Newspace parameters
| Level: | \( N \) | \(=\) | \( 5225 = 5^{2} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5225.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(41.7218350561\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.131947641.1 |
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| Defining polynomial: |
\( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1045) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} + 7\nu - 3 ) / 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} - 3\nu^{2} + 7\nu + 9 ) / 3 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 10\nu^{3} - 22\nu + 3 ) / 3 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 6\beta_{3} + 7\beta_{2} + \beta _1 + 21 \)
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| \(\nu^{5}\) | \(=\) |
\( 7\beta_{4} + 10\beta_{2} + 28\beta _1 + 3 \)
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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 |
|
−2.56745 | −0.903918 | 4.59179 | 0 | 2.32076 | −2.16848 | −6.65430 | −2.18293 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.65636 | −3.32622 | 0.743534 | 0 | 5.50942 | −2.81120 | 2.08116 | 8.06374 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.759131 | 2.25829 | −1.42372 | 0 | −1.71434 | −4.95189 | 2.59905 | 2.09989 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.577704 | 0.746709 | −1.66626 | 0 | 0.431377 | 4.19297 | −2.11801 | −2.44243 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.04201 | 1.09835 | 2.16980 | 0 | 2.24284 | 0.469142 | 0.346728 | −1.79363 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.36323 | −2.87321 | 3.58485 | 0 | −6.79006 | 0.269449 | 3.74537 | 5.25535 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 5225.2.a.k | 6 | |
| 5.b | even | 2 | 1 | 1045.2.a.g | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 9405.2.a.w | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1045.2.a.g | ✓ | 6 | 5.b | even | 2 | 1 | |
| 5225.2.a.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 9405.2.a.w | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):
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\( T_{2}^{6} - 10T_{2}^{4} + 25T_{2}^{2} + 3T_{2} - 9 \)
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\( T_{7}^{6} + 5T_{7}^{5} - 15T_{7}^{4} - 90T_{7}^{3} - 55T_{7}^{2} + 81T_{7} - 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 10 T^{4} + \cdots - 9 \)
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| $3$ |
\( T^{6} + 3 T^{5} + \cdots - 16 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} + 5 T^{5} + \cdots - 16 \)
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| $11$ |
\( (T + 1)^{6} \)
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| $13$ |
\( T^{6} - 9 T^{5} + \cdots - 14 \)
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| $17$ |
\( T^{6} - 5 T^{5} + \cdots - 6290 \)
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| $19$ |
\( (T + 1)^{6} \)
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| $23$ |
\( T^{6} + 8 T^{5} + \cdots + 92 \)
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| $29$ |
\( T^{6} + 5 T^{5} + \cdots + 482 \)
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| $31$ |
\( T^{6} + T^{5} + \cdots - 3240 \)
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| $37$ |
\( T^{6} + 9 T^{5} + \cdots + 1906 \)
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| $41$ |
\( T^{6} - 25 T^{5} + \cdots + 9526 \)
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| $43$ |
\( T^{6} + 15 T^{5} + \cdots + 4460 \)
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| $47$ |
\( T^{6} + 24 T^{5} + \cdots - 7268 \)
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| $53$ |
\( T^{6} + 5 T^{5} + \cdots - 627158 \)
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| $59$ |
\( T^{6} - 39 T^{5} + \cdots + 35420 \)
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| $61$ |
\( T^{6} + 11 T^{5} + \cdots - 156150 \)
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| $67$ |
\( T^{6} + 24 T^{5} + \cdots - 123184 \)
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| $71$ |
\( T^{6} + 24 T^{5} + \cdots - 56840 \)
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| $73$ |
\( T^{6} - 26 T^{5} + \cdots + 34902 \)
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| $79$ |
\( T^{6} - 11 T^{5} + \cdots + 8152 \)
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| $83$ |
\( T^{6} + 39 T^{5} + \cdots + 2100044 \)
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| $89$ |
\( T^{6} - 22 T^{5} + \cdots + 554 \)
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| $97$ |
\( T^{6} + 22 T^{5} + \cdots + 506 \)
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