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Properties

Label 960.2.o.e
Level $960$
Weight $2$
Character orbit 960.o
Analytic conductor $7.666$
Analytic rank $0$
Dimension $24$
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(959,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.959");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.o (of order \(2\), degree \(1\), 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: \(24\)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{21} + 8 q^{25} + 24 q^{45} + 40 q^{49} - 32 q^{61} + 56 q^{69} + 8 q^{81} + 32 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
959.1 0 −1.69977 − 0.332823i 0 −1.59038 + 1.57184i 0 1.60011 0 2.77846 + 1.13145i 0
959.2 0 −1.69977 − 0.332823i 0 1.59038 + 1.57184i 0 1.60011 0 2.77846 + 1.13145i 0
959.3 0 −1.69977 + 0.332823i 0 −1.59038 − 1.57184i 0 1.60011 0 2.77846 − 1.13145i 0
959.4 0 −1.69977 + 0.332823i 0 1.59038 − 1.57184i 0 1.60011 0 2.77846 − 1.13145i 0
959.5 0 −1.16422 − 1.28241i 0 −2.19399 − 0.431733i 0 −4.22289 0 −0.289169 + 2.98603i 0
959.6 0 −1.16422 − 1.28241i 0 2.19399 − 0.431733i 0 −4.22289 0 −0.289169 + 2.98603i 0
959.7 0 −1.16422 + 1.28241i 0 −2.19399 + 0.431733i 0 −4.22289 0 −0.289169 − 2.98603i 0
959.8 0 −1.16422 + 1.28241i 0 2.19399 + 0.431733i 0 −4.22289 0 −0.289169 − 2.98603i 0
959.9 0 −0.505327 − 1.65670i 0 −0.810603 − 2.08397i 0 2.36789 0 −2.48929 + 1.67435i 0
959.10 0 −0.505327 − 1.65670i 0 0.810603 − 2.08397i 0 2.36789 0 −2.48929 + 1.67435i 0
959.11 0 −0.505327 + 1.65670i 0 −0.810603 + 2.08397i 0 2.36789 0 −2.48929 − 1.67435i 0
959.12 0 −0.505327 + 1.65670i 0 0.810603 + 2.08397i 0 2.36789 0 −2.48929 − 1.67435i 0
959.13 0 0.505327 − 1.65670i 0 −0.810603 + 2.08397i 0 −2.36789 0 −2.48929 − 1.67435i 0
959.14 0 0.505327 − 1.65670i 0 0.810603 + 2.08397i 0 −2.36789 0 −2.48929 − 1.67435i 0
959.15 0 0.505327 + 1.65670i 0 −0.810603 − 2.08397i 0 −2.36789 0 −2.48929 + 1.67435i 0
959.16 0 0.505327 + 1.65670i 0 0.810603 − 2.08397i 0 −2.36789 0 −2.48929 + 1.67435i 0
959.17 0 1.16422 − 1.28241i 0 −2.19399 + 0.431733i 0 4.22289 0 −0.289169 − 2.98603i 0
959.18 0 1.16422 − 1.28241i 0 2.19399 + 0.431733i 0 4.22289 0 −0.289169 − 2.98603i 0
959.19 0 1.16422 + 1.28241i 0 −2.19399 − 0.431733i 0 4.22289 0 −0.289169 + 2.98603i 0
959.20 0 1.16422 + 1.28241i 0 2.19399 − 0.431733i 0 4.22289 0 −0.289169 + 2.98603i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 959.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.o.e 24
3.b odd 2 1 inner 960.2.o.e 24
4.b odd 2 1 inner 960.2.o.e 24
5.b even 2 1 inner 960.2.o.e 24
8.b even 2 1 480.2.o.a ✓ 24
8.d odd 2 1 480.2.o.a ✓ 24
12.b even 2 1 inner 960.2.o.e 24
15.d odd 2 1 inner 960.2.o.e 24
20.d odd 2 1 inner 960.2.o.e 24
24.f even 2 1 480.2.o.a ✓ 24
24.h odd 2 1 480.2.o.a ✓ 24
40.e odd 2 1 480.2.o.a ✓ 24
40.f even 2 1 480.2.o.a ✓ 24
40.i odd 4 2 2400.2.h.h 24
40.k even 4 2 2400.2.h.h 24
60.h even 2 1 inner 960.2.o.e 24
120.i odd 2 1 480.2.o.a ✓ 24
120.m even 2 1 480.2.o.a ✓ 24
120.q odd 4 2 2400.2.h.h 24
120.w even 4 2 2400.2.h.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.o.a ✓ 24 8.b even 2 1
480.2.o.a ✓ 24 8.d odd 2 1
480.2.o.a ✓ 24 24.f even 2 1
480.2.o.a ✓ 24 24.h odd 2 1
480.2.o.a ✓ 24 40.e odd 2 1
480.2.o.a ✓ 24 40.f even 2 1
480.2.o.a ✓ 24 120.i odd 2 1
480.2.o.a ✓ 24 120.m even 2 1
960.2.o.e 24 1.a even 1 1 trivial
960.2.o.e 24 3.b odd 2 1 inner
960.2.o.e 24 4.b odd 2 1 inner
960.2.o.e 24 5.b even 2 1 inner
960.2.o.e 24 12.b even 2 1 inner
960.2.o.e 24 15.d odd 2 1 inner
960.2.o.e 24 20.d odd 2 1 inner
960.2.o.e 24 60.h even 2 1 inner
2400.2.h.h 24 40.i odd 4 2
2400.2.h.h 24 40.k even 4 2
2400.2.h.h 24 120.q odd 4 2
2400.2.h.h 24 120.w even 4 2

Hecke kernels

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

\( T_{7}^{6} - 26T_{7}^{4} + 160T_{7}^{2} - 256 \) Copy content Toggle raw display
\( T_{11}^{6} - 40T_{11}^{4} + 288T_{11}^{2} - 512 \) Copy content Toggle raw display