# Properties

 Label 825.6.a.a.1.1 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} +9.00000 q^{6} +26.0000 q^{7} +63.0000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} +9.00000 q^{6} +26.0000 q^{7} +63.0000 q^{8} +81.0000 q^{9} +121.000 q^{11} +279.000 q^{12} +692.000 q^{13} -26.0000 q^{14} +929.000 q^{16} +1442.00 q^{17} -81.0000 q^{18} +2160.00 q^{19} -234.000 q^{21} -121.000 q^{22} +1582.00 q^{23} -567.000 q^{24} -692.000 q^{26} -729.000 q^{27} -806.000 q^{28} -5526.00 q^{29} +4792.00 q^{31} -2945.00 q^{32} -1089.00 q^{33} -1442.00 q^{34} -2511.00 q^{36} +10194.0 q^{37} -2160.00 q^{38} -6228.00 q^{39} -10622.0 q^{41} +234.000 q^{42} -8580.00 q^{43} -3751.00 q^{44} -1582.00 q^{46} +2362.00 q^{47} -8361.00 q^{48} -16131.0 q^{49} -12978.0 q^{51} -21452.0 q^{52} +30804.0 q^{53} +729.000 q^{54} +1638.00 q^{56} -19440.0 q^{57} +5526.00 q^{58} +6416.00 q^{59} +42096.0 q^{61} -4792.00 q^{62} +2106.00 q^{63} -26783.0 q^{64} +1089.00 q^{66} +28444.0 q^{67} -44702.0 q^{68} -14238.0 q^{69} +45690.0 q^{71} +5103.00 q^{72} +18374.0 q^{73} -10194.0 q^{74} -66960.0 q^{76} +3146.00 q^{77} +6228.00 q^{78} -105214. q^{79} +6561.00 q^{81} +10622.0 q^{82} -62292.0 q^{83} +7254.00 q^{84} +8580.00 q^{86} +49734.0 q^{87} +7623.00 q^{88} -72246.0 q^{89} +17992.0 q^{91} -49042.0 q^{92} -43128.0 q^{93} -2362.00 q^{94} +26505.0 q^{96} -79262.0 q^{97} +16131.0 q^{98} +9801.00 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.176777 −0.0883883 0.996086i $$-0.528172\pi$$
−0.0883883 + 0.996086i $$0.528172\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ −31.0000 −0.968750
$$5$$ 0 0
$$6$$ 9.00000 0.102062
$$7$$ 26.0000 0.200553 0.100276 0.994960i $$-0.468027\pi$$
0.100276 + 0.994960i $$0.468027\pi$$
$$8$$ 63.0000 0.348029
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ 121.000 0.301511
$$12$$ 279.000 0.559308
$$13$$ 692.000 1.13566 0.567829 0.823146i $$-0.307783\pi$$
0.567829 + 0.823146i $$0.307783\pi$$
$$14$$ −26.0000 −0.0354530
$$15$$ 0 0
$$16$$ 929.000 0.907227
$$17$$ 1442.00 1.21016 0.605080 0.796165i $$-0.293141\pi$$
0.605080 + 0.796165i $$0.293141\pi$$
$$18$$ −81.0000 −0.0589256
$$19$$ 2160.00 1.37268 0.686341 0.727280i $$-0.259216\pi$$
0.686341 + 0.727280i $$0.259216\pi$$
$$20$$ 0 0
$$21$$ −234.000 −0.115789
$$22$$ −121.000 −0.0533002
$$23$$ 1582.00 0.623572 0.311786 0.950152i $$-0.399073\pi$$
0.311786 + 0.950152i $$0.399073\pi$$
$$24$$ −567.000 −0.200935
$$25$$ 0 0
$$26$$ −692.000 −0.200758
$$27$$ −729.000 −0.192450
$$28$$ −806.000 −0.194285
$$29$$ −5526.00 −1.22016 −0.610079 0.792341i $$-0.708862\pi$$
−0.610079 + 0.792341i $$0.708862\pi$$
$$30$$ 0 0
$$31$$ 4792.00 0.895597 0.447798 0.894135i $$-0.352208\pi$$
0.447798 + 0.894135i $$0.352208\pi$$
$$32$$ −2945.00 −0.508406
$$33$$ −1089.00 −0.174078
$$34$$ −1442.00 −0.213928
$$35$$ 0 0
$$36$$ −2511.00 −0.322917
$$37$$ 10194.0 1.22417 0.612083 0.790794i $$-0.290332\pi$$
0.612083 + 0.790794i $$0.290332\pi$$
$$38$$ −2160.00 −0.242658
$$39$$ −6228.00 −0.655673
$$40$$ 0 0
$$41$$ −10622.0 −0.986840 −0.493420 0.869791i $$-0.664253\pi$$
−0.493420 + 0.869791i $$0.664253\pi$$
$$42$$ 234.000 0.0204688
$$43$$ −8580.00 −0.707646 −0.353823 0.935312i $$-0.615119\pi$$
−0.353823 + 0.935312i $$0.615119\pi$$
$$44$$ −3751.00 −0.292089
$$45$$ 0 0
$$46$$ −1582.00 −0.110233
$$47$$ 2362.00 0.155968 0.0779840 0.996955i $$-0.475152\pi$$
0.0779840 + 0.996955i $$0.475152\pi$$
$$48$$ −8361.00 −0.523788
$$49$$ −16131.0 −0.959779
$$50$$ 0 0
$$51$$ −12978.0 −0.698686
$$52$$ −21452.0 −1.10017
$$53$$ 30804.0 1.50632 0.753160 0.657837i $$-0.228528\pi$$
0.753160 + 0.657837i $$0.228528\pi$$
$$54$$ 729.000 0.0340207
$$55$$ 0 0
$$56$$ 1638.00 0.0697981
$$57$$ −19440.0 −0.792518
$$58$$ 5526.00 0.215695
$$59$$ 6416.00 0.239957 0.119979 0.992776i $$-0.461717\pi$$
0.119979 + 0.992776i $$0.461717\pi$$
$$60$$ 0 0
$$61$$ 42096.0 1.44849 0.724246 0.689541i $$-0.242188\pi$$
0.724246 + 0.689541i $$0.242188\pi$$
$$62$$ −4792.00 −0.158321
$$63$$ 2106.00 0.0668509
$$64$$ −26783.0 −0.817352
$$65$$ 0 0
$$66$$ 1089.00 0.0307729
$$67$$ 28444.0 0.774112 0.387056 0.922056i $$-0.373492\pi$$
0.387056 + 0.922056i $$0.373492\pi$$
$$68$$ −44702.0 −1.17234
$$69$$ −14238.0 −0.360020
$$70$$ 0 0
$$71$$ 45690.0 1.07566 0.537830 0.843053i $$-0.319244\pi$$
0.537830 + 0.843053i $$0.319244\pi$$
$$72$$ 5103.00 0.116010
$$73$$ 18374.0 0.403549 0.201775 0.979432i $$-0.435329\pi$$
0.201775 + 0.979432i $$0.435329\pi$$
$$74$$ −10194.0 −0.216404
$$75$$ 0 0
$$76$$ −66960.0 −1.32979
$$77$$ 3146.00 0.0604689
$$78$$ 6228.00 0.115908
$$79$$ −105214. −1.89673 −0.948366 0.317179i $$-0.897264\pi$$
−0.948366 + 0.317179i $$0.897264\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 10622.0 0.174450
$$83$$ −62292.0 −0.992515 −0.496257 0.868175i $$-0.665293\pi$$
−0.496257 + 0.868175i $$0.665293\pi$$
