# Properties

 Label 4200.2.a.bo.1.3 Level $4200$ Weight $2$ Character 4200.1 Self dual yes Analytic conductor $33.537$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.5371688489$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 840) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 4200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +5.41855 q^{11} -4.34017 q^{13} -1.07838 q^{17} +4.34017 q^{19} +1.00000 q^{21} -6.34017 q^{23} -1.00000 q^{27} -8.83710 q^{29} -4.34017 q^{31} -5.41855 q^{33} +8.68035 q^{37} +4.34017 q^{39} +8.34017 q^{41} -6.15676 q^{43} +6.83710 q^{47} +1.00000 q^{49} +1.07838 q^{51} -6.18342 q^{53} -4.34017 q^{57} -6.83710 q^{59} -4.52359 q^{61} -1.00000 q^{63} +6.34017 q^{69} -14.0989 q^{71} +11.1773 q^{73} -5.41855 q^{77} +0.680346 q^{79} +1.00000 q^{81} +6.83710 q^{83} +8.83710 q^{87} +6.49693 q^{89} +4.34017 q^{91} +4.34017 q^{93} +10.4969 q^{97} +5.41855 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{19} + 3 q^{21} - 8 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} + 14 q^{41} - 12 q^{43} - 8 q^{47} + 3 q^{49} - 14 q^{53} - 2 q^{57} + 8 q^{59} + 2 q^{61} - 3 q^{63} + 8 q^{69} - 6 q^{71} - 6 q^{73} - 2 q^{77} - 20 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{87} + 2 q^{89} + 2 q^{91} + 2 q^{93} + 14 q^{97} + 2 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^19 + 3 * q^21 - 8 * q^23 - 3 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^37 + 2 * q^39 + 14 * q^41 - 12 * q^43 - 8 * q^47 + 3 * q^49 - 14 * q^53 - 2 * q^57 + 8 * q^59 + 2 * q^61 - 3 * q^63 + 8 * q^69 - 6 * q^71 - 6 * q^73 - 2 * q^77 - 20 * q^79 + 3 * q^81 - 8 * q^83 - 2 * q^87 + 2 * q^89 + 2 * q^91 + 2 * q^93 + 14 * 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.41855 1.63375 0.816877 0.576812i $$-0.195703\pi$$
0.816877 + 0.576812i $$0.195703\pi$$
$$12$$ 0 0
$$13$$ −4.34017 −1.20375 −0.601874 0.798591i $$-0.705579\pi$$
−0.601874 + 0.798591i $$0.705579\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.07838 −0.261545 −0.130773 0.991412i $$-0.541746\pi$$
−0.130773 + 0.991412i $$0.541746\pi$$
$$18$$ 0 0
$$19$$ 4.34017 0.995704 0.497852 0.867262i $$-0.334122\pi$$
0.497852 + 0.867262i $$0.334122\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −6.34017 −1.32202 −0.661009 0.750378i $$-0.729871\pi$$
−0.661009 + 0.750378i $$0.729871\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −8.83710 −1.64101 −0.820504 0.571640i $$-0.806307\pi$$
−0.820504 + 0.571640i $$0.806307\pi$$
$$30$$ 0 0
$$31$$ −4.34017 −0.779518 −0.389759 0.920917i $$-0.627442\pi$$
−0.389759 + 0.920917i $$0.627442\pi$$
$$32$$ 0 0
$$33$$ −5.41855 −0.943249
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.68035 1.42704 0.713520 0.700635i $$-0.247100\pi$$
0.713520 + 0.700635i $$0.247100\pi$$
$$38$$ 0 0
$$39$$ 4.34017 0.694984
$$40$$ 0 0
$$41$$ 8.34017 1.30252 0.651258 0.758856i $$-0.274242\pi$$
0.651258 + 0.758856i $$0.274242\pi$$
$$42$$ 0 0
$$43$$ −6.15676 −0.938896 −0.469448 0.882960i $$-0.655547\pi$$
−0.469448 + 0.882960i $$0.655547\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.83710 0.997294 0.498647 0.866805i $$-0.333830\pi$$
0.498647 + 0.866805i $$0.333830\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 1.07838 0.151003
$$52$$ 0 0
$$53$$ −6.18342 −0.849358 −0.424679 0.905344i $$-0.639613\pi$$
−0.424679 + 0.905344i $$0.639613\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −4.34017 −0.574870
$$58$$ 0 0
$$59$$ −6.83710 −0.890115 −0.445057 0.895502i $$-0.646817\pi$$
−0.445057 + 0.895502i $$0.646817\pi$$
$$60$$ 0 0
$$61$$ −4.52359 −0.579186 −0.289593 0.957150i $$-0.593520\pi$$
−0.289593 + 0.957150i $$0.593520\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 6.34017 0.763267
$$70$$ 0 0
$$71$$ −14.0989 −1.67323 −0.836616 0.547790i $$-0.815469\pi$$
−0.836616 + 0.547790i $$0.815469\pi$$
$$72$$ 0 0
$$73$$ 11.1773 1.30820 0.654101 0.756408i $$-0.273047\pi$$
0.654101 + 0.756408i $$0.273047\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −5.41855 −0.617501
$$78$$ 0 0
$$79$$ 0.680346 0.0765449 0.0382724 0.999267i $$-0.487815\pi$$
