# Properties

 Label 546.2.a.j Level $546$ Weight $2$ Character orbit 546.a Self dual yes Analytic conductor $4.360$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + ( - \beta + 2) q^{5} + q^{6} - q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + (-b + 2) * q^5 + q^6 - q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + ( - \beta + 2) q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + ( - \beta + 2) q^{10} + \beta q^{11} + q^{12} + q^{13} - q^{14} + ( - \beta + 2) q^{15} + q^{16} + (3 \beta - 2) q^{17} + q^{18} + 3 \beta q^{19} + ( - \beta + 2) q^{20} - q^{21} + \beta q^{22} + (\beta - 4) q^{23} + q^{24} + ( - 3 \beta + 3) q^{25} + q^{26} + q^{27} - q^{28} + ( - 3 \beta + 2) q^{29} + ( - \beta + 2) q^{30} - 4 \beta q^{31} + q^{32} + \beta q^{33} + (3 \beta - 2) q^{34} + (\beta - 2) q^{35} + q^{36} + (\beta - 6) q^{37} + 3 \beta q^{38} + q^{39} + ( - \beta + 2) q^{40} + (2 \beta + 2) q^{41} - q^{42} + ( - \beta - 8) q^{43} + \beta q^{44} + ( - \beta + 2) q^{45} + (\beta - 4) q^{46} + q^{48} + q^{49} + ( - 3 \beta + 3) q^{50} + (3 \beta - 2) q^{51} + q^{52} + ( - 4 \beta + 6) q^{53} + q^{54} + (\beta - 4) q^{55} - q^{56} + 3 \beta q^{57} + ( - 3 \beta + 2) q^{58} + ( - 4 \beta - 4) q^{59} + ( - \beta + 2) q^{60} + ( - 3 \beta + 2) q^{61} - 4 \beta q^{62} - q^{63} + q^{64} + ( - \beta + 2) q^{65} + \beta q^{66} + (2 \beta - 4) q^{67} + (3 \beta - 2) q^{68} + (\beta - 4) q^{69} + (\beta - 2) q^{70} + 8 q^{71} + q^{72} + (\beta - 2) q^{73} + (\beta - 6) q^{74} + ( - 3 \beta + 3) q^{75} + 3 \beta q^{76} - \beta q^{77} + q^{78} + (2 \beta - 8) q^{79} + ( - \beta + 2) q^{80} + q^{81} + (2 \beta + 2) q^{82} + (2 \beta + 12) q^{83} - q^{84} + (5 \beta - 16) q^{85} + ( - \beta - 8) q^{86} + ( - 3 \beta + 2) q^{87} + \beta q^{88} + 10 q^{89} + ( - \beta + 2) q^{90} - q^{91} + (\beta - 4) q^{92} - 4 \beta q^{93} + (3 \beta - 12) q^{95} + q^{96} + ( - 8 \beta + 2) q^{97} + q^{98} + \beta q^{99} +O(q^{100})$$ q + q^2 + q^3 + q^4 + (-b + 2) * q^5 + q^6 - q^7 + q^8 + q^9 + (-b + 2) * q^10 + b * q^11 + q^12 + q^13 - q^14 + (-b + 2) * q^15 + q^16 + (3*b - 2) * q^17 + q^18 + 3*b * q^19 + (-b + 2) * q^20 - q^21 + b * q^22 + (b - 4) * q^23 + q^24 + (-3*b + 3) * q^25 + q^26 + q^27 - q^28 + (-3*b + 2) * q^29 + (-b + 2) * q^30 - 4*b * q^31 + q^32 + b * q^33 + (3*b - 2) * q^34 + (b - 2) * q^35 + q^36 + (b - 6) * q^37 + 3*b * q^38 + q^39 + (-b + 2) * q^40 + (2*b + 2) * q^41 - q^42 + (-b - 8) * q^43 + b * q^44 + (-b + 2) * q^45 + (b - 4) * q^46 + q^48 + q^49 + (-3*b + 3) * q^50 + (3*b - 2) * q^51 + q^52 + (-4*b + 6) * q^53 + q^54 + (b - 4) * q^55 - q^56 + 3*b * q^57 + (-3*b + 2) * q^58 + (-4*b - 4) * q^59 + (-b + 2) * q^60 + (-3*b + 2) * q^61 - 4*b * q^62 - q^63 + q^64 + (-b + 2) * q^65 + b * q^66 + (2*b - 4) * q^67 + (3*b - 2) * q^68 + (b - 4) * q^69 + (b - 2) * q^70 + 8 * q^71 + q^72 + (b - 2) * q^73 + (b - 6) * q^74 + (-3*b + 3) * q^75 + 3*b * q^76 - b * q^77 + q^78 + (2*b - 8) * q^79 + (-b + 2) * q^80 + q^81 + (2*b + 2) * q^82 + (2*b + 12) * q^83 - q^84 + (5*b - 16) * q^85 + (-b - 8) * q^86 + (-3*b + 2) * q^87 + b * q^88 + 10 * q^89 + (-b + 2) * q^90 - q^91 + (b - 4) * q^92 - 4*b * q^93 + (3*b - 12) * q^95 + q^96 + (-8*b + 2) * q^97 + q^98 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{10} + q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 