# Properties

 Label 7360.2.a.bz.1.3 Level $7360$ Weight $2$ Character 7360.1 Self dual yes Analytic conductor $58.770$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.11903$$ of defining polynomial Character $$\chi$$ $$=$$ 7360.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.11903 q^{3} +1.00000 q^{5} +4.50973 q^{7} +6.72833 q^{9} +O(q^{10})$$ $$q+3.11903 q^{3} +1.00000 q^{5} +4.50973 q^{7} +6.72833 q^{9} -4.33763 q^{11} +3.72833 q^{13} +3.11903 q^{15} +1.11903 q^{17} -4.50973 q^{19} +14.0660 q^{21} -1.00000 q^{23} +1.00000 q^{25} +11.6288 q^{27} +8.23805 q^{29} +1.72833 q^{31} -13.5292 q^{33} +4.50973 q^{35} +0.781399 q^{37} +11.6288 q^{39} +3.90043 q^{41} -8.00000 q^{43} +6.72833 q^{45} -11.4567 q^{47} +13.3376 q^{49} +3.49027 q^{51} +6.00000 q^{53} -4.33763 q^{55} -14.0660 q^{57} +2.23805 q^{59} -3.55623 q^{61} +30.3429 q^{63} +3.72833 q^{65} -2.43720 q^{67} -3.11903 q^{69} +7.11903 q^{71} -9.45665 q^{73} +3.11903 q^{75} -19.5615 q^{77} -14.9133 q^{79} +16.0854 q^{81} -2.78140 q^{83} +1.11903 q^{85} +25.6947 q^{87} -7.69471 q^{89} +16.8137 q^{91} +5.39070 q^{93} -4.50973 q^{95} -0.642920 q^{97} -29.1850 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{3} + 3q^{5} + 3q^{7} + 10q^{9} + O(q^{10})$$ $$3q - q^{3} + 3q^{5} + 3q^{7} + 10q^{9} - 3q^{11} + q^{13} - q^{15} - 7q^{17} - 3q^{19} + 22q^{21} - 3q^{23} + 3q^{25} + 14q^{27} + 4q^{29} - 5q^{31} - 9q^{33} + 3q^{35} + 2q^{37} + 14q^{39} + q^{41} - 24q^{43} + 10q^{45} - 14q^{47} + 30q^{49} + 21q^{51} + 18q^{53} - 3q^{55} - 22q^{57} - 14q^{59} - q^{61} + 8q^{63} + q^{65} - 8q^{67} + q^{69} + 11q^{71} - 8q^{73} - q^{75} + 24q^{77} - 4q^{79} + 7q^{81} - 8q^{83} - 7q^{85} + 36q^{87} + 18q^{89} - q^{91} + 16q^{93} - 3q^{95} - 33q^{97} - 57q^{99} + O(q^{100})$$

## 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
$$3$$ 3.11903 1.80077 0.900385 0.435093i $$-0.143285\pi$$
0.900385 + 0.435093i $$0.143285\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.50973 1.70452 0.852258 0.523122i $$-0.175233\pi$$
0.852258 + 0.523122i $$0.175233\pi$$
$$8$$ 0 0
$$9$$ 6.72833 2.24278
$$10$$ 0 0
$$11$$ −4.33763 −1.30784 −0.653922 0.756562i $$-0.726878\pi$$
−0.653922 + 0.756562i $$0.726878\pi$$
$$12$$ 0 0
$$13$$ 3.72833 1.03405 0.517026 0.855970i $$-0.327039\pi$$
0.517026 + 0.855970i $$0.327039\pi$$
$$14$$ 0 0
$$15$$ 3.11903 0.805329
$$16$$ 0 0
$$17$$ 1.11903 0.271404 0.135702 0.990750i $$-0.456671\pi$$
0.135702 + 0.990750i $$0.456671\pi$$
$$18$$ 0 0
$$19$$ −4.50973 −1.03460 −0.517301 0.855803i $$-0.673063\pi$$
−0.517301 + 0.855803i $$0.673063\pi$$
$$20$$ 0 0
$$21$$ 14.0660 3.06944
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 11.6288 2.23795
$$28$$ 0 0
$$29$$ 8.23805 1.52977 0.764884 0.644168i $$-0.222796\pi$$
0.764884 + 0.644168i $$0.222796\pi$$
$$30$$ 0 0
$$31$$ 1.72833 0.310417 0.155208 0.987882i $$-0.450395\pi$$
0.155208 + 0.987882i $$0.450395\pi$$
$$32$$ 0 0
$$33$$ −13.5292 −2.35513
$$34$$ 0 0
$$35$$ 4.50973 0.762283
$$36$$ 0 0
$$37$$ 0.781399 0.128461 0.0642306 0.997935i $$-0.479541\pi$$
0.0642306 + 0.997935i $$0.479541\pi$$
$$38$$ 0 0
$$39$$ 11.6288 1.86209
$$40$$ 0 0
$$41$$ 3.90043 0.609144 0.304572 0.952489i $$-0.401487\pi$$
0.304572 + 0.952489i $$0.401487\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 6.72833 1.00300
$$46$$ 0 0
$$47$$ −11.4567 −1.67112 −0.835562 0.549396i $$-0.814858\pi$$
−0.835562 + 0.549396i $$0.814858\pi$$
$$48$$ 0 0
$$49$$ 13.3376 1.90538
$$50$$ 0 0
$$51$$ 3.49027 0.488736
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −4.33763 −0.584886
$$56$$ 0 0
$$57$$ −14.0660 −1.86308
$$58$$ 0 0
$$59$$ 2.23805 0.291370 0.145685 0.989331i $$-0.453461\pi$$
0.145685 + 0.989331i $$0.453461\pi$$
$$60$$ 0 0
$$61$$ −3.55623 −0.455329 −0.227664 0.973740i $$-0.573109\pi$$
−0.227664 + 0.973740i $$0.573109\pi$$
$$62$$ 0 0
$$63$$ 30.3429 3.82285
$$64$$ 0 0
$$65$$ 3.72833 0.462442
$$66$$ 0 0
$$67$$ −2.43720 −0.297752 −0.148876 0.988856i $$-0.547565\pi$$
−0.148876 + 0.988856i $$0.547565\pi$$
$$68$$ 0 0
$$69$$ −3.11903 −0.375487
$$70$$ 0 0
$$71$$ 7.11903 0.844873 0.422437 0.906393i $$-0.361175\pi$$
0.422437 + 0.906393i $$0.361175\pi$$
$$72$$ 0 0
