# Properties

 Label 165.2.a.c.1.3 Level $165$ Weight $2$ Character 165.1 Self dual yes Analytic conductor $1.318$ Analytic rank $0$ 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] = [165,2,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, 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("165.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.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: $$1.31753163335$$ Analytic rank: $$0$$ 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 165.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.90321 q^{2} +1.00000 q^{3} +1.62222 q^{4} +1.00000 q^{5} +1.90321 q^{6} -4.42864 q^{7} -0.719004 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.90321 q^{2} +1.00000 q^{3} +1.62222 q^{4} +1.00000 q^{5} +1.90321 q^{6} -4.42864 q^{7} -0.719004 q^{8} +1.00000 q^{9} +1.90321 q^{10} +1.00000 q^{11} +1.62222 q^{12} -0.622216 q^{13} -8.42864 q^{14} +1.00000 q^{15} -4.61285 q^{16} -5.18421 q^{17} +1.90321 q^{18} +7.05086 q^{19} +1.62222 q^{20} -4.42864 q^{21} +1.90321 q^{22} +8.85728 q^{23} -0.719004 q^{24} +1.00000 q^{25} -1.18421 q^{26} +1.00000 q^{27} -7.18421 q^{28} -7.80642 q^{29} +1.90321 q^{30} +2.75557 q^{31} -7.34122 q^{32} +1.00000 q^{33} -9.86665 q^{34} -4.42864 q^{35} +1.62222 q^{36} -2.00000 q^{37} +13.4193 q^{38} -0.622216 q^{39} -0.719004 q^{40} -0.193576 q^{41} -8.42864 q^{42} +5.67307 q^{43} +1.62222 q^{44} +1.00000 q^{45} +16.8573 q^{46} -2.75557 q^{47} -4.61285 q^{48} +12.6128 q^{49} +1.90321 q^{50} -5.18421 q^{51} -1.00937 q^{52} -10.8573 q^{53} +1.90321 q^{54} +1.00000 q^{55} +3.18421 q^{56} +7.05086 q^{57} -14.8573 q^{58} -4.85728 q^{59} +1.62222 q^{60} +6.85728 q^{61} +5.24443 q^{62} -4.42864 q^{63} -4.74620 q^{64} -0.622216 q^{65} +1.90321 q^{66} -1.24443 q^{67} -8.40990 q^{68} +8.85728 q^{69} -8.42864 q^{70} +2.75557 q^{71} -0.719004 q^{72} +4.23506 q^{73} -3.80642 q^{74} +1.00000 q^{75} +11.4380 q^{76} -4.42864 q^{77} -1.18421 q^{78} +8.56199 q^{79} -4.61285 q^{80} +1.00000 q^{81} -0.368416 q^{82} +0.133353 q^{83} -7.18421 q^{84} -5.18421 q^{85} +10.7971 q^{86} -7.80642 q^{87} -0.719004 q^{88} +5.61285 q^{89} +1.90321 q^{90} +2.75557 q^{91} +14.3684 q^{92} +2.75557 q^{93} -5.24443 q^{94} +7.05086 q^{95} -7.34122 q^{96} +7.24443 q^{97} +24.0049 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 - 9 * q^8 + 3 * q^9 $$3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 9 q^{8} + 3 q^{9} - q^{10} + 3 q^{11} + 5 q^{12} - 2 q^{13} - 12 q^{14} + 3 q^{15} + 13 q^{16} - 2 q^{17} - q^{18} + 8 q^{19} + 5 q^{20} - q^{22} - 9 q^{24} + 3 q^{25} + 10 q^{26} + 3 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} + 8 q^{31} - 29 q^{32} + 3 q^{33} - 30 q^{34} + 5 q^{36} - 6 q^{37} - 2 q^{39} - 9 q^{40} - 14 q^{41} - 12 q^{42} + 4 q^{43} + 5 q^{44} + 3 q^{45} + 24 q^{46} - 8 q^{47} + 13 q^{48} + 11 q^{49} - q^{50} - 2 q^{51} - 30 q^{52} - 6 q^{53} - q^{54} + 3 q^{55} - 4 q^{56} + 8 q^{57} - 18 q^{58} + 12 q^{59} + 5 q^{60} - 6 q^{61} + 16 q^{62} + 13 q^{64} - 2 q^{65} - q^{66} - 4 q^{67} + 42 q^{68} - 12 q^{70} + 8 q^{71} - 9 q^{72} - 14 q^{73} + 2 q^{74} + 3 q^{75} + 48 q^{76} + 10 q^{78} + 12 q^{79} + 13 q^{80} + 3 q^{81} + 26 q^{82} - 8 q^{84} - 2 q^{85} - 8 q^{86} - 10 q^{87} - 