Properties

 Label 3360.2.a.bg.1.1 Level $3360$ Weight $2$ Character 3360.1 Self dual yes Analytic conductor $26.830$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3360.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$26.8297350792$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 3360.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.12311 q^{11} +5.12311 q^{13} -1.00000 q^{15} +2.00000 q^{17} +7.12311 q^{19} +1.00000 q^{21} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -3.12311 q^{31} -3.12311 q^{33} -1.00000 q^{35} -2.00000 q^{37} +5.12311 q^{39} +2.00000 q^{41} -6.24621 q^{43} -1.00000 q^{45} +1.00000 q^{49} +2.00000 q^{51} -11.3693 q^{53} +3.12311 q^{55} +7.12311 q^{57} -4.00000 q^{59} +8.24621 q^{61} +1.00000 q^{63} -5.12311 q^{65} +14.2462 q^{67} +3.12311 q^{71} +6.87689 q^{73} +1.00000 q^{75} -3.12311 q^{77} +14.2462 q^{79} +1.00000 q^{81} -4.00000 q^{83} -2.00000 q^{85} -2.00000 q^{87} +16.2462 q^{89} +5.12311 q^{91} -3.12311 q^{93} -7.12311 q^{95} +13.1231 q^{97} -3.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 2 q^{31} + 2 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} + 4 q^{41} + 4 q^{43} - 2 q^{45} + 2 q^{49} + 4 q^{51} + 2 q^{53} - 2 q^{55} + 6 q^{57} - 8 q^{59} + 2 q^{63} - 2 q^{65} + 12 q^{67} - 2 q^{71} + 22 q^{73} + 2 q^{75} + 2 q^{77} + 12 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} - 4 q^{87} + 16 q^{89} + 2 q^{91} + 2 q^{93} - 6 q^{95} + 18 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 + 4 * q^17 + 6 * q^19 + 2 * q^21 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 2 * q^31 + 2 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 + 4 * q^41 + 4 * q^43 - 2 * q^45 + 2 * q^49 + 4 * q^51 + 2 * q^53 - 2 * q^55 + 6 * q^57 - 8 * q^59 + 2 * q^63 - 2 * q^65 + 12 * q^67 - 2 * q^71 + 22 * q^73 + 2 * q^75 + 2 * q^77 + 12 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^85 - 4 * q^87 + 16 * q^89 + 2 * q^91 + 2 * q^93 - 6 * q^95 + 18 * q^97 + 2 * q^99

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.12311 −0.941652 −0.470826 0.882226i $$-0.656044\pi$$
−0.470826 + 0.882226i $$0.656044\pi$$
$$12$$ 0 0
$$13$$ 5.12311 1.42089 0.710447 0.703751i $$-0.248493\pi$$
0.710447 + 0.703751i $$0.248493\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 7.12311 1.63415 0.817076 0.576530i $$-0.195593\pi$$
0.817076 + 0.576530i $$0.195593\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −3.12311 −0.560926 −0.280463 0.959865i $$-0.590488\pi$$
−0.280463 + 0.959865i $$0.590488\pi$$
$$32$$ 0 0
$$33$$ −3.12311 −0.543663
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 5.12311 0.820353
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −6.24621 −0.952538 −0.476269 0.879300i $$-0.658011\pi$$
−0.476269 + 0.879300i $$0.658011\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ −11.3693 −1.56170 −0.780848 0.624721i $$-0.785213\pi$$
−0.780848 + 0.624721i $$0.785213\pi$$
$$54$$ 0 0
$$55$$ 3.12311 0.421119
$$56$$ 0 0
$$57$$ 7.12311 0.943478
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 8.24621 1.05582 0.527910 0.849301i $$-0.322976\pi$$
0.527910 + 0.849301i $$0.322976\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −5.12311 −0.635443
$$66$$ 0 0
$$67$$ 14.2462 1.74045 0.870226 0.492653i $$-0.163973\pi$$
0.870226 + 0.492653i $$0.163973\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 3.12311 0.370644 0.185322 0.982678i $$-0.440667\pi$$
0.185322 + 0.982678i $$0.440667\pi$$
$$72$$ 0 0
$$73$$ 6.87689 0.804880 0.402440 0.915446i $$-0.368162\pi$$
0.402440 + 0.915446i $$0.368162\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −3.12311 −0.355911
$$78$$ 0 0
$$79$$ 14.2462 1.60282 0.801412 0.598113i $$-0.204082\pi$$
