↑

Properties

Label 153.3.p.b
Level $153$
Weight $3$
Character orbit 153.p
Analytic conductor $4.169$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

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 = [8,8,0,0,-16,0,8,24,0,16] 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.

\(f(q)\) \(=\) \( q + (\zeta_{16}^{7} - \zeta_{16}^{6} + \cdots + 1) q^{2} + (2 \zeta_{16}^{7} + \cdots - 2 \zeta_{16}) q^{4} + ( - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{4} + \cdots - 2) q^{5} + (\zeta_{16}^{7} - 5 \zeta_{16}^{6} + \cdots + 1) q^{7}+ \cdots + ( - 9 \zeta_{16}^{7} + 9 \zeta_{16}^{5} + \cdots + 15) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 16 q^{5} + 8 q^{7} + 24 q^{8} + 16 q^{10} + 8 q^{11} + 16 q^{13} - 8 q^{14} + 80 q^{20} - 104 q^{22} + 56 q^{23} + 64 q^{25} - 176 q^{26} + 152 q^{28} - 48 q^{29} + 24 q^{31} - 88 q^{32}+ \cdots + 120 q^{98}+O(q^{100}) \) 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
1.15851 − 2.79690i 0 −3.65205 − 3.65205i −7.87510 + 1.56645i 0 −7.70353 − 1.53233i −3.25778 + 1.34942i 0 −4.74219 + 23.8406i
28.1 3.23044 + 1.33809i 0 5.81684 + 5.81684i −0.268761 − 0.179580i 0 5.55967 − 3.71485i 5.65512 + 13.6527i 0 −0.627922 − 0.939752i
37.1 −1.23044 + 0.509666i 0 −1.57420 + 1.57420i −0.902812 − 1.35115i 0 4.92562 − 7.37170i 3.17331 − 7.66104i 0 1.79949 + 1.20238i
46.1 1.15851 + 2.79690i 0 −3.65205 + 3.65205i −7.87510 − 1.56645i 0 −7.70353 + 1.53233i −3.25778 − 1.34942i 0 −4.74219 − 23.8406i
73.1 0.841487 + 2.03153i 0 −0.590587 + 0.590587i 1.04667 − 5.26197i 0 1.21824 + 6.12453i 6.42935 + 2.66313i 0 11.5706 − 2.30154i
82.1 3.23044 − 1.33809i 0 5.81684 − 5.81684i −0.268761 + 0.179580i 0 5.55967 + 3.71485i 5.65512 − 13.6527i 0 −0.627922 + 0.939752i
91.1 −1.23044 − 0.509666i 0 −1.57420 − 1.57420i −0.902812 + 1.35115i 0 4.92562 + 7.37170i 3.17331 + 7.66104i 0 1.79949 − 1.20238i
109.1 0.841487 − 2.03153i 0 −0.590587 − 0.590587i 1.04667 + 5.26197i 0 1.21824 − 6.12453i 6.42935 − 2.66313i 0 11.5706 + 2.30154i
\(n\): 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.b 8
3.b odd 2 1 17.3.e.a ✓ 8
12.b even 2 1 272.3.bh.c 8
15.d odd 2 1 425.3.u.b 8
15.e even 4 1 425.3.t.a 8
15.e even 4 1 425.3.t.c 8
17.e odd 16 1 inner 153.3.p.b 8
51.c odd 2 1 289.3.e.c 8
51.f odd 4 1 289.3.e.i 8
51.f odd 4 1 289.3.e.m 8
51.g odd 8 1 289.3.e.b 8
51.g odd 8 1 289.3.e.d 8
51.g odd 8 1 289.3.e.k 8
51.g odd 8 1 289.3.e.l 8
51.i even 16 1 17.3.e.a ✓ 8
51.i even 16 1 289.3.e.b 8
51.i even 16 1 289.3.e.c 8
51.i even 16 1 289.3.e.d 8
51.i even 16 1 289.3.e.i 8
51.i even 16 1 289.3.e.k 8
51.i even 16 1 289.3.e.l 8
51.i even 16 1 289.3.e.m 8
204.t odd 16 1 272.3.bh.c 8
255.bc odd 16 1 425.3.t.a 8
255.be even 16 1 425.3.u.b 8
255.bj odd 16 1 425.3.t.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.3.e.a ✓ 8 3.b odd 2 1
17.3.e.a ✓ 8 51.i even 16 1
153.3.p.b 8 1.a even 1 1 trivial
153.3.p.b 8 17.e odd 16 1 inner
272.3.bh.c 8 12.b even 2 1
272.3.bh.c 8 204.t odd 16 1
289.3.e.b 8 51.g odd 8 1
289.3.e.b 8 51.i even 16 1
289.3.e.c 8 51.c odd 2 1
289.3.e.c 8 51.i even 16 1
289.3.e.d 8 51.g odd 8 1
289.3.e.d 8 51.i even 16 1
289.3.e.i 8 51.f odd 4 1
289.3.e.i 8 51.i even 16 1
289.3.e.k 8 51.g odd 8 1
289.3.e.k 8 51.i even 16 1
289.3.e.l 8 51.g odd 8 1
289.3.e.l 8 51.i even 16 1
289.3.e.m 8 51.f odd 4 1
289.3.e.m 8 51.i even 16 1
425.3.t.a 8 15.e even 4 1
425.3.t.a 8 255.bc odd 16 1
425.3.t.c 8 15.e even 4 1
425.3.t.c 8 255.bj odd 16 1
425.3.u.b 8 15.d odd 2 1
425.3.u.b 8 255.be even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 8T_{2}^{7} + 32T_{2}^{6} - 72T_{2}^{5} + 64T_{2}^{4} + 120T_{2}^{3} - 192T_{2}^{2} + 248T_{2} + 961 \) acting on \(S_{3}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 8 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 16 T^{7} + \cdots + 512 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 8454272 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + \cdots + 27572738 \) Copy content Toggle raw display
$13$ \( T^{8} - 16 T^{7} + \cdots + 9048064 \) Copy content Toggle raw display
$17$ \( T^{8} + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{8} + 368 T^{6} + \cdots + 929296 \) Copy content Toggle raw display
$23$ \( T^{8} - 56 T^{7} + \cdots + 859299968 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 4240836608 \) Copy content Toggle raw display
$31$ \( T^{8} - 24 T^{7} + \cdots + 14536832 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 120057840128 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 59058658562 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 6201305218564 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 2259754549504 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 26754490624 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 2675455605124 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 106680004247552 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 83643718741636 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 71246079996032 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 51301810562 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 3948319764608 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 117588822570244 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 21682310986562 \) Copy content Toggle raw display
show more
show less