Properties

 Label 5328.2.a.bf.1.2 Level $5328$ Weight $2$ Character 5328.1 Self dual yes Analytic conductor $42.544$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$5328 = 2^{4} \cdot 3^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5328.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$42.5442941969$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 5328.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+2.30278 q^{5} +2.60555 q^{7} +O(q^{10})$$ $$q+2.30278 q^{5} +2.60555 q^{7} -2.30278 q^{11} +1.30278 q^{13} +6.00000 q^{17} -2.00000 q^{19} +3.90833 q^{23} +0.302776 q^{25} +3.90833 q^{29} +0.302776 q^{31} +6.00000 q^{35} +1.00000 q^{37} -9.90833 q^{41} -0.605551 q^{43} +4.60555 q^{47} -0.211103 q^{49} +6.00000 q^{53} -5.30278 q^{55} +10.6056 q^{59} +7.51388 q^{61} +3.00000 q^{65} +3.51388 q^{67} +6.00000 q^{71} -12.3028 q^{73} -6.00000 q^{77} -9.11943 q^{79} +2.78890 q^{83} +13.8167 q^{85} +9.21110 q^{89} +3.39445 q^{91} -4.60555 q^{95} -16.4222 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + q^5 - 2 * q^7 $$2 q + q^{5} - 2 q^{7} - q^{11} - q^{13} + 12 q^{17} - 4 q^{19} - 3 q^{23} - 3 q^{25} - 3 q^{29} - 3 q^{31} + 12 q^{35} + 2 q^{37} - 9 q^{41} + 6 q^{43} + 2 q^{47} + 14 q^{49} + 12 q^{53} - 7 q^{55} + 14 q^{59} - 3 q^{61} + 6 q^{65} - 11 q^{67} + 12 q^{71} - 21 q^{73} - 12 q^{77} + 7 q^{79} + 20 q^{83} + 6 q^{85} + 4 q^{89} + 14 q^{91} - 2 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q + q^5 - 2 * q^7 - q^11 - q^13 + 12 * q^17 - 4 * q^19 - 3 * q^23 - 3 * q^25 - 3 * q^29 - 3 * q^31 + 12 * q^35 + 2 * q^37 - 9 * q^41 + 6 * q^43 + 2 * q^47 + 14 * q^49 + 12 * q^53 - 7 * q^55 + 14 * q^59 - 3 * q^61 + 6 * q^65 - 11 * q^67 + 12 * q^71 - 21 * q^73 - 12 * q^77 + 7 * q^79 + 20 * q^83 + 6 * q^85 + 4 * q^89 + 14 * q^91 - 2 * q^95 - 4 * q^97

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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 2.30278 1.02983 0.514916 0.857240i $$-0.327823\pi$$
0.514916 + 0.857240i $$0.327823\pi$$
$$6$$ 0 0
$$7$$ 2.60555 0.984806 0.492403 0.870367i $$-0.336119\pi$$
0.492403 + 0.870367i $$0.336119\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.30278 −0.694313 −0.347156 0.937807i $$-0.612853\pi$$
−0.347156 + 0.937807i $$0.612853\pi$$
$$12$$ 0 0
$$13$$ 1.30278 0.361325 0.180662 0.983545i $$-0.442176\pi$$
0.180662 + 0.983545i $$0.442176\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.90833 0.814942 0.407471 0.913218i $$-0.366411\pi$$
0.407471 + 0.913218i $$0.366411\pi$$
$$24$$ 0 0
$$25$$ 0.302776 0.0605551
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.90833 0.725758 0.362879 0.931836i $$-0.381794\pi$$
0.362879 + 0.931836i $$0.381794\pi$$
$$30$$ 0 0
$$31$$ 0.302776 0.0543801 0.0271901 0.999630i $$-0.491344\pi$$
0.0271901 + 0.999630i $$0.491344\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 6.00000 1.01419
$$36$$ 0 0
$$37$$ 1.00000 0.164399
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −9.90833 −1.54742 −0.773710 0.633540i $$-0.781601\pi$$
−0.773710 + 0.633540i $$0.781601\pi$$
$$42$$ 0 0
$$43$$ −0.605551 −0.0923457 −0.0461729 0.998933i $$-0.514703\pi$$
−0.0461729 + 0.998933i $$0.514703\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.60555 0.671789 0.335894 0.941900i $$-0.390961\pi$$
0.335894 + 0.941900i $$0.390961\pi$$
$$48$$ 0 0
$$49$$ −0.211103 −0.0301575
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −5.30278 −0.715026
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 10.6056 1.38073 0.690363 0.723464i $$-0.257451\pi$$
0.690363 + 0.723464i $$0.257451\pi$$
$$60$$ 0 0
$$61$$ 7.51388 0.962054 0.481027 0.876706i $$-0.340264\pi$$
0.481027 + 0.876706i $$0.340264\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 3.00000 0.372104
$$66$$ 0 0
$$67$$ 3.51388 0.429289 0.214644 0.976692i $$-0.431141\pi$$
