Newspace parameters
Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 160.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(9.44030560092\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{13}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x - 3 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{13}\). 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 | −7.21110 | 0 | −5.00000 | 0 | −7.21110 | 0 | 25.0000 | 0 | ||||||||||||||||||||||||
1.2 | 0 | 7.21110 | 0 | −5.00000 | 0 | 7.21110 | 0 | 25.0000 | 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 | 160.4.a.e | ✓ | 2 |
3.b | odd | 2 | 1 | 1440.4.a.bd | 2 | ||
4.b | odd | 2 | 1 | inner | 160.4.a.e | ✓ | 2 |
5.b | even | 2 | 1 | 800.4.a.q | 2 | ||
5.c | odd | 4 | 2 | 800.4.c.h | 4 | ||
8.b | even | 2 | 1 | 320.4.a.r | 2 | ||
8.d | odd | 2 | 1 | 320.4.a.r | 2 | ||
12.b | even | 2 | 1 | 1440.4.a.bd | 2 | ||
16.e | even | 4 | 2 | 1280.4.d.t | 4 | ||
16.f | odd | 4 | 2 | 1280.4.d.t | 4 | ||
20.d | odd | 2 | 1 | 800.4.a.q | 2 | ||
20.e | even | 4 | 2 | 800.4.c.h | 4 | ||
40.e | odd | 2 | 1 | 1600.4.a.ci | 2 | ||
40.f | even | 2 | 1 | 1600.4.a.ci | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
160.4.a.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
160.4.a.e | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
320.4.a.r | 2 | 8.b | even | 2 | 1 | ||
320.4.a.r | 2 | 8.d | odd | 2 | 1 | ||
800.4.a.q | 2 | 5.b | even | 2 | 1 | ||
800.4.a.q | 2 | 20.d | odd | 2 | 1 | ||
800.4.c.h | 4 | 5.c | odd | 4 | 2 | ||
800.4.c.h | 4 | 20.e | even | 4 | 2 | ||
1280.4.d.t | 4 | 16.e | even | 4 | 2 | ||
1280.4.d.t | 4 | 16.f | odd | 4 | 2 | ||
1440.4.a.bd | 2 | 3.b | odd | 2 | 1 | ||
1440.4.a.bd | 2 | 12.b | even | 2 | 1 | ||
1600.4.a.ci | 2 | 40.e | odd | 2 | 1 | ||
1600.4.a.ci | 2 | 40.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 52 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(160))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 52 \)
$5$
\( (T + 5)^{2} \)
$7$
\( T^{2} - 52 \)
$11$
\( T^{2} - 1872 \)
$13$
\( (T - 34)^{2} \)
$17$
\( (T - 114)^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} - 43732 \)
$29$
\( (T + 26)^{2} \)
$31$
\( T^{2} - 10192 \)
$37$
\( (T + 150)^{2} \)
$41$
\( (T - 342)^{2} \)
$43$
\( T^{2} - 206388 \)
$47$
\( T^{2} - 341172 \)
$53$
\( (T + 262)^{2} \)
$59$
\( T^{2} - 240448 \)
$61$
\( (T + 262)^{2} \)
$67$
\( T^{2} - 247572 \)
$71$
\( T^{2} - 1108432 \)
$73$
\( (T - 682)^{2} \)
$79$
\( T^{2} - 40768 \)
$83$
\( T^{2} - 22932 \)
$89$
\( (T + 630)^{2} \)
$97$
\( (T + 966)^{2} \)
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