[N,k,chi] = [2013,2,Mod(1,2013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, 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("2013.1");
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
\(3\)
\(1\)
\(11\)
\(1\)
\(61\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - T_{2}^{11} - 16 T_{2}^{10} + 13 T_{2}^{9} + 93 T_{2}^{8} - 59 T_{2}^{7} - 238 T_{2}^{6} + 108 T_{2}^{5} + 257 T_{2}^{4} - 71 T_{2}^{3} - 93 T_{2}^{2} + 13 T_{2} + 1 \)
T2^12 - T2^11 - 16*T2^10 + 13*T2^9 + 93*T2^8 - 59*T2^7 - 238*T2^6 + 108*T2^5 + 257*T2^4 - 71*T2^3 - 93*T2^2 + 13*T2 + 1
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\).
$p$
$F_p(T)$
$2$
\( T^{12} - T^{11} - 16 T^{10} + 13 T^{9} + \cdots + 1 \)
T^12 - T^11 - 16*T^10 + 13*T^9 + 93*T^8 - 59*T^7 - 238*T^6 + 108*T^5 + 257*T^4 - 71*T^3 - 93*T^2 + 13*T + 1
$3$
\( (T + 1)^{12} \)
(T + 1)^12
$5$
\( T^{12} + 3 T^{11} - 27 T^{10} - 74 T^{9} + \cdots + 2 \)
T^12 + 3*T^11 - 27*T^10 - 74*T^9 + 264*T^8 + 631*T^7 - 1154*T^6 - 2153*T^5 + 2181*T^4 + 2303*T^3 - 1285*T^2 + 105*T + 2
$7$
\( T^{12} + 9 T^{11} - 9 T^{10} - 267 T^{9} + \cdots + 5012 \)
T^12 + 9*T^11 - 9*T^10 - 267*T^9 - 347*T^8 + 2572*T^7 + 5844*T^6 - 7175*T^5 - 27042*T^4 - 10594*T^3 + 22396*T^2 + 21329*T + 5012
$11$
\( (T + 1)^{12} \)
(T + 1)^12
$13$
\( T^{12} + T^{11} - 59 T^{10} - 51 T^{9} + \cdots + 266 \)
T^12 + T^11 - 59*T^10 - 51*T^9 + 1061*T^8 + 601*T^7 - 6894*T^6 - 3175*T^5 + 15154*T^4 + 6086*T^3 - 7885*T^2 - 1897*T + 266
$17$
\( T^{12} - 9 T^{11} - 31 T^{10} + \cdots + 4454 \)
T^12 - 9*T^11 - 31*T^10 + 428*T^9 - 84*T^8 - 6162*T^7 + 6456*T^6 + 33091*T^5 - 33431*T^4 - 74147*T^3 + 33646*T^2 + 51069*T + 4454
$19$
\( T^{12} + 20 T^{11} + 61 T^{10} + \cdots + 122006 \)
T^12 + 20*T^11 + 61*T^10 - 1056*T^9 - 6532*T^8 + 12114*T^7 + 121499*T^6 - 83588*T^5 - 811339*T^4 + 904128*T^3 + 884074*T^2 - 1211517*T + 122006
$23$
\( T^{12} + 9 T^{11} - 102 T^{10} - 1076 T^{9} + \cdots - 16 \)
T^12 + 9*T^11 - 102*T^10 - 1076*T^9 + 1979*T^8 + 39439*T^7 + 55960*T^6 - 375883*T^5 - 1286616*T^4 - 1265900*T^3 - 230113*T^2 + 66227*T - 16
$29$
\( T^{12} - 18 T^{11} - 22 T^{10} + \cdots + 14631718 \)
T^12 - 18*T^11 - 22*T^10 + 2120*T^9 - 9942*T^8 - 44121*T^7 + 396083*T^6 - 225195*T^5 - 4084656*T^4 + 8597842*T^3 + 6108998*T^2 - 27029247*T + 14631718
$31$
\( T^{12} + 21 T^{11} + \cdots - 1061001166 \)
T^12 + 21*T^11 - 39*T^10 - 3182*T^9 - 8884*T^8 + 175332*T^7 + 759815*T^6 - 4310518*T^5 - 22318654*T^4 + 45188833*T^3 + 267916289*T^2 - 146460127*T - 1061001166
$37$
\( T^{12} + 18 T^{11} + 20 T^{10} + \cdots - 58978 \)
T^12 + 18*T^11 + 20*T^10 - 1217*T^9 - 5775*T^8 + 16079*T^7 + 153347*T^6 + 198386*T^5 - 672573*T^4 - 2141305*T^3 - 2028385*T^2 - 658397*T - 58978
$41$
