# Properties

 Label 7800.2.a.bw.1.1 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.10304.1 Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 8x + 8$$ x^4 - 2*x^3 - 7*x^2 + 8*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1560) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.692297$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} -2.97906 q^{11} -1.00000 q^{13} +1.44383 q^{17} +4.17113 q^{19} -2.82843 q^{21} +6.21302 q^{23} +1.00000 q^{27} -0.828427 q^{29} +1.65685 q^{31} -2.97906 q^{33} -0.0418875 q^{37} -1.00000 q^{39} -4.36365 q^{41} -8.99956 q^{43} -4.40554 q^{47} +1.00000 q^{49} +1.44383 q^{51} -6.72730 q^{53} +4.17113 q^{57} -2.97906 q^{59} +6.27226 q^{61} -2.82843 q^{63} +0.727302 q^{67} +6.21302 q^{69} -8.32176 q^{71} -6.87031 q^{73} +8.42604 q^{77} +6.11190 q^{79} +1.00000 q^{81} +14.7059 q^{83} -0.828427 q^{87} +14.7478 q^{89} +2.82843 q^{91} +1.65685 q^{93} -6.78654 q^{97} -2.97906 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 4 * q^9 $$4 q + 4 q^{3} + 4 q^{9} - 8 q^{11} - 4 q^{13} + 4 q^{17} - 12 q^{19} + 4 q^{23} + 4 q^{27} + 8 q^{29} - 16 q^{31} - 8 q^{33} - 8 q^{37} - 4 q^{39} - 4 q^{41} + 4 q^{43} - 12 q^{47} + 4 q^{49} + 4 q^{51} - 12 q^{57} - 8 q^{59} + 12 q^{61} - 24 q^{67} + 4 q^{69} - 12 q^{71} - 24 q^{73} - 8 q^{77} - 12 q^{79} + 4 q^{81} - 12 q^{83} + 8 q^{87} - 4 q^{89} - 16 q^{93} - 8 q^{97} - 8 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 + 4 * q^9 - 8 * q^11 - 4 * q^13 + 4 * q^17 - 12 * q^19 + 4 * q^23 + 4 * q^27 + 8 * q^29 - 16 * q^31 - 8 * q^33 - 8 * q^37 - 4 * q^39 - 4 * q^41 + 4 * q^43 - 12 * q^47 + 4 * q^49 + 4 * q^51 - 12 * q^57 - 8 * q^59 + 12 * q^61 - 24 * q^67 + 4 * q^69 - 12 * q^71 - 24 * q^73 - 8 * q^77 - 12 * q^79 + 4 * q^81 - 12 * q^83 + 8 * q^87 - 4 * q^89 - 16 * q^93 - 8 * q^97 - 8 * 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
$$7$$ −2.82843 −1.06904 −0.534522 0.845154i $$-0.679509\pi$$
−0.534522 + 0.845154i $$0.679509\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.97906 −0.898219 −0.449110 0.893477i $$-0.648259\pi$$
−0.449110 + 0.893477i $$0.648259\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.44383 0.350181 0.175090 0.984552i $$-0.443978\pi$$
0.175090 + 0.984552i $$0.443978\pi$$
$$18$$ 0 0
$$19$$ 4.17113 0.956924 0.478462 0.878108i $$-0.341194\pi$$
0.478462 + 0.878108i $$0.341194\pi$$
$$20$$ 0 0
$$21$$ −2.82843 −0.617213
$$22$$ 0 0
$$23$$ 6.21302 1.29550 0.647752 0.761851i $$-0.275709\pi$$
0.647752 + 0.761851i $$0.275709\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −0.828427 −0.153835 −0.0769175 0.997037i $$-0.524508\pi$$
−0.0769175 + 0.997037i $$0.524508\pi$$
$$30$$ 0 0
$$31$$ 1.65685 0.297580 0.148790 0.988869i $$-0.452462\pi$$
0.148790 + 0.988869i $$0.452462\pi$$
$$32$$ 0 0
$$33$$ −2.97906 −0.518587
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −0.0418875 −0.00688626 −0.00344313 0.999994i $$-0.501096\pi$$
−0.00344313 + 0.999994i $$0.501096\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −4.36365 −0.681488 −0.340744 0.940156i $$-0.610679\pi$$
−0.340744 + 0.940156i $$0.610679\pi$$
$$42$$ 0 0
$$43$$ −8.99956 −1.37242 −0.686210 0.727403i $$-0.740727\pi$$
−0.686210 + 0.727403i $$0.740727\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.40554 −0.642614 −0.321307 0.946975i $$-0.604122\pi$$
−0.321307 + 0.946975i $$0.604122\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 1.44383 0.202177
$$52$$ 0 0
$$53$$ −6.72730 −0.924066 −0.462033 0.886863i $$-0.652880\pi$$
−0.462033 + 0.886863i $$0.652880\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.17113 0.552480
$$58$$ 0 0
$$59$$ −2.97906 −0.387840 −0.193920 0.981017i $$-0.562120\pi$$
−0.193920 + 0.981017i $$0.562120\pi$$
$$60$$ 0 0
$$61$$ 6.27226 0.803081 0.401540 0.915841i $$-0.368475\pi$$
0.401540 + 0.915841i $$0.368475\pi$$
$$62$$ 0 0
$$63$$ −2.82843 −0.356348
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.727302 0.0888540 0.0444270 0.999013i $$-0.485854\pi$$
0.0444270 + 0.999013i $$0.485854\pi$$
$$68$$ 0 0
$$69$$ 6.21302 0.747960
$$70$$ 0 0
$$71$$ −8.32176 −0.987612 −0.493806 0.869572i $$-0.664395\pi$$
