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Properties

Label 880.2.bz.d
Level $880$
Weight $2$
Character orbit 880.bz
Analytic conductor $7.027$
Analytic rank $0$
Dimension $32$
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] = [880,2,Mod(271,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bz (of order \(10\), degree \(4\), 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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{5} - 6 q^{9} - 30 q^{13} + 30 q^{17} - 8 q^{25} + 70 q^{29} + 24 q^{33} + 30 q^{41} + 36 q^{45} - 32 q^{49} + 6 q^{53} + 100 q^{57} + 20 q^{61} - 42 q^{69} - 10 q^{73} + 32 q^{77} - 94 q^{81} + 10 q^{85} - 16 q^{89} - 6 q^{93} - 78 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1 0 −2.82257 − 0.917107i 0 0.809017 − 0.587785i 0 −0.458792 − 1.41202i 0 4.69874 + 3.41384i 0
271.2 0 −1.52460 − 0.495373i 0 0.809017 − 0.587785i 0 1.26256 + 3.88575i 0 −0.348040 − 0.252866i 0
271.3 0 −1.02316 − 0.332444i 0 0.809017 − 0.587785i 0 −1.10944 − 3.41452i 0 −1.49072 − 1.08307i 0
271.4 0 −0.743264 − 0.241501i 0 0.809017 − 0.587785i 0 0.611903 + 1.88324i 0 −1.93293 − 1.40436i 0
271.5 0 0.743264 + 0.241501i 0 0.809017 − 0.587785i 0 −0.611903 − 1.88324i 0 −1.93293 − 1.40436i 0
271.6 0 1.02316 + 0.332444i 0 0.809017 − 0.587785i 0 1.10944 + 3.41452i 0 −1.49072 − 1.08307i 0
271.7 0 1.52460 + 0.495373i 0 0.809017 − 0.587785i 0 −1.26256 − 3.88575i 0 −0.348040 − 0.252866i 0
271.8 0 2.82257 + 0.917107i 0 0.809017 − 0.587785i 0 0.458792 + 1.41202i 0 4.69874 + 3.41384i 0
431.1 0 −2.02564 − 2.78806i 0 −0.309017 − 0.951057i 0 −3.05958 − 2.22291i 0 −2.74299 + 8.44204i 0
431.2 0 −1.26430 − 1.74016i 0 −0.309017 − 0.951057i 0 −1.48063 − 1.07574i 0 −0.502647 + 1.54699i 0
431.3 0 −1.06156 − 1.46111i 0 −0.309017 − 0.951057i 0 1.73562 + 1.26100i 0 −0.0808819 + 0.248929i 0
431.4 0 −0.175622 − 0.241723i 0 −0.309017 − 0.951057i 0 0.156663 + 0.113822i 0 0.899464 − 2.76827i 0
431.5 0 0.175622 + 0.241723i 0 −0.309017 − 0.951057i 0 −0.156663 − 0.113822i 0 0.899464 − 2.76827i 0
431.6 0 1.06156 + 1.46111i 0 −0.309017 − 0.951057i 0 −1.73562 − 1.26100i 0 −0.0808819 + 0.248929i 0
431.7 0 1.26430 + 1.74016i 0 −0.309017 − 0.951057i 0 1.48063 + 1.07574i 0 −0.502647 + 1.54699i 0
431.8 0 2.02564 + 2.78806i 0 −0.309017 − 0.951057i 0 3.05958 + 2.22291i 0 −2.74299 + 8.44204i 0
591.1 0 −2.82257 + 0.917107i 0 0.809017 + 0.587785i 0 −0.458792 + 1.41202i 0 4.69874 − 3.41384i 0
591.2 0 −1.52460 + 0.495373i 0 0.809017 + 0.587785i 0 1.26256 − 3.88575i 0 −0.348040 + 0.252866i 0
591.3 0 −1.02316 + 0.332444i 0 0.809017 + 0.587785i 0 −1.10944 + 3.41452i 0 −1.49072 + 1.08307i 0
591.4 0 −0.743264 + 0.241501i 0 0.809017 + 0.587785i 0 0.611903 − 1.88324i 0 −1.93293 + 1.40436i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bz.d ✓ 32
4.b odd 2 1 inner 880.2.bz.d ✓ 32
11.d odd 10 1 inner 880.2.bz.d ✓ 32
44.g even 10 1 inner 880.2.bz.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.bz.d ✓ 32 1.a even 1 1 trivial
880.2.bz.d ✓ 32 4.b odd 2 1 inner
880.2.bz.d ✓ 32 11.d odd 10 1 inner
880.2.bz.d ✓ 32 44.g even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 9 T_{3}^{30} + 154 T_{3}^{28} - 1974 T_{3}^{26} + 18081 T_{3}^{24} - 77082 T_{3}^{22} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display