# 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: $$x^{4} - x^{2} + 1$$ 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.

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

## 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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{2} + 12$$ T5^2 + 12 $$T_{11}^{4} - 16T_{11}^{2} + 256$$ T11^4 - 16*T11^2 + 256

## Hecke characteristic polynomials

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