# Properties

 Label 8085.2.a.bk.1.3 Level $8085$ Weight $2$ Character 8085.1 Self dual yes Analytic conductor $64.559$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8085,2,Mod(1,8085)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8085, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8085.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8085.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: $$64.5590500342$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 8085.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} -1.00000 q^{5} -1.90321 q^{6} -0.719004 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} -1.00000 q^{5} -1.90321 q^{6} -0.719004 q^{8} +1.00000 q^{9} -1.90321 q^{10} +1.00000 q^{11} -1.62222 q^{12} +0.622216 q^{13} +1.00000 q^{15} -4.61285 q^{16} +5.18421 q^{17} +1.90321 q^{18} -7.05086 q^{19} -1.62222 q^{20} +1.90321 q^{22} +8.85728 q^{23} +0.719004 q^{24} +1.00000 q^{25} +1.18421 q^{26} -1.00000 q^{27} -7.80642 q^{29} +1.90321 q^{30} -2.75557 q^{31} -7.34122 q^{32} -1.00000 q^{33} +9.86665 q^{34} +1.62222 q^{36} -2.00000 q^{37} -13.4193 q^{38} -0.622216 q^{39} +0.719004 q^{40} +0.193576 q^{41} +5.67307 q^{43} +1.62222 q^{44} -1.00000 q^{45} +16.8573 q^{46} +2.75557 q^{47} +4.61285 q^{48} +1.90321 q^{50} -5.18421 q^{51} +1.00937 q^{52} -10.8573 q^{53} -1.90321 q^{54} -1.00000 q^{55} +7.05086 q^{57} -14.8573 q^{58} +4.85728 q^{59} +1.62222 q^{60} -6.85728 q^{61} -5.24443 q^{62} -4.74620 q^{64} -0.622216 q^{65} -1.90321 q^{66} -1.24443 q^{67} +8.40990 q^{68} -8.85728 q^{69} +2.75557 q^{71} -0.719004 q^{72} -4.23506 q^{73} -3.80642 q^{74} -1.00000 q^{75} -11.4380 q^{76} -1.18421 q^{78} +8.56199 q^{79} +4.61285 q^{80} +1.00000 q^{81} +0.368416 q^{82} -0.133353 q^{83} -5.18421 q^{85} +10.7971 q^{86} +7.80642 q^{87} -0.719004 q^{88} -5.61285 q^{89} -1.90321 q^{90} +14.3684 q^{92} +2.75557 q^{93} +5.24443 q^{94} +7.05086 q^{95} +7.34122 q^{96} -7.24443 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 - 3 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 9 * q^8 + 3 * q^9 $$3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} - 5 q^{12} + 2 q^{13} + 3 q^{15} + 13 q^{16} + 2 q^{17} - q^{18} - 8 q^{19} - 5 q^{20} - q^{22} + 9 q^{24} + 3 q^{25} - 10 q^{26} - 3 q^{27} - 10 q^{29} - q^{30} - 8 q^{31} - 29 q^{32} - 3 q^{33} + 30 q^{34} + 5 q^{36} - 6 q^{37} - 2 q^{39} + 9 q^{40} + 14 q^{41} + 4 q^{43} + 5 q^{44} - 3 q^{45} + 24 q^{46} + 8 q^{47} - 13 q^{48} - q^{50} - 2 q^{51} + 30 q^{52} - 6 q^{53} + q^{54} - 3 q^{55} + 8 q^{57} - 18 q^{58} - 12 q^{59} + 5 q^{60} + 6 q^{61} - 16 q^{62} + 13 q^{64} - 2 q^{65} + q^{66} - 4 q^{67} - 42 q^{68} + 8 q^{71} - 9 q^{72} + 14 q^{73} + 2 q^{74} - 3 q^{75} - 48 q^{76} + 10 q^{78} + 12 q^{79} - 13 q^{80} + 3 q^{81} - 26 q^{82} - 2 q^{85} - 8 q^{86} + 10 q^{87} - 9 q^{88} + 10 q^{89} + q^{90} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 8 q^{95} + 29 q^{96} - 22 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q - q^2 - 3 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 9 * q^8 + 3 * q^9 + q^10 + 3 * q^11 - 5 * q^12 + 2 * q^13 + 3 * q^15 + 13 * q^16 + 2 * q^17 - q^18 - 8 * q^19 - 5 * q^20 - q^22 + 9 * q^24 + 3 * q^25 - 10 * q^26 - 3 * q^27 - 10 * q^29 - q^30 - 8 * q^31 - 29 * q^32 - 3 * q^33 + 30 * q^34 + 5 * q^36 - 6 * q^37 - 2 * q^39 + 9 * q^40 + 14 * q^41 + 4 * q^43 + 5 * q^44 - 3 * q^45 + 24 * q^46 + 8 * q^47 - 13 * q^48 - q^50 - 2 * q^51 + 30 * q^52 - 6 * q^53 + q^54 - 3 * q^55 + 8 * q^57 - 18 * q^58 - 12 * q^59 + 5 * q^60 + 6 * q^61 - 16 * q^62 + 13 * q^64 - 2 * q^65 + q^66 - 4 * q^67 - 42 * q^68 + 8 * q^71 - 9 * q^72 + 14 * q^73 + 2 * q^74 - 3 * q^75 - 48 * q^76 + 10 * q^78 + 12 * q^79 - 13 * q^80 + 3 * q^81 - 26 * q^82 - 2 * q^85 - 8 * q^86 + 10 * q^87 - 9 * q^88 + 10 * q^89 + q^90 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 8 * q^95 + 29 * q^96 - 22 * q^97 + 3 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.90321 1.34577 0.672887 0.739745i $$-0.265054\pi$$
0.672887 + 0.739745i $$0.265054\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 1.62222 0.811108
$$5$$ −1.00000 −0.447214
$$6$$ −1.90321 −0.776983
$$7$$ 0 0
$$8$$ −0.719004 −0.254206
$$9$$ 1.00000 0.333333
$$10$$ −1.90321 −0.601848
$$11$$ 1.00000 0.301511
$$12$$ −1.62222 −0.468293
$$13$$ 0.622216 0.172572 0.0862858 0.996270i $$-0.472500\pi$$
0.0862858 + 0.996270i $$0.472500\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ −4.61285 −1.15321
$$17$$ 5.18421 1.25736 0.628678 0.777666i $$-0.283597\pi$$
0.628678 + 0.777666i $$0.283597\pi$$
$$18$$ 1.90321 0.448591
$$19$$ −7.05086 −1.61758 −0.808789 0.588100i $$-0.799876\pi$$
−0.808789 + 0.588100i $$0.799876\pi$$
$$20$$ −1.62222 −0.362738
$$21$$ 0 0
$$22$$ 1.90321 0.405766
$$23$$ 8.85728 1.84687 0.923435 0.383754i $$-0.125369\pi$$
0.923435 + 0.383754i $$0.125369\pi$$
$$24$$ 0.719004 0.146766
$$25$$ 1.00000 0.200000
$$26$$ 1.18421 0.232242
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −7.80642 −1.44962 −0.724808 0.688951i $$-0.758072\pi$$
−0.724808 + 0.688951i $$0.758072\pi$$
$$30$$ 1.90321 0.347477
$$31$$ −2.75557 −0.494915 −0.247457 0.968899i $$-0.579595\pi$$