0.214644 + 0.976692i $$0.431141\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ −12.3028 −1.43993 −0.719965 0.694010i $$-0.755842\pi$$
−0.719965 + 0.694010i $$0.755842\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ −9.11943 −1.02602 −0.513008 0.858384i $$-0.671469\pi$$
−0.513008 + 0.858384i $$0.671469\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 2.78890 0.306121 0.153061 0.988217i $$-0.451087\pi$$
0.153061 + 0.988217i $$0.451087\pi$$
$$84$$ 0 0
$$85$$ 13.8167 1.49863
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 9.21110 0.976375 0.488187 0.872739i $$-0.337658\pi$$
0.488187 + 0.872739i $$0.337658\pi$$
$$90$$ 0 0
$$91$$ 3.39445 0.355835
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −4.60555 −0.472520
$$96$$ 0 0
$$97$$ −16.4222 −1.66742 −0.833711 0.552201i $$-0.813788\pi$$
−0.833711 + 0.552201i $$0.813788\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.4222 1.23606 0.618028 0.786156i $$-0.287932\pi$$
0.618028 + 0.786156i $$0.287932\pi$$
$$102$$ 0 0
$$103$$ 0.302776 0.0298334 0.0149167 0.999889i $$-0.495252\pi$$
0.0149167 + 0.999889i $$0.495252\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0.697224 0.0674032 0.0337016 0.999432i $$-0.489270\pi$$
0.0337016 + 0.999432i $$0.489270\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 3.21110 0.302075 0.151038 0.988528i $$-0.451739\pi$$
0.151038 + 0.988528i $$0.451739\pi$$
$$114$$ 0 0
$$115$$ 9.00000 0.839254
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 15.6333 1.43310
$$120$$ 0 0
$$121$$ −5.69722 −0.517929
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −10.8167 −0.967471
$$126$$ 0 0
$$127$$ 19.2111 1.70471 0.852355 0.522964i $$-0.175174\pi$$
0.852355 + 0.522964i $$0.175174\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.6056 0.926611 0.463306 0.886199i $$-0.346663\pi$$
0.463306 + 0.886199i $$0.346663\pi$$
$$132$$ 0 0
$$133$$ −5.21110 −0.451860
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −0.908327 −0.0776036 −0.0388018 0.999247i $$-0.512354\pi$$
−0.0388018 + 0.999247i $$0.512354\pi$$
$$138$$ 0 0
$$139$$ 1.90833 0.161862 0.0809311 0.996720i $$-0.474211\pi$$
0.0809311 + 0.996720i $$0.474211\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −3.00000 −0.250873
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −19.8167 −1.62344 −0.811722 0.584044i $$-0.801469\pi$$
−0.811722 + 0.584044i $$0.801469\pi$$
$$150$$ 0 0
$$151$$ 20.6056 1.67686 0.838428 0.545012i $$-0.183475\pi$$
0.838428 + 0.545012i $$0.183475\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0.697224 0.0560024
$$156$$ 0 0
$$157$$ −7.21110 −0.575509 −0.287754 0.957704i $$-0.592909\pi$$
−0.287754 + 0.957704i $$0.592909\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 10.1833 0.802560
$$162$$ 0 0
$$163$$ −8.42221 −0.659678 −0.329839 0.944037i $$-0.606994\pi$$
−0.329839 + 0.944037i $$0.606994\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −5.51388 −0.426677 −0.213338 0.976978i $$-0.568434\pi$$
−0.213338 + 0.976978i $$0.568434\pi$$
$$168$$ 0 0
$$169$$ −11.3028 −0.869444
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 8.78890 0.668207 0.334104 0.942536i $$-0.391566\pi$$
0.334104 + 0.942536i $$0.391566\pi$$
$$174$$ 0 0
$$175$$ 0.788897 0.0596350
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −13.8167 −1.03271 −0.516353 0.856376i $$-0.672711\pi$$
−0.516353 + 0.856376i $$0.672711\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 2.30278 0.169303
$$186$$ 0 0
$$187$$ −13.8167 −1.01037
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −5.51388 −0.398970 −0.199485 0.979901i $$-0.563927\pi$$
−0.199485 + 0.979901i $$0.563927\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −26.4222 −1.87302 −0.936510 0.350640i $$-0.885964\pi$$
−0.936510 + 0.350640i $$0.885964\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 10.1833 0.714731
$$204$$ 0 0
$$205$$ −22.8167 −1.59358
