# Properties

 Label 1078.2.a.w.1.1 Level $1078$ Weight $2$ Character 1078.1 Self dual yes Analytic conductor $8.608$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 1078.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} -1.23607 q^{6} +1.00000 q^{8} -1.47214 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} -1.23607 q^{6} +1.00000 q^{8} -1.47214 q^{9} +1.23607 q^{10} +1.00000 q^{11} -1.23607 q^{12} +3.23607 q^{13} -1.52786 q^{15} +1.00000 q^{16} -2.47214 q^{17} -1.47214 q^{18} +7.23607 q^{19} +1.23607 q^{20} +1.00000 q^{22} +4.00000 q^{23} -1.23607 q^{24} -3.47214 q^{25} +3.23607 q^{26} +5.52786 q^{27} +4.47214 q^{29} -1.52786 q^{30} -2.00000 q^{31} +1.00000 q^{32} -1.23607 q^{33} -2.47214 q^{34} -1.47214 q^{36} +6.94427 q^{37} +7.23607 q^{38} -4.00000 q^{39} +1.23607 q^{40} +2.47214 q^{41} -10.4721 q^{43} +1.00000 q^{44} -1.81966 q^{45} +4.00000 q^{46} +2.00000 q^{47} -1.23607 q^{48} -3.47214 q^{50} +3.05573 q^{51} +3.23607 q^{52} +8.47214 q^{53} +5.52786 q^{54} +1.23607 q^{55} -8.94427 q^{57} +4.47214 q^{58} -2.76393 q^{59} -1.52786 q^{60} +0.763932 q^{61} -2.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -1.23607 q^{66} +11.4164 q^{67} -2.47214 q^{68} -4.94427 q^{69} +6.47214 q^{71} -1.47214 q^{72} -12.9443 q^{73} +6.94427 q^{74} +4.29180 q^{75} +7.23607 q^{76} -4.00000 q^{78} +1.23607 q^{80} -2.41641 q^{81} +2.47214 q^{82} +12.1803 q^{83} -3.05573 q^{85} -10.4721 q^{86} -5.52786 q^{87} +1.00000 q^{88} -10.0000 q^{89} -1.81966 q^{90} +4.00000 q^{92} +2.47214 q^{93} +2.00000 q^{94} +8.94427 q^{95} -1.23607 q^{96} -12.4721 q^{97} -1.47214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 6 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 6 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} - 12 q^{15} + 2 q^{16} + 4 q^{17} + 6 q^{18} + 10 q^{19} - 2 q^{20} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 20 q^{27} - 12 q^{30} - 4 q^{31} + 2 q^{32} + 2 q^{33} + 4 q^{34} + 6 q^{36} - 4 q^{37} + 10 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} - 12 q^{43} + 2 q^{44} - 26 q^{45} + 8 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{50} + 24 q^{51} + 2 q^{52} + 8 q^{53} + 20 q^{54} - 2 q^{55} - 10 q^{59} - 12 q^{60} + 6 q^{61} - 4 q^{62} + 2 q^{64} + 8 q^{65} + 2 q^{66} - 4 q^{67} + 4 q^{68} + 8 q^{69} + 4 q^{71} + 6 q^{72} - 8 q^{73} - 4 q^{74} + 22 q^{75} + 10 q^{76} - 8 q^{78} - 2 q^{80} + 22 q^{81} - 4 q^{82} + 2 q^{83} - 24 q^{85} - 12 q^{86} - 20 q^{87} + 2 q^{88} - 20 q^{89} - 26 q^{90} + 8 q^{92} - 4 q^{93} + 4 q^{94} + 2 q^{96} - 16 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 6 * q^9 - 2 * q^10 + 2 * q^11 + 2 * q^12 + 2 * q^13 - 12 * q^15 + 2 * q^16 + 4 * q^17 + 6 * q^18 + 10 * q^19 - 2 * q^20 + 2 * q^22 + 8 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 + 20 * q^27 - 12 * q^30 - 4 * q^31 + 2 * q^32 + 2 * q^33 + 4 * q^34 + 6 * q^36 - 4 * q^37 + 10 * q^38 - 8 * q^39 - 2 * q^40 - 4 * q^41 - 12 * q^43 + 2 * q^44 - 26 * q^45 + 8 * q^46 + 4 * q^47 + 2 * q^48 + 2 * q^50 + 24 * q^51 + 2 * q^52 + 8 * q^53 + 20 * q^54 - 2 * q^55 - 10 * q^59 - 12 * q^60 + 6 * q^61 - 4 * q^62 + 2 * q^64 + 8 * q^65 + 2 * q^66 - 4 * q^67 + 4 * q^68 + 8 * q^69 + 4 * q^71 + 6 * q^72 - 8 * q^73 - 4 * q^74 + 22 * q^75 + 10 * q^76 - 8 * q^78 - 2 * q^80 + 22 * q^81 - 4 * q^82 + 2 * q^83 - 24 * q^85 - 12 * q^86 - 20 * q^87 + 2 * q^88 - 20 * q^89 - 26 * q^90 + 8 * q^92 - 4 * q^93 + 4 * q^94 + 2 * q^96 - 16 * q^97 + 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.23607 −0.713644 −0.356822 0.934172i $$-0.616140\pi$$
−0.356822 + 0.934172i $$0.616140\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.23607 0.552786 0.276393 0.961045i $$-0.410861\pi$$
0.276393 + 0.961045i $$0.410861\pi$$
$$6$$ −1.23607 −0.504623
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ −1.47214 −0.490712
$$10$$ 1.23607 0.390879
$$11$$ 1.00000 0.301511
$$12$$ −1.23607 −0.356822
$$13$$ 3.23607 0.897524 0.448762 0.893651i $$-0.351865\pi$$
0.448762 + 0.893651i $$0.351865\pi$$
$$14$$ 0 0
$$15$$ −1.52786 −0.394493
$$16$$ 1.00000 0.250000
$$17$$ −2.47214 −0.599581 −0.299791 0.954005i $$-0.596917\pi$$
−0.299791 + 0.954005i $$0.596917\pi$$
$$18$$ −1.47214 −0.346986
$$19$$ 7.23607 1.66007 0.830034 0.557713i $$-0.188321\pi$$
0.830034 + 0.557713i $$0.188321\pi$$
$$20$$ 1.23607 0.276393
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −1.23607 −0.252311
$$25$$ −3.47214 −0.694427
$$26$$ 3.23607 0.634645
$$27$$ 5.52786 1.06384
$$28$$ 0 0
$$29$$ 4.47214 0.830455 0.415227 0.909718i $$-0.363702\pi$$
0.415227 + 0.909718i $$0.363702\pi$$
$$30$$ −1.52786 −0.278949
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −1.23607 −0.215172
$$34$$ −2.47214 −0.423968
$$35$$ 0 0
$$36$$ −1.47214 −0.245356
$$37$$ 6.94427 1.14163 0.570816 0.821078i $$-0.306627\pi$$
0.570816 + 0.821078i $$0.306627\pi$$
$$38$$ 7.23607 1.17385
$$39$$ −4.00000 −0.640513
$$40$$ 1.23607 0.195440
$$41$$ 2.47214 0.386083 0.193041 0.981191i $$-0.438165\pi$$
0.193041 + 0.981191i $$0.438165\pi$$
$$42$$ 0 0
