LMFDB - Embedded newform 387.8.a.d.1.13

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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + \cdots + 1551032970660 $ x^13 - 3x^12 - 1279x^11 + 3765x^10 + 598742x^9 - 1518614x^8 - 124677082x^7 + 193428526x^6 + 11160446785x^5 + 1754605765x^4 - 349352939351x^3 - 481872751923x^2 + 2098464001560x + 1551032970660
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-21.3781$ 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)

$f(q)$	$=$	$q+20.3781 q^{2} +287.267 q^{4} -405.085 q^{5} +52.5965 q^{7} +3245.57 q^{8} +O(q^{10})$	\(q+20.3781 q^{2} +287.267 q^{4} -405.085 q^{5} +52.5965 q^{7} +3245.57 q^{8} -8254.88 q^{10} -2544.97 q^{11} +7138.95 q^{13} +1071.82 q^{14} +29368.3 q^{16} +22507.9 q^{17} -6396.31 q^{19} -116368. q^{20} -51861.7 q^{22} -99842.9 q^{23} +85969.2 q^{25} +145478. q^{26} +15109.2 q^{28} -5608.53 q^{29} -163095. q^{31} +183038. q^{32} +458669. q^{34} -21306.1 q^{35} +198188. q^{37} -130345. q^{38} -1.31473e6 q^{40} -742519. q^{41} -79507.0 q^{43} -731087. q^{44} -2.03461e6 q^{46} -786601. q^{47} -820777. q^{49} +1.75189e6 q^{50} +2.05079e6 q^{52} -2.08070e6 q^{53} +1.03093e6 q^{55} +170705. q^{56} -114291. q^{58} -607486. q^{59} +2.59833e6 q^{61} -3.32357e6 q^{62} -29175.0 q^{64} -2.89188e6 q^{65} -755375. q^{67} +6.46580e6 q^{68} -434177. q^{70} +383737. q^{71} +2.94854e6 q^{73} +4.03869e6 q^{74} -1.83745e6 q^{76} -133856. q^{77} -7.28812e6 q^{79} -1.18967e7 q^{80} -1.51311e7 q^{82} +1.91613e6 q^{83} -9.11764e6 q^{85} -1.62020e6 q^{86} -8.25988e6 q^{88} -1.05744e7 q^{89} +375483. q^{91} -2.86816e7 q^{92} -1.60294e7 q^{94} +2.59105e6 q^{95} +1.18419e7 q^{97} -1.67259e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8}+O(q^{10}) $ 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8} + 4667 q^{10} - 1620 q^{11} + 13550 q^{13} - 44160 q^{14} + 114026 q^{16} - 110880 q^{17} + 105058 q^{19} - 167251 q^{20} + 201504 q^{22} - 160184 q^{23} + 270149 q^{25} - 272104 q^{26} + 208172 q^{28} - 285546 q^{29} - 99616 q^{31} - 200126 q^{32} - 80941 q^{34} + 187104 q^{35} + 176038 q^{37} - 652165 q^{38} - 895387 q^{40} + 410260 q^{41} - 1033591 q^{43} + 2177076 q^{44} - 3975765 q^{46} + 424556 q^{47} - 1561359 q^{49} + 4063801 q^{50} - 4172312 q^{52} - 3992458 q^{53} + 406960 q^{55} - 1559556 q^{56} - 4052005 q^{58} - 2248836 q^{59} + 6210394 q^{61} - 885317 q^{62} - 3096318 q^{64} - 5600420 q^{65} - 1993648 q^{67} - 9327135 q^{68} - 1105098 q^{70} - 4978064 q^{71} + 8224814 q^{73} + 3613563 q^{74} + 10687121 q^{76} - 17261892 q^{77} + 6945708 q^{79} - 15822799 q^{80} - 508449 q^{82} - 22937328 q^{83} - 575532 q^{85} + 1272112 q^{86} + 11202656 q^{88} - 9291302 q^{89} + 25581108 q^{91} + 14388137 q^{92} - 24645805 q^{94} - 30750464 q^{95} + 10001852 q^{97} + 32304856 q^{98}+O(q^{100}) \) 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8 + 4667 * q^10 - 1620 * q^11 + 13550 * q^13 - 44160 * q^14 + 114026 * q^16 - 110880 * q^17 + 105058 * q^19 - 167251 * q^20 + 201504 * q^22 - 160184 * q^23 + 270149 * q^25 - 272104 * q^26 + 208172 * q^28 - 285546 * q^29 - 99616 * q^31 - 200126 * q^32 - 80941 * q^34 + 187104 * q^35 + 176038 * q^37 - 652165 * q^38 - 895387 * q^40 + 410260 * q^41 - 1033591 * q^43 + 2177076 * q^44 - 3975765 * q^46 + 424556 * q^47 - 1561359 * q^49 + 4063801 * q^50 - 4172312 * q^52 - 3992458 * q^53 + 406960 * q^55 - 1559556 * q^56 - 4052005 * q^58 - 2248836 * q^59 + 6210394 * q^61 - 885317 * q^62 - 3096318 * q^64 - 5600420 * q^65 - 1993648 * q^67 - 9327135 * q^68 - 1105098 * q^70 - 4978064 * q^71 + 8224814 * q^73 + 3613563 * q^74 + 10687121 * q^76 - 17261892 * q^77 + 6945708 * q^79 - 15822799 * q^80 - 508449 * q^82 - 22937328 * q^83 - 575532 * q^85 + 1272112 * q^86 + 11202656 * q^88 - 9291302 * q^89 + 25581108 * q^91 + 14388137 * q^92 - 24645805 * q^94 - 30750464 * q^95 + 10001852 * q^97 + 32304856 * 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 $

$2$	20.3781	1.80119	0.900594	−	0.434662i	$-0.143132\pi$
$2$	20.3781	1.80119	0.900594	+	0.434662i	$0.143132\pi$
$3$	0	0
$3$	0	0
$4$	287.267	2.24428
$5$	−405.085	−1.44928	−0.724639	−	0.689129i	$-0.757993\pi$
$5$	−405.085	−1.44928	−0.724639	+	0.689129i	$0.757993\pi$
$6$	0	0
$7$	52.5965	0.0579580	0.0289790	−	0.999580i	$-0.490774\pi$
$7$	52.5965	0.0579580	0.0289790	+	0.999580i	$0.490774\pi$
$8$	3245.57	2.24117
$9$	0	0
$10$	−8254.88	−2.61042
$11$	−2544.97	−0.576512	−0.288256	−	0.957553i	$-0.593075\pi$
$11$	−2544.97	−0.576512	−0.288256	+	0.957553i	$0.593075\pi$
$12$	0	0
$13$	7138.95	0.901224	0.450612	−	0.892720i	$-0.351206\pi$
$13$	7138.95	0.901224	0.450612	+	0.892720i	$0.351206\pi$
$14$	1071.82	0.104393
$15$	0	0
$16$	29368.3	1.79250
$17$	22507.9	1.11113	0.555565	−	0.831473i	$-0.312502\pi$
$17$	22507.9	1.11113	0.555565	+	0.831473i	$0.312502\pi$
$18$	0	0
$19$	−6396.31	−0.213940	−0.106970	−	0.994262i	$-0.534115\pi$
$19$	−6396.31	−0.213940	−0.106970	+	0.994262i	$0.534115\pi$
$20$	−116368.	−3.25258
$21$	0	0
$22$	−51861.7	−1.03841
$23$	−99842.9	−1.71108	−0.855539	−	0.517739i	$-0.826774\pi$
$23$	−99842.9	−1.71108	−0.855539	+	0.517739i	$0.826774\pi$
$24$	0	0
$25$	85969.2	1.10041
$26$	145478.	1.62327
$27$	0	0
$28$	15109.2	0.130074
$29$	−5608.53	−0.0427028	−0.0213514	−	0.999772i	$-0.506797\pi$
$29$	−5608.53	−0.0427028	−0.0213514	+	0.999772i	$0.506797\pi$
$30$	0	0
$31$	−163095.	−0.983274	−0.491637	−	0.870800i	$-0.663601\pi$
$31$	−163095.	−0.983274	−0.491637	+	0.870800i	$0.663601\pi$
$32$	183038.	0.987453
$33$	0	0
$34$	458669.	2.00135
$35$	−21306.1	−0.0839972
$36$	0	0
$37$	198188.	0.643236	0.321618	−	0.946869i	$-0.395773\pi$
