Properties

Label 96.8.f.b
Level $96$
Weight $8$
Character orbit 96.f
Analytic conductor $29.989$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,8,Mod(47,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.47");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 96.f (of order \(2\), degree \(1\), 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: \(29.9889624465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta_1 + 9) q^{3} - 5 \beta_{2} q^{5} + 31 \beta_{3} q^{7} + (162 \beta_1 - 2025) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_1 + 9) q^{3} - 5 \beta_{2} q^{5} + 31 \beta_{3} q^{7} + (162 \beta_1 - 2025) q^{9} + 350 \beta_1 q^{11} - 203 \beta_{3} q^{13} + ( - 90 \beta_{3} - 45 \beta_{2}) q^{15} + 5404 \beta_1 q^{17} - 11570 q^{19} + (279 \beta_{3} - 3627 \beta_{2}) q^{21} + 2834 \beta_{2} q^{23} - 68525 q^{25} + ( - 16767 \beta_1 - 56133) q^{27} + 2795 \beta_{2} q^{29} - 1435 \beta_{3} q^{31} + (3150 \beta_1 - 81900) q^{33} - 29760 \beta_1 q^{35} - 5661 \beta_{3} q^{37} + ( - 1827 \beta_{3} + 23751 \beta_{2}) q^{39} - 79160 \beta_1 q^{41} - 495062 q^{43} + ( - 1620 \beta_{3} + 10125 \beta_{2}) q^{45} - 57968 \beta_{2} q^{47} - 1575113 q^{49} + (48636 \beta_1 - 1264536) q^{51} + 29211 \beta_{2} q^{53} - 3500 \beta_{3} q^{55} + ( - 104130 \beta_1 - 104130) q^{57} + 282506 \beta_1 q^{59} - 35755 \beta_{3} q^{61} + ( - 62775 \beta_{3} - 65286 \beta_{2}) q^{63} + 194880 \beta_1 q^{65} - 1400126 q^{67} + (51012 \beta_{3} + 25506 \beta_{2}) q^{69} + 182154 \beta_{2} q^{71} - 2223598 q^{73} + ( - 616725 \beta_1 - 616725) q^{75} - 141050 \beta_{2} q^{77} + 111901 \beta_{3} q^{79} + ( - 656100 \beta_1 + 3418281) q^{81} - 590962 \beta_1 q^{83} - 54040 \beta_{3} q^{85} + (50310 \beta_{3} + 25155 \beta_{2}) q^{87} + 1146364 \beta_1 q^{89} + 15707328 q^{91} + ( - 12915 \beta_{3} + 167895 \beta_{2}) q^{93} + 57850 \beta_{2} q^{95} + 6867926 q^{97} + ( - 708750 \beta_1 - 1474200) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{3} - 8100 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{3} - 8100 q^{9} - 46280 q^{19} - 274100 q^{25} - 224532 q^{27} - 327600 q^{33} - 1980248 q^{43} - 6300452 q^{49} - 5058144 q^{51} - 416520 q^{57} - 5600504 q^{67} - 8894392 q^{73} - 2466900 q^{75} + 13673124 q^{81} + 62829312 q^{91} + 27471704 q^{97} - 5896800 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 10x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 18\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} - 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} + 40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 8\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 40 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{2} - 8\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.22474 2.54951i
−1.22474 2.54951i
1.22474 + 2.54951i
−1.22474 + 2.54951i
0 9.00000 45.8912i 0 −97.9796 0 1548.76i 0 −2025.00 826.041i 0
47.2 0 9.00000 45.8912i 0 97.9796 0 1548.76i 0 −2025.00 826.041i 0
47.3 0 9.00000 + 45.8912i 0 −97.9796 0 1548.76i 0 −2025.00 + 826.041i 0
47.4 0 9.00000 + 45.8912i 0 97.9796 0 1548.76i 0 −2025.00 + 826.041i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.8.f.b 4
3.b odd 2 1 inner 96.8.f.b 4
4.b odd 2 1 24.8.f.b 4
8.b even 2 1 24.8.f.b 4
8.d odd 2 1 inner 96.8.f.b 4
12.b even 2 1 24.8.f.b 4
24.f even 2 1 inner 96.8.f.b 4
24.h odd 2 1 24.8.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.f.b 4 4.b odd 2 1
24.8.f.b 4 8.b even 2 1
24.8.f.b 4 12.b even 2 1
24.8.f.b 4 24.h odd 2 1
96.8.f.b 4 1.a even 1 1 trivial
96.8.f.b 4 3.b odd 2 1 inner
96.8.f.b 4 8.d odd 2 1 inner
96.8.f.b 4 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 9600 \) acting on \(S_{8}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 18 T + 2187)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 9600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2398656)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3185000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 102857664)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 759283616)^{2} \) Copy content Toggle raw display
$19$ \( (T + 11570)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 3084117504)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2999817600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5139825600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 79989114816)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 162923945600)^{2} \) Copy content Toggle raw display
$43$ \( (T + 495062)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 1290350985216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 327660488064)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2075050640936)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3190936382400)^{2} \) Copy content Toggle raw display
$67$ \( (T + 1400126)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12741150610944)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2223598)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 31254497167296)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 9080138221544)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 34167910932896)^{2} \) Copy content Toggle raw display
$97$ \( (T - 6867926)^{4} \) Copy content Toggle raw display
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