−0.493806 + 0.869572i $$0.664395\pi$$
$$72$$ 0 0
$$73$$ −6.87031 −0.804110 −0.402055 0.915616i $$-0.631704\pi$$
−0.402055 + 0.915616i $$0.631704\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.42604 0.960237
$$78$$ 0 0
$$79$$ 6.11190 0.687642 0.343821 0.939035i $$-0.388279\pi$$
0.343821 + 0.939035i $$0.388279\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 14.7059 1.61418 0.807092 0.590425i $$-0.201040\pi$$
0.807092 + 0.590425i $$0.201040\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −0.828427 −0.0888167
$$88$$ 0 0
$$89$$ 14.7478 1.56326 0.781632 0.623740i $$-0.214387\pi$$
0.781632 + 0.623740i $$0.214387\pi$$
$$90$$ 0 0
$$91$$ 2.82843 0.296500
$$92$$ 0 0
$$93$$ 1.65685 0.171808
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.78654 −0.689069 −0.344534 0.938774i $$-0.611963\pi$$
−0.344534 + 0.938774i $$0.611963\pi$$
$$98$$ 0 0
$$99$$ −2.97906 −0.299406
$$100$$ 0 0
$$101$$ 12.1421 1.20819 0.604094 0.796913i $$-0.293535\pi$$
0.604094 + 0.796913i $$0.293535\pi$$
$$102$$ 0 0
$$103$$ 2.81108 0.276984 0.138492 0.990364i $$-0.455775\pi$$
0.138492 + 0.990364i $$0.455775\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −18.3842 −1.77726 −0.888632 0.458621i $$-0.848343\pi$$
−0.888632 + 0.458621i $$0.848343\pi$$
$$108$$ 0 0
$$109$$ 1.44383 0.138294 0.0691470 0.997606i $$-0.477972\pi$$
0.0691470 + 0.997606i $$0.477972\pi$$
$$110$$ 0 0
$$111$$ −0.0418875 −0.00397579
$$112$$ 0 0
$$113$$ −3.40194 −0.320028 −0.160014 0.987115i $$-0.551154\pi$$
−0.160014 + 0.987115i $$0.551154\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ −4.08378 −0.374359
$$120$$ 0 0
$$121$$ −2.12522 −0.193202
$$122$$ 0 0
$$123$$ −4.36365 −0.393457
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −15.8106 −1.40297 −0.701484 0.712686i $$-0.747479\pi$$
−0.701484 + 0.712686i $$0.747479\pi$$
$$128$$ 0 0
$$129$$ −8.99956 −0.792367
$$130$$ 0 0
$$131$$ −19.1707 −1.67495 −0.837476 0.546475i $$-0.815970\pi$$
−0.837476 + 0.546475i $$0.815970\pi$$
$$132$$ 0 0
$$133$$ −11.7977 −1.02299
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1.67824 −0.143381 −0.0716907 0.997427i $$-0.522839\pi$$
−0.0716907 + 0.997427i $$0.522839\pi$$
$$138$$ 0 0
$$139$$ −12.3423 −1.04686 −0.523429 0.852069i $$-0.675347\pi$$
−0.523429 + 0.852069i $$0.675347\pi$$
$$140$$ 0 0
$$141$$ −4.40554 −0.371013
$$142$$ 0 0
$$143$$ 2.97906 0.249121
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ 0.677798 0.0555274 0.0277637 0.999615i $$-0.491161\pi$$
0.0277637 + 0.999615i $$0.491161\pi$$
$$150$$ 0 0
$$151$$ −17.9991 −1.46475 −0.732374 0.680903i $$-0.761588\pi$$
−0.732374 + 0.680903i $$0.761588\pi$$
$$152$$ 0 0
$$153$$ 1.44383 0.116727
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 12.8392 1.02468 0.512340 0.858783i $$-0.328779\pi$$
0.512340 + 0.858783i $$0.328779\pi$$
$$158$$ 0 0
$$159$$ −6.72730 −0.533510
$$160$$ 0 0
$$161$$ −17.5731 −1.38495
$$162$$ 0 0
$$163$$ −15.3137 −1.19946 −0.599731 0.800202i $$-0.704726\pi$$
−0.599731 + 0.800202i $$0.704726\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −15.3922 −1.19109 −0.595543 0.803324i $$-0.703063\pi$$
−0.595543 + 0.803324i $$0.703063\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 4.17113 0.318975
$$172$$ 0 0
$$173$$ 14.9706 1.13819 0.569095 0.822272i $$-0.307294\pi$$
0.569095 + 0.822272i $$0.307294\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −2.97906 −0.223920
$$178$$ 0 0
$$179$$ −7.63950 −0.571003 −0.285502 0.958378i $$-0.592160\pi$$
−0.285502 + 0.958378i $$0.592160\pi$$
$$180$$ 0 0
$$181$$ 15.0286 1.11706 0.558532 0.829483i $$-0.311365\pi$$
0.558532 + 0.829483i $$0.311365\pi$$
$$182$$ 0 0
$$183$$ 6.27226 0.463659
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4.30126 −0.314539
$$188$$ 0 0
$$189$$ −2.82843 −0.205738
$$190$$ 0 0
$$191$$ −10.8877 −0.787804 −0.393902 0.919152i $$-0.628875\pi$$
−0.393902 + 0.919152i $$0.628875\pi$$
$$192$$ 0 0
$$193$$ −4.94690 −0.356086 −0.178043 0.984023i $$-0.556977\pi$$
−0.178043 + 0.984023i $$0.556977\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −26.3628 −1.87827 −0.939135 0.343549i $$-0.888371\pi$$
