Properties

Label 126.2.g.a
Level $126$
Weight $2$
Character orbit 126.g
Analytic conductor $1.006$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} + q^{8} -3 \zeta_{6} q^{10} + 3 \zeta_{6} q^{11} + 2 q^{13} + ( -1 - 2 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} -6 \zeta_{6} q^{17} + ( -2 + 2 \zeta_{6} ) q^{19} + 3 q^{20} -3 q^{22} + ( 6 - 6 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} + ( -2 + 2 \zeta_{6} ) q^{26} + ( 3 - \zeta_{6} ) q^{28} + 9 q^{29} + 7 \zeta_{6} q^{31} -\zeta_{6} q^{32} + 6 q^{34} + ( -3 - 6 \zeta_{6} ) q^{35} + ( 10 - 10 \zeta_{6} ) q^{37} -2 \zeta_{6} q^{38} + ( -3 + 3 \zeta_{6} ) q^{40} -4 q^{43} + ( 3 - 3 \zeta_{6} ) q^{44} + 6 \zeta_{6} q^{46} + ( -12 + 12 \zeta_{6} ) q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + 4 q^{50} -2 \zeta_{6} q^{52} + 3 \zeta_{6} q^{53} -9 q^{55} + ( -2 + 3 \zeta_{6} ) q^{56} + ( -9 + 9 \zeta_{6} ) q^{58} + 3 \zeta_{6} q^{59} + ( 4 - 4 \zeta_{6} ) q^{61} -7 q^{62} + q^{64} + ( -6 + 6 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} + ( -6 + 6 \zeta_{6} ) q^{68} + ( 9 - 3 \zeta_{6} ) q^{70} -2 \zeta_{6} q^{73} + 10 \zeta_{6} q^{74} + 2 q^{76} + ( -9 + 3 \zeta_{6} ) q^{77} + ( -5 + 5 \zeta_{6} ) q^{79} -3 \zeta_{6} q^{80} + 9 q^{83} + 18 q^{85} + ( 4 - 4 \zeta_{6} ) q^{86} + 3 \zeta_{6} q^{88} + ( 6 - 6 \zeta_{6} ) q^{89} + ( -4 + 6 \zeta_{6} ) q^{91} -6 q^{92} -12 \zeta_{6} q^{94} -6 \zeta_{6} q^{95} -13 q^{97} + ( 8 - 5 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 3q^{5} - q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 3q^{5} - q^{7} + 2q^{8} - 3q^{10} + 3q^{11} + 4q^{13} - 4q^{14} - q^{16} - 6q^{17} - 2q^{19} + 6q^{20} - 6q^{22} + 6q^{23} - 4q^{25} - 2q^{26} + 5q^{28} + 18q^{29} + 7q^{31} - q^{32} + 12q^{34} - 12q^{35} + 10q^{37} - 2q^{38} - 3q^{40} - 8q^{43} + 3q^{44} + 6q^{46} - 12q^{47} - 13q^{49} + 8q^{50} - 2q^{52} + 3q^{53} - 18q^{55} - q^{56} - 9q^{58} + 3q^{59} + 4q^{61} - 14q^{62} + 2q^{64} - 6q^{65} - 2q^{67} - 6q^{68} + 15q^{70} - 2q^{73} + 10q^{74} + 4q^{76} - 15q^{77} - 5q^{79} - 3q^{80} + 18q^{83} + 36q^{85} + 4q^{86} + 3q^{88} + 6q^{89} - 2q^{91} - 12q^{92} - 12q^{94} - 6q^{95} - 26q^{97} + 11q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −1.50000 + 2.59808i 0 −0.500000 + 2.59808i 1.00000 0 −1.50000 2.59808i
109.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.50000 2.59808i 0 −0.500000 2.59808i 1.00000 0 −1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.g.a 2
3.b odd 2 1 126.2.g.d yes 2
4.b odd 2 1 1008.2.s.b 2
7.b odd 2 1 882.2.g.e 2
7.c even 3 1 inner 126.2.g.a 2
7.c even 3 1 882.2.a.j 1
7.d odd 6 1 882.2.a.h 1
7.d odd 6 1 882.2.g.e 2
9.c even 3 1 1134.2.e.k 2
9.c even 3 1 1134.2.h.f 2
9.d odd 6 1 1134.2.e.g 2
9.d odd 6 1 1134.2.h.j 2
12.b even 2 1 1008.2.s.o 2
21.c even 2 1 882.2.g.g 2
21.g even 6 1 882.2.a.e 1
21.g even 6 1 882.2.g.g 2
21.h odd 6 1 126.2.g.d yes 2
21.h odd 6 1 882.2.a.a 1
28.f even 6 1 7056.2.a.h 1
28.g odd 6 1 1008.2.s.b 2
28.g odd 6 1 7056.2.a.by 1
63.g even 3 1 1134.2.e.k 2
63.h even 3 1 1134.2.h.f 2
63.j odd 6 1 1134.2.h.j 2
63.n odd 6 1 1134.2.e.g 2
84.j odd 6 1 7056.2.a.bx 1
84.n even 6 1 1008.2.s.o 2
84.n even 6 1 7056.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 1.a even 1 1 trivial
126.2.g.a 2 7.c even 3 1 inner
126.2.g.d yes 2 3.b odd 2 1
126.2.g.d yes 2 21.h odd 6 1
882.2.a.a 1 21.h odd 6 1
882.2.a.e 1 21.g even 6 1
882.2.a.h 1 7.d odd 6 1
882.2.a.j 1 7.c even 3 1
882.2.g.e 2 7.b odd 2 1
882.2.g.e 2 7.d odd 6 1
882.2.g.g 2 21.c even 2 1
882.2.g.g 2 21.g even 6 1
1008.2.s.b 2 4.b odd 2 1
1008.2.s.b 2 28.g odd 6 1
1008.2.s.o 2 12.b even 2 1
1008.2.s.o 2 84.n even 6 1
1134.2.e.g 2 9.d odd 6 1
1134.2.e.g 2 63.n odd 6 1
1134.2.e.k 2 9.c even 3 1
1134.2.e.k 2 63.g even 3 1
1134.2.h.f 2 9.c even 3 1
1134.2.h.f 2 63.h even 3 1
1134.2.h.j 2 9.d odd 6 1
1134.2.h.j 2 63.j odd 6 1
7056.2.a.e 1 84.n even 6 1
7056.2.a.h 1 28.f even 6 1
7056.2.a.bx 1 84.j odd 6 1
7056.2.a.by 1 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\):

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)