LMFDB - Embedded newform 387.8.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.7
Root			$6.31419$ 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)

$f(q)$	$=$	$q+8.31419 q^{2} -58.8742 q^{4} -148.086 q^{5} +122.189 q^{7} -1553.71 q^{8} +O(q^{10})$	\(q+8.31419 q^{2} -58.8742 q^{4} -148.086 q^{5} +122.189 q^{7} -1553.71 q^{8} -1231.21 q^{10} -5634.96 q^{11} -7738.55 q^{13} +1015.90 q^{14} -5381.93 q^{16} -1712.43 q^{17} -51155.1 q^{19} +8718.44 q^{20} -46850.2 q^{22} +77057.3 q^{23} -56195.6 q^{25} -64339.8 q^{26} -7193.77 q^{28} +48088.8 q^{29} -192432. q^{31} +154128. q^{32} -14237.5 q^{34} -18094.4 q^{35} +461302. q^{37} -425313. q^{38} +230082. q^{40} -688771. q^{41} +79507.0 q^{43} +331754. q^{44} +640669. q^{46} +619513. q^{47} -808613. q^{49} -467221. q^{50} +455601. q^{52} -574333. q^{53} +834458. q^{55} -189846. q^{56} +399820. q^{58} +1.01363e6 q^{59} -3.23249e6 q^{61} -1.59991e6 q^{62} +1.97034e6 q^{64} +1.14597e6 q^{65} +3.15895e6 q^{67} +100818. q^{68} -150441. q^{70} -2.46302e6 q^{71} -4.02731e6 q^{73} +3.83535e6 q^{74} +3.01172e6 q^{76} -688530. q^{77} -463495. q^{79} +796987. q^{80} -5.72657e6 q^{82} +5.49654e6 q^{83} +253587. q^{85} +661036. q^{86} +8.75509e6 q^{88} +8.18997e6 q^{89} -945564. q^{91} -4.53669e6 q^{92} +5.15075e6 q^{94} +7.57535e6 q^{95} +1.20009e7 q^{97} -6.72296e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) $ 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8 - 1333 * q^10 - 1333 * q^11 - 17967 * q^13 + 22352 * q^14 - 34406 * q^16 + 63095 * q^17 - 54524 * q^19 + 280995 * q^20 - 289358 * q^22 + 138139 * q^23 + 3455 * q^25 + 132946 * q^26 - 12704 * q^28 + 308658 * q^29 - 209523 * q^31 + 644934 * q^32 + 762435 * q^34 + 578892 * q^35 - 298472 * q^37 + 369707 * q^38 + 2633173 * q^40 + 1346735 * q^41 + 874577 * q^43 - 3134292 * q^44 + 3588111 * q^46 - 499284 * q^47 + 2544563 * q^49 - 3049745 * q^50 + 983088 * q^52 + 2210495 * q^53 - 1855072 * q^55 + 469976 * q^56 + 4397067 * q^58 + 5824216 * q^59 - 4453034 * q^61 - 1002789 * q^62 + 4757538 * q^64 + 2162872 * q^65 - 6859513 * q^67 + 9397005 * q^68 + 845078 * q^70 + 10726554 * q^71 - 4456898 * q^73 - 1046637 * q^74 + 5861267 * q^76 + 17019816 * q^77 - 15541320 * q^79 + 15680911 * q^80 + 20233655 * q^82 + 11146767 * q^83 - 12471976 * q^85 + 1908168 * q^86 - 24463544 * q^88 + 13531356 * q^89 - 19746448 * q^91 + 26023161 * q^92 + 20288857 * q^94 + 12291624 * q^95 - 10999901 * q^97 - 29909168 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.31419	0.734878	0.367439	−	0.930048i	$-0.380235\pi$
$2$	8.31419	0.734878	0.367439	+	0.930048i	$0.380235\pi$
$3$	0	0
$3$	0	0
$4$	−58.8742	−0.459955
$5$	−148.086	−0.529808	−0.264904	−	0.964275i	$-0.585340\pi$
$5$	−148.086	−0.529808	−0.264904	+	0.964275i	$0.585340\pi$
$6$	0	0
$7$	122.189	0.134644	0.0673222	−	0.997731i	$-0.478554\pi$
$7$	122.189	0.134644	0.0673222	+	0.997731i	$0.478554\pi$
$8$	−1553.71	−1.07289
$9$	0	0
$10$	−1231.21	−0.389344
$11$	−5634.96	−1.27649	−0.638244	−	0.769834i	$-0.720339\pi$
$11$	−5634.96	−1.27649	−0.638244	+	0.769834i	$0.720339\pi$
$12$	0	0
$13$	−7738.55	−0.976917	−0.488459	−	0.872587i	$-0.662441\pi$
$13$	−7738.55	−0.976917	−0.488459	+	0.872587i	$0.662441\pi$
$14$	1015.90	0.0989472
$15$	0	0
$16$	−5381.93	−0.328487
$17$	−1712.43	−0.0845360	−0.0422680	−	0.999106i	$-0.513458\pi$
$17$	−1712.43	−0.0845360	−0.0422680	+	0.999106i	$0.513458\pi$
$18$	0	0
$19$	−51155.1	−1.71101	−0.855503	−	0.517798i	$-0.826752\pi$
$19$	−51155.1	−1.71101	−0.855503	+	0.517798i	$0.826752\pi$
$20$	8718.44	0.243688
$21$	0	0
$22$	−46850.2	−0.938062
$23$	77057.3	1.32058	0.660292	−	0.751009i	$-0.270432\pi$
$23$	77057.3	1.32058	0.660292	+	0.751009i	$0.270432\pi$
$24$	0	0
$25$	−56195.6	−0.719304
$26$	−64339.8	−0.717915
$27$	0	0
$28$	−7193.77	−0.0619304
$29$	48088.8	0.366143	0.183072	−	0.983100i	$-0.441396\pi$
$29$	48088.8	0.366143	0.183072	+	0.983100i	$0.441396\pi$
$30$	0	0
$31$	−192432.	−1.16014	−0.580070	−	0.814566i	$-0.696975\pi$
$31$	−192432.	−1.16014	−0.580070	+	0.814566i	$0.696975\pi$
$32$	154128.	0.831491
$33$	0	0
$34$	−14237.5	−0.0621236
$35$	−18094.4	−0.0713357
$36$	0	0
$37$	461302.	1.49720	0.748599	−	0.663023i	$-0.230727\pi$
$37$	461302.	1.49720	0.748599	+	0.663023i	$0.230727\pi$
$38$	−425313.	−1.25738
$39$	0	0
$40$	230082.	0.568425
$41$	−688771.	−1.56074	−0.780371	−	0.625316i	$-0.784970\pi$
$41$	−688771.	−1.56074	−0.780371	+	0.625316i	$0.784970\pi$
$42$	0	0
$43$	79507.0	0.152499
$43$	79507.0	0.152499
$44$	331754.	0.587127
$45$	0	0
$46$	640669.	0.970468
$47$	619513.	0.870379	0.435189	−	0.900339i	$-0.356681\pi$
$47$	619513.	0.870379	0.435189	+	0.900339i	$0.356681\pi$
$48$	0	0
$49$	−808613.	−0.981871
$50$	−467221.	−0.528600
$51$	0	0
$52$	455601.	0.449338
$53$	−574333.	−0.529905	−0.264952	−	0.964261i	$-0.585356\pi$
$53$	−574333.	−0.529905	−0.264952	+	0.964261i	$0.585356\pi$
$54$	0	0
$55$	834458.	0.676293
$56$	−189846.	−0.144458
$57$	0	0
$58$	399820.	0.269071
$59$	1.01363e6	0.642539	0.321269	−	0.946988i	$-0.395891\pi$
$59$	1.01363e6	0.642539	0.321269	+	0.946988i	$0.395891\pi$
$60$	0	0
$61$	−3.23249e6	−1.82340	−0.911702	−	0.410852i	$-0.865231\pi$
$61$	−3.23249e6	−1.82340	−0.911702	+	0.410852i	$0.865231\pi$
$62$	−1.59991e6	−0.852561
$63$	0	0
$64$	1.97034e6	0.939531
$65$	1.14597e6	0.517579
$66$	0	0
