Newspace parameters
Level: | \( N \) | \(=\) | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1575.x (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.786027394897\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\zeta_{6})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{6}\) |
Projective field: | Galois closure of 6.2.283618125.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(451\) | \(1226\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{6}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
451.1 |
|
0 | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −0.500000 | + | 0.866025i | 0 | 0 | 0 | ||||||||||||||||||||
901.1 | 0 | 0 | 0.500000 | + | 0.866025i | 0 | 0 | −0.500000 | − | 0.866025i | 0 | 0 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1575.1.x.a | ✓ | 2 |
3.b | odd | 2 | 1 | CM | 1575.1.x.a | ✓ | 2 |
5.b | even | 2 | 1 | 1575.1.x.b | yes | 2 | |
5.c | odd | 4 | 2 | 1575.1.bj.a | 4 | ||
7.d | odd | 6 | 1 | inner | 1575.1.x.a | ✓ | 2 |
15.d | odd | 2 | 1 | 1575.1.x.b | yes | 2 | |
15.e | even | 4 | 2 | 1575.1.bj.a | 4 | ||
21.g | even | 6 | 1 | inner | 1575.1.x.a | ✓ | 2 |
35.i | odd | 6 | 1 | 1575.1.x.b | yes | 2 | |
35.k | even | 12 | 2 | 1575.1.bj.a | 4 | ||
105.p | even | 6 | 1 | 1575.1.x.b | yes | 2 | |
105.w | odd | 12 | 2 | 1575.1.bj.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1575.1.x.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
1575.1.x.a | ✓ | 2 | 3.b | odd | 2 | 1 | CM |
1575.1.x.a | ✓ | 2 | 7.d | odd | 6 | 1 | inner |
1575.1.x.a | ✓ | 2 | 21.g | even | 6 | 1 | inner |
1575.1.x.b | yes | 2 | 5.b | even | 2 | 1 | |
1575.1.x.b | yes | 2 | 15.d | odd | 2 | 1 | |
1575.1.x.b | yes | 2 | 35.i | odd | 6 | 1 | |
1575.1.x.b | yes | 2 | 105.p | even | 6 | 1 | |
1575.1.bj.a | 4 | 5.c | odd | 4 | 2 | ||
1575.1.bj.a | 4 | 15.e | even | 4 | 2 | ||
1575.1.bj.a | 4 | 35.k | even | 12 | 2 | ||
1575.1.bj.a | 4 | 105.w | odd | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{37}^{2} - T_{37} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} \)
$7$
\( T^{2} + T + 1 \)
$11$
\( T^{2} \)
$13$
\( T^{2} + 3 \)
$17$
\( T^{2} \)
$19$
\( T^{2} - 3T + 3 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} - T + 1 \)
$41$
\( T^{2} \)
$43$
\( (T - 2)^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 3T + 3 \)
$67$
\( T^{2} + T + 1 \)
$71$
\( T^{2} \)
$73$
\( T^{2} + 3T + 3 \)
$79$
\( T^{2} - T + 1 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} + 3 \)
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