9 q^{88} - 10 q^{89} - q^{90} + 8 q^{91} + 16 q^{92} + 8 q^{93} - 16 q^{94} + 8 q^{95} - 29 q^{96} + 22 q^{97} + 39 q^{98} + 3 q^{99}+O(q^{100})$$ 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 - 12 * q^14 + 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 - 8 * q^28 - 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 - 12 * q^42 + 4 * q^43 + 5 * q^44 + 3 * q^45 + 24 * q^46 - 8 * q^47 + 13 * q^48 + 11 * q^49 - q^50 - 2 * q^51 - 30 * q^52 - 6 * q^53 - q^54 + 3 * q^55 - 4 * q^56 + 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 - 12 * q^70 + 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 - 8 * q^84 - 2 * q^85 - 8 * q^86 - 10 * q^87 - 9 * q^88 - 10 * q^89 - q^90 + 8 * q^91 + 16 * q^92 + 8 * q^93 - 16 * q^94 + 8 * q^95 - 29 * q^96 + 22 * q^97 + 39 * q^98 + 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
$$7$$ −4.42864 −1.67387 −0.836934 0.547304i $$-0.815654\pi$$
−0.836934 + 0.547304i $$0.815654\pi$$
$$8$$ −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 $$-0.527500\pi$$
−0.0862858 + 0.996270i $$0.527500\pi$$
$$14$$ −8.42864 −2.25265
$$15$$ 1.00000 0.258199
$$16$$ −4.61285 −1.15321
$$17$$ −5.18421 −1.25736 −0.628678 0.777666i $$-0.716403\pi$$
−0.628678 + 0.777666i $$0.716403\pi$$
$$18$$ 1.90321 0.448591
$$19$$ 7.05086 1.61758 0.808789 0.588100i $$-0.200124\pi$$
0.808789 + 0.588100i $$0.200124\pi$$
$$20$$ 1.62222 0.362738
$$21$$ −4.42864 −0.966408
$$22$$ 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
$$28$$ −7.18421 −1.35769
$$29$$ −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 $$-0.420405\pi$$
0.247457 + 0.968899i $$0.420405\pi$$
$$32$$ −7.34122 −1.29776
$$33$$ 1.00000 0.174078
$$34$$ −9.86665 −1.69212
$$35$$ −4.42864 −0.748577
$$36$$ 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 $$-0.504812\pi$$
−0.0151158 + 0.999886i $$0.504812\pi$$
$$42$$ −8.42864 −1.30057
$$43$$ 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 $$-0.564410\pi$$
−0.200971 + 0.979597i $$0.564410\pi$$
$$48$$ −4.61285 −0.665807
$$49$$ 12.6128 1.80184
$$50$$ 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
$$56$$ 3.18421 0.425508
$$57$$ 7.05086 0.933909
$$58$$ −14.8573 −1.95086
$$59$$ −4.85728 −0.632364 −0.316182 0.948699i $$-0.602401\pi$$
−0.316182 + 0.948699i $$0.602401\pi$$
$$60$$ 1.62222 0.209427
$$61$$ 6.85728 0.877985 0.438992 0.898491i $$-0.355336\pi$$
0.438992 + 0.898491i $$0.355336\pi$$
$$62$$ 5.24443 0.666043
$$63$$ −4.42864 −0.557956
$$64$$ −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
$$70$$ −8.42864 −1.00742
$$71$$ 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 $$-0.420280\pi$$
0.247838 + 0.968801i $$0.420280\pi$$
$$74$$ −3.80642 −0.442488
$$75$$ 1.00000 0.115470
$$76$$ 11.4380 1.31203
$$77$$ −4.42864 −0.504690
$$78$$ −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 $$-0.497670\pi$$
0.00731870 + 0.999973i $$0.497670\pi$$
$$84$$ −7.18421 −0.783861
$$85$$ −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 $$-0.403854\pi$$
0.297480 + 0.954728i $$0.403854\pi$$
$$90$$ 1.90321 0.200616
$$91$$ 2.75557 0.288862
$$92$$ 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 $$-0.380118\pi$$
0.367780 + 0.929913i $$0.380118\pi$$
$$98$$ 24.0049 2.42486
$$99$$ 1.00000 0.100504
$$100$$ 1.62222 0.162222
