Properties

Label 1176.2.q.j
Level $1176$
Weight $2$
Character orbit 1176.q
Analytic conductor $9.390$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.39040727770\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} -\zeta_{6} q^{9} + 2 q^{13} + 2 q^{15} + ( 6 - 6 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} - q^{27} + 6 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + 10 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{39} + 10 q^{41} + 12 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} -8 \zeta_{6} q^{47} -6 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{53} -4 q^{57} + ( 4 - 4 \zeta_{6} ) q^{59} -10 \zeta_{6} q^{61} + 4 \zeta_{6} q^{65} + ( -12 + 12 \zeta_{6} ) q^{67} + 4 q^{69} + 4 q^{71} + ( 2 - 2 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} -8 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -4 q^{83} + 12 q^{85} + ( 6 - 6 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} + 8 \zeta_{6} q^{93} + ( 8 - 8 \zeta_{6} ) q^{95} -10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{5} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{5} - q^{9} + 4q^{13} + 4q^{15} + 6q^{17} - 4q^{19} + 4q^{23} + q^{25} - 2q^{27} + 12q^{29} - 8q^{31} + 10q^{37} + 2q^{39} + 20q^{41} + 24q^{43} + 2q^{45} - 8q^{47} - 6q^{51} - 6q^{53} - 8q^{57} + 4q^{59} - 10q^{61} + 4q^{65} - 12q^{67} + 8q^{69} + 8q^{71} + 2q^{73} - q^{75} - 8q^{79} - q^{81} - 8q^{83} + 24q^{85} + 6q^{87} + 6q^{89} + 8q^{93} + 8q^{95} - 20q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 1.00000 + 1.73205i 0 0 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 1.00000 1.73205i 0 0 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.2.q.j 2
3.b odd 2 1 3528.2.s.h 2
4.b odd 2 1 2352.2.q.j 2
7.b odd 2 1 1176.2.q.b 2
7.c even 3 1 1176.2.a.a 1
7.c even 3 1 inner 1176.2.q.j 2
7.d odd 6 1 168.2.a.b 1
7.d odd 6 1 1176.2.q.b 2
21.c even 2 1 3528.2.s.v 2
21.g even 6 1 504.2.a.b 1
21.g even 6 1 3528.2.s.v 2
21.h odd 6 1 3528.2.a.w 1
21.h odd 6 1 3528.2.s.h 2
28.d even 2 1 2352.2.q.o 2
28.f even 6 1 336.2.a.c 1
28.f even 6 1 2352.2.q.o 2
28.g odd 6 1 2352.2.a.q 1
28.g odd 6 1 2352.2.q.j 2
35.i odd 6 1 4200.2.a.i 1
35.k even 12 2 4200.2.t.m 2
56.j odd 6 1 1344.2.a.c 1
56.k odd 6 1 9408.2.a.bc 1
56.m even 6 1 1344.2.a.n 1
56.p even 6 1 9408.2.a.cy 1
84.j odd 6 1 1008.2.a.e 1
84.n even 6 1 7056.2.a.br 1
112.v even 12 2 5376.2.c.f 2
112.x odd 12 2 5376.2.c.bd 2
140.s even 6 1 8400.2.a.bx 1
168.ba even 6 1 4032.2.a.be 1
168.be odd 6 1 4032.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.b 1 7.d odd 6 1
336.2.a.c 1 28.f even 6 1
504.2.a.b 1 21.g even 6 1
1008.2.a.e 1 84.j odd 6 1
1176.2.a.a 1 7.c even 3 1
1176.2.q.b 2 7.b odd 2 1
1176.2.q.b 2 7.d odd 6 1
1176.2.q.j 2 1.a even 1 1 trivial
1176.2.q.j 2 7.c even 3 1 inner
1344.2.a.c 1 56.j odd 6 1
1344.2.a.n 1 56.m even 6 1
2352.2.a.q 1 28.g odd 6 1
2352.2.q.j 2 4.b odd 2 1
2352.2.q.j 2 28.g odd 6 1
2352.2.q.o 2 28.d even 2 1
2352.2.q.o 2 28.f even 6 1
3528.2.a.w 1 21.h odd 6 1
3528.2.s.h 2 3.b odd 2 1
3528.2.s.h 2 21.h odd 6 1
3528.2.s.v 2 21.c even 2 1
3528.2.s.v 2 21.g even 6 1
4032.2.a.be 1 168.ba even 6 1
4032.2.a.bj 1 168.be odd 6 1
4200.2.a.i 1 35.i odd 6 1
4200.2.t.m 2 35.k even 12 2
5376.2.c.f 2 112.v even 12 2
5376.2.c.bd 2 112.x odd 12 2
7056.2.a.br 1 84.n even 6 1
8400.2.a.bx 1 140.s even 6 1
9408.2.a.bc 1 56.k odd 6 1
9408.2.a.cy 1 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1176, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11} \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \)
$41$ \( ( 1 - 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 4 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2} \)
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