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Properties

Label 960.2.bf.a
Level $960$
Weight $2$
Character orbit 960.bf
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(17,960)]
 
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, 3, 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("960.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.bf (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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{3} - 8 q^{13} + 4 q^{15} - 8 q^{19} - 4 q^{21} + 4 q^{27} + 16 q^{31} - 4 q^{33} - 8 q^{37} + 24 q^{39} + 8 q^{45} + 4 q^{51} - 12 q^{57} - 24 q^{61} + 32 q^{63} + 12 q^{69} + 36 q^{75} - 8 q^{81} + 16 q^{85} - 12 q^{87} + 8 q^{91} - 16 q^{93} - 8 q^{97} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −1.73139 − 0.0477618i 0 1.82438 + 1.29291i 0 −2.39784 + 2.39784i 0 2.99544 + 0.165389i 0
17.2 0 −1.73139 + 0.0477618i 0 −1.82438 − 1.29291i 0 −2.39784 + 2.39784i 0 2.99544 − 0.165389i 0
17.3 0 −1.72806 − 0.117548i 0 −1.91324 + 1.15738i 0 0.912923 − 0.912923i 0 2.97237 + 0.406258i 0
17.4 0 −1.72806 + 0.117548i 0 1.91324 − 1.15738i 0 0.912923 − 0.912923i 0 2.97237 − 0.406258i 0
17.5 0 −1.59602 − 0.672846i 0 1.06348 + 1.96698i 0 3.58885 − 3.58885i 0 2.09456 + 2.14775i 0
17.6 0 −1.59602 + 0.672846i 0 −1.06348 − 1.96698i 0 3.58885 − 3.58885i 0 2.09456 − 2.14775i 0
17.7 0 −1.51078 − 0.847085i 0 −0.206476 − 2.22651i 0 0.209149 − 0.209149i 0 1.56489 + 2.55951i 0
17.8 0 −1.51078 + 0.847085i 0 0.206476 + 2.22651i 0 0.209149 − 0.209149i 0 1.56489 − 2.55951i 0
17.9 0 −1.27877 − 1.16822i 0 0.219616 − 2.22526i 0 −1.05780 + 1.05780i 0 0.270509 + 2.98778i 0
17.10 0 −1.27877 + 1.16822i 0 −0.219616 + 2.22526i 0 −1.05780 + 1.05780i 0 0.270509 − 2.98778i 0
17.11 0 −1.10758 − 1.33164i 0 −1.60566 + 1.55623i 0 −0.854868 + 0.854868i 0 −0.546521 + 2.94980i 0
17.12 0 −1.10758 + 1.33164i 0 1.60566 − 1.55623i 0 −0.854868 + 0.854868i 0 −0.546521 − 2.94980i 0
17.13 0 −0.983076 − 1.42603i 0 −2.21263 − 0.322899i 0 1.41445 − 1.41445i 0 −1.06712 + 2.80379i 0
17.14 0 −0.983076 + 1.42603i 0 2.21263 + 0.322899i 0 1.41445 − 1.41445i 0 −1.06712 − 2.80379i 0
17.15 0 −0.801371 − 1.53551i 0 1.04254 + 1.97816i 0 −0.592869 + 0.592869i 0 −1.71561 + 2.46103i 0
17.16 0 −0.801371 + 1.53551i 0 −1.04254 − 1.97816i 0 −0.592869 + 0.592869i 0 −1.71561 − 2.46103i 0
17.17 0 −0.510164 − 1.65521i 0 2.20538 − 0.369196i 0 −2.84513 + 2.84513i 0 −2.47947 + 1.68886i 0
17.18 0 −0.510164 + 1.65521i 0 −2.20538 + 0.369196i 0 −2.84513 + 2.84513i 0 −2.47947 − 1.68886i 0
17.19 0 −0.356415 − 1.69498i 0 1.95643 − 1.08277i 0 2.05875 − 2.05875i 0 −2.74594 + 1.20823i 0
17.20 0 −0.356415 + 1.69498i 0 −1.95643 + 1.08277i 0 2.05875 − 2.05875i 0 −2.74594 − 1.20823i 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.44
Significant digits:
Format:

Inner twists

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

Hecke kernels

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