↑

Properties

Label 1133.1.s.a
Level $1133$
Weight $1$
Character orbit 1133.s
Analytic conductor $0.565$
Analytic rank $0$
Dimension $16$
Projective image $D_{17}$
CM discriminant -11
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1133,1,Mod(76,1133)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1133, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([17, 22]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1133.76");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1133 = 11 \cdot 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1133.s (of order \(34\), degree \(16\), 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: \(0.565440659313\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{17}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{17} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{34}^{9} - \zeta_{34}^{5}) q^{3} - \zeta_{34}^{11} q^{4} + (\zeta_{34}^{10} - \zeta_{34}^{3}) q^{5} + (\zeta_{34}^{14} + \zeta_{34}^{10} - \zeta_{34}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{34}^{9} - \zeta_{34}^{5}) q^{3} - \zeta_{34}^{11} q^{4} + (\zeta_{34}^{10} - \zeta_{34}^{3}) q^{5} + (\zeta_{34}^{14} + \zeta_{34}^{10} - \zeta_{34}) q^{9} + \zeta_{34}^{14} q^{11} + (\zeta_{34}^{16} - \zeta_{34}^{3}) q^{12} + ( - \zeta_{34}^{15} + \cdots + \zeta_{34}^{2}) q^{15} + \cdots + ( - \zeta_{34}^{15} + \cdots - \zeta_{34}^{7}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} - q^{4} - 2 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} - q^{4} - 2 q^{5} - 3 q^{9} - q^{11} - 2 q^{12} - 4 q^{15} - q^{16} - 2 q^{20} - 2 q^{23} - 3 q^{25} - 4 q^{27} - 2 q^{31} - 2 q^{33} - 3 q^{36} + 15 q^{37} - q^{44} + 11 q^{45} - 2 q^{47} - 2 q^{48} - q^{49} - 2 q^{53} + 15 q^{55} - 2 q^{59} - 4 q^{60} - q^{64} + 15 q^{67} - 4 q^{69} - 2 q^{71} - 6 q^{75} - 2 q^{80} - 5 q^{81} - 2 q^{89} - 2 q^{92} - 4 q^{93} - 2 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1133\mathbb{Z}\right)^\times\).

\(n\) \(310\) \(1035\)
\(\chi(n)\) \(-1\) \(-\zeta_{34}^{13}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0.850217 + 0.526432i
0.850217 − 0.526432i
−0.0922684 − 0.995734i
−0.445738 + 0.895163i
0.273663 + 0.961826i
−0.932472 − 0.361242i
0.602635 − 0.798017i
−0.0922684 + 0.995734i
−0.932472 + 0.361242i
0.982973 − 0.183750i
0.982973 + 0.183750i
−0.445738 − 0.895163i
0.273663 − 0.961826i
−0.739009 − 0.673696i
0.602635 + 0.798017i
−0.739009 + 0.673696i
0 0.658809 + 0.600584i −0.982973 + 0.183750i 0.831277 − 1.66943i 0 0 0 −0.0189399 − 0.204394i 0
164.1 0 0.658809 − 0.600584i −0.982973 − 0.183750i 0.831277 + 1.66943i 0 0 0 −0.0189399 + 0.204394i 0
175.1 0 1.18475 + 1.56886i −0.850217 − 0.526432i −0.876298 − 0.163808i 0 0 0 −0.784029 + 2.75558i 0
219.1 0 −0.111208 + 1.20013i 0.932472 + 0.361242i −0.890705 + 1.17948i 0 0 0 −0.444966 − 0.0831786i 0
285.1 0 −1.58561 + 0.614268i 0.0922684 − 0.995734i 1.67148 + 1.03494i 0 0 0 1.39782 − 1.27428i 0
318.1 0 −1.25664 + 0.778076i −0.602635 − 0.798017i −0.404479 + 0.368731i 0 0 0 0.527993 − 1.06035i 0
373.1 0 0.538007 − 0.100571i 0.739009 − 0.673696i −0.0505009 + 0.177492i 0 0 0 −0.653136 + 0.253026i 0
505.1 0 1.18475 − 1.56886i −0.850217 + 0.526432i −0.876298 + 0.163808i 0 0 0 −0.784029 − 2.75558i 0
538.1 0 −1.25664 − 0.778076i −0.602635 + 0.798017i −0.404479 − 0.368731i 0 0 0 0.527993 + 1.06035i 0
549.1 0 −0.510366 + 1.79375i 0.445738 + 0.895163i −1.12388 − 0.435393i 0 0 0 −2.10685 − 1.30451i 0
615.1 0 −0.510366 − 1.79375i 0.445738 − 0.895163i −1.12388 + 0.435393i 0 0 0 −2.10685 + 1.30451i 0
626.1 0 −0.111208 − 1.20013i 0.932472 − 0.361242i −0.890705 − 1.17948i 0 0 0 −0.444966 + 0.0831786i 0
648.1 0 −1.58561 − 0.614268i 0.0922684 + 0.995734i 1.67148 − 1.03494i 0 0 0 1.39782 + 1.27428i 0
802.1 0 0.0822551 − 0.165190i −0.273663 + 0.961826i −0.156896 + 1.69318i 0 0 0 0.582113 + 0.770842i 0
890.1 0 0.538007 + 0.100571i 0.739009 + 0.673696i −0.0505009 − 0.177492i 0 0 0 −0.653136 − 0.253026i 0
1044.1 0 0.0822551 + 0.165190i −0.273663 − 0.961826i −0.156896 − 1.69318i 0 0 0 0.582113 − 0.770842i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
103.e even 17 1 inner
1133.s odd 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1133.1.s.a ✓ 16
11.b odd 2 1 CM 1133.1.s.a ✓ 16
103.e even 17 1 inner 1133.1.s.a ✓ 16
1133.s odd 34 1 inner 1133.1.s.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1133.1.s.a ✓ 16 1.a even 1 1 trivial
1133.1.s.a ✓ 16 11.b odd 2 1 CM
1133.1.s.a ✓ 16 103.e even 17 1 inner
1133.1.s.a ✓ 16 1133.s odd 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1133, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + T^{15} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{16} - 15 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( (T^{8} + T^{7} - 7 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} - 15 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
show more
show less