Properties

Label 1134.2.l.f
Level 1134
Weight 2
Character orbit 1134.l
Analytic conductor 9.055
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1134.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{6} q^{2} - q^{4} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})\) \( q + \zeta_{24}^{6} q^{2} - q^{4} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} + ( 1 - 2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{10} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{11} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{13} + ( \zeta_{24} - \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{14} + q^{16} + ( -2 \zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 4 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{17} + ( -4 - \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{19} + ( 2 \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{20} + 3 \zeta_{24}^{4} q^{22} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{23} + ( 6 \zeta_{24} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{7} ) q^{25} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{26} + ( \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{28} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{29} + ( -1 + 3 \zeta_{24} - 3 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{31} + \zeta_{24}^{6} q^{32} + ( -2 - 2 \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{34} + ( -4 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 7 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{35} + ( 3 \zeta_{24} - 4 \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{37} + ( \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{38} + ( -1 + 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{40} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{41} + ( -4 + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{43} + ( -3 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{44} + ( 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{46} + ( \zeta_{24} + 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{47} + ( 5 - 4 \zeta_{24} - 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{49} + ( -6 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{50} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{52} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{53} + ( 3 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{55} + ( -\zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{56} + ( 3 + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{58} + ( -4 \zeta_{24} + 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{59} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{61} + ( -3 \zeta_{24} - 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{62} - q^{64} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{65} -10 q^{67} + ( 2 \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 4 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{68} + ( 5 - 4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{70} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{71} + ( 2 - 4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{73} + ( -3 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{74} + ( 4 + \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{76} + ( -6 \zeta_{24} + 3 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{77} + ( -7 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{79} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{80} + ( 8 + 2 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{82} + ( -4 \zeta_{24} - \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{83} + ( -3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{85} + ( -4 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{86} -3 \zeta_{24}^{4} q^{88} + ( 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + ( -6 + 2 \zeta_{24} + \zeta_{24}^{3} + 6 \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{91} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{92} + ( -2 + \zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{94} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{95} + ( 5 + 4 \zeta_{24} + 2 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} + 5 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + 4q^{7} + O(q^{10}) \) \( 8q - 8q^{4} + 4q^{7} + 12q^{10} + 8q^{16} - 24q^{19} + 12q^{22} - 16q^{25} - 4q^{28} - 24q^{34} - 16q^{37} - 12q^{40} - 16q^{43} + 20q^{49} + 12q^{58} - 8q^{64} - 80q^{67} + 48q^{70} + 24q^{73} + 24q^{76} - 56q^{79} + 48q^{82} - 12q^{88} - 24q^{91} + 60q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(\zeta_{24}^{2}\) \(\zeta_{24}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
1.00000i 0 −1.00000 −0.358719 0.621320i 0 −1.62132 2.09077i 1.00000i 0 −0.621320 + 0.358719i
215.2 1.00000i 0 −1.00000 2.09077 + 3.62132i 0 2.62132 + 0.358719i 1.00000i 0 3.62132 2.09077i
215.3 1.00000i 0 −1.00000 −2.09077 3.62132i 0 2.62132 + 0.358719i 1.00000i 0 3.62132 2.09077i
215.4 1.00000i 0 −1.00000 0.358719 + 0.621320i 0 −1.62132 2.09077i 1.00000i 0 −0.621320 + 0.358719i
269.1 1.00000i 0 −1.00000 −2.09077 + 3.62132i 0 2.62132 0.358719i 1.00000i 0 3.62132 + 2.09077i
269.2 1.00000i 0 −1.00000 0.358719 0.621320i 0 −1.62132 + 2.09077i 1.00000i 0 −0.621320 0.358719i
269.3 1.00000i 0 −1.00000 −0.358719 + 0.621320i 0 −1.62132 + 2.09077i 1.00000i 0 −0.621320 0.358719i
269.4 1.00000i 0 −1.00000 2.09077 3.62132i 0 2.62132 0.358719i 1.00000i 0 3.62132 + 2.09077i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.i even 6 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.l.f 8
3.b odd 2 1 inner 1134.2.l.f 8
7.d odd 6 1 1134.2.t.e 8
9.c even 3 1 126.2.k.a 8
9.c even 3 1 1134.2.t.e 8
9.d odd 6 1 126.2.k.a 8
9.d odd 6 1 1134.2.t.e 8
21.g even 6 1 1134.2.t.e 8
36.f odd 6 1 1008.2.bt.c 8
36.h even 6 1 1008.2.bt.c 8
45.h odd 6 1 3150.2.bf.a 8
45.j even 6 1 3150.2.bf.a 8
45.k odd 12 1 3150.2.bp.b 8
45.k odd 12 1 3150.2.bp.e 8
45.l even 12 1 3150.2.bp.b 8
45.l even 12 1 3150.2.bp.e 8
63.g even 3 1 882.2.k.a 8
63.h even 3 1 882.2.d.a 8
63.i even 6 1 882.2.d.a 8
63.i even 6 1 inner 1134.2.l.f 8
63.j odd 6 1 882.2.d.a 8
63.k odd 6 1 126.2.k.a 8
63.l odd 6 1 882.2.k.a 8
63.n odd 6 1 882.2.k.a 8
63.o even 6 1 882.2.k.a 8
63.s even 6 1 126.2.k.a 8
63.t odd 6 1 882.2.d.a 8
63.t odd 6 1 inner 1134.2.l.f 8
252.n even 6 1 1008.2.bt.c 8
252.r odd 6 1 7056.2.k.f 8
252.u odd 6 1 7056.2.k.f 8
252.bb even 6 1 7056.2.k.f 8
252.bj even 6 1 7056.2.k.f 8
252.bn odd 6 1 1008.2.bt.c 8
315.u even 6 1 3150.2.bf.a 8
315.bn odd 6 1 3150.2.bf.a 8
315.bw odd 12 1 3150.2.bp.b 8
315.bw odd 12 1 3150.2.bp.e 8
315.cg even 12 1 3150.2.bp.b 8
315.cg even 12 1 3150.2.bp.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.k.a 8 9.c even 3 1
126.2.k.a 8 9.d odd 6 1
126.2.k.a 8 63.k odd 6 1
126.2.k.a 8 63.s even 6 1
882.2.d.a 8 63.h even 3 1
882.2.d.a 8 63.i even 6 1
882.2.d.a 8 63.j odd 6 1
882.2.d.a 8 63.t odd 6 1
882.2.k.a 8 63.g even 3 1
882.2.k.a 8 63.l odd 6 1
882.2.k.a 8 63.n odd 6 1
882.2.k.a 8 63.o even 6 1
1008.2.bt.c 8 36.f odd 6 1
1008.2.bt.c 8 36.h even 6 1
1008.2.bt.c 8 252.n even 6 1
1008.2.bt.c 8 252.bn odd 6 1
1134.2.l.f 8 1.a even 1 1 trivial
1134.2.l.f 8 3.b odd 2 1 inner
1134.2.l.f 8 63.i even 6 1 inner
1134.2.l.f 8 63.t odd 6 1 inner
1134.2.t.e 8 7.d odd 6 1
1134.2.t.e 8 9.c even 3 1
1134.2.t.e 8 9.d odd 6 1
1134.2.t.e 8 21.g even 6 1
3150.2.bf.a 8 45.h odd 6 1
3150.2.bf.a 8 45.j even 6 1
3150.2.bf.a 8 315.u even 6 1
3150.2.bf.a 8 315.bn odd 6 1
3150.2.bp.b 8 45.k odd 12 1
3150.2.bp.b 8 45.l even 12 1
3150.2.bp.b 8 315.bw odd 12 1
3150.2.bp.b 8 315.cg even 12 1
3150.2.bp.e 8 45.k odd 12 1
3150.2.bp.e 8 45.l even 12 1
3150.2.bp.e 8 315.bw odd 12 1
3150.2.bp.e 8 315.cg even 12 1
7056.2.k.f 8 252.r odd 6 1
7056.2.k.f 8 252.u odd 6 1
7056.2.k.f 8 252.bb even 6 1
7056.2.k.f 8 252.bj even 6 1

