# Properties

 Label 150.6.a.j.1.1 Level $150$ Weight $6$ Character 150.1 Self dual yes Analytic conductor $24.058$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.0575729719$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 150.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} -4.00000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} -4.00000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -500.000 q^{11} -144.000 q^{12} -288.000 q^{13} -16.0000 q^{14} +256.000 q^{16} +1516.00 q^{17} +324.000 q^{18} -1344.00 q^{19} +36.0000 q^{21} -2000.00 q^{22} -4100.00 q^{23} -576.000 q^{24} -1152.00 q^{26} -729.000 q^{27} -64.0000 q^{28} -2646.00 q^{29} -5612.00 q^{31} +1024.00 q^{32} +4500.00 q^{33} +6064.00 q^{34} +1296.00 q^{36} -7288.00 q^{37} -5376.00 q^{38} +2592.00 q^{39} -18986.0 q^{41} +144.000 q^{42} -2404.00 q^{43} -8000.00 q^{44} -16400.0 q^{46} +8900.00 q^{47} -2304.00 q^{48} -16791.0 q^{49} -13644.0 q^{51} -4608.00 q^{52} +39804.0 q^{53} -2916.00 q^{54} -256.000 q^{56} +12096.0 q^{57} -10584.0 q^{58} -28300.0 q^{59} +18290.0 q^{61} -22448.0 q^{62} -324.000 q^{63} +4096.00 q^{64} +18000.0 q^{66} +65956.0 q^{67} +24256.0 q^{68} +36900.0 q^{69} -28800.0 q^{71} +5184.00 q^{72} -30808.0 q^{73} -29152.0 q^{74} -21504.0 q^{76} +2000.00 q^{77} +10368.0 q^{78} +60228.0 q^{79} +6561.00 q^{81} -75944.0 q^{82} -2468.00 q^{83} +576.000 q^{84} -9616.00 q^{86} +23814.0 q^{87} -32000.0 q^{88} +22678.0 q^{89} +1152.00 q^{91} -65600.0 q^{92} +50508.0 q^{93} +35600.0 q^{94} -9216.00 q^{96} -36968.0 q^{97} -67164.0 q^{98} -40500.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 0.707107
$$3$$ −9.00000 −0.577350
$$4$$ 16.0000 0.500000
$$5$$ 0 0
$$6$$ −36.0000 −0.408248
$$7$$ −4.00000 −0.0308542 −0.0154271 0.999881i $$-0.504911\pi$$
−0.0154271 + 0.999881i $$0.504911\pi$$
$$8$$ 64.0000 0.353553
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ −500.000 −1.24591 −0.622957 0.782256i $$-0.714069\pi$$
−0.622957 + 0.782256i $$0.714069\pi$$
$$12$$ −144.000 −0.288675
$$13$$ −288.000 −0.472644 −0.236322 0.971675i $$-0.575942\pi$$
−0.236322 + 0.971675i $$0.575942\pi$$
$$14$$ −16.0000 −0.0218172
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 1516.00 1.27226 0.636132 0.771581i $$-0.280534\pi$$
0.636132 + 0.771581i $$0.280534\pi$$
$$18$$ 324.000 0.235702
$$19$$ −1344.00 −0.854113 −0.427056 0.904225i $$-0.640449\pi$$
−0.427056 + 0.904225i $$0.640449\pi$$
$$20$$ 0 0
$$21$$ 36.0000 0.0178137
$$22$$ −2000.00 −0.880995
$$23$$ −4100.00 −1.61609 −0.808043 0.589124i $$-0.799473\pi$$
−0.808043 + 0.589124i $$0.799473\pi$$
$$24$$ −576.000 −0.204124
$$25$$ 0 0
$$26$$ −1152.00 −0.334210
$$27$$ −729.000 −0.192450
$$28$$ −64.0000 −0.0154271
$$29$$ −2646.00 −0.584245 −0.292122 0.956381i $$-0.594361\pi$$
−0.292122 + 0.956381i $$0.594361\pi$$
$$30$$ 0 0
$$31$$ −5612.00 −1.04885 −0.524425 0.851457i $$-0.675720\pi$$
−0.524425 + 0.851457i $$0.675720\pi$$
$$32$$ 1024.00 0.176777
$$33$$ 4500.00 0.719329
$$34$$ 6064.00 0.899626
$$35$$ 0 0
$$36$$ 1296.00 0.166667
$$37$$ −7288.00 −0.875193 −0.437597 0.899171i $$-0.644170\pi$$
−0.437597 + 0.899171i $$0.644170\pi$$
$$38$$ −5376.00 −0.603949
$$39$$ 2592.00 0.272881
$$40$$ 0 0
$$41$$ −18986.0 −1.76390 −0.881950 0.471343i $$-0.843769\pi$$
−0.881950 + 0.471343i $$0.843769\pi$$
$$42$$ 144.000 0.0125962
$$43$$ −2404.00 −0.198273 −0.0991364 0.995074i $$-0.531608\pi$$
−0.0991364 + 0.995074i $$0.531608\pi$$
$$44$$ −8000.00 −0.622957
$$45$$ 0 0
$$46$$ −16400.0 −1.14274
$$47$$ 8900.00 0.587686 0.293843 0.955854i $$-0.405066\pi$$
0.293843 + 0.955854i $$0.405066\pi$$
$$48$$ −2304.00 −0.144338
$$49$$ −16791.0 −0.999048
$$50$$ 0 0
$$51$$ −13644.0 −0.734541
$$52$$ −4608.00 −0.236322
$$53$$ 39804.0 1.94642 0.973211 0.229913i $$-0.0738443\pi$$
0.973211 + 0.229913i $$0.0738443\pi$$
$$54$$ −2916.00 −0.136083
$$55$$ 0 0
$$56$$ −256.000 −0.0109086
$$57$$ 12096.0 0.493122
$$58$$ −10584.0 −0.413123
$$59$$ −28300.0 −1.05842 −0.529208 0.848492i $$-0.677511\pi$$
−0.529208 + 0.848492i $$0.677511\pi$$
$$60$$ 0 0
$$61$$ 18290.0 0.629345 0.314673 0.949200i $$-0.398105\pi$$
0.314673 + 0.949200i $$0.398105\pi$$
$$62$$ −22448.0 −0.741649
$$63$$ −324.000 −0.0102847
$$64$$ 4096.00 0.125000
$$65$$ 0 0
$$66$$ 18000.0 0.508643
$$67$$ 65956.0 1.79501 0.897506 0.441002i $$-0.145377\pi$$
0.897506 + 0.441002i $$0.145377\pi$$
$$68$$ 24256.0 0.636132
$$69$$ 36900.0 0.933047
$$70$$ 0 0
$$71$$ −28800.0 −0.678026 −0.339013 0.940782i $$-0.610093\pi$$
−0.339013 + 0.940782i $$0.610093\pi$$
$$72$$ 5184.00 0.117851
$$73$$ −30808.0 −0.676638 −0.338319 0.941031i $$-0.609858\pi$$
−0.338319 + 0.941031i $$0.609858\pi$$
$$74$$ −29152.0 −0.618855
$$75$$ 0 0
$$76$$ −21504.0 −0.427056
$$77$$ 2000.00 0.0384418
$$78$$ 10368.0 0.192956
$$79$$ 60228.0 1.08575 0.542876 0.839813i $$-0.317335\pi$$
0.542876 + 0.839813i $$0.317335\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ −75944.0 −1.24727
