# Properties

 Label 126.2.k.a Level $126$ Weight $2$ Character orbit 126.k Analytic conductor $1.006$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 126.k (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.00611506547$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{2} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{5} + ( 1 - 2 \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{2} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{5} + ( 1 - 2 \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{7} -\zeta_{24}^{6} q^{8} + ( 1 + 2 \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{10} -3 \zeta_{24}^{2} q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{14} -\zeta_{24}^{4} q^{16} + ( 2 \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 4 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{17} + ( -4 + \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{19} + ( \zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{20} -3 q^{22} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{23} + ( -4 - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{25} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{26} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{28} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{29} + ( -1 + 6 \zeta_{24} + 3 \zeta_{24}^{3} - \zeta_{24}^{4} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{31} -\zeta_{24}^{2} q^{32} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{34} + ( 4 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 7 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{35} + ( -3 \zeta_{24} - 4 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{37} + ( 2 \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 4 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{38} + ( 2 + \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{40} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{41} + ( 4 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{43} + ( -3 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{44} + ( -3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{46} + ( -\zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{47} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} + 5 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{50} + ( -2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{52} + ( -3 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{53} + ( 3 - 3 \zeta_{24} + 3 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{55} + ( -2 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{56} + ( -3 \zeta_{24} + 3 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{58} + ( -8 \zeta_{24} - \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{59} + ( 4 + \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{61} + ( 3 \zeta_{24} - 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{62} - q^{64} + ( 3 \zeta_{24} + 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{65} + ( 10 - 10 \zeta_{24}^{4} ) q^{67} + ( \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{68} + ( 2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 7 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{70} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{71} + ( 2 + 4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{73} + ( -4 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{74} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{76} + ( -3 \zeta_{24} - 3 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{77} + ( 3 \zeta_{24} + 7 \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{79} + ( 2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{80} + ( 8 - 2 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{82} + ( -2 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{83} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{85} + ( 3 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{86} + ( -3 + 3 \zeta_{24}^{4} ) q^{88} + ( 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + ( -6 - 2 \zeta_{24} - \zeta_{24}^{3} + 6 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{91} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{92} + ( -2 + 2 \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{94} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{95} + ( 5 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{97} + ( -4 \zeta_{24} + 5 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + 4q^{7} + O(q^{10})$$ $$8q + 4q^{4} + 4q^{7} + 12q^{10} - 4q^{16} - 24q^{19} - 24q^{22} - 16q^{25} - 4q^{28} - 12q^{31} - 16q^{37} + 12q^{40} + 32q^{43} + 20q^{49} + 12q^{58} + 24q^{61} - 8q^{64} + 40q^{67} - 12q^{70} + 24q^{73} + 28q^{79} + 48q^{82} - 12q^{88} - 24q^{91} - 24q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$-1$$ $$\zeta_{24}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i
