# Properties

 Label 441.6.a.q.1.1 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ Defining polynomial: $$x^{2} - x - 48$$ x^2 - x - 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-6.44622$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.44622 q^{2} -2.33867 q^{4} -36.0000 q^{5} +187.016 q^{8} +O(q^{10})$$ $$q-5.44622 q^{2} -2.33867 q^{4} -36.0000 q^{5} +187.016 q^{8} +196.064 q^{10} -184.430 q^{11} +147.872 q^{13} -943.693 q^{16} -1968.38 q^{17} +1892.51 q^{19} +84.1920 q^{20} +1004.45 q^{22} -136.988 q^{23} -1829.00 q^{25} -805.344 q^{26} +1259.58 q^{29} +8969.02 q^{31} -844.949 q^{32} +10720.3 q^{34} +12897.2 q^{37} -10307.0 q^{38} -6732.58 q^{40} -8975.62 q^{41} +13538.9 q^{43} +431.321 q^{44} +746.069 q^{46} +20046.1 q^{47} +9961.14 q^{50} -345.823 q^{52} -9334.33 q^{53} +6639.49 q^{55} -6859.96 q^{58} -8866.46 q^{59} -41148.3 q^{61} -48847.3 q^{62} +34800.0 q^{64} -5323.39 q^{65} -55351.5 q^{67} +4603.39 q^{68} +63866.8 q^{71} +41299.3 q^{73} -70241.1 q^{74} -4425.95 q^{76} +16963.5 q^{79} +33973.0 q^{80} +48883.2 q^{82} +101693. q^{83} +70861.8 q^{85} -73736.0 q^{86} -34491.4 q^{88} -87102.5 q^{89} +320.370 q^{92} -109176. q^{94} -68130.4 q^{95} -118107. q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 37 q^{4} - 72 q^{5} + 249 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 37 * q^4 - 72 * q^5 + 249 * q^8 $$2 q + 3 q^{2} + 37 q^{4} - 72 q^{5} + 249 q^{8} - 108 q^{10} - 480 q^{11} + 1296 q^{13} - 1679 q^{16} - 936 q^{17} - 216 q^{19} - 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} + 8892 q^{26} - 6372 q^{29} + 9936 q^{31} - 9039 q^{32} + 19440 q^{34} + 11124 q^{37} - 28116 q^{38} - 8964 q^{40} - 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} - 7920 q^{47} - 5487 q^{50} + 44820 q^{52} - 2220 q^{53} + 17280 q^{55} - 71318 q^{58} - 29736 q^{59} - 17280 q^{61} - 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} + 45216 q^{68} + 92280 q^{71} + 56592 q^{73} - 85218 q^{74} - 87372 q^{76} - 56096 q^{79} + 60444 q^{80} - 52272 q^{82} + 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} - 123192 q^{89} + 25536 q^{92} - 345384 q^{94} + 7776 q^{95} + 35856 q^{97}+O(q^{100})$$ 2 * q + 3 * q^2 + 37 * q^4 - 72 * q^5 + 249 * q^8 - 108 * q^10 - 480 * q^11 + 1296 * q^13 - 1679 * q^16 - 936 * q^17 - 216 * q^19 - 1332 * q^20 - 1492 * q^22 + 504 * q^23 - 3658 * q^25 + 8892 * q^26 - 6372 * q^29 + 9936 * q^31 - 9039 * q^32 + 19440 * q^34 + 11124 * q^37 - 28116 * q^38 - 8964 * q^40 - 20952 * q^41 - 6264 * q^43 - 11196 * q^44 + 6160 * q^46 - 7920 * q^47 - 5487 * q^50 + 44820 * q^52 - 2220 * q^53 + 17280 * q^55 - 71318 * q^58 - 29736 * q^59 - 17280 * q^61 - 40680 * q^62 - 10879 * q^64 - 46656 * q^65 - 20680 * q^67 + 45216 * q^68 + 92280 * q^71 + 56592 * q^73 - 85218 * q^74 - 87372 * q^76 - 56096 * q^79 + 60444 * q^80 - 52272 * q^82 + 71352 * q^83 + 33696 * q^85 - 240996 * q^86 - 52812 * q^88 - 123192 * q^89 + 25536 * q^92 - 345384 * q^94 + 7776 * q^95 + 35856 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.44622 −0.962765 −0.481383 0.876511i $$-0.659865\pi$$
−0.481383 + 0.876511i $$0.659865\pi$$
$$3$$ 0 0
$$4$$ −2.33867 −0.0730833
$$5$$ −36.0000 −0.643988 −0.321994 0.946742i $$-0.604353\pi$$
−0.321994 + 0.946742i $$0.604353\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 187.016 1.03313
$$9$$ 0 0
$$10$$ 196.064 0.620009
$$11$$ −184.430 −0.459569 −0.229784 0.973242i $$-0.573802\pi$$
−0.229784 + 0.973242i $$0.573802\pi$$
$$12$$ 0 0
$$13$$ 147.872 0.242676 0.121338 0.992611i $$-0.461281\pi$$
0.121338 + 0.992611i $$0.461281\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −943.693 −0.921576
$$17$$ −1968.38 −1.65191 −0.825957 0.563733i $$-0.809365\pi$$
−0.825957 + 0.563733i $$0.809365\pi$$
$$18$$ 0 0
$$19$$ 1892.51 1.20269 0.601346 0.798989i $$-0.294631\pi$$
0.601346 + 0.798989i $$0.294631\pi$$
$$20$$ 84.1920 0.0470647
$$21$$ 0 0
$$22$$ 1004.45 0.442457
$$23$$ −136.988 −0.0539963 −0.0269982 0.999635i $$-0.508595\pi$$
−0.0269982 + 0.999635i $$0.508595\pi$$
$$24$$ 0 0
$$25$$ −1829.00 −0.585280
$$26$$ −805.344 −0.233640
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1259.58 0.278120 0.139060 0.990284i $$-0.455592\pi$$
0.139060 + 0.990284i $$0.455592\pi$$
$$30$$ 0 0
$$31$$ 8969.02 1.67626 0.838129 0.545472i $$-0.183650\pi$$
0.838129 + 0.545472i $$0.183650\pi$$
$$32$$ −844.949 −0.145866
$$33$$ 0 0
$$34$$ 10720.3 1.59041
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 12897.2 1.54879 0.774393 0.632705i $$-0.218055\pi$$
0.774393 + 0.632705i $$0.218055\pi$$
$$38$$ −10307.0 −1.15791
$$39$$ 0 0
$$40$$ −6732.58 −0.665321
$$41$$ −8975.62 −0.833882 −0.416941 0.908934i $$-0.636898\pi$$
−0.416941 + 0.908934i $$0.636898\pi$$
$$42$$ 0 0
$$43$$ 13538.9 1.11664 0.558320 0.829626i $$-0.311446\pi$$
0.558320 + 0.829626i $$0.311446\pi$$