$$83$$ −2468.00 −0.0393233 −0.0196616 0.999807i $$-0.506259\pi$$
−0.0196616 + 0.999807i $$0.506259\pi$$
$$84$$ 576.000 0.00890685
$$85$$ 0 0
$$86$$ −9616.00 −0.140200
$$87$$ 23814.0 0.337314
$$88$$ −32000.0 −0.440497
$$89$$ 22678.0 0.303480 0.151740 0.988420i $$-0.451512\pi$$
0.151740 + 0.988420i $$0.451512\pi$$
$$90$$ 0 0
$$91$$ 1152.00 0.0145831
$$92$$ −65600.0 −0.808043
$$93$$ 50508.0 0.605554
$$94$$ 35600.0 0.415557
$$95$$ 0 0
$$96$$ −9216.00 −0.102062
$$97$$ −36968.0 −0.398930 −0.199465 0.979905i $$-0.563920\pi$$
−0.199465 + 0.979905i $$0.563920\pi$$
$$98$$ −67164.0 −0.706434
$$99$$ −40500.0 −0.415305
$$100$$ 0 0
$$101$$ 167918. 1.63792 0.818962 0.573848i $$-0.194550\pi$$
0.818962 + 0.573848i $$0.194550\pi$$
$$102$$ −54576.0 −0.519399
$$103$$ −154364. −1.43368 −0.716841 0.697236i $$-0.754413\pi$$
−0.716841 + 0.697236i $$0.754413\pi$$
$$104$$ −18432.0 −0.167105
$$105$$ 0 0
$$106$$ 159216. 1.37633
$$107$$ 136788. 1.15502 0.577509 0.816385i $$-0.304025\pi$$
0.577509 + 0.816385i $$0.304025\pi$$
$$108$$ −11664.0 −0.0962250
$$109$$ −53810.0 −0.433807 −0.216904 0.976193i $$-0.569596\pi$$
−0.216904 + 0.976193i $$0.569596\pi$$
$$110$$ 0 0
$$111$$ 65592.0 0.505293
$$112$$ −1024.00 −0.00771356
$$113$$ 82692.0 0.609211 0.304605 0.952479i $$-0.401475\pi$$
0.304605 + 0.952479i $$0.401475\pi$$
$$114$$ 48384.0 0.348690
$$115$$ 0 0
$$116$$ −42336.0 −0.292122
$$117$$ −23328.0 −0.157548
$$118$$ −113200. −0.748413
$$119$$ −6064.00 −0.0392547
$$120$$ 0 0
$$121$$ 88949.0 0.552303
$$122$$ 73160.0 0.445014
$$123$$ 170874. 1.01839
$$124$$ −89792.0 −0.524425
$$125$$ 0 0
$$126$$ −1296.00 −0.00727241
$$127$$ −211780. −1.16513 −0.582567 0.812783i $$-0.697952\pi$$
−0.582567 + 0.812783i $$0.697952\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 21636.0 0.114473
$$130$$ 0 0
$$131$$ 169500. 0.862962 0.431481 0.902122i $$-0.357991\pi$$
0.431481 + 0.902122i $$0.357991\pi$$
$$132$$ 72000.0 0.359665
$$133$$ 5376.00 0.0263530
$$134$$ 263824. 1.26927
$$135$$ 0 0
$$136$$ 97024.0 0.449813
$$137$$ 252036. 1.14726 0.573629 0.819115i $$-0.305535\pi$$
0.573629 + 0.819115i $$0.305535\pi$$
$$138$$ 147600. 0.659764
$$139$$ 192016. 0.842947 0.421474 0.906841i $$-0.361513\pi$$
0.421474 + 0.906841i $$0.361513\pi$$
$$140$$ 0 0
$$141$$ −80100.0 −0.339301
$$142$$ −115200. −0.479437
$$143$$ 144000. 0.588874
$$144$$ 20736.0 0.0833333
$$145$$ 0 0
$$146$$ −123232. −0.478455
$$147$$ 151119. 0.576801
$$148$$ −116608. −0.437597
$$149$$ 235694. 0.869727 0.434863 0.900496i $$-0.356797\pi$$
0.434863 + 0.900496i $$0.356797\pi$$
$$150$$ 0 0
$$151$$ −371492. −1.32589 −0.662944 0.748669i $$-0.730693\pi$$
−0.662944 + 0.748669i $$0.730693\pi$$
$$152$$ −86016.0 −0.301975
$$153$$ 122796. 0.424088
$$154$$ 8000.00 0.0271824
$$155$$ 0 0
$$156$$ 41472.0 0.136441
$$157$$ 264952. 0.857863 0.428932 0.903337i $$-0.358890\pi$$
0.428932 + 0.903337i $$0.358890\pi$$
$$158$$ 240912. 0.767743
$$159$$ −358236. −1.12377
$$160$$ 0 0
$$161$$ 16400.0 0.0498631
$$162$$ 26244.0 0.0785674
$$163$$ −403124. −1.18842 −0.594210 0.804310i $$-0.702535\pi$$
−0.594210 + 0.804310i $$0.702535\pi$$
$$164$$ −303776. −0.881950
$$165$$ 0 0
$$166$$ −9872.00 −0.0278058
$$167$$ −261900. −0.726682 −0.363341 0.931656i $$-0.618364\pi$$
−0.363341 + 0.931656i $$0.618364\pi$$
$$168$$ 2304.00 0.00629810
$$169$$ −288349. −0.776608
$$170$$ 0 0
$$171$$ −108864. −0.284704
$$172$$ −38464.0 −0.0991364
$$173$$ 326228. 0.828716 0.414358 0.910114i $$-0.364006\pi$$
0.414358 + 0.910114i $$0.364006\pi$$
$$174$$ 95256.0 0.238517
$$175$$ 0 0
$$176$$ −128000. −0.311479
$$177$$ 254700. 0.611077
$$178$$ 90712.0 0.214593
$$179$$ −109516. −0.255473 −0.127736 0.991808i $$-0.540771\pi$$
−0.127736 + 0.991808i $$0.540771\pi$$
$$180$$ 0 0
$$181$$ −53146.0 −0.120580 −0.0602898 0.998181i $$-0.519202\pi$$
−0.0602898 + 0.998181i $$0.519202\pi$$
$$182$$ 4608.00 0.0103118
$$183$$ −164610. −0.363353
$$184$$ −262400. −0.571372
$$185$$ 0 0
$$186$$ 202032. 0.428191
$$187$$ −758000. −1.58513
$$188$$ 142400. 0.293843
$$189$$ 2916.00 0.00593790
$$190$$ 0 0
$$191$$ 232056. 0.460267 0.230133 0.973159i $$-0.426084\pi$$
0.230133 + 0.973159i $$0.426084\pi$$
$$192$$ −36864.0 −0.0721688
$$193$$ −1.03067e6 −1.99172 −0.995858 0.0909274i $$-0.971017\pi$$
−0.995858 + 0.0909274i $$0.971017\pi$$
$$194$$ −147872. −0.282086
$$195$$ 0 0
$$196$$ −268656. −0.499524
$$197$$ −522796. −0.959769 −0.479884 0.877332i $$-0.659321\pi$$
−0.479884 + 0.877332i $$0.659321\pi$$
$$198$$ −162000. −0.293665
$$199$$ −215292. −0.385385 −0.192693 0.981259i $$-0.561722\pi$$
−0.192693 + 0.981259i $$0.561722\pi$$
$$200$$ 0 0
$$201$$ −593604. −1.03635
$$202$$ 671672. 1.15819
$$203$$ 10584.0 0.0180264
$$204$$ −218304. −0.367271
$$205$$ 0 0
$$206$$ −617456. −1.01377
$$207$$ −332100. −0.538695
$$208$$ −73728.0 −0.118161
$$209$$ 672000. 1.06415
$$210$$ 0 0
$$211$$ −1.03008e6 −1.59281 −0.796407 0.604762i $$-0.793268\pi$$
