Properties

 Label 1950.2.f.m.649.4 Level $1950$ Weight $2$ Character 1950.649 Analytic conductor $15.571$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.f (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 649.4 Root $$-2.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.649 Dual form 1950.2.f.m.649.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +4.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +4.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} +1.00000i q^{12} -3.60555i q^{13} -4.60555 q^{14} +1.00000 q^{16} +4.60555i q^{17} +1.00000 q^{18} +4.60555i q^{19} +4.60555i q^{21} +1.39445i q^{23} -1.00000i q^{24} +3.60555i q^{26} -1.00000i q^{27} +4.60555 q^{28} -4.60555 q^{29} +6.00000i q^{31} -1.00000 q^{32} -4.60555i q^{34} -1.00000 q^{36} +9.21110 q^{37} -4.60555i q^{38} +3.60555 q^{39} +3.21110i q^{41} -4.60555i q^{42} -8.00000i q^{43} -1.39445i q^{46} +9.21110 q^{47} +1.00000i q^{48} +14.2111 q^{49} -4.60555 q^{51} -3.60555i q^{52} -6.00000i q^{53} +1.00000i q^{54} -4.60555 q^{56} -4.60555 q^{57} +4.60555 q^{58} +9.21110i q^{59} -11.2111 q^{61} -6.00000i q^{62} -4.60555 q^{63} +1.00000 q^{64} +3.21110 q^{67} +4.60555i q^{68} -1.39445 q^{69} +9.21110i q^{71} +1.00000 q^{72} -1.39445 q^{73} -9.21110 q^{74} +4.60555i q^{76} -3.60555 q^{78} +14.4222 q^{79} +1.00000 q^{81} -3.21110i q^{82} +2.78890 q^{83} +4.60555i q^{84} +8.00000i q^{86} -4.60555i q^{87} -15.2111i q^{89} -16.6056i q^{91} +1.39445i q^{92} -6.00000 q^{93} -9.21110 q^{94} -1.00000i q^{96} -1.39445 q^{97} -14.2111 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 4 * q^9 $$4 q - 4 q^{2} + 4 q^{4} + 4 q^{7} - 4 q^{8} - 4 q^{9} - 4 q^{14} + 4 q^{16} + 4 q^{18} + 4 q^{28} - 4 q^{29} - 4 q^{32} - 4 q^{36} + 8 q^{37} + 8 q^{47} + 28 q^{49} - 4 q^{51} - 4 q^{56} - 4 q^{57} + 4 q^{58} - 16 q^{61} - 4 q^{63} + 4 q^{64} - 16 q^{67} - 20 q^{69} + 4 q^{72} - 20 q^{73} - 8 q^{74} + 4 q^{81} + 40 q^{83} - 24 q^{93} - 8 q^{94} - 20 q^{97} - 28 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^7 - 4 * q^8 - 4 * q^9 - 4 * q^14 + 4 * q^16 + 4 * q^18 + 4 * q^28 - 4 * q^29 - 4 * q^32 - 4 * q^36 + 8 * q^37 + 8 * q^47 + 28 * q^49 - 4 * q^51 - 4 * q^56 - 4 * q^57 + 4 * q^58 - 16 * q^61 - 4 * q^63 + 4 * q^64 - 16 * q^67 - 20 * q^69 + 4 * q^72 - 20 * q^73 - 8 * q^74 + 4 * q^81 + 40 * q^83 - 24 * q^93 - 8 * q^94 - 20 * q^97 - 28 * q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000i 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ − 1.00000i − 0.408248i
$$7$$ 4.60555 1.74073 0.870367 0.492403i $$-0.163881\pi$$
0.870367 + 0.492403i $$0.163881\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 3.60555i − 1.00000i
$$14$$ −4.60555 −1.23089
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.60555i 1.11701i 0.829501 + 0.558505i $$0.188625\pi$$
−0.829501 + 0.558505i $$0.811375\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 4.60555i 1.05659i 0.849062 + 0.528293i $$0.177168\pi$$
−0.849062 + 0.528293i $$0.822832\pi$$
$$20$$ 0 0
$$21$$ 4.60555i 1.00501i
$$22$$ 0 0
$$23$$ 1.39445i 0.290763i 0.989376 + 0.145381i $$0.0464409\pi$$
−0.989376 + 0.145381i $$0.953559\pi$$
$$24$$ − 1.00000i − 0.204124i
$$25$$ 0 0
$$26$$ 3.60555i 0.707107i
$$27$$ − 1.00000i − 0.192450i
$$28$$ 4.60555 0.870367
$$29$$ −4.60555 −0.855229 −0.427615 0.903961i $$-0.640646\pi$$
−0.427615 + 0.903961i $$0.640646\pi$$
$$30$$ 0 0
$$31$$ 6.00000i 1.07763i 0.842424 + 0.538816i $$0.181128\pi$$
−0.842424 + 0.538816i $$0.818872\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ − 4.60555i − 0.789846i
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 9.21110 1.51430 0.757148 0.653243i $$-0.226592\pi$$
0.757148 + 0.653243i $$0.226592\pi$$
$$38$$ − 4.60555i − 0.747119i
$$39$$ 3.60555 0.577350
$$40$$ 0 0
$$41$$ 3.21110i 0.501490i 0.968053 + 0.250745i $$0.0806756\pi$$
−0.968053 + 0.250745i $$0.919324\pi$$
$$42$$ − 4.60555i − 0.710652i
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ − 1.39445i − 0.205600i
$$47$$ 9.21110 1.34358 0.671789 0.740743i $$-0.265526\pi$$
0.671789 + 0.740743i $$0.265526\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 14.2111 2.03016
$$50$$ 0 0
$$51$$ −4.60555 −0.644906
$$52$$ − 3.60555i − 0.500000i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 1.00000i 0.136083i
$$55$$ 0 0
$$56$$ −4.60555 −0.615443
$$57$$ −4.60555 −0.610020
$$58$$ 4.60555 0.604739
$$59$$ 9.21110i 1.19918i 0.800306 + 0.599592i $$0.204670\pi$$
−0.800306 + 0.599592i $$0.795330\pi$$
$$60$$ 0 0
$$61$$ −11.2111 −1.43543 −0.717717 0.696335i $$-0.754813\pi$$
−0.717717 + 0.696335i $$0.754813\pi$$
$$62$$ − 6.00000i − 0.762001i
$$63$$ −4.60555 −0.580245
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.21110 0.392299 0.196149 0.980574i $$-0.437156\pi$$
0.196149 + 0.980574i $$0.437156\pi$$
$$68$$ 4.60555i 0.558505i
