# Properties

 Label 285.2.a.g.1.1 Level $285$ Weight $2$ Character 285.1 Self dual yes Analytic conductor $2.276$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(1,285)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$2.27573645761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 285.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{6} +1.41421 q^{7} +1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} -0.414214 q^{6} +1.41421 q^{7} +1.58579 q^{8} +1.00000 q^{9} +0.414214 q^{10} +6.24264 q^{11} -1.82843 q^{12} -0.585786 q^{13} -0.585786 q^{14} -1.00000 q^{15} +3.00000 q^{16} +6.82843 q^{17} -0.414214 q^{18} -1.00000 q^{19} +1.82843 q^{20} +1.41421 q^{21} -2.58579 q^{22} -3.65685 q^{23} +1.58579 q^{24} +1.00000 q^{25} +0.242641 q^{26} +1.00000 q^{27} -2.58579 q^{28} -1.41421 q^{29} +0.414214 q^{30} -8.82843 q^{31} -4.41421 q^{32} +6.24264 q^{33} -2.82843 q^{34} -1.41421 q^{35} -1.82843 q^{36} -0.585786 q^{37} +0.414214 q^{38} -0.585786 q^{39} -1.58579 q^{40} +8.24264 q^{41} -0.585786 q^{42} +3.75736 q^{43} -11.4142 q^{44} -1.00000 q^{45} +1.51472 q^{46} +3.65685 q^{47} +3.00000 q^{48} -5.00000 q^{49} -0.414214 q^{50} +6.82843 q^{51} +1.07107 q^{52} +8.00000 q^{53} -0.414214 q^{54} -6.24264 q^{55} +2.24264 q^{56} -1.00000 q^{57} +0.585786 q^{58} -4.48528 q^{59} +1.82843 q^{60} -15.3137 q^{61} +3.65685 q^{62} +1.41421 q^{63} -4.17157 q^{64} +0.585786 q^{65} -2.58579 q^{66} +1.65685 q^{67} -12.4853 q^{68} -3.65685 q^{69} +0.585786 q^{70} -5.17157 q^{71} +1.58579 q^{72} +3.65685 q^{73} +0.242641 q^{74} +1.00000 q^{75} +1.82843 q^{76} +8.82843 q^{77} +0.242641 q^{78} -3.00000 q^{80} +1.00000 q^{81} -3.41421 q^{82} +7.17157 q^{83} -2.58579 q^{84} -6.82843 q^{85} -1.55635 q^{86} -1.41421 q^{87} +9.89949 q^{88} -13.8995 q^{89} +0.414214 q^{90} -0.828427 q^{91} +6.68629 q^{92} -8.82843 q^{93} -1.51472 q^{94} +1.00000 q^{95} -4.41421 q^{96} -18.2426 q^{97} +2.07107 q^{98} +6.24264 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{13} - 4 q^{14} - 2 q^{15} + 6 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 8 q^{22} + 4 q^{23} + 6 q^{24} + 2 q^{25} - 8 q^{26} + 2 q^{27} - 8 q^{28} - 2 q^{30} - 12 q^{31} - 6 q^{32} + 4 q^{33} + 2 q^{36} - 4 q^{37} - 2 q^{38} - 4 q^{39} - 6 q^{40} + 8 q^{41} - 4 q^{42} + 16 q^{43} - 20 q^{44} - 2 q^{45} + 20 q^{46} - 4 q^{47} + 6 q^{48} - 10 q^{49} + 2 q^{50} + 8 q^{51} - 12 q^{52} + 16 q^{53} + 2 q^{54} - 4 q^{55} - 4 q^{56} - 2 q^{57} + 4 q^{58} + 8 q^{59} - 2 q^{60} - 8 q^{61} - 4 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{66} - 8 q^{67} - 8 q^{68} + 4 q^{69} + 4 q^{70} - 16 q^{71} + 6 q^{72} - 4 q^{73} - 8 q^{74} + 2 q^{75} - 2 q^{76} + 12 q^{77} - 8 q^{78} - 6 q^{80} + 2 q^{81} - 4 q^{82} + 20 q^{83} - 8 q^{84} - 8 q^{85} + 28 q^{86} - 8 q^{89} - 2 q^{90} + 4 q^{91} + 36 q^{92} - 12 q^{93} - 20 q^{94} + 2 q^{95} - 6 q^{96} - 28 q^{97} - 10 q^{98} + 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 + 2 * q^12 - 4 * q^13 - 4 * q^14 - 2 * q^15 + 6 * q^16 + 8 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 8 * q^22 + 4 * q^23 + 6 * q^24 + 2 * q^25 - 8 * q^26 + 2 * q^27 - 8 * q^28 - 2 * q^30 - 12 * q^31 - 6 * q^32 + 4 * q^33 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 4 * q^39 - 6 * q^40 + 8 * q^41 - 4 * q^42 + 16 * q^43 - 20 * q^44 - 2 * q^45 + 20 * q^46 - 4 * q^47 + 6 * q^48 - 10 * q^49 + 2 * q^50 + 8 * q^51 - 12 * q^52 + 16 * q^53 + 2 * q^54 - 4 * q^55 - 4 * q^56 - 2 * q^57 + 4 * q^58 + 8 * q^59 - 2 * q^60 - 8 * q^61 - 4 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^66 - 8 * q^67 - 8 * q^68 + 4 * q^69 + 4 * q^70 - 16 * q^71 + 6 * q^72 - 4 * q^73 - 8 * q^74 + 2 * q^75 - 2 * q^76 + 12 * q^77 - 8 * q^78 - 6 * q^80 + 2 * q^81 - 4 * q^82 + 20 * q^83 - 8 * q^84 - 8 * q^85 + 28 * q^86 - 8 * q^89 - 2 * q^90 + 4 * q^91 + 36 * q^92 - 12 * q^93 - 20 * q^94 + 2 * q^95 - 6 * q^96 - 28 * q^97 - 10 * q^98 + 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.414214 −0.292893 −0.146447 0.989219i $$-0.546784\pi$$
−0.146447 + 0.989219i $$0.546784\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.82843 −0.914214
$$5$$ −1.00000 −0.447214
$$6$$ −0.414214 −0.169102
$$7$$ 1.41421 0.534522 0.267261 0.963624i $$-0.413881\pi$$
0.267261 + 0.963624i $$0.413881\pi$$
$$8$$ 1.58579 0.560660
$$9$$ 1.00000 0.333333
$$10$$ 0.414214 0.130986
$$11$$ 6.24264 1.88223 0.941113 0.338091i $$-0.109781\pi$$
0.941113 + 0.338091i $$0.109781\pi$$
$$12$$ −1.82843 −0.527821
$$13$$ −0.585786 −0.162468 −0.0812340 0.996695i $$-0.525886\pi$$
−0.0812340 + 0.996695i $$0.525886\pi$$
$$14$$ −0.585786 −0.156558
$$15$$ −1.00000 −0.258199
$$16$$ 3.00000 0.750000
$$17$$ 6.82843 1.65614 0.828068 0.560627i $$-0.189440\pi$$
0.828068 + 0.560627i $$0.189440\pi$$
$$18$$ −0.414214 −0.0976311
$$19$$ −1.00000 −0.229416
$$20$$ 1.82843 0.408849
$$21$$ 1.41421 0.308607
$$22$$ −2.58579 −0.551292
$$23$$ −3.65685 −0.762507 −0.381253 0.924471i $$-0.624507\pi$$
−0.381253 + 0.924471i $$0.624507\pi$$
$$24$$ 1.58579 0.323697
$$25$$ 1.00000 0.200000
$$26$$ 0.242641 0.0475858
$$27$$ 1.00000 0.192450
$$28$$ −2.58579 −0.488668
$$29$$ −1.41421 −0.262613 −0.131306 0.991342i $$-0.541917\pi$$
−0.131306 + 0.991342i $$0.541917\pi$$
$$30$$ 0.414214 0.0756247
$$31$$ −8.82843 −1.58563 −0.792816 0.609461i $$-0.791386\pi$$
−0.792816 + 0.609461i $$0.791386\pi$$
