# Properties

 Label 546.2.i.k Level $546$ Weight $2$ Character orbit 546.i Analytic conductor $4.360$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.21870000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73$$ x^6 - 3*x^5 + 24*x^4 - 43*x^3 + 138*x^2 - 117*x + 73 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} - \beta_{2} q^{3} + \beta_{2} q^{4} + (\beta_{3} - \beta_{2} - \beta_1) q^{5} + q^{6} + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{7} - q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10})$$ q + (b2 + 1) * q^2 - b2 * q^3 + b2 * q^4 + (b3 - b2 - b1) * q^5 + q^6 + (b5 - b4 + b3 + 1) * q^7 - q^8 + (-b2 - 1) * q^9 $$q + (\beta_{2} + 1) q^{2} - \beta_{2} q^{3} + \beta_{2} q^{4} + (\beta_{3} - \beta_{2} - \beta_1) q^{5} + q^{6} + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{7} - q^{8} + ( - \beta_{2} - 1) q^{9} + (\beta_{4} - \beta_{2} - \beta_1) q^{10} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{11} + (\beta_{2} + 1) q^{12} + q^{13} + (\beta_{3} + \beta_{2} + 1) q^{14} + ( - \beta_{4} + \beta_{3}) q^{15} + ( - \beta_{2} - 1) q^{16} + (2 \beta_{5} - \beta_{4} - 3 \beta_1) q^{17} - \beta_{2} q^{18} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - 2) q^{19} + (\beta_{4} - \beta_{3}) q^{20} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{21} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{22}+ \cdots + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^2 - b2 * q^3 + b2 * q^4 + (b3 - b2 - b1) * q^5 + q^6 + (b5 - b4 + b3 + 1) * q^7 - q^8 + (-b2 - 1) * q^9 + (b4 - b2 - b1) * q^10 + (b5 - 2*b4 - 2*b2) * q^11 + (b2 + 1) * q^12 + q^13 + (b3 + b2 + 1) * q^14 + (-b4 + b3) * q^15 + (-b2 - 1) * q^16 + (2*b5 - b4 - 3*b1) * q^17 - b2 * q^18 + (b5 + b3 - 2*b2 - 2) * q^19 + (b4 - b3) * q^20 + (b5 - b4 - b2) * q^21 + (2*b5 - 2*b4 + 2*b3 - b2 - b1 + 2) * q^22 + (b5 + 2*b3 - b1 + 1) * q^23 + b2 * q^24 + (b5 - b4 + 5*b2 - b1) * q^25 + (b2 + 1) * q^26 - q^27 + (-b5 + b4 + b2) * q^28 - 3 * q^29 + (b3 - b2 - b1) * q^30 + (-b5 + b4 - 4*b2 + b1) * q^31 - b2 * q^32 + (-b5 - 2*b3 - b2 + b1 - 2) * q^33 + (4*b5 - b4 + b3 - 2*b2 - 2*b1 - 3) * q^34 + (-b5 - b4 - b3 + 2*b2 - b1 + 7) * q^35 + q^36 + (-b5 - 2*b2 - b1 - 1) * q^37 + (-b5 + b4 - b2 + b1) * q^38 - b2 * q^39 + (-b3 + b2 + b1) * q^40 + (2*b5 - b2 - b1 + 5) * q^41 + (b5 - b4 + b3 + 1) * q^42 + (2*b4 - 2*b3 - 6) * q^43 + (b5 + 2*b3 + b2 - b1 + 2) * q^44 + (-b4 + b2 + b1) * q^45 + (-b5 + 2*b4 + b2) * q^46 + (b5 + 3*b3 + 2*b2 - 2*b1 + 4) * q^47 - q^48 + (-2*b5 + b4 - b3 + 2*b2 + 3*b1 + 3) * q^49 + (2*b5 - b4 + b3 - b2 - b1 - 6) * q^50 + (-2*b5 - b3 + 2*b2 - b1 + 3) * q^51 + b2 * q^52 + (-2*b5 + 3*b2 + 4*b1) * q^53 + (-b2 - 1) * q^54 + (-8*b5 + 3*b4 - 3*b3 + 4*b2 + 4*b1 + 10) * q^55 + (-b5 + b4 - b3 - 1) * q^56 + (2*b5 - b4 + b3 - b2 - b1 - 2) * q^57 + (-3*b2 - 3) * q^58 + (-b5 + 2*b1) * q^59 + (b4 - b2 - b1) * q^60 + (-2*b5 - 3*b3 + b1 - 1) * q^61 + (-2*b5 + b4 - b3 + b2 + b1 + 5) * q^62 + (-b3 - b2 - 1) * q^63 + q^64 + (b3 - b2 - b1) * q^65 + (b5 - 2*b4 - 2*b2) * q^66 + (-3*b5 + 3*b4 + 5*b2 + 3*b1) * q^67 + (2*b5 + b3 - 2*b2 + b1 - 3) * q^68 + (2*b5 - 2*b4 + 2*b3 - b2 - b1 + 1) * q^69 + (3*b5 - 2*b4 + b3 + 4*b2 - 3*b1 + 4) * q^70 + (2*b5 - b4 + b3 - b2 - b1 - 12) * q^71 + (b2 + 1) * q^72 + (-b5 + 2*b4 + 9*b2) * q^73 + (b5 - 3*b2 - 2*b1) * q^74 + (-b5 - b3 + 6*b2 + 6) * q^75 + (-2*b5 + b4 - b3 + b2 + b1 + 2) * q^76 + (b5 + b4 - 10*b2 + b1 - 1) * q^77 + q^78 + (b5 - b3 - 5*b2 + 2*b1 - 7) * q^79 + (-b4 + b2 + b1) * q^80 + b2 * q^81 + (b5 + 6*b2 + b1 + 5) * q^82 + (6*b5 - 3*b4 + 3*b3 - 3*b2 - 3*b1 + 3) * q^83 + (b3 + b2 + 1) * q^84 + (-10*b5 + 10*b4 - 10*b3 + 5*b2 + 5*b1 - 13) * q^85 + (-2*b3 - 4*b2 + 2*b1 - 6) * q^86 + 3*b2 * q^87 + (-b5 + 2*b4 + 2*b2) * q^88 + (b5 - 10*b2 + b1 - 11) * q^89 + (-b4 + b3) * q^90 + (b5 - b4 + b3 + 1) * q^91 + (-2*b5 + 2*b4 - 2*b3 + b2 + b1 - 1) * q^92 + (b5 + b3 - 5*b2 - 5) * q^93 + (-b5 + 3*b4 + 3*b2 - b1) * q^94 + (3*b5 - 6*b4 - b2) * q^95 + (-b2 - 1) * q^96 + (4*b5 - b4 + b3 - 2*b2 - 2*b1) * q^97 + (-3*b5 - b3 + 5*b2 + 2*b1 + 4) * q^98 + (-2*b5 + 2*b4 - 2*b3 + b2 + b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} + 3 q^{7} - 6 q^{8} - 3 q^{9}+O(q^{10})$$ 6 * q + 3 * q^2 + 3 * q^3 - 3 * q^4 - 3 * q^5 + 6 * q^6 + 3 * q^7 - 6 * q^8 - 3 * q^9 $$6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} + 3 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{12} + 6 q^{13} - 6 q^{15} - 3 q^{16} - 6 q^{17} + 3 q^{18} - 6 q^{19} + 6 q^{20} + 3 q^{21} + 6 q^{22} - 3 q^{24} - 18 q^{25} + 3 q^{26} - 6 q^{27} - 3 q^{28} - 18 q^{29} - 3 q^{30} + 15 q^{31} + 3 q^{32} - 3 q^{33} - 12 q^{34} + 30 q^{35} + 6 q^{36} - 6 q^{37} + 6 q^{38} + 3 q^{39} + 3 q^{40} + 36 q^{41} + 3 q^{42} - 24 q^{43} + 3 q^{44} - 3 q^{45} + 6 q^{47} - 6 q^{48} + 21 q^{49} - 36 q^{50} + 6 q^{51} - 3 q^{52} - 3 q^{53} - 3 q^{54} + 54 q^{55} - 3 q^{56} - 12 q^{57} - 9 q^{58} + 3 q^{59} + 3 q^{60} + 30 q^{62} + 6 q^{64} - 3 q^{65} + 3 q^{66} - 6 q^{67} - 6 q^{68} + 3 q^{70} - 72 q^{71} + 3 q^{72} - 24 q^{73} + 6 q^{74} + 18 q^{75} + 12 q^{76} + 33 q^{77} + 6 q^{78} - 15 q^{79} - 3 q^{80} - 3 q^{81} + 18 q^{82} + 18 q^{83} - 48 q^{85} - 