[N,k,chi] = [103,3,Mod(102,103)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(103, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("103.102");
S:= CuspForms(chi, 3);
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
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/103\mathbb{Z}\right)^\times\).
\(n\)
\(5\)
\(\chi(n)\)
\(-1\)
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
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} - 15T_{2}^{4} - 10T_{2}^{3} + 55T_{2}^{2} + 9T_{2} - 25 \)
T2^6 + T2^5 - 15*T2^4 - 10*T2^3 + 55*T2^2 + 9*T2 - 25
acting on \(S_{3}^{\mathrm{new}}(103, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{6} + T^{5} - 15 T^{4} - 10 T^{3} + 55 T^{2} + \cdots - 25)^{2} \)
(T^6 + T^5 - 15*T^4 - 10*T^3 + 55*T^2 + 9*T - 25)^2
$3$
\( T^{12} + 92 T^{10} + 3344 T^{8} + \cdots + 3274879 \)
T^12 + 92*T^10 + 3344*T^8 + 60552*T^6 + 561888*T^4 + 2410584*T^2 + 3274879
$5$
\( T^{12} + 271 T^{10} + \cdots + 265265199 \)
T^12 + 271*T^10 + 27964*T^8 + 1359151*T^6 + 30654328*T^4 + 255566691*T^2 + 265265199
$7$
\( (T^{6} - 124 T^{4} - 454 T^{3} - 240 T^{2} + \cdots + 5)^{2} \)
(T^6 - 124*T^4 - 454*T^3 - 240*T^2 + 28*T + 5)^2
$11$
\( T^{12} + 1001 T^{10} + \cdots + 1697697273600 \)
T^12 + 1001*T^10 + 378215*T^8 + 67147288*T^6 + 5642408800*T^4 + 197225148096*T^2 + 1697697273600
$13$
\( (T^{6} - 15 T^{5} - 307 T^{4} + \cdots - 262480)^{2} \)
(T^6 - 15*T^5 - 307*T^4 + 3380*T^3 + 29772*T^2 - 126368*T - 262480)^2
$17$
\( (T^{6} + 7 T^{5} - 562 T^{4} + \cdots + 226885)^{2} \)
(T^6 + 7*T^5 - 562*T^4 - 5725*T^3 + 55858*T^2 + 647163*T + 226885)^2
$19$
\( (T^{6} + 5 T^{5} - 747 T^{4} + \cdots - 2887376)^{2} \)
(T^6 + 5*T^5 - 747*T^4 - 3748*T^3 + 129308*T^2 + 706784*T - 2887376)^2
$23$
\( (T^{6} - 24 T^{5} - 1127 T^{4} + \cdots - 2308060)^{2} \)
(T^6 - 24*T^5 - 1127*T^4 + 22700*T^3 + 117893*T^2 - 1994676*T - 2308060)^2
$29$
\( (T^{6} - 48 T^{5} - 2212 T^{4} + \cdots + 56410112)^{2} \)
(T^6 - 48*T^5 - 2212*T^4 + 102832*T^3 + 82736*T^2 - 21885952*T + 56410112)^2
$31$
\( T^{12} + 5924 T^{10} + \cdots + 85\!\cdots\!00 \)
T^12 + 5924*T^10 + 11028175*T^8 + 8718840144*T^6 + 2891766692464*T^4 + 291793363001856*T^2 + 8558091956217600
$37$
\( T^{12} + 10164 T^{10} + \cdots + 67\!\cdots\!91 \)
T^12 + 10164*T^10 + 31495672*T^8 + 29341681376*T^6 + 10636758645112*T^4 + 1469484373655736*T^2 + 67458746296439991
$41$
\( (T^{6} + 20 T^{5} - 1841 T^{4} + \cdots + 3565468)^{2} \)
(T^6 + 20*T^5 - 1841*T^4 - 40492*T^3 + 140673*T^2 + 3645918*T + 3565468)^2
$43$
\( T^{12} + 10812 T^{10} + \cdots + 13\!\cdots\!79 \)
T^12 + 10812*T^10 + 40123772*T^8 + 63053511456*T^6 + 44692225368316*T^4 + 13439683691934984*T^2 + 1398572913954171879
$47$
\( T^{12} + 13319 T^{10} + \cdots + 11\!\cdots\!96 \)
T^12 + 13319*T^10 + 57793367*T^8 + 91703879872*T^6 + 52568578850224*T^4 + 9057410571891456*T^2 + 114235265317015296
$53$
\( T^{12} + 15497 T^{10} + \cdots + 14\!\cdots\!39 \)
T^12 + 15497*T^10 + 77908524*T^8 + 149442336433*T^6 + 92964035995480*T^4 + 7492854027429621*T^2 + 148607283862755639
$59$
\( (T^{6} - 97 T^{5} - 7281 T^{4} + \cdots - 18095168848)^{2} \)
(T^6 - 97*T^5 - 7281*T^4 + 694096*T^3 + 17196936*T^2 - 1187978416*T - 18095168848)^2
$61$
\( (T^{6} + 75 T^{5} - 7235 T^{4} + \cdots - 19383425104)^{2} \)
(T^6 + 75*T^5 - 7235*T^4 - 385956*T^3 + 20658576*T^2 + 448805984*T - 19383425104)^2
$67$
\( T^{12} + 39292 T^{10} + \cdots + 48\!\cdots\!71 \)
T^12 + 39292*T^10 + 625155800*T^8 + 5152569101816*T^6 + 23141168212915840*T^4 + 53400079989874999080*T^2 + 48949322471677538876871
$71$
\( T^{12} + 40631 T^{10} + \cdots + 28\!\cdots\!00 \)
T^12 + 40631*T^10 + 591333447*T^8 + 3665198985328*T^6 + 8664247075782448*T^4 + 3007044265669922304*T^2 + 280109906402790240000
$73$
\( T^{12} + 42187 T^{10} + \cdots + 10\!\cdots\!84 \)
T^12 + 42187*T^10 + 666886895*T^8 + 4739399704592*T^6 + 13595092154965408*T^4 + 7105791128281049088*T^2 + 105048588110655907584
$79$
\( (T^{6} - 79 T^{5} + \cdots - 205105099109)^{2} \)
(T^6 - 79*T^5 - 19920*T^4 + 1247657*T^3 + 120107612*T^2 - 4892929219*T - 205105099109)^2
$83$
\( (T^{6} - 81 T^{5} + \cdots - 167129829040)^{2} \)
(T^6 - 81*T^5 - 16159*T^4 + 907336*T^3 + 91171724*T^2 - 2348243872*T - 167129829040)^2
$89$
\( T^{12} + 28828 T^{10} + \cdots + 71\!\cdots\!00 \)
T^12 + 28828*T^10 + 194808464*T^8 + 540669448832*T^6 + 682686671107840*T^4 + 372572634716568576*T^2 + 71904378978679910400
$97$
\( (T^{6} - 96 T^{5} - 16423 T^{4} + \cdots - 28741698880)^{2} \)
(T^6 - 96*T^5 - 16423*T^4 + 1289360*T^3 + 50543253*T^2 - 1610892704*T - 28741698880)^2
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