Properties

Label 1134.2.t.a
Level $1134$
Weight $2$
Character orbit 1134.t
Analytic conductor $9.055$
Analytic rank $0$
Dimension $4$
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.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 - \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 - \zeta_{12}^{2} ) q^{7} -\zeta_{12}^{3} q^{8} -3 \zeta_{12}^{3} q^{11} + ( 4 - 2 \zeta_{12}^{2} ) q^{13} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{14} -\zeta_{12}^{2} q^{16} -3 \zeta_{12}^{2} q^{22} -5 q^{25} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{26} + ( -3 + 2 \zeta_{12}^{2} ) q^{28} -9 \zeta_{12} q^{29} + ( 1 + \zeta_{12}^{2} ) q^{31} -\zeta_{12} q^{32} + ( 8 - 8 \zeta_{12}^{2} ) q^{37} + ( -6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{41} + ( -4 + 4 \zeta_{12}^{2} ) q^{43} -3 \zeta_{12} q^{44} + ( -6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} + ( 3 + 5 \zeta_{12}^{2} ) q^{49} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{50} + ( 2 - 4 \zeta_{12}^{2} ) q^{52} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{56} -9 q^{58} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{59} + ( 16 - 8 \zeta_{12}^{2} ) q^{61} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{62} - q^{64} + ( -2 + 2 \zeta_{12}^{2} ) q^{67} + 12 \zeta_{12}^{3} q^{71} + ( 6 - 3 \zeta_{12}^{2} ) q^{73} -8 \zeta_{12}^{3} q^{74} + ( -3 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{77} + 13 \zeta_{12}^{2} q^{79} + ( -6 - 6 \zeta_{12}^{2} ) q^{82} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{83} + 4 \zeta_{12}^{3} q^{86} -3 q^{88} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{89} + ( -10 + 2 \zeta_{12}^{2} ) q^{91} + ( -6 - 6 \zeta_{12}^{2} ) q^{94} + ( 5 + 5 \zeta_{12}^{2} ) q^{97} + ( 8 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 10 q^{7} + O(q^{10}) \) \( 4 q + 2 q^{4} - 10 q^{7} + 12 q^{13} - 2 q^{16} - 6 q^{22} - 20 q^{25} - 8 q^{28} + 6 q^{31} + 16 q^{37} - 8 q^{43} + 22 q^{49} - 36 q^{58} + 48 q^{61} - 4 q^{64} - 4 q^{67} + 18 q^{73} + 26 q^{79} - 36 q^{82} - 12 q^{88} - 36 q^{91} - 36 q^{94} + 30 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −2.50000 0.866025i 1.00000i 0 0
593.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −2.50000 0.866025i 1.00000i 0 0
1025.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −2.50000 + 0.866025i 1.00000i 0 0
1025.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −2.50000 + 0.866025i 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.t.a 4
3.b odd 2 1 inner 1134.2.t.a 4
7.d odd 6 1 1134.2.l.d 4
9.c even 3 1 378.2.k.a 4
9.c even 3 1 1134.2.l.d 4
9.d odd 6 1 378.2.k.a 4
9.d odd 6 1 1134.2.l.d 4
21.g even 6 1 1134.2.l.d 4
63.g even 3 1 2646.2.d.c 4
63.i even 6 1 378.2.k.a 4
63.k odd 6 1 inner 1134.2.t.a 4
63.k odd 6 1 2646.2.d.c 4
63.n odd 6 1 2646.2.d.c 4
63.s even 6 1 inner 1134.2.t.a 4
63.s even 6 1 2646.2.d.c 4
63.t odd 6 1 378.2.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.a 4 9.c even 3 1
378.2.k.a 4 9.d odd 6 1
378.2.k.a 4 63.i even 6 1
378.2.k.a 4 63.t odd 6 1
1134.2.l.d 4 7.d odd 6 1
1134.2.l.d 4 9.c even 3 1
1134.2.l.d 4 9.d odd 6 1
1134.2.l.d 4 21.g even 6 1
1134.2.t.a 4 1.a even 1 1 trivial
1134.2.t.a 4 3.b odd 2 1 inner
1134.2.t.a 4 63.k odd 6 1 inner
1134.2.t.a 4 63.s even 6 1 inner
2646.2.d.c 4 63.g even 3 1
2646.2.d.c 4 63.k odd 6 1
2646.2.d.c 4 63.n odd 6 1
2646.2.d.c 4 63.s even 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{11}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 7 + 5 T + T^{2} )^{2} \)
$11$ \( ( 9 + T^{2} )^{2} \)
$13$ \( ( 12 - 6 T + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( 6561 - 81 T^{2} + T^{4} \)
$31$ \( ( 3 - 3 T + T^{2} )^{2} \)
$37$ \( ( 64 - 8 T + T^{2} )^{2} \)
$41$ \( 11664 + 108 T^{2} + T^{4} \)
$43$ \( ( 16 + 4 T + T^{2} )^{2} \)
$47$ \( 11664 + 108 T^{2} + T^{4} \)
$53$ \( 1296 - 36 T^{2} + T^{4} \)
$59$ \( 729 + 27 T^{2} + T^{4} \)
$61$ \( ( 192 - 24 T + T^{2} )^{2} \)
$67$ \( ( 4 + 2 T + T^{2} )^{2} \)
$71$ \( ( 144 + T^{2} )^{2} \)
$73$ \( ( 27 - 9 T + T^{2} )^{2} \)
$79$ \( ( 169 - 13 T + T^{2} )^{2} \)
$83$ \( 729 + 27 T^{2} + T^{4} \)
$89$ \( 11664 + 108 T^{2} + T^{4} \)
$97$ \( ( 75 - 15 T + T^{2} )^{2} \)
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