# Properties

 Label 546.2.i.d 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} - q^{6} + ( - \zeta_{6} - 2) q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 - q^6 + (-z - 2) * q^7 + q^8 - z * q^9 $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} - q^{6} + ( - \zeta_{6} - 2) q^{7} + q^{8} - \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} + \zeta_{6} q^{12} - q^{13} + (3 \zeta_{6} - 1) q^{14} - \zeta_{6} q^{16} + (7 \zeta_{6} - 7) q^{17} + (\zeta_{6} - 1) q^{18} - 7 \zeta_{6} q^{19} + (2 \zeta_{6} - 3) q^{21} - 5 q^{22} - 2 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} + ( - 5 \zeta_{6} + 5) q^{25} + \zeta_{6} q^{26} - q^{27} + ( - 2 \zeta_{6} + 3) q^{28} - 9 q^{29} + (\zeta_{6} - 1) q^{32} - 5 \zeta_{6} q^{33} + 7 q^{34} + q^{36} - 4 \zeta_{6} q^{37} + (7 \zeta_{6} - 7) q^{38} + (\zeta_{6} - 1) q^{39} + 4 q^{41} + (\zeta_{6} + 2) q^{42} + 2 q^{43} + 5 \zeta_{6} q^{44} + (2 \zeta_{6} - 2) q^{46} + 3 \zeta_{6} q^{47} - q^{48} + (5 \zeta_{6} + 3) q^{49} - 5 q^{50} + 7 \zeta_{6} q^{51} + ( - \zeta_{6} + 1) q^{52} + (\zeta_{6} - 1) q^{53} + \zeta_{6} q^{54} + ( - \zeta_{6} - 2) q^{56} - 7 q^{57} + 9 \zeta_{6} q^{58} + (7 \zeta_{6} - 7) q^{59} - 13 \zeta_{6} q^{61} + (3 \zeta_{6} - 1) q^{63} + q^{64} + (5 \zeta_{6} - 5) q^{66} + (3 \zeta_{6} - 3) q^{67} - 7 \zeta_{6} q^{68} - 2 q^{69} + 9 q^{71} - \zeta_{6} q^{72} + ( - 10 \zeta_{6} + 10) q^{73} + (4 \zeta_{6} - 4) q^{74} - 5 \zeta_{6} q^{75} + 7 q^{76} + (10 \zeta_{6} - 15) q^{77} + q^{78} - 14 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 4 \zeta_{6} q^{82} + 16 q^{83} + ( - 3 \zeta_{6} + 1) q^{84} - 2 \zeta_{6} q^{86} + (9 \zeta_{6} - 9) q^{87} + ( - 5 \zeta_{6} + 5) q^{88} + 12 \zeta_{6} q^{89} + (\zeta_{6} + 2) q^{91} + 2 q^{92} + ( - 3 \zeta_{6} + 3) q^{94} + \zeta_{6} q^{96} + 6 q^{97} + ( - 8 \zeta_{6} + 5) q^{98} - 5 q^{99} +O(q^{100})$$ q - z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 - q^6 + (-z - 2) * q^7 + q^8 - z * q^9 + (-5*z + 5) * q^11 + z * q^12 - q^13 + (3*z - 1) * q^14 - z * q^16 + (7*z - 7) * q^17 + (z - 1) * q^18 - 7*z * q^19 + (2*z - 3) * q^21 - 5 * q^22 - 2*z * q^23 + (-z + 1) * q^24 + (-5*z + 5) * q^25 + z * q^26 - q^27 + (-2*z + 3) * q^28 - 9 * q^29 + (z - 1) * q^32 - 5*z * q^33 + 7 * q^34 + q^36 - 4*z * q^37 + (7*z - 7) * q^38 + (z - 1) * q^39 + 4 * q^41 + (z + 2) * q^42 + 2 * q^43 + 5*z * q^44 + (2*z - 2) * q^46 + 3*z * q^47 - q^48 + (5*z + 3) * q^49 - 5 * q^50 + 7*z * q^51 + (-z + 1) * q^52 + (z - 1) * q^53 + z * q^54 + (-z - 2) * q^56 - 7 * q^57 + 9*z * q^58 + (7*z - 7) * q^59 - 13*z * q^61 + (3*z - 1) * q^63 + q^64 + (5*z - 5) * q^66 + (3*z - 3) * q^67 - 7*z * q^68 - 2 * q^69 + 9 * q^71 - z * q^72 + (-10*z + 10) * q^73 + (4*z - 4) * q^74 - 5*z * q^75 + 7 * q^76 + (10*z - 15) * q^77 + q^78 - 14*z * q^79 + (z - 1) * q^81 - 4*z * q^82 + 16 * q^83 + (-3*z + 1) * q^84 - 2*z * q^86 + (9*z - 9) * q^87 + (-5*z + 5) * q^88 + 12*z * q^89 + (z + 2) * q^91 + 2 * q^92 + (-3*z + 3) * q^94 + z * q^96 + 6 * q^97 + (-8*z + 5) * q^98 - 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} - q^{4} - 2 q^{6} - 5 q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 + q^3 - q^4 - 2 * q^6 - 5 * q^7 + 2 * q^8 - q^9 $$2 q - q^{2} + q^{3} - q^{4} - 2 q^{6} - 5 q^{7} + 2 q^{8} - q^{9} + 5 q^{11} + q^{12} - 2 q^{13} + q^{14} - q^{16} - 7 q^{17} - q^{18} - 7 q^{19} - 4 q^{21} - 10 q^{22} - 2 q^{23} + q^{24} + 5 q^{25} + q^{26} - 2 q^{27} + 4 q^{28} - 18 q^{29} - q^{32} - 5 q^{33} + 14 q^{34} + 2 q^{36} - 4 q^{37} - 7 q^{38} - q^{39} + 8 q^{41} + 5 q^{42} + 4 q^{43} + 5 q^{44} - 2 q^{46} + 3 q^{47} - 2 q^{48} + 11 q^{49} - 10 q^{50} + 7 q^{51} + q^{52} - q^{53} + q^{54} - 5 q^{56} - 14 q^{57} + 9 q^{58} - 7 q^{59} - 13 q^{61} + q^{63} + 2 q^{64} - 5 q^{66} - 3 q^{67} - 7 q^{68} - 4 q^{69} + 18 q^{71} - q^{72} + 10 q^{73} - 4 q^{74} - 5 q^{75} + 14 q^{76} - 20 q^{77} + 2 q^{78} - 14 q^{79} - q^{81} - 4 q^{82} + 32 q^{83} - q^{84} - 2 q^{86} - 9 q^{87} + 5 q^{88} + 12 q^{89} + 5 q^{91} + 4 q^{92} + 3 q^{94} + q^{96} + 12 q^{97} + 2 q^{98} - 10 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^3 - q^4 - 2 * q^6 - 5 * q^7 + 2 * q^8 - q^9 + 5 * q^11 + q^12 - 2 * q^13 + q^14 - q^16 - 7 * q^17 - q^18 - 7 * q^19 - 4 * q^21 - 10 * q^22 - 2 * q^23 + q^24 + 5 * q^25 + q^26 - 2 * q^27 + 4 * q^28 - 18 * q^29 - q^32 - 5 * q^33 + 14 * q^34 + 2 * q^36 - 4 * q^37 - 7 * q^38 - q^39 + 8 * q^41 + 5 * q^42 + 4 * q^43 + 5 * q^44 - 2 * q^46 + 3 * q^47 - 2 * q^48 + 11 * q^49 - 10 * q^50 + 7 * q^51 + q^52 - q^53 + q^54 - 5 * q^56 - 14 * q^57 + 9 * q^58 - 7 * q^59 - 13 * q^61 + q^63 + 2 * q^64 - 5 * q^66 - 3 * q^67 - 7 * q^68 - 4 * q^69 + 18 * q^71 - q^72 + 10 * q^73 - 4 * q^74 - 5 * q^75 + 14 * q^76 - 20 * q^77 + 2 * q^78 - 14 * q^79 - q^81 - 4 * q^82 + 32 * q^83 - q^84 - 2 * q^86 - 9 * q^87 + 5 * q^88 + 12 * q^89 + 5 * q^91 + 4 * q^92 + 3 * q^94 + q^96 + 12 * q^97 + 2 * q^98 - 10 * 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 −1.00000 −2.50000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0
235.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 −2.50000 0.866025i 1.00000 −0.500000 0.866025i 0
 $$n$$: 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.d 2
3.b odd 2 1 1638.2.j.i 2
7.c even 3 1 inner 546.2.i.d 2
7.c even 3 1 3822.2.a.u 1
7.d odd 6 1 3822.2.a.bf 1
21.h odd 6 1 1638.2.j.i 2

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

 $$T_{5}$$ T5 $$T_{17}^{2} + 7T_{17} + 49$$ T17^2 + 7*T17 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 7T + 49$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$T^{2} + 2T + 4$$
$29$ $$(T + 9)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4T + 16$$
$41$ $$(T - 4)^{2}$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2} + T + 1$$
$59$ $$T^{2} + 7T + 49$$
$61$ $$T^{2} + 13T + 169$$
$67$ $$T^{2} + 3T + 9$$
$71$ $$(T - 9)^{2}$$
$73$ $$T^{2} - 10T + 100$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$(T - 16)^{2}$$
$89$ $$T^{2} - 12T + 144$$
$97$ $$(T - 6)^{2}$$