Newspace parameters
Level: | \( N \) | \(=\) | \( 1340 = 2^{2} \cdot 5 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1340.bb (of order \(22\), degree \(10\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.668747116928\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Coefficient field: | \(\Q(\zeta_{22})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{11}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1340\mathbb{Z}\right)^\times\).
\(n\) | \(537\) | \(671\) | \(1141\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(\zeta_{22}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
59.1 |
|
0.142315 | − | 0.989821i | −1.25667 | − | 1.45027i | −0.959493 | − | 0.281733i | 0.841254 | + | 0.540641i | −1.61435 | + | 1.03748i | −0.186393 | + | 1.29639i | −0.415415 | + | 0.909632i | −0.381761 | + | 2.65520i | 0.654861 | − | 0.755750i | ||||||||||||||||||||||||||||||
159.1 | 0.142315 | + | 0.989821i | −1.25667 | + | 1.45027i | −0.959493 | + | 0.281733i | 0.841254 | − | 0.540641i | −1.61435 | − | 1.03748i | −0.186393 | − | 1.29639i | −0.415415 | − | 0.909632i | −0.381761 | − | 2.65520i | 0.654861 | + | 0.755750i | |||||||||||||||||||||||||||||||
359.1 | −0.415415 | + | 0.909632i | 1.10181 | − | 0.708089i | −0.654861 | − | 0.755750i | −0.142315 | + | 0.989821i | 0.186393 | + | 1.29639i | −0.698939 | + | 1.53046i | 0.959493 | − | 0.281733i | 0.297176 | − | 0.650724i | −0.841254 | − | 0.540641i | |||||||||||||||||||||||||||||||
399.1 | 0.959493 | + | 0.281733i | 0.239446 | − | 1.66538i | 0.841254 | + | 0.540641i | 0.415415 | + | 0.909632i | 0.698939 | − | 1.53046i | −0.273100 | − | 0.0801894i | 0.654861 | + | 0.755750i | −1.75667 | − | 0.515804i | 0.142315 | + | 0.989821i | |||||||||||||||||||||||||||||||
679.1 | −0.841254 | − | 0.540641i | 0.797176 | + | 0.234072i | 0.415415 | + | 0.909632i | −0.654861 | + | 0.755750i | −0.544078 | − | 0.627899i | 1.61435 | + | 1.03748i | 0.142315 | − | 0.989821i | −0.260554 | − | 0.167448i | 0.959493 | − | 0.281733i | |||||||||||||||||||||||||||||||
759.1 | 0.959493 | − | 0.281733i | 0.239446 | + | 1.66538i | 0.841254 | − | 0.540641i | 0.415415 | − | 0.909632i | 0.698939 | + | 1.53046i | −0.273100 | + | 0.0801894i | 0.654861 | − | 0.755750i | −1.75667 | + | 0.515804i | 0.142315 | − | 0.989821i | |||||||||||||||||||||||||||||||
799.1 | 0.654861 | − | 0.755750i | 0.118239 | + | 0.258908i | −0.142315 | − | 0.989821i | −0.959493 | + | 0.281733i | 0.273100 | + | 0.0801894i | 0.544078 | − | 0.627899i | −0.841254 | − | 0.540641i | 0.601808 | − | 0.694523i | −0.415415 | + | 0.909632i | |||||||||||||||||||||||||||||||
819.1 | −0.841254 | + | 0.540641i | 0.797176 | − | 0.234072i | 0.415415 | − | 0.909632i | −0.654861 | − | 0.755750i | −0.544078 | + | 0.627899i | 1.61435 | − | 1.03748i | 0.142315 | + | 0.989821i | −0.260554 | + | 0.167448i | 0.959493 | + | 0.281733i | |||||||||||||||||||||||||||||||
1019.1 | −0.415415 | − | 0.909632i | 1.10181 | + | 0.708089i | −0.654861 | + | 0.755750i | −0.142315 | − | 0.989821i | 0.186393 | − | 1.29639i | −0.698939 | − | 1.53046i | 0.959493 | + | 0.281733i | 0.297176 | + | 0.650724i | −0.841254 | + | 0.540641i | |||||||||||||||||||||||||||||||
1179.1 | 0.654861 | + | 0.755750i | 0.118239 | − | 0.258908i | −0.142315 | + | 0.989821i | −0.959493 | − | 0.281733i | 0.273100 | − | 0.0801894i | 0.544078 | + | 0.627899i | −0.841254 | + | 0.540641i | 0.601808 | + | 0.694523i | −0.415415 | − | 0.909632i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
67.e | even | 11 | 1 | inner |
1340.bb | odd | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1340.1.bb.b | yes | 10 |
4.b | odd | 2 | 1 | 1340.1.bb.a | ✓ | 10 | |
5.b | even | 2 | 1 | 1340.1.bb.a | ✓ | 10 | |
20.d | odd | 2 | 1 | CM | 1340.1.bb.b | yes | 10 |
67.e | even | 11 | 1 | inner | 1340.1.bb.b | yes | 10 |
268.k | odd | 22 | 1 | 1340.1.bb.a | ✓ | 10 | |
335.o | even | 22 | 1 | 1340.1.bb.a | ✓ | 10 | |
1340.bb | odd | 22 | 1 | inner | 1340.1.bb.b | yes | 10 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1340.1.bb.a | ✓ | 10 | 4.b | odd | 2 | 1 | |
1340.1.bb.a | ✓ | 10 | 5.b | even | 2 | 1 | |
1340.1.bb.a | ✓ | 10 | 268.k | odd | 22 | 1 | |
1340.1.bb.a | ✓ | 10 | 335.o | even | 22 | 1 | |
1340.1.bb.b | yes | 10 | 1.a | even | 1 | 1 | trivial |
1340.1.bb.b | yes | 10 | 20.d | odd | 2 | 1 | CM |
1340.1.bb.b | yes | 10 | 67.e | even | 11 | 1 | inner |
1340.1.bb.b | yes | 10 | 1340.bb | odd | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 2T_{3}^{9} + 4T_{3}^{8} - 8T_{3}^{7} + 16T_{3}^{6} - 32T_{3}^{5} + 53T_{3}^{4} - 51T_{3}^{3} + 25T_{3}^{2} - 6T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(1340, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \)
$3$
\( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \)
$5$
\( T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1 \)
$7$
\( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \)
$11$
\( T^{10} \)
$13$
\( T^{10} \)
$17$
\( T^{10} \)
$19$
\( T^{10} \)
$23$
\( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1024 \)
$29$
\( (T^{5} + T^{4} - 4 T^{3} - 3 T^{2} + 3 T + 1)^{2} \)
$31$
\( T^{10} \)
$37$
\( T^{10} \)
$41$
\( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \)
$43$
\( T^{10} + 9 T^{9} + 37 T^{8} + 91 T^{7} + \cdots + 1 \)
$47$
\( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \)
$53$
\( T^{10} \)
$59$
\( T^{10} \)
$61$
\( T^{10} - 9 T^{9} + 37 T^{8} - 91 T^{7} + \cdots + 1 \)
$67$
\( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \)
$71$
\( T^{10} \)
$73$
\( T^{10} \)
$79$
\( T^{10} \)
$83$
\( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \)
$89$
\( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \)
$97$
\( T^{10} \)
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