# Properties

 Label 8281.2.a.z.1.1 Level $8281$ Weight $2$ Character 8281.1 Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$66.1241179138$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 8281.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.381966 q^{2} +2.23607 q^{3} -1.85410 q^{4} +2.23607 q^{5} +0.854102 q^{6} -1.47214 q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q+0.381966 q^{2} +2.23607 q^{3} -1.85410 q^{4} +2.23607 q^{5} +0.854102 q^{6} -1.47214 q^{8} +2.00000 q^{9} +0.854102 q^{10} +3.00000 q^{11} -4.14590 q^{12} +5.00000 q^{15} +3.14590 q^{16} -7.47214 q^{17} +0.763932 q^{18} -3.00000 q^{19} -4.14590 q^{20} +1.14590 q^{22} -3.76393 q^{23} -3.29180 q^{24} -2.23607 q^{27} -4.47214 q^{29} +1.90983 q^{30} -5.00000 q^{31} +4.14590 q^{32} +6.70820 q^{33} -2.85410 q^{34} -3.70820 q^{36} +8.70820 q^{37} -1.14590 q^{38} -3.29180 q^{40} -4.47214 q^{41} -8.00000 q^{43} -5.56231 q^{44} +4.47214 q^{45} -1.43769 q^{46} -1.47214 q^{47} +7.03444 q^{48} -16.7082 q^{51} +1.47214 q^{53} -0.854102 q^{54} +6.70820 q^{55} -6.70820 q^{57} -1.70820 q^{58} -7.47214 q^{59} -9.27051 q^{60} +3.00000 q^{61} -1.90983 q^{62} -4.70820 q^{64} +2.56231 q^{66} +3.00000 q^{67} +13.8541 q^{68} -8.41641 q^{69} -8.94427 q^{71} -2.94427 q^{72} -10.7082 q^{73} +3.32624 q^{74} +5.56231 q^{76} +10.7082 q^{79} +7.03444 q^{80} -11.0000 q^{81} -1.70820 q^{82} -16.7082 q^{85} -3.05573 q^{86} -10.0000 q^{87} -4.41641 q^{88} +2.23607 q^{89} +1.70820 q^{90} +6.97871 q^{92} -11.1803 q^{93} -0.562306 q^{94} -6.70820 q^{95} +9.27051 q^{96} +17.4164 q^{97} +6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9 $$2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9} - 5 q^{10} + 6 q^{11} - 15 q^{12} + 10 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} - 6 q^{19} - 15 q^{20} + 9 q^{22} - 12 q^{23} - 20 q^{24} + 15 q^{30} - 10 q^{31} + 15 q^{32} + q^{34} + 6 q^{36} + 4 q^{37} - 9 q^{38} - 20 q^{40} - 16 q^{43} + 9 q^{44} - 23 q^{46} + 6 q^{47} - 15 q^{48} - 20 q^{51} - 6 q^{53} + 5 q^{54} + 10 q^{58} - 6 q^{59} + 15 q^{60} + 6 q^{61} - 15 q^{62} + 4 q^{64} - 15 q^{66} + 6 q^{67} + 21 q^{68} + 10 q^{69} + 12 q^{72} - 8 q^{73} - 9 q^{74} - 9 q^{76} + 8 q^{79} - 15 q^{80} - 22 q^{81} + 10 q^{82} - 20 q^{85} - 24 q^{86} - 20 q^{87} + 18 q^{88} - 10 q^{90} - 33 q^{92} + 19 q^{94} - 15 q^{96} + 8 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9 - 5 * q^10 + 6 * q^11 - 15 * q^12 + 10 * q^15 + 13 * q^16 - 6 * q^17 + 6 * q^18 - 6 * q^19 - 15 * q^20 + 9 * q^22 - 12 * q^23 - 20 * q^24 + 15 * q^30 - 10 * q^31 + 15 * q^32 + q^34 + 6 * q^36 + 4 * q^37 - 9 * q^38 - 20 * q^40 - 16 * q^43 + 9 * q^44 - 23 * q^46 + 6 * q^47 - 15 * q^48 - 20 * q^51 - 6 * q^53 + 5 * q^54 + 10 * q^58 - 6 * q^59 + 15 * q^60 + 6 * q^61 - 15 * q^62 + 4 * q^64 - 15 * q^66 + 6 * q^67 + 21 * q^68 + 10 * q^69 + 12 * q^72 - 8 * q^73 - 9 * q^74 - 9 * q^76 + 8 * q^79 - 15 * q^80 - 22 * q^81 + 10 * q^82 - 20 * q^85 - 24 * q^86 - 20 * q^87 + 18 * q^88 - 10 * q^90 - 33 * q^92 + 19 * q^94 - 15 * q^96 + 8 * q^97 + 12 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.381966 0.270091 0.135045 0.990839i $$-0.456882\pi$$
0.135045 + 0.990839i $$0.456882\pi$$
$$3$$ 2.23607 1.29099 0.645497 0.763763i $$-0.276650\pi$$
0.645497 + 0.763763i $$0.276650\pi$$
$$4$$ −1.85410 −0.927051
$$5$$ 2.23607 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$6$$ 0.854102 0.348686
$$7$$ 0 0
$$8$$ −1.47214 −0.520479
$$9$$ 2.00000 0.666667
$$10$$ 0.854102 0.270091
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ −4.14590 −1.19682
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 5.00000 1.29099
$$16$$ 3.14590 0.786475
$$17$$ −7.47214 −1.81226 −0.906130 0.423000i $$-0.860977\pi$$
−0.906130 + 0.423000i $$0.860977\pi$$
$$18$$ 0.763932 0.180061
$$19$$ −3.00000 −0.688247 −0.344124 0.938924i $$-0.611824\pi$$
−0.344124 + 0.938924i $$0.611824\pi$$
$$20$$ −4.14590 −0.927051
$$21$$ 0 0
$$22$$ 1.14590 0.244306
$$23$$ −3.76393 −0.784834 −0.392417 0.919787i $$-0.628361\pi$$
−0.392417 + 0.919787i $$0.628361\pi$$
$$24$$ −3.29180 −0.671935
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −2.23607 −0.430331
$$28$$ 0 0
$$29$$ −4.47214 −0.830455 −0.415227 0.909718i $$-0.636298\pi$$
−0.415227 + 0.909718i $$0.636298\pi$$
$$30$$ 1.90983 0.348686
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 4.14590 0.732898
$$33$$ 6.70820 1.16775
$$34$$ −2.85410 −0.489474
$$35$$ 0 0
$$36$$ −3.70820 −0.618034
$$37$$ 8.70820 1.43162 0.715810 0.698295i $$-0.246058\pi$$
0.715810 + 0.698295i $$0.246058\pi$$
$$38$$ −1.14590 −0.185889
$$39$$ 0 0
$$40$$ −3.29180 −0.520479
$$41$$ −4.47214 −0.698430 −0.349215 0.937043i $$-0.613552\pi$$
−0.349215 + 0.937043i $$0.613552\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −5.56231 −0.838549
$$45$$ 4.47214 0.666667
$$46$$ −1.43769 −0.211976
$$47$$ −1.47214 −0.214733 −0.107367 0.994220i $$-0.534242\pi$$
