# Properties

 Label 546.2.l.l Level $546$ Weight $2$ Character orbit 546.l Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ 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.l (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 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

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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
211.1
 1.28078 − 2.21837i −0.780776 + 1.35234i 1.28078 + 2.21837i −0.780776 − 1.35234i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.56155 0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 1.28078 + 2.21837i
211.2 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.56155 0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.780776 1.35234i
295.1 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −2.56155 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 1.28078 2.21837i
295.2 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 1.56155 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.780776 + 1.35234i
 $$n$$: 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.l 4
3.b odd 2 1 1638.2.r.y 4
13.c even 3 1 inner 546.2.l.l 4
13.c even 3 1 7098.2.a.bt 2
13.e even 6 1 7098.2.a.bi 2
39.i odd 6 1 1638.2.r.y 4

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

 $$T_{5}^{2} + T_{5} - 4$$ T5^2 + T5 - 4 $$T_{11}^{4} - T_{11}^{3} + 5T_{11}^{2} + 4T_{11} + 16$$ T11^4 - T11^3 + 5*T11^2 + 4*T11 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + T - 4)^{2}$$
$7$ $$(T^{2} + T + 1)^{2}$$
$11$ $$T^{4} - T^{3} + 5 T^{2} + 4 T + 16$$
$13$ $$(T^{2} + T + 13)^{2}$$
$17$ $$T^{4} - 8 T^{3} + 65 T^{2} + 8 T + 1$$
$19$ $$T^{4} - T^{3} + 5 T^{2} + 4 T + 16$$
$23$ $$T^{4} + 6 T^{3} + 44 T^{2} - 48 T + 64$$
$29$ $$T^{4} - 4 T^{3} + 29 T^{2} + 52 T + 169$$
$31$ $$T^{4}$$
$37$ $$T^{4} - 9 T^{3} + 65 T^{2} - 144 T + 256$$
$41$ $$T^{4} - 6 T^{3} + 95 T^{2} + \cdots + 3481$$
$43$ $$(T^{2} - 8 T + 64)^{2}$$
$47$ $$(T^{2} + 13 T + 4)^{2}$$
$53$ $$(T - 7)^{4}$$
$59$ $$T^{4} + 2 T^{3} + 20 T^{2} - 32 T + 256$$
$61$ $$T^{4} - 6 T^{3} + 95 T^{2} + \cdots + 3481$$
$67$ $$T^{4} + 10 T^{3} + 92 T^{2} + 80 T + 64$$
$71$ $$T^{4} + 2 T^{3} + 156 T^{2} + \cdots + 23104$$
$73$ $$(T^{2} + 9 T + 16)^{2}$$
$79$ $$(T^{2} + 9 T + 16)^{2}$$
$83$ $$(T^{2} + 2 T - 16)^{2}$$
$89$ $$T^{4} + 11 T^{3} + 95 T^{2} + \cdots + 676$$
$97$ $$T^{4} - 10 T^{3} + 92 T^{2} - 80 T + 64$$