−0.796407 + 0.604762i $$0.793268\pi$$
$$212$$ 636864. 0.973211
$$213$$ 259200. 0.391459
$$214$$ 547152. 0.816721
$$215$$ 0 0
$$216$$ −46656.0 −0.0680414
$$217$$ 22448.0 0.0323615
$$218$$ −215240. −0.306748
$$219$$ 277272. 0.390657
$$220$$ 0 0
$$221$$ −436608. −0.601327
$$222$$ 262368. 0.357296
$$223$$ 456020. 0.614075 0.307038 0.951697i $$-0.400662\pi$$
0.307038 + 0.951697i $$0.400662\pi$$
$$224$$ −4096.00 −0.00545431
$$225$$ 0 0
$$226$$ 330768. 0.430777
$$227$$ −434252. −0.559342 −0.279671 0.960096i $$-0.590225\pi$$
−0.279671 + 0.960096i $$0.590225\pi$$
$$228$$ 193536. 0.246561
$$229$$ −722710. −0.910700 −0.455350 0.890313i $$-0.650486\pi$$
−0.455350 + 0.890313i $$0.650486\pi$$
$$230$$ 0 0
$$231$$ −18000.0 −0.0221944
$$232$$ −169344. −0.206562
$$233$$ −565348. −0.682223 −0.341111 0.940023i $$-0.610803\pi$$
−0.341111 + 0.940023i $$0.610803\pi$$
$$234$$ −93312.0 −0.111403
$$235$$ 0 0
$$236$$ −452800. −0.529208
$$237$$ −542052. −0.626859
$$238$$ −24256.0 −0.0277573
$$239$$ 324904. 0.367926 0.183963 0.982933i $$-0.441107\pi$$
0.183963 + 0.982933i $$0.441107\pi$$
$$240$$ 0 0
$$241$$ 915262. 1.01509 0.507543 0.861626i $$-0.330554\pi$$
0.507543 + 0.861626i $$0.330554\pi$$
$$242$$ 355796. 0.390537
$$243$$ −59049.0 −0.0641500
$$244$$ 292640. 0.314673
$$245$$ 0 0
$$246$$ 683496. 0.720109
$$247$$ 387072. 0.403691
$$248$$ −359168. −0.370825
$$249$$ 22212.0 0.0227033
$$250$$ 0 0
$$251$$ 1.36708e6 1.36965 0.684823 0.728709i $$-0.259879\pi$$
0.684823 + 0.728709i $$0.259879\pi$$
$$252$$ −5184.00 −0.00514237
$$253$$ 2.05000e6 2.01350
$$254$$ −847120. −0.823874
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 892932. 0.843307 0.421653 0.906757i $$-0.361450\pi$$
0.421653 + 0.906757i $$0.361450\pi$$
$$258$$ 86544.0 0.0809446
$$259$$ 29152.0 0.0270034
$$260$$ 0 0
$$261$$ −214326. −0.194748
$$262$$ 678000. 0.610206
$$263$$ −1.86650e6 −1.66394 −0.831972 0.554818i $$-0.812788\pi$$
−0.831972 + 0.554818i $$0.812788\pi$$
$$264$$ 288000. 0.254321
$$265$$ 0 0
$$266$$ 21504.0 0.0186344
$$267$$ −204102. −0.175214
$$268$$ 1.05530e6 0.897506
$$269$$ −1.37227e6 −1.15627 −0.578133 0.815943i $$-0.696218\pi$$
−0.578133 + 0.815943i $$0.696218\pi$$
$$270$$ 0 0
$$271$$ 458644. 0.379361 0.189680 0.981846i $$-0.439255\pi$$
0.189680 + 0.981846i $$0.439255\pi$$
$$272$$ 388096. 0.318066
$$273$$ −10368.0 −0.00841954
$$274$$ 1.00814e6 0.811234
$$275$$ 0 0
$$276$$ 590400. 0.466524
$$277$$ −985408. −0.771643 −0.385822 0.922573i $$-0.626082\pi$$
−0.385822 + 0.922573i $$0.626082\pi$$
$$278$$ 768064. 0.596054
$$279$$ −454572. −0.349617
$$280$$ 0 0
$$281$$ 165798. 0.125260 0.0626302 0.998037i $$-0.480051\pi$$
0.0626302 + 0.998037i $$0.480051\pi$$
$$282$$ −320400. −0.239922
$$283$$ −1.66471e6 −1.23558 −0.617792 0.786342i $$-0.711972\pi$$
−0.617792 + 0.786342i $$0.711972\pi$$
$$284$$ −460800. −0.339013
$$285$$ 0 0
$$286$$ 576000. 0.416397
$$287$$ 75944.0 0.0544238
$$288$$ 82944.0 0.0589256
$$289$$ 878399. 0.618653
$$290$$ 0 0
$$291$$ 332712. 0.230322
$$292$$ −492928. −0.338319
$$293$$ 2.55104e6 1.73600 0.867998 0.496567i $$-0.165406\pi$$
0.867998 + 0.496567i $$0.165406\pi$$
$$294$$ 604476. 0.407860
$$295$$ 0 0
$$296$$ −466432. −0.309428
$$297$$ 364500. 0.239776
$$298$$ 942776. 0.614990
$$299$$ 1.18080e6 0.763833
$$300$$ 0 0
$$301$$ 9616.00 0.00611756
$$302$$ −1.48597e6 −0.937545
$$303$$ −1.51126e6 −0.945656
$$304$$ −344064. −0.213528
$$305$$ 0 0
$$306$$ 491184. 0.299875
$$307$$ 736020. 0.445701 0.222851 0.974853i $$-0.428464\pi$$
0.222851 + 0.974853i $$0.428464\pi$$
$$308$$ 32000.0 0.0192209
$$309$$ 1.38928e6 0.827737
$$310$$ 0 0
$$311$$ 1.71660e6 1.00639 0.503197 0.864172i $$-0.332157\pi$$
0.503197 + 0.864172i $$0.332157\pi$$
$$312$$ 165888. 0.0964780
$$313$$ 2.83851e6 1.63768 0.818842 0.574020i $$-0.194617\pi$$
0.818842 + 0.574020i $$0.194617\pi$$
$$314$$ 1.05981e6 0.606601
$$315$$ 0 0
$$316$$ 963648. 0.542876
$$317$$ −1.27605e6 −0.713215 −0.356607 0.934254i $$-0.616067\pi$$
−0.356607 + 0.934254i $$0.616067\pi$$
$$318$$ −1.43294e6 −0.794624
$$319$$ 1.32300e6 0.727919
$$320$$ 0 0
$$321$$ −1.23109e6 −0.666850
$$322$$ 65600.0 0.0352585
$$323$$ −2.03750e6 −1.08666
$$324$$ 104976. 0.0555556
$$325$$ 0 0
$$326$$ −1.61250e6 −0.840339
$$327$$ 484290. 0.250459
$$328$$ −1.21510e6 −0.623633
$$329$$ −35600.0 −0.0181326
$$330$$ 0 0
$$331$$ 443992. 0.222744 0.111372 0.993779i $$-0.464476\pi$$
0.111372 + 0.993779i $$0.464476\pi$$
$$332$$ −39488.0 −0.0196616
$$333$$ −590328. −0.291731
$$334$$ −1.04760e6 −0.513842
$$335$$ 0 0
$$336$$ 9216.00 0.00445343
$$337$$ 2.71326e6 1.30142 0.650708 0.759328i $$-0.274472\pi$$
0.650708 + 0.759328i $$0.274472\pi$$
$$338$$ −1.15340e6 −0.549145
$$339$$ −744228. −0.351728
$$340$$ 0 0
$$341$$ 2.80600e6 1.30678
$$342$$ −435456. −0.201316
$$343$$ 134392. 0.0616791
$$344$$ −153856. −0.0701001
$$345$$ 0 0
$$346$$ 1.30491e6 0.585991
$$347$$ −1.31051e6 −0.584273 −0.292137 0.956377i $$-0.594366\pi$$
