# Properties

 Label 169.4.a.f.1.2 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 1.2 Root $$-1.56155$$ 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 $$-0.524696\pi$$
−0.0775072 + 0.996992i $$0.524696\pi$$
$$3$$ −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 $$-0.793258\pi$$
−0.796387 + 0.604787i $$0.793258\pi$$
$$6$$ 1.61553 0.109923
$$7$$ 5.43845 0.293649 0.146824 0.989163i $$-0.453095\pi$$
0.146824 + 0.989163i $$0.453095\pi$$
$$8$$ 6.93087 0.306304
$$9$$ −13.4233 −0.497159
$$10$$ 7.80776 0.246903
$$11$$ −22.4233 −0.614625 −0.307313 0.951609i $$-0.599430\pi$$
−0.307313 + 0.951609i $$0.599430\pi$$
$$12$$ 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 $$-0.662221\pi$$
−0.487857 + 0.872923i $$0.662221\pi$$
$$20$$ 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 $$-0.795158\pi$$
−0.799983 + 0.600023i $$0.795158\pi$$
$$32$$ −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 $$-0.503035\pi$$
−0.00953442 + 0.999955i $$0.503035\pi$$
$$38$$ 35.4299 0.151250
$$39$$ 0 0
$$40$$ −123.423 −0.487873
$$41$$ 227.769 0.867598 0.433799 0.901010i $$-0.357173\pi$$
0.433799 + 0.901010i $$0.357173\pi$$
$$42$$ 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 $$-0.335382\pi$$
0.494416 + 0.869225i $$0.335382\pi$$
$$48$$ −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 $$-0.604081\pi$$
−0.321186 + 0.947016i $$0.604081\pi$$
$$60$$ −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 $$-0.626685\pi$$
−0.387570 + 0.921840i $$0.626685\pi$$
$$68$$ −530.810 −0.946619
$$69$$ −517.810 −0.903434
$$70$$ 42.4621 0.0725028
$$71$$ −152.963 −0.255681 −0.127841 0.991795i $$-0.540805\pi$$
−0.127841 + 0.991795i $$0.540805\pi$$
$$72$$ −93.0351 −0.152282
$$73$$ 117.268 0.188016 0.0940081 0.995571i $$-0.470032\pi$$
0.0940081 + 0.995571i $$0.470032\pi$$
$$74$$ 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 $$-0.428651\pi$$
0.222276 + 0.974984i $$0.428651\pi$$
$$84$$ 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 $$-0.359328\pi$$
0.427688 + 0.903927i $$0.359328\pi$$
$$90$$ −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 $$-0.369901\pi$$
0.397434 + 0.917631i $$0.369901\pi$$
$$98$$ 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 $$-0.475360\pi$$
0.0773307 + 0.997005i $$0.475360\pi$$
$$110$$ −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 $$-0.431181\pi$$
0.214520 + 0.976720i $$0.431181\pi$$
$$138$$ 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 $$-0.682740\pi$$
−0.543074 + 0.839685i $$0.682740\pi$$
$$150$$ 310.370 0.168944
$$151$$ 1803.24 0.971824 0.485912 0.874008i $$-0.338487\pi$$
0.485912 + 0.874008i $$0.338487\pi$$
$$152$$ −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 $$-0.427375\pi$$
0.226183 + 0.974085i $$0.427375\pi$$
$$164$$ −1778.37 −0.846750
$$165$$ −1471.32 −0.694193
$$166$$ −147.386 −0.0689120
$$167$$ 3680.43 1.70539 0.852696 0.522408i $$-0.174966\pi$$
0.852696 + 0.522408i $$0.174966\pi$$
$$168$$ −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 $$-0.619783\pi$$
−0.367491 + 0.930027i $$0.619783\pi$$
$$194$$ −332.943 −0.123216
$$195$$ 0 0
$$196$$ 2447.14 0.891813
$$197$$ −4016.05 −1.45244 −0.726222 0.687460i $$-0.758726\pi$$
−0.726222 + 0.687460i $$0.758726\pi$$
$$198$$ −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 $$-0.550851\pi$$
−0.159075 + 0.987267i $$0.550851\pi$$
$$224$$ −443.239 −0.132210
$$225$$ −2578.84 −0.764099
$$226$$ −554.584 −0.163232
$$227$$ 3464.19 1.01289 0.506446 0.862272i $$-0.330959\pi$$
0.506446 + 0.862272i $$0.330959\pi$$
$$228$$ −2324.75 −0.675266
$$229$$ −2324.64 −0.670815 −0.335407 0.942073i $$-0.608874\pi$$
−0.335407 + 0.942073i $$0.608874\pi$$
$$230$$ 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 $$-0.195106\pi$$
0.817958 + 0.575278i $$0.195106\pi$$
$$240$$ 3899.09 1.04869
$$241$$ −5173.96 −1.38292 −0.691461 0.722414i $$-0.743033\pi$$
−0.691461 + 0.722414i $$0.743033\pi$$
$$242$$ 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 $$-0.397050\pi$$
0.317819 + 0.948151i $$0.397050\pi$$
$$272$$ 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 $$-0.919629\pi$$
