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Properties

Label 9898.2.a.bm
Level $9898$
Weight $2$
Character orbit 9898.a
Self dual yes
Analytic conductor $79.036$
Analytic rank $1$
Dimension $24$
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] = [9898,2,Mod(1,9898)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9898, 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("9898.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9898 = 2 \cdot 7^{2} \cdot 101 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9898.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: \(79.0359279207\)
Analytic rank: \(1\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{2} - 6 q^{3} + 24 q^{4} - 18 q^{5} - 6 q^{6} + 24 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{2} - 6 q^{3} + 24 q^{4} - 18 q^{5} - 6 q^{6} + 24 q^{8} + 26 q^{9} - 18 q^{10} + 2 q^{11} - 6 q^{12} - 30 q^{13} + 24 q^{16} - 44 q^{17} + 26 q^{18} - 14 q^{19} - 18 q^{20} + 2 q^{22} + 4 q^{23} - 6 q^{24} + 14 q^{25} - 30 q^{26} - 18 q^{27} + 2 q^{29} - 32 q^{31} + 24 q^{32} - 48 q^{33} - 44 q^{34} + 26 q^{36} - 22 q^{37} - 14 q^{38} - 18 q^{40} - 26 q^{41} - 2 q^{43} + 2 q^{44} - 62 q^{45} + 4 q^{46} - 64 q^{47} - 6 q^{48} + 14 q^{50} - 4 q^{51} - 30 q^{52} - 10 q^{53} - 18 q^{54} - 24 q^{55} - 4 q^{57} + 2 q^{58} - 38 q^{59} - 42 q^{61} - 32 q^{62} + 24 q^{64} + 12 q^{65} - 48 q^{66} + 10 q^{67} - 44 q^{68} - 22 q^{69} + 8 q^{71} + 26 q^{72} - 56 q^{73} - 22 q^{74} - 30 q^{75} - 14 q^{76} + 8 q^{79} - 18 q^{80} + 12 q^{81} - 26 q^{82} - 54 q^{83} + 16 q^{85} - 2 q^{86} - 60 q^{87} + 2 q^{88} - 60 q^{89} - 62 q^{90} + 4 q^{92} + 12 q^{93} - 64 q^{94} - 16 q^{95} - 6 q^{96} - 32 q^{97} - 46 q^{99}+O(q^{100}) \) 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 1.00000 −3.26191 1.00000 −2.46600 −3.26191 0 1.00000 7.64004 −2.46600
1.2 1.00000 −2.97909 1.00000 −1.98334 −2.97909 0 1.00000 5.87497 −1.98334
1.3 1.00000 −2.84144 1.00000 2.84606 −2.84144 0 1.00000 5.07375 2.84606
1.4 1.00000 −2.80041 1.00000 −0.891726 −2.80041 0 1.00000 4.84227 −0.891726
1.5 1.00000 −2.45491 1.00000 2.10999 −2.45491 0 1.00000 3.02660 2.10999
1.6 1.00000 −2.27372 1.00000 −2.22619 −2.27372 0 1.00000 2.16980 −2.22619
1.7 1.00000 −2.11491 1.00000 −4.31248 −2.11491 0 1.00000 1.47284 −4.31248
1.8 1.00000 −1.64301 1.00000 −3.81324 −1.64301 0 1.00000 −0.300515 −3.81324
1.9 1.00000 −1.48371 1.00000 −0.966099 −1.48371 0 1.00000 −0.798594 −0.966099
1.10 1.00000 −1.35330 1.00000 2.31101 −1.35330 0 1.00000 −1.16858 2.31101
1.11 1.00000 −0.647717 1.00000 −1.25075 −0.647717 0 1.00000 −2.58046 −1.25075
1.12 1.00000 −0.393042 1.00000 1.54934 −0.393042 0 1.00000 −2.84552 1.54934
1.13 1.00000 0.222113 1.00000 1.18973 0.222113 0 1.00000 −2.95067 1.18973
1.14 1.00000 0.245221 1.00000 −2.35579 0.245221 0 1.00000 −2.93987 −2.35579
1.15 1.00000 0.660380 1.00000 2.94437 0.660380 0 1.00000 −2.56390 2.94437
1.16 1.00000 0.670107 1.00000 −0.715955 0.670107 0 1.00000 −2.55096 −0.715955
1.17 1.00000 0.944776 1.00000 0.197305 0.944776 0 1.00000 −2.10740 0.197305
1.18 1.00000 0.951378 1.00000 −4.24638 0.951378 0 1.00000 −2.09488 −4.24638
1.19 1.00000 1.57098 1.00000 2.70688 1.57098 0 1.00000 −0.532019 2.70688
1.20 1.00000 2.08158 1.00000 0.354203 2.08158 0 1.00000 1.33296 0.354203
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
\(2\) \( -1 \)
\(7\) \( +1 \)
\(101\) \( -1 \)

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 9898.2.a.bm ✓ 24
7.b odd 2 1 9898.2.a.bn yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9898.2.a.bm ✓ 24 1.a even 1 1 trivial
9898.2.a.bn yes 24 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9898))\):

\( T_{3}^{24} + 6 T_{3}^{23} - 31 T_{3}^{22} - 240 T_{3}^{21} + 311 T_{3}^{20} + 4022 T_{3}^{19} + \cdots + 2936 \) Copy content Toggle raw display
\( T_{5}^{24} + 18 T_{5}^{23} + 95 T_{5}^{22} - 162 T_{5}^{21} - 3280 T_{5}^{20} - 7556 T_{5}^{19} + \cdots - 478574 \) Copy content Toggle raw display