Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,34,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, 34, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.24");
S:= CuspForms(chi, 34);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 34 \) |
| 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: | \(172.457072203\) |
| 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: |
\( x^{12} + 20265063301 x^{10} + \cdots + 39\!\cdots\!96 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{47}\cdot 3^{10}\cdot 5^{36}\cdot 7^{2}\cdot 11^{4} \) |
| 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
\( x^{12} + 20265063301 x^{10} + \cdots + 39\!\cdots\!96 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 78\!\cdots\!81 \nu^{10} + \cdots + 78\!\cdots\!36 ) / 65\!\cdots\!20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 78\!\cdots\!81 \nu^{10} + \cdots - 10\!\cdots\!84 ) / 65\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 77\!\cdots\!97 \nu^{10} + \cdots - 15\!\cdots\!80 ) / 49\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 35\!\cdots\!83 \nu^{10} + \cdots + 26\!\cdots\!28 ) / 31\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 12\!\cdots\!75 \nu^{10} + \cdots - 71\!\cdots\!72 ) / 49\!\cdots\!64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 30\!\cdots\!07 \nu^{11} + \cdots + 12\!\cdots\!32 \nu ) / 28\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 30\!\cdots\!07 \nu^{11} + \cdots + 18\!\cdots\!72 \nu ) / 28\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 19\!\cdots\!65 \nu^{11} + \cdots - 23\!\cdots\!56 \nu ) / 11\!\cdots\!96 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 63\!\cdots\!57 \nu^{11} + \cdots + 95\!\cdots\!08 \nu ) / 19\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 15\!\cdots\!49 \nu^{11} + \cdots - 20\!\cdots\!04 \nu ) / 10\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 38\!\cdots\!33 \nu^{11} + \cdots + 35\!\cdots\!28 \nu ) / 19\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{6} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + \beta _1 - 13510042201 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 11\beta_{11} + 65\beta_{10} + 19\beta_{9} + 3873134\beta_{8} - 20313792650\beta_{7} + 34223724430\beta_{6} ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 353598 \beta_{5} - 1315257 \beta_{4} + 86372478 \beta_{3} + 11281974130 \beta_{2} + \cdots + 71\!\cdots\!39 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 64432913685 \beta_{11} - 713201255281 \beta_{10} - 194802365037 \beta_{9} + \cdots - 27\!\cdots\!14 \beta_{6} ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 38\!\cdots\!10 \beta_{5} + \cdots - 45\!\cdots\!59 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 30\!\cdots\!09 \beta_{11} + \cdots + 21\!\cdots\!06 \beta_{6} ) / 8 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 34\!\cdots\!62 \beta_{5} + \cdots + 31\!\cdots\!47 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 13\!\cdots\!65 \beta_{11} + \cdots - 17\!\cdots\!26 \beta_{6} ) / 8 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 29\!\cdots\!90 \beta_{5} + \cdots - 23\!\cdots\!99 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 53\!\cdots\!05 \beta_{11} + \cdots + 13\!\cdots\!94 \beta_{6} ) / 8 \)
|
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 |
|
− | 178863.i | 8.55112e7i | −2.34019e10 | 0 | 1.52948e13 | 8.09549e13i | 2.64931e15i | −1.75311e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.2 | − | 132922.i | − | 1.17526e8i | −9.07819e9 | 0 | −1.56217e13 | 1.34853e13i | 6.49000e13i | −8.25322e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.3 | − | 126401.i | 8.63211e7i | −7.38725e9 | 0 | 1.09111e13 | 7.78650e13i | − | 1.52021e14i | −1.89228e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.4 | − | 96184.0i | − | 1.07272e8i | −6.61431e8 | 0 | −1.03179e13 | − | 8.61302e13i | − | 7.62595e14i | −5.94832e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.5 | − | 76409.2i | 7.43191e7i | 2.75157e9 | 0 | 5.67866e12 | − | 1.36987e14i | − | 8.66595e14i | 3.57381e13 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.6 | − | 18258.4i | 3.67420e7i | 8.25656e9 | 0 | 6.70852e11 | − | 1.51681e13i | − | 3.07591e14i | 4.20908e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.7 | 18258.4i | − | 3.67420e7i | 8.25656e9 | 0 | 6.70852e11 | 1.51681e13i | 3.07591e14i | 4.20908e15 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.8 | 76409.2i | − | 7.43191e7i | 2.75157e9 | 0 | 5.67866e12 | 1.36987e14i | 8.66595e14i | 3.57381e13 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.9 | 96184.0i | 1.07272e8i | −6.61431e8 | 0 | −1.03179e13 | 8.61302e13i | 7.62595e14i | −5.94832e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.10 | 126401.i | − | 8.63211e7i | −7.38725e9 | 0 | 1.09111e13 | − | 7.78650e13i | 1.52021e14i | −1.89228e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.11 | 132922.i | 1.17526e8i | −9.07819e9 | 0 | −1.56217e13 | − | 1.34853e13i | − | 6.49000e13i | −8.25322e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24.12 | 178863.i | − | 8.55112e7i | −2.34019e10 | 0 | 1.52948e13 | − | 8.09549e13i | − | 2.64931e15i | −1.75311e15 | 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 | 25.34.b.c | 12 | |
| 5.b | even | 2 | 1 | inner | 25.34.b.c | 12 | |
| 5.c | odd | 4 | 1 | 5.34.a.b | ✓ | 6 | |
| 5.c | odd | 4 | 1 | 25.34.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.34.a.b | ✓ | 6 | 5.c | odd | 4 | 1 | |
| 25.34.a.c | 6 | 5.c | odd | 4 | 1 | ||
| 25.34.b.c | 12 | 1.a | even | 1 | 1 | trivial | |
| 25.34.b.c | 12 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 81060253204 T_{2}^{10} + \cdots + 16\!\cdots\!16 \)
acting on \(S_{34}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 16\!\cdots\!16 \)
|
| $3$ |
\( T^{12} + \cdots + 64\!\cdots\!96 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} + \cdots + 23\!\cdots\!36 \)
|
| $11$ |
\( (T^{6} + \cdots + 24\!\cdots\!24)^{2} \)
|
| $13$ |
\( T^{12} + \cdots + 26\!\cdots\!56 \)
|
| $17$ |
\( T^{12} + \cdots + 16\!\cdots\!76 \)
|
| $19$ |
\( (T^{6} + \cdots - 28\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 43\!\cdots\!16 \)
|
| $29$ |
\( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots + 11\!\cdots\!44)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 51\!\cdots\!56 \)
|
| $41$ |
\( (T^{6} + \cdots + 96\!\cdots\!04)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 52\!\cdots\!36 \)
|
| $47$ |
\( T^{12} + \cdots + 44\!\cdots\!96 \)
|
| $53$ |
\( T^{12} + \cdots + 47\!\cdots\!96 \)
|
| $59$ |
\( (T^{6} + \cdots + 95\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{6} + \cdots + 77\!\cdots\!24)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 32\!\cdots\!76 \)
|
| $71$ |
\( (T^{6} + \cdots + 14\!\cdots\!84)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 16\!\cdots\!16 \)
|
| $79$ |
\( (T^{6} + \cdots - 48\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 15\!\cdots\!76 \)
|
| $89$ |
\( (T^{6} + \cdots + 81\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 89\!\cdots\!96 \)
|
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