# Properties

 Label 169.4.b.e.168.3 Level $169$ Weight $4$ Character 169.168 Analytic conductor $9.971$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(168,169)]

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([1]))

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

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 168.3 Root $$-2.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 169.168 Dual form 169.4.b.e.168.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.438447i q^{2} -3.68466 q^{3} +7.80776 q^{4} +17.8078i q^{5} -1.61553i q^{6} +5.43845i q^{7} +6.93087i q^{8} -13.4233 q^{9} +O(q^{10})$$ $$q+0.438447i q^{2} -3.68466 q^{3} +7.80776 q^{4} +17.8078i q^{5} -1.61553i q^{6} +5.43845i q^{7} +6.93087i q^{8} -13.4233 q^{9} -7.80776 q^{10} -22.4233i q^{11} -28.7689 q^{12} -2.38447 q^{14} -65.6155i q^{15} +59.4233 q^{16} -67.9848 q^{17} -5.88540i q^{18} +80.8078i q^{19} +139.039i q^{20} -20.0388i q^{21} +9.83143 q^{22} -140.531 q^{23} -25.5379i q^{24} -192.116 q^{25} +148.946 q^{27} +42.4621i q^{28} -106.693 q^{29} +28.7689 q^{30} +276.155i q^{31} +81.5009i q^{32} +82.6222i q^{33} -29.8078i q^{34} -96.8466 q^{35} -104.806 q^{36} -4.29168i q^{37} -35.4299 q^{38} -123.423 q^{40} -227.769i q^{41} +8.78596 q^{42} -27.5294 q^{43} -175.076i q^{44} -239.039i q^{45} -61.6155i q^{46} +318.617i q^{47} -218.955 q^{48} +313.423 q^{49} -84.2329i q^{50} +250.501 q^{51} -67.6562 q^{53} +65.3050i q^{54} +399.309 q^{55} -37.6932 q^{56} -297.749i q^{57} -46.7793i q^{58} -291.115i q^{59} -512.311i q^{60} +663.311 q^{61} -121.080 q^{62} -73.0019i q^{63} +439.652 q^{64} -36.2255 q^{66} +425.101i q^{67} -530.810 q^{68} +517.810 q^{69} -42.4621i q^{70} +152.963i q^{71} -93.0351i q^{72} +117.268i q^{73} +1.88167 q^{74} +707.884 q^{75} +630.928i q^{76} +121.948 q^{77} +202.462 q^{79} +1058.20i q^{80} -186.386 q^{81} +99.8647 q^{82} -336.155i q^{83} -156.458i q^{84} -1210.66i q^{85} -12.0702i q^{86} +393.128 q^{87} +155.413 q^{88} +718.194i q^{89} +104.806 q^{90} -1097.23 q^{92} -1017.54i q^{93} -139.697 q^{94} -1439.01 q^{95} -300.303i q^{96} -759.368i q^{97} +137.420i q^{98} +300.994i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{3} - 10 q^{4} + 70 q^{9}+O(q^{10})$$ 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9 $$4 q + 10 q^{3} - 10 q^{4} + 70 q^{9} + 10 q^{10} - 280 q^{12} - 92 q^{14} + 114 q^{16} - 140 q^{17} - 340 q^{22} - 290 q^{23} - 150 q^{25} + 670 q^{27} + 68 q^{29} + 280 q^{30} - 140 q^{35} - 1450 q^{36} - 620 q^{38} - 370 q^{40} - 740 q^{42} - 910 q^{43} - 480 q^{48} + 1130 q^{49} + 466 q^{51} + 1090 q^{53} + 1020 q^{55} + 344 q^{56} + 1004 q^{61} + 1000 q^{62} + 2542 q^{64} - 3196 q^{66} - 1010 q^{68} + 958 q^{69} + 1698 q^{74} + 3450 q^{75} - 510 q^{77} + 480 q^{79} + 244 q^{81} + 3030 q^{82} + 3230 q^{87} + 2040 q^{88} + 1450 q^{90} - 2080 q^{92} + 2080 q^{94} - 2540 q^{95}+O(q^{100})$$ 