$$44$$ 431.321 0.0335868
$$45$$ 0 0
$$46$$ 746.069 0.0519858
$$47$$ 20046.1 1.32369 0.661845 0.749641i $$-0.269774\pi$$
0.661845 + 0.749641i $$0.269774\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 9961.14 0.563487
$$51$$ 0 0
$$52$$ −345.823 −0.0177356
$$53$$ −9334.33 −0.456450 −0.228225 0.973608i $$-0.573292\pi$$
−0.228225 + 0.973608i $$0.573292\pi$$
$$54$$ 0 0
$$55$$ 6639.49 0.295956
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6859.96 −0.267764
$$59$$ −8866.46 −0.331605 −0.165802 0.986159i $$-0.553021\pi$$
−0.165802 + 0.986159i $$0.553021\pi$$
$$60$$ 0 0
$$61$$ −41148.3 −1.41588 −0.707942 0.706271i $$-0.750376\pi$$
−0.707942 + 0.706271i $$0.750376\pi$$
$$62$$ −48847.3 −1.61384
$$63$$ 0 0
$$64$$ 34800.0 1.06201
$$65$$ −5323.39 −0.156281
$$66$$ 0 0
$$67$$ −55351.5 −1.50641 −0.753204 0.657787i $$-0.771493\pi$$
−0.753204 + 0.657787i $$0.771493\pi$$
$$68$$ 4603.39 0.120727
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 63866.8 1.50359 0.751794 0.659398i $$-0.229189\pi$$
0.751794 + 0.659398i $$0.229189\pi$$
$$72$$ 0 0
$$73$$ 41299.3 0.907060 0.453530 0.891241i $$-0.350165\pi$$
0.453530 + 0.891241i $$0.350165\pi$$
$$74$$ −70241.1 −1.49112
$$75$$ 0 0
$$76$$ −4425.95 −0.0878968
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16963.5 0.305808 0.152904 0.988241i $$-0.451138\pi$$
0.152904 + 0.988241i $$0.451138\pi$$
$$80$$ 33973.0 0.593483
$$81$$ 0 0
$$82$$ 48883.2 0.802833
$$83$$ 101693. 1.62030 0.810150 0.586223i $$-0.199386\pi$$
0.810150 + 0.586223i $$0.199386\pi$$
$$84$$ 0 0
$$85$$ 70861.8 1.06381
$$86$$ −73736.0 −1.07506
$$87$$ 0 0
$$88$$ −34491.4 −0.474793
$$89$$ −87102.5 −1.16562 −0.582808 0.812610i $$-0.698046\pi$$
−0.582808 + 0.812610i $$0.698046\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 320.370 0.00394623
$$93$$ 0 0
$$94$$ −109176. −1.27440
$$95$$ −68130.4 −0.774519
$$96$$ 0 0
$$97$$ −118107. −1.27452 −0.637258 0.770650i $$-0.719932\pi$$
−0.637258 + 0.770650i $$0.719932\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 4277.42 0.0427742
$$101$$ −22232.7 −0.216865 −0.108432 0.994104i $$-0.534583\pi$$
−0.108432 + 0.994104i $$0.534583\pi$$
$$102$$ 0 0
$$103$$ 135931. 1.26248 0.631239 0.775588i $$-0.282547\pi$$
0.631239 + 0.775588i $$0.282547\pi$$
$$104$$ 27654.4 0.250716
$$105$$ 0 0
$$106$$ 50836.8 0.439454
$$107$$ −117626. −0.993218 −0.496609 0.867974i $$-0.665422\pi$$
−0.496609 + 0.867974i $$0.665422\pi$$
$$108$$ 0 0
$$109$$ −34664.2 −0.279457 −0.139729 0.990190i $$-0.544623\pi$$
−0.139729 + 0.990190i $$0.544623\pi$$
$$110$$ −36160.1 −0.284937
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 26584.5 0.195854 0.0979269 0.995194i $$-0.468779\pi$$
0.0979269 + 0.995194i $$0.468779\pi$$
$$114$$ 0 0
$$115$$ 4931.58 0.0347730
$$116$$ −2945.74 −0.0203259
$$117$$ 0 0
$$118$$ 48288.7 0.319257
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −127036. −0.788797
$$122$$ 224103. 1.36316
$$123$$ 0 0
$$124$$ −20975.6 −0.122507
$$125$$ 178344. 1.02090
$$126$$ 0 0
$$127$$ −137111. −0.754334 −0.377167 0.926145i $$-0.623102\pi$$
−0.377167 + 0.926145i $$0.623102\pi$$
$$128$$ −162490. −0.876600
$$129$$ 0 0
$$130$$ 28992.4 0.150462
$$131$$ 54089.2 0.275380 0.137690 0.990475i $$-0.456032\pi$$
0.137690 + 0.990475i $$0.456032\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 301457. 1.45032
$$135$$ 0 0
$$136$$ −368119. −1.70664
$$137$$ −422849. −1.92479 −0.962395 0.271653i $$-0.912430\pi$$
−0.962395 + 0.271653i $$0.912430\pi$$
$$138$$ 0 0
$$139$$ −9913.38 −0.0435196 −0.0217598 0.999763i $$-0.506927\pi$$
−0.0217598 + 0.999763i $$0.506927\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −347833. −1.44760
$$143$$ −27272.1 −0.111526
$$144$$ 0 0
$$145$$ −45345.0 −0.179106
$$146$$ −224925. −0.873285
$$147$$ 0 0
$$148$$ −30162.3 −0.113190
$$149$$ −505462. −1.86519 −0.932595 0.360925i $$-0.882461\pi$$
−0.932595 + 0.360925i $$0.882461\pi$$
$$150$$ 0 0
$$151$$ −193103. −0.689201 −0.344601 0.938749i $$-0.611986\pi$$
−0.344601 + 0.938749i $$0.611986\pi$$
$$152$$ 353930. 1.24253
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −322885. −1.07949
$$156$$ 0 0
$$157$$ 264923. 0.857770 0.428885 0.903359i $$-0.358907\pi$$
0.428885 + 0.903359i $$0.358907\pi$$
$$158$$ −92387.1 −0.294421
$$159$$ 0 0
$$160$$ 30418.1 0.0939362
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −539093. −1.58926 −0.794629 0.607095i $$-0.792335\pi$$
−0.794629 + 0.607095i $$0.792335\pi$$
$$164$$ 20991.0 0.0609429
$$165$$ 0 0
$$166$$ −553842. −1.55997
$$167$$ −218748. −0.606949 −0.303475 0.952840i $$-0.598147\pi$$
−0.303475 + 0.952840i $$0.598147\pi$$
$$168$$ 0 0
$$169$$ −349427. −0.941108
$$170$$ −385929. −1.02420
$$171$$ 0 0
$$172$$ −31663.0 −0.0816078
$$173$$ −590201. −1.49929 −0.749644 0.661842i $$-0.769775\pi$$
−0.749644 + 0.661842i $$0.769775\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 174046. 0.423527
