# Properties

 Label 7800.2.a.bb.1.1 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$62.2833135766$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1560) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -3.37228 q^{7} +1.00000 q^{9} -3.37228 q^{11} +1.00000 q^{13} +1.37228 q^{17} -6.74456 q^{19} -3.37228 q^{21} +0.627719 q^{23} +1.00000 q^{27} -2.00000 q^{29} +6.74456 q^{31} -3.37228 q^{33} -5.37228 q^{37} +1.00000 q^{39} -1.37228 q^{41} +4.00000 q^{43} +1.25544 q^{47} +4.37228 q^{49} +1.37228 q^{51} +9.37228 q^{53} -6.74456 q^{57} -8.00000 q^{59} +8.11684 q^{61} -3.37228 q^{63} +4.00000 q^{67} +0.627719 q^{69} -11.3723 q^{71} +15.4891 q^{73} +11.3723 q^{77} -16.8614 q^{79} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{87} +13.3723 q^{89} -3.37228 q^{91} +6.74456 q^{93} -2.62772 q^{97} -3.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - q^7 + 2 * q^9 $$2 q + 2 q^{3} - q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{17} - 2 q^{19} - q^{21} + 7 q^{23} + 2 q^{27} - 4 q^{29} + 2 q^{31} - q^{33} - 5 q^{37} + 2 q^{39} + 3 q^{41} + 8 q^{43} + 14 q^{47} + 3 q^{49} - 3 q^{51} + 13 q^{53} - 2 q^{57} - 16 q^{59} - q^{61} - q^{63} + 8 q^{67} + 7 q^{69} - 17 q^{71} + 8 q^{73} + 17 q^{77} - 5 q^{79} + 2 q^{81} + 24 q^{83} - 4 q^{87} + 21 q^{89} - q^{91} + 2 q^{93} - 11 q^{97} - q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - q^7 + 2 * q^9 - q^11 + 2 * q^13 - 3 * q^17 - 2 * q^19 - q^21 + 7 * q^23 + 2 * q^27 - 4 * q^29 + 2 * q^31 - q^33 - 5 * q^37 + 2 * q^39 + 3 * q^41 + 8 * q^43 + 14 * q^47 + 3 * q^49 - 3 * q^51 + 13 * q^53 - 2 * q^57 - 16 * q^59 - q^61 - q^63 + 8 * q^67 + 7 * q^69 - 17 * q^71 + 8 * q^73 + 17 * q^77 - 5 * q^79 + 2 * q^81 + 24 * q^83 - 4 * q^87 + 21 * q^89 - q^91 + 2 * q^93 - 11 * q^97 - q^99

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.37228 −1.27460 −0.637301 0.770615i $$-0.719949\pi$$
−0.637301 + 0.770615i $$0.719949\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.37228 −1.01678 −0.508391 0.861127i $$-0.669759\pi$$
−0.508391 + 0.861127i $$0.669759\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.37228 0.332827 0.166414 0.986056i $$-0.446781\pi$$
0.166414 + 0.986056i $$0.446781\pi$$
$$18$$ 0 0
$$19$$ −6.74456 −1.54731 −0.773654 0.633608i $$-0.781573\pi$$
−0.773654 + 0.633608i $$0.781573\pi$$
$$20$$ 0 0
$$21$$ −3.37228 −0.735892
$$22$$ 0 0
$$23$$ 0.627719 0.130888 0.0654442 0.997856i $$-0.479154\pi$$
0.0654442 + 0.997856i $$0.479154\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 6.74456 1.21136 0.605680 0.795709i $$-0.292901\pi$$
0.605680 + 0.795709i $$0.292901\pi$$
$$32$$ 0 0
$$33$$ −3.37228 −0.587039
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.37228 −0.883198 −0.441599 0.897213i $$-0.645589\pi$$
−0.441599 + 0.897213i $$0.645589\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −1.37228 −0.214314 −0.107157 0.994242i $$-0.534175\pi$$
−0.107157 + 0.994242i $$0.534175\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.25544 0.183124 0.0915622 0.995799i $$-0.470814\pi$$
0.0915622 + 0.995799i $$0.470814\pi$$
$$48$$ 0 0
$$49$$ 4.37228 0.624612
$$50$$ 0 0
$$51$$ 1.37228 0.192158
$$52$$ 0 0
$$53$$ 9.37228 1.28738 0.643691 0.765286i $$-0.277402\pi$$
0.643691 + 0.765286i $$0.277402\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −6.74456 −0.893339
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 8.11684 1.03926 0.519628 0.854393i $$-0.326071\pi$$
0.519628 + 0.854393i $$0.326071\pi$$
$$62$$ 0 0
$$63$$ −3.37228 −0.424868
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 0.627719 0.0755684
$$70$$ 0 0
$$71$$ −11.3723 −1.34964 −0.674821 0.737982i $$-0.735779\pi$$
−0.674821 + 0.737982i $$0.735779\pi$$
$$72$$ 0 0
