# Properties

 Label 5520.2.a.bj.1.2 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: $$\Q(\sqrt{15})$$ Defining polynomial: $$x^{2} - 15$$ x^2 - 15 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1380) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-3.87298$$ of defining polynomial Character $$\chi$$ $$=$$ 5520.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +2.87298 q^{11} +4.87298 q^{13} -1.00000 q^{15} -3.87298 q^{17} -4.87298 q^{19} +3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.87298 q^{29} -3.00000 q^{31} -2.87298 q^{33} -3.00000 q^{35} +1.00000 q^{37} -4.87298 q^{39} +1.87298 q^{41} +11.7460 q^{43} +1.00000 q^{45} -0.872983 q^{47} +2.00000 q^{49} +3.87298 q^{51} +3.87298 q^{53} +2.87298 q^{55} +4.87298 q^{57} +1.87298 q^{59} +1.12702 q^{61} -3.00000 q^{63} +4.87298 q^{65} +4.74597 q^{67} -1.00000 q^{69} -9.61895 q^{71} +4.87298 q^{73} -1.00000 q^{75} -8.61895 q^{77} -4.00000 q^{79} +1.00000 q^{81} +7.87298 q^{83} -3.87298 q^{85} -1.87298 q^{87} -13.7460 q^{89} -14.6190 q^{91} +3.00000 q^{93} -4.87298 q^{95} +8.00000 q^{97} +2.87298 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 2 q^{19} + 6 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 6 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} - 2 q^{39} - 4 q^{41} + 8 q^{43} + 2 q^{45} + 6 q^{47} + 4 q^{49} - 2 q^{55} + 2 q^{57} - 4 q^{59} + 10 q^{61} - 6 q^{63} + 2 q^{65} - 6 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 6 q^{77} - 8 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{87} - 12 q^{89} - 6 q^{91} + 6 q^{93} - 2 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^15 - 2 * 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 + 2 * q^37 - 2 * q^39 - 4 * q^41 + 8 * q^43 + 2 * q^45 + 6 * q^47 + 4 * q^49 - 2 * q^55 + 2 * q^57 - 4 * q^59 + 10 * 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 - 8 * q^79 + 2 * q^81 + 8 * q^83 + 4 * q^87 - 12 * q^89 - 6 * q^91 + 6 * q^93 - 2 * q^95 + 16 * 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 $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.87298 0.866237 0.433119 0.901337i $$-0.357413\pi$$
0.433119 + 0.901337i $$0.357413\pi$$
$$12$$ 0 0
$$13$$ 4.87298 1.35152 0.675761 0.737121i $$-0.263815\pi$$
0.675761 + 0.737121i $$0.263815\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −3.87298 −0.939336 −0.469668 0.882843i $$-0.655626\pi$$
−0.469668 + 0.882843i $$0.655626\pi$$
$$18$$ 0 0
$$19$$ −4.87298 −1.11794 −0.558970 0.829188i $$-0.688803\pi$$
−0.558970 + 0.829188i $$0.688803\pi$$
$$20$$ 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
$$29$$ 1.87298 0.347804 0.173902 0.984763i $$-0.444362\pi$$
0.173902 + 0.984763i $$0.444362\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 0 0
$$33$$ −2.87298 −0.500122
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ 1.00000 0.164399 0.0821995 0.996616i $$-0.473806\pi$$
0.0821995 + 0.996616i $$0.473806\pi$$
$$38$$ 0 0
$$39$$ −4.87298 −0.780302
$$40$$ 0 0
$$41$$ 1.87298 0.292511 0.146255 0.989247i $$-0.453278\pi$$
0.146255 + 0.989247i $$0.453278\pi$$
$$42$$ 0 0
$$43$$ 11.7460 1.79124 0.895622 0.444817i $$-0.146731\pi$$
0.895622 + 0.444817i $$0.146731\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −0.872983 −0.127338 −0.0636689 0.997971i $$-0.520280\pi$$
−0.0636689 + 0.997971i $$0.520280\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 3.87298 0.542326
$$52$$ 0 0
$$53$$ 3.87298 0.531995 0.265998 0.963974i $$-0.414299\pi$$
0.265998 + 0.963974i $$0.414299\pi$$
$$54$$ 0 0
$$55$$ 2.87298 0.387393
$$56$$ 0 0
$$57$$ 4.87298 0.645442
$$58$$ 0 0
$$59$$ 1.87298 0.243842 0.121921 0.992540i $$-0.461095\pi$$
0.121921 + 0.992540i $$0.461095\pi$$
$$60$$ 0 0
$$61$$ 1.12702 0.144300 0.0721498 0.997394i $$-0.477014\pi$$
0.0721498 + 0.997394i $$0.477014\pi$$
$$62$$ 0 0
$$63$$ −3.00000 −0.377964
$$64$$ 0 0
$$65$$ 4.87298 0.604419
$$66$$ 0 0
$$67$$ 4.74597 0.579812 0.289906 0.957055i $$-0.406376\pi$$
0.289906 + 0.957055i $$0.406376\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −9.61895 −1.14156 −0.570780 0.821103i $$-0.693359\pi$$
