# Properties

 Label 5520.2.a.bx.1.2 Level $5520$ Weight $2$ Character 5520.1 Self dual yes Analytic conductor $44.077$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5520.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.0774219157$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2760) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.81361$$ of defining polynomial Character $$\chi$$ $$=$$ 5520.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} +1.71083 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} +1.71083 q^{7} +1.00000 q^{9} -2.52444 q^{11} -6.72999 q^{13} +1.00000 q^{15} +4.44082 q^{17} +3.10278 q^{19} -1.71083 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.01916 q^{29} +1.13249 q^{31} +2.52444 q^{33} -1.71083 q^{35} +2.49472 q^{37} +6.72999 q^{39} -7.39194 q^{41} +6.78389 q^{43} -1.00000 q^{45} +13.3083 q^{47} -4.07306 q^{49} -4.44082 q^{51} -12.4408 q^{53} +2.52444 q^{55} -3.10278 q^{57} -11.0192 q^{59} -2.89722 q^{61} +1.71083 q^{63} +6.72999 q^{65} +0.0836184 q^{67} +1.00000 q^{69} +13.8030 q^{71} -5.57331 q^{73} -1.00000 q^{75} -4.31889 q^{77} +1.00000 q^{81} -1.97028 q^{83} -4.44082 q^{85} -5.01916 q^{87} -7.62721 q^{89} -11.5139 q^{91} -1.13249 q^{93} -3.10278 q^{95} +9.83276 q^{97} -2.52444 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{15} - 6 q^{17} + 2 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 6 q^{35} - 8 q^{37} - 14 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} + 5 q^{49} + 6 q^{51} - 18 q^{53} + 2 q^{55} - 2 q^{57} - 12 q^{59} - 16 q^{61} + 6 q^{63} + 14 q^{67} + 3 q^{69} + 4 q^{71} - 3 q^{75} - 22 q^{77} + 3 q^{81} + 4 q^{83} + 6 q^{85} + 6 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 2 * q^11 + 3 * q^15 - 6 * q^17 + 2 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 + 6 * q^31 + 2 * q^33 - 6 * q^35 - 8 * q^37 - 14 * q^41 + 4 * q^43 - 3 * q^45 + 18 * q^47 + 5 * q^49 + 6 * q^51 - 18 * q^53 + 2 * q^55 - 2 * q^57 - 12 * q^59 - 16 * q^61 + 6 * q^63 + 14 * q^67 + 3 * q^69 + 4 * q^71 - 3 * q^75 - 22 * q^77 + 3 * q^81 + 4 * q^83 + 6 * q^85 + 6 * q^87 - 10 * q^89 + 2 * q^91 - 6 * q^93 - 2 * q^95 + 2 * q^97 - 2 * 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.71083 0.646634 0.323317 0.946291i $$-0.395202\pi$$
0.323317 + 0.946291i $$0.395202\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.52444 −0.761147 −0.380573 0.924751i $$-0.624273\pi$$
−0.380573 + 0.924751i $$0.624273\pi$$
$$12$$ 0 0
$$13$$ −6.72999 −1.86656 −0.933281 0.359146i $$-0.883068\pi$$
−0.933281 + 0.359146i $$0.883068\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 4.44082 1.07706 0.538528 0.842607i $$-0.318980\pi$$
0.538528 + 0.842607i $$0.318980\pi$$
$$18$$ 0 0
$$19$$ 3.10278 0.711825 0.355913 0.934519i $$-0.384170\pi$$
0.355913 + 0.934519i $$0.384170\pi$$
$$20$$ 0 0
$$21$$ −1.71083 −0.373334
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 5.01916 0.932034 0.466017 0.884776i $$-0.345689\pi$$
0.466017 + 0.884776i $$0.345689\pi$$
$$30$$ 0 0
$$31$$ 1.13249 0.203402 0.101701 0.994815i $$-0.467572\pi$$
0.101701 + 0.994815i $$0.467572\pi$$
$$32$$ 0 0
$$33$$ 2.52444 0.439448
$$34$$ 0 0
$$35$$ −1.71083 −0.289183
$$36$$ 0 0
$$37$$ 2.49472 0.410129 0.205065 0.978748i $$-0.434260\pi$$
0.205065 + 0.978748i $$0.434260\pi$$
$$38$$ 0 0
$$39$$ 6.72999 1.07766
$$40$$ 0 0
$$41$$ −7.39194 −1.15443 −0.577214 0.816593i $$-0.695860\pi$$
−0.577214 + 0.816593i $$0.695860\pi$$
$$42$$ 0 0
$$43$$ 6.78389 1.03453 0.517267 0.855824i $$-0.326950\pi$$
0.517267 + 0.855824i $$0.326950\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 13.3083 1.94122 0.970609 0.240660i $$-0.0773640\pi$$
0.970609 + 0.240660i $$0.0773640\pi$$
$$48$$ 0 0
$$49$$ −4.07306 −0.581865
$$50$$ 0 0
$$51$$ −4.44082 −0.621839
$$52$$ 0 0
$$53$$ −12.4408 −1.70888 −0.854439 0.519552i $$-0.826099\pi$$
−0.854439 + 0.519552i $$0.826099\pi$$
$$54$$ 0 0
$$55$$ 2.52444 0.340395
$$56$$ 0 0
$$57$$ −3.10278 −0.410973
$$58$$ 0 0
$$59$$ −11.0192 −1.43457 −0.717286 0.696779i $$-0.754616\pi$$
−0.717286 + 0.696779i $$0.754616\pi$$
$$60$$ 0 0
$$61$$ −2.89722 −0.370952 −0.185476 0.982649i $$-0.559383\pi$$
−0.185476 + 0.982649i $$0.559383\pi$$
