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Properties

Label 1156.1.g.a
Level $1156$
Weight $1$
Character orbit 1156.g
Analytic conductor $0.577$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -4, -68, 17
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1156,1,Mod(155,1156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1156, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1156.155");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1156.g (of order \(8\), degree \(4\), 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: \(0.576919154604\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{17})\)
Artin image: $\OD_{32}$
Artin field: Galois closure of 16.8.732780301186512843008.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{8} - \zeta_{8} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{8} - \zeta_{8} q^{9} - 2 \zeta_{8}^{2} q^{13} - q^{16} + q^{18} - \zeta_{8} q^{25} + 2 \zeta_{8} q^{26} - \zeta_{8}^{3} q^{32} + \zeta_{8}^{3} q^{36} - \zeta_{8}^{3} q^{49} + q^{50} - 2 q^{52} - 2 \zeta_{8}^{3} q^{53} + \zeta_{8}^{2} q^{64} - \zeta_{8}^{2} q^{72} + \zeta_{8}^{2} q^{81} + 2 \zeta_{8}^{2} q^{89} + \zeta_{8}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{16} + 4 q^{18} + 4 q^{50} - 8 q^{52}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
−0.707107 + 0.707107i
−0.707107 − 0.707107i
0.707107 + 0.707107i
0.707107 − 0.707107i
0.707107 + 0.707107i 0 1.00000i 0 0 0 −0.707107 + 0.707107i 0.707107 − 0.707107i 0
179.1 0.707107 − 0.707107i 0 − 1.00000i 0 0 0 −0.707107 − 0.707107i 0.707107 + 0.707107i 0
399.1 −0.707107 + 0.707107i 0 − 1.00000i 0 0 0 0.707107 + 0.707107i −0.707107 − 0.707107i 0
423.1 −0.707107 − 0.707107i 0 1.00000i 0 0 0 0.707107 − 0.707107i −0.707107 + 0.707107i 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 CM by \(\Q(\sqrt{-1}) \)
17.b even 2 1 RM by \(\Q(\sqrt{17}) \)
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
17.c even 4 2 inner
17.d even 8 4 inner
68.f odd 4 2 inner
68.g odd 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.1.g.a 4
4.b odd 2 1 CM 1156.1.g.a 4
17.b even 2 1 RM 1156.1.g.a 4
17.c even 4 2 inner 1156.1.g.a 4
17.d even 8 4 inner 1156.1.g.a 4
17.e odd 16 2 68.1.d.a ✓ 1
17.e odd 16 2 1156.1.c.a 1
17.e odd 16 4 1156.1.f.a 2
51.i even 16 2 612.1.e.a 1
68.d odd 2 1 CM 1156.1.g.a 4
68.f odd 4 2 inner 1156.1.g.a 4
68.g odd 8 4 inner 1156.1.g.a 4
68.i even 16 2 68.1.d.a ✓ 1
68.i even 16 2 1156.1.c.a 1
68.i even 16 4 1156.1.f.a 2
85.o even 16 2 1700.1.d.b 2
85.p odd 16 2 1700.1.h.d 1
85.r even 16 2 1700.1.d.b 2
119.p even 16 2 3332.1.g.a 1
119.s even 48 4 3332.1.o.d 2
119.t odd 48 4 3332.1.o.c 2
136.q odd 16 2 1088.1.g.a 1
136.s even 16 2 1088.1.g.a 1
204.t odd 16 2 612.1.e.a 1
340.bc odd 16 2 1700.1.d.b 2
340.bg even 16 2 1700.1.h.d 1
340.bj odd 16 2 1700.1.d.b 2
476.bf odd 16 2 3332.1.g.a 1
476.bk odd 48 4 3332.1.o.d 2
476.bm even 48 4 3332.1.o.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a ✓ 1 17.e odd 16 2
68.1.d.a ✓ 1 68.i even 16 2
612.1.e.a 1 51.i even 16 2
612.1.e.a 1 204.t odd 16 2
1088.1.g.a 1 136.q odd 16 2
1088.1.g.a 1 136.s even 16 2
1156.1.c.a 1 17.e odd 16 2
1156.1.c.a 1 68.i even 16 2
1156.1.f.a 2 17.e odd 16 4
1156.1.f.a 2 68.i even 16 4
1156.1.g.a 4 1.a even 1 1 trivial
1156.1.g.a 4 4.b odd 2 1 CM
1156.1.g.a 4 17.b even 2 1 RM
1156.1.g.a 4 17.c even 4 2 inner
1156.1.g.a 4 17.d even 8 4 inner
1156.1.g.a 4 68.d odd 2 1 CM
1156.1.g.a 4 68.f odd 4 2 inner
1156.1.g.a 4 68.g odd 8 4 inner
1700.1.d.b 2 85.o even 16 2
1700.1.d.b 2 85.r even 16 2
1700.1.d.b 2 340.bc odd 16 2
1700.1.d.b 2 340.bj odd 16 2
1700.1.h.d 1 85.p odd 16 2
1700.1.h.d 1 340.bg even 16 2
3332.1.g.a 1 119.p even 16 2
3332.1.g.a 1 476.bf odd 16 2
3332.1.o.c 2 119.t odd 48 4
3332.1.o.c 2 476.bm even 48 4
3332.1.o.d 2 119.s even 48 4
3332.1.o.d 2 476.bk odd 48 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{1}^{\mathrm{new}}(1156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) 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
$53$ \( T^{4} + 16 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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