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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{177}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−16.1521 | 27.0000 | 132.889 | −269.779 | −436.106 | 941.418 | −78.9720 | 729.000 | 4357.48 | ||||||||||||||||||||||||
| 1.2 | −2.84793 | 27.0000 | −119.889 | 235.779 | −76.8942 | −1107.42 | 705.972 | 729.000 | −671.481 | |||||||||||||||||||||||||
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.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 99.8.a.d | 2 | ||
| 4.b | odd | 2 | 1 | 528.8.a.f | 2 | ||
| 11.b | odd | 2 | 1 | 363.8.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.8.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 99.8.a.d | 2 | 3.b | odd | 2 | 1 | ||
| 363.8.a.d | 2 | 11.b | odd | 2 | 1 | ||
| 528.8.a.f | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 19T_{2} + 46 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 19T + 46 \)
$3$
\( (T - 27)^{2} \)
$5$
\( T^{2} + 34T - 63608 \)
$7$
\( T^{2} + 166 T - 1042544 \)
$11$
\( (T + 1331)^{2} \)
$13$
\( T^{2} + 12670 T + 6020608 \)
$17$
\( T^{2} + 62344 T + 927796876 \)
$19$
\( T^{2} + 37980 T + 328498848 \)
$23$
\( T^{2} + 90686 T + 282938824 \)
$29$
\( T^{2} - 13992 T - 38836484052 \)
$31$
\( T^{2} - 245000 T + 1286906368 \)
$37$
\( T^{2} - 327852 T - 6524218716 \)
$41$
\( T^{2} + 275932 T - 322827909452 \)
$43$
\( T^{2} + 244104 T - 207765596544 \)
$47$
\( T^{2} - 536926 T - 176077767704 \)
$53$
\( T^{2} + 1821882 T + 397572445128 \)
$59$
\( T^{2} - 2502028 T - 24663435104 \)
$61$
\( T^{2} + 2191098 T - 604169699352 \)
$67$
\( T^{2} - 1674784 T - 8829893772944 \)
$71$
\( T^{2} + 368310 T - 24503996328 \)
$73$
\( T^{2} + 3336604 T - 5666837293628 \)
$79$
\( T^{2} + 1682618 T - 21513626700752 \)
$83$
\( T^{2} + 8376504 T + 8122054073616 \)
$89$
\( T^{2} - 9027204 T - 15754651515708 \)
$97$
\( T^{2} + 16703552 T + 69751828200124 \)
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