−0.107367 + 0.994220i $$0.534242\pi$$
$$48$$ 7.03444 1.01533
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −16.7082 −2.33962
$$52$$ 0 0
$$53$$ 1.47214 0.202213 0.101107 0.994876i $$-0.467762\pi$$
0.101107 + 0.994876i $$0.467762\pi$$
$$54$$ −0.854102 −0.116229
$$55$$ 6.70820 0.904534
$$56$$ 0 0
$$57$$ −6.70820 −0.888523
$$58$$ −1.70820 −0.224298
$$59$$ −7.47214 −0.972789 −0.486395 0.873739i $$-0.661688\pi$$
−0.486395 + 0.873739i $$0.661688\pi$$
$$60$$ −9.27051 −1.19682
$$61$$ 3.00000 0.384111 0.192055 0.981384i $$-0.438485\pi$$
0.192055 + 0.981384i $$0.438485\pi$$
$$62$$ −1.90983 −0.242549
$$63$$ 0 0
$$64$$ −4.70820 −0.588525
$$65$$ 0 0
$$66$$ 2.56231 0.315398
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ 13.8541 1.68006
$$69$$ −8.41641 −1.01322
$$70$$ 0 0
$$71$$ −8.94427 −1.06149 −0.530745 0.847532i $$-0.678088\pi$$
−0.530745 + 0.847532i $$0.678088\pi$$
$$72$$ −2.94427 −0.346986
$$73$$ −10.7082 −1.25330 −0.626650 0.779301i $$-0.715575\pi$$
−0.626650 + 0.779301i $$0.715575\pi$$
$$74$$ 3.32624 0.386667
$$75$$ 0 0
$$76$$ 5.56231 0.638040
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.7082 1.20477 0.602384 0.798207i $$-0.294218\pi$$
0.602384 + 0.798207i $$0.294218\pi$$
$$80$$ 7.03444 0.786475
$$81$$ −11.0000 −1.22222
$$82$$ −1.70820 −0.188640
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −16.7082 −1.81226
$$86$$ −3.05573 −0.329508
$$87$$ −10.0000 −1.07211
$$88$$ −4.41641 −0.470791
$$89$$ 2.23607 0.237023 0.118511 0.992953i $$-0.462188\pi$$
0.118511 + 0.992953i $$0.462188\pi$$
$$90$$ 1.70820 0.180061
$$91$$ 0 0
$$92$$ 6.97871 0.727581
$$93$$ −11.1803 −1.15935
$$94$$ −0.562306 −0.0579974
$$95$$ −6.70820 −0.688247
$$96$$ 9.27051 0.946167
$$97$$ 17.4164 1.76837 0.884184 0.467139i $$-0.154715\pi$$
0.884184 + 0.467139i $$0.154715\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ −6.38197 −0.631909
$$103$$ 10.7082 1.05511 0.527555 0.849521i $$-0.323109\pi$$
0.527555 + 0.849521i $$0.323109\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0.562306 0.0546160
$$107$$ −14.2361 −1.37625 −0.688126 0.725591i $$-0.741567\pi$$
−0.688126 + 0.725591i $$0.741567\pi$$
$$108$$ 4.14590 0.398939
$$109$$ −10.7082 −1.02566 −0.512830 0.858490i $$-0.671403\pi$$
−0.512830 + 0.858490i $$0.671403\pi$$
$$110$$ 2.56231 0.244306
$$111$$ 19.4721 1.84821
$$112$$ 0 0
$$113$$ −14.9443 −1.40584 −0.702919 0.711269i $$-0.748121\pi$$
−0.702919 + 0.711269i $$0.748121\pi$$
$$114$$ −2.56231 −0.239982
$$115$$ −8.41641 −0.784834
$$116$$ 8.29180 0.769874
$$117$$ 0 0
$$118$$ −2.85410 −0.262741
$$119$$ 0 0
$$120$$ −7.36068 −0.671935
$$121$$ −2.00000 −0.181818
$$122$$ 1.14590 0.103745
$$123$$ −10.0000 −0.901670
$$124$$ 9.27051 0.832516
$$125$$ −11.1803 −1.00000
$$126$$ 0 0
$$127$$ 15.4164 1.36798 0.683992 0.729489i $$-0.260242\pi$$
0.683992 + 0.729489i $$0.260242\pi$$
$$128$$ −10.0902 −0.891853
$$129$$ −17.8885 −1.57500
$$130$$ 0 0
$$131$$ −3.76393 −0.328856 −0.164428 0.986389i $$-0.552578\pi$$
−0.164428 + 0.986389i $$0.552578\pi$$
$$132$$ −12.4377 −1.08256
$$133$$ 0 0
$$134$$ 1.14590 0.0989905
$$135$$ −5.00000 −0.430331
$$136$$ 11.0000 0.943242
$$137$$ 3.76393 0.321574 0.160787 0.986989i $$-0.448597\pi$$
0.160787 + 0.986989i $$0.448597\pi$$
$$138$$ −3.21478 −0.273660
$$139$$ 3.41641 0.289776 0.144888 0.989448i $$-0.453718\pi$$
0.144888 + 0.989448i $$0.453718\pi$$
$$140$$ 0 0
$$141$$ −3.29180 −0.277219
$$142$$ −3.41641 −0.286699
$$143$$ 0 0
$$144$$ 6.29180 0.524316
$$145$$ −10.0000 −0.830455
$$146$$ −4.09017 −0.338505
$$147$$ 0 0
$$148$$ −16.1459 −1.32718
$$149$$ 12.7082 1.04110 0.520548 0.853832i $$-0.325728\pi$$
0.520548 + 0.853832i $$0.325728\pi$$
$$150$$ 0 0
$$151$$ 6.41641 0.522160 0.261080 0.965317i $$-0.415921\pi$$
0.261080 + 0.965317i $$0.415921\pi$$
$$152$$ 4.41641 0.358218
$$153$$ −14.9443 −1.20817
$$154$$ 0 0
$$155$$ −11.1803 −0.898027
$$156$$ 0 0
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ 4.09017 0.325396
$$159$$ 3.29180 0.261056
$$160$$ 9.27051 0.732898
$$161$$ 0 0
$$162$$ −4.20163 −0.330111
$$163$$ −10.4164 −0.815876 −0.407938 0.913010i $$-0.633752\pi$$
−0.407938 + 0.913010i $$0.633752\pi$$
$$164$$ 8.29180 0.647480
$$165$$ 15.0000 1.16775
$$166$$ 0 0
$$167$$ −13.5279 −1.04682 −0.523409 0.852082i $$-0.675340\pi$$
−0.523409 + 0.852082i $$0.675340\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −6.38197 −0.489474
$$171$$ −6.00000 −0.458831
$$172$$ 14.8328 1.13099
$$173$$ 10.4164 0.791945 0.395972 0.918262i $$-0.370408\pi$$
0.395972 + 0.918262i $$0.370408\pi$$
$$174$$ −3.81966 −0.289568
$$175$$ 0 0
$$176$$ 9.43769 0.711393
$$177$$ −16.7082 −1.25587
$$178$$ 0.854102 0.0640176
$$179$$ 20.1246 1.50418 0.752092 0.659058i $$-0.229045\pi$$
0.752092 + 0.659058i $$0.229045\pi$$
$$180$$ −8.29180 −0.618034
$$181$$ 1.41641 0.105281 0.0526404 0.998614i $$-0.483236\pi$$
0.0526404 + 0.998614i $$0.483236\pi$$
$$182$$ 0 0
