# Properties

 Label 7360.2.a.bx.1.2 Level $7360$ Weight $2$ Character 7360.1 Self dual yes Analytic conductor $58.770$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7360 = 2^{6} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7360.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$58.7698958877$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1573.1 Defining polynomial: $$x^{3} - x^{2} - 7 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3680) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.277754$$ of defining polynomial Character $$\chi$$ $$=$$ 7360.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.277754 q^{3} -1.00000 q^{5} -1.27775 q^{7} -2.92285 q^{9} +O(q^{10})$$ $$q-0.277754 q^{3} -1.00000 q^{5} -1.27775 q^{7} -2.92285 q^{9} -0.277754 q^{11} +3.92285 q^{13} +0.277754 q^{15} +5.47836 q^{17} -3.92285 q^{19} +0.354902 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.64510 q^{27} -0.799393 q^{29} -5.27775 q^{31} +0.0771475 q^{33} +1.27775 q^{35} +1.75612 q^{37} -1.08959 q^{39} +10.0339 q^{41} -4.00000 q^{43} +2.92285 q^{45} +8.95672 q^{47} -5.36734 q^{49} -1.52164 q^{51} -10.6451 q^{53} +0.277754 q^{55} +1.08959 q^{57} +8.08959 q^{59} -15.0124 q^{61} +3.73469 q^{63} -3.92285 q^{65} +12.6451 q^{67} +0.277754 q^{69} -9.32407 q^{71} +4.55551 q^{73} -0.277754 q^{75} +0.354902 q^{77} +8.00000 q^{79} +8.31162 q^{81} -2.64510 q^{83} -5.47836 q^{85} +0.222035 q^{87} +13.2902 q^{89} -5.01244 q^{91} +1.46592 q^{93} +3.92285 q^{95} +3.16674 q^{97} +0.811835 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{5} - 4 q^{7} + 6 q^{9} + O(q^{10})$$ $$3 q - q^{3} - 3 q^{5} - 4 q^{7} + 6 q^{9} - q^{11} - 3 q^{13} + q^{15} + 2 q^{17} + 3 q^{19} + 16 q^{21} - 3 q^{23} + 3 q^{25} - 10 q^{27} - 17 q^{29} - 16 q^{31} + 15 q^{33} + 4 q^{35} - 9 q^{37} + 12 q^{39} + 16 q^{41} - 12 q^{43} - 6 q^{45} - 2 q^{47} - q^{49} - 19 q^{51} - 17 q^{53} + q^{55} - 12 q^{57} + 9 q^{59} - 15 q^{61} - 19 q^{63} + 3 q^{65} + 23 q^{67} + q^{69} + 16 q^{71} + 14 q^{73} - q^{75} + 16 q^{77} + 24 q^{79} + 11 q^{81} + 7 q^{83} - 2 q^{85} + 2 q^{87} + 10 q^{89} + 15 q^{91} + 20 q^{93} - 3 q^{95} + 9 q^{97} - 13 q^{99} + O(q^{100})$$

## 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
$$3$$ −0.277754 −0.160362 −0.0801808 0.996780i $$-0.525550\pi$$
−0.0801808 + 0.996780i $$0.525550\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −1.27775 −0.482946 −0.241473 0.970408i $$-0.577630\pi$$
−0.241473 + 0.970408i $$0.577630\pi$$
$$8$$ 0 0
$$9$$ −2.92285 −0.974284
$$10$$ 0 0
$$11$$ −0.277754 −0.0837461 −0.0418730 0.999123i $$-0.513333\pi$$
−0.0418730 + 0.999123i $$0.513333\pi$$
$$12$$ 0 0
$$13$$ 3.92285 1.08800 0.544002 0.839084i $$-0.316908\pi$$
0.544002 + 0.839084i $$0.316908\pi$$
$$14$$ 0 0
$$15$$ 0.277754 0.0717159
$$16$$ 0 0
$$17$$ 5.47836 1.32870 0.664349 0.747423i $$-0.268709\pi$$
0.664349 + 0.747423i $$0.268709\pi$$
$$18$$ 0 0
$$19$$ −3.92285 −0.899964 −0.449982 0.893038i $$-0.648570\pi$$
−0.449982 + 0.893038i $$0.648570\pi$$
$$20$$ 0 0
$$21$$ 0.354902 0.0774459
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.64510 0.316599
$$28$$ 0 0
$$29$$ −0.799393 −0.148444 −0.0742218 0.997242i $$-0.523647\pi$$
−0.0742218 + 0.997242i $$0.523647\pi$$
$$30$$ 0 0
$$31$$ −5.27775 −0.947913 −0.473956 0.880548i $$-0.657175\pi$$
−0.473956 + 0.880548i $$0.657175\pi$$
$$32$$ 0 0
$$33$$ 0.0771475 0.0134297
$$34$$ 0 0
$$35$$ 1.27775 0.215980
$$36$$ 0 0
$$37$$ 1.75612 0.288704 0.144352 0.989526i $$-0.453890\pi$$
0.144352 + 0.989526i $$0.453890\pi$$
$$38$$ 0 0
$$39$$ −1.08959 −0.174474
$$40$$ 0 0
$$41$$ 10.0339 1.56703 0.783514 0.621375i $$-0.213425\pi$$
0.783514 + 0.621375i $$0.213425\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 2.92285 0.435713
$$46$$ 0 0
$$47$$ 8.95672 1.30647 0.653236 0.757154i $$-0.273411\pi$$
0.653236 + 0.757154i $$0.273411\pi$$
$$48$$ 0 0
$$49$$ −5.36734 −0.766763
$$50$$ 0 0
$$51$$ −1.52164 −0.213072
$$52$$ 0 0
$$53$$ −10.6451 −1.46222 −0.731108 0.682261i $$-0.760997\pi$$
−0.731108 + 0.682261i $$0.760997\pi$$
$$54$$ 0 0
$$55$$ 0.277754 0.0374524
$$56$$ 0 0
$$57$$ 1.08959 0.144320
$$58$$ 0 0
$$59$$ 8.08959 1.05317 0.526587 0.850121i $$-0.323471\pi$$
0.526587 + 0.850121i $$0.323471\pi$$
