Properties

Label 84.2.f.a
Level $84$
Weight $2$
Character orbit 84.f
Analytic conductor $0.671$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 84.f (of order \(2\), degree \(1\), 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: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 3 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 3 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( -4 + 8 \zeta_{6} ) q^{13} + ( -2 + 4 \zeta_{6} ) q^{19} + ( -1 - 4 \zeta_{6} ) q^{21} -5 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 6 - 12 \zeta_{6} ) q^{31} + 10 q^{37} + 12 q^{39} -8 q^{43} + ( 5 - 8 \zeta_{6} ) q^{49} + 6 q^{57} + ( 4 - 8 \zeta_{6} ) q^{61} + ( -9 + 6 \zeta_{6} ) q^{63} -16 q^{67} + ( -8 + 16 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} -4 q^{79} + 9 q^{81} + ( 4 + 16 \zeta_{6} ) q^{91} -18 q^{93} + ( 8 - 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 4q^{7} - 6q^{9} - 6q^{21} - 10q^{25} + 20q^{37} + 24q^{39} - 16q^{43} + 2q^{49} + 12q^{57} - 12q^{63} - 32q^{67} - 8q^{79} + 18q^{81} + 24q^{91} - 36q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 0 0 2.00000 1.73205i 0 −3.00000 0
41.2 0 1.73205i 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.f.a 2
3.b odd 2 1 CM 84.2.f.a 2
4.b odd 2 1 336.2.k.a 2
5.b even 2 1 2100.2.d.b 2
5.c odd 4 2 2100.2.f.e 4
7.b odd 2 1 inner 84.2.f.a 2
7.c even 3 1 588.2.k.a 2
7.c even 3 1 588.2.k.e 2
7.d odd 6 1 588.2.k.a 2
7.d odd 6 1 588.2.k.e 2
8.b even 2 1 1344.2.k.b 2
8.d odd 2 1 1344.2.k.a 2
9.c even 3 1 2268.2.x.c 2
9.c even 3 1 2268.2.x.e 2
9.d odd 6 1 2268.2.x.c 2
9.d odd 6 1 2268.2.x.e 2
12.b even 2 1 336.2.k.a 2
15.d odd 2 1 2100.2.d.b 2
15.e even 4 2 2100.2.f.e 4
21.c even 2 1 inner 84.2.f.a 2
21.g even 6 1 588.2.k.a 2
21.g even 6 1 588.2.k.e 2
21.h odd 6 1 588.2.k.a 2
21.h odd 6 1 588.2.k.e 2
24.f even 2 1 1344.2.k.a 2
24.h odd 2 1 1344.2.k.b 2
28.d even 2 1 336.2.k.a 2
35.c odd 2 1 2100.2.d.b 2
35.f even 4 2 2100.2.f.e 4
56.e even 2 1 1344.2.k.a 2
56.h odd 2 1 1344.2.k.b 2
63.l odd 6 1 2268.2.x.c 2
63.l odd 6 1 2268.2.x.e 2
63.o even 6 1 2268.2.x.c 2
63.o even 6 1 2268.2.x.e 2
84.h odd 2 1 336.2.k.a 2
105.g even 2 1 2100.2.d.b 2
105.k odd 4 2 2100.2.f.e 4
168.e odd 2 1 1344.2.k.a 2
168.i even 2 1 1344.2.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.f.a 2 1.a even 1 1 trivial
84.2.f.a 2 3.b odd 2 1 CM
84.2.f.a 2 7.b odd 2 1 inner
84.2.f.a 2 21.c even 2 1 inner
336.2.k.a 2 4.b odd 2 1
336.2.k.a 2 12.b even 2 1
336.2.k.a 2 28.d even 2 1
336.2.k.a 2 84.h odd 2 1
588.2.k.a 2 7.c even 3 1
588.2.k.a 2 7.d odd 6 1
588.2.k.a 2 21.g even 6 1
588.2.k.a 2 21.h odd 6 1
588.2.k.e 2 7.c even 3 1
588.2.k.e 2 7.d odd 6 1
588.2.k.e 2 21.g even 6 1
588.2.k.e 2 21.h odd 6 1
1344.2.k.a 2 8.d odd 2 1
1344.2.k.a 2 24.f even 2 1
1344.2.k.a 2 56.e even 2 1
1344.2.k.a 2 168.e odd 2 1
1344.2.k.b 2 8.b even 2 1
1344.2.k.b 2 24.h odd 2 1
1344.2.k.b 2 56.h odd 2 1
1344.2.k.b 2 168.i even 2 1
2100.2.d.b 2 5.b even 2 1
2100.2.d.b 2 15.d odd 2 1
2100.2.d.b 2 35.c odd 2 1
2100.2.d.b 2 105.g even 2 1
2100.2.f.e 4 5.c odd 4 2
2100.2.f.e 4 15.e even 4 2
2100.2.f.e 4 35.f even 4 2
2100.2.f.e 4 105.k odd 4 2
2268.2.x.c 2 9.c even 3 1
2268.2.x.c 2 9.d odd 6 1
2268.2.x.c 2 63.l odd 6 1
2268.2.x.c 2 63.o even 6 1
2268.2.x.e 2 9.c even 3 1
2268.2.x.e 2 9.d odd 6 1
2268.2.x.e 2 63.l odd 6 1
2268.2.x.e 2 63.o even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 48 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 12 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 108 + T^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 48 + T^{2} \)
$67$ \( ( 16 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 192 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 192 + T^{2} \)
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