# Properties

 Label 585.2.bs.b Level $585$ Weight $2$ Character orbit 585.bs Analytic conductor $4.671$ Analytic rank $0$ Dimension $24$ 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.bs (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 8 q^{4} - 4 q^{5}+O(q^{10})$$ 24 * q + 8 * q^4 - 4 * q^5 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 8 q^{4} - 4 q^{5} - 4 q^{10} - 4 q^{11} - 24 q^{14} + 16 q^{16} - 16 q^{19} + 16 q^{20} - 16 q^{25} + 48 q^{26} + 12 q^{29} + 8 q^{31} - 32 q^{34} - 10 q^{35} - 48 q^{40} + 40 q^{41} - 40 q^{44} - 24 q^{46} - 16 q^{49} - 20 q^{50} + 20 q^{55} + 24 q^{56} - 12 q^{59} + 20 q^{61} + 48 q^{64} - 14 q^{65} - 56 q^{70} - 4 q^{71} + 12 q^{74} + 8 q^{76} + 136 q^{79} + 4 q^{80} - 4 q^{85} - 48 q^{86} + 64 q^{89} + 60 q^{91} - 48 q^{94} + 28 q^{95}+O(q^{100})$$ 24 * q + 8 * q^4 - 4 * q^5 - 4 * q^10 - 4 * q^11 - 24 * q^14 + 16 * q^16 - 16 * q^19 + 16 * q^20 - 16 * q^25 + 48 * q^26 + 12 * q^29 + 8 * q^31 - 32 * q^34 - 10 * q^35 - 48 * q^40 + 40 * q^41 - 40 * q^44 - 24 * q^46 - 16 * q^49 - 20 * q^50 + 20 * q^55 + 24 * q^56 - 12 * q^59 + 20 * q^61 + 48 * q^64 - 14 * q^65 - 56 * q^70 - 4 * q^71 + 12 * q^74 + 8 * q^76 + 136 * q^79 + 4 * q^80 - 4 * q^85 - 48 * q^86 + 64 * q^89 + 60 * q^91 - 48 * q^94 + 28 * q^95

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1 −2.01317 + 1.16230i 0 1.70191 2.94779i 0.174568 + 2.22924i 0 0.473110 + 0.273150i 3.26331i 0 −2.94250 4.28495i
289.2 −1.85914 + 1.07337i 0 1.30426 2.25904i 2.16557 + 0.557052i 0 0.635729 + 0.367038i 1.30633i 0 −4.62401 + 1.28883i
289.3 −1.52669 + 0.881436i 0 0.553860 0.959313i −1.52636 1.63408i 0 −1.92736 1.11276i 1.57298i 0 3.77062 + 1.14935i
289.4 −1.16430 + 0.672211i 0 −0.0962645 + 0.166735i 0.868136 2.06066i 0 3.39681 + 1.96115i 2.94768i 0 0.374428 + 2.98281i
289.5 −0.729738 + 0.421315i 0 −0.644988 + 1.11715i −2.23540 0.0545741i 0 −0.347589 0.200681i 2.77223i 0 1.65425 0.901983i
289.6 −0.521384 + 0.301021i 0 −0.818772 + 1.41816i −0.446511 + 2.19103i 0 3.08191 + 1.77934i 2.18996i 0 −0.426744 1.27678i
289.7 0.521384 0.301021i 0 −0.818772 + 1.41816i −0.446511 2.19103i 0 −3.08191 1.77934i 2.18996i 0 −0.892352 1.00796i
289.8 0.729738 0.421315i 0 −0.644988 + 1.11715i −2.23540 + 0.0545741i 0 0.347589 + 0.200681i 2.77223i 0 −1.60827 + 0.981632i
289.9 1.16430 0.672211i 0 −0.0962645 + 0.166735i 0.868136 + 2.06066i 0 −3.39681 1.96115i 2.94768i 0 2.39598 + 1.81567i
289.10 1.52669 0.881436i 0 0.553860 0.959313i −1.52636 + 1.63408i 0 1.92736 + 1.11276i 1.57298i 0 −0.889946 + 3.84013i
289.11 1.85914 1.07337i 0 1.30426 2.25904i 2.16557 0.557052i 0 −0.635729 0.367038i 1.30633i 0 3.42817 3.36010i
289.12 2.01317 1.16230i 0 1.70191 2.94779i 0.174568 2.22924i 0 −0.473110 0.273150i 3.26331i 0 −2.23963 4.69075i
334.1 −2.01317 1.16230i 0 1.70191 + 2.94779i 0.174568 2.22924i 0 0.473110 0.273150i 3.26331i 0 −2.94250 + 4.28495i
334.2 −1.85914 1.07337i 0 1.30426 + 2.25904i 2.16557 0.557052i 0 0.635729 0.367038i 1.30633i 0 −4.62401 1.28883i
334.3 −1.52669 0.881436i 0 0.553860 + 0.959313i −1.52636 + 1.63408i 0 −1.92736 + 1.11276i 1.57298i 0 3.77062 1.14935i
334.4 −1.16430 0.672211i 0 −0.0962645 0.166735i 0.868136 + 2.06066i 0 3.39681 1.96115i 2.94768i 0 0.374428 2.98281i
334.5 −0.729738 0.421315i 0 −0.644988 1.11715i −2.23540 + 0.0545741i 0 −0.347589 + 0.200681i 2.77223i 0 1.65425 + 0.901983i
334.6 −0.521384 0.301021i 0 −0.818772 1.41816i −0.446511 2.19103i 0 3.08191 1.77934i 2.18996i 0 −0.426744 + 1.27678i
334.7 0.521384 + 0.301021i 0 −0.818772 1.41816i −0.446511 + 2.19103i 0 −3.08191 + 1.77934i 2.18996i 0 −0.892352 + 1.00796i
334.8 0.729738 + 0.421315i 0 −0.644988 1.11715i −2.23540 0.0545741i 0 0.347589 0.200681i 2.77223i 0 −1.60827 0.981632i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 334.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n 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.bs.b 24
3.b odd 2 1 195.2.ba.a 24
5.b even 2 1 inner 585.2.bs.b 24
13.c even 3 1 inner 585.2.bs.b 24
15.d odd 2 1 195.2.ba.a 24
15.e even 4 1 975.2.i.o 12
15.e even 4 1 975.2.i.q 12
39.i odd 6 1 195.2.ba.a 24
65.n even 6 1 inner 585.2.bs.b 24
195.x odd 6 1 195.2.ba.a 24
195.bl even 12 1 975.2.i.o 12
195.bl even 12 1 975.2.i.q 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.ba.a 24 3.b odd 2 1
195.2.ba.a 24 15.d odd 2 1
195.2.ba.a 24 39.i odd 6 1
195.2.ba.a 24 195.x odd 6 1
585.2.bs.b 24 1.a even 1 1 trivial
585.2.bs.b 24 5.b even 2 1 inner
585.2.bs.b 24 13.c even 3 1 inner
585.2.bs.b 24 65.n even 6 1 inner
975.2.i.o 12 15.e even 4 1
975.2.i.o 12 195.bl even 12 1
975.2.i.q 12 15.e even 4 1
975.2.i.q 12 195.bl even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - 16 T_{2}^{22} + 160 T_{2}^{20} - 1000 T_{2}^{18} + 4576 T_{2}^{16} - 14660 T_{2}^{14} + 34968 T_{2}^{12} - 57280 T_{2}^{10} + 67920 T_{2}^{8} - 49696 T_{2}^{6} + 25744 T_{2}^{4} - 7056 T_{2}^{2} + \cdots + 1296$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.