Newspace parameters
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.29266605383\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{313}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 78 \)
|
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{313})\). 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 |
|
−8.34590 | −9.00000 | 37.6541 | −107.459 | 75.1131 | −26.6918 | −47.1885 | 81.0000 | 896.843 | ||||||||||||||||||||||||
| 1.2 | 9.34590 | −9.00000 | 55.3459 | 69.4590 | −84.1131 | 8.69181 | 218.189 | 81.0000 | 649.157 | |||||||||||||||||||||||||
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 | 33.6.a.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 99.6.a.e | 2 | ||
| 4.b | odd | 2 | 1 | 528.6.a.q | 2 | ||
| 5.b | even | 2 | 1 | 825.6.a.d | 2 | ||
| 11.b | odd | 2 | 1 | 363.6.a.g | 2 | ||
| 33.d | even | 2 | 1 | 1089.6.a.o | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.6.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 99.6.a.e | 2 | 3.b | odd | 2 | 1 | ||
| 363.6.a.g | 2 | 11.b | odd | 2 | 1 | ||
| 528.6.a.q | 2 | 4.b | odd | 2 | 1 | ||
| 825.6.a.d | 2 | 5.b | even | 2 | 1 | ||
| 1089.6.a.o | 2 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 78 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - T - 78 \)
$3$
\( (T + 9)^{2} \)
$5$
\( T^{2} + 38T - 7464 \)
$7$
\( T^{2} + 18T - 232 \)
$11$
\( (T - 121)^{2} \)
$13$
\( T^{2} + 66T - 878128 \)
$17$
\( T^{2} + 920T + 210348 \)
$19$
\( T^{2} + 2932 T + 2147904 \)
$23$
\( T^{2} - 5246 T + 6827232 \)
$29$
\( T^{2} + 12600 T + 38326572 \)
$31$
\( T^{2} - 9936 T - 4245184 \)
$37$
\( T^{2} - 5996 T + 975204 \)
$41$
\( T^{2} - 24244 T + 105471636 \)
$43$
\( T^{2} - 20360 T - 16684800 \)
$47$
\( T^{2} + 5806 T - 38936064 \)
$53$
\( T^{2} - 40770 T + 387938808 \)
$59$
\( T^{2} - 18212 T - 974471136 \)
$61$
\( T^{2} + 11398 T - 557566776 \)
$67$
\( T^{2} - 65368 T + 1022090128 \)
$71$
\( T^{2} - 61446 T + 190949616 \)
$73$
\( T^{2} - 53412 T + 705197636 \)
$79$
\( T^{2} - 17122 T - 281721704 \)
$83$
\( T^{2} + 14304 T - 922809744 \)
$89$
\( T^{2} + 58140 T + 129501828 \)
$97$
\( T^{2} + 183056 T + 8284064476 \)
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