LMFDB - Embedded newform 387.8.a.d.1.9

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			1.9
Root			$-4.99525$ 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)

$f(q)$	$=$	$q+3.99525 q^{2} -112.038 q^{4} +383.451 q^{5} +1003.83 q^{7} -959.011 q^{8} +O(q^{10})$	\(q+3.99525 q^{2} -112.038 q^{4} +383.451 q^{5} +1003.83 q^{7} -959.011 q^{8} +1531.98 q^{10} -1620.93 q^{11} +7962.52 q^{13} +4010.57 q^{14} +10509.4 q^{16} -36412.1 q^{17} -40353.3 q^{19} -42961.1 q^{20} -6476.03 q^{22} -13321.7 q^{23} +68909.7 q^{25} +31812.2 q^{26} -112468. q^{28} -119451. q^{29} -103590. q^{31} +164741. q^{32} -145476. q^{34} +384921. q^{35} +191874. q^{37} -161221. q^{38} -367734. q^{40} +240868. q^{41} -79507.0 q^{43} +181606. q^{44} -53223.7 q^{46} -502478. q^{47} +184140. q^{49} +275311. q^{50} -892105. q^{52} -1.09942e6 q^{53} -621548. q^{55} -962688. q^{56} -477236. q^{58} +2.43199e6 q^{59} -620335. q^{61} -413868. q^{62} -687019. q^{64} +3.05324e6 q^{65} -4.73308e6 q^{67} +4.07954e6 q^{68} +1.53786e6 q^{70} +2.81874e6 q^{71} +2.78345e6 q^{73} +766582. q^{74} +4.52110e6 q^{76} -1.62715e6 q^{77} +6.00989e6 q^{79} +4.02983e6 q^{80} +962329. q^{82} -4.68609e6 q^{83} -1.39623e7 q^{85} -317650. q^{86} +1.55449e6 q^{88} -2.94676e6 q^{89} +7.99305e6 q^{91} +1.49254e6 q^{92} -2.00753e6 q^{94} -1.54735e7 q^{95} -1.41877e7 q^{97} +735685. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.99525	0.353133	0.176567	−	0.984289i	$-0.443501\pi$
$2$	3.99525	0.353133	0.176567	+	0.984289i	$0.443501\pi$
$3$	0	0
$3$	0	0
$4$	−112.038	−0.875297
$5$	383.451	1.37188	0.685938	−	0.727660i	$-0.259392\pi$
$5$	383.451	1.37188	0.685938	+	0.727660i	$0.259392\pi$
$6$	0	0
$7$	1003.83	1.10616	0.553081	−	0.833127i	$-0.313452\pi$
$7$	1003.83	1.10616	0.553081	+	0.833127i	$0.313452\pi$
$8$	−959.011	−0.662230
$9$	0	0
$10$	1531.98	0.484455
$11$	−1620.93	−0.367190	−0.183595	−	0.983002i	$-0.558773\pi$
$11$	−1620.93	−0.367190	−0.183595	+	0.983002i	$0.558773\pi$
$12$	0	0
$13$	7962.52	1.00519	0.502596	−	0.864522i	$-0.332378\pi$
$13$	7962.52	1.00519	0.502596	+	0.864522i	$0.332378\pi$
$14$	4010.57	0.390623
$15$	0	0
$16$	10509.4	0.641441
$17$	−36412.1	−1.79753	−0.898763	−	0.438435i	$-0.855533\pi$
$17$	−36412.1	−1.79753	−0.898763	+	0.438435i	$0.855533\pi$
$18$	0	0
$19$	−40353.3	−1.34971	−0.674856	−	0.737950i	$-0.735794\pi$
$19$	−40353.3	−1.34971	−0.674856	+	0.737950i	$0.735794\pi$
$20$	−42961.1	−1.20080
$21$	0	0
$22$	−6476.03	−0.129667
$23$	−13321.7	−0.228304	−0.114152	−	0.993463i	$-0.536415\pi$
$23$	−13321.7	−0.228304	−0.114152	+	0.993463i	$0.536415\pi$
$24$	0	0
$25$	68909.7	0.882044
$26$	31812.2	0.354967
$27$	0	0
$28$	−112468.	−0.968220
$29$	−119451.	−0.909487	−0.454743	−	0.890623i	$-0.650269\pi$
$29$	−119451.	−0.909487	−0.454743	+	0.890623i	$0.650269\pi$
$30$	0	0
$31$	−103590.	−0.624529	−0.312264	−	0.949995i	$-0.601087\pi$
$31$	−103590.	−0.624529	−0.312264	+	0.949995i	$0.601087\pi$
$32$	164741.	0.888744
$33$	0	0
$34$	−145476.	−0.634766
$35$	384921.	1.51752
$36$	0	0
$37$	191874.	0.622743	0.311372	−	0.950288i	$-0.399212\pi$
$37$	191874.	0.622743	0.311372	+	0.950288i	$0.399212\pi$
$38$	−161221.	−0.476628
$39$	0	0
$40$	−367734.	−0.908497
$41$	240868.	0.545804	0.272902	−	0.962042i	$-0.412017\pi$
$41$	240868.	0.545804	0.272902	+	0.962042i	$0.412017\pi$
$42$	0	0
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	181606.	0.321400
$45$	0	0
$46$	−53223.7	−0.0806218
$47$	−502478.	−0.705951	−0.352976	−	0.935632i	$-0.614830\pi$
$47$	−502478.	−0.705951	−0.352976	+	0.935632i	$0.614830\pi$
$48$	0	0
$49$	184140.	0.223595
$50$	275311.	0.311479
$51$	0	0
$52$	−892105.	−0.879841
$53$	−1.09942e6	−1.01437	−0.507187	−	0.861836i	$-0.669315\pi$
$53$	−1.09942e6	−1.01437	−0.507187	+	0.861836i	$0.669315\pi$
$54$	0	0
$55$	−621548.	−0.503739
$56$	−962688.	−0.732534
$57$	0	0
$58$	−477236.	−0.321170
$59$	2.43199e6	1.54163	0.770813	−	0.637061i	$-0.219850\pi$
$59$	2.43199e6	1.54163	0.770813	+	0.637061i	$0.219850\pi$
$60$	0	0
$61$	−620335.	−0.349923	−0.174961	−	0.984575i	$-0.555980\pi$
$61$	−620335.	−0.349923	−0.174961	+	0.984575i	$0.555980\pi$
$62$	−413868.	−0.220542
$63$	0	0
$64$	−687019.	−0.327596
$65$	3.05324e6	1.37900
$66$	0	0
$67$	−4.73308e6	−1.92257	−0.961285	−	0.275555i	$-0.911138\pi$
$67$	−4.73308e6	−1.92257	−0.961285	+	0.275555i	$0.911138\pi$
$68$	4.07954e6	1.57337
$69$	0	0
$70$	1.53786e6	0.535886
$71$	2.81874e6	0.934655	0.467327	−	0.884084i	$-0.345217\pi$
$71$	2.81874e6	0.934655	0.467327	+	0.884084i	$0.345217\pi$
$72$	0	0
$73$	2.78345e6	0.837440	0.418720	−	0.908115i	$-0.362479\pi$
$73$	2.78345e6	0.837440	0.418720	+	0.908115i	$0.362479\pi$
$74$	766582.	0.219911
$75$	0	0
$76$	4.52110e6	1.18140
$77$	−1.62715e6	−0.406172
$78$	0	0
$79$	6.00989e6	1.37142	0.685712	−	0.727873i	$-0.259491\pi$
$79$	6.00989e6	1.37142	0.685712	+	0.727873i	$0.259491\pi$
$80$	4.02983e6	0.879978
$81$	0	0
$82$	962329.	0.192741
$83$	−4.68609e6	−0.899574	−0.449787	−	0.893136i	$-0.648500\pi$
$83$	−4.68609e6	−0.899574	−0.449787	+	0.893136i	$0.648500\pi$
$84$	0	0
$85$	−1.39623e7	−2.46598
