Properties

Label 2205.4.a.g
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 35)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{2} + q^{4} + 5 q^{5} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} + q^{4} + 5 q^{5} + 21 q^{8} - 15 q^{10} + 45 q^{11} - 59 q^{13} - 71 q^{16} - 54 q^{17} + 121 q^{19} + 5 q^{20} - 135 q^{22} - 69 q^{23} + 25 q^{25} + 177 q^{26} + 162 q^{29} + 88 q^{31} + 45 q^{32} + 162 q^{34} - 259 q^{37} - 363 q^{38} + 105 q^{40} + 195 q^{41} - 286 q^{43} + 45 q^{44} + 207 q^{46} + 45 q^{47} - 75 q^{50} - 59 q^{52} - 597 q^{53} + 225 q^{55} - 486 q^{58} - 360 q^{59} - 392 q^{61} - 264 q^{62} + 433 q^{64} - 295 q^{65} - 280 q^{67} - 54 q^{68} - 48 q^{71} - 668 q^{73} + 777 q^{74} + 121 q^{76} + 782 q^{79} - 355 q^{80} - 585 q^{82} + 768 q^{83} - 270 q^{85} + 858 q^{86} + 945 q^{88} - 1194 q^{89} - 69 q^{92} - 135 q^{94} + 605 q^{95} - 902 q^{97}+O(q^{100}) \) 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
−3.00000 0 1.00000 5.00000 0 0 21.0000 0 −15.0000
\(n\): 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.g 1
3.b odd 2 1 245.4.a.f 1
7.b odd 2 1 2205.4.a.e 1
7.d odd 6 2 315.4.j.b 2
15.d odd 2 1 1225.4.a.a 1
21.c even 2 1 245.4.a.e 1
21.g even 6 2 35.4.e.a 2
21.h odd 6 2 245.4.e.a 2
84.j odd 6 2 560.4.q.b 2
105.g even 2 1 1225.4.a.b 1
105.p even 6 2 175.4.e.b 2
105.w odd 12 4 175.4.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 21.g even 6 2
175.4.e.b 2 105.p even 6 2
175.4.k.b 4 105.w odd 12 4
245.4.a.e 1 21.c even 2 1
245.4.a.f 1 3.b odd 2 1
245.4.e.a 2 21.h odd 6 2
315.4.j.b 2 7.d odd 6 2
560.4.q.b 2 84.j odd 6 2
1225.4.a.a 1 15.d odd 2 1
1225.4.a.b 1 105.g even 2 1
2205.4.a.e 1 7.b odd 2 1
2205.4.a.g 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2} + 3 \) Copy content Toggle raw display
\( T_{11} - 45 \) Copy content Toggle raw display
\( T_{13} + 59 \) Copy content Toggle raw display
\( T_{17} + 54 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) 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
$11$ \( T - 45 \) Copy content Toggle raw display
$13$ \( T + 59 \) Copy content Toggle raw display
$17$ \( T + 54 \) Copy content Toggle raw display
$19$ \( T - 121 \) Copy content Toggle raw display
$23$ \( T + 69 \) Copy content Toggle raw display
$29$ \( T - 162 \) Copy content Toggle raw display
$31$ \( T - 88 \) Copy content Toggle raw display
$37$ \( T + 259 \) Copy content Toggle raw display
$41$ \( T - 195 \) Copy content Toggle raw display
$43$ \( T + 286 \) Copy content Toggle raw display
$47$ \( T - 45 \) Copy content Toggle raw display
$53$ \( T + 597 \) Copy content Toggle raw display
$59$ \( T + 360 \) Copy content Toggle raw display
$61$ \( T + 392 \) Copy content Toggle raw display
$67$ \( T + 280 \) Copy content Toggle raw display
$71$ \( T + 48 \) Copy content Toggle raw display
$73$ \( T + 668 \) Copy content Toggle raw display
$79$ \( T - 782 \) Copy content Toggle raw display
$83$ \( T - 768 \) Copy content Toggle raw display
$89$ \( T + 1194 \) Copy content Toggle raw display
$97$ \( T + 902 \) Copy content Toggle raw display
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