Newspace parameters
| Level: | \( N \) | \(=\) | \( 476 = 2^{2} \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 476.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.80087913621\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{13}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). 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 |
|
0 | −2.30278 | 0 | −1.30278 | 0 | −1.00000 | 0 | 2.30278 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 1.30278 | 0 | 2.30278 | 0 | −1.00000 | 0 | −1.30278 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(17\) | \(1\) |
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 | 476.2.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 4284.2.a.l | 2 | ||
| 4.b | odd | 2 | 1 | 1904.2.a.k | 2 | ||
| 7.b | odd | 2 | 1 | 3332.2.a.k | 2 | ||
| 8.b | even | 2 | 1 | 7616.2.a.t | 2 | ||
| 8.d | odd | 2 | 1 | 7616.2.a.o | 2 | ||
| 17.b | even | 2 | 1 | 8092.2.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 476.2.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 1904.2.a.k | 2 | 4.b | odd | 2 | 1 | ||
| 3332.2.a.k | 2 | 7.b | odd | 2 | 1 | ||
| 4284.2.a.l | 2 | 3.b | odd | 2 | 1 | ||
| 7616.2.a.o | 2 | 8.d | odd | 2 | 1 | ||
| 7616.2.a.t | 2 | 8.b | even | 2 | 1 | ||
| 8092.2.a.l | 2 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + T_{3} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(476))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + T - 3 \)
$5$
\( T^{2} - T - 3 \)
$7$
\( (T + 1)^{2} \)
$11$
\( (T - 4)^{2} \)
$13$
\( T^{2} + 2T - 12 \)
$17$
\( (T + 1)^{2} \)
$19$
\( T^{2} - 10T + 12 \)
$23$
\( (T - 4)^{2} \)
$29$
\( T^{2} - 4T - 48 \)
$31$
\( T^{2} - 11T + 27 \)
$37$
\( T^{2} + 2T - 116 \)
$41$
\( T^{2} + 5T - 75 \)
$43$
\( T^{2} + 5T + 3 \)
$47$
\( T^{2} - 2T - 12 \)
$53$
\( T^{2} - 5T + 3 \)
$59$
\( (T + 8)^{2} \)
$61$
\( T^{2} - 13T - 39 \)
$67$
\( T^{2} - 9T + 17 \)
$71$
\( T^{2} + 14T + 36 \)
$73$
\( T^{2} + 3T - 79 \)
$79$
\( T^{2} - 2T - 12 \)
$83$
\( T^{2} + 8T - 36 \)
$89$
\( T^{2} + 6T - 108 \)
$97$
\( T^{2} + 23T + 129 \)
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