# Properties

 Label 6720.2.a.cp.1.2 Level $6720$ Weight $2$ Character 6720.1 Self dual yes Analytic conductor $53.659$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6720.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$53.6594701583$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 3360) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 6720.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.47214 q^{13} +1.00000 q^{15} -4.47214 q^{17} -1.00000 q^{21} +6.47214 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.47214 q^{29} -6.47214 q^{31} -1.00000 q^{35} -8.47214 q^{37} -4.47214 q^{39} +10.9443 q^{41} -10.4721 q^{43} -1.00000 q^{45} -2.47214 q^{47} +1.00000 q^{49} +4.47214 q^{51} -2.00000 q^{53} +4.00000 q^{59} -12.4721 q^{61} +1.00000 q^{63} -4.47214 q^{65} -2.47214 q^{67} -6.47214 q^{69} +15.4164 q^{71} +8.47214 q^{73} -1.00000 q^{75} +12.9443 q^{79} +1.00000 q^{81} +16.9443 q^{83} +4.47214 q^{85} +8.47214 q^{87} -6.94427 q^{89} +4.47214 q^{91} +6.47214 q^{93} -4.47214 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} - 2 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 4 q^{31} - 2 q^{35} - 8 q^{37} + 4 q^{41} - 12 q^{43} - 2 q^{45} + 4 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{59} - 16 q^{61} + 2 q^{63} + 4 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} + 16 q^{83} + 8 q^{87} + 4 q^{89} + 4 q^{93}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^15 - 2 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 8 * q^29 - 4 * q^31 - 2 * q^35 - 8 * q^37 + 4 * q^41 - 12 * q^43 - 2 * q^45 + 4 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^59 - 16 * q^61 + 2 * q^63 + 4 * q^67 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 + 8 * q^79 + 2 * q^81 + 16 * q^83 + 8 * q^87 + 4 * q^89 + 4 * q^93

## 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
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 4.47214 1.24035 0.620174 0.784465i $$-0.287062\pi$$
0.620174 + 0.784465i $$0.287062\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −4.47214 −1.08465 −0.542326 0.840168i $$-0.682456\pi$$
−0.542326 + 0.840168i $$0.682456\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 6.47214 1.34953 0.674767 0.738031i $$-0.264244\pi$$
0.674767 + 0.738031i $$0.264244\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −8.47214 −1.57324 −0.786618 0.617440i $$-0.788170\pi$$
−0.786618 + 0.617440i $$0.788170\pi$$
$$30$$ 0 0
$$31$$ −6.47214 −1.16243 −0.581215 0.813750i $$-0.697422\pi$$
−0.581215 + 0.813750i $$0.697422\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ −8.47214 −1.39281 −0.696405 0.717649i $$-0.745218\pi$$
−0.696405 + 0.717649i $$0.745218\pi$$
$$38$$ 0 0
$$39$$ −4.47214 −0.716115
$$40$$ 0 0
$$41$$ 10.9443 1.70921 0.854604 0.519280i $$-0.173800\pi$$
0.854604 + 0.519280i $$0.173800\pi$$
$$42$$ 0 0
$$43$$ −10.4721 −1.59699 −0.798493 0.602004i $$-0.794369\pi$$
−0.798493 + 0.602004i $$0.794369\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −2.47214 −0.360598 −0.180299 0.983612i $$-0.557707\pi$$
−0.180299 + 0.983612i $$0.557707\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 4.47214 0.626224
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −12.4721 −1.59689 −0.798447 0.602066i $$-0.794345\pi$$
−0.798447 + 0.602066i $$0.794345\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −4.47214 −0.554700
$$66$$ 0 0
$$67$$ −2.47214 −0.302019 −0.151010 0.988532i $$-0.548252\pi$$
−0.151010 + 0.988532i $$0.548252\pi$$
$$68$$ 0 0
$$69$$ −6.47214 −0.779154
$$70$$ 0 0
$$71$$ 15.4164 1.82959 0.914796 0.403917i $$-0.132352\pi$$
0.914796 + 0.403917i $$0.132352\pi$$
$$72$$ 0 0
$$73$$ 8.47214 0.991589 0.495794 0.868440i $$-0.334877\pi$$