$$177$$ 0 0
$$178$$ 474380. 1.12222
$$179$$ 217352. 0.507026 0.253513 0.967332i $$-0.418414\pi$$
0.253513 + 0.967332i $$0.418414\pi$$
$$180$$ 0 0
$$181$$ 188109. 0.426790 0.213395 0.976966i $$-0.431548\pi$$
0.213395 + 0.976966i $$0.431548\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −25619.0 −0.0557851
$$185$$ −464300. −0.997399
$$186$$ 0 0
$$187$$ 363029. 0.759168
$$188$$ −46881.2 −0.0967396
$$189$$ 0 0
$$190$$ 371053. 0.745680
$$191$$ −173523. −0.344170 −0.172085 0.985082i $$-0.555050\pi$$
−0.172085 + 0.985082i $$0.555050\pi$$
$$192$$ 0 0
$$193$$ 117781. 0.227606 0.113803 0.993503i $$-0.463697\pi$$
0.113803 + 0.993503i $$0.463697\pi$$
$$194$$ 643236. 1.22706
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −224734. −0.412575 −0.206288 0.978491i $$-0.566138\pi$$
−0.206288 + 0.978491i $$0.566138\pi$$
$$198$$ 0 0
$$199$$ 740273. 1.32513 0.662567 0.749003i $$-0.269467\pi$$
0.662567 + 0.749003i $$0.269467\pi$$
$$200$$ −342052. −0.604669
$$201$$ 0 0
$$202$$ 121084. 0.208790
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 323122. 0.537010
$$206$$ −740308. −1.21547
$$207$$ 0 0
$$208$$ −139546. −0.223645
$$209$$ −349036. −0.552720
$$210$$ 0 0
$$211$$ −705896. −1.09153 −0.545763 0.837939i $$-0.683760\pi$$
−0.545763 + 0.837939i $$0.683760\pi$$
$$212$$ 21829.9 0.0333589
$$213$$ 0 0
$$214$$ 640618. 0.956236
$$215$$ −487402. −0.719102
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 188789. 0.269052
$$219$$ 0 0
$$220$$ −15527.5 −0.0216295
$$221$$ −291069. −0.400881
$$222$$ 0 0
$$223$$ 42214.3 0.0568456 0.0284228 0.999596i $$-0.490952\pi$$
0.0284228 + 0.999596i $$0.490952\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −144785. −0.188561
$$227$$ −742403. −0.956258 −0.478129 0.878290i $$-0.658685\pi$$
−0.478129 + 0.878290i $$0.658685\pi$$
$$228$$ 0 0
$$229$$ 941236. 1.18607 0.593034 0.805177i $$-0.297930\pi$$
0.593034 + 0.805177i $$0.297930\pi$$
$$230$$ −26858.5 −0.0334782
$$231$$ 0 0
$$232$$ 235562. 0.287333
$$233$$ 512331. 0.618246 0.309123 0.951022i $$-0.399965\pi$$
0.309123 + 0.951022i $$0.399965\pi$$
$$234$$ 0 0
$$235$$ −721661. −0.852440
$$236$$ 20735.7 0.0242348
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 115323. 0.130593 0.0652966 0.997866i $$-0.479201\pi$$
0.0652966 + 0.997866i $$0.479201\pi$$
$$240$$ 0 0
$$241$$ −909864. −1.00910 −0.504549 0.863383i $$-0.668341\pi$$
−0.504549 + 0.863383i $$0.668341\pi$$
$$242$$ 691869. 0.759426
$$243$$ 0 0
$$244$$ 96232.2 0.103477
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 279850. 0.291865
$$248$$ 1.67735e6 1.73179
$$249$$ 0 0
$$250$$ −971301. −0.982888
$$251$$ 321264. 0.321868 0.160934 0.986965i $$-0.448549\pi$$
0.160934 + 0.986965i $$0.448549\pi$$
$$252$$ 0 0
$$253$$ 25264.8 0.0248150
$$254$$ 746738. 0.726246
$$255$$ 0 0
$$256$$ −228642. −0.218050
$$257$$ 556492. 0.525565 0.262782 0.964855i $$-0.415360\pi$$
0.262782 + 0.964855i $$0.415360\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 12449.6 0.0114215
$$261$$ 0 0
$$262$$ −294582. −0.265126
$$263$$ 233379. 0.208052 0.104026 0.994575i $$-0.466827\pi$$
0.104026 + 0.994575i $$0.466827\pi$$
$$264$$ 0 0
$$265$$ 336036. 0.293948
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 129449. 0.110093
$$269$$ 1.47857e6 1.24584 0.622919 0.782286i $$-0.285947\pi$$
0.622919 + 0.782286i $$0.285947\pi$$
$$270$$ 0 0
$$271$$ 177762. 0.147033 0.0735165 0.997294i $$-0.476578\pi$$
0.0735165 + 0.997294i $$0.476578\pi$$
$$272$$ 1.85755e6 1.52236
$$273$$ 0 0
$$274$$ 2.30293e6 1.85312
$$275$$ 337323. 0.268976
$$276$$ 0 0
$$277$$ −1.22252e6 −0.957318 −0.478659 0.878001i $$-0.658877\pi$$
−0.478659 + 0.878001i $$0.658877\pi$$
$$278$$ 53990.5 0.0418991
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −177799. −0.134327 −0.0671636 0.997742i $$-0.521395\pi$$
−0.0671636 + 0.997742i $$0.521395\pi$$
$$282$$ 0 0
$$283$$ −1.09052e6 −0.809406 −0.404703 0.914448i $$-0.632625\pi$$
−0.404703 + 0.914448i $$0.632625\pi$$
$$284$$ −149363. −0.109887
$$285$$ 0 0
$$286$$ 148530. 0.107374
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2.45468e6 1.72882
$$290$$ 246959. 0.172437
$$291$$ 0 0
$$292$$ −96585.3 −0.0662909
$$293$$ −545742. −0.371380 −0.185690 0.982608i $$-0.559452\pi$$
−0.185690 + 0.982608i $$0.559452\pi$$
$$294$$ 0 0
$$295$$ 319193. 0.213549
$$296$$ 2.41198e6 1.60009
$$297$$ 0 0
$$298$$ 2.75286e6 1.79574
$$299$$ −20256.8 −0.0131036
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 1.05168e6 0.663539
$$303$$ 0 0
$$304$$ −1.78595e6 −1.10837
$$305$$ 1.48134e6 0.911811
$$306$$ 0 0
$$307$$ −3.29000e6 −1.99228 −0.996138 0.0878052i $$-0.972015\pi$$
−0.996138 + 0.0878052i $$0.972015\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 1.75850e6 1.03929
$$311$$ 699712. 0.410221 0.205111 0.978739i $$-0.434245\pi$$
0.205111 + 0.978739i $$0.434245\pi$$
