Properties

Label 84.2.k.b
Level $84$
Weight $2$
Character orbit 84.k
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.k (of order \(6\), 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 3 + 3 \zeta_{6} ) q^{11} + ( -6 + 3 \zeta_{6} ) q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + ( 2 - \zeta_{6} ) q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} + ( -6 + 3 \zeta_{6} ) q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -1 - \zeta_{6} ) q^{31} + ( 9 - 9 \zeta_{6} ) q^{33} + ( 6 - 9 \zeta_{6} ) q^{35} -7 \zeta_{6} q^{37} + 6 q^{41} + 4 q^{43} + 9 \zeta_{6} q^{45} -3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 3 + 3 \zeta_{6} ) q^{51} + ( -3 - 3 \zeta_{6} ) q^{53} + ( 9 - 18 \zeta_{6} ) q^{55} -3 \zeta_{6} q^{57} + ( 3 - 3 \zeta_{6} ) q^{59} + ( -14 + 7 \zeta_{6} ) q^{61} + ( -3 - 6 \zeta_{6} ) q^{63} + ( -5 + 5 \zeta_{6} ) q^{67} + 9 \zeta_{6} q^{69} + ( -6 + 12 \zeta_{6} ) q^{71} + ( -7 - 7 \zeta_{6} ) q^{73} + ( 4 + 4 \zeta_{6} ) q^{75} + ( -3 + 15 \zeta_{6} ) q^{77} + \zeta_{6} q^{79} + 9 q^{81} -12 q^{83} + 9 q^{85} + 9 \zeta_{6} q^{89} + ( -3 + 3 \zeta_{6} ) q^{93} + ( -3 - 3 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( -9 - 9 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} + 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{5} + 4q^{7} - 6q^{9} + 9q^{11} - 9q^{15} - 3q^{17} + 3q^{19} + 6q^{21} - 9q^{23} - 4q^{25} - 3q^{31} + 9q^{33} + 3q^{35} - 7q^{37} + 12q^{41} + 8q^{43} + 9q^{45} - 3q^{47} + 2q^{49} + 9q^{51} - 9q^{53} - 3q^{57} + 3q^{59} - 21q^{61} - 12q^{63} - 5q^{67} + 9q^{69} - 21q^{73} + 12q^{75} + 9q^{77} + q^{79} + 18q^{81} - 24q^{83} + 18q^{85} + 9q^{89} - 3q^{93} - 9q^{95} - 27q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 −1.50000 + 2.59808i 0 2.00000 1.73205i 0 −3.00000 0
17.1 0 1.73205i 0 −1.50000 2.59808i 0 2.00000 + 1.73205i 0 −3.00000 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
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.k.b yes 2
3.b odd 2 1 84.2.k.a 2
4.b odd 2 1 336.2.bc.b 2
5.b even 2 1 2100.2.bi.e 2
5.c odd 4 2 2100.2.bo.f 4
7.b odd 2 1 588.2.k.c 2
7.c even 3 1 588.2.f.a 2
7.c even 3 1 588.2.k.d 2
7.d odd 6 1 84.2.k.a 2
7.d odd 6 1 588.2.f.c 2
9.c even 3 1 2268.2.w.a 2
9.c even 3 1 2268.2.bm.f 2
9.d odd 6 1 2268.2.w.f 2
9.d odd 6 1 2268.2.bm.a 2
12.b even 2 1 336.2.bc.d 2
15.d odd 2 1 2100.2.bi.f 2
15.e even 4 2 2100.2.bo.a 4
21.c even 2 1 588.2.k.d 2
21.g even 6 1 inner 84.2.k.b yes 2
21.g even 6 1 588.2.f.a 2
21.h odd 6 1 588.2.f.c 2
21.h odd 6 1 588.2.k.c 2
28.f even 6 1 336.2.bc.d 2
28.f even 6 1 2352.2.k.a 2
28.g odd 6 1 2352.2.k.d 2
35.i odd 6 1 2100.2.bi.f 2
35.k even 12 2 2100.2.bo.a 4
63.i even 6 1 2268.2.bm.f 2
63.k odd 6 1 2268.2.w.f 2
63.s even 6 1 2268.2.w.a 2
63.t odd 6 1 2268.2.bm.a 2
84.j odd 6 1 336.2.bc.b 2
84.j odd 6 1 2352.2.k.d 2
84.n even 6 1 2352.2.k.a 2
105.p even 6 1 2100.2.bi.e 2
105.w odd 12 2 2100.2.bo.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.k.a 2 3.b odd 2 1
84.2.k.a 2 7.d odd 6 1
84.2.k.b yes 2 1.a even 1 1 trivial
84.2.k.b yes 2 21.g even 6 1 inner
336.2.bc.b 2 4.b odd 2 1
336.2.bc.b 2 84.j odd 6 1
336.2.bc.d 2 12.b even 2 1
336.2.bc.d 2 28.f even 6 1
588.2.f.a 2 7.c even 3 1
588.2.f.a 2 21.g even 6 1
588.2.f.c 2 7.d odd 6 1
588.2.f.c 2 21.h odd 6 1
588.2.k.c 2 7.b odd 2 1
588.2.k.c 2 21.h odd 6 1
588.2.k.d 2 7.c even 3 1
588.2.k.d 2 21.c even 2 1
2100.2.bi.e 2 5.b even 2 1
2100.2.bi.e 2 105.p even 6 1
2100.2.bi.f 2 15.d odd 2 1
2100.2.bi.f 2 35.i odd 6 1
2100.2.bo.a 4 15.e even 4 2
2100.2.bo.a 4 35.k even 12 2
2100.2.bo.f 4 5.c odd 4 2
2100.2.bo.f 4 105.w odd 12 2
2268.2.w.a 2 9.c even 3 1
2268.2.w.a 2 63.s even 6 1
2268.2.w.f 2 9.d odd 6 1
2268.2.w.f 2 63.k odd 6 1
2268.2.bm.a 2 9.d odd 6 1
2268.2.bm.a 2 63.t odd 6 1
2268.2.bm.f 2 9.c even 3 1
2268.2.bm.f 2 63.i even 6 1
2352.2.k.a 2 28.f even 6 1
2352.2.k.a 2 84.n even 6 1
2352.2.k.d 2 28.g odd 6 1
2352.2.k.d 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(84, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( 9 + 3 T + T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( 27 - 9 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 3 - 3 T + T^{2} \)
$23$ \( 27 + 9 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3 + 3 T + T^{2} \)
$37$ \( 49 + 7 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 9 + 3 T + T^{2} \)
$53$ \( 27 + 9 T + T^{2} \)
$59$ \( 9 - 3 T + T^{2} \)
$61$ \( 147 + 21 T + T^{2} \)
$67$ \( 25 + 5 T + T^{2} \)
$71$ \( 108 + T^{2} \)
$73$ \( 147 + 21 T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( 81 - 9 T + T^{2} \)
$97$ \( 48 + T^{2} \)
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