# Properties

 Label 400.6.a.u.1.2 Level $400$ Weight $6$ Character 400.1 Self dual yes Analytic conductor $64.154$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1535279252$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ x^2 - x - 60 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 200) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$8.26209$$ of defining polynomial Character $$\chi$$ $$=$$ 400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+19.5242 q^{3} -35.0483 q^{7} +138.193 q^{9} +O(q^{10})$$ $$q+19.5242 q^{3} -35.0483 q^{7} +138.193 q^{9} +426.008 q^{11} +1103.26 q^{13} +109.387 q^{17} -495.926 q^{19} -684.290 q^{21} -2497.37 q^{23} -2046.26 q^{27} -42.4221 q^{29} +7999.56 q^{31} +8317.45 q^{33} -13763.7 q^{37} +21540.2 q^{39} +11863.6 q^{41} +16816.0 q^{43} +13036.0 q^{47} -15578.6 q^{49} +2135.69 q^{51} +17817.3 q^{53} -9682.55 q^{57} +47346.1 q^{59} -22782.9 q^{61} -4843.45 q^{63} +39418.3 q^{67} -48759.1 q^{69} +1616.32 q^{71} +53293.3 q^{73} -14930.9 q^{77} +6516.98 q^{79} -73532.6 q^{81} -46016.4 q^{83} -828.257 q^{87} +113488. q^{89} -38667.3 q^{91} +156185. q^{93} +107418. q^{97} +58871.4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{3} - 8 q^{7} + 28 q^{9}+O(q^{10})$$ 2 * q + 8 * q^3 - 8 * q^7 + 28 * q^9 $$2 q + 8 q^{3} - 8 q^{7} + 28 q^{9} + 200 q^{11} + 592 q^{13} - 278 q^{17} + 840 q^{19} - 996 q^{21} - 1952 q^{23} + 2024 q^{27} - 4680 q^{29} + 5008 q^{31} + 10922 q^{33} - 12500 q^{37} + 27432 q^{39} - 5334 q^{41} + 224 q^{43} + 26072 q^{47} - 31654 q^{49} + 6600 q^{51} + 46812 q^{53} - 25078 q^{57} + 81776 q^{59} - 46932 q^{61} - 7824 q^{63} + 68808 q^{67} - 55044 q^{69} - 7448 q^{71} + 108822 q^{73} - 21044 q^{77} + 108104 q^{79} - 93662 q^{81} + 27224 q^{83} + 52616 q^{87} + 70990 q^{89} - 52496 q^{91} + 190660 q^{93} + 96852 q^{97} + 83776 q^{99}+O(q^{100})$$ 2 * q + 8 * q^3 - 8 * q^7 + 28 * q^9 + 200 * q^11 + 592 * q^13 - 278 * q^17 + 840 * q^19 - 996 * q^21 - 1952 * q^23 + 2024 * q^27 - 4680 * q^29 + 5008 * q^31 + 10922 * q^33 - 12500 * q^37 + 27432 * q^39 - 5334 * q^41 + 224 * q^43 + 26072 * q^47 - 31654 * q^49 + 6600 * q^51 + 46812 * q^53 - 25078 * q^57 + 81776 * q^59 - 46932 * q^61 - 7824 * q^63 + 68808 * q^67 - 55044 * q^69 - 7448 * q^71 + 108822 * q^73 - 21044 * q^77 + 108104 * q^79 - 93662 * q^81 + 27224 * q^83 + 52616 * q^87 + 70990 * q^89 - 52496 * q^91 + 190660 * q^93 + 96852 * q^97 + 83776 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 19.5242 1.25248 0.626238 0.779632i $$-0.284594\pi$$
0.626238 + 0.779632i $$0.284594\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −35.0483 −0.270348 −0.135174 0.990822i $$-0.543159\pi$$
−0.135174 + 0.990822i $$0.543159\pi$$
$$8$$ 0 0
$$9$$ 138.193 0.568697
$$10$$ 0 0
$$11$$ 426.008 1.06154 0.530769 0.847516i $$-0.321903\pi$$
0.530769 + 0.847516i $$0.321903\pi$$
$$12$$ 0 0
$$13$$ 1103.26 1.81058 0.905291 0.424791i $$-0.139653\pi$$
0.905291 + 0.424791i $$0.139653\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 109.387 0.0918000 0.0459000 0.998946i $$-0.485384\pi$$
0.0459000 + 0.998946i $$0.485384\pi$$
$$18$$ 0 0
$$19$$ −495.926 −0.315161 −0.157581 0.987506i $$-0.550369\pi$$
−0.157581 + 0.987506i $$0.550369\pi$$
$$20$$ 0 0
$$21$$ −684.290 −0.338604
$$22$$ 0 0
$$23$$ −2497.37 −0.984381 −0.492190 0.870488i $$-0.663804\pi$$
−0.492190 + 0.870488i $$0.663804\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −2046.26 −0.540197
$$28$$ 0 0
$$29$$ −42.4221 −0.00936694 −0.00468347 0.999989i $$-0.501491\pi$$
−0.00468347 + 0.999989i $$0.501491\pi$$
$$30$$ 0 0
$$31$$ 7999.56 1.49507 0.747535 0.664222i $$-0.231237\pi$$
0.747535 + 0.664222i $$0.231237\pi$$
$$32$$ 0 0
$$33$$ 8317.45 1.32955
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −13763.7 −1.65284 −0.826420 0.563054i $$-0.809626\pi$$
−0.826420 + 0.563054i $$0.809626\pi$$
$$38$$ 0 0
$$39$$ 21540.2 2.26771
$$40$$ 0 0
$$41$$ 11863.6 1.10219 0.551097 0.834441i $$-0.314210\pi$$
0.551097 + 0.834441i $$0.314210\pi$$
$$42$$ 0 0
$$43$$ 16816.0 1.38692 0.693461 0.720494i $$-0.256085\pi$$
0.693461 + 0.720494i $$0.256085\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 13036.0 0.860795 0.430397 0.902639i $$-0.358373\pi$$
0.430397 + 0.902639i $$0.358373\pi$$
$$48$$ 0 0
$$49$$ −15578.6 −0.926912
$$50$$ 0 0