−0.247457 + 0.968899i $$0.579595\pi$$
$$32$$ −7.34122 −1.29776
$$33$$ −1.00000 −0.174078
$$34$$ 9.86665 1.69212
$$35$$ 0 0
$$36$$ 1.62222 0.270369
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −13.4193 −2.17689
$$39$$ −0.622216 −0.0996342
$$40$$ 0.719004 0.113684
$$41$$ 0.193576 0.0302315 0.0151158 0.999886i $$-0.495188\pi$$
0.0151158 + 0.999886i $$0.495188\pi$$
$$42$$ 0 0
$$43$$ 5.67307 0.865135 0.432568 0.901602i $$-0.357608\pi$$
0.432568 + 0.901602i $$0.357608\pi$$
$$44$$ 1.62222 0.244558
$$45$$ −1.00000 −0.149071
$$46$$ 16.8573 2.48547
$$47$$ 2.75557 0.401941 0.200971 0.979597i $$-0.435590\pi$$
0.200971 + 0.979597i $$0.435590\pi$$
$$48$$ 4.61285 0.665807
$$49$$ 0 0
$$50$$ 1.90321 0.269155
$$51$$ −5.18421 −0.725934
$$52$$ 1.00937 0.139974
$$53$$ −10.8573 −1.49136 −0.745681 0.666303i $$-0.767876\pi$$
−0.745681 + 0.666303i $$0.767876\pi$$
$$54$$ −1.90321 −0.258994
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 7.05086 0.933909
$$58$$ −14.8573 −1.95086
$$59$$ 4.85728 0.632364 0.316182 0.948699i $$-0.397599\pi$$
0.316182 + 0.948699i $$0.397599\pi$$
$$60$$ 1.62222 0.209427
$$61$$ −6.85728 −0.877985 −0.438992 0.898491i $$-0.644664\pi$$
−0.438992 + 0.898491i $$0.644664\pi$$
$$62$$ −5.24443 −0.666043
$$63$$ 0 0
$$64$$ −4.74620 −0.593275
$$65$$ −0.622216 −0.0771764
$$66$$ −1.90321 −0.234269
$$67$$ −1.24443 −0.152031 −0.0760157 0.997107i $$-0.524220\pi$$
−0.0760157 + 0.997107i $$0.524220\pi$$
$$68$$ 8.40990 1.01985
$$69$$ −8.85728 −1.06629
$$70$$ 0 0
$$71$$ 2.75557 0.327026 0.163513 0.986541i $$-0.447717\pi$$
0.163513 + 0.986541i $$0.447717\pi$$
$$72$$ −0.719004 −0.0847354
$$73$$ −4.23506 −0.495677 −0.247838 0.968801i $$-0.579720\pi$$
−0.247838 + 0.968801i $$0.579720\pi$$
$$74$$ −3.80642 −0.442488
$$75$$ −1.00000 −0.115470
$$76$$ −11.4380 −1.31203
$$77$$ 0 0
$$78$$ −1.18421 −0.134085
$$79$$ 8.56199 0.963299 0.481650 0.876364i $$-0.340038\pi$$
0.481650 + 0.876364i $$0.340038\pi$$
$$80$$ 4.61285 0.515732
$$81$$ 1.00000 0.111111
$$82$$ 0.368416 0.0406848
$$83$$ −0.133353 −0.0146374 −0.00731870 0.999973i $$-0.502330\pi$$
−0.00731870 + 0.999973i $$0.502330\pi$$
$$84$$ 0 0
$$85$$ −5.18421 −0.562306
$$86$$ 10.7971 1.16428
$$87$$ 7.80642 0.836936
$$88$$ −0.719004 −0.0766461
$$89$$ −5.61285 −0.594961 −0.297480 0.954728i $$-0.596146\pi$$
−0.297480 + 0.954728i $$0.596146\pi$$
$$90$$ −1.90321 −0.200616
$$91$$ 0 0
$$92$$ 14.3684 1.49801
$$93$$ 2.75557 0.285739
$$94$$ 5.24443 0.540922
$$95$$ 7.05086 0.723402
$$96$$ 7.34122 0.749260
$$97$$ −7.24443 −0.735561 −0.367780 0.929913i $$-0.619882\pi$$
−0.367780 + 0.929913i $$0.619882\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ 1.62222 0.162222
$$101$$ −4.66370 −0.464056 −0.232028 0.972709i $$-0.574536\pi$$
−0.232028 + 0.972709i $$0.574536\pi$$
$$102$$ −9.86665 −0.976944
$$103$$ 11.6128 1.14425 0.572124 0.820167i $$-0.306120\pi$$
0.572124 + 0.820167i $$0.306120\pi$$
$$104$$ −0.447375 −0.0438688
$$105$$ 0 0
$$106$$ −20.6637 −2.00704
$$107$$ 2.62222 0.253499 0.126750 0.991935i $$-0.459546\pi$$
0.126750 + 0.991935i $$0.459546\pi$$
$$108$$ −1.62222 −0.156098
$$109$$ −19.7146 −1.88831 −0.944156 0.329499i $$-0.893120\pi$$
−0.944156 + 0.329499i $$0.893120\pi$$
$$110$$ −1.90321 −0.181464
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 13.4193 1.25683
$$115$$ −8.85728 −0.825946
$$116$$ −12.6637 −1.17580
$$117$$ 0.622216 0.0575239
$$118$$ 9.24443 0.851019
$$119$$ 0 0
$$120$$ −0.719004 −0.0656358
$$121$$ 1.00000 0.0909091
$$122$$ −13.0509 −1.18157
$$123$$ −0.193576 −0.0174542
$$124$$ −4.47013 −0.401429
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −15.1842 −1.34738 −0.673690 0.739014i $$-0.735292\pi$$
−0.673690 + 0.739014i $$0.735292\pi$$
$$128$$ 5.64941 0.499342
$$129$$ −5.67307 −0.499486
$$130$$ −1.18421 −0.103862
$$131$$ −1.24443 −0.108726 −0.0543632 0.998521i $$-0.517313\pi$$
−0.0543632 + 0.998521i $$0.517313\pi$$
$$132$$ −1.62222 −0.141196
$$133$$ 0 0
$$134$$ −2.36842 −0.204600
$$135$$ 1.00000 0.0860663
$$136$$ −3.72746 −0.319627
$$137$$ −0.488863 −0.0417663 −0.0208832 0.999782i $$-0.506648\pi$$
−0.0208832 + 0.999782i $$0.506648\pi$$
$$138$$ −16.8573 −1.43499
$$139$$ −17.8064 −1.51032 −0.755161 0.655540i $$-0.772441\pi$$
−0.755161 + 0.655540i $$0.772441\pi$$
$$140$$ 0 0
$$141$$ −2.75557 −0.232061
$$142$$ 5.24443 0.440103
$$143$$ 0.622216 0.0520323
$$144$$ −4.61285 −0.384404
$$145$$ 7.80642 0.648288
$$146$$ −8.06022 −0.667069
$$147$$ 0 0
$$148$$ −3.24443 −0.266691
$$149$$ −1.43801 −0.117806 −0.0589031 0.998264i $$-0.518760\pi$$
−0.0589031 + 0.998264i $$0.518760\pi$$
$$150$$ −1.90321 −0.155397
$$151$$ −12.1748 −0.990774 −0.495387 0.868672i $$-0.664974\pi$$
−0.495387 + 0.868672i $$0.664974\pi$$
$$152$$ 5.06959 0.411198
$$153$$ 5.18421 0.419118
$$154$$ 0 0
$$155$$ 2.75557 0.221333
$$156$$ −1.00937 −0.0808141
$$157$$ −18.4701 −1.47408 −0.737038 0.675851i $$-0.763776\pi$$
−0.737038 + 0.675851i $$0.763776\pi$$
$$158$$ 16.2953 1.29638
$$159$$ 10.8573 0.861038
$$160$$ 7.34122 0.580374
$$161$$ 0 0
$$162$$ 1.90321 0.149530
$$163$$ −10.1017 −0.791227 −0.395614 0.918417i $$-0.629468\pi$$
−0.395614 + 0.918417i $$0.629468\pi$$
$$164$$ 0.314022 0.0245210