$$84$$ 7254.00 0.112171
$$85$$ 0 0
$$86$$ 8580.00 0.125095
$$87$$ 49734.0 0.704458
$$88$$ 7623.00 0.104935
$$89$$ −72246.0 −0.966805 −0.483402 0.875398i $$-0.660599\pi$$
−0.483402 + 0.875398i $$0.660599\pi$$
$$90$$ 0 0
$$91$$ 17992.0 0.227759
$$92$$ −49042.0 −0.604086
$$93$$ −43128.0 −0.517073
$$94$$ −2362.00 −0.0275715
$$95$$ 0 0
$$96$$ 26505.0 0.293528
$$97$$ −79262.0 −0.855334 −0.427667 0.903936i $$-0.640664\pi$$
−0.427667 + 0.903936i $$0.640664\pi$$
$$98$$ 16131.0 0.169667
$$99$$ 9801.00 0.100504
$$100$$ 0 0
$$101$$ −24958.0 −0.243448 −0.121724 0.992564i $$-0.538842\pi$$
−0.121724 + 0.992564i $$0.538842\pi$$
$$102$$ 12978.0 0.123511
$$103$$ 56812.0 0.527651 0.263826 0.964570i $$-0.415016\pi$$
0.263826 + 0.964570i $$0.415016\pi$$
$$104$$ 43596.0 0.395242
$$105$$ 0 0
$$106$$ −30804.0 −0.266282
$$107$$ 12492.0 0.105481 0.0527403 0.998608i $$-0.483204\pi$$
0.0527403 + 0.998608i $$0.483204\pi$$
$$108$$ 22599.0 0.186436
$$109$$ 198748. 1.60227 0.801137 0.598482i $$-0.204229\pi$$
0.801137 + 0.598482i $$0.204229\pi$$
$$110$$ 0 0
$$111$$ −91746.0 −0.706773
$$112$$ 24154.0 0.181947
$$113$$ −166554. −1.22704 −0.613520 0.789679i $$-0.710247\pi$$
−0.613520 + 0.789679i $$0.710247\pi$$
$$114$$ 19440.0 0.140099
$$115$$ 0 0
$$116$$ 171306. 1.18203
$$117$$ 56052.0 0.378553
$$118$$ −6416.00 −0.0424189
$$119$$ 37492.0 0.242701
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ −42096.0 −0.256060
$$123$$ 95598.0 0.569752
$$124$$ −148552. −0.867609
$$125$$ 0 0
$$126$$ −2106.00 −0.0118177
$$127$$ −304226. −1.67374 −0.836868 0.547405i $$-0.815616\pi$$
−0.836868 + 0.547405i $$0.815616\pi$$
$$128$$ 121023. 0.652894
$$129$$ 77220.0 0.408560
$$130$$ 0 0
$$131$$ 274428. 1.39717 0.698586 0.715526i $$-0.253813\pi$$
0.698586 + 0.715526i $$0.253813\pi$$
$$132$$ 33759.0 0.168638
$$133$$ 56160.0 0.275295
$$134$$ −28444.0 −0.136845
$$135$$ 0 0
$$136$$ 90846.0 0.421171
$$137$$ 245458. 1.11732 0.558658 0.829398i $$-0.311317\pi$$
0.558658 + 0.829398i $$0.311317\pi$$
$$138$$ 14238.0 0.0636431
$$139$$ −59888.0 −0.262907 −0.131454 0.991322i $$-0.541964\pi$$
−0.131454 + 0.991322i $$0.541964\pi$$
$$140$$ 0 0
$$141$$ −21258.0 −0.0900481
$$142$$ −45690.0 −0.190152
$$143$$ 83732.0 0.342414
$$144$$ 75249.0 0.302409
$$145$$ 0 0
$$146$$ −18374.0 −0.0713381
$$147$$ 145179. 0.554128
$$148$$ −316014. −1.18591
$$149$$ 72038.0 0.265825 0.132913 0.991128i $$-0.457567\pi$$
0.132913 + 0.991128i $$0.457567\pi$$
$$150$$ 0 0
$$151$$ −323110. −1.15321 −0.576605 0.817023i $$-0.695623\pi$$
−0.576605 + 0.817023i $$0.695623\pi$$
$$152$$ 136080. 0.477733
$$153$$ 116802. 0.403387
$$154$$ −3146.00 −0.0106895
$$155$$ 0 0
$$156$$ 193068. 0.635183
$$157$$ 318766. 1.03210 0.516051 0.856558i $$-0.327401\pi$$
0.516051 + 0.856558i $$0.327401\pi$$
$$158$$ 105214. 0.335298
$$159$$ −277236. −0.869675
$$160$$ 0 0
$$161$$ 41132.0 0.125059
$$162$$ −6561.00 −0.0196419
$$163$$ 431996. 1.27353 0.636767 0.771056i $$-0.280271\pi$$
0.636767 + 0.771056i $$0.280271\pi$$
$$164$$ 329282. 0.956001
$$165$$ 0 0
$$166$$ 62292.0 0.175454
$$167$$ 251580. 0.698047 0.349024 0.937114i $$-0.386513\pi$$
0.349024 + 0.937114i $$0.386513\pi$$
$$168$$ −14742.0 −0.0402980
$$169$$ 107571. 0.289720
$$170$$ 0 0
$$171$$ 174960. 0.457560
$$172$$ 265980. 0.685532
$$173$$ −476634. −1.21079 −0.605396 0.795924i $$-0.706985\pi$$
−0.605396 + 0.795924i $$0.706985\pi$$
$$174$$ −49734.0 −0.124532
$$175$$ 0 0
$$176$$ 112409. 0.273539
$$177$$ −57744.0 −0.138540
$$178$$ 72246.0 0.170909
$$179$$ 90192.0 0.210395 0.105198 0.994451i $$-0.466453\pi$$
0.105198 + 0.994451i $$0.466453\pi$$
$$180$$ 0 0
$$181$$ 248002. 0.562676 0.281338 0.959609i $$-0.409222\pi$$
0.281338 + 0.959609i $$0.409222\pi$$
$$182$$ −17992.0 −0.0402625
$$183$$ −378864. −0.836288
$$184$$ 99666.0 0.217021
$$185$$ 0 0
$$186$$ 43128.0 0.0914065
$$187$$ 174482. 0.364877
$$188$$ −73222.0 −0.151094
$$189$$ −18954.0 −0.0385964
$$190$$ 0 0
$$191$$ −156802. −0.311006 −0.155503 0.987835i $$-0.549700\pi$$
−0.155503 + 0.987835i $$0.549700\pi$$
$$192$$ 241047. 0.471899
$$193$$ 431234. 0.833335 0.416668 0.909059i $$-0.363198\pi$$
0.416668 + 0.909059i $$0.363198\pi$$
$$194$$ 79262.0 0.151203
$$195$$ 0 0
$$196$$ 500061. 0.929786
$$197$$ 864974. 1.58795 0.793976 0.607949i $$-0.208007\pi$$
0.793976 + 0.607949i $$0.208007\pi$$
$$198$$ −9801.00 −0.0177667
$$199$$ −480060. −0.859336 −0.429668 0.902987i $$-0.641369\pi$$
−0.429668 + 0.902987i $$0.641369\pi$$
$$200$$ 0 0
$$201$$ −255996. −0.446934
$$202$$ 24958.0 0.0430359
$$203$$ −143676. −0.244706
$$204$$ 402318. 0.676853
$$205$$ 0 0
$$206$$ −56812.0 −0.0932765
$$207$$ 128142. 0.207857
$$208$$ 642868. 1.03030
$$209$$ 261360. 0.413879
$$210$$ 0 0
$$211$$ 525900. 0.813199 0.406600 0.913606i $$-0.366714\pi$$
0.406600 + 0.913606i $$0.366714\pi$$
$$212$$ −954924. −1.45925
$$213$$ −411210. −0.621033
