# Properties

 Label 2640.2.a.be.1.1 Level $2640$ Weight $2$ Character 2640.1 Self dual yes Analytic conductor $21.081$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2640.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.0805061336$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 2640.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} -3.35026 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} -3.35026 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.96239 q^{13} -1.00000 q^{15} -4.57452 q^{17} +4.31265 q^{19} +3.35026 q^{21} +6.70052 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.61213 q^{29} -9.92478 q^{31} +1.00000 q^{33} -3.35026 q^{35} -2.00000 q^{37} -2.96239 q^{39} -4.38787 q^{41} +9.27504 q^{43} +1.00000 q^{45} +9.92478 q^{47} +4.22425 q^{49} +4.57452 q^{51} +4.70052 q^{53} -1.00000 q^{55} -4.31265 q^{57} -10.7005 q^{59} -8.70052 q^{61} -3.35026 q^{63} +2.96239 q^{65} -5.92478 q^{67} -6.70052 q^{69} -9.92478 q^{71} -7.73813 q^{73} -1.00000 q^{75} +3.35026 q^{77} -11.5369 q^{79} +1.00000 q^{81} -10.8872 q^{83} -4.57452 q^{85} +3.61213 q^{87} -2.77575 q^{89} -9.92478 q^{91} +9.92478 q^{93} +4.31265 q^{95} +0.0752228 q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{15} - 2 q^{17} - 8 q^{19} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{37} + 2 q^{39} - 14 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47} + 11 q^{49} + 2 q^{51} - 6 q^{53} - 3 q^{55} + 8 q^{57} - 12 q^{59} - 6 q^{61} - 2 q^{65} + 4 q^{67} - 8 q^{71} - 14 q^{73} - 3 q^{75} - 12 q^{79} + 3 q^{81} - 2 q^{85} + 10 q^{87} - 10 q^{89} - 8 q^{91} + 8 q^{93} - 8 q^{95} + 22 q^{97} - 3 q^{99} + O(q^{100})$$

## 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
$$7$$ −3.35026 −1.26628 −0.633140 0.774037i $$-0.718234\pi$$
−0.633140 + 0.774037i $$0.718234\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 2.96239 0.821619 0.410809 0.911721i $$-0.365246\pi$$
0.410809 + 0.911721i $$0.365246\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −4.57452 −1.10948 −0.554741 0.832023i $$-0.687183\pi$$
−0.554741 + 0.832023i $$0.687183\pi$$
$$18$$ 0 0
$$19$$ 4.31265 0.989390 0.494695 0.869067i $$-0.335280\pi$$
0.494695 + 0.869067i $$0.335280\pi$$
$$20$$ 0 0
$$21$$ 3.35026 0.731087
$$22$$ 0 0
$$23$$ 6.70052 1.39716 0.698578 0.715534i $$-0.253817\pi$$
0.698578 + 0.715534i $$0.253817\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −3.61213 −0.670755 −0.335378 0.942084i $$-0.608864\pi$$
−0.335378 + 0.942084i $$0.608864\pi$$
$$30$$ 0 0
$$31$$ −9.92478 −1.78254 −0.891271 0.453470i $$-0.850186\pi$$
−0.891271 + 0.453470i $$0.850186\pi$$
$$32$$ 0 0
$$33$$ 1.00000 0.174078
$$34$$ 0 0
$$35$$ −3.35026 −0.566298
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ −2.96239 −0.474362
$$40$$ 0 0
$$41$$ −4.38787 −0.685271 −0.342635 0.939468i $$-0.611320\pi$$
−0.342635 + 0.939468i $$0.611320\pi$$
$$42$$ 0 0
$$43$$ 9.27504 1.41443 0.707215 0.706998i $$-0.249951\pi$$
0.707215 + 0.706998i $$0.249951\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 9.92478 1.44768 0.723839 0.689969i $$-0.242376\pi$$
0.723839 + 0.689969i $$0.242376\pi$$
$$48$$ 0 0
$$49$$ 4.22425 0.603465
$$50$$ 0 0
$$51$$ 4.57452 0.640560
$$52$$ 0 0
$$53$$ 4.70052 0.645667 0.322833 0.946456i $$-0.395365\pi$$
0.322833 + 0.946456i $$0.395365\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ −4.31265 −0.571224
$$58$$ 0 0
$$59$$ −10.7005 −1.39309 −0.696545 0.717513i $$-0.745280\pi$$
−0.696545 + 0.717513i $$0.745280\pi$$
$$60$$ 0 0
$$61$$ −8.70052 −1.11399 −0.556994 0.830517i $$-0.688045\pi$$
−0.556994 + 0.830517i $$0.688045\pi$$
$$62$$ 0 0
$$63$$ −3.35026 −0.422093
$$64$$ 0 0
$$65$$ 2.96239 0.367439
$$66$$ 0 0
$$67$$ −5.92478 −0.723827 −0.361913 0.932212i $$-0.617876\pi$$
−0.361913 + 0.932212i $$0.617876\pi$$
$$68$$ 0 0
$$69$$ −6.70052 −0.806648
$$70$$ 0 0
$$71$$ −9.92478 −1.17785 −0.588927 0.808186i $$-0.700450\pi$$
−0.588927 + 0.808186i $$0.700450\pi$$
$$72$$ 0 0
$$73$$ −7.73813 −0.905680 −0.452840 0.891592i $$-0.649589\pi$$
−0.452840 + 0.891592i $$0.649589\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 3.35026 0.381798
$$78$$ 0 0
$$79$$ −11.5369 −1.29800 −0.649002 0.760787i $$-0.724813\pi$$
