# Properties

 Label 7800.2.a.br.1.2 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.788.1 Defining polynomial: $$x^{3} - x^{2} - 7x - 3$$ x^3 - x^2 - 7*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.476452$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -0.476452 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -0.476452 q^{7} +1.00000 q^{9} -3.34364 q^{11} -1.00000 q^{13} -1.00000 q^{17} -6.77299 q^{19} -0.476452 q^{21} +4.77299 q^{23} +1.00000 q^{27} +7.59308 q^{29} +7.93672 q^{31} -3.34364 q^{33} -7.82009 q^{37} -1.00000 q^{39} +9.46027 q^{41} -1.82009 q^{43} +9.06953 q^{47} -6.77299 q^{49} -1.00000 q^{51} -9.50736 q^{53} -6.77299 q^{57} +3.16373 q^{59} -9.59308 q^{61} -0.476452 q^{63} +1.34364 q^{67} +4.77299 q^{69} -4.86719 q^{71} +14.4132 q^{73} +1.59308 q^{77} -2.17991 q^{79} +1.00000 q^{81} -9.16373 q^{83} +7.59308 q^{87} -10.5931 q^{89} +0.476452 q^{91} +7.93672 q^{93} -7.04710 q^{97} -3.34364 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + q^7 + 3 * q^9 $$3 q + 3 q^{3} + q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 3 q^{17} - 6 q^{19} + q^{21} + 3 q^{27} - q^{29} - 7 q^{31} - 3 q^{33} - 14 q^{37} - 3 q^{39} + 4 q^{43} + q^{47} - 6 q^{49} - 3 q^{51} - 5 q^{53} - 6 q^{57} - 7 q^{59} - 5 q^{61} + q^{63} - 3 q^{67} - 10 q^{71} + 10 q^{73} - 19 q^{77} - 16 q^{79} + 3 q^{81} - 11 q^{83} - q^{87} - 8 q^{89} - q^{91} - 7 q^{93} - 26 q^{97} - 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 - 3 * q^17 - 6 * q^19 + q^21 + 3 * q^27 - q^29 - 7 * q^31 - 3 * q^33 - 14 * q^37 - 3 * q^39 + 4 * q^43 + q^47 - 6 * q^49 - 3 * q^51 - 5 * q^53 - 6 * q^57 - 7 * q^59 - 5 * q^61 + q^63 - 3 * q^67 - 10 * q^71 + 10 * q^73 - 19 * q^77 - 16 * q^79 + 3 * q^81 - 11 * q^83 - q^87 - 8 * q^89 - q^91 - 7 * q^93 - 26 * q^97 - 3 * 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
$$7$$ −0.476452 −0.180082 −0.0900410 0.995938i $$-0.528700\pi$$
−0.0900410 + 0.995938i $$0.528700\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.34364 −1.00814 −0.504072 0.863661i $$-0.668165\pi$$
−0.504072 + 0.863661i $$0.668165\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 0 0
$$19$$ −6.77299 −1.55383 −0.776916 0.629605i $$-0.783217\pi$$
−0.776916 + 0.629605i $$0.783217\pi$$
$$20$$ 0 0
$$21$$ −0.476452 −0.103970
$$22$$ 0 0
$$23$$ 4.77299 0.995238 0.497619 0.867396i $$-0.334208\pi$$
0.497619 + 0.867396i $$0.334208\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 7.59308 1.41000 0.705000 0.709207i $$-0.250947\pi$$
0.705000 + 0.709207i $$0.250947\pi$$
$$30$$ 0 0
$$31$$ 7.93672 1.42548 0.712739 0.701430i $$-0.247455\pi$$
0.712739 + 0.701430i $$0.247455\pi$$
$$32$$ 0 0
$$33$$ −3.34364 −0.582053
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.82009 −1.28561 −0.642807 0.766028i $$-0.722230\pi$$
−0.642807 + 0.766028i $$0.722230\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 9.46027 1.47745 0.738723 0.674009i $$-0.235429\pi$$
0.738723 + 0.674009i $$0.235429\pi$$
$$42$$ 0 0
$$43$$ −1.82009 −0.277561 −0.138781 0.990323i $$-0.544318\pi$$
−0.138781 + 0.990323i $$0.544318\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 9.06953 1.32293 0.661464 0.749977i $$-0.269936\pi$$
0.661464 + 0.749977i $$0.269936\pi$$
$$48$$ 0 0
$$49$$ −6.77299 −0.967570
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 0 0
$$53$$ −9.50736 −1.30594 −0.652968 0.757385i $$-0.726477\pi$$
−0.652968 + 0.757385i $$0.726477\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −6.77299 −0.897105
$$58$$ 0 0
$$59$$ 3.16373 0.411882 0.205941 0.978564i $$-0.433974\pi$$
0.205941 + 0.978564i $$0.433974\pi$$
$$60$$ 0 0
$$61$$ −9.59308 −1.22827 −0.614134 0.789202i $$-0.710495\pi$$
−0.614134 + 0.789202i $$0.710495\pi$$
$$62$$ 0 0
$$63$$ −0.476452 −0.0600273
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1.34364 0.164151 0.0820757 0.996626i $$-0.473845\pi$$
0.0820757 + 0.996626i $$0.473845\pi$$
$$68$$ 0 0
$$69$$ 4.77299 0.574601
$$70$$ 0 0
$$71$$ −4.86719 −0.577629 −0.288814 0.957385i $$-0.593261\pi$$
−0.288814 + 0.957385i $$0.593261\pi$$