−0.292137 + 0.956377i $$0.594366\pi$$
$$348$$ 381024. 0.168657
$$349$$ −298910. −0.131364 −0.0656821 0.997841i $$-0.520922\pi$$
−0.0656821 + 0.997841i $$0.520922\pi$$
$$350$$ 0 0
$$351$$ 209952. 0.0909604
$$352$$ −512000. −0.220249
$$353$$ 737996. 0.315223 0.157611 0.987501i $$-0.449621\pi$$
0.157611 + 0.987501i $$0.449621\pi$$
$$354$$ 1.01880e6 0.432097
$$355$$ 0 0
$$356$$ 362848. 0.151740
$$357$$ 54576.0 0.0226637
$$358$$ −438064. −0.180647
$$359$$ −2.34074e6 −0.958557 −0.479278 0.877663i $$-0.659102\pi$$
−0.479278 + 0.877663i $$0.659102\pi$$
$$360$$ 0 0
$$361$$ −669763. −0.270491
$$362$$ −212584. −0.0852627
$$363$$ −800541. −0.318872
$$364$$ 18432.0 0.00729154
$$365$$ 0 0
$$366$$ −658440. −0.256929
$$367$$ 127292. 0.0493328 0.0246664 0.999696i $$-0.492148\pi$$
0.0246664 + 0.999696i $$0.492148\pi$$
$$368$$ −1.04960e6 −0.404021
$$369$$ −1.53787e6 −0.587966
$$370$$ 0 0
$$371$$ −159216. −0.0600554
$$372$$ 808128. 0.302777
$$373$$ 4.03870e6 1.50303 0.751517 0.659713i $$-0.229322\pi$$
0.751517 + 0.659713i $$0.229322\pi$$
$$374$$ −3.03200e6 −1.12086
$$375$$ 0 0
$$376$$ 569600. 0.207778
$$377$$ 762048. 0.276140
$$378$$ 11664.0 0.00419873
$$379$$ 1.01214e6 0.361944 0.180972 0.983488i $$-0.442076\pi$$
0.180972 + 0.983488i $$0.442076\pi$$
$$380$$ 0 0
$$381$$ 1.90602e6 0.672690
$$382$$ 928224. 0.325458
$$383$$ −2.37610e6 −0.827690 −0.413845 0.910347i $$-0.635814\pi$$
−0.413845 + 0.910347i $$0.635814\pi$$
$$384$$ −147456. −0.0510310
$$385$$ 0 0
$$386$$ −4.12269e6 −1.40836
$$387$$ −194724. −0.0660910
$$388$$ −591488. −0.199465
$$389$$ 1.42497e6 0.477456 0.238728 0.971087i $$-0.423270\pi$$
0.238728 + 0.971087i $$0.423270\pi$$
$$390$$ 0 0
$$391$$ −6.21560e6 −2.05609
$$392$$ −1.07462e6 −0.353217
$$393$$ −1.52550e6 −0.498231
$$394$$ −2.09118e6 −0.678659
$$395$$ 0 0
$$396$$ −648000. −0.207652
$$397$$ 1.69345e6 0.539257 0.269628 0.962964i $$-0.413099\pi$$
0.269628 + 0.962964i $$0.413099\pi$$
$$398$$ −861168. −0.272509
$$399$$ −48384.0 −0.0152149
$$400$$ 0 0
$$401$$ −2.84501e6 −0.883532 −0.441766 0.897130i $$-0.645648\pi$$
−0.441766 + 0.897130i $$0.645648\pi$$
$$402$$ −2.37442e6 −0.732810
$$403$$ 1.61626e6 0.495733
$$404$$ 2.68669e6 0.818962
$$405$$ 0 0
$$406$$ 42336.0 0.0127466
$$407$$ 3.64400e6 1.09042
$$408$$ −873216. −0.259700
$$409$$ −1.89069e6 −0.558873 −0.279436 0.960164i $$-0.590148\pi$$
−0.279436 + 0.960164i $$0.590148\pi$$
$$410$$ 0 0
$$411$$ −2.26832e6 −0.662370
$$412$$ −2.46982e6 −0.716841
$$413$$ 113200. 0.0326566
$$414$$ −1.32840e6 −0.380915
$$415$$ 0 0
$$416$$ −294912. −0.0835524
$$417$$ −1.72814e6 −0.486676
$$418$$ 2.68800e6 0.752469
$$419$$ 4.60930e6 1.28263 0.641313 0.767280i $$-0.278390\pi$$
0.641313 + 0.767280i $$0.278390\pi$$
$$420$$ 0 0
$$421$$ −6.04151e6 −1.66127 −0.830635 0.556817i $$-0.812022\pi$$
−0.830635 + 0.556817i $$0.812022\pi$$
$$422$$ −4.12032e6 −1.12629
$$423$$ 720900. 0.195895
$$424$$ 2.54746e6 0.688164
$$425$$ 0 0
$$426$$ 1.03680e6 0.276803
$$427$$ −73160.0 −0.0194180
$$428$$ 2.18861e6 0.577509
$$429$$ −1.29600e6 −0.339987
$$430$$ 0 0
$$431$$ −3800.00 −0.000985350 0 −0.000492675 1.00000i $$-0.500157\pi$$
−0.000492675 1.00000i $$0.500157\pi$$
$$432$$ −186624. −0.0481125
$$433$$ 250736. 0.0642683 0.0321342 0.999484i $$-0.489770\pi$$
0.0321342 + 0.999484i $$0.489770\pi$$
$$434$$ 89792.0 0.0228830
$$435$$ 0 0
$$436$$ −860960. −0.216904
$$437$$ 5.51040e6 1.38032
$$438$$ 1.10909e6 0.276236
$$439$$ −3.58873e6 −0.888750 −0.444375 0.895841i $$-0.646574\pi$$
−0.444375 + 0.895841i $$0.646574\pi$$
$$440$$ 0 0
$$441$$ −1.36007e6 −0.333016
$$442$$ −1.74643e6 −0.425203
$$443$$ −1.41479e6 −0.342517 −0.171258 0.985226i $$-0.554783\pi$$
−0.171258 + 0.985226i $$0.554783\pi$$
$$444$$ 1.04947e6 0.252647
$$445$$ 0 0
$$446$$ 1.82408e6 0.434217
$$447$$ −2.12125e6 −0.502137
$$448$$ −16384.0 −0.00385678
$$449$$ −829806. −0.194250 −0.0971249 0.995272i $$-0.530965\pi$$
−0.0971249 + 0.995272i $$0.530965\pi$$
$$450$$ 0 0
$$451$$ 9.49300e6 2.19767
$$452$$ 1.32307e6 0.304605
$$453$$ 3.34343e6 0.765502
$$454$$ −1.73701e6 −0.395514
$$455$$ 0 0
$$456$$ 774144. 0.174345
$$457$$ 4.68198e6 1.04867 0.524335 0.851512i $$-0.324314\pi$$
0.524335 + 0.851512i $$0.324314\pi$$
$$458$$ −2.89084e6 −0.643962
$$459$$ −1.10516e6 −0.244847
$$460$$ 0 0
$$461$$ −141930. −0.0311044 −0.0155522 0.999879i $$-0.504951\pi$$
−0.0155522 + 0.999879i $$0.504951\pi$$
$$462$$ −72000.0 −0.0156938
$$463$$ 727476. 0.157713 0.0788563 0.996886i $$-0.474873\pi$$
0.0788563 + 0.996886i $$0.474873\pi$$
$$464$$ −677376. −0.146061
$$465$$ 0 0
$$466$$ −2.26139e6 −0.482404
$$467$$ −4.47640e6 −0.949809 −0.474905 0.880037i $$-0.657517\pi$$
−0.474905 + 0.880037i $$0.657517\pi$$
$$468$$ −373248. −0.0787740
$$469$$ −263824. −0.0553837
$$470$$ 0 0
$$471$$ −2.38457e6 −0.495288
$$472$$ −1.81120e6 −0.374207
$$473$$ 1.20200e6 0.247031
$$474$$ −2.16821e6 −0.443256
$$475$$ 0 0
$$476$$ −97024.0 −0.0196274
