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Properties

Label 96.8.f.a
Level $96$
Weight $8$
Character orbit 96.f
Analytic conductor $29.989$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [96,8,Mod(47,96)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 96.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.9889624465\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 13 \beta + 43) q^{3} + ( - 1118 \beta + 1511) q^{9} + 362 \beta q^{11} - 23972 \beta q^{17} + 59722 q^{19} - 78125 q^{25} + ( - 67717 \beta + 35905) q^{27} + (15566 \beta + 9412) q^{33} - 601208 \beta q^{41} + \cdots + (546982 \beta + 809432) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 86 q^{3} + 3022 q^{9} + 119444 q^{19} - 156250 q^{25} + 71810 q^{27} + 18824 q^{33} + 441020 q^{43} + 1647086 q^{49} - 1246544 q^{51} + 5136092 q^{57} + 7702604 q^{67} - 9731228 q^{73} - 6718750 q^{75}+ \cdots + 1618864 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.41421i
− 1.41421i
0 43.0000 − 18.3848i 0 0 0 0 0 1511.00 − 1581.09i 0
47.2 0 43.0000 + 18.3848i 0 0 0 0 0 1511.00 + 1581.09i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
3.b 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.a 2
3.b odd 2 1 inner 96.8.f.a 2
4.b odd 2 1 24.8.f.a ✓ 2
8.b even 2 1 24.8.f.a ✓ 2
8.d odd 2 1 CM 96.8.f.a 2
12.b even 2 1 24.8.f.a ✓ 2
24.f even 2 1 inner 96.8.f.a 2
24.h odd 2 1 24.8.f.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.f.a ✓ 2 4.b odd 2 1
24.8.f.a ✓ 2 8.b even 2 1
24.8.f.a ✓ 2 12.b even 2 1
24.8.f.a ✓ 2 24.h odd 2 1
96.8.f.a 2 1.a even 1 1 trivial
96.8.f.a 2 3.b odd 2 1 inner
96.8.f.a 2 8.d odd 2 1 CM
96.8.f.a 2 24.f even 2 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 86T + 2187 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 262088 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1149313568 \) Copy content Toggle raw display
$19$ \( (T - 59722)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 722902118528 \) Copy content Toggle raw display
$43$ \( (T - 220510)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8891797521800 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T - 3851302)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 4865614)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 85418357742152 \) Copy content Toggle raw display
$89$ \( T^{2} + 126896341340192 \) Copy content Toggle raw display
$97$ \( (T + 9938890)^{2} \) Copy content Toggle raw display
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