# Properties

 Label 169.4.a.j.1.1 Level $169$ Weight $4$ Character 169.1 Self dual yes Analytic conductor $9.971$ 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] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.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: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 169.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} +17.8078 q^{5} -1.61553 q^{6} -5.43845 q^{7} -6.93087 q^{8} -13.4233 q^{9} +O(q^{10})$$ $$q+0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} +17.8078 q^{5} -1.61553 q^{6} -5.43845 q^{7} -6.93087 q^{8} -13.4233 q^{9} +7.80776 q^{10} +22.4233 q^{11} +28.7689 q^{12} -2.38447 q^{14} -65.6155 q^{15} +59.4233 q^{16} +67.9848 q^{17} -5.88540 q^{18} +80.8078 q^{19} -139.039 q^{20} +20.0388 q^{21} +9.83143 q^{22} +140.531 q^{23} +25.5379 q^{24} +192.116 q^{25} +148.946 q^{27} +42.4621 q^{28} -106.693 q^{29} -28.7689 q^{30} +276.155 q^{31} +81.5009 q^{32} -82.6222 q^{33} +29.8078 q^{34} -96.8466 q^{35} +104.806 q^{36} +4.29168 q^{37} +35.4299 q^{38} -123.423 q^{40} -227.769 q^{41} +8.78596 q^{42} +27.5294 q^{43} -175.076 q^{44} -239.039 q^{45} +61.6155 q^{46} -318.617 q^{47} -218.955 q^{48} -313.423 q^{49} +84.2329 q^{50} -250.501 q^{51} -67.6562 q^{53} +65.3050 q^{54} +399.309 q^{55} +37.6932 q^{56} -297.749 q^{57} -46.7793 q^{58} +291.115 q^{59} +512.311 q^{60} +663.311 q^{61} +121.080 q^{62} +73.0019 q^{63} -439.652 q^{64} -36.2255 q^{66} +425.101 q^{67} -530.810 q^{68} -517.810 q^{69} -42.4621 q^{70} +152.963 q^{71} +93.0351 q^{72} -117.268 q^{73} +1.88167 q^{74} -707.884 q^{75} -630.928 q^{76} -121.948 q^{77} +202.462 q^{79} +1058.20 q^{80} -186.386 q^{81} -99.8647 q^{82} -336.155 q^{83} -156.458 q^{84} +1210.66 q^{85} +12.0702 q^{86} +393.128 q^{87} -155.413 q^{88} -718.194 q^{89} -104.806 q^{90} -1097.23 q^{92} -1017.54 q^{93} -139.697 q^{94} +1439.01 q^{95} -300.303 q^{96} -759.368 q^{97} -137.420 q^{98} -300.994 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9}+O(q^{10})$$ 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9 $$2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9} - 5 q^{10} - 17 q^{11} + 140 q^{12} - 46 q^{14} - 90 q^{15} + 57 q^{16} + 70 q^{17} + 215 q^{18} + 141 q^{19} - 175 q^{20} - 63 q^{21} - 170 q^{22} + 145 q^{23} + 216 q^{24} + 75 q^{25} + 335 q^{27} - 80 q^{28} + 34 q^{29} - 140 q^{30} + 140 q^{31} - 105 q^{32} - 425 q^{33} + 39 q^{34} - 70 q^{35} + 725 q^{36} + 190 q^{37} + 310 q^{38} - 185 q^{40} - 538 q^{41} - 370 q^{42} + 455 q^{43} - 680 q^{44} - 375 q^{45} + 82 q^{46} - 60 q^{47} - 240 q^{48} - 565 q^{49} - 450 q^{50} - 233 q^{51} + 545 q^{53} + 914 q^{54} + 510 q^{55} - 172 q^{56} + 225 q^{57} + 595 q^{58} + 809 q^{59} + 200 q^{60} + 502 q^{61} - 500 q^{62} - 390 q^{63} - 1271 q^{64} - 1598 q^{66} + 475 q^{67} - 505 q^{68} - 479 q^{69} + 80 q^{70} - 127 q^{71} + 1155 q^{72} - 585 q^{73} + 849 q^{74} - 1725 q^{75} + 140 q^{76} + 255 q^{77} + 240 q^{79} + 1065 q^{80} + 122 q^{81} - 1515 q^{82} - 260 q^{83} - 1220 q^{84} + 1205 q^{85} + 1962 q^{86} + 1615 q^{87} - 1020 q^{88} - 921 q^{89} - 725 q^{90} - 1040 q^{92} - 2200 q^{93} + 1040 q^{94} + 1270 q^{95} - 1920 q^{96} + 415 q^{97} - 1285 q^{98} - 2210 q^{99}+O(q^{100})$$ 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9 - 5 * q^10 - 17 * q^11 + 140 * q^12 - 46 * q^14 - 90 * q^15 + 57 * q^16 + 70 * q^17 + 215 * q^18 + 141 * q^19 - 175 * q^20 - 63 * q^21 - 170 * q^22 + 145 * q^23 + 216 * q^24 + 75 * q^25 + 335 * q^27 - 80 * q^28 + 34 * q^29 - 140 * q^30 + 140 * q^31 - 105 * q^32 - 425 * q^33 + 39 * q^34 - 70 * q^35 + 725 * q^36 + 190 * q^37 + 310 * q^38 - 185 * q^40 - 538 * q^41 - 370 * q^42 + 455 * q^43 - 680 * q^44 - 375 * q^45 + 82 * q^46 - 60 * q^47 - 240 * q^48 - 565 * q^49 - 450 * q^50 - 233 * q^51 + 545 * q^53 + 914 * q^54 + 510 * q^55 - 172 * q^56 + 225 * q^57 + 595 * q^58 + 809 * q^59 + 200 * q^60 + 502 * q^61 - 500 * q^62 - 390 * q^63 - 1271 * q^64 - 1598 * q^66 + 475 * q^67 - 505 * q^68 - 479 * q^69 + 80 * q^70 - 127 * q^71 + 1155 * q^72 - 585 * q^73 + 849 * q^74 - 1725 * q^75 + 140 * q^76 + 255 * q^77 + 240 * q^79 + 1065 * q^80 + 122 * q^81 - 1515 * q^82 - 260 * q^83 - 1220 * q^84 + 1205 * q^85 + 1962 * q^86 + 1615 * q^87 - 1020 * q^88 - 921 * q^89 - 725 * q^90 - 1040 * q^92 - 2200 * q^93 + 1040 * q^94 + 1270 * q^95 - 1920 * q^96 + 415 * q^97 - 1285 * q^98 - 2210 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.438447 0.155014 0.0775072 0.996992i $$-0.475304\pi$$
0.0775072 + 0.996992i $$0.475304\pi$$
$$3$$ −3.68466 −0.709113 −0.354556 0.935035i $$-0.615368\pi$$
−0.354556 + 0.935035i $$0.615368\pi$$
$$4$$ −7.80776 −0.975971
$$5$$ 17.8078 1.59277 0.796387 0.604787i $$-0.206742\pi$$
0.796387 + 0.604787i $$0.206742\pi$$
$$6$$ −1.61553 −0.109923
$$7$$ −5.43845 −0.293649 −0.146824 0.989163i $$-0.546905\pi$$
−0.146824 + 0.989163i $$0.546905\pi$$
$$8$$ −6.93087 −0.306304
$$9$$ −13.4233 −0.497159
$$10$$ 7.80776 0.246903
$$11$$ 22.4233 0.614625 0.307313 0.951609i $$-0.400570\pi$$
0.307313 + 0.951609i $$0.400570\pi$$
$$12$$ 28.7689 0.692073
$$13$$ 0 0
$$14$$ −2.38447 −0.0455198
$$15$$ −65.6155 −1.12946
$$16$$ 59.4233 0.928489
$$17$$ 67.9848 0.969926 0.484963 0.874535i $$-0.338833\pi$$
0.484963 + 0.874535i $$0.338833\pi$$
$$18$$ −5.88540 −0.0770668
$$19$$ 80.8078 0.975714 0.487857 0.872923i $$-0.337779\pi$$
0.487857 + 0.872923i $$0.337779\pi$$
