Newspace parameters
Level: | \( N \) | \(=\) | \( 592 = 2^{4} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 592.g (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.72714379966\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 37) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).
\(n\) | \(113\) | \(149\) | \(223\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
369.1 |
|
0 | 1.00000 | 0 | − | 2.00000i | 0 | −3.00000 | 0 | −2.00000 | 0 | |||||||||||||||||||||||
369.2 | 0 | 1.00000 | 0 | 2.00000i | 0 | −3.00000 | 0 | −2.00000 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 592.2.g.b | 2 | |
3.b | odd | 2 | 1 | 5328.2.h.c | 2 | ||
4.b | odd | 2 | 1 | 37.2.b.a | ✓ | 2 | |
8.b | even | 2 | 1 | 2368.2.g.b | 2 | ||
8.d | odd | 2 | 1 | 2368.2.g.f | 2 | ||
12.b | even | 2 | 1 | 333.2.c.a | 2 | ||
20.d | odd | 2 | 1 | 925.2.c.b | 2 | ||
20.e | even | 4 | 1 | 925.2.d.a | 2 | ||
20.e | even | 4 | 1 | 925.2.d.d | 2 | ||
37.b | even | 2 | 1 | inner | 592.2.g.b | 2 | |
111.d | odd | 2 | 1 | 5328.2.h.c | 2 | ||
148.b | odd | 2 | 1 | 37.2.b.a | ✓ | 2 | |
148.g | even | 4 | 1 | 1369.2.a.a | 1 | ||
148.g | even | 4 | 1 | 1369.2.a.f | 1 | ||
296.e | even | 2 | 1 | 2368.2.g.b | 2 | ||
296.h | odd | 2 | 1 | 2368.2.g.f | 2 | ||
444.g | even | 2 | 1 | 333.2.c.a | 2 | ||
740.g | odd | 2 | 1 | 925.2.c.b | 2 | ||
740.m | even | 4 | 1 | 925.2.d.a | 2 | ||
740.m | even | 4 | 1 | 925.2.d.d | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.2.b.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
37.2.b.a | ✓ | 2 | 148.b | odd | 2 | 1 | |
333.2.c.a | 2 | 12.b | even | 2 | 1 | ||
333.2.c.a | 2 | 444.g | even | 2 | 1 | ||
592.2.g.b | 2 | 1.a | even | 1 | 1 | trivial | |
592.2.g.b | 2 | 37.b | even | 2 | 1 | inner | |
925.2.c.b | 2 | 20.d | odd | 2 | 1 | ||
925.2.c.b | 2 | 740.g | odd | 2 | 1 | ||
925.2.d.a | 2 | 20.e | even | 4 | 1 | ||
925.2.d.a | 2 | 740.m | even | 4 | 1 | ||
925.2.d.d | 2 | 20.e | even | 4 | 1 | ||
925.2.d.d | 2 | 740.m | even | 4 | 1 | ||
1369.2.a.a | 1 | 148.g | even | 4 | 1 | ||
1369.2.a.f | 1 | 148.g | even | 4 | 1 | ||
2368.2.g.b | 2 | 8.b | even | 2 | 1 | ||
2368.2.g.b | 2 | 296.e | even | 2 | 1 | ||
2368.2.g.f | 2 | 8.d | odd | 2 | 1 | ||
2368.2.g.f | 2 | 296.h | odd | 2 | 1 | ||
5328.2.h.c | 2 | 3.b | odd | 2 | 1 | ||
5328.2.h.c | 2 | 111.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T - 1)^{2} \)
$5$
\( T^{2} + 4 \)
$7$
\( (T + 3)^{2} \)
$11$
\( (T - 3)^{2} \)
$13$
\( T^{2} + 36 \)
$17$
\( T^{2} + 4 \)
$19$
\( T^{2} + 36 \)
$23$
\( T^{2} + 16 \)
$29$
\( T^{2} + 16 \)
$31$
\( T^{2} \)
$37$
\( T^{2} + 2T + 37 \)
$41$
\( (T + 3)^{2} \)
$43$
\( T^{2} + 36 \)
$47$
\( (T + 3)^{2} \)
$53$
\( (T - 9)^{2} \)
$59$
\( T^{2} + 16 \)
$61$
\( T^{2} \)
$67$
\( (T - 12)^{2} \)
$71$
\( (T - 3)^{2} \)
$73$
\( (T - 9)^{2} \)
$79$
\( T^{2} + 36 \)
$83$
\( (T + 9)^{2} \)
$89$
\( T^{2} + 196 \)
$97$
\( T^{2} + 144 \)
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