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4.60555 0.318573
$$210$$ 0 0
$$211$$ −10.3028 −0.709272 −0.354636 0.935004i $$-0.615395\pi$$
−0.354636 + 0.935004i $$0.615395\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1.39445 −0.0951006
$$216$$ 0 0
$$217$$ 0.788897 0.0535538
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 7.81665 0.525805
$$222$$ 0 0
$$223$$ 5.81665 0.389512 0.194756 0.980852i $$-0.437609\pi$$
0.194756 + 0.980852i $$0.437609\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 13.8167 0.917044 0.458522 0.888683i $$-0.348379\pi$$
0.458522 + 0.888683i $$0.348379\pi$$
$$228$$ 0 0
$$229$$ 24.6056 1.62598 0.812990 0.582277i $$-0.197838\pi$$
0.812990 + 0.582277i $$0.197838\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.51388 −0.557763 −0.278881 0.960326i $$-0.589964\pi$$
−0.278881 + 0.960326i $$0.589964\pi$$
$$234$$ 0 0
$$235$$ 10.6056 0.691830
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −17.5139 −1.13288 −0.566439 0.824103i $$-0.691679\pi$$
−0.566439 + 0.824103i $$0.691679\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −0.486122 −0.0310572
$$246$$ 0 0
$$247$$ −2.60555 −0.165787
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −21.2111 −1.33883 −0.669416 0.742887i $$-0.733456\pi$$
−0.669416 + 0.742887i $$0.733456\pi$$
$$252$$ 0 0
$$253$$ −9.00000 −0.565825
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −3.21110 −0.200303 −0.100152 0.994972i $$-0.531933\pi$$
−0.100152 + 0.994972i $$0.531933\pi$$
$$258$$ 0 0
$$259$$ 2.60555 0.161901
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 13.8167 0.851971 0.425986 0.904730i $$-0.359927\pi$$
0.425986 + 0.904730i $$0.359927\pi$$
$$264$$ 0 0
$$265$$ 13.8167 0.848750
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 21.2111 1.29326 0.646632 0.762802i $$-0.276177\pi$$
0.646632 + 0.762802i $$0.276177\pi$$
$$270$$ 0 0
$$271$$ 22.4222 1.36205 0.681026 0.732259i $$-0.261534\pi$$
0.681026 + 0.732259i $$0.261534\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −0.697224 −0.0420442
$$276$$ 0 0
$$277$$ 0.119429 0.00717582 0.00358791 0.999994i $$-0.498858\pi$$
0.00358791 + 0.999994i $$0.498858\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 0 0
$$283$$ −24.6056 −1.46265 −0.731324 0.682030i $$-0.761097\pi$$
−0.731324 + 0.682030i $$0.761097\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −25.8167 −1.52391
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −11.0278 −0.644248 −0.322124 0.946697i $$-0.604397\pi$$
−0.322124 + 0.946697i $$0.604397\pi$$
$$294$$ 0 0
$$295$$ 24.4222 1.42192
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 5.09167 0.294459
$$300$$ 0 0
$$301$$ −1.57779 −0.0909426
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 17.3028 0.990754
$$306$$ 0 0
$$307$$ −17.9083 −1.02208 −0.511041 0.859556i $$-0.670740\pi$$
−0.511041 + 0.859556i $$0.670740\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 15.9083 0.902078 0.451039 0.892504i $$-0.351053\pi$$
0.451039 + 0.892504i $$0.351053\pi$$
$$312$$ 0 0
$$313$$ −9.02776 −0.510279 −0.255139 0.966904i $$-0.582121\pi$$
−0.255139 + 0.966904i $$0.582121\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −9.21110 −0.517347 −0.258674 0.965965i $$-0.583285\pi$$
−0.258674 + 0.965965i $$0.583285\pi$$
$$318$$ 0 0
$$319$$ −9.00000 −0.503903
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ 0 0
$$325$$ 0.394449 0.0218801
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 13.2111 0.726148 0.363074 0.931760i $$-0.381727\pi$$
0.363074 + 0.931760i $$0.381727\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 8.09167 0.442095
$$336$$ 0 0
$$337$$ 6.11943 0.333347 0.166673 0.986012i $$-0.446697\pi$$
0.166673 + 0.986012i $$0.446697\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −0.697224 −0.0377568
$$342$$ 0 0
$$343$$ −18.7889 −1.01451
