Newspace parameters
Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 189.p (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.50917259820\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
Defining polynomial: |
\( x^{4} - 5x^{2} + 25 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 5x^{2} + 25 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( 5\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(136\) |
\(\chi(n)\) | \(-1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 |
|
−1.93649 | − | 1.11803i | 0 | 1.50000 | + | 2.59808i | 0 | 0 | −2.00000 | + | 1.73205i | − | 2.23607i | 0 | 0 | |||||||||||||||||||||||
26.2 | 1.93649 | + | 1.11803i | 0 | 1.50000 | + | 2.59808i | 0 | 0 | −2.00000 | + | 1.73205i | 2.23607i | 0 | 0 | |||||||||||||||||||||||||
80.1 | −1.93649 | + | 1.11803i | 0 | 1.50000 | − | 2.59808i | 0 | 0 | −2.00000 | − | 1.73205i | 2.23607i | 0 | 0 | |||||||||||||||||||||||||
80.2 | 1.93649 | − | 1.11803i | 0 | 1.50000 | − | 2.59808i | 0 | 0 | −2.00000 | − | 1.73205i | − | 2.23607i | 0 | 0 | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
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 | 189.2.p.c | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 189.2.p.c | ✓ | 4 |
7.c | even | 3 | 1 | 1323.2.c.b | 4 | ||
7.d | odd | 6 | 1 | inner | 189.2.p.c | ✓ | 4 |
7.d | odd | 6 | 1 | 1323.2.c.b | 4 | ||
9.c | even | 3 | 1 | 567.2.i.c | 4 | ||
9.c | even | 3 | 1 | 567.2.s.e | 4 | ||
9.d | odd | 6 | 1 | 567.2.i.c | 4 | ||
9.d | odd | 6 | 1 | 567.2.s.e | 4 | ||
21.g | even | 6 | 1 | inner | 189.2.p.c | ✓ | 4 |
21.g | even | 6 | 1 | 1323.2.c.b | 4 | ||
21.h | odd | 6 | 1 | 1323.2.c.b | 4 | ||
63.i | even | 6 | 1 | 567.2.s.e | 4 | ||
63.k | odd | 6 | 1 | 567.2.i.c | 4 | ||
63.s | even | 6 | 1 | 567.2.i.c | 4 | ||
63.t | odd | 6 | 1 | 567.2.s.e | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
189.2.p.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
189.2.p.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
189.2.p.c | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
189.2.p.c | ✓ | 4 | 21.g | even | 6 | 1 | inner |
567.2.i.c | 4 | 9.c | even | 3 | 1 | ||
567.2.i.c | 4 | 9.d | odd | 6 | 1 | ||
567.2.i.c | 4 | 63.k | odd | 6 | 1 | ||
567.2.i.c | 4 | 63.s | even | 6 | 1 | ||
567.2.s.e | 4 | 9.c | even | 3 | 1 | ||
567.2.s.e | 4 | 9.d | odd | 6 | 1 | ||
567.2.s.e | 4 | 63.i | even | 6 | 1 | ||
567.2.s.e | 4 | 63.t | odd | 6 | 1 | ||
1323.2.c.b | 4 | 7.c | even | 3 | 1 | ||
1323.2.c.b | 4 | 7.d | odd | 6 | 1 | ||
1323.2.c.b | 4 | 21.g | even | 6 | 1 | ||
1323.2.c.b | 4 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 5T^{2} + 25 \)
$3$
\( T^{4} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 4 T + 7)^{2} \)
$11$
\( T^{4} - 20T^{2} + 400 \)
$13$
\( (T^{2} + 3)^{2} \)
$17$
\( T^{4} + 60T^{2} + 3600 \)
$19$
\( (T^{2} + 6 T + 12)^{2} \)
$23$
\( T^{4} - 20T^{2} + 400 \)
$29$
\( (T^{2} + 20)^{2} \)
$31$
\( (T^{2} + 3 T + 3)^{2} \)
$37$
\( (T^{2} + 5 T + 25)^{2} \)
$41$
\( (T^{2} - 60)^{2} \)
$43$
\( (T + 7)^{4} \)
$47$
\( T^{4} + 60T^{2} + 3600 \)
$53$
\( T^{4} - 20T^{2} + 400 \)
$59$
\( T^{4} + 60T^{2} + 3600 \)
$61$
\( (T^{2} - 15 T + 75)^{2} \)
$67$
\( (T^{2} - T + 1)^{2} \)
$71$
\( (T^{2} + 80)^{2} \)
$73$
\( (T^{2} - 12 T + 48)^{2} \)
$79$
\( (T^{2} + 11 T + 121)^{2} \)
$83$
\( (T^{2} - 60)^{2} \)
$89$
\( T^{4} + 240 T^{2} + 57600 \)
$97$
\( (T^{2} + 3)^{2} \)
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