$$73$$ −9.45665 −1.10682 −0.553409 0.832910i $$-0.686673\pi$$
−0.553409 + 0.832910i $$0.686673\pi$$
$$74$$ 0 0
$$75$$ 3.11903 0.360154
$$76$$ 0 0
$$77$$ −19.5615 −2.22924
$$78$$ 0 0
$$79$$ −14.9133 −1.67788 −0.838939 0.544225i $$-0.816824\pi$$
−0.838939 + 0.544225i $$0.816824\pi$$
$$80$$ 0 0
$$81$$ 16.0854 1.78727
$$82$$ 0 0
$$83$$ −2.78140 −0.305298 −0.152649 0.988280i $$-0.548780\pi$$
−0.152649 + 0.988280i $$0.548780\pi$$
$$84$$ 0 0
$$85$$ 1.11903 0.121375
$$86$$ 0 0
$$87$$ 25.6947 2.75476
$$88$$ 0 0
$$89$$ −7.69471 −0.815637 −0.407819 0.913063i $$-0.633710\pi$$
−0.407819 + 0.913063i $$0.633710\pi$$
$$90$$ 0 0
$$91$$ 16.8137 1.76256
$$92$$ 0 0
$$93$$ 5.39070 0.558989
$$94$$ 0 0
$$95$$ −4.50973 −0.462688
$$96$$ 0 0
$$97$$ −0.642920 −0.0652786 −0.0326393 0.999467i $$-0.510391\pi$$
−0.0326393 + 0.999467i $$0.510391\pi$$
$$98$$ 0 0
$$99$$ −29.1850 −2.93320
$$100$$ 0 0
$$101$$ 8.23805 0.819717 0.409858 0.912149i $$-0.365578\pi$$
0.409858 + 0.912149i $$0.365578\pi$$
$$102$$ 0 0
$$103$$ 12.3376 1.21566 0.607831 0.794066i $$-0.292040\pi$$
0.607831 + 0.794066i $$0.292040\pi$$
$$104$$ 0 0
$$105$$ 14.0660 1.37270
$$106$$ 0 0
$$107$$ 15.9328 1.54028 0.770139 0.637876i $$-0.220187\pi$$
0.770139 + 0.637876i $$0.220187\pi$$
$$108$$ 0 0
$$109$$ 1.49027 0.142742 0.0713712 0.997450i $$-0.477263\pi$$
0.0713712 + 0.997450i $$0.477263\pi$$
$$110$$ 0 0
$$111$$ 2.43720 0.231329
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ 25.0854 2.31915
$$118$$ 0 0
$$119$$ 5.04650 0.462612
$$120$$ 0 0
$$121$$ 7.81502 0.710456
$$122$$ 0 0
$$123$$ 12.1655 1.09693
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −0.675256 −0.0599193 −0.0299597 0.999551i $$-0.509538\pi$$
−0.0299597 + 0.999551i $$0.509538\pi$$
$$128$$ 0 0
$$129$$ −24.9522 −2.19692
$$130$$ 0 0
$$131$$ 13.6947 1.19651 0.598256 0.801305i $$-0.295861\pi$$
0.598256 + 0.801305i $$0.295861\pi$$
$$132$$ 0 0
$$133$$ −20.3376 −1.76350
$$134$$ 0 0
$$135$$ 11.6288 1.00084
$$136$$ 0 0
$$137$$ 7.52918 0.643261 0.321631 0.946865i $$-0.395769\pi$$
0.321631 + 0.946865i $$0.395769\pi$$
$$138$$ 0 0
$$139$$ −4.67526 −0.396550 −0.198275 0.980146i $$-0.563534\pi$$
−0.198275 + 0.980146i $$0.563534\pi$$
$$140$$ 0 0
$$141$$ −35.7336 −3.00931
$$142$$ 0 0
$$143$$ −16.1721 −1.35238
$$144$$ 0 0
$$145$$ 8.23805 0.684133
$$146$$ 0 0
$$147$$ 41.6004 3.43114
$$148$$ 0 0
$$149$$ −7.52918 −0.616814 −0.308407 0.951254i $$-0.599796\pi$$
−0.308407 + 0.951254i $$0.599796\pi$$
$$150$$ 0 0
$$151$$ −13.3571 −1.08698 −0.543492 0.839414i $$-0.682898\pi$$
−0.543492 + 0.839414i $$0.682898\pi$$
$$152$$ 0 0
$$153$$ 7.52918 0.608698
$$154$$ 0 0
$$155$$ 1.72833 0.138823
$$156$$ 0 0
$$157$$ −16.2381 −1.29594 −0.647969 0.761667i $$-0.724381\pi$$
−0.647969 + 0.761667i $$0.724381\pi$$
$$158$$ 0 0
$$159$$ 18.7142 1.48413
$$160$$ 0 0
$$161$$ −4.50973 −0.355416
$$162$$ 0 0
$$163$$ 3.29112 0.257781 0.128890 0.991659i $$-0.458858\pi$$
0.128890 + 0.991659i $$0.458858\pi$$
$$164$$ 0 0
$$165$$ −13.5292 −1.05325
$$166$$ 0 0
$$167$$ 22.9133 1.77309 0.886543 0.462647i $$-0.153100\pi$$
0.886543 + 0.462647i $$0.153100\pi$$
$$168$$ 0 0
$$169$$ 0.900425 0.0692635
$$170$$ 0 0
$$171$$ −30.3429 −2.32038
$$172$$ 0 0
$$173$$ −0.575681 −0.0437683 −0.0218841 0.999761i $$-0.506966\pi$$
−0.0218841 + 0.999761i $$0.506966\pi$$
$$174$$ 0 0
$$175$$ 4.50973 0.340903
$$176$$ 0 0
$$177$$ 6.98055 0.524690
$$178$$ 0 0
$$179$$ −5.01945 −0.375171 −0.187586 0.982248i $$-0.560066\pi$$
−0.187586 + 0.982248i $$0.560066\pi$$
$$180$$ 0 0
$$181$$ 11.5292 0.856957 0.428479 0.903552i $$-0.359050\pi$$
0.428479 + 0.903552i $$0.359050\pi$$
$$182$$ 0 0
$$183$$ −11.0920 −0.819942
$$184$$ 0 0
$$185$$ 0.781399 0.0574496
$$186$$ 0 0
$$187$$ −4.85392 −0.354954
$$188$$ 0 0
$$189$$ 52.4425 3.81463
$$190$$ 0 0
$$191$$ −18.7142 −1.35411 −0.677055 0.735933i $$-0.736744\pi$$
−0.677055 + 0.735933i $$0.736744\pi$$
$$192$$ 0 0
$$193$$ 23.4956 1.69125 0.845624 0.533780i $$-0.179229\pi$$
0.845624 + 0.533780i $$0.179229\pi$$
$$194$$ 0 0
$$195$$ 11.6288 0.832752
$$196$$ 0 0
$$197$$ 18.1385 1.29231 0.646157 0.763205i $$-0.276375\pi$$
0.646157 + 0.763205i $$0.276375\pi$$
$$198$$ 0 0
$$199$$ −23.2575 −1.64868 −0.824340 0.566094i $$-0.808454\pi$$
