# Properties

 Label 6840.2.a.bf.1.2 Level $6840$ Weight $2$ Character 6840.1 Self dual yes Analytic conductor $54.618$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$54.6176749826$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.81361$$ of defining polynomial Character $$\chi$$ $$=$$ 6840.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -2.52444 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} -2.52444 q^{7} +3.10278 q^{11} +6.72999 q^{13} +2.57834 q^{17} -1.00000 q^{19} +4.57834 q^{23} +1.00000 q^{25} +1.10278 q^{29} +7.83276 q^{31} +2.52444 q^{35} -4.52444 q^{37} -6.15165 q^{41} -7.68111 q^{43} +3.42166 q^{47} -0.627213 q^{49} -2.57834 q^{53} -3.10278 q^{55} +10.2056 q^{59} -14.8816 q^{61} -6.72999 q^{65} +8.41110 q^{67} -9.45998 q^{71} -9.25443 q^{73} -7.83276 q^{77} +4.47054 q^{83} -2.57834 q^{85} +14.5628 q^{89} -16.9894 q^{91} +1.00000 q^{95} -12.9355 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} - 2 q^{7}+O(q^{10})$$ 3 * q - 3 * q^5 - 2 * q^7 $$3 q - 3 q^{5} - 2 q^{7} + 2 q^{11} + 6 q^{17} - 3 q^{19} + 12 q^{23} + 3 q^{25} - 4 q^{29} - 4 q^{31} + 2 q^{35} - 8 q^{37} - 14 q^{43} + 12 q^{47} + 11 q^{49} - 6 q^{53} - 2 q^{55} + 16 q^{59} - 6 q^{61} - 4 q^{67} + 12 q^{71} - 2 q^{73} + 4 q^{77} + 4 q^{83} - 6 q^{85} - 4 q^{89} - 20 q^{91} + 3 q^{95} - 4 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 - 2 * q^7 + 2 * q^11 + 6 * q^17 - 3 * q^19 + 12 * q^23 + 3 * q^25 - 4 * q^29 - 4 * q^31 + 2 * q^35 - 8 * q^37 - 14 * q^43 + 12 * q^47 + 11 * q^49 - 6 * q^53 - 2 * q^55 + 16 * q^59 - 6 * q^61 - 4 * q^67 + 12 * q^71 - 2 * q^73 + 4 * q^77 + 4 * q^83 - 6 * q^85 - 4 * q^89 - 20 * q^91 + 3 * q^95 - 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.52444 −0.954148 −0.477074 0.878863i $$-0.658303\pi$$
−0.477074 + 0.878863i $$0.658303\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.10278 0.935522 0.467761 0.883855i $$-0.345061\pi$$
0.467761 + 0.883855i $$0.345061\pi$$
$$12$$ 0 0
$$13$$ 6.72999 1.86656 0.933281 0.359146i $$-0.116932\pi$$
0.933281 + 0.359146i $$0.116932\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.57834 0.625339 0.312669 0.949862i $$-0.398777\pi$$
0.312669 + 0.949862i $$0.398777\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.57834 0.954649 0.477325 0.878727i $$-0.341607\pi$$
0.477325 + 0.878727i $$0.341607\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.10278 0.204780 0.102390 0.994744i $$-0.467351\pi$$
0.102390 + 0.994744i $$0.467351\pi$$
$$30$$ 0 0
$$31$$ 7.83276 1.40681 0.703403 0.710791i $$-0.251663\pi$$
0.703403 + 0.710791i $$0.251663\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.52444 0.426708
$$36$$ 0 0
$$37$$ −4.52444 −0.743813 −0.371907 0.928270i $$-0.621296\pi$$
−0.371907 + 0.928270i $$0.621296\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.15165 −0.960726 −0.480363 0.877070i $$-0.659495\pi$$
−0.480363 + 0.877070i $$0.659495\pi$$
$$42$$ 0 0
$$43$$ −7.68111 −1.17136 −0.585679 0.810543i $$-0.699172\pi$$
−0.585679 + 0.810543i $$0.699172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.42166 0.499101 0.249550 0.968362i $$-0.419717\pi$$
0.249550 + 0.968362i $$0.419717\pi$$
$$48$$ 0 0
$$49$$ −0.627213 −0.0896019
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −2.57834 −0.354162 −0.177081 0.984196i $$-0.556665\pi$$
−0.177081 + 0.984196i $$0.556665\pi$$
$$54$$ 0 0
$$55$$ −3.10278 −0.418378
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 10.2056 1.32865 0.664325 0.747444i $$-0.268719\pi$$
0.664325 + 0.747444i $$0.268719\pi$$
$$60$$ 0 0
$$61$$ −14.8816 −1.90540 −0.952699 0.303914i $$-0.901706\pi$$
−0.952699 + 0.303914i $$0.901706\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −6.72999 −0.834752
$$66$$ 0 0
$$67$$ 8.41110 1.02758 0.513790 0.857916i $$-0.328241\pi$$
0.513790 + 0.857916i $$0.328241\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −9.45998 −1.12269 −0.561346 0.827581i $$-0.689716\pi$$
