Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1232,4,Mod(1,1232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1232.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1232 = 2^{4} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1232.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: | \(72.6903531271\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 145x^{5} - 10x^{4} + 4790x^{3} - 2452x^{2} - 1496x + 320 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 616) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 145x^{5} - 10x^{4} + 4790x^{3} - 2452x^{2} - 1496x + 320 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 41\nu^{6} + 14\nu^{5} - 5961\nu^{4} - 2300\nu^{3} + 196706\nu^{2} - 44544\nu - 43900 ) / 2556 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 41\nu^{6} + 14\nu^{5} - 5961\nu^{4} - 2300\nu^{3} + 199262\nu^{2} - 44544\nu - 151252 ) / 2556 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -59\nu^{6} - 34\nu^{5} + 8391\nu^{4} + 6032\nu^{3} - 266094\nu^{2} - 49036\nu + 236 ) / 2556 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8\nu^{6} + \nu^{5} - 1151\nu^{4} - 357\nu^{3} + 37663\nu^{2} - 4906\nu - 14161 ) / 213 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -23\nu^{6} + 6\nu^{5} + 3318\nu^{4} - 580\nu^{3} - 109000\nu^{2} + 81040\nu + 23451 ) / 639 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 79\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{6} - 8\beta_{5} - 16\beta_{4} + 94\beta_{3} - 114\beta_{2} + 48\beta _1 + 3331 \)
|
| \(\nu^{5}\) | \(=\) |
\( -102\beta_{6} - 285\beta_{5} - 165\beta_{4} + 349\beta_{3} - 148\beta_{2} + 6709\beta _1 + 2921 \)
|
| \(\nu^{6}\) | \(=\) |
\( -1039\beta_{6} - 1178\beta_{5} - 2326\beta_{4} + 8862\beta_{3} - 11720\beta_{2} + 10206\beta _1 + 283145 \)
|
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 | −9.14775 | 0 | 12.3118 | 0 | 7.00000 | 0 | 56.6814 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.90093 | 0 | −12.7900 | 0 | 7.00000 | 0 | 35.4247 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.466684 | 0 | −6.69970 | 0 | 7.00000 | 0 | −26.7822 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.179568 | 0 | 18.8308 | 0 | 7.00000 | 0 | −26.9678 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.812616 | 0 | −16.9890 | 0 | 7.00000 | 0 | −26.3397 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 6.46220 | 0 | −3.00579 | 0 | 7.00000 | 0 | 14.7600 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 10.0610 | 0 | 14.3419 | 0 | 7.00000 | 0 | 74.2235 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1232.4.a.bd | 7 | |
| 4.b | odd | 2 | 1 | 616.4.a.j | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 616.4.a.j | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 1232.4.a.bd | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):
|
\( T_{3}^{7} - 145T_{3}^{5} - 10T_{3}^{4} + 4790T_{3}^{3} - 2452T_{3}^{2} - 1496T_{3} + 320 \)
|
|
\( T_{5}^{7} - 6T_{5}^{6} - 591T_{5}^{5} + 2228T_{5}^{4} + 107076T_{5}^{3} - 111752T_{5}^{2} - 6037280T_{5} - 14549536 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 145 T^{5} + \cdots + 320 \)
|
| $5$ |
\( T^{7} - 6 T^{6} + \cdots - 14549536 \)
|
| $7$ |
\( (T - 7)^{7} \)
|
| $11$ |
\( (T - 11)^{7} \)
|
| $13$ |
\( T^{7} + \cdots - 79161659136 \)
|
| $17$ |
\( T^{7} + \cdots + 151755555840 \)
|
| $19$ |
\( T^{7} + \cdots + 15743217343488 \)
|
| $23$ |
\( T^{7} + \cdots - 120191659904000 \)
|
| $29$ |
\( T^{7} + \cdots + 2542483952512 \)
|
| $31$ |
\( T^{7} + \cdots + 5980457097664 \)
|
| $37$ |
\( T^{7} + \cdots - 16\!\cdots\!72 \)
|
| $41$ |
\( T^{7} + \cdots - 45\!\cdots\!36 \)
|
| $43$ |
\( T^{7} + \cdots - 32\!\cdots\!28 \)
|
| $47$ |
\( T^{7} + \cdots - 62\!\cdots\!68 \)
|
| $53$ |
\( T^{7} + \cdots + 47\!\cdots\!96 \)
|
| $59$ |
\( T^{7} + \cdots + 11\!\cdots\!76 \)
|
| $61$ |
\( T^{7} + \cdots + 49\!\cdots\!96 \)
|
| $67$ |
\( T^{7} + \cdots - 55\!\cdots\!44 \)
|
| $71$ |
\( T^{7} + \cdots - 10\!\cdots\!44 \)
|
| $73$ |
\( T^{7} + \cdots - 90\!\cdots\!40 \)
|
| $79$ |
\( T^{7} + \cdots + 26\!\cdots\!88 \)
|
| $83$ |
\( T^{7} + \cdots - 17\!\cdots\!60 \)
|
| $89$ |
\( T^{7} + \cdots + 93\!\cdots\!64 \)
|
| $97$ |
\( T^{7} + \cdots - 18\!\cdots\!96 \)
|
| show more | |
| show less |