−0.824340 + 0.566094i $$0.808454\pi$$
$$200$$ 0 0
$$201$$ −7.60170 −0.536183
$$202$$ 0 0
$$203$$ 37.1514 2.60751
$$204$$ 0 0
$$205$$ 3.90043 0.272418
$$206$$ 0 0
$$207$$ −6.72833 −0.467651
$$208$$ 0 0
$$209$$ 19.5615 1.35310
$$210$$ 0 0
$$211$$ −4.34420 −0.299067 −0.149533 0.988757i $$-0.547777\pi$$
−0.149533 + 0.988757i $$0.547777\pi$$
$$212$$ 0 0
$$213$$ 22.2044 1.52142
$$214$$ 0 0
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 7.79428 0.529110
$$218$$ 0 0
$$219$$ −29.4956 −1.99313
$$220$$ 0 0
$$221$$ 4.17210 0.280646
$$222$$ 0 0
$$223$$ 12.4761 0.835462 0.417731 0.908571i $$-0.362825\pi$$
0.417731 + 0.908571i $$0.362825\pi$$
$$224$$ 0 0
$$225$$ 6.72833 0.448555
$$226$$ 0 0
$$227$$ −15.9328 −1.05749 −0.528747 0.848779i $$-0.677338\pi$$
−0.528747 + 0.848779i $$0.677338\pi$$
$$228$$ 0 0
$$229$$ 3.56280 0.235436 0.117718 0.993047i $$-0.462442\pi$$
0.117718 + 0.993047i $$0.462442\pi$$
$$230$$ 0 0
$$231$$ −61.0129 −4.01435
$$232$$ 0 0
$$233$$ −27.4956 −1.80129 −0.900647 0.434552i $$-0.856907\pi$$
−0.900647 + 0.434552i $$0.856907\pi$$
$$234$$ 0 0
$$235$$ −11.4567 −0.747350
$$236$$ 0 0
$$237$$ −46.5150 −3.02147
$$238$$ 0 0
$$239$$ 10.0389 0.649363 0.324681 0.945823i $$-0.394743\pi$$
0.324681 + 0.945823i $$0.394743\pi$$
$$240$$ 0 0
$$241$$ −23.6947 −1.52631 −0.763155 0.646215i $$-0.776351\pi$$
−0.763155 + 0.646215i $$0.776351\pi$$
$$242$$ 0 0
$$243$$ 15.2846 0.980505
$$244$$ 0 0
$$245$$ 13.3376 0.852110
$$246$$ 0 0
$$247$$ −16.8137 −1.06983
$$248$$ 0 0
$$249$$ −8.67526 −0.549772
$$250$$ 0 0
$$251$$ −12.4425 −0.785363 −0.392681 0.919675i $$-0.628452\pi$$
−0.392681 + 0.919675i $$0.628452\pi$$
$$252$$ 0 0
$$253$$ 4.33763 0.272704
$$254$$ 0 0
$$255$$ 3.49027 0.218569
$$256$$ 0 0
$$257$$ 5.45665 0.340377 0.170188 0.985412i $$-0.445562\pi$$
0.170188 + 0.985412i $$0.445562\pi$$
$$258$$ 0 0
$$259$$ 3.52389 0.218964
$$260$$ 0 0
$$261$$ 55.4283 3.43093
$$262$$ 0 0
$$263$$ 0.138479 0.00853895 0.00426948 0.999991i $$-0.498641\pi$$
0.00426948 + 0.999991i $$0.498641\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ −24.0000 −1.46878
$$268$$ 0 0
$$269$$ −14.6753 −0.894766 −0.447383 0.894342i $$-0.647644\pi$$
−0.447383 + 0.894342i $$0.647644\pi$$
$$270$$ 0 0
$$271$$ −8.31058 −0.504832 −0.252416 0.967619i $$-0.581225\pi$$
−0.252416 + 0.967619i $$0.581225\pi$$
$$272$$ 0 0
$$273$$ 52.4425 3.17396
$$274$$ 0 0
$$275$$ −4.33763 −0.261569
$$276$$ 0 0
$$277$$ −12.9133 −0.775886 −0.387943 0.921683i $$-0.626814\pi$$
−0.387943 + 0.921683i $$0.626814\pi$$
$$278$$ 0 0
$$279$$ 11.6288 0.696195
$$280$$ 0 0
$$281$$ 2.67526 0.159592 0.0797962 0.996811i $$-0.474573\pi$$
0.0797962 + 0.996811i $$0.474573\pi$$
$$282$$ 0 0
$$283$$ −0.742495 −0.0441367 −0.0220684 0.999756i $$-0.507025\pi$$
−0.0220684 + 0.999756i $$0.507025\pi$$
$$284$$ 0 0
$$285$$ −14.0660 −0.833195
$$286$$ 0 0
$$287$$ 17.5898 1.03830
$$288$$ 0 0
$$289$$ −15.7478 −0.926340
$$290$$ 0 0
$$291$$ −2.00528 −0.117552
$$292$$ 0 0
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ 2.23805 0.130305
$$296$$ 0 0
$$297$$ −50.4412 −2.92690
$$298$$ 0 0
$$299$$ −3.72833 −0.215615
$$300$$ 0 0
$$301$$ −36.0778 −2.07949
$$302$$ 0 0
$$303$$ 25.6947 1.47612
$$304$$ 0 0
$$305$$ −3.55623 −0.203629
$$306$$ 0 0
$$307$$ −30.5084 −1.74121 −0.870604 0.491984i $$-0.836272\pi$$
−0.870604 + 0.491984i $$0.836272\pi$$
$$308$$ 0 0
$$309$$ 38.4814 2.18913
$$310$$ 0 0
$$311$$ 5.56280 0.315437 0.157719 0.987484i $$-0.449586\pi$$
0.157719 + 0.987484i $$0.449586\pi$$
$$312$$ 0 0
$$313$$ 4.07252 0.230193 0.115096 0.993354i $$-0.463282\pi$$
0.115096 + 0.993354i $$0.463282\pi$$
$$314$$ 0 0
$$315$$ 30.3429 1.70963
$$316$$ 0 0
$$317$$ 6.16553 0.346291 0.173145 0.984896i $$-0.444607\pi$$
0.173145 + 0.984896i $$0.444607\pi$$
$$318$$ 0 0
$$319$$ −35.7336 −2.00070
$$320$$ 0 0
$$321$$ 49.6947 2.77369
$$322$$ 0 0
$$323$$ −5.04650 −0.280795
$$324$$ 0 0
$$325$$ 3.72833 0.206810
$$326$$ 0 0
$$327$$ 4.64820 0.257046
$$328$$ 0 0
$$329$$ −51.6664 −2.84846
$$330$$ 0 0
$$331$$ −27.5886 −1.51640 −0.758202 0.652019i $$-0.773922\pi$$
−0.758202 + 0.652019i $$0.773922\pi$$
$$332$$ 0 0
$$333$$ 5.25751 0.288110
$$334$$ 0 0
