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Properties

Label 9898.2.a.bk
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} + 24 q^{4} - 14 q^{5} - 24 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} + 24 q^{4} - 14 q^{5} - 24 q^{8} + 20 q^{9} + 14 q^{10} - 2 q^{11} - 2 q^{13} + 12 q^{15} + 24 q^{16} - 16 q^{17} - 20 q^{18} - 2 q^{19} - 14 q^{20} + 2 q^{22} + 6 q^{23} + 26 q^{25} + 2 q^{26} - 12 q^{27} - 22 q^{29} - 12 q^{30} - 24 q^{32} - 4 q^{33} + 16 q^{34} + 20 q^{36} - 8 q^{37} + 2 q^{38} + 4 q^{39} + 14 q^{40} - 46 q^{41} + 20 q^{43} - 2 q^{44} - 46 q^{45} - 6 q^{46} - 40 q^{47} - 26 q^{50} - 20 q^{51} - 2 q^{52} + 2 q^{53} + 12 q^{54} - 32 q^{55} - 44 q^{57} + 22 q^{58} - 32 q^{59} + 12 q^{60} + 32 q^{61} + 24 q^{64} + 16 q^{65} + 4 q^{66} - 2 q^{67} - 16 q^{68} - 32 q^{69} + 22 q^{71} - 20 q^{72} - 22 q^{73} + 8 q^{74} - 24 q^{75} - 2 q^{76} - 4 q^{78} + 38 q^{79} - 14 q^{80} + 8 q^{81} + 46 q^{82} - 140 q^{83} + 12 q^{85} - 20 q^{86} + 12 q^{87} + 2 q^{88} + 14 q^{89} + 46 q^{90} + 6 q^{92} - 4 q^{93} + 40 q^{94} + 28 q^{95} - 16 q^{97} - 14 q^{99}+O(q^{100}) \) 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 −3.11696 1.00000 −3.05158 3.11696 0 −1.00000 6.71543 3.05158
1.2 −1.00000 −3.03099 1.00000 1.97359 3.03099 0 −1.00000 6.18688 −1.97359
1.3 −1.00000 −2.93008 1.00000 −3.85712 2.93008 0 −1.00000 5.58536 3.85712
1.4 −1.00000 −2.53794 1.00000 −1.52856 2.53794 0 −1.00000 3.44115 1.52856
1.5 −1.00000 −2.13302 1.00000 −2.12368 2.13302 0 −1.00000 1.54979 2.12368
1.6 −1.00000 −1.73958 1.00000 2.43324 1.73958 0 −1.00000 0.0261212 −2.43324
1.7 −1.00000 −1.54390 1.00000 −4.16546 1.54390 0 −1.00000 −0.616373 4.16546
1.8 −1.00000 −1.27029 1.00000 1.54270 1.27029 0 −1.00000 −1.38636 −1.54270
1.9 −1.00000 −1.10725 1.00000 0.697719 1.10725 0 −1.00000 −1.77401 −0.697719
1.10 −1.00000 −0.370821 1.00000 0.567456 0.370821 0 −1.00000 −2.86249 −0.567456
1.11 −1.00000 −0.308132 1.00000 2.10367 0.308132 0 −1.00000 −2.90505 −2.10367
1.12 −1.00000 −0.285559 1.00000 −0.188249 0.285559 0 −1.00000 −2.91846 0.188249
1.13 −1.00000 0.0290086 1.00000 −3.55340 −0.0290086 0 −1.00000 −2.99916 3.55340
1.14 −1.00000 0.574047 1.00000 −2.26381 −0.574047 0 −1.00000 −2.67047 2.26381
1.15 −1.00000 0.623671 1.00000 −1.90874 −0.623671 0 −1.00000 −2.61103 1.90874
1.16 −1.00000 1.03879 1.00000 3.34323 −1.03879 0 −1.00000 −1.92093 −3.34323
1.17 −1.00000 1.51182 1.00000 2.72179 −1.51182 0 −1.00000 −0.714414 −2.72179
1.18 −1.00000 1.82303 1.00000 −1.62022 −1.82303 0 −1.00000 0.323452 1.62022
1.19 −1.00000 2.04298 1.00000 0.448739 −2.04298 0 −1.00000 1.17377 −0.448739
1.20 −1.00000 2.12277 1.00000 −4.39906 −2.12277 0 −1.00000 1.50616 4.39906
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.bk ✓ 24
7.b odd 2 1 9898.2.a.bl yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9898.2.a.bk ✓ 24 1.a even 1 1 trivial
9898.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9898))\):

\( T_{3}^{24} - 46 T_{3}^{22} + 4 T_{3}^{21} + 903 T_{3}^{20} - 152 T_{3}^{19} - 9914 T_{3}^{18} + \cdots + 112 \) Copy content Toggle raw display
\( T_{5}^{24} + 14 T_{5}^{23} + 25 T_{5}^{22} - 474 T_{5}^{21} - 2038 T_{5}^{20} + 5736 T_{5}^{19} + \cdots + 26948 \) Copy content Toggle raw display