$$477$$ 3.22412e6 0.648807
$$478$$ 1.29962e6 0.260163
$$479$$ −1.32718e6 −0.264297 −0.132149 0.991230i $$-0.542188\pi$$
−0.132149 + 0.991230i $$0.542188\pi$$
$$480$$ 0 0
$$481$$ 2.09894e6 0.413655
$$482$$ 3.66105e6 0.717774
$$483$$ −147600. −0.0287885
$$484$$ 1.42318e6 0.276152
$$485$$ 0 0
$$486$$ −236196. −0.0453609
$$487$$ −4.11647e6 −0.786507 −0.393253 0.919430i $$-0.628650\pi$$
−0.393253 + 0.919430i $$0.628650\pi$$
$$488$$ 1.17056e6 0.222507
$$489$$ 3.62812e6 0.686134
$$490$$ 0 0
$$491$$ −6.12316e6 −1.14623 −0.573115 0.819475i $$-0.694265\pi$$
−0.573115 + 0.819475i $$0.694265\pi$$
$$492$$ 2.73398e6 0.509194
$$493$$ −4.01134e6 −0.743313
$$494$$ 1.54829e6 0.285453
$$495$$ 0 0
$$496$$ −1.43667e6 −0.262213
$$497$$ 115200. 0.0209200
$$498$$ 88848.0 0.0160537
$$499$$ −7.90490e6 −1.42117 −0.710584 0.703613i $$-0.751569\pi$$
−0.710584 + 0.703613i $$0.751569\pi$$
$$500$$ 0 0
$$501$$ 2.35710e6 0.419550
$$502$$ 5.46830e6 0.968486
$$503$$ 3.97628e6 0.700741 0.350370 0.936611i $$-0.386056\pi$$
0.350370 + 0.936611i $$0.386056\pi$$
$$504$$ −20736.0 −0.00363621
$$505$$ 0 0
$$506$$ 8.20000e6 1.42376
$$507$$ 2.59514e6 0.448375
$$508$$ −3.38848e6 −0.582567
$$509$$ 781914. 0.133772 0.0668859 0.997761i $$-0.478694\pi$$
0.0668859 + 0.997761i $$0.478694\pi$$
$$510$$ 0 0
$$511$$ 123232. 0.0208772
$$512$$ 262144. 0.0441942
$$513$$ 979776. 0.164374
$$514$$ 3.57173e6 0.596308
$$515$$ 0 0
$$516$$ 346176. 0.0572365
$$517$$ −4.45000e6 −0.732207
$$518$$ 116608. 0.0190943
$$519$$ −2.93605e6 −0.478460
$$520$$ 0 0
$$521$$ 5.82694e6 0.940472 0.470236 0.882541i $$-0.344169\pi$$
0.470236 + 0.882541i $$0.344169\pi$$
$$522$$ −857304. −0.137708
$$523$$ −9.78938e6 −1.56495 −0.782476 0.622681i $$-0.786043\pi$$
−0.782476 + 0.622681i $$0.786043\pi$$
$$524$$ 2.71200e6 0.431481
$$525$$ 0 0
$$526$$ −7.46600e6 −1.17659
$$527$$ −8.50779e6 −1.33441
$$528$$ 1.15200e6 0.179832
$$529$$ 1.03737e7 1.61173
$$530$$ 0 0
$$531$$ −2.29230e6 −0.352805
$$532$$ 86016.0 0.0131765
$$533$$ 5.46797e6 0.833696
$$534$$ −816408. −0.123895
$$535$$ 0 0
$$536$$ 4.22118e6 0.634633
$$537$$ 985644. 0.147497
$$538$$ −5.48906e6 −0.817603
$$539$$ 8.39550e6 1.24473
$$540$$ 0 0
$$541$$ 4.76059e6 0.699307 0.349653 0.936879i $$-0.386299\pi$$
0.349653 + 0.936879i $$0.386299\pi$$
$$542$$ 1.83458e6 0.268249
$$543$$ 478314. 0.0696167
$$544$$ 1.55238e6 0.224906
$$545$$ 0 0
$$546$$ −41472.0 −0.00595351
$$547$$ −1.16595e6 −0.166614 −0.0833069 0.996524i $$-0.526548\pi$$
−0.0833069 + 0.996524i $$0.526548\pi$$
$$548$$ 4.03258e6 0.573629
$$549$$ 1.48149e6 0.209782
$$550$$ 0 0
$$551$$ 3.55622e6 0.499011
$$552$$ 2.36160e6 0.329882
$$553$$ −240912. −0.0335001
$$554$$ −3.94163e6 −0.545634
$$555$$ 0 0
$$556$$ 3.07226e6 0.421474
$$557$$ −1.61293e6 −0.220282 −0.110141 0.993916i $$-0.535130\pi$$
−0.110141 + 0.993916i $$0.535130\pi$$
$$558$$ −1.81829e6 −0.247216
$$559$$ 692352. 0.0937125
$$560$$ 0 0
$$561$$ 6.82200e6 0.915176
$$562$$ 663192. 0.0885724
$$563$$ 3.40603e6 0.452874 0.226437 0.974026i $$-0.427292\pi$$
0.226437 + 0.974026i $$0.427292\pi$$
$$564$$ −1.28160e6 −0.169650
$$565$$ 0 0
$$566$$ −6.65883e6 −0.873689
$$567$$ −26244.0 −0.00342825
$$568$$ −1.84320e6 −0.239719
$$569$$ −1.44009e7 −1.86470 −0.932350 0.361557i $$-0.882245\pi$$
−0.932350 + 0.361557i $$0.882245\pi$$
$$570$$ 0 0
$$571$$ 4.74772e6 0.609389 0.304695 0.952450i $$-0.401446\pi$$
0.304695 + 0.952450i $$0.401446\pi$$
$$572$$ 2.30400e6 0.294437
$$573$$ −2.08850e6 −0.265735
$$574$$ 303776. 0.0384834
$$575$$ 0 0
$$576$$ 331776. 0.0416667
$$577$$ −1.09094e7 −1.36415 −0.682074 0.731283i $$-0.738922\pi$$
−0.682074 + 0.731283i $$0.738922\pi$$
$$578$$ 3.51360e6 0.437454
$$579$$ 9.27605e6 1.14992
$$580$$ 0 0
$$581$$ 9872.00 0.00121329
$$582$$ 1.33085e6 0.162862
$$583$$ −1.99020e7 −2.42508
$$584$$ −1.97171e6 −0.239228
$$585$$ 0 0
$$586$$ 1.02042e7 1.22754
$$587$$ 8.53223e6 1.02204 0.511019 0.859569i $$-0.329268\pi$$
0.511019 + 0.859569i $$0.329268\pi$$
$$588$$ 2.41790e6 0.288400
$$589$$ 7.54253e6 0.895836
$$590$$ 0 0
$$591$$ 4.70516e6 0.554123
$$592$$ −1.86573e6 −0.218798
$$593$$ −4.63182e6 −0.540897 −0.270449 0.962734i $$-0.587172\pi$$
−0.270449 + 0.962734i $$0.587172\pi$$
$$594$$ 1.45800e6 0.169548
$$595$$ 0 0
$$596$$ 3.77110e6 0.434863
$$597$$ 1.93763e6 0.222502
$$598$$ 4.72320e6 0.540111
$$599$$ 6.27598e6 0.714684 0.357342 0.933974i $$-0.383683\pi$$
0.357342 + 0.933974i $$0.383683\pi$$
$$600$$ 0 0
$$601$$ 7.71988e6 0.871815 0.435907 0.899992i $$-0.356428\pi$$
0.435907 + 0.899992i $$0.356428\pi$$
$$602$$ 38464.0 0.00432577
$$603$$ 5.34244e6 0.598337
$$604$$ −5.94387e6 −0.662944
$$605$$ 0 0
$$606$$ −6.04505e6 −0.668680
$$607$$ −6.06160e6 −0.667753 −0.333876 0.942617i $$-0.608357\pi$$
−0.333876 + 0.942617i $$0.608357\pi$$
$$608$$ −1.37626e6 −0.150987
$$609$$ −95256.0 −0.0104076
$$610$$ 0 0
$$611$$ −2.56320e6 −0.277766
$$612$$ 1.96474e6 0.212044
$$613$$ −3.66489e6 −0.393921 −0.196961 0.980411i $$-0.563107\pi$$
