Newspace parameters
Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 70.i (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.558952814149\) |
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 \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(57\) |
\(\chi(n)\) | \(-1 + \zeta_{12}^{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 |
|
−0.866025 | − | 0.500000i | 2.59808 | − | 1.50000i | 0.500000 | + | 0.866025i | −2.23205 | − | 0.133975i | −3.00000 | −0.866025 | + | 2.50000i | − | 1.00000i | 3.00000 | − | 5.19615i | 1.86603 | + | 1.23205i | |||||||||||||||
9.2 | 0.866025 | + | 0.500000i | −2.59808 | + | 1.50000i | 0.500000 | + | 0.866025i | 1.23205 | + | 1.86603i | −3.00000 | 0.866025 | − | 2.50000i | 1.00000i | 3.00000 | − | 5.19615i | 0.133975 | + | 2.23205i | |||||||||||||||||
39.1 | −0.866025 | + | 0.500000i | 2.59808 | + | 1.50000i | 0.500000 | − | 0.866025i | −2.23205 | + | 0.133975i | −3.00000 | −0.866025 | − | 2.50000i | 1.00000i | 3.00000 | + | 5.19615i | 1.86603 | − | 1.23205i | |||||||||||||||||
39.2 | 0.866025 | − | 0.500000i | −2.59808 | − | 1.50000i | 0.500000 | − | 0.866025i | 1.23205 | − | 1.86603i | −3.00000 | 0.866025 | + | 2.50000i | − | 1.00000i | 3.00000 | + | 5.19615i | 0.133975 | − | 2.23205i | ||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
35.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 70.2.i.a | ✓ | 4 |
3.b | odd | 2 | 1 | 630.2.u.b | 4 | ||
4.b | odd | 2 | 1 | 560.2.bw.c | 4 | ||
5.b | even | 2 | 1 | inner | 70.2.i.a | ✓ | 4 |
5.c | odd | 4 | 1 | 350.2.e.f | 2 | ||
5.c | odd | 4 | 1 | 350.2.e.g | 2 | ||
7.b | odd | 2 | 1 | 490.2.i.b | 4 | ||
7.c | even | 3 | 1 | inner | 70.2.i.a | ✓ | 4 |
7.c | even | 3 | 1 | 490.2.c.c | 2 | ||
7.d | odd | 6 | 1 | 490.2.c.b | 2 | ||
7.d | odd | 6 | 1 | 490.2.i.b | 4 | ||
15.d | odd | 2 | 1 | 630.2.u.b | 4 | ||
20.d | odd | 2 | 1 | 560.2.bw.c | 4 | ||
21.h | odd | 6 | 1 | 630.2.u.b | 4 | ||
28.g | odd | 6 | 1 | 560.2.bw.c | 4 | ||
35.c | odd | 2 | 1 | 490.2.i.b | 4 | ||
35.i | odd | 6 | 1 | 490.2.c.b | 2 | ||
35.i | odd | 6 | 1 | 490.2.i.b | 4 | ||
35.j | even | 6 | 1 | inner | 70.2.i.a | ✓ | 4 |
35.j | even | 6 | 1 | 490.2.c.c | 2 | ||
35.k | even | 12 | 1 | 2450.2.a.c | 1 | ||
35.k | even | 12 | 1 | 2450.2.a.bh | 1 | ||
35.l | odd | 12 | 1 | 350.2.e.f | 2 | ||
35.l | odd | 12 | 1 | 350.2.e.g | 2 | ||
35.l | odd | 12 | 1 | 2450.2.a.r | 1 | ||
35.l | odd | 12 | 1 | 2450.2.a.s | 1 | ||
105.o | odd | 6 | 1 | 630.2.u.b | 4 | ||
140.p | odd | 6 | 1 | 560.2.bw.c | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
70.2.i.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
70.2.i.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
70.2.i.a | ✓ | 4 | 7.c | even | 3 | 1 | inner |
70.2.i.a | ✓ | 4 | 35.j | even | 6 | 1 | inner |
350.2.e.f | 2 | 5.c | odd | 4 | 1 | ||
350.2.e.f | 2 | 35.l | odd | 12 | 1 | ||
350.2.e.g | 2 | 5.c | odd | 4 | 1 | ||
350.2.e.g | 2 | 35.l | odd | 12 | 1 | ||
490.2.c.b | 2 | 7.d | odd | 6 | 1 | ||
490.2.c.b | 2 | 35.i | odd | 6 | 1 | ||
490.2.c.c | 2 | 7.c | even | 3 | 1 | ||
490.2.c.c | 2 | 35.j | even | 6 | 1 | ||
490.2.i.b | 4 | 7.b | odd | 2 | 1 | ||
490.2.i.b | 4 | 7.d | odd | 6 | 1 | ||
490.2.i.b | 4 | 35.c | odd | 2 | 1 | ||
490.2.i.b | 4 | 35.i | odd | 6 | 1 | ||
560.2.bw.c | 4 | 4.b | odd | 2 | 1 | ||
560.2.bw.c | 4 | 20.d | odd | 2 | 1 | ||
560.2.bw.c | 4 | 28.g | odd | 6 | 1 | ||
560.2.bw.c | 4 | 140.p | odd | 6 | 1 | ||
630.2.u.b | 4 | 3.b | odd | 2 | 1 | ||
630.2.u.b | 4 | 15.d | odd | 2 | 1 | ||
630.2.u.b | 4 | 21.h | odd | 6 | 1 | ||
630.2.u.b | 4 | 105.o | odd | 6 | 1 | ||
2450.2.a.c | 1 | 35.k | even | 12 | 1 | ||
2450.2.a.r | 1 | 35.l | odd | 12 | 1 | ||
2450.2.a.s | 1 | 35.l | odd | 12 | 1 | ||
2450.2.a.bh | 1 | 35.k | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 9T_{3}^{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(70, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{2} + 1 \)
$3$
\( T^{4} - 9T^{2} + 81 \)
$5$
\( T^{4} + 2 T^{3} - T^{2} + 10 T + 25 \)
$7$
\( T^{4} + 11T^{2} + 49 \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + 4)^{2} \)
$17$
\( T^{4} - 4T^{2} + 16 \)
$19$
\( (T^{2} + 2 T + 4)^{2} \)
$23$
\( T^{4} - T^{2} + 1 \)
$29$
\( (T - 1)^{4} \)
$31$
\( (T^{2} + 10 T + 100)^{2} \)
$37$
\( T^{4} - 64T^{2} + 4096 \)
$41$
\( (T + 3)^{4} \)
$43$
\( (T^{2} + 25)^{2} \)
$47$
\( T^{4} - 64T^{2} + 4096 \)
$53$
\( T^{4} - 36T^{2} + 1296 \)
$59$
\( (T^{2} - 2 T + 4)^{2} \)
$61$
\( (T^{2} - 9 T + 81)^{2} \)
$67$
\( T^{4} - 49T^{2} + 2401 \)
$71$
\( (T - 6)^{4} \)
$73$
\( T^{4} - 100 T^{2} + 10000 \)
$79$
\( (T^{2} + 10 T + 100)^{2} \)
$83$
\( (T^{2} + 81)^{2} \)
$89$
\( (T^{2} + 7 T + 49)^{2} \)
$97$
\( T^{4} \)
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