Properties

Label 2205.4.a.n
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 245)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - 7 q^{4} + 5 q^{5} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 7 q^{4} + 5 q^{5} + 15 q^{8} - 5 q^{10} + 44 q^{11} + 6 q^{13} + 41 q^{16} + 24 q^{17} - 114 q^{19} - 35 q^{20} - 44 q^{22} + 52 q^{23} + 25 q^{25} - 6 q^{26} - 146 q^{29} - 276 q^{31} - 161 q^{32} - 24 q^{34} - 210 q^{37} + 114 q^{38} + 75 q^{40} - 444 q^{41} + 492 q^{43} - 308 q^{44} - 52 q^{46} + 612 q^{47} - 25 q^{50} - 42 q^{52} - 50 q^{53} + 220 q^{55} + 146 q^{58} - 294 q^{59} + 450 q^{61} + 276 q^{62} - 167 q^{64} + 30 q^{65} - 668 q^{67} - 168 q^{68} + 308 q^{71} + 12 q^{73} + 210 q^{74} + 798 q^{76} + 596 q^{79} + 205 q^{80} + 444 q^{82} + 966 q^{83} + 120 q^{85} - 492 q^{86} + 660 q^{88} + 408 q^{89} - 364 q^{92} - 612 q^{94} - 570 q^{95} - 1200 q^{97}+O(q^{100}) \) 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
−1.00000 0 −7.00000 5.00000 0 0 15.0000 0 −5.00000
\(n\): 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.n 1
3.b odd 2 1 245.4.a.c yes 1
7.b odd 2 1 2205.4.a.k 1
15.d odd 2 1 1225.4.a.f 1
21.c even 2 1 245.4.a.b 1
21.g even 6 2 245.4.e.d 2
21.h odd 6 2 245.4.e.c 2
105.g even 2 1 1225.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.b 1 21.c even 2 1
245.4.a.c yes 1 3.b odd 2 1
245.4.e.c 2 21.h odd 6 2
245.4.e.d 2 21.g even 6 2
1225.4.a.f 1 15.d odd 2 1
1225.4.a.g 1 105.g even 2 1
2205.4.a.k 1 7.b odd 2 1
2205.4.a.n 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 44 \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display
\( T_{17} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) 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
$11$ \( T - 44 \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T - 24 \) Copy content Toggle raw display
$19$ \( T + 114 \) Copy content Toggle raw display
$23$ \( T - 52 \) Copy content Toggle raw display
$29$ \( T + 146 \) Copy content Toggle raw display
$31$ \( T + 276 \) Copy content Toggle raw display
$37$ \( T + 210 \) Copy content Toggle raw display
$41$ \( T + 444 \) Copy content Toggle raw display
$43$ \( T - 492 \) Copy content Toggle raw display
$47$ \( T - 612 \) Copy content Toggle raw display
$53$ \( T + 50 \) Copy content Toggle raw display
$59$ \( T + 294 \) Copy content Toggle raw display
$61$ \( T - 450 \) Copy content Toggle raw display
$67$ \( T + 668 \) Copy content Toggle raw display
$71$ \( T - 308 \) Copy content Toggle raw display
$73$ \( T - 12 \) Copy content Toggle raw display
$79$ \( T - 596 \) Copy content Toggle raw display
$83$ \( T - 966 \) Copy content Toggle raw display
$89$ \( T - 408 \) Copy content Toggle raw display
$97$ \( T + 1200 \) Copy content Toggle raw display
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