$$165$$ 1.00000 0.0778499
$$166$$ −0.253799 −0.0196986
$$167$$ 16.3368 1.26418 0.632089 0.774896i $$-0.282198\pi$$
0.632089 + 0.774896i $$0.282198\pi$$
$$168$$ 0 0
$$169$$ −12.6128 −0.970219
$$170$$ −9.86665 −0.756737
$$171$$ −7.05086 −0.539192
$$172$$ 9.20294 0.701718
$$173$$ 9.18421 0.698262 0.349131 0.937074i $$-0.386477\pi$$
0.349131 + 0.937074i $$0.386477\pi$$
$$174$$ 14.8573 1.12633
$$175$$ 0 0
$$176$$ −4.61285 −0.347706
$$177$$ −4.85728 −0.365095
$$178$$ −10.6824 −0.800683
$$179$$ −25.3274 −1.89306 −0.946530 0.322617i $$-0.895437\pi$$
−0.946530 + 0.322617i $$0.895437\pi$$
$$180$$ −1.62222 −0.120913
$$181$$ 13.6128 1.01184 0.505918 0.862582i $$-0.331154\pi$$
0.505918 + 0.862582i $$0.331154\pi$$
$$182$$ 0 0
$$183$$ 6.85728 0.506905
$$184$$ −6.36842 −0.469486
$$185$$ 2.00000 0.147043
$$186$$ 5.24443 0.384540
$$187$$ 5.18421 0.379107
$$188$$ 4.47013 0.326017
$$189$$ 0 0
$$190$$ 13.4193 0.973536
$$191$$ 6.10171 0.441504 0.220752 0.975330i $$-0.429149\pi$$
0.220752 + 0.975330i $$0.429149\pi$$
$$192$$ 4.74620 0.342528
$$193$$ 18.3368 1.31991 0.659955 0.751305i $$-0.270575\pi$$
0.659955 + 0.751305i $$0.270575\pi$$
$$194$$ −13.7877 −0.989898
$$195$$ 0.622216 0.0445578
$$196$$ 0 0
$$197$$ −6.69535 −0.477024 −0.238512 0.971140i $$-0.576660\pi$$
−0.238512 + 0.971140i $$0.576660\pi$$
$$198$$ 1.90321 0.135255
$$199$$ −14.1017 −0.999644 −0.499822 0.866128i $$-0.666601\pi$$
−0.499822 + 0.866128i $$0.666601\pi$$
$$200$$ −0.719004 −0.0508412
$$201$$ 1.24443 0.0877754
$$202$$ −8.87601 −0.624514
$$203$$ 0 0
$$204$$ −8.40990 −0.588811
$$205$$ −0.193576 −0.0135199
$$206$$ 22.1017 1.53990
$$207$$ 8.85728 0.615623
$$208$$ −2.87019 −0.199012
$$209$$ −7.05086 −0.487718
$$210$$ 0 0
$$211$$ −10.6637 −0.734120 −0.367060 0.930197i $$-0.619636\pi$$
−0.367060 + 0.930197i $$0.619636\pi$$
$$212$$ −17.6128 −1.20966
$$213$$ −2.75557 −0.188808
$$214$$ 4.99063 0.341153
$$215$$ −5.67307 −0.386900
$$216$$ 0.719004 0.0489220
$$217$$ 0 0
$$218$$ −37.5210 −2.54124
$$219$$ 4.23506 0.286179
$$220$$ −1.62222 −0.109370
$$221$$ 3.22570 0.216984
$$222$$ 3.80642 0.255470
$$223$$ −8.85728 −0.593127 −0.296564 0.955013i $$-0.595841\pi$$
−0.296564 + 0.955013i $$0.595841\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ −11.4193 −0.759599
$$227$$ −13.3778 −0.887915 −0.443957 0.896048i $$-0.646426\pi$$
−0.443957 + 0.896048i $$0.646426\pi$$
$$228$$ 11.4380 0.757501
$$229$$ −11.5111 −0.760677 −0.380339 0.924847i $$-0.624193\pi$$
−0.380339 + 0.924847i $$0.624193\pi$$
$$230$$ −16.8573 −1.11154
$$231$$ 0 0
$$232$$ 5.61285 0.368502
$$233$$ −4.32693 −0.283467 −0.141733 0.989905i $$-0.545268\pi$$
−0.141733 + 0.989905i $$0.545268\pi$$
$$234$$ 1.18421 0.0774141
$$235$$ −2.75557 −0.179753
$$236$$ 7.87955 0.512915
$$237$$ −8.56199 −0.556161
$$238$$ 0 0
$$239$$ −3.34614 −0.216444 −0.108222 0.994127i $$-0.534516\pi$$
−0.108222 + 0.994127i $$0.534516\pi$$
$$240$$ −4.61285 −0.297758
$$241$$ 1.34614 0.0867126 0.0433563 0.999060i $$-0.486195\pi$$
0.0433563 + 0.999060i $$0.486195\pi$$
$$242$$ 1.90321 0.122343
$$243$$ −1.00000 −0.0641500
$$244$$ −11.1240 −0.712140
$$245$$ 0 0
$$246$$ −0.368416 −0.0234894
$$247$$ −4.38715 −0.279148
$$248$$ 1.98126 0.125810
$$249$$ 0.133353 0.00845091
$$250$$ −1.90321 −0.120370
$$251$$ −22.7556 −1.43632 −0.718159 0.695879i $$-0.755015\pi$$
−0.718159 + 0.695879i $$0.755015\pi$$
$$252$$ 0 0
$$253$$ 8.85728 0.556852
$$254$$ −28.8988 −1.81327
$$255$$ 5.18421 0.324648
$$256$$ 20.2444 1.26528
$$257$$ 6.85728 0.427745 0.213873 0.976862i $$-0.431392\pi$$
0.213873 + 0.976862i $$0.431392\pi$$
$$258$$ −10.7971 −0.672195
$$259$$ 0 0
$$260$$ −1.00937 −0.0625983
$$261$$ −7.80642 −0.483206
$$262$$ −2.36842 −0.146321
$$263$$ −29.5812 −1.82406 −0.912028 0.410129i $$-0.865484\pi$$
−0.912028 + 0.410129i $$0.865484\pi$$
$$264$$ 0.719004 0.0442516
$$265$$ 10.8573 0.666957
$$266$$ 0 0
$$267$$ 5.61285 0.343501
$$268$$ −2.01874 −0.123314
$$269$$ −8.48886 −0.517575 −0.258788 0.965934i $$-0.583323\pi$$
−0.258788 + 0.965934i $$0.583323\pi$$
$$270$$ 1.90321 0.115826
$$271$$ −14.6637 −0.890757 −0.445378 0.895343i $$-0.646931\pi$$
−0.445378 + 0.895343i $$0.646931\pi$$
$$272$$ −23.9140 −1.45000
$$273$$ 0 0
$$274$$ −0.930409 −0.0562081
$$275$$ 1.00000 0.0603023
$$276$$ −14.3684 −0.864877
$$277$$ 14.6035 0.877438 0.438719 0.898624i $$-0.355432\pi$$
0.438719 + 0.898624i $$0.355432\pi$$
$$278$$ −33.8894 −2.03255
$$279$$ −2.75557 −0.164972
$$280$$ 0 0
$$281$$ −0.193576 −0.0115478 −0.00577389 0.999983i $$-0.501838\pi$$
−0.00577389 + 0.999983i $$0.501838\pi$$
$$282$$ −5.24443 −0.312301
$$283$$ −27.1842 −1.61593 −0.807967 0.589228i $$-0.799432\pi$$
−0.807967 + 0.589228i $$0.799432\pi$$
$$284$$ 4.47013 0.265253
$$285$$ −7.05086 −0.417657
$$286$$ 1.18421 0.0700237
$$287$$ 0 0
$$288$$ −7.34122 −0.432585
$$289$$ 9.87601 0.580942
$$290$$ 14.8573 0.872449
$$291$$ 7.24443 0.424676
$$292$$ −6.87019 −0.402047
$$293$$ 2.81579 0.164500 0.0822502 0.996612i $$-0.473789\pi$$
0.0822502 + 0.996612i $$0.473789\pi$$
$$294$$ 0 0
$$295$$ −4.85728 −0.282802
$$296$$ 1.43801 0.0835825
$$297$$ −1.00000 −0.0580259
$$298$$ −2.73683 −0.158540
$$299$$ 5.51114 0.318717
$$300$$ −1.62222 −0.0936587
$$301$$ 0 0