$86$	−317650.	−0.0538523
$87$	0	0
$88$	1.55449e6	0.243164
$89$	−2.94676e6	−0.443077	−0.221539	−	0.975152i	$-0.571108\pi$
$89$	−2.94676e6	−0.443077	−0.221539	+	0.975152i	$0.571108\pi$
$90$	0	0
$91$	7.99305e6	1.11191
$92$	1.49254e6	0.199834
$93$	0	0
$94$	−2.00753e6	−0.249295
$95$	−1.54735e7	−1.85164
$96$	0	0
$97$	−1.41877e7	−1.57838	−0.789192	−	0.614147i	$-0.789500\pi$
$97$	−1.41877e7	−1.57838	−0.789192	+	0.614147i	$0.789500\pi$
$98$	735685.	0.0789588
$99$	0	0
$100$	−7.72051e6	−0.772051
$101$	−1.46728e7	−1.41706	−0.708531	−	0.705680i	$-0.750642\pi$
$101$	−1.46728e7	−1.41706	−0.708531	+	0.705680i	$0.750642\pi$
$102$	0	0
$103$	−1.29765e7	−1.17011	−0.585057	−	0.810992i	$-0.698928\pi$
$103$	−1.29765e7	−1.17011	−0.585057	+	0.810992i	$0.698928\pi$
$104$	−7.63615e6	−0.665668
$105$	0	0
$106$	−4.39246e6	−0.358209
$107$	1.57757e7	1.24493	0.622464	−	0.782648i	$-0.286132\pi$
$107$	1.57757e7	1.24493	0.622464	+	0.782648i	$0.286132\pi$
$108$	0	0
$109$	1.93156e7	1.42861	0.714307	−	0.699833i	$-0.246742\pi$
$109$	1.93156e7	1.42861	0.714307	+	0.699833i	$0.246742\pi$
$110$	−2.48324e6	−0.177887
$111$	0	0
$112$	1.05497e7	0.709538
$113$	2.22839e6	0.145283	0.0726417	−	0.997358i	$-0.476857\pi$
$113$	2.22839e6	0.145283	0.0726417	+	0.997358i	$0.476857\pi$
$114$	0	0
$115$	−5.10824e6	−0.313205
$116$	1.33830e7	0.796071
$117$	0	0
$118$	9.71639e6	0.544400
$119$	−3.65518e7	−1.98835
$120$	0	0
$121$	−1.68597e7	−0.865172
$122$	−2.47839e6	−0.123569
$123$	0	0
$124$	1.16060e7	0.546648
$125$	−3.53361e6	−0.161821
$126$	0	0
$127$	−3.40735e7	−1.47606	−0.738028	−	0.674770i	$-0.764243\pi$
$127$	−3.40735e7	−1.47606	−0.738028	+	0.674770i	$0.764243\pi$
$128$	−2.38317e7	−1.00443
$129$	0	0
$130$	1.21984e7	0.486970
$131$	−3.55332e7	−1.38097	−0.690487	−	0.723345i	$-0.742604\pi$
$131$	−3.55332e7	−1.38097	−0.690487	+	0.723345i	$0.742604\pi$
$132$	0	0
$133$	−4.05080e7	−1.49300
$134$	−1.89098e7	−0.678924
$135$	0	0
$136$	3.49197e7	1.19037
$137$	3.77219e7	1.25335	0.626674	−	0.779281i	$-0.284416\pi$
$137$	3.77219e7	1.25335	0.626674	+	0.779281i	$0.284416\pi$
$138$	0	0
$139$	1.44974e7	0.457867	0.228934	−	0.973442i	$-0.426476\pi$
$139$	1.44974e7	0.457867	0.228934	+	0.973442i	$0.426476\pi$
$140$	−4.31258e7	−1.32828
$141$	0	0
$142$	1.12616e7	0.330058
$143$	−1.29067e7	−0.369096
$144$	0	0
$145$	−4.58036e7	−1.24770
$146$	1.11206e7	0.295728
$147$	0	0
$148$	−2.14971e7	−0.545085
$149$	−2.17133e7	−0.537743	−0.268871	−	0.963176i	$-0.586651\pi$
$149$	−2.17133e7	−0.537743	−0.268871	+	0.963176i	$0.586651\pi$
$150$	0	0
$151$	−1.73559e7	−0.410231	−0.205115	−	0.978738i	$-0.565757\pi$
$151$	−1.73559e7	−0.410231	−0.205115	+	0.978738i	$0.565757\pi$
$152$	3.86992e7	0.893819
$153$	0	0
$154$	−6.50086e6	−0.143433
$155$	−3.97217e7	−0.856776
$156$	0	0
$157$	−2.86214e6	−0.0590258	−0.0295129	−	0.999564i	$-0.509396\pi$
$157$	−2.86214e6	−0.0590258	−0.0295129	+	0.999564i	$0.509396\pi$
$158$	2.40110e7	0.484296
$159$	0	0
$160$	6.31701e7	1.21925
$161$	−1.33728e7	−0.252541
$162$	0	0
$163$	2.66759e7	0.482461	0.241231	−	0.970468i	$-0.422449\pi$
$163$	2.66759e7	0.482461	0.241231	+	0.970468i	$0.422449\pi$
$164$	−2.69864e7	−0.477740
$165$	0	0
$166$	−1.87221e7	−0.317670
$167$	4.84714e7	0.805337	0.402669	−	0.915346i	$-0.368083\pi$
$167$	4.84714e7	0.805337	0.402669	+	0.915346i	$0.368083\pi$
$168$	0	0
$169$	653231.	0.0104103
$170$	−5.57827e7	−0.870821
$171$	0	0
$172$	8.90781e6	0.133482
$173$	−1.18312e8	−1.73726	−0.868632	−	0.495458i	$-0.835000\pi$
$173$	−1.18312e8	−1.73726	−0.868632	+	0.495458i	$0.835000\pi$
$174$	0	0
$175$	6.91739e7	0.975684
$176$	−1.70350e7	−0.235531
$177$	0	0
$178$	−1.17730e7	−0.156465
$179$	3.21297e7	0.418718	0.209359	−	0.977839i	$-0.432862\pi$
$179$	3.21297e7	0.418718	0.209359	+	0.977839i	$0.432862\pi$
$180$	0	0
$181$	−1.19731e8	−1.50083	−0.750413	−	0.660969i	$-0.770145\pi$
$181$	−1.19731e8	−1.50083	−0.750413	+	0.660969i	$0.770145\pi$
$182$	3.19342e7	0.392651
$183$	0	0
$184$	1.27757e7	0.151190
$185$	7.35741e7	0.854327
$186$	0	0
$187$	5.90216e7	0.660033
$188$	5.62967e7	0.617917
$189$	0	0
$190$	−6.18205e7	−0.653875
$191$	2.39670e7	0.248884	0.124442	−	0.992227i	$-0.460286\pi$
$191$	2.39670e7	0.248884	0.124442	+	0.992227i	$0.460286\pi$
$192$	0	0
$193$	5.81894e7	0.582631	0.291315	−	0.956627i	$-0.405907\pi$
$193$	5.81894e7	0.582631	0.291315	+	0.956627i	$0.405907\pi$
$194$	−5.66836e7	−0.557380
$195$	0	0
$196$	−2.06307e7	−0.195712
$197$	1.66210e6	0.0154890	0.00774452	−	0.999970i	$-0.497535\pi$
$197$	1.66210e6	0.0154890	0.00774452	+	0.999970i	$0.497535\pi$
$198$	0	0
$199$	2.39093e7	0.215070	0.107535	−	0.994201i	$-0.465704\pi$
$199$	2.39093e7	0.215070	0.107535	+	0.994201i	$0.465704\pi$
$200$	−6.60852e7	−0.584116
$201$	0	0
$202$	−5.86216e7	−0.500412
$203$	−1.19909e8	−1.00604
$204$	0	0
$205$	9.23612e7	0.748775
$206$	−5.18444e7	−0.413206
$207$	0	0
$208$	8.36811e7	0.644772
$209$	6.54099e7	0.495600
$210$	0	0
$211$	−1.89004e7	−0.138510	−0.0692550	−	0.997599i	$-0.522062\pi$
$211$	−1.89004e7	−0.138510	−0.0692550	+	0.997599i	$0.522062\pi$
$212$	1.23177e8	0.887879
$213$	0	0