−0.561346 + 0.827581i $$0.689716\pi$$
$$72$$ 0 0
$$73$$ −9.25443 −1.08315 −0.541574 0.840653i $$-0.682172\pi$$
−0.541574 + 0.840653i $$0.682172\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −7.83276 −0.892626
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 4.47054 0.490705 0.245353 0.969434i $$-0.421096\pi$$
0.245353 + 0.969434i $$0.421096\pi$$
$$84$$ 0 0
$$85$$ −2.57834 −0.279660
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 14.5628 1.54365 0.771824 0.635836i $$-0.219344\pi$$
0.771824 + 0.635836i $$0.219344\pi$$
$$90$$ 0 0
$$91$$ −16.9894 −1.78098
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −12.9355 −1.31340 −0.656702 0.754150i $$-0.728049\pi$$
−0.656702 + 0.754150i $$0.728049\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 11.4600 1.14031 0.570155 0.821537i $$-0.306883\pi$$
0.570155 + 0.821537i $$0.306883\pi$$
$$102$$ 0 0
$$103$$ 14.5089 1.42960 0.714800 0.699329i $$-0.246518\pi$$
0.714800 + 0.699329i $$0.246518\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.2544 1.47470 0.737351 0.675510i $$-0.236077\pi$$
0.737351 + 0.675510i $$0.236077\pi$$
$$108$$ 0 0
$$109$$ 16.6167 1.59159 0.795793 0.605568i $$-0.207054\pi$$
0.795793 + 0.605568i $$0.207054\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 4.37279 0.411357 0.205679 0.978620i $$-0.434060\pi$$
0.205679 + 0.978620i $$0.434060\pi$$
$$114$$ 0 0
$$115$$ −4.57834 −0.426932
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −6.50885 −0.596665
$$120$$ 0 0
$$121$$ −1.37279 −0.124799
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −19.2544 −1.70855 −0.854277 0.519818i $$-0.826000\pi$$
−0.854277 + 0.519818i $$0.826000\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −11.4061 −0.996554 −0.498277 0.867018i $$-0.666034\pi$$
−0.498277 + 0.867018i $$0.666034\pi$$
$$132$$ 0 0
$$133$$ 2.52444 0.218897
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.74557 0.234570 0.117285 0.993098i $$-0.462581\pi$$
0.117285 + 0.993098i $$0.462581\pi$$
$$138$$ 0 0
$$139$$ 5.79445 0.491479 0.245739 0.969336i $$-0.420969\pi$$
0.245739 + 0.969336i $$0.420969\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 20.8816 1.74621
$$144$$ 0 0
$$145$$ −1.10278 −0.0915805
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 4.09775 0.335701 0.167850 0.985812i $$-0.446317\pi$$
0.167850 + 0.985812i $$0.446317\pi$$
$$150$$ 0 0
$$151$$ −6.67609 −0.543292 −0.271646 0.962397i $$-0.587568\pi$$
−0.271646 + 0.962397i $$0.587568\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −7.83276 −0.629143
$$156$$ 0 0
$$157$$ −19.4600 −1.55308 −0.776538 0.630071i $$-0.783026\pi$$
−0.776538 + 0.630071i $$0.783026\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −11.5577 −0.910877
$$162$$ 0 0
$$163$$ 12.7300 0.997090 0.498545 0.866864i $$-0.333868\pi$$
0.498545 + 0.866864i $$0.333868\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 20.8816 1.61587 0.807935 0.589272i $$-0.200585\pi$$
0.807935 + 0.589272i $$0.200585\pi$$
$$168$$ 0 0
$$169$$ 32.2927 2.48406
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0.372787 0.0283425 0.0141712 0.999900i $$-0.495489\pi$$
0.0141712 + 0.999900i $$0.495489\pi$$
$$174$$ 0 0
$$175$$ −2.52444 −0.190830
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −1.04888 −0.0783967 −0.0391983 0.999231i $$-0.512480\pi$$
−0.0391983 + 0.999231i $$0.512480\pi$$
$$180$$ 0 0
$$181$$ 5.36222 0.398571 0.199285 0.979941i $$-0.436138\pi$$
0.199285 + 0.979941i $$0.436138\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.52444 0.332643
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 25.7194 1.86099 0.930496 0.366302i $$-0.119376\pi$$
0.930496 + 0.366302i $$0.119376\pi$$
$$192$$ 0 0
$$193$$ 7.77886 0.559935 0.279967 0.960009i $$-0.409676\pi$$
0.279967 + 0.960009i $$0.409676\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.42166 0.386277 0.193139 0.981171i $$-0.438133\pi$$
0.193139 + 0.981171i $$0.438133\pi$$
$$198$$ 0 0
