Newspace parameters
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(10.3087058410\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.115512.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 70x - 194 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 70x - 194 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 6\nu - 45 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 6\nu - 92 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 47 \)
|
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 |
|
−6.51124 | −27.0000 | −85.6037 | −22.9029 | 175.804 | −1466.13 | 1390.83 | 729.000 | 149.126 | |||||||||||||||||||||||||||
| 1.2 | −2.40077 | −27.0000 | −122.236 | −505.769 | 64.8207 | 1401.07 | 600.759 | 729.000 | 1214.23 | ||||||||||||||||||||||||||||
| 1.3 | 17.9120 | −27.0000 | 192.840 | 84.6717 | −483.624 | 1679.06 | 1161.42 | 729.000 | 1516.64 | ||||||||||||||||||||||||||||
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 | 33.8.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 99.8.a.e | 3 | ||
| 4.b | odd | 2 | 1 | 528.8.a.o | 3 | ||
| 11.b | odd | 2 | 1 | 363.8.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.8.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 99.8.a.e | 3 | 3.b | odd | 2 | 1 | ||
| 363.8.a.e | 3 | 11.b | odd | 2 | 1 | ||
| 528.8.a.o | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 9T_{2}^{2} - 144T_{2} - 280 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 9 T^{2} - 144 T - 280 \)
$3$
\( (T + 27)^{3} \)
$5$
\( T^{3} + 444 T^{2} - 33180 T - 980800 \)
$7$
\( T^{3} - 1614 T^{2} + \cdots + 3449053112 \)
$11$
\( (T + 1331)^{3} \)
$13$
\( T^{3} - 20772 T^{2} + \cdots + 665759180384 \)
$17$
\( T^{3} + 14538 T^{2} + \cdots - 11730861043168 \)
$19$
\( T^{3} - 24492 T^{2} + \cdots - 5608943166816 \)
$23$
\( T^{3} + \cdots + 200462606267008 \)
$29$
\( T^{3} + 179862 T^{2} + \cdots + 61407931779072 \)
$31$
\( T^{3} - 288888 T^{2} + \cdots + 19\!\cdots\!96 \)
$37$
\( T^{3} - 107562 T^{2} + \cdots + 11\!\cdots\!64 \)
$41$
\( T^{3} + 135198 T^{2} + \cdots - 12\!\cdots\!52 \)
$43$
\( T^{3} - 193536 T^{2} + \cdots - 20\!\cdots\!12 \)
$47$
\( T^{3} + 591486 T^{2} + \cdots + 94\!\cdots\!80 \)
$53$
\( T^{3} - 79044 T^{2} + \cdots + 29\!\cdots\!44 \)
$59$
\( T^{3} - 2532768 T^{2} + \cdots + 38\!\cdots\!00 \)
$61$
\( T^{3} - 6678792 T^{2} + \cdots - 45\!\cdots\!60 \)
$67$
\( T^{3} - 7150356 T^{2} + \cdots - 13\!\cdots\!00 \)
$71$
\( T^{3} - 1390398 T^{2} + \cdots + 13\!\cdots\!84 \)
$73$
\( T^{3} + 6429114 T^{2} + \cdots + 26\!\cdots\!96 \)
$79$
\( T^{3} - 6873186 T^{2} + \cdots - 11\!\cdots\!12 \)
$83$
\( T^{3} - 6505596 T^{2} + \cdots + 81\!\cdots\!28 \)
$89$
\( T^{3} + 8842962 T^{2} + \cdots - 26\!\cdots\!48 \)
$97$
\( T^{3} + 1764774 T^{2} + \cdots + 24\!\cdots\!84 \)
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