# Properties

 Label 546.2.i.b Level $546$ Weight $2$ Character orbit 546.i Analytic conductor $4.360$ 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] = [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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + q^{6} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (z - 1) * q^3 + (z - 1) * q^4 + z * q^5 + q^6 + (-3*z + 1) * q^7 + q^8 - z * q^9 $$q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + q^{6} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + ( - \zeta_{6} + 1) q^{11} - \zeta_{6} q^{12} - q^{13} + (2 \zeta_{6} - 3) q^{14} - q^{15} - \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + (\zeta_{6} - 1) q^{18} + 4 \zeta_{6} q^{19} - q^{20} + (\zeta_{6} + 2) q^{21} - q^{22} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} + ( - 4 \zeta_{6} + 4) q^{25} + \zeta_{6} q^{26} + q^{27} + (\zeta_{6} + 2) q^{28} + 3 q^{29} + \zeta_{6} q^{30} + ( - 11 \zeta_{6} + 11) q^{31} + (\zeta_{6} - 1) q^{32} + \zeta_{6} q^{33} - 6 q^{34} + ( - 2 \zeta_{6} + 3) q^{35} + q^{36} - 4 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + ( - \zeta_{6} + 1) q^{39} + \zeta_{6} q^{40} + 12 q^{41} + ( - 3 \zeta_{6} + 1) q^{42} - 8 q^{43} + \zeta_{6} q^{44} + ( - \zeta_{6} + 1) q^{45} + ( - 6 \zeta_{6} + 6) q^{46} + 8 \zeta_{6} q^{47} + q^{48} + (3 \zeta_{6} - 8) q^{49} - 4 q^{50} + 6 \zeta_{6} q^{51} + ( - \zeta_{6} + 1) q^{52} + ( - 5 \zeta_{6} + 5) q^{53} - \zeta_{6} q^{54} + q^{55} + ( - 3 \zeta_{6} + 1) q^{56} - 4 q^{57} - 3 \zeta_{6} q^{58} + ( - 5 \zeta_{6} + 5) q^{59} + ( - \zeta_{6} + 1) q^{60} - 12 \zeta_{6} q^{61} - 11 q^{62} + (2 \zeta_{6} - 3) q^{63} + q^{64} - \zeta_{6} q^{65} + ( - \zeta_{6} + 1) q^{66} + (16 \zeta_{6} - 16) q^{67} + 6 \zeta_{6} q^{68} - 6 q^{69} + ( - \zeta_{6} - 2) q^{70} + 6 q^{71} - \zeta_{6} q^{72} + ( - 10 \zeta_{6} + 10) q^{73} + (4 \zeta_{6} - 4) q^{74} + 4 \zeta_{6} q^{75} - 4 q^{76} + ( - \zeta_{6} - 2) q^{77} - q^{78} - 7 \zeta_{6} q^{79} + ( - \zeta_{6} + 1) q^{80} + (\zeta_{6} - 1) q^{81} - 12 \zeta_{6} q^{82} - 17 q^{83} + (2 \zeta_{6} - 3) q^{84} + 6 q^{85} + 8 \zeta_{6} q^{86} + (3 \zeta_{6} - 3) q^{87} + ( - \zeta_{6} + 1) q^{88} + 12 \zeta_{6} q^{89} - q^{90} + (3 \zeta_{6} - 1) q^{91} - 6 q^{92} + 11 \zeta_{6} q^{93} + ( - 8 \zeta_{6} + 8) q^{94} + (4 \zeta_{6} - 4) q^{95} - \zeta_{6} q^{96} + 13 q^{97} + (5 \zeta_{6} + 3) q^{98} - q^{99} +O(q^{100})$$ q - z * q^2 + (z - 1) * q^3 + (z - 1) * q^4 + z * q^5 + q^6 + (-3*z + 1) * q^7 + q^8 - z * q^9 + (-z + 1) * q^10 + (-z + 1) * q^11 - z * q^12 - q^13 + (2*z - 3) * q^14 - q^15 - z * q^16 + (-6*z + 6) * q^17 + (z - 1) * q^18 + 4*z * q^19 - q^20 + (z + 2) * q^21 - q^22 + 6*z * q^23 + (z - 1) * q^24 + (-4*z + 4) * q^25 + z * q^26 + q^27 + (z + 2) * q^28 + 3 * q^29 + z * q^30 + (-11*z + 11) * q^31 + (z - 1) * q^32 + z * q^33 - 6 * q^34 + (-2*z + 3) * q^35 + q^36 - 4*z * q^37 + (-4*z + 4) * q^38 + (-z + 1) * q^39 + z * q^40 + 12 * q^41 + (-3*z + 1) * q^42 - 8 * q^43 + z * q^44 + (-z + 1) * q^45 + (-6*z + 6) * q^46 + 8*z * q^47 + q^48 + (3*z - 8) * q^49 - 4 * q^50 + 6*z * q^51 + (-z + 1) * q^52 + (-5*z + 5) * q^53 - z * q^54 + q^55 + (-3*z + 1) * q^56 - 4 * q^57 - 3*z * q^58 + (-5*z + 5) * q^59 + (-z + 1) * q^60 - 12*z * q^61 - 11 * q^62 + (2*z - 3) * q^63 + q^64 - z * q^65 + (-z + 1) * q^66 + (16*z - 16) * q^67 + 6*z * q^68 - 6 * q^69 + (-z - 2) * q^70 + 6 * q^71 - z * q^72 + (-10*z + 10) * q^73 + (4*z - 4) * q^74 + 4*z * q^75 - 4 * q^76 + (-z - 2) * q^77 - q^78 - 7*z * q^79 + (-z + 1) * q^80 + (z - 1) * q^81 - 