# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ 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: 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( 2 - 3 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 1 - \zeta_{6} ) q^{5} + ( 2 - 3 \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{10} + 5 \zeta_{6} q^{11} + ( -1 - 2 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} -4 \zeta_{6} q^{17} + ( -8 + 8 \zeta_{6} ) q^{19} - q^{20} + 5 q^{22} + ( -4 + 4 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + ( -3 + \zeta_{6} ) q^{28} + 5 q^{29} -3 \zeta_{6} q^{31} + \zeta_{6} q^{32} -4 q^{34} + ( -1 - 2 \zeta_{6} ) q^{35} + ( 4 - 4 \zeta_{6} ) q^{37} + 8 \zeta_{6} q^{38} + ( -1 + \zeta_{6} ) q^{40} + 2 q^{43} + ( 5 - 5 \zeta_{6} ) q^{44} + 4 \zeta_{6} q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + 4 q^{50} -9 \zeta_{6} q^{53} + 5 q^{55} + ( -2 + 3 \zeta_{6} ) q^{56} + ( 5 - 5 \zeta_{6} ) q^{58} -11 \zeta_{6} q^{59} + ( 6 - 6 \zeta_{6} ) q^{61} -3 q^{62} + q^{64} + 2 \zeta_{6} q^{67} + ( -4 + 4 \zeta_{6} ) q^{68} + ( -3 + \zeta_{6} ) q^{70} -2 q^{71} -10 \zeta_{6} q^{73} -4 \zeta_{6} q^{74} + 8 q^{76} + ( 15 - 5 \zeta_{6} ) q^{77} + ( -3 + 3 \zeta_{6} ) q^{79} + \zeta_{6} q^{80} + 7 q^{83} -4 q^{85} + ( 2 - 2 \zeta_{6} ) q^{86} -5 \zeta_{6} q^{88} + ( -6 + 6 \zeta_{6} ) q^{89} + 4 q^{92} + 6 \zeta_{6} q^{94} + 8 \zeta_{6} q^{95} + 7 q^{97} + ( -8 + 5 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} + q^{5} + q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} + q^{5} + q^{7} - 2q^{8} - q^{10} + 5q^{11} - 4q^{14} - q^{16} - 4q^{17} - 8q^{19} - 2q^{20} + 10q^{22} - 4q^{23} + 4q^{25} - 5q^{28} + 10q^{29} - 3q^{31} + q^{32} - 8q^{34} - 4q^{35} + 4q^{37} + 8q^{38} - q^{40} + 4q^{43} + 5q^{44} + 4q^{46} - 6q^{47} - 13q^{49} + 8q^{50} - 9q^{53} + 10q^{55} - q^{56} + 5q^{58} - 11q^{59} + 6q^{61} - 6q^{62} + 2q^{64} + 2q^{67} - 4q^{68} - 5q^{70} - 4q^{71} - 10q^{73} - 4q^{74} + 16q^{76} + 25q^{77} - 3q^{79} + q^{80} + 14q^{83} - 8q^{85} + 2q^{86} - 5q^{88} - 6q^{89} + 8q^{92} + 6q^{94} + 8q^{95} + 14q^{97} - 11q^{98} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 0.500000 2.59808i −1.00000 0 −0.500000 0.866025i
109.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.500000 + 2.59808i −1.00000 0 −0.500000 + 0.866025i
 $$n$$: 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 126.2.g.c 2
3.b odd 2 1 42.2.e.a 2
4.b odd 2 1 1008.2.s.k 2
7.b odd 2 1 882.2.g.i 2
7.c even 3 1 inner 126.2.g.c 2
7.c even 3 1 882.2.a.c 1
7.d odd 6 1 882.2.a.d 1
7.d odd 6 1 882.2.g.i 2
9.c even 3 1 1134.2.e.e 2
9.c even 3 1 1134.2.h.l 2
9.d odd 6 1 1134.2.e.l 2