$$20$$ −139.039 −1.55450
$$21$$ 20.0388 0.208230
$$22$$ 9.83143 0.0952758
$$23$$ 140.531 1.27403 0.637017 0.770850i $$-0.280168\pi$$
0.637017 + 0.770850i $$0.280168\pi$$
$$24$$ 25.5379 0.217204
$$25$$ 192.116 1.53693
$$26$$ 0 0
$$27$$ 148.946 1.06165
$$28$$ 42.4621 0.286592
$$29$$ −106.693 −0.683187 −0.341594 0.939848i $$-0.610967\pi$$
−0.341594 + 0.939848i $$0.610967\pi$$
$$30$$ −28.7689 −0.175082
$$31$$ 276.155 1.59997 0.799983 0.600023i $$-0.204842\pi$$
0.799983 + 0.600023i $$0.204842\pi$$
$$32$$ 81.5009 0.450233
$$33$$ −82.6222 −0.435839
$$34$$ 29.8078 0.150353
$$35$$ −96.8466 −0.467716
$$36$$ 104.806 0.485212
$$37$$ 4.29168 0.0190688 0.00953442 0.999955i $$-0.496965\pi$$
0.00953442 + 0.999955i $$0.496965\pi$$
$$38$$ 35.4299 0.151250
$$39$$ 0 0
$$40$$ −123.423 −0.487873
$$41$$ −227.769 −0.867598 −0.433799 0.901010i $$-0.642827\pi$$
−0.433799 + 0.901010i $$0.642827\pi$$
$$42$$ 8.78596 0.0322787
$$43$$ 27.5294 0.0976323 0.0488162 0.998808i $$-0.484455\pi$$
0.0488162 + 0.998808i $$0.484455\pi$$
$$44$$ −175.076 −0.599856
$$45$$ −239.039 −0.791862
$$46$$ 61.6155 0.197494
$$47$$ −318.617 −0.988832 −0.494416 0.869225i $$-0.664618\pi$$
−0.494416 + 0.869225i $$0.664618\pi$$
$$48$$ −218.955 −0.658403
$$49$$ −313.423 −0.913771
$$50$$ 84.2329 0.238247
$$51$$ −250.501 −0.687787
$$52$$ 0 0
$$53$$ −67.6562 −0.175345 −0.0876726 0.996149i $$-0.527943\pi$$
−0.0876726 + 0.996149i $$0.527943\pi$$
$$54$$ 65.3050 0.164572
$$55$$ 399.309 0.978960
$$56$$ 37.6932 0.0899457
$$57$$ −297.749 −0.691892
$$58$$ −46.7793 −0.105904
$$59$$ 291.115 0.642371 0.321186 0.947016i $$-0.395919\pi$$
0.321186 + 0.947016i $$0.395919\pi$$
$$60$$ 512.311 1.10232
$$61$$ 663.311 1.39227 0.696133 0.717913i $$-0.254902\pi$$
0.696133 + 0.717913i $$0.254902\pi$$
$$62$$ 121.080 0.248018
$$63$$ 73.0019 0.145990
$$64$$ −439.652 −0.858696
$$65$$ 0 0
$$66$$ −36.2255 −0.0675613
$$67$$ 425.101 0.775140 0.387570 0.921840i $$-0.373315\pi$$
0.387570 + 0.921840i $$0.373315\pi$$
$$68$$ −530.810 −0.946619
$$69$$ −517.810 −0.903434
$$70$$ −42.4621 −0.0725028
$$71$$ 152.963 0.255681 0.127841 0.991795i $$-0.459195\pi$$
0.127841 + 0.991795i $$0.459195\pi$$
$$72$$ 93.0351 0.152282
$$73$$ −117.268 −0.188016 −0.0940081 0.995571i $$-0.529968\pi$$
−0.0940081 + 0.995571i $$0.529968\pi$$
$$74$$ 1.88167 0.00295595
$$75$$ −707.884 −1.08986
$$76$$ −630.928 −0.952268
$$77$$ −121.948 −0.180484
$$78$$ 0 0
$$79$$ 202.462 0.288339 0.144169 0.989553i $$-0.453949\pi$$
0.144169 + 0.989553i $$0.453949\pi$$
$$80$$ 1058.20 1.47887
$$81$$ −186.386 −0.255674
$$82$$ −99.8647 −0.134490
$$83$$ −336.155 −0.444552 −0.222276 0.974984i $$-0.571349\pi$$
−0.222276 + 0.974984i $$0.571349\pi$$
$$84$$ −156.458 −0.203226
$$85$$ 1210.66 1.54487
$$86$$ 12.0702 0.0151344
$$87$$ 393.128 0.484457
$$88$$ −155.413 −0.188262
$$89$$ −718.194 −0.855376 −0.427688 0.903927i $$-0.640672\pi$$
−0.427688 + 0.903927i $$0.640672\pi$$
$$90$$ −104.806 −0.122750
$$91$$ 0 0
$$92$$ −1097.23 −1.24342
$$93$$ −1017.54 −1.13456
$$94$$ −139.697 −0.153283
$$95$$ 1439.01 1.55409
$$96$$ −300.303 −0.319266
$$97$$ −759.368 −0.794868 −0.397434 0.917631i $$-0.630099\pi$$
−0.397434 + 0.917631i $$0.630099\pi$$
$$98$$ −137.420 −0.141648
$$99$$ −300.994 −0.305566
$$100$$ −1500.00 −1.50000
$$101$$ −348.697 −0.343531 −0.171766 0.985138i $$-0.554947\pi$$
−0.171766 + 0.985138i $$0.554947\pi$$
$$102$$ −109.831 −0.106617
$$103$$ −580.303 −0.555136 −0.277568 0.960706i $$-0.589528\pi$$
−0.277568 + 0.960706i $$0.589528\pi$$
$$104$$ 0 0
$$105$$ 356.847 0.331663
$$106$$ −29.6637 −0.0271810
$$107$$ 571.493 0.516340 0.258170 0.966100i $$-0.416881\pi$$
0.258170 + 0.966100i $$0.416881\pi$$
$$108$$ −1162.94 −1.03614
$$109$$ −176.004 −0.154661 −0.0773307 0.997005i $$-0.524640\pi$$
−0.0773307 + 0.997005i $$0.524640\pi$$
$$110$$ 175.076 0.151753
$$111$$ −15.8134 −0.0135220
$$112$$ −323.170 −0.272649
$$113$$ 1264.88 1.05301 0.526505 0.850172i $$-0.323502\pi$$
0.526505 + 0.850172i $$0.323502\pi$$
$$114$$ −130.547 −0.107253
$$115$$ 2502.55 2.02925
$$116$$ 833.035 0.666770
$$117$$ 0 0
$$118$$ 127.638 0.0995768
$$119$$ −369.732 −0.284817
$$120$$ 454.773 0.345957
$$121$$ −828.196 −0.622236
$$122$$ 290.827 0.215821
$$123$$ 839.251 0.615225
$$124$$ −2156.16 −1.56152
$$125$$ 1195.19 0.855211
$$126$$ 32.0075 0.0226306
$$127$$ 2604.11 1.81950 0.909752 0.415151i $$-0.136271\pi$$
0.909752 + 0.415151i $$0.136271\pi$$
$$128$$ −844.772 −0.583344
$$129$$ −101.436 −0.0692323
$$130$$ 0 0
$$131$$ 2131.70 1.42174 0.710870 0.703324i $$-0.248302\pi$$
0.710870 + 0.703324i $$0.248302\pi$$
$$132$$ 645.094 0.425366
$$133$$ −439.469 −0.286517
$$134$$ 186.384 0.120158
$$135$$ 2652.40 1.69098
$$136$$ −471.194 −0.297092
$$137$$ −687.985 −0.429040 −0.214520 0.976720i $$-0.568819\pi$$
−0.214520 + 0.976720i $$0.568819\pi$$
$$138$$ −227.032 −0.140045
$$139$$ −679.580 −0.414685 −0.207343 0.978268i $$-0.566482\pi$$
−0.207343 + 0.978268i $$0.566482\pi$$
$$140$$ 756.155 0.456477
$$141$$ 1174.00 0.701194
$$142$$ 67.0662 0.0396343
$$143$$ 0 0
$$144$$ −797.656 −0.461607
$$145$$ −1899.97 −1.08816
$$146$$ −51.4158 −0.0291452
$$147$$ 1154.86 0.647966
$$148$$ −33.5084 −0.0186106
$$149$$ 1975.46 1.08615 0.543074 0.839685i $$-0.317260\pi$$
0.543074 + 0.839685i $$0.317260\pi$$
$$150$$ −310.370 −0.168944
$$151$$ −1803.24 −0.971824 −0.485912 0.874008i $$-0.661513\pi$$
−0.485912 + 0.874008i $$0.661513\pi$$
$$152$$ −560.068 −0.298865
$$153$$ −912.580 −0.482208