$$302$$ −23.1713 −1.33336
$$303$$ 4.66370 0.267923
$$304$$ 32.5245 1.86541
$$305$$ 6.85728 0.392647
$$306$$ 9.86665 0.564039
$$307$$ 24.4286 1.39422 0.697108 0.716966i $$-0.254470\pi$$
0.697108 + 0.716966i $$0.254470\pi$$
$$308$$ 0 0
$$309$$ −11.6128 −0.660632
$$310$$ 5.24443 0.297864
$$311$$ −19.8796 −1.12727 −0.563633 0.826025i $$-0.690597\pi$$
−0.563633 + 0.826025i $$0.690597\pi$$
$$312$$ 0.447375 0.0253276
$$313$$ 15.7146 0.888239 0.444120 0.895967i $$-0.353517\pi$$
0.444120 + 0.895967i $$0.353517\pi$$
$$314$$ −35.1526 −1.98377
$$315$$ 0 0
$$316$$ 13.8894 0.781340
$$317$$ 16.4889 0.926107 0.463053 0.886330i $$-0.346754\pi$$
0.463053 + 0.886330i $$0.346754\pi$$
$$318$$ 20.6637 1.15876
$$319$$ −7.80642 −0.437076
$$320$$ 4.74620 0.265321
$$321$$ −2.62222 −0.146358
$$322$$ 0 0
$$323$$ −36.5531 −2.03387
$$324$$ 1.62222 0.0901231
$$325$$ 0.622216 0.0345143
$$326$$ −19.2257 −1.06481
$$327$$ 19.7146 1.09022
$$328$$ −0.139182 −0.00768504
$$329$$ 0 0
$$330$$ 1.90321 0.104768
$$331$$ 15.3461 0.843500 0.421750 0.906712i $$-0.361416\pi$$
0.421750 + 0.906712i $$0.361416\pi$$
$$332$$ −0.216327 −0.0118725
$$333$$ −2.00000 −0.109599
$$334$$ 31.0923 1.70130
$$335$$ 1.24443 0.0679905
$$336$$ 0 0
$$337$$ 28.2351 1.53806 0.769031 0.639212i $$-0.220739\pi$$
0.769031 + 0.639212i $$0.220739\pi$$
$$338$$ −24.0049 −1.30570
$$339$$ 6.00000 0.325875
$$340$$ −8.40990 −0.456091
$$341$$ −2.75557 −0.149222
$$342$$ −13.4193 −0.725631
$$343$$ 0 0
$$344$$ −4.07896 −0.219923
$$345$$ 8.85728 0.476860
$$346$$ 17.4795 0.939703
$$347$$ 2.62222 0.140768 0.0703840 0.997520i $$-0.477578\pi$$
0.0703840 + 0.997520i $$0.477578\pi$$
$$348$$ 12.6637 0.678846
$$349$$ −5.14272 −0.275284 −0.137642 0.990482i $$-0.543952\pi$$
−0.137642 + 0.990482i $$0.543952\pi$$
$$350$$ 0 0
$$351$$ −0.622216 −0.0332114
$$352$$ −7.34122 −0.391288
$$353$$ 9.34614 0.497445 0.248722 0.968575i $$-0.419989\pi$$
0.248722 + 0.968575i $$0.419989\pi$$
$$354$$ −9.24443 −0.491336
$$355$$ −2.75557 −0.146250
$$356$$ −9.10525 −0.482577
$$357$$ 0 0
$$358$$ −48.2034 −2.54763
$$359$$ 10.7556 0.567657 0.283829 0.958875i $$-0.408395\pi$$
0.283829 + 0.958875i $$0.408395\pi$$
$$360$$ 0.719004 0.0378948
$$361$$ 30.7146 1.61656
$$362$$ 25.9081 1.36170
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 4.23506 0.221673
$$366$$ 13.0509 0.682179
$$367$$ −33.7975 −1.76422 −0.882108 0.471046i $$-0.843876\pi$$
−0.882108 + 0.471046i $$0.843876\pi$$
$$368$$ −40.8573 −2.12983
$$369$$ 0.193576 0.0100772
$$370$$ 3.80642 0.197887
$$371$$ 0 0
$$372$$ 4.47013 0.231765
$$373$$ 33.9496 1.75784 0.878922 0.476965i $$-0.158263\pi$$
0.878922 + 0.476965i $$0.158263\pi$$
$$374$$ 9.86665 0.510192
$$375$$ 1.00000 0.0516398
$$376$$ −1.98126 −0.102176
$$377$$ −4.85728 −0.250163
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 11.4380 0.586757
$$381$$ 15.1842 0.777911
$$382$$ 11.6128 0.594165
$$383$$ 14.6351 0.747820 0.373910 0.927465i $$-0.378017\pi$$
0.373910 + 0.927465i $$0.378017\pi$$
$$384$$ −5.64941 −0.288295
$$385$$ 0 0
$$386$$ 34.8988 1.77630
$$387$$ 5.67307 0.288378
$$388$$ −11.7520 −0.596619
$$389$$ −5.61285 −0.284583 −0.142291 0.989825i $$-0.545447\pi$$
−0.142291 + 0.989825i $$0.545447\pi$$
$$390$$ 1.18421 0.0599647
$$391$$ 45.9180 2.32217
$$392$$ 0 0
$$393$$ 1.24443 0.0627733
$$394$$ −12.7427 −0.641966
$$395$$ −8.56199 −0.430801
$$396$$ 1.62222 0.0815194
$$397$$ 12.7556 0.640184 0.320092 0.947387i $$-0.396286\pi$$
0.320092 + 0.947387i $$0.396286\pi$$
$$398$$ −26.8385 −1.34529
$$399$$ 0 0
$$400$$ −4.61285 −0.230642
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 2.36842 0.118126
$$403$$ −1.71456 −0.0854082
$$404$$ −7.56553 −0.376399
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −2.00000 −0.0991363
$$408$$ 3.72746 0.184537
$$409$$ 7.12399 0.352258 0.176129 0.984367i $$-0.443642\pi$$
0.176129 + 0.984367i $$0.443642\pi$$
$$410$$ −0.368416 −0.0181948
$$411$$ 0.488863 0.0241138
$$412$$ 18.8385 0.928108
$$413$$ 0 0
$$414$$ 16.8573 0.828490
$$415$$ 0.133353 0.00654605
$$416$$ −4.56782 −0.223956
$$417$$ 17.8064 0.871984
$$418$$ −13.4193 −0.656358
$$419$$ −15.6128 −0.762738 −0.381369 0.924423i $$-0.624547\pi$$
−0.381369 + 0.924423i $$0.624547\pi$$
$$420$$ 0 0
$$421$$ 7.89829 0.384939 0.192470 0.981303i $$-0.438350\pi$$
0.192470 + 0.981303i $$0.438350\pi$$
$$422$$ −20.2953 −0.987959
$$423$$ 2.75557 0.133980
$$424$$ 7.80642 0.379113
$$425$$ 5.18421 0.251471
$$426$$ −5.24443 −0.254094
$$427$$ 0 0
$$428$$ 4.25380 0.205615
$$429$$ −0.622216 −0.0300409
$$430$$ −10.7971 −0.520680
$$431$$ 34.3051 1.65242 0.826210 0.563362i $$-0.190492\pi$$
0.826210 + 0.563362i $$0.190492\pi$$
$$432$$ 4.61285 0.221936
$$433$$ −14.4701 −0.695390 −0.347695 0.937608i $$-0.613036\pi$$
−0.347695 + 0.937608i $$0.613036\pi$$
$$434$$ 0 0
$$435$$ −7.80642 −0.374289
$$436$$ −31.9813 −1.53162
$$437$$ −62.4514 −2.98746
$$438$$ 8.06022 0.385132
$$439$$ 19.3176 0.921977 0.460988 0.887406i $$-0.347495\pi$$
0.460988 + 0.887406i $$0.347495\pi$$
$$440$$ 0.719004 0.0342772
$$441$$ 0 0
$$442$$ 6.13918 0.292011
$$443$$ −13.1240 −0.623539 −0.311770 0.950158i $$-0.600922\pi$$
−0.311770 + 0.950158i $$0.600922\pi$$
$$444$$ 3.24443 0.153974
