Properties

Label 176.6.a.b
Level $176$
Weight $6$
Character orbit 176.a
Self dual yes
Analytic conductor $28.228$
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] = [176,6,Mod(1,176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 176.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: \(28.2275522871\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
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)
 
\(f(q)\) \(=\) \( q - q^{3} - 51 q^{5} + 166 q^{7} - 242 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 51 q^{5} + 166 q^{7} - 242 q^{9} + 121 q^{11} + 692 q^{13} + 51 q^{15} - 738 q^{17} - 1424 q^{19} - 166 q^{21} + 1779 q^{23} - 524 q^{25} + 485 q^{27} - 2064 q^{29} - 6245 q^{31} - 121 q^{33} - 8466 q^{35} - 14785 q^{37} - 692 q^{39} + 5304 q^{41} - 17798 q^{43} + 12342 q^{45} + 17184 q^{47} + 10749 q^{49} + 738 q^{51} - 30726 q^{53} - 6171 q^{55} + 1424 q^{57} + 34989 q^{59} - 45940 q^{61} - 40172 q^{63} - 35292 q^{65} - 25343 q^{67} - 1779 q^{69} - 13311 q^{71} - 53260 q^{73} + 524 q^{75} + 20086 q^{77} - 77234 q^{79} + 58321 q^{81} - 55014 q^{83} + 37638 q^{85} + 2064 q^{87} + 125415 q^{89} + 114872 q^{91} + 6245 q^{93} + 72624 q^{95} - 88807 q^{97} - 29282 q^{99}+O(q^{100}) \) 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 −1.00000 0 −51.0000 0 166.000 0 −242.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)

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 176.6.a.b 1
4.b odd 2 1 22.6.a.b 1
8.b even 2 1 704.6.a.f 1
8.d odd 2 1 704.6.a.e 1
12.b even 2 1 198.6.a.i 1
20.d odd 2 1 550.6.a.f 1
20.e even 4 2 550.6.b.f 2
28.d even 2 1 1078.6.a.a 1
44.c even 2 1 242.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.b 1 4.b odd 2 1
176.6.a.b 1 1.a even 1 1 trivial
198.6.a.i 1 12.b even 2 1
242.6.a.d 1 44.c even 2 1
550.6.a.f 1 20.d odd 2 1
550.6.b.f 2 20.e even 4 2
704.6.a.e 1 8.d odd 2 1
704.6.a.f 1 8.b even 2 1
1078.6.a.a 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(176))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 51 \) Copy content Toggle raw display
$7$ \( T - 166 \) Copy content Toggle raw display
$11$ \( T - 121 \) Copy content Toggle raw display
$13$ \( T - 692 \) Copy content Toggle raw display
$17$ \( T + 738 \) Copy content Toggle raw display
$19$ \( T + 1424 \) Copy content Toggle raw display
$23$ \( T - 1779 \) Copy content Toggle raw display
$29$ \( T + 2064 \) Copy content Toggle raw display
$31$ \( T + 6245 \) Copy content Toggle raw display
$37$ \( T + 14785 \) Copy content Toggle raw display
$41$ \( T - 5304 \) Copy content Toggle raw display
$43$ \( T + 17798 \) Copy content Toggle raw display
$47$ \( T - 17184 \) Copy content Toggle raw display
$53$ \( T + 30726 \) Copy content Toggle raw display
$59$ \( T - 34989 \) Copy content Toggle raw display
$61$ \( T + 45940 \) Copy content Toggle raw display
$67$ \( T + 25343 \) Copy content Toggle raw display
$71$ \( T + 13311 \) Copy content Toggle raw display
$73$ \( T + 53260 \) Copy content Toggle raw display
$79$ \( T + 77234 \) Copy content Toggle raw display
$83$ \( T + 55014 \) Copy content Toggle raw display
$89$ \( T - 125415 \) Copy content Toggle raw display
$97$ \( T + 88807 \) Copy content Toggle raw display
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