# Properties

 Label 1078.2.a.x.1.2 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: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1078.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.41421 q^{3} +1.00000 q^{4} +0.585786 q^{5} +2.41421 q^{6} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.41421 q^{3} +1.00000 q^{4} +0.585786 q^{5} +2.41421 q^{6} +1.00000 q^{8} +2.82843 q^{9} +0.585786 q^{10} +1.00000 q^{11} +2.41421 q^{12} +3.82843 q^{13} +1.41421 q^{15} +1.00000 q^{16} -3.65685 q^{17} +2.82843 q^{18} +0.585786 q^{19} +0.585786 q^{20} +1.00000 q^{22} -6.24264 q^{23} +2.41421 q^{24} -4.65685 q^{25} +3.82843 q^{26} -0.414214 q^{27} +2.65685 q^{29} +1.41421 q^{30} +4.00000 q^{31} +1.00000 q^{32} +2.41421 q^{33} -3.65685 q^{34} +2.82843 q^{36} -9.41421 q^{37} +0.585786 q^{38} +9.24264 q^{39} +0.585786 q^{40} +5.41421 q^{41} -5.65685 q^{43} +1.00000 q^{44} +1.65685 q^{45} -6.24264 q^{46} +10.4853 q^{47} +2.41421 q^{48} -4.65685 q^{50} -8.82843 q^{51} +3.82843 q^{52} +7.89949 q^{53} -0.414214 q^{54} +0.585786 q^{55} +1.41421 q^{57} +2.65685 q^{58} +5.58579 q^{59} +1.41421 q^{60} -11.8284 q^{61} +4.00000 q^{62} +1.00000 q^{64} +2.24264 q^{65} +2.41421 q^{66} +2.75736 q^{67} -3.65685 q^{68} -15.0711 q^{69} -11.0711 q^{71} +2.82843 q^{72} +9.41421 q^{73} -9.41421 q^{74} -11.2426 q^{75} +0.585786 q^{76} +9.24264 q^{78} -13.2426 q^{79} +0.585786 q^{80} -9.48528 q^{81} +5.41421 q^{82} +12.1421 q^{83} -2.14214 q^{85} -5.65685 q^{86} +6.41421 q^{87} +1.00000 q^{88} -12.4853 q^{89} +1.65685 q^{90} -6.24264 q^{92} +9.65685 q^{93} +10.4853 q^{94} +0.343146 q^{95} +2.41421 q^{96} +3.82843 q^{97} +2.82843 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 2 q^{27} - 6 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{33} + 4 q^{34} - 16 q^{37} + 4 q^{38} + 10 q^{39} + 4 q^{40} + 8 q^{41} + 2 q^{44} - 8 q^{45} - 4 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{58} + 14 q^{59} - 18 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} + 2 q^{66} + 14 q^{67} + 4 q^{68} - 16 q^{69} - 8 q^{71} + 16 q^{73} - 16 q^{74} - 14 q^{75} + 4 q^{76} + 10 q^{78} - 18 q^{79} + 4 q^{80} - 2 q^{81} + 8 q^{82} - 4 q^{83} + 24 q^{85} + 10 q^{87} + 2 q^{88} - 8 q^{89} - 8 q^{90} - 4 q^{92} + 8 q^{93} + 4 q^{94} + 12 q^{95} + 2 q^{96} + 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 + 4 * q^10 + 2 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 + 2 * q^27 - 6 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^33 + 4 * q^34 - 16 * q^37 + 4 * q^38 + 10 * q^39 + 4 * q^40 + 8 * q^41 + 2 * q^44 - 8 * q^45 - 4 * q^46 + 4 * q^47 + 2 * q^48 + 2 * q^50 - 12 * q^51 + 2 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 6 * q^58 + 14 * q^59 - 18 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^65 + 2 * q^66 + 14 * q^67 + 4 * q^68 - 16 * q^69 - 8 * q^71 + 16 * q^73 - 16 * q^74 - 14 * q^75 + 4 * q^76 + 10 * q^78 - 18 * q^79 + 4 * q^80 - 2 * q^81 + 8 * q^82 - 4 * q^83 + 24 * q^85 + 10 * q^87 + 2 * q^88 - 8 * q^89 - 8 * q^90 - 4 * q^92 + 8 * q^93 + 4 * q^94 + 12 * q^95 + 2 * q^96 + 2 * q^97

## 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
$$3$$ 2.41421 1.39385 0.696923 0.717146i $$-0.254552\pi$$
0.696923 + 0.717146i $$0.254552\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0.585786 0.261972 0.130986 0.991384i $$-0.458186\pi$$
0.130986 + 0.991384i $$0.458186\pi$$
$$6$$ 2.41421 0.985599
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 2.82843 0.942809
$$10$$ 0.585786 0.185242
$$11$$ 1.00000 0.301511
$$12$$ 2.41421 0.696923
$$13$$ 3.82843 1.06181 0.530907 0.847430i $$-0.321851\pi$$
0.530907 + 0.847430i $$0.321851\pi$$
$$14$$ 0 0
$$15$$ 1.41421 0.365148
$$16$$ 1.00000 0.250000
$$17$$ −3.65685 −0.886917 −0.443459 0.896295i $$-0.646249\pi$$
−0.443459 + 0.896295i $$0.646249\pi$$
$$18$$ 2.82843 0.666667
$$19$$ 0.585786 0.134389 0.0671943 0.997740i $$-0.478595\pi$$
0.0671943 + 0.997740i $$0.478595\pi$$
$$20$$ 0.585786 0.130986
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ −6.24264 −1.30168 −0.650840 0.759215i $$-0.725583\pi$$
−0.650840 + 0.759215i $$0.725583\pi$$
$$24$$ 2.41421 0.492799
$$25$$ −4.65685 −0.931371
$$26$$ 3.82843 0.750816
$$27$$ −0.414214 −0.0797154
$$28$$ 0 0
$$29$$ 2.65685 0.493365 0.246683 0.969096i $$-0.420659\pi$$
0.246683 + 0.969096i $$0.420659\pi$$
$$30$$ 1.41421 0.258199
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.41421 0.420261
$$34$$ −3.65685 −0.627145
$$35$$ 0 0
$$36$$ 2.82843 0.471405
$$37$$ −9.41421 −1.54769 −0.773844 0.633377i $$-0.781668\pi$$
−0.773844 + 0.633377i $$0.781668\pi$$
$$38$$ 0.585786 0.0950271
$$39$$ 9.24264 1.48001
$$40$$ 0.585786 0.0926210
$$41$$ 5.41421 0.845558 0.422779 0.906233i $$-0.361055\pi$$
0.422779 + 0.906233i $$0.361055\pi$$
$$42$$ 0 0
$$43$$ −5.65685 −0.862662 −0.431331 0.902194i $$-0.641956\pi$$
−0.431331 + 0.902194i $$0.641956\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 1.65685 0.246989
$$46$$ −6.24264 −0.920427
$$47$$ 10.4853 1.52944 0.764718 0.644365i $$-0.222878\pi$$