$$72$$ 0 0
$$73$$ 14.4132 1.68693 0.843467 0.537181i $$-0.180511\pi$$
0.843467 + 0.537181i $$0.180511\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.59308 0.181549
$$78$$ 0 0
$$79$$ −2.17991 −0.245259 −0.122630 0.992453i $$-0.539133\pi$$
−0.122630 + 0.992453i $$0.539133\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −9.16373 −1.00585 −0.502925 0.864330i $$-0.667743\pi$$
−0.502925 + 0.864330i $$0.667743\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 7.59308 0.814064
$$88$$ 0 0
$$89$$ −10.5931 −1.12286 −0.561432 0.827523i $$-0.689749\pi$$
−0.561432 + 0.827523i $$0.689749\pi$$
$$90$$ 0 0
$$91$$ 0.476452 0.0499457
$$92$$ 0 0
$$93$$ 7.93672 0.823000
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.04710 −0.715524 −0.357762 0.933813i $$-0.616460\pi$$
−0.357762 + 0.933813i $$0.616460\pi$$
$$98$$ 0 0
$$99$$ −3.34364 −0.336048
$$100$$ 0 0
$$101$$ −18.3661 −1.82749 −0.913746 0.406285i $$-0.866824\pi$$
−0.913746 + 0.406285i $$0.866824\pi$$
$$102$$ 0 0
$$103$$ 13.5545 1.33556 0.667780 0.744358i $$-0.267245\pi$$
0.667780 + 0.744358i $$0.267245\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.2333 −1.56933 −0.784664 0.619921i $$-0.787165\pi$$
−0.784664 + 0.619921i $$0.787165\pi$$
$$108$$ 0 0
$$109$$ −16.4132 −1.57210 −0.786048 0.618165i $$-0.787876\pi$$
−0.786048 + 0.618165i $$0.787876\pi$$
$$110$$ 0 0
$$111$$ −7.82009 −0.742250
$$112$$ 0 0
$$113$$ −8.59308 −0.808369 −0.404185 0.914677i $$-0.632445\pi$$
−0.404185 + 0.914677i $$0.632445\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 0.476452 0.0436763
$$120$$ 0 0
$$121$$ 0.179911 0.0163555
$$122$$ 0 0
$$123$$ 9.46027 0.853004
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2.86719 −0.254422 −0.127211 0.991876i $$-0.540602\pi$$
−0.127211 + 0.991876i $$0.540602\pi$$
$$128$$ 0 0
$$129$$ −1.82009 −0.160250
$$130$$ 0 0
$$131$$ −16.1475 −1.41082 −0.705409 0.708801i $$-0.749237\pi$$
−0.705409 + 0.708801i $$0.749237\pi$$
$$132$$ 0 0
$$133$$ 3.22701 0.279817
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −22.2333 −1.89952 −0.949758 0.312986i $$-0.898671\pi$$
−0.949758 + 0.312986i $$0.898671\pi$$
$$138$$ 0 0
$$139$$ 9.28036 0.787150 0.393575 0.919293i $$-0.371238\pi$$
0.393575 + 0.919293i $$0.371238\pi$$
$$140$$ 0 0
$$141$$ 9.06953 0.763793
$$142$$ 0 0
$$143$$ 3.34364 0.279609
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −6.77299 −0.558627
$$148$$ 0 0
$$149$$ −22.1391 −1.81370 −0.906852 0.421450i $$-0.861521\pi$$
−0.906852 + 0.421450i $$0.861521\pi$$
$$150$$ 0 0
$$151$$ −16.0224 −1.30389 −0.651944 0.758267i $$-0.726046\pi$$
−0.651944 + 0.758267i $$0.726046\pi$$
$$152$$ 0 0
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.22701 −0.496969 −0.248485 0.968636i $$-0.579933\pi$$
−0.248485 + 0.968636i $$0.579933\pi$$
$$158$$ 0 0
$$159$$ −9.50736 −0.753983
$$160$$ 0 0
$$161$$ −2.27410 −0.179224
$$162$$ 0 0
$$163$$ −17.6935 −1.38586 −0.692932 0.721003i $$-0.743681\pi$$
−0.692932 + 0.721003i $$0.743681\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −0.179911 −0.0139219 −0.00696095 0.999976i $$-0.502216\pi$$
−0.00696095 + 0.999976i $$0.502216\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −6.77299 −0.517944
$$172$$ 0 0
$$173$$ 12.4603 0.947337 0.473668 0.880703i $$-0.342930\pi$$
0.473668 + 0.880703i $$0.342930\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3.16373 0.237800
$$178$$ 0 0
$$179$$ −5.55446 −0.415160 −0.207580 0.978218i $$-0.566559\pi$$
−0.207580 + 0.978218i $$0.566559\pi$$
$$180$$ 0 0
$$181$$ 24.0063 1.78437 0.892185 0.451669i $$-0.149171\pi$$
0.892185 + 0.451669i $$0.149171\pi$$
$$182$$ 0 0
$$183$$ −9.59308 −0.709141
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.34364 0.244511
$$188$$ 0 0
$$189$$ −0.476452 −0.0346568
$$190$$ 0 0
$$191$$ 3.18617 0.230543 0.115271 0.993334i $$-0.463226\pi$$
0.115271 + 0.993334i $$0.463226\pi$$
$$192$$ 0 0
$$193$$ 14.8672 1.07016 0.535082 0.844800i $$-0.320281\pi$$
0.535082 + 0.844800i $$0.320281\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −23.7259 −1.69040 −0.845200 0.534450i $$-0.820519\pi$$