$$312$$ 0 0
$$313$$ 623219. 0.359567 0.179783 0.983706i $$-0.442460\pi$$
0.179783 + 0.983706i $$0.442460\pi$$
$$314$$ −1.44283e6 −0.825831
$$315$$ 0 0
$$316$$ −39672.0 −0.0223494
$$317$$ 639244. 0.357288 0.178644 0.983914i $$-0.442829\pi$$
0.178644 + 0.983914i $$0.442829\pi$$
$$318$$ 0 0
$$319$$ −232305. −0.127815
$$320$$ −1.25280e6 −0.683922
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −3.72519e6 −1.98675
$$324$$ 0 0
$$325$$ −270458. −0.142034
$$326$$ 2.93602e6 1.53008
$$327$$ 0 0
$$328$$ −1.67858e6 −0.861506
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1.40963e6 0.707190 0.353595 0.935399i $$-0.384959\pi$$
0.353595 + 0.935399i $$0.384959\pi$$
$$332$$ −237826. −0.118417
$$333$$ 0 0
$$334$$ 1.19135e6 0.584350
$$335$$ 1.99265e6 0.970108
$$336$$ 0 0
$$337$$ −1.55677e6 −0.746704 −0.373352 0.927690i $$-0.621792\pi$$
−0.373352 + 0.927690i $$0.621792\pi$$
$$338$$ 1.90306e6 0.906066
$$339$$ 0 0
$$340$$ −165722. −0.0777469
$$341$$ −1.65416e6 −0.770356
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2.53200e6 1.15363
$$345$$ 0 0
$$346$$ 3.21437e6 1.44346
$$347$$ −248297. −0.110700 −0.0553500 0.998467i $$-0.517627\pi$$
−0.0553500 + 0.998467i $$0.517627\pi$$
$$348$$ 0 0
$$349$$ 1.86169e6 0.818171 0.409086 0.912496i $$-0.365848\pi$$
0.409086 + 0.912496i $$0.365848\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 155834. 0.0670356
$$353$$ −1.77128e6 −0.756574 −0.378287 0.925688i $$-0.623487\pi$$
−0.378287 + 0.925688i $$0.623487\pi$$
$$354$$ 0 0
$$355$$ −2.29920e6 −0.968292
$$356$$ 203704. 0.0851871
$$357$$ 0 0
$$358$$ −1.18375e6 −0.488147
$$359$$ 4.47560e6 1.83280 0.916401 0.400262i $$-0.131081\pi$$
0.916401 + 0.400262i $$0.131081\pi$$
$$360$$ 0 0
$$361$$ 1.10550e6 0.446469
$$362$$ −1.02449e6 −0.410898
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1.48678e6 −0.584135
$$366$$ 0 0
$$367$$ 3.17097e6 1.22893 0.614465 0.788944i $$-0.289372\pi$$
0.614465 + 0.788944i $$0.289372\pi$$
$$368$$ 129275. 0.0497617
$$369$$ 0 0
$$370$$ 2.52868e6 0.960261
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −2.31179e6 −0.860353 −0.430177 0.902745i $$-0.641549\pi$$
−0.430177 + 0.902745i $$0.641549\pi$$
$$374$$ −1.97714e6 −0.730900
$$375$$ 0 0
$$376$$ 3.74895e6 1.36754
$$377$$ 186257. 0.0674931
$$378$$ 0 0
$$379$$ −591840. −0.211644 −0.105822 0.994385i $$-0.533747\pi$$
−0.105822 + 0.994385i $$0.533747\pi$$
$$380$$ 159334. 0.0566044
$$381$$ 0 0
$$382$$ 945042. 0.331354
$$383$$ 3.59766e6 1.25321 0.626605 0.779337i $$-0.284444\pi$$
0.626605 + 0.779337i $$0.284444\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −641464. −0.219131
$$387$$ 0 0
$$388$$ 276212. 0.0931459
$$389$$ 2.24746e6 0.753040 0.376520 0.926409i $$-0.377121\pi$$
0.376520 + 0.926409i $$0.377121\pi$$
$$390$$ 0 0
$$391$$ 269646. 0.0891973
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 1.22395e6 0.397213
$$395$$ −610687. −0.196936
$$396$$ 0 0
$$397$$ −4.58670e6 −1.46058 −0.730288 0.683139i $$-0.760614\pi$$
−0.730288 + 0.683139i $$0.760614\pi$$
$$398$$ −4.03169e6 −1.27579
$$399$$ 0 0
$$400$$ 1.72602e6 0.539380
$$401$$ −6.00432e6 −1.86468 −0.932338 0.361589i $$-0.882234\pi$$
−0.932338 + 0.361589i $$0.882234\pi$$
$$402$$ 0 0
$$403$$ 1.32627e6 0.406788
$$404$$ 51994.9 0.0158492
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.37864e6 −0.711774
$$408$$ 0 0
$$409$$ −3.55340e6 −1.05035 −0.525177 0.850993i $$-0.676001\pi$$
−0.525177 + 0.850993i $$0.676001\pi$$
$$410$$ −1.75980e6 −0.517014
$$411$$ 0 0
$$412$$ −317896. −0.0922661
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −3.66094e6 −1.04345
$$416$$ −124944. −0.0353983
$$417$$ 0 0
$$418$$ 1.90093e6 0.532139
$$419$$ −2.01375e6 −0.560365 −0.280182 0.959947i $$-0.590395\pi$$
−0.280182 + 0.959947i $$0.590395\pi$$
$$420$$ 0 0
$$421$$ −5.89987e6 −1.62232 −0.811161 0.584823i $$-0.801164\pi$$
−0.811161 + 0.584823i $$0.801164\pi$$
$$422$$ 3.84446e6 1.05088
$$423$$ 0 0
$$424$$ −1.74567e6 −0.471571
$$425$$ 3.60017e6 0.966832
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 275088. 0.0725877
$$429$$ 0 0
$$430$$ 2.65450e6 0.692327
$$431$$ 3.81048e6 0.988066 0.494033 0.869443i $$-0.335522\pi$$
0.494033 + 0.869443i $$0.335522\pi$$
$$432$$ 0 0
$$433$$ 6.59449e6 1.69029 0.845146 0.534536i $$-0.179513\pi$$
0.845146 + 0.534536i $$0.179513\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 81068.1 0.0204237
$$437$$ −259252. −0.0649410
$$438$$ 0 0
$$439$$ −4.55028e6 −1.12688 −0.563438 0.826158i $$-0.690522\pi$$
−0.563438 + 0.826158i $$0.690522\pi$$
$$440$$ 1.24169e6 0.305761
$$441$$ 0 0
$$442$$ 1.58523e6 0.385954
$$443$$ −5.11537e6 −1.23842 −0.619210 0.785225i $$-0.712547\pi$$
−0.619210 + 0.785225i $$0.712547\pi$$
$$444$$ 0 0
$$445$$ 3.13569e6 0.750643
$$446$$ −229908. −0.0547290
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −5.95600e6 −1.39424 −0.697122 0.716953i $$-0.745536\pi$$
