# Properties

 Label 5520.2.a.bu.1.1 Level $5520$ Weight $2$ Character 5520.1 Self dual yes Analytic conductor $44.077$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 345) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 5520.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -0.732051 q^{11} -4.19615 q^{13} +1.00000 q^{15} -5.73205 q^{17} +6.73205 q^{19} +3.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.267949 q^{29} -3.92820 q^{31} -0.732051 q^{33} +3.00000 q^{35} +11.9282 q^{37} -4.19615 q^{39} -0.267949 q^{41} +4.53590 q^{43} +1.00000 q^{45} +9.66025 q^{47} +2.00000 q^{49} -5.73205 q^{51} -2.26795 q^{53} -0.732051 q^{55} +6.73205 q^{57} +3.19615 q^{59} +13.1244 q^{61} +3.00000 q^{63} -4.19615 q^{65} -0.464102 q^{67} -1.00000 q^{69} +0.267949 q^{71} +9.66025 q^{73} +1.00000 q^{75} -2.19615 q^{77} +6.92820 q^{79} +1.00000 q^{81} +10.2679 q^{83} -5.73205 q^{85} -0.267949 q^{87} +9.46410 q^{89} -12.5885 q^{91} -3.92820 q^{93} +6.73205 q^{95} -4.00000 q^{97} -0.732051 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} - 8 q^{17} + 10 q^{19} + 6 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 6 q^{35} + 10 q^{37} + 2 q^{39} - 4 q^{41} + 16 q^{43} + 2 q^{45} + 2 q^{47} + 4 q^{49} - 8 q^{51} - 8 q^{53} + 2 q^{55} + 10 q^{57} - 4 q^{59} + 2 q^{61} + 6 q^{63} + 2 q^{65} + 6 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} + 2 q^{75} + 6 q^{77} + 2 q^{81} + 24 q^{83} - 8 q^{85} - 4 q^{87} + 12 q^{89} + 6 q^{91} + 6 q^{93} + 10 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 - 8 * q^17 + 10 * q^19 + 6 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 6 * q^31 + 2 * q^33 + 6 * q^35 + 10 * q^37 + 2 * q^39 - 4 * q^41 + 16 * q^43 + 2 * q^45 + 2 * q^47 + 4 * q^49 - 8 * q^51 - 8 * q^53 + 2 * q^55 + 10 * q^57 - 4 * q^59 + 2 * q^61 + 6 * q^63 + 2 * q^65 + 6 * q^67 - 2 * q^69 + 4 * q^71 + 2 * q^73 + 2 * q^75 + 6 * q^77 + 2 * q^81 + 24 * q^83 - 8 * q^85 - 4 * q^87 + 12 * q^89 + 6 * q^91 + 6 * q^93 + 10 * q^95 - 8 * 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
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −0.732051 −0.220722 −0.110361 0.993892i $$-0.535201\pi$$
−0.110361 + 0.993892i $$0.535201\pi$$
$$12$$ 0 0
$$13$$ −4.19615 −1.16380 −0.581902 0.813259i $$-0.697691\pi$$
−0.581902 + 0.813259i $$0.697691\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −5.73205 −1.39023 −0.695113 0.718900i $$-0.744646\pi$$
−0.695113 + 0.718900i $$0.744646\pi$$
$$18$$ 0 0
$$19$$ 6.73205 1.54444 0.772219 0.635356i $$-0.219147\pi$$
0.772219 + 0.635356i $$0.219147\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 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
$$29$$ −0.267949 −0.0497569 −0.0248785 0.999690i $$-0.507920\pi$$
−0.0248785 + 0.999690i $$0.507920\pi$$
$$30$$ 0 0
$$31$$ −3.92820 −0.705526 −0.352763 0.935713i $$-0.614758\pi$$
−0.352763 + 0.935713i $$0.614758\pi$$
$$32$$ 0 0
$$33$$ −0.732051 −0.127434
$$34$$ 0 0
$$35$$ 3.00000 0.507093
$$36$$ 0 0
$$37$$ 11.9282 1.96098 0.980492 0.196558i $$-0.0629763\pi$$
0.980492 + 0.196558i $$0.0629763\pi$$
$$38$$ 0 0
$$39$$ −4.19615 −0.671922
$$40$$ 0 0
$$41$$ −0.267949 −0.0418466 −0.0209233 0.999781i $$-0.506661\pi$$
−0.0209233 + 0.999781i $$0.506661\pi$$
$$42$$ 0 0
$$43$$ 4.53590 0.691718 0.345859 0.938286i $$-0.387588\pi$$
0.345859 + 0.938286i $$0.387588\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 9.66025 1.40909 0.704546 0.709658i $$-0.251150\pi$$
0.704546 + 0.709658i $$0.251150\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −5.73205 −0.802648
$$52$$ 0 0
$$53$$ −2.26795 −0.311527 −0.155763 0.987794i $$-0.549784\pi$$
−0.155763 + 0.987794i $$0.549784\pi$$
$$54$$ 0 0
$$55$$ −0.732051 −0.0987097
$$56$$ 0 0
$$57$$ 6.73205 0.891682
$$58$$ 0 0
$$59$$ 3.19615 0.416104 0.208052 0.978118i $$-0.433288\pi$$
0.208052 + 0.978118i $$0.433288\pi$$
$$60$$ 0 0
$$61$$ 13.1244 1.68040 0.840201 0.542275i $$-0.182437\pi$$
0.840201 + 0.542275i $$0.182437\pi$$
$$62$$ 0 0
$$63$$ 3.00000 0.377964
$$64$$ 0 0
$$65$$ −4.19615 −0.520469
$$66$$ 0 0
$$67$$ −0.464102 −0.0566990 −0.0283495 0.999598i $$-0.509025\pi$$