$$73$$ 15.4891 1.81286 0.906432 0.422351i $$-0.138795\pi$$
0.906432 + 0.422351i $$0.138795\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 11.3723 1.29599
$$78$$ 0 0
$$79$$ −16.8614 −1.89706 −0.948528 0.316693i $$-0.897428\pi$$
−0.948528 + 0.316693i $$0.897428\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ 13.3723 1.41746 0.708729 0.705480i $$-0.249269\pi$$
0.708729 + 0.705480i $$0.249269\pi$$
$$90$$ 0 0
$$91$$ −3.37228 −0.353511
$$92$$ 0 0
$$93$$ 6.74456 0.699379
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.62772 −0.266804 −0.133402 0.991062i $$-0.542590\pi$$
−0.133402 + 0.991062i $$0.542590\pi$$
$$98$$ 0 0
$$99$$ −3.37228 −0.338927
$$100$$ 0 0
$$101$$ 4.74456 0.472102 0.236051 0.971741i $$-0.424147\pi$$
0.236051 + 0.971741i $$0.424147\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.62772 0.834073 0.417037 0.908890i $$-0.363069\pi$$
0.417037 + 0.908890i $$0.363069\pi$$
$$108$$ 0 0
$$109$$ 8.74456 0.837577 0.418789 0.908084i $$-0.362455\pi$$
0.418789 + 0.908084i $$0.362455\pi$$
$$110$$ 0 0
$$111$$ −5.37228 −0.509914
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −4.62772 −0.424222
$$120$$ 0 0
$$121$$ 0.372281 0.0338438
$$122$$ 0 0
$$123$$ −1.37228 −0.123734
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.2337 1.44051 0.720253 0.693711i $$-0.244026\pi$$
0.720253 + 0.693711i $$0.244026\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −18.7446 −1.63772 −0.818860 0.573993i $$-0.805394\pi$$
−0.818860 + 0.573993i $$0.805394\pi$$
$$132$$ 0 0
$$133$$ 22.7446 1.97220
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −11.4891 −0.981582 −0.490791 0.871277i $$-0.663292\pi$$
−0.490791 + 0.871277i $$0.663292\pi$$
$$138$$ 0 0
$$139$$ 7.37228 0.625309 0.312654 0.949867i $$-0.398782\pi$$
0.312654 + 0.949867i $$0.398782\pi$$
$$140$$ 0 0
$$141$$ 1.25544 0.105727
$$142$$ 0 0
$$143$$ −3.37228 −0.282004
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.37228 0.360620
$$148$$ 0 0
$$149$$ 13.3723 1.09550 0.547750 0.836642i $$-0.315485\pi$$
0.547750 + 0.836642i $$0.315485\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 1.37228 0.110942
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ 9.37228 0.743270
$$160$$ 0 0
$$161$$ −2.11684 −0.166831
$$162$$ 0 0
$$163$$ 0.627719 0.0491667 0.0245834 0.999698i $$-0.492174\pi$$
0.0245834 + 0.999698i $$0.492174\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −6.74456 −0.515770
$$172$$ 0 0
$$173$$ −7.48913 −0.569388 −0.284694 0.958618i $$-0.591892\pi$$
−0.284694 + 0.958618i $$0.591892\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −8.00000 −0.601317
$$178$$ 0 0
$$179$$ 24.2337 1.81131 0.905655 0.424014i $$-0.139379\pi$$
0.905655 + 0.424014i $$0.139379\pi$$
$$180$$ 0 0
$$181$$ −5.37228 −0.399319 −0.199659 0.979865i $$-0.563984\pi$$
−0.199659 + 0.979865i $$0.563984\pi$$
$$182$$ 0 0
$$183$$ 8.11684 0.600014
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4.62772 −0.338412
$$188$$ 0 0
$$189$$ −3.37228 −0.245297
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ −9.37228 −0.674632 −0.337316 0.941392i $$-0.609519\pi$$
−0.337316 + 0.941392i $$0.609519\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 10.2337 0.729120 0.364560 0.931180i $$-0.381219\pi$$
0.364560 + 0.931180i $$0.381219\pi$$
$$198$$ 0 0
$$199$$ 21.4891 1.52332 0.761662 0.647975i $$-0.224384\pi$$
0.761662 + 0.647975i $$0.224384\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 6.74456 0.473375
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0.627719 0.0436295
$$208$$ 0 0
$$209$$ 22.7446 1.57327
$$210$$ 0 0
$$211$$ 22.9783 1.58189 0.790944 0.611889i $$-0.209590\pi$$
0.790944 + 0.611889i $$0.209590\pi$$
$$212$$ 0 0
$$213$$ −11.3723 −0.779216