$$154$$ −53.4677 −0.0279776
$$155$$ 4917.71 2.54839
$$156$$ 0 0
$$157$$ −397.168 −0.201894 −0.100947 0.994892i $$-0.532187\pi$$
−0.100947 + 0.994892i $$0.532187\pi$$
$$158$$ 88.7689 0.0446967
$$159$$ 249.290 0.124340
$$160$$ 1451.35 0.717120
$$161$$ −764.272 −0.374118
$$162$$ −81.7206 −0.0396332
$$163$$ −941.393 −0.452365 −0.226183 0.974085i $$-0.572625\pi$$
−0.226183 + 0.974085i $$0.572625\pi$$
$$164$$ 1778.37 0.846750
$$165$$ −1471.32 −0.694193
$$166$$ −147.386 −0.0689120
$$167$$ −3680.43 −1.70539 −0.852696 0.522408i $$-0.825034\pi$$
−0.852696 + 0.522408i $$0.825034\pi$$
$$168$$ −138.886 −0.0637817
$$169$$ 0 0
$$170$$ 530.810 0.239478
$$171$$ −1084.71 −0.485085
$$172$$ −214.943 −0.0952863
$$173$$ 1422.77 0.625269 0.312634 0.949874i $$-0.398789\pi$$
0.312634 + 0.949874i $$0.398789\pi$$
$$174$$ 172.366 0.0750978
$$175$$ −1044.82 −0.451318
$$176$$ 1332.47 0.570673
$$177$$ −1072.66 −0.455514
$$178$$ −314.890 −0.132596
$$179$$ 1167.89 0.487666 0.243833 0.969817i $$-0.421595\pi$$
0.243833 + 0.969817i $$0.421595\pi$$
$$180$$ 1866.36 0.772834
$$181$$ −1133.96 −0.465673 −0.232836 0.972516i $$-0.574801\pi$$
−0.232836 + 0.972516i $$0.574801\pi$$
$$182$$ 0 0
$$183$$ −2444.07 −0.987274
$$184$$ −974.004 −0.390242
$$185$$ 76.4252 0.0303724
$$186$$ −446.137 −0.175873
$$187$$ 1524.44 0.596141
$$188$$ 2487.69 0.965071
$$189$$ −810.035 −0.311753
$$190$$ 630.928 0.240907
$$191$$ 2682.12 1.01608 0.508040 0.861333i $$-0.330370\pi$$
0.508040 + 0.861333i $$0.330370\pi$$
$$192$$ 1619.97 0.608913
$$193$$ 1970.67 0.734983 0.367491 0.930027i $$-0.380217\pi$$
0.367491 + 0.930027i $$0.380217\pi$$
$$194$$ −332.943 −0.123216
$$195$$ 0 0
$$196$$ 2447.14 0.891813
$$197$$ 4016.05 1.45244 0.726222 0.687460i $$-0.241274\pi$$
0.726222 + 0.687460i $$0.241274\pi$$
$$198$$ −131.970 −0.0473672
$$199$$ −4226.06 −1.50541 −0.752707 0.658356i $$-0.771252\pi$$
−0.752707 + 0.658356i $$0.771252\pi$$
$$200$$ −1331.53 −0.470768
$$201$$ −1566.35 −0.549662
$$202$$ −152.885 −0.0532523
$$203$$ 580.245 0.200617
$$204$$ 1955.85 0.671260
$$205$$ −4056.06 −1.38189
$$206$$ −254.432 −0.0860541
$$207$$ −1886.39 −0.633398
$$208$$ 0 0
$$209$$ 1811.98 0.599699
$$210$$ 156.458 0.0514126
$$211$$ 1364.67 0.445249 0.222625 0.974904i $$-0.428538\pi$$
0.222625 + 0.974904i $$0.428538\pi$$
$$212$$ 528.244 0.171132
$$213$$ −563.617 −0.181307
$$214$$ 250.570 0.0800401
$$215$$ 490.237 0.155506
$$216$$ −1032.33 −0.325189
$$217$$ −1501.86 −0.469828
$$218$$ −77.1683 −0.0239748
$$219$$ 432.093 0.133325
$$220$$ −3117.71 −0.955436
$$221$$ 0 0
$$222$$ −6.93332 −0.00209610
$$223$$ 1059.47 0.318149 0.159075 0.987267i $$-0.449149\pi$$
0.159075 + 0.987267i $$0.449149\pi$$
$$224$$ −443.239 −0.132210
$$225$$ −2578.84 −0.764099
$$226$$ 554.584 0.163232
$$227$$ −3464.19 −1.01289 −0.506446 0.862272i $$-0.669041\pi$$
−0.506446 + 0.862272i $$0.669041\pi$$
$$228$$ 2324.75 0.675266
$$229$$ 2324.64 0.670815 0.335407 0.942073i $$-0.391126\pi$$
0.335407 + 0.942073i $$0.391126\pi$$
$$230$$ 1097.23 0.314563
$$231$$ 449.336 0.127983
$$232$$ 739.476 0.209263
$$233$$ −3731.01 −1.04904 −0.524521 0.851398i $$-0.675755\pi$$
−0.524521 + 0.851398i $$0.675755\pi$$
$$234$$ 0 0
$$235$$ −5673.86 −1.57499
$$236$$ −2272.95 −0.626935
$$237$$ −746.004 −0.204465
$$238$$ −162.108 −0.0441508
$$239$$ −6044.47 −1.63592 −0.817958 0.575278i $$-0.804894\pi$$
−0.817958 + 0.575278i $$0.804894\pi$$
$$240$$ −3899.09 −1.04869
$$241$$ 5173.96 1.38292 0.691461 0.722414i $$-0.256967\pi$$
0.691461 + 0.722414i $$0.256967\pi$$
$$242$$ −363.120 −0.0964556
$$243$$ −3334.77 −0.880353
$$244$$ −5178.97 −1.35881
$$245$$ −5581.37 −1.45543
$$246$$ 367.967 0.0953688
$$247$$ 0 0
$$248$$ −1914.00 −0.490076
$$249$$ 1238.62 0.315238
$$250$$ 524.029 0.132570
$$251$$ 5620.73 1.41346 0.706728 0.707486i $$-0.250171\pi$$
0.706728 + 0.707486i $$0.250171\pi$$
$$252$$ −569.981 −0.142482
$$253$$ 3151.17 0.783054
$$254$$ 1141.76 0.282050
$$255$$ −4460.86 −1.09549
$$256$$ 3146.83 0.768270
$$257$$ −1674.14 −0.406342 −0.203171 0.979143i $$-0.565125\pi$$
−0.203171 + 0.979143i $$0.565125\pi$$
$$258$$ −44.4745 −0.0107320
$$259$$ −23.3401 −0.00559954
$$260$$ 0 0
$$261$$ 1432.17 0.339653
$$262$$ 934.640 0.220390
$$263$$ −6309.18 −1.47924 −0.739622 0.673023i $$-0.764996\pi$$
−0.739622 + 0.673023i $$0.764996\pi$$
$$264$$ 572.644 0.133499
$$265$$ −1204.81 −0.279285
$$266$$ −192.684 −0.0444143
$$267$$ 2646.30 0.606558
$$268$$ −3319.09 −0.756514
$$269$$ −2482.73 −0.562731 −0.281366 0.959601i $$-0.590787\pi$$
−0.281366 + 0.959601i $$0.590787\pi$$
$$270$$ 1162.94 0.262126
$$271$$ −2835.72 −0.635638 −0.317819 0.948151i $$-0.602950\pi$$
−0.317819 + 0.948151i $$0.602950\pi$$
$$272$$ 4039.88 0.900566
$$273$$ 0 0
$$274$$ −301.645 −0.0665075
$$275$$ 4307.88 0.944637
$$276$$ 4042.94 0.881725
$$277$$ −3837.51 −0.832396 −0.416198 0.909274i $$-0.636638\pi$$
−0.416198 + 0.909274i $$0.636638\pi$$
$$278$$ −297.960 −0.0642822
$$279$$ −3706.91 −0.795438
$$280$$ 671.231 0.143263
$$281$$ 9122.13 1.93659 0.968293 0.249819i $$-0.0803712\pi$$
0.968293 + 0.249819i $$0.0803712\pi$$
$$282$$ 514.735 0.108695
$$283$$ 2127.85 0.446952 0.223476 0.974709i $$-0.428260\pi$$
0.223476 + 0.974709i $$0.428260\pi$$
$$284$$ −1194.30 −0.249537
$$285$$ −5302.24 −1.10203
$$286$$ 0 0
$$287$$ 1238.71 0.254769
$$288$$ −1094.01 −0.223838
$$289$$ −291.061 −0.0592430
$$290$$ −833.035 −0.168681
$$291$$ 2798.01 0.563651
$$292$$ 915.601 0.183498
