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Properties

Label 2003.4.a.b
Level $2003$
Weight $4$
Character orbit 2003.a
Self dual yes
Analytic conductor $118.181$
Analytic rank $0$
Dimension $259$
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] = [2003,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2003.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2003 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(118.180825741\)
Analytic rank: \(0\)
Dimension: \(259\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 259 q + 16 q^{2} + 57 q^{3} + 1088 q^{4} + 135 q^{5} + 120 q^{6} + 306 q^{7} + 192 q^{8} + 2468 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 259 q + 16 q^{2} + 57 q^{3} + 1088 q^{4} + 135 q^{5} + 120 q^{6} + 306 q^{7} + 192 q^{8} + 2468 q^{9} + 422 q^{10} + 161 q^{11} + 629 q^{12} + 1457 q^{13} + 131 q^{14} + 234 q^{15} + 4776 q^{16} + 1038 q^{17} + 747 q^{18} + 985 q^{19} + 993 q^{20} + 1244 q^{21} + 1501 q^{22} + 722 q^{23} + 1565 q^{24} + 7992 q^{25} + 723 q^{26} + 2040 q^{27} + 3648 q^{28} + 1669 q^{29} + 1190 q^{30} + 1214 q^{31} + 1892 q^{32} + 1824 q^{33} + 1952 q^{34} + 1236 q^{35} + 10938 q^{36} + 5187 q^{37} + 1608 q^{38} + 1506 q^{39} + 5347 q^{40} + 1652 q^{41} + 2372 q^{42} + 3845 q^{43} + 1895 q^{44} + 6003 q^{45} + 2422 q^{46} + 1662 q^{47} + 5128 q^{48} + 16095 q^{49} + 2294 q^{50} + 2040 q^{51} + 12318 q^{52} + 3765 q^{53} + 3421 q^{54} + 6198 q^{55} + 1289 q^{56} + 4322 q^{57} + 3375 q^{58} + 1181 q^{59} + 2295 q^{60} + 14601 q^{61} + 4611 q^{62} + 7268 q^{63} + 20910 q^{64} + 3558 q^{65} + 3466 q^{66} + 4111 q^{67} + 11099 q^{68} + 7954 q^{69} + 4962 q^{70} + 2764 q^{71} + 9339 q^{72} + 14156 q^{73} + 2336 q^{74} + 6595 q^{75} + 11474 q^{76} + 8586 q^{77} + 3315 q^{78} + 7532 q^{79} + 7106 q^{80} + 26691 q^{81} + 12060 q^{82} + 6143 q^{83} + 10930 q^{84} + 18024 q^{85} + 1417 q^{86} + 9724 q^{87} + 16454 q^{88} + 5420 q^{89} + 7204 q^{90} + 7304 q^{91} + 6899 q^{92} + 11236 q^{93} + 8521 q^{94} + 6878 q^{95} + 12792 q^{96} + 21020 q^{97} + 5550 q^{98} + 10027 q^{99}+O(q^{100}) \) 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 −5.63809 −1.91613 23.7881 −19.2537 10.8033 21.3668 −89.0145 −23.3284 108.554
1.2 −5.60580 −0.561367 23.4250 19.4069 3.14691 1.29469 −86.4693 −26.6849 −108.791
1.3 −5.49768 −3.05673 22.2245 4.83646 16.8049 −2.93638 −78.2019 −17.6564 −26.5894
1.4 −5.42693 4.73725 21.4516 4.33262 −25.7088 9.03172 −73.0010 −4.55844 −23.5129
1.5 −5.42447 7.72837 21.4249 −17.5661 −41.9223 −15.3084 −72.8227 32.7277 95.2866
1.6 −5.42361 −8.40037 21.4156 8.43112 45.5604 −29.8444 −72.7608 43.5662 −45.7271
1.7 −5.42109 −5.18541 21.3882 −2.82552 28.1106 13.7560 −72.5784 −0.111478 15.3174
1.8 −5.41048 1.71267 21.2733 −14.7093 −9.26636 27.3198 −71.8152 −24.0668 79.5846
1.9 −5.31725 −8.63348 20.2732 −4.39603 45.9064 22.4677 −65.2596 47.5370 23.3748
1.10 −5.30350 9.83838 20.1271 −9.71085 −52.1779 32.5014 −64.3163 69.7938 51.5015
1.11 −5.27804 −8.66164 19.8577 15.1622 45.7165 16.1816 −62.5857 48.0239 −80.0266
1.12 −5.21305 −2.23348 19.1758 −8.39062 11.6432 −21.9745 −58.2602 −22.0116 43.7407
1.13 −5.19180 −5.39423 18.9547 −19.8797 28.0057 −8.96529 −56.8748 2.09772 103.212
1.14 −5.18047 7.17185 18.8373 −9.31926 −37.1536 23.6574 −56.1422 24.4355 48.2782
1.15 −5.15167 0.879545 18.5397 −7.16288 −4.53113 1.45774 −54.2972 −26.2264 36.9008
1.16 −5.13302 8.38696 18.3479 21.3198 −43.0505 26.8765 −53.1161 43.3412 −109.435
1.17 −5.09138 0.700678 17.9222 14.3435 −3.56742 −22.6064 −50.5177 −26.5091 −73.0284
1.18 −5.00899 2.98364 17.0900 9.30464 −14.9450 −6.31248 −45.5318 −18.0979 −46.6069
1.19 −4.99655 7.90281 16.9655 5.95111 −39.4868 −0.427170 −44.7965 35.4544 −29.7350
1.20 −4.94518 −2.76742 16.4548 14.7945 13.6854 −19.9856 −41.8107 −19.3414 −73.1616
See next 80 embeddings (of 259 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.259
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 2003.4.a.b ✓ 259
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2003.4.a.b ✓ 259 1.a even 1 1 trivial