0.801412 + 0.598113i $$0.204082\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ 16.2462 1.72209 0.861047 0.508525i $$-0.169809\pi$$
0.861047 + 0.508525i $$0.169809\pi$$
$$90$$ 0 0
$$91$$ 5.12311 0.537047
$$92$$ 0 0
$$93$$ −3.12311 −0.323851
$$94$$ 0 0
$$95$$ −7.12311 −0.730815
$$96$$ 0 0
$$97$$ 13.1231 1.33245 0.666225 0.745751i $$-0.267909\pi$$
0.666225 + 0.745751i $$0.267909\pi$$
$$98$$ 0 0
$$99$$ −3.12311 −0.313884
$$100$$ 0 0
$$101$$ 16.2462 1.61656 0.808279 0.588799i $$-0.200399\pi$$
0.808279 + 0.588799i $$0.200399\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ −8.24621 −0.789844 −0.394922 0.918715i $$-0.629228\pi$$
−0.394922 + 0.918715i $$0.629228\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 19.3693 1.82211 0.911056 0.412283i $$-0.135268\pi$$
0.911056 + 0.412283i $$0.135268\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 5.12311 0.473631
$$118$$ 0 0
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ −1.24621 −0.113292
$$122$$ 0 0
$$123$$ 2.00000 0.180334
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −12.4924 −1.10852 −0.554262 0.832343i $$-0.686999\pi$$
−0.554262 + 0.832343i $$0.686999\pi$$
$$128$$ 0 0
$$129$$ −6.24621 −0.549948
$$130$$ 0 0
$$131$$ 18.2462 1.59418 0.797089 0.603861i $$-0.206372\pi$$
0.797089 + 0.603861i $$0.206372\pi$$
$$132$$ 0 0
$$133$$ 7.12311 0.617652
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 21.1231 1.80467 0.902334 0.431037i $$-0.141852\pi$$
0.902334 + 0.431037i $$0.141852\pi$$
$$138$$ 0 0
$$139$$ −0.876894 −0.0743772 −0.0371886 0.999308i $$-0.511840\pi$$
−0.0371886 + 0.999308i $$0.511840\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −16.0000 −1.33799
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ 4.24621 0.347863 0.173932 0.984758i $$-0.444353\pi$$
0.173932 + 0.984758i $$0.444353\pi$$
$$150$$ 0 0
$$151$$ −12.4924 −1.01662 −0.508309 0.861174i $$-0.669729\pi$$
−0.508309 + 0.861174i $$0.669729\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 3.12311 0.250854
$$156$$ 0 0
$$157$$ −1.12311 −0.0896336 −0.0448168 0.998995i $$-0.514270\pi$$
−0.0448168 + 0.998995i $$0.514270\pi$$
$$158$$ 0 0
$$159$$ −11.3693 −0.901645
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −6.24621 −0.489241 −0.244621 0.969619i $$-0.578663\pi$$
−0.244621 + 0.969619i $$0.578663\pi$$
$$164$$ 0 0
$$165$$ 3.12311 0.243133
$$166$$ 0 0
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ 13.2462 1.01894
$$170$$ 0 0
$$171$$ 7.12311 0.544718
$$172$$ 0 0
$$173$$ 16.2462 1.23518 0.617588 0.786502i $$-0.288110\pi$$
0.617588 + 0.786502i $$0.288110\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ −11.1231 −0.831380 −0.415690 0.909506i $$-0.636460\pi$$
−0.415690 + 0.909506i $$0.636460\pi$$
$$180$$ 0 0
$$181$$ −10.4924 −0.779896 −0.389948 0.920837i $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$182$$ 0 0
$$183$$ 8.24621 0.609577
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ −6.24621 −0.456768
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −3.12311 −0.225980 −0.112990 0.993596i $$-0.536043\pi$$
−0.112990 + 0.993596i $$0.536043\pi$$
$$192$$ 0 0
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ 0 0
$$195$$ −5.12311 −0.366873
$$196$$ 0 0
$$197$$ −17.6155 −1.25505 −0.627527 0.778595i $$-0.715933\pi$$
−0.627527 + 0.778595i $$0.715933\pi$$
$$198$$ 0 0
$$199$$ −3.12311 −0.221391 −0.110696 0.993854i $$-0.535308\pi$$
−0.110696 + 0.993854i $$0.535308\pi$$
$$200$$ 0 0
$$201$$ 14.2462 1.00485
$$202$$ 0 0
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −22.2462 −1.53880
$$210$$ 0 0
$$211$$ 22.2462 1.53149 0.765746 0.643143i $$-0.222370\pi$$
0.765746 + 0.643143i $$0.222370\pi$$
$$212$$ 0 0
$$213$$ 3.12311 0.213992
$$214$$ 0 0
$$215$$ 6.24621 0.425988
$$216$$ 0 0
$$217$$ −3.12311 −0.212010
$$218$$ 0 0