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −22.7446 −1.54400
$$218$$ 0 0
$$219$$ 15.4891 1.04666
$$220$$ 0 0
$$221$$ 1.37228 0.0923096
$$222$$ 0 0
$$223$$ −18.9783 −1.27088 −0.635439 0.772151i $$-0.719181\pi$$
−0.635439 + 0.772151i $$0.719181\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 18.7446 1.24412 0.622060 0.782969i $$-0.286296\pi$$
0.622060 + 0.782969i $$0.286296\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 11.3723 0.748241
$$232$$ 0 0
$$233$$ 18.6277 1.22034 0.610171 0.792270i $$-0.291101\pi$$
0.610171 + 0.792270i $$0.291101\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −16.8614 −1.09527
$$238$$ 0 0
$$239$$ −22.3505 −1.44574 −0.722868 0.690986i $$-0.757176\pi$$
−0.722868 + 0.690986i $$0.757176\pi$$
$$240$$ 0 0
$$241$$ −24.9783 −1.60899 −0.804495 0.593959i $$-0.797564\pi$$
−0.804495 + 0.593959i $$0.797564\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.74456 −0.429146
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 17.4891 1.10390 0.551952 0.833876i $$-0.313883\pi$$
0.551952 + 0.833876i $$0.313883\pi$$
$$252$$ 0 0
$$253$$ −2.11684 −0.133085
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −10.0000 −0.623783 −0.311891 0.950118i $$-0.600963\pi$$
−0.311891 + 0.950118i $$0.600963\pi$$
$$258$$ 0 0
$$259$$ 18.1168 1.12573
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ 17.4891 1.07843 0.539213 0.842170i $$-0.318722\pi$$
0.539213 + 0.842170i $$0.318722\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 13.3723 0.818370
$$268$$ 0 0
$$269$$ 12.7446 0.777050 0.388525 0.921438i $$-0.372985\pi$$
0.388525 + 0.921438i $$0.372985\pi$$
$$270$$ 0 0
$$271$$ 17.2554 1.04819 0.524097 0.851659i $$-0.324403\pi$$
0.524097 + 0.851659i $$0.324403\pi$$
$$272$$ 0 0
$$273$$ −3.37228 −0.204100
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 24.9783 1.50080 0.750399 0.660985i $$-0.229861\pi$$
0.750399 + 0.660985i $$0.229861\pi$$
$$278$$ 0 0
$$279$$ 6.74456 0.403786
$$280$$ 0 0
$$281$$ 12.9783 0.774218 0.387109 0.922034i $$-0.373474\pi$$
0.387109 + 0.922034i $$0.373474\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 4.62772 0.273166
$$288$$ 0 0
$$289$$ −15.1168 −0.889226
$$290$$ 0 0
$$291$$ −2.62772 −0.154040
$$292$$ 0 0
$$293$$ 15.2554 0.891232 0.445616 0.895224i $$-0.352985\pi$$
0.445616 + 0.895224i $$0.352985\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.37228 −0.195680
$$298$$ 0 0
$$299$$ 0.627719 0.0363019
$$300$$ 0 0
$$301$$ −13.4891 −0.777500
$$302$$ 0 0
$$303$$ 4.74456 0.272568
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −15.3723 −0.877342 −0.438671 0.898648i $$-0.644551\pi$$
−0.438671 + 0.898648i $$0.644551\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 18.2337 1.02411 0.512053 0.858954i $$-0.328885\pi$$
0.512053 + 0.858954i $$0.328885\pi$$
$$318$$ 0 0
$$319$$ 6.74456 0.377623
$$320$$ 0 0
$$321$$ 8.62772 0.481552
$$322$$ 0 0
$$323$$ −9.25544 −0.514986
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 8.74456 0.483575
$$328$$ 0 0
$$329$$ −4.23369 −0.233411
$$330$$ 0 0
$$331$$ −21.4891 −1.18115 −0.590575 0.806983i $$-0.701099\pi$$
−0.590575 + 0.806983i $$0.701099\pi$$
$$332$$ 0 0
$$333$$ −5.37228 −0.294399
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 18.2337 0.993252 0.496626 0.867965i $$-0.334572\pi$$
0.496626 + 0.867965i $$0.334572\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −22.7446 −1.23169
$$342$$ 0 0
$$343$$ 8.86141 0.478471
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −20.8614 −1.11990 −0.559949 0.828527i $$-0.689179\pi$$
−0.559949 + 0.828527i $$0.689179\pi$$
$$348$$ 0 0
$$349$$ −2.23369 −0.119567 −0.0597833 0.998211i $$-0.519041\pi$$
