# Properties

 Label 1083.6.a.b.1.1 Level $1083$ Weight $6$ Character 1083.1 Self dual yes Analytic conductor $173.696$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1083,6,Mod(1,1083)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1083, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1083.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1083 = 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1083.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: $$173.695676857$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1083.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -9.00000 q^{3} -28.0000 q^{4} -98.0000 q^{5} -18.0000 q^{6} +240.000 q^{7} -120.000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -9.00000 q^{3} -28.0000 q^{4} -98.0000 q^{5} -18.0000 q^{6} +240.000 q^{7} -120.000 q^{8} +81.0000 q^{9} -196.000 q^{10} +336.000 q^{11} +252.000 q^{12} -342.000 q^{13} +480.000 q^{14} +882.000 q^{15} +656.000 q^{16} -6.00000 q^{17} +162.000 q^{18} +2744.00 q^{20} -2160.00 q^{21} +672.000 q^{22} +2836.00 q^{23} +1080.00 q^{24} +6479.00 q^{25} -684.000 q^{26} -729.000 q^{27} -6720.00 q^{28} +5902.00 q^{29} +1764.00 q^{30} -2744.00 q^{31} +5152.00 q^{32} -3024.00 q^{33} -12.0000 q^{34} -23520.0 q^{35} -2268.00 q^{36} -13670.0 q^{37} +3078.00 q^{39} +11760.0 q^{40} -10990.0 q^{41} -4320.00 q^{42} -4996.00 q^{43} -9408.00 q^{44} -7938.00 q^{45} +5672.00 q^{46} -17124.0 q^{47} -5904.00 q^{48} +40793.0 q^{49} +12958.0 q^{50} +54.0000 q^{51} +9576.00 q^{52} +4470.00 q^{53} -1458.00 q^{54} -32928.0 q^{55} -28800.0 q^{56} +11804.0 q^{58} -26292.0 q^{59} -24696.0 q^{60} +29134.0 q^{61} -5488.00 q^{62} +19440.0 q^{63} -10688.0 q^{64} +33516.0 q^{65} -6048.00 q^{66} +42052.0 q^{67} +168.000 q^{68} -25524.0 q^{69} -47040.0 q^{70} +26112.0 q^{71} -9720.00 q^{72} -49046.0 q^{73} -27340.0 q^{74} -58311.0 q^{75} +80640.0 q^{77} +6156.00 q^{78} -79056.0 q^{79} -64288.0 q^{80} +6561.00 q^{81} -21980.0 q^{82} +9472.00 q^{83} +60480.0 q^{84} +588.000 q^{85} -9992.00 q^{86} -53118.0 q^{87} -40320.0 q^{88} -82894.0 q^{89} -15876.0 q^{90} -82080.0 q^{91} -79408.0 q^{92} +24696.0 q^{93} -34248.0 q^{94} -46368.0 q^{96} -39850.0 q^{97} +81586.0 q^{98} +27216.0 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.353553 0.176777 0.984251i $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ −28.0000 −0.875000
$$5$$ −98.0000 −1.75308 −0.876539 0.481331i $$-0.840153\pi$$
−0.876539 + 0.481331i $$0.840153\pi$$
$$6$$ −18.0000 −0.204124
$$7$$ 240.000 1.85125 0.925627 0.378436i $$-0.123538\pi$$
0.925627 + 0.378436i $$0.123538\pi$$
$$8$$ −120.000 −0.662913
$$9$$ 81.0000 0.333333
$$10$$ −196.000 −0.619806
$$11$$ 336.000 0.837255 0.418627 0.908158i $$-0.362511\pi$$
0.418627 + 0.908158i $$0.362511\pi$$
$$12$$ 252.000 0.505181
$$13$$ −342.000 −0.561265 −0.280632 0.959815i $$-0.590544\pi$$
−0.280632 + 0.959815i $$0.590544\pi$$
$$14$$ 480.000 0.654517
$$15$$ 882.000 1.01214
$$16$$ 656.000 0.640625
$$17$$ −6.00000 −0.00503534 −0.00251767 0.999997i $$-0.500801\pi$$
−0.00251767 + 0.999997i $$0.500801\pi$$
$$18$$ 162.000 0.117851
$$19$$ 0 0
$$20$$ 2744.00 1.53394
$$21$$ −2160.00 −1.06882
$$22$$ 672.000 0.296014
$$23$$ 2836.00 1.11786 0.558929 0.829216i $$-0.311212\pi$$
0.558929 + 0.829216i $$0.311212\pi$$
$$24$$ 1080.00 0.382733
$$25$$ 6479.00 2.07328
$$26$$ −684.000 −0.198437
$$27$$ −729.000 −0.192450
$$28$$ −6720.00 −1.61985
$$29$$ 5902.00 1.30318 0.651590 0.758572i $$-0.274102\pi$$
0.651590 + 0.758572i $$0.274102\pi$$
$$30$$ 1764.00 0.357845
$$31$$ −2744.00 −0.512838 −0.256419 0.966566i $$-0.582543\pi$$
−0.256419 + 0.966566i $$0.582543\pi$$
$$32$$ 5152.00 0.889408
$$33$$ −3024.00 −0.483389
$$34$$ −12.0000 −0.00178026
$$35$$ −23520.0 −3.24539
$$36$$ −2268.00 −0.291667
$$37$$ −13670.0 −1.64159 −0.820794 0.571224i $$-0.806469\pi$$
−0.820794 + 0.571224i $$0.806469\pi$$
$$38$$ 0 0
$$39$$ 3078.00 0.324046
$$40$$ 11760.0 1.16214
$$41$$ −10990.0 −1.02103 −0.510514 0.859869i $$-0.670545\pi$$
−0.510514 + 0.859869i $$0.670545\pi$$
$$42$$ −4320.00 −0.377886
$$43$$ −4996.00 −0.412051 −0.206026 0.978547i $$-0.566053\pi$$
−0.206026 + 0.978547i $$0.566053\pi$$
$$44$$ −9408.00 −0.732598
$$45$$ −7938.00 −0.584359
$$46$$ 5672.00 0.395222
$$47$$ −17124.0 −1.13073 −0.565367 0.824839i $$-0.691266\pi$$
−0.565367 + 0.824839i $$0.691266\pi$$
$$48$$ −5904.00 −0.369865
$$49$$ 40793.0 2.42714
$$50$$ 12958.0 0.733015
$$51$$ 54.0000 0.00290716
$$52$$ 9576.00 0.491107
$$53$$ 4470.00 0.218584 0.109292 0.994010i $$-0.465142\pi$$
0.109292 + 0.994010i $$0.465142\pi$$
$$54$$ −1458.00 −0.0680414
$$55$$ −32928.0 −1.46777
$$56$$ −28800.0 −1.22722
$$57$$ 0 0
$$58$$ 11804.0 0.460743
$$59$$ −26292.0 −0.983317 −0.491659 0.870788i $$-0.663609\pi$$
−0.491659 + 0.870788i $$0.663609\pi$$
$$60$$ −24696.0 −0.885622
$$61$$ 29134.0 1.00248 0.501240 0.865308i $$-0.332877\pi$$
0.501240 + 0.865308i $$0.332877\pi$$
$$62$$ −5488.00 −0.181315
$$63$$ 19440.0 0.617085
$$64$$ −10688.0 −0.326172
$$65$$ 33516.0 0.983940
$$66$$ −6048.00 −0.170904
$$67$$ 42052.0 1.14446 0.572229 0.820094i $$-0.306079\pi$$
0.572229 + 0.820094i $$0.306079\pi$$
$$68$$ 168.000 0.00440592
$$69$$ −25524.0 −0.645396
