Newspace parameters
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(53.8097419252\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4914253.1 |
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|
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| Defining polynomial: |
\( x^{4} - 48x^{2} - 25x + 342 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 456) |
| 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
\( x^{4} - 48x^{2} - 25x + 342 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - \nu^{2} - 73\nu - 18 ) / 9 \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 5\nu - 24 ) / 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{2} + 2\nu + 48 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} ) / 4 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{3} + 2\beta_{2} + 96 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 17\beta_{3} + 37\beta_{2} + 9\beta _1 + 42 ) / 2 \)
|
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 | −3.00000 | 0 | −20.6478 | 0 | −20.9397 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −7.97189 | 0 | 5.78057 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 4.81161 | 0 | 34.6916 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | 7.80807 | 0 | −5.53251 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(19\) | \( -1 \) |
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 | 912.4.a.r | 4 | |
| 4.b | odd | 2 | 1 | 456.4.a.e | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1368.4.a.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 456.4.a.e | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 912.4.a.r | 4 | 1.a | even | 1 | 1 | trivial | |
| 1368.4.a.j | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(912))\):
|
\( T_{5}^{4} + 16T_{5}^{3} - 159T_{5}^{2} - 1002T_{5} + 6184 \)
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\( T_{7}^{4} - 14T_{7}^{3} - 755T_{7}^{2} + 620T_{7} + 23232 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T + 3)^{4} \)
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| $5$ |
\( T^{4} + 16 T^{3} + \cdots + 6184 \)
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| $7$ |
\( T^{4} - 14 T^{3} + \cdots + 23232 \)
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| $11$ |
\( T^{4} - 38 T^{3} + \cdots + 1416000 \)
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| $13$ |
\( T^{4} + 62 T^{3} + \cdots - 3035680 \)
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| $17$ |
\( T^{4} + 150 T^{3} + \cdots - 92131124 \)
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| $19$ |
\( (T - 19)^{4} \)
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| $23$ |
\( T^{4} - 146 T^{3} + \cdots - 63221376 \)
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| $29$ |
\( T^{4} + 212 T^{3} + \cdots - 174135440 \)
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| $31$ |
\( T^{4} + \cdots - 1036065600 \)
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| $37$ |
\( T^{4} + 502 T^{3} + \cdots + 170720768 \)
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| $41$ |
\( T^{4} + 592 T^{3} + \cdots - 291598080 \)
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| $43$ |
\( T^{4} + \cdots + 1266194768 \)
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| $47$ |
\( T^{4} + \cdots - 23744617320 \)
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| $53$ |
\( T^{4} + \cdots + 15426933904 \)
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| $59$ |
\( T^{4} - 28 T^{3} + \cdots + 327376640 \)
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| $61$ |
\( T^{4} + \cdots - 22839663788 \)
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| $67$ |
\( T^{4} + \cdots - 283945462272 \)
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| $71$ |
\( T^{4} + \cdots + 8949399552 \)
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| $73$ |
\( T^{4} + \cdots + 12687615660 \)
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| $79$ |
\( T^{4} + \cdots + 107529574400 \)
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| $83$ |
\( T^{4} + \cdots - 804862025728 \)
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| $89$ |
\( T^{4} + \cdots + 394732712592 \)
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| $97$ |
\( T^{4} + \cdots - 90848070064 \)
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