LMFDB - Embedded newform 387.8.a.d.1.6

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			1.6
Root			$2.37903$ 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)

$f(q)$	$=$	$q-3.37903 q^{2} -116.582 q^{4} +241.175 q^{5} -173.086 q^{7} +826.450 q^{8} +O(q^{10})$	\(q-3.37903 q^{2} -116.582 q^{4} +241.175 q^{5} -173.086 q^{7} +826.450 q^{8} -814.939 q^{10} -682.590 q^{11} +168.040 q^{13} +584.863 q^{14} +12129.9 q^{16} -5562.09 q^{17} -13825.2 q^{19} -28116.8 q^{20} +2306.49 q^{22} +24641.2 q^{23} -19959.4 q^{25} -567.813 q^{26} +20178.7 q^{28} +33649.4 q^{29} +22707.6 q^{31} -146773. q^{32} +18794.5 q^{34} -41744.1 q^{35} +181778. q^{37} +46715.6 q^{38} +199320. q^{40} -335134. q^{41} -79507.0 q^{43} +79577.8 q^{44} -83263.2 q^{46} +743538. q^{47} -793584. q^{49} +67443.4 q^{50} -19590.5 q^{52} +12693.7 q^{53} -164624. q^{55} -143047. q^{56} -113702. q^{58} +830705. q^{59} +2.22282e6 q^{61} -76729.7 q^{62} -1.05668e6 q^{64} +40527.2 q^{65} +3.88681e6 q^{67} +648441. q^{68} +141055. q^{70} -3.88756e6 q^{71} +5.22595e6 q^{73} -614232. q^{74} +1.61177e6 q^{76} +118147. q^{77} -2.64353e6 q^{79} +2.92544e6 q^{80} +1.13243e6 q^{82} -6.76661e6 q^{83} -1.34144e6 q^{85} +268657. q^{86} -564127. q^{88} -327679. q^{89} -29085.4 q^{91} -2.87272e6 q^{92} -2.51244e6 q^{94} -3.33429e6 q^{95} +5.05740e6 q^{97} +2.68154e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.37903	−0.298667	−0.149333	−	0.988787i	$-0.547713\pi$
$2$	−3.37903	−0.298667	−0.149333	+	0.988787i	$0.547713\pi$
$3$	0	0
$3$	0	0
$4$	−116.582	−0.910798
$5$	241.175	0.862856	0.431428	−	0.902147i	$-0.358010\pi$
$5$	241.175	0.862856	0.431428	+	0.902147i	$0.358010\pi$
$6$	0	0
$7$	−173.086	−0.190730	−0.0953650	−	0.995442i	$-0.530402\pi$
$7$	−173.086	−0.190730	−0.0953650	+	0.995442i	$0.530402\pi$
$8$	826.450	0.570692
$9$	0	0
$10$	−814.939	−0.257706
$11$	−682.590	−0.154627	−0.0773135	−	0.997007i	$-0.524634\pi$
$11$	−682.590	−0.154627	−0.0773135	+	0.997007i	$0.524634\pi$
$12$	0	0
$13$	168.040	0.0212135	0.0106067	−	0.999944i	$-0.496624\pi$
$13$	168.040	0.0212135	0.0106067	+	0.999944i	$0.496624\pi$
$14$	584.863	0.0569647
$15$	0	0
$16$	12129.9	0.740351
$17$	−5562.09	−0.274579	−0.137289	−	0.990531i	$-0.543839\pi$
$17$	−5562.09	−0.274579	−0.137289	+	0.990531i	$0.543839\pi$
$18$	0	0
$19$	−13825.2	−0.462416	−0.231208	−	0.972904i	$-0.574268\pi$
$19$	−13825.2	−0.462416	−0.231208	+	0.972904i	$0.574268\pi$
$20$	−28116.8	−0.785887
$21$	0	0
$22$	2306.49	0.0461820
$23$	24641.2	0.422293	0.211146	−	0.977454i	$-0.432280\pi$
$23$	24641.2	0.422293	0.211146	+	0.977454i	$0.432280\pi$
$24$	0	0
$25$	−19959.4	−0.255480
$26$	−567.813	−0.00633576
$27$	0	0
$28$	20178.7	0.173716
$29$	33649.4	0.256203	0.128102	−	0.991761i	$-0.459112\pi$
$29$	33649.4	0.256203	0.128102	+	0.991761i	$0.459112\pi$
$30$	0	0
$31$	22707.6	0.136901	0.0684503	−	0.997655i	$-0.478195\pi$
$31$	22707.6	0.136901	0.0684503	+	0.997655i	$0.478195\pi$
$32$	−146773.	−0.791811
$33$	0	0
$34$	18794.5	0.0820076
$35$	−41744.1	−0.164572
$36$	0	0
$37$	181778.	0.589976	0.294988	−	0.955501i	$-0.404684\pi$
$37$	181778.	0.589976	0.294988	+	0.955501i	$0.404684\pi$
$38$	46715.6	0.138108
$39$	0	0
$40$	199320.	0.492425
$41$	−335134.	−0.759407	−0.379704	−	0.925108i	$-0.623974\pi$
$41$	−335134.	−0.759407	−0.379704	+	0.925108i	$0.623974\pi$
$42$	0	0
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	79577.8	0.140834
$45$	0	0
$46$	−83263.2	−0.126125
$47$	743538.	1.04463	0.522313	−	0.852754i	$-0.325069\pi$
$47$	743538.	1.04463	0.522313	+	0.852754i	$0.325069\pi$
$48$	0	0
$49$	−793584.	−0.963622
$50$	67443.4	0.0763035
$51$	0	0
$52$	−19590.5	−0.0193212
$53$	12693.7	0.0117118	0.00585591	−	0.999983i	$-0.498136\pi$
$53$	12693.7	0.0117118	0.00585591	+	0.999983i	$0.498136\pi$
$54$	0	0
$55$	−164624.	−0.133421
$56$	−143047.	−0.108848
$57$	0	0
$58$	−113702.	−0.0765194
$59$	830705.	0.526581	0.263290	−	0.964717i	$-0.415192\pi$
$59$	830705.	0.526581	0.263290	+	0.964717i	$0.415192\pi$
$60$	0	0
$61$	2.22282e6	1.25386	0.626931	−	0.779075i	$-0.284311\pi$
$61$	2.22282e6	1.25386	0.626931	+	0.779075i	$0.284311\pi$
$62$	−76729.7	−0.0408877
$63$	0	0
$64$	−1.05668e6	−0.503864
$65$	40527.2	0.0183042
$66$	0	0
$67$	3.88681e6	1.57882	0.789409	−	0.613868i	$-0.210387\pi$
$67$	3.88681e6	1.57882	0.789409	+	0.613868i	$0.210387\pi$
$68$	648441.	0.250086
$69$	0	0
$70$	141055.	0.0491523
$71$	−3.88756e6	−1.28906	−0.644530	−	0.764579i	$-0.722947\pi$
$71$	−3.88756e6	−1.28906	−0.644530	+	0.764579i	$0.722947\pi$
$72$	0	0
$73$	5.22595e6	1.57230	0.786150	−	0.618035i	$-0.212071\pi$
$73$	5.22595e6	1.57230	0.786150	+	0.618035i	$0.212071\pi$
$74$	−614232.	−0.176206
$75$	0	0
$76$	1.61177e6	0.421167
$77$	118147.	0.0294920
$78$	0	0
$79$	−2.64353e6	−0.603240	−0.301620	−	0.953428i	$-0.597527\pi$
$79$	−2.64353e6	−0.603240	−0.301620	+	0.953428i	$0.597527\pi$
$80$	2.92544e6	0.638816
$81$	0	0
$82$	1.13243e6	0.226810
$83$	−6.76661e6	−1.29897	−0.649483	−	0.760376i	$-0.725015\pi$
$83$	−6.76661e6	−1.29897	−0.649483	+	0.760376i	$0.725015\pi$
$84$	0	0
$85$	−1.34144e6	−0.236922
