LMFDB - Embedded newform 405.8.a.i.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,8,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	405.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:	$126.515935321$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 7 x^{14} - 1479 x^{13} + 9623 x^{12} + 858424 x^{11} - 5043114 x^{10} - 248945154 x^{9} + \cdots + 784812676793472 $ x^15 - 7x^14 - 1479x^13 + 9623x^12 + 858424x^11 - 5043114x^10 - 248945154x^9 + 1250661006x^8 + 38068075521x^7 - 145949082795x^6 - 2962070172819x^5 + 6595509983499x^4 + 100284834644406x^3 - 44138883317076x^2 - 768967569531540x + 784812676793472
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{33} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$15.1697$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.1697 q^{2} +72.7792 q^{4} -125.000 q^{5} -1587.51 q^{7} -782.460 q^{8} +O(q^{10})$	\(q+14.1697 q^{2} +72.7792 q^{4} -125.000 q^{5} -1587.51 q^{7} -782.460 q^{8} -1771.21 q^{10} -5446.11 q^{11} +8848.89 q^{13} -22494.4 q^{14} -20402.9 q^{16} -28524.0 q^{17} +17605.5 q^{19} -9097.40 q^{20} -77169.5 q^{22} -31995.5 q^{23} +15625.0 q^{25} +125386. q^{26} -115538. q^{28} +61571.2 q^{29} +254866. q^{31} -188948. q^{32} -404175. q^{34} +198438. q^{35} -544384. q^{37} +249464. q^{38} +97807.5 q^{40} +103050. q^{41} +578298. q^{43} -396363. q^{44} -453365. q^{46} -700969. q^{47} +1.69664e6 q^{49} +221401. q^{50} +644015. q^{52} -373781. q^{53} +680763. q^{55} +1.24216e6 q^{56} +872443. q^{58} +122387. q^{59} +260717. q^{61} +3.61136e6 q^{62} -65748.1 q^{64} -1.10611e6 q^{65} +456611. q^{67} -2.07595e6 q^{68} +2.81180e6 q^{70} +4.98797e6 q^{71} +3.69501e6 q^{73} -7.71373e6 q^{74} +1.28131e6 q^{76} +8.64573e6 q^{77} -6.34040e6 q^{79} +2.55037e6 q^{80} +1.46018e6 q^{82} +4.13316e6 q^{83} +3.56550e6 q^{85} +8.19428e6 q^{86} +4.26136e6 q^{88} +671500. q^{89} -1.40477e7 q^{91} -2.32860e6 q^{92} -9.93249e6 q^{94} -2.20069e6 q^{95} -2.66280e6 q^{97} +2.40408e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 8 q^{2} + 1088 q^{4} - 1875 q^{5} + 1289 q^{7} - 4434 q^{8}+O(q^{10}) $ 15 * q - 8 * q^2 + 1088 * q^4 - 1875 * q^5 + 1289 * q^7 - 4434 * q^8	\( 15 q - 8 q^{2} + 1088 q^{4} - 1875 q^{5} + 1289 q^{7} - 4434 q^{8} + 1000 q^{10} - 1658 q^{11} + 9874 q^{13} - 7695 q^{14} + 62612 q^{16} + 3598 q^{17} + 21376 q^{19} - 136000 q^{20} - 52519 q^{22} - 96441 q^{23} + 234375 q^{25} - 126146 q^{26} + 174449 q^{28} - 259297 q^{29} + 373568 q^{31} - 1134550 q^{32} + 423851 q^{34} - 161125 q^{35} + 517872 q^{37} - 690059 q^{38} + 554250 q^{40} - 520501 q^{41} + 1898836 q^{43} - 1277707 q^{44} + 3154677 q^{46} - 2259041 q^{47} + 4316308 q^{49} - 125000 q^{50} + 5398554 q^{52} + 102274 q^{53} + 207250 q^{55} + 504621 q^{56} + 3190987 q^{58} + 1680874 q^{59} - 1066457 q^{61} + 274110 q^{62} + 6541980 q^{64} - 1234250 q^{65} + 6522389 q^{67} - 1420717 q^{68} + 961875 q^{70} + 32786 q^{71} + 5359102 q^{73} + 4045556 q^{74} + 4649241 q^{76} - 2586078 q^{77} + 9319346 q^{79} - 7826500 q^{80} + 7460620 q^{82} + 12758277 q^{83} - 449750 q^{85} + 20044675 q^{86} + 6691143 q^{88} + 18776241 q^{89} + 9244102 q^{91} + 13862829 q^{92} + 25905119 q^{94} - 2672000 q^{95} + 2788224 q^{97} + 1679531 q^{98}+O(q^{100}) \) 15 * q - 8 * q^2 + 1088 * q^4 - 1875 * q^5 + 1289 * q^7 - 4434 * q^8 + 1000 * q^10 - 1658 * q^11 + 9874 * q^13 - 7695 * q^14 + 62612 * q^16 + 3598 * q^17 + 21376 * q^19 - 136000 * q^20 - 52519 * q^22 - 96441 * q^23 + 234375 * q^25 - 126146 * q^26 + 174449 * q^28 - 259297 * q^29 + 373568 * q^31 - 1134550 * q^32 + 423851 * q^34 - 161125 * q^35 + 517872 * q^37 - 690059 * q^38 + 554250 * q^40 - 520501 * q^41 + 1898836 * q^43 - 1277707 * q^44 + 3154677 * q^46 - 2259041 * q^47 + 4316308 * q^49 - 125000 * q^50 + 5398554 * q^52 + 102274 * q^53 + 207250 * q^55 + 504621 * q^56 + 3190987 * q^58 + 1680874 * q^59 - 1066457 * q^61 + 274110 * q^62 + 6541980 * q^64 - 1234250 * q^65 + 6522389 * q^67 - 1420717 * q^68 + 961875 * q^70 + 32786 * q^71 + 5359102 * q^73 + 4045556 * q^74 + 4649241 * q^76 - 2586078 * q^77 + 9319346 * q^79 - 7826500 * q^80 + 7460620 * q^82 + 12758277 * q^83 - 449750 * q^85 + 20044675 * q^86 + 6691143 * q^88 + 18776241 * q^89 + 9244102 * q^91 + 13862829 * q^92 + 25905119 * q^94 - 2672000 * q^95 + 2788224 * q^97 + 1679531 * 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 $

$2$	14.1697	1.25243	0.626216	−	0.779649i	$-0.284603\pi$
$2$	14.1697	1.25243	0.626216	+	0.779649i	$0.284603\pi$
$3$	0	0
$3$	0	0
$4$	72.7792	0.568587
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−1587.51	−1.74933	−0.874667	−	0.484725i	$-0.838920\pi$
$7$	−1587.51	−1.74933	−0.874667	+	0.484725i	$0.838920\pi$
$8$	−782.460	−0.540315
$9$	0	0
$10$	−1771.21	−0.560105
$11$	−5446.11	−1.23371	−0.616853	−	0.787078i	$-0.711593\pi$
$11$	−5446.11	−1.23371	−0.616853	+	0.787078i	$0.711593\pi$
$12$	0	0
$13$	8848.89	1.11709	0.558543	−	0.829475i	$-0.311360\pi$
$13$	8848.89	1.11709	0.558543	+	0.829475i	$0.311360\pi$
$14$	−22494.4	−2.19092
$15$	0	0
$16$	−20402.9	−1.24530
$17$	−28524.0	−1.40812	−0.704059	−	0.710141i	$-0.748631\pi$
$17$	−28524.0	−1.40812	−0.704059	+	0.710141i	$0.748631\pi$
$18$	0	0
$19$	17605.5	0.588858	0.294429	−	0.955673i	$-0.404871\pi$
$19$	17605.5	0.588858	0.294429	+	0.955673i	$0.404871\pi$
$20$	−9097.40	−0.254280
$21$	0	0
$22$	−77169.5	−1.54513
$23$	−31995.5	−0.548329	−0.274164	−	0.961683i	$-0.588401\pi$
$23$	−31995.5	−0.548329	−0.274164	+	0.961683i	$0.588401\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	125386.	1.39908
$27$	0	0
$28$	−115538.	−0.994649
$29$	61571.2	0.468797	0.234399	−	0.972141i	$-0.424688\pi$
$29$	61571.2	0.468797	0.234399	+	0.972141i	$0.424688\pi$