0.764718 + 0.644365i $$0.222878\pi$$
$$48$$ 2.41421 0.348462
$$49$$ 0 0
$$50$$ −4.65685 −0.658579
$$51$$ −8.82843 −1.23623
$$52$$ 3.82843 0.530907
$$53$$ 7.89949 1.08508 0.542540 0.840030i $$-0.317463\pi$$
0.542540 + 0.840030i $$0.317463\pi$$
$$54$$ −0.414214 −0.0563673
$$55$$ 0.585786 0.0789874
$$56$$ 0 0
$$57$$ 1.41421 0.187317
$$58$$ 2.65685 0.348862
$$59$$ 5.58579 0.727207 0.363604 0.931554i $$-0.381546\pi$$
0.363604 + 0.931554i $$0.381546\pi$$
$$60$$ 1.41421 0.182574
$$61$$ −11.8284 −1.51447 −0.757237 0.653140i $$-0.773451\pi$$
−0.757237 + 0.653140i $$0.773451\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 2.24264 0.278165
$$66$$ 2.41421 0.297169
$$67$$ 2.75736 0.336865 0.168433 0.985713i $$-0.446129\pi$$
0.168433 + 0.985713i $$0.446129\pi$$
$$68$$ −3.65685 −0.443459
$$69$$ −15.0711 −1.81434
$$70$$ 0 0
$$71$$ −11.0711 −1.31389 −0.656947 0.753937i $$-0.728152\pi$$
−0.656947 + 0.753937i $$0.728152\pi$$
$$72$$ 2.82843 0.333333
$$73$$ 9.41421 1.10185 0.550925 0.834555i $$-0.314275\pi$$
0.550925 + 0.834555i $$0.314275\pi$$
$$74$$ −9.41421 −1.09438
$$75$$ −11.2426 −1.29819
$$76$$ 0.585786 0.0671943
$$77$$ 0 0
$$78$$ 9.24264 1.04652
$$79$$ −13.2426 −1.48991 −0.744957 0.667113i $$-0.767530\pi$$
−0.744957 + 0.667113i $$0.767530\pi$$
$$80$$ 0.585786 0.0654929
$$81$$ −9.48528 −1.05392
$$82$$ 5.41421 0.597900
$$83$$ 12.1421 1.33277 0.666386 0.745607i $$-0.267840\pi$$
0.666386 + 0.745607i $$0.267840\pi$$
$$84$$ 0 0
$$85$$ −2.14214 −0.232347
$$86$$ −5.65685 −0.609994
$$87$$ 6.41421 0.687676
$$88$$ 1.00000 0.106600
$$89$$ −12.4853 −1.32344 −0.661719 0.749752i $$-0.730173\pi$$
−0.661719 + 0.749752i $$0.730173\pi$$
$$90$$ 1.65685 0.174648
$$91$$ 0 0
$$92$$ −6.24264 −0.650840
$$93$$ 9.65685 1.00137
$$94$$ 10.4853 1.08147
$$95$$ 0.343146 0.0352060
$$96$$ 2.41421 0.246400
$$97$$ 3.82843 0.388718 0.194359 0.980930i $$-0.437737\pi$$
0.194359 + 0.980930i $$0.437737\pi$$
$$98$$ 0 0
$$99$$ 2.82843 0.284268
$$100$$ −4.65685 −0.465685
$$101$$ −6.17157 −0.614094 −0.307047 0.951694i $$-0.599341\pi$$
−0.307047 + 0.951694i $$0.599341\pi$$
$$102$$ −8.82843 −0.874145
$$103$$ −13.4142 −1.32174 −0.660871 0.750500i $$-0.729813\pi$$
−0.660871 + 0.750500i $$0.729813\pi$$
$$104$$ 3.82843 0.375408
$$105$$ 0 0
$$106$$ 7.89949 0.767267
$$107$$ −3.07107 −0.296891 −0.148446 0.988921i $$-0.547427\pi$$
−0.148446 + 0.988921i $$0.547427\pi$$
$$108$$ −0.414214 −0.0398577
$$109$$ 16.4853 1.57900 0.789502 0.613748i $$-0.210339\pi$$
0.789502 + 0.613748i $$0.210339\pi$$
$$110$$ 0.585786 0.0558525
$$111$$ −22.7279 −2.15724
$$112$$ 0 0
$$113$$ −8.17157 −0.768717 −0.384358 0.923184i $$-0.625577\pi$$
−0.384358 + 0.923184i $$0.625577\pi$$
$$114$$ 1.41421 0.132453
$$115$$ −3.65685 −0.341003
$$116$$ 2.65685 0.246683
$$117$$ 10.8284 1.00109
$$118$$ 5.58579 0.514213
$$119$$ 0 0
$$120$$ 1.41421 0.129099
$$121$$ 1.00000 0.0909091
$$122$$ −11.8284 −1.07090
$$123$$ 13.0711 1.17858
$$124$$ 4.00000 0.359211
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ 15.7279 1.39563 0.697814 0.716279i $$-0.254156\pi$$
0.697814 + 0.716279i $$0.254156\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −13.6569 −1.20242
$$130$$ 2.24264 0.196693
$$131$$ 0.585786 0.0511804 0.0255902 0.999673i $$-0.491853\pi$$
0.0255902 + 0.999673i $$0.491853\pi$$
$$132$$ 2.41421 0.210130
$$133$$ 0 0
$$134$$ 2.75736 0.238200
$$135$$ −0.242641 −0.0208832
$$136$$ −3.65685 −0.313573
$$137$$ −16.6569 −1.42309 −0.711546 0.702640i $$-0.752004\pi$$
−0.711546 + 0.702640i $$0.752004\pi$$
$$138$$ −15.0711 −1.28293
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 25.3137 2.13180
$$142$$ −11.0711 −0.929063
$$143$$ 3.82843 0.320149
$$144$$ 2.82843 0.235702
$$145$$ 1.55635 0.129248
$$146$$ 9.41421 0.779126
$$147$$ 0 0
$$148$$ −9.41421 −0.773844
$$149$$ 17.6569 1.44651 0.723253 0.690583i $$-0.242646\pi$$
0.723253 + 0.690583i $$0.242646\pi$$
$$150$$ −11.2426 −0.917958
$$151$$ −15.7279 −1.27992 −0.639960 0.768408i $$-0.721049\pi$$
−0.639960 + 0.768408i $$0.721049\pi$$
$$152$$ 0.585786 0.0475136
$$153$$ −10.3431 −0.836194
$$154$$ 0 0
$$155$$ 2.34315 0.188206
$$156$$ 9.24264 0.740003
$$157$$ −17.6569 −1.40917 −0.704585 0.709619i $$-0.748867\pi$$
−0.704585 + 0.709619i $$0.748867\pi$$
$$158$$ −13.2426 −1.05353
$$159$$ 19.0711 1.51243
$$160$$ 0.585786 0.0463105
$$161$$ 0 0
$$162$$ −9.48528 −0.745234
$$163$$ 9.72792 0.761950 0.380975 0.924585i $$-0.375588\pi$$
0.380975 + 0.924585i $$0.375588\pi$$
$$164$$ 5.41421 0.422779
$$165$$ 1.41421 0.110096
$$166$$ 12.1421 0.942412
$$167$$ −13.7279 −1.06230 −0.531149 0.847278i $$-0.678240\pi$$
−0.531149 + 0.847278i $$0.678240\pi$$
$$168$$ 0 0
$$169$$ 1.65685 0.127450
$$170$$ −2.14214 −0.164294
$$171$$ 1.65685 0.126703
$$172$$ −5.65685 −0.431331
$$173$$ 9.82843 0.747241 0.373621 0.927582i $$-0.378116\pi$$
0.373621 + 0.927582i $$0.378116\pi$$
$$174$$ 6.41421 0.486260
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 13.4853 1.01362
$$178$$ −12.4853 −0.935811
$$179$$ 0.899495 0.0672314 0.0336157 0.999435i $$-0.489298\pi$$