$67$	3.15895e6	1.28316	0.641580	−	0.767056i	$-0.278279\pi$
$67$	3.15895e6	1.28316	0.641580	+	0.767056i	$0.278279\pi$
$68$	100818.	0.0388827
$69$	0	0
$70$	−150441.	−0.0524230
$71$	−2.46302e6	−0.816701	−0.408350	−	0.912825i	$-0.633896\pi$
$71$	−2.46302e6	−0.816701	−0.408350	+	0.912825i	$0.633896\pi$
$72$	0	0
$73$	−4.02731e6	−1.21167	−0.605836	−	0.795590i	$-0.707161\pi$
$73$	−4.02731e6	−1.21167	−0.605836	+	0.795590i	$0.707161\pi$
$74$	3.83535e6	1.10026
$75$	0	0
$76$	3.01172e6	0.786985
$77$	−688530.	−0.171872
$78$	0	0
$79$	−463495.	−0.105767	−0.0528835	−	0.998601i	$-0.516841\pi$
$79$	−463495.	−0.105767	−0.0528835	+	0.998601i	$0.516841\pi$
$80$	796987.	0.174035
$81$	0	0
$82$	−5.72657e6	−1.14696
$83$	5.49654e6	1.05516	0.527578	−	0.849507i	$-0.323100\pi$
$83$	5.49654e6	1.05516	0.527578	+	0.849507i	$0.323100\pi$
$84$	0	0
$85$	253587.	0.0447878
$86$	661036.	0.112068
$87$	0	0
$88$	8.75509e6	1.36953
$89$	8.18997e6	1.23145	0.615726	−	0.787960i	$-0.288863\pi$
$89$	8.18997e6	1.23145	0.615726	+	0.787960i	$0.288863\pi$
$90$	0	0
$91$	−945564.	−0.131536
$92$	−4.53669e6	−0.607409
$93$	0	0
$94$	5.15075e6	0.639622
$95$	7.57535e6	0.906505
$96$	0	0
$97$	1.20009e7	1.33510	0.667548	−	0.744567i	$-0.267344\pi$
$97$	1.20009e7	1.33510	0.667548	+	0.744567i	$0.267344\pi$
$98$	−6.72296e6	−0.721555
$99$	0	0
$100$	3.30847e6	0.330847
$101$	1.45741e7	1.40752	0.703762	−	0.710436i	$-0.251502\pi$
$101$	1.45741e7	1.40752	0.703762	+	0.710436i	$0.251502\pi$
$102$	0	0
$103$	3.29091e6	0.296747	0.148373	−	0.988931i	$-0.452596\pi$
$103$	3.29091e6	0.296747	0.148373	+	0.988931i	$0.452596\pi$
$104$	1.20234e7	1.04812
$105$	0	0
$106$	−4.77511e6	−0.389415
$107$	−2.34488e7	−1.85045	−0.925226	−	0.379416i	$-0.876125\pi$
$107$	−2.34488e7	−1.85045	−0.925226	+	0.379416i	$0.876125\pi$
$108$	0	0
$109$	2.05051e7	1.51659	0.758296	−	0.651910i	$-0.226032\pi$
$109$	2.05051e7	1.51659	0.758296	+	0.651910i	$0.226032\pi$
$110$	6.93785e6	0.496993
$111$	0	0
$112$	−657611.	−0.0442289
$113$	4.02775e6	0.262596	0.131298	−	0.991343i	$-0.458086\pi$
$113$	4.02775e6	0.262596	0.131298	+	0.991343i	$0.458086\pi$
$114$	0	0
$115$	−1.14111e7	−0.699656
$116$	−2.83119e6	−0.168409
$117$	0	0
$118$	8.42755e6	0.472187
$119$	−209240.	−0.0113823
$120$	0	0
$121$	1.22656e7	0.629421
$122$	−2.68755e7	−1.33998
$123$	0	0
$124$	1.13293e7	0.533612
$125$	1.98910e7	0.910901
$126$	0	0
$127$	−1.44813e7	−0.627328	−0.313664	−	0.949534i	$-0.601557\pi$
$127$	−1.44813e7	−0.627328	−0.313664	+	0.949534i	$0.601557\pi$
$128$	−3.34664e6	−0.141050
$129$	0	0
$130$	9.52781e6	0.380357
$131$	1.55714e7	0.605170	0.302585	−	0.953122i	$-0.402150\pi$
$131$	1.55714e7	0.605170	0.302585	+	0.953122i	$0.402150\pi$
$132$	0	0
$133$	−6.25058e6	−0.230377
$134$	2.62641e7	0.942966
$135$	0	0
$136$	2.66062e6	0.0906976
$137$	2.42862e7	0.806932	0.403466	−	0.914995i	$-0.367805\pi$
$137$	2.42862e7	0.806932	0.403466	+	0.914995i	$0.367805\pi$
$138$	0	0
$139$	5.15045e7	1.62665	0.813323	−	0.581812i	$-0.197656\pi$
$139$	5.15045e7	1.62665	0.813323	+	0.581812i	$0.197656\pi$
$140$	1.06530e6	0.0328112
$141$	0	0
$142$	−2.04780e7	−0.600175
$143$	4.36064e7	1.24702
$144$	0	0
$145$	−7.12128e6	−0.193986
$146$	−3.34838e7	−0.890431
$147$	0	0
$148$	−2.71588e7	−0.688644
$149$	−2.03855e7	−0.504858	−0.252429	−	0.967615i	$-0.581229\pi$
$149$	−2.03855e7	−0.504858	−0.252429	+	0.967615i	$0.581229\pi$
$150$	0	0
$151$	−1.46726e7	−0.346806	−0.173403	−	0.984851i	$-0.555476\pi$
$151$	−1.46726e7	−0.346806	−0.173403	+	0.984851i	$0.555476\pi$
$152$	7.94801e7	1.83572
$153$	0	0
$154$	−5.72457e6	−0.126305
$155$	2.84964e7	0.614652
$156$	0	0
$157$	−1.07322e7	−0.221329	−0.110665	−	0.993858i	$-0.535298\pi$
$157$	−1.07322e7	−0.221329	−0.110665	+	0.993858i	$0.535298\pi$
$158$	−3.85358e6	−0.0777257
$159$	0	0
$160$	−2.28242e7	−0.440530
$161$	9.41554e6	0.177809
$162$	0	0
$163$	−3.08777e7	−0.558454	−0.279227	−	0.960225i	$-0.590078\pi$
$163$	−3.08777e7	−0.558454	−0.279227	+	0.960225i	$0.590078\pi$
$164$	4.05509e7	0.717871
$165$	0	0
$166$	4.56993e7	0.775410
$167$	9.20853e6	0.152997	0.0764985	−	0.997070i	$-0.475626\pi$
$167$	9.20853e6	0.152997	0.0764985	+	0.997070i	$0.475626\pi$
$168$	0	0
$169$	−2.86337e6	−0.0456325
$170$	2.10837e6	0.0329136
$171$	0	0
$172$	−4.68091e6	−0.0701425
$173$	−1.49747e7	−0.219885	−0.109943	−	0.993938i	$-0.535067\pi$
$173$	−1.49747e7	−0.219885	−0.109943	+	0.993938i	$0.535067\pi$
$174$	0	0
$175$	−6.86647e6	−0.0968502
$176$	3.03270e7	0.419309
$177$	0	0
$178$	6.80930e7	0.904967
$179$	−8.89995e7	−1.15985	−0.579925	−	0.814670i	$-0.696918\pi$
$179$	−8.89995e7	−1.15985	−0.579925	+	0.814670i	$0.696918\pi$
$180$	0	0
$181$	5.85628e7	0.734086	0.367043	−	0.930204i	$-0.380370\pi$
$181$	5.85628e7	0.734086	0.367043	+	0.930204i	$0.380370\pi$
$182$	−7.86160e6	−0.0966632
$183$	0	0
$184$	−1.19725e8	−1.41684
$185$	−6.83123e7	−0.793228
$186$	0	0
$187$	9.64948e6	0.107909
$188$	−3.64734e7	−0.400335
$189$	0	0
$190$	6.29829e7	0.666170
$191$	−2.81969e7	−0.292809	−0.146404	−	0.989225i	$-0.546770\pi$
$191$	−2.81969e7	−0.292809	−0.146404	+	0.989225i	$0.546770\pi$
$192$	0	0
$193$	−1.42485e8	−1.42665	−0.713325	−	0.700833i	$-0.752812\pi$
$193$	−1.42485e8	−1.42665	−0.713325	+	0.700833i	$0.752812\pi$
$194$	9.97778e7	0.981133
$195$	0	0
$196$	4.76064e7	0.451616
$197$	−1.07088e8	−0.997951	−0.498976	−	0.866616i	$-0.666290\pi$