−0.939135 + 0.343549i $$0.888371\pi$$
$$198$$ 0 0
$$199$$ 6.11190 0.433261 0.216630 0.976254i $$-0.430493\pi$$
0.216630 + 0.976254i $$0.430493\pi$$
$$200$$ 0 0
$$201$$ 0.727302 0.0512999
$$202$$ 0 0
$$203$$ 2.34315 0.164457
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.21302 0.431835
$$208$$ 0 0
$$209$$ −12.4260 −0.859527
$$210$$ 0 0
$$211$$ 9.68585 0.666802 0.333401 0.942785i $$-0.391804\pi$$
0.333401 + 0.942785i $$0.391804\pi$$
$$212$$ 0 0
$$213$$ −8.32176 −0.570198
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4.68629 −0.318126
$$218$$ 0 0
$$219$$ −6.87031 −0.464253
$$220$$ 0 0
$$221$$ −1.44383 −0.0971227
$$222$$ 0 0
$$223$$ −9.21258 −0.616920 −0.308460 0.951237i $$-0.599814\pi$$
−0.308460 + 0.951237i $$0.599814\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −20.3218 −1.34880 −0.674401 0.738365i $$-0.735598\pi$$
−0.674401 + 0.738365i $$0.735598\pi$$
$$228$$ 0 0
$$229$$ −16.4715 −1.08847 −0.544234 0.838933i $$-0.683180\pi$$
−0.544234 + 0.838933i $$0.683180\pi$$
$$230$$ 0 0
$$231$$ 8.42604 0.554393
$$232$$ 0 0
$$233$$ −2.37339 −0.155486 −0.0777428 0.996973i $$-0.524771\pi$$
−0.0777428 + 0.996973i $$0.524771\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 6.11190 0.397010
$$238$$ 0 0
$$239$$ −13.6363 −0.882062 −0.441031 0.897492i $$-0.645387\pi$$
−0.441031 + 0.897492i $$0.645387\pi$$
$$240$$ 0 0
$$241$$ 1.07045 0.0689536 0.0344768 0.999405i $$-0.489024\pi$$
0.0344768 + 0.999405i $$0.489024\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.17113 −0.265403
$$248$$ 0 0
$$249$$ 14.7059 0.931950
$$250$$ 0 0
$$251$$ 14.6262 0.923196 0.461598 0.887089i $$-0.347276\pi$$
0.461598 + 0.887089i $$0.347276\pi$$
$$252$$ 0 0
$$253$$ −18.5089 −1.16365
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −24.9403 −1.55573 −0.777867 0.628429i $$-0.783698\pi$$
−0.777867 + 0.628429i $$0.783698\pi$$
$$258$$ 0 0
$$259$$ 0.118476 0.00736173
$$260$$ 0 0
$$261$$ −0.828427 −0.0512784
$$262$$ 0 0
$$263$$ 9.01691 0.556007 0.278003 0.960580i $$-0.410327\pi$$
0.278003 + 0.960580i $$0.410327\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 14.7478 0.902551
$$268$$ 0 0
$$269$$ 8.28391 0.505079 0.252539 0.967587i $$-0.418734\pi$$
0.252539 + 0.967587i $$0.418734\pi$$
$$270$$ 0 0
$$271$$ −19.9572 −1.21232 −0.606158 0.795344i $$-0.707290\pi$$
−0.606158 + 0.795344i $$0.707290\pi$$
$$272$$ 0 0
$$273$$ 2.82843 0.171184
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.6145 1.35878 0.679388 0.733780i $$-0.262245\pi$$
0.679388 + 0.733780i $$0.262245\pi$$
$$278$$ 0 0
$$279$$ 1.65685 0.0991933
$$280$$ 0 0
$$281$$ −1.89572 −0.113089 −0.0565446 0.998400i $$-0.518008\pi$$
−0.0565446 + 0.998400i $$0.518008\pi$$
$$282$$ 0 0
$$283$$ 11.2299 0.667550 0.333775 0.942653i $$-0.391677\pi$$
0.333775 + 0.942653i $$0.391677\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.3423 0.728541
$$288$$ 0 0
$$289$$ −14.9153 −0.877373
$$290$$ 0 0
$$291$$ −6.78654 −0.397834
$$292$$ 0 0
$$293$$ −27.6774 −1.61693 −0.808464 0.588545i $$-0.799701\pi$$
−0.808464 + 0.588545i $$0.799701\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2.97906 −0.172862
$$298$$ 0 0
$$299$$ −6.21302 −0.359308
$$300$$ 0 0
$$301$$ 25.4546 1.46718
$$302$$ 0 0
$$303$$ 12.1421 0.697547
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −24.9858 −1.42601 −0.713007 0.701157i $$-0.752667\pi$$
−0.713007 + 0.701157i $$0.752667\pi$$
$$308$$ 0 0
$$309$$ 2.81108 0.159917
$$310$$ 0 0
$$311$$ 18.3842 1.04247 0.521235 0.853413i $$-0.325472\pi$$
0.521235 + 0.853413i $$0.325472\pi$$
$$312$$ 0 0
$$313$$ 8.61453 0.486922 0.243461 0.969911i $$-0.421717\pi$$
0.243461 + 0.969911i $$0.421717\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −15.0491 −0.845240 −0.422620 0.906307i $$-0.638889\pi$$
−0.422620 + 0.906307i $$0.638889\pi$$
$$318$$ 0 0
$$319$$ 2.46793 0.138178
$$320$$ 0 0
$$321$$ −18.3842 −1.02610
$$322$$ 0 0
$$323$$ 6.02242 0.335096
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 1.44383 0.0798441
$$328$$ 0 0
$$329$$ 12.4607 0.686983
$$330$$ 0 0
$$331$$ 22.6738 1.24626 0.623131 0.782117i $$-0.285860\pi$$