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 10.1833 0.546671 0.273335 0.961919i $$-0.411873\pi$$
0.273335 + 0.961919i $$0.411873\pi$$
$$348$$ 0 0
$$349$$ 28.2389 1.51159 0.755796 0.654807i $$-0.227250\pi$$
0.755796 + 0.654807i $$0.227250\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −10.1833 −0.542005 −0.271002 0.962579i $$-0.587355\pi$$
−0.271002 + 0.962579i $$0.587355\pi$$
$$354$$ 0 0
$$355$$ 13.8167 0.733312
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 3.21110 0.169476 0.0847378 0.996403i $$-0.472995\pi$$
0.0847378 + 0.996403i $$0.472995\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −28.3305 −1.48289
$$366$$ 0 0
$$367$$ −3.81665 −0.199228 −0.0996139 0.995026i $$-0.531761\pi$$
−0.0996139 + 0.995026i $$0.531761\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 15.6333 0.811641
$$372$$ 0 0
$$373$$ −17.8167 −0.922511 −0.461256 0.887267i $$-0.652601\pi$$
−0.461256 + 0.887267i $$0.652601\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 5.09167 0.262235
$$378$$ 0 0
$$379$$ −24.3305 −1.24978 −0.624888 0.780715i $$-0.714855\pi$$
−0.624888 + 0.780715i $$0.714855\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −36.8444 −1.88266 −0.941331 0.337486i $$-0.890424\pi$$
−0.941331 + 0.337486i $$0.890424\pi$$
$$384$$ 0 0
$$385$$ −13.8167 −0.704162
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 37.1194 1.88203 0.941015 0.338365i $$-0.109874\pi$$
0.941015 + 0.338365i $$0.109874\pi$$
$$390$$ 0 0
$$391$$ 23.4500 1.18592
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −21.0000 −1.05662
$$396$$ 0 0
$$397$$ 6.18335 0.310333 0.155167 0.987888i $$-0.450409\pi$$
0.155167 + 0.987888i $$0.450409\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7.81665 0.390345 0.195173 0.980769i $$-0.437473\pi$$
0.195173 + 0.980769i $$0.437473\pi$$
$$402$$ 0 0
$$403$$ 0.394449 0.0196489
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.30278 −0.114144
$$408$$ 0 0
$$409$$ 31.0278 1.53422 0.767112 0.641513i $$-0.221693\pi$$
0.767112 + 0.641513i $$0.221693\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 27.6333 1.35975
$$414$$ 0 0
$$415$$ 6.42221 0.315254
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 36.1472 1.76591 0.882953 0.469462i $$-0.155552\pi$$
0.882953 + 0.469462i $$0.155552\pi$$
$$420$$ 0 0
$$421$$ −3.72498 −0.181544 −0.0907722 0.995872i $$-0.528934\pi$$
−0.0907722 + 0.995872i $$0.528934\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1.81665 0.0881207
$$426$$ 0 0
$$427$$ 19.5778 0.947436
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 9.21110 0.443683 0.221842 0.975083i $$-0.428793\pi$$
0.221842 + 0.975083i $$0.428793\pi$$
$$432$$ 0 0
$$433$$ 34.9361 1.67892 0.839461 0.543421i $$-0.182871\pi$$
0.839461 + 0.543421i $$0.182871\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −7.81665 −0.373921
$$438$$ 0 0
$$439$$ −30.3305 −1.44760 −0.723799 0.690011i $$-0.757606\pi$$
−0.723799 + 0.690011i $$0.757606\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 32.7250 1.55481 0.777405 0.629000i $$-0.216535\pi$$
0.777405 + 0.629000i $$0.216535\pi$$
$$444$$ 0 0
$$445$$ 21.2111 1.00550
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 15.2111 0.717856 0.358928 0.933365i $$-0.383142\pi$$
0.358928 + 0.933365i $$0.383142\pi$$
$$450$$ 0 0
$$451$$ 22.8167 1.07439
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 7.81665 0.366450
$$456$$ 0 0
$$457$$ −2.60555 −0.121883 −0.0609413 0.998141i $$-0.519410\pi$$
−0.0609413 + 0.998141i $$0.519410\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −12.4222 −0.578560 −0.289280 0.957245i $$-0.593416\pi$$
−0.289280 + 0.957245i $$0.593416\pi$$
$$462$$ 0 0
$$463$$ −26.6972 −1.24073 −0.620363 0.784315i $$-0.713015\pi$$
−0.620363 + 0.784315i $$0.713015\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 9.15559 0.422766
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1.39445 0.0641168