$30$	0	0
$31$	254866.	1.53654	0.768272	−	0.640123i	$-0.221117\pi$
$31$	254866.	1.53654	0.768272	+	0.640123i	$0.221117\pi$
$32$	−188948.	−1.01933
$33$	0	0
$34$	−404175.	−1.76357
$35$	198438.	0.782326
$36$	0	0
$37$	−544384.	−1.76685	−0.883424	−	0.468575i	$-0.844768\pi$
$37$	−544384.	−1.76685	−0.883424	+	0.468575i	$0.844768\pi$
$38$	249464.	0.737505
$39$	0	0
$40$	97807.5	0.241636
$41$	103050.	0.233510	0.116755	−	0.993161i	$-0.462751\pi$
$41$	103050.	0.233510	0.116755	+	0.993161i	$0.462751\pi$
$42$	0	0
$43$	578298.	1.10921	0.554603	−	0.832115i	$-0.312870\pi$
$43$	578298.	1.10921	0.554603	+	0.832115i	$0.312870\pi$
$44$	−396363.	−0.701470
$45$	0	0
$46$	−453365.	−0.686745
$47$	−700969.	−0.984818	−0.492409	−	0.870364i	$-0.663884\pi$
$47$	−700969.	−0.984818	−0.492409	+	0.870364i	$0.663884\pi$
$48$	0	0
$49$	1.69664e6	2.06017
$50$	221401.	0.250487
$51$	0	0
$52$	644015.	0.635162
$53$	−373781.	−0.344867	−0.172434	−	0.985021i	$-0.555163\pi$
$53$	−373781.	−0.344867	−0.172434	+	0.985021i	$0.555163\pi$
$54$	0	0
$55$	680763.	0.551730
$56$	1.24216e6	0.945192
$57$	0	0
$58$	872443.	0.587137
$59$	122387.	0.0775805	0.0387903	−	0.999247i	$-0.487650\pi$
$59$	122387.	0.0775805	0.0387903	+	0.999247i	$0.487650\pi$
$60$	0	0
$61$	260717.	0.147067	0.0735336	−	0.997293i	$-0.476572\pi$
$61$	260717.	0.147067	0.0735336	+	0.997293i	$0.476572\pi$
$62$	3.61136e6	1.92442
$63$	0	0
$64$	−65748.1	−0.0313511
$65$	−1.10611e6	−0.499576
$66$	0	0
$67$	456611.	0.185475	0.0927374	−	0.995691i	$-0.470438\pi$
$67$	456611.	0.185475	0.0927374	+	0.995691i	$0.470438\pi$
$68$	−2.07595e6	−0.800639
$69$	0	0
$70$	2.81180e6	0.979810
$71$	4.98797e6	1.65394	0.826970	−	0.562246i	$-0.190062\pi$
$71$	4.98797e6	1.65394	0.826970	+	0.562246i	$0.190062\pi$
$72$	0	0
$73$	3.69501e6	1.11169	0.555847	−	0.831285i	$-0.312394\pi$
$73$	3.69501e6	1.11169	0.555847	+	0.831285i	$0.312394\pi$
$74$	−7.71373e6	−2.21286
$75$	0	0
$76$	1.28131e6	0.334817
$77$	8.64573e6	2.15816
$78$	0	0
$79$	−6.34040e6	−1.44684	−0.723422	−	0.690406i	$-0.757432\pi$
$79$	−6.34040e6	−1.44684	−0.723422	+	0.690406i	$0.757432\pi$
$80$	2.55037e6	0.556913
$81$	0	0
$82$	1.46018e6	0.292455
$83$	4.13316e6	0.793430	0.396715	−	0.917942i	$-0.370150\pi$
$83$	4.13316e6	0.793430	0.396715	+	0.917942i	$0.370150\pi$
$84$	0	0
$85$	3.56550e6	0.629730
$86$	8.19428e6	1.38921
$87$	0	0
$88$	4.26136e6	0.666590
$89$	671500.	0.100967	0.0504837	−	0.998725i	$-0.483924\pi$
$89$	671500.	0.100967	0.0504837	+	0.998725i	$0.483924\pi$
$90$	0	0
$91$	−1.40477e7	−1.95416
$92$	−2.32860e6	−0.311773
$93$	0	0
$94$	−9.93249e6	−1.23342
$95$	−2.20069e6	−0.263345
$96$	0	0
$97$	−2.66280e6	−0.296236	−0.148118	−	0.988970i	$-0.547322\pi$
$97$	−2.66280e6	−0.296236	−0.148118	+	0.988970i	$0.547322\pi$
$98$	2.40408e7	2.58022
$99$	0	0
$100$	1.13717e6	0.113717
$101$	4.48325e6	0.432980	0.216490	−	0.976285i	$-0.430539\pi$
$101$	4.48325e6	0.432980	0.216490	+	0.976285i	$0.430539\pi$
$102$	0	0
$103$	−6.60142e6	−0.595261	−0.297630	−	0.954681i	$-0.596196\pi$
$103$	−6.60142e6	−0.595261	−0.297630	+	0.954681i	$0.596196\pi$
$104$	−6.92390e6	−0.603579
$105$	0	0
$106$	−5.29635e6	−0.431923
$107$	−2.06756e7	−1.63161	−0.815804	−	0.578329i	$-0.803705\pi$
$107$	−2.06756e7	−1.63161	−0.815804	+	0.578329i	$0.803705\pi$
$108$	0	0
$109$	3.20823e6	0.237286	0.118643	−	0.992937i	$-0.462146\pi$
$109$	3.20823e6	0.237286	0.118643	+	0.992937i	$0.462146\pi$
$110$	9.64618e6	0.691005
$111$	0	0
$112$	3.23898e7	2.17844
$113$	−1.69033e6	−0.110204	−0.0551018	−	0.998481i	$-0.517548\pi$
$113$	−1.69033e6	−0.110204	−0.0551018	+	0.998481i	$0.517548\pi$
$114$	0	0
$115$	3.99943e6	0.245220
$116$	4.48110e6	0.266552
$117$	0	0
$118$	1.73418e6	0.0971644
$119$	4.52821e7	2.46327
$120$	0	0
$121$	1.01729e7	0.522031
$122$	3.69428e6	0.184192
$123$	0	0
$124$	1.85489e7	0.873660
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−2.13533e7	−0.925023	−0.462511	−	0.886613i	$-0.653052\pi$
$127$	−2.13533e7	−0.925023	−0.462511	+	0.886613i	$0.653052\pi$
$128$	2.32537e7	0.980069
$129$	0	0
$130$	−1.56732e7	−0.625686
$131$	2.83548e7	1.10199	0.550994	−	0.834509i	$-0.314249\pi$
$131$	2.83548e7	1.10199	0.550994	+	0.834509i	$0.314249\pi$
$132$	0	0
$133$	−2.79488e7	−1.03011
$134$	6.47003e6	0.232295
$135$	0	0
$136$	2.23189e7	0.760828
$137$	−3.59741e7	−1.19527	−0.597637	−	0.801767i	$-0.703894\pi$
$137$	−3.59741e7	−1.19527	−0.597637	+	0.801767i	$0.703894\pi$
$138$	0	0
$139$	1.43690e7	0.453812	0.226906	−	0.973917i	$-0.427139\pi$
$139$	1.43690e7	0.453812	0.226906	+	0.973917i	$0.427139\pi$
$140$	1.44422e7	0.444821
$141$	0	0
$142$	7.06779e7	2.07145
$143$	−4.81920e7	−1.37816
$144$	0	0
$145$	−7.69640e6	−0.209652
$146$	5.23570e7	1.39232
$147$	0	0
$148$	−3.96198e7	−1.00461
$149$	−2.57862e6	−0.0638610	−0.0319305	−	0.999490i	$-0.510166\pi$
$149$	−2.57862e6	−0.0638610	−0.0319305	+	0.999490i	$0.510166\pi$
$150$	0	0
$151$	−2.25337e7	−0.532614	−0.266307	−	0.963888i	$-0.585804\pi$
$151$	−2.25337e7	−0.532614	−0.266307	+	0.963888i	$0.585804\pi$
$152$	−1.37756e7	−0.318169
$153$	0	0
$154$	1.22507e8	2.70295
$155$	−3.18582e7	−0.687164
$156$	0	0
$157$	−2.87711e7	−0.593345	−0.296673	−	0.954979i	$-0.595877\pi$
$157$	−2.87711e7	−0.593345	−0.296673	+	0.954979i	$0.595877\pi$
$158$	−8.98413e7	−1.81207
$159$	0	0
$160$	2.36184e7	0.455860
$161$	5.07930e7	0.959210
$162$	0	0
$163$	7.75003e7	1.40167	0.700837	−	0.713322i	$-0.252810\pi$
$163$	7.75003e7	1.40167	0.700837	+	0.713322i	$0.252810\pi$
$164$	7.49990e6	0.132771
$165$	0	0
$166$	5.85654e7	0.993717
$167$	1.37770e7	0.228901	0.114451	−	0.993429i	$-0.463489\pi$
$167$	1.37770e7	0.228901	0.114451	+	0.993429i	$0.463489\pi$
$168$	0	0
$169$	1.55543e7	0.247883
$170$	5.05219e7	0.788694
$171$	0	0
$172$	4.20881e7	0.630680
