# Properties

 Label 1850.2.a.q.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.37228 q^{11} -2.00000 q^{12} +4.74456 q^{13} +1.37228 q^{14} +1.00000 q^{16} +5.37228 q^{17} -1.00000 q^{18} -2.00000 q^{19} +2.74456 q^{21} +3.37228 q^{22} -6.74456 q^{23} +2.00000 q^{24} -4.74456 q^{26} +4.00000 q^{27} -1.37228 q^{28} +8.11684 q^{29} -2.62772 q^{31} -1.00000 q^{32} +6.74456 q^{33} -5.37228 q^{34} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} -9.48913 q^{39} +5.37228 q^{41} -2.74456 q^{42} -7.37228 q^{43} -3.37228 q^{44} +6.74456 q^{46} +8.74456 q^{47} -2.00000 q^{48} -5.11684 q^{49} -10.7446 q^{51} +4.74456 q^{52} -1.37228 q^{53} -4.00000 q^{54} +1.37228 q^{56} +4.00000 q^{57} -8.11684 q^{58} +12.7446 q^{59} -5.37228 q^{61} +2.62772 q^{62} -1.37228 q^{63} +1.00000 q^{64} -6.74456 q^{66} -4.74456 q^{67} +5.37228 q^{68} +13.4891 q^{69} -6.74456 q^{71} -1.00000 q^{72} -8.74456 q^{73} +1.00000 q^{74} -2.00000 q^{76} +4.62772 q^{77} +9.48913 q^{78} -4.74456 q^{79} -11.0000 q^{81} -5.37228 q^{82} +0.744563 q^{83} +2.74456 q^{84} +7.37228 q^{86} -16.2337 q^{87} +3.37228 q^{88} +10.0000 q^{89} -6.51087 q^{91} -6.74456 q^{92} +5.25544 q^{93} -8.74456 q^{94} +2.00000 q^{96} +0.116844 q^{97} +5.11684 q^{98} -3.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} - q^{11} - 4 q^{12} - 2 q^{13} - 3 q^{14} + 2 q^{16} + 5 q^{17} - 2 q^{18} - 4 q^{19} - 6 q^{21} + q^{22} - 2 q^{23} + 4 q^{24} + 2 q^{26} + 8 q^{27} + 3 q^{28} - q^{29} - 11 q^{31} - 2 q^{32} + 2 q^{33} - 5 q^{34} + 2 q^{36} - 2 q^{37} + 4 q^{38} + 4 q^{39} + 5 q^{41} + 6 q^{42} - 9 q^{43} - q^{44} + 2 q^{46} + 6 q^{47} - 4 q^{48} + 7 q^{49} - 10 q^{51} - 2 q^{52} + 3 q^{53} - 8 q^{54} - 3 q^{56} + 8 q^{57} + q^{58} + 14 q^{59} - 5 q^{61} + 11 q^{62} + 3 q^{63} + 2 q^{64} - 2 q^{66} + 2 q^{67} + 5 q^{68} + 4 q^{69} - 2 q^{71} - 2 q^{72} - 6 q^{73} + 2 q^{74} - 4 q^{76} + 15 q^{77} - 4 q^{78} + 2 q^{79} - 22 q^{81} - 5 q^{82} - 10 q^{83} - 6 q^{84} + 9 q^{86} + 2 q^{87} + q^{88} + 20 q^{89} - 36 q^{91} - 2 q^{92} + 22 q^{93} - 6 q^{94} + 4 q^{96} - 17 q^{97} - 7 q^{98} - q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 - q^11 - 4 * q^12 - 2 * q^13 - 3 * q^14 + 2 * q^16 + 5 * q^17 - 2 * q^18 - 4 * q^19 - 6 * q^21 + q^22 - 2 * q^23 + 4 * q^24 + 2 * q^26 + 8 * q^27 + 3 * q^28 - q^29 - 11 * q^31 - 2 * q^32 + 2 * q^33 - 5 * q^34 + 2 * q^36 - 2 * q^37 + 4 * q^38 + 4 * q^39 + 5 * q^41 + 6 * q^42 - 9 * q^43 - q^44 + 2 * q^46 + 6 * q^47 - 4 * q^48 + 7 * q^49 - 10 * q^51 - 2 * q^52 + 3 * q^53 - 8 * q^54 - 3 * q^56 + 8 * q^57 + q^58 + 14 * q^59 - 5 * q^61 + 11 * q^62 + 3 * q^63 + 2 * q^64 - 2 * q^66 + 2 * q^67 + 5 * q^68 + 4 * q^69 - 2 * q^71 - 2 * q^72 - 6 * q^73 + 2 * q^74 - 4 * q^76 + 15 * q^77 - 4 * q^78 + 2 * q^79 - 22 * q^81 - 5 * q^82 - 10 * q^83 - 6 * q^84 + 9 * q^86 + 2 * q^87 + q^88 + 20 * q^89 - 36 * q^91 - 2 * q^92 + 22 * q^93 - 6 * q^94 + 4 * q^96 - 17 * q^97 - 7 * q^98 - 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
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ −1.37228 −0.518674 −0.259337 0.965787i $$-0.583504\pi$$
−0.259337 + 0.965787i $$0.583504\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.37228 −1.01678 −0.508391 0.861127i $$-0.669759\pi$$
−0.508391 + 0.861127i $$0.669759\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ 4.74456 1.31590 0.657952 0.753059i $$-0.271423\pi$$
0.657952 + 0.753059i $$0.271423\pi$$
$$14$$ 1.37228 0.366758
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.37228 1.30297 0.651485 0.758662i $$-0.274146\pi$$
0.651485 + 0.758662i $$0.274146\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 2.74456 0.598913
$$22$$ 3.37228 0.718973
$$23$$ −6.74456 −1.40634 −0.703169 0.711022i $$-0.748232\pi$$
−0.703169 + 0.711022i $$0.748232\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 0 0
$$26$$ −4.74456 −0.930485
$$27$$ 4.00000 0.769800
$$28$$ −1.37228 −0.259337
$$29$$ 8.11684 1.50726 0.753630 0.657299i $$-0.228301\pi$$
0.753630 + 0.657299i $$0.228301\pi$$
$$30$$ 0 0
$$31$$ −2.62772 −0.471952 −0.235976 0.971759i $$-0.575829\pi$$
−0.235976 + 0.971759i $$0.575829\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 6.74456 1.17408
$$34$$ −5.37228 −0.921339
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −1.00000 −0.164399
$$38$$ 2.00000 0.324443
$$39$$ −9.48913 −1.51948
$$40$$ 0 0
$$41$$ 5.37228 0.839009 0.419505 0.907753i $$-0.362204\pi$$
0.419505 + 0.907753i $$0.362204\pi$$
$$42$$ −2.74456 −0.423495
$$43$$ −7.37228 −1.12426 −0.562131 0.827048i $$-0.690018\pi$$
−0.562131 + 0.827048i $$0.690018\pi$$
$$44$$ −3.37228 −0.508391
$$45$$ 0 0
$$46$$ 6.74456 0.994432
$$47$$ 8.74456 1.27553 0.637763 0.770233i $$-0.279860\pi$$
0.637763 + 0.770233i $$0.279860\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ −5.11684 −0.730978
