Newspace parameters
| Level: | \( N \) | \(=\) | \( 119 = 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 119.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.02122729068\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.68557.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 6x^{2} + 5x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 6x^{2} + 5x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 5\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} + 4\beta_{2} + 7\beta _1 + 17 ) / 2 \)
|
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 |
|
−4.05760 | −6.73668 | 8.46411 | 0.0480909 | 27.3347 | 7.00000 | −1.88319 | 18.3828 | −0.195134 | ||||||||||||||||||||||||||||||
| 1.2 | −2.11753 | 3.14786 | −3.51605 | −1.42528 | −6.66570 | 7.00000 | 24.3856 | −17.0910 | 3.01809 | |||||||||||||||||||||||||||||||
| 1.3 | 1.67764 | −4.69642 | −5.18552 | 11.9352 | −7.87891 | 7.00000 | −22.1206 | −4.94368 | 20.0230 | |||||||||||||||||||||||||||||||
| 1.4 | 2.49749 | 1.28523 | −1.76254 | −19.5580 | 3.20986 | 7.00000 | −24.3819 | −25.3482 | −48.8460 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 119.4.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1071.4.a.h | 4 | ||
| 4.b | odd | 2 | 1 | 1904.4.a.k | 4 | ||
| 7.b | odd | 2 | 1 | 833.4.a.e | 4 | ||
| 17.b | even | 2 | 1 | 2023.4.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 119.4.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 833.4.a.e | 4 | 7.b | odd | 2 | 1 | ||
| 1071.4.a.h | 4 | 3.b | odd | 2 | 1 | ||
| 1904.4.a.k | 4 | 4.b | odd | 2 | 1 | ||
| 2023.4.a.f | 4 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} - 13T_{2}^{2} - 10T_{2} + 36 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(119))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 36 \)
|
| $3$ |
\( T^{4} + 7 T^{3} + \cdots + 128 \)
|
| $5$ |
\( T^{4} + 9 T^{3} + \cdots + 16 \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( T^{4} + 26 T^{3} + \cdots + 284048 \)
|
| $13$ |
\( T^{4} + 88 T^{3} + \cdots + 619904 \)
|
| $17$ |
\( (T + 17)^{4} \)
|
| $19$ |
\( T^{4} + 258 T^{3} + \cdots + 5985472 \)
|
| $23$ |
\( T^{4} - 14 T^{3} + \cdots + 3541376 \)
|
| $29$ |
\( T^{4} - 198 T^{3} + \cdots + 66432 \)
|
| $31$ |
\( T^{4} + 299 T^{3} + \cdots - 141944832 \)
|
| $37$ |
\( T^{4} + 360 T^{3} + \cdots - 442304768 \)
|
| $41$ |
\( T^{4} + \cdots - 1955297756 \)
|
| $43$ |
\( T^{4} - 121 T^{3} + \cdots - 269580256 \)
|
| $47$ |
\( T^{4} + 584 T^{3} + \cdots + 608891904 \)
|
| $53$ |
\( T^{4} - 317 T^{3} + \cdots - 5003748 \)
|
| $59$ |
\( T^{4} + \cdots + 1111403392 \)
|
| $61$ |
\( T^{4} + 1359 T^{3} + \cdots - 156781656 \)
|
| $67$ |
\( T^{4} + \cdots + 95078496768 \)
|
| $71$ |
\( T^{4} + \cdots - 4894576576 \)
|
| $73$ |
\( T^{4} + \cdots - 67996830348 \)
|
| $79$ |
\( T^{4} + \cdots + 314824499904 \)
|
| $83$ |
\( T^{4} + \cdots - 116777879424 \)
|
| $89$ |
\( T^{4} + \cdots + 31539023856 \)
|
| $97$ |
\( T^{4} + \cdots + 2011586283188 \)
|
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