# Properties

 Label 546.2.i.g Level $546$ Weight $2$ Character orbit 546.i Analytic conductor $4.360$ 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] = [546,2,Mod(79,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("546.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.i (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: $$4.35983195036$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + 2 \zeta_{6} q^{5} + q^{6} + (3 \zeta_{6} - 2) q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + 2*z * q^5 + q^6 + (3*z - 2) * q^7 - q^8 - z * q^9 $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + 2 \zeta_{6} q^{5} + q^{6} + (3 \zeta_{6} - 2) q^{7} - q^{8} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} + (3 \zeta_{6} - 3) q^{11} + \zeta_{6} q^{12} - q^{13} + (\zeta_{6} - 3) q^{14} + 2 q^{15} - \zeta_{6} q^{16} + (\zeta_{6} - 1) q^{17} + ( - \zeta_{6} + 1) q^{18} + 3 \zeta_{6} q^{19} - 2 q^{20} + (2 \zeta_{6} + 1) q^{21} - 3 q^{22} + (\zeta_{6} - 1) q^{24} + ( - \zeta_{6} + 1) q^{25} - \zeta_{6} q^{26} - q^{27} + ( - 2 \zeta_{6} - 1) q^{28} + 3 q^{29} + 2 \zeta_{6} q^{30} + ( - 4 \zeta_{6} + 4) q^{31} + ( - \zeta_{6} + 1) q^{32} + 3 \zeta_{6} q^{33} - q^{34} + (2 \zeta_{6} - 6) q^{35} + q^{36} + 2 \zeta_{6} q^{37} + (3 \zeta_{6} - 3) q^{38} + (\zeta_{6} - 1) q^{39} - 2 \zeta_{6} q^{40} + 2 q^{41} + (3 \zeta_{6} - 2) q^{42} + 6 q^{43} - 3 \zeta_{6} q^{44} + ( - 2 \zeta_{6} + 2) q^{45} + 9 \zeta_{6} q^{47} - q^{48} + ( - 3 \zeta_{6} - 5) q^{49} + q^{50} + \zeta_{6} q^{51} + ( - \zeta_{6} + 1) q^{52} + (\zeta_{6} - 1) q^{53} - \zeta_{6} q^{54} - 6 q^{55} + ( - 3 \zeta_{6} + 2) q^{56} + 3 q^{57} + 3 \zeta_{6} q^{58} + (11 \zeta_{6} - 11) q^{59} + (2 \zeta_{6} - 2) q^{60} - 11 \zeta_{6} q^{61} + 4 q^{62} + ( - \zeta_{6} + 3) q^{63} + q^{64} - 2 \zeta_{6} q^{65} + (3 \zeta_{6} - 3) q^{66} + ( - 7 \zeta_{6} + 7) q^{67} - \zeta_{6} q^{68} + ( - 4 \zeta_{6} - 2) q^{70} + 15 q^{71} + \zeta_{6} q^{72} + ( - 12 \zeta_{6} + 12) q^{73} + (2 \zeta_{6} - 2) q^{74} - \zeta_{6} q^{75} - 3 q^{76} + ( - 6 \zeta_{6} - 3) q^{77} - q^{78} - 2 \zeta_{6} q^{79} + ( - 2 \zeta_{6} + 2) q^{80} + (\zeta_{6} - 1) q^{81} + 2 \zeta_{6} q^{82} + (\zeta_{6} - 3) q^{84} - 2 q^{85} + 6 \zeta_{6} q^{86} + ( - 3 \zeta_{6} + 3) q^{87} + ( - 3 \zeta_{6} + 3) q^{88} - 10 \zeta_{6} q^{89} + 2 q^{90} + ( - 3 \zeta_{6} + 2) q^{91} - 4 \zeta_{6} q^{93} + (9 \zeta_{6} - 9) q^{94} + (6 \zeta_{6} - 