# Properties

 Label 546.2.q.a Level $546$ Weight $2$ Character orbit 546.q Analytic conductor $4.360$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{6}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i −1.50000 0.866025i −0.500000 0.866025i 3.46410i 1.50000 0.866025i 0.500000 2.59808i 1.00000 1.50000 + 2.59808i 3.00000 + 1.73205i
335.1 −0.500000 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i 3.46410i 1.50000 + 0.866025i 0.500000 + 2.59808i 1.00000 1.50000 2.59808i 3.00000 1.73205i
 $$n$$: 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
273.u even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.q.a 2
3.b odd 2 1 546.2.q.c yes 2
7.b odd 2 1 546.2.q.b yes 2
13.e even 6 1 546.2.q.d yes 2
21.c even 2 1 546.2.q.d yes 2
39.h odd 6 1 546.2.q.b yes 2
91.t odd 6 1 546.2.q.c yes 2
273.u even 6 1 inner 546.2.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.q.a 2 1.a even 1 1 trivial
546.2.q.a 2 273.u even 6 1 inner
546.2.q.b yes 2 7.b odd 2 1
546.2.q.b yes 2 39.h odd 6 1
546.2.q.c yes 2 3.b odd 2 1
546.2.q.c yes 2 91.t odd 6 1
546.2.q.d yes 2 13.e even 6 1
546.2.q.d yes 2 21.c even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 12$$ T5^2 + 12 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9 $$T_{17}^{2} + 3T_{17} + 9$$ T17^2 + 3*T17 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} + 12$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + 7T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$T^{2} + 12T + 48$$
$29$ $$T^{2} - 3T + 3$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 12T + 48$$
$41$ $$T^{2} + 9T + 27$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} + 75$$
$53$ $$T^{2} + 75$$
$59$ $$T^{2} - 18T + 108$$
$61$ $$T^{2} + 9T + 27$$
$67$ $$T^{2}$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$(T - 4)^{2}$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 15T + 75$$
$97$ $$T^{2} + 2T + 4$$