Newspace parameters
Level: | \( N \) | \(=\) | \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3360.ft (of order \(12\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.67685844245\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 840) |
Projective image: | \(D_{12}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).
\(n\) | \(421\) | \(1121\) | \(1471\) | \(1921\) | \(2017\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) | \(\zeta_{24}^{4}\) | \(-\zeta_{24}^{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 |
|
0 | −0.965926 | + | 0.258819i | 0 | −0.707107 | + | 0.707107i | 0 | −0.866025 | − | 0.500000i | 0 | 0.866025 | − | 0.500000i | 0 | ||||||||||||||||||||||||||||||||||
17.2 | 0 | 0.965926 | − | 0.258819i | 0 | 0.707107 | − | 0.707107i | 0 | −0.866025 | − | 0.500000i | 0 | 0.866025 | − | 0.500000i | 0 | |||||||||||||||||||||||||||||||||||
593.1 | 0 | −0.965926 | − | 0.258819i | 0 | −0.707107 | − | 0.707107i | 0 | −0.866025 | + | 0.500000i | 0 | 0.866025 | + | 0.500000i | 0 | |||||||||||||||||||||||||||||||||||
593.2 | 0 | 0.965926 | + | 0.258819i | 0 | 0.707107 | + | 0.707107i | 0 | −0.866025 | + | 0.500000i | 0 | 0.866025 | + | 0.500000i | 0 | |||||||||||||||||||||||||||||||||||
1937.1 | 0 | −0.258819 | + | 0.965926i | 0 | 0.707107 | − | 0.707107i | 0 | 0.866025 | − | 0.500000i | 0 | −0.866025 | − | 0.500000i | 0 | |||||||||||||||||||||||||||||||||||
1937.2 | 0 | 0.258819 | − | 0.965926i | 0 | −0.707107 | + | 0.707107i | 0 | 0.866025 | − | 0.500000i | 0 | −0.866025 | − | 0.500000i | 0 | |||||||||||||||||||||||||||||||||||
2033.1 | 0 | −0.258819 | − | 0.965926i | 0 | 0.707107 | + | 0.707107i | 0 | 0.866025 | + | 0.500000i | 0 | −0.866025 | + | 0.500000i | 0 | |||||||||||||||||||||||||||||||||||
2033.2 | 0 | 0.258819 | + | 0.965926i | 0 | −0.707107 | − | 0.707107i | 0 | 0.866025 | + | 0.500000i | 0 | −0.866025 | + | 0.500000i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
24.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-6}) \) |
3.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
35.k | even | 12 | 1 | inner |
105.w | odd | 12 | 1 | inner |
280.bv | even | 12 | 1 | inner |
840.dh | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3360.1.ft.b | 8 | |
3.b | odd | 2 | 1 | inner | 3360.1.ft.b | 8 | |
4.b | odd | 2 | 1 | 840.1.dh.a | ✓ | 8 | |
5.c | odd | 4 | 1 | 3360.1.ft.a | 8 | ||
7.d | odd | 6 | 1 | 3360.1.ft.a | 8 | ||
8.b | even | 2 | 1 | inner | 3360.1.ft.b | 8 | |
8.d | odd | 2 | 1 | 840.1.dh.a | ✓ | 8 | |
12.b | even | 2 | 1 | 840.1.dh.a | ✓ | 8 | |
15.e | even | 4 | 1 | 3360.1.ft.a | 8 | ||
20.e | even | 4 | 1 | 840.1.dh.b | yes | 8 | |
21.g | even | 6 | 1 | 3360.1.ft.a | 8 | ||
24.f | even | 2 | 1 | 840.1.dh.a | ✓ | 8 | |
24.h | odd | 2 | 1 | CM | 3360.1.ft.b | 8 | |
28.f | even | 6 | 1 | 840.1.dh.b | yes | 8 | |
35.k | even | 12 | 1 | inner | 3360.1.ft.b | 8 | |
40.i | odd | 4 | 1 | 3360.1.ft.a | 8 | ||
40.k | even | 4 | 1 | 840.1.dh.b | yes | 8 | |
56.j | odd | 6 | 1 | 3360.1.ft.a | 8 | ||
56.m | even | 6 | 1 | 840.1.dh.b | yes | 8 | |
60.l | odd | 4 | 1 | 840.1.dh.b | yes | 8 | |