$$60$$ 0 0
$$61$$ −15.0124 −1.92215 −0.961073 0.276294i $$-0.910894\pi$$
−0.961073 + 0.276294i $$0.910894\pi$$
$$62$$ 0 0
$$63$$ 3.73469 0.470526
$$64$$ 0 0
$$65$$ −3.92285 −0.486570
$$66$$ 0 0
$$67$$ 12.6451 1.54484 0.772422 0.635109i $$-0.219045\pi$$
0.772422 + 0.635109i $$0.219045\pi$$
$$68$$ 0 0
$$69$$ 0.277754 0.0334377
$$70$$ 0 0
$$71$$ −9.32407 −1.10656 −0.553282 0.832994i $$-0.686625\pi$$
−0.553282 + 0.832994i $$0.686625\pi$$
$$72$$ 0 0
$$73$$ 4.55551 0.533182 0.266591 0.963810i $$-0.414103\pi$$
0.266591 + 0.963810i $$0.414103\pi$$
$$74$$ 0 0
$$75$$ −0.277754 −0.0320723
$$76$$ 0 0
$$77$$ 0.354902 0.0404448
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 8.31162 0.923514
$$82$$ 0 0
$$83$$ −2.64510 −0.290337 −0.145169 0.989407i $$-0.546372\pi$$
−0.145169 + 0.989407i $$0.546372\pi$$
$$84$$ 0 0
$$85$$ −5.47836 −0.594212
$$86$$ 0 0
$$87$$ 0.222035 0.0238046
$$88$$ 0 0
$$89$$ 13.2902 1.40876 0.704379 0.709824i $$-0.251226\pi$$
0.704379 + 0.709824i $$0.251226\pi$$
$$90$$ 0 0
$$91$$ −5.01244 −0.525447
$$92$$ 0 0
$$93$$ 1.46592 0.152009
$$94$$ 0 0
$$95$$ 3.92285 0.402476
$$96$$ 0 0
$$97$$ 3.16674 0.321533 0.160767 0.986992i $$-0.448603\pi$$
0.160767 + 0.986992i $$0.448603\pi$$
$$98$$ 0 0
$$99$$ 0.811835 0.0815925
$$100$$ 0 0
$$101$$ −5.60182 −0.557402 −0.278701 0.960378i $$-0.589904\pi$$
−0.278701 + 0.960378i $$0.589904\pi$$
$$102$$ 0 0
$$103$$ −2.83326 −0.279170 −0.139585 0.990210i $$-0.544577\pi$$
−0.139585 + 0.990210i $$0.544577\pi$$
$$104$$ 0 0
$$105$$ −0.354902 −0.0346349
$$106$$ 0 0
$$107$$ −19.2006 −1.85619 −0.928096 0.372340i $$-0.878556\pi$$
−0.928096 + 0.372340i $$0.878556\pi$$
$$108$$ 0 0
$$109$$ 4.07715 0.390520 0.195260 0.980752i $$-0.437445\pi$$
0.195260 + 0.980752i $$0.437445\pi$$
$$110$$ 0 0
$$111$$ −0.487769 −0.0462970
$$112$$ 0 0
$$113$$ 11.7561 1.10592 0.552961 0.833207i $$-0.313498\pi$$
0.552961 + 0.833207i $$0.313498\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ −11.4659 −1.06002
$$118$$ 0 0
$$119$$ −7.00000 −0.641689
$$120$$ 0 0
$$121$$ −10.9229 −0.992987
$$122$$ 0 0
$$123$$ −2.78695 −0.251291
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 17.8457 1.58355 0.791775 0.610813i $$-0.209157\pi$$
0.791775 + 0.610813i $$0.209157\pi$$
$$128$$ 0 0
$$129$$ 1.11102 0.0978196
$$130$$ 0 0
$$131$$ −12.4012 −1.08350 −0.541750 0.840540i $$-0.682238\pi$$
−0.541750 + 0.840540i $$0.682238\pi$$
$$132$$ 0 0
$$133$$ 5.01244 0.434634
$$134$$ 0 0
$$135$$ −1.64510 −0.141588
$$136$$ 0 0
$$137$$ 15.0339 1.28443 0.642215 0.766524i $$-0.278016\pi$$
0.642215 + 0.766524i $$0.278016\pi$$
$$138$$ 0 0
$$139$$ −18.0030 −1.52700 −0.763499 0.645809i $$-0.776520\pi$$
−0.763499 + 0.645809i $$0.776520\pi$$
$$140$$ 0 0
$$141$$ −2.48777 −0.209508
$$142$$ 0 0
$$143$$ −1.08959 −0.0911160
$$144$$ 0 0
$$145$$ 0.799393 0.0663860
$$146$$ 0 0
$$147$$ 1.49080 0.122959
$$148$$ 0 0
$$149$$ 14.3241 1.17347 0.586737 0.809778i $$-0.300412\pi$$
0.586737 + 0.809778i $$0.300412\pi$$
$$150$$ 0 0
$$151$$ −24.1912 −1.96865 −0.984326 0.176359i $$-0.943568\pi$$
−0.984326 + 0.176359i $$0.943568\pi$$
$$152$$ 0 0
$$153$$ −16.0124 −1.29453
$$154$$ 0 0
$$155$$ 5.27775 0.423919
$$156$$ 0 0
$$157$$ −12.7994 −1.02150 −0.510751 0.859728i $$-0.670633\pi$$
−0.510751 + 0.859728i $$0.670633\pi$$
$$158$$ 0 0
$$159$$ 2.95672 0.234483
$$160$$ 0 0
$$161$$ 1.27775 0.100701
$$162$$ 0 0
$$163$$ 19.4351 1.52227 0.761137 0.648592i $$-0.224642\pi$$
0.761137 + 0.648592i $$0.224642\pi$$
$$164$$ 0 0
$$165$$ −0.0771475 −0.00600592
$$166$$ 0 0
$$167$$ −24.8024 −1.91927 −0.959635 0.281249i $$-0.909251\pi$$
−0.959635 + 0.281249i $$0.909251\pi$$
$$168$$ 0 0
$$169$$ 2.38877 0.183752
$$170$$ 0 0
$$171$$ 11.4659 0.876821
$$172$$ 0 0
$$173$$ 12.6790 0.963964 0.481982 0.876181i $$-0.339917\pi$$
0.481982 + 0.876181i $$0.339917\pi$$
$$174$$ 0 0
$$175$$ −1.27775 −0.0965892
$$176$$ 0 0
$$177$$ −2.24692 −0.168889
$$178$$ 0 0
$$179$$ −0.154295 −0.0115325 −0.00576627 0.999983i $$-0.501835\pi$$
−0.00576627 + 0.999983i $$0.501835\pi$$
$$180$$ 0 0
$$181$$ −0.410621 −0.0305212 −0.0152606 0.999884i $$-0.504858\pi$$
−0.0152606 + 0.999884i $$0.504858\pi$$
$$182$$ 0 0
$$183$$ 4.16977 0.308238
$$184$$ 0 0
$$185$$ −1.75612 −0.129112
$$186$$ 0 0
$$187$$ −1.52164 −0.111273
$$188$$ 0 0
