Properties

Label 96.3.e.a
Level $96$
Weight $3$
Character orbit 96.e
Analytic conductor $2.616$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,3,Mod(65,96)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 96.e (of order \(2\), degree \(1\), 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: \(2.61581053786\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + 2 \beta_1) q^{7} + ( - 3 \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + 2 \beta_1) q^{7} + ( - 3 \beta_{2} - 3) q^{9} + (\beta_{3} + 4 \beta_1) q^{11} - 6 q^{13} - \beta_{3} q^{15} + 8 \beta_{2} q^{17} + (\beta_{3} - 2 \beta_1) q^{19} + ( - 9 \beta_{2} + 18) q^{21} + ( - 2 \beta_{3} - 8 \beta_1) q^{23} + 17 q^{25} + (3 \beta_{3} - 3 \beta_1) q^{27} + 11 \beta_{2} q^{29} + (3 \beta_{3} - 6 \beta_1) q^{31} + ( - 9 \beta_{2} - 36) q^{33} + ( - 2 \beta_{3} - 8 \beta_1) q^{35} - 38 q^{37} - 6 \beta_1 q^{39} - 2 \beta_{2} q^{41} + (\beta_{3} - 2 \beta_1) q^{43} + ( - 3 \beta_{2} + 24) q^{45} + (4 \beta_{3} + 16 \beta_1) q^{47} + 59 q^{49} - 8 \beta_{3} q^{51} + 5 \beta_{2} q^{53} + ( - 4 \beta_{3} + 8 \beta_1) q^{55} + (9 \beta_{2} - 18) q^{57} + (\beta_{3} + 4 \beta_1) q^{59} - 22 q^{61} + (9 \beta_{3} + 18 \beta_1) q^{63} - 6 \beta_{2} q^{65} + ( - 11 \beta_{3} + 22 \beta_1) q^{67} + (18 \beta_{2} + 72) q^{69} + (2 \beta_{3} + 8 \beta_1) q^{71} - 30 q^{73} + 17 \beta_1 q^{75} - 54 \beta_{2} q^{77} + (3 \beta_{3} - 6 \beta_1) q^{79} + (18 \beta_{2} - 63) q^{81} + ( - 5 \beta_{3} - 20 \beta_1) q^{83} - 64 q^{85} - 11 \beta_{3} q^{87} - 2 \beta_{2} q^{89} + (6 \beta_{3} - 12 \beta_1) q^{91} + (27 \beta_{2} - 54) q^{93} + (2 \beta_{3} + 8 \beta_1) q^{95} + 90 q^{97} + (9 \beta_{3} - 36 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} - 24 q^{13} + 72 q^{21} + 68 q^{25} - 144 q^{33} - 152 q^{37} + 96 q^{45} + 236 q^{49} - 72 q^{57} - 88 q^{61} + 288 q^{69} - 120 q^{73} - 252 q^{81} - 256 q^{85} - 216 q^{93} + 360 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + \nu^{2} + 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} - 4\nu^{2} + 10\nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 3\beta_{2} + 4\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta _1 - 12 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} - 5\beta_{2} - 4\beta_1 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.93185i
1.93185i
0.517638i
0.517638i
0 −1.73205 2.44949i 0 2.82843i 0 −10.3923 0 −3.00000 + 8.48528i 0
65.2 0 −1.73205 + 2.44949i 0 2.82843i 0 −10.3923 0 −3.00000 8.48528i 0
65.3 0 1.73205 2.44949i 0 2.82843i 0 10.3923 0 −3.00000 8.48528i 0
65.4 0 1.73205 + 2.44949i 0 2.82843i 0 10.3923 0 −3.00000 + 8.48528i 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
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.3.e.a 4
3.b odd 2 1 inner 96.3.e.a 4
4.b odd 2 1 inner 96.3.e.a 4
8.b even 2 1 192.3.e.e 4
8.d odd 2 1 192.3.e.e 4
12.b even 2 1 inner 96.3.e.a 4
16.e even 4 2 768.3.h.f 8
16.f odd 4 2 768.3.h.f 8
24.f even 2 1 192.3.e.e 4
24.h odd 2 1 192.3.e.e 4
48.i odd 4 2 768.3.h.f 8
48.k even 4 2 768.3.h.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.e.a 4 1.a even 1 1 trivial
96.3.e.a 4 3.b odd 2 1 inner
96.3.e.a 4 4.b odd 2 1 inner
96.3.e.a 4 12.b even 2 1 inner
192.3.e.e 4 8.b even 2 1
192.3.e.e 4 8.d odd 2 1
192.3.e.e 4 24.f even 2 1
192.3.e.e 4 24.h odd 2 1
768.3.h.f 8 16.e even 4 2
768.3.h.f 8 16.f odd 4 2
768.3.h.f 8 48.i odd 4 2
768.3.h.f 8 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8 \) acting on \(S_{3}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$13$ \( (T + 6)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 512)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 864)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 968)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 972)^{2} \) Copy content Toggle raw display
$37$ \( (T + 38)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 3456)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 200)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$61$ \( (T + 22)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 13068)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 864)^{2} \) Copy content Toggle raw display
$73$ \( (T + 30)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 972)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 5400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$97$ \( (T - 90)^{4} \) Copy content Toggle raw display
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