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Properties

Label 960.2.bi.g
Level $960$
Weight $2$
Character orbit 960.bi
Analytic conductor $7.666$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,2,Mod(353,960)]
 
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, 2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.353");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.bi (of order \(4\), degree \(2\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3} - 16 q^{19} + 24 q^{25} - 4 q^{27} + 24 q^{33} + 40 q^{43} + 32 q^{57} - 72 q^{67} - 48 q^{73} + 4 q^{75} - 16 q^{81} - 48 q^{97} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1 0 −1.71224 − 0.261188i 0 −2.19583 + 0.422277i 0 −2.94127 + 2.94127i 0 2.86356 + 0.894434i 0
353.2 0 −1.71224 − 0.261188i 0 2.19583 − 0.422277i 0 2.94127 − 2.94127i 0 2.86356 + 0.894434i 0
353.3 0 −1.53356 + 0.805097i 0 −0.217242 − 2.22549i 0 2.05938 − 2.05938i 0 1.70364 − 2.46934i 0
353.4 0 −1.53356 + 0.805097i 0 0.217242 + 2.22549i 0 −2.05938 + 2.05938i 0 1.70364 − 2.46934i 0
353.5 0 −0.937364 − 1.45648i 0 −1.84309 − 1.26611i 0 0.158660 − 0.158660i 0 −1.24270 + 2.73051i 0
353.6 0 −0.937364 − 1.45648i 0 1.84309 + 1.26611i 0 −0.158660 + 0.158660i 0 −1.24270 + 2.73051i 0
353.7 0 −0.805097 + 1.53356i 0 −0.217242 − 2.22549i 0 −2.05938 + 2.05938i 0 −1.70364 − 2.46934i 0
353.8 0 −0.805097 + 1.53356i 0 0.217242 + 2.22549i 0 2.05938 − 2.05938i 0 −1.70364 − 2.46934i 0
353.9 0 0.261188 + 1.71224i 0 −2.19583 + 0.422277i 0 2.94127 − 2.94127i 0 −2.86356 + 0.894434i 0
353.10 0 0.261188 + 1.71224i 0 2.19583 − 0.422277i 0 −2.94127 + 2.94127i 0 −2.86356 + 0.894434i 0
353.11 0 0.675846 − 1.59475i 0 −1.79838 + 1.32885i 0 1.04055 − 1.04055i 0 −2.08646 − 2.15561i 0
353.12 0 0.675846 − 1.59475i 0 1.79838 − 1.32885i 0 −1.04055 + 1.04055i 0 −2.08646 − 2.15561i 0
353.13 0 1.45648 + 0.937364i 0 −1.84309 − 1.26611i 0 −0.158660 + 0.158660i 0 1.24270 + 2.73051i 0
353.14 0 1.45648 + 0.937364i 0 1.84309 + 1.26611i 0 0.158660 − 0.158660i 0 1.24270 + 2.73051i 0
353.15 0 1.59475 − 0.675846i 0 −1.79838 + 1.32885i 0 −1.04055 + 1.04055i 0 2.08646 − 2.15561i 0
353.16 0 1.59475 − 0.675846i 0 1.79838 − 1.32885i 0 1.04055 − 1.04055i 0 2.08646 − 2.15561i 0
737.1 0 −1.71224 + 0.261188i 0 −2.19583 − 0.422277i 0 −2.94127 − 2.94127i 0 2.86356 − 0.894434i 0
737.2 0 −1.71224 + 0.261188i 0 2.19583 + 0.422277i 0 2.94127 + 2.94127i 0 2.86356 − 0.894434i 0
737.3 0 −1.53356 − 0.805097i 0 −0.217242 + 2.22549i 0 2.05938 + 2.05938i 0 1.70364 + 2.46934i 0
737.4 0 −1.53356 − 0.805097i 0 0.217242 − 2.22549i 0 −2.05938 − 2.05938i 0 1.70364 + 2.46934i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
20.e even 4 1 inner
24.f even 2 1 inner
40.i odd 4 1 inner
60.l odd 4 1 inner
120.w 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.bi.g ✓ 32
3.b odd 2 1 inner 960.2.bi.g ✓ 32
4.b odd 2 1 960.2.bi.h yes 32
5.c odd 4 1 960.2.bi.h yes 32
8.b even 2 1 960.2.bi.h yes 32
8.d odd 2 1 inner 960.2.bi.g ✓ 32
12.b even 2 1 960.2.bi.h yes 32
15.e even 4 1 960.2.bi.h yes 32
20.e even 4 1 inner 960.2.bi.g ✓ 32
24.f even 2 1 inner 960.2.bi.g ✓ 32
24.h odd 2 1 960.2.bi.h yes 32
40.i odd 4 1 inner 960.2.bi.g ✓ 32
40.k even 4 1 960.2.bi.h yes 32
60.l odd 4 1 inner 960.2.bi.g ✓ 32
120.q odd 4 1 960.2.bi.h yes 32
120.w even 4 1 inner 960.2.bi.g ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.bi.g ✓ 32 1.a even 1 1 trivial
960.2.bi.g ✓ 32 3.b odd 2 1 inner
960.2.bi.g ✓ 32 8.d odd 2 1 inner
960.2.bi.g ✓ 32 20.e even 4 1 inner
960.2.bi.g ✓ 32 24.f even 2 1 inner
960.2.bi.g ✓ 32 40.i odd 4 1 inner
960.2.bi.g ✓ 32 60.l odd 4 1 inner
960.2.bi.g ✓ 32 120.w even 4 1 inner
960.2.bi.h yes 32 4.b odd 2 1
960.2.bi.h yes 32 5.c odd 4 1
960.2.bi.h yes 32 8.b even 2 1
960.2.bi.h yes 32 12.b even 2 1
960.2.bi.h yes 32 15.e even 4 1
960.2.bi.h yes 32 24.h odd 2 1
960.2.bi.h yes 32 40.k even 4 1
960.2.bi.h yes 32 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{16} + 376T_{7}^{12} + 23280T_{7}^{8} + 101056T_{7}^{4} + 256 \) Copy content Toggle raw display
\( T_{11}^{8} - 54T_{11}^{6} + 816T_{11}^{4} - 4320T_{11}^{2} + 5760 \) Copy content Toggle raw display
\( T_{19}^{4} + 2T_{19}^{3} - 48T_{19}^{2} + 56T_{19} + 64 \) Copy content Toggle raw display
\( T_{23}^{16} + 4468T_{23}^{12} + 890976T_{23}^{8} + 24777280T_{23}^{4} + 104857600 \) Copy content Toggle raw display