Properties

Label 129.1.l.a
Level 129
Weight 1
Character orbit 129.l
Analytic conductor 0.064
Analytic rank 0
Dimension 6
Projective image \(D_{7}\)
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 129 = 3 \cdot 43 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 129.l (of order \(14\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0643793866297\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.170676802323.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{14}^{5} q^{3} \) \( + \zeta_{14}^{4} q^{4} \) \( + ( -\zeta_{14} + \zeta_{14}^{6} ) q^{7} \) \( -\zeta_{14}^{3} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{14}^{5} q^{3} \) \( + \zeta_{14}^{4} q^{4} \) \( + ( -\zeta_{14} + \zeta_{14}^{6} ) q^{7} \) \( -\zeta_{14}^{3} q^{9} \) \( + \zeta_{14}^{2} q^{12} \) \( + ( \zeta_{14}^{2} - \zeta_{14}^{3} ) q^{13} \) \( -\zeta_{14} q^{16} \) \( + ( \zeta_{14}^{2} + \zeta_{14}^{6} ) q^{19} \) \( + ( \zeta_{14}^{4} + \zeta_{14}^{6} ) q^{21} \) \( -\zeta_{14}^{5} q^{25} \) \( -\zeta_{14} q^{27} \) \( + ( -\zeta_{14}^{3} - \zeta_{14}^{5} ) q^{28} \) \( + ( 1 + \zeta_{14}^{4} ) q^{31} \) \(+ q^{36}\) \( + ( -\zeta_{14}^{3} + \zeta_{14}^{4} ) q^{37} \) \( + ( 1 - \zeta_{14} ) q^{39} \) \( -\zeta_{14}^{5} q^{43} \) \( + \zeta_{14}^{6} q^{48} \) \( + ( 1 + \zeta_{14}^{2} - \zeta_{14}^{5} ) q^{49} \) \( + ( 1 + \zeta_{14}^{6} ) q^{52} \) \( + ( 1 + \zeta_{14}^{4} ) q^{57} \) \( + ( -\zeta_{14} + \zeta_{14}^{2} ) q^{61} \) \( + ( \zeta_{14}^{2} + \zeta_{14}^{4} ) q^{63} \) \( -\zeta_{14}^{5} q^{64} \) \( + ( -\zeta_{14}^{3} - \zeta_{14}^{5} ) q^{67} \) \( + ( -\zeta_{14} + \zeta_{14}^{4} ) q^{73} \) \( -\zeta_{14}^{3} q^{75} \) \( + ( -\zeta_{14}^{3} + \zeta_{14}^{6} ) q^{76} \) \( + ( -\zeta_{14} + \zeta_{14}^{6} ) q^{79} \) \( + \zeta_{14}^{6} q^{81} \) \( + ( -\zeta_{14} - \zeta_{14}^{3} ) q^{84} \) \( + ( -\zeta_{14} + \zeta_{14}^{2} - \zeta_{14}^{3} + \zeta_{14}^{4} ) q^{91} \) \( + ( \zeta_{14}^{2} - \zeta_{14}^{5} ) q^{93} \) \( + ( \zeta_{14}^{2} + \zeta_{14}^{4} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut -\mathstrut q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut -\mathstrut q^{27} \) \(\mathstrut -\mathstrut 2q^{28} \) \(\mathstrut +\mathstrut 5q^{31} \) \(\mathstrut +\mathstrut 6q^{36} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 5q^{39} \) \(\mathstrut -\mathstrut q^{43} \) \(\mathstrut -\mathstrut q^{48} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 5q^{52} \) \(\mathstrut +\mathstrut 5q^{57} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut q^{64} \) \(\mathstrut -\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut -\mathstrut 2q^{76} \) \(\mathstrut -\mathstrut 2q^{79} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut -\mathstrut 2q^{84} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 2q^{93} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/129\mathbb{Z}\right)^\times\).

\(n\) \(44\) \(46\)
\(\chi(n)\) \(-1\) \(-\zeta_{14}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.900969 + 0.433884i
−0.623490 0.781831i
0.222521 + 0.974928i
0.900969 0.433884i
−0.623490 + 0.781831i
0.222521 0.974928i
0 0.623490 0.781831i −0.222521 + 0.974928i 0 0 −1.80194 0 −0.222521 0.974928i 0
35.1 0 −0.222521 0.974928i −0.900969 0.433884i 0 0 1.24698 0 −0.900969 + 0.433884i 0
41.1 0 −0.900969 0.433884i 0.623490 0.781831i 0 0 −0.445042 0 0.623490 + 0.781831i 0
47.1 0 0.623490 + 0.781831i −0.222521 0.974928i 0 0 −1.80194 0 −0.222521 + 0.974928i 0
59.1 0 −0.222521 + 0.974928i −0.900969 + 0.433884i 0 0 1.24698 0 −0.900969 0.433884i 0
107.1 0 −0.900969 + 0.433884i 0.623490 + 0.781831i 0 0 −0.445042 0 0.623490 0.781831i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.1
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
43.e Even 1 yes
129.l Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(129, [\chi])\).