# Properties

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.f 2 1.a even 1 1 trivial
546.2.i.f 2 7.c even 3 1 inner
1638.2.j.e 2 3.b odd 2 1
1638.2.j.e 2 21.h odd 6 1
3822.2.a.f 1 7.c even 3 1
3822.2.a.l 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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$T^{2} - 8T + 64$$
$41$ $$(T + 4)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + T + 1$$
$59$ $$T^{2} - 7T + 49$$
$61$ $$T^{2} - 4T + 16$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 6T + 36$$
$79$ $$T^{2} - 13T + 169$$
$83$ $$(T + 3)^{2}$$
$89$ $$T^{2} - 8T + 64$$
$97$ $$(T + 15)^{2}$$