Properties

Label 984.2.f.b
Level $984$
Weight $2$
Character orbit 984.f
Analytic conductor $7.857$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [984,2,Mod(493,984)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("984.493"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 984 = 2^{3} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 984.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,2,0,0,0,2,4,2,-40,-2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.85727955889\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{2} + 2 q^{6} + 4 q^{7} + 2 q^{8} - 40 q^{9} - 2 q^{10} - 24 q^{15} + 20 q^{16} - 2 q^{18} - 16 q^{20} + 28 q^{22} + 24 q^{23} - 2 q^{24} - 40 q^{25} - 2 q^{26} - 32 q^{28} + 8 q^{30} + 8 q^{31}+ \cdots + 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
493.1 −1.39204 − 0.249421i 1.00000i 1.87558 + 0.694411i − 1.00551i 0.249421 − 1.39204i −2.60962 −2.43769 − 1.43446i −1.00000 −0.250795 + 1.39971i
493.2 −1.39204 + 0.249421i − 1.00000i 1.87558 − 0.694411i 1.00551i 0.249421 + 1.39204i −2.60962 −2.43769 + 1.43446i −1.00000 −0.250795 − 1.39971i
493.3 −1.38355 − 0.292912i 1.00000i 1.82840 + 0.810516i 3.87937i 0.292912 − 1.38355i −0.342027 −2.29227 − 1.65695i −1.00000 1.13632 − 5.36729i
493.4 −1.38355 + 0.292912i − 1.00000i 1.82840 − 0.810516i − 3.87937i 0.292912 + 1.38355i −0.342027 −2.29227 + 1.65695i −1.00000 1.13632 + 5.36729i
493.5 −1.38179 − 0.301070i − 1.00000i 1.81871 + 0.832035i 1.93380i −0.301070 + 1.38179i 2.57085 −2.26259 − 1.69726i −1.00000 0.582209 − 2.67211i
493.6 −1.38179 + 0.301070i 1.00000i 1.81871 − 0.832035i − 1.93380i −0.301070 − 1.38179i 2.57085 −2.26259 + 1.69726i −1.00000 0.582209 + 2.67211i
493.7 −1.14409 − 0.831297i − 1.00000i 0.617889 + 1.90216i − 1.56855i −0.831297 + 1.14409i −4.24351 0.874339 − 2.68989i −1.00000 −1.30394 + 1.79457i
493.8 −1.14409 + 0.831297i 1.00000i 0.617889 − 1.90216i 1.56855i −0.831297 − 1.14409i −4.24351 0.874339 + 2.68989i −1.00000 −1.30394 − 1.79457i
493.9 −1.03128 − 0.967709i 1.00000i 0.127077 + 1.99596i 3.88105i 0.967709 − 1.03128i −0.372389 1.80046 − 2.18137i −1.00000 3.75573 − 4.00245i
493.10 −1.03128 + 0.967709i − 1.00000i 0.127077 − 1.99596i − 3.88105i 0.967709 + 1.03128i −0.372389 1.80046 + 2.18137i −1.00000 3.75573 + 4.00245i
493.11 −0.804080 − 1.16338i − 1.00000i −0.706909 + 1.87090i − 1.42710i −1.16338 + 0.804080i 3.46856 2.74498 − 0.681952i −1.00000 −1.66026 + 1.14750i
493.12 −0.804080 + 1.16338i 1.00000i −0.706909 − 1.87090i 1.42710i −1.16338 − 0.804080i 3.46856 2.74498 + 0.681952i −1.00000 −1.66026 − 1.14750i
493.13 −0.589421 − 1.28553i 1.00000i −1.30516 + 1.51544i − 0.836655i 1.28553 − 0.589421i 0.923115 2.71743 + 0.784595i −1.00000 −1.07554 + 0.493142i
493.14 −0.589421 + 1.28553i − 1.00000i −1.30516 − 1.51544i 0.836655i 1.28553 + 0.589421i 0.923115 2.71743 − 0.784595i −1.00000 −1.07554 − 0.493142i
493.15 −0.336995 − 1.37348i − 1.00000i −1.77287 + 0.925708i 1.10707i −1.37348 + 0.336995i 0.345097 1.86888 + 2.12303i −1.00000 1.52053 − 0.373075i
493.16 −0.336995 + 1.37348i 1.00000i −1.77287 − 0.925708i − 1.10707i −1.37348 − 0.336995i 0.345097 1.86888 − 2.12303i −1.00000 1.52053 + 0.373075i
493.17 −0.262443 − 1.38965i 1.00000i −1.86225 + 0.729406i 3.21468i 1.38965 − 0.262443i 2.98084 1.50235 + 2.39644i −1.00000 4.46728 − 0.843669i
493.18 −0.262443 + 1.38965i − 1.00000i −1.86225 − 0.729406i − 3.21468i 1.38965 + 0.262443i 2.98084 1.50235 − 2.39644i −1.00000 4.46728 + 0.843669i
493.19 −0.0624481 − 1.41283i 1.00000i −1.99220 + 0.176458i − 2.37442i 1.41283 − 0.0624481i −3.42077 0.373714 + 2.80363i −1.00000 −3.35466 + 0.148278i
493.20 −0.0624481 + 1.41283i − 1.00000i −1.99220 − 0.176458i 2.37442i 1.41283 + 0.0624481i −3.42077 0.373714 − 2.80363i −1.00000 −3.35466 − 0.148278i
See all 40 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 493.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 984.2.f.b ✓ 40
4.b odd 2 1 3936.2.f.b 40
8.b even 2 1 inner 984.2.f.b ✓ 40
8.d odd 2 1 3936.2.f.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
984.2.f.b ✓ 40 1.a even 1 1 trivial
984.2.f.b ✓ 40 8.b even 2 1 inner
3936.2.f.b 40 4.b odd 2 1
3936.2.f.b 40 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + 120 T_{5}^{38} + 6550 T_{5}^{36} + 215576 T_{5}^{34} + 4782625 T_{5}^{32} + \cdots + 12005146624 \) acting on \(S_{2}^{\mathrm{new}}(984, [\chi])\). Copy content Toggle raw display