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Properties

Label 726.2.m.d
Level $726$
Weight $2$
Character orbit 726.m
Analytic conductor $5.797$
Analytic rank $0$
Dimension $240$
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] = [726,2,Mod(25,726)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(726, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([0, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("726.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.m (of order \(55\), degree \(40\), 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: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(6\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{2} + 60 q^{3} + 6 q^{4} - 8 q^{5} - 6 q^{6} + 5 q^{7} + 6 q^{8} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{2} + 60 q^{3} + 6 q^{4} - 8 q^{5} - 6 q^{6} + 5 q^{7} + 6 q^{8} - 60 q^{9} + 13 q^{10} + 15 q^{11} + 24 q^{12} + 27 q^{13} + 16 q^{14} - 7 q^{15} + 6 q^{16} + 6 q^{18} - 6 q^{19} - 8 q^{20} + 10 q^{21} + 26 q^{23} - 6 q^{24} + 2 q^{25} - 5 q^{26} + 60 q^{27} + q^{29} + 37 q^{30} + 57 q^{31} - 24 q^{32} - 20 q^{33} - 15 q^{34} + 87 q^{35} + 6 q^{36} + 37 q^{37} - 18 q^{38} + 28 q^{39} + 7 q^{40} - 27 q^{41} - 11 q^{42} - 84 q^{43} - q^{44} + 2 q^{45} - 62 q^{46} + 10 q^{47} - 6 q^{48} + 13 q^{49} + 17 q^{50} - 35 q^{51} - 5 q^{52} + 73 q^{53} + 24 q^{54} + 27 q^{55} + q^{56} - 4 q^{57} - 21 q^{58} + 156 q^{59} + 8 q^{60} + 24 q^{61} - 4 q^{62} + 5 q^{63} + 6 q^{64} + 100 q^{65} + 45 q^{66} - 8 q^{67} + 11 q^{68} + 39 q^{69} - 6 q^{70} + 40 q^{71} + 6 q^{72} + 36 q^{73} - 18 q^{74} - 17 q^{75} + 4 q^{76} - 193 q^{77} + 50 q^{79} + 7 q^{80} - 60 q^{81} + 16 q^{82} + 21 q^{83} - 5 q^{84} - 45 q^{85} + 4 q^{86} - 6 q^{87} - 2 q^{88} - 6 q^{89} + 3 q^{90} - 125 q^{91} + 38 q^{92} + 163 q^{93} + 188 q^{94} + 45 q^{95} - 6 q^{96} + 41 q^{97} - 78 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.466667 − 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i −0.580649 + 2.86562i −0.696938 + 0.717132i −0.511474 − 1.94585i 0.993482 + 0.113991i −0.809017 + 0.587785i 2.80542 − 0.823745i
25.2 −0.466667 − 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i −0.255452 + 1.26071i −0.696938 + 0.717132i 0.171972 + 0.654251i 0.993482 + 0.113991i −0.809017 + 0.587785i 1.23422 − 0.362400i
25.3 −0.466667 − 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i −0.115459 + 0.569811i −0.696938 + 0.717132i −0.184974 − 0.703714i 0.993482 + 0.113991i −0.809017 + 0.587785i 0.557840 − 0.163797i
25.4 −0.466667 − 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i 0.236866 − 1.16898i −0.696938 + 0.717132i 0.714620 + 2.71870i 0.993482 + 0.113991i −0.809017 + 0.587785i −1.14442 + 0.336032i
25.5 −0.466667 − 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i 0.652530 − 3.22036i −0.696938 + 0.717132i 0.562313 + 2.13926i 0.993482 + 0.113991i −0.809017 + 0.587785i −3.15271 + 0.925719i
25.6 −0.466667 − 0.884433i −0.309017 − 0.951057i −0.564443 + 0.825472i 0.707198 − 3.49016i −0.696938 + 0.717132i −1.14414 − 4.35276i 0.993482 + 0.113991i −0.809017 + 0.587785i −3.41684 + 1.00327i
