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: | \(0\) |
| 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: |
\( x^{4} - x^{3} - 28008x^{2} - 426124x + 63795376 \)
|
| 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
\( x^{4} - x^{3} - 28008x^{2} - 426124x + 63795376 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 27\nu^{2} - 22824\nu + 43214 ) / 30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - 23\nu - 14001 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{3} + 23\beta _1 + 14001 \)
|
| \(\nu^{3}\) | \(=\) |
\( 135\beta_{3} + 30\beta_{2} + 23445\beta _1 + 334813 \)
|
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 |
|
−138.687 | 729.000 | 11042.0 | −19288.1 | −101103. | −364424. | −395257. | 531441. | 2.67501e6 | ||||||||||||||||||||||||||||||
| 1.2 | −50.8281 | 729.000 | −5608.51 | 40325.0 | −37053.7 | 161391. | 701453. | 531441. | −2.04964e6 | |||||||||||||||||||||||||||||||
| 1.3 | 51.8107 | 729.000 | −5507.65 | −32573.1 | 37770.0 | 273357. | −709788. | 531441. | −1.68763e6 | |||||||||||||||||||||||||||||||
| 1.4 | 178.704 | 729.000 | 23743.2 | −26885.8 | 130275. | −28325.8 | 2.77906e6 | 531441. | −4.80460e6 | |||||||||||||||||||||||||||||||
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.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 99.14.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.14.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 99.14.a.b | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 41T_{2}^{3} - 27378T_{2}^{2} + 129736T_{2} + 65266816 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 41 T^{3} - 27378 T^{2} + \cdots + 65266816 \)
$3$
\( (T - 729)^{4} \)
$5$
\( T^{4} + 38422 T^{3} + \cdots - 68\!\cdots\!00 \)
$7$
\( T^{4} - 41998 T^{3} + \cdots + 45\!\cdots\!84 \)
$11$
\( (T + 1771561)^{4} \)
$13$
\( T^{4} - 50891230 T^{3} + \cdots - 13\!\cdots\!64 \)
$17$
\( T^{4} - 192739820 T^{3} + \cdots - 42\!\cdots\!68 \)
$19$
\( T^{4} + 183220188 T^{3} + \cdots + 41\!\cdots\!84 \)
$23$
\( T^{4} + 14728250 T^{3} + \cdots + 46\!\cdots\!76 \)
$29$
\( T^{4} + 548077356 T^{3} + \cdots + 23\!\cdots\!48 \)
$31$
\( T^{4} + 681968600 T^{3} + \cdots + 16\!\cdots\!00 \)
$37$
\( T^{4} + 5154946872 T^{3} + \cdots + 52\!\cdots\!44 \)
$41$
\( T^{4} - 51238857992 T^{3} + \cdots - 11\!\cdots\!72 \)
$43$
\( T^{4} - 74347734480 T^{3} + \cdots - 19\!\cdots\!68 \)
$47$
\( T^{4} - 274474225690 T^{3} + \cdots - 10\!\cdots\!56 \)
$53$
\( T^{4} - 213982590882 T^{3} + \cdots + 10\!\cdots\!52 \)
$59$
\( T^{4} - 1321318473964 T^{3} + \cdots - 21\!\cdots\!64 \)
$61$
\( T^{4} + 103045963950 T^{3} + \cdots - 46\!\cdots\!88 \)
$67$
\( T^{4} - 564819557912 T^{3} + \cdots + 24\!\cdots\!72 \)
$71$
\( T^{4} - 755239009806 T^{3} + \cdots + 67\!\cdots\!24 \)
$73$
\( T^{4} - 1553343364552 T^{3} + \cdots - 81\!\cdots\!64 \)
$79$
\( T^{4} + 409630665982 T^{3} + \cdots - 20\!\cdots\!56 \)
$83$
\( T^{4} + 2873448789168 T^{3} + \cdots + 18\!\cdots\!00 \)
$89$
\( T^{4} + 2345798585688 T^{3} + \cdots + 85\!\cdots\!44 \)
$97$
\( T^{4} + 325787932204 T^{3} + \cdots + 64\!\cdots\!84 \)
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