Properties

Label 189.2.c.a
Level 189
Weight 2
Character orbit 189.c
Analytic conductor 1.509
Analytic rank 0
Dimension 2
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 189.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.5091725982\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + 3\sqrt{-3})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 2 q^{4} \) \( + \beta q^{7} \) \(+O(q^{10})\) \( q\) \( + 2 q^{4} \) \( + \beta q^{7} \) \( + ( 1 - 2 \beta ) q^{13} \) \( + 4 q^{16} \) \( + ( -1 + 2 \beta ) q^{19} \) \( -5 q^{25} \) \( + 2 \beta q^{28} \) \( + ( 2 - 4 \beta ) q^{31} \) \( -11 q^{37} \) \( -8 q^{43} \) \( + ( -7 + \beta ) q^{49} \) \( + ( 2 - 4 \beta ) q^{52} \) \( + ( -3 + 6 \beta ) q^{61} \) \( + 8 q^{64} \) \( + 5 q^{67} \) \( + ( 3 - 6 \beta ) q^{73} \) \( + ( -2 + 4 \beta ) q^{76} \) \( + 17 q^{79} \) \( + ( 14 - \beta ) q^{91} \) \( + ( 1 - 2 \beta ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 10q^{25} \) \(\mathstrut +\mathstrut 2q^{28} \) \(\mathstrut -\mathstrut 22q^{37} \) \(\mathstrut -\mathstrut 16q^{43} \) \(\mathstrut -\mathstrut 13q^{49} \) \(\mathstrut +\mathstrut 16q^{64} \) \(\mathstrut +\mathstrut 10q^{67} \) \(\mathstrut +\mathstrut 34q^{79} \) \(\mathstrut +\mathstrut 27q^{91} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 0 2.00000 0 0 0.500000 2.59808i 0 0 0
188.2 0 0 2.00000 0 0 0.500000 + 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
7.b Odd 1 yes
21.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).