\( T^{12} - 15 T^{11} - 223 T^{10} + \cdots + 19339376 \)
T^12 - 15*T^11 - 223*T^10 + 4606*T^9 - 372*T^8 - 381457*T^7 + 2101868*T^6 + 2861503*T^5 - 59782503*T^4 + 203383851*T^3 - 269089627*T^2 + 95308479*T + 19339376
$43$
\( T^{12} + 33 T^{11} + 270 T^{10} + \cdots - 6414256 \)
T^12 + 33*T^11 + 270*T^10 - 2154*T^9 - 46229*T^8 - 229638*T^7 + 484*T^6 + 2715347*T^5 + 3337081*T^4 - 10993511*T^3 - 9470558*T^2 + 22015447*T - 6414256
$47$
\( T^{12} + 20 T^{11} - 45 T^{10} + \cdots + 62497568 \)
T^12 + 20*T^11 - 45*T^10 - 3066*T^9 - 11794*T^8 + 119203*T^7 + 763848*T^6 - 1252534*T^5 - 15524903*T^4 - 10384202*T^3 + 100150914*T^2 + 181421857*T + 62497568
$53$
\( T^{12} - 250 T^{10} + \cdots + 94719784 \)
T^12 - 250*T^10 - 427*T^9 + 20357*T^8 + 52786*T^7 - 644232*T^6 - 1825560*T^5 + 8127369*T^4 + 20746337*T^3 - 43496537*T^2 - 70743231*T + 94719784
$59$
\( T^{12} + 21 T^{11} + \cdots - 120384652 \)
T^12 + 21*T^11 - 115*T^10 - 5149*T^9 - 20374*T^8 + 291966*T^7 + 2561593*T^6 + 2876941*T^5 - 29228756*T^4 - 75657779*T^3 + 56877309*T^2 + 182005393*T - 120384652
$61$
\( (T + 1)^{12} \)
(T + 1)^12
$67$
\( T^{12} + 34 T^{11} + \cdots + 686361986 \)
T^12 + 34*T^11 + 211*T^10 - 4217*T^9 - 52033*T^8 + 75313*T^7 + 2881259*T^6 + 4315604*T^5 - 59266951*T^4 - 146703421*T^3 + 371962508*T^2 + 1126349197*T + 686361986
$71$
\( T^{12} + 5 T^{11} + \cdots + 14666313456 \)
T^12 + 5*T^11 - 416*T^10 - 1244*T^9 + 64024*T^8 + 110045*T^7 - 4641243*T^6 - 4099247*T^5 + 164659178*T^4 + 58352955*T^3 - 2651197164*T^2 - 109018377*T + 14666313456
$73$
\( T^{12} + 2 T^{11} - 281 T^{10} + \cdots - 112159142 \)
T^12 + 2*T^11 - 281*T^10 - 1008*T^9 + 25081*T^8 + 120596*T^7 - 720161*T^6 - 4169147*T^5 + 4857102*T^4 + 42810056*T^3 + 10625359*T^2 - 130858397*T - 112159142
$79$
\( T^{12} + 31 T^{11} + \cdots + 291529652504 \)
T^12 + 31*T^11 - 257*T^10 - 16472*T^9 - 67462*T^8 + 2733392*T^7 + 26092741*T^6 - 114431644*T^5 - 2266294027*T^4 - 5300338156*T^3 + 40015555239*T^2 + 218188067861*T + 291529652504
$83$
\( T^{12} + 32 T^{11} + \cdots - 267389284 \)
T^12 + 32*T^11 - 41*T^10 - 11050*T^9 - 91719*T^8 + 761522*T^7 + 11192069*T^6 + 246054*T^5 - 372560993*T^4 - 591280150*T^3 + 3162205087*T^2 + 194420021*T - 267389284
$89$
\( T^{12} - 27 T^{11} + \cdots + 2510062488 \)
T^12 - 27*T^11 - 221*T^10 + 10861*T^9 - 37717*T^8 - 1071814*T^7 + 9231022*T^6 - 5773669*T^5 - 164058608*T^4 + 478447146*T^3 + 300412632*T^2 - 2587469553*T + 2510062488
$97$
\( T^{12} + 19 T^{11} + \cdots - 29402967834 \)
T^12 + 19*T^11 - 513*T^10 - 10437*T^9 + 75080*T^8 + 1923086*T^7 - 1881073*T^6 - 138008561*T^5 - 213843902*T^4 + 3341989086*T^3 + 6686099748*T^2 - 23884128333*T - 29402967834
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