# Properties

 Label 126.2.h.a Level 126 Weight 2 Character orbit 126.h Analytic conductor 1.006 Analytic rank 1 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.h (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.00611506547$$ Analytic rank: $$1$$ 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i −3.00000 1.73205i −0.500000 2.59808i 1.00000 1.50000 2.59808i 1.50000 2.59808i
79.1 −0.500000 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i −3.00000 1.73205i −0.500000 + 2.59808i 1.00000 1.50000 + 2.59808i 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
63.g 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.h.a yes 2
3.b odd 2 1 378.2.h.b 2
4.b odd 2 1 1008.2.t.c 2
7.b odd 2 1 882.2.h.e 2
7.c even 3 1 126.2.e.b 2
7.c even 3 1 882.2.f.e 2
7.d odd 6 1 882.2.e.h 2
7.d odd 6 1 882.2.f.a 2
9.c even 3 1 126.2.e.b 2
9.c even 3 1 1134.2.g.d 2
9.d odd 6 1 378.2.e.a 2
9.d odd 6 1 1134.2.g.f 2
12.b even 2 1 3024.2.t.f 2
21.c even 2 1 2646.2.h.f 2
21.g even 6 1 2646.2.e.e 2
21.g even 6 1 2646.2.f.i 2
21.h odd 6 1 378.2.e.a 2
21.h odd 6 1 2646.2.f.e 2
28.g odd 6 1 1008.2.q.e 2
36.f odd 6 1 1008.2.q.e 2
36.h even 6 1 3024.2.q.a 2
63.g even 3 1 inner 126.2.h.a yes 2
63.g even 3 1 7938.2.a.r 1
63.h even 3 1 882.2.f.e 2
63.h even 3 1 1134.2.g.d 2
63.i even 6 1 2646.2.f.i 2
63.j odd 6 1 1134.2.g.f 2
63.j odd 6 1 2646.2.f.e 2
63.k odd 6 1 882.2.h.e 2
63.k odd 6 1 7938.2.a.bd 1
63.l odd 6 1 882.2.e.h 2
63.n odd 6 1 378.2.h.b 2
63.n odd 6 1 7938.2.a.o 1
63.o even 6 1 2646.2.e.e 2
63.s even 6 1 2646.2.h.f 2
63.s even 6 1 7938.2.a.c 1
63.t odd 6 1 882.2.f.a 2
84.n even 6 1 3024.2.q.a 2
252.o even 6 1 3024.2.t.f 2
252.bl odd 6 1 1008.2.t.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.b 2 7.c even 3 1
126.2.e.b 2 9.c even 3 1
126.2.h.a yes 2 1.a even 1 1 trivial
126.2.h.a yes 2 63.g even 3 1 inner
378.2.e.a 2 9.d odd 6 1
378.2.e.a 2 21.h odd 6 1
378.2.h.b 2 3.b odd 2 1
378.2.h.b 2 63.n odd 6 1
882.2.e.h 2 7.d odd 6 1
882.2.e.h 2 63.l odd 6 1
882.2.f.a 2 7.d odd 6 1
882.2.f.a 2 63.t odd 6 1
882.2.f.e 2 7.c even 3 1
882.2.f.e 2 63.h even 3 1
882.2.h.e 2 7.b odd 2 1
882.2.h.e 2 63.k odd 6 1
1008.2.q.e 2 28.g odd 6 1
1008.2.q.e 2 36.f odd 6 1
1008.2.t.c 2 4.b odd 2 1
1008.2.t.c 2 252.bl odd 6 1
1134.2.g.d 2 9.c even 3 1
1134.2.g.d 2 63.h even 3 1
1134.2.g.f 2 9.d odd 6 1
1134.2.g.f 2 63.j odd 6 1
2646.2.e.e 2 21.g even 6 1
2646.2.e.e 2 63.o even 6 1
2646.2.f.e 2 21.h odd 6 1
2646.2.f.e 2 63.j odd 6 1
2646.2.f.i 2 21.g even 6 1
2646.2.f.i 2 63.i even 6 1
2646.2.h.f 2 21.c even 2 1
2646.2.h.f 2 63.s even 6 1
3024.2.q.a 2 36.h even 6 1
3024.2.q.a 2 84.n even 6 1
3024.2.t.f 2 12.b even 2 1
3024.2.t.f 2 252.o even 6 1
7938.2.a.c 1 63.s even 6 1
7938.2.a.o 1 63.n odd 6 1
7938.2.a.r 1 63.g even 3 1
7938.2.a.bd 1 63.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 3$$ acting on $$S_{2}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$( 1 + 3 T + 5 T^{2} )^{2}$$
$7$ $$1 + T + 7 T^{2}$$
$11$ $$( 1 + 3 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$( 1 + 3 T + 23 T^{2} )^{2}$$
$29$ $$1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4}$$
$41$ $$1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4}$$
$43$ $$1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4}$$
$67$ $$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4}$$
$79$ $$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$1 + 3 T - 74 T^{2} + 249 T^{3} + 6889 T^{4}$$
$89$ $$1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4}$$
$97$ $$1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4}$$