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Properties

Label 1003.6.a.c
Level $1003$
Weight $6$
Character orbit 1003.a
Self dual yes
Analytic conductor $160.865$
Analytic rank $0$
Dimension $98$
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: \(98\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 98 q + 25 q^{2} + 45 q^{3} + 1611 q^{4} + 541 q^{5} + 305 q^{6} + 247 q^{7} + 1425 q^{8} + 8873 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 98 q + 25 q^{2} + 45 q^{3} + 1611 q^{4} + 541 q^{5} + 305 q^{6} + 247 q^{7} + 1425 q^{8} + 8873 q^{9} + 1000 q^{10} + 2842 q^{11} + 2304 q^{12} + 3246 q^{13} + 2245 q^{14} + 999 q^{15} + 29551 q^{16} + 28322 q^{17} + 1540 q^{18} + 932 q^{19} + 18297 q^{20} + 1112 q^{21} + 7918 q^{22} + 12457 q^{23} + 18405 q^{24} + 65623 q^{25} + 16159 q^{26} + 12633 q^{27} + 23731 q^{28} + 28451 q^{29} - 19285 q^{30} + 21493 q^{31} + 52885 q^{32} + 39707 q^{33} + 7225 q^{34} + 20313 q^{35} + 155100 q^{36} + 71740 q^{37} + 78949 q^{38} + 42612 q^{39} + 9854 q^{40} + 39545 q^{41} - 113891 q^{42} + 10477 q^{43} + 129131 q^{44} + 212203 q^{45} - 8801 q^{46} + 92561 q^{47} + 42263 q^{48} + 242447 q^{49} - 24646 q^{50} + 13005 q^{51} + 126145 q^{52} + 197573 q^{53} + 138637 q^{54} + 56894 q^{55} + 65301 q^{56} + 154798 q^{57} + 46454 q^{58} - 341138 q^{59} - 62712 q^{60} + 200139 q^{61} + 337089 q^{62} + 72251 q^{63} + 437269 q^{64} + 352383 q^{65} + 342440 q^{66} + 25262 q^{67} + 465579 q^{68} + 233418 q^{69} + 435463 q^{70} + 95619 q^{71} + 215863 q^{72} + 372862 q^{73} + 231761 q^{74} - 225892 q^{75} - 156705 q^{76} + 515725 q^{77} + 140085 q^{78} + 262158 q^{79} + 294401 q^{80} + 933702 q^{81} + 466955 q^{82} + 219868 q^{83} - 213492 q^{84} + 156349 q^{85} + 89392 q^{86} + 236040 q^{87} + 389548 q^{88} + 388400 q^{89} - 227892 q^{90} + 331873 q^{91} + 380277 q^{92} + 398803 q^{93} + 423436 q^{94} + 224444 q^{95} + 664910 q^{96} + 487367 q^{97} + 510307 q^{98} + 524962 q^{99}+O(q^{100}) \) 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 −10.9232 7.20612 87.3155 78.4833 −78.7137 24.7188 −604.221 −191.072 −857.285
1.2 −10.8138 −12.8399 84.9380 35.7538 138.848 −159.251 −572.460 −78.1367 −386.634
1.3 −10.7119 −21.3093 82.7447 −35.7771 228.263 245.326 −543.572 211.086 383.240
1.4 −10.6035 18.4446 80.4332 −31.2915 −195.576 23.4602 −513.559 97.2028 331.798
1.5 −10.2275 0.736474 72.6019 9.11164 −7.53229 −185.065 −415.257 −242.458 −93.1894
1.6 −10.1584 9.46727 71.1933 45.9403 −96.1724 78.1161 −398.141 −153.371 −466.680
1.7 −10.0368 4.95431 68.7369 −99.7208 −49.7253 45.5241 −368.720 −218.455 1000.88
1.8 −10.0355 −17.6355 68.7121 −73.0782 176.982 −111.523 −368.426 68.0120 733.379
1.9 −9.72561 −30.3930 62.5875 103.603 295.591 32.5861 −297.483 680.735 −1007.60
1.10 −9.56102 28.4062 59.4131 99.4331 −271.592 −44.4244 −262.098 563.913 −950.682
1.11 −9.48141 28.9431 57.8971 −25.5004 −274.422 161.442 −245.541 594.705 241.780
1.12 −9.19129 −17.7439 52.4799 −11.2089 163.090 34.6364 −188.236 71.8470 103.024
1.13 −8.48713 21.4915 40.0314 −91.1966 −182.401 −98.5224 −68.1633 218.886 773.997
1.14 −8.12511 −29.5839 34.0173 −50.3920 240.372 −150.508 −16.3911 632.208 409.441
1.15 −8.07457 7.61526 33.1987 −34.6632 −61.4900 123.139 −9.67904 −185.008 279.890
1.16 −8.06402 25.0583 33.0284 13.6048 −202.071 208.151 −8.29333 384.918 −109.709
1.17 −7.63205 2.78346 26.2482 69.3207 −21.2435 −199.961 43.8978 −235.252 −529.059
1.18 −7.52040 0.365549 24.5564 84.2078 −2.74907 −86.4777 55.9785 −242.866 −633.276
1.19 −7.45548 −27.0971 23.5842 44.6208 202.022 209.173 62.7439 491.254 −332.670
1.20 −7.37334 −9.69688 22.3662 −64.7469 71.4985 205.936 71.0333 −148.970 477.401
See all 98 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.98
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.c ✓ 98
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.6.a.c ✓ 98 1.a even 1 1 trivial