↑

Properties

Label 624.2.cv.a
Level $624$
Weight $2$
Character orbit 624.cv
Analytic conductor $4.983$
Analytic rank $0$
Dimension $224$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(61,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.cv (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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 16 q^{12} - 32 q^{14} + 16 q^{20} - 20 q^{22} + 40 q^{26} + 8 q^{28} + 8 q^{30} - 40 q^{34} - 40 q^{38} - 20 q^{42} + 16 q^{43} + 32 q^{44} - 52 q^{46} + 112 q^{49} - 20 q^{50} + 56 q^{52} + 32 q^{59} - 48 q^{60} - 60 q^{62} + 16 q^{65} + 16 q^{68} + 56 q^{70} + 40 q^{74} - 16 q^{75} - 80 q^{76} + 12 q^{78} - 160 q^{79} + 112 q^{81} - 60 q^{82} + 80 q^{83} - 128 q^{86} + 8 q^{88} - 16 q^{90} - 88 q^{91} + 96 q^{92} - 20 q^{94} - 36 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.41363 − 0.0406242i 0.965926 − 0.258819i 1.99670 + 0.114855i −1.61258 + 1.61258i −1.37598 + 0.326634i 0.529159 − 0.305510i −2.81793 − 0.243477i 0.866025 − 0.500000i 2.34511 − 2.21409i
61.2 −1.41167 − 0.0847271i −0.965926 + 0.258819i 1.98564 + 0.239214i 1.26601 − 1.26601i 1.38550 − 0.283528i 3.30755 − 1.90962i −2.78281 − 0.505930i 0.866025 − 0.500000i −1.89446 + 1.67993i
61.3 −1.40818 + 0.130486i 0.965926 − 0.258819i 1.96595 − 0.367495i 1.98544 − 1.98544i −1.32643 + 0.490504i 2.36496 − 1.36541i −2.72046 + 0.774027i 0.866025 − 0.500000i −2.53679 + 3.05493i
61.4 −1.40626 − 0.149771i 0.965926 − 0.258819i 1.95514 + 0.421234i −0.233265 + 0.233265i −1.39711 + 0.219299i −2.43513 + 1.40592i −2.68634 − 0.885187i 0.866025 − 0.500000i 0.362967 − 0.293095i
61.5 −1.38118 + 0.303888i −0.965926 + 0.258819i 1.81530 − 0.839447i −1.86912 + 1.86912i 1.25546 − 0.651009i 1.37908 − 0.796211i −2.25216 + 1.71108i 0.866025 − 0.500000i 2.01358 − 3.14959i
61.6 −1.36611 − 0.365727i −0.965926 + 0.258819i 1.73249 + 0.999243i −1.20366 + 1.20366i 1.41421 0.000308994i 0.299830 − 0.173107i −2.00131 − 1.99869i 0.866025 − 0.500000i 2.08454 − 1.20412i
61.7 −1.36150 + 0.382513i −0.965926 + 0.258819i 1.70737 − 1.04158i 2.72385 − 2.72385i 1.21611 − 0.721861i −1.41400 + 0.816371i −1.92617 + 2.07120i 0.866025 − 0.500000i −2.66662 + 4.75043i
61.8 −1.31711 − 0.514996i −0.965926 + 0.258819i 1.46956 + 1.35661i 0.983203 − 0.983203i 1.40552 + 0.156554i −4.25268 + 2.45529i −1.23692 − 2.54362i 0.866025 − 0.500000i −1.80133 + 0.788641i
61.9 −1.24975 + 0.661901i 0.965926 − 0.258819i 1.12377 − 1.65443i −0.315413 + 0.315413i −1.03586 + 0.962808i 3.15416 − 1.82106i −0.309375 + 2.81146i 0.866025 − 0.500000i 0.185417 − 0.602960i
61.10 −1.19909 − 0.749791i 0.965926 − 0.258819i 0.875626 + 1.79813i −1.93060 + 1.93060i −1.35229 − 0.413896i 4.15519 − 2.39900i 0.298271 − 2.81266i 0.866025 − 0.500000i 3.76251 − 0.867413i
61.11 −1.17776 − 0.782865i 0.965926 − 0.258819i 0.774246 + 1.84406i 2.44914 − 2.44914i −1.34025 − 0.451362i −1.73112 + 0.999460i 0.531769 − 2.77799i 0.866025 − 0.500000i −4.80185 + 0.967158i
61.12 −1.15688 − 0.813403i −0.965926 + 0.258819i 0.676752 + 1.88202i 1.39577 − 1.39577i 1.32799 + 0.486264i 2.51623 − 1.45274i 0.747919 − 2.72775i 0.866025 − 0.500000i −2.75006 + 0.479417i
61.13 −1.04085 + 0.957404i 0.965926 − 0.258819i 0.166757 − 1.99304i −3.13230 + 3.13230i −0.757594 + 1.19417i 0.239413 − 0.138225i 1.73457 + 2.23411i 0.866025 − 0.500000i 0.261393 − 6.25914i
61.14 −1.02507 + 0.974284i 0.965926 − 0.258819i 0.101541 − 1.99742i 0.812158 − 0.812158i −0.737979 + 1.20639i −3.19729 + 1.84596i 1.84197 + 2.14643i 0.866025 − 0.500000i −0.0412469 + 1.62379i
61.15 −0.986624 + 1.01320i −0.965926 + 0.258819i −0.0531474 − 1.99929i −0.635996 + 0.635996i 0.690770 − 1.23403i 0.305415 − 0.176331i 2.07812 + 1.91870i 0.866025 − 0.500000i −0.0169023 − 1.27188i
61.16 −0.967294 + 1.03167i −0.965926 + 0.258819i −0.128683 − 1.99586i 0.314722 − 0.314722i 0.667319 − 1.24687i −1.77783 + 1.02643i 2.18354 + 1.79782i 0.866025 − 0.500000i 0.0202603 + 0.629118i
61.17 −0.758198 − 1.19379i 0.965926 − 0.258819i −0.850272 + 1.81026i 0.957625 − 0.957625i −1.04134 − 0.956877i −2.66803 + 1.54039i 2.80574 − 0.357487i 0.866025 − 0.500000i −1.86927 − 0.417135i
61.18 −0.742255 − 1.20377i −0.965926 + 0.258819i −0.898115 + 1.78701i −0.255802 + 0.255802i 1.02852 + 0.970641i 0.623423 − 0.359933i 2.81777 − 0.245292i 0.866025 − 0.500000i 0.497797 + 0.118056i
61.19 −0.727971 + 1.21246i 0.965926 − 0.258819i −0.940118 − 1.76527i 2.55500 − 2.55500i −0.389358 + 1.35956i 1.70986 − 0.987186i 2.82470 + 0.145210i 0.866025 − 0.500000i 1.23787 + 4.95779i
61.20 −0.547055 − 1.30412i −0.965926 + 0.258819i −1.40146 + 1.42685i −2.61994 + 2.61994i 0.865946 + 1.11810i −4.00792 + 2.31397i 2.62746 + 1.04711i 0.866025 − 0.500000i 4.84997 + 1.98347i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
16.e even 4 1 inner
208.bj even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.cv.a ✓ 224
13.c even 3 1 inner 624.2.cv.a ✓ 224
16.e even 4 1 inner 624.2.cv.a ✓ 224
208.bj even 12 1 inner 624.2.cv.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.cv.a ✓ 224 1.a even 1 1 trivial
624.2.cv.a ✓ 224 13.c even 3 1 inner
624.2.cv.a ✓ 224 16.e even 4 1 inner
624.2.cv.a ✓ 224 208.bj even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(624, [\chi])\).