$173$	1.98619e7	0.291649	0.145824	−	0.989311i	$-0.453417\pi$
$173$	1.98619e7	0.291649	0.145824	+	0.989311i	$0.453417\pi$
$174$	0	0
$175$	−2.48048e7	−0.349867
$176$	1.11117e8	1.53633
$177$	0	0
$178$	9.51493e6	0.126455
$179$	−1.24838e8	−1.62690	−0.813448	−	0.581638i	$-0.802412\pi$
$179$	−1.24838e8	−1.62690	−0.813448	+	0.581638i	$0.802412\pi$
$180$	0	0
$181$	8.67803e7	1.08779	0.543896	−	0.839152i	$-0.316948\pi$
$181$	8.67803e7	1.08779	0.543896	+	0.839152i	$0.316948\pi$
$182$	−1.99051e8	−2.44745
$183$	0	0
$184$	2.50352e7	0.296270
$185$	6.80480e7	0.790158
$186$	0	0
$187$	1.55345e8	1.73720
$188$	−5.10159e7	−0.559955
$189$	0	0
$190$	−3.11830e7	−0.329822
$191$	1.07588e8	1.11724	0.558622	−	0.829422i	$-0.311330\pi$
$191$	1.07588e8	1.11724	0.558622	+	0.829422i	$0.311330\pi$
$192$	0	0
$193$	6.11202e7	0.611976	0.305988	−	0.952035i	$-0.401013\pi$
$193$	6.11202e7	0.611976	0.305988	+	0.952035i	$0.401013\pi$
$194$	−3.77310e7	−0.371016
$195$	0	0
$196$	1.23480e8	1.17139
$197$	1.34089e7	0.124957	0.0624786	−	0.998046i	$-0.480099\pi$
$197$	1.34089e7	0.124957	0.0624786	+	0.998046i	$0.480099\pi$
$198$	0	0
$199$	2.02539e8	1.82190	0.910949	−	0.412520i	$-0.135351\pi$
$199$	2.02539e8	1.82190	0.910949	+	0.412520i	$0.135351\pi$
$200$	−1.22259e7	−0.108063
$201$	0	0
$202$	6.35261e7	0.542278
$203$	−9.77448e7	−0.820082
$204$	0	0
$205$	−1.28813e7	−0.104429
$206$	−9.35399e7	−0.745524
$207$	0	0
$208$	−1.80543e8	−1.39110
$209$	−9.58813e7	−0.726477
$210$	0	0
$211$	−7.98458e7	−0.585145	−0.292573	−	0.956243i	$-0.594511\pi$
$211$	−7.98458e7	−0.585145	−0.292573	+	0.956243i	$0.594511\pi$
$212$	−2.72035e7	−0.196087
$213$	0	0
$214$	−2.92967e8	−2.04348
$215$	−7.22872e7	−0.496052
$216$	0	0
$217$	−4.04601e8	−2.68793
$218$	4.54595e7	0.297185
$219$	0	0
$220$	4.95454e7	0.313707
$221$	−2.52406e8	−1.57299
$222$	0	0
$223$	8.28221e7	0.500126	0.250063	−	0.968230i	$-0.419549\pi$
$223$	8.28221e7	0.500126	0.250063	+	0.968230i	$0.419549\pi$
$224$	2.99956e8	1.78315
$225$	0	0
$226$	−2.39513e7	−0.138023
$227$	2.51035e7	0.142444	0.0712218	−	0.997461i	$-0.477310\pi$
$227$	2.51035e7	0.142444	0.0712218	+	0.997461i	$0.477310\pi$
$228$	0	0
$229$	7.72730e6	0.0425210	0.0212605	−	0.999774i	$-0.493232\pi$
$229$	7.72730e6	0.0425210	0.0212605	+	0.999774i	$0.493232\pi$
$230$	5.66706e7	0.307122
$231$	0	0
$232$	−4.81770e7	−0.253298
$233$	−1.81671e8	−0.940890	−0.470445	−	0.882429i	$-0.655907\pi$
$233$	−1.81671e8	−0.940890	−0.470445	+	0.882429i	$0.655907\pi$
$234$	0	0
$235$	8.76211e7	0.440424
$236$	8.90721e6	0.0441113
$237$	0	0
$238$	6.41632e8	3.08508
$239$	−6.91830e7	−0.327798	−0.163899	−	0.986477i	$-0.552407\pi$
$239$	−6.91830e7	−0.327798	−0.163899	+	0.986477i	$0.552407\pi$
$240$	0	0
$241$	5.46590e7	0.251537	0.125769	−	0.992060i	$-0.459860\pi$
$241$	5.46590e7	0.251537	0.125769	+	0.992060i	$0.459860\pi$
$242$	1.44147e8	0.653808
$243$	0	0
$244$	1.89748e7	0.0836205
$245$	−2.12080e8	−0.921335
$246$	0	0
$247$	1.55789e8	0.657805
$248$	−1.99422e8	−0.830218
$249$	0	0
$250$	−2.76751e7	−0.112021
$251$	1.67326e8	0.667892	0.333946	−	0.942592i	$-0.391620\pi$
$251$	1.67326e8	0.667892	0.333946	+	0.942592i	$0.391620\pi$
$252$	0	0
$253$	1.74251e8	0.676476
$254$	−3.02569e8	−1.15853
$255$	0	0
$256$	3.37912e8	1.25882
$257$	1.92976e8	0.709147	0.354574	−	0.935028i	$-0.384626\pi$
$257$	1.92976e8	0.709147	0.354574	+	0.935028i	$0.384626\pi$
$258$	0	0
$259$	8.64213e8	3.09081
$260$	−8.05019e7	−0.284053
$261$	0	0
$262$	4.01778e8	1.38017
$263$	−2.64898e8	−0.897912	−0.448956	−	0.893554i	$-0.648204\pi$
$263$	−2.64898e8	−0.897912	−0.448956	+	0.893554i	$0.648204\pi$
$264$	0	0
$265$	4.67227e7	0.154229
$266$	−3.96025e8	−1.29014
$267$	0	0
$268$	3.32318e7	0.105459
$269$	−1.43614e8	−0.449846	−0.224923	−	0.974376i	$-0.572213\pi$
$269$	−1.43614e8	−0.449846	−0.224923	+	0.974376i	$0.572213\pi$
$270$	0	0
$271$	−3.01559e8	−0.920405	−0.460203	−	0.887814i	$-0.652223\pi$
$271$	−3.01559e8	−0.920405	−0.460203	+	0.887814i	$0.652223\pi$
$272$	5.81973e8	1.75352
$273$	0	0
$274$	−5.09740e8	−1.49700
$275$	−8.50954e7	−0.246741
$276$	0	0
$277$	−8.55477e7	−0.241841	−0.120920	−	0.992662i	$-0.538585\pi$
$277$	−8.55477e7	−0.241841	−0.120920	+	0.992662i	$0.538585\pi$
$278$	2.03604e8	0.568369
$279$	0	0
$280$	−1.55270e8	−0.422702
$281$	−4.59839e8	−1.23633	−0.618163	−	0.786050i	$-0.712123\pi$
$281$	−4.59839e8	−1.23633	−0.618163	+	0.786050i	$0.712123\pi$
$282$	0	0
$283$	5.28600e8	1.38636	0.693178	−	0.720767i	$-0.256210\pi$
$283$	5.28600e8	1.38636	0.693178	+	0.720767i	$0.256210\pi$
$284$	3.63021e8	0.940410
$285$	0	0
$286$	−6.82864e8	−1.72605
$287$	−1.63593e8	−0.408486
$288$	0	0
$289$	4.03280e8	0.982799
$290$	−1.09055e8	−0.262576
$291$	0	0
$292$	2.68920e8	0.632095
$293$	2.28126e8	0.529832	0.264916	−	0.964271i	$-0.414656\pi$
$293$	2.28126e8	0.529832	0.264916	+	0.964271i	$0.414656\pi$
$294$	0	0
$295$	−1.52984e7	−0.0346951
$296$	4.25959e8	0.954655
$297$	0	0
$298$	−3.65382e7	−0.0799816
$299$	−2.83124e8	−0.612531
$300$	0	0
$301$	−9.18052e8	−1.94037
$302$	−3.19294e8	−0.667063
$303$	0	0
$304$	−3.59203e8	−0.733302
$305$	−3.25897e7	−0.0657704
$306$	0	0
$307$	7.37110e8	1.45394	0.726972	−	0.686667i	$-0.240927\pi$
$307$	7.37110e8	1.45394	0.726972	+	0.686667i	$0.240927\pi$
$308$	6.29230e8	1.22710
$309$	0	0
$310$	−4.51420e8	−0.860626
$311$	6.23750e7	0.117584	0.0587921	−	0.998270i	$-0.481275\pi$
$311$	6.23750e7	0.117584	0.0587921	+	0.998270i	$0.481275\pi$
$312$	0	0
$313$	4.06042e8	0.748455	0.374228	−	0.927337i	$-0.377908\pi$
$313$	4.06042e8	0.748455	0.374228	+	0.927337i	$0.377908\pi$
$314$	−4.07676e8	−0.743125
$315$	0	0
$316$	−4.61449e8	−0.822657
$317$	−4.58904e8	−0.809123	−0.404562	−	0.914511i	$-0.632576\pi$
$317$	−4.58904e8	−0.809123	−0.404562	+	0.914511i	$0.632576\pi$
$318$	0	0
$319$	−3.35323e8	−0.578358
