Newspace parameters
Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 78.i (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.622833135766\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).
\(n\) | \(53\) | \(67\) |
\(\chi(n)\) | \(1\) | \(\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 |
|
−0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | − | 1.73205i | −0.866025 | + | 0.500000i | 1.09808 | − | 0.633975i | − | 1.00000i | −0.500000 | − | 0.866025i | −0.866025 | + | 1.50000i | ||||||||||||||
43.2 | 0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | − | 1.73205i | 0.866025 | − | 0.500000i | −4.09808 | + | 2.36603i | 1.00000i | −0.500000 | − | 0.866025i | 0.866025 | − | 1.50000i | ||||||||||||||||
49.1 | −0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 1.73205i | −0.866025 | − | 0.500000i | 1.09808 | + | 0.633975i | 1.00000i | −0.500000 | + | 0.866025i | −0.866025 | − | 1.50000i | |||||||||||||||||
49.2 | 0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 1.73205i | 0.866025 | + | 0.500000i | −4.09808 | − | 2.36603i | − | 1.00000i | −0.500000 | + | 0.866025i | 0.866025 | + | 1.50000i | ||||||||||||||||
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 | 78.2.i.b | ✓ | 4 |
3.b | odd | 2 | 1 | 234.2.l.a | 4 | ||
4.b | odd | 2 | 1 | 624.2.bv.d | 4 | ||
5.b | even | 2 | 1 | 1950.2.bc.c | 4 | ||
5.c | odd | 4 | 1 | 1950.2.y.a | 4 | ||
5.c | odd | 4 | 1 | 1950.2.y.h | 4 | ||
12.b | even | 2 | 1 | 1872.2.by.k | 4 | ||
13.b | even | 2 | 1 | 1014.2.i.f | 4 | ||
13.c | even | 3 | 1 | 1014.2.b.d | 4 | ||
13.c | even | 3 | 1 | 1014.2.i.f | 4 | ||
13.d | odd | 4 | 1 | 1014.2.e.h | 4 | ||
13.d | odd | 4 | 1 | 1014.2.e.j | 4 | ||
13.e | even | 6 | 1 | inner | 78.2.i.b | ✓ | 4 |
13.e | even | 6 | 1 | 1014.2.b.d | 4 | ||
13.f | odd | 12 | 1 | 1014.2.a.h | 2 | ||
13.f | odd | 12 | 1 | 1014.2.a.j | 2 | ||
13.f | odd | 12 | 1 | 1014.2.e.h | 4 | ||
13.f | odd | 12 | 1 | 1014.2.e.j | 4 | ||
39.h | odd | 6 | 1 | 234.2.l.a | 4 | ||
39.h | odd | 6 | 1 | 3042.2.b.l | 4 | ||
39.i | odd | 6 | 1 | 3042.2.b.l | 4 | ||
39.k | even | 12 | 1 | 3042.2.a.s | 2 | ||
39.k | even | 12 | 1 | 3042.2.a.v | 2 | ||
52.i | odd | 6 | 1 | 624.2.bv.d | 4 | ||
52.l | even | 12 | 1 | 8112.2.a.bq | 2 | ||
52.l | even | 12 | 1 | 8112.2.a.bx | 2 | ||
65.l | even | 6 | 1 | 1950.2.bc.c | 4 | ||
65.r | odd | 12 | 1 | 1950.2.y.a | 4 | ||
65.r | odd | 12 | 1 | 1950.2.y.h | 4 | ||
156.r | even | 6 | 1 | 1872.2.by.k | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
78.2.i.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
78.2.i.b | ✓ | 4 | 13.e | even | 6 | 1 | inner |
234.2.l.a | 4 | 3.b | odd | 2 | 1 | ||
234.2.l.a | 4 | 39.h | odd | 6 | 1 | ||
624.2.bv.d | 4 | 4.b | odd | 2 | 1 | ||
624.2.bv.d | 4 | 52.i | odd | 6 | 1 | ||
1014.2.a.h | 2 | 13.f | odd | 12 | 1 | ||
1014.2.a.j | 2 | 13.f | odd | 12 | 1 | ||
1014.2.b.d | 4 | 13.c | even | 3 | 1 | ||
1014.2.b.d | 4 | 13.e | even | 6 | 1 | ||
1014.2.e.h | 4 | 13.d | odd | 4 | 1 | ||
1014.2.e.h | 4 | 13.f | odd | 12 | 1 | ||
1014.2.e.j | 4 | 13.d | odd | 4 | 1 | ||
1014.2.e.j | 4 | 13.f | odd | 12 | 1 | ||
1014.2.i.f | 4 | 13.b | even | 2 | 1 | ||
1014.2.i.f | 4 | 13.c | even | 3 | 1 | ||
1872.2.by.k | 4 | 12.b | even | 2 | 1 | ||
1872.2.by.k | 4 | 156.r | even | 6 | 1 | ||
1950.2.y.a | 4 | 5.c | odd | 4 | 1 | ||
1950.2.y.a | 4 | 65.r | odd | 12 | 1 | ||
1950.2.y.h | 4 | 5.c | odd | 4 | 1 | ||
1950.2.y.h | 4 | 65.r | odd | 12 | 1 | ||
1950.2.bc.c | 4 | 5.b | even | 2 | 1 | ||
1950.2.bc.c | 4 | 65.l | even | 6 | 1 | ||
3042.2.a.s | 2 | 39.k | even | 12 | 1 | ||
3042.2.a.v | 2 | 39.k | even | 12 | 1 | ||
3042.2.b.l | 4 | 39.h | odd | 6 | 1 | ||
3042.2.b.l | 4 | 39.i | odd | 6 | 1 | ||
8112.2.a.bq | 2 | 52.l | even | 12 | 1 | ||
8112.2.a.bx | 2 | 52.l | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{2} + 1 \)
$3$
\( (T^{2} - T + 1)^{2} \)
$5$
\( (T^{2} + 3)^{2} \)
$7$
\( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \)
$11$
\( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \)
$13$
\( T^{4} + 4 T^{3} + 3 T^{2} + 52 T + 169 \)
$17$
\( T^{4} + 27T^{2} + 729 \)
$19$
\( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \)
$23$
\( T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324 \)
$29$
\( (T^{2} - 3 T + 9)^{2} \)
$31$
\( T^{4} + 96T^{2} + 576 \)
$37$
\( T^{4} - 9T^{2} + 81 \)
$41$
\( T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9 \)
$43$
\( T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676 \)
$47$
\( T^{4} + 24T^{2} + 36 \)
$53$
\( (T - 3)^{4} \)
$59$
\( (T^{2} + 24 T + 192)^{2} \)
$61$
\( T^{4} + 20 T^{3} + 327 T^{2} + \cdots + 5329 \)
$67$
\( T^{4} + 6 T^{3} - 66 T^{2} + \cdots + 6084 \)
$71$
\( T^{4} - 18 T^{3} + 126 T^{2} + \cdots + 324 \)
$73$
\( (T^{2} + 147)^{2} \)
$79$
\( (T^{2} + 4 T - 104)^{2} \)
$83$
\( T^{4} + 168T^{2} + 4356 \)
$89$
\( T^{4} + 12 T^{3} + 24 T^{2} + \cdots + 576 \)
$97$
\( T^{4} - 36T^{2} + 1296 \)
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