$$69$$ −1.39445 −0.167872
$$70$$ 0 0
$$71$$ 9.21110i 1.09316i 0.837408 + 0.546578i $$0.184070\pi$$
−0.837408 + 0.546578i $$0.815930\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −1.39445 −0.163208 −0.0816039 0.996665i $$-0.526004\pi$$
−0.0816039 + 0.996665i $$0.526004\pi$$
$$74$$ −9.21110 −1.07077
$$75$$ 0 0
$$76$$ 4.60555i 0.528293i
$$77$$ 0 0
$$78$$ −3.60555 −0.408248
$$79$$ 14.4222 1.62262 0.811312 0.584613i $$-0.198754\pi$$
0.811312 + 0.584613i $$0.198754\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 3.21110i − 0.354607i
$$83$$ 2.78890 0.306121 0.153061 0.988217i $$-0.451087\pi$$
0.153061 + 0.988217i $$0.451087\pi$$
$$84$$ 4.60555i 0.502507i
$$85$$ 0 0
$$86$$ 8.00000i 0.862662i
$$87$$ − 4.60555i − 0.493767i
$$88$$ 0 0
$$89$$ − 15.2111i − 1.61237i −0.591661 0.806187i $$-0.701528\pi$$
0.591661 0.806187i $$-0.298472\pi$$
$$90$$ 0 0
$$91$$ − 16.6056i − 1.74073i
$$92$$ 1.39445i 0.145381i
$$93$$ −6.00000 −0.622171
$$94$$ −9.21110 −0.950053
$$95$$ 0 0
$$96$$ − 1.00000i − 0.102062i
$$97$$ −1.39445 −0.141585 −0.0707924 0.997491i $$-0.522553\pi$$
−0.0707924 + 0.997491i $$0.522553\pi$$
$$98$$ −14.2111 −1.43554
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −7.39445 −0.735775 −0.367888 0.929870i $$-0.619919\pi$$
−0.367888 + 0.929870i $$0.619919\pi$$
$$102$$ 4.60555 0.456018
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 3.60555i 0.353553i
$$105$$ 0 0
$$106$$ 6.00000i 0.582772i
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ − 1.39445i − 0.133564i −0.997768 0.0667820i $$-0.978727\pi$$
0.997768 0.0667820i $$-0.0212732\pi$$
$$110$$ 0 0
$$111$$ 9.21110i 0.874279i
$$112$$ 4.60555 0.435184
$$113$$ 13.8167i 1.29976i 0.760036 + 0.649881i $$0.225181\pi$$
−0.760036 + 0.649881i $$0.774819\pi$$
$$114$$ 4.60555 0.431349
$$115$$ 0 0
$$116$$ −4.60555 −0.427615
$$117$$ 3.60555i 0.333333i
$$118$$ − 9.21110i − 0.847951i
$$119$$ 21.2111i 1.94442i
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 11.2111 1.01501
$$123$$ −3.21110 −0.289535
$$124$$ 6.00000i 0.538816i
$$125$$ 0 0
$$126$$ 4.60555 0.410295
$$127$$ 1.21110i 0.107468i 0.998555 + 0.0537340i $$0.0171123\pi$$
−0.998555 + 0.0537340i $$0.982888\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 22.6056 1.97506 0.987528 0.157443i $$-0.0503250\pi$$
0.987528 + 0.157443i $$0.0503250\pi$$
$$132$$ 0 0
$$133$$ 21.2111i 1.83924i
$$134$$ −3.21110 −0.277397
$$135$$ 0 0
$$136$$ − 4.60555i − 0.394923i
$$137$$ −3.21110 −0.274343 −0.137172 0.990547i $$-0.543801\pi$$
−0.137172 + 0.990547i $$0.543801\pi$$
$$138$$ 1.39445 0.118703
$$139$$ −17.2111 −1.45983 −0.729913 0.683540i $$-0.760440\pi$$
−0.729913 + 0.683540i $$0.760440\pi$$
$$140$$ 0 0
$$141$$ 9.21110i 0.775715i
$$142$$ − 9.21110i − 0.772979i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 1.39445 0.115405
$$147$$ 14.2111i 1.17211i
$$148$$ 9.21110 0.757148
$$149$$ − 15.2111i − 1.24614i −0.782165 0.623071i $$-0.785885\pi$$
0.782165 0.623071i $$-0.214115\pi$$
$$150$$ 0 0
$$151$$ 6.00000i 0.488273i 0.969741 + 0.244137i $$0.0785045\pi$$
−0.969741 + 0.244137i $$0.921495\pi$$
$$152$$ − 4.60555i − 0.373560i
$$153$$ − 4.60555i − 0.372337i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 3.60555 0.288675
$$157$$ 20.4222i 1.62987i 0.579553 + 0.814935i $$0.303227\pi$$
−0.579553 + 0.814935i $$0.696773\pi$$
$$158$$ −14.4222 −1.14737
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 6.42221i 0.506141i
$$162$$ −1.00000 −0.0785674
$$163$$ 24.4222 1.91289 0.956447 0.291905i $$-0.0942891\pi$$
0.956447 + 0.291905i $$0.0942891\pi$$
$$164$$ 3.21110i 0.250745i
$$165$$ 0 0
$$166$$ −2.78890 −0.216460
$$167$$ −9.21110 −0.712777 −0.356388 0.934338i $$-0.615992\pi$$
−0.356388 + 0.934338i $$0.615992\pi$$
$$168$$ − 4.60555i − 0.355326i
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ − 4.60555i − 0.352195i
$$172$$ − 8.00000i − 0.609994i
$$173$$ 12.4222i 0.944443i 0.881480 + 0.472221i $$0.156548\pi$$
−0.881480 + 0.472221i $$0.843452\pi$$
$$174$$ 4.60555i 0.349146i
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −9.21110 −0.692349
$$178$$ 15.2111i 1.14012i
$$179$$ 19.8167 1.48117 0.740583 0.671965i $$-0.234549\pi$$
0.740583 + 0.671965i $$0.234549\pi$$
$$180$$ 0 0
$$181$$ 8.42221 0.626018 0.313009 0.949750i $$-0.398663\pi$$
0.313009 + 0.949750i $$0.398663\pi$$
$$182$$ 16.6056i 1.23089i
$$183$$ − 11.2111i − 0.828749i
$$184$$ − 1.39445i − 0.102800i
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ 0 0
$$188$$ 9.21110 0.671789
$$189$$ − 4.60555i − 0.335005i
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 7.81665 0.562655 0.281328 0.959612i $$-0.409225\pi$$
0.281328 + 0.959612i $$0.409225\pi$$
$$194$$ 1.39445 0.100116
$$195$$ 0 0
$$196$$ 14.2111 1.01508
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −22.4222 −1.58947 −0.794734 0.606958i $$-0.792390\pi$$