−0.845200 + 0.534450i $$0.820519\pi$$
$$198$$ 0 0
$$199$$ −9.64018 −0.683374 −0.341687 0.939814i $$-0.610998\pi$$
−0.341687 + 0.939814i $$0.610998\pi$$
$$200$$ 0 0
$$201$$ 1.34364 0.0947729
$$202$$ 0 0
$$203$$ −3.61774 −0.253916
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.77299 0.331746
$$208$$ 0 0
$$209$$ 22.6464 1.56649
$$210$$ 0 0
$$211$$ −10.8672 −0.748128 −0.374064 0.927403i $$-0.622036\pi$$
−0.374064 + 0.927403i $$0.622036\pi$$
$$212$$ 0 0
$$213$$ −4.86719 −0.333494
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.78147 −0.256703
$$218$$ 0 0
$$219$$ 14.4132 0.973952
$$220$$ 0 0
$$221$$ 1.00000 0.0672673
$$222$$ 0 0
$$223$$ 22.2417 1.48942 0.744708 0.667390i $$-0.232589\pi$$
0.744708 + 0.667390i $$0.232589\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −23.4827 −1.55860 −0.779301 0.626650i $$-0.784426\pi$$
−0.779301 + 0.626650i $$0.784426\pi$$
$$228$$ 0 0
$$229$$ 28.3723 1.87490 0.937448 0.348125i $$-0.113181\pi$$
0.937448 + 0.348125i $$0.113181\pi$$
$$230$$ 0 0
$$231$$ 1.59308 0.104817
$$232$$ 0 0
$$233$$ 16.5931 1.08705 0.543524 0.839393i $$-0.317089\pi$$
0.543524 + 0.839393i $$0.317089\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −2.17991 −0.141600
$$238$$ 0 0
$$239$$ 12.0224 0.777667 0.388833 0.921308i $$-0.372878\pi$$
0.388833 + 0.921308i $$0.372878\pi$$
$$240$$ 0 0
$$241$$ −23.6935 −1.52623 −0.763117 0.646260i $$-0.776332\pi$$
−0.763117 + 0.646260i $$0.776332\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.77299 0.430955
$$248$$ 0 0
$$249$$ −9.16373 −0.580728
$$250$$ 0 0
$$251$$ 18.7406 1.18290 0.591449 0.806342i $$-0.298556\pi$$
0.591449 + 0.806342i $$0.298556\pi$$
$$252$$ 0 0
$$253$$ −15.9592 −1.00334
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 9.24173 0.576484 0.288242 0.957558i $$-0.406929\pi$$
0.288242 + 0.957558i $$0.406929\pi$$
$$258$$ 0 0
$$259$$ 3.72590 0.231516
$$260$$ 0 0
$$261$$ 7.59308 0.470000
$$262$$ 0 0
$$263$$ 17.9143 1.10464 0.552321 0.833632i $$-0.313742\pi$$
0.552321 + 0.833632i $$0.313742\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −10.5931 −0.648286
$$268$$ 0 0
$$269$$ −6.28036 −0.382920 −0.191460 0.981500i $$-0.561322\pi$$
−0.191460 + 0.981500i $$0.561322\pi$$
$$270$$ 0 0
$$271$$ −18.2881 −1.11092 −0.555461 0.831543i $$-0.687458\pi$$
−0.555461 + 0.831543i $$0.687458\pi$$
$$272$$ 0 0
$$273$$ 0.476452 0.0288362
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −12.7645 −0.766946 −0.383473 0.923552i $$-0.625272\pi$$
−0.383473 + 0.923552i $$0.625272\pi$$
$$278$$ 0 0
$$279$$ 7.93672 0.475159
$$280$$ 0 0
$$281$$ −26.8263 −1.60033 −0.800163 0.599783i $$-0.795254\pi$$
−0.800163 + 0.599783i $$0.795254\pi$$
$$282$$ 0 0
$$283$$ 21.4518 1.27518 0.637588 0.770377i $$-0.279932\pi$$
0.637588 + 0.770377i $$0.279932\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.50736 −0.266061
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ −7.04710 −0.413108
$$292$$ 0 0
$$293$$ 11.3745 0.664508 0.332254 0.943190i $$-0.392191\pi$$
0.332254 + 0.943190i $$0.392191\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.34364 −0.194018
$$298$$ 0 0
$$299$$ −4.77299 −0.276029
$$300$$ 0 0
$$301$$ 0.867185 0.0499837
$$302$$ 0 0
$$303$$ −18.3661 −1.05510
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 14.0857 0.803914 0.401957 0.915658i $$-0.368330\pi$$
0.401957 + 0.915658i $$0.368330\pi$$
$$308$$ 0 0
$$309$$ 13.5545 0.771086
$$310$$ 0 0
$$311$$ −17.4518 −0.989601 −0.494800 0.869007i $$-0.664759\pi$$
−0.494800 + 0.869007i $$0.664759\pi$$
$$312$$ 0 0
$$313$$ 6.39844 0.361661 0.180831 0.983514i $$-0.442121\pi$$
0.180831 + 0.983514i $$0.442121\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5.58683 −0.313788 −0.156894 0.987615i $$-0.550148\pi$$
−0.156894 + 0.987615i $$0.550148\pi$$
$$318$$ 0 0
$$319$$ −25.3885 −1.42148
$$320$$ 0 0
$$321$$ −16.2333 −0.906052
$$322$$ 0 0
$$323$$ 6.77299 0.376859
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −16.4132 −0.907650
$$328$$ 0 0
$$329$$ −4.32120 −0.238235