−0.570780 + 0.821103i $$0.693359\pi$$
$$72$$ 0 0
$$73$$ 4.87298 0.570340 0.285170 0.958477i $$-0.407950\pi$$
0.285170 + 0.958477i $$0.407950\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −8.61895 −0.982221
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 7.87298 0.864172 0.432086 0.901832i $$-0.357778\pi$$
0.432086 + 0.901832i $$0.357778\pi$$
$$84$$ 0 0
$$85$$ −3.87298 −0.420084
$$86$$ 0 0
$$87$$ −1.87298 −0.200805
$$88$$ 0 0
$$89$$ −13.7460 −1.45707 −0.728535 0.685009i $$-0.759798\pi$$
−0.728535 + 0.685009i $$0.759798\pi$$
$$90$$ 0 0
$$91$$ −14.6190 −1.53248
$$92$$ 0 0
$$93$$ 3.00000 0.311086
$$94$$ 0 0
$$95$$ −4.87298 −0.499958
$$96$$ 0 0
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ 0 0
$$99$$ 2.87298 0.288746
$$100$$ 0 0
$$101$$ 3.87298 0.385376 0.192688 0.981260i $$-0.438279\pi$$
0.192688 + 0.981260i $$0.438279\pi$$
$$102$$ 0 0
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ 0 0
$$107$$ −17.6190 −1.70329 −0.851644 0.524121i $$-0.824394\pi$$
−0.851644 + 0.524121i $$0.824394\pi$$
$$108$$ 0 0
$$109$$ −6.61895 −0.633980 −0.316990 0.948429i $$-0.602672\pi$$
−0.316990 + 0.948429i $$0.602672\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ 0 0
$$113$$ 3.87298 0.364340 0.182170 0.983267i $$-0.441688\pi$$
0.182170 + 0.983267i $$0.441688\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 4.87298 0.450507
$$118$$ 0 0
$$119$$ 11.6190 1.06511
$$120$$ 0 0
$$121$$ −2.74597 −0.249633
$$122$$ 0 0
$$123$$ −1.87298 −0.168881
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −16.6190 −1.47469 −0.737347 0.675515i $$-0.763922\pi$$
−0.737347 + 0.675515i $$0.763922\pi$$
$$128$$ 0 0
$$129$$ −11.7460 −1.03417
$$130$$ 0 0
$$131$$ 19.4919 1.70302 0.851509 0.524340i $$-0.175688\pi$$
0.851509 + 0.524340i $$0.175688\pi$$
$$132$$ 0 0
$$133$$ 14.6190 1.26762
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 8.00000 0.683486 0.341743 0.939793i $$-0.388983\pi$$
0.341743 + 0.939793i $$0.388983\pi$$
$$138$$ 0 0
$$139$$ −10.4919 −0.889914 −0.444957 0.895552i $$-0.646781\pi$$
−0.444957 + 0.895552i $$0.646781\pi$$
$$140$$ 0 0
$$141$$ 0.872983 0.0735185
$$142$$ 0 0
$$143$$ 14.0000 1.17074
$$144$$ 0 0
$$145$$ 1.87298 0.155543
$$146$$ 0 0
$$147$$ −2.00000 −0.164957
$$148$$ 0 0
$$149$$ −3.12702 −0.256175 −0.128088 0.991763i $$-0.540884\pi$$
−0.128088 + 0.991763i $$0.540884\pi$$
$$150$$ 0 0
$$151$$ 11.7460 0.955873 0.477937 0.878394i $$-0.341385\pi$$
0.477937 + 0.878394i $$0.341385\pi$$
$$152$$ 0 0
$$153$$ −3.87298 −0.313112
$$154$$ 0 0
$$155$$ −3.00000 −0.240966
$$156$$ 0 0
$$157$$ 17.0000 1.35675 0.678374 0.734717i $$-0.262685\pi$$
0.678374 + 0.734717i $$0.262685\pi$$
$$158$$ 0 0
$$159$$ −3.87298 −0.307148
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ 22.0000 1.72317 0.861586 0.507611i $$-0.169471\pi$$
0.861586 + 0.507611i $$0.169471\pi$$
$$164$$ 0 0
$$165$$ −2.87298 −0.223661
$$166$$ 0 0
$$167$$ 16.8730 1.30567 0.652835 0.757500i $$-0.273579\pi$$
0.652835 + 0.757500i $$0.273579\pi$$
$$168$$ 0 0
$$169$$ 10.7460 0.826613
$$170$$ 0 0
$$171$$ −4.87298 −0.372646
$$172$$ 0 0
$$173$$ −7.49193 −0.569601 −0.284801 0.958587i $$-0.591927\pi$$
−0.284801 + 0.958587i $$0.591927\pi$$
$$174$$ 0 0
$$175$$ −3.00000 −0.226779
$$176$$ 0 0
$$177$$ −1.87298 −0.140782
$$178$$ 0 0
$$179$$ 0.254033 0.0189873 0.00949367 0.999955i $$-0.496978\pi$$
0.00949367 + 0.999955i $$0.496978\pi$$
$$180$$ 0 0
$$181$$ −6.25403 −0.464859 −0.232429 0.972613i $$-0.574667\pi$$
−0.232429 + 0.972613i $$0.574667\pi$$
$$182$$ 0 0
$$183$$ −1.12702 −0.0833115
$$184$$ 0 0
$$185$$ 1.00000 0.0735215
$$186$$ 0 0
$$187$$ −11.1270 −0.813688
$$188$$ 0 0
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ 11.1270 0.805123 0.402561 0.915393i $$-0.368120\pi$$
0.402561 + 0.915393i $$0.368120\pi$$
$$192$$ 0 0
$$193$$ −5.74597 −0.413604 −0.206802 0.978383i $$-0.566306\pi$$
−0.206802 + 0.978383i $$0.566306\pi$$
$$194$$ 0 0
$$195$$ −4.87298 −0.348962
$$196$$ 0 0
$$197$$ 19.7460 1.40684 0.703421 0.710774i $$-0.251655\pi$$
0.703421 + 0.710774i $$0.251655\pi$$
