# Properties

 Label 280.2.w.a Level $280$ Weight $2$ Character orbit 280.w Analytic conductor $2.236$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.w (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$36$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 8q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 8q^{6} + 16q^{10} - 16q^{12} - 8q^{16} + 8q^{17} - 28q^{18} - 20q^{20} - 36q^{22} - 8q^{25} - 32q^{26} - 4q^{30} + 40q^{32} + 64q^{36} - 4q^{40} + 20q^{42} - 64q^{43} + 48q^{46} - 80q^{48} - 32q^{51} + 16q^{52} - 24q^{56} + 4q^{58} - 80q^{60} - 40q^{62} - 8q^{65} + 32q^{66} - 24q^{68} + 40q^{72} - 40q^{73} + 112q^{75} - 8q^{76} + 28q^{78} - 20q^{80} - 72q^{81} + 24q^{82} + 80q^{83} + 8q^{86} - 88q^{88} + 136q^{90} - 96q^{92} - 56q^{96} - 8q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −1.41051 + 0.102291i −0.977895 0.977895i 1.97907 0.288564i −2.23508 0.0665977i 1.47936 + 1.27930i 0.707107 + 0.707107i −2.76198 + 0.609464i 1.08744i 3.15941 0.134691i
43.2 −1.34202 0.446087i −1.58247 1.58247i 1.60201 + 1.19731i −0.315812 2.21365i 1.41778 + 2.82962i −0.707107 0.707107i −1.61582 2.32145i 2.00843i −0.563657 + 3.11164i
43.3 −1.33794 0.458170i 2.03802 + 2.03802i 1.58016 + 1.22601i −1.56810 + 1.59407i −1.79299 3.66051i 0.707107 + 0.707107i −1.55244 2.36430i 5.30708i 2.82837 1.41432i
43.4 −1.33699 + 0.460930i −2.36821 2.36821i 1.57509 1.23252i 2.04939 + 0.894439i 4.25785 + 2.07469i 0.707107 + 0.707107i −1.53777 + 2.37387i 8.21680i −3.15228 0.251234i
43.5 −1.27334 0.615319i −1.25131 1.25131i 1.24277 + 1.56701i −0.429301 + 2.19447i 0.823380 + 2.36328i −0.707107 0.707107i −0.618244 2.76003i 0.131530i 1.89694 2.53014i
43.6 −1.25279 + 0.656136i 0.683238 + 0.683238i 1.13897 1.64400i −2.03938 0.917023i −1.30425 0.407657i −0.707107 0.707107i −0.348203 + 2.80691i 2.06637i 3.15661 0.189272i
43.7 −1.23432 + 0.690254i 2.03800 + 2.03800i 1.04710 1.70399i 1.22676 1.86951i −3.92228 1.10881i 0.707107 + 0.707107i −0.116271 + 2.82604i 5.30689i −0.223778 + 3.15435i
43.8 −1.15591 + 0.814786i −0.824980 0.824980i 0.672247 1.88364i 1.57833 1.58394i 1.62578 + 0.281419i −0.707107 0.707107i 0.757704 + 2.72505i 1.63882i −0.533826 + 3.11689i
43.9 −1.13903 0.838219i 1.33262 + 1.33262i 0.594776 + 1.90951i 2.22159 0.254081i −0.400865 2.63492i −0.707107 0.707107i 0.923123 2.67355i 0.551751i −2.74343 1.57277i
43.10 −0.977609 1.02190i 0.532572 + 0.532572i −0.0885601 + 1.99804i −1.25149 1.85304i 0.0235881 1.06488i 0.707107 + 0.707107i 2.12837 1.86280i 2.43273i −0.670153 + 3.09045i
43.11 −0.814786 + 1.15591i −0.824980 0.824980i −0.672247 1.88364i −1.57833 + 1.58394i 1.62578 0.281419i 0.707107 + 0.707107i 2.72505 + 0.757704i 1.63882i −0.544895 3.11498i
43.12 −0.700587 1.22849i 0.311152 + 0.311152i −1.01836 + 1.72132i −1.92300 + 1.14109i 0.164256 0.600234i −0.707107 0.707107i 2.82807 + 0.0450980i 2.80637i 2.74904 + 1.56295i
43.13 −0.690254 + 1.23432i 2.03800 + 2.03800i −1.04710 1.70399i −1.22676 + 1.86951i −3.92228 + 1.10881i −0.707107 0.707107i 2.82604 0.116271i 5.30689i −1.46080 2.80465i
43.14 −0.656136 + 1.25279i 0.683238 + 0.683238i −1.13897 1.64400i 2.03938 + 0.917023i −1.30425 + 0.407657i 0.707107 + 0.707107i 2.80691 0.348203i 2.06637i −2.48695 + 1.95322i
43.15 −0.460930 + 1.33699i −2.36821 2.36821i −1.57509 1.23252i −2.04939 0.894439i 4.25785 2.07469i −0.707107 0.707107i 2.37387 1.53777i 8.21680i 2.14048 2.32774i
43.16 −0.253372 1.39133i −0.161197 0.161197i −1.87161 + 0.705048i 0.0283969 + 2.23589i −0.183435 + 0.265120i 0.707107 + 0.707107i 1.45517 + 2.42538i 2.94803i 3.10367 0.606020i
43.17 −0.205776 1.39916i 1.94776 + 1.94776i −1.91531 + 0.575828i 2.03914 + 0.917560i 2.32443 3.12603i 0.707107 + 0.707107i 1.19980 + 2.56134i 4.58750i 0.864211 3.04190i
43.18 −0.102291 + 1.41051i −0.977895 0.977895i −1.97907 0.288564i 2.23508 + 0.0665977i 1.47936 1.27930i −0.707107 0.707107i 0.609464 2.76198i 1.08744i −0.322565 + 3.14578i
43.19 0.212809 1.39811i 0.570077 + 0.570077i −1.90942 0.595061i 0.776798 2.09680i 0.918349 0.675713i −0.707107 0.707107i −1.23830 + 2.54295i 2.35002i −2.76625 1.53227i
43.20 0.246308 1.39260i −2.09095 2.09095i −1.87867 0.686016i 0.291596 + 2.21697i −3.42686 + 2.39683i −0.707107 0.707107i −1.41807 + 2.44726i 5.74411i 3.15918 + 0.139981i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 267.36 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.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.w.a 72
4.b odd 2 1 1120.2.bi.a 72
5.c odd 4 1 inner 280.2.w.a 72
8.b even 2 1 1120.2.bi.a 72
8.d odd 2 1 inner 280.2.w.a 72
20.e even 4 1 1120.2.bi.a 72
40.i odd 4 1 1120.2.bi.a 72
40.k even 4 1 inner 280.2.w.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.w.a 72 1.a even 1 1 trivial
280.2.w.a 72 5.c odd 4 1 inner
280.2.w.a 72 8.d odd 2 1 inner
280.2.w.a 72 40.k even 4 1 inner
1120.2.bi.a 72 4.b odd 2 1
1120.2.bi.a 72 8.b even 2 1
1120.2.bi.a 72 20.e even 4 1
1120.2.bi.a 72 40.i odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(280, [\chi])$$.