$	0	0
$412$	−4.83742e9	−3.40780
$413$	−6.83881e8	−0.477700
$414$	0	0
$415$	−1.15285e8	−0.0791779
$416$	−5.66465e9	−3.85786
$417$	0	0
$418$	−4.11959e9	−2.75891
$419$	−2.86738e9	−1.90430	−0.952152	−	0.305625i	$-0.901135\pi$
$419$	−2.86738e9	−1.90430	−0.952152	+	0.305625i	$0.901135\pi$
$420$	0	0
$421$	1.98494e9	1.29646	0.648231	−	0.761444i	$-0.275509\pi$
$421$	1.98494e9	1.29646	0.648231	+	0.761444i	$0.275509\pi$
$422$	−4.47706e9	−2.90001
$423$	0	0
$424$	−1.74077e9	−1.10908
$425$	1.44863e9	0.915371
$426$	0	0
$427$	−7.31331e7	−0.0454586
$428$	−1.57038e9	−0.968172
$429$	0	0
$430$	7.10046e8	0.430672
$431$	−7.78770e7	−0.0468531	−0.0234266	−	0.999726i	$-0.507458\pi$
$431$	−7.78770e7	−0.0468531	−0.0234266	+	0.999726i	$0.507458\pi$
$432$	0	0
$433$	−1.06362e9	−0.629619	−0.314810	−	0.949155i	$-0.601941\pi$
$433$	−1.06362e9	−0.629619	−0.314810	+	0.949155i	$0.601941\pi$
$434$	−1.76088e9	−1.03399
$435$	0	0
$436$	4.97537e9	2.87490
$437$	1.90328e9	1.09098
$438$	0	0
$439$	−1.17090e9	−0.660531	−0.330266	−	0.943888i	$-0.607138\pi$
$439$	−1.17090e9	−0.660531	−0.330266	+	0.943888i	$0.607138\pi$
$440$	−1.34088e10	−7.50424
$441$	0	0
$442$	−2.90515e9	−1.60026
$443$	1.61819e8	0.0884335	0.0442167	−	0.999022i	$-0.485921\pi$
$443$	1.61819e8	0.0884335	0.0442167	+	0.999022i	$0.485921\pi$
$444$	0	0
$445$	−3.85718e9	−2.07496
$446$	−1.91111e9	−1.02003
$447$	0	0
$448$	3.48255e9	1.82989
$449$	−1.17696e9	−0.613618	−0.306809	−	0.951771i	$-0.599261\pi$
$449$	−1.17696e9	−0.613618	−0.306809	+	0.951771i	$0.599261\pi$
$450$	0	0
$451$	−5.58959e9	−2.86921
$452$	4.41989e9	2.25127
$453$	0	0
$454$	4.28773e9	2.15046
$455$	−1.61689e9	−0.804714
$456$	0	0
$457$	7.63423e8	0.374161	0.187081	−	0.982345i	$-0.440097\pi$
$457$	7.63423e8	0.374161	0.187081	+	0.982345i	$0.440097\pi$
$458$	−4.17428e9	−2.03026
$459$	0	0
$460$	9.82190e9	4.70482
$461$	3.36981e9	1.60196	0.800980	−	0.598691i	$-0.204312\pi$
$461$	3.36981e9	1.60196	0.800980	+	0.598691i	$0.204312\pi$
$462$	0	0
$463$	−2.92488e9	−1.36954	−0.684769	−	0.728761i	$-0.740097\pi$
$463$	−2.92488e9	−1.36954	−0.684769	+	0.728761i	$0.740097\pi$
$464$	6.62251e9	3.07758
$465$	0	0
$466$	1.63007e9	0.746202
$467$	2.12116e9	0.963748	0.481874	−	0.876240i	$-0.339956\pi$
$467$	2.12116e9	0.963748	0.481874	+	0.876240i	$0.339956\pi$
$468$	0	0
$469$	−6.09746e8	−0.272925
$470$	5.57875e9	2.47853
$471$	0	0
$472$	−6.83507e9	−2.99189
$473$	−5.46014e8	−0.237241
$474$	0	0
$475$	2.47569e9	1.05991
$476$	2.66154e9	1.13112
$477$	0	0
$478$	−6.74992e9	−2.82684
$479$	−3.77482e8	−0.156936	−0.0784678	−	0.996917i	$-0.525003\pi$
$479$	−3.77482e8	−0.156936	−0.0784678	+	0.996917i	$0.525003\pi$
$480$	0	0
$481$	4.97444e8	0.203815
$482$	−4.99616e9	−2.03223
$483$	0	0
$484$	9.59315e9	3.84594
$485$	−2.78012e9	−1.10654
$486$	0	0
$487$	−5.49674e8	−0.215652	−0.107826	−	0.994170i	$-0.534389\pi$
$487$	−5.49674e8	−0.215652	−0.107826	+	0.994170i	$0.534389\pi$
$488$	−7.30931e8	−0.284713
$489$	0	0
$490$	−5.32642e9	−2.04526