$214$	6.30277e7	0.439626
$215$	−3.04870e7	−0.209209
$216$	0	0
$217$	−1.03987e8	−0.690830
$218$	7.71705e7	0.504491
$219$	0	0
$220$	6.96370e7	0.440921
$221$	−2.89933e8	−1.80686
$222$	0	0
$223$	−2.65364e8	−1.60241	−0.801207	−	0.598387i	$-0.795808\pi$
$223$	−2.65364e8	−1.60241	−0.801207	+	0.598387i	$0.795808\pi$
$224$	1.65373e8	0.983095
$225$	0	0
$226$	8.90296e6	0.0513044
$227$	−2.54830e8	−1.44597	−0.722986	−	0.690862i	$-0.757231\pi$
$227$	−2.54830e8	−1.44597	−0.722986	+	0.690862i	$0.757231\pi$
$228$	0	0
$229$	−1.45333e8	−0.799727	−0.399863	−	0.916575i	$-0.630942\pi$
$229$	−1.45333e8	−0.799727	−0.399863	+	0.916575i	$0.630942\pi$
$230$	−2.04087e7	−0.110603
$231$	0	0
$232$	1.14555e8	0.602289
$233$	−6.81068e7	−0.352732	−0.176366	−	0.984325i	$-0.556434\pi$
$233$	−6.81068e7	−0.352732	−0.176366	+	0.984325i	$0.556434\pi$
$234$	0	0
$235$	−1.92676e8	−0.968478
$236$	−2.72475e8	−1.34938
$237$	0	0
$238$	−1.46033e8	−0.702154
$239$	2.19022e8	1.03776	0.518878	−	0.854848i	$-0.326350\pi$
$239$	2.19022e8	1.03776	0.518878	+	0.854848i	$0.326350\pi$
$240$	0	0
$241$	1.44105e7	0.0663160	0.0331580	−	0.999450i	$-0.489444\pi$
$241$	1.44105e7	0.0663160	0.0331580	+	0.999450i	$0.489444\pi$
$242$	−6.73589e7	−0.305521
$243$	0	0
$244$	6.95011e7	0.306286
$245$	7.06087e7	0.306745
$246$	0	0
$247$	−3.21314e8	−1.35672
$248$	9.93441e7	0.413581
$249$	0	0
$250$	−1.41177e7	−0.0571442
$251$	2.09847e8	0.837615	0.418808	−	0.908075i	$-0.362448\pi$
$251$	2.09847e8	0.837615	0.418808	+	0.908075i	$0.362448\pi$
$252$	0	0
$253$	2.15937e7	0.0838309
$254$	−1.36132e8	−0.521245
$255$	0	0
$256$	−7.27492e6	−0.0271012
$257$	2.67717e8	0.983808	0.491904	−	0.870649i	$-0.336301\pi$
$257$	2.67717e8	0.983808	0.491904	+	0.870649i	$0.336301\pi$
$258$	0	0
$259$	1.92609e8	0.688855
$260$	−3.42079e8	−1.20703
$261$	0	0
$262$	−1.41964e8	−0.487668
$263$	1.00008e8	0.338993	0.169497	−	0.985531i	$-0.445786\pi$
$263$	1.00008e8	0.338993	0.169497	+	0.985531i	$0.445786\pi$
$264$	0	0
$265$	−4.21574e8	−1.39160
$266$	−1.61839e8	−0.527228
$267$	0	0
$268$	5.30285e8	1.68282
$269$	−8.77731e7	−0.274934	−0.137467	−	0.990506i	$-0.543896\pi$
$269$	−8.77731e7	−0.274934	−0.137467	+	0.990506i	$0.543896\pi$
$270$	0	0
$271$	−2.14577e8	−0.654924	−0.327462	−	0.944864i	$-0.606193\pi$
$271$	−2.14577e8	−0.654924	−0.327462	+	0.944864i	$0.606193\pi$
$272$	−3.82669e8	−1.15301
$273$	0	0
$274$	1.50708e8	0.442599
$275$	−1.11698e8	−0.323878
$276$	0	0
$277$	−3.67563e8	−1.03909	−0.519544	−	0.854443i	$-0.673898\pi$
$277$	−3.67563e8	−1.03909	−0.519544	+	0.854443i	$0.673898\pi$
$278$	5.79208e7	0.161688
$279$	0	0
$280$	−3.69144e8	−1.00495
$281$	3.86156e8	1.03822	0.519111	−	0.854707i	$-0.326263\pi$
$281$	3.86156e8	1.03822	0.519111	+	0.854707i	$0.326263\pi$
$282$	0	0
$283$	2.07849e7	0.0545125	0.0272562	−	0.999628i	$-0.491323\pi$
$283$	2.07849e7	0.0545125	0.0272562	+	0.999628i	$0.491323\pi$
$284$	−3.15806e8	−0.818100
$285$	0	0
$286$	−5.15655e7	−0.130340
$287$	2.41792e8	0.603747
$288$	0	0
$289$	9.15506e8	2.23110
$290$	−1.82997e8	−0.440606
$291$	0	0
$292$	−3.11852e8	−0.733008
$293$	−9.72174e7	−0.225791	−0.112896	−	0.993607i	$-0.536013\pi$
$293$	−9.72174e7	−0.225791	−0.112896	+	0.993607i	$0.536013\pi$
$294$	0	0
$295$	9.32548e8	2.11492
$296$	−1.84009e8	−0.412399
$297$	0	0
$298$	−8.67501e7	−0.189895
$299$	−1.06075e8	−0.229489
$300$	0	0
$301$	−7.98118e7	−0.168688
$302$	−6.93412e7	−0.144866
$303$	0	0
$304$	−4.24088e8	−0.865761
$305$	−2.37868e8	−0.480051
$306$	0	0
$307$	−8.13750e8	−1.60512	−0.802558	−	0.596573i	$-0.796528\pi$
$307$	−8.13750e8	−1.60512	−0.802558	+	0.596573i	$0.796528\pi$
$308$	1.82302e8	0.355521
$309$	0	0
$310$	−1.58698e8	−0.302556
$311$	4.77734e7	0.0900585	0.0450293	−	0.998986i	$-0.485662\pi$
$311$	4.77734e7	0.0900585	0.0450293	+	0.998986i	$0.485662\pi$
$312$	0	0
$313$	4.14127e7	0.0763357	0.0381679	−	0.999271i	$-0.487848\pi$
$313$	4.14127e7	0.0763357	0.0381679	+	0.999271i	$0.487848\pi$
$314$	−1.14349e7	−0.0208440
$315$	0	0
$316$	−6.73336e8	−1.20040
$317$	−1.29531e8	−0.228385	−0.114192	−	0.993459i	$-0.536428\pi$
$317$	−1.29531e8	−0.228385	−0.114192	+	0.993459i	$0.536428\pi$
$318$	0	0
$319$	1.93622e8	0.333954
$320$	−2.63438e8	−0.449422
$321$	0	0
$322$	−5.34277e7	−0.0891808
$323$	1.46935e9	2.42614
$324$	0	0
$325$	5.48695e8	0.886624
$326$	1.06577e8	0.170373
$327$	0	0
$328$	−2.30995e8	−0.361447
$329$	−5.04405e8	−0.780897
$330$	0	0
$331$	1.06545e9	1.61486	0.807431	−	0.589962i	$-0.200857\pi$
$331$	1.06545e9	1.61486	0.807431	+	0.589962i	$0.200857\pi$
$332$	5.25020e8	0.787395
$333$	0	0
$334$	1.93655e8	0.284391
$335$	−1.81491e9	−2.63753
$336$	0	0
$337$	−2.29211e8	−0.326235	−0.163118	−	0.986607i	$-0.552155\pi$
$337$	−2.29211e8	−0.326235	−0.163118	+	0.986607i	$0.552155\pi$
$338$	2.60982e6	0.00367622
$339$	0	0
$340$	1.56431e9	2.15847
$341$	1.67913e8	0.229321
$342$	0	0
$343$	−6.41855e8	−0.858830
$344$	7.62481e7	0.100989
$345$	0	0
$346$	−4.72684e8	−0.613486
$347$	5.28413e8	0.678923	0.339461	−	0.940620i	$-0.389755\pi$
$347$	5.28413e8	0.678923	0.339461	+	0.940620i	$0.389755\pi$
$348$	0	0
