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Properties

Label 550.2.k.e
Level $550$
Weight $2$
Character orbit 550.k
Analytic conductor $4.392$
Analytic rank $0$
Dimension $28$
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] = [550,2,Mod(111,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.k (of order \(5\), 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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 7 q^{2} + 2 q^{3} - 7 q^{4} + 3 q^{5} - 2 q^{6} + 7 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 7 q^{2} + 2 q^{3} - 7 q^{4} + 3 q^{5} - 2 q^{6} + 7 q^{8} - 7 q^{9} + 2 q^{10} + 7 q^{11} + 2 q^{12} - 6 q^{13} - 9 q^{15} - 7 q^{16} + 13 q^{17} - 28 q^{18} + q^{19} + 3 q^{20} - 21 q^{21} - 7 q^{22} - 3 q^{23} + 8 q^{24} + 3 q^{25} + 6 q^{26} + 5 q^{27} - 16 q^{29} - q^{30} - 7 q^{31} - 28 q^{32} - 2 q^{33} - 13 q^{34} + 14 q^{35} - 7 q^{36} + 12 q^{37} + 9 q^{38} - 23 q^{39} + 2 q^{40} - 28 q^{41} + 21 q^{42} + 24 q^{43} + 7 q^{44} - 26 q^{45} - 17 q^{46} - 22 q^{47} + 2 q^{48} + 52 q^{49} + 22 q^{50} + 38 q^{51} - 6 q^{52} - 6 q^{53} - 20 q^{54} + 2 q^{55} - 30 q^{57} + 16 q^{58} + 33 q^{59} - 4 q^{60} + 10 q^{61} + 7 q^{62} - 14 q^{63} - 7 q^{64} - 34 q^{65} + 2 q^{66} + 33 q^{67} - 52 q^{68} - 9 q^{69} - 14 q^{70} - 53 q^{71} + 7 q^{72} - 35 q^{73} - 2 q^{74} + 61 q^{75} + 16 q^{76} - 2 q^{78} - 13 q^{79} - 2 q^{80} + 29 q^{81} - 32 q^{82} - 13 q^{83} + 14 q^{84} - 17 q^{85} + 11 q^{86} - 12 q^{87} - 7 q^{88} + 3 q^{89} - 49 q^{90} - 47 q^{91} + 17 q^{92} + 64 q^{93} + 22 q^{94} - 42 q^{95} - 2 q^{96} + 7 q^{97} + 53 q^{98} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1 0.809017 − 0.587785i −0.914265 + 2.81382i 0.309017 − 0.951057i −0.178597 + 2.22892i 0.914265 + 2.81382i −3.95733 −0.309017 − 0.951057i −4.65464 − 3.38180i 1.16564 + 1.90821i
111.2 0.809017 − 0.587785i −0.867379 + 2.66952i 0.309017 − 0.951057i 2.20676 − 0.360833i 0.867379 + 2.66952i 4.83292 −0.309017 − 0.951057i −3.94692 − 2.86761i 1.57322 − 1.58902i
111.3 0.809017 − 0.587785i −0.292625 + 0.900607i 0.309017 − 0.951057i −2.20530 + 0.369657i 0.292625 + 0.900607i 0.0430906 −0.309017 − 0.951057i 1.70159 + 1.23628i −1.56685 + 1.59530i
111.4 0.809017 − 0.587785i 0.0101960 − 0.0313800i 0.309017 − 0.951057i 2.13334 + 0.669986i −0.0101960 − 0.0313800i −0.656054 −0.309017 − 0.951057i 2.42617 + 1.76272i 2.11971 − 0.711913i
111.5 0.809017 − 0.587785i 0.155836 − 0.479613i 0.309017 − 0.951057i −1.44626 − 1.70538i −0.155836 − 0.479613i 4.09095 −0.309017 − 0.951057i 2.22131 + 1.61387i −2.17245 − 0.529595i
111.6 0.809017 − 0.587785i 0.377514 − 1.16187i 0.309017 − 0.951057i −0.259398 − 2.22097i −0.377514 − 1.16187i −4.84382 −0.309017 − 0.951057i 1.21963 + 0.886113i −1.51531 − 1.64433i
111.7 0.809017 − 0.587785i 0.912689 − 2.80897i 0.309017 − 0.951057i 1.05848 + 1.96968i −0.912689 − 2.80897i 0.490238 −0.309017 − 0.951057i −4.63025 − 3.36407i 2.01407 + 0.971347i
