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Properties

Label 570.2.k.a
Level $570$
Weight $2$
Character orbit 570.k
Analytic conductor $4.551$
Analytic rank $0$
Dimension $36$
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(77,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(570, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("570.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.k (of order \(4\), degree \(2\), 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: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{3} + 4 q^{6} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{3} + 4 q^{6} - 12 q^{7} - 4 q^{10} - 4 q^{12} + 8 q^{13} + 4 q^{15} - 36 q^{16} - 32 q^{21} - 4 q^{22} + 32 q^{25} + 28 q^{27} - 12 q^{28} - 8 q^{30} + 8 q^{31} + 36 q^{33} + 4 q^{36} - 32 q^{37} - 8 q^{40} + 12 q^{42} - 24 q^{43} - 28 q^{45} - 16 q^{46} - 4 q^{48} - 40 q^{51} - 8 q^{52} - 4 q^{55} + 4 q^{57} - 4 q^{58} - 24 q^{60} + 200 q^{61} + 28 q^{63} + 12 q^{70} - 68 q^{73} - 36 q^{75} + 36 q^{76} + 24 q^{78} - 92 q^{81} + 24 q^{82} + 24 q^{85} + 28 q^{87} - 4 q^{88} - 68 q^{90} + 64 q^{91} + 16 q^{93} - 4 q^{96} - 148 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −0.707107 + 0.707107i −1.49908 − 0.867616i − 1.00000i −0.850241 − 2.06811i 1.67351 − 0.446512i 0.811234 + 0.811234i 0.707107 + 0.707107i 1.49448 + 2.60125i 2.06359 + 0.861165i
77.2 −0.707107 + 0.707107i −1.35438 − 1.07966i − 1.00000i 1.25473 + 1.85086i 1.72112 − 0.194257i −3.56316 − 3.56316i 0.707107 + 0.707107i 0.668679 + 2.92453i −2.19598 − 0.421528i
77.3 −0.707107 + 0.707107i −1.10477 + 1.33397i − 1.00000i −2.20748 − 0.356413i −0.162066 − 1.72445i 1.29452 + 1.29452i 0.707107 + 0.707107i −0.558949 − 2.94747i 1.81295 − 1.30890i
77.4 −0.707107 + 0.707107i −0.752859 + 1.55987i − 1.00000i 1.50962 + 1.64956i −0.570645 − 1.63535i 0.306664 + 0.306664i 0.707107 + 0.707107i −1.86641 − 2.34873i −2.23388 − 0.0989549i
77.5 −0.707107 + 0.707107i −0.274986 − 1.71008i − 1.00000i 2.05001 + 0.893014i 1.40366 + 1.01477i 2.76585 + 2.76585i 0.707107 + 0.707107i −2.84877 + 0.940499i −2.08103 + 0.818117i
77.6 −0.707107 + 0.707107i 0.668751 − 1.59774i − 1.00000i −0.595769 + 2.15524i 0.656894 + 1.60265i −0.0702822 − 0.0702822i 0.707107 + 0.707107i −2.10554 − 2.13698i −1.10271 − 1.94526i
77.7 −0.707107 + 0.707107i 1.43759 + 0.966096i − 1.00000i −1.74702 − 1.39568i −1.69966 + 0.333395i 0.371042 + 0.371042i 0.707107 + 0.707107i 1.13332 + 2.77770i 2.22222 − 0.248430i
77.8 −0.707107 + 0.707107i 1.52000 − 0.830420i − 1.00000i −2.23234 + 0.129088i −0.487608 + 1.66200i −2.47472 − 2.47472i 0.707107 + 0.707107i 1.62081 − 2.52448i 1.48722 − 1.66978i
