# Properties

 Label 126.2.e.b Level $126$ Weight $2$ Character orbit 126.e 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.e (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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.73205i 1.00000 1.50000 2.59808i 1.73205i −2.00000 + 1.73205i 1.00000 −3.00000 1.50000 2.59808i
121.1 1.00000 1.73205i 1.00000 1.50000 + 2.59808i 1.73205i −2.00000 1.73205i 1.00000 −3.00000 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.h 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.e.b 2
3.b odd 2 1 378.2.e.a 2
4.b odd 2 1 1008.2.q.e 2
7.b odd 2 1 882.2.e.h 2
7.c even 3 1 126.2.h.a yes 2
7.c even 3 1 882.2.f.e 2
7.d odd 6 1 882.2.f.a 2
7.d odd 6 1 882.2.h.e 2
9.c even 3 1 126.2.h.a yes 2
9.c even 3 1 1134.2.g.d 2
9.d odd 6 1 378.2.h.b 2
9.d odd 6 1 1134.2.g.f 2
12.b even 2 1 3024.2.q.a 2
21.c even 2 1 2646.2.e.e 2
21.g even 6 1 2646.2.f.i 2
21.g even 6 1 2646.2.h.f 2
21.h odd 6 1 378.2.h.b 2
21.h odd 6 1 2646.2.f.e 2
28.g odd 6 1 1008.2.t.c 2
36.f odd 6 1 1008.2.t.c 2
36.h even 6 1 3024.2.t.f 2
63.g even 3 1 882.2.f.e 2
63.g even 3 1 1134.2.g.d 2
63.h even 3 1 inner 126.2.e.b 2
63.h even 3 1 7938.2.a.r 1
63.i even 6 1 2646.2.e.e 2
63.i even 6 1 7938.2.a.c 1
63.j odd 6 1 378.2.e.a 2
63.j odd 6 1 7938.2.a.o 1
63.k odd 6 1 882.2.f.a 2
63.l odd 6 1 882.2.h.e 2
63.n odd 6 1 1134.2.g.f 2
63.n odd 6 1 2646.2.f.e 2
63.o even 6 1 2646.2.h.f 2
63.s even 6 1 2646.2.f.i 2
63.t odd 6 1 882.2.e.h 2
63.t odd 6 1 7938.2.a.bd 1
84.n even 6 1 3024.2.t.f 2
252.u odd 6 1 1008.2.q.e 2
252.bb even 6 1 3024.2.q.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 5T + 25$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} - 3T + 9$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - 7T + 49$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$T^{2} + 11T + 121$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 3T + 9$$
$89$ $$T^{2} + 15T + 225$$
$97$ $$T^{2} - T + 1$$