$$330$$ 0 0
$$331$$ 10.4132 0.572360 0.286180 0.958176i $$-0.407615\pi$$
0.286180 + 0.958176i $$0.407615\pi$$
$$332$$ 0 0
$$333$$ −7.82009 −0.428538
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 16.3661 0.891517 0.445758 0.895153i $$-0.352934\pi$$
0.445758 + 0.895153i $$0.352934\pi$$
$$338$$ 0 0
$$339$$ −8.59308 −0.466712
$$340$$ 0 0
$$341$$ −26.5375 −1.43709
$$342$$ 0 0
$$343$$ 6.56217 0.354324
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 36.3723 1.95257 0.976285 0.216491i $$-0.0694613\pi$$
0.976285 + 0.216491i $$0.0694613\pi$$
$$348$$ 0 0
$$349$$ −17.7259 −0.948846 −0.474423 0.880297i $$-0.657343\pi$$
−0.474423 + 0.880297i $$0.657343\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −19.5375 −1.03988 −0.519938 0.854204i $$-0.674045\pi$$
−0.519938 + 0.854204i $$0.674045\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0.476452 0.0252165
$$358$$ 0 0
$$359$$ −5.92825 −0.312881 −0.156440 0.987687i $$-0.550002\pi$$
−0.156440 + 0.987687i $$0.550002\pi$$
$$360$$ 0 0
$$361$$ 26.8734 1.41439
$$362$$ 0 0
$$363$$ 0.179911 0.00944286
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 27.2804 1.42402 0.712012 0.702168i $$-0.247784\pi$$
0.712012 + 0.702168i $$0.247784\pi$$
$$368$$ 0 0
$$369$$ 9.46027 0.492482
$$370$$ 0 0
$$371$$ 4.52980 0.235176
$$372$$ 0 0
$$373$$ 27.2866 1.41285 0.706424 0.707789i $$-0.250307\pi$$
0.706424 + 0.707789i $$0.250307\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −7.59308 −0.391064
$$378$$ 0 0
$$379$$ −33.4503 −1.71823 −0.859114 0.511784i $$-0.828985\pi$$
−0.859114 + 0.511784i $$0.828985\pi$$
$$380$$ 0 0
$$381$$ −2.86719 −0.146890
$$382$$ 0 0
$$383$$ −2.60156 −0.132933 −0.0664666 0.997789i $$-0.521173\pi$$
−0.0664666 + 0.997789i $$0.521173\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.82009 −0.0925203
$$388$$ 0 0
$$389$$ −0.781467 −0.0396219 −0.0198110 0.999804i $$-0.506306\pi$$
−0.0198110 + 0.999804i $$0.506306\pi$$
$$390$$ 0 0
$$391$$ −4.77299 −0.241381
$$392$$ 0 0
$$393$$ −16.1475 −0.814536
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6.49889 0.326170 0.163085 0.986612i $$-0.447856\pi$$
0.163085 + 0.986612i $$0.447856\pi$$
$$398$$ 0 0
$$399$$ 3.22701 0.161552
$$400$$ 0 0
$$401$$ 21.0063 1.04900 0.524501 0.851410i $$-0.324252\pi$$
0.524501 + 0.851410i $$0.324252\pi$$
$$402$$ 0 0
$$403$$ −7.93672 −0.395356
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 26.1475 1.29609
$$408$$ 0 0
$$409$$ −8.08572 −0.399813 −0.199907 0.979815i $$-0.564064\pi$$
−0.199907 + 0.979815i $$0.564064\pi$$
$$410$$ 0 0
$$411$$ −22.2333 −1.09669
$$412$$ 0 0
$$413$$ −1.50736 −0.0741725
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 9.28036 0.454461
$$418$$ 0 0
$$419$$ 1.91428 0.0935188 0.0467594 0.998906i $$-0.485111\pi$$
0.0467594 + 0.998906i $$0.485111\pi$$
$$420$$ 0 0
$$421$$ −27.1089 −1.32121 −0.660604 0.750735i $$-0.729700\pi$$
−0.660604 + 0.750735i $$0.729700\pi$$
$$422$$ 0 0
$$423$$ 9.06953 0.440976
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.57064 0.221189
$$428$$ 0 0
$$429$$ 3.34364 0.161432
$$430$$ 0 0
$$431$$ −13.0386 −0.628048 −0.314024 0.949415i $$-0.601677\pi$$
−0.314024 + 0.949415i $$0.601677\pi$$
$$432$$ 0 0
$$433$$ 26.3275 1.26522 0.632608 0.774472i $$-0.281984\pi$$
0.632608 + 0.774472i $$0.281984\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −32.3275 −1.54643
$$438$$ 0 0
$$439$$ 7.69353 0.367192 0.183596 0.983002i $$-0.441226\pi$$
0.183596 + 0.983002i $$0.441226\pi$$
$$440$$ 0 0
$$441$$ −6.77299 −0.322523
$$442$$ 0 0
$$443$$ −23.7259 −1.12725 −0.563626 0.826030i $$-0.690594\pi$$
−0.563626 + 0.826030i $$0.690594\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −22.1391 −1.04714
$$448$$ 0 0
$$449$$ 9.73437 0.459393 0.229697 0.973262i $$-0.426227\pi$$
0.229697 + 0.973262i $$0.426227\pi$$
$$450$$ 0 0
$$451$$ −31.6317 −1.48948
$$452$$ 0 0
$$453$$ −16.0224 −0.752800
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.64865 0.264233 0.132116 0.991234i $$-0.457823\pi$$