$$183$$ 6.70820 0.495885
$$184$$ 5.54102 0.408489
$$185$$ 19.4721 1.43162
$$186$$ −4.27051 −0.313129
$$187$$ −22.4164 −1.63925
$$188$$ 2.72949 0.199069
$$189$$ 0 0
$$190$$ −2.56231 −0.185889
$$191$$ −11.1803 −0.808981 −0.404491 0.914542i $$-0.632551\pi$$
−0.404491 + 0.914542i $$0.632551\pi$$
$$192$$ −10.5279 −0.759783
$$193$$ 12.7082 0.914757 0.457378 0.889272i $$-0.348789\pi$$
0.457378 + 0.889272i $$0.348789\pi$$
$$194$$ 6.65248 0.477620
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 26.9443 1.91970 0.959850 0.280514i $$-0.0905049\pi$$
0.959850 + 0.280514i $$0.0905049\pi$$
$$198$$ 2.29180 0.162871
$$199$$ −7.29180 −0.516902 −0.258451 0.966024i $$-0.583212\pi$$
−0.258451 + 0.966024i $$0.583212\pi$$
$$200$$ 0 0
$$201$$ 6.70820 0.473160
$$202$$ −3.43769 −0.241875
$$203$$ 0 0
$$204$$ 30.9787 2.16894
$$205$$ −10.0000 −0.698430
$$206$$ 4.09017 0.284976
$$207$$ −7.52786 −0.523223
$$208$$ 0 0
$$209$$ −9.00000 −0.622543
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −2.72949 −0.187462
$$213$$ −20.0000 −1.37038
$$214$$ −5.43769 −0.371713
$$215$$ −17.8885 −1.21999
$$216$$ 3.29180 0.223978
$$217$$ 0 0
$$218$$ −4.09017 −0.277021
$$219$$ −23.9443 −1.61800
$$220$$ −12.4377 −0.838549
$$221$$ 0 0
$$222$$ 7.43769 0.499185
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −5.70820 −0.379704
$$227$$ 11.9443 0.792769 0.396385 0.918085i $$-0.370265\pi$$
0.396385 + 0.918085i $$0.370265\pi$$
$$228$$ 12.4377 0.823706
$$229$$ 16.1246 1.06554 0.532772 0.846259i $$-0.321150\pi$$
0.532772 + 0.846259i $$0.321150\pi$$
$$230$$ −3.21478 −0.211976
$$231$$ 0 0
$$232$$ 6.58359 0.432234
$$233$$ −5.94427 −0.389422 −0.194711 0.980861i $$-0.562377\pi$$
−0.194711 + 0.980861i $$0.562377\pi$$
$$234$$ 0 0
$$235$$ −3.29180 −0.214733
$$236$$ 13.8541 0.901825
$$237$$ 23.9443 1.55535
$$238$$ 0 0
$$239$$ 7.41641 0.479728 0.239864 0.970807i $$-0.422897\pi$$
0.239864 + 0.970807i $$0.422897\pi$$
$$240$$ 15.7295 1.01533
$$241$$ 8.70820 0.560945 0.280472 0.959862i $$-0.409509\pi$$
0.280472 + 0.959862i $$0.409509\pi$$
$$242$$ −0.763932 −0.0491074
$$243$$ −17.8885 −1.14755
$$244$$ −5.56231 −0.356090
$$245$$ 0 0
$$246$$ −3.81966 −0.243533
$$247$$ 0 0
$$248$$ 7.36068 0.467404
$$249$$ 0 0
$$250$$ −4.27051 −0.270091
$$251$$ 10.4721 0.660995 0.330498 0.943807i $$-0.392783\pi$$
0.330498 + 0.943807i $$0.392783\pi$$
$$252$$ 0 0
$$253$$ −11.2918 −0.709909
$$254$$ 5.88854 0.369480
$$255$$ −37.3607 −2.33962
$$256$$ 5.56231 0.347644
$$257$$ −17.9443 −1.11933 −0.559666 0.828718i $$-0.689071\pi$$
−0.559666 + 0.828718i $$0.689071\pi$$
$$258$$ −6.83282 −0.425393
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −8.94427 −0.553637
$$262$$ −1.43769 −0.0888210
$$263$$ −14.1246 −0.870961 −0.435480 0.900198i $$-0.643421\pi$$
−0.435480 + 0.900198i $$0.643421\pi$$
$$264$$ −9.87539 −0.607788
$$265$$ 3.29180 0.202213
$$266$$ 0 0
$$267$$ 5.00000 0.305995
$$268$$ −5.56231 −0.339772
$$269$$ 4.52786 0.276069 0.138034 0.990427i $$-0.455922\pi$$
0.138034 + 0.990427i $$0.455922\pi$$
$$270$$ −1.90983 −0.116229
$$271$$ −6.41641 −0.389769 −0.194885 0.980826i $$-0.562433\pi$$
−0.194885 + 0.980826i $$0.562433\pi$$
$$272$$ −23.5066 −1.42530
$$273$$ 0 0
$$274$$ 1.43769 0.0868543
$$275$$ 0 0
$$276$$ 15.6049 0.939303
$$277$$ 26.4164 1.58721 0.793604 0.608435i $$-0.208202\pi$$
0.793604 + 0.608435i $$0.208202\pi$$
$$278$$ 1.30495 0.0782658
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ −9.05573 −0.540219 −0.270110 0.962830i $$-0.587060\pi$$
−0.270110 + 0.962830i $$0.587060\pi$$
$$282$$ −1.25735 −0.0748744
$$283$$ −14.1246 −0.839621 −0.419811 0.907612i $$-0.637903\pi$$
−0.419811 + 0.907612i $$0.637903\pi$$
$$284$$ 16.5836 0.984055
$$285$$ −15.0000 −0.888523
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 8.29180 0.488599
$$289$$ 38.8328 2.28428
$$290$$ −3.81966 −0.224298
$$291$$ 38.9443 2.28295
$$292$$ 19.8541 1.16187
$$293$$ −2.94427 −0.172006 −0.0860031 0.996295i $$-0.527409\pi$$
−0.0860031 + 0.996295i $$0.527409\pi$$
$$294$$ 0 0
$$295$$ −16.7082 −0.972789
$$296$$ −12.8197 −0.745128
$$297$$ −6.70820 −0.389249
$$298$$ 4.85410 0.281191
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 2.45085 0.141031
$$303$$ −20.1246 −1.15613
$$304$$ −9.43769 −0.541289
$$305$$ 6.70820 0.384111
$$306$$ −5.70820 −0.326316
$$307$$ 7.41641 0.423277 0.211638 0.977348i $$-0.432120\pi$$
0.211638 + 0.977348i $$0.432120\pi$$
$$308$$ 0 0
$$309$$ 23.9443 1.36214
$$310$$ −4.27051 −0.242549
$$311$$ −32.2361 −1.82794 −0.913970 0.405782i $$-0.866999\pi$$
−0.913970 + 0.405782i $$0.866999\pi$$
$$312$$ 0 0
$$313$$ −32.4164 −1.83228 −0.916142 0.400854i $$-0.868713\pi$$
−0.916142 + 0.400854i $$0.868713\pi$$
$$314$$ 2.67376 0.150889
$$315$$ 0 0
$$316$$ −19.8541 −1.11688
$$317$$ 3.76393 0.211403 0.105702 0.994398i $$-0.466291\pi$$
0.105702 + 0.994398i $$0.466291\pi$$
$$318$$ 1.25735 0.0705089
$$319$$ −13.4164 −0.751175
