Properties

Label 1099.1.b.d
Level $1099$
Weight $1$
Character orbit 1099.b
Analytic conductor $0.548$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1099,1,Mod(1098,1099)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1099, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1099.1098");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1099 = 7 \cdot 157 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1099.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: no
Analytic conductor: \(0.548472448884\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.1099.1
Artin image: $\GL(2,3)$
Artin field: Galois closure of 8.2.1327373299.2

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - \beta q^{3} - q^{4} + q^{5} - 2 q^{6} + q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - \beta q^{3} - q^{4} + q^{5} - 2 q^{6} + q^{7} - q^{9} - \beta q^{10} - q^{11} + \beta q^{12} + \beta q^{13} - \beta q^{14} - \beta q^{15} - q^{16} + \beta q^{17} + \beta q^{18} - \beta q^{19} - q^{20} - \beta q^{21} + \beta q^{22} - \beta q^{23} + 2 q^{26} - q^{28} - 2 q^{30} + \beta q^{31} + \beta q^{32} + \beta q^{33} + 2 q^{34} + q^{35} + q^{36} - q^{37} - 2 q^{38} + 2 q^{39} - 2 q^{42} + \beta q^{43} + q^{44} - q^{45} - 2 q^{46} + \beta q^{48} + q^{49} + 2 q^{51} - \beta q^{52} - q^{55} - 2 q^{57} + q^{59} + \beta q^{60} + q^{61} + 2 q^{62} - q^{63} + q^{64} + \beta q^{65} + 2 q^{66} + q^{67} - \beta q^{68} - 2 q^{69} - \beta q^{70} - q^{71} + q^{73} + \beta q^{74} + \beta q^{76} - q^{77} - 2 \beta q^{78} + \beta q^{79} - q^{80} - q^{81} + \beta q^{84} + \beta q^{85} + 2 q^{86} + \beta q^{90} + \beta q^{91} + \beta q^{92} + 2 q^{93} - \beta q^{95} + 2 q^{96} - q^{97} - \beta q^{98} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{5} - 4 q^{6} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{5} - 4 q^{6} + 2 q^{7} - 2 q^{9} - 2 q^{11} - 2 q^{16} - 2 q^{20} + 4 q^{26} - 2 q^{28} - 4 q^{30} + 4 q^{34} + 2 q^{35} + 2 q^{36} - 2 q^{37} - 4 q^{38} + 4 q^{39} - 4 q^{42} + 2 q^{44} - 2 q^{45} - 4 q^{46} + 2 q^{49} + 4 q^{51} - 2 q^{55} - 4 q^{57} + 2 q^{59} + 2 q^{61} + 4 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{66} + 2 q^{67} - 4 q^{69} - 2 q^{71} + 2 q^{73} - 2 q^{77} - 2 q^{80} - 2 q^{81} + 4 q^{86} + 4 q^{93} + 4 q^{96} - 2 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1099\mathbb{Z}\right)^\times\).

\(n\) \(162\) \(472\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1098.1
1.41421i
1.41421i
1.41421i 1.41421i −1.00000 1.00000 −2.00000 1.00000 0 −1.00000 1.41421i
1098.2 1.41421i 1.41421i −1.00000 1.00000 −2.00000 1.00000 0 −1.00000 1.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
1099.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1099.1.b.d yes 2
7.b odd 2 1 1099.1.b.c 2
157.b even 2 1 1099.1.b.c 2
1099.b odd 2 1 inner 1099.1.b.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1099.1.b.c 2 7.b odd 2 1
1099.1.b.c 2 157.b even 2 1
1099.1.b.d yes 2 1.a even 1 1 trivial
1099.1.b.d yes 2 1099.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1099, [\chi])\):

\( T_{2}^{2} + 2 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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