$$219$$ 6.87689 0.464697
$$220$$ 0 0
$$221$$ 10.2462 0.689235
$$222$$ 0 0
$$223$$ −1.75379 −0.117442 −0.0587212 0.998274i $$-0.518702\pi$$
−0.0587212 + 0.998274i $$0.518702\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ −20.2462 −1.33791 −0.668954 0.743304i $$-0.733258\pi$$
−0.668954 + 0.743304i $$0.733258\pi$$
$$230$$ 0 0
$$231$$ −3.12311 −0.205485
$$232$$ 0 0
$$233$$ 14.8769 0.974618 0.487309 0.873230i $$-0.337979\pi$$
0.487309 + 0.873230i $$0.337979\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 14.2462 0.925391
$$238$$ 0 0
$$239$$ −4.87689 −0.315460 −0.157730 0.987482i $$-0.550418\pi$$
−0.157730 + 0.987482i $$0.550418\pi$$
$$240$$ 0 0
$$241$$ 3.75379 0.241803 0.120901 0.992665i $$-0.461422\pi$$
0.120901 + 0.992665i $$0.461422\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ 36.4924 2.32196
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −2.00000 −0.125245
$$256$$ 0 0
$$257$$ 3.75379 0.234155 0.117077 0.993123i $$-0.462647\pi$$
0.117077 + 0.993123i $$0.462647\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ −1.75379 −0.108143 −0.0540716 0.998537i $$-0.517220\pi$$
−0.0540716 + 0.998537i $$0.517220\pi$$
$$264$$ 0 0
$$265$$ 11.3693 0.698412
$$266$$ 0 0
$$267$$ 16.2462 0.994252
$$268$$ 0 0
$$269$$ 0.246211 0.0150118 0.00750588 0.999972i $$-0.497611\pi$$
0.00750588 + 0.999972i $$0.497611\pi$$
$$270$$ 0 0
$$271$$ 23.6155 1.43454 0.717271 0.696795i $$-0.245391\pi$$
0.717271 + 0.696795i $$0.245391\pi$$
$$272$$ 0 0
$$273$$ 5.12311 0.310064
$$274$$ 0 0
$$275$$ −3.12311 −0.188330
$$276$$ 0 0
$$277$$ 18.4924 1.11110 0.555551 0.831482i $$-0.312507\pi$$
0.555551 + 0.831482i $$0.312507\pi$$
$$278$$ 0 0
$$279$$ −3.12311 −0.186975
$$280$$ 0 0
$$281$$ −24.7386 −1.47578 −0.737892 0.674919i $$-0.764178\pi$$
−0.737892 + 0.674919i $$0.764178\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 0 0
$$285$$ −7.12311 −0.421936
$$286$$ 0 0
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 13.1231 0.769290
$$292$$ 0 0
$$293$$ 0.246211 0.0143838 0.00719191 0.999974i $$-0.497711\pi$$
0.00719191 + 0.999974i $$0.497711\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 0 0
$$297$$ −3.12311 −0.181221
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −6.24621 −0.360026
$$302$$ 0 0
$$303$$ 16.2462 0.933320
$$304$$ 0 0
$$305$$ −8.24621 −0.472177
$$306$$ 0 0
$$307$$ −26.2462 −1.49795 −0.748975 0.662598i $$-0.769454\pi$$
−0.748975 + 0.662598i $$0.769454\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ 0 0
$$313$$ 8.63068 0.487835 0.243918 0.969796i $$-0.421567\pi$$
0.243918 + 0.969796i $$0.421567\pi$$
$$314$$ 0 0
$$315$$ −1.00000 −0.0563436
$$316$$ 0 0
$$317$$ −6.87689 −0.386245 −0.193122 0.981175i $$-0.561861\pi$$
−0.193122 + 0.981175i $$0.561861\pi$$
$$318$$ 0 0
$$319$$ 6.24621 0.349721
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 0 0
$$323$$ 14.2462 0.792680
$$324$$ 0 0
$$325$$ 5.12311 0.284179
$$326$$ 0 0
$$327$$ −8.24621 −0.456017
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −14.2462 −0.783043 −0.391521 0.920169i $$-0.628051\pi$$
−0.391521 + 0.920169i $$0.628051\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ −14.2462 −0.778354
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 19.3693 1.05200
$$340$$ 0 0
$$341$$ 9.75379 0.528197
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −9.75379 −0.523611 −0.261805 0.965121i $$-0.584318\pi$$
−0.261805 + 0.965121i $$0.584318\pi$$
$$348$$ 0 0
$$349$$ −28.2462 −1.51199 −0.755993 0.654580i $$-0.772845\pi$$
−0.755993 + 0.654580i $$0.772845\pi$$
$$350$$ 0 0
$$351$$ 5.12311 0.273451
$$352$$ 0 0
$$353$$ 22.4924 1.19715 0.598575 0.801066i $$-0.295734\pi$$
0.598575 + 0.801066i $$0.295734\pi$$
$$354$$ 0 0
$$355$$ −3.12311 −0.165757
$$356$$ 0 0
$$357$$ 2.00000 0.105851
$$358$$ 0 0
