# Properties

 Label 240.2.t.b Level $240$ Weight $2$ Character orbit 240.t Analytic conductor $1.916$ Analytic rank $0$ Dimension $80$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 240.t (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 8 q^{4} - 4 q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80 q - 8 q^{4} - 4 q^{6} + 16 q^{10} - 32 q^{16} + 16 q^{19} + 8 q^{21} - 56 q^{24} + 8 q^{30} - 16 q^{34} - 40 q^{36} - 8 q^{39} - 8 q^{40} - 12 q^{45} - 24 q^{46} - 112 q^{49} + 32 q^{51} + 60 q^{54} - 40 q^{55} + 40 q^{60} + 16 q^{61} - 32 q^{64} - 56 q^{66} - 40 q^{69} + 48 q^{70} - 36 q^{75} - 128 q^{76} + 64 q^{81} + 128 q^{84} - 64 q^{85} + 72 q^{90} + 48 q^{91} + 56 q^{94} - 24 q^{96} - 8 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −1.41336 + 0.0490154i 0.209895 + 1.71929i 1.99519 0.138553i −0.768175 2.09998i −0.380929 2.41969i 4.08340i −2.81315 + 0.293621i −2.91189 + 0.721739i 1.18864 + 2.93038i
59.2 −1.41336 + 0.0490154i 1.71929 + 0.209895i 1.99519 0.138553i −2.09998 0.768175i −2.44026 0.212386i 4.08340i −2.81315 + 0.293621i 2.91189 + 0.721739i 3.00569 + 0.982779i
59.3 −1.27807 0.605427i −1.58930 + 0.688572i 1.26692 + 1.54755i −1.54468 + 1.61678i 2.44811 + 0.0821618i 0.970036i −0.682276 2.74490i 2.05174 2.18869i 2.95304 1.13116i
59.4 −1.27807 0.605427i 0.688572 1.58930i 1.26692 + 1.54755i 1.61678 1.54468i −1.84225 + 1.61435i 0.970036i −0.682276 2.74490i −2.05174 2.18869i −3.00154 + 0.995364i
59.5 −1.26832 + 0.625594i −0.262098 + 1.71211i 1.21726 1.58691i −0.0667377 + 2.23507i −0.738660 2.33546i 0.598963i −0.551120 + 2.77421i −2.86261 0.897478i −1.31360 2.87653i
59.6 −1.26832 + 0.625594i 1.71211 0.262098i 1.21726 1.58691i 2.23507 0.0667377i −2.00753 + 1.40351i 0.598963i −0.551120 + 2.77421i 2.86261 0.897478i −2.79303 + 1.48289i
59.7 −1.22048 0.714442i 0.591463 + 1.62793i 0.979146 + 1.74392i 2.22111 0.258204i 0.441196 2.40943i 4.18600i 0.0509042 2.82797i −2.30034 + 1.92573i −2.89529 1.27172i
59.8 −1.22048 0.714442i 1.62793 + 0.591463i 0.979146 + 1.74392i −0.258204 + 2.22111i −1.56430 1.88493i 4.18600i 0.0509042 2.82797i 2.30034 + 1.92573i 1.90199 2.52635i
59.9 −1.03022 + 0.968834i −1.72231 0.183389i 0.122722 1.99623i −2.23605 0.00763900i 1.95204 1.47971i 1.25117i 1.80759 + 2.17546i 2.93274 + 0.631707i 2.31104 2.15850i
59.10 −1.03022 + 0.968834i −0.183389 1.72231i 0.122722 1.99623i −0.00763900 2.23605i 1.85757 + 1.59670i 1.25117i 1.80759 + 2.17546i −2.93274 + 0.631707i 2.17424 + 2.29624i
59.11 −0.788977 1.17368i −1.64998 0.526861i −0.755031 + 1.85201i 2.11651 0.721367i 0.683429 + 2.35222i 2.57173i 2.76936 0.575028i 2.44484 + 1.73861i −2.51653 1.91496i