−0.0283495 + 0.999598i $$0.509025\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 0.267949 0.0317997 0.0158999 0.999874i $$-0.494939\pi$$
0.0158999 + 0.999874i $$0.494939\pi$$
$$72$$ 0 0
$$73$$ 9.66025 1.13065 0.565324 0.824869i $$-0.308751\pi$$
0.565324 + 0.824869i $$0.308751\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −2.19615 −0.250275
$$78$$ 0 0
$$79$$ 6.92820 0.779484 0.389742 0.920924i $$-0.372564\pi$$
0.389742 + 0.920924i $$0.372564\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 10.2679 1.12705 0.563527 0.826098i $$-0.309444\pi$$
0.563527 + 0.826098i $$0.309444\pi$$
$$84$$ 0 0
$$85$$ −5.73205 −0.621728
$$86$$ 0 0
$$87$$ −0.267949 −0.0287272
$$88$$ 0 0
$$89$$ 9.46410 1.00319 0.501596 0.865102i $$-0.332746\pi$$
0.501596 + 0.865102i $$0.332746\pi$$
$$90$$ 0 0
$$91$$ −12.5885 −1.31963
$$92$$ 0 0
$$93$$ −3.92820 −0.407336
$$94$$ 0 0
$$95$$ 6.73205 0.690694
$$96$$ 0 0
$$97$$ −4.00000 −0.406138 −0.203069 0.979164i $$-0.565092\pi$$
−0.203069 + 0.979164i $$0.565092\pi$$
$$98$$ 0 0
$$99$$ −0.732051 −0.0735739
$$100$$ 0 0
$$101$$ −5.19615 −0.517036 −0.258518 0.966006i $$-0.583234\pi$$
−0.258518 + 0.966006i $$0.583234\pi$$
$$102$$ 0 0
$$103$$ 11.8564 1.16825 0.584123 0.811665i $$-0.301438\pi$$
0.584123 + 0.811665i $$0.301438\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ 0 0
$$107$$ 18.1244 1.75215 0.876074 0.482177i $$-0.160154\pi$$
0.876074 + 0.482177i $$0.160154\pi$$
$$108$$ 0 0
$$109$$ 6.73205 0.644814 0.322407 0.946601i $$-0.395508\pi$$
0.322407 + 0.946601i $$0.395508\pi$$
$$110$$ 0 0
$$111$$ 11.9282 1.13217
$$112$$ 0 0
$$113$$ −9.19615 −0.865101 −0.432551 0.901610i $$-0.642386\pi$$
−0.432551 + 0.901610i $$0.642386\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ −4.19615 −0.387934
$$118$$ 0 0
$$119$$ −17.1962 −1.57637
$$120$$ 0 0
$$121$$ −10.4641 −0.951282
$$122$$ 0 0
$$123$$ −0.267949 −0.0241602
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −1.80385 −0.160066 −0.0800328 0.996792i $$-0.525503\pi$$
−0.0800328 + 0.996792i $$0.525503\pi$$
$$128$$ 0 0
$$129$$ 4.53590 0.399364
$$130$$ 0 0
$$131$$ −5.07180 −0.443125 −0.221562 0.975146i $$-0.571116\pi$$
−0.221562 + 0.975146i $$0.571116\pi$$
$$132$$ 0 0
$$133$$ 20.1962 1.75123
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 20.7846 1.77575 0.887875 0.460086i $$-0.152181\pi$$
0.887875 + 0.460086i $$0.152181\pi$$
$$138$$ 0 0
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ 9.66025 0.813540
$$142$$ 0 0
$$143$$ 3.07180 0.256877
$$144$$ 0 0
$$145$$ −0.267949 −0.0222520
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 0 0
$$149$$ −16.1962 −1.32684 −0.663420 0.748247i $$-0.730896\pi$$
−0.663420 + 0.748247i $$0.730896\pi$$
$$150$$ 0 0
$$151$$ 18.3923 1.49674 0.748372 0.663279i $$-0.230836\pi$$
0.748372 + 0.663279i $$0.230836\pi$$
$$152$$ 0 0
$$153$$ −5.73205 −0.463409
$$154$$ 0 0
$$155$$ −3.92820 −0.315521
$$156$$ 0 0
$$157$$ −13.9282 −1.11159 −0.555796 0.831319i $$-0.687586\pi$$
−0.555796 + 0.831319i $$0.687586\pi$$
$$158$$ 0 0
$$159$$ −2.26795 −0.179860
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ −14.0000 −1.09656 −0.548282 0.836293i $$-0.684718\pi$$
−0.548282 + 0.836293i $$0.684718\pi$$
$$164$$ 0 0
$$165$$ −0.732051 −0.0569901
$$166$$ 0 0
$$167$$ 6.33975 0.490584 0.245292 0.969449i $$-0.421116\pi$$
0.245292 + 0.969449i $$0.421116\pi$$
$$168$$ 0 0
$$169$$ 4.60770 0.354438
$$170$$ 0 0
$$171$$ 6.73205 0.514813
$$172$$ 0 0
$$173$$ −25.8564 −1.96583 −0.982913 0.184070i $$-0.941073\pi$$
−0.982913 + 0.184070i $$0.941073\pi$$
$$174$$ 0 0
$$175$$ 3.00000 0.226779
$$176$$ 0 0
$$177$$ 3.19615 0.240238
$$178$$ 0 0
$$179$$ −11.4641 −0.856867 −0.428434 0.903573i $$-0.640934\pi$$
−0.428434 + 0.903573i $$0.640934\pi$$
$$180$$ 0 0
$$181$$ −8.39230 −0.623795 −0.311898 0.950116i $$-0.600965\pi$$
−0.311898 + 0.950116i $$0.600965\pi$$
$$182$$ 0 0
$$183$$ 13.1244 0.970180
$$184$$ 0 0
$$185$$ 11.9282 0.876979
$$186$$ 0 0
$$187$$ 4.19615 0.306853
$$188$$ 0 0
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ 13.6603 0.988421 0.494211 0.869342i $$-0.335457\pi$$
0.494211 + 0.869342i $$0.335457\pi$$
$$192$$ 0 0
$$193$$ 17.4641 1.25709 0.628547 0.777772i $$-0.283650\pi$$