$$474$$ 0 0
$$475$$ −0.605551 −0.0277846
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −13.1194 −0.599442 −0.299721 0.954027i $$-0.596894\pi$$
−0.299721 + 0.954027i $$0.596894\pi$$
$$480$$ 0 0
$$481$$ 1.30278 0.0594015
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −37.8167 −1.71717
$$486$$ 0 0
$$487$$ 37.2111 1.68620 0.843098 0.537760i $$-0.180729\pi$$
0.843098 + 0.537760i $$0.180729\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 17.7250 0.799917 0.399959 0.916533i $$-0.369025\pi$$
0.399959 + 0.916533i $$0.369025\pi$$
$$492$$ 0 0
$$493$$ 23.4500 1.05613
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 15.6333 0.701250
$$498$$ 0 0
$$499$$ 42.2389 1.89087 0.945436 0.325809i $$-0.105637\pi$$
0.945436 + 0.325809i $$0.105637\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −6.48612 −0.289202 −0.144601 0.989490i $$-0.546190\pi$$
−0.144601 + 0.989490i $$0.546190\pi$$
$$504$$ 0 0
$$505$$ 28.6056 1.27293
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 4.18335 0.185424 0.0927118 0.995693i $$-0.470446\pi$$
0.0927118 + 0.995693i $$0.470446\pi$$
$$510$$ 0 0
$$511$$ −32.0555 −1.41805
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0.697224 0.0307234
$$516$$ 0 0
$$517$$ −10.6056 −0.466432
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −33.6333 −1.47350 −0.736751 0.676164i $$-0.763641\pi$$
−0.736751 + 0.676164i $$0.763641\pi$$
$$522$$ 0 0
$$523$$ 18.2389 0.797530 0.398765 0.917053i $$-0.369439\pi$$
0.398765 + 0.917053i $$0.369439\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.81665 0.0791347
$$528$$ 0 0
$$529$$ −7.72498 −0.335869
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −12.9083 −0.559122
$$534$$ 0 0
$$535$$ 1.60555 0.0694140
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0.486122 0.0209387
$$540$$ 0 0
$$541$$ 25.9361 1.11508 0.557540 0.830150i $$-0.311745\pi$$
0.557540 + 0.830150i $$0.311745\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 4.60555 0.197280
$$546$$ 0 0
$$547$$ 20.6056 0.881030 0.440515 0.897745i $$-0.354796\pi$$
0.440515 + 0.897745i $$0.354796\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −7.81665 −0.333001
$$552$$ 0 0
$$553$$ −23.7611 −1.01043
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −11.5139 −0.487859 −0.243929 0.969793i $$-0.578436\pi$$
−0.243929 + 0.969793i $$0.578436\pi$$
$$558$$ 0 0
$$559$$ −0.788897 −0.0333668
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −28.0555 −1.18240 −0.591199 0.806525i $$-0.701345\pi$$
−0.591199 + 0.806525i $$0.701345\pi$$
$$564$$ 0 0
$$565$$ 7.39445 0.311087
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −18.4222 −0.772299 −0.386150 0.922436i $$-0.626195\pi$$
−0.386150 + 0.922436i $$0.626195\pi$$
$$570$$ 0 0
$$571$$ 16.6972 0.698757 0.349379 0.936982i $$-0.386393\pi$$
0.349379 + 0.936982i $$0.386393\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1.18335 0.0493489
$$576$$ 0 0
$$577$$ 22.2389 0.925816 0.462908 0.886406i $$-0.346806\pi$$
0.462908 + 0.886406i $$0.346806\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 7.26662 0.301470
$$582$$ 0 0
$$583$$ −13.8167 −0.572227
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −45.6333 −1.88349 −0.941744 0.336330i $$-0.890814\pi$$
−0.941744 + 0.336330i $$0.890814\pi$$
$$588$$ 0 0
$$589$$ −0.605551 −0.0249513
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −18.4861 −0.759134 −0.379567 0.925164i $$-0.623927\pi$$
−0.379567 + 0.925164i $$0.623927\pi$$
$$594$$ 0 0
$$595$$ 36.0000 1.47586
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −20.7889 −0.849411 −0.424706 0.905331i $$-0.639622\pi$$
−0.424706 + 0.905331i $$0.639622\pi$$
$$600$$ 0 0
$$601$$ −24.3028 −0.991331 −0.495665 0.868514i $$-0.665076\pi$$
−0.495665 + 0.868514i $$0.665076\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −13.1194 −0.533381