59.12 −0.788977 1.17368i −0.526861 1.64998i −0.755031 + 1.85201i −0.721367 + 2.11651i −1.52086 + 1.92016i 2.57173i 2.76936 0.575028i −2.44484 + 1.73861i 3.05324 0.823229i
59.13 −0.743883 1.20276i −1.02525 + 1.39602i −0.893276 + 1.78943i −0.920160 2.03797i 2.44174 + 0.194661i 0.440232i 2.81675 0.256728i −0.897721 2.86253i −1.76670 + 2.62274i
59.14 −0.743883 1.20276i 1.39602 1.02525i −0.893276 + 1.78943i −2.03797 0.920160i −2.27161 0.916409i 0.440232i 2.81675 0.256728i 0.897721 2.86253i 0.409274 + 3.13568i
59.15 −0.676744 + 1.24178i −1.34702 + 1.08883i −1.08403 1.68073i 2.16547 0.557438i −0.440501 2.40956i 4.04743i 2.82072 0.208705i 0.628903 2.93334i −0.773254 + 3.06628i
59.16 −0.676744 + 1.24178i 1.08883 1.34702i −1.08403 1.68073i −0.557438 + 2.16547i 0.935838 + 2.26367i 4.04743i 2.82072 0.208705i −0.628903 2.93334i −2.31179 2.15769i
59.17 −0.265026 + 1.38916i −0.571191 + 1.63516i −1.85952 0.736326i −1.86979 1.22634i −2.12011 1.22683i 3.84936i 1.51570 2.38803i −2.34748 1.86798i 2.19912 2.27242i
59.18 −0.265026 + 1.38916i 1.63516 0.571191i −1.85952 0.736326i −1.22634 1.86979i 0.360116 + 2.42287i 3.84936i 1.51570 2.38803i 2.34748 1.86798i 2.92244 1.20803i
59.19 −0.0728709 1.41233i 0.938373 + 1.45584i −1.98938 + 0.205836i −1.83519 + 1.27753i 1.98775 1.43139i 2.98852i 0.435678 + 2.79467i −1.23891 + 2.73223i 1.93803 + 2.49880i
59.20 −0.0728709 1.41233i 1.45584 + 0.938373i −1.98938 + 0.205836i 1.27753 1.83519i 1.21921 2.12451i 2.98852i 0.435678 + 2.79467i 1.23891 + 2.73223i −2.68499 1.67057i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner
80.k odd 4 1 inner
240.t even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.t.b 80
3.b odd 2 1 inner 240.2.t.b 80
4.b odd 2 1 960.2.t.b 80
5.b even 2 1 inner 240.2.t.b 80
12.b even 2 1 960.2.t.b 80
15.d odd 2 1 inner 240.2.t.b 80
16.e even 4 1 960.2.t.b 80
16.f odd 4 1 inner 240.2.t.b 80
20.d odd 2 1 960.2.t.b 80
48.i odd 4 1 960.2.t.b 80
48.k even 4 1 inner 240.2.t.b 80
60.h even 2 1 960.2.t.b 80
80.k odd 4 1 inner 240.2.t.b 80
80.q even 4 1 960.2.t.b 80
240.t even 4 1 inner 240.2.t.b 80
240.bm odd 4 1 960.2.t.b 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.t.b 80 1.a even 1 1 trivial
240.2.t.b 80 3.b odd 2 1 inner
240.2.t.b 80 5.b even 2 1 inner
240.2.t.b 80 15.d odd 2 1 inner
240.2.t.b 80 16.f odd 4 1 inner
240.2.t.b 80 48.k even 4 1 inner
240.2.t.b 80 80.k odd 4 1 inner
240.2.t.b 80 240.t even 4 1 inner
960.2.t.b 80 4.b odd 2 1
960.2.t.b 80 12.b even 2 1
960.2.t.b 80 16.e even 4 1
960.2.t.b 80 20.d odd 2 1
960.2.t.b 80 48.i odd 4 1
960.2.t.b 80 60.h even 2 1
960.2.t.b 80 80.q even 4 1
960.2.t.b 80 240.bm odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{20} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$.