Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [176,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
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: no
Analytic conductor: \(4.79565265274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 22)
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 \(\beta = 6\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + \beta q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + \beta q^{7} - 8 q^{9} + ( - \beta - 7) q^{11} + \beta q^{13} + q^{15} + 3 \beta q^{17} + 3 \beta q^{19} - \beta q^{21} - 17 q^{23} - 24 q^{25} + 17 q^{27} - 4 \beta q^{29} - 17 q^{31} + (\beta + 7) q^{33} - \beta q^{35} + 47 q^{37} - \beta q^{39} + \beta q^{41} - 2 \beta q^{43} + 8 q^{45} + 58 q^{47} - 23 q^{49} - 3 \beta q^{51} + 2 q^{53} + (\beta + 7) q^{55} - 3 \beta q^{57} + 55 q^{59} - 10 \beta q^{61} - 8 \beta q^{63} - \beta q^{65} - 89 q^{67} + 17 q^{69} + 7 q^{71} + 15 \beta q^{73} + 24 q^{75} + ( - 7 \beta + 72) q^{77} + 4 \beta q^{79} + 55 q^{81} + 4 \beta q^{83} - 3 \beta q^{85} + 4 \beta q^{87} - 97 q^{89} - 72 q^{91} + 17 q^{93} - 3 \beta q^{95} - 121 q^{97} + (8 \beta + 56) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} - 16 q^{9} - 14 q^{11} + 2 q^{15} - 34 q^{23} - 48 q^{25} + 34 q^{27} - 34 q^{31} + 14 q^{33} + 94 q^{37} + 16 q^{45} + 116 q^{47} - 46 q^{49} + 4 q^{53} + 14 q^{55} + 110 q^{59} - 178 q^{67} + 34 q^{69} + 14 q^{71} + 48 q^{75} + 144 q^{77} + 110 q^{81} - 194 q^{89} - 144 q^{91} + 34 q^{93} - 242 q^{97} + 112 q^{99}+O(q^{100}) \) 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
1.41421i
1.41421i
0 −1.00000 0 −1.00000 0 8.48528i 0 −8.00000 0
65.2 0 −1.00000 0 −1.00000 0 8.48528i 0 −8.00000 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
11.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.h.c 2
3.b odd 2 1 1584.3.j.d 2
4.b odd 2 1 22.3.b.a 2
8.b even 2 1 704.3.h.e 2
8.d odd 2 1 704.3.h.d 2
11.b odd 2 1 inner 176.3.h.c 2
12.b even 2 1 198.3.d.b 2
20.d odd 2 1 550.3.d.a 2
20.e even 4 2 550.3.c.a 4
28.d even 2 1 1078.3.d.a 2
33.d even 2 1 1584.3.j.d 2
44.c even 2 1 22.3.b.a 2
44.g even 10 4 242.3.d.b 8
44.h odd 10 4 242.3.d.b 8
88.b odd 2 1 704.3.h.e 2
88.g even 2 1 704.3.h.d 2
132.d odd 2 1 198.3.d.b 2
220.g even 2 1 550.3.d.a 2
220.i odd 4 2 550.3.c.a 4
308.g odd 2 1 1078.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.b.a 2 4.b odd 2 1
22.3.b.a 2 44.c even 2 1
176.3.h.c 2 1.a even 1 1 trivial
176.3.h.c 2 11.b odd 2 1 inner
198.3.d.b 2 12.b even 2 1
198.3.d.b 2 132.d odd 2 1
242.3.d.b 8 44.g even 10 4
242.3.d.b 8 44.h odd 10 4
550.3.c.a 4 20.e even 4 2
550.3.c.a 4 220.i odd 4 2
550.3.d.a 2 20.d odd 2 1
550.3.d.a 2 220.g even 2 1
704.3.h.d 2 8.d odd 2 1
704.3.h.d 2 88.g even 2 1
704.3.h.e 2 8.b even 2 1
704.3.h.e 2 88.b odd 2 1
1078.3.d.a 2 28.d even 2 1
1078.3.d.a 2 308.g odd 2 1
1584.3.j.d 2 3.b odd 2 1
1584.3.j.d 2 33.d even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 72 \) Copy content Toggle raw display
$11$ \( T^{2} + 14T + 121 \) Copy content Toggle raw display
$13$ \( T^{2} + 72 \) Copy content Toggle raw display
$17$ \( T^{2} + 648 \) Copy content Toggle raw display
$19$ \( T^{2} + 648 \) Copy content Toggle raw display
$23$ \( (T + 17)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1152 \) Copy content Toggle raw display
$31$ \( (T + 17)^{2} \) Copy content Toggle raw display
$37$ \( (T - 47)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 72 \) Copy content Toggle raw display
$43$ \( T^{2} + 288 \) Copy content Toggle raw display
$47$ \( (T - 58)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T - 55)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 7200 \) Copy content Toggle raw display
$67$ \( (T + 89)^{2} \) Copy content Toggle raw display
$71$ \( (T - 7)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16200 \) Copy content Toggle raw display
$79$ \( T^{2} + 1152 \) Copy content Toggle raw display
$83$ \( T^{2} + 1152 \) Copy content Toggle raw display
$89$ \( (T + 97)^{2} \) Copy content Toggle raw display
$97$ \( (T + 121)^{2} \) Copy content Toggle raw display
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