12*z * q^82 - 17 * q^83 + (2*z - 3) * q^84 + 6 * q^85 + 8*z * q^86 + (3*z - 3) * q^87 + (-z + 1) * q^88 + 12*z * q^89 - q^90 + (3*z - 1) * q^91 - 6 * q^92 + 11*z * q^93 + (-8*z + 8) * q^94 + (4*z - 4) * q^95 - z * q^96 + 13 * q^97 + (5*z + 3) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 - q^3 - q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 - q^9 $$2 q - q^{2} - q^{3} - q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} - q^{9} + q^{10} + q^{11} - q^{12} - 2 q^{13} - 4 q^{14} - 2 q^{15} - q^{16} + 6 q^{17} - q^{18} + 4 q^{19} - 2 q^{20} + 5 q^{21} - 2 q^{22} + 6 q^{23} - q^{24} + 4 q^{25} + q^{26} + 2 q^{27} + 5 q^{28} + 6 q^{29} + q^{30} + 11 q^{31} - q^{32} + q^{33} - 12 q^{34} + 4 q^{35} + 2 q^{36} - 4 q^{37} + 4 q^{38} + q^{39} + q^{40} + 24 q^{41} - q^{42} - 16 q^{43} + q^{44} + q^{45} + 6 q^{46} + 8 q^{47} + 2 q^{48} - 13 q^{49} - 8 q^{50} + 6 q^{51} + q^{52} + 5 q^{53} - q^{54} + 2 q^{55} - q^{56} - 8 q^{57} - 3 q^{58} + 5 q^{59} + q^{60} - 12 q^{61} - 22 q^{62} - 4 q^{63} + 2 q^{64} - q^{65} + q^{66} - 16 q^{67} + 6 q^{68} - 12 q^{69} - 5 q^{70} + 12 q^{71} - q^{72} + 10 q^{73} - 4 q^{74} + 4 q^{75} - 8 q^{76} - 5 q^{77} - 2 q^{78} - 7 q^{79} + q^{80} - q^{81} - 12 q^{82} - 34 q^{83} - 4 q^{84} + 12 q^{85} + 8 q^{86} - 3 q^{87} + q^{88} + 12 q^{89} - 2 q^{90} + q^{91} - 12 q^{92} + 11 q^{93} + 8 q^{94} - 4 q^{95} - q^{96} + 26 q^{97} + 11 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^3 - q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 - q^9 + q^10 + q^11 - q^12 - 2 * q^13 - 4 * q^14 - 2 * q^15 - q^16 + 6 * q^17 - q^18 + 4 * q^19 - 2 * q^20 + 5 * q^21 - 2 * q^22 + 6 * q^23 - q^24 + 4 * q^25 + q^26 + 2 * q^27 + 5 * q^28 + 6 * q^29 + q^30 + 11 * q^31 - q^32 + q^33 - 12 * q^34 + 4 * q^35 + 2 * q^36 - 4 * q^37 + 4 * q^38 + q^39 + q^40 + 24 * q^41 - q^42 - 16 * q^43 + q^44 + q^45 + 6 * q^46 + 8 * q^47 + 2 * q^48 - 13 * q^49 - 8 * q^50 + 6 * q^51 + q^52 + 5 * q^53 - q^54 + 2 * q^55 - q^56 - 8 * q^57 - 3 * q^58 + 5 * q^59 + q^60 - 12 * q^61 - 22 * q^62 - 4 * q^63 + 2 * q^64 - q^65 + q^66 - 16 * q^67 + 6 * q^68 - 12 * q^69 - 5 * q^70 + 12 * q^71 - q^72 + 10 * q^73 - 4 * q^74 + 4 * q^75 - 8 * q^76 - 5 * q^77 - 2 * q^78 - 7 * q^79 + q^80 - q^81 - 12 * q^82 - 34 * q^83 - 4 * q^84 + 12 * q^85 + 8 * q^86 - 3 * q^87 + q^88 + 12 * q^89 - 2 * q^90 + q^91 - 12 * q^92 + 11 * q^93 + 8 * q^94 - 4 * q^95 - q^96 + 26 * q^97 + 11 * q^98 - 2 * 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 0.500000 0.866025i 1.00000 −0.500000 + 2.59808i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
235.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 2.59808i 1.00000 −0.500000 0.866025i 0.500000 0.866025i
 $$n$$: 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.b 2
3.b odd 2 1 1638.2.j.h 2
7.c even 3 1 inner 546.2.i.b 2
7.c even 3 1 3822.2.a.be 1
7.d odd 6 1 3822.2.a.v 1
21.h odd 6 1 1638.2.j.h 2

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

## Hecke kernels

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

 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1 $$T_{17}^{2} - 6T_{17} + 36$$ T17^2 - 6*T17 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} - 11T + 121$$
$37$ $$T^{2} + 4T + 16$$
$41$ $$(T - 12)^{2}$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} - 5T + 25$$
$59$ $$T^{2} - 5T + 25$$
$61$ $$T^{2} + 12T + 144$$
$67$ $$T^{2} + 16T + 256$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} - 10T + 100$$
$79$ $$T^{2} + 7T + 49$$
$83$ $$(T + 17)^{2}$$
$89$ $$T^{2} - 12T + 144$$
$97$ $$(T - 13)^{2}$$