$$198$$ 0 0
$$199$$ 27.2379 1.93084 0.965422 0.260693i $$-0.0839510\pi$$
0.965422 + 0.260693i $$0.0839510\pi$$
$$200$$ 0 0
$$201$$ −4.74597 −0.334755
$$202$$ 0 0
$$203$$ −5.61895 −0.394373
$$204$$ 0 0
$$205$$ 1.87298 0.130815
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ −14.0000 −0.968400
$$210$$ 0 0
$$211$$ −1.00000 −0.0688428 −0.0344214 0.999407i $$-0.510959\pi$$
−0.0344214 + 0.999407i $$0.510959\pi$$
$$212$$ 0 0
$$213$$ 9.61895 0.659080
$$214$$ 0 0
$$215$$ 11.7460 0.801068
$$216$$ 0 0
$$217$$ 9.00000 0.610960
$$218$$ 0 0
$$219$$ −4.87298 −0.329286
$$220$$ 0 0
$$221$$ −18.8730 −1.26953
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 23.4919 1.55921 0.779607 0.626269i $$-0.215419\pi$$
0.779607 + 0.626269i $$0.215419\pi$$
$$228$$ 0 0
$$229$$ 9.74597 0.644032 0.322016 0.946734i $$-0.395640\pi$$
0.322016 + 0.946734i $$0.395640\pi$$
$$230$$ 0 0
$$231$$ 8.61895 0.567085
$$232$$ 0 0
$$233$$ −15.4919 −1.01491 −0.507455 0.861678i $$-0.669414\pi$$
−0.507455 + 0.861678i $$0.669414\pi$$
$$234$$ 0 0
$$235$$ −0.872983 −0.0569472
$$236$$ 0 0
$$237$$ 4.00000 0.259828
$$238$$ 0 0
$$239$$ 10.1270 0.655062 0.327531 0.944840i $$-0.393783\pi$$
0.327531 + 0.944840i $$0.393783\pi$$
$$240$$ 0 0
$$241$$ −12.8730 −0.829222 −0.414611 0.909999i $$-0.636082\pi$$
−0.414611 + 0.909999i $$0.636082\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ −23.7460 −1.51092
$$248$$ 0 0
$$249$$ −7.87298 −0.498930
$$250$$ 0 0
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ 0 0
$$253$$ 2.87298 0.180623
$$254$$ 0 0
$$255$$ 3.87298 0.242536
$$256$$ 0 0
$$257$$ 6.61895 0.412879 0.206439 0.978459i $$-0.433812\pi$$
0.206439 + 0.978459i $$0.433812\pi$$
$$258$$ 0 0
$$259$$ −3.00000 −0.186411
$$260$$ 0 0
$$261$$ 1.87298 0.115935
$$262$$ 0 0
$$263$$ 14.1270 0.871109 0.435555 0.900162i $$-0.356552\pi$$
0.435555 + 0.900162i $$0.356552\pi$$
$$264$$ 0 0
$$265$$ 3.87298 0.237915
$$266$$ 0 0
$$267$$ 13.7460 0.841239
$$268$$ 0 0
$$269$$ 15.3649 0.936816 0.468408 0.883512i $$-0.344828\pi$$
0.468408 + 0.883512i $$0.344828\pi$$
$$270$$ 0 0
$$271$$ 22.7460 1.38172 0.690860 0.722989i $$-0.257232\pi$$
0.690860 + 0.722989i $$0.257232\pi$$
$$272$$ 0 0
$$273$$ 14.6190 0.884779
$$274$$ 0 0
$$275$$ 2.87298 0.173247
$$276$$ 0 0
$$277$$ 14.0000 0.841178 0.420589 0.907251i $$-0.361823\pi$$
0.420589 + 0.907251i $$0.361823\pi$$
$$278$$ 0 0
$$279$$ −3.00000 −0.179605
$$280$$ 0 0
$$281$$ 22.3649 1.33418 0.667090 0.744978i $$-0.267540\pi$$
0.667090 + 0.744978i $$0.267540\pi$$
$$282$$ 0 0
$$283$$ 24.2379 1.44079 0.720397 0.693562i $$-0.243960\pi$$
0.720397 + 0.693562i $$0.243960\pi$$
$$284$$ 0 0
$$285$$ 4.87298 0.288651
$$286$$ 0 0
$$287$$ −5.61895 −0.331676
$$288$$ 0 0
$$289$$ −2.00000 −0.117647
$$290$$ 0 0
$$291$$ −8.00000 −0.468968
$$292$$ 0 0
$$293$$ 9.61895 0.561945 0.280973 0.959716i $$-0.409343\pi$$
0.280973 + 0.959716i $$0.409343\pi$$
$$294$$ 0 0
$$295$$ 1.87298 0.109049
$$296$$ 0 0
$$297$$ −2.87298 −0.166707
$$298$$ 0 0
$$299$$ 4.87298 0.281812
$$300$$ 0 0
$$301$$ −35.2379 −2.03108
$$302$$ 0 0
$$303$$ −3.87298 −0.222497
$$304$$ 0 0
$$305$$ 1.12702 0.0645328
$$306$$ 0 0
$$307$$ −16.8730 −0.962992 −0.481496 0.876448i $$-0.659906\pi$$
−0.481496 + 0.876448i $$0.659906\pi$$
$$308$$ 0 0
$$309$$ −6.00000 −0.341328
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 0 0
$$313$$ 7.00000 0.395663 0.197832 0.980236i $$-0.436610\pi$$
0.197832 + 0.980236i $$0.436610\pi$$
$$314$$ 0 0
$$315$$ −3.00000 −0.169031
$$316$$ 0 0
$$317$$ −16.3649 −0.919145 −0.459573 0.888140i $$-0.651997\pi$$
−0.459573 + 0.888140i $$0.651997\pi$$
$$318$$ 0 0
$$319$$ 5.38105 0.301281
$$320$$ 0 0
$$321$$ 17.6190 0.983394
$$322$$ 0 0
$$323$$ 18.8730 1.05012
$$324$$ 0 0
$$325$$ 4.87298 0.270304
$$326$$ 0 0
$$327$$ 6.61895 0.366029
$$328$$ 0 0
$$329$$ 2.61895 0.144387
$$330$$ 0 0
$$331$$ −24.2379 −1.33224 −0.666118 0.745847i $$-0.732045\pi$$
−0.666118 + 0.745847i $$0.732045\pi$$
$$332$$ 0 0
$$333$$ 1.00000 0.0547997
$$334$$ 0 0
$$335$$ 4.74597 0.259300
$$336$$ 0 0
$$337$$ 29.4919 1.60653 0.803264 0.595623i $$-0.203095\pi$$
