# Properties

 Label 585.2.bt.a Level $585$ Weight $2$ Character orbit 585.bt Analytic conductor $4.671$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.bt (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
376.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−1.73205 1.00000i 1.50000 0.866025i 1.00000 + 1.73205i −0.866025 + 0.500000i −3.46410 −1.73205 1.00000i 0 1.50000 2.59808i 2.00000
376.2 1.73205 + 1.00000i 1.50000 0.866025i 1.00000 + 1.73205i 0.866025 0.500000i 3.46410 1.73205 + 1.00000i 0 1.50000 2.59808i 2.00000
571.1 −1.73205 + 1.00000i 1.50000 + 0.866025i 1.00000 1.73205i −0.866025 0.500000i −3.46410 −1.73205 + 1.00000i 0 1.50000 + 2.59808i 2.00000
571.2 1.73205 1.00000i 1.50000 + 0.866025i 1.00000 1.73205i 0.866025 + 0.500000i 3.46410 1.73205 1.00000i 0 1.50000 + 2.59808i 2.00000
 $$n$$: 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
9.c even 3 1 inner
13.b even 2 1 inner
117.t even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bt.a 4
9.c even 3 1 inner 585.2.bt.a 4
13.b even 2 1 inner 585.2.bt.a 4
117.t even 6 1 inner 585.2.bt.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.bt.a 4 1.a even 1 1 trivial
585.2.bt.a 4 9.c even 3 1 inner
585.2.bt.a 4 13.b even 2 1 inner
585.2.bt.a 4 117.t even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4T_{2}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$(T^{2} - 3 T + 3)^{2}$$
$5$ $$T^{4} - T^{2} + 1$$
$7$ $$T^{4} - 4T^{2} + 16$$
$11$ $$T^{4} - 36T^{2} + 1296$$
$13$ $$T^{4} - 6 T^{3} + 23 T^{2} - 78 T + 169$$
$17$ $$(T + 1)^{4}$$
$19$ $$(T^{2} + 64)^{2}$$
$23$ $$(T^{2} + T + 1)^{2}$$
$29$ $$(T^{2} + 2 T + 4)^{2}$$
$31$ $$T^{4} - 64T^{2} + 4096$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$T^{4} - 4T^{2} + 16$$
$43$ $$(T^{2} + 5 T + 25)^{2}$$
$47$ $$T^{4} - 100 T^{2} + 10000$$
$53$ $$(T - 9)^{4}$$
$59$ $$T^{4} - 36T^{2} + 1296$$
$61$ $$(T^{2} - 5 T + 25)^{2}$$
$67$ $$T^{4}$$
$71$ $$(T^{2} + 100)^{2}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T^{2} + T + 1)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} - 100 T^{2} + 10000$$