$$32$$ −4.41421 −0.780330
$$33$$ 6.24264 1.08670
$$34$$ −2.82843 −0.485071
$$35$$ −1.41421 −0.239046
$$36$$ −1.82843 −0.304738
$$37$$ −0.585786 −0.0963027 −0.0481513 0.998840i $$-0.515333\pi$$
−0.0481513 + 0.998840i $$0.515333\pi$$
$$38$$ 0.414214 0.0671943
$$39$$ −0.585786 −0.0938009
$$40$$ −1.58579 −0.250735
$$41$$ 8.24264 1.28728 0.643642 0.765327i $$-0.277423\pi$$
0.643642 + 0.765327i $$0.277423\pi$$
$$42$$ −0.585786 −0.0903888
$$43$$ 3.75736 0.572992 0.286496 0.958081i $$-0.407509\pi$$
0.286496 + 0.958081i $$0.407509\pi$$
$$44$$ −11.4142 −1.72076
$$45$$ −1.00000 −0.149071
$$46$$ 1.51472 0.223333
$$47$$ 3.65685 0.533407 0.266704 0.963779i $$-0.414066\pi$$
0.266704 + 0.963779i $$0.414066\pi$$
$$48$$ 3.00000 0.433013
$$49$$ −5.00000 −0.714286
$$50$$ −0.414214 −0.0585786
$$51$$ 6.82843 0.956171
$$52$$ 1.07107 0.148530
$$53$$ 8.00000 1.09888 0.549442 0.835532i $$-0.314840\pi$$
0.549442 + 0.835532i $$0.314840\pi$$
$$54$$ −0.414214 −0.0563673
$$55$$ −6.24264 −0.841757
$$56$$ 2.24264 0.299685
$$57$$ −1.00000 −0.132453
$$58$$ 0.585786 0.0769175
$$59$$ −4.48528 −0.583934 −0.291967 0.956428i $$-0.594310\pi$$
−0.291967 + 0.956428i $$0.594310\pi$$
$$60$$ 1.82843 0.236049
$$61$$ −15.3137 −1.96072 −0.980360 0.197218i $$-0.936809\pi$$
−0.980360 + 0.197218i $$0.936809\pi$$
$$62$$ 3.65685 0.464421
$$63$$ 1.41421 0.178174
$$64$$ −4.17157 −0.521447
$$65$$ 0.585786 0.0726579
$$66$$ −2.58579 −0.318288
$$67$$ 1.65685 0.202417 0.101208 0.994865i $$-0.467729\pi$$
0.101208 + 0.994865i $$0.467729\pi$$
$$68$$ −12.4853 −1.51406
$$69$$ −3.65685 −0.440234
$$70$$ 0.585786 0.0700149
$$71$$ −5.17157 −0.613753 −0.306876 0.951749i $$-0.599284\pi$$
−0.306876 + 0.951749i $$0.599284\pi$$
$$72$$ 1.58579 0.186887
$$73$$ 3.65685 0.428002 0.214001 0.976833i $$-0.431350\pi$$
0.214001 + 0.976833i $$0.431350\pi$$
$$74$$ 0.242641 0.0282064
$$75$$ 1.00000 0.115470
$$76$$ 1.82843 0.209735
$$77$$ 8.82843 1.00609
$$78$$ 0.242641 0.0274736
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 1.00000 0.111111
$$82$$ −3.41421 −0.377037
$$83$$ 7.17157 0.787182 0.393591 0.919286i $$-0.371233\pi$$
0.393591 + 0.919286i $$0.371233\pi$$
$$84$$ −2.58579 −0.282132
$$85$$ −6.82843 −0.740647
$$86$$ −1.55635 −0.167825
$$87$$ −1.41421 −0.151620
$$88$$ 9.89949 1.05529
$$89$$ −13.8995 −1.47334 −0.736672 0.676250i $$-0.763604\pi$$
−0.736672 + 0.676250i $$0.763604\pi$$
$$90$$ 0.414214 0.0436619
$$91$$ −0.828427 −0.0868428
$$92$$ 6.68629 0.697094
$$93$$ −8.82843 −0.915465
$$94$$ −1.51472 −0.156231
$$95$$ 1.00000 0.102598
$$96$$ −4.41421 −0.450524
$$97$$ −18.2426 −1.85226 −0.926130 0.377205i $$-0.876885\pi$$
−0.926130 + 0.377205i $$0.876885\pi$$
$$98$$ 2.07107 0.209209
$$99$$ 6.24264 0.627409
$$100$$ −1.82843 −0.182843
$$101$$ −8.82843 −0.878461 −0.439231 0.898374i $$-0.644749\pi$$
−0.439231 + 0.898374i $$0.644749\pi$$
$$102$$ −2.82843 −0.280056
$$103$$ 15.3137 1.50890 0.754452 0.656355i $$-0.227903\pi$$
0.754452 + 0.656355i $$0.227903\pi$$
$$104$$ −0.928932 −0.0910893
$$105$$ −1.41421 −0.138013
$$106$$ −3.31371 −0.321856
$$107$$ −3.31371 −0.320348 −0.160174 0.987089i $$-0.551206\pi$$
−0.160174 + 0.987089i $$0.551206\pi$$
$$108$$ −1.82843 −0.175940
$$109$$ 10.4853 1.00431 0.502154 0.864778i $$-0.332541\pi$$
0.502154 + 0.864778i $$0.332541\pi$$
$$110$$ 2.58579 0.246545
$$111$$ −0.585786 −0.0556004
$$112$$ 4.24264 0.400892
$$113$$ −18.1421 −1.70667 −0.853334 0.521364i $$-0.825423\pi$$
−0.853334 + 0.521364i $$0.825423\pi$$
$$114$$ 0.414214 0.0387947
$$115$$ 3.65685 0.341003
$$116$$ 2.58579 0.240084
$$117$$ −0.585786 −0.0541560
$$118$$ 1.85786 0.171030
$$119$$ 9.65685 0.885242
$$120$$ −1.58579 −0.144762
$$121$$ 27.9706 2.54278
$$122$$ 6.34315 0.574281
$$123$$ 8.24264 0.743214
$$124$$ 16.1421 1.44961
$$125$$ −1.00000 −0.0894427
$$126$$ −0.585786 −0.0521860
$$127$$ −3.31371 −0.294044 −0.147022 0.989133i $$-0.546969\pi$$
−0.147022 + 0.989133i $$0.546969\pi$$
$$128$$ 10.5563 0.933058
$$129$$ 3.75736 0.330817
$$130$$ −0.242641 −0.0212810
$$131$$ −11.4142 −0.997264 −0.498632 0.866814i $$-0.666164\pi$$
−0.498632 + 0.866814i $$0.666164\pi$$
$$132$$ −11.4142 −0.993480
$$133$$ −1.41421 −0.122628
$$134$$ −0.686292 −0.0592866
$$135$$ −1.00000 −0.0860663
$$136$$ 10.8284 0.928530
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 1.51472 0.128941
$$139$$ 14.1421 1.19952 0.599760 0.800180i $$-0.295263\pi$$
0.599760 + 0.800180i $$0.295263\pi$$
$$140$$ 2.58579 0.218539
$$141$$ 3.65685 0.307963
$$142$$ 2.14214 0.179764
$$143$$ −3.65685 −0.305802
$$144$$ 3.00000 0.250000
$$145$$ 1.41421 0.117444
$$146$$ −1.51472 −0.125359
$$147$$ −5.00000 −0.412393
$$148$$ 1.07107 0.0880412
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ −0.414214 −0.0338204
$$151$$ 10.4853 0.853280 0.426640 0.904422i $$-0.359697\pi$$
0.426640 + 0.904422i $$0.359697\pi$$
$$152$$ −1.58579 −0.128624
$$153$$ 6.82843 0.552046
$$154$$ −3.65685 −0.294678
$$155$$ 8.82843 0.709116
$$156$$ 1.07107 0.0857541
$$157$$ −6.48528 −0.517582 −0.258791 0.965933i $$-0.583324\pi$$
−0.258791 + 0.965933i $$0.583324\pi$$
$$158$$ 0 0
$$159$$ 8.00000 0.634441
$$160$$ 4.41421 0.348974
$$161$$ −5.17157 −0.407577
$$162$$ −0.414214 −0.0325437
$$163$$ 2.10051 0.164524 0.0822621 0.996611i $$-0.473786\pi$$
0.0822621 + 0.996611i $$0.473786\pi$$
$$164$$ −15.0711 −1.17685
$$165$$ −6.24264 −0.485989
$$166$$ −2.97056 −0.230560