$$189$$ −2.10203 −0.152900
$$190$$ 0 0
$$191$$ 7.44449 0.538664 0.269332 0.963047i $$-0.413197\pi$$
0.269332 + 0.963047i $$0.413197\pi$$
$$192$$ 0 0
$$193$$ −8.06774 −0.580729 −0.290364 0.956916i $$-0.593776\pi$$
−0.290364 + 0.956916i $$0.593776\pi$$
$$194$$ 0 0
$$195$$ 1.08959 0.0780271
$$196$$ 0 0
$$197$$ −18.4569 −1.31500 −0.657501 0.753454i $$-0.728386\pi$$
−0.657501 + 0.753454i $$0.728386\pi$$
$$198$$ 0 0
$$199$$ −3.29020 −0.233236 −0.116618 0.993177i $$-0.537205\pi$$
−0.116618 + 0.993177i $$0.537205\pi$$
$$200$$ 0 0
$$201$$ −3.51223 −0.247734
$$202$$ 0 0
$$203$$ 1.02143 0.0716902
$$204$$ 0 0
$$205$$ −10.0339 −0.700796
$$206$$ 0 0
$$207$$ 2.92285 0.203152
$$208$$ 0 0
$$209$$ 1.08959 0.0753685
$$210$$ 0 0
$$211$$ −26.3365 −1.81308 −0.906540 0.422120i $$-0.861286\pi$$
−0.906540 + 0.422120i $$0.861286\pi$$
$$212$$ 0 0
$$213$$ 2.58980 0.177450
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 6.74367 0.457790
$$218$$ 0 0
$$219$$ −1.26531 −0.0855019
$$220$$ 0 0
$$221$$ 21.4908 1.44563
$$222$$ 0 0
$$223$$ −25.9134 −1.73529 −0.867646 0.497182i $$-0.834368\pi$$
−0.867646 + 0.497182i $$0.834368\pi$$
$$224$$ 0 0
$$225$$ −2.92285 −0.194857
$$226$$ 0 0
$$227$$ −10.7347 −0.712486 −0.356243 0.934393i $$-0.615943\pi$$
−0.356243 + 0.934393i $$0.615943\pi$$
$$228$$ 0 0
$$229$$ −4.22203 −0.279000 −0.139500 0.990222i $$-0.544550\pi$$
−0.139500 + 0.990222i $$0.544550\pi$$
$$230$$ 0 0
$$231$$ −0.0985755 −0.00648579
$$232$$ 0 0
$$233$$ −4.55551 −0.298441 −0.149221 0.988804i $$-0.547676\pi$$
−0.149221 + 0.988804i $$0.547676\pi$$
$$234$$ 0 0
$$235$$ −8.95672 −0.584272
$$236$$ 0 0
$$237$$ −2.22203 −0.144337
$$238$$ 0 0
$$239$$ −24.3365 −1.57420 −0.787099 0.616827i $$-0.788418\pi$$
−0.787099 + 0.616827i $$0.788418\pi$$
$$240$$ 0 0
$$241$$ 15.6914 1.01077 0.505386 0.862893i $$-0.331350\pi$$
0.505386 + 0.862893i $$0.331350\pi$$
$$242$$ 0 0
$$243$$ −7.24388 −0.464695
$$244$$ 0 0
$$245$$ 5.36734 0.342907
$$246$$ 0 0
$$247$$ −15.3888 −0.979164
$$248$$ 0 0
$$249$$ 0.734688 0.0465589
$$250$$ 0 0
$$251$$ −30.6575 −1.93509 −0.967543 0.252705i $$-0.918680\pi$$
−0.967543 + 0.252705i $$0.918680\pi$$
$$252$$ 0 0
$$253$$ 0.277754 0.0174623
$$254$$ 0 0
$$255$$ 1.52164 0.0952887
$$256$$ 0 0
$$257$$ 4.24692 0.264916 0.132458 0.991189i $$-0.457713\pi$$
0.132458 + 0.991189i $$0.457713\pi$$
$$258$$ 0 0
$$259$$ −2.24388 −0.139428
$$260$$ 0 0
$$261$$ 2.33651 0.144626
$$262$$ 0 0
$$263$$ −20.2130 −1.24639 −0.623195 0.782066i $$-0.714166\pi$$
−0.623195 + 0.782066i $$0.714166\pi$$
$$264$$ 0 0
$$265$$ 10.6451 0.653923
$$266$$ 0 0
$$267$$ −3.69141 −0.225911
$$268$$ 0 0
$$269$$ −6.08959 −0.371289 −0.185644 0.982617i $$-0.559437\pi$$
−0.185644 + 0.982617i $$0.559437\pi$$
$$270$$ 0 0
$$271$$ −8.78999 −0.533954 −0.266977 0.963703i $$-0.586025\pi$$
−0.266977 + 0.963703i $$0.586025\pi$$
$$272$$ 0 0
$$273$$ 1.39223 0.0842614
$$274$$ 0 0
$$275$$ −0.277754 −0.0167492
$$276$$ 0 0
$$277$$ −22.8024 −1.37007 −0.685033 0.728512i $$-0.740212\pi$$
−0.685033 + 0.728512i $$0.740212\pi$$
$$278$$ 0 0
$$279$$ 15.4261 0.923536
$$280$$ 0 0
$$281$$ −23.3579 −1.39342 −0.696709 0.717354i $$-0.745353\pi$$
−0.696709 + 0.717354i $$0.745353\pi$$
$$282$$ 0 0
$$283$$ −12.2439 −0.727823 −0.363912 0.931433i $$-0.618559\pi$$
−0.363912 + 0.931433i $$0.618559\pi$$
$$284$$ 0 0
$$285$$ −1.08959 −0.0645417
$$286$$ 0 0
$$287$$ −12.8208 −0.756789
$$288$$ 0 0
$$289$$ 13.0124 0.765438
$$290$$ 0 0
$$291$$ −0.879575 −0.0515616
$$292$$ 0 0
$$293$$ −26.1573 −1.52813 −0.764064 0.645141i $$-0.776799\pi$$
−0.764064 + 0.645141i $$0.776799\pi$$
$$294$$ 0 0
$$295$$ −8.08959 −0.470994
$$296$$ 0 0
$$297$$ −0.456933 −0.0265140
$$298$$ 0 0
$$299$$ −3.92285 −0.226864
$$300$$ 0 0
$$301$$ 5.11102 0.294594
$$302$$ 0 0
$$303$$ 1.55593 0.0893859
$$304$$ 0 0
$$305$$ 15.0124 0.859610
$$306$$ 0 0
$$307$$ 5.38877 0.307553 0.153777 0.988106i $$-0.450856\pi$$
0.153777 + 0.988106i $$0.450856\pi$$
$$308$$ 0 0
$$309$$ 0.786951 0.0447681
$$310$$ 0 0
$$311$$ −26.5804 −1.50724 −0.753618 0.657313i $$-0.771693\pi$$
−0.753618 + 0.657313i $$0.771693\pi$$
$$312$$ 0 0
$$313$$ 25.1235 1.42006 0.710031 0.704170i $$-0.248681\pi$$
0.710031 + 0.704170i $$0.248681\pi$$
$$314$$ 0 0
$$315$$ −3.73469 −0.210426
$$316$$ 0 0
