Properties

Label 168.2.q.b
Level $168$
Weight $2$
Character orbit 168.q
Analytic conductor $1.341$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34148675396\)
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: yes
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} + ( -2 + 2 \zeta_{6} ) q^{5} + ( 3 - \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{3} + ( -2 + 2 \zeta_{6} ) q^{5} + ( 3 - \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} + 6 \zeta_{6} q^{11} -3 q^{13} -2 q^{15} -4 \zeta_{6} q^{17} + ( 5 - 5 \zeta_{6} ) q^{19} + ( 1 + 2 \zeta_{6} ) q^{21} + ( 4 - 4 \zeta_{6} ) q^{23} + \zeta_{6} q^{25} - q^{27} -4 q^{29} -7 \zeta_{6} q^{31} + ( -6 + 6 \zeta_{6} ) q^{33} + ( -4 + 6 \zeta_{6} ) q^{35} + ( 9 - 9 \zeta_{6} ) q^{37} -3 \zeta_{6} q^{39} -2 q^{41} - q^{43} -2 \zeta_{6} q^{45} + ( -2 + 2 \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{51} -8 \zeta_{6} q^{53} -12 q^{55} + 5 q^{57} + ( -10 + 10 \zeta_{6} ) q^{61} + ( -2 + 3 \zeta_{6} ) q^{63} + ( 6 - 6 \zeta_{6} ) q^{65} + 15 \zeta_{6} q^{67} + 4 q^{69} -6 q^{71} + 11 \zeta_{6} q^{73} + ( -1 + \zeta_{6} ) q^{75} + ( 6 + 12 \zeta_{6} ) q^{77} + ( -1 + \zeta_{6} ) q^{79} -\zeta_{6} q^{81} + 6 q^{83} + 8 q^{85} -4 \zeta_{6} q^{87} + ( 8 - 8 \zeta_{6} ) q^{89} + ( -9 + 3 \zeta_{6} ) q^{91} + ( 7 - 7 \zeta_{6} ) q^{93} + 10 \zeta_{6} q^{95} -14 q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 2q^{5} + 5q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - 2q^{5} + 5q^{7} - q^{9} + 6q^{11} - 6q^{13} - 4q^{15} - 4q^{17} + 5q^{19} + 4q^{21} + 4q^{23} + q^{25} - 2q^{27} - 8q^{29} - 7q^{31} - 6q^{33} - 2q^{35} + 9q^{37} - 3q^{39} - 4q^{41} - 2q^{43} - 2q^{45} - 2q^{47} + 11q^{49} + 4q^{51} - 8q^{53} - 24q^{55} + 10q^{57} - 10q^{61} - q^{63} + 6q^{65} + 15q^{67} + 8q^{69} - 12q^{71} + 11q^{73} - q^{75} + 24q^{77} - q^{79} - q^{81} + 12q^{83} + 16q^{85} - 4q^{87} + 8q^{89} - 15q^{91} + 7q^{93} + 10q^{95} - 28q^{97} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 0.866025i 0 −1.00000 1.73205i 0 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
121.1 0 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 2.50000 0.866025i 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 168.2.q.b 2
3.b odd 2 1 504.2.s.g 2
4.b odd 2 1 336.2.q.a 2
7.b odd 2 1 1176.2.q.e 2
7.c even 3 1 inner 168.2.q.b 2
7.c even 3 1 1176.2.a.d 1
7.d odd 6 1 1176.2.a.e 1
7.d odd 6 1 1176.2.q.e 2
8.b even 2 1 1344.2.q.i 2
8.d odd 2 1 1344.2.q.t 2
12.b even 2 1 1008.2.s.m 2
21.c even 2 1 3528.2.s.d 2
21.g even 6 1 3528.2.a.y 1
21.g even 6 1 3528.2.s.d 2
21.h odd 6 1 504.2.s.g 2
21.h odd 6 1 3528.2.a.f 1
28.d even 2 1 2352.2.q.v 2
28.f even 6 1 2352.2.a.e 1
28.f even 6 1 2352.2.q.v 2
28.g odd 6 1 336.2.q.a 2
28.g odd 6 1 2352.2.a.x 1
56.j odd 6 1 9408.2.a.bk 1
56.k odd 6 1 1344.2.q.t 2
56.k odd 6 1 9408.2.a.f 1
56.m even 6 1 9408.2.a.cs 1
56.p even 6 1 1344.2.q.i 2
56.p even 6 1 9408.2.a.cd 1
84.j odd 6 1 7056.2.a.bn 1
84.n even 6 1 1008.2.s.m 2
84.n even 6 1 7056.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.b 2 1.a even 1 1 trivial
168.2.q.b 2 7.c even 3 1 inner
336.2.q.a 2 4.b odd 2 1
336.2.q.a 2 28.g odd 6 1
504.2.s.g 2 3.b odd 2 1
504.2.s.g 2 21.h odd 6 1
1008.2.s.m 2 12.b even 2 1
1008.2.s.m 2 84.n even 6 1
1176.2.a.d 1 7.c even 3 1
1176.2.a.e 1 7.d odd 6 1
1176.2.q.e 2 7.b odd 2 1
1176.2.q.e 2 7.d odd 6 1
1344.2.q.i 2 8.b even 2 1
1344.2.q.i 2 56.p even 6 1
1344.2.q.t 2 8.d odd 2 1
1344.2.q.t 2 56.k odd 6 1
2352.2.a.e 1 28.f even 6 1
2352.2.a.x 1 28.g odd 6 1
2352.2.q.v 2 28.d even 2 1
2352.2.q.v 2 28.f even 6 1
3528.2.a.f 1 21.h odd 6 1
3528.2.a.y 1 21.g even 6 1
3528.2.s.d 2 21.c even 2 1
3528.2.s.d 2 21.g even 6 1
7056.2.a.i 1 84.n even 6 1
7056.2.a.bn 1 84.j odd 6 1
9408.2.a.f 1 56.k odd 6 1
9408.2.a.bk 1 56.j odd 6 1
9408.2.a.cd 1 56.p even 6 1
9408.2.a.cs 1 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2 T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \)
$19$ \( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( 1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + T + 43 T^{2} )^{2} \)
$47$ \( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 8 T + 11 T^{2} + 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 15 T + 158 T^{2} - 1005 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 8 T - 25 T^{2} - 712 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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