Properties

Label 84.2.i.a
Level $84$
Weight $2$
Character orbit 84.i
Analytic conductor $0.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 84.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.670743376979\)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} + ( 2 - 2 \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{3} + ( 2 - 2 \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} -2 \zeta_{6} q^{11} -3 q^{13} + 2 q^{15} -8 \zeta_{6} q^{17} + ( 1 - \zeta_{6} ) q^{19} + ( -3 + 2 \zeta_{6} ) q^{21} + ( -8 + 8 \zeta_{6} ) q^{23} + \zeta_{6} q^{25} - q^{27} + 4 q^{29} -3 \zeta_{6} q^{31} + ( 2 - 2 \zeta_{6} ) q^{33} + ( 4 + 2 \zeta_{6} ) q^{35} + ( 1 - \zeta_{6} ) q^{37} -3 \zeta_{6} q^{39} + 6 q^{41} + 11 q^{43} + 2 \zeta_{6} q^{45} + ( -6 + 6 \zeta_{6} ) q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 8 - 8 \zeta_{6} ) q^{51} + 12 \zeta_{6} q^{53} -4 q^{55} + q^{57} -4 \zeta_{6} q^{59} + ( 6 - 6 \zeta_{6} ) q^{61} + ( -2 - \zeta_{6} ) q^{63} + ( -6 + 6 \zeta_{6} ) q^{65} -13 \zeta_{6} q^{67} -8 q^{69} -10 q^{71} + 11 \zeta_{6} q^{73} + ( -1 + \zeta_{6} ) q^{75} + ( 6 - 4 \zeta_{6} ) q^{77} + ( 3 - 3 \zeta_{6} ) q^{79} -\zeta_{6} q^{81} + 2 q^{83} -16 q^{85} + 4 \zeta_{6} q^{87} + ( 3 - 9 \zeta_{6} ) q^{91} + ( 3 - 3 \zeta_{6} ) q^{93} -2 \zeta_{6} q^{95} + 10 q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{5} + q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{5} + q^{7} - q^{9} - 2q^{11} - 6q^{13} + 4q^{15} - 8q^{17} + q^{19} - 4q^{21} - 8q^{23} + q^{25} - 2q^{27} + 8q^{29} - 3q^{31} + 2q^{33} + 10q^{35} + q^{37} - 3q^{39} + 12q^{41} + 22q^{43} + 2q^{45} - 6q^{47} - 13q^{49} + 8q^{51} + 12q^{53} - 8q^{55} + 2q^{57} - 4q^{59} + 6q^{61} - 5q^{63} - 6q^{65} - 13q^{67} - 16q^{69} - 20q^{71} + 11q^{73} - q^{75} + 8q^{77} + 3q^{79} - q^{81} + 4q^{83} - 32q^{85} + 4q^{87} - 3q^{91} + 3q^{93} - 2q^{95} + 20q^{97} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 0.866025i 0 1.00000 + 1.73205i 0 0.500000 2.59808i 0 −0.500000 0.866025i 0
37.1 0 0.500000 + 0.866025i 0 1.00000 1.73205i 0 0.500000 + 2.59808i 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 84.2.i.a 2
3.b odd 2 1 252.2.k.a 2
4.b odd 2 1 336.2.q.c 2
5.b even 2 1 2100.2.q.b 2
5.c odd 4 2 2100.2.bc.a 4
7.b odd 2 1 588.2.i.b 2
7.c even 3 1 inner 84.2.i.a 2
7.c even 3 1 588.2.a.a 1
7.d odd 6 1 588.2.a.f 1
7.d odd 6 1 588.2.i.b 2
8.b even 2 1 1344.2.q.b 2
8.d odd 2 1 1344.2.q.n 2
9.c even 3 1 2268.2.i.g 2
9.c even 3 1 2268.2.l.b 2
9.d odd 6 1 2268.2.i.b 2
9.d odd 6 1 2268.2.l.g 2
12.b even 2 1 1008.2.s.c 2
21.c even 2 1 1764.2.k.j 2
21.g even 6 1 1764.2.a.c 1
21.g even 6 1 1764.2.k.j 2
21.h odd 6 1 252.2.k.a 2
21.h odd 6 1 1764.2.a.h 1
28.d even 2 1 2352.2.q.q 2
28.f even 6 1 2352.2.a.k 1
28.f even 6 1 2352.2.q.q 2
28.g odd 6 1 336.2.q.c 2
28.g odd 6 1 2352.2.a.o 1
35.j even 6 1 2100.2.q.b 2
35.l odd 12 2 2100.2.bc.a 4
56.j odd 6 1 9408.2.a.i 1
56.k odd 6 1 1344.2.q.n 2
56.k odd 6 1 9408.2.a.bi 1
56.m even 6 1 9408.2.a.bx 1
56.p even 6 1 1344.2.q.b 2
56.p even 6 1 9408.2.a.cx 1
63.g even 3 1 2268.2.i.g 2
63.h even 3 1 2268.2.l.b 2
63.j odd 6 1 2268.2.l.g 2
63.n odd 6 1 2268.2.i.b 2
84.j odd 6 1 7056.2.a.o 1
84.n even 6 1 1008.2.s.c 2
84.n even 6 1 7056.2.a.bs 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 1.a even 1 1 trivial
84.2.i.a 2 7.c even 3 1 inner
252.2.k.a 2 3.b odd 2 1
252.2.k.a 2 21.h odd 6 1
336.2.q.c 2 4.b odd 2 1
336.2.q.c 2 28.g odd 6 1
588.2.a.a 1 7.c even 3 1
588.2.a.f 1 7.d odd 6 1
588.2.i.b 2 7.b odd 2 1
588.2.i.b 2 7.d odd 6 1
1008.2.s.c 2 12.b even 2 1
1008.2.s.c 2 84.n even 6 1
1344.2.q.b 2 8.b even 2 1
1344.2.q.b 2 56.p even 6 1
1344.2.q.n 2 8.d odd 2 1
1344.2.q.n 2 56.k odd 6 1
1764.2.a.c 1 21.g even 6 1
1764.2.a.h 1 21.h odd 6 1
1764.2.k.j 2 21.c even 2 1
1764.2.k.j 2 21.g even 6 1
2100.2.q.b 2 5.b even 2 1
2100.2.q.b 2 35.j even 6 1
2100.2.bc.a 4 5.c odd 4 2
2100.2.bc.a 4 35.l odd 12 2
2268.2.i.b 2 9.d odd 6 1
2268.2.i.b 2 63.n odd 6 1
2268.2.i.g 2 9.c even 3 1
2268.2.i.g 2 63.g even 3 1
2268.2.l.b 2 9.c even 3 1
2268.2.l.b 2 63.h even 3 1
2268.2.l.g 2 9.d odd 6 1
2268.2.l.g 2 63.j odd 6 1
2352.2.a.k 1 28.f even 6 1
2352.2.a.o 1 28.g odd 6 1
2352.2.q.q 2 28.d even 2 1
2352.2.q.q 2 28.f even 6 1
7056.2.a.o 1 84.j odd 6 1
7056.2.a.bs 1 84.n even 6 1
9408.2.a.i 1 56.j odd 6 1
9408.2.a.bi 1 56.k odd 6 1
9408.2.a.bx 1 56.m even 6 1
9408.2.a.cx 1 56.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(84, [\chi])\).