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Properties

Label 220.2.r.c
Level $220$
Weight $2$
Character orbit 220.r
Analytic conductor $1.757$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,2,Mod(51,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 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("220.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 220.r (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: \(1.75670884447\)
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{2} - 2 q^{4} + 8 q^{5} - 15 q^{6} - 10 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{2} - 2 q^{4} + 8 q^{5} - 15 q^{6} - 10 q^{8} + 10 q^{9} + 6 q^{12} - 10 q^{13} + 21 q^{14} - 34 q^{16} + 10 q^{17} - 25 q^{18} + 2 q^{20} - 36 q^{22} - 15 q^{24} - 8 q^{25} + 12 q^{26} + 30 q^{28} - 10 q^{29} + 10 q^{30} + 36 q^{33} + 18 q^{34} + 15 q^{36} - 44 q^{37} + 29 q^{38} - 10 q^{40} - 30 q^{41} + 18 q^{42} + 9 q^{44} + 60 q^{45} - 10 q^{46} + 38 q^{48} + 12 q^{49} + 10 q^{50} + 10 q^{52} - 14 q^{53} - 34 q^{56} - 60 q^{57} + 10 q^{58} - 11 q^{60} + 20 q^{61} + 40 q^{62} - 74 q^{64} - 118 q^{66} - 20 q^{68} - 22 q^{69} - 21 q^{70} + 70 q^{72} + 10 q^{73} + 20 q^{74} + 44 q^{77} - 92 q^{78} - 46 q^{80} - 58 q^{81} + 37 q^{82} + 100 q^{84} - 10 q^{85} + 57 q^{86} + 19 q^{88} - 72 q^{89} + 10 q^{90} - 63 q^{92} - 26 q^{93} - 25 q^{94} + 94 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −1.26745 − 0.627351i −1.88633 − 0.612906i 1.21286 + 1.59027i 0.809017 − 0.587785i 2.00632 + 1.96022i 0.668539 + 2.05755i −0.539585 − 2.77648i 0.755537 + 0.548930i −1.39414 + 0.237451i
51.2 −1.26745 − 0.627351i 2.42619 + 0.788316i 1.21286 + 1.59027i 0.809017 − 0.587785i −2.58052 − 2.52122i −1.33130 − 4.09732i −0.539585 − 2.77648i 2.83789 + 2.06185i −1.39414 + 0.237451i
51.3 −1.21274 + 0.727497i −2.51735 − 0.817938i 0.941497 − 1.76453i 0.809017 − 0.587785i 3.64795 − 0.839417i 0.0373322 + 0.114897i 0.141898 + 2.82487i 3.24099 + 2.35472i −0.553519 + 1.30139i
51.4 −1.21274 + 0.727497i 0.919215 + 0.298671i 0.941497 − 1.76453i 0.809017 − 0.587785i −1.33205 + 0.306514i 0.819704 + 2.52279i 0.141898 + 2.82487i −1.67130 − 1.21427i −0.553519 + 1.30139i
51.5 −0.317132 + 1.37820i −0.919215 − 0.298671i −1.79885 − 0.874140i 0.809017 − 0.587785i 0.703140 − 1.17214i −0.819704 − 2.52279i 1.77521 − 2.20196i −1.67130 − 1.21427i 0.553519 + 1.30139i
51.6 −0.317132 + 1.37820i 2.51735 + 0.817938i −1.79885 − 0.874140i 0.809017 − 0.587785i −1.92561 + 3.21001i −0.0373322 − 0.114897i 1.77521 − 2.20196i 3.24099 + 2.35472i 0.553519 + 1.30139i
51.7 0.988310 + 1.01156i −2.42619 − 0.788316i −0.0464877 + 1.99946i 0.809017 − 0.587785i −1.60040 − 3.23332i 1.33130 + 4.09732i −2.06851 + 1.92906i 2.83789 + 2.06185i 1.39414 + 0.237451i
51.8 0.988310 + 1.01156i 1.88633 + 0.612906i −0.0464877 + 1.99946i 0.809017 − 0.587785i 1.24429 + 2.51387i −0.668539 − 2.05755i −2.06851 + 1.92906i 0.755537 + 0.548930i 1.39414 + 0.237451i