9.d odd 6 1 1134.2.h.e 2
12.b even 2 1 336.2.q.b 2
15.d odd 2 1 1050.2.i.l 2
15.e even 4 2 1050.2.o.a 4
21.c even 2 1 294.2.e.b 2
21.g even 6 1 294.2.a.f 1
21.g even 6 1 294.2.e.b 2
21.h odd 6 1 42.2.e.a 2
21.h odd 6 1 294.2.a.e 1
24.f even 2 1 1344.2.q.s 2
24.h odd 2 1 1344.2.q.g 2
28.f even 6 1 7056.2.a.bl 1
28.g odd 6 1 1008.2.s.k 2
28.g odd 6 1 7056.2.a.w 1
63.g even 3 1 1134.2.e.e 2
63.h even 3 1 1134.2.h.l 2
63.j odd 6 1 1134.2.h.e 2
63.n odd 6 1 1134.2.e.l 2
84.h odd 2 1 2352.2.q.u 2
84.j odd 6 1 2352.2.a.f 1
84.j odd 6 1 2352.2.q.u 2
84.n even 6 1 336.2.q.b 2
84.n even 6 1 2352.2.a.t 1
105.o odd 6 1 1050.2.i.l 2
105.o odd 6 1 7350.2.a.bl 1
105.p even 6 1 7350.2.a.q 1
105.x even 12 2 1050.2.o.a 4
168.s odd 6 1 1344.2.q.g 2
168.s odd 6 1 9408.2.a.ce 1
168.v even 6 1 1344.2.q.s 2
168.v even 6 1 9408.2.a.q 1
168.ba even 6 1 9408.2.a.z 1
168.be odd 6 1 9408.2.a.cr 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 3.b odd 2 1
42.2.e.a 2 21.h odd 6 1
126.2.g.c 2 1.a even 1 1 trivial
126.2.g.c 2 7.c even 3 1 inner
294.2.a.e 1 21.h odd 6 1
294.2.a.f 1 21.g even 6 1
294.2.e.b 2 21.c even 2 1
294.2.e.b 2 21.g even 6 1
336.2.q.b 2 12.b even 2 1
336.2.q.b 2 84.n even 6 1
882.2.a.c 1 7.c even 3 1
882.2.a.d 1 7.d odd 6 1
882.2.g.i 2 7.b odd 2 1
882.2.g.i 2 7.d odd 6 1
1008.2.s.k 2 4.b odd 2 1
1008.2.s.k 2 28.g odd 6 1
1050.2.i.l 2 15.d odd 2 1
1050.2.i.l 2 105.o odd 6 1
1050.2.o.a 4 15.e even 4 2
1050.2.o.a 4 105.x even 12 2
1134.2.e.e 2 9.c even 3 1
1134.2.e.e 2 63.g even 3 1
1134.2.e.l 2 9.d odd 6 1
1134.2.e.l 2 63.n odd 6 1
1134.2.h.e 2 9.d odd 6 1
1134.2.h.e 2 63.j odd 6 1
1134.2.h.l 2 9.c even 3 1
1134.2.h.l 2 63.h even 3 1
1344.2.q.g 2 24.h odd 2 1
1344.2.q.g 2 168.s odd 6 1
1344.2.q.s 2 24.f even 2 1
1344.2.q.s 2 168.v even 6 1
2352.2.a.f 1 84.j odd 6 1
2352.2.a.t 1 84.n even 6 1
2352.2.q.u 2 84.h odd 2 1
2352.2.q.u 2 84.j odd 6 1
7056.2.a.w 1 28.g odd 6 1
7056.2.a.bl 1 28.f even 6 1
7350.2.a.q 1 105.p even 6 1
7350.2.a.bl 1 105.o odd 6 1
9408.2.a.q 1 168.v even 6 1
9408.2.a.z 1 168.ba even 6 1
9408.2.a.ce 1 168.s odd 6 1
9408.2.a.cr 1 168.be odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - T_{5} + 1$$ $$T_{11}^{2} - 5 T_{11} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$16 + 4 T + T^{2}$$
$19$ $$64 + 8 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$9 + 3 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$121 + 11 T + T^{2}$$
$61$ $$36 - 6 T + T^{2}$$
$67$ $$4 - 2 T + T^{2}$$
$71$ $$( 2 + T )^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$9 + 3 T + T^{2}$$
$83$ $$( -7 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( -7 + T )^{2}$$