−0.649002 + 0.760787i $$0.724813\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −10.8872 −1.19502 −0.597511 0.801861i $$-0.703844\pi$$
−0.597511 + 0.801861i $$0.703844\pi$$
$$84$$ 0 0
$$85$$ −4.57452 −0.496176
$$86$$ 0 0
$$87$$ 3.61213 0.387261
$$88$$ 0 0
$$89$$ −2.77575 −0.294229 −0.147114 0.989120i $$-0.546999\pi$$
−0.147114 + 0.989120i $$0.546999\pi$$
$$90$$ 0 0
$$91$$ −9.92478 −1.04040
$$92$$ 0 0
$$93$$ 9.92478 1.02915
$$94$$ 0 0
$$95$$ 4.31265 0.442469
$$96$$ 0 0
$$97$$ 0.0752228 0.00763772 0.00381886 0.999993i $$-0.498784\pi$$
0.00381886 + 0.999993i $$0.498784\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −15.0884 −1.50135 −0.750676 0.660671i $$-0.770272\pi$$
−0.750676 + 0.660671i $$0.770272\pi$$
$$102$$ 0 0
$$103$$ 3.22425 0.317695 0.158848 0.987303i $$-0.449222\pi$$
0.158848 + 0.987303i $$0.449222\pi$$
$$104$$ 0 0
$$105$$ 3.35026 0.326952
$$106$$ 0 0
$$107$$ 0.962389 0.0930376 0.0465188 0.998917i $$-0.485187\pi$$
0.0465188 + 0.998917i $$0.485187\pi$$
$$108$$ 0 0
$$109$$ 11.4010 1.09202 0.546011 0.837778i $$-0.316146\pi$$
0.546011 + 0.837778i $$0.316146\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 6.70052 0.624827
$$116$$ 0 0
$$117$$ 2.96239 0.273873
$$118$$ 0 0
$$119$$ 15.3258 1.40492
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 4.38787 0.395641
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 14.5745 1.29328 0.646640 0.762796i $$-0.276174\pi$$
0.646640 + 0.762796i $$0.276174\pi$$
$$128$$ 0 0
$$129$$ −9.27504 −0.816622
$$130$$ 0 0
$$131$$ 5.92478 0.517650 0.258825 0.965924i $$-0.416665\pi$$
0.258825 + 0.965924i $$0.416665\pi$$
$$132$$ 0 0
$$133$$ −14.4485 −1.25284
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 13.8496 1.18325 0.591624 0.806214i $$-0.298487\pi$$
0.591624 + 0.806214i $$0.298487\pi$$
$$138$$ 0 0
$$139$$ −13.6121 −1.15457 −0.577283 0.816544i $$-0.695887\pi$$
−0.577283 + 0.816544i $$0.695887\pi$$
$$140$$ 0 0
$$141$$ −9.92478 −0.835817
$$142$$ 0 0
$$143$$ −2.96239 −0.247727
$$144$$ 0 0
$$145$$ −3.61213 −0.299971
$$146$$ 0 0
$$147$$ −4.22425 −0.348411
$$148$$ 0 0
$$149$$ 1.53690 0.125908 0.0629540 0.998016i $$-0.479948\pi$$
0.0629540 + 0.998016i $$0.479948\pi$$
$$150$$ 0 0
$$151$$ 6.76116 0.550215 0.275108 0.961413i $$-0.411287\pi$$
0.275108 + 0.961413i $$0.411287\pi$$
$$152$$ 0 0
$$153$$ −4.57452 −0.369828
$$154$$ 0 0
$$155$$ −9.92478 −0.797177
$$156$$ 0 0
$$157$$ −5.47627 −0.437054 −0.218527 0.975831i $$-0.570125\pi$$
−0.218527 + 0.975831i $$0.570125\pi$$
$$158$$ 0 0
$$159$$ −4.70052 −0.372776
$$160$$ 0 0
$$161$$ −22.4485 −1.76919
$$162$$ 0 0
$$163$$ −12.6253 −0.988890 −0.494445 0.869209i $$-0.664629\pi$$
−0.494445 + 0.869209i $$0.664629\pi$$
$$164$$ 0 0
$$165$$ 1.00000 0.0778499
$$166$$ 0 0
$$167$$ −18.3634 −1.42101 −0.710503 0.703695i $$-0.751532\pi$$
−0.710503 + 0.703695i $$0.751532\pi$$
$$168$$ 0 0
$$169$$ −4.22425 −0.324943
$$170$$ 0 0
$$171$$ 4.31265 0.329797
$$172$$ 0 0
$$173$$ −8.57452 −0.651908 −0.325954 0.945386i $$-0.605686\pi$$
−0.325954 + 0.945386i $$0.605686\pi$$
$$174$$ 0 0
$$175$$ −3.35026 −0.253256
$$176$$ 0 0
$$177$$ 10.7005 0.804301
$$178$$ 0 0
$$179$$ −14.1768 −1.05962 −0.529812 0.848115i $$-0.677737\pi$$
−0.529812 + 0.848115i $$0.677737\pi$$
$$180$$ 0 0
$$181$$ −5.22425 −0.388316 −0.194158 0.980970i $$-0.562197\pi$$
−0.194158 + 0.980970i $$0.562197\pi$$
$$182$$ 0 0
$$183$$ 8.70052 0.643161
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ 4.57452 0.334522
$$188$$ 0 0
$$189$$ 3.35026 0.243696
$$190$$ 0 0
$$191$$ 16.6253 1.20296 0.601482 0.798886i $$-0.294577\pi$$
0.601482 + 0.798886i $$0.294577\pi$$
$$192$$ 0 0
$$193$$ −16.3634 −1.17787 −0.588933 0.808182i $$-0.700452\pi$$
−0.588933 + 0.808182i $$0.700452\pi$$
$$194$$ 0 0
$$195$$ −2.96239 −0.212141
$$196$$ 0 0
$$197$$ −20.4241 −1.45515 −0.727577 0.686026i $$-0.759354\pi$$
−0.727577 + 0.686026i $$0.759354\pi$$
$$198$$ 0 0
$$199$$ 8.62530 0.611431 0.305716 0.952123i $$-0.401104\pi$$
0.305716 + 0.952123i $$0.401104\pi$$
$$200$$ 0 0
$$201$$ 5.92478 0.417902
$$202$$ 0 0
$$203$$ 12.1016 0.849364
$$204$$ 0 0
$$205$$ −4.38787 −0.306462
$$206$$ 0 0
$$207$$ 6.70052 0.465719
$$208$$ 0 0
