Newspace parameters
| Level: | \( N \) | \(=\) | \( 108 = 2^{2} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 108.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.94278685509\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(55\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
−1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 7.00000 | 0 | 8.66025i | 8.00000 | 0 | −7.00000 | − | 12.1244i | ||||||||||||||||||
| 55.2 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 7.00000 | 0 | − | 8.66025i | 8.00000 | 0 | −7.00000 | + | 12.1244i | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 108.3.d.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 108.3.d.b | yes | 2 | |
| 4.b | odd | 2 | 1 | inner | 108.3.d.a | ✓ | 2 |
| 8.b | even | 2 | 1 | 1728.3.g.a | 2 | ||
| 8.d | odd | 2 | 1 | 1728.3.g.a | 2 | ||
| 9.c | even | 3 | 1 | 324.3.f.c | 2 | ||
| 9.c | even | 3 | 1 | 324.3.f.i | 2 | ||
| 9.d | odd | 6 | 1 | 324.3.f.b | 2 | ||
| 9.d | odd | 6 | 1 | 324.3.f.h | 2 | ||
| 12.b | even | 2 | 1 | 108.3.d.b | yes | 2 | |
| 24.f | even | 2 | 1 | 1728.3.g.f | 2 | ||
| 24.h | odd | 2 | 1 | 1728.3.g.f | 2 | ||
| 36.f | odd | 6 | 1 | 324.3.f.c | 2 | ||
| 36.f | odd | 6 | 1 | 324.3.f.i | 2 | ||
| 36.h | even | 6 | 1 | 324.3.f.b | 2 | ||
| 36.h | even | 6 | 1 | 324.3.f.h | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 108.3.d.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 108.3.d.a | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
| 108.3.d.b | yes | 2 | 3.b | odd | 2 | 1 | |
| 108.3.d.b | yes | 2 | 12.b | even | 2 | 1 | |
| 324.3.f.b | 2 | 9.d | odd | 6 | 1 | ||
| 324.3.f.b | 2 | 36.h | even | 6 | 1 | ||
| 324.3.f.c | 2 | 9.c | even | 3 | 1 | ||
| 324.3.f.c | 2 | 36.f | odd | 6 | 1 | ||
| 324.3.f.h | 2 | 9.d | odd | 6 | 1 | ||
| 324.3.f.h | 2 | 36.h | even | 6 | 1 | ||
| 324.3.f.i | 2 | 9.c | even | 3 | 1 | ||
| 324.3.f.i | 2 | 36.f | odd | 6 | 1 | ||
| 1728.3.g.a | 2 | 8.b | even | 2 | 1 | ||
| 1728.3.g.a | 2 | 8.d | odd | 2 | 1 | ||
| 1728.3.g.f | 2 | 24.f | even | 2 | 1 | ||
| 1728.3.g.f | 2 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 7 \)
acting on \(S_{3}^{\mathrm{new}}(108, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T + 4 \)
$3$
\( T^{2} \)
$5$
\( (T - 7)^{2} \)
$7$
\( T^{2} + 75 \)
$11$
\( T^{2} + 75 \)
$13$
\( (T - 20)^{2} \)
$17$
\( (T + 8)^{2} \)
$19$
\( T^{2} + 108 \)
$23$
\( T^{2} + 12 \)
$29$
\( (T - 10)^{2} \)
$31$
\( T^{2} + 2883 \)
$37$
\( (T + 10)^{2} \)
$41$
\( (T + 50)^{2} \)
$43$
\( T^{2} + 300 \)
$47$
\( T^{2} + 7500 \)
$53$
\( (T + 47)^{2} \)
$59$
\( T^{2} + 1200 \)
$61$
\( (T + 64)^{2} \)
$67$
\( T^{2} + 7500 \)
$71$
\( T^{2} \)
$73$
\( (T + 55)^{2} \)
$79$
\( T^{2} + 48 \)
$83$
\( T^{2} + 867 \)
$89$
\( (T - 10)^{2} \)
$97$
\( (T + 25)^{2} \)
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