# Properties

 Label 546.2.l.j Level $546$ Weight $2$ Character orbit 546.l Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ 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(211,546)]

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, 0, 2]))

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

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

chi := DirichletCharacter("546.211");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.l (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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20$$ (v^3 + 4*v^2 - 4*v - 25) / 20 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5$$ (-v^3 + v^2 + 9*v + 5) / 5 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10$$ (3*v^3 + 2*v^2 + 8*v - 25) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3$$ (b3 + b2 - 2*b1 - 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3$$ (-b3 + 2*b2 + 14*b1 + 13) / 3 $$\nu^{3}$$ $$=$$ $$( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3$$ (8*b3 - 4*b2 - 4*b1 + 19) / 3

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
211.1
 −1.63746 + 1.52274i 2.13746 − 0.656712i −1.63746 − 1.52274i 2.13746 + 0.656712i
−0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −3.27492 −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 1.63746 + 2.83616i
211.2 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 4.27492 −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −2.13746 3.70219i
295.1 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −3.27492 −0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 1.63746 2.83616i
295.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 4.27492 −0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i −2.13746 + 3.70219i
 $$n$$: 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
13.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.l.j 4
3.b odd 2 1 1638.2.r.x 4
13.c even 3 1 inner 546.2.l.j 4
13.c even 3 1 7098.2.a.ca 2
13.e even 6 1 7098.2.a.bm 2
39.i odd 6 1 1638.2.r.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.j 4 1.a even 1 1 trivial
546.2.l.j 4 13.c even 3 1 inner
1638.2.r.x 4 3.b odd 2 1
1638.2.r.x 4 39.i odd 6 1
7098.2.a.bm 2 13.e even 6 1
7098.2.a.ca 2 13.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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{2} - T_{5} - 14$$ T5^2 - T5 - 14 $$T_{11}^{2} - 4T_{11} + 16$$ T11^2 - 4*T11 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} - T - 14)^{2}$$
$7$ $$(T^{2} + T + 1)^{2}$$
$11$ $$(T^{2} - 4 T + 16)^{2}$$
$13$ $$(T^{2} - 7 T + 13)^{2}$$
$17$ $$(T^{2} + 3 T + 9)^{2}$$
$19$ $$T^{4} - 2 T^{3} + \cdots + 3136$$
$23$ $$T^{4} - 3 T^{3} + \cdots + 144$$
$29$ $$T^{4} + 9 T^{3} + \cdots + 36$$
$31$ $$(T^{2} - 5 T - 8)^{2}$$
$37$ $$T^{4} - 7 T^{3} + \cdots + 4$$
$41$ $$T^{4} + 9 T^{3} + \cdots + 36$$
$43$ $$T^{4} - T^{3} + \cdots + 16384$$
$47$ $$(T^{2} - 2 T - 56)^{2}$$
$53$ $$(T^{2} - 8 T - 41)^{2}$$
$59$ $$T^{4} - T^{3} + \cdots + 16384$$
$61$ $$(T^{2} - 9 T + 81)^{2}$$
$67$ $$T^{4} - 13 T^{3} + \cdots + 784$$
$71$ $$T^{4} + 3 T^{3} + \cdots + 144$$
$73$ $$(T^{2} - 19 T + 76)^{2}$$
$79$ $$(T + 8)^{4}$$
$83$ $$(T^{2} + T - 128)^{2}$$
$89$ $$T^{4} + 9 T^{3} + \cdots + 36$$
$97$ $$T^{4} + 14 T^{3} + \cdots + 64$$