$$43$$ −10.4721 −1.59699 −0.798493 0.602004i $$-0.794369\pi$$
−0.798493 + 0.602004i $$0.794369\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −1.81966 −0.271259
$$46$$ 4.00000 0.589768
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ −1.23607 −0.178411
$$49$$ 0 0
$$50$$ −3.47214 −0.491034
$$51$$ 3.05573 0.427888
$$52$$ 3.23607 0.448762
$$53$$ 8.47214 1.16374 0.581869 0.813283i $$-0.302322\pi$$
0.581869 + 0.813283i $$0.302322\pi$$
$$54$$ 5.52786 0.752247
$$55$$ 1.23607 0.166671
$$56$$ 0 0
$$57$$ −8.94427 −1.18470
$$58$$ 4.47214 0.587220
$$59$$ −2.76393 −0.359833 −0.179917 0.983682i $$-0.557583\pi$$
−0.179917 + 0.983682i $$0.557583\pi$$
$$60$$ −1.52786 −0.197246
$$61$$ 0.763932 0.0978115 0.0489057 0.998803i $$-0.484427\pi$$
0.0489057 + 0.998803i $$0.484427\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 4.00000 0.496139
$$66$$ −1.23607 −0.152149
$$67$$ 11.4164 1.39474 0.697368 0.716713i $$-0.254354\pi$$
0.697368 + 0.716713i $$0.254354\pi$$
$$68$$ −2.47214 −0.299791
$$69$$ −4.94427 −0.595220
$$70$$ 0 0
$$71$$ 6.47214 0.768101 0.384051 0.923312i $$-0.374529\pi$$
0.384051 + 0.923312i $$0.374529\pi$$
$$72$$ −1.47214 −0.173493
$$73$$ −12.9443 −1.51501 −0.757506 0.652828i $$-0.773582\pi$$
−0.757506 + 0.652828i $$0.773582\pi$$
$$74$$ 6.94427 0.807255
$$75$$ 4.29180 0.495574
$$76$$ 7.23607 0.830034
$$77$$ 0 0
$$78$$ −4.00000 −0.452911
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 1.23607 0.138197
$$81$$ −2.41641 −0.268490
$$82$$ 2.47214 0.273002
$$83$$ 12.1803 1.33697 0.668483 0.743727i $$-0.266944\pi$$
0.668483 + 0.743727i $$0.266944\pi$$
$$84$$ 0 0
$$85$$ −3.05573 −0.331440
$$86$$ −10.4721 −1.12924
$$87$$ −5.52786 −0.592649
$$88$$ 1.00000 0.106600
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ −1.81966 −0.191809
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 2.47214 0.256349
$$94$$ 2.00000 0.206284
$$95$$ 8.94427 0.917663
$$96$$ −1.23607 −0.126156
$$97$$ −12.4721 −1.26635 −0.633177 0.774007i $$-0.718249\pi$$
−0.633177 + 0.774007i $$0.718249\pi$$
$$98$$ 0 0
$$99$$ −1.47214 −0.147955
$$100$$ −3.47214 −0.347214
$$101$$ −8.18034 −0.813974 −0.406987 0.913434i $$-0.633421\pi$$
−0.406987 + 0.913434i $$0.633421\pi$$
$$102$$ 3.05573 0.302562
$$103$$ 14.9443 1.47250 0.736251 0.676708i $$-0.236594\pi$$
0.736251 + 0.676708i $$0.236594\pi$$
$$104$$ 3.23607 0.317323
$$105$$ 0 0
$$106$$ 8.47214 0.822887
$$107$$ 2.47214 0.238990 0.119495 0.992835i $$-0.461872\pi$$
0.119495 + 0.992835i $$0.461872\pi$$
$$108$$ 5.52786 0.531919
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 1.23607 0.117854
$$111$$ −8.58359 −0.814719
$$112$$ 0 0
$$113$$ −0.472136 −0.0444148 −0.0222074 0.999753i $$-0.507069\pi$$
−0.0222074 + 0.999753i $$0.507069\pi$$
$$114$$ −8.94427 −0.837708
$$115$$ 4.94427 0.461056
$$116$$ 4.47214 0.415227
$$117$$ −4.76393 −0.440426
$$118$$ −2.76393 −0.254441
$$119$$ 0 0
$$120$$ −1.52786 −0.139474
$$121$$ 1.00000 0.0909091
$$122$$ 0.763932 0.0691632
$$123$$ −3.05573 −0.275526
$$124$$ −2.00000 −0.179605
$$125$$ −10.4721 −0.936656
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 12.9443 1.13968
$$130$$ 4.00000 0.350823
$$131$$ −4.76393 −0.416227 −0.208113 0.978105i $$-0.566732\pi$$
−0.208113 + 0.978105i $$0.566732\pi$$
$$132$$ −1.23607 −0.107586
$$133$$ 0 0
$$134$$ 11.4164 0.986227
$$135$$ 6.83282 0.588075
$$136$$ −2.47214 −0.211984
$$137$$ −19.8885 −1.69919 −0.849596 0.527433i $$-0.823154\pi$$
−0.849596 + 0.527433i $$0.823154\pi$$
$$138$$ −4.94427 −0.420884
$$139$$ 21.7082 1.84127 0.920633 0.390429i $$-0.127673\pi$$
0.920633 + 0.390429i $$0.127673\pi$$
$$140$$ 0 0
$$141$$ −2.47214 −0.208191
$$142$$ 6.47214 0.543130
$$143$$ 3.23607 0.270614
$$144$$ −1.47214 −0.122678
$$145$$ 5.52786 0.459064
$$146$$ −12.9443 −1.07128
$$147$$ 0 0
$$148$$ 6.94427 0.570816
$$149$$ −22.3607 −1.83186 −0.915929 0.401340i $$-0.868545\pi$$
−0.915929 + 0.401340i $$0.868545\pi$$
$$150$$ 4.29180 0.350424
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 7.23607 0.586923
$$153$$ 3.63932 0.294222
$$154$$ 0 0
$$155$$ −2.47214 −0.198567
$$156$$ −4.00000 −0.320256
$$157$$ 12.6525 1.00978 0.504889 0.863184i $$-0.331534\pi$$
0.504889 + 0.863184i $$0.331534\pi$$
$$158$$ 0 0
$$159$$ −10.4721 −0.830494
$$160$$ 1.23607 0.0977198
$$161$$ 0 0
$$162$$ −2.41641 −0.189851
$$163$$ −19.4164 −1.52081 −0.760405 0.649449i $$-0.775000\pi$$
−0.760405 + 0.649449i $$0.775000\pi$$
$$164$$ 2.47214 0.193041
$$165$$ −1.52786 −0.118944
$$166$$ 12.1803 0.945378
$$167$$ −11.4164 −0.883428 −0.441714 0.897156i $$-0.645629\pi$$
−0.441714 + 0.897156i $$0.645629\pi$$
$$168$$ 0 0
$$169$$ −2.52786 −0.194451
$$170$$ −3.05573 −0.234364
$$171$$ −10.6525 −0.814615
$$172$$ −10.4721 −0.798493
$$173$$ 3.23607 0.246034 0.123017 0.992405i $$-0.460743\pi$$
0.123017 + 0.992405i $$0.460743\pi$$
$$174$$ −5.52786 −0.419066
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 3.41641 0.256793
$$178$$ −10.0000 −0.749532
$$179$$ −8.94427 −0.668526 −0.334263 0.942480i $$-0.608487\pi$$
−0.334263 + 0.942480i $$0.608487\pi$$
$$180$$ −1.81966 −0.135629
