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Properties

Label 192.2.c.b
Level $192$
Weight $2$
Character orbit 192.c
Analytic conductor $1.533$
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] = [192,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.53312771881\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 1) q^{9} + (\beta_{2} + \beta_1) q^{11} + 2 q^{13} + (3 \beta_{2} - \beta_1) q^{15} + ( - 3 \beta_{2} + 3 \beta_1) q^{19} + (\beta_{3} + 2) q^{21} + ( - 2 \beta_{2} - 2 \beta_1) q^{23} - 3 q^{25} + (3 \beta_{2} - 2 \beta_1) q^{27} + \beta_{3} q^{29} + ( - \beta_{2} + \beta_1) q^{31} + (\beta_{3} - 4) q^{33} + ( - 2 \beta_{2} - 2 \beta_1) q^{35} - 6 q^{37} - 2 \beta_1 q^{39} + 2 \beta_{3} q^{41} + (\beta_{2} - \beta_1) q^{43} + ( - \beta_{3} - 8) q^{45} + (4 \beta_{2} + 4 \beta_1) q^{47} + 3 q^{49} + 3 \beta_{3} q^{53} + ( - 4 \beta_{2} + 4 \beta_1) q^{55} + (3 \beta_{3} + 6) q^{57} + (\beta_{2} + \beta_1) q^{59} + 2 q^{61} + ( - 3 \beta_{2} - \beta_1) q^{63} - 2 \beta_{3} q^{65} + (\beta_{2} - \beta_1) q^{67} + ( - 2 \beta_{3} + 8) q^{69} + (2 \beta_{2} + 2 \beta_1) q^{71} - 6 q^{73} + 3 \beta_1 q^{75} - 2 \beta_{3} q^{77} + (7 \beta_{2} - 7 \beta_1) q^{79} + ( - 2 \beta_{3} - 7) q^{81} + ( - \beta_{2} - \beta_1) q^{83} + ( - 3 \beta_{2} + \beta_1) q^{87} - 6 \beta_{3} q^{89} + ( - 2 \beta_{2} + 2 \beta_1) q^{91} + (\beta_{3} + 2) q^{93} + ( - 6 \beta_{2} - 6 \beta_1) q^{95} + 10 q^{97} + ( - 3 \beta_{2} + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} + 8 q^{13} + 8 q^{21} - 12 q^{25} - 16 q^{33} - 24 q^{37} - 32 q^{45} + 12 q^{49} + 24 q^{57} + 8 q^{61} + 32 q^{69} - 24 q^{73} - 28 q^{81} + 8 q^{93} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.707107 − 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 − 0.707107i
0 −1.41421 − 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 + 2.82843i 0
191.2 0 −1.41421 + 1.00000i 0 − 2.82843i 0 − 2.00000i 0 1.00000 − 2.82843i 0
191.3 0 1.41421 − 1.00000i 0 − 2.82843i 0 2.00000i 0 1.00000 − 2.82843i 0
191.4 0 1.41421 + 1.00000i 0 2.82843i 0 − 2.00000i 0 1.00000 + 2.82843i 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 192.2.c.b 4
3.b odd 2 1 inner 192.2.c.b 4
4.b odd 2 1 inner 192.2.c.b 4
8.b even 2 1 96.2.c.a ✓ 4
8.d odd 2 1 96.2.c.a ✓ 4
12.b even 2 1 inner 192.2.c.b 4
16.e even 4 1 768.2.f.a 4
16.e even 4 1 768.2.f.g 4
16.f odd 4 1 768.2.f.a 4
16.f odd 4 1 768.2.f.g 4
24.f even 2 1 96.2.c.a ✓ 4
24.h odd 2 1 96.2.c.a ✓ 4
40.e odd 2 1 2400.2.h.c 4
40.f even 2 1 2400.2.h.c 4
40.i odd 4 1 2400.2.o.a 4
40.i odd 4 1 2400.2.o.h 4
40.k even 4 1 2400.2.o.a 4
40.k even 4 1 2400.2.o.h 4
48.i odd 4 1 768.2.f.a 4
48.i odd 4 1 768.2.f.g 4
48.k even 4 1 768.2.f.a 4
48.k even 4 1 768.2.f.g 4
72.j odd 6 2 2592.2.s.e 8
72.l even 6 2 2592.2.s.e 8
72.n even 6 2 2592.2.s.e 8
72.p odd 6 2 2592.2.s.e 8
120.i odd 2 1 2400.2.h.c 4
120.m even 2 1 2400.2.h.c 4
120.q odd 4 1 2400.2.o.a 4
120.q odd 4 1 2400.2.o.h 4
120.w even 4 1 2400.2.o.a 4
120.w even 4 1 2400.2.o.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.c.a ✓ 4 8.b even 2 1
96.2.c.a ✓ 4 8.d odd 2 1
96.2.c.a ✓ 4 24.f even 2 1
96.2.c.a ✓ 4 24.h odd 2 1
192.2.c.b 4 1.a even 1 1 trivial
192.2.c.b 4 3.b odd 2 1 inner
192.2.c.b 4 4.b odd 2 1 inner
192.2.c.b 4 12.b even 2 1 inner
768.2.f.a 4 16.e even 4 1
768.2.f.a 4 16.f odd 4 1
768.2.f.a 4 48.i odd 4 1
768.2.f.a 4 48.k even 4 1
768.2.f.g 4 16.e even 4 1
768.2.f.g 4 16.f odd 4 1
768.2.f.g 4 48.i odd 4 1
768.2.f.g 4 48.k even 4 1
2400.2.h.c 4 40.e odd 2 1
2400.2.h.c 4 40.f even 2 1
2400.2.h.c 4 120.i odd 2 1
2400.2.h.c 4 120.m even 2 1
2400.2.o.a 4 40.i odd 4 1
2400.2.o.a 4 40.k even 4 1
2400.2.o.a 4 120.q odd 4 1
2400.2.o.a 4 120.w even 4 1
2400.2.o.h 4 40.i odd 4 1
2400.2.o.h 4 40.k even 4 1
2400.2.o.h 4 120.q odd 4 1
2400.2.o.h 4 120.w even 4 1
2592.2.s.e 8 72.j odd 6 2
2592.2.s.e 8 72.l even 6 2
2592.2.s.e 8 72.n even 6 2
2592.2.s.e 8 72.p odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T + 6)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$73$ \( (T + 6)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{4} \) Copy content Toggle raw display
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