Properties

 Label 4560.2.a.bv.1.2 Level $4560$ Weight $2$ Character 4560.1 Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ 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 $$\chi$$ $$=$$ 4560.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} +2.52444 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} +2.52444 q^{7} +1.00000 q^{9} +3.10278 q^{11} +6.72999 q^{13} +1.00000 q^{15} -2.57834 q^{17} +1.00000 q^{19} +2.52444 q^{21} +4.57834 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.10278 q^{29} -7.83276 q^{31} +3.10278 q^{33} +2.52444 q^{35} -4.52444 q^{37} +6.72999 q^{39} +6.15165 q^{41} +7.68111 q^{43} +1.00000 q^{45} +3.42166 q^{47} -0.627213 q^{49} -2.57834 q^{51} +2.57834 q^{53} +3.10278 q^{55} +1.00000 q^{57} +10.2056 q^{59} -14.8816 q^{61} +2.52444 q^{63} +6.72999 q^{65} -8.41110 q^{67} +4.57834 q^{69} -9.45998 q^{71} -9.25443 q^{73} +1.00000 q^{75} +7.83276 q^{77} +1.00000 q^{81} +4.47054 q^{83} -2.57834 q^{85} -1.10278 q^{87} -14.5628 q^{89} +16.9894 q^{91} -7.83276 q^{93} +1.00000 q^{95} -12.9355 q^{97} +3.10278 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{15} - 6 q^{17} + 3 q^{19} + 2 q^{21} + 12 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 14 q^{43} + 3 q^{45} + 12 q^{47} + 11 q^{49} - 6 q^{51} + 6 q^{53} + 2 q^{55} + 3 q^{57} + 16 q^{59} - 6 q^{61} + 2 q^{63} + 4 q^{67} + 12 q^{69} + 12 q^{71} - 2 q^{73} + 3 q^{75} - 4 q^{77} + 3 q^{81} + 4 q^{83} - 6 q^{85} + 4 q^{87} + 4 q^{89} + 20 q^{91} + 4 q^{93} + 3 q^{95} - 4 q^{97} + 2 q^{99} + O(q^{100})$$

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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.52444 0.954148 0.477074 0.878863i $$-0.341697\pi$$
0.477074 + 0.878863i $$0.341697\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 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
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −2.57834 −0.625339 −0.312669 0.949862i $$-0.601223\pi$$
−0.312669 + 0.949862i $$0.601223\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 2.52444 0.550878
$$22$$ 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
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −1.10278 −0.204780 −0.102390 0.994744i $$-0.532649\pi$$
−0.102390 + 0.994744i $$0.532649\pi$$
$$30$$ 0 0
$$31$$ −7.83276 −1.40681 −0.703403 0.710791i $$-0.748337\pi$$
−0.703403 + 0.710791i $$0.748337\pi$$
$$32$$ 0 0
$$33$$ 3.10278 0.540124
$$34$$ 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
$$39$$ 6.72999 1.07766
$$40$$ 0 0
$$41$$ 6.15165 0.960726 0.480363 0.877070i $$-0.340505\pi$$
0.480363 + 0.877070i $$0.340505\pi$$
$$42$$ 0 0
$$43$$ 7.68111 1.17136 0.585679 0.810543i $$-0.300828\pi$$
0.585679 + 0.810543i $$0.300828\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 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
$$51$$ −2.57834 −0.361039
$$52$$ 0 0
$$53$$ 2.57834 0.354162 0.177081 0.984196i $$-0.443335\pi$$
0.177081 + 0.984196i $$0.443335\pi$$
$$54$$ 0 0
$$55$$ 3.10278 0.418378
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 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
$$63$$ 2.52444 0.318049
$$64$$ 0 0
$$65$$ 6.72999 0.834752
$$66$$ 0 0
