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Properties

Label 5225.2.a.bc
Level $5225$
Weight $2$
Character orbit 5225.a
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $30$
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: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46} + 60 q^{49} + 64 q^{51} + 62 q^{54} - 32 q^{56} - 14 q^{59} + 78 q^{61} + 90 q^{64} + 12 q^{66} - 28 q^{69} + 20 q^{71} + 42 q^{74} + 42 q^{76} + 102 q^{79} + 42 q^{81} + 98 q^{84} - 52 q^{86} - 8 q^{89} + 56 q^{91} + 40 q^{94} - 74 q^{96} + 40 q^{99}+O(q^{100}) \) 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 −2.79219 −1.83665 5.79631 0 5.12828 0.971452 −10.6000 0.373298 0
1.2 −2.77451 2.74433 5.69792 0 −7.61418 3.20844 −10.2599 4.53135 0
1.3 −2.44393 −0.661177 3.97280 0 1.61587 −2.05193 −4.82139 −2.56284 0
1.4 −2.39831 0.881728 3.75187 0 −2.11465 −2.42713 −4.20151 −2.22256 0
1.5 −2.39110 −3.31399 3.71736 0 7.92409 0.907652 −4.10637 7.98255 0
1.6 −2.14697 −2.93729 2.60947 0 6.30628 −3.64055 −1.30851 5.62770 0
1.7 −1.98136 2.35379 1.92578 0 −4.66371 −1.46477 0.147060 2.54034 0
1.8 −1.66433 1.25863 0.769991 0 −2.09478 4.13429 2.04714 −1.41584 0
1.9 −1.65738 −1.46266 0.746921 0 2.42418 5.18504 2.07683 −0.860633 0
1.10 −1.45151 0.0791235 0.106883 0 −0.114849 −2.96150 2.74788 −2.99374 0
1.11 −0.961626 −0.510054 −1.07528 0 0.490481 −0.688271 2.95726 −2.73984 0
1.12 −0.661223 3.01131 −1.56278 0 −1.99115 0.592449 2.35579 6.06799 0
1.13 −0.598278 1.31473 −1.64206 0 −0.786572 −0.976473 2.17897 −1.27149 0
1.14 −0.379302 −2.60759 −1.85613 0 0.989065 −4.15654 1.46264 3.79954 0
1.15 −0.202376 −2.47875 −1.95904 0 0.501638 5.06314 0.801215 3.14418 0
1.16 0.202376 2.47875 −1.95904 0 0.501638 −5.06314 −0.801215 3.14418 0
1.17 0.379302 2.60759 −1.85613 0 0.989065 4.15654 −1.46264 3.79954 0
1.18 0.598278 −1.31473 −1.64206 0 −0.786572 0.976473 −2.17897 −1.27149 0
1.19 0.661223 −3.01131 −1.56278 0 −1.99115 −0.592449 −2.35579 6.06799 0
1.20 0.961626 0.510054 −1.07528 0 0.490481 0.688271 −2.95726 −2.73984 0
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
\(5\) \(-1\)
\(11\) \(-1\)
\(19\) \(-1\)

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.bc 30
5.b even 2 1 inner 5225.2.a.bc 30
5.c odd 4 2 1045.2.b.e ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.e ✓ 30 5.c odd 4 2
5225.2.a.bc 30 1.a even 1 1 trivial
5225.2.a.bc 30 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{30} - 51 T_{2}^{28} + 1161 T_{2}^{26} - 15571 T_{2}^{24} + 136754 T_{2}^{22} - 826847 T_{2}^{20} + \cdots - 2916 \) Copy content Toggle raw display
\( T_{7}^{30} - 135 T_{7}^{28} + 7918 T_{7}^{26} - 265743 T_{7}^{24} + 5660203 T_{7}^{22} + \cdots - 1597441024 \) Copy content Toggle raw display