$86$	268657.	0.0455463
$87$	0	0
$88$	−564127.	−0.0882444
$89$	−327679.	−0.0492701	−0.0246350	−	0.999697i	$-0.507842\pi$
$89$	−327679.	−0.0492701	−0.0246350	+	0.999697i	$0.507842\pi$
$90$	0	0
$91$	−29085.4	−0.00404604
$92$	−2.87272e6	−0.384624
$93$	0	0
$94$	−2.51244e6	−0.311995
$95$	−3.33429e6	−0.398998
$96$	0	0
$97$	5.05740e6	0.562634	0.281317	−	0.959615i	$-0.409229\pi$
$97$	5.05740e6	0.562634	0.281317	+	0.959615i	$0.409229\pi$
$98$	2.68154e6	0.287802
$99$	0	0
$100$	2.32691e6	0.232691
$101$	−1.31064e7	−1.26578	−0.632889	−	0.774242i	$-0.718131\pi$
$101$	−1.31064e7	−1.26578	−0.632889	+	0.774242i	$0.718131\pi$
$102$	0	0
$103$	−1.35656e7	−1.22324	−0.611618	−	0.791153i	$-0.709481\pi$
$103$	−1.35656e7	−1.22324	−0.611618	+	0.791153i	$0.709481\pi$
$104$	138877.	0.0121064
$105$	0	0
$106$	−42892.6	−0.00349793
$107$	1.06605e7	0.841269	0.420635	−	0.907230i	$-0.361807\pi$
$107$	1.06605e7	0.841269	0.420635	+	0.907230i	$0.361807\pi$
$108$	0	0
$109$	9.24482e6	0.683763	0.341881	−	0.939743i	$-0.388936\pi$
$109$	9.24482e6	0.683763	0.341881	+	0.939743i	$0.388936\pi$
$110$	556269.	0.0398484
$111$	0	0
$112$	−2.09952e6	−0.141207
$113$	4.05844e6	0.264597	0.132299	−	0.991210i	$-0.457764\pi$
$113$	4.05844e6	0.264597	0.132299	+	0.991210i	$0.457764\pi$
$114$	0	0
$115$	5.94285e6	0.364378
$116$	−3.92292e6	−0.233349
$117$	0	0
$118$	−2.80698e6	−0.157272
$119$	962721.	0.0523704
$120$	0	0
$121$	−1.90212e7	−0.976090
$122$	−7.51098e6	−0.374487
$123$	0	0
$124$	−2.64730e6	−0.124689
$125$	−2.36555e7	−1.08330
$126$	0	0
$127$	−2.45396e7	−1.06305	−0.531525	−	0.847043i	$-0.678381\pi$
$127$	−2.45396e7	−1.06305	−0.531525	+	0.847043i	$0.678381\pi$
$128$	2.23575e7	0.942298
$129$	0	0
$130$	−136943.	−0.00546684
$131$	2.67897e7	1.04116	0.520581	−	0.853812i	$-0.325715\pi$
$131$	2.67897e7	1.04116	0.520581	+	0.853812i	$0.325715\pi$
$132$	0	0
$133$	2.39294e6	0.0881965
$134$	−1.31337e7	−0.471540
$135$	0	0
$136$	−4.59679e6	−0.156700
$137$	−4.80765e7	−1.59739	−0.798695	−	0.601736i	$-0.794476\pi$
$137$	−4.80765e7	−1.59739	−0.798695	+	0.601736i	$0.794476\pi$
$138$	0	0
$139$	−5.94300e6	−0.187696	−0.0938478	−	0.995587i	$-0.529917\pi$
$139$	−5.94300e6	−0.187696	−0.0938478	+	0.995587i	$0.529917\pi$
$140$	4.86662e6	0.149892
$141$	0	0
$142$	1.31362e7	0.385000
$143$	−114703.	−0.00328017
$144$	0	0
$145$	8.11541e6	0.221066
$146$	−1.76587e7	−0.469594
$147$	0	0
$148$	−2.11920e7	−0.537349
$149$	−4.65127e7	−1.15191	−0.575956	−	0.817481i	$-0.695370\pi$
$149$	−4.65127e7	−1.15191	−0.575956	+	0.817481i	$0.695370\pi$
$150$	0	0
$151$	−1.54586e7	−0.365385	−0.182693	−	0.983170i	$-0.558481\pi$
$151$	−1.54586e7	−0.365385	−0.182693	+	0.983170i	$0.558481\pi$
$152$	−1.14258e7	−0.263897
$153$	0	0
$154$	−399221.	−0.00880828
$155$	5.47652e6	0.118125
$156$	0	0
$157$	2.72583e6	0.0562147	0.0281074	−	0.999605i	$-0.491052\pi$
$157$	2.72583e6	0.0562147	0.0281074	+	0.999605i	$0.491052\pi$
$158$	8.93258e6	0.180168
$159$	0	0
$160$	−3.53980e7	−0.683218
$161$	−4.26504e6	−0.0805439
$162$	0	0
$163$	−5.29222e7	−0.957152	−0.478576	−	0.878046i	$-0.658847\pi$
$163$	−5.29222e7	−0.957152	−0.478576	+	0.878046i	$0.658847\pi$
$164$	3.90706e7	0.691667
$165$	0	0
$166$	2.28646e7	0.387958
$167$	−3.70870e7	−0.616189	−0.308095	−	0.951356i	$-0.599691\pi$
$167$	−3.70870e7	−0.616189	−0.308095	+	0.951356i	$0.599691\pi$
$168$	0	0
$169$	−6.27203e7	−0.999550
$170$	4.53277e6	0.0707607
$171$	0	0
$172$	9.26910e6	0.138895
$173$	3.93055e7	0.577154	0.288577	−	0.957457i	$-0.406818\pi$
$173$	3.93055e7	0.577154	0.288577	+	0.957457i	$0.406818\pi$
$174$	0	0
$175$	3.45469e6	0.0487277
$176$	−8.27975e6	−0.114478
$177$	0	0
$178$	1.10724e6	0.0147153
$179$	4.76706e7	0.621248	0.310624	−	0.950533i	$-0.399462\pi$
$179$	4.76706e7	0.621248	0.310624	+	0.950533i	$0.399462\pi$
$180$	0	0
$181$	2.04178e7	0.255938	0.127969	−	0.991778i	$-0.459154\pi$
$181$	2.04178e7	0.255938	0.127969	+	0.991778i	$0.459154\pi$
$182$	98280.5	0.00120842
$183$	0	0
$184$	2.03647e7	0.240999
$185$	4.38403e7	0.509064
$186$	0	0
$187$	3.79663e6	0.0424573
$188$	−8.66833e7	−0.951444
$189$	0	0
$190$	1.12667e7	0.119167
$191$	−7.99910e7	−0.830662	−0.415331	−	0.909670i	$-0.636334\pi$
$191$	−7.99910e7	−0.830662	−0.415331	+	0.909670i	$0.636334\pi$
$192$	0	0
$193$	−3.82721e7	−0.383205	−0.191603	−	0.981473i	$-0.561368\pi$
$193$	−3.82721e7	−0.383205	−0.191603	+	0.981473i	$0.561368\pi$
$194$	−1.70891e7	−0.168040
$195$	0	0
$196$	9.25178e7	0.877665
$197$	−1.23397e8	−1.14993	−0.574967	−	0.818177i	$-0.694985\pi$
$197$	−1.23397e8	−1.14993	−0.574967	+	0.818177i	$0.694985\pi$
$198$	0	0
$199$	−7.95502e7	−0.715576	−0.357788	−	0.933803i	$-0.616469\pi$
$199$	−7.95502e7	−0.715576	−0.357788	+	0.933803i	$0.616469\pi$
$200$	−1.64954e7	−0.145801
$201$	0	0
$202$	4.42868e7	0.378046
$203$	−5.82424e6	−0.0488656
$204$	0	0
$205$	−8.08260e7	−0.655259
$206$	4.58387e7	0.365340
$207$	0	0
$208$	2.03831e6	0.0157054
$209$	9.43691e6	0.0715019
$210$	0	0
$211$	−2.30074e8	−1.68608	−0.843041	−	0.537849i	$-0.819237\pi$
$211$	−2.30074e8	−1.68608	−0.843041	+	0.537849i	$0.819237\pi$
$212$	−1.47986e6	−0.0106671
$213$	0	0