0.628547 + 0.777772i $$0.283650\pi$$
$$194$$ 0 0
$$195$$ −4.19615 −0.300493
$$196$$ 0 0
$$197$$ −22.3923 −1.59539 −0.797693 0.603064i $$-0.793946\pi$$
−0.797693 + 0.603064i $$0.793946\pi$$
$$198$$ 0 0
$$199$$ 4.53590 0.321541 0.160771 0.986992i $$-0.448602\pi$$
0.160771 + 0.986992i $$0.448602\pi$$
$$200$$ 0 0
$$201$$ −0.464102 −0.0327352
$$202$$ 0 0
$$203$$ −0.803848 −0.0564190
$$204$$ 0 0
$$205$$ −0.267949 −0.0187144
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ −4.92820 −0.340891
$$210$$ 0 0
$$211$$ −19.7846 −1.36203 −0.681014 0.732270i $$-0.738461\pi$$
−0.681014 + 0.732270i $$0.738461\pi$$
$$212$$ 0 0
$$213$$ 0.267949 0.0183596
$$214$$ 0 0
$$215$$ 4.53590 0.309346
$$216$$ 0 0
$$217$$ −11.7846 −0.799991
$$218$$ 0 0
$$219$$ 9.66025 0.652779
$$220$$ 0 0
$$221$$ 24.0526 1.61795
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −9.07180 −0.602116 −0.301058 0.953606i $$-0.597340\pi$$
−0.301058 + 0.953606i $$0.597340\pi$$
$$228$$ 0 0
$$229$$ −11.3205 −0.748080 −0.374040 0.927413i $$-0.622028\pi$$
−0.374040 + 0.927413i $$0.622028\pi$$
$$230$$ 0 0
$$231$$ −2.19615 −0.144496
$$232$$ 0 0
$$233$$ 25.8564 1.69391 0.846955 0.531665i $$-0.178433\pi$$
0.846955 + 0.531665i $$0.178433\pi$$
$$234$$ 0 0
$$235$$ 9.66025 0.630165
$$236$$ 0 0
$$237$$ 6.92820 0.450035
$$238$$ 0 0
$$239$$ −21.0526 −1.36178 −0.680888 0.732387i $$-0.738406\pi$$
−0.680888 + 0.732387i $$0.738406\pi$$
$$240$$ 0 0
$$241$$ −30.7321 −1.97963 −0.989813 0.142376i $$-0.954526\pi$$
−0.989813 + 0.142376i $$0.954526\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ −28.2487 −1.79742
$$248$$ 0 0
$$249$$ 10.2679 0.650705
$$250$$ 0 0
$$251$$ −19.8564 −1.25333 −0.626663 0.779291i $$-0.715580\pi$$
−0.626663 + 0.779291i $$0.715580\pi$$
$$252$$ 0 0
$$253$$ 0.732051 0.0460236
$$254$$ 0 0
$$255$$ −5.73205 −0.358955
$$256$$ 0 0
$$257$$ −5.66025 −0.353077 −0.176538 0.984294i $$-0.556490\pi$$
−0.176538 + 0.984294i $$0.556490\pi$$
$$258$$ 0 0
$$259$$ 35.7846 2.22355
$$260$$ 0 0
$$261$$ −0.267949 −0.0165856
$$262$$ 0 0
$$263$$ −8.26795 −0.509824 −0.254912 0.966964i $$-0.582046\pi$$
−0.254912 + 0.966964i $$0.582046\pi$$
$$264$$ 0 0
$$265$$ −2.26795 −0.139319
$$266$$ 0 0
$$267$$ 9.46410 0.579194
$$268$$ 0 0
$$269$$ 28.6603 1.74745 0.873723 0.486423i $$-0.161699\pi$$
0.873723 + 0.486423i $$0.161699\pi$$
$$270$$ 0 0
$$271$$ −5.39230 −0.327559 −0.163780 0.986497i $$-0.552369\pi$$
−0.163780 + 0.986497i $$0.552369\pi$$
$$272$$ 0 0
$$273$$ −12.5885 −0.761888
$$274$$ 0 0
$$275$$ −0.732051 −0.0441443
$$276$$ 0 0
$$277$$ 14.7846 0.888321 0.444161 0.895947i $$-0.353502\pi$$
0.444161 + 0.895947i $$0.353502\pi$$
$$278$$ 0 0
$$279$$ −3.92820 −0.235175
$$280$$ 0 0
$$281$$ −16.0526 −0.957615 −0.478808 0.877920i $$-0.658931\pi$$
−0.478808 + 0.877920i $$0.658931\pi$$
$$282$$ 0 0
$$283$$ 6.46410 0.384251 0.192125 0.981370i $$-0.438462\pi$$
0.192125 + 0.981370i $$0.438462\pi$$
$$284$$ 0 0
$$285$$ 6.73205 0.398772
$$286$$ 0 0
$$287$$ −0.803848 −0.0474496
$$288$$ 0 0
$$289$$ 15.8564 0.932730
$$290$$ 0 0
$$291$$ −4.00000 −0.234484
$$292$$ 0 0
$$293$$ −7.73205 −0.451711 −0.225856 0.974161i $$-0.572518\pi$$
−0.225856 + 0.974161i $$0.572518\pi$$
$$294$$ 0 0
$$295$$ 3.19615 0.186087
$$296$$ 0 0
$$297$$ −0.732051 −0.0424779
$$298$$ 0 0
$$299$$ 4.19615 0.242670
$$300$$ 0 0
$$301$$ 13.6077 0.784335
$$302$$ 0 0
$$303$$ −5.19615 −0.298511
$$304$$ 0 0
$$305$$ 13.1244 0.751498
$$306$$ 0 0
$$307$$ −24.1962 −1.38095 −0.690474 0.723358i $$-0.742598\pi$$
−0.690474 + 0.723358i $$0.742598\pi$$
$$308$$ 0 0
$$309$$ 11.8564 0.674487
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 5.92820 0.335082 0.167541 0.985865i $$-0.446417\pi$$
0.167541 + 0.985865i $$0.446417\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ −27.5167 −1.54549 −0.772745 0.634717i $$-0.781117\pi$$
−0.772745 + 0.634717i $$0.781117\pi$$
$$318$$ 0 0
$$319$$ 0.196152 0.0109824
$$320$$ 0 0
$$321$$ 18.1244 1.01160
$$322$$ 0 0
$$323$$ −38.5885 −2.14712
$$324$$ 0 0
$$325$$ −4.19615 −0.232761
$$326$$ 0 0
$$327$$ 6.73205 0.372283
$$328$$ 0 0
$$329$$ 28.9808 1.59776
$$330$$ 0 0
$$331$$ 14.3205 0.787126 0.393563 0.919298i $$-0.371242\pi$$