$$293$$ −8274.77 −1.64989 −0.824944 0.565215i $$-0.808793\pi$$
−0.824944 + 0.565215i $$0.808793\pi$$
$$294$$ 506.344 0.100444
$$295$$ 5184.10 1.02315
$$296$$ −29.7450 −0.00584086
$$297$$ 3339.86 0.652520
$$298$$ 866.136 0.168369
$$299$$ 0 0
$$300$$ 5526.99 1.06367
$$301$$ −149.717 −0.0286696
$$302$$ −790.625 −0.150647
$$303$$ 1284.83 0.243602
$$304$$ 4801.86 0.905940
$$305$$ 11812.1 2.21757
$$306$$ −400.118 −0.0747492
$$307$$ 3610.49 0.671211 0.335605 0.942003i $$-0.391059\pi$$
0.335605 + 0.942003i $$0.391059\pi$$
$$308$$ 952.140 0.176147
$$309$$ 2138.22 0.393654
$$310$$ 2156.16 0.395037
$$311$$ 3331.06 0.607354 0.303677 0.952775i $$-0.401786\pi$$
0.303677 + 0.952775i $$0.401786\pi$$
$$312$$ 0 0
$$313$$ −358.125 −0.0646724 −0.0323362 0.999477i $$-0.510295\pi$$
−0.0323362 + 0.999477i $$0.510295\pi$$
$$314$$ −174.137 −0.0312966
$$315$$ 1300.00 0.232529
$$316$$ −1580.78 −0.281410
$$317$$ −3047.46 −0.539944 −0.269972 0.962868i $$-0.587014\pi$$
−0.269972 + 0.962868i $$0.587014\pi$$
$$318$$ 109.301 0.0192744
$$319$$ −2392.41 −0.419904
$$320$$ −7829.23 −1.36771
$$321$$ −2105.76 −0.366143
$$322$$ −335.093 −0.0579938
$$323$$ 5493.70 0.946371
$$324$$ 1455.26 0.249530
$$325$$ 0 0
$$326$$ −412.751 −0.0701232
$$327$$ 648.514 0.109672
$$328$$ 1578.64 0.265749
$$329$$ 1732.78 0.290369
$$330$$ −645.094 −0.107610
$$331$$ −7694.77 −1.27777 −0.638887 0.769301i $$-0.720605\pi$$
−0.638887 + 0.769301i $$0.720605\pi$$
$$332$$ 2624.62 0.433870
$$333$$ −57.6084 −0.00948025
$$334$$ −1613.68 −0.264360
$$335$$ 7570.10 1.23462
$$336$$ 1190.77 0.193339
$$337$$ 4712.21 0.761693 0.380846 0.924638i $$-0.375633\pi$$
0.380846 + 0.924638i $$0.375633\pi$$
$$338$$ 0 0
$$339$$ −4660.66 −0.746703
$$340$$ −9452.53 −1.50775
$$341$$ 6192.31 0.983380
$$342$$ −475.586 −0.0751952
$$343$$ 3569.92 0.561976
$$344$$ −190.803 −0.0299052
$$345$$ −9221.03 −1.43897
$$346$$ 623.811 0.0969257
$$347$$ −5261.98 −0.814058 −0.407029 0.913415i $$-0.633435\pi$$
−0.407029 + 0.913415i $$0.633435\pi$$
$$348$$ −3069.45 −0.472815
$$349$$ −50.3345 −0.00772018 −0.00386009 0.999993i $$-0.501229\pi$$
−0.00386009 + 0.999993i $$0.501229\pi$$
$$350$$ −458.096 −0.0699608
$$351$$ 0 0
$$352$$ 1827.52 0.276725
$$353$$ −9057.64 −1.36569 −0.682846 0.730562i $$-0.739258\pi$$
−0.682846 + 0.730562i $$0.739258\pi$$
$$354$$ −470.304 −0.0706112
$$355$$ 2723.93 0.407243
$$356$$ 5607.49 0.834821
$$357$$ 1362.34 0.201968
$$358$$ 512.059 0.0755953
$$359$$ −7177.86 −1.05525 −0.527623 0.849479i $$-0.676917\pi$$
−0.527623 + 0.849479i $$0.676917\pi$$
$$360$$ 1656.75 0.242551
$$361$$ −329.105 −0.0479815
$$362$$ −497.183 −0.0721861
$$363$$ 3051.62 0.441235
$$364$$ 0 0
$$365$$ −2088.28 −0.299467
$$366$$ −1071.60 −0.153042
$$367$$ 4004.14 0.569522 0.284761 0.958599i $$-0.408086\pi$$
0.284761 + 0.958599i $$0.408086\pi$$
$$368$$ 8350.83 1.18293
$$369$$ 3057.41 0.431334
$$370$$ 33.5084 0.00470816
$$371$$ 367.945 0.0514899
$$372$$ 7944.70 1.10729
$$373$$ −10014.2 −1.39012 −0.695060 0.718952i $$-0.744622\pi$$
−0.695060 + 0.718952i $$0.744622\pi$$
$$374$$ 668.388 0.0924105
$$375$$ −4403.88 −0.606441
$$376$$ 2208.30 0.302883
$$377$$ 0 0
$$378$$ −355.158 −0.0483263
$$379$$ 8169.12 1.10717 0.553587 0.832791i $$-0.313258\pi$$
0.553587 + 0.832791i $$0.313258\pi$$
$$380$$ −11235.4 −1.51675
$$381$$ −9595.24 −1.29023
$$382$$ 1175.97 0.157507
$$383$$ 7310.25 0.975290 0.487645 0.873042i $$-0.337856\pi$$
0.487645 + 0.873042i $$0.337856\pi$$
$$384$$ 3112.70 0.413656
$$385$$ −2171.62 −0.287470
$$386$$ 864.033 0.113933
$$387$$ −369.535 −0.0485388
$$388$$ 5928.97 0.775767
$$389$$ 8785.47 1.14509 0.572546 0.819872i $$-0.305956\pi$$
0.572546 + 0.819872i $$0.305956\pi$$
$$390$$ 0 0
$$391$$ 9553.99 1.23572
$$392$$ 2172.30 0.279892
$$393$$ −7854.60 −1.00817
$$394$$ 1760.82 0.225150
$$395$$ 3605.40 0.459259
$$396$$ 2350.09 0.298224
$$397$$ −11266.8 −1.42434 −0.712171 0.702006i $$-0.752288\pi$$
−0.712171 + 0.702006i $$0.752288\pi$$
$$398$$ −1852.90 −0.233361
$$399$$ 1619.29 0.203173
$$400$$ 11416.2 1.42702
$$401$$ −1576.23 −0.196293 −0.0981464 0.995172i $$-0.531291\pi$$
−0.0981464 + 0.995172i $$0.531291\pi$$
$$402$$ −686.763 −0.0852055
$$403$$ 0 0
$$404$$ 2722.54 0.335276
$$405$$ −3319.12 −0.407231
$$406$$ 254.407 0.0310985
$$407$$ 96.2335 0.0117202
$$408$$ 1736.19 0.210672
$$409$$ −6755.78 −0.816753 −0.408377 0.912814i $$-0.633905\pi$$
−0.408377 + 0.912814i $$0.633905\pi$$
$$410$$ −1778.37 −0.214213
$$411$$ 2534.99 0.304238
$$412$$ 4530.87 0.541796
$$413$$ −1583.21 −0.188631
$$414$$ −827.083 −0.0981858
$$415$$ −5986.17 −0.708072
$$416$$ 0 0
$$417$$ 2504.02 0.294059
$$418$$ 794.456 0.0929620
$$419$$ 10756.2 1.25411 0.627057 0.778973i $$-0.284259\pi$$
0.627057 + 0.778973i $$0.284259\pi$$
$$420$$ −2786.17 −0.323694
$$421$$ −7886.03 −0.912925 −0.456463 0.889743i $$-0.650884\pi$$
−0.456463 + 0.889743i $$0.650884\pi$$
$$422$$ 598.335 0.0690201
$$423$$ 4276.89 0.491607
$$424$$ 468.916 0.0537089
$$425$$ 13061.0 1.49071
$$426$$ −247.116 −0.0281052
$$427$$ −3607.38 −0.408837
$$428$$ −4462.08 −0.503932
$$429$$ 0 0
$$430$$ 214.943 0.0241057
$$431$$ 14084.6 1.57409 0.787044 0.616897i $$-0.211610\pi$$
0.787044 + 0.616897i $$0.211610\pi$$
$$432$$ 8850.86 0.985735
$$433$$ 1864.14 0.206894 0.103447 0.994635i $$-0.467013\pi$$
0.103447 + 0.994635i $$0.467013\pi$$
$$434$$ −658.485 −0.0728301
$$435$$ 7000.73 0.771630
$$436$$ 1374.20 0.150945
$$437$$ 11356.0 1.24309
$$438$$ 189.450 0.0206673