$$199$$ 14.2056 1.00700 0.503502 0.863994i $$-0.332045\pi$$
0.503502 + 0.863994i $$0.332045\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −2.78389 −0.195391
$$204$$ 0 0
$$205$$ 6.15165 0.429650
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −3.10278 −0.214623
$$210$$ 0 0
$$211$$ 6.09775 0.419787 0.209893 0.977724i $$-0.432688\pi$$
0.209893 + 0.977724i $$0.432688\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 7.68111 0.523848
$$216$$ 0 0
$$217$$ −19.7733 −1.34230
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 17.3522 1.16723
$$222$$ 0 0
$$223$$ −24.8222 −1.66222 −0.831109 0.556110i $$-0.812293\pi$$
−0.831109 + 0.556110i $$0.812293\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 22.7839 1.51222 0.756110 0.654445i $$-0.227098\pi$$
0.756110 + 0.654445i $$0.227098\pi$$
$$228$$ 0 0
$$229$$ 6.47054 0.427585 0.213793 0.976879i $$-0.431418\pi$$
0.213793 + 0.976879i $$0.431418\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 19.3522 1.26780 0.633902 0.773414i $$-0.281452\pi$$
0.633902 + 0.773414i $$0.281452\pi$$
$$234$$ 0 0
$$235$$ −3.42166 −0.223205
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −23.9250 −1.54758 −0.773789 0.633443i $$-0.781641\pi$$
−0.773789 + 0.633443i $$0.781641\pi$$
$$240$$ 0 0
$$241$$ −8.09775 −0.521622 −0.260811 0.965390i $$-0.583990\pi$$
−0.260811 + 0.965390i $$0.583990\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0.627213 0.0400712
$$246$$ 0 0
$$247$$ −6.72999 −0.428219
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.00502 −0.0634365 −0.0317183 0.999497i $$-0.510098\pi$$
−0.0317183 + 0.999497i $$0.510098\pi$$
$$252$$ 0 0
$$253$$ 14.2056 0.893095
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 16.7839 1.04695 0.523475 0.852041i $$-0.324635\pi$$
0.523475 + 0.852041i $$0.324635\pi$$
$$258$$ 0 0
$$259$$ 11.4217 0.709708
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 22.0383 1.35894 0.679470 0.733703i $$-0.262210\pi$$
0.679470 + 0.733703i $$0.262210\pi$$
$$264$$ 0 0
$$265$$ 2.57834 0.158386
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −32.3572 −1.97285 −0.986427 0.164202i $$-0.947495\pi$$
−0.986427 + 0.164202i $$0.947495\pi$$
$$270$$ 0 0
$$271$$ −23.1466 −1.40606 −0.703029 0.711161i $$-0.748169\pi$$
−0.703029 + 0.711161i $$0.748169\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 3.10278 0.187104
$$276$$ 0 0
$$277$$ 30.8222 1.85193 0.925963 0.377614i $$-0.123255\pi$$
0.925963 + 0.377614i $$0.123255\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 3.94610 0.235405 0.117702 0.993049i $$-0.462447\pi$$
0.117702 + 0.993049i $$0.462447\pi$$
$$282$$ 0 0
$$283$$ 17.4756 1.03881 0.519407 0.854527i $$-0.326153\pi$$
0.519407 + 0.854527i $$0.326153\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 15.5295 0.916675
$$288$$ 0 0
$$289$$ −10.3522 −0.608952
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −2.05944 −0.120314 −0.0601568 0.998189i $$-0.519160\pi$$
−0.0601568 + 0.998189i $$0.519160\pi$$
$$294$$ 0 0
$$295$$ −10.2056 −0.594190
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 30.8122 1.78191
$$300$$ 0 0
$$301$$ 19.3905 1.11765
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 14.8816 0.852120
$$306$$ 0 0
$$307$$ 10.2056 0.582462 0.291231 0.956653i $$-0.405935\pi$$
0.291231 + 0.956653i $$0.405935\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 1.20053 0.0680756 0.0340378 0.999421i $$-0.489163\pi$$
0.0340378 + 0.999421i $$0.489163\pi$$
$$312$$ 0 0
$$313$$ −7.45998 −0.421663 −0.210831 0.977522i $$-0.567617\pi$$
−0.210831 + 0.977522i $$0.567617\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 4.48059 0.251655 0.125827 0.992052i $$-0.459841\pi$$
0.125827 + 0.992052i $$0.459841\pi$$
$$318$$ 0 0
$$319$$ 3.42166 0.191576
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2.57834 −0.143463
$$324$$ 0 0
$$325$$ 6.72999 0.373313
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −8.63778 −0.476216
$$330$$ 0 0