$$359$$ −7.61553 −0.401932 −0.200966 0.979598i $$-0.564408\pi$$
−0.200966 + 0.979598i $$0.564408\pi$$
$$360$$ 0 0
$$361$$ 31.7386 1.67045
$$362$$ 0 0
$$363$$ −1.24621 −0.0654091
$$364$$ 0 0
$$365$$ −6.87689 −0.359953
$$366$$ 0 0
$$367$$ 30.2462 1.57884 0.789420 0.613854i $$-0.210382\pi$$
0.789420 + 0.613854i $$0.210382\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ −11.3693 −0.590266
$$372$$ 0 0
$$373$$ −3.75379 −0.194364 −0.0971819 0.995267i $$-0.530983\pi$$
−0.0971819 + 0.995267i $$0.530983\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −10.2462 −0.527707
$$378$$ 0 0
$$379$$ −20.4924 −1.05263 −0.526313 0.850291i $$-0.676426\pi$$
−0.526313 + 0.850291i $$0.676426\pi$$
$$380$$ 0 0
$$381$$ −12.4924 −0.640006
$$382$$ 0 0
$$383$$ −28.4924 −1.45589 −0.727947 0.685633i $$-0.759526\pi$$
−0.727947 + 0.685633i $$0.759526\pi$$
$$384$$ 0 0
$$385$$ 3.12311 0.159168
$$386$$ 0 0
$$387$$ −6.24621 −0.317513
$$388$$ 0 0
$$389$$ −3.75379 −0.190325 −0.0951623 0.995462i $$-0.530337\pi$$
−0.0951623 + 0.995462i $$0.530337\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 18.2462 0.920400
$$394$$ 0 0
$$395$$ −14.2462 −0.716805
$$396$$ 0 0
$$397$$ −10.8769 −0.545896 −0.272948 0.962029i $$-0.587999\pi$$
−0.272948 + 0.962029i $$0.587999\pi$$
$$398$$ 0 0
$$399$$ 7.12311 0.356601
$$400$$ 0 0
$$401$$ 26.9848 1.34756 0.673779 0.738933i $$-0.264670\pi$$
0.673779 + 0.738933i $$0.264670\pi$$
$$402$$ 0 0
$$403$$ −16.0000 −0.797017
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 6.24621 0.309613
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 21.1231 1.04193
$$412$$ 0 0
$$413$$ −4.00000 −0.196827
$$414$$ 0 0
$$415$$ 4.00000 0.196352
$$416$$ 0 0
$$417$$ −0.876894 −0.0429417
$$418$$ 0 0
$$419$$ −34.2462 −1.67304 −0.836518 0.547939i $$-0.815413\pi$$
−0.836518 + 0.547939i $$0.815413\pi$$
$$420$$ 0 0
$$421$$ 26.4924 1.29116 0.645581 0.763692i $$-0.276615\pi$$
0.645581 + 0.763692i $$0.276615\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 8.24621 0.399062
$$428$$ 0 0
$$429$$ −16.0000 −0.772487
$$430$$ 0 0
$$431$$ 7.61553 0.366827 0.183414 0.983036i $$-0.441285\pi$$
0.183414 + 0.983036i $$0.441285\pi$$
$$432$$ 0 0
$$433$$ 13.1231 0.630656 0.315328 0.948983i $$-0.397885\pi$$
0.315328 + 0.948983i $$0.397885\pi$$
$$434$$ 0 0
$$435$$ 2.00000 0.0958927
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −7.61553 −0.363469 −0.181735 0.983348i $$-0.558171\pi$$
−0.181735 + 0.983348i $$0.558171\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 26.7386 1.27039 0.635195 0.772351i $$-0.280920\pi$$
0.635195 + 0.772351i $$0.280920\pi$$
$$444$$ 0 0
$$445$$ −16.2462 −0.770144
$$446$$ 0 0
$$447$$ 4.24621 0.200839
$$448$$ 0 0
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ −6.24621 −0.294123
$$452$$ 0 0
$$453$$ −12.4924 −0.586945
$$454$$ 0 0
$$455$$ −5.12311 −0.240175
$$456$$ 0 0
$$457$$ −12.2462 −0.572854 −0.286427 0.958102i $$-0.592468\pi$$
−0.286427 + 0.958102i $$0.592468\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −18.4924 −0.861278 −0.430639 0.902524i $$-0.641712\pi$$
−0.430639 + 0.902524i $$0.641712\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 0 0
$$465$$ 3.12311 0.144831
$$466$$ 0 0
$$467$$ −28.9848 −1.34126 −0.670629 0.741793i $$-0.733976\pi$$
−0.670629 + 0.741793i $$0.733976\pi$$
$$468$$ 0 0
$$469$$ 14.2462 0.657829
$$470$$ 0 0
$$471$$ −1.12311 −0.0517500
$$472$$ 0 0
$$473$$ 19.5076 0.896959
$$474$$ 0 0
$$475$$ 7.12311 0.326831
$$476$$ 0 0
$$477$$ −11.3693 −0.520565
$$478$$ 0 0
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ −10.2462 −0.467187
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −13.1231 −0.595890
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 0 0
$$489$$ −6.24621 −0.282463
$$490$$ 0 0
