# Properties

 Label 546.2.a.c Level $546$ Weight $2$ Character orbit 546.a Self dual yes Analytic conductor $4.360$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,2,Mod(1,546)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("546.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.a (trivial)

## 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: yes Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 1.00000 1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$7$$ $$+1$$
$$13$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.a.c 1
3.b odd 2 1 1638.2.a.o 1
4.b odd 2 1 4368.2.a.h 1
7.b odd 2 1 3822.2.a.e 1
13.b even 2 1 7098.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.c 1 1.a even 1 1 trivial
1638.2.a.o 1 3.b odd 2 1
3822.2.a.e 1 7.b odd 2 1
4368.2.a.h 1 4.b odd 2 1
7098.2.a.z 1 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(546))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T + 1$$
$17$ $$T - 5$$
$19$ $$T - 1$$
$23$ $$T - 3$$
$29$ $$T - 5$$
$31$ $$T - 4$$
$37$ $$T + 5$$
$41$ $$T + 8$$
$43$ $$T + 1$$
$47$ $$T - 8$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T - 13$$
$67$ $$T + 10$$
$71$ $$T - 8$$
$73$ $$T + 15$$
$79$ $$T - 6$$
$83$ $$T + 2$$
$89$ $$T + 2$$
$97$ $$T + 2$$