Newspace parameters
Level: | \( N \) | \(=\) | \( 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 13.e (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.767024830075\) |
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: |
\( x^{2} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 |
|
1.50000 | + | 0.866025i | −1.00000 | + | 1.73205i | −2.50000 | − | 4.33013i | − | 1.73205i | −3.00000 | + | 1.73205i | −12.0000 | + | 6.92820i | − | 22.5167i | 11.5000 | + | 19.9186i | 1.50000 | − | 2.59808i | ||||||||
10.1 | 1.50000 | − | 0.866025i | −1.00000 | − | 1.73205i | −2.50000 | + | 4.33013i | 1.73205i | −3.00000 | − | 1.73205i | −12.0000 | − | 6.92820i | 22.5167i | 11.5000 | − | 19.9186i | 1.50000 | + | 2.59808i | |||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 13.4.e.b | ✓ | 2 |
3.b | odd | 2 | 1 | 117.4.q.a | 2 | ||
4.b | odd | 2 | 1 | 208.4.w.b | 2 | ||
13.b | even | 2 | 1 | 169.4.e.a | 2 | ||
13.c | even | 3 | 1 | 169.4.b.d | 2 | ||
13.c | even | 3 | 1 | 169.4.e.a | 2 | ||
13.d | odd | 4 | 2 | 169.4.c.h | 4 | ||
13.e | even | 6 | 1 | inner | 13.4.e.b | ✓ | 2 |
13.e | even | 6 | 1 | 169.4.b.d | 2 | ||
13.f | odd | 12 | 2 | 169.4.a.i | 2 | ||
13.f | odd | 12 | 2 | 169.4.c.h | 4 | ||
39.h | odd | 6 | 1 | 117.4.q.a | 2 | ||
39.k | even | 12 | 2 | 1521.4.a.o | 2 | ||
52.i | odd | 6 | 1 | 208.4.w.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
13.4.e.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
13.4.e.b | ✓ | 2 | 13.e | even | 6 | 1 | inner |
117.4.q.a | 2 | 3.b | odd | 2 | 1 | ||
117.4.q.a | 2 | 39.h | odd | 6 | 1 | ||
169.4.a.i | 2 | 13.f | odd | 12 | 2 | ||
169.4.b.d | 2 | 13.c | even | 3 | 1 | ||
169.4.b.d | 2 | 13.e | even | 6 | 1 | ||
169.4.c.h | 4 | 13.d | odd | 4 | 2 | ||
169.4.c.h | 4 | 13.f | odd | 12 | 2 | ||
169.4.e.a | 2 | 13.b | even | 2 | 1 | ||
169.4.e.a | 2 | 13.c | even | 3 | 1 | ||
208.4.w.b | 2 | 4.b | odd | 2 | 1 | ||
208.4.w.b | 2 | 52.i | odd | 6 | 1 | ||
1521.4.a.o | 2 | 39.k | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} + 3 \)
acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 3T + 3 \)
$3$
\( T^{2} + 2T + 4 \)
$5$
\( T^{2} + 3 \)
$7$
\( T^{2} + 24T + 192 \)
$11$
\( T^{2} - 24T + 192 \)
$13$
\( T^{2} - 91T + 2197 \)
$17$
\( T^{2} + 117T + 13689 \)
$19$
\( T^{2} + 198T + 13068 \)
$23$
\( T^{2} - 78T + 6084 \)
$29$
\( T^{2} - 141T + 19881 \)
$31$
\( T^{2} + 24300 \)
$37$
\( T^{2} + 249T + 20667 \)
$41$
\( T^{2} - 471T + 73947 \)
$43$
\( T^{2} + 104T + 10816 \)
$47$
\( T^{2} + 90828 \)
$53$
\( (T - 93)^{2} \)
$59$
\( T^{2} + 492T + 80688 \)
$61$
\( T^{2} + 145T + 21025 \)
$67$
\( T^{2} + 1362 T + 618348 \)
$71$
\( T^{2} - 1830 T + 1116300 \)
$73$
\( T^{2} + 210675 \)
$79$
\( (T - 1276)^{2} \)
$83$
\( T^{2} + 623808 \)
$89$
\( T^{2} + 1692 T + 954288 \)
$97$
\( T^{2} - 348T + 40368 \)
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