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: | \(1\) |
| 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: |
\( x^{6} - x^{5} - 47x^{4} + 10x^{3} + 612x^{2} + 240x - 1440 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 47x^{4} + 10x^{3} + 612x^{2} + 240x - 1440 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 49\nu^{4} - 73\nu^{3} - 970\nu^{2} - 180\nu + 216 ) / 372 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 59\nu^{4} + 193\nu^{3} - 1688\nu^{2} - 1644\nu + 6288 ) / 372 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 49\nu^{4} + 73\nu^{3} + 1714\nu^{2} - 564\nu - 11748 ) / 372 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} - 71\nu^{3} + 111\nu^{2} + 608\nu - 308 ) / 62 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} - 28\nu^{4} - 100\nu^{3} + 665\nu^{2} - 92\nu - 2630 ) / 62 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + 3\beta_{4} + 2\beta_{3} + 2\beta_{2} - 2\beta _1 + 3 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 3\beta_{4} + 12\beta_{3} + 2\beta_{2} + 8\beta _1 + 313 ) / 20 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{5} + 19\beta_{4} + 26\beta_{3} + 36\beta_{2} - 16\beta _1 + 189 ) / 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -31\beta_{5} + 153\beta_{4} + 352\beta_{3} + 202\beta_{2} + 248\beta _1 + 7003 ) / 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 69\beta_{5} + 1273\beta_{4} + 1452\beta_{3} + 2342\beta_{2} - 352\beta _1 + 15723 ) / 20 \)
|
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 | −9.27387 | 0 | 5.95305 | 0 | −7.00000 | 0 | 59.0046 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.73933 | 0 | 19.7791 | 0 | −7.00000 | 0 | −13.0174 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.60307 | 0 | −19.7567 | 0 | −7.00000 | 0 | −14.0179 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.96505 | 0 | 3.10102 | 0 | −7.00000 | 0 | −18.2085 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 4.94660 | 0 | 4.15482 | 0 | −7.00000 | 0 | −2.53111 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 8.70461 | 0 | −11.2312 | 0 | −7.00000 | 0 | 48.7703 | 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.bb | 6 | |
| 4.b | odd | 2 | 1 | 616.4.a.i | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 616.4.a.i | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 1232.4.a.bb | 6 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):
|
\( T_{3}^{6} - 111T_{3}^{4} + 30T_{3}^{3} + 2616T_{3}^{2} + 24T_{3} - 15952 \)
|
|
\( T_{5}^{6} - 2T_{5}^{5} - 483T_{5}^{4} + 1328T_{5}^{3} + 35184T_{5}^{2} - 216128T_{5} + 336624 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 111 T^{4} + \cdots - 15952 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 336624 \)
|
| $7$ |
\( (T + 7)^{6} \)
|
| $11$ |
\( (T - 11)^{6} \)
|
| $13$ |
\( T^{6} + 6 T^{5} + \cdots - 325632 \)
|
| $17$ |
\( T^{6} + \cdots - 28913335168 \)
|
| $19$ |
\( T^{6} + \cdots + 16168602112 \)
|
| $23$ |
\( T^{6} + \cdots + 41994606848 \)
|
| $29$ |
\( T^{6} + \cdots + 1667440361024 \)
|
| $31$ |
\( T^{6} + \cdots + 54893114800 \)
|
| $37$ |
\( T^{6} + \cdots + 58488779493744 \)
|
| $41$ |
\( T^{6} + \cdots + 61200631406976 \)
|
| $43$ |
\( T^{6} + \cdots - 75562012164096 \)
|
| $47$ |
\( T^{6} + \cdots + 62336281709568 \)
|
| $53$ |
\( T^{6} + \cdots + 366629128251072 \)
|
| $59$ |
\( T^{6} + \cdots + 234132668007984 \)
|
| $61$ |
\( T^{6} + \cdots + 42\!\cdots\!68 \)
|
| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots - 33\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 23\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots + 773893110835200 \)
|
| $83$ |
\( T^{6} + \cdots - 31\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots - 67\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 20\!\cdots\!44 \)
|
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