Newspace parameters
Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 70.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.13013370040\) |
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: | \(\Z[a_1, \ldots, a_{5}]\) |
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 \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(57\) |
\(\chi(n)\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 |
|
1.00000 | + | 1.73205i | −5.00000 | + | 8.66025i | −2.00000 | + | 3.46410i | 2.50000 | + | 4.33013i | −20.0000 | 14.0000 | − | 12.1244i | −8.00000 | −36.5000 | − | 63.2199i | −5.00000 | + | 8.66025i | ||||||||||
51.1 | 1.00000 | − | 1.73205i | −5.00000 | − | 8.66025i | −2.00000 | − | 3.46410i | 2.50000 | − | 4.33013i | −20.0000 | 14.0000 | + | 12.1244i | −8.00000 | −36.5000 | + | 63.2199i | −5.00000 | − | 8.66025i | |||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 70.4.e.b | ✓ | 2 |
3.b | odd | 2 | 1 | 630.4.k.c | 2 | ||
4.b | odd | 2 | 1 | 560.4.q.g | 2 | ||
5.b | even | 2 | 1 | 350.4.e.d | 2 | ||
5.c | odd | 4 | 2 | 350.4.j.a | 4 | ||
7.b | odd | 2 | 1 | 490.4.e.s | 2 | ||
7.c | even | 3 | 1 | inner | 70.4.e.b | ✓ | 2 |
7.c | even | 3 | 1 | 490.4.a.h | 1 | ||
7.d | odd | 6 | 1 | 490.4.a.a | 1 | ||
7.d | odd | 6 | 1 | 490.4.e.s | 2 | ||
21.h | odd | 6 | 1 | 630.4.k.c | 2 | ||
28.g | odd | 6 | 1 | 560.4.q.g | 2 | ||
35.i | odd | 6 | 1 | 2450.4.a.bq | 1 | ||
35.j | even | 6 | 1 | 350.4.e.d | 2 | ||
35.j | even | 6 | 1 | 2450.4.a.v | 1 | ||
35.l | odd | 12 | 2 | 350.4.j.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
70.4.e.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
70.4.e.b | ✓ | 2 | 7.c | even | 3 | 1 | inner |
350.4.e.d | 2 | 5.b | even | 2 | 1 | ||
350.4.e.d | 2 | 35.j | even | 6 | 1 | ||
350.4.j.a | 4 | 5.c | odd | 4 | 2 | ||
350.4.j.a | 4 | 35.l | odd | 12 | 2 | ||
490.4.a.a | 1 | 7.d | odd | 6 | 1 | ||
490.4.a.h | 1 | 7.c | even | 3 | 1 | ||
490.4.e.s | 2 | 7.b | odd | 2 | 1 | ||
490.4.e.s | 2 | 7.d | odd | 6 | 1 | ||
560.4.q.g | 2 | 4.b | odd | 2 | 1 | ||
560.4.q.g | 2 | 28.g | odd | 6 | 1 | ||
630.4.k.c | 2 | 3.b | odd | 2 | 1 | ||
630.4.k.c | 2 | 21.h | odd | 6 | 1 | ||
2450.4.a.v | 1 | 35.j | even | 6 | 1 | ||
2450.4.a.bq | 1 | 35.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 10T_{3} + 100 \)
acting on \(S_{4}^{\mathrm{new}}(70, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 2T + 4 \)
$3$
\( T^{2} + 10T + 100 \)
$5$
\( T^{2} - 5T + 25 \)
$7$
\( T^{2} - 28T + 343 \)
$11$
\( T^{2} + 53T + 2809 \)
$13$
\( (T - 25)^{2} \)
$17$
\( T^{2} + 14T + 196 \)
$19$
\( T^{2} - 95T + 9025 \)
$23$
\( T^{2} + T + 1 \)
$29$
\( (T + 206)^{2} \)
$31$
\( T^{2} + 108T + 11664 \)
$37$
\( T^{2} - 57T + 3249 \)
$41$
\( (T - 243)^{2} \)
$43$
\( (T - 434)^{2} \)
$47$
\( T^{2} - 231T + 53361 \)
$53$
\( T^{2} + 263T + 69169 \)
$59$
\( T^{2} + 24T + 576 \)
$61$
\( T^{2} + 116T + 13456 \)
$67$
\( T^{2} - 204T + 41616 \)
$71$
\( (T - 484)^{2} \)
$73$
\( T^{2} - 692T + 478864 \)
$79$
\( T^{2} + 466T + 217156 \)
$83$
\( (T - 228)^{2} \)
$89$
\( T^{2} - 362T + 131044 \)
$97$
\( (T - 854)^{2} \)
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