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Properties

Label 8003.2.a.a
Level $8003$
Weight $2$
Character orbit 8003.a
Self dual yes
Analytic conductor $63.904$
Analytic rank $1$
Dimension $147$
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] = [8003,2,Mod(1,8003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8003, 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("8003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8003 = 53 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8003.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: \(63.9042767376\)
Analytic rank: \(1\)
Dimension: \(147\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 147 q - 6 q^{2} - 23 q^{3} + 130 q^{4} - 25 q^{5} - 18 q^{6} - 33 q^{7} - 15 q^{8} + 114 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 147 q - 6 q^{2} - 23 q^{3} + 130 q^{4} - 25 q^{5} - 18 q^{6} - 33 q^{7} - 15 q^{8} + 114 q^{9} - 24 q^{10} - 13 q^{11} - 52 q^{12} - 119 q^{13} - 6 q^{14} - 22 q^{15} + 100 q^{16} - 42 q^{17} - 6 q^{18} - 31 q^{19} - 50 q^{20} - 44 q^{21} - 38 q^{22} - 16 q^{23} - 30 q^{24} + 80 q^{25} - 18 q^{26} - 68 q^{27} - 72 q^{28} - 40 q^{29} - 3 q^{30} - 44 q^{31} - 41 q^{32} - 71 q^{33} - 55 q^{34} - 24 q^{35} + 108 q^{36} - 145 q^{37} - 39 q^{38} + 28 q^{39} - 81 q^{40} - 28 q^{41} - 54 q^{42} - 47 q^{43} - 33 q^{44} - 89 q^{45} - 43 q^{46} - 65 q^{47} - 82 q^{48} + 52 q^{49} - 43 q^{50} + 7 q^{51} - 215 q^{52} + 147 q^{53} - 88 q^{54} - 48 q^{55} + 19 q^{56} - 66 q^{57} - 110 q^{58} - 66 q^{59} - 118 q^{60} - 80 q^{61} - 41 q^{62} - 52 q^{63} + 33 q^{64} - 17 q^{65} - 86 q^{66} - 114 q^{67} - 77 q^{68} - 49 q^{69} - 56 q^{70} + 10 q^{71} + 5 q^{72} - 143 q^{73} - 30 q^{74} - 97 q^{75} - 104 q^{76} - 116 q^{77} - 33 q^{78} - 31 q^{79} - 83 q^{80} - q^{81} - 72 q^{82} - 70 q^{83} - 64 q^{84} - 85 q^{85} - 43 q^{87} - 149 q^{88} - 130 q^{89} + 42 q^{90} - 35 q^{91} - 31 q^{92} - 149 q^{93} - 94 q^{94} - 9 q^{95} + 6 q^{96} - 211 q^{97} - 18 q^{98} - 19 q^{99}+O(q^{100}) \) 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 −2.76716 −2.20267 5.65717 0.836026 6.09513 −1.25752 −10.1200 1.85174 −2.31342
1.2 −2.74141 2.30375 5.51534 −1.31886 −6.31554 1.25952 −9.63701 2.30728 3.61554
1.3 −2.73133 −0.575375 5.46014 −2.64139 1.57154 −4.90025 −9.45076 −2.66894 7.21448
1.4 −2.69892 1.36364 5.28418 3.25523 −3.68037 0.105698 −8.86375 −1.14047 −8.78561
1.5 −2.69176 −2.97285 5.24555 2.64194 8.00219 −3.02993 −8.73623 5.83784 −7.11145
1.6 −2.68460 −1.10941 5.20709 3.69885 2.97832 2.40865 −8.60976 −1.76921 −9.92993
1.7 −2.63479 0.411921 4.94210 −2.62173 −1.08532 0.446803 −7.75181 −2.83032 6.90770
1.8 −2.57093 1.68353 4.60966 0.0447393 −4.32822 −3.24015 −6.70923 −0.165740 −0.115021
1.9 −2.46705 3.03255 4.08634 −1.27620 −7.48147 0.505683 −5.14712 6.19639 3.14846
1.10 −2.46471 −1.16892 4.07480 −2.61176 2.88105 −0.765646 −5.11379 −1.63363 6.43723
1.11 −2.45879 −0.163481 4.04563 2.51114 0.401964 −0.639591 −5.02975 −2.97327 −6.17435
1.12 −2.44772 −0.651188 3.99131 −1.37302 1.59392 0.683234 −4.87416 −2.57595 3.36077
1.13 −2.41146 1.80060 3.81513 0.537438 −4.34207 4.37842 −4.37711 0.242159 −1.29601
1.14 −2.40138 1.08484 3.76663 −1.87801 −2.60511 −2.74065 −4.24236 −1.82313 4.50981
1.15 −2.37028 −0.677898 3.61822 2.58676 1.60681 −3.88263 −3.83563 −2.54045 −6.13133
1.16 −2.33017 −2.83003 3.42967 −2.62223 6.59443 3.30505 −3.33137 5.00905 6.11023
1.17 −2.31195 2.01743 3.34512 −2.72339 −4.66419 −3.92860 −3.10985 1.07001 6.29634
1.18 −2.30251 −2.68436 3.30157 −0.273715 6.18077 3.09928 −2.99689 4.20577 0.630233
1.19 −2.28070 −3.18519 3.20158 −4.19393 7.26446 −3.02488 −2.74045 7.14546 9.56509
1.20 −2.26111 −1.67317 3.11264 0.0442572 3.78322 2.69750 −2.51581 −0.200518 −0.100071
See next 80 embeddings (of 147 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.147
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(53\) \(-1\)
\(151\) \(-1\)

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 8003.2.a.a ✓ 147
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8003.2.a.a ✓ 147 1.a even 1 1 trivial