−0.196961 + 0.980411i $$0.563107\pi$$
$$614$$ 2.94408e6 0.315158
$$615$$ 0 0
$$616$$ 128000. 0.0135912
$$617$$ −9.32522e6 −0.986157 −0.493079 0.869985i $$-0.664129\pi$$
−0.493079 + 0.869985i $$0.664129\pi$$
$$618$$ 5.55710e6 0.585298
$$619$$ −7.40162e6 −0.776426 −0.388213 0.921570i $$-0.626907\pi$$
−0.388213 + 0.921570i $$0.626907\pi$$
$$620$$ 0 0
$$621$$ 2.98890e6 0.311016
$$622$$ 6.86640e6 0.711628
$$623$$ −90712.0 −0.00936364
$$624$$ 663552. 0.0682203
$$625$$ 0 0
$$626$$ 1.13540e7 1.15802
$$627$$ −6.04800e6 −0.614388
$$628$$ 4.23923e6 0.428932
$$629$$ −1.10486e7 −1.11348
$$630$$ 0 0
$$631$$ 160052. 0.0160025 0.00800125 0.999968i $$-0.497453\pi$$
0.00800125 + 0.999968i $$0.497453\pi$$
$$632$$ 3.85459e6 0.383871
$$633$$ 9.27072e6 0.919611
$$634$$ −5.10421e6 −0.504319
$$635$$ 0 0
$$636$$ −5.73178e6 −0.561884
$$637$$ 4.83581e6 0.472194
$$638$$ 5.29200e6 0.514717
$$639$$ −2.33280e6 −0.226009
$$640$$ 0 0
$$641$$ −1.69565e7 −1.63002 −0.815008 0.579450i $$-0.803268\pi$$
−0.815008 + 0.579450i $$0.803268\pi$$
$$642$$ −4.92437e6 −0.471534
$$643$$ 1.10128e7 1.05044 0.525219 0.850967i $$-0.323984\pi$$
0.525219 + 0.850967i $$0.323984\pi$$
$$644$$ 262400. 0.0249315
$$645$$ 0 0
$$646$$ −8.15002e6 −0.768382
$$647$$ 3.33848e6 0.313537 0.156768 0.987635i $$-0.449892\pi$$
0.156768 + 0.987635i $$0.449892\pi$$
$$648$$ 419904. 0.0392837
$$649$$ 1.41500e7 1.31870
$$650$$ 0 0
$$651$$ −202032. −0.0186839
$$652$$ −6.44998e6 −0.594210
$$653$$ −4.76181e6 −0.437008 −0.218504 0.975836i $$-0.570118\pi$$
−0.218504 + 0.975836i $$0.570118\pi$$
$$654$$ 1.93716e6 0.177101
$$655$$ 0 0
$$656$$ −4.86042e6 −0.440975
$$657$$ −2.49545e6 −0.225546
$$658$$ −142400. −0.0128217
$$659$$ 798188. 0.0715965 0.0357982 0.999359i $$-0.488603\pi$$
0.0357982 + 0.999359i $$0.488603\pi$$
$$660$$ 0 0
$$661$$ −1.54048e7 −1.37136 −0.685682 0.727901i $$-0.740496\pi$$
−0.685682 + 0.727901i $$0.740496\pi$$
$$662$$ 1.77597e6 0.157503
$$663$$ 3.92947e6 0.347177
$$664$$ −157952. −0.0139029
$$665$$ 0 0
$$666$$ −2.36131e6 −0.206285
$$667$$ 1.08486e7 0.944189
$$668$$ −4.19040e6 −0.363341
$$669$$ −4.10418e6 −0.354537
$$670$$ 0 0
$$671$$ −9.14500e6 −0.784111
$$672$$ 36864.0 0.00314905
$$673$$ −976704. −0.0831238 −0.0415619 0.999136i $$-0.513233\pi$$
−0.0415619 + 0.999136i $$0.513233\pi$$
$$674$$ 1.08530e7 0.920240
$$675$$ 0 0
$$676$$ −4.61358e6 −0.388304
$$677$$ 1.93885e7 1.62582 0.812911 0.582388i $$-0.197881\pi$$
0.812911 + 0.582388i $$0.197881\pi$$
$$678$$ −2.97691e6 −0.248709
$$679$$ 147872. 0.0123087
$$680$$ 0 0
$$681$$ 3.90827e6 0.322936
$$682$$ 1.12240e7 0.924031
$$683$$ −5.25573e6 −0.431103 −0.215552 0.976492i $$-0.569155\pi$$
−0.215552 + 0.976492i $$0.569155\pi$$
$$684$$ −1.74182e6 −0.142352
$$685$$ 0 0
$$686$$ 537568. 0.0436137
$$687$$ 6.50439e6 0.525793
$$688$$ −615424. −0.0495682
$$689$$ −1.14636e7 −0.919965
$$690$$ 0 0
$$691$$ −5.45034e6 −0.434238 −0.217119 0.976145i $$-0.569666\pi$$
−0.217119 + 0.976145i $$0.569666\pi$$
$$692$$ 5.21965e6 0.414358
$$693$$ 162000. 0.0128139
$$694$$ −5.24203e6 −0.413144
$$695$$ 0 0
$$696$$ 1.52410e6 0.119258
$$697$$ −2.87828e7 −2.24414
$$698$$ −1.19564e6 −0.0928885
$$699$$ 5.08813e6 0.393881
$$700$$ 0 0
$$701$$ −4.43961e6 −0.341232 −0.170616 0.985338i $$-0.554576\pi$$
−0.170616 + 0.985338i $$0.554576\pi$$
$$702$$ 839808. 0.0643187
$$703$$ 9.79507e6 0.747514
$$704$$ −2.04800e6 −0.155739
$$705$$ 0 0
$$706$$ 2.95198e6 0.222896
$$707$$ −671672. −0.0505369
$$708$$ 4.07520e6 0.305538
$$709$$ 4.55918e6 0.340621 0.170310 0.985390i $$-0.445523\pi$$
0.170310 + 0.985390i $$0.445523\pi$$
$$710$$ 0 0
$$711$$ 4.87847e6 0.361917
$$712$$ 1.45139e6 0.107296
$$713$$ 2.30092e7 1.69503
$$714$$ 218304. 0.0160257
$$715$$ 0 0
$$716$$ −1.75226e6 −0.127736
$$717$$ −2.92414e6 −0.212422
$$718$$ −9.36298e6 −0.677802
$$719$$ −2.06630e7 −1.49063 −0.745317 0.666710i $$-0.767702\pi$$
−0.745317 + 0.666710i $$0.767702\pi$$
$$720$$ 0 0
$$721$$ 617456. 0.0442352
$$722$$ −2.67905e6 −0.191266
$$723$$ −8.23736e6 −0.586060
$$724$$ −850336. −0.0602898
$$725$$ 0 0
$$726$$ −3.20216e6 −0.225477
$$727$$ 5.48161e6 0.384656 0.192328 0.981331i $$-0.438396\pi$$
0.192328 + 0.981331i $$0.438396\pi$$
$$728$$ 73728.0 0.00515589
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ −3.64446e6 −0.252255
$$732$$ −2.63376e6 −0.181676
$$733$$ −8.55579e6 −0.588166 −0.294083 0.955780i $$-0.595014\pi$$
−0.294083 + 0.955780i $$0.595014\pi$$
$$734$$ 509168. 0.0348836
$$735$$ 0 0
$$736$$ −4.19840e6 −0.285686
$$737$$ −3.29780e7 −2.23643
$$738$$ −6.15146e6 −0.415755
$$739$$ −5.29119e6 −0.356404 −0.178202 0.983994i $$-0.557028\pi$$
−0.178202 + 0.983994i $$0.557028\pi$$
$$740$$ 0 0
$$741$$ −3.48365e6 −0.233071
$$742$$ −636864. −0.0424656
$$743$$ −2.36432e6 −0.157121 −0.0785606 0.996909i $$-0.525032\pi$$
−0.0785606 + 0.996909i $$0.525032\pi$$
$$744$$ 3.23251e6 0.214096
$$745$$ 0 0
$$746$$ 1.61548e7 1.06281
$$747$$ −199908. −0.0131078
