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Properties

Label 320.3.i.a
Level $320$
Weight $3$
Character orbit 320.i
Analytic conductor $8.719$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,3,Mod(177,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.177");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.i (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: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 80)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{5} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{5} - 108 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{17} - 32 q^{19} - 4 q^{21} + 8 q^{31} - 4 q^{33} - 96 q^{35} - 72 q^{39} - 124 q^{43} - 34 q^{45} + 4 q^{47} + 100 q^{51} - 4 q^{53} + 36 q^{57} - 64 q^{59} - 36 q^{61} + 200 q^{63} - 4 q^{65} + 292 q^{67} - 60 q^{69} + 48 q^{73} - 96 q^{75} + 192 q^{77} + 100 q^{81} + 48 q^{85} - 36 q^{87} - 188 q^{91} - 380 q^{95} - 4 q^{97} + 92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1 0 − 5.30326i 0 1.70109 − 4.70173i 0 −7.26221 + 7.26221i 0 −19.1246 0
177.2 0 − 4.95045i 0 −3.96683 − 3.04373i 0 7.61189 − 7.61189i 0 −15.5069 0
177.3 0 − 4.94472i 0 −3.37360 + 3.69037i 0 −3.22480 + 3.22480i 0 −15.4503 0
177.4 0 − 4.50609i 0 1.81773 + 4.65788i 0 −1.52625 + 1.52625i 0 −11.3048 0
177.5 0 − 3.80597i 0 4.26786 + 2.60487i 0 5.17093 − 5.17093i 0 −5.48540 0
177.6 0 − 2.50699i 0 2.43881 − 4.36488i 0 7.18571 − 7.18571i 0 2.71500 0
177.7 0 − 2.05195i 0 4.87776 + 1.09885i 0 −6.87250 + 6.87250i 0 4.78950 0
177.8 0 − 1.96075i 0 −4.38831 + 2.39640i 0 −2.51657 + 2.51657i 0 5.15546 0
177.9 0 − 1.50709i 0 3.69249 − 3.37127i 0 1.28182 − 1.28182i 0 6.72868 0
177.10 0 − 1.24645i 0 −4.76958 + 1.50036i 0 3.62600 − 3.62600i 0 7.44635 0
177.11 0 − 0.616720i 0 −2.01201 − 4.57731i 0 −3.63369 + 3.63369i 0 8.61966 0
177.12 0 − 0.119786i 0 −3.85807 − 3.18046i 0 −4.73972 + 4.73972i 0 8.98565 0
177.13 0 0.390820i 0 0.192979 + 4.99627i 0 6.36907 − 6.36907i 0 8.84726 0
177.14 0 1.70661i 0 4.39780 + 2.37894i 0 0.332763 − 0.332763i 0 6.08749 0
177.15 0 1.90859i 0 −0.530759 + 4.97175i 0 −8.62025 + 8.62025i 0 5.35728 0
177.16 0 2.77329i 0 4.55712 − 2.05735i 0 5.39242 − 5.39242i 0 1.30888 0
177.17 0 2.88135i 0 0.975566 − 4.90390i 0 −2.87444 + 2.87444i 0 0.697817 0
177.18 0 3.32036i 0 −4.88691 − 1.05740i 0 9.08173 − 9.08173i 0 −2.02480 0
177.19 0 3.83124i 0 0.146974 + 4.99784i 0 1.69668 − 1.69668i 0 −5.67842 0
177.20 0 4.38426i 0 −4.75384 + 1.54952i 0 −3.84157 + 3.84157i 0 −10.2217 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.i.a 44
4.b odd 2 1 80.3.i.a ✓ 44
5.c odd 4 1 320.3.t.a 44
8.b even 2 1 640.3.i.a 44
8.d odd 2 1 640.3.i.b 44
16.e even 4 1 320.3.t.a 44
16.e even 4 1 640.3.t.a 44
16.f odd 4 1 80.3.t.a yes 44
16.f odd 4 1 640.3.t.b 44
20.d odd 2 1 400.3.i.b 44
20.e even 4 1 80.3.t.a yes 44
20.e even 4 1 400.3.t.b 44
40.i odd 4 1 640.3.t.a 44
40.k even 4 1 640.3.t.b 44
80.i odd 4 1 inner 320.3.i.a 44
80.j even 4 1 400.3.i.b 44
80.j even 4 1 640.3.i.b 44
80.k odd 4 1 400.3.t.b 44
80.s even 4 1 80.3.i.a ✓ 44
80.t odd 4 1 640.3.i.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.i.a ✓ 44 4.b odd 2 1
80.3.i.a ✓ 44 80.s even 4 1
80.3.t.a yes 44 16.f odd 4 1
80.3.t.a yes 44 20.e even 4 1
320.3.i.a 44 1.a even 1 1 trivial
320.3.i.a 44 80.i odd 4 1 inner
320.3.t.a 44 5.c odd 4 1
320.3.t.a 44 16.e even 4 1
400.3.i.b 44 20.d odd 2 1
400.3.i.b 44 80.j even 4 1
400.3.t.b 44 20.e even 4 1
400.3.t.b 44 80.k odd 4 1
640.3.i.a 44 8.b even 2 1
640.3.i.a 44 80.t odd 4 1
640.3.i.b 44 8.d odd 2 1
640.3.i.b 44 80.j even 4 1
640.3.t.a 44 16.e even 4 1
640.3.t.a 44 40.i odd 4 1
640.3.t.b 44 16.f odd 4 1
640.3.t.b 44 40.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(320, [\chi])\).