Properties

Label 189.2.e.c
Level 189
Weight 2
Character orbit 189.e
Analytic conductor 1.509
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \zeta_{6} q^{2} \) \( + ( 1 - \zeta_{6} ) q^{4} \) \( -4 \zeta_{6} q^{5} \) \( + ( 1 + 2 \zeta_{6} ) q^{7} \) \( + 3 q^{8} \) \(+O(q^{10})\) \( q\) \( + \zeta_{6} q^{2} \) \( + ( 1 - \zeta_{6} ) q^{4} \) \( -4 \zeta_{6} q^{5} \) \( + ( 1 + 2 \zeta_{6} ) q^{7} \) \( + 3 q^{8} \) \( + ( 4 - 4 \zeta_{6} ) q^{10} \) \( + ( 2 - 2 \zeta_{6} ) q^{11} \) \(+ q^{13}\) \( + ( -2 + 3 \zeta_{6} ) q^{14} \) \( + \zeta_{6} q^{16} \) \( + ( -6 + 6 \zeta_{6} ) q^{17} \) \( -4 \zeta_{6} q^{19} \) \( -4 q^{20} \) \( + 2 q^{22} \) \( + 6 \zeta_{6} q^{23} \) \( + ( -11 + 11 \zeta_{6} ) q^{25} \) \( + \zeta_{6} q^{26} \) \( + ( 3 - \zeta_{6} ) q^{28} \) \( -2 q^{29} \) \( + ( -3 + 3 \zeta_{6} ) q^{31} \) \( + ( 5 - 5 \zeta_{6} ) q^{32} \) \( -6 q^{34} \) \( + ( 8 - 12 \zeta_{6} ) q^{35} \) \( -3 \zeta_{6} q^{37} \) \( + ( 4 - 4 \zeta_{6} ) q^{38} \) \( -12 \zeta_{6} q^{40} \) \( + 2 q^{41} \) \(- q^{43}\) \( -2 \zeta_{6} q^{44} \) \( + ( -6 + 6 \zeta_{6} ) q^{46} \) \( + 6 \zeta_{6} q^{47} \) \( + ( -3 + 8 \zeta_{6} ) q^{49} \) \( -11 q^{50} \) \( + ( 1 - \zeta_{6} ) q^{52} \) \( + ( -6 + 6 \zeta_{6} ) q^{53} \) \( -8 q^{55} \) \( + ( 3 + 6 \zeta_{6} ) q^{56} \) \( -2 \zeta_{6} q^{58} \) \( + ( 6 - 6 \zeta_{6} ) q^{59} \) \( + 5 \zeta_{6} q^{61} \) \( -3 q^{62} \) \( + 7 q^{64} \) \( -4 \zeta_{6} q^{65} \) \( + ( -7 + 7 \zeta_{6} ) q^{67} \) \( + 6 \zeta_{6} q^{68} \) \( + ( 12 - 4 \zeta_{6} ) q^{70} \) \( + ( 6 - 6 \zeta_{6} ) q^{73} \) \( + ( 3 - 3 \zeta_{6} ) q^{74} \) \( -4 q^{76} \) \( + ( 6 - 2 \zeta_{6} ) q^{77} \) \( -11 \zeta_{6} q^{79} \) \( + ( 4 - 4 \zeta_{6} ) q^{80} \) \( + 2 \zeta_{6} q^{82} \) \( -6 q^{83} \) \( + 24 q^{85} \) \( -\zeta_{6} q^{86} \) \( + ( 6 - 6 \zeta_{6} ) q^{88} \) \( -4 \zeta_{6} q^{89} \) \( + ( 1 + 2 \zeta_{6} ) q^{91} \) \( + 6 q^{92} \) \( + ( -6 + 6 \zeta_{6} ) q^{94} \) \( + ( -16 + 16 \zeta_{6} ) q^{95} \) \( + 9 q^{97} \) \( + ( -8 + 5 \zeta_{6} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut q^{14} \) \(\mathstrut +\mathstrut q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 8q^{20} \) \(\mathstrut +\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 6q^{23} \) \(\mathstrut -\mathstrut 11q^{25} \) \(\mathstrut +\mathstrut q^{26} \) \(\mathstrut +\mathstrut 5q^{28} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 3q^{31} \) \(\mathstrut +\mathstrut 5q^{32} \) \(\mathstrut -\mathstrut 12q^{34} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut +\mathstrut 4q^{38} \) \(\mathstrut -\mathstrut 12q^{40} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 6q^{46} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 22q^{50} \) \(\mathstrut +\mathstrut q^{52} \) \(\mathstrut -\mathstrut 6q^{53} \) \(\mathstrut -\mathstrut 16q^{55} \) \(\mathstrut +\mathstrut 12q^{56} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut 6q^{59} \) \(\mathstrut +\mathstrut 5q^{61} \) \(\mathstrut -\mathstrut 6q^{62} \) \(\mathstrut +\mathstrut 14q^{64} \) \(\mathstrut -\mathstrut 4q^{65} \) \(\mathstrut -\mathstrut 7q^{67} \) \(\mathstrut +\mathstrut 6q^{68} \) \(\mathstrut +\mathstrut 20q^{70} \) \(\mathstrut +\mathstrut 6q^{73} \) \(\mathstrut +\mathstrut 3q^{74} \) \(\mathstrut -\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 10q^{77} \) \(\mathstrut -\mathstrut 11q^{79} \) \(\mathstrut +\mathstrut 4q^{80} \) \(\mathstrut +\mathstrut 2q^{82} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut +\mathstrut 48q^{85} \) \(\mathstrut -\mathstrut q^{86} \) \(\mathstrut +\mathstrut 6q^{88} \) \(\mathstrut -\mathstrut 4q^{89} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 12q^{92} \) \(\mathstrut -\mathstrut 6q^{94} \) \(\mathstrut -\mathstrut 16q^{95} \) \(\mathstrut +\mathstrut 18q^{97} \) \(\mathstrut -\mathstrut 11q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0.500000 + 0.866025i 0 0.500000 0.866025i −2.00000 3.46410i 0 2.00000 + 1.73205i 3.00000 0 2.00000 3.46410i
163.1 0.500000 0.866025i 0 0.500000 + 0.866025i −2.00000 + 3.46410i 0 2.00000 1.73205i 3.00000 0 2.00000 + 3.46410i
\(n\): 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
7.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).