$$748$$ −1.21280e7 −0.792566
$$749$$ −547152. −0.0356372
$$750$$ 0 0
$$751$$ −8.79694e6 −0.569157 −0.284578 0.958653i $$-0.591854\pi$$
−0.284578 + 0.958653i $$0.591854\pi$$
$$752$$ 2.27840e6 0.146922
$$753$$ −1.23037e7 −0.790766
$$754$$ 3.04819e6 0.195260
$$755$$ 0 0
$$756$$ 46656.0 0.00296895
$$757$$ 2.95808e7 1.87616 0.938079 0.346421i $$-0.112603\pi$$
0.938079 + 0.346421i $$0.112603\pi$$
$$758$$ 4.04854e6 0.255933
$$759$$ −1.84500e7 −1.16250
$$760$$ 0 0
$$761$$ −1.26296e7 −0.790549 −0.395274 0.918563i $$-0.629351\pi$$
−0.395274 + 0.918563i $$0.629351\pi$$
$$762$$ 7.62408e6 0.475664
$$763$$ 215240. 0.0133848
$$764$$ 3.71290e6 0.230133
$$765$$ 0 0
$$766$$ −9.50440e6 −0.585265
$$767$$ 8.15040e6 0.500254
$$768$$ −589824. −0.0360844
$$769$$ −2.32186e7 −1.41586 −0.707929 0.706283i $$-0.750370\pi$$
−0.707929 + 0.706283i $$0.750370\pi$$
$$770$$ 0 0
$$771$$ −8.03639e6 −0.486883
$$772$$ −1.64908e7 −0.995858
$$773$$ −1.73201e7 −1.04256 −0.521280 0.853386i $$-0.674545\pi$$
−0.521280 + 0.853386i $$0.674545\pi$$
$$774$$ −778896. −0.0467334
$$775$$ 0 0
$$776$$ −2.36595e6 −0.141043
$$777$$ −262368. −0.0155904
$$778$$ 5.69990e6 0.337612
$$779$$ 2.55172e7 1.50657
$$780$$ 0 0
$$781$$ 1.44000e7 0.844763
$$782$$ −2.48624e7 −1.45387
$$783$$ 1.92893e6 0.112438
$$784$$ −4.29850e6 −0.249762
$$785$$ 0 0
$$786$$ −6.10200e6 −0.352303
$$787$$ 556676. 0.0320380 0.0160190 0.999872i $$-0.494901\pi$$
0.0160190 + 0.999872i $$0.494901\pi$$
$$788$$ −8.36474e6 −0.479884
$$789$$ 1.67985e7 0.960678
$$790$$ 0 0
$$791$$ −330768. −0.0187967
$$792$$ −2.59200e6 −0.146832
$$793$$ −5.26752e6 −0.297456
$$794$$ 6.77379e6 0.381312
$$795$$ 0 0
$$796$$ −3.44467e6 −0.192693
$$797$$ 3.00562e6 0.167606 0.0838028 0.996482i $$-0.473293\pi$$
0.0838028 + 0.996482i $$0.473293\pi$$
$$798$$ −193536. −0.0107586
$$799$$ 1.34924e7 0.747691
$$800$$ 0 0
$$801$$ 1.83692e6 0.101160
$$802$$ −1.13800e7 −0.624751
$$803$$ 1.54040e7 0.843033
$$804$$ −9.49766e6 −0.518175
$$805$$ 0 0
$$806$$ 6.46502e6 0.350536
$$807$$ 1.23504e7 0.667570
$$808$$ 1.07468e7 0.579094
$$809$$ 2.23153e6 0.119876 0.0599378 0.998202i $$-0.480910\pi$$
0.0599378 + 0.998202i $$0.480910\pi$$
$$810$$ 0 0
$$811$$ 2.24862e7 1.20051 0.600253 0.799810i $$-0.295067\pi$$
0.600253 + 0.799810i $$0.295067\pi$$
$$812$$ 169344. 0.00901322
$$813$$ −4.12780e6 −0.219024
$$814$$ 1.45760e7 0.771041
$$815$$ 0 0
$$816$$ −3.49286e6 −0.183635
$$817$$ 3.23098e6 0.169347
$$818$$ −7.56278e6 −0.395183
$$819$$ 93312.0 0.00486102
$$820$$ 0 0
$$821$$ −1.65921e7 −0.859098 −0.429549 0.903044i $$-0.641327\pi$$
−0.429549 + 0.903044i $$0.641327\pi$$
$$822$$ −9.07330e6 −0.468366
$$823$$ 1.47544e7 0.759316 0.379658 0.925127i $$-0.376042\pi$$
0.379658 + 0.925127i $$0.376042\pi$$
$$824$$ −9.87930e6 −0.506883
$$825$$ 0 0
$$826$$ 452800. 0.0230917
$$827$$ −3.39475e6 −0.172601 −0.0863006 0.996269i $$-0.527505\pi$$
−0.0863006 + 0.996269i $$0.527505\pi$$
$$828$$ −5.31360e6 −0.269348
$$829$$ −509442. −0.0257459 −0.0128730 0.999917i $$-0.504098\pi$$
−0.0128730 + 0.999917i $$0.504098\pi$$
$$830$$ 0 0
$$831$$ 8.86867e6 0.445509
$$832$$ −1.17965e6 −0.0590805
$$833$$ −2.54552e7 −1.27105
$$834$$ −6.91258e6 −0.344132
$$835$$ 0 0
$$836$$ 1.07520e7 0.532076
$$837$$ 4.09115e6 0.201851
$$838$$ 1.84372e7 0.906953
$$839$$ 4.00609e7 1.96479 0.982394 0.186819i $$-0.0598178\pi$$
0.982394 + 0.186819i $$0.0598178\pi$$
$$840$$ 0 0
$$841$$ −1.35098e7 −0.658658
$$842$$ −2.41660e7 −1.17470
$$843$$ −1.49218e6 −0.0723191
$$844$$ −1.64813e7 −0.796407
$$845$$ 0 0
$$846$$ 2.88360e6 0.138519
$$847$$ −355796. −0.0170409
$$848$$ 1.01898e7 0.486606
$$849$$ 1.49824e7 0.713364
$$850$$ 0 0
$$851$$ 2.98808e7 1.41439
$$852$$ 4.14720e6 0.195729
$$853$$ 9.67506e6 0.455283 0.227641 0.973745i $$-0.426899\pi$$
0.227641 + 0.973745i $$0.426899\pi$$
$$854$$ −292640. −0.0137306
$$855$$ 0 0
$$856$$ 8.75443e6 0.408360
$$857$$ 3.27535e7 1.52337 0.761686 0.647946i $$-0.224372\pi$$
0.761686 + 0.647946i $$0.224372\pi$$
$$858$$ −5.18400e6 −0.240407
$$859$$ −2.17420e7 −1.00535 −0.502675 0.864476i $$-0.667651\pi$$
−0.502675 + 0.864476i $$0.667651\pi$$
$$860$$ 0 0
$$861$$ −683496. −0.0314216
$$862$$ −15200.0 −0.000696748 0
$$863$$ 2.08744e7 0.954087 0.477043 0.878880i $$-0.341708\pi$$
0.477043 + 0.878880i $$0.341708\pi$$
$$864$$ −746496. −0.0340207
$$865$$ 0 0
$$866$$ 1.00294e6 0.0454446
$$867$$ −7.90559e6 −0.357180
$$868$$ 359168. 0.0161807
$$869$$ −3.01140e7 −1.35275
$$870$$ 0 0
$$871$$ −1.89953e7 −0.848401
$$872$$ −3.44384e6 −0.153374
$$873$$ −2.99441e6 −0.132977
$$874$$ 2.20416e7 0.976033
$$875$$ 0 0
$$876$$ 4.43635e6 0.195329
$$877$$ 3.96804e7 1.74212 0.871058 0.491181i $$-0.163434\pi$$
0.871058 + 0.491181i $$0.163434\pi$$
$$878$$ −1.43549e7 −0.628441
$$879$$ −2.29594e7 −1.00228
$$880$$ 0 0
$$881$$ 2.60742e7 1.13180 0.565902 0.824472i $$-0.308528\pi$$
0.565902 + 0.824472i $$0.308528\pi$$
$$882$$ −5.44028e6 −0.235478
