Newspace parameters
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.j (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.79289409601\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 95) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\(\beta_{1}\) | \(=\) |
\( 2\zeta_{12} \)
|
\(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
\(\beta_{3}\) | \(=\) |
\( 2\zeta_{12}^{3} \)
|
\(\zeta_{12}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
\(\zeta_{12}^{2}\) | \(=\) |
\( \beta_{2} \)
|
\(\zeta_{12}^{3}\) | \(=\) |
\( ( \beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).
\(n\) | \(77\) | \(401\) |
\(\chi(n)\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
0 | −1.73205 | − | 1.00000i | −1.00000 | − | 1.73205i | 0 | 0 | − | 4.00000i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||
49.2 | 0 | 1.73205 | + | 1.00000i | −1.00000 | − | 1.73205i | 0 | 0 | 4.00000i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||
349.1 | 0 | −1.73205 | + | 1.00000i | −1.00000 | + | 1.73205i | 0 | 0 | 4.00000i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||
349.2 | 0 | 1.73205 | − | 1.00000i | −1.00000 | + | 1.73205i | 0 | 0 | − | 4.00000i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
95.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 475.2.j.a | 4 | |
5.b | even | 2 | 1 | inner | 475.2.j.a | 4 | |
5.c | odd | 4 | 1 | 95.2.e.a | ✓ | 2 | |
5.c | odd | 4 | 1 | 475.2.e.b | 2 | ||
15.e | even | 4 | 1 | 855.2.k.b | 2 | ||
19.c | even | 3 | 1 | inner | 475.2.j.a | 4 | |
20.e | even | 4 | 1 | 1520.2.q.c | 2 | ||
95.i | even | 6 | 1 | inner | 475.2.j.a | 4 | |
95.l | even | 12 | 1 | 1805.2.a.b | 1 | ||
95.l | even | 12 | 1 | 9025.2.a.e | 1 | ||
95.m | odd | 12 | 1 | 95.2.e.a | ✓ | 2 | |
95.m | odd | 12 | 1 | 475.2.e.b | 2 | ||
95.m | odd | 12 | 1 | 1805.2.a.a | 1 | ||
95.m | odd | 12 | 1 | 9025.2.a.g | 1 | ||
285.v | even | 12 | 1 | 855.2.k.b | 2 | ||
380.v | even | 12 | 1 | 1520.2.q.c | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
95.2.e.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
95.2.e.a | ✓ | 2 | 95.m | odd | 12 | 1 | |
475.2.e.b | 2 | 5.c | odd | 4 | 1 | ||
475.2.e.b | 2 | 95.m | odd | 12 | 1 | ||
475.2.j.a | 4 | 1.a | even | 1 | 1 | trivial | |
475.2.j.a | 4 | 5.b | even | 2 | 1 | inner | |
475.2.j.a | 4 | 19.c | even | 3 | 1 | inner | |
475.2.j.a | 4 | 95.i | even | 6 | 1 | inner | |
855.2.k.b | 2 | 15.e | even | 4 | 1 | ||
855.2.k.b | 2 | 285.v | even | 12 | 1 | ||
1520.2.q.c | 2 | 20.e | even | 4 | 1 | ||
1520.2.q.c | 2 | 380.v | even | 12 | 1 | ||
1805.2.a.a | 1 | 95.m | odd | 12 | 1 | ||
1805.2.a.b | 1 | 95.l | even | 12 | 1 | ||
9025.2.a.e | 1 | 95.l | even | 12 | 1 | ||
9025.2.a.g | 1 | 95.m | odd | 12 | 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
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - 4T^{2} + 16 \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 16)^{2} \)
$11$
\( (T - 3)^{4} \)
$13$
\( T^{4} - 4T^{2} + 16 \)
$17$
\( T^{4} - 36T^{2} + 1296 \)
$19$
\( (T^{2} - 7 T + 19)^{2} \)
$23$
\( T^{4} \)
$29$
\( (T^{2} + 3 T + 9)^{2} \)
$31$
\( (T + 7)^{4} \)
$37$
\( (T^{2} + 64)^{2} \)
$41$
\( (T^{2} - 6 T + 36)^{2} \)
$43$
\( T^{4} - 16T^{2} + 256 \)
$47$
\( T^{4} - 36T^{2} + 1296 \)
$53$
\( T^{4} - 36T^{2} + 1296 \)
$59$
\( (T^{2} + 15 T + 225)^{2} \)
$61$
\( (T^{2} + 5 T + 25)^{2} \)
$67$
\( T^{4} - 4T^{2} + 16 \)
$71$
\( (T^{2} - 3 T + 9)^{2} \)
$73$
\( T^{4} - 64T^{2} + 4096 \)
$79$
\( (T^{2} - 5 T + 25)^{2} \)
$83$
\( (T^{2} + 144)^{2} \)
$89$
\( (T^{2} + 15 T + 225)^{2} \)
$97$
\( T^{4} - 64T^{2} + 4096 \)
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