Newspace parameters
Level: | \( N \) | \(=\) | \( 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 13.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.767024830075\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{17}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x - 4 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). 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 |
|
−1.56155 | 8.68466 | −5.56155 | −3.56155 | −13.5616 | −27.1771 | 21.1771 | 48.4233 | 5.56155 | ||||||||||||||||||||||||
1.2 | 2.56155 | −3.68466 | −1.43845 | 0.561553 | −9.43845 | 18.1771 | −24.1771 | −13.4233 | 1.43845 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(13\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 13.4.a.b | ✓ | 2 |
3.b | odd | 2 | 1 | 117.4.a.d | 2 | ||
4.b | odd | 2 | 1 | 208.4.a.h | 2 | ||
5.b | even | 2 | 1 | 325.4.a.f | 2 | ||
5.c | odd | 4 | 2 | 325.4.b.e | 4 | ||
7.b | odd | 2 | 1 | 637.4.a.b | 2 | ||
8.b | even | 2 | 1 | 832.4.a.s | 2 | ||
8.d | odd | 2 | 1 | 832.4.a.z | 2 | ||
11.b | odd | 2 | 1 | 1573.4.a.b | 2 | ||
12.b | even | 2 | 1 | 1872.4.a.bb | 2 | ||
13.b | even | 2 | 1 | 169.4.a.g | 2 | ||
13.c | even | 3 | 2 | 169.4.c.g | 4 | ||
13.d | odd | 4 | 2 | 169.4.b.f | 4 | ||
13.e | even | 6 | 2 | 169.4.c.j | 4 | ||
13.f | odd | 12 | 4 | 169.4.e.f | 8 | ||
39.d | odd | 2 | 1 | 1521.4.a.r | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
117.4.a.d | 2 | 3.b | odd | 2 | 1 | ||
169.4.a.g | 2 | 13.b | even | 2 | 1 | ||
169.4.b.f | 4 | 13.d | odd | 4 | 2 | ||
169.4.c.g | 4 | 13.c | even | 3 | 2 | ||
169.4.c.j | 4 | 13.e | even | 6 | 2 | ||
169.4.e.f | 8 | 13.f | odd | 12 | 4 | ||
208.4.a.h | 2 | 4.b | odd | 2 | 1 | ||
325.4.a.f | 2 | 5.b | even | 2 | 1 | ||
325.4.b.e | 4 | 5.c | odd | 4 | 2 | ||
637.4.a.b | 2 | 7.b | odd | 2 | 1 | ||
832.4.a.s | 2 | 8.b | even | 2 | 1 | ||
832.4.a.z | 2 | 8.d | odd | 2 | 1 | ||
1521.4.a.r | 2 | 39.d | odd | 2 | 1 | ||
1573.4.a.b | 2 | 11.b | odd | 2 | 1 | ||
1872.4.a.bb | 2 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 4 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - T - 4 \)
$3$
\( T^{2} - 5T - 32 \)
$5$
\( T^{2} + 3T - 2 \)
$7$
\( T^{2} + 9T - 494 \)
$11$
\( T^{2} - 80T + 988 \)
$13$
\( (T + 13)^{2} \)
$17$
\( T^{2} - 19T - 1138 \)
$19$
\( T^{2} + 84T - 2588 \)
$23$
\( T^{2} - 196T + 8992 \)
$29$
\( T^{2} + 44T - 38684 \)
$31$
\( T^{2} + 86T - 3064 \)
$37$
\( T^{2} - 209T + 10814 \)
$41$
\( T^{2} + 230T + 11168 \)
$43$
\( T^{2} - 287T - 66316 \)
$47$
\( T^{2} - 435T - 14918 \)
$53$
\( T^{2} + 118T - 344 \)
$59$
\( T^{2} + 368T - 31492 \)
$61$
\( T^{2} + 1058 T + 126416 \)
$67$
\( T^{2} - 68T - 227596 \)
$71$
\( T^{2} + 131T - 222494 \)
$73$
\( T^{2} - 456T - 235316 \)
$79$
\( T^{2} + 1008 T + 247216 \)
$83$
\( T^{2} - 1958 T + 817664 \)
$89$
\( T^{2} + 720T - 510212 \)
$97$
\( T^{2} + 928T - 881476 \)
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