Properties

Label 1156.2.h.d
Level $1156$
Weight $2$
Character orbit 1156.h
Analytic conductor $9.231$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1156,2,Mod(733,1156)]
 
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([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1156.733");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1156.h (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: \(9.23070647366\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + 2 \beta_{6} q^{5} + 3 \beta_{7} q^{7} - \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + 2 \beta_{6} q^{5} + 3 \beta_{7} q^{7} - \beta_{4} q^{9} + \beta_{5} q^{11} + 4 \beta_{2} q^{13} + 4 \beta_1 q^{15} - 4 \beta_1 q^{19} + 6 \beta_{2} q^{21} - \beta_{5} q^{23} - 3 \beta_{4} q^{25} + 4 \beta_{7} q^{27} - 2 \beta_{6} q^{29} - 3 \beta_{3} q^{31} - 2 q^{33} + 12 q^{35} - 6 \beta_{3} q^{37} - 4 \beta_{6} q^{39} - 8 \beta_{7} q^{41} - 8 \beta_{4} q^{43} - 2 \beta_{5} q^{45} + 12 \beta_{2} q^{47} + 11 \beta_1 q^{49} - 6 \beta_1 q^{53} + 4 \beta_{2} q^{55} - 4 \beta_{5} q^{57} - 6 \beta_{7} q^{61} + 3 \beta_{6} q^{63} + 8 \beta_{3} q^{65} + 4 q^{67} + 2 q^{69} + 5 \beta_{3} q^{71} + 3 \beta_{7} q^{75} + 6 \beta_{4} q^{77} - 3 \beta_{5} q^{79} + 5 \beta_{2} q^{81} - 4 \beta_1 q^{87} + 12 \beta_{2} q^{89} + 12 \beta_{5} q^{91} - 6 \beta_{4} q^{93} - 8 \beta_{7} q^{95} + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{33} + 96 q^{35} + 32 q^{67} + 16 q^{69}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{16}^{5} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{16}^{5} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{16}^{7} + \zeta_{16}^{3} \) Copy content Toggle raw display
\(\zeta_{16}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{5} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(579\) \(581\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
733.1
−0.382683 + 0.923880i
0.382683 0.923880i
−0.382683 0.923880i
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.923880 0.382683i
−0.923880 0.382683i
0.923880 + 0.382683i
0 −1.30656 + 0.541196i 0 1.08239 + 2.61313i 0 1.62359 3.91969i 0 −0.707107 + 0.707107i 0
733.2 0 1.30656 0.541196i 0 −1.08239 2.61313i 0 −1.62359 + 3.91969i 0 −0.707107 + 0.707107i 0
757.1 0 −1.30656 0.541196i 0 1.08239 2.61313i 0 1.62359 + 3.91969i 0 −0.707107 0.707107i 0
757.2 0 1.30656 + 0.541196i 0 −1.08239 + 2.61313i 0 −1.62359 3.91969i 0 −0.707107 0.707107i 0
977.1 0 −0.541196 + 1.30656i 0 −2.61313 1.08239i 0 −3.91969 + 1.62359i 0 0.707107 + 0.707107i 0
977.2 0 0.541196 1.30656i 0 2.61313 + 1.08239i 0 3.91969 1.62359i 0 0.707107 + 0.707107i 0
1001.1 0 −0.541196 1.30656i 0 −2.61313 + 1.08239i 0 −3.91969 1.62359i 0 0.707107 0.707107i 0
1001.2 0 0.541196 + 1.30656i 0 2.61313 1.08239i 0 3.91969 + 1.62359i 0 0.707107 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 733.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.2.h.d 8
17.b even 2 1 inner 1156.2.h.d 8
17.c even 4 2 inner 1156.2.h.d 8
17.d even 8 4 inner 1156.2.h.d 8
17.e odd 16 2 68.2.b.a 2
17.e odd 16 2 1156.2.a.c 2
17.e odd 16 2 1156.2.e.a 2
17.e odd 16 2 1156.2.e.b 2
51.i even 16 2 612.2.b.a 2
68.i even 16 2 272.2.b.c 2
68.i even 16 2 4624.2.a.n 2
85.o even 16 2 1700.2.g.a 4
85.p odd 16 2 1700.2.c.a 2
85.r even 16 2 1700.2.g.a 4
119.p even 16 2 3332.2.b.a 2
136.q odd 16 2 1088.2.b.e 2
136.s even 16 2 1088.2.b.f 2
204.t odd 16 2 2448.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.b.a 2 17.e odd 16 2
272.2.b.c 2 68.i even 16 2
612.2.b.a 2 51.i even 16 2
1088.2.b.e 2 136.q odd 16 2
1088.2.b.f 2 136.s even 16 2
1156.2.a.c 2 17.e odd 16 2
1156.2.e.a 2 17.e odd 16 2
1156.2.e.b 2 17.e odd 16 2
1156.2.h.d 8 1.a even 1 1 trivial
1156.2.h.d 8 17.b even 2 1 inner
1156.2.h.d 8 17.c even 4 2 inner
1156.2.h.d 8 17.d even 8 4 inner
1700.2.c.a 2 85.p odd 16 2
1700.2.g.a 4 85.o even 16 2
1700.2.g.a 4 85.r even 16 2
2448.2.c.d 2 204.t odd 16 2
3332.2.b.a 2 119.p even 16 2
4624.2.a.n 2 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 16 \) Copy content Toggle raw display
$5$ \( T^{8} + 4096 \) Copy content Toggle raw display
$7$ \( T^{8} + 104976 \) Copy content Toggle raw display
$11$ \( T^{8} + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 16 \) Copy content Toggle raw display
$29$ \( T^{8} + 4096 \) Copy content Toggle raw display
$31$ \( T^{8} + 104976 \) Copy content Toggle raw display
$37$ \( T^{8} + 26873856 \) Copy content Toggle raw display
$41$ \( T^{8} + 268435456 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 26873856 \) Copy content Toggle raw display
$67$ \( (T - 4)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 6250000 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} + 104976 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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