Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,34,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, 34, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.1");
S:= CuspForms(chi, 34);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 34 \) |
| 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: | \(172.457072203\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 26286285043 x^{14} + \cdots + 16\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | multiple of \( 2^{85}\cdot 3^{26}\cdot 5^{66}\cdot 7^{4}\cdot 11^{8} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 26286285043 x^{14} + \cdots + 16\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 13143142522 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 23\!\cdots\!27 \nu^{14} + \cdots + 10\!\cdots\!20 ) / 17\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 39\!\cdots\!63 \nu^{14} + \cdots + 22\!\cdots\!08 ) / 57\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 77\!\cdots\!41 \nu^{14} + \cdots - 11\!\cdots\!84 ) / 42\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 40\!\cdots\!83 \nu^{14} + \cdots - 25\!\cdots\!72 ) / 57\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 19\!\cdots\!57 \nu^{14} + \cdots + 11\!\cdots\!92 ) / 28\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 13\!\cdots\!33 \nu^{14} + \cdots - 52\!\cdots\!56 ) / 17\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 37\!\cdots\!75 \nu^{15} + \cdots + 74\!\cdots\!40 \nu ) / 22\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 29\!\cdots\!35 \nu^{15} + \cdots + 29\!\cdots\!72 \nu ) / 33\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 25\!\cdots\!45 \nu^{15} + \cdots - 71\!\cdots\!84 \nu ) / 66\!\cdots\!20 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 33\!\cdots\!81 \nu^{15} + \cdots - 14\!\cdots\!24 \nu ) / 33\!\cdots\!60 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 39\!\cdots\!53 \nu^{15} + \cdots + 13\!\cdots\!44 \nu ) / 22\!\cdots\!40 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 24\!\cdots\!05 \nu^{15} + \cdots - 38\!\cdots\!76 \nu ) / 11\!\cdots\!20 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 23\!\cdots\!37 \nu^{15} + \cdots - 49\!\cdots\!88 \nu ) / 66\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 13143142522 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{11} + \beta_{10} + 1699587\beta_{9} + 20788434710\beta_1 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{7} - 359 \beta_{6} + 171 \beta_{5} - 945 \beta_{4} + 69225 \beta_{3} + 27411674260 \beta_{2} + 27\!\cdots\!16 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 10660 \beta_{15} - 164886 \beta_{14} + 371924 \beta_{13} + 16214668 \beta_{12} + \cdots + 12\!\cdots\!60 \beta_1 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2840318544 \beta_{8} + 10382932155 \beta_{7} - 3358385182845 \beta_{6} + 2428074969321 \beta_{5} + \cdots + 15\!\cdots\!00 ) / 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 136499115736620 \beta_{15} + \cdots + 73\!\cdots\!72 \beta_1 ) / 8 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 42\!\cdots\!64 \beta_{8} + \cdots + 96\!\cdots\!76 ) / 16 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 12\!\cdots\!16 \beta_{15} + \cdots + 45\!\cdots\!04 \beta_1 ) / 8 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 43\!\cdots\!40 \beta_{8} + \cdots + 59\!\cdots\!36 ) / 16 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 10\!\cdots\!00 \beta_{15} + \cdots + 28\!\cdots\!44 \beta_1 ) / 8 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 37\!\cdots\!20 \beta_{8} + \cdots + 37\!\cdots\!28 ) / 16 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 85\!\cdots\!20 \beta_{15} + \cdots + 17\!\cdots\!20 \beta_1 ) / 8 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 30\!\cdots\!00 \beta_{8} + \cdots + 23\!\cdots\!76 ) / 16 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 65\!\cdots\!60 \beta_{15} + \cdots + 11\!\cdots\!20 \beta_1 ) / 8 \)
|
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 |
|