$$62$$ 0 0
$$63$$ 1.71083 0.215545
$$64$$ 0 0
$$65$$ 6.72999 0.834752
$$66$$ 0 0
$$67$$ 0.0836184 0.0102156 0.00510781 0.999987i $$-0.498374\pi$$
0.00510781 + 0.999987i $$0.498374\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 13.8030 1.63812 0.819060 0.573708i $$-0.194496\pi$$
0.819060 + 0.573708i $$0.194496\pi$$
$$72$$ 0 0
$$73$$ −5.57331 −0.652307 −0.326154 0.945317i $$-0.605753\pi$$
−0.326154 + 0.945317i $$0.605753\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −4.31889 −0.492183
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −1.97028 −0.216266 −0.108133 0.994136i $$-0.534487\pi$$
−0.108133 + 0.994136i $$0.534487\pi$$
$$84$$ 0 0
$$85$$ −4.44082 −0.481675
$$86$$ 0 0
$$87$$ −5.01916 −0.538110
$$88$$ 0 0
$$89$$ −7.62721 −0.808483 −0.404241 0.914652i $$-0.632464\pi$$
−0.404241 + 0.914652i $$0.632464\pi$$
$$90$$ 0 0
$$91$$ −11.5139 −1.20698
$$92$$ 0 0
$$93$$ −1.13249 −0.117434
$$94$$ 0 0
$$95$$ −3.10278 −0.318338
$$96$$ 0 0
$$97$$ 9.83276 0.998366 0.499183 0.866497i $$-0.333634\pi$$
0.499183 + 0.866497i $$0.333634\pi$$
$$98$$ 0 0
$$99$$ −2.52444 −0.253716
$$100$$ 0 0
$$101$$ −10.8136 −1.07599 −0.537997 0.842947i $$-0.680819\pi$$
−0.537997 + 0.842947i $$0.680819\pi$$
$$102$$ 0 0
$$103$$ −15.0872 −1.48658 −0.743292 0.668967i $$-0.766737\pi$$
−0.743292 + 0.668967i $$0.766737\pi$$
$$104$$ 0 0
$$105$$ 1.71083 0.166960
$$106$$ 0 0
$$107$$ 13.8625 1.34014 0.670068 0.742299i $$-0.266265\pi$$
0.670068 + 0.742299i $$0.266265\pi$$
$$108$$ 0 0
$$109$$ 16.8277 1.61181 0.805903 0.592048i $$-0.201680\pi$$
0.805903 + 0.592048i $$0.201680\pi$$
$$110$$ 0 0
$$111$$ −2.49472 −0.236788
$$112$$ 0 0
$$113$$ −19.6952 −1.85277 −0.926386 0.376574i $$-0.877102\pi$$
−0.926386 + 0.376574i $$0.877102\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ −6.72999 −0.622188
$$118$$ 0 0
$$119$$ 7.59749 0.696461
$$120$$ 0 0
$$121$$ −4.62721 −0.420656
$$122$$ 0 0
$$123$$ 7.39194 0.666509
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 18.9355 1.68026 0.840129 0.542387i $$-0.182479\pi$$
0.840129 + 0.542387i $$0.182479\pi$$
$$128$$ 0 0
$$129$$ −6.78389 −0.597288
$$130$$ 0 0
$$131$$ −12.4111 −1.08436 −0.542181 0.840261i $$-0.682401\pi$$
−0.542181 + 0.840261i $$0.682401\pi$$
$$132$$ 0 0
$$133$$ 5.30833 0.460290
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −13.4217 −1.14669 −0.573345 0.819314i $$-0.694355\pi$$
−0.573345 + 0.819314i $$0.694355\pi$$
$$138$$ 0 0
$$139$$ −15.2786 −1.29591 −0.647957 0.761677i $$-0.724376\pi$$
−0.647957 + 0.761677i $$0.724376\pi$$
$$140$$ 0 0
$$141$$ −13.3083 −1.12076
$$142$$ 0 0
$$143$$ 16.9894 1.42073
$$144$$ 0 0
$$145$$ −5.01916 −0.416818
$$146$$ 0 0
$$147$$ 4.07306 0.335940
$$148$$ 0 0
$$149$$ −1.68111 −0.137722 −0.0688610 0.997626i $$-0.521937\pi$$
−0.0688610 + 0.997626i $$0.521937\pi$$
$$150$$ 0 0
$$151$$ 2.78389 0.226550 0.113275 0.993564i $$-0.463866\pi$$
0.113275 + 0.993564i $$0.463866\pi$$
$$152$$ 0 0
$$153$$ 4.44082 0.359019
$$154$$ 0 0
$$155$$ −1.13249 −0.0909641
$$156$$ 0 0
$$157$$ −9.91638 −0.791413 −0.395707 0.918377i $$-0.629500\pi$$
−0.395707 + 0.918377i $$0.629500\pi$$
$$158$$ 0 0
$$159$$ 12.4408 0.986621
$$160$$ 0 0
$$161$$ −1.71083 −0.134832
$$162$$ 0 0
$$163$$ −7.42166 −0.581310 −0.290655 0.956828i $$-0.593873\pi$$
−0.290655 + 0.956828i $$0.593873\pi$$
$$164$$ 0 0
$$165$$ −2.52444 −0.196527
$$166$$ 0 0
$$167$$ 5.94610 0.460123 0.230062 0.973176i $$-0.426107\pi$$
0.230062 + 0.973176i $$0.426107\pi$$
$$168$$ 0 0
$$169$$ 32.2927 2.48406
$$170$$ 0 0
$$171$$ 3.10278 0.237275
$$172$$ 0 0
$$173$$ 13.0872 0.995001 0.497500 0.867464i $$-0.334251\pi$$
0.497500 + 0.867464i $$0.334251\pi$$
$$174$$ 0 0
$$175$$ 1.71083 0.129327
$$176$$ 0 0
$$177$$ 11.0192 0.828251
$$178$$ 0 0
$$179$$ −26.4494 −1.97692 −0.988461 0.151476i $$-0.951597\pi$$
−0.988461 + 0.151476i $$0.951597\pi$$
$$180$$ 0 0
$$181$$ −24.0383 −1.78675 −0.893377 0.449308i $$-0.851671\pi$$
−0.893377 + 0.449308i $$0.851671\pi$$
$$182$$ 0 0
$$183$$ 2.89722 0.214169
$$184$$ 0 0
$$185$$ −2.49472 −0.183415
$$186$$ 0 0
$$187$$ −11.2106 −0.819798
$$188$$ 0 0
$$189$$ −1.71083 −0.124445
$$190$$ 0 0
$$191$$ −25.7194 −1.86099 −0.930496 0.366302i $$-0.880624\pi$$