$320$	8.21851e6	0.0140207
$321$	0	0
$322$	7.19720e8	1.20135
$323$	−5.02179e8	−0.829182
$324$	0	0
$325$	1.38264e8	0.223417
$326$	1.09815e9	1.75550
$327$	0	0
$328$	−8.06326e7	−0.126169
$329$	1.11279e9	1.72278
$330$	0	0
$331$	2.07571e8	0.314607	0.157304	−	0.987550i	$-0.449720\pi$
$331$	2.07571e8	0.314607	0.157304	+	0.987550i	$0.449720\pi$
$332$	3.00808e8	0.451134
$333$	0	0
$334$	1.95216e8	0.286684
$335$	−5.70764e7	−0.0829469
$336$	0	0
$337$	−2.16938e8	−0.308767	−0.154383	−	0.988011i	$-0.549339\pi$
$337$	−2.16938e8	−0.308767	−0.154383	+	0.988011i	$0.549339\pi$
$338$	2.20399e8	0.310457
$339$	0	0
$340$	2.59494e8	0.358057
$341$	−1.38802e9	−1.89564
$342$	0	0
$343$	−1.38604e9	−1.85459
$344$	−4.52495e8	−0.599321
$345$	0	0
$346$	2.81437e8	0.365270
$347$	4.86302e8	0.624817	0.312409	−	0.949948i	$-0.398864\pi$
$347$	4.86302e8	0.624817	0.312409	+	0.949948i	$0.398864\pi$
$348$	0	0
$349$	−2.29156e7	−0.0288565	−0.0144282	−	0.999896i	$-0.504593\pi$
$349$	−2.29156e7	−0.0288565	−0.0144282	+	0.999896i	$0.504593\pi$
$350$	−3.51476e8	−0.438184
$351$	0	0
$352$	1.02903e9	1.25756
$353$	4.34061e8	0.525217	0.262609	−	0.964902i	$-0.415417\pi$
$353$	4.34061e8	0.525217	0.262609	+	0.964902i	$0.415417\pi$
$354$	0	0
$355$	−6.23497e8	−0.739665
$356$	4.88713e7	0.0574088
$357$	0	0
$358$	−1.76891e9	−2.03758
$359$	1.52370e9	1.73808	0.869038	−	0.494746i	$-0.164739\pi$
$359$	1.52370e9	1.73808	0.869038	+	0.494746i	$0.164739\pi$
$360$	0	0
$361$	−5.83919e8	−0.653247
$362$	1.22965e9	1.36239
$363$	0	0
$364$	−1.02238e9	−1.11111
$365$	−4.61876e8	−0.497165
$366$	0	0
$367$	−2.04186e8	−0.215622	−0.107811	−	0.994171i	$-0.534384\pi$
$367$	−2.04186e8	−0.215622	−0.107811	+	0.994171i	$0.534384\pi$
$368$	6.52801e8	0.682831
$369$	0	0
$370$	9.64216e8	0.989620
$371$	5.93381e8	0.603288
$372$	0	0
$373$	−4.11491e8	−0.410562	−0.205281	−	0.978703i	$-0.565811\pi$
$373$	−4.11491e8	−0.410562	−0.205281	+	0.978703i	$0.565811\pi$
$374$	2.20118e9	2.17573
$375$	0	0
$376$	5.48480e8	0.532112
$377$	5.44837e8	0.523687
$378$	0	0
$379$	2.16590e8	0.204363	0.102181	−	0.994766i	$-0.467418\pi$
$379$	2.16590e8	0.204363	0.102181	+	0.994766i	$0.467418\pi$
$380$	−1.60164e8	−0.149735
$381$	0	0
$382$	1.52449e9	1.39927
$383$	−1.86980e9	−1.70059	−0.850296	−	0.526305i	$-0.823577\pi$
$383$	−1.86980e9	−1.70059	−0.850296	+	0.526305i	$0.823577\pi$
$384$	0	0
$385$	−1.08072e9	−0.965160
$386$	8.66053e8	0.766459
$387$	0	0
$388$	−1.93797e8	−0.168436
$389$	8.80182e8	0.758139	0.379070	−	0.925368i	$-0.376244\pi$
$389$	8.80182e8	0.758139	0.379070	+	0.925368i	$0.376244\pi$
$390$	0	0
$391$	9.12639e8	0.772112
$392$	−1.32755e9	−1.11314
$393$	0	0
$394$	1.89999e8	0.156500
$395$	7.92550e8	0.647048
$396$	0	0
$397$	1.80894e9	1.45097	0.725484	−	0.688239i	$-0.241616\pi$
$397$	1.80894e9	1.45097	0.725484	+	0.688239i	$0.241616\pi$
$398$	2.86991e9	2.28180
$399$	0	0
$400$	−3.18796e8	−0.249059
$401$	−1.18998e8	−0.0921585	−0.0460792	−	0.998938i	$-0.514673\pi$
$401$	−1.18998e8	−0.0921585	−0.0460792	+	0.998938i	$0.514673\pi$
$402$	0	0
$403$	2.25528e9	1.71645
$404$	3.26287e8	0.246187
$405$	0	0
$406$	−1.38501e9	−1.02710
$407$	2.96477e9	2.17977
$408$	0	0
$409$	1.93782e8	0.140050	0.0700248	−	0.997545i	$-0.477692\pi$
$409$	1.93782e8	0.140050	0.0700248	+	0.997545i	$0.477692\pi$
$410$	−1.82523e8	−0.130790
$411$	0	0
$412$	−4.80446e8	−0.338458
$413$	−1.94290e8	−0.135714
$414$	0	0
$415$	−5.16645e8	−0.354833
$416$	−1.67198e9	−1.13868
$417$	0	0
$418$	−1.35861e9	−0.909864
$419$	2.59371e9	1.72255	0.861274	−	0.508140i	$-0.169667\pi$
$419$	2.59371e9	1.72255	0.861274	+	0.508140i	$0.169667\pi$
$420$	0	0
$421$	−2.58032e9	−1.68533	−0.842666	−	0.538437i	$-0.819015\pi$
$421$	−2.58032e9	−1.68533	−0.842666	+	0.538437i	$0.819015\pi$
$422$	−1.13139e9	−0.732855
$423$	0	0
$424$	2.92469e8	0.186337
$425$	−4.45688e8	−0.281624
$426$	0	0
$427$	−4.13891e8	−0.257269
$428$	−1.50476e9	−0.927711
$429$	0	0
$430$	−1.02429e9	−0.621272
$431$	−2.38743e9	−1.43635	−0.718175	−	0.695863i	$-0.755022\pi$
$431$	−2.38743e9	−1.43635	−0.718175	+	0.695863i	$0.755022\pi$
$432$	0	0
$433$	−2.28952e9	−1.35530	−0.677651	−	0.735384i	$-0.737002\pi$
$433$	−2.28952e9	−1.35530	−0.677651	+	0.735384i	$0.737002\pi$
$434$	−5.73306e9	−3.36645
$435$	0	0
$436$	2.33492e8	0.134918
$437$	−5.63296e8	−0.322888
$438$	0	0
$439$	3.06968e9	1.73168	0.865838	−	0.500325i	$-0.166786\pi$
$439$	3.06968e9	1.73168	0.865838	+	0.500325i	$0.166786\pi$
$440$	−5.32670e8	−0.298108
$441$	0	0
$442$	−3.57650e9	−1.97007
$443$	−2.69061e9	−1.47041	−0.735205	−	0.677845i	$-0.762914\pi$
$443$	−2.69061e9	−1.47041	−0.735205	+	0.677845i	$0.762914\pi$
$444$	0	0
$445$	−8.39376e7	−0.0451540
$446$	1.17356e9	0.626374
$447$	0	0
$448$	1.04376e8	0.0548436
$449$	7.49434e8	0.390725	0.195363	−	0.980731i	$-0.437412\pi$
$449$	7.49434e8	0.390725	0.195363	+	0.980731i	$0.437412\pi$
$450$	0	0
$451$	−5.61222e8	−0.288082
$452$	−1.23021e8	−0.0626604
$453$	0	0
$454$	3.55708e8	0.178401
$455$	1.75596e9	0.873926
$456$	0	0
$457$	−3.10016e9	−1.51942	−0.759709	−	0.650264i	$-0.774658\pi$
$457$	−3.10016e9	−1.51942	−0.759709	+	0.650264i	$0.774658\pi$
$458$	1.09493e8	0.0532547
$459$	0	0
$460$	2.91075e8	0.139429
$461$	−4.00381e8	−0.190336	−0.0951679	−	0.995461i	$-0.530339\pi$
$461$	−4.00381e8	−0.190336	−0.0951679	+	0.995461i	$0.530339\pi$
$462$	0	0
$463$	2.61953e9	1.22656	0.613282	−	0.789864i	$-0.289849\pi$
$463$	2.61953e9	1.22656	0.613282	+	0.789864i	$0.289849\pi$
$464$	−1.25623e9	−0.583791
$465$	0	0
$466$	−2.57421e9	−1.17840
$467$	−2.13796e9	−0.971381	−0.485690	−	0.874131i	$-0.661432\pi$
$467$	−2.13796e9	−0.971381	−0.485690	+	0.874131i	$0.661432\pi$
$468$	0	0
$469$	−7.24874e8	−0.324457
$470$	1.24156e9	0.551602
$471$	0	0
$472$	−9.57628e7	−0.0419179
$473$	−3.14947e9	−1.36843
