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Properties

Label 550.2.k.d
Level $550$
Weight $2$
Character orbit 550.k
Analytic conductor $4.392$
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] = [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: \(24\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} - 2 q^{3} - 6 q^{4} + 6 q^{5} - 2 q^{6} - 6 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} - 2 q^{3} - 6 q^{4} + 6 q^{5} - 2 q^{6} - 6 q^{8} - 14 q^{9} + q^{10} + 6 q^{11} - 2 q^{12} - 9 q^{15} - 6 q^{16} + 13 q^{17} + 16 q^{18} - 11 q^{19} - 9 q^{20} + 33 q^{21} + 6 q^{22} - 17 q^{23} + 8 q^{24} - 16 q^{25} - 11 q^{27} + 10 q^{29} + q^{30} - 11 q^{31} + 24 q^{32} + 2 q^{33} - 12 q^{34} + 4 q^{35} - 14 q^{36} - 15 q^{37} - q^{38} + 3 q^{39} + q^{40} + 28 q^{41} + 33 q^{42} + 16 q^{43} + 6 q^{44} + 13 q^{45} + 3 q^{46} - 28 q^{47} - 2 q^{48} + 16 q^{49} - q^{50} + 26 q^{51} - 57 q^{53} + 4 q^{54} - q^{55} + 18 q^{57} + 10 q^{58} - 7 q^{59} + 46 q^{60} - q^{62} - 30 q^{63} - 6 q^{64} - 23 q^{65} + 2 q^{66} + 23 q^{67} - 2 q^{68} - 45 q^{69} - 6 q^{70} + 29 q^{71} + 6 q^{72} - 27 q^{73} + 40 q^{74} + 19 q^{75} + 24 q^{76} - 2 q^{78} - q^{79} + q^{80} - 26 q^{81} - 2 q^{82} - 47 q^{83} - 32 q^{84} + 12 q^{85} - 9 q^{86} - 14 q^{87} + 6 q^{88} - 4 q^{89} - 32 q^{90} + 27 q^{91} + 3 q^{92} + 148 q^{93} - 28 q^{94} - 6 q^{95} - 2 q^{96} - 31 q^{97} + 36 q^{98} - 16 q^{99}+O(q^{100}) \) 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 −0.731897 + 2.25255i 0.309017 − 0.951057i 1.29607 − 1.82214i −0.731897 − 2.25255i −2.12543 0.309017 + 0.951057i −2.11125 − 1.53391i 0.0224812 + 2.23595i
111.2 −0.809017 + 0.587785i −0.661910 + 2.03715i 0.309017 − 0.951057i 1.97046 + 1.05702i −0.661910 − 2.03715i 0.241386 0.309017 + 0.951057i −1.28480 − 0.933462i −2.21544 + 0.303061i
111.3 −0.809017 + 0.587785i −0.0620893 + 0.191091i 0.309017 − 0.951057i −1.64020 − 1.51978i −0.0620893 − 0.191091i −2.69650 0.309017 + 0.951057i 2.39439 + 1.73963i 2.22026 + 0.265440i
111.4 −0.809017 + 0.587785i 0.469168 − 1.44395i 0.309017 − 0.951057i 0.945392 − 2.02638i 0.469168 + 1.44395i 3.66578 0.309017 + 0.951057i 0.562180 + 0.408448i 0.426241 + 2.19507i
111.5 −0.809017 + 0.587785i 0.656675 − 2.02104i 0.309017 − 0.951057i 0.0395752 + 2.23572i 0.656675 + 2.02104i −0.639946 0.309017 + 0.951057i −1.22633 − 0.890978i −1.34614 − 1.78547i
111.6 −0.809017 + 0.587785i 0.948087 − 2.91791i 0.309017 − 0.951057i −2.22933 + 0.173454i 0.948087 + 2.91791i 1.55472 0.309017 + 0.951057i −5.18830 − 3.76952i 1.70161 − 1.45069i
221.1 0.309017 − 0.951057i −2.36415 − 1.71765i −0.809017 − 0.587785i −0.175584 − 2.22916i −2.36415 + 1.71765i −4.24497 −0.809017 + 0.587785i 1.71181 + 5.26841i −2.17432 − 0.521859i
