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Properties

Label 5225.2.a.bb
Level $5225$
Weight $2$
Character orbit 5225.a
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 1045)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 32 q^{4} - 12 q^{6} + 34 q^{9} - 22 q^{11} - 8 q^{14} + 40 q^{16} - 22 q^{19} - 22 q^{21} - 22 q^{24} + 16 q^{26} - 10 q^{29} + 76 q^{31} + 56 q^{34} + 104 q^{36} - 8 q^{39} + 6 q^{41} - 32 q^{44} + 88 q^{46} + 28 q^{49} + 8 q^{51} + 38 q^{54} + 44 q^{56} + 40 q^{59} - 6 q^{61} + 140 q^{64} + 12 q^{66} + 74 q^{69} + 62 q^{71} - 26 q^{74} - 32 q^{76} + 102 q^{79} + 94 q^{81} - 38 q^{84} + 28 q^{86} + 54 q^{89} + 88 q^{91} + 36 q^{94} + 2 q^{96} - 34 q^{99}+O(q^{100}) \) 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 −2.74370 2.61424 5.52788 0 −7.17269 −3.68386 −9.67944 3.83426 0
1.2 −2.71685 −2.46712 5.38125 0 6.70278 −0.267393 −9.18634 3.08666 0
1.3 −2.58163 0.787674 4.66484 0 −2.03349 0.906789 −6.87964 −2.37957 0
1.4 −2.06343 3.28666 2.25776 0 −6.78181 3.51972 −0.531875 7.80213 0
1.5 −1.84475 −3.33093 1.40312 0 6.14475 3.04962 1.10110 8.09513 0
1.6 −1.66420 1.15649 0.769547 0 −1.92462 3.92678 2.04771 −1.66254 0
1.7 −1.60597 0.776012 0.579148 0 −1.24625 0.953282 2.28185 −2.39780 0
1.8 −1.28593 −0.496540 −0.346393 0 0.638514 −3.51780 3.01729 −2.75345 0
1.9 −1.06190 2.29282 −0.872373 0 −2.43474 −3.97400 3.05017 2.25702 0
1.10 −0.790175 −2.57534 −1.37562 0 2.03497 −0.0669205 2.66733 3.63236 0
1.11 −0.104144 −0.696995 −1.98915 0 0.0725876 3.21657 0.415445 −2.51420 0
1.12 0.104144 0.696995 −1.98915 0 0.0725876 −3.21657 −0.415445 −2.51420 0
1.13 0.790175 2.57534 −1.37562 0 2.03497 0.0669205 −2.66733 3.63236 0
1.14 1.06190 −2.29282 −0.872373 0 −2.43474 3.97400 −3.05017 2.25702 0
1.15 1.28593 0.496540 −0.346393 0 0.638514 3.51780 −3.01729 −2.75345 0
1.16 1.60597 −0.776012 0.579148 0 −1.24625 −0.953282 −2.28185 −2.39780 0
1.17 1.66420 −1.15649 0.769547 0 −1.92462 −3.92678 −2.04771 −1.66254 0
1.18 1.84475 3.33093 1.40312 0 6.14475 −3.04962 −1.10110 8.09513 0
1.19 2.06343 −3.28666 2.25776 0 −6.78181 −3.51972 0.531875 7.80213 0
1.20 2.58163 −0.787674 4.66484 0 −2.03349 −0.906789 6.87964 −2.37957 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)
\(19\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5225.2.a.bb 22
5.b even 2 1 inner 5225.2.a.bb 22
5.c odd 4 2 1045.2.b.d ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.d ✓ 22 5.c odd 4 2
5225.2.a.bb 22 1.a even 1 1 trivial
5225.2.a.bb 22 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{22} - 38 T_{2}^{20} + 620 T_{2}^{18} - 5698 T_{2}^{16} + 32546 T_{2}^{14} - 120279 T_{2}^{12} + \cdots - 484 \) Copy content Toggle raw display
\( T_{7}^{22} - 91 T_{7}^{20} + 3554 T_{7}^{18} - 77495 T_{7}^{16} + 1025919 T_{7}^{14} - 8370354 T_{7}^{12} + \cdots - 11664 \) Copy content Toggle raw display