Newspace parameters
Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 42.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.73612043215\) |
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 |
|
2.00000 | + | 3.46410i | 4.50000 | − | 7.79423i | −8.00000 | + | 13.8564i | −43.0000 | − | 74.4782i | 36.0000 | 24.5000 | − | 127.306i | −64.0000 | −40.5000 | − | 70.1481i | 172.000 | − | 297.913i | ||||||||||
37.1 | 2.00000 | − | 3.46410i | 4.50000 | + | 7.79423i | −8.00000 | − | 13.8564i | −43.0000 | + | 74.4782i | 36.0000 | 24.5000 | + | 127.306i | −64.0000 | −40.5000 | + | 70.1481i | 172.000 | + | 297.913i | |||||||||||
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 | 42.6.e.b | ✓ | 2 |
3.b | odd | 2 | 1 | 126.6.g.b | 2 | ||
4.b | odd | 2 | 1 | 336.6.q.a | 2 | ||
7.b | odd | 2 | 1 | 294.6.e.m | 2 | ||
7.c | even | 3 | 1 | inner | 42.6.e.b | ✓ | 2 |
7.c | even | 3 | 1 | 294.6.a.d | 1 | ||
7.d | odd | 6 | 1 | 294.6.a.e | 1 | ||
7.d | odd | 6 | 1 | 294.6.e.m | 2 | ||
21.g | even | 6 | 1 | 882.6.a.y | 1 | ||
21.h | odd | 6 | 1 | 126.6.g.b | 2 | ||
21.h | odd | 6 | 1 | 882.6.a.m | 1 | ||
28.g | odd | 6 | 1 | 336.6.q.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
42.6.e.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
42.6.e.b | ✓ | 2 | 7.c | even | 3 | 1 | inner |
126.6.g.b | 2 | 3.b | odd | 2 | 1 | ||
126.6.g.b | 2 | 21.h | odd | 6 | 1 | ||
294.6.a.d | 1 | 7.c | even | 3 | 1 | ||
294.6.a.e | 1 | 7.d | odd | 6 | 1 | ||
294.6.e.m | 2 | 7.b | odd | 2 | 1 | ||
294.6.e.m | 2 | 7.d | odd | 6 | 1 | ||
336.6.q.a | 2 | 4.b | odd | 2 | 1 | ||
336.6.q.a | 2 | 28.g | odd | 6 | 1 | ||
882.6.a.m | 1 | 21.h | odd | 6 | 1 | ||
882.6.a.y | 1 | 21.g | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 86T_{5} + 7396 \)
acting on \(S_{6}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 4T + 16 \)
$3$
\( T^{2} - 9T + 81 \)
$5$
\( T^{2} + 86T + 7396 \)
$7$
\( T^{2} - 49T + 16807 \)
$11$
\( T^{2} + 34T + 1156 \)
$13$
\( (T + 3)^{2} \)
$17$
\( T^{2} - 1904 T + 3625216 \)
$19$
\( T^{2} - 1489 T + 2217121 \)
$23$
\( T^{2} - 224T + 50176 \)
$29$
\( (T + 6508)^{2} \)
$31$
\( T^{2} + 1731 T + 2996361 \)
$37$
\( T^{2} - 7633 T + 58262689 \)
$41$
\( (T - 15414)^{2} \)
$43$
\( (T - 18491)^{2} \)
$47$
\( T^{2} + 18462 T + 340845444 \)
$53$
\( T^{2} - 19956 T + 398241936 \)
$59$
\( T^{2} - 31828 T + 1013021584 \)
$61$
\( T^{2} - 57654 T + 3323983716 \)
$67$
\( T^{2} - 60563 T + 3667876969 \)
$71$
\( (T + 44834)^{2} \)
$73$
\( T^{2} + 20821 T + 433514041 \)
$79$
\( T^{2} - 30531 T + 932141961 \)
$83$
\( (T - 110602)^{2} \)
$89$
\( T^{2} - 58992 T + 3480056064 \)
$97$
\( (T + 119846)^{2} \)
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