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Properties

Label 168.2.q.c
Level $168$
Weight $2$
Character orbit 168.q
Analytic conductor $1.341$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( -2 + \beta_{1} - \beta_{3} ) q^{7} -\beta_{2} q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( -2 + \beta_{1} - \beta_{3} ) q^{7} -\beta_{2} q^{9} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{11} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{15} + ( 4 - 4 \beta_{2} ) q^{17} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{19} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{21} -4 \beta_{2} q^{23} + ( -10 + \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{25} - q^{27} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{2} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( -8 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{35} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{37} + ( 2 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{39} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{45} + 6 \beta_{2} q^{47} + ( -1 - 3 \beta_{1} + 3 \beta_{3} ) q^{49} -4 \beta_{2} q^{51} + ( -5 - \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 14 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{55} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{57} + ( -3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{59} -10 \beta_{2} q^{61} + ( 2 \beta_{2} + \beta_{3} ) q^{63} + ( -2 + 4 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -4 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{67} -4 q^{69} + 2 q^{71} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{73} + ( -1 + 2 \beta_{1} + 10 \beta_{2} - \beta_{3} ) q^{75} + ( 3 + 2 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{77} + ( -2 + 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{79} + ( -1 + \beta_{2} ) q^{81} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{83} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{85} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{87} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -1 + 4 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{91} -\beta_{2} q^{93} + ( 12 + 2 \beta_{1} - 14 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 12 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{97} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + q^{5} - 6q^{7} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + q^{5} - 6q^{7} - 2q^{9} + q^{11} + 10q^{13} + 2q^{15} + 8q^{17} + 5q^{19} - 3q^{21} - 8q^{23} - 19q^{25} - 4q^{27} + 6q^{29} + 2q^{31} - q^{33} - 30q^{35} - 3q^{37} + 5q^{39} + 12q^{41} - 14q^{43} + q^{45} + 12q^{47} - 10q^{49} - 8q^{51} - 11q^{53} + 58q^{55} + 10q^{57} - 5q^{59} - 20q^{61} + 3q^{63} - 26q^{65} - 7q^{67} - 16q^{69} + 8q^{71} + q^{73} + 19q^{75} + 27q^{77} - 8q^{79} - 2q^{81} + 14q^{83} + 8q^{85} + 3q^{87} + 6q^{89} - 15q^{91} - 2q^{93} + 26q^{95} + 50q^{97} - 2q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 5 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu + 5 \)\()/4\)

Display \(\nu^j\) in terms of \(\beta_i\)

\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} + 4 \beta_{1} + 5\)

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\) \(1\)

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
−1.63746 − 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 − 0.656712i
0 0.500000 − 0.866025i 0 −1.63746 − 2.83616i 0 −1.50000 − 2.17945i 0 −0.500000 − 0.866025i 0
25.2 0 0.500000 − 0.866025i 0 2.13746 + 3.70219i 0 −1.50000 + 2.17945i 0 −0.500000 − 0.866025i 0
121.1 0 0.500000 + 0.866025i 0 −1.63746 + 2.83616i 0 −1.50000 + 2.17945i 0 −0.500000 + 0.866025i 0
121.2 0 0.500000 + 0.866025i 0 2.13746 − 3.70219i 0 −1.50000 − 2.17945i 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.c ✓ 4
3.b odd 2 1 504.2.s.i 4
4.b odd 2 1 336.2.q.g 4
7.b odd 2 1 1176.2.q.l 4
7.c even 3 1 inner 168.2.q.c ✓ 4
7.c even 3 1 1176.2.a.k 2
7.d odd 6 1 1176.2.a.n 2
7.d odd 6 1 1176.2.q.l 4
8.b even 2 1 1344.2.q.w 4
8.d odd 2 1 1344.2.q.x 4
12.b even 2 1 1008.2.s.r 4
21.c even 2 1 3528.2.s.bk 4
21.g even 6 1 3528.2.a.bd 2
21.g even 6 1 3528.2.s.bk 4
21.h odd 6 1 504.2.s.i 4
21.h odd 6 1 3528.2.a.bk 2
28.d even 2 1 2352.2.q.bf 4
28.f even 6 1 2352.2.a.ba 2
28.f even 6 1 2352.2.q.bf 4
28.g odd 6 1 336.2.q.g 4
28.g odd 6 1 2352.2.a.bf 2
56.j odd 6 1 9408.2.a.dj 2
56.k odd 6 1 1344.2.q.x 4
56.k odd 6 1 9408.2.a.dp 2
56.m even 6 1 9408.2.a.dw 2
56.p even 6 1 1344.2.q.w 4
56.p even 6 1 9408.2.a.ec 2
84.j odd 6 1 7056.2.a.ch 2
84.n even 6 1 1008.2.s.r 4
84.n even 6 1 7056.2.a.cu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c ✓ 4 1.a even 1 1 trivial
168.2.q.c ✓ 4 7.c even 3 1 inner
336.2.q.g 4 4.b odd 2 1
336.2.q.g 4 28.g odd 6 1
504.2.s.i 4 3.b odd 2 1
504.2.s.i 4 21.h odd 6 1
1008.2.s.r 4 12.b even 2 1
1008.2.s.r 4 84.n even 6 1
1176.2.a.k 2 7.c even 3 1
1176.2.a.n 2 7.d odd 6 1
1176.2.q.l 4 7.b odd 2 1
1176.2.q.l 4 7.d odd 6 1
1344.2.q.w 4 8.b even 2 1
1344.2.q.w 4 56.p even 6 1
1344.2.q.x 4 8.d odd 2 1
1344.2.q.x 4 56.k odd 6 1
2352.2.a.ba 2 28.f even 6 1
2352.2.a.bf 2 28.g odd 6 1
2352.2.q.bf 4 28.d even 2 1
2352.2.q.bf 4 28.f even 6 1
3528.2.a.bd 2 21.g even 6 1
3528.2.a.bk 2 21.h odd 6 1
3528.2.s.bk 4 21.c even 2 1
3528.2.s.bk 4 21.g even 6 1
7056.2.a.ch 2 84.j odd 6 1
7056.2.a.cu 2 84.n even 6 1
9408.2.a.dj 2 56.j odd 6 1
9408.2.a.dp 2 56.k odd 6 1
9408.2.a.dw 2 56.m even 6 1
9408.2.a.ec 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - T_{5}^{3} + 15 T_{5}^{2} + 14 T_{5} + 196 \) acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\).