Properties

Label 189.2.a.f
Level 189
Weight 2
Character orbit 189.a
Self dual Yes
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.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.5091725982\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + 5 q^{4} \) \( - \beta q^{5} \) \(- q^{7}\) \( + 3 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + 5 q^{4} \) \( - \beta q^{5} \) \(- q^{7}\) \( + 3 \beta q^{8} \) \( -7 q^{10} \) \( - \beta q^{11} \) \( -2 q^{13} \) \( - \beta q^{14} \) \( + 11 q^{16} \) \( + 7 q^{19} \) \( -5 \beta q^{20} \) \( -7 q^{22} \) \( -3 \beta q^{23} \) \( + 2 q^{25} \) \( -2 \beta q^{26} \) \( -5 q^{28} \) \( + 2 \beta q^{29} \) \( + 3 q^{31} \) \( + 5 \beta q^{32} \) \( + \beta q^{35} \) \( -3 q^{37} \) \( + 7 \beta q^{38} \) \( -21 q^{40} \) \( + \beta q^{41} \) \( + 8 q^{43} \) \( -5 \beta q^{44} \) \( -21 q^{46} \) \(+ q^{49}\) \( + 2 \beta q^{50} \) \( -10 q^{52} \) \( + 7 q^{55} \) \( -3 \beta q^{56} \) \( + 14 q^{58} \) \( -8 q^{61} \) \( + 3 \beta q^{62} \) \( + 13 q^{64} \) \( + 2 \beta q^{65} \) \( -2 q^{67} \) \( + 7 q^{70} \) \( + 3 \beta q^{71} \) \( -3 \beta q^{74} \) \( + 35 q^{76} \) \( + \beta q^{77} \) \( -4 q^{79} \) \( -11 \beta q^{80} \) \( + 7 q^{82} \) \( + 6 \beta q^{83} \) \( + 8 \beta q^{86} \) \( -21 q^{88} \) \( -7 \beta q^{89} \) \( + 2 q^{91} \) \( -15 \beta q^{92} \) \( -7 \beta q^{95} \) \( -12 q^{97} \) \( + \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 14q^{10} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 22q^{16} \) \(\mathstrut +\mathstrut 14q^{19} \) \(\mathstrut -\mathstrut 14q^{22} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 10q^{28} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 42q^{40} \) \(\mathstrut +\mathstrut 16q^{43} \) \(\mathstrut -\mathstrut 42q^{46} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 20q^{52} \) \(\mathstrut +\mathstrut 14q^{55} \) \(\mathstrut +\mathstrut 28q^{58} \) \(\mathstrut -\mathstrut 16q^{61} \) \(\mathstrut +\mathstrut 26q^{64} \) \(\mathstrut -\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 14q^{70} \) \(\mathstrut +\mathstrut 70q^{76} \) \(\mathstrut -\mathstrut 8q^{79} \) \(\mathstrut +\mathstrut 14q^{82} \) \(\mathstrut -\mathstrut 42q^{88} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 24q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.64575
2.64575
−2.64575 0 5.00000 2.64575 0 −1.00000 −7.93725 0 −7.00000
1.2 2.64575 0 5.00000 −2.64575 0 −1.00000 7.93725 0 −7.00000
\(n\): 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 yes

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(189))\):

\(T_{2}^{2} \) \(\mathstrut -\mathstrut 7 \)
\(T_{5}^{2} \) \(\mathstrut -\mathstrut 7 \)