221.2 0.309017 − 0.951057i −1.01112 − 0.734621i −0.809017 − 0.587785i −1.65653 − 1.50197i −1.01112 + 0.734621i 2.50630 −0.809017 + 0.587785i −0.444358 − 1.36759i −1.94036 + 1.11131i
221.3 0.309017 − 0.951057i −0.698057 − 0.507168i −0.809017 − 0.587785i 0.738087 + 2.11074i −0.698057 + 0.507168i −3.52352 −0.809017 + 0.587785i −0.696987 − 2.14511i 2.23552 − 0.0497071i
221.4 0.309017 − 0.951057i −0.150258 − 0.109169i −0.809017 − 0.587785i 1.98642 − 1.02671i −0.150258 + 0.109169i 4.69829 −0.809017 + 0.587785i −0.916391 − 2.82036i −0.362623 − 2.20647i
221.5 0.309017 − 0.951057i 0.611300 + 0.444136i −0.809017 − 0.587785i 2.11332 − 0.730668i 0.611300 − 0.444136i −1.28954 −0.809017 + 0.587785i −0.750620 − 2.31017i −0.0418545 − 2.23568i
221.6 0.309017 − 0.951057i 1.99425 + 1.44891i −0.809017 − 0.587785i −0.387683 + 2.20220i 1.99425 − 1.44891i 1.85343 −0.809017 + 0.587785i 0.950646 + 2.92579i 1.97462 + 1.04923i
331.1 0.309017 + 0.951057i −2.36415 + 1.71765i −0.809017 + 0.587785i −0.175584 + 2.22916i −2.36415 − 1.71765i −4.24497 −0.809017 − 0.587785i 1.71181 − 5.26841i −2.17432 + 0.521859i
331.2 0.309017 + 0.951057i −1.01112 + 0.734621i −0.809017 + 0.587785i −1.65653 + 1.50197i −1.01112 − 0.734621i 2.50630 −0.809017 − 0.587785i −0.444358 + 1.36759i −1.94036 − 1.11131i
331.3 0.309017 + 0.951057i −0.698057 + 0.507168i −0.809017 + 0.587785i 0.738087 − 2.11074i −0.698057 − 0.507168i −3.52352 −0.809017 − 0.587785i −0.696987 + 2.14511i 2.23552 + 0.0497071i
331.4 0.309017 + 0.951057i −0.150258 + 0.109169i −0.809017 + 0.587785i 1.98642 + 1.02671i −0.150258 − 0.109169i 4.69829 −0.809017 − 0.587785i −0.916391 + 2.82036i −0.362623 + 2.20647i
331.5 0.309017 + 0.951057i 0.611300 − 0.444136i −0.809017 + 0.587785i 2.11332 + 0.730668i 0.611300 + 0.444136i −1.28954 −0.809017 − 0.587785i −0.750620 + 2.31017i −0.0418545 + 2.23568i
331.6 0.309017 + 0.951057i 1.99425 − 1.44891i −0.809017 + 0.587785i −0.387683 − 2.20220i 1.99425 + 1.44891i 1.85343 −0.809017 − 0.587785i 0.950646 − 2.92579i 1.97462 − 1.04923i
441.1 −0.809017 − 0.587785i −0.731897 − 2.25255i 0.309017 + 0.951057i 1.29607 + 1.82214i −0.731897 + 2.25255i −2.12543 0.309017 − 0.951057i −2.11125 + 1.53391i 0.0224812 − 2.23595i
441.2 −0.809017 − 0.587785i −0.661910 − 2.03715i 0.309017 + 0.951057i 1.97046 − 1.05702i −0.661910 + 2.03715i 0.241386 0.309017 − 0.951057i −1.28480 + 0.933462i −2.21544 − 0.303061i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.6
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.d ✓ 24
25.d even 5 1 inner 550.2.k.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.k.d ✓ 24 1.a even 1 1 trivial
550.2.k.d ✓ 24 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 2 T_{3}^{23} + 18 T_{3}^{22} + 41 T_{3}^{21} + 196 T_{3}^{20} + 324 T_{3}^{19} + 1436 T_{3}^{18} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display