$214$	−3.60222e7	−0.251259
$215$	−1.91751e7	−0.131584
$216$	0	0
$217$	−3.93037e6	−0.0261110
$218$	−3.12385e7	−0.204217
$219$	0	0
$220$	1.91922e7	0.121519
$221$	−934655.	−0.00582477
$222$	0	0
$223$	1.94009e8	1.17153	0.585767	−	0.810479i	$-0.300793\pi$
$223$	1.94009e8	1.17153	0.585767	+	0.810479i	$0.300793\pi$
$224$	2.54044e7	0.151022
$225$	0	0
$226$	−1.37136e7	−0.0790264
$227$	−2.48770e8	−1.41159	−0.705793	−	0.708418i	$-0.749409\pi$
$227$	−2.48770e8	−1.41159	−0.705793	+	0.708418i	$0.749409\pi$
$228$	0	0
$229$	2.61570e8	1.43934	0.719670	−	0.694316i	$-0.244293\pi$
$229$	2.61570e8	1.43934	0.719670	+	0.694316i	$0.244293\pi$
$230$	−2.00811e7	−0.108828
$231$	0	0
$232$	2.78096e7	0.146213
$233$	−1.70926e8	−0.885244	−0.442622	−	0.896708i	$-0.645952\pi$
$233$	−1.70926e8	−0.885244	−0.442622	+	0.896708i	$0.645952\pi$
$234$	0	0
$235$	1.79323e8	0.901362
$236$	−9.68454e7	−0.479609
$237$	0	0
$238$	−3.25306e6	−0.0156413
$239$	−8.66501e7	−0.410560	−0.205280	−	0.978703i	$-0.565810\pi$
$239$	−8.66501e7	−0.410560	−0.205280	+	0.978703i	$0.565810\pi$
$240$	0	0
$241$	−1.19315e8	−0.549079	−0.274540	−	0.961576i	$-0.588525\pi$
$241$	−1.19315e8	−0.549079	−0.274540	+	0.961576i	$0.588525\pi$
$242$	6.42733e7	0.291526
$243$	0	0
$244$	−2.59141e8	−1.14202
$245$	−1.91393e8	−0.831467
$246$	0	0
$247$	−2.32318e6	−0.00980944
$248$	1.87667e7	0.0781281
$249$	0	0
$250$	7.99328e7	0.323545
$251$	−4.55167e8	−1.81682	−0.908412	−	0.418077i	$-0.862704\pi$
$251$	−4.55167e8	−1.81682	−0.908412	+	0.418077i	$0.862704\pi$
$252$	0	0
$253$	−1.68198e7	−0.0652979
$254$	8.29199e7	0.317498
$255$	0	0
$256$	5.97082e7	0.222431
$257$	−5.12052e8	−1.88169	−0.940845	−	0.338837i	$-0.889967\pi$
$257$	−5.12052e8	−1.88169	−0.940845	+	0.338837i	$0.889967\pi$
$258$	0	0
$259$	−3.14632e7	−0.112526
$260$	−4.72475e6	−0.0166714
$261$	0	0
$262$	−9.05232e7	−0.310961
$263$	−9.38421e7	−0.318092	−0.159046	−	0.987271i	$-0.550842\pi$
$263$	−9.38421e7	−0.318092	−0.159046	+	0.987271i	$0.550842\pi$
$264$	0	0
$265$	3.06142e6	0.0101056
$266$	−8.08582e6	−0.0263414
$267$	0	0
$268$	−4.53133e8	−1.43798
$269$	5.02612e6	0.0157434	0.00787172	−	0.999969i	$-0.497494\pi$
$269$	5.02612e6	0.0157434	0.00787172	+	0.999969i	$0.497494\pi$
$270$	0	0
$271$	−4.90350e8	−1.49663	−0.748314	−	0.663345i	$-0.769136\pi$
$271$	−4.90350e8	−1.49663	−0.748314	+	0.663345i	$0.769136\pi$
$272$	−6.74677e7	−0.203285
$273$	0	0
$274$	1.62452e8	0.477088
$275$	1.36241e7	0.0395041
$276$	0	0
$277$	5.00692e7	0.141544	0.0707721	−	0.997493i	$-0.477454\pi$
$277$	5.00692e7	0.141544	0.0707721	+	0.997493i	$0.477454\pi$
$278$	2.00816e7	0.0560584
$279$	0	0
$280$	−3.44994e7	−0.0939202
$281$	−5.08518e8	−1.36721	−0.683604	−	0.729853i	$-0.739588\pi$
$281$	−5.08518e8	−1.36721	−0.683604	+	0.729853i	$0.739588\pi$
$282$	0	0
$283$	9.23331e7	0.242161	0.121081	−	0.992643i	$-0.461364\pi$
$283$	9.23331e7	0.242161	0.121081	+	0.992643i	$0.461364\pi$
$284$	4.53220e8	1.17407
$285$	0	0
$286$	387583.	0.000979679 0
$287$	5.80070e7	0.144842
$288$	0	0
$289$	−3.79402e8	−0.924606
$290$	−2.74222e7	−0.0660252
$291$	0	0
$292$	−6.09253e8	−1.43205
$293$	2.46782e8	0.573162	0.286581	−	0.958056i	$-0.407481\pi$
$293$	2.46782e8	0.573162	0.286581	+	0.958056i	$0.407481\pi$
$294$	0	0
$295$	2.00346e8	0.454363
$296$	1.50230e8	0.336695
$297$	0	0
$298$	1.57168e8	0.344038
$299$	4.14071e6	0.00895830
$300$	0	0
$301$	1.37616e7	0.0290860
$302$	5.22351e7	0.109128
$303$	0	0
$304$	−1.67698e8	−0.342350
$305$	5.36090e8	1.08190
$306$	0	0
$307$	3.84464e8	0.758352	0.379176	−	0.925325i	$-0.376207\pi$
$307$	3.84464e8	0.758352	0.379176	+	0.925325i	$0.376207\pi$
$308$	−1.37738e7	−0.0268613
$309$	0	0
$310$	−1.85053e7	−0.0352802
$311$	5.62608e8	1.06058	0.530291	−	0.847815i	$-0.322083\pi$
$311$	5.62608e8	1.06058	0.530291	+	0.847815i	$0.322083\pi$
$312$	0	0
$313$	−2.64846e8	−0.488188	−0.244094	−	0.969752i	$-0.578491\pi$
$313$	−2.64846e8	−0.488188	−0.244094	+	0.969752i	$0.578491\pi$
$314$	−9.21066e6	−0.0167895
$315$	0	0
$316$	3.08189e8	0.549429
$317$	−9.81092e7	−0.172983	−0.0864913	−	0.996253i	$-0.527565\pi$
$317$	−9.81092e7	−0.172983	−0.0864913	+	0.996253i	$0.527565\pi$
$318$	0	0
$319$	−2.29687e7	−0.0396159
$320$	−2.54845e8	−0.434762
$321$	0	0
$322$	1.44117e7	0.0240558
$323$	7.68968e7	0.126970
$324$	0	0
$325$	−3.35398e6	−0.00541962
$326$	1.78826e8	0.285870
$327$	0	0
$328$	−2.76971e8	−0.433388
$329$	−1.28696e8	−0.199241
$330$	0	0
$331$	6.54036e8	0.991296	0.495648	−	0.868523i	$-0.334931\pi$
$331$	6.54036e8	0.991296	0.495648	+	0.868523i	$0.334931\pi$
$332$	7.88866e8	1.18310
$333$	0	0
$334$	1.25318e8	0.184035
$335$	9.37404e8	1.36229
$336$	0	0
$337$	6.62977e8	0.943613	0.471806	−	0.881702i	$-0.343602\pi$
$337$	6.62977e8	0.943613	0.471806	+	0.881702i	$0.343602\pi$
$338$	2.11934e8	0.298532
$339$	0	0
$340$	1.56388e8	0.215788
$341$	−1.55000e7	−0.0211685
$342$	0	0
$343$	2.79902e8	0.374522
$344$	−6.57086e7	−0.0870297
$345$	0	0
$346$	−1.32814e8	−0.172377
$347$	5.98992e8	0.769605	0.384802	−	0.922999i	$-0.374270\pi$
$347$	5.98992e8	0.769605	0.384802	+	0.922999i	$0.374270\pi$
$348$	0	0
