Newspace parameters
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.36806021607\) |
| 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 \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(1\) | \(-1 + \zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
1.00000 | + | 1.73205i | 4.50000 | − | 7.79423i | 14.0000 | − | 24.2487i | −5.50000 | − | 9.52628i | 18.0000 | 129.500 | − | 6.06218i | 120.000 | −40.5000 | − | 70.1481i | 11.0000 | − | 19.0526i | ||||||||||
| 16.1 | 1.00000 | − | 1.73205i | 4.50000 | + | 7.79423i | 14.0000 | + | 24.2487i | −5.50000 | + | 9.52628i | 18.0000 | 129.500 | + | 6.06218i | 120.000 | −40.5000 | + | 70.1481i | 11.0000 | + | 19.0526i | |||||||||||
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 | 21.6.e.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 63.6.e.a | 2 | ||
| 4.b | odd | 2 | 1 | 336.6.q.b | 2 | ||
| 7.b | odd | 2 | 1 | 147.6.e.g | 2 | ||
| 7.c | even | 3 | 1 | inner | 21.6.e.a | ✓ | 2 |
| 7.c | even | 3 | 1 | 147.6.a.c | 1 | ||
| 7.d | odd | 6 | 1 | 147.6.a.d | 1 | ||
| 7.d | odd | 6 | 1 | 147.6.e.g | 2 | ||
| 21.g | even | 6 | 1 | 441.6.a.h | 1 | ||
| 21.h | odd | 6 | 1 | 63.6.e.a | 2 | ||
| 21.h | odd | 6 | 1 | 441.6.a.g | 1 | ||
| 28.g | odd | 6 | 1 | 336.6.q.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.6.e.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 21.6.e.a | ✓ | 2 | 7.c | even | 3 | 1 | inner |
| 63.6.e.a | 2 | 3.b | odd | 2 | 1 | ||
| 63.6.e.a | 2 | 21.h | odd | 6 | 1 | ||
| 147.6.a.c | 1 | 7.c | even | 3 | 1 | ||
| 147.6.a.d | 1 | 7.d | odd | 6 | 1 | ||
| 147.6.e.g | 2 | 7.b | odd | 2 | 1 | ||
| 147.6.e.g | 2 | 7.d | odd | 6 | 1 | ||
| 336.6.q.b | 2 | 4.b | odd | 2 | 1 | ||
| 336.6.q.b | 2 | 28.g | odd | 6 | 1 | ||
| 441.6.a.g | 1 | 21.h | odd | 6 | 1 | ||
| 441.6.a.h | 1 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2T_{2} + 4 \)
acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 2T + 4 \)
$3$
\( T^{2} - 9T + 81 \)
$5$
\( T^{2} + 11T + 121 \)
$7$
\( T^{2} - 259T + 16807 \)
$11$
\( T^{2} + 269T + 72361 \)
$13$
\( (T + 308)^{2} \)
$17$
\( T^{2} + 1896 T + 3594816 \)
$19$
\( T^{2} - 164T + 26896 \)
$23$
\( T^{2} - 3264 T + 10653696 \)
$29$
\( (T - 2417)^{2} \)
$31$
\( T^{2} + 2841 T + 8071281 \)
$37$
\( T^{2} - 11328 T + 128323584 \)
$41$
\( (T + 16856)^{2} \)
$43$
\( (T + 7894)^{2} \)
$47$
\( T^{2} + 21102 T + 445294404 \)
$53$
\( T^{2} - 29691 T + 881555481 \)
$59$
\( T^{2} - 8163 T + 66634569 \)
$61$
\( T^{2} + 15166 T + 230007556 \)
$67$
\( T^{2} - 32078 T + 1028998084 \)
$71$
\( (T + 38274)^{2} \)
$73$
\( T^{2} + 34866 T + 1215637956 \)
$79$
\( T^{2} + 13529 T + 183033841 \)
$83$
\( (T + 68103)^{2} \)
$89$
\( T^{2} - 114922 T + 13207066084 \)
$97$
\( (T - 154959)^{2} \)
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