# Properties

 Label 240.2.bl.a Level $240$ Weight $2$ Character orbit 240.bl Analytic conductor $1.916$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48 q + 12 q^{10} - 16 q^{14} - 4 q^{16} + 8 q^{19} - 4 q^{24} - 40 q^{26} - 8 q^{30} - 48 q^{31} - 28 q^{34} + 24 q^{35} - 4 q^{36} - 16 q^{40} - 40 q^{44} - 4 q^{46} + 48 q^{49} - 32 q^{50} + 8 q^{51} - 4 q^{54} + 48 q^{56} - 32 q^{59} - 24 q^{60} + 16 q^{61} + 48 q^{64} + 16 q^{65} + 24 q^{66} - 16 q^{69} + 40 q^{74} - 16 q^{75} + 60 q^{76} - 96 q^{79} + 72 q^{80} - 48 q^{81} + 16 q^{86} + 8 q^{90} - 32 q^{91} + 44 q^{94} - 48 q^{95} - 40 q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −1.40976 0.112110i −0.707107 + 0.707107i 1.97486 + 0.316097i 2.18678 + 0.466917i 1.07613 0.917579i 1.00010 −2.74865 0.667024i 1.00000i −3.03049 0.903402i
109.2 −1.39033 0.258835i 0.707107 0.707107i 1.86601 + 0.719729i 0.404088 2.19925i −1.16613 + 0.800085i 1.81567 −2.40807 1.48364i 1.00000i −1.13106 + 2.95308i
109.3 −1.36038 + 0.386472i 0.707107 0.707107i 1.70128 1.05150i −1.98421 + 1.03097i −0.688658 + 1.23521i −3.91927 −1.90801 + 2.08794i 1.00000i 2.30085 2.16936i
109.4 −1.22294 0.710230i −0.707107 + 0.707107i 0.991146 + 1.73713i −2.15195 0.607542i 1.36696 0.362538i 2.25286 0.0216568 2.82834i 1.00000i 2.20020 + 2.27137i
109.5 −1.20386 + 0.742113i −0.707107 + 0.707107i 0.898535 1.78679i −2.06370 0.860885i 0.326501 1.37601i 0.707398 0.244298 + 2.81786i 1.00000i 3.12328 0.495122i
109.6 −1.06224 0.933621i 0.707107 0.707107i 0.256702 + 1.98346i −1.86022 + 1.24079i −1.41129 + 0.0909462i 1.58988 1.57912 2.34657i 1.00000i 3.13443 + 0.418728i
109.7 −0.903247 + 1.08818i −0.707107 + 0.707107i −0.368289 1.96580i 0.770325 + 2.09919i −0.130770 1.40815i −3.05002 2.47181 + 1.37484i 1.00000i −2.98010 1.05783i
109.8 −0.750333 1.19875i −0.707107 + 0.707107i −0.874002 + 1.79892i 1.07735 1.95942i 1.37821 + 0.317079i −1.22137 2.81225 0.302081i 1.00000i −3.15722 + 0.178735i
109.9 −0.550383 + 1.30272i 0.707107 0.707107i −1.39416 1.43399i 0.162008 + 2.23019i 0.531983 + 1.31034i 2.93661 2.63541 1.02695i 1.00000i −2.99448 1.01641i
109.10 −0.456856 + 1.33839i 0.707107 0.707107i −1.58257 1.22290i −1.50085 1.65754i 0.623338 + 1.26943i −2.58977 2.35972 1.55940i 1.00000i 2.90411 1.25146i
109.11 −0.382275 + 1.36157i −0.707107 + 0.707107i −1.70773 1.04099i 1.38805 1.75308i −0.692464 1.23308i 4.66030 2.07020 1.92725i 1.00000i 1.85633 + 2.56009i
109.12 −0.345118 1.37146i 0.707107 0.707107i −1.76179 + 0.946629i 2.16437 + 0.561697i −1.21380 0.725731i 4.51614 1.90629 + 2.08952i 1.00000i 0.0233792 3.16219i
