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Properties

Label 525.2.n.c
Level $525$
Weight $2$
Character orbit 525.n
Analytic conductor $4.192$
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] = [525,2,Mod(106,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.n (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: \(4.19214610612\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} - 6 q^{3} - 9 q^{4} - q^{5} + q^{6} + 24 q^{7} + 9 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} - 6 q^{3} - 9 q^{4} - q^{5} + q^{6} + 24 q^{7} + 9 q^{8} - 6 q^{9} - 16 q^{10} + 2 q^{11} + q^{12} + 8 q^{13} + q^{14} - 6 q^{15} + 13 q^{16} + 12 q^{17} - 4 q^{18} + 19 q^{19} + 11 q^{20} - 6 q^{21} - 19 q^{22} - 6 q^{24} + 9 q^{25} - 14 q^{26} - 6 q^{27} - 9 q^{28} + 5 q^{29} - 6 q^{30} + 17 q^{31} - 26 q^{32} + 7 q^{33} - 7 q^{34} - q^{35} + q^{36} + 22 q^{37} - 16 q^{38} + 8 q^{39} + 3 q^{40} + 37 q^{41} + q^{42} + 8 q^{43} + 13 q^{44} + 4 q^{45} + 24 q^{46} - 24 q^{47} + 13 q^{48} + 24 q^{49} - 21 q^{50} - 8 q^{51} + 23 q^{52} - 24 q^{53} + q^{54} - 55 q^{55} + 9 q^{56} - 26 q^{57} + 8 q^{58} - 39 q^{60} + 24 q^{62} - 6 q^{63} - q^{64} - 34 q^{65} + 16 q^{66} + 34 q^{67} + 22 q^{68} + 10 q^{69} - 16 q^{70} - 24 q^{71} + 9 q^{72} + 46 q^{73} + 10 q^{74} + 24 q^{75} - 20 q^{76} + 2 q^{77} - 14 q^{78} + 10 q^{79} + 6 q^{80} - 6 q^{81} - 78 q^{82} + 42 q^{83} + q^{84} - 22 q^{85} - 96 q^{86} - 10 q^{87} - 39 q^{88} + 29 q^{89} + 14 q^{90} + 8 q^{91} + 42 q^{92} - 58 q^{93} + 54 q^{94} - 42 q^{95} + 9 q^{96} - 32 q^{97} + q^{98} - 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −0.758114 − 2.33323i −0.809017 + 0.587785i −3.25121 + 2.36214i 0.447435 − 2.19084i 1.98477 + 1.44202i 1.00000 4.00669 + 2.91103i 0.309017 − 0.951057i −5.45096 + 0.616938i
106.2 −0.612004 − 1.88356i −0.809017 + 0.587785i −1.55520 + 1.12992i −2.21075 + 0.335565i 1.60225 + 1.16410i 1.00000 −0.124444 − 0.0904141i 0.309017 − 0.951057i 1.98504 + 3.95870i
106.3 −0.262997 − 0.809422i −0.809017 + 0.587785i 1.03204 − 0.749819i 2.19857 + 0.407800i 0.688536 + 0.500250i 1.00000 −2.25541 − 1.63865i 0.309017 − 0.951057i −0.248135 − 1.88682i
106.4 0.139991 + 0.430847i −0.809017 + 0.587785i 1.45200 − 1.05494i −0.891879 + 2.05050i −0.366500 − 0.266278i 1.00000 1.39078 + 1.01046i 0.309017 − 0.951057i −1.00831 − 0.0972121i
106.5 0.558268 + 1.71817i −0.809017 + 0.587785i −1.02242 + 0.742831i 0.925434 − 2.03558i −1.46156 − 1.06189i 1.00000 1.07603 + 0.781785i 0.309017 − 0.951057i 4.01411 + 0.453657i
106.6 0.625840 + 1.92614i −0.809017 + 0.587785i −1.70029 + 1.23534i 0.958238 + 2.02034i −1.63847 − 1.19042i 1.00000 −0.166599 − 0.121041i 0.309017 − 0.951057i −3.29175 + 3.11011i
211.1 −1.66257 + 1.20793i 0.309017 − 0.951057i 0.687021 − 2.11443i 2.21818 − 0.282242i 0.635047 + 1.95447i 1.00000 0.141773 + 0.436331i −0.809017 − 0.587785i −3.34697 + 3.14866i