77.9 −0.707107 + 0.707107i 1.65263 + 0.518471i − 1.00000i 2.11139 − 0.736229i −1.53520 + 0.801972i −2.44114 − 2.44114i 0.707107 + 0.707107i 2.46238 + 1.71368i −0.972386 + 2.01357i
77.10 0.707107 − 0.707107i −1.55987 + 0.752859i − 1.00000i −1.50962 − 1.64956i −0.570645 + 1.63535i 0.306664 + 0.306664i −0.707107 − 0.707107i 1.86641 − 2.34873i −2.23388 − 0.0989549i
77.11 0.707107 − 0.707107i −1.33397 + 1.10477i − 1.00000i 2.20748 + 0.356413i −0.162066 + 1.72445i 1.29452 + 1.29452i −0.707107 − 0.707107i 0.558949 − 2.94747i 1.81295 − 1.30890i
77.12 0.707107 − 0.707107i −0.966096 − 1.43759i − 1.00000i 1.74702 + 1.39568i −1.69966 − 0.333395i 0.371042 + 0.371042i −0.707107 − 0.707107i −1.13332 + 2.77770i 2.22222 − 0.248430i
77.13 0.707107 − 0.707107i −0.518471 − 1.65263i − 1.00000i −2.11139 + 0.736229i −1.53520 − 0.801972i −2.44114 − 2.44114i −0.707107 − 0.707107i −2.46238 + 1.71368i −0.972386 + 2.01357i
77.14 0.707107 − 0.707107i 0.830420 − 1.52000i − 1.00000i 2.23234 − 0.129088i −0.487608 − 1.66200i −2.47472 − 2.47472i −0.707107 − 0.707107i −1.62081 − 2.52448i 1.48722 − 1.66978i
77.15 0.707107 − 0.707107i 0.867616 + 1.49908i − 1.00000i 0.850241 + 2.06811i 1.67351 + 0.446512i 0.811234 + 0.811234i −0.707107 − 0.707107i −1.49448 + 2.60125i 2.06359 + 0.861165i
77.16 0.707107 − 0.707107i 1.07966 + 1.35438i − 1.00000i −1.25473 − 1.85086i 1.72112 + 0.194257i −3.56316 − 3.56316i −0.707107 − 0.707107i −0.668679 + 2.92453i −2.19598 − 0.421528i
77.17 0.707107 − 0.707107i 1.59774 − 0.668751i − 1.00000i 0.595769 − 2.15524i 0.656894 − 1.60265i −0.0702822 − 0.0702822i −0.707107 − 0.707107i 2.10554 − 2.13698i −1.10271 − 1.94526i
77.18 0.707107 − 0.707107i 1.71008 + 0.274986i − 1.00000i −2.05001 − 0.893014i 1.40366 − 1.01477i 2.76585 + 2.76585i −0.707107 − 0.707107i 2.84877 + 0.940499i −2.08103 + 0.818117i
533.1 −0.707107 − 0.707107i −1.49908 + 0.867616i 1.00000i −0.850241 + 2.06811i 1.67351 + 0.446512i 0.811234 − 0.811234i 0.707107 − 0.707107i 1.49448 − 2.60125i 2.06359 − 0.861165i
533.2 −0.707107 − 0.707107i −1.35438 + 1.07966i 1.00000i 1.25473 − 1.85086i 1.72112 + 0.194257i −3.56316 + 3.56316i 0.707107 − 0.707107i 0.668679 − 2.92453i −2.19598 + 0.421528i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.k.a ✓ 36
3.b odd 2 1 inner 570.2.k.a ✓ 36
5.c odd 4 1 inner 570.2.k.a ✓ 36
15.e even 4 1 inner 570.2.k.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.k.a ✓ 36 1.a even 1 1 trivial
570.2.k.a ✓ 36 3.b odd 2 1 inner
570.2.k.a ✓ 36 5.c odd 4 1 inner
570.2.k.a ✓ 36 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{18} + 6 T_{7}^{17} + 18 T_{7}^{16} - 32 T_{7}^{15} + 238 T_{7}^{14} + 1452 T_{7}^{13} + \cdots + 128 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display