# Properties

 Label 126.2.f.b Level 126 Weight 2 Character orbit 126.f 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.f (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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 2 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + ( -2 + 2 \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{6} + \zeta_{6} q^{7} - q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 2 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + ( -2 + 2 \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{6} + \zeta_{6} q^{7} - q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} -2 q^{10} -\zeta_{6} q^{11} + ( -1 + 2 \zeta_{6} ) q^{12} + ( 6 - 6 \zeta_{6} ) q^{13} + ( -1 + \zeta_{6} ) q^{14} + ( -2 + 4 \zeta_{6} ) q^{15} -\zeta_{6} q^{16} -5 q^{17} + 3 q^{18} -7 q^{19} -2 \zeta_{6} q^{20} + ( 1 + \zeta_{6} ) q^{21} + ( 1 - \zeta_{6} ) q^{22} + ( -4 + 4 \zeta_{6} ) q^{23} + ( -2 + \zeta_{6} ) q^{24} + \zeta_{6} q^{25} + 6 q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} - q^{28} + 4 \zeta_{6} q^{29} + ( -4 + 2 \zeta_{6} ) q^{30} + ( 6 - 6 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + ( -1 - \zeta_{6} ) q^{33} -5 \zeta_{6} q^{34} -2 q^{35} + 3 \zeta_{6} q^{36} + 2 q^{37} -7 \zeta_{6} q^{38} + ( 6 - 12 \zeta_{6} ) q^{39} + ( 2 - 2 \zeta_{6} ) q^{40} + ( -3 + 3 \zeta_{6} ) q^{41} + ( -1 + 2 \zeta_{6} ) q^{42} + \zeta_{6} q^{43} + q^{44} + 6 \zeta_{6} q^{45} -4 q^{46} + ( -1 - \zeta_{6} ) q^{48} + ( -1 + \zeta_{6} ) q^{49} + ( -1 + \zeta_{6} ) q^{50} + ( -10 + 5 \zeta_{6} ) q^{51} + 6 \zeta_{6} q^{52} + 12 q^{53} + ( 6 - 3 \zeta_{6} ) q^{54} + 2 q^{55} -\zeta_{6} q^{56} + ( -14 + 7 \zeta_{6} ) q^{57} + ( -4 + 4 \zeta_{6} ) q^{58} + ( 7 - 7 \zeta_{6} ) q^{59} + ( -2 - 2 \zeta_{6} ) q^{60} + 12 \zeta_{6} q^{61} + 6 q^{62} + 3 q^{63} + q^{64} + 12 \zeta_{6} q^{65} + ( 1 - 2 \zeta_{6} ) q^{66} + ( -13 + 13 \zeta_{6} ) q^{67} + ( 5 - 5 \zeta_{6} ) q^{68} + ( -4 + 8 \zeta_{6} ) q^{69} -2 \zeta_{6} q^{70} -8 q^{71} + ( -3 + 3 \zeta_{6} ) q^{72} + q^{73} + 2 \zeta_{6} q^{74} + ( 1 + \zeta_{6} ) q^{75} + ( 7 - 7 \zeta_{6} ) q^{76} + ( 1 - \zeta_{6} ) q^{77} + ( 12 - 6 \zeta_{6} ) q^{78} + 6 \zeta_{6} q^{79} + 2 q^{80} -9 \zeta_{6} q^{81} -3 q^{82} -16 \zeta_{6} q^{83} + ( -2 + \zeta_{6} ) q^{84} + ( 10 - 10 \zeta_{6} ) q^{85} + ( -1 + \zeta_{6} ) q^{86} + ( 4 + 4 \zeta_{6} ) q^{87} + \zeta_{6} q^{88} -6 q^{89} + ( -6 + 6 \zeta_{6} ) q^{90} + 6 q^{91} -4 \zeta_{6} q^{92} + ( 6 - 12 \zeta_{6} ) q^{93} + ( 14 - 14 \zeta_{6} ) q^{95} + ( 1 - 2 \zeta_{6} ) q^{96} + 5 \zeta_{6} q^{97} - q^{98} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 3q^{3} - q^{4} - 2q^{5} + 3q^{6} + q^{7} - 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q + q^{2} + 3q^{3} - q^{4} - 2q^{5} + 3q^{6} + q^{7} - 2q^{8} + 3q^{9} - 4q^{10} - q^{11} + 6q^{13} - q^{14} - q^{16} - 10q^{17} + 6q^{18} - 14q^{19} - 2q^{20} + 3q^{21} + q^{22} - 4q^{23} - 3q^{24} + q^{25} + 12q^{26} - 2q^{28} + 4q^{29} - 6q^{30} + 6q^{31} + q^{32} - 3q^{33} - 5q^{34} - 4q^{35} + 3q^{36} + 4q^{37} - 7q^{38} + 2q^{40} - 3q^{41} + q^{43} + 2q^{44} + 6q^{45} - 8q^{46} - 3q^{48} - q^{49} - q^{50} - 15q^{51} + 6q^{52} + 24q^{53} + 9q^{54} + 4q^{55} - q^{56} - 21q^{57} - 4q^{58} + 7q^{59} - 6q^{60} + 12q^{61} + 12q^{62} + 6q^{63} + 2q^{64} + 12q^{65} - 13q^{67} + 5q^{68} - 2q^{70} - 16q^{71} - 3q^{72} + 2q^{73} + 2q^{74} + 3q^{75} + 7q^{76} + q^{77} + 18q^{78} + 6q^{79} + 4q^{80} - 9q^{81} - 6q^{82} - 16q^{83} - 3q^{84} + 10q^{85} - q^{86} + 12q^{87} + q^{88} - 12q^{89} - 6q^{90} + 12q^{91} - 4q^{92} + 14q^{95} + 5q^{97} - 2q^{98} - 6q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i 1.50000 + 0.866025i −0.500000 0.866025i −1.00000 1.73205i 1.50000 0.866025i 0.500000 0.866025i −1.00000 1.50000 + 2.59808i −2.00000