−0.697122 + 0.716953i $$0.745536\pi$$
$$450$$ 0 0
$$451$$ 1.65537e6 0.383226
$$452$$ −62172.2 −0.0143136
$$453$$ 0 0
$$454$$ 4.04329e6 0.920652
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −2.59834e6 −0.581976 −0.290988 0.956727i $$-0.593984\pi$$
−0.290988 + 0.956727i $$0.593984\pi$$
$$458$$ −5.12618e6 −1.14191
$$459$$ 0 0
$$460$$ −11533.3 −0.00254132
$$461$$ −4.51513e6 −0.989505 −0.494752 0.869034i $$-0.664741\pi$$
−0.494752 + 0.869034i $$0.664741\pi$$
$$462$$ 0 0
$$463$$ −5.55129e6 −1.20349 −0.601744 0.798689i $$-0.705527\pi$$
−0.601744 + 0.798689i $$0.705527\pi$$
$$464$$ −1.18866e6 −0.256308
$$465$$ 0 0
$$466$$ −2.79027e6 −0.595225
$$467$$ 7.95350e6 1.68759 0.843794 0.536668i $$-0.180317\pi$$
0.843794 + 0.536668i $$0.180317\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 3.93033e6 0.820699
$$471$$ 0 0
$$472$$ −1.65817e6 −0.342590
$$473$$ −2.49699e6 −0.513173
$$474$$ 0 0
$$475$$ −3.46140e6 −0.703912
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −628074. −0.125731
$$479$$ 6.56370e6 1.30710 0.653552 0.756882i $$-0.273278\pi$$
0.653552 + 0.756882i $$0.273278\pi$$
$$480$$ 0 0
$$481$$ 1.90714e6 0.375854
$$482$$ 4.95532e6 0.971525
$$483$$ 0 0
$$484$$ 297096. 0.0576479
$$485$$ 4.25185e6 0.820773
$$486$$ 0 0
$$487$$ 4.66370e6 0.891062 0.445531 0.895266i $$-0.353015\pi$$
0.445531 + 0.895266i $$0.353015\pi$$
$$488$$ −7.69539e6 −1.46279
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 917227. 0.171701 0.0858506 0.996308i $$-0.472639\pi$$
0.0858506 + 0.996308i $$0.472639\pi$$
$$492$$ 0 0
$$493$$ −2.47934e6 −0.459430
$$494$$ −1.52412e6 −0.280998
$$495$$ 0 0
$$496$$ −8.46401e6 −1.54480
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 313436. 0.0563504 0.0281752 0.999603i $$-0.491030\pi$$
0.0281752 + 0.999603i $$0.491030\pi$$
$$500$$ −417087. −0.0746108
$$501$$ 0 0
$$502$$ −1.74967e6 −0.309883
$$503$$ 3.74110e6 0.659294 0.329647 0.944104i $$-0.393070\pi$$
0.329647 + 0.944104i $$0.393070\pi$$
$$504$$ 0 0
$$505$$ 800378. 0.139658
$$506$$ −137598. −0.0238910
$$507$$ 0 0
$$508$$ 320657. 0.0551292
$$509$$ −1.03017e7 −1.76244 −0.881220 0.472707i $$-0.843277\pi$$
−0.881220 + 0.472707i $$0.843277\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 6.44492e6 1.08653
$$513$$ 0 0
$$514$$ −3.03078e6 −0.505995
$$515$$ −4.89350e6 −0.813021
$$516$$ 0 0
$$517$$ −3.69711e6 −0.608326
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −995560. −0.161458
$$521$$ 6.57325e6 1.06093 0.530464 0.847707i $$-0.322018\pi$$
0.530464 + 0.847707i $$0.322018\pi$$
$$522$$ 0 0
$$523$$ 8.36532e6 1.33730 0.668650 0.743578i $$-0.266873\pi$$
0.668650 + 0.743578i $$0.266873\pi$$
$$524$$ −126497. −0.0201257
$$525$$ 0 0
$$526$$ −1.27103e6 −0.200306
$$527$$ −1.76545e7 −2.76904
$$528$$ 0 0
$$529$$ −6.41758e6 −0.997084
$$530$$ −1.83013e6 −0.283003
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1.32724e6 −0.202364
$$534$$ 0 0
$$535$$ 4.23454e6 0.639620
$$536$$ −1.03516e7 −1.55631
$$537$$ 0 0
$$538$$ −8.05263e6 −1.19945
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −8.06623e6 −1.18489 −0.592444 0.805612i $$-0.701837\pi$$
−0.592444 + 0.805612i $$0.701837\pi$$
$$542$$ −968129. −0.141558
$$543$$ 0 0
$$544$$ 1.66318e6 0.240959
$$545$$ 1.24791e6 0.179967
$$546$$ 0 0
$$547$$ 3.90775e6 0.558416 0.279208 0.960231i $$-0.409928\pi$$
0.279208 + 0.960231i $$0.409928\pi$$
$$548$$ 988902. 0.140670
$$549$$ 0 0
$$550$$ −1.83714e6 −0.258961
$$551$$ 2.38377e6 0.334492
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 6.65811e6 0.921672
$$555$$ 0 0
$$556$$ 23184.1 0.00318056
$$557$$ 1.15862e7 1.58235 0.791177 0.611588i $$-0.209469\pi$$
0.791177 + 0.611588i $$0.209469\pi$$
$$558$$ 0 0
$$559$$ 2.00203e6 0.270982
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 968334. 0.129326
$$563$$ −1.91176e6 −0.254192 −0.127096 0.991890i $$-0.540566\pi$$
−0.127096 + 0.991890i $$0.540566\pi$$
$$564$$ 0 0
$$565$$ −957041. −0.126127
$$566$$ 5.93920e6 0.779268
$$567$$ 0 0
$$568$$ 1.19441e7 1.55340
$$569$$ −8.16817e6 −1.05766 −0.528828 0.848729i $$-0.677368\pi$$
−0.528828 + 0.848729i $$0.677368\pi$$
$$570$$ 0 0
$$571$$ −7.46593e6 −0.958283 −0.479141 0.877738i $$-0.659052\pi$$
−0.479141 + 0.877738i $$0.659052\pi$$
$$572$$ 63780.3 0.00815072
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 250552. 0.0316030
$$576$$ 0 0
$$577$$ −6.88438e6 −0.860845 −0.430423 0.902627i $$-0.641635\pi$$
−0.430423 + 0.902627i $$0.641635\pi$$
$$578$$ −1.33687e7 −1.66445
$$579$$ 0 0
$$580$$ 106047. 0.0130896
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 1.72153e6 0.209770
$$584$$ 7.72363e6 0.937108
$$585$$ 0 0
$$586$$ 2.97223e6 0.357551
$$587$$ 8.91086e6 1.06739 0.533696 0.845676i $$-0.320803\pi$$
0.533696 + 0.845676i $$0.320803\pi$$
$$588$$ 0 0
$$589$$ 1.69740e7 2.01602
$$590$$ −1.73839e6 −0.205598
$$591$$ 0 0
