Newspace parameters
| Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 475.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(28.0259072527\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3144.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 16x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 19) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 16x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 10 \)
|
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 |
|
−5.07177 | 8.66998 | 17.7229 | 0 | −43.9722 | 15.1058 | −49.3121 | 48.1686 | 0 | |||||||||||||||||||||||||||
| 1.2 | −1.89080 | −2.95388 | −4.42486 | 0 | 5.58521 | −5.94196 | 23.4930 | −18.2746 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 3.96257 | −6.71610 | 7.70200 | 0 | −26.6130 | 25.8362 | −1.18085 | 18.1060 | 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 | 475.4.a.f | 3 | |
| 5.b | even | 2 | 1 | 19.4.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 2 | 475.4.b.f | 6 | ||
| 15.d | odd | 2 | 1 | 171.4.a.f | 3 | ||
| 20.d | odd | 2 | 1 | 304.4.a.i | 3 | ||
| 35.c | odd | 2 | 1 | 931.4.a.c | 3 | ||
| 40.e | odd | 2 | 1 | 1216.4.a.u | 3 | ||
| 40.f | even | 2 | 1 | 1216.4.a.s | 3 | ||
| 55.d | odd | 2 | 1 | 2299.4.a.h | 3 | ||
| 95.d | odd | 2 | 1 | 361.4.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.4.a.b | ✓ | 3 | 5.b | even | 2 | 1 | |
| 171.4.a.f | 3 | 15.d | odd | 2 | 1 | ||
| 304.4.a.i | 3 | 20.d | odd | 2 | 1 | ||
| 361.4.a.i | 3 | 95.d | odd | 2 | 1 | ||
| 475.4.a.f | 3 | 1.a | even | 1 | 1 | trivial | |
| 475.4.b.f | 6 | 5.c | odd | 4 | 2 | ||
| 931.4.a.c | 3 | 35.c | odd | 2 | 1 | ||
| 1216.4.a.s | 3 | 40.f | even | 2 | 1 | ||
| 1216.4.a.u | 3 | 40.e | odd | 2 | 1 | ||
| 2299.4.a.h | 3 | 55.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(475))\):
|
\( T_{2}^{3} + 3T_{2}^{2} - 18T_{2} - 38 \)
|
|
\( T_{3}^{3} + T_{3}^{2} - 64T_{3} - 172 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 3 T^{2} + \cdots - 38 \)
|
| $3$ |
\( T^{3} + T^{2} + \cdots - 172 \)
|
| $5$ |
\( T^{3} \)
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| $7$ |
\( T^{3} - 35 T^{2} + \cdots + 2319 \)
|
| $11$ |
\( T^{3} - 16 T^{2} + \cdots + 1182 \)
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| $13$ |
\( T^{3} + 65 T^{2} + \cdots - 4848 \)
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| $17$ |
\( T^{3} + 29 T^{2} + \cdots + 218619 \)
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| $19$ |
\( (T + 19)^{3} \)
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| $23$ |
\( T^{3} - 101 T^{2} + \cdots + 378176 \)
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| $29$ |
\( T^{3} - 377 T^{2} + \cdots + 4544396 \)
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| $31$ |
\( T^{3} + 140 T^{2} + \cdots - 2444352 \)
|
| $37$ |
\( T^{3} - 290 T^{2} + \cdots + 10001448 \)
|
| $41$ |
\( T^{3} - 956 T^{2} + \cdots - 31578144 \)
|
| $43$ |
\( T^{3} - 570 T^{2} + \cdots + 65963504 \)
|
| $47$ |
\( T^{3} + 66 T^{2} + \cdots - 2940624 \)
|
| $53$ |
\( T^{3} + 817 T^{2} + \cdots + 16824816 \)
|
| $59$ |
\( T^{3} - 265 T^{2} + \cdots + 31557612 \)
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| $61$ |
\( T^{3} - 988 T^{2} + \cdots + 76875874 \)
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| $67$ |
\( T^{3} - 207 T^{2} + \cdots + 7515248 \)
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| $71$ |
\( T^{3} - 846 T^{2} + \cdots + 1727928 \)
|
| $73$ |
\( T^{3} + 627 T^{2} + \cdots - 145581839 \)
|
| $79$ |
\( T^{3} - 382 T^{2} + \cdots - 56023488 \)
|
| $83$ |
\( T^{3} - 766 T^{2} + \cdots + 78728352 \)
|
| $89$ |
\( T^{3} + 172 T^{2} + \cdots - 76923456 \)
|
| $97$ |
\( T^{3} - 2450 T^{2} + \cdots - 196438912 \)
|
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