Newspace parameters
| Level: | \( N \) | \(=\) | \( 46 = 2 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 46.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(14.3697111723\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.285765.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 121x + 46 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 121x + 46 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 5\nu - 81 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 5\nu - 81 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{2} + 5\beta _1 + 162 ) / 2 \)
|
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 |
|
8.00000 | −30.3959 | 64.0000 | 115.226 | −243.167 | −1058.60 | 512.000 | −1263.09 | 921.805 | |||||||||||||||||||||||||||
| 1.2 | 8.00000 | −20.0329 | 64.0000 | −288.133 | −160.263 | 1349.86 | 512.000 | −1785.68 | −2305.07 | ||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 38.4288 | 64.0000 | −397.092 | 307.430 | −1673.26 | 512.000 | −710.227 | −3176.74 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 46.8.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 414.8.a.d | 3 | ||
| 4.b | odd | 2 | 1 | 368.8.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 46.8.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 368.8.a.b | 3 | 4.b | odd | 2 | 1 | ||
| 414.8.a.d | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 12T_{3}^{2} - 1329T_{3} - 23400 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(46))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{3} \)
|
| $3$ |
\( T^{3} + 12 T^{2} + \cdots - 23400 \)
|
| $5$ |
\( T^{3} + 570 T^{2} + \cdots - 13183600 \)
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| $7$ |
\( T^{3} + \cdots - 2391036528 \)
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| $11$ |
\( T^{3} + \cdots - 77567772648 \)
|
| $13$ |
\( T^{3} + \cdots - 164166584566 \)
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| $17$ |
\( T^{3} + \cdots - 14898856912840 \)
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| $19$ |
\( T^{3} + \cdots - 556058270392 \)
|
| $23$ |
\( (T + 12167)^{3} \)
|
| $29$ |
\( T^{3} + \cdots - 33\!\cdots\!02 \)
|
| $31$ |
\( T^{3} + \cdots - 829548394660080 \)
|
| $37$ |
\( T^{3} + \cdots - 18\!\cdots\!12 \)
|
| $41$ |
\( T^{3} + \cdots - 19\!\cdots\!74 \)
|
| $43$ |
\( T^{3} + \cdots - 14\!\cdots\!60 \)
|
| $47$ |
\( T^{3} + \cdots + 23\!\cdots\!00 \)
|
| $53$ |
\( T^{3} + \cdots + 11\!\cdots\!72 \)
|
| $59$ |
\( T^{3} + \cdots + 19\!\cdots\!72 \)
|
| $61$ |
\( T^{3} + \cdots - 23\!\cdots\!88 \)
|
| $67$ |
\( T^{3} + \cdots + 84\!\cdots\!80 \)
|
| $71$ |
\( T^{3} + \cdots - 54\!\cdots\!20 \)
|
| $73$ |
\( T^{3} + \cdots + 16\!\cdots\!82 \)
|
| $79$ |
\( T^{3} + \cdots - 43\!\cdots\!12 \)
|
| $83$ |
\( T^{3} + \cdots - 77\!\cdots\!88 \)
|
| $89$ |
\( T^{3} + \cdots + 49\!\cdots\!96 \)
|
| $97$ |
\( T^{3} + \cdots + 59\!\cdots\!72 \)
|
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