# Properties

 Label 7728.2.a.bf.1.2 Level $7728$ Weight $2$ Character 7728.1 Self dual yes Analytic conductor $61.708$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$61.7083906820$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1932) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 7728.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.60555 q^{11} +2.30278 q^{13} -0.697224 q^{15} -0.394449 q^{17} +0.394449 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.51388 q^{25} +1.00000 q^{27} -5.60555 q^{29} +3.60555 q^{31} -5.60555 q^{33} -0.697224 q^{35} +5.60555 q^{37} +2.30278 q^{39} +3.60555 q^{41} -4.30278 q^{43} -0.697224 q^{45} +4.60555 q^{47} +1.00000 q^{49} -0.394449 q^{51} +5.90833 q^{53} +3.90833 q^{55} +0.394449 q^{57} -3.90833 q^{59} +4.90833 q^{61} +1.00000 q^{63} -1.60555 q^{65} -13.3028 q^{67} +1.00000 q^{69} -12.9083 q^{71} +11.0000 q^{73} -4.51388 q^{75} -5.60555 q^{77} -13.4222 q^{79} +1.00000 q^{81} -16.2111 q^{83} +0.275019 q^{85} -5.60555 q^{87} -15.5139 q^{89} +2.30278 q^{91} +3.60555 q^{93} -0.275019 q^{95} +3.78890 q^{97} -5.60555 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} + q^{13} - 5 q^{15} - 8 q^{17} + 8 q^{19} + 2 q^{21} + 2 q^{23} + 9 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{33} - 5 q^{35} + 4 q^{37} + q^{39} - 5 q^{43} - 5 q^{45} + 2 q^{47} + 2 q^{49} - 8 q^{51} + q^{53} - 3 q^{55} + 8 q^{57} + 3 q^{59} - q^{61} + 2 q^{63} + 4 q^{65} - 23 q^{67} + 2 q^{69} - 15 q^{71} + 22 q^{73} + 9 q^{75} - 4 q^{77} + 2 q^{79} + 2 q^{81} - 18 q^{83} + 33 q^{85} - 4 q^{87} - 13 q^{89} + q^{91} - 33 q^{95} + 22 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 + q^13 - 5 * q^15 - 8 * q^17 + 8 * q^19 + 2 * q^21 + 2 * q^23 + 9 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^33 - 5 * q^35 + 4 * q^37 + q^39 - 5 * q^43 - 5 * q^45 + 2 * q^47 + 2 * q^49 - 8 * q^51 + q^53 - 3 * q^55 + 8 * q^57 + 3 * q^59 - q^61 + 2 * q^63 + 4 * q^65 - 23 * q^67 + 2 * q^69 - 15 * q^71 + 22 * q^73 + 9 * q^75 - 4 * q^77 + 2 * q^79 + 2 * q^81 - 18 * q^83 + 33 * q^85 - 4 * q^87 - 13 * q^89 + q^91 - 33 * q^95 + 22 * q^97 - 4 * 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
$$5$$ −0.697224 −0.311808 −0.155904 0.987772i $$-0.549829\pi$$
−0.155904 + 0.987772i $$0.549829\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.60555 −1.69014 −0.845069 0.534658i $$-0.820441\pi$$
−0.845069 + 0.534658i $$0.820441\pi$$
$$12$$ 0 0
$$13$$ 2.30278 0.638675 0.319338 0.947641i $$-0.396540\pi$$
0.319338 + 0.947641i $$0.396540\pi$$
$$14$$ 0 0
$$15$$ −0.697224 −0.180023
$$16$$ 0 0
$$17$$ −0.394449 −0.0956679 −0.0478339 0.998855i $$-0.515232\pi$$
−0.0478339 + 0.998855i $$0.515232\pi$$
$$18$$ 0 0
$$19$$ 0.394449 0.0904927 0.0452464 0.998976i $$-0.485593\pi$$
0.0452464 + 0.998976i $$0.485593\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ −4.51388 −0.902776
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −5.60555 −1.04092 −0.520462 0.853885i $$-0.674240\pi$$
−0.520462 + 0.853885i $$0.674240\pi$$
$$30$$ 0 0
$$31$$ 3.60555 0.647576 0.323788 0.946130i $$-0.395044\pi$$
0.323788 + 0.946130i $$0.395044\pi$$
$$32$$ 0 0
$$33$$ −5.60555 −0.975801
$$34$$ 0 0
$$35$$ −0.697224 −0.117852
$$36$$ 0 0
$$37$$ 5.60555 0.921547 0.460773 0.887518i $$-0.347572\pi$$
0.460773 + 0.887518i $$0.347572\pi$$
$$38$$ 0 0
$$39$$ 2.30278 0.368739
$$40$$ 0 0
$$41$$ 3.60555 0.563093 0.281546 0.959548i $$-0.409153\pi$$
0.281546 + 0.959548i $$0.409153\pi$$
$$42$$ 0 0
$$43$$ −4.30278 −0.656167 −0.328084 0.944649i $$-0.606403\pi$$
−0.328084 + 0.944649i $$0.606403\pi$$
$$44$$ 0 0
$$45$$ −0.697224 −0.103936
$$46$$ 0 0
$$47$$ 4.60555 0.671789 0.335894 0.941900i $$-0.390961\pi$$
0.335894 + 0.941900i $$0.390961\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −0.394449 −0.0552339
$$52$$ 0 0
$$53$$ 5.90833 0.811571 0.405786 0.913968i $$-0.366998\pi$$
0.405786 + 0.913968i $$0.366998\pi$$
$$54$$ 0 0
$$55$$ 3.90833 0.526999
$$56$$ 0 0
$$57$$ 0.394449 0.0522460
$$58$$ 0 0
$$59$$ −3.90833 −0.508821 −0.254410 0.967096i $$-0.581881\pi$$
−0.254410 + 0.967096i $$0.581881\pi$$
$$60$$ 0 0
$$61$$ 4.90833 0.628447 0.314223 0.949349i $$-0.398256\pi$$
