Newspace parameters
| Level: | \( N \) | \(=\) | \( 7406 = 2 \cdot 7 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7406.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(59.1372077370\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.84770496.1 |
|
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 10x^{4} + 24x^{3} + 6x^{2} - 28x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 2x^{5} - 10x^{4} + 24x^{3} + 6x^{2} - 28x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 9\nu^{3} - 3\nu^{2} + 13\nu + 3 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + 10\nu^{3} - 4\nu^{2} - 15\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 10\nu^{3} + 5\nu^{2} + 15\nu - 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{5} - 19\nu^{3} + 10\nu^{2} + 23\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} + 7\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + 9\beta_{4} + 8\beta_{3} + 2\beta_{2} - 5\beta _1 + 29 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{5} - 24\beta_{4} - 5\beta_{3} + 55\beta _1 - 56 \)
|
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 |
|
1.00000 | −3.13793 | 1.00000 | 3.90332 | −3.13793 | 1.00000 | 1.00000 | 6.84662 | 3.90332 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.06455 | 1.00000 | −2.60497 | −1.06455 | 1.00000 | 1.00000 | −1.86674 | −2.60497 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.0360320 | 1.00000 | −2.45350 | 0.0360320 | 1.00000 | 1.00000 | −2.99870 | −2.45350 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.58560 | 1.00000 | 3.51290 | 1.58560 | 1.00000 | 1.00000 | −0.485859 | 3.51290 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.21099 | 1.00000 | −0.639974 | 2.21099 | 1.00000 | 1.00000 | 1.88850 | −0.639974 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.36985 | 1.00000 | 2.28223 | 2.36985 | 1.00000 | 1.00000 | 2.61619 | 2.28223 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 7406.2.a.bl | yes | 6 |
| 23.b | odd | 2 | 1 | 7406.2.a.bk | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7406.2.a.bk | ✓ | 6 | 23.b | odd | 2 | 1 | |
| 7406.2.a.bl | yes | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 24T_{3}^{3} + 6T_{3}^{2} - 28T_{3} + 1 \)
|
|
\( T_{5}^{6} - 4T_{5}^{5} - 15T_{5}^{4} + 54T_{5}^{3} + 77T_{5}^{2} - 176T_{5} - 128 \)
|
|
\( T_{11}^{6} - 16T_{11}^{5} + 95T_{11}^{4} - 264T_{11}^{3} + 357T_{11}^{2} - 224T_{11} + 52 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 1 \)
|
| $5$ |
\( T^{6} - 4 T^{5} + \cdots - 128 \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} - 16 T^{5} + \cdots + 52 \)
|
| $13$ |
\( T^{6} - 51 T^{4} + \cdots - 1404 \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots + 1417 \)
|
| $19$ |
\( T^{6} - 18 T^{5} + \cdots - 431 \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( T^{6} - 4 T^{5} + \cdots + 256 \)
|
| $31$ |
\( T^{6} + 2 T^{5} + \cdots - 18716 \)
|
| $37$ |
\( T^{6} - 4 T^{5} + \cdots + 10816 \)
|
| $41$ |
\( T^{6} - 107 T^{4} + \cdots + 4084 \)
|
| $43$ |
\( T^{6} - 8 T^{5} + \cdots - 1067 \)
|
| $47$ |
\( T^{6} - 8 T^{5} + \cdots + 10816 \)
|
| $53$ |
\( T^{6} + 8 T^{5} + \cdots - 62912 \)
|
| $59$ |
\( T^{6} - 2 T^{5} + \cdots - 368 \)
|
| $61$ |
\( T^{6} - 28 T^{5} + \cdots + 10816 \)
|
| $67$ |
\( T^{6} - 16 T^{5} + \cdots + 2704 \)
|
| $71$ |
\( T^{6} + 10 T^{5} + \cdots + 192148 \)
|
| $73$ |
\( T^{6} + 28 T^{5} + \cdots - 9683 \)
|
| $79$ |
\( T^{6} - 16 T^{5} + \cdots + 180304 \)
|
| $83$ |
\( T^{6} - 24 T^{5} + \cdots + 158041 \)
|
| $89$ |
\( T^{6} + 46 T^{5} + \cdots + 14992 \)
|
| $97$ |
\( T^{6} - 2 T^{5} + \cdots + 349456 \)
|
| show more | |
| show less |