0.0336157 + 0.999435i $$0.489298\pi$$
$$180$$ 1.65685 0.123495
$$181$$ 7.65685 0.569129 0.284565 0.958657i $$-0.408151\pi$$
0.284565 + 0.958657i $$0.408151\pi$$
$$182$$ 0 0
$$183$$ −28.5563 −2.11095
$$184$$ −6.24264 −0.460214
$$185$$ −5.51472 −0.405450
$$186$$ 9.65685 0.708075
$$187$$ −3.65685 −0.267416
$$188$$ 10.4853 0.764718
$$189$$ 0 0
$$190$$ 0.343146 0.0248944
$$191$$ −7.17157 −0.518917 −0.259458 0.965754i $$-0.583544\pi$$
−0.259458 + 0.965754i $$0.583544\pi$$
$$192$$ 2.41421 0.174231
$$193$$ 21.8995 1.57636 0.788180 0.615445i $$-0.211024\pi$$
0.788180 + 0.615445i $$0.211024\pi$$
$$194$$ 3.82843 0.274865
$$195$$ 5.41421 0.387720
$$196$$ 0 0
$$197$$ −0.514719 −0.0366722 −0.0183361 0.999832i $$-0.505837\pi$$
−0.0183361 + 0.999832i $$0.505837\pi$$
$$198$$ 2.82843 0.201008
$$199$$ −0.100505 −0.00712462 −0.00356231 0.999994i $$-0.501134\pi$$
−0.00356231 + 0.999994i $$0.501134\pi$$
$$200$$ −4.65685 −0.329289
$$201$$ 6.65685 0.469538
$$202$$ −6.17157 −0.434230
$$203$$ 0 0
$$204$$ −8.82843 −0.618114
$$205$$ 3.17157 0.221512
$$206$$ −13.4142 −0.934613
$$207$$ −17.6569 −1.22724
$$208$$ 3.82843 0.265454
$$209$$ 0.585786 0.0405197
$$210$$ 0 0
$$211$$ 7.41421 0.510416 0.255208 0.966886i $$-0.417856\pi$$
0.255208 + 0.966886i $$0.417856\pi$$
$$212$$ 7.89949 0.542540
$$213$$ −26.7279 −1.83137
$$214$$ −3.07107 −0.209934
$$215$$ −3.31371 −0.225993
$$216$$ −0.414214 −0.0281837
$$217$$ 0 0
$$218$$ 16.4853 1.11652
$$219$$ 22.7279 1.53581
$$220$$ 0.585786 0.0394937
$$221$$ −14.0000 −0.941742
$$222$$ −22.7279 −1.52540
$$223$$ −8.58579 −0.574947 −0.287473 0.957789i $$-0.592815\pi$$
−0.287473 + 0.957789i $$0.592815\pi$$
$$224$$ 0 0
$$225$$ −13.1716 −0.878105
$$226$$ −8.17157 −0.543565
$$227$$ 28.8284 1.91341 0.956705 0.291059i $$-0.0940078\pi$$
0.956705 + 0.291059i $$0.0940078\pi$$
$$228$$ 1.41421 0.0936586
$$229$$ −23.3137 −1.54061 −0.770307 0.637674i $$-0.779897\pi$$
−0.770307 + 0.637674i $$0.779897\pi$$
$$230$$ −3.65685 −0.241126
$$231$$ 0 0
$$232$$ 2.65685 0.174431
$$233$$ 1.41421 0.0926482 0.0463241 0.998926i $$-0.485249\pi$$
0.0463241 + 0.998926i $$0.485249\pi$$
$$234$$ 10.8284 0.707876
$$235$$ 6.14214 0.400669
$$236$$ 5.58579 0.363604
$$237$$ −31.9706 −2.07671
$$238$$ 0 0
$$239$$ 20.2132 1.30748 0.653742 0.756718i $$-0.273198\pi$$
0.653742 + 0.756718i $$0.273198\pi$$
$$240$$ 1.41421 0.0912871
$$241$$ −12.2426 −0.788618 −0.394309 0.918978i $$-0.629016\pi$$
−0.394309 + 0.918978i $$0.629016\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −21.6569 −1.38929
$$244$$ −11.8284 −0.757237
$$245$$ 0 0
$$246$$ 13.0711 0.833381
$$247$$ 2.24264 0.142696
$$248$$ 4.00000 0.254000
$$249$$ 29.3137 1.85768
$$250$$ −5.65685 −0.357771
$$251$$ 26.1421 1.65008 0.825038 0.565077i $$-0.191153\pi$$
0.825038 + 0.565077i $$0.191153\pi$$
$$252$$ 0 0
$$253$$ −6.24264 −0.392471
$$254$$ 15.7279 0.986858
$$255$$ −5.17157 −0.323856
$$256$$ 1.00000 0.0625000
$$257$$ 25.1421 1.56832 0.784162 0.620557i $$-0.213093\pi$$
0.784162 + 0.620557i $$0.213093\pi$$
$$258$$ −13.6569 −0.850239
$$259$$ 0 0
$$260$$ 2.24264 0.139083
$$261$$ 7.51472 0.465149
$$262$$ 0.585786 0.0361900
$$263$$ −17.0416 −1.05083 −0.525416 0.850845i $$-0.676090\pi$$
−0.525416 + 0.850845i $$0.676090\pi$$
$$264$$ 2.41421 0.148585
$$265$$ 4.62742 0.284260
$$266$$ 0 0
$$267$$ −30.1421 −1.84467
$$268$$ 2.75736 0.168433
$$269$$ −2.34315 −0.142864 −0.0714321 0.997445i $$-0.522757\pi$$
−0.0714321 + 0.997445i $$0.522757\pi$$
$$270$$ −0.242641 −0.0147666
$$271$$ −4.55635 −0.276779 −0.138389 0.990378i $$-0.544192\pi$$
−0.138389 + 0.990378i $$0.544192\pi$$
$$272$$ −3.65685 −0.221729
$$273$$ 0 0
$$274$$ −16.6569 −1.00628
$$275$$ −4.65685 −0.280819
$$276$$ −15.0711 −0.907172
$$277$$ 1.82843 0.109860 0.0549298 0.998490i $$-0.482507\pi$$
0.0549298 + 0.998490i $$0.482507\pi$$
$$278$$ 0 0
$$279$$ 11.3137 0.677334
$$280$$ 0 0
$$281$$ 8.72792 0.520664 0.260332 0.965519i $$-0.416168\pi$$
0.260332 + 0.965519i $$0.416168\pi$$
$$282$$ 25.3137 1.50741
$$283$$ 23.4142 1.39183 0.695915 0.718124i $$-0.254999\pi$$
0.695915 + 0.718124i $$0.254999\pi$$
$$284$$ −11.0711 −0.656947
$$285$$ 0.828427 0.0490718
$$286$$ 3.82843 0.226380
$$287$$ 0 0
$$288$$ 2.82843 0.166667
$$289$$ −3.62742 −0.213377
$$290$$ 1.55635 0.0913920
$$291$$ 9.24264 0.541813
$$292$$ 9.41421 0.550925
$$293$$ 10.8284 0.632603 0.316302 0.948659i $$-0.397559\pi$$
0.316302 + 0.948659i $$0.397559\pi$$
$$294$$ 0 0
$$295$$ 3.27208 0.190508
$$296$$ −9.41421 −0.547190
$$297$$ −0.414214 −0.0240351
$$298$$ 17.6569 1.02283
$$299$$ −23.8995 −1.38214
$$300$$ −11.2426 −0.649094
$$301$$ 0 0
$$302$$ −15.7279 −0.905040
$$303$$ −14.8995 −0.855954
$$304$$ 0.585786 0.0335972
$$305$$ −6.92893 −0.396750
$$306$$ −10.3431 −0.591278
$$307$$ −9.89949 −0.564994 −0.282497 0.959268i $$-0.591163\pi$$
−0.282497 + 0.959268i $$0.591163\pi$$
$$308$$ 0 0
$$309$$ −32.3848 −1.84231
$$310$$ 2.34315 0.133082
$$311$$ −16.7279 −0.948553 −0.474277 0.880376i $$-0.657290\pi$$
−0.474277 + 0.880376i $$0.657290\pi$$
$$312$$ 9.24264 0.523261