−0.0597833 + 0.998211i $$0.519041\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4.62772 −0.244925
$$358$$ 0 0
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 26.4891 1.39416
$$362$$ 0 0
$$363$$ 0.372281 0.0195397
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 9.48913 0.495328 0.247664 0.968846i $$-0.420337\pi$$
0.247664 + 0.968846i $$0.420337\pi$$
$$368$$ 0 0
$$369$$ −1.37228 −0.0714381
$$370$$ 0 0
$$371$$ −31.6060 −1.64090
$$372$$ 0 0
$$373$$ 15.2554 0.789897 0.394948 0.918703i $$-0.370763\pi$$
0.394948 + 0.918703i $$0.370763\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −2.00000 −0.103005
$$378$$ 0 0
$$379$$ 14.7446 0.757377 0.378689 0.925524i $$-0.376375\pi$$
0.378689 + 0.925524i $$0.376375\pi$$
$$380$$ 0 0
$$381$$ 16.2337 0.831677
$$382$$ 0 0
$$383$$ −9.25544 −0.472931 −0.236465 0.971640i $$-0.575989\pi$$
−0.236465 + 0.971640i $$0.575989\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ 0 0
$$389$$ 26.2337 1.33010 0.665050 0.746798i $$-0.268410\pi$$
0.665050 + 0.746798i $$0.268410\pi$$
$$390$$ 0 0
$$391$$ 0.861407 0.0435632
$$392$$ 0 0
$$393$$ −18.7446 −0.945538
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −24.3505 −1.22212 −0.611059 0.791585i $$-0.709256\pi$$
−0.611059 + 0.791585i $$0.709256\pi$$
$$398$$ 0 0
$$399$$ 22.7446 1.13865
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ 6.74456 0.335971
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 18.1168 0.898019
$$408$$ 0 0
$$409$$ 12.9783 0.641733 0.320867 0.947124i $$-0.396026\pi$$
0.320867 + 0.947124i $$0.396026\pi$$
$$410$$ 0 0
$$411$$ −11.4891 −0.566717
$$412$$ 0 0
$$413$$ 26.9783 1.32751
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 7.37228 0.361022
$$418$$ 0 0
$$419$$ −32.2337 −1.57472 −0.787359 0.616494i $$-0.788552\pi$$
−0.787359 + 0.616494i $$0.788552\pi$$
$$420$$ 0 0
$$421$$ 27.7228 1.35113 0.675564 0.737302i $$-0.263900\pi$$
0.675564 + 0.737302i $$0.263900\pi$$
$$422$$ 0 0
$$423$$ 1.25544 0.0610415
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −27.3723 −1.32464
$$428$$ 0 0
$$429$$ −3.37228 −0.162815
$$430$$ 0 0
$$431$$ 18.9783 0.914150 0.457075 0.889428i $$-0.348897\pi$$
0.457075 + 0.889428i $$0.348897\pi$$
$$432$$ 0 0
$$433$$ −6.23369 −0.299572 −0.149786 0.988718i $$-0.547858\pi$$
−0.149786 + 0.988718i $$0.547858\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.23369 −0.202525
$$438$$ 0 0
$$439$$ 0.394031 0.0188061 0.00940303 0.999956i $$-0.497007\pi$$
0.00940303 + 0.999956i $$0.497007\pi$$
$$440$$ 0 0
$$441$$ 4.37228 0.208204
$$442$$ 0 0
$$443$$ −18.3505 −0.871860 −0.435930 0.899981i $$-0.643580\pi$$
−0.435930 + 0.899981i $$0.643580\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 13.3723 0.632487
$$448$$ 0 0
$$449$$ −1.37228 −0.0647620 −0.0323810 0.999476i $$-0.510309\pi$$
−0.0323810 + 0.999476i $$0.510309\pi$$
$$450$$ 0 0
$$451$$ 4.62772 0.217911
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −40.1168 −1.87659 −0.938293 0.345840i $$-0.887594\pi$$
−0.938293 + 0.345840i $$0.887594\pi$$
$$458$$ 0 0
$$459$$ 1.37228 0.0640526
$$460$$ 0 0
$$461$$ −20.3505 −0.947819 −0.473909 0.880574i $$-0.657158\pi$$
−0.473909 + 0.880574i $$0.657158\pi$$
$$462$$ 0 0
$$463$$ 31.6060 1.46885 0.734427 0.678688i $$-0.237451\pi$$
0.734427 + 0.678688i $$0.237451\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −10.3505 −0.478965 −0.239483 0.970901i $$-0.576978\pi$$
−0.239483 + 0.970901i $$0.576978\pi$$
$$468$$ 0 0
$$469$$ −13.4891 −0.622870
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ −13.4891 −0.620231
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 9.37228 0.429127
$$478$$ 0 0
$$479$$ 13.8832 0.634338 0.317169 0.948369i $$-0.397268\pi$$
0.317169 + 0.948369i $$0.397268\pi$$
$$480$$ 0 0
$$481$$ −5.37228 −0.244955
