# Properties

 Label 280.6.a.g.1.3 Level $280$ Weight $6$ Character 280.1 Self dual yes Analytic conductor $44.907$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 280.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9074695476$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2 x^{3} - 232 x^{2} + 60 x + 5808$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$15.2911$$ of defining polynomial Character $$\chi$$ $$=$$ 280.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+6.08191 q^{3} +25.0000 q^{5} +49.0000 q^{7} -206.010 q^{9} +O(q^{10})$$ $$q+6.08191 q^{3} +25.0000 q^{5} +49.0000 q^{7} -206.010 q^{9} +407.671 q^{11} -1075.68 q^{13} +152.048 q^{15} -1775.88 q^{17} +2156.68 q^{19} +298.014 q^{21} -4540.75 q^{23} +625.000 q^{25} -2730.84 q^{27} +6114.50 q^{29} -2856.57 q^{31} +2479.42 q^{33} +1225.00 q^{35} -7036.59 q^{37} -6542.17 q^{39} -11206.3 q^{41} +10684.8 q^{43} -5150.26 q^{45} -7637.19 q^{47} +2401.00 q^{49} -10800.7 q^{51} +19261.2 q^{53} +10191.8 q^{55} +13116.8 q^{57} -9188.24 q^{59} -33709.8 q^{61} -10094.5 q^{63} -26891.9 q^{65} -29762.1 q^{67} -27616.4 q^{69} -29860.8 q^{71} -40725.0 q^{73} +3801.19 q^{75} +19975.9 q^{77} -39180.1 q^{79} +33451.8 q^{81} +34423.7 q^{83} -44396.9 q^{85} +37187.8 q^{87} -7995.49 q^{89} -52708.2 q^{91} -17373.4 q^{93} +53917.1 q^{95} -58573.9 q^{97} -83984.5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 13 q^{3} + 100 q^{5} + 196 q^{7} - 193 q^{9} + O(q^{10})$$ $$4 q - 13 q^{3} + 100 q^{5} + 196 q^{7} - 193 q^{9} - 595 q^{11} - 969 q^{13} - 325 q^{15} - 1315 q^{17} - 1090 q^{19} - 637 q^{21} - 1534 q^{23} + 2500 q^{25} + 173 q^{27} + 4099 q^{29} - 4820 q^{31} + 4149 q^{33} + 4900 q^{35} + 7692 q^{37} - 6371 q^{39} - 9722 q^{41} - 20610 q^{43} - 4825 q^{45} - 1661 q^{47} + 9604 q^{49} - 73361 q^{51} - 28898 q^{53} - 14875 q^{55} - 21246 q^{57} - 101872 q^{59} - 24742 q^{61} - 9457 q^{63} - 24225 q^{65} - 82060 q^{67} + 16914 q^{69} - 102784 q^{71} - 80652 q^{73} - 8125 q^{75} - 29155 q^{77} - 117801 q^{79} - 141052 q^{81} - 155440 q^{83} - 32875 q^{85} - 82519 q^{87} + 56426 q^{89} - 47481 q^{91} - 17332 q^{93} - 27250 q^{95} - 261031 q^{97} - 61686 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 6.08191 0.390155 0.195077 0.980788i $$-0.437504\pi$$
0.195077 + 0.980788i $$0.437504\pi$$
$$4$$ 0 0
$$5$$ 25.0000 0.447214
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 0 0
$$9$$ −206.010 −0.847779
$$10$$ 0 0
$$11$$ 407.671 1.01585 0.507924 0.861402i $$-0.330413\pi$$
0.507924 + 0.861402i $$0.330413\pi$$
$$12$$ 0 0
$$13$$ −1075.68 −1.76532 −0.882661 0.470011i $$-0.844250\pi$$
−0.882661 + 0.470011i $$0.844250\pi$$
$$14$$ 0 0
$$15$$ 152.048 0.174482
$$16$$ 0 0
$$17$$ −1775.88 −1.49036 −0.745178 0.666865i $$-0.767636\pi$$
−0.745178 + 0.666865i $$0.767636\pi$$
$$18$$ 0 0
$$19$$ 2156.68 1.37057 0.685287 0.728273i $$-0.259677\pi$$
0.685287 + 0.728273i $$0.259677\pi$$
$$20$$ 0 0
$$21$$ 298.014 0.147465
$$22$$ 0 0
$$23$$ −4540.75 −1.78981 −0.894907 0.446253i $$-0.852758\pi$$
−0.894907 + 0.446253i $$0.852758\pi$$
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ 0 0
$$27$$ −2730.84 −0.720920
$$28$$ 0 0
$$29$$ 6114.50 1.35010 0.675050 0.737772i $$-0.264122\pi$$
0.675050 + 0.737772i $$0.264122\pi$$
$$30$$ 0 0
$$31$$ −2856.57 −0.533876 −0.266938 0.963714i $$-0.586012\pi$$
−0.266938 + 0.963714i $$0.586012\pi$$
$$32$$ 0 0
$$33$$ 2479.42 0.396337
$$34$$ 0 0
$$35$$ 1225.00 0.169031
$$36$$ 0 0
$$37$$ −7036.59 −0.845002 −0.422501 0.906362i $$-0.638848\pi$$
−0.422501 + 0.906362i $$0.638848\pi$$
$$38$$ 0 0
$$39$$ −6542.17 −0.688748
$$40$$ 0 0
$$41$$ −11206.3 −1.04113 −0.520563 0.853824i $$-0.674278\pi$$
−0.520563 + 0.853824i $$0.674278\pi$$
$$42$$ 0 0
$$43$$ 10684.8 0.881241 0.440620 0.897694i $$-0.354758\pi$$
0.440620 + 0.897694i $$0.354758\pi$$
$$44$$ 0 0
$$45$$ −5150.26 −0.379138
$$46$$ 0 0
$$47$$ −7637.19 −0.504300 −0.252150 0.967688i $$-0.581138\pi$$
−0.252150 + 0.967688i $$0.581138\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ −10800.7 −0.581469
$$52$$ 0 0
$$53$$ 19261.2 0.941875 0.470937 0.882167i $$-0.343916\pi$$
0.470937 + 0.882167i $$0.343916\pi$$
$$54$$ 0 0
$$55$$ 10191.8 0.454301
$$56$$ 0 0
$$57$$ 13116.8 0.534736
$$58$$ 0 0
$$59$$ −9188.24 −0.343639 −0.171819 0.985128i $$-0.554965\pi$$
−0.171819 + 0.985128i $$0.554965\pi$$
