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Properties

Label 644.2.k.a
Level $644$
Weight $2$
Character orbit 644.k
Analytic conductor $5.142$
Analytic rank $0$
Dimension $176$
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] = [644,2,Mod(47,644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(644, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("644.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.k (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: \(5.14236589017\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(88\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 12 q^{8} - 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q + 12 q^{8} - 88 q^{9} - 18 q^{10} - 30 q^{12} - 12 q^{14} - 8 q^{16} - 10 q^{18} - 16 q^{21} - 24 q^{22} + 36 q^{24} + 80 q^{25} + 6 q^{28} + 20 q^{30} - 30 q^{32} - 24 q^{33} - 40 q^{36} - 8 q^{37} + 18 q^{38} - 42 q^{42} - 14 q^{44} - 16 q^{49} - 28 q^{50} - 36 q^{52} + 8 q^{53} + 54 q^{54} - 16 q^{56} + 48 q^{57} + 38 q^{58} + 24 q^{60} + 24 q^{61} - 36 q^{64} - 72 q^{66} - 36 q^{68} - 46 q^{70} + 12 q^{72} - 72 q^{73} + 6 q^{74} - 8 q^{77} - 100 q^{78} + 96 q^{80} - 88 q^{81} + 80 q^{84} - 6 q^{86} + 2 q^{88} + 8 q^{93} + 108 q^{94} + 84 q^{96} + 66 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −1.41321 − 0.0532815i −1.52376 − 2.63923i 1.99432 + 0.150596i −3.64539 − 2.10467i 2.01277 + 3.81098i −0.316833 + 2.62671i −2.81037 − 0.319084i −3.14370 + 5.44504i 5.03957 + 3.16857i
47.2 −1.40647 + 0.147816i −0.887781 − 1.53768i 1.95630 − 0.415796i 1.56345 + 0.902661i 1.47593 + 2.03147i −2.58128 + 0.580511i −2.69001 + 0.873975i −0.0763119 + 0.132176i −2.33238 − 1.03846i
47.3 −1.39828 + 0.211716i −0.723234 − 1.25268i 1.91035 − 0.592075i 3.51050 + 2.02679i 1.27649 + 1.59847i 1.93890 − 1.80018i −2.54585 + 1.23234i 0.453866 − 0.786119i −5.33776 − 2.09078i
47.4 −1.38632 − 0.279475i 1.01070 + 1.75058i 1.84379 + 0.774886i 3.03255 + 1.75085i −0.911910 − 2.70933i −1.74067 + 1.99251i −2.33952 − 1.58954i −0.543018 + 0.940535i −3.71478 − 3.27476i
47.5 −1.38519 − 0.285045i −0.0183010 − 0.0316982i 1.83750 + 0.789681i −1.42741 − 0.824117i 0.0163149 + 0.0491246i 2.33854 − 1.23742i −2.32019 − 1.61763i 1.49933 − 2.59692i 1.74233 + 1.54843i
47.6 −1.37294 − 0.339158i −1.48570 − 2.57331i 1.76994 + 0.931290i 1.30773 + 0.755018i 1.16702 + 4.03690i 2.54121 + 0.736381i −2.11418 − 1.87890i −2.91463 + 5.04828i −1.53937 − 1.48012i
47.7 −1.37124 − 0.345964i 0.568795 + 0.985181i 1.76062 + 0.948802i −0.838896 − 0.484337i −0.439119 − 1.54771i 1.72507 + 2.00602i −2.08598 − 1.91015i 0.852945 − 1.47734i 0.982767 + 0.954372i
47.8 −1.37065 + 0.348312i −0.200106 − 0.346594i 1.75736 − 0.954827i −0.494349 − 0.285412i 0.394998 + 0.405360i −0.633989 + 2.56867i −2.07614 + 1.92084i 1.41991 − 2.45936i 0.776991 + 0.219013i