$$101$$ 4.66370 0.464056 0.232028 0.972709i $$-0.425464\pi$$
0.232028 + 0.972709i $$0.425464\pi$$
$$102$$ −9.86665 −0.976944
$$103$$ −11.6128 −1.14425 −0.572124 0.820167i $$-0.693880\pi$$
−0.572124 + 0.820167i $$0.693880\pi$$
$$104$$ 0.447375 0.0438688
$$105$$ −4.42864 −0.432191
$$106$$ −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
$$112$$ 20.4286 1.93032
$$113$$ −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
$$119$$ 22.9590 2.10465
$$120$$ −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
$$126$$ −8.42864 −0.750883
$$127$$ −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 $$-0.482687\pi$$
0.0543632 + 0.998521i $$0.482687\pi$$
$$132$$ 1.62222 0.141196
$$133$$ −31.2257 −2.70761
$$134$$ −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 $$-0.227559\pi$$
0.755161 + 0.655540i $$0.227559\pi$$
$$140$$ −7.18421 −0.607176
$$141$$ −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
$$147$$ 12.6128 1.04029
$$148$$ −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
$$154$$ −8.42864 −0.679199
$$155$$ 2.75557 0.221333
$$156$$ −1.00937 −0.0808141
$$157$$ 18.4701 1.47408 0.737038 0.675851i $$-0.236224\pi$$
0.737038 + 0.675851i $$0.236224\pi$$
$$158$$ 16.2953 1.29638
$$159$$ −10.8573 −0.861038
$$160$$ −7.34122 −0.580374
$$161$$ −39.2257 −3.09142
$$162$$ 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 $$-0.717802\pi$$
−0.632089 + 0.774896i $$0.717802\pi$$
$$168$$ 3.18421 0.245667
$$169$$ −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 $$-0.613523\pi$$
−0.349131 + 0.937074i $$0.613523\pi$$
$$174$$ −14.8573 −1.12633
$$175$$ −4.42864 −0.334774
$$176$$ −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 $$-0.668846\pi$$
−0.505918 + 0.862582i $$0.668846\pi$$
$$182$$ 5.24443 0.388743
$$183$$ 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
$$189$$ −4.42864 −0.322136
$$190$$ 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
$$196$$ 20.4608 1.46148
$$197$$ −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 $$-0.333399\pi$$
0.499822 + 0.866128i $$0.333399\pi$$
$$200$$ −0.719004 −0.0508412
$$201$$ −1.24443 −0.0877754
$$202$$ 8.87601 0.624514
$$203$$ 34.5718 2.42647
$$204$$ −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
$$210$$ −8.42864 −0.581631
$$211$$ −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
$$217$$ −12.2034 −0.828422
$$218$$ −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 $$-0.404159\pi$$
0.296564 + 0.955013i $$0.404159\pi$$
$$224$$ 32.5116 2.17227
$$225$$ 1.00000 0.0666667
$$226$$ −11.4193 −0.759599
$$227$$ 13.3778 0.887915 0.443957 0.896048i $$-0.353574\pi$$
0.443957 + 0.896048i $$0.353574\pi$$
$$228$$ 11.4380 0.757501
$$229$$ 11.5111 0.760677 0.380339 0.924847i $$-0.375807\pi$$
0.380339 + 0.924847i $$0.375807\pi$$
$$230$$ 16.8573 1.11154
$$231$$ −4.42864 −0.291383
$$232$$ 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
$$238$$ 43.6958 2.83238
$$239$$ −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 $$-0.513805\pi$$
−0.0433563 + 0.999060i $$0.513805\pi$$
$$242$$ 1.90321 0.122343
$$243$$ 1.00000 0.0641500
$$244$$ 11.1240 0.712140
$$245$$ 12.6128 0.805805
$$246$$ −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 $$-0.244985\pi$$
0.718159 + 0.695879i $$0.244985\pi$$
