Properties

Label 192.3.e.c
Level $192$
Weight $3$
Character orbit 192.e
Analytic conductor $5.232$
Analytic rank $0$
Dimension $2$
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(65,192)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.e (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: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 24)
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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} - 2 \beta q^{5} - 6 q^{7} + ( - 2 \beta - 7) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{3} - 2 \beta q^{5} - 6 q^{7} + ( - 2 \beta - 7) q^{9} - 2 \beta q^{11} - 10 q^{13} + (2 \beta + 16) q^{15} - 8 \beta q^{17} - 2 q^{19} + ( - 6 \beta + 6) q^{21} - 4 \beta q^{23} - 7 q^{25} + ( - 5 \beta + 23) q^{27} - 6 \beta q^{29} - 22 q^{31} + (2 \beta + 16) q^{33} + 12 \beta q^{35} + 6 q^{37} + ( - 10 \beta + 10) q^{39} + 12 \beta q^{41} - 82 q^{43} + (14 \beta - 32) q^{45} + 24 \beta q^{47} - 13 q^{49} + (8 \beta + 64) q^{51} + 22 \beta q^{53} - 32 q^{55} + ( - 2 \beta + 2) q^{57} - 26 \beta q^{59} + 86 q^{61} + (12 \beta + 42) q^{63} + 20 \beta q^{65} - 2 q^{67} + (4 \beta + 32) q^{69} - 44 \beta q^{71} + 82 q^{73} + ( - 7 \beta + 7) q^{75} + 12 \beta q^{77} + 10 q^{79} + (28 \beta + 17) q^{81} + 26 \beta q^{83} - 128 q^{85} + (6 \beta + 48) q^{87} - 12 \beta q^{89} + 60 q^{91} + ( - 22 \beta + 22) q^{93} + 4 \beta q^{95} - 94 q^{97} + (14 \beta - 32) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 12 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 12 q^{7} - 14 q^{9} - 20 q^{13} + 32 q^{15} - 4 q^{19} + 12 q^{21} - 14 q^{25} + 46 q^{27} - 44 q^{31} + 32 q^{33} + 12 q^{37} + 20 q^{39} - 164 q^{43} - 64 q^{45} - 26 q^{49} + 128 q^{51} - 64 q^{55} + 4 q^{57} + 172 q^{61} + 84 q^{63} - 4 q^{67} + 64 q^{69} + 164 q^{73} + 14 q^{75} + 20 q^{79} + 34 q^{81} - 256 q^{85} + 96 q^{87} + 120 q^{91} + 44 q^{93} - 188 q^{97} - 64 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.41421i
1.41421i
0 −1.00000 2.82843i 0 5.65685i 0 −6.00000 0 −7.00000 + 5.65685i 0
65.2 0 −1.00000 + 2.82843i 0 5.65685i 0 −6.00000 0 −7.00000 5.65685i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.3.e.c 2
3.b odd 2 1 inner 192.3.e.c 2
4.b odd 2 1 192.3.e.d 2
8.b even 2 1 24.3.e.a 2
8.d odd 2 1 48.3.e.b 2
12.b even 2 1 192.3.e.d 2
16.e even 4 2 768.3.h.d 4
16.f odd 4 2 768.3.h.c 4
24.f even 2 1 48.3.e.b 2
24.h odd 2 1 24.3.e.a 2
40.e odd 2 1 1200.3.l.n 2
40.f even 2 1 600.3.l.b 2
40.i odd 4 2 600.3.c.a 4
40.k even 4 2 1200.3.c.i 4
48.i odd 4 2 768.3.h.d 4
48.k even 4 2 768.3.h.c 4
56.h odd 2 1 1176.3.d.a 2
72.j odd 6 2 648.3.m.d 4
72.l even 6 2 1296.3.q.e 4
72.n even 6 2 648.3.m.d 4
72.p odd 6 2 1296.3.q.e 4
120.i odd 2 1 600.3.l.b 2
120.m even 2 1 1200.3.l.n 2
120.q odd 4 2 1200.3.c.i 4
120.w even 4 2 600.3.c.a 4
168.i even 2 1 1176.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 8.b even 2 1
24.3.e.a 2 24.h odd 2 1
48.3.e.b 2 8.d odd 2 1
48.3.e.b 2 24.f even 2 1
192.3.e.c 2 1.a even 1 1 trivial
192.3.e.c 2 3.b odd 2 1 inner
192.3.e.d 2 4.b odd 2 1
192.3.e.d 2 12.b even 2 1
600.3.c.a 4 40.i odd 4 2
600.3.c.a 4 120.w even 4 2
600.3.l.b 2 40.f even 2 1
600.3.l.b 2 120.i odd 2 1
648.3.m.d 4 72.j odd 6 2
648.3.m.d 4 72.n even 6 2
768.3.h.c 4 16.f odd 4 2
768.3.h.c 4 48.k even 4 2
768.3.h.d 4 16.e even 4 2
768.3.h.d 4 48.i odd 4 2
1176.3.d.a 2 56.h odd 2 1
1176.3.d.a 2 168.i even 2 1
1200.3.c.i 4 40.k even 4 2
1200.3.c.i 4 120.q odd 4 2
1200.3.l.n 2 40.e odd 2 1
1200.3.l.n 2 120.m even 2 1
1296.3.q.e 4 72.l even 6 2
1296.3.q.e 4 72.p odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{2} + 32 \) Copy content Toggle raw display
\( T_{7} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 32 \) Copy content Toggle raw display
$7$ \( (T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 32 \) Copy content Toggle raw display
$13$ \( (T + 10)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 512 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 128 \) Copy content Toggle raw display
$29$ \( T^{2} + 288 \) Copy content Toggle raw display
$31$ \( (T + 22)^{2} \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1152 \) Copy content Toggle raw display
$43$ \( (T + 82)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4608 \) Copy content Toggle raw display
$53$ \( T^{2} + 3872 \) Copy content Toggle raw display
$59$ \( T^{2} + 5408 \) Copy content Toggle raw display
$61$ \( (T - 86)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 15488 \) Copy content Toggle raw display
$73$ \( (T - 82)^{2} \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5408 \) Copy content Toggle raw display
$89$ \( T^{2} + 1152 \) Copy content Toggle raw display
$97$ \( (T + 94)^{2} \) Copy content Toggle raw display
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