Properties

Label 2205.4.a.a
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,4,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

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: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 5 q^{2} + 17 q^{4} - 5 q^{5} - 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} + 17 q^{4} - 5 q^{5} - 45 q^{8} + 25 q^{10} - 50 q^{11} + 20 q^{13} + 89 q^{16} - 10 q^{17} + 44 q^{19} - 85 q^{20} + 250 q^{22} - 120 q^{23} + 25 q^{25} - 100 q^{26} + 50 q^{29} - 108 q^{31} - 85 q^{32} + 50 q^{34} - 40 q^{37} - 220 q^{38} + 225 q^{40} + 400 q^{41} + 280 q^{43} - 850 q^{44} + 600 q^{46} - 280 q^{47} - 125 q^{50} + 340 q^{52} + 610 q^{53} + 250 q^{55} - 250 q^{58} + 50 q^{59} + 518 q^{61} + 540 q^{62} - 287 q^{64} - 100 q^{65} - 180 q^{67} - 170 q^{68} - 700 q^{71} + 410 q^{73} + 200 q^{74} + 748 q^{76} - 516 q^{79} - 445 q^{80} - 2000 q^{82} + 660 q^{83} + 50 q^{85} - 1400 q^{86} + 2250 q^{88} - 1500 q^{89} - 2040 q^{92} + 1400 q^{94} - 220 q^{95} + 1630 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−5.00000 0 17.0000 −5.00000 0 0 −45.0000 0 25.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.4.a.a 1
3.b odd 2 1 2205.4.a.t 1
7.b odd 2 1 45.4.a.a 1
21.c even 2 1 45.4.a.e yes 1
28.d even 2 1 720.4.a.bc 1
35.c odd 2 1 225.4.a.h 1
35.f even 4 2 225.4.b.a 2
63.l odd 6 2 405.4.e.n 2
63.o even 6 2 405.4.e.b 2
84.h odd 2 1 720.4.a.o 1
105.g even 2 1 225.4.a.a 1
105.k odd 4 2 225.4.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 7.b odd 2 1
45.4.a.e yes 1 21.c even 2 1
225.4.a.a 1 105.g even 2 1
225.4.a.h 1 35.c odd 2 1
225.4.b.a 2 35.f even 4 2
225.4.b.b 2 105.k odd 4 2
405.4.e.b 2 63.o even 6 2
405.4.e.n 2 63.l odd 6 2
720.4.a.o 1 84.h odd 2 1
720.4.a.bc 1 28.d even 2 1
2205.4.a.a 1 1.a even 1 1 trivial
2205.4.a.t 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2} + 5 \) Copy content Toggle raw display
\( T_{11} + 50 \) Copy content Toggle raw display
\( T_{13} - 20 \) Copy content Toggle raw display
\( T_{17} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 50 \) Copy content Toggle raw display
$13$ \( T - 20 \) Copy content Toggle raw display
$17$ \( T + 10 \) Copy content Toggle raw display
$19$ \( T - 44 \) Copy content Toggle raw display
$23$ \( T + 120 \) Copy content Toggle raw display
$29$ \( T - 50 \) Copy content Toggle raw display
$31$ \( T + 108 \) Copy content Toggle raw display
$37$ \( T + 40 \) Copy content Toggle raw display
$41$ \( T - 400 \) Copy content Toggle raw display
$43$ \( T - 280 \) Copy content Toggle raw display
$47$ \( T + 280 \) Copy content Toggle raw display
$53$ \( T - 610 \) Copy content Toggle raw display
$59$ \( T - 50 \) Copy content Toggle raw display
$61$ \( T - 518 \) Copy content Toggle raw display
$67$ \( T + 180 \) Copy content Toggle raw display
$71$ \( T + 700 \) Copy content Toggle raw display
$73$ \( T - 410 \) Copy content Toggle raw display
$79$ \( T + 516 \) Copy content Toggle raw display
$83$ \( T - 660 \) Copy content Toggle raw display
$89$ \( T + 1500 \) Copy content Toggle raw display
$97$ \( T - 1630 \) Copy content Toggle raw display
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