Properties

Label 1134.2.a.l
Level $1134$
Weight $2$
Character orbit 1134.a
Self dual yes
Analytic conductor $9.055$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( 2 + \beta ) q^{5} - q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( 2 + \beta ) q^{5} - q^{7} - q^{8} + ( -2 - \beta ) q^{10} + ( 1 - 3 \beta ) q^{11} + ( -3 + 2 \beta ) q^{13} + q^{14} + q^{16} + 7 q^{17} + ( -1 - \beta ) q^{19} + ( 2 + \beta ) q^{20} + ( -1 + 3 \beta ) q^{22} + ( 1 + 3 \beta ) q^{23} + ( 2 + 4 \beta ) q^{25} + ( 3 - 2 \beta ) q^{26} - q^{28} + ( 5 + 2 \beta ) q^{29} + ( 3 - 3 \beta ) q^{31} - q^{32} -7 q^{34} + ( -2 - \beta ) q^{35} + ( 2 - 5 \beta ) q^{37} + ( 1 + \beta ) q^{38} + ( -2 - \beta ) q^{40} + ( 6 + 2 \beta ) q^{41} + ( 2 + 2 \beta ) q^{43} + ( 1 - 3 \beta ) q^{44} + ( -1 - 3 \beta ) q^{46} + ( 3 - \beta ) q^{47} + q^{49} + ( -2 - 4 \beta ) q^{50} + ( -3 + 2 \beta ) q^{52} + ( -6 + 2 \beta ) q^{53} + ( -7 - 5 \beta ) q^{55} + q^{56} + ( -5 - 2 \beta ) q^{58} + ( -1 - 3 \beta ) q^{59} + ( -3 - 4 \beta ) q^{61} + ( -3 + 3 \beta ) q^{62} + q^{64} + \beta q^{65} + ( -5 + \beta ) q^{67} + 7 q^{68} + ( 2 + \beta ) q^{70} + ( 10 + 2 \beta ) q^{71} + ( 10 + \beta ) q^{73} + ( -2 + 5 \beta ) q^{74} + ( -1 - \beta ) q^{76} + ( -1 + 3 \beta ) q^{77} + ( 3 + 7 \beta ) q^{79} + ( 2 + \beta ) q^{80} + ( -6 - 2 \beta ) q^{82} + ( 1 - 9 \beta ) q^{83} + ( 14 + 7 \beta ) q^{85} + ( -2 - 2 \beta ) q^{86} + ( -1 + 3 \beta ) q^{88} + ( 3 - 4 \beta ) q^{89} + ( 3 - 2 \beta ) q^{91} + ( 1 + 3 \beta ) q^{92} + ( -3 + \beta ) q^{94} + ( -5 - 3 \beta ) q^{95} + ( 4 - 4 \beta ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 4q^{5} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 4q^{5} - 2q^{7} - 2q^{8} - 4q^{10} + 2q^{11} - 6q^{13} + 2q^{14} + 2q^{16} + 14q^{17} - 2q^{19} + 4q^{20} - 2q^{22} + 2q^{23} + 4q^{25} + 6q^{26} - 2q^{28} + 10q^{29} + 6q^{31} - 2q^{32} - 14q^{34} - 4q^{35} + 4q^{37} + 2q^{38} - 4q^{40} + 12q^{41} + 4q^{43} + 2q^{44} - 2q^{46} + 6q^{47} + 2q^{49} - 4q^{50} - 6q^{52} - 12q^{53} - 14q^{55} + 2q^{56} - 10q^{58} - 2q^{59} - 6q^{61} - 6q^{62} + 2q^{64} - 10q^{67} + 14q^{68} + 4q^{70} + 20q^{71} + 20q^{73} - 4q^{74} - 2q^{76} - 2q^{77} + 6q^{79} + 4q^{80} - 12q^{82} + 2q^{83} + 28q^{85} - 4q^{86} - 2q^{88} + 6q^{89} + 6q^{91} + 2q^{92} - 6q^{94} - 10q^{95} + 8q^{97} - 2q^{98} + O(q^{100}) \)

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
−1.00000 0 1.00000 0.267949 0 −1.00000 −1.00000 0 −0.267949
1.2 −1.00000 0 1.00000 3.73205 0 −1.00000 −1.00000 0 −3.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.a.l 2
3.b odd 2 1 1134.2.a.m yes 2
4.b odd 2 1 9072.2.a.bp 2
7.b odd 2 1 7938.2.a.bg 2
9.c even 3 2 1134.2.f.s 4
9.d odd 6 2 1134.2.f.r 4
12.b even 2 1 9072.2.a.y 2
21.c even 2 1 7938.2.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.l 2 1.a even 1 1 trivial
1134.2.a.m yes 2 3.b odd 2 1
1134.2.f.r 4 9.d odd 6 2
1134.2.f.s 4 9.c even 3 2
7938.2.a.bg 2 7.b odd 2 1
7938.2.a.bt 2 21.c even 2 1
9072.2.a.y 2 12.b even 2 1
9072.2.a.bp 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1134))\):

\( T_{5}^{2} - 4 T_{5} + 1 \)
\( T_{11}^{2} - 2 T_{11} - 26 \)
\( T_{13}^{2} + 6 T_{13} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - 4 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -26 - 2 T + T^{2} \)
$13$ \( -3 + 6 T + T^{2} \)
$17$ \( ( -7 + T )^{2} \)
$19$ \( -2 + 2 T + T^{2} \)
$23$ \( -26 - 2 T + T^{2} \)
$29$ \( 13 - 10 T + T^{2} \)
$31$ \( -18 - 6 T + T^{2} \)
$37$ \( -71 - 4 T + T^{2} \)
$41$ \( 24 - 12 T + T^{2} \)
$43$ \( -8 - 4 T + T^{2} \)
$47$ \( 6 - 6 T + T^{2} \)
$53$ \( 24 + 12 T + T^{2} \)
$59$ \( -26 + 2 T + T^{2} \)
$61$ \( -39 + 6 T + T^{2} \)
$67$ \( 22 + 10 T + T^{2} \)
$71$ \( 88 - 20 T + T^{2} \)
$73$ \( 97 - 20 T + T^{2} \)
$79$ \( -138 - 6 T + T^{2} \)
$83$ \( -242 - 2 T + T^{2} \)
$89$ \( -39 - 6 T + T^{2} \)
$97$ \( -32 - 8 T + T^{2} \)
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