Properties

Label 1134.2.f.r
Level $1134$
Weight $2$
Character orbit 1134.f
Analytic conductor $9.055$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0.133975 + 0.232051i 0 0.500000 0.866025i 1.00000 0 −0.267949
379.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.86603 + 3.23205i 0 0.500000 0.866025i 1.00000 0 −3.73205
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.133975 0.232051i 0 0.500000 + 0.866025i 1.00000 0 −0.267949
757.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.86603 3.23205i 0 0.500000 + 0.866025i 1.00000 0 −3.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.f.r 4
3.b odd 2 1 1134.2.f.s 4
9.c even 3 1 1134.2.a.m yes 2
9.c even 3 1 inner 1134.2.f.r 4
9.d odd 6 1 1134.2.a.l 2
9.d odd 6 1 1134.2.f.s 4
36.f odd 6 1 9072.2.a.y 2
36.h even 6 1 9072.2.a.bp 2
63.l odd 6 1 7938.2.a.bt 2
63.o even 6 1 7938.2.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.l 2 9.d odd 6 1
1134.2.a.m yes 2 9.c even 3 1
1134.2.f.r 4 1.a even 1 1 trivial
1134.2.f.r 4 9.c even 3 1 inner
1134.2.f.s 4 3.b odd 2 1
1134.2.f.s 4 9.d odd 6 1
7938.2.a.bg 2 63.o even 6 1
7938.2.a.bt 2 63.l odd 6 1
9072.2.a.y 2 36.f odd 6 1
9072.2.a.bp 2 36.h even 6 1

Hecke kernels

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

\( T_{5}^{4} - 4 T_{5}^{3} + 15 T_{5}^{2} - 4 T_{5} + 1 \)
\( T_{11}^{4} - 2 T_{11}^{3} + 30 T_{11}^{2} + 52 T_{11} + 676 \)
\( T_{13}^{4} - 6 T_{13}^{3} + 39 T_{13}^{2} + 18 T_{13} + 9 \)
\( T_{17} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( ( 1 - T + T^{2} )^{2} \)
$11$ \( 676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( ( 7 + T )^{4} \)
$19$ \( ( -2 + 2 T + T^{2} )^{2} \)
$23$ \( 676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( 169 - 130 T + 87 T^{2} - 10 T^{3} + T^{4} \)
$31$ \( 324 - 108 T + 54 T^{2} + 6 T^{3} + T^{4} \)
$37$ \( ( -71 - 4 T + T^{2} )^{2} \)
$41$ \( 576 - 288 T + 120 T^{2} - 12 T^{3} + T^{4} \)
$43$ \( 64 - 32 T + 24 T^{2} + 4 T^{3} + T^{4} \)
$47$ \( 36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( ( 24 - 12 T + T^{2} )^{2} \)
$59$ \( 676 - 52 T + 30 T^{2} + 2 T^{3} + T^{4} \)
$61$ \( 1521 + 234 T + 75 T^{2} - 6 T^{3} + T^{4} \)
$67$ \( 484 - 220 T + 78 T^{2} - 10 T^{3} + T^{4} \)
$71$ \( ( 88 + 20 T + T^{2} )^{2} \)
$73$ \( ( 97 - 20 T + T^{2} )^{2} \)
$79$ \( 19044 - 828 T + 174 T^{2} + 6 T^{3} + T^{4} \)
$83$ \( 58564 + 484 T + 246 T^{2} - 2 T^{3} + T^{4} \)
$89$ \( ( -39 + 6 T + T^{2} )^{2} \)
$97$ \( 1024 - 256 T + 96 T^{2} + 8 T^{3} + T^{4} \)
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