0.803264 + 0.595623i $$0.203095\pi$$
$$338$$ 0 0
$$339$$ −3.87298 −0.210352
$$340$$ 0 0
$$341$$ −8.61895 −0.466742
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ 0 0
$$347$$ 10.2540 0.550465 0.275233 0.961378i $$-0.411245\pi$$
0.275233 + 0.961378i $$0.411245\pi$$
$$348$$ 0 0
$$349$$ 0.745967 0.0399307 0.0199653 0.999801i $$-0.493644\pi$$
0.0199653 + 0.999801i $$0.493644\pi$$
$$350$$ 0 0
$$351$$ −4.87298 −0.260101
$$352$$ 0 0
$$353$$ −8.87298 −0.472261 −0.236131 0.971721i $$-0.575879\pi$$
−0.236131 + 0.971721i $$0.575879\pi$$
$$354$$ 0 0
$$355$$ −9.61895 −0.510521
$$356$$ 0 0
$$357$$ −11.6190 −0.614940
$$358$$ 0 0
$$359$$ 28.8730 1.52386 0.761929 0.647661i $$-0.224253\pi$$
0.761929 + 0.647661i $$0.224253\pi$$
$$360$$ 0 0
$$361$$ 4.74597 0.249788
$$362$$ 0 0
$$363$$ 2.74597 0.144126
$$364$$ 0 0
$$365$$ 4.87298 0.255064
$$366$$ 0 0
$$367$$ 35.9839 1.87834 0.939171 0.343449i $$-0.111595\pi$$
0.939171 + 0.343449i $$0.111595\pi$$
$$368$$ 0 0
$$369$$ 1.87298 0.0975036
$$370$$ 0 0
$$371$$ −11.6190 −0.603226
$$372$$ 0 0
$$373$$ −13.7460 −0.711739 −0.355870 0.934536i $$-0.615815\pi$$
−0.355870 + 0.934536i $$0.615815\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 9.12702 0.470065
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 0 0
$$381$$ 16.6190 0.851415
$$382$$ 0 0
$$383$$ 2.38105 0.121666 0.0608330 0.998148i $$-0.480624\pi$$
0.0608330 + 0.998148i $$0.480624\pi$$
$$384$$ 0 0
$$385$$ −8.61895 −0.439262
$$386$$ 0 0
$$387$$ 11.7460 0.597081
$$388$$ 0 0
$$389$$ −11.7460 −0.595544 −0.297772 0.954637i $$-0.596244\pi$$
−0.297772 + 0.954637i $$0.596244\pi$$
$$390$$ 0 0
$$391$$ −3.87298 −0.195865
$$392$$ 0 0
$$393$$ −19.4919 −0.983238
$$394$$ 0 0
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ −11.4919 −0.576764 −0.288382 0.957516i $$-0.593117\pi$$
−0.288382 + 0.957516i $$0.593117\pi$$
$$398$$ 0 0
$$399$$ −14.6190 −0.731863
$$400$$ 0 0
$$401$$ −19.2379 −0.960695 −0.480347 0.877078i $$-0.659489\pi$$
−0.480347 + 0.877078i $$0.659489\pi$$
$$402$$ 0 0
$$403$$ −14.6190 −0.728222
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 2.87298 0.142408
$$408$$ 0 0
$$409$$ −24.2379 −1.19849 −0.599244 0.800567i $$-0.704532\pi$$
−0.599244 + 0.800567i $$0.704532\pi$$
$$410$$ 0 0
$$411$$ −8.00000 −0.394611
$$412$$ 0 0
$$413$$ −5.61895 −0.276490
$$414$$ 0 0
$$415$$ 7.87298 0.386470
$$416$$ 0 0
$$417$$ 10.4919 0.513792
$$418$$ 0 0
$$419$$ 20.6190 1.00730 0.503651 0.863907i $$-0.331990\pi$$
0.503651 + 0.863907i $$0.331990\pi$$
$$420$$ 0 0
$$421$$ −31.1270 −1.51704 −0.758519 0.651651i $$-0.774077\pi$$
−0.758519 + 0.651651i $$0.774077\pi$$
$$422$$ 0 0
$$423$$ −0.872983 −0.0424459
$$424$$ 0 0
$$425$$ −3.87298 −0.187867
$$426$$ 0 0
$$427$$ −3.38105 −0.163620
$$428$$ 0 0
$$429$$ −14.0000 −0.675926
$$430$$ 0 0
$$431$$ 9.74597 0.469447 0.234723 0.972062i $$-0.424582\pi$$
0.234723 + 0.972062i $$0.424582\pi$$
$$432$$ 0 0
$$433$$ 19.0000 0.913082 0.456541 0.889702i $$-0.349088\pi$$
0.456541 + 0.889702i $$0.349088\pi$$
$$434$$ 0 0
$$435$$ −1.87298 −0.0898027
$$436$$ 0 0
$$437$$ −4.87298 −0.233106
$$438$$ 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
$$443$$ 1.38105 0.0656157 0.0328078 0.999462i $$-0.489555\pi$$
0.0328078 + 0.999462i $$0.489555\pi$$
$$444$$ 0 0
$$445$$ −13.7460 −0.651621
$$446$$ 0 0
$$447$$ 3.12702 0.147903
$$448$$ 0 0
$$449$$ −14.1270 −0.666695 −0.333348 0.942804i $$-0.608178\pi$$
−0.333348 + 0.942804i $$0.608178\pi$$
$$450$$ 0 0
$$451$$ 5.38105 0.253384
$$452$$ 0 0
$$453$$ −11.7460 −0.551874
$$454$$ 0 0
$$455$$ −14.6190 −0.685347
$$456$$ 0 0
$$457$$ −17.0000 −0.795226 −0.397613 0.917553i $$-0.630161\pi$$
−0.397613 + 0.917553i $$0.630161\pi$$
$$458$$ 0 0
$$459$$ 3.87298 0.180775
$$460$$ 0 0
$$461$$ 16.2540 0.757026 0.378513 0.925596i $$-0.376436\pi$$
0.378513 + 0.925596i $$0.376436\pi$$
$$462$$ 0 0
$$463$$ −27.1270 −1.26070 −0.630350 0.776311i $$-0.717088\pi$$
−0.630350 + 0.776311i $$0.717088\pi$$
$$464$$ 0 0
$$465$$ 3.00000 0.139122
$$466$$ 0 0
$$467$$ −17.6190 −0.815308 −0.407654 0.913137i $$-0.633653\pi$$
