[N,k,chi] = [6012,2,Mod(1,6012)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6012, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6012.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
\( p \)
Sign
\(2\)
\(-1\)
\(3\)
\(1\)
\(167\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 6 T_{5}^{9} - 19 T_{5}^{8} - 152 T_{5}^{7} + 55 T_{5}^{6} + 1182 T_{5}^{5} + 404 T_{5}^{4} - 2728 T_{5}^{3} - 587 T_{5}^{2} + 1196 T_{5} + 399 \)
T5^10 + 6*T5^9 - 19*T5^8 - 152*T5^7 + 55*T5^6 + 1182*T5^5 + 404*T5^4 - 2728*T5^3 - 587*T5^2 + 1196*T5 + 399
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).
$p$
$F_p(T)$
$2$
\( T^{10} \)
T^10
$3$
\( T^{10} \)
T^10
$5$
\( T^{10} + 6 T^{9} - 19 T^{8} - 152 T^{7} + \cdots + 399 \)
T^10 + 6*T^9 - 19*T^8 - 152*T^7 + 55*T^6 + 1182*T^5 + 404*T^4 - 2728*T^3 - 587*T^2 + 1196*T + 399
$7$
\( T^{10} - 4 T^{9} - 26 T^{8} + 82 T^{7} + \cdots + 9 \)
T^10 - 4*T^9 - 26*T^8 + 82*T^7 + 211*T^6 - 340*T^5 - 593*T^4 + 192*T^3 + 423*T^2 + 126*T + 9
$11$
\( T^{10} + 8 T^{9} - 37 T^{8} + \cdots - 50841 \)
T^10 + 8*T^9 - 37*T^8 - 424*T^7 - 75*T^6 + 6884*T^5 + 14188*T^4 - 25488*T^3 - 117001*T^2 - 135692*T - 50841
$13$
\( T^{10} + 2 T^{9} - 58 T^{8} + \cdots + 9289 \)
T^10 + 2*T^9 - 58*T^8 - 48*T^7 + 1223*T^6 - 246*T^5 - 10490*T^4 + 10028*T^3 + 27274*T^2 - 40164*T + 9289
$17$
\( T^{10} + 6 T^{9} - 97 T^{8} + \cdots - 655837 \)
T^10 + 6*T^9 - 97*T^8 - 640*T^7 + 2602*T^6 + 19124*T^5 - 26850*T^4 - 211136*T^3 + 160245*T^2 + 807334*T - 655837
$19$
\( T^{10} - 118 T^{8} + 4201 T^{6} + \cdots - 62927 \)
T^10 - 118*T^8 + 4201*T^6 + 642*T^5 - 51688*T^4 - 21682*T^3 + 143460*T^2 + 43712*T - 62927
$23$
\( T^{10} + 20 T^{9} + 97 T^{8} + \cdots - 2401 \)
T^10 + 20*T^9 + 97*T^8 - 370*T^7 - 3673*T^6 - 2284*T^5 + 29236*T^4 + 45624*T^3 - 5891*T^2 - 26180*T - 2401
$29$
\( T^{10} + 8 T^{9} - 99 T^{8} + \cdots + 199927 \)
T^10 + 8*T^9 - 99*T^8 - 860*T^7 + 1995*T^6 + 24948*T^5 + 5686*T^4 - 229740*T^3 - 265733*T^2 + 333944*T + 199927
$31$
\( T^{10} + 4 T^{9} - 168 T^{8} + \cdots - 496399 \)
T^10 + 4*T^9 - 168*T^8 - 658*T^7 + 9424*T^6 + 34786*T^5 - 205391*T^4 - 687628*T^3 + 1518122*T^2 + 4595216*T - 496399
$37$
\( T^{10} + 4 T^{9} - 167 T^{8} + \cdots - 7450623 \)
T^10 + 4*T^9 - 167*T^8 - 478*T^7 + 10635*T^6 + 18522*T^5 - 310140*T^4 - 214038*T^3 + 3703563*T^2 - 716688*T - 7450623
$41$
\( T^{10} - 14 T^{9} - 120 T^{8} + \cdots + 1233827 \)
