Properties

Label 1134.2.e.k
Level 1134
Weight 2
Character orbit 1134.e
Analytic conductor 9.055
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( 2 + \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( 2 + \zeta_{6} ) q^{7} + q^{8} + ( -3 + 3 \zeta_{6} ) q^{10} + 3 \zeta_{6} q^{11} -2 \zeta_{6} q^{13} + ( 2 + \zeta_{6} ) q^{14} + q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} + ( -3 + 3 \zeta_{6} ) q^{20} + 3 \zeta_{6} q^{22} + ( 6 - 6 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} -2 \zeta_{6} q^{26} + ( 2 + \zeta_{6} ) q^{28} + ( -9 + 9 \zeta_{6} ) q^{29} -7 q^{31} + q^{32} + ( -6 + 6 \zeta_{6} ) q^{34} + ( -9 + 6 \zeta_{6} ) q^{35} + 10 \zeta_{6} q^{37} -2 \zeta_{6} q^{38} + ( -3 + 3 \zeta_{6} ) q^{40} + ( 4 - 4 \zeta_{6} ) q^{43} + 3 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} + 12 q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{50} -2 \zeta_{6} q^{52} + ( 3 - 3 \zeta_{6} ) q^{53} -9 q^{55} + ( 2 + \zeta_{6} ) q^{56} + ( -9 + 9 \zeta_{6} ) q^{58} -3 q^{59} -4 q^{61} -7 q^{62} + q^{64} + 6 q^{65} + 2 q^{67} + ( -6 + 6 \zeta_{6} ) q^{68} + ( -9 + 6 \zeta_{6} ) q^{70} + ( -2 + 2 \zeta_{6} ) q^{73} + 10 \zeta_{6} q^{74} -2 \zeta_{6} q^{76} + ( -3 + 9 \zeta_{6} ) q^{77} + 5 q^{79} + ( -3 + 3 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{83} -18 \zeta_{6} q^{85} + ( 4 - 4 \zeta_{6} ) q^{86} + 3 \zeta_{6} q^{88} + 6 \zeta_{6} q^{89} + ( 2 - 6 \zeta_{6} ) q^{91} + ( 6 - 6 \zeta_{6} ) q^{92} + 12 q^{94} + 6 q^{95} + ( 13 - 13 \zeta_{6} ) q^{97} + ( 3 + 5 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 3q^{5} + 5q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 3q^{5} + 5q^{7} + 2q^{8} - 3q^{10} + 3q^{11} - 2q^{13} + 5q^{14} + 2q^{16} - 6q^{17} - 2q^{19} - 3q^{20} + 3q^{22} + 6q^{23} - 4q^{25} - 2q^{26} + 5q^{28} - 9q^{29} - 14q^{31} + 2q^{32} - 6q^{34} - 12q^{35} + 10q^{37} - 2q^{38} - 3q^{40} + 4q^{43} + 3q^{44} + 6q^{46} + 24q^{47} + 11q^{49} - 4q^{50} - 2q^{52} + 3q^{53} - 18q^{55} + 5q^{56} - 9q^{58} - 6q^{59} - 8q^{61} - 14q^{62} + 2q^{64} + 12q^{65} + 4q^{67} - 6q^{68} - 12q^{70} - 2q^{73} + 10q^{74} - 2q^{76} + 3q^{77} + 10q^{79} - 3q^{80} - 9q^{83} - 18q^{85} + 4q^{86} + 3q^{88} + 6q^{89} - 2q^{91} + 6q^{92} + 24q^{94} + 12q^{95} + 13q^{97} + 11q^{98} + O(q^{100}) \)

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 −1.50000 + 2.59808i 0 2.50000 + 0.866025i 1.00000 0 −1.50000 + 2.59808i
919.1 1.00000 0 1.00000 −1.50000 2.59808i 0 2.50000 0.866025i 1.00000 0 −1.50000 2.59808i
\(n\): 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.k 2
3.b odd 2 1 1134.2.e.g 2
7.c even 3 1 1134.2.h.f 2
9.c even 3 1 126.2.g.a 2
9.c even 3 1 1134.2.h.f 2
9.d odd 6 1 126.2.g.d yes 2
9.d odd 6 1 1134.2.h.j 2
21.h odd 6 1 1134.2.h.j 2
36.f odd 6 1 1008.2.s.b 2
36.h even 6 1 1008.2.s.o 2
63.g even 3 1 126.2.g.a 2
63.h even 3 1 882.2.a.j 1
63.h even 3 1 inner 1134.2.e.k 2
63.i even 6 1 882.2.a.e 1
63.j odd 6 1 882.2.a.a 1
63.j odd 6 1 1134.2.e.g 2
63.k odd 6 1 882.2.g.e 2
63.l odd 6 1 882.2.g.e 2
63.n odd 6 1 126.2.g.d yes 2
63.o even 6 1 882.2.g.g 2
63.s even 6 1 882.2.g.g 2
63.t odd 6 1 882.2.a.h 1
252.o even 6 1 1008.2.s.o 2
252.r odd 6 1 7056.2.a.bx 1
252.u odd 6 1 7056.2.a.by 1
252.bb even 6 1 7056.2.a.e 1
252.bj even 6 1 7056.2.a.h 1
252.bl odd 6 1 1008.2.s.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 9.c even 3 1
126.2.g.a 2 63.g even 3 1
126.2.g.d yes 2 9.d odd 6 1
126.2.g.d yes 2 63.n odd 6 1
882.2.a.a 1 63.j odd 6 1
882.2.a.e 1 63.i even 6 1
882.2.a.h 1 63.t odd 6 1
882.2.a.j 1 63.h even 3 1
882.2.g.e 2 63.k odd 6 1
882.2.g.e 2 63.l odd 6 1
882.2.g.g 2 63.o even 6 1
882.2.g.g 2 63.s even 6 1
1008.2.s.b 2 36.f odd 6 1
1008.2.s.b 2 252.bl odd 6 1
1008.2.s.o 2 36.h even 6 1
1008.2.s.o 2 252.o even 6 1
1134.2.e.g 2 3.b odd 2 1
1134.2.e.g 2 63.j odd 6 1
1134.2.e.k 2 1.a even 1 1 trivial
1134.2.e.k 2 63.h even 3 1 inner
1134.2.h.f 2 7.c even 3 1
1134.2.h.f 2 9.c even 3 1
1134.2.h.j 2 9.d odd 6 1
1134.2.h.j 2 21.h odd 6 1
7056.2.a.e 1 252.bb even 6 1
7056.2.a.h 1 252.bj even 6 1
7056.2.a.bx 1 252.r odd 6 1
7056.2.a.by 1 252.u odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{17}^{2} + 6 T_{17} + 36 \)
\( T_{23}^{2} - 6 T_{23} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ 1
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \)
$23$ \( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \)
$41$ \( 1 - 41 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 - 12 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 3 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 4 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 2 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 5 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4} \)
show more
show less