$37$	198188.	0.643236	0.321618	+	0.946869i	$0.395773\pi$
$38$	−130345.	−0.385346
$39$	0	0
$40$	−1.31473e6	−3.24808
$41$	−742519.	−1.68254	−0.841268	−	0.540619i	$-0.818190\pi$
$41$	−742519.	−1.68254	−0.841268	+	0.540619i	$0.818190\pi$
$42$	0	0
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	−731087.	−1.29385
$45$	0	0
$46$	−2.03461e6	−3.08197
$47$	−786601.	−1.10513	−0.552563	−	0.833471i	$-0.686350\pi$
$47$	−786601.	−1.10513	−0.552563	+	0.833471i	$0.686350\pi$
$48$	0	0
$49$	−820777.	−0.996641
$50$	1.75189e6	1.98204
$51$	0	0
$52$	2.05079e6	2.02259
$53$	−2.08070e6	−1.91975	−0.959875	−	0.280427i	$-0.909524\pi$
$53$	−2.08070e6	−1.91975	−0.959875	+	0.280427i	$0.909524\pi$
$54$	0	0
$55$	1.03093e6	0.835526
$56$	170705.	0.129894
$57$	0	0
$58$	−114291.	−0.0769157
$59$	−607486.	−0.385083	−0.192542	−	0.981289i	$-0.561673\pi$
$59$	−607486.	−0.385083	−0.192542	+	0.981289i	$0.561673\pi$
$60$	0	0
$61$	2.59833e6	1.46568	0.732841	−	0.680400i	$-0.238194\pi$
$61$	2.59833e6	1.46568	0.732841	+	0.680400i	$0.238194\pi$
$62$	−3.32357e6	−1.77106
$63$	0	0
$64$	−29175.0	−0.0139117
$65$	−2.89188e6	−1.30612
$66$	0	0
$67$	−755375.	−0.306832	−0.153416	−	0.988162i	$-0.549027\pi$
$67$	−755375.	−0.306832	−0.153416	+	0.988162i	$0.549027\pi$
$68$	6.46580e6	2.49368
$69$	0	0
$70$	−434177.	−0.151295
$71$	383737.	0.127242	0.0636209	−	0.997974i	$-0.479735\pi$
$71$	383737.	0.127242	0.0636209	+	0.997974i	$0.479735\pi$
$72$	0	0
$73$	2.94854e6	0.887109	0.443555	−	0.896247i	$-0.353717\pi$
$73$	2.94854e6	0.887109	0.443555	+	0.896247i	$0.353717\pi$
$74$	4.03869e6	1.15859
$75$	0	0
$76$	−1.83745e6	−0.480141
$77$	−133856.	−0.0334135
$78$	0	0
$79$	−7.28812e6	−1.66311	−0.831555	−	0.555443i	$-0.812549\pi$
$79$	−7.28812e6	−1.66311	−0.831555	+	0.555443i	$0.812549\pi$
$80$	−1.18967e7	−2.59783
$81$	0	0
$82$	−1.51311e7	−3.03056
$83$	1.91613e6	0.367834	0.183917	−	0.982942i	$-0.441122\pi$
$83$	1.91613e6	0.367834	0.183917	+	0.982942i	$0.441122\pi$
$84$	0	0
$85$	−9.11764e6	−1.61034
$86$	−1.62020e6	−0.274679
$87$	0	0
$88$	−8.25988e6	−1.29206
$89$	−1.05744e7	−1.58997	−0.794986	−	0.606627i	$-0.792522\pi$
$89$	−1.05744e7	−1.58997	−0.794986	+	0.606627i	$0.792522\pi$
$90$	0	0
$91$	375483.	0.0522331
$92$	−2.86816e7	−3.84013
$93$	0	0
$94$	−1.60294e7	−1.99054
$95$	2.59105e6	0.310059
$96$	0	0
$97$	1.18419e7	1.31741	0.658705	−	0.752402i	$-0.271105\pi$
$97$	1.18419e7	1.31741	0.658705	+	0.752402i	$0.271105\pi$
$98$	−1.67259e7	−1.79514
$99$	0	0
$100$	2.46961e7	2.46961
$101$	725420.	0.0700591	0.0350296	−	0.999386i	$-0.488847\pi$
$101$	725420.	0.0700591	0.0350296	+	0.999386i	$0.488847\pi$
$102$	0	0
$103$	−3.68761e6	−0.332518	−0.166259	−	0.986082i	$-0.553169\pi$
$103$	−3.68761e6	−0.332518	−0.166259	+	0.986082i	$0.553169\pi$
$104$	2.31699e7	2.01980
$105$	0	0
$106$	−4.24008e7	−3.45783
$107$	2.07350e7	1.63629	0.818145	−	0.575012i	$-0.195003\pi$
$107$	2.07350e7	1.63629	0.818145	+	0.575012i	$0.195003\pi$
$108$	0	0
$109$	−3.76731e6	−0.278637	−0.139319	−	0.990248i	$-0.544491\pi$
$109$	−3.76731e6	−0.278637	−0.139319	+	0.990248i	$0.544491\pi$
$110$	2.10084e7	1.50494
$111$	0	0
$112$	1.54467e6	0.103890
$113$	−2.24379e6	−0.146287	−0.0731437	−	0.997321i	$-0.523303\pi$
$113$	−2.24379e6	−0.146287	−0.0731437	+	0.997321i	$0.523303\pi$
$114$	0	0
$115$	4.04449e7	2.47983
$116$	−1.61115e6	−0.0958368
$117$	0	0
$118$	−1.23794e7	−0.693607
$119$	1.18384e6	0.0643988
$120$	0	0
$121$	−1.30103e7	−0.667634
$122$	5.29491e7	2.63997
$123$	0	0
$124$	−4.68519e7	−2.20674
$125$	−3.17758e6	−0.145516
$126$	0	0
$127$	3.84940e7	1.66755	0.833777	−	0.552101i	$-0.186174\pi$
$127$	3.84940e7	1.66755	0.833777	+	0.552101i	$0.186174\pi$
$128$	−2.40234e7	−1.01251
$129$	0	0
$130$	−5.89311e7	−2.35257
$131$	1.04514e6	0.0406188	0.0203094	−	0.999794i	$-0.493535\pi$
$131$	1.04514e6	0.0406188	0.0203094	+	0.999794i	$0.493535\pi$
$132$	0	0
$133$	−336423.	−0.0123995
$134$	−1.53931e7	−0.552662
$135$	0	0
$136$	7.30511e7	2.49024
$137$	−3.36576e7	−1.11831	−0.559154	−	0.829064i	$-0.688874\pi$
$137$	−3.36576e7	−1.11831	−0.559154	+	0.829064i	$0.688874\pi$
$138$	0	0
$139$	−3.30283e7	−1.04312	−0.521559	−	0.853215i	$-0.674650\pi$
$139$	−3.30283e7	−1.04312	−0.521559	+	0.853215i	$0.674650\pi$
$140$	−6.12054e6	−0.188513
$141$	0	0
$142$	7.81984e6	0.229186
$143$	−1.81684e7	−0.519566
$144$	0	0
$145$	2.27193e6	0.0618881
$146$	6.00857e7	1.59785
$147$	0	0
$148$	5.69328e7	1.44360
$149$	−3.23900e6	−0.0802157	−0.0401079	−	0.999195i	$-0.512770\pi$
$149$	−3.23900e6	−0.0802157	−0.0401079	+	0.999195i	$0.512770\pi$
$150$	0	0
$151$	6.37837e7	1.50761	0.753807	−	0.657096i	$-0.228215\pi$
$151$	6.37837e7	1.50761	0.753807	+	0.657096i	$0.228215\pi$
$152$	−2.07597e7	−0.479477
$153$	0	0
$154$	−2.72774e6	−0.0601840
$155$	6.60674e7	1.42504
$156$	0	0
$157$	−5.75867e7	−1.18761	−0.593804	−	0.804610i	$-0.702375\pi$
$157$	−5.75867e7	−1.18761	−0.593804	+	0.804610i	$0.702375\pi$
$158$	−1.48518e8	−2.99557
$159$	0	0
$160$	−7.41460e7	−1.43109
$161$	−5.25138e6	−0.0991706
$162$	0	0
$163$	9.59822e7	1.73594	0.867969	−	0.496619i	$-0.165425\pi$
$163$	9.59822e7	1.73594	0.867969	+	0.496619i	$0.165425\pi$
$164$	−2.13302e8	−3.77607
$165$	0	0
$166$	3.90471e7	0.662538
$167$	5.45343e7	0.906070	0.453035	−	0.891493i	$-0.350341\pi$
$167$	5.45343e7	0.906070	0.453035	+	0.891493i	$0.350341\pi$
$168$	0	0
$169$	−1.17839e7	−0.187796
$170$	−1.85800e8	−2.90052
$171$	0	0
$172$	−2.28398e7	−0.342249
$173$	5.48114e6	0.0804841	0.0402420	−	0.999190i	$-0.487187\pi$
$173$	5.48114e6	0.0804841	0.0402420	+	0.999190i	$0.487187\pi$
$174$	0	0
$175$	4.52168e6	0.0637773
$176$	−7.47415e7	−1.03340
$177$	0	0
$178$	−2.15486e8	−2.86384
