Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,36,Mod(1,25)]
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([0]))
N = Newforms(chi, 36, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.1");
S:= CuspForms(chi, 36);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 36 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.a (trivial) |
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: | yes |
| Analytic conductor: | \(193.987826584\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 5 x^{11} - 330549745601 x^{10} + \cdots - 37\!\cdots\!04 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{53}\cdot 3^{16}\cdot 5^{36}\cdot 7^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{12} - 5 x^{11} - 330549745601 x^{10} + \cdots - 37\!\cdots\!04 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 24\!\cdots\!71 \nu^{11} + \cdots + 73\!\cdots\!80 ) / 34\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 24\!\cdots\!71 \nu^{11} + \cdots - 78\!\cdots\!00 ) / 15\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\!\cdots\!15 \nu^{11} + \cdots + 15\!\cdots\!68 ) / 17\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 87\!\cdots\!77 \nu^{11} + \cdots - 48\!\cdots\!08 ) / 71\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20\!\cdots\!13 \nu^{11} + \cdots - 59\!\cdots\!40 ) / 26\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 38\!\cdots\!19 \nu^{11} + \cdots + 13\!\cdots\!20 ) / 26\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 17\!\cdots\!95 \nu^{11} + \cdots - 87\!\cdots\!28 ) / 24\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 39\!\cdots\!97 \nu^{11} + \cdots - 16\!\cdots\!20 ) / 39\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 16\!\cdots\!35 \nu^{11} + \cdots + 11\!\cdots\!92 ) / 78\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 71\!\cdots\!31 \nu^{11} + \cdots + 19\!\cdots\!36 ) / 40\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 22\beta_{2} - 9237\beta _1 + 55091628123 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - 5\beta_{5} + 10\beta_{4} - 32\beta_{3} - 3206984\beta_{2} + 86065641846\beta _1 - 509327550965594 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 6 \beta_{11} + 3 \beta_{10} - 40 \beta_{9} - 155 \beta_{8} + 4155 \beta_{7} - 36413 \beta_{6} + \cdots + 47\!\cdots\!11 \)
|
| \(\nu^{5}\) | \(=\) |
\( 696900 \beta_{11} - 13783218 \beta_{10} + 433136 \beta_{9} + 9168546 \beta_{8} - 1612320194 \beta_{7} + \cdots - 67\!\cdots\!24 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1155643070304 \beta_{11} + 1969365106752 \beta_{10} - 6616982218240 \beta_{9} - 24528247422368 \beta_{8} + \cdots + 46\!\cdots\!75 \)
|
| \(\nu^{7}\) | \(=\) |
\( 16\!\cdots\!20 \beta_{11} + \cdots - 10\!\cdots\!18 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 16\!\cdots\!94 \beta_{11} + \cdots + 50\!\cdots\!83 \)
|
| \(\nu^{9}\) | \(=\) |
\( 28\!\cdots\!20 \beta_{11} + \cdots - 17\!\cdots\!16 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 22\!\cdots\!60 \beta_{11} + \cdots + 55\!\cdots\!91 \)
|
| \(\nu^{11}\) | \(=\) |
\( 43\!\cdots\!00 \beta_{11} + \cdots - 30\!\cdots\!30 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−356194. | 3.59397e7 | 9.25143e10 | 0 | −1.28015e13 | 9.69021e14 | −2.07143e16 | −4.87399e16 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −272551. | −4.02907e8 | 3.99241e10 | 0 | 1.09813e14 | −2.78877e14 | −1.51656e15 | 1.12303e17 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −265648. | 6.66452e6 | 3.62091e10 | 0 | −1.77042e12 | −1.05903e15 | −4.91284e14 | −4.99871e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −222317. | 3.22816e8 | 1.50651e10 | 0 | −7.17675e13 | 3.73782e13 | 4.28952e15 | 5.41787e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −123413. | −1.26104e8 | −1.91289e10 | 0 | 1.55629e13 | 6.19248e14 | 6.60121e15 | −3.41294e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3550.41 | −1.83995e8 | −3.43471e10 | 0 | −6.53257e11 | −6.47363e14 | −2.43938e14 | −1.61776e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 29195.1 | 2.59226e8 | −3.35074e10 | 0 | 7.56813e12 | −8.96973e13 | −1.98139e15 | 1.71666e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 135533. | −4.31580e8 | −1.59904e10 | 0 | −5.84935e13 | 5.62337e14 | −6.82413e15 | 1.36230e17 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 197151. | −1.00963e8 | 4.50864e9 | 0 | −1.99048e13 | −5.35358e14 | −5.88516e15 | −3.98381e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 229500. | 1.62248e8 | 1.83103e10 | 0 | 3.72358e13 | 9.79379e14 | −3.68334e15 | −2.37072e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 315996. | 3.77837e8 | 6.54936e10 | 0 | 1.19395e14 | −1.09689e15 | 9.83816e15 | 9.27292e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 337783. | −1.90785e8 | 7.97375e10 | 0 | −6.44440e13 | 7.12698e13 | 1.53278e16 | −1.36326e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.36.a.e | yes | 12 |
| 5.b | even | 2 | 1 | 25.36.a.d | ✓ | 12 | |
| 5.c | odd | 4 | 2 | 25.36.b.d | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.36.a.d | ✓ | 12 | 5.b | even | 2 | 1 | |
| 25.36.a.e | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 25.36.b.d | 24 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 8585 T_{2}^{11} - 330515965426 T_{2}^{10} + \cdots - 48\!\cdots\!04 \)
acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots - 48\!\cdots\!04 \)
|
| $3$ |
\( T^{12} + \cdots + 95\!\cdots\!41 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} + \cdots + 88\!\cdots\!36 \)
|
| $11$ |
\( T^{12} + \cdots + 53\!\cdots\!21 \)
|
| $13$ |
\( T^{12} + \cdots - 25\!\cdots\!04 \)
|
| $17$ |
\( T^{12} + \cdots + 43\!\cdots\!41 \)
|
| $19$ |
\( T^{12} + \cdots + 58\!\cdots\!25 \)
|
| $23$ |
\( T^{12} + \cdots - 47\!\cdots\!24 \)
|
| $29$ |
\( T^{12} + \cdots + 28\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 38\!\cdots\!96 \)
|
| $37$ |
\( T^{12} + \cdots - 88\!\cdots\!24 \)
|
| $41$ |
\( T^{12} + \cdots + 77\!\cdots\!21 \)
|
| $43$ |
\( T^{12} + \cdots + 68\!\cdots\!36 \)
|
| $47$ |
\( T^{12} + \cdots - 45\!\cdots\!44 \)
|
| $53$ |
\( T^{12} + \cdots + 94\!\cdots\!16 \)
|
| $59$ |
\( T^{12} + \cdots - 10\!\cdots\!00 \)
|
| $61$ |
\( T^{12} + \cdots - 63\!\cdots\!04 \)
|
| $67$ |
\( T^{12} + \cdots + 12\!\cdots\!41 \)
|
| $71$ |
\( T^{12} + \cdots - 16\!\cdots\!04 \)
|
| $73$ |
\( T^{12} + \cdots - 31\!\cdots\!99 \)
|
| $79$ |
\( T^{12} + \cdots - 68\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 80\!\cdots\!81 \)
|
| $89$ |
\( T^{12} + \cdots + 85\!\cdots\!25 \)
|
| $97$ |
\( T^{12} + \cdots + 72\!\cdots\!56 \)
|
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