$197$	−1.07088e8	−0.997951	−0.498976	+	0.866616i	$0.666290\pi$
$198$	0	0
$199$	1.37599e8	1.23774	0.618871	−	0.785493i	$-0.287590\pi$
$199$	1.37599e8	1.23774	0.618871	+	0.785493i	$0.287590\pi$
$200$	8.73115e7	0.771732
$201$	0	0
$202$	1.21171e8	1.03436
$203$	5.87592e6	0.0492992
$204$	0	0
$205$	1.01997e8	0.826894
$206$	2.73613e7	0.218073
$207$	0	0
$208$	4.16483e7	0.320904
$209$	2.88257e8	2.18408
$210$	0	0
$211$	1.52091e8	1.11459	0.557295	−	0.830314i	$-0.311839\pi$
$211$	1.52091e8	1.11459	0.557295	+	0.830314i	$0.311839\pi$
$212$	3.38134e7	0.243732
$213$	0	0
$214$	−1.94958e8	−1.35986
$215$	−1.17739e7	−0.0807950
$216$	0	0
$217$	−2.35130e7	−0.156206
$218$	1.70483e8	1.11451
$219$	0	0
$220$	−4.91281e7	−0.311064
$221$	1.32517e7	0.0825846
$222$	0	0
$223$	−1.58102e8	−0.954709	−0.477354	−	0.878711i	$-0.658404\pi$
$223$	−1.58102e8	−0.954709	−0.477354	+	0.878711i	$0.658404\pi$
$224$	1.88328e7	0.111956
$225$	0	0
$226$	3.34875e7	0.192976
$227$	−1.42553e8	−0.808886	−0.404443	−	0.914563i	$-0.632535\pi$
$227$	−1.42553e8	−0.808886	−0.404443	+	0.914563i	$0.632535\pi$
$228$	0	0
$229$	−1.08019e7	−0.0594395	−0.0297198	−	0.999558i	$-0.509461\pi$
$229$	−1.08019e7	−0.0594395	−0.0297198	+	0.999558i	$0.509461\pi$
$230$	−9.48740e7	−0.514162
$231$	0	0
$232$	−7.47160e7	−0.392831
$233$	−4.31324e7	−0.223387	−0.111694	−	0.993743i	$-0.535627\pi$
$233$	−4.31324e7	−0.223387	−0.111694	+	0.993743i	$0.535627\pi$
$234$	0	0
$235$	−9.17412e7	−0.461134
$236$	−5.96769e7	−0.295539
$237$	0	0
$238$	−1.73966e6	−0.00836459
$239$	1.51247e8	0.716631	0.358315	−	0.933601i	$-0.383351\pi$
$239$	1.51247e8	0.716631	0.358315	+	0.933601i	$0.383351\pi$
$240$	0	0
$241$	−1.16354e7	−0.0535453	−0.0267726	−	0.999642i	$-0.508523\pi$
$241$	−1.16354e7	−0.0535453	−0.0267726	+	0.999642i	$0.508523\pi$
$242$	1.01979e8	0.462548
$243$	0	0
$244$	1.90310e8	0.838683
$245$	1.19744e8	0.520203
$246$	0	0
$247$	3.95866e8	1.67151
$248$	2.98983e8	1.24470
$249$	0	0
$250$	1.65377e8	0.669401
$251$	1.76350e8	0.703910	0.351955	−	0.936017i	$-0.385517\pi$
$251$	1.76350e8	0.703910	0.351955	+	0.936017i	$0.385517\pi$
$252$	0	0
$253$	−4.34215e8	−1.68571
$254$	−1.20400e8	−0.461009
$255$	0	0
$256$	−2.80028e8	−1.04319
$257$	3.46637e8	1.27382	0.636912	−	0.770937i	$-0.280211\pi$
$257$	3.46637e8	1.27382	0.636912	+	0.770937i	$0.280211\pi$
$258$	0	0
$259$	5.63660e7	0.201589
$260$	−6.74681e7	−0.238063
$261$	0	0
$262$	1.29463e8	0.444726
$263$	−3.71975e8	−1.26087	−0.630433	−	0.776244i	$-0.717122\pi$
$263$	−3.71975e8	−1.26087	−0.630433	+	0.776244i	$0.717122\pi$
$264$	0	0
$265$	8.50505e7	0.280748
$266$	−5.19686e7	−0.169299
$267$	0	0
$268$	−1.85981e8	−0.590196
$269$	−3.76594e8	−1.17962	−0.589808	−	0.807544i	$-0.700797\pi$
$269$	−3.76594e8	−1.17962	−0.589808	+	0.807544i	$0.700797\pi$
$270$	0	0
$271$	1.87035e8	0.570862	0.285431	−	0.958399i	$-0.407863\pi$
$271$	1.87035e8	0.570862	0.285431	+	0.958399i	$0.407863\pi$
$272$	9.21617e6	0.0277689
$273$	0	0
$274$	2.01920e8	0.592996
$275$	3.16660e8	0.918182
$276$	0	0
$277$	−2.58454e8	−0.730642	−0.365321	−	0.930882i	$-0.619041\pi$
$277$	−2.58454e8	−0.730642	−0.365321	+	0.930882i	$0.619041\pi$
$278$	4.28218e8	1.19539
$279$	0	0
$280$	2.81135e7	0.0765352
$281$	3.07159e8	0.825831	0.412916	−	0.910769i	$-0.364510\pi$
$281$	3.07159e8	0.825831	0.412916	+	0.910769i	$0.364510\pi$
$282$	0	0
$283$	−6.04080e8	−1.58432	−0.792159	−	0.610315i	$-0.791043\pi$
$283$	−6.04080e8	−1.58432	−0.792159	+	0.610315i	$0.791043\pi$
$284$	1.45008e8	0.375646
$285$	0	0
$286$	3.62552e8	0.916409
$287$	−8.41601e7	−0.210145
$288$	0	0
$289$	−4.07406e8	−0.992854
$290$	−5.92077e7	−0.142556
$291$	0	0
$292$	2.37105e8	0.557314
$293$	1.79377e8	0.416610	0.208305	−	0.978064i	$-0.433205\pi$
$293$	1.79377e8	0.416610	0.208305	+	0.978064i	$0.433205\pi$
$294$	0	0
$295$	−1.50105e8	−0.340422
$296$	−7.16729e8	−1.60633
$297$	0	0
$298$	−1.69489e8	−0.371009
$299$	−5.96312e8	−1.29010
$300$	0	0
$301$	9.71487e6	0.0205331
$302$	−1.21991e8	−0.254860
$303$	0	0
$304$	2.75313e8	0.562043
$305$	4.78686e8	0.966054
$306$	0	0
$307$	−1.45485e8	−0.286969	−0.143484	−	0.989653i	$-0.545831\pi$
$307$	−1.45485e8	−0.286969	−0.143484	+	0.989653i	$0.545831\pi$
$308$	4.05366e7	0.0790533
$309$	0	0
$310$	2.36925e8	0.451694
$311$	−3.56832e8	−0.672670	−0.336335	−	0.941742i	$-0.609187\pi$
$311$	−3.56832e8	−0.672670	−0.336335	+	0.941742i	$0.609187\pi$
$312$	0	0
$313$	6.50132e8	1.19838	0.599192	−	0.800605i	$-0.295489\pi$
$313$	6.50132e8	1.19838	0.599192	+	0.800605i	$0.295489\pi$
$314$	−8.92293e7	−0.162650
$315$	0	0
$316$	2.72879e7	0.0486480
$317$	1.34462e8	0.237078	0.118539	−	0.992949i	$-0.462179\pi$
$317$	1.34462e8	0.237078	0.118539	+	0.992949i	$0.462179\pi$
$318$	0	0
$319$	−2.70979e8	−0.467378
$320$	−2.91779e8	−0.497771
$321$	0	0
$322$	7.82826e7	0.130668
$323$	8.75995e7	0.144642
$324$	0	0
$325$	4.34872e8	0.702700
$326$	−2.56723e8	−0.410396
$327$	0	0
$328$	1.07015e9	1.67450
$329$	7.56976e7	0.117192
$330$	0	0
$331$	6.61036e8	1.00191	0.500953	−	0.865474i	$-0.332983\pi$
$331$	6.61036e8	1.00191	0.500953	+	0.865474i	$0.332983\pi$
$332$	−3.23605e8	−0.485324
$333$	0	0
$334$	7.65615e7	0.112434
$335$	−4.67796e8	−0.679828
$336$	0	0
$337$	−2.64722e8	−0.376778	−0.188389	−	0.982094i	$-0.560327\pi$
$337$	−2.64722e8	−0.376778	−0.188389	+	0.982094i	$0.560327\pi$
$338$	−2.38066e7	−0.0335343
$339$	0	0
$340$	−1.49297e7	−0.0206004