$179$	7.23802e7	0.943266	0.471633	−	0.881795i	$-0.343665\pi$
$179$	7.23802e7	0.943266	0.471633	+	0.881795i	$0.343665\pi$
$180$	0	0
$181$	−7.78319e7	−0.975624	−0.487812	−	0.872949i	$-0.662205\pi$
$181$	−7.78319e7	−0.975624	−0.487812	+	0.872949i	$0.662205\pi$
$182$	7.65164e6	0.0940816
$183$	0	0
$184$	−3.24047e8	−3.83482
$185$	−8.02829e7	−0.932228
$186$	0	0
$187$	−5.72821e7	−0.640580
$188$	−2.25965e8	−2.48021
$189$	0	0
$190$	5.28008e7	0.558474
$191$	−9.82338e6	−0.102010	−0.0510052	−	0.998698i	$-0.516243\pi$
$191$	−9.82338e6	−0.102010	−0.0510052	+	0.998698i	$0.516243\pi$
$192$	0	0
$193$	7.35110e7	0.736040	0.368020	−	0.929818i	$-0.380036\pi$
$193$	7.35110e7	0.736040	0.368020	+	0.929818i	$0.380036\pi$
$194$	2.41316e8	2.37290
$195$	0	0
$196$	−2.35782e8	−2.23674
$197$	3.73612e7	0.348168	0.174084	−	0.984731i	$-0.444304\pi$
$197$	3.73612e7	0.348168	0.174084	+	0.984731i	$0.444304\pi$
$198$	0	0
$199$	2.37151e7	0.213323	0.106662	−	0.994295i	$-0.465984\pi$
$199$	2.37151e7	0.213323	0.106662	+	0.994295i	$0.465984\pi$
$200$	2.79019e8	2.46620
$201$	0	0
$202$	1.47827e7	0.126190
$203$	−294989.	−0.00247497
$204$	0	0
$205$	3.00784e8	2.43846
$206$	−7.51466e7	−0.598927
$207$	0	0
$208$	2.09659e8	1.61544
$209$	1.62784e7	0.123339
$210$	0	0
$211$	9.22491e7	0.676042	0.338021	−	0.941139i	$-0.390242\pi$
$211$	9.22491e7	0.676042	0.338021	+	0.941139i	$0.390242\pi$
$212$	−5.97719e8	−4.30845
$213$	0	0
$214$	4.22540e8	2.94727
$215$	3.22071e7	0.221013
$216$	0	0
$217$	−8.57822e6	−0.0569886
$218$	−7.67707e7	−0.501877
$219$	0	0
$220$	2.96153e8	1.87515
$221$	1.60683e8	1.00138
$222$	0	0
$223$	−2.83890e8	−1.71428	−0.857141	−	0.515082i	$-0.827762\pi$
$223$	−2.83890e8	−1.71428	−0.857141	+	0.515082i	$0.827762\pi$
$224$	9.62715e6	0.0572308
$225$	0	0
$226$	−4.57241e7	−0.263491
$227$	−4.18699e7	−0.237581	−0.118790	−	0.992919i	$-0.537902\pi$
$227$	−4.18699e7	−0.237581	−0.118790	+	0.992919i	$0.537902\pi$
$228$	0	0
$229$	−8.20861e7	−0.451695	−0.225848	−	0.974163i	$-0.572515\pi$
$229$	−8.20861e7	−0.451695	−0.225848	+	0.974163i	$0.572515\pi$
$230$	8.24191e8	4.46663
$231$	0	0
$232$	−1.82029e7	−0.0957043
$233$	1.98224e8	1.02662	0.513312	−	0.858202i	$-0.328418\pi$
$233$	1.98224e8	1.02662	0.513312	+	0.858202i	$0.328418\pi$
$234$	0	0
$235$	3.18640e8	1.60163
$236$	−1.74511e8	−0.864233
$237$	0	0
$238$	2.41244e7	0.115994
$239$	4.10134e8	1.94327	0.971635	−	0.236487i	$-0.0759962\pi$
$239$	4.10134e8	1.94327	0.971635	+	0.236487i	$0.0759962\pi$
$240$	0	0
$241$	−2.36729e8	−1.08941	−0.544704	−	0.838628i	$-0.683358\pi$
$241$	−2.36729e8	−1.08941	−0.544704	+	0.838628i	$0.683358\pi$
$242$	−2.65125e8	−1.20253
$243$	0	0
$244$	7.46415e8	3.28940
$245$	3.32485e8	1.44441
$246$	0	0
$247$	−4.56629e7	−0.192808
$248$	−5.29336e8	−2.20369
$249$	0	0
$250$	−6.47530e7	−0.262102
$251$	1.19914e8	0.478644	0.239322	−	0.970940i	$-0.423075\pi$
$251$	1.19914e8	0.478644	0.239322	+	0.970940i	$0.423075\pi$
$252$	0	0
$253$	2.54097e8	0.986457
$254$	7.84435e8	3.00358
$255$	0	0
$256$	−4.85817e8	−1.80981
$257$	−4.20140e8	−1.54393	−0.771965	−	0.635664i	$-0.780726\pi$
$257$	−4.20140e8	−1.54393	−0.771965	+	0.635664i	$0.780726\pi$
$258$	0	0
$259$	1.04240e7	0.0372807
$260$	−8.30744e8	−2.93130
$261$	0	0
$262$	2.12981e7	0.0731620
$263$	2.09315e7	0.0709506	0.0354753	−	0.999371i	$-0.488705\pi$
$263$	2.09315e7	0.0709506	0.0354753	+	0.999371i	$0.488705\pi$
$264$	0	0
$265$	8.42863e8	2.78225
$266$	−6.85567e6	−0.0223339
$267$	0	0
$268$	−2.16995e8	−0.688616
$269$	−4.20833e8	−1.31819	−0.659093	−	0.752062i	$-0.729060\pi$
$269$	−4.20833e8	−1.31819	−0.659093	+	0.752062i	$0.729060\pi$
$270$	0	0
$271$	−1.41459e8	−0.431757	−0.215878	−	0.976420i	$-0.569261\pi$
$271$	−1.41459e8	−0.431757	−0.215878	+	0.976420i	$0.569261\pi$
$272$	6.61020e8	1.99170
$273$	0	0
$274$	−6.85879e8	−2.01428
$275$	−2.18789e8	−0.634398
$276$	0	0
$277$	−4.46880e8	−1.26332	−0.631658	−	0.775248i	$-0.717625\pi$
$277$	−4.46880e8	−1.26332	−0.631658	+	0.775248i	$0.717625\pi$
$278$	−6.73053e8	−1.87885
$279$	0	0
$280$	−6.91503e7	−0.188252
$281$	−4.25457e8	−1.14389	−0.571944	−	0.820293i	$-0.693811\pi$
$281$	−4.25457e8	−1.14389	−0.571944	+	0.820293i	$0.693811\pi$
$282$	0	0
$283$	−3.36158e7	−0.0881638	−0.0440819	−	0.999028i	$-0.514036\pi$
$283$	−3.36158e7	−0.0881638	−0.0440819	+	0.999028i	$0.514036\pi$
$284$	1.10235e8	0.285566
$285$	0	0
$286$	−3.70238e8	−0.935837
$287$	−3.90539e7	−0.0975164
$288$	0	0
$289$	9.62690e7	0.234609
$290$	4.62977e7	0.111472
$291$	0	0
$292$	8.47019e8	1.99092
$293$	4.64940e8	1.07984	0.539921	−	0.841715i	$-0.318454\pi$
$293$	4.64940e8	1.07984	0.539921	+	0.841715i	$0.318454\pi$
$294$	0	0
$295$	2.46084e8	0.558092
$296$	6.43231e8	1.44160
$297$	0	0
$298$	−6.60048e7	−0.144484
$299$	−7.12773e8	−1.54206
$300$	0	0
$301$	−4.18179e6	−0.00883851
$302$	1.29979e9	2.71550
$303$	0	0
$304$	−1.87849e8	−0.383487
$305$	−1.05255e9	−2.12418
$306$	0	0
$307$	−9.90322e8	−1.95340	−0.976702	−	0.214600i	$-0.931155\pi$
$307$	−9.90322e8	−1.95340	−0.976702	+	0.214600i	$0.931155\pi$
$308$	−3.84526e7	−0.0749891
$309$	0	0
$310$	1.34633e9	2.56676
$311$	1.57593e8	0.297081	0.148540	−	0.988906i	$-0.452543\pi$
$311$	1.57593e8	0.297081	0.148540	+	0.988906i	$0.452543\pi$
$312$	0	0
$313$	8.72665e8	1.60858	0.804290	−	0.594237i	$-0.202546\pi$
$313$	8.72665e8	1.60858	0.804290	+	0.594237i	$0.202546\pi$
$314$	−1.17351e9	−2.13911
$315$	0	0
$316$	−2.09364e9	−3.73248
$317$	−1.63595e8	−0.288444	−0.144222	−	0.989545i	$-0.546068\pi$
$317$	−1.63595e8	−0.288444	−0.144222	+	0.989545i	$0.546068\pi$
$318$	0	0
$319$	1.42735e7	0.0246187
$320$	1.18184e7	0.0201620
$321$	0	0
$322$	−1.07013e8	−0.178625
$323$	−1.43968e8	−0.237715
$324$	0	0
$325$	6.13730e8	0.991712
