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: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\Q(\zeta_{22})^+\) |
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| Defining polynomial: |
\( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 322) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
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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 | −2.39788 | 1.00000 | 2.07324 | 2.39788 | −1.00000 | −1.00000 | 2.74982 | −2.07324 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.54620 | 1.00000 | −2.27686 | 1.54620 | −1.00000 | −1.00000 | −0.609264 | 2.27686 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.37279 | 1.00000 | 2.44009 | 1.37279 | −1.00000 | −1.00000 | −1.11546 | −2.44009 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.0881559 | 1.00000 | 3.79797 | −0.0881559 | −1.00000 | −1.00000 | −2.99223 | −3.79797 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.22871 | 1.00000 | −2.03445 | −2.22871 | −1.00000 | −1.00000 | 1.96714 | 2.03445 | ||||||||||||||||||||||||||||||||||
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.bf | 5 | |
| 23.b | odd | 2 | 1 | 7406.2.a.be | 5 | ||
| 23.c | even | 11 | 2 | 322.2.i.b | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.2.i.b | ✓ | 10 | 23.c | even | 11 | 2 | |
| 7406.2.a.be | 5 | 23.b | odd | 2 | 1 | ||
| 7406.2.a.bf | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):
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\( T_{3}^{5} + 3T_{3}^{4} - 3T_{3}^{3} - 15T_{3}^{2} - 10T_{3} + 1 \)
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\( T_{5}^{5} - 4T_{5}^{4} - 9T_{5}^{3} + 38T_{5}^{2} + 20T_{5} - 89 \)
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\( T_{11}^{5} - 22T_{11}^{3} - 11T_{11}^{2} + 44T_{11} + 11 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
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| $3$ |
\( T^{5} + 3 T^{4} + \cdots + 1 \)
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| $5$ |
\( T^{5} - 4 T^{4} + \cdots - 89 \)
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| $7$ |
\( (T + 1)^{5} \)
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| $11$ |
\( T^{5} - 22 T^{3} + \cdots + 11 \)
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| $13$ |
\( T^{5} + 3 T^{4} + \cdots + 1 \)
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| $17$ |
\( T^{5} + T^{4} + \cdots + 67 \)
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| $19$ |
\( T^{5} - 4 T^{4} + \cdots - 1 \)
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| $23$ |
\( T^{5} \)
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| $29$ |
\( T^{5} + 10 T^{4} + \cdots - 353 \)
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| $31$ |
\( T^{5} + 15 T^{4} + \cdots - 109 \)
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| $37$ |
\( T^{5} + 3 T^{4} + \cdots + 89 \)
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| $41$ |
\( T^{5} + 6 T^{4} + \cdots + 8579 \)
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| $43$ |
\( T^{5} - 16 T^{4} + \cdots - 67 \)
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| $47$ |
\( T^{5} - 19 T^{4} + \cdots + 4643 \)
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| $53$ |
\( T^{5} - 14 T^{4} + \cdots - 7019 \)
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| $59$ |
\( T^{5} + T^{4} + \cdots - 131 \)
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| $61$ |
\( T^{5} + 3 T^{4} + \cdots + 9461 \)
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| $67$ |
\( T^{5} - 20 T^{4} + \cdots - 23 \)
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| $71$ |
\( T^{5} - 25 T^{4} + \cdots + 17621 \)
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| $73$ |
\( T^{5} - T^{4} + \cdots - 28337 \)
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| $79$ |
\( T^{5} - 25 T^{4} + \cdots - 13751 \)
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| $83$ |
\( T^{5} + 37 T^{4} + \cdots + 8933 \)
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| $89$ |
\( T^{5} + 26 T^{4} + \cdots - 5543 \)
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| $97$ |
\( T^{5} + 25 T^{4} + \cdots + 661 \)
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