$$331$$ 24.2439 1.33256 0.666282 0.745700i $$-0.267885\pi$$
0.666282 + 0.745700i $$0.267885\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −8.41110 −0.459547
$$336$$ 0 0
$$337$$ 23.4444 1.27710 0.638549 0.769581i $$-0.279535\pi$$
0.638549 + 0.769581i $$0.279535\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 24.3033 1.31610
$$342$$ 0 0
$$343$$ 19.2544 1.03964
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −0.881639 −0.0473289 −0.0236644 0.999720i $$-0.507533\pi$$
−0.0236644 + 0.999720i $$0.507533\pi$$
$$348$$ 0 0
$$349$$ −1.66553 −0.0891536 −0.0445768 0.999006i $$-0.514194\pi$$
−0.0445768 + 0.999006i $$0.514194\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ 0 0
$$355$$ 9.45998 0.502083
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 17.7194 0.935196 0.467598 0.883941i $$-0.345120\pi$$
0.467598 + 0.883941i $$0.345120\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 9.25443 0.484399
$$366$$ 0 0
$$367$$ −6.93554 −0.362032 −0.181016 0.983480i $$-0.557939\pi$$
−0.181016 + 0.983480i $$0.557939\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.50885 0.337923
$$372$$ 0 0
$$373$$ 3.25997 0.168795 0.0843973 0.996432i $$-0.473104\pi$$
0.0843973 + 0.996432i $$0.473104\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.42166 0.382235
$$378$$ 0 0
$$379$$ 14.1461 0.726637 0.363318 0.931665i $$-0.381644\pi$$
0.363318 + 0.931665i $$0.381644\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −28.0766 −1.43465 −0.717324 0.696739i $$-0.754633\pi$$
−0.717324 + 0.696739i $$0.754633\pi$$
$$384$$ 0 0
$$385$$ 7.83276 0.399195
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −35.3522 −1.79243 −0.896213 0.443623i $$-0.853693\pi$$
−0.896213 + 0.443623i $$0.853693\pi$$
$$390$$ 0 0
$$391$$ 11.8045 0.596979
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −21.2544 −1.06673 −0.533365 0.845885i $$-0.679073\pi$$
−0.533365 + 0.845885i $$0.679073\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −23.8272 −1.18987 −0.594937 0.803772i $$-0.702823\pi$$
−0.594937 + 0.803772i $$0.702823\pi$$
$$402$$ 0 0
$$403$$ 52.7144 2.62589
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −14.0383 −0.695853
$$408$$ 0 0
$$409$$ 15.9900 0.790652 0.395326 0.918541i $$-0.370632\pi$$
0.395326 + 0.918541i $$0.370632\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −25.7633 −1.26773
$$414$$ 0 0
$$415$$ −4.47054 −0.219450
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −1.30833 −0.0639159 −0.0319579 0.999489i $$-0.510174\pi$$
−0.0319579 + 0.999489i $$0.510174\pi$$
$$420$$ 0 0
$$421$$ −39.2333 −1.91211 −0.956057 0.293181i $$-0.905286\pi$$
−0.956057 + 0.293181i $$0.905286\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2.57834 0.125068
$$426$$ 0 0
$$427$$ 37.5678 1.81803
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −27.5577 −1.32741 −0.663705 0.747995i $$-0.731017\pi$$
−0.663705 + 0.747995i $$0.731017\pi$$
$$432$$ 0 0
$$433$$ 3.58336 0.172205 0.0861027 0.996286i $$-0.472559\pi$$
0.0861027 + 0.996286i $$0.472559\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.57834 −0.219012
$$438$$ 0 0
$$439$$ 5.68665 0.271409 0.135705 0.990749i $$-0.456670\pi$$
0.135705 + 0.990749i $$0.456670\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −0.362741 −0.0172343 −0.00861716 0.999963i $$-0.502743\pi$$
−0.00861716 + 0.999963i $$0.502743\pi$$
$$444$$ 0 0
$$445$$ −14.5628 −0.690341
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −0.799473 −0.0377295 −0.0188647 0.999822i $$-0.506005\pi$$
−0.0188647 + 0.999822i $$0.506005\pi$$
$$450$$ 0 0
$$451$$ −19.0872 −0.898781
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 16.9894 0.796477
$$456$$ 0 0
$$457$$ 24.2056 1.13229 0.566144 0.824306i $$-0.308435\pi$$
0.566144 + 0.824306i $$0.308435\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −23.7633 −1.10677 −0.553383 0.832927i $$-0.686663\pi$$
−0.553383 + 0.832927i $$0.686663\pi$$
$$462$$ 0 0