$$317$$ −19.5894 −1.10025 −0.550125 0.835083i $$-0.685420\pi$$
−0.550125 + 0.835083i $$0.685420\pi$$
$$318$$ 0 0
$$319$$ 0.222035 0.0124316
$$320$$ 0 0
$$321$$ 5.33305 0.297662
$$322$$ 0 0
$$323$$ −21.4908 −1.19578
$$324$$ 0 0
$$325$$ 3.92285 0.217601
$$326$$ 0 0
$$327$$ −1.13245 −0.0626244
$$328$$ 0 0
$$329$$ −11.4445 −0.630955
$$330$$ 0 0
$$331$$ 31.1141 1.71018 0.855091 0.518477i $$-0.173501\pi$$
0.855091 + 0.518477i $$0.173501\pi$$
$$332$$ 0 0
$$333$$ −5.13287 −0.281279
$$334$$ 0 0
$$335$$ −12.6451 −0.690876
$$336$$ 0 0
$$337$$ 17.9229 0.976320 0.488160 0.872754i $$-0.337668\pi$$
0.488160 + 0.872754i $$0.337668\pi$$
$$338$$ 0 0
$$339$$ −3.26531 −0.177347
$$340$$ 0 0
$$341$$ 1.46592 0.0793840
$$342$$ 0 0
$$343$$ 15.8024 0.853251
$$344$$ 0 0
$$345$$ −0.277754 −0.0149538
$$346$$ 0 0
$$347$$ −7.16674 −0.384731 −0.192365 0.981323i $$-0.561616\pi$$
−0.192365 + 0.981323i $$0.561616\pi$$
$$348$$ 0 0
$$349$$ −7.93529 −0.424767 −0.212383 0.977186i $$-0.568123\pi$$
−0.212383 + 0.977186i $$0.568123\pi$$
$$350$$ 0 0
$$351$$ 6.45348 0.344461
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 9.32407 0.494870
$$356$$ 0 0
$$357$$ 1.94428 0.102902
$$358$$ 0 0
$$359$$ 27.6049 1.45693 0.728464 0.685084i $$-0.240234\pi$$
0.728464 + 0.685084i $$0.240234\pi$$
$$360$$ 0 0
$$361$$ −3.61123 −0.190065
$$362$$ 0 0
$$363$$ 3.03387 0.159237
$$364$$ 0 0
$$365$$ −4.55551 −0.238446
$$366$$ 0 0
$$367$$ 11.5341 0.602074 0.301037 0.953612i $$-0.402667\pi$$
0.301037 + 0.953612i $$0.402667\pi$$
$$368$$ 0 0
$$369$$ −29.3275 −1.52673
$$370$$ 0 0
$$371$$ 13.6018 0.706171
$$372$$ 0 0
$$373$$ 11.6237 0.601851 0.300925 0.953648i $$-0.402704\pi$$
0.300925 + 0.953648i $$0.402704\pi$$
$$374$$ 0 0
$$375$$ 0.277754 0.0143432
$$376$$ 0 0
$$377$$ −3.13590 −0.161507
$$378$$ 0 0
$$379$$ 12.2778 0.630666 0.315333 0.948981i $$-0.397884\pi$$
0.315333 + 0.948981i $$0.397884\pi$$
$$380$$ 0 0
$$381$$ −4.95672 −0.253941
$$382$$ 0 0
$$383$$ 4.64510 0.237353 0.118677 0.992933i $$-0.462135\pi$$
0.118677 + 0.992933i $$0.462135\pi$$
$$384$$ 0 0
$$385$$ −0.354902 −0.0180875
$$386$$ 0 0
$$387$$ 11.6914 0.594308
$$388$$ 0 0
$$389$$ 9.90142 0.502022 0.251011 0.967984i $$-0.419237\pi$$
0.251011 + 0.967984i $$0.419237\pi$$
$$390$$ 0 0
$$391$$ −5.47836 −0.277053
$$392$$ 0 0
$$393$$ 3.44449 0.173752
$$394$$ 0 0
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ 6.34549 0.318471 0.159236 0.987241i $$-0.449097\pi$$
0.159236 + 0.987241i $$0.449097\pi$$
$$398$$ 0 0
$$399$$ −1.39223 −0.0696986
$$400$$ 0 0
$$401$$ −5.84571 −0.291921 −0.145960 0.989290i $$-0.546627\pi$$
−0.145960 + 0.989290i $$0.546627\pi$$
$$402$$ 0 0
$$403$$ −20.7039 −1.03133
$$404$$ 0 0
$$405$$ −8.31162 −0.413008
$$406$$ 0 0
$$407$$ −0.487769 −0.0241778
$$408$$ 0 0
$$409$$ 7.52467 0.372071 0.186036 0.982543i $$-0.440436\pi$$
0.186036 + 0.982543i $$0.440436\pi$$
$$410$$ 0 0
$$411$$ −4.17572 −0.205973
$$412$$ 0 0
$$413$$ −10.3365 −0.508626
$$414$$ 0 0
$$415$$ 2.64510 0.129843
$$416$$ 0 0
$$417$$ 5.00042 0.244872
$$418$$ 0 0
$$419$$ 14.4012 0.703545 0.351773 0.936085i $$-0.385579\pi$$
0.351773 + 0.936085i $$0.385579\pi$$
$$420$$ 0 0
$$421$$ 18.4784 0.900580 0.450290 0.892882i $$-0.351321\pi$$
0.450290 + 0.892882i $$0.351321\pi$$
$$422$$ 0 0
$$423$$ −26.1792 −1.27288
$$424$$ 0 0
$$425$$ 5.47836 0.265740
$$426$$ 0 0
$$427$$ 19.1822 0.928292
$$428$$ 0 0
$$429$$ 0.302638 0.0146115
$$430$$ 0 0
$$431$$ 13.1359 0.632734 0.316367 0.948637i $$-0.397537\pi$$
0.316367 + 0.948637i $$0.397537\pi$$
$$432$$ 0 0
$$433$$ −36.1479 −1.73716 −0.868579 0.495550i $$-0.834966\pi$$
−0.868579 + 0.495550i $$0.834966\pi$$
$$434$$ 0 0
$$435$$ −0.222035 −0.0106458
$$436$$ 0 0
$$437$$ 3.92285 0.187655
$$438$$ 0 0
$$439$$ 8.16977 0.389922 0.194961 0.980811i $$-0.437542\pi$$
0.194961 + 0.980811i $$0.437542\pi$$
$$440$$ 0 0
$$441$$ 15.6880 0.747045
$$442$$ 0 0
$$443$$ 1.36734 0.0649645 0.0324822 0.999472i $$-0.489659\pi$$
0.0324822 + 0.999472i $$0.489659\pi$$
$$444$$ 0 0
$$445$$ −13.2902 −0.630016
$$446$$ 0 0
$$447$$ −3.97857 −0.188180
$$448$$ 0 0
$$449$$ −0.944281 −0.0445634 −0.0222817 0.999752i $$-0.507093\pi$$
−0.0222817 + 0.999752i $$0.507093\pi$$
$$450$$ 0 0
$$451$$ −2.78695 −0.131232
$$452$$ 0 0
$$453$$ 6.71921 0.315696
$$454$$ 0 0
$$455$$ 5.01244 0.234987