0.132116 + 0.991234i $$0.457823\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −12.4540 −0.580041 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$462$$ 0 0
$$463$$ −14.2642 −0.662912 −0.331456 0.943471i $$-0.607540\pi$$
−0.331456 + 0.943471i $$0.607540\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 18.7815 0.869103 0.434551 0.900647i $$-0.356907\pi$$
0.434551 + 0.900647i $$0.356907\pi$$
$$468$$ 0 0
$$469$$ −0.640179 −0.0295607
$$470$$ 0 0
$$471$$ −6.22701 −0.286925
$$472$$ 0 0
$$473$$ 6.08572 0.279822
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −9.50736 −0.435312
$$478$$ 0 0
$$479$$ −12.7954 −0.584638 −0.292319 0.956321i $$-0.594427\pi$$
−0.292319 + 0.956321i $$0.594427\pi$$
$$480$$ 0 0
$$481$$ 7.82009 0.356565
$$482$$ 0 0
$$483$$ −2.27410 −0.103475
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −15.4378 −0.699555 −0.349777 0.936833i $$-0.613743\pi$$
−0.349777 + 0.936833i $$0.613743\pi$$
$$488$$ 0 0
$$489$$ −17.6935 −0.800129
$$490$$ 0 0
$$491$$ −11.3745 −0.513326 −0.256663 0.966501i $$-0.582623\pi$$
−0.256663 + 0.966501i $$0.582623\pi$$
$$492$$ 0 0
$$493$$ −7.59308 −0.341975
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.31898 0.104020
$$498$$ 0 0
$$499$$ 43.7075 1.95662 0.978308 0.207155i $$-0.0664205\pi$$
0.978308 + 0.207155i $$0.0664205\pi$$
$$500$$ 0 0
$$501$$ −0.179911 −0.00803781
$$502$$ 0 0
$$503$$ −10.5846 −0.471944 −0.235972 0.971760i $$-0.575827\pi$$
−0.235972 + 0.971760i $$0.575827\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ 4.05335 0.179662 0.0898308 0.995957i $$-0.471367\pi$$
0.0898308 + 0.995957i $$0.471367\pi$$
$$510$$ 0 0
$$511$$ −6.86719 −0.303786
$$512$$ 0 0
$$513$$ −6.77299 −0.299035
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −30.3252 −1.33370
$$518$$ 0 0
$$519$$ 12.4603 0.546945
$$520$$ 0 0
$$521$$ 19.1862 0.840561 0.420281 0.907394i $$-0.361932\pi$$
0.420281 + 0.907394i $$0.361932\pi$$
$$522$$ 0 0
$$523$$ 40.2333 1.75928 0.879639 0.475642i $$-0.157784\pi$$
0.879639 + 0.475642i $$0.157784\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −7.93672 −0.345729
$$528$$ 0 0
$$529$$ −0.218533 −0.00950146
$$530$$ 0 0
$$531$$ 3.16373 0.137294
$$532$$ 0 0
$$533$$ −9.46027 −0.409770
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −5.55446 −0.239693
$$538$$ 0 0
$$539$$ 22.6464 0.975451
$$540$$ 0 0
$$541$$ 43.7708 1.88185 0.940926 0.338611i $$-0.109957\pi$$
0.940926 + 0.338611i $$0.109957\pi$$
$$542$$ 0 0
$$543$$ 24.0063 1.03021
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.72590 −0.0737940 −0.0368970 0.999319i $$-0.511747\pi$$
−0.0368970 + 0.999319i $$0.511747\pi$$
$$548$$ 0 0
$$549$$ −9.59308 −0.409423
$$550$$ 0 0
$$551$$ −51.4279 −2.19090
$$552$$ 0 0
$$553$$ 1.03862 0.0441667
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −16.6016 −0.703430 −0.351715 0.936107i $$-0.614401\pi$$
−0.351715 + 0.936107i $$0.614401\pi$$
$$558$$ 0 0
$$559$$ 1.82009 0.0769816
$$560$$ 0 0
$$561$$ 3.34364 0.141168
$$562$$ 0 0
$$563$$ 27.1453 1.14404 0.572020 0.820240i $$-0.306160\pi$$
0.572020 + 0.820240i $$0.306160\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −0.476452 −0.0200091
$$568$$ 0 0
$$569$$ −24.0232 −1.00710 −0.503552 0.863965i $$-0.667974\pi$$
−0.503552 + 0.863965i $$0.667974\pi$$
$$570$$ 0 0
$$571$$ 37.9592 1.58854 0.794271 0.607564i $$-0.207853\pi$$
0.794271 + 0.607564i $$0.207853\pi$$
$$572$$ 0 0
$$573$$ 3.18617 0.133104
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −39.7020 −1.65282 −0.826408 0.563072i $$-0.809619\pi$$
−0.826408 + 0.563072i $$0.809619\pi$$
$$578$$ 0 0
$$579$$ 14.8672 0.617859
$$580$$ 0 0
$$581$$ 4.36608 0.181135
$$582$$ 0 0
$$583$$ 31.7892 1.31657
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 32.6070 1.34584 0.672918 0.739717i $$-0.265040\pi$$
0.672918 + 0.739717i $$0.265040\pi$$
$$588$$ 0 0
$$589$$ −53.7554 −2.21495
$$590$$ 0 0
$$591$$ −23.7259 −0.975953
$$592$$ 0 0
$$593$$ 0.454013 0.0186441 0.00932204 0.999957i $$-0.497033\pi$$
0.00932204 + 0.999957i $$0.497033\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −9.64018 −0.394546
$$598$$ 0 0
