Newspace parameters
Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 21.h (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.17076922476\) |
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: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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 \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
\(n\) | \(8\) | \(10\) |
\(\chi(n)\) | \(-1\) | \(-1 + \zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 |
|
0 | −4.50000 | − | 7.79423i | −8.00000 | − | 13.8564i | 0 | 0 | 35.5000 | − | 33.7750i | 0 | −40.5000 | + | 70.1481i | 0 | ||||||||||||||||
11.1 | 0 | −4.50000 | + | 7.79423i | −8.00000 | + | 13.8564i | 0 | 0 | 35.5000 | + | 33.7750i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
7.c | even | 3 | 1 | inner |
21.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 21.5.h.a | ✓ | 2 |
3.b | odd | 2 | 1 | CM | 21.5.h.a | ✓ | 2 |
7.b | odd | 2 | 1 | 147.5.h.a | 2 | ||
7.c | even | 3 | 1 | inner | 21.5.h.a | ✓ | 2 |
7.c | even | 3 | 1 | 147.5.b.b | 1 | ||
7.d | odd | 6 | 1 | 147.5.b.a | 1 | ||
7.d | odd | 6 | 1 | 147.5.h.a | 2 | ||
21.c | even | 2 | 1 | 147.5.h.a | 2 | ||
21.g | even | 6 | 1 | 147.5.b.a | 1 | ||
21.g | even | 6 | 1 | 147.5.h.a | 2 | ||
21.h | odd | 6 | 1 | inner | 21.5.h.a | ✓ | 2 |
21.h | odd | 6 | 1 | 147.5.b.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
21.5.h.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
21.5.h.a | ✓ | 2 | 3.b | odd | 2 | 1 | CM |
21.5.h.a | ✓ | 2 | 7.c | even | 3 | 1 | inner |
21.5.h.a | ✓ | 2 | 21.h | odd | 6 | 1 | inner |
147.5.b.a | 1 | 7.d | odd | 6 | 1 | ||
147.5.b.a | 1 | 21.g | even | 6 | 1 | ||
147.5.b.b | 1 | 7.c | even | 3 | 1 | ||
147.5.b.b | 1 | 21.h | odd | 6 | 1 | ||
147.5.h.a | 2 | 7.b | odd | 2 | 1 | ||
147.5.h.a | 2 | 7.d | odd | 6 | 1 | ||
147.5.h.a | 2 | 21.c | even | 2 | 1 | ||
147.5.h.a | 2 | 21.g | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{5}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 9T + 81 \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 71T + 2401 \)
$11$
\( T^{2} \)
$13$
\( (T - 191)^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} - 601T + 361201 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} - 1753 T + 3073009 \)
$37$
\( T^{2} + 2591 T + 6713281 \)
$41$
\( T^{2} \)
$43$
\( (T - 23)^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} - 1966 T + 3865156 \)
$67$
\( T^{2} - 8809 T + 77598481 \)
$71$
\( T^{2} \)
$73$
\( T^{2} - 1249 T + 1560001 \)
$79$
\( T^{2} - 12361 T + 152794321 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( (T + 18814)^{2} \)
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