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Properties

Label 5571.2.a.g
Level $5571$
Weight $2$
Character orbit 5571.a
Self dual yes
Analytic conductor $44.485$
Analytic rank $1$
Dimension $30$
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] = [5571,2,Mod(1,5571)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5571, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5571.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5571 = 3^{2} \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5571.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: \(44.4846589661\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 619)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 9 q^{2} + 33 q^{4} - 21 q^{5} + 2 q^{7} - 27 q^{8} + 5 q^{10} - 23 q^{11} + 9 q^{13} - 7 q^{14} + 35 q^{16} - 4 q^{17} - q^{19} - 29 q^{20} - 4 q^{23} + 35 q^{25} - q^{26} - 13 q^{28} - 90 q^{29} + 2 q^{31} - 43 q^{32} - 9 q^{34} - 9 q^{35} + 19 q^{37} - 5 q^{38} - 12 q^{40} - 59 q^{41} - 4 q^{43} - 52 q^{44} - q^{46} - 4 q^{47} + 30 q^{49} - 31 q^{50} - 12 q^{52} - 34 q^{53} - 17 q^{55} - 2 q^{56} + 6 q^{58} - 13 q^{59} + 16 q^{61} - 28 q^{62} + 37 q^{64} - 31 q^{65} - 11 q^{67} + 52 q^{68} - 40 q^{70} - 42 q^{71} - 4 q^{73} - 16 q^{74} - 42 q^{76} - 29 q^{77} + 3 q^{79} - 21 q^{80} - 43 q^{82} + 11 q^{83} + 19 q^{85} + 11 q^{86} - 47 q^{88} - 58 q^{89} - 39 q^{91} + 7 q^{92} - 46 q^{94} - 23 q^{95} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.81795 0 5.94086 2.73112 0 −1.24727 −11.1052 0 −7.69618
1.2 −2.70888 0 5.33801 −4.10380 0 −3.88170 −9.04227 0 11.1167
1.3 −2.58096 0 4.66134 1.70302 0 3.16609 −6.86880 0 −4.39543
1.4 −2.46595 0 4.08090 −2.37765 0 4.70879 −5.13138 0 5.86316
1.5 −2.38907 0 3.70768 −0.568936 0 −1.24185 −4.07977 0 1.35923
1.6 −2.34691 0 3.50799 −2.53565 0 −0.903819 −3.53912 0 5.95095
1.7 −2.07978 0 2.32549 −0.563031 0 −3.36630 −0.676957 0 1.17098
1.8 −1.95943 0 1.83936 1.58763 0 2.21007 0.314771 0 −3.11084
1.9 −1.77256 0 1.14198 −2.86335 0 2.04983 1.52089 0 5.07548
1.10 −1.54997 0 0.402413 −0.324059 0 2.73608 2.47622 0 0.502282
1.11 −1.45503 0 0.117111 2.69944 0 −1.44249 2.73966 0 −3.92777
1.12 −1.39793 0 −0.0457880 −3.90881 0 −1.82243 2.85987 0 5.46425
1.13 −0.993756 0 −1.01245 −2.26626 0 3.26853 2.99364 0 2.25211
1.14 −0.431096 0 −1.81416 −1.50079 0 −4.83036 1.64427 0 0.646986
1.15 −0.271769 0 −1.92614 −4.11968 0 2.70884 1.06700 0 1.11960
1.16 −0.262115 0 −1.93130 2.40890 0 1.00913 1.03045 0 −0.631409
1.17 −0.258847 0 −1.93300 2.46163 0 1.88069 1.01804 0 −0.637185
1.18 0.192901 0 −1.96279 −2.33409 0 −3.05253 −0.764427 0 −0.450249
1.19 0.386745 0 −1.85043 3.75067 0 −2.84697 −1.48913 0 1.45055
1.20 0.394547 0 −1.84433 −1.92856 0 3.31459 −1.51677 0 −0.760909
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(619\) \(-1\)

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 5571.2.a.g 30
3.b odd 2 1 619.2.a.b ✓ 30
12.b even 2 1 9904.2.a.n 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
619.2.a.b ✓ 30 3.b odd 2 1
5571.2.a.g 30 1.a even 1 1 trivial
9904.2.a.n 30 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 9 T_{2}^{29} - 6 T_{2}^{28} - 276 T_{2}^{27} - 458 T_{2}^{26} + 3470 T_{2}^{25} + \cdots - 288 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5571))\). Copy content Toggle raw display