$349$	−4.51049e6	−0.00567982	−0.00283991	−	0.999996i	$-0.500904\pi$
$349$	−4.51049e6	−0.00567982	−0.00283991	+	0.999996i	$0.500904\pi$
$350$	2.76367e8	0.344547
$351$	0	0
$352$	−2.67034e8	−0.326338
$353$	−2.41469e8	−0.292179	−0.146090	−	0.989271i	$-0.546669\pi$
$353$	−2.41469e8	−0.292179	−0.146090	+	0.989271i	$0.546669\pi$
$354$	0	0
$355$	1.08085e9	1.28223
$356$	3.30149e8	0.387824
$357$	0	0
$358$	1.28366e8	0.147863
$359$	7.82115e8	0.892154	0.446077	−	0.894995i	$-0.352821\pi$
$359$	7.82115e8	0.892154	0.446077	+	0.894995i	$0.352821\pi$
$360$	0	0
$361$	7.34514e8	0.821722
$362$	−4.78353e8	−0.529992
$363$	0	0
$364$	−8.95525e8	−0.973247
$365$	1.06732e9	1.14886
$366$	0	0
$367$	−2.09454e8	−0.221186	−0.110593	−	0.993866i	$-0.535275\pi$
$367$	−2.09454e8	−0.221186	−0.110593	+	0.993866i	$0.535275\pi$
$368$	−1.40003e8	−0.146444
$369$	0	0
$370$	2.93947e8	0.301691
$371$	−1.10364e9	−1.12206
$372$	0	0
$373$	7.65743e8	0.764015	0.382008	−	0.924159i	$-0.375233\pi$
$373$	7.65743e8	0.764015	0.382008	+	0.924159i	$0.375233\pi$
$374$	2.35806e8	0.233080
$375$	0	0
$376$	4.81882e8	0.467502
$377$	−9.51130e8	−0.914208
$378$	0	0
$379$	5.91930e8	0.558513	0.279257	−	0.960217i	$-0.409912\pi$
$379$	5.91930e8	0.558513	0.279257	+	0.960217i	$0.409912\pi$
$380$	1.73362e9	1.62073
$381$	0	0
$382$	9.57543e7	0.0878894
$383$	−1.93100e9	−1.75625	−0.878125	−	0.478432i	$-0.841205\pi$
$383$	−1.93100e9	−1.75625	−0.878125	+	0.478432i	$0.841205\pi$
$384$	0	0
$385$	−6.23932e8	−0.557217
$386$	2.32481e8	0.205746
$387$	0	0
$388$	1.58957e9	1.38155
$389$	2.18340e8	0.188066	0.0940330	−	0.995569i	$-0.470024\pi$
$389$	2.18340e8	0.188066	0.0940330	+	0.995569i	$0.470024\pi$
$390$	0	0
$391$	4.85073e8	0.410382
$392$	−1.76592e8	−0.148071
$393$	0	0
$394$	6.64048e6	0.00546969
$395$	2.30450e9	1.88142
$396$	0	0
$397$	1.18371e9	0.949464	0.474732	−	0.880130i	$-0.342545\pi$
$397$	1.18371e9	0.949464	0.474732	+	0.880130i	$0.342545\pi$
$398$	9.55235e7	0.0759485
$399$	0	0
$400$	7.24198e8	0.565780
$401$	1.56318e9	1.21061	0.605304	−	0.795994i	$-0.293052\pi$
$401$	1.56318e9	1.21061	0.605304	+	0.795994i	$0.293052\pi$
$402$	0	0
$403$	−8.24838e8	−0.627771
$404$	1.64391e9	1.24035
$405$	0	0
$406$	−4.79066e8	−0.355266
$407$	−3.11014e8	−0.228665
$408$	0	0
$409$	−2.43312e8	−0.175846	−0.0879230	−	0.996127i	$-0.528023\pi$
$409$	−2.43312e8	−0.175846	−0.0879230	+	0.996127i	$0.528023\pi$
$410$	3.69006e8	0.264417
$411$	0	0
$412$	1.45386e9	1.02420
$413$	2.44131e9	1.70529
$414$	0	0
$415$	−1.79689e9	−1.23410
$416$	1.31175e9	0.893358
$417$	0	0
$418$	2.61329e8	0.175013
$419$	8.76950e8	0.582406	0.291203	−	0.956661i	$-0.405944\pi$
$419$	8.76950e8	0.582406	0.291203	+	0.956661i	$0.405944\pi$
$420$	0	0
$421$	4.62306e7	0.0301955	0.0150977	−	0.999886i	$-0.495194\pi$
$421$	4.62306e7	0.0301955	0.0150977	+	0.999886i	$0.495194\pi$
$422$	−7.55116e7	−0.0489125
$423$	0	0
$424$	1.05436e9	0.671749
$425$	−2.50915e9	−1.58550
$426$	0	0
$427$	−6.22714e8	−0.387071
$428$	−1.76747e9	−1.08968
$429$	0	0
$430$	−1.21803e8	−0.0738787
$431$	−2.41795e9	−1.45471	−0.727355	−	0.686261i	$-0.759251\pi$
$431$	−2.41795e9	−1.45471	−0.727355	+	0.686261i	$0.759251\pi$
$432$	0	0
$433$	−2.05635e9	−1.21728	−0.608638	−	0.793448i	$-0.708284\pi$
$433$	−2.05635e9	−1.21728	−0.608638	+	0.793448i	$0.708284\pi$
$434$	−4.15455e8	−0.243955
$435$	0	0
$436$	−2.16408e9	−1.25046
$437$	5.37576e8	0.308145
$438$	0	0
$439$	−3.96658e8	−0.223764	−0.111882	−	0.993722i	$-0.535688\pi$
$439$	−3.96658e8	−0.223764	−0.111882	+	0.993722i	$0.535688\pi$
$440$	5.96072e8	0.333591
$441$	0	0
$442$	−1.15835e9	−0.638062
$443$	2.23365e9	1.22068	0.610340	−	0.792139i	$-0.291033\pi$
$443$	2.23365e9	1.22068	0.610340	+	0.792139i	$0.291033\pi$
$444$	0	0
$445$	−1.12994e9	−0.607847
$446$	−1.06019e9	−0.565866
$447$	0	0
$448$	−6.89653e8	−0.362375
$449$	1.08951e9	0.568027	0.284014	−	0.958820i	$-0.408334\pi$
$449$	1.08951e9	0.568027	0.284014	+	0.958820i	$0.408334\pi$
$450$	0	0
$451$	−3.90432e8	−0.200413
$452$	−2.49664e8	−0.127166
$453$	0	0
$454$	−1.01811e9	−0.510621
$455$	3.06494e9	1.52540
$456$	0	0
$457$	9.19833e8	0.450819	0.225410	−	0.974264i	$-0.427628\pi$
$457$	9.19833e8	0.450819	0.225410	+	0.974264i	$0.427628\pi$
$458$	−5.80643e8	−0.282410
$459$	0	0
$460$	5.72317e8	0.274147
$461$	−3.88554e8	−0.184713	−0.0923567	−	0.995726i	$-0.529440\pi$
$461$	−3.88554e8	−0.184713	−0.0923567	+	0.995726i	$0.529440\pi$
$462$	0	0
$463$	1.15813e7	0.00542279	0.00271140	−	0.999996i	$-0.499137\pi$
$463$	1.15813e7	0.00542279	0.00271140	+	0.999996i	$0.499137\pi$
$464$	−1.25535e9	−0.583382
$465$	0	0
$466$	−2.72104e8	−0.124561
$467$	1.80641e9	0.820742	0.410371	−	0.911919i	$-0.365399\pi$
$467$	1.80641e9	0.820742	0.410371	+	0.911919i	$0.365399\pi$
$468$	0	0
$469$	−4.75123e9	−2.12667
$470$	−7.69788e8	−0.342002
$471$	0	0
$472$	−2.33230e9	−1.02091
$473$	1.28876e8	0.0559959
$474$	0	0
$475$	−2.78073e9	−1.19051
$476$	4.09519e9	1.74040
$477$	0	0
$478$	8.75047e8	0.366466
$479$	−4.53886e8	−0.188700	−0.0943500	−	0.995539i	$-0.530077\pi$
$479$	−4.53886e8	−0.188700	−0.0943500	+	0.995539i	$0.530077\pi$
$480$	0	0