$491$	3.90286e9	1.48798	0.743991	−	0.668190i	$-0.232931\pi$
$491$	3.90286e9	1.48798	0.743991	+	0.668190i	$0.232931\pi$
$492$	0	0
$493$	1.79628e9	0.675164
$494$	−4.96485e9	−1.85294
$495$	0	0
$496$	−1.00742e10	−3.70703
$497$	−3.91365e8	−0.142999
$498$	0	0
$499$	5.05727e9	1.82207	0.911033	−	0.412334i	$-0.135286\pi$
$499$	5.05727e9	1.82207	0.911033	+	0.412334i	$0.135286\pi$
$500$	1.67490e9	0.599229
$501$	0	0
$502$	−1.61917e9	−0.571255
$503$	−2.90093e9	−1.01636	−0.508182	−	0.861249i	$-0.669682\pi$
$503$	−2.90093e9	−1.01636	−0.508182	+	0.861249i	$0.669682\pi$
$504$	0	0
$505$	−5.65402e9	−1.95361
$506$	−1.03419e10	−3.54875
$507$	0	0
$508$	−3.95450e9	−1.33835
$509$	1.02191e9	0.343479	0.171739	−	0.985142i	$-0.445061\pi$
$509$	1.02191e9	0.343479	0.171739	+	0.985142i	$0.445061\pi$
$510$	0	0
$511$	1.94557e9	0.645022
$512$	4.43967e9	1.46186
$513$	0	0
$514$	−1.79061e9	−0.581608
$515$	−5.72071e9	−1.84554
$516$	0	0
$517$	−4.28997e9	−1.36533
$518$	−6.24020e8	−0.197262
$519$	0	0
$520$	−1.61601e10	−5.04002
$521$	5.15409e8	0.159669	0.0798344	−	0.996808i	$-0.474561\pi$
$521$	5.15409e8	0.159669	0.0798344	+	0.996808i	$0.474561\pi$
$522$	0	0
$523$	4.99074e8	0.152549	0.0762745	−	0.997087i	$-0.475697\pi$
$523$	4.99074e8	0.152549	0.0762745	+	0.997087i	$0.475697\pi$
$524$	−1.20760e9	−0.366659
$525$	0	0
$526$	−1.03247e10	−3.09332
$527$	−2.73252e9	−0.813253
$528$	0	0
$529$	1.37322e9	0.403316
$530$	−3.26387e9	−0.952286
$531$	0	0
$532$	4.54854e9	1.30973
$533$	−6.73647e9	−1.92702
$534$	0	0
$535$	−1.85713e9	−0.524328
$536$	−6.09412e9	−1.70936
$537$	0	0
$538$	3.52641e9	0.976325
$539$	4.09594e9	1.12666
$540$	0	0
$541$	3.98160e9	1.08110	0.540551	−	0.841311i	$-0.318216\pi$
$541$	3.98160e9	1.08110	0.540551	+	0.841311i	$0.318216\pi$
$542$	2.08292e9	0.561920
$543$	0	0
$544$	1.10271e10	2.93673
$545$	5.88385e9	1.55695
$546$	0	0
$547$	4.26163e9	1.11332	0.556660	−	0.830740i	$-0.312083\pi$
$547$	4.26163e9	1.11332	0.556660	+	0.830740i	$0.312083\pi$
$548$	−8.73935e9	−2.26854
$549$	0	0
$550$	−1.34523e10	−3.44768
$551$	3.06981e9	0.781774
$552$	0	0
$553$	−3.21240e9	−0.807778
$554$	1.25859e10	3.14484
$555$	0	0
$556$	8.54570e9	2.10856
$557$	6.93171e9	1.69960	0.849800	−	0.527104i	$-0.176722\pi$
$557$	6.93171e9	1.69960	0.849800	+	0.527104i	$0.176722\pi$
$558$	0	0
$559$	−6.58047e8	−0.159337
$560$	1.16043e10	2.79229
$561$	0	0
$562$	−3.75250e9	−0.891752
$563$	−8.38570e8	−0.198043	−0.0990216	−	0.995085i	$-0.531571\pi$
$563$	−8.38570e8	−0.198043	−0.0990216	+	0.995085i	$0.531571\pi$
$564$	0	0
$565$	5.22693e9	1.21921
$566$	−7.26503e9	−1.68415
$567$	0	0
$568$	−3.91150e9	−0.895622
$569$	−7.43736e9	−1.69249	−0.846244	−	0.532795i	$-0.821142\pi$
$569$	−7.43736e9	−1.69249	−0.846244	+	0.532795i	$0.821142\pi$
$570$	0	0
$571$	−2.43113e9	−0.546490	−0.273245	−	0.961944i	$-0.588097\pi$
$571$	−2.43113e9	−0.546490	−0.273245	+	0.961944i	$0.588097\pi$
$572$	1.97023e10	4.40181
$573$	0	0
$574$	8.45058e9	1.86507
$575$	6.21505e9	1.36335
$576$	0	0