0.623131 + 0.782117i $$0.285860\pi$$
$$332$$ 0 0
$$333$$ −0.0418875 −0.00229542
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −27.9282 −1.52135 −0.760674 0.649134i $$-0.775131\pi$$
−0.760674 + 0.649134i $$0.775131\pi$$
$$338$$ 0 0
$$339$$ −3.40194 −0.184768
$$340$$ 0 0
$$341$$ −4.93586 −0.267292
$$342$$ 0 0
$$343$$ 16.9706 0.916324
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −19.7398 −1.05969 −0.529843 0.848096i $$-0.677749\pi$$
−0.529843 + 0.848096i $$0.677749\pi$$
$$348$$ 0 0
$$349$$ 30.7157 1.64417 0.822086 0.569364i $$-0.192810\pi$$
0.822086 + 0.569364i $$0.192810\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ 17.0072 0.905201 0.452600 0.891713i $$-0.350496\pi$$
0.452600 + 0.891713i $$0.350496\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4.08378 −0.216136
$$358$$ 0 0
$$359$$ −7.90203 −0.417053 −0.208527 0.978017i $$-0.566867\pi$$
−0.208527 + 0.978017i $$0.566867\pi$$
$$360$$ 0 0
$$361$$ −1.60164 −0.0842968
$$362$$ 0 0
$$363$$ −2.12522 −0.111545
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 34.1810 1.78424 0.892118 0.451803i $$-0.149219\pi$$
0.892118 + 0.451803i $$0.149219\pi$$
$$368$$ 0 0
$$369$$ −4.36365 −0.227163
$$370$$ 0 0
$$371$$ 19.0277 0.987868
$$372$$ 0 0
$$373$$ −18.5098 −0.958402 −0.479201 0.877705i $$-0.659074\pi$$
−0.479201 + 0.877705i $$0.659074\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0.828427 0.0426662
$$378$$ 0 0
$$379$$ −12.7994 −0.657462 −0.328731 0.944424i $$-0.606621\pi$$
−0.328731 + 0.944424i $$0.606621\pi$$
$$380$$ 0 0
$$381$$ −15.8106 −0.810004
$$382$$ 0 0
$$383$$ 0.664032 0.0339304 0.0169652 0.999856i $$-0.494600\pi$$
0.0169652 + 0.999856i $$0.494600\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.99956 −0.457473
$$388$$ 0 0
$$389$$ 2.32580 0.117923 0.0589613 0.998260i $$-0.481221\pi$$
0.0589613 + 0.998260i $$0.481221\pi$$
$$390$$ 0 0
$$391$$ 8.97056 0.453661
$$392$$ 0 0
$$393$$ −19.1707 −0.967034
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −18.2433 −0.915603 −0.457802 0.889054i $$-0.651363\pi$$
−0.457802 + 0.889054i $$0.651363\pi$$
$$398$$ 0 0
$$399$$ −11.7977 −0.590626
$$400$$ 0 0
$$401$$ 16.9919 0.848537 0.424269 0.905536i $$-0.360531\pi$$
0.424269 + 0.905536i $$0.360531\pi$$
$$402$$ 0 0
$$403$$ −1.65685 −0.0825338
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0.124785 0.00618538
$$408$$ 0 0
$$409$$ −10.8111 −0.534573 −0.267287 0.963617i $$-0.586127\pi$$
−0.267287 + 0.963617i $$0.586127\pi$$
$$410$$ 0 0
$$411$$ −1.67824 −0.0827813
$$412$$ 0 0
$$413$$ 8.42604 0.414618
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −12.3423 −0.604403
$$418$$ 0 0
$$419$$ −5.08780 −0.248555 −0.124278 0.992247i $$-0.539661\pi$$
−0.124278 + 0.992247i $$0.539661\pi$$
$$420$$ 0 0
$$421$$ 12.0945 0.589452 0.294726 0.955582i $$-0.404772\pi$$
0.294726 + 0.955582i $$0.404772\pi$$
$$422$$ 0 0
$$423$$ −4.40554 −0.214205
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −17.7406 −0.858529
$$428$$ 0 0
$$429$$ 2.97906 0.143830
$$430$$ 0 0
$$431$$ −8.48931 −0.408916 −0.204458 0.978875i $$-0.565543\pi$$
−0.204458 + 0.978875i $$0.565543\pi$$
$$432$$ 0 0
$$433$$ 13.8453 0.665365 0.332682 0.943039i $$-0.392046\pi$$
0.332682 + 0.943039i $$0.392046\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 25.9153 1.23970
$$438$$ 0 0
$$439$$ 0.602516 0.0287565 0.0143783 0.999897i $$-0.495423\pi$$
0.0143783 + 0.999897i $$0.495423\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −17.6721 −0.839626 −0.419813 0.907611i $$-0.637904\pi$$
−0.419813 + 0.907611i $$0.637904\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0.677798 0.0320587
$$448$$ 0 0
$$449$$ 3.33509 0.157393 0.0786963 0.996899i $$-0.474924\pi$$
0.0786963 + 0.996899i $$0.474924\pi$$
$$450$$ 0 0
$$451$$ 12.9996 0.612125
$$452$$ 0 0
$$453$$ −17.9991 −0.845673
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −38.9266 −1.82091 −0.910454 0.413611i $$-0.864267\pi$$
−0.910454 + 0.413611i $$0.864267\pi$$
$$458$$ 0 0
$$459$$ 1.44383 0.0673923
$$460$$ 0 0
$$461$$ 28.3918 1.32234 0.661168 0.750238i $$-0.270061\pi$$
0.661168 + 0.750238i $$0.270061\pi$$
$$462$$ 0 0