0.314223 + 0.949349i $$0.398256\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −1.60555 −0.199144
$$66$$ 0 0
$$67$$ −13.3028 −1.62519 −0.812596 0.582827i $$-0.801947\pi$$
−0.812596 + 0.582827i $$0.801947\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −12.9083 −1.53194 −0.765968 0.642878i $$-0.777740\pi$$
−0.765968 + 0.642878i $$0.777740\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 0 0
$$75$$ −4.51388 −0.521218
$$76$$ 0 0
$$77$$ −5.60555 −0.638812
$$78$$ 0 0
$$79$$ −13.4222 −1.51012 −0.755058 0.655658i $$-0.772391\pi$$
−0.755058 + 0.655658i $$0.772391\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −16.2111 −1.77940 −0.889700 0.456546i $$-0.849086\pi$$
−0.889700 + 0.456546i $$0.849086\pi$$
$$84$$ 0 0
$$85$$ 0.275019 0.0298300
$$86$$ 0 0
$$87$$ −5.60555 −0.600978
$$88$$ 0 0
$$89$$ −15.5139 −1.64447 −0.822234 0.569150i $$-0.807272\pi$$
−0.822234 + 0.569150i $$0.807272\pi$$
$$90$$ 0 0
$$91$$ 2.30278 0.241396
$$92$$ 0 0
$$93$$ 3.60555 0.373878
$$94$$ 0 0
$$95$$ −0.275019 −0.0282164
$$96$$ 0 0
$$97$$ 3.78890 0.384704 0.192352 0.981326i $$-0.438388\pi$$
0.192352 + 0.981326i $$0.438388\pi$$
$$98$$ 0 0
$$99$$ −5.60555 −0.563379
$$100$$ 0 0
$$101$$ −12.5139 −1.24518 −0.622589 0.782549i $$-0.713919\pi$$
−0.622589 + 0.782549i $$0.713919\pi$$
$$102$$ 0 0
$$103$$ −2.78890 −0.274798 −0.137399 0.990516i $$-0.543874\pi$$
−0.137399 + 0.990516i $$0.543874\pi$$
$$104$$ 0 0
$$105$$ −0.697224 −0.0680421
$$106$$ 0 0
$$107$$ −0.302776 −0.0292704 −0.0146352 0.999893i $$-0.504659\pi$$
−0.0146352 + 0.999893i $$0.504659\pi$$
$$108$$ 0 0
$$109$$ −4.51388 −0.432351 −0.216176 0.976355i $$-0.569358\pi$$
−0.216176 + 0.976355i $$0.569358\pi$$
$$110$$ 0 0
$$111$$ 5.60555 0.532055
$$112$$ 0 0
$$113$$ 5.30278 0.498843 0.249422 0.968395i $$-0.419760\pi$$
0.249422 + 0.968395i $$0.419760\pi$$
$$114$$ 0 0
$$115$$ −0.697224 −0.0650165
$$116$$ 0 0
$$117$$ 2.30278 0.212892
$$118$$ 0 0
$$119$$ −0.394449 −0.0361591
$$120$$ 0 0
$$121$$ 20.4222 1.85656
$$122$$ 0 0
$$123$$ 3.60555 0.325102
$$124$$ 0 0
$$125$$ 6.63331 0.593301
$$126$$ 0 0
$$127$$ −18.1194 −1.60784 −0.803920 0.594738i $$-0.797256\pi$$
−0.803920 + 0.594738i $$0.797256\pi$$
$$128$$ 0 0
$$129$$ −4.30278 −0.378838
$$130$$ 0 0
$$131$$ 6.81665 0.595574 0.297787 0.954632i $$-0.403752\pi$$
0.297787 + 0.954632i $$0.403752\pi$$
$$132$$ 0 0
$$133$$ 0.394449 0.0342030
$$134$$ 0 0
$$135$$ −0.697224 −0.0600075
$$136$$ 0 0
$$137$$ −5.60555 −0.478915 −0.239457 0.970907i $$-0.576970\pi$$
−0.239457 + 0.970907i $$0.576970\pi$$
$$138$$ 0 0
$$139$$ 15.1194 1.28241 0.641207 0.767368i $$-0.278434\pi$$
0.641207 + 0.767368i $$0.278434\pi$$
$$140$$ 0 0
$$141$$ 4.60555 0.387857
$$142$$ 0 0
$$143$$ −12.9083 −1.07945
$$144$$ 0 0
$$145$$ 3.90833 0.324569
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ −15.3944 −1.26116 −0.630581 0.776123i $$-0.717183\pi$$
−0.630581 + 0.776123i $$0.717183\pi$$
$$150$$ 0 0
$$151$$ −8.60555 −0.700310 −0.350155 0.936692i $$-0.613871\pi$$
−0.350155 + 0.936692i $$0.613871\pi$$
$$152$$ 0 0
$$153$$ −0.394449 −0.0318893
$$154$$ 0 0
$$155$$ −2.51388 −0.201920
$$156$$ 0 0
$$157$$ 11.8167 0.943072 0.471536 0.881847i $$-0.343700\pi$$
0.471536 + 0.881847i $$0.343700\pi$$
$$158$$ 0 0
$$159$$ 5.90833 0.468561
$$160$$ 0 0
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ 9.11943 0.714289 0.357144 0.934049i $$-0.383750\pi$$
0.357144 + 0.934049i $$0.383750\pi$$
$$164$$ 0 0
$$165$$ 3.90833 0.304263
$$166$$ 0 0
$$167$$ 14.0278 1.08550 0.542750 0.839894i $$-0.317383\pi$$
0.542750 + 0.839894i $$0.317383\pi$$
$$168$$ 0 0
$$169$$ −7.69722 −0.592094
$$170$$ 0 0
$$171$$ 0.394449 0.0301642
$$172$$ 0 0
$$173$$ −6.39445 −0.486161 −0.243080 0.970006i $$-0.578158\pi$$
−0.243080 + 0.970006i $$0.578158\pi$$
$$174$$ 0 0
$$175$$ −4.51388 −0.341217
$$176$$ 0 0
$$177$$ −3.90833 −0.293768
$$178$$ 0 0
$$179$$ −24.9361 −1.86381 −0.931905 0.362702i $$-0.881854\pi$$
−0.931905 + 0.362702i $$0.881854\pi$$
$$180$$ 0 0
$$181$$ −19.0000 −1.41226 −0.706129 0.708083i $$-0.749560\pi$$
−0.706129 + 0.708083i $$0.749560\pi$$
$$182$$ 0 0
$$183$$ 4.90833 0.362834
$$184$$ 0 0
$$185$$ −3.90833 −0.287346
$$186$$ 0 0
$$187$$ 2.21110 0.161692
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 8.60555 0.622676 0.311338 0.950299i $$-0.399223\pi$$
