Newspace parameters
Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 18.c (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.06203438010\) |
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: | \(\Z[a_1, \ldots, a_{5}]\) |
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 \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
\(n\) | \(11\) |
\(\chi(n)\) | \(-1 + \zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
|
−1.00000 | + | 1.73205i | 5.19615i | −2.00000 | − | 3.46410i | 4.50000 | + | 7.79423i | −9.00000 | − | 5.19615i | 15.5000 | − | 26.8468i | 8.00000 | −27.0000 | −18.0000 | ||||||||||||||
13.1 | −1.00000 | − | 1.73205i | − | 5.19615i | −2.00000 | + | 3.46410i | 4.50000 | − | 7.79423i | −9.00000 | + | 5.19615i | 15.5000 | + | 26.8468i | 8.00000 | −27.0000 | −18.0000 | ||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 18.4.c.a | ✓ | 2 |
3.b | odd | 2 | 1 | 54.4.c.a | 2 | ||
4.b | odd | 2 | 1 | 144.4.i.a | 2 | ||
9.c | even | 3 | 1 | inner | 18.4.c.a | ✓ | 2 |
9.c | even | 3 | 1 | 162.4.a.d | 1 | ||
9.d | odd | 6 | 1 | 54.4.c.a | 2 | ||
9.d | odd | 6 | 1 | 162.4.a.a | 1 | ||
12.b | even | 2 | 1 | 432.4.i.a | 2 | ||
36.f | odd | 6 | 1 | 144.4.i.a | 2 | ||
36.f | odd | 6 | 1 | 1296.4.a.b | 1 | ||
36.h | even | 6 | 1 | 432.4.i.a | 2 | ||
36.h | even | 6 | 1 | 1296.4.a.g | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
18.4.c.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
18.4.c.a | ✓ | 2 | 9.c | even | 3 | 1 | inner |
54.4.c.a | 2 | 3.b | odd | 2 | 1 | ||
54.4.c.a | 2 | 9.d | odd | 6 | 1 | ||
144.4.i.a | 2 | 4.b | odd | 2 | 1 | ||
144.4.i.a | 2 | 36.f | odd | 6 | 1 | ||
162.4.a.a | 1 | 9.d | odd | 6 | 1 | ||
162.4.a.d | 1 | 9.c | even | 3 | 1 | ||
432.4.i.a | 2 | 12.b | even | 2 | 1 | ||
432.4.i.a | 2 | 36.h | even | 6 | 1 | ||
1296.4.a.b | 1 | 36.f | odd | 6 | 1 | ||
1296.4.a.g | 1 | 36.h | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 9T_{5} + 81 \)
acting on \(S_{4}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T + 4 \)
$3$
\( T^{2} + 27 \)
$5$
\( T^{2} - 9T + 81 \)
$7$
\( T^{2} - 31T + 961 \)
$11$
\( T^{2} - 15T + 225 \)
$13$
\( T^{2} - 37T + 1369 \)
$17$
\( (T + 42)^{2} \)
$19$
\( (T + 28)^{2} \)
$23$
\( T^{2} + 195T + 38025 \)
$29$
\( T^{2} + 111T + 12321 \)
$31$
\( T^{2} - 205T + 42025 \)
$37$
\( (T + 166)^{2} \)
$41$
\( T^{2} - 261T + 68121 \)
$43$
\( T^{2} - 43T + 1849 \)
$47$
\( T^{2} + 177T + 31329 \)
$53$
\( (T - 114)^{2} \)
$59$
\( T^{2} + 159T + 25281 \)
$61$
\( T^{2} + 191T + 36481 \)
$67$
\( T^{2} - 421T + 177241 \)
$71$
\( (T - 156)^{2} \)
$73$
\( (T - 182)^{2} \)
$79$
\( T^{2} + 1133 T + 1283689 \)
$83$
\( T^{2} - 1083 T + 1172889 \)
$89$
\( (T + 1050)^{2} \)
$97$
\( T^{2} - 901T + 811801 \)
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