$$67$$ −8.41110 −1.02758 −0.513790 0.857916i $$-0.671759\pi$$
−0.513790 + 0.857916i $$0.671759\pi$$
$$68$$ 0 0
$$69$$ 4.57834 0.551167
$$70$$ 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
$$75$$ 1.00000 0.115470
$$76$$ 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
$$81$$ 1.00000 0.111111
$$82$$ 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
$$87$$ −1.10278 −0.118230
$$88$$ 0 0
$$89$$ −14.5628 −1.54365 −0.771824 0.635836i $$-0.780656\pi$$
−0.771824 + 0.635836i $$0.780656\pi$$
$$90$$ 0 0
$$91$$ 16.9894 1.78098
$$92$$ 0 0
$$93$$ −7.83276 −0.812220
$$94$$ 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
$$99$$ 3.10278 0.311841
$$100$$ 0 0
$$101$$ −11.4600 −1.14031 −0.570155 0.821537i $$-0.693117\pi$$
−0.570155 + 0.821537i $$0.693117\pi$$
$$102$$ 0 0
$$103$$ −14.5089 −1.42960 −0.714800 0.699329i $$-0.753482\pi$$
−0.714800 + 0.699329i $$0.753482\pi$$
$$104$$ 0 0
$$105$$ 2.52444 0.246360
$$106$$ 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
$$111$$ −4.52444 −0.429441
$$112$$ 0 0
$$113$$ −4.37279 −0.411357 −0.205679 0.978620i $$-0.565940\pi$$
−0.205679 + 0.978620i $$0.565940\pi$$
$$114$$ 0 0
$$115$$ 4.57834 0.426932
$$116$$ 0 0
$$117$$ 6.72999 0.622188
$$118$$ 0 0
$$119$$ −6.50885 −0.596665
$$120$$ 0 0
$$121$$ −1.37279 −0.124799
$$122$$ 0 0
$$123$$ 6.15165 0.554676
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 19.2544 1.70855 0.854277 0.519818i $$-0.174000\pi$$
0.854277 + 0.519818i $$0.174000\pi$$
$$128$$ 0 0
$$129$$ 7.68111 0.676284
$$130$$ 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
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −2.74557 −0.234570 −0.117285 0.993098i $$-0.537419\pi$$
−0.117285 + 0.993098i $$0.537419\pi$$
$$138$$ 0 0
$$139$$ −5.79445 −0.491479 −0.245739 0.969336i $$-0.579031\pi$$
−0.245739 + 0.969336i $$0.579031\pi$$
$$140$$ 0 0
$$141$$ 3.42166 0.288156
$$142$$ 0 0
$$143$$ 20.8816 1.74621
$$144$$ 0 0
$$145$$ −1.10278 −0.0915805
$$146$$ 0 0
$$147$$ −0.627213 −0.0517317
$$148$$ 0 0
$$149$$ −4.09775 −0.335701 −0.167850 0.985812i $$-0.553683\pi$$
−0.167850 + 0.985812i $$0.553683\pi$$
$$150$$ 0 0
$$151$$ 6.67609 0.543292 0.271646 0.962397i $$-0.412432\pi$$
0.271646 + 0.962397i $$0.412432\pi$$
$$152$$ 0 0
$$153$$ −2.57834 −0.208446
$$154$$ 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
$$159$$ 2.57834 0.204475
$$160$$ 0 0
$$161$$ 11.5577 0.910877
$$162$$ 0 0
$$163$$ −12.7300 −0.997090 −0.498545 0.866864i $$-0.666132\pi$$
−0.498545 + 0.866864i $$0.666132\pi$$
$$164$$ 0 0
$$165$$ 3.10278 0.241551
$$166$$ 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
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −0.372787 −0.0283425 −0.0141712 0.999900i $$-0.504511\pi$$
−0.0141712 + 0.999900i $$0.504511\pi$$
$$174$$ 0 0
$$175$$ 2.52444 0.190830
$$176$$ 0 0
$$177$$ 10.2056 0.767096
$$178$$ 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
$$183$$ −14.8816 −1.10008
$$184$$ 0 0
$$185$$ −4.52444 −0.332643
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ 0 0
$$189$$ 2.52444 0.183626
$$190$$ 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
$$195$$ 6.72999 0.481944
$$196$$ 0 0
$$197$$ −5.42166 −0.386277 −0.193139 0.981171i $$-0.561867\pi$$