$$463$$ 10.2259 0.475238 0.237619 0.971358i $$-0.423633\pi$$
0.237619 + 0.971358i $$0.423633\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −5.61584 −0.259870 −0.129935 0.991522i $$-0.541477\pi$$
−0.129935 + 0.991522i $$0.541477\pi$$
$$468$$ 0 0
$$469$$ −2.05712 −0.0949890
$$470$$ 0 0
$$471$$ 12.8392 0.591599
$$472$$ 0 0
$$473$$ 26.8102 1.23273
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −6.72730 −0.308022
$$478$$ 0 0
$$479$$ −8.95006 −0.408939 −0.204469 0.978873i $$-0.565547\pi$$
−0.204469 + 0.978873i $$0.565547\pi$$
$$480$$ 0 0
$$481$$ 0.0418875 0.00190991
$$482$$ 0 0
$$483$$ −17.5731 −0.799603
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 18.3596 0.831954 0.415977 0.909375i $$-0.363440\pi$$
0.415977 + 0.909375i $$0.363440\pi$$
$$488$$ 0 0
$$489$$ −15.3137 −0.692510
$$490$$ 0 0
$$491$$ −33.7981 −1.52529 −0.762644 0.646819i $$-0.776099\pi$$
−0.762644 + 0.646819i $$0.776099\pi$$
$$492$$ 0 0
$$493$$ −1.19611 −0.0538701
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 23.5375 1.05580
$$498$$ 0 0
$$499$$ 5.72898 0.256464 0.128232 0.991744i $$-0.459070\pi$$
0.128232 + 0.991744i $$0.459070\pi$$
$$500$$ 0 0
$$501$$ −15.3922 −0.687673
$$502$$ 0 0
$$503$$ −20.2121 −0.901215 −0.450607 0.892722i $$-0.648793\pi$$
−0.450607 + 0.892722i $$0.648793\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ 6.02708 0.267146 0.133573 0.991039i $$-0.457355\pi$$
0.133573 + 0.991039i $$0.457355\pi$$
$$510$$ 0 0
$$511$$ 19.4322 0.859629
$$512$$ 0 0
$$513$$ 4.17113 0.184160
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 13.1243 0.577208
$$518$$ 0 0
$$519$$ 14.9706 0.657135
$$520$$ 0 0
$$521$$ −1.95899 −0.0858249 −0.0429125 0.999079i $$-0.513664\pi$$
−0.0429125 + 0.999079i $$0.513664\pi$$
$$522$$ 0 0
$$523$$ −21.6859 −0.948256 −0.474128 0.880456i $$-0.657237\pi$$
−0.474128 + 0.880456i $$0.657237\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2.39222 0.104207
$$528$$ 0 0
$$529$$ 15.6016 0.678332
$$530$$ 0 0
$$531$$ −2.97906 −0.129280
$$532$$ 0 0
$$533$$ 4.36365 0.189011
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −7.63950 −0.329669
$$538$$ 0 0
$$539$$ −2.97906 −0.128317
$$540$$ 0 0
$$541$$ 39.3245 1.69069 0.845346 0.534220i $$-0.179394\pi$$
0.845346 + 0.534220i $$0.179394\pi$$
$$542$$ 0 0
$$543$$ 15.0286 0.644937
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 14.1466 0.604865 0.302432 0.953171i $$-0.402201\pi$$
0.302432 + 0.953171i $$0.402201\pi$$
$$548$$ 0 0
$$549$$ 6.27226 0.267694
$$550$$ 0 0
$$551$$ −3.45548 −0.147208
$$552$$ 0 0
$$553$$ −17.2871 −0.735120
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −29.7603 −1.26098 −0.630491 0.776196i $$-0.717147\pi$$
−0.630491 + 0.776196i $$0.717147\pi$$
$$558$$ 0 0
$$559$$ 8.99956 0.380641
$$560$$ 0 0
$$561$$ −4.30126 −0.181599
$$562$$ 0 0
$$563$$ −28.6498 −1.20745 −0.603723 0.797194i $$-0.706317\pi$$
−0.603723 + 0.797194i $$0.706317\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −2.82843 −0.118783
$$568$$ 0 0
$$569$$ 39.7326 1.66568 0.832838 0.553517i $$-0.186715\pi$$
0.832838 + 0.553517i $$0.186715\pi$$
$$570$$ 0 0
$$571$$ 30.5726 1.27943 0.639713 0.768614i $$-0.279053\pi$$
0.639713 + 0.768614i $$0.279053\pi$$
$$572$$ 0 0
$$573$$ −10.8877 −0.454839
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −35.3116 −1.47004 −0.735020 0.678045i $$-0.762827\pi$$
−0.735020 + 0.678045i $$0.762827\pi$$
$$578$$ 0 0
$$579$$ −4.94690 −0.205586
$$580$$ 0 0
$$581$$ −41.5946 −1.72564
$$582$$ 0 0
$$583$$ 20.0410 0.830014
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 19.5517 0.806985 0.403492 0.914983i $$-0.367796\pi$$
0.403492 + 0.914983i $$0.367796\pi$$
$$588$$ 0 0
$$589$$ 6.91096 0.284761
$$590$$ 0 0
$$591$$ −26.3628 −1.08442
$$592$$ 0 0
$$593$$ 4.66491 0.191565 0.0957824 0.995402i $$-0.469465\pi$$
0.0957824 + 0.995402i $$0.469465\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 6.11190 0.250143
$$598$$ 0 0
$$599$$ −19.6497 −0.802864 −0.401432 0.915889i $$-0.631487\pi$$
−0.401432 + 0.915889i $$0.631487\pi$$
$$600$$ 0 0
$$601$$ 44.2714 1.80587 0.902934 0.429780i $$-0.141409\pi$$
0.902934 + 0.429780i $$0.141409\pi$$