$481$	1.52780e9	0.625976
$482$	5.75734e7	0.0234184
$483$	0	0
$484$	1.88893e9	0.757282
$485$	−5.44031e9	−2.16535
$486$	0	0
$487$	3.04275e9	1.19375	0.596877	−	0.802333i	$-0.296408\pi$
$487$	3.04275e9	1.19375	0.596877	+	0.802333i	$0.296408\pi$
$488$	5.94909e8	0.231729
$489$	0	0
$490$	2.82099e8	0.108322
$491$	−6.19673e8	−0.236253	−0.118126	−	0.992999i	$-0.537689\pi$
$491$	−6.19673e8	−0.236253	−0.118126	+	0.992999i	$0.537689\pi$
$492$	0	0
$493$	4.34946e9	1.63483
$494$	−1.28373e9	−0.479103
$495$	0	0
$496$	−1.08867e9	−0.400598
$497$	2.82955e9	1.03388
$498$	0	0
$499$	4.07486e9	1.46812	0.734059	−	0.679086i	$-0.237624\pi$
$499$	4.07486e9	1.46812	0.734059	+	0.679086i	$0.237624\pi$
$500$	3.95899e8	0.141641
$501$	0	0
$502$	8.38390e8	0.295790
$503$	3.44575e8	0.120725	0.0603623	−	0.998177i	$-0.480774\pi$
$503$	3.44575e8	0.120725	0.0603623	+	0.998177i	$0.480774\pi$
$504$	0	0
$505$	−5.62631e9	−1.94403
$506$	8.62720e7	0.0296035
$507$	0	0
$508$	3.81752e9	1.29199
$509$	−5.78650e8	−0.194493	−0.0972464	−	0.995260i	$-0.531003\pi$
$509$	−5.78650e8	−0.194493	−0.0972464	+	0.995260i	$0.531003\pi$
$510$	0	0
$511$	2.79412e9	0.926344
$512$	3.02139e9	0.994859
$513$	0	0
$514$	1.06960e9	0.347415
$515$	−4.97586e9	−1.60525
$516$	0	0
$517$	8.14484e8	0.259218
$518$	7.69521e8	0.243258
$519$	0	0
$520$	−2.92809e9	−0.913214
$521$	−3.77073e9	−1.16814	−0.584068	−	0.811705i	$-0.698540\pi$
$521$	−3.77073e9	−1.16814	−0.584068	+	0.811705i	$0.698540\pi$
$522$	0	0
$523$	5.11275e8	0.156278	0.0781391	−	0.996942i	$-0.475102\pi$
$523$	5.11275e8	0.156278	0.0781391	+	0.996942i	$0.475102\pi$
$524$	3.98107e9	1.20876
$525$	0	0
$526$	3.99558e8	0.119710
$527$	3.77194e9	1.12261
$528$	0	0
$529$	−3.22736e9	−0.947877
$530$	−1.68429e9	−0.491419
$531$	0	0
$532$	4.53843e9	1.30682
$533$	1.91792e9	0.548637
$534$	0	0
$535$	6.04920e9	1.70789
$536$	4.53908e9	1.27318
$537$	0	0
$538$	−3.50675e8	−0.0970884
$539$	−2.98479e8	−0.0821018
$540$	0	0
$541$	4.17611e9	1.13392	0.566959	−	0.823746i	$-0.308120\pi$
$541$	4.17611e9	1.13392	0.566959	+	0.823746i	$0.308120\pi$
$542$	−8.57289e8	−0.231276
$543$	0	0
$544$	−5.99857e9	−1.59754
$545$	7.40657e9	1.95988
$546$	0	0
$547$	1.14489e9	0.299093	0.149547	−	0.988755i	$-0.452219\pi$
$547$	1.14489e9	0.299093	0.149547	+	0.988755i	$0.452219\pi$
$548$	−4.22629e9	−1.09705
$549$	0	0
$550$	−4.46261e8	−0.114372
$551$	4.82023e9	1.22754
$552$	0	0
$553$	6.03294e9	1.51702
$554$	−1.46851e9	−0.366937
$555$	0	0
$556$	−1.62426e9	−0.400770
$557$	5.14104e9	1.26054	0.630271	−	0.776375i	$-0.282944\pi$
$557$	5.14104e9	1.26054	0.630271	+	0.776375i	$0.282944\pi$
$558$	0	0
$559$	−6.33076e8	−0.153290
$560$	4.04528e9	0.973399
$561$	0	0
$562$	1.54279e9	0.366631
$563$	9.25393e8	0.218548	0.109274	−	0.994012i	$-0.465147\pi$
$563$	9.25393e8	0.218548	0.109274	+	0.994012i	$0.465147\pi$
$564$	0	0
$565$	8.54478e8	0.199311
$566$	8.30409e7	0.0192502
$567$	0	0
$568$	−2.70321e9	−0.618956
$569$	−3.80230e9	−0.865273	−0.432637	−	0.901568i	$-0.642417\pi$
$569$	−3.80230e9	−0.865273	−0.432637	+	0.901568i	$0.642417\pi$
$570$	0	0
$571$	3.02988e9	0.681082	0.340541	−	0.940230i	$-0.389390\pi$
$571$	3.02988e9	0.681082	0.340541	+	0.940230i	$0.389390\pi$
$572$	1.44604e9	0.323069
$573$	0	0
$574$	9.66019e8	0.213203
$575$	−9.17998e8	−0.201374
$576$	0	0
$577$	4.05303e9	0.878344	0.439172	−	0.898403i	$-0.355272\pi$
$577$	4.05303e9	0.878344	0.439172	+	0.898403i	$0.355272\pi$
$578$	3.65767e9	0.787875
$579$	0	0
$580$	5.13174e9	1.09211
$581$	−4.70406e9	−0.995075
$582$	0	0
$583$	1.78209e9	0.372468
$584$	−2.66936e9	−0.554578
$585$	0	0
$586$	−3.88407e8	−0.0797344
$587$	−1.99299e9	−0.406697	−0.203348	−	0.979106i	$-0.565182\pi$
$587$	−1.99299e9	−0.406697	−0.203348	+	0.979106i	$0.565182\pi$
$588$	0	0
$589$	4.18020e9	0.842933
$590$	3.72576e9	0.746849
$591$	0	0
$592$	2.01647e9	0.399453
$593$	−6.10682e9	−1.20261	−0.601303	−	0.799021i	$-0.705352\pi$
$593$	−6.10682e9	−1.20261	−0.601303	+	0.799021i	$0.705352\pi$
$594$	0	0
$595$	−1.40158e10	−2.72778
$596$	2.43272e9	0.470684
$597$	0	0
$598$	−4.23795e8	−0.0810403
$599$	−6.06702e9	−1.15340	−0.576702	−	0.816954i	$-0.695661\pi$
$599$	−6.06702e9	−1.15340	−0.576702	+	0.816954i	$0.695661\pi$
$600$	0	0
$601$	−3.62416e9	−0.681000	−0.340500	−	0.940245i	$-0.610596\pi$
$601$	−3.62416e9	−0.681000	−0.340500	+	0.940245i	$0.610596\pi$
$602$	−3.18868e8	−0.0595694
$603$	0	0
$604$	1.94452e9	0.359074
$605$	−6.46489e9	−1.18691
$606$	0	0
$607$	6.80418e9	1.23485	0.617427	−	0.786629i	$-0.288175\pi$
$607$	6.80418e9	1.23485	0.617427	+	0.786629i	$0.288175\pi$
$608$	−6.64784e9	−1.19955
$609$	0	0
$610$	−9.50343e8	−0.169522
$611$	−4.00099e9	−0.709616
$612$	0	0
$613$	−5.90679e9	−1.03571	−0.517857	−	0.855467i	$-0.673270\pi$
$613$	−5.90679e9	−1.03571	−0.517857	+	0.855467i	$0.673270\pi$
$614$	−3.25113e9	−0.566820
$615$	0	0
$616$	1.56045e9	0.268979
$617$	3.23829e9	0.555031	0.277516	−	0.960721i	$-0.410489\pi$
$617$	3.23829e9	0.555031	0.277516	+	0.960721i	$0.410489\pi$
$618$	0	0
$619$	−3.39944e9	−0.576090	−0.288045	−	0.957617i	$-0.593005\pi$
$619$	−3.39944e9	−0.576090	−0.288045	+	0.957617i	$0.593005\pi$