$$445$$ 5.61285 0.266074
$$446$$ −16.8573 −0.798215
$$447$$ 1.43801 0.0680154
$$448$$ 0 0
$$449$$ −32.3051 −1.52457 −0.762287 0.647240i $$-0.775923\pi$$
−0.762287 + 0.647240i $$0.775923\pi$$
$$450$$ 1.90321 0.0897183
$$451$$ 0.193576 0.00911514
$$452$$ −9.73329 −0.457816
$$453$$ 12.1748 0.572024
$$454$$ −25.4608 −1.19493
$$455$$ 0 0
$$456$$ −5.06959 −0.237405
$$457$$ −23.4608 −1.09745 −0.548724 0.836004i $$-0.684886\pi$$
−0.548724 + 0.836004i $$0.684886\pi$$
$$458$$ −21.9081 −1.02370
$$459$$ −5.18421 −0.241978
$$460$$ −14.3684 −0.669931
$$461$$ 28.8671 1.34448 0.672238 0.740335i $$-0.265333\pi$$
0.672238 + 0.740335i $$0.265333\pi$$
$$462$$ 0 0
$$463$$ −19.3461 −0.899091 −0.449546 0.893257i $$-0.648414\pi$$
−0.449546 + 0.893257i $$0.648414\pi$$
$$464$$ 36.0098 1.67172
$$465$$ −2.75557 −0.127786
$$466$$ −8.23506 −0.381482
$$467$$ −3.14272 −0.145428 −0.0727139 0.997353i $$-0.523166\pi$$
−0.0727139 + 0.997353i $$0.523166\pi$$
$$468$$ 1.00937 0.0466580
$$469$$ 0 0
$$470$$ −5.24443 −0.241908
$$471$$ 18.4701 0.851059
$$472$$ −3.49240 −0.160751
$$473$$ 5.67307 0.260848
$$474$$ −16.2953 −0.748467
$$475$$ −7.05086 −0.323515
$$476$$ 0 0
$$477$$ −10.8573 −0.497121
$$478$$ −6.36842 −0.291285
$$479$$ 24.8573 1.13576 0.567879 0.823112i $$-0.307764\pi$$
0.567879 + 0.823112i $$0.307764\pi$$
$$480$$ −7.34122 −0.335079
$$481$$ −1.24443 −0.0567412
$$482$$ 2.56199 0.116696
$$483$$ 0 0
$$484$$ 1.62222 0.0737371
$$485$$ 7.24443 0.328953
$$486$$ −1.90321 −0.0863314
$$487$$ −11.5299 −0.522468 −0.261234 0.965275i $$-0.584129\pi$$
−0.261234 + 0.965275i $$0.584129\pi$$
$$488$$ 4.93041 0.223189
$$489$$ 10.1017 0.456815
$$490$$ 0 0
$$491$$ 16.3872 0.739542 0.369771 0.929123i $$-0.379436\pi$$
0.369771 + 0.929123i $$0.379436\pi$$
$$492$$ −0.314022 −0.0141572
$$493$$ −40.4701 −1.82268
$$494$$ −8.34968 −0.375670
$$495$$ −1.00000 −0.0449467
$$496$$ 12.7110 0.570742
$$497$$ 0 0
$$498$$ 0.253799 0.0113730
$$499$$ −25.3274 −1.13381 −0.566905 0.823783i $$-0.691859\pi$$
−0.566905 + 0.823783i $$0.691859\pi$$
$$500$$ −1.62222 −0.0725477
$$501$$ −16.3368 −0.729873
$$502$$ −43.3087 −1.93296
$$503$$ −19.0923 −0.851285 −0.425643 0.904891i $$-0.639952\pi$$
−0.425643 + 0.904891i $$0.639952\pi$$
$$504$$ 0 0
$$505$$ 4.66370 0.207532
$$506$$ 16.8573 0.749397
$$507$$ 12.6128 0.560156
$$508$$ −24.6321 −1.09287
$$509$$ 32.4514 1.43838 0.719191 0.694812i $$-0.244512\pi$$
0.719191 + 0.694812i $$0.244512\pi$$
$$510$$ 9.86665 0.436902
$$511$$ 0 0
$$512$$ 27.2306 1.20343
$$513$$ 7.05086 0.311303
$$514$$ 13.0509 0.575649
$$515$$ −11.6128 −0.511723
$$516$$ −9.20294 −0.405137
$$517$$ 2.75557 0.121190
$$518$$ 0 0
$$519$$ −9.18421 −0.403142
$$520$$ 0.447375 0.0196187
$$521$$ 29.2257 1.28040 0.640200 0.768208i $$-0.278851\pi$$
0.640200 + 0.768208i $$0.278851\pi$$
$$522$$ −14.8573 −0.650285
$$523$$ −6.71408 −0.293586 −0.146793 0.989167i $$-0.546895\pi$$
−0.146793 + 0.989167i $$0.546895\pi$$
$$524$$ −2.01874 −0.0881889
$$525$$ 0 0
$$526$$ −56.2993 −2.45477
$$527$$ −14.2854 −0.622284
$$528$$ 4.61285 0.200748
$$529$$ 55.4514 2.41093
$$530$$ 20.6637 0.897574
$$531$$ 4.85728 0.210788
$$532$$ 0 0
$$533$$ 0.120446 0.00521710
$$534$$ 10.6824 0.462274
$$535$$ −2.62222 −0.113368
$$536$$ 0.894751 0.0386473
$$537$$ 25.3274 1.09296
$$538$$ −16.1561 −0.696539
$$539$$ 0 0
$$540$$ 1.62222 0.0698090
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ −27.9081 −1.19876
$$543$$ −13.6128 −0.584183
$$544$$ −38.0584 −1.63174
$$545$$ 19.7146 0.844479
$$546$$ 0 0
$$547$$ 41.3689 1.76881 0.884403 0.466724i $$-0.154566\pi$$
0.884403 + 0.466724i $$0.154566\pi$$
$$548$$ −0.793040 −0.0338770
$$549$$ −6.85728 −0.292662
$$550$$ 1.90321 0.0811532
$$551$$ 55.0420 2.34487
$$552$$ 6.36842 0.271058
$$553$$ 0 0
$$554$$ 27.7935 1.18083
$$555$$ −2.00000 −0.0848953
$$556$$ −28.8859 −1.22503
$$557$$ −20.7971 −0.881200 −0.440600 0.897704i $$-0.645234\pi$$
−0.440600 + 0.897704i $$0.645234\pi$$
$$558$$ −5.24443 −0.222014
$$559$$ 3.52987 0.149298
$$560$$ 0 0
$$561$$ −5.18421 −0.218877
$$562$$ −0.368416 −0.0155407
$$563$$ −37.7275 −1.59002 −0.795012 0.606594i $$-0.792535\pi$$
−0.795012 + 0.606594i $$0.792535\pi$$
$$564$$ −4.47013 −0.188226
$$565$$ 6.00000 0.252422
$$566$$ −51.7373 −2.17468
$$567$$ 0 0
$$568$$ −1.98126 −0.0831320
$$569$$ −7.33630 −0.307554 −0.153777 0.988106i $$-0.549144\pi$$
−0.153777 + 0.988106i $$0.549144\pi$$
$$570$$ −13.4193 −0.562071
$$571$$ 36.6450 1.53354 0.766772 0.641919i $$-0.221862\pi$$
0.766772 + 0.641919i $$0.221862\pi$$
$$572$$ 1.00937 0.0422038
$$573$$ −6.10171 −0.254903
$$574$$ 0 0
$$575$$ 8.85728 0.369374
$$576$$ −4.74620 −0.197758
$$577$$ −4.22216 −0.175771 −0.0878853 0.996131i $$-0.528011\pi$$
−0.0878853 + 0.996131i $$0.528011\pi$$
$$578$$ 18.7961 0.781817
$$579$$ −18.3368 −0.762050
$$580$$ 12.6637 0.525832
$$581$$ 0 0
$$582$$ 13.7877 0.571518
$$583$$ −10.8573 −0.449663
$$584$$ 3.04503 0.126004
$$585$$ −0.622216 −0.0257255
$$586$$ 5.35905 0.221380
$$587$$ −34.3684 −1.41854 −0.709268 0.704939i $$-0.750974\pi$$
−0.709268 + 0.704939i $$0.750974\pi$$
$$588$$ 0 0
$$589$$ 19.4291 0.800563
$$590$$ −9.24443 −0.380587
$$591$$ 6.69535 0.275410
$$592$$ 9.22570 0.379174