$$599$$ −8.31898 −0.339904 −0.169952 0.985452i $$-0.554361\pi$$
−0.169952 + 0.985452i $$0.554361\pi$$
$$600$$ 0 0
$$601$$ 36.3337 1.48208 0.741041 0.671459i $$-0.234332\pi$$
0.741041 + 0.671459i $$0.234332\pi$$
$$602$$ 0 0
$$603$$ 1.34364 0.0547171
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −34.9654 −1.41920 −0.709601 0.704604i $$-0.751125\pi$$
−0.709601 + 0.704604i $$0.751125\pi$$
$$608$$ 0 0
$$609$$ −3.61774 −0.146598
$$610$$ 0 0
$$611$$ −9.06953 −0.366914
$$612$$ 0 0
$$613$$ −15.3745 −0.620972 −0.310486 0.950578i $$-0.600492\pi$$
−0.310486 + 0.950578i $$0.600492\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11.8734 0.478007 0.239003 0.971019i $$-0.423179\pi$$
0.239003 + 0.971019i $$0.423179\pi$$
$$618$$ 0 0
$$619$$ 13.1946 0.530337 0.265169 0.964202i $$-0.414572\pi$$
0.265169 + 0.964202i $$0.414572\pi$$
$$620$$ 0 0
$$621$$ 4.77299 0.191534
$$622$$ 0 0
$$623$$ 5.04710 0.202208
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 22.6464 0.904411
$$628$$ 0 0
$$629$$ 7.82009 0.311807
$$630$$ 0 0
$$631$$ −2.63392 −0.104855 −0.0524274 0.998625i $$-0.516696\pi$$
−0.0524274 + 0.998625i $$0.516696\pi$$
$$632$$ 0 0
$$633$$ −10.8672 −0.431932
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.77299 0.268356
$$638$$ 0 0
$$639$$ −4.86719 −0.192543
$$640$$ 0 0
$$641$$ −18.4687 −0.729471 −0.364736 0.931111i $$-0.618841\pi$$
−0.364736 + 0.931111i $$0.618841\pi$$
$$642$$ 0 0
$$643$$ 36.5074 1.43971 0.719855 0.694125i $$-0.244208\pi$$
0.719855 + 0.694125i $$0.244208\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 1.40692 0.0553116 0.0276558 0.999618i $$-0.491196\pi$$
0.0276558 + 0.999618i $$0.491196\pi$$
$$648$$ 0 0
$$649$$ −10.5784 −0.415237
$$650$$ 0 0
$$651$$ −3.78147 −0.148207
$$652$$ 0 0
$$653$$ 38.7469 1.51628 0.758141 0.652090i $$-0.226108\pi$$
0.758141 + 0.652090i $$0.226108\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 14.4132 0.562311
$$658$$ 0 0
$$659$$ 28.0857 1.09406 0.547032 0.837112i $$-0.315758\pi$$
0.547032 + 0.837112i $$0.315758\pi$$
$$660$$ 0 0
$$661$$ −45.2071 −1.75835 −0.879177 0.476495i $$-0.841907\pi$$
−0.879177 + 0.476495i $$0.841907\pi$$
$$662$$ 0 0
$$663$$ 1.00000 0.0388368
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 36.2417 1.40329
$$668$$ 0 0
$$669$$ 22.2417 0.859915
$$670$$ 0 0
$$671$$ 32.0758 1.23827
$$672$$ 0 0
$$673$$ 21.2741 0.820056 0.410028 0.912073i $$-0.365519\pi$$
0.410028 + 0.912073i $$0.365519\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −19.5460 −0.751213 −0.375607 0.926779i $$-0.622566\pi$$
−0.375607 + 0.926779i $$0.622566\pi$$
$$678$$ 0 0
$$679$$ 3.35760 0.128853
$$680$$ 0 0
$$681$$ −23.4827 −0.899859
$$682$$ 0 0
$$683$$ −7.79765 −0.298369 −0.149184 0.988809i $$-0.547665\pi$$
−0.149184 + 0.988809i $$0.547665\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 28.3723 1.08247
$$688$$ 0 0
$$689$$ 9.50736 0.362202
$$690$$ 0 0
$$691$$ −20.7267 −0.788479 −0.394240 0.919008i $$-0.628992\pi$$
−0.394240 + 0.919008i $$0.628992\pi$$
$$692$$ 0 0
$$693$$ 1.59308 0.0605162
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −9.46027 −0.358333
$$698$$ 0 0
$$699$$ 16.5931 0.627608
$$700$$ 0 0
$$701$$ −32.7792 −1.23806 −0.619028 0.785369i $$-0.712473\pi$$
−0.619028 + 0.785369i $$0.712473\pi$$
$$702$$ 0 0
$$703$$ 52.9654 1.99763
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.75055 0.329098
$$708$$ 0 0
$$709$$ 4.08572 0.153442 0.0767212 0.997053i $$-0.475555\pi$$
0.0767212 + 0.997053i $$0.475555\pi$$
$$710$$ 0 0
$$711$$ −2.17991 −0.0817530
$$712$$ 0 0
$$713$$ 37.8819 1.41869
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 12.0224 0.448986
$$718$$ 0 0
$$719$$ 30.0857 1.12201 0.561004 0.827813i $$-0.310415\pi$$
0.561004 + 0.827813i $$0.310415\pi$$
$$720$$ 0 0
$$721$$ −6.45805 −0.240510
$$722$$ 0 0
$$723$$ −23.6935 −0.881172
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −18.9121 −0.701410 −0.350705 0.936486i $$-0.614058\pi$$
−0.350705 + 0.936486i $$0.614058\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.82009 0.0673184
$$732$$ 0 0