$$214$$ −12492.0 −0.0186465
$$215$$ 0 0
$$216$$ −45927.0 −0.0669782
$$217$$ 124592. 0.179614
$$218$$ −198748. −0.283245
$$219$$ −165366. −0.232989
$$220$$ 0 0
$$221$$ 997864. 1.37433
$$222$$ 91746.0 0.124941
$$223$$ 245264. 0.330272 0.165136 0.986271i $$-0.447194\pi$$
0.165136 + 0.986271i $$0.447194\pi$$
$$224$$ −76570.0 −0.101962
$$225$$ 0 0
$$226$$ 166554. 0.216912
$$227$$ 799308. 1.02955 0.514777 0.857324i $$-0.327875\pi$$
0.514777 + 0.857324i $$0.327875\pi$$
$$228$$ 602640. 0.767752
$$229$$ −1.53989e6 −1.94045 −0.970224 0.242208i $$-0.922128\pi$$
−0.970224 + 0.242208i $$0.922128\pi$$
$$230$$ 0 0
$$231$$ −28314.0 −0.0349117
$$232$$ −348138. −0.424650
$$233$$ 721830. 0.871054 0.435527 0.900176i $$-0.356562\pi$$
0.435527 + 0.900176i $$0.356562\pi$$
$$234$$ −56052.0 −0.0669193
$$235$$ 0 0
$$236$$ −198896. −0.232459
$$237$$ 946926. 1.09508
$$238$$ −37492.0 −0.0429038
$$239$$ −638436. −0.722974 −0.361487 0.932377i $$-0.617731\pi$$
−0.361487 + 0.932377i $$0.617731\pi$$
$$240$$ 0 0
$$241$$ 220990. 0.245092 0.122546 0.992463i $$-0.460894\pi$$
0.122546 + 0.992463i $$0.460894\pi$$
$$242$$ −14641.0 −0.0160706
$$243$$ −59049.0 −0.0641500
$$244$$ −1.30498e6 −1.40323
$$245$$ 0 0
$$246$$ −95598.0 −0.100719
$$247$$ 1.49472e6 1.55890
$$248$$ 301896. 0.311694
$$249$$ 560628. 0.573029
$$250$$ 0 0
$$251$$ 627304. 0.628483 0.314242 0.949343i $$-0.398250\pi$$
0.314242 + 0.949343i $$0.398250\pi$$
$$252$$ −65286.0 −0.0647618
$$253$$ 191422. 0.188014
$$254$$ 304226. 0.295878
$$255$$ 0 0
$$256$$ 736033. 0.701936
$$257$$ 468014. 0.442004 0.221002 0.975273i $$-0.429067\pi$$
0.221002 + 0.975273i $$0.429067\pi$$
$$258$$ −77220.0 −0.0722238
$$259$$ 265044. 0.245510
$$260$$ 0 0
$$261$$ −447606. −0.406719
$$262$$ −274428. −0.246988
$$263$$ −1.54510e6 −1.37743 −0.688713 0.725034i $$-0.741824\pi$$
−0.688713 + 0.725034i $$0.741824\pi$$
$$264$$ −68607.0 −0.0605841
$$265$$ 0 0
$$266$$ −56160.0 −0.0486657
$$267$$ 650214. 0.558185
$$268$$ −881764. −0.749921
$$269$$ −1.07457e6 −0.905430 −0.452715 0.891655i $$-0.649544\pi$$
−0.452715 + 0.891655i $$0.649544\pi$$
$$270$$ 0 0
$$271$$ 1.58723e6 1.31285 0.656427 0.754389i $$-0.272067\pi$$
0.656427 + 0.754389i $$0.272067\pi$$
$$272$$ 1.33962e6 1.09789
$$273$$ −161928. −0.131497
$$274$$ −245458. −0.197515
$$275$$ 0 0
$$276$$ 441378. 0.348769
$$277$$ −692704. −0.542436 −0.271218 0.962518i $$-0.587426\pi$$
−0.271218 + 0.962518i $$0.587426\pi$$
$$278$$ 59888.0 0.0464759
$$279$$ 388152. 0.298532
$$280$$ 0 0
$$281$$ −567018. −0.428382 −0.214191 0.976792i $$-0.568711\pi$$
−0.214191 + 0.976792i $$0.568711\pi$$
$$282$$ 21258.0 0.0159184
$$283$$ −714916. −0.530626 −0.265313 0.964162i $$-0.585475\pi$$
−0.265313 + 0.964162i $$0.585475\pi$$
$$284$$ −1.41639e6 −1.04205
$$285$$ 0 0
$$286$$ −83732.0 −0.0605308
$$287$$ −276172. −0.197913
$$288$$ −238545. −0.169469
$$289$$ 659507. 0.464488
$$290$$ 0 0
$$291$$ 713358. 0.493827
$$292$$ −569594. −0.390938
$$293$$ −2.14409e6 −1.45907 −0.729533 0.683946i $$-0.760262\pi$$
−0.729533 + 0.683946i $$0.760262\pi$$
$$294$$ −145179. −0.0979570
$$295$$ 0 0
$$296$$ 642222. 0.426045
$$297$$ −88209.0 −0.0580259
$$298$$ −72038.0 −0.0469917
$$299$$ 1.09474e6 0.708165
$$300$$ 0 0
$$301$$ −223080. −0.141920
$$302$$ 323110. 0.203860
$$303$$ 224622. 0.140555
$$304$$ 2.00664e6 1.24533
$$305$$ 0 0
$$306$$ −116802. −0.0713094
$$307$$ 588808. 0.356556 0.178278 0.983980i $$-0.442947\pi$$
0.178278 + 0.983980i $$0.442947\pi$$
$$308$$ −97526.0 −0.0585792
$$309$$ −511308. −0.304640
$$310$$ 0 0
$$311$$ 2.51827e6 1.47639 0.738194 0.674588i $$-0.235679\pi$$
0.738194 + 0.674588i $$0.235679\pi$$
$$312$$ −392364. −0.228193
$$313$$ 2.23562e6 1.28985 0.644923 0.764248i $$-0.276890\pi$$
0.644923 + 0.764248i $$0.276890\pi$$
$$314$$ −318766. −0.182452
$$315$$ 0 0
$$316$$ 3.26163e6 1.83746
$$317$$ −1.06079e6 −0.592901 −0.296450 0.955048i $$-0.595803\pi$$
−0.296450 + 0.955048i $$0.595803\pi$$
$$318$$ 277236. 0.153738
$$319$$ −668646. −0.367891
$$320$$ 0 0
$$321$$ −112428. −0.0608992
$$322$$ −41132.0 −0.0221075
$$323$$ 3.11472e6 1.66116
$$324$$ −203391. −0.107639
$$325$$ 0 0
$$326$$ −431996. −0.225131
$$327$$ −1.78873e6 −0.925073
$$328$$ −669186. −0.343449
$$329$$ 61412.0 0.0312798
$$330$$ 0 0
$$331$$ −2.34566e6 −1.17678 −0.588390 0.808577i $$-0.700238\pi$$
−0.588390 + 0.808577i $$0.700238\pi$$
$$332$$ 1.93105e6 0.961499
$$333$$ 825714. 0.408055
$$334$$ −251580. −0.123399
$$335$$ 0 0
$$336$$ −217386. −0.105047
$$337$$ −839978. −0.402896 −0.201448 0.979499i $$-0.564565\pi$$
−0.201448 + 0.979499i $$0.564565\pi$$
$$338$$ −107571. −0.0512157
$$339$$ 1.49899e6 0.708432
$$340$$ 0 0
$$341$$ 579832. 0.270033
$$342$$ −174960. −0.0808860
$$343$$ −856388. −0.393039
$$344$$ −540540. −0.246281
$$345$$ 0 0
$$346$$ 476634. 0.214040
$$347$$ 2.02560e6 0.903086 0.451543 0.892249i $$-0.350874\pi$$