$$883$$ 4.10486e7 1.77172 0.885862 0.463949i $$-0.153568\pi$$
0.885862 + 0.463949i $$0.153568\pi$$
$$884$$ −6.98573e6 −0.300664
$$885$$ 0 0
$$886$$ −5.65915e6 −0.242196
$$887$$ 1.37553e7 0.587031 0.293515 0.955954i $$-0.405175\pi$$
0.293515 + 0.955954i $$0.405175\pi$$
$$888$$ 4.19789e6 0.178648
$$889$$ 847120. 0.0359493
$$890$$ 0 0
$$891$$ −3.28050e6 −0.138435
$$892$$ 7.29632e6 0.307038
$$893$$ −1.19616e7 −0.501950
$$894$$ −8.48498e6 −0.355064
$$895$$ 0 0
$$896$$ −65536.0 −0.00272716
$$897$$ −1.06272e7 −0.440999
$$898$$ −3.31922e6 −0.137355
$$899$$ 1.48494e7 0.612785
$$900$$ 0 0
$$901$$ 6.03429e7 2.47636
$$902$$ 3.79720e7 1.55399
$$903$$ −86544.0 −0.00353197
$$904$$ 5.29229e6 0.215388
$$905$$ 0 0
$$906$$ 1.33737e7 0.541292
$$907$$ −5.86936e6 −0.236904 −0.118452 0.992960i $$-0.537793\pi$$
−0.118452 + 0.992960i $$0.537793\pi$$
$$908$$ −6.94803e6 −0.279671
$$909$$ 1.36014e7 0.545975
$$910$$ 0 0
$$911$$ 4.63982e7 1.85227 0.926137 0.377188i $$-0.123109\pi$$
0.926137 + 0.377188i $$0.123109\pi$$
$$912$$ 3.09658e6 0.123281
$$913$$ 1.23400e6 0.0489935
$$914$$ 1.87279e7 0.741521
$$915$$ 0 0
$$916$$ −1.15634e7 −0.455350
$$917$$ −678000. −0.0266260
$$918$$ −4.42066e6 −0.173133
$$919$$ 2.27859e7 0.889975 0.444988 0.895537i $$-0.353208\pi$$
0.444988 + 0.895537i $$0.353208\pi$$
$$920$$ 0 0
$$921$$ −6.62418e6 −0.257326
$$922$$ −567720. −0.0219941
$$923$$ 8.29440e6 0.320465
$$924$$ −288000. −0.0110972
$$925$$ 0 0
$$926$$ 2.90990e6 0.111520
$$927$$ −1.25035e7 −0.477894
$$928$$ −2.70950e6 −0.103281
$$929$$ 2.70352e7 1.02775 0.513877 0.857864i $$-0.328209\pi$$
0.513877 + 0.857864i $$0.328209\pi$$
$$930$$ 0 0
$$931$$ 2.25671e7 0.853300
$$932$$ −9.04557e6 −0.341111
$$933$$ −1.54494e7 −0.581042
$$934$$ −1.79056e7 −0.671616
$$935$$ 0 0
$$936$$ −1.49299e6 −0.0557016
$$937$$ −2.86149e7 −1.06474 −0.532370 0.846512i $$-0.678699\pi$$
−0.532370 + 0.846512i $$0.678699\pi$$
$$938$$ −1.05530e6 −0.0391622
$$939$$ −2.55466e7 −0.945517
$$940$$ 0 0
$$941$$ −3.67892e7 −1.35440 −0.677200 0.735799i $$-0.736807\pi$$
−0.677200 + 0.735799i $$0.736807\pi$$
$$942$$ −9.53827e6 −0.350221
$$943$$ 7.78426e7 2.85061
$$944$$ −7.24480e6 −0.264604
$$945$$ 0 0
$$946$$ 4.80800e6 0.174677
$$947$$ 7.96828e6 0.288728 0.144364 0.989525i $$-0.453886\pi$$
0.144364 + 0.989525i $$0.453886\pi$$
$$948$$ −8.67283e6 −0.313430
$$949$$ 8.87270e6 0.319809
$$950$$ 0 0
$$951$$ 1.14845e7 0.411775
$$952$$ −388096. −0.0138786
$$953$$ −4.82202e7 −1.71987 −0.859937 0.510400i $$-0.829497\pi$$
−0.859937 + 0.510400i $$0.829497\pi$$
$$954$$ 1.28965e7 0.458776
$$955$$ 0 0
$$956$$ 5.19846e6 0.183963
$$957$$ −1.19070e7 −0.420264
$$958$$ −5.30874e6 −0.186886
$$959$$ −1.00814e6 −0.0353978
$$960$$ 0 0
$$961$$ 2.86539e6 0.100087
$$962$$ 8.39578e6 0.292498
$$963$$ 1.10798e7 0.385006
$$964$$ 1.46442e7 0.507543
$$965$$ 0 0
$$966$$ −590400. −0.0203565
$$967$$ 4.83510e7 1.66280 0.831398 0.555678i $$-0.187541\pi$$
0.831398 + 0.555678i $$0.187541\pi$$
$$968$$ 5.69274e6 0.195269
$$969$$ 1.83375e7 0.627381
$$970$$ 0 0
$$971$$ −4.05515e7 −1.38025 −0.690127 0.723688i $$-0.742445\pi$$
−0.690127 + 0.723688i $$0.742445\pi$$
$$972$$ −944784. −0.0320750
$$973$$ −768064. −0.0260085
$$974$$ −1.64659e7 −0.556144
$$975$$ 0 0
$$976$$ 4.68224e6 0.157336
$$977$$ −4.34929e7 −1.45775 −0.728874 0.684648i $$-0.759956\pi$$
−0.728874 + 0.684648i $$0.759956\pi$$
$$978$$ 1.45125e7 0.485170
$$979$$ −1.13390e7 −0.378110
$$980$$ 0 0
$$981$$ −4.35861e6 −0.144602
$$982$$ −2.44926e7 −0.810507
$$983$$ −3.34896e6 −0.110542 −0.0552709 0.998471i $$-0.517602\pi$$
−0.0552709 + 0.998471i $$0.517602\pi$$
$$984$$ 1.09359e7 0.360054
$$985$$ 0 0
$$986$$ −1.60453e7 −0.525602
$$987$$ 320400. 0.0104689
$$988$$ 6.19315e6 0.201846
$$989$$ 9.85640e6 0.320426
$$990$$ 0 0
$$991$$ −5.55726e7 −1.79753 −0.898766 0.438429i $$-0.855535\pi$$
−0.898766 + 0.438429i $$0.855535\pi$$
$$992$$ −5.74669e6 −0.185412
$$993$$ −3.99593e6 −0.128601
$$994$$ 460800. 0.0147927
$$995$$ 0 0
$$996$$ 355392. 0.0113517
$$997$$ −1.27342e7 −0.405726 −0.202863 0.979207i $$-0.565025\pi$$
−0.202863 + 0.979207i $$0.565025\pi$$
$$998$$ −3.16196e7 −1.00492
$$999$$ 5.31295e6 0.168431
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 150.6.a.j.1.1 1
3.2 odd 2 450.6.a.f.1.1 1
5.2 odd 4 30.6.c.a.19.2 yes 2
5.3 odd 4 30.6.c.a.19.1 2
5.4 even 2 150.6.a.f.1.1 1
15.2 even 4 90.6.c.b.19.1 2
15.8 even 4 90.6.c.b.19.2 2
15.14 odd 2 450.6.a.s.1.1 1
20.3 even 4 240.6.f.a.49.2 2
20.7 even 4 240.6.f.a.49.1 2
60.23 odd 4 720.6.f.g.289.2 2
60.47 odd 4 720.6.f.g.289.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
30.6.c.a.19.1 2 5.3 odd 4
30.6.c.a.19.2 yes 2 5.2 odd 4
90.6.c.b.19.1 2 15.2 even 4
90.6.c.b.19.2 2 15.8 even 4
150.6.a.f.1.1 1 5.4 even 2
150.6.a.j.1.1 1 1.1 even 1 trivial
240.6.f.a.49.1 2 20.7 even 4
240.6.f.a.49.2 2 20.3 even 4
450.6.a.f.1.1 1 3.2 odd 2
450.6.a.s.1.1 1 15.14 odd 2
720.6.f.g.289.1 2 60.47 odd 4
720.6.f.g.289.2 2 60.23 odd 4