Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,36,Mod(24,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 36, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.24");
S:= CuspForms(chi, 36);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 36 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.b (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(193.987826584\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 78600033344 x^{10} + \cdots + 67\!\cdots\!76 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{59}\cdot 3^{12}\cdot 5^{36}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{12} + 78600033344 x^{10} + \cdots + 67\!\cdots\!76 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 24\!\cdots\!41 \nu^{10} + \cdots - 76\!\cdots\!64 ) / 18\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 24\!\cdots\!41 \nu^{10} + \cdots + 17\!\cdots\!64 ) / 18\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!07 \nu^{10} + \cdots + 19\!\cdots\!12 ) / 23\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 83\!\cdots\!39 \nu^{10} + \cdots - 93\!\cdots\!56 ) / 12\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16\!\cdots\!53 \nu^{10} + \cdots + 50\!\cdots\!32 ) / 11\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 57\!\cdots\!93 \nu^{11} + \cdots - 68\!\cdots\!32 \nu ) / 19\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 15\!\cdots\!59 \nu^{11} + \cdots + 71\!\cdots\!92 \nu ) / 57\!\cdots\!68 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 17\!\cdots\!47 \nu^{11} + \cdots - 41\!\cdots\!52 \nu ) / 11\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 32\!\cdots\!49 \nu^{11} + \cdots - 14\!\cdots\!24 \nu ) / 11\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 18\!\cdots\!23 \nu^{11} + \cdots - 26\!\cdots\!28 \nu ) / 11\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 52400022230 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -71\beta_{11} - 60\beta_{10} + 250\beta_{9} + 9563698\beta_{8} + 54286801011\beta_{7} - 75338425202\beta_1 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 127839 \beta_{6} + 7598562 \beta_{5} + 1374158004 \beta_{4} - 45637095735 \beta_{3} + \cdots + 19\!\cdots\!35 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1134533383095 \beta_{11} - 161780045228 \beta_{10} - 729477812072 \beta_{9} + \cdots + 75\!\cdots\!04 \beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 31\!\cdots\!48 \beta_{6} + \cdots - 49\!\cdots\!38 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 15\!\cdots\!61 \beta_{11} + \cdots - 78\!\cdots\!34 \beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 58\!\cdots\!47 \beta_{6} + \cdots + 20\!\cdots\!77 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( 18\!\cdots\!05 \beta_{11} + \cdots + 82\!\cdots\!64 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 12\!\cdots\!20 \beta_{6} + \cdots - 21\!\cdots\!44 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 44\!\cdots\!10 \beta_{11} + \cdots - 17\!\cdots\!92 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
Embeddings
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 24.1 |
|
− | 297764.i | 3.03721e8i | −5.43035e10 | 0 | 9.04372e13 | − | 8.94007e13i | 5.93852e15i | −4.22152e16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.2 | − | 297171.i | − | 3.24715e8i | −5.39508e10 | 0 | −9.64960e13 | − | 9.94825e14i | 5.82190e15i | −5.54086e16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.3 | − | 269208.i | 7.85708e7i | −3.81130e10 | 0 | 2.11519e13 | 8.11724e14i | 1.01042e15i | 4.38582e16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.4 | − | 223239.i | − | 3.16690e8i | −1.54758e10 | 0 | −7.06976e13 | 1.11832e15i | − | 4.21562e15i | −5.02612e16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.5 | − | 120157.i | − | 5.36787e7i | 1.99219e10 | 0 | −6.44989e12 | − | 9.33159e14i | − | 6.52235e15i | 4.71501e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.6 | − | 26081.3i | − | 1.04938e8i | 3.36795e10 | 0 | −2.73691e12 | − | 6.09797e14i | − | 1.77455e15i | 3.90197e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.7 | 26081.3i | 1.04938e8i | 3.36795e10 | 0 | −2.73691e12 | 6.09797e14i | 1.77455e15i | 3.90197e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.8 | 120157.i | 5.36787e7i | 1.99219e10 | 0 | −6.44989e12 | 9.33159e14i | 6.52235e15i | 4.71501e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.9 | 223239.i | 3.16690e8i | −1.54758e10 | 0 | −7.06976e13 | − | 1.11832e15i | 4.21562e15i | −5.02612e16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.10 | 269208.i | − | 7.85708e7i | −3.81130e10 | 0 | 2.11519e13 | − | 8.11724e14i | − | 1.01042e15i | 4.38582e16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.11 | 297171.i | 3.24715e8i | −5.39508e10 | 0 | −9.64960e13 | 9.94825e14i | − | 5.82190e15i | −5.54086e16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.12 | 297764.i | − | 3.03721e8i | −5.43035e10 | 0 | 9.04372e13 | 8.94007e13i | − | 5.93852e15i | −4.22152e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.36.b.c | 12 | |
| 5.b | even | 2 | 1 | inner | 25.36.b.c | 12 | |
| 5.c | odd | 4 | 1 | 5.36.a.b | ✓ | 6 | |
| 5.c | odd | 4 | 1 | 25.36.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.36.a.b | ✓ | 6 | 5.c | odd | 4 | 1 | |
| 25.36.a.c | 6 | 5.c | odd | 4 | 1 | ||
| 25.36.b.c | 12 | 1.a | even | 1 | 1 | trivial | |
| 25.36.b.c | 12 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 314400133376 T_{2}^{10} + \cdots + 27\!\cdots\!96 \)
acting on \(S_{36}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 27\!\cdots\!96 \)
|
| $3$ |
\( T^{12} + \cdots + 19\!\cdots\!16 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} + \cdots + 21\!\cdots\!36 \)
|
| $11$ |
\( (T^{6} + \cdots - 29\!\cdots\!36)^{2} \)
|
| $13$ |
\( T^{12} + \cdots + 31\!\cdots\!96 \)
|
| $17$ |
\( T^{12} + \cdots + 93\!\cdots\!16 \)
|
| $19$ |
\( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 64\!\cdots\!76 \)
|
| $29$ |
\( (T^{6} + \cdots - 33\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots - 12\!\cdots\!36)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 58\!\cdots\!76 \)
|
| $41$ |
\( (T^{6} + \cdots + 21\!\cdots\!64)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 40\!\cdots\!36 \)
|
| $47$ |
\( T^{12} + \cdots + 14\!\cdots\!56 \)
|
| $53$ |
\( T^{12} + \cdots + 15\!\cdots\!16 \)
|
| $59$ |
\( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{6} + \cdots - 18\!\cdots\!36)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 62\!\cdots\!16 \)
|
| $71$ |
\( (T^{6} + \cdots + 10\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 56\!\cdots\!76 \)
|
| $79$ |
\( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 53\!\cdots\!56 \)
|
| $89$ |
\( (T^{6} + \cdots - 29\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 41\!\cdots\!56 \)
|
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