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Properties

Label 960.2.bk.b
Level $960$
Weight $2$
Character orbit 960.bk
Analytic conductor $7.666$
Analytic rank $0$
Dimension $60$
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] = [960,2,Mod(431,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.bk (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: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{3} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{3} - 16 q^{7} + 16 q^{13} - 20 q^{19} - 16 q^{21} + 20 q^{27} + 24 q^{37} + 80 q^{39} - 8 q^{45} + 28 q^{49} + 36 q^{51} + 20 q^{61} + 40 q^{67} + 12 q^{69} - 4 q^{75} - 28 q^{81} - 20 q^{85} + 24 q^{87} + 16 q^{91} - 48 q^{93} - 8 q^{97} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1 0 −1.73074 + 0.0673020i 0 −0.707107 − 0.707107i 0 −0.271924 0 2.99094 − 0.232965i 0
431.2 0 −1.72837 − 0.112911i 0 0.707107 + 0.707107i 0 2.33502 0 2.97450 + 0.390304i 0
431.3 0 −1.67583 − 0.437717i 0 0.707107 + 0.707107i 0 −2.03158 0 2.61681 + 1.46708i 0
431.4 0 −1.55407 + 0.764773i 0 −0.707107 − 0.707107i 0 −3.70081 0 1.83024 − 2.37702i 0
431.5 0 −1.50264 − 0.861442i 0 −0.707107 − 0.707107i 0 −0.563322 0 1.51583 + 2.58887i 0
431.6 0 −1.44431 + 0.956016i 0 0.707107 + 0.707107i 0 2.89523 0 1.17207 − 2.76157i 0
431.7 0 −1.36262 + 1.06924i 0 0.707107 + 0.707107i 0 −1.05330 0 0.713439 − 2.91393i 0
431.8 0 −1.25946 − 1.18902i 0 −0.707107 − 0.707107i 0 2.92811 0 0.172464 + 2.99504i 0
431.9 0 −1.06924 + 1.36262i 0 −0.707107 − 0.707107i 0 −1.05330 0 −0.713439 − 2.91393i 0
431.10 0 −0.956016 + 1.44431i 0 −0.707107 − 0.707107i 0 2.89523 0 −1.17207 − 2.76157i 0
431.11 0 −0.807747 − 1.53217i 0 0.707107 + 0.707107i 0 5.10839 0 −1.69509 + 2.47521i 0
431.12 0 −0.795457 − 1.53859i 0 0.707107 + 0.707107i 0 −3.51098 0 −1.73450 + 2.44776i 0
431.13 0 −0.773899 − 1.54954i 0 −0.707107 − 0.707107i 0 −1.63342 0 −1.80216 + 2.39838i 0
431.14 0 −0.764773 + 1.55407i 0 0.707107 + 0.707107i 0 −3.70081 0 −1.83024 − 2.37702i 0
431.15 0 −0.148181 − 1.72570i 0 0.707107 + 0.707107i 0 1.12315 0 −2.95608 + 0.511431i 0
431.16 0 −0.0673020 + 1.73074i 0 0.707107 + 0.707107i 0 −0.271924 0 −2.99094 − 0.232965i 0
431.17 0 0.112911 + 1.72837i 0 −0.707107 − 0.707107i 0 2.33502 0 −2.97450 + 0.390304i 0
431.18 0 0.427993 − 1.67834i 0 0.707107 + 0.707107i 0 −2.37757 0 −2.63364 − 1.43663i 0
431.19 0 0.437717 + 1.67583i 0 −0.707107 − 0.707107i 0 −2.03158 0 −2.61681 + 1.46708i 0
431.20 0 0.516584 − 1.65322i 0 −0.707107 − 0.707107i 0 1.29120 0 −2.46628 − 1.70806i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k 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.bk.b 60
3.b odd 2 1 inner 960.2.bk.b 60
4.b odd 2 1 240.2.bk.b ✓ 60
12.b even 2 1 240.2.bk.b ✓ 60
16.e even 4 1 240.2.bk.b ✓ 60
16.f odd 4 1 inner 960.2.bk.b 60
48.i odd 4 1 240.2.bk.b ✓ 60
48.k even 4 1 inner 960.2.bk.b 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.bk.b ✓ 60 4.b odd 2 1
240.2.bk.b ✓ 60 12.b even 2 1
240.2.bk.b ✓ 60 16.e even 4 1
240.2.bk.b ✓ 60 48.i odd 4 1
960.2.bk.b 60 1.a even 1 1 trivial
960.2.bk.b 60 3.b odd 2 1 inner
960.2.bk.b 60 16.f odd 4 1 inner
960.2.bk.b 60 48.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{15} + 4 T_{7}^{14} - 48 T_{7}^{13} - 208 T_{7}^{12} + 732 T_{7}^{11} + 3608 T_{7}^{10} + \cdots + 11008 \) acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\). Copy content Toggle raw display