Newspace parameters
Level: | \( N \) | \(=\) | \( 896 = 2^{7} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 896.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.447162251319\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{2}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} - 2 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{4}\) |
Projective field: | Galois closure of 4.0.1568.1 |
Artin image: | $D_8$ |
Artin field: | Galois closure of 8.0.1258815488.3 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(129\) | \(645\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
321.1 |
|
0 | −1.41421 | 0 | 1.41421 | 0 | 1.00000 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||
321.2 | 0 | 1.41421 | 0 | −1.41421 | 0 | 1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
56.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-14}) \) |
7.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 896.1.h.b | yes | 2 |
4.b | odd | 2 | 1 | 896.1.h.a | ✓ | 2 | |
7.b | odd | 2 | 1 | inner | 896.1.h.b | yes | 2 |
8.b | even | 2 | 1 | inner | 896.1.h.b | yes | 2 |
8.d | odd | 2 | 1 | 896.1.h.a | ✓ | 2 | |
16.e | even | 4 | 2 | 1792.1.c.c | 2 | ||
16.f | odd | 4 | 2 | 1792.1.c.d | 2 | ||
28.d | even | 2 | 1 | 896.1.h.a | ✓ | 2 | |
56.e | even | 2 | 1 | 896.1.h.a | ✓ | 2 | |
56.h | odd | 2 | 1 | CM | 896.1.h.b | yes | 2 |
112.j | even | 4 | 2 | 1792.1.c.d | 2 | ||
112.l | odd | 4 | 2 | 1792.1.c.c | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
896.1.h.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
896.1.h.a | ✓ | 2 | 8.d | odd | 2 | 1 | |
896.1.h.a | ✓ | 2 | 28.d | even | 2 | 1 | |
896.1.h.a | ✓ | 2 | 56.e | even | 2 | 1 | |
896.1.h.b | yes | 2 | 1.a | even | 1 | 1 | trivial |
896.1.h.b | yes | 2 | 7.b | odd | 2 | 1 | inner |
896.1.h.b | yes | 2 | 8.b | even | 2 | 1 | inner |
896.1.h.b | yes | 2 | 56.h | odd | 2 | 1 | CM |
1792.1.c.c | 2 | 16.e | even | 4 | 2 | ||
1792.1.c.c | 2 | 112.l | odd | 4 | 2 | ||
1792.1.c.d | 2 | 16.f | odd | 4 | 2 | ||
1792.1.c.d | 2 | 112.j | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{71} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(896, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 2 \)
$5$
\( T^{2} - 2 \)
$7$
\( (T - 1)^{2} \)
$11$
\( T^{2} \)
$13$
\( T^{2} - 2 \)
$17$
\( T^{2} \)
$19$
\( T^{2} - 2 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} - 2 \)
$61$
\( T^{2} - 2 \)
$67$
\( T^{2} \)
$71$
\( (T + 2)^{2} \)
$73$
\( T^{2} \)
$79$
\( (T + 2)^{2} \)
$83$
\( T^{2} - 2 \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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