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Properties

Label 531.2.i.c
Level $531$
Weight $2$
Character orbit 531.i
Analytic conductor $4.240$
Analytic rank $0$
Dimension $140$
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] = [531,2,Mod(19,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(58))
 
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("531.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 531.i (of order \(29\), degree \(28\), 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: \(4.24005634733\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(5\) over \(\Q(\zeta_{29})\)
Twist minimal: no (minimal twist has level 177)
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 140 q + q^{2} - 9 q^{4} + 2 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 140 q + q^{2} - 9 q^{4} + 2 q^{5} - 2 q^{7} + 9 q^{8} + 88 q^{10} + 14 q^{11} - 12 q^{13} + q^{14} - 41 q^{16} + 16 q^{17} - 10 q^{19} + 32 q^{20} - 26 q^{22} + 22 q^{23} + 27 q^{25} + 56 q^{26} - 50 q^{28} + 24 q^{29} - 24 q^{31} - 106 q^{32} - 54 q^{34} + 70 q^{35} - 28 q^{37} + 80 q^{38} - 50 q^{40} + 40 q^{41} + 4 q^{43} + 104 q^{44} - 28 q^{46} - 31 q^{47} - q^{49} - 39 q^{50} + 62 q^{52} - 4 q^{53} + 5 q^{55} - 96 q^{56} + 128 q^{58} + q^{59} - 16 q^{61} - 223 q^{62} + 97 q^{64} - 121 q^{65} - 12 q^{67} - 10 q^{68} - 2 q^{70} + 22 q^{71} + 179 q^{73} + 38 q^{74} + 112 q^{76} + 62 q^{77} - 84 q^{79} - 204 q^{80} - 152 q^{82} + 88 q^{83} - 118 q^{85} + 118 q^{86} + 18 q^{88} + 86 q^{89} + 78 q^{91} + 174 q^{92} - 164 q^{94} - 218 q^{95} - 84 q^{97} - 129 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.31462 + 2.47964i 0 −3.29802 − 4.86421i −1.27307 + 0.280224i 0 1.14406 − 0.869689i 10.8169 − 1.17641i 0 0.978753 − 3.52515i
19.2 −0.655840 + 1.23705i 0 0.0222185 + 0.0327698i −2.32817 + 0.512469i 0 −1.36308 + 1.03619i −2.83899 + 0.308758i 0 0.892959 − 3.21615i
19.3 −0.0292574 + 0.0551852i 0 1.12018 + 1.65215i 2.41343 − 0.531236i 0 −2.91820 + 2.21836i −0.248138 + 0.0269866i 0 −0.0412942 + 0.148728i
19.4 0.454038 − 0.856407i 0 0.595092 + 0.877695i −3.60206 + 0.792874i 0 1.56865 − 1.19245i 2.94914 − 0.320738i 0 −0.956451 + 3.44483i
19.5 1.07727 − 2.03195i 0 −1.84595 − 2.72257i 3.23491 − 0.712057i 0 1.31099 − 0.996592i −2.94796 + 0.320610i 0 2.03802 − 7.34027i
28.1 −1.31462 − 2.47964i 0 −3.29802 + 4.86421i −1.27307 − 0.280224i 0 1.14406 + 0.869689i 10.8169 + 1.17641i 0 0.978753 + 3.52515i
28.2 −0.655840 − 1.23705i 0 0.0222185 − 0.0327698i −2.32817 − 0.512469i 0 −1.36308 − 1.03619i −2.83899 − 0.308758i 0 0.892959 + 3.21615i
28.3 −0.0292574 − 0.0551852i 0 1.12018 − 1.65215i 2.41343 + 0.531236i 0 −2.91820 − 2.21836i −0.248138 − 0.0269866i 0 −0.0412942 − 0.148728i
28.4 0.454038 + 0.856407i 0 0.595092 − 0.877695i −3.60206 − 0.792874i 0 1.56865 + 1.19245i 2.94914 + 0.320738i 0 −0.956451 − 3.44483i
28.5 1.07727 + 2.03195i 0 −1.84595 + 2.72257i 3.23491 + 0.712057i 0 1.31099 + 0.996592i −2.94796 − 0.320610i 0 2.03802 + 7.34027i
46.1 −1.65338 − 1.94651i 0 −0.731672 + 4.46300i −0.865278 + 1.63209i 0 −2.96873 + 0.322869i 5.52030 − 3.32145i 0 4.60751 − 1.01419i
46.2 −1.15330 − 1.35777i 0 −0.189873 + 1.15817i 1.46043 − 2.75467i 0 1.72018 − 0.187081i −1.26142 + 0.758972i 0 −5.42452 + 1.19403i
46.3 −0.398193 − 0.468789i 0 0.262358 − 1.60032i −0.780962 + 1.47305i 0 −2.98728 + 0.324887i −1.90875 + 1.14846i 0 1.00152 − 0.220452i
46.4 0.814427 + 0.958818i 0 0.0675241 − 0.411879i −0.588484 + 1.11000i 0 2.16620 − 0.235588i 2.60580 − 1.56786i 0 −1.54356 + 0.339764i
46.5 1.74306 + 2.05209i 0 −0.849247 + 5.18018i 1.70562 − 3.21713i 0 3.51311 − 0.382074i −7.49637 + 4.51042i 0 9.57483 − 2.10758i
64.1 −2.28036 − 0.768343i 0 3.01751 + 2.29385i −0.739682 + 1.85646i 0 −0.807669 + 0.373667i −2.41776 − 3.56592i 0 3.11314 − 3.66507i
64.2 −0.827908 − 0.278955i 0 −0.984570 − 0.748451i −0.302390 + 0.758941i 0 3.54386 − 1.63956i 1.58690 + 2.34050i 0 0.462061 − 0.543981i
64.3 −0.288668 − 0.0972636i 0 −1.51832 − 1.15419i 1.30154 − 3.26662i 0 −3.34578 + 1.54792i 0.667919 + 0.985107i 0 −0.693437 + 0.816377i
64.4 1.73089 + 0.583204i 0 1.06366 + 0.808574i 0.835093 − 2.09593i 0 2.04844 − 0.947709i −0.680501 − 1.00366i 0 2.66781 − 3.14078i
64.5 2.61370 + 0.880659i 0 4.46370 + 3.39321i −0.422705 + 1.06091i 0 −3.24336 + 1.50054i 5.58292 + 8.23419i 0 −2.03912 + 2.40064i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.2.i.c 140
3.b odd 2 1 177.2.e.a ✓ 140
59.c even 29 1 inner 531.2.i.c 140
177.h odd 58 1 177.2.e.a ✓ 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.e.a ✓ 140 3.b odd 2 1
177.2.e.a ✓ 140 177.h odd 58 1
531.2.i.c 140 1.a even 1 1 trivial
531.2.i.c 140 59.c even 29 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{140} - T_{2}^{139} + 10 T_{2}^{138} - 14 T_{2}^{137} + 88 T_{2}^{136} - 121 T_{2}^{135} + \cdots + 23694752761 \) acting on \(S_{2}^{\mathrm{new}}(531, [\chi])\). Copy content Toggle raw display