$474$	0	0
$475$	2.75086e8	0.117772
$476$	3.29559e9	1.40058
$477$	0	0
$478$	−9.80299e8	−0.410545
$479$	−7.25094e8	−0.301453	−0.150727	−	0.988575i	$-0.548161\pi$
$479$	−7.25094e8	−0.301453	−0.150727	+	0.988575i	$0.548161\pi$
$480$	0	0
$481$	−4.81719e9	−1.97372
$482$	7.74500e8	0.315034
$483$	0	0
$484$	7.40376e8	0.296820
$485$	3.32850e8	0.132481
$486$	0	0
$487$	4.04706e8	0.158777	0.0793886	−	0.996844i	$-0.474703\pi$
$487$	4.04706e8	0.158777	0.0793886	+	0.996844i	$0.474703\pi$
$488$	−2.04001e8	−0.0794626
$489$	0	0
$490$	−3.00510e9	−1.15391
$491$	4.69323e8	0.178931	0.0894657	−	0.995990i	$-0.471484\pi$
$491$	4.69323e8	0.178931	0.0894657	+	0.995990i	$0.471484\pi$
$492$	0	0
$493$	−1.75626e9	−0.660122
$494$	2.20748e9	0.823857
$495$	0	0
$496$	−5.20000e9	−1.91345
$497$	−7.91844e9	−2.89329
$498$	0	0
$499$	−3.63323e9	−1.30901	−0.654503	−	0.756060i	$-0.727122\pi$
$499$	−3.63323e9	−1.30901	−0.654503	+	0.756060i	$0.727122\pi$
$500$	−1.42147e8	−0.0508560
$501$	0	0
$502$	2.37096e9	0.836490
$503$	3.55566e9	1.24575	0.622877	−	0.782320i	$-0.285964\pi$
$503$	3.55566e9	1.24575	0.622877	+	0.782320i	$0.285964\pi$
$504$	0	0
$505$	−5.60406e8	−0.193635
$506$	2.46907e9	0.847241
$507$	0	0
$508$	−1.55408e9	−0.525956
$509$	−4.65150e9	−1.56344	−0.781719	−	0.623631i	$-0.785657\pi$
$509$	−4.65150e9	−1.56344	−0.781719	+	0.623631i	$0.785657\pi$
$510$	0	0
$511$	−5.86585e9	−1.94472
$512$	1.81163e9	0.596520
$513$	0	0
$514$	2.73440e9	0.888159
$515$	8.25178e8	0.266209
$516$	0	0
$517$	3.81755e9	1.21498
$518$	1.22456e10	3.87103
$519$	0	0
$520$	8.65488e8	0.269929
$521$	−1.25080e9	−0.387485	−0.193743	−	0.981052i	$-0.562063\pi$
$521$	−1.25080e9	−0.387485	−0.193743	+	0.981052i	$0.562063\pi$
$522$	0	0
$523$	5.71916e9	1.74814	0.874071	−	0.485799i	$-0.161471\pi$
$523$	5.71916e9	1.74814	0.874071	+	0.485799i	$0.161471\pi$
$524$	2.06364e9	0.626577
$525$	0	0
$526$	−3.75351e9	−1.12457
$527$	−7.26979e9	−2.16364
$528$	0	0
$529$	−2.38112e9	−0.699336
$530$	6.62044e8	0.193162
$531$	0	0
$532$	−2.03409e9	−0.585707
$533$	9.11879e8	0.260851
$534$	0	0
$535$	2.58445e9	0.729677
$536$	−3.57280e8	−0.100215
$537$	0	0
$538$	−2.03496e9	−0.563402
$539$	−9.24007e9	−2.54164
$540$	0	0
$541$	−5.92476e9	−1.60872	−0.804360	−	0.594143i	$-0.797491\pi$
$541$	−5.92476e9	−1.60872	−0.804360	+	0.594143i	$0.797491\pi$
$542$	−4.27298e9	−1.15275
$543$	0	0
$544$	5.38954e9	1.43534
$545$	−4.01029e8	−0.106118
$546$	0	0
$547$	3.15966e9	0.825439	0.412719	−	0.910858i	$-0.364579\pi$
$547$	3.15966e9	0.825439	0.412719	+	0.910858i	$0.364579\pi$
$548$	−2.61816e9	−0.679618
$549$	0	0
$550$	−1.20577e9	−0.309027
$551$	1.08399e9	0.276055
$552$	0	0
$553$	1.00654e10	2.53101
$554$	−1.21218e9	−0.302889
$555$	0	0
$556$	1.04577e9	0.258032
$557$	−2.67265e9	−0.655313	−0.327656	−	0.944797i	$-0.606259\pi$
$557$	−2.67265e9	−0.655313	−0.327656	+	0.944797i	$0.606259\pi$
$558$	0	0
$559$	5.11729e9	1.23908
$560$	−4.04872e9	−0.974227
$561$	0	0
$562$	−6.51576e9	−1.54842
$563$	−4.84830e9	−1.14501	−0.572506	−	0.819900i	$-0.694029\pi$
$563$	−4.84830e9	−1.14501	−0.572506	+	0.819900i	$0.694029\pi$
$564$	0	0
$565$	2.11291e8	0.0492846
$566$	7.49008e9	1.73632
$567$	0	0
$568$	−3.90289e9	−0.893649
$569$	6.81940e9	1.55186	0.775931	−	0.630818i	$-0.217281\pi$
$569$	6.81940e9	1.55186	0.775931	+	0.630818i	$0.217281\pi$
$570$	0	0
$571$	−6.16570e8	−0.138598	−0.0692989	−	0.997596i	$-0.522076\pi$
$571$	−6.16570e8	−0.138598	−0.0692989	+	0.997596i	$0.522076\pi$
$572$	−3.50737e9	−0.783603
$573$	0	0
$574$	−2.31805e9	−0.511602
$575$	−4.99929e8	−0.109666
$576$	0	0
$577$	9.00936e9	1.95244	0.976222	−	0.216771i	$-0.0695525\pi$
$577$	9.00936e9	1.95244	0.976222	+	0.216771i	$0.0695525\pi$
$578$	5.71435e9	1.23089
$579$	0	0
$580$	−5.60138e8	−0.119206
$581$	−6.56142e9	−1.38797
$582$	0	0
$583$	2.03565e9	0.425465
$584$	−2.89120e9	−0.600665
$585$	0	0
$586$	3.23247e9	0.663579
$587$	−8.80343e9	−1.79646	−0.898232	−	0.439522i	$-0.855148\pi$
$587$	−8.80343e9	−1.79646	−0.898232	+	0.439522i	$0.855148\pi$
$588$	0	0
$589$	4.48703e9	0.904806
$590$	−2.16772e8	−0.0434532
$591$	0	0
$592$	1.11070e10	2.20025
$593$	−1.43818e9	−0.283220	−0.141610	−	0.989923i	$-0.545228\pi$
$593$	−1.43818e9	−0.283220	−0.141610	+	0.989923i	$0.545228\pi$
$594$	0	0
$595$	−5.66026e9	−1.10161
$596$	−1.87670e8	−0.0363106
$597$	0	0
$598$	−4.01177e9	−0.767153
$599$	−6.89028e8	−0.130991	−0.0654957	−	0.997853i	$-0.520863\pi$
$599$	−6.89028e8	−0.130991	−0.0654957	+	0.997853i	$0.520863\pi$
$600$	0	0
$601$	4.06928e9	0.764639	0.382320	−	0.924030i	$-0.375125\pi$
$601$	4.06928e9	0.764639	0.382320	+	0.924030i	$0.375125\pi$
$602$	−1.30085e10	−2.43018
$603$	0	0
$604$	−1.63998e9	−0.302838
$605$	−1.27161e9	−0.233459
$606$	0	0
$607$	9.28000e9	1.68418	0.842088	−	0.539340i	$-0.181326\pi$
$607$	9.28000e9	1.68418	0.842088	+	0.539340i	$0.181326\pi$
$608$	−3.32651e9	−0.600243
$609$	0	0
$610$	−4.61784e8	−0.0823730
$611$	−6.20279e9	−1.10013
$612$	0	0
$613$	9.72207e8	0.170470	0.0852348	−	0.996361i	$-0.472836\pi$
$613$	9.72207e8	0.170470	0.0852348	+	0.996361i	$0.472836\pi$
$614$	1.04446e10	1.82097
$615$	0	0
$616$	−6.76494e9	−1.16609
$617$	1.11807e10	1.91633	0.958164	−	0.286221i	$-0.0923992\pi$
$617$	1.11807e10	1.91633	0.958164	+	0.286221i	$0.0923992\pi$
$618$	0	0
$619$	−1.09554e9	−0.185656	−0.0928281	−	0.995682i	$-0.529591\pi$
$619$	−1.09554e9	−0.185656	−0.0928281	+	0.995682i	$0.529591\pi$
$620$	−2.31861e9	−0.390713
$621$	0	0
$622$	8.83832e8	0.147266
$623$	−1.06601e9	−0.176626
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	5.75348e9	0.937390
$627$	0	0
$628$	−2.09394e9	−0.337369
$629$	1.55280e10	2.48793
$630$	0	0
$631$	8.06380e9	1.27772	0.638862	−	0.769321i	$-0.279405\pi$
$631$	8.06380e9	1.27772	0.638862	+	0.769321i	$0.279405\pi$
