Properties

Label 1134.2.e.o
Level $1134$
Weight $2$
Character orbit 1134.e
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.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( 2 - 2 \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( 2 - 2 \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} + q^{8} + ( 2 - 2 \zeta_{6} ) q^{10} -5 \zeta_{6} q^{11} -6 \zeta_{6} q^{13} + ( 1 + 2 \zeta_{6} ) q^{14} + q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + ( 2 - 2 \zeta_{6} ) q^{20} -5 \zeta_{6} q^{22} + ( 4 - 4 \zeta_{6} ) q^{23} + \zeta_{6} q^{25} -6 \zeta_{6} q^{26} + ( 1 + 2 \zeta_{6} ) q^{28} + ( -7 + 7 \zeta_{6} ) q^{29} + 3 q^{31} + q^{32} + ( 4 - 4 \zeta_{6} ) q^{34} + ( 6 - 2 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} + 4 \zeta_{6} q^{38} + ( 2 - 2 \zeta_{6} ) q^{40} + 6 \zeta_{6} q^{41} + ( -8 + 8 \zeta_{6} ) q^{43} -5 \zeta_{6} q^{44} + ( 4 - 4 \zeta_{6} ) q^{46} + 6 q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + \zeta_{6} q^{50} -6 \zeta_{6} q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} -10 q^{55} + ( 1 + 2 \zeta_{6} ) q^{56} + ( -7 + 7 \zeta_{6} ) q^{58} + 7 q^{59} + 3 q^{62} + q^{64} -12 q^{65} + 10 q^{67} + ( 4 - 4 \zeta_{6} ) q^{68} + ( 6 - 2 \zeta_{6} ) q^{70} -4 q^{71} + ( -13 + 13 \zeta_{6} ) q^{73} -8 \zeta_{6} q^{74} + 4 \zeta_{6} q^{76} + ( 10 - 15 \zeta_{6} ) q^{77} -3 q^{79} + ( 2 - 2 \zeta_{6} ) q^{80} + 6 \zeta_{6} q^{82} + ( 7 - 7 \zeta_{6} ) q^{83} -8 \zeta_{6} q^{85} + ( -8 + 8 \zeta_{6} ) q^{86} -5 \zeta_{6} q^{88} -6 \zeta_{6} q^{89} + ( 12 - 18 \zeta_{6} ) q^{91} + ( 4 - 4 \zeta_{6} ) q^{92} + 6 q^{94} + 8 q^{95} + ( 5 - 5 \zeta_{6} ) q^{97} + ( -3 + 8 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} + 2q^{8} + 2q^{10} - 5q^{11} - 6q^{13} + 4q^{14} + 2q^{16} + 4q^{17} + 4q^{19} + 2q^{20} - 5q^{22} + 4q^{23} + q^{25} - 6q^{26} + 4q^{28} - 7q^{29} + 6q^{31} + 2q^{32} + 4q^{34} + 10q^{35} - 8q^{37} + 4q^{38} + 2q^{40} + 6q^{41} - 8q^{43} - 5q^{44} + 4q^{46} + 12q^{47} + 2q^{49} + q^{50} - 6q^{52} + 6q^{53} - 20q^{55} + 4q^{56} - 7q^{58} + 14q^{59} + 6q^{62} + 2q^{64} - 24q^{65} + 20q^{67} + 4q^{68} + 10q^{70} - 8q^{71} - 13q^{73} - 8q^{74} + 4q^{76} + 5q^{77} - 6q^{79} + 2q^{80} + 6q^{82} + 7q^{83} - 8q^{85} - 8q^{86} - 5q^{88} - 6q^{89} + 6q^{91} + 4q^{92} + 12q^{94} + 16q^{95} + 5q^{97} + 2q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 0 1.00000 1.00000 1.73205i 0 2.00000 + 1.73205i 1.00000 0 1.00000 1.73205i
919.1 1.00000 0 1.00000 1.00000 + 1.73205i 0 2.00000 1.73205i 1.00000 0 1.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h 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.e.o 2
3.b odd 2 1 1134.2.e.b 2
7.c even 3 1 1134.2.h.b 2
9.c even 3 1 378.2.g.c 2
9.c even 3 1 1134.2.h.b 2
9.d odd 6 1 378.2.g.d yes 2
9.d odd 6 1 1134.2.h.o 2
21.h odd 6 1 1134.2.h.o 2
63.g even 3 1 378.2.g.c 2
63.h even 3 1 inner 1134.2.e.o 2
63.h even 3 1 2646.2.a.t 1
63.i even 6 1 2646.2.a.c 1
63.j odd 6 1 1134.2.e.b 2
63.j odd 6 1 2646.2.a.k 1
63.n odd 6 1 378.2.g.d yes 2
63.t odd 6 1 2646.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.c 2 9.c even 3 1
378.2.g.c 2 63.g even 3 1
378.2.g.d yes 2 9.d odd 6 1
378.2.g.d yes 2 63.n odd 6 1
1134.2.e.b 2 3.b odd 2 1
1134.2.e.b 2 63.j odd 6 1
1134.2.e.o 2 1.a even 1 1 trivial
1134.2.e.o 2 63.h even 3 1 inner
1134.2.h.b 2 7.c even 3 1
1134.2.h.b 2 9.c even 3 1
1134.2.h.o 2 9.d odd 6 1
1134.2.h.o 2 21.h odd 6 1
2646.2.a.c 1 63.i even 6 1
2646.2.a.k 1 63.j odd 6 1
2646.2.a.t 1 63.h even 3 1
2646.2.a.bb 1 63.t odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} + 5 T_{11} + 25 \)
\( T_{17}^{2} - 4 T_{17} + 16 \)
\( T_{23}^{2} - 4 T_{23} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( 25 + 5 T + T^{2} \)
$13$ \( 36 + 6 T + T^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( 16 - 4 T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( 49 + 7 T + T^{2} \)
$31$ \( ( -3 + T )^{2} \)
$37$ \( 64 + 8 T + T^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 64 + 8 T + T^{2} \)
$47$ \( ( -6 + T )^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( ( -7 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( ( -10 + T )^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( 169 + 13 T + T^{2} \)
$79$ \( ( 3 + T )^{2} \)
$83$ \( 49 - 7 T + T^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( 25 - 5 T + T^{2} \)
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