Properties

Label 5577.2.a.a
Level $5577$
Weight $2$
Character orbit 5577.a
Self dual yes
Analytic conductor $44.533$
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] = [5577,2,Mod(1,5577)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5577, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5577.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5577.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: \(44.5325692073\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9} - 2 q^{10} - q^{11} + q^{12} + 4 q^{14} - 2 q^{15} - q^{16} - 2 q^{17} - q^{18} - 2 q^{20} + 4 q^{21} + q^{22} + 8 q^{23} - 3 q^{24} - q^{25} - q^{27} + 4 q^{28} - 6 q^{29} + 2 q^{30} + 8 q^{31} - 5 q^{32} + q^{33} + 2 q^{34} - 8 q^{35} - q^{36} - 6 q^{37} + 6 q^{40} + 2 q^{41} - 4 q^{42} + q^{44} + 2 q^{45} - 8 q^{46} - 8 q^{47} + q^{48} + 9 q^{49} + q^{50} + 2 q^{51} + 6 q^{53} + q^{54} - 2 q^{55} - 12 q^{56} + 6 q^{58} + 4 q^{59} + 2 q^{60} + 6 q^{61} - 8 q^{62} - 4 q^{63} + 7 q^{64} - q^{66} + 4 q^{67} + 2 q^{68} - 8 q^{69} + 8 q^{70} + 3 q^{72} + 14 q^{73} + 6 q^{74} + q^{75} + 4 q^{77} - 4 q^{79} - 2 q^{80} + q^{81} - 2 q^{82} - 12 q^{83} - 4 q^{84} - 4 q^{85} + 6 q^{87} - 3 q^{88} + 6 q^{89} - 2 q^{90} - 8 q^{92} - 8 q^{93} + 8 q^{94} + 5 q^{96} - 2 q^{97} - 9 q^{98} - q^{99}+O(q^{100}) \) 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 −1.00000 −1.00000 2.00000 1.00000 −4.00000 3.00000 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(11\) \( +1 \)
\(13\) \( +1 \)

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 5577.2.a.a 1
13.b even 2 1 33.2.a.a 1
39.d odd 2 1 99.2.a.b 1
52.b odd 2 1 528.2.a.g 1
65.d even 2 1 825.2.a.a 1
65.h odd 4 2 825.2.c.a 2
91.b odd 2 1 1617.2.a.j 1
104.e even 2 1 2112.2.a.bb 1
104.h odd 2 1 2112.2.a.j 1
117.n odd 6 2 891.2.e.g 2
117.t even 6 2 891.2.e.e 2
143.d odd 2 1 363.2.a.b 1
143.l odd 10 4 363.2.e.g 4
143.n even 10 4 363.2.e.e 4
156.h even 2 1 1584.2.a.o 1
195.e odd 2 1 2475.2.a.g 1
195.s even 4 2 2475.2.c.d 2
221.b even 2 1 9537.2.a.m 1
273.g even 2 1 4851.2.a.b 1
312.b odd 2 1 6336.2.a.x 1
312.h even 2 1 6336.2.a.n 1
429.e even 2 1 1089.2.a.j 1
572.b even 2 1 5808.2.a.t 1
715.c odd 2 1 9075.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 13.b even 2 1
99.2.a.b 1 39.d odd 2 1
363.2.a.b 1 143.d odd 2 1
363.2.e.e 4 143.n even 10 4
363.2.e.g 4 143.l odd 10 4
528.2.a.g 1 52.b odd 2 1
825.2.a.a 1 65.d even 2 1
825.2.c.a 2 65.h odd 4 2
891.2.e.e 2 117.t even 6 2
891.2.e.g 2 117.n odd 6 2
1089.2.a.j 1 429.e even 2 1
1584.2.a.o 1 156.h even 2 1
1617.2.a.j 1 91.b odd 2 1
2112.2.a.j 1 104.h odd 2 1
2112.2.a.bb 1 104.e even 2 1
2475.2.a.g 1 195.e odd 2 1
2475.2.c.d 2 195.s even 4 2
4851.2.a.b 1 273.g even 2 1
5577.2.a.a 1 1.a even 1 1 trivial
5808.2.a.t 1 572.b even 2 1
6336.2.a.n 1 312.h even 2 1
6336.2.a.x 1 312.b odd 2 1
9075.2.a.q 1 715.c odd 2 1
9537.2.a.m 1 221.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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