−0.407654 + 0.913137i $$0.633653\pi$$
$$468$$ 0 0
$$469$$ −14.2379 −0.657445
$$470$$ 0 0
$$471$$ −17.0000 −0.783319
$$472$$ 0 0
$$473$$ 33.7460 1.55164
$$474$$ 0 0
$$475$$ −4.87298 −0.223588
$$476$$ 0 0
$$477$$ 3.87298 0.177332
$$478$$ 0 0
$$479$$ 27.1270 1.23947 0.619733 0.784813i $$-0.287241\pi$$
0.619733 + 0.784813i $$0.287241\pi$$
$$480$$ 0 0
$$481$$ 4.87298 0.222189
$$482$$ 0 0
$$483$$ 3.00000 0.136505
$$484$$ 0 0
$$485$$ 8.00000 0.363261
$$486$$ 0 0
$$487$$ −28.1109 −1.27383 −0.636913 0.770936i $$-0.719789\pi$$
−0.636913 + 0.770936i $$0.719789\pi$$
$$488$$ 0 0
$$489$$ −22.0000 −0.994874
$$490$$ 0 0
$$491$$ −35.8730 −1.61893 −0.809463 0.587172i $$-0.800241\pi$$
−0.809463 + 0.587172i $$0.800241\pi$$
$$492$$ 0 0
$$493$$ −7.25403 −0.326705
$$494$$ 0 0
$$495$$ 2.87298 0.129131
$$496$$ 0 0
$$497$$ 28.8569 1.29441
$$498$$ 0 0
$$499$$ −18.7460 −0.839185 −0.419592 0.907713i $$-0.637827\pi$$
−0.419592 + 0.907713i $$0.637827\pi$$
$$500$$ 0 0
$$501$$ −16.8730 −0.753829
$$502$$ 0 0
$$503$$ −17.1109 −0.762937 −0.381468 0.924382i $$-0.624581\pi$$
−0.381468 + 0.924382i $$0.624581\pi$$
$$504$$ 0 0
$$505$$ 3.87298 0.172345
$$506$$ 0 0
$$507$$ −10.7460 −0.477245
$$508$$ 0 0
$$509$$ 20.0000 0.886484 0.443242 0.896402i $$-0.353828\pi$$
0.443242 + 0.896402i $$0.353828\pi$$
$$510$$ 0 0
$$511$$ −14.6190 −0.646704
$$512$$ 0 0
$$513$$ 4.87298 0.215147
$$514$$ 0 0
$$515$$ 6.00000 0.264392
$$516$$ 0 0
$$517$$ −2.50807 −0.110305
$$518$$ 0 0
$$519$$ 7.49193 0.328859
$$520$$ 0 0
$$521$$ 33.1270 1.45132 0.725660 0.688053i $$-0.241534\pi$$
0.725660 + 0.688053i $$0.241534\pi$$
$$522$$ 0 0
$$523$$ −42.9839 −1.87955 −0.939777 0.341789i $$-0.888967\pi$$
−0.939777 + 0.341789i $$0.888967\pi$$
$$524$$ 0 0
$$525$$ 3.00000 0.130931
$$526$$ 0 0
$$527$$ 11.6190 0.506129
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 1.87298 0.0812806
$$532$$ 0 0
$$533$$ 9.12702 0.395335
$$534$$ 0 0
$$535$$ −17.6190 −0.761734
$$536$$ 0 0
$$537$$ −0.254033 −0.0109623
$$538$$ 0 0
$$539$$ 5.74597 0.247496
$$540$$ 0 0
$$541$$ −40.0000 −1.71973 −0.859867 0.510518i $$-0.829454\pi$$
−0.859867 + 0.510518i $$0.829454\pi$$
$$542$$ 0 0
$$543$$ 6.25403 0.268386
$$544$$ 0 0
$$545$$ −6.61895 −0.283525
$$546$$ 0 0
$$547$$ −16.0000 −0.684111 −0.342055 0.939680i $$-0.611123\pi$$
−0.342055 + 0.939680i $$0.611123\pi$$
$$548$$ 0 0
$$549$$ 1.12702 0.0480999
$$550$$ 0 0
$$551$$ −9.12702 −0.388824
$$552$$ 0 0
$$553$$ 12.0000 0.510292
$$554$$ 0 0
$$555$$ −1.00000 −0.0424476
$$556$$ 0 0
$$557$$ 35.8730 1.51999 0.759994 0.649931i $$-0.225202\pi$$
0.759994 + 0.649931i $$0.225202\pi$$
$$558$$ 0 0
$$559$$ 57.2379 2.42091
$$560$$ 0 0
$$561$$ 11.1270 0.469783
$$562$$ 0 0
$$563$$ 21.1109 0.889718 0.444859 0.895601i $$-0.353254\pi$$
0.444859 + 0.895601i $$0.353254\pi$$
$$564$$ 0 0
$$565$$ 3.87298 0.162938
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ 19.7460 0.827794 0.413897 0.910324i $$-0.364167\pi$$
0.413897 + 0.910324i $$0.364167\pi$$
$$570$$ 0 0
$$571$$ 2.36492 0.0989687 0.0494843 0.998775i $$-0.484242\pi$$
0.0494843 + 0.998775i $$0.484242\pi$$
$$572$$ 0 0
$$573$$ −11.1270 −0.464838
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ 8.00000 0.333044 0.166522 0.986038i $$-0.446746\pi$$
0.166522 + 0.986038i $$0.446746\pi$$
$$578$$ 0 0
$$579$$ 5.74597 0.238794
$$580$$ 0 0
$$581$$ −23.6190 −0.979879
$$582$$ 0 0
$$583$$ 11.1270 0.460834
$$584$$ 0 0
$$585$$ 4.87298 0.201473
$$586$$ 0 0
$$587$$ 17.4919 0.721969 0.360985 0.932572i $$-0.382441\pi$$
0.360985 + 0.932572i $$0.382441\pi$$
$$588$$ 0 0
$$589$$ 14.6190 0.602363
$$590$$ 0 0
$$591$$ −19.7460 −0.812241
$$592$$ 0 0
$$593$$ 21.3810 0.878014 0.439007 0.898484i $$-0.355330\pi$$
0.439007 + 0.898484i $$0.355330\pi$$
$$594$$ 0 0
$$595$$ 11.6190 0.476331
$$596$$ 0 0
$$597$$ −27.2379 −1.11477
$$598$$ 0 0
$$599$$ −6.25403 −0.255533 −0.127766 0.991804i $$-0.540781\pi$$
−0.127766 + 0.991804i $$0.540781\pi$$
$$600$$ 0 0
$$601$$ 2.74597 0.112010 0.0560052 0.998430i $$-0.482164\pi$$
0.0560052 + 0.998430i $$0.482164\pi$$
$$602$$ 0 0
$$603$$ 4.74597 0.193271