$577$	−2.78106e8	−0.0602692	−0.0301346	−	0.999546i	$-0.509594\pi$
$577$	−2.78106e8	−0.0602692	−0.0301346	+	0.999546i	$0.509594\pi$
$578$	−3.28434e9	−0.707459
$579$	0	0
$580$	1.58418e10	3.37137
$581$	−1.34028e8	−0.0283518
$582$	0	0
$583$	2.50987e9	0.524579
$584$	1.94451e10	4.03985
$585$	0	0
$586$	6.46945e9	1.32809
$587$	4.32874e9	0.883340	0.441670	−	0.897178i	$-0.354386\pi$
$587$	4.32874e9	0.883340	0.441670	+	0.897178i	$0.354386\pi$
$588$	0	0
$589$	−4.66983e9	−0.941668
$590$	−1.28155e10	−2.56893
$591$	0	0
$592$	−3.57011e9	−0.707221
$593$	1.16343e9	0.229112	0.114556	−	0.993417i	$-0.463456\pi$
$593$	1.16343e9	0.229112	0.114556	+	0.993417i	$0.463456\pi$
$594$	0	0
$595$	3.14753e9	0.612576
$596$	1.48282e10	2.86897
$597$	0	0
$598$	−1.24639e10	−2.38342
$599$	6.78491e9	1.28988	0.644941	−	0.764232i	$-0.276882\pi$
$599$	6.78491e9	1.28988	0.644941	+	0.764232i	$0.276882\pi$
$600$	0	0
$601$	2.61686e9	0.491723	0.245862	−	0.969305i	$-0.420929\pi$
$601$	2.61686e9	0.491723	0.245862	+	0.969305i	$0.420929\pi$
$602$	8.25488e8	0.154214
$603$	0	0
$604$	2.09310e10	3.86509
$605$	1.13448e10	2.08283
$606$	0	0
$607$	1.41096e9	0.256067	0.128034	−	0.991770i	$-0.459133\pi$
$607$	1.41096e9	0.256067	0.128034	+	0.991770i	$0.459133\pi$
$608$	1.88451e10	3.40045
$609$	0	0
$610$	−1.37046e9	−0.244463
$611$	−5.17020e9	−0.916986
$612$	0	0
$613$	2.52910e9	0.443459	0.221730	−	0.975108i	$-0.428830\pi$
$613$	2.52910e9	0.443459	0.221730	+	0.975108i	$0.428830\pi$
$614$	6.90558e9	1.20396
$615$	0	0
$616$	−1.55889e10	−2.68710
$617$	3.77565e9	0.647133	0.323567	−	0.946205i	$-0.395118\pi$
$617$	3.77565e9	0.647133	0.323567	+	0.946205i	$0.395118\pi$
$618$	0	0
$619$	−9.20238e9	−1.55949	−0.779746	−	0.626096i	$-0.784652\pi$
$619$	−9.20238e9	−1.55949	−0.779746	+	0.626096i	$0.784652\pi$
$620$	−2.40987e10	−4.06091
$621$	0	0
$622$	1.91750e10	3.19498
$623$	−4.48430e9	−0.742996
$624$	0	0
$625$	−5.04368e9	−0.826357
$626$	−1.80790e10	−2.94554
$627$	0	0
$628$	−1.97901e10	−3.18852
$629$	−9.68349e8	−0.155151
$630$	0	0
$631$	−4.06379e9	−0.643915	−0.321958	−	0.946754i	$-0.604341\pi$
$631$	−4.06379e9	−0.643915	−0.321958	+	0.946754i	$0.604341\pi$
$632$	−3.21065e10	−5.05921
$633$	0	0
$634$	4.44597e9	0.692874
$635$	−4.67657e9	−0.724802
$636$	0	0
$637$	4.93635e9	0.756690
$638$	−1.66806e10	−2.54296
$639$	0	0
$640$	2.93490e10	4.42552
$641$	−4.61167e9	−0.691599	−0.345800	−	0.938308i	$-0.612392\pi$
$641$	−4.61167e9	−0.691599	−0.345800	+	0.938308i	$0.612392\pi$
$642$	0	0
$643$	−8.37065e9	−1.24171	−0.620856	−	0.783925i	$-0.713215\pi$
$643$	−8.37065e9	−1.24171	−0.620856	+	0.783925i	$0.713215\pi$
$644$	1.14188e10	1.68469
$645$	0	0
$646$	9.66483e9	1.41052
$647$	−1.08886e10	−1.58054	−0.790272	−	0.612756i	$-0.790061\pi$
$647$	−1.08886e10	−1.58054	−0.790272	+	0.612756i	$0.790061\pi$
$648$	0	0
$649$	9.85489e9	1.41513
$650$	−1.62125e10	−2.31554
$651$	0	0
$652$	−7.03849e9	−0.994518
$653$	−4.93139e9	−0.693064	−0.346532	−	0.938038i	$-0.612641\pi$
