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Properties

Label 618.2.f.b
Level $618$
Weight $2$
Character orbit 618.f
Analytic conductor $4.935$
Analytic rank $0$
Dimension $64$
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] = [618,2,Mod(47,618)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(618, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("618.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 618 = 2 \cdot 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 618.f (of order \(6\), 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: \(4.93475484492\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} - 12 q^{6} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} - 12 q^{6} + 4 q^{7} + 4 q^{9} + 6 q^{12} + 8 q^{13} + 4 q^{15} - 32 q^{16} + 8 q^{19} - 6 q^{21} - 68 q^{25} - 4 q^{28} + 4 q^{30} - 8 q^{33} + 2 q^{36} - 36 q^{43} - 18 q^{45} - 32 q^{46} + 6 q^{48} - 56 q^{49} + 4 q^{52} - 18 q^{54} - 24 q^{55} + 12 q^{57} - 8 q^{58} - 4 q^{60} + 120 q^{61} - 38 q^{63} - 64 q^{64} + 48 q^{66} + 84 q^{67} + 60 q^{70} + 30 q^{75} + 16 q^{76} + 66 q^{78} + 56 q^{79} - 12 q^{81} + 4 q^{82} - 6 q^{84} - 24 q^{85} - 42 q^{87} + 12 q^{88} + 44 q^{91} - 80 q^{93} + 12 q^{96} + 32 q^{97} - 108 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −0.866025 − 0.500000i −1.62567 + 0.597649i 0.500000 + 0.866025i −2.13616 − 3.69994i 1.70670 + 0.295257i −2.46709 − 4.27312i − 1.00000i 2.28563 − 1.94317i 4.27232i
47.2 −0.866025 − 0.500000i −1.56170 + 0.749069i 0.500000 + 0.866025i 1.98122 + 3.43157i 1.72700 + 0.132135i 1.89199 + 3.27702i − 1.00000i 1.87779 − 2.33964i − 3.96243i
47.3 −0.866025 − 0.500000i −1.42125 − 0.989969i 0.500000 + 0.866025i 0.936027 + 1.62125i 0.735857 + 1.56796i −1.59107 − 2.75582i − 1.00000i 1.03992 + 2.81399i − 1.87205i
47.4 −0.866025 − 0.500000i −1.27074 − 1.17695i 0.500000 + 0.866025i −1.57024 − 2.71974i 0.512016 + 1.65464i 1.02765 + 1.77994i − 1.00000i 0.229560 + 2.99120i 3.14048i
47.5 −0.866025 − 0.500000i −0.740829 + 1.56562i 0.500000 + 0.866025i −0.0664562 − 0.115106i 1.42439 − 0.985454i 0.0452166 + 0.0783174i − 1.00000i −1.90234 − 2.31972i 0.132912i
47.6 −0.866025 − 0.500000i −0.491076 − 1.66098i 0.500000 + 0.866025i 0.00270572 + 0.00468644i −0.405204 + 1.68399i 2.15117 + 3.72594i − 1.00000i −2.51769 + 1.63133i − 0.00541144i
47.7 −0.866025 − 0.500000i −0.0565392 − 1.73113i 0.500000 + 0.866025i 1.93638 + 3.35391i −0.816599 + 1.52747i 0.374278 + 0.648269i − 1.00000i −2.99361 + 0.195753i − 3.87276i
47.8 −0.866025 − 0.500000i 0.314478 + 1.70326i 0.500000 + 0.866025i −1.17731 − 2.03915i 0.579285 − 1.63231i 0.0943620 + 0.163440i − 1.00000i −2.80221 + 1.07128i 2.35461i
47.9 −0.866025 − 0.500000i 0.537755 − 1.64646i 0.500000 + 0.866025i −1.10151 − 1.90788i −1.28894 + 1.15700i −1.04899 − 1.81690i − 1.00000i −2.42164 − 1.77078i 2.20303i
47.10 −0.866025 − 0.500000i 0.545013 + 1.64407i 0.500000 + 0.866025i 1.42067 + 2.46066i 0.350039 − 1.69631i −0.338738 − 0.586712i − 1.00000i −2.40592 + 1.79208i − 2.84133i
47.11 −0.866025 − 0.500000i 1.33176 − 1.10744i 0.500000 + 0.866025i −1.94973 − 3.37703i −1.70705 + 0.293195i 0.587571 + 1.01770i − 1.00000i 0.547144 − 2.94968i 3.89946i
47.12 −0.866025 − 0.500000i 1.44514 + 0.954765i 0.500000 + 0.866025i −1.00410 − 1.73915i −0.774144 − 1.54942i 2.56001 + 4.43407i − 1.00000i 1.17685 + 2.75953i 2.00820i
47.13 −0.866025 − 0.500000i 1.55892 − 0.754826i 0.500000 + 0.866025i 0.735035 + 1.27312i −1.72748 − 0.125762i −2.22523 − 3.85420i − 1.00000i 1.86047 − 2.35343i − 1.47007i
47.14 −0.866025 − 0.500000i 1.58506 − 0.698263i 0.500000 + 0.866025i 1.11787 + 1.93620i −1.72184 − 0.187818i 0.707608 + 1.22561i − 1.00000i 2.02486 − 2.21358i − 2.23573i
47.15 −0.866025 − 0.500000i 1.58616 + 0.695769i 0.500000 + 0.866025i 1.24874 + 2.16288i −1.02577 − 1.39563i 0.861797 + 1.49268i − 1.00000i 2.03181 + 2.20720i − 2.49748i
47.16 −0.866025 − 0.500000i 1.72762 + 0.123761i 0.500000 + 0.866025i −0.373125 − 0.646272i −1.43429 − 0.970992i −1.63054 − 2.82418i − 1.00000i 2.96937 + 0.427624i 0.746250i
47.17 0.866025 + 0.500000i −1.72762 + 0.123761i 0.500000 + 0.866025i 0.373125 + 0.646272i −1.55805 − 0.756632i −1.63054 − 2.82418i 1.00000i 2.96937 − 0.427624i 0.746250i
47.18 0.866025 + 0.500000i −1.58616 + 0.695769i 0.500000 + 0.866025i −1.24874 − 2.16288i −1.72154 − 0.190527i 0.861797 + 1.49268i 1.00000i 2.03181 − 2.20720i − 2.49748i
47.19 0.866025 + 0.500000i −1.58506 − 0.698263i 0.500000 + 0.866025i −1.11787 − 1.93620i −1.02357 − 1.39725i 0.707608 + 1.22561i 1.00000i 2.02486 + 2.21358i − 2.23573i
47.20 0.866025 + 0.500000i −1.55892 − 0.754826i 0.500000 + 0.866025i −0.735035 − 1.27312i −0.972653 − 1.43316i −2.22523 − 3.85420i 1.00000i 1.86047 + 2.35343i − 1.47007i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
103.d odd 6 1 inner
309.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 618.2.f.b ✓ 64
3.b odd 2 1 inner 618.2.f.b ✓ 64
103.d odd 6 1 inner 618.2.f.b ✓ 64
309.g even 6 1 inner 618.2.f.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
618.2.f.b ✓ 64 1.a even 1 1 trivial
618.2.f.b ✓ 64 3.b odd 2 1 inner
618.2.f.b ✓ 64 103.d odd 6 1 inner
618.2.f.b ✓ 64 309.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} + 114 T_{5}^{62} + 7173 T_{5}^{60} + 311142 T_{5}^{58} + 10273252 T_{5}^{56} + \cdots + 1475789056 \) acting on \(S_{2}^{\mathrm{new}}(618, [\chi])\). Copy content Toggle raw display