$326$	1.95594e9	3.12675
$327$	0	0
$328$	−2.40990e9	−3.77086
$329$	−4.13724e7	−0.0640509
$330$	0	0
$331$	6.41815e8	0.972773	0.486387	−	0.873744i	$-0.338315\pi$
$331$	6.41815e8	0.972773	0.486387	+	0.873744i	$0.338315\pi$
$332$	5.50442e8	0.825521
$333$	0	0
$334$	1.11131e9	1.63200
$335$	3.05991e8	0.444685
$336$	0	0
$337$	−2.16017e8	−0.307457	−0.153728	−	0.988113i	$-0.549128\pi$
$337$	−2.16017e8	−0.307457	−0.153728	+	0.988113i	$0.549128\pi$
$338$	−2.40134e8	−0.338256
$339$	0	0
$340$	−2.61920e9	−3.61404
$341$	4.15072e8	0.566870
$342$	0	0
$343$	−8.64854e7	−0.115721
$344$	−2.58045e8	−0.341776
$345$	0	0
$346$	1.11695e8	0.144967
$347$	1.18322e9	1.52024	0.760118	−	0.649785i	$-0.225141\pi$
$347$	1.18322e9	1.52024	0.760118	+	0.649785i	$0.225141\pi$
$348$	0	0
$349$	−4.71635e8	−0.593905	−0.296953	−	0.954892i	$-0.595970\pi$
$349$	−4.71635e8	−0.593905	−0.296953	+	0.954892i	$0.595970\pi$
$350$	9.21432e7	0.114875
$351$	0	0
$352$	−4.65827e8	−0.569279
$353$	2.14674e8	0.259757	0.129879	−	0.991530i	$-0.458541\pi$
$353$	2.14674e8	0.259757	0.129879	+	0.991530i	$0.458541\pi$
$354$	0	0
$355$	−1.55446e8	−0.184409
$356$	−3.03767e9	−3.56834
$357$	0	0
$358$	1.47497e9	1.69900
$359$	−5.07607e8	−0.579024	−0.289512	−	0.957174i	$-0.593493\pi$
$359$	−5.07607e8	−0.579024	−0.289512	+	0.957174i	$0.593493\pi$
$360$	0	0
$361$	−8.52959e8	−0.954230
$362$	−1.58607e9	−1.75728
$363$	0	0
$364$	1.07864e8	0.117226
$365$	−1.19441e9	−1.28567
$366$	0	0
$367$	−7.17068e8	−0.757232	−0.378616	−	0.925554i	$-0.623600\pi$
$367$	−7.17068e8	−0.757232	−0.378616	+	0.925554i	$0.623600\pi$
$368$	−2.93222e9	−3.06711
$369$	0	0
$370$	−1.63601e9	−1.67912
$371$	−1.09438e8	−0.111265
$372$	0	0
$373$	1.47530e9	1.47197	0.735985	−	0.676998i	$-0.236720\pi$
$373$	1.47530e9	1.47197	0.735985	+	0.676998i	$0.236720\pi$
$374$	−1.16730e9	−1.15380
$375$	0	0
$376$	−2.55297e9	−2.47678
$377$	−4.00390e7	−0.0384847
$378$	0	0
$379$	1.57868e9	1.48956	0.744778	−	0.667313i	$-0.232556\pi$
$379$	1.57868e9	1.48956	0.744778	+	0.667313i	$0.232556\pi$
$380$	7.44325e8	0.695857
$381$	0	0
$382$	−2.00182e8	−0.183740
$383$	−6.74713e8	−0.613653	−0.306827	−	0.951765i	$-0.599267\pi$
$383$	−6.74713e8	−0.613653	−0.306827	+	0.951765i	$0.599267\pi$
$384$	0	0
$385$	5.42233e7	0.0484254
$386$	1.49802e9	1.32575
$387$	0	0
$388$	3.40179e9	2.95663
$389$	5.60683e8	0.482941	0.241470	−	0.970408i	$-0.422370\pi$
$389$	5.60683e8	0.482941	0.241470	+	0.970408i	$0.422370\pi$
$390$	0	0
$391$	−2.24726e9	−1.90123
$392$	−2.66389e9	−2.23365
$393$	0	0
$394$	7.61350e8	0.627116
$395$	2.95231e9	2.41031
$396$	0	0
$397$	3.50283e8	0.280965	0.140482	−	0.990083i	$-0.455135\pi$
$397$	3.50283e8	0.280965	0.140482	+	0.990083i	$0.455135\pi$
$398$	4.83268e8	0.384236
$399$	0	0
$400$	2.52477e9	1.97248
$401$	−1.19625e9	−0.926440	−0.463220	−	0.886243i	$-0.653306\pi$
$401$	−1.19625e9	−0.926440	−0.463220	+	0.886243i	$0.653306\pi$
$402$	0	0
$403$	−1.16433e9	−0.886150
$404$	2.08389e8	0.157232
$405$	0	0
$406$	−6.01131e6	−0.00445788
$407$	−5.04382e8	−0.370834
$408$	0	0
$409$	1.57416e9	1.13767	0.568836	−	0.822451i	$-0.307394\pi$
$409$	1.57416e9	1.13767	0.568836	+	0.822451i	$0.307394\pi$
$410$	6.12940e9	4.39213
$411$	0	0
$412$	−1.05933e9	−0.746262
$413$	−3.19516e7	−0.0223186
$414$	0	0
$415$	−7.76197e8	−0.533094
$416$	1.30670e9	0.889916
$417$	0	0
$418$	3.31724e8	0.222157
$419$	−2.03488e9	−1.35142	−0.675710	−	0.737167i	$-0.736163\pi$
$419$	−2.03488e9	−1.35142	−0.675710	+	0.737167i	$0.736163\pi$
$420$	0	0
$421$	−1.11312e9	−0.727032	−0.363516	−	0.931588i	$-0.618424\pi$
$421$	−1.11312e9	−0.727032	−0.363516	+	0.931588i	$0.618424\pi$
$422$	1.87986e9	1.21768
$423$	0	0
$424$	−6.75307e9	−4.30250
$425$	1.93499e9	1.22269
$426$	0	0
$427$	1.36663e8	0.0849480
$428$	5.95648e9	3.67229
$429$	0	0
$430$	6.56320e8	0.398085
$431$	2.21785e9	1.33432	0.667161	−	0.744913i	$-0.267509\pi$
$431$	2.21785e9	1.33432	0.667161	+	0.744913i	$0.267509\pi$
$432$	0	0
$433$	1.61665e9	0.956992	0.478496	−	0.878090i	$-0.341182\pi$
$433$	1.61665e9	0.956992	0.478496	+	0.878090i	$0.341182\pi$
$434$	−1.74808e8	−0.102647
$435$	0	0
$436$	−1.08223e9	−0.625338
$437$	6.38626e8	0.366068
$438$	0	0
$439$	−2.27405e9	−1.28285	−0.641424	−	0.767187i	$-0.721656\pi$
$439$	−2.27405e9	−1.28285	−0.641424	+	0.767187i	$0.721656\pi$
$440$	3.34596e9	1.87256
$441$	0	0
$442$	3.27442e9	1.80367
$443$	1.06718e9	0.583209	0.291605	−	0.956539i	$-0.405811\pi$
$443$	1.06718e9	0.583209	0.291605	+	0.956539i	$0.405811\pi$
$444$	0	0
$445$	4.28353e9	2.30431
$446$	−5.78513e9	−3.08774
$447$	0	0
$448$	−1.53450e6	−0.000806296 0
$449$	1.72761e9	0.900707	0.450354	−	0.892850i	$-0.351298\pi$
$449$	1.72761e9	0.900707	0.450354	+	0.892850i	$0.351298\pi$
$450$	0	0
$451$	1.88969e9	0.970002
$452$	−6.44567e8	−0.328309
$453$	0	0
$454$	−8.53229e8	−0.427927
$455$	−1.52103e8	−0.0757003
$456$	0	0
$457$	3.85685e8	0.189028	0.0945139	−	0.995524i	$-0.469870\pi$
$457$	3.85685e8	0.189028	0.0945139	+	0.995524i	$0.469870\pi$
$458$	−1.67276e9	−0.813587
$459$	0	0
$460$	1.16185e10	5.56542
$461$	2.53341e9	1.20435	0.602174	−	0.798365i	$-0.294301\pi$
$461$	2.53341e9	1.20435	0.602174	+	0.798365i	$0.294301\pi$
$462$	0	0
$463$	−5.21077e8	−0.243988	−0.121994	−	0.992531i	$-0.538929\pi$
$463$	−5.21077e8	−0.243988	−0.121994	+	0.992531i	$0.538929\pi$
$464$	−1.64713e8	−0.0765447
$465$	0	0
$466$	4.03944e9	1.84914
$467$	1.97610e9	0.897842	0.448921	−	0.893571i	$-0.351808\pi$
$467$	1.97610e9	0.897842	0.448921	+	0.893571i	$0.351808\pi$
$468$	0	0
$469$	−3.97300e7	−0.0177834
$470$	6.49329e9	2.88484
$471$	0	0
$472$	−1.97164e9	−0.863039
$473$	2.02343e8	0.0879173
$474$	0	0
$475$	−5.49886e8	−0.235421
$476$	3.40078e8	0.144529