$$50$$ 0 0
$$51$$ −10.7446 −1.50454
$$52$$ 4.74456 0.657952
$$53$$ −1.37228 −0.188497 −0.0942487 0.995549i $$-0.530045\pi$$
−0.0942487 + 0.995549i $$0.530045\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ 1.37228 0.183379
$$57$$ 4.00000 0.529813
$$58$$ −8.11684 −1.06579
$$59$$ 12.7446 1.65920 0.829600 0.558358i $$-0.188568\pi$$
0.829600 + 0.558358i $$0.188568\pi$$
$$60$$ 0 0
$$61$$ −5.37228 −0.687850 −0.343925 0.938997i $$-0.611757\pi$$
−0.343925 + 0.938997i $$0.611757\pi$$
$$62$$ 2.62772 0.333721
$$63$$ −1.37228 −0.172891
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −6.74456 −0.830198
$$67$$ −4.74456 −0.579641 −0.289820 0.957081i $$-0.593596\pi$$
−0.289820 + 0.957081i $$0.593596\pi$$
$$68$$ 5.37228 0.651485
$$69$$ 13.4891 1.62390
$$70$$ 0 0
$$71$$ −6.74456 −0.800432 −0.400216 0.916421i $$-0.631065\pi$$
−0.400216 + 0.916421i $$0.631065\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −8.74456 −1.02347 −0.511737 0.859142i $$-0.670998\pi$$
−0.511737 + 0.859142i $$0.670998\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 4.62772 0.527377
$$78$$ 9.48913 1.07443
$$79$$ −4.74456 −0.533805 −0.266903 0.963724i $$-0.586000\pi$$
−0.266903 + 0.963724i $$0.586000\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ −5.37228 −0.593269
$$83$$ 0.744563 0.0817264 0.0408632 0.999165i $$-0.486989\pi$$
0.0408632 + 0.999165i $$0.486989\pi$$
$$84$$ 2.74456 0.299456
$$85$$ 0 0
$$86$$ 7.37228 0.794974
$$87$$ −16.2337 −1.74043
$$88$$ 3.37228 0.359486
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −6.51087 −0.682525
$$92$$ −6.74456 −0.703169
$$93$$ 5.25544 0.544963
$$94$$ −8.74456 −0.901933
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ 0.116844 0.0118637 0.00593185 0.999982i $$-0.498112\pi$$
0.00593185 + 0.999982i $$0.498112\pi$$
$$98$$ 5.11684 0.516879
$$99$$ −3.37228 −0.338927
$$100$$ 0 0
$$101$$ −11.4891 −1.14321 −0.571605 0.820529i $$-0.693679\pi$$
−0.571605 + 0.820529i $$0.693679\pi$$
$$102$$ 10.7446 1.06387
$$103$$ 9.48913 0.934991 0.467496 0.883995i $$-0.345156\pi$$
0.467496 + 0.883995i $$0.345156\pi$$
$$104$$ −4.74456 −0.465243
$$105$$ 0 0
$$106$$ 1.37228 0.133288
$$107$$ 3.48913 0.337306 0.168653 0.985675i $$-0.446058\pi$$
0.168653 + 0.985675i $$0.446058\pi$$
$$108$$ 4.00000 0.384900
$$109$$ 0.116844 0.0111916 0.00559581 0.999984i $$-0.498219\pi$$
0.00559581 + 0.999984i $$0.498219\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ −1.37228 −0.129668
$$113$$ −17.3723 −1.63425 −0.817123 0.576463i $$-0.804433\pi$$
−0.817123 + 0.576463i $$0.804433\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 8.11684 0.753630
$$117$$ 4.74456 0.438635
$$118$$ −12.7446 −1.17323
$$119$$ −7.37228 −0.675816
$$120$$ 0 0
$$121$$ 0.372281 0.0338438
$$122$$ 5.37228 0.486383
$$123$$ −10.7446 −0.968805
$$124$$ −2.62772 −0.235976
$$125$$ 0 0
$$126$$ 1.37228 0.122253
$$127$$ 16.7446 1.48584 0.742920 0.669380i $$-0.233440\pi$$
0.742920 + 0.669380i $$0.233440\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 14.7446 1.29819
$$130$$ 0 0
$$131$$ −3.25544 −0.284429 −0.142214 0.989836i $$-0.545422\pi$$
−0.142214 + 0.989836i $$0.545422\pi$$
$$132$$ 6.74456 0.587039
$$133$$ 2.74456 0.237984
$$134$$ 4.74456 0.409868
$$135$$ 0 0
$$136$$ −5.37228 −0.460669
$$137$$ −16.7446 −1.43058 −0.715292 0.698825i $$-0.753706\pi$$
−0.715292 + 0.698825i $$0.753706\pi$$
$$138$$ −13.4891 −1.14827
$$139$$ −1.88316 −0.159727 −0.0798636 0.996806i $$-0.525448\pi$$
−0.0798636 + 0.996806i $$0.525448\pi$$
$$140$$ 0 0
$$141$$ −17.4891 −1.47285
$$142$$ 6.74456 0.565991
$$143$$ −16.0000 −1.33799
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 8.74456 0.723705
$$147$$ 10.2337 0.844060
$$148$$ −1.00000 −0.0821995
$$149$$ −11.4891 −0.941226 −0.470613 0.882340i $$-0.655967\pi$$
−0.470613 + 0.882340i $$0.655967\pi$$
$$150$$ 0 0
$$151$$ −20.0000 −1.62758 −0.813788 0.581161i $$-0.802599\pi$$
−0.813788 + 0.581161i $$0.802599\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 5.37228 0.434323
$$154$$ −4.62772 −0.372912
$$155$$ 0 0
$$156$$ −9.48913 −0.759738
$$157$$ −17.3723 −1.38646 −0.693229 0.720717i $$-0.743813\pi$$
−0.693229 + 0.720717i $$0.743813\pi$$
$$158$$ 4.74456 0.377457
$$159$$ 2.74456 0.217658
$$160$$ 0 0
$$161$$ 9.25544 0.729431
$$162$$ 11.0000 0.864242
$$163$$ −19.3723 −1.51735 −0.758677 0.651467i $$-0.774154\pi$$
−0.758677 + 0.651467i $$0.774154\pi$$
$$164$$ 5.37228 0.419505
$$165$$ 0 0
$$166$$ −0.744563 −0.0577893
$$167$$ −1.48913 −0.115232 −0.0576160 0.998339i $$-0.518350\pi$$
−0.0576160 + 0.998339i $$0.518350\pi$$
$$168$$ −2.74456 −0.211748
$$169$$ 9.51087 0.731606
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ −7.37228 −0.562131
$$173$$ −22.8614 −1.73812 −0.869060 0.494706i $$-0.835276\pi$$
−0.869060 + 0.494706i $$0.835276\pi$$
$$174$$ 16.2337 1.23067