$$181$$ −9.23607 −0.686512 −0.343256 0.939242i $$-0.611530\pi$$
−0.343256 + 0.939242i $$0.611530\pi$$
$$182$$ 0 0
$$183$$ −0.944272 −0.0698026
$$184$$ 4.00000 0.294884
$$185$$ 8.58359 0.631078
$$186$$ 2.47214 0.181266
$$187$$ −2.47214 −0.180780
$$188$$ 2.00000 0.145865
$$189$$ 0 0
$$190$$ 8.94427 0.648886
$$191$$ −2.47214 −0.178877 −0.0894387 0.995992i $$-0.528507\pi$$
−0.0894387 + 0.995992i $$0.528507\pi$$
$$192$$ −1.23607 −0.0892055
$$193$$ −14.9443 −1.07571 −0.537856 0.843037i $$-0.680766\pi$$
−0.537856 + 0.843037i $$0.680766\pi$$
$$194$$ −12.4721 −0.895447
$$195$$ −4.94427 −0.354067
$$196$$ 0 0
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ −1.47214 −0.104620
$$199$$ −18.9443 −1.34292 −0.671462 0.741039i $$-0.734333\pi$$
−0.671462 + 0.741039i $$0.734333\pi$$
$$200$$ −3.47214 −0.245517
$$201$$ −14.1115 −0.995345
$$202$$ −8.18034 −0.575567
$$203$$ 0 0
$$204$$ 3.05573 0.213944
$$205$$ 3.05573 0.213421
$$206$$ 14.9443 1.04122
$$207$$ −5.88854 −0.409282
$$208$$ 3.23607 0.224381
$$209$$ 7.23607 0.500529
$$210$$ 0 0
$$211$$ −13.5279 −0.931297 −0.465648 0.884970i $$-0.654179\pi$$
−0.465648 + 0.884970i $$0.654179\pi$$
$$212$$ 8.47214 0.581869
$$213$$ −8.00000 −0.548151
$$214$$ 2.47214 0.168992
$$215$$ −12.9443 −0.882792
$$216$$ 5.52786 0.376124
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ 16.0000 1.08118
$$220$$ 1.23607 0.0833357
$$221$$ −8.00000 −0.538138
$$222$$ −8.58359 −0.576093
$$223$$ 0.472136 0.0316166 0.0158083 0.999875i $$-0.494968\pi$$
0.0158083 + 0.999875i $$0.494968\pi$$
$$224$$ 0 0
$$225$$ 5.11146 0.340764
$$226$$ −0.472136 −0.0314060
$$227$$ 19.2361 1.27674 0.638371 0.769729i $$-0.279608\pi$$
0.638371 + 0.769729i $$0.279608\pi$$
$$228$$ −8.94427 −0.592349
$$229$$ −17.2361 −1.13899 −0.569496 0.821994i $$-0.692861\pi$$
−0.569496 + 0.821994i $$0.692861\pi$$
$$230$$ 4.94427 0.326016
$$231$$ 0 0
$$232$$ 4.47214 0.293610
$$233$$ −14.9443 −0.979032 −0.489516 0.871994i $$-0.662826\pi$$
−0.489516 + 0.871994i $$0.662826\pi$$
$$234$$ −4.76393 −0.311428
$$235$$ 2.47214 0.161264
$$236$$ −2.76393 −0.179917
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ −1.52786 −0.0986232
$$241$$ −15.4164 −0.993058 −0.496529 0.868020i $$-0.665392\pi$$
−0.496529 + 0.868020i $$0.665392\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −13.5967 −0.872232
$$244$$ 0.763932 0.0489057
$$245$$ 0 0
$$246$$ −3.05573 −0.194826
$$247$$ 23.4164 1.48995
$$248$$ −2.00000 −0.127000
$$249$$ −15.0557 −0.954118
$$250$$ −10.4721 −0.662316
$$251$$ −29.2361 −1.84536 −0.922682 0.385562i $$-0.874008\pi$$
−0.922682 + 0.385562i $$0.874008\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ −12.0000 −0.752947
$$255$$ 3.77709 0.236530
$$256$$ 1.00000 0.0625000
$$257$$ −6.94427 −0.433172 −0.216586 0.976264i $$-0.569492\pi$$
−0.216586 + 0.976264i $$0.569492\pi$$
$$258$$ 12.9443 0.805875
$$259$$ 0 0
$$260$$ 4.00000 0.248069
$$261$$ −6.58359 −0.407514
$$262$$ −4.76393 −0.294317
$$263$$ −4.94427 −0.304877 −0.152438 0.988313i $$-0.548713\pi$$
−0.152438 + 0.988313i $$0.548713\pi$$
$$264$$ −1.23607 −0.0760747
$$265$$ 10.4721 0.643298
$$266$$ 0 0
$$267$$ 12.3607 0.756461
$$268$$ 11.4164 0.697368
$$269$$ 22.7639 1.38794 0.693971 0.720003i $$-0.255860\pi$$
0.693971 + 0.720003i $$0.255860\pi$$
$$270$$ 6.83282 0.415832
$$271$$ −0.944272 −0.0573604 −0.0286802 0.999589i $$-0.509130\pi$$
−0.0286802 + 0.999589i $$0.509130\pi$$
$$272$$ −2.47214 −0.149895
$$273$$ 0 0
$$274$$ −19.8885 −1.20151
$$275$$ −3.47214 −0.209378
$$276$$ −4.94427 −0.297610
$$277$$ 3.52786 0.211969 0.105984 0.994368i $$-0.466201\pi$$
0.105984 + 0.994368i $$0.466201\pi$$
$$278$$ 21.7082 1.30197
$$279$$ 2.94427 0.176269
$$280$$ 0 0
$$281$$ 28.8328 1.72002 0.860011 0.510276i $$-0.170457\pi$$
0.860011 + 0.510276i $$0.170457\pi$$
$$282$$ −2.47214 −0.147214
$$283$$ −14.6525 −0.870999 −0.435500 0.900189i $$-0.643428\pi$$
−0.435500 + 0.900189i $$0.643428\pi$$
$$284$$ 6.47214 0.384051
$$285$$ −11.0557 −0.654885
$$286$$ 3.23607 0.191353
$$287$$ 0 0
$$288$$ −1.47214 −0.0867464
$$289$$ −10.8885 −0.640503
$$290$$ 5.52786 0.324607
$$291$$ 15.4164 0.903726
$$292$$ −12.9443 −0.757506
$$293$$ 26.6525 1.55705 0.778527 0.627611i $$-0.215967\pi$$
0.778527 + 0.627611i $$0.215967\pi$$
$$294$$ 0 0
$$295$$ −3.41641 −0.198911
$$296$$ 6.94427 0.403628
$$297$$ 5.52786 0.320759
$$298$$ −22.3607 −1.29532
$$299$$ 12.9443 0.748587
$$300$$ 4.29180 0.247787
$$301$$ 0 0
$$302$$ 12.0000 0.690522
$$303$$ 10.1115 0.580888
$$304$$ 7.23607 0.415017
$$305$$ 0.944272 0.0540689
$$306$$ 3.63932 0.208046
$$307$$ 26.0689 1.48783 0.743915 0.668274i $$-0.232967\pi$$
0.743915 + 0.668274i $$0.232967\pi$$
$$308$$ 0 0
$$309$$ −18.4721 −1.05084
$$310$$ −2.47214 −0.140408
$$311$$ 21.4164 1.21441 0.607207 0.794544i $$-0.292290\pi$$
0.607207 + 0.794544i $$0.292290\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ −19.5279 −1.10378 −0.551890 0.833917i $$-0.686093\pi$$
−0.551890 + 0.833917i $$0.686093\pi$$
$$314$$ 12.6525 0.714021
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −30.9443 −1.73800 −0.869002 0.494809i $$-0.835238\pi$$