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9 + 10 * q^10 - 280 * q^12 - 92 * q^14 + 114 * q^16 - 140 * q^17 - 340 * q^22 - 290 * q^23 - 150 * q^25 + 670 * q^27 + 68 * q^29 + 280 * q^30 - 140 * q^35 - 1450 * q^36 - 620 * q^38 - 370 * q^40 - 740 * q^42 - 910 * q^43 - 480 * q^48 + 1130 * q^49 + 466 * q^51 + 1090 * q^53 + 1020 * q^55 + 344 * q^56 + 1004 * q^61 + 1000 * q^62 + 2542 * q^64 - 3196 * q^66 - 1010 * q^68 + 958 * q^69 + 1698 * q^74 + 3450 * q^75 - 510 * q^77 + 480 * q^79 + 244 * q^81 + 3030 * q^82 + 3230 * q^87 + 2040 * q^88 + 1450 * q^90 - 2080 * q^92 + 2080 * q^94 - 2540 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.438447i 0.155014i 0.996992 + 0.0775072i $$0.0246961\pi$$
−0.996992 + 0.0775072i $$0.975304\pi$$
$$3$$ −3.68466 −0.709113 −0.354556 0.935035i $$-0.615368\pi$$
−0.354556 + 0.935035i $$0.615368\pi$$
$$4$$ 7.80776 0.975971
$$5$$ 17.8078i 1.59277i 0.604787 + 0.796387i $$0.293258\pi$$
−0.604787 + 0.796387i $$0.706742\pi$$
$$6$$ − 1.61553i − 0.109923i
$$7$$ 5.43845i 0.293649i 0.989163 + 0.146824i $$0.0469052\pi$$
−0.989163 + 0.146824i $$0.953095\pi$$
$$8$$ 6.93087i 0.306304i
$$9$$ −13.4233 −0.497159
$$10$$ −7.80776 −0.246903
$$11$$ − 22.4233i − 0.614625i −0.951609 0.307313i $$-0.900570\pi$$
0.951609 0.307313i $$-0.0994297\pi$$
$$12$$ −28.7689 −0.692073
$$13$$ 0 0
$$14$$ −2.38447 −0.0455198
$$15$$ − 65.6155i − 1.12946i
$$16$$ 59.4233 0.928489
$$17$$ −67.9848 −0.969926 −0.484963 0.874535i $$-0.661167\pi$$
−0.484963 + 0.874535i $$0.661167\pi$$
$$18$$ − 5.88540i − 0.0770668i
$$19$$ 80.8078i 0.975714i 0.872923 + 0.487857i $$0.162221\pi$$
−0.872923 + 0.487857i $$0.837779\pi$$
$$20$$ 139.039i 1.55450i
$$21$$ − 20.0388i − 0.208230i
$$22$$ 9.83143 0.0952758
$$23$$ −140.531 −1.27403 −0.637017 0.770850i $$-0.719832\pi$$
−0.637017 + 0.770850i $$0.719832\pi$$
$$24$$ − 25.5379i − 0.217204i
$$25$$ −192.116 −1.53693
$$26$$ 0 0
$$27$$ 148.946 1.06165
$$28$$ 42.4621i 0.286592i
$$29$$ −106.693 −0.683187 −0.341594 0.939848i $$-0.610967\pi$$
−0.341594 + 0.939848i $$0.610967\pi$$
$$30$$ 28.7689 0.175082
$$31$$ 276.155i 1.59997i 0.600023 + 0.799983i $$0.295158\pi$$
−0.600023 + 0.799983i $$0.704842\pi$$
$$32$$ 81.5009i 0.450233i
$$33$$ 82.6222i 0.435839i
$$34$$ − 29.8078i − 0.150353i
$$35$$ −96.8466 −0.467716
$$36$$ −104.806 −0.485212
$$37$$ − 4.29168i − 0.0190688i −0.999955 0.00953442i $$-0.996965\pi$$
0.999955 0.00953442i $$-0.00303495\pi$$
$$38$$ −35.4299 −0.151250
$$39$$ 0 0
$$40$$ −123.423 −0.487873
$$41$$ − 227.769i − 0.867598i −0.901010 0.433799i $$-0.857173\pi$$
0.901010 0.433799i $$-0.142827\pi$$
$$42$$ 8.78596 0.0322787
$$43$$ −27.5294 −0.0976323 −0.0488162 0.998808i $$-0.515545\pi$$
−0.0488162 + 0.998808i $$0.515545\pi$$
$$44$$ − 175.076i − 0.599856i
$$45$$ − 239.039i − 0.791862i
$$46$$ − 61.6155i − 0.197494i
$$47$$ 318.617i 0.988832i 0.869225 + 0.494416i $$0.164618\pi$$
−0.869225 + 0.494416i $$0.835382\pi$$
$$48$$ −218.955 −0.658403
$$49$$ 313.423 0.913771