$632$	4.96111e9	0.781752
$633$	0	0
$634$	−6.50251e9	−1.01337
$635$	2.66917e9	0.413683
$636$	0	0
$637$	1.50133e10	2.30139
$638$	−4.75142e9	−0.724354
$639$	0	0
$640$	−2.90671e9	−0.438300
$641$	1.20472e10	1.80669	0.903343	−	0.428919i	$-0.141105\pi$
$641$	1.20472e10	1.80669	0.903343	+	0.428919i	$0.141105\pi$
$642$	0	0
$643$	−6.78073e9	−1.00586	−0.502931	−	0.864327i	$-0.667745\pi$
$643$	−6.78073e9	−1.00586	−0.502931	+	0.864327i	$0.667745\pi$
$644$	3.69668e9	0.545395
$645$	0	0
$646$	−7.11570e9	−1.03849
$647$	7.37841e9	1.07102	0.535510	−	0.844529i	$-0.320119\pi$
$647$	7.37841e9	1.07102	0.535510	+	0.844529i	$0.320119\pi$
$648$	0	0
$649$	−6.66532e8	−0.0957116
$650$	1.95915e9	0.279815
$651$	0	0
$652$	5.64041e9	0.796974
$653$	1.24944e10	1.75597	0.877987	−	0.478685i	$-0.158886\pi$
$653$	1.24944e10	1.75597	0.877987	+	0.478685i	$0.158886\pi$
$654$	0	0
$655$	−3.54435e9	−0.492824
$656$	−2.10252e9	−0.290789
$657$	0	0
$658$	1.57679e10	2.15766
$659$	−1.36321e9	−0.185551	−0.0927754	−	0.995687i	$-0.529574\pi$
$659$	−1.36321e9	−0.185551	−0.0927754	+	0.995687i	$0.529574\pi$
$660$	0	0
$661$	−5.86014e9	−0.789228	−0.394614	−	0.918847i	$-0.629122\pi$
$661$	−5.86014e9	−0.789228	−0.394614	+	0.918847i	$0.629122\pi$
$662$	2.94121e9	0.394024
$663$	0	0
$664$	−3.23403e9	−0.428702
$665$	3.49360e9	0.460679
$666$	0	0
$667$	−1.97000e9	−0.257055
$668$	1.00268e9	0.130150
$669$	0	0
$670$	−8.08754e8	−0.103885
$671$	−1.41989e9	−0.181438
$672$	0	0
$673$	−5.63137e9	−0.712133	−0.356066	−	0.934461i	$-0.615882\pi$
$673$	−5.63137e9	−0.712133	−0.356066	+	0.934461i	$0.615882\pi$
$674$	−3.07394e9	−0.386710
$675$	0	0
$676$	1.13203e9	0.140943
$677$	4.29962e9	0.532562	0.266281	−	0.963895i	$-0.414205\pi$
$677$	4.29962e9	0.532562	0.266281	+	0.963895i	$0.414205\pi$
$678$	0	0
$679$	4.22722e9	0.518216
$680$	−2.78986e9	−0.340253
$681$	0	0
$682$	−1.96678e10	−2.37417
$683$	8.18926e9	0.983495	0.491748	−	0.870738i	$-0.336358\pi$
$683$	8.18926e9	0.983495	0.491748	+	0.870738i	$0.336358\pi$
$684$	0	0
$685$	4.49676e9	0.534543
$686$	−1.96398e10	−2.32275
$687$	0	0
$688$	−1.17990e10	−1.38129
$689$	−3.30755e9	−0.385247
$690$	0	0
$691$	1.15361e10	1.33011	0.665053	−	0.746796i	$-0.268409\pi$
$691$	1.15361e10	1.33011	0.665053	+	0.746796i	$0.268409\pi$
$692$	1.44553e9	0.165828
$693$	0	0
$694$	6.89073e9	0.782541
$695$	−1.79613e9	−0.202951
$696$	0	0
$697$	−2.93940e9	−0.328810
$698$	−3.24707e8	−0.0361408
$699$	0	0
$700$	−1.80527e9	−0.198930
$701$	1.50937e9	0.165494	0.0827470	−	0.996571i	$-0.473631\pi$
$701$	1.50937e9	0.165494	0.0827470	+	0.996571i	$0.473631\pi$
$702$	0	0
$703$	−9.58414e9	−1.04042
$704$	3.58071e8	0.0386781
$705$	0	0
$706$	6.15049e9	0.657799
$707$	−7.11719e9	−0.757427
$708$	0	0
$709$	−2.71527e9	−0.286122	−0.143061	−	0.989714i	$-0.545695\pi$
$709$	−2.71527e9	−0.286122	−0.143061	+	0.989714i	$0.545695\pi$
$710$	−8.83473e9	−0.926380
$711$	0	0
$712$	−5.25422e8	−0.0545542
$713$	−8.15454e9	−0.842531
$714$	0	0
$715$	6.02400e9	0.616331
$716$	−9.08558e9	−0.925032
$717$	0	0
$718$	2.15903e10	2.17682
$719$	1.01787e10	1.02127	0.510634	−	0.859798i	$-0.329411\pi$
$719$	1.01787e10	1.02127	0.510634	+	0.859798i	$0.329411\pi$
$720$	0	0
$721$	1.04798e10	1.04131
$722$	−8.27393e9	−0.818147
$723$	0	0
$724$	6.31580e9	0.618505
$725$	9.62051e8	0.0937594
$726$	0	0
$727$	9.99017e9	0.964278	0.482139	−	0.876095i	$-0.339860\pi$
$727$	9.99017e9	0.964278	0.482139	+	0.876095i	$0.339860\pi$
$728$	1.09917e10	1.05586
$729$	0	0
$730$	−6.54462e9	−0.622665
$731$	−1.64954e10	−1.56189
$732$	0	0
$733$	−8.15662e9	−0.764973	−0.382487	−	0.923961i	$-0.624932\pi$
$733$	−8.15662e9	−0.764973	−0.382487	+	0.923961i	$0.624932\pi$
$734$	−2.89324e9	−0.270052
$735$	0	0
$736$	6.04546e9	0.558930
$737$	−2.48675e9	−0.228821
$738$	0	0
$739$	−5.79589e9	−0.528280	−0.264140	−	0.964484i	$-0.585088\pi$
$739$	−5.79589e9	−0.528280	−0.264140	+	0.964484i	$0.585088\pi$
$740$	4.95248e9	0.449274
$741$	0	0
$742$	8.40800e9	0.755578
$743$	2.00709e10	1.79517	0.897587	−	0.440836i	$-0.145318\pi$
$743$	2.00709e10	1.79517	0.897587	+	0.440836i	$0.145318\pi$
$744$	0	0
$745$	3.22328e8	0.0285595
$746$	−5.83068e9	−0.514201
$747$	0	0
$748$	1.13059e10	0.987753
$749$	3.28227e10	2.85423
$750$	0	0
$751$	7.64646e9	0.658750	0.329375	−	0.944199i	$-0.393162\pi$
$751$	7.64646e9	0.658750	0.329375	+	0.944199i	$0.393162\pi$
$752$	1.43018e10	1.22639
$753$	0	0
$754$	7.72015e9	0.655883
$755$	2.81671e9	0.238192
$756$	0	0
$757$	1.50730e10	1.26289	0.631443	−	0.775422i	$-0.282463\pi$
$757$	1.50730e10	1.26289	0.631443	+	0.775422i	$0.282463\pi$
$758$	3.06901e9	0.255951
$759$	0	0
$760$	1.72195e9	0.142289
$761$	6.16862e9	0.507390	0.253695	−	0.967284i	$-0.418354\pi$
$761$	6.16862e9	0.507390	0.253695	+	0.967284i	$0.418354\pi$
$762$	0	0
$763$	−5.09309e9	−0.415093
$764$	7.83018e9	0.635251
$765$	0	0
$766$	−2.64945e10	−2.12988
$767$	1.08299e9	0.0866642
$768$	0	0
$769$	2.02154e10	1.60302	0.801510	−	0.597981i	$-0.204030\pi$
$769$	2.02154e10	1.60302	0.801510	+	0.597981i	$0.204030\pi$
$770$	−1.53134e10	−1.20880
$771$	0	0
$772$	4.44828e9	0.347962
$773$	8.79891e9	0.685174	0.342587	−	0.939486i	$-0.388697\pi$
$773$	8.79891e9	0.685174	0.342587	+	0.939486i	$0.388697\pi$
$774$	0	0
$775$	3.98227e9	0.307309
$776$	2.08354e9	0.160061
$777$	0	0
$778$	1.24719e10	0.949518
$779$	1.81425e9	0.137504
$780$	0	0
$781$	−2.71650e10	−2.04048
$782$	1.29318e10	0.967018
$783$	0	0
$784$	−3.46164e10	−2.56552
$785$	3.59639e9	0.265352
$786$	0	0
$787$	1.14191e10	0.835062	0.417531	−	0.908663i	$-0.362895\pi$
$787$	1.14191e10	0.835062	0.417531	+	0.908663i	$0.362895\pi$
$788$	9.75888e8	0.0710491
$789$	0	0
$790$	1.12302e10	0.810385
$791$	2.68341e9	0.192783
$792$	0	0
$793$	2.30706e9	0.164287
$794$	2.56321e10	1.81724
$795$	0	0
$796$	1.47407e10	1.03591
