Newspace parameters
| Level: | \( N \) | \(=\) | \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9075.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(72.4642398343\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5725.1 |
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| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} + 6x + 11 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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
\( x^{4} - x^{3} - 8x^{2} + 6x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 5\nu + 1 ) / 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{2} + 5\beta _1 - 1 \)
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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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−2.55157 | 1.00000 | 4.51049 | 0 | −2.55157 | 4.19499 | −6.40567 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.86205 | 1.00000 | 1.46722 | 0 | −1.86205 | −2.63089 | 0.992053 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0.933531 | 1.00000 | −1.12852 | 0 | 0.933531 | 2.04108 | −2.92057 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.48008 | 1.00000 | 4.15081 | 0 | 2.48008 | 4.39482 | 5.33418 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 9075.2.a.cq | 4 | |
| 5.b | even | 2 | 1 | 1815.2.a.v | 4 | ||
| 11.b | odd | 2 | 1 | 9075.2.a.dg | 4 | ||
| 11.c | even | 5 | 2 | 825.2.n.i | 8 | ||
| 15.d | odd | 2 | 1 | 5445.2.a.bk | 4 | ||
| 55.d | odd | 2 | 1 | 1815.2.a.r | 4 | ||
| 55.j | even | 10 | 2 | 165.2.m.b | ✓ | 8 | |
| 55.k | odd | 20 | 4 | 825.2.bx.g | 16 | ||
| 165.d | even | 2 | 1 | 5445.2.a.br | 4 | ||
| 165.o | odd | 10 | 2 | 495.2.n.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.2.m.b | ✓ | 8 | 55.j | even | 10 | 2 | |
| 495.2.n.b | 8 | 165.o | odd | 10 | 2 | ||
| 825.2.n.i | 8 | 11.c | even | 5 | 2 | ||
| 825.2.bx.g | 16 | 55.k | odd | 20 | 4 | ||
| 1815.2.a.r | 4 | 55.d | odd | 2 | 1 | ||
| 1815.2.a.v | 4 | 5.b | even | 2 | 1 | ||
| 5445.2.a.bk | 4 | 15.d | odd | 2 | 1 | ||
| 5445.2.a.br | 4 | 165.d | even | 2 | 1 | ||
| 9075.2.a.cq | 4 | 1.a | even | 1 | 1 | trivial | |
| 9075.2.a.dg | 4 | 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):
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\( T_{2}^{4} + T_{2}^{3} - 8T_{2}^{2} - 6T_{2} + 11 \)
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\( T_{7}^{4} - 8T_{7}^{3} + 8T_{7}^{2} + 57T_{7} - 99 \)
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\( T_{13}^{4} - 15T_{13}^{3} + 76T_{13}^{2} - 150T_{13} + 99 \)
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\( T_{17}^{4} - 6T_{17}^{3} - 20T_{17}^{2} + 102T_{17} + 99 \)
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\( T_{19}^{4} + T_{19}^{3} - 10T_{19}^{2} - 12T_{19} + 9 \)
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\( T_{23}^{4} - T_{23}^{3} - 39T_{23}^{2} + 29T_{23} + 341 \)
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\( T_{37}^{4} - T_{37}^{3} - 83T_{37}^{2} + 21T_{37} + 1151 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} + \cdots + 11 \)
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| $3$ |
\( (T - 1)^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 8 T^{3} + \cdots - 99 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} - 15 T^{3} + \cdots + 99 \)
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| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 99 \)
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| $19$ |
\( T^{4} + T^{3} - 10 T^{2} + \cdots + 9 \)
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| $23$ |
\( T^{4} - T^{3} + \cdots + 341 \)
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| $29$ |
\( T^{4} - 17 T^{3} + \cdots - 619 \)
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| $31$ |
\( T^{4} - 15 T^{3} + \cdots - 25 \)
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| $37$ |
\( T^{4} - T^{3} + \cdots + 1151 \)
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| $41$ |
\( T^{4} + 12 T^{3} + \cdots + 11 \)
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| $43$ |
\( T^{4} - 14 T^{3} + \cdots + 9 \)
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| $47$ |
\( T^{4} + 14 T^{3} + \cdots - 3509 \)
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| $53$ |
\( T^{4} + 2 T^{3} + \cdots + 1711 \)
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| $59$ |
\( T^{4} + 11 T^{3} + \cdots - 99 \)
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| $61$ |
\( T^{4} - T^{3} + \cdots + 1111 \)
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| $67$ |
\( T^{4} + 5 T^{3} + \cdots + 1049 \)
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| $71$ |
\( T^{4} + 3 T^{3} + \cdots - 821 \)
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| $73$ |
\( T^{4} - 45 T^{3} + \cdots - 151 \)
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| $79$ |
\( T^{4} - 229 T^{2} + \cdots - 841 \)
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| $83$ |
\( T^{4} + 15 T^{3} + \cdots - 1975 \)
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| $89$ |
\( T^{4} - 2 T^{3} + \cdots + 99 \)
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| $97$ |
\( T^{4} - 26 T^{3} + \cdots + 891 \)
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