$$209$$ −4.31265 −0.298312
$$210$$ 0 0
$$211$$ −9.08840 −0.625671 −0.312836 0.949807i $$-0.601279\pi$$
−0.312836 + 0.949807i $$0.601279\pi$$
$$212$$ 0 0
$$213$$ 9.92478 0.680035
$$214$$ 0 0
$$215$$ 9.27504 0.632552
$$216$$ 0 0
$$217$$ 33.2506 2.25720
$$218$$ 0 0
$$219$$ 7.73813 0.522895
$$220$$ 0 0
$$221$$ −13.5515 −0.911572
$$222$$ 0 0
$$223$$ 6.70052 0.448700 0.224350 0.974509i $$-0.427974\pi$$
0.224350 + 0.974509i $$0.427974\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −16.9624 −1.12583 −0.562917 0.826514i $$-0.690321\pi$$
−0.562917 + 0.826514i $$0.690321\pi$$
$$228$$ 0 0
$$229$$ 25.8496 1.70819 0.854093 0.520120i $$-0.174113\pi$$
0.854093 + 0.520120i $$0.174113\pi$$
$$230$$ 0 0
$$231$$ −3.35026 −0.220431
$$232$$ 0 0
$$233$$ −19.2750 −1.26275 −0.631375 0.775478i $$-0.717509\pi$$
−0.631375 + 0.775478i $$0.717509\pi$$
$$234$$ 0 0
$$235$$ 9.92478 0.647421
$$236$$ 0 0
$$237$$ 11.5369 0.749402
$$238$$ 0 0
$$239$$ −26.5501 −1.71738 −0.858691 0.512494i $$-0.828722\pi$$
−0.858691 + 0.512494i $$0.828722\pi$$
$$240$$ 0 0
$$241$$ 28.5501 1.83907 0.919536 0.393006i $$-0.128565\pi$$
0.919536 + 0.393006i $$0.128565\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 4.22425 0.269878
$$246$$ 0 0
$$247$$ 12.7757 0.812901
$$248$$ 0 0
$$249$$ 10.8872 0.689946
$$250$$ 0 0
$$251$$ −29.9248 −1.88884 −0.944418 0.328748i $$-0.893373\pi$$
−0.944418 + 0.328748i $$0.893373\pi$$
$$252$$ 0 0
$$253$$ −6.70052 −0.421258
$$254$$ 0 0
$$255$$ 4.57452 0.286467
$$256$$ 0 0
$$257$$ 8.70052 0.542724 0.271362 0.962477i $$-0.412526\pi$$
0.271362 + 0.962477i $$0.412526\pi$$
$$258$$ 0 0
$$259$$ 6.70052 0.416350
$$260$$ 0 0
$$261$$ −3.61213 −0.223585
$$262$$ 0 0
$$263$$ −12.2882 −0.757724 −0.378862 0.925453i $$-0.623684\pi$$
−0.378862 + 0.925453i $$0.623684\pi$$
$$264$$ 0 0
$$265$$ 4.70052 0.288751
$$266$$ 0 0
$$267$$ 2.77575 0.169873
$$268$$ 0 0
$$269$$ −5.84955 −0.356654 −0.178327 0.983971i $$-0.557068\pi$$
−0.178327 + 0.983971i $$0.557068\pi$$
$$270$$ 0 0
$$271$$ 5.08840 0.309098 0.154549 0.987985i $$-0.450608\pi$$
0.154549 + 0.987985i $$0.450608\pi$$
$$272$$ 0 0
$$273$$ 9.92478 0.600675
$$274$$ 0 0
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ 1.41090 0.0847725 0.0423863 0.999101i $$-0.486504\pi$$
0.0423863 + 0.999101i $$0.486504\pi$$
$$278$$ 0 0
$$279$$ −9.92478 −0.594181
$$280$$ 0 0
$$281$$ −4.38787 −0.261759 −0.130879 0.991398i $$-0.541780\pi$$
−0.130879 + 0.991398i $$0.541780\pi$$
$$282$$ 0 0
$$283$$ −26.5745 −1.57969 −0.789845 0.613306i $$-0.789839\pi$$
−0.789845 + 0.613306i $$0.789839\pi$$
$$284$$ 0 0
$$285$$ −4.31265 −0.255459
$$286$$ 0 0
$$287$$ 14.7005 0.867744
$$288$$ 0 0
$$289$$ 3.92619 0.230952
$$290$$ 0 0
$$291$$ −0.0752228 −0.00440964
$$292$$ 0 0
$$293$$ −3.42548 −0.200119 −0.100059 0.994981i $$-0.531903\pi$$
−0.100059 + 0.994981i $$0.531903\pi$$
$$294$$ 0 0
$$295$$ −10.7005 −0.623009
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ 0 0
$$299$$ 19.8496 1.14793
$$300$$ 0 0
$$301$$ −31.0738 −1.79106
$$302$$ 0 0
$$303$$ 15.0884 0.866806
$$304$$ 0 0
$$305$$ −8.70052 −0.498191
$$306$$ 0 0
$$307$$ 16.6497 0.950251 0.475125 0.879918i $$-0.342403\pi$$
0.475125 + 0.879918i $$0.342403\pi$$
$$308$$ 0 0
$$309$$ −3.22425 −0.183421
$$310$$ 0 0
$$311$$ −32.9986 −1.87118 −0.935589 0.353091i $$-0.885131\pi$$
−0.935589 + 0.353091i $$0.885131\pi$$
$$312$$ 0 0
$$313$$ 15.4010 0.870519 0.435259 0.900305i $$-0.356657\pi$$
0.435259 + 0.900305i $$0.356657\pi$$
$$314$$ 0 0
$$315$$ −3.35026 −0.188766
$$316$$ 0 0
$$317$$ 2.15045 0.120781 0.0603905 0.998175i $$-0.480765\pi$$
0.0603905 + 0.998175i $$0.480765\pi$$
$$318$$ 0 0
$$319$$ 3.61213 0.202240
$$320$$ 0 0
$$321$$ −0.962389 −0.0537153
$$322$$ 0 0
$$323$$ −19.7283 −1.09771
$$324$$ 0 0
$$325$$ 2.96239 0.164324
$$326$$ 0 0
$$327$$ −11.4010 −0.630479
$$328$$ 0 0
$$329$$ −33.2506 −1.83316
$$330$$ 0 0
$$331$$ 14.5501 0.799745 0.399872 0.916571i $$-0.369054\pi$$
0.399872 + 0.916571i $$0.369054\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ −5.92478 −0.323705
$$336$$ 0 0
$$337$$ 16.2619 0.885840 0.442920 0.896561i $$-0.353943\pi$$
0.442920 + 0.896561i $$0.353943\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 9.92478 0.537457
