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 $2$

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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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} + 16 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 9)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
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