$$51$$ 2135.69 0.114977
$$52$$ 0 0
$$53$$ 17817.3 0.871269 0.435634 0.900124i $$-0.356524\pi$$
0.435634 + 0.900124i $$0.356524\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −9682.55 −0.394732
$$58$$ 0 0
$$59$$ 47346.1 1.77074 0.885368 0.464891i $$-0.153906\pi$$
0.885368 + 0.464891i $$0.153906\pi$$
$$60$$ 0 0
$$61$$ −22782.9 −0.783944 −0.391972 0.919977i $$-0.628207\pi$$
−0.391972 + 0.919977i $$0.628207\pi$$
$$62$$ 0 0
$$63$$ −4843.45 −0.153746
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 39418.3 1.07278 0.536390 0.843970i $$-0.319787\pi$$
0.536390 + 0.843970i $$0.319787\pi$$
$$68$$ 0 0
$$69$$ −48759.1 −1.23291
$$70$$ 0 0
$$71$$ 1616.32 0.0380523 0.0190261 0.999819i $$-0.493943\pi$$
0.0190261 + 0.999819i $$0.493943\pi$$
$$72$$ 0 0
$$73$$ 53293.3 1.17048 0.585242 0.810859i $$-0.301000\pi$$
0.585242 + 0.810859i $$0.301000\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −14930.9 −0.286984
$$78$$ 0 0
$$79$$ 6516.98 0.117484 0.0587420 0.998273i $$-0.481291\pi$$
0.0587420 + 0.998273i $$0.481291\pi$$
$$80$$ 0 0
$$81$$ −73532.6 −1.24528
$$82$$ 0 0
$$83$$ −46016.4 −0.733191 −0.366595 0.930380i $$-0.619477\pi$$
−0.366595 + 0.930380i $$0.619477\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −828.257 −0.0117319
$$88$$ 0 0
$$89$$ 113488. 1.51872 0.759358 0.650673i $$-0.225513\pi$$
0.759358 + 0.650673i $$0.225513\pi$$
$$90$$ 0 0
$$91$$ −38667.3 −0.489487
$$92$$ 0 0
$$93$$ 156185. 1.87254
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 107418. 1.15917 0.579585 0.814912i $$-0.303215\pi$$
0.579585 + 0.814912i $$0.303215\pi$$
$$98$$ 0 0
$$99$$ 58871.4 0.603694
$$100$$ 0 0
$$101$$ −197554. −1.92700 −0.963502 0.267703i $$-0.913736\pi$$
−0.963502 + 0.267703i $$0.913736\pi$$
$$102$$ 0 0
$$103$$ 81026.0 0.752543 0.376272 0.926509i $$-0.377206\pi$$
0.376272 + 0.926509i $$0.377206\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 137442. 1.16054 0.580271 0.814423i $$-0.302946\pi$$
0.580271 + 0.814423i $$0.302946\pi$$
$$108$$ 0 0
$$109$$ −68693.1 −0.553792 −0.276896 0.960900i $$-0.589306\pi$$
−0.276896 + 0.960900i $$0.589306\pi$$
$$110$$ 0 0
$$111$$ −268725. −2.07014
$$112$$ 0 0
$$113$$ −139632. −1.02870 −0.514352 0.857579i $$-0.671967\pi$$
−0.514352 + 0.857579i $$0.671967\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 152463. 1.02967
$$118$$ 0 0
$$119$$ −3833.83 −0.0248179
$$120$$ 0 0
$$121$$ 20431.5 0.126864
$$122$$ 0 0
$$123$$ 231628. 1.38047
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 65230.1 0.358871 0.179436 0.983770i $$-0.442573\pi$$
0.179436 + 0.983770i $$0.442573\pi$$
$$128$$ 0 0
$$129$$ 328319. 1.73709
$$130$$ 0 0
$$131$$ −118542. −0.603524 −0.301762 0.953383i $$-0.597575\pi$$
−0.301762 + 0.953383i $$0.597575\pi$$
$$132$$ 0 0
$$133$$ 17381.4 0.0852031
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 84757.8 0.385814 0.192907 0.981217i $$-0.438208\pi$$
0.192907 + 0.981217i $$0.438208\pi$$
$$138$$ 0 0
$$139$$ −168334. −0.738982 −0.369491 0.929234i $$-0.620468\pi$$
−0.369491 + 0.929234i $$0.620468\pi$$
$$140$$ 0 0
$$141$$ 254517. 1.07813
$$142$$ 0 0
$$143$$ 469996. 1.92200
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −304160. −1.16094
$$148$$ 0 0
$$149$$ 67628.6 0.249554 0.124777 0.992185i $$-0.460178\pi$$
0.124777 + 0.992185i $$0.460178\pi$$
$$150$$ 0 0
$$151$$ 65622.3 0.234212 0.117106 0.993119i $$-0.462638\pi$$
0.117106 + 0.993119i $$0.462638\pi$$
$$152$$ 0 0
$$153$$ 15116.5 0.0522064
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −297885. −0.964495 −0.482247 0.876035i $$-0.660179\pi$$
−0.482247 + 0.876035i $$0.660179\pi$$
$$158$$ 0 0
$$159$$ 347868. 1.09124
$$160$$ 0 0
$$161$$ 87528.7 0.266125
$$162$$ 0 0
$$163$$ 473879. 1.39701 0.698503 0.715607i $$-0.253850\pi$$
0.698503 + 0.715607i $$0.253850\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −267673. −0.742701 −0.371351 0.928493i $$-0.621105\pi$$
−0.371351 + 0.928493i $$0.621105\pi$$
$$168$$ 0 0
$$169$$ 845883. 2.27821
$$170$$ 0 0
$$171$$ −68533.7 −0.179231
$$172$$ 0 0
$$173$$ 703284. 1.78655 0.893276 0.449509i $$-0.148401\pi$$
0.893276 + 0.449509i $$0.148401\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 924393. 2.21780
$$178$$ 0 0
$$179$$ −635322. −1.48204 −0.741022 0.671481i $$-0.765659\pi$$
−0.741022 + 0.671481i $$0.765659\pi$$
$$180$$ 0 0
