# Properties

 Label 845.2.l.f Level $845$ Weight $2$ Character orbit 845.l Analytic conductor $6.747$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 65) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 8 q^{4} + 12 q^{9}+O(q^{10})$$ 24 * q - 8 * q^4 + 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 8 q^{4} + 12 q^{9} - 14 q^{10} - 88 q^{14} - 32 q^{16} + 4 q^{25} + 36 q^{29} - 8 q^{30} + 20 q^{35} + 4 q^{36} + 140 q^{40} - 12 q^{49} - 48 q^{51} - 52 q^{55} + 32 q^{56} + 12 q^{61} + 24 q^{64} + 8 q^{66} + 48 q^{69} + 16 q^{74} - 4 q^{75} - 208 q^{79} + 28 q^{81} - 124 q^{90} + 112 q^{94} - 40 q^{95}+O(q^{100})$$ 24 * q - 8 * q^4 + 12 * q^9 - 14 * q^10 - 88 * q^14 - 32 * q^16 + 4 * q^25 + 36 * q^29 - 8 * q^30 + 20 * q^35 + 4 * q^36 + 140 * q^40 - 12 * q^49 - 48 * q^51 - 52 * q^55 + 32 * q^56 + 12 * q^61 + 24 * q^64 + 8 * q^66 + 48 * q^69 + 16 * q^74 - 4 * q^75 - 208 * q^79 + 28 * q^81 - 124 * q^90 + 112 * q^94 - 40 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
654.1 −1.27287 + 2.20467i −1.86449 1.07646i −2.24039 3.88048i 2.08125 0.817544i 4.74650 2.74039i 1.46928 + 2.54486i 6.31544 0.817544 + 1.41603i −0.846746 + 5.62912i
654.2 −1.27287 + 2.20467i 1.86449 + 1.07646i −2.24039 3.88048i 2.08125 + 0.817544i −4.74650 + 2.74039i 1.46928 + 2.54486i 6.31544 0.817544 + 1.41603i −4.45158 + 3.54786i
654.3 −0.593667 + 1.02826i −0.298874 0.172555i 0.295120 + 0.511162i −1.71029 + 1.44045i 0.354863 0.204880i 1.01478 + 1.75765i −3.07548 −1.44045 2.49493i −0.465813 2.61378i
654.4 −0.593667 + 1.02826i 0.298874 + 0.172555i 0.295120 + 0.511162i −1.71029 1.44045i −0.354863 + 0.204880i 1.01478 + 1.75765i −3.07548 −1.44045 2.49493i 2.49650 0.903481i
654.5 −0.165418 + 0.286513i −2.33117 1.34590i 0.945274 + 1.63726i 0.702335 + 2.12291i 0.771236 0.445274i 1.67674 + 2.90420i −1.28714 2.12291 + 3.67698i −0.724419 0.149939i
654.6 −0.165418 + 0.286513i 2.33117 + 1.34590i 0.945274 + 1.63726i 0.702335 2.12291i −0.771236 + 0.445274i 1.67674 + 2.90420i −1.28714 2.12291 + 3.67698i 0.492061 + 0.552395i
654.7 0.165418 0.286513i −2.33117 1.34590i 0.945274 + 1.63726i −0.702335 2.12291i −0.771236 + 0.445274i −1.67674 2.90420i 1.28714 2.12291 + 3.67698i −0.724419 0.149939i
654.8 0.165418 0.286513i 2.33117 + 1.34590i 0.945274 + 1.63726i −0.702335 + 2.12291i 0.771236 0.445274i −1.67674 2.90420i 1.28714 2.12291 + 3.67698i 0.492061 + 0.552395i
654.9 0.593667 1.02826i −0.298874 0.172555i 0.295120 + 0.511162i 1.71029 1.44045i −0.354863 + 0.204880i −1.01478 1.75765i 3.07548 −1.44045 2.49493i −0.465813 2.61378i
654.10 0.593667 1.02826i 0.298874 + 0.172555i 0.295120 + 0.511162i 1.71029 + 1.44045i 0.354863 0.204880i −1.01478 1.75765i 3.07548 −1.44045 2.49493i 2.49650 0.903481i
654.11 1.27287 2.20467i −1.86449 1.07646i −2.24039 3.88048i −2.08125 + 0.817544i −4.74650 + 2.74039i −1.46928 2.54486i −6.31544 0.817544 + 1.41603i −0.846746 + 5.62912i
654.12 1.27287 2.20467i 1.86449 + 1.07646i −2.24039 3.88048i −2.08125 0.817544i 4.74650 2.74039i −1.46928 2.54486i −6.31544 0.817544 + 1.41603i −4.45158 + 3.54786i
699.1 −1.27287 2.20467i −1.86449 + 1.07646i −2.24039 + 3.88048i 2.08125 + 0.817544i 4.74650 + 2.74039i 1.46928 2.54486i 6.31544 0.817544 1.41603i −0.846746 5.62912i
699.2 −1.27287 2.20467i 1.86449 1.07646i −2.24039 + 3.88048i 2.08125 0.817544i −4.74650 2.74039i 1.46928 2.54486i 6.31544 0.817544 1.41603i −4.45158 3.54786i
699.3 −0.593667 1.02826i −0.298874 + 0.172555i 0.295120 0.511162i −1.71029 1.44045i 0.354863 + 0.204880i 1.01478 1.75765i −3.07548 −1.44045 + 2.49493i −0.465813 + 2.61378i
