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Properties

Label 880.2.bz.c
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
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{5} - 6 q^{9} + 10 q^{13} - 30 q^{17} - 8 q^{25} - 10 q^{29} - 16 q^{33} + 24 q^{37} + 70 q^{41} - 36 q^{45} + 16 q^{49} + 42 q^{53} - 60 q^{57} - 60 q^{61} + 30 q^{69} + 70 q^{73} - 100 q^{77} - 14 q^{81} - 10 q^{85} + 48 q^{89} + 14 q^{93} - 62 q^{97}+O(q^{100}) \) 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 −2.41776 − 0.785576i 0 −0.809017 + 0.587785i 0 −0.219916 − 0.676833i 0 2.80136 + 2.03531i 0
271.2 0 −2.26520 − 0.736009i 0 −0.809017 + 0.587785i 0 −1.30671 − 4.02163i 0 2.16239 + 1.57107i 0
271.3 0 −0.923473 − 0.300054i 0 −0.809017 + 0.587785i 0 0.121956 + 0.375343i 0 −1.66428 − 1.20917i 0
271.4 0 −0.247153 − 0.0803050i 0 −0.809017 + 0.587785i 0 0.964835 + 2.96946i 0 −2.37242 − 1.72366i 0
271.5 0 0.247153 + 0.0803050i 0 −0.809017 + 0.587785i 0 −0.964835 − 2.96946i 0 −2.37242 − 1.72366i 0
271.6 0 0.923473 + 0.300054i 0 −0.809017 + 0.587785i 0 −0.121956 − 0.375343i 0 −1.66428 − 1.20917i 0
271.7 0 2.26520 + 0.736009i 0 −0.809017 + 0.587785i 0 1.30671 + 4.02163i 0 2.16239 + 1.57107i 0
271.8 0 2.41776 + 0.785576i 0 −0.809017 + 0.587785i 0 0.219916 + 0.676833i 0 2.80136 + 2.03531i 0
431.1 0 −1.77368 − 2.44126i 0 0.309017 + 0.951057i 0 2.82668 + 2.05370i 0 −1.88677 + 5.80687i 0
431.2 0 −1.57943 − 2.17390i 0 0.309017 + 0.951057i 0 0.234746 + 0.170553i 0 −1.30419 + 4.01388i 0
431.3 0 −1.01220 − 1.39318i 0 0.309017 + 0.951057i 0 −4.11502 − 2.98973i 0 0.0106619 − 0.0328139i 0
431.4 0 −0.440826 − 0.606744i 0 0.309017 + 0.951057i 0 1.52308 + 1.10658i 0 0.753240 − 2.31823i 0
431.5 0 0.440826 + 0.606744i 0 0.309017 + 0.951057i 0 −1.52308 − 1.10658i 0 0.753240 − 2.31823i 0
431.6 0 1.01220 + 1.39318i 0 0.309017 + 0.951057i 0 4.11502 + 2.98973i 0 0.0106619 − 0.0328139i 0
431.7 0 1.57943 + 2.17390i 0 0.309017 + 0.951057i 0 −0.234746 − 0.170553i 0 −1.30419 + 4.01388i 0
431.8 0 1.77368 + 2.44126i 0 0.309017 + 0.951057i 0 −2.82668 − 2.05370i 0 −1.88677 + 5.80687i 0
591.1 0 −2.41776 + 0.785576i 0 −0.809017 − 0.587785i 0 −0.219916 + 0.676833i 0 2.80136 − 2.03531i 0
591.2 0 −2.26520 + 0.736009i 0 −0.809017 − 0.587785i 0 −1.30671 + 4.02163i 0 2.16239 − 1.57107i 0
591.3 0 −0.923473 + 0.300054i 0 −0.809017 − 0.587785i 0 0.121956 − 0.375343i 0 −1.66428 + 1.20917i 0
591.4 0 −0.247153 + 0.0803050i 0 −0.809017 − 0.587785i 0 0.964835 − 2.96946i 0 −2.37242 + 1.72366i 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.c ✓ 32
4.b odd 2 1 inner 880.2.bz.c ✓ 32
11.d odd 10 1 inner 880.2.bz.c ✓ 32
44.g even 10 1 inner 880.2.bz.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.bz.c ✓ 32 1.a even 1 1 trivial
880.2.bz.c ✓ 32 4.b odd 2 1 inner
880.2.bz.c ✓ 32 11.d odd 10 1 inner
880.2.bz.c ✓ 32 44.g even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 9 T_{3}^{30} + 134 T_{3}^{28} - 1494 T_{3}^{26} + 12841 T_{3}^{24} - 79182 T_{3}^{22} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display