$341$	1.08435e9	1.48091
$342$	0	0
$343$	−1.99431e8	−0.266848
$344$	−1.23531e8	−0.163614
$345$	0	0
$346$	−1.24502e8	−0.161589
$347$	3.92072e8	0.503748	0.251874	−	0.967760i	$-0.418953\pi$
$347$	3.92072e8	0.503748	0.251874	+	0.967760i	$0.418953\pi$
$348$	0	0
$349$	−2.58464e8	−0.325470	−0.162735	−	0.986670i	$-0.552032\pi$
$349$	−2.58464e8	−0.325470	−0.162735	+	0.986670i	$0.552032\pi$
$350$	−5.70892e7	−0.0711731
$351$	0	0
$352$	−8.68507e8	−1.06139
$353$	2.36908e8	0.286661	0.143331	−	0.989675i	$-0.454219\pi$
$353$	2.36908e8	0.286661	0.143331	+	0.989675i	$0.454219\pi$
$354$	0	0
$355$	3.64738e8	0.432695
$356$	−4.82178e8	−0.566412
$357$	0	0
$358$	−7.39959e8	−0.852348
$359$	1.22528e9	1.39767	0.698836	−	0.715282i	$-0.253702\pi$
$359$	1.22528e9	1.39767	0.698836	+	0.715282i	$0.253702\pi$
$360$	0	0
$361$	1.72297e9	1.92754
$362$	4.86903e8	0.539464
$363$	0	0
$364$	5.56694e7	0.0605008
$365$	5.96388e8	0.641953
$366$	0	0
$367$	−4.80402e8	−0.507311	−0.253655	−	0.967295i	$-0.581633\pi$
$367$	−4.80402e8	−0.507311	−0.253655	+	0.967295i	$0.581633\pi$
$368$	−4.14717e8	−0.433795
$369$	0	0
$370$	−5.67962e8	−0.582925
$371$	−7.01770e7	−0.0713488
$372$	0	0
$373$	−1.06518e9	−1.06278	−0.531389	−	0.847128i	$-0.678330\pi$
$373$	−1.06518e9	−1.06278	−0.531389	+	0.847128i	$0.678330\pi$
$374$	8.02276e7	0.0793000
$375$	0	0
$376$	−9.62543e8	−0.933819
$377$	−3.72138e8	−0.357692
$378$	0	0
$379$	1.57397e9	1.48511	0.742554	−	0.669786i	$-0.233614\pi$
$379$	1.57397e9	1.48511	0.742554	+	0.669786i	$0.233614\pi$
$380$	−4.45993e8	−0.416951
$381$	0	0
$382$	−2.34434e8	−0.215179
$383$	8.92980e8	0.812168	0.406084	−	0.913836i	$-0.366894\pi$
$383$	8.92980e8	0.812168	0.406084	+	0.913836i	$0.366894\pi$
$384$	0	0
$385$	1.01961e8	0.0910591
$386$	−1.18464e9	−1.04841
$387$	0	0
$388$	−7.06543e8	−0.614084
$389$	−2.17007e9	−1.86917	−0.934586	−	0.355737i	$-0.884230\pi$
$389$	−2.17007e9	−1.86917	−0.934586	+	0.355737i	$0.884230\pi$
$390$	0	0
$391$	−1.31955e8	−0.111637
$392$	1.25635e9	1.05344
$393$	0	0
$394$	−8.90351e8	−0.733372
$395$	6.86370e7	0.0560362
$396$	0	0
$397$	−1.33189e9	−1.06832	−0.534162	−	0.845382i	$-0.679373\pi$
$397$	−1.33189e9	−1.06832	−0.534162	+	0.845382i	$0.679373\pi$
$398$	1.14403e9	0.909589
$399$	0	0
$400$	3.02441e8	0.236282
$401$	−3.57769e8	−0.277075	−0.138538	−	0.990357i	$-0.544240\pi$
$401$	−3.57769e8	−0.277075	−0.138538	+	0.990357i	$0.544240\pi$
$402$	0	0
$403$	1.48914e9	1.13336
$404$	−8.58036e8	−0.647397
$405$	0	0
$406$	4.88535e7	0.0362289
$407$	−2.59942e9	−1.91116
$408$	0	0
$409$	7.47840e8	0.540477	0.270238	−	0.962793i	$-0.412897\pi$
$409$	7.47840e8	0.540477	0.270238	+	0.962793i	$0.412897\pi$
$410$	8.48024e8	0.607666
$411$	0	0
$412$	−1.93750e8	−0.136490
$413$	1.23855e8	0.0865143
$414$	0	0
$415$	−8.13960e8	−0.559030
$416$	−1.19273e9	−0.812298
$417$	0	0
$418$	2.39663e9	1.60503
$419$	−9.00703e8	−0.598181	−0.299091	−	0.954225i	$-0.596683\pi$
$419$	−9.00703e8	−0.598181	−0.299091	+	0.954225i	$0.596683\pi$
$420$	0	0
$421$	−1.70587e9	−1.11419	−0.557094	−	0.830449i	$-0.688084\pi$
$421$	−1.70587e9	−1.11419	−0.557094	+	0.830449i	$0.688084\pi$
$422$	1.26451e9	0.819088
$423$	0	0
$424$	8.92345e8	0.568529
$425$	9.62310e7	0.0608070
$426$	0	0
$427$	−3.94974e8	−0.245511
$428$	1.38053e9	0.851124
$429$	0	0
$430$	−9.78901e7	−0.0593744
$431$	−1.88962e9	−1.13685	−0.568425	−	0.822735i	$-0.692447\pi$
$431$	−1.88962e9	−1.13685	−0.568425	+	0.822735i	$0.692447\pi$
$432$	0	0
$433$	7.46212e7	0.0441728	0.0220864	−	0.999756i	$-0.492969\pi$
$433$	7.46212e7	0.0441728	0.0220864	+	0.999756i	$0.492969\pi$
$434$	−1.95492e8	−0.114793
$435$	0	0
$436$	−1.20722e9	−0.697564
$437$	−3.94188e9	−2.25953
$438$	0	0
$439$	−1.72954e9	−0.975677	−0.487838	−	0.872934i	$-0.662214\pi$
$439$	−1.72954e9	−0.975677	−0.487838	+	0.872934i	$0.662214\pi$
$440$	−1.29650e9	−0.725587
$441$	0	0
$442$	1.10177e8	0.0606896
$443$	−3.58748e9	−1.96054	−0.980270	−	0.197661i	$-0.936665\pi$
$443$	−3.58748e9	−1.96054	−0.980270	+	0.197661i	$0.936665\pi$
$444$	0	0
$445$	−1.21282e9	−0.652433
$446$	−1.31449e9	−0.701594
$447$	0	0
$448$	2.40753e8	0.126503
$449$	3.27797e9	1.70900	0.854502	−	0.519447i	$-0.173862\pi$
$449$	3.27797e9	1.70900	0.854502	+	0.519447i	$0.173862\pi$
$450$	0	0
$451$	3.88120e9	1.99227
$452$	−2.37130e8	−0.120782
$453$	0	0
$454$	−1.18522e9	−0.594432
$455$	1.40025e8	0.0696891
$456$	0	0
$457$	−1.77071e9	−0.867842	−0.433921	−	0.900951i	$-0.642870\pi$
$457$	−1.77071e9	−0.867842	−0.433921	+	0.900951i	$0.642870\pi$
$458$	−8.98089e7	−0.0436808
$459$	0	0
$460$	6.71819e8	0.321810
$461$	−2.35457e9	−1.11933	−0.559664	−	0.828719i	$-0.689070\pi$
$461$	−2.35457e9	−1.11933	−0.559664	+	0.828719i	$0.689070\pi$
$462$	0	0
$463$	−4.94391e8	−0.231493	−0.115746	−	0.993279i	$-0.536926\pi$
$463$	−4.94391e8	−0.231493	−0.115746	+	0.993279i	$0.536926\pi$
$464$	−2.58811e8	−0.120273
$465$	0	0
$466$	−3.58611e8	−0.164162
$467$	3.45960e9	1.57187	0.785935	−	0.618309i	$-0.212182\pi$
$467$	3.45960e9	1.57187	0.785935	+	0.618309i	$0.212182\pi$
$468$	0	0
$469$	3.85988e8	0.172770
$470$	−7.62754e8	−0.338877
$471$	0	0
$472$	−1.57489e9	−0.689372
$473$	−4.48019e8	−0.194663
$474$	0	0
$475$	2.87469e9	1.23073
$476$	1.23188e7	0.00523534
$477$	0	0
$478$	1.25750e9	0.526636
$479$	2.26891e9	0.943284	0.471642	−	0.881790i	$-0.343662\pi$
$479$	2.26891e9	0.943284	0.471642	+	0.881790i	$0.343662\pi$
