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Properties

Label 8033.2.a.e
Level $8033$
Weight $2$
Character orbit 8033.a
Self dual yes
Analytic conductor $64.144$
Analytic rank $0$
Dimension $169$
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] = [8033,2,Mod(1,8033)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8033, 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("8033.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8033 = 29 \cdot 277 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8033.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: \(64.1438279437\)
Analytic rank: \(0\)
Dimension: \(169\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 169 q + 3 q^{2} + 8 q^{3} + 183 q^{4} + 13 q^{5} + 17 q^{6} + 76 q^{7} + 6 q^{8} + 181 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 169 q + 3 q^{2} + 8 q^{3} + 183 q^{4} + 13 q^{5} + 17 q^{6} + 76 q^{7} + 6 q^{8} + 181 q^{9} + 30 q^{10} + 2 q^{11} + 25 q^{12} + 63 q^{13} - 3 q^{14} + 26 q^{15} + 219 q^{16} + 14 q^{17} + 31 q^{18} + 51 q^{19} + 49 q^{20} + 16 q^{21} + 53 q^{22} + 50 q^{23} + 43 q^{24} + 214 q^{25} - 2 q^{26} + 23 q^{27} + 149 q^{28} - 169 q^{29} + 26 q^{30} + 65 q^{31} + 25 q^{32} + 57 q^{33} + 60 q^{34} + 60 q^{35} + 218 q^{36} + 52 q^{37} + 35 q^{38} + 49 q^{39} + 72 q^{40} + 3 q^{41} + 78 q^{42} + 132 q^{43} + 64 q^{45} + 32 q^{46} + 54 q^{47} + 76 q^{48} + 245 q^{49} - 6 q^{50} + 44 q^{51} + 193 q^{52} + 58 q^{53} + 54 q^{54} + 213 q^{55} + 12 q^{56} + 52 q^{57} - 3 q^{58} + 25 q^{59} + 32 q^{60} + 100 q^{61} + 78 q^{62} + 227 q^{63} + 292 q^{64} + 37 q^{65} + 59 q^{66} + 110 q^{67} + 24 q^{68} - 20 q^{69} + 47 q^{70} + 44 q^{71} + 71 q^{72} + 139 q^{73} + 35 q^{74} + 53 q^{75} + 92 q^{76} + 22 q^{77} + 83 q^{78} + 137 q^{79} + 90 q^{80} + 177 q^{81} + 91 q^{82} + 126 q^{83} + 36 q^{84} + 95 q^{85} + 21 q^{86} - 8 q^{87} + 75 q^{88} + 19 q^{89} + 130 q^{90} + 102 q^{91} + 68 q^{92} + 22 q^{93} + 82 q^{94} + 37 q^{95} + 107 q^{96} + 116 q^{97} - 13 q^{98} - 11 q^{99}+O(q^{100}) \) 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 −2.81826 0.318379 5.94259 3.64952 −0.897274 3.95184 −11.1113 −2.89864 −10.2853
1.2 −2.78256 −1.98361 5.74263 −0.893629 5.51951 2.13334 −10.4141 0.934708 2.48658
1.3 −2.77688 −0.579971 5.71104 −3.81454 1.61051 −2.45600 −10.3051 −2.66363 10.5925
1.4 −2.75139 −3.04076 5.57013 1.50173 8.36631 3.15937 −9.82282 6.24621 −4.13185
1.5 −2.74782 1.78651 5.55052 1.77314 −4.90901 −4.26973 −9.75621 0.191622 −4.87228
1.6 −2.68231 3.00048 5.19481 2.85837 −8.04822 2.57996 −8.56948 6.00286 −7.66705
1.7 −2.64249 2.27007 4.98277 1.28777 −5.99863 4.91822 −7.88196 2.15320 −3.40294
1.8 −2.64021 2.12352 4.97070 −3.10978 −5.60653 2.54835 −7.84326 1.50933 8.21046
1.9 −2.61964 −0.130833 4.86253 −0.647747 0.342735 −1.29837 −7.49880 −2.98288 1.69687
1.10 −2.61493 −1.04206 4.83784 −2.72711 2.72492 4.43987 −7.42074 −1.91410 7.13119
1.11 −2.59038 2.80961 4.71008 −2.74249 −7.27797 −2.16078 −7.02013 4.89393 7.10409
1.12 −2.56941 −2.98680 4.60185 −2.62071 7.67431 −0.147137 −6.68522 5.92098 6.73366
1.13 −2.52594 −2.31351 4.38038 1.97796 5.84380 −1.35042 −6.01269 2.35235 −4.99621
1.14 −2.49641 −2.16208 4.23205 4.42906 5.39744 1.14877 −5.57210 1.67460 −11.0567
1.15 −2.48374 0.850952 4.16898 −1.59831 −2.11355 3.64335 −5.38719 −2.27588 3.96978
1.16 −2.44776 0.458025 3.99150 1.76580 −1.12113 −1.33543 −4.87472 −2.79021 −4.32226
1.17 −2.36338 −0.800416 3.58557 2.26233 1.89169 3.86578 −3.74731 −2.35933 −5.34676
1.18 −2.31656 −0.263356 3.36647 −0.903221 0.610081 2.39669 −3.16552 −2.93064 2.09237
1.19 −2.30642 0.966593 3.31956 1.65248 −2.22937 −3.04982 −3.04346 −2.06570 −3.81132
1.20 −2.30030 0.878714 3.29139 2.14800 −2.02131 −2.80339 −2.97058 −2.22786 −4.94106
See next 80 embeddings (of 169 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.169
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(1\)
\(277\) \(-1\)

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 8033.2.a.e ✓ 169
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8033.2.a.e ✓ 169 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{169} - 3 T_{2}^{168} - 256 T_{2}^{167} + 771 T_{2}^{166} + 32112 T_{2}^{165} - 97109 T_{2}^{164} + \cdots + 712050240 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8033))\). Copy content Toggle raw display