Newspace parameters
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.l (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.54586299101\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{13})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 315) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 7 ) / 4 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) |
\(\nu^{3}\) | \(=\) | \( 4\beta_{3} - 7 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).
\(n\) | \(136\) | \(596\) | \(757\) |
\(\chi(n)\) | \(-1 - \beta_{2}\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
46.1 |
|
−1.30278 | 0 | −0.302776 | −0.500000 | + | 0.866025i | 0 | 2.50000 | + | 0.866025i | 3.00000 | 0 | 0.651388 | − | 1.12824i | ||||||||||||||||||||||||
46.2 | 2.30278 | 0 | 3.30278 | −0.500000 | + | 0.866025i | 0 | 2.50000 | + | 0.866025i | 3.00000 | 0 | −1.15139 | + | 1.99426i | |||||||||||||||||||||||||
226.1 | −1.30278 | 0 | −0.302776 | −0.500000 | − | 0.866025i | 0 | 2.50000 | − | 0.866025i | 3.00000 | 0 | 0.651388 | + | 1.12824i | |||||||||||||||||||||||||
226.2 | 2.30278 | 0 | 3.30278 | −0.500000 | − | 0.866025i | 0 | 2.50000 | − | 0.866025i | 3.00000 | 0 | −1.15139 | − | 1.99426i | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.l.a | 4 | |
3.b | odd | 2 | 1 | 315.2.l.a | yes | 4 | |
7.c | even | 3 | 1 | 945.2.k.a | 4 | ||
9.c | even | 3 | 1 | 945.2.k.a | 4 | ||
9.d | odd | 6 | 1 | 315.2.k.a | ✓ | 4 | |
21.h | odd | 6 | 1 | 315.2.k.a | ✓ | 4 | |
63.h | even | 3 | 1 | inner | 945.2.l.a | 4 | |
63.j | odd | 6 | 1 | 315.2.l.a | yes | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.k.a | ✓ | 4 | 9.d | odd | 6 | 1 | |
315.2.k.a | ✓ | 4 | 21.h | odd | 6 | 1 | |
315.2.l.a | yes | 4 | 3.b | odd | 2 | 1 | |
315.2.l.a | yes | 4 | 63.j | odd | 6 | 1 | |
945.2.k.a | 4 | 7.c | even | 3 | 1 | ||
945.2.k.a | 4 | 9.c | even | 3 | 1 | ||
945.2.l.a | 4 | 1.a | even | 1 | 1 | trivial | |
945.2.l.a | 4 | 63.h | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T - 3)^{2} \)
$3$
\( T^{4} \)
$5$
\( (T^{2} + T + 1)^{2} \)
$7$
\( (T^{2} - 5 T + 7)^{2} \)
$11$
\( T^{4} \)
$13$
\( T^{4} - 4 T^{3} + 25 T^{2} + 36 T + 81 \)
$17$
\( T^{4} - 4 T^{3} + 25 T^{2} + 36 T + 81 \)
$19$
\( T^{4} + 13T^{2} + 169 \)
$23$
\( T^{4} - 4 T^{3} + 64 T^{2} + \cdots + 2304 \)
$29$
\( T^{4} - 2 T^{3} + 55 T^{2} + \cdots + 2601 \)
$31$
\( (T^{2} - 13)^{2} \)
$37$
\( T^{4} + 13T^{2} + 169 \)
$41$
\( (T^{2} - 3 T + 9)^{2} \)
$43$
\( T^{4} - 6 T^{3} + 79 T^{2} + \cdots + 1849 \)
$47$
\( (T^{2} - 2 T - 51)^{2} \)
$53$
\( T^{4} - 8 T^{3} + 61 T^{2} - 24 T + 9 \)
$59$
\( (T^{2} - 16 T + 51)^{2} \)
$61$
\( (T^{2} + 6 T - 43)^{2} \)
$67$
\( (T^{2} + 6 T - 43)^{2} \)
$71$
\( T^{4} \)
$73$
\( T^{4} + 13T^{2} + 169 \)
$79$
\( (T^{2} - 4 T - 113)^{2} \)
$83$
\( T^{4} + 14 T^{3} + 199 T^{2} - 42 T + 9 \)
$89$
\( (T^{2} - 3 T + 9)^{2} \)
$97$
\( T^{4} + 20 T^{3} + 313 T^{2} + \cdots + 7569 \)
show more
show less