$$181$$ −547906. −1.24311 −0.621555 0.783370i $$-0.713499\pi$$
−0.621555 + 0.783370i $$0.713499\pi$$
$$182$$ 0 0
$$183$$ −444818. −0.981872
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 46599.6 0.0974492
$$188$$ 0 0
$$189$$ 71718.1 0.146041
$$190$$ 0 0
$$191$$ 80607.9 0.159880 0.0799400 0.996800i $$-0.474527\pi$$
0.0799400 + 0.996800i $$0.474527\pi$$
$$192$$ 0 0
$$193$$ −26549.3 −0.0513049 −0.0256525 0.999671i $$-0.508166\pi$$
−0.0256525 + 0.999671i $$0.508166\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −703676. −1.29184 −0.645918 0.763407i $$-0.723525\pi$$
−0.645918 + 0.763407i $$0.723525\pi$$
$$198$$ 0 0
$$199$$ −73599.5 −0.131747 −0.0658737 0.997828i $$-0.520983\pi$$
−0.0658737 + 0.997828i $$0.520983\pi$$
$$200$$ 0 0
$$201$$ 769610. 1.34363
$$202$$ 0 0
$$203$$ 1486.83 0.00253233
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −345120. −0.559815
$$208$$ 0 0
$$209$$ −211268. −0.334556
$$210$$ 0 0
$$211$$ 1.00808e6 1.55879 0.779395 0.626533i $$-0.215526\pi$$
0.779395 + 0.626533i $$0.215526\pi$$
$$212$$ 0 0
$$213$$ 31557.2 0.0476596
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −280371. −0.404189
$$218$$ 0 0
$$219$$ 1.04051e6 1.46600
$$220$$ 0 0
$$221$$ 120682. 0.166211
$$222$$ 0 0
$$223$$ −141067. −0.189961 −0.0949805 0.995479i $$-0.530279\pi$$
−0.0949805 + 0.995479i $$0.530279\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1.14301e6 1.47226 0.736129 0.676842i $$-0.236652\pi$$
0.736129 + 0.676842i $$0.236652\pi$$
$$228$$ 0 0
$$229$$ −1.29731e6 −1.63476 −0.817380 0.576099i $$-0.804574\pi$$
−0.817380 + 0.576099i $$0.804574\pi$$
$$230$$ 0 0
$$231$$ −291513. −0.359441
$$232$$ 0 0
$$233$$ −626232. −0.755693 −0.377846 0.925868i $$-0.623335\pi$$
−0.377846 + 0.925868i $$0.623335\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 127239. 0.147146
$$238$$ 0 0
$$239$$ −270281. −0.306070 −0.153035 0.988221i $$-0.548905\pi$$
−0.153035 + 0.988221i $$0.548905\pi$$
$$240$$ 0 0
$$241$$ −694339. −0.770068 −0.385034 0.922902i $$-0.625810\pi$$
−0.385034 + 0.922902i $$0.625810\pi$$
$$242$$ 0 0
$$243$$ −938421. −1.01949
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −547134. −0.570626
$$248$$ 0 0
$$249$$ −898431. −0.918304
$$250$$ 0 0
$$251$$ −864085. −0.865710 −0.432855 0.901464i $$-0.642494\pi$$
−0.432855 + 0.901464i $$0.642494\pi$$
$$252$$ 0 0
$$253$$ −1.06390e6 −1.04496
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1.00660e6 −0.950660 −0.475330 0.879808i $$-0.657671\pi$$
−0.475330 + 0.879808i $$0.657671\pi$$
$$258$$ 0 0
$$259$$ 482395. 0.446841
$$260$$ 0 0
$$261$$ −5862.46 −0.00532695
$$262$$ 0 0
$$263$$ −931373. −0.830299 −0.415149 0.909753i $$-0.636271\pi$$
−0.415149 + 0.909753i $$0.636271\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 2.21577e6 1.90216
$$268$$ 0 0
$$269$$ −1.96236e6 −1.65348 −0.826738 0.562587i $$-0.809806\pi$$
−0.826738 + 0.562587i $$0.809806\pi$$
$$270$$ 0 0
$$271$$ −1.12152e6 −0.927649 −0.463825 0.885927i $$-0.653523\pi$$
−0.463825 + 0.885927i $$0.653523\pi$$
$$272$$ 0 0
$$273$$ −754948. −0.613070
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −105917. −0.0829401 −0.0414701 0.999140i $$-0.513204\pi$$
−0.0414701 + 0.999140i $$0.513204\pi$$
$$278$$ 0 0
$$279$$ 1.10549e6 0.850242
$$280$$ 0 0
$$281$$ −1.05276e6 −0.795362 −0.397681 0.917524i $$-0.630185\pi$$
−0.397681 + 0.917524i $$0.630185\pi$$
$$282$$ 0 0
$$283$$ −950210. −0.705267 −0.352633 0.935762i $$-0.614714\pi$$
−0.352633 + 0.935762i $$0.614714\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −415801. −0.297975
$$288$$ 0 0
$$289$$ −1.40789e6 −0.991573
$$290$$ 0 0
$$291$$ 2.09725e6 1.45183
$$292$$ 0 0
$$293$$ 381801. 0.259817 0.129909 0.991526i $$-0.458532\pi$$
0.129909 + 0.991526i $$0.458532\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −871723. −0.573440
$$298$$ 0 0
$$299$$ −2.75524e6 −1.78230
$$300$$ 0 0
$$301$$ −589373. −0.374951
$$302$$ 0 0
$$303$$ −3.85708e6 −2.41353
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −1.61048e6 −0.975238 −0.487619 0.873057i $$-0.662134\pi$$
−0.487619 + 0.873057i $$0.662134\pi$$
$$308$$ 0 0
$$309$$ 1.58197e6 0.942543
$$310$$ 0 0
$$311$$ 2.15652e6 1.26431 0.632153 0.774843i $$-0.282171\pi$$
0.632153 + 0.774843i $$0.282171\pi$$
$$312$$ 0 0
$$313$$ 364108. 0.210073 0.105036 0.994468i $$-0.466504\pi$$