$$313$$ 20.6569 1.16759 0.583797 0.811900i $$-0.301566\pi$$
0.583797 + 0.811900i $$0.301566\pi$$
$$314$$ −17.6569 −0.996434
$$315$$ 0 0
$$316$$ −13.2426 −0.744957
$$317$$ 8.68629 0.487871 0.243935 0.969791i $$-0.421562\pi$$
0.243935 + 0.969791i $$0.421562\pi$$
$$318$$ 19.0711 1.06945
$$319$$ 2.65685 0.148755
$$320$$ 0.585786 0.0327465
$$321$$ −7.41421 −0.413821
$$322$$ 0 0
$$323$$ −2.14214 −0.119192
$$324$$ −9.48528 −0.526960
$$325$$ −17.8284 −0.988943
$$326$$ 9.72792 0.538780
$$327$$ 39.7990 2.20089
$$328$$ 5.41421 0.298950
$$329$$ 0 0
$$330$$ 1.41421 0.0778499
$$331$$ −24.0711 −1.32307 −0.661533 0.749916i $$-0.730094\pi$$
−0.661533 + 0.749916i $$0.730094\pi$$
$$332$$ 12.1421 0.666386
$$333$$ −26.6274 −1.45917
$$334$$ −13.7279 −0.751158
$$335$$ 1.61522 0.0882491
$$336$$ 0 0
$$337$$ −28.2426 −1.53847 −0.769237 0.638963i $$-0.779364\pi$$
−0.769237 + 0.638963i $$0.779364\pi$$
$$338$$ 1.65685 0.0901210
$$339$$ −19.7279 −1.07147
$$340$$ −2.14214 −0.116174
$$341$$ 4.00000 0.216612
$$342$$ 1.65685 0.0895924
$$343$$ 0 0
$$344$$ −5.65685 −0.304997
$$345$$ −8.82843 −0.475307
$$346$$ 9.82843 0.528380
$$347$$ 17.4142 0.934844 0.467422 0.884034i $$-0.345183\pi$$
0.467422 + 0.884034i $$0.345183\pi$$
$$348$$ 6.41421 0.343838
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ −1.58579 −0.0846430
$$352$$ 1.00000 0.0533002
$$353$$ 2.68629 0.142977 0.0714884 0.997441i $$-0.477225\pi$$
0.0714884 + 0.997441i $$0.477225\pi$$
$$354$$ 13.4853 0.716735
$$355$$ −6.48528 −0.344203
$$356$$ −12.4853 −0.661719
$$357$$ 0 0
$$358$$ 0.899495 0.0475398
$$359$$ 19.2426 1.01559 0.507794 0.861479i $$-0.330461\pi$$
0.507794 + 0.861479i $$0.330461\pi$$
$$360$$ 1.65685 0.0873239
$$361$$ −18.6569 −0.981940
$$362$$ 7.65685 0.402435
$$363$$ 2.41421 0.126713
$$364$$ 0 0
$$365$$ 5.51472 0.288654
$$366$$ −28.5563 −1.49266
$$367$$ 10.7279 0.559993 0.279996 0.960001i $$-0.409667\pi$$
0.279996 + 0.960001i $$0.409667\pi$$
$$368$$ −6.24264 −0.325420
$$369$$ 15.3137 0.797200
$$370$$ −5.51472 −0.286697
$$371$$ 0 0
$$372$$ 9.65685 0.500685
$$373$$ −19.9706 −1.03404 −0.517018 0.855974i $$-0.672958\pi$$
−0.517018 + 0.855974i $$0.672958\pi$$
$$374$$ −3.65685 −0.189091
$$375$$ −13.6569 −0.705237
$$376$$ 10.4853 0.540737
$$377$$ 10.1716 0.523863
$$378$$ 0 0
$$379$$ 27.8701 1.43159 0.715794 0.698311i $$-0.246065\pi$$
0.715794 + 0.698311i $$0.246065\pi$$
$$380$$ 0.343146 0.0176030
$$381$$ 37.9706 1.94529
$$382$$ −7.17157 −0.366930
$$383$$ 6.38478 0.326247 0.163123 0.986606i $$-0.447843\pi$$
0.163123 + 0.986606i $$0.447843\pi$$
$$384$$ 2.41421 0.123200
$$385$$ 0 0
$$386$$ 21.8995 1.11465
$$387$$ −16.0000 −0.813326
$$388$$ 3.82843 0.194359
$$389$$ −22.7279 −1.15235 −0.576176 0.817326i $$-0.695456\pi$$
−0.576176 + 0.817326i $$0.695456\pi$$
$$390$$ 5.41421 0.274159
$$391$$ 22.8284 1.15448
$$392$$ 0 0
$$393$$ 1.41421 0.0713376
$$394$$ −0.514719 −0.0259311
$$395$$ −7.75736 −0.390315
$$396$$ 2.82843 0.142134
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ −0.100505 −0.00503786
$$399$$ 0 0
$$400$$ −4.65685 −0.232843
$$401$$ 18.3137 0.914543 0.457271 0.889327i $$-0.348827\pi$$
0.457271 + 0.889327i $$0.348827\pi$$
$$402$$ 6.65685 0.332014
$$403$$ 15.3137 0.762830
$$404$$ −6.17157 −0.307047
$$405$$ −5.55635 −0.276097
$$406$$ 0 0
$$407$$ −9.41421 −0.466645
$$408$$ −8.82843 −0.437072
$$409$$ −2.72792 −0.134887 −0.0674435 0.997723i $$-0.521484\pi$$
−0.0674435 + 0.997723i $$0.521484\pi$$
$$410$$ 3.17157 0.156633
$$411$$ −40.2132 −1.98357
$$412$$ −13.4142 −0.660871
$$413$$ 0 0
$$414$$ −17.6569 −0.867787
$$415$$ 7.11270 0.349149
$$416$$ 3.82843 0.187704
$$417$$ 0 0
$$418$$ 0.585786 0.0286518
$$419$$ 26.1421 1.27713 0.638563 0.769569i $$-0.279529\pi$$
0.638563 + 0.769569i $$0.279529\pi$$
$$420$$ 0 0
$$421$$ 0.686292 0.0334478 0.0167239 0.999860i $$-0.494676\pi$$
0.0167239 + 0.999860i $$0.494676\pi$$
$$422$$ 7.41421 0.360918
$$423$$ 29.6569 1.44197
$$424$$ 7.89949 0.383633
$$425$$ 17.0294 0.826049
$$426$$ −26.7279 −1.29497
$$427$$ 0 0
$$428$$ −3.07107 −0.148446
$$429$$ 9.24264 0.446239
$$430$$ −3.31371 −0.159801
$$431$$ −17.5858 −0.847078 −0.423539 0.905878i $$-0.639212\pi$$
−0.423539 + 0.905878i $$0.639212\pi$$
$$432$$ −0.414214 −0.0199289
$$433$$ −26.1421 −1.25631 −0.628155 0.778088i $$-0.716190\pi$$
−0.628155 + 0.778088i $$0.716190\pi$$
$$434$$ 0 0
$$435$$ 3.75736 0.180152
$$436$$ 16.4853 0.789502
$$437$$ −3.65685 −0.174931
$$438$$ 22.7279 1.08598
$$439$$ 27.3848 1.30700 0.653502 0.756925i $$-0.273299\pi$$
0.653502 + 0.756925i $$0.273299\pi$$
$$440$$ 0.585786 0.0279263
$$441$$ 0 0
$$442$$ −14.0000 −0.665912
$$443$$ 12.6274 0.599947 0.299973 0.953948i $$-0.403022\pi$$
0.299973 + 0.953948i $$0.403022\pi$$
$$444$$ −22.7279 −1.07862
$$445$$ −7.31371 −0.346703
$$446$$ −8.58579 −0.406549
$$447$$ 42.6274 2.01621
$$448$$ 0 0
$$449$$ 22.3431 1.05444 0.527219 0.849729i $$-0.323235\pi$$
0.527219 + 0.849729i $$0.323235\pi$$
$$450$$ −13.1716 −0.620914
$$451$$ 5.41421 0.254945
$$452$$ −8.17157 −0.384358
$$453$$ −37.9706 −1.78401
$$454$$ 28.8284 1.35299