0.311338 + 0.950299i $$0.399223\pi$$
$$192$$ 0 0
$$193$$ 21.0278 1.51361 0.756806 0.653640i $$-0.226759\pi$$
0.756806 + 0.653640i $$0.226759\pi$$
$$194$$ 0 0
$$195$$ −1.60555 −0.114976
$$196$$ 0 0
$$197$$ −25.9361 −1.84787 −0.923935 0.382550i $$-0.875046\pi$$
−0.923935 + 0.382550i $$0.875046\pi$$
$$198$$ 0 0
$$199$$ 1.90833 0.135278 0.0676389 0.997710i $$-0.478453\pi$$
0.0676389 + 0.997710i $$0.478453\pi$$
$$200$$ 0 0
$$201$$ −13.3028 −0.938305
$$202$$ 0 0
$$203$$ −5.60555 −0.393433
$$204$$ 0 0
$$205$$ −2.51388 −0.175577
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ −2.21110 −0.152945
$$210$$ 0 0
$$211$$ −25.6056 −1.76276 −0.881379 0.472409i $$-0.843385\pi$$
−0.881379 + 0.472409i $$0.843385\pi$$
$$212$$ 0 0
$$213$$ −12.9083 −0.884464
$$214$$ 0 0
$$215$$ 3.00000 0.204598
$$216$$ 0 0
$$217$$ 3.60555 0.244761
$$218$$ 0 0
$$219$$ 11.0000 0.743311
$$220$$ 0 0
$$221$$ −0.908327 −0.0611007
$$222$$ 0 0
$$223$$ −17.7250 −1.18695 −0.593476 0.804852i $$-0.702245\pi$$
−0.593476 + 0.804852i $$0.702245\pi$$
$$224$$ 0 0
$$225$$ −4.51388 −0.300925
$$226$$ 0 0
$$227$$ −11.9083 −0.790383 −0.395192 0.918599i $$-0.629322\pi$$
−0.395192 + 0.918599i $$0.629322\pi$$
$$228$$ 0 0
$$229$$ 3.11943 0.206138 0.103069 0.994674i $$-0.467134\pi$$
0.103069 + 0.994674i $$0.467134\pi$$
$$230$$ 0 0
$$231$$ −5.60555 −0.368818
$$232$$ 0 0
$$233$$ 7.11943 0.466409 0.233205 0.972428i $$-0.425079\pi$$
0.233205 + 0.972428i $$0.425079\pi$$
$$234$$ 0 0
$$235$$ −3.21110 −0.209469
$$236$$ 0 0
$$237$$ −13.4222 −0.871866
$$238$$ 0 0
$$239$$ −5.90833 −0.382178 −0.191089 0.981573i $$-0.561202\pi$$
−0.191089 + 0.981573i $$0.561202\pi$$
$$240$$ 0 0
$$241$$ 8.21110 0.528924 0.264462 0.964396i $$-0.414806\pi$$
0.264462 + 0.964396i $$0.414806\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −0.697224 −0.0445440
$$246$$ 0 0
$$247$$ 0.908327 0.0577955
$$248$$ 0 0
$$249$$ −16.2111 −1.02734
$$250$$ 0 0
$$251$$ −4.18335 −0.264050 −0.132025 0.991246i $$-0.542148\pi$$
−0.132025 + 0.991246i $$0.542148\pi$$
$$252$$ 0 0
$$253$$ −5.60555 −0.352418
$$254$$ 0 0
$$255$$ 0.275019 0.0172224
$$256$$ 0 0
$$257$$ 25.8167 1.61040 0.805199 0.593004i $$-0.202058\pi$$
0.805199 + 0.593004i $$0.202058\pi$$
$$258$$ 0 0
$$259$$ 5.60555 0.348312
$$260$$ 0 0
$$261$$ −5.60555 −0.346975
$$262$$ 0 0
$$263$$ 9.42221 0.580998 0.290499 0.956875i $$-0.406179\pi$$
0.290499 + 0.956875i $$0.406179\pi$$
$$264$$ 0 0
$$265$$ −4.11943 −0.253055
$$266$$ 0 0
$$267$$ −15.5139 −0.949434
$$268$$ 0 0
$$269$$ 12.1194 0.738935 0.369467 0.929244i $$-0.379540\pi$$
0.369467 + 0.929244i $$0.379540\pi$$
$$270$$ 0 0
$$271$$ 0.577795 0.0350985 0.0175493 0.999846i $$-0.494414\pi$$
0.0175493 + 0.999846i $$0.494414\pi$$
$$272$$ 0 0
$$273$$ 2.30278 0.139370
$$274$$ 0 0
$$275$$ 25.3028 1.52581
$$276$$ 0 0
$$277$$ 24.5416 1.47456 0.737282 0.675585i $$-0.236109\pi$$
0.737282 + 0.675585i $$0.236109\pi$$
$$278$$ 0 0
$$279$$ 3.60555 0.215859
$$280$$ 0 0
$$281$$ 6.60555 0.394054 0.197027 0.980398i $$-0.436871\pi$$
0.197027 + 0.980398i $$0.436871\pi$$
$$282$$ 0 0
$$283$$ −25.3028 −1.50409 −0.752047 0.659110i $$-0.770933\pi$$
−0.752047 + 0.659110i $$0.770933\pi$$
$$284$$ 0 0
$$285$$ −0.275019 −0.0162907
$$286$$ 0 0
$$287$$ 3.60555 0.212829
$$288$$ 0 0
$$289$$ −16.8444 −0.990848
$$290$$ 0 0
$$291$$ 3.78890 0.222109
$$292$$ 0 0
$$293$$ −18.7889 −1.09766 −0.548830 0.835934i $$-0.684926\pi$$
−0.548830 + 0.835934i $$0.684926\pi$$
$$294$$ 0 0
$$295$$ 2.72498 0.158655
$$296$$ 0 0
$$297$$ −5.60555 −0.325267
$$298$$ 0 0
$$299$$ 2.30278 0.133173
$$300$$ 0 0
$$301$$ −4.30278 −0.248008
$$302$$ 0 0
$$303$$ −12.5139 −0.718904
$$304$$ 0 0
$$305$$ −3.42221 −0.195955
$$306$$ 0 0
$$307$$ 25.6056 1.46139 0.730693 0.682706i $$-0.239197\pi$$
0.730693 + 0.682706i $$0.239197\pi$$
$$308$$ 0 0
$$309$$ −2.78890 −0.158655
$$310$$ 0 0
$$311$$ −10.1194 −0.573820 −0.286910 0.957958i $$-0.592628\pi$$
−0.286910 + 0.957958i $$0.592628\pi$$
$$312$$ 0 0
$$313$$ −30.4222 −1.71956 −0.859782 0.510661i $$-0.829401\pi$$
−0.859782 + 0.510661i $$0.829401\pi$$
$$314$$ 0 0
$$315$$ −0.697224 −0.0392841
$$316$$ 0 0
$$317$$ −12.6972 −0.713147 −0.356574 0.934267i $$-0.616055\pi$$