0.0382724 + 0.999267i $$0.487815\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.83710 0.750469 0.375235 0.926930i $$-0.377562\pi$$
0.375235 + 0.926930i $$0.377562\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 8.83710 0.947437
$$88$$ 0 0
$$89$$ 6.49693 0.688673 0.344337 0.938846i $$-0.388104\pi$$
0.344337 + 0.938846i $$0.388104\pi$$
$$90$$ 0 0
$$91$$ 4.34017 0.454974
$$92$$ 0 0
$$93$$ 4.34017 0.450055
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.4969 1.06580 0.532901 0.846178i $$-0.321102\pi$$
0.532901 + 0.846178i $$0.321102\pi$$
$$98$$ 0 0
$$99$$ 5.41855 0.544585
$$100$$ 0 0
$$101$$ 18.8638 1.87701 0.938507 0.345259i $$-0.112209\pi$$
0.938507 + 0.345259i $$0.112209\pi$$
$$102$$ 0 0
$$103$$ −10.1568 −1.00077 −0.500387 0.865802i $$-0.666809\pi$$
−0.500387 + 0.865802i $$0.666809\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −14.6537 −1.41663 −0.708313 0.705899i $$-0.750543\pi$$
−0.708313 + 0.705899i $$0.750543\pi$$
$$108$$ 0 0
$$109$$ −12.8371 −1.22957 −0.614786 0.788694i $$-0.710757\pi$$
−0.614786 + 0.788694i $$0.710757\pi$$
$$110$$ 0 0
$$111$$ −8.68035 −0.823902
$$112$$ 0 0
$$113$$ −1.50307 −0.141397 −0.0706985 0.997498i $$-0.522523\pi$$
−0.0706985 + 0.997498i $$0.522523\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −4.34017 −0.401249
$$118$$ 0 0
$$119$$ 1.07838 0.0988547
$$120$$ 0 0
$$121$$ 18.3607 1.66915
$$122$$ 0 0
$$123$$ −8.34017 −0.752008
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −19.2039 −1.70407 −0.852037 0.523482i $$-0.824633\pi$$
−0.852037 + 0.523482i $$0.824633\pi$$
$$128$$ 0 0
$$129$$ 6.15676 0.542072
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ −4.34017 −0.376341
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −8.65368 −0.739334 −0.369667 0.929164i $$-0.620528\pi$$
−0.369667 + 0.929164i $$0.620528\pi$$
$$138$$ 0 0
$$139$$ −6.18342 −0.524471 −0.262235 0.965004i $$-0.584460\pi$$
−0.262235 + 0.965004i $$0.584460\pi$$
$$140$$ 0 0
$$141$$ −6.83710 −0.575788
$$142$$ 0 0
$$143$$ −23.5174 −1.96663
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ −13.2039 −1.08171 −0.540854 0.841116i $$-0.681899\pi$$
−0.540854 + 0.841116i $$0.681899\pi$$
$$150$$ 0 0
$$151$$ −18.1568 −1.47758 −0.738788 0.673938i $$-0.764601\pi$$
−0.738788 + 0.673938i $$0.764601\pi$$
$$152$$ 0 0
$$153$$ −1.07838 −0.0871817
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −15.1773 −1.21128 −0.605639 0.795739i $$-0.707083\pi$$
−0.605639 + 0.795739i $$0.707083\pi$$
$$158$$ 0 0
$$159$$ 6.18342 0.490377
$$160$$ 0 0
$$161$$ 6.34017 0.499676
$$162$$ 0 0
$$163$$ 2.83710 0.222219 0.111109 0.993808i $$-0.464560\pi$$
0.111109 + 0.993808i $$0.464560\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −13.3607 −1.03388 −0.516941 0.856021i $$-0.672929\pi$$
−0.516941 + 0.856021i $$0.672929\pi$$
$$168$$ 0 0
$$169$$ 5.83710 0.449008
$$170$$ 0 0
$$171$$ 4.34017 0.331901
$$172$$ 0 0
$$173$$ −2.55479 −0.194237 −0.0971184 0.995273i $$-0.530963\pi$$
−0.0971184 + 0.995273i $$0.530963\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.83710 0.513908
$$178$$ 0 0
$$179$$ −11.9421 −0.892598 −0.446299 0.894884i $$-0.647258\pi$$
−0.446299 + 0.894884i $$0.647258\pi$$
$$180$$ 0 0
$$181$$ 4.15676 0.308969 0.154485 0.987995i $$-0.450628\pi$$
0.154485 + 0.987995i $$0.450628\pi$$
$$182$$ 0 0
$$183$$ 4.52359 0.334393
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −5.84324 −0.427300
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 6.09890 0.441301 0.220650 0.975353i $$-0.429182\pi$$
0.220650 + 0.975353i $$0.429182\pi$$
$$192$$ 0 0
$$193$$ −12.6803 −0.912751 −0.456376 0.889787i $$-0.650853\pi$$
−0.456376 + 0.889787i $$0.650853\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 11.8576 0.844820 0.422410 0.906405i $$-0.361184\pi$$
0.422410 + 0.906405i $$0.361184\pi$$
$$198$$ 0 0
$$199$$ −5.50307 −0.390102 −0.195051 0.980793i $$-0.562487\pi$$
−0.195051 + 0.980793i $$0.562487\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 8.83710 0.620243
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −6.34017 −0.440672
