# Properties

 Label 126.2.f.a Level $126$ Weight $2$ Character orbit 126.f Analytic conductor $1.006$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,2,Mod(43,126)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("126.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.00611506547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

## Character values

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

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 1.73205i −0.500000 0.866025i 1.50000 + 2.59808i 1.50000 + 0.866025i −0.500000 + 0.866025i −1.00000 −3.00000 3.00000
85.1 0.500000 + 0.866025i 1.73205i −0.500000 + 0.866025i 1.50000 2.59808i 1.50000 0.866025i −0.500000 0.866025i −1.00000 −3.00000 3.00000
 $$n$$: 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.a 2
3.b odd 2 1 378.2.f.a 2
4.b odd 2 1 1008.2.r.d 2
7.b odd 2 1 882.2.f.h 2
7.c even 3 1 882.2.e.b 2
7.c even 3 1 882.2.h.j 2
7.d odd 6 1 882.2.e.d 2
7.d odd 6 1 882.2.h.f 2
9.c even 3 1 inner 126.2.f.a 2
9.c even 3 1 1134.2.a.a 1
9.d odd 6 1 378.2.f.a 2
9.d odd 6 1 1134.2.a.h 1
12.b even 2 1 3024.2.r.a 2
21.c even 2 1 2646.2.f.c 2
21.g even 6 1 2646.2.e.j 2
21.g even 6 1 2646.2.h.a 2
21.h odd 6 1 2646.2.e.f 2
21.h odd 6 1 2646.2.h.e 2
36.f odd 6 1 1008.2.r.d 2
36.f odd 6 1 9072.2.a.c 1
36.h even 6 1 3024.2.r.a 2
36.h even 6 1 9072.2.a.w 1
63.g even 3 1 882.2.e.b 2
63.h even 3 1 882.2.h.j 2
63.i even 6 1 2646.2.h.a 2
63.j odd 6 1 2646.2.h.e 2
63.k odd 6 1 882.2.e.d 2
63.l odd 6 1 882.2.f.h 2
63.l odd 6 1 7938.2.a.l 1
63.n odd 6 1 2646.2.e.f 2
63.o even 6 1 2646.2.f.c 2
63.o even 6 1 7938.2.a.u 1
63.s even 6 1 2646.2.e.j 2
63.t odd 6 1 882.2.h.f 2

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

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$(T - 6)^{2}$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} + 2T + 4$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 2T + 4$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} + 8T + 64$$
$71$ $$(T - 3)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$T^{2} + 12T + 144$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 2T + 4$$