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Properties

Label 525.2.n.d
Level $525$
Weight $2$
Character orbit 525.n
Analytic conductor $4.192$
Analytic rank $0$
Dimension $32$
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: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{2} - 8 q^{3} - 15 q^{4} - 3 q^{5} + q^{6} - 32 q^{7} - 3 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{2} - 8 q^{3} - 15 q^{4} - 3 q^{5} + q^{6} - 32 q^{7} - 3 q^{8} - 8 q^{9} + 8 q^{10} + 2 q^{11} - 5 q^{12} + 12 q^{13} - q^{14} - 8 q^{15} - 17 q^{16} + 12 q^{17} - 4 q^{18} - 13 q^{19} - 27 q^{20} + 8 q^{21} - 21 q^{22} - 12 q^{23} - 18 q^{24} + 11 q^{25} - 2 q^{26} - 8 q^{27} + 15 q^{28} + 21 q^{29} - 12 q^{30} + 3 q^{31} - 50 q^{32} - 13 q^{33} - 41 q^{34} + 3 q^{35} - 5 q^{36} - 22 q^{37} + 44 q^{38} + 12 q^{39} - 39 q^{40} - 3 q^{41} - q^{42} + 24 q^{43} - 43 q^{44} + 2 q^{45} + 10 q^{46} + 8 q^{47} - 17 q^{48} + 32 q^{49} + 19 q^{50} - 8 q^{51} + 53 q^{52} + 18 q^{53} + q^{54} + 23 q^{55} + 3 q^{56} + 42 q^{57} - 32 q^{58} + 28 q^{59} + 73 q^{60} + 36 q^{61} + 10 q^{62} + 8 q^{63} + 9 q^{64} - 34 q^{65} + 4 q^{66} - 22 q^{67} - 78 q^{68} - 2 q^{69} - 8 q^{70} - 40 q^{71} - 3 q^{72} - 10 q^{73} - 34 q^{74} + 6 q^{75} + 132 q^{76} - 2 q^{77} + 28 q^{78} + 18 q^{79} + 148 q^{80} - 8 q^{81} + 102 q^{82} + 16 q^{83} + 5 q^{84} + 18 q^{85} + 16 q^{86} - 34 q^{87} + 13 q^{88} - 17 q^{89} - 2 q^{90} - 12 q^{91} - 106 q^{92} + 18 q^{93} - 20 q^{94} - 92 q^{95} - 15 q^{96} + 30 q^{97} + q^{98} + 22 q^{99}+O(q^{100}) \) 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 −0.864713 − 2.66131i −0.809017 + 0.587785i −4.71683 + 3.42698i −1.60184 + 1.56017i 2.26385 + 1.64478i −1.00000 8.67128 + 6.30005i 0.309017 − 0.951057i 5.53723 + 2.91389i
106.2 −0.590098 − 1.81614i −0.809017 + 0.587785i −1.33210 + 0.967825i 1.37624 + 1.76238i 1.54490 + 1.12243i −1.00000 −0.546024 − 0.396709i 0.309017 − 0.951057i 2.38860 − 3.53941i
106.3 −0.573156 − 1.76399i −0.809017 + 0.587785i −1.16513 + 0.846516i 1.88413 − 1.20418i 1.50054 + 1.09021i −1.00000 −0.840036 − 0.610322i 0.309017 − 0.951057i −3.20407 − 2.63341i
106.4 −0.198286 − 0.610262i −0.809017 + 0.587785i 1.28493 − 0.933558i −1.63030 + 1.53040i 0.519120 + 0.377162i −1.00000 −1.86274 − 1.35336i 0.309017 − 0.951057i 1.25721 + 0.691449i
106.5 0.143867 + 0.442776i −0.809017 + 0.587785i 1.44268 − 1.04817i 0.251880 − 2.22184i −0.376647 − 0.273650i −1.00000 1.42495 + 1.03529i 0.309017 − 0.951057i 1.02001 − 0.208121i
106.6 0.276260 + 0.850242i −0.809017 + 0.587785i 0.971442 − 0.705794i −1.96138 − 1.07378i −0.723259 − 0.525478i −1.00000 2.31498 + 1.68193i 0.309017 − 0.951057i 0.371124 − 1.96429i
106.7 0.662392 + 2.03863i −0.809017 + 0.587785i −2.09923 + 1.52518i 2.11075 − 0.738048i −1.73416 − 1.25994i −1.00000 −1.03146 − 0.749399i 0.309017 − 0.951057i 2.90276 + 3.81418i