$$335$$ −2.43720 −0.133159
$$336$$ 0 0
$$337$$ −17.4230 −0.949093 −0.474547 0.880230i $$-0.657388\pi$$
−0.474547 + 0.880230i $$0.657388\pi$$
$$338$$ 0 0
$$339$$ −18.7142 −1.01641
$$340$$ 0 0
$$341$$ −7.49684 −0.405977
$$342$$ 0 0
$$343$$ 28.5810 1.54323
$$344$$ 0 0
$$345$$ −3.11903 −0.167923
$$346$$ 0 0
$$347$$ −4.88097 −0.262024 −0.131012 0.991381i $$-0.541823\pi$$
−0.131012 + 0.991381i $$0.541823\pi$$
$$348$$ 0 0
$$349$$ −24.0389 −1.28677 −0.643387 0.765542i $$-0.722471\pi$$
−0.643387 + 0.765542i $$0.722471\pi$$
$$350$$ 0 0
$$351$$ 43.3558 2.31416
$$352$$ 0 0
$$353$$ −14.3442 −0.763464 −0.381732 0.924273i $$-0.624672\pi$$
−0.381732 + 0.924273i $$0.624672\pi$$
$$354$$ 0 0
$$355$$ 7.11903 0.377839
$$356$$ 0 0
$$357$$ 15.7402 0.833059
$$358$$ 0 0
$$359$$ 26.7814 1.41347 0.706734 0.707479i $$-0.250168\pi$$
0.706734 + 0.707479i $$0.250168\pi$$
$$360$$ 0 0
$$361$$ 1.33763 0.0704015
$$362$$ 0 0
$$363$$ 24.3752 1.27937
$$364$$ 0 0
$$365$$ −9.45665 −0.494984
$$366$$ 0 0
$$367$$ −20.4761 −1.06884 −0.534422 0.845218i $$-0.679471\pi$$
−0.534422 + 0.845218i $$0.679471\pi$$
$$368$$ 0 0
$$369$$ 26.2433 1.36617
$$370$$ 0 0
$$371$$ 27.0584 1.40480
$$372$$ 0 0
$$373$$ 3.89386 0.201616 0.100808 0.994906i $$-0.467857\pi$$
0.100808 + 0.994906i $$0.467857\pi$$
$$374$$ 0 0
$$375$$ 3.11903 0.161066
$$376$$ 0 0
$$377$$ 30.7142 1.58186
$$378$$ 0 0
$$379$$ 30.3765 1.56034 0.780169 0.625569i $$-0.215133\pi$$
0.780169 + 0.625569i $$0.215133\pi$$
$$380$$ 0 0
$$381$$ −2.10614 −0.107901
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −19.5615 −0.996947
$$386$$ 0 0
$$387$$ −53.8266 −2.73616
$$388$$ 0 0
$$389$$ 18.6818 0.947206 0.473603 0.880738i $$-0.342953\pi$$
0.473603 + 0.880738i $$0.342953\pi$$
$$390$$ 0 0
$$391$$ −1.11903 −0.0565916
$$392$$ 0 0
$$393$$ 42.7142 2.15464
$$394$$ 0 0
$$395$$ −14.9133 −0.750370
$$396$$ 0 0
$$397$$ 28.5757 1.43417 0.717086 0.696985i $$-0.245475\pi$$
0.717086 + 0.696985i $$0.245475\pi$$
$$398$$ 0 0
$$399$$ −63.4336 −3.17565
$$400$$ 0 0
$$401$$ 12.1061 0.604552 0.302276 0.953220i $$-0.402254\pi$$
0.302276 + 0.953220i $$0.402254\pi$$
$$402$$ 0 0
$$403$$ 6.44377 0.320987
$$404$$ 0 0
$$405$$ 16.0854 0.799290
$$406$$ 0 0
$$407$$ −3.38942 −0.168007
$$408$$ 0 0
$$409$$ 25.2911 1.25057 0.625283 0.780398i $$-0.284984\pi$$
0.625283 + 0.780398i $$0.284984\pi$$
$$410$$ 0 0
$$411$$ 23.4837 1.15837
$$412$$ 0 0
$$413$$ 10.0930 0.496644
$$414$$ 0 0
$$415$$ −2.78140 −0.136533
$$416$$ 0 0
$$417$$ −14.5822 −0.714096
$$418$$ 0 0
$$419$$ 17.3505 0.847628 0.423814 0.905749i $$-0.360691\pi$$
0.423814 + 0.905749i $$0.360691\pi$$
$$420$$ 0 0
$$421$$ −21.4230 −1.04409 −0.522047 0.852916i $$-0.674832\pi$$
−0.522047 + 0.852916i $$0.674832\pi$$
$$422$$ 0 0
$$423$$ −77.0841 −3.74796
$$424$$ 0 0
$$425$$ 1.11903 0.0542808
$$426$$ 0 0
$$427$$ −16.0376 −0.776115
$$428$$ 0 0
$$429$$ −50.4412 −2.43532
$$430$$ 0 0
$$431$$ 22.5822 1.08775 0.543874 0.839167i $$-0.316957\pi$$
0.543874 + 0.839167i $$0.316957\pi$$
$$432$$ 0 0
$$433$$ 1.01417 0.0487378 0.0243689 0.999703i $$-0.492242\pi$$
0.0243689 + 0.999703i $$0.492242\pi$$
$$434$$ 0 0
$$435$$ 25.6947 1.23197
$$436$$ 0 0
$$437$$ 4.50973 0.215729
$$438$$ 0 0
$$439$$ −26.7478 −1.27660 −0.638301 0.769787i $$-0.720362\pi$$
−0.638301 + 0.769787i $$0.720362\pi$$
$$440$$ 0 0
$$441$$ 89.7399 4.27333
$$442$$ 0 0
$$443$$ −10.2044 −0.484827 −0.242414 0.970173i $$-0.577939\pi$$
−0.242414 + 0.970173i $$0.577939\pi$$
$$444$$ 0 0
$$445$$ −7.69471 −0.364764
$$446$$ 0 0
$$447$$ −23.4837 −1.11074
$$448$$ 0 0
$$449$$ 38.7867 1.83046 0.915228 0.402936i $$-0.132010\pi$$
0.915228 + 0.402936i $$0.132010\pi$$
$$450$$ 0 0
$$451$$ −16.9186 −0.796665
$$452$$ 0 0
$$453$$ −41.6611 −1.95741
$$454$$ 0 0
$$455$$ 16.8137 0.788240
$$456$$ 0 0
$$457$$ 34.9522 1.63500 0.817498 0.575932i $$-0.195361\pi$$
0.817498 + 0.575932i $$0.195361\pi$$
$$458$$ 0 0
$$459$$ 13.0129 0.607389
$$460$$ 0 0
$$461$$ −16.3700 −0.762425 −0.381213 0.924487i $$-0.624493\pi$$
−0.381213 + 0.924487i $$0.624493\pi$$
$$462$$ 0 0
$$463$$ 29.2186 1.35790 0.678952 0.734183i $$-0.262435\pi$$
0.678952 + 0.734183i $$0.262435\pi$$
$$464$$ 0 0
$$465$$ 5.39070 0.249988
$$466$$ 0 0