$$604$$ 0 0
$$605$$ −2.74597 −0.111639
$$606$$ 0 0
$$607$$ −38.8730 −1.57781 −0.788903 0.614518i $$-0.789351\pi$$
−0.788903 + 0.614518i $$0.789351\pi$$
$$608$$ 0 0
$$609$$ 5.61895 0.227691
$$610$$ 0 0
$$611$$ −4.25403 −0.172100
$$612$$ 0 0
$$613$$ −12.9839 −0.524413 −0.262207 0.965012i $$-0.584450\pi$$
−0.262207 + 0.965012i $$0.584450\pi$$
$$614$$ 0 0
$$615$$ −1.87298 −0.0755260
$$616$$ 0 0
$$617$$ 5.61895 0.226210 0.113105 0.993583i $$-0.463920\pi$$
0.113105 + 0.993583i $$0.463920\pi$$
$$618$$ 0 0
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 0 0
$$623$$ 41.2379 1.65216
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 14.0000 0.559106
$$628$$ 0 0
$$629$$ −3.87298 −0.154426
$$630$$ 0 0
$$631$$ 6.36492 0.253383 0.126692 0.991942i $$-0.459564\pi$$
0.126692 + 0.991942i $$0.459564\pi$$
$$632$$ 0 0
$$633$$ 1.00000 0.0397464
$$634$$ 0 0
$$635$$ −16.6190 −0.659503
$$636$$ 0 0
$$637$$ 9.74597 0.386149
$$638$$ 0 0
$$639$$ −9.61895 −0.380520
$$640$$ 0 0
$$641$$ 44.1109 1.74228 0.871138 0.491039i $$-0.163383\pi$$
0.871138 + 0.491039i $$0.163383\pi$$
$$642$$ 0 0
$$643$$ −8.49193 −0.334889 −0.167445 0.985881i $$-0.553552\pi$$
−0.167445 + 0.985881i $$0.553552\pi$$
$$644$$ 0 0
$$645$$ −11.7460 −0.462497
$$646$$ 0 0
$$647$$ 41.8569 1.64556 0.822781 0.568358i $$-0.192421\pi$$
0.822781 + 0.568358i $$0.192421\pi$$
$$648$$ 0 0
$$649$$ 5.38105 0.211225
$$650$$ 0 0
$$651$$ −9.00000 −0.352738
$$652$$ 0 0
$$653$$ 44.1109 1.72619 0.863096 0.505040i $$-0.168522\pi$$
0.863096 + 0.505040i $$0.168522\pi$$
$$654$$ 0 0
$$655$$ 19.4919 0.761613
$$656$$ 0 0
$$657$$ 4.87298 0.190113
$$658$$ 0 0
$$659$$ −21.1270 −0.822992 −0.411496 0.911412i $$-0.634994\pi$$
−0.411496 + 0.911412i $$0.634994\pi$$
$$660$$ 0 0
$$661$$ 3.23790 0.125940 0.0629699 0.998015i $$-0.479943\pi$$
0.0629699 + 0.998015i $$0.479943\pi$$
$$662$$ 0 0
$$663$$ 18.8730 0.732966
$$664$$ 0 0
$$665$$ 14.6190 0.566899
$$666$$ 0 0
$$667$$ 1.87298 0.0725222
$$668$$ 0 0
$$669$$ −2.00000 −0.0773245
$$670$$ 0 0
$$671$$ 3.23790 0.124998
$$672$$ 0 0
$$673$$ 26.8730 1.03588 0.517939 0.855418i $$-0.326700\pi$$
0.517939 + 0.855418i $$0.326700\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 47.1109 1.81062 0.905309 0.424753i $$-0.139639\pi$$
0.905309 + 0.424753i $$0.139639\pi$$
$$678$$ 0 0
$$679$$ −24.0000 −0.921035
$$680$$ 0 0
$$681$$ −23.4919 −0.900213
$$682$$ 0 0
$$683$$ −24.8730 −0.951738 −0.475869 0.879516i $$-0.657866\pi$$
−0.475869 + 0.879516i $$0.657866\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ 0 0
$$687$$ −9.74597 −0.371832
$$688$$ 0 0
$$689$$ 18.8730 0.719003
$$690$$ 0 0
$$691$$ 30.7298 1.16902 0.584509 0.811387i $$-0.301287\pi$$
0.584509 + 0.811387i $$0.301287\pi$$
$$692$$ 0 0
$$693$$ −8.61895 −0.327407
$$694$$ 0 0
$$695$$ −10.4919 −0.397982
$$696$$ 0 0
$$697$$ −7.25403 −0.274766
$$698$$ 0 0
$$699$$ 15.4919 0.585959
$$700$$ 0 0
$$701$$ −39.8569 −1.50537 −0.752686 0.658379i $$-0.771242\pi$$
−0.752686 + 0.658379i $$0.771242\pi$$
$$702$$ 0 0
$$703$$ −4.87298 −0.183788
$$704$$ 0 0
$$705$$ 0.872983 0.0328785
$$706$$ 0 0
$$707$$ −11.6190 −0.436976
$$708$$ 0 0
$$709$$ 16.6190 0.624138 0.312069 0.950059i $$-0.398978\pi$$
0.312069 + 0.950059i $$0.398978\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ −3.00000 −0.112351
$$714$$ 0 0
$$715$$ 14.0000 0.523570
$$716$$ 0 0
$$717$$ −10.1270 −0.378200
$$718$$ 0 0
$$719$$ 0.635083 0.0236846 0.0118423 0.999930i $$-0.496230\pi$$
0.0118423 + 0.999930i $$0.496230\pi$$
$$720$$ 0 0
$$721$$ −18.0000 −0.670355
$$722$$ 0 0
$$723$$ 12.8730 0.478751
$$724$$ 0 0
$$725$$ 1.87298 0.0695609
$$726$$ 0 0
$$727$$ −26.7460 −0.991953 −0.495976 0.868336i $$-0.665190\pi$$
−0.495976 + 0.868336i $$0.665190\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −45.4919 −1.68258
$$732$$ 0 0
$$733$$ 7.25403 0.267934 0.133967 0.990986i $$-0.457228\pi$$
0.133967 + 0.990986i $$0.457228\pi$$
$$734$$ 0 0
$$735$$ −2.00000 −0.0737711
$$736$$ 0 0
$$737$$ 13.6351 0.502255
$$738$$ 0 0
$$739$$ −28.7460 −1.05744 −0.528719 0.848797i $$-0.677327\pi$$