$$439$$ 6154.49 0.669106 0.334553 0.942377i $$-0.391415\pi$$
0.334553 + 0.942377i $$0.391415\pi$$
$$440$$ −2767.56 −0.299859
$$441$$ 4207.17 0.454289
$$442$$ 0 0
$$443$$ −14539.3 −1.55933 −0.779663 0.626200i $$-0.784609\pi$$
−0.779663 + 0.626200i $$0.784609\pi$$
$$444$$ 123.467 0.0131970
$$445$$ −12789.4 −1.36242
$$446$$ 464.521 0.0493177
$$447$$ −7278.90 −0.770202
$$448$$ 2391.03 0.252155
$$449$$ 7043.87 0.740358 0.370179 0.928960i $$-0.379296\pi$$
0.370179 + 0.928960i $$0.379296\pi$$
$$450$$ −1130.68 −0.118446
$$451$$ −5107.33 −0.533248
$$452$$ −9875.90 −1.02771
$$453$$ 6644.32 0.689133
$$454$$ −1518.87 −0.157013
$$455$$ 0 0
$$456$$ 2063.66 0.211929
$$457$$ −14098.9 −1.44314 −0.721572 0.692340i $$-0.756580\pi$$
−0.721572 + 0.692340i $$0.756580\pi$$
$$458$$ 1019.23 0.103986
$$459$$ 10126.1 1.02973
$$460$$ −19539.3 −1.98049
$$461$$ −14449.7 −1.45985 −0.729924 0.683529i $$-0.760444\pi$$
−0.729924 + 0.683529i $$0.760444\pi$$
$$462$$ 197.010 0.0198393
$$463$$ −15806.5 −1.58659 −0.793293 0.608840i $$-0.791635\pi$$
−0.793293 + 0.608840i $$0.791635\pi$$
$$464$$ −6340.06 −0.634332
$$465$$ −18120.1 −1.80709
$$466$$ −1635.85 −0.162617
$$467$$ −15071.3 −1.49340 −0.746699 0.665162i $$-0.768362\pi$$
−0.746699 + 0.665162i $$0.768362\pi$$
$$468$$ 0 0
$$469$$ −2311.89 −0.227619
$$470$$ −2487.69 −0.244146
$$471$$ 1463.43 0.143166
$$472$$ −2017.68 −0.196761
$$473$$ 617.299 0.0600073
$$474$$ −327.083 −0.0316950
$$475$$ 15524.5 1.49961
$$476$$ 2886.78 0.277973
$$477$$ 908.169 0.0871744
$$478$$ −2650.18 −0.253591
$$479$$ 392.545 0.0374443 0.0187222 0.999825i $$-0.494040\pi$$
0.0187222 + 0.999825i $$0.494040\pi$$
$$480$$ −5347.73 −0.508519
$$481$$ 0 0
$$482$$ 2268.51 0.214373
$$483$$ 2816.08 0.265292
$$484$$ 6466.36 0.607284
$$485$$ −13522.7 −1.26605
$$486$$ −1462.12 −0.136467
$$487$$ −9497.89 −0.883758 −0.441879 0.897075i $$-0.645688\pi$$
−0.441879 + 0.897075i $$0.645688\pi$$
$$488$$ −4597.32 −0.426457
$$489$$ 3468.71 0.320778
$$490$$ −2447.14 −0.225613
$$491$$ −1893.82 −0.174067 −0.0870337 0.996205i $$-0.527739\pi$$
−0.0870337 + 0.996205i $$0.527739\pi$$
$$492$$ −6552.67 −0.600442
$$493$$ −7253.52 −0.662641
$$494$$ 0 0
$$495$$ −5360.04 −0.486699
$$496$$ 16410.1 1.48555
$$497$$ −831.881 −0.0750804
$$498$$ 543.068 0.0488664
$$499$$ 13370.1 1.19945 0.599727 0.800205i $$-0.295276\pi$$
0.599727 + 0.800205i $$0.295276\pi$$
$$500$$ −9331.79 −0.834661
$$501$$ 13561.1 1.20932
$$502$$ 2464.39 0.219106
$$503$$ 5554.71 0.492391 0.246195 0.969220i $$-0.420820\pi$$
0.246195 + 0.969220i $$0.420820\pi$$
$$504$$ −505.966 −0.0447173
$$505$$ −6209.51 −0.547168
$$506$$ 1381.62 0.121385
$$507$$ 0 0
$$508$$ −20332.3 −1.77578
$$509$$ −2197.55 −0.191365 −0.0956824 0.995412i $$-0.530503\pi$$
−0.0956824 + 0.995412i $$0.530503\pi$$
$$510$$ −1955.85 −0.169817
$$511$$ 637.756 0.0552107
$$512$$ 8137.89 0.702437
$$513$$ 12036.0 1.03587
$$514$$ −734.022 −0.0629890
$$515$$ −10333.9 −0.884206
$$516$$ 791.991 0.0675687
$$517$$ −7144.45 −0.607761
$$518$$ −10.2334 −0.000868010 0
$$519$$ −5242.44 −0.443386
$$520$$ 0 0
$$521$$ 17005.2 1.42997 0.714983 0.699142i $$-0.246435\pi$$
0.714983 + 0.699142i $$0.246435\pi$$
$$522$$ 627.932 0.0526511
$$523$$ −14486.2 −1.21116 −0.605581 0.795783i $$-0.707059\pi$$
−0.605581 + 0.795783i $$0.707059\pi$$
$$524$$ −16643.8 −1.38758
$$525$$ 3849.79 0.320035
$$526$$ −2766.24 −0.229304
$$527$$ 18774.4 1.55185
$$528$$ −4909.68 −0.404671
$$529$$ 7582.03 0.623163
$$530$$ −528.244 −0.0432933
$$531$$ −3907.72 −0.319361
$$532$$ 3431.27 0.279632
$$533$$ 0 0
$$534$$ 1160.26 0.0940252
$$535$$ 10177.0 0.822413
$$536$$ −2946.32 −0.237429
$$537$$ −4303.28 −0.345810
$$538$$ −1088.55 −0.0872315
$$539$$ −7027.98 −0.561626
$$540$$ −20709.3 −1.65034
$$541$$ 15266.7 1.21325 0.606623 0.794990i $$-0.292524\pi$$
0.606623 + 0.794990i $$0.292524\pi$$
$$542$$ −1243.31 −0.0985330
$$543$$ 4178.27 0.330215
$$544$$ 5540.83 0.436693
$$545$$ −3134.23 −0.246341
$$546$$ 0 0
$$547$$ 15260.5 1.19286 0.596430 0.802665i $$-0.296586\pi$$
0.596430 + 0.802665i $$0.296586\pi$$
$$548$$ 5371.62 0.418731
$$549$$ −8903.81 −0.692177
$$550$$ 1888.78 0.146432
$$551$$ −8621.64 −0.666595
$$552$$ 3588.87 0.276726
$$553$$ −1101.08 −0.0846703
$$554$$ −1682.55 −0.129033
$$555$$ −281.601 −0.0215374
$$556$$ 5306.00 0.404721
$$557$$ 10442.1 0.794337 0.397169 0.917746i $$-0.369993\pi$$
0.397169 + 0.917746i $$0.369993\pi$$
$$558$$ −1625.29 −0.123304
$$559$$ 0 0
$$560$$ −5754.94 −0.434269
$$561$$ −5617.06 −0.422731
$$562$$ 3999.57 0.300199
$$563$$ 7145.26 0.534879 0.267440 0.963575i $$-0.413822\pi$$
0.267440 + 0.963575i $$0.413822\pi$$
$$564$$ −9166.29 −0.684344
$$565$$ 22524.7 1.67721
$$566$$ 932.950 0.0692841
$$567$$ 1013.65 0.0750783
$$568$$ −1060.17 −0.0783162
$$569$$ 4438.86 0.327042 0.163521 0.986540i $$-0.447715\pi$$
0.163521 + 0.986540i $$0.447715\pi$$
$$570$$ −2324.75 −0.170830
$$571$$ 10117.3 0.741497 0.370748 0.928733i $$-0.379101\pi$$
0.370748 + 0.928733i $$0.379101\pi$$
$$572$$ 0 0
$$573$$ −9882.70 −0.720516
$$574$$ 543.109 0.0394929
$$575$$ 26998.4 1.95810
$$576$$ 5901.58 0.426909
$$577$$ −3105.60 −0.224069 −0.112035 0.993704i $$-0.535737\pi$$
−0.112035 + 0.993704i $$0.535737\pi$$
$$578$$ −127.615 −0.00918352
$$579$$ −7261.23 −0.521186
$$580$$ 14834.5 1.06202
$$581$$ 1828.16 0.130542
$$582$$ 1226.78 0.0873741
$$583$$ −1517.08 −0.107772
$$584$$ 812.769 0.0575901
$$585$$ 0 0
$$586$$ −3628.05 −0.255757
