Newspace parameters
| Level: | \( N \) | \(=\) | \( 7616 = 2^{6} \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7616.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(60.8140661794\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.109859312.1 |
|
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 7x^{4} + 15x^{3} + 13x^{2} - 9x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 3808) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 7x^{4} + 15x^{3} + 13x^{2} - 9x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} + 11\nu^{3} - 8\nu^{2} - 8\nu - 19 ) / 11 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} + 22\nu^{3} - 30\nu^{2} - 63\nu + 3 ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 8\nu^{4} - 11\nu^{3} - 37\nu^{2} + 73\nu + 18 ) / 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 13\nu^{4} + 11\nu^{3} - 67\nu^{2} - 12\nu + 32 ) / 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{5} + 10\nu^{4} + 33\nu^{3} - 49\nu^{2} - 60\nu + 17 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} + \beta_{3} + 3\beta_{2} + 2\beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4\beta_{5} - 7\beta_{4} + 7\beta_{3} + 13\beta_{2} + 2\beta _1 + 15 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -24\beta_{5} - 13\beta_{4} + 17\beta_{3} + 43\beta_{2} + 16\beta _1 + 63 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -74\beta_{5} - 63\beta_{4} + 73\beta_{3} + 163\beta_{2} + 32\beta _1 + 189 ) / 2 \)
|
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.
| 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 | −3.38207 | 0 | −2.62360 | 0 | 1.00000 | 0 | 8.43840 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.59220 | 0 | 3.54237 | 0 | 1.00000 | 0 | 3.71950 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.07243 | 0 | −3.18060 | 0 | 1.00000 | 0 | −1.84989 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.531974 | 0 | −0.110568 | 0 | 1.00000 | 0 | −2.71700 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.73670 | 0 | 1.54368 | 0 | 1.00000 | 0 | 0.0161353 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.84197 | 0 | −3.17127 | 0 | 1.00000 | 0 | 0.392864 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 7616.2.a.bu | 6 | |
| 4.b | odd | 2 | 1 | 7616.2.a.cc | 6 | ||
| 8.b | even | 2 | 1 | 3808.2.a.p | yes | 6 | |
| 8.d | odd | 2 | 1 | 3808.2.a.h | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3808.2.a.h | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 3808.2.a.p | yes | 6 | 8.b | even | 2 | 1 | |
| 7616.2.a.bu | 6 | 1.a | even | 1 | 1 | trivial | |
| 7616.2.a.cc | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7616))\):
|
\( T_{3}^{6} + 4T_{3}^{5} - 5T_{3}^{4} - 26T_{3}^{3} + 3T_{3}^{2} + 38T_{3} + 16 \)
|
|
\( T_{5}^{6} + 4T_{5}^{5} - 13T_{5}^{4} - 62T_{5}^{3} + 5T_{5}^{2} + 146T_{5} + 16 \)
|
|
\( T_{11}^{6} + 8T_{11}^{5} - 2T_{11}^{4} - 124T_{11}^{3} - 156T_{11}^{2} + 184T_{11} - 32 \)
|
|
\( T_{19}^{6} + 6T_{19}^{5} - 8T_{19}^{4} - 60T_{19}^{3} - 4T_{19}^{2} + 64T_{19} - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 4 T^{5} + \cdots + 16 \)
|
| $5$ |
\( T^{6} + 4 T^{5} + \cdots + 16 \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} + 8 T^{5} + \cdots - 32 \)
|
| $13$ |
\( T^{6} + 8 T^{5} + \cdots - 256 \)
|
| $17$ |
\( (T + 1)^{6} \)
|
| $19$ |
\( T^{6} + 6 T^{5} + \cdots - 16 \)
|
| $23$ |
\( T^{6} - 18 T^{5} + \cdots - 256 \)
|
| $29$ |
\( T^{6} - 8 T^{5} + \cdots - 40864 \)
|
| $31$ |
\( T^{6} + 4 T^{5} + \cdots - 11584 \)
|
| $37$ |
\( T^{6} - 8 T^{5} + \cdots + 1088 \)
|
| $41$ |
\( T^{6} - 16 T^{5} + \cdots - 85252 \)
|
| $43$ |
\( T^{6} + 12 T^{5} + \cdots - 7936 \)
|
| $47$ |
\( T^{6} - 18 T^{5} + \cdots + 20096 \)
|
| $53$ |
\( T^{6} - 2 T^{5} + \cdots - 25652 \)
|
| $59$ |
\( T^{6} + 16 T^{5} + \cdots + 256 \)
|
| $61$ |
\( T^{6} + 6 T^{5} + \cdots + 41248 \)
|
| $67$ |
\( T^{6} + 12 T^{5} + \cdots - 1888 \)
|
| $71$ |
\( T^{6} + 2 T^{5} + \cdots + 161024 \)
|
| $73$ |
\( T^{6} - 8 T^{5} + \cdots - 120964 \)
|
| $79$ |
\( T^{6} + 18 T^{5} + \cdots + 741376 \)
|
| $83$ |
\( T^{6} + 28 T^{5} + \cdots + 46768 \)
|
| $89$ |
\( T^{6} + 2 T^{5} + \cdots - 66544 \)
|
| $97$ |
\( T^{6} + 18 T^{5} + \cdots - 4516 \)
|
| show more | |
| show less |