Properties

Label 192.8.a.a
Level $192$
Weight $8$
Character orbit 192.a
Self dual yes
Analytic conductor $59.978$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,8,Mod(1,192)]
 
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, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 192.a (trivial)

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: yes
Analytic conductor: \(59.9779248930\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 27 q^{3} - 390 q^{5} + 64 q^{7} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 27 q^{3} - 390 q^{5} + 64 q^{7} + 729 q^{9} - 948 q^{11} + 5098 q^{13} + 10530 q^{15} + 28386 q^{17} - 8620 q^{19} - 1728 q^{21} + 15288 q^{23} + 73975 q^{25} - 19683 q^{27} - 36510 q^{29} + 276808 q^{31} + 25596 q^{33} - 24960 q^{35} - 268526 q^{37} - 137646 q^{39} - 629718 q^{41} + 685772 q^{43} - 284310 q^{45} - 583296 q^{47} - 819447 q^{49} - 766422 q^{51} + 428058 q^{53} + 369720 q^{55} + 232740 q^{57} + 1306380 q^{59} - 300662 q^{61} + 46656 q^{63} - 1988220 q^{65} - 507244 q^{67} - 412776 q^{69} - 5560632 q^{71} + 1369082 q^{73} - 1997325 q^{75} - 60672 q^{77} + 6913720 q^{79} + 531441 q^{81} - 4376748 q^{83} - 11070540 q^{85} + 985770 q^{87} - 8528310 q^{89} + 326272 q^{91} - 7473816 q^{93} + 3361800 q^{95} - 8826814 q^{97} - 691092 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −27.0000 0 −390.000 0 64.0000 0 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.8.a.a 1
3.b odd 2 1 576.8.a.x 1
4.b odd 2 1 192.8.a.i 1
8.b even 2 1 48.8.a.g 1
8.d odd 2 1 3.8.a.a 1
12.b even 2 1 576.8.a.w 1
24.f even 2 1 9.8.a.a 1
24.h odd 2 1 144.8.a.b 1
40.e odd 2 1 75.8.a.a 1
40.k even 4 2 75.8.b.c 2
56.e even 2 1 147.8.a.b 1
56.k odd 6 2 147.8.e.b 2
56.m even 6 2 147.8.e.a 2
72.l even 6 2 81.8.c.c 2
72.p odd 6 2 81.8.c.a 2
88.g even 2 1 363.8.a.b 1
104.h odd 2 1 507.8.a.a 1
120.m even 2 1 225.8.a.i 1
120.q odd 4 2 225.8.b.f 2
168.e odd 2 1 441.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.8.a.a 1 8.d odd 2 1
9.8.a.a 1 24.f even 2 1
48.8.a.g 1 8.b even 2 1
75.8.a.a 1 40.e odd 2 1
75.8.b.c 2 40.k even 4 2
81.8.c.a 2 72.p odd 6 2
81.8.c.c 2 72.l even 6 2
144.8.a.b 1 24.h odd 2 1
147.8.a.b 1 56.e even 2 1
147.8.e.a 2 56.m even 6 2
147.8.e.b 2 56.k odd 6 2
192.8.a.a 1 1.a even 1 1 trivial
192.8.a.i 1 4.b odd 2 1
225.8.a.i 1 120.m even 2 1
225.8.b.f 2 120.q odd 4 2
363.8.a.b 1 88.g even 2 1
441.8.a.a 1 168.e odd 2 1
507.8.a.a 1 104.h odd 2 1
576.8.a.w 1 12.b even 2 1
576.8.a.x 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(192))\):

\( T_{5} + 390 \) Copy content Toggle raw display
\( T_{7} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 27 \) Copy content Toggle raw display
$5$ \( T + 390 \) Copy content Toggle raw display
$7$ \( T - 64 \) Copy content Toggle raw display
$11$ \( T + 948 \) Copy content Toggle raw display
$13$ \( T - 5098 \) Copy content Toggle raw display
$17$ \( T - 28386 \) Copy content Toggle raw display
$19$ \( T + 8620 \) Copy content Toggle raw display
$23$ \( T - 15288 \) Copy content Toggle raw display
$29$ \( T + 36510 \) Copy content Toggle raw display
$31$ \( T - 276808 \) Copy content Toggle raw display
$37$ \( T + 268526 \) Copy content Toggle raw display
$41$ \( T + 629718 \) Copy content Toggle raw display
$43$ \( T - 685772 \) Copy content Toggle raw display
$47$ \( T + 583296 \) Copy content Toggle raw display
$53$ \( T - 428058 \) Copy content Toggle raw display
$59$ \( T - 1306380 \) Copy content Toggle raw display
$61$ \( T + 300662 \) Copy content Toggle raw display
$67$ \( T + 507244 \) Copy content Toggle raw display
$71$ \( T + 5560632 \) Copy content Toggle raw display
$73$ \( T - 1369082 \) Copy content Toggle raw display
$79$ \( T - 6913720 \) Copy content Toggle raw display
$83$ \( T + 4376748 \) Copy content Toggle raw display
$89$ \( T + 8528310 \) Copy content Toggle raw display
$97$ \( T + 8826814 \) Copy content Toggle raw display
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