$$320$$ −10.5279 −0.588525
$$321$$ −31.8328 −1.77673
$$322$$ 0 0
$$323$$ 22.4164 1.24728
$$324$$ 20.3951 1.13306
$$325$$ 0 0
$$326$$ −3.97871 −0.220361
$$327$$ −23.9443 −1.32412
$$328$$ 6.58359 0.363518
$$329$$ 0 0
$$330$$ 5.72949 0.315398
$$331$$ 28.4164 1.56191 0.780954 0.624589i $$-0.214734\pi$$
0.780954 + 0.624589i $$0.214734\pi$$
$$332$$ 0 0
$$333$$ 17.4164 0.954413
$$334$$ −5.16718 −0.282736
$$335$$ 6.70820 0.366508
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ 0 0
$$339$$ −33.4164 −1.81493
$$340$$ 30.9787 1.68006
$$341$$ −15.0000 −0.812296
$$342$$ −2.29180 −0.123926
$$343$$ 0 0
$$344$$ 11.7771 0.634978
$$345$$ −18.8197 −1.01322
$$346$$ 3.97871 0.213897
$$347$$ −35.0689 −1.88260 −0.941298 0.337576i $$-0.890393\pi$$
−0.941298 + 0.337576i $$0.890393\pi$$
$$348$$ 18.5410 0.993903
$$349$$ 2.58359 0.138297 0.0691483 0.997606i $$-0.477972\pi$$
0.0691483 + 0.997606i $$0.477972\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 12.4377 0.662931
$$353$$ −30.7082 −1.63443 −0.817216 0.576331i $$-0.804484\pi$$
−0.817216 + 0.576331i $$0.804484\pi$$
$$354$$ −6.38197 −0.339198
$$355$$ −20.0000 −1.06149
$$356$$ −4.14590 −0.219732
$$357$$ 0 0
$$358$$ 7.68692 0.406266
$$359$$ 5.94427 0.313727 0.156863 0.987620i $$-0.449862\pi$$
0.156863 + 0.987620i $$0.449862\pi$$
$$360$$ −6.58359 −0.346986
$$361$$ −10.0000 −0.526316
$$362$$ 0.541020 0.0284354
$$363$$ −4.47214 −0.234726
$$364$$ 0 0
$$365$$ −23.9443 −1.25330
$$366$$ 2.56231 0.133934
$$367$$ 0.708204 0.0369679 0.0184840 0.999829i $$-0.494116\pi$$
0.0184840 + 0.999829i $$0.494116\pi$$
$$368$$ −11.8409 −0.617252
$$369$$ −8.94427 −0.465620
$$370$$ 7.43769 0.386667
$$371$$ 0 0
$$372$$ 20.7295 1.07477
$$373$$ −28.4164 −1.47135 −0.735673 0.677337i $$-0.763134\pi$$
−0.735673 + 0.677337i $$0.763134\pi$$
$$374$$ −8.56231 −0.442746
$$375$$ −25.0000 −1.29099
$$376$$ 2.16718 0.111764
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −11.4164 −0.586421 −0.293211 0.956048i $$-0.594724\pi$$
−0.293211 + 0.956048i $$0.594724\pi$$
$$380$$ 12.4377 0.638040
$$381$$ 34.4721 1.76606
$$382$$ −4.27051 −0.218498
$$383$$ −15.0000 −0.766464 −0.383232 0.923652i $$-0.625189\pi$$
−0.383232 + 0.923652i $$0.625189\pi$$
$$384$$ −22.5623 −1.15138
$$385$$ 0 0
$$386$$ 4.85410 0.247067
$$387$$ −16.0000 −0.813326
$$388$$ −32.2918 −1.63937
$$389$$ −7.47214 −0.378852 −0.189426 0.981895i $$-0.560663\pi$$
−0.189426 + 0.981895i $$0.560663\pi$$
$$390$$ 0 0
$$391$$ 28.1246 1.42232
$$392$$ 0 0
$$393$$ −8.41641 −0.424552
$$394$$ 10.2918 0.518493
$$395$$ 23.9443 1.20477
$$396$$ −11.1246 −0.559033
$$397$$ −14.1246 −0.708894 −0.354447 0.935076i $$-0.615331\pi$$
−0.354447 + 0.935076i $$0.615331\pi$$
$$398$$ −2.78522 −0.139610
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −9.76393 −0.487587 −0.243794 0.969827i $$-0.578392\pi$$
−0.243794 + 0.969827i $$0.578392\pi$$
$$402$$ 2.56231 0.127796
$$403$$ 0 0
$$404$$ 16.6869 0.830205
$$405$$ −24.5967 −1.22222
$$406$$ 0 0
$$407$$ 26.1246 1.29495
$$408$$ 24.5967 1.21772
$$409$$ 4.70820 0.232806 0.116403 0.993202i $$-0.462864\pi$$
0.116403 + 0.993202i $$0.462864\pi$$
$$410$$ −3.81966 −0.188640
$$411$$ 8.41641 0.415151
$$412$$ −19.8541 −0.978141
$$413$$ 0 0
$$414$$ −2.87539 −0.141318
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 7.63932 0.374099
$$418$$ −3.43769 −0.168143
$$419$$ 15.0557 0.735520 0.367760 0.929921i $$-0.380125\pi$$
0.367760 + 0.929921i $$0.380125\pi$$
$$420$$ 0 0
$$421$$ 13.4164 0.653876 0.326938 0.945046i $$-0.393983\pi$$
0.326938 + 0.945046i $$0.393983\pi$$
$$422$$ 1.52786 0.0743753
$$423$$ −2.94427 −0.143155
$$424$$ −2.16718 −0.105248
$$425$$ 0 0
$$426$$ −7.63932 −0.370126
$$427$$ 0 0
$$428$$ 26.3951 1.27586
$$429$$ 0 0
$$430$$ −6.83282 −0.329508
$$431$$ −13.3607 −0.643561 −0.321781 0.946814i $$-0.604281\pi$$
−0.321781 + 0.946814i $$0.604281\pi$$
$$432$$ −7.03444 −0.338445
$$433$$ 2.58359 0.124160 0.0620798 0.998071i $$-0.480227\pi$$
0.0620798 + 0.998071i $$0.480227\pi$$
$$434$$ 0 0
$$435$$ −22.3607 −1.07211
$$436$$ 19.8541 0.950839
$$437$$ 11.2918 0.540160
$$438$$ −9.14590 −0.437008
$$439$$ −16.1246 −0.769586 −0.384793 0.923003i $$-0.625727\pi$$
−0.384793 + 0.923003i $$0.625727\pi$$
$$440$$ −9.87539 −0.470791
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −2.23607 −0.106239 −0.0531194 0.998588i $$-0.516916\pi$$
−0.0531194 + 0.998588i $$0.516916\pi$$
$$444$$ −36.1033 −1.71339
$$445$$ 5.00000 0.237023
$$446$$ 1.52786 0.0723465
$$447$$ 28.4164 1.34405
$$448$$ 0 0
$$449$$ −10.3607 −0.488951 −0.244475 0.969656i $$-0.578616\pi$$
−0.244475 + 0.969656i $$0.578616\pi$$
$$450$$ 0 0
$$451$$ −13.4164 −0.631754
$$452$$ 27.7082 1.30328
$$453$$ 14.3475 0.674105
$$454$$ 4.56231 0.214120
$$455$$ 0 0
$$456$$ 9.87539 0.462457
$$457$$ −34.1246 −1.59628 −0.798141 0.602471i $$-0.794183\pi$$
−0.798141 + 0.602471i $$0.794183\pi$$
$$458$$ 6.15905 0.287794
$$459$$ 16.7082 0.779872