85.1 0.500000 + 0.866025i 1.50000 0.866025i −0.500000 + 0.866025i −1.00000 + 1.73205i 1.50000 + 0.866025i 0.500000 + 0.866025i −1.00000 1.50000 2.59808i −2.00000
 $$n$$: 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
9.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.f.b 2
3.b odd 2 1 378.2.f.b 2
4.b odd 2 1 1008.2.r.a 2
7.b odd 2 1 882.2.f.f 2
7.c even 3 1 882.2.e.a 2
7.c even 3 1 882.2.h.h 2
7.d odd 6 1 882.2.e.e 2
7.d odd 6 1 882.2.h.g 2
9.c even 3 1 inner 126.2.f.b 2
9.c even 3 1 1134.2.a.c 1
9.d odd 6 1 378.2.f.b 2
9.d odd 6 1 1134.2.a.f 1
12.b even 2 1 3024.2.r.c 2
21.c even 2 1 2646.2.f.b 2
21.g even 6 1 2646.2.e.h 2
21.g even 6 1 2646.2.h.c 2
21.h odd 6 1 2646.2.e.i 2
21.h odd 6 1 2646.2.h.b 2
36.f odd 6 1 1008.2.r.a 2
36.f odd 6 1 9072.2.a.t 1
36.h even 6 1 3024.2.r.c 2
36.h even 6 1 9072.2.a.f 1
63.g even 3 1 882.2.e.a 2
63.h even 3 1 882.2.h.h 2
63.i even 6 1 2646.2.h.c 2
63.j odd 6 1 2646.2.h.b 2
63.k odd 6 1 882.2.e.e 2
63.l odd 6 1 882.2.f.f 2
63.l odd 6 1 7938.2.a.e 1
63.n odd 6 1 2646.2.e.i 2
63.o even 6 1 2646.2.f.b 2
63.o even 6 1 7938.2.a.bb 1
63.s even 6 1 2646.2.e.h 2
63.t odd 6 1 882.2.h.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.b 2 1.a even 1 1 trivial
126.2.f.b 2 9.c even 3 1 inner
378.2.f.b 2 3.b odd 2 1
378.2.f.b 2 9.d odd 6 1
882.2.e.a 2 7.c even 3 1
882.2.e.a 2 63.g even 3 1
882.2.e.e 2 7.d odd 6 1
882.2.e.e 2 63.k odd 6 1
882.2.f.f 2 7.b odd 2 1
882.2.f.f 2 63.l odd 6 1
882.2.h.g 2 7.d odd 6 1
882.2.h.g 2 63.t odd 6 1
882.2.h.h 2 7.c even 3 1
882.2.h.h 2 63.h even 3 1
1008.2.r.a 2 4.b odd 2 1
1008.2.r.a 2 36.f odd 6 1
1134.2.a.c 1 9.c even 3 1
1134.2.a.f 1 9.d odd 6 1
2646.2.e.h 2 21.g even 6 1
2646.2.e.h 2 63.s even 6 1
2646.2.e.i 2 21.h odd 6 1
2646.2.e.i 2 63.n odd 6 1
2646.2.f.b 2 21.c even 2 1
2646.2.f.b 2 63.o even 6 1
2646.2.h.b 2 21.h odd 6 1
2646.2.h.b 2 63.j odd 6 1
2646.2.h.c 2 21.g even 6 1
2646.2.h.c 2 63.i even 6 1
3024.2.r.c 2 12.b even 2 1
3024.2.r.c 2 36.h even 6 1
7938.2.a.e 1 63.l odd 6 1
7938.2.a.bb 1 63.o even 6 1
9072.2.a.f 1 36.h even 6 1
9072.2.a.t 1 36.f odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - 3 T + 3 T^{2}$$
$5$ $$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$1 - T + T^{2}$$
$11$ $$1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 5 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4}$$
$31$ $$1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$
$43$ $$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 12 T + 53 T^{2} )^{2}$$
$59$ $$1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4}$$
$61$ $$1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4}$$
$67$ $$1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - T + 73 T^{2} )^{2}$$
$79$ $$1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4}$$
$83$ $$1 + 16 T + 173 T^{2} + 1328 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$