Newspace parameters
Level: | \( N \) | \(=\) | \( 1449 = 3^{2} \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1449.bq (of order \(22\), degree \(10\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.723145203305\) |
Analytic rank: | \(0\) |
Dimension: | \(20\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{22})\) |
Coefficient field: | \(\Q(\zeta_{44})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{22}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1449\mathbb{Z}\right)^\times\).
\(n\) | \(442\) | \(829\) | \(1289\) |
\(\chi(n)\) | \(-\zeta_{44}^{6}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 |
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−1.03748 | + | 0.304632i | 0 | 0.142315 | − | 0.0914602i | 0 | 0 | −0.415415 | + | 0.909632i | 0.588302 | − | 0.678936i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
55.2 | 1.03748 | − | 0.304632i | 0 | 0.142315 | − | 0.0914602i | 0 | 0 | −0.415415 | + | 0.909632i | −0.588302 | + | 0.678936i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.1 | −0.0801894 | − | 0.557730i | 0 | 0.654861 | − | 0.192284i | 0 | 0 | −0.841254 | + | 0.540641i | −0.393828 | − | 0.862362i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.2 | 0.0801894 | + | 0.557730i | 0 | 0.654861 | − | 0.192284i | 0 | 0 | −0.841254 | + | 0.540641i | 0.393828 | + | 0.862362i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
307.1 | −0.0801894 | + | 0.557730i | 0 | 0.654861 | + | 0.192284i | 0 | 0 | −0.841254 | − | 0.540641i | −0.393828 | + | 0.862362i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
307.2 | 0.0801894 | − | 0.557730i | 0 | 0.654861 | + | 0.192284i | 0 | 0 | −0.841254 | − | 0.540641i | 0.393828 | − | 0.862362i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
370.1 | −1.53046 | − | 0.983568i | 0 | 0.959493 | + | 2.10100i | 0 | 0 | 0.654861 | − | 0.755750i | 0.339098 | − | 2.35848i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
370.2 | 1.53046 | + | 0.983568i | 0 | 0.959493 | + | 2.10100i | 0 | 0 | 0.654861 | − | 0.755750i | −0.339098 | + | 2.35848i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
496.1 | −1.29639 | − | 1.49611i | 0 | −0.415415 | + | 2.88927i | 0 | 0 | 0.959493 | + | 0.281733i | 3.19584 | − | 2.05384i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
496.2 | 1.29639 | + | 1.49611i | 0 | −0.415415 | + | 2.88927i | 0 | 0 | 0.959493 | + | 0.281733i | −3.19584 | + | 2.05384i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
685.1 | −1.03748 | − | 0.304632i | 0 | 0.142315 | + | 0.0914602i | 0 | 0 | −0.415415 | − | 0.909632i | 0.588302 | + | 0.678936i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
685.2 | 1.03748 | + | 0.304632i | 0 | 0.142315 | + | 0.0914602i | 0 | 0 | −0.415415 | − | 0.909632i | −0.588302 | − | 0.678936i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
748.1 | −1.53046 | + | 0.983568i | 0 | 0.959493 | − | 2.10100i | 0 | 0 | 0.654861 | + | 0.755750i | 0.339098 | + | 2.35848i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
748.2 | 1.53046 | − | 0.983568i | 0 | 0.959493 | − | 2.10100i | 0 | 0 | 0.654861 | + | 0.755750i | −0.339098 | − | 2.35848i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
811.1 | −0.627899 | + | 1.37491i | 0 | −0.841254 | − | 0.970858i | 0 | 0 | 0.142315 | − | 0.989821i | 0.412791 | − | 0.121206i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