$480$	0	0
$481$	−3.56981e9	−1.46264
$482$	−9.67389e7	−0.0393492
$483$	0	0
$484$	−7.22130e8	−0.289505
$485$	−1.77716e9	−0.707345
$486$	0	0
$487$	−2.51843e9	−0.988051	−0.494025	−	0.869447i	$-0.664475\pi$
$487$	−2.51843e9	−0.988051	−0.494025	+	0.869447i	$0.664475\pi$
$488$	5.02235e9	1.95631
$489$	0	0
$490$	9.95576e8	0.382286
$491$	−7.20363e8	−0.274642	−0.137321	−	0.990527i	$-0.543849\pi$
$491$	−7.20363e8	−0.274642	−0.137321	+	0.990527i	$0.543849\pi$
$492$	0	0
$493$	−8.23487e7	−0.0309523
$494$	3.29131e9	1.22836
$495$	0	0
$496$	1.03565e9	0.381091
$497$	−3.00953e8	−0.109964
$498$	0	0
$499$	−2.15427e9	−0.776154	−0.388077	−	0.921627i	$-0.626861\pi$
$499$	−2.15427e9	−0.776154	−0.388077	+	0.921627i	$0.626861\pi$
$500$	−1.17107e9	−0.418973
$501$	0	0
$502$	1.46621e9	0.517288
$503$	−2.80736e9	−0.983579	−0.491790	−	0.870714i	$-0.663657\pi$
$503$	−2.80736e9	−0.983579	−0.491790	+	0.870714i	$0.663657\pi$
$504$	0	0
$505$	−2.15821e9	−0.745717
$506$	−3.61015e9	−1.23879
$507$	0	0
$508$	8.52575e8	0.288542
$509$	2.99574e9	1.00691	0.503457	−	0.864020i	$-0.332061\pi$
$509$	2.99574e9	1.00691	0.503457	+	0.864020i	$0.332061\pi$
$510$	0	0
$511$	−4.92092e8	−0.163145
$512$	−1.89984e9	−0.625563
$513$	0	0
$514$	2.88201e9	0.936105
$515$	−4.87338e8	−0.157219
$516$	0	0
$517$	−3.49094e9	−1.11103
$518$	4.68637e8	0.148144
$519$	0	0
$520$	−1.78050e9	−0.555304
$521$	−2.58005e9	−0.799275	−0.399637	−	0.916673i	$-0.630864\pi$
$521$	−2.58005e9	−0.799275	−0.399637	+	0.916673i	$0.630864\pi$
$522$	0	0
$523$	1.70818e9	0.522129	0.261064	−	0.965321i	$-0.415926\pi$
$523$	1.70818e9	0.522129	0.261064	+	0.965321i	$0.415926\pi$
$524$	−9.16753e8	−0.278351
$525$	0	0
$526$	−3.09267e9	−0.926582
$527$	3.29526e8	0.0980736
$528$	0	0
$529$	2.53300e9	0.743944
$530$	7.07126e8	0.206315
$531$	0	0
$532$	3.67998e8	0.105963
$533$	5.33009e9	1.52472
$534$	0	0
$535$	3.47244e9	0.980384
$536$	−4.90809e9	−1.37669
$537$	0	0
$538$	−3.13108e9	−0.866873
$539$	4.55650e9	1.25335
$540$	0	0
$541$	−4.88301e9	−1.32586	−0.662929	−	0.748682i	$-0.730687\pi$
$541$	−4.88301e9	−1.32586	−0.662929	+	0.748682i	$0.730687\pi$
$542$	1.55505e9	0.419514
$543$	0	0
$544$	−2.63934e8	−0.0702909
$545$	−3.03651e9	−0.803503
$546$	0	0
$547$	−6.00910e9	−1.56984	−0.784918	−	0.619600i	$-0.787295\pi$
$547$	−6.00910e9	−1.56984	−0.784918	+	0.619600i	$0.787295\pi$
$548$	−1.42983e9	−0.371152
$549$	0	0
$550$	2.63277e9	0.674752
$551$	−2.45999e9	−0.626474
$552$	0	0
$553$	−5.66339e7	−0.0142409
$554$	−2.14884e9	−0.536932
$555$	0	0
$556$	−3.03229e9	−0.748184
$557$	3.64266e9	0.893152	0.446576	−	0.894746i	$-0.352643\pi$
$557$	3.64266e9	0.893152	0.446576	+	0.894746i	$0.352643\pi$
$558$	0	0
$559$	−6.15269e8	−0.148979
$560$	9.73829e7	0.0234328
$561$	0	0
$562$	2.55378e9	0.606885
$563$	−9.78473e8	−0.231084	−0.115542	−	0.993303i	$-0.536860\pi$
$563$	−9.78473e8	−0.231084	−0.115542	+	0.993303i	$0.536860\pi$
$564$	0	0
$565$	−5.96452e8	−0.139125
$566$	−5.02244e9	−1.16428
$567$	0	0
$568$	3.82681e9	0.876229
$569$	2.98433e9	0.679131	0.339566	−	0.940582i	$-0.389720\pi$
$569$	2.98433e9	0.679131	0.339566	+	0.940582i	$0.389720\pi$
$570$	0	0
$571$	4.09176e9	0.919779	0.459890	−	0.887976i	$-0.347889\pi$
$571$	4.09176e9	0.919779	0.459890	+	0.887976i	$0.347889\pi$
$572$	−2.56729e9	−0.573574
$573$	0	0
$574$	−6.99723e8	−0.154431
$575$	−4.33028e9	−0.949901
$576$	0	0
$577$	−3.21664e9	−0.697087	−0.348544	−	0.937293i	$-0.613324\pi$
$577$	−3.21664e9	−0.697087	−0.348544	+	0.937293i	$0.613324\pi$
$578$	−3.38725e9	−0.729626
$579$	0	0
$580$	4.19260e8	0.0892247
$581$	6.71616e8	0.142071
$582$	0	0
$583$	3.23634e9	0.676417
$584$	6.25726e9	1.29999
$585$	0	0
$586$	1.49137e9	0.306157
$587$	3.18322e9	0.649582	0.324791	−	0.945786i	$-0.394706\pi$
$587$	3.18322e9	0.649582	0.324791	+	0.945786i	$0.394706\pi$
$588$	0	0
$589$	9.84387e9	1.98501
$590$	−1.24800e9	−0.250169
$591$	0	0
$592$	−2.48269e9	−0.491810
$593$	1.62368e9	0.319749	0.159875	−	0.987137i	$-0.448891\pi$
$593$	1.62368e9	0.319749	0.159875	+	0.987137i	$0.448891\pi$
$594$	0	0
$595$	3.09854e7	0.00603043
$596$	1.20018e9	0.232212
$597$	0	0
$598$	−4.95785e9	−0.948067
$599$	3.68608e9	0.700763	0.350382	−	0.936607i	$-0.386052\pi$
$599$	3.68608e9	0.700763	0.350382	+	0.936607i	$0.386052\pi$
$600$	0	0
$601$	−2.91768e9	−0.548248	−0.274124	−	0.961694i	$-0.588388\pi$
$601$	−2.91768e9	−0.548248	−0.274124	+	0.961694i	$0.588388\pi$
$602$	8.07713e7	0.0150893
$603$	0	0
$604$	8.63837e8	0.159515
$605$	−1.81637e9	−0.333472
$606$	0	0
$607$	−3.35727e9	−0.609292	−0.304646	−	0.952466i	$-0.598538\pi$
$607$	−3.35727e9	−0.609292	−0.304646	+	0.952466i	$0.598538\pi$
$608$	−7.88445e9	−1.42269
$609$	0	0
$610$	3.97989e9	0.709931
$611$	−4.79414e9	−0.850288
$612$	0	0
$613$	1.51150e9	0.265030	0.132515	−	0.991181i	$-0.457695\pi$
$613$	1.51150e9	0.265030	0.132515	+	0.991181i	$0.457695\pi$
$614$	−1.20959e9	−0.210887
$615$	0	0
$616$	1.06977e9	0.184399
$617$	5.64788e9	0.968026	0.484013	−	0.875061i	$-0.339179\pi$
$617$	5.64788e9	0.968026	0.484013	+	0.875061i	$0.339179\pi$
$618$	0	0
$619$	−2.42139e9	−0.410343	−0.205172	−	0.978726i	$-0.565775\pi$
$619$	−2.42139e9	−0.410343	−0.205172	+	0.978726i	$0.565775\pi$
$620$	−1.67770e9	−0.282712
$621$	0	0
$622$	−2.96677e9	−0.494330
$623$	1.00072e9	0.165808
$624$	0	0
$625$	1.44471e9	0.236701
$626$	5.40532e9	0.880666
$627$	0	0
