Properties

Label 11.9.b.a
Level $11$
Weight $9$
Character orbit 11.b
Self dual yes
Analytic conductor $4.481$
Analytic rank $0$
Dimension $1$
CM discriminant -11
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [11,9,Mod(10,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 11.b (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: yes
Analytic conductor: \(4.48116471067\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 113 q^{3} + 256 q^{4} + 1151 q^{5} + 6208 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 113 q^{3} + 256 q^{4} + 1151 q^{5} + 6208 q^{9} + 14641 q^{11} - 28928 q^{12} - 130063 q^{15} + 65536 q^{16} + 294656 q^{20} - 531793 q^{23} + 934176 q^{25} + 39889 q^{27} - 1541233 q^{31} - 1654433 q^{33} + 1589248 q^{36} + 716447 q^{37} + 3748096 q^{44} + 7145408 q^{45} - 6080638 q^{47} - 7405568 q^{48} + 5764801 q^{49} - 15265438 q^{53} + 16851791 q^{55} - 4101553 q^{59} - 33296128 q^{60} + 16777216 q^{64} + 19806767 q^{67} + 60092609 q^{69} + 7043087 q^{71} - 105561888 q^{75} + 75431936 q^{80} - 45238145 q^{81} - 84100993 q^{89} - 136139008 q^{92} + 174159329 q^{93} - 81155713 q^{97} + 90891328 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0
0 −113.000 256.000 1151.00 0 0 0 6208.00 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.9.b.a 1
3.b odd 2 1 99.9.c.a 1
4.b odd 2 1 176.9.h.a 1
11.b odd 2 1 CM 11.9.b.a 1
33.d even 2 1 99.9.c.a 1
44.c even 2 1 176.9.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.9.b.a 1 1.a even 1 1 trivial
11.9.b.a 1 11.b odd 2 1 CM
99.9.c.a 1 3.b odd 2 1
99.9.c.a 1 33.d even 2 1
176.9.h.a 1 4.b odd 2 1
176.9.h.a 1 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{9}^{\mathrm{new}}(11, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 113 \) Copy content Toggle raw display
$5$ \( T - 1151 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 14641 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 531793 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 1541233 \) Copy content Toggle raw display
$37$ \( T - 716447 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 6080638 \) Copy content Toggle raw display
$53$ \( T + 15265438 \) Copy content Toggle raw display
$59$ \( T + 4101553 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 19806767 \) Copy content Toggle raw display
$71$ \( T - 7043087 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 84100993 \) Copy content Toggle raw display
$97$ \( T + 81155713 \) Copy content Toggle raw display
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