Properties

Label 84.1.p.a
Level 84
Weight 1
Character orbit 84.p
Analytic conductor 0.042
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 84.p (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0419214610612\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.588.1
Artin image size \(18\)
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.21168.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 0 0 −0.500000 0.866025i 0 −0.500000 0.866025i 0
65.1 0 −0.500000 0.866025i 0 0 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
7.c Even 1 yes
21.h Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(84, [\chi])\).