Properties

Label 189.2.e.b
Level 189
Weight 2
Character orbit 189.e
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.e (of order \(3\) and degree \(2\))

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: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 2 - 2 \zeta_{6} ) q^{4} \) \( + ( 1 - 3 \zeta_{6} ) q^{7} \) \(+O(q^{10})\) \( q\) \( + ( 2 - 2 \zeta_{6} ) q^{4} \) \( + ( 1 - 3 \zeta_{6} ) q^{7} \) \( + 2 q^{13} \) \( -4 \zeta_{6} q^{16} \) \( + 7 \zeta_{6} q^{19} \) \( + ( 5 - 5 \zeta_{6} ) q^{25} \) \( + ( -4 - 2 \zeta_{6} ) q^{28} \) \( + ( -11 + 11 \zeta_{6} ) q^{31} \) \( + 10 \zeta_{6} q^{37} \) \( -13 q^{43} \) \( + ( -8 + 3 \zeta_{6} ) q^{49} \) \( + ( 4 - 4 \zeta_{6} ) q^{52} \) \( + 13 \zeta_{6} q^{61} \) \( -8 q^{64} \) \( + ( 16 - 16 \zeta_{6} ) q^{67} \) \( + ( 7 - 7 \zeta_{6} ) q^{73} \) \( + 14 q^{76} \) \( + 4 \zeta_{6} q^{79} \) \( + ( 2 - 6 \zeta_{6} ) q^{91} \) \( + 5 q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 4q^{16} \) \(\mathstrut +\mathstrut 7q^{19} \) \(\mathstrut +\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 10q^{28} \) \(\mathstrut -\mathstrut 11q^{31} \) \(\mathstrut +\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 26q^{43} \) \(\mathstrut -\mathstrut 13q^{49} \) \(\mathstrut +\mathstrut 4q^{52} \) \(\mathstrut +\mathstrut 13q^{61} \) \(\mathstrut -\mathstrut 16q^{64} \) \(\mathstrut +\mathstrut 16q^{67} \) \(\mathstrut +\mathstrut 7q^{73} \) \(\mathstrut +\mathstrut 28q^{76} \) \(\mathstrut +\mathstrut 4q^{79} \) \(\mathstrut -\mathstrut 2q^{91} \) \(\mathstrut +\mathstrut 10q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 1.00000 1.73205i 0 0 −0.500000 2.59808i 0 0 0
163.1 0 0 1.00000 + 1.73205i 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.c Even 1 yes
21.h Odd 1 yes

Hecke kernels

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