↑

Properties

Label 384.4.f.i
Level $384$
Weight $4$
Character orbit 384.f
Analytic conductor $22.657$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

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: 8.0.77720518656.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 119x^{4} - 300x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{3} + \beta_{3} q^{5} + \beta_{7} q^{7} + (3 \beta_{4} + 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{3} + \beta_{3} q^{5} + \beta_{7} q^{7} + (3 \beta_{4} + 21) q^{9} - 7 \beta_1 q^{11} + \beta_{6} q^{13} + (\beta_{7} + \beta_{5}) q^{15} + 14 \beta_{4} q^{17} + (2 \beta_{2} + \beta_1) q^{19} + ( - 3 \beta_{6} - 3 \beta_{3}) q^{21} - 2 \beta_{5} q^{23} + 139 q^{25} + ( - 15 \beta_{2} - 42 \beta_1) q^{27} - 7 \beta_{3} q^{29} - 7 \beta_{7} q^{31} + (21 \beta_{4} - 42) q^{33} - 132 \beta_1 q^{35} + 7 \beta_{6} q^{37} + ( - 8 \beta_{7} + \beta_{5}) q^{39} - 56 \beta_{4} q^{41} + ( - 98 \beta_{2} - 49 \beta_1) q^{43} + ( - 6 \beta_{6} + 21 \beta_{3}) q^{45} - 449 q^{49} + (28 \beta_{2} - 98 \beta_1) q^{51} + 21 \beta_{3} q^{53} + 14 \beta_{7} q^{55} + ( - 3 \beta_{4} - 48) q^{57} - 197 \beta_1 q^{59} + \beta_{6} q^{61} + (21 \beta_{7} - 6 \beta_{5}) q^{63} - 132 \beta_{4} q^{65} + (70 \beta_{2} + 35 \beta_1) q^{67} + (6 \beta_{6} - 48 \beta_{3}) q^{69} + 14 \beta_{5} q^{71} - 98 q^{73} + ( - 139 \beta_{2} - 139 \beta_1) q^{75} - 42 \beta_{3} q^{77} + 5 \beta_{7} q^{79} + (126 \beta_{4} + 153) q^{81} - 267 \beta_1 q^{83} - 28 \beta_{6} q^{85} + ( - 7 \beta_{7} - 7 \beta_{5}) q^{87} - 70 \beta_{4} q^{89} + ( - 264 \beta_{2} - 132 \beta_1) q^{91} + (21 \beta_{6} + 21 \beta_{3}) q^{93} - 2 \beta_{5} q^{95} + 994 q^{97} + (84 \beta_{2} - 105 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 168 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 168 q^{9} + 1112 q^{25} - 336 q^{33} - 3592 q^{49} - 384 q^{57} - 784 q^{73} + 1224 q^{81} + 7952 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} + 119x^{4} - 300x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 48\nu^{6} - 476\nu^{4} + 5712\nu^{2} - 8450 ) / 2975 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -68\nu^{7} - 120\nu^{6} + 1666\nu^{5} + 1190\nu^{4} - 14042\nu^{3} - 14280\nu^{2} + 62050\nu + 21125 ) / 14875 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -44\nu^{7} + 578\nu^{5} - 6086\nu^{3} + 27650\nu ) / 2125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 48\nu^{7} - 476\nu^{5} + 4012\nu^{3} - 2500\nu ) / 2125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -24\nu^{6} - 9936 ) / 119 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{6} + 96\nu^{4} - 752\nu^{2} + 1200 ) / 25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -72\nu^{7} + 714\nu^{5} - 7518\nu^{3} + 3750\nu ) / 875 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} - 3\beta_{4} + 6\beta_{3} - 6\beta_{2} - 3\beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} + \beta_{5} + 72\beta _1 + 144 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -14\beta_{7} - 51\beta_{4} ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{6} - \beta_{5} + 47\beta _1 - 94 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -118\beta_{7} - 537\beta_{4} - 354\beta_{3} + 1074\beta_{2} + 537\beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -119\beta_{5} - 9936 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1066\beta_{7} + 5169\beta_{4} - 3198\beta_{3} + 10338\beta_{2} + 5169\beta_1 ) / 48 \) 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
−2.64338 − 1.52616i
1.41864 + 0.819051i
−2.64338 + 1.52616i
1.41864 − 0.819051i
−1.41864 − 0.819051i
2.64338 + 1.52616i
−1.41864 + 0.819051i
2.64338 − 1.52616i
0 −4.89898 − 1.73205i 0 −16.2481 0 28.1425i 0 21.0000 + 16.9706i 0
191.2 0 −4.89898 − 1.73205i 0 16.2481 0 − 28.1425i 0 21.0000 + 16.9706i 0
191.3 0 −4.89898 + 1.73205i 0 −16.2481 0 − 28.1425i 0 21.0000 − 16.9706i 0
191.4 0 −4.89898 + 1.73205i 0 16.2481 0 28.1425i 0 21.0000 − 16.9706i 0
191.5 0 4.89898 − 1.73205i 0 −16.2481 0 28.1425i 0 21.0000 − 16.9706i 0
191.6 0 4.89898 − 1.73205i 0 16.2481 0 − 28.1425i 0 21.0000 − 16.9706i 0
191.7 0 4.89898 + 1.73205i 0 −16.2481 0 − 28.1425i 0 21.0000 + 16.9706i 0
191.8 0 4.89898 + 1.73205i 0 16.2481 0 28.1425i 0 21.0000 + 16.9706i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 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 384.4.f.i ✓ 8
3.b odd 2 1 inner 384.4.f.i ✓ 8
4.b odd 2 1 inner 384.4.f.i ✓ 8
8.b even 2 1 inner 384.4.f.i ✓ 8
8.d odd 2 1 inner 384.4.f.i ✓ 8
12.b even 2 1 inner 384.4.f.i ✓ 8
16.e even 4 2 768.4.c.s 8
16.f odd 4 2 768.4.c.s 8
24.f even 2 1 inner 384.4.f.i ✓ 8
24.h odd 2 1 inner 384.4.f.i ✓ 8
48.i odd 4 2 768.4.c.s 8
48.k even 4 2 768.4.c.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.i ✓ 8 1.a even 1 1 trivial
384.4.f.i ✓ 8 3.b odd 2 1 inner
384.4.f.i ✓ 8 4.b odd 2 1 inner
384.4.f.i ✓ 8 8.b even 2 1 inner
384.4.f.i ✓ 8 8.d odd 2 1 inner
384.4.f.i ✓ 8 12.b even 2 1 inner
384.4.f.i ✓ 8 24.f even 2 1 inner
384.4.f.i ✓ 8 24.h odd 2 1 inner
768.4.c.s 8 16.e even 4 2
768.4.c.s 8 16.f odd 4 2
768.4.c.s 8 48.i odd 4 2
768.4.c.s 8 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} - 264 \) Copy content Toggle raw display
\( T_{23}^{2} - 25344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 42 T^{2} + 729)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 264)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 792)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 588)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2112)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6272)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 25344)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 12936)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 38808)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 103488)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 100352)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 230496)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{2} - 116424)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 465708)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2112)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 117600)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1241856)^{4} \) Copy content Toggle raw display
$73$ \( (T + 98)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 19800)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 855468)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 156800)^{4} \) Copy content Toggle raw display
$97$ \( (T - 994)^{8} \) Copy content Toggle raw display
show more
show less