0.105036 + 0.994468i $$0.466504\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.03846e6 0.580420 0.290210 0.956963i $$-0.406275\pi$$
0.290210 + 0.956963i $$0.406275\pi$$
$$318$$ 0 0
$$319$$ −18072.2 −0.00994336
$$320$$ 0 0
$$321$$ 2.68345e6 1.45355
$$322$$ 0 0
$$323$$ −54247.8 −0.0289318
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −1.34118e6 −0.693612
$$328$$ 0 0
$$329$$ −456890. −0.232714
$$330$$ 0 0
$$331$$ 2.82942e6 1.41948 0.709738 0.704466i $$-0.248814\pi$$
0.709738 + 0.704466i $$0.248814\pi$$
$$332$$ 0 0
$$333$$ −1.90205e6 −0.939966
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −804843. −0.386043 −0.193022 0.981194i $$-0.561829\pi$$
−0.193022 + 0.981194i $$0.561829\pi$$
$$338$$ 0 0
$$339$$ −2.72621e6 −1.28843
$$340$$ 0 0
$$341$$ 3.40787e6 1.58708
$$342$$ 0 0
$$343$$ 1.13506e6 0.520936
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 759889. 0.338787 0.169393 0.985549i $$-0.445819\pi$$
0.169393 + 0.985549i $$0.445819\pi$$
$$348$$ 0 0
$$349$$ 580673. 0.255193 0.127596 0.991826i $$-0.459274\pi$$
0.127596 + 0.991826i $$0.459274\pi$$
$$350$$ 0 0
$$351$$ −2.25755e6 −0.978071
$$352$$ 0 0
$$353$$ 1.64210e6 0.701393 0.350697 0.936489i $$-0.385945\pi$$
0.350697 + 0.936489i $$0.385945\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −74852.3 −0.0310838
$$358$$ 0 0
$$359$$ 2.88181e6 1.18013 0.590065 0.807356i $$-0.299102\pi$$
0.590065 + 0.807356i $$0.299102\pi$$
$$360$$ 0 0
$$361$$ −2.23016e6 −0.900673
$$362$$ 0 0
$$363$$ 398909. 0.158894
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3.83418e6 −1.48596 −0.742980 0.669314i $$-0.766588\pi$$
−0.742980 + 0.669314i $$0.766588\pi$$
$$368$$ 0 0
$$369$$ 1.63947e6 0.626814
$$370$$ 0 0
$$371$$ −624467. −0.235545
$$372$$ 0 0
$$373$$ −3.19341e6 −1.18845 −0.594227 0.804298i $$-0.702542\pi$$
−0.594227 + 0.804298i $$0.702542\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −46802.5 −0.0169596
$$378$$ 0 0
$$379$$ −3.47338e6 −1.24209 −0.621046 0.783774i $$-0.713292\pi$$
−0.621046 + 0.783774i $$0.713292\pi$$
$$380$$ 0 0
$$381$$ 1.27356e6 0.449478
$$382$$ 0 0
$$383$$ 1.56341e6 0.544599 0.272299 0.962213i $$-0.412216\pi$$
0.272299 + 0.962213i $$0.412216\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.32386e6 0.788738
$$388$$ 0 0
$$389$$ 4.98980e6 1.67190 0.835949 0.548808i $$-0.184918\pi$$
0.835949 + 0.548808i $$0.184918\pi$$
$$390$$ 0 0
$$391$$ −273179. −0.0903661
$$392$$ 0 0
$$393$$ −2.31444e6 −0.755899
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.37245e6 0.437038 0.218519 0.975833i $$-0.429877\pi$$
0.218519 + 0.975833i $$0.429877\pi$$
$$398$$ 0 0
$$399$$ 339357. 0.106715
$$400$$ 0 0
$$401$$ 5.44247e6 1.69019 0.845094 0.534617i $$-0.179544\pi$$
0.845094 + 0.534617i $$0.179544\pi$$
$$402$$ 0 0
$$403$$ 8.82557e6 2.70695
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −5.86344e6 −1.75455
$$408$$ 0 0
$$409$$ −6.03830e6 −1.78487 −0.892435 0.451176i $$-0.851005\pi$$
−0.892435 + 0.451176i $$0.851005\pi$$
$$410$$ 0 0
$$411$$ 1.65483e6 0.483223
$$412$$ 0 0
$$413$$ −1.65940e6 −0.478714
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −3.28657e6 −0.925557
$$418$$ 0 0
$$419$$ −770366. −0.214369 −0.107184 0.994239i $$-0.534184\pi$$
−0.107184 + 0.994239i $$0.534184\pi$$
$$420$$ 0 0
$$421$$ 2.00249e6 0.550636 0.275318 0.961353i $$-0.411217\pi$$
0.275318 + 0.961353i $$0.411217\pi$$
$$422$$ 0 0
$$423$$ 1.80149e6 0.489532
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 798504. 0.211937
$$428$$ 0 0
$$429$$ 9.17628e6 2.40726
$$430$$ 0 0
$$431$$ −5.56690e6 −1.44351 −0.721756 0.692148i $$-0.756665\pi$$
−0.721756 + 0.692148i $$0.756665\pi$$
$$432$$ 0 0
$$433$$ 2.22941e6 0.571441 0.285720 0.958313i $$-0.407767\pi$$
0.285720 + 0.958313i $$0.407767\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.23851e6 0.310239
$$438$$ 0 0
$$439$$ −543061. −0.134489 −0.0672446 0.997737i $$-0.521421\pi$$
−0.0672446 + 0.997737i $$0.521421\pi$$
$$440$$ 0 0
$$441$$ −2.15286e6 −0.527132
$$442$$ 0 0
$$443$$ −4.53358e6 −1.09757 −0.548785 0.835964i $$-0.684909\pi$$
−0.548785 + 0.835964i $$0.684909\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 1.32039e6 0.312561
$$448$$ 0 0
$$449$$ −3.39468e6 −0.794662 −0.397331 0.917675i $$-0.630064\pi$$
−0.397331 + 0.917675i $$0.630064\pi$$
$$450$$ 0 0
$$451$$ 5.05400e6 1.17002