0.451543 + 0.892249i $$0.350874\pi$$
$$348$$ −1.54175e6 −0.682444
$$349$$ −378924. −0.166528 −0.0832642 0.996528i $$-0.526535\pi$$
−0.0832642 + 0.996528i $$0.526535\pi$$
$$350$$ 0 0
$$351$$ −504468. −0.218558
$$352$$ −356345. −0.153290
$$353$$ 1.98730e6 0.848842 0.424421 0.905465i $$-0.360478\pi$$
0.424421 + 0.905465i $$0.360478\pi$$
$$354$$ 57744.0 0.0244906
$$355$$ 0 0
$$356$$ 2.23963e6 0.936592
$$357$$ −337428. −0.140123
$$358$$ −90192.0 −0.0371929
$$359$$ −3.43975e6 −1.40861 −0.704305 0.709898i $$-0.748741\pi$$
−0.704305 + 0.709898i $$0.748741\pi$$
$$360$$ 0 0
$$361$$ 2.18950e6 0.884254
$$362$$ −248002. −0.0994681
$$363$$ −131769. −0.0524864
$$364$$ −557752. −0.220642
$$365$$ 0 0
$$366$$ 378864. 0.147836
$$367$$ 1.79679e6 0.696358 0.348179 0.937428i $$-0.386800\pi$$
0.348179 + 0.937428i $$0.386800\pi$$
$$368$$ 1.46968e6 0.565721
$$369$$ −860382. −0.328947
$$370$$ 0 0
$$371$$ 800904. 0.302096
$$372$$ 1.33697e6 0.500915
$$373$$ 1.43541e6 0.534201 0.267100 0.963669i $$-0.413934\pi$$
0.267100 + 0.963669i $$0.413934\pi$$
$$374$$ −174482. −0.0645018
$$375$$ 0 0
$$376$$ 148806. 0.0542814
$$377$$ −3.82399e6 −1.38568
$$378$$ 18954.0 0.00682294
$$379$$ 2.66235e6 0.952065 0.476033 0.879428i $$-0.342074\pi$$
0.476033 + 0.879428i $$0.342074\pi$$
$$380$$ 0 0
$$381$$ 2.73803e6 0.966332
$$382$$ 156802. 0.0549785
$$383$$ −2.04091e6 −0.710932 −0.355466 0.934689i $$-0.615678\pi$$
−0.355466 + 0.934689i $$0.615678\pi$$
$$384$$ −1.08921e6 −0.376949
$$385$$ 0 0
$$386$$ −431234. −0.147314
$$387$$ −694980. −0.235882
$$388$$ 2.45712e6 0.828605
$$389$$ −4.29947e6 −1.44059 −0.720296 0.693667i $$-0.755994\pi$$
−0.720296 + 0.693667i $$0.755994\pi$$
$$390$$ 0 0
$$391$$ 2.28124e6 0.754623
$$392$$ −1.01625e6 −0.334031
$$393$$ −2.46985e6 −0.806658
$$394$$ −864974. −0.280713
$$395$$ 0 0
$$396$$ −303831. −0.0973630
$$397$$ −728818. −0.232083 −0.116041 0.993244i $$-0.537021\pi$$
−0.116041 + 0.993244i $$0.537021\pi$$
$$398$$ 480060. 0.151911
$$399$$ −505440. −0.158942
$$400$$ 0 0
$$401$$ −5.92515e6 −1.84009 −0.920044 0.391814i $$-0.871848\pi$$
−0.920044 + 0.391814i $$0.871848\pi$$
$$402$$ 255996. 0.0790075
$$403$$ 3.31606e6 1.01709
$$404$$ 773698. 0.235840
$$405$$ 0 0
$$406$$ 143676. 0.0432583
$$407$$ 1.23347e6 0.369100
$$408$$ −817614. −0.243163
$$409$$ 1.38212e6 0.408542 0.204271 0.978914i $$-0.434518\pi$$
0.204271 + 0.978914i $$0.434518\pi$$
$$410$$ 0 0
$$411$$ −2.20912e6 −0.645082
$$412$$ −1.76117e6 −0.511162
$$413$$ 166816. 0.0481241
$$414$$ −128142. −0.0367444
$$415$$ 0 0
$$416$$ −2.03794e6 −0.577375
$$417$$ 538992. 0.151790
$$418$$ −261360. −0.0731642
$$419$$ 5.47794e6 1.52434 0.762170 0.647377i $$-0.224134\pi$$
0.762170 + 0.647377i $$0.224134\pi$$
$$420$$ 0 0
$$421$$ 1.02873e6 0.282877 0.141439 0.989947i $$-0.454827\pi$$
0.141439 + 0.989947i $$0.454827\pi$$
$$422$$ −525900. −0.143755
$$423$$ 191322. 0.0519893
$$424$$ 1.94065e6 0.524243
$$425$$ 0 0
$$426$$ 411210. 0.109784
$$427$$ 1.09450e6 0.290499
$$428$$ −387252. −0.102184
$$429$$ −753588. −0.197693
$$430$$ 0 0
$$431$$ −5.14310e6 −1.33362 −0.666810 0.745228i $$-0.732341\pi$$
−0.666810 + 0.745228i $$0.732341\pi$$
$$432$$ −677241. −0.174596
$$433$$ −412954. −0.105848 −0.0529239 0.998599i $$-0.516854\pi$$
−0.0529239 + 0.998599i $$0.516854\pi$$
$$434$$ −124592. −0.0317516
$$435$$ 0 0
$$436$$ −6.16119e6 −1.55220
$$437$$ 3.41712e6 0.855966
$$438$$ 165366. 0.0411871
$$439$$ 5.96365e6 1.47690 0.738450 0.674309i $$-0.235558\pi$$
0.738450 + 0.674309i $$0.235558\pi$$
$$440$$ 0 0
$$441$$ −1.30661e6 −0.319926
$$442$$ −997864. −0.242949
$$443$$ −2.18433e6 −0.528821 −0.264410 0.964410i $$-0.585177\pi$$
−0.264410 + 0.964410i $$0.585177\pi$$
$$444$$ 2.84413e6 0.684686
$$445$$ 0 0
$$446$$ −245264. −0.0583844
$$447$$ −648342. −0.153474
$$448$$ −696358. −0.163922
$$449$$ −7858.00 −0.00183948 −0.000919742 1.00000i $$-0.500293\pi$$
−0.000919742 1.00000i $$0.500293\pi$$
$$450$$ 0 0
$$451$$ −1.28526e6 −0.297543
$$452$$ 5.16317e6 1.18870
$$453$$ 2.90799e6 0.665806
$$454$$ −799308. −0.182001
$$455$$ 0 0
$$456$$ −1.22472e6 −0.275819
$$457$$ 899922. 0.201565 0.100782 0.994908i $$-0.467865\pi$$
0.100782 + 0.994908i $$0.467865\pi$$
$$458$$ 1.53989e6 0.343026
$$459$$ −1.05122e6 −0.232895
$$460$$ 0 0
$$461$$ 1.13619e6 0.249000 0.124500 0.992220i $$-0.460267\pi$$
0.124500 + 0.992220i $$0.460267\pi$$
$$462$$ 28314.0 0.00617158
$$463$$ 7.38964e6 1.60203 0.801016 0.598643i $$-0.204293\pi$$
0.801016 + 0.598643i $$0.204293\pi$$
$$464$$ −5.13365e6 −1.10696
$$465$$ 0 0
$$466$$ −721830. −0.153982
$$467$$ −4.20851e6 −0.892968 −0.446484 0.894792i $$-0.647324\pi$$
−0.446484 + 0.894792i $$0.647324\pi$$
$$468$$ −1.73761e6 −0.366723
$$469$$ 739544. 0.155250
$$470$$ 0 0
$$471$$ −2.86889e6 −0.595885
$$472$$ 404208. 0.0835122
$$473$$ −1.03818e6 −0.213363
$$474$$ −946926. −0.193584
$$475$$ 0 0
