Properties

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

Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{12}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.46410i 0.866025 0.500000i 0.866025 0.500000i 1.00000i −0.500000 0.866025i −1.73205 + 3.00000i
43.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.46410i −0.866025 + 0.500000i −0.866025 + 0.500000i 1.00000i −0.500000 0.866025i 1.73205 3.00000i
127.1 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 3.46410i 0.866025 + 0.500000i 0.866025 + 0.500000i 1.00000i −0.500000 + 0.866025i −1.73205 3.00000i
127.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 3.46410i −0.866025 0.500000i −0.866025 0.500000i 1.00000i −0.500000 + 0.866025i 1.73205 + 3.00000i
 $$n$$: 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.e 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.s.c 4
3.b odd 2 1 1638.2.bj.e 4
13.e even 6 1 inner 546.2.s.c 4
13.f odd 12 1 7098.2.a.bn 2
13.f odd 12 1 7098.2.a.bz 2
39.h odd 6 1 1638.2.bj.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.c 4 1.a even 1 1 trivial
546.2.s.c 4 13.e even 6 1 inner
1638.2.bj.e 4 3.b odd 2 1
1638.2.bj.e 4 39.h odd 6 1
7098.2.a.bn 2 13.f odd 12 1
7098.2.a.bz 2 13.f odd 12 1

Hecke kernels

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

 $$T_{5}^{2} + 12$$ $$T_{11}^{4} - 16 T_{11}^{2} + 256$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 12 + T^{2} )^{2}$$
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$256 - 16 T^{2} + T^{4}$$
$13$ $$169 - T^{2} + T^{4}$$
$17$ $$9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4}$$
$19$ $$16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4}$$
$23$ $$169 - 104 T + 51 T^{2} - 8 T^{3} + T^{4}$$
$29$ $$( 64 + 8 T + T^{2} )^{2}$$
$31$ $$9 + 42 T^{2} + T^{4}$$
$37$ $$1936 + 1056 T + 236 T^{2} + 24 T^{3} + T^{4}$$
$41$ $$( 48 + 12 T + T^{2} )^{2}$$
$43$ $$121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4}$$
$47$ $$16 + 56 T^{2} + T^{4}$$
$53$ $$( 97 + 20 T + T^{2} )^{2}$$
$59$ $$3721 + 366 T - 49 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$729 + 27 T^{2} + T^{4}$$
$67$ $$529 - 414 T + 131 T^{2} - 18 T^{3} + T^{4}$$
$71$ $$529 - 552 T + 215 T^{2} - 24 T^{3} + T^{4}$$
$73$ $$1024 + 128 T^{2} + T^{4}$$
$79$ $$( 24 + 12 T + T^{2} )^{2}$$
$83$ $$( 3 + T^{2} )^{2}$$
$89$ $$9409 - 582 T - 85 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$10816 - 3744 T + 536 T^{2} - 36 T^{3} + T^{4}$$