$$342$$ 0 0
$$343$$ 9.29948 0.502125
$$344$$ 0 0
$$345$$ −6.70052 −0.360744
$$346$$ 0 0
$$347$$ 0.962389 0.0516637 0.0258319 0.999666i $$-0.491777\pi$$
0.0258319 + 0.999666i $$0.491777\pi$$
$$348$$ 0 0
$$349$$ 20.7005 1.10807 0.554037 0.832492i $$-0.313087\pi$$
0.554037 + 0.832492i $$0.313087\pi$$
$$350$$ 0 0
$$351$$ −2.96239 −0.158121
$$352$$ 0 0
$$353$$ 20.5501 1.09377 0.546885 0.837208i $$-0.315813\pi$$
0.546885 + 0.837208i $$0.315813\pi$$
$$354$$ 0 0
$$355$$ −9.92478 −0.526752
$$356$$ 0 0
$$357$$ −15.3258 −0.811129
$$358$$ 0 0
$$359$$ −17.9248 −0.946034 −0.473017 0.881053i $$-0.656835\pi$$
−0.473017 + 0.881053i $$0.656835\pi$$
$$360$$ 0 0
$$361$$ −0.401047 −0.0211077
$$362$$ 0 0
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ −7.73813 −0.405032
$$366$$ 0 0
$$367$$ 29.6531 1.54788 0.773939 0.633261i $$-0.218284\pi$$
0.773939 + 0.633261i $$0.218284\pi$$
$$368$$ 0 0
$$369$$ −4.38787 −0.228424
$$370$$ 0 0
$$371$$ −15.7480 −0.817595
$$372$$ 0 0
$$373$$ −9.13918 −0.473209 −0.236604 0.971606i $$-0.576035\pi$$
−0.236604 + 0.971606i $$0.576035\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −10.7005 −0.551105
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −14.5745 −0.746675
$$382$$ 0 0
$$383$$ 34.9234 1.78450 0.892250 0.451541i $$-0.149126\pi$$
0.892250 + 0.451541i $$0.149126\pi$$
$$384$$ 0 0
$$385$$ 3.35026 0.170745
$$386$$ 0 0
$$387$$ 9.27504 0.471477
$$388$$ 0 0
$$389$$ 2.77575 0.140736 0.0703680 0.997521i $$-0.477583\pi$$
0.0703680 + 0.997521i $$0.477583\pi$$
$$390$$ 0 0
$$391$$ −30.6516 −1.55012
$$392$$ 0 0
$$393$$ −5.92478 −0.298865
$$394$$ 0 0
$$395$$ −11.5369 −0.580485
$$396$$ 0 0
$$397$$ −19.9248 −0.999996 −0.499998 0.866027i $$-0.666666\pi$$
−0.499998 + 0.866027i $$0.666666\pi$$
$$398$$ 0 0
$$399$$ 14.4485 0.723330
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ −29.4010 −1.46457
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 2.00000 0.0991363
$$408$$ 0 0
$$409$$ −13.0738 −0.646458 −0.323229 0.946321i $$-0.604768\pi$$
−0.323229 + 0.946321i $$0.604768\pi$$
$$410$$ 0 0
$$411$$ −13.8496 −0.683148
$$412$$ 0 0
$$413$$ 35.8496 1.76404
$$414$$ 0 0
$$415$$ −10.8872 −0.534430
$$416$$ 0 0
$$417$$ 13.6121 0.666589
$$418$$ 0 0
$$419$$ −7.22425 −0.352928 −0.176464 0.984307i $$-0.556466\pi$$
−0.176464 + 0.984307i $$0.556466\pi$$
$$420$$ 0 0
$$421$$ 30.6253 1.49259 0.746293 0.665618i $$-0.231832\pi$$
0.746293 + 0.665618i $$0.231832\pi$$
$$422$$ 0 0
$$423$$ 9.92478 0.482559
$$424$$ 0 0
$$425$$ −4.57452 −0.221897
$$426$$ 0 0
$$427$$ 29.1490 1.41062
$$428$$ 0 0
$$429$$ 2.96239 0.143025
$$430$$ 0 0
$$431$$ 33.8759 1.63174 0.815872 0.578232i $$-0.196257\pi$$
0.815872 + 0.578232i $$0.196257\pi$$
$$432$$ 0 0
$$433$$ −9.47627 −0.455400 −0.227700 0.973731i $$-0.573121\pi$$
−0.227700 + 0.973731i $$0.573121\pi$$
$$434$$ 0 0
$$435$$ 3.61213 0.173188
$$436$$ 0 0
$$437$$ 28.8970 1.38233
$$438$$ 0 0
$$439$$ 29.4617 1.40613 0.703065 0.711126i $$-0.251814\pi$$
0.703065 + 0.711126i $$0.251814\pi$$
$$440$$ 0 0
$$441$$ 4.22425 0.201155
$$442$$ 0 0
$$443$$ 19.0738 0.906224 0.453112 0.891454i $$-0.350314\pi$$
0.453112 + 0.891454i $$0.350314\pi$$
$$444$$ 0 0
$$445$$ −2.77575 −0.131583
$$446$$ 0 0
$$447$$ −1.53690 −0.0726931
$$448$$ 0 0
$$449$$ 35.8759 1.69309 0.846544 0.532318i $$-0.178679\pi$$
0.846544 + 0.532318i $$0.178679\pi$$
$$450$$ 0 0
$$451$$ 4.38787 0.206617
$$452$$ 0 0
$$453$$ −6.76116 −0.317667
$$454$$ 0 0
$$455$$ −9.92478 −0.465281
$$456$$ 0 0
$$457$$ 5.28963 0.247438 0.123719 0.992317i $$-0.460518\pi$$
0.123719 + 0.992317i $$0.460518\pi$$
$$458$$ 0 0
$$459$$ 4.57452 0.213520
$$460$$ 0 0
$$461$$ 36.3390 1.69248 0.846238 0.532805i $$-0.178862\pi$$
0.846238 + 0.532805i $$0.178862\pi$$
$$462$$ 0 0
$$463$$ −10.5501 −0.490304 −0.245152 0.969485i $$-0.578838\pi$$
−0.245152 + 0.969485i $$0.578838\pi$$
$$464$$ 0 0
$$465$$ 9.92478 0.460251
$$466$$ 0 0
$$467$$ −18.7005 −0.865357 −0.432679 0.901548i $$-0.642431\pi$$
−0.432679 + 0.901548i $$0.642431\pi$$
$$468$$ 0 0
$$469$$ 19.8496 0.916567
$$470$$ 0 0
$$471$$ 5.47627 0.252333
$$472$$ 0 0
$$473$$ −9.27504 −0.426467
$$474$$ 0 0
$$475$$ 4.31265 0.197878
$$476$$ 0 0
