Properties

Label 1134.2.h.a
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 42)
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{10} -3 q^{11} + ( 4 - 4 \zeta_{6} ) q^{13} + ( 3 - \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + 4 \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} + ( 3 - 3 \zeta_{6} ) q^{22} + 4 q^{25} + 4 \zeta_{6} q^{26} + ( -2 + 3 \zeta_{6} ) q^{28} + 9 \zeta_{6} q^{29} + \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( 3 + 6 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} -4 q^{38} -3 q^{40} + 10 \zeta_{6} q^{43} + 3 \zeta_{6} q^{44} + ( -6 + 6 \zeta_{6} ) q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -4 + 4 \zeta_{6} ) q^{50} -4 q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( -1 - 2 \zeta_{6} ) q^{56} -9 q^{58} + 3 \zeta_{6} q^{59} + ( 10 - 10 \zeta_{6} ) q^{61} - q^{62} + q^{64} + ( -12 + 12 \zeta_{6} ) q^{65} + 10 \zeta_{6} q^{67} + ( -9 + 3 \zeta_{6} ) q^{70} + 6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} + 8 q^{74} + ( 4 - 4 \zeta_{6} ) q^{76} + ( 3 + 6 \zeta_{6} ) q^{77} + ( 1 - \zeta_{6} ) q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} -9 \zeta_{6} q^{83} -10 q^{86} -3 q^{88} + 6 \zeta_{6} q^{89} + ( -12 + 4 \zeta_{6} ) q^{91} -6 \zeta_{6} q^{94} -12 \zeta_{6} q^{95} + \zeta_{6} q^{97} + ( -5 - 3 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 6q^{5} - 4q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 6q^{5} - 4q^{7} + 2q^{8} + 3q^{10} - 6q^{11} + 4q^{13} + 5q^{14} - q^{16} + 4q^{19} + 3q^{20} + 3q^{22} + 8q^{25} + 4q^{26} - q^{28} + 9q^{29} + q^{31} - q^{32} + 12q^{35} - 8q^{37} - 8q^{38} - 6q^{40} + 10q^{43} + 3q^{44} - 6q^{47} + 2q^{49} - 4q^{50} - 8q^{52} - 3q^{53} + 18q^{55} - 4q^{56} - 18q^{58} + 3q^{59} + 10q^{61} - 2q^{62} + 2q^{64} - 12q^{65} + 10q^{67} - 15q^{70} + 12q^{71} - 2q^{73} + 16q^{74} + 4q^{76} + 12q^{77} + q^{79} + 3q^{80} - 9q^{83} - 20q^{86} - 6q^{88} + 6q^{89} - 20q^{91} - 6q^{94} - 12q^{95} + q^{97} - 13q^{98} + O(q^{100}) \)

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 −3.00000 0 −2.00000 1.73205i 1.00000 0 1.50000 2.59808i
541.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −3.00000 0 −2.00000 + 1.73205i 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.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.a 2
3.b odd 2 1 1134.2.h.p 2
7.c even 3 1 1134.2.e.p 2
9.c even 3 1 126.2.g.b 2
9.c even 3 1 1134.2.e.p 2
9.d odd 6 1 42.2.e.b 2
9.d odd 6 1 1134.2.e.a 2
21.h odd 6 1 1134.2.e.a 2
36.f odd 6 1 1008.2.s.n 2
36.h even 6 1 336.2.q.d 2
45.h odd 6 1 1050.2.i.e 2
45.l even 12 2 1050.2.o.b 4
63.g even 3 1 882.2.a.g 1
63.g even 3 1 inner 1134.2.h.a 2
63.h even 3 1 126.2.g.b 2
63.i even 6 1 294.2.e.f 2
63.j odd 6 1 42.2.e.b 2
63.k odd 6 1 882.2.a.k 1
63.l odd 6 1 882.2.g.b 2
63.n odd 6 1 294.2.a.d 1
63.n odd 6 1 1134.2.h.p 2
63.o even 6 1 294.2.e.f 2
63.s even 6 1 294.2.a.a 1
63.t odd 6 1 882.2.g.b 2
72.j odd 6 1 1344.2.q.v 2
72.l even 6 1 1344.2.q.j 2
252.n even 6 1 7056.2.a.bz 1
252.o even 6 1 2352.2.a.m 1
252.r odd 6 1 2352.2.q.m 2
252.s odd 6 1 2352.2.q.m 2
252.u odd 6 1 1008.2.s.n 2
252.bb even 6 1 336.2.q.d 2
252.bl odd 6 1 7056.2.a.g 1
252.bn odd 6 1 2352.2.a.n 1
315.u even 6 1 7350.2.a.cw 1
315.v odd 6 1 7350.2.a.ce 1
315.br odd 6 1 1050.2.i.e 2
315.bv even 12 2 1050.2.o.b 4
504.u odd 6 1 9408.2.a.bm 1
504.y even 6 1 9408.2.a.db 1
504.bi odd 6 1 1344.2.q.v 2
504.bt even 6 1 1344.2.q.j 2
504.cy even 6 1 9408.2.a.bu 1
504.db odd 6 1 9408.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 9.d odd 6 1
42.2.e.b 2 63.j odd 6 1
126.2.g.b 2 9.c even 3 1
126.2.g.b 2 63.h even 3 1
294.2.a.a 1 63.s even 6 1
294.2.a.d 1 63.n odd 6 1
294.2.e.f 2 63.i even 6 1
294.2.e.f 2 63.o even 6 1
336.2.q.d 2 36.h even 6 1
336.2.q.d 2 252.bb even 6 1
882.2.a.g 1 63.g even 3 1
882.2.a.k 1 63.k odd 6 1
882.2.g.b 2 63.l odd 6 1
882.2.g.b 2 63.t odd 6 1
1008.2.s.n 2 36.f odd 6 1
1008.2.s.n 2 252.u odd 6 1
1050.2.i.e 2 45.h odd 6 1
1050.2.i.e 2 315.br odd 6 1
1050.2.o.b 4 45.l even 12 2
1050.2.o.b 4 315.bv even 12 2
1134.2.e.a 2 9.d odd 6 1
1134.2.e.a 2 21.h odd 6 1
1134.2.e.p 2 7.c even 3 1
1134.2.e.p 2 9.c even 3 1
1134.2.h.a 2 1.a even 1 1 trivial
1134.2.h.a 2 63.g even 3 1 inner
1134.2.h.p 2 3.b odd 2 1
1134.2.h.p 2 63.n odd 6 1
1344.2.q.j 2 72.l even 6 1
1344.2.q.j 2 504.bt even 6 1
1344.2.q.v 2 72.j odd 6 1
1344.2.q.v 2 504.bi odd 6 1
2352.2.a.m 1 252.o even 6 1
2352.2.a.n 1 252.bn odd 6 1
2352.2.q.m 2 252.r odd 6 1
2352.2.q.m 2 252.s odd 6 1
7056.2.a.g 1 252.bl odd 6 1
7056.2.a.bz 1 252.n even 6 1
7350.2.a.ce 1 315.v odd 6 1
7350.2.a.cw 1 315.u even 6 1
9408.2.a.d 1 504.db odd 6 1
9408.2.a.bm 1 504.u odd 6 1
9408.2.a.bu 1 504.cy even 6 1
9408.2.a.db 1 504.y even 6 1

Hecke kernels

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

\( T_{5} + 3 \)
\( T_{11} + 3 \)
\( T_{17} \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ 1
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( ( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \)
$31$ \( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \)
$37$ \( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 41 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \)
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