Newspace parameters
| Level: | \( N \) | \(=\) | \( 9350 = 2 \cdot 5^{2} \cdot 11 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9350.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(74.6601258899\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.3484752.1 |
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| Defining polynomial: |
\( x^{5} - x^{4} - 8x^{3} + 4x^{2} + 12x - 6 \)
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| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - x^{4} - 8x^{3} + 4x^{2} + 12x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 4\nu + 5 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + 2\nu + 7 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 6\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 9\beta_{2} + 11\beta _1 + 15 \)
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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 | −1.88032 | 1.00000 | 0 | −1.88032 | −2.09827 | 1.00000 | 0.535597 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −0.270590 | 1.00000 | 0 | −0.270590 | −2.53051 | 1.00000 | −2.92678 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.496807 | 1.00000 | 0 | 0.496807 | 2.10504 | 1.00000 | −2.75318 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 2.54426 | 1.00000 | 0 | 2.54426 | 4.26913 | 1.00000 | 3.47328 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 3.10984 | 1.00000 | 0 | 3.10984 | −2.74539 | 1.00000 | 6.67109 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 9350.2.a.cw | yes | 5 |
| 5.b | even | 2 | 1 | 9350.2.a.cm | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9350.2.a.cm | ✓ | 5 | 5.b | even | 2 | 1 | |
| 9350.2.a.cw | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9350))\):
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\( T_{3}^{5} - 4T_{3}^{4} - 2T_{3}^{3} + 16T_{3}^{2} - 3T_{3} - 2 \)
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\( T_{7}^{5} + T_{7}^{4} - 20T_{7}^{3} - 34T_{7}^{2} + 69T_{7} + 131 \)
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\( T_{13}^{5} + 2T_{13}^{4} - 24T_{13}^{3} + 14T_{13}^{2} + 19T_{13} - 8 \)
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\( T_{19}^{5} + 8T_{19}^{4} - 12T_{19}^{3} - 160T_{19}^{2} - 200T_{19} + 112 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
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| $3$ |
\( T^{5} - 4 T^{4} + \cdots - 2 \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} + T^{4} + \cdots + 131 \)
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| $11$ |
\( (T - 1)^{5} \)
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| $13$ |
\( T^{5} + 2 T^{4} + \cdots - 8 \)
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| $17$ |
\( (T + 1)^{5} \)
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| $19$ |
\( T^{5} + 8 T^{4} + \cdots + 112 \)
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| $23$ |
\( T^{5} - 2 T^{4} + \cdots + 24 \)
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| $29$ |
\( T^{5} + 5 T^{4} + \cdots + 24 \)
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| $31$ |
\( T^{5} - 8 T^{4} + \cdots + 76 \)
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| $37$ |
\( T^{5} - 16 T^{4} + \cdots - 2608 \)
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| $41$ |
\( T^{5} - 9 T^{4} + \cdots + 162 \)
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| $43$ |
\( T^{5} - 8 T^{4} + \cdots - 12800 \)
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| $47$ |
\( T^{5} - T^{4} + \cdots - 12978 \)
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| $53$ |
\( T^{5} - 34 T^{4} + \cdots + 7218 \)
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| $59$ |
\( T^{5} + 7 T^{4} + \cdots + 1944 \)
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| $61$ |
\( T^{5} + 21 T^{4} + \cdots + 50 \)
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| $67$ |
\( T^{5} - 19 T^{4} + \cdots + 1728 \)
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| $71$ |
\( T^{5} - 24 T^{4} + \cdots + 3516 \)
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| $73$ |
\( T^{5} - 12 T^{4} + \cdots - 10708 \)
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| $79$ |
\( T^{5} - 21 T^{4} + \cdots - 1249 \)
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| $83$ |
\( T^{5} - 12 T^{4} + \cdots - 2268 \)
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| $89$ |
\( T^{5} - 9 T^{4} + \cdots - 6804 \)
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| $97$ |
\( T^{5} - 2 T^{4} + \cdots - 5144 \)
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