−0.193139 + 0.981171i $$0.561867\pi$$
$$198$$ 0 0
$$199$$ −14.2056 −1.00700 −0.503502 0.863994i $$-0.667955\pi$$
−0.503502 + 0.863994i $$0.667955\pi$$
$$200$$ 0 0
$$201$$ −8.41110 −0.593273
$$202$$ 0 0
$$203$$ −2.78389 −0.195391
$$204$$ 0 0
$$205$$ 6.15165 0.429650
$$206$$ 0 0
$$207$$ 4.57834 0.318216
$$208$$ 0 0
$$209$$ 3.10278 0.214623
$$210$$ 0 0
$$211$$ −6.09775 −0.419787 −0.209893 0.977724i $$-0.567312\pi$$
−0.209893 + 0.977724i $$0.567312\pi$$
$$212$$ 0 0
$$213$$ −9.45998 −0.648187
$$214$$ 0 0
$$215$$ 7.68111 0.523848
$$216$$ 0 0
$$217$$ −19.7733 −1.34230
$$218$$ 0 0
$$219$$ −9.25443 −0.625356
$$220$$ 0 0
$$221$$ −17.3522 −1.16723
$$222$$ 0 0
$$223$$ 24.8222 1.66222 0.831109 0.556110i $$-0.187707\pi$$
0.831109 + 0.556110i $$0.187707\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 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
$$231$$ 7.83276 0.515358
$$232$$ 0 0
$$233$$ −19.3522 −1.26780 −0.633902 0.773414i $$-0.718548\pi$$
−0.633902 + 0.773414i $$0.718548\pi$$
$$234$$ 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
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −0.627213 −0.0400712
$$246$$ 0 0
$$247$$ 6.72999 0.428219
$$248$$ 0 0
$$249$$ 4.47054 0.283309
$$250$$ 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
$$255$$ −2.57834 −0.161462
$$256$$ 0 0
$$257$$ −16.7839 −1.04695 −0.523475 0.852041i $$-0.675365\pi$$
−0.523475 + 0.852041i $$0.675365\pi$$
$$258$$ 0 0
$$259$$ −11.4217 −0.709708
$$260$$ 0 0
$$261$$ −1.10278 −0.0682601
$$262$$ 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
$$267$$ −14.5628 −0.891226
$$268$$ 0 0
$$269$$ 32.3572 1.97285 0.986427 0.164202i $$-0.0525050\pi$$
0.986427 + 0.164202i $$0.0525050\pi$$
$$270$$ 0 0
$$271$$ 23.1466 1.40606 0.703029 0.711161i $$-0.251831\pi$$
0.703029 + 0.711161i $$0.251831\pi$$
$$272$$ 0 0
$$273$$ 16.9894 1.02825
$$274$$ 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
$$279$$ −7.83276 −0.468935
$$280$$ 0 0
$$281$$ −3.94610 −0.235405 −0.117702 0.993049i $$-0.537553\pi$$
−0.117702 + 0.993049i $$0.537553\pi$$
$$282$$ 0 0
$$283$$ −17.4756 −1.03881 −0.519407 0.854527i $$-0.673847\pi$$
−0.519407 + 0.854527i $$0.673847\pi$$
$$284$$ 0 0
$$285$$ 1.00000 0.0592349
$$286$$ 0 0
$$287$$ 15.5295 0.916675
$$288$$ 0 0
$$289$$ −10.3522 −0.608952
$$290$$ 0 0
$$291$$ −12.9355 −0.758295
$$292$$ 0 0
$$293$$ 2.05944 0.120314 0.0601568 0.998189i $$-0.480840\pi$$
0.0601568 + 0.998189i $$0.480840\pi$$
$$294$$ 0 0
$$295$$ 10.2056 0.594190
$$296$$ 0 0
$$297$$ 3.10278 0.180041
$$298$$ 0 0
$$299$$ 30.8122 1.78191
$$300$$ 0 0
$$301$$ 19.3905 1.11765
$$302$$ 0 0
$$303$$ −11.4600 −0.658358
$$304$$ 0 0
$$305$$ −14.8816 −0.852120
$$306$$ 0 0
$$307$$ −10.2056 −0.582462 −0.291231 0.956653i $$-0.594065\pi$$
−0.291231 + 0.956653i $$0.594065\pi$$
$$308$$ 0 0
$$309$$ −14.5089 −0.825380
$$310$$ 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
$$315$$ 2.52444 0.142236
$$316$$ 0 0
$$317$$ −4.48059 −0.251655 −0.125827 0.992052i $$-0.540159\pi$$
−0.125827 + 0.992052i $$0.540159\pi$$
$$318$$ 0 0
$$319$$ −3.42166 −0.191576
$$320$$ 0 0
$$321$$ 15.2544 0.851419
$$322$$ 0 0
$$323$$ −2.57834 −0.143463
$$324$$ 0 0
$$325$$ 6.72999 0.373313
