Properties

Label 1134.2.f.o
Level 1134
Weight 2
Character orbit 1134.f
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.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 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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} - q^{8} + 3 q^{10} + ( -3 + 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -6 q^{17} -7 q^{19} + ( 3 - 3 \zeta_{6} ) q^{20} + 3 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 4 q^{26} + q^{28} -5 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( -6 + 6 \zeta_{6} ) q^{34} -3 q^{35} -7 q^{37} + ( -7 + 7 \zeta_{6} ) q^{38} -3 \zeta_{6} q^{40} + 9 \zeta_{6} q^{41} + ( 10 - 10 \zeta_{6} ) q^{43} + 3 q^{44} + 3 q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + ( 4 - 4 \zeta_{6} ) q^{52} + 12 q^{53} -9 q^{55} + ( 1 - \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} -5 q^{62} + q^{64} + ( -12 + 12 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} + ( -3 + 3 \zeta_{6} ) q^{70} + 9 q^{71} + 2 q^{73} + ( -7 + 7 \zeta_{6} ) q^{74} + 7 \zeta_{6} q^{76} -3 \zeta_{6} q^{77} + ( 10 - 10 \zeta_{6} ) q^{79} -3 q^{80} + 9 q^{82} -18 \zeta_{6} q^{85} -10 \zeta_{6} q^{86} + ( 3 - 3 \zeta_{6} ) q^{88} + 15 q^{89} -4 q^{91} + ( 3 - 3 \zeta_{6} ) q^{92} + 6 \zeta_{6} q^{94} -21 \zeta_{6} q^{95} + ( -8 + 8 \zeta_{6} ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + 3q^{5} - q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + 3q^{5} - q^{7} - 2q^{8} + 6q^{10} - 3q^{11} + 4q^{13} + q^{14} - q^{16} - 12q^{17} - 14q^{19} + 3q^{20} + 3q^{22} + 3q^{23} - 4q^{25} + 8q^{26} + 2q^{28} - 5q^{31} + q^{32} - 6q^{34} - 6q^{35} - 14q^{37} - 7q^{38} - 3q^{40} + 9q^{41} + 10q^{43} + 6q^{44} + 6q^{46} - 6q^{47} - q^{49} + 4q^{50} + 4q^{52} + 24q^{53} - 18q^{55} + q^{56} + 6q^{59} - 8q^{61} - 10q^{62} + 2q^{64} - 12q^{65} + 4q^{67} + 6q^{68} - 3q^{70} + 18q^{71} + 4q^{73} - 7q^{74} + 7q^{76} - 3q^{77} + 10q^{79} - 6q^{80} + 18q^{82} - 18q^{85} - 10q^{86} + 3q^{88} + 30q^{89} - 8q^{91} + 3q^{92} + 6q^{94} - 21q^{95} - 8q^{97} - 2q^{98} + O(q^{100}) \)

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 1.50000 + 2.59808i 0 −0.500000 + 0.866025i −1.00000 0 3.00000
757.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 −0.500000 0.866025i −1.00000 0 3.00000
\(n\): 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.o 2
3.b odd 2 1 1134.2.f.b 2
9.c even 3 1 378.2.a.b 1
9.c even 3 1 inner 1134.2.f.o 2
9.d odd 6 1 378.2.a.g yes 1
9.d odd 6 1 1134.2.f.b 2
36.f odd 6 1 3024.2.a.c 1
36.h even 6 1 3024.2.a.bb 1
45.h odd 6 1 9450.2.a.h 1
45.j even 6 1 9450.2.a.cu 1
63.l odd 6 1 2646.2.a.n 1
63.o even 6 1 2646.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.b 1 9.c even 3 1
378.2.a.g yes 1 9.d odd 6 1
1134.2.f.b 2 3.b odd 2 1
1134.2.f.b 2 9.d odd 6 1
1134.2.f.o 2 1.a even 1 1 trivial
1134.2.f.o 2 9.c even 3 1 inner
2646.2.a.n 1 63.l odd 6 1
2646.2.a.q 1 63.o even 6 1
3024.2.a.c 1 36.f odd 6 1
3024.2.a.bb 1 36.h even 6 1
9450.2.a.h 1 45.h odd 6 1
9450.2.a.cu 1 45.j even 6 1

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 9 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13}^{2} - 4 T_{13} + 16 \)
\( T_{17} + 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( \)
$5$ \( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( 1 - 29 T^{2} + 841 T^{4} \)
$31$ \( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 7 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 12 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 9 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 2 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 83 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 15 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 8 T - 33 T^{2} + 776 T^{3} + 9409 T^{4} \)
show more
show less