−0.356574 + 0.934267i $$0.616055\pi$$
$$318$$ 0 0
$$319$$ 31.4222 1.75931
$$320$$ 0 0
$$321$$ −0.302776 −0.0168993
$$322$$ 0 0
$$323$$ −0.155590 −0.00865725
$$324$$ 0 0
$$325$$ −10.3944 −0.576580
$$326$$ 0 0
$$327$$ −4.51388 −0.249618
$$328$$ 0 0
$$329$$ 4.60555 0.253912
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ 0 0
$$333$$ 5.60555 0.307182
$$334$$ 0 0
$$335$$ 9.27502 0.506748
$$336$$ 0 0
$$337$$ 16.7250 0.911068 0.455534 0.890218i $$-0.349448\pi$$
0.455534 + 0.890218i $$0.349448\pi$$
$$338$$ 0 0
$$339$$ 5.30278 0.288007
$$340$$ 0 0
$$341$$ −20.2111 −1.09449
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −0.697224 −0.0375373
$$346$$ 0 0
$$347$$ −0.211103 −0.0113326 −0.00566629 0.999984i $$-0.501804\pi$$
−0.00566629 + 0.999984i $$0.501804\pi$$
$$348$$ 0 0
$$349$$ 0.697224 0.0373216 0.0186608 0.999826i $$-0.494060\pi$$
0.0186608 + 0.999826i $$0.494060\pi$$
$$350$$ 0 0
$$351$$ 2.30278 0.122913
$$352$$ 0 0
$$353$$ 15.6056 0.830600 0.415300 0.909685i $$-0.363677\pi$$
0.415300 + 0.909685i $$0.363677\pi$$
$$354$$ 0 0
$$355$$ 9.00000 0.477670
$$356$$ 0 0
$$357$$ −0.394449 −0.0208764
$$358$$ 0 0
$$359$$ −9.90833 −0.522941 −0.261471 0.965211i $$-0.584207\pi$$
−0.261471 + 0.965211i $$0.584207\pi$$
$$360$$ 0 0
$$361$$ −18.8444 −0.991811
$$362$$ 0 0
$$363$$ 20.4222 1.07189
$$364$$ 0 0
$$365$$ −7.66947 −0.401438
$$366$$ 0 0
$$367$$ −25.5139 −1.33181 −0.665907 0.746035i $$-0.731955\pi$$
−0.665907 + 0.746035i $$0.731955\pi$$
$$368$$ 0 0
$$369$$ 3.60555 0.187698
$$370$$ 0 0
$$371$$ 5.90833 0.306745
$$372$$ 0 0
$$373$$ 11.1833 0.579052 0.289526 0.957170i $$-0.406502\pi$$
0.289526 + 0.957170i $$0.406502\pi$$
$$374$$ 0 0
$$375$$ 6.63331 0.342543
$$376$$ 0 0
$$377$$ −12.9083 −0.664813
$$378$$ 0 0
$$379$$ 10.4222 0.535353 0.267676 0.963509i $$-0.413744\pi$$
0.267676 + 0.963509i $$0.413744\pi$$
$$380$$ 0 0
$$381$$ −18.1194 −0.928286
$$382$$ 0 0
$$383$$ 26.6333 1.36090 0.680449 0.732795i $$-0.261785\pi$$
0.680449 + 0.732795i $$0.261785\pi$$
$$384$$ 0 0
$$385$$ 3.90833 0.199187
$$386$$ 0 0
$$387$$ −4.30278 −0.218722
$$388$$ 0 0
$$389$$ −12.6333 −0.640534 −0.320267 0.947327i $$-0.603773\pi$$
−0.320267 + 0.947327i $$0.603773\pi$$
$$390$$ 0 0
$$391$$ −0.394449 −0.0199481
$$392$$ 0 0
$$393$$ 6.81665 0.343855
$$394$$ 0 0
$$395$$ 9.35829 0.470867
$$396$$ 0 0
$$397$$ −8.60555 −0.431900 −0.215950 0.976404i $$-0.569285\pi$$
−0.215950 + 0.976404i $$0.569285\pi$$
$$398$$ 0 0
$$399$$ 0.394449 0.0197471
$$400$$ 0 0
$$401$$ −26.6333 −1.33000 −0.665002 0.746842i $$-0.731569\pi$$
−0.665002 + 0.746842i $$0.731569\pi$$
$$402$$ 0 0
$$403$$ 8.30278 0.413591
$$404$$ 0 0
$$405$$ −0.697224 −0.0346454
$$406$$ 0 0
$$407$$ −31.4222 −1.55754
$$408$$ 0 0
$$409$$ 8.39445 0.415079 0.207539 0.978227i $$-0.433454\pi$$
0.207539 + 0.978227i $$0.433454\pi$$
$$410$$ 0 0
$$411$$ −5.60555 −0.276501
$$412$$ 0 0
$$413$$ −3.90833 −0.192316
$$414$$ 0 0
$$415$$ 11.3028 0.554831
$$416$$ 0 0
$$417$$ 15.1194 0.740402
$$418$$ 0 0
$$419$$ 11.9083 0.581760 0.290880 0.956760i $$-0.406052\pi$$
0.290880 + 0.956760i $$0.406052\pi$$
$$420$$ 0 0
$$421$$ −7.09167 −0.345627 −0.172813 0.984955i $$-0.555286\pi$$
−0.172813 + 0.984955i $$0.555286\pi$$
$$422$$ 0 0
$$423$$ 4.60555 0.223930
$$424$$ 0 0
$$425$$ 1.78049 0.0863666
$$426$$ 0 0
$$427$$ 4.90833 0.237531
$$428$$ 0 0
$$429$$ −12.9083 −0.623220
$$430$$ 0 0
$$431$$ 4.30278 0.207257 0.103629 0.994616i $$-0.466955\pi$$
0.103629 + 0.994616i $$0.466955\pi$$
$$432$$ 0 0
$$433$$ 23.8167 1.14456 0.572278 0.820060i $$-0.306060\pi$$
0.572278 + 0.820060i $$0.306060\pi$$
$$434$$ 0 0
$$435$$ 3.90833 0.187390
$$436$$ 0 0
$$437$$ 0.394449 0.0188690
$$438$$ 0 0
$$439$$ 27.6056 1.31754 0.658771 0.752344i $$-0.271077\pi$$
0.658771 + 0.752344i $$0.271077\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 6.78890 0.322550 0.161275 0.986909i $$-0.448439\pi$$
0.161275 + 0.986909i $$0.448439\pi$$
$$444$$ 0 0
$$445$$ 10.8167 0.512759
$$446$$ 0 0
$$447$$ −15.3944 −0.728132
$$448$$ 0 0
$$449$$ −7.11943 −0.335987 −0.167993 0.985788i $$-0.553729\pi$$
−0.167993 + 0.985788i $$0.553729\pi$$
$$450$$ 0 0
$$451$$ −20.2111 −0.951704
$$452$$ 0 0
$$453$$ −8.60555 −0.404324
$$454$$ 0 0
$$455$$ −1.60555 −0.0752694
$$456$$ 0 0
