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Properties

Label 272.3.bh.f
Level $272$
Weight $3$
Character orbit 272.bh
Analytic conductor $7.411$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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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: \(32\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{3} - 8 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{3} - 8 q^{7} + 16 q^{9} + 24 q^{11} + 48 q^{13} - 96 q^{15} - 40 q^{19} - 80 q^{21} + 48 q^{23} + 112 q^{25} + 80 q^{27} + 56 q^{29} + 24 q^{31} + 96 q^{35} + 48 q^{37} + 72 q^{39} - 160 q^{41} - 112 q^{43} - 504 q^{45} - 48 q^{47} + 208 q^{49} + 400 q^{51} + 304 q^{53} + 368 q^{55} - 264 q^{57} - 192 q^{59} - 288 q^{61} - 56 q^{63} + 8 q^{65} + 32 q^{69} - 352 q^{71} - 184 q^{73} - 24 q^{75} + 688 q^{77} + 424 q^{79} + 312 q^{81} - 600 q^{83} - 512 q^{85} - 1336 q^{87} - 144 q^{89} + 24 q^{91} + 944 q^{93} + 256 q^{95} + 416 q^{97} - 128 q^{99}+O(q^{100}) \) 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 −0.685983 + 3.44867i 0 4.33868 − 2.89902i 0 4.47041 + 2.98703i 0 −3.10782 − 1.28730i 0
65.2 0 −0.0843794 + 0.424204i 0 −1.82111 + 1.21683i 0 −5.14990 − 3.44105i 0 8.14209 + 3.37256i 0
65.3 0 0.533765 − 2.68342i 0 −4.10611 + 2.74362i 0 5.75863 + 3.84780i 0 1.39909 + 0.579524i 0
65.4 0 0.912174 − 4.58581i 0 7.13034 − 4.76434i 0 −7.38570 − 4.93497i 0 −11.8827 − 4.92196i 0
97.1 0 −2.88082 + 1.92490i 0 2.22013 + 0.441611i 0 −3.75824 + 0.747561i 0 1.14971 − 2.77565i 0
97.2 0 −0.705713 + 0.471542i 0 −6.65787 − 1.32433i 0 11.2240 − 2.23259i 0 −3.16847 + 7.64937i 0
97.3 0 2.30540 − 1.54042i 0 5.10009 + 1.01447i 0 1.70555 − 0.339255i 0 −0.502166 + 1.21234i 0
97.4 0 3.91211 − 2.61399i 0 −7.19369 − 1.43091i 0 −10.7125 + 2.13084i 0 5.02753 − 12.1375i 0
113.1 0 −0.685983 − 3.44867i 0 4.33868 + 2.89902i 0 4.47041 − 2.98703i 0 −3.10782 + 1.28730i 0
113.2 0 −0.0843794 − 0.424204i 0 −1.82111 − 1.21683i 0 −5.14990 + 3.44105i 0 8.14209 − 3.37256i 0
113.3 0 0.533765 + 2.68342i 0 −4.10611 − 2.74362i 0 5.75863 − 3.84780i 0 1.39909 − 0.579524i 0
113.4 0 0.912174 + 4.58581i 0 7.13034 + 4.76434i 0 −7.38570 + 4.93497i 0 −11.8827 + 4.92196i 0
129.1 0 −2.88082 − 1.92490i 0 2.22013 − 0.441611i 0 −3.75824 − 0.747561i 0 1.14971 + 2.77565i 0
129.2 0 −0.705713 − 0.471542i 0 −6.65787 + 1.32433i 0 11.2240 + 2.23259i 0 −3.16847 − 7.64937i 0
129.3 0 2.30540 + 1.54042i 0 5.10009 − 1.01447i 0 1.70555 + 0.339255i 0 −0.502166 − 1.21234i 0
129.4 0 3.91211 + 2.61399i 0 −7.19369 + 1.43091i 0 −10.7125 − 2.13084i 0 5.02753 + 12.1375i 0
177.1 0 −2.24344 + 3.35755i 0 −0.109963 − 0.552824i 0 −1.03372 + 5.19685i 0 −2.79594 − 6.74999i 0
177.2 0 −0.265432 + 0.397247i 0 0.373441 + 1.87742i 0 2.10274 − 10.5712i 0 3.35680 + 8.10403i 0
177.3 0 1.13351 − 1.69642i 0 0.797063 + 4.00711i 0 −0.788753 + 3.96533i 0 1.85116 + 4.46910i 0
177.4 0 2.15859 − 3.23056i 0 −1.60027 − 8.04508i 0 −0.739073 + 3.71557i 0 −2.33284 − 5.63197i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.4
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.f 32
4.b odd 2 1 136.3.t.a ✓ 32
17.e odd 16 1 inner 272.3.bh.f 32
68.i even 16 1 136.3.t.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.t.a ✓ 32 4.b odd 2 1
136.3.t.a ✓ 32 68.i even 16 1
272.3.bh.f 32 1.a even 1 1 trivial
272.3.bh.f 32 17.e odd 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 8 T_{3}^{31} + 24 T_{3}^{30} - 410 T_{3}^{28} + 2008 T_{3}^{27} - 3452 T_{3}^{26} + \cdots + 68254697536 \) acting on \(S_{3}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display