Properties

Label 384.4.f.c
Level $384$
Weight $4$
Character orbit 384.f
Analytic conductor $22.657$
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] = [384,4,Mod(191,384)]
 
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, 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("384.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.f (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: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (2 \beta_{2} - \beta_1) q^{3} - 2 \beta_{2} q^{5} + 2 \beta_{3} q^{7} + (3 \beta_{3} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} - \beta_1) q^{3} - 2 \beta_{2} q^{5} + 2 \beta_{3} q^{7} + (3 \beta_{3} + 9) q^{9} + ( - 5 \beta_{2} + 10 \beta_1) q^{11} + (12 \beta_{2} - 24 \beta_1) q^{13} + ( - 2 \beta_{3} - 24) q^{15} - 6 \beta_{3} q^{17} + 9 \beta_{2} q^{19} + ( - 6 \beta_{2} - 24 \beta_1) q^{21} - 144 q^{23} - 93 q^{25} + (9 \beta_{2} - 45 \beta_1) q^{27} - 2 \beta_{2} q^{29} + 26 \beta_{3} q^{31} + ( - 15 \beta_{3} + 90) q^{33} + ( - 16 \beta_{2} + 32 \beta_1) q^{35} + ( - 12 \beta_{2} + 24 \beta_1) q^{37} + (36 \beta_{3} - 216) q^{39} + 36 \beta_{3} q^{41} - 81 \beta_{2} q^{43} + ( - 42 \beta_{2} + 48 \beta_1) q^{45} - 576 q^{47} + 55 q^{49} + (18 \beta_{2} + 72 \beta_1) q^{51} + 182 \beta_{2} q^{53} + 20 \beta_{3} q^{55} + (9 \beta_{3} + 108) q^{57} + (69 \beta_{2} - 138 \beta_1) q^{59} + ( - 84 \beta_{2} + 168 \beta_1) q^{61} + (18 \beta_{3} - 432) q^{63} - 48 \beta_{3} q^{65} + 279 \beta_{2} q^{67} + ( - 288 \beta_{2} + 144 \beta_1) q^{69} - 720 q^{71} - 178 q^{73} + ( - 186 \beta_{2} + 93 \beta_1) q^{75} + 180 \beta_{2} q^{77} + 114 \beta_{3} q^{79} + (54 \beta_{3} - 567) q^{81} + ( - 73 \beta_{2} + 146 \beta_1) q^{83} + (48 \beta_{2} - 96 \beta_1) q^{85} + ( - 2 \beta_{3} - 24) q^{87} - 102 \beta_{3} q^{89} - 432 \beta_{2} q^{91} + ( - 78 \beta_{2} - 312 \beta_1) q^{93} - 144 q^{95} + 650 q^{97} + (225 \beta_{2} + 90 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{9} - 96 q^{15} - 576 q^{23} - 372 q^{25} + 360 q^{33} - 864 q^{39} - 2304 q^{47} + 220 q^{49} + 432 q^{57} - 1728 q^{63} - 2880 q^{71} - 712 q^{73} - 2268 q^{81} - 96 q^{87} - 576 q^{95} + 2600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{8}^{3} - 3\zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 6\zeta_{8}^{3} + 6\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + 3\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} - 3\beta_{2} ) / 12 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 −4.24264 3.00000i 0 5.65685 0 16.9706i 0 9.00000 + 25.4558i 0
191.2 0 −4.24264 + 3.00000i 0 5.65685 0 16.9706i 0 9.00000 25.4558i 0
191.3 0 4.24264 3.00000i 0 −5.65685 0 16.9706i 0 9.00000 25.4558i 0
191.4 0 4.24264 + 3.00000i 0 −5.65685 0 16.9706i 0 9.00000 + 25.4558i 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.b even 2 1 inner
12.b even 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 384.4.f.c 4
3.b odd 2 1 384.4.f.d yes 4
4.b odd 2 1 384.4.f.d yes 4
8.b even 2 1 inner 384.4.f.c 4
8.d odd 2 1 384.4.f.d yes 4
12.b even 2 1 inner 384.4.f.c 4
16.e even 4 1 768.4.c.b 2
16.e even 4 1 768.4.c.i 2
16.f odd 4 1 768.4.c.c 2
16.f odd 4 1 768.4.c.h 2
24.f even 2 1 inner 384.4.f.c 4
24.h odd 2 1 384.4.f.d yes 4
48.i odd 4 1 768.4.c.c 2
48.i odd 4 1 768.4.c.h 2
48.k even 4 1 768.4.c.b 2
48.k even 4 1 768.4.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.c 4 1.a even 1 1 trivial
384.4.f.c 4 8.b even 2 1 inner
384.4.f.c 4 12.b even 2 1 inner
384.4.f.c 4 24.f even 2 1 inner
384.4.f.d yes 4 3.b odd 2 1
384.4.f.d yes 4 4.b odd 2 1
384.4.f.d yes 4 8.d odd 2 1
384.4.f.d yes 4 24.h odd 2 1
768.4.c.b 2 16.e even 4 1
768.4.c.b 2 48.k even 4 1
768.4.c.c 2 16.f odd 4 1
768.4.c.c 2 48.i odd 4 1
768.4.c.h 2 16.f odd 4 1
768.4.c.h 2 48.i odd 4 1
768.4.c.i 2 16.e even 4 1
768.4.c.i 2 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 32 \) Copy content Toggle raw display
\( T_{23} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 18T^{2} + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 900)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 5184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 648)^{2} \) Copy content Toggle raw display
$23$ \( (T + 144)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 48672)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 5184)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 93312)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 52488)^{2} \) Copy content Toggle raw display
$47$ \( (T + 576)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 264992)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 171396)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 254016)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 622728)^{2} \) Copy content Toggle raw display
$71$ \( (T + 720)^{4} \) Copy content Toggle raw display
$73$ \( (T + 178)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 935712)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 191844)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 749088)^{2} \) Copy content Toggle raw display
$97$ \( (T - 650)^{4} \) Copy content Toggle raw display
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