$$70$$ −47040.0 −1.14742
$$71$$ 26112.0 0.614744 0.307372 0.951589i $$-0.400550\pi$$
0.307372 + 0.951589i $$0.400550\pi$$
$$72$$ −9720.00 −0.220971
$$73$$ −49046.0 −1.07720 −0.538600 0.842562i $$-0.681047\pi$$
−0.538600 + 0.842562i $$0.681047\pi$$
$$74$$ −27340.0 −0.580389
$$75$$ −58311.0 −1.19701
$$76$$ 0 0
$$77$$ 80640.0 1.54997
$$78$$ 6156.00 0.114568
$$79$$ −79056.0 −1.42517 −0.712586 0.701585i $$-0.752476\pi$$
−0.712586 + 0.701585i $$0.752476\pi$$
$$80$$ −64288.0 −1.12307
$$81$$ 6561.00 0.111111
$$82$$ −21980.0 −0.360988
$$83$$ 9472.00 0.150920 0.0754599 0.997149i $$-0.475958\pi$$
0.0754599 + 0.997149i $$0.475958\pi$$
$$84$$ 60480.0 0.935220
$$85$$ 588.000 0.00882734
$$86$$ −9992.00 −0.145682
$$87$$ −53118.0 −0.752391
$$88$$ −40320.0 −0.555027
$$89$$ −82894.0 −1.10930 −0.554649 0.832085i $$-0.687147\pi$$
−0.554649 + 0.832085i $$0.687147\pi$$
$$90$$ −15876.0 −0.206602
$$91$$ −82080.0 −1.03904
$$92$$ −79408.0 −0.978126
$$93$$ 24696.0 0.296087
$$94$$ −34248.0 −0.399775
$$95$$ 0 0
$$96$$ −46368.0 −0.513500
$$97$$ −39850.0 −0.430030 −0.215015 0.976611i $$-0.568980\pi$$
−0.215015 + 0.976611i $$0.568980\pi$$
$$98$$ 81586.0 0.858125
$$99$$ 27216.0 0.279085
$$100$$ −181412. −1.81412
$$101$$ 99542.0 0.970964 0.485482 0.874247i $$-0.338644\pi$$
0.485482 + 0.874247i $$0.338644\pi$$
$$102$$ 108.000 0.00102783
$$103$$ −69368.0 −0.644267 −0.322134 0.946694i $$-0.604400\pi$$
−0.322134 + 0.946694i $$0.604400\pi$$
$$104$$ 41040.0 0.372069
$$105$$ 211680. 1.87373
$$106$$ 8940.00 0.0772810
$$107$$ 110204. 0.930546 0.465273 0.885167i $$-0.345956\pi$$
0.465273 + 0.885167i $$0.345956\pi$$
$$108$$ 20412.0 0.168394
$$109$$ 217786. 1.75575 0.877877 0.478886i $$-0.158959\pi$$
0.877877 + 0.478886i $$0.158959\pi$$
$$110$$ −65856.0 −0.518936
$$111$$ 123030. 0.947771
$$112$$ 157440. 1.18596
$$113$$ 84618.0 0.623400 0.311700 0.950181i $$-0.399102\pi$$
0.311700 + 0.950181i $$0.399102\pi$$
$$114$$ 0 0
$$115$$ −277928. −1.95969
$$116$$ −165256. −1.14028
$$117$$ −27702.0 −0.187088
$$118$$ −52584.0 −0.347655
$$119$$ −1440.00 −0.00932170
$$120$$ −105840. −0.670960
$$121$$ −48155.0 −0.299005
$$122$$ 58268.0 0.354430
$$123$$ 98910.0 0.589491
$$124$$ 76832.0 0.448733
$$125$$ −328692. −1.88154
$$126$$ 38880.0 0.218172
$$127$$ −89608.0 −0.492989 −0.246495 0.969144i $$-0.579279\pi$$
−0.246495 + 0.969144i $$0.579279\pi$$
$$128$$ −186240. −1.00473
$$129$$ 44964.0 0.237898
$$130$$ 67032.0 0.347875
$$131$$ 255992. 1.30331 0.651656 0.758515i $$-0.274075\pi$$
0.651656 + 0.758515i $$0.274075\pi$$
$$132$$ 84672.0 0.422966
$$133$$ 0 0
$$134$$ 84104.0 0.404627
$$135$$ 71442.0 0.337380
$$136$$ 720.000 0.00333799
$$137$$ 50226.0 0.228627 0.114313 0.993445i $$-0.463533\pi$$
0.114313 + 0.993445i $$0.463533\pi$$
$$138$$ −51048.0 −0.228182
$$139$$ −242108. −1.06285 −0.531425 0.847105i $$-0.678343\pi$$
−0.531425 + 0.847105i $$0.678343\pi$$
$$140$$ 658560. 2.83972
$$141$$ 154116. 0.652830
$$142$$ 52224.0 0.217345
$$143$$ −114912. −0.469921
$$144$$ 53136.0 0.213542
$$145$$ −578396. −2.28457
$$146$$ −98092.0 −0.380848
$$147$$ −367137. −1.40131
$$148$$ 382760. 1.43639
$$149$$ −352506. −1.30077 −0.650385 0.759604i $$-0.725393\pi$$
−0.650385 + 0.759604i $$0.725393\pi$$
$$150$$ −116622. −0.423207
$$151$$ 272120. 0.971221 0.485611 0.874175i $$-0.338597\pi$$
0.485611 + 0.874175i $$0.338597\pi$$
$$152$$ 0 0
$$153$$ −486.000 −0.00167845
$$154$$ 161280. 0.547998
$$155$$ 268912. 0.899044
$$156$$ −86184.0 −0.283541
$$157$$ 124942. 0.404538 0.202269 0.979330i $$-0.435168\pi$$
0.202269 + 0.979330i $$0.435168\pi$$
$$158$$ −158112. −0.503874
$$159$$ −40230.0 −0.126199
$$160$$ −504896. −1.55920
$$161$$ 680640. 2.06944
$$162$$ 13122.0 0.0392837
$$163$$ −28900.0 −0.0851979 −0.0425989 0.999092i $$-0.513564\pi$$
−0.0425989 + 0.999092i $$0.513564\pi$$
$$164$$ 307720. 0.893400
$$165$$ 296352. 0.847419
$$166$$ 18944.0 0.0533582
$$167$$ −370200. −1.02718 −0.513588 0.858037i $$-0.671684\pi$$
−0.513588 + 0.858037i $$0.671684\pi$$
$$168$$ 259200. 0.708536
$$169$$ −254329. −0.684982
$$170$$ 1176.00 0.00312094
$$171$$ 0 0
$$172$$ 139888. 0.360545
$$173$$ 277414. 0.704714 0.352357 0.935866i $$-0.385380\pi$$
0.352357 + 0.935866i $$0.385380\pi$$
$$174$$ −106236. −0.266010
$$175$$ 1.55496e6 3.83817
$$176$$ 220416. 0.536366
$$177$$ 236628. 0.567718
$$178$$ −165788. −0.392196
$$179$$ 439108. 1.02433 0.512164 0.858888i $$-0.328844\pi$$
0.512164 + 0.858888i $$0.328844\pi$$
$$180$$ 222264. 0.511314
$$181$$ −25406.0 −0.0576421 −0.0288211 0.999585i $$-0.509175\pi$$
−0.0288211 + 0.999585i $$0.509175\pi$$
$$182$$ −164160. −0.367357
$$183$$ −262206. −0.578782
$$184$$ −340320. −0.741042
$$185$$ 1.33966e6 2.87783
$$186$$ 49392.0 0.104683
$$187$$ −2016.00 −0.00421586
$$188$$ 479472. 0.989393
$$189$$ −174960. −0.356274
$$190$$ 0 0
$$191$$ 640644. 1.27067 0.635336 0.772236i $$-0.280862\pi$$
0.635336 + 0.772236i $$0.280862\pi$$
$$192$$ 96192.0 0.188315
$$193$$ 224846. 0.434502 0.217251 0.976116i $$-0.430291\pi$$
0.217251 + 0.976116i $$0.430291\pi$$
$$194$$ −79700.0 −0.152039
$$195$$ −301644. −0.568078
$$196$$ −1.14220e6 −2.12375
$$197$$ 325438. 0.597452 0.298726 0.954339i $$-0.403438\pi$$
0.298726 + 0.954339i $$0.403438\pi$$
$$198$$ 54432.0 0.0986714
