# Properties

 Label 546.2.bk.a Level $546$ Weight $2$ Character orbit 546.bk 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.bk (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.3317760000.2 Defining polynomial: $$x^{8} - 25 x^{4} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 25 x^{4} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/25$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/25$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/125$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/125$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$25 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$25 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$125 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$125 \beta_{7}$$

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1 + \beta_{4}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −0.578737 − 2.15988i 0.578737 + 2.15988i −2.15988 + 0.578737i 2.15988 − 0.578737i −0.578737 + 2.15988i 0.578737 − 2.15988i −2.15988 − 0.578737i 2.15988 + 0.578737i
−0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −3.60464 + 2.08114i 1.00000i −2.59808 + 0.500000i 1.00000i −0.500000 0.866025i 2.08114 3.60464i
25.2 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 1.87259 1.08114i 1.00000i −2.59808 + 0.500000i 1.00000i −0.500000 0.866025i −1.08114 + 1.87259i
25.3 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −1.87259 + 1.08114i 1.00000i 2.59808 0.500000i 1.00000i −0.500000 0.866025i −1.08114 + 1.87259i
25.4 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 3.60464 2.08114i 1.00000i 2.59808 0.500000i 1.00000i −0.500000 0.866025i 2.08114 3.60464i
415.1 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −3.60464 2.08114i 1.00000i −2.59808 0.500000i 1.00000i −0.500000 + 0.866025i 2.08114 + 3.60464i
415.2 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 1.87259 + 1.08114i 1.00000i −2.59808 0.500000i 1.00000i −0.500000 + 0.866025i −1.08114 1.87259i
415.3 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −1.87259 1.08114i 1.00000i 2.59808 + 0.500000i 1.00000i −0.500000 + 0.866025i −1.08114 1.87259i
415.4 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 3.60464 + 2.08114i 1.00000i 2.59808 + 0.500000i 1.00000i −0.500000 + 0.866025i 2.08114 + 3.60464i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 415.4 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
13.b even 2 1 inner
91.r 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.bk.a 8
3.b odd 2 1 1638.2.dm.b 8
7.c even 3 1 inner 546.2.bk.a 8
7.c even 3 1 3822.2.c.g 4
7.d odd 6 1 3822.2.c.f 4
13.b even 2 1 inner 546.2.bk.a 8
21.h odd 6 1 1638.2.dm.b 8
39.d odd 2 1 1638.2.dm.b 8
91.r even 6 1 inner 546.2.bk.a 8
91.r even 6 1 3822.2.c.g 4
91.s odd 6 1 3822.2.c.f 4
273.w odd 6 1 1638.2.dm.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bk.a 8 1.a even 1 1 trivial
546.2.bk.a 8 7.c even 3 1 inner
546.2.bk.a 8 13.b even 2 1 inner
546.2.bk.a 8 91.r even 6 1 inner
1638.2.dm.b 8 3.b odd 2 1
1638.2.dm.b 8 21.h odd 6 1
1638.2.dm.b 8 39.d odd 2 1
1638.2.dm.b 8 273.w odd 6 1
3822.2.c.f 4 7.d odd 6 1
3822.2.c.f 4 91.s odd 6 1
3822.2.c.g 4 7.c even 3 1
3822.2.c.g 4 91.r even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 22 T_{5}^{6} + 403 T_{5}^{4} - 1782 T_{5}^{2} + 6561$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{4}$$
$5$ $$6561 - 1782 T^{2} + 403 T^{4} - 22 T^{6} + T^{8}$$
$7$ $$( 49 - 13 T^{2} + T^{4} )^{2}$$
$11$ $$6561 - 1782 T^{2} + 403 T^{4} - 22 T^{6} + T^{8}$$
$13$ $$( 169 + 104 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$17$ $$( 36 - 24 T + 22 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$19$ $$( 1600 - 40 T^{2} + T^{4} )^{2}$$
$23$ $$( 36 + 24 T + 22 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$29$ $$( -9 - 2 T + T^{2} )^{4}$$
$31$ $$( 6561 - 81 T^{2} + T^{4} )^{2}$$
$37$ $$( 1600 - 40 T^{2} + T^{4} )^{2}$$
$41$ $$T^{8}$$
$43$ $$( 26 + 12 T + T^{2} )^{4}$$
$47$ $$29986576 - 1160912 T^{2} + 39468 T^{4} - 212 T^{6} + T^{8}$$
$53$ $$( 1 + 6 T + 37 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$59$ $$50625 - 15750 T^{2} + 4675 T^{4} - 70 T^{6} + T^{8}$$
$61$ $$( 2916 - 648 T + 198 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$67$ $$456976 - 62192 T^{2} + 7788 T^{4} - 92 T^{6} + T^{8}$$
$71$ $$( 3600 + 280 T^{2} + T^{4} )^{2}$$
$73$ $$( 1600 - 40 T^{2} + T^{4} )^{2}$$
$79$ $$( 81 + 9 T + T^{2} )^{4}$$
$83$ $$( 62001 + 502 T^{2} + T^{4} )^{2}$$
$89$ $$( 100 - 10 T^{2} + T^{4} )^{2}$$
$97$ $$( 121 + T^{2} )^{4}$$