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Properties

Label 960.2.bb.a
Level $960$
Weight $2$
Character orbit 960.bb
Analytic conductor $7.666$
Analytic rank $0$
Dimension $88$
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(497,960)]
 
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, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.497");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.bb (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: \(88\)
Relative dimension: \(44\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{15} + 8 q^{19} - 4 q^{21} + 16 q^{31} - 4 q^{33} - 24 q^{39} + 40 q^{43} + 8 q^{45} + 4 q^{51} + 12 q^{57} - 24 q^{61} + 32 q^{63} + 8 q^{67} - 12 q^{69} + 24 q^{75} - 8 q^{81} - 24 q^{85} + 12 q^{87} + 8 q^{91} - 8 q^{97} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
497.1 0 −1.72946 + 0.0947166i 0 −1.85549 − 1.24786i 0 −0.907692 + 0.907692i 0 2.98206 − 0.327617i 0
497.2 0 −1.70651 − 0.296326i 0 2.17873 − 0.503115i 0 2.12944 − 2.12944i 0 2.82438 + 1.01137i 0
497.3 0 −1.69498 + 0.356415i 0 1.08277 + 1.95643i 0 −2.05875 + 2.05875i 0 2.74594 − 1.20823i 0
497.4 0 −1.66945 − 0.461441i 0 1.37523 − 1.76316i 0 0.367568 − 0.367568i 0 2.57414 + 1.54071i 0
497.5 0 −1.65521 + 0.510164i 0 0.369196 + 2.20538i 0 2.84513 − 2.84513i 0 2.47947 − 1.68886i 0
497.6 0 −1.61586 − 0.623684i 0 −0.435524 + 2.19324i 0 −2.31966 + 2.31966i 0 2.22204 + 2.01558i 0
497.7 0 −1.54229 − 0.788258i 0 −2.21092 + 0.334403i 0 2.87827 − 2.87827i 0 1.75730 + 2.43144i 0
497.8 0 −1.53551 + 0.801371i 0 −1.97816 + 1.04254i 0 0.592869 − 0.592869i 0 1.71561 − 2.46103i 0
497.9 0 −1.42603 + 0.983076i 0 0.322899 − 2.21263i 0 −1.41445 + 1.41445i 0 1.06712 − 2.80379i 0
497.10 0 −1.33164 + 1.10758i 0 −1.55623 − 1.60566i 0 0.854868 − 0.854868i 0 0.546521 − 2.94980i 0
497.11 0 −1.16822 + 1.27877i 0 2.22526 + 0.219616i 0 1.05780 − 1.05780i 0 −0.270509 − 2.98778i 0
497.12 0 −1.13657 − 1.30699i 0 −0.583249 − 2.15866i 0 2.32976 − 2.32976i 0 −0.416423 + 2.97096i 0
497.13 0 −1.09486 − 1.34211i 0 −1.71343 − 1.43671i 0 −3.11495 + 3.11495i 0 −0.602544 + 2.93887i 0
497.14 0 −1.05053 − 1.37710i 0 −0.966676 + 2.01632i 0 −0.166503 + 0.166503i 0 −0.792787 + 2.89335i 0
497.15 0 −0.856888 − 1.50524i 0 1.41205 − 1.73381i 0 −1.42263 + 1.42263i 0 −1.53149 + 2.57964i 0
497.16 0 −0.847085 + 1.51078i 0 2.22651 − 0.206476i 0 −0.209149 + 0.209149i 0 −1.56489 − 2.55951i 0
497.17 0 −0.720802 − 1.57494i 0 2.13443 + 0.666497i 0 −2.25736 + 2.25736i 0 −1.96089 + 2.27044i 0
497.18 0 −0.672846 + 1.59602i 0 −1.96698 + 1.06348i 0 −3.58885 + 3.58885i 0 −2.09456 − 2.14775i 0
497.19 0 −0.349092 − 1.69651i 0 0.111699 + 2.23328i 0 1.82036 − 1.82036i 0 −2.75627 + 1.18447i 0
497.20 0 −0.147625 − 1.72575i 0 2.12868 + 0.684619i 0 1.09901 − 1.09901i 0 −2.95641 + 0.509529i 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 497.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
80.i odd 4 1 inner
240.bb 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.bb.a 88
3.b odd 2 1 inner 960.2.bb.a 88
4.b odd 2 1 240.2.bb.a ✓ 88
5.c odd 4 1 960.2.bf.a 88
12.b even 2 1 240.2.bb.a ✓ 88
15.e even 4 1 960.2.bf.a 88
16.e even 4 1 960.2.bf.a 88
16.f odd 4 1 240.2.bf.a yes 88
20.e even 4 1 240.2.bf.a yes 88
48.i odd 4 1 960.2.bf.a 88
48.k even 4 1 240.2.bf.a yes 88
60.l odd 4 1 240.2.bf.a yes 88
80.i odd 4 1 inner 960.2.bb.a 88
80.s even 4 1 240.2.bb.a ✓ 88
240.z odd 4 1 240.2.bb.a ✓ 88
240.bb even 4 1 inner 960.2.bb.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.bb.a ✓ 88 4.b odd 2 1
240.2.bb.a ✓ 88 12.b even 2 1
240.2.bb.a ✓ 88 80.s even 4 1
240.2.bb.a ✓ 88 240.z odd 4 1
240.2.bf.a yes 88 16.f odd 4 1
240.2.bf.a yes 88 20.e even 4 1
240.2.bf.a yes 88 48.k even 4 1
240.2.bf.a yes 88 60.l odd 4 1
960.2.bb.a 88 1.a even 1 1 trivial
960.2.bb.a 88 3.b odd 2 1 inner
960.2.bb.a 88 80.i odd 4 1 inner
960.2.bb.a 88 240.bb even 4 1 inner
960.2.bf.a 88 5.c odd 4 1
960.2.bf.a 88 15.e even 4 1
960.2.bf.a 88 16.e even 4 1
960.2.bf.a 88 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(960, [\chi])\).