$349$	1.09709e9	1.38151	0.690753	−	0.723091i	$-0.257279\pi$
$349$	1.09709e9	1.38151	0.690753	+	0.723091i	$0.257279\pi$
$350$	−1.16735e7	−0.0145534
$351$	0	0
$352$	1.00186e8	0.122435
$353$	−1.20482e9	−1.45784	−0.728921	−	0.684598i	$-0.759978\pi$
$353$	−1.20482e9	−1.45784	−0.728921	+	0.684598i	$0.759978\pi$
$354$	0	0
$355$	−9.37585e8	−1.11227
$356$	3.82015e7	0.0448751
$357$	0	0
$358$	−1.61080e8	−0.185546
$359$	5.38564e8	0.614338	0.307169	−	0.951655i	$-0.400618\pi$
$359$	5.38564e8	0.614338	0.307169	+	0.951655i	$0.400618\pi$
$360$	0	0
$361$	−7.02737e8	−0.786172
$362$	−6.89924e7	−0.0764401
$363$	0	0
$364$	3.39084e6	0.00368513
$365$	1.26037e9	1.35667
$366$	0	0
$367$	1.10179e9	1.16350	0.581750	−	0.813368i	$-0.302368\pi$
$367$	1.10179e9	1.16350	0.581750	+	0.813368i	$0.302368\pi$
$368$	2.98895e8	0.312645
$369$	0	0
$370$	−1.48138e8	−0.152041
$371$	−2.19711e6	−0.00223379
$372$	0	0
$373$	1.71818e9	1.71431	0.857154	−	0.515060i	$-0.172231\pi$
$373$	1.71818e9	1.71431	0.857154	+	0.515060i	$0.172231\pi$
$374$	−1.28289e7	−0.0126806
$375$	0	0
$376$	6.14498e8	0.596160
$377$	5.65446e6	0.00543496
$378$	0	0
$379$	1.07347e8	0.101287	0.0506433	−	0.998717i	$-0.483873\pi$
$379$	1.07347e8	0.101287	0.0506433	+	0.998717i	$0.483873\pi$
$380$	3.88719e8	0.363407
$381$	0	0
$382$	2.70292e8	0.248091
$383$	6.46792e8	0.588259	0.294130	−	0.955766i	$-0.404970\pi$
$383$	6.46792e8	0.588259	0.294130	+	0.955766i	$0.404970\pi$
$384$	0	0
$385$	2.84941e7	0.0254473
$386$	1.29322e8	0.114451
$387$	0	0
$388$	−5.89603e8	−0.512446
$389$	5.05273e7	0.0435214	0.0217607	−	0.999763i	$-0.493073\pi$
$389$	5.05273e7	0.0435214	0.0217607	+	0.999763i	$0.493073\pi$
$390$	0	0
$391$	−1.37056e8	−0.115953
$392$	−6.55858e8	−0.549932
$393$	0	0
$394$	4.16962e8	0.343447
$395$	−6.37555e8	−0.520509
$396$	0	0
$397$	−1.12623e9	−0.903359	−0.451679	−	0.892180i	$-0.649175\pi$
$397$	−1.12623e9	−0.903359	−0.451679	+	0.892180i	$0.649175\pi$
$398$	2.68803e8	0.213719
$399$	0	0
$400$	−2.42106e8	−0.189145
$401$	90676.9	7.02250e−5 0	3.51125e−5	−	1.00000i	$-0.499989\pi$
$401$	90676.9	7.02250e−5 0	3.51125e−5		1.00000i	$0.499989\pi$
$402$	0	0
$403$	3.81579e6	0.00290413
$404$	1.52797e9	1.15287
$405$	0	0
$406$	1.96803e7	0.0145945
$407$	−1.24080e8	−0.0912262
$408$	0	0
$409$	−3.28251e8	−0.237233	−0.118616	−	0.992940i	$-0.537846\pi$
$409$	−3.28251e8	−0.237233	−0.118616	+	0.992940i	$0.537846\pi$
$410$	2.73114e8	0.195704
$411$	0	0
$412$	1.58151e9	1.11412
$413$	−1.43783e8	−0.100435
$414$	0	0
$415$	−1.63194e9	−1.12082
$416$	−2.46638e7	−0.0167970
$417$	0	0
$418$	−3.18876e7	−0.0213553
$419$	−2.00896e9	−1.33421	−0.667103	−	0.744965i	$-0.732466\pi$
$419$	−2.00896e9	−1.33421	−0.667103	+	0.744965i	$0.732466\pi$
$420$	0	0
$421$	−1.17015e9	−0.764283	−0.382141	−	0.924104i	$-0.624813\pi$
$421$	−1.17015e9	−0.764283	−0.382141	+	0.924104i	$0.624813\pi$
$422$	7.77427e8	0.503577
$423$	0	0
$424$	1.04908e7	0.00668384
$425$	1.11016e8	0.0701495
$426$	0	0
$427$	−3.84739e8	−0.239149
$428$	−1.24283e9	−0.766227
$429$	0	0
$430$	6.47934e7	0.0392999
$431$	2.84974e9	1.71449	0.857245	−	0.514909i	$-0.172174\pi$
$431$	2.84974e9	1.71449	0.857245	+	0.514909i	$0.172174\pi$
$432$	0	0
$433$	1.92069e9	1.13697	0.568486	−	0.822693i	$-0.307529\pi$
$433$	1.92069e9	1.13697	0.568486	+	0.822693i	$0.307529\pi$
$434$	1.32808e7	0.00779850
$435$	0	0
$436$	−1.07778e9	−0.622770
$437$	−3.40668e8	−0.195275
$438$	0	0
$439$	7.86877e8	0.443896	0.221948	−	0.975059i	$-0.428759\pi$
$439$	7.86877e8	0.443896	0.221948	+	0.975059i	$0.428759\pi$
$440$	−1.36053e8	−0.0761422
$441$	0	0
$442$	3.15823e6	0.00173966
$443$	1.97285e9	1.07816	0.539078	−	0.842256i	$-0.318773\pi$
$443$	1.97285e9	1.07816	0.539078	+	0.842256i	$0.318773\pi$
$444$	0	0
$445$	−7.90281e7	−0.0425130
$446$	−6.55562e8	−0.349898
$447$	0	0
$448$	1.82896e8	0.0961019
$449$	−5.76219e7	−0.0300417	−0.0150209	−	0.999887i	$-0.504781\pi$
$449$	−5.76219e7	−0.0300417	−0.0150209	+	0.999887i	$0.504781\pi$
$450$	0	0
$451$	2.28759e8	0.117425
$452$	−4.73142e8	−0.240995
$453$	0	0
$454$	8.40601e8	0.421594
$455$	−7.01469e6	−0.00349115
$456$	0	0
$457$	4.53691e8	0.222358	0.111179	−	0.993800i	$-0.464537\pi$
$457$	4.53691e8	0.222358	0.111179	+	0.993800i	$0.464537\pi$
$458$	−8.83852e8	−0.429883
$459$	0	0
$460$	−6.92830e8	−0.331875
$461$	−2.09843e9	−0.997563	−0.498781	−	0.866728i	$-0.666219\pi$
$461$	−2.09843e9	−0.997563	−0.498781	+	0.866728i	$0.666219\pi$
$462$	0	0
$463$	−2.12353e8	−0.0994318	−0.0497159	−	0.998763i	$-0.515832\pi$
$463$	−2.12353e8	−0.0994318	−0.0497159	+	0.998763i	$0.515832\pi$
$464$	4.08165e8	0.189680
$465$	0	0
$466$	5.77565e8	0.264393
$467$	−1.04433e9	−0.474491	−0.237246	−	0.971450i	$-0.576245\pi$
$467$	−1.04433e9	−0.474491	−0.237246	+	0.971450i	$0.576245\pi$
$468$	0	0
$469$	−6.72753e8	−0.301128
$470$	−6.05939e8	−0.269207
$471$	0	0
$472$	6.86537e8	0.300515
$473$	5.42707e7	0.0235804
$474$	0	0
$475$	2.75942e8	0.118138
$476$	−1.12236e8	−0.0476989
$477$	0	0
$478$	2.92793e8	0.122621
$479$	−1.22092e9	−0.507589	−0.253794	−	0.967258i	$-0.581679\pi$
$479$	−1.22092e9	−0.507589	−0.253794	+	0.967258i	$0.581679\pi$
$480$	0	0
