Newspace parameters
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(132.316651346\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.307532.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 76x + 168 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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
\( x^{3} - x^{2} - 76x + 168 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2\nu - 52 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2\beta _1 + 52 \)
|
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 |
|
−9.91848 | 9.00000 | 66.3762 | 0 | −89.2663 | −92.6461 | −340.959 | 81.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | −4.30119 | 9.00000 | −13.4998 | 0 | −38.7107 | 148.216 | 195.703 | 81.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 7.21967 | 9.00000 | 20.1236 | 0 | 64.9770 | −147.570 | −85.7438 | 81.0000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
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 | 825.6.a.f | 3 | |
| 5.b | even | 2 | 1 | 165.6.a.e | ✓ | 3 | |
| 15.d | odd | 2 | 1 | 495.6.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.6.a.e | ✓ | 3 | 5.b | even | 2 | 1 | |
| 495.6.a.a | 3 | 15.d | odd | 2 | 1 | ||
| 825.6.a.f | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 7T_{2}^{2} - 60T_{2} - 308 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} + 7 T^{2} - 60 T - 308 \)
$3$
\( (T - 9)^{3} \)
$5$
\( T^{3} \)
$7$
\( T^{3} + 92 T^{2} - 21932 T - 2026368 \)
$11$
\( (T - 121)^{3} \)
$13$
\( T^{3} - 90 T^{2} + \cdots - 210673232 \)
$17$
\( T^{3} + 1934 T^{2} + \cdots - 123909408 \)
$19$
\( T^{3} - 2084 T^{2} + \cdots + 14437440 \)
$23$
\( T^{3} + 1220 T^{2} + \cdots - 2404328128 \)
$29$
\( T^{3} - 4402 T^{2} + \cdots + 80705567064 \)
$31$
\( T^{3} + 10688 T^{2} + \cdots + 593236224 \)
$37$
\( T^{3} - 8190 T^{2} + \cdots + 165140256344 \)
$41$
\( T^{3} - 5974 T^{2} + \cdots + 127610311752 \)
$43$
\( T^{3} + 18868 T^{2} + \cdots - 5672527691040 \)
$47$
\( T^{3} + 55500 T^{2} + \cdots + 5576540180928 \)
$53$
\( T^{3} + 9206 T^{2} + \cdots + 850686588776 \)
$59$
\( T^{3} + 59196 T^{2} + \cdots + 7185357358784 \)
$61$
\( T^{3} - 79902 T^{2} + \cdots - 2993338614376 \)
$67$
\( T^{3} + 4468 T^{2} + \cdots + 6432661987328 \)
$71$
\( T^{3} + 75164 T^{2} + \cdots - 31171869026560 \)
$73$
\( T^{3} + \cdots + 152306824713328 \)
$79$
\( T^{3} + \cdots - 283704612543488 \)
$83$
\( T^{3} + \cdots - 200710881230832 \)
$89$
\( T^{3} + \cdots + 340522035911288 \)
$97$
\( T^{3} + 33486 T^{2} + \cdots + 93893760682568 \)
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