Newspace parameters
Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 189.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.14987699641\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{7})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 8x^{2} + 9 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 3 \) |
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 \( x^{4} + 8x^{2} + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 4 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} + 6\nu \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 4 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 6\beta_1 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
134.1 |
|
− | 2.57794i | 0 | −2.64575 | − | 1.66471i | 0 | −2.64575 | − | 3.49117i | 0 | −4.29150 | |||||||||||||||||||||||||||
134.2 | − | 1.16372i | 0 | 2.64575 | 5.40636i | 0 | 2.64575 | − | 7.73381i | 0 | 6.29150 | |||||||||||||||||||||||||||||
134.3 | 1.16372i | 0 | 2.64575 | − | 5.40636i | 0 | 2.64575 | 7.73381i | 0 | 6.29150 | ||||||||||||||||||||||||||||||
134.4 | 2.57794i | 0 | −2.64575 | 1.66471i | 0 | −2.64575 | 3.49117i | 0 | −4.29150 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 189.3.b.b | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 189.3.b.b | ✓ | 4 |
4.b | odd | 2 | 1 | 3024.3.d.c | 4 | ||
9.c | even | 3 | 2 | 567.3.r.b | 8 | ||
9.d | odd | 6 | 2 | 567.3.r.b | 8 | ||
12.b | even | 2 | 1 | 3024.3.d.c | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
189.3.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
189.3.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
567.3.r.b | 8 | 9.c | even | 3 | 2 | ||
567.3.r.b | 8 | 9.d | odd | 6 | 2 | ||
3024.3.d.c | 4 | 4.b | odd | 2 | 1 | ||
3024.3.d.c | 4 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 8T_{2}^{2} + 9 \)
acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 8T^{2} + 9 \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 32T^{2} + 81 \)
$7$
\( (T^{2} - 7)^{2} \)
$11$
\( T^{4} + 536 T^{2} + 68121 \)
$13$
\( (T^{2} + 6 T - 166)^{2} \)
$17$
\( (T^{2} + 18)^{2} \)
$19$
\( (T^{2} - 12 T + 29)^{2} \)
$23$
\( T^{4} + 1152 T^{2} + 263169 \)
$29$
\( T^{4} + 2804 T^{2} + \cdots + 1954404 \)
$31$
\( (T^{2} - 32 T - 87)^{2} \)
$37$
\( (T^{2} + 46 T - 843)^{2} \)
$41$
\( T^{4} + 6512 T^{2} + \cdots + 6922161 \)
$43$
\( (T^{2} + 14 T - 1526)^{2} \)
$47$
\( T^{4} + 1044 T^{2} + 236196 \)
$53$
\( T^{4} + 6624 T^{2} + \cdots + 8928144 \)
$59$
\( T^{4} + 13644 T^{2} + \cdots + 30536676 \)
$61$
\( (T^{2} - 48 T - 7516)^{2} \)
$67$
\( (T^{2} - 78 T + 1514)^{2} \)
$71$
\( T^{4} + 15192 T^{2} + 729 \)
$73$
\( (T^{2} - 98 T - 1302)^{2} \)
$79$
\( (T^{2} + 242 T + 14074)^{2} \)
$83$
\( T^{4} + 24588 T^{2} + \cdots + 145009764 \)
$89$
\( T^{4} + 8600 T^{2} + 5625 \)
$97$
\( (T^{2} - 8 T - 10092)^{2} \)
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