Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6256,2,Mod(1,6256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6256, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6256.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6256 = 2^{4} \cdot 17 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6256.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: | \(49.9544115045\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 3x^{9} - 15x^{8} + 43x^{7} + 74x^{6} - 192x^{5} - 136x^{4} + 268x^{3} + 88x^{2} - 56x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3128) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3x^{9} - 15x^{8} + 43x^{7} + 74x^{6} - 192x^{5} - 136x^{4} + 268x^{3} + 88x^{2} - 56x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 6\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 11\nu^{4} + 16\nu^{3} + 32\nu^{2} - 24\nu - 16 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} - 2\nu^{8} - 15\nu^{7} + 24\nu^{6} + 72\nu^{5} - 80\nu^{4} - 116\nu^{3} + 56\nu^{2} + 40\nu + 16 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{9} + 4\nu^{8} + 11\nu^{7} - 54\nu^{6} - 24\nu^{5} + 224\nu^{4} - 44\nu^{3} - 288\nu^{2} + 72\nu + 64 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{9} + 3\nu^{8} + 15\nu^{7} - 43\nu^{6} - 72\nu^{5} + 190\nu^{4} + 112\nu^{3} - 256\nu^{2} - 16\nu + 48 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{9} - 3\nu^{8} - 15\nu^{7} + 43\nu^{6} + 74\nu^{5} - 192\nu^{4} - 132\nu^{3} + 264\nu^{2} + 56\nu - 40 ) / 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{9} - 10\nu^{8} - 41\nu^{7} + 140\nu^{6} + 168\nu^{5} - 596\nu^{4} - 188\nu^{3} + 736\nu^{2} - 8\nu - 96 ) / 16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -3\nu^{9} + 10\nu^{8} + 41\nu^{7} - 140\nu^{6} - 168\nu^{5} + 604\nu^{4} + 180\nu^{3} - 784\nu^{2} + 40\nu + 96 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - \beta_{6} - \beta_{5} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{6} - \beta_{5} + 2\beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{9} + 2\beta_{8} - 7\beta_{6} - 7\beta_{5} + 2\beta_{2} + 2\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15\beta_{9} + 2\beta_{8} + 4\beta_{7} - 9\beta_{6} - 13\beta_{5} + 22\beta_{2} + 42\beta _1 + 26 \)
|
| \(\nu^{6}\) | \(=\) |
\( 81\beta_{9} + 26\beta_{8} + 8\beta_{7} - 47\beta_{6} - 55\beta_{5} + 4\beta_{3} + 34\beta_{2} + 34\beta _1 + 194 \)
|
| \(\nu^{7}\) | \(=\) |
\( 175 \beta_{9} + 38 \beta_{8} + 64 \beta_{7} - 69 \beta_{6} - 141 \beta_{5} + 4 \beta_{4} + 8 \beta_{3} + \cdots + 286 \)
|
| \(\nu^{8}\) | \(=\) |
\( 753 \beta_{9} + 274 \beta_{8} + 152 \beta_{7} - 319 \beta_{6} - 479 \beta_{5} + 16 \beta_{4} + \cdots + 1570 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1887 \beta_{9} + 510 \beta_{8} + 784 \beta_{7} - 517 \beta_{6} - 1437 \beta_{5} + 108 \beta_{4} + \cdots + 2974 \)
|
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 | −2.59535 | 0 | −1.13855 | 0 | 0.0332599 | 0 | 3.73583 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.28607 | 0 | 2.85356 | 0 | 4.93148 | 0 | 2.22612 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.44115 | 0 | −1.63472 | 0 | −1.17083 | 0 | −0.923094 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.475740 | 0 | 1.03980 | 0 | 2.64119 | 0 | −2.77367 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.274167 | 0 | 0.692600 | 0 | −4.54993 | 0 | −2.92483 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.512647 | 0 | 2.05672 | 0 | −0.760014 | 0 | −2.73719 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.46387 | 0 | −2.74420 | 0 | −1.77336 | 0 | −0.857087 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.27842 | 0 | 4.33314 | 0 | 2.86280 | 0 | 2.19118 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.64337 | 0 | −3.04516 | 0 | 3.85261 | 0 | 3.98741 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 3.17417 | 0 | 1.58679 | 0 | 0.932795 | 0 | 7.07534 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(17\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 6256.2.a.be | 10 | |
| 4.b | odd | 2 | 1 | 3128.2.a.h | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3128.2.a.h | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 6256.2.a.be | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6256))\):
|
\( T_{3}^{10} - 3T_{3}^{9} - 15T_{3}^{8} + 43T_{3}^{7} + 74T_{3}^{6} - 192T_{3}^{5} - 136T_{3}^{4} + 268T_{3}^{3} + 88T_{3}^{2} - 56T_{3} - 16 \)
|
|
\( T_{5}^{10} - 4 T_{5}^{9} - 20 T_{5}^{8} + 80 T_{5}^{7} + 115 T_{5}^{6} - 522 T_{5}^{5} - 119 T_{5}^{4} + \cdots + 452 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} - 3 T^{9} + \cdots - 16 \)
|
| $5$ |
\( T^{10} - 4 T^{9} + \cdots + 452 \)
|
| $7$ |
\( T^{10} - 7 T^{9} + \cdots + 32 \)
|
| $11$ |
\( T^{10} - 9 T^{9} + \cdots + 2032 \)
|
| $13$ |
\( T^{10} + 9 T^{9} + \cdots - 1078 \)
|
| $17$ |
\( (T + 1)^{10} \)
|
| $19$ |
\( T^{10} - 4 T^{9} + \cdots - 15968 \)
|
| $23$ |
\( (T - 1)^{10} \)
|
| $29$ |
\( T^{10} - 5 T^{9} + \cdots + 32416 \)
|
| $31$ |
\( T^{10} - 18 T^{9} + \cdots - 1080032 \)
|
| $37$ |
\( T^{10} + 11 T^{9} + \cdots - 43912 \)
|
| $41$ |
\( T^{10} + 9 T^{9} + \cdots - 27728 \)
|
| $43$ |
\( T^{10} + \cdots + 140113568 \)
|
| $47$ |
\( T^{10} - 2 T^{9} + \cdots - 13475744 \)
|
| $53$ |
\( T^{10} - 20 T^{9} + \cdots - 67933184 \)
|
| $59$ |
\( T^{10} - T^{9} + \cdots - 447808 \)
|
| $61$ |
\( T^{10} - 17 T^{9} + \cdots + 1855724 \)
|
| $67$ |
\( T^{10} - 19 T^{9} + \cdots + 10234784 \)
|
| $71$ |
\( T^{10} - 24 T^{9} + \cdots - 868576 \)
|
| $73$ |
\( T^{10} + \cdots + 157180144 \)
|
| $79$ |
\( T^{10} + \cdots + 139592288 \)
|
| $83$ |
\( T^{10} - 9 T^{9} + \cdots - 10030496 \)
|
| $89$ |
\( T^{10} - 12 T^{9} + \cdots - 1863104 \)
|
| $97$ |
\( T^{10} + 8 T^{9} + \cdots + 1715488 \)
|
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