$$199$$ −983576. −1.76066 −0.880329 0.474363i $$-0.842679\pi$$
−0.880329 + 0.474363i $$0.842679\pi$$
$$200$$ −777480. −1.37440
$$201$$ −378468. −0.660753
$$202$$ 199084. 0.343287
$$203$$ 1.41648e6 2.41252
$$204$$ −1512.00 −0.00254376
$$205$$ 1.07702e6 1.78994
$$206$$ −138736. −0.227783
$$207$$ 229716. 0.372619
$$208$$ −224352. −0.359560
$$209$$ 0 0
$$210$$ 423360. 0.662463
$$211$$ 866308. 1.33957 0.669786 0.742554i $$-0.266386\pi$$
0.669786 + 0.742554i $$0.266386\pi$$
$$212$$ −125160. −0.191261
$$213$$ −235008. −0.354923
$$214$$ 220408. 0.328998
$$215$$ 489608. 0.722358
$$216$$ 87480.0 0.127578
$$217$$ −658560. −0.949393
$$218$$ 435572. 0.620753
$$219$$ 441414. 0.621922
$$220$$ 921984. 1.28430
$$221$$ 2052.00 0.00282616
$$222$$ 246060. 0.335088
$$223$$ 76928.0 0.103591 0.0517955 0.998658i $$-0.483506\pi$$
0.0517955 + 0.998658i $$0.483506\pi$$
$$224$$ 1.23648e6 1.64652
$$225$$ 524799. 0.691093
$$226$$ 169236. 0.220405
$$227$$ 969612. 1.24892 0.624458 0.781058i $$-0.285320\pi$$
0.624458 + 0.781058i $$0.285320\pi$$
$$228$$ 0 0
$$229$$ 840150. 1.05869 0.529344 0.848407i $$-0.322438\pi$$
0.529344 + 0.848407i $$0.322438\pi$$
$$230$$ −555856. −0.692856
$$231$$ −725760. −0.894876
$$232$$ −708240. −0.863894
$$233$$ 575914. 0.694973 0.347486 0.937685i $$-0.387035\pi$$
0.347486 + 0.937685i $$0.387035\pi$$
$$234$$ −55404.0 −0.0661457
$$235$$ 1.67815e6 1.98226
$$236$$ 736176. 0.860402
$$237$$ 711504. 0.822823
$$238$$ −2880.00 −0.00329572
$$239$$ −352188. −0.398823 −0.199411 0.979916i $$-0.563903\pi$$
−0.199411 + 0.979916i $$0.563903\pi$$
$$240$$ 578592. 0.648402
$$241$$ −451290. −0.500510 −0.250255 0.968180i $$-0.580515\pi$$
−0.250255 + 0.968180i $$0.580515\pi$$
$$242$$ −96310.0 −0.105714
$$243$$ −59049.0 −0.0641500
$$244$$ −815752. −0.877170
$$245$$ −3.99771e6 −4.25497
$$246$$ 197820. 0.208417
$$247$$ 0 0
$$248$$ 329280. 0.339967
$$249$$ −85248.0 −0.0871336
$$250$$ −657384. −0.665226
$$251$$ 145752. 0.146026 0.0730130 0.997331i $$-0.476739\pi$$
0.0730130 + 0.997331i $$0.476739\pi$$
$$252$$ −544320. −0.539949
$$253$$ 952896. 0.935932
$$254$$ −179216. −0.174298
$$255$$ −5292.00 −0.00509647
$$256$$ −30464.0 −0.0290527
$$257$$ 1.87818e6 1.77380 0.886899 0.461964i $$-0.152855\pi$$
0.886899 + 0.461964i $$0.152855\pi$$
$$258$$ 89928.0 0.0841096
$$259$$ −3.28080e6 −3.03900
$$260$$ −938448. −0.860948
$$261$$ 478062. 0.434393
$$262$$ 511984. 0.460790
$$263$$ 895068. 0.797933 0.398967 0.916965i $$-0.369369\pi$$
0.398967 + 0.916965i $$0.369369\pi$$
$$264$$ 362880. 0.320445
$$265$$ −438060. −0.383194
$$266$$ 0 0
$$267$$ 746046. 0.640453
$$268$$ −1.17746e6 −1.00140
$$269$$ −1.27291e6 −1.07255 −0.536273 0.844045i $$-0.680168\pi$$
−0.536273 + 0.844045i $$0.680168\pi$$
$$270$$ 142884. 0.119282
$$271$$ 306672. 0.253659 0.126830 0.991925i $$-0.459520\pi$$
0.126830 + 0.991925i $$0.459520\pi$$
$$272$$ −3936.00 −0.00322577
$$273$$ 738720. 0.599892
$$274$$ 100452. 0.0808318
$$275$$ 2.17694e6 1.73586
$$276$$ 714672. 0.564721
$$277$$ −192026. −0.150370 −0.0751849 0.997170i $$-0.523955\pi$$
−0.0751849 + 0.997170i $$0.523955\pi$$
$$278$$ −484216. −0.375774
$$279$$ −222264. −0.170946
$$280$$ 2.82240e6 2.15141
$$281$$ −1.00055e6 −0.755915 −0.377958 0.925823i $$-0.623373\pi$$
−0.377958 + 0.925823i $$0.623373\pi$$
$$282$$ 308232. 0.230810
$$283$$ −1.26847e6 −0.941485 −0.470743 0.882271i $$-0.656014\pi$$
−0.470743 + 0.882271i $$0.656014\pi$$
$$284$$ −731136. −0.537901
$$285$$ 0 0
$$286$$ −229824. −0.166142
$$287$$ −2.63760e6 −1.89018
$$288$$ 417312. 0.296469
$$289$$ −1.41982e6 −0.999975
$$290$$ −1.15679e6 −0.807719
$$291$$ 358650. 0.248278
$$292$$ 1.37329e6 0.942550
$$293$$ 1.52560e6 1.03818 0.519088 0.854721i $$-0.326272\pi$$
0.519088 + 0.854721i $$0.326272\pi$$
$$294$$ −734274. −0.495439
$$295$$ 2.57662e6 1.72383
$$296$$ 1.64040e6 1.08823
$$297$$ −244944. −0.161130
$$298$$ −705012. −0.459892
$$299$$ −969912. −0.627414
$$300$$ 1.63271e6 1.04738
$$301$$ −1.19904e6 −0.762812
$$302$$ 544240. 0.343379
$$303$$ −895878. −0.560586
$$304$$ 0 0
$$305$$ −2.85513e6 −1.75742
$$306$$ −972.000 −0.000593421 0
$$307$$ 1.19665e6 0.724639 0.362320 0.932054i $$-0.381985\pi$$
0.362320 + 0.932054i $$0.381985\pi$$
$$308$$ −2.25792e6 −1.35623
$$309$$ 624312. 0.371968
$$310$$ 537824. 0.317860
$$311$$ 2.37144e6 1.39031 0.695155 0.718859i $$-0.255336\pi$$
0.695155 + 0.718859i $$0.255336\pi$$
$$312$$ −369360. −0.214814
$$313$$ 353738. 0.204090 0.102045 0.994780i $$-0.467461\pi$$
0.102045 + 0.994780i $$0.467461\pi$$
$$314$$ 249884. 0.143026
$$315$$ −1.90512e6 −1.08180
$$316$$ 2.21357e6 1.24703
$$317$$ −2.70427e6 −1.51148 −0.755738 0.654874i $$-0.772722\pi$$
−0.755738 + 0.654874i $$0.772722\pi$$
$$318$$ −80460.0 −0.0446182
$$319$$ 1.98307e6 1.09109
$$320$$ 1.04742e6 0.571805
$$321$$ −991836. −0.537251
$$322$$ 1.36128e6 0.731657
$$323$$ 0 0
$$324$$ −183708. −0.0972222
$$325$$ −2.21582e6 −1.16366
$$326$$ −57800.0 −0.0301220
$$327$$ −1.96007e6 −1.01369
$$328$$ 1.31880e6 0.676853
$$329$$ −4.10976e6 −2.09328
$$330$$ 592704. 0.299608
$$331$$ 327444. 0.164273 0.0821367 0.996621i $$-0.473826\pi$$
0.0821367 + 0.996621i $$0.473826\pi$$
$$332$$ −265216. −0.132055
$$333$$ −1.10727e6 −0.547196
$$334$$ −740400. −0.363162