$$452$$ 0 0
$$453$$ 1.28122e6 0.293345
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.34100e6 1.86822 0.934109 0.356988i $$-0.116196\pi$$
0.934109 + 0.356988i $$0.116196\pi$$
$$458$$ 0 0
$$459$$ −223834. −0.0495900
$$460$$ 0 0
$$461$$ −3.18453e6 −0.697900 −0.348950 0.937141i $$-0.613462\pi$$
−0.348950 + 0.937141i $$0.613462\pi$$
$$462$$ 0 0
$$463$$ −4.05943e6 −0.880061 −0.440030 0.897983i $$-0.645032\pi$$
−0.440030 + 0.897983i $$0.645032\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 7.98706e6 1.69471 0.847353 0.531029i $$-0.178195\pi$$
0.847353 + 0.531029i $$0.178195\pi$$
$$468$$ 0 0
$$469$$ −1.38155e6 −0.290024
$$470$$ 0 0
$$471$$ −5.81596e6 −1.20801
$$472$$ 0 0
$$473$$ 7.16375e6 1.47227
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2.46223e6 0.495488
$$478$$ 0 0
$$479$$ −5.27553e6 −1.05058 −0.525288 0.850924i $$-0.676042\pi$$
−0.525288 + 0.850924i $$0.676042\pi$$
$$480$$ 0 0
$$481$$ −1.51849e7 −2.99260
$$482$$ 0 0
$$483$$ 1.70892e6 0.333315
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −8.76969e6 −1.67557 −0.837784 0.546002i $$-0.816149\pi$$
−0.837784 + 0.546002i $$0.816149\pi$$
$$488$$ 0 0
$$489$$ 9.25209e6 1.74972
$$490$$ 0 0
$$491$$ −7.83080e6 −1.46589 −0.732946 0.680286i $$-0.761855\pi$$
−0.732946 + 0.680286i $$0.761855\pi$$
$$492$$ 0 0
$$493$$ −4640.42 −0.000859885 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −56649.2 −0.0102873
$$498$$ 0 0
$$499$$ 6.54560e6 1.17679 0.588394 0.808574i $$-0.299760\pi$$
0.588394 + 0.808574i $$0.299760\pi$$
$$500$$ 0 0
$$501$$ −5.22610e6 −0.930216
$$502$$ 0 0
$$503$$ 2.07773e6 0.366158 0.183079 0.983098i $$-0.441394\pi$$
0.183079 + 0.983098i $$0.441394\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.65152e7 2.85340
$$508$$ 0 0
$$509$$ −6.25093e6 −1.06942 −0.534712 0.845034i $$-0.679580\pi$$
−0.534712 + 0.845034i $$0.679580\pi$$
$$510$$ 0 0
$$511$$ −1.86784e6 −0.316437
$$512$$ 0 0
$$513$$ 1.01480e6 0.170249
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 5.55344e6 0.913767
$$518$$ 0 0
$$519$$ 1.37310e7 2.23761
$$520$$ 0 0
$$521$$ 2.19897e6 0.354916 0.177458 0.984128i $$-0.443213\pi$$
0.177458 + 0.984128i $$0.443213\pi$$
$$522$$ 0 0
$$523$$ 2.10298e6 0.336187 0.168093 0.985771i $$-0.446239\pi$$
0.168093 + 0.985771i $$0.446239\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 875046. 0.137247
$$528$$ 0 0
$$529$$ −199490. −0.0309944
$$530$$ 0 0
$$531$$ 6.54291e6 1.00701
$$532$$ 0 0
$$533$$ 1.30886e7 1.99561
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −1.24041e7 −1.85623
$$538$$ 0 0
$$539$$ −6.63661e6 −0.983953
$$540$$ 0 0
$$541$$ −1.83729e6 −0.269889 −0.134944 0.990853i $$-0.543086\pi$$
−0.134944 + 0.990853i $$0.543086\pi$$
$$542$$ 0 0
$$543$$ −1.06974e7 −1.55697
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −4.31872e6 −0.617145 −0.308572 0.951201i $$-0.599851\pi$$
−0.308572 + 0.951201i $$0.599851\pi$$
$$548$$ 0 0
$$549$$ −3.14845e6 −0.445827
$$550$$ 0 0
$$551$$ 21038.3 0.00295210
$$552$$ 0 0
$$553$$ −228409. −0.0317615
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −4.26837e6 −0.582940 −0.291470 0.956580i $$-0.594144\pi$$
−0.291470 + 0.956580i $$0.594144\pi$$
$$558$$ 0 0
$$559$$ 1.85524e7 2.51114
$$560$$ 0 0
$$561$$ 909819. 0.122053
$$562$$ 0 0
$$563$$ 1.26350e7 1.67998 0.839988 0.542605i $$-0.182562\pi$$
0.839988 + 0.542605i $$0.182562\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.57720e6 0.336659
$$568$$ 0 0
$$569$$ −8.67409e6 −1.12316 −0.561582 0.827421i $$-0.689807\pi$$
−0.561582 + 0.827421i $$0.689807\pi$$
$$570$$ 0 0
$$571$$ −3.28781e6 −0.422004 −0.211002 0.977486i $$-0.567673\pi$$
−0.211002 + 0.977486i $$0.567673\pi$$
$$572$$ 0 0
$$573$$ 1.57380e6 0.200246
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −117668. −0.0147135 −0.00735677 0.999973i $$-0.502342\pi$$
−0.00735677 + 0.999973i $$0.502342\pi$$
$$578$$ 0 0
$$579$$ −518352. −0.0642582
$$580$$ 0 0
$$581$$ 1.61280e6 0.198216
$$582$$ 0 0
$$583$$ 7.59031e6 0.924885
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1.10151e7 1.31945 0.659723 0.751509i $$-0.270674\pi$$
0.659723 + 0.751509i $$0.270674\pi$$
$$588$$ 0 0
$$589$$ −3.96719e6 −0.471189
$$590$$ 0 0
$$591$$ −1.37387e7 −1.61799
$$592$$ 0 0
$$593$$ −5.33879e6 −0.623456 −0.311728 0.950171i $$-0.600908\pi$$
