Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,14,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, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.24");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| 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: | \(26.8077322380\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 8933x^{4} + 19907716x^{2} + 350438400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 5^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 8933x^{4} + 19907716x^{2} + 350438400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 27653\nu^{3} + 330511964\nu ) / 217020960 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{5} + 138265\nu^{3} + 517649780\nu ) / 43404192 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 46\nu^{4} + 251854\nu^{2} + 136863720 ) / 34779 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 104\nu^{4} + 418196\nu^{2} - 140819427 ) / 34779 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -241\nu^{5} - 1841685\nu^{3} - 3374013284\nu ) / 7234032 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 25\beta_1 ) / 50 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -23\beta_{4} + 52\beta_{3} - 297759 ) / 100 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 75\beta_{5} + 17216\beta_{2} - 111850\beta_1 ) / 50 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 125927\beta_{4} - 209098\beta_{3} + 1332726591 ) / 100 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2073975\beta_{5} - 145562084\beta_{2} + 504739150\beta_1 ) / 50 \)
|
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 |
|
− | 176.217i | − | 573.185i | −22860.4 | 0 | −101005. | 201493.i | 2.58481e6i | 1.26578e6 | 0 | ||||||||||||||||||||||||||||||||||
| 24.2 | − | 90.6415i | 1125.79i | −23.8902 | 0 | 102044. | 324482.i | − | 740370.i | 326912. | 0 | |||||||||||||||||||||||||||||||||||
| 24.3 | − | 56.4248i | 2114.98i | 5008.25 | 0 | 119337. | − | 325303.i | − | 744821.i | −2.87881e6 | 0 | ||||||||||||||||||||||||||||||||||
| 24.4 | 56.4248i | − | 2114.98i | 5008.25 | 0 | 119337. | 325303.i | 744821.i | −2.87881e6 | 0 | ||||||||||||||||||||||||||||||||||||
| 24.5 | 90.6415i | − | 1125.79i | −23.8902 | 0 | 102044. | − | 324482.i | 740370.i | 326912. | 0 | |||||||||||||||||||||||||||||||||||
| 24.6 | 176.217i | 573.185i | −22860.4 | 0 | −101005. | − | 201493.i | − | 2.58481e6i | 1.26578e6 | 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 | 25.14.b.b | 6 | |
| 5.b | even | 2 | 1 | inner | 25.14.b.b | 6 | |
| 5.c | odd | 4 | 1 | 5.14.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 25.14.a.b | 3 | ||
| 15.e | even | 4 | 1 | 45.14.a.e | 3 | ||
| 20.e | even | 4 | 1 | 80.14.a.g | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.14.a.b | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 25.14.a.b | 3 | 5.c | odd | 4 | 1 | ||
| 25.14.b.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 25.14.b.b | 6 | 5.b | even | 2 | 1 | inner | |
| 45.14.a.e | 3 | 15.e | even | 4 | 1 | ||
| 80.14.a.g | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 42452T_{2}^{4} + 380143168T_{2}^{2} + 812247957504 \)
acting on \(S_{14}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 812247957504 \)
|
| $3$ |
\( T^{6} + \cdots + 18\!\cdots\!36 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 45\!\cdots\!44 \)
|
| $11$ |
\( (T^{3} + \cdots + 88\!\cdots\!32)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 24\!\cdots\!16 \)
|
| $17$ |
\( T^{6} + \cdots + 30\!\cdots\!24 \)
|
| $19$ |
\( (T^{3} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 17\!\cdots\!96 \)
|
| $29$ |
\( (T^{3} + \cdots - 18\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots + 61\!\cdots\!12)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 21\!\cdots\!84 \)
|
| $41$ |
\( (T^{3} + \cdots - 82\!\cdots\!48)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 12\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots + 35\!\cdots\!64 \)
|
| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!36 \)
|
| $59$ |
\( (T^{3} + \cdots - 18\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 83\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 61\!\cdots\!24 \)
|
| $71$ |
\( (T^{3} + \cdots - 40\!\cdots\!28)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 13\!\cdots\!96 \)
|
| $79$ |
\( (T^{3} + \cdots - 23\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 32\!\cdots\!76 \)
|
| $89$ |
\( (T^{3} + \cdots - 22\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 31\!\cdots\!64 \)
|
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