$$482$$ 0 0
$$483$$ −2.11684 −0.0963197
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −2.11684 −0.0959234 −0.0479617 0.998849i $$-0.515273\pi$$
−0.0479617 + 0.998849i $$0.515273\pi$$
$$488$$ 0 0
$$489$$ 0.627719 0.0283864
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −2.74456 −0.123609
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 38.3505 1.72026
$$498$$ 0 0
$$499$$ −14.7446 −0.660057 −0.330029 0.943971i $$-0.607058\pi$$
−0.330029 + 0.943971i $$0.607058\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 22.9783 1.02455 0.512275 0.858822i $$-0.328803\pi$$
0.512275 + 0.858822i $$0.328803\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ 34.8614 1.54520 0.772602 0.634890i $$-0.218955\pi$$
0.772602 + 0.634890i $$0.218955\pi$$
$$510$$ 0 0
$$511$$ −52.2337 −2.31068
$$512$$ 0 0
$$513$$ −6.74456 −0.297780
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −4.23369 −0.186197
$$518$$ 0 0
$$519$$ −7.48913 −0.328736
$$520$$ 0 0
$$521$$ 23.4891 1.02908 0.514539 0.857467i $$-0.327963\pi$$
0.514539 + 0.857467i $$0.327963\pi$$
$$522$$ 0 0
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 9.25544 0.403173
$$528$$ 0 0
$$529$$ −22.6060 −0.982868
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 0 0
$$533$$ −1.37228 −0.0594401
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 24.2337 1.04576
$$538$$ 0 0
$$539$$ −14.7446 −0.635093
$$540$$ 0 0
$$541$$ 23.4891 1.00988 0.504938 0.863156i $$-0.331515\pi$$
0.504938 + 0.863156i $$0.331515\pi$$
$$542$$ 0 0
$$543$$ −5.37228 −0.230547
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −6.97825 −0.298368 −0.149184 0.988809i $$-0.547665\pi$$
−0.149184 + 0.988809i $$0.547665\pi$$
$$548$$ 0 0
$$549$$ 8.11684 0.346418
$$550$$ 0 0
$$551$$ 13.4891 0.574656
$$552$$ 0 0
$$553$$ 56.8614 2.41799
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −12.9783 −0.549906 −0.274953 0.961458i $$-0.588662\pi$$
−0.274953 + 0.961458i $$0.588662\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ −4.62772 −0.195382
$$562$$ 0 0
$$563$$ −1.88316 −0.0793656 −0.0396828 0.999212i $$-0.512635\pi$$
−0.0396828 + 0.999212i $$0.512635\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −3.37228 −0.141623
$$568$$ 0 0
$$569$$ −23.7228 −0.994512 −0.497256 0.867604i $$-0.665659\pi$$
−0.497256 + 0.867604i $$0.665659\pi$$
$$570$$ 0 0
$$571$$ −34.3505 −1.43753 −0.718763 0.695256i $$-0.755291\pi$$
−0.718763 + 0.695256i $$0.755291\pi$$
$$572$$ 0 0
$$573$$ −8.00000 −0.334205
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −40.1168 −1.67009 −0.835043 0.550185i $$-0.814557\pi$$
−0.835043 + 0.550185i $$0.814557\pi$$
$$578$$ 0 0
$$579$$ −9.37228 −0.389499
$$580$$ 0 0
$$581$$ −40.4674 −1.67887
$$582$$ 0 0
$$583$$ −31.6060 −1.30899
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 2.74456 0.113280 0.0566401 0.998395i $$-0.481961\pi$$
0.0566401 + 0.998395i $$0.481961\pi$$
$$588$$ 0 0
$$589$$ −45.4891 −1.87435
$$590$$ 0 0
$$591$$ 10.2337 0.420958
$$592$$ 0 0
$$593$$ 11.2554 0.462205 0.231103 0.972929i $$-0.425767\pi$$
0.231103 + 0.972929i $$0.425767\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 21.4891 0.879491
$$598$$ 0 0
$$599$$ 5.48913 0.224280 0.112140 0.993692i $$-0.464230\pi$$
0.112140 + 0.993692i $$0.464230\pi$$
$$600$$ 0 0
$$601$$ −37.6060 −1.53398 −0.766990 0.641659i $$-0.778246\pi$$
−0.766990 + 0.641659i $$0.778246\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ 0 0
$$609$$ 6.74456 0.273303
$$610$$ 0 0
$$611$$ 1.25544 0.0507896
$$612$$ 0 0
$$613$$ 14.8614 0.600247 0.300123 0.953900i $$-0.402972\pi$$
0.300123 + 0.953900i $$0.402972\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −46.4674 −1.87071 −0.935353 0.353716i $$-0.884918\pi$$
