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 discriminant -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 \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0643793866297\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Defining polynomial: \(x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.170676802323.1

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
43.e even 7 1 inner
129.l odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.1.l.a 6
3.b odd 2 1 CM 129.1.l.a 6
4.b odd 2 1 2064.1.cj.a 6
5.b even 2 1 3225.1.bm.a 6
5.c odd 4 2 3225.1.bi.a 12
9.c even 3 2 3483.1.bs.a 12
9.d odd 6 2 3483.1.bs.a 12
12.b even 2 1 2064.1.cj.a 6
15.d odd 2 1 3225.1.bm.a 6
15.e even 4 2 3225.1.bi.a 12
43.e even 7 1 inner 129.1.l.a 6
129.l odd 14 1 inner 129.1.l.a 6
172.k odd 14 1 2064.1.cj.a 6
215.p even 14 1 3225.1.bm.a 6
215.s odd 28 2 3225.1.bi.a 12
387.ba even 21 2 3483.1.bs.a 12
387.bd odd 42 2 3483.1.bs.a 12
516.v even 14 1 2064.1.cj.a 6
645.ba odd 14 1 3225.1.bm.a 6
645.bk even 28 2 3225.1.bi.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.1.l.a 6 1.a even 1 1 trivial
129.1.l.a 6 3.b odd 2 1 CM
129.1.l.a 6 43.e even 7 1 inner
129.1.l.a 6 129.l odd 14 1 inner
2064.1.cj.a 6 4.b odd 2 1
2064.1.cj.a 6 12.b even 2 1
2064.1.cj.a 6 172.k odd 14 1
2064.1.cj.a 6 516.v even 14 1
3225.1.bi.a 12 5.c odd 4 2
3225.1.bi.a 12 15.e even 4 2
3225.1.bi.a 12 215.s odd 28 2
3225.1.bi.a 12 645.bk even 28 2
3225.1.bm.a 6 5.b even 2 1
3225.1.bm.a 6 15.d odd 2 1
3225.1.bm.a 6 215.p even 14 1
3225.1.bm.a 6 645.ba odd 14 1
3483.1.bs.a 12 9.c even 3 2
3483.1.bs.a 12 9.d odd 6 2
3483.1.bs.a 12 387.ba even 21 2
3483.1.bs.a 12 387.bd odd 42 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(129, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$11$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$13$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$17$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$19$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$23$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$29$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$31$ \( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$37$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$41$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$43$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$47$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$53$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$59$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$61$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$67$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$71$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$73$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$79$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$83$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$89$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \)
$97$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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