−0.930496 + 0.366302i $$0.880624\pi$$
$$192$$ 0 0
$$193$$ 26.1361 1.88132 0.940658 0.339357i $$-0.110210\pi$$
0.940658 + 0.339357i $$0.110210\pi$$
$$194$$ 0 0
$$195$$ −6.72999 −0.481944
$$196$$ 0 0
$$197$$ −14.3033 −1.01907 −0.509534 0.860451i $$-0.670182\pi$$
−0.509534 + 0.860451i $$0.670182\pi$$
$$198$$ 0 0
$$199$$ 25.2927 1.79295 0.896477 0.443090i $$-0.146118\pi$$
0.896477 + 0.443090i $$0.146118\pi$$
$$200$$ 0 0
$$201$$ −0.0836184 −0.00589799
$$202$$ 0 0
$$203$$ 8.58693 0.602684
$$204$$ 0 0
$$205$$ 7.39194 0.516276
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ −7.83276 −0.541804
$$210$$ 0 0
$$211$$ 21.7875 1.49991 0.749955 0.661489i $$-0.230075\pi$$
0.749955 + 0.661489i $$0.230075\pi$$
$$212$$ 0 0
$$213$$ −13.8030 −0.945769
$$214$$ 0 0
$$215$$ −6.78389 −0.462657
$$216$$ 0 0
$$217$$ 1.93751 0.131527
$$218$$ 0 0
$$219$$ 5.57331 0.376610
$$220$$ 0 0
$$221$$ −29.8867 −2.01039
$$222$$ 0 0
$$223$$ −20.7738 −1.39112 −0.695560 0.718468i $$-0.744843\pi$$
−0.695560 + 0.718468i $$0.744843\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ −9.19499 −0.607622 −0.303811 0.952732i $$-0.598259\pi$$
−0.303811 + 0.952732i $$0.598259\pi$$
$$230$$ 0 0
$$231$$ 4.31889 0.284162
$$232$$ 0 0
$$233$$ 6.98944 0.457893 0.228947 0.973439i $$-0.426472\pi$$
0.228947 + 0.973439i $$0.426472\pi$$
$$234$$ 0 0
$$235$$ −13.3083 −0.868139
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.2353 −1.05017 −0.525086 0.851049i $$-0.675967\pi$$
−0.525086 + 0.851049i $$0.675967\pi$$
$$240$$ 0 0
$$241$$ 7.77886 0.501081 0.250540 0.968106i $$-0.419392\pi$$
0.250540 + 0.968106i $$0.419392\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 4.07306 0.260218
$$246$$ 0 0
$$247$$ −20.8816 −1.32867
$$248$$ 0 0
$$249$$ 1.97028 0.124861
$$250$$ 0 0
$$251$$ 19.0872 1.20477 0.602386 0.798205i $$-0.294217\pi$$
0.602386 + 0.798205i $$0.294217\pi$$
$$252$$ 0 0
$$253$$ 2.52444 0.158710
$$254$$ 0 0
$$255$$ 4.44082 0.278095
$$256$$ 0 0
$$257$$ −27.2489 −1.69974 −0.849869 0.526993i $$-0.823319\pi$$
−0.849869 + 0.526993i $$0.823319\pi$$
$$258$$ 0 0
$$259$$ 4.26804 0.265203
$$260$$ 0 0
$$261$$ 5.01916 0.310678
$$262$$ 0 0
$$263$$ −15.4303 −0.951470 −0.475735 0.879589i $$-0.657818\pi$$
−0.475735 + 0.879589i $$0.657818\pi$$
$$264$$ 0 0
$$265$$ 12.4408 0.764233
$$266$$ 0 0
$$267$$ 7.62721 0.466778
$$268$$ 0 0
$$269$$ −12.9114 −0.787219 −0.393610 0.919278i $$-0.628774\pi$$
−0.393610 + 0.919278i $$0.628774\pi$$
$$270$$ 0 0
$$271$$ 3.74914 0.227744 0.113872 0.993495i $$-0.463675\pi$$
0.113872 + 0.993495i $$0.463675\pi$$
$$272$$ 0 0
$$273$$ 11.5139 0.696851
$$274$$ 0 0
$$275$$ −2.52444 −0.152229
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ 1.13249 0.0678007
$$280$$ 0 0
$$281$$ −1.84835 −0.110263 −0.0551316 0.998479i $$-0.517558\pi$$
−0.0551316 + 0.998479i $$0.517558\pi$$
$$282$$ 0 0
$$283$$ −1.40753 −0.0836689 −0.0418345 0.999125i $$-0.513320\pi$$
−0.0418345 + 0.999125i $$0.513320\pi$$
$$284$$ 0 0
$$285$$ 3.10278 0.183793
$$286$$ 0 0
$$287$$ −12.6464 −0.746492
$$288$$ 0 0
$$289$$ 2.72088 0.160052
$$290$$ 0 0
$$291$$ −9.83276 −0.576407
$$292$$ 0 0
$$293$$ 27.5280 1.60820 0.804102 0.594492i $$-0.202647\pi$$
0.804102 + 0.594492i $$0.202647\pi$$
$$294$$ 0 0
$$295$$ 11.0192 0.641560
$$296$$ 0 0
$$297$$ 2.52444 0.146483
$$298$$ 0 0
$$299$$ 6.72999 0.389205
$$300$$ 0 0
$$301$$ 11.6061 0.668964
$$302$$ 0 0
$$303$$ 10.8136 0.621225
$$304$$ 0 0
$$305$$ 2.89722 0.165895
$$306$$ 0 0
$$307$$ 4.56275 0.260410 0.130205 0.991487i $$-0.458436\pi$$
0.130205 + 0.991487i $$0.458436\pi$$
$$308$$ 0 0
$$309$$ 15.0872 0.858280
$$310$$ 0 0
$$311$$ −6.84333 −0.388049 −0.194025 0.980997i $$-0.562154\pi$$
−0.194025 + 0.980997i $$0.562154\pi$$
$$312$$ 0 0
$$313$$ −25.7491 −1.45543 −0.727714 0.685881i $$-0.759417\pi$$
−0.727714 + 0.685881i $$0.759417\pi$$
$$314$$ 0 0
$$315$$ −1.71083 −0.0963944
$$316$$ 0 0
$$317$$ 9.23884 0.518905 0.259452 0.965756i $$-0.416458\pi$$
0.259452 + 0.965756i $$0.416458\pi$$
$$318$$ 0 0
$$319$$ −12.6705 −0.709415
$$320$$ 0 0
$$321$$ −13.8625 −0.773728
$$322$$ 0 0
$$323$$ 13.7789 0.766676
$$324$$ 0 0
$$325$$ −6.72999 −0.373313
$$326$$ 0 0
