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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Newspace parameters

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

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 24q + q^{2} - 6q^{3} - 9q^{4} - q^{5} + q^{6} + 24q^{7} + 9q^{8} - 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + q^{2} - 6q^{3} - 9q^{4} - q^{5} + q^{6} + 24q^{7} + 9q^{8} - 6q^{9} - 16q^{10} + 2q^{11} + q^{12} + 8q^{13} + q^{14} - 6q^{15} + 13q^{16} + 12q^{17} - 4q^{18} + 19q^{19} + 11q^{20} - 6q^{21} - 19q^{22} - 6q^{24} + 9q^{25} - 14q^{26} - 6q^{27} - 9q^{28} + 5q^{29} - 6q^{30} + 17q^{31} - 26q^{32} + 7q^{33} - 7q^{34} - q^{35} + q^{36} + 22q^{37} - 16q^{38} + 8q^{39} + 3q^{40} + 37q^{41} + q^{42} + 8q^{43} + 13q^{44} + 4q^{45} + 24q^{46} - 24q^{47} + 13q^{48} + 24q^{49} - 21q^{50} - 8q^{51} + 23q^{52} - 24q^{53} + q^{54} - 55q^{55} + 9q^{56} - 26q^{57} + 8q^{58} - 39q^{60} + 24q^{62} - 6q^{63} - q^{64} - 34q^{65} + 16q^{66} + 34q^{67} + 22q^{68} + 10q^{69} - 16q^{70} - 24q^{71} + 9q^{72} + 46q^{73} + 10q^{74} + 24q^{75} - 20q^{76} + 2q^{77} - 14q^{78} + 10q^{79} + 6q^{80} - 6q^{81} - 78q^{82} + 42q^{83} + q^{84} - 22q^{85} - 96q^{86} - 10q^{87} - 39q^{88} + 29q^{89} + 14q^{90} + 8q^{91} + 42q^{92} - 58q^{93} + 54q^{94} - 42q^{95} + 9q^{96} - 32q^{97} + q^{98} - 18q^{99} + O(q^{100}) \)

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.

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 421.6
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 \(T_{2}^{24} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database