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Properties

Label 656.2.bs.d
Level $656$
Weight $2$
Character orbit 656.bs
Analytic conductor $5.238$
Analytic rank $0$
Dimension $24$
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] = [656,2,Mod(33,656)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(656, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("656.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 656 = 2^{4} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 656.bs (of order \(20\), degree \(8\), 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: \(5.23818637260\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 41)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} - 10 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} - 10 q^{5} + 8 q^{7} + 16 q^{11} - 8 q^{15} + 8 q^{17} - 16 q^{19} - 10 q^{21} - 12 q^{23} - 8 q^{25} + 6 q^{27} + 40 q^{29} + 12 q^{31} + 10 q^{33} + 36 q^{35} + 50 q^{39} - 4 q^{41} + 16 q^{45} + 12 q^{47} - 30 q^{49} + 24 q^{51} - 26 q^{53} - 20 q^{55} + 10 q^{57} - 6 q^{59} + 30 q^{61} - 92 q^{63} + 68 q^{65} + 22 q^{67} - 38 q^{69} - 4 q^{71} - 4 q^{75} - 20 q^{77} + 2 q^{79} + 28 q^{81} - 80 q^{83} - 56 q^{85} + 10 q^{87} - 72 q^{89} - 6 q^{93} + 40 q^{95} - 22 q^{97} - 14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1 0 −2.22848 + 2.22848i 0 −2.57687 + 0.837277i 0 0.252316 − 1.59306i 0 − 6.93221i 0
33.2 0 0.983583 − 0.983583i 0 1.02071 − 0.331648i 0 −0.625712 + 3.95059i 0 1.06513i 0
33.3 0 2.02366 − 2.02366i 0 0.110420 − 0.0358775i 0 0.422339 − 2.66654i 0 − 5.19040i 0
49.1 0 −1.24382 − 1.24382i 0 −1.15434 + 0.375067i 0 −1.19485 − 0.189245i 0 0.0941838i 0
49.2 0 0.242613 + 0.242613i 0 2.26179 − 0.734900i 0 4.85225 + 0.768522i 0 − 2.88228i 0
49.3 0 0.604406 + 0.604406i 0 −3.27974 + 1.06565i 0 −1.70635 − 0.270260i 0 − 2.26939i 0
225.1 0 −0.964912 + 0.964912i 0 −2.03721 + 2.80398i 0 −1.29765 − 0.661185i 0 1.13789i 0
225.2 0 1.55151 − 1.55151i 0 1.49595 − 2.05900i 0 1.01547 + 0.517405i 0 − 1.81438i 0
225.3 0 1.67347 − 1.67347i 0 −1.68856 + 2.32411i 0 1.86997 + 0.952797i 0 − 2.60102i 0
241.1 0 −1.24382 + 1.24382i 0 −1.15434 − 0.375067i 0 −1.19485 + 0.189245i 0 − 0.0941838i 0
241.2 0 0.242613 − 0.242613i 0 2.26179 + 0.734900i 0 4.85225 − 0.768522i 0 2.88228i 0
241.3 0 0.604406 − 0.604406i 0 −3.27974 − 1.06565i 0 −1.70635 + 0.270260i 0 2.26939i 0
289.1 0 −1.49921 + 1.49921i 0 1.72514 + 2.37446i 0 1.11329 + 2.18496i 0 − 1.49525i 0
289.2 0 −0.0432913 + 0.0432913i 0 −0.422124 − 0.581004i 0 1.42228 + 2.79137i 0 2.99625i 0
289.3 0 1.90046 − 1.90046i 0 −0.455161 − 0.626475i 0 −2.12336 − 4.16732i 0 − 4.22349i 0
449.1 0 −0.964912 − 0.964912i 0 −2.03721 − 2.80398i 0 −1.29765 + 0.661185i 0 − 1.13789i 0
449.2 0 1.55151 + 1.55151i 0 1.49595 + 2.05900i 0 1.01547 − 0.517405i 0 1.81438i 0
449.3 0 1.67347 + 1.67347i 0 −1.68856 − 2.32411i 0 1.86997 − 0.952797i 0 2.60102i 0
497.1 0 −2.22848 − 2.22848i 0 −2.57687 − 0.837277i 0 0.252316 + 1.59306i 0 6.93221i 0
497.2 0 0.983583 + 0.983583i 0 1.02071 + 0.331648i 0 −0.625712 − 3.95059i 0 − 1.06513i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 656.2.bs.d 24
4.b odd 2 1 41.2.g.a ✓ 24
12.b even 2 1 369.2.u.a 24
41.g even 20 1 inner 656.2.bs.d 24
164.n odd 20 1 41.2.g.a ✓ 24
164.o even 40 2 1681.2.a.m 24
492.y even 20 1 369.2.u.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.g.a ✓ 24 4.b odd 2 1
41.2.g.a ✓ 24 164.n odd 20 1
369.2.u.a 24 12.b even 2 1
369.2.u.a 24 492.y even 20 1
656.2.bs.d 24 1.a even 1 1 trivial
656.2.bs.d 24 41.g even 20 1 inner
1681.2.a.m 24 164.o even 40 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 6 T_{3}^{23} + 18 T_{3}^{22} - 26 T_{3}^{21} + 149 T_{3}^{20} - 790 T_{3}^{19} + 2396 T_{3}^{18} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(656, [\chi])\). Copy content Toggle raw display