0.393563 + 0.919298i $$0.371242\pi$$
$$332$$ 0 0
$$333$$ 11.9282 0.653662
$$334$$ 0 0
$$335$$ −0.464102 −0.0253566
$$336$$ 0 0
$$337$$ −22.7846 −1.24116 −0.620578 0.784144i $$-0.713102\pi$$
−0.620578 + 0.784144i $$0.713102\pi$$
$$338$$ 0 0
$$339$$ −9.19615 −0.499466
$$340$$ 0 0
$$341$$ 2.87564 0.155725
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ 0 0
$$347$$ −7.32051 −0.392985 −0.196493 0.980505i $$-0.562955\pi$$
−0.196493 + 0.980505i $$0.562955\pi$$
$$348$$ 0 0
$$349$$ 30.3205 1.62302 0.811510 0.584339i $$-0.198646\pi$$
0.811510 + 0.584339i $$0.198646\pi$$
$$350$$ 0 0
$$351$$ −4.19615 −0.223974
$$352$$ 0 0
$$353$$ 1.26795 0.0674861 0.0337431 0.999431i $$-0.489257\pi$$
0.0337431 + 0.999431i $$0.489257\pi$$
$$354$$ 0 0
$$355$$ 0.267949 0.0142213
$$356$$ 0 0
$$357$$ −17.1962 −0.910117
$$358$$ 0 0
$$359$$ 27.1244 1.43157 0.715784 0.698321i $$-0.246069\pi$$
0.715784 + 0.698321i $$0.246069\pi$$
$$360$$ 0 0
$$361$$ 26.3205 1.38529
$$362$$ 0 0
$$363$$ −10.4641 −0.549223
$$364$$ 0 0
$$365$$ 9.66025 0.505641
$$366$$ 0 0
$$367$$ 15.0000 0.782994 0.391497 0.920179i $$-0.371957\pi$$
0.391497 + 0.920179i $$0.371957\pi$$
$$368$$ 0 0
$$369$$ −0.267949 −0.0139489
$$370$$ 0 0
$$371$$ −6.80385 −0.353238
$$372$$ 0 0
$$373$$ 32.3923 1.67721 0.838605 0.544740i $$-0.183372\pi$$
0.838605 + 0.544740i $$0.183372\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 1.12436 0.0579073
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ −1.80385 −0.0924139
$$382$$ 0 0
$$383$$ 2.41154 0.123224 0.0616120 0.998100i $$-0.480376\pi$$
0.0616120 + 0.998100i $$0.480376\pi$$
$$384$$ 0 0
$$385$$ −2.19615 −0.111926
$$386$$ 0 0
$$387$$ 4.53590 0.230573
$$388$$ 0 0
$$389$$ 3.46410 0.175637 0.0878185 0.996136i $$-0.472010\pi$$
0.0878185 + 0.996136i $$0.472010\pi$$
$$390$$ 0 0
$$391$$ 5.73205 0.289882
$$392$$ 0 0
$$393$$ −5.07180 −0.255838
$$394$$ 0 0
$$395$$ 6.92820 0.348596
$$396$$ 0 0
$$397$$ 24.7846 1.24390 0.621952 0.783055i $$-0.286340\pi$$
0.621952 + 0.783055i $$0.286340\pi$$
$$398$$ 0 0
$$399$$ 20.1962 1.01107
$$400$$ 0 0
$$401$$ 5.32051 0.265693 0.132847 0.991137i $$-0.457588\pi$$
0.132847 + 0.991137i $$0.457588\pi$$
$$402$$ 0 0
$$403$$ 16.4833 0.821094
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −8.73205 −0.432832
$$408$$ 0 0
$$409$$ −11.3923 −0.563313 −0.281657 0.959515i $$-0.590884\pi$$
−0.281657 + 0.959515i $$0.590884\pi$$
$$410$$ 0 0
$$411$$ 20.7846 1.02523
$$412$$ 0 0
$$413$$ 9.58846 0.471817
$$414$$ 0 0
$$415$$ 10.2679 0.504034
$$416$$ 0 0
$$417$$ −5.00000 −0.244851
$$418$$ 0 0
$$419$$ −11.2679 −0.550475 −0.275238 0.961376i $$-0.588757\pi$$
−0.275238 + 0.961376i $$0.588757\pi$$
$$420$$ 0 0
$$421$$ 21.6603 1.05566 0.527828 0.849351i $$-0.323007\pi$$
0.527828 + 0.849351i $$0.323007\pi$$
$$422$$ 0 0
$$423$$ 9.66025 0.469698
$$424$$ 0 0
$$425$$ −5.73205 −0.278045
$$426$$ 0 0
$$427$$ 39.3731 1.90540
$$428$$ 0 0
$$429$$ 3.07180 0.148308
$$430$$ 0 0
$$431$$ 16.3923 0.789590 0.394795 0.918769i $$-0.370816\pi$$
0.394795 + 0.918769i $$0.370816\pi$$
$$432$$ 0 0
$$433$$ 24.0718 1.15682 0.578408 0.815747i $$-0.303674\pi$$
0.578408 + 0.815747i $$0.303674\pi$$
$$434$$ 0 0
$$435$$ −0.267949 −0.0128472
$$436$$ 0 0
$$437$$ −6.73205 −0.322038
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −19.5167 −0.927265 −0.463632 0.886028i $$-0.653454\pi$$
−0.463632 + 0.886028i $$0.653454\pi$$
$$444$$ 0 0
$$445$$ 9.46410 0.448641
$$446$$ 0 0
$$447$$ −16.1962 −0.766052
$$448$$ 0 0
$$449$$ −29.0526 −1.37108 −0.685538 0.728037i $$-0.740433\pi$$
−0.685538 + 0.728037i $$0.740433\pi$$
$$450$$ 0 0
$$451$$ 0.196152 0.00923646
$$452$$ 0 0
$$453$$ 18.3923 0.864146
$$454$$ 0 0
$$455$$ −12.5885 −0.590156
$$456$$ 0 0
$$457$$ −23.9282 −1.11931 −0.559657 0.828724i $$-0.689067\pi$$
−0.559657 + 0.828724i $$0.689067\pi$$
$$458$$ 0 0
$$459$$ −5.73205 −0.267549
$$460$$ 0 0
$$461$$ −27.4641 −1.27913 −0.639565 0.768737i $$-0.720886\pi$$
−0.639565 + 0.768737i $$0.720886\pi$$
$$462$$ 0 0
$$463$$ −17.6603 −0.820742 −0.410371 0.911919i $$-0.634601\pi$$
−0.410371 + 0.911919i $$0.634601\pi$$
$$464$$ 0 0
$$465$$ −3.92820 −0.182166
$$466$$ 0 0
$$467$$ 12.2679 0.567693 0.283846 0.958870i $$-0.408389\pi$$
