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Properties

Label 320.6.l.a
Level $320$
Weight $6$
Character orbit 320.l
Analytic conductor $51.323$
Analytic rank $0$
Dimension $80$
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] = [320,6,Mod(81,320)]
 
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, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.81");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.l (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: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 1208 q^{11} + 1800 q^{15} - 2360 q^{19} + 7464 q^{27} - 8144 q^{29} + 21296 q^{37} - 32072 q^{43} + 88360 q^{47} - 192080 q^{49} + 5920 q^{51} - 49456 q^{53} - 44984 q^{59} + 48080 q^{61} - 158760 q^{63} - 61160 q^{67} - 22320 q^{69} - 14896 q^{77} - 177680 q^{79} - 524880 q^{81} + 329240 q^{83} + 132400 q^{85} - 364832 q^{91} - 362352 q^{93} - 288800 q^{95} - 659000 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −21.2511 − 21.2511i 0 −17.6777 + 17.6777i 0 169.621i 0 660.222i 0
81.2 0 −21.1465 − 21.1465i 0 17.6777 − 17.6777i 0 100.160i 0 651.346i 0
81.3 0 −19.6483 − 19.6483i 0 −17.6777 + 17.6777i 0 − 152.043i 0 529.109i 0
81.4 0 −17.6894 − 17.6894i 0 17.6777 − 17.6777i 0 145.744i 0 382.828i 0
81.5 0 −16.3104 − 16.3104i 0 −17.6777 + 17.6777i 0 − 214.455i 0 289.056i 0
81.6 0 −16.2010 − 16.2010i 0 17.6777 − 17.6777i 0 − 159.660i 0 281.948i 0
81.7 0 −15.7717 − 15.7717i 0 −17.6777 + 17.6777i 0 − 14.9661i 0 254.492i 0
81.8 0 −14.9697 − 14.9697i 0 17.6777 − 17.6777i 0 − 66.6523i 0 205.184i 0
81.9 0 −11.6877 − 11.6877i 0 −17.6777 + 17.6777i 0 158.930i 0 30.2029i 0
81.10 0 −10.2463 − 10.2463i 0 −17.6777 + 17.6777i 0 107.195i 0 − 33.0270i 0
81.11 0 −9.88924 − 9.88924i 0 −17.6777 + 17.6777i 0 122.335i 0 − 47.4058i 0
81.12 0 −9.77316 − 9.77316i 0 17.6777 − 17.6777i 0 103.649i 0 − 51.9707i 0
81.13 0 −7.88400 − 7.88400i 0 17.6777 − 17.6777i 0 208.047i 0 − 118.685i 0
81.14 0 −6.55397 − 6.55397i 0 17.6777 − 17.6777i 0 − 188.280i 0 − 157.091i 0
81.15 0 −5.64986 − 5.64986i 0 −17.6777 + 17.6777i 0 − 143.077i 0 − 179.158i 0
81.16 0 −4.24643 − 4.24643i 0 17.6777 − 17.6777i 0 − 137.329i 0 − 206.936i 0
81.17 0 −3.86003 − 3.86003i 0 −17.6777 + 17.6777i 0 20.3706i 0 − 213.200i 0
81.18 0 −3.58258 − 3.58258i 0 17.6777 − 17.6777i 0 149.558i 0 − 217.330i 0
81.19 0 −2.82595 − 2.82595i 0 17.6777 − 17.6777i 0 − 197.416i 0 − 227.028i 0
81.20 0 −2.21237 − 2.21237i 0 −17.6777 + 17.6777i 0 21.9533i 0 − 233.211i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.l.a 80
4.b odd 2 1 80.6.l.a ✓ 80
16.e even 4 1 inner 320.6.l.a 80
16.f odd 4 1 80.6.l.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.l.a ✓ 80 4.b odd 2 1
80.6.l.a ✓ 80 16.f odd 4 1
320.6.l.a 80 1.a even 1 1 trivial
320.6.l.a 80 16.e even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(320, [\chi])\).