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Properties

Label 1156.2.e.d
Level $1156$
Weight $2$
Character orbit 1156.e
Analytic conductor $9.231$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1156,2,Mod(829,1156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1156, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1156.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1156.e (of order \(4\), degree \(2\), 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: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} + ( - 2 \beta_{2} + \beta_1) q^{5} + \beta_{4} q^{7} + ( - \beta_{5} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{3} + ( - 2 \beta_{2} + \beta_1) q^{5} + \beta_{4} q^{7} + ( - \beta_{5} + \beta_{3}) q^{9} + ( - 2 \beta_{7} + \beta_{4}) q^{11} + (\beta_{6} - 2) q^{13} + ( - \beta_{5} + 6 \beta_{3}) q^{15} + ( - \beta_{5} - 2 \beta_{3}) q^{19} + ( - \beta_{6} + 4) q^{21} + ( - 2 \beta_{7} + \beta_{4}) q^{23} + 7 \beta_{3} q^{25} - 2 \beta_{7} q^{27} + (2 \beta_{2} - \beta_1) q^{29} + ( - 3 \beta_{2} + \beta_1) q^{31} + \beta_{6} q^{33} + ( - \beta_{6} + 6) q^{35} + (2 \beta_{2} + 3 \beta_1) q^{37} + (4 \beta_{2} - 6 \beta_1) q^{39} + 3 \beta_{7} q^{41} + (3 \beta_{5} + 2 \beta_{3}) q^{43} + ( - 5 \beta_{7} - 2 \beta_{4}) q^{45} - 2 \beta_{6} q^{47} + (\beta_{5} + 3 \beta_{3}) q^{49} + (2 \beta_{5} - 6 \beta_{3}) q^{53} + (3 \beta_{6} + 6) q^{55} - 2 \beta_{7} q^{57} + (\beta_{5} + 6 \beta_{3}) q^{59} + (3 \beta_{7} - 2 \beta_{4}) q^{61} + ( - 3 \beta_{2} + 5 \beta_1) q^{63} + (4 \beta_{2} - 8 \beta_1) q^{65} + ( - 2 \beta_{6} + 8) q^{67} + \beta_{6} q^{69} + (3 \beta_{2} - 3 \beta_1) q^{71} - \beta_1 q^{73} - 7 \beta_{4} q^{75} - \beta_{5} q^{77} + ( - 2 \beta_{7} - 3 \beta_{4}) q^{79} + ( - \beta_{6} - 1) q^{81} + (\beta_{5} + 6 \beta_{3}) q^{83} + (\beta_{5} - 6 \beta_{3}) q^{87} + ( - \beta_{6} - 6) q^{89} + ( - 2 \beta_{7} - 4 \beta_{4}) q^{91} + ( - \beta_{5} + 8 \beta_{3}) q^{93} + ( - 8 \beta_{7} + 4 \beta_{4}) q^{95} + ( - 4 \beta_{2} + \beta_1) q^{97} + (5 \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{13} + 32 q^{21} + 48 q^{35} + 48 q^{55} + 64 q^{67} - 8 q^{81} - 48 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{7} - \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 4\zeta_{24}^{4} - 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(579\) \(581\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.258819 − 0.965926i
0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0 −1.93185 − 1.93185i 0 −2.44949 − 2.44949i 0 −1.93185 + 1.93185i 0 4.46410i 0
829.2 0 −0.517638 − 0.517638i 0 −2.44949 − 2.44949i 0 −0.517638 + 0.517638i 0 − 2.46410i 0
829.3 0 0.517638 + 0.517638i 0 2.44949 + 2.44949i 0 0.517638 − 0.517638i 0 − 2.46410i 0
829.4 0 1.93185 + 1.93185i 0 2.44949 + 2.44949i 0 1.93185 − 1.93185i 0 4.46410i 0
905.1 0 −1.93185 + 1.93185i 0 −2.44949 + 2.44949i 0 −1.93185 − 1.93185i 0 − 4.46410i 0
905.2 0 −0.517638 + 0.517638i 0 −2.44949 + 2.44949i 0 −0.517638 − 0.517638i 0 2.46410i 0
905.3 0 0.517638 − 0.517638i 0 2.44949 − 2.44949i 0 0.517638 + 0.517638i 0 2.46410i 0
905.4 0 1.93185 − 1.93185i 0 2.44949 − 2.44949i 0 1.93185 + 1.93185i 0 − 4.46410i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.4
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.2.e.d 8
17.b even 2 1 inner 1156.2.e.d 8
17.c even 4 2 inner 1156.2.e.d 8
17.d even 8 1 68.2.a.a ✓ 2
17.d even 8 1 1156.2.a.a 2
17.d even 8 2 1156.2.b.c 4
17.e odd 16 8 1156.2.h.f 16
51.g odd 8 1 612.2.a.e 2
68.g odd 8 1 272.2.a.e 2
68.g odd 8 1 4624.2.a.x 2
85.k odd 8 1 1700.2.e.c 4
85.m even 8 1 1700.2.a.d 2
85.n odd 8 1 1700.2.e.c 4
119.l odd 8 1 3332.2.a.h 2
136.o even 8 1 1088.2.a.p 2
136.p odd 8 1 1088.2.a.t 2
187.i odd 8 1 8228.2.a.k 2
204.p even 8 1 2448.2.a.y 2
340.ba odd 8 1 6800.2.a.bh 2
408.bd even 8 1 9792.2.a.cs 2
408.be odd 8 1 9792.2.a.cr 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.a.a ✓ 2 17.d even 8 1
272.2.a.e 2 68.g odd 8 1
612.2.a.e 2 51.g odd 8 1
1088.2.a.p 2 136.o even 8 1
1088.2.a.t 2 136.p odd 8 1
1156.2.a.a 2 17.d even 8 1
1156.2.b.c 4 17.d even 8 2
1156.2.e.d 8 1.a even 1 1 trivial
1156.2.e.d 8 17.b even 2 1 inner
1156.2.e.d 8 17.c even 4 2 inner
1156.2.h.f 16 17.e odd 16 8
1700.2.a.d 2 85.m even 8 1
1700.2.e.c 4 85.k odd 8 1
1700.2.e.c 4 85.n odd 8 1
2448.2.a.y 2 204.p even 8 1
3332.2.a.h 2 119.l odd 8 1
4624.2.a.x 2 68.g odd 8 1
6800.2.a.bh 2 340.ba odd 8 1
8228.2.a.k 2 187.i odd 8 1
9792.2.a.cr 2 408.be odd 8 1
9792.2.a.cs 2 408.bd even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 56T_{3}^{4} + 16 \) 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
$3$ \( T^{8} + 56T^{4} + 16 \) Copy content Toggle raw display
$5$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 56T^{4} + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + 504T^{4} + 1296 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 8)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 32 T^{2} + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 504T^{4} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 1784 T^{4} + 456976 \) Copy content Toggle raw display
$37$ \( T^{8} + 17696 T^{4} + 7311616 \) Copy content Toggle raw display
$41$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 224 T^{2} + 10816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 168 T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 3104T^{4} + 256 \) Copy content Toggle raw display
$67$ \( (T^{2} - 16 T + 16)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 4536 T^{4} + 104976 \) Copy content Toggle raw display
$73$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 22136 T^{4} + 234256 \) Copy content Toggle raw display
$83$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T + 24)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 6944 T^{4} + 3748096 \) Copy content Toggle raw display
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