$$60$$ 0 0
$$61$$ −33709.8 −1.15993 −0.579964 0.814642i $$-0.696933\pi$$
−0.579964 + 0.814642i $$0.696933\pi$$
$$62$$ 0 0
$$63$$ −10094.5 −0.320430
$$64$$ 0 0
$$65$$ −26891.9 −0.789476
$$66$$ 0 0
$$67$$ −29762.1 −0.809985 −0.404993 0.914320i $$-0.632726\pi$$
−0.404993 + 0.914320i $$0.632726\pi$$
$$68$$ 0 0
$$69$$ −27616.4 −0.698304
$$70$$ 0 0
$$71$$ −29860.8 −0.703001 −0.351500 0.936188i $$-0.614328\pi$$
−0.351500 + 0.936188i $$0.614328\pi$$
$$72$$ 0 0
$$73$$ −40725.0 −0.894445 −0.447222 0.894423i $$-0.647587\pi$$
−0.447222 + 0.894423i $$0.647587\pi$$
$$74$$ 0 0
$$75$$ 3801.19 0.0780309
$$76$$ 0 0
$$77$$ 19975.9 0.383954
$$78$$ 0 0
$$79$$ −39180.1 −0.706313 −0.353157 0.935564i $$-0.614892\pi$$
−0.353157 + 0.935564i $$0.614892\pi$$
$$80$$ 0 0
$$81$$ 33451.8 0.566509
$$82$$ 0 0
$$83$$ 34423.7 0.548483 0.274241 0.961661i $$-0.411573\pi$$
0.274241 + 0.961661i $$0.411573\pi$$
$$84$$ 0 0
$$85$$ −44396.9 −0.666508
$$86$$ 0 0
$$87$$ 37187.8 0.526748
$$88$$ 0 0
$$89$$ −7995.49 −0.106997 −0.0534983 0.998568i $$-0.517037\pi$$
−0.0534983 + 0.998568i $$0.517037\pi$$
$$90$$ 0 0
$$91$$ −52708.2 −0.667229
$$92$$ 0 0
$$93$$ −17373.4 −0.208294
$$94$$ 0 0
$$95$$ 53917.1 0.612940
$$96$$ 0 0
$$97$$ −58573.9 −0.632084 −0.316042 0.948745i $$-0.602354\pi$$
−0.316042 + 0.948745i $$0.602354\pi$$
$$98$$ 0 0
$$99$$ −83984.5 −0.861214
$$100$$ 0 0
$$101$$ 159709. 1.55785 0.778925 0.627117i $$-0.215765\pi$$
0.778925 + 0.627117i $$0.215765\pi$$
$$102$$ 0 0
$$103$$ −69774.5 −0.648043 −0.324021 0.946050i $$-0.605035\pi$$
−0.324021 + 0.946050i $$0.605035\pi$$
$$104$$ 0 0
$$105$$ 7450.34 0.0659482
$$106$$ 0 0
$$107$$ −136651. −1.15386 −0.576929 0.816794i $$-0.695749\pi$$
−0.576929 + 0.816794i $$0.695749\pi$$
$$108$$ 0 0
$$109$$ −164476. −1.32598 −0.662990 0.748628i $$-0.730713\pi$$
−0.662990 + 0.748628i $$0.730713\pi$$
$$110$$ 0 0
$$111$$ −42795.9 −0.329682
$$112$$ 0 0
$$113$$ −57417.3 −0.423006 −0.211503 0.977377i $$-0.567836\pi$$
−0.211503 + 0.977377i $$0.567836\pi$$
$$114$$ 0 0
$$115$$ −113519. −0.800429
$$116$$ 0 0
$$117$$ 221601. 1.49660
$$118$$ 0 0
$$119$$ −87017.9 −0.563302
$$120$$ 0 0
$$121$$ 5144.87 0.0319456
$$122$$ 0 0
$$123$$ −68155.7 −0.406200
$$124$$ 0 0
$$125$$ 15625.0 0.0894427
$$126$$ 0 0
$$127$$ −132464. −0.728766 −0.364383 0.931249i $$-0.618720\pi$$
−0.364383 + 0.931249i $$0.618720\pi$$
$$128$$ 0 0
$$129$$ 64983.9 0.343820
$$130$$ 0 0
$$131$$ 68982.8 0.351206 0.175603 0.984461i $$-0.443812\pi$$
0.175603 + 0.984461i $$0.443812\pi$$
$$132$$ 0 0
$$133$$ 105678. 0.518028
$$134$$ 0 0
$$135$$ −68271.0 −0.322405
$$136$$ 0 0
$$137$$ −379201. −1.72611 −0.863055 0.505110i $$-0.831452\pi$$
−0.863055 + 0.505110i $$0.831452\pi$$
$$138$$ 0 0
$$139$$ 87581.5 0.384481 0.192241 0.981348i $$-0.438425\pi$$
0.192241 + 0.981348i $$0.438425\pi$$
$$140$$ 0 0
$$141$$ −46448.7 −0.196755
$$142$$ 0 0
$$143$$ −438523. −1.79330
$$144$$ 0 0
$$145$$ 152863. 0.603783
$$146$$ 0 0
$$147$$ 14602.7 0.0557364
$$148$$ 0 0
$$149$$ 278024. 1.02593 0.512963 0.858411i $$-0.328548\pi$$
0.512963 + 0.858411i $$0.328548\pi$$
$$150$$ 0 0
$$151$$ 214016. 0.763843 0.381921 0.924195i $$-0.375263\pi$$
0.381921 + 0.924195i $$0.375263\pi$$
$$152$$ 0 0
$$153$$ 365849. 1.26349
$$154$$ 0 0
$$155$$ −71414.2 −0.238757
$$156$$ 0 0
$$157$$ 372088. 1.20475 0.602374 0.798214i $$-0.294222\pi$$
0.602374 + 0.798214i $$0.294222\pi$$
$$158$$ 0 0
$$159$$ 117145. 0.367477
$$160$$ 0 0
$$161$$ −222497. −0.676486
$$162$$ 0 0
$$163$$ 505449. 1.49008 0.745038 0.667022i $$-0.232432\pi$$
0.745038 + 0.667022i $$0.232432\pi$$
$$164$$ 0 0
$$165$$ 61985.5 0.177248
$$166$$ 0 0
$$167$$ −210214. −0.583271 −0.291635 0.956530i $$-0.594199\pi$$
−0.291635 + 0.956530i $$0.594199\pi$$
$$168$$ 0 0
$$169$$ 785790. 2.11636
$$170$$ 0 0
$$171$$ −444299. −1.16194
$$172$$ 0 0
$$173$$ 126094. 0.320316 0.160158 0.987091i $$-0.448800\pi$$
0.160158 + 0.987091i $$0.448800\pi$$
$$174$$ 0 0
$$175$$ 30625.0 0.0755929
$$176$$ 0 0
$$177$$ −55882.0 −0.134072
$$178$$ 0 0
$$179$$ −133191. −0.310701 −0.155350 0.987859i $$-0.549651\pi$$
−0.155350 + 0.987859i $$0.549651\pi$$
$$180$$ 0 0
$$181$$ 137432. 0.311810 0.155905 0.987772i $$-0.450171\pi$$
0.155905 + 0.987772i $$0.450171\pi$$
$$182$$ 0 0
$$183$$ −205020. −0.452551
$$184$$ 0 0
$$185$$ −175915. −0.377897
$$186$$ 0 0
$$187$$ −723973. −1.51397
