Newspace parameters
| Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 475.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.79289409601\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
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| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 19) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).
| \(n\) | \(77\) | \(401\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 324.1 |
|
0 | − | 2.00000i | 2.00000 | 0 | 0 | 1.00000i | 0 | −1.00000 | 0 | |||||||||||||||||||||||
| 324.2 | 0 | 2.00000i | 2.00000 | 0 | 0 | − | 1.00000i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 475.2.b.a | 2 | |
| 5.b | even | 2 | 1 | inner | 475.2.b.a | 2 | |
| 5.c | odd | 4 | 1 | 19.2.a.a | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 475.2.a.b | 1 | ||
| 15.e | even | 4 | 1 | 171.2.a.b | 1 | ||
| 15.e | even | 4 | 1 | 4275.2.a.i | 1 | ||
| 20.e | even | 4 | 1 | 304.2.a.f | 1 | ||
| 20.e | even | 4 | 1 | 7600.2.a.c | 1 | ||
| 35.f | even | 4 | 1 | 931.2.a.a | 1 | ||
| 35.k | even | 12 | 2 | 931.2.f.b | 2 | ||
| 35.l | odd | 12 | 2 | 931.2.f.c | 2 | ||
| 40.i | odd | 4 | 1 | 1216.2.a.o | 1 | ||
| 40.k | even | 4 | 1 | 1216.2.a.b | 1 | ||
| 55.e | even | 4 | 1 | 2299.2.a.b | 1 | ||
| 60.l | odd | 4 | 1 | 2736.2.a.c | 1 | ||
| 65.h | odd | 4 | 1 | 3211.2.a.a | 1 | ||
| 85.g | odd | 4 | 1 | 5491.2.a.b | 1 | ||
| 95.g | even | 4 | 1 | 361.2.a.b | 1 | ||
| 95.g | even | 4 | 1 | 9025.2.a.d | 1 | ||
| 95.l | even | 12 | 2 | 361.2.c.a | 2 | ||
| 95.m | odd | 12 | 2 | 361.2.c.c | 2 | ||
| 95.q | odd | 36 | 6 | 361.2.e.d | 6 | ||
| 95.r | even | 36 | 6 | 361.2.e.e | 6 | ||
| 105.k | odd | 4 | 1 | 8379.2.a.j | 1 | ||
| 285.j | odd | 4 | 1 | 3249.2.a.d | 1 | ||
| 380.j | odd | 4 | 1 | 5776.2.a.c | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.2.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 171.2.a.b | 1 | 15.e | even | 4 | 1 | ||
| 304.2.a.f | 1 | 20.e | even | 4 | 1 | ||
| 361.2.a.b | 1 | 95.g | even | 4 | 1 | ||
| 361.2.c.a | 2 | 95.l | even | 12 | 2 | ||
| 361.2.c.c | 2 | 95.m | odd | 12 | 2 | ||
| 361.2.e.d | 6 | 95.q | odd | 36 | 6 | ||
| 361.2.e.e | 6 | 95.r | even | 36 | 6 | ||
| 475.2.a.b | 1 | 5.c | odd | 4 | 1 | ||
| 475.2.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 475.2.b.a | 2 | 5.b | even | 2 | 1 | inner | |
| 931.2.a.a | 1 | 35.f | even | 4 | 1 | ||
| 931.2.f.b | 2 | 35.k | even | 12 | 2 | ||
| 931.2.f.c | 2 | 35.l | odd | 12 | 2 | ||
| 1216.2.a.b | 1 | 40.k | even | 4 | 1 | ||
| 1216.2.a.o | 1 | 40.i | odd | 4 | 1 | ||
| 2299.2.a.b | 1 | 55.e | even | 4 | 1 | ||
| 2736.2.a.c | 1 | 60.l | odd | 4 | 1 | ||
| 3211.2.a.a | 1 | 65.h | odd | 4 | 1 | ||
| 3249.2.a.d | 1 | 285.j | odd | 4 | 1 | ||
| 4275.2.a.i | 1 | 15.e | even | 4 | 1 | ||
| 5491.2.a.b | 1 | 85.g | odd | 4 | 1 | ||
| 5776.2.a.c | 1 | 380.j | odd | 4 | 1 | ||
| 7600.2.a.c | 1 | 20.e | even | 4 | 1 | ||
| 8379.2.a.j | 1 | 105.k | odd | 4 | 1 | ||
| 9025.2.a.d | 1 | 95.g | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 4 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 1 \)
|
| $11$ |
\( (T - 3)^{2} \)
|
| $13$ |
\( T^{2} + 16 \)
|
| $17$ |
\( T^{2} + 9 \)
|
| $19$ |
\( (T + 1)^{2} \)
|
| $23$ |
\( T^{2} \)
|
| $29$ |
\( (T + 6)^{2} \)
|
| $31$ |
\( (T + 4)^{2} \)
|
| $37$ |
\( T^{2} + 4 \)
|
| $41$ |
\( (T + 6)^{2} \)
|
| $43$ |
\( T^{2} + 1 \)
|
| $47$ |
\( T^{2} + 9 \)
|
| $53$ |
\( T^{2} + 144 \)
|
| $59$ |
\( (T - 6)^{2} \)
|
| $61$ |
\( (T + 1)^{2} \)
|
| $67$ |
\( T^{2} + 16 \)
|
| $71$ |
\( (T - 6)^{2} \)
|
| $73$ |
\( T^{2} + 49 \)
|
| $79$ |
\( (T + 8)^{2} \)
|
| $83$ |
\( T^{2} + 144 \)
|
| $89$ |
\( (T + 12)^{2} \)
|
| $97$ |
\( T^{2} + 64 \)
|
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