$$733$$ −30.3893 −1.12245 −0.561227 0.827662i $$-0.689670\pi$$
−0.561227 + 0.827662i $$0.689670\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −4.49264 −0.165488
$$738$$ 0 0
$$739$$ −5.60927 −0.206340 −0.103170 0.994664i $$-0.532899\pi$$
−0.103170 + 0.994664i $$0.532899\pi$$
$$740$$ 0 0
$$741$$ 6.77299 0.248812
$$742$$ 0 0
$$743$$ −9.51507 −0.349074 −0.174537 0.984651i $$-0.555843\pi$$
−0.174537 + 0.984651i $$0.555843\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −9.16373 −0.335283
$$748$$ 0 0
$$749$$ 7.73437 0.282608
$$750$$ 0 0
$$751$$ −31.5221 −1.15026 −0.575129 0.818063i $$-0.695048\pi$$
−0.575129 + 0.818063i $$0.695048\pi$$
$$752$$ 0 0
$$753$$ 18.7406 0.682946
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 25.5607 0.929020 0.464510 0.885568i $$-0.346230\pi$$
0.464510 + 0.885568i $$0.346230\pi$$
$$758$$ 0 0
$$759$$ −15.9592 −0.579281
$$760$$ 0 0
$$761$$ 10.4132 0.377477 0.188739 0.982027i $$-0.439560\pi$$
0.188739 + 0.982027i $$0.439560\pi$$
$$762$$ 0 0
$$763$$ 7.82009 0.283106
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −3.16373 −0.114236
$$768$$ 0 0
$$769$$ −2.95290 −0.106484 −0.0532422 0.998582i $$-0.516956\pi$$
−0.0532422 + 0.998582i $$0.516956\pi$$
$$770$$ 0 0
$$771$$ 9.24173 0.332833
$$772$$ 0 0
$$773$$ −11.0471 −0.397336 −0.198668 0.980067i $$-0.563662\pi$$
−0.198668 + 0.980067i $$0.563662\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 3.72590 0.133666
$$778$$ 0 0
$$779$$ −64.0743 −2.29570
$$780$$ 0 0
$$781$$ 16.2741 0.582333
$$782$$ 0 0
$$783$$ 7.59308 0.271355
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 15.1553 0.540226 0.270113 0.962829i $$-0.412939\pi$$
0.270113 + 0.962829i $$0.412939\pi$$
$$788$$ 0 0
$$789$$ 17.9143 0.637765
$$790$$ 0 0
$$791$$ 4.09419 0.145573
$$792$$ 0 0
$$793$$ 9.59308 0.340660
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −38.7962 −1.37423 −0.687116 0.726548i $$-0.741124\pi$$
−0.687116 + 0.726548i $$0.741124\pi$$
$$798$$ 0 0
$$799$$ −9.06953 −0.320857
$$800$$ 0 0
$$801$$ −10.5931 −0.374288
$$802$$ 0 0
$$803$$ −48.1924 −1.70067
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.28036 −0.221079
$$808$$ 0 0
$$809$$ −50.1391 −1.76280 −0.881398 0.472375i $$-0.843397\pi$$
−0.881398 + 0.472375i $$0.843397\pi$$
$$810$$ 0 0
$$811$$ −17.5684 −0.616911 −0.308455 0.951239i $$-0.599812\pi$$
−0.308455 + 0.951239i $$0.599812\pi$$
$$812$$ 0 0
$$813$$ −18.2881 −0.641391
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 12.3275 0.431283
$$818$$ 0 0
$$819$$ 0.476452 0.0166486
$$820$$ 0 0
$$821$$ 1.19464 0.0416932 0.0208466 0.999783i $$-0.493364\pi$$
0.0208466 + 0.999783i $$0.493364\pi$$
$$822$$ 0 0
$$823$$ −44.2458 −1.54231 −0.771155 0.636647i $$-0.780321\pi$$
−0.771155 + 0.636647i $$0.780321\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 6.84475 0.238015 0.119008 0.992893i $$-0.462029\pi$$
0.119008 + 0.992893i $$0.462029\pi$$
$$828$$ 0 0
$$829$$ 1.04488 0.0362901 0.0181451 0.999835i $$-0.494224\pi$$
0.0181451 + 0.999835i $$0.494224\pi$$
$$830$$ 0 0
$$831$$ −12.7645 −0.442796
$$832$$ 0 0
$$833$$ 6.77299 0.234670
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 7.93672 0.274333
$$838$$ 0 0
$$839$$ 38.2417 1.32025 0.660126 0.751155i $$-0.270503\pi$$
0.660126 + 0.751155i $$0.270503\pi$$
$$840$$ 0 0
$$841$$ 28.6549 0.988100
$$842$$ 0 0
$$843$$ −26.8263 −0.923948
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −0.0857188 −0.00294533
$$848$$ 0 0
$$849$$ 21.4518 0.736224
$$850$$ 0 0
$$851$$ −37.3252 −1.27949
$$852$$ 0 0
$$853$$ −44.9978 −1.54069 −0.770347 0.637624i $$-0.779917\pi$$
−0.770347 + 0.637624i $$0.779917\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −12.3723 −0.422631 −0.211315 0.977418i $$-0.567775\pi$$
−0.211315 + 0.977418i $$0.567775\pi$$
$$858$$ 0 0
$$859$$ −22.4750 −0.766837 −0.383418 0.923575i $$-0.625253\pi$$
−0.383418 + 0.923575i $$0.625253\pi$$
$$860$$ 0 0
$$861$$ −4.50736 −0.153611
$$862$$ 0 0
$$863$$ −12.1251 −0.412743 −0.206372 0.978474i $$-0.566166\pi$$