$$252$$ −7.18421 −0.452563
$$253$$ 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 $$-0.568608\pi$$
−0.213873 + 0.976862i $$0.568608\pi$$
$$258$$ 10.7971 0.672195
$$259$$ 8.85728 0.550365
$$260$$ −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
$$266$$ −59.4291 −3.64383
$$267$$ 5.61285 0.343501
$$268$$ −2.01874 −0.123314
$$269$$ 8.48886 0.517575 0.258788 0.965934i $$-0.416677\pi$$
0.258788 + 0.965934i $$0.416677\pi$$
$$270$$ 1.90321 0.115826
$$271$$ 14.6637 0.890757 0.445378 0.895343i $$-0.353069\pi$$
0.445378 + 0.895343i $$0.353069\pi$$
$$272$$ 23.9140 1.45000
$$273$$ 2.75557 0.166775
$$274$$ −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
$$280$$ 3.18421 0.190293
$$281$$ −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 $$-0.200568\pi$$
0.807967 + 0.589228i $$0.200568\pi$$
$$284$$ 4.47013 0.265253
$$285$$ 7.05086 0.417657
$$286$$ −1.18421 −0.0700237
$$287$$ 0.857279 0.0506036
$$288$$ −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 $$-0.526211\pi$$
−0.0822502 + 0.996612i $$0.526211\pi$$
$$294$$ 24.0049 1.40000
$$295$$ −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
$$301$$ −25.1240 −1.44812
$$302$$ −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 $$-0.745530\pi$$
−0.697108 + 0.716966i $$0.745530\pi$$
$$308$$ −7.18421 −0.409358
$$309$$ −11.6128 −0.660632
$$310$$ 5.24443 0.297864
$$311$$ 19.8796 1.12727 0.563633 0.826025i $$-0.309403\pi$$
0.563633 + 0.826025i $$0.309403\pi$$
$$312$$ 0.447375 0.0253276
$$313$$ −15.7146 −0.888239 −0.444120 0.895967i $$-0.646483\pi$$
−0.444120 + 0.895967i $$0.646483\pi$$
$$314$$ 35.1526 1.98377
$$315$$ −4.42864 −0.249526
$$316$$ 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
$$322$$ −74.6548 −4.16035
$$323$$ −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
$$329$$ 12.2034 0.672796
$$330$$ 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
$$336$$ 20.4286 1.11447
$$337$$ 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
$$343$$ −24.8573 −1.34217
$$344$$ −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 $$-0.456048\pi$$
0.137642 + 0.990482i $$0.456048\pi$$
$$350$$ −8.42864 −0.450530
$$351$$ −0.622216 −0.0332114
$$352$$ −7.34122 −0.391288
$$353$$ −9.34614 −0.497445 −0.248722 0.968575i $$-0.580011\pi$$
−0.248722 + 0.968575i $$0.580011\pi$$
$$354$$ −9.24443 −0.491336
$$355$$ 2.75557 0.146250
$$356$$ 9.10525 0.482577
$$357$$ 22.9590 1.21512
$$358$$ −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
$$364$$ 4.47013 0.234298
$$365$$ 4.23506 0.221673
$$366$$ 13.0509 0.682179
$$367$$ 33.7975 1.76422 0.882108 0.471046i $$-0.156124\pi$$
0.882108 + 0.471046i $$0.156124\pi$$
$$368$$ −40.8573 −2.12983
$$369$$ −0.193576 −0.0100772
$$370$$ −3.80642 −0.197887
$$371$$ 48.0830 2.49634
$$372$$ 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
$$378$$ −8.42864 −0.433522
$$379$$ −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 $$-0.621983\pi$$
−0.373910 + 0.927465i $$0.621983\pi$$
$$384$$ 5.64941 0.288295
$$385$$ −4.42864 −0.225704
$$386$$ 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
$$392$$ −9.06868 −0.458038
$$393$$ 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 $$-0.603714\pi$$
−0.320092 + 0.947387i $$0.603714\pi$$
$$398$$ 26.8385 1.34529
