# Properties

 Label 960.2.bl.a Level $960$ Weight $2$ Character orbit 960.bl Analytic conductor $7.666$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 960.bl (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 240) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48 q - 8 q^{19} + 48 q^{31} - 24 q^{35} + 48 q^{49} - 8 q^{51} + 32 q^{59} + 16 q^{61} + 16 q^{65} - 16 q^{69} + 16 q^{75} + 96 q^{79} - 48 q^{81} + 32 q^{91} + 48 q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 0 −0.707107 0.707107i 0 −2.09919 + 0.770325i 0 −3.05002 0 1.00000i 0
49.2 0 −0.707107 0.707107i 0 −1.98421 1.03097i 0 3.91927 0 1.00000i 0
49.3 0 −0.707107 0.707107i 0 −1.86022 1.24079i 0 −1.58988 0 1.00000i 0
49.4 0 −0.707107 0.707107i 0 −1.50085 + 1.65754i 0 2.58977 0 1.00000i 0
49.5 0 −0.707107 0.707107i 0 −0.466917 + 2.18678i 0 1.00010 0 1.00000i 0
49.6 0 −0.707107 0.707107i 0 0.162008 2.23019i 0 −2.93661 0 1.00000i 0
49.7 0 −0.707107 0.707107i 0 0.404088 + 2.19925i 0 −1.81567 0 1.00000i 0
49.8 0 −0.707107 0.707107i 0 0.607542 2.15195i 0 2.25286 0 1.00000i 0
49.9 0 −0.707107 0.707107i 0 0.860885 2.06370i 0 0.707398 0 1.00000i 0
49.10 0 −0.707107 0.707107i 0 1.75308 + 1.38805i 0 4.66030 0 1.00000i 0
49.11 0 −0.707107 0.707107i 0 1.95942 + 1.07735i 0 −1.22137 0 1.00000i 0
49.12 0 −0.707107 0.707107i 0 2.16437 0.561697i 0 −4.51614 0 1.00000i 0
49.13 0 0.707107 + 0.707107i 0 −2.23019 + 0.162008i 0 2.93661 0 1.00000i 0
49.14 0 0.707107 + 0.707107i 0 −2.15195 + 0.607542i 0 −2.25286 0 1.00000i 0
49.15 0 0.707107 + 0.707107i 0 −2.06370 + 0.860885i 0 −0.707398 0 1.00000i 0
49.16 0 0.707107 + 0.707107i 0 −1.24079 1.86022i 0 1.58988 0 1.00000i 0
49.17 0 0.707107 + 0.707107i 0 −1.03097 1.98421i 0 −3.91927 0 1.00000i 0
49.18 0 0.707107 + 0.707107i 0 −0.561697 + 2.16437i 0 4.51614 0 1.00000i 0
49.19 0 0.707107 + 0.707107i 0 0.770325 2.09919i 0 3.05002 0 1.00000i 0
49.20 0 0.707107 + 0.707107i 0 1.07735 + 1.95942i 0 1.22137 0 1.00000i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.24 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 960.2.bl.a 48
4.b odd 2 1 240.2.bl.a 48
5.b even 2 1 inner 960.2.bl.a 48
8.b even 2 1 1920.2.bl.b 48
8.d odd 2 1 1920.2.bl.a 48
12.b even 2 1 720.2.bm.h 48
16.e even 4 1 inner 960.2.bl.a 48
16.e even 4 1 1920.2.bl.b 48
16.f odd 4 1 240.2.bl.a 48
16.f odd 4 1 1920.2.bl.a 48
20.d odd 2 1 240.2.bl.a 48
40.e odd 2 1 1920.2.bl.a 48
40.f even 2 1 1920.2.bl.b 48
48.k even 4 1 720.2.bm.h 48
60.h even 2 1 720.2.bm.h 48
80.k odd 4 1 240.2.bl.a 48
80.k odd 4 1 1920.2.bl.a 48
80.q even 4 1 inner 960.2.bl.a 48
80.q even 4 1 1920.2.bl.b 48
240.t even 4 1 720.2.bm.h 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.bl.a 48 4.b odd 2 1
240.2.bl.a 48 16.f odd 4 1
240.2.bl.a 48 20.d odd 2 1
240.2.bl.a 48 80.k odd 4 1
720.2.bm.h 48 12.b even 2 1
720.2.bm.h 48 48.k even 4 1
720.2.bm.h 48 60.h even 2 1
720.2.bm.h 48 240.t even 4 1
960.2.bl.a 48 1.a even 1 1 trivial
960.2.bl.a 48 5.b even 2 1 inner
960.2.bl.a 48 16.e even 4 1 inner
960.2.bl.a 48 80.q even 4 1 inner
1920.2.bl.a 48 8.d odd 2 1
1920.2.bl.a 48 16.f odd 4 1
1920.2.bl.a 48 40.e odd 2 1
1920.2.bl.a 48 80.k odd 4 1
1920.2.bl.b 48 8.b even 2 1
1920.2.bl.b 48 16.e even 4 1
1920.2.bl.b 48 40.f even 2 1
1920.2.bl.b 48 80.q even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(960, [\chi])$$.