$$491$$ 4.87689 0.220091 0.110046 0.993927i $$-0.464900\pi$$
0.110046 + 0.993927i $$0.464900\pi$$
$$492$$ 0 0
$$493$$ −4.00000 −0.180151
$$494$$ 0 0
$$495$$ 3.12311 0.140373
$$496$$ 0 0
$$497$$ 3.12311 0.140090
$$498$$ 0 0
$$499$$ 12.4924 0.559238 0.279619 0.960111i $$-0.409792\pi$$
0.279619 + 0.960111i $$0.409792\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ 0 0
$$503$$ −36.4924 −1.62712 −0.813558 0.581483i $$-0.802473\pi$$
−0.813558 + 0.581483i $$0.802473\pi$$
$$504$$ 0 0
$$505$$ −16.2462 −0.722947
$$506$$ 0 0
$$507$$ 13.2462 0.588285
$$508$$ 0 0
$$509$$ −28.2462 −1.25199 −0.625996 0.779827i $$-0.715307\pi$$
−0.625996 + 0.779827i $$0.715307\pi$$
$$510$$ 0 0
$$511$$ 6.87689 0.304216
$$512$$ 0 0
$$513$$ 7.12311 0.314493
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 16.2462 0.713130
$$520$$ 0 0
$$521$$ −10.4924 −0.459681 −0.229841 0.973228i $$-0.573821\pi$$
−0.229841 + 0.973228i $$0.573821\pi$$
$$522$$ 0 0
$$523$$ 14.7386 0.644475 0.322238 0.946659i $$-0.395565\pi$$
0.322238 + 0.946659i $$0.395565\pi$$
$$524$$ 0 0
$$525$$ 1.00000 0.0436436
$$526$$ 0 0
$$527$$ −6.24621 −0.272089
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 10.2462 0.443813
$$534$$ 0 0
$$535$$ 8.00000 0.345870
$$536$$ 0 0
$$537$$ −11.1231 −0.479997
$$538$$ 0 0
$$539$$ −3.12311 −0.134522
$$540$$ 0 0
$$541$$ 18.4924 0.795051 0.397526 0.917591i $$-0.369869\pi$$
0.397526 + 0.917591i $$0.369869\pi$$
$$542$$ 0 0
$$543$$ −10.4924 −0.450273
$$544$$ 0 0
$$545$$ 8.24621 0.353229
$$546$$ 0 0
$$547$$ 38.2462 1.63529 0.817645 0.575723i $$-0.195279\pi$$
0.817645 + 0.575723i $$0.195279\pi$$
$$548$$ 0 0
$$549$$ 8.24621 0.351940
$$550$$ 0 0
$$551$$ −14.2462 −0.606909
$$552$$ 0 0
$$553$$ 14.2462 0.605811
$$554$$ 0 0
$$555$$ 2.00000 0.0848953
$$556$$ 0 0
$$557$$ −6.87689 −0.291383 −0.145692 0.989330i $$-0.546541\pi$$
−0.145692 + 0.989330i $$0.546541\pi$$
$$558$$ 0 0
$$559$$ −32.0000 −1.35346
$$560$$ 0 0
$$561$$ −6.24621 −0.263715
$$562$$ 0 0
$$563$$ 16.4924 0.695073 0.347536 0.937667i $$-0.387018\pi$$
0.347536 + 0.937667i $$0.387018\pi$$
$$564$$ 0 0
$$565$$ −19.3693 −0.814873
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −36.4924 −1.52716 −0.763580 0.645713i $$-0.776560\pi$$
−0.763580 + 0.645713i $$0.776560\pi$$
$$572$$ 0 0
$$573$$ −3.12311 −0.130470
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 17.6155 0.733344 0.366672 0.930350i $$-0.380497\pi$$
0.366672 + 0.930350i $$0.380497\pi$$
$$578$$ 0 0
$$579$$ 18.0000 0.748054
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ 35.5076 1.47057
$$584$$ 0 0
$$585$$ −5.12311 −0.211814
$$586$$ 0 0
$$587$$ 0.492423 0.0203245 0.0101622 0.999948i $$-0.496765\pi$$
0.0101622 + 0.999948i $$0.496765\pi$$
$$588$$ 0 0
$$589$$ −22.2462 −0.916639
$$590$$ 0 0
$$591$$ −17.6155 −0.724606
$$592$$ 0 0
$$593$$ −2.49242 −0.102352 −0.0511758 0.998690i $$-0.516297\pi$$
−0.0511758 + 0.998690i $$0.516297\pi$$
$$594$$ 0 0
$$595$$ −2.00000 −0.0819920
$$596$$ 0 0
$$597$$ −3.12311 −0.127820
$$598$$ 0 0
$$599$$ −3.12311 −0.127607 −0.0638033 0.997962i $$-0.520323\pi$$
−0.0638033 + 0.997962i $$0.520323\pi$$
$$600$$ 0 0
$$601$$ 0.246211 0.0100432 0.00502158 0.999987i $$-0.498402\pi$$
0.00502158 + 0.999987i $$0.498402\pi$$
$$602$$ 0 0
$$603$$ 14.2462 0.580151
$$604$$ 0 0
$$605$$ 1.24621 0.0506657
$$606$$ 0 0
$$607$$ 30.2462 1.22766 0.613828 0.789440i $$-0.289629\pi$$
0.613828 + 0.789440i $$0.289629\pi$$
$$608$$ 0 0
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −42.9848 −1.73614 −0.868071 0.496440i $$-0.834640\pi$$
−0.868071 + 0.496440i $$0.834640\pi$$
$$614$$ 0 0
$$615$$ −2.00000 −0.0806478
$$616$$ 0 0
$$617$$ 11.3693 0.457711 0.228856 0.973460i $$-0.426502\pi$$
0.228856 + 0.973460i $$0.426502\pi$$
$$618$$ 0 0
$$619$$ −8.87689 −0.356793 −0.178396 0.983959i $$-0.557091\pi$$
