Properties

Label 126.4.g.d
Level $126$
Weight $4$
Character orbit 126.g
Analytic conductor $7.434$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + (9 \zeta_{6} - 9) q^{5} + ( - 14 \zeta_{6} - 7) q^{7} - 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + (9 \zeta_{6} - 9) q^{5} + ( - 14 \zeta_{6} - 7) q^{7} - 8 q^{8} + 18 \zeta_{6} q^{10} - 57 \zeta_{6} q^{11} - 70 q^{13} + (14 \zeta_{6} - 42) q^{14} + (16 \zeta_{6} - 16) q^{16} + 51 \zeta_{6} q^{17} + (5 \zeta_{6} - 5) q^{19} + 36 q^{20} - 114 q^{22} + ( - 69 \zeta_{6} + 69) q^{23} + 44 \zeta_{6} q^{25} + (140 \zeta_{6} - 140) q^{26} + (84 \zeta_{6} - 56) q^{28} - 114 q^{29} - 23 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + 102 q^{34} + ( - 63 \zeta_{6} + 189) q^{35} + ( - 253 \zeta_{6} + 253) q^{37} + 10 \zeta_{6} q^{38} + ( - 72 \zeta_{6} + 72) q^{40} + 42 q^{41} - 124 q^{43} + (228 \zeta_{6} - 228) q^{44} - 138 \zeta_{6} q^{46} + ( - 201 \zeta_{6} + 201) q^{47} + (392 \zeta_{6} - 147) q^{49} + 88 q^{50} + 280 \zeta_{6} q^{52} - 393 \zeta_{6} q^{53} + 513 q^{55} + (112 \zeta_{6} + 56) q^{56} + (228 \zeta_{6} - 228) q^{58} + 219 \zeta_{6} q^{59} + ( - 709 \zeta_{6} + 709) q^{61} - 46 q^{62} + 64 q^{64} + ( - 630 \zeta_{6} + 630) q^{65} - 419 \zeta_{6} q^{67} + ( - 204 \zeta_{6} + 204) q^{68} + ( - 378 \zeta_{6} + 252) q^{70} + 96 q^{71} + 313 \zeta_{6} q^{73} - 506 \zeta_{6} q^{74} + 20 q^{76} + (1197 \zeta_{6} - 798) q^{77} + (461 \zeta_{6} - 461) q^{79} - 144 \zeta_{6} q^{80} + ( - 84 \zeta_{6} + 84) q^{82} + 588 q^{83} - 459 q^{85} + (248 \zeta_{6} - 248) q^{86} + 456 \zeta_{6} q^{88} + (1017 \zeta_{6} - 1017) q^{89} + (980 \zeta_{6} + 490) q^{91} - 276 q^{92} - 402 \zeta_{6} q^{94} - 45 \zeta_{6} q^{95} - 1834 q^{97} + (294 \zeta_{6} + 490) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} - 9 q^{5} - 28 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} - 9 q^{5} - 28 q^{7} - 16 q^{8} + 18 q^{10} - 57 q^{11} - 140 q^{13} - 70 q^{14} - 16 q^{16} + 51 q^{17} - 5 q^{19} + 72 q^{20} - 228 q^{22} + 69 q^{23} + 44 q^{25} - 140 q^{26} - 28 q^{28} - 228 q^{29} - 23 q^{31} + 32 q^{32} + 204 q^{34} + 315 q^{35} + 253 q^{37} + 10 q^{38} + 72 q^{40} + 84 q^{41} - 248 q^{43} - 228 q^{44} - 138 q^{46} + 201 q^{47} + 98 q^{49} + 176 q^{50} + 280 q^{52} - 393 q^{53} + 1026 q^{55} + 224 q^{56} - 228 q^{58} + 219 q^{59} + 709 q^{61} - 92 q^{62} + 128 q^{64} + 630 q^{65} - 419 q^{67} + 204 q^{68} + 126 q^{70} + 192 q^{71} + 313 q^{73} - 506 q^{74} + 40 q^{76} - 399 q^{77} - 461 q^{79} - 144 q^{80} + 84 q^{82} + 1176 q^{83} - 918 q^{85} - 248 q^{86} + 456 q^{88} - 1017 q^{89} + 1960 q^{91} - 552 q^{92} - 402 q^{94} - 45 q^{95} - 3668 q^{97} + 1274 q^{98}+O(q^{100}) \) 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 −4.50000 + 7.79423i 0 −14.0000 12.1244i −8.00000 0 9.00000 + 15.5885i