$$327$$ −16.8277 −0.930576
$$328$$ 0 0
$$329$$ 22.7683 1.25526
$$330$$ 0 0
$$331$$ −13.6030 −0.747690 −0.373845 0.927491i $$-0.621961\pi$$
−0.373845 + 0.927491i $$0.621961\pi$$
$$332$$ 0 0
$$333$$ 2.49472 0.136710
$$334$$ 0 0
$$335$$ −0.0836184 −0.00456856
$$336$$ 0 0
$$337$$ −9.25443 −0.504121 −0.252060 0.967712i $$-0.581108\pi$$
−0.252060 + 0.967712i $$0.581108\pi$$
$$338$$ 0 0
$$339$$ 19.6952 1.06970
$$340$$ 0 0
$$341$$ −2.85891 −0.154819
$$342$$ 0 0
$$343$$ −18.9441 −1.02289
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ 0 0
$$347$$ −22.9511 −1.23208 −0.616040 0.787715i $$-0.711264\pi$$
−0.616040 + 0.787715i $$0.711264\pi$$
$$348$$ 0 0
$$349$$ −2.96526 −0.158727 −0.0793633 0.996846i $$-0.525289\pi$$
−0.0793633 + 0.996846i $$0.525289\pi$$
$$350$$ 0 0
$$351$$ 6.72999 0.359220
$$352$$ 0 0
$$353$$ −19.2489 −1.02451 −0.512257 0.858832i $$-0.671191\pi$$
−0.512257 + 0.858832i $$0.671191\pi$$
$$354$$ 0 0
$$355$$ −13.8030 −0.732589
$$356$$ 0 0
$$357$$ −7.59749 −0.402102
$$358$$ 0 0
$$359$$ −21.6116 −1.14062 −0.570309 0.821430i $$-0.693177\pi$$
−0.570309 + 0.821430i $$0.693177\pi$$
$$360$$ 0 0
$$361$$ −9.37279 −0.493305
$$362$$ 0 0
$$363$$ 4.62721 0.242866
$$364$$ 0 0
$$365$$ 5.57331 0.291721
$$366$$ 0 0
$$367$$ 9.71083 0.506901 0.253451 0.967348i $$-0.418434\pi$$
0.253451 + 0.967348i $$0.418434\pi$$
$$368$$ 0 0
$$369$$ −7.39194 −0.384809
$$370$$ 0 0
$$371$$ −21.2841 −1.10502
$$372$$ 0 0
$$373$$ −32.4494 −1.68017 −0.840083 0.542457i $$-0.817494\pi$$
−0.840083 + 0.542457i $$0.817494\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −33.7789 −1.73970
$$378$$ 0 0
$$379$$ −28.8222 −1.48050 −0.740248 0.672333i $$-0.765292\pi$$
−0.740248 + 0.672333i $$0.765292\pi$$
$$380$$ 0 0
$$381$$ −18.9355 −0.970097
$$382$$ 0 0
$$383$$ −2.54862 −0.130228 −0.0651141 0.997878i $$-0.520741\pi$$
−0.0651141 + 0.997878i $$0.520741\pi$$
$$384$$ 0 0
$$385$$ 4.31889 0.220111
$$386$$ 0 0
$$387$$ 6.78389 0.344844
$$388$$ 0 0
$$389$$ −17.5577 −0.890212 −0.445106 0.895478i $$-0.646834\pi$$
−0.445106 + 0.895478i $$0.646834\pi$$
$$390$$ 0 0
$$391$$ −4.44082 −0.224582
$$392$$ 0 0
$$393$$ 12.4111 0.626057
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 29.0872 1.45984 0.729922 0.683530i $$-0.239556\pi$$
0.729922 + 0.683530i $$0.239556\pi$$
$$398$$ 0 0
$$399$$ −5.30833 −0.265749
$$400$$ 0 0
$$401$$ −0.616650 −0.0307940 −0.0153970 0.999881i $$-0.504901\pi$$
−0.0153970 + 0.999881i $$0.504901\pi$$
$$402$$ 0 0
$$403$$ −7.62167 −0.379663
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −6.29776 −0.312168
$$408$$ 0 0
$$409$$ −8.07357 −0.399212 −0.199606 0.979876i $$-0.563966\pi$$
−0.199606 + 0.979876i $$0.563966\pi$$
$$410$$ 0 0
$$411$$ 13.4217 0.662042
$$412$$ 0 0
$$413$$ −18.8519 −0.927642
$$414$$ 0 0
$$415$$ 1.97028 0.0967173
$$416$$ 0 0
$$417$$ 15.2786 0.748197
$$418$$ 0 0
$$419$$ −19.9844 −0.976303 −0.488151 0.872759i $$-0.662329\pi$$
−0.488151 + 0.872759i $$0.662329\pi$$
$$420$$ 0 0
$$421$$ 9.04334 0.440745 0.220373 0.975416i $$-0.429273\pi$$
0.220373 + 0.975416i $$0.429273\pi$$
$$422$$ 0 0
$$423$$ 13.3083 0.647073
$$424$$ 0 0
$$425$$ 4.44082 0.215411
$$426$$ 0 0
$$427$$ −4.95666 −0.239870
$$428$$ 0 0
$$429$$ −16.9894 −0.820258
$$430$$ 0 0
$$431$$ −8.63778 −0.416067 −0.208034 0.978122i $$-0.566706\pi$$
−0.208034 + 0.978122i $$0.566706\pi$$
$$432$$ 0 0
$$433$$ −34.6902 −1.66711 −0.833553 0.552440i $$-0.813697\pi$$
−0.833553 + 0.552440i $$0.813697\pi$$
$$434$$ 0 0
$$435$$ 5.01916 0.240650
$$436$$ 0 0
$$437$$ −3.10278 −0.148426
$$438$$ 0 0
$$439$$ −11.4217 −0.545126 −0.272563 0.962138i $$-0.587871\pi$$
−0.272563 + 0.962138i $$0.587871\pi$$
$$440$$ 0 0
$$441$$ −4.07306 −0.193955
$$442$$ 0 0
$$443$$ 8.78943 0.417598 0.208799 0.977959i $$-0.433045\pi$$
0.208799 + 0.977959i $$0.433045\pi$$
$$444$$ 0 0
$$445$$ 7.62721 0.361565
$$446$$ 0 0
$$447$$ 1.68111 0.0795139
$$448$$ 0 0
$$449$$ −11.5874 −0.546845 −0.273423 0.961894i $$-0.588156\pi$$
−0.273423 + 0.961894i $$0.588156\pi$$
$$450$$ 0 0
$$451$$ 18.6605 0.878689
$$452$$ 0 0
$$453$$ −2.78389 −0.130798
$$454$$ 0 0
$$455$$ 11.5139 0.539779
$$456$$ 0 0
$$457$$ 10.6620 0.498745 0.249373 0.968408i $$-0.419776\pi$$