31.1 0.774142 − 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i −3.97931 + 1.95650i 0.254218 − 0.967147i 0.0278107 + 0.323794i −0.466667 − 0.884433i 0.309017 − 0.951057i −1.84207 + 4.03356i
31.2 0.774142 − 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i −1.76552 + 0.868045i 0.254218 − 0.967147i −0.237615 − 2.76650i −0.466667 − 0.884433i 0.309017 − 0.951057i −0.817275 + 1.78958i
31.3 0.774142 − 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i −1.68449 + 0.828208i 0.254218 − 0.967147i 0.361318 + 4.20675i −0.466667 − 0.884433i 0.309017 − 0.951057i −0.779768 + 1.70745i
31.4 0.774142 − 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i 0.569721 − 0.280113i 0.254218 − 0.967147i −0.105754 − 1.23128i −0.466667 − 0.884433i 0.309017 − 0.951057i 0.263730 − 0.577488i
31.5 0.774142 − 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i 2.61323 − 1.28484i 0.254218 − 0.967147i −0.193727 − 2.25552i −0.466667 − 0.884433i 0.309017 − 0.951057i 1.20969 − 2.64885i
31.6 0.774142 − 0.633012i 0.809017 − 0.587785i 0.198590 − 0.980083i 2.82563 − 1.38927i 0.254218 − 0.967147i 0.0896264 + 1.04350i −0.466667 − 0.884433i 0.309017 − 0.951057i 1.30801 − 2.86415i
37.1 −0.362808 − 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i −2.23483 − 2.29958i 0.998369 − 0.0570888i −2.20363 + 3.65430i 0.897398 + 0.441221i −0.809017 − 0.587785i −1.33209 + 2.91686i
37.2 −0.362808 − 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i −0.588493 − 0.605545i 0.998369 − 0.0570888i −0.0643936 + 0.106785i 0.897398 + 0.441221i −0.809017 − 0.587785i −0.350776 + 0.768092i
37.3 −0.362808 − 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i 0.166896 + 0.171731i 0.998369 − 0.0570888i 0.937724 − 1.55504i 0.897398 + 0.441221i −0.809017 − 0.587785i 0.0994794 − 0.217829i
37.4 −0.362808 − 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i 0.366961 + 0.377594i 0.998369 − 0.0570888i 0.226828 − 0.376152i 0.897398 + 0.441221i −0.809017 − 0.587785i 0.218730 − 0.478952i
37.5 −0.362808 − 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i 2.59224 + 2.66735i 0.998369 − 0.0570888i −1.94250 + 3.22127i 0.897398 + 0.441221i −0.809017 − 0.587785i 1.54512 − 3.38334i
37.6 −0.362808 − 0.931864i −0.309017 + 0.951057i −0.736741 + 0.676175i 2.88813 + 2.97182i 0.998369 − 0.0570888i 2.62599 − 4.35472i 0.897398 + 0.441221i −0.809017 − 0.587785i 1.72149 − 3.76955i
49.1 −0.921124 − 0.389270i 0.809017 + 0.587785i 0.696938 + 0.717132i −0.964578 − 3.66963i −0.516397 − 0.856349i 3.10598 − 1.10821i −0.362808 − 0.931864i 0.309017 + 0.951057i −0.539984 + 3.75567i
49.2 −0.921124 − 0.389270i 0.809017 + 0.587785i 0.696938 + 0.717132i −0.561602 − 2.13656i −0.516397 − 0.856349i −1.33503 + 0.476337i −0.362808 − 0.931864i 0.309017 + 0.951057i −0.314392 + 2.18665i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.m.d ✓ 240
121.g even 55 1 inner 726.2.m.d ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
726.2.m.d ✓ 240 1.a even 1 1 trivial
726.2.m.d ✓ 240 121.g even 55 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{240} + 8 T_{5}^{239} + 16 T_{5}^{238} + 151 T_{5}^{237} + 1258 T_{5}^{236} + \cdots + 22\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\). Copy content Toggle raw display