$$602$$ 0 0
$$603$$ 0.727302 0.0296180
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 34.2648 1.39077 0.695383 0.718640i $$-0.255235\pi$$
0.695383 + 0.718640i $$0.255235\pi$$
$$608$$ 0 0
$$609$$ 2.34315 0.0949491
$$610$$ 0 0
$$611$$ 4.40554 0.178229
$$612$$ 0 0
$$613$$ 33.7203 1.36195 0.680975 0.732307i $$-0.261556\pi$$
0.680975 + 0.732307i $$0.261556\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −35.0901 −1.41267 −0.706337 0.707876i $$-0.749653\pi$$
−0.706337 + 0.707876i $$0.749653\pi$$
$$618$$ 0 0
$$619$$ 2.63907 0.106073 0.0530365 0.998593i $$-0.483110\pi$$
0.0530365 + 0.998593i $$0.483110\pi$$
$$620$$ 0 0
$$621$$ 6.21302 0.249320
$$622$$ 0 0
$$623$$ −41.7131 −1.67120
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −12.4260 −0.496248
$$628$$ 0 0
$$629$$ −0.0604785 −0.00241144
$$630$$ 0 0
$$631$$ −28.7683 −1.14525 −0.572624 0.819818i $$-0.694075\pi$$
−0.572624 + 0.819818i $$0.694075\pi$$
$$632$$ 0 0
$$633$$ 9.68585 0.384978
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ −8.32176 −0.329204
$$640$$ 0 0
$$641$$ 35.1944 1.39009 0.695047 0.718965i $$-0.255384\pi$$
0.695047 + 0.718965i $$0.255384\pi$$
$$642$$ 0 0
$$643$$ −27.2709 −1.07546 −0.537731 0.843117i $$-0.680718\pi$$
−0.537731 + 0.843117i $$0.680718\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −4.22913 −0.166264 −0.0831322 0.996539i $$-0.526492\pi$$
−0.0831322 + 0.996539i $$0.526492\pi$$
$$648$$ 0 0
$$649$$ 8.87478 0.348365
$$650$$ 0 0
$$651$$ −4.68629 −0.183670
$$652$$ 0 0
$$653$$ −22.1400 −0.866406 −0.433203 0.901296i $$-0.642617\pi$$
−0.433203 + 0.901296i $$0.642617\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −6.87031 −0.268037
$$658$$ 0 0
$$659$$ 27.0932 1.05540 0.527701 0.849430i $$-0.323054\pi$$
0.527701 + 0.849430i $$0.323054\pi$$
$$660$$ 0 0
$$661$$ 19.0597 0.741335 0.370668 0.928766i $$-0.379129\pi$$
0.370668 + 0.928766i $$0.379129\pi$$
$$662$$ 0 0
$$663$$ −1.44383 −0.0560738
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −5.14704 −0.199294
$$668$$ 0 0
$$669$$ −9.21258 −0.356179
$$670$$ 0 0
$$671$$ −18.6854 −0.721342
$$672$$ 0 0
$$673$$ 23.6213 0.910533 0.455267 0.890355i $$-0.349544\pi$$
0.455267 + 0.890355i $$0.349544\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18.8111 0.722968 0.361484 0.932378i $$-0.382270\pi$$
0.361484 + 0.932378i $$0.382270\pi$$
$$678$$ 0 0
$$679$$ 19.1952 0.736645
$$680$$ 0 0
$$681$$ −20.3218 −0.778732
$$682$$ 0 0
$$683$$ −33.7827 −1.29266 −0.646329 0.763059i $$-0.723697\pi$$
−0.646329 + 0.763059i $$0.723697\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −16.4715 −0.628428
$$688$$ 0 0
$$689$$ 6.72730 0.256290
$$690$$ 0 0
$$691$$ −47.3236 −1.80027 −0.900137 0.435606i $$-0.856534\pi$$
−0.900137 + 0.435606i $$0.856534\pi$$
$$692$$ 0 0
$$693$$ 8.42604 0.320079
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −6.30038 −0.238644
$$698$$ 0 0
$$699$$ −2.37339 −0.0897697
$$700$$ 0 0
$$701$$ −3.29636 −0.124502 −0.0622509 0.998061i $$-0.519828\pi$$
−0.0622509 + 0.998061i $$0.519828\pi$$
$$702$$ 0 0
$$703$$ −0.174718 −0.00658963
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −34.3431 −1.29161
$$708$$ 0 0
$$709$$ 46.8404 1.75913 0.879565 0.475779i $$-0.157834\pi$$
0.879565 + 0.475779i $$0.157834\pi$$
$$710$$ 0 0
$$711$$ 6.11190 0.229214
$$712$$ 0 0
$$713$$ 10.2941 0.385516
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −13.6363 −0.509259
$$718$$ 0 0
$$719$$ −20.0152 −0.746442 −0.373221 0.927742i $$-0.621747\pi$$
−0.373221 + 0.927742i $$0.621747\pi$$
$$720$$ 0 0
$$721$$ −7.95093 −0.296108
$$722$$ 0 0
$$723$$ 1.07045 0.0398104
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −10.8383 −0.401971 −0.200986 0.979594i $$-0.564414\pi$$
−0.200986 + 0.979594i $$0.564414\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −12.9939 −0.480595
$$732$$ 0 0
$$733$$ 21.1952 0.782864 0.391432 0.920207i $$-0.371980\pi$$
0.391432 + 0.920207i $$0.371980\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −2.16667 −0.0798104
$$738$$ 0 0
$$739$$ 7.07403 0.260222 0.130111 0.991499i $$-0.458467\pi$$
0.130111 + 0.991499i $$0.458467\pi$$