$$188$$ 0 0
$$189$$ −133811. −0.272482
$$190$$ 0 0
$$191$$ 223165. 0.442631 0.221316 0.975202i $$-0.428965\pi$$
0.221316 + 0.975202i $$0.428965\pi$$
$$192$$ 0 0
$$193$$ 849159. 1.64095 0.820476 0.571681i $$-0.193708\pi$$
0.820476 + 0.571681i $$0.193708\pi$$
$$194$$ 0 0
$$195$$ −163554. −0.308018
$$196$$ 0 0
$$197$$ −198502. −0.364418 −0.182209 0.983260i $$-0.558325\pi$$
−0.182209 + 0.983260i $$0.558325\pi$$
$$198$$ 0 0
$$199$$ −485537. −0.869139 −0.434570 0.900638i $$-0.643100\pi$$
−0.434570 + 0.900638i $$0.643100\pi$$
$$200$$ 0 0
$$201$$ −181011. −0.316020
$$202$$ 0 0
$$203$$ 299611. 0.510290
$$204$$ 0 0
$$205$$ −280158. −0.465605
$$206$$ 0 0
$$207$$ 935441. 1.51737
$$208$$ 0 0
$$209$$ 879218. 1.39229
$$210$$ 0 0
$$211$$ 1.03186e6 1.59557 0.797785 0.602943i $$-0.206005\pi$$
0.797785 + 0.602943i $$0.206005\pi$$
$$212$$ 0 0
$$213$$ −181611. −0.274279
$$214$$ 0 0
$$215$$ 267120. 0.394103
$$216$$ 0 0
$$217$$ −139972. −0.201786
$$218$$ 0 0
$$219$$ −247685. −0.348972
$$220$$ 0 0
$$221$$ 1.91027e6 2.63096
$$222$$ 0 0
$$223$$ 1.24200e6 1.67247 0.836235 0.548372i $$-0.184752\pi$$
0.836235 + 0.548372i $$0.184752\pi$$
$$224$$ 0 0
$$225$$ −128756. −0.169556
$$226$$ 0 0
$$227$$ 1.45644e6 1.87597 0.937987 0.346670i $$-0.112688\pi$$
0.937987 + 0.346670i $$0.112688\pi$$
$$228$$ 0 0
$$229$$ −602646. −0.759406 −0.379703 0.925109i $$-0.623974\pi$$
−0.379703 + 0.925109i $$0.623974\pi$$
$$230$$ 0 0
$$231$$ 121492. 0.149801
$$232$$ 0 0
$$233$$ 754395. 0.910351 0.455175 0.890402i $$-0.349577\pi$$
0.455175 + 0.890402i $$0.349577\pi$$
$$234$$ 0 0
$$235$$ −190930. −0.225530
$$236$$ 0 0
$$237$$ −238290. −0.275571
$$238$$ 0 0
$$239$$ −43290.4 −0.0490226 −0.0245113 0.999700i $$-0.507803\pi$$
−0.0245113 + 0.999700i $$0.507803\pi$$
$$240$$ 0 0
$$241$$ 696883. 0.772890 0.386445 0.922313i $$-0.373703\pi$$
0.386445 + 0.922313i $$0.373703\pi$$
$$242$$ 0 0
$$243$$ 867045. 0.941946
$$244$$ 0 0
$$245$$ 60025.0 0.0638877
$$246$$ 0 0
$$247$$ −2.31990e6 −2.41950
$$248$$ 0 0
$$249$$ 209362. 0.213993
$$250$$ 0 0
$$251$$ −1.98194e6 −1.98566 −0.992831 0.119529i $$-0.961862\pi$$
−0.992831 + 0.119529i $$0.961862\pi$$
$$252$$ 0 0
$$253$$ −1.85113e6 −1.81818
$$254$$ 0 0
$$255$$ −270018. −0.260041
$$256$$ 0 0
$$257$$ 160003. 0.151110 0.0755552 0.997142i $$-0.475927\pi$$
0.0755552 + 0.997142i $$0.475927\pi$$
$$258$$ 0 0
$$259$$ −344793. −0.319381
$$260$$ 0 0
$$261$$ −1.25965e6 −1.14459
$$262$$ 0 0
$$263$$ −913315. −0.814200 −0.407100 0.913384i $$-0.633460\pi$$
−0.407100 + 0.913384i $$0.633460\pi$$
$$264$$ 0 0
$$265$$ 481529. 0.421219
$$266$$ 0 0
$$267$$ −48627.8 −0.0417452
$$268$$ 0 0
$$269$$ 842402. 0.709805 0.354902 0.934903i $$-0.384514\pi$$
0.354902 + 0.934903i $$0.384514\pi$$
$$270$$ 0 0
$$271$$ −1.25079e6 −1.03457 −0.517285 0.855813i $$-0.673057\pi$$
−0.517285 + 0.855813i $$0.673057\pi$$
$$272$$ 0 0
$$273$$ −320567. −0.260322
$$274$$ 0 0
$$275$$ 254795. 0.203169
$$276$$ 0 0
$$277$$ −370062. −0.289784 −0.144892 0.989447i $$-0.546284\pi$$
−0.144892 + 0.989447i $$0.546284\pi$$
$$278$$ 0 0
$$279$$ 588483. 0.452609
$$280$$ 0 0
$$281$$ 1.08198e6 0.817435 0.408717 0.912661i $$-0.365976\pi$$
0.408717 + 0.912661i $$0.365976\pi$$
$$282$$ 0 0
$$283$$ 2.17327e6 1.61305 0.806525 0.591200i $$-0.201346\pi$$
0.806525 + 0.591200i $$0.201346\pi$$
$$284$$ 0 0
$$285$$ 327919. 0.239141
$$286$$ 0 0
$$287$$ −549109. −0.393508
$$288$$ 0 0
$$289$$ 1.73388e6 1.22116
$$290$$ 0 0
$$291$$ −356241. −0.246611
$$292$$ 0 0
$$293$$ −2.06185e6 −1.40310 −0.701550 0.712620i $$-0.747508\pi$$
−0.701550 + 0.712620i $$0.747508\pi$$
$$294$$ 0 0
$$295$$ −229706. −0.153680
$$296$$ 0 0
$$297$$ −1.11329e6 −0.732344
$$298$$ 0 0
$$299$$ 4.88438e6 3.15960
$$300$$ 0 0
$$301$$ 523554. 0.333078
$$302$$ 0 0
$$303$$ 971335. 0.607803
$$304$$ 0 0
$$305$$ −842744. −0.518736
$$306$$ 0 0
$$307$$ −1.45256e6 −0.879603 −0.439802 0.898095i $$-0.644951\pi$$
−0.439802 + 0.898095i $$0.644951\pi$$
$$308$$ 0 0
$$309$$ −424362. −0.252837
$$310$$ 0 0
$$311$$ −2.66514e6 −1.56250 −0.781248 0.624221i $$-0.785417\pi$$
−0.781248 + 0.624221i $$0.785417\pi$$
$$312$$ 0 0
$$313$$ −1.21556e6 −0.701317 −0.350658 0.936503i $$-0.614042\pi$$
−0.350658 + 0.936503i $$0.614042\pi$$
$$314$$ 0 0
$$315$$ −252363. −0.143301
$$316$$ 0 0
$$317$$ 2.41341e6 1.34891 0.674455 0.738316i $$-0.264379\pi$$
0.674455 + 0.738316i $$0.264379\pi$$