6) q^{95} - \zeta_{6} q^{96} - 12 q^{97} + ( - 8 \zeta_{6} + 3) q^{98} + 3 q^{99} +O(q^{100})$$ q + z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + 2*z * q^5 + q^6 + (3*z - 2) * q^7 - q^8 - z * q^9 + (2*z - 2) * q^10 + (3*z - 3) * q^11 + z * q^12 - q^13 + (z - 3) * q^14 + 2 * q^15 - z * q^16 + (z - 1) * q^17 + (-z + 1) * q^18 + 3*z * q^19 - 2 * q^20 + (2*z + 1) * q^21 - 3 * q^22 + (z - 1) * q^24 + (-z + 1) * q^25 - z * q^26 - q^27 + (-2*z - 1) * q^28 + 3 * q^29 + 2*z * q^30 + (-4*z + 4) * q^31 + (-z + 1) * q^32 + 3*z * q^33 - q^34 + (2*z - 6) * q^35 + q^36 + 2*z * q^37 + (3*z - 3) * q^38 + (z - 1) * q^39 - 2*z * q^40 + 2 * q^41 + (3*z - 2) * q^42 + 6 * q^43 - 3*z * q^44 + (-2*z + 2) * q^45 + 9*z * q^47 - q^48 + (-3*z - 5) * q^49 + q^50 + z * q^51 + (-z + 1) * q^52 + (z - 1) * q^53 - z * q^54 - 6 * q^55 + (-3*z + 2) * q^56 + 3 * q^57 + 3*z * q^58 + (11*z - 11) * q^59 + (2*z - 2) * q^60 - 11*z * q^61 + 4 * q^62 + (-z + 3) * q^63 + q^64 - 2*z * q^65 + (3*z - 3) * q^66 + (-7*z + 7) * q^67 - z * q^68 + (-4*z - 2) * q^70 + 15 * q^71 + z * q^72 + (-12*z + 12) * q^73 + (2*z - 2) * q^74 - z * q^75 - 3 * q^76 + (-6*z - 3) * q^77 - q^78 - 2*z * q^79 + (-2*z + 2) * q^80 + (z - 1) * q^81 + 2*z * q^82 + (z - 3) * q^84 - 2 * q^85 + 6*z * q^86 + (-3*z + 3) * q^87 + (-3*z + 3) * q^88 - 10*z * q^89 + 2 * q^90 + (-3*z + 2) * q^91 - 4*z * q^93 + (9*z - 9) * q^94 + (6*z - 6) * q^95 - z * q^96 - 12 * q^97 + (-8*z + 3) * q^98 + 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} - q^{4} + 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 - q^4 + 2 * q^5 + 2 * q^6 - q^7 - 2 * q^8 - q^9 $$2 q + q^{2} + q^{3} - q^{4} + 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} - q^{9} - 2 q^{10} - 3 q^{11} + q^{12} - 2 q^{13} - 5 q^{14} + 4 q^{15} - q^{16} - q^{17} + q^{18} + 3 q^{19} - 4 q^{20} + 4 q^{21} - 6 q^{22} - q^{24} + q^{25} - q^{26} - 2 q^{27} - 4 q^{28} + 6 q^{29} + 2 q^{30} + 4 q^{31} + q^{32} + 3 q^{33} - 2 q^{34} - 10 q^{35} + 2 q^{36} + 2 q^{37} - 3 q^{38} - q^{39} - 2 q^{40} + 4 q^{41} - q^{42} + 12 q^{43} - 3 q^{44} + 2 q^{45} + 9 q^{47} - 2 q^{48} - 13 q^{49} + 2 q^{50} + q^{51} + q^{52} - q^{53} - q^{54} - 12 q^{55} + q^{56} + 6 q^{57} + 3 q^{58} - 11 q^{59} - 2 q^{60} - 11 q^{61} + 8 q^{62} + 5 q^{63} + 2 q^{64} - 2 q^{65} - 3 q^{66} + 7 q^{67} - q^{68} - 8 q^{70} + 30 q^{71} + q^{72} + 12 q^{73} - 2 q^{74} - q^{75} - 6 q^{76} - 12 q^{77} - 2 q^{78} - 2 q^{79} + 2 q^{80} - q^{81} + 2 q^{82} - 5 