Newspace parameters
| Level: | \( N \) | \(=\) | \( 1104 = 2^{4} \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1104.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(177.063737074\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 75x^{2} - 42x + 736 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 69) |
| 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
\( x^{4} - 75x^{2} - 42x + 736 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 58\nu + 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 38 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta _1 + 152 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{3} + 16\beta_{2} + 59\beta _1 + 136 ) / 4 \)
|
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 | 9.00000 | 0 | −86.6918 | 0 | 64.0051 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | −0.408582 | 0 | 4.34307 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | 42.3660 | 0 | −191.647 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.00000 | 0 | 66.7344 | 0 | 185.299 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 1104.6.a.o | 4 | |
| 4.b | odd | 2 | 1 | 69.6.a.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 207.6.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.6.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 207.6.a.e | 4 | 12.b | even | 2 | 1 | ||
| 1104.6.a.o | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 22T_{5}^{3} - 6640T_{5}^{2} + 242392T_{5} + 100144 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1104))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} - 22 T^{3} + \cdots + 100144 \)
|
| $7$ |
\( T^{4} - 62 T^{3} + \cdots - 9871616 \)
|
| $11$ |
\( T^{4} + \cdots - 45159083072 \)
|
| $13$ |
\( T^{4} + \cdots + 16813724400 \)
|
| $17$ |
\( T^{4} + \cdots - 95458629376 \)
|
| $19$ |
\( T^{4} + \cdots + 3908943190016 \)
|
| $23$ |
\( (T - 529)^{4} \)
|
| $29$ |
\( T^{4} + \cdots + 61820529282864 \)
|
| $31$ |
\( T^{4} + \cdots + 378047008189440 \)
|
| $37$ |
\( T^{4} + \cdots - 684323468629888 \)
|
| $41$ |
\( T^{4} + \cdots - 12\!\cdots\!64 \)
|
| $43$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots + 12\!\cdots\!92 \)
|
| $53$ |
\( T^{4} + \cdots + 40\!\cdots\!32 \)
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| $59$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
|
| $61$ |
\( T^{4} + \cdots + 60\!\cdots\!56 \)
|
| $67$ |
\( T^{4} + \cdots - 73\!\cdots\!28 \)
|
| $71$ |
\( T^{4} + \cdots + 12\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots - 14\!\cdots\!88 \)
|
| $79$ |
\( T^{4} + \cdots + 75\!\cdots\!80 \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!52 \)
|
| $89$ |
\( T^{4} + \cdots - 75\!\cdots\!28 \)
|
| $97$ |
\( T^{4} + \cdots + 20\!\cdots\!20 \)
|
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