$$318$$ 0 0
$$319$$ 2.49271e6 1.37150
$$320$$ 0 0
$$321$$ −831097. −0.450183
$$322$$ 0 0
$$323$$ −3.83000e6 −2.04264
$$324$$ 0 0
$$325$$ −672299. −0.353064
$$326$$ 0 0
$$327$$ −1.00033e6 −0.517338
$$328$$ 0 0
$$329$$ −374223. −0.190608
$$330$$ 0 0
$$331$$ 1.47548e6 0.740226 0.370113 0.928987i $$-0.379319\pi$$
0.370113 + 0.928987i $$0.379319\pi$$
$$332$$ 0 0
$$333$$ 1.44961e6 0.716376
$$334$$ 0 0
$$335$$ −744054. −0.362236
$$336$$ 0 0
$$337$$ −1.98021e6 −0.949812 −0.474906 0.880037i $$-0.657518\pi$$
−0.474906 + 0.880037i $$0.657518\pi$$
$$338$$ 0 0
$$339$$ −349207. −0.165038
$$340$$ 0 0
$$341$$ −1.16454e6 −0.542337
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 0 0
$$345$$ −690410. −0.312291
$$346$$ 0 0
$$347$$ −212122. −0.0945720 −0.0472860 0.998881i $$-0.515057\pi$$
−0.0472860 + 0.998881i $$0.515057\pi$$
$$348$$ 0 0
$$349$$ −852633. −0.374713 −0.187356 0.982292i $$-0.559992\pi$$
−0.187356 + 0.982292i $$0.559992\pi$$
$$350$$ 0 0
$$351$$ 2.93750e6 1.27265
$$352$$ 0 0
$$353$$ −987395. −0.421749 −0.210875 0.977513i $$-0.567631\pi$$
−0.210875 + 0.977513i $$0.567631\pi$$
$$354$$ 0 0
$$355$$ −746520. −0.314391
$$356$$ 0 0
$$357$$ −529235. −0.219775
$$358$$ 0 0
$$359$$ −940935. −0.385322 −0.192661 0.981265i $$-0.561712\pi$$
−0.192661 + 0.981265i $$0.561712\pi$$
$$360$$ 0 0
$$361$$ 2.17519e6 0.878474
$$362$$ 0 0
$$363$$ 31290.6 0.0124637
$$364$$ 0 0
$$365$$ −1.01812e6 −0.400008
$$366$$ 0 0
$$367$$ −4.12708e6 −1.59948 −0.799739 0.600348i $$-0.795029\pi$$
−0.799739 + 0.600348i $$0.795029\pi$$
$$368$$ 0 0
$$369$$ 2.30862e6 0.882644
$$370$$ 0 0
$$371$$ 943798. 0.355995
$$372$$ 0 0
$$373$$ 2.26644e6 0.843476 0.421738 0.906718i $$-0.361420\pi$$
0.421738 + 0.906718i $$0.361420\pi$$
$$374$$ 0 0
$$375$$ 95029.8 0.0348965
$$376$$ 0 0
$$377$$ −6.57723e6 −2.38336
$$378$$ 0 0
$$379$$ −2.24173e6 −0.801653 −0.400826 0.916154i $$-0.631277\pi$$
−0.400826 + 0.916154i $$0.631277\pi$$
$$380$$ 0 0
$$381$$ −805633. −0.284331
$$382$$ 0 0
$$383$$ 1.03804e6 0.361591 0.180795 0.983521i $$-0.442133\pi$$
0.180795 + 0.983521i $$0.442133\pi$$
$$384$$ 0 0
$$385$$ 499397. 0.171710
$$386$$ 0 0
$$387$$ −2.20118e6 −0.747098
$$388$$ 0 0
$$389$$ −259366. −0.0869040 −0.0434520 0.999056i $$-0.513836\pi$$
−0.0434520 + 0.999056i $$0.513836\pi$$
$$390$$ 0 0
$$391$$ 8.06380e6 2.66746
$$392$$ 0 0
$$393$$ 419547. 0.137025
$$394$$ 0 0
$$395$$ −979502. −0.315873
$$396$$ 0 0
$$397$$ −2.75320e6 −0.876720 −0.438360 0.898799i $$-0.644441\pi$$
−0.438360 + 0.898799i $$0.644441\pi$$
$$398$$ 0 0
$$399$$ 642721. 0.202111
$$400$$ 0 0
$$401$$ −5.92473e6 −1.83996 −0.919979 0.391968i $$-0.871794\pi$$
−0.919979 + 0.391968i $$0.871794\pi$$
$$402$$ 0 0
$$403$$ 3.07275e6 0.942463
$$404$$ 0 0
$$405$$ 836295. 0.253351
$$406$$ 0 0
$$407$$ −2.86862e6 −0.858393
$$408$$ 0 0
$$409$$ 3.69799e6 1.09309 0.546547 0.837428i $$-0.315942\pi$$
0.546547 + 0.837428i $$0.315942\pi$$
$$410$$ 0 0
$$411$$ −2.30627e6 −0.673450
$$412$$ 0 0
$$413$$ −450224. −0.129883
$$414$$ 0 0
$$415$$ 860594. 0.245289
$$416$$ 0 0
$$417$$ 532662. 0.150007
$$418$$ 0 0
$$419$$ −4.15994e6 −1.15758 −0.578791 0.815476i $$-0.696475\pi$$
−0.578791 + 0.815476i $$0.696475\pi$$
$$420$$ 0 0
$$421$$ −5.20770e6 −1.43199 −0.715996 0.698104i $$-0.754027\pi$$
−0.715996 + 0.698104i $$0.754027\pi$$
$$422$$ 0 0
$$423$$ 1.57334e6 0.427535
$$424$$ 0 0
$$425$$ −1.10992e6 −0.298071
$$426$$ 0 0
$$427$$ −1.65178e6 −0.438412
$$428$$ 0 0
$$429$$ −2.66706e6 −0.699663
$$430$$ 0 0
$$431$$ 1.68636e6 0.437276 0.218638 0.975806i $$-0.429839\pi$$
0.218638 + 0.975806i $$0.429839\pi$$
$$432$$ 0 0
$$433$$ −3.10555e6 −0.796010 −0.398005 0.917383i $$-0.630297\pi$$
−0.398005 + 0.917383i $$0.630297\pi$$
$$434$$ 0 0
$$435$$ 929696. 0.235569
$$436$$ 0 0
$$437$$ −9.79296e6 −2.45307
$$438$$ 0 0
$$439$$ 1.22048e6 0.302252 0.151126 0.988515i $$-0.451710\pi$$
0.151126 + 0.988515i $$0.451710\pi$$
$$440$$ 0 0
$$441$$ −494631. −0.121111
$$442$$ 0 0
$$443$$ 2.25319e6 0.545493 0.272747 0.962086i $$-0.412068\pi$$
0.272747 + 0.962086i $$0.412068\pi$$
$$444$$ 0 0
$$445$$ −199887. −0.0478503
$$446$$ 0 0
$$447$$ 1.69091e6 0.400270
$$448$$ 0 0
$$449$$ 7.67885e6 1.79755 0.898774 0.438412i $$-0.144459\pi$$
0.898774 + 0.438412i $$0.144459\pi$$
$$450$$ 0 0
$$451$$ −4.56849e6 −1.05762
$$452$$ 0 0
$$453$$ 1.30163e6 0.298017
$$454$$ 0 0
