Properties

Label 176.11.h.a
Level $176$
Weight $11$
Character orbit 176.h
Self dual yes
Analytic conductor $111.823$
Analytic rank $0$
Dimension $1$
CM discriminant -11
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [176,11,Mod(65,176)]
 
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, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.65");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 176.h (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: yes
Analytic conductor: \(111.822876471\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 475 q^{3} - 3001 q^{5} + 166576 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 475 q^{3} - 3001 q^{5} + 166576 q^{9} + 161051 q^{11} + 1425475 q^{15} + 11910325 q^{23} - 759624 q^{25} - 51075325 q^{27} - 3192323 q^{31} - 76499225 q^{33} - 137082625 q^{37} - 499894576 q^{45} - 151795250 q^{47} + 282475249 q^{49} + 375066650 q^{53} - 483314051 q^{55} + 813567973 q^{59} - 2616638675 q^{67} - 5657404375 q^{69} - 783651827 q^{71} + 360821400 q^{75} + 14424633151 q^{81} - 2870912977 q^{89} + 1516353425 q^{93} + 9454010975 q^{97} + 26827231376 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
0
0 −475.000 0 −3001.00 0 0 0 166576. 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.11.h.a 1
4.b odd 2 1 11.11.b.a 1
11.b odd 2 1 CM 176.11.h.a 1
12.b even 2 1 99.11.c.a 1
44.c even 2 1 11.11.b.a 1
132.d odd 2 1 99.11.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.11.b.a 1 4.b odd 2 1
11.11.b.a 1 44.c even 2 1
99.11.c.a 1 12.b even 2 1
99.11.c.a 1 132.d odd 2 1
176.11.h.a 1 1.a even 1 1 trivial
176.11.h.a 1 11.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 475 \) acting on \(S_{11}^{\mathrm{new}}(176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 475 \) Copy content Toggle raw display
$5$ \( T + 3001 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 161051 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 11910325 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 3192323 \) Copy content Toggle raw display
$37$ \( T + 137082625 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 151795250 \) Copy content Toggle raw display
$53$ \( T - 375066650 \) Copy content Toggle raw display
$59$ \( T - 813567973 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 2616638675 \) Copy content Toggle raw display
$71$ \( T + 783651827 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 2870912977 \) Copy content Toggle raw display
$97$ \( T - 9454010975 \) Copy content Toggle raw display
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