Newspace parameters
Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 363.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(2.89856959337\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{3}, \sqrt{11})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 7x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
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 \( x^{4} - 7x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} - 5\nu ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{2} - 4 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + 4 \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{2} + 5\beta_1 \) |
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 |
|
−2.52434 | 1.00000 | 4.37228 | −2.37228 | −2.52434 | −0.792287 | −5.98844 | 1.00000 | 5.98844 | ||||||||||||||||||||||||||||||
1.2 | −0.792287 | 1.00000 | −1.37228 | 3.37228 | −0.792287 | −2.52434 | 2.67181 | 1.00000 | −2.67181 | |||||||||||||||||||||||||||||||
1.3 | 0.792287 | 1.00000 | −1.37228 | 3.37228 | 0.792287 | 2.52434 | −2.67181 | 1.00000 | 2.67181 | |||||||||||||||||||||||||||||||
1.4 | 2.52434 | 1.00000 | 4.37228 | −2.37228 | 2.52434 | 0.792287 | 5.98844 | 1.00000 | −5.98844 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(11\) | \(1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 363.2.a.j | ✓ | 4 |
3.b | odd | 2 | 1 | 1089.2.a.u | 4 | ||
4.b | odd | 2 | 1 | 5808.2.a.ck | 4 | ||
5.b | even | 2 | 1 | 9075.2.a.cv | 4 | ||
11.b | odd | 2 | 1 | inner | 363.2.a.j | ✓ | 4 |
11.c | even | 5 | 4 | 363.2.e.n | 16 | ||
11.d | odd | 10 | 4 | 363.2.e.n | 16 | ||
33.d | even | 2 | 1 | 1089.2.a.u | 4 | ||
44.c | even | 2 | 1 | 5808.2.a.ck | 4 | ||
55.d | odd | 2 | 1 | 9075.2.a.cv | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
363.2.a.j | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
363.2.a.j | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
363.2.e.n | 16 | 11.c | even | 5 | 4 | ||
363.2.e.n | 16 | 11.d | odd | 10 | 4 | ||
1089.2.a.u | 4 | 3.b | odd | 2 | 1 | ||
1089.2.a.u | 4 | 33.d | even | 2 | 1 | ||
5808.2.a.ck | 4 | 4.b | odd | 2 | 1 | ||
5808.2.a.ck | 4 | 44.c | even | 2 | 1 | ||
9075.2.a.cv | 4 | 5.b | even | 2 | 1 | ||
9075.2.a.cv | 4 | 55.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 7T_{2}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 7T^{2} + 4 \)
$3$
\( (T - 1)^{4} \)
$5$
\( (T^{2} - T - 8)^{2} \)
$7$
\( T^{4} - 7T^{2} + 4 \)
$11$
\( T^{4} \)
$13$
\( T^{4} - 51T^{2} + 576 \)
$17$
\( T^{4} - 43T^{2} + 256 \)
$19$
\( T^{4} - 19T^{2} + 16 \)
$23$
\( (T - 2)^{4} \)
$29$
\( T^{4} - 7T^{2} + 4 \)
$31$
\( (T^{2} - 9 T + 12)^{2} \)
$37$
\( (T - 5)^{4} \)
$41$
\( T^{4} - 151T^{2} + 3844 \)
$43$
\( (T^{2} - 44)^{2} \)
$47$
\( (T^{2} + 14 T + 16)^{2} \)
$53$
\( (T^{2} - 9 T - 54)^{2} \)
$59$
\( (T + 6)^{4} \)
$61$
\( T^{4} - 43T^{2} + 256 \)
$67$
\( (T^{2} - 15 T - 18)^{2} \)
$71$
\( (T^{2} + 10 T - 8)^{2} \)
$73$
\( T^{4} - 139T^{2} + 4624 \)
$79$
\( T^{4} - 51T^{2} + 576 \)
$83$
\( T^{4} - 76T^{2} + 256 \)
$89$
\( (T^{2} + 7 T + 4)^{2} \)
$97$
\( (T^{2} + 2 T - 131)^{2} \)
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