$$208$$ 0 0
$$209$$ 23.5174 1.62674
$$210$$ 0 0
$$211$$ −19.1506 −1.31838 −0.659191 0.751975i $$-0.729101\pi$$
−0.659191 + 0.751975i $$0.729101\pi$$
$$212$$ 0 0
$$213$$ 14.0989 0.966040
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.34017 0.294630
$$218$$ 0 0
$$219$$ −11.1773 −0.755290
$$220$$ 0 0
$$221$$ 4.68035 0.314834
$$222$$ 0 0
$$223$$ −12.3135 −0.824574 −0.412287 0.911054i $$-0.635270\pi$$
−0.412287 + 0.911054i $$0.635270\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 15.2039 1.00912 0.504560 0.863376i $$-0.331655\pi$$
0.504560 + 0.863376i $$0.331655\pi$$
$$228$$ 0 0
$$229$$ 5.20394 0.343886 0.171943 0.985107i $$-0.444996\pi$$
0.171943 + 0.985107i $$0.444996\pi$$
$$230$$ 0 0
$$231$$ 5.41855 0.356514
$$232$$ 0 0
$$233$$ −11.6598 −0.763861 −0.381930 0.924191i $$-0.624741\pi$$
−0.381930 + 0.924191i $$0.624741\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −0.680346 −0.0441932
$$238$$ 0 0
$$239$$ 20.6225 1.33396 0.666979 0.745077i $$-0.267587\pi$$
0.666979 + 0.745077i $$0.267587\pi$$
$$240$$ 0 0
$$241$$ −20.3545 −1.31115 −0.655576 0.755129i $$-0.727574\pi$$
−0.655576 + 0.755129i $$0.727574\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −18.8371 −1.19858
$$248$$ 0 0
$$249$$ −6.83710 −0.433284
$$250$$ 0 0
$$251$$ 10.5236 0.664243 0.332122 0.943237i $$-0.392236\pi$$
0.332122 + 0.943237i $$0.392236\pi$$
$$252$$ 0 0
$$253$$ −34.3545 −2.15985
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −22.8059 −1.42259 −0.711297 0.702892i $$-0.751892\pi$$
−0.711297 + 0.702892i $$0.751892\pi$$
$$258$$ 0 0
$$259$$ −8.68035 −0.539370
$$260$$ 0 0
$$261$$ −8.83710 −0.547003
$$262$$ 0 0
$$263$$ 28.0144 1.72744 0.863720 0.503972i $$-0.168128\pi$$
0.863720 + 0.503972i $$0.168128\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −6.49693 −0.397606
$$268$$ 0 0
$$269$$ 18.4969 1.12778 0.563889 0.825851i $$-0.309305\pi$$
0.563889 + 0.825851i $$0.309305\pi$$
$$270$$ 0 0
$$271$$ −29.0205 −1.76287 −0.881435 0.472304i $$-0.843422\pi$$
−0.881435 + 0.472304i $$0.843422\pi$$
$$272$$ 0 0
$$273$$ −4.34017 −0.262679
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8.68035 0.521551 0.260776 0.965399i $$-0.416022\pi$$
0.260776 + 0.965399i $$0.416022\pi$$
$$278$$ 0 0
$$279$$ −4.34017 −0.259839
$$280$$ 0 0
$$281$$ 5.63317 0.336046 0.168023 0.985783i $$-0.446262\pi$$
0.168023 + 0.985783i $$0.446262\pi$$
$$282$$ 0 0
$$283$$ 2.47027 0.146842 0.0734210 0.997301i $$-0.476608\pi$$
0.0734210 + 0.997301i $$0.476608\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8.34017 −0.492305
$$288$$ 0 0
$$289$$ −15.8371 −0.931594
$$290$$ 0 0
$$291$$ −10.4969 −0.615341
$$292$$ 0 0
$$293$$ 7.60197 0.444112 0.222056 0.975034i $$-0.428723\pi$$
0.222056 + 0.975034i $$0.428723\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −5.41855 −0.314416
$$298$$ 0 0
$$299$$ 27.5174 1.59138
$$300$$ 0 0
$$301$$ 6.15676 0.354869
$$302$$ 0 0
$$303$$ −18.8638 −1.08369
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −6.15676 −0.351385 −0.175692 0.984445i $$-0.556216\pi$$
−0.175692 + 0.984445i $$0.556216\pi$$
$$308$$ 0 0
$$309$$ 10.1568 0.577798
$$310$$ 0 0
$$311$$ −1.52973 −0.0867432 −0.0433716 0.999059i $$-0.513810\pi$$
−0.0433716 + 0.999059i $$0.513810\pi$$
$$312$$ 0 0
$$313$$ −11.9733 −0.676773 −0.338387 0.941007i $$-0.609881\pi$$
−0.338387 + 0.941007i $$0.609881\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −23.5441 −1.32237 −0.661184 0.750223i $$-0.729946\pi$$
−0.661184 + 0.750223i $$0.729946\pi$$
$$318$$ 0 0
$$319$$ −47.8843 −2.68101
$$320$$ 0 0
$$321$$ 14.6537 0.817889
$$322$$ 0 0
$$323$$ −4.68035 −0.260421
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 12.8371 0.709893
$$328$$ 0 0
$$329$$ −6.83710 −0.376942
$$330$$ 0 0
$$331$$ −9.16290 −0.503638 −0.251819 0.967774i $$-0.581029\pi$$
−0.251819 + 0.967774i $$0.581029\pi$$
$$332$$ 0 0
$$333$$ 8.68035 0.475680
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −16.0000 −0.871576 −0.435788 0.900049i $$-0.643530\pi$$
−0.435788 + 0.900049i $$0.643530\pi$$
$$338$$ 0 0
$$339$$ 1.50307 0.0816356