$$476$$ −1.16225e6 −0.235116
$$477$$ 2.49512e6 0.502107
$$478$$ 638436. 0.127805
$$479$$ 7.39441e6 1.47253 0.736266 0.676692i $$-0.236587\pi$$
0.736266 + 0.676692i $$0.236587\pi$$
$$480$$ 0 0
$$481$$ 7.05425e6 1.39023
$$482$$ −220990. −0.0433266
$$483$$ −370188. −0.0722029
$$484$$ −453871. −0.0880682
$$485$$ 0 0
$$486$$ 59049.0 0.0113402
$$487$$ 3.81644e6 0.729181 0.364591 0.931168i $$-0.381209\pi$$
0.364591 + 0.931168i $$0.381209\pi$$
$$488$$ 2.65205e6 0.504118
$$489$$ −3.88796e6 −0.735275
$$490$$ 0 0
$$491$$ 1.69716e6 0.317702 0.158851 0.987303i $$-0.449221\pi$$
0.158851 + 0.987303i $$0.449221\pi$$
$$492$$ −2.96354e6 −0.551947
$$493$$ −7.96849e6 −1.47659
$$494$$ −1.49472e6 −0.275577
$$495$$ 0 0
$$496$$ 4.45177e6 0.812509
$$497$$ 1.18794e6 0.215727
$$498$$ −560628. −0.101298
$$499$$ 6.95160e6 1.24978 0.624889 0.780713i $$-0.285144\pi$$
0.624889 + 0.780713i $$0.285144\pi$$
$$500$$ 0 0
$$501$$ −2.26422e6 −0.403018
$$502$$ −627304. −0.111101
$$503$$ −6.01023e6 −1.05918 −0.529591 0.848253i $$-0.677655\pi$$
−0.529591 + 0.848253i $$0.677655\pi$$
$$504$$ 132678. 0.0232660
$$505$$ 0 0
$$506$$ −191422. −0.0332365
$$507$$ −968139. −0.167270
$$508$$ 9.43101e6 1.62143
$$509$$ 624660. 0.106868 0.0534342 0.998571i $$-0.482983\pi$$
0.0534342 + 0.998571i $$0.482983\pi$$
$$510$$ 0 0
$$511$$ 477724. 0.0809328
$$512$$ −4.60877e6 −0.776980
$$513$$ −1.57464e6 −0.264173
$$514$$ −468014. −0.0781360
$$515$$ 0 0
$$516$$ −2.39382e6 −0.395792
$$517$$ 285802. 0.0470261
$$518$$ −265044. −0.0434004
$$519$$ 4.28971e6 0.699051
$$520$$ 0 0
$$521$$ −647490. −0.104505 −0.0522527 0.998634i $$-0.516640\pi$$
−0.0522527 + 0.998634i $$0.516640\pi$$
$$522$$ 447606. 0.0718985
$$523$$ 114676. 0.0183324 0.00916618 0.999958i $$-0.497082\pi$$
0.00916618 + 0.999958i $$0.497082\pi$$
$$524$$ −8.50727e6 −1.35351
$$525$$ 0 0
$$526$$ 1.54510e6 0.243497
$$527$$ 6.91006e6 1.08382
$$528$$ −1.01168e6 −0.157928
$$529$$ −3.93362e6 −0.611157
$$530$$ 0 0
$$531$$ 519696. 0.0799858
$$532$$ −1.74096e6 −0.266692
$$533$$ −7.35042e6 −1.12071
$$534$$ −650214. −0.0986741
$$535$$ 0 0
$$536$$ 1.79197e6 0.269413
$$537$$ −811728. −0.121472
$$538$$ 1.07457e6 0.160059
$$539$$ −1.95185e6 −0.289384
$$540$$ 0 0
$$541$$ −2.12404e6 −0.312011 −0.156006 0.987756i $$-0.549862\pi$$
−0.156006 + 0.987756i $$0.549862\pi$$
$$542$$ −1.58723e6 −0.232082
$$543$$ −2.23202e6 −0.324861
$$544$$ −4.24669e6 −0.615252
$$545$$ 0 0
$$546$$ 161928. 0.0232456
$$547$$ −1.22672e7 −1.75299 −0.876494 0.481413i $$-0.840124\pi$$
−0.876494 + 0.481413i $$0.840124\pi$$
$$548$$ −7.60920e6 −1.08240
$$549$$ 3.40978e6 0.482831
$$550$$ 0 0
$$551$$ −1.19362e7 −1.67489
$$552$$ −896994. −0.125297
$$553$$ −2.73556e6 −0.380394
$$554$$ 692704. 0.0958900
$$555$$ 0 0
$$556$$ 1.85653e6 0.254692
$$557$$ −1.10980e7 −1.51568 −0.757839 0.652442i $$-0.773745\pi$$
−0.757839 + 0.652442i $$0.773745\pi$$
$$558$$ −388152. −0.0527736
$$559$$ −5.93736e6 −0.803644
$$560$$ 0 0
$$561$$ −1.57034e6 −0.210662
$$562$$ 567018. 0.0757279
$$563$$ −4.61984e6 −0.614265 −0.307132 0.951667i $$-0.599369\pi$$
−0.307132 + 0.951667i $$0.599369\pi$$
$$564$$ 658998. 0.0872341
$$565$$ 0 0
$$566$$ 714916. 0.0938024
$$567$$ 170586. 0.0222836
$$568$$ 2.87847e6 0.374361
$$569$$ 1.01716e7 1.31707 0.658537 0.752548i $$-0.271176\pi$$
0.658537 + 0.752548i $$0.271176\pi$$
$$570$$ 0 0
$$571$$ −9.36866e6 −1.20251 −0.601253 0.799059i $$-0.705331\pi$$
−0.601253 + 0.799059i $$0.705331\pi$$
$$572$$ −2.59569e6 −0.331713
$$573$$ 1.41122e6 0.179559
$$574$$ 276172. 0.0349865
$$575$$ 0 0
$$576$$ −2.16942e6 −0.272451
$$577$$ 6.14973e6 0.768983 0.384491 0.923129i $$-0.374377\pi$$
0.384491 + 0.923129i $$0.374377\pi$$
$$578$$ −659507. −0.0821107
$$579$$ −3.88111e6 −0.481126
$$580$$ 0 0
$$581$$ −1.61959e6 −0.199051
$$582$$ −713358. −0.0872972
$$583$$ 3.72728e6 0.454173
$$584$$ 1.15756e6 0.140447
$$585$$ 0 0
$$586$$ 2.14409e6 0.257929
$$587$$ −1.04649e6 −0.125354 −0.0626771 0.998034i $$-0.519964\pi$$
−0.0626771 + 0.998034i $$0.519964\pi$$
$$588$$ −4.50055e6 −0.536812
$$589$$ 1.03507e7 1.22937
$$590$$ 0 0
$$591$$ −7.78477e6 −0.916805
$$592$$ 9.47023e6 1.11060
$$593$$ 3.31784e6 0.387453 0.193726 0.981056i $$-0.437943\pi$$
0.193726 + 0.981056i $$0.437943\pi$$
$$594$$ 88209.0 0.0102576
$$595$$ 0 0
$$596$$ −2.23318e6 −0.257518
$$597$$ 4.32054e6 0.496138
$$598$$ −1.09474e6 −0.125187
$$599$$ −1.73991e7 −1.98134 −0.990670 0.136280i $$-0.956485\pi$$
−0.990670 + 0.136280i $$0.956485\pi$$
$$600$$ 0 0
$$601$$ 7.13163e6 0.805383 0.402691 0.915336i $$-0.368075\pi$$
0.402691 + 0.915336i $$0.368075\pi$$
$$602$$ 223080. 0.0250882
$$603$$ 2.30396e6 0.258037
$$604$$ 1.00164e7 1.11717
$$605$$ 0 0
$$606$$ −224622. −0.0248468
$$607$$ 9.64617e6 1.06263 0.531317 0.847173i $$-0.321697\pi$$
0.531317 + 0.847173i $$0.321697\pi$$
$$608$$ −6.36120e6 −0.697879
$$609$$ 1.29308e6 0.141281
