Newspace parameters
Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 495.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(3.95259490005\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.4.48704.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 2x^{3} - 6x^{2} + 4x + 6 \) |
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 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 \( x^{4} - 2x^{3} - 6x^{2} + 4x + 6 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 2\nu - 3 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} - 3\nu^{2} - 3\nu + 5 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 2\beta _1 + 3 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} + 3\beta_{2} + 9\beta _1 + 4 \) |
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 |
|
−2.69696 | 0 | 5.27358 | −1.00000 | 0 | −4.13176 | −8.82872 | 0 | 2.69696 | ||||||||||||||||||||||||||||||
1.2 | −1.85206 | 0 | 1.43013 | −1.00000 | 0 | 4.90749 | 1.05543 | 0 | 1.85206 | |||||||||||||||||||||||||||||||
1.3 | 0.262696 | 0 | −1.93099 | −1.00000 | 0 | 0.704647 | −1.03266 | 0 | −0.262696 | |||||||||||||||||||||||||||||||
1.4 | 2.28632 | 0 | 3.22727 | −1.00000 | 0 | 2.51962 | 2.80595 | 0 | −2.28632 | |||||||||||||||||||||||||||||||
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 | 495.2.a.f | ✓ | 4 |
3.b | odd | 2 | 1 | 495.2.a.g | yes | 4 | |
4.b | odd | 2 | 1 | 7920.2.a.cm | 4 | ||
5.b | even | 2 | 1 | 2475.2.a.bj | 4 | ||
5.c | odd | 4 | 2 | 2475.2.c.t | 8 | ||
11.b | odd | 2 | 1 | 5445.2.a.bs | 4 | ||
12.b | even | 2 | 1 | 7920.2.a.cn | 4 | ||
15.d | odd | 2 | 1 | 2475.2.a.bf | 4 | ||
15.e | even | 4 | 2 | 2475.2.c.s | 8 | ||
33.d | even | 2 | 1 | 5445.2.a.bh | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
495.2.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
495.2.a.g | yes | 4 | 3.b | odd | 2 | 1 | |
2475.2.a.bf | 4 | 15.d | odd | 2 | 1 | ||
2475.2.a.bj | 4 | 5.b | even | 2 | 1 | ||
2475.2.c.s | 8 | 15.e | even | 4 | 2 | ||
2475.2.c.t | 8 | 5.c | odd | 4 | 2 | ||
5445.2.a.bh | 4 | 33.d | even | 2 | 1 | ||
5445.2.a.bs | 4 | 11.b | odd | 2 | 1 | ||
7920.2.a.cm | 4 | 4.b | odd | 2 | 1 | ||
7920.2.a.cn | 4 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} - 6T_{2}^{2} - 10T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(495))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 2 T^{3} - 6 T^{2} - 10 T + 3 \)
$3$
\( T^{4} \)
$5$
\( (T + 1)^{4} \)
$7$
\( T^{4} - 4 T^{3} - 16 T^{2} + 64 T - 36 \)
$11$
\( (T - 1)^{4} \)
$13$
\( T^{4} - 8 T^{3} - 8 T^{2} + 168 T - 292 \)
$17$
\( T^{4} + 4 T^{3} - 40 T^{2} - 240 T - 324 \)
$19$
\( T^{4} - 4 T^{3} - 56 T^{2} + 192 T + 288 \)
$23$
\( T^{4} - 8 T^{3} - 32 T^{2} + 256 T - 192 \)
$29$
\( T^{4} - 4 T^{3} - 88 T^{2} + 240 T - 144 \)
$31$
\( T^{4} - 88 T^{2} + 192 T + 144 \)
$37$
\( T^{4} - 8 T^{3} - 56 T^{2} + 224 T + 976 \)
$41$
\( T^{4} - 4 T^{3} - 120 T^{2} + \cdots + 2160 \)
$43$
\( T^{4} - 12 T^{3} + 32 T^{2} - 36 \)
$47$
\( T^{4} - 128 T^{2} - 576 T - 576 \)
$53$
\( T^{4} + 16 T^{3} + 40 T^{2} + \cdots - 240 \)
$59$
\( T^{4} - 24 T^{3} + 88 T^{2} + \cdots - 8496 \)
$61$
\( T^{4} - 8 T^{3} - 104 T^{2} + \cdots - 2224 \)
$67$
\( T^{4} - 224 T^{2} - 576 T + 6208 \)
$71$
\( T^{4} - 16 T^{3} + 40 T^{2} + \cdots - 240 \)
$73$
\( T^{4} - 8 T^{3} - 8 T^{2} + 168 T - 292 \)
$79$
\( T^{4} - 12 T^{3} - 8 T^{2} + 192 T + 160 \)
$83$
\( T^{4} + 8 T^{3} - 72 T^{2} + \cdots + 1836 \)
$89$
\( T^{4} - 16 T^{3} - 24 T^{2} + \cdots + 720 \)
$97$
\( T^{4} - 8 T^{3} - 104 T^{2} + \cdots - 2864 \)
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