0.495794 + 0.868440i $$0.334877\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.9443 1.45634 0.728172 0.685394i $$-0.240370\pi$$
0.728172 + 0.685394i $$0.240370\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 16.9443 1.85988 0.929938 0.367717i $$-0.119860\pi$$
0.929938 + 0.367717i $$0.119860\pi$$
$$84$$ 0 0
$$85$$ 4.47214 0.485071
$$86$$ 0 0
$$87$$ 8.47214 0.908308
$$88$$ 0 0
$$89$$ −6.94427 −0.736091 −0.368046 0.929808i $$-0.619973\pi$$
−0.368046 + 0.929808i $$0.619973\pi$$
$$90$$ 0 0
$$91$$ 4.47214 0.468807
$$92$$ 0 0
$$93$$ 6.47214 0.671129
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −4.47214 −0.454077 −0.227038 0.973886i $$-0.572904\pi$$
−0.227038 + 0.973886i $$0.572904\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.94427 0.690981 0.345490 0.938422i $$-0.387713\pi$$
0.345490 + 0.938422i $$0.387713\pi$$
$$102$$ 0 0
$$103$$ −17.8885 −1.76261 −0.881305 0.472547i $$-0.843335\pi$$
−0.881305 + 0.472547i $$0.843335\pi$$
$$104$$ 0 0
$$105$$ 1.00000 0.0975900
$$106$$ 0 0
$$107$$ 6.47214 0.625685 0.312842 0.949805i $$-0.398719\pi$$
0.312842 + 0.949805i $$0.398719\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 8.47214 0.804140
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ −6.47214 −0.603530
$$116$$ 0 0
$$117$$ 4.47214 0.413449
$$118$$ 0 0
$$119$$ −4.47214 −0.409960
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −10.9443 −0.986812
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 10.4721 0.922020
$$130$$ 0 0
$$131$$ −0.944272 −0.0825014 −0.0412507 0.999149i $$-0.513134\pi$$
−0.0412507 + 0.999149i $$0.513134\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 10.9443 0.935032 0.467516 0.883985i $$-0.345149\pi$$
0.467516 + 0.883985i $$0.345149\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 2.47214 0.208191
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 8.47214 0.703573
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ −3.52786 −0.289014 −0.144507 0.989504i $$-0.546160\pi$$
−0.144507 + 0.989504i $$0.546160\pi$$
$$150$$ 0 0
$$151$$ −20.9443 −1.70442 −0.852210 0.523199i $$-0.824738\pi$$
−0.852210 + 0.523199i $$0.824738\pi$$
$$152$$ 0 0
$$153$$ −4.47214 −0.361551
$$154$$ 0 0
$$155$$ 6.47214 0.519854
$$156$$ 0 0
$$157$$ −16.4721 −1.31462 −0.657310 0.753620i $$-0.728306\pi$$
−0.657310 + 0.753620i $$0.728306\pi$$
$$158$$ 0 0
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 6.47214 0.510076
$$162$$ 0 0
$$163$$ 2.47214 0.193633 0.0968163 0.995302i $$-0.469134\pi$$
0.0968163 + 0.995302i $$0.469134\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −10.4721 −0.810358 −0.405179 0.914237i $$-0.632791\pi$$
−0.405179 + 0.914237i $$0.632791\pi$$
$$168$$ 0 0
$$169$$ 7.00000 0.538462
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 0 0
$$181$$ 13.4164 0.997234 0.498617 0.866822i $$-0.333841\pi$$
0.498617 + 0.866822i $$0.333841\pi$$
$$182$$ 0 0
$$183$$ 12.4721 0.921967
$$184$$ 0 0
$$185$$ 8.47214 0.622884
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −23.4164 −1.69435 −0.847176 0.531313i $$-0.821699\pi$$
−0.847176 + 0.531313i $$0.821699\pi$$
$$192$$ 0 0
$$193$$ −18.9443 −1.36364 −0.681819 0.731521i $$-0.738811\pi$$
−0.681819 + 0.731521i $$0.738811\pi$$
$$194$$ 0 0
$$195$$ 4.47214 0.320256
$$196$$ 0 0
$$197$$ −14.9443 −1.06474 −0.532368 0.846513i $$-0.678698\pi$$
−0.532368 + 0.846513i $$0.678698\pi$$
$$198$$ 0 0
$$199$$ 11.4164 0.809288 0.404644 0.914474i $$-0.367395\pi$$
0.404644 + 0.914474i $$0.367395\pi$$
$$200$$ 0 0
$$201$$ 2.47214 0.174371
$$202$$ 0 0
$$203$$ −8.47214 −0.594627
$$204$$ 0 0
$$205$$ −10.9443 −0.764381
$$206$$ 0 0
$$207$$ 6.47214 0.449845
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 0 0
$$213$$ −15.4164 −1.05631
$$214$$ 0 0
$$215$$ 10.4721 0.714194
$$216$$ 0 0
$$217$$ −6.47214 −0.439357
$$218$$ 0 0
$$219$$ −8.47214 −0.572494