211.2 −1.14972 + 0.835319i 0.309017 − 0.951057i 0.00606019 − 0.0186513i −1.87514 + 1.21814i 0.439153 + 1.35158i 1.00000 −0.869694 − 2.67664i −0.809017 − 0.587785i 1.13834 − 2.96686i
211.3 −0.0144696 + 0.0105128i 0.309017 − 0.951057i −0.617935 + 1.90181i −1.54334 − 1.61805i 0.00552688 + 0.0170100i 1.00000 −0.0221058 − 0.0680345i −0.809017 − 0.587785i 0.0393417 + 0.00718776i
211.4 0.728341 − 0.529171i 0.309017 − 0.951057i −0.367575 + 1.13128i −2.01165 + 0.976343i −0.278202 − 0.856217i 1.00000 0.887323 + 2.73090i −0.809017 − 0.587785i −0.948519 + 1.77562i
211.5 1.16621 − 0.847303i 0.309017 − 0.951057i 0.0240954 − 0.0741580i 2.06863 + 0.848976i −0.445454 − 1.37097i 1.00000 0.856173 + 2.63503i −0.809017 − 0.587785i 3.13181 − 0.762672i
211.6 1.74122 − 1.26507i 0.309017 − 0.951057i 0.813418 − 2.50344i −0.783736 − 2.09422i −0.665089 − 2.04693i 1.00000 −0.420520 − 1.29423i −0.809017 − 0.587785i −4.01400 − 2.65502i
316.1 −1.66257 − 1.20793i 0.309017 + 0.951057i 0.687021 + 2.11443i 2.21818 + 0.282242i 0.635047 − 1.95447i 1.00000 0.141773 − 0.436331i −0.809017 + 0.587785i −3.34697 − 3.14866i
316.2 −1.14972 − 0.835319i 0.309017 + 0.951057i 0.00606019 + 0.0186513i −1.87514 − 1.21814i 0.439153 − 1.35158i 1.00000 −0.869694 + 2.67664i −0.809017 + 0.587785i 1.13834 + 2.96686i
316.3 −0.0144696 − 0.0105128i 0.309017 + 0.951057i −0.617935 − 1.90181i −1.54334 + 1.61805i 0.00552688 − 0.0170100i 1.00000 −0.0221058 + 0.0680345i −0.809017 + 0.587785i 0.0393417 − 0.00718776i
316.4 0.728341 + 0.529171i 0.309017 + 0.951057i −0.367575 − 1.13128i −2.01165 − 0.976343i −0.278202 + 0.856217i 1.00000 0.887323 − 2.73090i −0.809017 + 0.587785i −0.948519 − 1.77562i
316.5 1.16621 + 0.847303i 0.309017 + 0.951057i 0.0240954 + 0.0741580i 2.06863 − 0.848976i −0.445454 + 1.37097i 1.00000 0.856173 − 2.63503i −0.809017 + 0.587785i 3.13181 + 0.762672i
316.6 1.74122 + 1.26507i 0.309017 + 0.951057i 0.813418 + 2.50344i −0.783736 + 2.09422i −0.665089 + 2.04693i 1.00000 −0.420520 + 1.29423i −0.809017 + 0.587785i −4.01400 + 2.65502i
421.1 −0.758114 + 2.33323i −0.809017 − 0.587785i −3.25121 − 2.36214i 0.447435 + 2.19084i 1.98477 − 1.44202i 1.00000 4.00669 − 2.91103i 0.309017 + 0.951057i −5.45096 − 0.616938i
421.2 −0.612004 + 1.88356i −0.809017 − 0.587785i −1.55520 − 1.12992i −2.21075 − 0.335565i 1.60225 − 1.16410i 1.00000 −0.124444 + 0.0904141i 0.309017 + 0.951057i 1.98504 − 3.95870i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.6
Significant digits:
Format:

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 525.2.n.c ✓ 24
25.d even 5 1 inner 525.2.n.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.n.c ✓ 24 1.a even 1 1 trivial
525.2.n.c ✓ 24 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - T_{2}^{23} + 11 T_{2}^{22} - 15 T_{2}^{21} + 72 T_{2}^{20} - 81 T_{2}^{19} + 338 T_{2}^{18} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display