Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 38 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(26.0142114374\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 11777633936x^{2} - 35120319927360x + 11967042111800832000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 3^{10}\cdot 5\cdot 7 \) |
| 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
\( x^{4} - x^{3} - 11777633936x^{2} - 35120319927360x + 11967042111800832000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 27\nu^{3} - 128691\nu^{2} - 262552030440\nu + 46478406852080 ) / 680 \)
|
| \(\beta_{3}\) | \(=\) |
\( 36\nu^{2} - 161070\nu - 211997370590 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 26845\beta _1 + 211997397435 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 14299\beta_{3} + 2720\beta_{2} + 175418543615\beta _1 + 2845612193201705 ) / 108 \)
|
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 |
|
−520663. | 3.87420e8 | 1.33651e11 | −2.63441e12 | −2.01716e14 | 8.34393e15 | 1.97212e15 | 1.50095e17 | 1.37164e18 | ||||||||||||||||||||||||||||||
| 1.2 | −81067.1 | 3.87420e8 | −1.30867e11 | −7.16603e12 | −3.14071e13 | −5.96869e15 | 2.17508e16 | 1.50095e17 | 5.80929e17 | |||||||||||||||||||||||||||||||
| 1.3 | 322000. | 3.87420e8 | −3.37552e10 | 1.53382e13 | 1.24749e14 | 2.48687e15 | −5.51245e16 | 1.50095e17 | 4.93889e18 | |||||||||||||||||||||||||||||||
| 1.4 | 717293. | 3.87420e8 | 3.77070e11 | −9.63758e12 | 2.77894e14 | 1.74370e15 | 1.71886e17 | 1.50095e17 | −6.91297e18 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
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 | 3.38.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 9.38.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.38.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 9.38.a.c | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 437562T_{2}^{3} - 352197132768T_{2}^{2} + 95113282115764224T_{2} + 9748851418544266543104 \)
acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 97\!\cdots\!04 \)
|
| $3$ |
\( (T - 387420489)^{4} \)
|
| $5$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots - 21\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 11\!\cdots\!24 \)
|
| $13$ |
\( T^{4} + \cdots + 26\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots - 35\!\cdots\!24 \)
|
| $19$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 90\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 12\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 16\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 16\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 24\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 48\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 27\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots - 40\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots - 97\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 15\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 43\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 64\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 31\!\cdots\!36 \)
|
| show more | |
| show less |