$$175$$ 0 0
$$176$$ −3.37228 −0.254195
$$177$$ −25.4891 −1.91588
$$178$$ −10.0000 −0.749532
$$179$$ −26.2337 −1.96080 −0.980399 0.197023i $$-0.936873\pi$$
−0.980399 + 0.197023i $$0.936873\pi$$
$$180$$ 0 0
$$181$$ 7.48913 0.556662 0.278331 0.960485i $$-0.410219\pi$$
0.278331 + 0.960485i $$0.410219\pi$$
$$182$$ 6.51087 0.482618
$$183$$ 10.7446 0.794261
$$184$$ 6.74456 0.497216
$$185$$ 0 0
$$186$$ −5.25544 −0.385347
$$187$$ −18.1168 −1.32483
$$188$$ 8.74456 0.637763
$$189$$ −5.48913 −0.399275
$$190$$ 0 0
$$191$$ 18.6277 1.34785 0.673927 0.738798i $$-0.264606\pi$$
0.673927 + 0.738798i $$0.264606\pi$$
$$192$$ −2.00000 −0.144338
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −0.116844 −0.00838891
$$195$$ 0 0
$$196$$ −5.11684 −0.365489
$$197$$ −27.4891 −1.95852 −0.979260 0.202610i $$-0.935058\pi$$
−0.979260 + 0.202610i $$0.935058\pi$$
$$198$$ 3.37228 0.239658
$$199$$ −11.2554 −0.797877 −0.398938 0.916978i $$-0.630621\pi$$
−0.398938 + 0.916978i $$0.630621\pi$$
$$200$$ 0 0
$$201$$ 9.48913 0.669311
$$202$$ 11.4891 0.808372
$$203$$ −11.1386 −0.781776
$$204$$ −10.7446 −0.752270
$$205$$ 0 0
$$206$$ −9.48913 −0.661139
$$207$$ −6.74456 −0.468780
$$208$$ 4.74456 0.328976
$$209$$ 6.74456 0.466531
$$210$$ 0 0
$$211$$ −14.1168 −0.971844 −0.485922 0.874002i $$-0.661516\pi$$
−0.485922 + 0.874002i $$0.661516\pi$$
$$212$$ −1.37228 −0.0942487
$$213$$ 13.4891 0.924260
$$214$$ −3.48913 −0.238512
$$215$$ 0 0
$$216$$ −4.00000 −0.272166
$$217$$ 3.60597 0.244789
$$218$$ −0.116844 −0.00791367
$$219$$ 17.4891 1.18181
$$220$$ 0 0
$$221$$ 25.4891 1.71458
$$222$$ −2.00000 −0.134231
$$223$$ 1.37228 0.0918948 0.0459474 0.998944i $$-0.485369\pi$$
0.0459474 + 0.998944i $$0.485369\pi$$
$$224$$ 1.37228 0.0916894
$$225$$ 0 0
$$226$$ 17.3723 1.15559
$$227$$ 11.3723 0.754805 0.377402 0.926049i $$-0.376817\pi$$
0.377402 + 0.926049i $$0.376817\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ −9.25544 −0.608963
$$232$$ −8.11684 −0.532897
$$233$$ −6.23369 −0.408382 −0.204191 0.978931i $$-0.565456\pi$$
−0.204191 + 0.978931i $$0.565456\pi$$
$$234$$ −4.74456 −0.310162
$$235$$ 0 0
$$236$$ 12.7446 0.829600
$$237$$ 9.48913 0.616385
$$238$$ 7.37228 0.477874
$$239$$ 17.6060 1.13884 0.569418 0.822048i $$-0.307169\pi$$
0.569418 + 0.822048i $$0.307169\pi$$
$$240$$ 0 0
$$241$$ −10.2337 −0.659210 −0.329605 0.944119i $$-0.606916\pi$$
−0.329605 + 0.944119i $$0.606916\pi$$
$$242$$ −0.372281 −0.0239311
$$243$$ 10.0000 0.641500
$$244$$ −5.37228 −0.343925
$$245$$ 0 0
$$246$$ 10.7446 0.685048
$$247$$ −9.48913 −0.603779
$$248$$ 2.62772 0.166860
$$249$$ −1.48913 −0.0943695
$$250$$ 0 0
$$251$$ 11.4891 0.725187 0.362594 0.931947i $$-0.381891\pi$$
0.362594 + 0.931947i $$0.381891\pi$$
$$252$$ −1.37228 −0.0864456
$$253$$ 22.7446 1.42994
$$254$$ −16.7446 −1.05065
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 20.9783 1.30859 0.654294 0.756241i $$-0.272966\pi$$
0.654294 + 0.756241i $$0.272966\pi$$
$$258$$ −14.7446 −0.917956
$$259$$ 1.37228 0.0852694
$$260$$ 0 0
$$261$$ 8.11684 0.502420
$$262$$ 3.25544 0.201122
$$263$$ −0.116844 −0.00720491 −0.00360245 0.999994i $$-0.501147\pi$$
−0.00360245 + 0.999994i $$0.501147\pi$$
$$264$$ −6.74456 −0.415099
$$265$$ 0 0
$$266$$ −2.74456 −0.168280
$$267$$ −20.0000 −1.22398
$$268$$ −4.74456 −0.289820
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −6.74456 −0.409703 −0.204852 0.978793i $$-0.565671\pi$$
−0.204852 + 0.978793i $$0.565671\pi$$
$$272$$ 5.37228 0.325742
$$273$$ 13.0217 0.788112
$$274$$ 16.7446 1.01158
$$275$$ 0 0
$$276$$ 13.4891 0.811950
$$277$$ −0.744563 −0.0447364 −0.0223682 0.999750i $$-0.507121\pi$$
−0.0223682 + 0.999750i $$0.507121\pi$$
$$278$$ 1.88316 0.112944
$$279$$ −2.62772 −0.157317
$$280$$ 0 0
$$281$$ 24.7446 1.47614 0.738068 0.674726i $$-0.235738\pi$$
0.738068 + 0.674726i $$0.235738\pi$$
$$282$$ 17.4891 1.04146
$$283$$ −5.48913 −0.326295 −0.163147 0.986602i $$-0.552165\pi$$
−0.163147 + 0.986602i $$0.552165\pi$$
$$284$$ −6.74456 −0.400216
$$285$$ 0 0
$$286$$ 16.0000 0.946100
$$287$$ −7.37228 −0.435172
$$288$$ −1.00000 −0.0589256
$$289$$ 11.8614 0.697730
$$290$$ 0 0
$$291$$ −0.233688 −0.0136990
$$292$$ −8.74456 −0.511737
$$293$$ 18.8614 1.10190 0.550948 0.834540i $$-0.314266\pi$$
0.550948 + 0.834540i $$0.314266\pi$$
$$294$$ −10.2337 −0.596841
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ −13.4891 −0.782718
$$298$$ 11.4891 0.665547
$$299$$ −32.0000 −1.85061
$$300$$ 0 0
$$301$$ 10.1168 0.583125
$$302$$ 20.0000 1.15087
$$303$$ 22.9783 1.32007
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ −5.37228 −0.307113
$$307$$ 23.4891 1.34060 0.670298 0.742092i $$-0.266166\pi$$
0.670298 + 0.742092i $$0.266166\pi$$
$$308$$ 4.62772 0.263689
$$309$$ −18.9783 −1.07963
$$310$$ 0 0
$$311$$ −1.37228 −0.0778149 −0.0389075 0.999243i $$-0.512388\pi$$
