Newspace parameters
Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 320.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(8.71936845953\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 20) |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
319.1 |
|
0 | 4.00000 | 0 | 5.00000 | 0 | 4.00000 | 0 | 7.00000 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 320.3.h.b | 1 | |
4.b | odd | 2 | 1 | 320.3.h.a | 1 | ||
5.b | even | 2 | 1 | 320.3.h.a | 1 | ||
5.c | odd | 4 | 2 | 1600.3.b.f | 2 | ||
8.b | even | 2 | 1 | 20.3.d.b | yes | 1 | |
8.d | odd | 2 | 1 | 20.3.d.a | ✓ | 1 | |
16.e | even | 4 | 2 | 1280.3.e.b | 2 | ||
16.f | odd | 4 | 2 | 1280.3.e.c | 2 | ||
20.d | odd | 2 | 1 | CM | 320.3.h.b | 1 | |
20.e | even | 4 | 2 | 1600.3.b.f | 2 | ||
24.f | even | 2 | 1 | 180.3.f.b | 1 | ||
24.h | odd | 2 | 1 | 180.3.f.a | 1 | ||
40.e | odd | 2 | 1 | 20.3.d.b | yes | 1 | |
40.f | even | 2 | 1 | 20.3.d.a | ✓ | 1 | |
40.i | odd | 4 | 2 | 100.3.b.c | 2 | ||
40.k | even | 4 | 2 | 100.3.b.c | 2 | ||
80.k | odd | 4 | 2 | 1280.3.e.b | 2 | ||
80.q | even | 4 | 2 | 1280.3.e.c | 2 | ||
120.i | odd | 2 | 1 | 180.3.f.b | 1 | ||
120.m | even | 2 | 1 | 180.3.f.a | 1 | ||
120.q | odd | 4 | 2 | 900.3.c.h | 2 | ||
120.w | even | 4 | 2 | 900.3.c.h | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
20.3.d.a | ✓ | 1 | 8.d | odd | 2 | 1 | |
20.3.d.a | ✓ | 1 | 40.f | even | 2 | 1 | |
20.3.d.b | yes | 1 | 8.b | even | 2 | 1 | |
20.3.d.b | yes | 1 | 40.e | odd | 2 | 1 | |
100.3.b.c | 2 | 40.i | odd | 4 | 2 | ||
100.3.b.c | 2 | 40.k | even | 4 | 2 | ||
180.3.f.a | 1 | 24.h | odd | 2 | 1 | ||
180.3.f.a | 1 | 120.m | even | 2 | 1 | ||
180.3.f.b | 1 | 24.f | even | 2 | 1 | ||
180.3.f.b | 1 | 120.i | odd | 2 | 1 | ||
320.3.h.a | 1 | 4.b | odd | 2 | 1 | ||
320.3.h.a | 1 | 5.b | even | 2 | 1 | ||
320.3.h.b | 1 | 1.a | even | 1 | 1 | trivial | |
320.3.h.b | 1 | 20.d | odd | 2 | 1 | CM | |
900.3.c.h | 2 | 120.q | odd | 4 | 2 | ||
900.3.c.h | 2 | 120.w | even | 4 | 2 | ||
1280.3.e.b | 2 | 16.e | even | 4 | 2 | ||
1280.3.e.b | 2 | 80.k | odd | 4 | 2 | ||
1280.3.e.c | 2 | 16.f | odd | 4 | 2 | ||
1280.3.e.c | 2 | 80.q | even | 4 | 2 | ||
1600.3.b.f | 2 | 5.c | odd | 4 | 2 | ||
1600.3.b.f | 2 | 20.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 4 \)
acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 4 \)
$5$
\( T - 5 \)
$7$
\( T - 4 \)
$11$
\( T \)
$13$
\( T \)
$17$
\( T \)
$19$
\( T \)
$23$
\( T + 44 \)
$29$
\( T - 22 \)
$31$
\( T \)
$37$
\( T \)
$41$
\( T - 62 \)
$43$
\( T + 76 \)
$47$
\( T - 4 \)
$53$
\( T \)
$59$
\( T \)
$61$
\( T - 58 \)
$67$
\( T - 116 \)
$71$
\( T \)
$73$
\( T \)
$79$
\( T \)
$83$
\( T + 76 \)
$89$
\( T + 142 \)
$97$
\( T \)
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