Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4235,2,Mod(1,4235)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4235.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4235 = 5 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4235.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: | \(33.8166452560\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.32694400625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 16x^{6} + 44x^{5} + 83x^{4} - 200x^{3} - 132x^{2} + 288x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 385) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 16x^{6} + 44x^{5} + 83x^{4} - 200x^{3} - 132x^{2} + 288x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{7} + 45\nu^{6} + 6\nu^{5} - 452\nu^{4} + 337\nu^{3} + 546\nu^{2} - 1244\nu + 1712 ) / 784 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 23\nu^{7} - 109\nu^{6} - 204\nu^{5} + 1256\nu^{4} + 253\nu^{3} - 3472\nu^{2} + 744\nu + 200 ) / 784 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -6\nu^{7} + 5\nu^{6} + 115\nu^{5} - 72\nu^{4} - 654\nu^{3} + 273\nu^{2} + 1016\nu - 180 ) / 196 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17\nu^{7} - 6\nu^{6} - 285\nu^{5} - 90\nu^{4} + 1559\nu^{3} + 1015\nu^{2} - 2454\nu - 1156 ) / 392 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -10\nu^{7} + 41\nu^{6} + 61\nu^{5} - 414\nu^{4} + 282\nu^{3} + 945\nu^{2} - 1606\nu + 92 ) / 196 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -23\nu^{7} + 11\nu^{6} + 498\nu^{5} - 80\nu^{4} - 3389\nu^{3} + 42\nu^{2} + 6704\nu - 984 ) / 392 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{4} + 4\beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 11\beta_{7} + 11\beta_{6} + 10\beta_{5} - 11\beta_{4} + 14\beta_{3} - 4\beta_{2} + 2\beta _1 + 47 \)
|
| \(\nu^{5}\) | \(=\) |
\( 19\beta_{7} + 27\beta_{6} + 30\beta_{5} + 7\beta_{4} + 54\beta_{3} + 28\beta_{2} + 41\beta _1 + 55 \)
|
| \(\nu^{6}\) | \(=\) |
\( 118\beta_{7} + 114\beta_{6} + 111\beta_{5} - 108\beta_{4} + 159\beta_{3} + 7\beta_{2} + 31\beta _1 + 415 \)
|
| \(\nu^{7}\) | \(=\) |
\( 267\beta_{7} + 308\beta_{6} + 375\beta_{5} - 11\beta_{4} + 609\beta_{3} + 327\beta_{2} + 303\beta _1 + 752 \)
|
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 |
|
−1.35567 | −3.04321 | −0.162147 | −1.00000 | 4.12560 | 1.00000 | 2.93117 | 6.26111 | 1.35567 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.35567 | 2.20535 | −0.162147 | −1.00000 | −2.98974 | 1.00000 | 2.93117 | 1.86359 | 1.35567 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.477260 | −1.87300 | −1.77222 | −1.00000 | 0.893910 | 1.00000 | 1.80033 | 0.508144 | 0.477260 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.477260 | 2.64523 | −1.77222 | −1.00000 | −1.26246 | 1.00000 | 1.80033 | 3.99722 | 0.477260 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.737640 | −1.37726 | −1.45589 | −1.00000 | −1.01592 | 1.00000 | −2.54920 | −1.10316 | −0.737640 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.737640 | 1.83315 | −1.45589 | −1.00000 | 1.35220 | 1.00000 | −2.54920 | 0.360427 | −0.737640 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.09529 | −3.33308 | 2.39026 | −1.00000 | −6.98377 | 1.00000 | 0.817703 | 8.10940 | −2.09529 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.09529 | −0.0571808 | 2.39026 | −1.00000 | −0.119811 | 1.00000 | 0.817703 | −2.99673 | −2.09529 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 4235.2.a.bh | 8 | |
| 11.b | odd | 2 | 1 | 4235.2.a.bg | 8 | ||
| 11.d | odd | 10 | 2 | 385.2.n.d | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 385.2.n.d | ✓ | 16 | 11.d | odd | 10 | 2 | |
| 4235.2.a.bg | 8 | 11.b | odd | 2 | 1 | ||
| 4235.2.a.bh | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):
|
\( T_{2}^{4} - T_{2}^{3} - 3T_{2}^{2} + T_{2} + 1 \)
|
|
\( T_{3}^{8} + 3T_{3}^{7} - 16T_{3}^{6} - 44T_{3}^{5} + 83T_{3}^{4} + 200T_{3}^{3} - 132T_{3}^{2} - 288T_{3} - 16 \)
|
|
\( T_{13}^{8} - 34T_{13}^{6} + 5T_{13}^{5} + 337T_{13}^{4} - 70T_{13}^{3} - 816T_{13}^{2} - 240T_{13} + 176 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} - 3 T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} + 3 T^{7} + \cdots - 16 \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} - 34 T^{6} + \cdots + 176 \)
|
| $17$ |
\( T^{8} - 7 T^{7} + \cdots - 5776 \)
|
| $19$ |
\( T^{8} + 19 T^{7} + \cdots + 400 \)
|
| $23$ |
\( T^{8} + 16 T^{7} + \cdots + 44521 \)
|
| $29$ |
\( T^{8} + 8 T^{7} + \cdots + 22159 \)
|
| $31$ |
\( T^{8} + 6 T^{7} + \cdots - 1936 \)
|
| $37$ |
\( T^{8} + 26 T^{7} + \cdots + 78901 \)
|
| $41$ |
\( T^{8} + 9 T^{7} + \cdots + 202576 \)
|
| $43$ |
\( T^{8} - 4 T^{7} + \cdots + 177991 \)
|
| $47$ |
\( T^{8} - 9 T^{7} + \cdots + 1496944 \)
|
| $53$ |
\( T^{8} + 7 T^{7} + \cdots - 35101 \)
|
| $59$ |
\( T^{8} - 20 T^{7} + \cdots - 123856 \)
|
| $61$ |
\( T^{8} + 13 T^{7} + \cdots - 53776 \)
|
| $67$ |
\( T^{8} + 21 T^{7} + \cdots - 2443421 \)
|
| $71$ |
\( T^{8} - 3 T^{7} + \cdots + 9882025 \)
|
| $73$ |
\( T^{8} + 16 T^{7} + \cdots + 36784 \)
|
| $79$ |
\( T^{8} - 14 T^{7} + \cdots - 2842211 \)
|
| $83$ |
\( T^{8} + 4 T^{7} + \cdots - 1081616 \)
|
| $89$ |
\( T^{8} + 23 T^{7} + \cdots - 875536 \)
|
| $97$ |
\( T^{8} + 38 T^{7} + \cdots - 14917376 \)
|
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