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Properties

Label 561.2.m.f
Level $561$
Weight $2$
Character orbit 561.m
Analytic conductor $4.480$
Analytic rank $0$
Dimension $40$
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] = [561,2,Mod(103,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.m (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: \(4.47960755339\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 3 q^{2} - 10 q^{3} - 17 q^{4} - 6 q^{5} + 3 q^{6} + 3 q^{7} + 8 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 3 q^{2} - 10 q^{3} - 17 q^{4} - 6 q^{5} + 3 q^{6} + 3 q^{7} + 8 q^{8} - 10 q^{9} - 12 q^{10} - 8 q^{11} + 58 q^{12} + 4 q^{13} - 8 q^{14} - 11 q^{15} + 3 q^{16} + 10 q^{17} - 2 q^{18} + 8 q^{19} - 21 q^{20} - 2 q^{21} - 9 q^{22} + 72 q^{23} - 2 q^{24} + 8 q^{25} - 46 q^{26} - 10 q^{27} - 21 q^{28} - 2 q^{29} - 17 q^{30} + 9 q^{31} + 14 q^{32} - 3 q^{33} + 2 q^{34} + 9 q^{35} - 17 q^{36} - 6 q^{37} - 12 q^{38} + 4 q^{39} - 40 q^{40} - 3 q^{41} - 13 q^{42} + 12 q^{43} + 5 q^{44} + 34 q^{45} - 33 q^{46} - 11 q^{47} - 42 q^{48} - 15 q^{49} - 6 q^{50} + 10 q^{51} + 38 q^{52} - 65 q^{53} - 2 q^{54} - 27 q^{55} + 102 q^{56} - 2 q^{57} + 55 q^{58} - 6 q^{59} - 21 q^{60} - 34 q^{61} + 3 q^{63} + 30 q^{64} + 16 q^{65} + 21 q^{66} - 30 q^{67} + 12 q^{68} - 18 q^{69} - 2 q^{70} + 30 q^{71} - 2 q^{72} - 51 q^{73} - 29 q^{74} - 47 q^{75} - 22 q^{76} + 27 q^{77} + 34 q^{78} - 26 q^{79} + 39 q^{80} - 10 q^{81} + 30 q^{82} - 43 q^{83} + 14 q^{84} + 11 q^{85} - 81 q^{86} + 8 q^{87} - 53 q^{88} + 32 q^{89} + 23 q^{90} - 24 q^{91} - 103 q^{92} + 9 q^{93} + 28 q^{94} - 87 q^{95} - 31 q^{96} + 30 q^{97} + 138 q^{98} - 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −2.04014 − 1.48225i 0.309017 − 0.951057i 1.34709 + 4.14590i −2.62856 + 1.90976i −2.04014 + 1.48225i 0.354228 + 1.09020i 1.83849 − 5.65830i −0.809017 − 0.587785i 8.19338
103.2 −1.87147 − 1.35970i 0.309017 − 0.951057i 1.03558 + 3.18718i 2.83935 − 2.06291i −1.87147 + 1.35970i 1.29700 + 3.99177i 0.965889 − 2.97270i −0.809017 − 0.587785i −8.11871
103.3 −1.22998 − 0.893634i 0.309017 − 0.951057i 0.0962387 + 0.296192i −1.98664 + 1.44338i −1.22998 + 0.893634i −0.558929 − 1.72021i −0.793306 + 2.44155i −0.809017 − 0.587785i 3.73338
103.4 −0.868785 − 0.631209i 0.309017 − 0.951057i −0.261672 − 0.805344i 2.12378 − 1.54302i −0.868785 + 0.631209i −0.970684 − 2.98746i −0.944696 + 2.90748i −0.809017 − 0.587785i −2.81907
103.5 0.176223 + 0.128034i 0.309017 − 0.951057i −0.603372 − 1.85699i −0.428534 + 0.311348i 0.176223 − 0.128034i 0.296964 + 0.913962i 0.266051 − 0.818822i −0.809017 − 0.587785i −0.115381
103.6 0.481514 + 0.349840i 0.309017 − 0.951057i −0.508567 − 1.56521i −2.08130 + 1.51216i 0.481514 − 0.349840i 1.46469 + 4.50784i 0.670535 − 2.06369i −0.809017 − 0.587785i −1.53119
103.7 1.25381 + 0.910946i 0.309017 − 0.951057i 0.124181 + 0.382191i 2.18281 − 1.58591i 1.25381 − 0.910946i −0.298934 − 0.920023i 0.765370 − 2.35557i −0.809017 − 0.587785i 4.18150
