Properties

Label 1100.3.f.b
Level $1100$
Weight $3$
Character orbit 1100.f
Self dual yes
Analytic conductor $29.973$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,3,Mod(901,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.901");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1100.f (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: \(29.9728290796\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 220)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + 13) q^{9} + 11 q^{11} + ( - \beta_{2} + 5 \beta_1) q^{23} + (2 \beta_{2} + 13 \beta_1) q^{27} + ( - 3 \beta_{3} - 20) q^{31} + 11 \beta_1 q^{33} + ( - \beta_{2} - 7 \beta_1) q^{37} + ( - 3 \beta_{2} - 6 \beta_1) q^{47} + 49 q^{49} + (3 \beta_{2} + 6 \beta_1) q^{53} + ( - 3 \beta_{3} + 52) q^{59} + (2 \beta_{2} + 11 \beta_1) q^{67} + ( - \beta_{3} + 106) q^{69} + (3 \beta_{3} + 68) q^{71} + (16 \beta_{3} + 177) q^{81} + ( - 9 \beta_{3} + 44) q^{89} + ( - 6 \beta_{2} - 47 \beta_1) q^{93} + ( - 2 \beta_{2} - 23 \beta_1) q^{97} + (11 \beta_{3} + 143) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 50 q^{9} + 44 q^{11} - 74 q^{31} + 196 q^{49} + 214 q^{59} + 426 q^{69} + 266 q^{71} + 676 q^{81} + 194 q^{89} + 550 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -6\nu^{3} + 44\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\nu^{2} - 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 6\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 18 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 11\beta_1 ) / 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
−2.52434
0.792287
−0.792287
2.52434
0 −5.98844 0 0 0 0 0 26.8614 0
901.2 0 −2.67181 0 0 0 0 0 −1.86141 0
901.3 0 2.67181 0 0 0 0 0 −1.86141 0
901.4 0 5.98844 0 0 0 0 0 26.8614 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 \(\Q(\sqrt{-11}) \)
5.b even 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.3.f.b 4
5.b even 2 1 inner 1100.3.f.b 4
5.c odd 4 2 220.3.e.a 4
11.b odd 2 1 CM 1100.3.f.b 4
15.e even 4 2 1980.3.p.a 4
20.e even 4 2 880.3.i.e 4
55.d odd 2 1 inner 1100.3.f.b 4
55.e even 4 2 220.3.e.a 4
165.l odd 4 2 1980.3.p.a 4
220.i odd 4 2 880.3.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.e.a 4 5.c odd 4 2
220.3.e.a 4 55.e even 4 2
880.3.i.e 4 20.e even 4 2
880.3.i.e 4 220.i odd 4 2
1100.3.f.b 4 1.a even 1 1 trivial
1100.3.f.b 4 5.b even 2 1 inner
1100.3.f.b 4 11.b odd 2 1 CM
1100.3.f.b 4 55.d odd 2 1 inner
1980.3.p.a 4 15.e even 4 2
1980.3.p.a 4 165.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 43T_{3}^{2} + 256 \) acting on \(S_{3}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 43T^{2} + 256 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 2283 T^{2} + 484416 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 37 T - 1514)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 3363 T^{2} + 553536 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6336)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 6336)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 107 T + 1006)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 10203 T^{2} + 10653696 \) Copy content Toggle raw display
$71$ \( (T^{2} - 133 T + 2566)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 97 T - 14354)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 27843 T^{2} + 147456 \) Copy content Toggle raw display
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