$$587$$ −19662.3 −1.38254 −0.691270 0.722597i $$-0.742948\pi$$
−0.691270 + 0.722597i $$0.742948\pi$$
$$588$$ −9016.86 −0.632396
$$589$$ 22315.5 1.56111
$$590$$ 2272.95 0.158603
$$591$$ −14797.8 −1.02995
$$592$$ 255.026 0.0177052
$$593$$ −6395.51 −0.442888 −0.221444 0.975173i $$-0.571077\pi$$
−0.221444 + 0.975173i $$0.571077\pi$$
$$594$$ 1464.35 0.101150
$$595$$ −6584.10 −0.453650
$$596$$ −15423.9 −1.06005
$$597$$ 15571.6 1.06751
$$598$$ 0 0
$$599$$ 8878.48 0.605618 0.302809 0.953051i $$-0.402076\pi$$
0.302809 + 0.953051i $$0.402076\pi$$
$$600$$ 4906.25 0.333828
$$601$$ 19100.6 1.29639 0.648194 0.761475i $$-0.275525\pi$$
0.648194 + 0.761475i $$0.275525\pi$$
$$602$$ −65.6430 −0.00444420
$$603$$ −5706.26 −0.385368
$$604$$ 14079.3 0.948472
$$605$$ −14748.3 −0.991082
$$606$$ 563.330 0.0377619
$$607$$ 16595.8 1.10972 0.554861 0.831943i $$-0.312771\pi$$
0.554861 + 0.831943i $$0.312771\pi$$
$$608$$ 6585.91 0.439299
$$609$$ −2138.01 −0.142260
$$610$$ 5178.97 0.343755
$$611$$ 0 0
$$612$$ 7125.21 0.470620
$$613$$ −16469.2 −1.08513 −0.542564 0.840015i $$-0.682546\pi$$
−0.542564 + 0.840015i $$0.682546\pi$$
$$614$$ 1583.01 0.104047
$$615$$ 14945.2 0.979915
$$616$$ 845.205 0.0552829
$$617$$ −10116.0 −0.660055 −0.330027 0.943971i $$-0.607058\pi$$
−0.330027 + 0.943971i $$0.607058\pi$$
$$618$$ 937.496 0.0610220
$$619$$ −18854.8 −1.22430 −0.612148 0.790743i $$-0.709694\pi$$
−0.612148 + 0.790743i $$0.709694\pi$$
$$620$$ −38396.3 −2.48715
$$621$$ 20931.6 1.35258
$$622$$ 1460.49 0.0941487
$$623$$ 3905.86 0.251180
$$624$$ 0 0
$$625$$ −2730.82 −0.174773
$$626$$ −157.019 −0.0100252
$$627$$ −6676.51 −0.425254
$$628$$ 3100.99 0.197043
$$629$$ 291.769 0.0184954
$$630$$ 569.981 0.0360454
$$631$$ −18946.2 −1.19531 −0.597653 0.801755i $$-0.703900\pi$$
−0.597653 + 0.801755i $$0.703900\pi$$
$$632$$ −1403.24 −0.0883194
$$633$$ −5028.33 −0.315732
$$634$$ −1336.15 −0.0836991
$$635$$ 46373.3 2.89806
$$636$$ −1946.40 −0.121352
$$637$$ 0 0
$$638$$ −1048.95 −0.0650912
$$639$$ −2053.27 −0.127114
$$640$$ −15043.5 −0.929135
$$641$$ −23586.9 −1.45340 −0.726698 0.686957i $$-0.758946\pi$$
−0.726698 + 0.686957i $$0.758946\pi$$
$$642$$ −923.263 −0.0567575
$$643$$ 27153.0 1.66534 0.832669 0.553772i $$-0.186812\pi$$
0.832669 + 0.553772i $$0.186812\pi$$
$$644$$ 5967.25 0.365128
$$645$$ −1806.35 −0.110272
$$646$$ 2408.70 0.146701
$$647$$ 6856.72 0.416639 0.208319 0.978061i $$-0.433201\pi$$
0.208319 + 0.978061i $$0.433201\pi$$
$$648$$ 1291.82 0.0783140
$$649$$ 6527.75 0.394817
$$650$$ 0 0
$$651$$ 5533.83 0.333161
$$652$$ 7350.17 0.441495
$$653$$ −8073.89 −0.483853 −0.241926 0.970295i $$-0.577779\pi$$
−0.241926 + 0.970295i $$0.577779\pi$$
$$654$$ 284.339 0.0170008
$$655$$ 37960.9 2.26451
$$656$$ −13534.8 −0.805555
$$657$$ 1574.12 0.0934739
$$658$$ 759.734 0.0450114
$$659$$ 5305.73 0.313629 0.156815 0.987628i $$-0.449877\pi$$
0.156815 + 0.987628i $$0.449877\pi$$
$$660$$ 11487.7 0.677512
$$661$$ −25848.3 −1.52100 −0.760502 0.649336i $$-0.775047\pi$$
−0.760502 + 0.649336i $$0.775047\pi$$
$$662$$ −3373.75 −0.198073
$$663$$ 0 0
$$664$$ 2329.85 0.136168
$$665$$ −7825.96 −0.456357
$$666$$ −25.2583 −0.00146958
$$667$$ −14993.7 −0.870404
$$668$$ 28735.9 1.66441
$$669$$ −3903.78 −0.225604
$$670$$ 3319.09 0.191385
$$671$$ 14873.6 0.855722
$$672$$ 1633.18 0.0937521
$$673$$ −14529.1 −0.832177 −0.416089 0.909324i $$-0.636599\pi$$
−0.416089 + 0.909324i $$0.636599\pi$$
$$674$$ 2066.06 0.118073
$$675$$ 28615.0 1.63169
$$676$$ 0 0
$$677$$ 12058.1 0.684535 0.342267 0.939603i $$-0.388805\pi$$
0.342267 + 0.939603i $$0.388805\pi$$
$$678$$ −2043.45 −0.115750
$$679$$ 4129.78 0.233412
$$680$$ −8390.91 −0.473201
$$681$$ 12764.4 0.718255
$$682$$ 2715.00 0.152438
$$683$$ −30028.8 −1.68231 −0.841156 0.540792i $$-0.818125\pi$$
−0.841156 + 0.540792i $$0.818125\pi$$
$$684$$ 8469.13 0.473429
$$685$$ −12251.5 −0.683364
$$686$$ 1565.22 0.0871144
$$687$$ −8565.50 −0.475683
$$688$$ 1635.89 0.0906505
$$689$$ 0 0
$$690$$ −4042.94 −0.223061
$$691$$ −449.696 −0.0247572 −0.0123786 0.999923i $$-0.503940\pi$$
−0.0123786 + 0.999923i $$0.503940\pi$$
$$692$$ −11108.7 −0.610244
$$693$$ 1636.94 0.0897291
$$694$$ −2307.10 −0.126191
$$695$$ −12101.8 −0.660500
$$696$$ −2724.72 −0.148391
$$697$$ −15484.8 −0.841506
$$698$$ −22.0690 −0.00119674
$$699$$ 13747.5 0.743889
$$700$$ 8157.67 0.440473
$$701$$ −26986.0 −1.45399 −0.726994 0.686644i $$-0.759083\pi$$
−0.726994 + 0.686644i $$0.759083\pi$$
$$702$$ 0 0
$$703$$ 346.801 0.0186057
$$704$$ −9858.46 −0.527776
$$705$$ 20906.2 1.11684
$$706$$ −3971.29 −0.211702
$$707$$ 1896.37 0.100877
$$708$$ 8375.06 0.444568
$$709$$ 9098.87 0.481968 0.240984 0.970529i $$-0.422530\pi$$
0.240984 + 0.970529i $$0.422530\pi$$
$$710$$ 1194.30 0.0631285
$$711$$ −2717.71 −0.143350
$$712$$ 4977.71 0.262005
$$713$$ 38808.4 2.03841
$$714$$ 597.312 0.0313079
$$715$$ 0 0
$$716$$ −9118.62 −0.475948
$$717$$ 22271.8 1.16005
$$718$$ −3147.11 −0.163578
$$719$$ −6293.55 −0.326439 −0.163220 0.986590i $$-0.552188\pi$$
−0.163220 + 0.986590i $$0.552188\pi$$
$$720$$ −14204.5 −0.735235
$$721$$ 3155.95 0.163015
$$722$$ −144.295 −0.00743783
$$723$$ −19064.3 −0.980648
$$724$$ 8853.72 0.454483
$$725$$ −20497.5 −1.05001
$$726$$ 1337.97 0.0683979
$$727$$ 18070.7 0.921878 0.460939 0.887432i $$-0.347513\pi$$
0.460939 + 0.887432i $$0.347513\pi$$
$$728$$ 0 0
$$729$$ 17319.9 0.879944
$$730$$ −915.601 −0.0464218
$$731$$ 1871.58 0.0946962
$$732$$ 19082.7 0.963550