$797$	−1.31340e10	−0.918949	−0.459474	−	0.888191i	$-0.651962\pi$
$797$	−1.31340e10	−0.918949	−0.459474	+	0.888191i	$0.651962\pi$
$798$	0	0
$799$	1.99944e10	1.38674
$800$	−2.95231e9	−0.203867
$801$	0	0
$802$	−1.68616e9	−0.115422
$803$	−2.01234e10	−1.37150
$804$	0	0
$805$	−6.34913e9	−0.428972
$806$	3.19565e10	2.14974
$807$	0	0
$808$	−3.50796e9	−0.233946
$809$	2.49937e9	0.165963	0.0829814	−	0.996551i	$-0.473556\pi$
$809$	2.49937e9	0.165963	0.0829814	+	0.996551i	$0.473556\pi$
$810$	0	0
$811$	4.41362e9	0.290551	0.145275	−	0.989391i	$-0.453593\pi$
$811$	4.41362e9	0.290551	0.145275	+	0.989391i	$0.453593\pi$
$812$	−7.11379e9	−0.466289
$813$	0	0
$814$	4.20098e10	2.73002
$815$	−9.68754e9	−0.626847
$816$	0	0
$817$	1.01812e10	0.653164
$818$	2.74582e9	0.175403
$819$	0	0
$820$	−9.37488e8	−0.0593769
$821$	2.41435e10	1.52265	0.761323	−	0.648373i	$-0.224550\pi$
$821$	2.41435e10	1.52265	0.761323	+	0.648373i	$0.224550\pi$
$822$	0	0
$823$	2.50506e10	1.56646	0.783229	−	0.621734i	$-0.213571\pi$
$823$	2.50506e10	1.56646	0.783229	+	0.621734i	$0.213571\pi$
$824$	5.16535e9	0.321628
$825$	0	0
$826$	−2.75302e9	−0.169973
$827$	−6.73891e9	−0.414305	−0.207153	−	0.978309i	$-0.566420\pi$
$827$	−6.73891e9	−0.414305	−0.207153	+	0.978309i	$0.566420\pi$
$828$	0	0
$829$	8.81728e8	0.0537519	0.0268759	−	0.999639i	$-0.491444\pi$
$829$	8.81728e8	0.0537519	0.0268759	+	0.999639i	$0.491444\pi$
$830$	−7.32068e9	−0.444404
$831$	0	0
$832$	−5.81798e8	−0.0350219
$833$	−4.83949e10	−2.90096
$834$	0	0
$835$	−1.72213e9	−0.102368
$836$	−6.97817e9	−0.413066
$837$	0	0
$838$	3.67519e10	2.15738
$839$	−1.53663e10	−0.898259	−0.449129	−	0.893467i	$-0.648266\pi$
$839$	−1.53663e10	−0.898259	−0.449129	+	0.893467i	$0.648266\pi$
$840$	0	0
$841$	−1.34589e10	−0.780229
$842$	−3.65622e10	−2.11076
$843$	0	0
$844$	−5.81112e9	−0.332706
$845$	−1.94429e9	−0.110857
$846$	0	0
$847$	−1.61496e10	−0.913206
$848$	7.62623e9	0.429462
$849$	0	0
$850$	−6.31524e9	−0.352715
$851$	1.74178e10	0.968813
$852$	0	0
$853$	1.78907e10	0.986972	0.493486	−	0.869754i	$-0.335722\pi$
$853$	1.78907e10	0.986972	0.493486	+	0.869754i	$0.335722\pi$
$854$	−5.86469e9	−0.322213
$855$	0	0
$856$	1.61779e10	0.881582
$857$	−4.80615e9	−0.260834	−0.130417	−	0.991459i	$-0.541632\pi$
$857$	−4.80615e9	−0.260834	−0.130417	+	0.991459i	$0.541632\pi$
$858$	0	0
$859$	−2.57254e10	−1.38480	−0.692398	−	0.721516i	$-0.743446\pi$
$859$	−2.57254e10	−1.38480	−0.692398	+	0.721516i	$0.743446\pi$
$860$	−5.26101e9	−0.282049
$861$	0	0
$862$	−3.38291e10	−1.79893
$863$	−3.46286e9	−0.183399	−0.0916997	−	0.995787i	$-0.529230\pi$
$863$	−3.46286e9	−0.183399	−0.0916997	+	0.995787i	$0.529230\pi$
$864$	0	0
$865$	−2.48274e9	−0.130429
$866$	−3.24417e10	−1.69742
$867$	0	0
$868$	−2.94465e10	−1.52832
$869$	3.45305e10	1.78498
$870$	0	0
$871$	4.04050e9	0.207192
$872$	−2.51031e9	−0.128209
$873$	0	0
$874$	−7.98170e9	−0.404395
$875$	3.10060e9	0.156465
$876$	0	0
$877$	−3.63610e10	−1.82028	−0.910139	−	0.414303i	$-0.864025\pi$
$877$	−3.63610e10	−1.82028	−0.910139	+	0.414303i	$0.864025\pi$
$878$	4.34962e10	2.16881
$879$	0	0
$880$	−1.38896e10	−0.687067
$881$	−8.18644e9	−0.403348	−0.201674	−	0.979453i	$-0.564638\pi$
$881$	−8.18644e9	−0.403348	−0.201674	+	0.979453i	$0.564638\pi$
$882$	0	0
$883$	−6.82519e9	−0.333620	−0.166810	−	0.985989i	$-0.553347\pi$
$883$	−6.82519e9	−0.333620	−0.166810	+	0.985989i	$0.553347\pi$
$884$	−1.83699e10	−0.894383
$885$	0	0
$886$	−3.81251e10	−1.84159
$887$	−5.33390e9	−0.256633	−0.128316	−	0.991733i	$-0.540957\pi$
$887$	−5.33390e9	−0.256633	−0.128316	+	0.991733i	$0.540957\pi$
$888$	0	0
$889$	3.38986e10	1.61817
$890$	−1.18937e9	−0.0565524
$891$	0	0
$892$	6.02773e9	0.284365
$893$	−1.23409e10	−0.579918
$894$	0	0
$895$	1.56047e10	0.727570
$896$	−3.69154e10	−1.71447
$897$	0	0
$898$	1.06192e10	0.489357
$899$	1.56924e10	0.720328
$900$	0	0
$901$	1.06617e10	0.485614
$902$	−7.95232e9	−0.360804
$903$	0	0
$904$	1.32261e9	0.0595447
$905$	−1.08475e10	−0.486476
$906$	0	0
$907$	−1.26859e10	−0.564542	−0.282271	−	0.959335i	$-0.591088\pi$
$907$	−1.26859e10	−0.564542	−0.282271	+	0.959335i	$0.591088\pi$
$908$	1.82701e9	0.0809917
$909$	0	0
$910$	2.48813e10	1.09453
$911$	−8.19088e9	−0.358935	−0.179468	−	0.983764i	$-0.557438\pi$
$911$	−8.19088e9	−0.358935	−0.179468	+	0.983764i	$0.557438\pi$
$912$	0	0
$913$	−2.25096e10	−0.978859
$914$	−4.39282e10	−1.90297
$915$	0	0
$916$	5.62387e8	0.0241769
$917$	−4.50134e10	−1.92775
$918$	0	0
$919$	1.28182e10	0.544783	0.272391	−	0.962187i	$-0.412185\pi$
$919$	1.28182e10	0.544783	0.272391	+	0.962187i	$0.412185\pi$
$920$	−3.12940e9	−0.132496
$921$	0	0
$922$	−5.67327e9	−0.238383
$923$	4.41380e10	1.84760
$924$	0	0
$925$	−8.50600e9	−0.353370
$926$	3.71179e10	1.53619
$927$	0	0
$928$	−1.16337e10	−0.477861
$929$	8.69604e9	0.355849	0.177925	−	0.984044i	$-0.443062\pi$
$929$	8.69604e9	0.355849	0.177925	+	0.984044i	$0.443062\pi$
$930$	0	0
$931$	2.98701e10	1.21315
$932$	−1.32218e10	−0.534978
$933$	0	0
$934$	−3.02941e10	−1.21659
$935$	−1.94181e10	−0.776902
$936$	0	0
$937$	4.21585e10	1.67416	0.837080	−	0.547081i	$-0.184261\pi$
$937$	4.21585e10	1.67416	0.837080	+	0.547081i	$0.184261\pi$
$938$	−1.02712e10	−0.406361
$939$	0	0
$940$	6.37699e9	0.250420
$941$	1.44517e10	0.565399	0.282700	−	0.959209i	$-0.408770\pi$
$941$	1.44517e10	0.565399	0.282700	+	0.959209i	$0.408770\pi$
$942$	0	0
$943$	−3.29714e9	−0.128040
$944$	−2.49705e9	−0.0966107
$945$	0	0
$946$	−4.46269e10	−1.71387
$947$	4.14350e10	1.58541	0.792706	−	0.609604i	$-0.208671\pi$
$947$	4.14350e10	1.58541	0.792706	+	0.609604i	$0.208671\pi$
$948$	0	0
$949$	3.26967e10	1.24186
$950$	3.89787e9	0.147501
$951$	0	0
$952$	−3.54314e10	−1.33094
$953$	1.85697e9	0.0694992	0.0347496	−	0.999396i	$-0.488937\pi$