$$340$$ 0 0
$$341$$ −23.5174 −1.27354
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 23.0205 1.23581 0.617903 0.786254i $$-0.287982\pi$$
0.617903 + 0.786254i $$0.287982\pi$$
$$348$$ 0 0
$$349$$ −3.78992 −0.202870 −0.101435 0.994842i $$-0.532343\pi$$
−0.101435 + 0.994842i $$0.532343\pi$$
$$350$$ 0 0
$$351$$ 4.34017 0.231661
$$352$$ 0 0
$$353$$ −28.5958 −1.52200 −0.761001 0.648751i $$-0.775292\pi$$
−0.761001 + 0.648751i $$0.775292\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −1.07838 −0.0570738
$$358$$ 0 0
$$359$$ −11.2618 −0.594375 −0.297187 0.954819i $$-0.596049\pi$$
−0.297187 + 0.954819i $$0.596049\pi$$
$$360$$ 0 0
$$361$$ −0.162899 −0.00857361
$$362$$ 0 0
$$363$$ −18.3607 −0.963686
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 25.3607 1.32382 0.661909 0.749584i $$-0.269747\pi$$
0.661909 + 0.749584i $$0.269747\pi$$
$$368$$ 0 0
$$369$$ 8.34017 0.434172
$$370$$ 0 0
$$371$$ 6.18342 0.321027
$$372$$ 0 0
$$373$$ 21.3074 1.10325 0.551627 0.834091i $$-0.314007\pi$$
0.551627 + 0.834091i $$0.314007\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 38.3545 1.97536
$$378$$ 0 0
$$379$$ −1.84324 −0.0946811 −0.0473406 0.998879i $$-0.515075\pi$$
−0.0473406 + 0.998879i $$0.515075\pi$$
$$380$$ 0 0
$$381$$ 19.2039 0.983847
$$382$$ 0 0
$$383$$ −4.99386 −0.255174 −0.127587 0.991827i $$-0.540723\pi$$
−0.127587 + 0.991827i $$0.540723\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −6.15676 −0.312965
$$388$$ 0 0
$$389$$ 16.8371 0.853675 0.426837 0.904328i $$-0.359628\pi$$
0.426837 + 0.904328i $$0.359628\pi$$
$$390$$ 0 0
$$391$$ 6.83710 0.345767
$$392$$ 0 0
$$393$$ 4.00000 0.201773
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 36.8515 1.84952 0.924761 0.380548i $$-0.124264\pi$$
0.924761 + 0.380548i $$0.124264\pi$$
$$398$$ 0 0
$$399$$ 4.34017 0.217280
$$400$$ 0 0
$$401$$ −24.3545 −1.21621 −0.608104 0.793857i $$-0.708070\pi$$
−0.608104 + 0.793857i $$0.708070\pi$$
$$402$$ 0 0
$$403$$ 18.8371 0.938343
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 47.0349 2.33143
$$408$$ 0 0
$$409$$ −28.0410 −1.38654 −0.693270 0.720678i $$-0.743831\pi$$
−0.693270 + 0.720678i $$0.743831\pi$$
$$410$$ 0 0
$$411$$ 8.65368 0.426855
$$412$$ 0 0
$$413$$ 6.83710 0.336432
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 6.18342 0.302803
$$418$$ 0 0
$$419$$ 0.482553 0.0235742 0.0117871 0.999931i $$-0.496248\pi$$
0.0117871 + 0.999931i $$0.496248\pi$$
$$420$$ 0 0
$$421$$ 21.1506 1.03082 0.515409 0.856944i $$-0.327640\pi$$
0.515409 + 0.856944i $$0.327640\pi$$
$$422$$ 0 0
$$423$$ 6.83710 0.332431
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.52359 0.218912
$$428$$ 0 0
$$429$$ 23.5174 1.13543
$$430$$ 0 0
$$431$$ −28.7382 −1.38427 −0.692135 0.721768i $$-0.743330\pi$$
−0.692135 + 0.721768i $$0.743330\pi$$
$$432$$ 0 0
$$433$$ −9.02052 −0.433498 −0.216749 0.976227i $$-0.569545\pi$$
−0.216749 + 0.976227i $$0.569545\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −27.5174 −1.31634
$$438$$ 0 0
$$439$$ −17.8166 −0.850339 −0.425170 0.905114i $$-0.639786\pi$$
−0.425170 + 0.905114i $$0.639786\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −11.7009 −0.555925 −0.277962 0.960592i $$-0.589659\pi$$
−0.277962 + 0.960592i $$0.589659\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 13.2039 0.624525
$$448$$ 0 0
$$449$$ 10.7337 0.506553 0.253277 0.967394i $$-0.418492\pi$$
0.253277 + 0.967394i $$0.418492\pi$$
$$450$$ 0 0
$$451$$ 45.1917 2.12799
$$452$$ 0 0
$$453$$ 18.1568 0.853079
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −20.9939 −0.982051 −0.491026 0.871145i $$-0.663378\pi$$
−0.491026 + 0.871145i $$0.663378\pi$$
$$458$$ 0 0
$$459$$ 1.07838 0.0503344
$$460$$ 0 0
$$461$$ −15.3751 −0.716088 −0.358044 0.933705i $$-0.616556\pi$$
−0.358044 + 0.933705i $$0.616556\pi$$
$$462$$ 0 0
$$463$$ 29.1917 1.35665 0.678326 0.734762i $$-0.262706\pi$$
0.678326 + 0.734762i $$0.262706\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −15.2039 −0.703554 −0.351777 0.936084i $$-0.614423\pi$$
−0.351777 + 0.936084i $$0.614423\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 15.1773 0.699332
