Properties

Label 176.3.d.a
Level $176$
Weight $3$
Character orbit 176.d
Analytic conductor $4.796$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [176,3,Mod(111,176)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.111");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 176.d (of order \(2\), degree \(1\), 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: \(4.79565265274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(\beta = \sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + 5 q^{5} - 2 \beta q^{7} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + 5 q^{5} - 2 \beta q^{7} - 2 q^{9} + \beta q^{11} - 12 q^{13} - 5 \beta q^{15} + 14 q^{17} - 8 \beta q^{19} - 22 q^{21} - \beta q^{23} - 7 \beta q^{27} - 24 q^{29} + 11 \beta q^{31} + 11 q^{33} - 10 \beta q^{35} + 51 q^{37} + 12 \beta q^{39} + 40 q^{41} + 18 \beta q^{43} - 10 q^{45} + 28 \beta q^{47} + 5 q^{49} - 14 \beta q^{51} + 46 q^{53} + 5 \beta q^{55} - 88 q^{57} - 11 \beta q^{59} - 44 q^{61} + 4 \beta q^{63} - 60 q^{65} + 21 \beta q^{67} - 11 q^{69} + 17 \beta q^{71} + 44 q^{73} + 22 q^{77} + 6 \beta q^{79} - 95 q^{81} - 38 \beta q^{83} + 70 q^{85} + 24 \beta q^{87} - 89 q^{89} + 24 \beta q^{91} + 121 q^{93} - 40 \beta q^{95} + 185 q^{97} - 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} - 4 q^{9} - 24 q^{13} + 28 q^{17} - 44 q^{21} - 48 q^{29} + 22 q^{33} + 102 q^{37} + 80 q^{41} - 20 q^{45} + 10 q^{49} + 92 q^{53} - 176 q^{57} - 88 q^{61} - 120 q^{65} - 22 q^{69} + 88 q^{73} + 44 q^{77} - 190 q^{81} + 140 q^{85} - 178 q^{89} + 242 q^{93} + 370 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
0.500000 + 1.65831i
0.500000 1.65831i
0 3.31662i 0 5.00000 0 6.63325i 0 −2.00000 0
111.2 0 3.31662i 0 5.00000 0 6.63325i 0 −2.00000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.3.d.a 2
3.b odd 2 1 1584.3.l.a 2
4.b odd 2 1 inner 176.3.d.a 2
8.b even 2 1 704.3.d.a 2
8.d odd 2 1 704.3.d.a 2
12.b even 2 1 1584.3.l.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.3.d.a 2 1.a even 1 1 trivial
176.3.d.a 2 4.b odd 2 1 inner
704.3.d.a 2 8.b even 2 1
704.3.d.a 2 8.d odd 2 1
1584.3.l.a 2 3.b odd 2 1
1584.3.l.a 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 11 \) acting on \(S_{3}^{\mathrm{new}}(176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 11 \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 44 \) Copy content Toggle raw display
$11$ \( T^{2} + 11 \) Copy content Toggle raw display
$13$ \( (T + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T - 14)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 704 \) Copy content Toggle raw display
$23$ \( T^{2} + 11 \) Copy content Toggle raw display
$29$ \( (T + 24)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1331 \) Copy content Toggle raw display
$37$ \( (T - 51)^{2} \) Copy content Toggle raw display
$41$ \( (T - 40)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 3564 \) Copy content Toggle raw display
$47$ \( T^{2} + 8624 \) Copy content Toggle raw display
$53$ \( (T - 46)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1331 \) Copy content Toggle raw display
$61$ \( (T + 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4851 \) Copy content Toggle raw display
$71$ \( T^{2} + 3179 \) Copy content Toggle raw display
$73$ \( (T - 44)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 396 \) Copy content Toggle raw display
$83$ \( T^{2} + 15884 \) Copy content Toggle raw display
$89$ \( (T + 89)^{2} \) Copy content Toggle raw display
$97$ \( (T - 185)^{2} \) Copy content Toggle raw display
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