$$220$$ 0 0
$$221$$ −20.0000 −1.34535
$$222$$ 0 0
$$223$$ 25.8885 1.73363 0.866813 0.498634i $$-0.166165\pi$$
0.866813 + 0.498634i $$0.166165\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −0.944272 −0.0626735 −0.0313368 0.999509i $$-0.509976\pi$$
−0.0313368 + 0.999509i $$0.509976\pi$$
$$228$$ 0 0
$$229$$ −20.4721 −1.35284 −0.676418 0.736518i $$-0.736469\pi$$
−0.676418 + 0.736518i $$0.736469\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2.94427 0.192886 0.0964428 0.995339i $$-0.469254\pi$$
0.0964428 + 0.995339i $$0.469254\pi$$
$$234$$ 0 0
$$235$$ 2.47214 0.161264
$$236$$ 0 0
$$237$$ −12.9443 −0.840821
$$238$$ 0 0
$$239$$ −20.3607 −1.31702 −0.658511 0.752571i $$-0.728814\pi$$
−0.658511 + 0.752571i $$0.728814\pi$$
$$240$$ 0 0
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −16.9443 −1.07380
$$250$$ 0 0
$$251$$ 7.05573 0.445354 0.222677 0.974892i $$-0.428521\pi$$
0.222677 + 0.974892i $$0.428521\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −4.47214 −0.280056
$$256$$ 0 0
$$257$$ 8.47214 0.528477 0.264239 0.964457i $$-0.414879\pi$$
0.264239 + 0.964457i $$0.414879\pi$$
$$258$$ 0 0
$$259$$ −8.47214 −0.526433
$$260$$ 0 0
$$261$$ −8.47214 −0.524412
$$262$$ 0 0
$$263$$ −9.52786 −0.587513 −0.293757 0.955880i $$-0.594906\pi$$
−0.293757 + 0.955880i $$0.594906\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ 0 0
$$267$$ 6.94427 0.424983
$$268$$ 0 0
$$269$$ −22.0000 −1.34136 −0.670682 0.741745i $$-0.733998\pi$$
−0.670682 + 0.741745i $$0.733998\pi$$
$$270$$ 0 0
$$271$$ 24.3607 1.47981 0.739903 0.672714i $$-0.234871\pi$$
0.739903 + 0.672714i $$0.234871\pi$$
$$272$$ 0 0
$$273$$ −4.47214 −0.270666
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 4.47214 0.268705 0.134352 0.990934i $$-0.457105\pi$$
0.134352 + 0.990934i $$0.457105\pi$$
$$278$$ 0 0
$$279$$ −6.47214 −0.387477
$$280$$ 0 0
$$281$$ −31.8885 −1.90231 −0.951156 0.308712i $$-0.900102\pi$$
−0.951156 + 0.308712i $$0.900102\pi$$
$$282$$ 0 0
$$283$$ 5.88854 0.350038 0.175019 0.984565i $$-0.444001\pi$$
0.175019 + 0.984565i $$0.444001\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10.9443 0.646020
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ 4.47214 0.262161
$$292$$ 0 0
$$293$$ 5.05573 0.295359 0.147679 0.989035i $$-0.452820\pi$$
0.147679 + 0.989035i $$0.452820\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 28.9443 1.67389
$$300$$ 0 0
$$301$$ −10.4721 −0.603604
$$302$$ 0 0
$$303$$ −6.94427 −0.398938
$$304$$ 0 0
$$305$$ 12.4721 0.714152
$$306$$ 0 0
$$307$$ 24.9443 1.42364 0.711822 0.702360i $$-0.247870\pi$$
0.711822 + 0.702360i $$0.247870\pi$$
$$308$$ 0 0
$$309$$ 17.8885 1.01764
$$310$$ 0 0
$$311$$ −33.8885 −1.92164 −0.960822 0.277168i $$-0.910604\pi$$
−0.960822 + 0.277168i $$0.910604\pi$$
$$312$$ 0 0
$$313$$ 0.472136 0.0266867 0.0133434 0.999911i $$-0.495753\pi$$
0.0133434 + 0.999911i $$0.495753\pi$$
$$314$$ 0 0
$$315$$ −1.00000 −0.0563436
$$316$$ 0 0
$$317$$ 15.8885 0.892390 0.446195 0.894936i $$-0.352779\pi$$
0.446195 + 0.894936i $$0.352779\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −6.47214 −0.361239
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 4.47214 0.248069
$$326$$ 0 0
$$327$$ 14.0000 0.774202
$$328$$ 0 0
$$329$$ −2.47214 −0.136293
$$330$$ 0 0
$$331$$ −32.0000 −1.75888 −0.879440 0.476011i $$-0.842082\pi$$
−0.879440 + 0.476011i $$0.842082\pi$$
$$332$$ 0 0
$$333$$ −8.47214 −0.464270
$$334$$ 0 0
$$335$$ 2.47214 0.135067
$$336$$ 0 0
$$337$$ 6.94427 0.378279 0.189139 0.981950i $$-0.439430\pi$$
0.189139 + 0.981950i $$0.439430\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 6.47214 0.348448
$$346$$ 0 0
$$347$$ 9.52786 0.511483 0.255741 0.966745i $$-0.417680\pi$$
0.255741 + 0.966745i $$0.417680\pi$$
$$348$$ 0 0
