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Properties

Label 725.2.l.f
Level $725$
Weight $2$
Character orbit 725.l
Analytic conductor $5.789$
Analytic rank $0$
Dimension $54$
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] = [725,2,Mod(226,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.l (of order \(7\), degree \(6\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q - q^{2} - 7 q^{3} - 15 q^{4} + 2 q^{6} + 3 q^{7} - 9 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - q^{2} - 7 q^{3} - 15 q^{4} + 2 q^{6} + 3 q^{7} - 9 q^{8} - 14 q^{9} - 16 q^{11} - 40 q^{12} - 7 q^{13} + 2 q^{14} - 7 q^{16} + 12 q^{17} + 11 q^{18} + q^{19} + 35 q^{21} - 20 q^{22} + 10 q^{23} + 41 q^{24} + 15 q^{26} - q^{27} + 140 q^{28} + 15 q^{29} - 5 q^{31} + 36 q^{32} - 3 q^{33} + 9 q^{34} - q^{36} - 4 q^{37} + 11 q^{38} - 26 q^{39} - 37 q^{42} + 6 q^{43} + 82 q^{44} - 30 q^{46} + 28 q^{47} - 67 q^{48} + 4 q^{49} - 26 q^{51} + 71 q^{52} - 46 q^{53} - 22 q^{54} - 4 q^{56} + 110 q^{57} - q^{58} + 4 q^{59} - 33 q^{61} + 71 q^{62} + 39 q^{63} - 139 q^{64} - q^{66} + 66 q^{67} + 16 q^{68} + 64 q^{69} - 47 q^{71} - 22 q^{72} - 24 q^{73} - 27 q^{74} - 52 q^{76} - 71 q^{77} - 22 q^{78} - 36 q^{79} + 2 q^{81} + 7 q^{82} - 58 q^{83} - 140 q^{84} + 26 q^{86} - 58 q^{87} + 68 q^{88} - 68 q^{89} + 98 q^{91} - 33 q^{92} - 104 q^{93} + 20 q^{94} - 24 q^{96} - 15 q^{97} - 58 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1 −1.48325 + 1.85993i 0.528228 − 2.31432i −0.814285 − 3.56762i 0 3.52098 + 4.41517i −0.209649 + 0.918532i 3.55660 + 1.71277i −2.37414 − 1.14333i 0
226.2 −1.27375 + 1.59723i −0.660508 + 2.89387i −0.483669 − 2.11909i 0 −3.78087 − 4.74106i −0.0536618 + 0.235108i 0.319511 + 0.153868i −5.23533 − 2.52120i 0
226.3 −1.19379 + 1.49696i −0.0519596 + 0.227650i −0.370726 − 1.62426i 0 −0.278755 − 0.349547i −0.387748 + 1.69883i −0.576129 − 0.277449i 2.65378 + 1.27799i 0
226.4 −0.409717 + 0.513768i −0.257722 + 1.12915i 0.348952 + 1.52886i 0 −0.474530 − 0.595042i 0.129832 − 0.568830i −2.11257 − 1.01736i 1.49434 + 0.719638i 0
226.5 −0.0729620 + 0.0914914i 0.597501 − 2.61782i 0.441995 + 1.93651i 0 0.195913 + 0.245668i −0.940895 + 4.12233i −0.420289 − 0.202400i −3.79308 − 1.82665i 0
226.6 0.857966 − 1.07585i −0.402829 + 1.76491i 0.0236832 + 0.103763i 0 1.55317 + 1.94762i 0.780745 − 3.42067i 2.61155 + 1.25765i −0.249728 − 0.120263i 0
226.7 1.03499 − 1.29784i 0.188926 − 0.827738i −0.168134 − 0.736643i 0 −0.878733 − 1.10190i −0.319460 + 1.39965i 1.86115 + 0.896282i 2.05345 + 0.988889i 0