$$592$$ −1.21710e7 −1.42732
$$593$$ −1.23430e7 −1.44140 −0.720700 0.693247i $$-0.756180\pi$$
−0.720700 + 0.693247i $$0.756180\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1.18211e6 0.136314
$$597$$ 0 0
$$598$$ 110323. 0.0126157
$$599$$ −9.05732e6 −1.03141 −0.515707 0.856765i $$-0.672471\pi$$
−0.515707 + 0.856765i $$0.672471\pi$$
$$600$$ 0 0
$$601$$ 7.41700e6 0.837611 0.418805 0.908076i $$-0.362449\pi$$
0.418805 + 0.908076i $$0.362449\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 451603. 0.0503691
$$605$$ 4.57331e6 0.507975
$$606$$ 0 0
$$607$$ −5.77813e6 −0.636525 −0.318263 0.948003i $$-0.603099\pi$$
−0.318263 + 0.948003i $$0.603099\pi$$
$$608$$ −1.59908e6 −0.175432
$$609$$ 0 0
$$610$$ −8.06770e6 −0.877860
$$611$$ 2.96426e6 0.321228
$$612$$ 0 0
$$613$$ 6.29264e6 0.676366 0.338183 0.941080i $$-0.390188\pi$$
0.338183 + 0.941080i $$0.390188\pi$$
$$614$$ 1.79180e7 1.91809
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9.79133e6 1.03545 0.517725 0.855547i $$-0.326779\pi$$
0.517725 + 0.855547i $$0.326779\pi$$
$$618$$ 0 0
$$619$$ −1.32677e7 −1.39178 −0.695889 0.718150i $$-0.744989\pi$$
−0.695889 + 0.718150i $$0.744989\pi$$
$$620$$ 755120. 0.0788927
$$621$$ 0 0
$$622$$ −3.81079e6 −0.394947
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −704759. −0.0721673
$$626$$ −3.39419e6 −0.346178
$$627$$ 0 0
$$628$$ −619567. −0.0626886
$$629$$ −2.53867e7 −2.55846
$$630$$ 0 0
$$631$$ 4.41233e6 0.441158 0.220579 0.975369i $$-0.429205\pi$$
0.220579 + 0.975369i $$0.429205\pi$$
$$632$$ 3.17245e6 0.315938
$$633$$ 0 0
$$634$$ −3.48146e6 −0.343984
$$635$$ 4.93600e6 0.485782
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 1.26518e6 0.123056
$$639$$ 0 0
$$640$$ 5.84964e6 0.564520
$$641$$ −8.13035e6 −0.781564 −0.390782 0.920483i $$-0.627795\pi$$
−0.390782 + 0.920483i $$0.627795\pi$$
$$642$$ 0 0
$$643$$ −3.12961e6 −0.298513 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 2.02882e7 1.91277
$$647$$ −1.34660e7 −1.26467 −0.632336 0.774695i $$-0.717904\pi$$
−0.632336 + 0.774695i $$0.717904\pi$$
$$648$$ 0 0
$$649$$ 1.63524e6 0.152395
$$650$$ 1.47297e6 0.136745
$$651$$ 0 0
$$652$$ 1.26076e6 0.116148
$$653$$ −1.44772e7 −1.32862 −0.664312 0.747455i $$-0.731275\pi$$
−0.664312 + 0.747455i $$0.731275\pi$$
$$654$$ 0 0
$$655$$ −1.94721e6 −0.177341
$$656$$ 8.47023e6 0.768485
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −432708. −0.0388134 −0.0194067 0.999812i $$-0.506178\pi$$
−0.0194067 + 0.999812i $$0.506178\pi$$
$$660$$ 0 0
$$661$$ −29965.2 −0.00266756 −0.00133378 0.999999i $$-0.500425\pi$$
−0.00133378 + 0.999999i $$0.500425\pi$$
$$662$$ −7.67718e6 −0.680858
$$663$$ 0 0
$$664$$ 1.90182e7 1.67398
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −172548. −0.0150174
$$668$$ 511578. 0.0443579
$$669$$ 0 0
$$670$$ −1.08524e7 −0.933986
$$671$$ 7.58899e6 0.650696
$$672$$ 0 0
$$673$$ −6.71329e6 −0.571344 −0.285672 0.958327i $$-0.592217\pi$$
−0.285672 + 0.958327i $$0.592217\pi$$
$$674$$ 8.47849e6 0.718901
$$675$$ 0 0
$$676$$ 817193. 0.0687793
$$677$$ 4.45929e6 0.373933 0.186967 0.982366i $$-0.440134\pi$$
0.186967 + 0.982366i $$0.440134\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 1.32523e7 1.09905
$$681$$ 0 0
$$682$$ 9.00892e6 0.741672
$$683$$ 1.79839e7 1.47514 0.737569 0.675272i $$-0.235974\pi$$
0.737569 + 0.675272i $$0.235974\pi$$
$$684$$ 0 0
$$685$$ 1.52225e7 1.23954
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −1.27766e7 −1.02907
$$689$$ −1.38029e6 −0.110770
$$690$$ 0 0
$$691$$ 2.50935e6 0.199925 0.0999624 0.994991i $$-0.468128\pi$$
0.0999624 + 0.994991i $$0.468128\pi$$
$$692$$ 1.38028e6 0.109573
$$693$$ 0 0
$$694$$ 1.35228e6 0.106578
$$695$$ 356882. 0.0280261
$$696$$ 0 0
$$697$$ 1.76675e7 1.37750
$$698$$ −1.01392e7 −0.787707
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −9.68649e6 −0.744512 −0.372256 0.928130i $$-0.621416\pi$$
−0.372256 + 0.928130i $$0.621416\pi$$
$$702$$ 0 0
$$703$$ 2.44081e7 1.86271
$$704$$ −6.41817e6 −0.488067
$$705$$ 0 0
$$706$$ 9.64681e6 0.728403
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −9.52720e6 −0.711786 −0.355893 0.934527i $$-0.615823\pi$$
−0.355893 + 0.934527i $$0.615823\pi$$
$$710$$ 1.25220e7 0.932238
$$711$$ 0 0
$$712$$ −1.62896e7 −1.20423
$$713$$ −1.22865e6 −0.0905118
$$714$$ 0 0
$$715$$ 981794. 0.0718217
$$716$$ −508313. −0.0370552
$$717$$ 0 0
$$718$$ −2.43751e7 −1.76456
$$719$$ −757176. −0.0546229 −0.0273114 0.999627i $$-0.508695\pi$$
−0.0273114 + 0.999627i $$0.508695\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −6.02081e6 −0.429845
$$723$$ 0 0
$$724$$ −439925. −0.0311912
$$725$$ −2.30378e6 −0.162778
$$726$$ 0 0
$$727$$ −2.66570e7 −1.87058 −0.935288 0.353888i $$-0.884859\pi$$
−0.935288 + 0.353888i $$0.884859\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 8.09731e6 0.562385
$$731$$ −2.66498e7 −1.84459
$$732$$ 0 0
