# Properties

 Label 126.2.g.d 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} - q^{8} -3 \zeta_{6} q^{10} -3 \zeta_{6} q^{11} + 2 q^{13} + ( 1 + 2 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + 6 \zeta_{6} q^{17} + ( -2 + 2 \zeta_{6} ) q^{19} -3 q^{20} -3 q^{22} + ( -6 + 6 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} + ( 2 - 2 \zeta_{6} ) q^{26} + ( 3 - \zeta_{6} ) q^{28} -9 q^{29} + 7 \zeta_{6} q^{31} + \zeta_{6} q^{32} + 6 q^{34} + ( 3 + 6 \zeta_{6} ) q^{35} + ( 10 - 10 \zeta_{6} ) q^{37} + 2 \zeta_{6} q^{38} + ( -3 + 3 \zeta_{6} ) q^{40} -4 q^{43} + ( -3 + 3 \zeta_{6} ) q^{44} + 6 \zeta_{6} q^{46} + ( 12 - 12 \zeta_{6} ) q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} -4 q^{50} -2 \zeta_{6} q^{52} -3 \zeta_{6} q^{53} -9 q^{55} + ( 2 - 3 \zeta_{6} ) q^{56} + ( -9 + 9 \zeta_{6} ) q^{58} -3 \zeta_{6} q^{59} + ( 4 - 4 \zeta_{6} ) q^{61} + 7 q^{62} + q^{64} + ( 6 - 6 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} + ( 6 - 6 \zeta_{6} ) q^{68} + ( 9 - 3 \zeta_{6} ) q^{70} -2 \zeta_{6} q^{73} -10 \zeta_{6} q^{74} + 2 q^{76} + ( 9 - 3 \zeta_{6} ) q^{77} + ( -5 + 5 \zeta_{6} ) q^{79} + 3 \zeta_{6} q^{80} -9 q^{83} + 18 q^{85} + ( -4 + 4 \zeta_{6} ) q^{86} + 3 \zeta_{6} q^{88} + ( -6 + 6 \zeta_{6} ) q^{89} + ( -4 + 6 \zeta_{6} ) q^{91} + 6 q^{92} -12 \zeta_{6} q^{94} + 6 \zeta_{6} q^{95} -13 q^{97} + ( -8 + 5 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} + 3q^{5} - q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} + 3q^{5} - q^{7} - 2q^{8} - 3q^{10} - 3q^{11} + 4q^{13} + 4q^{14} - q^{16} + 6q^{17} - 2q^{19} - 6q^{20} - 6q^{22} - 6q^{23} - 4q^{25} + 2q^{26} + 5q^{28} - 18q^{29} + 7q^{31} + q^{32} + 12q^{34} + 12q^{35} + 10q^{37} + 2q^{38} - 3q^{40} - 8q^{43} - 3q^{44} + 6q^{46} + 12q^{47} - 13q^{49} - 8q^{50} - 2q^{52} - 3q^{53} - 18q^{55} + q^{56} - 9q^{58} - 3q^{59} + 4q^{61} + 14q^{62} + 2q^{64} + 6q^{65} - 2q^{67} + 6q^{68} + 15q^{70} - 2q^{73} - 10q^{74} + 4q^{76} + 15q^{77} - 5q^{79} + 3q^{80} - 18q^{83} + 36q^{85} - 4q^{86} + 3q^{88} - 6q^{89} - 2q^{91} + 12q^{92} - 12q^{94} + 6q^{95} - 26q^{97} - 11q^{98} + O(q^{100})$$

## 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 1.50000 2.59808i 0 −0.500000 + 2.59808i −1.00000 0 −1.50000 2.59808i
109.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.50000 + 2.59808i 0 −0.500000 2.59808i −1.00000 0 −1.50000 + 2.59808i
 $$n$$: 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.d yes 2
3.b odd 2 1 126.2.g.a 2
4.b odd 2 1 1008.2.s.o 2
7.b odd 2 1 882.2.g.g 2
7.c even 3 1 inner 126.2.g.d yes 2
7.c even 3 1 882.2.a.a 1
7.d odd 6 1 882.2.a.e 1
7.d odd 6 1 882.2.g.g 2
9.c even 3 1 1134.2.e.g 2
9.c even 3 1 1134.2.h.j 2
9.d odd 6 1 1134.2.e.k 2
9.d odd 6 1 1134.2.h.f 2
12.b even 2 1 1008.2.s.b 2
21.c even 2 1 882.2.g.e 2
21.g even 6 1 882.2.a.h 1
21.g even 6 1 882.2.g.e 2
21.h odd 6 1 126.2.g.a 2
21.h odd 6 1 882.2.a.j 1
28.f even 6 1 7056.2.a.bx 1
28.g odd 6 1 1008.2.s.o 2
28.g odd 6 1 7056.2.a.e 1
63.g even 3 1 1134.2.e.g 2
63.h even 3 1 1134.2.h.j 2
63.j odd 6 1 1134.2.h.f 2
63.n odd 6 1 1134.2.e.k 2
84.j odd 6 1 7056.2.a.h 1
84.n even 6 1 1008.2.s.b 2
84.n even 6 1 7056.2.a.by 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 3.b odd 2 1
126.2.g.a 2 21.h odd 6 1
126.2.g.d yes 2 1.a even 1 1 trivial
126.2.g.d yes 2 7.c even 3 1 inner
882.2.a.a 1 7.c even 3 1
882.2.a.e 1 7.d odd 6 1
882.2.a.h 1 21.g even 6 1
882.2.a.j 1 21.h odd 6 1
882.2.g.e 2 21.c even 2 1
882.2.g.e 2 21.g even 6 1
882.2.g.g 2 7.b odd 2 1
882.2.g.g 2 7.d odd 6 1
1008.2.s.b 2 12.b even 2 1
1008.2.s.b 2 84.n even 6 1
1008.2.s.o 2 4.b odd 2 1
1008.2.s.o 2 28.g odd 6 1
1134.2.e.g 2 9.c even 3 1
1134.2.e.g 2 63.g even 3 1
1134.2.e.k 2 9.d odd 6 1
1134.2.e.k 2 63.n odd 6 1
1134.2.h.f 2 9.d odd 6 1
1134.2.h.f 2 63.j odd 6 1
1134.2.h.j 2 9.c even 3 1
1134.2.h.j 2 63.h even 3 1
7056.2.a.e 1 28.g odd 6 1
7056.2.a.h 1 84.j odd 6 1
7056.2.a.bx 1 28.f even 6 1
7056.2.a.by 1 84.n even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{11}^{2} + 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$49 - 7 T + T^{2}$$
$37$ $$100 - 10 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$144 - 12 T + T^{2}$$
$53$ $$9 + 3 T + T^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$16 - 4 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$25 + 5 T + T^{2}$$
$83$ $$( 9 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( 13 + T )^{2}$$