Newspace parameters
| Level: | \( N \) | \(=\) | \( 182 = 2 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 182.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(29.1898552060\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 479x - 1701 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{3} - x^{2} - 479x - 1701 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 5\nu - 318 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + 5\beta _1 + 318 \)
|
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 |
|
−4.00000 | −24.2736 | 16.0000 | −58.9463 | 97.0943 | −49.0000 | −64.0000 | 346.207 | 235.785 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | −8.68385 | 16.0000 | 86.3367 | 34.7354 | −49.0000 | −64.0000 | −167.591 | −345.347 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 18.9574 | 16.0000 | −54.3904 | −75.8297 | −49.0000 | −64.0000 | 116.384 | 217.562 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 182.6.a.e | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 182.6.a.e | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 14T_{3}^{2} - 414T_{3} - 3996 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(182))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( T^{3} + 14 T^{2} + \cdots - 3996 \)
|
| $5$ |
\( T^{3} + 27 T^{2} + \cdots - 276805 \)
|
| $7$ |
\( (T + 49)^{3} \)
|
| $11$ |
\( T^{3} - 676 T^{2} + \cdots + 1638480 \)
|
| $13$ |
\( (T + 169)^{3} \)
|
| $17$ |
\( T^{3} + \cdots - 1095211500 \)
|
| $19$ |
\( T^{3} - 1777 T^{2} + \cdots + 126287223 \)
|
| $23$ |
\( T^{3} + 895 T^{2} + \cdots + 169410981 \)
|
| $29$ |
\( T^{3} + \cdots - 4279391855 \)
|
| $31$ |
\( T^{3} - 4041 T^{2} + \cdots - 859829725 \)
|
| $37$ |
\( T^{3} + \cdots - 86146687452 \)
|
| $41$ |
\( T^{3} + \cdots - 906535806600 \)
|
| $43$ |
\( T^{3} + \cdots + 869128777081 \)
|
| $47$ |
\( T^{3} + \cdots - 818817693275 \)
|
| $53$ |
\( T^{3} + \cdots + 9866912725521 \)
|
| $59$ |
\( T^{3} + \cdots - 7080764774784 \)
|
| $61$ |
\( T^{3} + \cdots - 2615768987320 \)
|
| $67$ |
\( T^{3} + \cdots + 54100087377400 \)
|
| $71$ |
\( T^{3} + \cdots - 29265347945708 \)
|
| $73$ |
\( T^{3} + \cdots + 26301681566013 \)
|
| $79$ |
\( T^{3} + \cdots - 108705227153559 \)
|
| $83$ |
\( T^{3} + \cdots - 25439489896347 \)
|
| $89$ |
\( T^{3} + \cdots - 216305663057175 \)
|
| $97$ |
\( T^{3} + \cdots - 421624326241975 \)
|
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