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: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−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:

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.f 2
3.b odd 2 1 inner 189.2.a.f 2
4.b odd 2 1 3024.2.a.bi 2
5.b even 2 1 4725.2.a.bb 2
7.b odd 2 1 1323.2.a.w 2
9.c even 3 2 567.2.f.i 4
9.d odd 6 2 567.2.f.i 4
12.b even 2 1 3024.2.a.bi 2
15.d odd 2 1 4725.2.a.bb 2
21.c even 2 1 1323.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.f 2 1.a even 1 1 trivial
189.2.a.f 2 3.b odd 2 1 inner
567.2.f.i 4 9.c even 3 2
567.2.f.i 4 9.d odd 6 2
1323.2.a.w 2 7.b odd 2 1
1323.2.a.w 2 21.c even 2 1
3024.2.a.bi 2 4.b odd 2 1
3024.2.a.bi 2 12.b even 2 1
4725.2.a.bb 2 5.b even 2 1
4725.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(189))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -7 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -7 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -7 + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -7 + T )^{2} \)
$23$ \( -63 + T^{2} \)
$29$ \( -28 + T^{2} \)
$31$ \( ( -3 + T )^{2} \)
$37$ \( ( 3 + T )^{2} \)
$41$ \( -7 + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( -63 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -252 + T^{2} \)
$89$ \( -343 + T^{2} \)
$97$ \( ( 12 + T )^{2} \)
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