$$593$$ −27.9398 −1.14735 −0.573675 0.819083i $$-0.694483\pi$$
−0.573675 + 0.819083i $$0.694483\pi$$
$$594$$ −1.90321 −0.0780897
$$595$$ 0 0
$$596$$ −2.33276 −0.0955535
$$597$$ 14.1017 0.577145
$$598$$ 10.4889 0.428921
$$599$$ −31.2257 −1.27585 −0.637924 0.770100i $$-0.720206\pi$$
−0.637924 + 0.770100i $$0.720206\pi$$
$$600$$ 0.719004 0.0293532
$$601$$ 8.75557 0.357147 0.178574 0.983927i $$-0.442852\pi$$
0.178574 + 0.983927i $$0.442852\pi$$
$$602$$ 0 0
$$603$$ −1.24443 −0.0506772
$$604$$ −19.7502 −0.803625
$$605$$ −1.00000 −0.0406558
$$606$$ 8.87601 0.360563
$$607$$ 15.1842 0.616308 0.308154 0.951336i $$-0.400289\pi$$
0.308154 + 0.951336i $$0.400289\pi$$
$$608$$ 51.7619 2.09922
$$609$$ 0 0
$$610$$ 13.0509 0.528414
$$611$$ 1.71456 0.0693636
$$612$$ 8.40990 0.339950
$$613$$ 42.7239 1.72560 0.862802 0.505543i $$-0.168708\pi$$
0.862802 + 0.505543i $$0.168708\pi$$
$$614$$ 46.4929 1.87630
$$615$$ 0.193576 0.00780574
$$616$$ 0 0
$$617$$ −3.51114 −0.141353 −0.0706765 0.997499i $$-0.522516\pi$$
−0.0706765 + 0.997499i $$0.522516\pi$$
$$618$$ −22.1017 −0.889061
$$619$$ −17.5941 −0.707167 −0.353584 0.935403i $$-0.615037\pi$$
−0.353584 + 0.935403i $$0.615037\pi$$
$$620$$ 4.47013 0.179525
$$621$$ −8.85728 −0.355430
$$622$$ −37.8350 −1.51705
$$623$$ 0 0
$$624$$ 2.87019 0.114899
$$625$$ 1.00000 0.0400000
$$626$$ 29.9081 1.19537
$$627$$ 7.05086 0.281584
$$628$$ −29.9625 −1.19564
$$629$$ −10.3684 −0.413416
$$630$$ 0 0
$$631$$ 15.8163 0.629636 0.314818 0.949152i $$-0.398057\pi$$
0.314818 + 0.949152i $$0.398057\pi$$
$$632$$ −6.15610 −0.244877
$$633$$ 10.6637 0.423844
$$634$$ 31.3818 1.24633
$$635$$ 15.1842 0.602567
$$636$$ 17.6128 0.698395
$$637$$ 0 0
$$638$$ −14.8573 −0.588205
$$639$$ 2.75557 0.109009
$$640$$ −5.64941 −0.223313
$$641$$ 25.8163 1.01968 0.509841 0.860269i $$-0.329704\pi$$
0.509841 + 0.860269i $$0.329704\pi$$
$$642$$ −4.99063 −0.196965
$$643$$ −18.1017 −0.713862 −0.356931 0.934131i $$-0.616177\pi$$
−0.356931 + 0.934131i $$0.616177\pi$$
$$644$$ 0 0
$$645$$ 5.67307 0.223377
$$646$$ −69.5683 −2.73713
$$647$$ 47.0420 1.84941 0.924705 0.380684i $$-0.124311\pi$$
0.924705 + 0.380684i $$0.124311\pi$$
$$648$$ −0.719004 −0.0282451
$$649$$ 4.85728 0.190665
$$650$$ 1.18421 0.0464485
$$651$$ 0 0
$$652$$ −16.3872 −0.641770
$$653$$ 30.0830 1.17724 0.588619 0.808411i $$-0.299672\pi$$
0.588619 + 0.808411i $$0.299672\pi$$
$$654$$ 37.5210 1.46719
$$655$$ 1.24443 0.0486240
$$656$$ −0.892937 −0.0348633
$$657$$ −4.23506 −0.165226
$$658$$ 0 0
$$659$$ −10.2854 −0.400664 −0.200332 0.979728i $$-0.564202\pi$$
−0.200332 + 0.979728i $$0.564202\pi$$
$$660$$ 1.62222 0.0631447
$$661$$ 27.7146 1.07797 0.538986 0.842315i $$-0.318808\pi$$
0.538986 + 0.842315i $$0.318808\pi$$
$$662$$ 29.2070 1.13516
$$663$$ −3.22570 −0.125276
$$664$$ 0.0958814 0.00372092
$$665$$ 0 0
$$666$$ −3.80642 −0.147496
$$667$$ −69.1437 −2.67725
$$668$$ 26.5018 1.02538
$$669$$ 8.85728 0.342442
$$670$$ 2.36842 0.0914999
$$671$$ −6.85728 −0.264722
$$672$$ 0 0
$$673$$ −9.86665 −0.380331 −0.190166 0.981752i $$-0.560903\pi$$
−0.190166 + 0.981752i $$0.560903\pi$$
$$674$$ 53.7373 2.06988
$$675$$ −1.00000 −0.0384900
$$676$$ −20.4608 −0.786952
$$677$$ 5.65433 0.217314 0.108657 0.994079i $$-0.465345\pi$$
0.108657 + 0.994079i $$0.465345\pi$$
$$678$$ 11.4193 0.438554
$$679$$ 0 0
$$680$$ 3.72746 0.142942
$$681$$ 13.3778 0.512638
$$682$$ −5.24443 −0.200820
$$683$$ 34.1847 1.30804 0.654020 0.756477i $$-0.273081\pi$$
0.654020 + 0.756477i $$0.273081\pi$$
$$684$$ −11.4380 −0.437343
$$685$$ 0.488863 0.0186785
$$686$$ 0 0
$$687$$ 11.5111 0.439177
$$688$$ −26.1690 −0.997684
$$689$$ −6.75557 −0.257367
$$690$$ 16.8573 0.641746
$$691$$ 19.2257 0.731380 0.365690 0.930737i $$-0.380833\pi$$
0.365690 + 0.930737i $$0.380833\pi$$
$$692$$ 14.8988 0.566366
$$693$$ 0 0
$$694$$ 4.99063 0.189442
$$695$$ 17.8064 0.675436
$$696$$ −5.61285 −0.212754
$$697$$ 1.00354 0.0380118
$$698$$ −9.78769 −0.370469
$$699$$ 4.32693 0.163659
$$700$$ 0 0
$$701$$ −29.9081 −1.12961 −0.564807 0.825223i $$-0.691050\pi$$
−0.564807 + 0.825223i $$0.691050\pi$$
$$702$$ −1.18421 −0.0446951
$$703$$ 14.1017 0.531856
$$704$$ −4.74620 −0.178879
$$705$$ 2.75557 0.103781
$$706$$ 17.7877 0.669448
$$707$$ 0 0
$$708$$ −7.87955 −0.296132
$$709$$ −15.3274 −0.575633 −0.287816 0.957686i $$-0.592929\pi$$
−0.287816 + 0.957686i $$0.592929\pi$$
$$710$$ −5.24443 −0.196820
$$711$$ 8.56199 0.321100
$$712$$ 4.03566 0.151243
$$713$$ −24.4068 −0.914043
$$714$$ 0 0
$$715$$ −0.622216 −0.0232695
$$716$$ −41.0865 −1.53548
$$717$$ 3.34614 0.124964
$$718$$ 20.4701 0.763938
$$719$$ −23.8163 −0.888197 −0.444098 0.895978i $$-0.646476\pi$$
−0.444098 + 0.895978i $$0.646476\pi$$
$$720$$ 4.61285 0.171911
$$721$$ 0 0
$$722$$ 58.4563 2.17552
$$723$$ −1.34614 −0.0500635
$$724$$ 22.0830 0.820707
$$725$$ −7.80642 −0.289923
$$726$$ −1.90321 −0.0706348
$$727$$ 32.9403 1.22169 0.610843 0.791752i $$-0.290831\pi$$
0.610843 + 0.791752i $$0.290831\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 8.06022 0.298322
$$731$$ 29.4104 1.08778
$$732$$ 11.1240 0.411154
$$733$$ 29.8666 1.10315 0.551575 0.834125i $$-0.314027\pi$$
0.551575 + 0.834125i $$0.314027\pi$$
$$734$$ −64.3239 −2.37424