$$335$$ −4.12110e6 −2.00632
$$336$$ −1.41696e6 −0.684714
$$337$$ −367946. −0.176486 −0.0882428 0.996099i $$-0.528125\pi$$
−0.0882428 + 0.996099i $$0.528125\pi$$
$$338$$ −508658. −0.242178
$$339$$ −761562. −0.359920
$$340$$ −16464.0 −0.00772393
$$341$$ −921984. −0.429376
$$342$$ 0 0
$$343$$ 5.75664e6 2.64201
$$344$$ 599520. 0.273154
$$345$$ 2.50135e6 1.13143
$$346$$ 554828. 0.249154
$$347$$ 566160. 0.252415 0.126208 0.992004i $$-0.459719\pi$$
0.126208 + 0.992004i $$0.459719\pi$$
$$348$$ 1.48730e6 0.658342
$$349$$ −4.50687e6 −1.98067 −0.990333 0.138713i $$-0.955703\pi$$
−0.990333 + 0.138713i $$0.955703\pi$$
$$350$$ 3.10992e6 1.35700
$$351$$ 249318. 0.108015
$$352$$ 1.73107e6 0.744661
$$353$$ −1.09778e6 −0.468899 −0.234450 0.972128i $$-0.575329\pi$$
−0.234450 + 0.972128i $$0.575329\pi$$
$$354$$ 473256. 0.200719
$$355$$ −2.55898e6 −1.07769
$$356$$ 2.32103e6 0.970635
$$357$$ 12960.0 0.00538189
$$358$$ 878216. 0.362154
$$359$$ −1.95223e6 −0.799456 −0.399728 0.916634i $$-0.630895\pi$$
−0.399728 + 0.916634i $$0.630895\pi$$
$$360$$ 952560. 0.387379
$$361$$ 0 0
$$362$$ −50812.0 −0.0203796
$$363$$ 433395. 0.172630
$$364$$ 2.29824e6 0.909163
$$365$$ 4.80651e6 1.88842
$$366$$ −524412. −0.204630
$$367$$ 1.49163e6 0.578091 0.289046 0.957315i $$-0.406662\pi$$
0.289046 + 0.957315i $$0.406662\pi$$
$$368$$ 1.86042e6 0.716128
$$369$$ −890190. −0.340343
$$370$$ 2.67932e6 1.01747
$$371$$ 1.07280e6 0.404654
$$372$$ −691488. −0.259076
$$373$$ −1.48305e6 −0.551928 −0.275964 0.961168i $$-0.588997\pi$$
−0.275964 + 0.961168i $$0.588997\pi$$
$$374$$ −4032.00 −0.00149053
$$375$$ 2.95823e6 1.08631
$$376$$ 2.05488e6 0.749578
$$377$$ −2.01848e6 −0.731429
$$378$$ −349920. −0.125962
$$379$$ 2.79436e6 0.999272 0.499636 0.866235i $$-0.333467\pi$$
0.499636 + 0.866235i $$0.333467\pi$$
$$380$$ 0 0
$$381$$ 806472. 0.284627
$$382$$ 1.28129e6 0.449250
$$383$$ −2.06910e6 −0.720748 −0.360374 0.932808i $$-0.617351\pi$$
−0.360374 + 0.932808i $$0.617351\pi$$
$$384$$ 1.67616e6 0.580079
$$385$$ −7.90272e6 −2.71722
$$386$$ 449692. 0.153620
$$387$$ −404676. −0.137350
$$388$$ 1.11580e6 0.376276
$$389$$ 4.61696e6 1.54697 0.773485 0.633815i $$-0.218512\pi$$
0.773485 + 0.633815i $$0.218512\pi$$
$$390$$ −603288. −0.200846
$$391$$ −17016.0 −0.00562880
$$392$$ −4.89516e6 −1.60898
$$393$$ −2.30393e6 −0.752467
$$394$$ 650876. 0.211231
$$395$$ 7.74749e6 2.49844
$$396$$ −762048. −0.244199
$$397$$ 875870. 0.278910 0.139455 0.990228i $$-0.455465\pi$$
0.139455 + 0.990228i $$0.455465\pi$$
$$398$$ −1.96715e6 −0.622487
$$399$$ 0 0
$$400$$ 4.25022e6 1.32820
$$401$$ 3.36615e6 1.04538 0.522689 0.852524i $$-0.324929\pi$$
0.522689 + 0.852524i $$0.324929\pi$$
$$402$$ −756936. −0.233611
$$403$$ 938448. 0.287838
$$404$$ −2.78718e6 −0.849593
$$405$$ −642978. −0.194786
$$406$$ 2.83296e6 0.852954
$$407$$ −4.59312e6 −1.37443
$$408$$ −6480.00 −0.00192719
$$409$$ −6.58655e6 −1.94693 −0.973463 0.228844i $$-0.926505\pi$$
−0.973463 + 0.228844i $$0.926505\pi$$
$$410$$ 2.15404e6 0.632840
$$411$$ −452034. −0.131998
$$412$$ 1.94230e6 0.563734
$$413$$ −6.31008e6 −1.82037
$$414$$ 459432. 0.131741
$$415$$ −928256. −0.264574
$$416$$ −1.76198e6 −0.499193
$$417$$ 2.17897e6 0.613637
$$418$$ 0 0
$$419$$ −1.06775e6 −0.297122 −0.148561 0.988903i $$-0.547464\pi$$
−0.148561 + 0.988903i $$0.547464\pi$$
$$420$$ −5.92704e6 −1.63951
$$421$$ −3.60621e6 −0.991620 −0.495810 0.868431i $$-0.665129\pi$$
−0.495810 + 0.868431i $$0.665129\pi$$
$$422$$ 1.73262e6 0.473610
$$423$$ −1.38704e6 −0.376911
$$424$$ −536400. −0.144902
$$425$$ −38874.0 −0.0104397
$$426$$ −470016. −0.125484
$$427$$ 6.99216e6 1.85584
$$428$$ −3.08571e6 −0.814228
$$429$$ 1.03421e6 0.271309
$$430$$ 979216. 0.255392
$$431$$ −1.05310e6 −0.273071 −0.136535 0.990635i $$-0.543597\pi$$
−0.136535 + 0.990635i $$0.543597\pi$$
$$432$$ −478224. −0.123288
$$433$$ 3.45697e6 0.886087 0.443044 0.896500i $$-0.353899\pi$$
0.443044 + 0.896500i $$0.353899\pi$$
$$434$$ −1.31712e6 −0.335661
$$435$$ 5.20556e6 1.31900
$$436$$ −6.09801e6 −1.53628
$$437$$ 0 0
$$438$$ 882828. 0.219883
$$439$$ 5.88610e6 1.45769 0.728847 0.684676i $$-0.240056\pi$$
0.728847 + 0.684676i $$0.240056\pi$$
$$440$$ 3.95136e6 0.973005
$$441$$ 3.30423e6 0.809048
$$442$$ 4104.00 0.000999198 0
$$443$$ −4.80216e6 −1.16259 −0.581296 0.813692i $$-0.697454\pi$$
−0.581296 + 0.813692i $$0.697454\pi$$
$$444$$ −3.44484e6 −0.829300
$$445$$ 8.12361e6 1.94468
$$446$$ 153856. 0.0366250
$$447$$ 3.17255e6 0.751000
$$448$$ −2.56512e6 −0.603827
$$449$$ 3.17967e6 0.744330 0.372165 0.928167i $$-0.378616\pi$$
0.372165 + 0.928167i $$0.378616\pi$$
$$450$$ 1.04960e6 0.244338
$$451$$ −3.69264e6 −0.854861
$$452$$ −2.36930e6 −0.545475
$$453$$ −2.44908e6 −0.560735
$$454$$ 1.93922e6 0.441559
$$455$$ 8.04384e6 1.82152
$$456$$ 0 0
$$457$$ 483546. 0.108305 0.0541523 0.998533i $$-0.482754\pi$$
0.0541523 + 0.998533i $$0.482754\pi$$
$$458$$ 1.68030e6 0.374303
$$459$$ 4374.00 0.000969052 0
$$460$$ 7.78198e6 1.71473
$$461$$ 707982. 0.155156 0.0775782 0.996986i $$-0.475281\pi$$
0.0775782 + 0.996986i $$0.475281\pi$$
$$462$$ −1.45152e6 −0.316387
$$463$$ 8.24091e6 1.78658 0.893291 0.449479i $$-0.148390\pi$$
0.893291 + 0.449479i $$0.148390\pi$$