$$457$$ −20.6972 −0.968175 −0.484088 0.875020i $$-0.660848\pi$$
−0.484088 + 0.875020i $$0.660848\pi$$
$$458$$ 0 0
$$459$$ −0.394449 −0.0184113
$$460$$ 0 0
$$461$$ −26.0917 −1.21521 −0.607605 0.794239i $$-0.707870\pi$$
−0.607605 + 0.794239i $$0.707870\pi$$
$$462$$ 0 0
$$463$$ −17.0000 −0.790057 −0.395029 0.918669i $$-0.629265\pi$$
−0.395029 + 0.918669i $$0.629265\pi$$
$$464$$ 0 0
$$465$$ −2.51388 −0.116578
$$466$$ 0 0
$$467$$ 23.8444 1.10339 0.551694 0.834047i $$-0.313982\pi$$
0.551694 + 0.834047i $$0.313982\pi$$
$$468$$ 0 0
$$469$$ −13.3028 −0.614265
$$470$$ 0 0
$$471$$ 11.8167 0.544483
$$472$$ 0 0
$$473$$ 24.1194 1.10901
$$474$$ 0 0
$$475$$ −1.78049 −0.0816946
$$476$$ 0 0
$$477$$ 5.90833 0.270524
$$478$$ 0 0
$$479$$ 27.2389 1.24458 0.622288 0.782789i $$-0.286203\pi$$
0.622288 + 0.782789i $$0.286203\pi$$
$$480$$ 0 0
$$481$$ 12.9083 0.588569
$$482$$ 0 0
$$483$$ 1.00000 0.0455016
$$484$$ 0 0
$$485$$ −2.64171 −0.119954
$$486$$ 0 0
$$487$$ −4.21110 −0.190823 −0.0954116 0.995438i $$-0.530417\pi$$
−0.0954116 + 0.995438i $$0.530417\pi$$
$$488$$ 0 0
$$489$$ 9.11943 0.412395
$$490$$ 0 0
$$491$$ −34.9083 −1.57539 −0.787695 0.616065i $$-0.788726\pi$$
−0.787695 + 0.616065i $$0.788726\pi$$
$$492$$ 0 0
$$493$$ 2.21110 0.0995831
$$494$$ 0 0
$$495$$ 3.90833 0.175666
$$496$$ 0 0
$$497$$ −12.9083 −0.579018
$$498$$ 0 0
$$499$$ −12.8806 −0.576614 −0.288307 0.957538i $$-0.593092\pi$$
−0.288307 + 0.957538i $$0.593092\pi$$
$$500$$ 0 0
$$501$$ 14.0278 0.626714
$$502$$ 0 0
$$503$$ −21.1194 −0.941669 −0.470834 0.882222i $$-0.656047\pi$$
−0.470834 + 0.882222i $$0.656047\pi$$
$$504$$ 0 0
$$505$$ 8.72498 0.388257
$$506$$ 0 0
$$507$$ −7.69722 −0.341846
$$508$$ 0 0
$$509$$ −19.3944 −0.859644 −0.429822 0.902914i $$-0.641424\pi$$
−0.429822 + 0.902914i $$0.641424\pi$$
$$510$$ 0 0
$$511$$ 11.0000 0.486611
$$512$$ 0 0
$$513$$ 0.394449 0.0174153
$$514$$ 0 0
$$515$$ 1.94449 0.0856843
$$516$$ 0 0
$$517$$ −25.8167 −1.13542
$$518$$ 0 0
$$519$$ −6.39445 −0.280685
$$520$$ 0 0
$$521$$ 4.78890 0.209805 0.104903 0.994482i $$-0.466547\pi$$
0.104903 + 0.994482i $$0.466547\pi$$
$$522$$ 0 0
$$523$$ −31.2111 −1.36477 −0.682383 0.730995i $$-0.739056\pi$$
−0.682383 + 0.730995i $$0.739056\pi$$
$$524$$ 0 0
$$525$$ −4.51388 −0.197002
$$526$$ 0 0
$$527$$ −1.42221 −0.0619522
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −3.90833 −0.169607
$$532$$ 0 0
$$533$$ 8.30278 0.359633
$$534$$ 0 0
$$535$$ 0.211103 0.00912676
$$536$$ 0 0
$$537$$ −24.9361 −1.07607
$$538$$ 0 0
$$539$$ −5.60555 −0.241448
$$540$$ 0 0
$$541$$ 27.0278 1.16201 0.581007 0.813899i $$-0.302659\pi$$
0.581007 + 0.813899i $$0.302659\pi$$
$$542$$ 0 0
$$543$$ −19.0000 −0.815368
$$544$$ 0 0
$$545$$ 3.14719 0.134811
$$546$$ 0 0
$$547$$ 28.3305 1.21133 0.605663 0.795721i $$-0.292908\pi$$
0.605663 + 0.795721i $$0.292908\pi$$
$$548$$ 0 0
$$549$$ 4.90833 0.209482
$$550$$ 0 0
$$551$$ −2.21110 −0.0941961
$$552$$ 0 0
$$553$$ −13.4222 −0.570770
$$554$$ 0 0
$$555$$ −3.90833 −0.165899
$$556$$ 0 0
$$557$$ 16.1833 0.685710 0.342855 0.939388i $$-0.388606\pi$$
0.342855 + 0.939388i $$0.388606\pi$$
$$558$$ 0 0
$$559$$ −9.90833 −0.419078
$$560$$ 0 0
$$561$$ 2.21110 0.0933528
$$562$$ 0 0
$$563$$ 38.5416 1.62434 0.812168 0.583423i $$-0.198287\pi$$
0.812168 + 0.583423i $$0.198287\pi$$
$$564$$ 0 0
$$565$$ −3.69722 −0.155543
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −14.7889 −0.619983 −0.309991 0.950739i $$-0.600326\pi$$
−0.309991 + 0.950739i $$0.600326\pi$$
$$570$$ 0 0
$$571$$ 29.7889 1.24663 0.623313 0.781972i $$-0.285786\pi$$
0.623313 + 0.781972i $$0.285786\pi$$
$$572$$ 0 0
$$573$$ 8.60555 0.359502
$$574$$ 0 0
$$575$$ −4.51388 −0.188242
$$576$$ 0 0
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 0 0
$$579$$ 21.0278 0.873884
$$580$$ 0 0
$$581$$ −16.2111 −0.672550
$$582$$ 0 0
$$583$$ −33.1194 −1.37167
$$584$$ 0 0
$$585$$ −1.60555 −0.0663814
$$586$$ 0 0
$$587$$ 35.5139 1.46581 0.732907 0.680329i $$-0.238163\pi$$
0.732907 + 0.680329i $$0.238163\pi$$
$$588$$ 0 0
$$589$$ 1.42221 0.0586009
$$590$$ 0 0
$$591$$ −25.9361 −1.06687
$$592$$ 0 0
$$593$$ 21.3944 0.878565 0.439282 0.898349i $$-0.355233\pi$$
0.439282 + 0.898349i $$0.355233\pi$$
$$594$$ 0 0
$$595$$ 0.275019 0.0112747