0.249373 + 0.968408i $$0.419776\pi$$
$$458$$ 0 0
$$459$$ −4.44082 −0.207280
$$460$$ 0 0
$$461$$ −22.3033 −1.03877 −0.519384 0.854541i $$-0.673839\pi$$
−0.519384 + 0.854541i $$0.673839\pi$$
$$462$$ 0 0
$$463$$ 27.9250 1.29778 0.648892 0.760881i $$-0.275233\pi$$
0.648892 + 0.760881i $$0.275233\pi$$
$$464$$ 0 0
$$465$$ 1.13249 0.0525182
$$466$$ 0 0
$$467$$ −24.4308 −1.13052 −0.565261 0.824912i $$-0.691224\pi$$
−0.565261 + 0.824912i $$0.691224\pi$$
$$468$$ 0 0
$$469$$ 0.143057 0.00660576
$$470$$ 0 0
$$471$$ 9.91638 0.456923
$$472$$ 0 0
$$473$$ −17.1255 −0.787431
$$474$$ 0 0
$$475$$ 3.10278 0.142365
$$476$$ 0 0
$$477$$ −12.4408 −0.569626
$$478$$ 0 0
$$479$$ −19.2106 −0.877753 −0.438877 0.898547i $$-0.644624\pi$$
−0.438877 + 0.898547i $$0.644624\pi$$
$$480$$ 0 0
$$481$$ −16.7894 −0.765532
$$482$$ 0 0
$$483$$ 1.71083 0.0778455
$$484$$ 0 0
$$485$$ −9.83276 −0.446483
$$486$$ 0 0
$$487$$ −22.4933 −1.01927 −0.509634 0.860392i $$-0.670219\pi$$
−0.509634 + 0.860392i $$0.670219\pi$$
$$488$$ 0 0
$$489$$ 7.42166 0.335619
$$490$$ 0 0
$$491$$ 8.53857 0.385340 0.192670 0.981264i $$-0.438285\pi$$
0.192670 + 0.981264i $$0.438285\pi$$
$$492$$ 0 0
$$493$$ 22.2892 1.00385
$$494$$ 0 0
$$495$$ 2.52444 0.113465
$$496$$ 0 0
$$497$$ 23.6147 1.05926
$$498$$ 0 0
$$499$$ −17.0247 −0.762130 −0.381065 0.924548i $$-0.624443\pi$$
−0.381065 + 0.924548i $$0.624443\pi$$
$$500$$ 0 0
$$501$$ −5.94610 −0.265652
$$502$$ 0 0
$$503$$ 33.8625 1.50985 0.754927 0.655809i $$-0.227672\pi$$
0.754927 + 0.655809i $$0.227672\pi$$
$$504$$ 0 0
$$505$$ 10.8136 0.481199
$$506$$ 0 0
$$507$$ −32.2927 −1.43417
$$508$$ 0 0
$$509$$ 34.7738 1.54132 0.770662 0.637244i $$-0.219926\pi$$
0.770662 + 0.637244i $$0.219926\pi$$
$$510$$ 0 0
$$511$$ −9.53500 −0.421804
$$512$$ 0 0
$$513$$ −3.10278 −0.136991
$$514$$ 0 0
$$515$$ 15.0872 0.664821
$$516$$ 0 0
$$517$$ −33.5960 −1.47755
$$518$$ 0 0
$$519$$ −13.0872 −0.574464
$$520$$ 0 0
$$521$$ −12.7995 −0.560755 −0.280378 0.959890i $$-0.590460\pi$$
−0.280378 + 0.959890i $$0.590460\pi$$
$$522$$ 0 0
$$523$$ 24.4111 1.06742 0.533711 0.845667i $$-0.320797\pi$$
0.533711 + 0.845667i $$0.320797\pi$$
$$524$$ 0 0
$$525$$ −1.71083 −0.0746668
$$526$$ 0 0
$$527$$ 5.02920 0.219076
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −11.0192 −0.478191
$$532$$ 0 0
$$533$$ 49.7477 2.15481
$$534$$ 0 0
$$535$$ −13.8625 −0.599327
$$536$$ 0 0
$$537$$ 26.4494 1.14138
$$538$$ 0 0
$$539$$ 10.2822 0.442885
$$540$$ 0 0
$$541$$ 27.1849 1.16877 0.584386 0.811476i $$-0.301335\pi$$
0.584386 + 0.811476i $$0.301335\pi$$
$$542$$ 0 0
$$543$$ 24.0383 1.03158
$$544$$ 0 0
$$545$$ −16.8277 −0.720821
$$546$$ 0 0
$$547$$ −26.8433 −1.14774 −0.573869 0.818947i $$-0.694558\pi$$
−0.573869 + 0.818947i $$0.694558\pi$$
$$548$$ 0 0
$$549$$ −2.89722 −0.123651
$$550$$ 0 0
$$551$$ 15.5733 0.663445
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 2.49472 0.105895
$$556$$ 0 0
$$557$$ −32.8519 −1.39198 −0.695990 0.718051i $$-0.745034\pi$$
−0.695990 + 0.718051i $$0.745034\pi$$
$$558$$ 0 0
$$559$$ −45.6555 −1.93102
$$560$$ 0 0
$$561$$ 11.2106 0.473311
$$562$$ 0 0
$$563$$ 26.7441 1.12713 0.563565 0.826072i $$-0.309429\pi$$
0.563565 + 0.826072i $$0.309429\pi$$
$$564$$ 0 0
$$565$$ 19.6952 0.828585
$$566$$ 0 0
$$567$$ 1.71083 0.0718482
$$568$$ 0 0
$$569$$ −34.3799 −1.44128 −0.720641 0.693309i $$-0.756152\pi$$
−0.720641 + 0.693309i $$0.756152\pi$$
$$570$$ 0 0
$$571$$ −39.9532 −1.67199 −0.835996 0.548736i $$-0.815109\pi$$
−0.835996 + 0.548736i $$0.815109\pi$$
$$572$$ 0 0
$$573$$ 25.7194 1.07444
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 23.5194 0.979126 0.489563 0.871968i $$-0.337156\pi$$
0.489563 + 0.871968i $$0.337156\pi$$
$$578$$ 0 0
$$579$$ −26.1361 −1.08618
$$580$$ 0 0
$$581$$ −3.37082 −0.139845
$$582$$ 0 0
$$583$$ 31.4061 1.30071
$$584$$ 0 0
$$585$$ 6.72999 0.278251
$$586$$ 0 0
$$587$$ 8.36274 0.345167 0.172584 0.984995i $$-0.444788\pi$$
0.172584 + 0.984995i $$0.444788\pi$$
$$588$$ 0 0
$$589$$ 3.51388 0.144787
$$590$$ 0 0
$$591$$ 14.3033 0.588359
$$592$$ 0 0
$$593$$ 5.87662 0.241324 0.120662 0.992694i $$-0.461498\pi$$
0.120662 + 0.992694i $$0.461498\pi$$
$$594$$ 0 0
$$595$$ −7.59749 −0.311467