$$460$$ 15.6049 0.727581
$$461$$ 10.3607 0.482545 0.241272 0.970457i $$-0.422435\pi$$
0.241272 + 0.970457i $$0.422435\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ −14.0689 −0.653132
$$465$$ −25.0000 −1.15935
$$466$$ −2.27051 −0.105179
$$467$$ −21.6525 −1.00196 −0.500979 0.865460i $$-0.667026\pi$$
−0.500979 + 0.865460i $$0.667026\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −1.25735 −0.0579974
$$471$$ 15.6525 0.721228
$$472$$ 11.0000 0.506316
$$473$$ −24.0000 −1.10352
$$474$$ 9.14590 0.420085
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2.94427 0.134809
$$478$$ 2.83282 0.129570
$$479$$ 29.8328 1.36310 0.681548 0.731773i $$-0.261307\pi$$
0.681548 + 0.731773i $$0.261307\pi$$
$$480$$ 20.7295 0.946167
$$481$$ 0 0
$$482$$ 3.32624 0.151506
$$483$$ 0 0
$$484$$ 3.70820 0.168555
$$485$$ 38.9443 1.76837
$$486$$ −6.83282 −0.309943
$$487$$ −31.8328 −1.44248 −0.721241 0.692684i $$-0.756428\pi$$
−0.721241 + 0.692684i $$0.756428\pi$$
$$488$$ −4.41641 −0.199921
$$489$$ −23.2918 −1.05329
$$490$$ 0 0
$$491$$ −34.4721 −1.55571 −0.777853 0.628446i $$-0.783691\pi$$
−0.777853 + 0.628446i $$0.783691\pi$$
$$492$$ 18.5410 0.835894
$$493$$ 33.4164 1.50500
$$494$$ 0 0
$$495$$ 13.4164 0.603023
$$496$$ −15.7295 −0.706275
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0.416408 0.0186410 0.00932049 0.999957i $$-0.497033\pi$$
0.00932049 + 0.999957i $$0.497033\pi$$
$$500$$ 20.7295 0.927051
$$501$$ −30.2492 −1.35144
$$502$$ 4.00000 0.178529
$$503$$ 3.05573 0.136248 0.0681241 0.997677i $$-0.478299\pi$$
0.0681241 + 0.997677i $$0.478299\pi$$
$$504$$ 0 0
$$505$$ −20.1246 −0.895533
$$506$$ −4.31308 −0.191740
$$507$$ 0 0
$$508$$ −28.5836 −1.26819
$$509$$ 15.7639 0.698724 0.349362 0.936988i $$-0.386398\pi$$
0.349362 + 0.936988i $$0.386398\pi$$
$$510$$ −14.2705 −0.631909
$$511$$ 0 0
$$512$$ 22.3050 0.985749
$$513$$ 6.70820 0.296174
$$514$$ −6.85410 −0.302321
$$515$$ 23.9443 1.05511
$$516$$ 33.1672 1.46010
$$517$$ −4.41641 −0.194233
$$518$$ 0 0
$$519$$ 23.2918 1.02240
$$520$$ 0 0
$$521$$ −0.0557281 −0.00244149 −0.00122075 0.999999i $$-0.500389\pi$$
−0.00122075 + 0.999999i $$0.500389\pi$$
$$522$$ −3.41641 −0.149532
$$523$$ 19.2918 0.843571 0.421786 0.906696i $$-0.361403\pi$$
0.421786 + 0.906696i $$0.361403\pi$$
$$524$$ 6.97871 0.304867
$$525$$ 0 0
$$526$$ −5.39512 −0.235238
$$527$$ 37.3607 1.62746
$$528$$ 21.1033 0.918404
$$529$$ −8.83282 −0.384035
$$530$$ 1.25735 0.0546160
$$531$$ −14.9443 −0.648526
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 1.90983 0.0826464
$$535$$ −31.8328 −1.37625
$$536$$ −4.41641 −0.190760
$$537$$ 45.0000 1.94189
$$538$$ 1.72949 0.0745636
$$539$$ 0 0
$$540$$ 9.27051 0.398939
$$541$$ −14.7082 −0.632355 −0.316178 0.948700i $$-0.602400\pi$$
−0.316178 + 0.948700i $$0.602400\pi$$
$$542$$ −2.45085 −0.105273
$$543$$ 3.16718 0.135917
$$544$$ −30.9787 −1.32820
$$545$$ −23.9443 −1.02566
$$546$$ 0 0
$$547$$ −31.4164 −1.34327 −0.671634 0.740883i $$-0.734407\pi$$
−0.671634 + 0.740883i $$0.734407\pi$$
$$548$$ −6.97871 −0.298116
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 13.4164 0.571558
$$552$$ 12.3901 0.527358
$$553$$ 0 0
$$554$$ 10.0902 0.428690
$$555$$ 43.5410 1.84821
$$556$$ −6.33437 −0.268637
$$557$$ 5.29180 0.224221 0.112110 0.993696i $$-0.464239\pi$$
0.112110 + 0.993696i $$0.464239\pi$$
$$558$$ −3.81966 −0.161699
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −50.1246 −2.11626
$$562$$ −3.45898 −0.145908
$$563$$ −36.5967 −1.54237 −0.771185 0.636612i $$-0.780335\pi$$
−0.771185 + 0.636612i $$0.780335\pi$$
$$564$$ 6.10333 0.256996
$$565$$ −33.4164 −1.40584
$$566$$ −5.39512 −0.226774
$$567$$ 0 0
$$568$$ 13.1672 0.552483
$$569$$ 16.5279 0.692884 0.346442 0.938071i $$-0.387390\pi$$
0.346442 + 0.938071i $$0.387390\pi$$
$$570$$ −5.72949 −0.239982
$$571$$ 4.12461 0.172610 0.0863048 0.996269i $$-0.472494\pi$$
0.0863048 + 0.996269i $$0.472494\pi$$
$$572$$ 0 0
$$573$$ −25.0000 −1.04439
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −9.41641 −0.392350
$$577$$ 32.7082 1.36166 0.680830 0.732441i $$-0.261619\pi$$
0.680830 + 0.732441i $$0.261619\pi$$
$$578$$ 14.8328 0.616964
$$579$$ 28.4164 1.18095
$$580$$ 18.5410 0.769874
$$581$$ 0 0
$$582$$ 14.8754 0.616605
$$583$$ 4.41641 0.182909
$$584$$ 15.7639 0.652316
$$585$$ 0 0
$$586$$ −1.12461 −0.0464573
$$587$$ 41.8885 1.72893 0.864463 0.502697i $$-0.167659\pi$$
0.864463 + 0.502697i $$0.167659\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ −6.38197 −0.262741
$$591$$ 60.2492 2.47832
$$592$$ 27.3951 1.12593
$$593$$ −32.2361 −1.32378 −0.661888 0.749602i $$-0.730245\pi$$
−0.661888 + 0.749602i $$0.730245\pi$$
$$594$$ −2.56231 −0.105133
$$595$$ 0 0
$$596$$ −23.5623 −0.965150
$$597$$ −16.3050 −0.667317
$$598$$ 0 0
$$599$$ 41.0689 1.67803 0.839015 0.544109i $$-0.183132\pi$$
0.839015 + 0.544109i $$0.183132\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ 6.00000 0.244339
