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Properties

Label 384.3.h.e
Level $384$
Weight $3$
Character orbit 384.h
Analytic conductor $10.463$
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] = [384,3,Mod(65,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.h (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: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - 4 q^{5} + ( - 2 \beta_{3} - 2 \beta_1) q^{7} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - 4 q^{5} + ( - 2 \beta_{3} - 2 \beta_1) q^{7} + ( - \beta_{2} + 1) q^{9} + ( - \beta_{3} - \beta_1) q^{11} + 2 \beta_{2} q^{13} - 4 \beta_{3} q^{15} + 2 \beta_{2} q^{17} + ( - 5 \beta_{3} + 5 \beta_1) q^{19} + (2 \beta_{2} - 20) q^{21} + (4 \beta_{3} - 4 \beta_1) q^{23} - 9 q^{25} + (2 \beta_{3} - 9 \beta_1) q^{27} - 52 q^{29} + (6 \beta_{3} + 6 \beta_1) q^{31} + (\beta_{2} - 10) q^{33} + (8 \beta_{3} + 8 \beta_1) q^{35} + 6 \beta_{2} q^{37} + ( - 2 \beta_{3} + 18 \beta_1) q^{39} - 4 \beta_{2} q^{41} + (9 \beta_{3} - 9 \beta_1) q^{43} + (4 \beta_{2} - 4) q^{45} + (16 \beta_{3} - 16 \beta_1) q^{47} + 31 q^{49} + ( - 2 \beta_{3} + 18 \beta_1) q^{51} - 20 q^{53} + (4 \beta_{3} + 4 \beta_1) q^{55} + (5 \beta_{2} + 40) q^{57} + ( - 23 \beta_{3} - 23 \beta_1) q^{59} + 2 \beta_{2} q^{61} + ( - 22 \beta_{3} + 18 \beta_1) q^{63} - 8 \beta_{2} q^{65} + ( - 11 \beta_{3} + 11 \beta_1) q^{67} + ( - 4 \beta_{2} - 32) q^{69} + (20 \beta_{3} - 20 \beta_1) q^{71} - 50 q^{73} - 9 \beta_{3} q^{75} + 40 q^{77} + ( - 18 \beta_{3} - 18 \beta_1) q^{79} + ( - 2 \beta_{2} - 79) q^{81} + (23 \beta_{3} + 23 \beta_1) q^{83} - 8 \beta_{2} q^{85} - 52 \beta_{3} q^{87} + 18 \beta_{2} q^{89} + (40 \beta_{3} - 40 \beta_1) q^{91} + ( - 6 \beta_{2} + 60) q^{93} + (20 \beta_{3} - 20 \beta_1) q^{95} + 50 q^{97} + ( - 11 \beta_{3} + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{5} + 4 q^{9} - 80 q^{21} - 36 q^{25} - 208 q^{29} - 40 q^{33} - 16 q^{45} + 124 q^{49} - 80 q^{53} + 160 q^{57} - 128 q^{69} - 200 q^{73} + 160 q^{77} - 316 q^{81} + 240 q^{93} + 200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{3} + 2\nu^{2} + 4\nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{3} + 16\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{3} + 2\nu^{2} - 4\nu + 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} - \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
− 1.61803i
1.61803i
0.618034i
− 0.618034i
0 −2.23607 − 2.00000i 0 −4.00000 0 8.94427 0 1.00000 + 8.94427i 0
65.2 0 −2.23607 + 2.00000i 0 −4.00000 0 8.94427 0 1.00000 − 8.94427i 0
65.3 0 2.23607 − 2.00000i 0 −4.00000 0 −8.94427 0 1.00000 − 8.94427i 0
65.4 0 2.23607 + 2.00000i 0 −4.00000 0 −8.94427 0 1.00000 + 8.94427i 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
4.b odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.h.e ✓ 4
3.b odd 2 1 384.3.h.f yes 4
4.b odd 2 1 inner 384.3.h.e ✓ 4
8.b even 2 1 384.3.h.f yes 4
8.d odd 2 1 384.3.h.f yes 4
12.b even 2 1 384.3.h.f yes 4
16.e even 4 1 768.3.e.h 4
16.e even 4 1 768.3.e.m 4
16.f odd 4 1 768.3.e.h 4
16.f odd 4 1 768.3.e.m 4
24.f even 2 1 inner 384.3.h.e ✓ 4
24.h odd 2 1 inner 384.3.h.e ✓ 4
48.i odd 4 1 768.3.e.h 4
48.i odd 4 1 768.3.e.m 4
48.k even 4 1 768.3.e.h 4
48.k even 4 1 768.3.e.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.e ✓ 4 1.a even 1 1 trivial
384.3.h.e ✓ 4 4.b odd 2 1 inner
384.3.h.e ✓ 4 24.f even 2 1 inner
384.3.h.e ✓ 4 24.h odd 2 1 inner
384.3.h.f yes 4 3.b odd 2 1
384.3.h.f yes 4 8.b even 2 1
384.3.h.f yes 4 8.d odd 2 1
384.3.h.f yes 4 12.b even 2 1
768.3.e.h 4 16.e even 4 1
768.3.e.h 4 16.f odd 4 1
768.3.e.h 4 48.i odd 4 1
768.3.e.h 4 48.k even 4 1
768.3.e.m 4 16.e even 4 1
768.3.e.m 4 16.f odd 4 1
768.3.e.m 4 48.i odd 4 1
768.3.e.m 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\):

\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2T^{2} + 81 \) Copy content Toggle raw display
$5$ \( (T + 4)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T + 52)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 720)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2880)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1280)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1296)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4096)^{2} \) Copy content Toggle raw display
$53$ \( (T + 20)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 10580)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1936)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 6400)^{2} \) Copy content Toggle raw display
$73$ \( (T + 50)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 6480)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 10580)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 25920)^{2} \) Copy content Toggle raw display
$97$ \( (T - 50)^{4} \) Copy content Toggle raw display
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