$$326$$ 0 0
$$327$$ 16.6167 0.918903
$$328$$ 0 0
$$329$$ 8.63778 0.476216
$$330$$ 0 0
$$331$$ −24.2439 −1.33256 −0.666282 0.745700i $$-0.732115\pi$$
−0.666282 + 0.745700i $$0.732115\pi$$
$$332$$ 0 0
$$333$$ −4.52444 −0.247938
$$334$$ 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
$$339$$ −4.37279 −0.237497
$$340$$ 0 0
$$341$$ −24.3033 −1.31610
$$342$$ 0 0
$$343$$ −19.2544 −1.03964
$$344$$ 0 0
$$345$$ 4.57834 0.246489
$$346$$ 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
$$351$$ 6.72999 0.359220
$$352$$ 0 0
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ 0 0
$$355$$ −9.45998 −0.502083
$$356$$ 0 0
$$357$$ −6.50885 −0.344485
$$358$$ 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
$$363$$ −1.37279 −0.0720526
$$364$$ 0 0
$$365$$ −9.25443 −0.484399
$$366$$ 0 0
$$367$$ 6.93554 0.362032 0.181016 0.983480i $$-0.442061\pi$$
0.181016 + 0.983480i $$0.442061\pi$$
$$368$$ 0 0
$$369$$ 6.15165 0.320242
$$370$$ 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
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −7.42166 −0.382235
$$378$$ 0 0
$$379$$ −14.1461 −0.726637 −0.363318 0.931665i $$-0.618356\pi$$
−0.363318 + 0.931665i $$0.618356\pi$$
$$380$$ 0 0
$$381$$ 19.2544 0.986434
$$382$$ 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
$$387$$ 7.68111 0.390453
$$388$$ 0 0
$$389$$ 35.3522 1.79243 0.896213 0.443623i $$-0.146307\pi$$
0.896213 + 0.443623i $$0.146307\pi$$
$$390$$ 0 0
$$391$$ −11.8045 −0.596979
$$392$$ 0 0
$$393$$ −11.4061 −0.575360
$$394$$ 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
$$399$$ 2.52444 0.126380
$$400$$ 0 0
$$401$$ 23.8272 1.18987 0.594937 0.803772i $$-0.297177\pi$$
0.594937 + 0.803772i $$0.297177\pi$$
$$402$$ 0 0
$$403$$ −52.7144 −2.62589
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 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
$$411$$ −2.74557 −0.135429
$$412$$ 0 0
$$413$$ 25.7633 1.26773
$$414$$ 0 0
$$415$$ 4.47054 0.219450
$$416$$ 0 0
$$417$$ −5.79445 −0.283755
$$418$$ 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
$$423$$ 3.42166 0.166367
$$424$$ 0 0
$$425$$ −2.57834 −0.125068
$$426$$ 0 0
$$427$$ −37.5678 −1.81803
$$428$$ 0 0
$$429$$ 20.8816 1.00818
$$430$$ 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
$$435$$ −1.10278 −0.0528740
$$436$$ 0 0
$$437$$ 4.57834 0.219012
$$438$$ 0 0
$$439$$ −5.68665 −0.271409 −0.135705 0.990749i $$-0.543330\pi$$
−0.135705 + 0.990749i $$0.543330\pi$$
$$440$$ 0 0
$$441$$ −0.627213 −0.0298673
$$442$$ 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
$$447$$ −4.09775 −0.193817
$$448$$ 0 0
$$449$$ 0.799473 0.0377295 0.0188647 0.999822i $$-0.493995\pi$$
0.0188647 + 0.999822i $$0.493995\pi$$
$$450$$ 0 0
$$451$$ 19.0872 0.898781
$$452$$ 0 0
$$453$$ 6.67609 0.313670
$$454$$ 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
$$459$$ −2.57834 −0.120346
$$460$$ 0 0
$$461$$ 23.7633 1.10677 0.553383 0.832927i $$-0.313337\pi$$
0.553383 + 0.832927i $$0.313337\pi$$
$$462$$ 0 0
$$463$$ 4.62219 0.214811 0.107406 0.994215i $$-0.465746\pi$$
0.107406 + 0.994215i $$0.465746\pi$$
$$464$$ 0 0
$$465$$ −7.83276 −0.363236
$$466$$ 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
