Newspace parameters
Level: | \( N \) | \(=\) | \( 3645 = 3^{6} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3645.n (of order \(18\), degree \(6\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.81909197105\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | \(\Q(\zeta_{18})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 1215) |
Projective image: | \(D_{9}\) |
Projective field: | Galois closure of 9.1.242137805625.3 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3645\mathbb{Z}\right)^\times\).
\(n\) | \(731\) | \(2917\) |
\(\chi(n)\) | \(\zeta_{18}^{7}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
404.1 |
|
−1.17365 | + | 0.984808i | 0 | 0.233956 | − | 1.32683i | 0.939693 | − | 0.342020i | 0 | 0 | 0.266044 | + | 0.460802i | 0 | −0.766044 | + | 1.32683i | ||||||||||||||||||||||||||
809.1 | −0.0603074 | + | 0.342020i | 0 | 0.826352 | + | 0.300767i | −0.766044 | + | 0.642788i | 0 | 0 | −0.326352 | + | 0.565258i | 0 | −0.173648 | − | 0.300767i | |||||||||||||||||||||||||||
1619.1 | −1.76604 | − | 0.642788i | 0 | 1.93969 | + | 1.62760i | −0.173648 | + | 0.984808i | 0 | 0 | −1.43969 | − | 2.49362i | 0 | 0.939693 | − | 1.62760i | |||||||||||||||||||||||||||
2024.1 | −1.76604 | + | 0.642788i | 0 | 1.93969 | − | 1.62760i | −0.173648 | − | 0.984808i | 0 | 0 | −1.43969 | + | 2.49362i | 0 | 0.939693 | + | 1.62760i | |||||||||||||||||||||||||||
2834.1 | −0.0603074 | − | 0.342020i | 0 | 0.826352 | − | 0.300767i | −0.766044 | − | 0.642788i | 0 | 0 | −0.326352 | − | 0.565258i | 0 | −0.173648 | + | 0.300767i | |||||||||||||||||||||||||||
3239.1 | −1.17365 | − | 0.984808i | 0 | 0.233956 | + | 1.32683i | 0.939693 | + | 0.342020i | 0 | 0 | 0.266044 | − | 0.460802i | 0 | −0.766044 | − | 1.32683i | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
15.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-15}) \) |
27.e | even | 9 | 1 | inner |
135.n | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3645.1.n.a | 6 | |
3.b | odd | 2 | 1 | 3645.1.n.h | 6 | ||
5.b | even | 2 | 1 | 3645.1.n.h | 6 | ||
9.c | even | 3 | 1 | 3645.1.n.f | 6 | ||
9.c | even | 3 | 1 | 3645.1.n.g | 6 | ||
9.d | odd | 6 | 1 | 3645.1.n.b | 6 | ||
9.d | odd | 6 | 1 | 3645.1.n.c | 6 | ||
15.d | odd | 2 | 1 | CM | 3645.1.n.a | 6 | |
27.e | even | 9 | 1 | 1215.1.d.a | ✓ | 3 | |
27.e | even | 9 | 2 | 1215.1.h.b | 6 | ||
27.e | even | 9 | 1 | inner | 3645.1.n.a | 6 | |
27.e | even | 9 | 1 | 3645.1.n.f | 6 | ||
27.e | even | 9 | 1 | 3645.1.n.g | 6 | ||
27.f | odd | 18 | 1 | 1215.1.d.b | yes | 3 | |
27.f | odd | 18 | 2 | 1215.1.h.a | 6 | ||
27.f | odd | 18 | 1 | 3645.1.n.b | 6 | ||
27.f | odd | 18 | 1 | 3645.1.n.c | 6 | ||
27.f | odd | 18 | 1 | 3645.1.n.h | 6 | ||
45.h | odd | 6 | 1 | 3645.1.n.f | 6 | ||
45.h | odd | 6 | 1 | 3645.1.n.g | 6 | ||
45.j | even | 6 | 1 | 3645.1.n.b | 6 | ||
45.j | even | 6 | 1 | 3645.1.n.c | 6 | ||
135.n | odd | 18 | 1 | 1215.1.d.a | ✓ | 3 | |