$$463$$ −4.62219 −0.214811 −0.107406 0.994215i $$-0.534254\pi$$
−0.107406 + 0.994215i $$0.534254\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 4.98944 0.230884 0.115442 0.993314i $$-0.463172\pi$$
0.115442 + 0.993314i $$0.463172\pi$$
$$468$$ 0 0
$$469$$ −21.2333 −0.980463
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −23.8328 −1.09583
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −12.9739 −0.592790 −0.296395 0.955065i $$-0.595785\pi$$
−0.296395 + 0.955065i $$0.595785\pi$$
$$480$$ 0 0
$$481$$ −30.4494 −1.38837
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 12.9355 0.587373
$$486$$ 0 0
$$487$$ 7.14663 0.323845 0.161922 0.986804i $$-0.448231\pi$$
0.161922 + 0.986804i $$0.448231\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −22.1416 −0.999237 −0.499618 0.866246i $$-0.666526\pi$$
−0.499618 + 0.866246i $$0.666526\pi$$
$$492$$ 0 0
$$493$$ 2.84333 0.128057
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 23.8811 1.07121
$$498$$ 0 0
$$499$$ −4.51890 −0.202294 −0.101147 0.994872i $$-0.532251\pi$$
−0.101147 + 0.994872i $$0.532251\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 8.35166 0.372382 0.186191 0.982514i $$-0.440386\pi$$
0.186191 + 0.982514i $$0.440386\pi$$
$$504$$ 0 0
$$505$$ −11.4600 −0.509962
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −4.28057 −0.189733 −0.0948666 0.995490i $$-0.530242\pi$$
−0.0948666 + 0.995490i $$0.530242\pi$$
$$510$$ 0 0
$$511$$ 23.3622 1.03348
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −14.5089 −0.639336
$$516$$ 0 0
$$517$$ 10.6167 0.466920
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −40.1305 −1.75815 −0.879075 0.476683i $$-0.841839\pi$$
−0.879075 + 0.476683i $$0.841839\pi$$
$$522$$ 0 0
$$523$$ 20.5189 0.897229 0.448614 0.893725i $$-0.351918\pi$$
0.448614 + 0.893725i $$0.351918\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 20.1955 0.879730
$$528$$ 0 0
$$529$$ −2.03883 −0.0886448
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −41.4005 −1.79326
$$534$$ 0 0
$$535$$ −15.2544 −0.659506
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1.94610 −0.0838245
$$540$$ 0 0
$$541$$ −15.6867 −0.674422 −0.337211 0.941429i $$-0.609484\pi$$
−0.337211 + 0.941429i $$0.609484\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −16.6167 −0.711779
$$546$$ 0 0
$$547$$ 23.8922 1.02156 0.510778 0.859712i $$-0.329357\pi$$
0.510778 + 0.859712i $$0.329357\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.10278 −0.0469798
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −22.8222 −0.967008 −0.483504 0.875342i $$-0.660636\pi$$
−0.483504 + 0.875342i $$0.660636\pi$$
$$558$$ 0 0
$$559$$ −51.6938 −2.18641
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 15.6061 0.657718 0.328859 0.944379i $$-0.393336\pi$$
0.328859 + 0.944379i $$0.393336\pi$$
$$564$$ 0 0
$$565$$ −4.37279 −0.183965
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0.691675 0.0289965 0.0144983 0.999895i $$-0.495385\pi$$
0.0144983 + 0.999895i $$0.495385\pi$$
$$570$$ 0 0
$$571$$ −44.7910 −1.87445 −0.937223 0.348730i $$-0.886613\pi$$
−0.937223 + 0.348730i $$0.886613\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.57834 0.190930
$$576$$ 0 0
$$577$$ 3.45998 0.144041 0.0720203 0.997403i $$-0.477055\pi$$
0.0720203 + 0.997403i $$0.477055\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −11.2856 −0.468205
$$582$$ 0 0
$$583$$ −8.00000 −0.331326
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −27.8328 −1.14878 −0.574391 0.818581i $$-0.694761\pi$$
−0.574391 + 0.818581i $$0.694761\pi$$
$$588$$ 0 0
$$589$$ −7.83276 −0.322743
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 17.5889 0.722290 0.361145 0.932510i $$-0.382386\pi$$
0.361145 + 0.932510i $$0.382386\pi$$
$$594$$ 0 0
$$595$$ 6.50885 0.266837
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 19.4700 0.795524 0.397762 0.917489i $$-0.369787\pi$$
0.397762 + 0.917489i $$0.369787\pi$$
$$600$$ 0 0