−0.794734 + 0.606958i $$0.792390\pi$$
$$200$$ 0 0
$$201$$ 3.21110i 0.226494i
$$202$$ 7.39445 0.520272
$$203$$ −21.2111 −1.48873
$$204$$ −4.60555 −0.322453
$$205$$ 0 0
$$206$$ − 4.00000i − 0.278693i
$$207$$ − 1.39445i − 0.0969209i
$$208$$ − 3.60555i − 0.250000i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −17.2111 −1.18486 −0.592431 0.805622i $$-0.701832\pi$$
−0.592431 + 0.805622i $$0.701832\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ −9.21110 −0.631134
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 1.00000i 0.0680414i
$$217$$ 27.6333i 1.87587i
$$218$$ 1.39445i 0.0944440i
$$219$$ − 1.39445i − 0.0942281i
$$220$$ 0 0
$$221$$ 16.6056 1.11701
$$222$$ − 9.21110i − 0.618209i
$$223$$ 1.81665 0.121652 0.0608261 0.998148i $$-0.480627\pi$$
0.0608261 + 0.998148i $$0.480627\pi$$
$$224$$ −4.60555 −0.307721
$$225$$ 0 0
$$226$$ − 13.8167i − 0.919070i
$$227$$ −24.0000 −1.59294 −0.796468 0.604681i $$-0.793301\pi$$
−0.796468 + 0.604681i $$0.793301\pi$$
$$228$$ −4.60555 −0.305010
$$229$$ 19.8167i 1.30952i 0.755836 + 0.654761i $$0.227231\pi$$
−0.755836 + 0.654761i $$0.772769\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 4.60555 0.302369
$$233$$ 1.81665i 0.119013i 0.998228 + 0.0595065i $$0.0189527\pi$$
−0.998228 + 0.0595065i $$0.981047\pi$$
$$234$$ − 3.60555i − 0.235702i
$$235$$ 0 0
$$236$$ 9.21110i 0.599592i
$$237$$ 14.4222i 0.936823i
$$238$$ − 21.2111i − 1.37491i
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ − 6.42221i − 0.413691i −0.978374 0.206845i $$-0.933680\pi$$
0.978374 0.206845i $$-0.0663197\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 1.00000i 0.0641500i
$$244$$ −11.2111 −0.717717
$$245$$ 0 0
$$246$$ 3.21110 0.204732
$$247$$ 16.6056 1.05659
$$248$$ − 6.00000i − 0.381000i
$$249$$ 2.78890i 0.176739i
$$250$$ 0 0
$$251$$ −13.3944 −0.845450 −0.422725 0.906258i $$-0.638926\pi$$
−0.422725 + 0.906258i $$0.638926\pi$$
$$252$$ −4.60555 −0.290122
$$253$$ 0 0
$$254$$ − 1.21110i − 0.0759913i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 28.6056i − 1.78437i −0.451675 0.892183i $$-0.649173\pi$$
0.451675 0.892183i $$-0.350827\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 42.4222 2.63599
$$260$$ 0 0
$$261$$ 4.60555 0.285076
$$262$$ −22.6056 −1.39658
$$263$$ 7.81665i 0.481996i 0.970526 + 0.240998i $$0.0774746\pi$$
−0.970526 + 0.240998i $$0.922525\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ − 21.2111i − 1.30054i
$$267$$ 15.2111 0.930904
$$268$$ 3.21110 0.196149
$$269$$ −25.8167 −1.57407 −0.787035 0.616909i $$-0.788385\pi$$
−0.787035 + 0.616909i $$0.788385\pi$$
$$270$$ 0 0
$$271$$ − 0.422205i − 0.0256471i −0.999918 0.0128236i $$-0.995918\pi$$
0.999918 0.0128236i $$-0.00408198\pi$$
$$272$$ 4.60555i 0.279253i
$$273$$ 16.6056 1.00501
$$274$$ 3.21110 0.193990
$$275$$ 0 0
$$276$$ −1.39445 −0.0839359
$$277$$ − 16.4222i − 0.986715i −0.869827 0.493357i $$-0.835770\pi$$
0.869827 0.493357i $$-0.164230\pi$$
$$278$$ 17.2111 1.03225
$$279$$ − 6.00000i − 0.359211i
$$280$$ 0 0
$$281$$ 27.2111i 1.62328i 0.584159 + 0.811639i $$0.301424\pi$$
−0.584159 + 0.811639i $$0.698576\pi$$
$$282$$ − 9.21110i − 0.548513i
$$283$$ − 10.4222i − 0.619536i −0.950812 0.309768i $$-0.899749\pi$$
0.950812 0.309768i $$-0.100251\pi$$
$$284$$ 9.21110i 0.546578i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 14.7889i 0.872961i
$$288$$ 1.00000 0.0589256
$$289$$ −4.21110 −0.247712
$$290$$ 0 0
$$291$$ − 1.39445i − 0.0817440i
$$292$$ −1.39445 −0.0816039
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ − 14.2111i − 0.828808i
$$295$$ 0 0
$$296$$ −9.21110 −0.535384
$$297$$ 0 0
$$298$$ 15.2111i 0.881156i
$$299$$ 5.02776 0.290763
$$300$$ 0 0
$$301$$ − 36.8444i − 2.12368i
$$302$$ − 6.00000i − 0.345261i
$$303$$ − 7.39445i − 0.424800i
$$304$$ 4.60555i 0.264146i
$$305$$ 0 0
$$306$$ 4.60555i 0.263282i
$$307$$ 8.78890 0.501609 0.250804 0.968038i $$-0.419305\pi$$
0.250804 + 0.968038i $$0.419305\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ −3.60555 −0.204124
$$313$$ − 3.57779i − 0.202229i −0.994875 0.101114i $$-0.967759\pi$$
0.994875 0.101114i $$-0.0322408\pi$$
$$314$$ − 20.4222i − 1.15249i
$$315$$ 0 0
$$316$$ 14.4222 0.811312
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 6.42221i − 0.357895i
$$323$$ −21.2111 −1.18022
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −24.4222 −1.35262
$$327$$ 1.39445 0.0771132
$$328$$ − 3.21110i − 0.177303i
$$329$$ 42.4222 2.33881
$$330$$ 0 0
$$331$$ − 16.6056i − 0.912724i −0.889794 0.456362i $$-0.849152\pi$$
0.889794 0.456362i $$-0.150848\pi$$
$$332$$ 2.78890 0.153061
$$333$$ −9.21110 −0.504765
$$334$$ 9.21110 0.504009
$$335$$ 0 0
$$336$$ 4.60555i 0.251253i
$$337$$ 13.6333i 0.742654i 0.928502 + 0.371327i $$0.121097\pi$$
−0.928502 + 0.371327i $$0.878903\pi$$
$$338$$ 13.0000 0.707107
$$339$$ −13.8167 −0.750418