$$464$$ 3.87171e6 0.834849
$$465$$ −2.42021e6 −0.519063
$$466$$ 1.15183e6 0.245710
$$467$$ 2.13640e6 0.453305 0.226652 0.973976i $$-0.427222\pi$$
0.226652 + 0.973976i $$0.427222\pi$$
$$468$$ 775656. 0.163702
$$469$$ 1.00925e7 2.11868
$$470$$ 3.35630e6 0.700836
$$471$$ −1.12448e6 −0.233560
$$472$$ 3.15504e6 0.651853
$$473$$ −1.67866e6 −0.344992
$$474$$ 1.42301e6 0.290912
$$475$$ 0 0
$$476$$ 40320.0 0.00815649
$$477$$ 362070. 0.0728612
$$478$$ −704376. −0.141005
$$479$$ 1.20736e6 0.240436 0.120218 0.992748i $$-0.461641\pi$$
0.120218 + 0.992748i $$0.461641\pi$$
$$480$$ 4.54406e6 0.900205
$$481$$ 4.67514e6 0.921365
$$482$$ −902580. −0.176957
$$483$$ −6.12576e6 −1.19479
$$484$$ 1.34834e6 0.261629
$$485$$ 3.90530e6 0.753876
$$486$$ −118098. −0.0226805
$$487$$ 8.77970e6 1.67748 0.838739 0.544533i $$-0.183293\pi$$
0.838739 + 0.544533i $$0.183293\pi$$
$$488$$ −3.49608e6 −0.664556
$$489$$ 260100. 0.0491890
$$490$$ −7.99543e6 −1.50436
$$491$$ 4.23400e6 0.792587 0.396294 0.918124i $$-0.370296\pi$$
0.396294 + 0.918124i $$0.370296\pi$$
$$492$$ −2.76948e6 −0.515805
$$493$$ −35412.0 −0.00656195
$$494$$ 0 0
$$495$$ −2.66717e6 −0.489257
$$496$$ −1.80006e6 −0.328537
$$497$$ 6.26688e6 1.13805
$$498$$ −170496. −0.0308064
$$499$$ −1.73450e6 −0.311834 −0.155917 0.987770i $$-0.549833\pi$$
−0.155917 + 0.987770i $$0.549833\pi$$
$$500$$ 9.20338e6 1.64635
$$501$$ 3.33180e6 0.593041
$$502$$ 291504. 0.0516280
$$503$$ −4.72352e6 −0.832427 −0.416214 0.909267i $$-0.636643\pi$$
−0.416214 + 0.909267i $$0.636643\pi$$
$$504$$ −2.33280e6 −0.409073
$$505$$ −9.75512e6 −1.70217
$$506$$ 1.90579e6 0.330902
$$507$$ 2.28896e6 0.395475
$$508$$ 2.50902e6 0.431366
$$509$$ 2.49741e6 0.427264 0.213632 0.976914i $$-0.431471\pi$$
0.213632 + 0.976914i $$0.431471\pi$$
$$510$$ −10584.0 −0.00180187
$$511$$ −1.17710e7 −1.99417
$$512$$ 5.89875e6 0.994455
$$513$$ 0 0
$$514$$ 3.75636e6 0.627132
$$515$$ 6.79806e6 1.12945
$$516$$ −1.25899e6 −0.208161
$$517$$ −5.75366e6 −0.946713
$$518$$ −6.56160e6 −1.07445
$$519$$ −2.49673e6 −0.406867
$$520$$ −4.02192e6 −0.652266
$$521$$ −2.86413e6 −0.462272 −0.231136 0.972921i $$-0.574244\pi$$
−0.231136 + 0.972921i $$0.574244\pi$$
$$522$$ 956124. 0.153581
$$523$$ −2.46228e6 −0.393625 −0.196812 0.980441i $$-0.563059\pi$$
−0.196812 + 0.980441i $$0.563059\pi$$
$$524$$ −7.16778e6 −1.14040
$$525$$ −1.39946e7 −2.21597
$$526$$ 1.79014e6 0.282112
$$527$$ 16464.0 0.00258231
$$528$$ −1.98374e6 −0.309671
$$529$$ 1.60655e6 0.249606
$$530$$ −876120. −0.135480
$$531$$ −2.12965e6 −0.327772
$$532$$ 0 0
$$533$$ 3.75858e6 0.573067
$$534$$ 1.49209e6 0.226434
$$535$$ −1.08000e7 −1.63132
$$536$$ −5.04624e6 −0.758675
$$537$$ −3.95197e6 −0.591396
$$538$$ −2.54581e6 −0.379202
$$539$$ 1.37064e7 2.03214
$$540$$ −2.00038e6 −0.295207
$$541$$ 6.88192e6 1.01092 0.505459 0.862850i $$-0.331323\pi$$
0.505459 + 0.862850i $$0.331323\pi$$
$$542$$ 613344. 0.0896821
$$543$$ 228654. 0.0332797
$$544$$ −30912.0 −0.00447847
$$545$$ −2.13430e7 −3.07797
$$546$$ 1.47744e6 0.212094
$$547$$ 4.99680e6 0.714041 0.357021 0.934097i $$-0.383793\pi$$
0.357021 + 0.934097i $$0.383793\pi$$
$$548$$ −1.40633e6 −0.200048
$$549$$ 2.35985e6 0.334160
$$550$$ 4.35389e6 0.613720
$$551$$ 0 0
$$552$$ 3.06288e6 0.427841
$$553$$ −1.89734e7 −2.63836
$$554$$ −384052. −0.0531638
$$555$$ −1.20569e7 −1.66152
$$556$$ 6.77902e6 0.929994
$$557$$ −8.42887e6 −1.15115 −0.575574 0.817750i $$-0.695221\pi$$
−0.575574 + 0.817750i $$0.695221\pi$$
$$558$$ −444528. −0.0604385
$$559$$ 1.70863e6 0.231270
$$560$$ −1.54291e7 −2.07908
$$561$$ 18144.0 0.00243403
$$562$$ −2.00110e6 −0.267256
$$563$$ 1.01781e7 1.35330 0.676652 0.736303i $$-0.263430\pi$$
0.676652 + 0.736303i $$0.263430\pi$$
$$564$$ −4.31525e6 −0.571226
$$565$$ −8.29256e6 −1.09287
$$566$$ −2.53694e6 −0.332865
$$567$$ 1.57464e6 0.205695
$$568$$ −3.13344e6 −0.407522
$$569$$ −1.13792e7 −1.47344 −0.736719 0.676199i $$-0.763626\pi$$
−0.736719 + 0.676199i $$0.763626\pi$$
$$570$$ 0 0
$$571$$ −7.57426e6 −0.972187 −0.486094 0.873907i $$-0.661579\pi$$
−0.486094 + 0.873907i $$0.661579\pi$$
$$572$$ 3.21754e6 0.411181
$$573$$ −5.76580e6 −0.733623
$$574$$ −5.27520e6 −0.668281
$$575$$ 1.83744e7 2.31763
$$576$$ −865728. −0.108724
$$577$$ −6.06488e6 −0.758372 −0.379186 0.925320i $$-0.623796\pi$$
−0.379186 + 0.925320i $$0.623796\pi$$
$$578$$ −2.83964e6 −0.353544
$$579$$ −2.02361e6 −0.250860
$$580$$ 1.61951e7 1.99900
$$581$$ 2.27328e6 0.279391
$$582$$ 717300. 0.0877796
$$583$$ 1.50192e6 0.183010
$$584$$ 5.88552e6 0.714090
$$585$$ 2.71480e6 0.327980
$$586$$ 3.05120e6 0.367051
$$587$$ 1.62011e7 1.94066 0.970330 0.241783i $$-0.0777323\pi$$
0.970330 + 0.241783i $$0.0777323\pi$$
$$588$$ 1.02798e7 1.22615
$$589$$ 0 0
$$590$$ 5.15323e6 0.609466
$$591$$ −2.92894e6 −0.344939
$$592$$ −8.96752e6 −1.05164
$$593$$ 9.92291e6 1.15878 0.579392 0.815049i $$-0.303290\pi$$
0.579392 + 0.815049i $$0.303290\pi$$
$$594$$ −489888. −0.0569680
$$595$$ 141120. 0.0163417
$$596$$ 9.87017e6 1.13817
$$597$$ 8.85218e6 1.01652
$$598$$ −1.93982e6 −0.221824
$$599$$ 9.55020e6 1.08754 0.543770 0.839234i $$-0.316996\pi$$
0.543770 + 0.839234i $$0.316996\pi$$
$$600$$ 6.99732e6 0.793512
$$601$$ −1.57009e6 −0.177312 −0.0886560 0.996062i $$-0.528257\pi$$