−0.178396 + 0.983959i $$0.557091\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 16.2462 0.650891
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −22.2462 −0.888428
$$628$$ 0 0
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 14.2462 0.567133 0.283566 0.958953i $$-0.408482\pi$$
0.283566 + 0.958953i $$0.408482\pi$$
$$632$$ 0 0
$$633$$ 22.2462 0.884208
$$634$$ 0 0
$$635$$ 12.4924 0.495747
$$636$$ 0 0
$$637$$ 5.12311 0.202985
$$638$$ 0 0
$$639$$ 3.12311 0.123548
$$640$$ 0 0
$$641$$ 8.24621 0.325706 0.162853 0.986650i $$-0.447930\pi$$
0.162853 + 0.986650i $$0.447930\pi$$
$$642$$ 0 0
$$643$$ −5.75379 −0.226907 −0.113454 0.993543i $$-0.536191\pi$$
−0.113454 + 0.993543i $$0.536191\pi$$
$$644$$ 0 0
$$645$$ 6.24621 0.245944
$$646$$ 0 0
$$647$$ 24.9848 0.982256 0.491128 0.871088i $$-0.336585\pi$$
0.491128 + 0.871088i $$0.336585\pi$$
$$648$$ 0 0
$$649$$ 12.4924 0.490370
$$650$$ 0 0
$$651$$ −3.12311 −0.122404
$$652$$ 0 0
$$653$$ −33.6155 −1.31548 −0.657739 0.753246i $$-0.728487\pi$$
−0.657739 + 0.753246i $$0.728487\pi$$
$$654$$ 0 0
$$655$$ −18.2462 −0.712938
$$656$$ 0 0
$$657$$ 6.87689 0.268293
$$658$$ 0 0
$$659$$ 31.6155 1.23157 0.615783 0.787916i $$-0.288840\pi$$
0.615783 + 0.787916i $$0.288840\pi$$
$$660$$ 0 0
$$661$$ 48.2462 1.87656 0.938280 0.345876i $$-0.112418\pi$$
0.938280 + 0.345876i $$0.112418\pi$$
$$662$$ 0 0
$$663$$ 10.2462 0.397930
$$664$$ 0 0
$$665$$ −7.12311 −0.276222
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −1.75379 −0.0678054
$$670$$ 0 0
$$671$$ −25.7538 −0.994214
$$672$$ 0 0
$$673$$ −45.2311 −1.74353 −0.871765 0.489925i $$-0.837024\pi$$
−0.871765 + 0.489925i $$0.837024\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 24.2462 0.931858 0.465929 0.884822i $$-0.345720\pi$$
0.465929 + 0.884822i $$0.345720\pi$$
$$678$$ 0 0
$$679$$ 13.1231 0.503619
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ −30.2462 −1.15734 −0.578670 0.815562i $$-0.696428\pi$$
−0.578670 + 0.815562i $$0.696428\pi$$
$$684$$ 0 0
$$685$$ −21.1231 −0.807072
$$686$$ 0 0
$$687$$ −20.2462 −0.772441
$$688$$ 0 0
$$689$$ −58.2462 −2.21900
$$690$$ 0 0
$$691$$ 45.3693 1.72593 0.862965 0.505264i $$-0.168605\pi$$
0.862965 + 0.505264i $$0.168605\pi$$
$$692$$ 0 0
$$693$$ −3.12311 −0.118637
$$694$$ 0 0
$$695$$ 0.876894 0.0332625
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 0 0
$$699$$ 14.8769 0.562696
$$700$$ 0 0
$$701$$ −5.50758 −0.208018 −0.104009 0.994576i $$-0.533167\pi$$
−0.104009 + 0.994576i $$0.533167\pi$$
$$702$$ 0 0
$$703$$ −14.2462 −0.537306
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 16.2462 0.611002
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 14.2462 0.534275
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 16.0000 0.598366
$$716$$ 0 0
$$717$$ −4.87689 −0.182131
$$718$$ 0 0
$$719$$ 39.2311 1.46307 0.731536 0.681803i $$-0.238804\pi$$
0.731536 + 0.681803i $$0.238804\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 3.75379 0.139605
$$724$$ 0 0
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ −18.7386 −0.694977 −0.347489 0.937684i $$-0.612966\pi$$
−0.347489 + 0.937684i $$0.612966\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −12.4924 −0.462049
$$732$$ 0 0
$$733$$ −6.38447 −0.235816 −0.117908 0.993025i $$-0.537619\pi$$
−0.117908 + 0.993025i $$0.537619\pi$$
$$734$$ 0 0
$$735$$ −1.00000 −0.0368856
$$736$$ 0 0
$$737$$ −44.4924 −1.63890
$$738$$ 0 0
$$739$$ −40.0000 −1.47142 −0.735712 0.677295i $$-0.763152\pi$$
−0.735712 + 0.677295i $$0.763152\pi$$
$$740$$ 0 0
$$741$$ 36.4924 1.34058
$$742$$ 0 0
$$743$$ −34.7386 −1.27444 −0.637218 0.770683i $$-0.719915\pi$$
−0.637218 + 0.770683i $$0.719915\pi$$
$$744$$ 0 0
$$745$$ −4.24621 −0.155569