$$477$$ 4.70052 0.215222
$$478$$ 0 0
$$479$$ 9.29948 0.424904 0.212452 0.977172i $$-0.431855\pi$$
0.212452 + 0.977172i $$0.431855\pi$$
$$480$$ 0 0
$$481$$ −5.92478 −0.270147
$$482$$ 0 0
$$483$$ 22.4485 1.02144
$$484$$ 0 0
$$485$$ 0.0752228 0.00341569
$$486$$ 0 0
$$487$$ 35.4763 1.60758 0.803792 0.594911i $$-0.202813\pi$$
0.803792 + 0.594911i $$0.202813\pi$$
$$488$$ 0 0
$$489$$ 12.6253 0.570936
$$490$$ 0 0
$$491$$ −24.7757 −1.11811 −0.559057 0.829129i $$-0.688837\pi$$
−0.559057 + 0.829129i $$0.688837\pi$$
$$492$$ 0 0
$$493$$ 16.5237 0.744191
$$494$$ 0 0
$$495$$ −1.00000 −0.0449467
$$496$$ 0 0
$$497$$ 33.2506 1.49149
$$498$$ 0 0
$$499$$ −14.1768 −0.634640 −0.317320 0.948318i $$-0.602783\pi$$
−0.317320 + 0.948318i $$0.602783\pi$$
$$500$$ 0 0
$$501$$ 18.3634 0.820418
$$502$$ 0 0
$$503$$ 8.43866 0.376261 0.188131 0.982144i $$-0.439757\pi$$
0.188131 + 0.982144i $$0.439757\pi$$
$$504$$ 0 0
$$505$$ −15.0884 −0.671425
$$506$$ 0 0
$$507$$ 4.22425 0.187606
$$508$$ 0 0
$$509$$ 1.10299 0.0488890 0.0244445 0.999701i $$-0.492218\pi$$
0.0244445 + 0.999701i $$0.492218\pi$$
$$510$$ 0 0
$$511$$ 25.9248 1.14684
$$512$$ 0 0
$$513$$ −4.31265 −0.190408
$$514$$ 0 0
$$515$$ 3.22425 0.142078
$$516$$ 0 0
$$517$$ −9.92478 −0.436491
$$518$$ 0 0
$$519$$ 8.57452 0.376379
$$520$$ 0 0
$$521$$ −12.4485 −0.545379 −0.272690 0.962102i $$-0.587913\pi$$
−0.272690 + 0.962102i $$0.587913\pi$$
$$522$$ 0 0
$$523$$ −30.0508 −1.31403 −0.657015 0.753878i $$-0.728181\pi$$
−0.657015 + 0.753878i $$0.728181\pi$$
$$524$$ 0 0
$$525$$ 3.35026 0.146217
$$526$$ 0 0
$$527$$ 45.4010 1.97770
$$528$$ 0 0
$$529$$ 21.8970 0.952044
$$530$$ 0 0
$$531$$ −10.7005 −0.464363
$$532$$ 0 0
$$533$$ −12.9986 −0.563031
$$534$$ 0 0
$$535$$ 0.962389 0.0416077
$$536$$ 0 0
$$537$$ 14.1768 0.611774
$$538$$ 0 0
$$539$$ −4.22425 −0.181951
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 5.22425 0.224194
$$544$$ 0 0
$$545$$ 11.4010 0.488367
$$546$$ 0 0
$$547$$ 14.3028 0.611544 0.305772 0.952105i $$-0.401086\pi$$
0.305772 + 0.952105i $$0.401086\pi$$
$$548$$ 0 0
$$549$$ −8.70052 −0.371329
$$550$$ 0 0
$$551$$ −15.5778 −0.663638
$$552$$ 0 0
$$553$$ 38.6516 1.64364
$$554$$ 0 0
$$555$$ 2.00000 0.0848953
$$556$$ 0 0
$$557$$ −11.7988 −0.499930 −0.249965 0.968255i $$-0.580419\pi$$
−0.249965 + 0.968255i $$0.580419\pi$$
$$558$$ 0 0
$$559$$ 27.4763 1.16212
$$560$$ 0 0
$$561$$ −4.57452 −0.193136
$$562$$ 0 0
$$563$$ −30.4847 −1.28478 −0.642389 0.766379i $$-0.722056\pi$$
−0.642389 + 0.766379i $$0.722056\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ −3.35026 −0.140698
$$568$$ 0 0
$$569$$ −27.0884 −1.13560 −0.567802 0.823165i $$-0.692206\pi$$
−0.567802 + 0.823165i $$0.692206\pi$$
$$570$$ 0 0
$$571$$ −7.28489 −0.304863 −0.152432 0.988314i $$-0.548710\pi$$
−0.152432 + 0.988314i $$0.548710\pi$$
$$572$$ 0 0
$$573$$ −16.6253 −0.694532
$$574$$ 0 0
$$575$$ 6.70052 0.279431
$$576$$ 0 0
$$577$$ −31.6239 −1.31652 −0.658260 0.752791i $$-0.728707\pi$$
−0.658260 + 0.752791i $$0.728707\pi$$
$$578$$ 0 0
$$579$$ 16.3634 0.680041
$$580$$ 0 0
$$581$$ 36.4749 1.51323
$$582$$ 0 0
$$583$$ −4.70052 −0.194676
$$584$$ 0 0
$$585$$ 2.96239 0.122480
$$586$$ 0 0
$$587$$ −33.1490 −1.36821 −0.684103 0.729385i $$-0.739806\pi$$
−0.684103 + 0.729385i $$0.739806\pi$$
$$588$$ 0 0
$$589$$ −42.8021 −1.76363
$$590$$ 0 0
$$591$$ 20.4241 0.840134
$$592$$ 0 0
$$593$$ 34.4993 1.41672 0.708358 0.705853i $$-0.249436\pi$$
0.708358 + 0.705853i $$0.249436\pi$$
$$594$$ 0 0
$$595$$ 15.3258 0.628298
$$596$$ 0 0
$$597$$ −8.62530 −0.353010
$$598$$ 0 0
$$599$$ 14.4485 0.590350 0.295175 0.955443i $$-0.404622\pi$$
0.295175 + 0.955443i $$0.404622\pi$$
$$600$$ 0 0
$$601$$ −15.9248 −0.649585 −0.324793 0.945785i $$-0.605295\pi$$
−0.324793 + 0.945785i $$0.605295\pi$$
$$602$$ 0 0
$$603$$ −5.92478 −0.241276
$$604$$ 0 0
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ 14.5745 0.591561 0.295781 0.955256i $$-0.404420\pi$$
0.295781 + 0.955256i $$0.404420\pi$$
$$608$$ 0 0
$$609$$ −12.1016 −0.490380
$$610$$ 0 0
$$611$$ 29.4010 1.18944
$$612$$ 0 0
$$613$$ 16.4123 0.662887 0.331443 0.943475i $$-0.392464\pi$$
0.331443 + 0.943475i $$0.392464\pi$$
$$614$$ 0 0
$$615$$ 4.38787 0.176936
$$616$$ 0 0