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.60555i 0.249040i
$$343$$ 33.2111 1.79323
$$344$$ 8.00000i 0.431331i
$$345$$ 0 0
$$346$$ − 12.4222i − 0.667822i
$$347$$ − 27.6333i − 1.48343i −0.670713 0.741717i $$-0.734012\pi$$
0.670713 0.741717i $$-0.265988\pi$$
$$348$$ − 4.60555i − 0.246883i
$$349$$ 7.81665i 0.418416i 0.977871 + 0.209208i $$0.0670886\pi$$
−0.977871 + 0.209208i $$0.932911\pi$$
$$350$$ 0 0
$$351$$ −3.60555 −0.192450
$$352$$ 0 0
$$353$$ −8.78890 −0.467786 −0.233893 0.972262i $$-0.575147\pi$$
−0.233893 + 0.972262i $$0.575147\pi$$
$$354$$ 9.21110 0.489565
$$355$$ 0 0
$$356$$ − 15.2111i − 0.806187i
$$357$$ −21.2111 −1.12261
$$358$$ −19.8167 −1.04734
$$359$$ − 15.6333i − 0.825094i −0.910936 0.412547i $$-0.864639\pi$$
0.910936 0.412547i $$-0.135361\pi$$
$$360$$ 0 0
$$361$$ −2.21110 −0.116374
$$362$$ −8.42221 −0.442661
$$363$$ 11.0000i 0.577350i
$$364$$ − 16.6056i − 0.870367i
$$365$$ 0 0
$$366$$ 11.2111i 0.586014i
$$367$$ − 19.6333i − 1.02485i −0.858732 0.512425i $$-0.828747\pi$$
0.858732 0.512425i $$-0.171253\pi$$
$$368$$ 1.39445i 0.0726907i
$$369$$ − 3.21110i − 0.167163i
$$370$$ 0 0
$$371$$ − 27.6333i − 1.43465i
$$372$$ −6.00000 −0.311086
$$373$$ 20.4222i 1.05742i 0.848802 + 0.528711i $$0.177324\pi$$
−0.848802 + 0.528711i $$0.822676\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −9.21110 −0.475026
$$377$$ 16.6056i 0.855229i
$$378$$ 4.60555i 0.236884i
$$379$$ − 35.0278i − 1.79925i −0.436658 0.899627i $$-0.643838\pi$$
0.436658 0.899627i $$-0.356162\pi$$
$$380$$ 0 0
$$381$$ −1.21110 −0.0620467
$$382$$ 12.0000 0.613973
$$383$$ −27.6333 −1.41200 −0.705998 0.708214i $$-0.749501\pi$$
−0.705998 + 0.708214i $$0.749501\pi$$
$$384$$ − 1.00000i − 0.0510310i
$$385$$ 0 0
$$386$$ −7.81665 −0.397857
$$387$$ 8.00000i 0.406663i
$$388$$ −1.39445 −0.0707924
$$389$$ −4.60555 −0.233511 −0.116755 0.993161i $$-0.537249\pi$$
−0.116755 + 0.993161i $$0.537249\pi$$
$$390$$ 0 0
$$391$$ −6.42221 −0.324785
$$392$$ −14.2111 −0.717769
$$393$$ 22.6056i 1.14030i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 3.63331 0.182350 0.0911752 0.995835i $$-0.470938\pi$$
0.0911752 + 0.995835i $$0.470938\pi$$
$$398$$ 22.4222 1.12392
$$399$$ −21.2111 −1.06188
$$400$$ 0 0
$$401$$ 8.78890i 0.438897i 0.975624 + 0.219448i $$0.0704257\pi$$
−0.975624 + 0.219448i $$0.929574\pi$$
$$402$$ − 3.21110i − 0.160155i
$$403$$ 21.6333 1.07763
$$404$$ −7.39445 −0.367888
$$405$$ 0 0
$$406$$ 21.2111 1.05269
$$407$$ 0 0
$$408$$ 4.60555 0.228009
$$409$$ 14.7889i 0.731264i 0.930760 + 0.365632i $$0.119147\pi$$
−0.930760 + 0.365632i $$0.880853\pi$$
$$410$$ 0 0
$$411$$ − 3.21110i − 0.158392i
$$412$$ 4.00000i 0.197066i
$$413$$ 42.4222i 2.08746i
$$414$$ 1.39445i 0.0685334i
$$415$$ 0 0
$$416$$ 3.60555i 0.176777i
$$417$$ − 17.2111i − 0.842831i
$$418$$ 0 0
$$419$$ −4.18335 −0.204370 −0.102185 0.994765i $$-0.532583\pi$$
−0.102185 + 0.994765i $$0.532583\pi$$
$$420$$ 0 0
$$421$$ − 19.8167i − 0.965805i −0.875674 0.482902i $$-0.839583\pi$$
0.875674 0.482902i $$-0.160417\pi$$
$$422$$ 17.2111 0.837823
$$423$$ −9.21110 −0.447859
$$424$$ 6.00000i 0.291386i
$$425$$ 0 0
$$426$$ 9.21110 0.446279
$$427$$ −51.6333 −2.49871
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000i 0.578020i 0.957326 + 0.289010i $$0.0933260\pi$$
−0.957326 + 0.289010i $$0.906674\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 19.2111i − 0.923227i −0.887081 0.461613i $$-0.847271\pi$$
0.887081 0.461613i $$-0.152729\pi$$
$$434$$ − 27.6333i − 1.32644i
$$435$$ 0 0
$$436$$ − 1.39445i − 0.0667820i
$$437$$ −6.42221 −0.307216
$$438$$ 1.39445i 0.0666293i
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −14.2111 −0.676719
$$442$$ −16.6056 −0.789846
$$443$$ − 15.6333i − 0.742761i −0.928481 0.371380i $$-0.878885\pi$$
0.928481 0.371380i $$-0.121115\pi$$
$$444$$ 9.21110i 0.437140i
$$445$$ 0 0
$$446$$ −1.81665 −0.0860211
$$447$$ 15.2111 0.719460
$$448$$ 4.60555 0.217592
$$449$$ − 33.6333i − 1.58725i −0.608405 0.793627i $$-0.708190\pi$$
0.608405 0.793627i $$-0.291810\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 13.8167i 0.649881i
$$453$$ −6.00000 −0.281905
$$454$$ 24.0000 1.12638
$$455$$ 0 0
$$456$$ 4.60555 0.215675
$$457$$ −38.2389 −1.78874 −0.894369 0.447330i $$-0.852375\pi$$
−0.894369 + 0.447330i $$0.852375\pi$$
$$458$$ − 19.8167i − 0.925971i
$$459$$ 4.60555 0.214969
$$460$$ 0 0
$$461$$ − 33.6333i − 1.56646i −0.621733 0.783230i $$-0.713571\pi$$
0.621733 0.783230i $$-0.286429\pi$$
$$462$$ 0 0
$$463$$ −31.3944 −1.45902 −0.729512 0.683968i $$-0.760253\pi$$
−0.729512 + 0.683968i $$0.760253\pi$$
$$464$$ −4.60555 −0.213807
$$465$$ 0 0
$$466$$ − 1.81665i − 0.0841549i
$$467$$ − 30.4222i − 1.40777i −0.710313 0.703886i $$-0.751447\pi$$
0.710313 0.703886i $$-0.248553\pi$$
$$468$$ 3.60555i 0.166667i
$$469$$ 14.7889 0.682888
$$470$$ 0 0
$$471$$ −20.4222 −0.941006
