# Properties

 Label 25.6.e.a Level $25$ Weight $6$ Character orbit 25.e Analytic conductor $4.010$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 25.e (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.00959549532$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 5q^{2} - 5q^{3} + 189q^{4} - 60q^{5} - 139q^{6} + 610q^{8} + 1031q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 5q^{2} - 5q^{3} + 189q^{4} - 60q^{5} - 139q^{6} + 610q^{8} + 1031q^{9} + 435q^{10} - 724q^{11} + 315q^{12} - 5q^{13} - 497q^{14} - 105q^{15} - 6967q^{16} - 960q^{17} + 4075q^{19} + 9155q^{20} + 2016q^{21} - 5450q^{22} - 10995q^{23} - 20720q^{24} - 16610q^{25} + 21666q^{26} - 13805q^{27} + 14235q^{28} + 17195q^{29} + 39595q^{30} + 8631q^{31} + 28460q^{33} - 24067q^{34} - 4145q^{35} - 43827q^{36} + 19980q^{37} - 77385q^{38} + 6037q^{39} + 93210q^{40} - 26514q^{41} + 182885q^{42} + 26748q^{44} - 94185q^{45} - 14559q^{46} - 7700q^{47} - 127830q^{48} - 130126q^{49} - 231115q^{50} + 121116q^{51} - 189260q^{52} + 45840q^{53} + 19985q^{54} + 79370q^{55} - 43280q^{56} + 337370q^{58} + 47640q^{59} + 71480q^{60} + 7791q^{61} + 135320q^{62} - 35790q^{63} + 68204q^{64} + 197485q^{65} - 14798q^{66} - 66180q^{67} + 17957q^{69} - 129930q^{70} - 46694q^{71} - 50545q^{72} - 18985q^{73} - 183292q^{74} - 202210q^{75} + 273360q^{76} - 132990q^{77} - 665225q^{78} - 54065q^{79} + 362810q^{80} - 382452q^{81} + 161155q^{83} + 474258q^{84} - 83075q^{85} + 74691q^{86} + 897680q^{87} + 578430q^{88} + 436475q^{89} + 775655q^{90} + 181046q^{91} - 1412870q^{92} - 336327q^{94} - 362785q^{95} - 766024q^{96} - 475210q^{97} - 361660q^{98} - 451848q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −9.71189 3.15559i −6.12321 + 8.42787i 58.4746 + 42.4843i −38.3382 + 40.6839i 86.0628 62.5283i 211.819i −241.764 332.759i 41.5558 + 127.895i 500.718 274.138i
4.2 −8.74738 2.84220i 16.5578 22.7899i 42.5500 + 30.9144i 53.9635 14.5924i −209.611 + 152.291i 122.989i −111.339 153.244i −170.126 523.593i −513.514 25.7300i
4.3 −6.91215 2.24589i 1.65280 2.27488i 16.8453 + 12.2388i −34.0654 44.3232i −16.5335 + 12.0123i 197.082i 47.7523 + 65.7253i 72.6478 + 223.587i 135.920 + 382.876i
4.4 −5.43609 1.76629i −17.9990 + 24.7735i 0.542750 + 0.394331i 37.4160 41.5336i 141.602 102.880i 28.6321i 105.256 + 144.873i −214.672 660.693i −276.757 + 159.693i
4.5 −4.17440 1.35635i −0.0707334 + 0.0973562i −10.3026 7.48525i 26.6799 + 49.1241i 0.427319 0.310465i 51.4533i 115.412 + 158.851i 75.0867 + 231.093i −44.7435 241.251i
4.6 −0.580095 0.188484i 15.1807 20.8945i −25.5876 18.5905i −52.0512 + 20.3881i −12.7445 + 9.25945i 10.2303i 22.8118 + 31.3978i −131.033 403.279i 34.0375 2.01618i