0.283846 + 0.958870i $$0.408389\pi$$
$$468$$ 0 0
$$469$$ −1.39230 −0.0642907
$$470$$ 0 0
$$471$$ −13.9282 −0.641778
$$472$$ 0 0
$$473$$ −3.32051 −0.152677
$$474$$ 0 0
$$475$$ 6.73205 0.308888
$$476$$ 0 0
$$477$$ −2.26795 −0.103842
$$478$$ 0 0
$$479$$ 38.4449 1.75659 0.878295 0.478119i $$-0.158681\pi$$
0.878295 + 0.478119i $$0.158681\pi$$
$$480$$ 0 0
$$481$$ −50.0526 −2.28220
$$482$$ 0 0
$$483$$ −3.00000 −0.136505
$$484$$ 0 0
$$485$$ −4.00000 −0.181631
$$486$$ 0 0
$$487$$ −7.66025 −0.347119 −0.173560 0.984823i $$-0.555527\pi$$
−0.173560 + 0.984823i $$0.555527\pi$$
$$488$$ 0 0
$$489$$ −14.0000 −0.633102
$$490$$ 0 0
$$491$$ 25.4449 1.14831 0.574155 0.818746i $$-0.305331\pi$$
0.574155 + 0.818746i $$0.305331\pi$$
$$492$$ 0 0
$$493$$ 1.53590 0.0691734
$$494$$ 0 0
$$495$$ −0.732051 −0.0329032
$$496$$ 0 0
$$497$$ 0.803848 0.0360575
$$498$$ 0 0
$$499$$ −14.6077 −0.653930 −0.326965 0.945036i $$-0.606026\pi$$
−0.326965 + 0.945036i $$0.606026\pi$$
$$500$$ 0 0
$$501$$ 6.33975 0.283239
$$502$$ 0 0
$$503$$ 5.33975 0.238088 0.119044 0.992889i $$-0.462017\pi$$
0.119044 + 0.992889i $$0.462017\pi$$
$$504$$ 0 0
$$505$$ −5.19615 −0.231226
$$506$$ 0 0
$$507$$ 4.60770 0.204635
$$508$$ 0 0
$$509$$ −34.9282 −1.54817 −0.774083 0.633084i $$-0.781789\pi$$
−0.774083 + 0.633084i $$0.781789\pi$$
$$510$$ 0 0
$$511$$ 28.9808 1.28203
$$512$$ 0 0
$$513$$ 6.73205 0.297227
$$514$$ 0 0
$$515$$ 11.8564 0.522456
$$516$$ 0 0
$$517$$ −7.07180 −0.311017
$$518$$ 0 0
$$519$$ −25.8564 −1.13497
$$520$$ 0 0
$$521$$ −12.7321 −0.557801 −0.278901 0.960320i $$-0.589970\pi$$
−0.278901 + 0.960320i $$0.589970\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ 3.00000 0.130931
$$526$$ 0 0
$$527$$ 22.5167 0.980841
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 3.19615 0.138701
$$532$$ 0 0
$$533$$ 1.12436 0.0487012
$$534$$ 0 0
$$535$$ 18.1244 0.783584
$$536$$ 0 0
$$537$$ −11.4641 −0.494713
$$538$$ 0 0
$$539$$ −1.46410 −0.0630633
$$540$$ 0 0
$$541$$ −1.85641 −0.0798131 −0.0399066 0.999203i $$-0.512706\pi$$
−0.0399066 + 0.999203i $$0.512706\pi$$
$$542$$ 0 0
$$543$$ −8.39230 −0.360148
$$544$$ 0 0
$$545$$ 6.73205 0.288369
$$546$$ 0 0
$$547$$ −34.9282 −1.49342 −0.746711 0.665149i $$-0.768368\pi$$
−0.746711 + 0.665149i $$0.768368\pi$$
$$548$$ 0 0
$$549$$ 13.1244 0.560134
$$550$$ 0 0
$$551$$ −1.80385 −0.0768465
$$552$$ 0 0
$$553$$ 20.7846 0.883852
$$554$$ 0 0
$$555$$ 11.9282 0.506324
$$556$$ 0 0
$$557$$ 3.58846 0.152048 0.0760239 0.997106i $$-0.475777\pi$$
0.0760239 + 0.997106i $$0.475777\pi$$
$$558$$ 0 0
$$559$$ −19.0333 −0.805024
$$560$$ 0 0
$$561$$ 4.19615 0.177162
$$562$$ 0 0
$$563$$ 4.51666 0.190355 0.0951773 0.995460i $$-0.469658\pi$$
0.0951773 + 0.995460i $$0.469658\pi$$
$$564$$ 0 0
$$565$$ −9.19615 −0.386885
$$566$$ 0 0
$$567$$ 3.00000 0.125988
$$568$$ 0 0
$$569$$ −17.3205 −0.726113 −0.363057 0.931767i $$-0.618267\pi$$
−0.363057 + 0.931767i $$0.618267\pi$$
$$570$$ 0 0
$$571$$ −14.5885 −0.610508 −0.305254 0.952271i $$-0.598741\pi$$
−0.305254 + 0.952271i $$0.598741\pi$$
$$572$$ 0 0
$$573$$ 13.6603 0.570665
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −10.9282 −0.454947 −0.227474 0.973784i $$-0.573047\pi$$
−0.227474 + 0.973784i $$0.573047\pi$$
$$578$$ 0 0
$$579$$ 17.4641 0.725783
$$580$$ 0 0
$$581$$ 30.8038 1.27796
$$582$$ 0 0
$$583$$ 1.66025 0.0687607
$$584$$ 0 0
$$585$$ −4.19615 −0.173490
$$586$$ 0 0
$$587$$ −19.8564 −0.819562 −0.409781 0.912184i $$-0.634395\pi$$
−0.409781 + 0.912184i $$0.634395\pi$$
$$588$$ 0 0
$$589$$ −26.4449 −1.08964
$$590$$ 0 0
$$591$$ −22.3923 −0.921096
$$592$$ 0 0
$$593$$ 4.58846 0.188425 0.0942127 0.995552i $$-0.469967\pi$$
0.0942127 + 0.995552i $$0.469967\pi$$
$$594$$ 0 0
$$595$$ −17.1962 −0.704974
$$596$$ 0 0
$$597$$ 4.53590 0.185642
$$598$$ 0 0
$$599$$ −27.3205 −1.11629 −0.558143 0.829745i $$-0.688486\pi$$
−0.558143 + 0.829745i $$0.688486\pi$$
$$600$$ 0 0
$$601$$ 2.46410 0.100513 0.0502564 0.998736i $$-0.483996\pi$$
0.0502564 + 0.998736i $$0.483996\pi$$
$$602$$ 0 0
$$603$$ −0.464102 −0.0188997
$$604$$ 0 0
$$605$$ −10.4641 −0.425426
$$606$$ 0 0
$$607$$ −37.1244 −1.50683 −0.753416 0.657545i $$-0.771595\pi$$
