# Properties

 Label 546.2.e.e Level $546$ Weight $2$ Character orbit 546.e Analytic conductor $4.360$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.303595776.1 Defining polynomial: $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 8 \nu^{4} + 16 \nu^{2} + 45$$$$)/24$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu$$$$)/54$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153$$$$)/72$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu$$$$)/216$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{6} - 16 \nu^{4} - 8 \nu^{2} - 81$$$$)/72$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 45 \nu$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} - 4 \beta_{5} - \beta_{3} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{4} + 5 \beta_{2} - 3$$ $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 8 \beta_{5} + 8 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{6} + 8 \beta_{4} - 16 \beta_{2} - 5$$ $$\nu^{7}$$ $$=$$ $$24 \beta_{5} - 24 \beta_{3} - 13 \beta_{1}$$

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
545.1
 −1.26217 − 1.18614i −1.26217 + 1.18614i −0.396143 − 1.68614i −0.396143 + 1.68614i 0.396143 − 1.68614i 0.396143 + 1.68614i 1.26217 − 1.18614i 1.26217 + 1.18614i
−1.00000 −1.26217 1.18614i 1.00000 3.37228i 1.26217 + 1.18614i 2.52434 + 0.792287i −1.00000 0.186141 + 2.99422i 3.37228i
545.2 −1.00000 −1.26217 + 1.18614i 1.00000 3.37228i 1.26217 1.18614i 2.52434 0.792287i −1.00000 0.186141 2.99422i 3.37228i
545.3 −1.00000 −0.396143 1.68614i 1.00000 2.37228i 0.396143 + 1.68614i 0.792287 2.52434i −1.00000 −2.68614 + 1.33591i 2.37228i
545.4 −1.00000 −0.396143 + 1.68614i 1.00000 2.37228i 0.396143 1.68614i 0.792287 + 2.52434i −1.00000 −2.68614 1.33591i 2.37228i
545.5 −1.00000 0.396143 1.68614i 1.00000 2.37228i −0.396143 + 1.68614i −0.792287 + 2.52434i −1.00000 −2.68614 1.33591i 2.37228i
545.6 −1.00000 0.396143 + 1.68614i 1.00000 2.37228i −0.396143 1.68614i −0.792287 2.52434i −1.00000 −2.68614 + 1.33591i 2.37228i
545.7 −1.00000 1.26217 1.18614i 1.00000 3.37228i −1.26217 + 1.18614i −2.52434 0.792287i −1.00000 0.186141 2.99422i 3.37228i
545.8 −1.00000 1.26217 + 1.18614i 1.00000 3.37228i −1.26217 1.18614i −2.52434 + 0.792287i −1.00000 0.186141 + 2.99422i 3.37228i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 545.8 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.b odd 2 1 inner
39.d odd 2 1 inner
273.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.e.e 8
3.b odd 2 1 546.2.e.g yes 8
7.b odd 2 1 inner 546.2.e.e 8
13.b even 2 1 546.2.e.g yes 8
21.c even 2 1 546.2.e.g yes 8
39.d odd 2 1 inner 546.2.e.e 8
91.b odd 2 1 546.2.e.g yes 8
273.g even 2 1 inner 546.2.e.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.e.e 8 1.a even 1 1 trivial
546.2.e.e 8 7.b odd 2 1 inner
546.2.e.e 8 39.d odd 2 1 inner
546.2.e.e 8 273.g even 2 1 inner
546.2.e.g yes 8 3.b odd 2 1
546.2.e.g yes 8 13.b even 2 1
546.2.e.g yes 8 21.c even 2 1
546.2.e.g yes 8 91.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 17 T_{5}^{2} + 64$$ $$T_{17}^{4} - 7 T_{17}^{2} + 4$$ $$T_{19}^{4} - 63 T_{19}^{2} + 324$$ $$T_{71} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{8}$$
$3$ $$81 + 45 T^{2} + 16 T^{4} + 5 T^{6} + T^{8}$$
$5$ $$( 64 + 17 T^{2} + T^{4} )^{2}$$
$7$ $$2401 - 34 T^{4} + T^{8}$$
$11$ $$( -33 + T^{2} )^{4}$$
$13$ $$( 169 - 22 T^{2} + T^{4} )^{2}$$
$17$ $$( 4 - 7 T^{2} + T^{4} )^{2}$$
$19$ $$( 324 - 63 T^{2} + T^{4} )^{2}$$
$23$ $$( 1 + 46 T^{2} + T^{4} )^{2}$$
$29$ $$( 324 + 63 T^{2} + T^{4} )^{2}$$
$31$ $$( 576 - 51 T^{2} + T^{4} )^{2}$$
$37$ $$( 1681 + 94 T^{2} + T^{4} )^{2}$$
$41$ $$( 484 + 77 T^{2} + T^{4} )^{2}$$
$43$ $$( 22 + 11 T + T^{2} )^{4}$$
$47$ $$( 2304 + 129 T^{2} + T^{4} )^{2}$$
$53$ $$( 1024 + 112 T^{2} + T^{4} )^{2}$$
$59$ $$( 576 + 84 T^{2} + T^{4} )^{2}$$
$61$ $$( 841 + 74 T^{2} + T^{4} )^{2}$$
$67$ $$( 256 + 43 T^{2} + T^{4} )^{2}$$
$71$ $$( -2 + T )^{8}$$
$73$ $$( 1369 - 118 T^{2} + T^{4} )^{2}$$
$79$ $$( 34 + 13 T + T^{2} )^{4}$$
$83$ $$( 1024 + 68 T^{2} + T^{4} )^{2}$$
$89$ $$( 100 + T^{2} )^{4}$$
$97$ $$( 4624 - 139 T^{2} + T^{4} )^{2}$$