Newspace parameters
| Level: | \( N \) | \(=\) | \( 5004 = 2^{2} \cdot 3^{2} \cdot 139 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5004.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(39.9571411714\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 12x^{5} + 27x^{4} + 51x^{3} - 51x^{2} - 64x - 13 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 556) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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
\( x^{7} - 3x^{6} - 12x^{5} + 27x^{4} + 51x^{3} - 51x^{2} - 64x - 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 4\nu^{4} + 19\nu^{3} + 4\nu^{2} + \nu - 9 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 8\nu^{4} + 35\nu^{3} + 16\nu^{2} - 67\nu - 5 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 6\nu^{5} + 2\nu^{4} + 33\nu^{3} - 28\nu^{2} - 23\nu + 5 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 8\nu^{4} + 35\nu^{3} + 24\nu^{2} - 75\nu - 37 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 13\nu^{4} - 32\nu^{3} - 50\nu^{2} + 77\nu + 38 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} - \beta_{4} + 6\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{6} + 15\beta_{5} - 4\beta_{4} - 5\beta_{3} + 2\beta_{2} + 10\beta _1 + 33 \)
|
| \(\nu^{5}\) | \(=\) |
\( 19\beta_{6} + 50\beta_{5} - 23\beta_{4} + \beta_{3} + 10\beta_{2} + 44\beta _1 + 77 \)
|
| \(\nu^{6}\) | \(=\) |
\( 73\beta_{6} + 199\beta_{5} - 89\beta_{4} - 12\beta_{3} + 56\beta_{2} + 97\beta _1 + 338 \)
|
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.
| 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 | 0 | 0 | −3.03644 | 0 | 2.24323 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −2.71725 | 0 | −2.01403 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −1.27411 | 0 | 3.91779 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | −0.435870 | 0 | 1.81286 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 0.712965 | 0 | −3.69084 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 3.73710 | 0 | 0.558925 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0 | 0 | 4.01360 | 0 | −1.82794 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(139\) | \( +1 \) |
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 | 5004.2.a.n | 7 | |
| 3.b | odd | 2 | 1 | 556.2.a.c | ✓ | 7 | |
| 12.b | even | 2 | 1 | 2224.2.a.n | 7 | ||
| 24.f | even | 2 | 1 | 8896.2.a.bf | 7 | ||
| 24.h | odd | 2 | 1 | 8896.2.a.bc | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 556.2.a.c | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 2224.2.a.n | 7 | 12.b | even | 2 | 1 | ||
| 5004.2.a.n | 7 | 1.a | even | 1 | 1 | trivial | |
| 8896.2.a.bc | 7 | 24.h | odd | 2 | 1 | ||
| 8896.2.a.bf | 7 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5004))\):
|
\( T_{5}^{7} - T_{5}^{6} - 24T_{5}^{5} + 2T_{5}^{4} + 161T_{5}^{3} + 117T_{5}^{2} - 91T_{5} - 49 \)
|
|
\( T_{7}^{7} - T_{7}^{6} - 22T_{7}^{5} + 18T_{7}^{4} + 125T_{7}^{3} - 85T_{7}^{2} - 209T_{7} + 121 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - T^{6} + \cdots - 49 \)
|
| $7$ |
\( T^{7} - T^{6} + \cdots + 121 \)
|
| $11$ |
\( T^{7} + 14 T^{6} + \cdots - 1 \)
|
| $13$ |
\( T^{7} + 2 T^{6} + \cdots + 919 \)
|
| $17$ |
\( T^{7} + 3 T^{6} + \cdots + 176 \)
|
| $19$ |
\( T^{7} - 8 T^{6} + \cdots + 17536 \)
|
| $23$ |
\( T^{7} + 23 T^{6} + \cdots + 88592 \)
|
| $29$ |
\( T^{7} - 2 T^{6} + \cdots + 8163 \)
|
| $31$ |
\( T^{7} - 6 T^{6} + \cdots + 8111 \)
|
| $37$ |
\( T^{7} + 6 T^{6} + \cdots - 100834 \)
|
| $41$ |
\( T^{7} + 3 T^{6} + \cdots + 1808 \)
|
| $43$ |
\( T^{7} + 2 T^{6} + \cdots - 505152 \)
|
| $47$ |
\( T^{7} - 5 T^{6} + \cdots + 26912 \)
|
| $53$ |
\( T^{7} - 4 T^{6} + \cdots + 4336 \)
|
| $59$ |
\( T^{7} + 22 T^{6} + \cdots + 17504 \)
|
| $61$ |
\( T^{7} + 22 T^{6} + \cdots - 225504 \)
|
| $67$ |
\( T^{7} + 13 T^{6} + \cdots - 246536 \)
|
| $71$ |
\( T^{7} + 42 T^{6} + \cdots + 923877 \)
|
| $73$ |
\( T^{7} + 21 T^{6} + \cdots - 690208 \)
|
| $79$ |
\( T^{7} - 4 T^{6} + \cdots - 4483 \)
|
| $83$ |
\( T^{7} + 19 T^{6} + \cdots - 3604061 \)
|
| $89$ |
\( T^{7} + 22 T^{6} + \cdots + 60867 \)
|
| $97$ |
\( T^{7} + 23 T^{6} + \cdots + 176 \)
|
| show more | |
| show less |