$653$	−4.93139e9	−0.693064	−0.346532	+	0.938038i	$0.612641\pi$
$654$	0	0
$655$	−1.42809e9	−0.198569
$656$	4.83470e10	6.68661
$657$	0	0
$658$	6.48576e9	0.887504
$659$	9.90535e9	1.34825	0.674126	−	0.738616i	$-0.264520\pi$
$659$	9.90535e9	1.34825	0.674126	+	0.738616i	$0.264520\pi$
$660$	0	0
$661$	−6.13396e9	−0.826107	−0.413053	−	0.910707i	$-0.635538\pi$
$661$	−6.13396e9	−0.826107	−0.413053	+	0.910707i	$0.635538\pi$
$662$	−9.66428e9	−1.29469
$663$	0	0
$664$	−1.33955e9	−0.177570
$665$	5.37907e9	0.709303
$666$	0	0
$667$	7.70655e9	1.00559
$668$	−1.10558e10	−1.43506
$669$	0	0
$670$	−1.14262e10	−1.46771
$671$	1.05387e9	0.134666
$672$	0	0
$673$	−5.75452e9	−0.727707	−0.363853	−	0.931456i	$-0.618539\pi$
$673$	−5.75452e9	−0.727707	−0.363853	+	0.931456i	$0.618539\pi$
$674$	2.08327e10	2.62081
$675$	0	0
$676$	1.99431e9	0.248301
$677$	3.89999e9	0.483062	0.241531	−	0.970393i	$-0.422350\pi$
$677$	3.89999e9	0.483062	0.241531	+	0.970393i	$0.422350\pi$
$678$	0	0
$679$	−3.23212e9	−0.396226
$680$	3.14580e10	3.83663
$681$	0	0
$682$	2.53747e10	3.06306
$683$	1.93635e8	0.0232547	0.0116273	−	0.999932i	$-0.496299\pi$
$683$	1.93635e8	0.0232547	0.0116273	+	0.999932i	$0.496299\pi$
$684$	0	0
$685$	−1.03351e10	−1.22856
$686$	−1.47429e10	−1.74361
$687$	0	0
$688$	4.72273e9	0.552884
$689$	3.02485e9	0.352319
$690$	0	0
$691$	1.72931e8	0.0199388	0.00996940	−	0.999950i	$-0.496827\pi$
$691$	1.72931e8	0.0199388	0.00996940	+	0.999950i	$0.496827\pi$
$692$	9.75676e9	1.11927
$693$	0	0
$694$	−1.57253e10	−1.78583
$695$	1.01061e10	1.14192
$696$	0	0
$697$	1.31135e10	1.46692
$698$	1.06204e10	1.18208
$699$	0	0
$700$	1.48530e10	1.63671
$701$	1.41045e10	1.54649	0.773243	−	0.634110i	$-0.218633\pi$
$701$	1.41045e10	1.54649	0.773243	+	0.634110i	$0.218633\pi$
$702$	0	0
$703$	−1.65489e9	−0.179650
$704$	−5.01844e10	−5.42082
$705$	0	0
$706$	−5.04811e9	−0.539898
$707$	−6.57327e9	−0.699541
$708$	0	0
$709$	7.26226e9	0.765262	0.382631	−	0.923901i	$-0.375018\pi$
$709$	7.26226e9	0.765262	0.382631	+	0.923901i	$0.375018\pi$
$710$	−7.33391e9	−0.769008
$711$	0	0
$712$	−4.48184e10	−4.65347
$713$	−1.17233e10	−1.21126
$714$	0	0
$715$	2.32998e10	2.38387
$716$	1.34226e9	0.136660
$717$	0	0
$718$	2.30620e9	0.232520
$719$	5.40870e9	0.542677	0.271339	−	0.962484i	$-0.412534\pi$
$719$	5.40870e9	0.542677	0.271339	+	0.962484i	$0.412534\pi$
$720$	0	0
$721$	−6.65080e9	−0.660846
$722$	−2.95688e9	−0.292384
$723$	0	0
$724$	−1.85239e10	−1.81405
$725$	1.00243e10	0.976947
$726$	0	0
$727$	−1.56561e10	−1.51116	−0.755582	−	0.655054i	$-0.772646\pi$
$727$	−1.56561e10	−1.51116	−0.755582	+	0.655054i	$0.772646\pi$
$728$	−1.87875e10	−1.80471
$729$	0	0
$730$	3.64587e10	3.46874
$731$	1.28099e9	0.121292
$732$	0	0
$733$	1.25157e10	1.17379	0.586895	−	0.809663i	$-0.300350\pi$
$733$	1.25157e10	1.17379	0.586895	+	0.809663i	$0.300350\pi$
$734$	4.42531e9	0.413055
$735$	0	0
$736$	4.73093e10	4.37396
$737$	8.78659e9	0.808507
$738$	0	0
$739$	−1.12760e10	−1.02778	−0.513888	−	0.857857i	$-0.671795\pi$
