Newspace parameters
Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 465.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(3.71304369399\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.4.8468.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 2 \) |
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 \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{3} - 3\nu - 1 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - \nu - 3 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} - \nu^{2} - 4\nu + 2 \) |
\(\nu\) | \(=\) | \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + \beta_{2} + \beta _1 + 6 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -3\beta_{3} - 3\beta_{2} + 5\beta _1 + 2 ) / 2 \) |
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 |
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−2.46793 | −1.00000 | 4.09069 | 1.00000 | 2.46793 | 2.31451 | −5.15968 | 1.00000 | −2.46793 | ||||||||||||||||||||||||||||||
1.2 | 0.0872450 | −1.00000 | −1.99239 | 1.00000 | −0.0872450 | −3.46958 | −0.348316 | 1.00000 | 0.0872450 | |||||||||||||||||||||||||||||||
1.3 | 1.79888 | −1.00000 | 1.23597 | 1.00000 | −1.79888 | 4.20813 | −1.37440 | 1.00000 | 1.79888 | |||||||||||||||||||||||||||||||
1.4 | 2.58181 | −1.00000 | 4.66573 | 1.00000 | −2.58181 | 0.946946 | 6.88240 | 1.00000 | 2.58181 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(5\) | \(-1\) |
\(31\) | \(1\) |
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 | 465.2.a.h | ✓ | 4 |
3.b | odd | 2 | 1 | 1395.2.a.k | 4 | ||
4.b | odd | 2 | 1 | 7440.2.a.bz | 4 | ||
5.b | even | 2 | 1 | 2325.2.a.v | 4 | ||
5.c | odd | 4 | 2 | 2325.2.c.p | 8 | ||
15.d | odd | 2 | 1 | 6975.2.a.bo | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
465.2.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
1395.2.a.k | 4 | 3.b | odd | 2 | 1 | ||
2325.2.a.v | 4 | 5.b | even | 2 | 1 | ||
2325.2.c.p | 8 | 5.c | odd | 4 | 2 | ||
6975.2.a.bo | 4 | 15.d | odd | 2 | 1 | ||
7440.2.a.bz | 4 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 12T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(465))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} - 6 T^{2} + 12 T - 1 \)
$3$
\( (T + 1)^{4} \)
$5$
\( (T - 1)^{4} \)
$7$
\( T^{4} - 4 T^{3} - 10 T^{2} + 46 T - 32 \)
$11$
\( T^{4} - 6 T^{3} - 12 T^{2} + 56 T + 32 \)
$13$
\( T^{4} - 2 T^{3} - 50 T^{2} + 94 T + 4 \)
$17$
\( T^{4} + 10 T^{3} + 20 T^{2} + \cdots - 136 \)
$19$
\( (T - 4)^{4} \)
$23$
\( T^{4} - 6 T^{3} - 84 T^{2} + \cdots + 2048 \)
$29$
\( T^{4} - 12 T^{2} + 14 T - 4 \)
$31$
\( (T + 1)^{4} \)
$37$
\( T^{4} - 8 T^{3} - 26 T^{2} + 174 T + 388 \)
$41$
\( T^{4} + 6 T^{3} - 64 T^{2} - 480 T - 704 \)
$43$
\( T^{4} - 2 T^{3} - 92 T^{2} + 232 T + 928 \)
$47$
\( T^{4} + 2 T^{3} - 108 T^{2} + \cdots - 736 \)
$53$
\( T^{4} + 6 T^{3} - 8 T^{2} - 20 T - 8 \)
$59$
\( T^{4} - 4 T^{3} - 108 T^{2} + \cdots - 808 \)
$61$
\( T^{4} - 168 T^{2} + 608 T + 848 \)
$67$
\( T^{4} + 2 T^{3} - 66 T^{2} - 202 T + 184 \)
$71$
\( T^{4} - 22 T^{3} + 88 T^{2} + \cdots - 1808 \)
$73$
\( T^{4} + 32 T^{3} + 342 T^{2} + \cdots + 788 \)
$79$
\( T^{4} - 24 T^{3} - 68 T^{2} + \cdots - 24064 \)
$83$
\( T^{4} + 6 T^{3} - 44 T^{2} - 340 T - 464 \)
$89$
\( T^{4} + 14 T^{3} - 60 T^{2} - 482 T - 92 \)
$97$
\( T^{4} + 14 T^{3} + 48 T^{2} - 64 \)
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