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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \( T_{5}^{2} - 7T_{5} + 49 \) acting on \(S_{4}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} + 20T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 35T + 1225 \) Copy content Toggle raw display
$13$ \( (T - 66)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 59T + 3481 \) Copy content Toggle raw display
$19$ \( T^{2} + 137T + 18769 \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$29$ \( (T + 106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 75T + 5625 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( (T - 498)^{2} \) Copy content Toggle raw display
$43$ \( (T - 260)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 171T + 29241 \) Copy content Toggle raw display
$53$ \( T^{2} + 417T + 173889 \) Copy content Toggle raw display
$59$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$61$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$67$ \( T^{2} + 439T + 192721 \) Copy content Toggle raw display
$71$ \( (T - 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 295T + 87025 \) Copy content Toggle raw display
$79$ \( T^{2} - 495T + 245025 \) Copy content Toggle raw display
$83$ \( (T + 932)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 873T + 762129 \) Copy content Toggle raw display
$97$ \( (T + 290)^{2} \) Copy content Toggle raw display
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