Newspace parameters
| Level: | \( N \) | \(=\) | \( 418 = 2 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 418.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(24.6627983824\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2213.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 13x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{3} - x^{2} - 13x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu - 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4\nu - 8 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 4\beta _1 + 28 ) / 3 \)
|
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 |
|
2.00000 | −8.79844 | 4.00000 | −12.5464 | −17.5969 | 15.1433 | 8.00000 | 50.4126 | −25.0928 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −5.17601 | 4.00000 | 10.3340 | −10.3520 | −14.9820 | 8.00000 | −0.208920 | 20.6680 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 3.97445 | 4.00000 | −9.78756 | 7.94891 | −13.1614 | 8.00000 | −11.2037 | −19.5751 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 418.4.a.d | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 418.4.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 10T_{3}^{2} - 10T_{3} - 181 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(418))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} + 10 T^{2} + \cdots - 181 \)
|
| $5$ |
\( T^{3} + 12 T^{2} + \cdots - 1269 \)
|
| $7$ |
\( T^{3} + 13 T^{2} + \cdots - 2986 \)
|
| $11$ |
\( (T - 11)^{3} \)
|
| $13$ |
\( T^{3} + 55 T^{2} + \cdots - 4500 \)
|
| $17$ |
\( T^{3} + 32 T^{2} + \cdots - 28856 \)
|
| $19$ |
\( (T - 19)^{3} \)
|
| $23$ |
\( T^{3} + 381 T^{2} + \cdots + 1149404 \)
|
| $29$ |
\( T^{3} + 207 T^{2} + \cdots + 14780 \)
|
| $31$ |
\( T^{3} - 4 T^{2} + \cdots - 14225 \)
|
| $37$ |
\( T^{3} + 181 T^{2} + \cdots - 7531200 \)
|
| $41$ |
\( T^{3} + 671 T^{2} + \cdots - 21881228 \)
|
| $43$ |
\( T^{3} + 399 T^{2} + \cdots + 2012018 \)
|
| $47$ |
\( T^{3} + 80 T^{2} + \cdots + 4515808 \)
|
| $53$ |
\( T^{3} + 48 T^{2} + \cdots - 18022392 \)
|
| $59$ |
\( T^{3} - 119 T^{2} + \cdots - 389520 \)
|
| $61$ |
\( T^{3} - 500 T^{2} + \cdots - 559392 \)
|
| $67$ |
\( T^{3} + 1346 T^{2} + \cdots + 36432243 \)
|
| $71$ |
\( T^{3} - 148 T^{2} + \cdots + 76542927 \)
|
| $73$ |
\( T^{3} + 338 T^{2} + \cdots + 27907104 \)
|
| $79$ |
\( T^{3} + 316 T^{2} + \cdots + 59818104 \)
|
| $83$ |
\( T^{3} - 1121 T^{2} + \cdots - 17877690 \)
|
| $89$ |
\( T^{3} - 481 T^{2} + \cdots - 494983404 \)
|
| $97$ |
\( T^{3} + 2499 T^{2} + \cdots + 349694896 \)
|
| show more | |
| show less |