Newspace parameters
| Level: | \( N \) | \(=\) | \( 1216 = 2^{6} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1216.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(71.7463225670\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3221.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 9x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 152) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 9x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{2} + 3\nu + 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} - \beta _1 + 26 ) / 4 \)
|
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 | −4.73435 | 0 | 18.0754 | 0 | 0.213413 | 0 | −4.58596 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.573746 | 0 | −5.92862 | 0 | −31.1522 | 0 | −26.6708 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 10.3081 | 0 | −14.1468 | 0 | −4.06119 | 0 | 79.2568 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 1216.4.a.x | 3 | |
| 4.b | odd | 2 | 1 | 1216.4.a.q | 3 | ||
| 8.b | even | 2 | 1 | 152.4.a.b | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 304.4.a.j | 3 | ||
| 24.h | odd | 2 | 1 | 1368.4.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.4.a.b | ✓ | 3 | 8.b | even | 2 | 1 | |
| 304.4.a.j | 3 | 8.d | odd | 2 | 1 | ||
| 1216.4.a.q | 3 | 4.b | odd | 2 | 1 | ||
| 1216.4.a.x | 3 | 1.a | even | 1 | 1 | trivial | |
| 1368.4.a.e | 3 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):
|
\( T_{3}^{3} - 5T_{3}^{2} - 52T_{3} - 28 \)
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\( T_{5}^{3} + 2T_{5}^{2} - 279T_{5} - 1516 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} - 5 T^{2} + \cdots - 28 \)
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| $5$ |
\( T^{3} + 2 T^{2} + \cdots - 1516 \)
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| $7$ |
\( T^{3} + 35 T^{2} + \cdots - 27 \)
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| $11$ |
\( T^{3} - 28 T^{2} + \cdots - 358 \)
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| $13$ |
\( T^{3} - 109 T^{2} + \cdots + 113456 \)
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| $17$ |
\( T^{3} + 123 T^{2} + \cdots + 6637 \)
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| $19$ |
\( (T + 19)^{3} \)
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| $23$ |
\( T^{3} + 193 T^{2} + \cdots - 246848 \)
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| $29$ |
\( T^{3} - 297 T^{2} + \cdots + 1167636 \)
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| $31$ |
\( T^{3} + 140 T^{2} + \cdots - 3668096 \)
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| $37$ |
\( T^{3} + 38 T^{2} + \cdots + 3429384 \)
|
| $41$ |
\( T^{3} - 736 T^{2} + \cdots - 7266624 \)
|
| $43$ |
\( T^{3} - 514 T^{2} + \cdots + 25097948 \)
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| $47$ |
\( T^{3} - 134 T^{2} + \cdots + 58888776 \)
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| $53$ |
\( T^{3} + 311 T^{2} + \cdots + 13619792 \)
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| $59$ |
\( T^{3} + 199 T^{2} + \cdots - 117609444 \)
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| $61$ |
\( T^{3} + 56 T^{2} + \cdots - 120500582 \)
|
| $67$ |
\( T^{3} - 509 T^{2} + \cdots + 177822064 \)
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| $71$ |
\( T^{3} + 874 T^{2} + \cdots - 112230216 \)
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| $73$ |
\( T^{3} - 203 T^{2} + \cdots + 474103 \)
|
| $79$ |
\( T^{3} - 242 T^{2} + \cdots + 201599456 \)
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| $83$ |
\( T^{3} - 62 T^{2} + \cdots - 83648992 \)
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| $89$ |
\( T^{3} - 1764 T^{2} + \cdots - 127322496 \)
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| $97$ |
\( T^{3} + 2178 T^{2} + \cdots + 99903104 \)
|
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