Newspace parameters
Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 320.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.55521286468\) |
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: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
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
\(\beta_{1}\) | \(=\) | \( \zeta_{12}^{3} \) |
\(\beta_{2}\) | \(=\) | \( 2\zeta_{12}^{2} - 1 \) |
\(\beta_{3}\) | \(=\) | \( -\zeta_{12}^{3} + 2\zeta_{12} \) |
\(\zeta_{12}\) | \(=\) | \( ( \beta_{3} + \beta_1 ) / 2 \) |
\(\zeta_{12}^{2}\) | \(=\) | \( ( \beta_{2} + 1 ) / 2 \) |
\(\zeta_{12}^{3}\) | \(=\) | \( \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(257\) | \(261\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
161.1 |
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0 | − | 2.73205i | 0 | − | 1.00000i | 0 | 4.73205 | 0 | −4.46410 | 0 | ||||||||||||||||||||||||||||
161.2 | 0 | − | 0.732051i | 0 | 1.00000i | 0 | 1.26795 | 0 | 2.46410 | 0 | ||||||||||||||||||||||||||||||
161.3 | 0 | 0.732051i | 0 | − | 1.00000i | 0 | 1.26795 | 0 | 2.46410 | 0 | ||||||||||||||||||||||||||||||
161.4 | 0 | 2.73205i | 0 | 1.00000i | 0 | 4.73205 | 0 | −4.46410 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
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 | 320.2.d.b | yes | 4 |
3.b | odd | 2 | 1 | 2880.2.k.l | 4 | ||
4.b | odd | 2 | 1 | 320.2.d.a | ✓ | 4 | |
5.b | even | 2 | 1 | 1600.2.d.b | 4 | ||
5.c | odd | 4 | 1 | 1600.2.f.d | 4 | ||
5.c | odd | 4 | 1 | 1600.2.f.h | 4 | ||
8.b | even | 2 | 1 | inner | 320.2.d.b | yes | 4 |
8.d | odd | 2 | 1 | 320.2.d.a | ✓ | 4 | |
12.b | even | 2 | 1 | 2880.2.k.e | 4 | ||
16.e | even | 4 | 1 | 1280.2.a.c | 2 | ||
16.e | even | 4 | 1 | 1280.2.a.m | 2 | ||
16.f | odd | 4 | 1 | 1280.2.a.b | 2 | ||
16.f | odd | 4 | 1 | 1280.2.a.p | 2 | ||
20.d | odd | 2 | 1 | 1600.2.d.h | 4 | ||
20.e | even | 4 | 1 | 1600.2.f.e | 4 | ||
20.e | even | 4 | 1 | 1600.2.f.i | 4 | ||
24.f | even | 2 | 1 | 2880.2.k.e | 4 | ||
24.h | odd | 2 | 1 | 2880.2.k.l | 4 | ||
40.e | odd | 2 | 1 | 1600.2.d.h | 4 | ||
40.f | even | 2 | 1 | 1600.2.d.b | 4 | ||
40.i | odd | 4 | 1 | 1600.2.f.d | 4 | ||
40.i | odd | 4 | 1 | 1600.2.f.h | 4 | ||
40.k | even | 4 | 1 | 1600.2.f.e | 4 | ||
40.k | even | 4 | 1 | 1600.2.f.i | 4 | ||
80.k | odd | 4 | 1 | 6400.2.a.y | 2 | ||
80.k | odd | 4 | 1 | 6400.2.a.cd | 2 | ||
80.q | even | 4 | 1 | 6400.2.a.bf | 2 | ||
80.q | even | 4 | 1 | 6400.2.a.ck | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
320.2.d.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
320.2.d.a | ✓ | 4 | 8.d | odd | 2 | 1 | |
320.2.d.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
320.2.d.b | yes | 4 | 8.b | even | 2 | 1 | inner |
1280.2.a.b | 2 | 16.f | odd | 4 | 1 | ||
1280.2.a.c | 2 | 16.e | even | 4 | 1 | ||
1280.2.a.m | 2 | 16.e | even | 4 | 1 | ||
1280.2.a.p | 2 | 16.f | odd | 4 | 1 | ||
1600.2.d.b | 4 | 5.b | even | 2 | 1 | ||
1600.2.d.b | 4 | 40.f | even | 2 | 1 | ||
1600.2.d.h | 4 | 20.d | odd | 2 | 1 | ||
1600.2.d.h | 4 | 40.e | odd | 2 | 1 | ||
1600.2.f.d | 4 | 5.c | odd | 4 | 1 | ||
1600.2.f.d | 4 | 40.i | odd | 4 | 1 | ||
1600.2.f.e | 4 | 20.e | even | 4 | 1 | ||
1600.2.f.e | 4 | 40.k | even | 4 | 1 | ||
1600.2.f.h | 4 | 5.c | odd | 4 | 1 | ||
1600.2.f.h | 4 | 40.i | odd | 4 | 1 | ||
1600.2.f.i | 4 | 20.e | even | 4 | 1 | ||
1600.2.f.i | 4 | 40.k | even | 4 | 1 | ||
2880.2.k.e | 4 | 12.b | even | 2 | 1 | ||
2880.2.k.e | 4 | 24.f | even | 2 | 1 | ||
2880.2.k.l | 4 | 3.b | odd | 2 | 1 | ||
2880.2.k.l | 4 | 24.h | odd | 2 | 1 | ||
6400.2.a.y | 2 | 80.k | odd | 4 | 1 | ||
6400.2.a.bf | 2 | 80.q | even | 4 | 1 | ||
6400.2.a.cd | 2 | 80.k | odd | 4 | 1 | ||
6400.2.a.ck | 2 | 80.q | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 6T_{7} + 6 \)
acting on \(S_{2}^{\mathrm{new}}(320, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 8T^{2} + 4 \)
$5$
\( (T^{2} + 1)^{2} \)
$7$
\( (T^{2} - 6 T + 6)^{2} \)
$11$
\( (T^{2} + 12)^{2} \)
$13$
\( (T^{2} + 12)^{2} \)
$17$
\( (T^{2} - 12)^{2} \)
$19$
\( (T^{2} + 4)^{2} \)
$23$
\( (T^{2} - 6 T - 18)^{2} \)
$29$
\( T^{4} \)
$31$
\( (T^{2} + 12 T + 24)^{2} \)
$37$
\( (T^{2} + 36)^{2} \)
$41$
\( (T^{2} - 12 T + 24)^{2} \)
$43$
\( T^{4} + 104T^{2} + 4 \)
$47$
\( (T^{2} + 6 T - 18)^{2} \)
$53$
\( (T^{2} + 108)^{2} \)
$59$
\( (T^{2} + 36)^{2} \)
$61$
\( T^{4} + 168T^{2} + 144 \)
$67$
\( T^{4} + 104T^{2} + 4 \)
$71$
\( (T^{2} - 12 T - 72)^{2} \)
$73$
\( (T^{2} + 8 T - 92)^{2} \)
$79$
\( (T + 12)^{4} \)
$83$
\( T^{4} + 24T^{2} + 36 \)
$89$
\( (T^{2} + 12 T - 12)^{2} \)
$97$
\( (T^{2} - 8 T - 92)^{2} \)
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