−0.869002 + 0.494809i $$0.835238\pi$$
$$318$$ −10.4721 −0.587248
$$319$$ 4.47214 0.250392
$$320$$ 1.23607 0.0690983
$$321$$ −3.05573 −0.170554
$$322$$ 0 0
$$323$$ −17.8885 −0.995345
$$324$$ −2.41641 −0.134245
$$325$$ −11.2361 −0.623265
$$326$$ −19.4164 −1.07538
$$327$$ 12.3607 0.683547
$$328$$ 2.47214 0.136501
$$329$$ 0 0
$$330$$ −1.52786 −0.0841061
$$331$$ −16.9443 −0.931341 −0.465671 0.884958i $$-0.654187\pi$$
−0.465671 + 0.884958i $$0.654187\pi$$
$$332$$ 12.1803 0.668483
$$333$$ −10.2229 −0.560212
$$334$$ −11.4164 −0.624678
$$335$$ 14.1115 0.770991
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ −2.52786 −0.137498
$$339$$ 0.583592 0.0316964
$$340$$ −3.05573 −0.165720
$$341$$ −2.00000 −0.108306
$$342$$ −10.6525 −0.576020
$$343$$ 0 0
$$344$$ −10.4721 −0.564620
$$345$$ −6.11146 −0.329030
$$346$$ 3.23607 0.173972
$$347$$ 2.47214 0.132711 0.0663556 0.997796i $$-0.478863\pi$$
0.0663556 + 0.997796i $$0.478863\pi$$
$$348$$ −5.52786 −0.296325
$$349$$ −21.7082 −1.16201 −0.581007 0.813899i $$-0.697341\pi$$
−0.581007 + 0.813899i $$0.697341\pi$$
$$350$$ 0 0
$$351$$ 17.8885 0.954820
$$352$$ 1.00000 0.0533002
$$353$$ 17.0557 0.907785 0.453892 0.891056i $$-0.350035\pi$$
0.453892 + 0.891056i $$0.350035\pi$$
$$354$$ 3.41641 0.181580
$$355$$ 8.00000 0.424596
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ −8.94427 −0.472719
$$359$$ 26.8328 1.41618 0.708091 0.706121i $$-0.249557\pi$$
0.708091 + 0.706121i $$0.249557\pi$$
$$360$$ −1.81966 −0.0959045
$$361$$ 33.3607 1.75583
$$362$$ −9.23607 −0.485437
$$363$$ −1.23607 −0.0648767
$$364$$ 0 0
$$365$$ −16.0000 −0.837478
$$366$$ −0.944272 −0.0493579
$$367$$ 5.41641 0.282734 0.141367 0.989957i $$-0.454850\pi$$
0.141367 + 0.989957i $$0.454850\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −3.63932 −0.189455
$$370$$ 8.58359 0.446240
$$371$$ 0 0
$$372$$ 2.47214 0.128174
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ −2.47214 −0.127831
$$375$$ 12.9443 0.668439
$$376$$ 2.00000 0.103142
$$377$$ 14.4721 0.745353
$$378$$ 0 0
$$379$$ 14.4721 0.743384 0.371692 0.928356i $$-0.378778\pi$$
0.371692 + 0.928356i $$0.378778\pi$$
$$380$$ 8.94427 0.458831
$$381$$ 14.8328 0.759908
$$382$$ −2.47214 −0.126485
$$383$$ 23.8885 1.22065 0.610324 0.792152i $$-0.291039\pi$$
0.610324 + 0.792152i $$0.291039\pi$$
$$384$$ −1.23607 −0.0630778
$$385$$ 0 0
$$386$$ −14.9443 −0.760643
$$387$$ 15.4164 0.783660
$$388$$ −12.4721 −0.633177
$$389$$ 33.4164 1.69428 0.847140 0.531370i $$-0.178323\pi$$
0.847140 + 0.531370i $$0.178323\pi$$
$$390$$ −4.94427 −0.250363
$$391$$ −9.88854 −0.500085
$$392$$ 0 0
$$393$$ 5.88854 0.297038
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ −1.47214 −0.0739776
$$397$$ 23.7082 1.18988 0.594940 0.803770i $$-0.297176\pi$$
0.594940 + 0.803770i $$0.297176\pi$$
$$398$$ −18.9443 −0.949591
$$399$$ 0 0
$$400$$ −3.47214 −0.173607
$$401$$ 14.3607 0.717138 0.358569 0.933503i $$-0.383265\pi$$
0.358569 + 0.933503i $$0.383265\pi$$
$$402$$ −14.1115 −0.703815
$$403$$ −6.47214 −0.322400
$$404$$ −8.18034 −0.406987
$$405$$ −2.98684 −0.148417
$$406$$ 0 0
$$407$$ 6.94427 0.344215
$$408$$ 3.05573 0.151281
$$409$$ −3.41641 −0.168930 −0.0844652 0.996426i $$-0.526918\pi$$
−0.0844652 + 0.996426i $$0.526918\pi$$
$$410$$ 3.05573 0.150912
$$411$$ 24.5836 1.21262
$$412$$ 14.9443 0.736251
$$413$$ 0 0
$$414$$ −5.88854 −0.289406
$$415$$ 15.0557 0.739057
$$416$$ 3.23607 0.158661
$$417$$ −26.8328 −1.31401
$$418$$ 7.23607 0.353928
$$419$$ −17.2361 −0.842037 −0.421019 0.907052i $$-0.638327\pi$$
−0.421019 + 0.907052i $$0.638327\pi$$
$$420$$ 0 0
$$421$$ 16.4721 0.802803 0.401401 0.915902i $$-0.368523\pi$$
0.401401 + 0.915902i $$0.368523\pi$$
$$422$$ −13.5279 −0.658526
$$423$$ −2.94427 −0.143155
$$424$$ 8.47214 0.411443
$$425$$ 8.58359 0.416365
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ 2.47214 0.119495
$$429$$ −4.00000 −0.193122
$$430$$ −12.9443 −0.624228
$$431$$ 23.0557 1.11056 0.555278 0.831665i $$-0.312612\pi$$
0.555278 + 0.831665i $$0.312612\pi$$
$$432$$ 5.52786 0.265959
$$433$$ −28.4721 −1.36828 −0.684142 0.729349i $$-0.739823\pi$$
−0.684142 + 0.729349i $$0.739823\pi$$
$$434$$ 0 0
$$435$$ −6.83282 −0.327608
$$436$$ −10.0000 −0.478913
$$437$$ 28.9443 1.38459
$$438$$ 16.0000 0.764510
$$439$$ −8.94427 −0.426887 −0.213443 0.976955i $$-0.568468\pi$$
−0.213443 + 0.976955i $$0.568468\pi$$
$$440$$ 1.23607 0.0589272
$$441$$ 0 0
$$442$$ −8.00000 −0.380521
$$443$$ −24.9443 −1.18514 −0.592569 0.805520i $$-0.701886\pi$$
−0.592569 + 0.805520i $$0.701886\pi$$
$$444$$ −8.58359 −0.407359
$$445$$ −12.3607 −0.585952
$$446$$ 0.472136 0.0223563
$$447$$ 27.6393 1.30729
$$448$$ 0 0
$$449$$ 18.9443 0.894035 0.447018 0.894525i $$-0.352486\pi$$
0.447018 + 0.894525i $$0.352486\pi$$
$$450$$ 5.11146 0.240956
$$451$$ 2.47214 0.116408
$$452$$ −0.472136 −0.0222074
$$453$$ −14.8328 −0.696906
$$454$$ 19.2361 0.902793
$$455$$ 0 0
$$456$$ −8.94427 −0.418854
$$457$$ 26.9443 1.26040 0.630200 0.776433i $$-0.282973\pi$$
0.630200 + 0.776433i $$0.282973\pi$$
$$458$$ −17.2361 −0.805389
$$459$$ −13.6656 −0.637857
