Newspace parameters
Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 160.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(1.27760643234\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{2}) \) |
Defining polynomial: |
\( x^{2} - 2 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
0 | −2.82843 | 0 | 1.00000 | 0 | 2.82843 | 0 | 5.00000 | 0 | ||||||||||||||||||||||||
1.2 | 0 | 2.82843 | 0 | 1.00000 | 0 | −2.82843 | 0 | 5.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(5\) | \(-1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 160.2.a.c | ✓ | 2 |
3.b | odd | 2 | 1 | 1440.2.a.o | 2 | ||
4.b | odd | 2 | 1 | inner | 160.2.a.c | ✓ | 2 |
5.b | even | 2 | 1 | 800.2.a.m | 2 | ||
5.c | odd | 4 | 2 | 800.2.c.f | 4 | ||
7.b | odd | 2 | 1 | 7840.2.a.bf | 2 | ||
8.b | even | 2 | 1 | 320.2.a.g | 2 | ||
8.d | odd | 2 | 1 | 320.2.a.g | 2 | ||
12.b | even | 2 | 1 | 1440.2.a.o | 2 | ||
15.d | odd | 2 | 1 | 7200.2.a.cm | 2 | ||
15.e | even | 4 | 2 | 7200.2.f.bh | 4 | ||
16.e | even | 4 | 2 | 1280.2.d.l | 4 | ||
16.f | odd | 4 | 2 | 1280.2.d.l | 4 | ||
20.d | odd | 2 | 1 | 800.2.a.m | 2 | ||
20.e | even | 4 | 2 | 800.2.c.f | 4 | ||
24.f | even | 2 | 1 | 2880.2.a.bk | 2 | ||
24.h | odd | 2 | 1 | 2880.2.a.bk | 2 | ||
28.d | even | 2 | 1 | 7840.2.a.bf | 2 | ||
40.e | odd | 2 | 1 | 1600.2.a.bc | 2 | ||
40.f | even | 2 | 1 | 1600.2.a.bc | 2 | ||
40.i | odd | 4 | 2 | 1600.2.c.n | 4 | ||
40.k | even | 4 | 2 | 1600.2.c.n | 4 | ||
60.h | even | 2 | 1 | 7200.2.a.cm | 2 | ||
60.l | odd | 4 | 2 | 7200.2.f.bh | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
160.2.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
160.2.a.c | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
320.2.a.g | 2 | 8.b | even | 2 | 1 | ||
320.2.a.g | 2 | 8.d | odd | 2 | 1 | ||
800.2.a.m | 2 | 5.b | even | 2 | 1 | ||
800.2.a.m | 2 | 20.d | odd | 2 | 1 | ||
800.2.c.f | 4 | 5.c | odd | 4 | 2 | ||
800.2.c.f | 4 | 20.e | even | 4 | 2 | ||
1280.2.d.l | 4 | 16.e | even | 4 | 2 | ||
1280.2.d.l | 4 | 16.f | odd | 4 | 2 | ||
1440.2.a.o | 2 | 3.b | odd | 2 | 1 | ||
1440.2.a.o | 2 | 12.b | even | 2 | 1 | ||
1600.2.a.bc | 2 | 40.e | odd | 2 | 1 | ||
1600.2.a.bc | 2 | 40.f | even | 2 | 1 | ||
1600.2.c.n | 4 | 40.i | odd | 4 | 2 | ||
1600.2.c.n | 4 | 40.k | even | 4 | 2 | ||
2880.2.a.bk | 2 | 24.f | even | 2 | 1 | ||
2880.2.a.bk | 2 | 24.h | odd | 2 | 1 | ||
7200.2.a.cm | 2 | 15.d | odd | 2 | 1 | ||
7200.2.a.cm | 2 | 60.h | even | 2 | 1 | ||
7200.2.f.bh | 4 | 15.e | even | 4 | 2 | ||
7200.2.f.bh | 4 | 60.l | odd | 4 | 2 | ||
7840.2.a.bf | 2 | 7.b | odd | 2 | 1 | ||
7840.2.a.bf | 2 | 28.d | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(160))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 8 \)
$5$
\( (T - 1)^{2} \)
$7$
\( T^{2} - 8 \)
$11$
\( T^{2} - 32 \)
$13$
\( (T + 2)^{2} \)
$17$
\( (T - 2)^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} - 8 \)
$29$
\( (T - 6)^{2} \)
$31$
\( T^{2} - 32 \)
$37$
\( (T + 10)^{2} \)
$41$
\( (T - 2)^{2} \)
$43$
\( T^{2} - 72 \)
$47$
\( T^{2} - 8 \)
$53$
\( (T - 6)^{2} \)
$59$
\( T^{2} - 128 \)
$61$
\( (T + 2)^{2} \)
$67$
\( T^{2} - 8 \)
$71$
\( T^{2} - 32 \)
$73$
\( (T + 6)^{2} \)
$79$
\( T^{2} - 128 \)
$83$
\( T^{2} - 8 \)
$89$
\( (T - 10)^{2} \)
$97$
\( (T - 2)^{2} \)
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