−0.968293 + 0.249819i $$0.919629\pi$$
$$282$$ 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 $$-0.191207\pi$$
0.824944 + 0.565215i $$0.191207\pi$$
$$294$$ −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 $$-0.608941\pi$$
−0.335605 + 0.942003i $$0.608941\pi$$
$$308$$ 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 $$-0.412986\pi$$
0.269972 + 0.962868i $$0.412986\pi$$
$$318$$ −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 $$-0.279395\pi$$
0.638887 + 0.769301i $$0.279395\pi$$
$$332$$ −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 $$-0.498771\pi$$
0.00386009 + 0.999993i $$0.498771\pi$$
$$350$$ −458.096 −0.0699608
$$351$$ 0 0
$$352$$ 1827.52 0.276725
$$353$$ 9057.64 1.36569 0.682846 0.730562i $$-0.260742\pi$$
0.682846 + 0.730562i $$0.260742\pi$$
$$354$$ −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 $$-0.323083\pi$$
0.527623 + 0.849479i $$0.323083\pi$$
$$360$$ 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 $$-0.686742\pi$$
−0.553587 + 0.832791i $$0.686742\pi$$
$$380$$ −11235.4 −1.51675
$$381$$ −9595.24 −1.29023
$$382$$ −1175.97 −0.157507
$$383$$ −7310.25 −0.975290 −0.487645 0.873042i $$-0.662144\pi$$
−0.487645 + 0.873042i $$0.662144\pi$$
$$384$$ −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 $$-0.247712\pi$$
0.712171 + 0.702006i $$0.247712\pi$$
$$398$$ 1852.90 0.233361
$$399$$ 1619.29 0.203173
$$400$$ 11416.2 1.42702
$$401$$ 1576.23 0.196293 0.0981464 0.995172i $$-0.468709\pi$$
0.0981464 + 0.995172i $$0.468709\pi$$
$$402$$ −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 $$-0.366095\pi$$
0.408377 + 0.912814i $$0.366095\pi$$
$$410$$ 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 $$-0.349116\pi$$
0.456463 + 0.889743i $$0.349116\pi$$
$$422$$ −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 $$-0.788390\pi$$
−0.787044 + 0.616897i $$0.788390\pi$$
$$432$$ 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 $$-0.620704\pi$$
−0.370179 + 0.928960i $$0.620704\pi$$
$$450$$ 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 $$-0.243420\pi$$
0.721572 + 0.692340i $$0.243420\pi$$
$$458$$ 1019.23 0.103986
$$459$$ 10126.1 1.02973
$$460$$ 19539.3 1.98049
$$461$$ 14449.7 1.45985 0.729924 0.683529i $$-0.239556\pi$$
0.729924 + 0.683529i $$0.239556\pi$$
$$462$$ −197.010 −0.0198393
$$463$$ 15806.5 1.58659 0.793293 0.608840i $$-0.208365\pi$$
0.793293 + 0.608840i $$0.208365\pi$$
$$464$$ −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 $$-0.505960\pi$$
−0.0187222 + 0.999825i $$0.505960\pi$$
$$480$$ −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 $$-0.354312\pi$$
0.441879 + 0.897075i $$0.354312\pi$$
$$488$$ 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 $$-0.704724\pi$$
−0.599727 + 0.800205i $$0.704724\pi$$
$$500$$ 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 $$-0.469497\pi$$
0.0956824 + 0.995412i $$0.469497\pi$$
$$510$$ −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 $$-0.707476\pi$$
−0.606623 + 0.794990i $$0.707476\pi$$
$$542$$ −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 $$-0.630007\pi$$
−0.397169 + 0.917746i $$0.630007\pi$$
$$558$$ −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 $$-0.464263\pi$$
0.112035 + 0.993704i $$0.464263\pi$$
$$578$$ 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 $$-0.257052\pi$$
0.691270 + 0.722597i $$0.257052\pi$$
$$588$$ −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 $$-0.428923\pi$$
0.221444 + 0.975173i $$0.428923\pi$$
$$594$$ 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 $$-0.317454\pi$$
0.542564 + 0.840015i $$0.317454\pi$$
$$614$$ 1583.01 0.104047
$$615$$ 14945.2 0.979915
$$616$$ −845.205 −0.0552829
$$617$$ 10116.0 0.660055 0.330027 0.943971i $$-0.392942\pi$$
0.330027 + 0.943971i $$0.392942\pi$$
$$618$$ −937.496 −0.0610220
$$619$$ 18854.8 1.22430 0.612148 0.790743i $$-0.290306\pi$$
0.612148 + 0.790743i $$0.290306\pi$$
$$620$$ −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 $$-0.296100\pi$$
0.597653 + 0.801755i $$0.296100\pi$$
$$632$$ 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 $$-0.813188\pi$$
−0.832669 + 0.553772i $$0.813188\pi$$