−0.0886560 + 0.996062i $$0.528257\pi$$
$$602$$ −2.39808e6 −0.269695
$$603$$ 3.40621e6 0.381486
$$604$$ −7.61936e6 −0.849818
$$605$$ 4.71919e6 0.524178
$$606$$ −1.79176e6 −0.198197
$$607$$ −8.83870e6 −0.973680 −0.486840 0.873491i $$-0.661851\pi$$
−0.486840 + 0.873491i $$0.661851\pi$$
$$608$$ 0 0
$$609$$ −1.27483e7 −1.39287
$$610$$ −5.71026e6 −0.621343
$$611$$ 5.85641e6 0.634641
$$612$$ 13608.0 0.00146864
$$613$$ −4.94068e6 −0.531050 −0.265525 0.964104i $$-0.585545\pi$$
−0.265525 + 0.964104i $$0.585545\pi$$
$$614$$ 2.39330e6 0.256199
$$615$$ −9.69318e6 −1.03342
$$616$$ −9.67680e6 −1.02750
$$617$$ −5.44608e6 −0.575932 −0.287966 0.957641i $$-0.592979\pi$$
−0.287966 + 0.957641i $$0.592979\pi$$
$$618$$ 1.24862e6 0.131511
$$619$$ 9.25648e6 0.971000 0.485500 0.874237i $$-0.338637\pi$$
0.485500 + 0.874237i $$0.338637\pi$$
$$620$$ −7.52954e6 −0.786663
$$621$$ −2.06744e6 −0.215132
$$622$$ 4.74289e6 0.491549
$$623$$ −1.98946e7 −2.05359
$$624$$ 2.01917e6 0.207592
$$625$$ 1.19649e7 1.22521
$$626$$ 707476. 0.0721566
$$627$$ 0 0
$$628$$ −3.49838e6 −0.353971
$$629$$ 82020.0 0.00826596
$$630$$ −3.81024e6 −0.382473
$$631$$ −1.84220e7 −1.84189 −0.920944 0.389694i $$-0.872581\pi$$
−0.920944 + 0.389694i $$0.872581\pi$$
$$632$$ 9.48672e6 0.944764
$$633$$ −7.79677e6 −0.773402
$$634$$ −5.40853e6 −0.534387
$$635$$ 8.78158e6 0.864248
$$636$$ 1.12644e6 0.110424
$$637$$ −1.39512e7 −1.36227
$$638$$ 3.96614e6 0.385760
$$639$$ 2.11507e6 0.204915
$$640$$ 1.82515e7 1.76136
$$641$$ 1.83640e7 1.76532 0.882660 0.470013i $$-0.155751\pi$$
0.882660 + 0.470013i $$0.155751\pi$$
$$642$$ −1.98367e6 −0.189947
$$643$$ 5.30824e6 0.506318 0.253159 0.967425i $$-0.418530\pi$$
0.253159 + 0.967425i $$0.418530\pi$$
$$644$$ −1.90579e7 −1.81076
$$645$$ −4.40647e6 −0.417053
$$646$$ 0 0
$$647$$ −3.32276e6 −0.312060 −0.156030 0.987752i $$-0.549870\pi$$
−0.156030 + 0.987752i $$0.549870\pi$$
$$648$$ −787320. −0.0736570
$$649$$ −8.83411e6 −0.823287
$$650$$ −4.43164e6 −0.411416
$$651$$ 5.92704e6 0.548132
$$652$$ 809200. 0.0745482
$$653$$ −5.81096e6 −0.533292 −0.266646 0.963795i $$-0.585915\pi$$
−0.266646 + 0.963795i $$0.585915\pi$$
$$654$$ −3.92015e6 −0.358392
$$655$$ −2.50872e7 −2.28481
$$656$$ −7.20944e6 −0.654097
$$657$$ −3.97273e6 −0.359067
$$658$$ −8.21952e6 −0.740085
$$659$$ 1.60066e7 1.43577 0.717886 0.696160i $$-0.245110\pi$$
0.717886 + 0.696160i $$0.245110\pi$$
$$660$$ −8.29786e6 −0.741491
$$661$$ 1.27998e7 1.13946 0.569731 0.821831i $$-0.307047\pi$$
0.569731 + 0.821831i $$0.307047\pi$$
$$662$$ 654888. 0.0580794
$$663$$ −18468.0 −0.00163168
$$664$$ −1.13664e6 −0.100047
$$665$$ 0 0
$$666$$ −2.21454e6 −0.193463
$$667$$ 1.67381e7 1.45677
$$668$$ 1.03656e7 0.898780
$$669$$ −692352. −0.0598083
$$670$$ −8.24219e6 −0.709342
$$671$$ 9.78902e6 0.839331
$$672$$ −1.11283e7 −0.950619
$$673$$ −9.83823e6 −0.837296 −0.418648 0.908149i $$-0.637496\pi$$
−0.418648 + 0.908149i $$0.637496\pi$$
$$674$$ −735892. −0.0623971
$$675$$ −4.72319e6 −0.399003
$$676$$ 7.12121e6 0.599359
$$677$$ 8.54417e6 0.716470 0.358235 0.933631i $$-0.383379\pi$$
0.358235 + 0.933631i $$0.383379\pi$$
$$678$$ −1.52312e6 −0.127251
$$679$$ −9.56400e6 −0.796095
$$680$$ −70560.0 −0.00585176
$$681$$ −8.72651e6 −0.721062
$$682$$ −1.84397e6 −0.151807
$$683$$ 1.00816e7 0.826947 0.413473 0.910516i $$-0.364315\pi$$
0.413473 + 0.910516i $$0.364315\pi$$
$$684$$ 0 0
$$685$$ −4.92215e6 −0.400800
$$686$$ 1.15133e7 0.934090
$$687$$ −7.56135e6 −0.611234
$$688$$ −3.27738e6 −0.263970
$$689$$ −1.52874e6 −0.122683
$$690$$ 5.00270e6 0.400020
$$691$$ −7.71320e6 −0.614525 −0.307262 0.951625i $$-0.599413\pi$$
−0.307262 + 0.951625i $$0.599413\pi$$
$$692$$ −7.76759e6 −0.616625
$$693$$ 6.53184e6 0.516657
$$694$$ 1.13232e6 0.0892422
$$695$$ 2.37266e7 1.86326
$$696$$ 6.37416e6 0.498769
$$697$$ 65940.0 0.00514123
$$698$$ −9.01373e6 −0.700271
$$699$$ −5.18323e6 −0.401243
$$700$$ −4.35389e7 −3.35840
$$701$$ 4.92440e6 0.378493 0.189247 0.981930i $$-0.439395\pi$$
0.189247 + 0.981930i $$0.439395\pi$$
$$702$$ 498636. 0.0381892
$$703$$ 0 0
$$704$$ −3.59117e6 −0.273089
$$705$$ −1.51034e7 −1.14446
$$706$$ −2.19556e6 −0.165781
$$707$$ 2.38901e7 1.79750
$$708$$ −6.62558e6 −0.496754
$$709$$ 5.35468e6 0.400053 0.200027 0.979790i $$-0.435897\pi$$
0.200027 + 0.979790i $$0.435897\pi$$
$$710$$ −5.11795e6 −0.381022
$$711$$ −6.40354e6 −0.475057
$$712$$ 9.94728e6 0.735367
$$713$$ −7.78198e6 −0.573280
$$714$$ 25920.0 0.00190278
$$715$$ 1.12614e7 0.823809
$$716$$ −1.22950e7 −0.896286
$$717$$ 3.16969e6 0.230260
$$718$$ −3.90446e6 −0.282650
$$719$$ 9.39507e6 0.677763 0.338881 0.940829i $$-0.389951\pi$$
0.338881 + 0.940829i $$0.389951\pi$$
$$720$$ −5.20733e6 −0.374355
$$721$$ −1.66483e7 −1.19270
$$722$$ 0 0
$$723$$ 4.06161e6 0.288970
$$724$$ 711368. 0.0504368
$$725$$ 3.82391e7 2.70186
$$726$$ 866790. 0.0610341
$$727$$ 6.83055e6 0.479314 0.239657 0.970858i $$-0.422965\pi$$
0.239657 + 0.970858i $$0.422965\pi$$
$$728$$ 9.84960e6 0.688795
$$729$$ 531441. 0.0370370
$$730$$ 9.61302e6 0.667656
$$731$$ 29976.0 0.00207482
$$732$$ 7.34177e6 0.506434
$$733$$ 1.29536e7 0.890491 0.445245 0.895409i $$-0.353116\pi$$
0.445245 + 0.895409i $$0.353116\pi$$
$$734$$ 2.98326e6 0.204386
$$735$$ 3.59794e7 2.45661