−0.0389075 + 0.999243i $$0.512388\pi$$
$$312$$ 9.48913 0.537216
$$313$$ 3.48913 0.197217 0.0986085 0.995126i $$-0.468561\pi$$
0.0986085 + 0.995126i $$0.468561\pi$$
$$314$$ 17.3723 0.980375
$$315$$ 0 0
$$316$$ −4.74456 −0.266903
$$317$$ −25.3723 −1.42505 −0.712525 0.701647i $$-0.752448\pi$$
−0.712525 + 0.701647i $$0.752448\pi$$
$$318$$ −2.74456 −0.153907
$$319$$ −27.3723 −1.53255
$$320$$ 0 0
$$321$$ −6.97825 −0.389488
$$322$$ −9.25544 −0.515785
$$323$$ −10.7446 −0.597843
$$324$$ −11.0000 −0.611111
$$325$$ 0 0
$$326$$ 19.3723 1.07293
$$327$$ −0.233688 −0.0129230
$$328$$ −5.37228 −0.296635
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ 23.4891 1.29108 0.645540 0.763727i $$-0.276633\pi$$
0.645540 + 0.763727i $$0.276633\pi$$
$$332$$ 0.744563 0.0408632
$$333$$ −1.00000 −0.0547997
$$334$$ 1.48913 0.0814813
$$335$$ 0 0
$$336$$ 2.74456 0.149728
$$337$$ 7.25544 0.395229 0.197614 0.980280i $$-0.436681\pi$$
0.197614 + 0.980280i $$0.436681\pi$$
$$338$$ −9.51087 −0.517323
$$339$$ 34.7446 1.88707
$$340$$ 0 0
$$341$$ 8.86141 0.479872
$$342$$ 2.00000 0.108148
$$343$$ 16.6277 0.897812
$$344$$ 7.37228 0.397487
$$345$$ 0 0
$$346$$ 22.8614 1.22904
$$347$$ 22.9783 1.23354 0.616769 0.787145i $$-0.288441\pi$$
0.616769 + 0.787145i $$0.288441\pi$$
$$348$$ −16.2337 −0.870217
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ 18.9783 1.01298
$$352$$ 3.37228 0.179743
$$353$$ 22.8614 1.21679 0.608395 0.793634i $$-0.291814\pi$$
0.608395 + 0.793634i $$0.291814\pi$$
$$354$$ 25.4891 1.35473
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 14.7446 0.780365
$$358$$ 26.2337 1.38649
$$359$$ −30.9783 −1.63497 −0.817485 0.575950i $$-0.804632\pi$$
−0.817485 + 0.575950i $$0.804632\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −7.48913 −0.393620
$$363$$ −0.744563 −0.0390794
$$364$$ −6.51087 −0.341263
$$365$$ 0 0
$$366$$ −10.7446 −0.561627
$$367$$ −3.88316 −0.202699 −0.101350 0.994851i $$-0.532316\pi$$
−0.101350 + 0.994851i $$0.532316\pi$$
$$368$$ −6.74456 −0.351585
$$369$$ 5.37228 0.279670
$$370$$ 0 0
$$371$$ 1.88316 0.0977686
$$372$$ 5.25544 0.272482
$$373$$ 31.4891 1.63045 0.815223 0.579148i $$-0.196615\pi$$
0.815223 + 0.579148i $$0.196615\pi$$
$$374$$ 18.1168 0.936800
$$375$$ 0 0
$$376$$ −8.74456 −0.450966
$$377$$ 38.5109 1.98341
$$378$$ 5.48913 0.282330
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ −33.4891 −1.71570
$$382$$ −18.6277 −0.953077
$$383$$ 13.4891 0.689262 0.344631 0.938738i $$-0.388004\pi$$
0.344631 + 0.938738i $$0.388004\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ −7.37228 −0.374754
$$388$$ 0.116844 0.00593185
$$389$$ 14.8614 0.753503 0.376752 0.926314i $$-0.377041\pi$$
0.376752 + 0.926314i $$0.377041\pi$$
$$390$$ 0 0
$$391$$ −36.2337 −1.83242
$$392$$ 5.11684 0.258440
$$393$$ 6.51087 0.328430
$$394$$ 27.4891 1.38488
$$395$$ 0 0
$$396$$ −3.37228 −0.169464
$$397$$ −24.9783 −1.25362 −0.626811 0.779171i $$-0.715640\pi$$
−0.626811 + 0.779171i $$0.715640\pi$$
$$398$$ 11.2554 0.564184
$$399$$ −5.48913 −0.274800
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ −9.48913 −0.473275
$$403$$ −12.4674 −0.621044
$$404$$ −11.4891 −0.571605
$$405$$ 0 0
$$406$$ 11.1386 0.552799
$$407$$ 3.37228 0.167158
$$408$$ 10.7446 0.531935
$$409$$ 6.23369 0.308236 0.154118 0.988052i $$-0.450746\pi$$
0.154118 + 0.988052i $$0.450746\pi$$
$$410$$ 0 0
$$411$$ 33.4891 1.65190
$$412$$ 9.48913 0.467496
$$413$$ −17.4891 −0.860584
$$414$$ 6.74456 0.331477
$$415$$ 0 0
$$416$$ −4.74456 −0.232621
$$417$$ 3.76631 0.184437
$$418$$ −6.74456 −0.329887
$$419$$ 13.4891 0.658987 0.329493 0.944158i $$-0.393122\pi$$
0.329493 + 0.944158i $$0.393122\pi$$
$$420$$ 0 0
$$421$$ −7.48913 −0.364998 −0.182499 0.983206i $$-0.558419\pi$$
−0.182499 + 0.983206i $$0.558419\pi$$
$$422$$ 14.1168 0.687197
$$423$$ 8.74456 0.425175
$$424$$ 1.37228 0.0666439
$$425$$ 0 0
$$426$$ −13.4891 −0.653550
$$427$$ 7.37228 0.356770
$$428$$ 3.48913 0.168653
$$429$$ 32.0000 1.54497
$$430$$ 0 0
$$431$$ −1.37228 −0.0661005 −0.0330502 0.999454i $$-0.510522\pi$$
−0.0330502 + 0.999454i $$0.510522\pi$$
$$432$$ 4.00000 0.192450
$$433$$ −12.9783 −0.623695 −0.311847 0.950132i $$-0.600948\pi$$
−0.311847 + 0.950132i $$0.600948\pi$$
$$434$$ −3.60597 −0.173092
$$435$$ 0 0
$$436$$ 0.116844 0.00559581
$$437$$ 13.4891 0.645272
$$438$$ −17.4891 −0.835663
$$439$$ −13.3723 −0.638224 −0.319112 0.947717i $$-0.603385\pi$$
−0.319112 + 0.947717i $$0.603385\pi$$
$$440$$ 0 0
$$441$$ −5.11684 −0.243659
$$442$$ −25.4891 −1.21239
$$443$$ 16.9783 0.806661 0.403331 0.915054i $$-0.367852\pi$$
0.403331 + 0.915054i $$0.367852\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ −1.37228 −0.0649794
$$447$$ 22.9783 1.08683
$$448$$ −1.37228 −0.0648342
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −18.1168 −0.853089