$$455$$ 0 0
$$456$$ 1.41421 0.0662266
$$457$$ −11.6569 −0.545285 −0.272642 0.962115i $$-0.587898\pi$$
−0.272642 + 0.962115i $$0.587898\pi$$
$$458$$ −23.3137 −1.08938
$$459$$ 1.51472 0.0707010
$$460$$ −3.65685 −0.170502
$$461$$ 8.31371 0.387208 0.193604 0.981080i $$-0.437982\pi$$
0.193604 + 0.981080i $$0.437982\pi$$
$$462$$ 0 0
$$463$$ 12.8284 0.596188 0.298094 0.954537i $$-0.403649\pi$$
0.298094 + 0.954537i $$0.403649\pi$$
$$464$$ 2.65685 0.123341
$$465$$ 5.65685 0.262330
$$466$$ 1.41421 0.0655122
$$467$$ 34.0000 1.57333 0.786666 0.617379i $$-0.211805\pi$$
0.786666 + 0.617379i $$0.211805\pi$$
$$468$$ 10.8284 0.500544
$$469$$ 0 0
$$470$$ 6.14214 0.283316
$$471$$ −42.6274 −1.96417
$$472$$ 5.58579 0.257107
$$473$$ −5.65685 −0.260102
$$474$$ −31.9706 −1.46846
$$475$$ −2.72792 −0.125166
$$476$$ 0 0
$$477$$ 22.3431 1.02302
$$478$$ 20.2132 0.924530
$$479$$ 11.9289 0.545047 0.272523 0.962149i $$-0.412142\pi$$
0.272523 + 0.962149i $$0.412142\pi$$
$$480$$ 1.41421 0.0645497
$$481$$ −36.0416 −1.64336
$$482$$ −12.2426 −0.557637
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 2.24264 0.101833
$$486$$ −21.6569 −0.982375
$$487$$ −9.65685 −0.437594 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$488$$ −11.8284 −0.535448
$$489$$ 23.4853 1.06204
$$490$$ 0 0
$$491$$ 19.1716 0.865201 0.432600 0.901586i $$-0.357596\pi$$
0.432600 + 0.901586i $$0.357596\pi$$
$$492$$ 13.0711 0.589289
$$493$$ −9.71573 −0.437574
$$494$$ 2.24264 0.100901
$$495$$ 1.65685 0.0744701
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 29.3137 1.31358
$$499$$ 22.1421 0.991218 0.495609 0.868546i $$-0.334945\pi$$
0.495609 + 0.868546i $$0.334945\pi$$
$$500$$ −5.65685 −0.252982
$$501$$ −33.1421 −1.48068
$$502$$ 26.1421 1.16678
$$503$$ −4.21320 −0.187857 −0.0939287 0.995579i $$-0.529943\pi$$
−0.0939287 + 0.995579i $$0.529943\pi$$
$$504$$ 0 0
$$505$$ −3.61522 −0.160875
$$506$$ −6.24264 −0.277519
$$507$$ 4.00000 0.177646
$$508$$ 15.7279 0.697814
$$509$$ −9.31371 −0.412823 −0.206411 0.978465i $$-0.566179\pi$$
−0.206411 + 0.978465i $$0.566179\pi$$
$$510$$ −5.17157 −0.229001
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −0.242641 −0.0107128
$$514$$ 25.1421 1.10897
$$515$$ −7.85786 −0.346259
$$516$$ −13.6569 −0.601209
$$517$$ 10.4853 0.461142
$$518$$ 0 0
$$519$$ 23.7279 1.04154
$$520$$ 2.24264 0.0983463
$$521$$ 28.2843 1.23916 0.619578 0.784935i $$-0.287304\pi$$
0.619578 + 0.784935i $$0.287304\pi$$
$$522$$ 7.51472 0.328910
$$523$$ 12.7279 0.556553 0.278277 0.960501i $$-0.410237\pi$$
0.278277 + 0.960501i $$0.410237\pi$$
$$524$$ 0.585786 0.0255902
$$525$$ 0 0
$$526$$ −17.0416 −0.743050
$$527$$ −14.6274 −0.637180
$$528$$ 2.41421 0.105065
$$529$$ 15.9706 0.694372
$$530$$ 4.62742 0.201002
$$531$$ 15.7990 0.685618
$$532$$ 0 0
$$533$$ 20.7279 0.897826
$$534$$ −30.1421 −1.30438
$$535$$ −1.79899 −0.0777771
$$536$$ 2.75736 0.119100
$$537$$ 2.17157 0.0937103
$$538$$ −2.34315 −0.101020
$$539$$ 0 0
$$540$$ −0.242641 −0.0104416
$$541$$ −12.8579 −0.552803 −0.276401 0.961042i $$-0.589142\pi$$
−0.276401 + 0.961042i $$0.589142\pi$$
$$542$$ −4.55635 −0.195712
$$543$$ 18.4853 0.793279
$$544$$ −3.65685 −0.156786
$$545$$ 9.65685 0.413654
$$546$$ 0 0
$$547$$ 34.8701 1.49094 0.745468 0.666541i $$-0.232226\pi$$
0.745468 + 0.666541i $$0.232226\pi$$
$$548$$ −16.6569 −0.711546
$$549$$ −33.4558 −1.42786
$$550$$ −4.65685 −0.198569
$$551$$ 1.55635 0.0663027
$$552$$ −15.0711 −0.641467
$$553$$ 0 0
$$554$$ 1.82843 0.0776824
$$555$$ −13.3137 −0.565135
$$556$$ 0 0
$$557$$ 7.51472 0.318409 0.159204 0.987246i $$-0.449107\pi$$
0.159204 + 0.987246i $$0.449107\pi$$
$$558$$ 11.3137 0.478947
$$559$$ −21.6569 −0.915987
$$560$$ 0 0
$$561$$ −8.82843 −0.372736
$$562$$ 8.72792 0.368165
$$563$$ −9.07107 −0.382300 −0.191150 0.981561i $$-0.561222\pi$$
−0.191150 + 0.981561i $$0.561222\pi$$
$$564$$ 25.3137 1.06590
$$565$$ −4.78680 −0.201382
$$566$$ 23.4142 0.984173
$$567$$ 0 0
$$568$$ −11.0711 −0.464532
$$569$$ −4.00000 −0.167689 −0.0838444 0.996479i $$-0.526720\pi$$
−0.0838444 + 0.996479i $$0.526720\pi$$
$$570$$ 0.828427 0.0346990
$$571$$ 26.3848 1.10417 0.552084 0.833788i $$-0.313833\pi$$
0.552084 + 0.833788i $$0.313833\pi$$
$$572$$ 3.82843 0.160075
$$573$$ −17.3137 −0.723291
$$574$$ 0 0
$$575$$ 29.0711 1.21235
$$576$$ 2.82843 0.117851
$$577$$ 9.68629 0.403246 0.201623 0.979463i $$-0.435378\pi$$
0.201623 + 0.979463i $$0.435378\pi$$
$$578$$ −3.62742 −0.150881
$$579$$ 52.8701 2.19720
$$580$$ 1.55635 0.0646239
$$581$$ 0 0
$$582$$ 9.24264 0.383120
$$583$$ 7.89949 0.327164
$$584$$ 9.41421 0.389563
$$585$$ 6.34315 0.262257
$$586$$ 10.8284 0.447318
$$587$$ 25.1005 1.03601 0.518004 0.855378i $$-0.326675\pi$$
0.518004 + 0.855378i $$0.326675\pi$$
$$588$$ 0 0
$$589$$ 2.34315 0.0965476
$$590$$ 3.27208 0.134709
$$591$$ −1.24264 −0.0511154
$$592$$ −9.41421 −0.386922
$$593$$ −23.6985 −0.973180 −0.486590 0.873630i $$-0.661759\pi$$
−0.486590 + 0.873630i $$0.661759\pi$$
$$594$$ −0.414214 −0.0169954
$$595$$ 0 0
$$596$$ 17.6569 0.723253
$$597$$ −0.242641 −0.00993062