$$349$$ −15.5279 −0.831188 −0.415594 0.909550i $$-0.636426\pi$$
−0.415594 + 0.909550i $$0.636426\pi$$
$$350$$ 0 0
$$351$$ −4.47214 −0.238705
$$352$$ 0 0
$$353$$ −17.4164 −0.926982 −0.463491 0.886102i $$-0.653403\pi$$
−0.463491 + 0.886102i $$0.653403\pi$$
$$354$$ 0 0
$$355$$ −15.4164 −0.818218
$$356$$ 0 0
$$357$$ 4.47214 0.236691
$$358$$ 0 0
$$359$$ 15.4164 0.813647 0.406823 0.913507i $$-0.366636\pi$$
0.406823 + 0.913507i $$0.366636\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ −8.47214 −0.443452
$$366$$ 0 0
$$367$$ −20.9443 −1.09328 −0.546641 0.837367i $$-0.684094\pi$$
−0.546641 + 0.837367i $$0.684094\pi$$
$$368$$ 0 0
$$369$$ 10.9443 0.569736
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ 2.58359 0.133773 0.0668867 0.997761i $$-0.478693\pi$$
0.0668867 + 0.997761i $$0.478693\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −37.8885 −1.95136
$$378$$ 0 0
$$379$$ −19.0557 −0.978827 −0.489414 0.872052i $$-0.662789\pi$$
−0.489414 + 0.872052i $$0.662789\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 23.4164 1.19652 0.598261 0.801301i $$-0.295858\pi$$
0.598261 + 0.801301i $$0.295858\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −10.4721 −0.532329
$$388$$ 0 0
$$389$$ −24.4721 −1.24079 −0.620393 0.784291i $$-0.713027\pi$$
−0.620393 + 0.784291i $$0.713027\pi$$
$$390$$ 0 0
$$391$$ −28.9443 −1.46377
$$392$$ 0 0
$$393$$ 0.944272 0.0476322
$$394$$ 0 0
$$395$$ −12.9443 −0.651297
$$396$$ 0 0
$$397$$ 1.41641 0.0710875 0.0355437 0.999368i $$-0.488684\pi$$
0.0355437 + 0.999368i $$0.488684\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −26.9443 −1.34553 −0.672766 0.739855i $$-0.734894\pi$$
−0.672766 + 0.739855i $$0.734894\pi$$
$$402$$ 0 0
$$403$$ −28.9443 −1.44182
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 5.05573 0.249990 0.124995 0.992157i $$-0.460109\pi$$
0.124995 + 0.992157i $$0.460109\pi$$
$$410$$ 0 0
$$411$$ −10.9443 −0.539841
$$412$$ 0 0
$$413$$ 4.00000 0.196827
$$414$$ 0 0
$$415$$ −16.9443 −0.831762
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 15.0557 0.735520 0.367760 0.929921i $$-0.380125\pi$$
0.367760 + 0.929921i $$0.380125\pi$$
$$420$$ 0 0
$$421$$ −23.8885 −1.16426 −0.582128 0.813097i $$-0.697780\pi$$
−0.582128 + 0.813097i $$0.697780\pi$$
$$422$$ 0 0
$$423$$ −2.47214 −0.120199
$$424$$ 0 0
$$425$$ −4.47214 −0.216930
$$426$$ 0 0
$$427$$ −12.4721 −0.603569
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.4721 −1.66046 −0.830232 0.557418i $$-0.811792\pi$$
−0.830232 + 0.557418i $$0.811792\pi$$
$$432$$ 0 0
$$433$$ 18.3607 0.882358 0.441179 0.897419i $$-0.354560\pi$$
0.441179 + 0.897419i $$0.354560\pi$$
$$434$$ 0 0
$$435$$ −8.47214 −0.406208
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 30.4721 1.45436 0.727178 0.686449i $$-0.240832\pi$$
0.727178 + 0.686449i $$0.240832\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 1.52786 0.0725910 0.0362955 0.999341i $$-0.488444\pi$$
0.0362955 + 0.999341i $$0.488444\pi$$
$$444$$ 0 0
$$445$$ 6.94427 0.329190
$$446$$ 0 0
$$447$$ 3.52786 0.166862
$$448$$ 0 0
$$449$$ 32.8328 1.54948 0.774738 0.632282i $$-0.217882\pi$$
0.774738 + 0.632282i $$0.217882\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 20.9443 0.984048
$$454$$ 0 0
$$455$$ −4.47214 −0.209657
$$456$$ 0 0
$$457$$ −34.9443 −1.63462 −0.817312 0.576195i $$-0.804537\pi$$
−0.817312 + 0.576195i $$0.804537\pi$$
$$458$$ 0 0
$$459$$ 4.47214 0.208741
$$460$$ 0 0
$$461$$ −28.8328 −1.34288 −0.671439 0.741060i $$-0.734323\pi$$
−0.671439 + 0.741060i $$0.734323\pi$$
$$462$$ 0 0
$$463$$ 3.05573 0.142012 0.0710059 0.997476i $$-0.477379\pi$$
0.0710059 + 0.997476i $$0.477379\pi$$
$$464$$ 0 0
$$465$$ −6.47214 −0.300138
$$466$$ 0 0
$$467$$ 29.8885 1.38308 0.691538 0.722340i $$-0.256933\pi$$
0.691538 + 0.722340i $$0.256933\pi$$
$$468$$ 0 0