$477$	0	0
$478$	8.35775e9	3.50019
$479$	1.86520e9	0.775444	0.387722	−	0.921776i	$-0.373262\pi$
$479$	1.86520e9	0.775444	0.387722	+	0.921776i	$0.373262\pi$
$480$	0	0
$481$	1.41485e9	0.579700
$482$	−4.82408e9	−1.96223
$483$	0	0
$484$	−3.73743e9	−1.49835
$485$	−4.79699e9	−1.90929
$486$	0	0
$487$	2.65232e9	1.04058	0.520288	−	0.853991i	$-0.325824\pi$
$487$	2.65232e9	1.04058	0.520288	+	0.853991i	$0.325824\pi$
$488$	8.43306e9	3.28485
$489$	0	0
$490$	6.77541e9	2.60165
$491$	−3.14043e9	−1.19730	−0.598651	−	0.801010i	$-0.704297\pi$
$491$	−3.14043e9	−1.19730	−0.598651	+	0.801010i	$0.704297\pi$
$492$	0	0
$493$	−1.26236e8	−0.0474483
$494$	−9.30525e8	−0.347283
$495$	0	0
$496$	−4.78982e9	−1.76252
$497$	2.01832e7	0.00737468
$498$	0	0
$499$	−1.74081e8	−0.0627191	−0.0313596	−	0.999508i	$-0.509984\pi$
$499$	−1.74081e8	−0.0627191	−0.0313596	+	0.999508i	$0.509984\pi$
$500$	−9.12814e8	−0.326578
$501$	0	0
$502$	2.44363e9	0.862128
$503$	−7.08807e8	−0.248336	−0.124168	−	0.992261i	$-0.539626\pi$
$503$	−7.08807e8	−0.248336	−0.124168	+	0.992261i	$0.539626\pi$
$504$	0	0
$505$	−2.93857e8	−0.101535
$506$	5.17802e9	1.77679
$507$	0	0
$508$	1.10581e10	3.74245
$509$	−5.15128e9	−1.73142	−0.865711	−	0.500545i	$-0.833133\pi$
$509$	−5.15128e9	−1.73142	−0.865711	+	0.500545i	$0.833133\pi$
$510$	0	0
$511$	1.55083e8	0.0514151
$512$	−6.82504e9	−2.24730
$513$	0	0
$514$	−8.56165e9	−2.78091
$515$	1.49380e9	0.481911
$516$	0	0
$517$	2.00188e9	0.637119
$518$	2.12421e8	0.0671495
$519$	0	0
$520$	−9.38581e9	−2.92725
$521$	3.29977e9	1.02224	0.511118	−	0.859510i	$-0.329231\pi$
$521$	3.29977e9	1.02224	0.511118	+	0.859510i	$0.329231\pi$
$522$	0	0
$523$	4.25818e9	1.30157	0.650786	−	0.759261i	$-0.274440\pi$
$523$	4.25818e9	1.30157	0.650786	+	0.759261i	$0.274440\pi$
$524$	3.00236e8	0.0911598
$525$	0	0
$526$	4.26545e8	0.127795
$527$	−3.67093e9	−1.09255
$528$	0	0
$529$	6.56378e9	1.92779
$530$	1.71760e10	5.01136
$531$	0	0
$532$	−9.66435e7	−0.0278280
$533$	−5.30081e9	−1.51634
$534$	0	0
$535$	−8.39944e9	−2.37144
$536$	−2.45162e9	−0.687664
$537$	0	0
$538$	−8.57578e9	−2.37430
$539$	2.08885e9	0.574576
$540$	0	0
$541$	−7.50896e8	−0.203887	−0.101943	−	0.994790i	$-0.532506\pi$
$541$	−7.50896e8	−0.203887	−0.101943	+	0.994790i	$0.532506\pi$
$542$	−2.88268e9	−0.777675
$543$	0	0
$544$	4.11981e9	1.09719
$545$	1.52608e9	0.403822
$546$	0	0
$547$	−6.03569e7	−0.0157678	−0.00788390	−	0.999969i	$-0.502510\pi$
$547$	−6.03569e7	−0.0157678	−0.00788390	+	0.999969i	$0.502510\pi$
$548$	−9.66874e9	−2.50979
$549$	0	0
$550$	−4.45851e9	−1.14267
$551$	3.58739e7	0.00913583
$552$	0	0
$553$	−3.83329e8	−0.0963905
$554$	−9.10657e9	−2.27547
$555$	0	0
$556$	−9.48794e9	−2.34105
$557$	−1.40575e9	−0.344678	−0.172339	−	0.985038i	$-0.555133\pi$
$557$	−1.40575e9	−0.344678	−0.172339	+	0.985038i	$0.555133\pi$
$558$	0	0
$559$	−5.67596e8	−0.137435
$560$	−6.25723e8	−0.150565
$561$	0	0
$562$	−8.67001e9	−2.06036
$563$	5.50487e9	1.30007	0.650036	−	0.759903i	$-0.274754\pi$
$563$	5.50487e9	1.30007	0.650036	+	0.759903i	$0.274754\pi$
$564$	0	0
$565$	9.08926e8	0.212011
$566$	−6.85026e8	−0.158800
$567$	0	0
$568$	1.24544e9	0.285171
$569$	−1.17013e9	−0.266283	−0.133141	−	0.991097i	$-0.542506\pi$
$569$	−1.17013e9	−0.266283	−0.133141	+	0.991097i	$0.542506\pi$
$570$	0	0
$571$	−1.88234e9	−0.423127	−0.211563	−	0.977364i	$-0.567855\pi$
$571$	−1.88234e9	−0.423127	−0.211563	+	0.977364i	$0.567855\pi$
$572$	−5.21919e9	−1.16605
$573$	0	0
$574$	−7.95844e8	−0.175645
$575$	−8.58342e9	−1.88288
$576$	0	0
$577$	−6.06026e9	−1.31334	−0.656668	−	0.754180i	$-0.728035\pi$
$577$	−6.06026e9	−1.31334	−0.656668	+	0.754180i	$0.728035\pi$
$578$	1.96178e9	0.422574
$579$	0	0
$580$	6.52652e8	0.138894
$581$	1.00782e8	0.0213189
$582$	0	0
$583$	5.29533e9	1.10676
$584$	9.56969e9	1.98817
$585$	0	0
$586$	9.47460e9	1.94500
$587$	−7.70428e9	−1.57217	−0.786083	−	0.618120i	$-0.787894\pi$
$587$	−7.70428e9	−1.57217	−0.786083	+	0.618120i	$0.787894\pi$
$588$	0	0
$589$	1.04321e9	0.210362
$590$	5.01472e9	1.00523
$591$	0	0
$592$	5.82044e9	1.15300
$593$	8.40982e9	1.65613	0.828067	−	0.560629i	$-0.189441\pi$
$593$	8.40982e9	1.65613	0.828067	+	0.560629i	$0.189441\pi$
$594$	0	0
$595$	−4.79556e8	−0.0933318
$596$	−9.30460e8	−0.180026
$597$	0	0
$598$	−1.45250e10	−2.77755
$599$	−2.66302e9	−0.506267	−0.253134	−	0.967431i	$-0.581461\pi$
$599$	−2.66302e9	−0.506267	−0.253134	+	0.967431i	$0.581461\pi$
$600$	0	0
$601$	5.05119e9	0.949146	0.474573	−	0.880216i	$-0.342602\pi$
$601$	5.05119e9	0.949146	0.474573	+	0.880216i	$0.342602\pi$
$602$	−8.52169e7	−0.0159198
$603$	0	0
$604$	1.83230e10	3.38350
$605$	5.27028e9	0.967587
$606$	0	0
$607$	1.98881e9	0.360938	0.180469	−	0.983581i	$-0.442238\pi$
$607$	1.98881e9	0.360938	0.180469	+	0.983581i	$0.442238\pi$
$608$	−1.17077e9	−0.211256
$609$	0	0
$610$	−2.14489e10	−3.82605
$611$	−5.61550e9	−0.995966
$612$	0	0
$613$	3.81895e9	0.669626	0.334813	−	0.942285i	$-0.391327\pi$
$613$	3.81895e9	0.669626	0.334813	+	0.942285i	$0.391327\pi$
$614$	−2.01809e10	−3.51845
$615$	0	0
$616$	−4.34440e8	−0.0748855
$617$	8.24101e9	1.41248	0.706241	−	0.707972i	$-0.250390\pi$
$617$	8.24101e9	1.41248	0.706241	+	0.707972i	$0.250390\pi$
$618$	0	0
$619$	2.93974e9	0.498185	0.249093	−	0.968480i	$-0.419868\pi$
$619$	2.93974e9	0.498185	0.249093	+	0.968480i	$0.419868\pi$
$620$	1.89790e10	3.19818
$621$	0	0
$622$	3.21144e9	0.535098
$623$	−5.56175e8	−0.0921516
$624$	0	0
$625$	−5.42915e9	−0.889513
$626$	1.77833e10	2.89735
$627$	0	0
$628$	−1.65428e10	−2.66532
$629$	4.46080e9	0.714719
$630$	0	0
$631$	−5.09030e9	−0.806567	−0.403283	−	0.915075i	$-0.632131\pi$
$631$	−5.09030e9	−0.806567	−0.403283	+	0.915075i	$0.632131\pi$