$$598$$ −23.8995 −0.977323
$$599$$ 42.6274 1.74171 0.870855 0.491541i $$-0.163566\pi$$
0.870855 + 0.491541i $$0.163566\pi$$
$$600$$ −11.2426 −0.458979
$$601$$ −31.9411 −1.30291 −0.651453 0.758689i $$-0.725840\pi$$
−0.651453 + 0.758689i $$0.725840\pi$$
$$602$$ 0 0
$$603$$ 7.79899 0.317599
$$604$$ −15.7279 −0.639960
$$605$$ 0.585786 0.0238156
$$606$$ −14.8995 −0.605251
$$607$$ 42.9706 1.74412 0.872061 0.489398i $$-0.162783\pi$$
0.872061 + 0.489398i $$0.162783\pi$$
$$608$$ 0.585786 0.0237568
$$609$$ 0 0
$$610$$ −6.92893 −0.280544
$$611$$ 40.1421 1.62398
$$612$$ −10.3431 −0.418097
$$613$$ 28.6274 1.15625 0.578125 0.815948i $$-0.303785\pi$$
0.578125 + 0.815948i $$0.303785\pi$$
$$614$$ −9.89949 −0.399511
$$615$$ 7.65685 0.308754
$$616$$ 0 0
$$617$$ −41.9706 −1.68967 −0.844836 0.535026i $$-0.820302\pi$$
−0.844836 + 0.535026i $$0.820302\pi$$
$$618$$ −32.3848 −1.30271
$$619$$ 21.9411 0.881888 0.440944 0.897535i $$-0.354644\pi$$
0.440944 + 0.897535i $$0.354644\pi$$
$$620$$ 2.34315 0.0941030
$$621$$ 2.58579 0.103764
$$622$$ −16.7279 −0.670729
$$623$$ 0 0
$$624$$ 9.24264 0.370002
$$625$$ 19.9706 0.798823
$$626$$ 20.6569 0.825614
$$627$$ 1.41421 0.0564782
$$628$$ −17.6569 −0.704585
$$629$$ 34.4264 1.37267
$$630$$ 0 0
$$631$$ 23.2721 0.926447 0.463223 0.886242i $$-0.346693\pi$$
0.463223 + 0.886242i $$0.346693\pi$$
$$632$$ −13.2426 −0.526764
$$633$$ 17.8995 0.711441
$$634$$ 8.68629 0.344977
$$635$$ 9.21320 0.365615
$$636$$ 19.0711 0.756217
$$637$$ 0 0
$$638$$ 2.65685 0.105186
$$639$$ −31.3137 −1.23875
$$640$$ 0.585786 0.0231552
$$641$$ 15.2843 0.603692 0.301846 0.953357i $$-0.402397\pi$$
0.301846 + 0.953357i $$0.402397\pi$$
$$642$$ −7.41421 −0.292616
$$643$$ −1.58579 −0.0625373 −0.0312687 0.999511i $$-0.509955\pi$$
−0.0312687 + 0.999511i $$0.509955\pi$$
$$644$$ 0 0
$$645$$ −8.00000 −0.315000
$$646$$ −2.14214 −0.0842812
$$647$$ −30.1838 −1.18665 −0.593323 0.804964i $$-0.702184\pi$$
−0.593323 + 0.804964i $$0.702184\pi$$
$$648$$ −9.48528 −0.372617
$$649$$ 5.58579 0.219261
$$650$$ −17.8284 −0.699288
$$651$$ 0 0
$$652$$ 9.72792 0.380975
$$653$$ −18.3848 −0.719452 −0.359726 0.933058i $$-0.617130\pi$$
−0.359726 + 0.933058i $$0.617130\pi$$
$$654$$ 39.7990 1.55626
$$655$$ 0.343146 0.0134078
$$656$$ 5.41421 0.211390
$$657$$ 26.6274 1.03883
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 1.41421 0.0550482
$$661$$ −18.9706 −0.737869 −0.368935 0.929455i $$-0.620277\pi$$
−0.368935 + 0.929455i $$0.620277\pi$$
$$662$$ −24.0711 −0.935549
$$663$$ −33.7990 −1.31264
$$664$$ 12.1421 0.471206
$$665$$ 0 0
$$666$$ −26.6274 −1.03179
$$667$$ −16.5858 −0.642204
$$668$$ −13.7279 −0.531149
$$669$$ −20.7279 −0.801388
$$670$$ 1.61522 0.0624015
$$671$$ −11.8284 −0.456631
$$672$$ 0 0
$$673$$ 5.55635 0.214182 0.107091 0.994249i $$-0.465846\pi$$
0.107091 + 0.994249i $$0.465846\pi$$
$$674$$ −28.2426 −1.08787
$$675$$ 1.92893 0.0742446
$$676$$ 1.65685 0.0637252
$$677$$ 35.3137 1.35722 0.678608 0.734501i $$-0.262584\pi$$
0.678608 + 0.734501i $$0.262584\pi$$
$$678$$ −19.7279 −0.757646
$$679$$ 0 0
$$680$$ −2.14214 −0.0821472
$$681$$ 69.5980 2.66700
$$682$$ 4.00000 0.153168
$$683$$ 13.5858 0.519846 0.259923 0.965629i $$-0.416303\pi$$
0.259923 + 0.965629i $$0.416303\pi$$
$$684$$ 1.65685 0.0633514
$$685$$ −9.75736 −0.372810
$$686$$ 0 0
$$687$$ −56.2843 −2.14738
$$688$$ −5.65685 −0.215666
$$689$$ 30.2426 1.15215
$$690$$ −8.82843 −0.336092
$$691$$ −28.0711 −1.06787 −0.533937 0.845524i $$-0.679288\pi$$
−0.533937 + 0.845524i $$0.679288\pi$$
$$692$$ 9.82843 0.373621
$$693$$ 0 0
$$694$$ 17.4142 0.661035
$$695$$ 0 0
$$696$$ 6.41421 0.243130
$$697$$ −19.7990 −0.749940
$$698$$ 0 0
$$699$$ 3.41421 0.129137
$$700$$ 0 0
$$701$$ −26.1127 −0.986263 −0.493132 0.869955i $$-0.664148\pi$$
−0.493132 + 0.869955i $$0.664148\pi$$
$$702$$ −1.58579 −0.0598517
$$703$$ −5.51472 −0.207992
$$704$$ 1.00000 0.0376889
$$705$$ 14.8284 0.558471
$$706$$ 2.68629 0.101100
$$707$$ 0 0
$$708$$ 13.4853 0.506808
$$709$$ −12.0416 −0.452233 −0.226116 0.974100i $$-0.572603\pi$$
−0.226116 + 0.974100i $$0.572603\pi$$
$$710$$ −6.48528 −0.243388
$$711$$ −37.4558 −1.40470
$$712$$ −12.4853 −0.467906
$$713$$ −24.9706 −0.935155
$$714$$ 0 0
$$715$$ 2.24264 0.0838700
$$716$$ 0.899495 0.0336157
$$717$$ 48.7990 1.82243
$$718$$ 19.2426 0.718129
$$719$$ 1.51472 0.0564895 0.0282447 0.999601i $$-0.491008\pi$$
0.0282447 + 0.999601i $$0.491008\pi$$
$$720$$ 1.65685 0.0617473
$$721$$ 0 0
$$722$$ −18.6569 −0.694336
$$723$$ −29.5563 −1.09921
$$724$$ 7.65685 0.284565
$$725$$ −12.3726 −0.459506
$$726$$ 2.41421 0.0895999
$$727$$ −36.4264 −1.35098 −0.675490 0.737369i $$-0.736068\pi$$
−0.675490 + 0.737369i $$0.736068\pi$$
$$728$$ 0 0
$$729$$ −23.8284 −0.882534
$$730$$ 5.51472 0.204109
$$731$$ 20.6863 0.765110
$$732$$ −28.5563 −1.05547
$$733$$ 13.0000 0.480166 0.240083 0.970752i $$-0.422825\pi$$
0.240083 + 0.970752i $$0.422825\pi$$
$$734$$ 10.7279 0.395975
$$735$$ 0 0
$$736$$ −6.24264 −0.230107
$$737$$ 2.75736 0.101569
$$738$$ 15.3137 0.563705
$$739$$ −52.4264 −1.92854 −0.964268 0.264928i $$-0.914652\pi$$
