Newspace parameters
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(35.3862065541\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 13257x^{2} - 346987x + 15531868 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| 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
\( x^{4} - x^{3} - 13257x^{2} - 346987x + 15531868 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 42\nu^{2} - 9375\nu + 10508 ) / 240 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 90\nu^{2} + 7311\nu - 328220 ) / 48 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} + 43\beta _1 + 6619 \)
|
| \(\nu^{3}\) | \(=\) |
\( 42\beta_{3} + 450\beta_{2} + 11181\beta _1 + 267490 \)
|
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 |
|
−83.2732 | 729.000 | −1257.57 | 1569.19 | −60706.2 | −149633. | 786896. | 531441. | −130671. | ||||||||||||||||||||||||||||||
| 1.2 | −68.7847 | 729.000 | −3460.66 | −69187.2 | −50144.0 | 284486. | 801525. | 531441. | 4.75902e6 | |||||||||||||||||||||||||||||||
| 1.3 | 20.8773 | 729.000 | −7756.14 | 22184.8 | 15219.6 | −59030.1 | −332955. | 531441. | 463160. | |||||||||||||||||||||||||||||||
| 1.4 | 120.181 | 729.000 | 6251.37 | −37998.8 | 87611.6 | −6170.97 | −233226. | 531441. | −4.56672e6 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 33.14.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 99.14.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.14.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 99.14.a.c | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 11T_{2}^{3} - 13212T_{2}^{2} - 426448T_{2} + 14371648 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 11 T^{3} - 13212 T^{2} + \cdots + 14371648 \)
$3$
\( (T - 729)^{4} \)
$5$
\( T^{4} + 83432 T^{3} + \cdots + 91\!\cdots\!00 \)
$7$
\( T^{4} - 69652 T^{3} + \cdots - 15\!\cdots\!88 \)
$11$
\( (T - 1771561)^{4} \)
$13$
\( T^{4} + 11391512 T^{3} + \cdots - 24\!\cdots\!24 \)
$17$
\( T^{4} + 166440956 T^{3} + \cdots + 12\!\cdots\!04 \)
$19$
\( T^{4} + 189590676 T^{3} + \cdots - 64\!\cdots\!00 \)
$23$
\( T^{4} + 505367152 T^{3} + \cdots - 82\!\cdots\!16 \)
$29$
\( T^{4} + 1129971084 T^{3} + \cdots - 12\!\cdots\!36 \)
$31$
\( T^{4} - 579322264 T^{3} + \cdots + 12\!\cdots\!32 \)
$37$
\( T^{4} + 25851986544 T^{3} + \cdots + 30\!\cdots\!36 \)
$41$
\( T^{4} + 44818054028 T^{3} + \cdots - 90\!\cdots\!16 \)
$43$
\( T^{4} + 90015055548 T^{3} + \cdots - 73\!\cdots\!00 \)
$47$
\( T^{4} + 99166057072 T^{3} + \cdots - 19\!\cdots\!64 \)
$53$
\( T^{4} + 135081202320 T^{3} + \cdots + 93\!\cdots\!24 \)
$59$
\( T^{4} + 397512944800 T^{3} + \cdots + 85\!\cdots\!16 \)
$61$
\( T^{4} + 354311482272 T^{3} + \cdots - 67\!\cdots\!04 \)
$67$
\( T^{4} + 467820853432 T^{3} + \cdots - 24\!\cdots\!48 \)
$71$
\( T^{4} + 1809393745344 T^{3} + \cdots + 85\!\cdots\!00 \)
$73$
\( T^{4} - 301866290176 T^{3} + \cdots + 41\!\cdots\!52 \)
$79$
\( T^{4} - 3418698303812 T^{3} + \cdots - 11\!\cdots\!72 \)
$83$
\( T^{4} - 2239472717328 T^{3} + \cdots + 27\!\cdots\!84 \)
$89$
\( T^{4} - 201191498616 T^{3} + \cdots + 55\!\cdots\!08 \)
$97$
\( T^{4} - 18754143066776 T^{3} + \cdots - 11\!\cdots\!96 \)
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