106.8 0.834718 + 2.56900i −0.809017 + 0.587785i −4.28496 + 3.11320i −0.620482 + 2.14826i −2.18532 − 1.58773i −1.00000 −7.20390 − 5.23394i 0.309017 − 0.951057i −6.03679 + 0.199170i
211.1 −2.01265 + 1.46227i 0.309017 − 0.951057i 1.29447 − 3.98396i −1.44782 + 1.70406i 0.768762 + 2.36601i −1.00000 1.68281 + 5.17916i −0.809017 − 0.587785i 0.422145 − 5.54678i
211.2 −1.24444 + 0.904140i 0.309017 − 0.951057i 0.113133 − 0.348187i −1.50439 − 1.65433i 0.475335 + 1.46293i −1.00000 −0.776647 − 2.39027i −0.809017 − 0.587785i 3.36787 + 0.698540i
211.3 −1.04196 + 0.757031i 0.309017 − 0.951057i −0.105441 + 0.324514i 1.64809 − 1.51123i 0.397995 + 1.22490i −1.00000 −0.931791 − 2.86776i −0.809017 − 0.587785i −0.573204 + 2.82230i
211.4 0.0219060 − 0.0159156i 0.309017 − 0.951057i −0.617807 + 1.90142i 1.28481 + 1.83010i −0.00836734 − 0.0257520i −1.00000 0.0334632 + 0.102989i −0.809017 − 0.587785i 0.0572722 + 0.0196417i
211.5 0.464153 − 0.337227i 0.309017 − 0.951057i −0.516318 + 1.58906i 1.08776 − 1.95366i −0.177291 − 0.545645i −1.00000 0.650806 + 2.00297i −0.809017 − 0.587785i −0.153940 − 1.27362i
211.6 0.546644 − 0.397160i 0.309017 − 0.951057i −0.476951 + 1.46790i −2.15335 − 0.602582i −0.208799 − 0.642618i −1.00000 0.739869 + 2.27708i −0.809017 − 0.587785i −1.41643 + 0.525825i
211.7 1.88023 − 1.36607i 0.309017 − 0.951057i 1.05109 − 3.23494i 1.98079 − 1.03753i −0.718185 − 2.21034i −1.00000 −1.00647 − 3.09761i −0.809017 − 0.587785i 2.30701 − 4.65669i
211.8 2.19514 − 1.59486i 0.309017 − 0.951057i 1.65701 − 5.09975i −2.20491 + 0.371995i −0.838467 − 2.58054i −1.00000 −2.81909 − 8.67626i −0.809017 − 0.587785i −4.24679 + 4.33310i
316.1 −2.01265 − 1.46227i 0.309017 + 0.951057i 1.29447 + 3.98396i −1.44782 − 1.70406i 0.768762 − 2.36601i −1.00000 1.68281 − 5.17916i −0.809017 + 0.587785i 0.422145 + 5.54678i
316.2 −1.24444 − 0.904140i 0.309017 + 0.951057i 0.113133 + 0.348187i −1.50439 + 1.65433i 0.475335 − 1.46293i −1.00000 −0.776647 + 2.39027i −0.809017 + 0.587785i 3.36787 − 0.698540i
316.3 −1.04196 − 0.757031i 0.309017 + 0.951057i −0.105441 − 0.324514i 1.64809 + 1.51123i 0.397995 − 1.22490i −1.00000 −0.931791 + 2.86776i −0.809017 + 0.587785i −0.573204 − 2.82230i
316.4 0.0219060 + 0.0159156i 0.309017 + 0.951057i −0.617807 − 1.90142i 1.28481 − 1.83010i −0.00836734 + 0.0257520i −1.00000 0.0334632 − 0.102989i −0.809017 + 0.587785i 0.0572722 − 0.0196417i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.8
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.d ✓ 32
25.d even 5 1 inner 525.2.n.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.n.d ✓ 32 1.a even 1 1 trivial
525.2.n.d ✓ 32 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - T_{2}^{31} + 16 T_{2}^{30} - 16 T_{2}^{29} + 156 T_{2}^{28} - 101 T_{2}^{27} + 1186 T_{2}^{26} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display