$$733$$ −34771.5 −1.75214 −0.876068 0.482188i $$-0.839842\pi$$
−0.876068 + 0.482188i $$0.839842\pi$$
$$734$$ 1755.61 0.0882842
$$735$$ 20565.4 1.03206
$$736$$ 11453.4 0.573613
$$737$$ 9532.17 0.476421
$$738$$ 1340.51 0.0668631
$$739$$ −23631.5 −1.17632 −0.588158 0.808746i $$-0.700147\pi$$
−0.588158 + 0.808746i $$0.700147\pi$$
$$740$$ −596.710 −0.0296425
$$741$$ 0 0
$$742$$ 161.324 0.00798167
$$743$$ −32502.8 −1.60486 −0.802431 0.596745i $$-0.796460\pi$$
−0.802431 + 0.596745i $$0.796460\pi$$
$$744$$ 7052.42 0.347519
$$745$$ 35178.6 1.72999
$$746$$ −4390.69 −0.215489
$$747$$ 4512.31 0.221013
$$748$$ −11902.5 −0.581816
$$749$$ −3108.04 −0.151622
$$750$$ −1930.87 −0.0940072
$$751$$ 2020.86 0.0981920 0.0490960 0.998794i $$-0.484366\pi$$
0.0490960 + 0.998794i $$0.484366\pi$$
$$752$$ −18933.3 −0.918120
$$753$$ −20710.5 −1.00230
$$754$$ 0 0
$$755$$ −32111.6 −1.54790
$$756$$ 6324.56 0.304262
$$757$$ 12568.2 0.603434 0.301717 0.953398i $$-0.402440\pi$$
0.301717 + 0.953398i $$0.402440\pi$$
$$758$$ 3581.73 0.171628
$$759$$ −11611.0 −0.555273
$$760$$ −9973.56 −0.476025
$$761$$ 8704.81 0.414651 0.207325 0.978272i $$-0.433524\pi$$
0.207325 + 0.978272i $$0.433524\pi$$
$$762$$ −4207.01 −0.200005
$$763$$ 957.187 0.0454161
$$764$$ −20941.4 −0.991665
$$765$$ −16251.0 −0.768048
$$766$$ 3205.16 0.151184
$$767$$ 0 0
$$768$$ −11595.0 −0.544790
$$769$$ 21915.9 1.02771 0.513853 0.857878i $$-0.328218\pi$$
0.513853 + 0.857878i $$0.328218\pi$$
$$770$$ −952.140 −0.0445620
$$771$$ 6168.63 0.288143
$$772$$ −15386.5 −0.717322
$$773$$ 23077.5 1.07379 0.536896 0.843649i $$-0.319597\pi$$
0.536896 + 0.843649i $$0.319597\pi$$
$$774$$ −162.022 −0.00752422
$$775$$ 53054.0 2.45904
$$776$$ 5263.08 0.243471
$$777$$ 86.0001 0.00397070
$$778$$ 3851.96 0.177506
$$779$$ −18405.5 −0.846528
$$780$$ 0 0
$$781$$ 3429.94 0.157148
$$782$$ 4188.92 0.191554
$$783$$ −15891.5 −0.725309
$$784$$ −18624.6 −0.848426
$$785$$ −7072.67 −0.321572
$$786$$ −3443.83 −0.156282
$$787$$ 16522.4 0.748362 0.374181 0.927356i $$-0.377924\pi$$
0.374181 + 0.927356i $$0.377924\pi$$
$$788$$ −31356.4 −1.41754
$$789$$ 23247.2 1.04895
$$790$$ 1580.78 0.0711918
$$791$$ −6879.00 −0.309215
$$792$$ 2086.15 0.0935962
$$793$$ 0 0
$$794$$ −4939.89 −0.220794
$$795$$ 4439.30 0.198045
$$796$$ 32996.1 1.46924
$$797$$ −11719.4 −0.520855 −0.260427 0.965493i $$-0.583863\pi$$
−0.260427 + 0.965493i $$0.583863\pi$$
$$798$$ 709.974 0.0314948
$$799$$ −21661.2 −0.959095
$$800$$ 15657.7 0.691978
$$801$$ 9640.53 0.425258
$$802$$ −691.096 −0.0304282
$$803$$ −2629.53 −0.115559
$$804$$ 12229.7 0.536454
$$805$$ −13610.0 −0.595886
$$806$$ 0 0
$$807$$ 9148.01 0.399040
$$808$$ 2416.77 0.105225
$$809$$ −24096.0 −1.04718 −0.523592 0.851969i $$-0.675408\pi$$
−0.523592 + 0.851969i $$0.675408\pi$$
$$810$$ −1455.26 −0.0631267
$$811$$ −16622.6 −0.719729 −0.359864 0.933005i $$-0.617177\pi$$
−0.359864 + 0.933005i $$0.617177\pi$$
$$812$$ −4530.42 −0.195796
$$813$$ 10448.7 0.450739
$$814$$ 42.1933 0.00181680
$$815$$ −16764.1 −0.720516
$$816$$ −14885.6 −0.638603
$$817$$ 2224.59 0.0952613
$$818$$ −2962.05 −0.126609
$$819$$ 0 0
$$820$$ 31668.7 1.34868
$$821$$ 38005.5 1.61559 0.807797 0.589461i $$-0.200660\pi$$
0.807797 + 0.589461i $$0.200660\pi$$
$$822$$ 1111.46 0.0471613
$$823$$ −15859.5 −0.671722 −0.335861 0.941912i $$-0.609027\pi$$
−0.335861 + 0.941912i $$0.609027\pi$$
$$824$$ 4022.01 0.170040
$$825$$ −15873.1 −0.669854
$$826$$ −694.155 −0.0292406
$$827$$ −12201.0 −0.513023 −0.256512 0.966541i $$-0.582573\pi$$
−0.256512 + 0.966541i $$0.582573\pi$$
$$828$$ 14728.5 0.618177
$$829$$ −5431.41 −0.227552 −0.113776 0.993506i $$-0.536295\pi$$
−0.113776 + 0.993506i $$0.536295\pi$$
$$830$$ −2624.62 −0.109761
$$831$$ 14139.9 0.590263
$$832$$ 0 0
$$833$$ −21308.0 −0.886290
$$834$$ 1097.88 0.0455834
$$835$$ −65540.3 −2.71630
$$836$$ −14147.5 −0.585288
$$837$$ 41132.2 1.69861
$$838$$ 4716.02 0.194406
$$839$$ 7960.90 0.327582 0.163791 0.986495i $$-0.447628\pi$$
0.163791 + 0.986495i $$0.447628\pi$$
$$840$$ −2473.26 −0.101590
$$841$$ −13005.6 −0.533255
$$842$$ −3457.61 −0.141517
$$843$$ −33611.9 −1.37326
$$844$$ −10655.0 −0.434550
$$845$$ 0 0
$$846$$ 1875.19 0.0762062
$$847$$ 4504.10 0.182719
$$848$$ −4020.35 −0.162806
$$849$$ −7840.40 −0.316940
$$850$$ 5726.56 0.231082
$$851$$ 603.115 0.0242944
$$852$$ 4400.59 0.176950
$$853$$ −13576.7 −0.544969 −0.272485 0.962160i $$-0.587845\pi$$
−0.272485 + 0.962160i $$0.587845\pi$$
$$854$$ −1581.65 −0.0633756
$$855$$ −19316.2 −0.772631
$$856$$ −3960.95 −0.158157
$$857$$ −31223.9 −1.24456 −0.622281 0.782794i $$-0.713794\pi$$
−0.622281 + 0.782794i $$0.713794\pi$$
$$858$$ 0 0
$$859$$ −11815.8 −0.469323 −0.234661 0.972077i $$-0.575398\pi$$
−0.234661 + 0.972077i $$0.575398\pi$$
$$860$$ −3827.65 −0.151770
$$861$$ −4564.22 −0.180660
$$862$$ 6175.36 0.244006
$$863$$ 1790.84 0.0706384 0.0353192 0.999376i $$-0.488755\pi$$
0.0353192 + 0.999376i $$0.488755\pi$$
$$864$$ 12139.2 0.477992
$$865$$ 25336.4 0.995912
$$866$$ 817.328 0.0320715
$$867$$ 1072.46 0.0420100
$$868$$ 11726.1 0.458538
$$869$$ 4539.87 0.177220
$$870$$ 3069.45 0.119614
$$871$$ 0 0
$$872$$ 1219.86 0.0473734
$$873$$ 10193.2 0.395176
$$874$$ 4979.01 0.192698
$$875$$ −6500.00 −0.251132
$$876$$ −3373.68 −0.130121
$$877$$ −43542.5 −1.67654 −0.838270 0.545255i $$-0.816433\pi$$
−0.838270 + 0.545255i $$0.816433\pi$$
$$878$$ 2698.42 0.103721
$$879$$ 30489.7 1.16996
$$880$$ 23728.2 0.908953