−0.311728 + 0.950171i $$0.600908\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.43697e6 −0.165011
$$598$$ 0 0
$$599$$ 7.17784e6 0.817385 0.408692 0.912672i $$-0.365985\pi$$
0.408692 + 0.912672i $$0.365985\pi$$
$$600$$ 0 0
$$601$$ −809127. −0.0913756 −0.0456878 0.998956i $$-0.514548\pi$$
−0.0456878 + 0.998956i $$0.514548\pi$$
$$602$$ 0 0
$$603$$ 5.44735e6 0.610087
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −591991. −0.0652143 −0.0326072 0.999468i $$-0.510381\pi$$
−0.0326072 + 0.999468i $$0.510381\pi$$
$$608$$ 0 0
$$609$$ 29029.1 0.00317168
$$610$$ 0 0
$$611$$ 1.43821e7 1.55854
$$612$$ 0 0
$$613$$ −4.01173e6 −0.431201 −0.215601 0.976482i $$-0.569171\pi$$
−0.215601 + 0.976482i $$0.569171\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8.24093e6 0.871491 0.435746 0.900070i $$-0.356485\pi$$
0.435746 + 0.900070i $$0.356485\pi$$
$$618$$ 0 0
$$619$$ −1.96985e6 −0.206636 −0.103318 0.994648i $$-0.532946\pi$$
−0.103318 + 0.994648i $$0.532946\pi$$
$$620$$ 0 0
$$621$$ 5.11027e6 0.531759
$$622$$ 0 0
$$623$$ −3.97758e6 −0.410581
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −4.12484e6 −0.419024
$$628$$ 0 0
$$629$$ −1.50557e6 −0.151731
$$630$$ 0 0
$$631$$ 1.82360e7 1.82329 0.911646 0.410977i $$-0.134813\pi$$
0.911646 + 0.410977i $$0.134813\pi$$
$$632$$ 0 0
$$633$$ 1.96819e7 1.95235
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1.71872e7 −1.67825
$$638$$ 0 0
$$639$$ 223364. 0.0216402
$$640$$ 0 0
$$641$$ 7.65378e6 0.735752 0.367876 0.929875i $$-0.380085\pi$$
0.367876 + 0.929875i $$0.380085\pi$$
$$642$$ 0 0
$$643$$ −2.47762e6 −0.236323 −0.118162 0.992994i $$-0.537700\pi$$
−0.118162 + 0.992994i $$0.537700\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −4.79533e6 −0.450358 −0.225179 0.974317i $$-0.572297\pi$$
−0.225179 + 0.974317i $$0.572297\pi$$
$$648$$ 0 0
$$649$$ 2.01698e7 1.87970
$$650$$ 0 0
$$651$$ −5.47402e6 −0.506237
$$652$$ 0 0
$$653$$ 1.67125e7 1.53376 0.766881 0.641789i $$-0.221807\pi$$
0.766881 + 0.641789i $$0.221807\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 7.36478e6 0.665650
$$658$$ 0 0
$$659$$ −3.28380e6 −0.294553 −0.147277 0.989095i $$-0.547051\pi$$
−0.147277 + 0.989095i $$0.547051\pi$$
$$660$$ 0 0
$$661$$ −1.98880e7 −1.77047 −0.885234 0.465145i $$-0.846002\pi$$
−0.885234 + 0.465145i $$0.846002\pi$$
$$662$$ 0 0
$$663$$ 2.35621e6 0.208176
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 105944. 0.00922063
$$668$$ 0 0
$$669$$ −2.75423e6 −0.237922
$$670$$ 0 0
$$671$$ −9.70571e6 −0.832187
$$672$$ 0 0
$$673$$ −1.39915e7 −1.19076 −0.595381 0.803443i $$-0.702999\pi$$
−0.595381 + 0.803443i $$0.702999\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 3.13032e6 0.262493 0.131246 0.991350i $$-0.458102\pi$$
0.131246 + 0.991350i $$0.458102\pi$$
$$678$$ 0 0
$$679$$ −3.76482e6 −0.313379
$$680$$ 0 0
$$681$$ 2.23162e7 1.84397
$$682$$ 0 0
$$683$$ −8.94034e6 −0.733335 −0.366667 0.930352i $$-0.619501\pi$$
−0.366667 + 0.930352i $$0.619501\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −2.53288e7 −2.04750
$$688$$ 0 0
$$689$$ 1.96571e7 1.57750
$$690$$ 0 0
$$691$$ 2.38378e7 1.89920 0.949599 0.313466i $$-0.101490\pi$$
0.949599 + 0.313466i $$0.101490\pi$$
$$692$$ 0 0
$$693$$ −2.06335e6 −0.163207
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.29772e6 0.101181
$$698$$ 0 0
$$699$$ −1.22267e7 −0.946487
$$700$$ 0 0
$$701$$ 1.57226e7 1.20845 0.604225 0.796814i $$-0.293483\pi$$
0.604225 + 0.796814i $$0.293483\pi$$
$$702$$ 0 0
$$703$$ 6.82578e6 0.520912
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6.92394e6 0.520961
$$708$$ 0 0
$$709$$ −3.05866e6 −0.228515 −0.114258 0.993451i $$-0.536449\pi$$
−0.114258 + 0.993451i $$0.536449\pi$$
$$710$$ 0 0
$$711$$ 900603. 0.0668128
$$712$$ 0 0
$$713$$ −1.99778e7 −1.47172
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −5.27701e6 −0.383345
$$718$$ 0 0
$$719$$ −2.16146e7 −1.55929 −0.779643 0.626224i $$-0.784600\pi$$
−0.779643 + 0.626224i $$0.784600\pi$$
$$720$$ 0 0
$$721$$ −2.83983e6 −0.203448
$$722$$ 0 0
$$723$$ −1.35564e7 −0.964492
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −2.73589e7 −1.91983 −0.959916 0.280287i $$-0.909570\pi$$
−0.959916 + 0.280287i $$0.909570\pi$$
$$728$$ 0 0
$$729$$ −453482. −0.0316039
$$730$$ 0 0
$$731$$ 1.83945e6 0.127319
$$732$$ 0 0