$628$	6.31848e8	0.101801
$629$	−7.89947e8	−0.126567
$630$	0	0
$631$	7.99860e9	1.26739	0.633696	−	0.773582i	$-0.281537\pi$
$631$	7.99860e9	1.26739	0.633696	+	0.773582i	$0.281537\pi$
$632$	7.20135e8	0.113476
$633$	0	0
$634$	1.11794e9	0.174223
$635$	2.14448e9	0.332363
$636$	0	0
$637$	6.25749e9	0.959207
$638$	−2.25297e9	−0.343465
$639$	0	0
$640$	4.95591e8	0.0747297
$641$	−9.69674e8	−0.145419	−0.0727097	−	0.997353i	$-0.523165\pi$
$641$	−9.69674e8	−0.145419	−0.0727097	+	0.997353i	$0.523165\pi$
$642$	0	0
$643$	3.12786e9	0.463990	0.231995	−	0.972717i	$-0.425475\pi$
$643$	3.12786e9	0.463990	0.231995	+	0.972717i	$0.425475\pi$
$644$	−5.54333e8	−0.0817843
$645$	0	0
$646$	7.28319e8	0.106294
$647$	1.05397e10	1.52990	0.764949	−	0.644091i	$-0.222764\pi$
$647$	1.05397e10	1.52990	0.764949	+	0.644091i	$0.222764\pi$
$648$	0	0
$649$	−5.71179e9	−0.820193
$650$	3.61561e9	0.516399
$651$	0	0
$652$	1.81790e9	0.256864
$653$	1.16184e10	1.63286	0.816430	−	0.577444i	$-0.195950\pi$
$653$	1.16184e10	1.63286	0.816430	+	0.577444i	$0.195950\pi$
$654$	0	0
$655$	−2.30590e9	−0.320624
$656$	3.70692e9	0.512683
$657$	0	0
$658$	6.29365e8	0.0861215
$659$	5.81607e9	0.791645	0.395822	−	0.918327i	$-0.370460\pi$
$659$	5.81607e9	0.791645	0.395822	+	0.918327i	$0.370460\pi$
$660$	0	0
$661$	8.71459e9	1.17366	0.586830	−	0.809710i	$-0.300376\pi$
$661$	8.71459e9	1.17366	0.586830	+	0.809710i	$0.300376\pi$
$662$	5.49598e9	0.736278
$663$	0	0
$664$	−8.54003e9	−1.13206
$665$	9.25623e8	0.122056
$666$	0	0
$667$	3.70560e9	0.483524
$668$	−5.42145e8	−0.0703717
$669$	0	0
$670$	−3.88934e9	−0.499591
$671$	1.82150e10	2.32755
$672$	0	0
$673$	1.52007e10	1.92226	0.961129	−	0.276098i	$-0.0890415\pi$
$673$	1.52007e10	1.92226	0.961129	+	0.276098i	$0.0890415\pi$
$674$	−2.20095e9	−0.276886
$675$	0	0
$676$	1.68579e8	0.0209889
$677$	7.66446e9	0.949339	0.474670	−	0.880164i	$-0.342568\pi$
$677$	7.66446e9	0.949339	0.474670	+	0.880164i	$0.342568\pi$
$678$	0	0
$679$	1.46638e9	0.179763
$680$	−3.93999e8	−0.0480523
$681$	0	0
$682$	9.01546e9	1.08828
$683$	5.86341e9	0.704170	0.352085	−	0.935968i	$-0.385473\pi$
$683$	5.86341e9	0.704170	0.352085	+	0.935968i	$0.385473\pi$
$684$	0	0
$685$	−3.59644e9	−0.427519
$686$	−1.65811e9	−0.196101
$687$	0	0
$688$	−4.27901e8	−0.0500938
$689$	4.44450e9	0.517673
$690$	0	0
$691$	−2.93527e9	−0.338435	−0.169218	−	0.985579i	$-0.554124\pi$
$691$	−2.93527e9	−0.338435	−0.169218	+	0.985579i	$0.554124\pi$
$692$	8.81623e8	0.101137
$693$	0	0
$694$	3.25976e9	0.370193
$695$	−7.62708e9	−0.861810
$696$	0	0
$697$	1.17947e9	0.131939
$698$	−2.14892e9	−0.239181
$699$	0	0
$700$	4.04258e8	0.0445467
$701$	1.59500e10	1.74883	0.874415	−	0.485178i	$-0.161245\pi$
$701$	1.59500e10	1.74883	0.874415	+	0.485178i	$0.161245\pi$
$702$	0	0
$703$	−2.35980e10	−2.56172
$704$	−1.11028e10	−1.19930
$705$	0	0
$706$	1.96970e9	0.210661
$707$	1.78079e9	0.189515
$708$	0	0
$709$	−5.26013e9	−0.554287	−0.277143	−	0.960829i	$-0.589388\pi$
$709$	−5.26013e9	−0.554287	−0.277143	+	0.960829i	$0.589388\pi$
$710$	3.03250e9	0.317978
$711$	0	0
$712$	−1.27248e10	−1.32121
$713$	−1.48283e10	−1.53206
$714$	0	0
$715$	−6.45750e9	−0.660683
$716$	5.23978e9	0.533479
$717$	0	0
$718$	1.01872e10	1.02712
$719$	−1.60266e10	−1.60802	−0.804008	−	0.594618i	$-0.797303\pi$
$719$	−1.60266e10	−1.60802	−0.804008	+	0.594618i	$0.797303\pi$
$720$	0	0
$721$	4.02113e8	0.0399553
$722$	1.43251e10	1.41651
$723$	0	0
$724$	−3.44784e9	−0.337647
$725$	−2.70238e9	−0.263368
$726$	0	0
$727$	1.07033e10	1.03311	0.516556	−	0.856253i	$-0.327214\pi$
$727$	1.07033e10	1.03311	0.516556	+	0.856253i	$0.327214\pi$
$728$	1.46913e9	0.141124
$729$	0	0
$730$	4.95848e9	0.471757
$731$	−1.36150e8	−0.0128916
$732$	0	0
$733$	1.43369e10	1.34460	0.672298	−	0.740280i	$-0.265307\pi$
$733$	1.43369e10	1.34460	0.672298	+	0.740280i	$0.265307\pi$
$734$	−3.99416e9	−0.372811
$735$	0	0
$736$	1.18767e10	1.09805
$737$	−1.78006e10	−1.63794
$738$	0	0
$739$	2.07369e10	1.89012	0.945060	−	0.326898i	$-0.106003\pi$
$739$	2.07369e10	1.89012	0.945060	+	0.326898i	$0.106003\pi$
$740$	4.02183e9	0.364849
$741$	0	0
$742$	−5.83465e8	−0.0524326
$743$	5.43590e9	0.486196	0.243098	−	0.970002i	$-0.421836\pi$
$743$	5.43590e9	0.486196	0.243098	+	0.970002i	$0.421836\pi$
$744$	0	0
$745$	3.01880e9	0.267478
$746$	−8.85612e9	−0.781012
$747$	0	0
$748$	−5.68105e8	−0.0496333
$749$	−2.86518e9	−0.249153
$750$	0	0
$751$	−3.49854e9	−0.301403	−0.150701	−	0.988579i	$-0.548153\pi$
$751$	−3.49854e9	−0.301403	−0.150701	+	0.988579i	$0.548153\pi$
$752$	−3.33418e9	−0.285908
$753$	0	0
$754$	−3.09403e9	−0.262860
$755$	2.17280e9	0.183741
$756$	0	0
$757$	−2.02699e10	−1.69831	−0.849153	−	0.528148i	$-0.822887\pi$
$757$	−2.02699e10	−1.69831	−0.849153	+	0.528148i	$0.822887\pi$
$758$	1.30862e10	1.09137
$759$	0	0
$760$	−1.17699e10	−0.972578
$761$	−1.16941e10	−0.961876	−0.480938	−	0.876755i	$-0.659704\pi$
$761$	−1.16941e10	−0.961876	−0.480938	+	0.876755i	$0.659704\pi$
$762$	0	0
$763$	2.50549e9	0.204201
$764$	1.66007e9	0.134679
$765$	0	0
$766$	7.42441e9	0.596844
$767$	−7.84406e9	−0.627707
$768$	0	0
$769$	1.64924e10	1.30780	0.653899	−	0.756582i	$-0.273132\pi$
$769$	1.64924e10	1.30780	0.653899	+	0.756582i	$0.273132\pi$
$770$	8.47727e8	0.0669173
$771$	0	0
$772$	8.38867e9	0.656195
$773$	4.04094e9	0.314669	0.157334	−	0.987545i	$-0.449710\pi$
$773$	4.04094e9	0.314669	0.157334	+	0.987545i	$0.449710\pi$