$$472$$ − 9.21110i − 0.423975i
$$473$$ 0 0
$$474$$ − 14.4222i − 0.662434i
$$475$$ 0 0
$$476$$ 21.2111i 0.972209i
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ − 5.57779i − 0.254856i −0.991848 0.127428i $$-0.959328\pi$$
0.991848 0.127428i $$-0.0406722\pi$$
$$480$$ 0 0
$$481$$ − 33.2111i − 1.51430i
$$482$$ 6.42221i 0.292523i
$$483$$ −6.42221 −0.292220
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 0.972244 0.0440566 0.0220283 0.999757i $$-0.492988\pi$$
0.0220283 + 0.999757i $$0.492988\pi$$
$$488$$ 11.2111 0.507503
$$489$$ 24.4222i 1.10441i
$$490$$ 0 0
$$491$$ 7.81665 0.352761 0.176380 0.984322i $$-0.443561\pi$$
0.176380 + 0.984322i $$0.443561\pi$$
$$492$$ −3.21110 −0.144768
$$493$$ − 21.2111i − 0.955300i
$$494$$ −16.6056 −0.747119
$$495$$ 0 0
$$496$$ 6.00000i 0.269408i
$$497$$ 42.4222i 1.90290i
$$498$$ − 2.78890i − 0.124973i
$$499$$ − 23.0278i − 1.03086i −0.856930 0.515432i $$-0.827631\pi$$
0.856930 0.515432i $$-0.172369\pi$$
$$500$$ 0 0
$$501$$ − 9.21110i − 0.411522i
$$502$$ 13.3944 0.597824
$$503$$ − 23.4500i − 1.04558i −0.852461 0.522791i $$-0.824891\pi$$
0.852461 0.522791i $$-0.175109\pi$$
$$504$$ 4.60555 0.205148
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 13.0000i − 0.577350i
$$508$$ 1.21110i 0.0537340i
$$509$$ 33.6333i 1.49077i 0.666634 + 0.745385i $$0.267734\pi$$
−0.666634 + 0.745385i $$0.732266\pi$$
$$510$$ 0 0
$$511$$ −6.42221 −0.284102
$$512$$ −1.00000 −0.0441942
$$513$$ 4.60555 0.203340
$$514$$ 28.6056i 1.26174i
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ −42.4222 −1.86392
$$519$$ −12.4222 −0.545274
$$520$$ 0 0
$$521$$ 21.6333 0.947772 0.473886 0.880586i $$-0.342851\pi$$
0.473886 + 0.880586i $$0.342851\pi$$
$$522$$ −4.60555 −0.201580
$$523$$ − 32.8444i − 1.43619i −0.695947 0.718093i $$-0.745015\pi$$
0.695947 0.718093i $$-0.254985\pi$$
$$524$$ 22.6056 0.987528
$$525$$ 0 0
$$526$$ − 7.81665i − 0.340822i
$$527$$ −27.6333 −1.20373
$$528$$ 0 0
$$529$$ 21.0555 0.915457
$$530$$ 0 0
$$531$$ − 9.21110i − 0.399728i
$$532$$ 21.2111i 0.919618i
$$533$$ 11.5778 0.501490
$$534$$ −15.2111 −0.658249
$$535$$ 0 0
$$536$$ −3.21110 −0.138699
$$537$$ 19.8167i 0.855152i
$$538$$ 25.8167 1.11303
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 6.97224i 0.299760i 0.988704 + 0.149880i $$0.0478888\pi$$
−0.988704 + 0.149880i $$0.952111\pi$$
$$542$$ 0.422205i 0.0181353i
$$543$$ 8.42221i 0.361431i
$$544$$ − 4.60555i − 0.197461i
$$545$$ 0 0
$$546$$ −16.6056 −0.710652
$$547$$ 14.4222i 0.616649i 0.951281 + 0.308324i $$0.0997682\pi$$
−0.951281 + 0.308324i $$0.900232\pi$$
$$548$$ −3.21110 −0.137172
$$549$$ 11.2111 0.478478
$$550$$ 0 0
$$551$$ − 21.2111i − 0.903623i
$$552$$ 1.39445 0.0593517
$$553$$ 66.4222 2.82456
$$554$$ 16.4222i 0.697713i
$$555$$ 0 0
$$556$$ −17.2111 −0.729913
$$557$$ −11.5778 −0.490567 −0.245283 0.969451i $$-0.578881\pi$$
−0.245283 + 0.969451i $$0.578881\pi$$
$$558$$ 6.00000i 0.254000i
$$559$$ −28.8444 −1.21999
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 27.2111i − 1.14783i
$$563$$ − 34.0555i − 1.43527i −0.696420 0.717634i $$-0.745225\pi$$
0.696420 0.717634i $$-0.254775\pi$$
$$564$$ 9.21110i 0.387857i
$$565$$ 0 0
$$566$$ 10.4222i 0.438078i
$$567$$ 4.60555 0.193415
$$568$$ − 9.21110i − 0.386489i
$$569$$ −33.6333 −1.40998 −0.704991 0.709216i $$-0.749049\pi$$
−0.704991 + 0.709216i $$0.749049\pi$$
$$570$$ 0 0
$$571$$ −30.0555 −1.25778 −0.628892 0.777493i $$-0.716491\pi$$
−0.628892 + 0.777493i $$0.716491\pi$$
$$572$$ 0 0
$$573$$ − 12.0000i − 0.501307i
$$574$$ − 14.7889i − 0.617277i
$$575$$ 0 0
$$576$$ −1.00000 −0.0416667
$$577$$ 37.3944 1.55675 0.778376 0.627799i $$-0.216044\pi$$
0.778376 + 0.627799i $$0.216044\pi$$
$$578$$ 4.21110 0.175159
$$579$$ 7.81665i 0.324849i
$$580$$ 0 0
$$581$$ 12.8444 0.532876
$$582$$ 1.39445i 0.0578018i
$$583$$ 0 0
$$584$$ 1.39445 0.0577027
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 6.42221 0.265073 0.132536 0.991178i $$-0.457688\pi$$
0.132536 + 0.991178i $$0.457688\pi$$
$$588$$ 14.2111i 0.586056i
$$589$$ −27.6333 −1.13861
$$590$$ 0 0
$$591$$ 6.00000i 0.246807i
$$592$$ 9.21110 0.378574
$$593$$ −24.4222 −1.00290 −0.501450 0.865187i $$-0.667200\pi$$
−0.501450 + 0.865187i $$0.667200\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ − 15.2111i − 0.623071i
$$597$$ − 22.4222i − 0.917680i
$$598$$ −5.02776 −0.205600
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 1.63331 0.0666240 0.0333120 0.999445i $$-0.489394\pi$$
0.0333120 + 0.999445i $$0.489394\pi$$
$$602$$ 36.8444i 1.50167i
$$603$$ −3.21110 −0.130766
$$604$$ 6.00000i 0.244137i
$$605$$ 0 0
$$606$$ 7.39445i 0.300379i
$$607$$ 17.2111i 0.698577i 0.937015 + 0.349289i $$0.113577\pi$$
−0.937015 + 0.349289i $$0.886423\pi$$
$$608$$ − 4.60555i − 0.186780i
$$609$$ − 21.2111i − 0.859517i
$$610$$ 0 0
$$611$$ − 33.2111i − 1.34358i
