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Properties

Label 384.2.c.c
Level $384$
Weight $2$
Character orbit 384.c
Analytic conductor $3.066$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,2,Mod(383,384)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.383");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.c (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: \(3.06625543762\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{3} + \beta_1 - 2) q^{11} + (2 \beta_{3} - 2 \beta_1 - 2) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{15} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{17} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{19} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{21} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{23} + ( - 2 \beta_{3} + 2 \beta_1 + 1) q^{25} + (\beta_{3} + 2 \beta_{2} + 3) q^{27} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{29} + (\beta_{3} - \beta_{2} + \beta_1) q^{31} + ( - \beta_{3} + \beta_{2} - 3 \beta_1) q^{33} - 4 q^{35} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{37} + (2 \beta_{3} - 2 \beta_{2} - 6) q^{39} + (2 \beta_{3} + 2 \beta_1) q^{41} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{43} + (\beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{45} + 8 q^{47} + (2 \beta_{3} - 2 \beta_1 - 1) q^{49} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{51}+ \cdots + (5 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 12 q^{11} - 12 q^{15} - 8 q^{21} + 8 q^{23} - 4 q^{25} + 14 q^{27} + 4 q^{33} - 16 q^{35} - 16 q^{37} - 20 q^{39} + 8 q^{45} + 32 q^{47} + 4 q^{49} + 16 q^{51} - 4 q^{57} - 4 q^{59} + 16 q^{61} - 8 q^{63} + 24 q^{69} + 8 q^{71} - 8 q^{73} + 18 q^{75} + 4 q^{81} - 20 q^{83} + 32 q^{85} - 4 q^{87} - 16 q^{93} - 8 q^{95} - 16 q^{97} + 20 q^{99}+O(q^{100}) \) 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

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

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + \beta_{2} - 2\beta_1 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
− 1.61803i
1.61803i
− 0.618034i
0.618034i
0 −0.618034 − 1.61803i 0 − 3.23607i 0 − 1.23607i 0 −2.23607 + 2.00000i 0
383.2 0 −0.618034 + 1.61803i 0 3.23607i 0 1.23607i 0 −2.23607 − 2.00000i 0
383.3 0 1.61803 − 0.618034i 0 − 1.23607i 0 − 3.23607i 0 2.23607 − 2.00000i 0
383.4 0 1.61803 + 0.618034i 0 1.23607i 0 3.23607i 0 2.23607 + 2.00000i 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
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 384.2.c.c yes 4
3.b odd 2 1 384.2.c.b yes 4
4.b odd 2 1 384.2.c.b yes 4
8.b even 2 1 384.2.c.a ✓ 4
8.d odd 2 1 384.2.c.d yes 4
12.b even 2 1 inner 384.2.c.c yes 4
16.e even 4 1 768.2.f.c 4
16.e even 4 1 768.2.f.e 4
16.f odd 4 1 768.2.f.b 4
16.f odd 4 1 768.2.f.f 4
24.f even 2 1 384.2.c.a ✓ 4
24.h odd 2 1 384.2.c.d yes 4
48.i odd 4 1 768.2.f.b 4
48.i odd 4 1 768.2.f.f 4
48.k even 4 1 768.2.f.c 4
48.k even 4 1 768.2.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.c.a ✓ 4 8.b even 2 1
384.2.c.a ✓ 4 24.f even 2 1
384.2.c.b yes 4 3.b odd 2 1
384.2.c.b yes 4 4.b odd 2 1
384.2.c.c yes 4 1.a even 1 1 trivial
384.2.c.c yes 4 12.b even 2 1 inner
384.2.c.d yes 4 8.d odd 2 1
384.2.c.d yes 4 24.h odd 2 1
768.2.f.b 4 16.f odd 4 1
768.2.f.b 4 48.i odd 4 1
768.2.f.c 4 16.e even 4 1
768.2.f.c 4 48.k even 4 1
768.2.f.e 4 16.e even 4 1
768.2.f.e 4 48.k even 4 1
768.2.f.f 4 16.f odd 4 1
768.2.f.f 4 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(384, [\chi])\):

\( T_{11}^{2} + 6T_{11} + 4 \) Copy content Toggle raw display
\( T_{23}^{2} - 4T_{23} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$19$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$31$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$43$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$47$ \( (T - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 176)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 188T^{2} + 16 \) Copy content Toggle raw display
$83$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
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