↑

Properties

Label 272.3.bh.g
Level $272$
Weight $3$
Character orbit 272.bh
Analytic conductor $7.411$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [272,3,Mod(65,272)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(272, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 0, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("272.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 272.bh (of order \(16\), degree \(8\), not 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: \(7.41146319060\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{3} + 8 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{3} + 8 q^{7} - 16 q^{9} - 24 q^{11} - 48 q^{13} + 96 q^{15} + 40 q^{19} + 80 q^{21} - 48 q^{23} + 48 q^{25} - 224 q^{27} + 24 q^{29} - 88 q^{31} - 32 q^{35} - 176 q^{37} + 120 q^{39} + 352 q^{43} + 264 q^{45} + 48 q^{47} - 208 q^{49} - 400 q^{51} - 472 q^{53} + 208 q^{55} + 24 q^{57} + 576 q^{59} + 632 q^{63} - 32 q^{65} + 160 q^{69} + 160 q^{71} + 256 q^{73} - 1128 q^{75} - 208 q^{77} - 1000 q^{79} + 24 q^{81} - 312 q^{83} + 1240 q^{85} + 664 q^{87} + 720 q^{89} - 664 q^{91} - 432 q^{93} - 736 q^{95} - 288 q^{97} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −0.980550 + 4.92956i 0 3.33211 − 2.22644i 0 −9.23443 − 6.17025i 0 −15.0241 − 6.22320i 0
65.2 0 −0.657237 + 3.30415i 0 −5.77013 + 3.85548i 0 10.9767 + 7.33440i 0 −2.17054 − 0.899066i 0
65.3 0 0.0253220 − 0.127302i 0 −2.11962 + 1.41628i 0 −5.83745 − 3.90046i 0 8.29935 + 3.43770i 0
65.4 0 0.262425 − 1.31930i 0 6.57938 − 4.39620i 0 6.01468 + 4.01888i 0 6.64323 + 2.75172i 0
65.5 0 1.10801 − 5.57033i 0 −3.32830 + 2.22390i 0 0.387063 + 0.258627i 0 −21.4860 − 8.89979i 0
97.1 0 −4.59051 + 3.06728i 0 −8.77163 − 1.74479i 0 −5.01798 + 0.998137i 0 8.22042 − 19.8459i 0
97.2 0 −3.29196 + 2.19962i 0 8.73068 + 1.73664i 0 3.91058 − 0.777863i 0 2.55452 − 6.16717i 0
97.3 0 0.504198 − 0.336895i 0 1.91245 + 0.380411i 0 −6.19362 + 1.23199i 0 −3.30343 + 7.97519i 0
97.4 0 0.938835 − 0.627309i 0 −4.64597 − 0.924140i 0 −0.473863 + 0.0942571i 0 −2.95626 + 7.13704i 0
97.5 0 4.45729 − 2.97827i 0 2.23327 + 0.444225i 0 9.31608 − 1.85308i 0 7.55324 − 18.2351i 0
113.1 0 −0.980550 − 4.92956i 0 3.33211 + 2.22644i 0 −9.23443 + 6.17025i 0 −15.0241 + 6.22320i 0
113.2 0 −0.657237 − 3.30415i 0 −5.77013 − 3.85548i 0 10.9767 − 7.33440i 0 −2.17054 + 0.899066i 0
113.3 0 0.0253220 + 0.127302i 0 −2.11962 − 1.41628i 0 −5.83745 + 3.90046i 0 8.29935 − 3.43770i 0
113.4 0 0.262425 + 1.31930i 0 6.57938 + 4.39620i 0 6.01468 − 4.01888i 0 6.64323 − 2.75172i 0
113.5 0 1.10801 + 5.57033i 0 −3.32830 − 2.22390i 0 0.387063 − 0.258627i 0 −21.4860 + 8.89979i 0
129.1 0 −4.59051 − 3.06728i 0 −8.77163 + 1.74479i 0 −5.01798 − 0.998137i 0 8.22042 + 19.8459i 0
129.2 0 −3.29196 − 2.19962i 0 8.73068 − 1.73664i 0 3.91058 + 0.777863i 0 2.55452 + 6.16717i 0
129.3 0 0.504198 + 0.336895i 0 1.91245 − 0.380411i 0 −6.19362 − 1.23199i 0 −3.30343 − 7.97519i 0
129.4 0 0.938835 + 0.627309i 0 −4.64597 + 0.924140i 0 −0.473863 − 0.0942571i 0 −2.95626 − 7.13704i 0
129.5 0 4.45729 + 2.97827i 0 2.23327 − 0.444225i 0 9.31608 + 1.85308i 0 7.55324 + 18.2351i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.3.bh.g 40
4.b odd 2 1 136.3.t.b ✓ 40
17.e odd 16 1 inner 272.3.bh.g 40
68.i even 16 1 136.3.t.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.t.b ✓ 40 4.b odd 2 1
136.3.t.b ✓ 40 68.i even 16 1
272.3.bh.g 40 1.a even 1 1 trivial
272.3.bh.g 40 17.e odd 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 8 T_{3}^{39} + 40 T_{3}^{38} + 272 T_{3}^{37} + 1542 T_{3}^{36} + 6216 T_{3}^{35} + \cdots + 558577609736192 \) acting on \(S_{3}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display