$$472$$ 0 0
$$473$$ −33.3607 −1.53393
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −6.18342 −0.283119
$$478$$ 0 0
$$479$$ −18.7838 −0.858253 −0.429126 0.903244i $$-0.641178\pi$$
−0.429126 + 0.903244i $$0.641178\pi$$
$$480$$ 0 0
$$481$$ −37.6742 −1.71780
$$482$$ 0 0
$$483$$ −6.34017 −0.288488
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −16.5113 −0.748199 −0.374099 0.927389i $$-0.622048\pi$$
−0.374099 + 0.927389i $$0.622048\pi$$
$$488$$ 0 0
$$489$$ −2.83710 −0.128298
$$490$$ 0 0
$$491$$ 39.4063 1.77838 0.889190 0.457538i $$-0.151269\pi$$
0.889190 + 0.457538i $$0.151269\pi$$
$$492$$ 0 0
$$493$$ 9.52973 0.429198
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 14.0989 0.632422
$$498$$ 0 0
$$499$$ −31.5174 −1.41091 −0.705457 0.708752i $$-0.749258\pi$$
−0.705457 + 0.708752i $$0.749258\pi$$
$$500$$ 0 0
$$501$$ 13.3607 0.596912
$$502$$ 0 0
$$503$$ 20.3668 0.908112 0.454056 0.890973i $$-0.349977\pi$$
0.454056 + 0.890973i $$0.349977\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −5.83710 −0.259235
$$508$$ 0 0
$$509$$ 11.1773 0.495424 0.247712 0.968834i $$-0.420321\pi$$
0.247712 + 0.968834i $$0.420321\pi$$
$$510$$ 0 0
$$511$$ −11.1773 −0.494454
$$512$$ 0 0
$$513$$ −4.34017 −0.191623
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 37.0472 1.62933
$$518$$ 0 0
$$519$$ 2.55479 0.112143
$$520$$ 0 0
$$521$$ 32.6537 1.43058 0.715292 0.698826i $$-0.246294\pi$$
0.715292 + 0.698826i $$0.246294\pi$$
$$522$$ 0 0
$$523$$ −15.6865 −0.685922 −0.342961 0.939350i $$-0.611430\pi$$
−0.342961 + 0.939350i $$0.611430\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4.68035 0.203879
$$528$$ 0 0
$$529$$ 17.1978 0.747730
$$530$$ 0 0
$$531$$ −6.83710 −0.296705
$$532$$ 0 0
$$533$$ −36.1978 −1.56790
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 11.9421 0.515341
$$538$$ 0 0
$$539$$ 5.41855 0.233394
$$540$$ 0 0
$$541$$ 30.1978 1.29830 0.649152 0.760658i $$-0.275124\pi$$
0.649152 + 0.760658i $$0.275124\pi$$
$$542$$ 0 0
$$543$$ −4.15676 −0.178383
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −9.36069 −0.400234 −0.200117 0.979772i $$-0.564132\pi$$
−0.200117 + 0.979772i $$0.564132\pi$$
$$548$$ 0 0
$$549$$ −4.52359 −0.193062
$$550$$ 0 0
$$551$$ −38.3545 −1.63396
$$552$$ 0 0
$$553$$ −0.680346 −0.0289313
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 7.49079 0.317395 0.158697 0.987327i $$-0.449271\pi$$
0.158697 + 0.987327i $$0.449271\pi$$
$$558$$ 0 0
$$559$$ 26.7214 1.13019
$$560$$ 0 0
$$561$$ 5.84324 0.246702
$$562$$ 0 0
$$563$$ −31.7152 −1.33664 −0.668319 0.743875i $$-0.732986\pi$$
−0.668319 + 0.743875i $$0.732986\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 30.6803 1.28619 0.643094 0.765788i $$-0.277651\pi$$
0.643094 + 0.765788i $$0.277651\pi$$
$$570$$ 0 0
$$571$$ −10.6393 −0.445241 −0.222621 0.974905i $$-0.571461\pi$$
−0.222621 + 0.974905i $$0.571461\pi$$
$$572$$ 0 0
$$573$$ −6.09890 −0.254785
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −30.0677 −1.25173 −0.625867 0.779930i $$-0.715255\pi$$
−0.625867 + 0.779930i $$0.715255\pi$$
$$578$$ 0 0
$$579$$ 12.6803 0.526977
$$580$$ 0 0
$$581$$ −6.83710 −0.283651
$$582$$ 0 0
$$583$$ −33.5052 −1.38764
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −10.0410 −0.414438 −0.207219 0.978295i $$-0.566441\pi$$
−0.207219 + 0.978295i $$0.566441\pi$$
$$588$$ 0 0
$$589$$ −18.8371 −0.776169
$$590$$ 0 0
$$591$$ −11.8576 −0.487757
$$592$$ 0 0
$$593$$ 24.2823 0.997155 0.498578 0.866845i $$-0.333856\pi$$
0.498578 + 0.866845i $$0.333856\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 5.50307 0.225226
$$598$$ 0 0
$$599$$ −9.90110 −0.404548 −0.202274 0.979329i $$-0.564833\pi$$
−0.202274 + 0.979329i $$0.564833\pi$$
$$600$$ 0 0
$$601$$ 17.6865 0.721447 0.360723 0.932673i $$-0.382530\pi$$
0.360723 + 0.932673i $$0.382530\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −28.3135 −1.14921 −0.574605 0.818431i $$-0.694844\pi$$
−0.574605 + 0.818431i $$0.694844\pi$$
$$608$$ 0 0
$$609$$ −8.83710 −0.358097
$$610$$ 0 0
$$611$$ −29.6742 −1.20049
