Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [232,4,Mod(1,232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(232, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("232.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 232 = 2^{3} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 232.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: | \(13.6884431213\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 9\nu^{4} + 43\nu^{3} + 447\nu^{2} + 671\nu - 113 ) / 14 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 12\nu^{4} + 22\nu^{3} - 365\nu^{2} - 400\nu + 321 ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} - 3\nu^{4} + 201\nu^{3} + 555\nu^{2} + 261\nu - 159 ) / 14 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 6\nu^{4} + 431\nu^{3} + 731\nu^{2} - 32\nu - 123 ) / 21 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -41\nu^{5} + 9\nu^{4} + 1595\nu^{3} + 3347\nu^{2} + 1093\nu - 1035 ) / 42 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{5} - 15\beta_{3} - 9\beta_{2} - \beta _1 + 107 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{5} + 17\beta_{4} - 32\beta_{3} - 16\beta_{2} + 5\beta _1 + 95 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 244\beta_{5} + 170\beta_{4} - 861\beta_{3} - 455\beta_{2} + 27\beta _1 + 4403 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1002\beta_{5} + 2736\beta_{4} - 6473\beta_{3} - 3351\beta_{2} + 729\beta _1 + 24309 ) / 8 \)
|
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 |
|
0 | −9.88017 | 0 | 2.64112 | 0 | 3.30580 | 0 | 70.6177 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.36242 | 0 | 8.72860 | 0 | −8.76166 | 0 | 13.4804 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.116789 | 0 | −14.4485 | 0 | 30.0998 | 0 | −26.9864 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.586661 | 0 | 15.4201 | 0 | −28.5777 | 0 | −26.6558 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.88453 | 0 | −3.33508 | 0 | 0.0162797 | 0 | −18.6795 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 7.88819 | 0 | −14.0062 | 0 | −34.0826 | 0 | 35.2236 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(29\) | \(1\) |
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 | 232.4.a.e | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2088.4.a.l | 6 | ||
| 4.b | odd | 2 | 1 | 464.4.a.n | 6 | ||
| 8.b | even | 2 | 1 | 1856.4.a.bd | 6 | ||
| 8.d | odd | 2 | 1 | 1856.4.a.bc | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 232.4.a.e | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 464.4.a.n | 6 | 4.b | odd | 2 | 1 | ||
| 1856.4.a.bc | 6 | 8.d | odd | 2 | 1 | ||
| 1856.4.a.bd | 6 | 8.b | even | 2 | 1 | ||
| 2088.4.a.l | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 5T_{3}^{5} - 92T_{3}^{4} - 266T_{3}^{3} + 1581T_{3}^{2} - 651T_{3} - 98 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(232))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 5 T^{5} + \cdots - 98 \)
|
| $5$ |
\( T^{6} + 5 T^{5} + \cdots - 239922 \)
|
| $7$ |
\( T^{6} + 38 T^{5} + \cdots - 13824 \)
|
| $11$ |
\( T^{6} + \cdots - 2827812238 \)
|
| $13$ |
\( T^{6} + \cdots - 7992521554 \)
|
| $17$ |
\( T^{6} + \cdots + 200391946752 \)
|
| $19$ |
\( T^{6} + \cdots + 1256352636928 \)
|
| $23$ |
\( T^{6} + \cdots + 4069096656896 \)
|
| $29$ |
\( (T + 29)^{6} \)
|
| $31$ |
\( T^{6} + \cdots + 766872337410 \)
|
| $37$ |
\( T^{6} + \cdots - 38445998866432 \)
|
| $41$ |
\( T^{6} + \cdots + 64375103462400 \)
|
| $43$ |
\( T^{6} + \cdots - 905147314238 \)
|
| $47$ |
\( T^{6} + \cdots - 154765125165962 \)
|
| $53$ |
\( T^{6} + \cdots - 36699799804014 \)
|
| $59$ |
\( T^{6} + \cdots - 110093808390144 \)
|
| $61$ |
\( T^{6} + \cdots + 40\!\cdots\!72 \)
|
| $67$ |
\( T^{6} + \cdots + 681378807496704 \)
|
| $71$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 21\!\cdots\!72 \)
|
| $79$ |
\( T^{6} + \cdots - 10\!\cdots\!74 \)
|
| $83$ |
\( T^{6} + \cdots + 47\!\cdots\!44 \)
|
| $89$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 83\!\cdots\!28 \)
|
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