Newspace parameters
Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 42.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.47808022024\) |
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: | \(\Z[a_1, a_2, a_3]\) |
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 \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(31\) |
\(\chi(n)\) | \(1\) | \(-1 + \zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 |
|
1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | 7.50000 | + | 12.9904i | 6.00000 | 17.5000 | − | 6.06218i | −8.00000 | −4.50000 | − | 7.79423i | −15.0000 | + | 25.9808i | ||||||||||
37.1 | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | 7.50000 | − | 12.9904i | 6.00000 | 17.5000 | + | 6.06218i | −8.00000 | −4.50000 | + | 7.79423i | −15.0000 | − | 25.9808i | |||||||||||
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 | 42.4.e.b | ✓ | 2 |
3.b | odd | 2 | 1 | 126.4.g.a | 2 | ||
4.b | odd | 2 | 1 | 336.4.q.d | 2 | ||
7.b | odd | 2 | 1 | 294.4.e.e | 2 | ||
7.c | even | 3 | 1 | inner | 42.4.e.b | ✓ | 2 |
7.c | even | 3 | 1 | 294.4.a.a | 1 | ||
7.d | odd | 6 | 1 | 294.4.a.g | 1 | ||
7.d | odd | 6 | 1 | 294.4.e.e | 2 | ||
21.c | even | 2 | 1 | 882.4.g.l | 2 | ||
21.g | even | 6 | 1 | 882.4.a.h | 1 | ||
21.g | even | 6 | 1 | 882.4.g.l | 2 | ||
21.h | odd | 6 | 1 | 126.4.g.a | 2 | ||
21.h | odd | 6 | 1 | 882.4.a.r | 1 | ||
28.f | even | 6 | 1 | 2352.4.a.q | 1 | ||
28.g | odd | 6 | 1 | 336.4.q.d | 2 | ||
28.g | odd | 6 | 1 | 2352.4.a.u | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
42.4.e.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
42.4.e.b | ✓ | 2 | 7.c | even | 3 | 1 | inner |
126.4.g.a | 2 | 3.b | odd | 2 | 1 | ||
126.4.g.a | 2 | 21.h | odd | 6 | 1 | ||
294.4.a.a | 1 | 7.c | even | 3 | 1 | ||
294.4.a.g | 1 | 7.d | odd | 6 | 1 | ||
294.4.e.e | 2 | 7.b | odd | 2 | 1 | ||
294.4.e.e | 2 | 7.d | odd | 6 | 1 | ||
336.4.q.d | 2 | 4.b | odd | 2 | 1 | ||
336.4.q.d | 2 | 28.g | odd | 6 | 1 | ||
882.4.a.h | 1 | 21.g | even | 6 | 1 | ||
882.4.a.r | 1 | 21.h | odd | 6 | 1 | ||
882.4.g.l | 2 | 21.c | even | 2 | 1 | ||
882.4.g.l | 2 | 21.g | even | 6 | 1 | ||
2352.4.a.q | 1 | 28.f | even | 6 | 1 | ||
2352.4.a.u | 1 | 28.g | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 15T_{5} + 225 \)
acting on \(S_{4}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 2T + 4 \)
$3$
\( T^{2} - 3T + 9 \)
$5$
\( T^{2} - 15T + 225 \)
$7$
\( T^{2} - 35T + 343 \)
$11$
\( T^{2} - 9T + 81 \)
$13$
\( (T + 88)^{2} \)
$17$
\( T^{2} - 84T + 7056 \)
$19$
\( T^{2} + 104T + 10816 \)
$23$
\( T^{2} - 84T + 7056 \)
$29$
\( (T - 51)^{2} \)
$31$
\( T^{2} + 185T + 34225 \)
$37$
\( T^{2} + 44T + 1936 \)
$41$
\( (T + 168)^{2} \)
$43$
\( (T - 326)^{2} \)
$47$
\( T^{2} - 138T + 19044 \)
$53$
\( T^{2} + 639T + 408321 \)
$59$
\( T^{2} + 159T + 25281 \)
$61$
\( T^{2} + 722T + 521284 \)
$67$
\( T^{2} - 166T + 27556 \)
$71$
\( (T - 1086)^{2} \)
$73$
\( T^{2} + 218T + 47524 \)
$79$
\( T^{2} - 583T + 339889 \)
$83$
\( (T + 597)^{2} \)
$89$
\( T^{2} - 1038 T + 1077444 \)
$97$
\( (T + 169)^{2} \)
show more
show less