$$612$$ 0 0
$$613$$ 23.6865 0.956688 0.478344 0.878172i $$-0.341237\pi$$
0.478344 + 0.878172i $$0.341237\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 13.7009 0.551576 0.275788 0.961218i $$-0.411061\pi$$
0.275788 + 0.961218i $$0.411061\pi$$
$$618$$ 0 0
$$619$$ −2.49693 −0.100360 −0.0501800 0.998740i $$-0.515980\pi$$
−0.0501800 + 0.998740i $$0.515980\pi$$
$$620$$ 0 0
$$621$$ 6.34017 0.254422
$$622$$ 0 0
$$623$$ −6.49693 −0.260294
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −23.5174 −0.939196
$$628$$ 0 0
$$629$$ −9.36069 −0.373235
$$630$$ 0 0
$$631$$ 8.68035 0.345559 0.172780 0.984961i $$-0.444725\pi$$
0.172780 + 0.984961i $$0.444725\pi$$
$$632$$ 0 0
$$633$$ 19.1506 0.761169
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −4.34017 −0.171964
$$638$$ 0 0
$$639$$ −14.0989 −0.557744
$$640$$ 0 0
$$641$$ 3.30737 0.130633 0.0653166 0.997865i $$-0.479194\pi$$
0.0653166 + 0.997865i $$0.479194\pi$$
$$642$$ 0 0
$$643$$ 6.15676 0.242799 0.121399 0.992604i $$-0.461262\pi$$
0.121399 + 0.992604i $$0.461262\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 6.95282 0.273344 0.136672 0.990616i $$-0.456359\pi$$
0.136672 + 0.990616i $$0.456359\pi$$
$$648$$ 0 0
$$649$$ −37.0472 −1.45423
$$650$$ 0 0
$$651$$ −4.34017 −0.170105
$$652$$ 0 0
$$653$$ 38.7480 1.51633 0.758164 0.652064i $$-0.226097\pi$$
0.758164 + 0.652064i $$0.226097\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 11.1773 0.436067
$$658$$ 0 0
$$659$$ 9.22076 0.359190 0.179595 0.983741i $$-0.442521\pi$$
0.179595 + 0.983741i $$0.442521\pi$$
$$660$$ 0 0
$$661$$ 25.8843 1.00678 0.503391 0.864059i $$-0.332086\pi$$
0.503391 + 0.864059i $$0.332086\pi$$
$$662$$ 0 0
$$663$$ −4.68035 −0.181770
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 56.0288 2.16944
$$668$$ 0 0
$$669$$ 12.3135 0.476068
$$670$$ 0 0
$$671$$ −24.5113 −0.946248
$$672$$ 0 0
$$673$$ 40.0821 1.54505 0.772525 0.634984i $$-0.218993\pi$$
0.772525 + 0.634984i $$0.218993\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −33.5897 −1.29096 −0.645478 0.763779i $$-0.723342\pi$$
−0.645478 + 0.763779i $$0.723342\pi$$
$$678$$ 0 0
$$679$$ −10.4969 −0.402835
$$680$$ 0 0
$$681$$ −15.2039 −0.582616
$$682$$ 0 0
$$683$$ 18.7070 0.715804 0.357902 0.933759i $$-0.383492\pi$$
0.357902 + 0.933759i $$0.383492\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −5.20394 −0.198543
$$688$$ 0 0
$$689$$ 26.8371 1.02241
$$690$$ 0 0
$$691$$ −19.1773 −0.729538 −0.364769 0.931098i $$-0.618852\pi$$
−0.364769 + 0.931098i $$0.618852\pi$$
$$692$$ 0 0
$$693$$ −5.41855 −0.205834
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −8.99386 −0.340667
$$698$$ 0 0
$$699$$ 11.6598 0.441015
$$700$$ 0 0
$$701$$ 21.4641 0.810689 0.405344 0.914164i $$-0.367152\pi$$
0.405344 + 0.914164i $$0.367152\pi$$
$$702$$ 0 0
$$703$$ 37.6742 1.42091
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −18.8638 −0.709445
$$708$$ 0 0
$$709$$ 2.62702 0.0986599 0.0493299 0.998783i $$-0.484291\pi$$
0.0493299 + 0.998783i $$0.484291\pi$$
$$710$$ 0 0
$$711$$ 0.680346 0.0255150
$$712$$ 0 0
$$713$$ 27.5174 1.03054
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −20.6225 −0.770161
$$718$$ 0 0
$$719$$ −4.36683 −0.162855 −0.0814277 0.996679i $$-0.525948\pi$$
−0.0814277 + 0.996679i $$0.525948\pi$$
$$720$$ 0 0
$$721$$ 10.1568 0.378257
$$722$$ 0 0
$$723$$ 20.3545 0.756994
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.1445 1.04382 0.521910 0.853000i $$-0.325220\pi$$
0.521910 + 0.853000i $$0.325220\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 6.63931 0.245564
$$732$$ 0 0
$$733$$ −26.7480 −0.987962 −0.493981 0.869473i $$-0.664459\pi$$
−0.493981 + 0.869473i $$0.664459\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 19.0349 0.700210 0.350105 0.936710i $$-0.386146\pi$$
0.350105 + 0.936710i $$0.386146\pi$$
$$740$$ 0 0
$$741$$ 18.8371 0.691998
$$742$$ 0 0
$$743$$ 33.6598 1.23486 0.617430 0.786626i $$-0.288174\pi$$
0.617430 + 0.786626i $$0.288174\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.83710 0.250156
$$748$$ 0 0
$$749$$ 14.6537 0.535434
$$750$$ 0 0