$481$	3.05460e7	0.0125154
$482$	4.03168e8	0.163992
$483$	0	0
$484$	2.21754e9	0.889021
$485$	1.21972e9	0.485472
$486$	0	0
$487$	−8.85642e8	−0.347462	−0.173731	−	0.984793i	$-0.555582\pi$
$487$	−8.85642e8	−0.347462	−0.173731	+	0.984793i	$0.555582\pi$
$488$	1.83705e9	0.715569
$489$	0	0
$490$	6.46723e8	0.248332
$491$	1.70646e9	0.650597	0.325298	−	0.945611i	$-0.394535\pi$
$491$	1.70646e9	0.650597	0.325298	+	0.945611i	$0.394535\pi$
$492$	0	0
$493$	−1.87161e8	−0.0703480
$494$	7.85010e6	0.00292975
$495$	0	0
$496$	2.75441e8	0.101355
$497$	6.72883e8	0.245862
$498$	0	0
$499$	−1.84137e9	−0.663422	−0.331711	−	0.943381i	$-0.607626\pi$
$499$	−1.84137e9	−0.663422	−0.331711	+	0.943381i	$0.607626\pi$
$500$	2.75781e9	0.986666
$501$	0	0
$502$	1.53802e9	0.542625
$503$	1.71136e9	0.599587	0.299794	−	0.954004i	$-0.403082\pi$
$503$	1.71136e9	0.599587	0.299794	+	0.954004i	$0.403082\pi$
$504$	0	0
$505$	−3.16093e9	−1.09218
$506$	5.68346e7	0.0195023
$507$	0	0
$508$	2.86087e9	0.968224
$509$	−2.79743e8	−0.0940257	−0.0470128	−	0.998894i	$-0.514970\pi$
$509$	−2.79743e8	−0.0940257	−0.0470128	+	0.998894i	$0.514970\pi$
$510$	0	0
$511$	−9.04540e8	−0.299885
$512$	−3.06352e9	−1.00873
$513$	0	0
$514$	1.73024e9	0.561999
$515$	−3.27170e9	−1.05548
$516$	0	0
$517$	−5.07532e8	−0.161527
$518$	1.06315e8	0.0336078
$519$	0	0
$520$	3.34937e7	0.0104460
$521$	3.76777e9	1.16722	0.583609	−	0.812035i	$-0.301640\pi$
$521$	3.76777e9	1.16722	0.583609	+	0.812035i	$0.301640\pi$
$522$	0	0
$523$	8.71991e8	0.266536	0.133268	−	0.991080i	$-0.457453\pi$
$523$	8.71991e8	0.266536	0.133268	+	0.991080i	$0.457453\pi$
$524$	−3.12320e9	−0.948288
$525$	0	0
$526$	3.17095e8	0.0950035
$527$	−1.26302e8	−0.0375900
$528$	0	0
$529$	−2.79764e9	−0.821669
$530$	−1.03446e7	−0.00301821
$531$	0	0
$532$	−2.78974e8	−0.0803292
$533$	−5.63159e7	−0.0161097
$534$	0	0
$535$	2.57106e9	0.725894
$536$	3.21226e9	0.901018
$537$	0	0
$538$	−1.69834e7	−0.00470205
$539$	5.41692e8	0.149002
$540$	0	0
$541$	−1.47997e9	−0.401848	−0.200924	−	0.979607i	$-0.564394\pi$
$541$	−1.47997e9	−0.401848	−0.200924	+	0.979607i	$0.564394\pi$
$542$	1.65691e9	0.446993
$543$	0	0
$544$	8.16365e8	0.217414
$545$	2.22962e9	0.589989
$546$	0	0
$547$	6.46090e9	1.68786	0.843932	−	0.536450i	$-0.180235\pi$
$547$	6.46090e9	1.68786	0.843932	+	0.536450i	$0.180235\pi$
$548$	5.60486e9	1.45490
$549$	0	0
$550$	−4.60362e7	−0.0117986
$551$	−4.65209e8	−0.118472
$552$	0	0
$553$	4.57559e8	0.115056
$554$	−1.69185e8	−0.0422745
$555$	0	0
$556$	6.92848e8	0.170953
$557$	−3.65569e9	−0.896347	−0.448174	−	0.893946i	$-0.647925\pi$
$557$	−3.65569e9	−0.896347	−0.448174	+	0.893946i	$0.647925\pi$
$558$	0	0
$559$	−1.33604e7	−0.00323502
$560$	−5.06352e8	−0.121841
$561$	0	0
$562$	1.71830e9	0.408340
$563$	−7.39782e9	−1.74713	−0.873564	−	0.486710i	$-0.838197\pi$
$563$	−7.39782e9	−1.74713	−0.873564	+	0.486710i	$0.838197\pi$
$564$	0	0
$565$	9.78797e8	0.228309
$566$	−3.11996e8	−0.0723256
$567$	0	0
$568$	−3.21288e9	−0.735657
$569$	1.65447e9	0.376502	0.188251	−	0.982121i	$-0.439718\pi$
$569$	1.65447e9	0.376502	0.188251	+	0.982121i	$0.439718\pi$
$570$	0	0
$571$	−1.49420e9	−0.335878	−0.167939	−	0.985797i	$-0.553711\pi$
$571$	−1.49420e9	−0.335878	−0.167939	+	0.985797i	$0.553711\pi$
$572$	1.33723e7	0.00298758
$573$	0	0
$574$	−1.96007e8	−0.0432594
$575$	−4.91823e8	−0.107888
$576$	0	0
$577$	−5.15169e9	−1.11644	−0.558219	−	0.829694i	$-0.688515\pi$
$577$	−5.15169e9	−1.11644	−0.558219	+	0.829694i	$0.688515\pi$
$578$	1.28201e9	0.276149
$579$	0	0
$580$	−9.46113e8	−0.201347
$581$	1.17121e9	0.247752
$582$	0	0
$583$	−8.66462e6	−0.00181096
$584$	4.31899e9	0.897300
$585$	0	0
$586$	−8.33885e8	−0.171184
$587$	7.65712e8	0.156254	0.0781272	−	0.996943i	$-0.475106\pi$
$587$	7.65712e8	0.156254	0.0781272	+	0.996943i	$0.475106\pi$
$588$	0	0
$589$	−3.13936e8	−0.0633050
$590$	−6.76974e8	−0.135703
$591$	0	0
$592$	2.20495e9	0.436790
$593$	−1.51945e9	−0.299222	−0.149611	−	0.988745i	$-0.547802\pi$
$593$	−1.51945e9	−0.299222	−0.149611	+	0.988745i	$0.547802\pi$
$594$	0	0
$595$	2.32185e8	0.0451881
$596$	5.42255e9	1.04916
$597$	0	0
$598$	−1.39916e7	−0.00267555
$599$	8.16357e9	1.55198	0.775991	−	0.630745i	$-0.217250\pi$
$599$	8.16357e9	1.55198	0.775991	+	0.630745i	$0.217250\pi$
$600$	0	0
$601$	−8.29079e9	−1.55788	−0.778942	−	0.627096i	$-0.784243\pi$
$601$	−8.29079e9	−1.55788	−0.778942	+	0.627096i	$0.784243\pi$
$602$	−4.65007e7	−0.00868704
$603$	0	0
$604$	1.80220e9	0.332792
$605$	−4.58746e9	−0.842225
$606$	0	0
$607$	−5.48624e9	−0.995668	−0.497834	−	0.867272i	$-0.665871\pi$
$607$	−5.48624e9	−0.995668	−0.497834	+	0.867272i	$0.665871\pi$
$608$	2.02916e9	0.366146
$609$	0	0
$610$	−1.81146e9	−0.323128
$611$	1.24944e8	0.0221601
$612$	0	0
$613$	−5.20382e8	−0.0912453	−0.0456227	−	0.998959i	$-0.514527\pi$
$613$	−5.20382e8	−0.0912453	−0.0456227	+	0.998959i	$0.514527\pi$
$614$	−1.29911e9	−0.226495
$615$	0	0
$616$	9.76424e7	0.0168309
$617$	−1.65572e9	−0.283784	−0.141892	−	0.989882i	$-0.545319\pi$
$617$	−1.65572e9	−0.283784	−0.141892	+	0.989882i	$0.545319\pi$
$618$	0	0
$619$	−4.63680e8	−0.0785781	−0.0392890	−	0.999228i	$-0.512509\pi$