$$471$$ −19.4600 −0.896668
$$472$$ 0 0
$$473$$ 23.8328 1.09583
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 2.57834 0.118054
$$478$$ 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
$$483$$ 11.5577 0.525895
$$484$$ 0 0
$$485$$ −12.9355 −0.587373
$$486$$ 0 0
$$487$$ −7.14663 −0.323845 −0.161922 0.986804i $$-0.551769\pi$$
−0.161922 + 0.986804i $$0.551769\pi$$
$$488$$ 0 0
$$489$$ −12.7300 −0.575670
$$490$$ 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
$$495$$ 3.10278 0.139459
$$496$$ 0 0
$$497$$ −23.8811 −1.07121
$$498$$ 0 0
$$499$$ 4.51890 0.202294 0.101147 0.994872i $$-0.467749\pi$$
0.101147 + 0.994872i $$0.467749\pi$$
$$500$$ 0 0
$$501$$ 20.8816 0.932923
$$502$$ 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
$$507$$ 32.2927 1.43417
$$508$$ 0 0
$$509$$ 4.28057 0.189733 0.0948666 0.995490i $$-0.469758\pi$$
0.0948666 + 0.995490i $$0.469758\pi$$
$$510$$ 0 0
$$511$$ −23.3622 −1.03348
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −14.5089 −0.639336
$$516$$ 0 0
$$517$$ 10.6167 0.466920
$$518$$ 0 0
$$519$$ −0.372787 −0.0163635
$$520$$ 0 0
$$521$$ 40.1305 1.75815 0.879075 0.476683i $$-0.158161\pi$$
0.879075 + 0.476683i $$0.158161\pi$$
$$522$$ 0 0
$$523$$ −20.5189 −0.897229 −0.448614 0.893725i $$-0.648082\pi$$
−0.448614 + 0.893725i $$0.648082\pi$$
$$524$$ 0 0
$$525$$ 2.52444 0.110176
$$526$$ 0 0
$$527$$ 20.1955 0.879730
$$528$$ 0 0
$$529$$ −2.03883 −0.0886448
$$530$$ 0 0
$$531$$ 10.2056 0.442883
$$532$$ 0 0
$$533$$ 41.4005 1.79326
$$534$$ 0 0
$$535$$ 15.2544 0.659506
$$536$$ 0 0
$$537$$ −1.04888 −0.0452623
$$538$$ 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
$$543$$ 5.36222 0.230115
$$544$$ 0 0
$$545$$ 16.6167 0.711779
$$546$$ 0 0
$$547$$ −23.8922 −1.02156 −0.510778 0.859712i $$-0.670643\pi$$
−0.510778 + 0.859712i $$0.670643\pi$$
$$548$$ 0 0
$$549$$ −14.8816 −0.635133
$$550$$ 0 0
$$551$$ −1.10278 −0.0469798
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −4.52444 −0.192052
$$556$$ 0 0
$$557$$ 22.8222 0.967008 0.483504 0.875342i $$-0.339364\pi$$
0.483504 + 0.875342i $$0.339364\pi$$
$$558$$ 0 0
$$559$$ 51.6938 2.18641
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 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
$$567$$ 2.52444 0.106016
$$568$$ 0 0
$$569$$ −0.691675 −0.0289965 −0.0144983 0.999895i $$-0.504615\pi$$
−0.0144983 + 0.999895i $$0.504615\pi$$
$$570$$ 0 0
$$571$$ 44.7910 1.87445 0.937223 0.348730i $$-0.113387\pi$$
0.937223 + 0.348730i $$0.113387\pi$$
$$572$$ 0 0
$$573$$ 25.7194 1.07444
$$574$$ 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
$$579$$ 7.77886 0.323279
$$580$$ 0 0
$$581$$ 11.2856 0.468205
$$582$$ 0 0
$$583$$ 8.00000 0.331326
$$584$$ 0 0
$$585$$ 6.72999 0.278251
$$586$$ 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
$$591$$ −5.42166 −0.223017
$$592$$ 0 0
$$593$$ −17.5889 −0.722290 −0.361145 0.932510i $$-0.617614\pi$$
−0.361145 + 0.932510i $$0.617614\pi$$
$$594$$ 0 0
$$595$$ −6.50885 −0.266837
$$596$$ 0 0
$$597$$ −14.2056 −0.581394
$$598$$ 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
$$603$$ −8.41110 −0.342526
$$604$$ 0 0
$$605$$ −1.37279 −0.0558117
$$606$$ 0 0
$$607$$ −4.71440 −0.191352 −0.0956758 0.995413i $$-0.530501\pi$$