$$733$$ −1.54693e7 −1.06344 −0.531718 0.846921i $$-0.678453\pi$$
−0.531718 + 0.846921i $$0.678453\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.67925e7 1.13880
$$738$$ 0 0
$$739$$ −5.56327e6 −0.374731 −0.187365 0.982290i $$-0.559995\pi$$
−0.187365 + 0.982290i $$0.559995\pi$$
$$740$$ 0 0
$$741$$ −1.06823e7 −0.714695
$$742$$ 0 0
$$743$$ −1.55089e7 −1.03064 −0.515322 0.856997i $$-0.672328\pi$$
−0.515322 + 0.856997i $$0.672328\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −6.35916e6 −0.416963
$$748$$ 0 0
$$749$$ −4.81713e6 −0.313750
$$750$$ 0 0
$$751$$ −2.20221e7 −1.42482 −0.712409 0.701765i $$-0.752396\pi$$
−0.712409 + 0.701765i $$0.752396\pi$$
$$752$$ 0 0
$$753$$ −1.68706e7 −1.08428
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.75653e6 0.111408 0.0557040 0.998447i $$-0.482260\pi$$
0.0557040 + 0.998447i $$0.482260\pi$$
$$758$$ 0 0
$$759$$ −2.07717e7 −1.30879
$$760$$ 0 0
$$761$$ −2.16974e6 −0.135814 −0.0679072 0.997692i $$-0.521632\pi$$
−0.0679072 + 0.997692i $$0.521632\pi$$
$$762$$ 0 0
$$763$$ 2.40758e6 0.149716
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 5.22349e7 3.20606
$$768$$ 0 0
$$769$$ −1.12131e7 −0.683769 −0.341884 0.939742i $$-0.611065\pi$$
−0.341884 + 0.939742i $$0.611065\pi$$
$$770$$ 0 0
$$771$$ −1.96531e7 −1.19068
$$772$$ 0 0
$$773$$ −1.43367e7 −0.862980 −0.431490 0.902118i $$-0.642012\pi$$
−0.431490 + 0.902118i $$0.642012\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 9.41836e6 0.559658
$$778$$ 0 0
$$779$$ −5.88348e6 −0.347369
$$780$$ 0 0
$$781$$ 688563. 0.0403939
$$782$$ 0 0
$$783$$ 86806.8 0.00505999
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 2.00549e6 0.115421 0.0577103 0.998333i $$-0.481620\pi$$
0.0577103 + 0.998333i $$0.481620\pi$$
$$788$$ 0 0
$$789$$ −1.81843e7 −1.03993
$$790$$ 0 0
$$791$$ 4.89389e6 0.278108
$$792$$ 0 0
$$793$$ −2.51354e7 −1.41940
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −448115. −0.0249887 −0.0124944 0.999922i $$-0.503977\pi$$
−0.0124944 + 0.999922i $$0.503977\pi$$
$$798$$ 0 0
$$799$$ 1.42597e6 0.0790210
$$800$$ 0 0
$$801$$ 1.56834e7 0.863690
$$802$$ 0 0
$$803$$ 2.27033e7 1.24251
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −3.83134e7 −2.07094
$$808$$ 0 0
$$809$$ 1.28425e7 0.689889 0.344944 0.938623i $$-0.387898\pi$$
0.344944 + 0.938623i $$0.387898\pi$$
$$810$$ 0 0
$$811$$ −1.78700e7 −0.954054 −0.477027 0.878889i $$-0.658286\pi$$
−0.477027 + 0.878889i $$0.658286\pi$$
$$812$$ 0 0
$$813$$ −2.18968e7 −1.16186
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.33950e6 −0.437104
$$818$$ 0 0
$$819$$ −5.34357e6 −0.278370
$$820$$ 0 0
$$821$$ 2.73336e7 1.41527 0.707634 0.706580i $$-0.249763\pi$$
0.707634 + 0.706580i $$0.249763\pi$$
$$822$$ 0 0
$$823$$ 3.05819e6 0.157385 0.0786927 0.996899i $$-0.474925\pi$$
0.0786927 + 0.996899i $$0.474925\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1.23961e7 0.630261 0.315131 0.949048i $$-0.397952\pi$$
0.315131 + 0.949048i $$0.397952\pi$$
$$828$$ 0 0
$$829$$ −399873. −0.0202086 −0.0101043 0.999949i $$-0.503216\pi$$
−0.0101043 + 0.999949i $$0.503216\pi$$
$$830$$ 0 0
$$831$$ −2.06793e6 −0.103881
$$832$$ 0 0
$$833$$ −1.70409e6 −0.0850905
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1.63692e7 −0.807632
$$838$$ 0 0
$$839$$ 2.08411e6 0.102215 0.0511075 0.998693i $$-0.483725\pi$$
0.0511075 + 0.998693i $$0.483725\pi$$
$$840$$ 0 0
$$841$$ −2.05093e7 −0.999912
$$842$$ 0 0
$$843$$ −2.05543e7 −0.996172
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −716092. −0.0342973
$$848$$ 0 0
$$849$$ −1.85521e7 −0.883330
$$850$$ 0 0
$$851$$ 3.43730e7 1.62702
$$852$$ 0 0
$$853$$ 2.14298e7 1.00843 0.504214 0.863579i $$-0.331782\pi$$
0.504214 + 0.863579i $$0.331782\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 3.49123e6 0.162377 0.0811887 0.996699i $$-0.474128\pi$$
0.0811887 + 0.996699i $$0.474128\pi$$
$$858$$ 0 0
$$859$$ 568043. 0.0262662 0.0131331 0.999914i $$-0.495819\pi$$
0.0131331 + 0.999914i $$0.495819\pi$$
$$860$$ 0 0
$$861$$ −8.11816e6 −0.373207
$$862$$ 0 0
$$863$$ −3.28678e7 −1.50226 −0.751128 0.660157i $$-0.770490\pi$$
−0.751128 + 0.660157i $$0.770490\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −2.74879e7 −1.24192
$$868$$ 0 0
$$869$$ 2.77628e6 0.124714
$$870$$ 0 0
$$871$$ 4.34885e7 1.94236
$$872$$ 0 0
$$873$$ 1.48444e7 0.659217