$$167$$ 5.31371 0.411187 0.205594 0.978637i $$-0.434088\pi$$
0.205594 + 0.978637i $$0.434088\pi$$
$$168$$ 2.24264 0.173023
$$169$$ −12.6569 −0.973604
$$170$$ 2.82843 0.216930
$$171$$ −1.00000 −0.0764719
$$172$$ −6.87006 −0.523837
$$173$$ 19.7990 1.50529 0.752645 0.658427i $$-0.228778\pi$$
0.752645 + 0.658427i $$0.228778\pi$$
$$174$$ 0.585786 0.0444084
$$175$$ 1.41421 0.106904
$$176$$ 18.7279 1.41167
$$177$$ −4.48528 −0.337134
$$178$$ 5.75736 0.431532
$$179$$ −0.485281 −0.0362716 −0.0181358 0.999836i $$-0.505773\pi$$
−0.0181358 + 0.999836i $$0.505773\pi$$
$$180$$ 1.82843 0.136283
$$181$$ −15.1716 −1.12769 −0.563847 0.825879i $$-0.690679\pi$$
−0.563847 + 0.825879i $$0.690679\pi$$
$$182$$ 0.343146 0.0254357
$$183$$ −15.3137 −1.13202
$$184$$ −5.79899 −0.427507
$$185$$ 0.585786 0.0430679
$$186$$ 3.65685 0.268134
$$187$$ 42.6274 3.11723
$$188$$ −6.68629 −0.487648
$$189$$ 1.41421 0.102869
$$190$$ −0.414214 −0.0300502
$$191$$ 1.75736 0.127158 0.0635790 0.997977i $$-0.479749\pi$$
0.0635790 + 0.997977i $$0.479749\pi$$
$$192$$ −4.17157 −0.301057
$$193$$ −9.07107 −0.652950 −0.326475 0.945206i $$-0.605861\pi$$
−0.326475 + 0.945206i $$0.605861\pi$$
$$194$$ 7.55635 0.542514
$$195$$ 0.585786 0.0419490
$$196$$ 9.14214 0.653010
$$197$$ 1.17157 0.0834711 0.0417356 0.999129i $$-0.486711\pi$$
0.0417356 + 0.999129i $$0.486711\pi$$
$$198$$ −2.58579 −0.183764
$$199$$ −10.1421 −0.718957 −0.359478 0.933153i $$-0.617045\pi$$
−0.359478 + 0.933153i $$0.617045\pi$$
$$200$$ 1.58579 0.112132
$$201$$ 1.65685 0.116865
$$202$$ 3.65685 0.257295
$$203$$ −2.00000 −0.140372
$$204$$ −12.4853 −0.874145
$$205$$ −8.24264 −0.575691
$$206$$ −6.34315 −0.441948
$$207$$ −3.65685 −0.254169
$$208$$ −1.75736 −0.121851
$$209$$ −6.24264 −0.431812
$$210$$ 0.585786 0.0404231
$$211$$ −15.3137 −1.05424 −0.527120 0.849791i $$-0.676728\pi$$
−0.527120 + 0.849791i $$0.676728\pi$$
$$212$$ −14.6274 −1.00462
$$213$$ −5.17157 −0.354350
$$214$$ 1.37258 0.0938278
$$215$$ −3.75736 −0.256250
$$216$$ 1.58579 0.107899
$$217$$ −12.4853 −0.847556
$$218$$ −4.34315 −0.294155
$$219$$ 3.65685 0.247107
$$220$$ 11.4142 0.769546
$$221$$ −4.00000 −0.269069
$$222$$ 0.242641 0.0162850
$$223$$ −26.6274 −1.78310 −0.891552 0.452919i $$-0.850383\pi$$
−0.891552 + 0.452919i $$0.850383\pi$$
$$224$$ −6.24264 −0.417104
$$225$$ 1.00000 0.0666667
$$226$$ 7.51472 0.499872
$$227$$ −18.9706 −1.25912 −0.629560 0.776952i $$-0.716765\pi$$
−0.629560 + 0.776952i $$0.716765\pi$$
$$228$$ 1.82843 0.121091
$$229$$ 22.6274 1.49526 0.747631 0.664114i $$-0.231191\pi$$
0.747631 + 0.664114i $$0.231191\pi$$
$$230$$ −1.51472 −0.0998776
$$231$$ 8.82843 0.580868
$$232$$ −2.24264 −0.147237
$$233$$ −11.6569 −0.763666 −0.381833 0.924231i $$-0.624707\pi$$
−0.381833 + 0.924231i $$0.624707\pi$$
$$234$$ 0.242641 0.0158619
$$235$$ −3.65685 −0.238547
$$236$$ 8.20101 0.533840
$$237$$ 0 0
$$238$$ −4.00000 −0.259281
$$239$$ −1.27208 −0.0822839 −0.0411419 0.999153i $$-0.513100\pi$$
−0.0411419 + 0.999153i $$0.513100\pi$$
$$240$$ −3.00000 −0.193649
$$241$$ −8.34315 −0.537429 −0.268715 0.963220i $$-0.586599\pi$$
−0.268715 + 0.963220i $$0.586599\pi$$
$$242$$ −11.5858 −0.744763
$$243$$ 1.00000 0.0641500
$$244$$ 28.0000 1.79252
$$245$$ 5.00000 0.319438
$$246$$ −3.41421 −0.217682
$$247$$ 0.585786 0.0372727
$$248$$ −14.0000 −0.889001
$$249$$ 7.17157 0.454480
$$250$$ 0.414214 0.0261972
$$251$$ −10.2426 −0.646510 −0.323255 0.946312i $$-0.604777\pi$$
−0.323255 + 0.946312i $$0.604777\pi$$
$$252$$ −2.58579 −0.162889
$$253$$ −22.8284 −1.43521
$$254$$ 1.37258 0.0861235
$$255$$ −6.82843 −0.427613
$$256$$ 3.97056 0.248160
$$257$$ 12.4853 0.778810 0.389405 0.921067i $$-0.372681\pi$$
0.389405 + 0.921067i $$0.372681\pi$$
$$258$$ −1.55635 −0.0968941
$$259$$ −0.828427 −0.0514760
$$260$$ −1.07107 −0.0664248
$$261$$ −1.41421 −0.0875376
$$262$$ 4.72792 0.292092
$$263$$ 27.4558 1.69300 0.846500 0.532389i $$-0.178706\pi$$
0.846500 + 0.532389i $$0.178706\pi$$
$$264$$ 9.89949 0.609272
$$265$$ −8.00000 −0.491436
$$266$$ 0.585786 0.0359169
$$267$$ −13.8995 −0.850635
$$268$$ −3.02944 −0.185052
$$269$$ 11.0711 0.675015 0.337507 0.941323i $$-0.390416\pi$$
0.337507 + 0.941323i $$0.390416\pi$$
$$270$$ 0.414214 0.0252082
$$271$$ −5.17157 −0.314151 −0.157075 0.987587i $$-0.550207\pi$$
−0.157075 + 0.987587i $$0.550207\pi$$
$$272$$ 20.4853 1.24210
$$273$$ −0.828427 −0.0501387
$$274$$ −4.14214 −0.250236
$$275$$ 6.24264 0.376445
$$276$$ 6.68629 0.402467
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ −5.85786 −0.351331
$$279$$ −8.82843 −0.528544
$$280$$ −2.24264 −0.134023
$$281$$ −17.4142 −1.03884 −0.519422 0.854518i $$-0.673853\pi$$
−0.519422 + 0.854518i $$0.673853\pi$$
$$282$$ −1.51472 −0.0902002
$$283$$ 26.3848 1.56841 0.784206 0.620500i $$-0.213071\pi$$
0.784206 + 0.620500i $$0.213071\pi$$
$$284$$ 9.45584 0.561101
$$285$$ 1.00000 0.0592349
$$286$$ 1.51472 0.0895672
$$287$$ 11.6569 0.688082
$$288$$ −4.41421 −0.260110
$$289$$ 29.6274 1.74279
$$290$$ −0.585786 −0.0343986
$$291$$ −18.2426 −1.06940
$$292$$ −6.68629 −0.391286
$$293$$ −28.4853 −1.66413 −0.832064 0.554680i $$-0.812841\pi$$
−0.832064 + 0.554680i $$0.812841\pi$$
$$294$$ 2.07107 0.120787
$$295$$ 4.48528 0.261143
$$296$$ −0.928932 −0.0539931
$$297$$ 6.24264 0.362235
$$298$$ 2.48528 0.143968
$$299$$ 2.14214 0.123883
$$300$$ −1.82843 −0.105564
$$301$$ 5.31371 0.306277