$$460$$ 4.94427 0.230528
$$461$$ −24.7639 −1.15337 −0.576686 0.816966i $$-0.695654\pi$$
−0.576686 + 0.816966i $$0.695654\pi$$
$$462$$ 0 0
$$463$$ −30.4721 −1.41616 −0.708080 0.706132i $$-0.750438\pi$$
−0.708080 + 0.706132i $$0.750438\pi$$
$$464$$ 4.47214 0.207614
$$465$$ 3.05573 0.141706
$$466$$ −14.9443 −0.692280
$$467$$ 27.1246 1.25518 0.627589 0.778545i $$-0.284042\pi$$
0.627589 + 0.778545i $$0.284042\pi$$
$$468$$ −4.76393 −0.220213
$$469$$ 0 0
$$470$$ 2.47214 0.114031
$$471$$ −15.6393 −0.720622
$$472$$ −2.76393 −0.127220
$$473$$ −10.4721 −0.481509
$$474$$ 0 0
$$475$$ −25.1246 −1.15280
$$476$$ 0 0
$$477$$ −12.4721 −0.571060
$$478$$ 20.0000 0.914779
$$479$$ −12.3607 −0.564774 −0.282387 0.959301i $$-0.591126\pi$$
−0.282387 + 0.959301i $$0.591126\pi$$
$$480$$ −1.52786 −0.0697371
$$481$$ 22.4721 1.02464
$$482$$ −15.4164 −0.702198
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −15.4164 −0.700023
$$486$$ −13.5967 −0.616761
$$487$$ 16.9443 0.767818 0.383909 0.923371i $$-0.374578\pi$$
0.383909 + 0.923371i $$0.374578\pi$$
$$488$$ 0.763932 0.0345816
$$489$$ 24.0000 1.08532
$$490$$ 0 0
$$491$$ −16.9443 −0.764684 −0.382342 0.924021i $$-0.624882\pi$$
−0.382342 + 0.924021i $$0.624882\pi$$
$$492$$ −3.05573 −0.137763
$$493$$ −11.0557 −0.497925
$$494$$ 23.4164 1.05355
$$495$$ −1.81966 −0.0817876
$$496$$ −2.00000 −0.0898027
$$497$$ 0 0
$$498$$ −15.0557 −0.674663
$$499$$ −32.3607 −1.44866 −0.724331 0.689452i $$-0.757851\pi$$
−0.724331 + 0.689452i $$0.757851\pi$$
$$500$$ −10.4721 −0.468328
$$501$$ 14.1115 0.630453
$$502$$ −29.2361 −1.30487
$$503$$ −4.00000 −0.178351 −0.0891756 0.996016i $$-0.528423\pi$$
−0.0891756 + 0.996016i $$0.528423\pi$$
$$504$$ 0 0
$$505$$ −10.1115 −0.449954
$$506$$ 4.00000 0.177822
$$507$$ 3.12461 0.138769
$$508$$ −12.0000 −0.532414
$$509$$ −24.0689 −1.06683 −0.533417 0.845852i $$-0.679092\pi$$
−0.533417 + 0.845852i $$0.679092\pi$$
$$510$$ 3.77709 0.167252
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 40.0000 1.76604
$$514$$ −6.94427 −0.306299
$$515$$ 18.4721 0.813980
$$516$$ 12.9443 0.569840
$$517$$ 2.00000 0.0879599
$$518$$ 0 0
$$519$$ −4.00000 −0.175581
$$520$$ 4.00000 0.175412
$$521$$ 10.3607 0.453910 0.226955 0.973905i $$-0.427123\pi$$
0.226955 + 0.973905i $$0.427123\pi$$
$$522$$ −6.58359 −0.288156
$$523$$ 14.2918 0.624937 0.312468 0.949928i $$-0.398844\pi$$
0.312468 + 0.949928i $$0.398844\pi$$
$$524$$ −4.76393 −0.208113
$$525$$ 0 0
$$526$$ −4.94427 −0.215580
$$527$$ 4.94427 0.215376
$$528$$ −1.23607 −0.0537930
$$529$$ −7.00000 −0.304348
$$530$$ 10.4721 0.454881
$$531$$ 4.06888 0.176575
$$532$$ 0 0
$$533$$ 8.00000 0.346518
$$534$$ 12.3607 0.534899
$$535$$ 3.05573 0.132111
$$536$$ 11.4164 0.493114
$$537$$ 11.0557 0.477090
$$538$$ 22.7639 0.981423
$$539$$ 0 0
$$540$$ 6.83282 0.294038
$$541$$ −26.9443 −1.15842 −0.579212 0.815177i $$-0.696640\pi$$
−0.579212 + 0.815177i $$0.696640\pi$$
$$542$$ −0.944272 −0.0405600
$$543$$ 11.4164 0.489925
$$544$$ −2.47214 −0.105992
$$545$$ −12.3607 −0.529473
$$546$$ 0 0
$$547$$ −0.944272 −0.0403742 −0.0201871 0.999796i $$-0.506426\pi$$
−0.0201871 + 0.999796i $$0.506426\pi$$
$$548$$ −19.8885 −0.849596
$$549$$ −1.12461 −0.0479973
$$550$$ −3.47214 −0.148052
$$551$$ 32.3607 1.37861
$$552$$ −4.94427 −0.210442
$$553$$ 0 0
$$554$$ 3.52786 0.149885
$$555$$ −10.6099 −0.450365
$$556$$ 21.7082 0.920633
$$557$$ 24.8328 1.05220 0.526100 0.850423i $$-0.323654\pi$$
0.526100 + 0.850423i $$0.323654\pi$$
$$558$$ 2.94427 0.124641
$$559$$ −33.8885 −1.43333
$$560$$ 0 0
$$561$$ 3.05573 0.129013
$$562$$ 28.8328 1.21624
$$563$$ −31.2361 −1.31644 −0.658222 0.752824i $$-0.728691\pi$$
−0.658222 + 0.752824i $$0.728691\pi$$
$$564$$ −2.47214 −0.104096
$$565$$ −0.583592 −0.0245519
$$566$$ −14.6525 −0.615889
$$567$$ 0 0
$$568$$ 6.47214 0.271565
$$569$$ −36.8328 −1.54411 −0.772056 0.635555i $$-0.780772\pi$$
−0.772056 + 0.635555i $$0.780772\pi$$
$$570$$ −11.0557 −0.463073
$$571$$ −10.1115 −0.423151 −0.211576 0.977362i $$-0.567859\pi$$
−0.211576 + 0.977362i $$0.567859\pi$$
$$572$$ 3.23607 0.135307
$$573$$ 3.05573 0.127655
$$574$$ 0 0
$$575$$ −13.8885 −0.579192
$$576$$ −1.47214 −0.0613390
$$577$$ −26.9443 −1.12170 −0.560852 0.827916i $$-0.689526\pi$$
−0.560852 + 0.827916i $$0.689526\pi$$
$$578$$ −10.8885 −0.452904
$$579$$ 18.4721 0.767676
$$580$$ 5.52786 0.229532
$$581$$ 0 0
$$582$$ 15.4164 0.639031
$$583$$ 8.47214 0.350880
$$584$$ −12.9443 −0.535638
$$585$$ −5.88854 −0.243461
$$586$$ 26.6525 1.10100
$$587$$ 5.81966 0.240203 0.120102 0.992762i $$-0.461678\pi$$
0.120102 + 0.992762i $$0.461678\pi$$
$$588$$ 0 0
$$589$$ −14.4721 −0.596314
$$590$$ −3.41641 −0.140651
$$591$$ −22.2492 −0.915211
$$592$$ 6.94427 0.285408
$$593$$ −24.0000 −0.985562 −0.492781 0.870153i $$-0.664020\pi$$
−0.492781 + 0.870153i $$0.664020\pi$$
$$594$$ 5.52786 0.226811
$$595$$ 0 0
$$596$$ −22.3607 −0.915929
$$597$$ 23.4164 0.958370
$$598$$ 12.9443 0.529331
$$599$$ −32.3607 −1.32222 −0.661111 0.750288i $$-0.729915\pi$$
−0.661111 + 0.750288i $$0.729915\pi$$
$$600$$ 4.29180 0.175212