4.7 0.835458 + 0.271457i 2.21975 3.05523i −25.2642 18.3555i 40.9701 38.0322i 2.68387 1.94995i 153.044i −32.6474 44.9353i 70.6840 + 217.543i 44.5529 20.6527i
4.8 2.22114 + 0.721693i −9.09397 + 12.5168i −21.4759 15.6032i −55.6985 4.76195i −29.2323 + 21.2385i 43.5518i −80.3681 110.617i 1.12167 + 3.45216i −120.278 50.7742i
4.9 5.71541 + 1.85705i −10.7874 + 14.8476i 3.32876 + 2.41849i 37.8738 + 41.1166i −89.2272 + 64.8274i 177.090i −98.5002 135.574i −28.9918 89.2276i 140.109 + 305.332i
4.10 6.57040 + 2.13485i 10.0602 13.8467i 12.7241 + 9.24458i 15.9125 53.5891i 95.6602 69.5012i 188.942i −66.0770 90.9472i −15.4317 47.4939i 218.956 318.131i
4.11 8.05151 + 2.61610i 8.11728 11.1725i 32.0944 + 23.3179i −0.713405 + 55.8971i 94.5847 68.7198i 150.001i 38.1710 + 52.5378i 16.1571 + 49.7264i −151.976 + 448.190i
4.12 10.3591 + 3.36586i −9.28713 + 12.7826i 70.0926 + 50.9253i −25.7064 49.6405i −139.231 + 101.157i 71.6578i 349.815 + 481.478i −2.05400 6.32156i −99.2113 600.754i
9.1 −6.42181 + 8.83886i 13.2720 4.31235i −26.9973 83.0892i −22.7744 51.0522i −47.1143 + 145.003i 134.700i 575.283 + 186.921i −39.0403 + 28.3645i 597.496 + 126.548i
9.2 −5.77566 + 7.94951i −24.0141 + 7.80267i −19.9480 61.3935i 11.5397 + 54.6977i 76.6701 235.966i 104.735i 304.214 + 98.8452i 319.207 231.917i −501.469 224.180i
9.3 −3.62228 + 4.98565i 10.7412 3.49003i −1.84718 5.68504i 55.8518 + 2.36063i −21.5076 + 66.1937i 133.004i −152.517 49.5557i −93.3979 + 67.8576i −214.080 + 269.907i
9.4 −3.32714 + 4.57941i 3.38178 1.09881i −0.0126365 0.0388913i −43.0053 + 35.7147i −6.21976 + 19.1425i 56.5747i −172.049 55.9023i −186.362 + 135.400i −20.4679 315.767i
9.5 −2.41181 + 3.31957i −18.6046 + 6.04499i 4.68583 + 14.4215i 9.93343 55.0121i 24.8039 76.3384i 138.174i −184.051 59.8017i 112.997 82.0971i 158.659 + 165.653i
9.6 0.123616 0.170142i 28.0358 9.10938i 9.87488 + 30.3917i −30.6717 46.7359i 1.91577 5.89614i 86.5992i 12.7921 + 4.15640i 506.434 367.946i −11.7433 0.558733i
9.7 0.958035 1.31862i −8.25369 + 2.68179i 9.06761 + 27.9072i −55.8091 + 3.21620i −4.37106 + 13.4527i 127.304i 95.0904 + 30.8967i −135.660 + 98.5625i −49.2261 + 76.6723i
9.8 1.61555 2.22361i 12.9949 4.22230i 7.55409 + 23.2491i 30.9389 + 46.5595i 11.6052 35.7170i 221.003i 147.549 + 47.9417i −45.5512 + 33.0949i 153.514 + 6.42313i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.12 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 25.6.e.a 48
25.e even 10 1 inner 25.6.e.a 48
25.f odd 20 2 625.6.a.g 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.e.a 48 1.a even 1 1 trivial
25.6.e.a 48 25.e even 10 1 inner
625.6.a.g 48 25.f odd 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database