−0.528719 + 0.848797i $$0.677327\pi$$
$$740$$ 0 0
$$741$$ 23.7460 0.872330
$$742$$ 0 0
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 0 0
$$745$$ −3.12702 −0.114565
$$746$$ 0 0
$$747$$ 7.87298 0.288057
$$748$$ 0 0
$$749$$ 52.8569 1.93135
$$750$$ 0 0
$$751$$ −39.8569 −1.45440 −0.727199 0.686427i $$-0.759178\pi$$
−0.727199 + 0.686427i $$0.759178\pi$$
$$752$$ 0 0
$$753$$ 2.00000 0.0728841
$$754$$ 0 0
$$755$$ 11.7460 0.427479
$$756$$ 0 0
$$757$$ 10.7460 0.390569 0.195284 0.980747i $$-0.437437\pi$$
0.195284 + 0.980747i $$0.437437\pi$$
$$758$$ 0 0
$$759$$ −2.87298 −0.104283
$$760$$ 0 0
$$761$$ 54.8569 1.98856 0.994280 0.106808i $$-0.0340631\pi$$
0.994280 + 0.106808i $$0.0340631\pi$$
$$762$$ 0 0
$$763$$ 19.8569 0.718866
$$764$$ 0 0
$$765$$ −3.87298 −0.140028
$$766$$ 0 0
$$767$$ 9.12702 0.329557
$$768$$ 0 0
$$769$$ 21.1270 0.761860 0.380930 0.924604i $$-0.375604\pi$$
0.380930 + 0.924604i $$0.375604\pi$$
$$770$$ 0 0
$$771$$ −6.61895 −0.238376
$$772$$ 0 0
$$773$$ 19.7460 0.710213 0.355107 0.934826i $$-0.384445\pi$$
0.355107 + 0.934826i $$0.384445\pi$$
$$774$$ 0 0
$$775$$ −3.00000 −0.107763
$$776$$ 0 0
$$777$$ 3.00000 0.107624
$$778$$ 0 0
$$779$$ −9.12702 −0.327009
$$780$$ 0 0
$$781$$ −27.6351 −0.988861
$$782$$ 0 0
$$783$$ −1.87298 −0.0669350
$$784$$ 0 0
$$785$$ 17.0000 0.606756
$$786$$ 0 0
$$787$$ −25.0000 −0.891154 −0.445577 0.895244i $$-0.647001\pi$$
−0.445577 + 0.895244i $$0.647001\pi$$
$$788$$ 0 0
$$789$$ −14.1270 −0.502935
$$790$$ 0 0
$$791$$ −11.6190 −0.413122
$$792$$ 0 0
$$793$$ 5.49193 0.195024
$$794$$ 0 0
$$795$$ −3.87298 −0.137361
$$796$$ 0 0
$$797$$ 4.12702 0.146186 0.0730932 0.997325i $$-0.476713\pi$$
0.0730932 + 0.997325i $$0.476713\pi$$
$$798$$ 0 0
$$799$$ 3.38105 0.119613
$$800$$ 0 0
$$801$$ −13.7460 −0.485690
$$802$$ 0 0
$$803$$ 14.0000 0.494049
$$804$$ 0 0
$$805$$ −3.00000 −0.105736
$$806$$ 0 0
$$807$$ −15.3649 −0.540871
$$808$$ 0 0
$$809$$ −44.8569 −1.57708 −0.788541 0.614982i $$-0.789163\pi$$
−0.788541 + 0.614982i $$0.789163\pi$$
$$810$$ 0 0
$$811$$ 2.49193 0.0875036 0.0437518 0.999042i $$-0.486069\pi$$
0.0437518 + 0.999042i $$0.486069\pi$$
$$812$$ 0 0
$$813$$ −22.7460 −0.797736
$$814$$ 0 0
$$815$$ 22.0000 0.770626
$$816$$ 0 0
$$817$$ −57.2379 −2.00250
$$818$$ 0 0
$$819$$ −14.6190 −0.510827
$$820$$ 0 0
$$821$$ 27.4919 0.959475 0.479738 0.877412i $$-0.340732\pi$$
0.479738 + 0.877412i $$0.340732\pi$$
$$822$$ 0 0
$$823$$ −44.0000 −1.53374 −0.766872 0.641800i $$-0.778188\pi$$
−0.766872 + 0.641800i $$0.778188\pi$$
$$824$$ 0 0
$$825$$ −2.87298 −0.100024
$$826$$ 0 0
$$827$$ −39.1109 −1.36002 −0.680009 0.733203i $$-0.738024\pi$$
−0.680009 + 0.733203i $$0.738024\pi$$
$$828$$ 0 0
$$829$$ 34.4919 1.19795 0.598977 0.800766i $$-0.295574\pi$$
0.598977 + 0.800766i $$0.295574\pi$$
$$830$$ 0 0
$$831$$ −14.0000 −0.485655
$$832$$ 0 0
$$833$$ −7.74597 −0.268382
$$834$$ 0 0
$$835$$ 16.8730 0.583914
$$836$$ 0 0
$$837$$ 3.00000 0.103695
$$838$$ 0 0
$$839$$ −18.0000 −0.621429 −0.310715 0.950503i $$-0.600568\pi$$
−0.310715 + 0.950503i $$0.600568\pi$$
$$840$$ 0 0
$$841$$ −25.4919 −0.879032
$$842$$ 0 0
$$843$$ −22.3649 −0.770289
$$844$$ 0 0
$$845$$ 10.7460 0.369672
$$846$$ 0 0
$$847$$ 8.23790 0.283058
$$848$$ 0 0
$$849$$ −24.2379 −0.831843
$$850$$ 0 0
$$851$$ 1.00000 0.0342796
$$852$$ 0 0
$$853$$ −28.2540 −0.967400 −0.483700 0.875234i $$-0.660707\pi$$
−0.483700 + 0.875234i $$0.660707\pi$$
$$854$$ 0 0
$$855$$ −4.87298 −0.166653
$$856$$ 0 0
$$857$$ −45.4919 −1.55397 −0.776987 0.629516i $$-0.783253\pi$$
−0.776987 + 0.629516i $$0.783253\pi$$
$$858$$ 0 0
$$859$$ −29.0000 −0.989467 −0.494734 0.869045i $$-0.664734\pi$$
−0.494734 + 0.869045i $$0.664734\pi$$
$$860$$ 0 0
$$861$$ 5.61895 0.191493
$$862$$ 0 0
$$863$$ −23.7460 −0.808322 −0.404161 0.914688i $$-0.632436\pi$$
−0.404161 + 0.914688i $$0.632436\pi$$
$$864$$ 0 0
$$865$$ −7.49193 −0.254733
$$866$$ 0 0
$$867$$ 2.00000 0.0679236
$$868$$ 0 0
$$869$$ −11.4919 −0.389837
$$870$$ 0 0
$$871$$ 23.1270 0.783629
$$872$$ 0 0
$$873$$ 8.00000 0.270759
$$874$$ 0 0
$$875$$ −3.00000 −0.101419
$$876$$ 0 0