$$604$$ −11.8967 −0.484069
$$605$$ −4.47214 −0.181818
$$606$$ −7.68692 −0.312260
$$607$$ −16.1246 −0.654478 −0.327239 0.944942i $$-0.606118\pi$$
−0.327239 + 0.944942i $$0.606118\pi$$
$$608$$ −12.4377 −0.504415
$$609$$ 0 0
$$610$$ 2.56231 0.103745
$$611$$ 0 0
$$612$$ 27.7082 1.12004
$$613$$ −22.1246 −0.893605 −0.446802 0.894633i $$-0.647437\pi$$
−0.446802 + 0.894633i $$0.647437\pi$$
$$614$$ 2.83282 0.114323
$$615$$ −22.3607 −0.901670
$$616$$ 0 0
$$617$$ −4.47214 −0.180041 −0.0900207 0.995940i $$-0.528693\pi$$
−0.0900207 + 0.995940i $$0.528693\pi$$
$$618$$ 9.14590 0.367902
$$619$$ −17.0000 −0.683288 −0.341644 0.939829i $$-0.610984\pi$$
−0.341644 + 0.939829i $$0.610984\pi$$
$$620$$ 20.7295 0.832516
$$621$$ 8.41641 0.337739
$$622$$ −12.3131 −0.493710
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −25.0000 −1.00000
$$626$$ −12.3820 −0.494883
$$627$$ −20.1246 −0.803700
$$628$$ −12.9787 −0.517907
$$629$$ −65.0689 −2.59447
$$630$$ 0 0
$$631$$ 30.8328 1.22744 0.613718 0.789526i $$-0.289673\pi$$
0.613718 + 0.789526i $$0.289673\pi$$
$$632$$ −15.7639 −0.627056
$$633$$ 8.94427 0.355503
$$634$$ 1.43769 0.0570981
$$635$$ 34.4721 1.36798
$$636$$ −6.10333 −0.242013
$$637$$ 0 0
$$638$$ −5.12461 −0.202885
$$639$$ −17.8885 −0.707660
$$640$$ −22.5623 −0.891853
$$641$$ −5.94427 −0.234785 −0.117392 0.993086i $$-0.537453\pi$$
−0.117392 + 0.993086i $$0.537453\pi$$
$$642$$ −12.1591 −0.479880
$$643$$ 18.8328 0.742694 0.371347 0.928494i $$-0.378896\pi$$
0.371347 + 0.928494i $$0.378896\pi$$
$$644$$ 0 0
$$645$$ −40.0000 −1.57500
$$646$$ 8.56231 0.336879
$$647$$ 15.7639 0.619744 0.309872 0.950778i $$-0.399714\pi$$
0.309872 + 0.950778i $$0.399714\pi$$
$$648$$ 16.1935 0.636141
$$649$$ −22.4164 −0.879921
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 19.3131 0.756359
$$653$$ 13.4721 0.527205 0.263603 0.964631i $$-0.415089\pi$$
0.263603 + 0.964631i $$0.415089\pi$$
$$654$$ −9.14590 −0.357633
$$655$$ −8.41641 −0.328856
$$656$$ −14.0689 −0.549298
$$657$$ −21.4164 −0.835534
$$658$$ 0 0
$$659$$ 8.94427 0.348419 0.174210 0.984709i $$-0.444263\pi$$
0.174210 + 0.984709i $$0.444263\pi$$
$$660$$ −27.8115 −1.08256
$$661$$ −6.70820 −0.260919 −0.130459 0.991454i $$-0.541645\pi$$
−0.130459 + 0.991454i $$0.541645\pi$$
$$662$$ 10.8541 0.421857
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 6.65248 0.257778
$$667$$ 16.8328 0.651769
$$668$$ 25.0820 0.970453
$$669$$ 8.94427 0.345806
$$670$$ 2.56231 0.0989905
$$671$$ 9.00000 0.347441
$$672$$ 0 0
$$673$$ 17.4164 0.671353 0.335677 0.941977i $$-0.391035\pi$$
0.335677 + 0.941977i $$0.391035\pi$$
$$674$$ 6.87539 0.264830
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 32.8885 1.26401 0.632005 0.774965i $$-0.282232\pi$$
0.632005 + 0.774965i $$0.282232\pi$$
$$678$$ −12.7639 −0.490196
$$679$$ 0 0
$$680$$ 24.5967 0.943242
$$681$$ 26.7082 1.02346
$$682$$ −5.72949 −0.219394
$$683$$ 4.52786 0.173254 0.0866270 0.996241i $$-0.472391\pi$$
0.0866270 + 0.996241i $$0.472391\pi$$
$$684$$ 11.1246 0.425360
$$685$$ 8.41641 0.321574
$$686$$ 0 0
$$687$$ 36.0557 1.37561
$$688$$ −25.1672 −0.959490
$$689$$ 0 0
$$690$$ −7.18847 −0.273660
$$691$$ −1.83282 −0.0697236 −0.0348618 0.999392i $$-0.511099\pi$$
−0.0348618 + 0.999392i $$0.511099\pi$$
$$692$$ −19.3131 −0.734173
$$693$$ 0 0
$$694$$ −13.3951 −0.508472
$$695$$ 7.63932 0.289776
$$696$$ 14.7214 0.558012
$$697$$ 33.4164 1.26574
$$698$$ 0.986844 0.0373526
$$699$$ −13.2918 −0.502742
$$700$$ 0 0
$$701$$ 22.3607 0.844551 0.422276 0.906467i $$-0.361231\pi$$
0.422276 + 0.906467i $$0.361231\pi$$
$$702$$ 0 0
$$703$$ −26.1246 −0.985308
$$704$$ −14.1246 −0.532341
$$705$$ −7.36068 −0.277219
$$706$$ −11.7295 −0.441445
$$707$$ 0 0
$$708$$ 30.9787 1.16425
$$709$$ 9.87539 0.370878 0.185439 0.982656i $$-0.440629\pi$$
0.185439 + 0.982656i $$0.440629\pi$$
$$710$$ −7.63932 −0.286699
$$711$$ 21.4164 0.803178
$$712$$ −3.29180 −0.123365
$$713$$ 18.8197 0.704802
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −37.3131 −1.39446
$$717$$ 16.5836 0.619326
$$718$$ 2.27051 0.0847347
$$719$$ 11.2918 0.421113 0.210556 0.977582i $$-0.432472\pi$$
0.210556 + 0.977582i $$0.432472\pi$$
$$720$$ 14.0689 0.524316
$$721$$ 0 0
$$722$$ −3.81966 −0.142153
$$723$$ 19.4721 0.724177
$$724$$ −2.62616 −0.0976006
$$725$$ 0 0
$$726$$ −1.70820 −0.0633974
$$727$$ 14.8328 0.550119 0.275059 0.961427i $$-0.411302\pi$$
0.275059 + 0.961427i $$0.411302\pi$$
$$728$$ 0 0
$$729$$ −7.00000 −0.259259
$$730$$ −9.14590 −0.338505
$$731$$ 59.7771 2.21094
$$732$$ −12.4377 −0.459710
$$733$$ −15.2918 −0.564815 −0.282408 0.959294i $$-0.591133\pi$$
−0.282408 + 0.959294i $$0.591133\pi$$
$$734$$ 0.270510 0.00998470
$$735$$ 0 0
$$736$$ −15.6049 −0.575203
$$737$$ 9.00000 0.331519
$$738$$ −3.41641 −0.125760
$$739$$ −35.8328 −1.31813 −0.659066 0.752085i $$-0.729048\pi$$
−0.659066 + 0.752085i $$0.729048\pi$$
$$740$$ −36.1033 −1.32718