3 q^{15} + 2 q^{16} - q^{17} + 2 q^{18} + 3 q^{19} + 3 q^{20} - 2 q^{21} + q^{22} - 7 q^{23} + 2 q^{24} + 3 q^{25} + 2 q^{26} + 2 q^{27} - 2 q^{28} + q^{29} + 3 q^{30} - 4 q^{31} + 2 q^{32} + q^{33} - q^{34} - 3 q^{35} + 2 q^{36} - 11 q^{37} + 3 q^{38} + 2 q^{39} + 3 q^{40} + 6 q^{41} - 2 q^{42} - 17 q^{43} + q^{44} + 3 q^{45} - 7 q^{46} + 2 q^{48} + 2 q^{49} + 3 q^{50} - q^{51} + 2 q^{52} + 8 q^{53} + 2 q^{54} - 7 q^{55} - 2 q^{56} + 3 q^{57} + q^{58} - 12 q^{59} + 3 q^{60} + q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} + 3 q^{65} + q^{66} - 6 q^{67} - q^{68} - 7 q^{69} - 3 q^{70} + 16 q^{71} + 2 q^{72} - 3 q^{73} - 11 q^{74} + 3 q^{75} + 3 q^{76} - q^{77} + 2 q^{78} - 14 q^{79} + 3 q^{80} + 2 q^{81} + 6 q^{82} + 26 q^{83} - 2 q^{84} - 27 q^{85} - 17 q^{86} + q^{87} + q^{88} + 20 q^{89} + 3 q^{90} - 2 q^{91} - 7 q^{92} - 4 q^{93} - 21 q^{95} + 2 q^{96} - 4 q^{97} + 2 q^{98} + q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 3 * q^10 + q^11 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 3 * q^15 + 2 * q^16 - q^17 + 2 * q^18 + 3 * q^19 + 3 * q^20 - 2 * q^21 + q^22 - 7 * q^23 + 2 * q^24 + 3 * q^25 + 2 * q^26 + 2 * q^27 - 2 * q^28 + q^29 + 3 * q^30 - 4 * q^31 + 2 * q^32 + q^33 - q^34 - 3 * q^35 + 2 * q^36 - 11 * q^37 + 3 * q^38 + 2 * q^39 + 3 * q^40 + 6 * q^41 - 2 * q^42 - 17 * q^43 + q^44 + 3 * q^45 - 7 * q^46 + 2 * q^48 + 2 * q^49 + 3 * q^50 - q^51 + 2 * q^52 + 8 * q^53 + 2 * q^54 - 7 * q^55 - 2 * q^56 + 3 * q^57 + q^58 - 12 * q^59 + 3 * q^60 + q^61 - 4 * q^62 - 2 * q^63 + 2 * q^64 + 3 * q^65 + q^66 - 6 * q^67 - q^68 - 7 * q^69 - 3 * q^70 + 16 * q^71 + 2 * q^72 - 3 * q^73 - 11 * q^74 + 3 * q^75 + 3 * q^76 - q^77 + 2 * q^78 - 14 * q^79 + 3 * q^80 + 2 * q^81 + 6 * q^82 + 26 * q^83 - 2 * q^84 - 27 * q^85 - 17 * q^86 + q^87 + q^88 + 20 * q^89 + 3 * q^90 - 2 * q^91 - 7 * q^92 - 4 * q^93 - 21 * q^95 + 2 * q^96 - 4 * q^97 + 2 * q^98 + 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
 2.56155 −1.56155
1.00000 1.00000 1.00000 −0.561553 1.00000 −1.00000 1.00000 1.00000 −0.561553
1.2 1.00000 1.00000 1.00000 3.56155 1.00000 −1.00000 1.00000 1.00000 3.56155
 $$n$$: 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.j 2
3.b odd 2 1 1638.2.a.u 2
4.b odd 2 1 4368.2.a.be 2
7.b odd 2 1 3822.2.a.bo 2
13.b even 2 1 7098.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.j 2 1.a even 1 1 trivial
1638.2.a.u 2 3.b odd 2 1
3822.2.a.bo 2 7.b odd 2 1
4368.2.a.be 2 4.b odd 2 1
7098.2.a.bl 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(546))$$:

 $$T_{5}^{2} - 3T_{5} - 2$$ T5^2 - 3*T5 - 2 $$T_{11}^{2} - T_{11} - 4$$ T11^2 - T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 3T - 2$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - T - 4$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + T - 38$$
$19$ $$T^{2} - 3T - 36$$
$23$ $$T^{2} + 7T + 8$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} + 4T - 64$$
$37$ $$T^{2} + 11T + 26$$
$41$ $$T^{2} - 6T - 8$$
$43$ $$T^{2} + 17T + 68$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 8T - 52$$
$59$ $$T^{2} + 12T - 32$$
$61$ $$T^{2} - T - 38$$
$67$ $$T^{2} + 6T - 8$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 3T - 2$$
$79$ $$T^{2} + 14T + 32$$
$83$ $$T^{2} - 26T + 152$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 4T - 268$$