−0.753416 + 0.657545i $$0.771595\pi$$
$$608$$ 0 0
$$609$$ −0.803848 −0.0325735
$$610$$ 0 0
$$611$$ −40.5359 −1.63991
$$612$$ 0 0
$$613$$ 20.1436 0.813592 0.406796 0.913519i $$-0.366646\pi$$
0.406796 + 0.913519i $$0.366646\pi$$
$$614$$ 0 0
$$615$$ −0.267949 −0.0108048
$$616$$ 0 0
$$617$$ −40.5167 −1.63114 −0.815570 0.578659i $$-0.803576\pi$$
−0.815570 + 0.578659i $$0.803576\pi$$
$$618$$ 0 0
$$619$$ −40.7846 −1.63927 −0.819636 0.572885i $$-0.805824\pi$$
−0.819636 + 0.572885i $$0.805824\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 0 0
$$623$$ 28.3923 1.13751
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −4.92820 −0.196813
$$628$$ 0 0
$$629$$ −68.3731 −2.72621
$$630$$ 0 0
$$631$$ 17.1244 0.681710 0.340855 0.940116i $$-0.389284\pi$$
0.340855 + 0.940116i $$0.389284\pi$$
$$632$$ 0 0
$$633$$ −19.7846 −0.786368
$$634$$ 0 0
$$635$$ −1.80385 −0.0715835
$$636$$ 0 0
$$637$$ −8.39230 −0.332515
$$638$$ 0 0
$$639$$ 0.267949 0.0105999
$$640$$ 0 0
$$641$$ −25.5167 −1.00785 −0.503924 0.863748i $$-0.668111\pi$$
−0.503924 + 0.863748i $$0.668111\pi$$
$$642$$ 0 0
$$643$$ 43.6410 1.72103 0.860517 0.509422i $$-0.170141\pi$$
0.860517 + 0.509422i $$0.170141\pi$$
$$644$$ 0 0
$$645$$ 4.53590 0.178601
$$646$$ 0 0
$$647$$ 4.33975 0.170613 0.0853065 0.996355i $$-0.472813\pi$$
0.0853065 + 0.996355i $$0.472813\pi$$
$$648$$ 0 0
$$649$$ −2.33975 −0.0918431
$$650$$ 0 0
$$651$$ −11.7846 −0.461875
$$652$$ 0 0
$$653$$ 9.41154 0.368302 0.184151 0.982898i $$-0.441046\pi$$
0.184151 + 0.982898i $$0.441046\pi$$
$$654$$ 0 0
$$655$$ −5.07180 −0.198171
$$656$$ 0 0
$$657$$ 9.66025 0.376882
$$658$$ 0 0
$$659$$ −5.80385 −0.226086 −0.113043 0.993590i $$-0.536060\pi$$
−0.113043 + 0.993590i $$0.536060\pi$$
$$660$$ 0 0
$$661$$ 9.32051 0.362526 0.181263 0.983435i $$-0.441982\pi$$
0.181263 + 0.983435i $$0.441982\pi$$
$$662$$ 0 0
$$663$$ 24.0526 0.934124
$$664$$ 0 0
$$665$$ 20.1962 0.783173
$$666$$ 0 0
$$667$$ 0.267949 0.0103750
$$668$$ 0 0
$$669$$ −2.00000 −0.0773245
$$670$$ 0 0
$$671$$ −9.60770 −0.370901
$$672$$ 0 0
$$673$$ 23.6603 0.912036 0.456018 0.889971i $$-0.349275\pi$$
0.456018 + 0.889971i $$0.349275\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 39.8372 1.53107 0.765533 0.643396i $$-0.222475\pi$$
0.765533 + 0.643396i $$0.222475\pi$$
$$678$$ 0 0
$$679$$ −12.0000 −0.460518
$$680$$ 0 0
$$681$$ −9.07180 −0.347632
$$682$$ 0 0
$$683$$ −10.0526 −0.384650 −0.192325 0.981331i $$-0.561603\pi$$
−0.192325 + 0.981331i $$0.561603\pi$$
$$684$$ 0 0
$$685$$ 20.7846 0.794139
$$686$$ 0 0
$$687$$ −11.3205 −0.431904
$$688$$ 0 0
$$689$$ 9.51666 0.362556
$$690$$ 0 0
$$691$$ 2.67949 0.101933 0.0509663 0.998700i $$-0.483770\pi$$
0.0509663 + 0.998700i $$0.483770\pi$$
$$692$$ 0 0
$$693$$ −2.19615 −0.0834249
$$694$$ 0 0
$$695$$ −5.00000 −0.189661
$$696$$ 0 0
$$697$$ 1.53590 0.0581763
$$698$$ 0 0
$$699$$ 25.8564 0.977979
$$700$$ 0 0
$$701$$ −15.5167 −0.586056 −0.293028 0.956104i $$-0.594663\pi$$
−0.293028 + 0.956104i $$0.594663\pi$$
$$702$$ 0 0
$$703$$ 80.3013 3.02862
$$704$$ 0 0
$$705$$ 9.66025 0.363826
$$706$$ 0 0
$$707$$ −15.5885 −0.586264
$$708$$ 0 0
$$709$$ −7.66025 −0.287687 −0.143843 0.989600i $$-0.545946\pi$$
−0.143843 + 0.989600i $$0.545946\pi$$
$$710$$ 0 0
$$711$$ 6.92820 0.259828
$$712$$ 0 0
$$713$$ 3.92820 0.147112
$$714$$ 0 0
$$715$$ 3.07180 0.114879
$$716$$ 0 0
$$717$$ −21.0526 −0.786222
$$718$$ 0 0
$$719$$ −5.19615 −0.193784 −0.0968919 0.995295i $$-0.530890\pi$$
−0.0968919 + 0.995295i $$0.530890\pi$$
$$720$$ 0 0
$$721$$ 35.5692 1.32467
$$722$$ 0 0
$$723$$ −30.7321 −1.14294
$$724$$ 0 0
$$725$$ −0.267949 −0.00995138
$$726$$ 0 0
$$727$$ 50.1769 1.86096 0.930479 0.366344i $$-0.119391\pi$$
0.930479 + 0.366344i $$0.119391\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −26.0000 −0.961645
$$732$$ 0 0
$$733$$ −27.1051 −1.00115 −0.500575 0.865693i $$-0.666878\pi$$
−0.500575 + 0.865693i $$0.666878\pi$$
$$734$$ 0 0
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ 0.339746 0.0125147
$$738$$ 0 0
$$739$$ −48.3205 −1.77750 −0.888749 0.458394i $$-0.848425\pi$$
−0.888749 + 0.458394i $$0.848425\pi$$
$$740$$ 0 0
$$741$$ −28.2487 −1.03774
$$742$$ 0 0