$$733$$ 2.70857e7 1.86200 0.931001 0.365017i $$-0.118937\pi$$
0.931001 + 0.365017i $$0.118937\pi$$
$$734$$ −1.72698e7 −1.18317
$$735$$ 0 0
$$736$$ 115748. 0.00787625
$$737$$ 1.02085e7 0.692298
$$738$$ 0 0
$$739$$ 2.19507e7 1.47855 0.739276 0.673402i $$-0.235168\pi$$
0.739276 + 0.673402i $$0.235168\pi$$
$$740$$ 1.08584e6 0.0728932
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −5.77370e6 −0.383691 −0.191846 0.981425i $$-0.561447\pi$$
−0.191846 + 0.981425i $$0.561447\pi$$
$$744$$ 0 0
$$745$$ 1.81966e7 1.20116
$$746$$ 1.25905e7 0.828318
$$747$$ 0 0
$$748$$ −849005. −0.0554825
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −8.37749e6 −0.542018 −0.271009 0.962577i $$-0.587357\pi$$
−0.271009 + 0.962577i $$0.587357\pi$$
$$752$$ −1.89174e7 −1.21988
$$753$$ 0 0
$$754$$ −1.01440e6 −0.0649800
$$755$$ 6.95170e6 0.443837
$$756$$ 0 0
$$757$$ 1.18828e7 0.753665 0.376833 0.926281i $$-0.377013\pi$$
0.376833 + 0.926281i $$0.377013\pi$$
$$758$$ 3.22329e6 0.203764
$$759$$ 0 0
$$760$$ −1.27415e7 −0.800177
$$761$$ −1.76940e7 −1.10755 −0.553777 0.832665i $$-0.686814\pi$$
−0.553777 + 0.832665i $$0.686814\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 405811. 0.0251531
$$765$$ 0 0
$$766$$ −1.95937e7 −1.20655
$$767$$ −1.31110e6 −0.0804726
$$768$$ 0 0
$$769$$ −4.12006e6 −0.251239 −0.125620 0.992078i $$-0.540092\pi$$
−0.125620 + 0.992078i $$0.540092\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −275452. −0.0166342
$$773$$ −1.26864e7 −0.763642 −0.381821 0.924236i $$-0.624703\pi$$
−0.381821 + 0.924236i $$0.624703\pi$$
$$774$$ 0 0
$$775$$ −1.64043e7 −0.981080
$$776$$ −2.20879e7 −1.31674
$$777$$ 0 0
$$778$$ −1.22402e7 −0.725001
$$779$$ −1.69865e7 −1.00290
$$780$$ 0 0
$$781$$ −1.17790e7 −0.691002
$$782$$ −1.46855e6 −0.0858761
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −9.53723e6 −0.552393
$$786$$ 0 0
$$787$$ 9.14809e6 0.526494 0.263247 0.964728i $$-0.415207\pi$$
0.263247 + 0.964728i $$0.415207\pi$$
$$788$$ 525578. 0.0301524
$$789$$ 0 0
$$790$$ 3.32594e6 0.189603
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −6.08468e6 −0.343602
$$794$$ 2.49802e7 1.40619
$$795$$ 0 0
$$796$$ −1.73125e6 −0.0968451
$$797$$ 1.10180e7 0.614408 0.307204 0.951644i $$-0.400607\pi$$
0.307204 + 0.951644i $$0.400607\pi$$
$$798$$ 0 0
$$799$$ −3.94585e7 −2.18662
$$800$$ 1.54541e6 0.0853727
$$801$$ 0 0
$$802$$ 3.27009e7 1.79524
$$803$$ −7.61684e6 −0.416856
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −7.22315e6 −0.391642
$$807$$ 0 0
$$808$$ −4.15788e6 −0.224049
$$809$$ 3.10273e7 1.66676 0.833378 0.552703i $$-0.186404\pi$$
0.833378 + 0.552703i $$0.186404\pi$$
$$810$$ 0 0
$$811$$ −2.94456e7 −1.57206 −0.786028 0.618191i $$-0.787866\pi$$
−0.786028 + 0.618191i $$0.787866\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 1.29546e7 0.685271
$$815$$ 1.94073e7 1.02346
$$816$$ 0 0
$$817$$ 2.56226e7 1.34297
$$818$$ 1.93526e7 1.01124
$$819$$ 0 0
$$820$$ −755675. −0.0392464
$$821$$ 5.23311e6 0.270958 0.135479 0.990780i $$-0.456743\pi$$
0.135479 + 0.990780i $$0.456743\pi$$
$$822$$ 0 0
$$823$$ 1.41631e7 0.728884 0.364442 0.931226i $$-0.381260\pi$$
0.364442 + 0.931226i $$0.381260\pi$$
$$824$$ 2.54212e7 1.30430
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −2.98006e7 −1.51517 −0.757585 0.652737i $$-0.773621\pi$$
−0.757585 + 0.652737i $$0.773621\pi$$
$$828$$ 0 0
$$829$$ −1.99304e7 −1.00723 −0.503617 0.863927i $$-0.667998\pi$$
−0.503617 + 0.863927i $$0.667998\pi$$
$$830$$ 1.99383e7 1.00460
$$831$$ 0 0
$$832$$ 5.14594e6 0.257725
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 7.87492e6 0.390868
$$836$$ 816280. 0.0403946
$$837$$ 0 0
$$838$$ 1.09673e7 0.539500
$$839$$ 1.88103e7 0.922552 0.461276 0.887257i $$-0.347392\pi$$
0.461276 + 0.887257i $$0.347392\pi$$
$$840$$ 0 0
$$841$$ −1.89246e7 −0.922650
$$842$$ 3.21320e7 1.56191
$$843$$ 0 0
$$844$$ 1.65085e6 0.0797724
$$845$$ 1.25794e7 0.606062
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 8.80874e6 0.420653
$$849$$ 0 0
$$850$$ −1.96073e7 −0.930833
$$851$$ −1.76677e6 −0.0836288
$$852$$ 0 0
$$853$$ 2.05980e7 0.969285 0.484643 0.874712i $$-0.338950\pi$$
0.484643 + 0.874712i $$0.338950\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −2.19980e7 −1.02612
$$857$$ 2.47572e7 1.15146 0.575731 0.817639i $$-0.304718\pi$$
0.575731 + 0.817639i $$0.304718\pi$$
$$858$$ 0 0
$$859$$ −3.91065e7 −1.80828 −0.904141 0.427234i $$-0.859488\pi$$
−0.904141 + 0.427234i $$0.859488\pi$$
$$860$$ 1.13987e6 0.0525544
$$861$$ 0 0
$$862$$ −2.07527e7 −0.951275
$$863$$ 3.93363e7 1.79790 0.898952 0.438048i $$-0.144330\pi$$
0.898952 + 0.438048i $$0.144330\pi$$
$$864$$ 0 0
$$865$$ 2.12472e7 0.965522
$$866$$ −3.59151e7 −1.62735
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3.12859e6 −0.140540
$$870$$ 0 0
$$871$$ −8.18494e6 −0.365570
$$872$$ −6.48277e6 −0.288715
$$873$$ 0 0