$632$	−2.36541e10	−3.72732
$633$	0	0
$634$	−3.33375e9	−0.519542
$635$	−1.55934e10	−2.41675
$636$	0	0
$637$	−5.85948e9	−0.898196
$638$	2.90868e8	0.0443428
$639$	0	0
$640$	9.73153e9	1.46741
$641$	1.31257e9	0.196843	0.0984216	−	0.995145i	$-0.468621\pi$
$641$	1.31257e9	0.196843	0.0984216	+	0.995145i	$0.468621\pi$
$642$	0	0
$643$	−6.98129e9	−1.03561	−0.517806	−	0.855498i	$-0.673251\pi$
$643$	−6.98129e9	−1.03561	−0.517806	+	0.855498i	$0.673251\pi$
$644$	−1.50855e9	−0.222566
$645$	0	0
$646$	−2.93379e9	−0.428169
$647$	4.67412e9	0.678476	0.339238	−	0.940700i	$-0.389831\pi$
$647$	4.67412e9	0.678476	0.339238	+	0.940700i	$0.389831\pi$
$648$	0	0
$649$	1.54604e9	0.222005
$650$	1.25067e10	1.78626
$651$	0	0
$652$	2.75725e10	3.89592
$653$	2.18418e9	0.306968	0.153484	−	0.988151i	$-0.450951\pi$
$653$	2.18418e9	0.306968	0.153484	+	0.988151i	$0.450951\pi$
$654$	0	0
$655$	−4.23373e8	−0.0588679
$656$	−2.18065e10	−3.01594
$657$	0	0
$658$	−8.43091e8	−0.115368
$659$	−3.50457e9	−0.477019	−0.238510	−	0.971140i	$-0.576659\pi$
$659$	−3.50457e9	−0.477019	−0.238510	+	0.971140i	$0.576659\pi$
$660$	0	0
$661$	−8.46214e9	−1.13966	−0.569830	−	0.821763i	$-0.692991\pi$
$661$	−8.46214e9	−1.13966	−0.569830	+	0.821763i	$0.692991\pi$
$662$	1.30790e10	1.75215
$663$	0	0
$664$	6.21893e9	0.824380
$665$	1.36280e8	0.0179704
$666$	0	0
$667$	5.59972e8	0.0730677
$668$	1.56659e10	2.03347
$669$	0	0
$670$	6.23552e9	0.800961
$671$	−6.61268e9	−0.844984
$672$	0	0
$673$	5.78313e9	0.731324	0.365662	−	0.930748i	$-0.380843\pi$
$673$	5.78313e9	0.731324	0.365662	+	0.930748i	$0.380843\pi$
$674$	−4.40203e9	−0.553787
$675$	0	0
$676$	−3.38514e9	−0.421466
$677$	2.36319e9	0.292710	0.146355	−	0.989232i	$-0.453246\pi$
$677$	2.36319e9	0.292710	0.146355	+	0.989232i	$0.453246\pi$
$678$	0	0
$679$	6.22843e8	0.0763544
$680$	−2.95919e10	−3.60904
$681$	0	0
$682$	8.45839e9	1.02104
$683$	5.06343e9	0.608096	0.304048	−	0.952657i	$-0.401662\pi$
$683$	5.06343e9	0.608096	0.304048	+	0.952657i	$0.401662\pi$
$684$	0	0
$685$	1.36342e10	1.62074
$686$	−1.76241e9	−0.208436
$687$	0	0
$688$	−2.33499e9	−0.273354
$689$	−1.48540e10	−1.73012
$690$	0	0
$691$	−2.24876e8	−0.0259281	−0.0129640	−	0.999916i	$-0.504127\pi$
$691$	−2.24876e8	−0.0259281	−0.0129640	+	0.999916i	$0.504127\pi$
$692$	1.57455e9	0.180628
$693$	0	0
$694$	2.41117e10	2.73823
$695$	1.33793e10	1.51177
$696$	0	0
$697$	−1.67126e10	−1.86951
$698$	−9.61103e9	−1.06973
$699$	0	0
$700$	1.29893e9	0.143134
$701$	−7.33460e9	−0.804198	−0.402099	−	0.915596i	$-0.631719\pi$
$701$	−7.33460e9	−0.804198	−0.402099	+	0.915596i	$0.631719\pi$
$702$	0	0
$703$	−1.26767e9	−0.137614
$704$	7.42496e7	0.00802028
$705$	0	0
$706$	4.37465e9	0.467871
$707$	3.81545e7	0.00406049
$708$	0	0
$709$	9.29450e8	0.0979409	0.0489704	−	0.998800i	$-0.484406\pi$
$709$	9.29450e8	0.0979409	0.0489704	+	0.998800i	$0.484406\pi$
$710$	−3.16770e9	−0.332155
$711$	0	0
$712$	−3.43199e10	−3.56341
$713$	1.62839e10	1.68246
$714$	0	0
$715$	7.35976e9	0.752996
$716$	2.07925e10	2.11695
$717$	0	0
$718$	−1.03441e10	−1.04293
$719$	−1.46064e10	−1.46552	−0.732759	−	0.680489i	$-0.761768\pi$
$719$	−1.46064e10	−1.46552	−0.732759	+	0.680489i	$0.761768\pi$
$720$	0	0
$721$	−1.93955e8	−0.0192721
$722$	−1.73817e10	−1.71875
$723$	0	0
$724$	−2.23586e10	−2.18957
$725$	−4.82161e8	−0.0469904
$726$	0	0
$727$	6.09535e9	0.588339	0.294170	−	0.955753i	$-0.404957\pi$
$727$	6.09535e9	0.588339	0.294170	+	0.955753i	$0.404957\pi$
$728$	1.21866e9	0.117064
$729$	0	0
$730$	−2.43398e10	−2.31573
$731$	−1.78954e9	−0.169446
$732$	0	0
$733$	−5.59722e9	−0.524939	−0.262469	−	0.964940i	$-0.584537\pi$
$733$	−5.59722e9	−0.524939	−0.262469	+	0.964940i	$0.584537\pi$
$734$	−1.46125e10	−1.36392
$735$	0	0
$736$	−1.82750e10	−1.68961
$737$	1.92241e9	0.176892
$738$	0	0
$739$	−8.34763e9	−0.760865	−0.380433	−	0.924809i	$-0.624225\pi$
$739$	−8.34763e9	−0.760865	−0.380433	+	0.924809i	$0.624225\pi$
$740$	−2.30627e10	−2.09218
$741$	0	0
$742$	−2.23013e9	−0.200409
$743$	1.62616e10	1.45446	0.727230	−	0.686393i	$-0.240807\pi$
$743$	1.62616e10	1.45446	0.727230	+	0.686393i	$0.240807\pi$
$744$	0	0
$745$	1.31207e9	0.116255
$746$	3.00638e10	2.65129
$747$	0	0
$748$	−1.64553e10	−1.43764
$749$	1.09059e9	0.0948361
$750$	0	0
$751$	−1.06354e10	−0.916253	−0.458127	−	0.888887i	$-0.651479\pi$
$751$	−1.06354e10	−0.916253	−0.458127	+	0.888887i	$0.651479\pi$
$752$	−2.31011e10	−1.98094
$753$	0	0
$754$	−8.15919e8	−0.0693182
$755$	−2.58378e10	−2.18495
$756$	0	0
$757$	−1.33873e10	−1.12165	−0.560826	−	0.827934i	$-0.689516\pi$
$757$	−1.33873e10	−1.12165	−0.560826	+	0.827934i	$0.689516\pi$
$758$	3.21705e10	2.68297
$759$	0	0
$760$	8.40944e9	0.694895
$761$	3.98898e9	0.328107	0.164054	−	0.986451i	$-0.447543\pi$
$761$	3.98898e9	0.328107	0.164054	+	0.986451i	$0.447543\pi$
$762$	0	0
$763$	−1.98147e8	−0.0161492
$764$	−2.82194e9	−0.228939
$765$	0	0
$766$	−1.37494e10	−1.10530
$767$	−4.33681e9	−0.347046
$768$	0	0
$769$	−1.33758e10	−1.06067	−0.530333	−	0.847789i	$-0.677933\pi$
$769$	−1.33758e10	−1.06067	−0.530333	+	0.847789i	$0.677933\pi$
$770$	1.10497e9	0.0872233
$771$	0	0
$772$	2.11173e10	1.65188
$773$	−1.88395e10	−1.46704	−0.733518	−	0.679670i	$-0.762123\pi$
$773$	−1.88395e10	−1.46704	−0.733518	+	0.679670i	$0.762123\pi$
$774$	0	0
$775$	−1.40211e10	−1.08200
$776$	3.84337e10	2.95254
$777$	0	0
$778$	1.14257e10	0.869867
$779$	4.74938e9	0.359962
$780$	0	0
$781$	−9.76600e8	−0.0733564
$782$	−4.57949e10	−3.42447
$783$	0	0
$784$	−2.41048e10	−1.78648
$785$	2.33275e10	1.72117
$786$	0	0
$787$	1.86675e8	0.0136513	0.00682566	−	0.999977i	$-0.497827\pi$
$787$	1.86675e8	0.0136513	0.00682566	+	0.999977i	$0.497827\pi$