$$467$$ −24.2770 −1.12340 −0.561702 0.827340i $$-0.689853\pi$$
−0.561702 + 0.827340i $$0.689853\pi$$
$$468$$ 0 0
$$469$$ −10.9911 −0.507523
$$470$$ 0 0
$$471$$ −50.6469 −2.33369
$$472$$ 0 0
$$473$$ 34.7010 1.59555
$$474$$ 0 0
$$475$$ −4.50973 −0.206920
$$476$$ 0 0
$$477$$ 40.3700 1.84841
$$478$$ 0 0
$$479$$ 24.6080 1.12437 0.562185 0.827012i $$-0.309961\pi$$
0.562185 + 0.827012i $$0.309961\pi$$
$$480$$ 0 0
$$481$$ 2.91331 0.132835
$$482$$ 0 0
$$483$$ −14.0660 −0.640023
$$484$$ 0 0
$$485$$ −0.642920 −0.0291935
$$486$$ 0 0
$$487$$ −30.2381 −1.37022 −0.685108 0.728441i $$-0.740245\pi$$
−0.685108 + 0.728441i $$0.740245\pi$$
$$488$$ 0 0
$$489$$ 10.2651 0.464204
$$490$$ 0 0
$$491$$ −12.3311 −0.556493 −0.278246 0.960510i $$-0.589753\pi$$
−0.278246 + 0.960510i $$0.589753\pi$$
$$492$$ 0 0
$$493$$ 9.21860 0.415185
$$494$$ 0 0
$$495$$ −29.1850 −1.31177
$$496$$ 0 0
$$497$$ 32.1049 1.44010
$$498$$ 0 0
$$499$$ 26.9133 1.20481 0.602403 0.798192i $$-0.294210\pi$$
0.602403 + 0.798192i $$0.294210\pi$$
$$500$$ 0 0
$$501$$ 71.4672 3.19292
$$502$$ 0 0
$$503$$ 20.5097 0.914483 0.457242 0.889342i $$-0.348837\pi$$
0.457242 + 0.889342i $$0.348837\pi$$
$$504$$ 0 0
$$505$$ 8.23805 0.366589
$$506$$ 0 0
$$507$$ 2.80845 0.124728
$$508$$ 0 0
$$509$$ 36.7142 1.62733 0.813663 0.581336i $$-0.197470\pi$$
0.813663 + 0.581336i $$0.197470\pi$$
$$510$$ 0 0
$$511$$ −42.6469 −1.88659
$$512$$ 0 0
$$513$$ −52.4425 −2.31539
$$514$$ 0 0
$$515$$ 12.3376 0.543661
$$516$$ 0 0
$$517$$ 49.6947 2.18557
$$518$$ 0 0
$$519$$ −1.79557 −0.0788166
$$520$$ 0 0
$$521$$ −4.91331 −0.215256 −0.107628 0.994191i $$-0.534326\pi$$
−0.107628 + 0.994191i $$0.534326\pi$$
$$522$$ 0 0
$$523$$ 0.344196 0.0150506 0.00752531 0.999972i $$-0.497605\pi$$
0.00752531 + 0.999972i $$0.497605\pi$$
$$524$$ 0 0
$$525$$ 14.0660 0.613889
$$526$$ 0 0
$$527$$ 1.93404 0.0842483
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 15.0584 0.653477
$$532$$ 0 0
$$533$$ 14.5421 0.629887
$$534$$ 0 0
$$535$$ 15.9328 0.688833
$$536$$ 0 0
$$537$$ −15.6558 −0.675598
$$538$$ 0 0
$$539$$ −57.8537 −2.49193
$$540$$ 0 0
$$541$$ 6.13191 0.263631 0.131816 0.991274i $$-0.457919\pi$$
0.131816 + 0.991274i $$0.457919\pi$$
$$542$$ 0 0
$$543$$ 35.9598 1.54318
$$544$$ 0 0
$$545$$ 1.49027 0.0638363
$$546$$ 0 0
$$547$$ 9.18498 0.392721 0.196361 0.980532i $$-0.437088\pi$$
0.196361 + 0.980532i $$0.437088\pi$$
$$548$$ 0 0
$$549$$ −23.9275 −1.02120
$$550$$ 0 0
$$551$$ −37.1514 −1.58270
$$552$$ 0 0
$$553$$ −67.2549 −2.85997
$$554$$ 0 0
$$555$$ 2.43720 0.103454
$$556$$ 0 0
$$557$$ 4.30529 0.182421 0.0912105 0.995832i $$-0.470926\pi$$
0.0912105 + 0.995832i $$0.470926\pi$$
$$558$$ 0 0
$$559$$ −29.8266 −1.26153
$$560$$ 0 0
$$561$$ −15.1395 −0.639191
$$562$$ 0 0
$$563$$ −11.1256 −0.468888 −0.234444 0.972130i $$-0.575327\pi$$
−0.234444 + 0.972130i $$0.575327\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 72.5408 3.04643
$$568$$ 0 0
$$569$$ −16.0389 −0.672386 −0.336193 0.941793i $$-0.609139\pi$$
−0.336193 + 0.941793i $$0.609139\pi$$
$$570$$ 0 0
$$571$$ −17.9004 −0.749109 −0.374555 0.927205i $$-0.622204\pi$$
−0.374555 + 0.927205i $$0.622204\pi$$
$$572$$ 0 0
$$573$$ −58.3700 −2.43844
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −9.12559 −0.379903 −0.189952 0.981793i $$-0.560833\pi$$
−0.189952 + 0.981793i $$0.560833\pi$$
$$578$$ 0 0
$$579$$ 73.2833 3.04555
$$580$$ 0 0
$$581$$ −12.5433 −0.520386
$$582$$ 0 0
$$583$$ −26.0258 −1.07788
$$584$$ 0 0
$$585$$ 25.0854 1.03715
$$586$$ 0 0
$$587$$ 33.6340 1.38823 0.694113 0.719866i $$-0.255797\pi$$
0.694113 + 0.719866i $$0.255797\pi$$
$$588$$ 0 0
$$589$$ −7.79428 −0.321158
$$590$$ 0 0
$$591$$ 56.5744 2.32716
$$592$$ 0 0
$$593$$ −17.4567 −0.716859 −0.358429 0.933557i $$-0.616688\pi$$
−0.358429 + 0.933557i $$0.616688\pi$$
$$594$$ 0 0
$$595$$ 5.04650 0.206886
$$596$$ 0 0
$$597$$ −72.5408 −2.96890
$$598$$ 0 0
$$599$$ −11.5951 −0.473764 −0.236882 0.971538i $$-0.576126\pi$$
−0.236882 + 0.971538i $$0.576126\pi$$
$$600$$ 0 0
$$601$$ −31.6611 −1.29148 −0.645741 0.763556i $$-0.723452\pi$$
−0.645741 + 0.763556i $$0.723452\pi$$
$$602$$ 0 0
$$603$$ −16.3983 −0.667790
$$604$$ 0 0
$$605$$ 7.81502 0.317726
$$606$$ 0 0
