Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8040,2,Mod(1,8040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8040, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8040.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8040.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: | \(64.1997232251\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 9x^{6} + 21x^{5} + 23x^{4} - 40x^{3} - 11x^{2} + 20x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 9x^{6} + 21x^{5} + 23x^{4} - 40x^{3} - 11x^{2} + 20x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{7} + 19\nu^{6} + 10\nu^{5} - 116\nu^{4} + 53\nu^{3} + 118\nu^{2} - 148\nu + 63 ) / 29 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 6\nu^{7} - 14\nu^{6} - 44\nu^{5} + 29\nu^{4} + 51\nu^{3} + 142\nu^{2} + 106\nu - 109 ) / 29 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 18\nu^{7} - 42\nu^{6} - 190\nu^{5} + 261\nu^{4} + 559\nu^{3} - 386\nu^{2} - 378\nu + 137 ) / 29 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15\nu^{7} - 35\nu^{6} - 168\nu^{5} + 232\nu^{4} + 577\nu^{3} - 399\nu^{2} - 518\nu + 177 ) / 29 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -22\nu^{7} + 61\nu^{6} + 200\nu^{5} - 377\nu^{4} - 535\nu^{3} + 591\nu^{2} + 346\nu - 219 ) / 29 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 24\nu^{7} - 56\nu^{6} - 263\nu^{5} + 377\nu^{4} + 813\nu^{3} - 679\nu^{2} - 562\nu + 318 ) / 29 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -28\nu^{7} + 75\nu^{6} + 273\nu^{5} - 493\nu^{4} - 760\nu^{3} + 797\nu^{2} + 472\nu - 284 ) / 29 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} + 3\beta_{3} - \beta_{2} - 2\beta _1 + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{7} + 4\beta_{6} - 2\beta_{5} + 7\beta_{3} - 3\beta_{2} - 8\beta _1 + 12 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 32\beta_{7} + 6\beta_{6} - 6\beta_{5} - 4\beta_{4} + 37\beta_{3} - 15\beta_{2} - 26\beta _1 + 52 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 126\beta_{7} + 34\beta_{6} - 32\beta_{5} - 12\beta_{4} + 117\beta_{3} - 49\beta_{2} - 94\beta _1 + 160 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 446\beta_{7} + 100\beta_{6} - 106\beta_{5} - 64\beta_{4} + 475\beta_{3} - 195\beta_{2} - 334\beta _1 + 610 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1660\beta_{7} + 402\beta_{6} - 436\beta_{5} - 218\beta_{4} + 1657\beta_{3} - 683\beta_{2} - 1210\beta _1 + 2120 ) / 2 \)
|
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 | −1.00000 | 0 | 1.00000 | 0 | −3.46554 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | 1.00000 | 0 | −2.71846 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | 1.00000 | 0 | −1.80547 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 1.00000 | 0 | 0.653863 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 1.00000 | 0 | 0.986082 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 1.00000 | 0 | 1.52573 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −1.00000 | 0 | 1.00000 | 0 | 3.85394 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | −1.00000 | 0 | 1.00000 | 0 | 3.96985 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(67\) | \(-1\) |
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 | 8040.2.a.v | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8040.2.a.v | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):
|
\( T_{7}^{8} - 3T_{7}^{7} - 24T_{7}^{6} + 62T_{7}^{5} + 167T_{7}^{4} - 358T_{7}^{3} - 230T_{7}^{2} + 640T_{7} - 256 \)
|
|
\( T_{11}^{8} - 7T_{11}^{7} - 18T_{11}^{6} + 162T_{11}^{5} + 147T_{11}^{4} - 1188T_{11}^{3} - 952T_{11}^{2} + 2128T_{11} + 1648 \)
|
|
\( T_{13}^{8} + 5T_{13}^{7} - 33T_{13}^{6} - 197T_{13}^{5} - 9T_{13}^{4} + 1268T_{13}^{3} + 2158T_{13}^{2} + 1240T_{13} + 200 \)
|
|
\( T_{17}^{8} + 6T_{17}^{7} - 54T_{17}^{6} - 333T_{17}^{5} + 47T_{17}^{4} + 2344T_{17}^{3} + 3216T_{17}^{2} + 512T_{17} - 512 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} - 3 T^{7} + \cdots - 256 \)
|
| $11$ |
\( T^{8} - 7 T^{7} + \cdots + 1648 \)
|
| $13$ |
\( T^{8} + 5 T^{7} + \cdots + 200 \)
|
| $17$ |
\( T^{8} + 6 T^{7} + \cdots - 512 \)
|
| $19$ |
\( T^{8} - 3 T^{7} + \cdots - 31168 \)
|
| $23$ |
\( T^{8} - 14 T^{7} + \cdots + 24032 \)
|
| $29$ |
\( T^{8} + 3 T^{7} + \cdots - 6392 \)
|
| $31$ |
\( T^{8} - 15 T^{7} + \cdots - 512488 \)
|
| $37$ |
\( T^{8} + 10 T^{7} + \cdots - 21632 \)
|
| $41$ |
\( T^{8} - 7 T^{7} + \cdots + 7424 \)
|
| $43$ |
\( T^{8} - 13 T^{7} + \cdots + 512 \)
|
| $47$ |
\( T^{8} - 19 T^{7} + \cdots + 349952 \)
|
| $53$ |
\( T^{8} + 11 T^{7} + \cdots + 688 \)
|
| $59$ |
\( T^{8} - 13 T^{7} + \cdots + 782912 \)
|
| $61$ |
\( T^{8} - 9 T^{7} + \cdots - 203584 \)
|
| $67$ |
\( (T - 1)^{8} \)
|
| $71$ |
\( T^{8} - 32 T^{7} + \cdots - 506624 \)
|
| $73$ |
\( T^{8} - 378 T^{6} + \cdots - 982864 \)
|
| $79$ |
\( T^{8} - 27 T^{7} + \cdots - 88984 \)
|
| $83$ |
\( T^{8} - 22 T^{7} + \cdots - 24365056 \)
|
| $89$ |
\( T^{8} - 20 T^{7} + \cdots + 197392 \)
|
| $97$ |
\( T^{8} + 9 T^{7} + \cdots + 11668672 \)
|
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