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Properties

Label 9025.2.a.ct
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $24$
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] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, 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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 95)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9} + 12 q^{11} - 24 q^{14} + 6 q^{16} - 6 q^{21} - 42 q^{24} - 12 q^{26} - 36 q^{29} - 42 q^{31} - 6 q^{34} - 6 q^{36} + 24 q^{39} - 60 q^{41} - 30 q^{44} - 6 q^{46} + 12 q^{49} - 30 q^{51} - 24 q^{54} - 18 q^{56} - 60 q^{59} + 30 q^{61} + 36 q^{66} - 66 q^{69} - 96 q^{71} + 24 q^{74} - 72 q^{79} - 96 q^{81} + 54 q^{84} - 108 q^{86} - 84 q^{89} - 96 q^{91} - 36 q^{94} - 120 q^{96}+O(q^{100}) \) 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 −2.68669 2.14658 5.21828 0 −5.76720 2.78303 −8.64651 1.60782 0
1.2 −2.37097 2.28512 3.62149 0 −5.41794 −1.63677 −3.84450 2.22176 0
1.3 −2.32462 −1.14858 3.40387 0 2.67001 −0.143302 −3.26348 −1.68077 0
1.4 −1.96177 −0.187708 1.84854 0 0.368240 0.677067 0.297123 −2.96477 0
1.5 −1.78468 −2.38377 1.18508 0 4.25426 4.23911 1.45438 2.68235 0
1.6 −1.61907 −1.18857 0.621387 0 1.92438 −2.23190 2.23207 −1.58730 0
1.7 −1.47917 2.48321 0.187941 0 −3.67308 3.24988 2.68034 3.16631 0
1.8 −1.22159 −0.804421 −0.507728 0 0.982669 −3.79180 3.06341 −2.35291 0
1.9 −1.04740 0.531453 −0.902948 0 −0.556645 2.74033 3.04056 −2.71756 0
1.10 −0.449373 1.95684 −1.79806 0 −0.879352 2.06079 1.70675 0.829224 0
1.11 −0.249751 2.30156 −1.93762 0 −0.574816 −3.96043 0.983424 2.29718 0
1.12 −0.244477 −2.73837 −1.94023 0 0.669469 1.94027 0.963297 4.49866 0
1.13 0.244477 2.73837 −1.94023 0 0.669469 −1.94027 −0.963297 4.49866 0
1.14 0.249751 −2.30156 −1.93762 0 −0.574816 3.96043 −0.983424 2.29718 0
1.15 0.449373 −1.95684 −1.79806 0 −0.879352 −2.06079 −1.70675 0.829224 0
1.16 1.04740 −0.531453 −0.902948 0 −0.556645 −2.74033 −3.04056 −2.71756 0
1.17 1.22159 0.804421 −0.507728 0 0.982669 3.79180 −3.06341 −2.35291 0
1.18 1.47917 −2.48321 0.187941 0 −3.67308 −3.24988 −2.68034 3.16631 0
1.19 1.61907 1.18857 0.621387 0 1.92438 2.23190 −2.23207 −1.58730 0
1.20 1.78468 2.38377 1.18508 0 4.25426 −4.23911 −1.45438 2.68235 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

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

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 9025.2.a.ct 24
5.b even 2 1 inner 9025.2.a.ct 24
5.c odd 4 2 1805.2.b.l 24
19.b odd 2 1 9025.2.a.cu 24
19.f odd 18 2 475.2.l.f 48
95.d odd 2 1 9025.2.a.cu 24
95.g even 4 2 1805.2.b.k 24
95.o odd 18 2 475.2.l.f 48
95.r even 36 4 95.2.p.a ✓ 48
285.bj odd 36 4 855.2.da.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.p.a ✓ 48 95.r even 36 4
475.2.l.f 48 19.f odd 18 2
475.2.l.f 48 95.o odd 18 2
855.2.da.b 48 285.bj odd 36 4
1805.2.b.k 24 95.g even 4 2
1805.2.b.l 24 5.c odd 4 2
9025.2.a.ct 24 1.a even 1 1 trivial
9025.2.a.ct 24 5.b even 2 1 inner
9025.2.a.cu 24 19.b odd 2 1
9025.2.a.cu 24 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{24} - 33 T_{2}^{22} + 468 T_{2}^{20} - 3743 T_{2}^{18} + 18618 T_{2}^{16} - 59871 T_{2}^{14} + \cdots + 19 \) Copy content Toggle raw display
\( T_{3}^{24} - 42 T_{3}^{22} + 771 T_{3}^{20} - 8116 T_{3}^{18} + 54009 T_{3}^{16} - 236172 T_{3}^{14} + \cdots + 1539 \) Copy content Toggle raw display
\( T_{7}^{24} - 90 T_{7}^{22} + 3516 T_{7}^{20} - 78431 T_{7}^{18} + 1105830 T_{7}^{16} - 10303185 T_{7}^{14} + \cdots + 5000211 \) Copy content Toggle raw display
\( T_{11}^{12} - 6 T_{11}^{11} - 39 T_{11}^{10} + 237 T_{11}^{9} + 570 T_{11}^{8} - 3315 T_{11}^{7} + \cdots + 18981 \) Copy content Toggle raw display
\( T_{29}^{12} + 18 T_{29}^{11} + 45 T_{29}^{10} - 756 T_{29}^{9} - 3600 T_{29}^{8} + 9369 T_{29}^{7} + \cdots - 263169 \) Copy content Toggle raw display