$$607$$ −36.0778 −1.46435 −0.732177 0.681115i $$-0.761495\pi$$
−0.732177 + 0.681115i $$0.761495\pi$$
$$608$$ 0 0
$$609$$ 115.876 4.69554
$$610$$ 0 0
$$611$$ −42.7142 −1.72803
$$612$$ 0 0
$$613$$ 32.0389 1.29404 0.647020 0.762473i $$-0.276015\pi$$
0.647020 + 0.762473i $$0.276015\pi$$
$$614$$ 0 0
$$615$$ 12.1655 0.490562
$$616$$ 0 0
$$617$$ −13.3960 −0.539302 −0.269651 0.962958i $$-0.586908\pi$$
−0.269651 + 0.962958i $$0.586908\pi$$
$$618$$ 0 0
$$619$$ −37.1309 −1.49242 −0.746208 0.665713i $$-0.768128\pi$$
−0.746208 + 0.665713i $$0.768128\pi$$
$$620$$ 0 0
$$621$$ −11.6288 −0.466646
$$622$$ 0 0
$$623$$ −34.7010 −1.39027
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 61.0129 2.43662
$$628$$ 0 0
$$629$$ 0.874406 0.0348648
$$630$$ 0 0
$$631$$ 11.1125 0.442380 0.221190 0.975231i $$-0.429006\pi$$
0.221190 + 0.975231i $$0.429006\pi$$
$$632$$ 0 0
$$633$$ −13.5497 −0.538551
$$634$$ 0 0
$$635$$ −0.675256 −0.0267967
$$636$$ 0 0
$$637$$ 49.7270 1.97026
$$638$$ 0 0
$$639$$ 47.8991 1.89486
$$640$$ 0 0
$$641$$ 12.3831 0.489103 0.244552 0.969636i $$-0.421359\pi$$
0.244552 + 0.969636i $$0.421359\pi$$
$$642$$ 0 0
$$643$$ 37.4956 1.47868 0.739340 0.673332i $$-0.235138\pi$$
0.739340 + 0.673332i $$0.235138\pi$$
$$644$$ 0 0
$$645$$ −24.9522 −0.982492
$$646$$ 0 0
$$647$$ 14.5691 0.572771 0.286385 0.958114i $$-0.407546\pi$$
0.286385 + 0.958114i $$0.407546\pi$$
$$648$$ 0 0
$$649$$ −9.70784 −0.381066
$$650$$ 0 0
$$651$$ 24.3106 0.952807
$$652$$ 0 0
$$653$$ 4.41672 0.172840 0.0864198 0.996259i $$-0.472457\pi$$
0.0864198 + 0.996259i $$0.472457\pi$$
$$654$$ 0 0
$$655$$ 13.6947 0.535097
$$656$$ 0 0
$$657$$ −63.6275 −2.48234
$$658$$ 0 0
$$659$$ 31.8655 1.24130 0.620652 0.784086i $$-0.286868\pi$$
0.620652 + 0.784086i $$0.286868\pi$$
$$660$$ 0 0
$$661$$ −33.1190 −1.28818 −0.644090 0.764949i $$-0.722764\pi$$
−0.644090 + 0.764949i $$0.722764\pi$$
$$662$$ 0 0
$$663$$ 13.0129 0.505379
$$664$$ 0 0
$$665$$ −20.3376 −0.788659
$$666$$ 0 0
$$667$$ −8.23805 −0.318979
$$668$$ 0 0
$$669$$ 38.9133 1.50448
$$670$$ 0 0
$$671$$ 15.4256 0.595499
$$672$$ 0 0
$$673$$ 19.3505 0.745907 0.372954 0.927850i $$-0.378345\pi$$
0.372954 + 0.927850i $$0.378345\pi$$
$$674$$ 0 0
$$675$$ 11.6288 0.447591
$$676$$ 0 0
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ −2.89939 −0.111268
$$680$$ 0 0
$$681$$ −49.6947 −1.90431
$$682$$ 0 0
$$683$$ −38.3495 −1.46740 −0.733701 0.679472i $$-0.762209\pi$$
−0.733701 + 0.679472i $$0.762209\pi$$
$$684$$ 0 0
$$685$$ 7.52918 0.287675
$$686$$ 0 0
$$687$$ 11.1125 0.423967
$$688$$ 0 0
$$689$$ 22.3700 0.852228
$$690$$ 0 0
$$691$$ 21.0195 0.799618 0.399809 0.916599i $$-0.369077\pi$$
0.399809 + 0.916599i $$0.369077\pi$$
$$692$$ 0 0
$$693$$ −131.616 −4.99969
$$694$$ 0 0
$$695$$ −4.67526 −0.177343
$$696$$ 0 0
$$697$$ 4.36468 0.165324
$$698$$ 0 0
$$699$$ −85.7594 −3.24372
$$700$$ 0 0
$$701$$ −19.3169 −0.729589 −0.364794 0.931088i $$-0.618861\pi$$
−0.364794 + 0.931088i $$0.618861\pi$$
$$702$$ 0 0
$$703$$ −3.52389 −0.132906
$$704$$ 0 0
$$705$$ −35.7336 −1.34581
$$706$$ 0 0
$$707$$ 37.1514 1.39722
$$708$$ 0 0
$$709$$ −12.2315 −0.459363 −0.229682 0.973266i $$-0.573768\pi$$
−0.229682 + 0.973266i $$0.573768\pi$$
$$710$$ 0 0
$$711$$ −100.342 −3.76311
$$712$$ 0 0
$$713$$ −1.72833 −0.0647264
$$714$$ 0 0
$$715$$ −16.1721 −0.604802
$$716$$ 0 0
$$717$$ 31.3116 1.16935
$$718$$ 0 0
$$719$$ −40.6416 −1.51568 −0.757839 0.652442i $$-0.773745\pi$$
−0.757839 + 0.652442i $$0.773745\pi$$
$$720$$ 0 0
$$721$$ 55.6393 2.07212
$$722$$ 0 0
$$723$$ −73.9044 −2.74854
$$724$$ 0 0
$$725$$ 8.23805 0.305954
$$726$$ 0 0
$$727$$ −23.4501 −0.869716 −0.434858 0.900499i $$-0.643201\pi$$
−0.434858 + 0.900499i $$0.643201\pi$$
$$728$$ 0 0
$$729$$ −0.583281 −0.0216030
$$730$$ 0 0
$$731$$ −8.95221 −0.331110
$$732$$ 0 0
$$733$$ 12.5150 0.462252 0.231126 0.972924i $$-0.425759\pi$$
0.231126 + 0.972924i $$0.425759\pi$$
$$734$$ 0 0
$$735$$ 41.6004 1.53445
$$736$$ 0 0
$$737$$ 10.5717 0.389413
$$738$$ 0 0
$$739$$ 21.3505 0.785391 0.392696 0.919668i $$-0.371543\pi$$
0.392696 + 0.919668i $$0.371543\pi$$
$$740$$ 0 0
$$741$$ −52.4425 −1.92652
$$742$$ 0 0
$$743$$ 24.9858 0.916641 0.458321 0.888787i $$-0.348451\pi$$