$619$	−4.63680e8	−0.0785781	−0.0392890	+	0.999228i	$0.512509\pi$
$620$	−6.38464e8	−0.107588
$621$	0	0
$622$	−1.90107e9	−0.316761
$623$	5.67166e7	0.00939728
$624$	0	0
$625$	−4.14581e9	−0.679250
$626$	8.94921e8	0.145806
$627$	0	0
$628$	−3.17783e8	−0.0512003
$629$	−1.01106e9	−0.161995
$630$	0	0
$631$	−5.00414e8	−0.0792915	−0.0396458	−	0.999214i	$-0.512623\pi$
$631$	−5.00414e8	−0.0792915	−0.0396458	+	0.999214i	$0.512623\pi$
$632$	−2.18475e9	−0.344264
$633$	0	0
$634$	3.31514e8	0.0516642
$635$	−5.91834e9	−0.917258
$636$	0	0
$637$	−1.33354e8	−0.0204418
$638$	7.76121e7	0.0118320
$639$	0	0
$640$	5.39208e9	0.813067
$641$	−6.72118e9	−1.00796	−0.503979	−	0.863716i	$-0.668131\pi$
$641$	−6.72118e9	−1.00796	−0.503979	+	0.863716i	$0.668131\pi$
$642$	0	0
$643$	−2.00229e9	−0.297021	−0.148511	−	0.988911i	$-0.547448\pi$
$643$	−2.00229e9	−0.297021	−0.148511	+	0.988911i	$0.547448\pi$
$644$	4.97228e8	0.0733592
$645$	0	0
$646$	−2.59837e8	−0.0379216
$647$	−1.06244e10	−1.54219	−0.771095	−	0.636720i	$-0.780291\pi$
$647$	−1.06244e10	−1.54219	−0.771095	+	0.636720i	$0.780291\pi$
$648$	0	0
$649$	−5.67031e8	−0.0814236
$650$	1.13332e7	0.00161866
$651$	0	0
$652$	6.16978e9	0.871772
$653$	2.49361e9	0.350456	0.175228	−	0.984528i	$-0.443934\pi$
$653$	2.49361e9	0.350456	0.175228	+	0.984528i	$0.443934\pi$
$654$	0	0
$655$	6.46102e9	0.898372
$656$	−4.06514e9	−0.562228
$657$	0	0
$658$	4.34868e8	0.0595068
$659$	7.14359e9	0.972339	0.486170	−	0.873864i	$-0.338394\pi$
$659$	7.14359e9	0.972339	0.486170	+	0.873864i	$0.338394\pi$
$660$	0	0
$661$	−1.06348e10	−1.43227	−0.716134	−	0.697963i	$-0.754090\pi$
$661$	−1.06348e10	−1.43227	−0.716134	+	0.697963i	$0.754090\pi$
$662$	−2.21001e9	−0.296067
$663$	0	0
$664$	−5.59227e9	−0.741310
$665$	5.77119e8	0.0761009
$666$	0	0
$667$	8.29161e8	0.108193
$668$	4.32368e9	0.561224
$669$	0	0
$670$	−3.16752e9	−0.406871
$671$	−1.51727e9	−0.193881
$672$	0	0
$673$	6.11664e9	0.773500	0.386750	−	0.922185i	$-0.373598\pi$
$673$	6.11664e9	0.773500	0.386750	+	0.922185i	$0.373598\pi$
$674$	−2.24022e9	−0.281826
$675$	0	0
$676$	7.31207e9	0.910388
$677$	−9.09572e9	−1.12662	−0.563309	−	0.826246i	$-0.690472\pi$
$677$	−9.09572e9	−1.12662	−0.563309	+	0.826246i	$0.690472\pi$
$678$	0	0
$679$	−8.75366e8	−0.107311
$680$	−1.10863e9	−0.135209
$681$	0	0
$682$	5.23749e7	0.00632234
$683$	−7.39960e9	−0.888660	−0.444330	−	0.895863i	$-0.646558\pi$
$683$	−7.39960e9	−0.888660	−0.444330	+	0.895863i	$0.646558\pi$
$684$	0	0
$685$	−1.15949e10	−1.37832
$686$	−9.45798e8	−0.111857
$687$	0	0
$688$	−9.64413e8	−0.112903
$689$	2.13306e6	0.000248448 0
$690$	0	0
$691$	7.73691e9	0.892061	0.446030	−	0.895018i	$-0.352837\pi$
$691$	7.73691e9	0.892061	0.446030	+	0.895018i	$0.352837\pi$
$692$	−4.58232e9	−0.525671
$693$	0	0
$694$	−2.02401e9	−0.229855
$695$	−1.43331e9	−0.161954
$696$	0	0
$697$	1.86405e9	0.208517
$698$	−3.70710e9	−0.412610
$699$	0	0
$700$	−4.02755e8	−0.0443811
$701$	−9.43506e9	−1.03450	−0.517251	−	0.855833i	$-0.673045\pi$
$701$	−9.43506e9	−1.03450	−0.517251	+	0.855833i	$0.673045\pi$
$702$	0	0
$703$	−2.51311e9	−0.272814
$704$	7.21278e8	0.0779109
$705$	0	0
$706$	4.07112e9	0.435409
$707$	2.26853e9	0.241422
$708$	0	0
$709$	−1.29011e10	−1.35945	−0.679726	−	0.733466i	$-0.737901\pi$
$709$	−1.29011e10	−1.35945	−0.679726	+	0.733466i	$0.737901\pi$
$710$	3.16813e9	0.332199
$711$	0	0
$712$	−2.70810e8	−0.0281180
$713$	5.59542e8	0.0578122
$714$	0	0
$715$	−2.76634e7	−0.00283032
$716$	−5.55754e9	−0.565831
$717$	0	0
$718$	−1.81983e9	−0.183482
$719$	−1.56677e10	−1.57201	−0.786004	−	0.618222i	$-0.787853\pi$
$719$	−1.56677e10	−1.57201	−0.786004	+	0.618222i	$0.787853\pi$
$720$	0	0
$721$	2.34802e9	0.233308
$722$	2.37457e9	0.234803
$723$	0	0
$724$	−2.38035e9	−0.233108
$725$	−6.71622e8	−0.0654549
$726$	0	0
$727$	1.29468e10	1.24966	0.624830	−	0.780761i	$-0.285168\pi$
$727$	1.29468e10	1.24966	0.624830	+	0.780761i	$0.285168\pi$
$728$	−2.40377e7	−0.00230904
$729$	0	0
$730$	−4.25884e9	−0.405192
$731$	4.42225e8	0.0418729
$732$	0	0
$733$	1.96470e10	1.84261	0.921305	−	0.388842i	$-0.127125\pi$
$733$	1.96470e10	1.84261	0.921305	+	0.388842i	$0.127125\pi$
$734$	−3.72297e9	−0.347499
$735$	0	0
$736$	−3.61666e9	−0.334376
$737$	−2.65310e9	−0.244128
$738$	0	0
$739$	5.46414e9	0.498042	0.249021	−	0.968498i	$-0.419891\pi$
$739$	5.46414e9	0.498042	0.249021	+	0.968498i	$0.419891\pi$
$740$	−5.11100e9	−0.463655
$741$	0	0
$742$	7.42410e6	0.000667160 0
$743$	−1.10547e10	−0.988750	−0.494375	−	0.869249i	$-0.664603\pi$
$743$	−1.10547e10	−0.988750	−0.494375	+	0.869249i	$0.664603\pi$
$744$	0	0
$745$	−1.12177e10	−0.993934
$746$	−5.80580e9	−0.512007
$747$	0	0
$748$	−4.42619e8	−0.0386700
$749$	−1.84519e9	−0.160455
$750$	0	0
$751$	−1.34448e10	−1.15828	−0.579141	−	0.815227i	$-0.696612\pi$
$751$	−1.34448e10	−1.15828	−0.579141	+	0.815227i	$0.696612\pi$
$752$	9.01906e9	0.773390
$753$	0	0
$754$	−1.91066e7	−0.00162324
$755$	−3.72824e9	−0.315275
$756$	0	0
$757$	4.89813e9	0.410388	0.205194	−	0.978721i	$-0.434217\pi$
$757$	4.89813e9	0.410388	0.205194	+	0.978721i	$0.434217\pi$
$758$	−3.62728e8	−0.0302509