$$596$$ 0 0
$$597$$ −25.2927 −1.03516
$$598$$ 0 0
$$599$$ 9.12550 0.372858 0.186429 0.982468i $$-0.440309\pi$$
0.186429 + 0.982468i $$0.440309\pi$$
$$600$$ 0 0
$$601$$ 15.3764 0.627215 0.313607 0.949553i $$-0.398462\pi$$
0.313607 + 0.949553i $$0.398462\pi$$
$$602$$ 0 0
$$603$$ 0.0836184 0.00340521
$$604$$ 0 0
$$605$$ 4.62721 0.188123
$$606$$ 0 0
$$607$$ −11.1511 −0.452611 −0.226305 0.974056i $$-0.572665\pi$$
−0.226305 + 0.974056i $$0.572665\pi$$
$$608$$ 0 0
$$609$$ −8.58693 −0.347960
$$610$$ 0 0
$$611$$ −89.5649 −3.62341
$$612$$ 0 0
$$613$$ 5.47002 0.220932 0.110466 0.993880i $$-0.464766\pi$$
0.110466 + 0.993880i $$0.464766\pi$$
$$614$$ 0 0
$$615$$ −7.39194 −0.298072
$$616$$ 0 0
$$617$$ 11.1169 0.447550 0.223775 0.974641i $$-0.428162\pi$$
0.223775 + 0.974641i $$0.428162\pi$$
$$618$$ 0 0
$$619$$ 18.8433 0.757377 0.378689 0.925524i $$-0.376375\pi$$
0.378689 + 0.925524i $$0.376375\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ −13.0489 −0.522792
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 7.83276 0.312810
$$628$$ 0 0
$$629$$ 11.0786 0.441733
$$630$$ 0 0
$$631$$ 4.10329 0.163349 0.0816747 0.996659i $$-0.473973\pi$$
0.0816747 + 0.996659i $$0.473973\pi$$
$$632$$ 0 0
$$633$$ −21.7875 −0.865974
$$634$$ 0 0
$$635$$ −18.9355 −0.751434
$$636$$ 0 0
$$637$$ 27.4116 1.08609
$$638$$ 0 0
$$639$$ 13.8030 0.546040
$$640$$ 0 0
$$641$$ 15.6116 0.616622 0.308311 0.951286i $$-0.400236\pi$$
0.308311 + 0.951286i $$0.400236\pi$$
$$642$$ 0 0
$$643$$ 0.602517 0.0237609 0.0118805 0.999929i $$-0.496218\pi$$
0.0118805 + 0.999929i $$0.496218\pi$$
$$644$$ 0 0
$$645$$ 6.78389 0.267115
$$646$$ 0 0
$$647$$ 27.3466 1.07511 0.537554 0.843230i $$-0.319349\pi$$
0.537554 + 0.843230i $$0.319349\pi$$
$$648$$ 0 0
$$649$$ 27.8172 1.09192
$$650$$ 0 0
$$651$$ −1.93751 −0.0759369
$$652$$ 0 0
$$653$$ 45.8172 1.79296 0.896482 0.443079i $$-0.146114\pi$$
0.896482 + 0.443079i $$0.146114\pi$$
$$654$$ 0 0
$$655$$ 12.4111 0.484942
$$656$$ 0 0
$$657$$ −5.57331 −0.217436
$$658$$ 0 0
$$659$$ −4.83779 −0.188453 −0.0942267 0.995551i $$-0.530038\pi$$
−0.0942267 + 0.995551i $$0.530038\pi$$
$$660$$ 0 0
$$661$$ 36.6933 1.42720 0.713602 0.700552i $$-0.247063\pi$$
0.713602 + 0.700552i $$0.247063\pi$$
$$662$$ 0 0
$$663$$ 29.8867 1.16070
$$664$$ 0 0
$$665$$ −5.30833 −0.205848
$$666$$ 0 0
$$667$$ −5.01916 −0.194343
$$668$$ 0 0
$$669$$ 20.7738 0.803163
$$670$$ 0 0
$$671$$ 7.31386 0.282349
$$672$$ 0 0
$$673$$ 37.2983 1.43774 0.718871 0.695143i $$-0.244659\pi$$
0.718871 + 0.695143i $$0.244659\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 30.1275 1.15789 0.578946 0.815366i $$-0.303464\pi$$
0.578946 + 0.815366i $$0.303464\pi$$
$$678$$ 0 0
$$679$$ 16.8222 0.645577
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ 20.8972 0.799610 0.399805 0.916600i $$-0.369078\pi$$
0.399805 + 0.916600i $$0.369078\pi$$
$$684$$ 0 0
$$685$$ 13.4217 0.512815
$$686$$ 0 0
$$687$$ 9.19499 0.350811
$$688$$ 0 0
$$689$$ 83.7266 3.18973
$$690$$ 0 0
$$691$$ −1.21611 −0.0462631 −0.0231316 0.999732i $$-0.507364\pi$$
−0.0231316 + 0.999732i $$0.507364\pi$$
$$692$$ 0 0
$$693$$ −4.31889 −0.164061
$$694$$ 0 0
$$695$$ 15.2786 0.579551
$$696$$ 0 0
$$697$$ −32.8263 −1.24338
$$698$$ 0 0
$$699$$ −6.98944 −0.264365
$$700$$ 0 0
$$701$$ 2.42669 0.0916547 0.0458273 0.998949i $$-0.485408\pi$$
0.0458273 + 0.998949i $$0.485408\pi$$
$$702$$ 0 0
$$703$$ 7.74055 0.291940
$$704$$ 0 0
$$705$$ 13.3083 0.501221
$$706$$ 0 0
$$707$$ −18.5003 −0.695774
$$708$$ 0 0
$$709$$ 3.00502 0.112856 0.0564280 0.998407i $$-0.482029\pi$$
0.0564280 + 0.998407i $$0.482029\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −1.13249 −0.0424122
$$714$$ 0 0
$$715$$ −16.9894 −0.635369
$$716$$ 0 0
$$717$$ 16.2353 0.606317
$$718$$ 0 0
$$719$$ −18.3814 −0.685510 −0.342755 0.939425i $$-0.611360\pi$$
−0.342755 + 0.939425i $$0.611360\pi$$
$$720$$ 0 0
$$721$$ −25.8116 −0.961276
$$722$$ 0 0
$$723$$ −7.77886 −0.289299
$$724$$ 0 0
$$725$$ 5.01916 0.186407
$$726$$ 0 0
$$727$$ 23.5335 0.872811 0.436405 0.899750i $$-0.356251\pi$$
0.436405 + 0.899750i $$0.356251\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 30.1260 1.11425
$$732$$ 0 0