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ −6.24621 −0.227927 −0.113964 0.993485i $$-0.536355\pi$$
−0.113964 + 0.993485i $$0.536355\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ 12.4924 0.454646
$$756$$ 0 0
$$757$$ 52.2462 1.89892 0.949460 0.313887i $$-0.101631\pi$$
0.949460 + 0.313887i $$0.101631\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 20.7386 0.751775 0.375887 0.926665i $$-0.377338\pi$$
0.375887 + 0.926665i $$0.377338\pi$$
$$762$$ 0 0
$$763$$ −8.24621 −0.298533
$$764$$ 0 0
$$765$$ −2.00000 −0.0723102
$$766$$ 0 0
$$767$$ −20.4924 −0.739938
$$768$$ 0 0
$$769$$ −15.7538 −0.568096 −0.284048 0.958810i $$-0.591678\pi$$
−0.284048 + 0.958810i $$0.591678\pi$$
$$770$$ 0 0
$$771$$ 3.75379 0.135189
$$772$$ 0 0
$$773$$ 0.246211 0.00885560 0.00442780 0.999990i $$-0.498591\pi$$
0.00442780 + 0.999990i $$0.498591\pi$$
$$774$$ 0 0
$$775$$ −3.12311 −0.112185
$$776$$ 0 0
$$777$$ −2.00000 −0.0717496
$$778$$ 0 0
$$779$$ 14.2462 0.510423
$$780$$ 0 0
$$781$$ −9.75379 −0.349018
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ 1.12311 0.0400854
$$786$$ 0 0
$$787$$ −51.2311 −1.82619 −0.913095 0.407747i $$-0.866315\pi$$
−0.913095 + 0.407747i $$0.866315\pi$$
$$788$$ 0 0
$$789$$ −1.75379 −0.0624365
$$790$$ 0 0
$$791$$ 19.3693 0.688694
$$792$$ 0 0
$$793$$ 42.2462 1.50021
$$794$$ 0 0
$$795$$ 11.3693 0.403228
$$796$$ 0 0
$$797$$ 12.7386 0.451226 0.225613 0.974217i $$-0.427562\pi$$
0.225613 + 0.974217i $$0.427562\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 16.2462 0.574032
$$802$$ 0 0
$$803$$ −21.4773 −0.757916
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0.246211 0.00866705
$$808$$ 0 0
$$809$$ 3.75379 0.131976 0.0659881 0.997820i $$-0.478980\pi$$
0.0659881 + 0.997820i $$0.478980\pi$$
$$810$$ 0 0
$$811$$ −12.3845 −0.434878 −0.217439 0.976074i $$-0.569770\pi$$
−0.217439 + 0.976074i $$0.569770\pi$$
$$812$$ 0 0
$$813$$ 23.6155 0.828233
$$814$$ 0 0
$$815$$ 6.24621 0.218795
$$816$$ 0 0
$$817$$ −44.4924 −1.55659
$$818$$ 0 0
$$819$$ 5.12311 0.179016
$$820$$ 0 0
$$821$$ −10.9848 −0.383374 −0.191687 0.981456i $$-0.561396\pi$$
−0.191687 + 0.981456i $$0.561396\pi$$
$$822$$ 0 0
$$823$$ 20.4924 0.714321 0.357160 0.934043i $$-0.383745\pi$$
0.357160 + 0.934043i $$0.383745\pi$$
$$824$$ 0 0
$$825$$ −3.12311 −0.108733
$$826$$ 0 0
$$827$$ −16.9848 −0.590621 −0.295310 0.955401i $$-0.595423\pi$$
−0.295310 + 0.955401i $$0.595423\pi$$
$$828$$ 0 0
$$829$$ −26.4924 −0.920120 −0.460060 0.887888i $$-0.652172\pi$$
−0.460060 + 0.887888i $$0.652172\pi$$
$$830$$ 0 0
$$831$$ 18.4924 0.641495
$$832$$ 0 0
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ 16.0000 0.553703
$$836$$ 0 0
$$837$$ −3.12311 −0.107950
$$838$$ 0 0
$$839$$ 10.7386 0.370739 0.185369 0.982669i $$-0.440652\pi$$
0.185369 + 0.982669i $$0.440652\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −24.7386 −0.852044
$$844$$ 0 0
$$845$$ −13.2462 −0.455684
$$846$$ 0 0
$$847$$ −1.24621 −0.0428203
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 41.6155 1.42489 0.712444 0.701729i $$-0.247588\pi$$
0.712444 + 0.701729i $$0.247588\pi$$
$$854$$ 0 0
$$855$$ −7.12311 −0.243605
$$856$$ 0 0
$$857$$ −31.7538 −1.08469 −0.542344 0.840156i $$-0.682463\pi$$
−0.542344 + 0.840156i $$0.682463\pi$$
$$858$$ 0 0
$$859$$ −19.6155 −0.669273 −0.334637 0.942347i $$-0.608614\pi$$
−0.334637 + 0.942347i $$0.608614\pi$$
$$860$$ 0 0
$$861$$ 2.00000 0.0681598
$$862$$ 0 0
$$863$$ 40.0000 1.36162 0.680808 0.732462i $$-0.261629\pi$$
0.680808 + 0.732462i $$0.261629\pi$$
$$864$$ 0 0
$$865$$ −16.2462 −0.552388
$$866$$ 0 0
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ −44.4924 −1.50930
$$870$$ 0 0
$$871$$ 72.9848 2.47300
$$872$$ 0 0
$$873$$ 13.1231 0.444150
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ −22.4924 −0.759515 −0.379758 0.925086i $$-0.623993\pi$$