$$612$$ − 4.60555i − 0.186168i
$$613$$ −33.2111 −1.34138 −0.670692 0.741736i $$-0.734003\pi$$
−0.670692 + 0.741736i $$0.734003\pi$$
$$614$$ −8.78890 −0.354691
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.4222 −0.500099 −0.250050 0.968233i $$-0.580447\pi$$
−0.250050 + 0.968233i $$0.580447\pi$$
$$618$$ 4.00000 0.160904
$$619$$ − 25.8167i − 1.03766i −0.854878 0.518829i $$-0.826368\pi$$
0.854878 0.518829i $$-0.173632\pi$$
$$620$$ 0 0
$$621$$ 1.39445 0.0559573
$$622$$ −12.0000 −0.481156
$$623$$ − 70.0555i − 2.80671i
$$624$$ 3.60555 0.144338
$$625$$ 0 0
$$626$$ 3.57779i 0.142997i
$$627$$ 0 0
$$628$$ 20.4222i 0.814935i
$$629$$ 42.4222i 1.69148i
$$630$$ 0 0
$$631$$ 3.21110i 0.127832i 0.997955 + 0.0639160i $$0.0203590\pi$$
−0.997955 + 0.0639160i $$0.979641\pi$$
$$632$$ −14.4222 −0.573685
$$633$$ − 17.2111i − 0.684080i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ − 51.2389i − 2.03016i
$$638$$ 0 0
$$639$$ − 9.21110i − 0.364386i
$$640$$ 0 0
$$641$$ 0.422205 0.0166761 0.00833805 0.999965i $$-0.497346\pi$$
0.00833805 + 0.999965i $$0.497346\pi$$
$$642$$ 0 0
$$643$$ 9.63331 0.379901 0.189950 0.981794i $$-0.439167\pi$$
0.189950 + 0.981794i $$0.439167\pi$$
$$644$$ 6.42221i 0.253070i
$$645$$ 0 0
$$646$$ 21.2111 0.834540
$$647$$ − 34.6056i − 1.36048i −0.732987 0.680242i $$-0.761875\pi$$
0.732987 0.680242i $$-0.238125\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −27.6333 −1.08303
$$652$$ 24.4222 0.956447
$$653$$ − 39.2111i − 1.53445i −0.641379 0.767225i $$-0.721637\pi$$
0.641379 0.767225i $$-0.278363\pi$$
$$654$$ −1.39445 −0.0545273
$$655$$ 0 0
$$656$$ 3.21110i 0.125372i
$$657$$ 1.39445 0.0544026
$$658$$ −42.4222 −1.65379
$$659$$ 26.2389 1.02212 0.511060 0.859545i $$-0.329253\pi$$
0.511060 + 0.859545i $$0.329253\pi$$
$$660$$ 0 0
$$661$$ 50.2389i 1.95407i 0.213090 + 0.977033i $$0.431647\pi$$
−0.213090 + 0.977033i $$0.568353\pi$$
$$662$$ 16.6056i 0.645393i
$$663$$ 16.6056i 0.644906i
$$664$$ −2.78890 −0.108230
$$665$$ 0 0
$$666$$ 9.21110 0.356923
$$667$$ − 6.42221i − 0.248669i
$$668$$ −9.21110 −0.356388
$$669$$ 1.81665i 0.0702359i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 4.60555i − 0.177663i
$$673$$ 37.6333i 1.45066i 0.688403 + 0.725329i $$0.258312\pi$$
−0.688403 + 0.725329i $$0.741688\pi$$
$$674$$ − 13.6333i − 0.525135i
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ − 28.0555i − 1.07826i −0.842222 0.539130i $$-0.818753\pi$$
0.842222 0.539130i $$-0.181247\pi$$
$$678$$ 13.8167 0.530625
$$679$$ −6.42221 −0.246462
$$680$$ 0 0
$$681$$ − 24.0000i − 0.919682i
$$682$$ 0 0
$$683$$ 9.21110 0.352453 0.176227 0.984350i $$-0.443611\pi$$
0.176227 + 0.984350i $$0.443611\pi$$
$$684$$ − 4.60555i − 0.176098i
$$685$$ 0 0
$$686$$ −33.2111 −1.26801
$$687$$ −19.8167 −0.756053
$$688$$ − 8.00000i − 0.304997i
$$689$$ −21.6333 −0.824163
$$690$$ 0 0
$$691$$ 20.2389i 0.769922i 0.922933 + 0.384961i $$0.125785\pi$$
−0.922933 + 0.384961i $$0.874215\pi$$
$$692$$ 12.4222i 0.472221i
$$693$$ 0 0
$$694$$ 27.6333i 1.04895i
$$695$$ 0 0
$$696$$ 4.60555i 0.174573i
$$697$$ −14.7889 −0.560169
$$698$$ − 7.81665i − 0.295865i
$$699$$ −1.81665 −0.0687122
$$700$$ 0 0
$$701$$ 47.0278 1.77621 0.888107 0.459637i $$-0.152020\pi$$
0.888107 + 0.459637i $$0.152020\pi$$
$$702$$ 3.60555 0.136083
$$703$$ 42.4222i 1.59998i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 8.78890 0.330775
$$707$$ −34.0555 −1.28079
$$708$$ −9.21110 −0.346174
$$709$$ 1.39445i 0.0523696i 0.999657 + 0.0261848i $$0.00833584\pi$$
−0.999657 + 0.0261848i $$0.991664\pi$$
$$710$$ 0 0
$$711$$ −14.4222 −0.540875
$$712$$ 15.2111i 0.570060i
$$713$$ −8.36669 −0.313335
$$714$$ 21.2111 0.793806
$$715$$ 0 0
$$716$$ 19.8167 0.740583
$$717$$ 0 0
$$718$$ 15.6333i 0.583430i
$$719$$ 51.6333 1.92560 0.962799 0.270220i $$-0.0870963\pi$$
0.962799 + 0.270220i $$0.0870963\pi$$
$$720$$ 0 0
$$721$$ 18.4222i 0.686079i
$$722$$ 2.21110 0.0822887
$$723$$ 6.42221 0.238844
$$724$$ 8.42221 0.313009
$$725$$ 0 0
$$726$$ − 11.0000i − 0.408248i
$$727$$ − 14.4222i − 0.534890i −0.963573 0.267445i $$-0.913821\pi$$
0.963573 0.267445i $$-0.0861794\pi$$
$$728$$ 16.6056i 0.615443i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 36.8444 1.36274
$$732$$ − 11.2111i − 0.414374i
$$733$$ −34.0555 −1.25787 −0.628935 0.777458i $$-0.716509\pi$$
−0.628935 + 0.777458i $$0.716509\pi$$
$$734$$ 19.6333i 0.724679i
$$735$$ 0 0
$$736$$ − 1.39445i − 0.0514001i
$$737$$ 0 0
$$738$$ 3.21110i 0.118202i
$$739$$ − 20.2389i − 0.744498i −0.928133 0.372249i $$-0.878587\pi$$
0.928133 0.372249i $$-0.121413\pi$$
$$740$$ 0 0
$$741$$ 16.6056i 0.610020i
$$742$$ 27.6333i 1.01445i
$$743$$ 36.8444 1.35169 0.675845 0.737044i $$-0.263779\pi$$
0.675845 + 0.737044i $$0.263779\pi$$
$$744$$ 6.00000 0.219971
$$745$$ 0 0
$$746$$ − 20.4222i − 0.747710i
$$747$$ −2.78890 −0.102040