$$50$$ − 84.2329i − 0.238247i
$$51$$ 250.501 0.687787
$$52$$ 0 0
$$53$$ −67.6562 −0.175345 −0.0876726 0.996149i $$-0.527943\pi$$
−0.0876726 + 0.996149i $$0.527943\pi$$
$$54$$ 65.3050i 0.164572i
$$55$$ 399.309 0.978960
$$56$$ −37.6932 −0.0899457
$$57$$ − 297.749i − 0.691892i
$$58$$ − 46.7793i − 0.105904i
$$59$$ − 291.115i − 0.642371i −0.947016 0.321186i $$-0.895919\pi$$
0.947016 0.321186i $$-0.104081\pi$$
$$60$$ − 512.311i − 1.10232i
$$61$$ 663.311 1.39227 0.696133 0.717913i $$-0.254902\pi$$
0.696133 + 0.717913i $$0.254902\pi$$
$$62$$ −121.080 −0.248018
$$63$$ − 73.0019i − 0.145990i
$$64$$ 439.652 0.858696
$$65$$ 0 0
$$66$$ −36.2255 −0.0675613
$$67$$ 425.101i 0.775140i 0.921840 + 0.387570i $$0.126685\pi$$
−0.921840 + 0.387570i $$0.873315\pi$$
$$68$$ −530.810 −0.946619
$$69$$ 517.810 0.903434
$$70$$ − 42.4621i − 0.0725028i
$$71$$ 152.963i 0.255681i 0.991795 + 0.127841i $$0.0408046\pi$$
−0.991795 + 0.127841i $$0.959195\pi$$
$$72$$ − 93.0351i − 0.152282i
$$73$$ 117.268i 0.188016i 0.995571 + 0.0940081i $$0.0299680\pi$$
−0.995571 + 0.0940081i $$0.970032\pi$$
$$74$$ 1.88167 0.00295595
$$75$$ 707.884 1.08986
$$76$$ 630.928i 0.952268i
$$77$$ 121.948 0.180484
$$78$$ 0 0
$$79$$ 202.462 0.288339 0.144169 0.989553i $$-0.453949\pi$$
0.144169 + 0.989553i $$0.453949\pi$$
$$80$$ 1058.20i 1.47887i
$$81$$ −186.386 −0.255674
$$82$$ 99.8647 0.134490
$$83$$ − 336.155i − 0.444552i −0.974984 0.222276i $$-0.928651\pi$$
0.974984 0.222276i $$-0.0713486\pi$$
$$84$$ − 156.458i − 0.203226i
$$85$$ − 1210.66i − 1.54487i
$$86$$ − 12.0702i − 0.0151344i
$$87$$ 393.128 0.484457
$$88$$ 155.413 0.188262
$$89$$ 718.194i 0.855376i 0.903927 + 0.427688i $$0.140672\pi$$
−0.903927 + 0.427688i $$0.859328\pi$$
$$90$$ 104.806 0.122750
$$91$$ 0 0
$$92$$ −1097.23 −1.24342
$$93$$ − 1017.54i − 1.13456i
$$94$$ −139.697 −0.153283
$$95$$ −1439.01 −1.55409
$$96$$ − 300.303i − 0.319266i
$$97$$ − 759.368i − 0.794868i −0.917631 0.397434i $$-0.869901\pi$$
0.917631 0.397434i $$-0.130099\pi$$
$$98$$ 137.420i 0.141648i
$$99$$ 300.994i 0.305566i
$$100$$ −1500.00 −1.50000
$$101$$ 348.697 0.343531 0.171766 0.985138i $$-0.445053\pi$$
0.171766 + 0.985138i $$0.445053\pi$$
$$102$$ 109.831i 0.106617i
$$103$$ 580.303 0.555136 0.277568 0.960706i $$-0.410472\pi$$
0.277568 + 0.960706i $$0.410472\pi$$
$$104$$ 0 0
$$105$$ 356.847 0.331663
$$106$$ − 29.6637i − 0.0271810i
$$107$$ 571.493 0.516340 0.258170 0.966100i $$-0.416881\pi$$
0.258170 + 0.966100i $$0.416881\pi$$
$$108$$ 1162.94 1.03614
$$109$$ − 176.004i − 0.154661i −0.997005 0.0773307i $$-0.975360\pi$$
0.997005 0.0773307i $$-0.0246397\pi$$
$$110$$ 175.076i 0.151753i
$$111$$ 15.8134i 0.0135220i
$$112$$ 323.170i 0.272649i
$$113$$ 1264.88 1.05301 0.526505 0.850172i $$-0.323502\pi$$
0.526505 + 0.850172i $$0.323502\pi$$
$$114$$ 130.547 0.107253
$$115$$ − 2502.55i − 2.02925i
$$116$$ −833.035 −0.666770
$$117$$ 0 0
$$118$$ 127.638 0.0995768
$$119$$ − 369.732i − 0.284817i
