# Properties

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

# Related objects

## 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( -6 + 6 \zeta_{6} ) q^{5} + ( 7 - 21 \zeta_{6} ) q^{7} + 8 q^{8} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( -6 + 6 \zeta_{6} ) q^{5} + ( 7 - 21 \zeta_{6} ) q^{7} + 8 q^{8} -12 \zeta_{6} q^{10} -30 \zeta_{6} q^{11} + 53 q^{13} + ( 28 + 14 \zeta_{6} ) q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} -84 \zeta_{6} q^{17} + ( 97 - 97 \zeta_{6} ) q^{19} + 24 q^{20} + 60 q^{22} + ( 84 - 84 \zeta_{6} ) q^{23} + 89 \zeta_{6} q^{25} + ( -106 + 106 \zeta_{6} ) q^{26} + ( -84 + 56 \zeta_{6} ) q^{28} + 180 q^{29} -179 \zeta_{6} q^{31} -32 \zeta_{6} q^{32} + 168 q^{34} + ( 84 + 42 \zeta_{6} ) q^{35} + ( 145 - 145 \zeta_{6} ) q^{37} + 194 \zeta_{6} q^{38} + ( -48 + 48 \zeta_{6} ) q^{40} -126 q^{41} -325 q^{43} + ( -120 + 120 \zeta_{6} ) q^{44} + 168 \zeta_{6} q^{46} + ( -366 + 366 \zeta_{6} ) q^{47} + ( -392 + 147 \zeta_{6} ) q^{49} -178 q^{50} -212 \zeta_{6} q^{52} -768 \zeta_{6} q^{53} + 180 q^{55} + ( 56 - 168 \zeta_{6} ) q^{56} + ( -360 + 360 \zeta_{6} ) q^{58} -264 \zeta_{6} q^{59} + ( -818 + 818 \zeta_{6} ) q^{61} + 358 q^{62} + 64 q^{64} + ( -318 + 318 \zeta_{6} ) q^{65} + 523 \zeta_{6} q^{67} + ( -336 + 336 \zeta_{6} ) q^{68} + ( -252 + 168 \zeta_{6} ) q^{70} + 342 q^{71} + 43 \zeta_{6} q^{73} + 290 \zeta_{6} q^{74} -388 q^{76} + ( -630 + 420 \zeta_{6} ) q^{77} + ( 1171 - 1171 \zeta_{6} ) q^{79} -96 \zeta_{6} q^{80} + ( 252 - 252 \zeta_{6} ) q^{82} + 810 q^{83} + 504 q^{85} + ( 650 - 650 \zeta_{6} ) q^{86} -240 \zeta_{6} q^{88} + ( -600 + 600 \zeta_{6} ) q^{89} + ( 371 - 1113 \zeta_{6} ) q^{91} -336 q^{92} -732 \zeta_{6} q^{94} + 582 \zeta_{6} q^{95} + 386 q^{97} + ( 490 - 784 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 4q^{4} - 6q^{5} - 7q^{7} + 16q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 4q^{4} - 6q^{5} - 7q^{7} + 16q^{8} - 12q^{10} - 30q^{11} + 106q^{13} + 70q^{14} - 16q^{16} - 84q^{17} + 97q^{19} + 48q^{20} + 120q^{22} + 84q^{23} + 89q^{25} - 106q^{26} - 112q^{28} + 360q^{29} - 179q^{31} - 32q^{32} + 336q^{34} + 210q^{35} + 145q^{37} + 194q^{38} - 48q^{40} - 252q^{41} - 650q^{43} - 120q^{44} + 168q^{46} - 366q^{47} - 637q^{49} - 356q^{50} - 212q^{52} - 768q^{53} + 360q^{55} - 56q^{56} - 360q^{58} - 264q^{59} - 818q^{61} + 716q^{62} + 128q^{64} - 318q^{65} + 523q^{67} - 336q^{68} - 336q^{70} + 684q^{71} + 43q^{73} + 290q^{74} - 776q^{76} - 840q^{77} + 1171q^{79} - 96q^{80} + 252q^{82} + 1620q^{83} + 1008q^{85} + 650q^{86} - 240q^{88} - 600q^{89} - 371q^{91} - 672q^{92} - 732q^{94} + 582q^{95} + 772q^{97} + 196q^{98} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 + 1.73205i 0 −2.00000 3.46410i −3.00000 + 5.19615i 0 −3.50000 18.1865i 8.00000 0 −6.00000 10.3923i
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −3.00000 5.19615i 0 −3.50000 + 18.1865i 8.00000 0 −6.00000 + 10.3923i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b 2
3.b odd 2 1 42.4.e.a 2
7.b odd 2 1 882.4.g.g 2
7.c even 3 1 inner 126.4.g.b 2
7.c even 3 1 882.4.a.o 1
7.d odd 6 1 882.4.a.l 1
7.d odd 6 1 882.4.g.g 2
12.b even 2 1 336.4.q.f 2
21.c even 2 1 294.4.e.i 2
21.g even 6 1 294.4.a.c 1
21.g even 6 1 294.4.e.i 2
21.h odd 6 1 42.4.e.a 2
21.h odd 6 1 294.4.a.d 1
84.j odd 6 1 2352.4.a.bf 1
84.n even 6 1 336.4.q.f 2
84.n even 6 1 2352.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 3.b odd 2 1
42.4.e.a 2 21.h odd 6 1
126.4.g.b 2 1.a even 1 1 trivial
126.4.g.b 2 7.c even 3 1 inner
294.4.a.c 1 21.g even 6 1
294.4.a.d 1 21.h odd 6 1
294.4.e.i 2 21.c even 2 1
294.4.e.i 2 21.g even 6 1
336.4.q.f 2 12.b even 2 1
336.4.q.f 2 84.n even 6 1
882.4.a.l 1 7.d odd 6 1
882.4.a.o 1 7.c even 3 1
882.4.g.g 2 7.b odd 2 1
882.4.g.g 2 7.d odd 6 1
2352.4.a.f 1 84.n even 6 1
2352.4.a.bf 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 6 T_{5} + 36$$ acting on $$S_{4}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$36 + 6 T + T^{2}$$
$7$ $$343 + 7 T + T^{2}$$
$11$ $$900 + 30 T + T^{2}$$
$13$ $$( -53 + T )^{2}$$
$17$ $$7056 + 84 T + T^{2}$$
$19$ $$9409 - 97 T + T^{2}$$
$23$ $$7056 - 84 T + T^{2}$$
$29$ $$( -180 + T )^{2}$$
$31$ $$32041 + 179 T + T^{2}$$
$37$ $$21025 - 145 T + T^{2}$$
$41$ $$( 126 + T )^{2}$$
$43$ $$( 325 + T )^{2}$$
$47$ $$133956 + 366 T + T^{2}$$
$53$ $$589824 + 768 T + T^{2}$$
$59$ $$69696 + 264 T + T^{2}$$
$61$ $$669124 + 818 T + T^{2}$$
$67$ $$273529 - 523 T + T^{2}$$
$71$ $$( -342 + T )^{2}$$
$73$ $$1849 - 43 T + T^{2}$$
$79$ $$1371241 - 1171 T + T^{2}$$
$83$ $$( -810 + T )^{2}$$
$89$ $$360000 + 600 T + T^{2}$$
$97$ $$( -386 + T )^{2}$$