$774$	0	0
$775$	1.08138e10	0.834493
$776$	−1.86459e10	−1.43241
$777$	0	0
$778$	−1.80423e10	−1.37361
$779$	3.52342e10	2.67044
$780$	0	0
$781$	1.38790e10	1.04251
$782$	−1.09710e9	−0.0820395
$783$	0	0
$784$	4.35190e9	0.322532
$785$	1.58928e9	0.117262
$786$	0	0
$787$	2.46052e10	1.79935	0.899675	−	0.436560i	$-0.143803\pi$
$787$	2.46052e10	1.79935	0.899675	+	0.436560i	$0.143803\pi$
$788$	6.30473e9	0.459012
$789$	0	0
$790$	5.70661e8	0.0411797
$791$	4.92146e8	0.0353571
$792$	0	0
$793$	2.50148e10	1.78131
$794$	−1.10736e10	−0.785087
$795$	0	0
$796$	−8.10105e9	−0.569305
$797$	1.84665e10	1.29205	0.646026	−	0.763316i	$-0.276430\pi$
$797$	1.84665e10	1.29205	0.646026	+	0.763316i	$0.276430\pi$
$798$	0	0
$799$	−1.06087e9	−0.0735783
$800$	−8.66133e9	−0.598094
$801$	0	0
$802$	−2.97456e9	−0.203616
$803$	2.26937e10	1.54668
$804$	0	0
$805$	−1.39431e9	−0.0942048
$806$	1.23810e10	0.832882
$807$	0	0
$808$	−2.26438e10	−1.51012
$809$	−1.80517e10	−1.19867	−0.599333	−	0.800500i	$-0.704567\pi$
$809$	−1.80517e10	−1.19867	−0.599333	+	0.800500i	$0.704567\pi$
$810$	0	0
$811$	1.23449e10	0.812673	0.406337	−	0.913723i	$-0.366806\pi$
$811$	1.23449e10	0.812673	0.406337	+	0.913723i	$0.366806\pi$
$812$	−3.45940e8	−0.0226754
$813$	0	0
$814$	−2.16121e10	−1.40447
$815$	4.57254e9	0.295874
$816$	0	0
$817$	−4.06719e9	−0.260926
$818$	6.21768e9	0.397184
$819$	0	0
$820$	−6.00501e9	−0.380334
$821$	1.36091e10	0.858276	0.429138	−	0.903239i	$-0.358817\pi$
$821$	1.36091e10	0.858276	0.429138	+	0.903239i	$0.358817\pi$
$822$	0	0
$823$	−1.55999e10	−0.975490	−0.487745	−	0.872986i	$-0.662180\pi$
$823$	−1.55999e10	−0.975490	−0.487745	+	0.872986i	$0.662180\pi$
$824$	−5.11312e9	−0.318376
$825$	0	0
$826$	1.02975e9	0.0635774
$827$	1.54413e10	0.949325	0.474662	−	0.880168i	$-0.342570\pi$
$827$	1.54413e10	0.949325	0.474662	+	0.880168i	$0.342570\pi$
$828$	0	0
$829$	−5.48114e9	−0.334141	−0.167071	−	0.985945i	$-0.553431\pi$
$829$	−5.48114e9	−0.334141	−0.167071	+	0.985945i	$0.553431\pi$
$830$	−6.76742e9	−0.410818
$831$	0	0
$832$	−1.52476e10	−0.917844
$833$	1.38469e9	0.0830034
$834$	0	0
$835$	−1.36365e9	−0.0810590
$836$	−1.69709e10	−1.00458
$837$	0	0
$838$	−7.48862e9	−0.439590
$839$	9.70181e9	0.567134	0.283567	−	0.958952i	$-0.408482\pi$
$839$	9.70181e9	0.567134	0.283567	+	0.958952i	$0.408482\pi$
$840$	0	0
$841$	−1.49373e10	−0.865939
$842$	−1.41829e10	−0.818792
$843$	0	0
$844$	−8.95425e9	−0.512661
$845$	4.24025e8	0.0241764
$846$	0	0
$847$	1.49872e9	0.0847481
$848$	3.09102e9	0.174067
$849$	0	0
$850$	8.00083e8	0.0446857
$851$	3.55467e10	1.97718
$852$	0	0
$853$	7.25754e9	0.400376	0.200188	−	0.979758i	$-0.435845\pi$
$853$	7.25754e9	0.400376	0.200188	+	0.979758i	$0.435845\pi$
$854$	−3.28389e9	−0.180421
$855$	0	0
$856$	3.64326e10	1.98533
$857$	−2.87404e10	−1.55977	−0.779884	−	0.625923i	$-0.784722\pi$
$857$	−2.87404e10	−1.55977	−0.779884	+	0.625923i	$0.784722\pi$
$858$	0	0
$859$	−7.30578e9	−0.393270	−0.196635	−	0.980477i	$-0.563001\pi$
$859$	−7.30578e9	−0.393270	−0.196635	+	0.980477i	$0.563001\pi$
$860$	6.93177e8	0.0371620
$861$	0	0
$862$	−1.57106e10	−0.835446
$863$	−3.12604e10	−1.65560	−0.827802	−	0.561021i	$-0.810409\pi$
$863$	−3.12604e10	−1.65560	−0.827802	+	0.561021i	$0.810409\pi$
$864$	0	0
$865$	2.21754e9	0.116497
$866$	6.20415e8	0.0324616
$867$	0	0
$868$	1.38431e9	0.0718479
$869$	2.61177e9	0.135010
$870$	0	0
$871$	−2.44457e10	−1.25354
$872$	−3.18589e10	−1.62713
$873$	0	0
$874$	−3.27735e10	−1.66048
$875$	2.43046e9	0.122648
$876$	0	0
$877$	−1.16138e10	−0.581403	−0.290702	−	0.956814i	$-0.593889\pi$
$877$	−1.16138e10	−0.581403	−0.290702	+	0.956814i	$0.593889\pi$
$878$	−1.43798e10	−0.717003
$879$	0	0
$880$	−4.49099e9	−0.222153
$881$	5.59437e9	0.275636	0.137818	−	0.990458i	$-0.455991\pi$
$881$	5.59437e9	0.275636	0.137818	+	0.990458i	$0.455991\pi$
$882$	0	0
$883$	2.50609e10	1.22500	0.612498	−	0.790472i	$-0.290165\pi$
$883$	2.50609e10	1.22500	0.612498	+	0.790472i	$0.290165\pi$
$884$	−7.80185e8	−0.0379852
$885$	0	0
$886$	−2.98270e10	−1.44076
$887$	1.82957e10	0.880271	0.440135	−	0.897931i	$-0.354930\pi$
$887$	1.82957e10	0.880271	0.440135	+	0.897931i	$0.354930\pi$
$888$	0	0
$889$	−1.76945e9	−0.0844662
$890$	−1.00836e10	−0.479459
$891$	0	0
$892$	9.30815e9	0.439123
$893$	−3.16913e10	−1.48922
$894$	0	0
$895$	1.31796e10	0.614498
$896$	−4.08923e8	−0.0189917
$897$	0	0
$898$	2.72537e10	1.25591
$899$	−9.25382e9	−0.424778
$900$	0	0
$901$	9.83504e8	0.0447960
$902$	3.22690e10	1.46407
$903$	0	0
$904$	−6.25794e9	−0.281736
$905$	−8.67233e9	−0.388925
$906$	0	0
$907$	−7.13828e9	−0.317664	−0.158832	−	0.987306i	$-0.550773\pi$
$907$	−7.13828e9	−0.317664	−0.158832	+	0.987306i	$0.550773\pi$
$908$	8.39272e9	0.372051
$909$	0	0
$910$	1.16419e9	0.0512129
$911$	−3.00662e10	−1.31754	−0.658770	−	0.752344i	$-0.728923\pi$
$911$	−3.00662e10	−1.31754	−0.658770	+	0.752344i	$0.728923\pi$
$912$	0	0
$913$	−3.09728e10	−1.34689
$914$	−1.47220e10	−0.637758
$915$	0	0
$916$	6.35952e8	0.0273395
$917$	1.90265e9	0.0814828
$918$	0	0
$919$	1.16122e10	0.493524	0.246762	−	0.969076i	$-0.420633\pi$
$919$	1.16122e10	0.493524	0.246762	+	0.969076i	$0.420633\pi$
$920$	1.77295e10	0.750653
$921$	0	0
$922$	−1.95763e10	−0.822570
$923$	1.90602e10	0.797849
$924$	0	0
$925$	−2.59231e10	−1.07694
$926$	−4.11046e9	−0.170119
$927$	0	0
$928$	7.41185e9	0.304445
