Properties

Label 133.1.r.a
Level 133
Weight 1
Character orbit 133.r
Analytic conductor 0.066
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -19
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.0663756466802\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.931.1
Artin image size \(18\)
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.336091.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{6} q^{4} \) \( -\zeta_{6}^{2} q^{5} \) \( + \zeta_{6}^{2} q^{7} \) \( + \zeta_{6}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{6} q^{4} \) \( -\zeta_{6}^{2} q^{5} \) \( + \zeta_{6}^{2} q^{7} \) \( + \zeta_{6}^{2} q^{9} \) \( + \zeta_{6} q^{11} \) \( + \zeta_{6}^{2} q^{16} \) \( -2 \zeta_{6} q^{17} \) \( + \zeta_{6}^{2} q^{19} \) \(- q^{20}\) \( -\zeta_{6}^{2} q^{23} \) \(+ q^{28}\) \( + \zeta_{6} q^{35} \) \(+ q^{36}\) \(- q^{43}\) \( -\zeta_{6}^{2} q^{44} \) \( + \zeta_{6} q^{45} \) \( -\zeta_{6}^{2} q^{47} \) \( -\zeta_{6} q^{49} \) \(+ q^{55}\) \( -\zeta_{6}^{2} q^{61} \) \( -\zeta_{6} q^{63} \) \(+ q^{64}\) \( + 2 \zeta_{6}^{2} q^{68} \) \( + \zeta_{6} q^{73} \) \(+ q^{76}\) \(- q^{77}\) \( + \zeta_{6} q^{80} \) \( -\zeta_{6} q^{81} \) \(- q^{83}\) \( -2 q^{85} \) \(- q^{92}\) \( + \zeta_{6} q^{95} \) \(- q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut -\mathstrut 2q^{20} \) \(\mathstrut +\mathstrut q^{23} \) \(\mathstrut +\mathstrut 2q^{28} \) \(\mathstrut +\mathstrut q^{35} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut q^{44} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut -\mathstrut q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 2q^{68} \) \(\mathstrut +\mathstrut q^{73} \) \(\mathstrut +\mathstrut 2q^{76} \) \(\mathstrut -\mathstrut 2q^{77} \) \(\mathstrut +\mathstrut q^{80} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut -\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 2q^{92} \) \(\mathstrut +\mathstrut q^{95} \) \(\mathstrut -\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(78\) \(115\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −0.500000 0.866025i 0
37.1 0 0 −0.500000 0.866025i 0.500000 0.866025i 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
19.b Odd 1 CM by \(\Q(\sqrt{-19}) \) yes
7.c Even 1 yes
133.r Odd 1 yes

Hecke kernels

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