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −4.50000 7.79423i 0 −14.0000 + 12.1244i −8.00000 0 9.00000 15.5885i
\(n\): 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.d 2
3.b odd 2 1 14.4.c.a 2
7.b odd 2 1 882.4.g.u 2
7.c even 3 1 inner 126.4.g.d 2
7.c even 3 1 882.4.a.f 1
7.d odd 6 1 882.4.a.c 1
7.d odd 6 1 882.4.g.u 2
12.b even 2 1 112.4.i.a 2
15.d odd 2 1 350.4.e.e 2
15.e even 4 2 350.4.j.b 4
21.c even 2 1 98.4.c.a 2
21.g even 6 1 98.4.a.f 1
21.g even 6 1 98.4.c.a 2
21.h odd 6 1 14.4.c.a 2
21.h odd 6 1 98.4.a.d 1
24.f even 2 1 448.4.i.e 2
24.h odd 2 1 448.4.i.b 2
84.j odd 6 1 784.4.a.c 1
84.n even 6 1 112.4.i.a 2
84.n even 6 1 784.4.a.p 1
105.o odd 6 1 350.4.e.e 2
105.o odd 6 1 2450.4.a.q 1
105.p even 6 1 2450.4.a.d 1
105.x even 12 2 350.4.j.b 4
168.s odd 6 1 448.4.i.b 2
168.v even 6 1 448.4.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 3.b odd 2 1
14.4.c.a 2 21.h odd 6 1
98.4.a.d 1 21.h odd 6 1
98.4.a.f 1 21.g even 6 1
98.4.c.a 2 21.c even 2 1
98.4.c.a 2 21.g even 6 1
112.4.i.a 2 12.b even 2 1
112.4.i.a 2 84.n even 6 1
126.4.g.d 2 1.a even 1 1 trivial
126.4.g.d 2 7.c even 3 1 inner
350.4.e.e 2 15.d odd 2 1
350.4.e.e 2 105.o odd 6 1
350.4.j.b 4 15.e even 4 2
350.4.j.b 4 105.x even 12 2
448.4.i.b 2 24.h odd 2 1
448.4.i.b 2 168.s odd 6 1
448.4.i.e 2 24.f even 2 1
448.4.i.e 2 168.v even 6 1
784.4.a.c 1 84.j odd 6 1
784.4.a.p 1 84.n even 6 1
882.4.a.c 1 7.d odd 6 1
882.4.a.f 1 7.c even 3 1
882.4.g.u 2 7.b odd 2 1
882.4.g.u 2 7.d odd 6 1
2450.4.a.d 1 105.p even 6 1
2450.4.a.q 1 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 9T_{5} + 81 \) 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
$5$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 57T + 3249 \) Copy content Toggle raw display
$13$ \( (T + 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T + 114)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 23T + 529 \) Copy content Toggle raw display
$37$ \( T^{2} - 253T + 64009 \) Copy content Toggle raw display
$41$ \( (T - 42)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 201T + 40401 \) Copy content Toggle raw display
$53$ \( T^{2} + 393T + 154449 \) Copy content Toggle raw display
$59$ \( T^{2} - 219T + 47961 \) Copy content Toggle raw display
$61$ \( T^{2} - 709T + 502681 \) Copy content Toggle raw display
$67$ \( T^{2} + 419T + 175561 \) Copy content Toggle raw display
$71$ \( (T - 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 313T + 97969 \) Copy content Toggle raw display
$79$ \( T^{2} + 461T + 212521 \) Copy content Toggle raw display
$83$ \( (T - 588)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1017 T + 1034289 \) Copy content Toggle raw display
$97$ \( (T + 1834)^{2} \) Copy content Toggle raw display
show more
show less