$$399$$ −31.2257 −1.56324
$$400$$ −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
$$406$$ 65.7975 3.26548
$$407$$ −2.00000 −0.0991363
$$408$$ 3.72746 0.184537
$$409$$ −7.12399 −0.352258 −0.176129 0.984367i $$-0.556358\pi$$
−0.176129 + 0.984367i $$0.556358\pi$$
$$410$$ −0.368416 −0.0181948
$$411$$ −0.488863 −0.0241138
$$412$$ −18.8385 −0.928108
$$413$$ 21.5111 1.05849
$$414$$ 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 $$-0.375453\pi$$
0.381369 + 0.924423i $$0.375453\pi$$
$$420$$ −7.18421 −0.350553
$$421$$ 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
$$427$$ −30.3684 −1.46963
$$428$$ 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 $$-0.386964\pi$$
0.347695 + 0.937608i $$0.386964\pi$$
$$434$$ −23.2257 −1.11487
$$435$$ −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 $$-0.652505\pi$$
−0.460988 + 0.887406i $$0.652505\pi$$
$$440$$ −0.719004 −0.0342772
$$441$$ 12.6128 0.600612
$$442$$ 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
$$448$$ 21.0192 0.993064
$$449$$ −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
$$455$$ 2.75557 0.129183
$$456$$ −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 $$-0.734667\pi$$
−0.672238 + 0.740335i $$0.734667\pi$$
$$462$$ −8.42864 −0.392136
$$463$$ −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 $$-0.476834\pi$$
0.0727139 + 0.997353i $$0.476834\pi$$
$$468$$ −1.00937 −0.0466580
$$469$$ 5.51114 0.254481
$$470$$ −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
$$476$$ 37.2444 1.70710
$$477$$ −10.8573 −0.497121
$$478$$ −6.36842 −0.291285
$$479$$ −24.8573 −1.13576 −0.567879 0.823112i $$-0.692236\pi$$
−0.567879 + 0.823112i $$0.692236\pi$$
$$480$$ −7.34122 −0.335079
$$481$$ 1.24443 0.0567412
$$482$$ −2.56199 −0.116696
$$483$$ −39.2257 −1.78483
$$484$$ 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
$$490$$ 24.0049 1.08443
$$491$$ 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
$$497$$ −12.2034 −0.547398
$$498$$ 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 $$-0.360048\pi$$
0.425643 + 0.904891i $$0.360048\pi$$
$$504$$ 3.18421 0.141836
$$505$$ 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 $$-0.755488\pi$$
−0.719191 + 0.694812i $$0.755488\pi$$
$$510$$ −9.86665 −0.436902
$$511$$ −18.7556 −0.829698
$$512$$ 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
$$518$$ 16.8573 0.740666
$$519$$ −9.18421 −0.403142
$$520$$ 0.447375 0.0196187
$$521$$ −29.2257 −1.28040 −0.640200 0.768208i $$-0.721149\pi$$
−0.640200 + 0.768208i $$0.721149\pi$$
$$522$$ −14.8573 −0.650285
$$523$$ 6.71408 0.293586 0.146793 0.989167i $$-0.453105\pi$$
0.146793 + 0.989167i $$0.453105\pi$$
$$524$$ 2.01874 0.0881889
$$525$$ −4.42864 −0.193282
$$526$$ −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
$$532$$ −50.6548 −2.19616
$$533$$ 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
$$539$$ 12.6128 0.543274
$$540$$ 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
$$546$$ 5.24443 0.224441
$$547$$ 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
$$553$$ −37.9180 −1.61244
$$554$$ 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
$$560$$ 20.4286 0.863268
$$561$$ −5.18421 −0.218877
$$562$$ −0.368416 −0.0155407
$$563$$ 37.7275 1.59002 0.795012 0.606594i $$-0.207465\pi$$
