Newspace parameters
Level: | \( N \) | \(=\) | \( 896 = 2^{7} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 896.ba (of order \(12\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.15459602111\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 112) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(645\) |
\(\chi(n)\) | \(1\) | \(-1 + \zeta_{12}^{2}\) | \(\zeta_{12}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 |
|
0 | −0.133975 | + | 0.500000i | 0 | −0.232051 | − | 0.866025i | 0 | 1.73205 | − | 2.00000i | 0 | 2.36603 | + | 1.36603i | 0 | ||||||||||||||||||||||
417.1 | 0 | −1.86603 | + | 0.500000i | 0 | 3.23205 | + | 0.866025i | 0 | −1.73205 | − | 2.00000i | 0 | 0.633975 | − | 0.366025i | 0 | |||||||||||||||||||||||
737.1 | 0 | −1.86603 | − | 0.500000i | 0 | 3.23205 | − | 0.866025i | 0 | −1.73205 | + | 2.00000i | 0 | 0.633975 | + | 0.366025i | 0 | |||||||||||||||||||||||
865.1 | 0 | −0.133975 | − | 0.500000i | 0 | −0.232051 | + | 0.866025i | 0 | 1.73205 | + | 2.00000i | 0 | 2.36603 | − | 1.36603i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
112.w | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 896.2.ba.a | 4 | |
4.b | odd | 2 | 1 | 896.2.ba.d | 4 | ||
7.c | even | 3 | 1 | 896.2.ba.c | 4 | ||
8.b | even | 2 | 1 | 448.2.ba.b | 4 | ||
8.d | odd | 2 | 1 | 112.2.w.a | ✓ | 4 | |
16.e | even | 4 | 1 | 448.2.ba.a | 4 | ||
16.e | even | 4 | 1 | 896.2.ba.c | 4 | ||
16.f | odd | 4 | 1 | 112.2.w.b | yes | 4 | |
16.f | odd | 4 | 1 | 896.2.ba.b | 4 | ||
28.g | odd | 6 | 1 | 896.2.ba.b | 4 | ||
56.e | even | 2 | 1 | 784.2.x.a | 4 | ||
56.k | odd | 6 | 1 | 112.2.w.b | yes | 4 | |
56.k | odd | 6 | 1 | 784.2.m.e | 4 | ||
56.m | even | 6 | 1 | 784.2.m.d | 4 | ||
56.m | even | 6 | 1 | 784.2.x.h | 4 | ||
56.p | even | 6 | 1 | 448.2.ba.a | 4 | ||
112.j | even | 4 | 1 | 784.2.x.h | 4 | ||
112.u | odd | 12 | 1 | 112.2.w.a | ✓ | 4 | |
112.u | odd | 12 | 1 | 784.2.m.e | 4 | ||
112.u | odd | 12 | 1 | 896.2.ba.d | 4 | ||
112.v | even | 12 | 1 | 784.2.m.d | 4 | ||
112.v | even | 12 | 1 | 784.2.x.a | 4 | ||
112.w | even | 12 | 1 | 448.2.ba.b | 4 | ||
112.w | even | 12 | 1 | inner | 896.2.ba.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
112.2.w.a | ✓ | 4 | 8.d | odd | 2 | 1 | |
112.2.w.a | ✓ | 4 | 112.u | odd | 12 | 1 | |
112.2.w.b | yes | 4 | 16.f | odd | 4 | 1 | |
112.2.w.b | yes | 4 | 56.k | odd | 6 | 1 | |
448.2.ba.a | 4 | 16.e | even | 4 | 1 | ||
448.2.ba.a | 4 | 56.p | even | 6 | 1 | ||
448.2.ba.b | 4 | 8.b | even | 2 | 1 | ||
448.2.ba.b | 4 | 112.w | even | 12 | 1 | ||
784.2.m.d | 4 | 56.m | even | 6 | 1 | ||
784.2.m.d | 4 | 112.v | even | 12 | 1 | ||
784.2.m.e | 4 | 56.k | odd | 6 | 1 | ||
784.2.m.e | 4 | 112.u | odd | 12 | 1 | ||
784.2.x.a | 4 | 56.e | even | 2 | 1 | ||
784.2.x.a | 4 | 112.v | even | 12 | 1 | ||
784.2.x.h | 4 | 56.m | even | 6 | 1 | ||
784.2.x.h | 4 | 112.j | even | 4 | 1 | ||
896.2.ba.a | 4 | 1.a | even | 1 | 1 | trivial | |
896.2.ba.a | 4 | 112.w | even | 12 | 1 | inner | |
896.2.ba.b | 4 | 16.f | odd | 4 | 1 | ||
896.2.ba.b | 4 | 28.g | odd | 6 | 1 | ||
896.2.ba.c | 4 | 7.c | even | 3 | 1 | ||
896.2.ba.c | 4 | 16.e | even | 4 | 1 | ||
896.2.ba.d | 4 | 4.b | odd | 2 | 1 | ||
896.2.ba.d | 4 | 112.u | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4T_{3}^{3} + 5T_{3}^{2} + 2T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1 \)
$5$
\( T^{4} - 6 T^{3} + 9 T^{2} + 9 \)
$7$
\( T^{4} + 2T^{2} + 49 \)
$11$
\( T^{4} - 8 T^{3} + 41 T^{2} - 130 T + 169 \)
$13$
\( T^{4} + 8 T^{3} + 32 T^{2} + 16 T + 4 \)
$17$
\( T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9 \)
$19$
\( T^{4} + 8 T^{3} + 41 T^{2} + 130 T + 169 \)
$23$
\( T^{4} - 12 T^{3} + 59 T^{2} + \cdots + 121 \)
$29$
\( T^{4} - 8 T^{3} + 32 T^{2} - 16 T + 4 \)
$31$
\( T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1 \)
$37$
\( T^{4} + 22 T^{3} + 137 T^{2} + \cdots + 169 \)
$41$
\( T^{4} + 104T^{2} + 1936 \)
$43$
\( T^{4} - 12 T^{3} + 72 T^{2} + 72 T + 36 \)
$47$
\( T^{4} - 12 T^{3} + 111 T^{2} + \cdots + 1089 \)
$53$
\( T^{4} + 2 T^{3} + 101 T^{2} + \cdots + 2209 \)
$59$
\( T^{4} - 28 T^{3} + 365 T^{2} + \cdots + 14641 \)
$61$
\( T^{4} - 14 T^{3} + 53 T^{2} - 4 T + 1 \)
$67$
\( T^{4} + 12 T^{3} + 45 T^{2} + \cdots + 1521 \)
$71$
\( T^{4} + 56T^{2} + 16 \)
$73$
\( T^{4} + 18 T^{3} + 131 T^{2} + \cdots + 529 \)
$79$
\( T^{4} + 16 T^{3} + 267 T^{2} + \cdots + 121 \)
$83$
\( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 676 \)
$89$
\( (T^{2} - 9 T + 27)^{2} \)
$97$
\( (T^{2} - 8 T - 32)^{2} \)
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