Newspace parameters
| Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 84.k (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(64.5408271670\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(43\) | \(73\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
0 | 364.500 | − | 210.444i | 0 | 0 | 0 | 38442.5 | + | 22349.5i | 0 | 88573.5 | − | 153414.i | 0 | ||||||||||||||||||
| 17.1 | 0 | 364.500 | + | 210.444i | 0 | 0 | 0 | 38442.5 | − | 22349.5i | 0 | 88573.5 | + | 153414.i | 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.d | odd | 6 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 84.12.k.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | CM | 84.12.k.a | ✓ | 2 |
| 7.d | odd | 6 | 1 | inner | 84.12.k.a | ✓ | 2 |
| 21.g | even | 6 | 1 | inner | 84.12.k.a | ✓ | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.12.k.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 84.12.k.a | ✓ | 2 | 3.b | odd | 2 | 1 | CM |
| 84.12.k.a | ✓ | 2 | 7.d | odd | 6 | 1 | inner |
| 84.12.k.a | ✓ | 2 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{12}^{\mathrm{new}}(84, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 729T + 177147 \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 76885 T + 1977326743 \)
$11$
\( T^{2} \)
$13$
\( T^{2} + 2112372157923 \)
$17$
\( T^{2} \)
$19$
\( T^{2} + 11271165 T + 42346386819075 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} - 546211707 T + 99\!\cdots\!83 \)
$37$
\( T^{2} - 119374607 T + 14\!\cdots\!49 \)
$41$
\( T^{2} \)
$43$
\( (T - 218924719)^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 21523645452 T + 15\!\cdots\!68 \)
$67$
\( T^{2} + 5951291615 T + 35\!\cdots\!25 \)
$71$
\( T^{2} \)
$73$
\( T^{2} - 55144201461 T + 10\!\cdots\!07 \)
$79$
\( T^{2} - 54296224537 T + 29\!\cdots\!69 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} + 15\!\cdots\!48 \)
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