$620$	4.45034e9	0.749933
$621$	0	0
$622$	1.90867e8	0.0318027
$623$	−2.95806e9	−0.490115
$624$	0	0
$625$	−6.73854e9	−1.10404
$626$	1.65454e8	0.0269567
$627$	0	0
$628$	3.20668e8	0.0516651
$629$	−6.98653e9	−1.11940
$630$	0	0
$631$	1.54777e9	0.245247	0.122623	−	0.992453i	$-0.460869\pi$
$631$	1.54777e9	0.245247	0.122623	+	0.992453i	$0.460869\pi$
$632$	−5.76356e9	−0.908198
$633$	0	0
$634$	−5.17509e8	−0.0806503
$635$	−1.30655e10	−2.02497
$636$	0	0
$637$	1.46622e9	0.224756
$638$	7.73567e8	0.117930
$639$	0	0
$640$	−9.13827e9	−1.37795
$641$	7.31802e9	1.09747	0.548733	−	0.835998i	$-0.315111\pi$
$641$	7.31802e9	1.09747	0.548733	+	0.835998i	$0.315111\pi$
$642$	0	0
$643$	−1.05084e10	−1.55883	−0.779413	−	0.626511i	$-0.784482\pi$
$643$	−1.05084e10	−1.55883	−0.779413	+	0.626511i	$0.784482\pi$
$644$	1.49826e9	0.221049
$645$	0	0
$646$	5.87041e9	0.856751
$647$	1.82079e9	0.264299	0.132150	−	0.991230i	$-0.457812\pi$
$647$	1.82079e9	0.264299	0.132150	+	0.991230i	$0.457812\pi$
$648$	0	0
$649$	−3.94209e9	−0.566070
$650$	2.19217e9	0.313096
$651$	0	0
$652$	−2.98872e9	−0.422297
$653$	−1.05192e10	−1.47838	−0.739190	−	0.673497i	$-0.764792\pi$
$653$	−1.05192e10	−1.47838	−0.739190	+	0.673497i	$0.764792\pi$
$654$	0	0
$655$	−1.36253e10	−1.89452
$656$	2.53138e9	0.350101
$657$	0	0
$658$	−2.01522e9	−0.275761
$659$	−7.91936e9	−1.07793	−0.538965	−	0.842328i	$-0.681185\pi$
$659$	−7.91936e9	−1.07793	−0.538965	+	0.842328i	$0.681185\pi$
$660$	0	0
$661$	1.09291e10	1.47191	0.735954	−	0.677031i	$-0.236734\pi$
$661$	1.09291e10	1.47191	0.735954	+	0.677031i	$0.236734\pi$
$662$	4.25674e9	0.570262
$663$	0	0
$664$	4.49401e9	0.595725
$665$	−1.55328e10	−2.04821
$666$	0	0
$667$	1.59129e9	0.207640
$668$	−5.43064e9	−0.704909
$669$	0	0
$670$	−7.25100e9	−0.931399
$671$	1.00552e9	0.128488
$672$	0	0
$673$	8.76741e9	1.10871	0.554356	−	0.832280i	$-0.312965\pi$
$673$	8.76741e9	1.10871	0.554356	+	0.832280i	$0.312965\pi$
$674$	−9.15754e8	−0.115204
$675$	0	0
$676$	−7.31866e7	−0.00911210
$677$	3.98192e8	0.0493211	0.0246605	−	0.999696i	$-0.492150\pi$
$677$	3.98192e8	0.0493211	0.0246605	+	0.999696i	$0.492150\pi$
$678$	0	0
$679$	−1.42421e10	−1.74595
$680$	1.33900e10	1.63305
$681$	0	0
$682$	6.70852e8	0.0809807
$683$	−4.45535e9	−0.535068	−0.267534	−	0.963548i	$-0.586209\pi$
$683$	−4.45535e9	−0.535068	−0.267534	+	0.963548i	$0.586209\pi$
$684$	0	0
$685$	1.44645e10	1.71944
$686$	−2.56437e9	−0.303281
$687$	0	0
$688$	−8.35569e8	−0.0978189
$689$	−8.75416e9	−1.01964
$690$	0	0
$691$	−1.49029e10	−1.71829	−0.859146	−	0.511730i	$-0.829005\pi$
$691$	−1.49029e10	−1.71829	−0.859146	+	0.511730i	$0.829005\pi$
$692$	1.32554e10	1.52062
$693$	0	0
$694$	2.11114e9	0.239750
$695$	5.55906e9	0.628137
$696$	0	0
$697$	−8.77054e9	−0.981096
$698$	−1.80205e7	−0.00200573
$699$	0	0
$700$	−7.75011e9	−0.854013
$701$	−4.40614e9	−0.483109	−0.241555	−	0.970387i	$-0.577657\pi$
$701$	−4.40614e9	−0.483109	−0.241555	+	0.970387i	$0.577657\pi$
$702$	0	0
$703$	−7.74272e9	−0.840524
$704$	1.11361e9	0.120290
$705$	0	0
$706$	−9.64727e8	−0.103178
$707$	−1.47291e10	−1.56750
$708$	0	0
$709$	−7.42152e9	−0.782043	−0.391022	−	0.920381i	$-0.627878\pi$
$709$	−7.42152e9	−0.782043	−0.391022	+	0.920381i	$0.627878\pi$
$710$	4.31826e9	0.452798
$711$	0	0
$712$	2.82597e9	0.293419
$713$	1.38000e9	0.142582
$714$	0	0
$715$	−4.94909e9	−0.506354
$716$	−3.59975e9	−0.366503
$717$	0	0
$718$	3.12474e9	0.315049
$719$	7.84543e9	0.787165	0.393582	−	0.919289i	$-0.371236\pi$
$719$	7.84543e9	0.787165	0.393582	+	0.919289i	$0.371236\pi$
$720$	0	0
$721$	−1.30263e10	−1.29434
$722$	2.93456e9	0.290177
$723$	0	0
$724$	1.34144e10	1.31367
$725$	−8.23132e9	−0.802207
$726$	0	0
$727$	1.92504e10	1.85810	0.929052	−	0.369949i	$-0.120625\pi$
$727$	1.92504e10	1.85810	0.929052	+	0.369949i	$0.120625\pi$
$728$	−7.66543e9	−0.736337
$729$	0	0
$730$	4.26420e9	0.405702
$731$	2.89502e9	0.274120
$732$	0	0
$733$	−1.25342e10	−1.17553	−0.587766	−	0.809031i	$-0.699992\pi$
$733$	−1.25342e10	−1.17553	−0.587766	+	0.809031i	$0.699992\pi$
$734$	−8.36820e8	−0.0781081
$735$	0	0
$736$	−2.19464e9	−0.202904
$737$	7.67201e9	0.705948
$738$	0	0
$739$	3.51649e9	0.320519	0.160259	−	0.987075i	$-0.448767\pi$
$739$	3.51649e9	0.320519	0.160259	+	0.987075i	$0.448767\pi$
$740$	−8.24310e9	−0.747789
$741$	0	0
$742$	−4.40930e9	−0.396238
$743$	−3.44470e9	−0.308099	−0.154050	−	0.988063i	$-0.549232\pi$
$743$	−3.44470e9	−0.308099	−0.154050	+	0.988063i	$0.549232\pi$
$744$	0	0
$745$	−8.32600e9	−0.737716
$746$	3.05933e9	0.269799
$747$	0	0
$748$	−6.61267e9	−0.577725
$749$	1.58362e10	1.37709
$750$	0	0
$751$	5.87692e9	0.506302	0.253151	−	0.967427i	$-0.418533\pi$
$751$	5.87692e9	0.506302	0.253151	+	0.967427i	$0.418533\pi$
$752$	−5.28073e9	−0.452826
$753$	0	0
$754$	−3.80000e9	−0.322837
$755$	−6.65514e9	−0.562786
$756$	0	0
$757$	−8.53566e9	−0.715157	−0.357579	−	0.933883i	$-0.616398\pi$
$757$	−8.53566e9	−0.715157	−0.357579	+	0.933883i	$0.616398\pi$
$758$	2.36491e9	0.197230
$759$	0	0
$760$	1.48393e10	1.22621
$761$	−1.24999e10	−1.02815	−0.514077	−	0.857744i	$-0.671866\pi$