−0.935353 + 0.353716i $$0.884918\pi$$
$$618$$ 0 0
$$619$$ 41.7228 1.67698 0.838491 0.544916i $$-0.183438\pi$$
0.838491 + 0.544916i $$0.183438\pi$$
$$620$$ 0 0
$$621$$ 0.627719 0.0251895
$$622$$ 0 0
$$623$$ −45.0951 −1.80670
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 22.7446 0.908330
$$628$$ 0 0
$$629$$ −7.37228 −0.293952
$$630$$ 0 0
$$631$$ −18.5109 −0.736906 −0.368453 0.929646i $$-0.620113\pi$$
−0.368453 + 0.929646i $$0.620113\pi$$
$$632$$ 0 0
$$633$$ 22.9783 0.913303
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4.37228 0.173236
$$638$$ 0 0
$$639$$ −11.3723 −0.449880
$$640$$ 0 0
$$641$$ 22.2337 0.878178 0.439089 0.898444i $$-0.355301\pi$$
0.439089 + 0.898444i $$0.355301\pi$$
$$642$$ 0 0
$$643$$ 6.11684 0.241225 0.120612 0.992700i $$-0.461514\pi$$
0.120612 + 0.992700i $$0.461514\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 4.86141 0.191122 0.0955608 0.995424i $$-0.469536\pi$$
0.0955608 + 0.995424i $$0.469536\pi$$
$$648$$ 0 0
$$649$$ 26.9783 1.05899
$$650$$ 0 0
$$651$$ −22.7446 −0.891430
$$652$$ 0 0
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 15.4891 0.604288
$$658$$ 0 0
$$659$$ −14.5109 −0.565263 −0.282632 0.959229i $$-0.591207\pi$$
−0.282632 + 0.959229i $$0.591207\pi$$
$$660$$ 0 0
$$661$$ −34.2337 −1.33154 −0.665768 0.746159i $$-0.731896\pi$$
−0.665768 + 0.746159i $$0.731896\pi$$
$$662$$ 0 0
$$663$$ 1.37228 0.0532950
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1.25544 −0.0486107
$$668$$ 0 0
$$669$$ −18.9783 −0.733742
$$670$$ 0 0
$$671$$ −27.3723 −1.05670
$$672$$ 0 0
$$673$$ 8.51087 0.328070 0.164035 0.986455i $$-0.447549\pi$$
0.164035 + 0.986455i $$0.447549\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 17.3723 0.667671 0.333836 0.942631i $$-0.391657\pi$$
0.333836 + 0.942631i $$0.391657\pi$$
$$678$$ 0 0
$$679$$ 8.86141 0.340070
$$680$$ 0 0
$$681$$ 18.7446 0.718293
$$682$$ 0 0
$$683$$ −1.02175 −0.0390962 −0.0195481 0.999809i $$-0.506223\pi$$
−0.0195481 + 0.999809i $$0.506223\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 2.00000 0.0763048
$$688$$ 0 0
$$689$$ 9.37228 0.357055
$$690$$ 0 0
$$691$$ −33.7228 −1.28288 −0.641438 0.767175i $$-0.721662\pi$$
−0.641438 + 0.767175i $$0.721662\pi$$
$$692$$ 0 0
$$693$$ 11.3723 0.431997
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −1.88316 −0.0713296
$$698$$ 0 0
$$699$$ 18.6277 0.704565
$$700$$ 0 0
$$701$$ −16.7446 −0.632433 −0.316217 0.948687i $$-0.602413\pi$$
−0.316217 + 0.948687i $$0.602413\pi$$
$$702$$ 0 0
$$703$$ 36.2337 1.36658
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −16.0000 −0.601742
$$708$$ 0 0
$$709$$ −39.2554 −1.47427 −0.737134 0.675746i $$-0.763822\pi$$
−0.737134 + 0.675746i $$0.763822\pi$$
$$710$$ 0 0
$$711$$ −16.8614 −0.632352
$$712$$ 0 0
$$713$$ 4.23369 0.158553
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −22.3505 −0.834696
$$718$$ 0 0
$$719$$ 50.9783 1.90117 0.950584 0.310468i $$-0.100486\pi$$
0.950584 + 0.310468i $$0.100486\pi$$
$$720$$ 0 0
$$721$$ −13.4891 −0.502361
$$722$$ 0 0
$$723$$ −24.9783 −0.928951
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 10.7446 0.398494 0.199247 0.979949i $$-0.436150\pi$$
0.199247 + 0.979949i $$0.436150\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 5.48913 0.203023
$$732$$ 0 0
$$733$$ 43.0951 1.59175 0.795877 0.605459i $$-0.207010\pi$$
0.795877 + 0.605459i $$0.207010\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −13.4891 −0.496878
$$738$$ 0 0
$$739$$ 28.2337 1.03859 0.519296 0.854594i $$-0.326194\pi$$
0.519296 + 0.854594i $$0.326194\pi$$
$$740$$ 0 0
$$741$$ −6.74456 −0.247768
$$742$$ 0 0
$$743$$ 12.2337 0.448810 0.224405 0.974496i $$-0.427956\pi$$
0.224405 + 0.974496i $$0.427956\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ −29.0951 −1.06311
