Properties

Label 1134.2.f.g
Level 1134
Weight 2
Character orbit 1134.f
Analytic conductor 9.055
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 + 1.73205i 0 0.500000 0.866025i 1.00000 0 −2.00000
757.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 1.73205i 0 0.500000 + 0.866025i 1.00000 0 −2.00000
\(n\): 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.g 2
3.b odd 2 1 1134.2.f.j 2
9.c even 3 1 42.2.a.a 1
9.c even 3 1 inner 1134.2.f.g 2
9.d odd 6 1 126.2.a.a 1
9.d odd 6 1 1134.2.f.j 2
36.f odd 6 1 336.2.a.d 1
36.h even 6 1 1008.2.a.j 1
45.h odd 6 1 3150.2.a.bo 1
45.j even 6 1 1050.2.a.i 1
45.k odd 12 2 1050.2.g.a 2
45.l even 12 2 3150.2.g.r 2
63.g even 3 1 294.2.e.c 2
63.h even 3 1 294.2.e.c 2
63.i even 6 1 882.2.g.j 2
63.j odd 6 1 882.2.g.h 2
63.k odd 6 1 294.2.e.a 2
63.l odd 6 1 294.2.a.g 1
63.n odd 6 1 882.2.g.h 2
63.o even 6 1 882.2.a.b 1
63.s even 6 1 882.2.g.j 2
63.t odd 6 1 294.2.e.a 2
72.j odd 6 1 4032.2.a.e 1
72.l even 6 1 4032.2.a.m 1
72.n even 6 1 1344.2.a.q 1
72.p odd 6 1 1344.2.a.i 1
99.h odd 6 1 5082.2.a.d 1
117.t even 6 1 7098.2.a.f 1
144.v odd 12 2 5376.2.c.e 2
144.x even 12 2 5376.2.c.bc 2
180.p odd 6 1 8400.2.a.k 1
252.n even 6 1 2352.2.q.n 2
252.s odd 6 1 7056.2.a.k 1
252.u odd 6 1 2352.2.q.i 2
252.bi even 6 1 2352.2.a.l 1
252.bj even 6 1 2352.2.q.n 2
252.bl odd 6 1 2352.2.q.i 2
315.bg odd 6 1 7350.2.a.f 1
504.be even 6 1 9408.2.a.bw 1
504.bn odd 6 1 9408.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 9.c even 3 1
126.2.a.a 1 9.d odd 6 1
294.2.a.g 1 63.l odd 6 1
294.2.e.a 2 63.k odd 6 1
294.2.e.a 2 63.t odd 6 1
294.2.e.c 2 63.g even 3 1
294.2.e.c 2 63.h even 3 1
336.2.a.d 1 36.f odd 6 1
882.2.a.b 1 63.o even 6 1
882.2.g.h 2 63.j odd 6 1
882.2.g.h 2 63.n odd 6 1
882.2.g.j 2 63.i even 6 1
882.2.g.j 2 63.s even 6 1
1008.2.a.j 1 36.h even 6 1
1050.2.a.i 1 45.j even 6 1
1050.2.g.a 2 45.k odd 12 2
1134.2.f.g 2 1.a even 1 1 trivial
1134.2.f.g 2 9.c even 3 1 inner
1134.2.f.j 2 3.b odd 2 1
1134.2.f.j 2 9.d odd 6 1
1344.2.a.i 1 72.p odd 6 1
1344.2.a.q 1 72.n even 6 1
2352.2.a.l 1 252.bi even 6 1
2352.2.q.i 2 252.u odd 6 1
2352.2.q.i 2 252.bl odd 6 1
2352.2.q.n 2 252.n even 6 1
2352.2.q.n 2 252.bj even 6 1
3150.2.a.bo 1 45.h odd 6 1
3150.2.g.r 2 45.l even 12 2
4032.2.a.e 1 72.j odd 6 1
4032.2.a.m 1 72.l even 6 1
5082.2.a.d 1 99.h odd 6 1
5376.2.c.e 2 144.v odd 12 2
5376.2.c.bc 2 144.x even 12 2
7056.2.a.k 1 252.s odd 6 1
7098.2.a.f 1 117.t even 6 1
7350.2.a.f 1 315.bg odd 6 1
8400.2.a.k 1 180.p odd 6 1
9408.2.a.n 1 504.bn odd 6 1
9408.2.a.bw 1 504.be even 6 1

Hecke kernels

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

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} - 4 T_{11} + 16 \)
\( T_{13}^{2} + 6 T_{13} + 36 \)
\( T_{17} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ 1
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \)
$29$ \( 1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( ( 1 + 10 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 79 T^{2} + 6241 T^{4} \)
$83$ \( 1 - 4 T - 67 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 19 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \)
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