Newspace parameters
Level: | \( N \) | \(=\) | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2205.ba (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.10043835286\) |
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 6.2.168781725.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).
\(n\) | \(442\) | \(1081\) | \(1226\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-\zeta_{6}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
344.1 |
|
0 | −1.00000 | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 0 | 0 | 0 | 1.00000 | 0 | ||||||||||||||||||||
1814.1 | 0 | −1.00000 | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 0 | 0 | 0 | 1.00000 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
35.c | odd | 2 | 1 | CM by \(\Q(\sqrt{-35}) \) |
45.h | odd | 6 | 1 | inner |
63.o | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2205.1.ba.a | ✓ | 2 |
5.b | even | 2 | 1 | 2205.1.ba.b | yes | 2 | |
7.b | odd | 2 | 1 | 2205.1.ba.b | yes | 2 | |
7.c | even | 3 | 1 | 2205.1.v.b | 2 | ||
7.c | even | 3 | 1 | 2205.1.br.b | 2 | ||
7.d | odd | 6 | 1 | 2205.1.v.a | 2 | ||
7.d | odd | 6 | 1 | 2205.1.br.a | 2 | ||
9.d | odd | 6 | 1 | 2205.1.ba.b | yes | 2 | |
35.c | odd | 2 | 1 | CM | 2205.1.ba.a | ✓ | 2 |
35.i | odd | 6 | 1 | 2205.1.v.b | 2 | ||
35.i | odd | 6 | 1 | 2205.1.br.b | 2 | ||
35.j | even | 6 | 1 | 2205.1.v.a | 2 | ||
35.j | even | 6 | 1 | 2205.1.br.a | 2 | ||
45.h | odd | 6 | 1 | inner | 2205.1.ba.a | ✓ | 2 |
63.i | even | 6 | 1 | 2205.1.v.b | 2 | ||
63.j | odd | 6 | 1 | 2205.1.v.a | 2 | ||
63.n | odd | 6 | 1 | 2205.1.br.a | 2 | ||
63.o | even | 6 | 1 | inner | 2205.1.ba.a | ✓ | 2 |
63.s | even | 6 | 1 | 2205.1.br.b | 2 | ||
315.u | even | 6 | 1 | 2205.1.br.a | 2 | ||
315.v | odd | 6 | 1 | 2205.1.br.b | 2 | ||
315.z | even | 6 | 1 | 2205.1.ba.b | yes | 2 | |
315.bq | even | 6 | 1 | 2205.1.v.a | 2 | ||
315.br | odd | 6 | 1 | 2205.1.v.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2205.1.v.a | 2 | 7.d | odd | 6 | 1 | ||
2205.1.v.a | 2 | 35.j | even | 6 | 1 | ||
2205.1.v.a | 2 | 63.j | odd | 6 | 1 | ||
2205.1.v.a | 2 | 315.bq | even | 6 | 1 | ||
2205.1.v.b | 2 | 7.c | even | 3 | 1 | ||
2205.1.v.b | 2 | 35.i | odd | 6 | 1 | ||
2205.1.v.b | 2 | 63.i | even | 6 | 1 | ||
2205.1.v.b | 2 | 315.br | odd | 6 | 1 | ||
2205.1.ba.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
2205.1.ba.a | ✓ | 2 | 35.c | odd | 2 | 1 | CM |
2205.1.ba.a | ✓ | 2 | 45.h | odd | 6 | 1 | inner |
2205.1.ba.a | ✓ | 2 | 63.o | even | 6 | 1 | inner |
2205.1.ba.b | yes | 2 | 5.b | even | 2 | 1 | |
2205.1.ba.b | yes | 2 | 7.b | odd | 2 | 1 | |
2205.1.ba.b | yes | 2 | 9.d | odd | 6 | 1 | |
2205.1.ba.b | yes | 2 | 315.z | even | 6 | 1 | |
2205.1.br.a | 2 | 7.d | odd | 6 | 1 | ||
2205.1.br.a | 2 | 35.j | even | 6 | 1 | ||
2205.1.br.a | 2 | 63.n | odd | 6 | 1 | ||
2205.1.br.a | 2 | 315.u | even | 6 | 1 | ||
2205.1.br.b | 2 | 7.c | even | 3 | 1 | ||
2205.1.br.b | 2 | 35.i | odd | 6 | 1 | ||
2205.1.br.b | 2 | 63.s | even | 6 | 1 | ||
2205.1.br.b | 2 | 315.v | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{2} + 3T_{13} + 3 \)
acting on \(S_{1}^{\mathrm{new}}(2205, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T + 1)^{2} \)
$5$
\( T^{2} - T + 1 \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 3T + 3 \)
$13$
\( T^{2} + 3T + 3 \)
$17$
\( (T + 1)^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} \)
$47$
\( T^{2} - T + 1 \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} \)
$67$
\( T^{2} \)
$71$
\( T^{2} + 3 \)
$73$
\( T^{2} + 3 \)
$79$
\( T^{2} + T + 1 \)
$83$
\( T^{2} - T + 1 \)
$89$
\( T^{2} \)
$97$
\( T^{2} - 3T + 3 \)
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