$$877$$ −22.7298 −0.767532 −0.383766 0.923430i $$-0.625373\pi$$
−0.383766 + 0.923430i $$0.625373\pi$$
$$878$$ 0 0
$$879$$ −9.61895 −0.324439
$$880$$ 0 0
$$881$$ −50.1109 −1.68828 −0.844139 0.536124i $$-0.819888\pi$$
−0.844139 + 0.536124i $$0.819888\pi$$
$$882$$ 0 0
$$883$$ 45.3488 1.52611 0.763054 0.646335i $$-0.223699\pi$$
0.763054 + 0.646335i $$0.223699\pi$$
$$884$$ 0 0
$$885$$ −1.87298 −0.0629596
$$886$$ 0 0
$$887$$ −4.25403 −0.142836 −0.0714182 0.997446i $$-0.522752\pi$$
−0.0714182 + 0.997446i $$0.522752\pi$$
$$888$$ 0 0
$$889$$ 49.8569 1.67215
$$890$$ 0 0
$$891$$ 2.87298 0.0962486
$$892$$ 0 0
$$893$$ 4.25403 0.142356
$$894$$ 0 0
$$895$$ 0.254033 0.00849140
$$896$$ 0 0
$$897$$ −4.87298 −0.162704
$$898$$ 0 0
$$899$$ −5.61895 −0.187402
$$900$$ 0 0
$$901$$ −15.0000 −0.499722
$$902$$ 0 0
$$903$$ 35.2379 1.17264
$$904$$ 0 0
$$905$$ −6.25403 −0.207891
$$906$$ 0 0
$$907$$ −15.2540 −0.506502 −0.253251 0.967401i $$-0.581500\pi$$
−0.253251 + 0.967401i $$0.581500\pi$$
$$908$$ 0 0
$$909$$ 3.87298 0.128459
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ 22.6190 0.748578
$$914$$ 0 0
$$915$$ −1.12702 −0.0372580
$$916$$ 0 0
$$917$$ −58.4758 −1.93104
$$918$$ 0 0
$$919$$ 44.0000 1.45143 0.725713 0.687998i $$-0.241510\pi$$
0.725713 + 0.687998i $$0.241510\pi$$
$$920$$ 0 0
$$921$$ 16.8730 0.555984
$$922$$ 0 0
$$923$$ −46.8730 −1.54284
$$924$$ 0 0
$$925$$ 1.00000 0.0328798
$$926$$ 0 0
$$927$$ 6.00000 0.197066
$$928$$ 0 0
$$929$$ −7.36492 −0.241635 −0.120818 0.992675i $$-0.538552\pi$$
−0.120818 + 0.992675i $$0.538552\pi$$
$$930$$ 0 0
$$931$$ −9.74597 −0.319411
$$932$$ 0 0
$$933$$ −4.00000 −0.130954
$$934$$ 0 0
$$935$$ −11.1270 −0.363892
$$936$$ 0 0
$$937$$ −38.4758 −1.25695 −0.628475 0.777830i $$-0.716320\pi$$
−0.628475 + 0.777830i $$0.716320\pi$$
$$938$$ 0 0
$$939$$ −7.00000 −0.228436
$$940$$ 0 0
$$941$$ 38.1109 1.24238 0.621190 0.783660i $$-0.286650\pi$$
0.621190 + 0.783660i $$0.286650\pi$$
$$942$$ 0 0
$$943$$ 1.87298 0.0609927
$$944$$ 0 0
$$945$$ 3.00000 0.0975900
$$946$$ 0 0
$$947$$ −21.2379 −0.690139 −0.345070 0.938577i $$-0.612145\pi$$
−0.345070 + 0.938577i $$0.612145\pi$$
$$948$$ 0 0
$$949$$ 23.7460 0.770827
$$950$$ 0 0
$$951$$ 16.3649 0.530669
$$952$$ 0 0
$$953$$ −52.9839 −1.71632 −0.858158 0.513386i $$-0.828391\pi$$
−0.858158 + 0.513386i $$0.828391\pi$$
$$954$$ 0 0
$$955$$ 11.1270 0.360062
$$956$$ 0 0
$$957$$ −5.38105 −0.173945
$$958$$ 0 0
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ −17.6190 −0.567763
$$964$$ 0 0
$$965$$ −5.74597 −0.184969
$$966$$ 0 0
$$967$$ −44.6190 −1.43485 −0.717424 0.696636i $$-0.754679\pi$$
−0.717424 + 0.696636i $$0.754679\pi$$
$$968$$ 0 0
$$969$$ −18.8730 −0.606288
$$970$$ 0 0
$$971$$ −38.2540 −1.22763 −0.613815 0.789450i $$-0.710366\pi$$
−0.613815 + 0.789450i $$0.710366\pi$$
$$972$$ 0 0
$$973$$ 31.4758 1.00907
$$974$$ 0 0
$$975$$ −4.87298 −0.156060
$$976$$ 0 0
$$977$$ −44.1270 −1.41175 −0.705874 0.708337i $$-0.749446\pi$$
−0.705874 + 0.708337i $$0.749446\pi$$
$$978$$ 0 0
$$979$$ −39.4919 −1.26217
$$980$$ 0 0
$$981$$ −6.61895 −0.211327
$$982$$ 0 0
$$983$$ 31.3649 1.00039 0.500193 0.865914i $$-0.333262\pi$$
0.500193 + 0.865914i $$0.333262\pi$$
$$984$$ 0 0
$$985$$ 19.7460 0.629159
$$986$$ 0 0
$$987$$ −2.61895 −0.0833621
$$988$$ 0 0
$$989$$ 11.7460 0.373500
$$990$$ 0 0
$$991$$ 42.4919 1.34980 0.674900 0.737909i $$-0.264187\pi$$
0.674900 + 0.737909i $$0.264187\pi$$
$$992$$ 0 0
$$993$$ 24.2379 0.769167
$$994$$ 0 0
$$995$$ 27.2379 0.863499
$$996$$ 0 0
$$997$$ −26.2540 −0.831474 −0.415737 0.909485i $$-0.636476\pi$$
−0.415737 + 0.909485i $$0.636476\pi$$
$$998$$ 0 0
$$999$$ −1.00000 −0.0316386
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.bj.1.2 2
4.3 odd 2 1380.2.a.i.1.1 2
12.11 even 2 4140.2.a.p.1.2 2
20.3 even 4 6900.2.f.o.6349.3 4
20.7 even 4 6900.2.f.o.6349.1 4
20.19 odd 2 6900.2.a.j.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.i.1.1 2 4.3 odd 2
4140.2.a.p.1.2 2 12.11 even 2
5520.2.a.bj.1.2 2 1.1 even 1 trivial
6900.2.a.j.1.1 2 20.19 odd 2
6900.2.f.o.6349.1 4 20.7 even 4
6900.2.f.o.6349.3 4 20.3 even 4