$$736$$ 1.46111e7 0.994232
$$737$$ 1.41295e7 0.958202
$$738$$ −1.78038e6 −0.120329
$$739$$ 1.15292e7 0.776582 0.388291 0.921537i $$-0.373066\pi$$
0.388291 + 0.921537i $$0.373066\pi$$
$$740$$ −3.75105e7 −2.51810
$$741$$ 0 0
$$742$$ 2.14560e6 0.143067
$$743$$ −3.20131e6 −0.212743 −0.106372 0.994326i $$-0.533923\pi$$
−0.106372 + 0.994326i $$0.533923\pi$$
$$744$$ −2.96352e6 −0.196280
$$745$$ 3.45456e7 2.28035
$$746$$ −2.96609e6 −0.195136
$$747$$ 767232. 0.0503066
$$748$$ 56448.0 0.00368888
$$749$$ 2.64490e7 1.72268
$$750$$ 5.91646e6 0.384068
$$751$$ −5.52822e6 −0.357673 −0.178836 0.983879i $$-0.557233\pi$$
−0.178836 + 0.983879i $$0.557233\pi$$
$$752$$ −1.12333e7 −0.724377
$$753$$ −1.31177e6 −0.0843082
$$754$$ −4.03697e6 −0.258599
$$755$$ −2.66678e7 −1.70263
$$756$$ 4.89888e6 0.311740
$$757$$ 1.62708e7 1.03198 0.515989 0.856595i $$-0.327425\pi$$
0.515989 + 0.856595i $$0.327425\pi$$
$$758$$ 5.58871e6 0.353296
$$759$$ −8.57606e6 −0.540360
$$760$$ 0 0
$$761$$ −8.40043e6 −0.525823 −0.262912 0.964820i $$-0.584683\pi$$
−0.262912 + 0.964820i $$0.584683\pi$$
$$762$$ 1.61294e6 0.100631
$$763$$ 5.22686e7 3.25035
$$764$$ −1.79380e7 −1.11184
$$765$$ 47628.0 0.00294245
$$766$$ −4.13819e6 −0.254823
$$767$$ 8.99186e6 0.551901
$$768$$ 274176. 0.0167736
$$769$$ 7.66543e6 0.467434 0.233717 0.972305i $$-0.424911\pi$$
0.233717 + 0.972305i $$0.424911\pi$$
$$770$$ −1.58054e7 −0.960682
$$771$$ −1.69036e7 −1.02410
$$772$$ −6.29569e6 −0.380189
$$773$$ 2.59829e7 1.56401 0.782004 0.623273i $$-0.214198\pi$$
0.782004 + 0.623273i $$0.214198\pi$$
$$774$$ −809352. −0.0485607
$$775$$ −1.77784e7 −1.06326
$$776$$ 4.78200e6 0.285072
$$777$$ 2.95272e7 1.75457
$$778$$ 9.23392e6 0.546937
$$779$$ 0 0
$$780$$ 8.44603e6 0.497068
$$781$$ 8.77363e6 0.514697
$$782$$ −34032.0 −0.00199008
$$783$$ −4.30256e6 −0.250797
$$784$$ 2.67602e7 1.55489
$$785$$ −1.22443e7 −0.709186
$$786$$ −4.60786e6 −0.266037
$$787$$ −2.59845e7 −1.49547 −0.747734 0.663999i $$-0.768858\pi$$
−0.747734 + 0.663999i $$0.768858\pi$$
$$788$$ −9.11226e6 −0.522770
$$789$$ −8.05561e6 −0.460687
$$790$$ 1.54950e7 0.883330
$$791$$ 2.03083e7 1.15407
$$792$$ −3.26592e6 −0.185009
$$793$$ −9.96383e6 −0.562656
$$794$$ 1.75174e6 0.0986094
$$795$$ 3.94254e6 0.221237
$$796$$ 2.75401e7 1.54058
$$797$$ −2.12011e7 −1.18226 −0.591129 0.806577i $$-0.701317\pi$$
−0.591129 + 0.806577i $$0.701317\pi$$
$$798$$ 0 0
$$799$$ 102744. 0.00569363
$$800$$ 3.33798e7 1.84399
$$801$$ −6.71441e6 −0.369766
$$802$$ 6.73231e6 0.369597
$$803$$ −1.64795e7 −0.901891
$$804$$ 1.05971e7 0.578159
$$805$$ −6.67027e7 −3.62789
$$806$$ 1.87690e6 0.101766
$$807$$ 1.14562e7 0.619234
$$808$$ −1.19450e7 −0.643664
$$809$$ −1.06374e7 −0.571429 −0.285714 0.958315i $$-0.592231\pi$$
−0.285714 + 0.958315i $$0.592231\pi$$
$$810$$ −1.28596e6 −0.0688674
$$811$$ 1.19205e7 0.636416 0.318208 0.948021i $$-0.396919\pi$$
0.318208 + 0.948021i $$0.396919\pi$$
$$812$$ −3.96614e7 −2.11095
$$813$$ −2.76005e6 −0.146450
$$814$$ −9.18624e6 −0.485933
$$815$$ 2.83220e6 0.149358
$$816$$ 35424.0 0.00186240
$$817$$ 0 0
$$818$$ −1.31731e7 −0.688342
$$819$$ −6.64848e6 −0.346348
$$820$$ −3.01566e7 −1.56620
$$821$$ 8.14431e6 0.421693 0.210847 0.977519i $$-0.432378\pi$$
0.210847 + 0.977519i $$0.432378\pi$$
$$822$$ −904068. −0.0466683
$$823$$ −9.57675e6 −0.492854 −0.246427 0.969161i $$-0.579257\pi$$
−0.246427 + 0.969161i $$0.579257\pi$$
$$824$$ 8.32416e6 0.427093
$$825$$ −1.95925e7 −1.00220
$$826$$ −1.26202e7 −0.643598
$$827$$ 2.95107e6 0.150043 0.0750214 0.997182i $$-0.476097\pi$$
0.0750214 + 0.997182i $$0.476097\pi$$
$$828$$ −6.43205e6 −0.326042
$$829$$ −8.79546e6 −0.444501 −0.222250 0.974990i $$-0.571340\pi$$
−0.222250 + 0.974990i $$0.571340\pi$$
$$830$$ −1.85651e6 −0.0935411
$$831$$ 1.72823e6 0.0868160
$$832$$ 3.65530e6 0.183069
$$833$$ −244758. −0.0122215
$$834$$ 4.35794e6 0.216953
$$835$$ 3.62796e7 1.80072
$$836$$ 0 0
$$837$$ 2.00038e6 0.0986956
$$838$$ −2.13550e6 −0.105049
$$839$$ 1.46842e6 0.0720185 0.0360093 0.999351i $$-0.488535\pi$$
0.0360093 + 0.999351i $$0.488535\pi$$
$$840$$ −2.54016e7 −1.24212
$$841$$ 1.43225e7 0.698277
$$842$$ −7.21241e6 −0.350591
$$843$$ 9.00495e6 0.436428
$$844$$ −2.42566e7 −1.17213
$$845$$ 2.49242e7 1.20083
$$846$$ −2.77409e6 −0.133258
$$847$$ −1.15572e7 −0.553534
$$848$$ 2.93232e6 0.140030
$$849$$ 1.14162e7 0.543567
$$850$$ −77748.0 −0.00369098
$$851$$ −3.87681e7 −1.83506
$$852$$ 6.58022e6 0.310557
$$853$$ 2.37789e7 1.11897 0.559486 0.828840i $$-0.310999\pi$$
0.559486 + 0.828840i $$0.310999\pi$$
$$854$$ 1.39843e7 0.656140
$$855$$ 0 0
$$856$$ −1.32245e7 −0.616871
$$857$$ 4.09013e7 1.90233 0.951163 0.308688i $$-0.0998900\pi$$
0.951163 + 0.308688i $$0.0998900\pi$$
$$858$$ 2.06842e6 0.0959223
$$859$$ −7.99731e6 −0.369795 −0.184897 0.982758i $$-0.559195\pi$$
−0.184897 + 0.982758i $$0.559195\pi$$
$$860$$ −1.37090e7 −0.632063
$$861$$ 2.37384e7 1.09130
$$862$$ −2.10619e6 −0.0965450
$$863$$ 4.40634e6 0.201396 0.100698 0.994917i $$-0.467892\pi$$
0.100698 + 0.994917i $$0.467892\pi$$
$$864$$ −3.75581e6 −0.171167
$$865$$ −2.71866e7 −1.23542
$$866$$ 6.91395e6 0.313279
$$867$$ 1.27784e7 0.577336
$$868$$ 1.84397e7 0.830719
$$869$$ −2.65628e7 −1.19323
$$870$$ 1.04111e7 0.466337
