Newspace parameters
| Level: | \( N \) | \(=\) | \( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9522.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(76.0335528047\) |
| Analytic rank: | \(0\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 46) |
| 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 |
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1.00000 | 0 | 1.00000 | −3.22871 | 0 | 1.19647 | 1.00000 | 0 | −3.22871 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −1.08816 | 0 | 0.863693 | 1.00000 | 0 | −1.08816 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | 0.372786 | 0 | −2.05954 | 1.00000 | 0 | 0.372786 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 0.546200 | 0 | 4.70760 | 1.00000 | 0 | 0.546200 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 1.39788 | 0 | 4.29177 | 1.00000 | 0 | 1.39788 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 9522.2.a.bw | 5 | |
| 3.b | odd | 2 | 1 | 1058.2.a.k | 5 | ||
| 12.b | even | 2 | 1 | 8464.2.a.bv | 5 | ||
| 23.b | odd | 2 | 1 | 9522.2.a.bz | 5 | ||
| 23.d | odd | 22 | 2 | 414.2.i.c | 10 | ||
| 69.c | even | 2 | 1 | 1058.2.a.j | 5 | ||
| 69.g | even | 22 | 2 | 46.2.c.b | ✓ | 10 | |
| 276.h | odd | 2 | 1 | 8464.2.a.bu | 5 | ||
| 276.j | odd | 22 | 2 | 368.2.m.a | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 46.2.c.b | ✓ | 10 | 69.g | even | 22 | 2 | |
| 368.2.m.a | 10 | 276.j | odd | 22 | 2 | ||
| 414.2.i.c | 10 | 23.d | odd | 22 | 2 | ||
| 1058.2.a.j | 5 | 69.c | even | 2 | 1 | ||
| 1058.2.a.k | 5 | 3.b | odd | 2 | 1 | ||
| 8464.2.a.bu | 5 | 276.h | odd | 2 | 1 | ||
| 8464.2.a.bv | 5 | 12.b | even | 2 | 1 | ||
| 9522.2.a.bw | 5 | 1.a | even | 1 | 1 | trivial | |
| 9522.2.a.bz | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9522))\):
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\( T_{5}^{5} + 2T_{5}^{4} - 5T_{5}^{3} - 2T_{5}^{2} + 4T_{5} - 1 \)
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\( T_{7}^{5} - 9T_{7}^{4} + 17T_{7}^{3} + 31T_{7}^{2} - 84T_{7} + 43 \)
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\( T_{11}^{5} + T_{11}^{4} - 15T_{11}^{3} + 19T_{11}^{2} - 8T_{11} + 1 \)
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\( T_{29}^{5} + 10T_{29}^{4} + 7T_{29}^{3} - 129T_{29}^{2} - 239T_{29} - 23 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} + 2 T^{4} + \cdots - 1 \)
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| $7$ |
\( T^{5} - 9 T^{4} + \cdots + 43 \)
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| $11$ |
\( T^{5} + T^{4} - 15 T^{3} + \cdots + 1 \)
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| $13$ |
\( T^{5} - T^{4} + \cdots - 661 \)
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| $17$ |
\( T^{5} - T^{4} - 37 T^{3} + \cdots - 1 \)
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| $19$ |
\( T^{5} - 10 T^{4} + \cdots - 373 \)
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| $23$ |
\( T^{5} \)
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| $29$ |
\( T^{5} + 10 T^{4} + \cdots - 23 \)
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| $31$ |
\( T^{5} - 66 T^{3} + \cdots + 1441 \)
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| $37$ |
\( T^{5} - 20 T^{4} + \cdots + 3827 \)
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| $41$ |
\( T^{5} - 3 T^{4} + \cdots - 1409 \)
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| $43$ |
\( T^{5} - 27 T^{4} + \cdots + 7303 \)
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| $47$ |
\( T^{5} - 11 T^{4} + \cdots - 3883 \)
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| $53$ |
\( T^{5} - 2 T^{4} + \cdots - 43 \)
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| $59$ |
\( T^{5} - 17 T^{4} + \cdots - 2881 \)
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| $61$ |
\( T^{5} - 7 T^{4} + \cdots + 23 \)
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| $67$ |
\( T^{5} - 28 T^{4} + \cdots + 397 \)
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| $71$ |
\( T^{5} - 66 T^{3} + \cdots + 1441 \)
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| $73$ |
\( T^{5} - 8 T^{4} + \cdots + 5897 \)
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| $79$ |
\( T^{5} - 16 T^{4} + \cdots - 10891 \)
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| $83$ |
\( T^{5} + 18 T^{4} + \cdots - 30383 \)
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| $89$ |
\( T^{5} - 14 T^{4} + \cdots - 617 \)
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| $97$ |
\( T^{5} - 47 T^{4} + \cdots - 28073 \)
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