$$881$$ −1020.04 −0.0390080 −0.0195040 0.999810i $$-0.506209\pi$$
−0.0195040 + 0.999810i $$0.506209\pi$$
$$882$$ 1844.62 0.0704214
$$883$$ −34781.9 −1.32560 −0.662800 0.748797i $$-0.730632\pi$$
−0.662800 + 0.748797i $$0.730632\pi$$
$$884$$ 0 0
$$885$$ −19101.6 −0.725531
$$886$$ −6374.70 −0.241718
$$887$$ 49785.1 1.88458 0.942288 0.334802i $$-0.108670\pi$$
0.942288 + 0.334802i $$0.108670\pi$$
$$888$$ 109.600 0.00414183
$$889$$ −14162.3 −0.534295
$$890$$ −5607.49 −0.211195
$$891$$ −4179.40 −0.157144
$$892$$ −8272.08 −0.310504
$$893$$ −25746.8 −0.964818
$$894$$ −3191.41 −0.119392
$$895$$ 20797.5 0.776743
$$896$$ 4594.25 0.171298
$$897$$ 0 0
$$898$$ 3088.36 0.114766
$$899$$ −29463.9 −1.09308
$$900$$ 20134.9 0.745738
$$901$$ −4599.60 −0.170072
$$902$$ −2239.29 −0.0826611
$$903$$ 551.656 0.0203300
$$904$$ −8766.74 −0.322541
$$905$$ −20193.3 −0.741712
$$906$$ 2913.18 0.106826
$$907$$ 17389.9 0.636627 0.318314 0.947985i $$-0.396884\pi$$
0.318314 + 0.947985i $$0.396884\pi$$
$$908$$ 27047.6 0.988553
$$909$$ 4680.66 0.170790
$$910$$ 0 0
$$911$$ 20419.5 0.742621 0.371311 0.928509i $$-0.378909\pi$$
0.371311 + 0.928509i $$0.378909\pi$$
$$912$$ −17693.2 −0.642414
$$913$$ −7537.71 −0.273233
$$914$$ −6181.60 −0.223708
$$915$$ −43523.5 −1.57250
$$916$$ −18150.2 −0.654695
$$917$$ −11593.2 −0.417492
$$918$$ 4439.75 0.159623
$$919$$ −33231.8 −1.19283 −0.596417 0.802674i $$-0.703410\pi$$
−0.596417 + 0.802674i $$0.703410\pi$$
$$920$$ −17344.8 −0.621567
$$921$$ −13303.4 −0.475964
$$922$$ −6335.43 −0.226297
$$923$$ 0 0
$$924$$ −3508.31 −0.124908
$$925$$ 824.502 0.0293075
$$926$$ −6930.31 −0.245944
$$927$$ 7789.58 0.275991
$$928$$ −8695.59 −0.307594
$$929$$ 25222.8 0.890780 0.445390 0.895337i $$-0.353065\pi$$
0.445390 + 0.895337i $$0.353065\pi$$
$$930$$ −7944.70 −0.280126
$$931$$ −25327.0 −0.891579
$$932$$ 29130.9 1.02383
$$933$$ −12273.8 −0.430683
$$934$$ −6607.97 −0.231498
$$935$$ 27146.9 0.949519
$$936$$ 0 0
$$937$$ −26979.4 −0.940639 −0.470319 0.882496i $$-0.655861\pi$$
−0.470319 + 0.882496i $$0.655861\pi$$
$$938$$ −1013.64 −0.0352842
$$939$$ 1319.57 0.0458600
$$940$$ 44300.2 1.53714
$$941$$ −7641.67 −0.264730 −0.132365 0.991201i $$-0.542257\pi$$
−0.132365 + 0.991201i $$0.542257\pi$$
$$942$$ 641.635 0.0221928
$$943$$ −32008.7 −1.10535
$$944$$ 17299.0 0.596434
$$945$$ −14424.9 −0.496553
$$946$$ 270.653 0.00930200
$$947$$ 2869.32 0.0984587 0.0492293 0.998788i $$-0.484323\pi$$
0.0492293 + 0.998788i $$0.484323\pi$$
$$948$$ 5824.62 0.199552
$$949$$ 0 0
$$950$$ 6806.67 0.232461
$$951$$ 11228.8 0.382881
$$952$$ 2562.56 0.0872407
$$953$$ 12313.6 0.418548 0.209274 0.977857i $$-0.432890\pi$$
0.209274 + 0.977857i $$0.432890\pi$$
$$954$$ 398.184 0.0135133
$$955$$ 47762.6 1.61839
$$956$$ 47193.8 1.59661
$$957$$ 8815.22 0.297759
$$958$$ 172.110 0.00580442
$$959$$ 3741.57 0.125987
$$960$$ 28848.0 0.969861
$$961$$ 46470.7 1.55989
$$962$$ 0 0
$$963$$ −7671.32 −0.256703
$$964$$ −40397.1 −1.34969
$$965$$ 35093.2 1.17066
$$966$$ 1234.70 0.0411241
$$967$$ 17838.0 0.593207 0.296603 0.955001i $$-0.404146\pi$$
0.296603 + 0.955001i $$0.404146\pi$$
$$968$$ 5740.12 0.190593
$$969$$ −20242.4 −0.671084
$$970$$ −5928.97 −0.196255
$$971$$ 41525.3 1.37241 0.686206 0.727408i $$-0.259275\pi$$
0.686206 + 0.727408i $$0.259275\pi$$
$$972$$ 26037.1 0.859199
$$973$$ 3695.86 0.121772
$$974$$ −4164.32 −0.136995
$$975$$ 0 0
$$976$$ 39416.1 1.29270
$$977$$ 31654.4 1.03655 0.518277 0.855213i $$-0.326574\pi$$
0.518277 + 0.855213i $$0.326574\pi$$
$$978$$ 1520.85 0.0497253
$$979$$ −16104.3 −0.525735
$$980$$ 43578.0 1.42046
$$981$$ 2362.55 0.0768913
$$982$$ −830.342 −0.0269830
$$983$$ 39913.2 1.29505 0.647525 0.762045i $$-0.275804\pi$$
0.647525 + 0.762045i $$0.275804\pi$$
$$984$$ −5816.74 −0.188446
$$985$$ 71516.8 2.31342
$$986$$ −3180.28 −0.102719
$$987$$ −6384.72 −0.205905
$$988$$ 0 0
$$989$$ 3868.74 0.124387
$$990$$ −2350.09 −0.0754453
$$991$$ −2700.94 −0.0865773 −0.0432887 0.999063i $$-0.513784\pi$$
−0.0432887 + 0.999063i $$0.513784\pi$$
$$992$$ 22506.9 0.720358
$$993$$ 28352.6 0.906085
$$994$$ −364.736 −0.0116386
$$995$$ −75256.6 −2.39778
$$996$$ −9670.83 −0.307663
$$997$$ −9729.08 −0.309050 −0.154525 0.987989i $$-0.549385\pi$$
−0.154525 + 0.987989i $$0.549385\pi$$
$$998$$ 5862.08 0.185933
$$999$$ 639.228 0.0202445
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 169.4.a.j.1.1 2
3.2 odd 2 1521.4.a.l.1.2 2
13.2 odd 12 169.4.e.g.147.3 8
13.3 even 3 169.4.c.f.22.2 4
13.4 even 6 13.4.c.b.3.1 4
13.5 odd 4 169.4.b.e.168.2 4
13.6 odd 12 169.4.e.g.23.2 8
13.7 odd 12 169.4.e.g.23.3 8
13.8 odd 4 169.4.b.e.168.3 4
13.9 even 3 169.4.c.f.146.2 4
13.10 even 6 13.4.c.b.9.1 yes 4
13.11 odd 12 169.4.e.g.147.2 8
13.12 even 2 169.4.a.f.1.2 2
39.17 odd 6 117.4.g.d.55.2 4
39.23 odd 6 117.4.g.d.100.2 4
39.38 odd 2 1521.4.a.t.1.1 2
52.23 odd 6 208.4.i.e.113.1 4
52.43 odd 6 208.4.i.e.81.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.b.3.1 4 13.4 even 6
13.4.c.b.9.1 yes 4 13.10 even 6
117.4.g.d.55.2 4 39.17 odd 6
117.4.g.d.100.2 4 39.23 odd 6
169.4.a.f.1.2 2 13.12 even 2
169.4.a.j.1.1 2 1.1 even 1 trivial
169.4.b.e.168.2 4 13.5 odd 4
169.4.b.e.168.3 4 13.8 odd 4
169.4.c.f.22.2 4 13.3 even 3
169.4.c.f.146.2 4 13.9 even 3
169.4.e.g.23.2 8 13.6 odd 12
169.4.e.g.23.3 8 13.7 odd 12
169.4.e.g.147.2 8 13.11 odd 12
169.4.e.g.147.3 8 13.2 odd 12
208.4.i.e.81.1 4 52.43 odd 6
208.4.i.e.113.1 4 52.23 odd 6
1521.4.a.l.1.2 2 3.2 odd 2
1521.4.a.t.1.1 2 39.38 odd 2