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Properties

Label 9251.2.a.ba
Level $9251$
Weight $2$
Character orbit 9251.a
Self dual yes
Analytic conductor $73.870$
Analytic rank $1$
Dimension $40$
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] = [9251,2,Mod(1,9251)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9251, 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("9251.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9251 = 11 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9251.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: \(73.8696069099\)
Analytic rank: \(1\)
Dimension: \(40\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 5 q^{3} + 28 q^{4} - 12 q^{5} - 8 q^{6} - 15 q^{7} - 3 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 5 q^{3} + 28 q^{4} - 12 q^{5} - 8 q^{6} - 15 q^{7} - 3 q^{8} + 25 q^{9} - 25 q^{10} + 40 q^{11} - 17 q^{12} - 35 q^{13} + 3 q^{14} + 15 q^{15} - 6 q^{17} + 24 q^{18} + 2 q^{19} - 6 q^{20} - 5 q^{21} + 8 q^{23} - 18 q^{24} + 20 q^{25} - 20 q^{26} + q^{27} - 50 q^{28} - 5 q^{30} - 12 q^{31} - 6 q^{32} - 5 q^{33} - 26 q^{34} - 28 q^{35} - 22 q^{36} - 17 q^{37} - 12 q^{38} - 30 q^{39} + 30 q^{40} + 9 q^{41} - 34 q^{42} + 6 q^{43} + 28 q^{44} - 89 q^{45} - 7 q^{46} - 8 q^{47} + 33 q^{48} + q^{49} + 17 q^{50} - 52 q^{51} - 65 q^{52} - 51 q^{53} + 5 q^{54} - 12 q^{55} - 4 q^{56} - 49 q^{57} - 56 q^{59} + 15 q^{60} - 39 q^{61} + 53 q^{63} - 13 q^{64} - 13 q^{65} - 8 q^{66} - 68 q^{67} - 107 q^{68} - 31 q^{69} + 51 q^{70} - 47 q^{71} + 71 q^{72} + 19 q^{73} - 54 q^{74} - 22 q^{75} + 54 q^{76} - 15 q^{77} + 28 q^{78} + 10 q^{79} + 10 q^{80} - 4 q^{81} + 34 q^{82} - 40 q^{83} + 11 q^{84} + 26 q^{85} - 46 q^{86} - 3 q^{88} + 29 q^{89} - 100 q^{90} - 50 q^{91} + 76 q^{92} - 73 q^{93} - 116 q^{94} + 5 q^{95} + 13 q^{96} - 22 q^{97} + 102 q^{98} + 25 q^{99}+O(q^{100}) \) 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 −2.58991 −1.19982 4.70764 −1.86126 3.10741 −0.316317 −7.01255 −1.56044 4.82050
1.2 −2.54144 0.626545 4.45893 −0.217422 −1.59233 −0.803024 −6.24924 −2.60744 0.552566
1.3 −2.49635 0.791807 4.23177 −1.63232 −1.97663 −1.61450 −5.57129 −2.37304 4.07485
1.4 −2.32804 −0.113798 3.41976 1.83347 0.264925 −3.17671 −3.30524 −2.98705 −4.26838
1.5 −2.27836 −0.958669 3.19093 3.82900 2.18419 −3.39107 −2.71337 −2.08095 −8.72384
1.6 −2.21557 2.66785 2.90873 −0.0303185 −5.91079 2.76647 −2.01335 4.11741 0.0671726
1.7 −1.78406 −1.44186 1.18289 3.10385 2.57237 −2.24082 1.45778 −0.921049 −5.53747
1.8 −1.62004 −2.62880 0.624541 −2.73623 4.25878 0.785723 2.22830 3.91061 4.43281
1.9 −1.61832 −2.31596 0.618969 −3.33009 3.74798 2.98584 2.23495 2.36368 5.38916
1.10 −1.52030 0.979935 0.311315 −1.77021 −1.48980 −2.37307 2.56731 −2.03973 2.69125
1.11 −1.47508 2.74115 0.175860 1.34040 −4.04341 −0.0559519 2.69075 4.51390 −1.97719
1.12 −1.39682 1.99241 −0.0488806 1.81401 −2.78305 3.01968 2.86193 0.969698 −2.53386
1.13 −1.38566 −2.26395 −0.0799528 2.13001 3.13706 −0.998015 2.88210 2.12548 −2.95147
1.14 −0.982505 −2.74351 −1.03468 −0.473482 2.69551 0.734426 2.98159 4.52686 0.465198
1.15 −0.853251 0.944865 −1.27196 −2.43921 −0.806207 −5.16758 2.79181 −2.10723 2.08126
1.16 −0.667928 2.16237 −1.55387 3.76965 −1.44431 −2.15374 2.37373 1.67584 −2.51785
1.17 −0.468952 −2.79675 −1.78008 −1.84513 1.31154 −1.35451 1.77268 4.82182 0.865275
1.18 −0.271965 0.120724 −1.92604 0.336187 −0.0328326 1.58806 1.06774 −2.98543 −0.0914310
1.19 −0.248963 2.33934 −1.93802 −1.48502 −0.582409 2.77156 0.980420 2.47252 0.369716
1.20 −0.245772 −1.67672 −1.93960 1.54159 0.412090 0.822896 0.968242 −0.188622 −0.378881
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(29\) \(-1\)

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 9251.2.a.ba ✓ 40
29.b even 2 1 9251.2.a.bb yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9251.2.a.ba ✓ 40 1.a even 1 1 trivial
9251.2.a.bb yes 40 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 54 T_{2}^{38} + T_{2}^{37} + 1336 T_{2}^{36} - 48 T_{2}^{35} - 20088 T_{2}^{34} + \cdots - 279 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9251))\). Copy content Toggle raw display