Newspace parameters
Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 405.h (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.202121330116\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\zeta_{6})\) |
Defining polynomial: |
\( x^{2} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 135) |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.135.1 |
Artin image: | $C_6\times S_3$ |
Artin field: | Galois closure of 12.0.6053445140625.2 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
\(n\) | \(82\) | \(326\) |
\(\chi(n)\) | \(-1\) | \(-\zeta_{6}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
134.1 |
|
−0.500000 | − | 0.866025i | 0 | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −1.00000 | 0 | −1.00000 | ||||||||||||||||||||
269.1 | −0.500000 | + | 0.866025i | 0 | 0 | 0.500000 | + | 0.866025i | 0 | 0 | −1.00000 | 0 | −1.00000 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
15.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-15}) \) |
9.c | even | 3 | 1 | inner |
45.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 405.1.h.a | 2 | |
3.b | odd | 2 | 1 | 405.1.h.b | 2 | ||
5.b | even | 2 | 1 | 405.1.h.b | 2 | ||
5.c | odd | 4 | 2 | 2025.1.j.c | 4 | ||
9.c | even | 3 | 1 | 135.1.d.b | yes | 1 | |
9.c | even | 3 | 1 | inner | 405.1.h.a | 2 | |
9.d | odd | 6 | 1 | 135.1.d.a | ✓ | 1 | |
9.d | odd | 6 | 1 | 405.1.h.b | 2 | ||
15.d | odd | 2 | 1 | CM | 405.1.h.a | 2 | |
15.e | even | 4 | 2 | 2025.1.j.c | 4 | ||
27.e | even | 9 | 6 | 3645.1.n.e | 6 | ||
27.f | odd | 18 | 6 | 3645.1.n.d | 6 | ||
36.f | odd | 6 | 1 | 2160.1.c.a | 1 | ||
36.h | even | 6 | 1 | 2160.1.c.b | 1 | ||
45.h | odd | 6 | 1 | 135.1.d.b | yes | 1 | |
45.h | odd | 6 | 1 | inner | 405.1.h.a | 2 | |
45.j | even | 6 | 1 | 135.1.d.a | ✓ | 1 | |
45.j | even | 6 | 1 | 405.1.h.b | 2 | ||
45.k | odd | 12 | 2 | 675.1.c.c | 2 | ||
45.k | odd | 12 | 2 | 2025.1.j.c | 4 | ||
45.l | even | 12 | 2 | 675.1.c.c | 2 | ||
45.l | even | 12 | 2 | 2025.1.j.c | 4 | ||
135.n | odd | 18 | 6 | 3645.1.n.e | 6 | ||
135.p | even | 18 | 6 | 3645.1.n.d | 6 | ||
180.n | even | 6 | 1 | 2160.1.c.a | 1 | ||
180.p | odd | 6 | 1 | 2160.1.c.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
135.1.d.a | ✓ | 1 | 9.d | odd | 6 | 1 | |
135.1.d.a | ✓ | 1 | 45.j | even | 6 | 1 | |
135.1.d.b | yes | 1 | 9.c | even | 3 | 1 | |
135.1.d.b | yes | 1 | 45.h | odd | 6 | 1 | |
405.1.h.a | 2 | 1.a | even | 1 | 1 | trivial | |
405.1.h.a | 2 | 9.c | even | 3 | 1 | inner | |
405.1.h.a | 2 | 15.d | odd | 2 | 1 | CM | |
405.1.h.a | 2 | 45.h | odd | 6 | 1 | inner | |
405.1.h.b | 2 | 3.b | odd | 2 | 1 | ||
405.1.h.b | 2 | 5.b | even | 2 | 1 | ||
405.1.h.b | 2 | 9.d | odd | 6 | 1 | ||
405.1.h.b | 2 | 45.j | even | 6 | 1 | ||
675.1.c.c | 2 | 45.k | odd | 12 | 2 | ||
675.1.c.c | 2 | 45.l | even | 12 | 2 | ||
2025.1.j.c | 4 | 5.c | odd | 4 | 2 | ||
2025.1.j.c | 4 | 15.e | even | 4 | 2 | ||
2025.1.j.c | 4 | 45.k | odd | 12 | 2 | ||
2025.1.j.c | 4 | 45.l | even | 12 | 2 | ||
2160.1.c.a | 1 | 36.f | odd | 6 | 1 | ||
2160.1.c.a | 1 | 180.n | even | 6 | 1 | ||
2160.1.c.b | 1 | 36.h | even | 6 | 1 | ||
2160.1.c.b | 1 | 180.p | odd | 6 | 1 | ||
3645.1.n.d | 6 | 27.f | odd | 18 | 6 | ||
3645.1.n.d | 6 | 135.p | even | 18 | 6 | ||
3645.1.n.e | 6 | 27.e | even | 9 | 6 | ||
3645.1.n.e | 6 | 135.n | odd | 18 | 6 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(405, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + T + 1 \)
$3$
\( T^{2} \)
$5$
\( T^{2} - T + 1 \)
$7$
\( T^{2} \)
$11$
\( T^{2} \)
$13$
\( T^{2} \)
$17$
\( (T - 1)^{2} \)
$19$
\( (T + 1)^{2} \)
$23$
\( T^{2} + T + 1 \)
$29$
\( T^{2} \)
$31$
\( T^{2} - T + 1 \)
$37$
\( T^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} \)
$47$
\( T^{2} - 2T + 4 \)
$53$
\( (T - 1)^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} - T + 1 \)
$67$
\( T^{2} \)
$71$
\( T^{2} \)
$73$
\( T^{2} \)
$79$
\( T^{2} - T + 1 \)
$83$
\( T^{2} + T + 1 \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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