Newspace parameters
Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 19 \) |
Character orbit: | \([\chi]\) | \(=\) | 39.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(80.1005937068\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - 3 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 3^{3}\cdot 17 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 459\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
\(n\) | \(14\) | \(28\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
38.1 |
|
−795.011 | 19683.0 | 369899. | −2.22285e6 | −1.56482e7 | 0 | −8.56664e7 | 3.87420e8 | 1.76719e9 | ||||||||||||||||||||||||
38.2 | 795.011 | 19683.0 | 369899. | 2.22285e6 | 1.56482e7 | 0 | 8.56664e7 | 3.87420e8 | 1.76719e9 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
39.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-39}) \) |
3.b | odd | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 39.19.d.c | ✓ | 2 |
3.b | odd | 2 | 1 | inner | 39.19.d.c | ✓ | 2 |
13.b | even | 2 | 1 | inner | 39.19.d.c | ✓ | 2 |
39.d | odd | 2 | 1 | CM | 39.19.d.c | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
39.19.d.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
39.19.d.c | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
39.19.d.c | ✓ | 2 | 13.b | even | 2 | 1 | inner |
39.19.d.c | ✓ | 2 | 39.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 632043 \)
acting on \(S_{19}^{\mathrm{new}}(39, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 632043 \)
$3$
\( (T - 19683)^{2} \)
$5$
\( T^{2} - 4941069469488 \)
$7$
\( T^{2} \)
$11$
\( T^{2} - 21\!\cdots\!72 \)
$13$
\( (T + 10604499373)^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( T^{2} - 23\!\cdots\!72 \)
$43$
\( (T + 765985469578630)^{2} \)
$47$
\( T^{2} - 19\!\cdots\!48 \)
$53$
\( T^{2} \)
$59$
\( T^{2} - 21\!\cdots\!32 \)
$61$
\( (T + 16\!\cdots\!70)^{2} \)
$67$
\( T^{2} \)
$71$
\( T^{2} - 28\!\cdots\!32 \)
$73$
\( T^{2} \)
$79$
\( (T + 57\!\cdots\!50)^{2} \)
$83$
\( T^{2} - 18\!\cdots\!48 \)
$89$
\( T^{2} - 13\!\cdots\!12 \)
$97$
\( T^{2} \)
show more
show less