$$456$$ 0 0
$$457$$ −7.75612 −0.362816 −0.181408 0.983408i $$-0.558065\pi$$
−0.181408 + 0.983408i $$0.558065\pi$$
$$458$$ 0 0
$$459$$ 9.01244 0.420665
$$460$$ 0 0
$$461$$ −9.84571 −0.458560 −0.229280 0.973360i $$-0.573637\pi$$
−0.229280 + 0.973360i $$0.573637\pi$$
$$462$$ 0 0
$$463$$ −7.29020 −0.338804 −0.169402 0.985547i $$-0.554184\pi$$
−0.169402 + 0.985547i $$0.554184\pi$$
$$464$$ 0 0
$$465$$ −1.46592 −0.0679804
$$466$$ 0 0
$$467$$ −37.5153 −1.73600 −0.868000 0.496565i $$-0.834595\pi$$
−0.868000 + 0.496565i $$0.834595\pi$$
$$468$$ 0 0
$$469$$ −16.1573 −0.746076
$$470$$ 0 0
$$471$$ 3.55509 0.163810
$$472$$ 0 0
$$473$$ 1.11102 0.0510846
$$474$$ 0 0
$$475$$ −3.92285 −0.179993
$$476$$ 0 0
$$477$$ 31.1141 1.42461
$$478$$ 0 0
$$479$$ 26.1792 1.19616 0.598079 0.801437i $$-0.295931\pi$$
0.598079 + 0.801437i $$0.295931\pi$$
$$480$$ 0 0
$$481$$ 6.88898 0.314111
$$482$$ 0 0
$$483$$ −0.354902 −0.0161486
$$484$$ 0 0
$$485$$ −3.16674 −0.143794
$$486$$ 0 0
$$487$$ 10.9567 0.496496 0.248248 0.968696i $$-0.420145\pi$$
0.248248 + 0.968696i $$0.420145\pi$$
$$488$$ 0 0
$$489$$ −5.39818 −0.244114
$$490$$ 0 0
$$491$$ −19.6884 −0.888524 −0.444262 0.895897i $$-0.646534\pi$$
−0.444262 + 0.895897i $$0.646534\pi$$
$$492$$ 0 0
$$493$$ −4.37936 −0.197237
$$494$$ 0 0
$$495$$ −0.811835 −0.0364893
$$496$$ 0 0
$$497$$ 11.9139 0.534410
$$498$$ 0 0
$$499$$ 32.8243 1.46942 0.734708 0.678383i $$-0.237319\pi$$
0.734708 + 0.678383i $$0.237319\pi$$
$$500$$ 0 0
$$501$$ 6.88898 0.307777
$$502$$ 0 0
$$503$$ −17.9010 −0.798166 −0.399083 0.916915i $$-0.630672\pi$$
−0.399083 + 0.916915i $$0.630672\pi$$
$$504$$ 0 0
$$505$$ 5.60182 0.249278
$$506$$ 0 0
$$507$$ −0.663492 −0.0294667
$$508$$ 0 0
$$509$$ 43.3579 1.92181 0.960903 0.276884i $$-0.0893018\pi$$
0.960903 + 0.276884i $$0.0893018\pi$$
$$510$$ 0 0
$$511$$ −5.82082 −0.257498
$$512$$ 0 0
$$513$$ −6.45348 −0.284928
$$514$$ 0 0
$$515$$ 2.83326 0.124848
$$516$$ 0 0
$$517$$ −2.48777 −0.109412
$$518$$ 0 0
$$519$$ −3.52164 −0.154583
$$520$$ 0 0
$$521$$ −25.5122 −1.11771 −0.558856 0.829265i $$-0.688759\pi$$
−0.558856 + 0.829265i $$0.688759\pi$$
$$522$$ 0 0
$$523$$ −24.4012 −1.06699 −0.533495 0.845803i $$-0.679122\pi$$
−0.533495 + 0.845803i $$0.679122\pi$$
$$524$$ 0 0
$$525$$ 0.354902 0.0154892
$$526$$ 0 0
$$527$$ −28.9134 −1.25949
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −23.6447 −1.02609
$$532$$ 0 0
$$533$$ 39.3614 1.70493
$$534$$ 0 0
$$535$$ 19.2006 0.830115
$$536$$ 0 0
$$537$$ 0.0428561 0.00184938
$$538$$ 0 0
$$539$$ 1.49080 0.0642134
$$540$$ 0 0
$$541$$ 16.3147 0.701422 0.350711 0.936484i $$-0.385940\pi$$
0.350711 + 0.936484i $$0.385940\pi$$
$$542$$ 0 0
$$543$$ 0.114052 0.00489443
$$544$$ 0 0
$$545$$ −4.07715 −0.174646
$$546$$ 0 0
$$547$$ 11.1016 0.474671 0.237335 0.971428i $$-0.423726\pi$$
0.237335 + 0.971428i $$0.423726\pi$$
$$548$$ 0 0
$$549$$ 43.8792 1.87272
$$550$$ 0 0
$$551$$ 3.13590 0.133594
$$552$$ 0 0
$$553$$ −10.2220 −0.434685
$$554$$ 0 0
$$555$$ 0.487769 0.0207046
$$556$$ 0 0
$$557$$ −39.8487 −1.68845 −0.844223 0.535993i $$-0.819937\pi$$
−0.844223 + 0.535993i $$0.819937\pi$$
$$558$$ 0 0
$$559$$ −15.6914 −0.663676
$$560$$ 0 0
$$561$$ 0.422642 0.0178440
$$562$$ 0 0
$$563$$ 11.0463 0.465547 0.232773 0.972531i $$-0.425220\pi$$
0.232773 + 0.972531i $$0.425220\pi$$
$$564$$ 0 0
$$565$$ −11.7561 −0.494584
$$566$$ 0 0
$$567$$ −10.6202 −0.446007
$$568$$ 0 0
$$569$$ −38.4938 −1.61375 −0.806873 0.590725i $$-0.798842\pi$$
−0.806873 + 0.590725i $$0.798842\pi$$
$$570$$ 0 0
$$571$$ −20.2778 −0.848598 −0.424299 0.905522i $$-0.639479\pi$$
−0.424299 + 0.905522i $$0.639479\pi$$
$$572$$ 0 0
$$573$$ −2.06774 −0.0863811
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −14.8024 −0.616233 −0.308117 0.951349i $$-0.599699\pi$$
−0.308117 + 0.951349i $$0.599699\pi$$
$$578$$ 0 0
$$579$$ 2.24085 0.0931265
$$580$$ 0 0
$$581$$ 3.37979 0.140217
$$582$$ 0 0
$$583$$ 2.95672 0.122455
$$584$$ 0 0
$$585$$ 11.4659 0.474057
$$586$$ 0 0
$$587$$ −36.7716 −1.51773 −0.758863 0.651250i $$-0.774245\pi$$
−0.758863 + 0.651250i $$0.774245\pi$$
$$588$$ 0 0
$$589$$ 20.7039 0.853087
$$590$$ 0 0
$$591$$ 5.12649 0.210876
$$592$$ 0 0
$$593$$ −8.73469 −0.358691 −0.179345 0.983786i $$-0.557398\pi$$
−0.179345 + 0.983786i $$0.557398\pi$$
