Properties

Label 176.4.a.a
Level $176$
Weight $4$
Character orbit 176.a
Self dual yes
Analytic conductor $10.384$
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] = [176,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(10.3843361610\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
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)
 
\(f(q)\) \(=\) \( q - 7 q^{3} + 9 q^{5} - 2 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 7 q^{3} + 9 q^{5} - 2 q^{7} + 22 q^{9} + 11 q^{11} - 63 q^{15} - 38 q^{17} - 44 q^{19} + 14 q^{21} - 175 q^{23} - 44 q^{25} + 35 q^{27} - 264 q^{29} - 159 q^{31} - 77 q^{33} - 18 q^{35} - 173 q^{37} - 220 q^{41} + 542 q^{43} + 198 q^{45} + 264 q^{47} - 339 q^{49} + 266 q^{51} + 682 q^{53} + 99 q^{55} + 308 q^{57} - 421 q^{59} + 308 q^{61} - 44 q^{63} - 177 q^{67} + 1225 q^{69} - 365 q^{71} - 528 q^{73} + 308 q^{75} - 22 q^{77} - 686 q^{79} - 839 q^{81} - 698 q^{83} - 342 q^{85} + 1848 q^{87} + 967 q^{89} + 1113 q^{93} - 396 q^{95} - 1127 q^{97} + 242 q^{99}+O(q^{100}) \) 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 −7.00000 0 9.00000 0 −2.00000 0 22.0000 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 176.4.a.a 1
3.b odd 2 1 1584.4.a.g 1
4.b odd 2 1 88.4.a.b 1
8.b even 2 1 704.4.a.k 1
8.d odd 2 1 704.4.a.a 1
11.b odd 2 1 1936.4.a.b 1
12.b even 2 1 792.4.a.b 1
20.d odd 2 1 2200.4.a.a 1
44.c even 2 1 968.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.4.a.b 1 4.b odd 2 1
176.4.a.a 1 1.a even 1 1 trivial
704.4.a.a 1 8.d odd 2 1
704.4.a.k 1 8.b even 2 1
792.4.a.b 1 12.b even 2 1
968.4.a.e 1 44.c even 2 1
1584.4.a.g 1 3.b odd 2 1
1936.4.a.b 1 11.b odd 2 1
2200.4.a.a 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(176))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 7 \) Copy content Toggle raw display
$5$ \( T - 9 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 38 \) Copy content Toggle raw display
$19$ \( T + 44 \) Copy content Toggle raw display
$23$ \( T + 175 \) Copy content Toggle raw display
$29$ \( T + 264 \) Copy content Toggle raw display
$31$ \( T + 159 \) Copy content Toggle raw display
$37$ \( T + 173 \) Copy content Toggle raw display
$41$ \( T + 220 \) Copy content Toggle raw display
$43$ \( T - 542 \) Copy content Toggle raw display
$47$ \( T - 264 \) Copy content Toggle raw display
$53$ \( T - 682 \) Copy content Toggle raw display
$59$ \( T + 421 \) Copy content Toggle raw display
$61$ \( T - 308 \) Copy content Toggle raw display
$67$ \( T + 177 \) Copy content Toggle raw display
$71$ \( T + 365 \) Copy content Toggle raw display
$73$ \( T + 528 \) Copy content Toggle raw display
$79$ \( T + 686 \) Copy content Toggle raw display
$83$ \( T + 698 \) Copy content Toggle raw display
$89$ \( T - 967 \) Copy content Toggle raw display
$97$ \( T + 1127 \) Copy content Toggle raw display
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