$$644$$ −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 $$-0.224953\pi$$
0.760502 + 0.649336i $$0.224953\pi$$
$$662$$ −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 $$-0.181875\pi$$
0.841156 + 0.540792i $$0.181875\pi$$
$$684$$ −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 $$-0.496060\pi$$
0.0123786 + 0.999923i $$0.496060\pi$$
$$692$$ −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 $$-0.577470\pi$$
−0.240984 + 0.970529i $$0.577470\pi$$
$$710$$ −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 $$-0.160158\pi$$
0.876068 + 0.482188i $$0.160158\pi$$
$$734$$ −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 $$-0.299853\pi$$
0.588158 + 0.808746i $$0.299853\pi$$
$$740$$ −596.710 −0.0296425
$$741$$ 0 0
$$742$$ 161.324 0.00798167
$$743$$ 32502.8 1.60486 0.802431 0.596745i $$-0.203540\pi$$
0.802431 + 0.596745i $$0.203540\pi$$
$$744$$ 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 $$-0.566476\pi$$
−0.207325 + 0.978272i $$0.566476\pi$$
$$762$$ 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 $$-0.671782\pi$$
−0.513853 + 0.857878i $$0.671782\pi$$
$$770$$ −952.140 −0.0445620
$$771$$ 6168.63 0.288143
$$772$$ 15386.5 0.717322
$$773$$ −23077.5 −1.07379 −0.536896 0.843649i $$-0.680403\pi$$
−0.536896 + 0.843649i $$0.680403\pi$$
$$774$$ 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 $$-0.622076\pi$$
−0.374181 + 0.927356i $$0.622076\pi$$
$$788$$ 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 $$-0.382823\pi$$
0.359864 + 0.933005i $$0.382823\pi$$
$$812$$ 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 $$-0.799340\pi$$
−0.807797 + 0.589461i $$0.799340\pi$$
$$822$$ 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 $$-0.417427\pi$$
0.256512 + 0.966541i $$0.417427\pi$$
$$828$$ 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 $$-0.552372\pi$$
−0.163791 + 0.986495i $$0.552372\pi$$
$$840$$ 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 $$-0.412155\pi$$
0.272485 + 0.962160i $$0.412155\pi$$
$$854$$ −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 $$-0.511245\pi$$
−0.0353192 + 0.999376i $$0.511245\pi$$
$$864$$ −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 $$-0.183567\pi$$
0.838270 + 0.545255i $$0.183567\pi$$
$$878$$ −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 $$-0.646935\pi$$
−0.445390 + 0.895337i $$0.646935\pi$$
$$930$$ 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 $$-0.457743\pi$$
0.132365 + 0.991201i $$0.457743\pi$$
$$942$$ −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 $$-0.515677\pi$$
−0.0492293 + 0.998788i $$0.515677\pi$$
$$948$$ 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 $$-0.595854\pi$$
−0.296603 + 0.955001i $$0.595854\pi$$
$$968$$ −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 $$-0.673426\pi$$
−0.518277 + 0.855213i $$0.673426\pi$$
$$978$$ 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 $$-0.724196\pi$$
−0.647525 + 0.762045i $$0.724196\pi$$
$$984$$ −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.f.1.2 2
3.2 odd 2 1521.4.a.t.1.1 2
13.2 odd 12 169.4.e.g.147.2 8
13.3 even 3 13.4.c.b.9.1 yes 4
13.4 even 6 169.4.c.f.146.2 4
13.5 odd 4 169.4.b.e.168.3 4
13.6 odd 12 169.4.e.g.23.3 8
13.7 odd 12 169.4.e.g.23.2 8
13.8 odd 4 169.4.b.e.168.2 4
13.9 even 3 13.4.c.b.3.1 4
13.10 even 6 169.4.c.f.22.2 4
13.11 odd 12 169.4.e.g.147.3 8
13.12 even 2 169.4.a.j.1.1 2
39.29 odd 6 117.4.g.d.100.2 4
39.35 odd 6 117.4.g.d.55.2 4
39.38 odd 2 1521.4.a.l.1.2 2
52.3 odd 6 208.4.i.e.113.1 4
52.35 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 13.9 even 3
13.4.c.b.9.1 yes 4 13.3 even 3
117.4.g.d.55.2 4 39.35 odd 6
117.4.g.d.100.2 4 39.29 odd 6
169.4.a.f.1.2 2 1.1 even 1 trivial
169.4.a.j.1.1 2 13.12 even 2
169.4.b.e.168.2 4 13.8 odd 4
169.4.b.e.168.3 4 13.5 odd 4
169.4.c.f.22.2 4 13.10 even 6
169.4.c.f.146.2 4 13.4 even 6
169.4.e.g.23.2 8 13.7 odd 12
169.4.e.g.23.3 8 13.6 odd 12
169.4.e.g.147.2 8 13.2 odd 12
169.4.e.g.147.3 8 13.11 odd 12
208.4.i.e.81.1 4 52.35 odd 6
208.4.i.e.113.1 4 52.3 odd 6
1521.4.a.l.1.2 2 39.38 odd 2
1521.4.a.t.1.1 2 3.2 odd 2