Newspace parameters
| Level: | \( N \) | \(=\) | \( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4212.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(33.6329893314\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.113688.1 |
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| Defining polynomial: |
\( x^{4} - x^{3} - 14x^{2} + 3x + 39 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 14x^{2} + 3x + 39 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 8\nu + 7 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} + 6\nu - 21 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 7 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} + 10\beta _1 + 7 \)
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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 |
|
0 | 0 | 0 | −3.61097 | 0 | 4.38402 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −3.07250 | 0 | −0.971250 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 0.909921 | 0 | 2.39406 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 2.77356 | 0 | −4.80682 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 4212.2.a.f | ✓ | 4 |
| 3.b | odd | 2 | 1 | 4212.2.a.h | yes | 4 | |
| 9.c | even | 3 | 2 | 4212.2.i.x | 8 | ||
| 9.d | odd | 6 | 2 | 4212.2.i.v | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4212.2.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 4212.2.a.h | yes | 4 | 3.b | odd | 2 | 1 | |
| 4212.2.i.v | 8 | 9.d | odd | 6 | 2 | ||
| 4212.2.i.x | 8 | 9.c | even | 3 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4212))\):
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\( T_{5}^{4} + 3T_{5}^{3} - 11T_{5}^{2} - 24T_{5} + 28 \)
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\( T_{11}^{4} - 3T_{11}^{3} - 26T_{11}^{2} + 123T_{11} - 137 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} + 3 T^{3} + \cdots + 28 \)
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| $7$ |
\( T^{4} - T^{3} + \cdots + 49 \)
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| $11$ |
\( T^{4} - 3 T^{3} + \cdots - 137 \)
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| $13$ |
\( (T + 1)^{4} \)
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| $17$ |
\( T^{4} - 45 T^{2} + \cdots - 36 \)
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| $19$ |
\( T^{4} + 6 T^{3} + \cdots + 909 \)
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| $23$ |
\( T^{4} + 3 T^{3} + \cdots + 136 \)
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| $29$ |
\( T^{4} - 9 T^{3} + \cdots - 1313 \)
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| $31$ |
\( T^{4} - T^{3} - 87 T^{2} + \cdots + 4 \)
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| $37$ |
\( T^{4} - 15 T^{3} + \cdots - 576 \)
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| $41$ |
\( T^{4} + 3 T^{3} + \cdots + 504 \)
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| $43$ |
\( T^{4} - 10 T^{3} + \cdots - 1604 \)
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| $47$ |
\( T^{4} - 24 T^{3} + \cdots - 6308 \)
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| $53$ |
\( T^{4} + 12 T^{3} + \cdots - 36 \)
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| $59$ |
\( T^{4} - 6 T^{3} + \cdots + 1 \)
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| $61$ |
\( T^{4} - 13 T^{3} + \cdots - 1481 \)
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| $67$ |
\( T^{4} - T^{3} + \cdots + 28 \)
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| $71$ |
\( T^{4} + 30 T^{3} + \cdots + 2341 \)
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| $73$ |
\( T^{4} - 8 T^{3} + \cdots - 416 \)
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| $79$ |
\( T^{4} - 14 T^{3} + \cdots - 5036 \)
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| $83$ |
\( T^{4} - 24 T^{3} + \cdots - 513 \)
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| $89$ |
\( T^{4} + 3 T^{3} + \cdots + 1404 \)
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| $97$ |
\( T^{4} - 8 T^{3} + \cdots + 20992 \)
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