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Properties

Label 320.3.t.a
Level $320$
Weight $3$
Character orbit 320.t
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(17,320)]
 
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, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.t (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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 4 q^{3} - 2 q^{5} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 4 q^{3} - 2 q^{5} + 108 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} - 4 q^{17} + 32 q^{19} - 4 q^{21} + 40 q^{27} + 8 q^{31} - 4 q^{33} + 4 q^{35} - 4 q^{37} + 72 q^{39} - 70 q^{45} + 4 q^{47} + 100 q^{51} - 36 q^{57} + 64 q^{59} - 36 q^{61} + 200 q^{63} - 4 q^{65} + 60 q^{69} - 48 q^{73} + 324 q^{75} + 100 q^{81} - 156 q^{83} - 52 q^{85} + 36 q^{87} - 188 q^{91} - 40 q^{93} - 380 q^{95} - 4 q^{97} - 92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −5.73309 0 −0.266013 − 4.99292i 0 3.79616 − 3.79616i 0 23.8683 0
17.2 0 −4.59062 0 −4.31900 + 2.51918i 0 −1.15913 + 1.15913i 0 12.0738 0
17.3 0 −4.38426 0 1.54952 + 4.75384i 0 3.84157 − 3.84157i 0 10.2217 0
17.4 0 −3.83124 0 4.99784 − 0.146974i 0 −1.69668 + 1.69668i 0 5.67842 0
17.5 0 −3.32036 0 −1.05740 + 4.88691i 0 −9.08173 + 9.08173i 0 2.02480 0
17.6 0 −2.88135 0 −4.90390 − 0.975566i 0 2.87444 − 2.87444i 0 −0.697817 0
17.7 0 −2.77329 0 −2.05735 − 4.55712i 0 −5.39242 + 5.39242i 0 −1.30888 0
17.8 0 −1.90859 0 4.97175 + 0.530759i 0 8.62025 − 8.62025i 0 −5.35728 0
17.9 0 −1.70661 0 2.37894 − 4.39780i 0 −0.332763 + 0.332763i 0 −6.08749 0
17.10 0 −0.390820 0 4.99627 − 0.192979i 0 −6.36907 + 6.36907i 0 −8.84726 0
17.11 0 0.119786 0 −3.18046 + 3.85807i 0 4.73972 − 4.73972i 0 −8.98565 0
17.12 0 0.616720 0 −4.57731 + 2.01201i 0 3.63369 − 3.63369i 0 −8.61966 0
17.13 0 1.24645 0 1.50036 + 4.76958i 0 −3.62600 + 3.62600i 0 −7.44635 0
17.14 0 1.50709 0 −3.37127 − 3.69249i 0 −1.28182 + 1.28182i 0 −6.72868 0
17.15 0 1.96075 0 2.39640 + 4.38831i 0 2.51657 − 2.51657i 0 −5.15546 0
17.16 0 2.05195 0 1.09885 − 4.87776i 0 6.87250 − 6.87250i 0 −4.78950 0
17.17 0 2.50699 0 −4.36488 − 2.43881i 0 −7.18571 + 7.18571i 0 −2.71500 0
17.18 0 3.80597 0 2.60487 − 4.26786i 0 −5.17093 + 5.17093i 0 5.48540 0
17.19 0 4.50609 0 4.65788 − 1.81773i 0 1.52625 − 1.52625i 0 11.3048 0
17.20 0 4.94472 0 3.69037 + 3.37360i 0 3.22480 − 3.22480i 0 15.4503 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.22
Significant digits:
Format:

Inner twists

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

Hecke kernels

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