103.8 1.33783 + 0.971988i 0.309017 − 0.951057i 0.226985 + 0.698588i −3.31674 + 2.40975i 1.33783 − 0.971988i −1.56084 − 4.80376i 0.646656 − 1.99020i −0.809017 − 0.587785i −6.77946
103.9 1.96697 + 1.42909i 0.309017 − 0.951057i 1.20864 + 3.71982i 0.429210 − 0.311839i 1.96697 − 1.42909i −0.498176 − 1.53323i −1.43595 + 4.41940i −0.809017 − 0.587785i 1.28989
103.10 2.10306 + 1.52796i 0.309017 − 0.951057i 1.47016 + 4.52467i −3.10552 + 2.25629i 2.10306 − 1.52796i 0.665659 + 2.04869i −2.21511 + 6.81740i −0.809017 − 0.587785i −9.97861
256.1 −2.04014 + 1.48225i 0.309017 + 0.951057i 1.34709 − 4.14590i −2.62856 − 1.90976i −2.04014 − 1.48225i 0.354228 − 1.09020i 1.83849 + 5.65830i −0.809017 + 0.587785i 8.19338
256.2 −1.87147 + 1.35970i 0.309017 + 0.951057i 1.03558 − 3.18718i 2.83935 + 2.06291i −1.87147 − 1.35970i 1.29700 − 3.99177i 0.965889 + 2.97270i −0.809017 + 0.587785i −8.11871
256.3 −1.22998 + 0.893634i 0.309017 + 0.951057i 0.0962387 − 0.296192i −1.98664 − 1.44338i −1.22998 − 0.893634i −0.558929 + 1.72021i −0.793306 − 2.44155i −0.809017 + 0.587785i 3.73338
256.4 −0.868785 + 0.631209i 0.309017 + 0.951057i −0.261672 + 0.805344i 2.12378 + 1.54302i −0.868785 − 0.631209i −0.970684 + 2.98746i −0.944696 − 2.90748i −0.809017 + 0.587785i −2.81907
256.5 0.176223 − 0.128034i 0.309017 + 0.951057i −0.603372 + 1.85699i −0.428534 − 0.311348i 0.176223 + 0.128034i 0.296964 − 0.913962i 0.266051 + 0.818822i −0.809017 + 0.587785i −0.115381
256.6 0.481514 − 0.349840i 0.309017 + 0.951057i −0.508567 + 1.56521i −2.08130 − 1.51216i 0.481514 + 0.349840i 1.46469 − 4.50784i 0.670535 + 2.06369i −0.809017 + 0.587785i −1.53119
256.7 1.25381 − 0.910946i 0.309017 + 0.951057i 0.124181 − 0.382191i 2.18281 + 1.58591i 1.25381 + 0.910946i −0.298934 + 0.920023i 0.765370 + 2.35557i −0.809017 + 0.587785i 4.18150
256.8 1.33783 − 0.971988i 0.309017 + 0.951057i 0.226985 − 0.698588i −3.31674 − 2.40975i 1.33783 + 0.971988i −1.56084 + 4.80376i 0.646656 + 1.99020i −0.809017 + 0.587785i −6.77946
256.9 1.96697 − 1.42909i 0.309017 + 0.951057i 1.20864 − 3.71982i 0.429210 + 0.311839i 1.96697 + 1.42909i −0.498176 + 1.53323i −1.43595 − 4.41940i −0.809017 + 0.587785i 1.28989
256.10 2.10306 − 1.52796i 0.309017 + 0.951057i 1.47016 − 4.52467i −3.10552 − 2.25629i 2.10306 + 1.52796i 0.665659 − 2.04869i −2.21511 − 6.81740i −0.809017 + 0.587785i −9.97861
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.m.f ✓ 40
11.c even 5 1 inner 561.2.m.f ✓ 40
11.c even 5 1 6171.2.a.bq 20
11.d odd 10 1 6171.2.a.br 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.m.f ✓ 40 1.a even 1 1 trivial
561.2.m.f ✓ 40 11.c even 5 1 inner
6171.2.a.bq 20 11.c even 5 1
6171.2.a.br 20 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 3 T_{2}^{39} + 23 T_{2}^{38} - 60 T_{2}^{37} + 295 T_{2}^{36} - 686 T_{2}^{35} + \cdots + 160000 \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\). Copy content Toggle raw display