$$740$$ 0 0
$$741$$ −4.17113 −0.153230
$$742$$ 0 0
$$743$$ 21.3913 0.784772 0.392386 0.919801i $$-0.371650\pi$$
0.392386 + 0.919801i $$0.371650\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 14.7059 0.538061
$$748$$ 0 0
$$749$$ 51.9982 1.89997
$$750$$ 0 0
$$751$$ 18.2247 0.665028 0.332514 0.943098i $$-0.392103\pi$$
0.332514 + 0.943098i $$0.392103\pi$$
$$752$$ 0 0
$$753$$ 14.6262 0.533007
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −39.3708 −1.43096 −0.715479 0.698635i $$-0.753791\pi$$
−0.715479 + 0.698635i $$0.753791\pi$$
$$758$$ 0 0
$$759$$ −18.5089 −0.671832
$$760$$ 0 0
$$761$$ −13.8610 −0.502462 −0.251231 0.967927i $$-0.580835\pi$$
−0.251231 + 0.967927i $$0.580835\pi$$
$$762$$ 0 0
$$763$$ −4.08378 −0.147843
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 2.97906 0.107567
$$768$$ 0 0
$$769$$ −12.6274 −0.455356 −0.227678 0.973736i $$-0.573113\pi$$
−0.227678 + 0.973736i $$0.573113\pi$$
$$770$$ 0 0
$$771$$ −24.9403 −0.898204
$$772$$ 0 0
$$773$$ −34.7825 −1.25104 −0.625520 0.780208i $$-0.715113\pi$$
−0.625520 + 0.780208i $$0.715113\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0.118476 0.00425029
$$778$$ 0 0
$$779$$ −18.2014 −0.652132
$$780$$ 0 0
$$781$$ 24.7910 0.887092
$$782$$ 0 0
$$783$$ −0.828427 −0.0296056
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 36.7835 1.31119 0.655596 0.755112i $$-0.272418\pi$$
0.655596 + 0.755112i $$0.272418\pi$$
$$788$$ 0 0
$$789$$ 9.01691 0.321011
$$790$$ 0 0
$$791$$ 9.62215 0.342124
$$792$$ 0 0
$$793$$ −6.27226 −0.222734
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 42.4413 1.50335 0.751674 0.659535i $$-0.229247\pi$$
0.751674 + 0.659535i $$0.229247\pi$$
$$798$$ 0 0
$$799$$ −6.36086 −0.225031
$$800$$ 0 0
$$801$$ 14.7478 0.521088
$$802$$ 0 0
$$803$$ 20.4671 0.722267
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 8.28391 0.291607
$$808$$ 0 0
$$809$$ 55.3771 1.94696 0.973478 0.228780i $$-0.0734735\pi$$
0.973478 + 0.228780i $$0.0734735\pi$$
$$810$$ 0 0
$$811$$ −17.1159 −0.601021 −0.300511 0.953778i $$-0.597157\pi$$
−0.300511 + 0.953778i $$0.597157\pi$$
$$812$$ 0 0
$$813$$ −19.9572 −0.699931
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −37.5384 −1.31330
$$818$$ 0 0
$$819$$ 2.82843 0.0988332
$$820$$ 0 0
$$821$$ 15.5460 0.542559 0.271279 0.962501i $$-0.412553\pi$$
0.271279 + 0.962501i $$0.412553\pi$$
$$822$$ 0 0
$$823$$ −0.0998847 −0.00348176 −0.00174088 0.999998i $$-0.500554\pi$$
−0.00174088 + 0.999998i $$0.500554\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 9.53311 0.331499 0.165749 0.986168i $$-0.446996\pi$$
0.165749 + 0.986168i $$0.446996\pi$$
$$828$$ 0 0
$$829$$ 50.6361 1.75866 0.879332 0.476210i $$-0.157990\pi$$
0.879332 + 0.476210i $$0.157990\pi$$
$$830$$ 0 0
$$831$$ 22.6145 0.784489
$$832$$ 0 0
$$833$$ 1.44383 0.0500258
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1.65685 0.0572693
$$838$$ 0 0
$$839$$ −43.2310 −1.49250 −0.746249 0.665666i $$-0.768147\pi$$
−0.746249 + 0.665666i $$0.768147\pi$$
$$840$$ 0 0
$$841$$ −28.3137 −0.976335
$$842$$ 0 0
$$843$$ −1.89572 −0.0652921
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 6.01104 0.206542
$$848$$ 0 0
$$849$$ 11.2299 0.385410
$$850$$ 0 0
$$851$$ −0.260248 −0.00892119
$$852$$ 0 0
$$853$$ 31.6979 1.08531 0.542657 0.839954i $$-0.317418\pi$$
0.542657 + 0.839954i $$0.317418\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 28.8912 0.986906 0.493453 0.869772i $$-0.335734\pi$$
0.493453 + 0.869772i $$0.335734\pi$$
$$858$$ 0 0
$$859$$ 49.7099 1.69608 0.848040 0.529933i $$-0.177783\pi$$
0.848040 + 0.529933i $$0.177783\pi$$
$$860$$ 0 0
$$861$$ 12.3423 0.420623
$$862$$ 0 0
$$863$$ 4.56502 0.155395 0.0776976 0.996977i $$-0.475243\pi$$
0.0776976 + 0.996977i $$0.475243\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −14.9153 −0.506552
$$868$$ 0 0
$$869$$ −18.2077 −0.617653
$$870$$ 0 0
$$871$$ −0.727302 −0.0246437
$$872$$ 0 0
$$873$$ −6.78654 −0.229690
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13.5312 −0.456916 −0.228458 0.973554i $$-0.573368\pi$$
−0.228458 + 0.973554i $$0.573368\pi$$
$$878$$ 0 0
$$879$$ −27.6774 −0.933534
