# Properties

 Label 50.6.d.a Level $50$ Weight $6$ Character orbit 50.d Analytic conductor $8.019$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 50.d (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.01919099065$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{2} - 11q^{3} - 96q^{4} - 120q^{5} - 44q^{6} + 548q^{7} - 384q^{8} - 893q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{2} - 11q^{3} - 96q^{4} - 120q^{5} - 44q^{6} + 548q^{7} - 384q^{8} - 893q^{9} + 320q^{10} - 327q^{11} - 96q^{12} - 1161q^{13} - 948q^{14} + 760q^{15} - 1536q^{16} + 83q^{17} + 7368q^{18} - 6250q^{19} + 2320q^{20} + 6193q^{21} + 852q^{22} + 2069q^{23} + 2176q^{24} - 14380q^{25} + 15976q^{26} - 18935q^{27} - 592q^{28} - 11775q^{29} - 9840q^{30} + 2103q^{31} + 24576q^{32} - 11747q^{33} + 972q^{34} + 33235q^{35} - 14288q^{36} + 4148q^{37} + 14260q^{38} - 12576q^{39} + 1600q^{40} - 26427q^{41} + 24772q^{42} + 13024q^{43} + 3408q^{44} - 65535q^{45} - 16964q^{46} + 29803q^{47} - 1536q^{48} + 38368q^{49} - 14020q^{50} + 194058q^{51} - 18576q^{52} - 29011q^{53} + 11820q^{54} - 4185q^{55} - 2368q^{56} - 78040q^{57} - 47100q^{58} + 31430q^{59} + 28160q^{60} - 18362q^{61} - 15828q^{62} - 95886q^{63} - 24576q^{64} - 193280q^{65} + 28992q^{66} - 71657q^{67} - 10432q^{68} + 245919q^{69} - 107460q^{70} - 153827q^{71} - 1792q^{72} + 108264q^{73} - 27648q^{74} + 126020q^{75} + 85920q^{76} + 96141q^{77} - 55724q^{78} + 66670q^{79} - 33280q^{80} + 17709q^{81} - 42408q^{82} - 30621q^{83} - 140352q^{84} - 64895q^{85} - 94444q^{86} - 265360q^{87} - 20928q^{88} + 351835q^{89} - 101720q^{90} + 19323q^{91} - 67856q^{92} + 707998q^{93} + 119212q^{94} + 158570q^{95} - 11264q^{96} + 11058q^{97} + 18532q^{98} - 455966q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −3.23607 + 2.35114i −6.86375 + 21.1244i 4.94427 15.2169i −38.8214 40.2232i −27.4550 84.4978i −78.8464 19.7771 + 60.8676i −202.540 147.154i 220.199 + 38.8904i
11.2 −3.23607 + 2.35114i −4.88978 + 15.0492i 4.94427 15.2169i 51.0006 22.8897i −19.5591 60.1968i 188.761 19.7771 + 60.8676i −5.97726 4.34273i −111.224 + 193.982i
11.3 −3.23607 + 2.35114i −3.10515 + 9.55668i 4.94427 15.2169i −0.387302 + 55.9004i −12.4206 38.2267i −100.787 19.7771 + 60.8676i 114.903 + 83.4819i −130.176 181.808i
11.4 −3.23607 + 2.35114i 2.33322 7.18091i 4.94427 15.2169i −54.1589 + 13.8497i 9.33287 + 28.7236i 54.5999 19.7771 + 60.8676i 150.470 + 109.323i 142.699 172.154i
11.5 −3.23607 + 2.35114i 5.67913 17.4786i 4.94427 15.2169i 6.69834 55.4989i 22.7165 + 69.9143i −86.1311 19.7771 + 60.8676i −76.6568 55.6945i 108.810 + 195.347i
11.6 −3.23607 + 2.35114i 9.12748 28.0915i 4.94427 15.2169i 30.2654 + 47.0001i 36.5099 + 112.366i 204.125 19.7771 + 60.8676i −509.231 369.978i −208.445 80.9373i
21.1 1.23607 3.80423i −19.7905 14.3786i −12.9443 9.40456i −30.4129 + 46.9047i −79.1620 + 57.5145i 49.2262 −51.7771 + 37.6183i 109.827 + 338.014i 140.844 + 173.675i