84.j | odd | 6 | 1 | 840.1.dh.b | yes | 8 | |
105.w | odd | 12 | 1 | inner | 3360.1.ft.b | 8 | |
120.q | odd | 4 | 1 | 840.1.dh.b | yes | 8 | |
120.w | even | 4 | 1 | 3360.1.ft.a | 8 | ||
140.x | odd | 12 | 1 | 840.1.dh.a | ✓ | 8 | |
168.ba | even | 6 | 1 | 3360.1.ft.a | 8 | ||
168.be | odd | 6 | 1 | 840.1.dh.b | yes | 8 | |
280.bp | odd | 12 | 1 | 840.1.dh.a | ✓ | 8 | |
280.bv | even | 12 | 1 | inner | 3360.1.ft.b | 8 | |
420.br | even | 12 | 1 | 840.1.dh.a | ✓ | 8 | |
840.dh | odd | 12 | 1 | inner | 3360.1.ft.b | 8 | |
840.dk | even | 12 | 1 | 840.1.dh.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
840.1.dh.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
840.1.dh.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
840.1.dh.a | ✓ | 8 | 12.b | even | 2 | 1 | |
840.1.dh.a | ✓ | 8 | 24.f | even | 2 | 1 | |
840.1.dh.a | ✓ | 8 | 140.x | odd | 12 | 1 | |
840.1.dh.a | ✓ | 8 | 280.bp | odd | 12 | 1 | |
840.1.dh.a | ✓ | 8 | 420.br | even | 12 | 1 | |
840.1.dh.a | ✓ | 8 | 840.dk | even | 12 | 1 | |
840.1.dh.b | yes | 8 | 20.e | even | 4 | 1 | |
840.1.dh.b | yes | 8 | 28.f | even | 6 | 1 | |
840.1.dh.b | yes | 8 | 40.k | even | 4 | 1 | |
840.1.dh.b | yes | 8 | 56.m | even | 6 | 1 | |
840.1.dh.b | yes | 8 | 60.l | odd | 4 | 1 | |
840.1.dh.b | yes | 8 | 84.j | odd | 6 | 1 | |
840.1.dh.b | yes | 8 | 120.q | odd | 4 | 1 | |
840.1.dh.b | yes | 8 | 168.be | odd | 6 | 1 | |
3360.1.ft.a | 8 | 5.c | odd | 4 | 1 | ||
3360.1.ft.a | 8 | 7.d | odd | 6 | 1 | ||
3360.1.ft.a | 8 | 15.e | even | 4 | 1 | ||
3360.1.ft.a | 8 | 21.g | even | 6 | 1 | ||
3360.1.ft.a | 8 | 40.i | odd | 4 | 1 | ||
3360.1.ft.a | 8 | 56.j | odd | 6 | 1 | ||
3360.1.ft.a | 8 | 120.w | even | 4 | 1 | ||
3360.1.ft.a | 8 | 168.ba | even | 6 | 1 | ||
3360.1.ft.b | 8 | 1.a | even | 1 | 1 | trivial | |
3360.1.ft.b | 8 | 3.b | odd | 2 | 1 | inner | |
3360.1.ft.b | 8 | 8.b | even | 2 | 1 | inner | |
3360.1.ft.b | 8 | 24.h | odd | 2 | 1 | CM | |
3360.1.ft.b | 8 | 35.k | even | 12 | 1 | inner | |
3360.1.ft.b | 8 | 105.w | odd | 12 | 1 | inner | |
3360.1.ft.b | 8 | 280.bv | even | 12 | 1 | inner | |
3360.1.ft.b | 8 | 840.dh | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{73}^{4} + 2T_{73}^{3} + 2T_{73}^{2} + 4T_{73} + 4 \)
acting on \(S_{1}^{\mathrm{new}}(3360, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} - T^{4} + 1 \)
$5$
\( (T^{4} + 1)^{2} \)
$7$
\( (T^{4} - T^{2} + 1)^{2} \)
$11$
\( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \)
$13$
\( T^{8} \)
$17$
\( T^{8} \)
$19$
\( T^{8} \)
$23$
\( T^{8} \)
$29$
\( (T^{4} + 4 T^{2} + 1)^{2} \)
$31$
\( (T^{4} - T^{2} + 1)^{2} \)
$37$
\( T^{8} \)
$41$
\( T^{8} \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( T^{8} - 9T^{4} + 81 \)
$59$
\( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \)
$61$
\( T^{8} \)
$67$
\( T^{8} \)
$71$
\( T^{8} \)
$73$
\( (T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2} \)
$79$
\( (T^{2} + 3 T + 3)^{4} \)
$83$
\( (T^{4} + 9)^{2} \)
$89$
\( T^{8} \)
$97$
\( (T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1)^{2} \)
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