$$880$$ 0 0
$$881$$ −28.0634 −0.945481 −0.472740 0.881202i $$-0.656735\pi$$
−0.472740 + 0.881202i $$0.656735\pi$$
$$882$$ 0 0
$$883$$ 47.3700 1.59413 0.797063 0.603896i $$-0.206386\pi$$
0.797063 + 0.603896i $$0.206386\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0.606101 0.0203509 0.0101754 0.999948i $$-0.496761\pi$$
0.0101754 + 0.999948i $$0.496761\pi$$
$$888$$ 0 0
$$889$$ 44.7192 1.49984
$$890$$ 0 0
$$891$$ −2.97906 −0.0998021
$$892$$ 0 0
$$893$$ −18.3761 −0.614932
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −6.21302 −0.207447
$$898$$ 0 0
$$899$$ −1.37258 −0.0457782
$$900$$ 0 0
$$901$$ −9.71310 −0.323590
$$902$$ 0 0
$$903$$ 25.4546 0.847076
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 11.8582 0.393746 0.196873 0.980429i $$-0.436921\pi$$
0.196873 + 0.980429i $$0.436921\pi$$
$$908$$ 0 0
$$909$$ 12.1421 0.402729
$$910$$ 0 0
$$911$$ −33.5642 −1.11203 −0.556015 0.831172i $$-0.687670\pi$$
−0.556015 + 0.831172i $$0.687670\pi$$
$$912$$ 0 0
$$913$$ −43.8098 −1.44989
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 54.2229 1.79060
$$918$$ 0 0
$$919$$ −19.2242 −0.634149 −0.317074 0.948401i $$-0.602701\pi$$
−0.317074 + 0.948401i $$0.602701\pi$$
$$920$$ 0 0
$$921$$ −24.9858 −0.823310
$$922$$ 0 0
$$923$$ 8.32176 0.273914
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2.81108 0.0923279
$$928$$ 0 0
$$929$$ −36.0349 −1.18227 −0.591133 0.806574i $$-0.701319\pi$$
−0.591133 + 0.806574i $$0.701319\pi$$
$$930$$ 0 0
$$931$$ 4.17113 0.136703
$$932$$ 0 0
$$933$$ 18.3842 0.601870
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 41.5851 1.35853 0.679263 0.733895i $$-0.262300\pi$$
0.679263 + 0.733895i $$0.262300\pi$$
$$938$$ 0 0
$$939$$ 8.61453 0.281124
$$940$$ 0 0
$$941$$ −56.4933 −1.84163 −0.920814 0.390002i $$-0.872474\pi$$
−0.920814 + 0.390002i $$0.872474\pi$$
$$942$$ 0 0
$$943$$ −27.1115 −0.882871
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −11.3922 −0.370197 −0.185099 0.982720i $$-0.559260\pi$$
−0.185099 + 0.982720i $$0.559260\pi$$
$$948$$ 0 0
$$949$$ 6.87031 0.223020
$$950$$ 0 0
$$951$$ −15.0491 −0.487999
$$952$$ 0 0
$$953$$ 56.7986 1.83989 0.919943 0.392053i $$-0.128235\pi$$
0.919943 + 0.392053i $$0.128235\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 2.46793 0.0797769
$$958$$ 0 0
$$959$$ 4.74677 0.153281
$$960$$ 0 0
$$961$$ −28.2548 −0.911446
$$962$$ 0 0
$$963$$ −18.3842 −0.592421
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −25.7224 −0.827177 −0.413588 0.910464i $$-0.635725\pi$$
−0.413588 + 0.910464i $$0.635725\pi$$
$$968$$ 0 0
$$969$$ 6.02242 0.193468
$$970$$ 0 0
$$971$$ −32.8703 −1.05486 −0.527429 0.849599i $$-0.676844\pi$$
−0.527429 + 0.849599i $$0.676844\pi$$
$$972$$ 0 0
$$973$$ 34.9092 1.11914
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 30.1453 0.964433 0.482217 0.876052i $$-0.339832\pi$$
0.482217 + 0.876052i $$0.339832\pi$$
$$978$$ 0 0
$$979$$ −43.9345 −1.40415
$$980$$ 0 0
$$981$$ 1.44383 0.0460980
$$982$$ 0 0
$$983$$ −32.4965 −1.03648 −0.518238 0.855236i $$-0.673412\pi$$
−0.518238 + 0.855236i $$0.673412\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 12.4607 0.396630
$$988$$ 0 0
$$989$$ −55.9145 −1.77798
$$990$$ 0 0
$$991$$ 37.4893 1.19089 0.595443 0.803397i $$-0.296976\pi$$
0.595443 + 0.803397i $$0.296976\pi$$
$$992$$ 0 0
$$993$$ 22.6738 0.719530
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −18.8659 −0.597488 −0.298744 0.954333i $$-0.596568\pi$$
−0.298744 + 0.954333i $$0.596568\pi$$
$$998$$ 0 0
$$999$$ −0.0418875 −0.00132526
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.bw.1.1 4
5.2 odd 4 1560.2.l.e.1249.3 8
5.3 odd 4 1560.2.l.e.1249.7 yes 8
5.4 even 2 7800.2.a.bv.1.3 4
15.2 even 4 4680.2.l.f.2809.4 8
15.8 even 4 4680.2.l.f.2809.3 8
20.3 even 4 3120.2.l.o.1249.3 8
20.7 even 4 3120.2.l.o.1249.7 8

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.e.1249.3 8 5.2 odd 4
1560.2.l.e.1249.7 yes 8 5.3 odd 4
3120.2.l.o.1249.3 8 20.3 even 4
3120.2.l.o.1249.7 8 20.7 even 4
4680.2.l.f.2809.3 8 15.8 even 4
4680.2.l.f.2809.4 8 15.2 even 4
7800.2.a.bv.1.3 4 5.4 even 2
7800.2.a.bw.1.1 4 1.1 even 1 trivial