$761$	−1.24999e10	−1.02815	−0.514077	+	0.857744i	$0.671866\pi$
$762$	0	0
$763$	1.93896e10	1.58028
$764$	−2.68522e9	−0.217848
$765$	0	0
$766$	−7.71482e9	−0.620190
$767$	1.93647e10	1.54963
$768$	0	0
$769$	−9.50817e9	−0.753971	−0.376985	−	0.926219i	$-0.623039\pi$
$769$	−9.50817e9	−0.753971	−0.376985	+	0.926219i	$0.623039\pi$
$770$	−2.49276e9	−0.196772
$771$	0	0
$772$	−6.51943e9	−0.509975
$773$	1.06727e10	0.831089	0.415544	−	0.909573i	$-0.363591\pi$
$773$	1.06727e10	0.831089	0.415544	+	0.909573i	$0.363591\pi$
$774$	0	0
$775$	−7.13836e9	−0.550862
$776$	1.36062e10	1.04525
$777$	0	0
$778$	8.72323e8	0.0664123
$779$	−9.71982e9	−0.736677
$780$	0	0
$781$	−4.56899e9	−0.343196
$782$	1.93799e9	0.144920
$783$	0	0
$784$	1.93520e9	0.143423
$785$	−1.09749e9	−0.0809760
$786$	0	0
$787$	2.51180e10	1.83685	0.918425	−	0.395596i	$-0.129462\pi$
$787$	2.51180e10	1.83685	0.918425	+	0.395596i	$0.129462\pi$
$788$	−1.86218e8	−0.0135575
$789$	0	0
$790$	9.20705e9	0.664394
$791$	2.23693e9	0.160707
$792$	0	0
$793$	−4.93943e9	−0.351739
$794$	4.72921e9	0.335287
$795$	0	0
$796$	−2.67875e9	−0.188250
$797$	−1.35480e10	−0.947917	−0.473959	−	0.880547i	$-0.657175\pi$
$797$	−1.35480e10	−0.947917	−0.473959	+	0.880547i	$0.657175\pi$
$798$	0	0
$799$	1.82963e10	1.26897
$800$	1.13523e10	0.783912
$801$	0	0
$802$	6.24529e9	0.427506
$803$	−4.51179e9	−0.307499
$804$	0	0
$805$	−5.12782e9	−0.346455
$806$	−3.29543e9	−0.221687
$807$	0	0
$808$	1.40714e10	0.938421
$809$	−1.21582e10	−0.807328	−0.403664	−	0.914907i	$-0.632264\pi$
$809$	−1.21582e10	−0.807328	−0.403664	+	0.914907i	$0.632264\pi$
$810$	0	0
$811$	2.69149e10	1.77182	0.885911	−	0.463855i	$-0.153534\pi$
$811$	2.69149e10	1.77182	0.885911	+	0.463855i	$0.153534\pi$
$812$	1.34343e10	0.880583
$813$	0	0
$814$	−1.24258e9	−0.0807492
$815$	1.02289e10	0.661877
$816$	0	0
$817$	3.20837e9	0.205829
$818$	−9.72093e8	−0.0620971
$819$	0	0
$820$	−1.03480e10	−0.655400
$821$	1.50522e10	0.949292	0.474646	−	0.880177i	$-0.342576\pi$
$821$	1.50522e10	0.949292	0.474646	+	0.880177i	$0.342576\pi$
$822$	0	0
$823$	−1.95484e10	−1.22240	−0.611198	−	0.791478i	$-0.709312\pi$
$823$	−1.95484e10	−1.22240	−0.611198	+	0.791478i	$0.709312\pi$
$824$	1.24446e10	0.774885
$825$	0	0
$826$	9.75364e9	0.602194
$827$	2.07950e9	0.127847	0.0639233	−	0.997955i	$-0.479639\pi$
$827$	2.07950e9	0.127847	0.0639233	+	0.997955i	$0.479639\pi$
$828$	0	0
$829$	2.13263e10	1.30009	0.650047	−	0.759894i	$-0.274749\pi$
$829$	2.13263e10	1.30009	0.650047	+	0.759894i	$0.274749\pi$
$830$	−7.17900e9	−0.435804
$831$	0	0
$832$	−5.47040e9	−0.329297
$833$	−6.70493e9	−0.401918
$834$	0	0
$835$	1.85864e10	1.10482
$836$	−7.32840e9	−0.433797
$837$	0	0
$838$	3.50363e9	0.205667
$839$	1.08997e10	0.637161	0.318580	−	0.947896i	$-0.396794\pi$
$839$	1.08997e10	0.637161	0.318580	+	0.947896i	$0.396794\pi$
$840$	0	0
$841$	−2.98137e9	−0.172834
$842$	1.84703e8	0.0106630
$843$	0	0
$844$	2.11756e9	0.121237
$845$	2.50482e8	0.0142816
$846$	0	0
$847$	−1.69244e10	−0.957020
$848$	−1.15542e10	−0.650662
$849$	0	0
$850$	−1.00247e10	−0.559892
$851$	−2.55609e9	−0.142175
$852$	0	0
$853$	−2.05206e10	−1.13206	−0.566028	−	0.824386i	$-0.691521\pi$
$853$	−2.05206e10	−1.13206	−0.566028	+	0.824386i	$0.691521\pi$
$854$	−2.48790e9	−0.136688
$855$	0	0
$856$	−1.51290e10	−0.824429
$857$	2.27910e10	1.23689	0.618443	−	0.785830i	$-0.287764\pi$
$857$	2.27910e10	1.23689	0.618443	+	0.785830i	$0.287764\pi$
$858$	0	0
$859$	−5.82236e9	−0.313418	−0.156709	−	0.987645i	$-0.550088\pi$
$859$	−5.82236e9	−0.313418	−0.156709	+	0.987645i	$0.550088\pi$
$860$	3.41571e9	0.183120
$861$	0	0
$862$	−9.66030e9	−0.513707
$863$	−1.77088e10	−0.937891	−0.468945	−	0.883227i	$-0.655366\pi$
$863$	−1.77088e10	−0.937891	−0.468945	+	0.883227i	$0.655366\pi$
$864$	0	0
$865$	−4.53667e10	−2.38331
$866$	−8.21562e9	−0.429861
$867$	0	0
$868$	1.16505e10	0.604681
$869$	−9.74163e9	−0.503573
$870$	0	0
$871$	−3.76873e10	−1.93255
$872$	−1.85238e10	−0.946070
$873$	0	0
$874$	2.14775e9	0.108816
$875$	−3.54716e9	−0.179000
$876$	0	0
$877$	2.41157e10	1.20726	0.603630	−	0.797264i	$-0.293720\pi$
$877$	2.41157e10	1.20726	0.603630	+	0.797264i	$0.293720\pi$
$878$	−1.58475e9	−0.0790185
$879$	0	0
$880$	−6.53209e9	−0.323119
$881$	−2.64320e10	−1.30231	−0.651155	−	0.758945i	$-0.725715\pi$
$881$	−2.64320e10	−1.30231	−0.651155	+	0.758945i	$0.725715\pi$
$882$	0	0
$883$	2.34325e10	1.14540	0.572699	−	0.819766i	$-0.305896\pi$
$883$	2.34325e10	1.14540	0.572699	+	0.819766i	$0.305896\pi$
$884$	3.24835e10	1.58154
$885$	0	0
$886$	8.92398e9	0.431063
$887$	2.01156e10	0.967832	0.483916	−	0.875114i	$-0.339214\pi$
$887$	2.01156e10	0.967832	0.483916	+	0.875114i	$0.339214\pi$
$888$	0	0
$889$	−3.42041e10	−1.63276
$890$	−4.51438e9	−0.214651
$891$	0	0
$892$	2.97308e10	1.40259
$893$	2.02766e10	0.952831
$894$	0	0
$895$	1.23202e10	0.574429
$896$	−2.39230e10	−1.11106
$897$	0	0
$898$	4.35286e9	0.200589
$899$	1.23739e10	0.568000
$900$	0	0
$901$	4.00323e10	1.82336
$902$	−1.55987e9	−0.0707727
$903$	0	0
$904$	−2.13705e9	−0.0962110
$905$	−4.59108e10	−2.05895
$906$	0	0
