Properties

Label 1134.2.a.p
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: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( 1 + \beta ) q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( 1 + \beta ) q^{5} - q^{7} + q^{8} + ( 1 + \beta ) q^{10} + 2 q^{11} -2 \beta q^{13} - q^{14} + q^{16} + 2 q^{17} + ( 5 + \beta ) q^{19} + ( 1 + \beta ) q^{20} + 2 q^{22} - q^{23} + ( 2 + 2 \beta ) q^{25} -2 \beta q^{26} - q^{28} + ( -2 + 2 \beta ) q^{29} + 6 q^{31} + q^{32} + 2 q^{34} + ( -1 - \beta ) q^{35} + ( 2 - 4 \beta ) q^{37} + ( 5 + \beta ) q^{38} + ( 1 + \beta ) q^{40} -4 \beta q^{41} + ( 2 - 2 \beta ) q^{43} + 2 q^{44} - q^{46} -4 \beta q^{47} + q^{49} + ( 2 + 2 \beta ) q^{50} -2 \beta q^{52} + ( -6 + 2 \beta ) q^{53} + ( 2 + 2 \beta ) q^{55} - q^{56} + ( -2 + 2 \beta ) q^{58} -2 q^{59} + ( 9 + \beta ) q^{61} + 6 q^{62} + q^{64} + ( -12 - 2 \beta ) q^{65} + ( -8 + 2 \beta ) q^{67} + 2 q^{68} + ( -1 - \beta ) q^{70} + ( 5 + 2 \beta ) q^{71} + ( -2 + 2 \beta ) q^{73} + ( 2 - 4 \beta ) q^{74} + ( 5 + \beta ) q^{76} -2 q^{77} + ( 3 + 2 \beta ) q^{79} + ( 1 + \beta ) q^{80} -4 \beta q^{82} + 2 q^{83} + ( 2 + 2 \beta ) q^{85} + ( 2 - 2 \beta ) q^{86} + 2 q^{88} + ( -12 + 2 \beta ) q^{89} + 2 \beta q^{91} - q^{92} -4 \beta q^{94} + ( 11 + 6 \beta ) q^{95} + ( -2 - 2 \beta ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + 2q^{10} + 4q^{11} - 2q^{14} + 2q^{16} + 4q^{17} + 10q^{19} + 2q^{20} + 4q^{22} - 2q^{23} + 4q^{25} - 2q^{28} - 4q^{29} + 12q^{31} + 2q^{32} + 4q^{34} - 2q^{35} + 4q^{37} + 10q^{38} + 2q^{40} + 4q^{43} + 4q^{44} - 2q^{46} + 2q^{49} + 4q^{50} - 12q^{53} + 4q^{55} - 2q^{56} - 4q^{58} - 4q^{59} + 18q^{61} + 12q^{62} + 2q^{64} - 24q^{65} - 16q^{67} + 4q^{68} - 2q^{70} + 10q^{71} - 4q^{73} + 4q^{74} + 10q^{76} - 4q^{77} + 6q^{79} + 2q^{80} + 4q^{83} + 4q^{85} + 4q^{86} + 4q^{88} - 24q^{89} - 2q^{92} + 22q^{95} - 4q^{97} + 2q^{98} + O(q^{100}) \)

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
−2.44949
2.44949
1.00000 0 1.00000 −1.44949 0 −1.00000 1.00000 0 −1.44949
1.2 1.00000 0 1.00000 3.44949 0 −1.00000 1.00000 0 3.44949
\(n\): 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.p 2
3.b odd 2 1 1134.2.a.i 2
4.b odd 2 1 9072.2.a.bk 2
7.b odd 2 1 7938.2.a.bn 2
9.c even 3 2 126.2.f.c 4
9.d odd 6 2 378.2.f.d 4
12.b even 2 1 9072.2.a.bd 2
21.c even 2 1 7938.2.a.bm 2
36.f odd 6 2 1008.2.r.e 4
36.h even 6 2 3024.2.r.e 4
63.g even 3 2 882.2.h.k 4
63.h even 3 2 882.2.e.m 4
63.i even 6 2 2646.2.e.k 4
63.j odd 6 2 2646.2.e.l 4
63.k odd 6 2 882.2.h.l 4
63.l odd 6 2 882.2.f.j 4
63.n odd 6 2 2646.2.h.m 4
63.o even 6 2 2646.2.f.k 4
63.s even 6 2 2646.2.h.n 4
63.t odd 6 2 882.2.e.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 9.c even 3 2
378.2.f.d 4 9.d odd 6 2
882.2.e.m 4 63.h even 3 2
882.2.e.n 4 63.t odd 6 2
882.2.f.j 4 63.l odd 6 2
882.2.h.k 4 63.g even 3 2
882.2.h.l 4 63.k odd 6 2
1008.2.r.e 4 36.f odd 6 2
1134.2.a.i 2 3.b odd 2 1
1134.2.a.p 2 1.a even 1 1 trivial
2646.2.e.k 4 63.i even 6 2
2646.2.e.l 4 63.j odd 6 2
2646.2.f.k 4 63.o even 6 2
2646.2.h.m 4 63.n odd 6 2
2646.2.h.n 4 63.s even 6 2
3024.2.r.e 4 36.h even 6 2
7938.2.a.bm 2 21.c even 2 1
7938.2.a.bn 2 7.b odd 2 1
9072.2.a.bd 2 12.b even 2 1
9072.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1134))\):

\( T_{5}^{2} - 2 T_{5} - 5 \)
\( T_{11} - 2 \)
\( T_{13}^{2} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -5 - 2 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -24 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 19 - 10 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -20 + 4 T + T^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( -92 - 4 T + T^{2} \)
$41$ \( -96 + T^{2} \)
$43$ \( -20 - 4 T + T^{2} \)
$47$ \( -96 + T^{2} \)
$53$ \( 12 + 12 T + T^{2} \)
$59$ \( ( 2 + T )^{2} \)
$61$ \( 75 - 18 T + T^{2} \)
$67$ \( 40 + 16 T + T^{2} \)
$71$ \( 1 - 10 T + T^{2} \)
$73$ \( -20 + 4 T + T^{2} \)
$79$ \( -15 - 6 T + T^{2} \)
$83$ \( ( -2 + T )^{2} \)
$89$ \( 120 + 24 T + T^{2} \)
$97$ \( -20 + 4 T + T^{2} \)
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