−0.206372 + 0.978474i $$0.566166\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ 7.28883 0.247257
$$870$$ 0 0
$$871$$ −1.34364 −0.0455274
$$872$$ 0 0
$$873$$ −7.04710 −0.238508
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 26.0449 0.879473 0.439737 0.898127i $$-0.355072\pi$$
0.439737 + 0.898127i $$0.355072\pi$$
$$878$$ 0 0
$$879$$ 11.3745 0.383654
$$880$$ 0 0
$$881$$ −13.4132 −0.451901 −0.225951 0.974139i $$-0.572549\pi$$
−0.225951 + 0.974139i $$0.572549\pi$$
$$882$$ 0 0
$$883$$ 28.5158 0.959634 0.479817 0.877369i $$-0.340703\pi$$
0.479817 + 0.877369i $$0.340703\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −30.6464 −1.02901 −0.514503 0.857488i $$-0.672024\pi$$
−0.514503 + 0.857488i $$0.672024\pi$$
$$888$$ 0 0
$$889$$ 1.36608 0.0458167
$$890$$ 0 0
$$891$$ −3.34364 −0.112016
$$892$$ 0 0
$$893$$ −61.4279 −2.05561
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −4.77299 −0.159366
$$898$$ 0 0
$$899$$ 60.2642 2.00992
$$900$$ 0 0
$$901$$ 9.50736 0.316736
$$902$$ 0 0
$$903$$ 0.867185 0.0288581
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 36.9978 1.22849 0.614246 0.789115i $$-0.289460\pi$$
0.614246 + 0.789115i $$0.289460\pi$$
$$908$$ 0 0
$$909$$ −18.3661 −0.609164
$$910$$ 0 0
$$911$$ 35.3337 1.17066 0.585329 0.810796i $$-0.300965\pi$$
0.585329 + 0.810796i $$0.300965\pi$$
$$912$$ 0 0
$$913$$ 30.6402 1.01404
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 7.69353 0.254063
$$918$$ 0 0
$$919$$ −35.6851 −1.17714 −0.588571 0.808446i $$-0.700309\pi$$
−0.588571 + 0.808446i $$0.700309\pi$$
$$920$$ 0 0
$$921$$ 14.0857 0.464140
$$922$$ 0 0
$$923$$ 4.86719 0.160205
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 13.5545 0.445187
$$928$$ 0 0
$$929$$ −48.2009 −1.58142 −0.790710 0.612191i $$-0.790288\pi$$
−0.790710 + 0.612191i $$0.790288\pi$$
$$930$$ 0 0
$$931$$ 45.8734 1.50344
$$932$$ 0 0
$$933$$ −17.4518 −0.571346
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −36.1538 −1.18109 −0.590547 0.807004i $$-0.701088\pi$$
−0.590547 + 0.807004i $$0.701088\pi$$
$$938$$ 0 0
$$939$$ 6.39844 0.208805
$$940$$ 0 0
$$941$$ −6.66629 −0.217315 −0.108657 0.994079i $$-0.534655\pi$$
−0.108657 + 0.994079i $$0.534655\pi$$
$$942$$ 0 0
$$943$$ 45.1538 1.47041
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 15.3760 0.499653 0.249827 0.968291i $$-0.419626\pi$$
0.249827 + 0.968291i $$0.419626\pi$$
$$948$$ 0 0
$$949$$ −14.4132 −0.467871
$$950$$ 0 0
$$951$$ −5.58683 −0.181165
$$952$$ 0 0
$$953$$ −7.10266 −0.230078 −0.115039 0.993361i $$-0.536699\pi$$
−0.115039 + 0.993361i $$0.536699\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −25.3885 −0.820694
$$958$$ 0 0
$$959$$ 10.5931 0.342068
$$960$$ 0 0
$$961$$ 31.9915 1.03198
$$962$$ 0 0
$$963$$ −16.2333 −0.523110
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 28.8573 0.927987 0.463993 0.885839i $$-0.346416\pi$$
0.463993 + 0.885839i $$0.346416\pi$$
$$968$$ 0 0
$$969$$ 6.77299 0.217580
$$970$$ 0 0
$$971$$ −25.5909 −0.821250 −0.410625 0.911804i $$-0.634689\pi$$
−0.410625 + 0.911804i $$0.634689\pi$$
$$972$$ 0 0
$$973$$ −4.42165 −0.141751
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −31.6402 −1.01226 −0.506129 0.862457i $$-0.668924\pi$$
−0.506129 + 0.862457i $$0.668924\pi$$
$$978$$ 0 0
$$979$$ 35.4194 1.13201
$$980$$ 0 0
$$981$$ −16.4132 −0.524032
$$982$$ 0 0
$$983$$ 5.12289 0.163395 0.0816973 0.996657i $$-0.473966\pi$$
0.0816973 + 0.996657i $$0.473966\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −4.32120 −0.137545
$$988$$ 0 0
$$989$$ −8.68727 −0.276239
$$990$$ 0 0
$$991$$ −26.3359 −0.836588 −0.418294 0.908312i $$-0.637372\pi$$
−0.418294 + 0.908312i $$0.637372\pi$$
$$992$$ 0 0
$$993$$ 10.4132 0.330452
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 42.0766 1.33258 0.666289 0.745694i $$-0.267882\pi$$
0.666289 + 0.745694i $$0.267882\pi$$
$$998$$ 0 0
$$999$$ −7.82009 −0.247417
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.br.1.2 yes 3
5.4 even 2 7800.2.a.bg.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bg.1.2 3 5.4 even 2
7800.2.a.br.1.2 yes 3 1.1 even 1 trivial