$$733$$ −22.6308 −0.835887 −0.417944 0.908473i $$-0.637249\pi$$
−0.417944 + 0.908473i $$0.637249\pi$$
$$734$$ 0 0
$$735$$ −4.07306 −0.150237
$$736$$ 0 0
$$737$$ −0.211090 −0.00777558
$$738$$ 0 0
$$739$$ 2.18137 0.0802430 0.0401215 0.999195i $$-0.487226\pi$$
0.0401215 + 0.999195i $$0.487226\pi$$
$$740$$ 0 0
$$741$$ 20.8816 0.767106
$$742$$ 0 0
$$743$$ 25.7633 0.945163 0.472582 0.881287i $$-0.343322\pi$$
0.472582 + 0.881287i $$0.343322\pi$$
$$744$$ 0 0
$$745$$ 1.68111 0.0615912
$$746$$ 0 0
$$747$$ −1.97028 −0.0720888
$$748$$ 0 0
$$749$$ 23.7164 0.866577
$$750$$ 0 0
$$751$$ −33.3850 −1.21823 −0.609117 0.793080i $$-0.708476\pi$$
−0.609117 + 0.793080i $$0.708476\pi$$
$$752$$ 0 0
$$753$$ −19.0872 −0.695576
$$754$$ 0 0
$$755$$ −2.78389 −0.101316
$$756$$ 0 0
$$757$$ 30.9935 1.12648 0.563239 0.826294i $$-0.309555\pi$$
0.563239 + 0.826294i $$0.309555\pi$$
$$758$$ 0 0
$$759$$ −2.52444 −0.0916313
$$760$$ 0 0
$$761$$ 8.44082 0.305979 0.152990 0.988228i $$-0.451110\pi$$
0.152990 + 0.988228i $$0.451110\pi$$
$$762$$ 0 0
$$763$$ 28.7894 1.04225
$$764$$ 0 0
$$765$$ −4.44082 −0.160558
$$766$$ 0 0
$$767$$ 74.1588 2.67772
$$768$$ 0 0
$$769$$ 25.4372 0.917291 0.458645 0.888619i $$-0.348335\pi$$
0.458645 + 0.888619i $$0.348335\pi$$
$$770$$ 0 0
$$771$$ 27.2489 0.981345
$$772$$ 0 0
$$773$$ −35.8711 −1.29019 −0.645096 0.764101i $$-0.723183\pi$$
−0.645096 + 0.764101i $$0.723183\pi$$
$$774$$ 0 0
$$775$$ 1.13249 0.0406804
$$776$$ 0 0
$$777$$ −4.26804 −0.153115
$$778$$ 0 0
$$779$$ −22.9355 −0.821751
$$780$$ 0 0
$$781$$ −34.8449 −1.24685
$$782$$ 0 0
$$783$$ −5.01916 −0.179370
$$784$$ 0 0
$$785$$ 9.91638 0.353931
$$786$$ 0 0
$$787$$ 24.0524 0.857377 0.428689 0.903452i $$-0.358976\pi$$
0.428689 + 0.903452i $$0.358976\pi$$
$$788$$ 0 0
$$789$$ 15.4303 0.549332
$$790$$ 0 0
$$791$$ −33.6952 −1.19807
$$792$$ 0 0
$$793$$ 19.4983 0.692405
$$794$$ 0 0
$$795$$ −12.4408 −0.441230
$$796$$ 0 0
$$797$$ −49.9291 −1.76858 −0.884289 0.466940i $$-0.845356\pi$$
−0.884289 + 0.466940i $$0.845356\pi$$
$$798$$ 0 0
$$799$$ 59.0999 2.09080
$$800$$ 0 0
$$801$$ −7.62721 −0.269494
$$802$$ 0 0
$$803$$ 14.0695 0.496501
$$804$$ 0 0
$$805$$ 1.71083 0.0602989
$$806$$ 0 0
$$807$$ 12.9114 0.454501
$$808$$ 0 0
$$809$$ 20.8207 0.732019 0.366009 0.930611i $$-0.380724\pi$$
0.366009 + 0.930611i $$0.380724\pi$$
$$810$$ 0 0
$$811$$ −11.0347 −0.387482 −0.193741 0.981053i $$-0.562062\pi$$
−0.193741 + 0.981053i $$0.562062\pi$$
$$812$$ 0 0
$$813$$ −3.74914 −0.131488
$$814$$ 0 0
$$815$$ 7.42166 0.259970
$$816$$ 0 0
$$817$$ 21.0489 0.736407
$$818$$ 0 0
$$819$$ −11.5139 −0.402327
$$820$$ 0 0
$$821$$ −26.3627 −0.920066 −0.460033 0.887902i $$-0.652162\pi$$
−0.460033 + 0.887902i $$0.652162\pi$$
$$822$$ 0 0
$$823$$ 17.0177 0.593200 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$824$$ 0 0
$$825$$ 2.52444 0.0878896
$$826$$ 0 0
$$827$$ 26.4408 0.919437 0.459719 0.888065i $$-0.347950\pi$$
0.459719 + 0.888065i $$0.347950\pi$$
$$828$$ 0 0
$$829$$ 14.1602 0.491806 0.245903 0.969294i $$-0.420916\pi$$
0.245903 + 0.969294i $$0.420916\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ 0 0
$$833$$ −18.0877 −0.626702
$$834$$ 0 0
$$835$$ −5.94610 −0.205773
$$836$$ 0 0
$$837$$ −1.13249 −0.0391447
$$838$$ 0 0
$$839$$ 3.08719 0.106582 0.0532908 0.998579i $$-0.483029\pi$$
0.0532908 + 0.998579i $$0.483029\pi$$
$$840$$ 0 0
$$841$$ −3.80807 −0.131313
$$842$$ 0 0
$$843$$ 1.84835 0.0636605
$$844$$ 0 0
$$845$$ −32.2927 −1.11090
$$846$$ 0 0
$$847$$ −7.91638 −0.272010
$$848$$ 0 0
$$849$$ 1.40753 0.0483063
$$850$$ 0 0
$$851$$ −2.49472 −0.0855179
$$852$$ 0 0
$$853$$ −16.2822 −0.557491 −0.278746 0.960365i $$-0.589919\pi$$
−0.278746 + 0.960365i $$0.589919\pi$$
$$854$$ 0 0
$$855$$ −3.10278 −0.106113
$$856$$ 0 0
$$857$$ −29.5889 −1.01074 −0.505369 0.862903i $$-0.668644\pi$$
−0.505369 + 0.862903i $$0.668644\pi$$
$$858$$ 0 0
$$859$$ −30.8958 −1.05415 −0.527075 0.849819i $$-0.676711\pi$$
−0.527075 + 0.849819i $$0.676711\pi$$
$$860$$ 0 0
$$861$$ 12.6464 0.430987
$$862$$ 0 0
$$863$$ −28.2127 −0.960371 −0.480186 0.877167i $$-0.659431\pi$$
−0.480186 + 0.877167i $$0.659431\pi$$
$$864$$ 0 0
$$865$$ −13.0872 −0.444978
$$866$$ 0 0
$$867$$ −2.72088 −0.0924059
