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Properties

Label 384.4.f.h
Level $384$
Weight $4$
Character orbit 384.f
Analytic conductor $22.657$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.12745506816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{6} - 2 \beta_{3}) q^{5} + (2 \beta_{5} - 3 \beta_{2}) q^{7} + (3 \beta_{5} - 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{6} - 2 \beta_{3}) q^{5} + (2 \beta_{5} - 3 \beta_{2}) q^{7} + (3 \beta_{5} - 15) q^{9} + ( - 2 \beta_{4} - \beta_{3} - 8 \beta_1) q^{11} + (8 \beta_{4} + 4 \beta_{3} + 7 \beta_1) q^{13} + ( - \beta_{7} + 6 \beta_{5} + \cdots + 24) q^{15}+ \cdots + (96 \beta_{6} + 30 \beta_{4} + \cdots + 120 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 120 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 120 q^{9} + 192 q^{15} + 640 q^{23} + 216 q^{25} + 336 q^{33} - 1344 q^{39} - 776 q^{49} + 672 q^{57} - 2688 q^{63} + 640 q^{71} - 1232 q^{73} - 2232 q^{81} - 1344 q^{87} + 9856 q^{95} + 5712 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 23x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{6} + 96\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{7} - 2\nu^{5} - 182\nu^{3} - 38\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{7} + 2\nu^{5} - 182\nu^{3} + 38\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} - 5\nu^{6} - \nu^{5} + 91\nu^{3} - 110\nu^{2} - 19\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\nu^{7} + 2\nu^{5} + 278\nu^{3} + 58\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -12\nu^{7} + 2\nu^{5} - 278\nu^{3} + 58\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{4} + 92 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{4} + 2\beta_{3} + 5\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{6} + 2\beta_{5} + 3\beta_{3} + 3\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} - 92 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{6} - 19\beta_{5} + 29\beta_{3} - 29\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -48\beta_{4} - 24\beta_{3} - 55\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 91\beta_{6} - 91\beta_{5} - 139\beta_{3} - 139\beta_{2} ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−1.54779 + 1.54779i
0.323042 + 0.323042i
−1.54779 − 1.54779i
0.323042 − 0.323042i
−0.323042 − 0.323042i
1.54779 − 1.54779i
−0.323042 + 0.323042i
1.54779 + 1.54779i
0 −2.44949 − 4.58258i 0 −17.2813 0 0.269691i 0 −15.0000 + 22.4499i 0
191.2 0 −2.44949 − 4.58258i 0 −2.31464 0 29.6636i 0 −15.0000 + 22.4499i 0
191.3 0 −2.44949 + 4.58258i 0 −17.2813 0 − 0.269691i 0 −15.0000 − 22.4499i 0
191.4 0 −2.44949 + 4.58258i 0 −2.31464 0 − 29.6636i 0 −15.0000 − 22.4499i 0
191.5 0 2.44949 − 4.58258i 0 2.31464 0 − 29.6636i 0 −15.0000 − 22.4499i 0
191.6 0 2.44949 − 4.58258i 0 17.2813 0 − 0.269691i 0 −15.0000 − 22.4499i 0
191.7 0 2.44949 + 4.58258i 0 2.31464 0 29.6636i 0 −15.0000 + 22.4499i 0
191.8 0 2.44949 + 4.58258i 0 17.2813 0 0.269691i 0 −15.0000 + 22.4499i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.8
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.h yes 8
3.b odd 2 1 384.4.f.g ✓ 8
4.b odd 2 1 384.4.f.g ✓ 8
8.b even 2 1 inner 384.4.f.h yes 8
8.d odd 2 1 384.4.f.g ✓ 8
12.b even 2 1 inner 384.4.f.h yes 8
16.e even 4 1 768.4.c.m 4
16.e even 4 1 768.4.c.p 4
16.f odd 4 1 768.4.c.n 4
16.f odd 4 1 768.4.c.o 4
24.f even 2 1 inner 384.4.f.h yes 8
24.h odd 2 1 384.4.f.g ✓ 8
48.i odd 4 1 768.4.c.n 4
48.i odd 4 1 768.4.c.o 4
48.k even 4 1 768.4.c.m 4
48.k even 4 1 768.4.c.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.g ✓ 8 3.b odd 2 1
384.4.f.g ✓ 8 4.b odd 2 1
384.4.f.g ✓ 8 8.d odd 2 1
384.4.f.g ✓ 8 24.h odd 2 1
384.4.f.h yes 8 1.a even 1 1 trivial
384.4.f.h yes 8 8.b even 2 1 inner
384.4.f.h yes 8 12.b even 2 1 inner
384.4.f.h yes 8 24.f even 2 1 inner
768.4.c.m 4 16.e even 4 1
768.4.c.m 4 48.k even 4 1
768.4.c.n 4 16.f odd 4 1
768.4.c.n 4 48.i odd 4 1
768.4.c.o 4 16.f odd 4 1
768.4.c.o 4 48.i odd 4 1
768.4.c.p 4 16.e even 4 1
768.4.c.p 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} - 304T_{5}^{2} + 1600 \) Copy content Toggle raw display
\( T_{23}^{2} - 160T_{23} + 5056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 30 T^{2} + 729)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 304 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 880 T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2216 T^{2} + 883600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4256 T^{2} + 313600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 12736 T^{2} + 35046400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 31024 T^{2} + 173185600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 160 T + 5056)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 68656 T^{2} + 621006400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 6384 T^{2} + 705600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 162464 T^{2} + 2548230400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 76800 T^{2} + 681836544)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 143920 T^{2} + 1873850944)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 86016)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 462000 T^{2} + 17640000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2100)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 778400 T^{2} + 150258567424)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 512176 T^{2} + 19993960000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 160 T - 833600)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 308 T - 169820)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1715056 T^{2} + 734586126400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1133160 T^{2} + 113291481744)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 862144 T^{2} + 126280729600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 1428 T + 423780)^{4} \) Copy content Toggle raw display
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