Newspace parameters
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(71.3923111069\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.52525.1 |
|
|
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 24x^{2} + 25x + 125 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 110) |
| 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
\( x^{4} - 2x^{3} - 24x^{2} + 25x + 125 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 15 ) / 5 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 15\nu ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} + \beta _1 + 15 \)
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| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} + 5\beta_{2} + 16\beta _1 + 15 \)
|
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 |
|
2.00000 | −8.01861 | 4.00000 | 5.00000 | −16.0372 | 0.227345 | 8.00000 | 37.2981 | 10.0000 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −1.93972 | 4.00000 | 5.00000 | −3.87944 | 2.87207 | 8.00000 | −23.2375 | 10.0000 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0.546474 | 4.00000 | 5.00000 | 1.09295 | −11.6093 | 8.00000 | −26.7014 | 10.0000 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 3.41185 | 4.00000 | 5.00000 | 6.82371 | −16.4901 | 8.00000 | −15.3593 | 10.0000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 1210.4.a.ba | 4 | |
| 11.b | odd | 2 | 1 | 1210.4.a.y | 4 | ||
| 11.d | odd | 10 | 2 | 110.4.g.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.g.a | ✓ | 8 | 11.d | odd | 10 | 2 | |
| 1210.4.a.y | 4 | 11.b | odd | 2 | 1 | ||
| 1210.4.a.ba | 4 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{4} + 6T_{3}^{3} - 22T_{3}^{2} - 43T_{3} + 29 \)
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\( T_{7}^{4} + 25T_{7}^{3} + 105T_{7}^{2} - 575T_{7} + 125 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
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| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 29 \)
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| $5$ |
\( (T - 5)^{4} \)
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| $7$ |
\( T^{4} + 25 T^{3} + \cdots + 125 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} + 59 T^{3} + \cdots + 4543781 \)
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| $17$ |
\( T^{4} + 23 T^{3} + \cdots + 2770889 \)
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| $19$ |
\( T^{4} - 134 T^{3} + \cdots + 23589989 \)
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| $23$ |
\( T^{4} + 259 T^{3} + \cdots - 3351821 \)
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| $29$ |
\( T^{4} + 236 T^{3} + \cdots + 77783945 \)
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| $31$ |
\( T^{4} + 405 T^{3} + \cdots + 3900181 \)
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| $37$ |
\( T^{4} + 55 T^{3} + \cdots + 561149611 \)
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| $41$ |
\( T^{4} + \cdots + 7390534509 \)
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| $43$ |
\( T^{4} - 147 T^{3} + \cdots + 959787745 \)
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| $47$ |
\( T^{4} + \cdots + 5588412559 \)
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| $53$ |
\( T^{4} + \cdots + 7826405995 \)
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| $59$ |
\( T^{4} + \cdots - 4300744001 \)
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| $61$ |
\( T^{4} + \cdots - 39139679205 \)
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| $67$ |
\( T^{4} + \cdots - 58505616781 \)
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| $71$ |
\( T^{4} + \cdots - 101540482255 \)
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| $73$ |
\( T^{4} + \cdots + 47681297019 \)
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| $79$ |
\( T^{4} + \cdots - 54476878595 \)
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| $83$ |
\( T^{4} + 1373 T^{3} + \cdots + 209588391 \)
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| $89$ |
\( T^{4} + \cdots - 24048117271 \)
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| $97$ |
\( T^{4} + \cdots - 1324233523521 \)
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