$$735$$ 0 0
$$736$$ −65.0232 −2.39679
$$737$$ −1.24443 −0.0458392
$$738$$ 0.368416 0.0135616
$$739$$ −5.06959 −0.186488 −0.0932440 0.995643i $$-0.529724\pi$$
−0.0932440 + 0.995643i $$0.529724\pi$$
$$740$$ 3.24443 0.119268
$$741$$ 4.38715 0.161166
$$742$$ 0 0
$$743$$ 22.4385 0.823188 0.411594 0.911367i $$-0.364972\pi$$
0.411594 + 0.911367i $$0.364972\pi$$
$$744$$ −1.98126 −0.0726367
$$745$$ 1.43801 0.0526845
$$746$$ 64.6133 2.36566
$$747$$ −0.133353 −0.00487913
$$748$$ 8.40990 0.307497
$$749$$ 0 0
$$750$$ 1.90321 0.0694955
$$751$$ −6.63512 −0.242119 −0.121060 0.992645i $$-0.538629\pi$$
−0.121060 + 0.992645i $$0.538629\pi$$
$$752$$ −12.7110 −0.463523
$$753$$ 22.7556 0.829259
$$754$$ −9.24443 −0.336662
$$755$$ 12.1748 0.443088
$$756$$ 0 0
$$757$$ 8.75557 0.318227 0.159113 0.987260i $$-0.449136\pi$$
0.159113 + 0.987260i $$0.449136\pi$$
$$758$$ −38.0642 −1.38256
$$759$$ −8.85728 −0.321499
$$760$$ −5.06959 −0.183893
$$761$$ −3.15257 −0.114280 −0.0571402 0.998366i $$-0.518198\pi$$
−0.0571402 + 0.998366i $$0.518198\pi$$
$$762$$ 28.8988 1.04689
$$763$$ 0 0
$$764$$ 9.89829 0.358108
$$765$$ −5.18421 −0.187435
$$766$$ 27.8537 1.00640
$$767$$ 3.02227 0.109128
$$768$$ −20.2444 −0.730508
$$769$$ 28.9590 1.04429 0.522144 0.852857i $$-0.325132\pi$$
0.522144 + 0.852857i $$0.325132\pi$$
$$770$$ 0 0
$$771$$ −6.85728 −0.246959
$$772$$ 29.7462 1.07059
$$773$$ −29.1427 −1.04819 −0.524095 0.851660i $$-0.675596\pi$$
−0.524095 + 0.851660i $$0.675596\pi$$
$$774$$ 10.7971 0.388092
$$775$$ −2.75557 −0.0989830
$$776$$ 5.20877 0.186984
$$777$$ 0 0
$$778$$ −10.6824 −0.382984
$$779$$ −1.36488 −0.0489018
$$780$$ 1.00937 0.0361412
$$781$$ 2.75557 0.0986020
$$782$$ 87.3916 3.12512
$$783$$ 7.80642 0.278979
$$784$$ 0 0
$$785$$ 18.4701 0.659227
$$786$$ 2.36842 0.0844786
$$787$$ 11.2672 0.401632 0.200816 0.979629i $$-0.435641\pi$$
0.200816 + 0.979629i $$0.435641\pi$$
$$788$$ −10.8613 −0.386918
$$789$$ 29.5812 1.05312
$$790$$ −16.2953 −0.579760
$$791$$ 0 0
$$792$$ −0.719004 −0.0255487
$$793$$ −4.26671 −0.151515
$$794$$ 24.2766 0.861543
$$795$$ −10.8573 −0.385068
$$796$$ −22.8760 −0.810819
$$797$$ −41.9625 −1.48639 −0.743195 0.669075i $$-0.766690\pi$$
−0.743195 + 0.669075i $$0.766690\pi$$
$$798$$ 0 0
$$799$$ 14.2854 0.505383
$$800$$ −7.34122 −0.259551
$$801$$ −5.61285 −0.198320
$$802$$ 3.80642 0.134409
$$803$$ −4.23506 −0.149452
$$804$$ 2.01874 0.0711953
$$805$$ 0 0
$$806$$ −3.26317 −0.114940
$$807$$ 8.48886 0.298822
$$808$$ 3.35322 0.117966
$$809$$ −27.8064 −0.977622 −0.488811 0.872390i $$-0.662569\pi$$
−0.488811 + 0.872390i $$0.662569\pi$$
$$810$$ −1.90321 −0.0668721
$$811$$ −6.78415 −0.238224 −0.119112 0.992881i $$-0.538005\pi$$
−0.119112 + 0.992881i $$0.538005\pi$$
$$812$$ 0 0
$$813$$ 14.6637 0.514279
$$814$$ −3.80642 −0.133415
$$815$$ 10.1017 0.353847
$$816$$ 23.9140 0.837156
$$817$$ −40.0000 −1.39942
$$818$$ 13.5585 0.474060
$$819$$ 0 0
$$820$$ −0.314022 −0.0109661
$$821$$ 3.62269 0.126433 0.0632164 0.998000i $$-0.479864\pi$$
0.0632164 + 0.998000i $$0.479864\pi$$
$$822$$ 0.930409 0.0324517
$$823$$ 42.0642 1.46627 0.733134 0.680085i $$-0.238057\pi$$
0.733134 + 0.680085i $$0.238057\pi$$
$$824$$ −8.34968 −0.290875
$$825$$ −1.00000 −0.0348155
$$826$$ 0 0
$$827$$ 30.8256 1.07191 0.535956 0.844246i $$-0.319951\pi$$
0.535956 + 0.844246i $$0.319951\pi$$
$$828$$ 14.3684 0.499337
$$829$$ −7.12399 −0.247426 −0.123713 0.992318i $$-0.539480\pi$$
−0.123713 + 0.992318i $$0.539480\pi$$
$$830$$ 0.253799 0.00880950
$$831$$ −14.6035 −0.506589
$$832$$ −2.95316 −0.102382
$$833$$ 0 0
$$834$$ 33.8894 1.17349
$$835$$ −16.3368 −0.565357
$$836$$ −11.4380 −0.395592
$$837$$ 2.75557 0.0952464
$$838$$ −29.7146 −1.02647
$$839$$ −3.34614 −0.115522 −0.0577608 0.998330i $$-0.518396\pi$$
−0.0577608 + 0.998330i $$0.518396\pi$$
$$840$$ 0 0
$$841$$ 31.9403 1.10139
$$842$$ 15.0321 0.518041
$$843$$ 0.193576 0.00666712
$$844$$ −17.2988 −0.595450
$$845$$ 12.6128 0.433895
$$846$$ 5.24443 0.180307
$$847$$ 0 0
$$848$$ 50.0830 1.71986
$$849$$ 27.1842 0.932960
$$850$$ 9.86665 0.338423
$$851$$ −17.7146 −0.607247
$$852$$ −4.47013 −0.153144
$$853$$ 26.4197 0.904595 0.452297 0.891867i $$-0.350605\pi$$
0.452297 + 0.891867i $$0.350605\pi$$
$$854$$ 0 0
$$855$$ 7.05086 0.241134
$$856$$ −1.88538 −0.0644411
$$857$$ −38.7783 −1.32464 −0.662321 0.749220i $$-0.730429\pi$$
−0.662321 + 0.749220i $$0.730429\pi$$
$$858$$ −1.18421 −0.0404282
$$859$$ 27.3087 0.931760 0.465880 0.884848i $$-0.345738\pi$$
0.465880 + 0.884848i $$0.345738\pi$$
$$860$$ −9.20294 −0.313818
$$861$$ 0 0
$$862$$ 65.2899 2.22378
$$863$$ −49.5308 −1.68605 −0.843024 0.537875i $$-0.819227\pi$$
−0.843024 + 0.537875i $$0.819227\pi$$
$$864$$ 7.34122 0.249753
$$865$$ −9.18421 −0.312272
$$866$$ −27.5397 −0.935838
$$867$$ −9.87601 −0.335407
$$868$$ 0 0
$$869$$ 8.56199 0.290446
$$870$$ −14.8573 −0.503709
$$871$$ −0.774305 −0.0262363
$$872$$ 14.1748 0.480021
$$873$$ −7.24443 −0.245187
$$874$$ −118.858 −4.02044
$$875$$ 0 0
$$876$$ 6.87019 0.232122
$$877$$ −4.50177 −0.152014 −0.0760070 0.997107i $$-0.524217\pi$$
−0.0760070 + 0.997107i $$0.524217\pi$$
$$878$$ 36.7654 1.24077
$$879$$ −2.81579 −0.0949743
$$880$$ 4.61285 0.155499
