Newform invariants
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 \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) |
\(8\) |
\(10\) |
| \(\chi(n)\) |
\(-1\) |
\(\zeta_{6}\) |
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.
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(21, [\chi])\).
| $p$ |
$F_p(T)$ |
| $2$
| \( 1 + 2 T^{2} + 4 T^{4} \)
|
| $3$
| \( 1 + 3 T + 3 T^{2} \)
|
| $5$
| \( 1 - 5 T^{2} + 25 T^{4} \)
|
| $7$
| \( 1 - T + 7 T^{2} \)
|
| $11$
| \( 1 + 11 T^{2} + 121 T^{4} \)
|
| $13$
| \( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
|
| $17$
| \( 1 - 17 T^{2} + 289 T^{4} \)
|
| $19$
| \( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
|
| $23$
| \( 1 + 23 T^{2} + 529 T^{4} \)
|
| $29$
| \( ( 1 - 29 T^{2} )^{2} \)
|
| $31$
| \( ( 1 - 11 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \)
|
| $37$
| \( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \)
|
| $41$
| \( ( 1 + 41 T^{2} )^{2} \)
|
| $43$
| \( ( 1 + 5 T + 43 T^{2} )^{2} \)
|
| $47$
| \( 1 - 47 T^{2} + 2209 T^{4} \)
|
| $53$
| \( 1 + 53 T^{2} + 2809 T^{4} \)
|
| $59$
| \( 1 - 59 T^{2} + 3481 T^{4} \)
|
| $61$
| \( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \)
|
| $67$
| \( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
|
| $71$
| \( ( 1 - 71 T^{2} )^{2} \)
|
| $73$
| \( ( 1 + 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
|
| $79$
| \( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \)
|
| $83$
| \( ( 1 + 83 T^{2} )^{2} \)
|
| $89$
| \( 1 - 89 T^{2} + 7921 T^{4} \)
|
| $97$
| \( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \)
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