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Properties

Label 503.8.a.a
Level $503$
Weight $8$
Character orbit 503.a
Self dual yes
Analytic conductor $157.130$
Analytic rank $1$
Dimension $136$
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] = [503,8,Mod(1,503)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(503, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("503.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 503.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: \(157.129667819\)
Analytic rank: \(1\)
Dimension: \(136\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 41 q^{2} - 271 q^{3} + 7973 q^{4} - 627 q^{5} - 1943 q^{6} - 5000 q^{7} - 7308 q^{8} + 84433 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 41 q^{2} - 271 q^{3} + 7973 q^{4} - 627 q^{5} - 1943 q^{6} - 5000 q^{7} - 7308 q^{8} + 84433 q^{9} - 14116 q^{10} - 13186 q^{11} - 54156 q^{12} - 59023 q^{13} - 17654 q^{14} - 61296 q^{15} + 409049 q^{16} - 106617 q^{17} - 122549 q^{18} - 102130 q^{19} - 138319 q^{20} - 145938 q^{21} - 421523 q^{22} - 345101 q^{23} - 526139 q^{24} + 1138635 q^{25} - 59352 q^{26} - 743179 q^{27} - 957255 q^{28} - 479329 q^{29} - 267660 q^{30} - 546388 q^{31} - 1250605 q^{32} - 1377461 q^{33} - 1032720 q^{34} - 550648 q^{35} + 3467642 q^{36} - 3631503 q^{37} - 819394 q^{38} - 1647851 q^{39} - 3167279 q^{40} - 1974324 q^{41} - 1435911 q^{42} - 3414691 q^{43} - 3072568 q^{44} - 1951870 q^{45} - 5281684 q^{46} - 3152134 q^{47} - 6352403 q^{48} + 7086172 q^{49} - 5466685 q^{50} - 2970287 q^{51} - 9418146 q^{52} - 5625580 q^{53} - 7151614 q^{54} - 6572253 q^{55} - 2712927 q^{56} - 14263019 q^{57} - 10673874 q^{58} - 3270135 q^{59} - 13091044 q^{60} - 11689181 q^{61} - 4442088 q^{62} - 11002880 q^{63} + 16990888 q^{64} - 12402222 q^{65} - 10541967 q^{66} - 18356829 q^{67} - 12652516 q^{68} - 14685041 q^{69} - 5068580 q^{70} - 9201925 q^{71} - 19787365 q^{72} - 36531179 q^{73} - 7247443 q^{74} - 21472395 q^{75} - 15442608 q^{76} - 20431540 q^{77} - 15534382 q^{78} - 24699482 q^{79} - 19206796 q^{80} + 27272280 q^{81} - 28699117 q^{82} - 9928753 q^{83} - 9779347 q^{84} - 49276270 q^{85} - 14999831 q^{86} - 17886335 q^{87} - 89693396 q^{88} - 25597729 q^{89} - 36691472 q^{90} - 33778565 q^{91} - 79292862 q^{92} - 43338268 q^{93} - 21241318 q^{94} - 31406553 q^{95} - 78053959 q^{96} - 124613513 q^{97} - 36783764 q^{98} - 44397059 q^{99}+O(q^{100}) \) 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 −22.0124 84.1673 356.547 30.7179 −1852.73 48.4720 −5030.87 4897.14 −676.176
1.2 −21.9042 −70.5787 351.792 307.470 1545.97 455.320 −4901.97 2794.35 −6734.86
1.3 −21.7676 −49.7382 345.829 282.300 1082.68 −395.249 −4741.63 286.889 −6144.99
1.4 −21.3338 −80.9859 327.129 −251.438 1727.73 −772.226 −4248.17 4371.71 5364.11
1.5 −21.2043 15.7612 321.623 −67.6975 −334.205 892.775 −4105.65 −1938.59 1435.48
1.6 −21.0422 −2.09674 314.776 468.211 44.1200 1352.20 −3930.18 −2182.60 −9852.20
1.7 −20.5430 42.3157 294.015 −489.207 −869.292 −1511.08 −3410.45 −396.382 10049.8
1.8 −20.2449 −27.0982 281.857 −404.371 548.601 −981.101 −3114.82 −1452.69 8186.47
1.9 −20.0053 53.3250 272.212 −225.286 −1066.78 17.0654 −2885.01 656.552 4506.91
1.10 −19.8000 −39.2817 264.042 −280.506 777.779 622.238 −2693.64 −643.950 5554.03
1.11 −19.6650 81.3188 258.712 −190.480 −1599.13 −940.985 −2570.46 4425.75 3745.79
1.12 −19.4151 47.3371 248.945 −331.092 −919.054 1443.01 −2348.15 53.8045 6428.18
1.13 −19.3175 −24.1850 245.164 128.612 467.192 −678.781 −2263.32 −1602.09 −2484.46
1.14 −18.8765 −17.8562 228.323 442.924 337.062 −55.3855 −1893.75 −1868.16 −8360.87
1.15 −18.8390 64.9996 226.909 334.316 −1224.53 1080.04 −1863.35 2037.95 −6298.20
1.16 −18.6819 −65.7923 221.013 479.144 1229.12 −1651.27 −1737.65 2141.62 −8951.31
1.17 −18.6500 −19.2958 219.824 −77.9961 359.867 1216.36 −1712.53 −1814.67 1454.63
1.18 −18.4503 −44.8367 212.413 53.9566 827.251 482.930 −1557.45 −176.667 −995.515
1.19 −17.7775 −6.91401 188.038 −241.197 122.914 36.3829 −1067.33 −2139.20 4287.87
1.20 −17.5449 −12.2594 179.824 469.573 215.090 49.8153 −909.240 −2036.71 −8238.61
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.136
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(503\) \(-1\)

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 503.8.a.a ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.8.a.a ✓ 136 1.a even 1 1 trivial