$907$	−2.12612e10	−0.946156	−0.473078	−	0.881021i	$-0.656857\pi$
$907$	−2.12612e10	−0.946156	−0.473078	+	0.881021i	$0.656857\pi$
$908$	2.85506e10	1.26566
$909$	0	0
$910$	1.22452e10	0.538668
$911$	−5.63102e9	−0.246759	−0.123379	−	0.992360i	$-0.539373\pi$
$911$	−5.63102e9	−0.246759	−0.123379	+	0.992360i	$0.539373\pi$
$912$	0	0
$913$	7.59583e9	0.330315
$914$	3.67496e9	0.159199
$915$	0	0
$916$	1.62829e10	0.699998
$917$	−3.56695e10	−1.52758
$918$	0	0
$919$	−1.37242e10	−0.583286	−0.291643	−	0.956527i	$-0.594202\pi$
$919$	−1.37242e10	−0.583286	−0.291643	+	0.956527i	$0.594202\pi$
$920$	4.89886e9	0.207414
$921$	0	0
$922$	−1.55237e9	−0.0652285
$923$	2.24443e10	0.939507
$924$	0	0
$925$	1.32220e10	0.549287
$926$	4.62701e7	0.00191497
$927$	0	0
$928$	−1.96785e10	−0.808301
$929$	−1.80192e10	−0.737363	−0.368682	−	0.929556i	$-0.620191\pi$
$929$	−1.80192e10	−0.737363	−0.368682	+	0.929556i	$0.620191\pi$
$930$	0	0
$931$	−7.43065e9	−0.301789
$932$	7.63055e9	0.308745
$933$	0	0
$934$	7.21705e9	0.289831
$935$	2.26319e10	0.905484
$936$	0	0
$937$	1.76207e10	0.699736	0.349868	−	0.936799i	$-0.386227\pi$
$937$	1.76207e10	0.699736	0.349868	+	0.936799i	$0.386227\pi$
$938$	−1.89823e10	−0.751000
$939$	0	0
$940$	2.15870e10	0.847706
$941$	3.45116e9	0.135021	0.0675105	−	0.997719i	$-0.478494\pi$
$941$	3.45116e9	0.135021	0.0675105	+	0.997719i	$0.478494\pi$
$942$	0	0
$943$	−3.20879e9	−0.124609
$944$	2.55587e10	0.988863
$945$	0	0
$946$	5.14890e8	0.0197740
$947$	−5.63093e9	−0.215454	−0.107727	−	0.994181i	$-0.534357\pi$
$947$	−5.63093e9	−0.215454	−0.107727	+	0.994181i	$0.534357\pi$
$948$	0	0
$949$	2.21633e10	0.841787
$950$	−1.11097e10	−0.420407
$951$	0	0
$952$	3.50535e10	1.31675
$953$	−4.39854e10	−1.64620	−0.823101	−	0.567896i	$-0.807758\pi$
$953$	−4.39854e10	−1.64620	−0.823101	+	0.567896i	$0.807758\pi$
$954$	0	0
$955$	9.19019e9	0.341439
$956$	−2.45388e10	−0.908344
$957$	0	0
$958$	−1.81339e9	−0.0666363
$959$	3.78666e10	1.38641
$960$	0	0
$961$	−1.67817e10	−0.609964
$962$	6.10393e9	0.221053
$963$	0	0
$964$	−1.61452e9	−0.0580462
$965$	2.23128e10	0.799298
$966$	0	0
$967$	3.74348e10	1.33132	0.665660	−	0.746255i	$-0.268150\pi$
$967$	3.74348e10	1.33132	0.665660	+	0.746255i	$0.268150\pi$
$968$	1.61687e10	0.572942
$969$	0	0
$970$	−2.17354e10	−0.764656
$971$	−3.42358e9	−0.120009	−0.0600044	−	0.998198i	$-0.519111\pi$
$971$	−3.42358e9	−0.120009	−0.0600044	+	0.998198i	$0.519111\pi$
$972$	0	0
$973$	1.45530e10	0.506475
$974$	1.21565e10	0.421554
$975$	0	0
$976$	−6.51934e9	−0.224455
$977$	5.03847e10	1.72849	0.864247	−	0.503068i	$-0.167795\pi$
$977$	5.03847e10	1.72849	0.864247	+	0.503068i	$0.167795\pi$
$978$	0	0
$979$	4.77650e9	0.162693
$980$	−7.91086e9	−0.268493
$981$	0	0
$982$	−2.47575e9	−0.0834288
$983$	3.95623e10	1.32845	0.664224	−	0.747534i	$-0.268762\pi$
$983$	3.95623e10	1.32845	0.664224	+	0.747534i	$0.268762\pi$
$984$	0	0
$985$	6.37332e8	0.0212490
$986$	1.73772e10	0.577311
$987$	0	0
$988$	3.59993e10	1.18753
$989$	1.05917e9	0.0348160
$990$	0	0
$991$	1.79657e10	0.586388	0.293194	−	0.956053i	$-0.405282\pi$
$991$	1.79657e10	0.586388	0.293194	+	0.956053i	$0.405282\pi$
$992$	−1.70655e10	−0.555046
$993$	0	0
$994$	1.13048e10	0.365097
$995$	9.16804e9	0.295050
$996$	0	0
$997$	1.97079e9	0.0629807	0.0314904	−	0.999504i	$-0.489975\pi$
$997$	1.97079e9	0.0629807	0.0314904	+	0.999504i	$0.489975\pi$
$998$	1.62801e10	0.518441
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.d.1.9		13
3.2	odd	2		43.8.a.b.1.5	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.b.1.5	✓	13	3.2	odd	2
387.8.a.d.1.9		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)

\(f(q)\)	\(=\)	\(q+3.99525 q^{2} -112.038 q^{4} +383.451 q^{5} +1003.83 q^{7} -959.011 q^{8} +O(q^{10})\)	\(q+3.99525 q^{2} -112.038 q^{4} +383.451 q^{5} +1003.83 q^{7} -959.011 q^{8} +1531.98 q^{10} -1620.93 q^{11} +7962.52 q^{13} +4010.57 q^{14} +10509.4 q^{16} -36412.1 q^{17} -40353.3 q^{19} -42961.1 q^{20} -6476.03 q^{22} -13321.7 q^{23} +68909.7 q^{25} +31812.2 q^{26} -112468. q^{28} -119451. q^{29} -103590. q^{31} +164741. q^{32} -145476. q^{34} +384921. q^{35} +191874. q^{37} -161221. q^{38} -367734. q^{40} +240868. q^{41} -79507.0 q^{43} +181606. q^{44} -53223.7 q^{46} -502478. q^{47} +184140. q^{49} +275311. q^{50} -892105. q^{52} -1.09942e6 q^{53} -621548. q^{55} -962688. q^{56} -477236. q^{58} +2.43199e6 q^{59} -620335. q^{61} -413868. q^{62} -687019. q^{64} +3.05324e6 q^{65} -4.73308e6 q^{67} +4.07954e6 q^{68} +1.53786e6 q^{70} +2.81874e6 q^{71} +2.78345e6 q^{73} +766582. q^{74} +4.52110e6 q^{76} -1.62715e6 q^{77} +6.00989e6 q^{79} +4.02983e6 q^{80} +962329. q^{82} -4.68609e6 q^{83} -1.39623e7 q^{85} -317650. q^{86} +1.55449e6 q^{88} -2.94676e6 q^{89} +7.99305e6 q^{91} +1.49254e6 q^{92} -2.00753e6 q^{94} -1.54735e7 q^{95} -1.41877e7 q^{97} +735685. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 16 q^{2} + 922 q^{4} - 998 q^{5} + 1360 q^{7} - 3870 q^{8}+O(q^{10}) \) 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

L-functions

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