$$601$$ 40.4777 1.65112 0.825560 0.564315i $$-0.190860\pi$$
0.825560 + 0.564315i $$0.190860\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 1.37279 0.0558117
$$606$$ 0 0
$$607$$ 4.71440 0.191352 0.0956758 0.995413i $$-0.469499\pi$$
0.0956758 + 0.995413i $$0.469499\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 23.0278 0.931603
$$612$$ 0 0
$$613$$ −9.48110 −0.382938 −0.191469 0.981499i $$-0.561325\pi$$
−0.191469 + 0.981499i $$0.561325\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −7.73501 −0.311400 −0.155700 0.987804i $$-0.549763\pi$$
−0.155700 + 0.987804i $$0.549763\pi$$
$$618$$ 0 0
$$619$$ 6.12892 0.246342 0.123171 0.992385i $$-0.460694\pi$$
0.123171 + 0.992385i $$0.460694\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −36.7628 −1.47287
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −11.6655 −0.465135
$$630$$ 0 0
$$631$$ 4.19550 0.167020 0.0835102 0.996507i $$-0.473387\pi$$
0.0835102 + 0.996507i $$0.473387\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 19.2544 0.764089
$$636$$ 0 0
$$637$$ −4.22114 −0.167247
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 47.3850 1.87159 0.935797 0.352541i $$-0.114682\pi$$
0.935797 + 0.352541i $$0.114682\pi$$
$$642$$ 0 0
$$643$$ 21.3678 0.842662 0.421331 0.906907i $$-0.361563\pi$$
0.421331 + 0.906907i $$0.361563\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −9.29274 −0.365335 −0.182668 0.983175i $$-0.558473\pi$$
−0.182668 + 0.983175i $$0.558473\pi$$
$$648$$ 0 0
$$649$$ 31.6655 1.24298
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 21.9094 0.857381 0.428690 0.903451i $$-0.358975\pi$$
0.428690 + 0.903451i $$0.358975\pi$$
$$654$$ 0 0
$$655$$ 11.4061 0.445672
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 8.52998 0.332281 0.166140 0.986102i $$-0.446870\pi$$
0.166140 + 0.986102i $$0.446870\pi$$
$$660$$ 0 0
$$661$$ 5.36222 0.208566 0.104283 0.994548i $$-0.466745\pi$$
0.104283 + 0.994548i $$0.466745\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −2.52444 −0.0978935
$$666$$ 0 0
$$667$$ 5.04888 0.195493
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −46.1744 −1.78254
$$672$$ 0 0
$$673$$ −2.31889 −0.0893866 −0.0446933 0.999001i $$-0.514231\pi$$
−0.0446933 + 0.999001i $$0.514231\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 35.9305 1.38092 0.690461 0.723370i $$-0.257408\pi$$
0.690461 + 0.723370i $$0.257408\pi$$
$$678$$ 0 0
$$679$$ 32.6550 1.25318
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 5.15667 0.197315 0.0986573 0.995121i $$-0.468545\pi$$
0.0986573 + 0.995121i $$0.468545\pi$$
$$684$$ 0 0
$$685$$ −2.74557 −0.104903
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −17.3522 −0.661065
$$690$$ 0 0
$$691$$ 19.9688 0.759650 0.379825 0.925058i $$-0.375984\pi$$
0.379825 + 0.925058i $$0.375984\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −5.79445 −0.219796
$$696$$ 0 0
$$697$$ −15.8610 −0.600779
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 39.2333 1.48182 0.740911 0.671604i $$-0.234394\pi$$
0.740911 + 0.671604i $$0.234394\pi$$
$$702$$ 0 0
$$703$$ 4.52444 0.170642
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −28.9300 −1.08802
$$708$$ 0 0
$$709$$ 36.8605 1.38433 0.692163 0.721741i $$-0.256658\pi$$
0.692163 + 0.721741i $$0.256658\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 35.8610 1.34301
$$714$$ 0 0
$$715$$ −20.8816 −0.780929
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 17.8383 0.665256 0.332628 0.943058i $$-0.392065\pi$$
0.332628 + 0.943058i $$0.392065\pi$$
$$720$$ 0 0
$$721$$ −36.6267 −1.36405
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1.10278 0.0409560
$$726$$ 0 0
$$727$$ −2.00554 −0.0743813 −0.0371907 0.999308i $$-0.511841\pi$$
−0.0371907 + 0.999308i $$0.511841\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −19.8045 −0.732496
$$732$$ 0 0
$$733$$ −43.4288 −1.60408 −0.802040 0.597271i $$-0.796252\pi$$