$$469$$ −2.47214 −0.114153
$$470$$ 0 0
$$471$$ 16.4721 0.758996
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ 0 0
$$479$$ 9.88854 0.451819 0.225910 0.974148i $$-0.427465\pi$$
0.225910 + 0.974148i $$0.427465\pi$$
$$480$$ 0 0
$$481$$ −37.8885 −1.72757
$$482$$ 0 0
$$483$$ −6.47214 −0.294492
$$484$$ 0 0
$$485$$ 4.47214 0.203069
$$486$$ 0 0
$$487$$ −19.0557 −0.863497 −0.431749 0.901994i $$-0.642103\pi$$
−0.431749 + 0.901994i $$0.642103\pi$$
$$488$$ 0 0
$$489$$ −2.47214 −0.111794
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 37.8885 1.70641
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 15.4164 0.691520
$$498$$ 0 0
$$499$$ −11.0557 −0.494922 −0.247461 0.968898i $$-0.579596\pi$$
−0.247461 + 0.968898i $$0.579596\pi$$
$$500$$ 0 0
$$501$$ 10.4721 0.467861
$$502$$ 0 0
$$503$$ −21.5279 −0.959880 −0.479940 0.877301i $$-0.659342\pi$$
−0.479940 + 0.877301i $$0.659342\pi$$
$$504$$ 0 0
$$505$$ −6.94427 −0.309016
$$506$$ 0 0
$$507$$ −7.00000 −0.310881
$$508$$ 0 0
$$509$$ −22.0000 −0.975133 −0.487566 0.873086i $$-0.662115\pi$$
−0.487566 + 0.873086i $$0.662115\pi$$
$$510$$ 0 0
$$511$$ 8.47214 0.374785
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 17.8885 0.788263
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −34.0000 −1.48957 −0.744784 0.667306i $$-0.767447\pi$$
−0.744784 + 0.667306i $$0.767447\pi$$
$$522$$ 0 0
$$523$$ −24.9443 −1.09074 −0.545368 0.838196i $$-0.683610\pi$$
−0.545368 + 0.838196i $$0.683610\pi$$
$$524$$ 0 0
$$525$$ −1.00000 −0.0436436
$$526$$ 0 0
$$527$$ 28.9443 1.26083
$$528$$ 0 0
$$529$$ 18.8885 0.821241
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ 48.9443 2.12001
$$534$$ 0 0
$$535$$ −6.47214 −0.279815
$$536$$ 0 0
$$537$$ −16.0000 −0.690451
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.9443 0.642504 0.321252 0.946994i $$-0.395896\pi$$
0.321252 + 0.946994i $$0.395896\pi$$
$$542$$ 0 0
$$543$$ −13.4164 −0.575753
$$544$$ 0 0
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ 15.4164 0.659158 0.329579 0.944128i $$-0.393093\pi$$
0.329579 + 0.944128i $$0.393093\pi$$
$$548$$ 0 0
$$549$$ −12.4721 −0.532298
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 12.9443 0.550446
$$554$$ 0 0
$$555$$ −8.47214 −0.359622
$$556$$ 0 0
$$557$$ −10.0000 −0.423714 −0.211857 0.977301i $$-0.567951\pi$$
−0.211857 + 0.977301i $$0.567951\pi$$
$$558$$ 0 0
$$559$$ −46.8328 −1.98082
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 32.8328 1.37642 0.688212 0.725510i $$-0.258396\pi$$
0.688212 + 0.725510i $$0.258396\pi$$
$$570$$ 0 0
$$571$$ 4.94427 0.206911 0.103456 0.994634i $$-0.467010\pi$$
0.103456 + 0.994634i $$0.467010\pi$$
$$572$$ 0 0
$$573$$ 23.4164 0.978234
$$574$$ 0 0
$$575$$ 6.47214 0.269907
$$576$$ 0 0
$$577$$ 42.3607 1.76350 0.881749 0.471719i $$-0.156366\pi$$
0.881749 + 0.471719i $$0.156366\pi$$
$$578$$ 0 0
$$579$$ 18.9443 0.787297
$$580$$ 0 0
$$581$$ 16.9443 0.702967
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −4.47214 −0.184900
$$586$$ 0 0
$$587$$ −5.88854 −0.243046 −0.121523 0.992589i $$-0.538778\pi$$
−0.121523 + 0.992589i $$0.538778\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 14.9443 0.614725
$$592$$ 0 0
$$593$$ −12.4721 −0.512169 −0.256085 0.966654i $$-0.582433\pi$$
−0.256085 + 0.966654i $$0.582433\pi$$
$$594$$ 0 0
$$595$$ 4.47214 0.183340
$$596$$ 0 0
$$597$$ −11.4164 −0.467242
$$598$$ 0 0
$$599$$ 2.47214 0.101009 0.0505044 0.998724i $$-0.483917\pi$$
0.0505044 + 0.998724i $$0.483917\pi$$
$$600$$ 0 0
$$601$$ −9.05573 −0.369391 −0.184695 0.982796i $$-0.559130\pi$$
−0.184695 + 0.982796i $$0.559130\pi$$
$$602$$ 0 0
$$603$$ −2.47214 −0.100673
$$604$$ 0 0
$$605$$ 11.0000 0.447214
$$606$$ 0 0
$$607$$ −9.88854 −0.401364 −0.200682 0.979656i $$-0.564316\pi$$
−0.200682 + 0.979656i $$0.564316\pi$$