0.458321 + 0.888787i $$0.348451\pi$$
$$744$$ 0 0
$$745$$ −7.52918 −0.275848
$$746$$ 0 0
$$747$$ −18.7142 −0.684715
$$748$$ 0 0
$$749$$ 71.8524 2.62543
$$750$$ 0 0
$$751$$ −33.6275 −1.22708 −0.613542 0.789662i $$-0.710256\pi$$
−0.613542 + 0.789662i $$0.710256\pi$$
$$752$$ 0 0
$$753$$ −38.8085 −1.41426
$$754$$ 0 0
$$755$$ −13.3571 −0.486114
$$756$$ 0 0
$$757$$ 37.1230 1.34926 0.674630 0.738156i $$-0.264303\pi$$
0.674630 + 0.738156i $$0.264303\pi$$
$$758$$ 0 0
$$759$$ 13.5292 0.491078
$$760$$ 0 0
$$761$$ −3.87337 −0.140410 −0.0702048 0.997533i $$-0.522365\pi$$
−0.0702048 + 0.997533i $$0.522365\pi$$
$$762$$ 0 0
$$763$$ 6.72073 0.243307
$$764$$ 0 0
$$765$$ 7.52918 0.272218
$$766$$ 0 0
$$767$$ 8.34420 0.301291
$$768$$ 0 0
$$769$$ 23.1645 0.835333 0.417667 0.908600i $$-0.362848\pi$$
0.417667 + 0.908600i $$0.362848\pi$$
$$770$$ 0 0
$$771$$ 17.0195 0.612941
$$772$$ 0 0
$$773$$ 8.78140 0.315845 0.157922 0.987452i $$-0.449520\pi$$
0.157922 + 0.987452i $$0.449520\pi$$
$$774$$ 0 0
$$775$$ 1.72833 0.0620834
$$776$$ 0 0
$$777$$ 10.9911 0.394304
$$778$$ 0 0
$$779$$ −17.5898 −0.630222
$$780$$ 0 0
$$781$$ −30.8797 −1.10496
$$782$$ 0 0
$$783$$ 95.7983 3.42355
$$784$$ 0 0
$$785$$ −16.2381 −0.579561
$$786$$ 0 0
$$787$$ −49.6275 −1.76903 −0.884514 0.466513i $$-0.845510\pi$$
−0.884514 + 0.466513i $$0.845510\pi$$
$$788$$ 0 0
$$789$$ 0.431918 0.0153767
$$790$$ 0 0
$$791$$ −27.0584 −0.962084
$$792$$ 0 0
$$793$$ −13.2588 −0.470833
$$794$$ 0 0
$$795$$ 18.7142 0.663723
$$796$$ 0 0
$$797$$ −18.3311 −0.649319 −0.324660 0.945831i $$-0.605250\pi$$
−0.324660 + 0.945831i $$0.605250\pi$$
$$798$$ 0 0
$$799$$ −12.8203 −0.453550
$$800$$ 0 0
$$801$$ −51.7725 −1.82929
$$802$$ 0 0
$$803$$ 41.0195 1.44755
$$804$$ 0 0
$$805$$ −4.50973 −0.158947
$$806$$ 0 0
$$807$$ −45.7725 −1.61127
$$808$$ 0 0
$$809$$ −1.93933 −0.0681832 −0.0340916 0.999419i $$-0.510854\pi$$
−0.0340916 + 0.999419i $$0.510854\pi$$
$$810$$ 0 0
$$811$$ 5.41775 0.190243 0.0951215 0.995466i $$-0.469676\pi$$
0.0951215 + 0.995466i $$0.469676\pi$$
$$812$$ 0 0
$$813$$ −25.9209 −0.909086
$$814$$ 0 0
$$815$$ 3.29112 0.115283
$$816$$ 0 0
$$817$$ 36.0778 1.26220
$$818$$ 0 0
$$819$$ 113.128 3.95302
$$820$$ 0 0
$$821$$ −37.8655 −1.32152 −0.660758 0.750599i $$-0.729765\pi$$
−0.660758 + 0.750599i $$0.729765\pi$$
$$822$$ 0 0
$$823$$ 43.7336 1.52446 0.762229 0.647308i $$-0.224105\pi$$
0.762229 + 0.647308i $$0.224105\pi$$
$$824$$ 0 0
$$825$$ −13.5292 −0.471026
$$826$$ 0 0
$$827$$ −28.1991 −0.980581 −0.490290 0.871559i $$-0.663109\pi$$
−0.490290 + 0.871559i $$0.663109\pi$$
$$828$$ 0 0
$$829$$ −1.12559 −0.0390935 −0.0195468 0.999809i $$-0.506222\pi$$
−0.0195468 + 0.999809i $$0.506222\pi$$
$$830$$ 0 0
$$831$$ −40.2770 −1.39719
$$832$$ 0 0
$$833$$ 14.9252 0.517126
$$834$$ 0 0
$$835$$ 22.9133 0.792948
$$836$$ 0 0
$$837$$ 20.0983 0.694699
$$838$$ 0 0
$$839$$ 39.9328 1.37863 0.689316 0.724461i $$-0.257911\pi$$
0.689316 + 0.724461i $$0.257911\pi$$
$$840$$ 0 0
$$841$$ 38.8655 1.34019
$$842$$ 0 0
$$843$$ 8.34420 0.287389
$$844$$ 0 0
$$845$$ 0.900425 0.0309756
$$846$$ 0 0
$$847$$ 35.2436 1.21098
$$848$$ 0 0
$$849$$ −2.31586 −0.0794801
$$850$$ 0 0
$$851$$ −0.781399 −0.0267860
$$852$$ 0 0
$$853$$ −47.9921 −1.64322 −0.821610 0.570050i $$-0.806924\pi$$
−0.821610 + 0.570050i $$0.806924\pi$$
$$854$$ 0 0
$$855$$ −30.3429 −1.03771
$$856$$ 0 0
$$857$$ −43.4283 −1.48348 −0.741742 0.670686i $$-0.766000\pi$$
−0.741742 + 0.670686i $$0.766000\pi$$
$$858$$ 0 0
$$859$$ −32.5433 −1.11036 −0.555182 0.831729i $$-0.687352\pi$$
−0.555182 + 0.831729i $$0.687352\pi$$
$$860$$ 0 0
$$861$$ 54.8632 1.86973
$$862$$ 0 0
$$863$$ 46.1036 1.56938 0.784692 0.619886i $$-0.212821\pi$$
0.784692 + 0.619886i $$0.212821\pi$$
$$864$$ 0 0
$$865$$ −0.575681 −0.0195738
$$866$$ 0 0
$$867$$ −49.1177 −1.66813
$$868$$ 0 0
$$869$$ 64.6884 2.19440
$$870$$ 0 0
$$871$$ −9.08669 −0.307891
$$872$$ 0 0
$$873$$ −4.32578 −0.146405
$$874$$ 0 0
$$875$$ 4.50973 0.152457
$$876$$ 0 0
$$877$$ 24.0996 0.813785 0.406892 0.913476i $$-0.366612\pi$$
0.406892 + 0.913476i $$0.366612\pi$$
$$878$$ 0 0
$$879$$ 18.7142 0.631213
$$880$$ 0 0
$$881$$ 2.34420 0.0789780 0.0394890 0.999220i $$-0.487427\pi$$