$$302$$ −4.34315 −0.249920
$$303$$ −8.82843 −0.507180
$$304$$ −3.00000 −0.172062
$$305$$ 15.3137 0.876860
$$306$$ −2.82843 −0.161690
$$307$$ 26.8284 1.53118 0.765590 0.643329i $$-0.222447\pi$$
0.765590 + 0.643329i $$0.222447\pi$$
$$308$$ −16.1421 −0.919784
$$309$$ 15.3137 0.871166
$$310$$ −3.65685 −0.207695
$$311$$ 2.24264 0.127168 0.0635842 0.997976i $$-0.479747\pi$$
0.0635842 + 0.997976i $$0.479747\pi$$
$$312$$ −0.928932 −0.0525904
$$313$$ −33.7990 −1.91043 −0.955216 0.295910i $$-0.904377\pi$$
−0.955216 + 0.295910i $$0.904377\pi$$
$$314$$ 2.68629 0.151596
$$315$$ −1.41421 −0.0796819
$$316$$ 0 0
$$317$$ 18.6274 1.04622 0.523110 0.852265i $$-0.324772\pi$$
0.523110 + 0.852265i $$0.324772\pi$$
$$318$$ −3.31371 −0.185824
$$319$$ −8.82843 −0.494297
$$320$$ 4.17157 0.233198
$$321$$ −3.31371 −0.184953
$$322$$ 2.14214 0.119377
$$323$$ −6.82843 −0.379944
$$324$$ −1.82843 −0.101579
$$325$$ −0.585786 −0.0324936
$$326$$ −0.870058 −0.0481880
$$327$$ 10.4853 0.579837
$$328$$ 13.0711 0.721729
$$329$$ 5.17157 0.285118
$$330$$ 2.58579 0.142343
$$331$$ −0.142136 −0.00781248 −0.00390624 0.999992i $$-0.501243\pi$$
−0.00390624 + 0.999992i $$0.501243\pi$$
$$332$$ −13.1127 −0.719653
$$333$$ −0.585786 −0.0321009
$$334$$ −2.20101 −0.120434
$$335$$ −1.65685 −0.0905236
$$336$$ 4.24264 0.231455
$$337$$ −27.4142 −1.49335 −0.746674 0.665191i $$-0.768350\pi$$
−0.746674 + 0.665191i $$0.768350\pi$$
$$338$$ 5.24264 0.285162
$$339$$ −18.1421 −0.985346
$$340$$ 12.4853 0.677109
$$341$$ −55.1127 −2.98452
$$342$$ 0.414214 0.0223981
$$343$$ −16.9706 −0.916324
$$344$$ 5.95837 0.321254
$$345$$ 3.65685 0.196878
$$346$$ −8.20101 −0.440889
$$347$$ 23.4558 1.25918 0.629588 0.776929i $$-0.283224\pi$$
0.629588 + 0.776929i $$0.283224\pi$$
$$348$$ 2.58579 0.138613
$$349$$ 18.0000 0.963518 0.481759 0.876304i $$-0.339998\pi$$
0.481759 + 0.876304i $$0.339998\pi$$
$$350$$ −0.585786 −0.0313116
$$351$$ −0.585786 −0.0312670
$$352$$ −27.5563 −1.46876
$$353$$ −7.65685 −0.407533 −0.203767 0.979019i $$-0.565318\pi$$
−0.203767 + 0.979019i $$0.565318\pi$$
$$354$$ 1.85786 0.0987444
$$355$$ 5.17157 0.274479
$$356$$ 25.4142 1.34695
$$357$$ 9.65685 0.511095
$$358$$ 0.201010 0.0106237
$$359$$ −28.8701 −1.52370 −0.761852 0.647752i $$-0.775710\pi$$
−0.761852 + 0.647752i $$0.775710\pi$$
$$360$$ −1.58579 −0.0835783
$$361$$ 1.00000 0.0526316
$$362$$ 6.28427 0.330294
$$363$$ 27.9706 1.46807
$$364$$ 1.51472 0.0793928
$$365$$ −3.65685 −0.191408
$$366$$ 6.34315 0.331562
$$367$$ 22.3848 1.16848 0.584238 0.811582i $$-0.301394\pi$$
0.584238 + 0.811582i $$0.301394\pi$$
$$368$$ −10.9706 −0.571880
$$369$$ 8.24264 0.429095
$$370$$ −0.242641 −0.0126143
$$371$$ 11.3137 0.587378
$$372$$ 16.1421 0.836931
$$373$$ 3.41421 0.176781 0.0883906 0.996086i $$-0.471828\pi$$
0.0883906 + 0.996086i $$0.471828\pi$$
$$374$$ −17.6569 −0.913014
$$375$$ −1.00000 −0.0516398
$$376$$ 5.79899 0.299060
$$377$$ 0.828427 0.0426662
$$378$$ −0.585786 −0.0301296
$$379$$ 8.82843 0.453486 0.226743 0.973955i $$-0.427192\pi$$
0.226743 + 0.973955i $$0.427192\pi$$
$$380$$ −1.82843 −0.0937963
$$381$$ −3.31371 −0.169766
$$382$$ −0.727922 −0.0372437
$$383$$ 28.0000 1.43073 0.715367 0.698749i $$-0.246260\pi$$
0.715367 + 0.698749i $$0.246260\pi$$
$$384$$ 10.5563 0.538701
$$385$$ −8.82843 −0.449938
$$386$$ 3.75736 0.191245
$$387$$ 3.75736 0.190997
$$388$$ 33.3553 1.69336
$$389$$ −2.97056 −0.150614 −0.0753068 0.997160i $$-0.523994\pi$$
−0.0753068 + 0.997160i $$0.523994\pi$$
$$390$$ −0.242641 −0.0122866
$$391$$ −24.9706 −1.26282
$$392$$ −7.92893 −0.400472
$$393$$ −11.4142 −0.575771
$$394$$ −0.485281 −0.0244481
$$395$$ 0 0
$$396$$ −11.4142 −0.573586
$$397$$ −16.6274 −0.834506 −0.417253 0.908790i $$-0.637007\pi$$
−0.417253 + 0.908790i $$0.637007\pi$$
$$398$$ 4.20101 0.210578
$$399$$ −1.41421 −0.0707992
$$400$$ 3.00000 0.150000
$$401$$ −16.2426 −0.811119 −0.405559 0.914069i $$-0.632923\pi$$
−0.405559 + 0.914069i $$0.632923\pi$$
$$402$$ −0.686292 −0.0342291
$$403$$ 5.17157 0.257614
$$404$$ 16.1421 0.803101
$$405$$ −1.00000 −0.0496904
$$406$$ 0.828427 0.0411141
$$407$$ −3.65685 −0.181264
$$408$$ 10.8284 0.536087
$$409$$ −7.17157 −0.354611 −0.177306 0.984156i $$-0.556738\pi$$
−0.177306 + 0.984156i $$0.556738\pi$$
$$410$$ 3.41421 0.168616
$$411$$ 10.0000 0.493264
$$412$$ −28.0000 −1.37946
$$413$$ −6.34315 −0.312126
$$414$$ 1.51472 0.0744444
$$415$$ −7.17157 −0.352039
$$416$$ 2.58579 0.126779
$$417$$ 14.1421 0.692543
$$418$$ 2.58579 0.126475
$$419$$ 0.585786 0.0286175 0.0143088 0.999898i $$-0.495445\pi$$
0.0143088 + 0.999898i $$0.495445\pi$$
$$420$$ 2.58579 0.126173
$$421$$ −13.3137 −0.648870 −0.324435 0.945908i $$-0.605174\pi$$
−0.324435 + 0.945908i $$0.605174\pi$$
$$422$$ 6.34315 0.308780
$$423$$ 3.65685 0.177802
$$424$$ 12.6863 0.616101
$$425$$ 6.82843 0.331227
$$426$$ 2.14214 0.103787
$$427$$ −21.6569 −1.04805
$$428$$ 6.05887 0.292867
$$429$$ −3.65685 −0.176555
$$430$$ 1.55635 0.0750538
$$431$$ 31.1127 1.49865 0.749323 0.662205i $$-0.230379\pi$$
0.749323 + 0.662205i $$0.230379\pi$$
$$432$$ 3.00000 0.144338
$$433$$ −25.0711 −1.20484 −0.602419 0.798180i $$-0.705796\pi$$
−0.602419 + 0.798180i $$0.705796\pi$$
$$434$$ 5.17157 0.248243
$$435$$ 1.41421 0.0678064
$$436$$ −19.1716 −0.918152
$$437$$ 3.65685 0.174931
$$438$$ −1.51472 −0.0723761
$$439$$ −10.3431 −0.493651 −0.246826 0.969060i $$-0.579388\pi$$
