Newspace parameters
Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 400.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(92.0216334479\) |
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: | \(\Z[a_1, \ldots, a_{31}]\) |
Coefficient ring index: | \( 2^{11} \) |
Twist minimal: | no (minimal twist has level 16) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( 2\zeta_{12}^{3} \) |
\(\beta_{2}\) | \(=\) | \( -16\zeta_{12}^{3} + 32\zeta_{12} \) |
\(\beta_{3}\) | \(=\) | \( 32\zeta_{12}^{2} - 16 \) |
\(\zeta_{12}\) | \(=\) | \( ( \beta_{2} + 8\beta_1 ) / 32 \) |
\(\zeta_{12}^{2}\) | \(=\) | \( ( \beta_{3} + 16 ) / 32 \) |
\(\zeta_{12}^{3}\) | \(=\) | \( ( \beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(177\) | \(351\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
399.1 |
|
0 | −27.7128 | 0 | 0 | 0 | 609.682 | 0 | 39.0000 | 0 | ||||||||||||||||||||||||||||||
399.2 | 0 | −27.7128 | 0 | 0 | 0 | 609.682 | 0 | 39.0000 | 0 | |||||||||||||||||||||||||||||||
399.3 | 0 | 27.7128 | 0 | 0 | 0 | −609.682 | 0 | 39.0000 | 0 | |||||||||||||||||||||||||||||||
399.4 | 0 | 27.7128 | 0 | 0 | 0 | −609.682 | 0 | 39.0000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 400.7.h.b | 4 | |
4.b | odd | 2 | 1 | inner | 400.7.h.b | 4 | |
5.b | even | 2 | 1 | inner | 400.7.h.b | 4 | |
5.c | odd | 4 | 1 | 16.7.c.b | ✓ | 2 | |
5.c | odd | 4 | 1 | 400.7.b.c | 2 | ||
15.e | even | 4 | 1 | 144.7.g.f | 2 | ||
20.d | odd | 2 | 1 | inner | 400.7.h.b | 4 | |
20.e | even | 4 | 1 | 16.7.c.b | ✓ | 2 | |
20.e | even | 4 | 1 | 400.7.b.c | 2 | ||
40.i | odd | 4 | 1 | 64.7.c.d | 2 | ||
40.k | even | 4 | 1 | 64.7.c.d | 2 | ||
60.l | odd | 4 | 1 | 144.7.g.f | 2 | ||
80.i | odd | 4 | 1 | 256.7.d.e | 4 | ||
80.j | even | 4 | 1 | 256.7.d.e | 4 | ||
80.s | even | 4 | 1 | 256.7.d.e | 4 | ||
80.t | odd | 4 | 1 | 256.7.d.e | 4 | ||
120.q | odd | 4 | 1 | 576.7.g.d | 2 | ||
120.w | even | 4 | 1 | 576.7.g.d | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
16.7.c.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
16.7.c.b | ✓ | 2 | 20.e | even | 4 | 1 | |
64.7.c.d | 2 | 40.i | odd | 4 | 1 | ||
64.7.c.d | 2 | 40.k | even | 4 | 1 | ||
144.7.g.f | 2 | 15.e | even | 4 | 1 | ||
144.7.g.f | 2 | 60.l | odd | 4 | 1 | ||
256.7.d.e | 4 | 80.i | odd | 4 | 1 | ||
256.7.d.e | 4 | 80.j | even | 4 | 1 | ||
256.7.d.e | 4 | 80.s | even | 4 | 1 | ||
256.7.d.e | 4 | 80.t | odd | 4 | 1 | ||
400.7.b.c | 2 | 5.c | odd | 4 | 1 | ||
400.7.b.c | 2 | 20.e | even | 4 | 1 | ||
400.7.h.b | 4 | 1.a | even | 1 | 1 | trivial | |
400.7.h.b | 4 | 4.b | odd | 2 | 1 | inner | |
400.7.h.b | 4 | 5.b | even | 2 | 1 | inner | |
400.7.h.b | 4 | 20.d | odd | 2 | 1 | inner | |
576.7.g.d | 2 | 120.q | odd | 4 | 1 | ||
576.7.g.d | 2 | 120.w | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 768 \)
acting on \(S_{7}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - 768)^{2} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} - 371712)^{2} \)
$11$
\( (T^{2} + 836352)^{2} \)
$13$
\( (T^{2} + 23716)^{2} \)
$17$
\( (T^{2} + 55621764)^{2} \)
$19$
\( (T^{2} + 4553472)^{2} \)
$23$
\( (T^{2} - 71912448)^{2} \)
$29$
\( (T - 10758)^{4} \)
$31$
\( (T^{2} + 3981312)^{2} \)
$37$
\( (T^{2} + 128822500)^{2} \)
$41$
\( (T - 67122)^{4} \)
$43$
\( (T^{2} - 6330348288)^{2} \)
$47$
\( (T^{2} - 4830769152)^{2} \)
$53$
\( (T^{2} + 12091641444)^{2} \)
$59$
\( (T^{2} + 93554203392)^{2} \)
$61$
\( (T - 306746)^{4} \)
$67$
\( (T^{2} - 48551731968)^{2} \)
$71$
\( (T^{2} + 138602769408)^{2} \)
$73$
\( (T^{2} + 27450525124)^{2} \)
$79$
\( (T^{2} + 580800000000)^{2} \)
$83$
\( (T^{2} - 244784339712)^{2} \)
$89$
\( (T + 471954)^{4} \)
$97$
\( (T^{2} + 829181432836)^{2} \)
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