# Properties

 Label 50.6.e.a Level $50$ Weight $6$ Character orbit 50.e Analytic conductor $8.019$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 50.e (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.01919099065$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 192q^{4} + 180q^{5} + 72q^{6} + 686q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 192q^{4} + 180q^{5} + 72q^{6} + 686q^{9} + 80q^{10} + 716q^{11} - 160q^{12} - 784q^{14} - 940q^{15} - 3072q^{16} + 1910q^{17} - 5990q^{19} - 640q^{20} - 4344q^{21} + 4720q^{22} + 12410q^{23} + 4608q^{24} + 23660q^{25} - 15248q^{26} + 27600q^{27} - 3040q^{28} - 17170q^{29} - 14720q^{30} - 18834q^{31} - 56930q^{33} + 11696q^{34} + 28450q^{35} - 10976q^{36} + 3540q^{37} + 13432q^{39} - 1280q^{40} + 37146q^{41} - 43560q^{42} + 1184q^{44} + 75240q^{45} + 28672q^{46} - 17830q^{47} - 2560q^{48} - 54656q^{49} + 8800q^{50} - 85984q^{51} + 22350q^{53} + 16080q^{54} - 64730q^{55} + 12544q^{56} + 18000q^{59} + 78720q^{60} + 67086q^{61} - 159600q^{62} + 132820q^{63} + 49152q^{64} - 195820q^{65} - 52256q^{66} - 90790q^{67} - 212608q^{69} + 180080q^{70} - 122874q^{71} + 121200q^{73} + 281456q^{74} + 672810q^{75} - 102400q^{76} + 434040q^{77} + 170720q^{78} + 113600q^{79} - 28160q^{80} + 115878q^{81} - 582750q^{83} - 174816q^{84} + 107770q^{85} - 26888q^{86} - 911580q^{87} - 56960q^{88} - 357730q^{89} - 282920q^{90} - 216324q^{91} + 451040q^{92} + 79136q^{94} + 135000q^{95} + 18432q^{96} - 123520q^{97} - 206240q^{98} + 235252q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −2.35114 + 3.23607i −19.7385 + 6.41342i −4.94427 15.2169i −52.1632 20.0999i 25.6537 78.9540i 5.87904i 60.8676 + 19.7771i 151.885 110.351i 187.687 121.546i
9.2 −2.35114 + 3.23607i −11.7770 + 3.82658i −4.94427 15.2169i 55.5298 6.43773i 15.3063 47.1080i 133.881i 60.8676 + 19.7771i −72.5363 + 52.7007i −109.725 + 194.834i
9.3 −2.35114 + 3.23607i −2.02410 + 0.657668i −4.94427 15.2169i 24.5492 50.2229i 2.63067 8.09638i 244.793i 60.8676 + 19.7771i −192.927 + 140.169i 104.806 + 197.524i
9.4 −2.35114 + 3.23607i −0.401275 + 0.130382i −4.94427 15.2169i −3.09691 + 55.8159i 0.521529 1.60510i 69.1609i 60.8676 + 19.7771i −196.447 + 142.727i −173.343 141.253i
9.5 −2.35114 + 3.23607i 17.7513 5.76774i −4.94427 15.2169i −50.3601 + 24.2664i −23.0710 + 71.0051i 135.417i 60.8676 + 19.7771i 85.2500 61.9377i 39.8760 220.023i
9.6 −2.35114 + 3.23607i 25.8671 8.40474i −4.94427 15.2169i 52.0281 + 20.4469i −33.6189 + 103.468i 51.8294i 60.8676 + 19.7771i 401.877 291.981i −188.493 + 120.293i
9.7 2.35114 3.23607i −27.7680 + 9.02236i −4.94427 15.2169i 12.4445 54.4989i −36.0894 + 111.072i 185.806i −60.8676 19.7771i 493.066 358.234i −147.103 168.406i
9.8 2.35114 3.23607i −12.2615 + 3.98399i −4.94427 15.2169i 55.8772 + 1.65580i −15.9360 + 49.0459i 145.423i −60.8676 19.7771i −62.1197 + 45.1326i 136.733 176.929i
9.9 2.35114 3.23607i −6.79071 + 2.20643i −4.94427 15.2169i 6.92330 + 55.4713i −8.82574 + 27.1628i 47.7219i −60.8676 19.7771i −155.346 + 112.865i 195.787 + 108.017i
9.10 2.35114 3.23607i −1.61240 + 0.523901i −4.94427 15.2169i −47.4044 29.6281i −2.09560 + 6.44960i 48.2855i −60.8676 19.7771i −194.266 + 141.142i −207.333 + 83.7438i
9.11 2.35114 3.23607i 18.4329 5.98921i −4.94427 15.2169i 49.2396 26.4662i 23.9569 73.7316i 28.3664i −60.8676 19.7771i 107.310 77.9654i 30.1231 221.569i
9.12 2.35114 3.23607i 22.5582 7.32959i −4.94427 15.2169i −54.0951 + 14.0970i 29.3184 90.2327i 229.539i −60.8676 19.7771i 258.557 187.853i −81.5663 + 208.199i
19.1 −3.80423 + 1.23607i −14.7520 20.3044i 12.9443 9.40456i −53.1272 17.3924i 81.2178 + 59.0082i 248.570i −37.6183 + 51.7771i −119.557 + 367.957i 223.606 + 0.495932i
19.2 −3.80423 + 1.23607i −9.38088 12.9117i 12.9443 9.40456i 48.5017 27.7954i 51.6467 + 37.5235i 40.6861i −37.6183 + 51.7771i −3.61930 + 11.1391i −150.154 + 165.691i
19.3 −3.80423 + 1.23607i −6.76547 9.31187i 12.9443 9.40456i −7.78168 + 55.3574i 37.2475 + 27.0619i 51.2744i −37.6183 + 51.7771i 34.1518 105.108i −38.8223 220.211i
19.4 −3.80423 + 1.23607i 4.87367 + 6.70803i 12.9443 9.40456i 44.0032 + 34.4778i −26.8321 19.4947i 63.1969i −37.6183 + 51.7771i 53.8461 165.721i −210.015 76.7702i
19.5 −3.80423 + 1.23607i 9.57478 + 13.1786i 12.9443 9.40456i −0.421737 55.9001i −52.7142 38.2991i 120.466i −37.6183 + 51.7771i −6.90673 + 21.2567i 70.7007 + 212.135i
19.6 −3.80423 + 1.23607i 10.0418 + 13.8214i 12.9443 9.40456i −52.7366 + 18.5432i −55.2856 40.1674i 107.711i −37.6183 + 51.7771i −15.1017 + 46.4781i 177.701 135.729i
19.7 3.80423 1.23607i −16.9585 23.3414i 12.9443 9.40456i −8.25144 55.2894i −93.3657 67.8341i 138.453i 37.6183 51.7771i −182.139 + 560.566i −99.7318 200.134i
19.8 3.80423 1.23607i −8.70645 11.9834i 12.9443 9.40456i 45.9076 + 31.8981i −47.9336 34.8258i 137.000i 37.6183 51.7771i 7.29147 22.4408i 214.071 + 64.6026i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 39.12 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.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.6.e.a 48
25.e even 10 1 inner 50.6.e.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.e.a 48 1.a even 1 1 trivial
50.6.e.a 48 25.e even 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(50, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database