12 q^{86} - 9 q^{87} - 3 q^{88} - 30 q^{89} - 6 q^{90} + 3 q^{91} - 15 q^{93} - 6 q^{94} - 6 q^{95} - 3 q^{96} + 6 q^{97} + 9 q^{98} - 6 q^{99}+O(q^{100})$$ 6 * q + 3 * q^2 + 3 * q^3 - 3 * q^4 - 3 * q^5 + 6 * q^6 + 3 * q^7 - 6 * q^8 - 3 * q^9 + 3 * q^10 + 3 * q^11 + 3 * q^12 + 6 * q^13 - 6 * q^15 - 3 * q^16 - 6 * q^17 + 3 * q^18 - 6 * q^19 + 6 * q^20 + 3 * q^21 + 6 * q^22 - 3 * q^24 - 18 * q^25 + 3 * q^26 - 6 * q^27 - 3 * q^28 - 18 * q^29 - 3 * q^30 + 15 * q^31 + 3 * q^32 - 3 * q^33 - 12 * q^34 + 30 * q^35 + 6 * q^36 - 6 * q^37 + 6 * q^38 + 3 * q^39 + 3 * q^40 + 36 * q^41 + 3 * q^42 - 24 * q^43 + 3 * q^44 - 3 * q^45 + 6 * q^47 - 6 * q^48 + 21 * q^49 - 36 * q^50 + 6 * q^51 - 3 * q^52 - 3 * q^53 - 3 * q^54 + 54 * q^55 - 3 * q^56 - 12 * q^57 - 9 * q^58 + 3 * q^59 + 3 * q^60 + 30 * q^62 + 6 * q^64 - 3 * q^65 + 3 * q^66 - 6 * q^67 - 6 * q^68 + 3 * q^70 - 72 * q^71 + 3 * q^72 - 24 * q^73 + 6 * q^74 + 18 * q^75 + 12 * q^76 + 33 * q^77 + 6 * q^78 - 15 * q^79 - 3 * q^80 - 3 * q^81 + 18 * q^82 + 18 * q^83 - 48 * q^85 - 12 * q^86 - 9 * q^87 - 3 * q^88 - 30 * q^89 - 6 * q^90 + 3 * q^91 - 15 * q^93 - 6 * q^94 - 6 * q^95 - 3 * q^96 + 6 * q^97 + 9 * q^98 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 30\nu^{3} + 40\nu^{2} - 70\nu + 13 ) / 31$$ (-2*v^5 + 5*v^4 - 30*v^3 + 40*v^2 - 70*v + 13) / 31 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + 18\nu^{4} - 15\nu^{3} + 144\nu^{2} + 151\nu - 164 ) / 62$$ (-v^5 + 18*v^4 - 15*v^3 + 144*v^2 + 151*v - 164) / 62 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} - 13\nu^{4} + 47\nu^{3} - 197\nu^{2} + 461\nu - 133 ) / 62$$ (-v^5 - 13*v^4 + 47*v^3 - 197*v^2 + 461*v - 133) / 62 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} - 13\nu^{4} + 16\nu^{3} - 166\nu^{2} + 151\nu - 102 ) / 31$$ (-v^5 - 13*v^4 + 16*v^3 - 166*v^2 + 151*v - 102) / 31
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-2\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 8$$ -2*b5 + 2*b4 - 2*b3 + b2 + 2*b1 - 8 $$\nu^{3}$$ $$=$$ $$-3\beta_{5} + 4\beta_{4} - 2\beta_{3} + \beta_{2} - 8\beta _1 - 7$$ -3*b5 + 4*b4 - 2*b3 + b2 - 8*b1 - 7 $$\nu^{4}$$ $$=$$ $$16\beta_{5} - 16\beta_{4} + 20\beta_{3} - 9\beta_{2} - 28\beta _1 + 75$$ 16*b5 - 16*b4 + 20*b3 - 9*b2 - 28*b1 + 75 $$\nu^{5}$$ $$=$$ $$45\beta_{5} - 60\beta_{4} + 40\beta_{3} - 33\beta_{2} + 55\beta _1 + 139$$ 45*b5 - 60*b4 + 40*b3 - 33*b2 + 55*b1 + 139

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$ $$1$$