$$452$$ −17.3723 −0.817123
$$453$$ 40.0000 1.87936
$$454$$ −11.3723 −0.533728
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ −18.8614 −0.882299 −0.441150 0.897434i $$-0.645429\pi$$
−0.441150 + 0.897434i $$0.645429\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 21.4891 1.00303
$$460$$ 0 0
$$461$$ −39.0951 −1.82084 −0.910420 0.413685i $$-0.864241\pi$$
−0.910420 + 0.413685i $$0.864241\pi$$
$$462$$ 9.25544 0.430602
$$463$$ 5.48913 0.255101 0.127551 0.991832i $$-0.459288\pi$$
0.127551 + 0.991832i $$0.459288\pi$$
$$464$$ 8.11684 0.376815
$$465$$ 0 0
$$466$$ 6.23369 0.288770
$$467$$ −30.3505 −1.40446 −0.702228 0.711953i $$-0.747811\pi$$
−0.702228 + 0.711953i $$0.747811\pi$$
$$468$$ 4.74456 0.219317
$$469$$ 6.51087 0.300644
$$470$$ 0 0
$$471$$ 34.7446 1.60094
$$472$$ −12.7446 −0.586616
$$473$$ 24.8614 1.14313
$$474$$ −9.48913 −0.435850
$$475$$ 0 0
$$476$$ −7.37228 −0.337908
$$477$$ −1.37228 −0.0628324
$$478$$ −17.6060 −0.805278
$$479$$ 3.25544 0.148745 0.0743724 0.997231i $$-0.476305\pi$$
0.0743724 + 0.997231i $$0.476305\pi$$
$$480$$ 0 0
$$481$$ −4.74456 −0.216333
$$482$$ 10.2337 0.466132
$$483$$ −18.5109 −0.842274
$$484$$ 0.372281 0.0169219
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ 37.7228 1.70938 0.854692 0.519136i $$-0.173746\pi$$
0.854692 + 0.519136i $$0.173746\pi$$
$$488$$ 5.37228 0.243192
$$489$$ 38.7446 1.75209
$$490$$ 0 0
$$491$$ 14.9783 0.675959 0.337979 0.941153i $$-0.390257\pi$$
0.337979 + 0.941153i $$0.390257\pi$$
$$492$$ −10.7446 −0.484402
$$493$$ 43.6060 1.96391
$$494$$ 9.48913 0.426936
$$495$$ 0 0
$$496$$ −2.62772 −0.117988
$$497$$ 9.25544 0.415163
$$498$$ 1.48913 0.0667293
$$499$$ −24.9783 −1.11818 −0.559090 0.829107i $$-0.688849\pi$$
−0.559090 + 0.829107i $$0.688849\pi$$
$$500$$ 0 0
$$501$$ 2.97825 0.133058
$$502$$ −11.4891 −0.512785
$$503$$ −29.4891 −1.31486 −0.657428 0.753518i $$-0.728355\pi$$
−0.657428 + 0.753518i $$0.728355\pi$$
$$504$$ 1.37228 0.0611263
$$505$$ 0 0
$$506$$ −22.7446 −1.01112
$$507$$ −19.0217 −0.844786
$$508$$ 16.7446 0.742920
$$509$$ −11.2554 −0.498888 −0.249444 0.968389i $$-0.580248\pi$$
−0.249444 + 0.968389i $$0.580248\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ −1.00000 −0.0441942
$$513$$ −8.00000 −0.353209
$$514$$ −20.9783 −0.925311
$$515$$ 0 0
$$516$$ 14.7446 0.649093
$$517$$ −29.4891 −1.29693
$$518$$ −1.37228 −0.0602946
$$519$$ 45.7228 2.00701
$$520$$ 0 0
$$521$$ 27.0951 1.18706 0.593529 0.804813i $$-0.297734\pi$$
0.593529 + 0.804813i $$0.297734\pi$$
$$522$$ −8.11684 −0.355265
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ −3.25544 −0.142214
$$525$$ 0 0
$$526$$ 0.116844 0.00509464
$$527$$ −14.1168 −0.614939
$$528$$ 6.74456 0.293519
$$529$$ 22.4891 0.977788
$$530$$ 0 0
$$531$$ 12.7446 0.553067
$$532$$ 2.74456 0.118992
$$533$$ 25.4891 1.10406
$$534$$ 20.0000 0.865485
$$535$$ 0 0
$$536$$ 4.74456 0.204934
$$537$$ 52.4674 2.26413
$$538$$ −10.0000 −0.431131
$$539$$ 17.2554 0.743244
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 6.74456 0.289704
$$543$$ −14.9783 −0.642778
$$544$$ −5.37228 −0.230335
$$545$$ 0 0
$$546$$ −13.0217 −0.557279
$$547$$ −39.3723 −1.68344 −0.841719 0.539916i $$-0.818456\pi$$
−0.841719 + 0.539916i $$0.818456\pi$$
$$548$$ −16.7446 −0.715292
$$549$$ −5.37228 −0.229283
$$550$$ 0 0
$$551$$ −16.2337 −0.691578
$$552$$ −13.4891 −0.574135
$$553$$ 6.51087 0.276871
$$554$$ 0.744563 0.0316334
$$555$$ 0 0
$$556$$ −1.88316 −0.0798636
$$557$$ −8.74456 −0.370519 −0.185260 0.982690i $$-0.559313\pi$$
−0.185260 + 0.982690i $$0.559313\pi$$
$$558$$ 2.62772 0.111240
$$559$$ −34.9783 −1.47942
$$560$$ 0 0
$$561$$ 36.2337 1.52979
$$562$$ −24.7446 −1.04379
$$563$$ −33.0951 −1.39479 −0.697396 0.716686i $$-0.745658\pi$$
−0.697396 + 0.716686i $$0.745658\pi$$
$$564$$ −17.4891 −0.736425
$$565$$ 0 0
$$566$$ 5.48913 0.230725
$$567$$ 15.0951 0.633934
$$568$$ 6.74456 0.282996
$$569$$ −32.9783 −1.38252 −0.691260 0.722606i $$-0.742944\pi$$
−0.691260 + 0.722606i $$0.742944\pi$$
$$570$$ 0 0
$$571$$ −27.6060 −1.15527 −0.577637 0.816294i $$-0.696025\pi$$
−0.577637 + 0.816294i $$0.696025\pi$$
$$572$$ −16.0000 −0.668994
$$573$$ −37.2554 −1.55637
$$574$$ 7.37228 0.307713
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 7.48913 0.311776 0.155888 0.987775i $$-0.450176\pi$$
0.155888 + 0.987775i $$0.450176\pi$$
$$578$$ −11.8614 −0.493369
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −1.02175 −0.0423893
$$582$$ 0.233688 0.00968668
$$583$$ 4.62772 0.191661
$$584$$ 8.74456 0.361853
$$585$$ 0 0
$$586$$ −18.8614 −0.779158
$$587$$ −47.8397 −1.97455 −0.987277 0.159010i $$-0.949170\pi$$
−0.987277 + 0.159010i $$0.949170\pi$$
$$588$$ 10.2337 0.422030
$$589$$ 5.25544 0.216547
$$590$$ 0 0
$$591$$ 54.9783 2.26150
$$592$$ −1.00000 −0.0410997
$$593$$ 10.2337 0.420247 0.210124 0.977675i $$-0.432613\pi$$