$$608$$ 0 0
$$609$$ 8.47214 0.343308
$$610$$ 0 0
$$611$$ −11.0557 −0.447267
$$612$$ 0 0
$$613$$ 23.5279 0.950281 0.475141 0.879910i $$-0.342397\pi$$
0.475141 + 0.879910i $$0.342397\pi$$
$$614$$ 0 0
$$615$$ 10.9443 0.441316
$$616$$ 0 0
$$617$$ −38.9443 −1.56784 −0.783919 0.620864i $$-0.786782\pi$$
−0.783919 + 0.620864i $$0.786782\pi$$
$$618$$ 0 0
$$619$$ −44.9443 −1.80646 −0.903231 0.429154i $$-0.858812\pi$$
−0.903231 + 0.429154i $$0.858812\pi$$
$$620$$ 0 0
$$621$$ −6.47214 −0.259718
$$622$$ 0 0
$$623$$ −6.94427 −0.278216
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 37.8885 1.51072
$$630$$ 0 0
$$631$$ 33.8885 1.34908 0.674541 0.738238i $$-0.264342\pi$$
0.674541 + 0.738238i $$0.264342\pi$$
$$632$$ 0 0
$$633$$ 8.00000 0.317971
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4.47214 0.177192
$$638$$ 0 0
$$639$$ 15.4164 0.609864
$$640$$ 0 0
$$641$$ 6.94427 0.274282 0.137141 0.990552i $$-0.456209\pi$$
0.137141 + 0.990552i $$0.456209\pi$$
$$642$$ 0 0
$$643$$ 12.0000 0.473234 0.236617 0.971603i $$-0.423961\pi$$
0.236617 + 0.971603i $$0.423961\pi$$
$$644$$ 0 0
$$645$$ −10.4721 −0.412340
$$646$$ 0 0
$$647$$ −31.4164 −1.23511 −0.617553 0.786529i $$-0.711876\pi$$
−0.617553 + 0.786529i $$0.711876\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 6.47214 0.253663
$$652$$ 0 0
$$653$$ 39.8885 1.56096 0.780480 0.625181i $$-0.214975\pi$$
0.780480 + 0.625181i $$0.214975\pi$$
$$654$$ 0 0
$$655$$ 0.944272 0.0368958
$$656$$ 0 0
$$657$$ 8.47214 0.330530
$$658$$ 0 0
$$659$$ −22.8328 −0.889440 −0.444720 0.895670i $$-0.646697\pi$$
−0.444720 + 0.895670i $$0.646697\pi$$
$$660$$ 0 0
$$661$$ 1.63932 0.0637622 0.0318811 0.999492i $$-0.489850\pi$$
0.0318811 + 0.999492i $$0.489850\pi$$
$$662$$ 0 0
$$663$$ 20.0000 0.776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −54.8328 −2.12314
$$668$$ 0 0
$$669$$ −25.8885 −1.00091
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −25.7771 −0.993634 −0.496817 0.867855i $$-0.665498\pi$$
−0.496817 + 0.867855i $$0.665498\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −23.8885 −0.918111 −0.459056 0.888408i $$-0.651812\pi$$
−0.459056 + 0.888408i $$0.651812\pi$$
$$678$$ 0 0
$$679$$ −4.47214 −0.171625
$$680$$ 0 0
$$681$$ 0.944272 0.0361846
$$682$$ 0 0
$$683$$ 32.3607 1.23825 0.619123 0.785294i $$-0.287488\pi$$
0.619123 + 0.785294i $$0.287488\pi$$
$$684$$ 0 0
$$685$$ −10.9443 −0.418159
$$686$$ 0 0
$$687$$ 20.4721 0.781061
$$688$$ 0 0
$$689$$ −8.94427 −0.340750
$$690$$ 0 0
$$691$$ 9.88854 0.376178 0.188089 0.982152i $$-0.439771\pi$$
0.188089 + 0.982152i $$0.439771\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −48.9443 −1.85390
$$698$$ 0 0
$$699$$ −2.94427 −0.111363
$$700$$ 0 0
$$701$$ 17.4164 0.657809 0.328904 0.944363i $$-0.393321\pi$$
0.328904 + 0.944363i $$0.393321\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −2.47214 −0.0931060
$$706$$ 0 0
$$707$$ 6.94427 0.261166
$$708$$ 0 0
$$709$$ 40.8328 1.53351 0.766754 0.641941i $$-0.221870\pi$$
0.766754 + 0.641941i $$0.221870\pi$$
$$710$$ 0 0
$$711$$ 12.9443 0.485448
$$712$$ 0 0
$$713$$ −41.8885 −1.56874
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 20.3607 0.760384
$$718$$ 0 0
$$719$$ 16.0000 0.596699 0.298350 0.954457i $$-0.403564\pi$$
0.298350 + 0.954457i $$0.403564\pi$$
$$720$$ 0 0
$$721$$ −17.8885 −0.666204
$$722$$ 0 0
$$723$$ 6.00000 0.223142
$$724$$ 0 0
$$725$$ −8.47214 −0.314647
$$726$$ 0 0
$$727$$ −11.0557 −0.410034 −0.205017 0.978758i $$-0.565725\pi$$
−0.205017 + 0.978758i $$0.565725\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 46.8328 1.73217
$$732$$ 0 0
$$733$$ 2.58359 0.0954272 0.0477136 0.998861i $$-0.484807\pi$$
0.0477136 + 0.998861i $$0.484807\pi$$
$$734$$ 0 0
$$735$$ 1.00000 0.0368856