−0.866025 + 0.500000i 0 0.500000 0.866025i −2.09077 3.62132i 0 −1.62132 2.09077i 1.00000i 0 3.62132 + 2.09077i
17.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.358719 + 0.621320i 0 2.62132 + 0.358719i 1.00000i 0 −0.621320 0.358719i
17.3 0.866025 0.500000i 0 0.500000 0.866025i −0.358719 0.621320i 0 2.62132 + 0.358719i 1.00000i 0 −0.621320 0.358719i
17.4 0.866025 0.500000i 0 0.500000 0.866025i 2.09077 + 3.62132i 0 −1.62132 2.09077i 1.00000i 0 3.62132 + 2.09077i
89.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.09077 + 3.62132i 0 −1.62132 + 2.09077i 1.00000i 0 3.62132 2.09077i
89.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.358719 0.621320i 0 2.62132 0.358719i 1.00000i 0 −0.621320 + 0.358719i
89.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.358719 + 0.621320i 0 2.62132 0.358719i 1.00000i 0 −0.621320 + 0.358719i
89.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.09077 3.62132i 0 −1.62132 + 2.09077i 1.00000i 0 3.62132 2.09077i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.k.a 8
3.b odd 2 1 inner 126.2.k.a 8
4.b odd 2 1 1008.2.bt.c 8
5.b even 2 1 3150.2.bf.a 8
5.c odd 4 1 3150.2.bp.b 8
5.c odd 4 1 3150.2.bp.e 8
7.b odd 2 1 882.2.k.a 8
7.c even 3 1 882.2.d.a 8
7.c even 3 1 882.2.k.a 8
7.d odd 6 1 inner 126.2.k.a 8
7.d odd 6 1 882.2.d.a 8
9.c even 3 1 1134.2.l.f 8
9.c even 3 1 1134.2.t.e 8
9.d odd 6 1 1134.2.l.f 8
9.d odd 6 1 1134.2.t.e 8
12.b even 2 1 1008.2.bt.c 8
15.d odd 2 1 3150.2.bf.a 8
15.e even 4 1 3150.2.bp.b 8
15.e even 4 1 3150.2.bp.e 8
21.c even 2 1 882.2.k.a 8
21.g even 6 1 inner 126.2.k.a 8
21.g even 6 1 882.2.d.a 8
21.h odd 6 1 882.2.d.a 8
21.h odd 6 1 882.2.k.a 8
28.f even 6 1 1008.2.bt.c 8
28.f even 6 1 7056.2.k.f 8
28.g odd 6 1 7056.2.k.f 8
35.i odd 6 1 3150.2.bf.a 8
35.k even 12 1 3150.2.bp.b 8
35.k even 12 1 3150.2.bp.e 8
63.i even 6 1 1134.2.t.e 8
63.k odd 6 1 1134.2.l.f 8
63.s even 6 1 1134.2.l.f 8
63.t odd 6 1 1134.2.t.e 8
84.j odd 6 1 1008.2.bt.c 8
84.j odd 6 1 7056.2.k.f 8
84.n even 6 1 7056.2.k.f 8
105.p even 6 1 3150.2.bf.a 8
105.w odd 12 1 3150.2.bp.b 8
105.w odd 12 1 3150.2.bp.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.k.a 8 1.a even 1 1 trivial
126.2.k.a 8 3.b odd 2 1 inner
126.2.k.a 8 7.d odd 6 1 inner
126.2.k.a 8 21.g even 6 1 inner
882.2.d.a 8 7.c even 3 1
882.2.d.a 8 7.d odd 6 1
882.2.d.a 8 21.g even 6 1
882.2.d.a 8 21.h odd 6 1
882.2.k.a 8 7.b odd 2 1
882.2.k.a 8 7.c even 3 1
882.2.k.a 8 21.c even 2 1
882.2.k.a 8 21.h odd 6 1
1008.2.bt.c 8 4.b odd 2 1
1008.2.bt.c 8 12.b even 2 1
1008.2.bt.c 8 28.f even 6 1
1008.2.bt.c 8 84.j odd 6 1
1134.2.l.f 8 9.c even 3 1
1134.2.l.f 8 9.d odd 6 1
1134.2.l.f 8 63.k odd 6 1
1134.2.l.f 8 63.s even 6 1
1134.2.t.e 8 9.c even 3 1
1134.2.t.e 8 9.d odd 6 1
1134.2.t.e 8 63.i even 6 1
1134.2.t.e 8 63.t odd 6 1
3150.2.bf.a 8 5.b even 2 1
3150.2.bf.a 8 15.d odd 2 1
3150.2.bf.a 8 35.i odd 6 1
3150.2.bf.a 8 105.p even 6 1
3150.2.bp.b 8 5.c odd 4 1
3150.2.bp.b 8 15.e even 4 1
3150.2.bp.b 8 35.k even 12 1
3150.2.bp.b 8 105.w odd 12 1
3150.2.bp.e 8 5.c odd 4 1
3150.2.bp.e 8 15.e even 4 1
3150.2.bp.e 8 35.k even 12 1
3150.2.bp.e 8 105.w odd 12 1
7056.2.k.f 8 28.f even 6 1
7056.2.k.f 8 28.g odd 6 1
7056.2.k.f 8 84.j odd 6 1
7056.2.k.f 8 84.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$81 + 162 T^{2} + 315 T^{4} + 18 T^{6} + T^{8}$$
$7$ $$( 49 - 14 T - 3 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$( 81 - 9 T^{2} + T^{4} )^{2}$$
$13$ $$( 6 + T^{2} )^{4}$$
$17$ $$1296 + 1296 T^{2} + 1260 T^{4} + 36 T^{6} + T^{8}$$
$19$ $$( 36 + 72 T + 54 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$23$ $$( 324 - 18 T^{2} + T^{4} )^{2}$$
$29$ $$( 81 + 54 T^{2} + T^{4} )^{2}$$
$31$ $$( 2601 - 306 T - 39 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$37$ $$( 4 - 16 T + 66 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$41$ $$( 576 - 144 T^{2} + T^{4} )^{2}$$
$43$ $$( -2 - 8 T + T^{2} )^{4}$$
$47$ $$1296 + 1296 T^{2} + 1260 T^{4} + 36 T^{6} + T^{8}$$
$53$ $$6561 - 4374 T^{2} + 2835 T^{4} - 54 T^{6} + T^{8}$$
$59$ $$74805201 + 1712502 T^{2} + 30555 T^{4} + 198 T^{6} + T^{8}$$
$61$ $$( 36 - 72 T + 54 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$67$ $$( 100 - 10 T + T^{2} )^{4}$$
$71$ $$( 324 + 108 T^{2} + T^{4} )^{2}$$
$73$ $$( 144 + 144 T + 36 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$79$ $$( 961 - 434 T + 165 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$83$ $$( 441 - 54 T^{2} + T^{4} )^{2}$$
$89$ $$( 11664 + 108 T^{2} + T^{4} )^{2}$$
$97$ $$( 2601 + 198 T^{2} + T^{4} )^{2}$$