Newspace parameters
| Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 475.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.79289409601\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.11344.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 95) |
| 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
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \)
|
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 |
|
−1.95594 | −0.296842 | 1.82571 | 0 | 0.580605 | 3.56331 | 0.340899 | −2.91188 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.816594 | −1.53844 | −1.33317 | 0 | 1.25628 | −5.03316 | 2.72185 | −0.633188 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 2.14243 | 2.87834 | 2.59002 | 0 | 6.16666 | −3.10482 | 1.26409 | 5.28487 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.63010 | −3.04306 | 4.91744 | 0 | −8.00355 | 0.574672 | 7.67316 | 6.26020 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(19\) | \(-1\) |
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 | 475.2.a.i | 4 | |
| 3.b | odd | 2 | 1 | 4275.2.a.bo | 4 | ||
| 4.b | odd | 2 | 1 | 7600.2.a.cf | 4 | ||
| 5.b | even | 2 | 1 | 95.2.a.b | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 475.2.b.e | 8 | ||
| 15.d | odd | 2 | 1 | 855.2.a.m | 4 | ||
| 19.b | odd | 2 | 1 | 9025.2.a.bf | 4 | ||
| 20.d | odd | 2 | 1 | 1520.2.a.t | 4 | ||
| 35.c | odd | 2 | 1 | 4655.2.a.y | 4 | ||
| 40.e | odd | 2 | 1 | 6080.2.a.ch | 4 | ||
| 40.f | even | 2 | 1 | 6080.2.a.cc | 4 | ||
| 95.d | odd | 2 | 1 | 1805.2.a.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.2.a.b | ✓ | 4 | 5.b | even | 2 | 1 | |
| 475.2.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 475.2.b.e | 8 | 5.c | odd | 4 | 2 | ||
| 855.2.a.m | 4 | 15.d | odd | 2 | 1 | ||
| 1520.2.a.t | 4 | 20.d | odd | 2 | 1 | ||
| 1805.2.a.p | 4 | 95.d | odd | 2 | 1 | ||
| 4275.2.a.bo | 4 | 3.b | odd | 2 | 1 | ||
| 4655.2.a.y | 4 | 35.c | odd | 2 | 1 | ||
| 6080.2.a.cc | 4 | 40.f | even | 2 | 1 | ||
| 6080.2.a.ch | 4 | 40.e | odd | 2 | 1 | ||
| 7600.2.a.cf | 4 | 4.b | odd | 2 | 1 | ||
| 9025.2.a.bf | 4 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 8T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} - 6 T^{2} + 8 T + 9 \)
$3$
\( T^{4} + 2 T^{3} - 8 T^{2} - 16 T - 4 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 4 T^{3} - 16 T^{2} - 48 T + 32 \)
$11$
\( T^{4} - 4 T^{3} - 16 T^{2} + 32 T + 48 \)
$13$
\( T^{4} + 2 T^{3} - 24 T^{2} - 32 T + 20 \)
$17$
\( T^{4} + 4 T^{3} - 32 T^{2} - 16 T + 48 \)
$19$
\( (T - 1)^{4} \)
$23$
\( T^{4} - 8 T^{3} - 24 T^{2} + 176 T + 288 \)
$29$
\( T^{4} - 4 T^{3} - 32 T^{2} + 16 T + 48 \)
$31$
\( T^{4} - 4 T^{3} - 80 T^{2} + 512 T - 640 \)
$37$
\( T^{4} - 6 T^{3} - 24 T^{2} + 40 T + 4 \)
$41$
\( T^{4} - 16 T^{3} + 56 T^{2} + \cdots - 240 \)
$43$
\( T^{4} + 4 T^{3} - 16 T^{2} - 48 T + 32 \)
$47$
\( T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 1056 \)
$53$
\( T^{4} - 10 T^{3} + 184 T - 348 \)
$59$
\( T^{4} - 64 T^{2} - 224 T - 192 \)
$61$
\( T^{4} - 20 T^{3} + 56 T^{2} + \cdots - 2656 \)
$67$
\( T^{4} - 18 T^{3} + 8 T^{2} + \cdots - 1076 \)
$71$
\( T^{4} + 20 T^{3} + 32 T^{2} + \cdots - 4224 \)
$73$
\( T^{4} + 28 T^{3} + 256 T^{2} + \cdots + 176 \)
$79$
\( T^{4} + 16 T^{3} + 32 T^{2} + \cdots - 1856 \)
$83$
\( T^{4} - 72 T^{2} + 112 T + 480 \)
$89$
\( T^{4} - 4 T^{3} - 144 T^{2} + \cdots + 240 \)
$97$
\( T^{4} + 30 T^{3} + 224 T^{2} + \cdots - 1388 \)
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