$739$	−1.12760e10	−1.02778	−0.513888	+	0.857857i	$0.671795\pi$
$740$	−8.54011e9	−0.774733
$741$	0	0
$742$	−3.79452e9	−0.340992
$743$	−6.56019e9	−0.586753	−0.293377	−	0.955997i	$-0.594779\pi$
$743$	−6.56019e9	−0.586753	−0.293377	+	0.955997i	$0.594779\pi$
$744$	0	0
$745$	1.75357e10	1.55373
$746$	−1.67847e10	−1.48022
$747$	0	0
$748$	−3.83535e10	−3.35081
$749$	−2.15907e9	−0.187750
$750$	0	0
$751$	1.28366e10	1.10589	0.552943	−	0.833219i	$-0.313505\pi$
$751$	1.28366e10	1.10589	0.552943	+	0.833219i	$0.313505\pi$
$752$	3.71060e10	3.18186
$753$	0	0
$754$	−2.01031e10	−1.70791
$755$	2.47528e10	2.09320
$756$	0	0
$757$	−3.54032e9	−0.296624	−0.148312	−	0.988941i	$-0.547384\pi$
$757$	−3.54032e9	−0.296624	−0.148312	+	0.988941i	$0.547384\pi$
$758$	2.13434e10	1.78000
$759$	0	0
$760$	5.37613e10	4.44244
$761$	1.14256e10	0.939795	0.469897	−	0.882721i	$-0.344291\pi$
$761$	1.14256e10	0.939795	0.469897	+	0.882721i	$0.344291\pi$
$762$	0	0
$763$	6.84047e9	0.557506
$764$	6.34144e9	0.514471
$765$	0	0
$766$	3.21154e10	2.58174
$767$	1.18769e10	0.950431
$768$	0	0
$769$	1.70491e10	1.35195	0.675974	−	0.736926i	$-0.263723\pi$
$769$	1.70491e10	1.35195	0.675974	+	0.736926i	$0.263723\pi$
$770$	−2.92285e10	−2.30722
$771$	0	0
$772$	2.85950e10	2.23681
$773$	1.14110e10	0.888576	0.444288	−	0.895884i	$-0.353457\pi$
$773$	1.14110e10	0.888576	0.444288	+	0.895884i	$0.353457\pi$
$774$	0	0
$775$	−1.52491e10	−1.17676
$776$	−3.23035e10	−2.48161
$777$	0	0
$778$	−2.50198e10	−1.90482
$779$	2.24108e10	1.69854
$780$	0	0
$781$	5.63966e9	0.423618
$782$	2.42629e10	1.81434
$783$	0	0
$784$	−3.54277e10	−2.62565
$785$	−2.34037e10	−1.72679
$786$	0	0
$787$	−1.06817e10	−0.781142	−0.390571	−	0.920573i	$-0.627722\pi$
$787$	−1.06817e10	−0.781142	−0.390571	+	0.920573i	$0.627722\pi$
$788$	−2.45483e10	−1.78722
$789$	0	0
$790$	−6.01982e10	−4.34399
$791$	6.07675e9	0.436570
$792$	0	0
$793$	1.27010e9	0.0904444
$794$	3.33497e10	2.36440
$795$	0	0
$796$	5.58174e10	3.92260
$797$	−1.80601e10	−1.26362	−0.631810	−	0.775123i	$-0.717688\pi$
$797$	−1.80601e10	−1.26362	−0.631810	+	0.775123i	$0.717688\pi$
$798$	0	0
$799$	1.00646e10	0.698041
$800$	6.15377e10	4.24939
$801$	0	0
$802$	1.10129e10	0.753862
$803$	−2.80362e10	−1.91080
$804$	0	0
$805$	1.35038e10	0.912368
$806$	3.05811e10	2.05722
$807$	0	0
$808$	−6.56967e10	−4.38131
$809$	−1.87987e10	−1.24827	−0.624134	−	0.781317i	$-0.714548\pi$
$809$	−1.87987e10	−1.24827	−0.624134	+	0.781317i	$0.714548\pi$
$810$	0	0
$811$	−1.93491e9	−0.127376	−0.0636881	−	0.997970i	$-0.520286\pi$
$811$	−1.93491e9	−0.127376	−0.0636881	+	0.997970i	$0.520286\pi$
$812$	1.84174e10	1.20721
$813$	0	0
$814$	8.99228e9	0.584365
$815$	−8.32368e9	−0.538596
$816$	0	0
$817$	2.18918e9	0.140445
$818$	3.50279e10	2.23758
$819$	0	0
$820$	1.15652e11	7.32492
$821$	−1.74599e10	−1.10113	−0.550567	−	0.834791i	$-0.685589\pi$
$821$	−1.74599e10	−1.10113	−0.550567	+	0.834791i	$0.685589\pi$
$822$	0	0
$823$	−1.78070e10	−1.11350	−0.556752	−	0.830679i	$-0.687953\pi$