0.795012 + 0.606594i $$0.207465\pi$$
$$564$$ −4.47013 −0.188226
$$565$$ −6.00000 −0.252422
$$566$$ 51.7373 2.17468
$$567$$ −4.42864 −0.185985
$$568$$ −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
$$574$$ 1.63158 0.0681010
$$575$$ 8.85728 0.369374
$$576$$ −4.74620 −0.197758
$$577$$ 4.22216 0.175771 0.0878853 0.996131i $$-0.471989\pi$$
0.0878853 + 0.996131i $$0.471989\pi$$
$$578$$ 18.7961 0.781817
$$579$$ 18.3368 0.762050
$$580$$ −12.6637 −0.525832
$$581$$ −0.590573 −0.0245011
$$582$$ 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 $$-0.249026\pi$$
0.709268 + 0.704939i $$0.249026\pi$$
$$588$$ 20.4608 0.843787
$$589$$ 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 $$-0.305517\pi$$
0.573675 + 0.819083i $$0.305517\pi$$
$$594$$ 1.90321 0.0780897
$$595$$ 22.9590 0.941227
$$596$$ −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 $$-0.557148\pi$$
−0.178574 + 0.983927i $$0.557148\pi$$
$$602$$ −47.8163 −1.94885
$$603$$ −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 $$-0.599711\pi$$
−0.308154 + 0.951336i $$0.599711\pi$$
$$608$$ −51.7619 −2.09922
$$609$$ 34.5718 1.40092
$$610$$ 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
$$616$$ 3.18421 0.128295
$$617$$ −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 $$-0.384963\pi$$
0.353584 + 0.935403i $$0.384963\pi$$
$$620$$ 4.47013 0.179525
$$621$$ 8.85728 0.355430
$$622$$ 37.8350 1.51705
$$623$$ −24.8573 −0.995886
$$624$$ 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
$$630$$ −8.42864 −0.335805
$$631$$ 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
$$637$$ −7.84791 −0.310946
$$638$$ −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 $$-0.383823\pi$$
0.356931 + 0.934131i $$0.383823\pi$$
$$644$$ −63.6325 −2.50747
$$645$$ 5.67307 0.223377
$$646$$ −69.5683 −2.73713
$$647$$ −47.0420 −1.84941 −0.924705 0.380684i $$-0.875689\pi$$
−0.924705 + 0.380684i $$0.875689\pi$$
$$648$$ −0.719004 −0.0282451
$$649$$ −4.85728 −0.190665
$$650$$ −1.18421 −0.0464485
$$651$$ −12.2034 −0.478290
$$652$$ −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
$$658$$ 23.2257 0.905432
$$659$$ −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 $$-0.681192\pi$$
−0.538986 + 0.842315i $$0.681192\pi$$
$$662$$ 29.2070 1.13516
$$663$$ 3.22570 0.125276
$$664$$ −0.0958814 −0.00372092
$$665$$ −31.2257 −1.21088
$$666$$ −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
$$672$$ 32.5116 1.25416
$$673$$ −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 $$-0.534655\pi$$
−0.108657 + 0.994079i $$0.534655\pi$$
$$678$$ −11.4193 −0.438554
$$679$$ −32.0830 −1.23123
$$680$$ 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
$$686$$ −47.3087 −1.80625
$$687$$ 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 $$-0.619167\pi$$
−0.365690 + 0.930737i $$0.619167\pi$$
$$692$$ −14.8988 −0.566366
$$693$$ −4.42864 −0.168230
$$694$$ 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
$$700$$ −7.18421 −0.271538
$$701$$ −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
$$707$$ −20.6539 −0.776768
$$708$$ −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
$$714$$ 43.6958 1.63528
$$715$$ −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 $$-0.353524\pi$$
0.444098 + 0.895978i $$0.353524\pi$$