$$601$$ 34.8328 1.42086 0.710430 0.703768i $$-0.248500\pi$$
0.710430 + 0.703768i $$0.248500\pi$$
$$602$$ 0 0
$$603$$ −16.8065 −0.684414
$$604$$ 12.0000 0.488273
$$605$$ 1.23607 0.0502533
$$606$$ 10.1115 0.410750
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 7.23607 0.293461
$$609$$ 0 0
$$610$$ 0.944272 0.0382325
$$611$$ 6.47214 0.261835
$$612$$ 3.63932 0.147111
$$613$$ 28.4721 1.14998 0.574989 0.818161i $$-0.305006\pi$$
0.574989 + 0.818161i $$0.305006\pi$$
$$614$$ 26.0689 1.05205
$$615$$ −3.77709 −0.152307
$$616$$ 0 0
$$617$$ 21.4164 0.862192 0.431096 0.902306i $$-0.358127\pi$$
0.431096 + 0.902306i $$0.358127\pi$$
$$618$$ −18.4721 −0.743058
$$619$$ −18.5410 −0.745227 −0.372613 0.927987i $$-0.621538\pi$$
−0.372613 + 0.927987i $$0.621538\pi$$
$$620$$ −2.47214 −0.0992834
$$621$$ 22.1115 0.887302
$$622$$ 21.4164 0.858720
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 4.41641 0.176656
$$626$$ −19.5279 −0.780490
$$627$$ −8.94427 −0.357200
$$628$$ 12.6525 0.504889
$$629$$ −17.1672 −0.684500
$$630$$ 0 0
$$631$$ −31.4164 −1.25067 −0.625334 0.780357i $$-0.715037\pi$$
−0.625334 + 0.780357i $$0.715037\pi$$
$$632$$ 0 0
$$633$$ 16.7214 0.664614
$$634$$ −30.9443 −1.22895
$$635$$ −14.8328 −0.588622
$$636$$ −10.4721 −0.415247
$$637$$ 0 0
$$638$$ 4.47214 0.177054
$$639$$ −9.52786 −0.376916
$$640$$ 1.23607 0.0488599
$$641$$ 27.5279 1.08729 0.543643 0.839317i $$-0.317045\pi$$
0.543643 + 0.839317i $$0.317045\pi$$
$$642$$ −3.05573 −0.120600
$$643$$ 18.7639 0.739977 0.369989 0.929036i $$-0.379362\pi$$
0.369989 + 0.929036i $$0.379362\pi$$
$$644$$ 0 0
$$645$$ 16.0000 0.629999
$$646$$ −17.8885 −0.703815
$$647$$ 28.8328 1.13353 0.566767 0.823878i $$-0.308194\pi$$
0.566767 + 0.823878i $$0.308194\pi$$
$$648$$ −2.41641 −0.0949255
$$649$$ −2.76393 −0.108494
$$650$$ −11.2361 −0.440715
$$651$$ 0 0
$$652$$ −19.4164 −0.760405
$$653$$ 46.3607 1.81423 0.907117 0.420879i $$-0.138278\pi$$
0.907117 + 0.420879i $$0.138278\pi$$
$$654$$ 12.3607 0.483341
$$655$$ −5.88854 −0.230084
$$656$$ 2.47214 0.0965207
$$657$$ 19.0557 0.743435
$$658$$ 0 0
$$659$$ 16.5836 0.646005 0.323003 0.946398i $$-0.395308\pi$$
0.323003 + 0.946398i $$0.395308\pi$$
$$660$$ −1.52786 −0.0594720
$$661$$ 3.12461 0.121533 0.0607667 0.998152i $$-0.480645\pi$$
0.0607667 + 0.998152i $$0.480645\pi$$
$$662$$ −16.9443 −0.658558
$$663$$ 9.88854 0.384039
$$664$$ 12.1803 0.472689
$$665$$ 0 0
$$666$$ −10.2229 −0.396130
$$667$$ 17.8885 0.692647
$$668$$ −11.4164 −0.441714
$$669$$ −0.583592 −0.0225630
$$670$$ 14.1115 0.545173
$$671$$ 0.763932 0.0294913
$$672$$ 0 0
$$673$$ −3.88854 −0.149892 −0.0749462 0.997188i $$-0.523878\pi$$
−0.0749462 + 0.997188i $$0.523878\pi$$
$$674$$ 18.0000 0.693334
$$675$$ −19.1935 −0.738758
$$676$$ −2.52786 −0.0972255
$$677$$ 26.0689 1.00191 0.500954 0.865474i $$-0.332982\pi$$
0.500954 + 0.865474i $$0.332982\pi$$
$$678$$ 0.583592 0.0224127
$$679$$ 0 0
$$680$$ −3.05573 −0.117182
$$681$$ −23.7771 −0.911140
$$682$$ −2.00000 −0.0765840
$$683$$ 32.9443 1.26058 0.630289 0.776361i $$-0.282936\pi$$
0.630289 + 0.776361i $$0.282936\pi$$
$$684$$ −10.6525 −0.407308
$$685$$ −24.5836 −0.939291
$$686$$ 0 0
$$687$$ 21.3050 0.812835
$$688$$ −10.4721 −0.399246
$$689$$ 27.4164 1.04448
$$690$$ −6.11146 −0.232659
$$691$$ −12.6525 −0.481323 −0.240661 0.970609i $$-0.577364\pi$$
−0.240661 + 0.970609i $$0.577364\pi$$
$$692$$ 3.23607 0.123017
$$693$$ 0 0
$$694$$ 2.47214 0.0938410
$$695$$ 26.8328 1.01783
$$696$$ −5.52786 −0.209533
$$697$$ −6.11146 −0.231488
$$698$$ −21.7082 −0.821668
$$699$$ 18.4721 0.698680
$$700$$ 0 0
$$701$$ −42.7214 −1.61356 −0.806782 0.590850i $$-0.798793\pi$$
−0.806782 + 0.590850i $$0.798793\pi$$
$$702$$ 17.8885 0.675160
$$703$$ 50.2492 1.89519
$$704$$ 1.00000 0.0376889
$$705$$ −3.05573 −0.115085
$$706$$ 17.0557 0.641901
$$707$$ 0 0
$$708$$ 3.41641 0.128396
$$709$$ 4.47214 0.167955 0.0839773 0.996468i $$-0.473238\pi$$
0.0839773 + 0.996468i $$0.473238\pi$$
$$710$$ 8.00000 0.300235
$$711$$ 0 0
$$712$$ −10.0000 −0.374766
$$713$$ −8.00000 −0.299602
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ −8.94427 −0.334263
$$717$$ −24.7214 −0.923236
$$718$$ 26.8328 1.00139
$$719$$ −16.8328 −0.627758 −0.313879 0.949463i $$-0.601629\pi$$
−0.313879 + 0.949463i $$0.601629\pi$$
$$720$$ −1.81966 −0.0678147
$$721$$ 0 0
$$722$$ 33.3607 1.24156
$$723$$ 19.0557 0.708690
$$724$$ −9.23607 −0.343256
$$725$$ −15.5279 −0.576690
$$726$$ −1.23607 −0.0458748
$$727$$ −18.0000 −0.667583 −0.333792 0.942647i $$-0.608328\pi$$
−0.333792 + 0.942647i $$0.608328\pi$$
$$728$$ 0 0
$$729$$ 24.0557 0.890953
$$730$$ −16.0000 −0.592187
$$731$$ 25.8885 0.957522
$$732$$ −0.944272 −0.0349013
$$733$$ −49.1246 −1.81446 −0.907229 0.420636i $$-0.861807\pi$$
−0.907229 + 0.420636i $$0.861807\pi$$
$$734$$ 5.41641 0.199923
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 11.4164 0.420529
$$738$$ −3.63932 −0.133965
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 8.58359 0.315539
$$741$$ −28.9443 −1.06329
$$742$$ 0 0
$$743$$ 21.8885 0.803013 0.401506 0.915856i $$-0.368487\pi$$
0.401506 + 0.915856i $$0.368487\pi$$
