Properties

Label 192.4.c.b
Level $192$
Weight $4$
Character orbit 192.c
Analytic conductor $11.328$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,4,Mod(191,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 192.c (of order \(2\), degree \(1\), not 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: \(11.3283667211\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 12)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + (3 \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + (3 \beta_{2} - 3) q^{9} + (5 \beta_{3} + 5 \beta_1) q^{11} + 10 q^{13} + (\beta_{3} + 9 \beta_1) q^{15} - 4 \beta_{2} q^{17} + ( - 9 \beta_{3} + 9 \beta_1) q^{19} + (3 \beta_{2} - 30) q^{21} + (14 \beta_{3} + 14 \beta_1) q^{23} + 45 q^{25} + (6 \beta_{3} + 27 \beta_1) q^{27} - 17 \beta_{2} q^{29} + ( - 29 \beta_{3} + 29 \beta_1) q^{31} + ( - 15 \beta_{2} - 120) q^{33} + (10 \beta_{3} + 10 \beta_1) q^{35} + 130 q^{37} - 10 \beta_{3} q^{39} + 14 \beta_{2} q^{41} + ( - 29 \beta_{3} + 29 \beta_1) q^{43} + ( - 3 \beta_{2} - 240) q^{45} + (28 \beta_{3} + 28 \beta_1) q^{47} + 283 q^{49} + ( - 4 \beta_{3} - 36 \beta_1) q^{51} + 61 \beta_{2} q^{53} + (40 \beta_{3} - 40 \beta_1) q^{55} + (27 \beta_{2} - 270) q^{57} + (25 \beta_{3} + 25 \beta_1) q^{59} + 442 q^{61} + (33 \beta_{3} + 27 \beta_1) q^{63} + 10 \beta_{2} q^{65} + (95 \beta_{3} - 95 \beta_1) q^{67} + ( - 42 \beta_{2} - 336) q^{69} + ( - 150 \beta_{3} - 150 \beta_1) q^{71} + 410 q^{73} - 45 \beta_{3} q^{75} - 30 \beta_{2} q^{77} + (11 \beta_{3} - 11 \beta_1) q^{79} + ( - 18 \beta_{2} - 711) q^{81} + ( - 181 \beta_{3} - 181 \beta_1) q^{83} + 320 q^{85} + ( - 17 \beta_{3} - 153 \beta_1) q^{87} - 94 \beta_{2} q^{89} + ( - 10 \beta_{3} + 10 \beta_1) q^{91} + (87 \beta_{2} - 870) q^{93} + (90 \beta_{3} + 90 \beta_1) q^{95} + 770 q^{97} + (105 \beta_{3} - 135 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} + 40 q^{13} - 120 q^{21} + 180 q^{25} - 480 q^{33} + 520 q^{37} - 960 q^{45} + 1132 q^{49} - 1080 q^{57} + 1768 q^{61} - 1344 q^{69} + 1640 q^{73} - 2844 q^{81} + 1280 q^{85} - 3480 q^{93} + 3080 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( -\nu^{3} - 2\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + \beta_{2} - 3\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
0.866025 + 1.11803i
0.866025 1.11803i
−0.866025 1.11803i
−0.866025 + 1.11803i
0 −3.46410 3.87298i 0 8.94427i 0 7.74597i 0 −3.00000 + 26.8328i 0
191.2 0 −3.46410 + 3.87298i 0 8.94427i 0 7.74597i 0 −3.00000 26.8328i 0
191.3 0 3.46410 3.87298i 0 8.94427i 0 7.74597i 0 −3.00000 26.8328i 0
191.4 0 3.46410 + 3.87298i 0 8.94427i 0 7.74597i 0 −3.00000 + 26.8328i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
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 192.4.c.b 4
3.b odd 2 1 inner 192.4.c.b 4
4.b odd 2 1 inner 192.4.c.b 4
8.b even 2 1 12.4.b.a 4
8.d odd 2 1 12.4.b.a 4
12.b even 2 1 inner 192.4.c.b 4
16.e even 4 2 768.4.f.c 8
16.f odd 4 2 768.4.f.c 8
24.f even 2 1 12.4.b.a 4
24.h odd 2 1 12.4.b.a 4
48.i odd 4 2 768.4.f.c 8
48.k even 4 2 768.4.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.b.a 4 8.b even 2 1
12.4.b.a 4 8.d odd 2 1
12.4.b.a 4 24.f even 2 1
12.4.b.a 4 24.h odd 2 1
192.4.c.b 4 1.a even 1 1 trivial
192.4.c.b 4 3.b odd 2 1 inner
192.4.c.b 4 4.b odd 2 1 inner
192.4.c.b 4 12.b even 2 1 inner
768.4.f.c 8 16.e even 4 2
768.4.f.c 8 16.f odd 4 2
768.4.f.c 8 48.i odd 4 2
768.4.f.c 8 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 80 \) acting on \(S_{4}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 1200)^{2} \) Copy content Toggle raw display
$13$ \( (T - 10)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1280)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4860)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 9408)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 23120)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 50460)^{2} \) Copy content Toggle raw display
$37$ \( (T - 130)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 15680)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 50460)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 37632)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 297680)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 30000)^{2} \) Copy content Toggle raw display
$61$ \( (T - 442)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 541500)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1080000)^{2} \) Copy content Toggle raw display
$73$ \( (T - 410)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 7260)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1572528)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 706880)^{2} \) Copy content Toggle raw display
$97$ \( (T - 770)^{4} \) Copy content Toggle raw display
show more
show less