$$881$$ 15.1240 0.509540 0.254770 0.967002i $$-0.418000\pi$$
0.254770 + 0.967002i $$0.418000\pi$$
$$882$$ 0 0
$$883$$ −30.2480 −1.01793 −0.508963 0.860789i $$-0.669971\pi$$
−0.508963 + 0.860789i $$0.669971\pi$$
$$884$$ 5.23277 0.175997
$$885$$ 4.85728 0.163276
$$886$$ −24.9777 −0.839143
$$887$$ −57.1941 −1.92039 −0.960194 0.279333i $$-0.909887\pi$$
−0.960194 + 0.279333i $$0.909887\pi$$
$$888$$ −1.43801 −0.0482564
$$889$$ 0 0
$$890$$ 10.6824 0.358076
$$891$$ 1.00000 0.0335013
$$892$$ −14.3684 −0.481090
$$893$$ −19.4291 −0.650171
$$894$$ 2.73683 0.0915334
$$895$$ 25.3274 0.846602
$$896$$ 0 0
$$897$$ −5.51114 −0.184012
$$898$$ −61.4835 −2.05173
$$899$$ 21.5111 0.717437
$$900$$ 1.62222 0.0540739
$$901$$ −56.2864 −1.87517
$$902$$ 0.368416 0.0122669
$$903$$ 0 0
$$904$$ 4.31402 0.143482
$$905$$ −13.6128 −0.452506
$$906$$ 23.1713 0.769815
$$907$$ −53.2641 −1.76861 −0.884303 0.466913i $$-0.845366\pi$$
−0.884303 + 0.466913i $$0.845366\pi$$
$$908$$ −21.7017 −0.720195
$$909$$ −4.66370 −0.154685
$$910$$ 0 0
$$911$$ −0.590573 −0.0195665 −0.00978327 0.999952i $$-0.503114\pi$$
−0.00978327 + 0.999952i $$0.503114\pi$$
$$912$$ −32.5245 −1.07699
$$913$$ −0.133353 −0.00441334
$$914$$ −44.6508 −1.47692
$$915$$ −6.85728 −0.226695
$$916$$ −18.6735 −0.616991
$$917$$ 0 0
$$918$$ −9.86665 −0.325648
$$919$$ 55.8707 1.84300 0.921502 0.388375i $$-0.126963\pi$$
0.921502 + 0.388375i $$0.126963\pi$$
$$920$$ 6.36842 0.209960
$$921$$ −24.4286 −0.804951
$$922$$ 54.9403 1.80936
$$923$$ 1.71456 0.0564354
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ −36.8198 −1.20997
$$927$$ 11.6128 0.381416
$$928$$ 57.3087 1.88125
$$929$$ −15.3274 −0.502876 −0.251438 0.967873i $$-0.580903\pi$$
−0.251438 + 0.967873i $$0.580903\pi$$
$$930$$ −5.24443 −0.171972
$$931$$ 0 0
$$932$$ −7.01921 −0.229922
$$933$$ 19.8796 0.650827
$$934$$ −5.98126 −0.195713
$$935$$ −5.18421 −0.169542
$$936$$ −0.447375 −0.0146229
$$937$$ 27.8479 0.909752 0.454876 0.890555i $$-0.349684\pi$$
0.454876 + 0.890555i $$0.349684\pi$$
$$938$$ 0 0
$$939$$ −15.7146 −0.512825
$$940$$ −4.47013 −0.145799
$$941$$ −10.4157 −0.339543 −0.169772 0.985483i $$-0.554303\pi$$
−0.169772 + 0.985483i $$0.554303\pi$$
$$942$$ 35.1526 1.14533
$$943$$ 1.71456 0.0558337
$$944$$ −22.4059 −0.729250
$$945$$ 0 0
$$946$$ 10.7971 0.351043
$$947$$ 8.47013 0.275242 0.137621 0.990485i $$-0.456054\pi$$
0.137621 + 0.990485i $$0.456054\pi$$
$$948$$ −13.8894 −0.451107
$$949$$ −2.63512 −0.0855397
$$950$$ −13.4193 −0.435379
$$951$$ −16.4889 −0.534688
$$952$$ 0 0
$$953$$ −8.71408 −0.282277 −0.141138 0.989990i $$-0.545076\pi$$
−0.141138 + 0.989990i $$0.545076\pi$$
$$954$$ −20.6637 −0.669012
$$955$$ −6.10171 −0.197447
$$956$$ −5.42816 −0.175559
$$957$$ 7.80642 0.252346
$$958$$ 47.3087 1.52847
$$959$$ 0 0
$$960$$ −4.74620 −0.153183
$$961$$ −23.4068 −0.755059
$$962$$ −2.36842 −0.0763608
$$963$$ 2.62222 0.0844997
$$964$$ 2.18373 0.0703333
$$965$$ −18.3368 −0.590282
$$966$$ 0 0
$$967$$ 44.2449 1.42282 0.711410 0.702777i $$-0.248057\pi$$
0.711410 + 0.702777i $$0.248057\pi$$
$$968$$ −0.719004 −0.0231097
$$969$$ 36.5531 1.17425
$$970$$ 13.7877 0.442696
$$971$$ 57.1437 1.83383 0.916914 0.399085i $$-0.130672\pi$$
0.916914 + 0.399085i $$0.130672\pi$$
$$972$$ −1.62222 −0.0520326
$$973$$ 0 0
$$974$$ −21.9438 −0.703124
$$975$$ −0.622216 −0.0199268
$$976$$ 31.6316 1.01250
$$977$$ −16.2480 −0.519819 −0.259909 0.965633i $$-0.583693\pi$$
−0.259909 + 0.965633i $$0.583693\pi$$
$$978$$ 19.2257 0.614770
$$979$$ −5.61285 −0.179387
$$980$$ 0 0
$$981$$ −19.7146 −0.629437
$$982$$ 31.1882 0.995256
$$983$$ −1.12399 −0.0358496 −0.0179248 0.999839i $$-0.505706\pi$$
−0.0179248 + 0.999839i $$0.505706\pi$$
$$984$$ 0.139182 0.00443696
$$985$$ 6.69535 0.213331
$$986$$ −77.0232 −2.45292
$$987$$ 0 0
$$988$$ −7.11691 −0.226419
$$989$$ 50.2480 1.59779
$$990$$ −1.90321 −0.0604880
$$991$$ −53.6513 −1.70429 −0.852144 0.523307i $$-0.824698\pi$$
−0.852144 + 0.523307i $$0.824698\pi$$
$$992$$ 20.2292 0.642279
$$993$$ −15.3461 −0.486995
$$994$$ 0 0
$$995$$ 14.1017 0.447054
$$996$$ 0.216327 0.00685460
$$997$$ 35.7275 1.13150 0.565750 0.824577i $$-0.308587\pi$$
0.565750 + 0.824577i $$0.308587\pi$$
$$998$$ −48.2034 −1.52585
$$999$$ 2.00000 0.0632772
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 8085.2.a.bk.1.3 3
7.6 odd 2 165.2.a.c.1.3 3
21.20 even 2 495.2.a.e.1.1 3
28.27 even 2 2640.2.a.be.1.3 3
35.13 even 4 825.2.c.g.199.2 6
35.27 even 4 825.2.c.g.199.5 6
35.34 odd 2 825.2.a.k.1.1 3
77.76 even 2 1815.2.a.m.1.1 3
84.83 odd 2 7920.2.a.cj.1.3 3
105.62 odd 4 2475.2.c.r.199.2 6
105.83 odd 4 2475.2.c.r.199.5 6
105.104 even 2 2475.2.a.bb.1.3 3
231.230 odd 2 5445.2.a.z.1.3 3
385.384 even 2 9075.2.a.cf.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 3 7.6 odd 2
495.2.a.e.1.1 3 21.20 even 2
825.2.a.k.1.1 3 35.34 odd 2
825.2.c.g.199.2 6 35.13 even 4
825.2.c.g.199.5 6 35.27 even 4
1815.2.a.m.1.1 3 77.76 even 2
2475.2.a.bb.1.3 3 105.104 even 2
2475.2.c.r.199.2 6 105.62 odd 4
2475.2.c.r.199.5 6 105.83 odd 4
2640.2.a.be.1.3 3 28.27 even 2
5445.2.a.z.1.3 3 231.230 odd 2
7920.2.a.cj.1.3 3 84.83 odd 2
8085.2.a.bk.1.3 3 1.1 even 1 trivial
9075.2.a.cf.1.3 3 385.384 even 2