$$743$$ 28.0000 1.02722 0.513610 0.858024i $$-0.328308\pi$$
0.513610 + 0.858024i $$0.328308\pi$$
$$744$$ 0 0
$$745$$ −16.1962 −0.593381
$$746$$ 0 0
$$747$$ 10.2679 0.375685
$$748$$ 0 0
$$749$$ 54.3731 1.98675
$$750$$ 0 0
$$751$$ −5.26795 −0.192230 −0.0961151 0.995370i $$-0.530642\pi$$
−0.0961151 + 0.995370i $$0.530642\pi$$
$$752$$ 0 0
$$753$$ −19.8564 −0.723608
$$754$$ 0 0
$$755$$ 18.3923 0.669365
$$756$$ 0 0
$$757$$ −18.6077 −0.676308 −0.338154 0.941091i $$-0.609803\pi$$
−0.338154 + 0.941091i $$0.609803\pi$$
$$758$$ 0 0
$$759$$ 0.732051 0.0265718
$$760$$ 0 0
$$761$$ −3.33975 −0.121066 −0.0605328 0.998166i $$-0.519280\pi$$
−0.0605328 + 0.998166i $$0.519280\pi$$
$$762$$ 0 0
$$763$$ 20.1962 0.731150
$$764$$ 0 0
$$765$$ −5.73205 −0.207243
$$766$$ 0 0
$$767$$ −13.4115 −0.484263
$$768$$ 0 0
$$769$$ −1.80385 −0.0650484 −0.0325242 0.999471i $$-0.510355\pi$$
−0.0325242 + 0.999471i $$0.510355\pi$$
$$770$$ 0 0
$$771$$ −5.66025 −0.203849
$$772$$ 0 0
$$773$$ 15.4641 0.556205 0.278103 0.960551i $$-0.410294\pi$$
0.278103 + 0.960551i $$0.410294\pi$$
$$774$$ 0 0
$$775$$ −3.92820 −0.141105
$$776$$ 0 0
$$777$$ 35.7846 1.28377
$$778$$ 0 0
$$779$$ −1.80385 −0.0646295
$$780$$ 0 0
$$781$$ −0.196152 −0.00701889
$$782$$ 0 0
$$783$$ −0.267949 −0.00957572
$$784$$ 0 0
$$785$$ −13.9282 −0.497119
$$786$$ 0 0
$$787$$ 17.0000 0.605985 0.302992 0.952993i $$-0.402014\pi$$
0.302992 + 0.952993i $$0.402014\pi$$
$$788$$ 0 0
$$789$$ −8.26795 −0.294347
$$790$$ 0 0
$$791$$ −27.5885 −0.980933
$$792$$ 0 0
$$793$$ −55.0718 −1.95566
$$794$$ 0 0
$$795$$ −2.26795 −0.0804359
$$796$$ 0 0
$$797$$ 29.9808 1.06197 0.530987 0.847380i $$-0.321821\pi$$
0.530987 + 0.847380i $$0.321821\pi$$
$$798$$ 0 0
$$799$$ −55.3731 −1.95896
$$800$$ 0 0
$$801$$ 9.46410 0.334398
$$802$$ 0 0
$$803$$ −7.07180 −0.249558
$$804$$ 0 0
$$805$$ −3.00000 −0.105736
$$806$$ 0 0
$$807$$ 28.6603 1.00889
$$808$$ 0 0
$$809$$ 10.1244 0.355953 0.177977 0.984035i $$-0.443045\pi$$
0.177977 + 0.984035i $$0.443045\pi$$
$$810$$ 0 0
$$811$$ −3.00000 −0.105344 −0.0526721 0.998612i $$-0.516774\pi$$
−0.0526721 + 0.998612i $$0.516774\pi$$
$$812$$ 0 0
$$813$$ −5.39230 −0.189116
$$814$$ 0 0
$$815$$ −14.0000 −0.490399
$$816$$ 0 0
$$817$$ 30.5359 1.06832
$$818$$ 0 0
$$819$$ −12.5885 −0.439876
$$820$$ 0 0
$$821$$ −39.7128 −1.38599 −0.692993 0.720944i $$-0.743708\pi$$
−0.692993 + 0.720944i $$0.743708\pi$$
$$822$$ 0 0
$$823$$ 46.9282 1.63581 0.817907 0.575350i $$-0.195134\pi$$
0.817907 + 0.575350i $$0.195134\pi$$
$$824$$ 0 0
$$825$$ −0.732051 −0.0254867
$$826$$ 0 0
$$827$$ −32.3731 −1.12572 −0.562861 0.826552i $$-0.690299\pi$$
−0.562861 + 0.826552i $$0.690299\pi$$
$$828$$ 0 0
$$829$$ −27.9282 −0.969987 −0.484993 0.874518i $$-0.661178\pi$$
−0.484993 + 0.874518i $$0.661178\pi$$
$$830$$ 0 0
$$831$$ 14.7846 0.512872
$$832$$ 0 0
$$833$$ −11.4641 −0.397208
$$834$$ 0 0
$$835$$ 6.33975 0.219396
$$836$$ 0 0
$$837$$ −3.92820 −0.135779
$$838$$ 0 0
$$839$$ −11.8564 −0.409329 −0.204664 0.978832i $$-0.565610\pi$$
−0.204664 + 0.978832i $$0.565610\pi$$
$$840$$ 0 0
$$841$$ −28.9282 −0.997524
$$842$$ 0 0
$$843$$ −16.0526 −0.552879
$$844$$ 0 0
$$845$$ 4.60770 0.158510
$$846$$ 0 0
$$847$$ −31.3923 −1.07865
$$848$$ 0 0
$$849$$ 6.46410 0.221847
$$850$$ 0 0
$$851$$ −11.9282 −0.408894
$$852$$ 0 0
$$853$$ 15.4641 0.529481 0.264740 0.964320i $$-0.414714\pi$$
0.264740 + 0.964320i $$0.414714\pi$$
$$854$$ 0 0
$$855$$ 6.73205 0.230231
$$856$$ 0 0
$$857$$ −4.92820 −0.168344 −0.0841721 0.996451i $$-0.526825\pi$$
−0.0841721 + 0.996451i $$0.526825\pi$$
$$858$$ 0 0
$$859$$ −36.0718 −1.23075 −0.615377 0.788233i $$-0.710996\pi$$
−0.615377 + 0.788233i $$0.710996\pi$$
$$860$$ 0 0
$$861$$ −0.803848 −0.0273951
$$862$$ 0 0
$$863$$ 6.39230 0.217597 0.108798 0.994064i $$-0.465300\pi$$
0.108798 + 0.994064i $$0.465300\pi$$
$$864$$ 0 0
$$865$$ −25.8564 −0.879144
$$866$$ 0 0
$$867$$ 15.8564 0.538512
$$868$$ 0 0
$$869$$ −5.07180 −0.172049
$$870$$ 0 0
$$871$$ 1.94744 0.0659865
$$872$$ 0 0
$$873$$ −4.00000 −0.135379
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ 20.5359 0.693448 0.346724 0.937967i $$-0.387294\pi$$
0.346724 + 0.937967i $$0.387294\pi$$
$$878$$ 0 0
$$879$$ −7.73205 −0.260796