$788$	1.07326e10	0.781385
$789$	0	0
$790$	6.01625e10	4.34142
$791$	−1.18015e8	−0.00847853
$792$	0	0
$793$	1.85493e10	1.32091
$794$	7.13810e9	0.506070
$795$	0	0
$796$	6.81257e9	0.478757
$797$	1.16617e10	0.815938	0.407969	−	0.912996i	$-0.366237\pi$
$797$	1.16617e10	0.815938	0.407969	+	0.912996i	$0.366237\pi$
$798$	0	0
$799$	−1.77048e10	−1.22794
$800$	1.57356e10	1.08660
$801$	0	0
$802$	−2.43773e10	−1.66869
$803$	−7.50395e9	−0.511429
$804$	0	0
$805$	2.12726e9	0.143726
$806$	−2.37268e10	−1.59612
$807$	0	0
$808$	2.35440e9	0.157015
$809$	−2.35019e10	−1.56057	−0.780286	−	0.625423i	$-0.784927\pi$
$809$	−2.35019e10	−1.56057	−0.780286	+	0.625423i	$0.784927\pi$
$810$	0	0
$811$	−2.42645e10	−1.59734	−0.798670	−	0.601769i	$-0.794463\pi$
$811$	−2.42645e10	−1.59734	−0.798670	+	0.601769i	$0.794463\pi$
$812$	−8.47406e7	−0.00555451
$813$	0	0
$814$	−1.02783e10	−0.667941
$815$	−3.88810e10	−2.51586
$816$	0	0
$817$	5.08552e8	0.0326255
$818$	3.20784e10	2.04916
$819$	0	0
$820$	8.64053e10	5.47258
$821$	−2.94003e10	−1.85417	−0.927087	−	0.374847i	$-0.877695\pi$
$821$	−2.94003e10	−1.85417	−0.927087	+	0.374847i	$0.877695\pi$
$822$	0	0
$823$	1.83212e10	1.14566	0.572829	−	0.819675i	$-0.305846\pi$
$823$	1.83212e10	1.14566	0.572829	+	0.819675i	$0.305846\pi$
$824$	−1.19684e10	−0.745231
$825$	0	0
$826$	−6.51114e8	−0.0402001
$827$	2.71066e10	1.66650	0.833251	−	0.552896i	$-0.186477\pi$
$827$	2.71066e10	1.66650	0.833251	+	0.552896i	$0.186477\pi$
$828$	0	0
$829$	−9.94304e9	−0.606147	−0.303074	−	0.952967i	$-0.598013\pi$
$829$	−9.94304e9	−0.606147	−0.303074	+	0.952967i	$0.598013\pi$
$830$	−1.58174e10	−0.960201
$831$	0	0
$832$	−2.08279e8	−0.0125376
$833$	−1.84740e10	−1.10740
$834$	0	0
$835$	−2.20910e10	−1.31315
$836$	4.67626e9	0.276807
$837$	0	0
$838$	−4.14671e10	−2.43416
$839$	2.24029e10	1.30960	0.654798	−	0.755804i	$-0.272754\pi$
$839$	2.24029e10	1.30960	0.654798	+	0.755804i	$0.272754\pi$
$840$	0	0
$841$	−1.72184e10	−0.998176
$842$	−2.26832e10	−1.30952
$843$	0	0
$844$	2.65002e10	1.51723
$845$	4.77350e9	0.272169
$846$	0	0
$847$	−6.84295e8	−0.0386947
$848$	−6.11068e10	−3.44115
$849$	0	0
$850$	3.94314e10	2.20230
$851$	−1.97876e10	−1.10063
$852$	0	0
$853$	5.08263e9	0.280393	0.140197	−	0.990124i	$-0.455227\pi$
$853$	5.08263e9	0.280393	0.140197	+	0.990124i	$0.455227\pi$
$854$	2.78493e9	0.153007
$855$	0	0
$856$	6.72968e10	3.66721
$857$	−1.75834e10	−0.954267	−0.477134	−	0.878831i	$-0.658324\pi$
$857$	−1.75834e10	−0.954267	−0.477134	+	0.878831i	$0.658324\pi$
$858$	0	0
$859$	3.92669e7	0.00211374	0.00105687	−	0.999999i	$-0.499664\pi$
$859$	3.92669e7	0.00211374	0.00105687	+	0.999999i	$0.499664\pi$
$860$	9.25206e9	0.496014
$861$	0	0
$862$	4.51955e10	2.40337
$863$	−2.83133e10	−1.49952	−0.749761	−	0.661709i	$-0.769832\pi$
$863$	−2.83133e10	−1.49952	−0.749761	+	0.661709i	$0.769832\pi$
$864$	0	0
$865$	−2.22033e9	−0.116644
$866$	3.29443e10	1.72372
$867$	0	0
$868$	−2.46424e9	−0.127898
$869$	1.85481e10	0.958803
$870$	0	0
$871$	−5.39258e9	−0.276524
$872$	−1.22271e10	−0.624474
$873$	0	0
$874$	1.30140e10	0.659357
$875$	−1.67129e8	−0.00843382
$876$	0	0
$877$	−2.63453e10	−1.31888	−0.659440	−	0.751757i	$-0.729207\pi$
$877$	−2.63453e10	−1.31888	−0.659440	+	0.751757i	$0.729207\pi$
$878$	−4.63409e10	−2.31065
$879$	0	0
$880$	3.02767e10	1.49768
$881$	7.36199e9	0.362727	0.181363	−	0.983416i	$-0.441949\pi$
$881$	7.36199e9	0.362727	0.181363	+	0.983416i	$0.441949\pi$
$882$	0	0
$883$	2.45713e10	1.20106	0.600531	−	0.799601i	$-0.294956\pi$
$883$	2.45713e10	1.20106	0.600531	+	0.799601i	$0.294956\pi$
$884$	4.61590e10	2.24736
$885$	0	0
$886$	2.17471e10	1.05047
$887$	−3.76588e9	−0.181190	−0.0905950	−	0.995888i	$-0.528877\pi$
$887$	−3.76588e9	−0.181190	−0.0905950	+	0.995888i	$0.528877\pi$
$888$	0	0
$889$	2.02465e9	0.0966481
$890$	8.72902e10	4.15050
$891$	0	0
$892$	−8.15522e10	−3.84732
$893$	5.03134e9	0.236431
$894$	0	0
$895$	−2.93202e10	−1.36705
$896$	−1.26355e9	−0.0586831
$897$	0	0
$898$	3.52054e10	1.62234
$899$	9.14723e8	0.0419885
$900$	0	0
$901$	−4.68324e10	−2.13309
$902$	3.85083e10	1.74716
$903$	0	0
$904$	−7.28236e9	−0.327856
$905$	3.15286e10	1.41395
$906$	0	0
$907$	−6.47573e9	−0.288180	−0.144090	−	0.989565i	$-0.546025\pi$
$907$	−6.47573e9	−0.288180	−0.144090	+	0.989565i	$0.546025\pi$
$908$	−1.20279e10	−0.533197
$909$	0	0
$910$	−3.09957e9	−0.136350
$911$	1.27336e10	0.558004	0.279002	−	0.960291i	$-0.409996\pi$
$911$	1.27336e10	0.558004	0.279002	+	0.960291i	$0.409996\pi$
$912$	0	0
$913$	−4.87650e9	−0.212061
$914$	7.85952e9	0.340474
$915$	0	0
$916$	−2.35807e10	−1.01373
$917$	5.49709e7	0.00235418
$918$	0	0
$919$	6.63174e8	0.0281853	0.0140927	−	0.999901i	$-0.495514\pi$
$919$	6.63174e8	0.0281853	0.0140927	+	0.999901i	$0.495514\pi$
$920$	1.31267e11	5.55773
$921$	0	0
$922$	5.16261e10	2.16926
$923$	2.73948e9	0.114673
$924$	0	0
$925$	1.70380e10	0.707821
$926$	−1.06186e10	−0.439468
$927$	0	0
$928$	−1.02657e9	−0.0421670
$929$	−3.43993e10	−1.40765	−0.703825	−	0.710373i	$-0.748526\pi$
$929$	−3.43993e10	−1.40765	−0.703825	+	0.710373i	$0.748526\pi$
$930$	0	0
$931$	5.24994e9	0.213221
$932$	5.69434e10	2.30403
$933$	0	0
$934$	4.02692e10	1.61718
$935$	2.32041e10	0.928378
$936$	0	0
$937$	−9.38341e9	−0.372625	−0.186313	−	0.982491i	$-0.559654\pi$
$937$	−9.38341e9	−0.372625	−0.186313	+	0.982491i	$0.559654\pi$
$938$	−8.09623e8	−0.0320312
$939$	0	0
$940$	9.15350e10	3.59451
$941$	−2.39000e10	−0.935050	−0.467525	−	0.883980i	$-0.654854\pi$
$941$	−2.39000e10	−0.935050	−0.467525	+	0.883980i	$0.654854\pi$
$942$	0	0
$943$	7.41353e10	2.87895
$944$	−1.78409e10	−0.690261
$945$	0	0
$946$	4.12337e9	0.158356
$947$	−4.09342e10	−1.56625	−0.783126	−	0.621863i	$-0.786376\pi$
