Properties

Label 124.1.i.a
Level 124
Weight 1
Character orbit 124.i
Analytic conductor 0.062
Analytic rank 0
Dimension 4
Projective image \(A_{4}\)
CM/RM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 124.i (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0618840615665\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.15376.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{12}^{3} q^{2} \) \( -\zeta_{12} q^{3} \) \(- q^{4}\) \( -\zeta_{12}^{2} q^{5} \) \( + \zeta_{12}^{4} q^{6} \) \( + \zeta_{12} q^{7} \) \( + \zeta_{12}^{3} q^{8} \) \(+O(q^{10})\) \( q\) \( -\zeta_{12}^{3} q^{2} \) \( -\zeta_{12} q^{3} \) \(- q^{4}\) \( -\zeta_{12}^{2} q^{5} \) \( + \zeta_{12}^{4} q^{6} \) \( + \zeta_{12} q^{7} \) \( + \zeta_{12}^{3} q^{8} \) \( + \zeta_{12}^{5} q^{10} \) \( -\zeta_{12}^{5} q^{11} \) \( + \zeta_{12} q^{12} \) \( + \zeta_{12}^{2} q^{13} \) \( -\zeta_{12}^{4} q^{14} \) \( + \zeta_{12}^{3} q^{15} \) \(+ q^{16}\) \( + \zeta_{12}^{4} q^{17} \) \( -\zeta_{12} q^{19} \) \( + \zeta_{12}^{2} q^{20} \) \( -\zeta_{12}^{2} q^{21} \) \( -\zeta_{12}^{2} q^{22} \) \( -\zeta_{12}^{4} q^{24} \) \( -\zeta_{12}^{5} q^{26} \) \( + \zeta_{12}^{3} q^{27} \) \( -\zeta_{12} q^{28} \) \(+ q^{30}\) \( + \zeta_{12}^{3} q^{31} \) \( -\zeta_{12}^{3} q^{32} \) \(- q^{33}\) \( + \zeta_{12} q^{34} \) \( -\zeta_{12}^{3} q^{35} \) \( + \zeta_{12}^{4} q^{37} \) \( + \zeta_{12}^{4} q^{38} \) \( -\zeta_{12}^{3} q^{39} \) \( -\zeta_{12}^{5} q^{40} \) \( -\zeta_{12}^{2} q^{41} \) \( + \zeta_{12}^{5} q^{42} \) \( -\zeta_{12} q^{43} \) \( + \zeta_{12}^{5} q^{44} \) \( -\zeta_{12} q^{48} \) \( -\zeta_{12}^{5} q^{51} \) \( -\zeta_{12}^{2} q^{52} \) \( + \zeta_{12}^{2} q^{53} \) \(+ q^{54}\) \( -\zeta_{12} q^{55} \) \( + \zeta_{12}^{4} q^{56} \) \( + \zeta_{12}^{2} q^{57} \) \( + \zeta_{12} q^{59} \) \( -\zeta_{12}^{3} q^{60} \) \(+ q^{62}\) \(- q^{64}\) \( -\zeta_{12}^{4} q^{65} \) \( + \zeta_{12}^{3} q^{66} \) \( -\zeta_{12}^{5} q^{67} \) \( -\zeta_{12}^{4} q^{68} \) \(- q^{70}\) \( + \zeta_{12}^{5} q^{71} \) \( -\zeta_{12}^{2} q^{73} \) \( + \zeta_{12} q^{74} \) \( + \zeta_{12} q^{76} \) \(+ q^{77}\) \(- q^{78}\) \( + \zeta_{12} q^{79} \) \( -\zeta_{12}^{2} q^{80} \) \( -\zeta_{12}^{4} q^{81} \) \( + \zeta_{12}^{5} q^{82} \) \( + \zeta_{12}^{5} q^{83} \) \( + \zeta_{12}^{2} q^{84} \) \(+ q^{85}\) \( + \zeta_{12}^{4} q^{86} \) \( + \zeta_{12}^{2} q^{88} \) \( + \zeta_{12}^{3} q^{91} \) \( -\zeta_{12}^{4} q^{93} \) \( + \zeta_{12}^{3} q^{95} \) \( + \zeta_{12}^{4} q^{96} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 2q^{20} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut 2q^{24} \) \(\mathstrut +\mathstrut 4q^{30} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 2q^{38} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 2q^{52} \) \(\mathstrut +\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut 4q^{54} \) \(\mathstrut -\mathstrut 2q^{56} \) \(\mathstrut +\mathstrut 2q^{57} \) \(\mathstrut +\mathstrut 4q^{62} \) \(\mathstrut -\mathstrut 4q^{64} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 2q^{68} \) \(\mathstrut -\mathstrut 4q^{70} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 4q^{78} \) \(\mathstrut -\mathstrut 2q^{80} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 2q^{84} \) \(\mathstrut +\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 2q^{86} \) \(\mathstrut +\mathstrut 2q^{88} \) \(\mathstrut +\mathstrut 2q^{93} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
1.00000i −0.866025 0.500000i −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i 0.866025 + 0.500000i 1.00000i 0 −0.866025 + 0.500000i
67.2 1.00000i 0.866025 + 0.500000i −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i −0.866025 0.500000i 1.00000i 0 0.866025 0.500000i
87.1 1.00000i 0.866025 0.500000i −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i −0.866025 + 0.500000i 1.00000i 0 0.866025 + 0.500000i
87.2 1.00000i −0.866025 + 0.500000i −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i 0.866025 0.500000i 1.00000i 0 −0.866025 0.500000i
\(n\): 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
4.b Odd 1 yes
31.c Even 1 yes
124.i Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(124, [\chi])\).