$$880$$ 0 0
$$881$$ −24.1962 −0.815189 −0.407595 0.913163i $$-0.633632\pi$$
−0.407595 + 0.913163i $$0.633632\pi$$
$$882$$ 0 0
$$883$$ 44.0526 1.48249 0.741243 0.671236i $$-0.234236\pi$$
0.741243 + 0.671236i $$0.234236\pi$$
$$884$$ 0 0
$$885$$ 3.19615 0.107437
$$886$$ 0 0
$$887$$ 3.46410 0.116313 0.0581566 0.998307i $$-0.481478\pi$$
0.0581566 + 0.998307i $$0.481478\pi$$
$$888$$ 0 0
$$889$$ −5.41154 −0.181497
$$890$$ 0 0
$$891$$ −0.732051 −0.0245246
$$892$$ 0 0
$$893$$ 65.0333 2.17626
$$894$$ 0 0
$$895$$ −11.4641 −0.383203
$$896$$ 0 0
$$897$$ 4.19615 0.140105
$$898$$ 0 0
$$899$$ 1.05256 0.0351048
$$900$$ 0 0
$$901$$ 13.0000 0.433093
$$902$$ 0 0
$$903$$ 13.6077 0.452836
$$904$$ 0 0
$$905$$ −8.39230 −0.278970
$$906$$ 0 0
$$907$$ −40.1769 −1.33405 −0.667026 0.745034i $$-0.732433\pi$$
−0.667026 + 0.745034i $$0.732433\pi$$
$$908$$ 0 0
$$909$$ −5.19615 −0.172345
$$910$$ 0 0
$$911$$ 34.9282 1.15722 0.578612 0.815603i $$-0.303595\pi$$
0.578612 + 0.815603i $$0.303595\pi$$
$$912$$ 0 0
$$913$$ −7.51666 −0.248765
$$914$$ 0 0
$$915$$ 13.1244 0.433878
$$916$$ 0 0
$$917$$ −15.2154 −0.502456
$$918$$ 0 0
$$919$$ 33.8564 1.11682 0.558410 0.829565i $$-0.311412\pi$$
0.558410 + 0.829565i $$0.311412\pi$$
$$920$$ 0 0
$$921$$ −24.1962 −0.797290
$$922$$ 0 0
$$923$$ −1.12436 −0.0370086
$$924$$ 0 0
$$925$$ 11.9282 0.392197
$$926$$ 0 0
$$927$$ 11.8564 0.389415
$$928$$ 0 0
$$929$$ 29.9808 0.983637 0.491818 0.870698i $$-0.336332\pi$$
0.491818 + 0.870698i $$0.336332\pi$$
$$930$$ 0 0
$$931$$ 13.4641 0.441268
$$932$$ 0 0
$$933$$ 12.0000 0.392862
$$934$$ 0 0
$$935$$ 4.19615 0.137229
$$936$$ 0 0
$$937$$ 36.4974 1.19232 0.596159 0.802866i $$-0.296693\pi$$
0.596159 + 0.802866i $$0.296693\pi$$
$$938$$ 0 0
$$939$$ 5.92820 0.193460
$$940$$ 0 0
$$941$$ 48.6936 1.58737 0.793683 0.608332i $$-0.208161\pi$$
0.793683 + 0.608332i $$0.208161\pi$$
$$942$$ 0 0
$$943$$ 0.267949 0.00872563
$$944$$ 0 0
$$945$$ 3.00000 0.0975900
$$946$$ 0 0
$$947$$ −31.6077 −1.02711 −0.513556 0.858056i $$-0.671672\pi$$
−0.513556 + 0.858056i $$0.671672\pi$$
$$948$$ 0 0
$$949$$ −40.5359 −1.31585
$$950$$ 0 0
$$951$$ −27.5167 −0.892289
$$952$$ 0 0
$$953$$ 12.1436 0.393370 0.196685 0.980467i $$-0.436982\pi$$
0.196685 + 0.980467i $$0.436982\pi$$
$$954$$ 0 0
$$955$$ 13.6603 0.442035
$$956$$ 0 0
$$957$$ 0.196152 0.00634071
$$958$$ 0 0
$$959$$ 62.3538 2.01351
$$960$$ 0 0
$$961$$ −15.5692 −0.502233
$$962$$ 0 0
$$963$$ 18.1244 0.584049
$$964$$ 0 0
$$965$$ 17.4641 0.562189
$$966$$ 0 0
$$967$$ −5.80385 −0.186639 −0.0933196 0.995636i $$-0.529748\pi$$
−0.0933196 + 0.995636i $$0.529748\pi$$
$$968$$ 0 0
$$969$$ −38.5885 −1.23964
$$970$$ 0 0
$$971$$ −2.53590 −0.0813809 −0.0406904 0.999172i $$-0.512956\pi$$
−0.0406904 + 0.999172i $$0.512956\pi$$
$$972$$ 0 0
$$973$$ −15.0000 −0.480878
$$974$$ 0 0
$$975$$ −4.19615 −0.134384
$$976$$ 0 0
$$977$$ −16.1244 −0.515864 −0.257932 0.966163i $$-0.583041\pi$$
−0.257932 + 0.966163i $$0.583041\pi$$
$$978$$ 0 0
$$979$$ −6.92820 −0.221426
$$980$$ 0 0
$$981$$ 6.73205 0.214938
$$982$$ 0 0
$$983$$ 13.9808 0.445917 0.222959 0.974828i $$-0.428429\pi$$
0.222959 + 0.974828i $$0.428429\pi$$
$$984$$ 0 0
$$985$$ −22.3923 −0.713478
$$986$$ 0 0
$$987$$ 28.9808 0.922468
$$988$$ 0 0
$$989$$ −4.53590 −0.144233
$$990$$ 0 0
$$991$$ 25.0000 0.794151 0.397076 0.917786i $$-0.370025\pi$$
0.397076 + 0.917786i $$0.370025\pi$$
$$992$$ 0 0
$$993$$ 14.3205 0.454448
$$994$$ 0 0
$$995$$ 4.53590 0.143798
$$996$$ 0 0
$$997$$ −57.4641 −1.81991 −0.909953 0.414711i $$-0.863883\pi$$
−0.909953 + 0.414711i $$0.863883\pi$$
$$998$$ 0 0
$$999$$ 11.9282 0.377392
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.bu.1.1 2
4.3 odd 2 345.2.a.g.1.2 2
12.11 even 2 1035.2.a.l.1.1 2
20.3 even 4 1725.2.b.p.1174.2 4
20.7 even 4 1725.2.b.p.1174.3 4
20.19 odd 2 1725.2.a.bd.1.1 2
60.59 even 2 5175.2.a.bd.1.2 2
92.91 even 2 7935.2.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.g.1.2 2 4.3 odd 2
1035.2.a.l.1.1 2 12.11 even 2
1725.2.a.bd.1.1 2 20.19 odd 2
1725.2.b.p.1174.2 4 20.3 even 4
1725.2.b.p.1174.3 4 20.7 even 4
5175.2.a.bd.1.2 2 60.59 even 2
5520.2.a.bu.1.1 2 1.1 even 1 trivial
7935.2.a.n.1.2 2 92.91 even 2