−0.802040 + 0.597271i $$0.796252\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 26.0978 0.961323
$$738$$ 0 0
$$739$$ −20.6066 −0.758026 −0.379013 0.925391i $$-0.623736\pi$$
−0.379013 + 0.925391i $$0.623736\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 47.4499 1.74077 0.870385 0.492373i $$-0.163870\pi$$
0.870385 + 0.492373i $$0.163870\pi$$
$$744$$ 0 0
$$745$$ −4.09775 −0.150130
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −38.5089 −1.40708
$$750$$ 0 0
$$751$$ −7.83276 −0.285822 −0.142911 0.989736i $$-0.545646\pi$$
−0.142911 + 0.989736i $$0.545646\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 6.67609 0.242968
$$756$$ 0 0
$$757$$ −43.4600 −1.57958 −0.789790 0.613378i $$-0.789810\pi$$
−0.789790 + 0.613378i $$0.789810\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 2.41110 0.0874023 0.0437012 0.999045i $$-0.486085\pi$$
0.0437012 + 0.999045i $$0.486085\pi$$
$$762$$ 0 0
$$763$$ −41.9477 −1.51861
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 68.6832 2.48001
$$768$$ 0 0
$$769$$ −8.09775 −0.292012 −0.146006 0.989284i $$-0.546642\pi$$
−0.146006 + 0.989284i $$0.546642\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −13.6172 −0.489775 −0.244888 0.969551i $$-0.578751\pi$$
−0.244888 + 0.969551i $$0.578751\pi$$
$$774$$ 0 0
$$775$$ 7.83276 0.281361
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6.15165 0.220406
$$780$$ 0 0
$$781$$ −29.3522 −1.05030
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 19.4600 0.694556
$$786$$ 0 0
$$787$$ 36.9099 1.31570 0.657848 0.753151i $$-0.271467\pi$$
0.657848 + 0.753151i $$0.271467\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −11.0388 −0.392496
$$792$$ 0 0
$$793$$ −100.153 −3.55655
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 26.4705 0.937635 0.468817 0.883295i $$-0.344680\pi$$
0.468817 + 0.883295i $$0.344680\pi$$
$$798$$ 0 0
$$799$$ 8.82220 0.312107
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −28.7144 −1.01331
$$804$$ 0 0
$$805$$ 11.5577 0.407356
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −29.2544 −1.02853 −0.514265 0.857631i $$-0.671935\pi$$
−0.514265 + 0.857631i $$0.671935\pi$$
$$810$$ 0 0
$$811$$ 38.9200 1.36666 0.683332 0.730108i $$-0.260530\pi$$
0.683332 + 0.730108i $$0.260530\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −12.7300 −0.445912
$$816$$ 0 0
$$817$$ 7.68111 0.268728
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 36.6167 1.27793 0.638965 0.769236i $$-0.279363\pi$$
0.638965 + 0.769236i $$0.279363\pi$$
$$822$$ 0 0
$$823$$ 41.5522 1.44842 0.724209 0.689580i $$-0.242205\pi$$
0.724209 + 0.689580i $$0.242205\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −25.0972 −0.872716 −0.436358 0.899773i $$-0.643732\pi$$
−0.436358 + 0.899773i $$0.643732\pi$$
$$828$$ 0 0
$$829$$ 16.9511 0.588737 0.294368 0.955692i $$-0.404891\pi$$
0.294368 + 0.955692i $$0.404891\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −1.61717 −0.0560315
$$834$$ 0 0
$$835$$ −20.8816 −0.722639
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 29.6555 1.02382 0.511910 0.859039i $$-0.328938\pi$$
0.511910 + 0.859039i $$0.328938\pi$$
$$840$$ 0 0
$$841$$ −27.7839 −0.958065
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −32.2927 −1.11090
$$846$$ 0 0
$$847$$ 3.46552 0.119077
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −20.7144 −0.710081
$$852$$ 0 0
$$853$$ 36.4777 1.24897 0.624486 0.781036i $$-0.285308\pi$$
0.624486 + 0.781036i $$0.285308\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 24.8716 0.849597 0.424799 0.905288i $$-0.360345\pi$$
0.424799 + 0.905288i $$0.360345\pi$$
$$858$$ 0 0
$$859$$ 15.0388 0.513118 0.256559 0.966529i $$-0.417411\pi$$
0.256559 + 0.966529i $$0.417411\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 40.4877 1.37822 0.689109 0.724658i $$-0.258002\pi$$
0.689109 + 0.724658i $$0.258002\pi$$
$$864$$ 0 0
$$865$$ −0.372787 −0.0126751
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 56.6066 1.91804