$$606$$ 0 0
$$607$$ 13.4861 0.547385 0.273692 0.961817i $$-0.411755\pi$$
0.273692 + 0.961817i $$0.411755\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6.00000 0.242734
$$612$$ 0 0
$$613$$ −29.8167 −1.20428 −0.602142 0.798389i $$-0.705686\pi$$
−0.602142 + 0.798389i $$0.705686\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.5694 1.71378 0.856890 0.515500i $$-0.172394\pi$$
0.856890 + 0.515500i $$0.172394\pi$$
$$618$$ 0 0
$$619$$ 6.30278 0.253330 0.126665 0.991946i $$-0.459573\pi$$
0.126665 + 0.991946i $$0.459573\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 24.0000 0.961540
$$624$$ 0 0
$$625$$ −26.4222 −1.05689
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ −14.6972 −0.585087 −0.292544 0.956252i $$-0.594502\pi$$
−0.292544 + 0.956252i $$0.594502\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 44.2389 1.75557
$$636$$ 0 0
$$637$$ −0.275019 −0.0108967
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 20.5139 0.810249 0.405125 0.914261i $$-0.367228\pi$$
0.405125 + 0.914261i $$0.367228\pi$$
$$642$$ 0 0
$$643$$ 8.18335 0.322720 0.161360 0.986896i $$-0.448412\pi$$
0.161360 + 0.986896i $$0.448412\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 20.9361 0.823082 0.411541 0.911391i $$-0.364991\pi$$
0.411541 + 0.911391i $$0.364991\pi$$
$$648$$ 0 0
$$649$$ −24.4222 −0.958655
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3.90833 0.152945 0.0764723 0.997072i $$-0.475634\pi$$
0.0764723 + 0.997072i $$0.475634\pi$$
$$654$$ 0 0
$$655$$ 24.4222 0.954255
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −16.8806 −0.657574 −0.328787 0.944404i $$-0.606640\pi$$
−0.328787 + 0.944404i $$0.606640\pi$$
$$660$$ 0 0
$$661$$ −30.5139 −1.18685 −0.593426 0.804888i $$-0.702225\pi$$
−0.593426 + 0.804888i $$0.702225\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −12.0000 −0.465340
$$666$$ 0 0
$$667$$ 15.2750 0.591451
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −17.3028 −0.667966
$$672$$ 0 0
$$673$$ 20.6972 0.797819 0.398910 0.916990i $$-0.369389\pi$$
0.398910 + 0.916990i $$0.369389\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −14.2389 −0.547244 −0.273622 0.961837i $$-0.588222\pi$$
−0.273622 + 0.961837i $$0.588222\pi$$
$$678$$ 0 0
$$679$$ −42.7889 −1.64209
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ −2.09167 −0.0799187
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 7.81665 0.297791
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.39445 0.166691
$$696$$ 0 0
$$697$$ −59.4500 −2.25183
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −40.1194 −1.51529 −0.757645 0.652667i $$-0.773650\pi$$
−0.757645 + 0.652667i $$0.773650\pi$$
$$702$$ 0 0
$$703$$ −2.00000 −0.0754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 32.3667 1.21727
$$708$$ 0 0
$$709$$ −41.3305 −1.55220 −0.776100 0.630609i $$-0.782805\pi$$
−0.776100 + 0.630609i $$0.782805\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1.18335 0.0443167
$$714$$ 0 0
$$715$$ −6.90833 −0.258357
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −51.6333 −1.92560 −0.962799 0.270220i $$-0.912904\pi$$
−0.962799 + 0.270220i $$0.912904\pi$$
$$720$$ 0 0
$$721$$ 0.788897 0.0293801
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1.18335 0.0439484
$$726$$ 0 0
$$727$$ −19.0917 −0.708071 −0.354035 0.935232i $$-0.615191\pi$$
−0.354035 + 0.935232i $$0.615191\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −3.63331 −0.134383
$$732$$ 0 0
$$733$$ −13.6333 −0.503558 −0.251779 0.967785i $$-0.581016\pi$$
−0.251779 + 0.967785i $$0.581016\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −8.09167 −0.298061
$$738$$ 0 0
$$739$$ 2.66947 0.0981980 0.0490990 0.998794i $$-0.484365\pi$$
0.0490990 + 0.998794i $$0.484365\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −29.4500 −1.08041 −0.540207 0.841532i $$-0.681654\pi$$
