Properties

Label 1134.2.h.n
Level $1134$
Weight $2$
Character orbit 1134.h
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.h (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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 2 + \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 2 + \zeta_{6} ) q^{7} - q^{8} + 6 q^{11} + ( -5 + 5 \zeta_{6} ) q^{13} + ( 3 - 2 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + ( 6 - 6 \zeta_{6} ) q^{22} + 6 q^{23} -5 q^{25} + 5 \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} -6 \zeta_{6} q^{29} + \zeta_{6} q^{31} + \zeta_{6} q^{32} + 6 \zeta_{6} q^{34} + \zeta_{6} q^{37} + 4 q^{38} + ( 6 - 6 \zeta_{6} ) q^{41} + \zeta_{6} q^{43} -6 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} + ( 6 - 6 \zeta_{6} ) q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( -5 + 5 \zeta_{6} ) q^{50} + 5 q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} + ( -2 - \zeta_{6} ) q^{56} -6 q^{58} + 6 \zeta_{6} q^{59} + ( 1 - \zeta_{6} ) q^{61} + q^{62} + q^{64} + \zeta_{6} q^{67} + 6 q^{68} + 12 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} + q^{74} + ( 4 - 4 \zeta_{6} ) q^{76} + ( 12 + 6 \zeta_{6} ) q^{77} + ( 1 - \zeta_{6} ) q^{79} -6 \zeta_{6} q^{82} -6 \zeta_{6} q^{83} + q^{86} -6 q^{88} + ( -15 + 10 \zeta_{6} ) q^{91} -6 \zeta_{6} q^{92} -6 \zeta_{6} q^{94} -17 \zeta_{6} q^{97} + ( 8 - 3 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + 5q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + 5q^{7} - 2q^{8} + 12q^{11} - 5q^{13} + 4q^{14} - q^{16} - 6q^{17} + 4q^{19} + 6q^{22} + 12q^{23} - 10q^{25} + 5q^{26} - q^{28} - 6q^{29} + q^{31} + q^{32} + 6q^{34} + q^{37} + 8q^{38} + 6q^{41} + q^{43} - 6q^{44} + 6q^{46} + 6q^{47} + 11q^{49} - 5q^{50} + 10q^{52} + 6q^{53} - 5q^{56} - 12q^{58} + 6q^{59} + q^{61} + 2q^{62} + 2q^{64} + q^{67} + 12q^{68} + 24q^{71} - 2q^{73} + 2q^{74} + 4q^{76} + 30q^{77} + q^{79} - 6q^{82} - 6q^{83} + 2q^{86} - 12q^{88} - 20q^{91} - 6q^{92} - 6q^{94} - 17q^{97} + 13q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 2.50000 + 0.866025i −1.00000 0 0
541.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 2.50000 0.866025i −1.00000 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
63.g 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.h.n 2
3.b odd 2 1 1134.2.h.d 2
7.c even 3 1 1134.2.e.c 2
9.c even 3 1 378.2.g.e yes 2
9.c even 3 1 1134.2.e.c 2
9.d odd 6 1 378.2.g.b 2
9.d odd 6 1 1134.2.e.m 2
21.h odd 6 1 1134.2.e.m 2
63.g even 3 1 inner 1134.2.h.n 2
63.g even 3 1 2646.2.a.h 1
63.h even 3 1 378.2.g.e yes 2
63.j odd 6 1 378.2.g.b 2
63.k odd 6 1 2646.2.a.g 1
63.n odd 6 1 1134.2.h.d 2
63.n odd 6 1 2646.2.a.x 1
63.s even 6 1 2646.2.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.b 2 9.d odd 6 1
378.2.g.b 2 63.j odd 6 1
378.2.g.e yes 2 9.c even 3 1
378.2.g.e yes 2 63.h even 3 1
1134.2.e.c 2 7.c even 3 1
1134.2.e.c 2 9.c even 3 1
1134.2.e.m 2 9.d odd 6 1
1134.2.e.m 2 21.h odd 6 1
1134.2.h.d 2 3.b odd 2 1
1134.2.h.d 2 63.n odd 6 1
1134.2.h.n 2 1.a even 1 1 trivial
1134.2.h.n 2 63.g even 3 1 inner
2646.2.a.g 1 63.k odd 6 1
2646.2.a.h 1 63.g even 3 1
2646.2.a.w 1 63.s even 6 1
2646.2.a.x 1 63.n odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - 5 T + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 16 - 4 T + T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 1 - T + T^{2} \)
$37$ \( 1 - T + T^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( 36 - 6 T + T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 1 - T + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 4 + 2 T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( 36 + 6 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 289 + 17 T + T^{2} \)
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