$823$	−1.78070e10	−1.11350	−0.556752	+	0.830679i	$0.687953\pi$
$824$	−6.64716e10	−4.13896
$825$	0	0
$826$	−1.48990e10	−0.919874
$827$	2.44106e10	1.50075	0.750377	−	0.661010i	$-0.229872\pi$
$827$	2.44106e10	1.50075	0.750377	+	0.661010i	$0.229872\pi$
$828$	0	0
$829$	1.80753e10	1.10191	0.550953	−	0.834536i	$-0.314264\pi$
$829$	1.80753e10	1.10191	0.550953	+	0.834536i	$0.314264\pi$
$830$	−2.51160e9	−0.152467
$831$	0	0
$832$	−6.04814e10	−3.64074
$833$	−9.60934e9	−0.576018
$834$	0	0
$835$	−1.30745e10	−0.777181
$836$	−6.55455e10	−3.87990
$837$	0	0
$838$	−6.24688e10	−3.66699
$839$	3.99758e9	0.233684	0.116842	−	0.993150i	$-0.462723\pi$
$839$	3.99758e9	0.233684	0.116842	+	0.993150i	$0.462723\pi$
$840$	0	0
$841$	−4.81993e9	−0.279418
$842$	4.32439e10	2.49651
$843$	0	0
$844$	−7.12331e10	−4.07834
$845$	2.35846e9	0.134471
$846$	0	0
$847$	1.31893e10	0.745812
$848$	−2.17090e10	−1.22252
$849$	0	0
$850$	3.15600e10	1.76267
$851$	−4.15450e9	−0.231081
$852$	0	0
$853$	1.49532e10	0.824923	0.412461	−	0.910975i	$-0.364669\pi$
$853$	1.49532e10	0.824923	0.412461	+	0.910975i	$0.364669\pi$
$854$	−1.59328e9	−0.0875366
$855$	0	0
$856$	−2.15788e10	−1.17590
$857$	2.77267e10	1.50475	0.752377	−	0.658732i	$-0.228907\pi$
$857$	2.77267e10	1.50475	0.752377	+	0.658732i	$0.228907\pi$
$858$	0	0
$859$	−1.83528e10	−0.987930	−0.493965	−	0.869482i	$-0.664453\pi$
$859$	−1.83528e10	−0.987930	−0.493965	+	0.869482i	$0.664453\pi$
$860$	1.12973e10	0.605663
$861$	0	0
$862$	−1.69663e9	−0.0902218
$863$	5.98092e9	0.316760	0.158380	−	0.987378i	$-0.449373\pi$
$863$	5.98092e9	0.316760	0.158380	+	0.987378i	$0.449373\pi$
$864$	0	0
$865$	1.15383e10	0.606157
$866$	−2.31720e10	−1.21241
$867$	0	0
$868$	−2.80168e10	−1.45412
$869$	4.62915e10	2.39294
$870$	0	0
$871$	1.05894e10	0.543012
$872$	6.83672e10	3.49173
$873$	0	0
$874$	4.14649e10	2.10083
$875$	2.30276e9	0.116204
$876$	0	0
$877$	2.39573e10	1.19933	0.599666	−	0.800251i	$-0.295300\pi$
$877$	2.39573e10	1.19933	0.599666	+	0.800251i	$0.295300\pi$
$878$	−2.55092e10	−1.27194
$879$	0	0
$880$	−1.67221e11	−8.27181
$881$	−1.57373e9	−0.0775380	−0.0387690	−	0.999248i	$-0.512344\pi$
$881$	−1.57373e9	−0.0775380	−0.0387690	+	0.999248i	$0.512344\pi$
$882$	0	0
$883$	1.75377e10	0.857254	0.428627	−	0.903482i	$-0.358998\pi$
$883$	1.75377e10	0.857254	0.428627	+	0.903482i	$0.358998\pi$
$884$	−4.62229e10	−2.25048
$885$	0	0
$886$	3.52539e9	0.170290
$887$	−4.02646e10	−1.93727	−0.968637	−	0.248479i	$-0.920069\pi$
$887$	−4.02646e10	−1.93727	−0.968637	+	0.248479i	$0.920069\pi$
$888$	0	0
$889$	−5.43691e9	−0.259535
$890$	−8.40326e10	−3.99561
$891$	0	0
$892$	−3.04070e10	−1.43449
$893$	1.72002e10	0.808263
$894$	0	0
$895$	1.58734e9	0.0740100
$896$	3.41207e10	1.58467
$897$	0	0
$898$	−2.56412e10	−1.18160
$899$	−1.89086e10	−0.867960
$900$	0	0
$901$	−5.88831e9	−0.268197
$902$	−1.21775e11	−5.52503
$903$	0	0
$904$	6.07342e10	2.73429
$905$	−2.19063e10	−0.982424
$906$	0	0
