Newspace parameters
Level: | \( N \) | \(=\) | \( 578 = 2 \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 578.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.61535323683\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{2}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - 2 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 34) |
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 |
|
1.00000 | −2.82843 | 1.00000 | 2.82843 | −2.82843 | 0 | 1.00000 | 5.00000 | 2.82843 | ||||||||||||||||||||||||
1.2 | 1.00000 | 2.82843 | 1.00000 | −2.82843 | 2.82843 | 0 | 1.00000 | 5.00000 | −2.82843 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(17\) | \(1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 578.2.a.d | 2 | |
3.b | odd | 2 | 1 | 5202.2.a.u | 2 | ||
4.b | odd | 2 | 1 | 4624.2.a.s | 2 | ||
17.b | even | 2 | 1 | inner | 578.2.a.d | 2 | |
17.c | even | 4 | 2 | 34.2.b.a | ✓ | 2 | |
17.d | even | 8 | 2 | 578.2.c.a | 2 | ||
17.d | even | 8 | 2 | 578.2.c.d | 2 | ||
17.e | odd | 16 | 8 | 578.2.d.g | 8 | ||
51.c | odd | 2 | 1 | 5202.2.a.u | 2 | ||
51.f | odd | 4 | 2 | 306.2.b.d | 2 | ||
68.d | odd | 2 | 1 | 4624.2.a.s | 2 | ||
68.f | odd | 4 | 2 | 272.2.b.a | 2 | ||
85.f | odd | 4 | 2 | 850.2.d.i | 4 | ||
85.i | odd | 4 | 2 | 850.2.d.i | 4 | ||
85.j | even | 4 | 2 | 850.2.b.f | 2 | ||
119.f | odd | 4 | 2 | 1666.2.b.c | 2 | ||
136.i | even | 4 | 2 | 1088.2.b.b | 2 | ||
136.j | odd | 4 | 2 | 1088.2.b.a | 2 | ||
204.l | even | 4 | 2 | 2448.2.c.n | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
34.2.b.a | ✓ | 2 | 17.c | even | 4 | 2 | |
272.2.b.a | 2 | 68.f | odd | 4 | 2 | ||
306.2.b.d | 2 | 51.f | odd | 4 | 2 | ||
578.2.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
578.2.a.d | 2 | 17.b | even | 2 | 1 | inner | |
578.2.c.a | 2 | 17.d | even | 8 | 2 | ||
578.2.c.d | 2 | 17.d | even | 8 | 2 | ||
578.2.d.g | 8 | 17.e | odd | 16 | 8 | ||
850.2.b.f | 2 | 85.j | even | 4 | 2 | ||
850.2.d.i | 4 | 85.f | odd | 4 | 2 | ||
850.2.d.i | 4 | 85.i | odd | 4 | 2 | ||
1088.2.b.a | 2 | 136.j | odd | 4 | 2 | ||
1088.2.b.b | 2 | 136.i | even | 4 | 2 | ||
1666.2.b.c | 2 | 119.f | odd | 4 | 2 | ||
2448.2.c.n | 2 | 204.l | even | 4 | 2 | ||
4624.2.a.s | 2 | 4.b | odd | 2 | 1 | ||
4624.2.a.s | 2 | 68.d | odd | 2 | 1 | ||
5202.2.a.u | 2 | 3.b | odd | 2 | 1 | ||
5202.2.a.u | 2 | 51.c | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(578))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{2} \)
$3$
\( T^{2} - 8 \)
$5$
\( T^{2} - 8 \)
$7$
\( T^{2} \)
$11$
\( T^{2} - 8 \)
$13$
\( (T - 2)^{2} \)
$17$
\( T^{2} \)
$19$
\( (T - 4)^{2} \)
$23$
\( T^{2} - 32 \)
$29$
\( T^{2} - 8 \)
$31$
\( T^{2} \)
$37$
\( T^{2} - 72 \)
$41$
\( T^{2} - 32 \)
$43$
\( (T - 4)^{2} \)
$47$
\( T^{2} \)
$53$
\( (T + 6)^{2} \)
$59$
\( (T + 12)^{2} \)
$61$
\( T^{2} - 72 \)
$67$
\( (T + 4)^{2} \)
$71$
\( T^{2} - 32 \)
$73$
\( T^{2} \)
$79$
\( T^{2} - 288 \)
$83$
\( (T - 12)^{2} \)
$89$
\( (T - 6)^{2} \)
$97$
\( T^{2} - 288 \)
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