811.2 | 0.627899 | − | 1.37491i | 0 | −0.841254 | − | 0.970858i | 0 | 0 | 0.142315 | − | 0.989821i | −0.412791 | + | 0.121206i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1189.1 | −1.29639 | + | 1.49611i | 0 | −0.415415 | − | 2.88927i | 0 | 0 | 0.959493 | − | 0.281733i | 3.19584 | + | 2.05384i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1189.2 | 1.29639 | − | 1.49611i | 0 | −0.415415 | − | 2.88927i | 0 | 0 | 0.959493 | − | 0.281733i | −3.19584 | − | 2.05384i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1315.1 | −0.627899 | − | 1.37491i | 0 | −0.841254 | + | 0.970858i | 0 | 0 | 0.142315 | + | 0.989821i | 0.412791 | + | 0.121206i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1315.2 | 0.627899 | + | 1.37491i | 0 | −0.841254 | + | 0.970858i | 0 | 0 | 0.142315 | + | 0.989821i | −0.412791 | − | 0.121206i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
3.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
23.c | even | 11 | 1 | inner |
69.h | odd | 22 | 1 | inner |
161.l | odd | 22 | 1 | inner |
483.v | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1449.1.bq.b | ✓ | 20 |
3.b | odd | 2 | 1 | inner | 1449.1.bq.b | ✓ | 20 |
7.b | odd | 2 | 1 | CM | 1449.1.bq.b | ✓ | 20 |
21.c | even | 2 | 1 | inner | 1449.1.bq.b | ✓ | 20 |
23.c | even | 11 | 1 | inner | 1449.1.bq.b | ✓ | 20 |
69.h | odd | 22 | 1 | inner | 1449.1.bq.b | ✓ | 20 |
161.l | odd | 22 | 1 | inner | 1449.1.bq.b | ✓ | 20 |
483.v | even | 22 | 1 | inner | 1449.1.bq.b | ✓ | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1449.1.bq.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
1449.1.bq.b | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
1449.1.bq.b | ✓ | 20 | 7.b | odd | 2 | 1 | CM |
1449.1.bq.b | ✓ | 20 | 21.c | even | 2 | 1 | inner |
1449.1.bq.b | ✓ | 20 | 23.c | even | 11 | 1 | inner |
1449.1.bq.b | ✓ | 20 | 69.h | odd | 22 | 1 | inner |
1449.1.bq.b | ✓ | 20 | 161.l | odd | 22 | 1 | inner |
1449.1.bq.b | ✓ | 20 | 483.v | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 22T_{2}^{16} + 165T_{2}^{12} + 99T_{2}^{10} + 484T_{2}^{8} - 968T_{2}^{6} + 484T_{2}^{4} + 605T_{2}^{2} + 121 \)
acting on \(S_{1}^{\mathrm{new}}(1449, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{20} + 22 T^{16} + 165 T^{12} + \cdots + 121 \)
$3$
\( T^{20} \)
$5$
\( T^{20} \)
$7$
\( (T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \)
$11$
\( T^{20} + 55 T^{14} - 264 T^{10} + \cdots + 121 \)
$13$
\( T^{20} \)
$17$
\( T^{20} \)
$19$
\( T^{20} \)
$23$
\( T^{20} - T^{18} + T^{16} - T^{14} + T^{12} - T^{10} + \cdots + 1 \)
$29$
\( T^{20} + 22 T^{12} + 462 T^{10} + \cdots + 121 \)
$31$
\( T^{20} \)
$37$
\( (T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} - 6 T^{6} + \cdots + 1)^{2} \)
$41$
\( T^{20} \)
$43$
\( (T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + 5 T^{6} + \cdots + 1)^{2} \)
$47$
\( T^{20} \)
$53$
\( T^{20} + 55 T^{14} - 264 T^{10} + \cdots + 121 \)
$59$
\( T^{20} \)
$61$
\( T^{20} \)
$67$
\( (T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + 5 T^{6} + \cdots + 1)^{2} \)
$71$
\( T^{20} + 55 T^{14} - 264 T^{10} + \cdots + 121 \)
$73$
\( T^{20} \)
$79$
\( (T^{10} + 2 T^{9} + 4 T^{8} - 3 T^{7} - 6 T^{6} + \cdots + 1)^{2} \)
$83$
\( T^{20} \)
$89$
\( T^{20} \)
$97$
\( T^{20} \)
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