Newspace parameters
Level: | \( N \) | \(=\) | \( 130 = 2 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 130.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.03805522628\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).
\(n\) | \(27\) | \(41\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
61.1 |
|
0.500000 | − | 0.866025i | 1.00000 | − | 1.73205i | −0.500000 | − | 0.866025i | −1.00000 | −1.00000 | − | 1.73205i | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | ||||||||||
81.1 | 0.500000 | + | 0.866025i | 1.00000 | + | 1.73205i | −0.500000 | + | 0.866025i | −1.00000 | −1.00000 | + | 1.73205i | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | |||||||||||
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 | 130.2.e.b | ✓ | 2 |
3.b | odd | 2 | 1 | 1170.2.i.f | 2 | ||
4.b | odd | 2 | 1 | 1040.2.q.c | 2 | ||
5.b | even | 2 | 1 | 650.2.e.a | 2 | ||
5.c | odd | 4 | 2 | 650.2.o.b | 4 | ||
13.b | even | 2 | 1 | 1690.2.e.e | 2 | ||
13.c | even | 3 | 1 | inner | 130.2.e.b | ✓ | 2 |
13.c | even | 3 | 1 | 1690.2.a.a | 1 | ||
13.d | odd | 4 | 2 | 1690.2.l.i | 4 | ||
13.e | even | 6 | 1 | 1690.2.a.g | 1 | ||
13.e | even | 6 | 1 | 1690.2.e.e | 2 | ||
13.f | odd | 12 | 2 | 1690.2.d.a | 2 | ||
13.f | odd | 12 | 2 | 1690.2.l.i | 4 | ||
39.i | odd | 6 | 1 | 1170.2.i.f | 2 | ||
52.j | odd | 6 | 1 | 1040.2.q.c | 2 | ||
65.l | even | 6 | 1 | 8450.2.a.k | 1 | ||
65.n | even | 6 | 1 | 650.2.e.a | 2 | ||
65.n | even | 6 | 1 | 8450.2.a.w | 1 | ||
65.q | odd | 12 | 2 | 650.2.o.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
130.2.e.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
130.2.e.b | ✓ | 2 | 13.c | even | 3 | 1 | inner |
650.2.e.a | 2 | 5.b | even | 2 | 1 | ||
650.2.e.a | 2 | 65.n | even | 6 | 1 | ||
650.2.o.b | 4 | 5.c | odd | 4 | 2 | ||
650.2.o.b | 4 | 65.q | odd | 12 | 2 | ||
1040.2.q.c | 2 | 4.b | odd | 2 | 1 | ||
1040.2.q.c | 2 | 52.j | odd | 6 | 1 | ||
1170.2.i.f | 2 | 3.b | odd | 2 | 1 | ||
1170.2.i.f | 2 | 39.i | odd | 6 | 1 | ||
1690.2.a.a | 1 | 13.c | even | 3 | 1 | ||
1690.2.a.g | 1 | 13.e | even | 6 | 1 | ||
1690.2.d.a | 2 | 13.f | odd | 12 | 2 | ||
1690.2.e.e | 2 | 13.b | even | 2 | 1 | ||
1690.2.e.e | 2 | 13.e | even | 6 | 1 | ||
1690.2.l.i | 4 | 13.d | odd | 4 | 2 | ||
1690.2.l.i | 4 | 13.f | odd | 12 | 2 | ||
8450.2.a.k | 1 | 65.l | even | 6 | 1 | ||
8450.2.a.w | 1 | 65.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(130, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - T + 1 \)
$3$
\( T^{2} - 2T + 4 \)
$5$
\( (T + 1)^{2} \)
$7$
\( T^{2} - T + 1 \)
$11$
\( T^{2} + 3T + 9 \)
$13$
\( T^{2} - 5T + 13 \)
$17$
\( T^{2} - 6T + 36 \)
$19$
\( T^{2} + 5T + 25 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( (T + 4)^{2} \)
$37$
\( T^{2} + 11T + 121 \)
$41$
\( T^{2} + 6T + 36 \)
$43$
\( T^{2} + 2T + 4 \)
$47$
\( (T + 3)^{2} \)
$53$
\( (T + 9)^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 8T + 64 \)
$67$
\( T^{2} - 16T + 256 \)
$71$
\( T^{2} + 6T + 36 \)
$73$
\( (T - 14)^{2} \)
$79$
\( (T + 16)^{2} \)
$83$
\( (T + 6)^{2} \)
$89$
\( T^{2} + 9T + 81 \)
$97$
\( T^{2} - 10T + 100 \)
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