$947$	−4.09342e10	−1.56625	−0.783126	+	0.621863i	$0.786376\pi$
$948$	0	0
$949$	2.10495e10	0.799484
$950$	−1.12056e10	−0.424037
$951$	0	0
$952$	3.84223e9	0.144329
$953$	2.39304e10	0.895621	0.447811	−	0.894128i	$-0.352204\pi$
$953$	2.39304e10	0.895621	0.447811	+	0.894128i	$0.352204\pi$
$954$	0	0
$955$	3.97931e9	0.147841
$956$	1.17818e11	4.36123
$957$	0	0
$958$	3.80092e10	1.39672
$959$	−1.77027e9	−0.0648149
$960$	0	0
$961$	−9.12637e8	−0.0331716
$962$	2.88320e10	1.04415
$963$	0	0
$964$	−6.80044e10	−2.44493
$965$	−2.97782e10	−1.06673
$966$	0	0
$967$	−9.99102e9	−0.355318	−0.177659	−	0.984092i	$-0.556852\pi$
$967$	−9.99102e9	−0.355318	−0.177659	+	0.984092i	$0.556852\pi$
$968$	−4.22258e10	−1.49628
$969$	0	0
$970$	−9.77535e10	−3.43899
$971$	−2.91846e10	−1.02303	−0.511513	−	0.859275i	$-0.670915\pi$
$971$	−2.91846e10	−1.02303	−0.511513	+	0.859275i	$0.670915\pi$
$972$	0	0
$973$	−1.73717e9	−0.0604571
$974$	5.40492e10	1.87427
$975$	0	0
$976$	7.63086e10	2.62724
$977$	1.72546e9	0.0591935	0.0295968	−	0.999562i	$-0.490578\pi$
$977$	1.72546e9	0.0591935	0.0295968	+	0.999562i	$0.490578\pi$
$978$	0	0
$979$	2.69115e10	0.916639
$980$	9.55120e10	3.24165
$981$	0	0
$982$	−6.39960e10	−2.15657
$983$	−5.02005e10	−1.68566	−0.842831	−	0.538178i	$-0.819113\pi$
$983$	−5.02005e10	−1.68566	−0.842831	+	0.538178i	$0.819113\pi$
$984$	0	0
$985$	−1.51345e10	−0.504592
$986$	−2.57246e9	−0.0854633
$987$	0	0
$988$	−1.31175e10	−0.432714
$989$	7.93821e9	0.260937
$990$	0	0
$991$	3.49646e10	1.14122	0.570612	−	0.821220i	$-0.306706\pi$
$991$	3.49646e10	1.14122	0.570612	+	0.821220i	$0.306706\pi$
$992$	−2.98526e10	−0.970937
$993$	0	0
$994$	4.11296e8	0.0132832
$995$	−9.60663e9	−0.309165
$996$	0	0
$997$	−2.66912e9	−0.0852972	−0.0426486	−	0.999090i	$-0.513580\pi$
$997$	−2.66912e9	−0.0852972	−0.0426486	+	0.999090i	$0.513580\pi$
$998$	−3.54745e9	−0.112969
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.d.1.13		13
3.2	odd	2		43.8.a.b.1.1	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.b.1.1	✓	13	3.2	odd	2
387.8.a.d.1.13		13	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)

\(f(q)\)	\(=\)	\(q+20.3781 q^{2} +287.267 q^{4} -405.085 q^{5} +52.5965 q^{7} +3245.57 q^{8} +O(q^{10})\)	\(q+20.3781 q^{2} +287.267 q^{4} -405.085 q^{5} +52.5965 q^{7} +3245.57 q^{8} -8254.88 q^{10} -2544.97 q^{11} +7138.95 q^{13} +1071.82 q^{14} +29368.3 q^{16} +22507.9 q^{17} -6396.31 q^{19} -116368. q^{20} -51861.7 q^{22} -99842.9 q^{23} +85969.2 q^{25} +145478. q^{26} +15109.2 q^{28} -5608.53 q^{29} -163095. q^{31} +183038. q^{32} +458669. q^{34} -21306.1 q^{35} +198188. q^{37} -130345. q^{38} -1.31473e6 q^{40} -742519. q^{41} -79507.0 q^{43} -731087. q^{44} -2.03461e6 q^{46} -786601. q^{47} -820777. q^{49} +1.75189e6 q^{50} +2.05079e6 q^{52} -2.08070e6 q^{53} +1.03093e6 q^{55} +170705. q^{56} -114291. q^{58} -607486. q^{59} +2.59833e6 q^{61} -3.32357e6 q^{62} -29175.0 q^{64} -2.89188e6 q^{65} -755375. q^{67} +6.46580e6 q^{68} -434177. q^{70} +383737. q^{71} +2.94854e6 q^{73} +4.03869e6 q^{74} -1.83745e6 q^{76} -133856. q^{77} -7.28812e6 q^{79} -1.18967e7 q^{80} -1.51311e7 q^{82} +1.91613e6 q^{83} -9.11764e6 q^{85} -1.62020e6 q^{86} -8.25988e6 q^{88} -1.05744e7 q^{89} +375483. q^{91} -2.86816e7 q^{92} -1.60294e7 q^{94} +2.59105e6 q^{95} +1.18419e7 q^{97} -1.67259e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8}+O(q^{10}) \) 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8} + 4667 q^{10} - 1620 q^{11} + 13550 q^{13} - 44160 q^{14} + 114026 q^{16} - 110880 q^{17} + 105058 q^{19} - 167251 q^{20} + 201504 q^{22} - 160184 q^{23} + 270149 q^{25} - 272104 q^{26} + 208172 q^{28} - 285546 q^{29} - 99616 q^{31} - 200126 q^{32} - 80941 q^{34} + 187104 q^{35} + 176038 q^{37} - 652165 q^{38} - 895387 q^{40} + 410260 q^{41} - 1033591 q^{43} + 2177076 q^{44} - 3975765 q^{46} + 424556 q^{47} - 1561359 q^{49} + 4063801 q^{50} - 4172312 q^{52} - 3992458 q^{53} + 406960 q^{55} - 1559556 q^{56} - 4052005 q^{58} - 2248836 q^{59} + 6210394 q^{61} - 885317 q^{62} - 3096318 q^{64} - 5600420 q^{65} - 1993648 q^{67} - 9327135 q^{68} - 1105098 q^{70} - 4978064 q^{71} + 8224814 q^{73} + 3613563 q^{74} + 10687121 q^{76} - 17261892 q^{77} + 6945708 q^{79} - 15822799 q^{80} - 508449 q^{82} - 22937328 q^{83} - 575532 q^{85} + 1272112 q^{86} + 11202656 q^{88} - 9291302 q^{89} + 25581108 q^{91} + 14388137 q^{92} - 24645805 q^{94} - 30750464 q^{95} + 10001852 q^{97} + 32304856 q^{98}+O(q^{100}) \) 13 * q - 16 * q^2 + 922 * q^4 - 998 * q^5 + 1360 * q^7 - 3870 * q^8 + 4667 * q^10 - 1620 * q^11 + 13550 * q^13 - 44160 * q^14 + 114026 * q^16 - 110880 * q^17 + 105058 * q^19 - 167251 * q^20 + 201504 * q^22 - 160184 * q^23 + 270149 * q^25 - 272104 * q^26 + 208172 * q^28 - 285546 * q^29 - 99616 * q^31 - 200126 * q^32 - 80941 * q^34 + 187104 * q^35 + 176038 * q^37 - 652165 * q^38 - 895387 * q^40 + 410260 * q^41 - 1033591 * q^43 + 2177076 * q^44 - 3975765 * q^46 + 424556 * q^47 - 1561359 * q^49 + 4063801 * q^50 - 4172312 * q^52 - 3992458 * q^53 + 406960 * q^55 - 1559556 * q^56 - 4052005 * q^58 - 2248836 * q^59 + 6210394 * q^61 - 885317 * q^62 - 3096318 * q^64 - 5600420 * q^65 - 1993648 * q^67 - 9327135 * q^68 - 1105098 * q^70 - 4978064 * q^71 + 8224814 * q^73 + 3613563 * q^74 + 10687121 * q^76 - 17261892 * q^77 + 6945708 * q^79 - 15822799 * q^80 - 508449 * q^82 - 22937328 * q^83 - 575532 * q^85 + 1272112 * q^86 + 11202656 * q^88 - 9291302 * q^89 + 25581108 * q^91 + 14388137 * q^92 - 24645805 * q^94 - 30750464 * q^95 + 10001852 * q^97 + 32304856 * q^98

Introduction

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