$$617$$ −17.8496 −0.718596 −0.359298 0.933223i $$-0.616984\pi$$
−0.359298 + 0.933223i $$0.616984\pi$$
$$618$$ 0 0
$$619$$ 0.402462 0.0161763 0.00808815 0.999967i $$-0.497425\pi$$
0.00808815 + 0.999967i $$0.497425\pi$$
$$620$$ 0 0
$$621$$ −6.70052 −0.268883
$$622$$ 0 0
$$623$$ 9.29948 0.372576
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 4.31265 0.172231
$$628$$ 0 0
$$629$$ 9.14903 0.364796
$$630$$ 0 0
$$631$$ 38.0263 1.51380 0.756902 0.653528i $$-0.226712\pi$$
0.756902 + 0.653528i $$0.226712\pi$$
$$632$$ 0 0
$$633$$ 9.08840 0.361231
$$634$$ 0 0
$$635$$ 14.5745 0.578372
$$636$$ 0 0
$$637$$ 12.5139 0.495818
$$638$$ 0 0
$$639$$ −9.92478 −0.392618
$$640$$ 0 0
$$641$$ −28.0263 −1.10697 −0.553487 0.832858i $$-0.686703\pi$$
−0.553487 + 0.832858i $$0.686703\pi$$
$$642$$ 0 0
$$643$$ 4.62530 0.182404 0.0912020 0.995832i $$-0.470929\pi$$
0.0912020 + 0.995832i $$0.470929\pi$$
$$644$$ 0 0
$$645$$ −9.27504 −0.365204
$$646$$ 0 0
$$647$$ −23.5778 −0.926941 −0.463470 0.886112i $$-0.653396\pi$$
−0.463470 + 0.886112i $$0.653396\pi$$
$$648$$ 0 0
$$649$$ 10.7005 0.420032
$$650$$ 0 0
$$651$$ −33.2506 −1.30319
$$652$$ 0 0
$$653$$ −2.25202 −0.0881282 −0.0440641 0.999029i $$-0.514031\pi$$
−0.0440641 + 0.999029i $$0.514031\pi$$
$$654$$ 0 0
$$655$$ 5.92478 0.231500
$$656$$ 0 0
$$657$$ −7.73813 −0.301893
$$658$$ 0 0
$$659$$ 41.4010 1.61276 0.806378 0.591401i $$-0.201425\pi$$
0.806378 + 0.591401i $$0.201425\pi$$
$$660$$ 0 0
$$661$$ 3.40105 0.132285 0.0661427 0.997810i $$-0.478931\pi$$
0.0661427 + 0.997810i $$0.478931\pi$$
$$662$$ 0 0
$$663$$ 13.5515 0.526296
$$664$$ 0 0
$$665$$ −14.4485 −0.560289
$$666$$ 0 0
$$667$$ −24.2031 −0.937149
$$668$$ 0 0
$$669$$ −6.70052 −0.259057
$$670$$ 0 0
$$671$$ 8.70052 0.335880
$$672$$ 0 0
$$673$$ 0.887166 0.0341977 0.0170989 0.999854i $$-0.494557\pi$$
0.0170989 + 0.999854i $$0.494557\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 18.9018 0.726453 0.363227 0.931701i $$-0.381675\pi$$
0.363227 + 0.931701i $$0.381675\pi$$
$$678$$ 0 0
$$679$$ −0.252016 −0.00967149
$$680$$ 0 0
$$681$$ 16.9624 0.650000
$$682$$ 0 0
$$683$$ 20.8773 0.798848 0.399424 0.916766i $$-0.369210\pi$$
0.399424 + 0.916766i $$0.369210\pi$$
$$684$$ 0 0
$$685$$ 13.8496 0.529164
$$686$$ 0 0
$$687$$ −25.8496 −0.986222
$$688$$ 0 0
$$689$$ 13.9248 0.530492
$$690$$ 0 0
$$691$$ 2.44851 0.0931456 0.0465728 0.998915i $$-0.485170\pi$$
0.0465728 + 0.998915i $$0.485170\pi$$
$$692$$ 0 0
$$693$$ 3.35026 0.127266
$$694$$ 0 0
$$695$$ −13.6121 −0.516337
$$696$$ 0 0
$$697$$ 20.0724 0.760296
$$698$$ 0 0
$$699$$ 19.2750 0.729049
$$700$$ 0 0
$$701$$ −2.98683 −0.112811 −0.0564054 0.998408i $$-0.517964\pi$$
−0.0564054 + 0.998408i $$0.517964\pi$$
$$702$$ 0 0
$$703$$ −8.62530 −0.325309
$$704$$ 0 0
$$705$$ −9.92478 −0.373789
$$706$$ 0 0
$$707$$ 50.5501 1.90113
$$708$$ 0 0
$$709$$ 24.1768 0.907979 0.453989 0.891007i $$-0.350000\pi$$
0.453989 + 0.891007i $$0.350000\pi$$
$$710$$ 0 0
$$711$$ −11.5369 −0.432668
$$712$$ 0 0
$$713$$ −66.5012 −2.49049
$$714$$ 0 0
$$715$$ −2.96239 −0.110787
$$716$$ 0 0
$$717$$ 26.5501 0.991531
$$718$$ 0 0
$$719$$ 30.0263 1.11979 0.559897 0.828562i $$-0.310841\pi$$
0.559897 + 0.828562i $$0.310841\pi$$
$$720$$ 0 0
$$721$$ −10.8021 −0.402291
$$722$$ 0 0
$$723$$ −28.5501 −1.06179
$$724$$ 0 0
$$725$$ −3.61213 −0.134151
$$726$$ 0 0
$$727$$ −14.9525 −0.554559 −0.277279 0.960789i $$-0.589433\pi$$
−0.277279 + 0.960789i $$0.589433\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −42.4288 −1.56929
$$732$$ 0 0
$$733$$ −19.1128 −0.705949 −0.352974 0.935633i $$-0.614830\pi$$
−0.352974 + 0.935633i $$0.614830\pi$$
$$734$$ 0 0
$$735$$ −4.22425 −0.155814
$$736$$ 0 0
$$737$$ 5.92478 0.218242
$$738$$ 0 0
$$739$$ 3.31406 0.121910 0.0609549 0.998141i $$-0.480585\pi$$
0.0609549 + 0.998141i $$0.480585\pi$$
$$740$$ 0 0
$$741$$ −12.7757 −0.469329
$$742$$ 0 0
$$743$$ 34.9887 1.28361 0.641806 0.766867i $$-0.278185\pi$$
0.641806 + 0.766867i $$0.278185\pi$$
$$744$$ 0 0
$$745$$ 1.53690 0.0563078
$$746$$ 0 0
$$747$$ −10.8872 −0.398341
$$748$$ 0 0
$$749$$ −3.22425 −0.117812
$$750$$ 0 0
$$751$$ 26.9234 0.982447 0.491224 0.871033i $$-0.336550\pi$$
0.491224 + 0.871033i $$0.336550\pi$$
$$752$$ 0 0