−0.379758 + 0.925086i $$0.623993\pi$$
$$878$$ 0 0
$$879$$ 0.246211 0.00830450
$$880$$ 0 0
$$881$$ −23.7538 −0.800285 −0.400143 0.916453i $$-0.631039\pi$$
−0.400143 + 0.916453i $$0.631039\pi$$
$$882$$ 0 0
$$883$$ 1.75379 0.0590197 0.0295098 0.999564i $$-0.490605\pi$$
0.0295098 + 0.999564i $$0.490605\pi$$
$$884$$ 0 0
$$885$$ 4.00000 0.134459
$$886$$ 0 0
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ −12.4924 −0.418982
$$890$$ 0 0
$$891$$ −3.12311 −0.104628
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 11.1231 0.371804
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 6.24621 0.208323
$$900$$ 0 0
$$901$$ −22.7386 −0.757534
$$902$$ 0 0
$$903$$ −6.24621 −0.207861
$$904$$ 0 0
$$905$$ 10.4924 0.348780
$$906$$ 0 0
$$907$$ 10.7386 0.356570 0.178285 0.983979i $$-0.442945\pi$$
0.178285 + 0.983979i $$0.442945\pi$$
$$908$$ 0 0
$$909$$ 16.2462 0.538853
$$910$$ 0 0
$$911$$ −39.6155 −1.31252 −0.656261 0.754534i $$-0.727863\pi$$
−0.656261 + 0.754534i $$0.727863\pi$$
$$912$$ 0 0
$$913$$ 12.4924 0.413439
$$914$$ 0 0
$$915$$ −8.24621 −0.272611
$$916$$ 0 0
$$917$$ 18.2462 0.602543
$$918$$ 0 0
$$919$$ −1.75379 −0.0578522 −0.0289261 0.999582i $$-0.509209\pi$$
−0.0289261 + 0.999582i $$0.509209\pi$$
$$920$$ 0 0
$$921$$ −26.2462 −0.864842
$$922$$ 0 0
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −28.2462 −0.926728 −0.463364 0.886168i $$-0.653358\pi$$
−0.463364 + 0.886168i $$0.653358\pi$$
$$930$$ 0 0
$$931$$ 7.12311 0.233450
$$932$$ 0 0
$$933$$ −16.0000 −0.523816
$$934$$ 0 0
$$935$$ 6.24621 0.204273
$$936$$ 0 0
$$937$$ −34.1080 −1.11426 −0.557129 0.830426i $$-0.688097\pi$$
−0.557129 + 0.830426i $$0.688097\pi$$
$$938$$ 0 0
$$939$$ 8.63068 0.281652
$$940$$ 0 0
$$941$$ −22.9848 −0.749285 −0.374642 0.927169i $$-0.622234\pi$$
−0.374642 + 0.927169i $$0.622234\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −1.00000 −0.0325300
$$946$$ 0 0
$$947$$ 31.2311 1.01487 0.507436 0.861689i $$-0.330593\pi$$
0.507436 + 0.861689i $$0.330593\pi$$
$$948$$ 0 0
$$949$$ 35.2311 1.14365
$$950$$ 0 0
$$951$$ −6.87689 −0.222999
$$952$$ 0 0
$$953$$ 27.3693 0.886579 0.443290 0.896378i $$-0.353811\pi$$
0.443290 + 0.896378i $$0.353811\pi$$
$$954$$ 0 0
$$955$$ 3.12311 0.101061
$$956$$ 0 0
$$957$$ 6.24621 0.201911
$$958$$ 0 0
$$959$$ 21.1231 0.682101
$$960$$ 0 0
$$961$$ −21.2462 −0.685362
$$962$$ 0 0
$$963$$ −8.00000 −0.257796
$$964$$ 0 0
$$965$$ −18.0000 −0.579441
$$966$$ 0 0
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ 0 0
$$969$$ 14.2462 0.457654
$$970$$ 0 0
$$971$$ 44.0000 1.41203 0.706014 0.708198i $$-0.250492\pi$$
0.706014 + 0.708198i $$0.250492\pi$$
$$972$$ 0 0
$$973$$ −0.876894 −0.0281119
$$974$$ 0 0
$$975$$ 5.12311 0.164071
$$976$$ 0 0
$$977$$ 35.3693 1.13156 0.565782 0.824555i $$-0.308574\pi$$
0.565782 + 0.824555i $$0.308574\pi$$
$$978$$ 0 0
$$979$$ −50.7386 −1.62161
$$980$$ 0 0
$$981$$ −8.24621 −0.263281
$$982$$ 0 0
$$983$$ −44.4924 −1.41909 −0.709544 0.704661i $$-0.751099\pi$$
−0.709544 + 0.704661i $$0.751099\pi$$
$$984$$ 0 0
$$985$$ 17.6155 0.561277
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −58.7386 −1.86589 −0.932947 0.360013i $$-0.882772\pi$$
−0.932947 + 0.360013i $$0.882772\pi$$
$$992$$ 0 0
$$993$$ −14.2462 −0.452090
$$994$$ 0 0
$$995$$ 3.12311 0.0990091
$$996$$ 0 0
$$997$$ −19.8617 −0.629028 −0.314514 0.949253i $$-0.601841\pi$$
−0.314514 + 0.949253i $$0.601841\pi$$
$$998$$ 0 0
$$999$$ −2.00000 −0.0632772
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 3360.2.a.bg.1.1 yes 2
4.3 odd 2 3360.2.a.ba.1.2 2
8.3 odd 2 6720.2.a.cw.1.1 2
8.5 even 2 6720.2.a.ct.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.ba.1.2 2 4.3 odd 2
3360.2.a.bg.1.1 yes 2 1.1 even 1 trivial
6720.2.a.ct.1.2 2 8.5 even 2
6720.2.a.cw.1.1 2 8.3 odd 2