# Properties

 Label 546.2.i.h Level $546$ Weight $2$ Character orbit 546.i 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.i (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: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 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.

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

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

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

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.707107 + 1.22474i 1.00000 −1.62132 2.09077i 1.00000 −0.500000 + 0.866025i −0.707107 1.22474i
79.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.707107 1.22474i 1.00000 2.62132 + 0.358719i 1.00000 −0.500000 + 0.866025i 0.707107 + 1.22474i
235.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.707107 1.22474i 1.00000 −1.62132 + 2.09077i 1.00000 −0.500000 0.866025i −0.707107 + 1.22474i
235.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.707107 + 1.22474i 1.00000 2.62132 0.358719i 1.00000 −0.500000 0.866025i 0.707107 1.22474i
 $$n$$: 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.h 4
3.b odd 2 1 1638.2.j.n 4
7.c even 3 1 inner 546.2.i.h 4
7.c even 3 1 3822.2.a.bs 2
7.d odd 6 1 3822.2.a.bp 2
21.h odd 6 1 1638.2.j.n 4

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

 $$T_{5}^{4} + 2T_{5}^{2} + 4$$ T5^4 + 2*T5^2 + 4 $$T_{17}^{4} + 2T_{17}^{3} + 21T_{17}^{2} - 34T_{17} + 289$$ T17^4 + 2*T17^3 + 21*T17^2 - 34*T17 + 289

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$T^{4} + 2T^{2} + 4$$
$7$ $$T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49$$
$11$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289$$
$19$ $$(T^{2} + 3 T + 9)^{2}$$
$23$ $$T^{4} - 4 T^{3} + 62 T^{2} + \cdots + 2116$$
$29$ $$(T - 5)^{4}$$
$31$ $$T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64$$
$37$ $$T^{4} + 4 T^{3} + 62 T^{2} + \cdots + 2116$$
$41$ $$(T^{2} - 12 T + 34)^{2}$$
$43$ $$(T^{2} - 4 T - 4)^{2}$$
$47$ $$(T^{2} + 9 T + 81)^{2}$$
$53$ $$T^{4} + 2 T^{3} + 131 T^{2} + \cdots + 16129$$
$59$ $$T^{4} + 18 T^{3} + 245 T^{2} + \cdots + 6241$$
$61$ $$T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289$$
$67$ $$T^{4} - 18 T^{3} + 275 T^{2} + \cdots + 2401$$
$71$ $$(T + 1)^{4}$$
$73$ $$T^{4} + 12 T^{3} + 158 T^{2} + \cdots + 196$$
$79$ $$T^{4} + 4 T^{3} + 140 T^{2} + \cdots + 15376$$
$83$ $$(T^{2} + 4 T - 28)^{2}$$
$89$ $$T^{4} - 4 T^{3} + 30 T^{2} + 56 T + 196$$
$97$ $$(T^{2} + 4 T - 94)^{2}$$