−0.246826 + 0.969060i $$0.579388\pi$$
$$440$$ −9.89949 −0.471940
$$441$$ −5.00000 −0.238095
$$442$$ 1.65685 0.0788085
$$443$$ 18.0000 0.855206 0.427603 0.903967i $$-0.359358\pi$$
0.427603 + 0.903967i $$0.359358\pi$$
$$444$$ 1.07107 0.0508306
$$445$$ 13.8995 0.658899
$$446$$ 11.0294 0.522259
$$447$$ −6.00000 −0.283790
$$448$$ −5.89949 −0.278725
$$449$$ 26.8701 1.26808 0.634038 0.773302i $$-0.281396\pi$$
0.634038 + 0.773302i $$0.281396\pi$$
$$450$$ −0.414214 −0.0195262
$$451$$ 51.4558 2.42296
$$452$$ 33.1716 1.56026
$$453$$ 10.4853 0.492641
$$454$$ 7.85786 0.368788
$$455$$ 0.828427 0.0388373
$$456$$ −1.58579 −0.0742613
$$457$$ 27.1716 1.27103 0.635516 0.772087i $$-0.280787\pi$$
0.635516 + 0.772087i $$0.280787\pi$$
$$458$$ −9.37258 −0.437952
$$459$$ 6.82843 0.318724
$$460$$ −6.68629 −0.311750
$$461$$ 8.34315 0.388579 0.194290 0.980944i $$-0.437760\pi$$
0.194290 + 0.980944i $$0.437760\pi$$
$$462$$ −3.65685 −0.170132
$$463$$ 15.7574 0.732307 0.366153 0.930555i $$-0.380675\pi$$
0.366153 + 0.930555i $$0.380675\pi$$
$$464$$ −4.24264 −0.196960
$$465$$ 8.82843 0.409409
$$466$$ 4.82843 0.223673
$$467$$ 24.3431 1.12647 0.563233 0.826298i $$-0.309557\pi$$
0.563233 + 0.826298i $$0.309557\pi$$
$$468$$ 1.07107 0.0495101
$$469$$ 2.34315 0.108196
$$470$$ 1.51472 0.0698688
$$471$$ −6.48528 −0.298826
$$472$$ −7.11270 −0.327388
$$473$$ 23.4558 1.07850
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ −17.6569 −0.809301
$$477$$ 8.00000 0.366295
$$478$$ 0.526912 0.0241004
$$479$$ 2.92893 0.133826 0.0669132 0.997759i $$-0.478685\pi$$
0.0669132 + 0.997759i $$0.478685\pi$$
$$480$$ 4.41421 0.201480
$$481$$ 0.343146 0.0156461
$$482$$ 3.45584 0.157409
$$483$$ −5.17157 −0.235315
$$484$$ −51.1421 −2.32464
$$485$$ 18.2426 0.828356
$$486$$ −0.414214 −0.0187891
$$487$$ 3.51472 0.159267 0.0796336 0.996824i $$-0.474625\pi$$
0.0796336 + 0.996824i $$0.474625\pi$$
$$488$$ −24.2843 −1.09930
$$489$$ 2.10051 0.0949881
$$490$$ −2.07107 −0.0935613
$$491$$ −10.2426 −0.462244 −0.231122 0.972925i $$-0.574240\pi$$
−0.231122 + 0.972925i $$0.574240\pi$$
$$492$$ −15.0711 −0.679456
$$493$$ −9.65685 −0.434923
$$494$$ −0.242641 −0.0109169
$$495$$ −6.24264 −0.280586
$$496$$ −26.4853 −1.18922
$$497$$ −7.31371 −0.328065
$$498$$ −2.97056 −0.133114
$$499$$ −10.8284 −0.484747 −0.242373 0.970183i $$-0.577926\pi$$
−0.242373 + 0.970183i $$0.577926\pi$$
$$500$$ 1.82843 0.0817697
$$501$$ 5.31371 0.237399
$$502$$ 4.24264 0.189358
$$503$$ −0.828427 −0.0369377 −0.0184689 0.999829i $$-0.505879\pi$$
−0.0184689 + 0.999829i $$0.505879\pi$$
$$504$$ 2.24264 0.0998952
$$505$$ 8.82843 0.392860
$$506$$ 9.45584 0.420364
$$507$$ −12.6569 −0.562111
$$508$$ 6.05887 0.268819
$$509$$ 24.7279 1.09605 0.548023 0.836463i $$-0.315381\pi$$
0.548023 + 0.836463i $$0.315381\pi$$
$$510$$ 2.82843 0.125245
$$511$$ 5.17157 0.228777
$$512$$ −22.7574 −1.00574
$$513$$ −1.00000 −0.0441511
$$514$$ −5.17157 −0.228108
$$515$$ −15.3137 −0.674803
$$516$$ −6.87006 −0.302437
$$517$$ 22.8284 1.00399
$$518$$ 0.343146 0.0150770
$$519$$ 19.7990 0.869079
$$520$$ 0.928932 0.0407364
$$521$$ 16.2426 0.711603 0.355802 0.934562i $$-0.384208\pi$$
0.355802 + 0.934562i $$0.384208\pi$$
$$522$$ 0.585786 0.0256392
$$523$$ −32.4853 −1.42048 −0.710241 0.703959i $$-0.751414\pi$$
−0.710241 + 0.703959i $$0.751414\pi$$
$$524$$ 20.8701 0.911713
$$525$$ 1.41421 0.0617213
$$526$$ −11.3726 −0.495868
$$527$$ −60.2843 −2.62602
$$528$$ 18.7279 0.815028
$$529$$ −9.62742 −0.418583
$$530$$ 3.31371 0.143938
$$531$$ −4.48528 −0.194645
$$532$$ 2.58579 0.112108
$$533$$ −4.82843 −0.209142
$$534$$ 5.75736 0.249145
$$535$$ 3.31371 0.143264
$$536$$ 2.62742 0.113487
$$537$$ −0.485281 −0.0209414
$$538$$ −4.58579 −0.197707
$$539$$ −31.2132 −1.34445
$$540$$ 1.82843 0.0786830
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 2.14214 0.0920126
$$543$$ −15.1716 −0.651075
$$544$$ −30.1421 −1.29233
$$545$$ −10.4853 −0.449140
$$546$$ 0.343146 0.0146853
$$547$$ 7.51472 0.321306 0.160653 0.987011i $$-0.448640\pi$$
0.160653 + 0.987011i $$0.448640\pi$$
$$548$$ −18.2843 −0.781065
$$549$$ −15.3137 −0.653573
$$550$$ −2.58579 −0.110258
$$551$$ 1.41421 0.0602475
$$552$$ −5.79899 −0.246821
$$553$$ 0 0
$$554$$ 9.11270 0.387161
$$555$$ 0.585786 0.0248652
$$556$$ −25.8579 −1.09662
$$557$$ −2.68629 −0.113822 −0.0569109 0.998379i $$-0.518125\pi$$
−0.0569109 + 0.998379i $$0.518125\pi$$
$$558$$ 3.65685 0.154807
$$559$$ −2.20101 −0.0930928
$$560$$ −4.24264 −0.179284
$$561$$ 42.6274 1.79973
$$562$$ 7.21320 0.304271
$$563$$ 26.2843 1.10775 0.553875 0.832600i $$-0.313149\pi$$
0.553875 + 0.832600i $$0.313149\pi$$
$$564$$ −6.68629 −0.281544
$$565$$ 18.1421 0.763245
$$566$$ −10.9289 −0.459377
$$567$$ 1.41421 0.0593914
$$568$$ −8.20101 −0.344107
$$569$$ 16.9289 0.709698 0.354849 0.934924i $$-0.384532\pi$$
0.354849 + 0.934924i $$0.384532\pi$$
$$570$$ −0.414214 −0.0173495
$$571$$ 14.8284 0.620550 0.310275 0.950647i $$-0.399579\pi$$
0.310275 + 0.950647i $$0.399579\pi$$
$$572$$ 6.68629 0.279568
$$573$$ 1.75736 0.0734147
$$574$$ −4.82843 −0.201535
$$575$$ −3.65685 −0.152501
$$576$$ −4.17157 −0.173816
$$577$$ −31.4558 −1.30952 −0.654762 0.755835i $$-0.727231\pi$$
−0.654762 + 0.755835i $$0.727231\pi$$
$$578$$ −12.2721 −0.510451
$$579$$ −9.07107 −0.376981
$$580$$ −2.58579 −0.107369
$$581$$ 10.1421 0.420767
$$582$$ 7.55635 0.313221
$$583$$ 49.9411 2.06835