−0.0956758 + 0.995413i $$0.530501\pi$$
$$608$$ 0 0
$$609$$ −2.78389 −0.112809
$$610$$ 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
$$615$$ 6.15165 0.248059
$$616$$ 0 0
$$617$$ 7.73501 0.311400 0.155700 0.987804i $$-0.450237\pi$$
0.155700 + 0.987804i $$0.450237\pi$$
$$618$$ 0 0
$$619$$ −6.12892 −0.246342 −0.123171 0.992385i $$-0.539306\pi$$
−0.123171 + 0.992385i $$0.539306\pi$$
$$620$$ 0 0
$$621$$ 4.57834 0.183722
$$622$$ 0 0
$$623$$ −36.7628 −1.47287
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 3.10278 0.123913
$$628$$ 0 0
$$629$$ 11.6655 0.465135
$$630$$ 0 0
$$631$$ −4.19550 −0.167020 −0.0835102 0.996507i $$-0.526613\pi$$
−0.0835102 + 0.996507i $$0.526613\pi$$
$$632$$ 0 0
$$633$$ −6.09775 −0.242364
$$634$$ 0 0
$$635$$ 19.2544 0.764089
$$636$$ 0 0
$$637$$ −4.22114 −0.167247
$$638$$ 0 0
$$639$$ −9.45998 −0.374231
$$640$$ 0 0
$$641$$ −47.3850 −1.87159 −0.935797 0.352541i $$-0.885318\pi$$
−0.935797 + 0.352541i $$0.885318\pi$$
$$642$$ 0 0
$$643$$ −21.3678 −0.842662 −0.421331 0.906907i $$-0.638437\pi$$
−0.421331 + 0.906907i $$0.638437\pi$$
$$644$$ 0 0
$$645$$ 7.68111 0.302443
$$646$$ 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
$$651$$ −19.7733 −0.774978
$$652$$ 0 0
$$653$$ −21.9094 −0.857381 −0.428690 0.903451i $$-0.641025\pi$$
−0.428690 + 0.903451i $$0.641025\pi$$
$$654$$ 0 0
$$655$$ −11.4061 −0.445672
$$656$$ 0 0
$$657$$ −9.25443 −0.361050
$$658$$ 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
$$663$$ −17.3522 −0.673903
$$664$$ 0 0
$$665$$ 2.52444 0.0978935
$$666$$ 0 0
$$667$$ −5.04888 −0.195493
$$668$$ 0 0
$$669$$ 24.8222 0.959682
$$670$$ 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
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −35.9305 −1.38092 −0.690461 0.723370i $$-0.742592\pi$$
−0.690461 + 0.723370i $$0.742592\pi$$
$$678$$ 0 0
$$679$$ −32.6550 −1.25318
$$680$$ 0 0
$$681$$ 22.7839 0.873080
$$682$$ 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
$$687$$ 6.47054 0.246866
$$688$$ 0 0
$$689$$ 17.3522 0.661065
$$690$$ 0 0
$$691$$ −19.9688 −0.759650 −0.379825 0.925058i $$-0.624016\pi$$
−0.379825 + 0.925058i $$0.624016\pi$$
$$692$$ 0 0
$$693$$ 7.83276 0.297542
$$694$$ 0 0
$$695$$ −5.79445 −0.219796
$$696$$ 0 0
$$697$$ −15.8610 −0.600779
$$698$$ 0 0
$$699$$ −19.3522 −0.731967
$$700$$ 0 0
$$701$$ −39.2333 −1.48182 −0.740911 0.671604i $$-0.765606\pi$$
−0.740911 + 0.671604i $$0.765606\pi$$
$$702$$ 0 0
$$703$$ −4.52444 −0.170642
$$704$$ 0 0
$$705$$ 3.42166 0.128867
$$706$$ 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
$$717$$ −23.9250 −0.893495
$$718$$ 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
$$723$$ −8.09775 −0.301159
$$724$$ 0 0
$$725$$ −1.10278 −0.0409560
$$726$$ 0 0
$$727$$ 2.00554 0.0743813 0.0371907 0.999308i $$-0.488159\pi$$
0.0371907 + 0.999308i $$0.488159\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 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
$$735$$ −0.627213 −0.0231351
$$736$$ 0 0
$$737$$ −26.0978 −0.961323
$$738$$ 0 0
$$739$$ 20.6066 0.758026 0.379013 0.925391i $$-0.376264\pi$$
0.379013 + 0.925391i $$0.376264\pi$$
$$740$$ 0 0
$$741$$ 6.72999 0.247232
$$742$$ 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