47.9 −1.37002 − 0.350776i 1.50642 + 2.60920i 1.75391 + 0.961140i −2.39443 − 1.38242i −1.14859 − 4.10308i −2.53168 + 0.768494i −2.06575 − 1.93201i −3.03862 + 5.26305i 2.79550 + 2.73386i
47.10 −1.35081 + 0.418709i 1.36452 + 2.36342i 1.64937 − 1.13119i 2.05969 + 1.18916i −2.83279 − 2.62119i 2.58996 − 0.540468i −1.75434 + 2.21863i −2.22383 + 3.85179i −3.28015 − 0.743918i
47.11 −1.31607 + 0.517655i −0.511235 − 0.885485i 1.46407 − 1.36254i −3.18853 − 1.84090i 1.13120 + 0.900714i 1.48589 − 2.18910i −1.22148 + 2.55107i 0.977278 − 1.69269i 5.14926 + 0.772186i
47.12 −1.27479 − 0.612291i −1.01704 − 1.76156i 1.25020 + 1.56109i −0.595562 − 0.343848i 0.217926 + 2.86835i −1.15701 − 2.37935i −0.637907 − 2.75555i −0.568729 + 0.985067i 0.548684 + 0.802992i
47.13 −1.25814 + 0.645823i 0.739694 + 1.28119i 1.16582 − 1.62507i 1.48612 + 0.858010i −1.75806 − 1.13420i 1.24467 + 2.33469i −0.417262 + 2.79748i 0.405707 − 0.702705i −2.42386 − 0.119727i
47.14 −1.23917 + 0.681509i −1.55875 − 2.69983i 1.07109 − 1.68901i −0.0407085 − 0.0235031i 3.77151 + 2.28325i −1.04500 − 2.43063i −0.176191 + 2.82293i −3.35938 + 5.81861i 0.0664624 + 0.00138114i
47.15 −1.23890 − 0.682010i −0.0127826 − 0.0221400i 1.06973 + 1.68988i −3.16842 − 1.82929i 0.000736536 0.0361470i −2.63703 + 0.214669i −0.172766 − 2.82315i 1.49967 − 2.59751i 2.67775 + 4.42719i
47.16 −1.22453 − 0.707487i 1.65734 + 2.87059i 0.998925 + 1.73267i 0.671604 + 0.387751i 0.00145429 − 4.68766i 1.31016 − 2.29858i 0.00263245 − 2.82843i −3.99354 + 6.91701i −0.548067 − 0.949961i
47.17 −1.18544 − 0.771194i −0.967133 − 1.67512i 0.810521 + 1.82840i 1.15002 + 0.663961i −0.145370 + 2.73160i −1.32455 + 2.29032i 0.449232 − 2.79252i −0.370693 + 0.642059i −0.851227 − 1.67397i
47.18 −1.08646 + 0.905322i 0.0212978 + 0.0368889i 0.360786 − 1.96719i 1.91276 + 1.10433i −0.0565355 − 0.0207969i −2.50217 − 0.859748i 1.38896 + 2.46390i 1.49909 − 2.59650i −3.07791 + 0.531851i
47.19 −1.05109 + 0.946151i 0.109351 + 0.189402i 0.209596 − 1.98899i 1.56933 + 0.906054i −0.294142 − 0.0956164i 1.95132 − 1.78671i 1.66158 + 2.28892i 1.47608 − 2.55665i −2.50678 + 0.532477i
47.20 −1.02215 − 0.977350i 0.0832547 + 0.144201i 0.0895745 + 1.99799i 2.88637 + 1.66644i 0.0558365 − 0.228764i 2.62089 + 0.361866i 1.86118 − 2.12979i 1.48614 − 2.57407i −1.32160 − 4.52434i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 644.2.k.a ✓ 176
4.b odd 2 1 inner 644.2.k.a ✓ 176
7.d odd 6 1 inner 644.2.k.a ✓ 176
28.f even 6 1 inner 644.2.k.a ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.k.a ✓ 176 1.a even 1 1 trivial
644.2.k.a ✓ 176 4.b odd 2 1 inner
644.2.k.a ✓ 176 7.d odd 6 1 inner
644.2.k.a ✓ 176 28.f even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(644, [\chi])\).