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 2.52444 0.0853416
$$876$$ 0 0
$$877$$ −40.0822 −1.35348 −0.676739 0.736223i $$-0.736608\pi$$
−0.676739 + 0.736223i $$0.736608\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 21.0278 0.708443 0.354221 0.935162i $$-0.384746\pi$$
0.354221 + 0.935162i $$0.384746\pi$$
$$882$$ 0 0
$$883$$ −0.287716 −0.00968241 −0.00484121 0.999988i $$-0.501541\pi$$
−0.00484121 + 0.999988i $$0.501541\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 20.4111 0.685338 0.342669 0.939456i $$-0.388669\pi$$
0.342669 + 0.939456i $$0.388669\pi$$
$$888$$ 0 0
$$889$$ 48.6066 1.63021
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −3.42166 −0.114502
$$894$$ 0 0
$$895$$ 1.04888 0.0350601
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 8.63778 0.288086
$$900$$ 0 0
$$901$$ −6.64782 −0.221471
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −5.36222 −0.178246
$$906$$ 0 0
$$907$$ −16.3799 −0.543887 −0.271943 0.962313i $$-0.587666\pi$$
−0.271943 + 0.962313i $$0.587666\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 50.5855 1.67597 0.837986 0.545692i $$-0.183733\pi$$
0.837986 + 0.545692i $$0.183733\pi$$
$$912$$ 0 0
$$913$$ 13.8711 0.459066
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 28.7939 0.950859
$$918$$ 0 0
$$919$$ −28.7456 −0.948229 −0.474114 0.880463i $$-0.657232\pi$$
−0.474114 + 0.880463i $$0.657232\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −63.6655 −2.09558
$$924$$ 0 0
$$925$$ −4.52444 −0.148763
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 49.7733 1.63301 0.816505 0.577339i $$-0.195909\pi$$
0.816505 + 0.577339i $$0.195909\pi$$
$$930$$ 0 0
$$931$$ 0.627213 0.0205561
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −8.00000 −0.261628
$$936$$ 0 0
$$937$$ 34.3033 1.12064 0.560320 0.828276i $$-0.310678\pi$$
0.560320 + 0.828276i $$0.310678\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −34.6705 −1.13023 −0.565114 0.825013i $$-0.691168\pi$$
−0.565114 + 0.825013i $$0.691168\pi$$
$$942$$ 0 0
$$943$$ −28.1643 −0.917157
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −5.96169 −0.193729 −0.0968644 0.995298i $$-0.530881\pi$$
−0.0968644 + 0.995298i $$0.530881\pi$$
$$948$$ 0 0
$$949$$ −62.2822 −2.02177
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −9.52946 −0.308690 −0.154345 0.988017i $$-0.549327\pi$$
−0.154345 + 0.988017i $$0.549327\pi$$
$$954$$ 0 0
$$955$$ −25.7194 −0.832261
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −6.93103 −0.223815
$$960$$ 0 0
$$961$$ 30.3522 0.979103
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −7.77886 −0.250410
$$966$$ 0 0
$$967$$ −19.9844 −0.642655 −0.321328 0.946968i $$-0.604129\pi$$
−0.321328 + 0.946968i $$0.604129\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −18.3900 −0.590162 −0.295081 0.955472i $$-0.595347\pi$$
−0.295081 + 0.955472i $$0.595347\pi$$
$$972$$ 0 0
$$973$$ −14.6277 −0.468943
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 38.4394 1.22978 0.614892 0.788611i $$-0.289200\pi$$
0.614892 + 0.788611i $$0.289200\pi$$
$$978$$ 0 0
$$979$$ 45.1849 1.44412
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −54.5260 −1.73911 −0.869555 0.493836i $$-0.835594\pi$$
−0.869555 + 0.493836i $$0.835594\pi$$
$$984$$ 0 0
$$985$$ −5.42166 −0.172749
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −35.1667 −1.11824
$$990$$ 0 0
$$991$$ 2.91995 0.0927553 0.0463777 0.998924i $$-0.485232\pi$$
0.0463777 + 0.998924i $$0.485232\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −14.2056 −0.450346
$$996$$ 0 0
$$997$$ −36.2822 −1.14907 −0.574534 0.818481i $$-0.694817\pi$$
−0.574534 + 0.818481i $$0.694817\pi$$
$$998$$ 0 0
$$999$$ 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 6840.2.a.bf.1.2 3
3.2 odd 2 2280.2.a.s.1.2 3
12.11 even 2 4560.2.a.bv.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.s.1.2 3 3.2 odd 2
4560.2.a.bv.1.2 3 12.11 even 2
6840.2.a.bf.1.2 3 1.1 even 1 trivial