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −19.4164 −0.712319 −0.356159 0.934425i $$-0.615914\pi$$
−0.356159 + 0.934425i $$0.615914\pi$$
$$744$$ 0 0
$$745$$ 3.52786 0.129251
$$746$$ 0 0
$$747$$ 16.9443 0.619958
$$748$$ 0 0
$$749$$ 6.47214 0.236487
$$750$$ 0 0
$$751$$ 28.9443 1.05619 0.528096 0.849185i $$-0.322906\pi$$
0.528096 + 0.849185i $$0.322906\pi$$
$$752$$ 0 0
$$753$$ −7.05573 −0.257125
$$754$$ 0 0
$$755$$ 20.9443 0.762240
$$756$$ 0 0
$$757$$ 20.4721 0.744072 0.372036 0.928218i $$-0.378660\pi$$
0.372036 + 0.928218i $$0.378660\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −0.111456 −0.00404028 −0.00202014 0.999998i $$-0.500643\pi$$
−0.00202014 + 0.999998i $$0.500643\pi$$
$$762$$ 0 0
$$763$$ −14.0000 −0.506834
$$764$$ 0 0
$$765$$ 4.47214 0.161690
$$766$$ 0 0
$$767$$ 17.8885 0.645918
$$768$$ 0 0
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 0 0
$$771$$ −8.47214 −0.305117
$$772$$ 0 0
$$773$$ 21.0557 0.757322 0.378661 0.925535i $$-0.376385\pi$$
0.378661 + 0.925535i $$0.376385\pi$$
$$774$$ 0 0
$$775$$ −6.47214 −0.232486
$$776$$ 0 0
$$777$$ 8.47214 0.303936
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 8.47214 0.302769
$$784$$ 0 0
$$785$$ 16.4721 0.587916
$$786$$ 0 0
$$787$$ −23.7771 −0.847562 −0.423781 0.905765i $$-0.639297\pi$$
−0.423781 + 0.905765i $$0.639297\pi$$
$$788$$ 0 0
$$789$$ 9.52786 0.339201
$$790$$ 0 0
$$791$$ −2.00000 −0.0711118
$$792$$ 0 0
$$793$$ −55.7771 −1.98070
$$794$$ 0 0
$$795$$ −2.00000 −0.0709327
$$796$$ 0 0
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ 11.0557 0.391124
$$800$$ 0 0
$$801$$ −6.94427 −0.245364
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −6.47214 −0.228113
$$806$$ 0 0
$$807$$ 22.0000 0.774437
$$808$$ 0 0
$$809$$ −12.8328 −0.451178 −0.225589 0.974223i $$-0.572431\pi$$
−0.225589 + 0.974223i $$0.572431\pi$$
$$810$$ 0 0
$$811$$ −27.7771 −0.975385 −0.487693 0.873015i $$-0.662161\pi$$
−0.487693 + 0.873015i $$0.662161\pi$$
$$812$$ 0 0
$$813$$ −24.3607 −0.854366
$$814$$ 0 0
$$815$$ −2.47214 −0.0865951
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 4.47214 0.156269
$$820$$ 0 0
$$821$$ −31.3050 −1.09255 −0.546275 0.837606i $$-0.683955\pi$$
−0.546275 + 0.837606i $$0.683955\pi$$
$$822$$ 0 0
$$823$$ −3.05573 −0.106516 −0.0532580 0.998581i $$-0.516961\pi$$
−0.0532580 + 0.998581i $$0.516961\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 11.4164 0.396987 0.198494 0.980102i $$-0.436395\pi$$
0.198494 + 0.980102i $$0.436395\pi$$
$$828$$ 0 0
$$829$$ −33.4164 −1.16060 −0.580300 0.814403i $$-0.697065\pi$$
−0.580300 + 0.814403i $$0.697065\pi$$
$$830$$ 0 0
$$831$$ −4.47214 −0.155137
$$832$$ 0 0
$$833$$ −4.47214 −0.154950
$$834$$ 0 0
$$835$$ 10.4721 0.362403
$$836$$ 0 0
$$837$$ 6.47214 0.223710
$$838$$ 0 0
$$839$$ −49.8885 −1.72234 −0.861172 0.508314i $$-0.830269\pi$$
−0.861172 + 0.508314i $$0.830269\pi$$
$$840$$ 0 0
$$841$$ 42.7771 1.47507
$$842$$ 0 0
$$843$$ 31.8885 1.09830
$$844$$ 0 0
$$845$$ −7.00000 −0.240807
$$846$$ 0 0
$$847$$ −11.0000 −0.377964
$$848$$ 0 0
$$849$$ −5.88854 −0.202094
$$850$$ 0 0
$$851$$ −54.8328 −1.87964
$$852$$ 0 0
$$853$$ 31.5279 1.07949 0.539747 0.841827i $$-0.318520\pi$$
0.539747 + 0.841827i $$0.318520\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18.3607 0.627189 0.313594 0.949557i $$-0.398467\pi$$
0.313594 + 0.949557i $$0.398467\pi$$
$$858$$ 0 0
$$859$$ 4.94427 0.168696 0.0843482 0.996436i $$-0.473119\pi$$
0.0843482 + 0.996436i $$0.473119\pi$$
$$860$$ 0 0
$$861$$ −10.9443 −0.372980
$$862$$ 0 0
$$863$$ −48.3607 −1.64622 −0.823108 0.567884i $$-0.807762\pi$$
−0.823108 + 0.567884i $$0.807762\pi$$
$$864$$ 0 0
$$865$$ 6.00000 0.204006
$$866$$ 0 0
$$867$$ −3.00000 −0.101885
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −11.0557 −0.374609
$$872$$ 0 0
$$873$$ −4.47214 −0.151359