$$720$$ −4.61285 −0.171911
$$721$$ 51.4291 1.91532
$$722$$ 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 $$-0.709169\pi$$
−0.610843 + 0.791752i $$0.709169\pi$$
$$728$$ −1.98126 −0.0734305
$$729$$ 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 $$-0.685973\pi$$
−0.551575 + 0.834125i $$0.685973\pi$$
$$734$$ 64.3239 2.37424
$$735$$ 12.6128 0.465232
$$736$$ −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
$$742$$ 91.5121 3.35951
$$743$$ 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
$$749$$ −11.6128 −0.424324
$$750$$ 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
$$756$$ −7.18421 −0.261287
$$757$$ 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 $$-0.481802\pi$$
0.0571402 + 0.998366i $$0.481802\pi$$
$$762$$ −28.8988 −1.04689
$$763$$ 87.3087 3.16079
$$764$$ 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 $$-0.674868\pi$$
−0.522144 + 0.852857i $$0.674868\pi$$
$$770$$ −8.42864 −0.303747
$$771$$ −6.85728 −0.246959
$$772$$ 29.7462 1.07059
$$773$$ 29.1427 1.04819 0.524095 0.851660i $$-0.324404\pi$$
0.524095 + 0.851660i $$0.324404\pi$$
$$774$$ 10.7971 0.388092
$$775$$ 2.75557 0.0989830
$$776$$ −5.20877 −0.186984
$$777$$ 8.85728 0.317753
$$778$$ −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
$$784$$ −58.1811 −2.07790
$$785$$ 18.4701 0.659227
$$786$$ 2.36842 0.0844786
$$787$$ −11.2672 −0.401632 −0.200816 0.979629i $$-0.564359\pi$$
−0.200816 + 0.979629i $$0.564359\pi$$
$$788$$ −10.8613 −0.386918
$$789$$ −29.5812 −1.05312
$$790$$ 16.2953 0.579760
$$791$$ 26.5718 0.944786
$$792$$ −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 $$-0.233310\pi$$
0.743195 + 0.669075i $$0.233310\pi$$
$$798$$ −59.4291 −2.10377
$$799$$ 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
$$805$$ −39.2257 −1.38252
$$806$$ −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 $$-0.461995\pi$$
0.119112 + 0.992881i $$0.461995\pi$$
$$812$$ 56.0830 1.96813
$$813$$ 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
$$819$$ 2.75557 0.0962874
$$820$$ −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
$$826$$ 40.9403 1.42449
$$827$$ 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 $$-0.460520\pi$$
0.123713 + 0.992318i $$0.460520\pi$$
$$830$$ 0.253799 0.00880950
$$831$$ 14.6035 0.506589
$$832$$ 2.95316 0.102382
$$833$$ −65.3876 −2.26555
$$834$$ 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 $$-0.481604\pi$$
0.0577608 + 0.998330i $$0.481604\pi$$
$$840$$ 3.18421 0.109866
$$841$$ 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
$$847$$ −4.42864 −0.152170
$$848$$ 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 $$-0.649395\pi$$
−0.452297 + 0.891867i $$0.649395\pi$$
$$854$$ −57.7975 −1.97779
$$855$$ 7.05086 0.241134
$$856$$ −1.88538 −0.0644411
$$857$$ 38.7783 1.32464 0.662321 0.749220i $$-0.269571\pi$$
0.662321 + 0.749220i $$0.269571\pi$$
$$858$$ −1.18421 −0.0404282
$$859$$ −27.3087 −0.931760 −0.465880 0.884848i $$-0.654262\pi$$
−0.465880 + 0.884848i $$0.654262\pi$$
$$860$$ 9.20294 0.313818
$$861$$ 0.857279 0.0292160
$$862$$ 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
$$868$$ −19.7966 −0.671940
$$869$$ 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
$$875$$ −4.42864 −0.149715
