# Properties

 Label 503.2.c.a Level $503$ Weight $2$ Character orbit 503.c Analytic conductor $4.016$ Analytic rank $0$ Dimension $10250$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 503.c (of order $$251$$, degree $$250$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.01647522167$$ Analytic rank: $$0$$ Dimension: $$10250$$ Relative dimension: $$41$$ over $$\Q(\zeta_{251})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{251}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$10250q - 249q^{2} - 249q^{3} - 285q^{4} - 245q^{5} - 237q^{6} - 247q^{7} - 239q^{8} - 280q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$10250q - 249q^{2} - 249q^{3} - 285q^{4} - 245q^{5} - 237q^{6} - 247q^{7} - 239q^{8} - 280q^{9} - 233q^{10} - 239q^{11} - 239q^{12} - 245q^{13} - 223q^{14} - 231q^{15} - 263q^{16} - 243q^{17} - 217q^{18} - 229q^{19} - 195q^{20} - 215q^{21} - 233q^{22} - 241q^{23} - 181q^{24} - 280q^{25} - 215q^{26} - 237q^{27} - 185q^{28} - 217q^{29} - 159q^{30} - 219q^{31} - 185q^{32} - 209q^{33} - 167q^{34} - 187q^{35} - 203q^{36} - 237q^{37} - 161q^{38} - 197q^{39} - 135q^{40} - 209q^{41} - 117q^{42} - 221q^{43} - 147q^{44} - 167q^{45} - 167q^{46} - 195q^{47} - 141q^{48} - 234q^{49} - 145q^{50} - 177q^{51} - 179q^{52} - 211q^{53} - 83q^{54} - 165q^{55} - 125q^{56} - 177q^{57} - 205q^{58} - 189q^{59} - 27q^{60} - 193q^{61} - 143q^{62} - 159q^{63} - 185q^{64} - 155q^{65} - 97q^{66} - 201q^{67} - 155q^{68} - 113q^{69} - 123q^{70} - 153q^{71} - 21q^{72} - 213q^{73} - 83q^{74} - 161q^{75} - 97q^{76} - 139q^{77} - 43q^{78} - 191q^{79} + 3q^{80} - 188q^{81} - 97q^{82} - 161q^{83} + 67q^{84} - 159q^{85} - 121q^{86} - 121q^{87} - 143q^{88} - 155q^{89} + 115q^{90} - 137q^{91} - 119q^{92} - 123q^{93} - 99q^{94} - 181q^{95} + 67q^{96} - 229q^{97} - 63q^{98} - 85q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −1.73497 2.13612i −2.22490 + 1.92567i −1.14279 + 5.45470i −0.109819 0.346829i 7.97361 + 1.41168i 0.0493067 0.143165i 8.74162 4.50242i 0.811649 5.59990i −0.550336 + 0.836323i
2.2 −1.72191 2.12004i 1.69918 1.47066i −1.11950 + 5.34354i −0.0846785 0.267431i −6.04369 1.07000i −1.43085 + 4.15454i 8.40007 4.32651i 0.294060 2.02884i −0.421156 + 0.640014i
2.3 −1.53474 1.88959i 0.404749 0.350314i −0.805040 + 3.84256i 1.19914 + 3.78712i −1.28313 0.227171i 1.44952 4.20876i 4.16812 2.14681i −0.389221 + 2.68540i 5.31575 8.07812i
2.4 −1.49378 1.83917i −0.567738 + 0.491382i −0.741049 + 3.53712i −0.276099 0.871972i 1.75181 + 0.310147i 0.530619 1.54068i 3.39955 1.75096i −0.349454 + 2.41102i −1.19127 + 1.81033i
2.5 −1.47995 1.82214i 1.66985 1.44527i −0.719831 + 3.43585i 0.408432 + 1.28990i −5.10478 0.903771i 0.254080 0.737735i 3.15213 1.62353i 0.269269 1.85780i 1.74593 2.65321i
2.6 −1.43081 1.76164i −1.19346 + 1.03295i −0.646042 + 3.08364i 0.511937 + 1.61679i 3.52731 + 0.624490i −1.13380 + 3.29205i 2.32143 1.19567i −0.0729611 + 0.503389i 2.11572 3.21517i