0.210124 + 0.977675i $$0.432613\pi$$
$$594$$ 13.4891 0.553466
$$595$$ 0 0
$$596$$ −11.4891 −0.470613
$$597$$ 22.5109 0.921309
$$598$$ 32.0000 1.30858
$$599$$ 17.4891 0.714586 0.357293 0.933992i $$-0.383700\pi$$
0.357293 + 0.933992i $$0.383700\pi$$
$$600$$ 0 0
$$601$$ −5.37228 −0.219140 −0.109570 0.993979i $$-0.534947\pi$$
−0.109570 + 0.993979i $$0.534947\pi$$
$$602$$ −10.1168 −0.412332
$$603$$ −4.74456 −0.193214
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ −22.9783 −0.933428
$$607$$ −17.7228 −0.719347 −0.359673 0.933078i $$-0.617112\pi$$
−0.359673 + 0.933078i $$0.617112\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 22.2772 0.902717
$$610$$ 0 0
$$611$$ 41.4891 1.67847
$$612$$ 5.37228 0.217162
$$613$$ −43.0951 −1.74059 −0.870297 0.492527i $$-0.836073\pi$$
−0.870297 + 0.492527i $$0.836073\pi$$
$$614$$ −23.4891 −0.947944
$$615$$ 0 0
$$616$$ −4.62772 −0.186456
$$617$$ 31.7228 1.27711 0.638556 0.769575i $$-0.279532\pi$$
0.638556 + 0.769575i $$0.279532\pi$$
$$618$$ 18.9783 0.763417
$$619$$ 30.1168 1.21050 0.605249 0.796036i $$-0.293074\pi$$
0.605249 + 0.796036i $$0.293074\pi$$
$$620$$ 0 0
$$621$$ −26.9783 −1.08260
$$622$$ 1.37228 0.0550235
$$623$$ −13.7228 −0.549793
$$624$$ −9.48913 −0.379869
$$625$$ 0 0
$$626$$ −3.48913 −0.139453
$$627$$ −13.4891 −0.538704
$$628$$ −17.3723 −0.693229
$$629$$ −5.37228 −0.214207
$$630$$ 0 0
$$631$$ −19.0951 −0.760164 −0.380082 0.924953i $$-0.624104\pi$$
−0.380082 + 0.924953i $$0.624104\pi$$
$$632$$ 4.74456 0.188729
$$633$$ 28.2337 1.12219
$$634$$ 25.3723 1.00766
$$635$$ 0 0
$$636$$ 2.74456 0.108829
$$637$$ −24.2772 −0.961897
$$638$$ 27.3723 1.08368
$$639$$ −6.74456 −0.266811
$$640$$ 0 0
$$641$$ −49.6060 −1.95932 −0.979659 0.200670i $$-0.935688\pi$$
−0.979659 + 0.200670i $$0.935688\pi$$
$$642$$ 6.97825 0.275410
$$643$$ −3.37228 −0.132990 −0.0664949 0.997787i $$-0.521182\pi$$
−0.0664949 + 0.997787i $$0.521182\pi$$
$$644$$ 9.25544 0.364715
$$645$$ 0 0
$$646$$ 10.7446 0.422739
$$647$$ 13.4891 0.530312 0.265156 0.964205i $$-0.414577\pi$$
0.265156 + 0.964205i $$0.414577\pi$$
$$648$$ 11.0000 0.432121
$$649$$ −42.9783 −1.68704
$$650$$ 0 0
$$651$$ −7.21194 −0.282658
$$652$$ −19.3723 −0.758677
$$653$$ −0.510875 −0.0199921 −0.00999604 0.999950i $$-0.503182\pi$$
−0.00999604 + 0.999950i $$0.503182\pi$$
$$654$$ 0.233688 0.00913792
$$655$$ 0 0
$$656$$ 5.37228 0.209752
$$657$$ −8.74456 −0.341158
$$658$$ 12.0000 0.467809
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 4.35053 0.169216 0.0846080 0.996414i $$-0.473036\pi$$
0.0846080 + 0.996414i $$0.473036\pi$$
$$662$$ −23.4891 −0.912931
$$663$$ −50.9783 −1.97983
$$664$$ −0.744563 −0.0288946
$$665$$ 0 0
$$666$$ 1.00000 0.0387492
$$667$$ −54.7446 −2.11972
$$668$$ −1.48913 −0.0576160
$$669$$ −2.74456 −0.106111
$$670$$ 0 0
$$671$$ 18.1168 0.699393
$$672$$ −2.74456 −0.105874
$$673$$ 8.51087 0.328070 0.164035 0.986455i $$-0.447549\pi$$
0.164035 + 0.986455i $$0.447549\pi$$
$$674$$ −7.25544 −0.279469
$$675$$ 0 0
$$676$$ 9.51087 0.365803
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ −34.7446 −1.33436
$$679$$ −0.160343 −0.00615339
$$680$$ 0 0
$$681$$ −22.7446 −0.871574
$$682$$ −8.86141 −0.339321
$$683$$ −8.62772 −0.330130 −0.165065 0.986283i $$-0.552783\pi$$
−0.165065 + 0.986283i $$0.552783\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ −16.6277 −0.634849
$$687$$ 20.0000 0.763048
$$688$$ −7.37228 −0.281066
$$689$$ −6.51087 −0.248045
$$690$$ 0 0
$$691$$ 22.3505 0.850254 0.425127 0.905134i $$-0.360229\pi$$
0.425127 + 0.905134i $$0.360229\pi$$
$$692$$ −22.8614 −0.869060
$$693$$ 4.62772 0.175792
$$694$$ −22.9783 −0.872242
$$695$$ 0 0
$$696$$ 16.2337 0.615336
$$697$$ 28.8614 1.09320
$$698$$ 22.0000 0.832712
$$699$$ 12.4674 0.471559
$$700$$ 0 0
$$701$$ −26.4674 −0.999659 −0.499829 0.866124i $$-0.666604\pi$$
−0.499829 + 0.866124i $$0.666604\pi$$
$$702$$ −18.9783 −0.716288
$$703$$ 2.00000 0.0754314
$$704$$ −3.37228 −0.127098
$$705$$ 0 0
$$706$$ −22.8614 −0.860400
$$707$$ 15.7663 0.592953
$$708$$ −25.4891 −0.957940
$$709$$ 40.1168 1.50662 0.753310 0.657666i $$-0.228456\pi$$
0.753310 + 0.657666i $$0.228456\pi$$
$$710$$ 0 0
$$711$$ −4.74456 −0.177935
$$712$$ −10.0000 −0.374766
$$713$$ 17.7228 0.663725
$$714$$ −14.7446 −0.551801
$$715$$ 0 0
$$716$$ −26.2337 −0.980399
$$717$$ −35.2119 −1.31501
$$718$$ 30.9783 1.15610
$$719$$ −34.7446 −1.29575 −0.647877 0.761745i $$-0.724343\pi$$
−0.647877 + 0.761745i $$0.724343\pi$$
$$720$$ 0 0
$$721$$ −13.0217 −0.484955
$$722$$ 15.0000 0.558242
$$723$$ 20.4674 0.761190
$$724$$ 7.48913 0.278331
$$725$$ 0 0
$$726$$ 0.744563 0.0276333
$$727$$ 48.0000 1.78022 0.890111 0.455744i $$-0.150627\pi$$
0.890111 + 0.455744i $$0.150627\pi$$
$$728$$ 6.51087 0.241309
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −39.6060 −1.46488
$$732$$ 10.7446 0.397130
