# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,2,Mod(25,126)]

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, 4]))

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

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

chi := DirichletCharacter("126.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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.a 2
3.b odd 2 1 378.2.e.b 2
4.b odd 2 1 1008.2.q.a 2
7.b odd 2 1 882.2.e.c 2
7.c even 3 1 126.2.h.b yes 2
7.c even 3 1 882.2.f.i 2
7.d odd 6 1 882.2.f.g 2
7.d odd 6 1 882.2.h.i 2
9.c even 3 1 126.2.h.b yes 2
9.c even 3 1 1134.2.g.e 2
9.d odd 6 1 378.2.h.a 2
9.d odd 6 1 1134.2.g.c 2
12.b even 2 1 3024.2.q.f 2
21.c even 2 1 2646.2.e.g 2
21.g even 6 1 2646.2.f.a 2
21.g even 6 1 2646.2.h.d 2
21.h odd 6 1 378.2.h.a 2
21.h odd 6 1 2646.2.f.d 2
28.g odd 6 1 1008.2.t.f 2
36.f odd 6 1 1008.2.t.f 2
36.h even 6 1 3024.2.t.a 2
63.g even 3 1 882.2.f.i 2
63.g even 3 1 1134.2.g.e 2
63.h even 3 1 inner 126.2.e.a 2
63.h even 3 1 7938.2.a.m 1
63.i even 6 1 2646.2.e.g 2
63.i even 6 1 7938.2.a.be 1
63.j odd 6 1 378.2.e.b 2
63.j odd 6 1 7938.2.a.t 1
63.k odd 6 1 882.2.f.g 2
63.l odd 6 1 882.2.h.i 2
63.n odd 6 1 1134.2.g.c 2
63.n odd 6 1 2646.2.f.d 2
63.o even 6 1 2646.2.h.d 2
63.s even 6 1 2646.2.f.a 2
63.t odd 6 1 882.2.e.c 2
63.t odd 6 1 7938.2.a.b 1
84.n even 6 1 3024.2.t.a 2
252.u odd 6 1 1008.2.q.a 2
252.bb even 6 1 3024.2.q.f 2

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

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} - T + 1$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$T^{2} - 9T + 81$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$T^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$(T + 4)^{2}$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} - 9T + 81$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$T^{2} - T + 1$$