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Properties

Label 1920.2.bl.b
Level $1920$
Weight $2$
Character orbit 1920.bl
Analytic conductor $15.331$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1920,2,Mod(289,1920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1920.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1920.bl (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.3312771881\)
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 −0.707107 + 0.707107i 0 −1.38805 + 1.75308i 0 −4.66030 0 − 1.00000i 0
289.2 0 −0.707107 + 0.707107i 0 1.03097 − 1.98421i 0 −3.91927 0 − 1.00000i 0
289.3 0 −0.707107 + 0.707107i 0 −1.65754 − 1.50085i 0 −2.58977 0 − 1.00000i 0
289.4 0 −0.707107 + 0.707107i 0 −2.19925 + 0.404088i 0 1.81567 0 − 1.00000i 0
289.5 0 −0.707107 + 0.707107i 0 1.24079 − 1.86022i 0 1.58988 0 − 1.00000i 0
289.6 0 −0.707107 + 0.707107i 0 −2.18678 − 0.466917i 0 −1.00010 0 − 1.00000i 0
289.7 0 −0.707107 + 0.707107i 0 2.06370 + 0.860885i 0 −0.707398 0 − 1.00000i 0
289.8 0 −0.707107 + 0.707107i 0 −1.07735 + 1.95942i 0 1.22137 0 − 1.00000i 0
289.9 0 −0.707107 + 0.707107i 0 2.15195 + 0.607542i 0 −2.25286 0 − 1.00000i 0
289.10 0 −0.707107 + 0.707107i 0 −0.770325 − 2.09919i 0 3.05002 0 − 1.00000i 0
289.11 0 −0.707107 + 0.707107i 0 2.23019 + 0.162008i 0 2.93661 0 − 1.00000i 0
289.12 0 −0.707107 + 0.707107i 0 0.561697 + 2.16437i 0 4.51614 0 − 1.00000i 0
289.13 0 0.707107 − 0.707107i 0 −2.16437 − 0.561697i 0 −4.51614 0 − 1.00000i 0
289.14 0 0.707107 − 0.707107i 0 −0.162008 − 2.23019i 0 −2.93661 0 − 1.00000i 0
289.15 0 0.707107 − 0.707107i 0 2.09919 + 0.770325i 0 −3.05002 0 − 1.00000i 0
289.16 0 0.707107 − 0.707107i 0 −0.607542 − 2.15195i 0 2.25286 0 − 1.00000i 0
289.17 0 0.707107 − 0.707107i 0 −1.95942 + 1.07735i 0 −1.22137 0 − 1.00000i 0
289.18 0 0.707107 − 0.707107i 0 −0.860885 − 2.06370i 0 0.707398 0 − 1.00000i 0
289.19 0 0.707107 − 0.707107i 0 0.466917 + 2.18678i 0 1.00010 0 − 1.00000i 0
289.20 0 0.707107 − 0.707107i 0 1.86022 − 1.24079i 0 −1.58988 0 − 1.00000i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.24
Significant digits:
Format:

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 1920.2.bl.b 48
4.b odd 2 1 1920.2.bl.a 48
5.b even 2 1 inner 1920.2.bl.b 48
8.b even 2 1 960.2.bl.a 48
8.d odd 2 1 240.2.bl.a ✓ 48
16.e even 4 1 960.2.bl.a 48
16.e even 4 1 inner 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 1920.2.bl.a 48
24.f even 2 1 720.2.bm.h 48
40.e odd 2 1 240.2.bl.a ✓ 48
40.f even 2 1 960.2.bl.a 48
48.k even 4 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 1 960.2.bl.a 48
80.q even 4 1 inner 1920.2.bl.b 48
120.m even 2 1 720.2.bm.h 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 8.d odd 2 1
240.2.bl.a ✓ 48 16.f odd 4 1
240.2.bl.a ✓ 48 40.e odd 2 1
240.2.bl.a ✓ 48 80.k odd 4 1
720.2.bm.h 48 24.f even 2 1
720.2.bm.h 48 48.k even 4 1
720.2.bm.h 48 120.m even 2 1
720.2.bm.h 48 240.t even 4 1
960.2.bl.a 48 8.b even 2 1
960.2.bl.a 48 16.e even 4 1
960.2.bl.a 48 40.f even 2 1
960.2.bl.a 48 80.q even 4 1
1920.2.bl.a 48 4.b odd 2 1
1920.2.bl.a 48 16.f odd 4 1
1920.2.bl.a 48 20.d odd 2 1
1920.2.bl.a 48 80.k odd 4 1
1920.2.bl.b 48 1.a even 1 1 trivial
1920.2.bl.b 48 5.b even 2 1 inner
1920.2.bl.b 48 16.e even 4 1 inner
1920.2.bl.b 48 80.q even 4 1 inner