Newspace parameters
Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 39.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.311416567883\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
Defining polynomial: |
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 4 ) / 5 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \)
|
\(\nu^{3}\) | \(=\) |
\( 5\beta_{3} - 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
\(n\) | \(14\) | \(28\) |
\(\chi(n)\) | \(1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 |
|
−1.28078 | − | 2.21837i | −0.500000 | − | 0.866025i | −2.28078 | + | 3.95042i | 0.561553 | −1.28078 | + | 2.21837i | 1.78078 | − | 3.08440i | 6.56155 | −0.500000 | + | 0.866025i | −0.719224 | − | 1.24573i | ||||||||||||||||
16.2 | 0.780776 | + | 1.35234i | −0.500000 | − | 0.866025i | −0.219224 | + | 0.379706i | −3.56155 | 0.780776 | − | 1.35234i | −0.280776 | + | 0.486319i | 2.43845 | −0.500000 | + | 0.866025i | −2.78078 | − | 4.81645i | |||||||||||||||||
22.1 | −1.28078 | + | 2.21837i | −0.500000 | + | 0.866025i | −2.28078 | − | 3.95042i | 0.561553 | −1.28078 | − | 2.21837i | 1.78078 | + | 3.08440i | 6.56155 | −0.500000 | − | 0.866025i | −0.719224 | + | 1.24573i | |||||||||||||||||
22.2 | 0.780776 | − | 1.35234i | −0.500000 | + | 0.866025i | −0.219224 | − | 0.379706i | −3.56155 | 0.780776 | + | 1.35234i | −0.280776 | − | 0.486319i | 2.43845 | −0.500000 | − | 0.866025i | −2.78078 | + | 4.81645i | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 39.2.e.b | ✓ | 4 |
3.b | odd | 2 | 1 | 117.2.g.c | 4 | ||
4.b | odd | 2 | 1 | 624.2.q.h | 4 | ||
5.b | even | 2 | 1 | 975.2.i.k | 4 | ||
5.c | odd | 4 | 2 | 975.2.bb.i | 8 | ||
12.b | even | 2 | 1 | 1872.2.t.r | 4 | ||
13.b | even | 2 | 1 | 507.2.e.g | 4 | ||
13.c | even | 3 | 1 | inner | 39.2.e.b | ✓ | 4 |
13.c | even | 3 | 1 | 507.2.a.g | 2 | ||
13.d | odd | 4 | 2 | 507.2.j.g | 8 | ||
13.e | even | 6 | 1 | 507.2.a.d | 2 | ||
13.e | even | 6 | 1 | 507.2.e.g | 4 | ||
13.f | odd | 12 | 2 | 507.2.b.d | 4 | ||
13.f | odd | 12 | 2 | 507.2.j.g | 8 | ||
39.h | odd | 6 | 1 | 1521.2.a.m | 2 | ||
39.i | odd | 6 | 1 | 117.2.g.c | 4 | ||
39.i | odd | 6 | 1 | 1521.2.a.g | 2 | ||
39.k | even | 12 | 2 | 1521.2.b.h | 4 | ||
52.i | odd | 6 | 1 | 8112.2.a.bo | 2 | ||
52.j | odd | 6 | 1 | 624.2.q.h | 4 | ||
52.j | odd | 6 | 1 | 8112.2.a.bk | 2 | ||
65.n | even | 6 | 1 | 975.2.i.k | 4 | ||
65.q | odd | 12 | 2 | 975.2.bb.i | 8 | ||
156.p | even | 6 | 1 | 1872.2.t.r | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
39.2.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
39.2.e.b | ✓ | 4 | 13.c | even | 3 | 1 | inner |
117.2.g.c | 4 | 3.b | odd | 2 | 1 | ||
117.2.g.c | 4 | 39.i | odd | 6 | 1 | ||
507.2.a.d | 2 | 13.e | even | 6 | 1 | ||
507.2.a.g | 2 | 13.c | even | 3 | 1 | ||
507.2.b.d | 4 | 13.f | odd | 12 | 2 | ||
507.2.e.g | 4 | 13.b | even | 2 | 1 | ||
507.2.e.g | 4 | 13.e | even | 6 | 1 | ||
507.2.j.g | 8 | 13.d | odd | 4 | 2 | ||
507.2.j.g | 8 | 13.f | odd | 12 | 2 | ||
624.2.q.h | 4 | 4.b | odd | 2 | 1 | ||
624.2.q.h | 4 | 52.j | odd | 6 | 1 | ||
975.2.i.k | 4 | 5.b | even | 2 | 1 | ||
975.2.i.k | 4 | 65.n | even | 6 | 1 | ||
975.2.bb.i | 8 | 5.c | odd | 4 | 2 | ||
975.2.bb.i | 8 | 65.q | odd | 12 | 2 | ||
1521.2.a.g | 2 | 39.i | odd | 6 | 1 | ||
1521.2.a.m | 2 | 39.h | odd | 6 | 1 | ||
1521.2.b.h | 4 | 39.k | even | 12 | 2 | ||
1872.2.t.r | 4 | 12.b | even | 2 | 1 | ||
1872.2.t.r | 4 | 156.p | even | 6 | 1 | ||
8112.2.a.bk | 2 | 52.j | odd | 6 | 1 | ||
8112.2.a.bo | 2 | 52.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + T_{2}^{3} + 5T_{2}^{2} - 4T_{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(39, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \)
$3$
\( (T^{2} + T + 1)^{2} \)
$5$
\( (T^{2} + 3 T - 2)^{2} \)
$7$
\( T^{4} - 3 T^{3} + 11 T^{2} + 6 T + 4 \)
$11$
\( (T^{2} - 2 T + 4)^{2} \)
$13$
\( (T^{2} - T + 13)^{2} \)
$17$
\( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \)
$19$
\( T^{4} + 6 T^{3} + 44 T^{2} - 48 T + 64 \)
$23$
\( (T^{2} + 2 T + 4)^{2} \)
$29$
\( T^{4} + T^{3} + 39 T^{2} - 38 T + 1444 \)
$31$
\( (T^{2} - T - 4)^{2} \)
$37$
\( T^{4} + 11 T^{3} + 95 T^{2} + \cdots + 676 \)
$41$
\( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \)
$43$
\( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \)
$47$
\( (T^{2} - 68)^{2} \)
$53$
\( (T^{2} - 11 T - 8)^{2} \)
$59$
\( T^{4} - 14 T^{3} + 164 T^{2} + \cdots + 1024 \)
$61$
\( T^{4} + 16 T^{3} + 209 T^{2} + \cdots + 2209 \)
$67$
\( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \)
$71$
\( (T^{2} + 14 T + 196)^{2} \)
$73$
\( (T^{2} + 12 T + 19)^{2} \)
$79$
\( (T^{2} - 15 T + 52)^{2} \)
$83$
\( (T^{2} + 10 T + 8)^{2} \)
$89$
\( T^{4} + 18 T^{3} + 260 T^{2} + \cdots + 4096 \)
$97$
\( T^{4} - 13 T^{3} + 131 T^{2} + \cdots + 1444 \)
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