$759$	0	0
$760$	−2.75562e9	−0.227705
$761$	−1.08629e10	−0.893512	−0.446756	−	0.894656i	$-0.647421\pi$
$761$	−1.08629e10	−0.893512	−0.446756	+	0.894656i	$0.647421\pi$
$762$	0	0
$763$	−1.60015e9	−0.130414
$764$	9.32552e9	0.756565
$765$	0	0
$766$	−2.18553e9	−0.175694
$767$	1.39592e8	0.0111706
$768$	0	0
$769$	1.74439e9	0.138325	0.0691627	−	0.997605i	$-0.477967\pi$
$769$	1.74439e9	0.138325	0.0691627	+	0.997605i	$0.477967\pi$
$770$	−9.62824e7	−0.00760028
$771$	0	0
$772$	4.46184e9	0.349022
$773$	2.29156e10	1.78444	0.892221	−	0.451598i	$-0.149146\pi$
$773$	2.29156e10	1.78444	0.892221	+	0.451598i	$0.149146\pi$
$774$	0	0
$775$	−4.53230e8	−0.0349754
$776$	4.17969e9	0.321091
$777$	0	0
$778$	−1.70733e8	−0.0129984
$779$	4.63328e9	0.351162
$780$	0	0
$781$	2.65361e9	0.199324
$782$	4.63118e8	0.0346312
$783$	0	0
$784$	−9.62611e9	−0.713419
$785$	6.57403e8	0.0485052
$786$	0	0
$787$	1.46509e10	1.07140	0.535701	−	0.844407i	$-0.320047\pi$
$787$	1.46509e10	1.07140	0.535701	+	0.844407i	$0.320047\pi$
$788$	1.43859e10	1.04736
$789$	0	0
$790$	2.15432e9	0.155459
$791$	−7.02460e8	−0.0504666
$792$	0	0
$793$	3.73523e8	0.0265988
$794$	3.80556e9	0.269803
$795$	0	0
$796$	9.27414e9	0.651745
$797$	1.25977e10	0.881426	0.440713	−	0.897648i	$-0.354726\pi$
$797$	1.25977e10	0.881426	0.440713	+	0.897648i	$0.354726\pi$
$798$	0	0
$799$	−4.13563e9	−0.286832
$800$	2.92950e9	0.202292
$801$	0	0
$802$	−306400.	−2.09739e−5 0
$803$	−3.56718e9	−0.243120
$804$	0	0
$805$	−1.02862e9	−0.0694978
$806$	−1.28937e7	−0.000867369 0
$807$	0	0
$808$	−1.08318e10	−0.722370
$809$	−2.20916e10	−1.46692	−0.733462	−	0.679731i	$-0.762097\pi$
$809$	−2.20916e10	−1.46692	−0.733462	+	0.679731i	$0.762097\pi$
$810$	0	0
$811$	−7.33733e9	−0.483020	−0.241510	−	0.970398i	$-0.577643\pi$
$811$	−7.33733e9	−0.483020	−0.241510	+	0.970398i	$0.577643\pi$
$812$	6.79003e8	0.0445067
$813$	0	0
$814$	4.19269e8	0.0272463
$815$	−1.27635e10	−0.825884
$816$	0	0
$817$	1.09920e9	0.0705177
$818$	1.10917e9	0.0708535
$819$	0	0
$820$	9.42287e9	0.596808
$821$	1.04735e10	0.660528	0.330264	−	0.943889i	$-0.392862\pi$
$821$	1.04735e10	0.660528	0.330264	+	0.943889i	$0.392862\pi$
$822$	0	0
$823$	2.13220e10	1.33330	0.666650	−	0.745371i	$-0.267727\pi$
$823$	2.13220e10	1.33330	0.666650	+	0.745371i	$0.267727\pi$
$824$	−1.12113e10	−0.698091
$825$	0	0
$826$	4.85849e8	0.0299965
$827$	−4.07306e9	−0.250410	−0.125205	−	0.992131i	$-0.539959\pi$
$827$	−4.07306e9	−0.250410	−0.125205	+	0.992131i	$0.539959\pi$
$828$	0	0
$829$	−4.97505e9	−0.303289	−0.151644	−	0.988435i	$-0.548457\pi$
$829$	−4.97505e9	−0.303289	−0.151644	+	0.988435i	$0.548457\pi$
$830$	5.51438e9	0.334752
$831$	0	0
$832$	−1.77565e8	−0.0106887
$833$	4.41399e9	0.264590
$834$	0	0
$835$	−8.94448e9	−0.531682
$836$	−1.10018e9	−0.0651238
$837$	0	0
$838$	6.78835e9	0.398483
$839$	−5.31627e9	−0.310771	−0.155386	−	0.987854i	$-0.549662\pi$
$839$	−5.31627e9	−0.310771	−0.155386	+	0.987854i	$0.549662\pi$
$840$	0	0
$841$	−1.61176e10	−0.934360
$842$	3.95397e9	0.228266
$843$	0	0
$844$	2.68225e10	1.53568
$845$	−1.51266e10	−0.862467
$846$	0	0
$847$	3.29231e9	0.186170
$848$	1.53974e8	0.00867086
$849$	0	0
$850$	−3.75126e8	−0.0209513
$851$	4.47921e9	0.249143
$852$	0	0
$853$	1.45978e10	0.805315	0.402658	−	0.915351i	$-0.368087\pi$
$853$	1.45978e10	0.805315	0.402658	+	0.915351i	$0.368087\pi$
$854$	1.30005e9	0.0714259
$855$	0	0
$856$	8.81039e9	0.480106
$857$	1.57329e10	0.853839	0.426919	−	0.904290i	$-0.359599\pi$
$857$	1.57329e10	0.853839	0.426919	+	0.904290i	$0.359599\pi$
$858$	0	0
$859$	1.26238e10	0.679540	0.339770	−	0.940509i	$-0.389651\pi$
$859$	1.26238e10	0.679540	0.339770	+	0.940509i	$0.389651\pi$
$860$	2.23548e9	0.119847
$861$	0	0
$862$	−9.62936e9	−0.512061
$863$	−5.25485e9	−0.278306	−0.139153	−	0.990271i	$-0.544438\pi$
$863$	−5.25485e9	−0.278306	−0.139153	+	0.990271i	$0.544438\pi$
$864$	0	0
$865$	9.47952e9	0.498000
$866$	−6.49008e9	−0.339576
$867$	0	0
$868$	4.58211e8	0.0237819
$869$	1.80445e9	0.0932771
$870$	0	0
$871$	6.53141e8	0.0334922
$872$	7.64038e9	0.390218
$873$	0	0
$874$	1.15113e9	0.0583221
$875$	4.09445e9	0.206617
$876$	0	0
$877$	−2.81828e10	−1.41086	−0.705432	−	0.708778i	$-0.749247\pi$
$877$	−2.81828e10	−1.41086	−0.705432	+	0.708778i	$0.749247\pi$
$878$	−2.65888e9	−0.132577
$879$	0	0
$880$	−1.99687e9	−0.0987782
$881$	−7.05789e9	−0.347744	−0.173872	−	0.984768i	$-0.555628\pi$
$881$	−7.05789e9	−0.347744	−0.173872	+	0.984768i	$0.555628\pi$
$882$	0	0
$883$	−3.54924e10	−1.73489	−0.867447	−	0.497530i	$-0.834240\pi$
$883$	−3.54924e10	−1.73489	−0.867447	+	0.497530i	$0.834240\pi$
$884$	1.08964e8	0.00530519
$885$	0	0
$886$	−6.66633e9	−0.322010
$887$	2.92725e10	1.40840	0.704202	−	0.709999i	$-0.251305\pi$
$887$	2.92725e10	1.40840	0.704202	+	0.709999i	$0.251305\pi$
$888$	0	0
$889$	4.24745e9	0.202755
$890$	2.67038e8	0.0126972
$891$	0	0
$892$	−2.26180e10	−1.06703
$893$	−1.02795e10	−0.483051
$894$	0	0
$895$	1.14970e10	0.536047
$896$	−3.86977e9	−0.179724
$897$	0	0
$898$	1.94706e8	0.00897248
$899$	7.64098e8	0.0350744
$900$	0	0
$901$	−7.06038e7	−0.00321582
$902$	−7.72983e8	−0.0350709