$$455$$ −1.31771e6 −0.298394
$$456$$ 0 0
$$457$$ 7.98138e6 1.78767 0.893835 0.448396i $$-0.148005\pi$$
0.893835 + 0.448396i $$0.148005\pi$$
$$458$$ 0 0
$$459$$ 4.84963e6 1.07443
$$460$$ 0 0
$$461$$ −6.10635e6 −1.33823 −0.669113 0.743161i $$-0.733326\pi$$
−0.669113 + 0.743161i $$0.733326\pi$$
$$462$$ 0 0
$$463$$ 7.69353e6 1.66791 0.833957 0.551830i $$-0.186070\pi$$
0.833957 + 0.551830i $$0.186070\pi$$
$$464$$ 0 0
$$465$$ −434335. −0.0931520
$$466$$ 0 0
$$467$$ 4.41567e6 0.936924 0.468462 0.883484i $$-0.344808\pi$$
0.468462 + 0.883484i $$0.344808\pi$$
$$468$$ 0 0
$$469$$ −1.45835e6 −0.306146
$$470$$ 0 0
$$471$$ 2.26300e6 0.470038
$$472$$ 0 0
$$473$$ 4.35588e6 0.895206
$$474$$ 0 0
$$475$$ 1.34793e6 0.274115
$$476$$ 0 0
$$477$$ −3.96800e6 −0.798502
$$478$$ 0 0
$$479$$ −2.46498e6 −0.490878 −0.245439 0.969412i $$-0.578932\pi$$
−0.245439 + 0.969412i $$0.578932\pi$$
$$480$$ 0 0
$$481$$ 7.56910e6 1.49170
$$482$$ 0 0
$$483$$ −1.35320e6 −0.263934
$$484$$ 0 0
$$485$$ −1.46435e6 −0.282677
$$486$$ 0 0
$$487$$ 29208.5 0.00558068 0.00279034 0.999996i $$-0.499112\pi$$
0.00279034 + 0.999996i $$0.499112\pi$$
$$488$$ 0 0
$$489$$ 3.07409e6 0.581360
$$490$$ 0 0
$$491$$ −1.41942e6 −0.265710 −0.132855 0.991135i $$-0.542414\pi$$
−0.132855 + 0.991135i $$0.542414\pi$$
$$492$$ 0 0
$$493$$ −1.08586e7 −2.01213
$$494$$ 0 0
$$495$$ −2.09961e6 −0.385147
$$496$$ 0 0
$$497$$ −1.46318e6 −0.265709
$$498$$ 0 0
$$499$$ 6.68747e6 1.20229 0.601147 0.799138i $$-0.294711\pi$$
0.601147 + 0.799138i $$0.294711\pi$$
$$500$$ 0 0
$$501$$ −1.27850e6 −0.227566
$$502$$ 0 0
$$503$$ −5.12274e6 −0.902782 −0.451391 0.892326i $$-0.649072\pi$$
−0.451391 + 0.892326i $$0.649072\pi$$
$$504$$ 0 0
$$505$$ 3.99272e6 0.696692
$$506$$ 0 0
$$507$$ 4.77910e6 0.825708
$$508$$ 0 0
$$509$$ 2.78484e6 0.476438 0.238219 0.971211i $$-0.423436\pi$$
0.238219 + 0.971211i $$0.423436\pi$$
$$510$$ 0 0
$$511$$ −1.99552e6 −0.338068
$$512$$ 0 0
$$513$$ −5.88956e6 −0.988074
$$514$$ 0 0
$$515$$ −1.74436e6 −0.289814
$$516$$ 0 0
$$517$$ −3.11346e6 −0.512292
$$518$$ 0 0
$$519$$ 766891. 0.124973
$$520$$ 0 0
$$521$$ 6.73984e6 1.08781 0.543907 0.839145i $$-0.316944\pi$$
0.543907 + 0.839145i $$0.316944\pi$$
$$522$$ 0 0
$$523$$ 6.94281e6 1.10989 0.554947 0.831886i $$-0.312739\pi$$
0.554947 + 0.831886i $$0.312739\pi$$
$$524$$ 0 0
$$525$$ 186258. 0.0294929
$$526$$ 0 0
$$527$$ 5.07291e6 0.795666
$$528$$ 0 0
$$529$$ 1.41820e7 2.20343
$$530$$ 0 0
$$531$$ 1.89287e6 0.291330
$$532$$ 0 0
$$533$$ 1.20544e7 1.83792
$$534$$ 0 0
$$535$$ −3.41627e6 −0.516021
$$536$$ 0 0
$$537$$ −810055. −0.121221
$$538$$ 0 0
$$539$$ 978819. 0.145121
$$540$$ 0 0
$$541$$ 5.25286e6 0.771619 0.385810 0.922578i $$-0.373922\pi$$
0.385810 + 0.922578i $$0.373922\pi$$
$$542$$ 0 0
$$543$$ 835847. 0.121654
$$544$$ 0 0
$$545$$ −4.11191e6 −0.592997
$$546$$ 0 0
$$547$$ −8.43518e6 −1.20539 −0.602693 0.797974i $$-0.705905\pi$$
−0.602693 + 0.797974i $$0.705905\pi$$
$$548$$ 0 0
$$549$$ 6.94456e6 0.983363
$$550$$ 0 0
$$551$$ 1.31871e7 1.85041
$$552$$ 0 0
$$553$$ −1.91982e6 −0.266961
$$554$$ 0 0
$$555$$ −1.06990e6 −0.147438
$$556$$ 0 0
$$557$$ −8.14320e6 −1.11213 −0.556067 0.831137i $$-0.687690\pi$$
−0.556067 + 0.831137i $$0.687690\pi$$
$$558$$ 0 0
$$559$$ −1.14934e7 −1.55567
$$560$$ 0 0
$$561$$ −4.40314e6 −0.590684
$$562$$ 0 0
$$563$$ 5.32372e6 0.707854 0.353927 0.935273i $$-0.384846\pi$$
0.353927 + 0.935273i $$0.384846\pi$$
$$564$$ 0 0
$$565$$ −1.43543e6 −0.189174
$$566$$ 0 0
$$567$$ 1.63914e6 0.214120
$$568$$ 0 0
$$569$$ 594659. 0.0769994 0.0384997 0.999259i $$-0.487742\pi$$
0.0384997 + 0.999259i $$0.487742\pi$$
$$570$$ 0 0
$$571$$ 5.08376e6 0.652521 0.326260 0.945280i $$-0.394211\pi$$
0.326260 + 0.945280i $$0.394211\pi$$
$$572$$ 0 0
$$573$$ 1.35727e6 0.172695
$$574$$ 0 0
$$575$$ −2.83797e6 −0.357963
$$576$$ 0 0
$$577$$ 6.83943e6 0.855225 0.427612 0.903962i $$-0.359355\pi$$
0.427612 + 0.903962i $$0.359355\pi$$
$$578$$ 0 0
$$579$$ 5.16451e6 0.640225
$$580$$ 0 0
$$581$$ 1.68676e6 0.207307
$$582$$ 0 0
$$583$$ 7.85223e6 0.956801
$$584$$ 0 0
$$585$$ 5.54002e6 0.669301
$$586$$ 0 0
$$587$$ −1.24602e7 −1.49256 −0.746278 0.665634i $$-0.768161\pi$$
−0.746278 + 0.665634i $$0.768161\pi$$
$$588$$ 0 0
$$589$$ −6.16072e6 −0.731717
$$590$$ 0 0
$$591$$ −1.20727e6 −0.142179
$$592$$ 0 0
$$593$$ −1.04774e7 −1.22353 −0.611765 0.791039i $$-0.709540\pi$$