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −15.0557 −0.552341 −0.276171 0.961109i $$-0.589065\pi$$
−0.276171 + 0.961109i $$0.589065\pi$$
$$744$$ 16.4590 0.603415
$$745$$ 28.4164 1.04110
$$746$$ −10.8541 −0.397397
$$747$$ 0 0
$$748$$ 41.5623 1.51967
$$749$$ 0 0
$$750$$ −9.54915 −0.348686
$$751$$ 30.1246 1.09926 0.549631 0.835407i $$-0.314768\pi$$
0.549631 + 0.835407i $$0.314768\pi$$
$$752$$ −4.63119 −0.168882
$$753$$ 23.4164 0.853341
$$754$$ 0 0
$$755$$ 14.3475 0.522160
$$756$$ 0 0
$$757$$ 0.832816 0.0302692 0.0151346 0.999885i $$-0.495182\pi$$
0.0151346 + 0.999885i $$0.495182\pi$$
$$758$$ −4.36068 −0.158387
$$759$$ −25.2492 −0.916489
$$760$$ 9.87539 0.358218
$$761$$ −33.5410 −1.21586 −0.607931 0.793990i $$-0.708000\pi$$
−0.607931 + 0.793990i $$0.708000\pi$$
$$762$$ 13.1672 0.476997
$$763$$ 0 0
$$764$$ 20.7295 0.749967
$$765$$ −33.4164 −1.20817
$$766$$ −5.72949 −0.207015
$$767$$ 0 0
$$768$$ 12.4377 0.448807
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 0 0
$$771$$ −40.1246 −1.44505
$$772$$ −23.5623 −0.848026
$$773$$ −11.0689 −0.398120 −0.199060 0.979987i $$-0.563789\pi$$
−0.199060 + 0.979987i $$0.563789\pi$$
$$774$$ −6.11146 −0.219672
$$775$$ 0 0
$$776$$ −25.6393 −0.920398
$$777$$ 0 0
$$778$$ −2.85410 −0.102325
$$779$$ 13.4164 0.480693
$$780$$ 0 0
$$781$$ −26.8328 −0.960154
$$782$$ 10.7426 0.384156
$$783$$ 10.0000 0.357371
$$784$$ 0 0
$$785$$ 15.6525 0.558661
$$786$$ −3.21478 −0.114667
$$787$$ 6.41641 0.228720 0.114360 0.993439i $$-0.463518\pi$$
0.114360 + 0.993439i $$0.463518\pi$$
$$788$$ −49.9574 −1.77966
$$789$$ −31.5836 −1.12441
$$790$$ 9.14590 0.325396
$$791$$ 0 0
$$792$$ −8.83282 −0.313860
$$793$$ 0 0
$$794$$ −5.39512 −0.191466
$$795$$ 7.36068 0.261056
$$796$$ 13.5197 0.479194
$$797$$ 26.9443 0.954415 0.477208 0.878791i $$-0.341649\pi$$
0.477208 + 0.878791i $$0.341649\pi$$
$$798$$ 0 0
$$799$$ 11.0000 0.389152
$$800$$ 0 0
$$801$$ 4.47214 0.158015
$$802$$ −3.72949 −0.131693
$$803$$ −32.1246 −1.13365
$$804$$ −12.4377 −0.438644
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 10.1246 0.356403
$$808$$ 13.2492 0.466106
$$809$$ −4.41641 −0.155273 −0.0776363 0.996982i $$-0.524737\pi$$
−0.0776363 + 0.996982i $$0.524737\pi$$
$$810$$ −9.39512 −0.330111
$$811$$ −38.8328 −1.36360 −0.681802 0.731536i $$-0.738804\pi$$
−0.681802 + 0.731536i $$0.738804\pi$$
$$812$$ 0 0
$$813$$ −14.3475 −0.503190
$$814$$ 9.97871 0.349754
$$815$$ −23.2918 −0.815876
$$816$$ −52.5623 −1.84005
$$817$$ 24.0000 0.839654
$$818$$ 1.79837 0.0628787
$$819$$ 0 0
$$820$$ 18.5410 0.647480
$$821$$ −33.7639 −1.17837 −0.589185 0.807998i $$-0.700551\pi$$
−0.589185 + 0.807998i $$0.700551\pi$$
$$822$$ 3.21478 0.112128
$$823$$ −6.12461 −0.213491 −0.106745 0.994286i $$-0.534043\pi$$
−0.106745 + 0.994286i $$0.534043\pi$$
$$824$$ −15.7639 −0.549163
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 26.8328 0.933068 0.466534 0.884503i $$-0.345502\pi$$
0.466534 + 0.884503i $$0.345502\pi$$
$$828$$ 13.9574 0.485054
$$829$$ −25.0000 −0.868286 −0.434143 0.900844i $$-0.642949\pi$$
−0.434143 + 0.900844i $$0.642949\pi$$
$$830$$ 0 0
$$831$$ 59.0689 2.04908
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 2.91796 0.101041
$$835$$ −30.2492 −1.04682
$$836$$ 16.6869 0.577129
$$837$$ 11.1803 0.386449
$$838$$ 5.75078 0.198657
$$839$$ −29.8885 −1.03187 −0.515934 0.856629i $$-0.672555\pi$$
−0.515934 + 0.856629i $$0.672555\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 5.12461 0.176606
$$843$$ −20.2492 −0.697420
$$844$$ −7.41641 −0.255283
$$845$$ 0 0
$$846$$ −1.12461 −0.0386650
$$847$$ 0 0
$$848$$ 4.63119 0.159036
$$849$$ −31.5836 −1.08395
$$850$$ 0 0
$$851$$ −32.7771 −1.12358
$$852$$ 37.0820 1.27041
$$853$$ −28.2492 −0.967235 −0.483617 0.875279i $$-0.660677\pi$$
−0.483617 + 0.875279i $$0.660677\pi$$
$$854$$ 0 0
$$855$$ −13.4164 −0.458831
$$856$$ 20.9574 0.716310
$$857$$ 43.3607 1.48117 0.740586 0.671961i $$-0.234548\pi$$
0.740586 + 0.671961i $$0.234548\pi$$
$$858$$ 0 0
$$859$$ 7.29180 0.248793 0.124396 0.992233i $$-0.460301\pi$$
0.124396 + 0.992233i $$0.460301\pi$$
$$860$$ 33.1672 1.13099
$$861$$ 0 0
$$862$$ −5.10333 −0.173820
$$863$$ −6.05573 −0.206139 −0.103070 0.994674i $$-0.532866\pi$$
−0.103070 + 0.994674i $$0.532866\pi$$
$$864$$ −9.27051 −0.315389
$$865$$ 23.2918 0.791945
$$866$$ 0.986844 0.0335343
$$867$$ 86.8328 2.94900
$$868$$ 0 0
$$869$$ 32.1246 1.08975
$$870$$ −8.54102 −0.289568
$$871$$ 0 0
$$872$$ 15.7639 0.533834
$$873$$ 34.8328 1.17891
$$874$$ 4.31308 0.145892
$$875$$ 0 0
$$876$$ 44.3951 1.49997
$$877$$ 8.12461 0.274349 0.137174 0.990547i $$-0.456198\pi$$
0.137174 + 0.990547i $$0.456198\pi$$
$$878$$ −6.15905 −0.207858
$$879$$ −6.58359 −0.222059
$$880$$ 21.1033 0.711393
$$881$$ −19.3050 −0.650400 −0.325200 0.945645i $$-0.605432\pi$$
−0.325200 + 0.945645i $$0.605432\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 0 0
$$885$$ −37.3607 −1.25587