$$596$$ 0 0
$$597$$ 1.90833 0.0781026
$$598$$ 0 0
$$599$$ −26.5416 −1.08446 −0.542231 0.840230i $$-0.682420\pi$$
−0.542231 + 0.840230i $$0.682420\pi$$
$$600$$ 0 0
$$601$$ −27.5416 −1.12345 −0.561723 0.827325i $$-0.689861\pi$$
−0.561723 + 0.827325i $$0.689861\pi$$
$$602$$ 0 0
$$603$$ −13.3028 −0.541731
$$604$$ 0 0
$$605$$ −14.2389 −0.578892
$$606$$ 0 0
$$607$$ 20.5416 0.833759 0.416880 0.908962i $$-0.363124\pi$$
0.416880 + 0.908962i $$0.363124\pi$$
$$608$$ 0 0
$$609$$ −5.60555 −0.227148
$$610$$ 0 0
$$611$$ 10.6056 0.429055
$$612$$ 0 0
$$613$$ −28.6333 −1.15649 −0.578244 0.815864i $$-0.696262\pi$$
−0.578244 + 0.815864i $$0.696262\pi$$
$$614$$ 0 0
$$615$$ −2.51388 −0.101369
$$616$$ 0 0
$$617$$ 20.3028 0.817359 0.408679 0.912678i $$-0.365989\pi$$
0.408679 + 0.912678i $$0.365989\pi$$
$$618$$ 0 0
$$619$$ −40.3028 −1.61991 −0.809953 0.586495i $$-0.800507\pi$$
−0.809953 + 0.586495i $$0.800507\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ −15.5139 −0.621550
$$624$$ 0 0
$$625$$ 17.9445 0.717779
$$626$$ 0 0
$$627$$ −2.21110 −0.0883029
$$628$$ 0 0
$$629$$ −2.21110 −0.0881624
$$630$$ 0 0
$$631$$ −16.3944 −0.652653 −0.326326 0.945257i $$-0.605811\pi$$
−0.326326 + 0.945257i $$0.605811\pi$$
$$632$$ 0 0
$$633$$ −25.6056 −1.01773
$$634$$ 0 0
$$635$$ 12.6333 0.501338
$$636$$ 0 0
$$637$$ 2.30278 0.0912393
$$638$$ 0 0
$$639$$ −12.9083 −0.510646
$$640$$ 0 0
$$641$$ −30.5416 −1.20632 −0.603161 0.797619i $$-0.706092\pi$$
−0.603161 + 0.797619i $$0.706092\pi$$
$$642$$ 0 0
$$643$$ −19.0917 −0.752902 −0.376451 0.926437i $$-0.622856\pi$$
−0.376451 + 0.926437i $$0.622856\pi$$
$$644$$ 0 0
$$645$$ 3.00000 0.118125
$$646$$ 0 0
$$647$$ 16.5139 0.649228 0.324614 0.945847i $$-0.394766\pi$$
0.324614 + 0.945847i $$0.394766\pi$$
$$648$$ 0 0
$$649$$ 21.9083 0.859977
$$650$$ 0 0
$$651$$ 3.60555 0.141313
$$652$$ 0 0
$$653$$ 17.9083 0.700807 0.350403 0.936599i $$-0.386044\pi$$
0.350403 + 0.936599i $$0.386044\pi$$
$$654$$ 0 0
$$655$$ −4.75274 −0.185705
$$656$$ 0 0
$$657$$ 11.0000 0.429151
$$658$$ 0 0
$$659$$ 13.1833 0.513550 0.256775 0.966471i $$-0.417340\pi$$
0.256775 + 0.966471i $$0.417340\pi$$
$$660$$ 0 0
$$661$$ −5.42221 −0.210899 −0.105450 0.994425i $$-0.533628\pi$$
−0.105450 + 0.994425i $$0.533628\pi$$
$$662$$ 0 0
$$663$$ −0.908327 −0.0352765
$$664$$ 0 0
$$665$$ −0.275019 −0.0106648
$$666$$ 0 0
$$667$$ −5.60555 −0.217048
$$668$$ 0 0
$$669$$ −17.7250 −0.685287
$$670$$ 0 0
$$671$$ −27.5139 −1.06216
$$672$$ 0 0
$$673$$ 5.00000 0.192736 0.0963679 0.995346i $$-0.469277\pi$$
0.0963679 + 0.995346i $$0.469277\pi$$
$$674$$ 0 0
$$675$$ −4.51388 −0.173739
$$676$$ 0 0
$$677$$ −30.1194 −1.15758 −0.578792 0.815475i $$-0.696476\pi$$
−0.578792 + 0.815475i $$0.696476\pi$$
$$678$$ 0 0
$$679$$ 3.78890 0.145405
$$680$$ 0 0
$$681$$ −11.9083 −0.456328
$$682$$ 0 0
$$683$$ 30.6333 1.17215 0.586075 0.810256i $$-0.300672\pi$$
0.586075 + 0.810256i $$0.300672\pi$$
$$684$$ 0 0
$$685$$ 3.90833 0.149329
$$686$$ 0 0
$$687$$ 3.11943 0.119014
$$688$$ 0 0
$$689$$ 13.6056 0.518330
$$690$$ 0 0
$$691$$ 3.09167 0.117613 0.0588064 0.998269i $$-0.481271\pi$$
0.0588064 + 0.998269i $$0.481271\pi$$
$$692$$ 0 0
$$693$$ −5.60555 −0.212937
$$694$$ 0 0
$$695$$ −10.5416 −0.399867
$$696$$ 0 0
$$697$$ −1.42221 −0.0538699
$$698$$ 0 0
$$699$$ 7.11943 0.269282
$$700$$ 0 0
$$701$$ 18.9361 0.715206 0.357603 0.933874i $$-0.383594\pi$$
0.357603 + 0.933874i $$0.383594\pi$$
$$702$$ 0 0
$$703$$ 2.21110 0.0833933
$$704$$ 0 0
$$705$$ −3.21110 −0.120937
$$706$$ 0 0
$$707$$ −12.5139 −0.470633
$$708$$ 0 0
$$709$$ −20.1194 −0.755601 −0.377801 0.925887i $$-0.623319\pi$$
−0.377801 + 0.925887i $$0.623319\pi$$
$$710$$ 0 0
$$711$$ −13.4222 −0.503372
$$712$$ 0 0
$$713$$ 3.60555 0.135029
$$714$$ 0 0
$$715$$ 9.00000 0.336581
$$716$$ 0 0
$$717$$ −5.90833 −0.220651
$$718$$ 0 0
$$719$$ −41.8444 −1.56053 −0.780267 0.625447i $$-0.784917\pi$$
−0.780267 + 0.625447i $$0.784917\pi$$
$$720$$ 0 0
$$721$$ −2.78890 −0.103864
$$722$$ 0 0
$$723$$ 8.21110 0.305374
$$724$$ 0 0
$$725$$ 25.3028 0.939721
$$726$$ 0 0
$$727$$ 6.39445 0.237157 0.118578 0.992945i $$-0.462166\pi$$
0.118578 + 0.992945i $$0.462166\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.69722 0.0627741
