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Properties

Label 192.3.g.c
Level $192$
Weight $3$
Character orbit 192.g
Analytic conductor $5.232$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,3,Mod(127,192)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.g (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: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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 96)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{3} + 2) q^{5} + (4 \beta_{2} - \beta_1) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{3} + 2) q^{5} + (4 \beta_{2} - \beta_1) q^{7} - 3 q^{9} + ( - 4 \beta_{2} - 2 \beta_1) q^{11} + ( - 2 \beta_{3} - 10) q^{13} + ( - 2 \beta_{2} + 3 \beta_1) q^{15} + ( - 2 \beta_{3} - 6) q^{17} + (4 \beta_{2} + 6 \beta_1) q^{19} + ( - \beta_{3} + 12) q^{21} - 2 \beta_1 q^{23} + ( - 4 \beta_{3} + 27) q^{25} + 3 \beta_{2} q^{27} + (\beta_{3} + 10) q^{29} + ( - 20 \beta_{2} - \beta_1) q^{31} + ( - 2 \beta_{3} - 12) q^{33} + (24 \beta_{2} - 14 \beta_1) q^{35} + (4 \beta_{3} - 18) q^{37} + (10 \beta_{2} + 6 \beta_1) q^{39} + (2 \beta_{3} - 22) q^{41} + ( - 20 \beta_{2} + 10 \beta_1) q^{43} + (3 \beta_{3} - 6) q^{45} + ( - 8 \beta_{2} - 14 \beta_1) q^{47} + (8 \beta_{3} - 15) q^{49} + (6 \beta_{2} + 6 \beta_1) q^{51} + (\beta_{3} - 6) q^{53} + (24 \beta_{2} + 8 \beta_1) q^{55} + (6 \beta_{3} + 12) q^{57} + ( - 44 \beta_{2} - 8 \beta_1) q^{59} + 14 q^{61} + ( - 12 \beta_{2} + 3 \beta_1) q^{63} + (6 \beta_{3} + 76) q^{65} + (28 \beta_{2} - 8 \beta_1) q^{67} - 2 \beta_{3} q^{69} + (48 \beta_{2} - 10 \beta_1) q^{71} + (8 \beta_{3} - 30) q^{73} + ( - 27 \beta_{2} + 12 \beta_1) q^{75} + (4 \beta_{3} + 16) q^{77} + (12 \beta_{2} + 19 \beta_1) q^{79} + 9 q^{81} + (4 \beta_{2} + 14 \beta_1) q^{83} + (2 \beta_{3} + 84) q^{85} + ( - 10 \beta_{2} - 3 \beta_1) q^{87} + (4 \beta_{3} - 78) q^{89} + ( - 8 \beta_{2} - 14 \beta_1) q^{91} + ( - \beta_{3} - 60) q^{93} - 88 \beta_{2} q^{95} + ( - 12 \beta_{3} - 62) q^{97} + (12 \beta_{2} + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{5} - 12 q^{9} - 40 q^{13} - 24 q^{17} + 48 q^{21} + 108 q^{25} + 40 q^{29} - 48 q^{33} - 72 q^{37} - 88 q^{41} - 24 q^{45} - 60 q^{49} - 24 q^{53} + 48 q^{57} + 56 q^{61} + 304 q^{65} - 120 q^{73} + 64 q^{77} + 36 q^{81} + 336 q^{85} - 312 q^{89} - 240 q^{93} - 248 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 4\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{12}^{3} + 8\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 − 1.73205i 0 −4.92820 0 2.92820i 0 −3.00000 0
127.2 0 − 1.73205i 0 8.92820 0 10.9282i 0 −3.00000 0
127.3 0 1.73205i 0 −4.92820 0 − 2.92820i 0 −3.00000 0
127.4 0 1.73205i 0 8.92820 0 − 10.9282i 0 −3.00000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.3.g.c 4
3.b odd 2 1 576.3.g.j 4
4.b odd 2 1 inner 192.3.g.c 4
8.b even 2 1 96.3.g.a ✓ 4
8.d odd 2 1 96.3.g.a ✓ 4
12.b even 2 1 576.3.g.j 4
16.e even 4 1 768.3.b.a 4
16.e even 4 1 768.3.b.d 4
16.f odd 4 1 768.3.b.a 4
16.f odd 4 1 768.3.b.d 4
24.f even 2 1 288.3.g.d 4
24.h odd 2 1 288.3.g.d 4
40.e odd 2 1 2400.3.e.a 4
40.f even 2 1 2400.3.e.a 4
40.i odd 4 1 2400.3.j.a 4
40.i odd 4 1 2400.3.j.b 4
40.k even 4 1 2400.3.j.a 4
40.k even 4 1 2400.3.j.b 4
48.i odd 4 1 2304.3.b.k 4
48.i odd 4 1 2304.3.b.o 4
48.k even 4 1 2304.3.b.k 4
48.k even 4 1 2304.3.b.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.g.a ✓ 4 8.b even 2 1
96.3.g.a ✓ 4 8.d odd 2 1
192.3.g.c 4 1.a even 1 1 trivial
192.3.g.c 4 4.b odd 2 1 inner
288.3.g.d 4 24.f even 2 1
288.3.g.d 4 24.h odd 2 1
576.3.g.j 4 3.b odd 2 1
576.3.g.j 4 12.b even 2 1
768.3.b.a 4 16.e even 4 1
768.3.b.a 4 16.f odd 4 1
768.3.b.d 4 16.e even 4 1
768.3.b.d 4 16.f odd 4 1
2304.3.b.k 4 48.i odd 4 1
2304.3.b.k 4 48.k even 4 1
2304.3.b.o 4 48.i odd 4 1
2304.3.b.o 4 48.k even 4 1
2400.3.e.a 4 40.e odd 2 1
2400.3.e.a 4 40.f even 2 1
2400.3.j.a 4 40.i odd 4 1
2400.3.j.a 4 40.k even 4 1
2400.3.j.b 4 40.i odd 4 1
2400.3.j.b 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 4T_{5} - 44 \) acting on \(S_{3}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$11$ \( T^{4} + 224T^{2} + 256 \) Copy content Toggle raw display
$13$ \( (T^{2} + 20 T - 92)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12 T - 156)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 1248 T^{2} + 278784 \) Copy content Toggle raw display
$23$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 20 T + 52)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2432 T^{2} + 1401856 \) Copy content Toggle raw display
$37$ \( (T^{2} + 36 T - 444)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 44 T + 292)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 5600 T^{2} + 160000 \) Copy content Toggle raw display
$47$ \( T^{4} + 6656 T^{2} + 8667136 \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 13664 T^{2} + 22886656 \) Copy content Toggle raw display
$61$ \( (T - 14)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 6752 T^{2} + 1763584 \) Copy content Toggle raw display
$71$ \( T^{4} + 17024 T^{2} + 28217344 \) Copy content Toggle raw display
$73$ \( (T^{2} + 60 T - 2172)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 12416 T^{2} + 28558336 \) Copy content Toggle raw display
$83$ \( T^{4} + 6368 T^{2} + 9535744 \) Copy content Toggle raw display
$89$ \( (T^{2} + 156 T + 5316)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 124 T - 3068)^{2} \) Copy content Toggle raw display
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