Hecke kernels

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

\( T_{5}^{8} + 18 T_{5}^{6} + 315 T_{5}^{4} + 162 T_{5}^{2} + 81 \)
\( T_{11}^{4} - 9 T_{11}^{2} + 81 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( \)
$5$ \( ( 1 - 2 T^{2} + 25 T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} ) \)
$7$ \( ( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( 1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 295936 T^{10} + 39923038 T^{12} - 772402208 T^{14} + 6975757441 T^{16} \)
$19$ \( ( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2487 T^{4} + 10032 T^{5} + 33212 T^{6} + 82308 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 28 T^{2} + 255 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( 1 + 62 T^{2} + 1849 T^{4} + 19406 T^{6} + 39940 T^{8} + 16320446 T^{10} + 1307762569 T^{12} + 36879045902 T^{14} + 500246412961 T^{16} \)
$31$ \( ( 1 - 10 T^{2} + 1299 T^{4} - 9610 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 8 T - 8 T^{2} - 16 T^{3} + 1447 T^{4} - 592 T^{5} - 10952 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( 1 - 20 T^{2} + 1546 T^{4} + 90160 T^{6} - 2184845 T^{8} + 151558960 T^{10} + 4368626506 T^{12} - 95002084820 T^{14} + 7984925229121 T^{16} \)
$43$ \( ( 1 + 8 T - 20 T^{2} - 16 T^{3} + 2455 T^{4} - 688 T^{5} - 36980 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 152 T^{2} + 9906 T^{4} + 335768 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( 1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 2482296446 T^{10} + 108517785193 T^{12} + 3501969058382 T^{14} + 62259690411361 T^{16} \)
$59$ \( ( 1 + 38 T^{2} + 6171 T^{4} + 132278 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 208 T^{2} + 17970 T^{4} - 773968 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 10 T + 67 T^{2} )^{8} \)
$71$ \( ( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 16131 T^{4} - 117384 T^{5} + 969878 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 14 T + 189 T^{2} + 1106 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( 1 - 278 T^{2} + 44473 T^{4} - 5291174 T^{6} + 496693924 T^{8} - 36450897686 T^{10} + 2110613909833 T^{12} - 90889423796582 T^{14} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 30 T + 545 T^{2} - 7350 T^{3} + 79716 T^{4} - 712950 T^{5} + 5127905 T^{6} - 27380190 T^{7} + 88529281 T^{8} )^{2} \)
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