Newspace parameters
| Level: | \( N \) | \(=\) | \( 9409 = 97^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9409.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(75.1312432618\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{24})^+\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 97) |
| 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 \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
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 |
|
−1.73205 | 2.73205 | 1.00000 | −0.896575 | −4.73205 | −1.41421 | 1.73205 | 4.46410 | 1.55291 | ||||||||||||||||||||||||||||||
| 1.2 | −1.73205 | 2.73205 | 1.00000 | 0.896575 | −4.73205 | 1.41421 | 1.73205 | 4.46410 | −1.55291 | |||||||||||||||||||||||||||||||
| 1.3 | 1.73205 | −0.732051 | 1.00000 | −3.34607 | −1.26795 | −1.41421 | −1.73205 | −2.46410 | −5.79555 | |||||||||||||||||||||||||||||||
| 1.4 | 1.73205 | −0.732051 | 1.00000 | 3.34607 | −1.26795 | 1.41421 | −1.73205 | −2.46410 | 5.79555 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(97\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 97.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 9409.2.a.b | 4 | |
| 97.b | even | 2 | 1 | inner | 9409.2.a.b | 4 | |
| 97.i | even | 24 | 2 | 97.2.g.a | ✓ | 4 | |
| 291.r | odd | 24 | 2 | 873.2.be.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 97.2.g.a | ✓ | 4 | 97.i | even | 24 | 2 | |
| 873.2.be.c | 4 | 291.r | odd | 24 | 2 | ||
| 9409.2.a.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 9409.2.a.b | 4 | 97.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9409))\):
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\( T_{2}^{2} - 3 \)
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\( T_{5}^{4} - 12T_{5}^{2} + 9 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 3)^{2} \)
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| $3$ |
\( (T^{2} - 2 T - 2)^{2} \)
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| $5$ |
\( T^{4} - 12T^{2} + 9 \)
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| $7$ |
\( (T^{2} - 2)^{2} \)
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| $11$ |
\( (T^{2} - 12)^{2} \)
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| $13$ |
\( T^{4} - 4T^{2} + 1 \)
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| $17$ |
\( T^{4} - 36T^{2} + 81 \)
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| $19$ |
\( T^{4} - 28T^{2} + 4 \)
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| $23$ |
\( (T^{2} - 72)^{2} \)
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| $29$ |
\( T^{4} - 84T^{2} + 36 \)
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| $31$ |
\( (T^{2} - 2 T - 26)^{2} \)
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| $37$ |
\( T^{4} - 148T^{2} + 2209 \)
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| $41$ |
\( (T^{2} - 54)^{2} \)
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| $43$ |
\( (T^{2} + 6 T - 18)^{2} \)
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| $47$ |
\( (T^{2} + 12 T - 12)^{2} \)
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| $53$ |
\( (T + 3)^{4} \)
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| $59$ |
\( (T^{2} - 24)^{2} \)
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| $61$ |
\( (T^{2} - 3)^{2} \)
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| $67$ |
\( T^{4} - 16T^{2} + 16 \)
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| $71$ |
\( T^{4} - 228T^{2} + 6084 \)
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| $73$ |
\( (T^{2} + 8 T - 92)^{2} \)
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| $79$ |
\( (T - 6)^{4} \)
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| $83$ |
\( T^{4} - 228T^{2} + 6084 \)
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| $89$ |
\( (T^{2} + 6 T - 3)^{2} \)
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| $97$ |
\( T^{4} \)
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