0.0394890 + 0.999220i $$0.487427\pi$$
$$882$$ 0 0
$$883$$ 41.0505 1.38146 0.690730 0.723113i $$-0.257289\pi$$
0.690730 + 0.723113i $$0.257289\pi$$
$$884$$ 0 0
$$885$$ 6.98055 0.234649
$$886$$ 0 0
$$887$$ 54.7788 1.83929 0.919647 0.392747i $$-0.128475\pi$$
0.919647 + 0.392747i $$0.128475\pi$$
$$888$$ 0 0
$$889$$ −3.04522 −0.102133
$$890$$ 0 0
$$891$$ −69.7725 −2.33747
$$892$$ 0 0
$$893$$ 51.6664 1.72895
$$894$$ 0 0
$$895$$ −5.01945 −0.167782
$$896$$ 0 0
$$897$$ −11.6288 −0.388273
$$898$$ 0 0
$$899$$ 14.2381 0.474866
$$900$$ 0 0
$$901$$ 6.71416 0.223681
$$902$$ 0 0
$$903$$ −112.528 −3.74469
$$904$$ 0 0
$$905$$ 11.5292 0.383243
$$906$$ 0 0
$$907$$ 10.1061 0.335569 0.167784 0.985824i $$-0.446339\pi$$
0.167784 + 0.985824i $$0.446339\pi$$
$$908$$ 0 0
$$909$$ 55.4283 1.83844
$$910$$ 0 0
$$911$$ −25.4178 −0.842128 −0.421064 0.907031i $$-0.638343\pi$$
−0.421064 + 0.907031i $$0.638343\pi$$
$$912$$ 0 0
$$913$$ 12.0647 0.399282
$$914$$ 0 0
$$915$$ −11.0920 −0.366689
$$916$$ 0 0
$$917$$ 61.7594 2.03947
$$918$$ 0 0
$$919$$ 23.6017 0.778548 0.389274 0.921122i $$-0.372726\pi$$
0.389274 + 0.921122i $$0.372726\pi$$
$$920$$ 0 0
$$921$$ −95.1566 −3.13552
$$922$$ 0 0
$$923$$ 26.5421 0.873643
$$924$$ 0 0
$$925$$ 0.781399 0.0256922
$$926$$ 0 0
$$927$$ 83.0116 2.72646
$$928$$ 0 0
$$929$$ −7.08669 −0.232507 −0.116253 0.993220i $$-0.537088\pi$$
−0.116253 + 0.993220i $$0.537088\pi$$
$$930$$ 0 0
$$931$$ −60.1490 −1.97131
$$932$$ 0 0
$$933$$ 17.3505 0.568030
$$934$$ 0 0
$$935$$ −4.85392 −0.158740
$$936$$ 0 0
$$937$$ 27.3169 0.892404 0.446202 0.894932i $$-0.352776\pi$$
0.446202 + 0.894932i $$0.352776\pi$$
$$938$$ 0 0
$$939$$ 12.7023 0.414524
$$940$$ 0 0
$$941$$ −55.8979 −1.82222 −0.911109 0.412165i $$-0.864773\pi$$
−0.911109 + 0.412165i $$0.864773\pi$$
$$942$$ 0 0
$$943$$ −3.90043 −0.127015
$$944$$ 0 0
$$945$$ 52.4425 1.70595
$$946$$ 0 0
$$947$$ −37.5939 −1.22164 −0.610818 0.791771i $$-0.709159\pi$$
−0.610818 + 0.791771i $$0.709159\pi$$
$$948$$ 0 0
$$949$$ −35.2575 −1.14451
$$950$$ 0 0
$$951$$ 19.2305 0.623590
$$952$$ 0 0
$$953$$ −29.3828 −0.951804 −0.475902 0.879498i $$-0.657878\pi$$
−0.475902 + 0.879498i $$0.657878\pi$$
$$954$$ 0 0
$$955$$ −18.7142 −0.605576
$$956$$ 0 0
$$957$$ −111.454 −3.60280
$$958$$ 0 0
$$959$$ 33.9545 1.09645
$$960$$ 0 0
$$961$$ −28.0129 −0.903641
$$962$$ 0 0
$$963$$ 107.201 3.45450
$$964$$ 0 0
$$965$$ 23.4956 0.756349
$$966$$ 0 0
$$967$$ −49.2292 −1.58310 −0.791552 0.611102i $$-0.790726\pi$$
−0.791552 + 0.611102i $$0.790726\pi$$
$$968$$ 0 0
$$969$$ −15.7402 −0.505647
$$970$$ 0 0
$$971$$ −9.62347 −0.308832 −0.154416 0.988006i $$-0.549350\pi$$
−0.154416 + 0.988006i $$0.549350\pi$$
$$972$$ 0 0
$$973$$ −21.0841 −0.675926
$$974$$ 0 0
$$975$$ 11.6288 0.372418
$$976$$ 0 0
$$977$$ 18.9858 0.607411 0.303705 0.952766i $$-0.401776\pi$$
0.303705 + 0.952766i $$0.401776\pi$$
$$978$$ 0 0
$$979$$ 33.3768 1.06673
$$980$$ 0 0
$$981$$ 10.0271 0.320139
$$982$$ 0 0
$$983$$ 4.33763 0.138349 0.0691744 0.997605i $$-0.477963\pi$$
0.0691744 + 0.997605i $$0.477963\pi$$
$$984$$ 0 0
$$985$$ 18.1385 0.577940
$$986$$ 0 0
$$987$$ −161.149 −5.12942
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 1.96766 0.0625049 0.0312524 0.999512i $$-0.490050\pi$$
0.0312524 + 0.999512i $$0.490050\pi$$
$$992$$ 0 0
$$993$$ −86.0495 −2.73070
$$994$$ 0 0
$$995$$ −23.2575 −0.737312
$$996$$ 0 0
$$997$$ −3.96110 −0.125449 −0.0627246 0.998031i $$-0.519979\pi$$
−0.0627246 + 0.998031i $$0.519979\pi$$
$$998$$ 0 0
$$999$$ 9.08669 0.287490
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.bz.1.3 3
4.3 odd 2 7360.2.a.ce.1.1 3
8.3 odd 2 1840.2.a.r.1.3 3
8.5 even 2 230.2.a.d.1.1 3
24.5 odd 2 2070.2.a.z.1.3 3
40.13 odd 4 1150.2.b.j.599.1 6
40.19 odd 2 9200.2.a.cf.1.1 3
40.29 even 2 1150.2.a.q.1.3 3
40.37 odd 4 1150.2.b.j.599.6 6
184.45 odd 2 5290.2.a.r.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 3 8.5 even 2
1150.2.a.q.1.3 3 40.29 even 2
1150.2.b.j.599.1 6 40.13 odd 4
1150.2.b.j.599.6 6 40.37 odd 4
1840.2.a.r.1.3 3 8.3 odd 2
2070.2.a.z.1.3 3 24.5 odd 2
5290.2.a.r.1.1 3 184.45 odd 2
7360.2.a.bz.1.3 3 1.1 even 1 trivial
7360.2.a.ce.1.1 3 4.3 odd 2
9200.2.a.cf.1.1 3 40.19 odd 2