$$750$$ 0 0
$$751$$ 8.86141 0.323357 0.161679 0.986843i $$-0.448309\pi$$
0.161679 + 0.986843i $$0.448309\pi$$
$$752$$ 0 0
$$753$$ 17.4891 0.637339
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 39.2554 1.42676 0.713382 0.700776i $$-0.247163\pi$$
0.713382 + 0.700776i $$0.247163\pi$$
$$758$$ 0 0
$$759$$ −2.11684 −0.0768366
$$760$$ 0 0
$$761$$ −0.510875 −0.0185192 −0.00925960 0.999957i $$-0.502947\pi$$
−0.00925960 + 0.999957i $$0.502947\pi$$
$$762$$ 0 0
$$763$$ −29.4891 −1.06758
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ 6.23369 0.224793 0.112396 0.993663i $$-0.464147\pi$$
0.112396 + 0.993663i $$0.464147\pi$$
$$770$$ 0 0
$$771$$ −10.0000 −0.360141
$$772$$ 0 0
$$773$$ −16.7446 −0.602260 −0.301130 0.953583i $$-0.597364\pi$$
−0.301130 + 0.953583i $$0.597364\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 18.1168 0.649938
$$778$$ 0 0
$$779$$ 9.25544 0.331610
$$780$$ 0 0
$$781$$ 38.3505 1.37229
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 6.97825 0.248748 0.124374 0.992235i $$-0.460308\pi$$
0.124374 + 0.992235i $$0.460308\pi$$
$$788$$ 0 0
$$789$$ 17.4891 0.622629
$$790$$ 0 0
$$791$$ −20.2337 −0.719427
$$792$$ 0 0
$$793$$ 8.11684 0.288238
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 10.6277 0.376453 0.188227 0.982126i $$-0.439726\pi$$
0.188227 + 0.982126i $$0.439726\pi$$
$$798$$ 0 0
$$799$$ 1.72281 0.0609488
$$800$$ 0 0
$$801$$ 13.3723 0.472486
$$802$$ 0 0
$$803$$ −52.2337 −1.84329
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 12.7446 0.448630
$$808$$ 0 0
$$809$$ 34.4674 1.21181 0.605904 0.795538i $$-0.292811\pi$$
0.605904 + 0.795538i $$0.292811\pi$$
$$810$$ 0 0
$$811$$ 29.4891 1.03550 0.517752 0.855531i $$-0.326769\pi$$
0.517752 + 0.855531i $$0.326769\pi$$
$$812$$ 0 0
$$813$$ 17.2554 0.605175
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −26.9783 −0.943850
$$818$$ 0 0
$$819$$ −3.37228 −0.117837
$$820$$ 0 0
$$821$$ 39.0951 1.36443 0.682214 0.731152i $$-0.261017\pi$$
0.682214 + 0.731152i $$0.261017\pi$$
$$822$$ 0 0
$$823$$ 44.0000 1.53374 0.766872 0.641800i $$-0.221812\pi$$
0.766872 + 0.641800i $$0.221812\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 32.2337 1.12088 0.560438 0.828197i $$-0.310633\pi$$
0.560438 + 0.828197i $$0.310633\pi$$
$$828$$ 0 0
$$829$$ −20.9783 −0.728605 −0.364302 0.931281i $$-0.618693\pi$$
−0.364302 + 0.931281i $$0.618693\pi$$
$$830$$ 0 0
$$831$$ 24.9783 0.866486
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 6.74456 0.233126
$$838$$ 0 0
$$839$$ −11.3723 −0.392615 −0.196307 0.980542i $$-0.562895\pi$$
−0.196307 + 0.980542i $$0.562895\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 12.9783 0.446995
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −1.25544 −0.0431373
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ −3.37228 −0.115600
$$852$$ 0 0
$$853$$ 3.88316 0.132957 0.0664784 0.997788i $$-0.478824\pi$$
0.0664784 + 0.997788i $$0.478824\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 21.6060 0.738046 0.369023 0.929420i $$-0.379692\pi$$
0.369023 + 0.929420i $$0.379692\pi$$
$$858$$ 0 0
$$859$$ −30.1168 −1.02757 −0.513787 0.857918i $$-0.671758\pi$$
−0.513787 + 0.857918i $$0.671758\pi$$
$$860$$ 0 0
$$861$$ 4.62772 0.157712
$$862$$ 0 0
$$863$$ −46.7446 −1.59120 −0.795602 0.605820i $$-0.792845\pi$$
−0.795602 + 0.605820i $$0.792845\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −15.1168 −0.513395
$$868$$ 0 0
$$869$$ 56.8614 1.92889
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ −2.62772 −0.0889348
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12.9783 −0.438244 −0.219122 0.975697i $$-0.570319\pi$$
−0.219122 + 0.975697i $$0.570319\pi$$