T^10 - 14*T^9 - 120*T^8 + 2310*T^7 - 872*T^6 - 95488*T^5 + 319559*T^4 + 256052*T^3 - 1522102*T^2 - 190930*T + 1233827
$43$
\( T^{10} - 20 T^{9} - 6 T^{8} + \cdots + 1501993 \)
T^10 - 20*T^9 - 6*T^8 + 1884*T^7 - 3522*T^6 - 68326*T^5 + 112427*T^4 + 1089960*T^3 - 789550*T^2 - 5296254*T + 1501993
$47$
\( T^{10} + 48 T^{9} + 913 T^{8} + \cdots - 720657 \)
T^10 + 48*T^9 + 913*T^8 + 8456*T^7 + 34667*T^6 - 2472*T^5 - 469629*T^4 - 953100*T^3 + 1262754*T^2 + 3637008*T - 720657
$53$
\( T^{10} + 22 T^{9} + 21 T^{8} + \cdots - 3701117 \)
T^10 + 22*T^9 + 21*T^8 - 2110*T^7 - 8589*T^6 + 57362*T^5 + 325717*T^4 - 162682*T^3 - 3374086*T^2 - 6393632*T - 3701117
$59$
\( T^{10} + 2 T^{9} - 275 T^{8} + \cdots + 304311 \)
T^10 + 2*T^9 - 275*T^8 - 702*T^7 + 17571*T^6 + 51366*T^5 - 229391*T^4 - 482238*T^3 + 493274*T^2 + 932152*T + 304311
$61$
\( T^{10} + 8 T^{9} - 354 T^{8} + \cdots - 64024443 \)
T^10 + 8*T^9 - 354*T^8 - 2376*T^7 + 41937*T^6 + 234042*T^5 - 1769864*T^4 - 8452494*T^3 + 16933248*T^2 + 48023712*T - 64024443
$67$
\( T^{10} + 6 T^{9} - 314 T^{8} + \cdots + 92973717 \)
T^10 + 6*T^9 - 314*T^8 - 1808*T^7 + 34935*T^6 + 196782*T^5 - 1620746*T^4 - 9253972*T^3 + 24725470*T^2 + 160633388*T + 92973717
$71$
\( T^{10} + 20 T^{9} + \cdots + 609118083 \)
T^10 + 20*T^9 - 297*T^8 - 7364*T^7 + 20292*T^6 + 893378*T^5 + 443937*T^4 - 42448404*T^3 - 56505321*T^2 + 674212374*T + 609118083
$73$
\( T^{10} - 20 T^{9} - 78 T^{8} + \cdots + 9538237 \)
T^10 - 20*T^9 - 78*T^8 + 3650*T^7 - 14133*T^6 - 107602*T^5 + 627484*T^4 + 567014*T^3 - 6826738*T^2 + 5055418*T + 9538237
$79$
\( T^{10} + 4 T^{9} - 106 T^{8} + \cdots + 189 \)
T^10 + 4*T^9 - 106*T^8 - 420*T^7 + 2686*T^6 + 10382*T^5 - 6585*T^4 - 52740*T^3 - 51318*T^2 - 9810*T + 189
$83$
\( T^{10} + 46 T^{9} + 797 T^{8} + \cdots + 3298771 \)
T^10 + 46*T^9 + 797*T^8 + 5584*T^7 - 4125*T^6 - 307022*T^5 - 1845175*T^4 - 4257930*T^3 - 2006974*T^2 + 4595294*T + 3298771
$89$
\( T^{10} - 8 T^{9} - 462 T^{8} + \cdots + 11998287 \)
T^10 - 8*T^9 - 462*T^8 + 3356*T^7 + 74129*T^6 - 469740*T^5 - 4684834*T^4 + 24275920*T^3 + 87857450*T^2 - 349555696*T + 11998287
$97$
\( T^{10} + 34 T^{9} - 51 T^{8} + \cdots + 95931549 \)
T^10 + 34*T^9 - 51*T^8 - 11934*T^7 - 70448*T^6 + 1270124*T^5 + 10866707*T^4 - 37122510*T^3 - 379507249*T^2 + 57882378*T + 95931549
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