Properties

Label 126.4.g.c
Level $126$
Weight $4$
Character orbit 126.g
Analytic conductor $7.434$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( 7 - 7 \zeta_{6} ) q^{5} + ( -19 + 18 \zeta_{6} ) q^{7} + 8 q^{8} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( 7 - 7 \zeta_{6} ) q^{5} + ( -19 + 18 \zeta_{6} ) q^{7} + 8 q^{8} + 14 \zeta_{6} q^{10} + 35 \zeta_{6} q^{11} + 66 q^{13} + ( 2 - 38 \zeta_{6} ) q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} + 59 \zeta_{6} q^{17} + ( -137 + 137 \zeta_{6} ) q^{19} -28 q^{20} -70 q^{22} + ( -7 + 7 \zeta_{6} ) q^{23} + 76 \zeta_{6} q^{25} + ( -132 + 132 \zeta_{6} ) q^{26} + ( 72 + 4 \zeta_{6} ) q^{28} -106 q^{29} -75 \zeta_{6} q^{31} -32 \zeta_{6} q^{32} -118 q^{34} + ( -7 + 133 \zeta_{6} ) q^{35} + ( -11 + 11 \zeta_{6} ) q^{37} -274 \zeta_{6} q^{38} + ( 56 - 56 \zeta_{6} ) q^{40} + 498 q^{41} + 260 q^{43} + ( 140 - 140 \zeta_{6} ) q^{44} -14 \zeta_{6} q^{46} + ( -171 + 171 \zeta_{6} ) q^{47} + ( 37 - 360 \zeta_{6} ) q^{49} -152 q^{50} -264 \zeta_{6} q^{52} -417 \zeta_{6} q^{53} + 245 q^{55} + ( -152 + 144 \zeta_{6} ) q^{56} + ( 212 - 212 \zeta_{6} ) q^{58} -17 \zeta_{6} q^{59} + ( -51 + 51 \zeta_{6} ) q^{61} + 150 q^{62} + 64 q^{64} + ( 462 - 462 \zeta_{6} ) q^{65} -439 \zeta_{6} q^{67} + ( 236 - 236 \zeta_{6} ) q^{68} + ( -252 - 14 \zeta_{6} ) q^{70} + 784 q^{71} -295 \zeta_{6} q^{73} -22 \zeta_{6} q^{74} + 548 q^{76} + ( -630 - 35 \zeta_{6} ) q^{77} + ( 495 - 495 \zeta_{6} ) q^{79} + 112 \zeta_{6} q^{80} + ( -996 + 996 \zeta_{6} ) q^{82} -932 q^{83} + 413 q^{85} + ( -520 + 520 \zeta_{6} ) q^{86} + 280 \zeta_{6} q^{88} + ( -873 + 873 \zeta_{6} ) q^{89} + ( -1254 + 1188 \zeta_{6} ) q^{91} + 28 q^{92} -342 \zeta_{6} q^{94} + 959 \zeta_{6} q^{95} -290 q^{97} + ( 646 + 74 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} + 7q^{5} - 20q^{7} + 16q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} + 7q^{5} - 20q^{7} + 16q^{8} + 14q^{10} + 35q^{11} + 132q^{13} - 34q^{14} - 16q^{16} + 59q^{17} - 137q^{19} - 56q^{20} - 140q^{22} - 7q^{23} + 76q^{25} - 132q^{26} + 148q^{28} - 212q^{29} - 75q^{31} - 32q^{32} - 236q^{34} + 119q^{35} - 11q^{37} - 274q^{38} + 56q^{40} + 996q^{41} + 520q^{43} + 140q^{44} - 14q^{46} - 171q^{47} - 286q^{49} - 304q^{50} - 264q^{52} - 417q^{53} + 490q^{55} - 160q^{56} + 212q^{58} - 17q^{59} - 51q^{61} + 300q^{62} + 128q^{64} + 462q^{65} - 439q^{67} + 236q^{68} - 518q^{70} + 1568q^{71} - 295q^{73} - 22q^{74} + 1096q^{76} - 1295q^{77} + 495q^{79} + 112q^{80} - 996q^{82} - 1864q^{83} + 826q^{85} - 520q^{86} + 280q^{88} - 873q^{89} - 1320q^{91} + 56q^{92} - 342q^{94} + 959q^{95} - 580q^{97} + 1366q^{98} + O(q^{100}) \)

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
−1.00000 + 1.73205i 0 −2.00000 3.46410i 3.50000 6.06218i 0 −10.0000 + 15.5885i 8.00000 0 7.00000 + 12.1244i
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i 3.50000 + 6.06218i 0 −10.0000 15.5885i 8.00000 0 7.00000 12.1244i
\(n\): 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 126.4.g.c 2
3.b odd 2 1 14.4.c.b 2
7.b odd 2 1 882.4.g.d 2
7.c even 3 1 inner 126.4.g.c 2
7.c even 3 1 882.4.a.k 1
7.d odd 6 1 882.4.a.p 1
7.d odd 6 1 882.4.g.d 2
12.b even 2 1 112.4.i.b 2
15.d odd 2 1 350.4.e.b 2
15.e even 4 2 350.4.j.d 4
21.c even 2 1 98.4.c.e 2
21.g even 6 1 98.4.a.c 1
21.g even 6 1 98.4.c.e 2
21.h odd 6 1 14.4.c.b 2
21.h odd 6 1 98.4.a.b 1
24.f even 2 1 448.4.i.d 2
24.h odd 2 1 448.4.i.c 2
84.j odd 6 1 784.4.a.j 1
84.n even 6 1 112.4.i.b 2
84.n even 6 1 784.4.a.l 1
105.o odd 6 1 350.4.e.b 2
105.o odd 6 1 2450.4.a.bh 1
105.p even 6 1 2450.4.a.bf 1
105.x even 12 2 350.4.j.d 4
168.s odd 6 1 448.4.i.c 2
168.v even 6 1 448.4.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 3.b odd 2 1
14.4.c.b 2 21.h odd 6 1
98.4.a.b 1 21.h odd 6 1
98.4.a.c 1 21.g even 6 1
98.4.c.e 2 21.c even 2 1
98.4.c.e 2 21.g even 6 1
112.4.i.b 2 12.b even 2 1
112.4.i.b 2 84.n even 6 1
126.4.g.c 2 1.a even 1 1 trivial
126.4.g.c 2 7.c even 3 1 inner
350.4.e.b 2 15.d odd 2 1
350.4.e.b 2 105.o odd 6 1
350.4.j.d 4 15.e even 4 2
350.4.j.d 4 105.x even 12 2
448.4.i.c 2 24.h odd 2 1
448.4.i.c 2 168.s odd 6 1
448.4.i.d 2 24.f even 2 1
448.4.i.d 2 168.v even 6 1
784.4.a.j 1 84.j odd 6 1
784.4.a.l 1 84.n even 6 1
882.4.a.k 1 7.c even 3 1
882.4.a.p 1 7.d odd 6 1
882.4.g.d 2 7.b odd 2 1
882.4.g.d 2 7.d odd 6 1
2450.4.a.bf 1 105.p even 6 1
2450.4.a.bh 1 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 7 T_{5} + 49 \) acting on \(S_{4}^{\mathrm{new}}(126, [\chi])\).