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: | \(4\) |
| Coefficient field: | 4.4.449797.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 10x^{2} - x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 10x^{2} - x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 9\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 9\beta _1 + 1 \)
|
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 | −3.03941 | 1.00000 | 2.03941 | −3.03941 | −1.00000 | 1.00000 | 6.23799 | 2.03941 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −0.703441 | 1.00000 | −0.296559 | −0.703441 | −1.00000 | 1.00000 | −2.50517 | −0.296559 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.594177 | 1.00000 | −1.59418 | 0.594177 | −1.00000 | 1.00000 | −2.64695 | −1.59418 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 3.14867 | 1.00000 | −4.14867 | 3.14867 | −1.00000 | 1.00000 | 6.91413 | −4.14867 | |||||||||||||||||||||||||||||||
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.ba | ✓ | 4 |
| 23.b | odd | 2 | 1 | 7406.2.a.bd | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7406.2.a.ba | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 7406.2.a.bd | yes | 4 | 23.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):
|
\( T_{3}^{4} - 10T_{3}^{2} - T_{3} + 4 \)
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\( T_{5}^{4} + 4T_{5}^{3} - 4T_{5}^{2} - 15T_{5} - 4 \)
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\( T_{11}^{4} - 3T_{11}^{3} - 31T_{11}^{2} + 68T_{11} + 64 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( T^{4} - 10T^{2} - T + 4 \)
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| $5$ |
\( T^{4} + 4 T^{3} + \cdots - 4 \)
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| $7$ |
\( (T + 1)^{4} \)
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| $11$ |
\( T^{4} - 3 T^{3} + \cdots + 64 \)
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| $13$ |
\( T^{4} + 8 T^{3} + \cdots + 286 \)
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| $17$ |
\( T^{4} - 5 T^{3} + \cdots + 498 \)
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| $19$ |
\( T^{4} + 3 T^{3} + \cdots + 24 \)
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| $23$ |
\( T^{4} \)
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| $29$ |
\( T^{4} - 9 T^{3} + \cdots + 352 \)
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| $31$ |
\( T^{4} + 4 T^{3} + \cdots + 272 \)
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| $37$ |
\( T^{4} + 4 T^{3} + \cdots + 144 \)
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| $41$ |
\( T^{4} - 76 T^{2} + \cdots - 258 \)
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| $43$ |
\( T^{4} + 13 T^{3} + \cdots + 2076 \)
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| $47$ |
\( T^{4} + T^{3} + \cdots - 384 \)
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| $53$ |
\( T^{4} + 17 T^{3} + \cdots + 24 \)
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| $59$ |
\( T^{4} + 7 T^{3} + \cdots - 864 \)
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| $61$ |
\( T^{4} + 33 T^{3} + \cdots + 1912 \)
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| $67$ |
\( T^{4} + T^{3} + \cdots - 256 \)
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| $71$ |
\( T^{4} + 35 T^{3} + \cdots + 1576 \)
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| $73$ |
\( T^{4} + 10 T^{3} + \cdots - 389 \)
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| $79$ |
\( T^{4} + 12 T^{3} + \cdots - 3968 \)
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| $83$ |
\( T^{4} + 12 T^{3} + \cdots + 136 \)
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| $89$ |
\( T^{4} + 3 T^{3} + \cdots + 186 \)
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| $97$ |
\( T^{4} - 5 T^{3} + \cdots + 25478 \)
|
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