−0.540207 + 0.841532i $$0.681654\pi$$
$$744$$ 0 0
$$745$$ −45.6333 −1.67188
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 1.81665 0.0663791
$$750$$ 0 0
$$751$$ −14.0000 −0.510867 −0.255434 0.966827i $$-0.582218\pi$$
−0.255434 + 0.966827i $$0.582218\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 47.4500 1.72688
$$756$$ 0 0
$$757$$ 5.69722 0.207069 0.103535 0.994626i $$-0.466985\pi$$
0.103535 + 0.994626i $$0.466985\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −16.8806 −0.611920 −0.305960 0.952044i $$-0.598977\pi$$
−0.305960 + 0.952044i $$0.598977\pi$$
$$762$$ 0 0
$$763$$ 5.21110 0.188655
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 13.8167 0.498890
$$768$$ 0 0
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −22.0555 −0.793282 −0.396641 0.917974i $$-0.629824\pi$$
−0.396641 + 0.917974i $$0.629824\pi$$
$$774$$ 0 0
$$775$$ 0.0916731 0.00329299
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 19.8167 0.710005
$$780$$ 0 0
$$781$$ −13.8167 −0.494399
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −16.6056 −0.592678
$$786$$ 0 0
$$787$$ −10.7889 −0.384583 −0.192291 0.981338i $$-0.561592\pi$$
−0.192291 + 0.981338i $$0.561592\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 8.36669 0.297485
$$792$$ 0 0
$$793$$ 9.78890 0.347614
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −22.3305 −0.790988 −0.395494 0.918469i $$-0.629427\pi$$
−0.395494 + 0.918469i $$0.629427\pi$$
$$798$$ 0 0
$$799$$ 27.6333 0.977596
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 28.3305 0.999763
$$804$$ 0 0
$$805$$ 23.4500 0.826503
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 35.4500 1.24635 0.623177 0.782081i $$-0.285842\pi$$
0.623177 + 0.782081i $$0.285842\pi$$
$$810$$ 0 0
$$811$$ 7.14719 0.250972 0.125486 0.992095i $$-0.459951\pi$$
0.125486 + 0.992095i $$0.459951\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −19.3944 −0.679358
$$816$$ 0 0
$$817$$ 1.21110 0.0423711
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3.21110 0.112068 0.0560341 0.998429i $$-0.482154\pi$$
0.0560341 + 0.998429i $$0.482154\pi$$
$$822$$ 0 0
$$823$$ −44.8444 −1.56318 −0.781589 0.623794i $$-0.785590\pi$$
−0.781589 + 0.623794i $$0.785590\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 34.6056 1.20335 0.601676 0.798740i $$-0.294500\pi$$
0.601676 + 0.798740i $$0.294500\pi$$
$$828$$ 0 0
$$829$$ −27.7250 −0.962928 −0.481464 0.876466i $$-0.659895\pi$$
−0.481464 + 0.876466i $$0.659895\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −1.26662 −0.0438856
$$834$$ 0 0
$$835$$ −12.6972 −0.439406
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −12.9722 −0.447852 −0.223926 0.974606i $$-0.571887\pi$$
−0.223926 + 0.974606i $$0.571887\pi$$
$$840$$ 0 0
$$841$$ −13.7250 −0.473275
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −26.0278 −0.895382
$$846$$ 0 0
$$847$$ −14.8444 −0.510060
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 3.90833 0.133976
$$852$$ 0 0
$$853$$ 42.5416 1.45660 0.728299 0.685260i $$-0.240311\pi$$
0.728299 + 0.685260i $$0.240311\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −42.8444 −1.46354 −0.731769 0.681553i $$-0.761305\pi$$
−0.731769 + 0.681553i $$0.761305\pi$$
$$858$$ 0 0
$$859$$ −48.0555 −1.63963 −0.819816 0.572626i $$-0.805925\pi$$
−0.819816 + 0.572626i $$0.805925\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 0 0
$$865$$ 20.2389 0.688142
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 21.0000 0.712376
$$870$$ 0 0
$$871$$ 4.57779 0.155113
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −28.1833 −0.952771
$$876$$ 0 0
$$877$$ −7.21110 −0.243502 −0.121751 0.992561i $$-0.538851\pi$$
−0.121751 + 0.992561i $$0.538851\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 28.5416 0.961592 0.480796 0.876832i $$-0.340348\pi$$
0.480796 + 0.876832i $$0.340348\pi$$
$$882$$ 0 0
