↑

Properties

Label 2003.2.a.b
Level $2003$
Weight $2$
Character orbit 2003.a
Self dual yes
Analytic conductor $15.994$
Analytic rank $0$
Dimension $92$
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] = [2003,2,Mod(1,2003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2003, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2003.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: \(15.9940355249\)
Analytic rank: \(0\)
Dimension: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q + 5 q^{2} + 16 q^{3} + 101 q^{4} + 24 q^{5} + 8 q^{6} + 38 q^{7} + 9 q^{8} + 102 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q + 5 q^{2} + 16 q^{3} + 101 q^{4} + 24 q^{5} + 8 q^{6} + 38 q^{7} + 9 q^{8} + 102 q^{9} + 37 q^{10} + 8 q^{11} + 41 q^{12} + 106 q^{13} + 9 q^{14} + 5 q^{15} + 115 q^{16} + 54 q^{17} + 10 q^{18} + 38 q^{19} + 30 q^{20} + 33 q^{21} + 42 q^{22} + 18 q^{23} + 5 q^{24} + 126 q^{25} - 3 q^{26} + 46 q^{27} + 84 q^{28} + 36 q^{29} - 12 q^{30} + 28 q^{31} + 13 q^{32} + 11 q^{33} + 12 q^{34} + 2 q^{35} + 87 q^{36} + 116 q^{37} - 12 q^{38} + 15 q^{39} + 90 q^{40} + 18 q^{41} + 8 q^{42} + 86 q^{43} - 6 q^{44} + 84 q^{45} + 6 q^{46} + 17 q^{47} + 58 q^{48} + 140 q^{49} - 8 q^{50} + 13 q^{51} + 164 q^{52} + 56 q^{53} + 9 q^{54} + 78 q^{55} - 5 q^{56} + 70 q^{57} + 20 q^{58} + 7 q^{59} - 17 q^{60} + 202 q^{61} + 27 q^{62} + 88 q^{63} + 117 q^{64} + 31 q^{65} - 30 q^{66} + 37 q^{67} + 91 q^{68} + 47 q^{69} - 26 q^{70} - 18 q^{72} + 159 q^{73} - 17 q^{74} + 32 q^{75} + 70 q^{76} + 72 q^{77} - 23 q^{78} + 58 q^{79} - 29 q^{80} + 88 q^{81} + 96 q^{82} + 46 q^{83} - 4 q^{84} + 163 q^{85} - 36 q^{86} + 31 q^{87} + 83 q^{88} + 25 q^{89} + 21 q^{90} + 47 q^{91} + 19 q^{92} + 35 q^{93} + 20 q^{94} - 5 q^{95} - 14 q^{96} + 161 q^{97} - 9 q^{98} - 32 q^{99}+O(q^{100}) \) 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 −2.79685 2.55624 5.82240 1.52335 −7.14944 3.17476 −10.6907 3.53438 −4.26058
1.2 −2.69367 0.504502 5.25585 −1.69646 −1.35896 3.17583 −8.77019 −2.74548 4.56971
1.3 −2.66739 1.38335 5.11496 −0.408041 −3.68993 −4.77020 −8.30880 −1.08635 1.08840
1.4 −2.66215 −1.77578 5.08704 −3.76298 4.72740 −0.528526 −8.21816 0.153404 10.0176
1.5 −2.65934 −2.92850 5.07212 −0.925639 7.78788 2.73667 −8.16981 5.57610 2.46159
1.6 −2.62205 2.86087 4.87513 −3.94065 −7.50134 −0.0235023 −7.53872 5.18459 10.3326
1.7 −2.49850 1.06891 4.24248 3.19347 −2.67066 −1.22322 −5.60283 −1.85744 −7.97887
1.8 −2.44592 −1.39509 3.98253 3.87659 3.41227 4.18212 −4.84911 −1.05373 −9.48183
1.9 −2.42491 −1.57099 3.88017 −0.729811 3.80950 3.00371 −4.55923 −0.531996 1.76972
1.10 −2.33994 −0.610052 3.47530 −1.89953 1.42748 −1.34283 −3.45211 −2.62784 4.44479
1.11 −2.27609 0.393100 3.18060 −3.34850 −0.894732 4.74168 −2.68716 −2.84547 7.62151
1.12 −2.24762 −0.577966 3.05178 1.23285 1.29905 −4.88739 −2.36400 −2.66596 −2.77097
1.13 −2.24372 2.48976 3.03430 2.01422 −5.58633 3.94117 −2.32068 3.19890 −4.51935
1.14 −2.23571 −2.21943 2.99839 2.50227 4.96199 −1.75296 −2.23211 1.92586 −5.59435
1.15 −2.01113 −2.77633 2.04465 3.87412 5.58356 −1.44380 −0.0897972 4.70799 −7.79136
1.16 −2.00629 3.45354 2.02522 3.32754 −6.92882 −1.41917 −0.0505888 8.92694 −6.67602
1.17 −1.99994 1.04340 1.99974 −3.45747 −2.08672 −2.04368 0.000513161 0 −1.91133 6.91471
1.18 −1.97294 2.68051 1.89248 1.04191 −5.28848 −1.51981 0.212129 4.18515 −2.05563
1.19 −1.92028 3.09782 1.68748 −2.32550 −5.94868 3.25741 0.600123 6.59646 4.46561
1.20 −1.82286 1.84680 1.32281 3.02734 −3.36646 0.731415 1.23442 0.410676 −5.51841
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.92
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2003\) \(-1\)

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 2003.2.a.b ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2003.2.a.b ✓ 92 1.a even 1 1 trivial