$$751$$ −49.8720 −1.81985 −0.909927 0.414767i $$-0.863863\pi$$
−0.909927 + 0.414767i $$0.863863\pi$$
$$752$$ 0 0
$$753$$ −10.5236 −0.383501
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −7.63317 −0.277432 −0.138716 0.990332i $$-0.544298\pi$$
−0.138716 + 0.990332i $$0.544298\pi$$
$$758$$ 0 0
$$759$$ 34.3545 1.24699
$$760$$ 0 0
$$761$$ 19.5974 0.710406 0.355203 0.934789i $$-0.384412\pi$$
0.355203 + 0.934789i $$0.384412\pi$$
$$762$$ 0 0
$$763$$ 12.8371 0.464734
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 29.6742 1.07147
$$768$$ 0 0
$$769$$ −32.4079 −1.16866 −0.584329 0.811517i $$-0.698642\pi$$
−0.584329 + 0.811517i $$0.698642\pi$$
$$770$$ 0 0
$$771$$ 22.8059 0.821335
$$772$$ 0 0
$$773$$ −12.0845 −0.434650 −0.217325 0.976099i $$-0.569733\pi$$
−0.217325 + 0.976099i $$0.569733\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 8.68035 0.311406
$$778$$ 0 0
$$779$$ 36.1978 1.29692
$$780$$ 0 0
$$781$$ −76.3956 −2.73365
$$782$$ 0 0
$$783$$ 8.83710 0.315812
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 25.0472 0.892836 0.446418 0.894825i $$-0.352700\pi$$
0.446418 + 0.894825i $$0.352700\pi$$
$$788$$ 0 0
$$789$$ −28.0144 −0.997338
$$790$$ 0 0
$$791$$ 1.50307 0.0534431
$$792$$ 0 0
$$793$$ 19.6332 0.697194
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −21.2762 −0.753641 −0.376820 0.926286i $$-0.622983\pi$$
−0.376820 + 0.926286i $$0.622983\pi$$
$$798$$ 0 0
$$799$$ −7.37298 −0.260837
$$800$$ 0 0
$$801$$ 6.49693 0.229558
$$802$$ 0 0
$$803$$ 60.5646 2.13728
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −18.4969 −0.651123
$$808$$ 0 0
$$809$$ −49.6619 −1.74602 −0.873010 0.487702i $$-0.837835\pi$$
−0.873010 + 0.487702i $$0.837835\pi$$
$$810$$ 0 0
$$811$$ 25.8166 0.906543 0.453271 0.891373i $$-0.350257\pi$$
0.453271 + 0.891373i $$0.350257\pi$$
$$812$$ 0 0
$$813$$ 29.0205 1.01779
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −26.7214 −0.934863
$$818$$ 0 0
$$819$$ 4.34017 0.151658
$$820$$ 0 0
$$821$$ 24.8904 0.868682 0.434341 0.900749i $$-0.356981\pi$$
0.434341 + 0.900749i $$0.356981\pi$$
$$822$$ 0 0
$$823$$ 18.4703 0.643833 0.321917 0.946768i $$-0.395673\pi$$
0.321917 + 0.946768i $$0.395673\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −21.6886 −0.754186 −0.377093 0.926175i $$-0.623076\pi$$
−0.377093 + 0.926175i $$0.623076\pi$$
$$828$$ 0 0
$$829$$ 47.5052 1.64992 0.824961 0.565189i $$-0.191197\pi$$
0.824961 + 0.565189i $$0.191197\pi$$
$$830$$ 0 0
$$831$$ −8.68035 −0.301118
$$832$$ 0 0
$$833$$ −1.07838 −0.0373636
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 4.34017 0.150018
$$838$$ 0 0
$$839$$ 11.4641 0.395785 0.197893 0.980224i $$-0.436590\pi$$
0.197893 + 0.980224i $$0.436590\pi$$
$$840$$ 0 0
$$841$$ 49.0944 1.69291
$$842$$ 0 0
$$843$$ −5.63317 −0.194017
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −18.3607 −0.630881
$$848$$ 0 0
$$849$$ −2.47027 −0.0847793
$$850$$ 0 0
$$851$$ −55.0349 −1.88657
$$852$$ 0 0
$$853$$ 39.8043 1.36287 0.681437 0.731877i $$-0.261356\pi$$
0.681437 + 0.731877i $$0.261356\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 16.9627 0.579433 0.289717 0.957112i $$-0.406439\pi$$
0.289717 + 0.957112i $$0.406439\pi$$
$$858$$ 0 0
$$859$$ −41.0493 −1.40058 −0.700292 0.713857i $$-0.746947\pi$$
−0.700292 + 0.713857i $$0.746947\pi$$
$$860$$ 0 0
$$861$$ 8.34017 0.284232
$$862$$ 0 0
$$863$$ 50.0554 1.70391 0.851953 0.523618i $$-0.175418\pi$$
0.851953 + 0.523618i $$0.175418\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 15.8371 0.537856
$$868$$ 0 0
$$869$$ 3.68649 0.125056
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 10.4969 0.355267
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 6.32580 0.213607 0.106803 0.994280i $$-0.465938\pi$$
0.106803 + 0.994280i $$0.465938\pi$$
$$878$$ 0 0
$$879$$ −7.60197 −0.256408
$$880$$ 0 0
$$881$$ 20.5380 0.691942 0.345971 0.938245i $$-0.387550\pi$$
0.345971 + 0.938245i $$0.387550\pi$$
$$882$$ 0 0
$$883$$ −5.30737 −0.178607 −0.0893036 0.996004i $$-0.528464\pi$$