$$886$$ −0.854102 −0.0286941
$$887$$ −15.7639 −0.529301 −0.264651 0.964344i $$-0.585257\pi$$
−0.264651 + 0.964344i $$0.585257\pi$$
$$888$$ −28.6656 −0.961956
$$889$$ 0 0
$$890$$ 1.90983 0.0640176
$$891$$ −33.0000 −1.10554
$$892$$ −7.41641 −0.248320
$$893$$ 4.41641 0.147789
$$894$$ 10.8541 0.363015
$$895$$ 45.0000 1.50418
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −3.95743 −0.132061
$$899$$ 22.3607 0.745770
$$900$$ 0 0
$$901$$ −11.0000 −0.366463
$$902$$ −5.12461 −0.170631
$$903$$ 0 0
$$904$$ 22.0000 0.731709
$$905$$ 3.16718 0.105281
$$906$$ 5.48027 0.182070
$$907$$ 5.29180 0.175711 0.0878556 0.996133i $$-0.471999\pi$$
0.0878556 + 0.996133i $$0.471999\pi$$
$$908$$ −22.1459 −0.734937
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ 46.2492 1.53231 0.766153 0.642659i $$-0.222169\pi$$
0.766153 + 0.642659i $$0.222169\pi$$
$$912$$ −21.1033 −0.698801
$$913$$ 0 0
$$914$$ −13.0344 −0.431141
$$915$$ 15.0000 0.495885
$$916$$ −29.8967 −0.987814
$$917$$ 0 0
$$918$$ 6.38197 0.210636
$$919$$ −20.1246 −0.663850 −0.331925 0.943306i $$-0.607698\pi$$
−0.331925 + 0.943306i $$0.607698\pi$$
$$920$$ 12.3901 0.408489
$$921$$ 16.5836 0.546448
$$922$$ 3.95743 0.130331
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 9.16718 0.301252
$$927$$ 21.4164 0.703407
$$928$$ −18.5410 −0.608639
$$929$$ 47.1803 1.54794 0.773968 0.633224i $$-0.218269\pi$$
0.773968 + 0.633224i $$0.218269\pi$$
$$930$$ −9.54915 −0.313129
$$931$$ 0 0
$$932$$ 11.0213 0.361014
$$933$$ −72.0820 −2.35986
$$934$$ −8.27051 −0.270619
$$935$$ −50.1246 −1.63925
$$936$$ 0 0
$$937$$ −1.41641 −0.0462720 −0.0231360 0.999732i $$-0.507365\pi$$
−0.0231360 + 0.999732i $$0.507365\pi$$
$$938$$ 0 0
$$939$$ −72.4853 −2.36547
$$940$$ 6.10333 0.199069
$$941$$ −45.7639 −1.49186 −0.745931 0.666023i $$-0.767995\pi$$
−0.745931 + 0.666023i $$0.767995\pi$$
$$942$$ 5.97871 0.194797
$$943$$ 16.8328 0.548152
$$944$$ −23.5066 −0.765074
$$945$$ 0 0
$$946$$ −9.16718 −0.298051
$$947$$ 31.4721 1.02271 0.511354 0.859370i $$-0.329144\pi$$
0.511354 + 0.859370i $$0.329144\pi$$
$$948$$ −44.3951 −1.44189
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 8.41641 0.272921
$$952$$ 0 0
$$953$$ 29.7771 0.964574 0.482287 0.876013i $$-0.339806\pi$$
0.482287 + 0.876013i $$0.339806\pi$$
$$954$$ 1.12461 0.0364107
$$955$$ −25.0000 −0.808981
$$956$$ −13.7508 −0.444732
$$957$$ −30.0000 −0.969762
$$958$$ 11.3951 0.368160
$$959$$ 0 0
$$960$$ −23.5410 −0.759783
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ −28.4721 −0.917502
$$964$$ −16.1459 −0.520024
$$965$$ 28.4164 0.914757
$$966$$ 0 0
$$967$$ −43.4164 −1.39618 −0.698089 0.716011i $$-0.745966\pi$$
−0.698089 + 0.716011i $$0.745966\pi$$
$$968$$ 2.94427 0.0946325
$$969$$ 50.1246 1.61023
$$970$$ 14.8754 0.477620
$$971$$ −24.7082 −0.792924 −0.396462 0.918051i $$-0.629762\pi$$
−0.396462 + 0.918051i $$0.629762\pi$$
$$972$$ 33.1672 1.06384
$$973$$ 0 0
$$974$$ −12.1591 −0.389601
$$975$$ 0 0
$$976$$ 9.43769 0.302093
$$977$$ 33.6525 1.07664 0.538319 0.842741i $$-0.319060\pi$$
0.538319 + 0.842741i $$0.319060\pi$$
$$978$$ −8.89667 −0.284484
$$979$$ 6.70820 0.214395
$$980$$ 0 0
$$981$$ −21.4164 −0.683773
$$982$$ −13.1672 −0.420182
$$983$$ 1.47214 0.0469538 0.0234769 0.999724i $$-0.492526\pi$$
0.0234769 + 0.999724i $$0.492526\pi$$
$$984$$ 14.7214 0.469300
$$985$$ 60.2492 1.91970
$$986$$ 12.7639 0.406486
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 30.1115 0.957489
$$990$$ 5.12461 0.162871
$$991$$ 17.2918 0.549292 0.274646 0.961545i $$-0.411439\pi$$
0.274646 + 0.961545i $$0.411439\pi$$
$$992$$ −20.7295 −0.658162
$$993$$ 63.5410 2.01641
$$994$$ 0 0
$$995$$ −16.3050 −0.516902
$$996$$ 0 0
$$997$$ −0.416408 −0.0131878 −0.00659388 0.999978i $$-0.502099\pi$$
−0.00659388 + 0.999978i $$0.502099\pi$$
$$998$$ 0.159054 0.00503476
$$999$$ −19.4721 −0.616071
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 8281.2.a.z.1.1 2
7.2 even 3 1183.2.e.d.508.2 4
7.4 even 3 1183.2.e.d.170.2 4
7.6 odd 2 8281.2.a.ba.1.1 2
13.12 even 2 637.2.a.f.1.2 2
39.38 odd 2 5733.2.a.v.1.1 2
91.12 odd 6 637.2.e.h.508.1 4
91.25 even 6 91.2.e.b.79.1 yes 4
91.38 odd 6 637.2.e.h.79.1 4
91.51 even 6 91.2.e.b.53.1 4
91.90 odd 2 637.2.a.e.1.2 2
273.116 odd 6 819.2.j.c.352.2 4
273.233 odd 6 819.2.j.c.235.2 4
273.272 even 2 5733.2.a.w.1.1 2
364.51 odd 6 1456.2.r.j.417.2 4
364.207 odd 6 1456.2.r.j.625.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.1 4 91.51 even 6
91.2.e.b.79.1 yes 4 91.25 even 6
637.2.a.e.1.2 2 91.90 odd 2
637.2.a.f.1.2 2 13.12 even 2
637.2.e.h.79.1 4 91.38 odd 6
637.2.e.h.508.1 4 91.12 odd 6
819.2.j.c.235.2 4 273.233 odd 6
819.2.j.c.352.2 4 273.116 odd 6
1183.2.e.d.170.2 4 7.4 even 3
1183.2.e.d.508.2 4 7.2 even 3
1456.2.r.j.417.2 4 364.51 odd 6
1456.2.r.j.625.2 4 364.207 odd 6
5733.2.a.v.1.1 2 39.38 odd 2
5733.2.a.w.1.1 2 273.272 even 2
8281.2.a.z.1.1 2 1.1 even 1 trivial
8281.2.a.ba.1.1 2 7.6 odd 2