# Properties

 Label 546.2.i.j Level $546$ Weight $2$ Character orbit 546.i Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## 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
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.822876 + 1.42526i −1.00000 −1.32288 + 2.29129i −1.00000 −0.500000 + 0.866025i 0.822876 + 1.42526i
79.2 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.82288 3.15731i −1.00000 1.32288 2.29129i −1.00000 −0.500000 + 0.866025i −1.82288 3.15731i
235.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.822876 1.42526i −1.00000 −1.32288 2.29129i −1.00000 −0.500000 0.866025i 0.822876 1.42526i
235.2 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.82288 + 3.15731i −1.00000 1.32288 + 2.29129i −1.00000 −0.500000 0.866025i −1.82288 + 3.15731i
 $$n$$: 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.j 4
3.b odd 2 1 1638.2.j.k 4
7.c even 3 1 inner 546.2.i.j 4
7.c even 3 1 3822.2.a.bk 2
7.d odd 6 1 3822.2.a.bi 2
21.h odd 6 1 1638.2.j.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.j 4 1.a even 1 1 trivial
546.2.i.j 4 7.c even 3 1 inner
1638.2.j.k 4 3.b odd 2 1
1638.2.j.k 4 21.h odd 6 1
3822.2.a.bi 2 7.d odd 6 1
3822.2.a.bk 2 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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{4} - 2T_{5}^{3} + 10T_{5}^{2} + 12T_{5} + 36$$ T5^4 - 2*T5^3 + 10*T5^2 + 12*T5 + 36 $$T_{17}^{4} - 8T_{17}^{3} + 55T_{17}^{2} - 72T_{17} + 81$$ T17^4 - 8*T17^3 + 55*T17^2 - 72*T17 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$T^{4} - 2 T^{3} + \cdots + 36$$
$7$ $$T^{4} + 7T^{2} + 49$$
$11$ $$T^{4} - 4 T^{3} + \cdots + 9$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} - 8 T^{3} + \cdots + 81$$
$19$ $$(T^{2} + 5 T + 25)^{2}$$
$23$ $$T^{4} - 10 T^{3} + \cdots + 324$$
$29$ $$(T^{2} + 2 T - 27)^{2}$$
$31$ $$T^{4} - 4 T^{3} + \cdots + 576$$
$37$ $$T^{4} + 6 T^{3} + \cdots + 4$$
$41$ $$(T^{2} + 10 T + 18)^{2}$$
$43$ $$(T^{2} - 28)^{2}$$
$47$ $$(T^{2} + 3 T + 9)^{2}$$
$53$ $$(T^{2} - 3 T + 9)^{2}$$
$59$ $$T^{4} + 63T^{2} + 3969$$
$61$ $$T^{4} - 8 T^{3} + \cdots + 2209$$
$67$ $$T^{4} - 6 T^{3} + \cdots + 10609$$
$71$ $$(T^{2} - 22 T + 93)^{2}$$
$73$ $$T^{4} - 22 T^{3} + \cdots + 12996$$
$79$ $$(T^{2} - 10 T + 100)^{2}$$
$83$ $$(T^{2} + 16 T + 36)^{2}$$
$89$ $$T^{4} + 18 T^{3} + \cdots + 324$$
$97$ $$(T^{2} + 14 T - 14)^{2}$$