21.2 1.23607 3.80423i −16.6068 12.0655i −12.9443 9.40456i 2.13641 55.8609i −66.4272 + 48.2622i 12.6460 −51.7771 + 37.6183i 55.1171 + 169.633i −209.867 77.1752i
21.3 1.23607 3.80423i −0.732003 0.531831i −12.9443 9.40456i 54.7131 + 11.4663i −2.92801 + 2.12732i 133.265 −51.7771 + 37.6183i −74.8381 230.328i 111.249 193.968i
21.4 1.23607 3.80423i 1.10044 + 0.799520i −12.9443 9.40456i −0.782112 + 55.8962i 4.40178 3.19808i −204.325 −51.7771 + 37.6183i −74.5194 229.347i 211.675 + 72.0669i
21.5 1.23607 3.80423i 8.60489 + 6.25182i −12.9443 9.40456i −38.0198 40.9817i 34.4196 25.0073i −106.678 −51.7771 + 37.6183i −40.1323 123.514i −202.899 + 93.9797i
21.6 1.23607 3.80423i 19.6428 + 14.2713i −12.9443 9.40456i −42.2315 + 36.6265i 78.5712 57.0853i 208.145 −51.7771 + 37.6183i 107.078 + 329.551i 87.1346 + 205.931i
31.1 1.23607 + 3.80423i −19.7905 + 14.3786i −12.9443 + 9.40456i −30.4129 46.9047i −79.1620 57.5145i 49.2262 −51.7771 37.6183i 109.827 338.014i 140.844 173.675i
31.2 1.23607 + 3.80423i −16.6068 + 12.0655i −12.9443 + 9.40456i 2.13641 + 55.8609i −66.4272 48.2622i 12.6460 −51.7771 37.6183i 55.1171 169.633i −209.867 + 77.1752i
31.3 1.23607 + 3.80423i −0.732003 + 0.531831i −12.9443 + 9.40456i 54.7131 11.4663i −2.92801 2.12732i 133.265 −51.7771 37.6183i −74.8381 + 230.328i 111.249 + 193.968i
31.4 1.23607 + 3.80423i 1.10044 0.799520i −12.9443 + 9.40456i −0.782112 55.8962i 4.40178 + 3.19808i −204.325 −51.7771 37.6183i −74.5194 + 229.347i 211.675 72.0669i
31.5 1.23607 + 3.80423i 8.60489 6.25182i −12.9443 + 9.40456i −38.0198 + 40.9817i 34.4196 + 25.0073i −106.678 −51.7771 37.6183i −40.1323 + 123.514i −202.899 93.9797i
31.6 1.23607 + 3.80423i 19.6428 14.2713i −12.9443 + 9.40456i −42.2315 36.6265i 78.5712 + 57.0853i 208.145 −51.7771 37.6183i 107.078 329.551i 87.1346 205.931i
41.1 −3.23607 2.35114i −6.86375 21.1244i 4.94427 + 15.2169i −38.8214 + 40.2232i −27.4550 + 84.4978i −78.8464 19.7771 60.8676i −202.540 + 147.154i 220.199 38.8904i
41.2 −3.23607 2.35114i −4.88978 15.0492i 4.94427 + 15.2169i 51.0006 + 22.8897i −19.5591 + 60.1968i 188.761 19.7771 60.8676i −5.97726 + 4.34273i −111.224 193.982i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 41.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.6.d.a 24
25.d even 5 1 inner 50.6.d.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.d.a 24 1.a even 1 1 trivial
50.6.d.a 24 25.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$38\!\cdots\!41$$$$T_{3}^{15} +$$$$15\!\cdots\!56$$$$T_{3}^{14} +$$$$10\!\cdots\!66$$$$T_{3}^{13} +$$$$40\!\cdots\!16$$$$T_{3}^{12} -$$$$28\!\cdots\!59$$$$T_{3}^{11} +$$$$70\!\cdots\!51$$$$T_{3}^{10} -$$$$52\!\cdots\!79$$$$T_{3}^{9} +$$$$11\!\cdots\!06$$$$T_{3}^{8} -$$$$58\!\cdots\!69$$$$T_{3}^{7} +$$$$10\!\cdots\!31$$$$T_{3}^{6} -$$$$36\!\cdots\!84$$$$T_{3}^{5} +$$$$36\!\cdots\!01$$$$T_{3}^{4} -$$$$23\!\cdots\!14$$$$T_{3}^{3} -$$$$20\!\cdots\!64$$$$T_{3}^{2} +$$$$27\!\cdots\!36$$$$T_{3} +$$$$53\!\cdots\!96$$">$$T_{3}^{24} + \cdots$$ acting on $$S_{6}^{\mathrm{new}}(50, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database