$953$	1.85697e9	0.0694992	0.0347496	+	0.999396i	$0.488937\pi$
$954$	0	0
$955$	−1.34485e10	−0.499647
$956$	−5.03508e9	−0.186382
$957$	0	0
$958$	−1.02743e10	−0.377550
$959$	5.71091e10	2.09093
$960$	0	0
$961$	3.74438e10	1.36097
$962$	−6.82579e10	−2.47195
$963$	0	0
$964$	3.97804e9	0.143021
$965$	−7.64003e9	−0.273684
$966$	0	0
$967$	−4.20681e10	−1.49610	−0.748050	−	0.663643i	$-0.769010\pi$
$967$	−4.20681e10	−1.49610	−0.748050	+	0.663643i	$0.769010\pi$
$968$	−7.95989e9	−0.282061
$969$	0	0
$970$	4.71637e9	0.165923
$971$	−4.48349e9	−0.157163	−0.0785813	−	0.996908i	$-0.525039\pi$
$971$	−4.48349e9	−0.157163	−0.0785813	+	0.996908i	$0.525039\pi$
$972$	0	0
$973$	−2.28109e10	−0.793868
$974$	5.73455e9	0.198858
$975$	0	0
$976$	−5.31940e9	−0.183142
$977$	−1.37025e10	−0.470077	−0.235039	−	0.971986i	$-0.575522\pi$
$977$	−1.37025e10	−0.470077	−0.235039	+	0.971986i	$0.575522\pi$
$978$	0	0
$979$	−3.65706e9	−0.124564
$980$	−1.54350e10	−0.523860
$981$	0	0
$982$	6.65014e9	0.224100
$983$	2.20524e10	0.740488	0.370244	−	0.928935i	$-0.379274\pi$
$983$	2.20524e10	0.740488	0.370244	+	0.928935i	$0.379274\pi$
$984$	0	0
$985$	−1.67611e9	−0.0558825
$986$	−2.48856e10	−0.826758
$987$	0	0
$988$	1.13382e10	0.374020
$989$	−1.85029e10	−0.608209
$990$	0	0
$991$	3.97974e10	1.29896	0.649481	−	0.760378i	$-0.274986\pi$
$991$	3.97974e10	1.29896	0.649481	+	0.760378i	$0.274986\pi$
$992$	−4.81562e10	−1.56625
$993$	0	0
$994$	−1.12202e11	−3.62366
$995$	−2.53174e10	−0.814777
$996$	0	0
$997$	−2.28904e10	−0.731510	−0.365755	−	0.930711i	$-0.619189\pi$
$997$	−2.28904e10	−0.731510	−0.365755	+	0.930711i	$0.619189\pi$
$998$	−5.14817e10	−1.63944
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.i.1.13		15
3.2	odd	2		405.8.a.j.1.3		15
9.2	odd	6		135.8.e.b.91.13		30
9.4	even	3		45.8.e.b.16.3	✓	30
9.5	odd	6		135.8.e.b.46.13		30
9.7	even	3		45.8.e.b.31.3	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.e.b.16.3	✓	30	9.4	even	3
45.8.e.b.31.3	yes	30	9.7	even	3
135.8.e.b.46.13		30	9.5	odd	6
135.8.e.b.91.13		30	9.2	odd	6
405.8.a.i.1.13		15	1.1	even	1	trivial
405.8.a.j.1.3		15	3.2	odd	2

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+14.1697 q^{2} +72.7792 q^{4} -125.000 q^{5} -1587.51 q^{7} -782.460 q^{8} +O(q^{10})\)	\(q+14.1697 q^{2} +72.7792 q^{4} -125.000 q^{5} -1587.51 q^{7} -782.460 q^{8} -1771.21 q^{10} -5446.11 q^{11} +8848.89 q^{13} -22494.4 q^{14} -20402.9 q^{16} -28524.0 q^{17} +17605.5 q^{19} -9097.40 q^{20} -77169.5 q^{22} -31995.5 q^{23} +15625.0 q^{25} +125386. q^{26} -115538. q^{28} +61571.2 q^{29} +254866. q^{31} -188948. q^{32} -404175. q^{34} +198438. q^{35} -544384. q^{37} +249464. q^{38} +97807.5 q^{40} +103050. q^{41} +578298. q^{43} -396363. q^{44} -453365. q^{46} -700969. q^{47} +1.69664e6 q^{49} +221401. q^{50} +644015. q^{52} -373781. q^{53} +680763. q^{55} +1.24216e6 q^{56} +872443. q^{58} +122387. q^{59} +260717. q^{61} +3.61136e6 q^{62} -65748.1 q^{64} -1.10611e6 q^{65} +456611. q^{67} -2.07595e6 q^{68} +2.81180e6 q^{70} +4.98797e6 q^{71} +3.69501e6 q^{73} -7.71373e6 q^{74} +1.28131e6 q^{76} +8.64573e6 q^{77} -6.34040e6 q^{79} +2.55037e6 q^{80} +1.46018e6 q^{82} +4.13316e6 q^{83} +3.56550e6 q^{85} +8.19428e6 q^{86} +4.26136e6 q^{88} +671500. q^{89} -1.40477e7 q^{91} -2.32860e6 q^{92} -9.93249e6 q^{94} -2.20069e6 q^{95} -2.66280e6 q^{97} +2.40408e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 8 q^{2} + 1088 q^{4} - 1875 q^{5} + 1289 q^{7} - 4434 q^{8}+O(q^{10}) \) 15 * q - 8 * q^2 + 1088 * q^4 - 1875 * q^5 + 1289 * q^7 - 4434 * q^8	\( 15 q - 8 q^{2} + 1088 q^{4} - 1875 q^{5} + 1289 q^{7} - 4434 q^{8} + 1000 q^{10} - 1658 q^{11} + 9874 q^{13} - 7695 q^{14} + 62612 q^{16} + 3598 q^{17} + 21376 q^{19} - 136000 q^{20} - 52519 q^{22} - 96441 q^{23} + 234375 q^{25} - 126146 q^{26} + 174449 q^{28} - 259297 q^{29} + 373568 q^{31} - 1134550 q^{32} + 423851 q^{34} - 161125 q^{35} + 517872 q^{37} - 690059 q^{38} + 554250 q^{40} - 520501 q^{41} + 1898836 q^{43} - 1277707 q^{44} + 3154677 q^{46} - 2259041 q^{47} + 4316308 q^{49} - 125000 q^{50} + 5398554 q^{52} + 102274 q^{53} + 207250 q^{55} + 504621 q^{56} + 3190987 q^{58} + 1680874 q^{59} - 1066457 q^{61} + 274110 q^{62} + 6541980 q^{64} - 1234250 q^{65} + 6522389 q^{67} - 1420717 q^{68} + 961875 q^{70} + 32786 q^{71} + 5359102 q^{73} + 4045556 q^{74} + 4649241 q^{76} - 2586078 q^{77} + 9319346 q^{79} - 7826500 q^{80} + 7460620 q^{82} + 12758277 q^{83} - 449750 q^{85} + 20044675 q^{86} + 6691143 q^{88} + 18776241 q^{89} + 9244102 q^{91} + 13862829 q^{92} + 25905119 q^{94} - 2672000 q^{95} + 2788224 q^{97} + 1679531 q^{98}+O(q^{100}) \) 15 * q - 8 * q^2 + 1088 * q^4 - 1875 * q^5 + 1289 * q^7 - 4434 * q^8 + 1000 * q^10 - 1658 * q^11 + 9874 * q^13 - 7695 * q^14 + 62612 * q^16 + 3598 * q^17 + 21376 * q^19 - 136000 * q^20 - 52519 * q^22 - 96441 * q^23 + 234375 * q^25 - 126146 * q^26 + 174449 * q^28 - 259297 * q^29 + 373568 * q^31 - 1134550 * q^32 + 423851 * q^34 - 161125 * q^35 + 517872 * q^37 - 690059 * q^38 + 554250 * q^40 - 520501 * q^41 + 1898836 * q^43 - 1277707 * q^44 + 3154677 * q^46 - 2259041 * q^47 + 4316308 * q^49 - 125000 * q^50 + 5398554 * q^52 + 102274 * q^53 + 207250 * q^55 + 504621 * q^56 + 3190987 * q^58 + 1680874 * q^59 - 1066457 * q^61 + 274110 * q^62 + 6541980 * q^64 - 1234250 * q^65 + 6522389 * q^67 - 1420717 * q^68 + 961875 * q^70 + 32786 * q^71 + 5359102 * q^73 + 4045556 * q^74 + 4649241 * q^76 - 2586078 * q^77 + 9319346 * q^79 - 7826500 * q^80 + 7460620 * q^82 + 12758277 * q^83 - 449750 * q^85 + 20044675 * q^86 + 6691143 * q^88 + 18776241 * q^89 + 9244102 * q^91 + 13862829 * q^92 + 25905119 * q^94 - 2672000 * q^95 + 2788224 * q^97 + 1679531 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more