Properties

Label 1176.1.n.e
Level $1176$
Weight $1$
Character orbit 1176.n
Analytic conductor $0.587$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -56
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1176,1,Mod(197,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1176.197");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1176.n (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: \(0.586900454856\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.4032.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{8}^{2} q^{2} - \zeta_{8} q^{3} - q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{5} + \zeta_{8}^{3} q^{6} + \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{2} q^{2} - \zeta_{8} q^{3} - q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{5} + \zeta_{8}^{3} q^{6} + \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{10} + \zeta_{8} q^{12} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{13} + ( - \zeta_{8}^{2} - 1) q^{15} + q^{16} + q^{18} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{19} + (\zeta_{8}^{3} - \zeta_{8}) q^{20} - \zeta_{8}^{3} q^{24} + q^{25} + (\zeta_{8}^{3} - \zeta_{8}) q^{26} - \zeta_{8}^{3} q^{27} + (\zeta_{8}^{2} - 1) q^{30} - \zeta_{8}^{2} q^{32} - \zeta_{8}^{2} q^{36} + (\zeta_{8}^{3} - \zeta_{8}) q^{38} + (\zeta_{8}^{2} - 1) q^{39} + (\zeta_{8}^{3} + \zeta_{8}) q^{40} + (\zeta_{8}^{3} + \zeta_{8}) q^{45} - \zeta_{8} q^{48} - \zeta_{8}^{2} q^{50} + (\zeta_{8}^{3} + \zeta_{8}) q^{52} - \zeta_{8} q^{54} + (\zeta_{8}^{2} - 1) q^{57} + (\zeta_{8}^{3} - \zeta_{8}) q^{59} + (\zeta_{8}^{2} + 1) q^{60} + (\zeta_{8}^{3} + \zeta_{8}) q^{61} - q^{64} - \zeta_{8}^{2} q^{65} + \zeta_{8}^{2} q^{71} - q^{72} - \zeta_{8} q^{75} + (\zeta_{8}^{3} + \zeta_{8}) q^{76} + (\zeta_{8}^{2} + 1) q^{78} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{80} - q^{81} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{83} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{90} - \zeta_{8}^{2} q^{95} + \zeta_{8}^{3} q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{15} + 4 q^{16} + 4 q^{18} + 4 q^{25} - 4 q^{30} - 4 q^{39} - 4 q^{57} + 4 q^{60} - 4 q^{64} - 4 q^{72} + 4 q^{78} - 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
1.00000i −0.707107 0.707107i −1.00000 1.41421 −0.707107 + 0.707107i 0 1.00000i 1.00000i 1.41421i
197.2 1.00000i 0.707107 + 0.707107i −1.00000 −1.41421 0.707107 0.707107i 0 1.00000i 1.00000i 1.41421i
197.3 1.00000i −0.707107 + 0.707107i −1.00000 1.41421 −0.707107 0.707107i 0 1.00000i 1.00000i 1.41421i
197.4 1.00000i 0.707107 0.707107i −1.00000 −1.41421 0.707107 + 0.707107i 0 1.00000i 1.00000i 1.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
168.i even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.1.n.e 4
3.b odd 2 1 inner 1176.1.n.e 4
7.b odd 2 1 inner 1176.1.n.e 4
7.c even 3 2 1176.1.s.c 8
7.d odd 6 2 1176.1.s.c 8
8.b even 2 1 inner 1176.1.n.e 4
21.c even 2 1 inner 1176.1.n.e 4
21.g even 6 2 1176.1.s.c 8
21.h odd 6 2 1176.1.s.c 8
24.h odd 2 1 inner 1176.1.n.e 4
56.h odd 2 1 CM 1176.1.n.e 4
56.j odd 6 2 1176.1.s.c 8
56.p even 6 2 1176.1.s.c 8
168.i even 2 1 inner 1176.1.n.e 4
168.s odd 6 2 1176.1.s.c 8
168.ba even 6 2 1176.1.s.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.1.n.e 4 1.a even 1 1 trivial
1176.1.n.e 4 3.b odd 2 1 inner
1176.1.n.e 4 7.b odd 2 1 inner
1176.1.n.e 4 8.b even 2 1 inner
1176.1.n.e 4 21.c even 2 1 inner
1176.1.n.e 4 24.h odd 2 1 inner
1176.1.n.e 4 56.h odd 2 1 CM
1176.1.n.e 4 168.i even 2 1 inner
1176.1.s.c 8 7.c even 3 2
1176.1.s.c 8 7.d odd 6 2
1176.1.s.c 8 21.g even 6 2
1176.1.s.c 8 21.h odd 6 2
1176.1.s.c 8 56.j odd 6 2
1176.1.s.c 8 56.p even 6 2
1176.1.s.c 8 168.s odd 6 2
1176.1.s.c 8 168.ba even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1176, [\chi])\):

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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