$$610$$ 0 0
$$611$$ 1.63450e6 0.177126
$$612$$ −3.62086e6 −0.390781
$$613$$ −3.68170e6 −0.395729 −0.197864 0.980229i $$-0.563401\pi$$
−0.197864 + 0.980229i $$0.563401\pi$$
$$614$$ −588808. −0.0630308
$$615$$ 0 0
$$616$$ 198198. 0.0210449
$$617$$ −1.83190e7 −1.93727 −0.968635 0.248489i $$-0.920066\pi$$
−0.968635 + 0.248489i $$0.920066\pi$$
$$618$$ 511308. 0.0538532
$$619$$ 1.09660e6 0.115033 0.0575166 0.998345i $$-0.481682\pi$$
0.0575166 + 0.998345i $$0.481682\pi$$
$$620$$ 0 0
$$621$$ −1.15328e6 −0.120007
$$622$$ −2.51827e6 −0.260991
$$623$$ −1.87840e6 −0.193895
$$624$$ −5.78581e6 −0.594844
$$625$$ 0 0
$$626$$ −2.23562e6 −0.228015
$$627$$ −2.35224e6 −0.238953
$$628$$ −9.88175e6 −0.999849
$$629$$ 1.46997e7 1.48144
$$630$$ 0 0
$$631$$ −9.58030e6 −0.957869 −0.478934 0.877851i $$-0.658977\pi$$
−0.478934 + 0.877851i $$0.658977\pi$$
$$632$$ −6.62848e6 −0.660118
$$633$$ −4.73310e6 −0.469501
$$634$$ 1.06079e6 0.104811
$$635$$ 0 0
$$636$$ 8.59432e6 0.842497
$$637$$ −1.11627e7 −1.08998
$$638$$ 668646. 0.0650346
$$639$$ 3.70089e6 0.358554
$$640$$ 0 0
$$641$$ −1.18062e7 −1.13492 −0.567462 0.823400i $$-0.692075\pi$$
−0.567462 + 0.823400i $$0.692075\pi$$
$$642$$ 112428. 0.0107656
$$643$$ 5.88298e6 0.561138 0.280569 0.959834i $$-0.409477\pi$$
0.280569 + 0.959834i $$0.409477\pi$$
$$644$$ −1.27509e6 −0.121151
$$645$$ 0 0
$$646$$ −3.11472e6 −0.293655
$$647$$ 3.62822e6 0.340748 0.170374 0.985379i $$-0.445502\pi$$
0.170374 + 0.985379i $$0.445502\pi$$
$$648$$ 413343. 0.0386699
$$649$$ 776336. 0.0723499
$$650$$ 0 0
$$651$$ −1.12133e6 −0.103700
$$652$$ −1.33919e7 −1.23374
$$653$$ 5.70795e6 0.523838 0.261919 0.965090i $$-0.415645\pi$$
0.261919 + 0.965090i $$0.415645\pi$$
$$654$$ 1.78873e6 0.163531
$$655$$ 0 0
$$656$$ −9.86784e6 −0.895287
$$657$$ 1.48829e6 0.134516
$$658$$ −61412.0 −0.00552953
$$659$$ 1.08205e7 0.970588 0.485294 0.874351i $$-0.338713\pi$$
0.485294 + 0.874351i $$0.338713\pi$$
$$660$$ 0 0
$$661$$ 1.14311e7 1.01762 0.508809 0.860879i $$-0.330086\pi$$
0.508809 + 0.860879i $$0.330086\pi$$
$$662$$ 2.34566e6 0.208027
$$663$$ −8.98078e6 −0.793469
$$664$$ −3.92440e6 −0.345424
$$665$$ 0 0
$$666$$ −825714. −0.0721347
$$667$$ −8.74213e6 −0.760857
$$668$$ −7.79898e6 −0.676233
$$669$$ −2.20738e6 −0.190683
$$670$$ 0 0
$$671$$ 5.09362e6 0.436737
$$672$$ 689130. 0.0588678
$$673$$ 2.03858e7 1.73496 0.867482 0.497468i $$-0.165737\pi$$
0.867482 + 0.497468i $$0.165737\pi$$
$$674$$ 839978. 0.0712227
$$675$$ 0 0
$$676$$ −3.33470e6 −0.280666
$$677$$ −6.09278e6 −0.510909 −0.255455 0.966821i $$-0.582225\pi$$
−0.255455 + 0.966821i $$0.582225\pi$$
$$678$$ −1.49899e6 −0.125234
$$679$$ −2.06081e6 −0.171539
$$680$$ 0 0
$$681$$ −7.19377e6 −0.594414
$$682$$ −579832. −0.0477355
$$683$$ −1.44978e7 −1.18918 −0.594592 0.804027i $$-0.702686\pi$$
−0.594592 + 0.804027i $$0.702686\pi$$
$$684$$ −5.42376e6 −0.443262
$$685$$ 0 0
$$686$$ 856388. 0.0694801
$$687$$ 1.38590e7 1.12032
$$688$$ −7.97082e6 −0.641995
$$689$$ 2.13164e7 1.71067
$$690$$ 0 0
$$691$$ 9.87069e6 0.786416 0.393208 0.919449i $$-0.371365\pi$$
0.393208 + 0.919449i $$0.371365\pi$$
$$692$$ 1.47757e7 1.17296
$$693$$ 254826. 0.0201563
$$694$$ −2.02560e6 −0.159645
$$695$$ 0 0
$$696$$ 3.13324e6 0.245172
$$697$$ −1.53169e7 −1.19423
$$698$$ 378924. 0.0294384
$$699$$ −6.49647e6 −0.502903
$$700$$ 0 0
$$701$$ 6.35411e6 0.488382 0.244191 0.969727i $$-0.421478\pi$$
0.244191 + 0.969727i $$0.421478\pi$$
$$702$$ 504468. 0.0386359
$$703$$ 2.20190e7 1.68039
$$704$$ −3.24074e6 −0.246441
$$705$$ 0 0
$$706$$ −1.98730e6 −0.150056
$$707$$ −648908. −0.0488241
$$708$$ 1.79006e6 0.134210
$$709$$ −411382. −0.0307348 −0.0153674 0.999882i $$-0.504892\pi$$
−0.0153674 + 0.999882i $$0.504892\pi$$
$$710$$ 0 0
$$711$$ −8.52233e6 −0.632244
$$712$$ −4.55150e6 −0.336476
$$713$$ 7.58094e6 0.558470
$$714$$ 337428. 0.0247705
$$715$$ 0 0
$$716$$ −2.79595e6 −0.203820
$$717$$ 5.74592e6 0.417409
$$718$$ 3.43975e6 0.249009
$$719$$ 6.29795e6 0.454336 0.227168 0.973856i $$-0.427053\pi$$
0.227168 + 0.973856i $$0.427053\pi$$
$$720$$ 0 0
$$721$$ 1.47711e6 0.105822
$$722$$ −2.18950e6 −0.156316
$$723$$ −1.98891e6 −0.141504
$$724$$ −7.68806e6 −0.545093
$$725$$ 0 0
$$726$$ 131769. 0.00927837
$$727$$ −1.14699e7 −0.804866 −0.402433 0.915449i $$-0.631835\pi$$
−0.402433 + 0.915449i $$0.631835\pi$$
$$728$$ 1.13350e6 0.0792668
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ −1.23724e7 −0.856365
$$732$$ 1.17448e7 0.810154
$$733$$ 1.87547e7 1.28929 0.644646 0.764481i $$-0.277005\pi$$
0.644646 + 0.764481i $$0.277005\pi$$
$$734$$ −1.79679e6 −0.123100
$$735$$ 0 0
$$736$$ −4.65899e6 −0.317028
$$737$$ 3.44172e6 0.233403
$$738$$ 860382. 0.0581501
$$739$$ 2.79727e6 0.188418 0.0942091 0.995552i $$-0.469968\pi$$
0.0942091 + 0.995552i $$0.469968\pi$$
$$740$$ 0 0
$$741$$ −1.34525e7 −0.900030
$$742$$ −800904. −0.0534036
