Newspace parameters
Level: | \( N \) | \(=\) | \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3332.w (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.66288462209\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{8}\) |
Projective field: | Galois closure of 8.2.3089659810545728.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).
\(n\) | \(785\) | \(885\) | \(1667\) |
\(\chi(n)\) | \(-\zeta_{8}^{3}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
491.1 |
|
0.707107 | + | 0.707107i | 0 | 1.00000i | −1.70711 | + | 0.707107i | 0 | 0 | −0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | −1.70711 | − | 0.707107i | ||||||||||||||||||||
1079.1 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −1.70711 | − | 0.707107i | 0 | 0 | −0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | −1.70711 | + | 0.707107i | ||||||||||||||||||||
1471.1 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | −0.292893 | + | 0.707107i | 0 | 0 | 0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | −0.292893 | − | 0.707107i | ||||||||||||||||||||
2059.1 | −0.707107 | − | 0.707107i | 0 | 1.00000i | −0.292893 | − | 0.707107i | 0 | 0 | 0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | −0.292893 | + | 0.707107i | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
17.d | even | 8 | 1 | inner |
68.g | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3332.1.w.a | ✓ | 4 |
4.b | odd | 2 | 1 | CM | 3332.1.w.a | ✓ | 4 |
7.b | odd | 2 | 1 | 3332.1.w.d | yes | 4 | |
7.c | even | 3 | 2 | 3332.1.bp.d | 8 | ||
7.d | odd | 6 | 2 | 3332.1.bp.a | 8 | ||
17.d | even | 8 | 1 | inner | 3332.1.w.a | ✓ | 4 |
28.d | even | 2 | 1 | 3332.1.w.d | yes | 4 | |
28.f | even | 6 | 2 | 3332.1.bp.a | 8 | ||
28.g | odd | 6 | 2 | 3332.1.bp.d | 8 | ||
68.g | odd | 8 | 1 | inner | 3332.1.w.a | ✓ | 4 |
119.l | odd | 8 | 1 | 3332.1.w.d | yes | 4 | |
119.q | even | 24 | 2 | 3332.1.bp.d | 8 | ||
119.r | odd | 24 | 2 | 3332.1.bp.a | 8 | ||
476.w | even | 8 | 1 | 3332.1.w.d | yes | 4 | |
476.bg | odd | 24 | 2 | 3332.1.bp.d | 8 | ||
476.bj | even | 24 | 2 | 3332.1.bp.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3332.1.w.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
3332.1.w.a | ✓ | 4 | 4.b | odd | 2 | 1 | CM |
3332.1.w.a | ✓ | 4 | 17.d | even | 8 | 1 | inner |
3332.1.w.a | ✓ | 4 | 68.g | odd | 8 | 1 | inner |
3332.1.w.d | yes | 4 | 7.b | odd | 2 | 1 | |
3332.1.w.d | yes | 4 | 28.d | even | 2 | 1 | |
3332.1.w.d | yes | 4 | 119.l | odd | 8 | 1 | |
3332.1.w.d | yes | 4 | 476.w | even | 8 | 1 | |
3332.1.bp.a | 8 | 7.d | odd | 6 | 2 | ||
3332.1.bp.a | 8 | 28.f | even | 6 | 2 | ||
3332.1.bp.a | 8 | 119.r | odd | 24 | 2 | ||
3332.1.bp.a | 8 | 476.bj | even | 24 | 2 | ||
3332.1.bp.d | 8 | 7.c | even | 3 | 2 | ||
3332.1.bp.d | 8 | 28.g | odd | 6 | 2 | ||
3332.1.bp.d | 8 | 119.q | even | 24 | 2 | ||
3332.1.bp.d | 8 | 476.bg | odd | 24 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 4T_{5}^{3} + 6T_{5}^{2} + 4T_{5} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 1 \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \)
$7$
\( T^{4} \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + 4)^{2} \)
$17$
\( T^{4} + 1 \)
$19$
\( T^{4} \)
$23$
\( T^{4} \)
$29$
\( T^{4} + 2 T^{2} + 4 T + 2 \)
$31$
\( T^{4} \)
$37$
\( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \)
$41$
\( T^{4} + 2 T^{2} - 4 T + 2 \)
$43$
\( T^{4} \)
$47$
\( T^{4} \)
$53$
\( (T^{2} + 2 T + 2)^{2} \)
$59$
\( T^{4} \)
$61$
\( T^{4} + 2 T^{2} + 4 T + 2 \)
$67$
\( T^{4} \)
$71$
\( T^{4} \)
$73$
\( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \)
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