−162009. | 1.50582e7 | 1.76568e10 | 0 | −2.43956e12 | −1.33679e14 | −1.46891e15 | −5.33231e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −156524. | −1.41317e8 | 1.59098e10 | 0 | 2.21195e13 | −8.25005e13 | −1.14573e15 | 1.44114e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −156068. | 6.96239e7 | 1.57674e10 | 0 | −1.08661e13 | 1.55193e14 | −1.12018e15 | −7.11573e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −114942. | −6.13071e7 | 4.62174e9 | 0 | 7.04677e12 | 1.47106e14 | 4.56113e14 | −1.80050e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −102456. | 8.77372e7 | 1.90737e9 | 0 | −8.98924e12 | −1.11815e14 | 6.84671e14 | 2.13876e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −62697.7 | −3.24591e7 | −4.65894e9 | 0 | 2.03511e12 | −2.03055e13 | 8.30673e14 | −4.50547e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −40404.6 | 1.28550e8 | −6.95740e9 | 0 | −5.19400e12 | 4.50276e13 | 6.28184e14 | 1.09659e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −27727.3 | −6.45007e7 | −7.82113e9 | 0 | 1.78843e12 | −3.09117e13 | 4.55035e14 | −1.39871e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 27727.3 | 6.45007e7 | −7.82113e9 | 0 | 1.78843e12 | 3.09117e13 | −4.55035e14 | −1.39871e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 40404.6 | −1.28550e8 | −6.95740e9 | 0 | −5.19400e12 | −4.50276e13 | −6.28184e14 | 1.09659e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 62697.7 | 3.24591e7 | −4.65894e9 | 0 | 2.03511e12 | 2.03055e13 | −8.30673e14 | −4.50547e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 102456. | −8.77372e7 | 1.90737e9 | 0 | −8.98924e12 | 1.11815e14 | −6.84671e14 | 2.13876e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 114942. | 6.13071e7 | 4.62174e9 | 0 | 7.04677e12 | −1.47106e14 | −4.56113e14 | −1.80050e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 156068. | −6.96239e7 | 1.57674e10 | 0 | −1.08661e13 | −1.55193e14 | 1.12018e15 | −7.11573e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 156524. | 1.41317e8 | 1.59098e10 | 0 | 2.21195e13 | 8.25005e13 | 1.14573e15 | 1.44114e16 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 162009. | −1.50582e7 | 1.76568e10 | 0 | −2.43956e12 | 1.33679e14 | 1.46891e15 | −5.33231e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
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 | 25.34.a.f | 16 | |
| 5.b | even | 2 | 1 | inner | 25.34.a.f | 16 | |
| 5.c | odd | 4 | 2 | 5.34.b.a | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.34.b.a | ✓ | 16 | 5.c | odd | 4 | 2 | |
| 25.34.a.f | 16 | 1.a | even | 1 | 1 | trivial | |
| 25.34.a.f | 16 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 105145140172 T_{2}^{14} + \cdots + 10\!\cdots\!96 \)
acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots + 10\!\cdots\!96 \)
|
| $3$ |
\( T^{16} + \cdots + 46\!\cdots\!36 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + \cdots + 63\!\cdots\!56 \)
|
| $11$ |
\( (T^{8} + \cdots + 26\!\cdots\!76)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 13\!\cdots\!16 \)
|
| $17$ |
\( T^{16} + \cdots + 58\!\cdots\!76 \)
|
| $19$ |
\( (T^{8} + \cdots - 29\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 77\!\cdots\!96 \)
|
| $29$ |
\( (T^{8} + \cdots - 78\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{8} + \cdots + 28\!\cdots\!16)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 41\!\cdots\!16 \)
|
| $41$ |
\( (T^{8} + \cdots - 88\!\cdots\!64)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 12\!\cdots\!56 \)
|
| $47$ |
\( T^{16} + \cdots + 27\!\cdots\!36 \)
|
| $53$ |
\( T^{16} + \cdots + 14\!\cdots\!36 \)
|
| $59$ |
\( (T^{8} + \cdots + 47\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{8} + \cdots + 85\!\cdots\!76)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 15\!\cdots\!76 \)
|
| $71$ |
\( (T^{8} + \cdots + 18\!\cdots\!96)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 16\!\cdots\!96 \)
|
| $79$ |
\( (T^{8} + \cdots - 17\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 22\!\cdots\!76 \)
|
| $89$ |
\( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 13\!\cdots\!36 \)
|
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