Properties

Label 96.4.f.a
Level $96$
Weight $4$
Character orbit 96.f
Analytic conductor $5.664$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.47");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(5.66418336055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

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

\(f(q)\) \(=\) \( q + (\beta - 5) q^{3} + ( - 10 \beta + 23) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 5) q^{3} + ( - 10 \beta + 23) q^{9} - 50 \beta q^{11} - 76 \beta q^{17} + 106 q^{19} - 125 q^{25} + (73 \beta - 95) q^{27} + (250 \beta + 100) q^{33} - 40 \beta q^{41} - 290 q^{43} + 343 q^{49} + (380 \beta + 152) q^{51} + (106 \beta - 530) q^{57} - 230 \beta q^{59} + 70 q^{67} - 430 q^{73} + ( - 125 \beta + 625) q^{75} + ( - 460 \beta + 329) q^{81} - 482 \beta q^{83} - 940 \beta q^{89} + 1910 q^{97} + ( - 1150 \beta - 1000) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{3} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{3} + 46 q^{9} + 212 q^{19} - 250 q^{25} - 190 q^{27} + 200 q^{33} - 580 q^{43} + 686 q^{49} + 304 q^{51} - 1060 q^{57} + 140 q^{67} - 860 q^{73} + 1250 q^{75} + 658 q^{81} + 3820 q^{97} - 2000 q^{99}+O(q^{100}) \) 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.

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
1.41421i
1.41421i
0 −5.00000 1.41421i 0 0 0 0 0 23.0000 + 14.1421i 0
47.2 0 −5.00000 + 1.41421i 0 0 0 0 0 23.0000 14.1421i 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 96.4.f.a 2
3.b odd 2 1 inner 96.4.f.a 2
4.b odd 2 1 24.4.f.a 2
8.b even 2 1 24.4.f.a 2
8.d odd 2 1 CM 96.4.f.a 2
12.b even 2 1 24.4.f.a 2
16.e even 4 2 768.4.c.l 4
16.f odd 4 2 768.4.c.l 4
24.f even 2 1 inner 96.4.f.a 2
24.h odd 2 1 24.4.f.a 2
48.i odd 4 2 768.4.c.l 4
48.k even 4 2 768.4.c.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.f.a 2 4.b odd 2 1
24.4.f.a 2 8.b even 2 1
24.4.f.a 2 12.b even 2 1
24.4.f.a 2 24.h odd 2 1
96.4.f.a 2 1.a even 1 1 trivial
96.4.f.a 2 3.b odd 2 1 inner
96.4.f.a 2 8.d odd 2 1 CM
96.4.f.a 2 24.f even 2 1 inner
768.4.c.l 4 16.e even 4 2
768.4.c.l 4 16.f odd 4 2
768.4.c.l 4 48.i odd 4 2
768.4.c.l 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{4}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 10T + 27 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5000 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 11552 \) Copy content Toggle raw display
$19$ \( (T - 106)^{2} \) 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
$41$ \( T^{2} + 3200 \) Copy content Toggle raw display
$43$ \( (T + 290)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 105800 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T - 70)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 430)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 464648 \) Copy content Toggle raw display
$89$ \( T^{2} + 1767200 \) Copy content Toggle raw display
$97$ \( (T - 1910)^{2} \) Copy content Toggle raw display
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