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Properties

Label 153.3.p.a
Level $153$
Weight $3$
Character orbit 153.p
Analytic conductor $4.169$
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] = [153,3,Mod(10,153)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(153, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 3])) 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("153.10"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 153.p (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8,24,0,-16,-16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16894804471\)
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: \(\Z[a_1, \ldots, a_{4}]\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{16}^{7} + \cdots + \zeta_{16}^{2}) q^{2} + (\zeta_{16}^{6} + \zeta_{16}^{4} + \cdots - 1) q^{4} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}^{6} + \cdots + 3) q^{5} + ( - \zeta_{16}^{7} + 2 \zeta_{16}^{6} + \cdots - 2) q^{7}+ \cdots + ( - 40 \zeta_{16}^{7} + 40 \zeta_{16}^{5} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 24 q^{5} - 16 q^{7} - 16 q^{8} - 40 q^{11} - 16 q^{14} + 16 q^{17} - 32 q^{19} + 40 q^{20} + 8 q^{23} + 16 q^{25} + 80 q^{28} - 24 q^{29} + 32 q^{31} + 24 q^{32} + 64 q^{34} - 80 q^{35}+ \cdots - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\)
\(\chi(n)\) \(\zeta_{16}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.923880 + 0.382683i
0.382683 − 0.923880i
−0.382683 + 0.923880i
−0.923880 − 0.382683i
0.923880 − 0.382683i
−0.382683 − 0.923880i
−0.324423 + 0.783227i 0 2.32023 + 2.32023i 6.70292 − 1.33329i 0 0.886687 + 0.176373i −5.70292 + 2.36223i 0 −1.13031 + 5.68246i
28.1 −0.216773 − 0.0897902i 0 −2.78950 − 2.78950i 0.286621 + 0.191514i 0 −5.96908 + 3.98841i 0.713379 + 1.72225i 0 −0.0449356 − 0.0672509i
37.1 1.63099 − 0.675577i 0 −0.624715 + 0.624715i 4.29916 + 6.43416i 0 −2.27356 + 3.40262i −3.29916 + 7.96489i 0 11.3586 + 7.58960i
46.1 −0.324423 − 0.783227i 0 2.32023 − 2.32023i 6.70292 + 1.33329i 0 0.886687 − 0.176373i −5.70292 − 2.36223i 0 −1.13031 − 5.68246i
73.1 −1.08979 − 2.63099i 0 −2.90602 + 2.90602i 0.711297 − 3.57593i 0 −0.644047 − 3.23784i 0.288703 + 0.119585i 0 −10.1834 + 2.02560i
82.1 −0.216773 + 0.0897902i 0 −2.78950 + 2.78950i 0.286621 − 0.191514i 0 −5.96908 − 3.98841i 0.713379 − 1.72225i 0 −0.0449356 + 0.0672509i
91.1 1.63099 + 0.675577i 0 −0.624715 − 0.624715i 4.29916 − 6.43416i 0 −2.27356 − 3.40262i −3.29916 − 7.96489i 0 11.3586 − 7.58960i
109.1 −1.08979 + 2.63099i 0 −2.90602 − 2.90602i 0.711297 + 3.57593i 0 −0.644047 + 3.23784i 0.288703 − 0.119585i 0 −10.1834 − 2.02560i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.1
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 153.3.p.a 8
3.b odd 2 1 17.3.e.b ✓ 8
12.b even 2 1 272.3.bh.b 8
15.d odd 2 1 425.3.u.a 8
15.e even 4 1 425.3.t.b 8
15.e even 4 1 425.3.t.d 8
17.e odd 16 1 inner 153.3.p.a 8
51.c odd 2 1 289.3.e.g 8
51.f odd 4 1 289.3.e.f 8
51.f odd 4 1 289.3.e.h 8
51.g odd 8 1 289.3.e.a 8
51.g odd 8 1 289.3.e.e 8
51.g odd 8 1 289.3.e.j 8
51.g odd 8 1 289.3.e.n 8
51.i even 16 1 17.3.e.b ✓ 8
51.i even 16 1 289.3.e.a 8
51.i even 16 1 289.3.e.e 8
51.i even 16 1 289.3.e.f 8
51.i even 16 1 289.3.e.g 8
51.i even 16 1 289.3.e.h 8
51.i even 16 1 289.3.e.j 8
51.i even 16 1 289.3.e.n 8
204.t odd 16 1 272.3.bh.b 8
255.bc odd 16 1 425.3.t.d 8
255.be even 16 1 425.3.u.a 8
255.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 3.b odd 2 1
17.3.e.b ✓ 8 51.i even 16 1
153.3.p.a 8 1.a even 1 1 trivial
153.3.p.a 8 17.e odd 16 1 inner
272.3.bh.b 8 12.b even 2 1
272.3.bh.b 8 204.t odd 16 1
289.3.e.a 8 51.g odd 8 1
289.3.e.a 8 51.i even 16 1
289.3.e.e 8 51.g odd 8 1
289.3.e.e 8 51.i even 16 1
289.3.e.f 8 51.f odd 4 1
289.3.e.f 8 51.i even 16 1
289.3.e.g 8 51.c odd 2 1
289.3.e.g 8 51.i even 16 1
289.3.e.h 8 51.f odd 4 1
289.3.e.h 8 51.i even 16 1
289.3.e.j 8 51.g odd 8 1
289.3.e.j 8 51.i even 16 1
289.3.e.n 8 51.g odd 8 1
289.3.e.n 8 51.i even 16 1
425.3.t.b 8 15.e even 4 1
425.3.t.b 8 255.bj odd 16 1
425.3.t.d 8 15.e even 4 1
425.3.t.d 8 255.bc odd 16 1
425.3.u.a 8 15.d odd 2 1
425.3.u.a 8 255.be even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{6} - 16T_{2}^{5} + 8T_{2}^{4} + 8T_{2}^{3} + 20T_{2}^{2} + 8T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) 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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