$$732$$ 0 0
$$733$$ −37.0278 −1.36765 −0.683826 0.729645i $$-0.739685\pi$$
−0.683826 + 0.729645i $$0.739685\pi$$
$$734$$ 0 0
$$735$$ −0.697224 −0.0257175
$$736$$ 0 0
$$737$$ 74.5694 2.74680
$$738$$ 0 0
$$739$$ 36.8167 1.35432 0.677161 0.735835i $$-0.263210\pi$$
0.677161 + 0.735835i $$0.263210\pi$$
$$740$$ 0 0
$$741$$ 0.908327 0.0333682
$$742$$ 0 0
$$743$$ −27.4861 −1.00837 −0.504184 0.863596i $$-0.668207\pi$$
−0.504184 + 0.863596i $$0.668207\pi$$
$$744$$ 0 0
$$745$$ 10.7334 0.393241
$$746$$ 0 0
$$747$$ −16.2111 −0.593133
$$748$$ 0 0
$$749$$ −0.302776 −0.0110632
$$750$$ 0 0
$$751$$ 42.9361 1.56676 0.783380 0.621543i $$-0.213494\pi$$
0.783380 + 0.621543i $$0.213494\pi$$
$$752$$ 0 0
$$753$$ −4.18335 −0.152450
$$754$$ 0 0
$$755$$ 6.00000 0.218362
$$756$$ 0 0
$$757$$ 20.8167 0.756594 0.378297 0.925684i $$-0.376510\pi$$
0.378297 + 0.925684i $$0.376510\pi$$
$$758$$ 0 0
$$759$$ −5.60555 −0.203469
$$760$$ 0 0
$$761$$ −15.6333 −0.566707 −0.283353 0.959016i $$-0.591447\pi$$
−0.283353 + 0.959016i $$0.591447\pi$$
$$762$$ 0 0
$$763$$ −4.51388 −0.163413
$$764$$ 0 0
$$765$$ 0.275019 0.00994334
$$766$$ 0 0
$$767$$ −9.00000 −0.324971
$$768$$ 0 0
$$769$$ 11.8167 0.426119 0.213060 0.977039i $$-0.431657\pi$$
0.213060 + 0.977039i $$0.431657\pi$$
$$770$$ 0 0
$$771$$ 25.8167 0.929764
$$772$$ 0 0
$$773$$ −2.21110 −0.0795278 −0.0397639 0.999209i $$-0.512661\pi$$
−0.0397639 + 0.999209i $$0.512661\pi$$
$$774$$ 0 0
$$775$$ −16.2750 −0.584616
$$776$$ 0 0
$$777$$ 5.60555 0.201098
$$778$$ 0 0
$$779$$ 1.42221 0.0509558
$$780$$ 0 0
$$781$$ 72.3583 2.58918
$$782$$ 0 0
$$783$$ −5.60555 −0.200326
$$784$$ 0 0
$$785$$ −8.23886 −0.294057
$$786$$ 0 0
$$787$$ 31.1194 1.10929 0.554644 0.832088i $$-0.312854\pi$$
0.554644 + 0.832088i $$0.312854\pi$$
$$788$$ 0 0
$$789$$ 9.42221 0.335439
$$790$$ 0 0
$$791$$ 5.30278 0.188545
$$792$$ 0 0
$$793$$ 11.3028 0.401373
$$794$$ 0 0
$$795$$ −4.11943 −0.146101
$$796$$ 0 0
$$797$$ −49.8167 −1.76460 −0.882298 0.470691i $$-0.844005\pi$$
−0.882298 + 0.470691i $$0.844005\pi$$
$$798$$ 0 0
$$799$$ −1.81665 −0.0642686
$$800$$ 0 0
$$801$$ −15.5139 −0.548156
$$802$$ 0 0
$$803$$ −61.6611 −2.17597
$$804$$ 0 0
$$805$$ −0.697224 −0.0245739
$$806$$ 0 0
$$807$$ 12.1194 0.426624
$$808$$ 0 0
$$809$$ 25.5139 0.897020 0.448510 0.893778i $$-0.351955\pi$$
0.448510 + 0.893778i $$0.351955\pi$$
$$810$$ 0 0
$$811$$ −10.6333 −0.373386 −0.186693 0.982418i $$-0.559777\pi$$
−0.186693 + 0.982418i $$0.559777\pi$$
$$812$$ 0 0
$$813$$ 0.577795 0.0202642
$$814$$ 0 0
$$815$$ −6.35829 −0.222721
$$816$$ 0 0
$$817$$ −1.69722 −0.0593784
$$818$$ 0 0
$$819$$ 2.30278 0.0804655
$$820$$ 0 0
$$821$$ −17.4500 −0.609008 −0.304504 0.952511i $$-0.598491\pi$$
−0.304504 + 0.952511i $$0.598491\pi$$
$$822$$ 0 0
$$823$$ 48.5139 1.69109 0.845544 0.533906i $$-0.179276\pi$$
0.845544 + 0.533906i $$0.179276\pi$$
$$824$$ 0 0
$$825$$ 25.3028 0.880930
$$826$$ 0 0
$$827$$ −22.6972 −0.789260 −0.394630 0.918840i $$-0.629127\pi$$
−0.394630 + 0.918840i $$0.629127\pi$$
$$828$$ 0 0
$$829$$ 1.18335 0.0410993 0.0205497 0.999789i $$-0.493458\pi$$
0.0205497 + 0.999789i $$0.493458\pi$$
$$830$$ 0 0
$$831$$ 24.5416 0.851340
$$832$$ 0 0
$$833$$ −0.394449 −0.0136668
$$834$$ 0 0
$$835$$ −9.78049 −0.338468
$$836$$ 0 0
$$837$$ 3.60555 0.124626
$$838$$ 0 0
$$839$$ 28.5416 0.985367 0.492683 0.870209i $$-0.336016\pi$$
0.492683 + 0.870209i $$0.336016\pi$$
$$840$$ 0 0
$$841$$ 2.42221 0.0835243
$$842$$ 0 0
$$843$$ 6.60555 0.227507
$$844$$ 0 0
$$845$$ 5.36669 0.184620
$$846$$ 0 0
$$847$$ 20.4222 0.701715
$$848$$ 0 0
$$849$$ −25.3028 −0.868389
$$850$$ 0 0
$$851$$ 5.60555 0.192156
$$852$$ 0 0
$$853$$ −2.60555 −0.0892124 −0.0446062 0.999005i $$-0.514203\pi$$
−0.0446062 + 0.999005i $$0.514203\pi$$
$$854$$ 0 0
$$855$$ −0.275019 −0.00940546
$$856$$ 0 0
$$857$$ 9.02776 0.308382 0.154191 0.988041i $$-0.450723\pi$$
0.154191 + 0.988041i $$0.450723\pi$$
$$858$$ 0 0
$$859$$ −28.0555 −0.957242 −0.478621 0.878022i $$-0.658863\pi$$
−0.478621 + 0.878022i $$0.658863\pi$$
$$860$$ 0 0
$$861$$ 3.60555 0.122877
$$862$$ 0 0
$$863$$ 11.8167 0.402244 0.201122 0.979566i $$-0.435541\pi$$
0.201122 + 0.979566i $$0.435541\pi$$
$$864$$ 0 0
$$865$$ 4.45837 0.151589
$$866$$ 0 0
$$867$$ −16.8444 −0.572066