$929$	−9.40477e8	−0.0384851	−0.0192426	−	0.999815i	$-0.506125\pi$
$929$	−9.40477e8	−0.0384851	−0.0192426	+	0.999815i	$0.506125\pi$
$930$	0	0
$931$	4.13647e10	1.67999
$932$	2.53939e9	0.102748
$933$	0	0
$934$	2.87638e10	1.15513
$935$	−1.42895e9	−0.0571711
$936$	0	0
$937$	4.42595e9	0.175759	0.0878796	−	0.996131i	$-0.471991\pi$
$937$	4.42595e9	0.175759	0.0878796	+	0.996131i	$0.471991\pi$
$938$	3.20918e9	0.126965
$939$	0	0
$940$	5.40119e9	0.212101
$941$	−1.67372e10	−0.654814	−0.327407	−	0.944883i	$-0.606175\pi$
$941$	−1.67372e10	−0.654814	−0.327407	+	0.944883i	$0.606175\pi$
$942$	0	0
$943$	−5.30748e10	−2.06109
$944$	−5.45531e9	−0.211065
$945$	0	0
$946$	−3.72492e9	−0.143053
$947$	−2.33893e10	−0.894935	−0.447468	−	0.894300i	$-0.647674\pi$
$947$	−2.33893e10	−0.894935	−0.447468	+	0.894300i	$0.647674\pi$
$948$	0	0
$949$	3.11655e10	1.18370
$950$	2.39007e10	0.904438
$951$	0	0
$952$	3.25098e8	0.0122119
$953$	−1.32227e10	−0.494875	−0.247437	−	0.968904i	$-0.579588\pi$
$953$	−1.32227e10	−0.494875	−0.247437	+	0.968904i	$0.579588\pi$
$954$	0	0
$955$	4.17556e9	0.155132
$956$	−8.90457e9	−0.329618
$957$	0	0
$958$	1.88641e10	0.693198
$959$	2.96750e9	0.108649
$960$	0	0
$961$	9.51734e9	0.345926
$962$	−2.96801e10	−1.07486
$963$	0	0
$964$	6.85024e8	0.0246284
$965$	2.11000e10	0.755851
$966$	0	0
$967$	−1.96959e10	−0.700459	−0.350229	−	0.936664i	$-0.613896\pi$
$967$	−1.96959e10	−0.700459	−0.350229	+	0.936664i	$0.613896\pi$
$968$	−1.90572e10	−0.675299
$969$	0	0
$970$	−1.47757e10	−0.519812
$971$	4.75127e10	1.66549	0.832746	−	0.553655i	$-0.186768\pi$
$971$	4.75127e10	1.66549	0.832746	+	0.553655i	$0.186768\pi$
$972$	0	0
$973$	6.29327e9	0.219019
$974$	−2.09387e10	−0.726097
$975$	0	0
$976$	1.73970e10	0.598964
$977$	−4.37334e10	−1.50031	−0.750157	−	0.661260i	$-0.770022\pi$
$977$	−4.37334e10	−1.50031	−0.750157	+	0.661260i	$0.770022\pi$
$978$	0	0
$979$	−4.61502e10	−1.57193
$980$	−7.04984e9	−0.239270
$981$	0	0
$982$	−5.98924e9	−0.201828
$983$	−1.27037e10	−0.426572	−0.213286	−	0.976990i	$-0.568417\pi$
$983$	−1.27037e10	−0.426572	−0.213286	+	0.976990i	$0.568417\pi$
$984$	0	0
$985$	1.58582e10	0.528722
$986$	−6.84663e8	−0.0227461
$987$	0	0
$988$	−2.33063e10	−0.768820
$989$	6.12659e9	0.201387
$990$	0	0
$991$	−2.85362e10	−0.931404	−0.465702	−	0.884941i	$-0.654198\pi$
$991$	−2.85362e10	−0.931404	−0.465702	+	0.884941i	$0.654198\pi$
$992$	−2.96592e10	−0.964646
$993$	0	0
$994$	−2.50218e9	−0.0808103
$995$	−2.03765e10	−0.655766
$996$	0	0
$997$	3.65067e10	1.16665	0.583323	−	0.812240i	$-0.301752\pi$
$997$	3.65067e10	1.16665	0.583323	+	0.812240i	$0.301752\pi$
$998$	−1.79110e10	−0.570378
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.b.1.7		11
3.2	odd	2		43.8.a.a.1.5	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.a.1.5	✓	11	3.2	odd	2
387.8.a.b.1.7		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+8.31419 q^{2} -58.8742 q^{4} -148.086 q^{5} +122.189 q^{7} -1553.71 q^{8} +O(q^{10})\)	\(q+8.31419 q^{2} -58.8742 q^{4} -148.086 q^{5} +122.189 q^{7} -1553.71 q^{8} -1231.21 q^{10} -5634.96 q^{11} -7738.55 q^{13} +1015.90 q^{14} -5381.93 q^{16} -1712.43 q^{17} -51155.1 q^{19} +8718.44 q^{20} -46850.2 q^{22} +77057.3 q^{23} -56195.6 q^{25} -64339.8 q^{26} -7193.77 q^{28} +48088.8 q^{29} -192432. q^{31} +154128. q^{32} -14237.5 q^{34} -18094.4 q^{35} +461302. q^{37} -425313. q^{38} +230082. q^{40} -688771. q^{41} +79507.0 q^{43} +331754. q^{44} +640669. q^{46} +619513. q^{47} -808613. q^{49} -467221. q^{50} +455601. q^{52} -574333. q^{53} +834458. q^{55} -189846. q^{56} +399820. q^{58} +1.01363e6 q^{59} -3.23249e6 q^{61} -1.59991e6 q^{62} +1.97034e6 q^{64} +1.14597e6 q^{65} +3.15895e6 q^{67} +100818. q^{68} -150441. q^{70} -2.46302e6 q^{71} -4.02731e6 q^{73} +3.83535e6 q^{74} +3.01172e6 q^{76} -688530. q^{77} -463495. q^{79} +796987. q^{80} -5.72657e6 q^{82} +5.49654e6 q^{83} +253587. q^{85} +661036. q^{86} +8.75509e6 q^{88} +8.18997e6 q^{89} -945564. q^{91} -4.53669e6 q^{92} +5.15075e6 q^{94} +7.57535e6 q^{95} +1.20009e7 q^{97} -6.72296e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8	\( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 11 * q + 24 * q^2 + 602 * q^4 + 752 * q^5 - 12 * q^7 + 3810 * q^8 - 1333 * q^10 - 1333 * q^11 - 17967 * q^13 + 22352 * q^14 - 34406 * q^16 + 63095 * q^17 - 54524 * q^19 + 280995 * q^20 - 289358 * q^22 + 138139 * q^23 + 3455 * q^25 + 132946 * q^26 - 12704 * q^28 + 308658 * q^29 - 209523 * q^31 + 644934 * q^32 + 762435 * q^34 + 578892 * q^35 - 298472 * q^37 + 369707 * q^38 + 2633173 * q^40 + 1346735 * q^41 + 874577 * q^43 - 3134292 * q^44 + 3588111 * q^46 - 499284 * q^47 + 2544563 * q^49 - 3049745 * q^50 + 983088 * q^52 + 2210495 * q^53 - 1855072 * q^55 + 469976 * q^56 + 4397067 * q^58 + 5824216 * q^59 - 4453034 * q^61 - 1002789 * q^62 + 4757538 * q^64 + 2162872 * q^65 - 6859513 * q^67 + 9397005 * q^68 + 845078 * q^70 + 10726554 * q^71 - 4456898 * q^73 - 1046637 * q^74 + 5861267 * q^76 + 17019816 * q^77 - 15541320 * q^79 + 15680911 * q^80 + 20233655 * q^82 + 11146767 * q^83 - 12471976 * q^85 + 1908168 * q^86 - 24463544 * q^88 + 13531356 * q^89 - 19746448 * q^91 + 26023161 * q^92 + 20288857 * q^94 + 12291624 * q^95 - 10999901 * q^97 - 29909168 * q^98

Introduction

L-functions

Modular forms

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