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.4222 −0.380312 −0.190156 0.981754i $$-0.560899\pi$$
−0.190156 + 0.981754i $$0.560899\pi$$
$$752$$ 9.21110 0.335894
$$753$$ − 13.3944i − 0.488121i
$$754$$ − 16.6056i − 0.604739i
$$755$$ 0 0
$$756$$ − 4.60555i − 0.167502i
$$757$$ 12.7889i 0.464820i 0.972618 + 0.232410i $$0.0746612\pi$$
−0.972618 + 0.232410i $$0.925339\pi$$
$$758$$ 35.0278i 1.27227i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.6333i 1.21921i 0.792707 + 0.609603i $$0.208671\pi$$
−0.792707 + 0.609603i $$0.791329\pi$$
$$762$$ 1.21110 0.0438736
$$763$$ − 6.42221i − 0.232499i
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 27.6333 0.998432
$$767$$ 33.2111 1.19918
$$768$$ 1.00000i 0.0360844i
$$769$$ 12.8444i 0.463181i 0.972813 + 0.231591i $$0.0743930\pi$$
−0.972813 + 0.231591i $$0.925607\pi$$
$$770$$ 0 0
$$771$$ 28.6056 1.03020
$$772$$ 7.81665 0.281328
$$773$$ 30.0000 1.07903 0.539513 0.841978i $$-0.318609\pi$$
0.539513 + 0.841978i $$0.318609\pi$$
$$774$$ − 8.00000i − 0.287554i
$$775$$ 0 0
$$776$$ 1.39445 0.0500578
$$777$$ 42.4222i 1.52189i
$$778$$ 4.60555 0.165117
$$779$$ −14.7889 −0.529867
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 6.42221 0.229658
$$783$$ 4.60555i 0.164589i
$$784$$ 14.2111 0.507539
$$785$$ 0 0
$$786$$ − 22.6056i − 0.806313i
$$787$$ −49.2666 −1.75617 −0.878083 0.478509i $$-0.841177\pi$$
−0.878083 + 0.478509i $$0.841177\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −7.81665 −0.278280
$$790$$ 0 0
$$791$$ 63.6333i 2.26254i
$$792$$ 0 0
$$793$$ 40.4222i 1.43543i
$$794$$ −3.63331 −0.128941
$$795$$ 0 0
$$796$$ −22.4222 −0.794734
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 21.2111 0.750865
$$799$$ 42.4222i 1.50079i
$$800$$ 0 0
$$801$$ 15.2111i 0.537458i
$$802$$ − 8.78890i − 0.310347i
$$803$$ 0 0
$$804$$ 3.21110i 0.113247i
$$805$$ 0 0
$$806$$ −21.6333 −0.762001
$$807$$ − 25.8167i − 0.908789i
$$808$$ 7.39445 0.260136
$$809$$ 6.84441 0.240637 0.120318 0.992735i $$-0.461608\pi$$
0.120318 + 0.992735i $$0.461608\pi$$
$$810$$ 0 0
$$811$$ 32.2389i 1.13206i 0.824385 + 0.566030i $$0.191521\pi$$
−0.824385 + 0.566030i $$0.808479\pi$$
$$812$$ −21.2111 −0.744364
$$813$$ 0.422205 0.0148074
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −4.60555 −0.161227
$$817$$ 36.8444 1.28902
$$818$$ − 14.7889i − 0.517082i
$$819$$ 16.6056i 0.580245i
$$820$$ 0 0
$$821$$ − 3.21110i − 0.112068i −0.998429 0.0560341i $$-0.982154\pi$$
0.998429 0.0560341i $$-0.0178456\pi$$
$$822$$ 3.21110i 0.112000i
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ − 4.00000i − 0.139347i
$$825$$ 0 0
$$826$$ − 42.4222i − 1.47606i
$$827$$ 27.6333 0.960904 0.480452 0.877021i $$-0.340473\pi$$
0.480452 + 0.877021i $$0.340473\pi$$
$$828$$ − 1.39445i − 0.0484604i
$$829$$ −46.8444 −1.62697 −0.813487 0.581583i $$-0.802433\pi$$
−0.813487 + 0.581583i $$0.802433\pi$$
$$830$$ 0 0
$$831$$ 16.4222 0.569680
$$832$$ − 3.60555i − 0.125000i
$$833$$ 65.4500i 2.26771i
$$834$$ 17.2111i 0.595972i
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 6.00000 0.207390
$$838$$ 4.18335 0.144511
$$839$$ 18.4222i 0.636005i 0.948090 + 0.318003i $$0.103012\pi$$
−0.948090 + 0.318003i $$0.896988\pi$$
$$840$$ 0 0
$$841$$ −7.78890 −0.268583
$$842$$ 19.8167i 0.682927i
$$843$$ −27.2111 −0.937200
$$844$$ −17.2111 −0.592431
$$845$$ 0 0
$$846$$ 9.21110 0.316684
$$847$$ 50.6611 1.74073
$$848$$ − 6.00000i − 0.206041i
$$849$$ 10.4222 0.357689
$$850$$ 0 0
$$851$$ 12.8444i 0.440301i
$$852$$ −9.21110 −0.315567
$$853$$ −14.7889 −0.506362 −0.253181 0.967419i $$-0.581477\pi$$
−0.253181 + 0.967419i $$0.581477\pi$$
$$854$$ 51.6333 1.76686
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 23.0278i − 0.786613i −0.919407 0.393307i $$-0.871331\pi$$
0.919407 0.393307i $$-0.128669\pi$$
$$858$$ 0 0
$$859$$ −25.2111 −0.860192 −0.430096 0.902783i $$-0.641520\pi$$
−0.430096 + 0.902783i $$0.641520\pi$$
$$860$$ 0 0
$$861$$ −14.7889 −0.504004
$$862$$ − 12.0000i − 0.408722i
$$863$$ 51.6333 1.75762 0.878809 0.477173i $$-0.158339\pi$$
0.878809 + 0.477173i $$0.158339\pi$$
$$864$$ 1.00000i 0.0340207i
$$865$$ 0 0
$$866$$ 19.2111i 0.652820i
$$867$$ − 4.21110i − 0.143017i
$$868$$ 27.6333i 0.937936i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ − 11.5778i − 0.392299i
$$872$$ 1.39445i 0.0472220i
$$873$$ 1.39445 0.0471949
$$874$$ 6.42221 0.217234
$$875$$ 0 0
$$876$$ − 1.39445i − 0.0471141i
$$877$$ 24.8444 0.838936 0.419468 0.907770i $$-0.362217\pi$$
0.419468 + 0.907770i $$0.362217\pi$$
$$878$$ 8.00000 0.269987
$$879$$ 18.0000i 0.607125i
$$880$$ 0 0
$$881$$ 39.2111 1.32106 0.660528 0.750802i $$-0.270333\pi$$
0.660528 + 0.750802i $$0.270333\pi$$
$$882$$ 14.2111 0.478513
$$883$$ − 9.57779i − 0.322318i −0.986928 0.161159i $$-0.948477\pi$$
0.986928 0.161159i $$-0.0515233\pi$$
$$884$$ 16.6056 0.558505
$$885$$ 0 0
$$886$$ 15.6333i 0.525211i
