Properties

Label 133.1.m.a
Level 133
Weight 1
Character orbit 133.m
Analytic conductor 0.066
Analytic rank 0
Dimension 4
Projective image \(A_{4}\)
CM/RM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 133 = 7 \cdot 19 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 133.m (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0663756466802\)
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 4.0.17689.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{12}^{2} q^{2} \) \( + \zeta_{12}^{5} q^{3} \) \( -\zeta_{12}^{5} q^{5} \) \( + \zeta_{12} q^{6} \) \( -\zeta_{12}^{3} q^{7} \) \(- q^{8}\) \(+O(q^{10})\) \( q\) \( -\zeta_{12}^{2} q^{2} \) \( + \zeta_{12}^{5} q^{3} \) \( -\zeta_{12}^{5} q^{5} \) \( + \zeta_{12} q^{6} \) \( -\zeta_{12}^{3} q^{7} \) \(- q^{8}\) \( -\zeta_{12} q^{10} \) \( + \zeta_{12} q^{13} \) \( + \zeta_{12}^{5} q^{14} \) \( + \zeta_{12}^{4} q^{15} \) \( + \zeta_{12}^{2} q^{16} \) \( + \zeta_{12}^{5} q^{17} \) \( + \zeta_{12}^{3} q^{19} \) \( + \zeta_{12}^{2} q^{21} \) \( + \zeta_{12}^{4} q^{23} \) \( -\zeta_{12}^{5} q^{24} \) \( -\zeta_{12}^{3} q^{26} \) \( -\zeta_{12}^{3} q^{27} \) \( -\zeta_{12}^{4} q^{29} \) \(+ q^{30}\) \( + \zeta_{12} q^{34} \) \( -\zeta_{12}^{2} q^{35} \) \( -\zeta_{12}^{5} q^{38} \) \(- q^{39}\) \( + \zeta_{12}^{5} q^{40} \) \( + \zeta_{12}^{5} q^{41} \) \( -\zeta_{12}^{4} q^{42} \) \( -\zeta_{12}^{2} q^{43} \) \(+ q^{46}\) \( + \zeta_{12} q^{47} \) \( -\zeta_{12} q^{48} \) \(- q^{49}\) \( -\zeta_{12}^{4} q^{51} \) \( -\zeta_{12}^{4} q^{53} \) \( + \zeta_{12}^{5} q^{54} \) \( + \zeta_{12}^{3} q^{56} \) \( -\zeta_{12}^{2} q^{57} \) \(- q^{58}\) \( -\zeta_{12}^{5} q^{59} \) \( -\zeta_{12} q^{61} \) \(+ q^{64}\) \(+ q^{65}\) \( + \zeta_{12}^{4} q^{67} \) \( -\zeta_{12}^{3} q^{69} \) \( + \zeta_{12}^{4} q^{70} \) \( + \zeta_{12}^{2} q^{71} \) \( -\zeta_{12}^{5} q^{73} \) \( + \zeta_{12}^{2} q^{78} \) \( -\zeta_{12}^{2} q^{79} \) \( + \zeta_{12} q^{80} \) \( + \zeta_{12}^{2} q^{81} \) \( + \zeta_{12} q^{82} \) \( + \zeta_{12}^{4} q^{85} \) \( + \zeta_{12}^{4} q^{86} \) \( + \zeta_{12}^{3} q^{87} \) \( -\zeta_{12} q^{89} \) \( -\zeta_{12}^{4} q^{91} \) \( -\zeta_{12}^{3} q^{94} \) \( + \zeta_{12}^{2} q^{95} \) \( + \zeta_{12}^{5} q^{97} \) \( + \zeta_{12}^{2} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 4q^{30} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut +\mathstrut 2q^{42} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 2q^{51} \) \(\mathstrut +\mathstrut 2q^{53} \) \(\mathstrut -\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 4q^{58} \) \(\mathstrut +\mathstrut 4q^{64} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut -\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 2q^{70} \) \(\mathstrut +\mathstrut 2q^{71} \) \(\mathstrut +\mathstrut 2q^{78} \) \(\mathstrut -\mathstrut 2q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 2q^{85} \) \(\mathstrut -\mathstrut 2q^{86} \) \(\mathstrut +\mathstrut 2q^{91} \) \(\mathstrut +\mathstrut 2q^{95} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(115\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.500000 0.866025i −0.866025 + 0.500000i 0 0.866025 0.500000i 0.866025 + 0.500000i 1.00000i −1.00000 0 −0.866025 0.500000i
83.2 −0.500000 0.866025i 0.866025 0.500000i 0 −0.866025 + 0.500000i −0.866025 0.500000i 1.00000i −1.00000 0 0.866025 + 0.500000i
125.1 −0.500000 + 0.866025i −0.866025 0.500000i 0 0.866025 + 0.500000i 0.866025 0.500000i 1.00000i −1.00000 0 −0.866025 + 0.500000i
125.2 −0.500000 + 0.866025i 0.866025 + 0.500000i 0 −0.866025 0.500000i −0.866025 + 0.500000i 1.00000i −1.00000 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
7.b Odd 1 yes
19.c Even 1 yes
133.m Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(133, [\chi])\).