$$871$$ −1.43818e7 −0.642344
$$872$$ −2.61343e7 −1.16391
$$873$$ −3.22785e6 −0.143343
$$874$$ 0 0
$$875$$ −7.88861e7 −3.48321
$$876$$ −1.23596e7 −0.544182
$$877$$ 2.55057e7 1.11980 0.559898 0.828561i $$-0.310840\pi$$
0.559898 + 0.828561i $$0.310840\pi$$
$$878$$ 1.17722e7 0.515373
$$879$$ −1.37304e7 −0.599391
$$880$$ −2.16008e7 −0.940292
$$881$$ −3.50822e7 −1.52281 −0.761407 0.648274i $$-0.775491\pi$$
−0.761407 + 0.648274i $$0.775491\pi$$
$$882$$ 6.60847e6 0.286042
$$883$$ −2.47814e7 −1.06961 −0.534803 0.844977i $$-0.679614\pi$$
−0.534803 + 0.844977i $$0.679614\pi$$
$$884$$ −57456.0 −0.00247289
$$885$$ −2.31895e7 −0.995254
$$886$$ −9.60432e6 −0.411038
$$887$$ 6.85463e6 0.292533 0.146267 0.989245i $$-0.453274\pi$$
0.146267 + 0.989245i $$0.453274\pi$$
$$888$$ −1.47636e7 −0.628290
$$889$$ −2.15059e7 −0.912649
$$890$$ 1.62472e7 0.687550
$$891$$ 2.20450e6 0.0930283
$$892$$ −2.15398e6 −0.0906422
$$893$$ 0 0
$$894$$ 6.34511e6 0.265519
$$895$$ −4.30326e7 −1.79573
$$896$$ −4.46976e7 −1.86001
$$897$$ 8.72921e6 0.362238
$$898$$ 6.35933e6 0.263160
$$899$$ −1.61951e7 −0.668319
$$900$$ −1.46944e7 −0.604707
$$901$$ −26820.0 −0.00110064
$$902$$ −7.38528e6 −0.302239
$$903$$ 1.07914e7 0.440410
$$904$$ −1.01542e7 −0.413260
$$905$$ 2.48979e6 0.101051
$$906$$ −4.89816e6 −0.198250
$$907$$ 3.64392e7 1.47079 0.735395 0.677639i $$-0.236997\pi$$
0.735395 + 0.677639i $$0.236997\pi$$
$$908$$ −2.71491e7 −1.09280
$$909$$ 8.06290e6 0.323655
$$910$$ 1.60877e7 0.644006
$$911$$ 2.83123e7 1.13026 0.565132 0.825001i $$-0.308825\pi$$
0.565132 + 0.825001i $$0.308825\pi$$
$$912$$ 0 0
$$913$$ 3.18259e6 0.126358
$$914$$ 967092. 0.0382915
$$915$$ 2.56962e7 1.01465
$$916$$ −2.35242e7 −0.926352
$$917$$ 6.14381e7 2.41276
$$918$$ 8748.00 0.000342612 0
$$919$$ −3.75587e6 −0.146697 −0.0733486 0.997306i $$-0.523369\pi$$
−0.0733486 + 0.997306i $$0.523369\pi$$
$$920$$ 3.33514e7 1.29910
$$921$$ −1.07699e7 −0.418371
$$922$$ 1.41596e6 0.0548561
$$923$$ −8.93030e6 −0.345034
$$924$$ 2.03213e7 0.783017
$$925$$ −8.85679e7 −3.40347
$$926$$ 1.64818e7 0.631652
$$927$$ −5.61881e6 −0.214756
$$928$$ 3.04071e7 1.15906
$$929$$ −5.55366e6 −0.211125 −0.105563 0.994413i $$-0.533664\pi$$
−0.105563 + 0.994413i $$0.533664\pi$$
$$930$$ −4.84042e6 −0.183517
$$931$$ 0 0
$$932$$ −1.61256e7 −0.608101
$$933$$ −2.13430e7 −0.802696
$$934$$ 4.27280e6 0.160267
$$935$$ 197568. 0.00739073
$$936$$ 3.32424e6 0.124023
$$937$$ 2.46166e7 0.915965 0.457982 0.888961i $$-0.348572\pi$$
0.457982 + 0.888961i $$0.348572\pi$$
$$938$$ 2.01850e7 0.749067
$$939$$ −3.18364e6 −0.117831
$$940$$ −4.69883e7 −1.73448
$$941$$ 3.89956e7 1.43563 0.717813 0.696236i $$-0.245143\pi$$
0.717813 + 0.696236i $$0.245143\pi$$
$$942$$ −2.24896e6 −0.0825760
$$943$$ −3.11676e7 −1.14137
$$944$$ −1.72476e7 −0.629938
$$945$$ 1.71461e7 0.624576
$$946$$ −3.35731e6 −0.121973
$$947$$ 2.16559e7 0.784696 0.392348 0.919817i $$-0.371663\pi$$
0.392348 + 0.919817i $$0.371663\pi$$
$$948$$ −1.99221e7 −0.719970
$$949$$ 1.67737e7 0.604594
$$950$$ 0 0
$$951$$ 2.43384e7 0.872651
$$952$$ 172800. 0.00617947
$$953$$ 1.94047e7 0.692109 0.346054 0.938214i $$-0.387521\pi$$
0.346054 + 0.938214i $$0.387521\pi$$
$$954$$ 724140. 0.0257603
$$955$$ −6.27831e7 −2.22759
$$956$$ 9.86126e6 0.348970
$$957$$ −1.78476e7 −0.629943
$$958$$ 2.41473e6 0.0850070
$$959$$ 1.20542e7 0.423246
$$960$$ −9.42682e6 −0.330131
$$961$$ −2.10996e7 −0.736998
$$962$$ 9.35028e6 0.325752
$$963$$ 8.92652e6 0.310182
$$964$$ 1.26361e7 0.437947
$$965$$ −2.20349e7 −0.761716
$$966$$ −1.22515e7 −0.422423
$$967$$ 7.26789e6 0.249944 0.124972 0.992160i $$-0.460116\pi$$
0.124972 + 0.992160i $$0.460116\pi$$
$$968$$ 5.77860e6 0.198214
$$969$$ 0 0
$$970$$ 7.81060e6 0.266536
$$971$$ 2.43391e7 0.828431 0.414216 0.910179i $$-0.364056\pi$$
0.414216 + 0.910179i $$0.364056\pi$$
$$972$$ 1.65337e6 0.0561313
$$973$$ −5.81059e7 −1.96761
$$974$$ 1.75594e7 0.593078
$$975$$ 1.99424e7 0.671839
$$976$$ 1.91119e7 0.642213
$$977$$ 3.09415e7 1.03706 0.518531 0.855059i $$-0.326479\pi$$
0.518531 + 0.855059i $$0.326479\pi$$
$$978$$ 520200. 0.0173909
$$979$$ −2.78524e7 −0.928765
$$980$$ 1.11936e8 3.72310
$$981$$ 1.76407e7 0.585251
$$982$$ 8.46800e6 0.280222
$$983$$ −3.53169e7 −1.16573 −0.582865 0.812569i $$-0.698069\pi$$
−0.582865 + 0.812569i $$0.698069\pi$$
$$984$$ −1.18692e7 −0.390781
$$985$$ −3.18929e7 −1.04738
$$986$$ −70824.0 −0.00232000
$$987$$ 3.69878e7 1.20855
$$988$$ 0 0
$$989$$ −1.41687e7 −0.460615
$$990$$ −5.33434e6 −0.172979
$$991$$ −4.96000e7 −1.60434 −0.802172 0.597092i $$-0.796323\pi$$
−0.802172 + 0.597092i $$0.796323\pi$$
$$992$$ −1.41371e7 −0.456122
$$993$$ −2.94700e6 −0.0948432
$$994$$ 1.25338e7 0.402361
$$995$$ 9.63904e7 3.08657
$$996$$ 2.38694e6 0.0762419
$$997$$ −5.33551e7 −1.69996 −0.849978 0.526818i $$-0.823385\pi$$
−0.849978 + 0.526818i $$0.823385\pi$$
$$998$$ −3.46900e6 −0.110250
$$999$$ 9.96543e6 0.315924
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1083.6.a.b.1.1 1
19.18 odd 2 57.6.a.a.1.1 1
57.56 even 2 171.6.a.c.1.1 1
76.75 even 2 912.6.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.6.a.a.1.1 1 19.18 odd 2
171.6.a.c.1.1 1 57.56 even 2
912.6.a.a.1.1 1 76.75 even 2
1083.6.a.b.1.1 1 1.1 even 1 trivial