$$594$$ 0 0
$$595$$ 7.00000 0.286972
$$596$$ 0 0
$$597$$ 0.913866 0.0374021
$$598$$ 0 0
$$599$$ 20.8581 0.852241 0.426120 0.904666i $$-0.359880\pi$$
0.426120 + 0.904666i $$0.359880\pi$$
$$600$$ 0 0
$$601$$ 42.7283 1.74292 0.871462 0.490463i $$-0.163172\pi$$
0.871462 + 0.490463i $$0.163172\pi$$
$$602$$ 0 0
$$603$$ −36.9598 −1.50512
$$604$$ 0 0
$$605$$ 10.9229 0.444077
$$606$$ 0 0
$$607$$ −1.29020 −0.0523675 −0.0261837 0.999657i $$-0.508335\pi$$
−0.0261837 + 0.999657i $$0.508335\pi$$
$$608$$ 0 0
$$609$$ −0.283706 −0.0114964
$$610$$ 0 0
$$611$$ 35.1359 1.42145
$$612$$ 0 0
$$613$$ −42.4938 −1.71631 −0.858155 0.513391i $$-0.828389\pi$$
−0.858155 + 0.513391i $$0.828389\pi$$
$$614$$ 0 0
$$615$$ 2.78695 0.112381
$$616$$ 0 0
$$617$$ −6.34246 −0.255338 −0.127669 0.991817i $$-0.540749\pi$$
−0.127669 + 0.991817i $$0.540749\pi$$
$$618$$ 0 0
$$619$$ 22.3241 0.897280 0.448640 0.893713i $$-0.351909\pi$$
0.448640 + 0.893713i $$0.351909\pi$$
$$620$$ 0 0
$$621$$ −1.64510 −0.0660155
$$622$$ 0 0
$$623$$ −16.9816 −0.680354
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −0.302638 −0.0120862
$$628$$ 0 0
$$629$$ 9.62064 0.383600
$$630$$ 0 0
$$631$$ 30.6481 1.22008 0.610041 0.792370i $$-0.291153\pi$$
0.610041 + 0.792370i $$0.291153\pi$$
$$632$$ 0 0
$$633$$ 7.31508 0.290748
$$634$$ 0 0
$$635$$ −17.8457 −0.708185
$$636$$ 0 0
$$637$$ −21.0553 −0.834241
$$638$$ 0 0
$$639$$ 27.2529 1.07811
$$640$$ 0 0
$$641$$ −24.3147 −0.960371 −0.480186 0.877167i $$-0.659431\pi$$
−0.480186 + 0.877167i $$0.659431\pi$$
$$642$$ 0 0
$$643$$ −23.6944 −0.934418 −0.467209 0.884147i $$-0.654740\pi$$
−0.467209 + 0.884147i $$0.654740\pi$$
$$644$$ 0 0
$$645$$ −1.11102 −0.0437463
$$646$$ 0 0
$$647$$ −32.8951 −1.29324 −0.646619 0.762813i $$-0.723818\pi$$
−0.646619 + 0.762813i $$0.723818\pi$$
$$648$$ 0 0
$$649$$ −2.24692 −0.0881993
$$650$$ 0 0
$$651$$ −1.87308 −0.0734120
$$652$$ 0 0
$$653$$ 48.9045 1.91378 0.956890 0.290452i $$-0.0938055\pi$$
0.956890 + 0.290452i $$0.0938055\pi$$
$$654$$ 0 0
$$655$$ 12.4012 0.484556
$$656$$ 0 0
$$657$$ −13.3151 −0.519471
$$658$$ 0 0
$$659$$ 21.9134 0.853627 0.426813 0.904340i $$-0.359636\pi$$
0.426813 + 0.904340i $$0.359636\pi$$
$$660$$ 0 0
$$661$$ −6.34549 −0.246811 −0.123406 0.992356i $$-0.539382\pi$$
−0.123406 + 0.992356i $$0.539382\pi$$
$$662$$ 0 0
$$663$$ −5.96916 −0.231823
$$664$$ 0 0
$$665$$ −5.01244 −0.194374
$$666$$ 0 0
$$667$$ 0.799393 0.0309526
$$668$$ 0 0
$$669$$ 7.19757 0.278274
$$670$$ 0 0
$$671$$ 4.16977 0.160972
$$672$$ 0 0
$$673$$ 22.9816 0.885876 0.442938 0.896552i $$-0.353936\pi$$
0.442938 + 0.896552i $$0.353936\pi$$
$$674$$ 0 0
$$675$$ 1.64510 0.0633199
$$676$$ 0 0
$$677$$ 9.97815 0.383491 0.191746 0.981445i $$-0.438585\pi$$
0.191746 + 0.981445i $$0.438585\pi$$
$$678$$ 0 0
$$679$$ −4.04631 −0.155283
$$680$$ 0 0
$$681$$ 2.98161 0.114255
$$682$$ 0 0
$$683$$ 15.1265 0.578799 0.289400 0.957208i $$-0.406544\pi$$
0.289400 + 0.957208i $$0.406544\pi$$
$$684$$ 0 0
$$685$$ −15.0339 −0.574415
$$686$$ 0 0
$$687$$ 1.17269 0.0447409
$$688$$ 0 0
$$689$$ −41.7592 −1.59090
$$690$$ 0 0
$$691$$ 35.8457 1.36363 0.681817 0.731522i $$-0.261190\pi$$
0.681817 + 0.731522i $$0.261190\pi$$
$$692$$ 0 0
$$693$$ −1.03733 −0.0394047
$$694$$ 0 0
$$695$$ 18.0030 0.682894
$$696$$ 0 0
$$697$$ 54.9692 2.08211
$$698$$ 0 0
$$699$$ 1.26531 0.0478585
$$700$$ 0 0
$$701$$ 13.0339 0.492282 0.246141 0.969234i $$-0.420837\pi$$
0.246141 + 0.969234i $$0.420837\pi$$
$$702$$ 0 0
$$703$$ −6.88898 −0.259823
$$704$$ 0 0
$$705$$ 2.48777 0.0936948
$$706$$ 0 0
$$707$$ 7.15775 0.269195
$$708$$ 0 0
$$709$$ 2.21001 0.0829988 0.0414994 0.999139i $$-0.486787\pi$$
0.0414994 + 0.999139i $$0.486787\pi$$
$$710$$ 0 0
$$711$$ −23.3828 −0.876924
$$712$$ 0 0
$$713$$ 5.27775 0.197653
$$714$$ 0 0
$$715$$ 1.08959 0.0407483
$$716$$ 0 0
$$717$$ 6.75957 0.252441
$$718$$ 0 0
$$719$$ −37.9259 −1.41440 −0.707198 0.707015i $$-0.750041\pi$$
−0.707198 + 0.707015i $$0.750041\pi$$
$$720$$ 0 0
$$721$$ 3.62021 0.134824
$$722$$ 0 0
$$723$$ −4.35836 −0.162089
$$724$$ 0 0
$$725$$ −0.799393 −0.0296887
$$726$$ 0 0
$$727$$ −26.5465 −0.984556 −0.492278 0.870438i $$-0.663836\pi$$
−0.492278 + 0.870438i $$0.663836\pi$$
$$728$$ 0 0
$$729$$ −22.9229 −0.848995
$$730$$ 0 0