221.1 −0.309017 + 0.951057i −2.02207 − 1.46912i −0.809017 − 0.587785i −0.574600 + 2.16098i 2.02207 − 1.46912i 1.63511 0.809017 − 0.587785i 1.00340 + 3.08814i −1.87765 − 1.21426i
221.2 −0.309017 + 0.951057i −1.25108 − 0.908964i −0.809017 − 0.587785i 1.08658 − 1.95431i 1.25108 − 0.908964i −2.95859 0.809017 − 0.587785i −0.188062 − 0.578795i 1.52289 + 1.63732i
221.3 −0.309017 + 0.951057i −0.933081 − 0.677923i −0.809017 − 0.587785i 2.09583 + 0.779409i 0.933081 − 0.677923i 2.31190 0.809017 − 0.587785i −0.515991 − 1.58806i −1.38891 + 1.75241i
221.4 −0.309017 + 0.951057i 0.143077 + 0.103952i −0.809017 − 0.587785i −2.23315 + 0.114269i −0.143077 + 0.103952i 3.36723 0.809017 − 0.587785i −0.917386 − 2.82342i 0.581403 − 2.15916i
221.5 −0.309017 + 0.951057i 1.07319 + 0.779721i −0.809017 − 0.587785i −1.29005 − 1.82641i −1.07319 + 0.779721i −2.02560 0.809017 − 0.587785i −0.383270 − 1.17958i 2.13567 − 0.662514i
221.6 −0.309017 + 0.951057i 2.22454 + 1.61622i −0.809017 − 0.587785i 2.13242 − 0.672907i −2.22454 + 1.61622i 0.977849 0.809017 − 0.587785i 1.40935 + 4.33752i −0.0189801 + 2.23599i
221.7 −0.309017 + 0.951057i 2.38345 + 1.73168i −0.809017 − 0.587785i −1.02606 + 1.98676i −2.38345 + 1.73168i −3.30790 0.809017 − 0.587785i 1.75508 + 5.40159i −1.57245 − 1.58978i
331.1 −0.309017 − 0.951057i −2.02207 + 1.46912i −0.809017 + 0.587785i −0.574600 − 2.16098i 2.02207 + 1.46912i 1.63511 0.809017 + 0.587785i 1.00340 − 3.08814i −1.87765 + 1.21426i
331.2 −0.309017 − 0.951057i −1.25108 + 0.908964i −0.809017 + 0.587785i 1.08658 + 1.95431i 1.25108 + 0.908964i −2.95859 0.809017 + 0.587785i −0.188062 + 0.578795i 1.52289 − 1.63732i
331.3 −0.309017 − 0.951057i −0.933081 + 0.677923i −0.809017 + 0.587785i 2.09583 − 0.779409i 0.933081 + 0.677923i 2.31190 0.809017 + 0.587785i −0.515991 + 1.58806i −1.38891 − 1.75241i
331.4 −0.309017 − 0.951057i 0.143077 − 0.103952i −0.809017 + 0.587785i −2.23315 − 0.114269i −0.143077 − 0.103952i 3.36723 0.809017 + 0.587785i −0.917386 + 2.82342i 0.581403 + 2.15916i
331.5 −0.309017 − 0.951057i 1.07319 − 0.779721i −0.809017 + 0.587785i −1.29005 + 1.82641i −1.07319 − 0.779721i −2.02560 0.809017 + 0.587785i −0.383270 + 1.17958i 2.13567 + 0.662514i
331.6 −0.309017 − 0.951057i 2.22454 − 1.61622i −0.809017 + 0.587785i 2.13242 + 0.672907i −2.22454 − 1.61622i 0.977849 0.809017 + 0.587785i 1.40935 − 4.33752i −0.0189801 − 2.23599i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.2.k.e ✓ 28
25.d even 5 1 inner 550.2.k.e ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.k.e ✓ 28 1.a even 1 1 trivial
550.2.k.e ✓ 28 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 2 T_{3}^{27} + 16 T_{3}^{26} - 35 T_{3}^{25} + 178 T_{3}^{24} - 206 T_{3}^{23} + 1302 T_{3}^{22} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display