$$743$$ 2.25651e7 1.49956 0.749781 0.661686i $$-0.230159\pi$$
0.749781 + 0.661686i $$0.230159\pi$$
$$744$$ −2.71706e6 −0.179956
$$745$$ 0 0
$$746$$ −1.43541e6 −0.0944342
$$747$$ −5.04565e6 −0.330838
$$748$$ −5.40894e6 −0.353475
$$749$$ 324792. 0.0211544
$$750$$ 0 0
$$751$$ −7.49233e6 −0.484749 −0.242375 0.970183i $$-0.577926\pi$$
−0.242375 + 0.970183i $$0.577926\pi$$
$$752$$ 2.19430e6 0.141498
$$753$$ −5.64574e6 −0.362855
$$754$$ 3.82399e6 0.244956
$$755$$ 0 0
$$756$$ 587574. 0.0373902
$$757$$ −2.88492e7 −1.82976 −0.914880 0.403727i $$-0.867715\pi$$
−0.914880 + 0.403727i $$0.867715\pi$$
$$758$$ −2.66235e6 −0.168303
$$759$$ −1.72280e6 −0.108550
$$760$$ 0 0
$$761$$ −9.56279e6 −0.598581 −0.299291 0.954162i $$-0.596750\pi$$
−0.299291 + 0.954162i $$0.596750\pi$$
$$762$$ −2.73803e6 −0.170825
$$763$$ 5.16745e6 0.321340
$$764$$ 4.86086e6 0.301287
$$765$$ 0 0
$$766$$ 2.04091e6 0.125676
$$767$$ 4.43987e6 0.272510
$$768$$ −6.62430e6 −0.405263
$$769$$ −744898. −0.0454235 −0.0227118 0.999742i $$-0.507230\pi$$
−0.0227118 + 0.999742i $$0.507230\pi$$
$$770$$ 0 0
$$771$$ −4.21213e6 −0.255191
$$772$$ −1.33683e7 −0.807293
$$773$$ −6.07336e6 −0.365578 −0.182789 0.983152i $$-0.558513\pi$$
−0.182789 + 0.983152i $$0.558513\pi$$
$$774$$ 694980. 0.0416984
$$775$$ 0 0
$$776$$ −4.99351e6 −0.297681
$$777$$ −2.38540e6 −0.141745
$$778$$ 4.29947e6 0.254663
$$779$$ −2.29435e7 −1.35462
$$780$$ 0 0
$$781$$ 5.52849e6 0.324324
$$782$$ −2.28124e6 −0.133400
$$783$$ 4.02845e6 0.234819
$$784$$ −1.49857e7 −0.870737
$$785$$ 0 0
$$786$$ 2.46985e6 0.142598
$$787$$ 1.47512e7 0.848966 0.424483 0.905436i $$-0.360456\pi$$
0.424483 + 0.905436i $$0.360456\pi$$
$$788$$ −2.68142e7 −1.53833
$$789$$ 1.39059e7 0.795257
$$790$$ 0 0
$$791$$ −4.33040e6 −0.246086
$$792$$ 617463. 0.0349782
$$793$$ 2.91304e7 1.64499
$$794$$ 728818. 0.0410268
$$795$$ 0 0
$$796$$ 1.48819e7 0.832481
$$797$$ 2.78359e7 1.55224 0.776121 0.630584i $$-0.217185\pi$$
0.776121 + 0.630584i $$0.217185\pi$$
$$798$$ 505440. 0.0280972
$$799$$ 3.40600e6 0.188746
$$800$$ 0 0
$$801$$ −5.85193e6 −0.322268
$$802$$ 5.92515e6 0.325285
$$803$$ 2.22325e6 0.121675
$$804$$ 7.93588e6 0.432967
$$805$$ 0 0
$$806$$ −3.31606e6 −0.179798
$$807$$ 9.67115e6 0.522750
$$808$$ −1.57235e6 −0.0847270
$$809$$ 2.54767e7 1.36859 0.684293 0.729207i $$-0.260111\pi$$
0.684293 + 0.729207i $$0.260111\pi$$
$$810$$ 0 0
$$811$$ −1.91915e7 −1.02460 −0.512302 0.858805i $$-0.671207\pi$$
−0.512302 + 0.858805i $$0.671207\pi$$
$$812$$ 4.45396e6 0.237059
$$813$$ −1.42851e7 −0.757977
$$814$$ −1.23347e6 −0.0652483
$$815$$ 0 0
$$816$$ −1.20566e7 −0.633867
$$817$$ −1.85328e7 −0.971373
$$818$$ −1.38212e6 −0.0722207
$$819$$ 1.45735e6 0.0759197
$$820$$ 0 0
$$821$$ 3.27107e6 0.169368 0.0846840 0.996408i $$-0.473012\pi$$
0.0846840 + 0.996408i $$0.473012\pi$$
$$822$$ 2.20912e6 0.114036
$$823$$ 3.19195e7 1.64269 0.821347 0.570430i $$-0.193223\pi$$
0.821347 + 0.570430i $$0.193223\pi$$
$$824$$ 3.57916e6 0.183638
$$825$$ 0 0
$$826$$ −166816. −0.00850722
$$827$$ −2.45556e7 −1.24850 −0.624248 0.781226i $$-0.714595\pi$$
−0.624248 + 0.781226i $$0.714595\pi$$
$$828$$ −3.97240e6 −0.201362
$$829$$ −1.40969e7 −0.712421 −0.356211 0.934406i $$-0.615931\pi$$
−0.356211 + 0.934406i $$0.615931\pi$$
$$830$$ 0 0
$$831$$ 6.23434e6 0.313175
$$832$$ −1.85338e7 −0.928233
$$833$$ −2.32609e7 −1.16149
$$834$$ −538992. −0.0268329
$$835$$ 0 0
$$836$$ −8.10216e6 −0.400945
$$837$$ −3.49337e6 −0.172358
$$838$$ −5.47794e6 −0.269468
$$839$$ −3.01443e6 −0.147843 −0.0739213 0.997264i $$-0.523551\pi$$
−0.0739213 + 0.997264i $$0.523551\pi$$
$$840$$ 0 0
$$841$$ 1.00255e7 0.488784
$$842$$ −1.02873e6 −0.0500061
$$843$$ 5.10316e6 0.247326
$$844$$ −1.63029e7 −0.787787
$$845$$ 0 0
$$846$$ −191322. −0.00919050
$$847$$ 380666. 0.0182321
$$848$$ 2.86169e7 1.36657
$$849$$ 6.43424e6 0.306357
$$850$$ 0 0
$$851$$ 1.61269e7 0.763356
$$852$$ 1.27475e7 0.601626
$$853$$ 1.67201e7 0.786806 0.393403 0.919366i $$-0.371298\pi$$
0.393403 + 0.919366i $$0.371298\pi$$
$$854$$ −1.09450e6 −0.0513534
$$855$$ 0 0
$$856$$ 786996. 0.0367103
$$857$$ 9.15871e6 0.425973 0.212987 0.977055i $$-0.431681\pi$$
0.212987 + 0.977055i $$0.431681\pi$$
$$858$$ 753588. 0.0349475
$$859$$ 1.51068e7 0.698536 0.349268 0.937023i $$-0.386430\pi$$
0.349268 + 0.937023i $$0.386430\pi$$
$$860$$ 0 0
$$861$$ 2.48555e6 0.114265
$$862$$ 5.14310e6 0.235753
$$863$$ −5.11568e6 −0.233817 −0.116909 0.993143i $$-0.537298\pi$$
−0.116909 + 0.993143i $$0.537298\pi$$
$$864$$ 2.14690e6 0.0978427
$$865$$ 0 0
$$866$$ 412954. 0.0187114
$$867$$ −5.93556e6 −0.268172
$$868$$ −3.86235e6 −0.174001
$$869$$ −1.27309e7 −0.571886
$$870$$ 0 0
$$871$$ 1.96832e7 0.879127
$$872$$ 1.25211e7 0.557638
$$873$$ −6.42022e6 −0.285111
$$874$$ −3.41712e6 −0.151315
$$875$$ 0 0
$$876$$ 5.12635e6 0.225708
$$877$$ 1.26998e7 0.557568 0.278784 0.960354i $$-0.410069\pi$$