$907$	2.96929e10	1.32138	0.660688	−	0.750660i	$-0.270264\pi$
$907$	2.96929e10	1.32138	0.660688	+	0.750660i	$0.270264\pi$
$908$	6.82207e10	3.02424
$909$	0	0
$910$	−3.52257e10	−1.54958
$911$	−6.10006e9	−0.267313	−0.133656	−	0.991028i	$-0.542672\pi$
$911$	−6.10006e9	−0.267313	−0.133656	+	0.991028i	$0.542672\pi$
$912$	0	0
$913$	1.93138e9	0.0839886
$914$	1.66320e10	0.720496
$915$	0	0
$916$	−6.64156e10	−2.85520
$917$	−1.66028e9	−0.0711031
$918$	0	0
$919$	−3.43035e10	−1.45792	−0.728960	−	0.684556i	$-0.759996\pi$
$919$	−3.43035e10	−1.45792	−0.728960	+	0.684556i	$0.759996\pi$
$920$	1.34964e11	5.71426
$921$	0	0
$922$	7.34147e10	3.08478
$923$	6.79682e9	0.284511
$924$	0	0
$925$	−5.40397e9	−0.224500
$926$	−6.37214e10	−2.63722
$927$	0	0
$928$	7.63056e10	3.13428
$929$	1.04991e10	0.429632	0.214816	−	0.976655i	$-0.431085\pi$
$929$	1.04991e10	0.429632	0.214816	+	0.976655i	$0.431085\pi$
$930$	0	0
$931$	−1.64222e10	−0.666972
$932$	2.59356e10	1.04940
$933$	0	0
$934$	4.62116e10	1.85582
$935$	−4.53566e10	−1.81468
$936$	0	0
$937$	3.00943e10	1.19507	0.597537	−	0.801841i	$-0.296146\pi$
$937$	3.00943e10	1.19507	0.597537	+	0.801841i	$0.296146\pi$
$938$	−1.32839e10	−0.525553
$939$	0	0
$940$	8.87617e10	3.48561
$941$	1.94953e10	0.762720	0.381360	−	0.924427i	$-0.375456\pi$
$941$	1.94953e10	0.762720	0.381360	+	0.924427i	$0.375456\pi$
$942$	0	0
$943$	5.62608e10	2.18482
$944$	−8.52396e10	−3.29791
$945$	0	0
$946$	−1.18955e10	−0.456839
$947$	−2.51681e10	−0.962998	−0.481499	−	0.876447i	$-0.659908\pi$
$947$	−2.51681e10	−0.962998	−0.481499	+	0.876447i	$0.659908\pi$
$948$	0	0
$949$	−3.37887e10	−1.28333
$950$	5.39355e10	2.04100
$951$	0	0
$952$	3.65726e10	1.37381
$953$	−1.34093e10	−0.501859	−0.250929	−	0.968005i	$-0.580736\pi$
$953$	−1.34093e10	−0.501859	−0.250929	+	0.968005i	$0.580736\pi$
$954$	0	0
$955$	7.49935e9	0.278620
$956$	−1.07396e11	−3.97544
$957$	0	0
$958$	−8.22383e9	−0.302200
$959$	−1.20154e10	−0.439921
$960$	0	0
$961$	1.25131e9	0.0454815
$962$	1.08373e10	0.392473
$963$	0	0
$964$	−7.94924e10	−2.85796
$965$	3.38163e10	1.21138
$966$	0	0
$967$	−4.92333e10	−1.75092	−0.875460	−	0.483291i	$-0.839441\pi$
$967$	−4.92333e10	−1.75092	−0.875460	+	0.483291i	$0.839441\pi$
$968$	1.31821e11	4.67111
$969$	0	0
$970$	−6.05677e10	−2.13079
$971$	−1.64288e10	−0.575889	−0.287945	−	0.957647i	$-0.592972\pi$
$971$	−1.64288e10	−0.575889	−0.287945	+	0.957647i	$0.592972\pi$
$972$	0	0
$973$	1.17492e10	0.408896
$974$	−1.19752e10	−0.415266
$975$	0	0
$976$	−9.11538e9	−0.313834
$977$	−1.72848e10	−0.592972	−0.296486	−	0.955037i	$-0.595815\pi$
$977$	−1.72848e10	−0.592972	−0.296486	+	0.955037i	$0.595815\pi$
$978$	0	0
$979$	6.46198e10	2.20103
$980$	−8.47471e10	−2.87630
$981$	0	0
$982$	8.50278e10	2.86530
$983$	8.75111e9	0.293850	0.146925	−	0.989148i	$-0.453062\pi$
$983$	8.75111e9	0.293850	0.146925	+	0.989148i	$0.453062\pi$
$984$	0	0
$985$	−2.90306e10	−0.967898
$986$	3.91338e10	1.30012
$987$	0	0
$988$	−7.89943e10	−2.60583
$989$	5.49580e9	0.180652
$990$	0	0
