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Properties

Label 272.3.bh.b
Level $272$
Weight $3$
Character orbit 272.bh
Analytic conductor $7.411$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [272,3,Mod(65,272)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.41146319060\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{16}^{7} + \cdots - 2 \zeta_{16}^{2}) q^{3} + (2 \zeta_{16}^{7} - 2 \zeta_{16}^{6} + \cdots - 3) q^{5} + (\zeta_{16}^{7} - 2 \zeta_{16}^{6} + \cdots + 2) q^{7} + (2 \zeta_{16}^{7} - 7 \zeta_{16}^{6} + \cdots + 1) q^{9}+ \cdots + ( - 18 \zeta_{16}^{7} + 15 \zeta_{16}^{6} + \cdots - 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{5} + 16 q^{7} + 8 q^{9} - 40 q^{11} - 32 q^{15} - 16 q^{17} + 32 q^{19} - 64 q^{21} + 8 q^{23} + 16 q^{25} - 96 q^{27} + 24 q^{29} - 32 q^{31} - 80 q^{35} - 168 q^{37} + 72 q^{39} - 96 q^{43}+ \cdots - 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{16}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.923880 − 0.382683i
0.382683 − 0.923880i
−0.923880 + 0.382683i
0.382683 + 0.923880i
−0.382683 − 0.923880i
0.923880 − 0.382683i
−0.382683 + 0.923880i
0.923880 + 0.382683i
0 −0.273561 + 1.37529i 0 −0.286621 + 0.191514i 0 5.96908 + 3.98841i 0 6.49834 + 2.69170i 0
97.1 0 1.35595 − 0.906019i 0 −6.70292 − 1.33329i 0 −0.886687 + 0.176373i 0 −2.42641 + 5.85788i 0
113.1 0 −0.273561 − 1.37529i 0 −0.286621 − 0.191514i 0 5.96908 − 3.98841i 0 6.49834 − 2.69170i 0
129.1 0 1.35595 + 0.906019i 0 −6.70292 + 1.33329i 0 −0.886687 − 0.176373i 0 −2.42641 − 5.85788i 0
177.1 0 2.88669 − 4.32023i 0 −0.711297 − 3.57593i 0 0.644047 − 3.23784i 0 −6.88730 − 16.6274i 0
193.1 0 −3.96908 + 0.789499i 0 −4.29916 + 6.43416i 0 2.27356 + 3.40262i 0 6.81537 − 2.82302i 0
209.1 0 2.88669 + 4.32023i 0 −0.711297 + 3.57593i 0 0.644047 + 3.23784i 0 −6.88730 + 16.6274i 0
241.1 0 −3.96908 − 0.789499i 0 −4.29916 − 6.43416i 0 2.27356 − 3.40262i 0 6.81537 + 2.82302i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.1
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.b 8
4.b odd 2 1 17.3.e.b ✓ 8
12.b even 2 1 153.3.p.a 8
17.e odd 16 1 inner 272.3.bh.b 8
20.d odd 2 1 425.3.u.a 8
20.e even 4 1 425.3.t.b 8
20.e even 4 1 425.3.t.d 8
68.d odd 2 1 289.3.e.g 8
68.f odd 4 1 289.3.e.f 8
68.f odd 4 1 289.3.e.h 8
68.g odd 8 1 289.3.e.a 8
68.g odd 8 1 289.3.e.e 8
68.g odd 8 1 289.3.e.j 8
68.g odd 8 1 289.3.e.n 8
68.i even 16 1 17.3.e.b ✓ 8
68.i even 16 1 289.3.e.a 8
68.i even 16 1 289.3.e.e 8
68.i even 16 1 289.3.e.f 8
68.i even 16 1 289.3.e.g 8
68.i even 16 1 289.3.e.h 8
68.i even 16 1 289.3.e.j 8
68.i even 16 1 289.3.e.n 8
204.t odd 16 1 153.3.p.a 8
340.bc odd 16 1 425.3.t.d 8
340.bg even 16 1 425.3.u.a 8
340.bj odd 16 1 425.3.t.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.3.e.b ✓ 8 4.b odd 2 1
17.3.e.b ✓ 8 68.i even 16 1
153.3.p.a 8 12.b even 2 1
153.3.p.a 8 204.t odd 16 1
272.3.bh.b 8 1.a even 1 1 trivial
272.3.bh.b 8 17.e odd 16 1 inner
289.3.e.a 8 68.g odd 8 1
289.3.e.a 8 68.i even 16 1
289.3.e.e 8 68.g odd 8 1
289.3.e.e 8 68.i even 16 1
289.3.e.f 8 68.f odd 4 1
289.3.e.f 8 68.i even 16 1
289.3.e.g 8 68.d odd 2 1
289.3.e.g 8 68.i even 16 1
289.3.e.h 8 68.f odd 4 1
289.3.e.h 8 68.i even 16 1
289.3.e.j 8 68.g odd 8 1
289.3.e.j 8 68.i even 16 1
289.3.e.n 8 68.g odd 8 1
289.3.e.n 8 68.i even 16 1
425.3.t.b 8 20.e even 4 1
425.3.t.b 8 340.bj odd 16 1
425.3.t.d 8 20.e even 4 1
425.3.t.d 8 340.bc odd 16 1
425.3.u.a 8 20.d odd 2 1
425.3.u.a 8 340.bg even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 4T_{3}^{6} + 128T_{3}^{5} + 172T_{3}^{4} - 560T_{3}^{3} + 912T_{3}^{2} - 1088T_{3} + 2312 \) acting on \(S_{3}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + \cdots + 2312 \) Copy content Toggle raw display
$5$ \( T^{8} + 24 T^{7} + \cdots + 4418 \) Copy content Toggle raw display
$7$ \( T^{8} - 16 T^{7} + \cdots + 7688 \) Copy content Toggle raw display
$11$ \( T^{8} + 40 T^{7} + \cdots + 2221832 \) Copy content Toggle raw display
$13$ \( T^{8} + 784 T^{5} + \cdots + 16273156 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 34090452496 \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} + \cdots + 564614408 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 55846825218 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 182700453128 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 368317129538 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 126445152962 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 6626611216 \) Copy content Toggle raw display
$47$ \( T^{8} - 80 T^{7} + \cdots + 298321984 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 41458660996 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 2347596304 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 22880163635522 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 883915868224 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 53847249369608 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 18825456624578 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 376288804594952 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 128947789048324 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 18\!\cdots\!42 \) Copy content Toggle raw display
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