−0.611765 + 0.791039i $$0.709540\pi$$
$$594$$ 0 0
$$595$$ −2.17545e6 −0.251916
$$596$$ 0 0
$$597$$ −2.95299e6 −0.339099
$$598$$ 0 0
$$599$$ 2.01252e6 0.229179 0.114589 0.993413i $$-0.463445\pi$$
0.114589 + 0.993413i $$0.463445\pi$$
$$600$$ 0 0
$$601$$ −9.45258e6 −1.06749 −0.533746 0.845645i $$-0.679216\pi$$
−0.533746 + 0.845645i $$0.679216\pi$$
$$602$$ 0 0
$$603$$ 6.13131e6 0.686689
$$604$$ 0 0
$$605$$ 128622. 0.0142865
$$606$$ 0 0
$$607$$ 1.58379e7 1.74473 0.872363 0.488859i $$-0.162587\pi$$
0.872363 + 0.488859i $$0.162587\pi$$
$$608$$ 0 0
$$609$$ 1.82220e6 0.199092
$$610$$ 0 0
$$611$$ 8.21516e6 0.890252
$$612$$ 0 0
$$613$$ 1.15645e7 1.24301 0.621505 0.783410i $$-0.286521\pi$$
0.621505 + 0.783410i $$0.286521\pi$$
$$614$$ 0 0
$$615$$ −1.70389e6 −0.181658
$$616$$ 0 0
$$617$$ −1.24967e6 −0.132154 −0.0660771 0.997815i $$-0.521048\pi$$
−0.0660771 + 0.997815i $$0.521048\pi$$
$$618$$ 0 0
$$619$$ −1.11632e6 −0.117101 −0.0585505 0.998284i $$-0.518648\pi$$
−0.0585505 + 0.998284i $$0.518648\pi$$
$$620$$ 0 0
$$621$$ 1.24001e7 1.29031
$$622$$ 0 0
$$623$$ −391779. −0.0404409
$$624$$ 0 0
$$625$$ 390625. 0.0400000
$$626$$ 0 0
$$627$$ 5.34733e6 0.543210
$$628$$ 0 0
$$629$$ 1.24961e7 1.25935
$$630$$ 0 0
$$631$$ −1.52038e7 −1.52012 −0.760062 0.649851i $$-0.774831\pi$$
−0.760062 + 0.649851i $$0.774831\pi$$
$$632$$ 0 0
$$633$$ 6.27569e6 0.622519
$$634$$ 0 0
$$635$$ −3.31160e6 −0.325914
$$636$$ 0 0
$$637$$ −2.58270e6 −0.252189
$$638$$ 0 0
$$639$$ 6.15164e6 0.595989
$$640$$ 0 0
$$641$$ 1.02239e7 0.982814 0.491407 0.870930i $$-0.336483\pi$$
0.491407 + 0.870930i $$0.336483\pi$$
$$642$$ 0 0
$$643$$ −1.70419e6 −0.162551 −0.0812757 0.996692i $$-0.525899\pi$$
−0.0812757 + 0.996692i $$0.525899\pi$$
$$644$$ 0 0
$$645$$ 1.62460e6 0.153761
$$646$$ 0 0
$$647$$ −1.34708e7 −1.26512 −0.632560 0.774512i $$-0.717996\pi$$
−0.632560 + 0.774512i $$0.717996\pi$$
$$648$$ 0 0
$$649$$ −3.74578e6 −0.349085
$$650$$ 0 0
$$651$$ −851296. −0.0787278
$$652$$ 0 0
$$653$$ −1.55928e7 −1.43100 −0.715500 0.698612i $$-0.753801\pi$$
−0.715500 + 0.698612i $$0.753801\pi$$
$$654$$ 0 0
$$655$$ 1.72457e6 0.157064
$$656$$ 0 0
$$657$$ 8.38976e6 0.758292
$$658$$ 0 0
$$659$$ −2.02390e7 −1.81542 −0.907708 0.419603i $$-0.862169\pi$$
−0.907708 + 0.419603i $$0.862169\pi$$
$$660$$ 0 0
$$661$$ −5.28966e6 −0.470895 −0.235448 0.971887i $$-0.575656\pi$$
−0.235448 + 0.971887i $$0.575656\pi$$
$$662$$ 0 0
$$663$$ 1.16181e7 1.02648
$$664$$ 0 0
$$665$$ 2.64194e6 0.231669
$$666$$ 0 0
$$667$$ −2.77644e7 −2.41643
$$668$$ 0 0
$$669$$ 7.55371e6 0.652522
$$670$$ 0 0
$$671$$ −1.37425e7 −1.17831
$$672$$ 0 0
$$673$$ 7.68769e6 0.654271 0.327136 0.944977i $$-0.393917\pi$$
0.327136 + 0.944977i $$0.393917\pi$$
$$674$$ 0 0
$$675$$ −1.70678e6 −0.144184
$$676$$ 0 0
$$677$$ −1.13880e7 −0.954939 −0.477469 0.878648i $$-0.658446\pi$$
−0.477469 + 0.878648i $$0.658446\pi$$
$$678$$ 0 0
$$679$$ −2.87012e6 −0.238905
$$680$$ 0 0
$$681$$ 8.85791e6 0.731920
$$682$$ 0 0
$$683$$ 1.15735e7 0.949320 0.474660 0.880169i $$-0.342571\pi$$
0.474660 + 0.880169i $$0.342571\pi$$
$$684$$ 0 0
$$685$$ −9.48003e6 −0.771940
$$686$$ 0 0
$$687$$ −3.66524e6 −0.296286
$$688$$ 0 0
$$689$$ −2.07188e7 −1.66271
$$690$$ 0 0
$$691$$ 1.33398e7 1.06281 0.531404 0.847119i $$-0.321665\pi$$
0.531404 + 0.847119i $$0.321665\pi$$
$$692$$ 0 0
$$693$$ −4.11524e6 −0.325508
$$694$$ 0 0
$$695$$ 2.18954e6 0.171945
$$696$$ 0 0
$$697$$ 1.99010e7 1.55165
$$698$$ 0 0
$$699$$ 4.58816e6 0.355178
$$700$$ 0 0
$$701$$ −4.67569e6 −0.359377 −0.179689 0.983724i $$-0.557509\pi$$
−0.179689 + 0.983724i $$0.557509\pi$$
$$702$$ 0 0
$$703$$ −1.51757e7 −1.15814
$$704$$ 0 0
$$705$$ −1.16122e6 −0.0879916
$$706$$ 0 0
$$707$$ 7.82574e6 0.588812
$$708$$ 0 0
$$709$$ 2.08491e7 1.55766 0.778829 0.627236i $$-0.215814\pi$$
0.778829 + 0.627236i $$0.215814\pi$$
$$710$$ 0 0
$$711$$ 8.07150e6 0.598798
$$712$$ 0 0
$$713$$ 1.29710e7 0.955539
$$714$$ 0 0
$$715$$ −1.09631e7 −0.801987
$$716$$ 0 0
$$717$$ −263288. −0.0191264
$$718$$ 0 0
$$719$$ −1.97250e7 −1.42297 −0.711485 0.702702i $$-0.751977\pi$$
−0.711485 + 0.702702i $$0.751977\pi$$
$$720$$ 0 0
$$721$$ −3.41895e6 −0.244937
$$722$$ 0 0
$$723$$ 4.23838e6 0.301547
$$724$$ 0 0
$$725$$ 3.82156e6 0.270020
$$726$$ 0 0
$$727$$ 1.84776e7 1.29661 0.648306 0.761380i $$-0.275478\pi$$
0.648306 + 0.761380i $$0.275478\pi$$
