Properties

Label 8993.2.a.x
Level $8993$
Weight $2$
Character orbit 8993.a
Self dual yes
Analytic conductor $71.809$
Analytic rank $0$
Dimension $90$
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] = [8993,2,Mod(1,8993)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8993, 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("8993.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8993 = 17 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8993.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: \(71.8094665377\)
Analytic rank: \(0\)
Dimension: \(90\)
Twist minimal: no (minimal twist has level 391)
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.

\(\operatorname{Tr}(f)(q) = \) \( 90 q + 9 q^{2} + 19 q^{3} + 109 q^{4} + 8 q^{5} + 23 q^{6} + 10 q^{7} + 42 q^{8} + 107 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 90 q + 9 q^{2} + 19 q^{3} + 109 q^{4} + 8 q^{5} + 23 q^{6} + 10 q^{7} + 42 q^{8} + 107 q^{9} + 45 q^{10} + 12 q^{11} + 52 q^{12} + 37 q^{13} + 7 q^{14} - 10 q^{15} + 135 q^{16} + 90 q^{17} + 38 q^{18} - 7 q^{19} + 17 q^{20} - 16 q^{21} - 10 q^{22} + 54 q^{24} + 138 q^{25} + 4 q^{26} + 100 q^{27} + 3 q^{28} + 5 q^{29} - 14 q^{30} + 85 q^{31} + 88 q^{32} + 9 q^{33} + 9 q^{34} + 18 q^{35} + 130 q^{36} - 16 q^{38} + 97 q^{39} + 85 q^{40} - 5 q^{41} - 58 q^{42} - 11 q^{43} + 51 q^{44} - 7 q^{45} + 89 q^{47} + 109 q^{48} + 118 q^{49} + 35 q^{50} + 19 q^{51} + 120 q^{52} - 40 q^{53} + 77 q^{54} + 152 q^{55} + 170 q^{56} - 16 q^{57} + 129 q^{58} + 52 q^{59} - 88 q^{60} + 18 q^{61} + 21 q^{62} + 101 q^{63} + 226 q^{64} - 23 q^{65} - 38 q^{66} + q^{67} + 109 q^{68} + 35 q^{70} + 81 q^{71} + 138 q^{72} + 64 q^{73} - 5 q^{74} + 89 q^{75} - 59 q^{76} + 70 q^{77} + 91 q^{78} + 33 q^{79} - 24 q^{80} + 110 q^{81} + 154 q^{82} - 44 q^{83} - 40 q^{84} + 8 q^{85} + 15 q^{86} + 165 q^{87} - 45 q^{88} - 63 q^{89} + 191 q^{90} + 75 q^{91} + 119 q^{93} + 42 q^{94} + 106 q^{95} + 101 q^{96} - 10 q^{97} + 50 q^{98} - 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.78943 0.152810 5.78091 0.458306 −0.426253 −4.47274 −10.5466 −2.97665 −1.27841
1.2 −2.75381 1.00406 5.58347 3.41558 −2.76500 −2.29121 −9.86820 −1.99185 −9.40585
1.3 −2.71401 3.16264 5.36584 −3.30716 −8.58342 −1.44850 −9.13490 7.00227 8.97566
1.4 −2.57651 −2.99955 4.63838 0.110603 7.72836 1.22146 −6.79781 5.99729 −0.284970
1.5 −2.55642 −1.52801 4.53529 −3.04971 3.90623 3.12865 −6.48128 −0.665191 7.79636
1.6 −2.49625 1.18938 4.23125 0.892664 −2.96899 −1.82513 −5.56974 −1.58537 −2.22831
1.7 −2.47925 −1.56756 4.14668 −4.32472 3.88638 −2.44570 −5.32215 −0.542751 10.7221
1.8 −2.45766 1.94047 4.04010 −3.93953 −4.76903 4.55543 −5.01388 0.765442 9.68204
1.9 −2.41552 −0.509829 3.83473 3.12056 1.23150 −2.26311 −4.43182 −2.74007 −7.53778
1.10 −2.37716 −2.02408 3.65089 −2.33294 4.81157 −4.16483 −3.92443 1.09690 5.54577
1.11 −2.24566 2.48698 3.04301 3.24575 −5.58493 3.64668 −2.34224 3.18509 −7.28886
1.12 −2.22165 −2.39230 2.93574 −0.626912 5.31486 1.71533 −2.07889 2.72311 1.39278
1.13 −2.08596 2.77361 2.35123 2.18691 −5.78563 4.72020 −0.732662 4.69289 −4.56182
1.14 −2.02693 1.67419 2.10845 −0.958934 −3.39346 −0.144869 −0.219810 −0.197092 1.94369
1.15 −2.00839 −0.268714 2.03362 −2.53833 0.539682 0.944934 −0.0675119 −2.92779 5.09794
1.16 −2.00548 −1.48196 2.02196 2.45276 2.97204 −1.23566 −0.0440356 −0.803797 −4.91896
1.17 −1.88676 3.01129 1.55987 −3.29272 −5.68159 −2.53571 0.830415 6.06785 6.21259
1.18 −1.80981 −3.24553 1.27540 −0.171831 5.87377 2.25177 1.31139 7.53345 0.310981
1.19 −1.80678 1.07261 1.26445 1.84980 −1.93797 0.873536 1.32898 −1.84950 −3.34218
1.20 −1.71858 0.352612 0.953522 0.359463 −0.605992 −3.68964 1.79846 −2.87567 −0.617767
See all 90 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(23\) \(1\)

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 8993.2.a.x 90
23.b odd 2 1 8993.2.a.w 90
23.c even 11 2 391.2.i.b 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
391.2.i.b 180 23.c even 11 2
8993.2.a.w 90 23.b odd 2 1
8993.2.a.x 90 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):

\( T_{2}^{90} - 9 T_{2}^{89} - 104 T_{2}^{88} + 1153 T_{2}^{87} + 4718 T_{2}^{86} - 70918 T_{2}^{85} - 110317 T_{2}^{84} + 2788637 T_{2}^{83} + 678497 T_{2}^{82} - 78753075 T_{2}^{81} + 46234714 T_{2}^{80} + \cdots - 4643539 \) Copy content Toggle raw display
\( T_{5}^{90} - 8 T_{5}^{89} - 262 T_{5}^{88} + 2231 T_{5}^{87} + 32569 T_{5}^{86} - 298317 T_{5}^{85} - 2550719 T_{5}^{84} + 25472398 T_{5}^{83} + 140806923 T_{5}^{82} - 1560650598 T_{5}^{81} + \cdots + 48\!\cdots\!48 \) Copy content Toggle raw display