0.278784 + 0.960354i $$0.410069\pi$$
$$878$$ −5.96365e6 −0.261081
$$879$$ 1.92968e7 0.842392
$$880$$ 0 0
$$881$$ −8.38173e6 −0.363826 −0.181913 0.983315i $$-0.558229\pi$$
−0.181913 + 0.983315i $$0.558229\pi$$
$$882$$ 1.30661e6 0.0565555
$$883$$ 1.69529e7 0.731715 0.365858 0.930671i $$-0.380776\pi$$
0.365858 + 0.930671i $$0.380776\pi$$
$$884$$ −3.09338e7 −1.33138
$$885$$ 0 0
$$886$$ 2.18433e6 0.0934832
$$887$$ 1.05143e7 0.448717 0.224359 0.974507i $$-0.427971\pi$$
0.224359 + 0.974507i $$0.427971\pi$$
$$888$$ −5.78000e6 −0.245977
$$889$$ −7.90988e6 −0.335672
$$890$$ 0 0
$$891$$ 793881. 0.0335013
$$892$$ −7.60318e6 −0.319951
$$893$$ 5.10192e6 0.214094
$$894$$ 648342. 0.0271307
$$895$$ 0 0
$$896$$ 3.14660e6 0.130940
$$897$$ −9.85270e6 −0.408859
$$898$$ 7858.00 0.000325178 0
$$899$$ −2.64806e7 −1.09277
$$900$$ 0 0
$$901$$ 4.44194e7 1.82289
$$902$$ 1.28526e6 0.0525987
$$903$$ 2.00772e6 0.0819377
$$904$$ −1.04929e7 −0.427046
$$905$$ 0 0
$$906$$ −2.90799e6 −0.117699
$$907$$ 1.53747e7 0.620569 0.310284 0.950644i $$-0.399576\pi$$
0.310284 + 0.950644i $$0.399576\pi$$
$$908$$ −2.47785e7 −0.997381
$$909$$ −2.02160e6 −0.0811494
$$910$$ 0 0
$$911$$ 1.25424e7 0.500708 0.250354 0.968154i $$-0.419453\pi$$
0.250354 + 0.968154i $$0.419453\pi$$
$$912$$ −1.80598e7 −0.718993
$$913$$ −7.53733e6 −0.299255
$$914$$ −899922. −0.0356319
$$915$$ 0 0
$$916$$ 4.77367e7 1.87981
$$917$$ 7.13513e6 0.280207
$$918$$ 1.05122e6 0.0411705
$$919$$ 3.31432e7 1.29451 0.647256 0.762273i $$-0.275916\pi$$
0.647256 + 0.762273i $$0.275916\pi$$
$$920$$ 0 0
$$921$$ −5.29927e6 −0.205858
$$922$$ −1.13619e6 −0.0440173
$$923$$ 3.16175e7 1.22158
$$924$$ 877734. 0.0338207
$$925$$ 0 0
$$926$$ −7.38964e6 −0.283202
$$927$$ 4.60177e6 0.175884
$$928$$ 1.62741e7 0.620335
$$929$$ 3.10442e7 1.18016 0.590080 0.807345i $$-0.299096\pi$$
0.590080 + 0.807345i $$0.299096\pi$$
$$930$$ 0 0
$$931$$ −3.48430e7 −1.31747
$$932$$ −2.23767e7 −0.843834
$$933$$ −2.26644e7 −0.852393
$$934$$ 4.20851e6 0.157856
$$935$$ 0 0
$$936$$ 3.53128e6 0.131747
$$937$$ 3.10737e7 1.15623 0.578115 0.815955i $$-0.303788\pi$$
0.578115 + 0.815955i $$0.303788\pi$$
$$938$$ −739544. −0.0274446
$$939$$ −2.01206e7 −0.744692
$$940$$ 0 0
$$941$$ −2.50349e7 −0.921664 −0.460832 0.887488i $$-0.652449\pi$$
−0.460832 + 0.887488i $$0.652449\pi$$
$$942$$ 2.86889e6 0.105339
$$943$$ −1.68040e7 −0.615366
$$944$$ 5.96046e6 0.217696
$$945$$ 0 0
$$946$$ 1.03818e6 0.0377177
$$947$$ 5.37383e6 0.194719 0.0973596 0.995249i $$-0.468960\pi$$
0.0973596 + 0.995249i $$0.468960\pi$$
$$948$$ −2.93547e7 −1.06086
$$949$$ 1.27148e7 0.458294
$$950$$ 0 0
$$951$$ 9.54713e6 0.342311
$$952$$ 2.36200e6 0.0844669
$$953$$ −7.26908e6 −0.259267 −0.129634 0.991562i $$-0.541380\pi$$
−0.129634 + 0.991562i $$0.541380\pi$$
$$954$$ −2.49512e6 −0.0887608
$$955$$ 0 0
$$956$$ 1.97915e7 0.700381
$$957$$ 6.01781e6 0.212402
$$958$$ −7.39441e6 −0.260309
$$959$$ 6.38191e6 0.224080
$$960$$ 0 0
$$961$$ −5.66589e6 −0.197906
$$962$$ −7.05425e6 −0.245761
$$963$$ 1.01185e6 0.0351602
$$964$$ −6.85069e6 −0.237433
$$965$$ 0 0
$$966$$ 370188. 0.0127638
$$967$$ 2.54428e7 0.874983 0.437491 0.899223i $$-0.355867\pi$$
0.437491 + 0.899223i $$0.355867\pi$$
$$968$$ 922383. 0.0316390
$$969$$ −2.80325e7 −0.959074
$$970$$ 0 0
$$971$$ 9.88213e6 0.336358 0.168179 0.985756i $$-0.446211\pi$$
0.168179 + 0.985756i $$0.446211\pi$$
$$972$$ 1.83052e6 0.0621453
$$973$$ −1.55709e6 −0.0527268
$$974$$ −3.81644e6 −0.128902
$$975$$ 0 0
$$976$$ 3.91072e7 1.31411
$$977$$ −2.22197e6 −0.0744736 −0.0372368 0.999306i $$-0.511856\pi$$
−0.0372368 + 0.999306i $$0.511856\pi$$
$$978$$ 3.88796e6 0.129980
$$979$$ −8.74177e6 −0.291503
$$980$$ 0 0
$$981$$ 1.60986e7 0.534091
$$982$$ −1.69716e6 −0.0561623
$$983$$ −2.53706e7 −0.837428 −0.418714 0.908118i $$-0.637519\pi$$
−0.418714 + 0.908118i $$0.637519\pi$$
$$984$$ 6.02267e6 0.198290
$$985$$ 0 0
$$986$$ 7.96849e6 0.261026
$$987$$ −552708. −0.0180594
$$988$$ −4.63363e7 −1.51018
$$989$$ −1.35736e7 −0.441269
$$990$$ 0 0
$$991$$ 3.24132e7 1.04843 0.524214 0.851587i $$-0.324359\pi$$
0.524214 + 0.851587i $$0.324359\pi$$
$$992$$ −1.41124e7 −0.455327
$$993$$ 2.11109e7 0.679414
$$994$$ −1.18794e6 −0.0381354
$$995$$ 0 0
$$996$$ −1.73795e7 −0.555122
$$997$$ 1.55048e6 0.0494000 0.0247000 0.999695i $$-0.492137\pi$$
0.0247000 + 0.999695i $$0.492137\pi$$
$$998$$ −6.95160e6 −0.220932
$$999$$ −7.43143e6 −0.235591
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 825.6.a.a.1.1 1
5.4 even 2 33.6.a.b.1.1 1
15.14 odd 2 99.6.a.a.1.1 1
20.19 odd 2 528.6.a.a.1.1 1
55.54 odd 2 363.6.a.b.1.1 1
165.164 even 2 1089.6.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.b.1.1 1 5.4 even 2
99.6.a.a.1.1 1 15.14 odd 2
363.6.a.b.1.1 1 55.54 odd 2
528.6.a.a.1.1 1 20.19 odd 2
825.6.a.a.1.1 1 1.1 even 1 trivial
1089.6.a.h.1.1 1 165.164 even 2