−0.0893036 + 0.996004i $$0.528464\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −3.15061 −0.105787 −0.0528936 0.998600i $$-0.516844\pi$$
−0.0528936 + 0.998600i $$0.516844\pi$$
$$888$$ 0 0
$$889$$ 19.2039 0.644079
$$890$$ 0 0
$$891$$ 5.41855 0.181528
$$892$$ 0 0
$$893$$ 29.6742 0.993009
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −27.5174 −0.918781
$$898$$ 0 0
$$899$$ 38.3545 1.27920
$$900$$ 0 0
$$901$$ 6.66806 0.222145
$$902$$ 0 0
$$903$$ −6.15676 −0.204884
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 11.9467 0.396683 0.198341 0.980133i $$-0.436445\pi$$
0.198341 + 0.980133i $$0.436445\pi$$
$$908$$ 0 0
$$909$$ 18.8638 0.625672
$$910$$ 0 0
$$911$$ 46.0989 1.52732 0.763662 0.645616i $$-0.223399\pi$$
0.763662 + 0.645616i $$0.223399\pi$$
$$912$$ 0 0
$$913$$ 37.0472 1.22608
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4.00000 0.132092
$$918$$ 0 0
$$919$$ 7.25726 0.239395 0.119697 0.992810i $$-0.461808\pi$$
0.119697 + 0.992810i $$0.461808\pi$$
$$920$$ 0 0
$$921$$ 6.15676 0.202872
$$922$$ 0 0
$$923$$ 61.1917 2.01415
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −10.1568 −0.333592
$$928$$ 0 0
$$929$$ −44.9048 −1.47328 −0.736639 0.676286i $$-0.763588\pi$$
−0.736639 + 0.676286i $$0.763588\pi$$
$$930$$ 0 0
$$931$$ 4.34017 0.142243
$$932$$ 0 0
$$933$$ 1.52973 0.0500812
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −43.6886 −1.42724 −0.713622 0.700531i $$-0.752946\pi$$
−0.713622 + 0.700531i $$0.752946\pi$$
$$938$$ 0 0
$$939$$ 11.9733 0.390735
$$940$$ 0 0
$$941$$ 14.6660 0.478097 0.239048 0.971008i $$-0.423165\pi$$
0.239048 + 0.971008i $$0.423165\pi$$
$$942$$ 0 0
$$943$$ −52.8781 −1.72195
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.96719 0.0964209 0.0482104 0.998837i $$-0.484648\pi$$
0.0482104 + 0.998837i $$0.484648\pi$$
$$948$$ 0 0
$$949$$ −48.5113 −1.57474
$$950$$ 0 0
$$951$$ 23.5441 0.763470
$$952$$ 0 0
$$953$$ −45.5851 −1.47665 −0.738324 0.674446i $$-0.764382\pi$$
−0.738324 + 0.674446i $$0.764382\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 47.8843 1.54788
$$958$$ 0 0
$$959$$ 8.65368 0.279442
$$960$$ 0 0
$$961$$ −12.1629 −0.392352
$$962$$ 0 0
$$963$$ −14.6537 −0.472208
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.3668 1.04085 0.520424 0.853908i $$-0.325774\pi$$
0.520424 + 0.853908i $$0.325774\pi$$
$$968$$ 0 0
$$969$$ 4.68035 0.150354
$$970$$ 0 0
$$971$$ −11.3197 −0.363265 −0.181632 0.983366i $$-0.558138\pi$$
−0.181632 + 0.983366i $$0.558138\pi$$
$$972$$ 0 0
$$973$$ 6.18342 0.198231
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 39.6886 1.26975 0.634875 0.772615i $$-0.281052\pi$$
0.634875 + 0.772615i $$0.281052\pi$$
$$978$$ 0 0
$$979$$ 35.2039 1.12512
$$980$$ 0 0
$$981$$ −12.8371 −0.409857
$$982$$ 0 0
$$983$$ 43.5174 1.38799 0.693996 0.719979i $$-0.255849\pi$$
0.693996 + 0.719979i $$0.255849\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 6.83710 0.217627
$$988$$ 0 0
$$989$$ 39.0349 1.24124
$$990$$ 0 0
$$991$$ −3.80221 −0.120781 −0.0603905 0.998175i $$-0.519235\pi$$
−0.0603905 + 0.998175i $$0.519235\pi$$
$$992$$ 0 0
$$993$$ 9.16290 0.290776
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 23.8043 0.753890 0.376945 0.926236i $$-0.376975\pi$$
0.376945 + 0.926236i $$0.376975\pi$$
$$998$$ 0 0
$$999$$ −8.68035 −0.274634
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 4200.2.a.bo.1.3 3
4.3 odd 2 8400.2.a.dk.1.1 3
5.2 odd 4 840.2.t.e.169.4 yes 6
5.3 odd 4 840.2.t.e.169.1 6
5.4 even 2 4200.2.a.bq.1.3 3
15.2 even 4 2520.2.t.j.1009.6 6
15.8 even 4 2520.2.t.j.1009.5 6
20.3 even 4 1680.2.t.i.1009.4 6
20.7 even 4 1680.2.t.i.1009.1 6
20.19 odd 2 8400.2.a.dh.1.1 3
60.23 odd 4 5040.2.t.ba.1009.5 6
60.47 odd 4 5040.2.t.ba.1009.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.e.169.1 6 5.3 odd 4
840.2.t.e.169.4 yes 6 5.2 odd 4
1680.2.t.i.1009.1 6 20.7 even 4
1680.2.t.i.1009.4 6 20.3 even 4
2520.2.t.j.1009.5 6 15.8 even 4
2520.2.t.j.1009.6 6 15.2 even 4
4200.2.a.bo.1.3 3 1.1 even 1 trivial
4200.2.a.bq.1.3 3 5.4 even 2
5040.2.t.ba.1009.5 6 60.23 odd 4
5040.2.t.ba.1009.6 6 60.47 odd 4
8400.2.a.dh.1.1 3 20.19 odd 2
8400.2.a.dk.1.1 3 4.3 odd 2