$$584$$ 5.79899 0.239964
$$585$$ 0.585786 0.0242193
$$586$$ 11.7990 0.487412
$$587$$ −23.6569 −0.976423 −0.488211 0.872725i $$-0.662351\pi$$
−0.488211 + 0.872725i $$0.662351\pi$$
$$588$$ 9.14214 0.377015
$$589$$ 8.82843 0.363769
$$590$$ −1.85786 −0.0764871
$$591$$ 1.17157 0.0481921
$$592$$ −1.75736 −0.0722270
$$593$$ −8.62742 −0.354286 −0.177143 0.984185i $$-0.556685\pi$$
−0.177143 + 0.984185i $$0.556685\pi$$
$$594$$ −2.58579 −0.106096
$$595$$ −9.65685 −0.395892
$$596$$ 10.9706 0.449372
$$597$$ −10.1421 −0.415090
$$598$$ −0.887302 −0.0362845
$$599$$ −36.9706 −1.51058 −0.755288 0.655393i $$-0.772503\pi$$
−0.755288 + 0.655393i $$0.772503\pi$$
$$600$$ 1.58579 0.0647395
$$601$$ −20.1421 −0.821615 −0.410807 0.911722i $$-0.634753\pi$$
−0.410807 + 0.911722i $$0.634753\pi$$
$$602$$ −2.20101 −0.0897065
$$603$$ 1.65685 0.0674723
$$604$$ −19.1716 −0.780080
$$605$$ −27.9706 −1.13717
$$606$$ 3.65685 0.148550
$$607$$ −0.485281 −0.0196970 −0.00984848 0.999952i $$-0.503135\pi$$
−0.00984848 + 0.999952i $$0.503135\pi$$
$$608$$ 4.41421 0.179020
$$609$$ −2.00000 −0.0810441
$$610$$ −6.34315 −0.256826
$$611$$ −2.14214 −0.0866615
$$612$$ −12.4853 −0.504688
$$613$$ 6.48528 0.261938 0.130969 0.991386i $$-0.458191\pi$$
0.130969 + 0.991386i $$0.458191\pi$$
$$614$$ −11.1127 −0.448472
$$615$$ −8.24264 −0.332375
$$616$$ 14.0000 0.564076
$$617$$ 11.5147 0.463565 0.231783 0.972768i $$-0.425544\pi$$
0.231783 + 0.972768i $$0.425544\pi$$
$$618$$ −6.34315 −0.255159
$$619$$ −3.51472 −0.141268 −0.0706342 0.997502i $$-0.522502\pi$$
−0.0706342 + 0.997502i $$0.522502\pi$$
$$620$$ −16.1421 −0.648284
$$621$$ −3.65685 −0.146745
$$622$$ −0.928932 −0.0372468
$$623$$ −19.6569 −0.787535
$$624$$ −1.75736 −0.0703507
$$625$$ 1.00000 0.0400000
$$626$$ 14.0000 0.559553
$$627$$ −6.24264 −0.249307
$$628$$ 11.8579 0.473180
$$629$$ −4.00000 −0.159490
$$630$$ 0.585786 0.0233383
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ −15.3137 −0.608665
$$634$$ −7.71573 −0.306431
$$635$$ 3.31371 0.131501
$$636$$ −14.6274 −0.580015
$$637$$ 2.92893 0.116049
$$638$$ 3.65685 0.144776
$$639$$ −5.17157 −0.204584
$$640$$ −10.5563 −0.417276
$$641$$ 39.3553 1.55444 0.777221 0.629227i $$-0.216629\pi$$
0.777221 + 0.629227i $$0.216629\pi$$
$$642$$ 1.37258 0.0541715
$$643$$ 5.61522 0.221443 0.110721 0.993851i $$-0.464684\pi$$
0.110721 + 0.993851i $$0.464684\pi$$
$$644$$ 9.45584 0.372612
$$645$$ −3.75736 −0.147946
$$646$$ 2.82843 0.111283
$$647$$ −7.85786 −0.308925 −0.154462 0.987999i $$-0.549365\pi$$
−0.154462 + 0.987999i $$0.549365\pi$$
$$648$$ 1.58579 0.0622956
$$649$$ −28.0000 −1.09910
$$650$$ 0.242641 0.00951715
$$651$$ −12.4853 −0.489337
$$652$$ −3.84062 −0.150410
$$653$$ 37.1716 1.45464 0.727318 0.686301i $$-0.240767\pi$$
0.727318 + 0.686301i $$0.240767\pi$$
$$654$$ −4.34315 −0.169830
$$655$$ 11.4142 0.445990
$$656$$ 24.7279 0.965463
$$657$$ 3.65685 0.142667
$$658$$ −2.14214 −0.0835091
$$659$$ 26.6274 1.03726 0.518628 0.855000i $$-0.326443\pi$$
0.518628 + 0.855000i $$0.326443\pi$$
$$660$$ 11.4142 0.444298
$$661$$ −43.4558 −1.69024 −0.845118 0.534579i $$-0.820470\pi$$
−0.845118 + 0.534579i $$0.820470\pi$$
$$662$$ 0.0588745 0.00228822
$$663$$ −4.00000 −0.155347
$$664$$ 11.3726 0.441342
$$665$$ 1.41421 0.0548408
$$666$$ 0.242641 0.00940214
$$667$$ 5.17157 0.200244
$$668$$ −9.71573 −0.375913
$$669$$ −26.6274 −1.02948
$$670$$ 0.686292 0.0265138
$$671$$ −95.5980 −3.69052
$$672$$ −6.24264 −0.240815
$$673$$ 32.1838 1.24059 0.620297 0.784367i $$-0.287012\pi$$
0.620297 + 0.784367i $$0.287012\pi$$
$$674$$ 11.3553 0.437391
$$675$$ 1.00000 0.0384900
$$676$$ 23.1421 0.890082
$$677$$ −16.9706 −0.652232 −0.326116 0.945330i $$-0.605740\pi$$
−0.326116 + 0.945330i $$0.605740\pi$$
$$678$$ 7.51472 0.288601
$$679$$ −25.7990 −0.990074
$$680$$ −10.8284 −0.415251
$$681$$ −18.9706 −0.726954
$$682$$ 22.8284 0.874146
$$683$$ 29.6569 1.13479 0.567394 0.823446i $$-0.307952\pi$$
0.567394 + 0.823446i $$0.307952\pi$$
$$684$$ 1.82843 0.0699117
$$685$$ −10.0000 −0.382080
$$686$$ 7.02944 0.268385
$$687$$ 22.6274 0.863290
$$688$$ 11.2721 0.429744
$$689$$ −4.68629 −0.178533
$$690$$ −1.51472 −0.0576644
$$691$$ 41.1716 1.56624 0.783120 0.621870i $$-0.213627\pi$$
0.783120 + 0.621870i $$0.213627\pi$$
$$692$$ −36.2010 −1.37616
$$693$$ 8.82843 0.335364
$$694$$ −9.71573 −0.368804
$$695$$ −14.1421 −0.536442
$$696$$ −2.24264 −0.0850071
$$697$$ 56.2843 2.13192
$$698$$ −7.45584 −0.282208
$$699$$ −11.6569 −0.440903
$$700$$ −2.58579 −0.0977335
$$701$$ 29.3137 1.10716 0.553582 0.832795i $$-0.313261\pi$$
0.553582 + 0.832795i $$0.313261\pi$$
$$702$$ 0.242641 0.00915788
$$703$$ 0.585786 0.0220934
$$704$$ −26.0416 −0.981481
$$705$$ −3.65685 −0.137725
$$706$$ 3.17157 0.119364
$$707$$ −12.4853 −0.469557
$$708$$ 8.20101 0.308213
$$709$$ −4.97056 −0.186673 −0.0933367 0.995635i $$-0.529753\pi$$
−0.0933367 + 0.995635i $$0.529753\pi$$
$$710$$ −2.14214 −0.0803929
$$711$$ 0 0
$$712$$ −22.0416 −0.826045
$$713$$ 32.2843 1.20906
$$714$$ −4.00000 −0.149696
$$715$$ 3.65685 0.136759
$$716$$ 0.887302 0.0331600
$$717$$ −1.27208 −0.0475066
$$718$$ 11.9584 0.446282
$$719$$ 13.0711 0.487469 0.243734 0.969842i $$-0.421628\pi$$
0.243734 + 0.969842i $$0.421628\pi$$
$$720$$ −3.00000 −0.111803
$$721$$ 21.6569 0.806543
$$722$$ −0.414214 −0.0154154
$$723$$ −8.34315 −0.310285
$$724$$ 27.7401 1.03095
$$725$$ −1.41421 −0.0525226
