Newspace parameters
Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 189.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.50917259820\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
Defining polynomial: |
\( x^{4} - 5x^{2} + 25 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - 5x^{2} + 25 \)
:
\(\beta_{1}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
\(\beta_{2}\) | \(=\) |
\( ( 2\nu^{2} - 5 ) / 5 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 10\nu ) / 5 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( ( 5\beta_{2} + 5 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(136\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
188.1 |
|
− | 2.23607i | 0 | −3.00000 | −3.87298 | 0 | −2.00000 | + | 1.73205i | 2.23607i | 0 | 8.66025i | |||||||||||||||||||||||||||
188.2 | − | 2.23607i | 0 | −3.00000 | 3.87298 | 0 | −2.00000 | − | 1.73205i | 2.23607i | 0 | − | 8.66025i | |||||||||||||||||||||||||||
188.3 | 2.23607i | 0 | −3.00000 | −3.87298 | 0 | −2.00000 | − | 1.73205i | − | 2.23607i | 0 | − | 8.66025i | |||||||||||||||||||||||||||
188.4 | 2.23607i | 0 | −3.00000 | 3.87298 | 0 | −2.00000 | + | 1.73205i | − | 2.23607i | 0 | 8.66025i | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 189.2.c.b | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 189.2.c.b | ✓ | 4 |
4.b | odd | 2 | 1 | 3024.2.k.i | 4 | ||
7.b | odd | 2 | 1 | inner | 189.2.c.b | ✓ | 4 |
9.c | even | 3 | 1 | 567.2.o.c | 4 | ||
9.c | even | 3 | 1 | 567.2.o.d | 4 | ||
9.d | odd | 6 | 1 | 567.2.o.c | 4 | ||
9.d | odd | 6 | 1 | 567.2.o.d | 4 | ||
12.b | even | 2 | 1 | 3024.2.k.i | 4 | ||
21.c | even | 2 | 1 | inner | 189.2.c.b | ✓ | 4 |
28.d | even | 2 | 1 | 3024.2.k.i | 4 | ||
63.l | odd | 6 | 1 | 567.2.o.c | 4 | ||
63.l | odd | 6 | 1 | 567.2.o.d | 4 | ||
63.o | even | 6 | 1 | 567.2.o.c | 4 | ||
63.o | even | 6 | 1 | 567.2.o.d | 4 | ||
84.h | odd | 2 | 1 | 3024.2.k.i | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
189.2.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
189.2.c.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
189.2.c.b | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
189.2.c.b | ✓ | 4 | 21.c | even | 2 | 1 | inner |
567.2.o.c | 4 | 9.c | even | 3 | 1 | ||
567.2.o.c | 4 | 9.d | odd | 6 | 1 | ||
567.2.o.c | 4 | 63.l | odd | 6 | 1 | ||
567.2.o.c | 4 | 63.o | even | 6 | 1 | ||
567.2.o.d | 4 | 9.c | even | 3 | 1 | ||
567.2.o.d | 4 | 9.d | odd | 6 | 1 | ||
567.2.o.d | 4 | 63.l | odd | 6 | 1 | ||
567.2.o.d | 4 | 63.o | even | 6 | 1 | ||
3024.2.k.i | 4 | 4.b | odd | 2 | 1 | ||
3024.2.k.i | 4 | 12.b | even | 2 | 1 | ||
3024.2.k.i | 4 | 28.d | even | 2 | 1 | ||
3024.2.k.i | 4 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 5 \)
acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 5)^{2} \)
$3$
\( T^{4} \)
$5$
\( (T^{2} - 15)^{2} \)
$7$
\( (T^{2} + 4 T + 7)^{2} \)
$11$
\( (T^{2} + 5)^{2} \)
$13$
\( (T^{2} + 12)^{2} \)
$17$
\( T^{4} \)
$19$
\( (T^{2} + 27)^{2} \)
$23$
\( (T^{2} + 5)^{2} \)
$29$
\( (T^{2} + 20)^{2} \)
$31$
\( (T^{2} + 3)^{2} \)
$37$
\( (T + 1)^{4} \)
$41$
\( (T^{2} - 15)^{2} \)
$43$
\( (T - 2)^{4} \)
$47$
\( (T^{2} - 60)^{2} \)
$53$
\( (T^{2} + 80)^{2} \)
$59$
\( (T^{2} - 60)^{2} \)
$61$
\( (T^{2} + 48)^{2} \)
$67$
\( (T + 10)^{4} \)
$71$
\( (T^{2} + 125)^{2} \)
$73$
\( (T^{2} + 108)^{2} \)
$79$
\( (T - 2)^{4} \)
$83$
\( (T^{2} - 60)^{2} \)
$89$
\( (T^{2} - 135)^{2} \)
$97$
\( (T^{2} + 192)^{2} \)
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