$$747$$ 4.47054 0.163568
$$748$$ 0 0
$$749$$ 38.5089 1.40708
$$750$$ 0 0
$$751$$ 7.83276 0.285822 0.142911 0.989736i $$-0.454354\pi$$
0.142911 + 0.989736i $$0.454354\pi$$
$$752$$ 0 0
$$753$$ −1.00502 −0.0366251
$$754$$ 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
$$759$$ 14.2056 0.515629
$$760$$ 0 0
$$761$$ −2.41110 −0.0874023 −0.0437012 0.999045i $$-0.513915\pi$$
−0.0437012 + 0.999045i $$0.513915\pi$$
$$762$$ 0 0
$$763$$ 41.9477 1.51861
$$764$$ 0 0
$$765$$ −2.57834 −0.0932200
$$766$$ 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
$$771$$ −16.7839 −0.604457
$$772$$ 0 0
$$773$$ 13.6172 0.489775 0.244888 0.969551i $$-0.421249\pi$$
0.244888 + 0.969551i $$0.421249\pi$$
$$774$$ 0 0
$$775$$ −7.83276 −0.281361
$$776$$ 0 0
$$777$$ −11.4217 −0.409750
$$778$$ 0 0
$$779$$ 6.15165 0.220406
$$780$$ 0 0
$$781$$ −29.3522 −1.05030
$$782$$ 0 0
$$783$$ −1.10278 −0.0394100
$$784$$ 0 0
$$785$$ −19.4600 −0.694556
$$786$$ 0 0
$$787$$ −36.9099 −1.31570 −0.657848 0.753151i $$-0.728533\pi$$
−0.657848 + 0.753151i $$0.728533\pi$$
$$788$$ 0 0
$$789$$ 22.0383 0.784585
$$790$$ 0 0
$$791$$ −11.0388 −0.392496
$$792$$ 0 0
$$793$$ −100.153 −3.55655
$$794$$ 0 0
$$795$$ 2.57834 0.0914442
$$796$$ 0 0
$$797$$ −26.4705 −0.937635 −0.468817 0.883295i $$-0.655320\pi$$
−0.468817 + 0.883295i $$0.655320\pi$$
$$798$$ 0 0
$$799$$ −8.82220 −0.312107
$$800$$ 0 0
$$801$$ −14.5628 −0.514550
$$802$$ 0 0
$$803$$ −28.7144 −1.01331
$$804$$ 0 0
$$805$$ 11.5577 0.407356
$$806$$ 0 0
$$807$$ 32.3572 1.13903
$$808$$ 0 0
$$809$$ 29.2544 1.02853 0.514265 0.857631i $$-0.328065\pi$$
0.514265 + 0.857631i $$0.328065\pi$$
$$810$$ 0 0
$$811$$ −38.9200 −1.36666 −0.683332 0.730108i $$-0.739470\pi$$
−0.683332 + 0.730108i $$0.739470\pi$$
$$812$$ 0 0
$$813$$ 23.1466 0.811788
$$814$$ 0 0
$$815$$ −12.7300 −0.445912
$$816$$ 0 0
$$817$$ 7.68111 0.268728
$$818$$ 0 0
$$819$$ 16.9894 0.593659
$$820$$ 0 0
$$821$$ −36.6167 −1.27793 −0.638965 0.769236i $$-0.720637\pi$$
−0.638965 + 0.769236i $$0.720637\pi$$
$$822$$ 0 0
$$823$$ −41.5522 −1.44842 −0.724209 0.689580i $$-0.757795\pi$$
−0.724209 + 0.689580i $$0.757795\pi$$
$$824$$ 0 0
$$825$$ 3.10278 0.108025
$$826$$ 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
$$831$$ 30.8222 1.06921
$$832$$ 0 0
$$833$$ 1.61717 0.0560315
$$834$$ 0 0
$$835$$ 20.8816 0.722639
$$836$$ 0 0
$$837$$ −7.83276 −0.270740
$$838$$ 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
$$843$$ −3.94610 −0.135911
$$844$$ 0 0
$$845$$ 32.2927 1.11090
$$846$$ 0 0
$$847$$ −3.46552 −0.119077
$$848$$ 0 0
$$849$$ −17.4756 −0.599760
$$850$$ 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
$$855$$ 1.00000 0.0341993
$$856$$ 0 0
$$857$$ −24.8716 −0.849597 −0.424799 0.905288i $$-0.639655\pi$$
−0.424799 + 0.905288i $$0.639655\pi$$
$$858$$ 0 0
$$859$$ −15.0388 −0.513118 −0.256559 0.966529i $$-0.582589\pi$$
−0.256559 + 0.966529i $$0.582589\pi$$
$$860$$ 0 0
$$861$$ 15.5295 0.529243
$$862$$ 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
$$867$$ −10.3522 −0.351578
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −56.6066 −1.91804
$$872$$ 0 0
$$873$$ −12.9355 −0.437802
$$874$$ 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