$$744$$ 2.47214 0.0906329
$$745$$ −27.6393 −1.01263
$$746$$ −6.00000 −0.219676
$$747$$ −17.9311 −0.656065
$$748$$ −2.47214 −0.0903902
$$749$$ 0 0
$$750$$ 12.9443 0.472658
$$751$$ −16.9443 −0.618305 −0.309153 0.951012i $$-0.600045\pi$$
−0.309153 + 0.951012i $$0.600045\pi$$
$$752$$ 2.00000 0.0729325
$$753$$ 36.1378 1.31693
$$754$$ 14.4721 0.527044
$$755$$ 14.8328 0.539821
$$756$$ 0 0
$$757$$ −23.3050 −0.847033 −0.423516 0.905888i $$-0.639204\pi$$
−0.423516 + 0.905888i $$0.639204\pi$$
$$758$$ 14.4721 0.525652
$$759$$ −4.94427 −0.179466
$$760$$ 8.94427 0.324443
$$761$$ 11.4164 0.413844 0.206922 0.978357i $$-0.433655\pi$$
0.206922 + 0.978357i $$0.433655\pi$$
$$762$$ 14.8328 0.537336
$$763$$ 0 0
$$764$$ −2.47214 −0.0894387
$$765$$ 4.49845 0.162642
$$766$$ 23.8885 0.863128
$$767$$ −8.94427 −0.322959
$$768$$ −1.23607 −0.0446028
$$769$$ 43.4164 1.56564 0.782818 0.622251i $$-0.213782\pi$$
0.782818 + 0.622251i $$0.213782\pi$$
$$770$$ 0 0
$$771$$ 8.58359 0.309131
$$772$$ −14.9443 −0.537856
$$773$$ −15.7082 −0.564985 −0.282492 0.959270i $$-0.591161\pi$$
−0.282492 + 0.959270i $$0.591161\pi$$
$$774$$ 15.4164 0.554131
$$775$$ 6.94427 0.249446
$$776$$ −12.4721 −0.447724
$$777$$ 0 0
$$778$$ 33.4164 1.19804
$$779$$ 17.8885 0.640924
$$780$$ −4.94427 −0.177033
$$781$$ 6.47214 0.231591
$$782$$ −9.88854 −0.353614
$$783$$ 24.7214 0.883469
$$784$$ 0 0
$$785$$ 15.6393 0.558191
$$786$$ 5.88854 0.210037
$$787$$ 28.1803 1.00452 0.502260 0.864716i $$-0.332502\pi$$
0.502260 + 0.864716i $$0.332502\pi$$
$$788$$ 18.0000 0.641223
$$789$$ 6.11146 0.217574
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −1.47214 −0.0523101
$$793$$ 2.47214 0.0877881
$$794$$ 23.7082 0.841373
$$795$$ −12.9443 −0.459086
$$796$$ −18.9443 −0.671462
$$797$$ 41.5967 1.47343 0.736716 0.676202i $$-0.236375\pi$$
0.736716 + 0.676202i $$0.236375\pi$$
$$798$$ 0 0
$$799$$ −4.94427 −0.174916
$$800$$ −3.47214 −0.122759
$$801$$ 14.7214 0.520154
$$802$$ 14.3607 0.507093
$$803$$ −12.9443 −0.456793
$$804$$ −14.1115 −0.497673
$$805$$ 0 0
$$806$$ −6.47214 −0.227971
$$807$$ −28.1378 −0.990496
$$808$$ −8.18034 −0.287783
$$809$$ −21.0557 −0.740280 −0.370140 0.928976i $$-0.620690\pi$$
−0.370140 + 0.928976i $$0.620690\pi$$
$$810$$ −2.98684 −0.104947
$$811$$ −4.76393 −0.167284 −0.0836421 0.996496i $$-0.526655\pi$$
−0.0836421 + 0.996496i $$0.526655\pi$$
$$812$$ 0 0
$$813$$ 1.16718 0.0409349
$$814$$ 6.94427 0.243397
$$815$$ −24.0000 −0.840683
$$816$$ 3.05573 0.106972
$$817$$ −75.7771 −2.65110
$$818$$ −3.41641 −0.119452
$$819$$ 0 0
$$820$$ 3.05573 0.106711
$$821$$ −1.41641 −0.0494330 −0.0247165 0.999695i $$-0.507868\pi$$
−0.0247165 + 0.999695i $$0.507868\pi$$
$$822$$ 24.5836 0.857451
$$823$$ −46.2492 −1.61215 −0.806073 0.591816i $$-0.798411\pi$$
−0.806073 + 0.591816i $$0.798411\pi$$
$$824$$ 14.9443 0.520608
$$825$$ 4.29180 0.149421
$$826$$ 0 0
$$827$$ 16.9443 0.589210 0.294605 0.955619i $$-0.404812\pi$$
0.294605 + 0.955619i $$0.404812\pi$$
$$828$$ −5.88854 −0.204641
$$829$$ −11.7082 −0.406643 −0.203321 0.979112i $$-0.565174\pi$$
−0.203321 + 0.979112i $$0.565174\pi$$
$$830$$ 15.0557 0.522592
$$831$$ −4.36068 −0.151270
$$832$$ 3.23607 0.112190
$$833$$ 0 0
$$834$$ −26.8328 −0.929144
$$835$$ −14.1115 −0.488347
$$836$$ 7.23607 0.250265
$$837$$ −11.0557 −0.382142
$$838$$ −17.2361 −0.595410
$$839$$ −16.8328 −0.581133 −0.290567 0.956855i $$-0.593844\pi$$
−0.290567 + 0.956855i $$0.593844\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 16.4721 0.567667
$$843$$ −35.6393 −1.22748
$$844$$ −13.5279 −0.465648
$$845$$ −3.12461 −0.107490
$$846$$ −2.94427 −0.101226
$$847$$ 0 0
$$848$$ 8.47214 0.290934
$$849$$ 18.1115 0.621584
$$850$$ 8.58359 0.294415
$$851$$ 27.7771 0.952186
$$852$$ −8.00000 −0.274075
$$853$$ −32.5410 −1.11418 −0.557092 0.830451i $$-0.688083\pi$$
−0.557092 + 0.830451i $$0.688083\pi$$
$$854$$ 0 0
$$855$$ −13.1672 −0.450308
$$856$$ 2.47214 0.0844959
$$857$$ 46.4721 1.58746 0.793729 0.608272i $$-0.208137\pi$$
0.793729 + 0.608272i $$0.208137\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ −15.1246 −0.516045 −0.258023 0.966139i $$-0.583071\pi$$
−0.258023 + 0.966139i $$0.583071\pi$$
$$860$$ −12.9443 −0.441396
$$861$$ 0 0
$$862$$ 23.0557 0.785281
$$863$$ 0.583592 0.0198657 0.00993285 0.999951i $$-0.496838\pi$$
0.00993285 + 0.999951i $$0.496838\pi$$
$$864$$ 5.52786 0.188062
$$865$$ 4.00000 0.136004
$$866$$ −28.4721 −0.967523
$$867$$ 13.4590 0.457091
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −6.83282 −0.231654
$$871$$ 36.9443 1.25181
$$872$$ −10.0000 −0.338643
$$873$$ 18.3607 0.621415
$$874$$ 28.9443 0.979055
$$875$$ 0 0
$$876$$ 16.0000 0.540590
$$877$$ 9.05573 0.305790 0.152895 0.988242i $$-0.451140\pi$$
0.152895 + 0.988242i $$0.451140\pi$$
$$878$$ −8.94427 −0.301855
$$879$$ −32.9443 −1.11118
$$880$$ 1.23607 0.0416678
$$881$$ −28.8328 −0.971402 −0.485701 0.874125i $$-0.661436\pi$$
−0.485701 + 0.874125i $$0.661436\pi$$
$$882$$ 0 0
$$883$$ −2.83282 −0.0953318 −0.0476659 0.998863i $$-0.515178\pi$$
−0.0476659 + 0.998863i $$0.515178\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 4.22291 0.141952