$$753$$ 29.9248 1.09052
$$754$$ 0 0
$$755$$ 6.76116 0.246064
$$756$$ 0 0
$$757$$ 15.9248 0.578796 0.289398 0.957209i $$-0.406545\pi$$
0.289398 + 0.957209i $$0.406545\pi$$
$$758$$ 0 0
$$759$$ 6.70052 0.243214
$$760$$ 0 0
$$761$$ −30.9380 −1.12150 −0.560750 0.827985i $$-0.689487\pi$$
−0.560750 + 0.827985i $$0.689487\pi$$
$$762$$ 0 0
$$763$$ −38.1965 −1.38281
$$764$$ 0 0
$$765$$ −4.57452 −0.165392
$$766$$ 0 0
$$767$$ −31.6991 −1.14459
$$768$$ 0 0
$$769$$ 9.32582 0.336298 0.168149 0.985762i $$-0.446221\pi$$
0.168149 + 0.985762i $$0.446221\pi$$
$$770$$ 0 0
$$771$$ −8.70052 −0.313342
$$772$$ 0 0
$$773$$ 44.7005 1.60777 0.803883 0.594787i $$-0.202764\pi$$
0.803883 + 0.594787i $$0.202764\pi$$
$$774$$ 0 0
$$775$$ −9.92478 −0.356509
$$776$$ 0 0
$$777$$ −6.70052 −0.240380
$$778$$ 0 0
$$779$$ −18.9234 −0.678000
$$780$$ 0 0
$$781$$ 9.92478 0.355136
$$782$$ 0 0
$$783$$ 3.61213 0.129087
$$784$$ 0 0
$$785$$ −5.47627 −0.195456
$$786$$ 0 0
$$787$$ −21.6775 −0.772719 −0.386360 0.922348i $$-0.626268\pi$$
−0.386360 + 0.922348i $$0.626268\pi$$
$$788$$ 0 0
$$789$$ 12.2882 0.437472
$$790$$ 0 0
$$791$$ 20.1016 0.714730
$$792$$ 0 0
$$793$$ −25.7743 −0.915273
$$794$$ 0 0
$$795$$ −4.70052 −0.166710
$$796$$ 0 0
$$797$$ 22.7466 0.805725 0.402862 0.915261i $$-0.368015\pi$$
0.402862 + 0.915261i $$0.368015\pi$$
$$798$$ 0 0
$$799$$ −45.4010 −1.60617
$$800$$ 0 0
$$801$$ −2.77575 −0.0980762
$$802$$ 0 0
$$803$$ 7.73813 0.273073
$$804$$ 0 0
$$805$$ −22.4485 −0.791206
$$806$$ 0 0
$$807$$ 5.84955 0.205914
$$808$$ 0 0
$$809$$ −23.6121 −0.830158 −0.415079 0.909785i $$-0.636246\pi$$
−0.415079 + 0.909785i $$0.636246\pi$$
$$810$$ 0 0
$$811$$ 26.0870 0.916038 0.458019 0.888942i $$-0.348559\pi$$
0.458019 + 0.888942i $$0.348559\pi$$
$$812$$ 0 0
$$813$$ −5.08840 −0.178458
$$814$$ 0 0
$$815$$ −12.6253 −0.442245
$$816$$ 0 0
$$817$$ 40.0000 1.39942
$$818$$ 0 0
$$819$$ −9.92478 −0.346800
$$820$$ 0 0
$$821$$ −54.4142 −1.89907 −0.949535 0.313662i $$-0.898444\pi$$
−0.949535 + 0.313662i $$0.898444\pi$$
$$822$$ 0 0
$$823$$ −0.121269 −0.00422716 −0.00211358 0.999998i $$-0.500673\pi$$
−0.00211358 + 0.999998i $$0.500673\pi$$
$$824$$ 0 0
$$825$$ 1.00000 0.0348155
$$826$$ 0 0
$$827$$ 18.2130 0.633328 0.316664 0.948538i $$-0.397437\pi$$
0.316664 + 0.948538i $$0.397437\pi$$
$$828$$ 0 0
$$829$$ 13.0738 0.454072 0.227036 0.973886i $$-0.427096\pi$$
0.227036 + 0.973886i $$0.427096\pi$$
$$830$$ 0 0
$$831$$ −1.41090 −0.0489434
$$832$$ 0 0
$$833$$ −19.3239 −0.669534
$$834$$ 0 0
$$835$$ −18.3634 −0.635493
$$836$$ 0 0
$$837$$ 9.92478 0.343050
$$838$$ 0 0
$$839$$ 26.5501 0.916610 0.458305 0.888795i $$-0.348457\pi$$
0.458305 + 0.888795i $$0.348457\pi$$
$$840$$ 0 0
$$841$$ −15.9525 −0.550088
$$842$$ 0 0
$$843$$ 4.38787 0.151126
$$844$$ 0 0
$$845$$ −4.22425 −0.145319
$$846$$ 0 0
$$847$$ −3.35026 −0.115116
$$848$$ 0 0
$$849$$ 26.5745 0.912035
$$850$$ 0 0
$$851$$ −13.4010 −0.459382
$$852$$ 0 0
$$853$$ 40.6155 1.39065 0.695323 0.718697i $$-0.255261\pi$$
0.695323 + 0.718697i $$0.255261\pi$$
$$854$$ 0 0
$$855$$ 4.31265 0.147490
$$856$$ 0 0
$$857$$ 20.1721 0.689064 0.344532 0.938775i $$-0.388038\pi$$
0.344532 + 0.938775i $$0.388038\pi$$
$$858$$ 0 0
$$859$$ −21.8035 −0.743926 −0.371963 0.928248i $$-0.621315\pi$$
−0.371963 + 0.928248i $$0.621315\pi$$
$$860$$ 0 0
$$861$$ −14.7005 −0.500993
$$862$$ 0 0
$$863$$ −35.4274 −1.20596 −0.602981 0.797755i $$-0.706021\pi$$
−0.602981 + 0.797755i $$0.706021\pi$$
$$864$$ 0 0
$$865$$ −8.57452 −0.291542
$$866$$ 0 0
$$867$$ −3.92619 −0.133340
$$868$$ 0 0
$$869$$ 11.5369 0.391363
$$870$$ 0 0
$$871$$ −17.5515 −0.594710
$$872$$ 0 0
$$873$$ 0.0752228 0.00254591
$$874$$ 0 0
$$875$$ −3.35026 −0.113260
$$876$$ 0 0
$$877$$ −14.0362 −0.473969 −0.236984 0.971513i $$-0.576159\pi$$
−0.236984 + 0.971513i $$0.576159\pi$$
$$878$$ 0 0
$$879$$ 3.42548 0.115539
$$880$$ 0 0
$$881$$ −21.0738 −0.709995 −0.354997 0.934867i $$-0.615518\pi$$
−0.354997 + 0.934867i $$0.615518\pi$$
$$882$$ 0 0
$$883$$ 42.1476 1.41838 0.709190 0.705017i $$-0.249061\pi$$
0.709190 + 0.705017i $$0.249061\pi$$
$$884$$ 0 0
$$885$$ 10.7005 0.359694
$$886$$ 0 0
$$887$$ −6.93604 −0.232889 −0.116445 0.993197i $$-0.537150\pi$$
