↑

Properties

Label 8993.2.a.r
Level $8993$
Weight $2$
Character orbit 8993.a
Self dual yes
Analytic conductor $71.809$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{3} + 32 q^{4} + 4 q^{5} - 10 q^{6} + 22 q^{7} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{3} + 32 q^{4} + 4 q^{5} - 10 q^{6} + 22 q^{7} + 32 q^{9} + 26 q^{10} + 8 q^{11} - 18 q^{12} - 8 q^{13} + 8 q^{14} + 36 q^{15} + 44 q^{16} - 30 q^{17} - 20 q^{18} + 16 q^{19} - 48 q^{20} - 8 q^{21} + 24 q^{22} + 20 q^{24} + 58 q^{25} + 10 q^{26} - 20 q^{27} + 48 q^{28} - 12 q^{29} + 64 q^{30} + 8 q^{31} + 10 q^{32} + 28 q^{33} - 10 q^{35} + 56 q^{36} + 48 q^{37} + 56 q^{38} - 16 q^{39} + 46 q^{40} - 24 q^{41} + 28 q^{43} + 6 q^{44} - 20 q^{45} - 8 q^{47} - 6 q^{48} + 16 q^{49} - 2 q^{50} + 2 q^{51} - 46 q^{52} + 32 q^{53} - 32 q^{54} + 36 q^{55} + 94 q^{56} - 48 q^{57} + 10 q^{58} - 18 q^{59} + 96 q^{60} - 2 q^{61} - 48 q^{62} + 66 q^{63} + 12 q^{64} - 4 q^{65} - 140 q^{66} + 68 q^{67} - 32 q^{68} + 66 q^{70} - 50 q^{71} + 6 q^{72} + 30 q^{73} + 8 q^{74} - 52 q^{75} + 92 q^{76} - 12 q^{77} - 36 q^{78} + 84 q^{79} - 92 q^{80} + 30 q^{81} + 30 q^{82} + 8 q^{83} - 24 q^{84} - 4 q^{85} + 28 q^{86} + 14 q^{87} - 20 q^{88} + 12 q^{89} - 30 q^{90} + 64 q^{91} + 28 q^{93} + 18 q^{94} - 34 q^{95} + 54 q^{96} + 48 q^{97} + 94 q^{98} + 66 q^{99}+O(q^{100}) \) 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 −2.76720 −2.14267 5.65738 −1.84792 5.92918 2.60613 −10.1207 1.59103 5.11356
1.2 −2.60403 1.33369 4.78098 −4.41803 −3.47298 −0.892298 −7.24177 −1.22126 11.5047
1.3 −2.38508 −0.408727 3.68860 0.00967905 0.974846 −2.88663 −4.02744 −2.83294 −0.0230853
1.4 −2.30517 3.12414 3.31382 0.0867068 −7.20167 −0.863813 −3.02859 6.76022 −0.199874
1.5 −2.29088 −3.21196 3.24814 −3.77568 7.35822 2.24462 −2.85934 7.31669 8.64963
1.6 −1.96553 2.42374 1.86332 −1.57868 −4.76394 3.29220 0.268658 2.87451 3.10295
1.7 −1.91087 0.348931 1.65143 3.62221 −0.666763 −0.492920 0.666064 −2.87825 −6.92158
1.8 −1.75404 −2.74123 1.07665 2.70042 4.80821 0.553753 1.61959 4.51432 −4.73664
1.9 −1.39198 2.13354 −0.0624033 −1.69976 −2.96984 4.61178 2.87082 1.55200 2.36603
1.10 −1.08392 −1.91703 −0.825125 2.61618 2.07790 −0.512236 3.06220 0.675006 −2.83572
1.11 −1.02363 1.61350 −0.952176 −0.883070 −1.65163 0.156564 3.02194 −0.396613 0.903939
1.12 −0.652947 0.698499 −1.57366 −2.21702 −0.456083 −0.408169 2.33341 −2.51210 1.44760
1.13 −0.407869 −0.866005 −1.83364 −1.64213 0.353216 0.727945 1.56362 −2.25004 0.669772
1.14 −0.345603 2.96435 −1.88056 4.08147 −1.02449 3.67692 1.34113 5.78739 −1.41057
1.15 0.116704 −1.38463 −1.98638 −3.58450 −0.161592 4.73187 −0.465228 −1.08281 −0.418327
1.16 0.158021 3.09398 −1.97503 2.84618 0.488914 −0.570149 −0.628140 6.57269 0.449757
1.17 0.177021 −2.84164 −1.96866 0.355074 −0.503029 −0.832752 −0.702536 5.07491 0.0628554
1.18 0.502817 −0.493019 −1.74718 3.14098 −0.247898 4.98310 −1.88414 −2.75693 1.57934
1.19 0.618383 0.521171 −1.61760 4.00420 0.322283 −4.26451 −2.23706 −2.72838 2.47613
1.20 0.945231 1.91303 −1.10654 −0.990516 1.80825 −1.87580 −2.93640 0.659677 −0.936267
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
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.r yes 30
23.b odd 2 1 8993.2.a.q ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8993.2.a.q ✓ 30 23.b odd 2 1
8993.2.a.r yes 30 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):

\( T_{2}^{30} - 46 T_{2}^{28} + 939 T_{2}^{26} - 2 T_{2}^{25} - 11216 T_{2}^{24} + 78 T_{2}^{23} + \cdots + 69 \) Copy content Toggle raw display
\( T_{5}^{30} - 4 T_{5}^{29} - 96 T_{5}^{28} + 368 T_{5}^{27} + 4098 T_{5}^{26} - 14662 T_{5}^{25} + \cdots + 5184 \) Copy content Toggle raw display