# Properties

 Label 546.2.e.f 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: 8.0.10070523904.11 Defining polynomial: $$x^{8} - 10 x^{4} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 10 x^{4} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - \nu^{2}$$$$)/18$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 7 \nu$$$$)/6$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 19 \nu^{2}$$$$)/18$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + 3 \nu^{5} + 9 \nu^{4} + \nu^{3} - \nu^{2} - 3 \nu - 45$$$$)/36$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 3 \nu^{5} + 9 \nu^{4} - \nu^{3} + \nu^{2} + 3 \nu - 45$$$$)/36$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} - 10 \nu^{3}$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{5} + 5$$ $$\nu^{5}$$ $$=$$ $$6 \beta_{3} + 7 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{4} + 19 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{7} + 20 \beta_{6} - 20 \beta_{5} + 20 \beta_{3} + 20 \beta_{2} + 20 \beta_{1}$$

## 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
 −1.68014 + 0.420861i −1.68014 − 0.420861i −0.420861 + 1.68014i −0.420861 − 1.68014i 0.420861 + 1.68014i 0.420861 − 1.68014i 1.68014 + 0.420861i 1.68014 − 0.420861i
−1.00000 −1.68014 0.420861i 1.00000 0.841723i 1.68014 + 0.420861i 0.595188 2.57794i −1.00000 2.64575 + 1.41421i 0.841723i
545.2 −1.00000 −1.68014 + 0.420861i 1.00000 0.841723i 1.68014 0.420861i 0.595188 + 2.57794i −1.00000 2.64575 1.41421i 0.841723i
545.3 −1.00000 −0.420861 1.68014i 1.00000 3.36028i 0.420861 + 1.68014i 2.37608 + 1.16372i −1.00000 −2.64575 + 1.41421i 3.36028i
545.4 −1.00000 −0.420861 + 1.68014i 1.00000 3.36028i 0.420861 1.68014i 2.37608 1.16372i −1.00000 −2.64575 1.41421i 3.36028i
545.5 −1.00000 0.420861 1.68014i 1.00000 3.36028i −0.420861 + 1.68014i −2.37608 1.16372i −1.00000 −2.64575 1.41421i 3.36028i
545.6 −1.00000 0.420861 + 1.68014i 1.00000 3.36028i −0.420861 1.68014i −2.37608 + 1.16372i −1.00000 −2.64575 + 1.41421i 3.36028i
545.7 −1.00000 1.68014 0.420861i 1.00000 0.841723i −1.68014 + 0.420861i −0.595188 + 2.57794i −1.00000 2.64575 1.41421i 0.841723i
545.8 −1.00000 1.68014 + 0.420861i 1.00000 0.841723i −1.68014 0.420861i −0.595188 2.57794i −1.00000 2.64575 + 1.41421i 0.841723i
 $$n$$: 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.f 8
3.b odd 2 1 546.2.e.h yes 8
7.b odd 2 1 inner 546.2.e.f 8
13.b even 2 1 546.2.e.h yes 8
21.c even 2 1 546.2.e.h yes 8
39.d odd 2 1 inner 546.2.e.f 8
91.b odd 2 1 546.2.e.h yes 8
273.g even 2 1 inner 546.2.e.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.e.f 8 1.a even 1 1 trivial
546.2.e.f 8 7.b odd 2 1 inner
546.2.e.f 8 39.d odd 2 1 inner
546.2.e.f 8 273.g even 2 1 inner
546.2.e.h yes 8 3.b odd 2 1
546.2.e.h yes 8 13.b even 2 1
546.2.e.h yes 8 21.c even 2 1
546.2.e.h 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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{4} + 12 T_{5}^{2} + 8$$ $$T_{17}^{4} - 80 T_{17}^{2} + 1152$$ $$T_{19}^{4} - 52 T_{19}^{2} + 648$$ $$T_{71}^{2} - 8 T_{71} - 96$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{8}$$
$3$ $$81 - 10 T^{4} + T^{8}$$
$5$ $$( 8 + 12 T^{2} + T^{4} )^{2}$$
$7$ $$2401 + 196 T^{2} - 10 T^{4} + 4 T^{6} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$28561 + 310 T^{4} + T^{8}$$
$17$ $$( 1152 - 80 T^{2} + T^{4} )^{2}$$
$19$ $$( 648 - 52 T^{2} + T^{4} )^{2}$$
$23$ $$( 16 + 64 T^{2} + T^{4} )^{2}$$
$29$ $$( 144 + 32 T^{2} + T^{4} )^{2}$$
$31$ $$( 288 - 40 T^{2} + T^{4} )^{2}$$
$37$ $$( 144 + 32 T^{2} + T^{4} )^{2}$$
$41$ $$( 11552 + 216 T^{2} + T^{4} )^{2}$$
$43$ $$( -8 + T )^{8}$$
$47$ $$( 1152 + 80 T^{2} + T^{4} )^{2}$$
$53$ $$( 16 + 64 T^{2} + T^{4} )^{2}$$
$59$ $$( 72 + 20 T^{2} + T^{4} )^{2}$$
$61$ $$( 648 + 52 T^{2} + T^{4} )^{2}$$
$67$ $$( 576 + 176 T^{2} + T^{4} )^{2}$$
$71$ $$( -96 - 8 T + T^{2} )^{4}$$
$73$ $$( 1568 - 168 T^{2} + T^{4} )^{2}$$
$79$ $$( 8 + 12 T + T^{2} )^{4}$$
$83$ $$( 6728 + 180 T^{2} + T^{4} )^{2}$$
$89$ $$( 1568 + 168 T^{2} + T^{4} )^{2}$$
$97$ $$( 11552 - 216 T^{2} + T^{4} )^{2}$$