Properties

Label 1003.6.a.d
Level $1003$
Weight $6$
Character orbit 1003.a
Self dual yes
Analytic conductor $160.865$
Analytic rank $0$
Dimension $100$
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] = [1003,6,Mod(1,1003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1003, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1003.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1003.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: \(160.864971272\)
Analytic rank: \(0\)
Dimension: \(100\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 25 q^{2} + 63 q^{3} + 1707 q^{4} + 509 q^{5} + 207 q^{6} + 247 q^{7} + 765 q^{8} + 9003 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q + 25 q^{2} + 63 q^{3} + 1707 q^{4} + 509 q^{5} + 207 q^{6} + 247 q^{7} + 765 q^{8} + 9003 q^{9} + 424 q^{11} + 2880 q^{12} + 2426 q^{13} + 1069 q^{14} + 3699 q^{15} + 25895 q^{16} - 28900 q^{17} + 6644 q^{18} + 2224 q^{19} + 11919 q^{20} + 12248 q^{21} + 1160 q^{22} + 4675 q^{23} + 6477 q^{24} + 76925 q^{25} + 26507 q^{26} + 18465 q^{27} + 7371 q^{28} + 13905 q^{29} + 42029 q^{30} + 11503 q^{31} + 31765 q^{32} + 34053 q^{33} - 7225 q^{34} + 26861 q^{35} + 210764 q^{36} - 4418 q^{37} + 55637 q^{38} - 5718 q^{39} + 24116 q^{40} + 50715 q^{41} + 145355 q^{42} + 36979 q^{43} - 6793 q^{44} + 88939 q^{45} + 23917 q^{46} + 162533 q^{47} + 219761 q^{48} + 276061 q^{49} + 204874 q^{50} - 18207 q^{51} + 73665 q^{52} + 144329 q^{53} + 112241 q^{54} + 63002 q^{55} + 234871 q^{56} + 94768 q^{57} + 22318 q^{58} + 348100 q^{59} + 390780 q^{60} - 45447 q^{61} + 146617 q^{62} + 88467 q^{63} + 580433 q^{64} - 49981 q^{65} - 14744 q^{66} + 113930 q^{67} - 493323 q^{68} + 49070 q^{69} + 86899 q^{70} + 138703 q^{71} + 319055 q^{72} + 174214 q^{73} - 139931 q^{74} + 295788 q^{75} + 272539 q^{76} + 642017 q^{77} + 93149 q^{78} - 240788 q^{79} + 582895 q^{80} + 690560 q^{81} + 164633 q^{82} + 324136 q^{83} + 775436 q^{84} - 147101 q^{85} + 113296 q^{86} + 612596 q^{87} - 227510 q^{88} + 348396 q^{89} + 750464 q^{90} - 100399 q^{91} + 453265 q^{92} + 660393 q^{93} + 183864 q^{94} + 341370 q^{95} + 209486 q^{96} + 288603 q^{97} + 1100905 q^{98} + 301794 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −11.0579 −11.4049 90.2781 52.1219 126.115 −53.9097 −644.436 −112.928 −576.361
1.2 −11.0372 11.4506 89.8190 107.836 −126.382 −170.958 −638.158 −111.883 −1190.20
1.3 −10.8798 27.3463 86.3711 30.4368 −297.524 91.8048 −591.550 504.821 −331.148
1.4 −10.6829 −2.44660 82.1240 3.37371 26.1367 100.320 −535.468 −237.014 −36.0409
1.5 −10.4782 −2.88738 77.7924 −75.5324 30.2545 −43.6540 −479.821 −234.663 791.442
1.6 −10.0929 −24.7282 69.8675 20.2324 249.580 −225.563 −382.195 368.482 −204.205
1.7 −9.96834 28.0559 67.3679 −85.7673 −279.671 39.1762 −352.559 544.135 854.958
1.8 −9.86265 −29.0101 65.2718 −27.8967 286.116 106.432 −328.148 598.586 275.136
1.9 −9.41592 22.5666 56.6595 −60.2165 −212.485 −93.2886 −232.192 266.253 566.994
1.10 −9.36177 7.91970 55.6428 15.2707 −74.1424 −60.9674 −221.338 −180.278 −142.961
1.11 −9.28879 6.79931 54.2816 55.3186 −63.1573 −109.817 −206.969 −196.769 −513.843
1.12 −9.24292 −25.2981 53.4317 16.6075 233.829 77.1380 −198.091 396.995 −153.502
1.13 −8.95289 10.5913 48.1542 −18.1795 −94.8223 196.308 −144.627 −130.825 162.759
1.14 −8.88996 18.7065 47.0315 107.329 −166.300 114.464 −133.629 106.932 −954.147
1.15 −8.25067 −17.5967 36.0736 −93.5213 145.184 115.074 −33.6097 66.6434 771.613
1.16 −8.21422 13.1577 35.4733 −55.6754 −108.080 30.1229 −28.5308 −69.8746 457.330
1.17 −8.03714 −10.5980 32.5955 −79.0989 85.1778 −104.629 −4.78643 −130.682 635.729
1.18 −7.89514 −10.2115 30.3332 56.9036 80.6212 230.167 13.1594 −138.725 −449.262
1.19 −7.79360 −16.6480 28.7401 −72.0478 129.748 −223.681 25.4060 34.1567 561.511
1.20 −7.61836 18.9618 26.0394 46.8788 −144.458 −223.496 45.4097 116.548 −357.139
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.100
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)
\(59\) \(-1\)

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 1003.6.a.d 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.6.a.d 100 1.a even 1 1 trivial