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.92945e7 1.28614 0.643069 0.765808i $$-0.277661\pi$$
0.643069 + 0.765808i $$0.277661\pi$$
$$878$$ 0 0
$$879$$ 7.45435e6 0.325415
$$880$$ 0 0
$$881$$ 2.88332e6 0.125156 0.0625782 0.998040i $$-0.480068\pi$$
0.0625782 + 0.998040i $$0.480068\pi$$
$$882$$ 0 0
$$883$$ −2.79477e7 −1.20627 −0.603135 0.797639i $$-0.706082\pi$$
−0.603135 + 0.797639i $$0.706082\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 2.89567e7 1.23578 0.617888 0.786266i $$-0.287989\pi$$
0.617888 + 0.786266i $$0.287989\pi$$
$$888$$ 0 0
$$889$$ −2.28621e6 −0.0970199
$$890$$ 0 0
$$891$$ −3.13254e7 −1.32191
$$892$$ 0 0
$$893$$ −6.46490e6 −0.271289
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −5.37938e7 −2.23229
$$898$$ 0 0
$$899$$ −339358. −0.0140042
$$900$$ 0 0
$$901$$ 1.94898e6 0.0799825
$$902$$ 0 0
$$903$$ −1.15070e7 −0.469617
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.18508e6 0.0478333 0.0239166 0.999714i $$-0.492386\pi$$
0.0239166 + 0.999714i $$0.492386\pi$$
$$908$$ 0 0
$$909$$ −2.73007e7 −1.09588
$$910$$ 0 0
$$911$$ 1.14346e7 0.456485 0.228242 0.973604i $$-0.426702\pi$$
0.228242 + 0.973604i $$0.426702\pi$$
$$912$$ 0 0
$$913$$ −1.96033e7 −0.778310
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4.15471e6 0.163161
$$918$$ 0 0
$$919$$ 1.48904e7 0.581589 0.290795 0.956785i $$-0.406080\pi$$
0.290795 + 0.956785i $$0.406080\pi$$
$$920$$ 0 0
$$921$$ −3.14434e7 −1.22146
$$922$$ 0 0
$$923$$ 1.78321e6 0.0688968
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 1.11973e7 0.427969
$$928$$ 0 0
$$929$$ −2.02580e7 −0.770117 −0.385058 0.922892i $$-0.625819\pi$$
−0.385058 + 0.922892i $$0.625819\pi$$
$$930$$ 0 0
$$931$$ 7.72584e6 0.292127
$$932$$ 0 0
$$933$$ 4.21043e7 1.58351
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1.96237e7 −0.730182 −0.365091 0.930972i $$-0.618962\pi$$
−0.365091 + 0.930972i $$0.618962\pi$$
$$938$$ 0 0
$$939$$ 7.10891e6 0.263111
$$940$$ 0 0
$$941$$ 7.62854e6 0.280846 0.140423 0.990092i $$-0.455154\pi$$
0.140423 + 0.990092i $$0.455154\pi$$
$$942$$ 0 0
$$943$$ −2.96279e7 −1.08498
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −3.93605e7 −1.42622 −0.713108 0.701054i $$-0.752713\pi$$
−0.713108 + 0.701054i $$0.752713\pi$$
$$948$$ 0 0
$$949$$ 5.87962e7 2.11926
$$950$$ 0 0
$$951$$ 2.02751e7 0.726963
$$952$$ 0 0
$$953$$ 5.09953e7 1.81886 0.909428 0.415861i $$-0.136520\pi$$
0.909428 + 0.415861i $$0.136520\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −352844. −0.0124538
$$958$$ 0 0
$$959$$ −2.97062e6 −0.104304
$$960$$ 0 0
$$961$$ 3.53638e7 1.23524
$$962$$ 0 0
$$963$$ 1.89936e7 0.659997
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 5.17740e7 1.78051 0.890257 0.455458i $$-0.150524\pi$$
0.890257 + 0.455458i $$0.150524\pi$$
$$968$$ 0 0
$$969$$ −1.05914e6 −0.0362364
$$970$$ 0 0
$$971$$ 3.69158e7 1.25651 0.628253 0.778009i $$-0.283770\pi$$
0.628253 + 0.778009i $$0.283770\pi$$
$$972$$ 0 0
$$973$$ 5.89981e6 0.199782
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 3.14468e7 1.05400 0.526999 0.849866i $$-0.323317\pi$$
0.526999 + 0.849866i $$0.323317\pi$$
$$978$$ 0 0
$$979$$ 4.83470e7 1.61218
$$980$$ 0 0
$$981$$ −9.49293e6 −0.314940
$$982$$ 0 0
$$983$$ 5.15823e7 1.70262 0.851308 0.524667i $$-0.175810\pi$$
0.851308 + 0.524667i $$0.175810\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −8.92041e6 −0.291469
$$988$$ 0 0
$$989$$ −4.19958e7 −1.36526
$$990$$ 0 0
$$991$$ 1.30562e7 0.422312 0.211156 0.977452i $$-0.432277\pi$$
0.211156 + 0.977452i $$0.432277\pi$$
$$992$$ 0 0
$$993$$ 5.52421e7 1.77786
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −4.97672e7 −1.58564 −0.792821 0.609454i $$-0.791389\pi$$
−0.792821 + 0.609454i $$0.791389\pi$$
$$998$$ 0 0
$$999$$ 2.81641e7 0.892859
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 400.6.a.u.1.2 2
4.3 odd 2 200.6.a.e.1.1 2
5.2 odd 4 400.6.c.o.49.1 4
5.3 odd 4 400.6.c.o.49.4 4
5.4 even 2 400.6.a.r.1.1 2
20.3 even 4 200.6.c.f.49.1 4
20.7 even 4 200.6.c.f.49.4 4
20.19 odd 2 200.6.a.f.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
200.6.a.e.1.1 2 4.3 odd 2
200.6.a.f.1.2 yes 2 20.19 odd 2
200.6.c.f.49.1 4 20.3 even 4
200.6.c.f.49.4 4 20.7 even 4
400.6.a.r.1.1 2 5.4 even 2
400.6.a.u.1.2 2 1.1 even 1 trivial
400.6.c.o.49.1 4 5.2 odd 4
400.6.c.o.49.4 4 5.3 odd 4