Newspace parameters
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(48.7566812231\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.976277.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 267x + 1118 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 152) |
| 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
\( x^{3} - x^{2} - 267x + 1118 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + \nu - 181 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{2} + 30\nu - 369 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 30\beta _1 + 723 ) / 4 \)
|
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 | −14.2684 | 0 | −2.91197 | 0 | 21.1761 | 0 | −39.4116 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.13601 | 0 | 91.3591 | 0 | 209.718 | 0 | −233.165 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 24.4044 | 0 | −30.4472 | 0 | −33.8943 | 0 | 352.577 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 304.6.a.i | 3 | |
| 4.b | odd | 2 | 1 | 152.6.a.b | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.6.a.b | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 304.6.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 7T_{3}^{2} - 380T_{3} - 1092 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 7 T^{2} + \cdots - 1092 \)
|
| $5$ |
\( T^{3} - 58 T^{2} + \cdots - 8100 \)
|
| $7$ |
\( T^{3} - 197 T^{2} + \cdots + 150525 \)
|
| $11$ |
\( T^{3} + 476 T^{2} + \cdots - 91411762 \)
|
| $13$ |
\( T^{3} + 1417 T^{2} + \cdots - 74786288 \)
|
| $17$ |
\( T^{3} + 2427 T^{2} + \cdots - 527349875 \)
|
| $19$ |
\( (T + 361)^{3} \)
|
| $23$ |
\( T^{3} + \cdots + 15151368128 \)
|
| $29$ |
\( T^{3} + \cdots + 23375387724 \)
|
| $31$ |
\( T^{3} + \cdots + 17670050944 \)
|
| $37$ |
\( T^{3} + \cdots - 1044363272 \)
|
| $41$ |
\( T^{3} + \cdots - 1121026940608 \)
|
| $43$ |
\( T^{3} + \cdots + 94489455476 \)
|
| $47$ |
\( T^{3} + \cdots - 398506731064 \)
|
| $53$ |
\( T^{3} + \cdots + 13389533414544 \)
|
| $59$ |
\( T^{3} + \cdots - 9582166971468 \)
|
| $61$ |
\( T^{3} + \cdots - 3319491792002 \)
|
| $67$ |
\( T^{3} + \cdots - 24881604307408 \)
|
| $71$ |
\( T^{3} + \cdots + 76294453923000 \)
|
| $73$ |
\( T^{3} + \cdots + 10326027907407 \)
|
| $79$ |
\( T^{3} + \cdots - 146922104678560 \)
|
| $83$ |
\( T^{3} + \cdots - 106877538478240 \)
|
| $89$ |
\( T^{3} + \cdots + 305618004982656 \)
|
| $97$ |
\( T^{3} + \cdots + 156598394824064 \)
|
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