$$868$$ 0 0
$$869$$ 75.2389 2.55230
$$870$$ 0 0
$$871$$ −30.6333 −1.03797
$$872$$ 0 0
$$873$$ 3.78890 0.128235
$$874$$ 0 0
$$875$$ 6.63331 0.224247
$$876$$ 0 0
$$877$$ −32.0000 −1.08056 −0.540282 0.841484i $$-0.681682\pi$$
−0.540282 + 0.841484i $$0.681682\pi$$
$$878$$ 0 0
$$879$$ −18.7889 −0.633734
$$880$$ 0 0
$$881$$ −2.23886 −0.0754291 −0.0377145 0.999289i $$-0.512008\pi$$
−0.0377145 + 0.999289i $$0.512008\pi$$
$$882$$ 0 0
$$883$$ 6.90833 0.232484 0.116242 0.993221i $$-0.462915\pi$$
0.116242 + 0.993221i $$0.462915\pi$$
$$884$$ 0 0
$$885$$ 2.72498 0.0915992
$$886$$ 0 0
$$887$$ 1.51388 0.0508311 0.0254155 0.999677i $$-0.491909\pi$$
0.0254155 + 0.999677i $$0.491909\pi$$
$$888$$ 0 0
$$889$$ −18.1194 −0.607706
$$890$$ 0 0
$$891$$ −5.60555 −0.187793
$$892$$ 0 0
$$893$$ 1.81665 0.0607920
$$894$$ 0 0
$$895$$ 17.3860 0.581151
$$896$$ 0 0
$$897$$ 2.30278 0.0768874
$$898$$ 0 0
$$899$$ −20.2111 −0.674078
$$900$$ 0 0
$$901$$ −2.33053 −0.0776413
$$902$$ 0 0
$$903$$ −4.30278 −0.143187
$$904$$ 0 0
$$905$$ 13.2473 0.440354
$$906$$ 0 0
$$907$$ 49.7250 1.65109 0.825545 0.564336i $$-0.190868\pi$$
0.825545 + 0.564336i $$0.190868\pi$$
$$908$$ 0 0
$$909$$ −12.5139 −0.415059
$$910$$ 0 0
$$911$$ −24.6056 −0.815218 −0.407609 0.913156i $$-0.633637\pi$$
−0.407609 + 0.913156i $$0.633637\pi$$
$$912$$ 0 0
$$913$$ 90.8722 3.00743
$$914$$ 0 0
$$915$$ −3.42221 −0.113135
$$916$$ 0 0
$$917$$ 6.81665 0.225106
$$918$$ 0 0
$$919$$ −8.02776 −0.264811 −0.132406 0.991196i $$-0.542270\pi$$
−0.132406 + 0.991196i $$0.542270\pi$$
$$920$$ 0 0
$$921$$ 25.6056 0.843732
$$922$$ 0 0
$$923$$ −29.7250 −0.978410
$$924$$ 0 0
$$925$$ −25.3028 −0.831950
$$926$$ 0 0
$$927$$ −2.78890 −0.0915994
$$928$$ 0 0
$$929$$ −34.6972 −1.13838 −0.569190 0.822206i $$-0.692743\pi$$
−0.569190 + 0.822206i $$0.692743\pi$$
$$930$$ 0 0
$$931$$ 0.394449 0.0129275
$$932$$ 0 0
$$933$$ −10.1194 −0.331295
$$934$$ 0 0
$$935$$ −1.54163 −0.0504168
$$936$$ 0 0
$$937$$ −5.97224 −0.195105 −0.0975523 0.995230i $$-0.531101\pi$$
−0.0975523 + 0.995230i $$0.531101\pi$$
$$938$$ 0 0
$$939$$ −30.4222 −0.992791
$$940$$ 0 0
$$941$$ −5.21110 −0.169877 −0.0849385 0.996386i $$-0.527069\pi$$
−0.0849385 + 0.996386i $$0.527069\pi$$
$$942$$ 0 0
$$943$$ 3.60555 0.117413
$$944$$ 0 0
$$945$$ −0.697224 −0.0226807
$$946$$ 0 0
$$947$$ 18.8444 0.612361 0.306181 0.951973i $$-0.400949\pi$$
0.306181 + 0.951973i $$0.400949\pi$$
$$948$$ 0 0
$$949$$ 25.3305 0.822264
$$950$$ 0 0
$$951$$ −12.6972 −0.411736
$$952$$ 0 0
$$953$$ 26.7250 0.865707 0.432854 0.901464i $$-0.357507\pi$$
0.432854 + 0.901464i $$0.357507\pi$$
$$954$$ 0 0
$$955$$ −6.00000 −0.194155
$$956$$ 0 0
$$957$$ 31.4222 1.01574
$$958$$ 0 0
$$959$$ −5.60555 −0.181013
$$960$$ 0 0
$$961$$ −18.0000 −0.580645
$$962$$ 0 0
$$963$$ −0.302776 −0.00975681
$$964$$ 0 0
$$965$$ −14.6611 −0.471956
$$966$$ 0 0
$$967$$ 16.0555 0.516310 0.258155 0.966103i $$-0.416885\pi$$
0.258155 + 0.966103i $$0.416885\pi$$
$$968$$ 0 0
$$969$$ −0.155590 −0.00499826
$$970$$ 0 0
$$971$$ −6.06392 −0.194600 −0.0973002 0.995255i $$-0.531021\pi$$
−0.0973002 + 0.995255i $$0.531021\pi$$
$$972$$ 0 0
$$973$$ 15.1194 0.484707
$$974$$ 0 0
$$975$$ −10.3944 −0.332889
$$976$$ 0 0
$$977$$ −13.6972 −0.438213 −0.219107 0.975701i $$-0.570314\pi$$
−0.219107 + 0.975701i $$0.570314\pi$$
$$978$$ 0 0
$$979$$ 86.9638 2.77938
$$980$$ 0 0
$$981$$ −4.51388 −0.144117
$$982$$ 0 0
$$983$$ −2.57779 −0.0822189 −0.0411094 0.999155i $$-0.513089\pi$$
−0.0411094 + 0.999155i $$0.513089\pi$$
$$984$$ 0 0
$$985$$ 18.0833 0.576181
$$986$$ 0 0
$$987$$ 4.60555 0.146596
$$988$$ 0 0
$$989$$ −4.30278 −0.136820
$$990$$ 0 0
$$991$$ 30.9638 0.983599 0.491799 0.870709i $$-0.336339\pi$$
0.491799 + 0.870709i $$0.336339\pi$$
$$992$$ 0 0
$$993$$ 2.00000 0.0634681
$$994$$ 0 0
$$995$$ −1.33053 −0.0421807
$$996$$ 0 0
$$997$$ −3.18335 −0.100818 −0.0504088 0.998729i $$-0.516052\pi$$
−0.0504088 + 0.998729i $$0.516052\pi$$
$$998$$ 0 0
$$999$$ 5.60555 0.177352
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 7728.2.a.bf.1.2 2
4.3 odd 2 1932.2.a.c.1.2 2
12.11 even 2 5796.2.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.c.1.2 2 4.3 odd 2
5796.2.a.m.1.1 2 12.11 even 2
7728.2.a.bf.1.2 2 1.1 even 1 trivial