226.8 1.51335 − 1.89768i −0.673020 + 2.94870i −0.865928 − 3.79388i 0 4.57718 + 5.73960i −1.01647 + 4.45346i −4.13632 − 1.99195i −5.53894 − 2.66741i 0
226.9 1.65064 − 2.06984i 0.354874 − 1.55480i −1.11458 − 4.88328i 0 −2.63242 − 3.30096i 0.146839 − 0.643345i −7.17688 − 3.45620i 0.411430 + 0.198134i 0
326.1 −0.514614 − 2.25467i −1.90081 − 0.915381i −3.01678 + 1.45281i 0 −1.08570 + 4.75677i −4.15499 − 2.00094i 1.94425 + 2.43801i 0.904678 + 1.13443i 0
326.2 −0.511030 − 2.23897i 1.93206 + 0.930433i −2.94988 + 1.42059i 0 1.09587 − 4.80131i 0.654869 + 0.315368i 1.82438 + 2.28770i 0.996697 + 1.24982i 0
326.3 −0.464177 − 2.03369i −0.787980 − 0.379471i −2.11850 + 1.02022i 0 −0.405965 + 1.77865i 3.63896 + 1.75243i 0.456978 + 0.573033i −1.39356 − 1.74746i 0
326.4 −0.122672 − 0.537459i −2.84935 − 1.37218i 1.52812 − 0.735906i 0 −0.387954 + 1.69974i 1.26743 + 0.610362i −1.27041 − 1.59305i 4.36548 + 5.47414i 0
326.5 −0.0631478 − 0.276669i 1.95130 + 0.939698i 1.72938 − 0.832825i 0 0.136764 − 0.599204i 2.38462 + 1.14837i −0.693495 − 0.869616i 1.05408 + 1.32178i 0
326.6 0.180868 + 0.792434i −1.39126 − 0.669997i 1.20670 − 0.581116i 0 0.279294 − 1.22367i −2.26885 − 1.09262i 1.69231 + 2.12209i −0.383752 − 0.481210i 0
326.7 0.249106 + 1.09140i −0.256346 − 0.123450i 0.672831 − 0.324018i 0 0.0708762 − 0.310529i 1.34658 + 0.648478i 1.91720 + 2.40409i −1.82000 − 2.28220i 0
326.8 0.430261 + 1.88510i 2.31856 + 1.11656i −1.56652 + 0.754398i 0 −1.10724 + 4.85112i 0.319579 + 0.153901i 0.315001 + 0.394999i 2.25855 + 2.83214i 0
326.9 0.592885 + 2.59760i −0.238700 − 0.114952i −4.59406 + 2.21238i 0 0.157077 − 0.688198i −2.52064 − 1.21387i −5.14818 − 6.45561i −1.82671 − 2.29062i 0
401.1 −1.48325 − 1.85993i 0.528228 + 2.31432i −0.814285 + 3.56762i 0 3.52098 − 4.41517i −0.209649 − 0.918532i 3.55660 − 1.71277i −2.37414 + 1.14333i 0
401.2 −1.27375 − 1.59723i −0.660508 − 2.89387i −0.483669 + 2.11909i 0 −3.78087 + 4.74106i −0.0536618 − 0.235108i 0.319511 − 0.153868i −5.23533 + 2.52120i 0
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.l.f ✓ 54
5.b even 2 1 725.2.l.g yes 54
5.c odd 4 2 725.2.r.e 108
29.d even 7 1 inner 725.2.l.f ✓ 54
145.n even 14 1 725.2.l.g yes 54
145.p odd 28 2 725.2.r.e 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.l.f ✓ 54 1.a even 1 1 trivial
725.2.l.f ✓ 54 29.d even 7 1 inner
725.2.l.g yes 54 5.b even 2 1
725.2.l.g yes 54 145.n even 14 1
725.2.r.e 108 5.c odd 4 2
725.2.r.e 108 145.p odd 28 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{54} + T_{2}^{53} + 17 T_{2}^{52} + 21 T_{2}^{51} + 191 T_{2}^{50} + 266 T_{2}^{49} + 1776 T_{2}^{48} + \cdots + 41209 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display