$$733$$ 29.3723 1.08489 0.542445 0.840091i $$-0.317499\pi$$
0.542445 + 0.840091i $$0.317499\pi$$
$$734$$ 3.88316 0.143330
$$735$$ 0 0
$$736$$ 6.74456 0.248608
$$737$$ 16.0000 0.589368
$$738$$ −5.37228 −0.197756
$$739$$ −36.8614 −1.35597 −0.677984 0.735076i $$-0.737146\pi$$
−0.677984 + 0.735076i $$0.737146\pi$$
$$740$$ 0 0
$$741$$ 18.9783 0.697183
$$742$$ −1.88316 −0.0691328
$$743$$ 5.37228 0.197090 0.0985449 0.995133i $$-0.468581\pi$$
0.0985449 + 0.995133i $$0.468581\pi$$
$$744$$ −5.25544 −0.192674
$$745$$ 0 0
$$746$$ −31.4891 −1.15290
$$747$$ 0.744563 0.0272421
$$748$$ −18.1168 −0.662417
$$749$$ −4.78806 −0.174952
$$750$$ 0 0
$$751$$ 10.7446 0.392075 0.196037 0.980596i $$-0.437193\pi$$
0.196037 + 0.980596i $$0.437193\pi$$
$$752$$ 8.74456 0.318881
$$753$$ −22.9783 −0.837374
$$754$$ −38.5109 −1.40248
$$755$$ 0 0
$$756$$ −5.48913 −0.199638
$$757$$ −43.9565 −1.59763 −0.798813 0.601579i $$-0.794538\pi$$
−0.798813 + 0.601579i $$0.794538\pi$$
$$758$$ 8.00000 0.290573
$$759$$ −45.4891 −1.65115
$$760$$ 0 0
$$761$$ 5.37228 0.194745 0.0973725 0.995248i $$-0.468956\pi$$
0.0973725 + 0.995248i $$0.468956\pi$$
$$762$$ 33.4891 1.21318
$$763$$ −0.160343 −0.00580480
$$764$$ 18.6277 0.673927
$$765$$ 0 0
$$766$$ −13.4891 −0.487382
$$767$$ 60.4674 2.18335
$$768$$ −2.00000 −0.0721688
$$769$$ 46.2337 1.66723 0.833615 0.552346i $$-0.186267\pi$$
0.833615 + 0.552346i $$0.186267\pi$$
$$770$$ 0 0
$$771$$ −41.9565 −1.51103
$$772$$ −2.00000 −0.0719816
$$773$$ 24.3505 0.875828 0.437914 0.899017i $$-0.355717\pi$$
0.437914 + 0.899017i $$0.355717\pi$$
$$774$$ 7.37228 0.264991
$$775$$ 0 0
$$776$$ −0.116844 −0.00419445
$$777$$ −2.74456 −0.0984606
$$778$$ −14.8614 −0.532807
$$779$$ −10.7446 −0.384964
$$780$$ 0 0
$$781$$ 22.7446 0.813864
$$782$$ 36.2337 1.29571
$$783$$ 32.4674 1.16029
$$784$$ −5.11684 −0.182744
$$785$$ 0 0
$$786$$ −6.51087 −0.232235
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ −27.4891 −0.979260
$$789$$ 0.233688 0.00831951
$$790$$ 0 0
$$791$$ 23.8397 0.847641
$$792$$ 3.37228 0.119829
$$793$$ −25.4891 −0.905145
$$794$$ 24.9783 0.886445
$$795$$ 0 0
$$796$$ −11.2554 −0.398938
$$797$$ 27.4891 0.973715 0.486857 0.873481i $$-0.338143\pi$$
0.486857 + 0.873481i $$0.338143\pi$$
$$798$$ 5.48913 0.194313
$$799$$ 46.9783 1.66197
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ −10.0000 −0.353112
$$803$$ 29.4891 1.04065
$$804$$ 9.48913 0.334656
$$805$$ 0 0
$$806$$ 12.4674 0.439145
$$807$$ −20.0000 −0.704033
$$808$$ 11.4891 0.404186
$$809$$ −51.4891 −1.81026 −0.905131 0.425134i $$-0.860227\pi$$
−0.905131 + 0.425134i $$0.860227\pi$$
$$810$$ 0 0
$$811$$ 49.4891 1.73780 0.868899 0.494989i $$-0.164828\pi$$
0.868899 + 0.494989i $$0.164828\pi$$
$$812$$ −11.1386 −0.390888
$$813$$ 13.4891 0.473084
$$814$$ −3.37228 −0.118198
$$815$$ 0 0
$$816$$ −10.7446 −0.376135
$$817$$ 14.7446 0.515847
$$818$$ −6.23369 −0.217956
$$819$$ −6.51087 −0.227508
$$820$$ 0 0
$$821$$ −32.7446 −1.14279 −0.571397 0.820674i $$-0.693598\pi$$
−0.571397 + 0.820674i $$0.693598\pi$$
$$822$$ −33.4891 −1.16807
$$823$$ 39.7228 1.38465 0.692325 0.721586i $$-0.256586\pi$$
0.692325 + 0.721586i $$0.256586\pi$$
$$824$$ −9.48913 −0.330569
$$825$$ 0 0
$$826$$ 17.4891 0.608524
$$827$$ 54.3505 1.88995 0.944977 0.327138i $$-0.106084\pi$$
0.944977 + 0.327138i $$0.106084\pi$$
$$828$$ −6.74456 −0.234390
$$829$$ −40.3505 −1.40143 −0.700716 0.713440i $$-0.747136\pi$$
−0.700716 + 0.713440i $$0.747136\pi$$
$$830$$ 0 0
$$831$$ 1.48913 0.0516572
$$832$$ 4.74456 0.164488
$$833$$ −27.4891 −0.952442
$$834$$ −3.76631 −0.130417
$$835$$ 0 0
$$836$$ 6.74456 0.233266
$$837$$ −10.5109 −0.363309
$$838$$ −13.4891 −0.465974
$$839$$ −29.4891 −1.01808 −0.509039 0.860744i $$-0.669999\pi$$
−0.509039 + 0.860744i $$0.669999\pi$$
$$840$$ 0 0
$$841$$ 36.8832 1.27183
$$842$$ 7.48913 0.258092
$$843$$ −49.4891 −1.70450
$$844$$ −14.1168 −0.485922
$$845$$ 0 0
$$846$$ −8.74456 −0.300644
$$847$$ −0.510875 −0.0175539
$$848$$ −1.37228 −0.0471243
$$849$$ 10.9783 0.376773
$$850$$ 0 0
$$851$$ 6.74456 0.231201
$$852$$ 13.4891 0.462130
$$853$$ −35.4891 −1.21512 −0.607562 0.794272i $$-0.707852\pi$$
−0.607562 + 0.794272i $$0.707852\pi$$
$$854$$ −7.37228 −0.252274
$$855$$ 0 0
$$856$$ −3.48913 −0.119256
$$857$$ 29.1386 0.995355 0.497678 0.867362i $$-0.334186\pi$$
0.497678 + 0.867362i $$0.334186\pi$$
$$858$$ −32.0000 −1.09246
$$859$$ −19.7228 −0.672934 −0.336467 0.941695i $$-0.609232\pi$$
−0.336467 + 0.941695i $$0.609232\pi$$
$$860$$ 0 0
$$861$$ 14.7446 0.502493
$$862$$ 1.37228 0.0467401
$$863$$ −53.8397 −1.83272 −0.916362 0.400352i $$-0.868888\pi$$
−0.916362 + 0.400352i $$0.868888\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ 12.9783 0.441019
$$867$$ −23.7228 −0.805669
$$868$$ 3.60597 0.122395
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ −22.5109 −0.762752