$$728$$ 0 0
$$729$$ −2.85550e6 −0.199005
$$730$$ 0 0
$$731$$ −1.89748e7 −1.31336
$$732$$ 0 0
$$733$$ 1.95578e6 0.134449 0.0672247 0.997738i $$-0.478586\pi$$
0.0672247 + 0.997738i $$0.478586\pi$$
$$734$$ 0 0
$$735$$ 365067. 0.0249261
$$736$$ 0 0
$$737$$ −1.21332e7 −0.822821
$$738$$ 0 0
$$739$$ −1.55560e7 −1.04782 −0.523911 0.851773i $$-0.675528\pi$$
−0.523911 + 0.851773i $$0.675528\pi$$
$$740$$ 0 0
$$741$$ −1.41094e7 −0.943981
$$742$$ 0 0
$$743$$ −1.21540e7 −0.807694 −0.403847 0.914827i $$-0.632327\pi$$
−0.403847 + 0.914827i $$0.632327\pi$$
$$744$$ 0 0
$$745$$ 6.95059e6 0.458808
$$746$$ 0 0
$$747$$ −7.09165e6 −0.464992
$$748$$ 0 0
$$749$$ −6.69588e6 −0.436117
$$750$$ 0 0
$$751$$ −7.81290e6 −0.505490 −0.252745 0.967533i $$-0.581333\pi$$
−0.252745 + 0.967533i $$0.581333\pi$$
$$752$$ 0 0
$$753$$ −1.20540e7 −0.774715
$$754$$ 0 0
$$755$$ 5.35040e6 0.341601
$$756$$ 0 0
$$757$$ 2.04844e7 1.29922 0.649610 0.760268i $$-0.274932\pi$$
0.649610 + 0.760268i $$0.274932\pi$$
$$758$$ 0 0
$$759$$ −1.12584e7 −0.709370
$$760$$ 0 0
$$761$$ −8.17274e6 −0.511571 −0.255786 0.966734i $$-0.582334\pi$$
−0.255786 + 0.966734i $$0.582334\pi$$
$$762$$ 0 0
$$763$$ −8.05934e6 −0.501174
$$764$$ 0 0
$$765$$ 9.14622e6 0.565051
$$766$$ 0 0
$$767$$ 9.88359e6 0.606633
$$768$$ 0 0
$$769$$ 6.09119e6 0.371438 0.185719 0.982603i $$-0.440539\pi$$
0.185719 + 0.982603i $$0.440539\pi$$
$$770$$ 0 0
$$771$$ 973122. 0.0589564
$$772$$ 0 0
$$773$$ −2.91960e7 −1.75741 −0.878707 0.477361i $$-0.841593\pi$$
−0.878707 + 0.477361i $$0.841593\pi$$
$$774$$ 0 0
$$775$$ −1.78536e6 −0.106775
$$776$$ 0 0
$$777$$ −2.09700e6 −0.124608
$$778$$ 0 0
$$779$$ −2.41685e7 −1.42694
$$780$$ 0 0
$$781$$ −1.21734e7 −0.714141
$$782$$ 0 0
$$783$$ −1.66977e7 −0.973314
$$784$$ 0 0
$$785$$ 9.30219e6 0.538780
$$786$$ 0 0
$$787$$ −9.27644e6 −0.533881 −0.266940 0.963713i $$-0.586013\pi$$
−0.266940 + 0.963713i $$0.586013\pi$$
$$788$$ 0 0
$$789$$ −5.55470e6 −0.317664
$$790$$ 0 0
$$791$$ −2.81345e6 −0.159881
$$792$$ 0 0
$$793$$ 3.62608e7 2.04765
$$794$$ 0 0
$$795$$ 2.92862e6 0.164341
$$796$$ 0 0
$$797$$ −3.37629e7 −1.88275 −0.941377 0.337356i $$-0.890467\pi$$
−0.941377 + 0.337356i $$0.890467\pi$$
$$798$$ 0 0
$$799$$ 1.35627e7 0.751587
$$800$$ 0 0
$$801$$ 1.64715e6 0.0907095
$$802$$ 0 0
$$803$$ −1.66024e7 −0.908619
$$804$$ 0 0
$$805$$ −5.56242e6 −0.302534
$$806$$ 0 0
$$807$$ 5.12341e6 0.276934
$$808$$ 0 0
$$809$$ 2.68549e7 1.44262 0.721312 0.692611i $$-0.243540\pi$$
0.721312 + 0.692611i $$0.243540\pi$$
$$810$$ 0 0
$$811$$ 8.99843e6 0.480413 0.240206 0.970722i $$-0.422785\pi$$
0.240206 + 0.970722i $$0.422785\pi$$
$$812$$ 0 0
$$813$$ −7.60717e6 −0.403642
$$814$$ 0 0
$$815$$ 1.26362e7 0.666382
$$816$$ 0 0
$$817$$ 2.30437e7 1.20781
$$818$$ 0 0
$$819$$ 1.08584e7 0.565663
$$820$$ 0 0
$$821$$ −2.53478e7 −1.31245 −0.656225 0.754565i $$-0.727848\pi$$
−0.656225 + 0.754565i $$0.727848\pi$$
$$822$$ 0 0
$$823$$ −2.80232e7 −1.44217 −0.721087 0.692844i $$-0.756357\pi$$
−0.721087 + 0.692844i $$0.756357\pi$$
$$824$$ 0 0
$$825$$ 1.54964e6 0.0792675
$$826$$ 0 0
$$827$$ −1.08632e7 −0.552325 −0.276162 0.961111i $$-0.589063\pi$$
−0.276162 + 0.961111i $$0.589063\pi$$
$$828$$ 0 0
$$829$$ 2.61760e7 1.32287 0.661435 0.750002i $$-0.269948\pi$$
0.661435 + 0.750002i $$0.269948\pi$$
$$830$$ 0 0
$$831$$ −2.25068e6 −0.113061
$$832$$ 0 0
$$833$$ −4.26388e6 −0.212908
$$834$$ 0 0
$$835$$ −5.25535e6 −0.260847
$$836$$ 0 0
$$837$$ 7.80083e6 0.384882
$$838$$ 0 0
$$839$$ −5.33086e6 −0.261452 −0.130726 0.991419i $$-0.541731\pi$$
−0.130726 + 0.991419i $$0.541731\pi$$
$$840$$ 0 0
$$841$$ 1.68760e7 0.822771
$$842$$ 0 0
$$843$$ 6.58050e6 0.318926
$$844$$ 0 0
$$845$$ 1.96447e7 0.946465
$$846$$ 0 0
$$847$$ 252098. 0.0120743
$$848$$ 0 0
$$849$$ 1.32176e7 0.629339
$$850$$ 0 0
$$851$$ 3.19514e7 1.51240
$$852$$ 0 0
$$853$$ −3.52267e7 −1.65768 −0.828838 0.559488i $$-0.810998\pi$$
−0.828838 + 0.559488i $$0.810998\pi$$
$$854$$ 0 0
$$855$$ −1.11075e7 −0.519638
$$856$$ 0 0
$$857$$ −1.93997e6 −0.0902283 −0.0451141 0.998982i $$-0.514365\pi$$
−0.0451141 + 0.998982i $$0.514365\pi$$
$$858$$ 0 0
$$859$$ −2.90322e7 −1.34245 −0.671223 0.741255i $$-0.734231\pi$$
−0.671223 + 0.741255i $$0.734231\pi$$
$$860$$ 0 0
$$861$$ −3.33963e6 −0.153529
$$862$$ 0 0
$$863$$ 3.76000e7 1.71854 0.859272 0.511518i $$-0.170917\pi$$
0.859272 + 0.511518i $$0.170917\pi$$