$$876$$ 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 $$-0.582000\pi$$
−0.254770 + 0.967002i $$0.582000\pi$$
$$882$$ 24.0049 0.808288
$$883$$ −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 $$-0.0901135\pi$$
0.960194 + 0.279333i $$0.0901135\pi$$
$$888$$ 1.43801 0.0482564
$$889$$ 67.2454 2.25534
$$890$$ 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
$$896$$ −25.0192 −0.835833
$$897$$ −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
$$903$$ −25.1240 −0.836074
$$904$$ 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
$$910$$ 5.24443 0.173851
$$911$$ −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
$$917$$ −5.51114 −0.181994
$$918$$ −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
$$924$$ −7.18421 −0.236343
$$925$$ −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 $$-0.419097\pi$$
0.251438 + 0.967873i $$0.419097\pi$$
$$930$$ 5.24443 0.171972
$$931$$ 88.9314 2.91461
$$932$$ −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 $$-0.650316\pi$$
−0.454876 + 0.890555i $$0.650316\pi$$
$$938$$ 10.4889 0.342474
$$939$$ −15.7146 −0.512825
$$940$$ −4.47013 −0.145799
$$941$$ 10.4157 0.339543 0.169772 0.985483i $$-0.445697\pi$$
0.169772 + 0.985483i $$0.445697\pi$$
$$942$$ 35.1526 1.14533
$$943$$ −1.71456 −0.0558337
$$944$$ 22.4059 0.729250
$$945$$ −4.42864 −0.144064
$$946$$ 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
$$952$$ −16.5076 −0.535014
$$953$$ −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
$$959$$ 2.16500 0.0699114
$$960$$ −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
$$966$$ −74.6548 −2.40198
$$967$$ 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 $$-0.869328\pi$$
−0.916914 + 0.399085i $$0.869328\pi$$
$$972$$ 1.62222 0.0520326
$$973$$ −78.8582 −2.52808
$$974$$ −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
$$980$$ 20.4608 0.653595
$$981$$ −19.7146 −0.629437
$$982$$ 31.1882 0.995256
$$983$$ 1.12399 0.0358496 0.0179248 0.999839i $$-0.494294\pi$$
0.0179248 + 0.999839i $$0.494294\pi$$
$$984$$ 0.139182 0.00443696
$$985$$ −6.69535 −0.213331
$$986$$ 77.0232 2.45292
$$987$$ 12.2034 0.388439
$$988$$ −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
$$994$$ −23.2257 −0.736674
$$995$$ 14.1017 0.447054
$$996$$ 0.216327 0.00685460
$$997$$ −35.7275 −1.13150 −0.565750 0.824577i $$-0.691413\pi$$
−0.565750 + 0.824577i $$0.691413\pi$$
$$998$$ −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 165.2.a.c.1.3 3
3.2 odd 2 495.2.a.e.1.1 3
4.3 odd 2 2640.2.a.be.1.3 3
5.2 odd 4 825.2.c.g.199.5 6
5.3 odd 4 825.2.c.g.199.2 6
5.4 even 2 825.2.a.k.1.1 3
7.6 odd 2 8085.2.a.bk.1.3 3
11.10 odd 2 1815.2.a.m.1.1 3
12.11 even 2 7920.2.a.cj.1.3 3
15.2 even 4 2475.2.c.r.199.2 6
15.8 even 4 2475.2.c.r.199.5 6
15.14 odd 2 2475.2.a.bb.1.3 3
33.32 even 2 5445.2.a.z.1.3 3
55.54 odd 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 1.1 even 1 trivial
495.2.a.e.1.1 3 3.2 odd 2
825.2.a.k.1.1 3 5.4 even 2
825.2.c.g.199.2 6 5.3 odd 4
825.2.c.g.199.5 6 5.2 odd 4
1815.2.a.m.1.1 3 11.10 odd 2
2475.2.a.bb.1.3 3 15.14 odd 2
2475.2.c.r.199.2 6 15.2 even 4
2475.2.c.r.199.5 6 15.8 even 4
2640.2.a.be.1.3 3 4.3 odd 2
5445.2.a.z.1.3 3 33.32 even 2
7920.2.a.cj.1.3 3 12.11 even 2
8085.2.a.bk.1.3 3 7.6 odd 2
9075.2.a.cf.1.3 3 55.54 odd 2