$$879$$ 2.05944 0.0694631
$$880$$ 0 0
$$881$$ −21.0278 −0.708443 −0.354221 0.935162i $$-0.615254\pi$$
−0.354221 + 0.935162i $$0.615254\pi$$
$$882$$ 0 0
$$883$$ 0.287716 0.00968241 0.00484121 0.999988i $$-0.498459\pi$$
0.00484121 + 0.999988i $$0.498459\pi$$
$$884$$ 0 0
$$885$$ 10.2056 0.343056
$$886$$ 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
$$891$$ 3.10278 0.103947
$$892$$ 0 0
$$893$$ 3.42166 0.114502
$$894$$ 0 0
$$895$$ −1.04888 −0.0350601
$$896$$ 0 0
$$897$$ 30.8122 1.02879
$$898$$ 0 0
$$899$$ 8.63778 0.288086
$$900$$ 0 0
$$901$$ −6.64782 −0.221471
$$902$$ 0 0
$$903$$ 19.3905 0.645275
$$904$$ 0 0
$$905$$ 5.36222 0.178246
$$906$$ 0 0
$$907$$ 16.3799 0.543887 0.271943 0.962313i $$-0.412334\pi$$
0.271943 + 0.962313i $$0.412334\pi$$
$$908$$ 0 0
$$909$$ −11.4600 −0.380103
$$910$$ 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
$$915$$ −14.8816 −0.491972
$$916$$ 0 0
$$917$$ −28.7939 −0.950859
$$918$$ 0 0
$$919$$ 28.7456 0.948229 0.474114 0.880463i $$-0.342768\pi$$
0.474114 + 0.880463i $$0.342768\pi$$
$$920$$ 0 0
$$921$$ −10.2056 −0.336284
$$922$$ 0 0
$$923$$ −63.6655 −2.09558
$$924$$ 0 0
$$925$$ −4.52444 −0.148763
$$926$$ 0 0
$$927$$ −14.5089 −0.476533
$$928$$ 0 0
$$929$$ −49.7733 −1.63301 −0.816505 0.577339i $$-0.804091\pi$$
−0.816505 + 0.577339i $$0.804091\pi$$
$$930$$ 0 0
$$931$$ −0.627213 −0.0205561
$$932$$ 0 0
$$933$$ 1.20053 0.0393035
$$934$$ 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
$$939$$ −7.45998 −0.243447
$$940$$ 0 0
$$941$$ 34.6705 1.13023 0.565114 0.825013i $$-0.308832\pi$$
0.565114 + 0.825013i $$0.308832\pi$$
$$942$$ 0 0
$$943$$ 28.1643 0.917157
$$944$$ 0 0
$$945$$ 2.52444 0.0821200
$$946$$ 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
$$951$$ −4.48059 −0.145293
$$952$$ 0 0
$$953$$ 9.52946 0.308690 0.154345 0.988017i $$-0.450673\pi$$
0.154345 + 0.988017i $$0.450673\pi$$
$$954$$ 0 0
$$955$$ 25.7194 0.832261
$$956$$ 0 0
$$957$$ −3.42166 −0.110607
$$958$$ 0 0
$$959$$ −6.93103 −0.223815
$$960$$ 0 0
$$961$$ 30.3522 0.979103
$$962$$ 0 0
$$963$$ 15.2544 0.491567
$$964$$ 0 0
$$965$$ 7.77886 0.250410
$$966$$ 0 0
$$967$$ 19.9844 0.642655 0.321328 0.946968i $$-0.395871\pi$$
0.321328 + 0.946968i $$0.395871\pi$$
$$968$$ 0 0
$$969$$ −2.57834 −0.0828281
$$970$$ 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
$$975$$ 6.72999 0.215532
$$976$$ 0 0
$$977$$ −38.4394 −1.22978 −0.614892 0.788611i $$-0.710800\pi$$
−0.614892 + 0.788611i $$0.710800\pi$$
$$978$$ 0 0
$$979$$ −45.1849 −1.44412
$$980$$ 0 0
$$981$$ 16.6167 0.530529
$$982$$ 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
$$987$$ 8.63778 0.274943
$$988$$ 0 0
$$989$$ 35.1667 1.11824
$$990$$ 0 0
$$991$$ −2.91995 −0.0927553 −0.0463777 0.998924i $$-0.514768\pi$$
−0.0463777 + 0.998924i $$0.514768\pi$$
$$992$$ 0 0
$$993$$ −24.2439 −0.769356
$$994$$ 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
$$999$$ −4.52444 −0.143147
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 4560.2.a.bv.1.2 3
4.3 odd 2 2280.2.a.s.1.2 3
12.11 even 2 6840.2.a.bf.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.s.1.2 3 4.3 odd 2
4560.2.a.bv.1.2 3 1.1 even 1 trivial
6840.2.a.bf.1.2 3 12.11 even 2