$$726$$ −11.5858 −0.429989
$$727$$ −23.3553 −0.866202 −0.433101 0.901345i $$-0.642581\pi$$
−0.433101 + 0.901345i $$0.642581\pi$$
$$728$$ −1.31371 −0.0486893
$$729$$ 1.00000 0.0370370
$$730$$ 1.51472 0.0560623
$$731$$ 25.6569 0.948953
$$732$$ 28.0000 1.03491
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ −9.27208 −0.342239
$$735$$ 5.00000 0.184428
$$736$$ 16.1421 0.595007
$$737$$ 10.3431 0.380995
$$738$$ −3.41421 −0.125679
$$739$$ 25.6569 0.943803 0.471901 0.881651i $$-0.343568\pi$$
0.471901 + 0.881651i $$0.343568\pi$$
$$740$$ −1.07107 −0.0393732
$$741$$ 0.585786 0.0215194
$$742$$ −4.68629 −0.172039
$$743$$ −4.00000 −0.146746 −0.0733729 0.997305i $$-0.523376\pi$$
−0.0733729 + 0.997305i $$0.523376\pi$$
$$744$$ −14.0000 −0.513265
$$745$$ 6.00000 0.219823
$$746$$ −1.41421 −0.0517780
$$747$$ 7.17157 0.262394
$$748$$ −77.9411 −2.84981
$$749$$ −4.68629 −0.171233
$$750$$ 0.414214 0.0151249
$$751$$ 4.14214 0.151149 0.0755743 0.997140i $$-0.475921\pi$$
0.0755743 + 0.997140i $$0.475921\pi$$
$$752$$ 10.9706 0.400055
$$753$$ −10.2426 −0.373263
$$754$$ −0.343146 −0.0124966
$$755$$ −10.4853 −0.381598
$$756$$ −2.58579 −0.0940441
$$757$$ 52.4264 1.90547 0.952735 0.303802i $$-0.0982562\pi$$
0.952735 + 0.303802i $$0.0982562\pi$$
$$758$$ −3.65685 −0.132823
$$759$$ −22.8284 −0.828619
$$760$$ 1.58579 0.0575225
$$761$$ −23.9411 −0.867865 −0.433933 0.900945i $$-0.642874\pi$$
−0.433933 + 0.900945i $$0.642874\pi$$
$$762$$ 1.37258 0.0497234
$$763$$ 14.8284 0.536825
$$764$$ −3.21320 −0.116250
$$765$$ −6.82843 −0.246882
$$766$$ −11.5980 −0.419052
$$767$$ 2.62742 0.0948705
$$768$$ 3.97056 0.143275
$$769$$ 9.02944 0.325610 0.162805 0.986658i $$-0.447946\pi$$
0.162805 + 0.986658i $$0.447946\pi$$
$$770$$ 3.65685 0.131784
$$771$$ 12.4853 0.449646
$$772$$ 16.5858 0.596936
$$773$$ −14.3431 −0.515887 −0.257944 0.966160i $$-0.583045\pi$$
−0.257944 + 0.966160i $$0.583045\pi$$
$$774$$ −1.55635 −0.0559418
$$775$$ −8.82843 −0.317126
$$776$$ −28.9289 −1.03849
$$777$$ −0.828427 −0.0297197
$$778$$ 1.23045 0.0441137
$$779$$ −8.24264 −0.295323
$$780$$ −1.07107 −0.0383504
$$781$$ −32.2843 −1.15522
$$782$$ 10.3431 0.369870
$$783$$ −1.41421 −0.0505399
$$784$$ −15.0000 −0.535714
$$785$$ 6.48528 0.231470
$$786$$ 4.72792 0.168639
$$787$$ −37.1716 −1.32502 −0.662512 0.749052i $$-0.730510\pi$$
−0.662512 + 0.749052i $$0.730510\pi$$
$$788$$ −2.14214 −0.0763104
$$789$$ 27.4558 0.977454
$$790$$ 0 0
$$791$$ −25.6569 −0.912253
$$792$$ 9.89949 0.351763
$$793$$ 8.97056 0.318554
$$794$$ 6.88730 0.244421
$$795$$ −8.00000 −0.283731
$$796$$ 18.5442 0.657280
$$797$$ 14.1421 0.500940 0.250470 0.968124i $$-0.419415\pi$$
0.250470 + 0.968124i $$0.419415\pi$$
$$798$$ 0.585786 0.0207366
$$799$$ 24.9706 0.883395
$$800$$ −4.41421 −0.156066
$$801$$ −13.8995 −0.491115
$$802$$ 6.72792 0.237571
$$803$$ 22.8284 0.805598
$$804$$ −3.02944 −0.106840
$$805$$ 5.17157 0.182274
$$806$$ −2.14214 −0.0754535
$$807$$ 11.0711 0.389720
$$808$$ −14.0000 −0.492518
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0.414214 0.0145540
$$811$$ 31.3137 1.09957 0.549787 0.835305i $$-0.314709\pi$$
0.549787 + 0.835305i $$0.314709\pi$$
$$812$$ 3.65685 0.128330
$$813$$ −5.17157 −0.181375
$$814$$ 1.51472 0.0530909
$$815$$ −2.10051 −0.0735775
$$816$$ 20.4853 0.717128
$$817$$ −3.75736 −0.131453
$$818$$ 2.97056 0.103863
$$819$$ −0.828427 −0.0289476
$$820$$ 15.0711 0.526305
$$821$$ −2.48528 −0.0867369 −0.0433685 0.999059i $$-0.513809\pi$$
−0.0433685 + 0.999059i $$0.513809\pi$$
$$822$$ −4.14214 −0.144474
$$823$$ 57.0122 1.98732 0.993660 0.112426i $$-0.0358622\pi$$
0.993660 + 0.112426i $$0.0358622\pi$$
$$824$$ 24.2843 0.845983
$$825$$ 6.24264 0.217341
$$826$$ 2.62742 0.0914195
$$827$$ −25.3137 −0.880244 −0.440122 0.897938i $$-0.645065\pi$$
−0.440122 + 0.897938i $$0.645065\pi$$
$$828$$ 6.68629 0.232365
$$829$$ 29.5147 1.02509 0.512544 0.858661i $$-0.328703\pi$$
0.512544 + 0.858661i $$0.328703\pi$$
$$830$$ 2.97056 0.103110
$$831$$ −22.0000 −0.763172
$$832$$ 2.44365 0.0847183
$$833$$ −34.1421 −1.18295
$$834$$ −5.85786 −0.202841
$$835$$ −5.31371 −0.183888
$$836$$ 11.4142 0.394769
$$837$$ −8.82843 −0.305155
$$838$$ −0.242641 −0.00838188
$$839$$ −38.1421 −1.31681 −0.658406 0.752663i $$-0.728769\pi$$
−0.658406 + 0.752663i $$0.728769\pi$$
$$840$$ −2.24264 −0.0773785
$$841$$ −27.0000 −0.931034
$$842$$ 5.51472 0.190050
$$843$$ −17.4142 −0.599777
$$844$$ 28.0000 0.963800
$$845$$ 12.6569 0.435409
$$846$$ −1.51472 −0.0520771
$$847$$ 39.5563 1.35917
$$848$$ 24.0000 0.824163
$$849$$ 26.3848 0.905523
$$850$$ −2.82843 −0.0970143
$$851$$ 2.14214 0.0734315
$$852$$ 9.45584 0.323952
$$853$$ −11.1716 −0.382507 −0.191254 0.981541i $$-0.561255\pi$$
−0.191254 + 0.981541i $$0.561255\pi$$
$$854$$ 8.97056 0.306966
$$855$$ 1.00000 0.0341993
$$856$$ −5.25483 −0.179607
$$857$$ 35.3137 1.20629 0.603147 0.797630i $$-0.293913\pi$$
0.603147 + 0.797630i $$0.293913\pi$$
$$858$$ 1.51472 0.0517116
$$859$$ 17.9411 0.612143 0.306072 0.952008i $$-0.400985\pi$$
0.306072 + 0.952008i $$0.400985\pi$$
$$860$$ 6.87006 0.234267
$$861$$ 11.6569 0.397265
$$862$$ −12.8873 −0.438943
$$863$$ 31.3137 1.06593 0.532966 0.846137i $$-0.321078\pi$$
0.532966 + 0.846137i $$0.321078\pi$$
$$864$$ −4.41421 −0.150175
$$865$$ −19.7990 −0.673186
$$866$$ 10.3848 0.352889
$$867$$ 29.6274 1.00620
$$868$$ 22.8284 0.774847
$$869$$ 0 0