$$878$$ 0 0
$$879$$ 15.2554 0.514553
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ 6.97825 0.234837 0.117418 0.993083i $$-0.462538\pi$$
0.117418 + 0.993083i $$0.462538\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 50.3505 1.69061 0.845303 0.534288i $$-0.179420\pi$$
0.845303 + 0.534288i $$0.179420\pi$$
$$888$$ 0 0
$$889$$ −54.7446 −1.83607
$$890$$ 0 0
$$891$$ −3.37228 −0.112976
$$892$$ 0 0
$$893$$ −8.46738 −0.283350
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0.627719 0.0209589
$$898$$ 0 0
$$899$$ −13.4891 −0.449888
$$900$$ 0 0
$$901$$ 12.8614 0.428476
$$902$$ 0 0
$$903$$ −13.4891 −0.448890
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −37.2554 −1.23705 −0.618523 0.785767i $$-0.712269\pi$$
−0.618523 + 0.785767i $$0.712269\pi$$
$$908$$ 0 0
$$909$$ 4.74456 0.157367
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ −40.4674 −1.33927
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 63.2119 2.08744
$$918$$ 0 0
$$919$$ −48.8614 −1.61179 −0.805895 0.592059i $$-0.798315\pi$$
−0.805895 + 0.592059i $$0.798315\pi$$
$$920$$ 0 0
$$921$$ −15.3723 −0.506534
$$922$$ 0 0
$$923$$ −11.3723 −0.374323
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ 10.3940 0.341017 0.170509 0.985356i $$-0.445459\pi$$
0.170509 + 0.985356i $$0.445459\pi$$
$$930$$ 0 0
$$931$$ −29.4891 −0.966467
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3.48913 0.113985 0.0569924 0.998375i $$-0.481849\pi$$
0.0569924 + 0.998375i $$0.481849\pi$$
$$938$$ 0 0
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 37.3723 1.21830 0.609151 0.793054i $$-0.291510\pi$$
0.609151 + 0.793054i $$0.291510\pi$$
$$942$$ 0 0
$$943$$ −0.861407 −0.0280513
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 46.9783 1.52659 0.763294 0.646051i $$-0.223581\pi$$
0.763294 + 0.646051i $$0.223581\pi$$
$$948$$ 0 0
$$949$$ 15.4891 0.502798
$$950$$ 0 0
$$951$$ 18.2337 0.591268
$$952$$ 0 0
$$953$$ 60.3505 1.95495 0.977473 0.211062i $$-0.0676921\pi$$
0.977473 + 0.211062i $$0.0676921\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6.74456 0.218021
$$958$$ 0 0
$$959$$ 38.7446 1.25113
$$960$$ 0 0
$$961$$ 14.4891 0.467391
$$962$$ 0 0
$$963$$ 8.62772 0.278024
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −34.9783 −1.12482 −0.562412 0.826857i $$-0.690127\pi$$
−0.562412 + 0.826857i $$0.690127\pi$$
$$968$$ 0 0
$$969$$ −9.25544 −0.297327
$$970$$ 0 0
$$971$$ 6.51087 0.208944 0.104472 0.994528i $$-0.466685\pi$$
0.104472 + 0.994528i $$0.466685\pi$$
$$972$$ 0 0
$$973$$ −24.8614 −0.797020
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −60.7446 −1.94339 −0.971695 0.236237i $$-0.924086\pi$$
−0.971695 + 0.236237i $$0.924086\pi$$
$$978$$ 0 0
$$979$$ −45.0951 −1.44125
$$980$$ 0 0
$$981$$ 8.74456 0.279192
$$982$$ 0 0
$$983$$ −5.48913 −0.175076 −0.0875380 0.996161i $$-0.527900\pi$$
−0.0875380 + 0.996161i $$0.527900\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −4.23369 −0.134760
$$988$$ 0 0
$$989$$ 2.51087 0.0798412
$$990$$ 0 0
$$991$$ −15.6060 −0.495740 −0.247870 0.968793i $$-0.579731\pi$$
−0.247870 + 0.968793i $$0.579731\pi$$
$$992$$ 0 0
$$993$$ −21.4891 −0.681937
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 23.7228 0.751309 0.375655 0.926760i $$-0.377418\pi$$
0.375655 + 0.926760i $$0.377418\pi$$
$$998$$ 0 0
$$999$$ −5.37228 −0.169971
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 7800.2.a.bb.1.1 2
5.4 even 2 1560.2.a.n.1.2 2
15.14 odd 2 4680.2.a.bf.1.2 2
20.19 odd 2 3120.2.a.bd.1.1 2
60.59 even 2 9360.2.a.co.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.n.1.2 2 5.4 even 2
3120.2.a.bd.1.1 2 20.19 odd 2
4680.2.a.bf.1.2 2 15.14 odd 2
7800.2.a.bb.1.1 2 1.1 even 1 trivial
9360.2.a.co.1.1 2 60.59 even 2