## 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 + 0.679547i 0.5 − 3.05087i 0.5 + 3.23735i 0.5 − 0.679547i 0.5 + 3.05087i 0.5 − 3.23735i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.90280 + 3.29575i 1.00000 −2.64411 0.0932392i −1.00000 −0.500000 + 0.866025i 1.90280 + 3.29575i
79.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.30661 + 2.26312i 1.00000 1.77890 1.95845i −1.00000 −0.500000 + 0.866025i 1.30661 + 2.26312i
79.3 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.70942 2.96080i 1.00000 2.36521 + 1.18566i −1.00000 −0.500000 + 0.866025i −1.70942 2.96080i
235.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.90280 3.29575i 1.00000 −2.64411 + 0.0932392i −1.00000 −0.500000 0.866025i 1.90280 3.29575i
235.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.30661 2.26312i 1.00000 1.77890 + 1.95845i −1.00000 −0.500000 0.866025i 1.30661 2.26312i
235.3 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.70942 + 2.96080i 1.00000 2.36521 1.18566i −1.00000 −0.500000 0.866025i −1.70942 + 2.96080i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.3 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.k 6
3.b odd 2 1 1638.2.j.q 6
7.c even 3 1 inner 546.2.i.k 6
7.c even 3 1 3822.2.a.bv 3
7.d odd 6 1 3822.2.a.bw 3
21.h odd 6 1 1638.2.j.q 6

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

 $$T_{5}^{6} + 3T_{5}^{5} + 21T_{5}^{4} + 32T_{5}^{3} + 246T_{5}^{2} + 408T_{5} + 1156$$ T5^6 + 3*T5^5 + 21*T5^4 + 32*T5^3 + 246*T5^2 + 408*T5 + 1156 $$T_{17}^{6} + 6T_{17}^{5} + 99T_{17}^{4} + 386T_{17}^{3} + 6261T_{17}^{2} + 24066T_{17} + 145924$$ T17^6 + 6*T17^5 + 99*T17^4 + 386*T17^3 + 6261*T17^2 + 24066*T17 + 145924

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{3}$$
$3$ $$(T^{2} - T + 1)^{3}$$
$5$ $$T^{6} + 3 T^{5} + \cdots + 1156$$
$7$ $$T^{6} - 3 T^{5} + \cdots + 343$$
$11$ $$T^{6} - 3 T^{5} + \cdots + 7921$$
$13$ $$(T - 1)^{6}$$
$17$ $$T^{6} + 6 T^{5} + \cdots + 145924$$
$19$ $$T^{6} + 6 T^{5} + \cdots + 1024$$
$23$ $$T^{6} + 30 T^{4} + \cdots + 3600$$
$29$ $$(T + 3)^{6}$$
$31$ $$T^{6} - 15 T^{5} + \cdots + 1600$$
$37$ $$T^{6} + 6 T^{5} + \cdots + 1024$$
$41$ $$(T^{3} - 18 T^{2} + \cdots - 56)^{2}$$
$43$ $$(T^{3} + 12 T^{2} + \cdots - 16)^{2}$$
$47$ $$T^{6} - 6 T^{5} + \cdots + 44944$$
$53$ $$T^{6} + 3 T^{5} + \cdots + 1681$$
$59$ $$T^{6} - 3 T^{5} + \cdots + 2401$$
$61$ $$T^{6} + 75 T^{4} + \cdots + 40000$$
$67$ $$T^{6} + 6 T^{5} + \cdots + 64$$
$71$ $$(T^{3} + 36 T^{2} + \cdots + 1538)^{2}$$
$73$ $$T^{6} + 24 T^{5} + \cdots + 44944$$
$79$ $$T^{6} + 15 T^{5} + \cdots + 10000$$
$83$ $$(T^{3} - 9 T^{2} + \cdots + 108)^{2}$$
$89$ $$T^{6} + 30 T^{5} + \cdots + 462400$$
$97$ $$(T^{3} - 3 T^{2} + \cdots - 166)^{2}$$