Properties

Label 189.2.a.e
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: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} + \beta q^{5} + q^{7} -\beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + q^{4} + \beta q^{5} + q^{7} -\beta q^{8} + 3 q^{10} -\beta q^{11} + 2 q^{13} + \beta q^{14} -5 q^{16} -4 \beta q^{17} + 5 q^{19} + \beta q^{20} -3 q^{22} + \beta q^{23} -2 q^{25} + 2 \beta q^{26} + q^{28} -6 \beta q^{29} + 5 q^{31} -3 \beta q^{32} -12 q^{34} + \beta q^{35} -7 q^{37} + 5 \beta q^{38} -3 q^{40} + 3 \beta q^{41} -4 q^{43} -\beta q^{44} + 3 q^{46} + 4 \beta q^{47} + q^{49} -2 \beta q^{50} + 2 q^{52} + 8 \beta q^{53} -3 q^{55} -\beta q^{56} -18 q^{58} -4 \beta q^{59} + 8 q^{61} + 5 \beta q^{62} + q^{64} + 2 \beta q^{65} + 14 q^{67} -4 \beta q^{68} + 3 q^{70} + 3 \beta q^{71} -4 q^{73} -7 \beta q^{74} + 5 q^{76} -\beta q^{77} + 8 q^{79} -5 \beta q^{80} + 9 q^{82} -6 \beta q^{83} -12 q^{85} -4 \beta q^{86} + 3 q^{88} -5 \beta q^{89} + 2 q^{91} + \beta q^{92} + 12 q^{94} + 5 \beta q^{95} -4 q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{7} + 6q^{10} + 4q^{13} - 10q^{16} + 10q^{19} - 6q^{22} - 4q^{25} + 2q^{28} + 10q^{31} - 24q^{34} - 14q^{37} - 6q^{40} - 8q^{43} + 6q^{46} + 2q^{49} + 4q^{52} - 6q^{55} - 36q^{58} + 16q^{61} + 2q^{64} + 28q^{67} + 6q^{70} - 8q^{73} + 10q^{76} + 16q^{79} + 18q^{82} - 24q^{85} + 6q^{88} + 4q^{91} + 24q^{94} - 8q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
−1.73205 0 1.00000 −1.73205 0 1.00000 1.73205 0 3.00000
1.2 1.73205 0 1.00000 1.73205 0 1.00000 −1.73205 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 189.2.a.e 2
3.b odd 2 1 inner 189.2.a.e 2
4.b odd 2 1 3024.2.a.bg 2
5.b even 2 1 4725.2.a.ba 2
7.b odd 2 1 1323.2.a.t 2
9.c even 3 2 567.2.f.k 4
9.d odd 6 2 567.2.f.k 4
12.b even 2 1 3024.2.a.bg 2
15.d odd 2 1 4725.2.a.ba 2
21.c even 2 1 1323.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.e 2 1.a even 1 1 trivial
189.2.a.e 2 3.b odd 2 1 inner
567.2.f.k 4 9.c even 3 2
567.2.f.k 4 9.d odd 6 2
1323.2.a.t 2 7.b odd 2 1
1323.2.a.t 2 21.c even 2 1
3024.2.a.bg 2 4.b odd 2 1
3024.2.a.bg 2 12.b even 2 1
4725.2.a.ba 2 5.b even 2 1
4725.2.a.ba 2 15.d odd 2 1

Hecke kernels

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

\( T_{2}^{2} - 3 \)
\( T_{5}^{2} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -3 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -3 + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -48 + T^{2} \)
$19$ \( ( -5 + T )^{2} \)
$23$ \( -3 + T^{2} \)
$29$ \( -108 + T^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( ( 7 + T )^{2} \)
$41$ \( -27 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( -192 + T^{2} \)
$59$ \( -48 + T^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( ( -14 + T )^{2} \)
$71$ \( -27 + T^{2} \)
$73$ \( ( 4 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( -108 + T^{2} \)
$89$ \( -75 + T^{2} \)
$97$ \( ( 4 + T )^{2} \)
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