$$887$$ 6.97224i 0.234105i 0.993126 + 0.117053i $$0.0373446\pi$$
−0.993126 + 0.117053i $$0.962655\pi$$
$$888$$ − 9.21110i − 0.309104i
$$889$$ 5.57779i 0.187073i
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 1.81665 0.0608261
$$893$$ 42.4222i 1.41960i
$$894$$ −15.2111 −0.508735
$$895$$ 0 0
$$896$$ −4.60555 −0.153861
$$897$$ 5.02776i 0.167872i
$$898$$ 33.6333i 1.12236i
$$899$$ − 27.6333i − 0.921622i
$$900$$ 0 0
$$901$$ 27.6333 0.920599
$$902$$ 0 0
$$903$$ 36.8444 1.22611
$$904$$ − 13.8167i − 0.459535i
$$905$$ 0 0
$$906$$ 6.00000 0.199337
$$907$$ 21.5778i 0.716479i 0.933630 + 0.358239i $$0.116623\pi$$
−0.933630 + 0.358239i $$0.883377\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 7.39445 0.245258
$$910$$ 0 0
$$911$$ −27.6333 −0.915532 −0.457766 0.889073i $$-0.651350\pi$$
−0.457766 + 0.889073i $$0.651350\pi$$
$$912$$ −4.60555 −0.152505
$$913$$ 0 0
$$914$$ 38.2389 1.26483
$$915$$ 0 0
$$916$$ 19.8167i 0.654761i
$$917$$ 104.111 3.43805
$$918$$ −4.60555 −0.152006
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 8.78890i 0.289604i
$$922$$ 33.6333i 1.10765i
$$923$$ 33.2111 1.09316
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 31.3944 1.03169
$$927$$ − 4.00000i − 0.131377i
$$928$$ 4.60555 0.151185
$$929$$ − 39.2111i − 1.28647i −0.765667 0.643237i $$-0.777591\pi$$
0.765667 0.643237i $$-0.222409\pi$$
$$930$$ 0 0
$$931$$ 65.4500i 2.14504i
$$932$$ 1.81665i 0.0595065i
$$933$$ 12.0000i 0.392862i
$$934$$ 30.4222i 0.995445i
$$935$$ 0 0
$$936$$ − 3.60555i − 0.117851i
$$937$$ − 10.3667i − 0.338665i −0.985559 0.169333i $$-0.945839\pi$$
0.985559 0.169333i $$-0.0541612\pi$$
$$938$$ −14.7889 −0.482875
$$939$$ 3.57779 0.116757
$$940$$ 0 0
$$941$$ − 54.0000i − 1.76035i −0.474650 0.880175i $$-0.657425\pi$$
0.474650 0.880175i $$-0.342575\pi$$
$$942$$ 20.4222 0.665391
$$943$$ −4.47772 −0.145815
$$944$$ 9.21110i 0.299796i
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 15.6333 0.508014 0.254007 0.967202i $$-0.418251\pi$$
0.254007 + 0.967202i $$0.418251\pi$$
$$948$$ 14.4222i 0.468411i
$$949$$ 5.02776i 0.163208i
$$950$$ 0 0
$$951$$ 18.0000i 0.583690i
$$952$$ − 21.2111i − 0.687456i
$$953$$ 20.2389i 0.655601i 0.944747 + 0.327800i $$0.106307\pi$$
−0.944747 + 0.327800i $$0.893693\pi$$
$$954$$ − 6.00000i − 0.194257i
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 5.57779i 0.180210i
$$959$$ −14.7889 −0.477558
$$960$$ 0 0
$$961$$ −5.00000 −0.161290
$$962$$ 33.2111i 1.07077i
$$963$$ 0 0
$$964$$ − 6.42221i − 0.206845i
$$965$$ 0 0
$$966$$ 6.42221 0.206631
$$967$$ 8.23886 0.264944 0.132472 0.991187i $$-0.457709\pi$$
0.132472 + 0.991187i $$0.457709\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ − 21.2111i − 0.681399i
$$970$$ 0 0
$$971$$ −53.0278 −1.70174 −0.850871 0.525375i $$-0.823925\pi$$
−0.850871 + 0.525375i $$0.823925\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ −79.2666 −2.54117
$$974$$ −0.972244 −0.0311527
$$975$$ 0 0
$$976$$ −11.2111 −0.358859
$$977$$ −18.8444 −0.602886 −0.301443 0.953484i $$-0.597468\pi$$
−0.301443 + 0.953484i $$0.597468\pi$$
$$978$$ − 24.4222i − 0.780936i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 1.39445i 0.0445213i
$$982$$ −7.81665 −0.249439
$$983$$ −42.4222 −1.35306 −0.676529 0.736416i $$-0.736517\pi$$
−0.676529 + 0.736416i $$0.736517\pi$$
$$984$$ 3.21110 0.102366
$$985$$ 0 0
$$986$$ 21.2111i 0.675499i
$$987$$ 42.4222i 1.35031i
$$988$$ 16.6056 0.528293
$$989$$ 11.1556 0.354727
$$990$$ 0 0
$$991$$ −22.4222 −0.712265 −0.356132 0.934436i $$-0.615905\pi$$
−0.356132 + 0.934436i $$0.615905\pi$$
$$992$$ − 6.00000i − 0.190500i
$$993$$ 16.6056 0.526961
$$994$$ − 42.4222i − 1.34555i
$$995$$ 0 0
$$996$$ 2.78890i 0.0883696i
$$997$$ − 16.4222i − 0.520096i −0.965596 0.260048i $$-0.916262\pi$$
0.965596 0.260048i $$-0.0837385\pi$$
$$998$$ 23.0278i 0.728931i
$$999$$ − 9.21110i − 0.291426i
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 1950.2.f.m.649.4 4
5.2 odd 4 1950.2.b.k.1351.2 4
5.3 odd 4 390.2.b.c.181.3 yes 4
5.4 even 2 1950.2.f.n.649.1 4
13.12 even 2 1950.2.f.n.649.3 4
15.8 even 4 1170.2.b.d.181.1 4
20.3 even 4 3120.2.g.q.961.4 4
65.8 even 4 5070.2.a.z.1.1 2
65.12 odd 4 1950.2.b.k.1351.3 4
65.18 even 4 5070.2.a.bf.1.2 2
65.38 odd 4 390.2.b.c.181.2 4
65.64 even 2 inner 1950.2.f.m.649.2 4
195.38 even 4 1170.2.b.d.181.4 4
260.103 even 4 3120.2.g.q.961.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 65.38 odd 4
390.2.b.c.181.3 yes 4 5.3 odd 4
1170.2.b.d.181.1 4 15.8 even 4
1170.2.b.d.181.4 4 195.38 even 4
1950.2.b.k.1351.2 4 5.2 odd 4
1950.2.b.k.1351.3 4 65.12 odd 4
1950.2.f.m.649.2 4 65.64 even 2 inner
1950.2.f.m.649.4 4 1.1 even 1 trivial
1950.2.f.n.649.1 4 5.4 even 2
1950.2.f.n.649.3 4 13.12 even 2
3120.2.g.q.961.1 4 260.103 even 4
3120.2.g.q.961.4 4 20.3 even 4
5070.2.a.z.1.1 2 65.8 even 4
5070.2.a.bf.1.2 2 65.18 even 4