q^{84} - 4 q^{85} + 6 q^{86} + 3 q^{87} + 3 q^{88} - 10 q^{89} + 4 q^{90} + q^{91} - 4 q^{93} - 9 q^{94} - 6 q^{95} - q^{96} - 24 q^{97} - 2 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 - q^4 + 2 * q^5 + 2 * q^6 - q^7 - 2 * q^8 - q^9 - 2 * q^10 - 3 * q^11 + q^12 - 2 * q^13 - 5 * q^14 + 4 * q^15 - q^16 - q^17 + q^18 + 3 * q^19 - 4 * q^20 + 4 * q^21 - 6 * q^22 - q^24 + q^25 - q^26 - 2 * q^27 - 4 * q^28 + 6 * q^29 + 2 * q^30 + 4 * q^31 + q^32 + 3 * q^33 - 2 * q^34 - 10 * q^35 + 2 * q^36 + 2 * q^37 - 3 * q^38 - q^39 - 2 * q^40 + 4 * q^41 - q^42 + 12 * q^43 - 3 * q^44 + 2 * q^45 + 9 * q^47 - 2 * q^48 - 13 * q^49 + 2 * q^50 + q^51 + q^52 - q^53 - q^54 - 12 * q^55 + q^56 + 6 * q^57 + 3 * q^58 - 11 * q^59 - 2 * q^60 - 11 * q^61 + 8 * q^62 + 5 * q^63 + 2 * q^64 - 2 * q^65 - 3 * q^66 + 7 * q^67 - q^68 - 8 * q^70 + 30 * q^71 + q^72 + 12 * q^73 - 2 * q^74 - q^75 - 6 * q^76 - 12 * q^77 - 2 * q^78 - 2 * q^79 + 2 * q^80 - q^81 + 2 * q^82 - 5 * q^84 - 4 * q^85 + 6 * q^86 + 3 * q^87 + 3 * q^88 - 10 * q^89 + 4 * q^90 + q^91 - 4 * q^93 - 9 * q^94 - 6 * q^95 - q^96 - 24 * q^97 - 2 * q^98 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$$.

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 1.73205i 1.00000 −0.500000 2.59808i −1.00000 −0.500000 + 0.866025i −1.00000 1.73205i
235.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.00000 + 1.73205i 1.00000 −0.500000 + 2.59808i −1.00000 −0.500000 0.866025i −1.00000 + 1.73205i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.i.g 2
3.b odd 2 1 1638.2.j.b 2
7.c even 3 1 inner 546.2.i.g 2
7.c even 3 1 3822.2.a.c 1
7.d odd 6 1 3822.2.a.q 1
21.h odd 6 1 1638.2.j.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.g 2 1.a even 1 1 trivial
546.2.i.g 2 7.c even 3 1 inner
1638.2.j.b 2 3.b odd 2 1
1638.2.j.b 2 21.h odd 6 1
3822.2.a.c 1 7.c even 3 1
3822.2.a.q 1 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4 $$T_{17}^{2} + T_{17} + 1$$ T17^2 + T17 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + T + 1$$
$19$ $$T^{2} - 3T + 9$$
$23$ $$T^{2}$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} - 2T + 4$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} - 9T + 81$$
$53$ $$T^{2} + T + 1$$
$59$ $$T^{2} + 11T + 121$$
$61$ $$T^{2} + 11T + 121$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$(T - 15)^{2}$$
$73$ $$T^{2} - 12T + 144$$
$79$ $$T^{2} + 2T + 4$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 10T + 100$$
$97$ $$(T + 12)^{2}$$