$$883$$ −26.4222 −0.889178 −0.444589 0.895735i $$-0.646650\pi$$
−0.444589 + 0.895735i $$0.646650\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −0.422205 −0.0141763 −0.00708813 0.999975i $$-0.502256\pi$$
−0.00708813 + 0.999975i $$0.502256\pi$$
$$888$$ 0 0
$$889$$ 50.0555 1.67881
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −9.21110 −0.308238
$$894$$ 0 0
$$895$$ −31.8167 −1.06351
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1.18335 0.0394668
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 46.0555 1.53094
$$906$$ 0 0
$$907$$ −26.0000 −0.863316 −0.431658 0.902037i $$-0.642071\pi$$
−0.431658 + 0.902037i $$0.642071\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −17.5778 −0.582378 −0.291189 0.956665i $$-0.594051\pi$$
−0.291189 + 0.956665i $$0.594051\pi$$
$$912$$ 0 0
$$913$$ −6.42221 −0.212544
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 27.6333 0.912532
$$918$$ 0 0
$$919$$ 9.57779 0.315942 0.157971 0.987444i $$-0.449505\pi$$
0.157971 + 0.987444i $$0.449505\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 7.81665 0.257288
$$924$$ 0 0
$$925$$ 0.302776 0.00995520
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 18.4861 0.606510 0.303255 0.952909i $$-0.401927\pi$$
0.303255 + 0.952909i $$0.401927\pi$$
$$930$$ 0 0
$$931$$ 0.422205 0.0138372
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −31.8167 −1.04052
$$936$$ 0 0
$$937$$ −18.0917 −0.591029 −0.295515 0.955338i $$-0.595491\pi$$
−0.295515 + 0.955338i $$0.595491\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −13.8167 −0.450410 −0.225205 0.974311i $$-0.572305\pi$$
−0.225205 + 0.974311i $$0.572305\pi$$
$$942$$ 0 0
$$943$$ −38.7250 −1.26106
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −3.63331 −0.118067 −0.0590333 0.998256i $$-0.518802\pi$$
−0.0590333 + 0.998256i $$0.518802\pi$$
$$948$$ 0 0
$$949$$ −16.0278 −0.520283
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −49.7527 −1.61165 −0.805825 0.592154i $$-0.798278\pi$$
−0.805825 + 0.592154i $$0.798278\pi$$
$$954$$ 0 0
$$955$$ −12.6972 −0.410873
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −2.36669 −0.0764245
$$960$$ 0 0
$$961$$ −30.9083 −0.997043
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −9.21110 −0.296516
$$966$$ 0 0
$$967$$ 6.72498 0.216261 0.108130 0.994137i $$-0.465514\pi$$
0.108130 + 0.994137i $$0.465514\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −22.5416 −0.723395 −0.361698 0.932295i $$-0.617803\pi$$
−0.361698 + 0.932295i $$0.617803\pi$$
$$972$$ 0 0
$$973$$ 4.97224 0.159403
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ −21.2111 −0.677910
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 0 0
$$985$$ 13.8167 0.440235
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −2.36669 −0.0752564
$$990$$ 0 0
$$991$$ −50.6972 −1.61045 −0.805225 0.592969i $$-0.797956\pi$$
−0.805225 + 0.592969i $$0.797956\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −60.8444 −1.92890
$$996$$ 0 0
$$997$$ −52.4222 −1.66023 −0.830114 0.557594i $$-0.811725\pi$$
−0.830114 + 0.557594i $$0.811725\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 5328.2.a.bf.1.2 2
3.2 odd 2 592.2.a.f.1.1 2
4.3 odd 2 666.2.a.j.1.2 2
12.11 even 2 74.2.a.a.1.2 2
24.5 odd 2 2368.2.a.ba.1.2 2
24.11 even 2 2368.2.a.s.1.1 2
60.23 odd 4 1850.2.b.i.149.4 4
60.47 odd 4 1850.2.b.i.149.1 4
60.59 even 2 1850.2.a.u.1.1 2
84.83 odd 2 3626.2.a.a.1.1 2
132.131 odd 2 8954.2.a.p.1.2 2
444.443 even 2 2738.2.a.l.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 12.11 even 2
592.2.a.f.1.1 2 3.2 odd 2
666.2.a.j.1.2 2 4.3 odd 2
1850.2.a.u.1.1 2 60.59 even 2
1850.2.b.i.149.1 4 60.47 odd 4
1850.2.b.i.149.4 4 60.23 odd 4
2368.2.a.s.1.1 2 24.11 even 2
2368.2.a.ba.1.2 2 24.5 odd 2
2738.2.a.l.1.2 2 444.443 even 2
3626.2.a.a.1.1 2 84.83 odd 2
5328.2.a.bf.1.2 2 1.1 even 1 trivial
8954.2.a.p.1.2 2 132.131 odd 2