−0.116445 + 0.993197i $$0.537150\pi$$
$$888$$ 0 0
$$889$$ −48.8284 −1.63765
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ 42.8021 1.43232
$$894$$ 0 0
$$895$$ −14.1768 −0.473878
$$896$$ 0 0
$$897$$ −19.8496 −0.662757
$$898$$ 0 0
$$899$$ 35.8496 1.19565
$$900$$ 0 0
$$901$$ −21.5026 −0.716356
$$902$$ 0 0
$$903$$ 31.0738 1.03407
$$904$$ 0 0
$$905$$ −5.22425 −0.173660
$$906$$ 0 0
$$907$$ −53.2017 −1.76653 −0.883267 0.468870i $$-0.844661\pi$$
−0.883267 + 0.468870i $$0.844661\pi$$
$$908$$ 0 0
$$909$$ −15.0884 −0.500451
$$910$$ 0 0
$$911$$ −36.4749 −1.20847 −0.604233 0.796808i $$-0.706520\pi$$
−0.604233 + 0.796808i $$0.706520\pi$$
$$912$$ 0 0
$$913$$ 10.8872 0.360313
$$914$$ 0 0
$$915$$ 8.70052 0.287630
$$916$$ 0 0
$$917$$ −19.8496 −0.655490
$$918$$ 0 0
$$919$$ −9.73340 −0.321075 −0.160538 0.987030i $$-0.551323\pi$$
−0.160538 + 0.987030i $$0.551323\pi$$
$$920$$ 0 0
$$921$$ −16.6497 −0.548628
$$922$$ 0 0
$$923$$ −29.4010 −0.967747
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 0 0
$$927$$ 3.22425 0.105898
$$928$$ 0 0
$$929$$ −24.1768 −0.793215 −0.396607 0.917988i $$-0.629813\pi$$
−0.396607 + 0.917988i $$0.629813\pi$$
$$930$$ 0 0
$$931$$ 18.2177 0.597062
$$932$$ 0 0
$$933$$ 32.9986 1.08033
$$934$$ 0 0
$$935$$ 4.57452 0.149603
$$936$$ 0 0
$$937$$ −7.48612 −0.244561 −0.122280 0.992496i $$-0.539021\pi$$
−0.122280 + 0.992496i $$0.539021\pi$$
$$938$$ 0 0
$$939$$ −15.4010 −0.502594
$$940$$ 0 0
$$941$$ −21.2360 −0.692274 −0.346137 0.938184i $$-0.612507\pi$$
−0.346137 + 0.938184i $$0.612507\pi$$
$$942$$ 0 0
$$943$$ −29.4010 −0.957430
$$944$$ 0 0
$$945$$ 3.35026 0.108984
$$946$$ 0 0
$$947$$ 15.4763 0.502911 0.251456 0.967869i $$-0.419091\pi$$
0.251456 + 0.967869i $$0.419091\pi$$
$$948$$ 0 0
$$949$$ −22.9234 −0.744124
$$950$$ 0 0
$$951$$ −2.15045 −0.0697330
$$952$$ 0 0
$$953$$ −32.0508 −1.03823 −0.519113 0.854705i $$-0.673738\pi$$
−0.519113 + 0.854705i $$0.673738\pi$$
$$954$$ 0 0
$$955$$ 16.6253 0.537982
$$956$$ 0 0
$$957$$ −3.61213 −0.116763
$$958$$ 0 0
$$959$$ −46.3996 −1.49832
$$960$$ 0 0
$$961$$ 67.5012 2.17746
$$962$$ 0 0
$$963$$ 0.962389 0.0310125
$$964$$ 0 0
$$965$$ −16.3634 −0.526758
$$966$$ 0 0
$$967$$ 17.3766 0.558794 0.279397 0.960176i $$-0.409865\pi$$
0.279397 + 0.960176i $$0.409865\pi$$
$$968$$ 0 0
$$969$$ 19.7283 0.633764
$$970$$ 0 0
$$971$$ −36.2031 −1.16181 −0.580907 0.813970i $$-0.697302\pi$$
−0.580907 + 0.813970i $$0.697302\pi$$
$$972$$ 0 0
$$973$$ 45.6042 1.46200
$$974$$ 0 0
$$975$$ −2.96239 −0.0948724
$$976$$ 0 0
$$977$$ −28.1476 −0.900522 −0.450261 0.892897i $$-0.648669\pi$$
−0.450261 + 0.892897i $$0.648669\pi$$
$$978$$ 0 0
$$979$$ 2.77575 0.0887132
$$980$$ 0 0
$$981$$ 11.4010 0.364007
$$982$$ 0 0
$$983$$ −7.07381 −0.225619 −0.112810 0.993617i $$-0.535985\pi$$
−0.112810 + 0.993617i $$0.535985\pi$$
$$984$$ 0 0
$$985$$ −20.4241 −0.650765
$$986$$ 0 0
$$987$$ 33.2506 1.05838
$$988$$ 0 0
$$989$$ 62.1476 1.97618
$$990$$ 0 0
$$991$$ −44.4260 −1.41124 −0.705619 0.708592i $$-0.749331\pi$$
−0.705619 + 0.708592i $$0.749331\pi$$
$$992$$ 0 0
$$993$$ −14.5501 −0.461733
$$994$$ 0 0
$$995$$ 8.62530 0.273440
$$996$$ 0 0
$$997$$ −28.4847 −0.902120 −0.451060 0.892494i $$-0.648954\pi$$
−0.451060 + 0.892494i $$0.648954\pi$$
$$998$$ 0 0
$$999$$ 2.00000 0.0632772
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 2640.2.a.be.1.1 3
3.2 odd 2 7920.2.a.cj.1.1 3
4.3 odd 2 165.2.a.c.1.2 3
12.11 even 2 495.2.a.e.1.2 3
20.3 even 4 825.2.c.g.199.4 6
20.7 even 4 825.2.c.g.199.3 6
20.19 odd 2 825.2.a.k.1.2 3
28.27 even 2 8085.2.a.bk.1.2 3
44.43 even 2 1815.2.a.m.1.2 3
60.23 odd 4 2475.2.c.r.199.3 6
60.47 odd 4 2475.2.c.r.199.4 6
60.59 even 2 2475.2.a.bb.1.2 3
132.131 odd 2 5445.2.a.z.1.2 3
220.219 even 2 9075.2.a.cf.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 4.3 odd 2
495.2.a.e.1.2 3 12.11 even 2
825.2.a.k.1.2 3 20.19 odd 2
825.2.c.g.199.3 6 20.7 even 4
825.2.c.g.199.4 6 20.3 even 4
1815.2.a.m.1.2 3 44.43 even 2
2475.2.a.bb.1.2 3 60.59 even 2
2475.2.c.r.199.3 6 60.23 odd 4
2475.2.c.r.199.4 6 60.47 odd 4
2640.2.a.be.1.1 3 1.1 even 1 trivial
5445.2.a.z.1.2 3 132.131 odd 2
7920.2.a.cj.1.1 3 3.2 odd 2
8085.2.a.bk.1.2 3 28.27 even 2
9075.2.a.cf.1.2 3 220.219 even 2