2.7 −1.35722 1.67103i 2.35620 2.03931i −0.540193 + 2.57841i −0.767603 2.42424i −6.60563 1.16949i 0.954936 2.77271i 1.21411 0.625336i 0.962553 6.64105i −3.00917 + 4.57291i
2.8 −1.31051 1.61352i −0.0751196 + 0.0650167i −0.475908 + 2.27157i −1.09154 3.44730i 0.203351 + 0.0360021i −1.53778 + 4.46504i 0.592995 0.305425i −0.428907 + 2.95921i −4.13181 + 6.27895i
2.9 −1.15193 1.41827i −2.28946 + 1.98155i −0.274449 + 1.30998i 1.06426 + 3.36113i 5.44767 + 0.964477i 0.658051 1.91069i −1.07463 + 0.553494i 0.884773 6.10441i 3.54105 5.38118i
2.10 −0.994976 1.22503i −1.98352 + 1.71676i −0.100616 + 0.480254i −1.27158 4.01588i 4.07664 + 0.721744i 1.44819 4.20490i −2.11761 + 1.09069i 0.556781 3.84146i −3.65440 + 5.55343i
2.11 −0.968162 1.19202i 0.996021 0.862065i −0.0734591 + 0.350630i −0.521680 1.64756i −1.99191 0.352655i 0.222510 0.646070i −2.24135 + 1.15442i −0.181422 + 1.25171i −1.45885 + 2.21696i
2.12 −0.948756 1.16812i 1.18005 1.02135i −0.0542683 + 0.259030i 0.841155 + 2.65653i −2.31264 0.409440i −1.01850 + 2.95726i −2.32163 + 1.19577i −0.0809476 + 0.558491i 2.30510 3.50297i
2.13 −0.945803 1.16449i −0.790504 + 0.684188i −0.0513814 + 0.245250i −0.107766 0.340345i 1.54439 + 0.273425i 0.296825 0.861848i −2.33318 + 1.20172i −0.273541 + 1.88727i −0.294402 + 0.447391i
2.14 −0.766891 0.944209i −2.44234 + 2.11387i 0.106699 0.509291i −0.375743 1.18667i 3.86894 + 0.684973i −1.33365 + 3.87232i −2.72550 + 1.40379i 1.06626 7.35660i −0.832307 + 1.26482i
2.15 −0.604817 0.744661i 1.49016 1.28975i 0.221392 1.05673i −1.03109 3.25639i −1.86170 0.329604i −0.182067 + 0.528642i −2.62653 + 1.35281i 0.126808 0.874904i −1.80128 + 2.73733i
2.16 −0.604152 0.743843i −0.138203 + 0.119616i 0.221807 1.05871i 0.634410 + 2.00359i 0.172471 + 0.0305350i 0.803660 2.33347i −2.62536 + 1.35221i −0.425531 + 2.93591i 1.10707 1.68237i
2.17 −0.448047 0.551642i 2.42081 2.09524i 0.306545 1.46318i −0.150619 0.475684i −2.24046 0.396660i −0.834374 + 2.42265i −2.20809 + 1.13729i 1.04000 7.17539i −0.194923 + 0.296217i
2.18 −0.380517 0.468498i −1.24266 + 1.07553i 0.335410 1.60096i −0.464781 1.46787i 0.976735 + 0.172925i −0.519527 + 1.50848i −1.95081 + 1.00478i −0.0428949 + 0.295950i −0.510836 + 0.776297i
2.19 −0.225034 0.277066i −0.186491 + 0.161409i 0.383983 1.83280i 0.927472 + 2.92913i 0.0866877 + 0.0153475i −0.576054 + 1.67260i −1.22886 + 0.632932i −0.421597 + 2.90877i 0.602849 0.916125i
2.20 −0.102711 0.126460i 0.592852 0.513119i 0.404666 1.93152i −0.990176 3.12716i −0.125782 0.0222689i 1.03743 3.01224i −0.575493 + 0.296411i −0.342141 + 2.36057i −0.293759 + 0.446413i
See next 80 embeddings (of 10250 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 498.41 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
503.c even 251 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 503.2.c.a 10250
503.c even 251 1 inner 503.2.c.a 10250

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.c.a 10250 1.a even 1 1 trivial
503.2.c.a 10250 503.c even 251 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(503, [\chi])$$.