$$870$$ −0.585786 −0.0198600
$$871$$ −0.970563 −0.0328863
$$872$$ 16.6274 0.563075
$$873$$ −18.2426 −0.617420
$$874$$ −1.51472 −0.0512361
$$875$$ −1.41421 −0.0478091
$$876$$ −6.68629 −0.225909
$$877$$ −49.0711 −1.65701 −0.828506 0.559980i $$-0.810809\pi$$
−0.828506 + 0.559980i $$0.810809\pi$$
$$878$$ 4.28427 0.144587
$$879$$ −28.4853 −0.960785
$$880$$ −18.7279 −0.631318
$$881$$ 33.7990 1.13872 0.569358 0.822089i $$-0.307192\pi$$
0.569358 + 0.822089i $$0.307192\pi$$
$$882$$ 2.07107 0.0697365
$$883$$ 36.0416 1.21290 0.606449 0.795123i $$-0.292594\pi$$
0.606449 + 0.795123i $$0.292594\pi$$
$$884$$ 7.31371 0.245987
$$885$$ 4.48528 0.150771
$$886$$ −7.45584 −0.250484
$$887$$ −41.9411 −1.40825 −0.704123 0.710078i $$-0.748659\pi$$
−0.704123 + 0.710078i $$0.748659\pi$$
$$888$$ −0.928932 −0.0311729
$$889$$ −4.68629 −0.157173
$$890$$ −5.75736 −0.192987
$$891$$ 6.24264 0.209136
$$892$$ 48.6863 1.63014
$$893$$ −3.65685 −0.122372
$$894$$ 2.48528 0.0831202
$$895$$ 0.485281 0.0162212
$$896$$ 14.9289 0.498741
$$897$$ 2.14214 0.0715238
$$898$$ −11.1299 −0.371411
$$899$$ 12.4853 0.416407
$$900$$ −1.82843 −0.0609476
$$901$$ 54.6274 1.81990
$$902$$ −21.3137 −0.709669
$$903$$ 5.31371 0.176829
$$904$$ −28.7696 −0.956861
$$905$$ 15.1716 0.504320
$$906$$ −4.34315 −0.144291
$$907$$ 10.1421 0.336764 0.168382 0.985722i $$-0.446146\pi$$
0.168382 + 0.985722i $$0.446146\pi$$
$$908$$ 34.6863 1.15111
$$909$$ −8.82843 −0.292820
$$910$$ −0.343146 −0.0113752
$$911$$ 31.3137 1.03747 0.518735 0.854935i $$-0.326403\pi$$
0.518735 + 0.854935i $$0.326403\pi$$
$$912$$ −3.00000 −0.0993399
$$913$$ 44.7696 1.48166
$$914$$ −11.2548 −0.372277
$$915$$ 15.3137 0.506256
$$916$$ −41.3726 −1.36699
$$917$$ −16.1421 −0.533060
$$918$$ −2.82843 −0.0933520
$$919$$ −12.0000 −0.395843 −0.197922 0.980218i $$-0.563419\pi$$
−0.197922 + 0.980218i $$0.563419\pi$$
$$920$$ 5.79899 0.191187
$$921$$ 26.8284 0.884027
$$922$$ −3.45584 −0.113812
$$923$$ 3.02944 0.0997151
$$924$$ −16.1421 −0.531037
$$925$$ −0.585786 −0.0192605
$$926$$ −6.52691 −0.214488
$$927$$ 15.3137 0.502968
$$928$$ 6.24264 0.204925
$$929$$ −29.1127 −0.955157 −0.477578 0.878589i $$-0.658485\pi$$
−0.477578 + 0.878589i $$0.658485\pi$$
$$930$$ −3.65685 −0.119913
$$931$$ 5.00000 0.163868
$$932$$ 21.3137 0.698154
$$933$$ 2.24264 0.0734208
$$934$$ −10.0833 −0.329934
$$935$$ −42.6274 −1.39407
$$936$$ −0.928932 −0.0303631
$$937$$ 9.79899 0.320119 0.160060 0.987107i $$-0.448831\pi$$
0.160060 + 0.987107i $$0.448831\pi$$
$$938$$ −0.970563 −0.0316900
$$939$$ −33.7990 −1.10299
$$940$$ 6.68629 0.218083
$$941$$ −30.8701 −1.00634 −0.503168 0.864189i $$-0.667832\pi$$
−0.503168 + 0.864189i $$0.667832\pi$$
$$942$$ 2.68629 0.0875241
$$943$$ −30.1421 −0.981563
$$944$$ −13.4558 −0.437950
$$945$$ −1.41421 −0.0460044
$$946$$ −9.71573 −0.315886
$$947$$ −52.8284 −1.71669 −0.858347 0.513070i $$-0.828508\pi$$
−0.858347 + 0.513070i $$0.828508\pi$$
$$948$$ 0 0
$$949$$ −2.14214 −0.0695367
$$950$$ 0.414214 0.0134389
$$951$$ 18.6274 0.604035
$$952$$ 15.3137 0.496320
$$953$$ −2.54416 −0.0824133 −0.0412066 0.999151i $$-0.513120\pi$$
−0.0412066 + 0.999151i $$0.513120\pi$$
$$954$$ −3.31371 −0.107285
$$955$$ −1.75736 −0.0568668
$$956$$ 2.32590 0.0752250
$$957$$ −8.82843 −0.285383
$$958$$ −1.21320 −0.0391968
$$959$$ 14.1421 0.456673
$$960$$ 4.17157 0.134637
$$961$$ 46.9411 1.51423
$$962$$ −0.142136 −0.00458264
$$963$$ −3.31371 −0.106783
$$964$$ 15.2548 0.491325
$$965$$ 9.07107 0.292008
$$966$$ 2.14214 0.0689221
$$967$$ −40.0416 −1.28765 −0.643826 0.765172i $$-0.722654\pi$$
−0.643826 + 0.765172i $$0.722654\pi$$
$$968$$ 44.3553 1.42563
$$969$$ −6.82843 −0.219361
$$970$$ −7.55635 −0.242620
$$971$$ 33.6569 1.08010 0.540050 0.841633i $$-0.318405\pi$$
0.540050 + 0.841633i $$0.318405\pi$$
$$972$$ −1.82843 −0.0586468
$$973$$ 20.0000 0.641171
$$974$$ −1.45584 −0.0466483
$$975$$ −0.585786 −0.0187602
$$976$$ −45.9411 −1.47054
$$977$$ −24.2843 −0.776923 −0.388461 0.921465i $$-0.626993\pi$$
−0.388461 + 0.921465i $$0.626993\pi$$
$$978$$ −0.870058 −0.0278214
$$979$$ −86.7696 −2.77317
$$980$$ −9.14214 −0.292035
$$981$$ 10.4853 0.334769
$$982$$ 4.24264 0.135388
$$983$$ 20.6274 0.657912 0.328956 0.944345i $$-0.393303\pi$$
0.328956 + 0.944345i $$0.393303\pi$$
$$984$$ 13.0711 0.416690
$$985$$ −1.17157 −0.0373294
$$986$$ 4.00000 0.127386
$$987$$ 5.17157 0.164613
$$988$$ −1.07107 −0.0340752
$$989$$ −13.7401 −0.436910
$$990$$ 2.58579 0.0821817
$$991$$ 13.6569 0.433824 0.216912 0.976191i $$-0.430401\pi$$
0.216912 + 0.976191i $$0.430401\pi$$
$$992$$ 38.9706 1.23732
$$993$$ −0.142136 −0.00451054
$$994$$ 3.02944 0.0960879
$$995$$ 10.1421 0.321527
$$996$$ −13.1127 −0.415492
$$997$$ 58.0833 1.83952 0.919758 0.392487i $$-0.128385\pi$$
0.919758 + 0.392487i $$0.128385\pi$$
$$998$$ 4.48528 0.141979
$$999$$ −0.585786 −0.0185335
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 285.2.a.g.1.1 2
3.2 odd 2 855.2.a.d.1.2 2
4.3 odd 2 4560.2.a.bf.1.1 2
5.2 odd 4 1425.2.c.l.799.2 4
5.3 odd 4 1425.2.c.l.799.3 4
5.4 even 2 1425.2.a.k.1.2 2
15.14 odd 2 4275.2.a.y.1.1 2
19.18 odd 2 5415.2.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.g.1.1 2 1.1 even 1 trivial
855.2.a.d.1.2 2 3.2 odd 2
1425.2.a.k.1.2 2 5.4 even 2
1425.2.c.l.799.2 4 5.2 odd 4
1425.2.c.l.799.3 4 5.3 odd 4
4275.2.a.y.1.1 2 15.14 odd 2
4560.2.a.bf.1.1 2 4.3 odd 2
5415.2.a.n.1.2 2 19.18 odd 2