Newspace parameters
Level: | \( N \) | \(=\) | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3600.j (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.79663404548\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Projective image: | \(D_{6}\) |
Projective field: | Galois closure of 6.2.4320000.2 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).
\(n\) | \(577\) | \(901\) | \(2801\) | \(3151\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1999.1 |
|
0 | 0 | 0 | 0 | 0 | −1.73205 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
1999.2 | 0 | 0 | 0 | 0 | 0 | −1.73205 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
1999.3 | 0 | 0 | 0 | 0 | 0 | 1.73205 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
1999.4 | 0 | 0 | 0 | 0 | 0 | 1.73205 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
60.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3600.1.j.b | 4 | |
3.b | odd | 2 | 1 | CM | 3600.1.j.b | 4 | |
4.b | odd | 2 | 1 | inner | 3600.1.j.b | 4 | |
5.b | even | 2 | 1 | inner | 3600.1.j.b | 4 | |
5.c | odd | 4 | 1 | 3600.1.e.c | ✓ | 2 | |
5.c | odd | 4 | 1 | 3600.1.e.d | yes | 2 | |
12.b | even | 2 | 1 | inner | 3600.1.j.b | 4 | |
15.d | odd | 2 | 1 | inner | 3600.1.j.b | 4 | |
15.e | even | 4 | 1 | 3600.1.e.c | ✓ | 2 | |
15.e | even | 4 | 1 | 3600.1.e.d | yes | 2 | |
20.d | odd | 2 | 1 | inner | 3600.1.j.b | 4 | |
20.e | even | 4 | 1 | 3600.1.e.c | ✓ | 2 | |
20.e | even | 4 | 1 | 3600.1.e.d | yes | 2 | |
60.h | even | 2 | 1 | inner | 3600.1.j.b | 4 | |
60.l | odd | 4 | 1 | 3600.1.e.c | ✓ | 2 | |
60.l | odd | 4 | 1 | 3600.1.e.d | yes | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3600.1.e.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
3600.1.e.c | ✓ | 2 | 15.e | even | 4 | 1 | |
3600.1.e.c | ✓ | 2 | 20.e | even | 4 | 1 | |
3600.1.e.c | ✓ | 2 | 60.l | odd | 4 | 1 | |
3600.1.e.d | yes | 2 | 5.c | odd | 4 | 1 | |
3600.1.e.d | yes | 2 | 15.e | even | 4 | 1 | |
3600.1.e.d | yes | 2 | 20.e | even | 4 | 1 | |
3600.1.e.d | yes | 2 | 60.l | odd | 4 | 1 | |
3600.1.j.b | 4 | 1.a | even | 1 | 1 | trivial | |
3600.1.j.b | 4 | 3.b | odd | 2 | 1 | CM | |
3600.1.j.b | 4 | 4.b | odd | 2 | 1 | inner | |
3600.1.j.b | 4 | 5.b | even | 2 | 1 | inner | |
3600.1.j.b | 4 | 12.b | even | 2 | 1 | inner | |
3600.1.j.b | 4 | 15.d | odd | 2 | 1 | inner | |
3600.1.j.b | 4 | 20.d | odd | 2 | 1 | inner | |
3600.1.j.b | 4 | 60.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 3 \)
acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} - 3)^{2} \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + 1)^{2} \)
$17$
\( T^{4} \)
$19$
\( (T^{2} + 3)^{2} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( (T^{2} + 3)^{2} \)
$37$
\( (T^{2} + 4)^{2} \)
$41$
\( T^{4} \)
$43$
\( (T^{2} - 3)^{2} \)
$47$
\( T^{4} \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( (T + 1)^{4} \)
$67$
\( (T^{2} - 3)^{2} \)
$71$
\( T^{4} \)
$73$
\( (T^{2} + 4)^{2} \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( (T^{2} + 1)^{2} \)
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