151.1 −1.26745 + 0.627351i −1.88633 + 0.612906i 1.21286 − 1.59027i 0.809017 + 0.587785i 2.00632 − 1.96022i 0.668539 − 2.05755i −0.539585 + 2.77648i 0.755537 − 0.548930i −1.39414 − 0.237451i
151.2 −1.26745 + 0.627351i 2.42619 − 0.788316i 1.21286 − 1.59027i 0.809017 + 0.587785i −2.58052 + 2.52122i −1.33130 + 4.09732i −0.539585 + 2.77648i 2.83789 − 2.06185i −1.39414 − 0.237451i
151.3 −1.21274 − 0.727497i −2.51735 + 0.817938i 0.941497 + 1.76453i 0.809017 + 0.587785i 3.64795 + 0.839417i 0.0373322 − 0.114897i 0.141898 − 2.82487i 3.24099 − 2.35472i −0.553519 − 1.30139i
151.4 −1.21274 − 0.727497i 0.919215 − 0.298671i 0.941497 + 1.76453i 0.809017 + 0.587785i −1.33205 − 0.306514i 0.819704 − 2.52279i 0.141898 − 2.82487i −1.67130 + 1.21427i −0.553519 − 1.30139i
151.5 −0.317132 − 1.37820i −0.919215 + 0.298671i −1.79885 + 0.874140i 0.809017 + 0.587785i 0.703140 + 1.17214i −0.819704 + 2.52279i 1.77521 + 2.20196i −1.67130 + 1.21427i 0.553519 − 1.30139i
151.6 −0.317132 − 1.37820i 2.51735 − 0.817938i −1.79885 + 0.874140i 0.809017 + 0.587785i −1.92561 − 3.21001i −0.0373322 + 0.114897i 1.77521 + 2.20196i 3.24099 − 2.35472i 0.553519 − 1.30139i
151.7 0.988310 − 1.01156i −2.42619 + 0.788316i −0.0464877 − 1.99946i 0.809017 + 0.587785i −1.60040 + 3.23332i 1.33130 − 4.09732i −2.06851 − 1.92906i 2.83789 − 2.06185i 1.39414 − 0.237451i
151.8 0.988310 − 1.01156i 1.88633 − 0.612906i −0.0464877 − 1.99946i 0.809017 + 0.587785i 1.24429 − 2.51387i −0.668539 + 2.05755i −2.06851 − 1.92906i 0.755537 − 0.548930i 1.39414 − 0.237451i
171.1 −1.32039 − 0.506519i 0.181311 − 0.249553i 1.48688 + 1.33761i −0.309017 + 0.951057i −0.365805 + 0.237671i −3.62272 + 2.63206i −1.28574 − 2.51930i 0.897648 + 2.76268i 0.889752 − 1.09925i
171.2 −1.32039 − 0.506519i 1.52406 − 2.09769i 1.48688 + 1.33761i −0.309017 + 0.951057i −3.07488 + 1.99781i 2.61302 − 1.89847i −1.28574 − 2.51930i −1.15049 − 3.54085i 0.889752 − 1.09925i
171.3 −0.770496 + 1.18589i −1.52406 + 2.09769i −0.812670 − 1.82745i −0.309017 + 0.951057i −1.31335 − 3.42363i −2.61302 + 1.89847i 2.79331 + 0.444304i −1.15049 − 3.54085i −0.889752 − 1.09925i
171.4 −0.770496 + 1.18589i −0.181311 + 0.249553i −0.812670 − 1.82745i −0.309017 + 0.951057i −0.156243 − 0.407294i 3.62272 − 2.63206i 2.79331 + 0.444304i 0.897648 + 2.76268i −0.889752 − 1.09925i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.8
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 220.2.r.c ✓ 32
4.b odd 2 1 inner 220.2.r.c ✓ 32
11.d odd 10 1 inner 220.2.r.c ✓ 32
44.g even 10 1 inner 220.2.r.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.r.c ✓ 32 1.a even 1 1 trivial
220.2.r.c ✓ 32 4.b odd 2 1 inner
220.2.r.c ✓ 32 11.d odd 10 1 inner
220.2.r.c ✓ 32 44.g even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 17 T_{3}^{30} + 240 T_{3}^{28} - 3030 T_{3}^{26} + 31755 T_{3}^{24} - 250386 T_{3}^{22} + \cdots + 16777216 \) acting on \(S_{2}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display