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −0.562751 −0.0190681
$$872$$ 0 0
$$873$$ 9.83276 0.332789
$$874$$ 0 0
$$875$$ −1.71083 −0.0578367
$$876$$ 0 0
$$877$$ 14.1078 0.476386 0.238193 0.971218i $$-0.423445\pi$$
0.238193 + 0.971218i $$0.423445\pi$$
$$878$$ 0 0
$$879$$ −27.5280 −0.928497
$$880$$ 0 0
$$881$$ −18.6222 −0.627398 −0.313699 0.949523i $$-0.601568\pi$$
−0.313699 + 0.949523i $$0.601568\pi$$
$$882$$ 0 0
$$883$$ 26.1133 0.878784 0.439392 0.898295i $$-0.355194\pi$$
0.439392 + 0.898295i $$0.355194\pi$$
$$884$$ 0 0
$$885$$ −11.0192 −0.370405
$$886$$ 0 0
$$887$$ −14.0383 −0.471360 −0.235680 0.971831i $$-0.575732\pi$$
−0.235680 + 0.971831i $$0.575732\pi$$
$$888$$ 0 0
$$889$$ 32.3955 1.08651
$$890$$ 0 0
$$891$$ −2.52444 −0.0845719
$$892$$ 0 0
$$893$$ 41.2927 1.38181
$$894$$ 0 0
$$895$$ 26.4494 0.884106
$$896$$ 0 0
$$897$$ −6.72999 −0.224708
$$898$$ 0 0
$$899$$ 5.68417 0.189578
$$900$$ 0 0
$$901$$ −55.2474 −1.84056
$$902$$ 0 0
$$903$$ −11.6061 −0.386226
$$904$$ 0 0
$$905$$ 24.0383 0.799061
$$906$$ 0 0
$$907$$ −33.8953 −1.12547 −0.562737 0.826636i $$-0.690252\pi$$
−0.562737 + 0.826636i $$0.690252\pi$$
$$908$$ 0 0
$$909$$ −10.8136 −0.358665
$$910$$ 0 0
$$911$$ −14.6277 −0.484638 −0.242319 0.970197i $$-0.577908\pi$$
−0.242319 + 0.970197i $$0.577908\pi$$
$$912$$ 0 0
$$913$$ 4.97385 0.164610
$$914$$ 0 0
$$915$$ −2.89722 −0.0957793
$$916$$ 0 0
$$917$$ −21.2333 −0.701185
$$918$$ 0 0
$$919$$ −5.68665 −0.187585 −0.0937927 0.995592i $$-0.529899\pi$$
−0.0937927 + 0.995592i $$0.529899\pi$$
$$920$$ 0 0
$$921$$ −4.56275 −0.150348
$$922$$ 0 0
$$923$$ −92.8943 −3.05765
$$924$$ 0 0
$$925$$ 2.49472 0.0820258
$$926$$ 0 0
$$927$$ −15.0872 −0.495528
$$928$$ 0 0
$$929$$ −27.7719 −0.911166 −0.455583 0.890193i $$-0.650569\pi$$
−0.455583 + 0.890193i $$0.650569\pi$$
$$930$$ 0 0
$$931$$ −12.6378 −0.414186
$$932$$ 0 0
$$933$$ 6.84333 0.224040
$$934$$ 0 0
$$935$$ 11.2106 0.366625
$$936$$ 0 0
$$937$$ −7.32391 −0.239262 −0.119631 0.992818i $$-0.538171\pi$$
−0.119631 + 0.992818i $$0.538171\pi$$
$$938$$ 0 0
$$939$$ 25.7491 0.840292
$$940$$ 0 0
$$941$$ 44.0711 1.43668 0.718338 0.695694i $$-0.244903\pi$$
0.718338 + 0.695694i $$0.244903\pi$$
$$942$$ 0 0
$$943$$ 7.39194 0.240715
$$944$$ 0 0
$$945$$ 1.71083 0.0556534
$$946$$ 0 0
$$947$$ 4.97225 0.161576 0.0807882 0.996731i $$-0.474256\pi$$
0.0807882 + 0.996731i $$0.474256\pi$$
$$948$$ 0 0
$$949$$ 37.5083 1.21757
$$950$$ 0 0
$$951$$ −9.23884 −0.299590
$$952$$ 0 0
$$953$$ −15.0177 −0.486471 −0.243236 0.969967i $$-0.578209\pi$$
−0.243236 + 0.969967i $$0.578209\pi$$
$$954$$ 0 0
$$955$$ 25.7194 0.832261
$$956$$ 0 0
$$957$$ 12.6705 0.409581
$$958$$ 0 0
$$959$$ −22.9622 −0.741488
$$960$$ 0 0
$$961$$ −29.7175 −0.958628
$$962$$ 0 0
$$963$$ 13.8625 0.446712
$$964$$ 0 0
$$965$$ −26.1361 −0.841350
$$966$$ 0 0
$$967$$ 29.5834 0.951337 0.475668 0.879625i $$-0.342206\pi$$
0.475668 + 0.879625i $$0.342206\pi$$
$$968$$ 0 0
$$969$$ −13.7789 −0.442641
$$970$$ 0 0
$$971$$ 33.2444 1.06686 0.533431 0.845843i $$-0.320902\pi$$
0.533431 + 0.845843i $$0.320902\pi$$
$$972$$ 0 0
$$973$$ −26.1391 −0.837982
$$974$$ 0 0
$$975$$ 6.72999 0.215532
$$976$$ 0 0
$$977$$ 4.91136 0.157128 0.0785641 0.996909i $$-0.474966\pi$$
0.0785641 + 0.996909i $$0.474966\pi$$
$$978$$ 0 0
$$979$$ 19.2544 0.615374
$$980$$ 0 0
$$981$$ 16.8277 0.537268
$$982$$ 0 0
$$983$$ 5.83422 0.186083 0.0930413 0.995662i $$-0.470341\pi$$
0.0930413 + 0.995662i $$0.470341\pi$$
$$984$$ 0 0
$$985$$ 14.3033 0.455741
$$986$$ 0 0
$$987$$ −22.7683 −0.724723
$$988$$ 0 0
$$989$$ −6.78389 −0.215715
$$990$$ 0 0
$$991$$ 12.4353 0.395020 0.197510 0.980301i $$-0.436715\pi$$
0.197510 + 0.980301i $$0.436715\pi$$
$$992$$ 0 0
$$993$$ 13.6030 0.431679
$$994$$ 0 0
$$995$$ −25.2927 −0.801834
$$996$$ 0 0
$$997$$ 14.8816 0.471306 0.235653 0.971837i $$-0.424277\pi$$
0.235653 + 0.971837i $$0.424277\pi$$
$$998$$ 0 0
$$999$$ −2.49472 −0.0789294
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 5520.2.a.bx.1.2 3
4.3 odd 2 2760.2.a.r.1.2 3
12.11 even 2 8280.2.a.bm.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.r.1.2 3 4.3 odd 2
5520.2.a.bx.1.2 3 1.1 even 1 trivial
8280.2.a.bm.1.2 3 12.11 even 2