$$864$$ 0 0
$$865$$ 3.15235e6 0.143250
$$866$$ 0 0
$$867$$ 1.05453e7 0.476442
$$868$$ 0 0
$$869$$ −1.59726e7 −0.717506
$$870$$ 0 0
$$871$$ 3.20145e7 1.42988
$$872$$ 0 0
$$873$$ 1.20668e7 0.535868
$$874$$ 0 0
$$875$$ 765625. 0.0338062
$$876$$ 0 0
$$877$$ −2.05117e7 −0.900539 −0.450269 0.892893i $$-0.648672\pi$$
−0.450269 + 0.892893i $$0.648672\pi$$
$$878$$ 0 0
$$879$$ −1.25400e7 −0.547426
$$880$$ 0 0
$$881$$ −1.36918e7 −0.594322 −0.297161 0.954827i $$-0.596040\pi$$
−0.297161 + 0.954827i $$0.596040\pi$$
$$882$$ 0 0
$$883$$ −2.73738e7 −1.18150 −0.590750 0.806855i $$-0.701168\pi$$
−0.590750 + 0.806855i $$0.701168\pi$$
$$884$$ 0 0
$$885$$ −1.39705e6 −0.0599590
$$886$$ 0 0
$$887$$ −1.44173e7 −0.615284 −0.307642 0.951502i $$-0.599540\pi$$
−0.307642 + 0.951502i $$0.599540\pi$$
$$888$$ 0 0
$$889$$ −6.49073e6 −0.275448
$$890$$ 0 0
$$891$$ 1.36373e7 0.575487
$$892$$ 0 0
$$893$$ −1.64710e7 −0.691181
$$894$$ 0 0
$$895$$ −3.32977e6 −0.138950
$$896$$ 0 0
$$897$$ 2.97064e7 1.23273
$$898$$ 0 0
$$899$$ −1.74665e7 −0.720786
$$900$$ 0 0
$$901$$ −3.42054e7 −1.40373
$$902$$ 0 0
$$903$$ 3.18421e6 0.129952
$$904$$ 0 0
$$905$$ 3.43579e6 0.139446
$$906$$ 0 0
$$907$$ 2.74482e7 1.10789 0.553944 0.832554i $$-0.313122\pi$$
0.553944 + 0.832554i $$0.313122\pi$$
$$908$$ 0 0
$$909$$ −3.29017e7 −1.32071
$$910$$ 0 0
$$911$$ −3.10414e7 −1.23921 −0.619607 0.784913i $$-0.712708\pi$$
−0.619607 + 0.784913i $$0.712708\pi$$
$$912$$ 0 0
$$913$$ 1.40336e7 0.557175
$$914$$ 0 0
$$915$$ −5.12549e6 −0.202387
$$916$$ 0 0
$$917$$ 3.38016e6 0.132744
$$918$$ 0 0
$$919$$ −1.13810e7 −0.444519 −0.222260 0.974988i $$-0.571343\pi$$
−0.222260 + 0.974988i $$0.571343\pi$$
$$920$$ 0 0
$$921$$ −8.83431e6 −0.343181
$$922$$ 0 0
$$923$$ 3.21206e7 1.24102
$$924$$ 0 0
$$925$$ −4.39787e6 −0.169000
$$926$$ 0 0
$$927$$ 1.43743e7 0.549397
$$928$$ 0 0
$$929$$ 1.77569e7 0.675038 0.337519 0.941319i $$-0.390412\pi$$
0.337519 + 0.941319i $$0.390412\pi$$
$$930$$ 0 0
$$931$$ 5.17820e6 0.195796
$$932$$ 0 0
$$933$$ −1.62091e7 −0.609615
$$934$$ 0 0
$$935$$ −1.80993e7 −0.677070
$$936$$ 0 0
$$937$$ −1.93662e7 −0.720602 −0.360301 0.932836i $$-0.617326\pi$$
−0.360301 + 0.932836i $$0.617326\pi$$
$$938$$ 0 0
$$939$$ −7.39290e6 −0.273622
$$940$$ 0 0
$$941$$ 2.75371e7 1.01378 0.506891 0.862010i $$-0.330795\pi$$
0.506891 + 0.862010i $$0.330795\pi$$
$$942$$ 0 0
$$943$$ 5.08850e7 1.86342
$$944$$ 0 0
$$945$$ −3.34528e6 −0.121858
$$946$$ 0 0
$$947$$ 4.71647e7 1.70900 0.854501 0.519450i $$-0.173863\pi$$
0.854501 + 0.519450i $$0.173863\pi$$
$$948$$ 0 0
$$949$$ 4.38069e7 1.57898
$$950$$ 0 0
$$951$$ 1.46781e7 0.526283
$$952$$ 0 0
$$953$$ −3.72865e7 −1.32990 −0.664951 0.746887i $$-0.731547\pi$$
−0.664951 + 0.746887i $$0.731547\pi$$
$$954$$ 0 0
$$955$$ 5.57912e6 0.197951
$$956$$ 0 0
$$957$$ 1.51604e7 0.535095
$$958$$ 0 0
$$959$$ −1.85809e7 −0.652408
$$960$$ 0 0
$$961$$ −2.04692e7 −0.714976
$$962$$ 0 0
$$963$$ 2.81515e7 0.978217
$$964$$ 0 0
$$965$$ 2.12290e7 0.733856
$$966$$ 0 0
$$967$$ −2.46193e7 −0.846662 −0.423331 0.905975i $$-0.639139\pi$$
−0.423331 + 0.905975i $$0.639139\pi$$
$$968$$ 0 0
$$969$$ −2.32937e7 −0.796947
$$970$$ 0 0
$$971$$ −2.00471e6 −0.0682344 −0.0341172 0.999418i $$-0.510862\pi$$
−0.0341172 + 0.999418i $$0.510862\pi$$
$$972$$ 0 0
$$973$$ 4.29149e6 0.145320
$$974$$ 0 0
$$975$$ −4.08886e6 −0.137750
$$976$$ 0 0
$$977$$ 2.77180e7 0.929022 0.464511 0.885567i $$-0.346230\pi$$
0.464511 + 0.885567i $$0.346230\pi$$
$$978$$ 0 0
$$979$$ −3.25953e6 −0.108692
$$980$$ 0 0
$$981$$ 3.38838e7 1.12414
$$982$$ 0 0
$$983$$ −1.79222e7 −0.591573 −0.295787 0.955254i $$-0.595582\pi$$
−0.295787 + 0.955254i $$0.595582\pi$$
$$984$$ 0 0
$$985$$ −4.96255e6 −0.162973
$$986$$ 0 0
$$987$$ −2.27599e6 −0.0743664
$$988$$ 0 0
$$989$$ −4.85169e7 −1.57726
$$990$$ 0 0
$$991$$ −1.38359e7 −0.447532 −0.223766 0.974643i $$-0.571835\pi$$
−0.223766 + 0.974643i $$0.571835\pi$$
$$992$$ 0 0
$$993$$ 8.97375e6 0.288803
$$994$$ 0 0
$$995$$ −1.21384e7 −0.388691
$$996$$ 0 0
$$997$$ 3.19988e7 1.01952 0.509761 0.860316i $$-0.329734\pi$$
0.509761 + 0.860316i $$0.329734\pi$$
$$998$$ 0 0
$$999$$ 1.92158e7 0.609179
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 280.6.a.g.1.3 4
4.3 odd 2 560.6.a.x.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
280.6.a.g.1.3 4 1.1 even 1 trivial
560.6.a.x.1.2 4 4.3 odd 2