$$731$$ −21.9134 −0.810498
$$732$$ 0 0
$$733$$ 8.37936 0.309499 0.154749 0.987954i $$-0.450543\pi$$
0.154749 + 0.987954i $$0.450543\pi$$
$$734$$ 0 0
$$735$$ −1.49080 −0.0549891
$$736$$ 0 0
$$737$$ −3.51223 −0.129375
$$738$$ 0 0
$$739$$ −23.4904 −0.864108 −0.432054 0.901848i $$-0.642211\pi$$
−0.432054 + 0.901848i $$0.642211\pi$$
$$740$$ 0 0
$$741$$ 4.27430 0.157020
$$742$$ 0 0
$$743$$ −28.7681 −1.05540 −0.527700 0.849431i $$-0.676946\pi$$
−0.527700 + 0.849431i $$0.676946\pi$$
$$744$$ 0 0
$$745$$ −14.3241 −0.524793
$$746$$ 0 0
$$747$$ 7.73123 0.282871
$$748$$ 0 0
$$749$$ 24.5337 0.896440
$$750$$ 0 0
$$751$$ 3.13590 0.114431 0.0572153 0.998362i $$-0.481778\pi$$
0.0572153 + 0.998362i $$0.481778\pi$$
$$752$$ 0 0
$$753$$ 8.51527 0.310314
$$754$$ 0 0
$$755$$ 24.1912 0.880408
$$756$$ 0 0
$$757$$ 0.0218495 0.000794132 0 0.000397066 1.00000i $$-0.499874\pi$$
0.000397066 1.00000i $$0.499874\pi$$
$$758$$ 0 0
$$759$$ −0.0771475 −0.00280028
$$760$$ 0 0
$$761$$ −8.83368 −0.320221 −0.160110 0.987099i $$-0.551185\pi$$
−0.160110 + 0.987099i $$0.551185\pi$$
$$762$$ 0 0
$$763$$ −5.20959 −0.188600
$$764$$ 0 0
$$765$$ 16.0124 0.578931
$$766$$ 0 0
$$767$$ 31.7343 1.14586
$$768$$ 0 0
$$769$$ −53.2036 −1.91857 −0.959286 0.282436i $$-0.908858\pi$$
−0.959286 + 0.282436i $$0.908858\pi$$
$$770$$ 0 0
$$771$$ −1.17960 −0.0424823
$$772$$ 0 0
$$773$$ 15.3768 0.553063 0.276532 0.961005i $$-0.410815\pi$$
0.276532 + 0.961005i $$0.410815\pi$$
$$774$$ 0 0
$$775$$ −5.27775 −0.189583
$$776$$ 0 0
$$777$$ 0.623249 0.0223589
$$778$$ 0 0
$$779$$ −39.3614 −1.41027
$$780$$ 0 0
$$781$$ 2.58980 0.0926703
$$782$$ 0 0
$$783$$ −1.31508 −0.0469971
$$784$$ 0 0
$$785$$ 12.7994 0.456830
$$786$$ 0 0
$$787$$ −36.0957 −1.28667 −0.643336 0.765584i $$-0.722450\pi$$
−0.643336 + 0.765584i $$0.722450\pi$$
$$788$$ 0 0
$$789$$ 5.61426 0.199873
$$790$$ 0 0
$$791$$ −15.0214 −0.534100
$$792$$ 0 0
$$793$$ −58.8916 −2.09130
$$794$$ 0 0
$$795$$ −2.95672 −0.104864
$$796$$ 0 0
$$797$$ 6.13245 0.217222 0.108611 0.994084i $$-0.465360\pi$$
0.108611 + 0.994084i $$0.465360\pi$$
$$798$$ 0 0
$$799$$ 49.0682 1.73591
$$800$$ 0 0
$$801$$ −38.8453 −1.37253
$$802$$ 0 0
$$803$$ −1.26531 −0.0446519
$$804$$ 0 0
$$805$$ −1.27775 −0.0450349
$$806$$ 0 0
$$807$$ 1.69141 0.0595405
$$808$$ 0 0
$$809$$ −30.7008 −1.07938 −0.539692 0.841863i $$-0.681459\pi$$
−0.539692 + 0.841863i $$0.681459\pi$$
$$810$$ 0 0
$$811$$ 0.490803 0.0172344 0.00861722 0.999963i $$-0.497257\pi$$
0.00861722 + 0.999963i $$0.497257\pi$$
$$812$$ 0 0
$$813$$ 2.44146 0.0856256
$$814$$ 0 0
$$815$$ −19.4351 −0.680781
$$816$$ 0 0
$$817$$ 15.6914 0.548973
$$818$$ 0 0
$$819$$ 14.6506 0.511934
$$820$$ 0 0
$$821$$ 18.8024 0.656209 0.328105 0.944641i $$-0.393590\pi$$
0.328105 + 0.944641i $$0.393590\pi$$
$$822$$ 0 0
$$823$$ −36.2718 −1.26436 −0.632178 0.774823i $$-0.717839\pi$$
−0.632178 + 0.774823i $$0.717839\pi$$
$$824$$ 0 0
$$825$$ 0.0771475 0.00268593
$$826$$ 0 0
$$827$$ 51.6018 1.79437 0.897186 0.441654i $$-0.145608\pi$$
0.897186 + 0.441654i $$0.145608\pi$$
$$828$$ 0 0
$$829$$ −17.2255 −0.598266 −0.299133 0.954211i $$-0.596697\pi$$
−0.299133 + 0.954211i $$0.596697\pi$$
$$830$$ 0 0
$$831$$ 6.33347 0.219706
$$832$$ 0 0
$$833$$ −29.4042 −1.01880
$$834$$ 0 0
$$835$$ 24.8024 0.858323
$$836$$ 0 0
$$837$$ −8.68242 −0.300108
$$838$$ 0 0
$$839$$ −13.7592 −0.475019 −0.237509 0.971385i $$-0.576331\pi$$
−0.237509 + 0.971385i $$0.576331\pi$$
$$840$$ 0 0
$$841$$ −28.3610 −0.977965
$$842$$ 0 0
$$843$$ 6.48777 0.223451
$$844$$ 0 0
$$845$$ −2.38877 −0.0821762
$$846$$ 0 0
$$847$$ 13.9567 0.479559
$$848$$ 0 0
$$849$$ 3.40079 0.116715
$$850$$ 0 0
$$851$$ −1.75612 −0.0601989
$$852$$ 0 0
$$853$$ 22.3490 0.765213 0.382607 0.923911i $$-0.375026\pi$$
0.382607 + 0.923911i $$0.375026\pi$$
$$854$$ 0 0
$$855$$ −11.4659 −0.392126
$$856$$ 0 0
$$857$$ 35.9134 1.22678 0.613390 0.789780i $$-0.289805\pi$$
0.613390 + 0.789780i $$0.289805\pi$$
$$858$$ 0 0
$$859$$ −5.64552 −0.192623 −0.0963113 0.995351i $$-0.530704\pi$$
−0.0963113 + 0.995351i $$0.530704\pi$$
$$860$$ 0 0
$$861$$ 3.56104 0.121360
$$862$$ 0 0
$$863$$ 36.5616 1.24457 0.622285 0.782791i $$-0.286204\pi$$
0.622285 + 0.782791i $$0.286204\pi$$
$$864$$ 0 0
$$865$$ −12.6790 −0.431098
$$866$$ 0 0
$$867$$ −3.61426 −0.122747
$$868$$ 0 0