$$874$$ 1.41195e6 0.0625229
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.09253e6 0.355292 0.177646 0.984094i $$-0.443152\pi$$
0.177646 + 0.984094i $$0.443152\pi$$
$$878$$ 2.47818e7 1.08492
$$879$$ 0 0
$$880$$ −6.26564e6 −0.272746
$$881$$ 4.05755e6 0.176126 0.0880631 0.996115i $$-0.471932\pi$$
0.0880631 + 0.996115i $$0.471932\pi$$
$$882$$ 0 0
$$883$$ 1.79813e7 0.776102 0.388051 0.921638i $$-0.373148\pi$$
0.388051 + 0.921638i $$0.373148\pi$$
$$884$$ 680713. 0.0292977
$$885$$ 0 0
$$886$$ 2.78595e7 1.19231
$$887$$ 1.53139e7 0.653547 0.326773 0.945103i $$-0.394039\pi$$
0.326773 + 0.945103i $$0.394039\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −1.70777e7 −0.722693
$$891$$ 0 0
$$892$$ −98725.0 −0.00415447
$$893$$ 3.79376e7 1.59199
$$894$$ 0 0
$$895$$ −7.82466e6 −0.326519
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 3.24377e7 1.34233
$$899$$ 1.12972e7 0.466200
$$900$$ 0 0
$$901$$ 1.83735e7 0.754016
$$902$$ −9.01554e6 −0.368957
$$903$$ 0 0
$$904$$ 4.97172e6 0.202342
$$905$$ −6.77194e6 −0.274847
$$906$$ 0 0
$$907$$ −2.17757e7 −0.878930 −0.439465 0.898260i $$-0.644832\pi$$
−0.439465 + 0.898260i $$0.644832\pi$$
$$908$$ 1.73623e6 0.0698865
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 5.35112e6 0.213623 0.106812 0.994279i $$-0.465936\pi$$
0.106812 + 0.994279i $$0.465936\pi$$
$$912$$ 0 0
$$913$$ −1.87552e7 −0.744639
$$914$$ 1.41511e7 0.560306
$$915$$ 0 0
$$916$$ −2.20124e6 −0.0866818
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −2.67858e7 −1.04620 −0.523102 0.852270i $$-0.675225\pi$$
−0.523102 + 0.852270i $$0.675225\pi$$
$$920$$ 922285. 0.0359249
$$921$$ 0 0
$$922$$ 2.45904e7 0.952661
$$923$$ 9.44411e6 0.364886
$$924$$ 0 0
$$925$$ −2.35890e7 −0.906474
$$926$$ 3.02336e7 1.15868
$$927$$ 0 0
$$928$$ −1.06428e6 −0.0405683
$$929$$ −739974. −0.0281305 −0.0140652 0.999901i $$-0.504477\pi$$
−0.0140652 + 0.999901i $$0.504477\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −1.19817e6 −0.0451834
$$933$$ 0 0
$$934$$ −4.33166e7 −1.62475
$$935$$ −1.30691e7 −0.488895
$$936$$ 0 0
$$937$$ −122654. −0.00456385 −0.00228193 0.999997i $$-0.500726\pi$$
−0.00228193 + 0.999997i $$0.500726\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 1.68772e6 0.0622991
$$941$$ −1.22466e7 −0.450861 −0.225431 0.974259i $$-0.572379\pi$$
−0.225431 + 0.974259i $$0.572379\pi$$
$$942$$ 0 0
$$943$$ 1.22956e6 0.0450266
$$944$$ 8.36722e6 0.305599
$$945$$ 0 0
$$946$$ 1.35992e7 0.494065
$$947$$ 3.71174e6 0.134494 0.0672470 0.997736i $$-0.478578\pi$$
0.0672470 + 0.997736i $$0.478578\pi$$
$$948$$ 0 0
$$949$$ 6.10701e6 0.220122
$$950$$ 1.88516e7 0.677702
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 2.32857e7 0.830533 0.415266 0.909700i $$-0.363688\pi$$
0.415266 + 0.909700i $$0.363688\pi$$
$$954$$ 0 0
$$955$$ 6.24681e6 0.221641
$$956$$ −269702. −0.00954418
$$957$$ 0 0
$$958$$ −3.57474e7 −1.25843
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5.18142e7 1.80984
$$962$$ −1.03867e7 −0.361859
$$963$$ 0 0
$$964$$ 2.12787e6 0.0737483
$$965$$ −4.24013e6 −0.146575
$$966$$ 0 0
$$967$$ 2.25257e7 0.774661 0.387330 0.921941i $$-0.373397\pi$$
0.387330 + 0.921941i $$0.373397\pi$$
$$968$$ −2.37579e7 −0.814927
$$969$$ 0 0
$$970$$ −2.31565e7 −0.790212
$$971$$ −1.99456e7 −0.678890 −0.339445 0.940626i $$-0.610239\pi$$
−0.339445 + 0.940626i $$0.610239\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −2.53995e7 −0.857884
$$975$$ 0 0
$$976$$ 3.88314e7 1.30484
$$977$$ 959342. 0.0321542 0.0160771 0.999871i $$-0.494882\pi$$
0.0160771 + 0.999871i $$0.494882\pi$$
$$978$$ 0 0
$$979$$ 1.60643e7 0.535681
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −4.99542e6 −0.165308
$$983$$ −5.22097e7 −1.72333 −0.861663 0.507481i $$-0.830577\pi$$
−0.861663 + 0.507481i $$0.830577\pi$$
$$984$$ 0 0
$$985$$ 8.09042e6 0.265693
$$986$$ 1.35030e7 0.442323
$$987$$ 0 0
$$988$$ −654475. −0.0213305
$$989$$ −1.85468e6 −0.0602945
$$990$$ 0 0
$$991$$ −1.76305e7 −0.570269 −0.285134 0.958488i $$-0.592038\pi$$
−0.285134 + 0.958488i $$0.592038\pi$$
$$992$$ −7.57836e6 −0.244510
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −2.66498e7 −0.853369
$$996$$ 0 0
$$997$$ 4.04875e7 1.28998 0.644990 0.764191i $$-0.276862\pi$$
0.644990 + 0.764191i $$0.276862\pi$$
$$998$$ −1.70704e6 −0.0542522
$$999$$ 0 0
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 441.6.a.q.1.1 2
3.2 odd 2 147.6.a.h.1.2 2
7.6 odd 2 441.6.a.r.1.1 2
21.2 odd 6 147.6.e.n.67.1 4
21.5 even 6 147.6.e.m.67.1 4
21.11 odd 6 147.6.e.n.79.1 4
21.17 even 6 147.6.e.m.79.1 4
21.20 even 2 147.6.a.j.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.h.1.2 2 3.2 odd 2
147.6.a.j.1.2 yes 2 21.20 even 2
147.6.e.m.67.1 4 21.5 even 6
147.6.e.m.79.1 4 21.17 even 6
147.6.e.n.67.1 4 21.2 odd 6
147.6.e.n.79.1 4 21.11 odd 6
441.6.a.q.1.1 2 1.1 even 1 trivial
441.6.a.r.1.1 2 7.6 odd 2