109.13 0.345118 + 1.37146i −0.707107 + 0.707107i −1.76179 + 0.946629i −0.561697 2.16437i −1.21380 0.725731i −4.51614 −1.90629 2.08952i 1.00000i 2.77449 1.51731i
109.14 0.382275 1.36157i 0.707107 0.707107i −1.70773 1.04099i 1.75308 1.38805i −0.692464 1.23308i −4.66030 −2.07020 + 1.92725i 1.00000i −1.21977 2.91756i
109.15 0.456856 1.33839i −0.707107 + 0.707107i −1.58257 1.22290i 1.65754 + 1.50085i 0.623338 + 1.26943i 2.58977 −2.35972 + 1.55940i 1.00000i 2.76598 1.53277i
109.16 0.550383 1.30272i −0.707107 + 0.707107i −1.39416 1.43399i −2.23019 0.162008i 0.531983 + 1.31034i −2.93661 −2.63541 + 1.02695i 1.00000i −1.43851 + 2.81615i
109.17 0.750333 + 1.19875i 0.707107 0.707107i −0.874002 + 1.79892i 1.95942 1.07735i 1.37821 + 0.317079i 1.22137 −2.81225 + 0.302081i 1.00000i 2.76169 + 1.54047i
109.18 0.903247 1.08818i 0.707107 0.707107i −0.368289 1.96580i −2.09919 0.770325i −0.130770 1.40815i 3.05002 −2.47181 1.37484i 1.00000i −2.73434 + 1.58851i
109.19 1.06224 + 0.933621i −0.707107 + 0.707107i 0.256702 + 1.98346i −1.24079 + 1.86022i −1.41129 + 0.0909462i −1.58988 −1.57912 + 2.34657i 1.00000i −3.05476 + 0.817572i
109.20 1.20386 0.742113i 0.707107 0.707107i 0.898535 1.78679i 0.860885 + 2.06370i 0.326501 1.37601i −0.707398 −0.244298 2.81786i 1.00000i 2.56788 + 1.84553i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 229.24 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
16.e even 4 1 inner
80.q 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.bl.a 48
3.b odd 2 1 720.2.bm.h 48
4.b odd 2 1 960.2.bl.a 48
5.b even 2 1 inner 240.2.bl.a 48
8.b even 2 1 1920.2.bl.a 48
8.d odd 2 1 1920.2.bl.b 48
15.d odd 2 1 720.2.bm.h 48
16.e even 4 1 inner 240.2.bl.a 48
16.e even 4 1 1920.2.bl.a 48
16.f odd 4 1 960.2.bl.a 48
16.f odd 4 1 1920.2.bl.b 48
20.d odd 2 1 960.2.bl.a 48
40.e odd 2 1 1920.2.bl.b 48
40.f even 2 1 1920.2.bl.a 48
48.i odd 4 1 720.2.bm.h 48
80.k odd 4 1 960.2.bl.a 48
80.k odd 4 1 1920.2.bl.b 48
80.q even 4 1 inner 240.2.bl.a 48
80.q even 4 1 1920.2.bl.a 48
240.bm odd 4 1 720.2.bm.h 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.bl.a 48 1.a even 1 1 trivial
240.2.bl.a 48 5.b even 2 1 inner
240.2.bl.a 48 16.e even 4 1 inner
240.2.bl.a 48 80.q even 4 1 inner
720.2.bm.h 48 3.b odd 2 1
720.2.bm.h 48 15.d odd 2 1
720.2.bm.h 48 48.i odd 4 1
720.2.bm.h 48 240.bm odd 4 1
960.2.bl.a 48 4.b odd 2 1
960.2.bl.a 48 16.f odd 4 1
960.2.bl.a 48 20.d odd 2 1
960.2.bl.a 48 80.k odd 4 1
1920.2.bl.a 48 8.b even 2 1
1920.2.bl.a 48 16.e even 4 1
1920.2.bl.a 48 40.f even 2 1
1920.2.bl.a 48 80.q even 4 1
1920.2.bl.b 48 8.d odd 2 1
1920.2.bl.b 48 16.f odd 4 1
1920.2.bl.b 48 40.e odd 2 1
1920.2.bl.b 48 80.k odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(240, [\chi])$$.