Newspace parameters
Level: | \( N \) | \(=\) | \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2340.gi (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.16781212956\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(S_{4}\) |
Projective field: | Galois closure of 4.0.17795700.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).
\(n\) | \(937\) | \(1081\) | \(1171\) | \(2081\) |
\(\chi(n)\) | \(-1\) | \(\zeta_{12}^{3}\) | \(1\) | \(\zeta_{12}^{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
229.1 |
|
0 | 0.866025 | + | 0.500000i | 0 | 0.866025 | + | 0.500000i | 0 | 0 | 0 | 0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||
889.1 | 0 | 0.866025 | − | 0.500000i | 0 | 0.866025 | − | 0.500000i | 0 | 0 | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||
1669.1 | 0 | −0.866025 | − | 0.500000i | 0 | −0.866025 | − | 0.500000i | 0 | 0 | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||
1789.1 | 0 | −0.866025 | + | 0.500000i | 0 | −0.866025 | + | 0.500000i | 0 | 0 | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
65.g | odd | 4 | 1 | inner |
585.db | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2340.1.gi.a | ✓ | 4 |
5.b | even | 2 | 1 | 2340.1.gi.b | yes | 4 | |
9.c | even | 3 | 1 | inner | 2340.1.gi.a | ✓ | 4 |
13.d | odd | 4 | 1 | 2340.1.gi.b | yes | 4 | |
45.j | even | 6 | 1 | 2340.1.gi.b | yes | 4 | |
65.g | odd | 4 | 1 | inner | 2340.1.gi.a | ✓ | 4 |
117.y | odd | 12 | 1 | 2340.1.gi.b | yes | 4 | |
585.db | odd | 12 | 1 | inner | 2340.1.gi.a | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2340.1.gi.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
2340.1.gi.a | ✓ | 4 | 9.c | even | 3 | 1 | inner |
2340.1.gi.a | ✓ | 4 | 65.g | odd | 4 | 1 | inner |
2340.1.gi.a | ✓ | 4 | 585.db | odd | 12 | 1 | inner |
2340.1.gi.b | yes | 4 | 5.b | even | 2 | 1 | |
2340.1.gi.b | yes | 4 | 13.d | odd | 4 | 1 | |
2340.1.gi.b | yes | 4 | 45.j | even | 6 | 1 | |
2340.1.gi.b | yes | 4 | 117.y | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{17} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2340, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - T^{2} + 1 \)
$5$
\( T^{4} - T^{2} + 1 \)
$7$
\( T^{4} \)
$11$
\( T^{4} \)
$13$
\( T^{4} - T^{2} + 1 \)
$17$
\( (T + 1)^{4} \)
$19$
\( (T^{2} - 2 T + 2)^{2} \)
$23$
\( (T^{2} - T + 1)^{2} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( T^{4} \)
$43$
\( (T^{2} + T + 1)^{2} \)
$47$
\( T^{4} \)
$53$
\( (T^{2} + 1)^{2} \)
$59$
\( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \)
$61$
\( (T^{2} + T + 1)^{2} \)
$67$
\( T^{4} \)
$71$
\( (T^{2} + 2 T + 2)^{2} \)
$73$
\( T^{4} \)
$79$
\( (T^{2} + T + 1)^{2} \)
$83$
\( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \)
$89$
\( (T^{2} + 2 T + 2)^{2} \)
$97$
\( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \)
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