$$886$$ −24.9443 −0.838019
$$887$$ 44.3607 1.48949 0.744743 0.667351i $$-0.232572\pi$$
0.744743 + 0.667351i $$0.232572\pi$$
$$888$$ −8.58359 −0.288046
$$889$$ 0 0
$$890$$ −12.3607 −0.414331
$$891$$ −2.41641 −0.0809527
$$892$$ 0.472136 0.0158083
$$893$$ 14.4721 0.484292
$$894$$ 27.6393 0.924397
$$895$$ −11.0557 −0.369552
$$896$$ 0 0
$$897$$ −16.0000 −0.534224
$$898$$ 18.9443 0.632179
$$899$$ −8.94427 −0.298308
$$900$$ 5.11146 0.170382
$$901$$ −20.9443 −0.697755
$$902$$ 2.47214 0.0823131
$$903$$ 0 0
$$904$$ −0.472136 −0.0157030
$$905$$ −11.4164 −0.379494
$$906$$ −14.8328 −0.492787
$$907$$ −24.3607 −0.808883 −0.404442 0.914564i $$-0.632534\pi$$
−0.404442 + 0.914564i $$0.632534\pi$$
$$908$$ 19.2361 0.638371
$$909$$ 12.0426 0.399427
$$910$$ 0 0
$$911$$ −28.0000 −0.927681 −0.463841 0.885919i $$-0.653529\pi$$
−0.463841 + 0.885919i $$0.653529\pi$$
$$912$$ −8.94427 −0.296174
$$913$$ 12.1803 0.403110
$$914$$ 26.9443 0.891237
$$915$$ −1.16718 −0.0385859
$$916$$ −17.2361 −0.569496
$$917$$ 0 0
$$918$$ −13.6656 −0.451033
$$919$$ −22.1115 −0.729390 −0.364695 0.931127i $$-0.618827\pi$$
−0.364695 + 0.931127i $$0.618827\pi$$
$$920$$ 4.94427 0.163008
$$921$$ −32.2229 −1.06178
$$922$$ −24.7639 −0.815557
$$923$$ 20.9443 0.689389
$$924$$ 0 0
$$925$$ −24.1115 −0.792780
$$926$$ −30.4721 −1.00138
$$927$$ −22.0000 −0.722575
$$928$$ 4.47214 0.146805
$$929$$ −40.2492 −1.32053 −0.660267 0.751031i $$-0.729557\pi$$
−0.660267 + 0.751031i $$0.729557\pi$$
$$930$$ 3.05573 0.100201
$$931$$ 0 0
$$932$$ −14.9443 −0.489516
$$933$$ −26.4721 −0.866659
$$934$$ 27.1246 0.887544
$$935$$ −3.05573 −0.0999330
$$936$$ −4.76393 −0.155714
$$937$$ 3.05573 0.0998263 0.0499131 0.998754i $$-0.484106\pi$$
0.0499131 + 0.998754i $$0.484106\pi$$
$$938$$ 0 0
$$939$$ 24.1378 0.787706
$$940$$ 2.47214 0.0806322
$$941$$ 11.8197 0.385310 0.192655 0.981267i $$-0.438290\pi$$
0.192655 + 0.981267i $$0.438290\pi$$
$$942$$ −15.6393 −0.509557
$$943$$ 9.88854 0.322015
$$944$$ −2.76393 −0.0899583
$$945$$ 0 0
$$946$$ −10.4721 −0.340479
$$947$$ 16.9443 0.550615 0.275307 0.961356i $$-0.411220\pi$$
0.275307 + 0.961356i $$0.411220\pi$$
$$948$$ 0 0
$$949$$ −41.8885 −1.35976
$$950$$ −25.1246 −0.815150
$$951$$ 38.2492 1.24032
$$952$$ 0 0
$$953$$ 22.9443 0.743238 0.371619 0.928385i $$-0.378803\pi$$
0.371619 + 0.928385i $$0.378803\pi$$
$$954$$ −12.4721 −0.403800
$$955$$ −3.05573 −0.0988810
$$956$$ 20.0000 0.646846
$$957$$ −5.52786 −0.178690
$$958$$ −12.3607 −0.399355
$$959$$ 0 0
$$960$$ −1.52786 −0.0493116
$$961$$ −27.0000 −0.870968
$$962$$ 22.4721 0.724531
$$963$$ −3.63932 −0.117275
$$964$$ −15.4164 −0.496529
$$965$$ −18.4721 −0.594639
$$966$$ 0 0
$$967$$ 45.8885 1.47568 0.737838 0.674978i $$-0.235847\pi$$
0.737838 + 0.674978i $$0.235847\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 22.1115 0.710322
$$970$$ −15.4164 −0.494991
$$971$$ −50.5410 −1.62194 −0.810969 0.585089i $$-0.801060\pi$$
−0.810969 + 0.585089i $$0.801060\pi$$
$$972$$ −13.5967 −0.436116
$$973$$ 0 0
$$974$$ 16.9443 0.542929
$$975$$ 13.8885 0.444789
$$976$$ 0.763932 0.0244529
$$977$$ −28.8328 −0.922443 −0.461222 0.887285i $$-0.652589\pi$$
−0.461222 + 0.887285i $$0.652589\pi$$
$$978$$ 24.0000 0.767435
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ 14.7214 0.470017
$$982$$ −16.9443 −0.540713
$$983$$ −14.0000 −0.446531 −0.223265 0.974758i $$-0.571672\pi$$
−0.223265 + 0.974758i $$0.571672\pi$$
$$984$$ −3.05573 −0.0974131
$$985$$ 22.2492 0.708919
$$986$$ −11.0557 −0.352086
$$987$$ 0 0
$$988$$ 23.4164 0.744975
$$989$$ −41.8885 −1.33198
$$990$$ −1.81966 −0.0578326
$$991$$ −0.360680 −0.0114574 −0.00572869 0.999984i $$-0.501824\pi$$
−0.00572869 + 0.999984i $$0.501824\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 20.9443 0.664646
$$994$$ 0 0
$$995$$ −23.4164 −0.742350
$$996$$ −15.0557 −0.477059
$$997$$ −24.1803 −0.765799 −0.382900 0.923790i $$-0.625074\pi$$
−0.382900 + 0.923790i $$0.625074\pi$$
$$998$$ −32.3607 −1.02436
$$999$$ 38.3870 1.21451
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 1078.2.a.w.1.1 2
3.2 odd 2 9702.2.a.cu.1.1 2
4.3 odd 2 8624.2.a.bf.1.2 2
7.2 even 3 1078.2.e.n.67.2 4
7.3 odd 6 1078.2.e.q.177.1 4
7.4 even 3 1078.2.e.n.177.2 4
7.5 odd 6 1078.2.e.q.67.1 4
7.6 odd 2 154.2.a.d.1.2 2
21.20 even 2 1386.2.a.m.1.2 2
28.27 even 2 1232.2.a.p.1.1 2
35.13 even 4 3850.2.c.q.1849.2 4
35.27 even 4 3850.2.c.q.1849.3 4
35.34 odd 2 3850.2.a.bj.1.1 2
56.13 odd 2 4928.2.a.bt.1.1 2
56.27 even 2 4928.2.a.bk.1.2 2
77.76 even 2 1694.2.a.l.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.2 2 7.6 odd 2
1078.2.a.w.1.1 2 1.1 even 1 trivial
1078.2.e.n.67.2 4 7.2 even 3
1078.2.e.n.177.2 4 7.4 even 3
1078.2.e.q.67.1 4 7.5 odd 6
1078.2.e.q.177.1 4 7.3 odd 6
1232.2.a.p.1.1 2 28.27 even 2
1386.2.a.m.1.2 2 21.20 even 2
1694.2.a.l.1.2 2 77.76 even 2
3850.2.a.bj.1.1 2 35.34 odd 2
3850.2.c.q.1849.2 4 35.13 even 4
3850.2.c.q.1849.3 4 35.27 even 4
4928.2.a.bk.1.2 2 56.27 even 2
4928.2.a.bt.1.1 2 56.13 odd 2
8624.2.a.bf.1.2 2 4.3 odd 2
9702.2.a.cu.1.1 2 3.2 odd 2