[N,k,chi] = [240,12,Mod(1,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.1");
S:= CuspForms(chi, 12);
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
sage: f.q_expansion() # note that sage often uses an isomorphic number field
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{1609}\).
We also show the integral \(q\)-expansion of the trace form .
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
\(2\)
\(-1\)
\(3\)
\(1\)
\(5\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 10864T_{7} - 2590228560 \)
T7^2 - 10864*T7 - 2590228560
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(240))\).
$p$
$F_p(T)$
$2$
\( T^{2} \)
T^2
$3$
\( (T + 243)^{2} \)
(T + 243)^2
$5$
\( (T + 3125)^{2} \)
(T + 3125)^2
$7$
\( T^{2} - 10864 T - 2590228560 \)
T^2 - 10864*T - 2590228560
$11$
\( T^{2} - 361792 T - 140771013888 \)
T^2 - 361792*T - 140771013888
$13$
\( T^{2} + 2133732 T + 1132296332852 \)
T^2 + 2133732*T + 1132296332852
$17$
\( T^{2} + 7804588 T + 7888201038036 \)
T^2 + 7804588*T + 7888201038036
$19$
\( T^{2} - 15562224 T - 9309926445520 \)
T^2 - 15562224*T - 9309926445520
$23$
\( T^{2} + \cdots - 419924260414080 \)
T^2 - 37450248*T - 419924260414080
$29$
\( T^{2} + 70320668 T - 14\!\cdots\!20 \)
T^2 + 70320668*T - 14124575609660220
$31$
\( T^{2} + 298584872 T + 20\!\cdots\!00 \)
T^2 + 298584872*T + 20296527485904000
$37$
\( T^{2} - 236000956 T - 11\!\cdots\!80 \)
T^2 - 236000956*T - 117189079914752780
$41$
\( T^{2} + 464942588 T - 74\!\cdots\!60 \)
T^2 + 464942588*T - 748100965202073660
$43$
\( T^{2} - 242208600 T - 26\!\cdots\!24 \)
T^2 - 242208600*T - 2698864356672448624
$47$
\( T^{2} - 4375796920 T + 41\!\cdots\!84 \)
T^2 - 4375796920*T + 4155703794513965184
$53$
\( T^{2} + 2189541388 T - 21\!\cdots\!20 \)
T^2 + 2189541388*T - 214261439167227420
$59$
\( T^{2} - 5480385856 T - 26\!\cdots\!60 \)
T^2 - 5480385856*T - 26929360173386261760
$61$
\( T^{2} - 14557903980 T + 52\!\cdots\!36 \)
T^2 - 14557903980*T + 52666563384646073636
$67$
\( T^{2} - 15918388888 T + 59\!\cdots\!36 \)
T^2 - 15918388888*T + 59234027414549371536
$71$
\( T^{2} + 1120561024 T - 14\!\cdots\!56 \)
T^2 + 1120561024*T - 140403392882259296256
$73$
\( T^{2} + 24521574348 T + 14\!\cdots\!00 \)
T^2 + 24521574348*T + 142094665877391270500
$79$
\( T^{2} - 79243055560 T + 15\!\cdots\!00 \)
T^2 - 79243055560*T + 1556693740161057696000
$83$
\( T^{2} + 9245226696 T - 30\!\cdots\!32 \)
T^2 + 9245226696*T - 3057189764193156514032
$89$
\( T^{2} - 22117321236 T - 34\!\cdots\!00 \)
T^2 - 22117321236*T - 34648801304681742300
$97$
\( T^{2} + 160363673468 T + 64\!\cdots\!56 \)
T^2 + 160363673468*T + 6403017683654700241156
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