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Properties

Label 960.4.s.b
Level $960$
Weight $4$
Character orbit 960.s
Analytic conductor $56.642$
Analytic rank $0$
Dimension $52$
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] = [960,4,Mod(241,960)]
 
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([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.241");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.s (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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 40 q^{11} + 780 q^{15} - 408 q^{17} - 316 q^{19} - 400 q^{29} + 528 q^{33} - 16 q^{37} + 808 q^{43} - 4116 q^{49} + 444 q^{51} - 752 q^{53} - 688 q^{59} - 260 q^{61} - 504 q^{63} - 408 q^{67} - 420 q^{69} + 2008 q^{77} + 200 q^{79} - 4212 q^{81} + 1104 q^{83} + 740 q^{85} - 2648 q^{91} - 2352 q^{93} + 5432 q^{97} - 360 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 − 9.24942i 0 − 9.00000i 0
241.2 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 5.83627i 0 − 9.00000i 0
241.3 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 − 5.35310i 0 − 9.00000i 0
241.4 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 − 5.78064i 0 − 9.00000i 0
241.5 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 − 10.4604i 0 − 9.00000i 0
241.6 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 − 14.0467i 0 − 9.00000i 0
241.7 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 − 17.3764i 0 − 9.00000i 0
241.8 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 20.1042i 0 − 9.00000i 0
241.9 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 24.5183i 0 − 9.00000i 0
241.10 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 28.2829i 0 − 9.00000i 0
241.11 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 − 31.8310i 0 − 9.00000i 0
241.12 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 − 32.5362i 0 − 9.00000i 0
241.13 0 −2.12132 + 2.12132i 0 −3.53553 − 3.53553i 0 33.8922i 0 − 9.00000i 0
241.14 0 2.12132 − 2.12132i 0 3.53553 + 3.53553i 0 − 31.9806i 0 − 9.00000i 0
241.15 0 2.12132 − 2.12132i 0 3.53553 + 3.53553i 0 13.9877i 0 − 9.00000i 0
241.16 0 2.12132 − 2.12132i 0 3.53553 + 3.53553i 0 − 9.91614i 0 − 9.00000i 0
241.17 0 2.12132 − 2.12132i 0 3.53553 + 3.53553i 0 − 5.34769i 0 − 9.00000i 0
241.18 0 2.12132 − 2.12132i 0 3.53553 + 3.53553i 0 − 0.299037i 0 − 9.00000i 0
241.19 0 2.12132 − 2.12132i 0 3.53553 + 3.53553i 0 6.51896i 0 − 9.00000i 0
241.20 0 2.12132 − 2.12132i 0 3.53553 + 3.53553i 0 12.2351i 0 − 9.00000i 0
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.26
Significant digits:
Format:

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 960.4.s.b 52
4.b odd 2 1 240.4.s.b ✓ 52
16.e even 4 1 inner 960.4.s.b 52
16.f odd 4 1 240.4.s.b ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.s.b ✓ 52 4.b odd 2 1
240.4.s.b ✓ 52 16.f odd 4 1
960.4.s.b 52 1.a even 1 1 trivial
960.4.s.b 52 16.e even 4 1 inner