Newspace parameters
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.23904011012\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 |
|
1.50000 | + | 2.59808i | −1.50000 | + | 2.59808i | −0.500000 | + | 0.866025i | 1.50000 | + | 2.59808i | −9.00000 | −3.50000 | − | 18.1865i | 21.0000 | −4.50000 | − | 7.79423i | −4.50000 | + | 7.79423i | ||||||||||
| 16.1 | 1.50000 | − | 2.59808i | −1.50000 | − | 2.59808i | −0.500000 | − | 0.866025i | 1.50000 | − | 2.59808i | −9.00000 | −3.50000 | + | 18.1865i | 21.0000 | −4.50000 | + | 7.79423i | −4.50000 | − | 7.79423i | |||||||||||
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 | 21.4.e.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 63.4.e.a | 2 | ||
| 4.b | odd | 2 | 1 | 336.4.q.e | 2 | ||
| 7.b | odd | 2 | 1 | 147.4.e.h | 2 | ||
| 7.c | even | 3 | 1 | inner | 21.4.e.a | ✓ | 2 |
| 7.c | even | 3 | 1 | 147.4.a.b | 1 | ||
| 7.d | odd | 6 | 1 | 147.4.a.a | 1 | ||
| 7.d | odd | 6 | 1 | 147.4.e.h | 2 | ||
| 21.c | even | 2 | 1 | 441.4.e.c | 2 | ||
| 21.g | even | 6 | 1 | 441.4.a.k | 1 | ||
| 21.g | even | 6 | 1 | 441.4.e.c | 2 | ||
| 21.h | odd | 6 | 1 | 63.4.e.a | 2 | ||
| 21.h | odd | 6 | 1 | 441.4.a.l | 1 | ||
| 28.f | even | 6 | 1 | 2352.4.a.bd | 1 | ||
| 28.g | odd | 6 | 1 | 336.4.q.e | 2 | ||
| 28.g | odd | 6 | 1 | 2352.4.a.i | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.4.e.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 21.4.e.a | ✓ | 2 | 7.c | even | 3 | 1 | inner |
| 63.4.e.a | 2 | 3.b | odd | 2 | 1 | ||
| 63.4.e.a | 2 | 21.h | odd | 6 | 1 | ||
| 147.4.a.a | 1 | 7.d | odd | 6 | 1 | ||
| 147.4.a.b | 1 | 7.c | even | 3 | 1 | ||
| 147.4.e.h | 2 | 7.b | odd | 2 | 1 | ||
| 147.4.e.h | 2 | 7.d | odd | 6 | 1 | ||
| 336.4.q.e | 2 | 4.b | odd | 2 | 1 | ||
| 336.4.q.e | 2 | 28.g | odd | 6 | 1 | ||
| 441.4.a.k | 1 | 21.g | even | 6 | 1 | ||
| 441.4.a.l | 1 | 21.h | odd | 6 | 1 | ||
| 441.4.e.c | 2 | 21.c | even | 2 | 1 | ||
| 441.4.e.c | 2 | 21.g | even | 6 | 1 | ||
| 2352.4.a.i | 1 | 28.g | odd | 6 | 1 | ||
| 2352.4.a.bd | 1 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 3T + 9 \)
|
| $3$ |
\( T^{2} + 3T + 9 \)
|
| $5$ |
\( T^{2} - 3T + 9 \)
|
| $7$ |
\( T^{2} + 7T + 343 \)
|
| $11$ |
\( T^{2} - 15T + 225 \)
|
| $13$ |
\( (T + 64)^{2} \)
|
| $17$ |
\( T^{2} + 84T + 7056 \)
|
| $19$ |
\( T^{2} - 16T + 256 \)
|
| $23$ |
\( T^{2} - 84T + 7056 \)
|
| $29$ |
\( (T + 297)^{2} \)
|
| $31$ |
\( T^{2} - 253T + 64009 \)
|
| $37$ |
\( T^{2} - 316T + 99856 \)
|
| $41$ |
\( (T - 360)^{2} \)
|
| $43$ |
\( (T - 26)^{2} \)
|
| $47$ |
\( T^{2} - 30T + 900 \)
|
| $53$ |
\( T^{2} + 363T + 131769 \)
|
| $59$ |
\( T^{2} - 15T + 225 \)
|
| $61$ |
\( T^{2} - 118T + 13924 \)
|
| $67$ |
\( T^{2} - 370T + 136900 \)
|
| $71$ |
\( (T + 342)^{2} \)
|
| $73$ |
\( T^{2} + 362T + 131044 \)
|
| $79$ |
\( T^{2} + 467T + 218089 \)
|
| $83$ |
\( (T - 477)^{2} \)
|
| $89$ |
\( T^{2} + 906T + 820836 \)
|
| $97$ |
\( (T - 503)^{2} \)
|
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