$903$	0	0
$904$	3.35410e9	0.151003
$905$	4.92428e9	0.220837
$906$	0	0
$907$	−3.76454e10	−1.67528	−0.837639	−	0.546224i	$-0.816065\pi$
$907$	−3.76454e10	−1.67528	−0.837639	+	0.546224i	$0.816065\pi$
$908$	2.90021e10	1.28567
$909$	0	0
$910$	2.37028e7	0.00104269
$911$	−6.61567e9	−0.289908	−0.144954	−	0.989438i	$-0.546303\pi$
$911$	−6.61567e9	−0.289908	−0.144954	+	0.989438i	$0.546303\pi$
$912$	0	0
$913$	4.61882e9	0.200855
$914$	−1.53303e9	−0.0664110
$915$	0	0
$916$	−3.04944e10	−1.31095
$917$	−4.63692e9	−0.198581
$918$	0	0
$919$	2.38719e10	1.01457	0.507285	−	0.861778i	$-0.330649\pi$
$919$	2.38719e10	1.01457	0.507285	+	0.861778i	$0.330649\pi$
$920$	4.91147e9	0.207948
$921$	0	0
$922$	7.09064e9	0.297939
$923$	−6.53267e8	−0.0273454
$924$	0	0
$925$	−3.62817e9	−0.150727
$926$	7.17548e8	0.0296970
$927$	0	0
$928$	−4.93883e9	−0.202864
$929$	2.31224e10	0.946190	0.473095	−	0.881012i	$-0.343137\pi$
$929$	2.31224e10	0.946190	0.473095	+	0.881012i	$0.343137\pi$
$930$	0	0
$931$	1.09714e10	0.445594
$932$	1.99270e10	0.806279
$933$	0	0
$934$	3.52882e9	0.141715
$935$	9.15653e8	0.0366345
$936$	0	0
$937$	−3.07105e10	−1.21954	−0.609772	−	0.792577i	$-0.708739\pi$
$937$	−3.07105e10	−1.21954	−0.609772	+	0.792577i	$0.708739\pi$
$938$	2.27325e9	0.0899369
$939$	0	0
$940$	−2.09059e10	−0.820958
$941$	−4.74404e8	−0.0185603	−0.00928015	−	0.999957i	$-0.502954\pi$
$941$	−4.74404e8	−0.0185603	−0.00928015	+	0.999957i	$0.502954\pi$
$942$	0	0
$943$	−8.25809e9	−0.320692
$944$	1.00764e10	0.389855
$945$	0	0
$946$	−1.83382e8	−0.00704268
$947$	3.03487e10	1.16122	0.580611	−	0.814181i	$-0.302814\pi$
$947$	3.03487e10	1.16122	0.580611	+	0.814181i	$0.302814\pi$
$948$	0	0
$949$	8.78171e8	0.0333539
$950$	−9.32416e8	−0.0352839
$951$	0	0
$952$	7.95641e8	0.0298874
$953$	−1.34857e10	−0.504717	−0.252359	−	0.967634i	$-0.581206\pi$
$953$	−1.34857e10	−0.504717	−0.252359	+	0.967634i	$0.581206\pi$
$954$	0	0
$955$	−1.92919e10	−0.716741
$956$	1.01019e10	0.373937
$957$	0	0
$958$	4.12552e9	0.151600
$959$	8.32137e9	0.304670
$960$	0	0
$961$	−2.69970e10	−0.981258
$962$	−1.03216e8	−0.00373795
$963$	0	0
$964$	1.39100e10	0.500100
$965$	−9.23028e9	−0.330651
$966$	0	0
$967$	5.31442e10	1.89001	0.945004	−	0.327060i	$-0.106058\pi$
$967$	5.31442e10	1.89001	0.945004	+	0.327060i	$0.106058\pi$
$968$	−1.57201e10	−0.557047
$969$	0	0
$970$	−4.12147e9	−0.144995
$971$	−1.88877e10	−0.662083	−0.331042	−	0.943616i	$-0.607400\pi$
$971$	−1.88877e10	−0.662083	−0.331042	+	0.943616i	$0.607400\pi$
$972$	0	0
$973$	1.02865e9	0.0357992
$974$	2.99261e9	0.103775
$975$	0	0
$976$	2.69626e10	0.928299
$977$	4.88964e9	0.167744	0.0838718	−	0.996477i	$-0.473271\pi$
$977$	4.88964e9	0.167744	0.0838718	+	0.996477i	$0.473271\pi$
$978$	0	0
$979$	2.23670e8	0.00761848
$980$	2.23130e10	0.757298
$981$	0	0
$982$	−5.76619e9	−0.194312
$983$	−4.22030e10	−1.41712	−0.708560	−	0.705651i	$-0.750655\pi$
$983$	−4.22030e10	−1.41712	−0.708560	+	0.705651i	$0.750655\pi$
$984$	0	0
$985$	−2.97603e10	−0.992227
$986$	6.32423e8	0.0210106
$987$	0	0
$988$	2.70842e8	0.00893442
$989$	−1.95915e9	−0.0643991
$990$	0	0
$991$	4.76499e10	1.55527	0.777633	−	0.628719i	$-0.216420\pi$
$991$	4.76499e10	1.55527	0.777633	+	0.628719i	$0.216420\pi$
$992$	−3.33286e9	−0.108399
$993$	0	0
$994$	−2.27369e9	−0.0734310
$995$	−1.91856e10	−0.617439
$996$	0	0
$997$	−6.81917e8	−0.0217921	−0.0108960	−	0.999941i	$-0.503468\pi$
$997$	−6.81917e8	−0.0217921	−0.0108960	+	0.999941i	$0.503468\pi$
$998$	6.22205e9	0.198142
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.d.1.6		13
3.2	odd	2		43.8.a.b.1.8	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.b.1.8	✓	13	3.2	odd	2
387.8.a.d.1.6		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)

\(f(q)\)	\(=\)	\(q-3.37903 q^{2} -116.582 q^{4} +241.175 q^{5} -173.086 q^{7} +826.450 q^{8} +O(q^{10})\)	\(q-3.37903 q^{2} -116.582 q^{4} +241.175 q^{5} -173.086 q^{7} +826.450 q^{8} -814.939 q^{10} -682.590 q^{11} +168.040 q^{13} +584.863 q^{14} +12129.9 q^{16} -5562.09 q^{17} -13825.2 q^{19} -28116.8 q^{20} +2306.49 q^{22} +24641.2 q^{23} -19959.4 q^{25} -567.813 q^{26} +20178.7 q^{28} +33649.4 q^{29} +22707.6 q^{31} -146773. q^{32} +18794.5 q^{34} -41744.1 q^{35} +181778. q^{37} +46715.6 q^{38} +199320. q^{40} -335134. q^{41} -79507.0 q^{43} +79577.8 q^{44} -83263.2 q^{46} +743538. q^{47} -793584. q^{49} +67443.4 q^{50} -19590.5 q^{52} +12693.7 q^{53} -164624. q^{55} -143047. q^{56} -113702. q^{58} +830705. q^{59} +2.22282e6 q^{61} -76729.7 q^{62} -1.05668e6 q^{64} +40527.2 q^{65} +3.88681e6 q^{67} +648441. q^{68} +141055. q^{70} -3.88756e6 q^{71} +5.22595e6 q^{73} -614232. q^{74} +1.61177e6 q^{76} +118147. q^{77} -2.64353e6 q^{79} +2.92544e6 q^{80} +1.13243e6 q^{82} -6.76661e6 q^{83} -1.34144e6 q^{85} +268657. q^{86} -564127. q^{88} -327679. q^{89} -29085.4 q^{91} -2.87272e6 q^{92} -2.51244e6 q^{94} -3.33429e6 q^{95} +5.05740e6 q^{97} +2.68154e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8}+O(q^{10}) \) 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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