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.

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

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:
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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 \(S_{1}^{\mathrm{new}}(124, [\chi])\).