$$869$$ −2.22203 −0.0753774
$$870$$ 0 0
$$871$$ 49.6049 1.68080
$$872$$ 0 0
$$873$$ −9.25590 −0.313265
$$874$$ 0 0
$$875$$ 1.27775 0.0431960
$$876$$ 0 0
$$877$$ −6.45693 −0.218035 −0.109018 0.994040i $$-0.534770\pi$$
−0.109018 + 0.994040i $$0.534770\pi$$
$$878$$ 0 0
$$879$$ 7.26531 0.245053
$$880$$ 0 0
$$881$$ 12.3147 0.414891 0.207446 0.978247i $$-0.433485\pi$$
0.207446 + 0.978247i $$0.433485\pi$$
$$882$$ 0 0
$$883$$ −7.43508 −0.250210 −0.125105 0.992143i $$-0.539927\pi$$
−0.125105 + 0.992143i $$0.539927\pi$$
$$884$$ 0 0
$$885$$ 2.24692 0.0755293
$$886$$ 0 0
$$887$$ −23.6914 −0.795480 −0.397740 0.917498i $$-0.630205\pi$$
−0.397740 + 0.917498i $$0.630205\pi$$
$$888$$ 0 0
$$889$$ −22.8024 −0.764769
$$890$$ 0 0
$$891$$ −2.30859 −0.0773407
$$892$$ 0 0
$$893$$ −35.1359 −1.17578
$$894$$ 0 0
$$895$$ 0.154295 0.00515751
$$896$$ 0 0
$$897$$ 1.08959 0.0363803
$$898$$ 0 0
$$899$$ 4.21900 0.140712
$$900$$ 0 0
$$901$$ −58.3177 −1.94284
$$902$$ 0 0
$$903$$ −1.41961 −0.0472416
$$904$$ 0 0
$$905$$ 0.410621 0.0136495
$$906$$ 0 0
$$907$$ −10.5337 −0.349764 −0.174882 0.984589i $$-0.555954\pi$$
−0.174882 + 0.984589i $$0.555954\pi$$
$$908$$ 0 0
$$909$$ 16.3733 0.543068
$$910$$ 0 0
$$911$$ −37.9383 −1.25695 −0.628476 0.777829i $$-0.716321\pi$$
−0.628476 + 0.777829i $$0.716321\pi$$
$$912$$ 0 0
$$913$$ 0.734688 0.0243146
$$914$$ 0 0
$$915$$ −4.16977 −0.137848
$$916$$ 0 0
$$917$$ 15.8457 0.523271
$$918$$ 0 0
$$919$$ −27.2036 −0.897365 −0.448683 0.893691i $$-0.648107\pi$$
−0.448683 + 0.893691i $$0.648107\pi$$
$$920$$ 0 0
$$921$$ −1.49675 −0.0493198
$$922$$ 0 0
$$923$$ −36.5769 −1.20394
$$924$$ 0 0
$$925$$ 1.75612 0.0577407
$$926$$ 0 0
$$927$$ 8.28121 0.271991
$$928$$ 0 0
$$929$$ 1.84874 0.0606552 0.0303276 0.999540i $$-0.490345\pi$$
0.0303276 + 0.999540i $$0.490345\pi$$
$$930$$ 0 0
$$931$$ 21.0553 0.690060
$$932$$ 0 0
$$933$$ 7.38282 0.241703
$$934$$ 0 0
$$935$$ 1.52164 0.0497629
$$936$$ 0 0
$$937$$ 51.8363 1.69342 0.846709 0.532056i $$-0.178581\pi$$
0.846709 + 0.532056i $$0.178581\pi$$
$$938$$ 0 0
$$939$$ −6.97815 −0.227723
$$940$$ 0 0
$$941$$ 30.4569 0.992868 0.496434 0.868075i $$-0.334643\pi$$
0.496434 + 0.868075i $$0.334643\pi$$
$$942$$ 0 0
$$943$$ −10.0339 −0.326748
$$944$$ 0 0
$$945$$ 2.10203 0.0683791
$$946$$ 0 0
$$947$$ −38.8118 −1.26122 −0.630608 0.776102i $$-0.717194\pi$$
−0.630608 + 0.776102i $$0.717194\pi$$
$$948$$ 0 0
$$949$$ 17.8706 0.580104
$$950$$ 0 0
$$951$$ 5.44104 0.176438
$$952$$ 0 0
$$953$$ 2.49979 0.0809761 0.0404881 0.999180i $$-0.487109\pi$$
0.0404881 + 0.999180i $$0.487109\pi$$
$$954$$ 0 0
$$955$$ −7.44449 −0.240898
$$956$$ 0 0
$$957$$ −0.0616712 −0.00199355
$$958$$ 0 0
$$959$$ −19.2096 −0.620310
$$960$$ 0 0
$$961$$ −3.14531 −0.101462
$$962$$ 0 0
$$963$$ 56.1205 1.80846
$$964$$ 0 0
$$965$$ 8.06774 0.259710
$$966$$ 0 0
$$967$$ −1.35794 −0.0436683 −0.0218341 0.999762i $$-0.506951\pi$$
−0.0218341 + 0.999762i $$0.506951\pi$$
$$968$$ 0 0
$$969$$ 5.96916 0.191757
$$970$$ 0 0
$$971$$ 4.59241 0.147378 0.0736888 0.997281i $$-0.476523\pi$$
0.0736888 + 0.997281i $$0.476523\pi$$
$$972$$ 0 0
$$973$$ 23.0035 0.737457
$$974$$ 0 0
$$975$$ −1.08959 −0.0348948
$$976$$ 0 0
$$977$$ −9.45693 −0.302554 −0.151277 0.988491i $$-0.548339\pi$$
−0.151277 + 0.988491i $$0.548339\pi$$
$$978$$ 0 0
$$979$$ −3.69141 −0.117978
$$980$$ 0 0
$$981$$ −11.9169 −0.380477
$$982$$ 0 0
$$983$$ 24.6571 0.786440 0.393220 0.919444i $$-0.371361\pi$$
0.393220 + 0.919444i $$0.371361\pi$$
$$984$$ 0 0
$$985$$ 18.4569 0.588087
$$986$$ 0 0
$$987$$ 3.17876 0.101181
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ 18.7437 0.595412 0.297706 0.954658i $$-0.403778\pi$$
0.297706 + 0.954658i $$0.403778\pi$$
$$992$$ 0 0
$$993$$ −8.64206 −0.274248
$$994$$ 0 0
$$995$$ 3.29020 0.104306
$$996$$ 0 0
$$997$$ −17.0682 −0.540554 −0.270277 0.962783i $$-0.587115\pi$$
−0.270277 + 0.962783i $$0.587115\pi$$
$$998$$ 0 0
$$999$$ 2.88898 0.0914034
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 7360.2.a.bx.1.2 3
4.3 odd 2 7360.2.a.cd.1.2 3
8.3 odd 2 3680.2.a.q.1.2 3
8.5 even 2 3680.2.a.r.1.2 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
3680.2.a.q.1.2 3 8.3 odd 2
3680.2.a.r.1.2 yes 3 8.5 even 2
7360.2.a.bx.1.2 3 1.1 even 1 trivial
7360.2.a.cd.1.2 3 4.3 odd 2