Properties

Label 4400.2.b.m
Level $4400$
Weight $2$
Character orbit 4400.b
Analytic conductor $35.134$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4400,2,Mod(4049,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4400.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), 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: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 440)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q - \beta q^{7} + 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{7} + 3 q^{9} + q^{11} - 8 q^{19} + 4 \beta q^{23} - 10 q^{29} - 8 q^{31} + 5 \beta q^{37} - 2 q^{41} + 3 \beta q^{43} - 4 \beta q^{47} + 3 q^{49} + 7 \beta q^{53} - 4 q^{59} + 10 q^{61} - 3 \beta q^{63} + 2 \beta q^{67} - 4 \beta q^{73} - \beta q^{77} - 4 q^{79} + 9 q^{81} - 5 \beta q^{83} - 6 q^{89} + 5 \beta q^{97} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{9} + 2 q^{11} - 16 q^{19} - 20 q^{29} - 16 q^{31} - 4 q^{41} + 6 q^{49} - 8 q^{59} + 20 q^{61} - 8 q^{79} + 18 q^{81} - 12 q^{89} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4049.1
1.00000i
1.00000i
0 0 0 0 0 2.00000i 0 3.00000 0
4049.2 0 0 0 0 0 2.00000i 0 3.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4400.2.b.m 2
4.b odd 2 1 2200.2.b.d 2
5.b even 2 1 inner 4400.2.b.m 2
5.c odd 4 1 880.2.a.f 1
5.c odd 4 1 4400.2.a.o 1
15.e even 4 1 7920.2.a.bg 1
20.d odd 2 1 2200.2.b.d 2
20.e even 4 1 440.2.a.a 1
20.e even 4 1 2200.2.a.f 1
40.i odd 4 1 3520.2.a.u 1
40.k even 4 1 3520.2.a.t 1
55.e even 4 1 9680.2.a.n 1
60.l odd 4 1 3960.2.a.p 1
220.i odd 4 1 4840.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.a 1 20.e even 4 1
880.2.a.f 1 5.c odd 4 1
2200.2.a.f 1 20.e even 4 1
2200.2.b.d 2 4.b odd 2 1
2200.2.b.d 2 20.d odd 2 1
3520.2.a.t 1 40.k even 4 1
3520.2.a.u 1 40.i odd 4 1
3960.2.a.p 1 60.l odd 4 1
4400.2.a.o 1 5.c odd 4 1
4400.2.b.m 2 1.a even 1 1 trivial
4400.2.b.m 2 5.b even 2 1 inner
4840.2.a.c 1 220.i odd 4 1
7920.2.a.bg 1 15.e even 4 1
9680.2.a.n 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4400, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T + 10)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 196 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 100 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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