−0.964268 + 0.264928i $$0.914652\pi$$
$$740$$ −5.51472 −0.202725
$$741$$ 5.41421 0.198896
$$742$$ 0 0
$$743$$ −13.3137 −0.488433 −0.244216 0.969721i $$-0.578531\pi$$
−0.244216 + 0.969721i $$0.578531\pi$$
$$744$$ 9.65685 0.354037
$$745$$ 10.3431 0.378944
$$746$$ −19.9706 −0.731174
$$747$$ 34.3431 1.25655
$$748$$ −3.65685 −0.133708
$$749$$ 0 0
$$750$$ −13.6569 −0.498678
$$751$$ 45.6569 1.66604 0.833021 0.553241i $$-0.186609\pi$$
0.833021 + 0.553241i $$0.186609\pi$$
$$752$$ 10.4853 0.382359
$$753$$ 63.1127 2.29995
$$754$$ 10.1716 0.370427
$$755$$ −9.21320 −0.335303
$$756$$ 0 0
$$757$$ 8.34315 0.303237 0.151618 0.988439i $$-0.451552\pi$$
0.151618 + 0.988439i $$0.451552\pi$$
$$758$$ 27.8701 1.01229
$$759$$ −15.0711 −0.547045
$$760$$ 0.343146 0.0124472
$$761$$ −30.9706 −1.12268 −0.561341 0.827585i $$-0.689714\pi$$
−0.561341 + 0.827585i $$0.689714\pi$$
$$762$$ 37.9706 1.37553
$$763$$ 0 0
$$764$$ −7.17157 −0.259458
$$765$$ −6.05887 −0.219059
$$766$$ 6.38478 0.230691
$$767$$ 21.3848 0.772160
$$768$$ 2.41421 0.0871154
$$769$$ −22.9706 −0.828340 −0.414170 0.910200i $$-0.635928\pi$$
−0.414170 + 0.910200i $$0.635928\pi$$
$$770$$ 0 0
$$771$$ 60.6985 2.18600
$$772$$ 21.8995 0.788180
$$773$$ 21.9411 0.789167 0.394584 0.918860i $$-0.370889\pi$$
0.394584 + 0.918860i $$0.370889\pi$$
$$774$$ −16.0000 −0.575108
$$775$$ −18.6274 −0.669117
$$776$$ 3.82843 0.137433
$$777$$ 0 0
$$778$$ −22.7279 −0.814835
$$779$$ 3.17157 0.113633
$$780$$ 5.41421 0.193860
$$781$$ −11.0711 −0.396154
$$782$$ 22.8284 0.816343
$$783$$ −1.10051 −0.0393288
$$784$$ 0 0
$$785$$ −10.3431 −0.369163
$$786$$ 1.41421 0.0504433
$$787$$ −13.5563 −0.483232 −0.241616 0.970372i $$-0.577677\pi$$
−0.241616 + 0.970372i $$0.577677\pi$$
$$788$$ −0.514719 −0.0183361
$$789$$ −41.1421 −1.46470
$$790$$ −7.75736 −0.275994
$$791$$ 0 0
$$792$$ 2.82843 0.100504
$$793$$ −45.2843 −1.60809
$$794$$ −22.0000 −0.780751
$$795$$ 11.1716 0.396215
$$796$$ −0.100505 −0.00356231
$$797$$ 7.65685 0.271220 0.135610 0.990762i $$-0.456701\pi$$
0.135610 + 0.990762i $$0.456701\pi$$
$$798$$ 0 0
$$799$$ −38.3431 −1.35648
$$800$$ −4.65685 −0.164645
$$801$$ −35.3137 −1.24775
$$802$$ 18.3137 0.646680
$$803$$ 9.41421 0.332220
$$804$$ 6.65685 0.234769
$$805$$ 0 0
$$806$$ 15.3137 0.539402
$$807$$ −5.65685 −0.199131
$$808$$ −6.17157 −0.217115
$$809$$ −46.6274 −1.63933 −0.819666 0.572841i $$-0.805841\pi$$
−0.819666 + 0.572841i $$0.805841\pi$$
$$810$$ −5.55635 −0.195230
$$811$$ 24.2843 0.852736 0.426368 0.904550i $$-0.359793\pi$$
0.426368 + 0.904550i $$0.359793\pi$$
$$812$$ 0 0
$$813$$ −11.0000 −0.385787
$$814$$ −9.41421 −0.329968
$$815$$ 5.69848 0.199609
$$816$$ −8.82843 −0.309057
$$817$$ −3.31371 −0.115932
$$818$$ −2.72792 −0.0953796
$$819$$ 0 0
$$820$$ 3.17157 0.110756
$$821$$ −29.4853 −1.02904 −0.514522 0.857477i $$-0.672031\pi$$
−0.514522 + 0.857477i $$0.672031\pi$$
$$822$$ −40.2132 −1.40260
$$823$$ −43.4558 −1.51478 −0.757388 0.652965i $$-0.773525\pi$$
−0.757388 + 0.652965i $$0.773525\pi$$
$$824$$ −13.4142 −0.467306
$$825$$ −11.2426 −0.391419
$$826$$ 0 0
$$827$$ −21.3553 −0.742598 −0.371299 0.928513i $$-0.621087\pi$$
−0.371299 + 0.928513i $$0.621087\pi$$
$$828$$ −17.6569 −0.613618
$$829$$ −21.7574 −0.755664 −0.377832 0.925874i $$-0.623330\pi$$
−0.377832 + 0.925874i $$0.623330\pi$$
$$830$$ 7.11270 0.246885
$$831$$ 4.41421 0.153127
$$832$$ 3.82843 0.132727
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −8.04163 −0.278292
$$836$$ 0.585786 0.0202598
$$837$$ −1.65685 −0.0572693
$$838$$ 26.1421 0.903065
$$839$$ 18.4853 0.638183 0.319091 0.947724i $$-0.396622\pi$$
0.319091 + 0.947724i $$0.396622\pi$$
$$840$$ 0 0
$$841$$ −21.9411 −0.756591
$$842$$ 0.686292 0.0236512
$$843$$ 21.0711 0.725726
$$844$$ 7.41421 0.255208
$$845$$ 0.970563 0.0333884
$$846$$ 29.6569 1.01962
$$847$$ 0 0
$$848$$ 7.89949 0.271270
$$849$$ 56.5269 1.94000
$$850$$ 17.0294 0.584105
$$851$$ 58.7696 2.01459
$$852$$ −26.7279 −0.915684
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 0 0
$$855$$ 0.970563 0.0331925
$$856$$ −3.07107 −0.104967
$$857$$ 34.6274 1.18285 0.591425 0.806360i $$-0.298566\pi$$
0.591425 + 0.806360i $$0.298566\pi$$
$$858$$ 9.24264 0.315539
$$859$$ 5.72792 0.195434 0.0977171 0.995214i $$-0.468846\pi$$
0.0977171 + 0.995214i $$0.468846\pi$$
$$860$$ −3.31371 −0.112997
$$861$$ 0 0
$$862$$ −17.5858 −0.598974
$$863$$ 35.2132 1.19867 0.599336 0.800498i $$-0.295431\pi$$
0.599336 + 0.800498i $$0.295431\pi$$
$$864$$ −0.414214 −0.0140918
$$865$$ 5.75736 0.195756
$$866$$ −26.1421 −0.888346
$$867$$ −8.75736 −0.297416
$$868$$ 0 0
$$869$$ −13.2426 −0.449226
$$870$$ 3.75736 0.127386
$$871$$ 10.5563 0.357688
$$872$$ 16.4853 0.558262
$$873$$ 10.8284 0.366487
$$874$$ −3.65685 −0.123695
$$875$$ 0 0
$$876$$ 22.7279 0.767905
$$877$$ 19.6274 0.662771 0.331385 0.943495i $$-0.392484\pi$$
0.331385 + 0.943495i $$0.392484\pi$$
$$878$$ 27.3848 0.924191
$$879$$ 26.1421 0.881752
$$880$$ 0.585786 0.0197469
$$881$$ −6.45584 −0.217503 −0.108751 0.994069i $$-0.534685\pi$$
−0.108751 + 0.994069i $$0.534685\pi$$
