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Properties

Label 570.2.x.b
Level $570$
Weight $2$
Character orbit 570.x
Analytic conductor $4.551$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [570,2,Mod(103,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.x (of order \(12\), 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: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 20 q^{6} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 20 q^{6} - 8 q^{7} + 12 q^{10} + 8 q^{11} + 24 q^{13} + 20 q^{16} + 36 q^{21} - 8 q^{23} + 16 q^{25} + 4 q^{28} - 4 q^{35} - 20 q^{36} + 24 q^{38} + 12 q^{41} - 4 q^{42} - 12 q^{43} - 16 q^{47} + 24 q^{52} + 72 q^{53} + 8 q^{55} - 24 q^{57} - 16 q^{58} + 12 q^{60} + 24 q^{61} - 4 q^{63} + 4 q^{66} - 36 q^{67} - 24 q^{70} - 72 q^{71} - 32 q^{73} + 8 q^{76} - 88 q^{77} + 24 q^{78} + 20 q^{81} + 8 q^{82} - 88 q^{83} - 24 q^{85} - 48 q^{86} + 16 q^{87} - 24 q^{91} + 8 q^{92} - 96 q^{95} + 40 q^{96} - 60 q^{97} - 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i −2.23391 − 0.0982997i 0.500000 + 0.866025i −0.368637 + 0.368637i −0.707107 + 0.707107i −0.866025 + 0.500000i 2.18323 − 0.483227i
103.2 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i −1.02196 − 1.98887i 0.500000 + 0.866025i −0.613926 + 0.613926i −0.707107 + 0.707107i −0.866025 + 0.500000i 1.50189 + 1.65660i
103.3 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i 0.0149760 + 2.23602i 0.500000 + 0.866025i 1.36100 − 1.36100i −0.707107 + 0.707107i −0.866025 + 0.500000i −0.593190 − 2.15595i
103.4 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i 0.392079 + 2.20143i 0.500000 + 0.866025i −3.01170 + 3.01170i −0.707107 + 0.707107i −0.866025 + 0.500000i −0.948490 − 2.02494i
103.5 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.866025 − 0.500000i 1.17578 − 1.90199i 0.500000 + 0.866025i −2.04097 + 2.04097i −0.707107 + 0.707107i −0.866025 + 0.500000i −0.643442 + 2.14149i
103.6 0.965926 − 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i −1.99201 − 1.01583i 0.500000 + 0.866025i −0.219288 + 0.219288i 0.707107 − 0.707107i −0.866025 + 0.500000i −2.18705 − 0.465647i
103.7 0.965926 − 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i −0.981839 − 2.00898i 0.500000 + 0.866025i 3.62953 − 3.62953i 0.707107 − 0.707107i −0.866025 + 0.500000i −1.46835 − 1.68640i
103.8 0.965926 − 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i 0.649262 + 2.13973i 0.500000 + 0.866025i −2.55332 + 2.55332i 0.707107 − 0.707107i −0.866025 + 0.500000i 1.18094 + 1.89878i
103.9 0.965926 − 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i 1.89255 + 1.19091i 0.500000 + 0.866025i 2.32404 − 2.32404i 0.707107 − 0.707107i −0.866025 + 0.500000i 2.13629 + 0.660499i
103.10 0.965926 − 0.258819i 0.258819 + 0.965926i 0.866025 − 0.500000i 2.10507 − 0.754119i 0.500000 + 0.866025i −0.506739 + 0.506739i 0.707107 − 0.707107i −0.866025 + 0.500000i 1.83816 − 1.27325i
217.1 −0.258819 − 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i −2.10253 + 0.761162i 0.500000 + 0.866025i −3.01170 − 3.01170i 0.707107 + 0.707107i 0.866025 − 0.500000i 1.27940 + 1.83388i
217.2 −0.258819 − 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i −1.94394 + 1.10504i 0.500000 + 0.866025i 1.36100 + 1.36100i 0.707107 + 0.707107i 0.866025 − 0.500000i 1.57051 + 1.59169i
217.3 −0.258819 − 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i 1.05928 − 1.96924i 0.500000 + 0.866025i −2.04097 − 2.04097i 0.707107 + 0.707107i 0.866025 − 0.500000i −2.17631 − 0.513508i
217.4 −0.258819 − 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i 1.20208 + 1.88547i 0.500000 + 0.866025i −0.368637 − 0.368637i 0.707107 + 0.707107i 0.866025 − 0.500000i 1.51010 − 1.64912i
217.5 −0.258819 − 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i 2.23339 − 0.109394i 0.500000 + 0.866025i −0.613926 − 0.613926i 0.707107 + 0.707107i 0.866025 − 0.500000i −0.683710 − 2.12898i
217.6 0.258819 + 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i −2.17769 + 0.507589i 0.500000 + 0.866025i −2.55332 − 2.55332i −0.707107 − 0.707107i 0.866025 − 0.500000i −1.05392 − 1.97212i
217.7 0.258819 + 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i −1.97763 − 1.04354i 0.500000 + 0.866025i 2.32404 + 2.32404i −0.707107 − 0.707107i 0.866025 − 0.500000i 0.496137 − 2.18033i
217.8 0.258819 + 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i −0.399447 − 2.20010i 0.500000 + 0.866025i −0.506739 − 0.506739i −0.707107 − 0.707107i 0.866025 − 0.500000i 2.02175 − 0.955264i
217.9 0.258819 + 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i 1.87574 + 1.21721i 0.500000 + 0.866025i −0.219288 − 0.219288i −0.707107 − 0.707107i 0.866025 − 0.500000i −0.690261 + 2.12686i
217.10 0.258819 + 0.965926i 0.965926 − 0.258819i −0.866025 + 0.500000i 2.23075 − 0.154191i 0.500000 + 0.866025i 3.62953 + 3.62953i −0.707107 − 0.707107i 0.866025 − 0.500000i 0.726296 + 2.11483i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.x.b ✓ 40
5.c odd 4 1 inner 570.2.x.b ✓ 40
19.d odd 6 1 inner 570.2.x.b ✓ 40
95.l even 12 1 inner 570.2.x.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.x.b ✓ 40 1.a even 1 1 trivial
570.2.x.b ✓ 40 5.c odd 4 1 inner
570.2.x.b ✓ 40 19.d odd 6 1 inner
570.2.x.b ✓ 40 95.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + 4 T_{7}^{19} + 8 T_{7}^{18} + 24 T_{7}^{17} + 761 T_{7}^{16} + 3400 T_{7}^{15} + \cdots + 21025 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display