$$872$$ −0.116844 −0.00395684
$$873$$ 0.116844 0.00395457
$$874$$ −13.4891 −0.456276
$$875$$ 0 0
$$876$$ 17.4891 0.590903
$$877$$ −49.3723 −1.66718 −0.833592 0.552381i $$-0.813719\pi$$
−0.833592 + 0.552381i $$0.813719\pi$$
$$878$$ 13.3723 0.451293
$$879$$ −37.7228 −1.27236
$$880$$ 0 0
$$881$$ 1.37228 0.0462333 0.0231167 0.999733i $$-0.492641\pi$$
0.0231167 + 0.999733i $$0.492641\pi$$
$$882$$ 5.11684 0.172293
$$883$$ 11.3723 0.382708 0.191354 0.981521i $$-0.438712\pi$$
0.191354 + 0.981521i $$0.438712\pi$$
$$884$$ 25.4891 0.857292
$$885$$ 0 0
$$886$$ −16.9783 −0.570395
$$887$$ 25.3723 0.851918 0.425959 0.904743i $$-0.359937\pi$$
0.425959 + 0.904743i $$0.359937\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ −22.9783 −0.770666
$$890$$ 0 0
$$891$$ 37.0951 1.24273
$$892$$ 1.37228 0.0459474
$$893$$ −17.4891 −0.585251
$$894$$ −22.9783 −0.768508
$$895$$ 0 0
$$896$$ 1.37228 0.0458447
$$897$$ 64.0000 2.13690
$$898$$ −18.0000 −0.600668
$$899$$ −21.3288 −0.711355
$$900$$ 0 0
$$901$$ −7.37228 −0.245606
$$902$$ 18.1168 0.603225
$$903$$ −20.2337 −0.673335
$$904$$ 17.3723 0.577793
$$905$$ 0 0
$$906$$ −40.0000 −1.32891
$$907$$ −40.4674 −1.34370 −0.671849 0.740689i $$-0.734499\pi$$
−0.671849 + 0.740689i $$0.734499\pi$$
$$908$$ 11.3723 0.377402
$$909$$ −11.4891 −0.381070
$$910$$ 0 0
$$911$$ 17.7663 0.588624 0.294312 0.955709i $$-0.404909\pi$$
0.294312 + 0.955709i $$0.404909\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −2.51087 −0.0830978
$$914$$ 18.8614 0.623880
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 4.46738 0.147526
$$918$$ −21.4891 −0.709247
$$919$$ 7.72281 0.254752 0.127376 0.991854i $$-0.459344\pi$$
0.127376 + 0.991854i $$0.459344\pi$$
$$920$$ 0 0
$$921$$ −46.9783 −1.54799
$$922$$ 39.0951 1.28753
$$923$$ −32.0000 −1.05329
$$924$$ −9.25544 −0.304482
$$925$$ 0 0
$$926$$ −5.48913 −0.180384
$$927$$ 9.48913 0.311664
$$928$$ −8.11684 −0.266448
$$929$$ 31.0951 1.02020 0.510098 0.860116i $$-0.329609\pi$$
0.510098 + 0.860116i $$0.329609\pi$$
$$930$$ 0 0
$$931$$ 10.2337 0.335396
$$932$$ −6.23369 −0.204191
$$933$$ 2.74456 0.0898529
$$934$$ 30.3505 0.993100
$$935$$ 0 0
$$936$$ −4.74456 −0.155081
$$937$$ −36.9783 −1.20803 −0.604013 0.796974i $$-0.706433\pi$$
−0.604013 + 0.796974i $$0.706433\pi$$
$$938$$ −6.51087 −0.212588
$$939$$ −6.97825 −0.227727
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ −34.7446 −1.13204
$$943$$ −36.2337 −1.17993
$$944$$ 12.7446 0.414800
$$945$$ 0 0
$$946$$ −24.8614 −0.808314
$$947$$ −37.0951 −1.20543 −0.602714 0.797957i $$-0.705914\pi$$
−0.602714 + 0.797957i $$0.705914\pi$$
$$948$$ 9.48913 0.308192
$$949$$ −41.4891 −1.34679
$$950$$ 0 0
$$951$$ 50.7446 1.64551
$$952$$ 7.37228 0.238937
$$953$$ −46.2337 −1.49766 −0.748828 0.662764i $$-0.769383\pi$$
−0.748828 + 0.662764i $$0.769383\pi$$
$$954$$ 1.37228 0.0444292
$$955$$ 0 0
$$956$$ 17.6060 0.569418
$$957$$ 54.7446 1.76964
$$958$$ −3.25544 −0.105178
$$959$$ 22.9783 0.742006
$$960$$ 0 0
$$961$$ −24.0951 −0.777261
$$962$$ 4.74456 0.152971
$$963$$ 3.48913 0.112435
$$964$$ −10.2337 −0.329605
$$965$$ 0 0
$$966$$ 18.5109 0.595578
$$967$$ 28.0000 0.900419 0.450210 0.892923i $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$968$$ −0.372281 −0.0119656
$$969$$ 21.4891 0.690330
$$970$$ 0 0
$$971$$ −19.6060 −0.629185 −0.314593 0.949227i $$-0.601868\pi$$
−0.314593 + 0.949227i $$0.601868\pi$$
$$972$$ 10.0000 0.320750
$$973$$ 2.58422 0.0828463
$$974$$ −37.7228 −1.20872
$$975$$ 0 0
$$976$$ −5.37228 −0.171963
$$977$$ 11.8832 0.380176 0.190088 0.981767i $$-0.439123\pi$$
0.190088 + 0.981767i $$0.439123\pi$$
$$978$$ −38.7446 −1.23891
$$979$$ −33.7228 −1.07779
$$980$$ 0 0
$$981$$ 0.116844 0.00373054
$$982$$ −14.9783 −0.477975
$$983$$ −30.8614 −0.984326 −0.492163 0.870503i $$-0.663794\pi$$
−0.492163 + 0.870503i $$0.663794\pi$$
$$984$$ 10.7446 0.342524
$$985$$ 0 0
$$986$$ −43.6060 −1.38870
$$987$$ 24.0000 0.763928
$$988$$ −9.48913 −0.301889
$$989$$ 49.7228 1.58109
$$990$$ 0 0
$$991$$ 25.6060 0.813400 0.406700 0.913562i $$-0.366679\pi$$
0.406700 + 0.913562i $$0.366679\pi$$
$$992$$ 2.62772 0.0834302
$$993$$ −46.9783 −1.49081
$$994$$ −9.25544 −0.293565
$$995$$ 0 0
$$996$$ −1.48913 −0.0471847
$$997$$ −26.2337 −0.830829 −0.415415 0.909632i $$-0.636363\pi$$
−0.415415 + 0.909632i $$0.636363\pi$$
$$998$$ 24.9783 0.790673
$$999$$ −4.00000 −0.126554
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 1850.2.a.q.1.1 2
5.2 odd 4 1850.2.b.m.149.1 4
5.3 odd 4 1850.2.b.m.149.4 4
5.4 even 2 370.2.a.f.1.2 2
15.14 odd 2 3330.2.a.bb.1.2 2
20.19 odd 2 2960.2.a.o.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.2 2 5.4 even 2
1850.2.a.q.1.1 2 1.1 even 1 trivial
1850.2.b.m.149.1 4 5.2 odd 4
1850.2.b.m.149.4 4 5.3 odd 4
2960.2.a.o.1.1 2 20.19 odd 2
3330.2.a.bb.1.2 2 15.14 odd 2