$$882$$ 0 0
$$883$$ 15.7279 0.529287 0.264643 0.964346i $$-0.414746\pi$$
0.264643 + 0.964346i $$0.414746\pi$$
$$884$$ −14.0000 −0.470871
$$885$$ 7.89949 0.265539
$$886$$ 12.6274 0.424226
$$887$$ 3.10051 0.104105 0.0520524 0.998644i $$-0.483424\pi$$
0.0520524 + 0.998644i $$0.483424\pi$$
$$888$$ −22.7279 −0.762699
$$889$$ 0 0
$$890$$ −7.31371 −0.245156
$$891$$ −9.48528 −0.317769
$$892$$ −8.58579 −0.287473
$$893$$ 6.14214 0.205539
$$894$$ 42.6274 1.42567
$$895$$ 0.526912 0.0176127
$$896$$ 0 0
$$897$$ −57.6985 −1.92650
$$898$$ 22.3431 0.745600
$$899$$ 10.6274 0.354444
$$900$$ −13.1716 −0.439052
$$901$$ −28.8873 −0.962376
$$902$$ 5.41421 0.180274
$$903$$ 0 0
$$904$$ −8.17157 −0.271782
$$905$$ 4.48528 0.149096
$$906$$ −37.9706 −1.26149
$$907$$ 10.6863 0.354832 0.177416 0.984136i $$-0.443226\pi$$
0.177416 + 0.984136i $$0.443226\pi$$
$$908$$ 28.8284 0.956705
$$909$$ −17.4558 −0.578974
$$910$$ 0 0
$$911$$ −30.4853 −1.01002 −0.505011 0.863113i $$-0.668512\pi$$
−0.505011 + 0.863113i $$0.668512\pi$$
$$912$$ 1.41421 0.0468293
$$913$$ 12.1421 0.401846
$$914$$ −11.6569 −0.385574
$$915$$ −16.7279 −0.553008
$$916$$ −23.3137 −0.770307
$$917$$ 0 0
$$918$$ 1.51472 0.0499932
$$919$$ 22.1421 0.730402 0.365201 0.930929i $$-0.381000\pi$$
0.365201 + 0.930929i $$0.381000\pi$$
$$920$$ −3.65685 −0.120563
$$921$$ −23.8995 −0.787515
$$922$$ 8.31371 0.273797
$$923$$ −42.3848 −1.39511
$$924$$ 0 0
$$925$$ 43.8406 1.44147
$$926$$ 12.8284 0.421568
$$927$$ −37.9411 −1.24615
$$928$$ 2.65685 0.0872155
$$929$$ 40.4558 1.32731 0.663657 0.748037i $$-0.269004\pi$$
0.663657 + 0.748037i $$0.269004\pi$$
$$930$$ 5.65685 0.185496
$$931$$ 0 0
$$932$$ 1.41421 0.0463241
$$933$$ −40.3848 −1.32214
$$934$$ 34.0000 1.11251
$$935$$ −2.14214 −0.0700553
$$936$$ 10.8284 0.353938
$$937$$ 19.4142 0.634235 0.317117 0.948386i $$-0.397285\pi$$
0.317117 + 0.948386i $$0.397285\pi$$
$$938$$ 0 0
$$939$$ 49.8701 1.62745
$$940$$ 6.14214 0.200334
$$941$$ −24.6569 −0.803790 −0.401895 0.915686i $$-0.631648\pi$$
−0.401895 + 0.915686i $$0.631648\pi$$
$$942$$ −42.6274 −1.38888
$$943$$ −33.7990 −1.10065
$$944$$ 5.58579 0.181802
$$945$$ 0 0
$$946$$ −5.65685 −0.183920
$$947$$ 34.8284 1.13177 0.565886 0.824484i $$-0.308534\pi$$
0.565886 + 0.824484i $$0.308534\pi$$
$$948$$ −31.9706 −1.03836
$$949$$ 36.0416 1.16996
$$950$$ −2.72792 −0.0885055
$$951$$ 20.9706 0.680017
$$952$$ 0 0
$$953$$ 5.55635 0.179988 0.0899939 0.995942i $$-0.471315\pi$$
0.0899939 + 0.995942i $$0.471315\pi$$
$$954$$ 22.3431 0.723386
$$955$$ −4.20101 −0.135941
$$956$$ 20.2132 0.653742
$$957$$ 6.41421 0.207342
$$958$$ 11.9289 0.385406
$$959$$ 0 0
$$960$$ 1.41421 0.0456435
$$961$$ −15.0000 −0.483871
$$962$$ −36.0416 −1.16203
$$963$$ −8.68629 −0.279912
$$964$$ −12.2426 −0.394309
$$965$$ 12.8284 0.412962
$$966$$ 0 0
$$967$$ −36.2843 −1.16682 −0.583412 0.812177i $$-0.698283\pi$$
−0.583412 + 0.812177i $$0.698283\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ −5.17157 −0.166135
$$970$$ 2.24264 0.0720069
$$971$$ −22.2721 −0.714745 −0.357372 0.933962i $$-0.616327\pi$$
−0.357372 + 0.933962i $$0.616327\pi$$
$$972$$ −21.6569 −0.694644
$$973$$ 0 0
$$974$$ −9.65685 −0.309426
$$975$$ −43.0416 −1.37844
$$976$$ −11.8284 −0.378619
$$977$$ 40.0000 1.27971 0.639857 0.768494i $$-0.278994\pi$$
0.639857 + 0.768494i $$0.278994\pi$$
$$978$$ 23.4853 0.750976
$$979$$ −12.4853 −0.399031
$$980$$ 0 0
$$981$$ 46.6274 1.48870
$$982$$ 19.1716 0.611789
$$983$$ −32.4264 −1.03424 −0.517121 0.855912i $$-0.672996\pi$$
−0.517121 + 0.855912i $$0.672996\pi$$
$$984$$ 13.0711 0.416690
$$985$$ −0.301515 −0.00960707
$$986$$ −9.71573 −0.309412
$$987$$ 0 0
$$988$$ 2.24264 0.0713479
$$989$$ 35.3137 1.12291
$$990$$ 1.65685 0.0526583
$$991$$ −1.61522 −0.0513093 −0.0256546 0.999671i $$-0.508167\pi$$
−0.0256546 + 0.999671i $$0.508167\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −58.1127 −1.84415
$$994$$ 0 0
$$995$$ −0.0588745 −0.00186645
$$996$$ 29.3137 0.928840
$$997$$ −13.1716 −0.417148 −0.208574 0.978007i $$-0.566882\pi$$
−0.208574 + 0.978007i $$0.566882\pi$$
$$998$$ 22.1421 0.700897
$$999$$ 3.89949 0.123375
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.x.1.2 2
3.2 odd 2 9702.2.a.ch.1.2 2
4.3 odd 2 8624.2.a.bh.1.1 2
7.2 even 3 1078.2.e.m.67.1 4
7.3 odd 6 154.2.e.e.23.2 4
7.4 even 3 1078.2.e.m.177.1 4
7.5 odd 6 154.2.e.e.67.2 yes 4
7.6 odd 2 1078.2.a.t.1.1 2
21.5 even 6 1386.2.k.t.991.2 4
21.17 even 6 1386.2.k.t.793.2 4
21.20 even 2 9702.2.a.cx.1.1 2
28.3 even 6 1232.2.q.f.177.1 4
28.19 even 6 1232.2.q.f.529.1 4
28.27 even 2 8624.2.a.cc.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.2 4 7.3 odd 6
154.2.e.e.67.2 yes 4 7.5 odd 6
1078.2.a.t.1.1 2 7.6 odd 2
1078.2.a.x.1.2 2 1.1 even 1 trivial
1078.2.e.m.67.1 4 7.2 even 3
1078.2.e.m.177.1 4 7.4 even 3
1232.2.q.f.177.1 4 28.3 even 6
1232.2.q.f.529.1 4 28.19 even 6
1386.2.k.t.793.2 4 21.17 even 6
1386.2.k.t.991.2 4 21.5 even 6
8624.2.a.bh.1.1 2 4.3 odd 2
8624.2.a.cc.1.2 2 28.27 even 2
9702.2.a.ch.1.2 2 3.2 odd 2
9702.2.a.cx.1.1 2 21.20 even 2