135.n | odd | 18 | 2 | 1215.1.h.b | 6 | ||
135.n | odd | 18 | 1 | inner | 3645.1.n.a | 6 | |
135.n | odd | 18 | 1 | 3645.1.n.f | 6 | ||
135.n | odd | 18 | 1 | 3645.1.n.g | 6 | ||
135.p | even | 18 | 1 | 1215.1.d.b | yes | 3 | |
135.p | even | 18 | 2 | 1215.1.h.a | 6 | ||
135.p | even | 18 | 1 | 3645.1.n.b | 6 | ||
135.p | even | 18 | 1 | 3645.1.n.c | 6 | ||
135.p | even | 18 | 1 | 3645.1.n.h | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1215.1.d.a | ✓ | 3 | 27.e | even | 9 | 1 | |
1215.1.d.a | ✓ | 3 | 135.n | odd | 18 | 1 | |
1215.1.d.b | yes | 3 | 27.f | odd | 18 | 1 | |
1215.1.d.b | yes | 3 | 135.p | even | 18 | 1 | |
1215.1.h.a | 6 | 27.f | odd | 18 | 2 | ||
1215.1.h.a | 6 | 135.p | even | 18 | 2 | ||
1215.1.h.b | 6 | 27.e | even | 9 | 2 | ||
1215.1.h.b | 6 | 135.n | odd | 18 | 2 | ||
3645.1.n.a | 6 | 1.a | even | 1 | 1 | trivial | |
3645.1.n.a | 6 | 15.d | odd | 2 | 1 | CM | |
3645.1.n.a | 6 | 27.e | even | 9 | 1 | inner | |
3645.1.n.a | 6 | 135.n | odd | 18 | 1 | inner | |
3645.1.n.b | 6 | 9.d | odd | 6 | 1 | ||
3645.1.n.b | 6 | 27.f | odd | 18 | 1 | ||
3645.1.n.b | 6 | 45.j | even | 6 | 1 | ||
3645.1.n.b | 6 | 135.p | even | 18 | 1 | ||
3645.1.n.c | 6 | 9.d | odd | 6 | 1 | ||
3645.1.n.c | 6 | 27.f | odd | 18 | 1 | ||
3645.1.n.c | 6 | 45.j | even | 6 | 1 | ||
3645.1.n.c | 6 | 135.p | even | 18 | 1 | ||
3645.1.n.f | 6 | 9.c | even | 3 | 1 | ||
3645.1.n.f | 6 | 27.e | even | 9 | 1 | ||
3645.1.n.f | 6 | 45.h | odd | 6 | 1 | ||
3645.1.n.f | 6 | 135.n | odd | 18 | 1 | ||
3645.1.n.g | 6 | 9.c | even | 3 | 1 | ||
3645.1.n.g | 6 | 27.e | even | 9 | 1 | ||
3645.1.n.g | 6 | 45.h | odd | 6 | 1 | ||
3645.1.n.g | 6 | 135.n | odd | 18 | 1 | ||
3645.1.n.h | 6 | 3.b | odd | 2 | 1 | ||
3645.1.n.h | 6 | 5.b | even | 2 | 1 | ||
3645.1.n.h | 6 | 27.f | odd | 18 | 1 | ||
3645.1.n.h | 6 | 135.p | even | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 6T_{2}^{5} + 15T_{2}^{4} + 19T_{2}^{3} + 12T_{2}^{2} + 3T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3645, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 \)
$3$
\( T^{6} \)
$5$
\( T^{6} - T^{3} + 1 \)
$7$
\( T^{6} \)
$11$
\( T^{6} \)
$13$
\( T^{6} \)
$17$
\( T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 \)
$19$
\( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \)
$23$
\( T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 \)
$29$
\( T^{6} \)
$31$
\( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \)
$37$
\( T^{6} \)
$41$
\( T^{6} \)
$43$
\( T^{6} \)
$47$
\( T^{6} + T^{3} + 1 \)
$53$
\( (T^{3} - 3 T - 1)^{2} \)
$59$
\( T^{6} \)
$61$
\( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \)
$67$
\( T^{6} \)
$71$
\( T^{6} \)
$73$
\( T^{6} \)
$79$
\( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \)
$83$
\( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \)
$89$
\( T^{6} \)
$97$
\( T^{6} \)
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