$991$	−2.33937e10	−0.763555	−0.381778	−	0.924254i	$-0.624688\pi$
$991$	−2.33937e10	−0.763555	−0.381778	+	0.924254i	$0.624688\pi$
$992$	−1.16077e11	−3.77533
$993$	0	0
$994$	−8.52628e9	−0.275364
$995$	6.60094e10	2.12434
$996$	0	0
$997$	−8.07234e9	−0.257968	−0.128984	−	0.991647i	$-0.541172\pi$
$997$	−8.07234e9	−0.257968	−0.128984	+	0.991647i	$0.541172\pi$
$998$	1.10178e11	3.50862
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.b.1.11		11
3.2	odd	2		43.8.a.a.1.1	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.a.1.1	✓	11	3.2	odd	2
387.8.a.b.1.11		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q+21.7860 q^{2} +346.631 q^{4} +409.924 q^{5} +476.571 q^{7} +4763.10 q^{8} +O(q^{10})\)	\(q+21.7860 q^{2} +346.631 q^{4} +409.924 q^{5} +476.571 q^{7} +4763.10 q^{8} +8930.61 q^{10} -6867.50 q^{11} -8276.59 q^{13} +10382.6 q^{14} +59400.2 q^{16} +16111.6 q^{17} +27534.5 q^{19} +142092. q^{20} -149616. q^{22} +69123.4 q^{23} +89912.4 q^{25} -180314. q^{26} +165194. q^{28} +111490. q^{29} -169599. q^{31} +684418. q^{32} +351008. q^{34} +195358. q^{35} -60102.6 q^{37} +599867. q^{38} +1.95251e6 q^{40} +813919. q^{41} +79507.0 q^{43} -2.38049e6 q^{44} +1.50592e6 q^{46} +624677. q^{47} -596423. q^{49} +1.95883e6 q^{50} -2.86892e6 q^{52} -365470. q^{53} -2.81515e6 q^{55} +2.26995e6 q^{56} +2.42892e6 q^{58} -1.43500e6 q^{59} -153457. q^{61} -3.69489e6 q^{62} +7.30752e6 q^{64} -3.39277e6 q^{65} -1.27944e6 q^{67} +5.58478e6 q^{68} +4.25607e6 q^{70} -821210. q^{71} +4.08244e6 q^{73} -1.30940e6 q^{74} +9.54431e6 q^{76} -3.27285e6 q^{77} -6.74067e6 q^{79} +2.43496e7 q^{80} +1.77321e7 q^{82} -281235. q^{83} +6.60453e6 q^{85} +1.73214e6 q^{86} -3.27106e7 q^{88} -9.40951e6 q^{89} -3.94438e6 q^{91} +2.39603e7 q^{92} +1.36092e7 q^{94} +1.12870e7 q^{95} -6.78204e6 q^{97} -1.29937e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8 - 1333 * q^10 - 1333 * q^11 - 17967 * q^13 + 22352 * q^14 - 34406 * q^16 + 63095 * q^17 - 54524 * q^19 + 280995 * q^20 - 289358 * q^22 + 138139 * q^23 + 3455 * q^25 + 132946 * q^26 - 12704 * q^28 + 308658 * q^29 - 209523 * q^31 + 644934 * q^32 + 762435 * q^34 + 578892 * q^35 - 298472 * q^37 + 369707 * q^38 + 2633173 * q^40 + 1346735 * q^41 + 874577 * q^43 - 3134292 * q^44 + 3588111 * q^46 - 499284 * q^47 + 2544563 * q^49 - 3049745 * q^50 + 983088 * q^52 + 2210495 * q^53 - 1855072 * q^55 + 469976 * q^56 + 4397067 * q^58 + 5824216 * q^59 - 4453034 * q^61 - 1002789 * q^62 + 4757538 * q^64 + 2162872 * q^65 - 6859513 * q^67 + 9397005 * q^68 + 845078 * q^70 + 10726554 * q^71 - 4456898 * q^73 - 1046637 * q^74 + 5861267 * q^76 + 17019816 * q^77 - 15541320 * q^79 + 15680911 * q^80 + 20233655 * q^82 + 11146767 * q^83 - 12471976 * q^85 + 1908168 * q^86 - 24463544 * q^88 + 13531356 * q^89 - 19746448 * q^91 + 26023161 * q^92 + 20288857 * q^94 + 12291624 * q^95 - 10999901 * q^97 - 29909168 * q^98

Introduction

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