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ −47.3050 −1.59737 −0.798687 0.601746i $$-0.794472\pi$$
−0.798687 + 0.601746i $$0.794472\pi$$
$$878$$ 0 0
$$879$$ −5.05573 −0.170525
$$880$$ 0 0
$$881$$ −32.8328 −1.10617 −0.553083 0.833126i $$-0.686549\pi$$
−0.553083 + 0.833126i $$0.686549\pi$$
$$882$$ 0 0
$$883$$ −25.3050 −0.851579 −0.425790 0.904822i $$-0.640004\pi$$
−0.425790 + 0.904822i $$0.640004\pi$$
$$884$$ 0 0
$$885$$ 4.00000 0.134459
$$886$$ 0 0
$$887$$ 24.5836 0.825436 0.412718 0.910859i $$-0.364579\pi$$
0.412718 + 0.910859i $$0.364579\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −16.0000 −0.534821
$$896$$ 0 0
$$897$$ −28.9443 −0.966421
$$898$$ 0 0
$$899$$ 54.8328 1.82878
$$900$$ 0 0
$$901$$ 8.94427 0.297977
$$902$$ 0 0
$$903$$ 10.4721 0.348491
$$904$$ 0 0
$$905$$ −13.4164 −0.445976
$$906$$ 0 0
$$907$$ 36.3607 1.20734 0.603668 0.797236i $$-0.293705\pi$$
0.603668 + 0.797236i $$0.293705\pi$$
$$908$$ 0 0
$$909$$ 6.94427 0.230327
$$910$$ 0 0
$$911$$ 0.583592 0.0193353 0.00966764 0.999953i $$-0.496923\pi$$
0.00966764 + 0.999953i $$0.496923\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −12.4721 −0.412316
$$916$$ 0 0
$$917$$ −0.944272 −0.0311826
$$918$$ 0 0
$$919$$ −56.7214 −1.87107 −0.935533 0.353241i $$-0.885080\pi$$
−0.935533 + 0.353241i $$0.885080\pi$$
$$920$$ 0 0
$$921$$ −24.9443 −0.821942
$$922$$ 0 0
$$923$$ 68.9443 2.26933
$$924$$ 0 0
$$925$$ −8.47214 −0.278562
$$926$$ 0 0
$$927$$ −17.8885 −0.587537
$$928$$ 0 0
$$929$$ 4.11146 0.134893 0.0674463 0.997723i $$-0.478515\pi$$
0.0674463 + 0.997723i $$0.478515\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 33.8885 1.10946
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 31.3050 1.02269 0.511344 0.859376i $$-0.329148\pi$$
0.511344 + 0.859376i $$0.329148\pi$$
$$938$$ 0 0
$$939$$ −0.472136 −0.0154076
$$940$$ 0 0
$$941$$ 2.00000 0.0651981 0.0325991 0.999469i $$-0.489622\pi$$
0.0325991 + 0.999469i $$0.489622\pi$$
$$942$$ 0 0
$$943$$ 70.8328 2.30663
$$944$$ 0 0
$$945$$ 1.00000 0.0325300
$$946$$ 0 0
$$947$$ −56.3607 −1.83148 −0.915738 0.401776i $$-0.868393\pi$$
−0.915738 + 0.401776i $$0.868393\pi$$
$$948$$ 0 0
$$949$$ 37.8885 1.22991
$$950$$ 0 0
$$951$$ −15.8885 −0.515221
$$952$$ 0 0
$$953$$ 41.7771 1.35329 0.676646 0.736308i $$-0.263433\pi$$
0.676646 + 0.736308i $$0.263433\pi$$
$$954$$ 0 0
$$955$$ 23.4164 0.757737
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 10.9443 0.353409
$$960$$ 0 0
$$961$$ 10.8885 0.351243
$$962$$ 0 0
$$963$$ 6.47214 0.208562
$$964$$ 0 0
$$965$$ 18.9443 0.609838
$$966$$ 0 0
$$967$$ −3.05573 −0.0982656 −0.0491328 0.998792i $$-0.515646\pi$$
−0.0491328 + 0.998792i $$0.515646\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −4.47214 −0.143223
$$976$$ 0 0
$$977$$ −24.8328 −0.794472 −0.397236 0.917716i $$-0.630031\pi$$
−0.397236 + 0.917716i $$0.630031\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 0 0
$$983$$ 30.2492 0.964800 0.482400 0.875951i $$-0.339765\pi$$
0.482400 + 0.875951i $$0.339765\pi$$
$$984$$ 0 0
$$985$$ 14.9443 0.476164
$$986$$ 0 0
$$987$$ 2.47214 0.0786890
$$988$$ 0 0
$$989$$ −67.7771 −2.15519
$$990$$ 0 0
$$991$$ −36.9443 −1.17357 −0.586787 0.809742i $$-0.699607\pi$$
−0.586787 + 0.809742i $$0.699607\pi$$
$$992$$ 0 0
$$993$$ 32.0000 1.01549
$$994$$ 0 0
$$995$$ −11.4164 −0.361924
$$996$$ 0 0
$$997$$ 17.4164 0.551583 0.275792 0.961217i $$-0.411060\pi$$
0.275792 + 0.961217i $$0.411060\pi$$
$$998$$ 0 0
$$999$$ 8.47214 0.268047
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 6720.2.a.cp.1.2 2
4.3 odd 2 6720.2.a.cv.1.2 2
8.3 odd 2 3360.2.a.bd.1.1 2
8.5 even 2 3360.2.a.bh.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.bd.1.1 2 8.3 odd 2
3360.2.a.bh.1.1 yes 2 8.5 even 2
6720.2.a.cp.1.2 2 1.1 even 1 trivial
6720.2.a.cv.1.2 2 4.3 odd 2