699.4 −0.593667 1.02826i 0.298874 0.172555i 0.295120 0.511162i −1.71029 + 1.44045i −0.354863 0.204880i 1.01478 1.75765i −3.07548 −1.44045 + 2.49493i 2.49650 + 0.903481i
699.5 −0.165418 0.286513i −2.33117 + 1.34590i 0.945274 1.63726i 0.702335 2.12291i 0.771236 + 0.445274i 1.67674 2.90420i −1.28714 2.12291 3.67698i −0.724419 + 0.149939i
699.6 −0.165418 0.286513i 2.33117 1.34590i 0.945274 1.63726i 0.702335 + 2.12291i −0.771236 0.445274i 1.67674 2.90420i −1.28714 2.12291 3.67698i 0.492061 0.552395i
699.7 0.165418 + 0.286513i −2.33117 + 1.34590i 0.945274 1.63726i −0.702335 + 2.12291i −0.771236 0.445274i −1.67674 + 2.90420i 1.28714 2.12291 3.67698i −0.724419 + 0.149939i
699.8 0.165418 + 0.286513i 2.33117 1.34590i 0.945274 1.63726i −0.702335 2.12291i 0.771236 + 0.445274i −1.67674 + 2.90420i 1.28714 2.12291 3.67698i 0.492061 0.552395i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 699.12 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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner
65.d even 2 1 inner
65.l even 6 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 845.2.l.f 24
5.b even 2 1 inner 845.2.l.f 24
13.b even 2 1 inner 845.2.l.f 24
13.c even 3 1 845.2.d.d 12
13.c even 3 1 inner 845.2.l.f 24
13.d odd 4 1 65.2.n.a 12
13.d odd 4 1 845.2.n.e 12
13.e even 6 1 845.2.d.d 12
13.e even 6 1 inner 845.2.l.f 24
13.f odd 12 1 65.2.n.a 12
13.f odd 12 1 845.2.b.d 6
13.f odd 12 1 845.2.b.e 6
13.f odd 12 1 845.2.n.e 12
39.f even 4 1 585.2.bs.a 12
39.k even 12 1 585.2.bs.a 12
52.f even 4 1 1040.2.dh.a 12
52.l even 12 1 1040.2.dh.a 12
65.d even 2 1 inner 845.2.l.f 24
65.f even 4 1 325.2.e.e 12
65.g odd 4 1 65.2.n.a 12
65.g odd 4 1 845.2.n.e 12
65.k even 4 1 325.2.e.e 12
65.l even 6 1 845.2.d.d 12
65.l even 6 1 inner 845.2.l.f 24
65.n even 6 1 845.2.d.d 12
65.n even 6 1 inner 845.2.l.f 24
65.o even 12 1 325.2.e.e 12
65.o even 12 1 4225.2.a.bq 6
65.o even 12 1 4225.2.a.br 6
65.s odd 12 1 65.2.n.a 12
65.s odd 12 1 845.2.b.d 6
65.s odd 12 1 845.2.b.e 6
65.s odd 12 1 845.2.n.e 12
65.t even 12 1 325.2.e.e 12
65.t even 12 1 4225.2.a.bq 6
65.t even 12 1 4225.2.a.br 6
195.n even 4 1 585.2.bs.a 12
195.bh even 12 1 585.2.bs.a 12
260.u even 4 1 1040.2.dh.a 12
260.bc even 12 1 1040.2.dh.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 13.d odd 4 1
65.2.n.a 12 13.f odd 12 1
65.2.n.a 12 65.g odd 4 1
65.2.n.a 12 65.s odd 12 1
325.2.e.e 12 65.f even 4 1
325.2.e.e 12 65.k even 4 1
325.2.e.e 12 65.o even 12 1
325.2.e.e 12 65.t even 12 1
585.2.bs.a 12 39.f even 4 1
585.2.bs.a 12 39.k even 12 1
585.2.bs.a 12 195.n even 4 1
585.2.bs.a 12 195.bh even 12 1
845.2.b.d 6 13.f odd 12 1
845.2.b.d 6 65.s odd 12 1
845.2.b.e 6 13.f odd 12 1
845.2.b.e 6 65.s odd 12 1
845.2.d.d 12 13.c even 3 1
845.2.d.d 12 13.e even 6 1
845.2.d.d 12 65.l even 6 1
845.2.d.d 12 65.n even 6 1
845.2.l.f 24 1.a even 1 1 trivial
845.2.l.f 24 5.b even 2 1 inner
845.2.l.f 24 13.b even 2 1 inner
845.2.l.f 24 13.c even 3 1 inner
845.2.l.f 24 13.e even 6 1 inner
845.2.l.f 24 65.d even 2 1 inner
845.2.l.f 24 65.l even 6 1 inner
845.2.l.f 24 65.n even 6 1 inner
845.2.n.e 12 13.d odd 4 1
845.2.n.e 12 13.f odd 12 1
845.2.n.e 12 65.g odd 4 1
845.2.n.e 12 65.s odd 12 1
1040.2.dh.a 12 52.f even 4 1
1040.2.dh.a 12 52.l even 12 1
1040.2.dh.a 12 260.u even 4 1
1040.2.dh.a 12 260.bc even 12 1
4225.2.a.bq 6 65.o even 12 1
4225.2.a.bq 6 65.t even 12 1
4225.2.a.br 6 65.o even 12 1
4225.2.a.br 6 65.t even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 8T_{2}^{10} + 54T_{2}^{8} + 78T_{2}^{6} + 92T_{2}^{4} + 10T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.