$$120$$ 454.773 0.345957
$$121$$ 828.196 0.622236
$$122$$ 290.827i 0.215821i
$$123$$ 839.251i 0.615225i
$$124$$ 2156.16i 1.56152i
$$125$$ − 1195.19i − 0.855211i
$$126$$ 32.0075 0.0226306
$$127$$ −2604.11 −1.81950 −0.909752 0.415151i $$-0.863729\pi$$
−0.909752 + 0.415151i $$0.863729\pi$$
$$128$$ 844.772i 0.583344i
$$129$$ 101.436 0.0692323
$$130$$ 0 0
$$131$$ 2131.70 1.42174 0.710870 0.703324i $$-0.248302\pi$$
0.710870 + 0.703324i $$0.248302\pi$$
$$132$$ 645.094i 0.425366i
$$133$$ −439.469 −0.286517
$$134$$ −186.384 −0.120158
$$135$$ 2652.40i 1.69098i
$$136$$ − 471.194i − 0.297092i
$$137$$ 687.985i 0.429040i 0.976720 + 0.214520i $$0.0688188\pi$$
−0.976720 + 0.214520i $$0.931181\pi$$
$$138$$ 227.032i 0.140045i
$$139$$ −679.580 −0.414685 −0.207343 0.978268i $$-0.566482\pi$$
−0.207343 + 0.978268i $$0.566482\pi$$
$$140$$ −756.155 −0.456477
$$141$$ − 1174.00i − 0.701194i
$$142$$ −67.0662 −0.0396343
$$143$$ 0 0
$$144$$ −797.656 −0.461607
$$145$$ − 1899.97i − 1.08816i
$$146$$ −51.4158 −0.0291452
$$147$$ −1154.86 −0.647966
$$148$$ − 33.5084i − 0.0186106i
$$149$$ 1975.46i 1.08615i 0.839685 + 0.543074i $$0.182740\pi$$
−0.839685 + 0.543074i $$0.817260\pi$$
$$150$$ 310.370i 0.168944i
$$151$$ 1803.24i 0.971824i 0.874008 + 0.485912i $$0.161513\pi$$
−0.874008 + 0.485912i $$0.838487\pi$$
$$152$$ −560.068 −0.298865
$$153$$ 912.580 0.482208
$$154$$ 53.4677i 0.0279776i
$$155$$ −4917.71 −2.54839
$$156$$ 0 0
$$157$$ −397.168 −0.201894 −0.100947 0.994892i $$-0.532187\pi$$
−0.100947 + 0.994892i $$0.532187\pi$$
$$158$$ 88.7689i 0.0446967i
$$159$$ 249.290 0.124340
$$160$$ −1451.35 −0.717120
$$161$$ − 764.272i − 0.374118i
$$162$$ − 81.7206i − 0.0396332i
$$163$$ 941.393i 0.452365i 0.974085 + 0.226183i $$0.0726246\pi$$
−0.974085 + 0.226183i $$0.927375\pi$$
$$164$$ − 1778.37i − 0.846750i
$$165$$ −1471.32 −0.694193
$$166$$ 147.386 0.0689120
$$167$$ 3680.43i 1.70539i 0.522408 + 0.852696i $$0.325034\pi$$
−0.522408 + 0.852696i $$0.674966\pi$$
$$168$$ 138.886 0.0637817
$$169$$ 0 0
$$170$$ 530.810 0.239478
$$171$$ − 1084.71i − 0.485085i
$$172$$ −214.943 −0.0952863
$$173$$ −1422.77 −0.625269 −0.312634 0.949874i $$-0.601211\pi$$
−0.312634 + 0.949874i $$0.601211\pi$$
$$174$$ 172.366i 0.0750978i
$$175$$ − 1044.82i − 0.451318i
$$176$$ − 1332.47i − 0.570673i
$$177$$ 1072.66i 0.455514i
$$178$$ −314.890 −0.132596
$$179$$ −1167.89 −0.487666 −0.243833 0.969817i $$-0.578405\pi$$
−0.243833 + 0.969817i $$0.578405\pi$$
$$180$$ − 1866.36i − 0.772834i
$$181$$ 1133.96 0.465673 0.232836 0.972516i $$-0.425199\pi$$
0.232836 + 0.972516i $$0.425199\pi$$
$$182$$ 0 0
$$183$$ −2444.07 −0.987274
$$184$$ − 974.004i − 0.390242i
$$185$$ 76.4252 0.0303724
$$186$$ 446.137 0.175873
$$187$$ 1524.44i 0.596141i
$$188$$ 2487.69i 0.965071i
$$189$$ 810.035i 0.311753i
$$190$$ − 630.928i − 0.240907i
$$191$$ 2682.12 1.01608 0.508040 0.861333i $$-0.330370\pi$$
0.508040 + 0.861333i $$0.330370\pi$$
$$192$$ −1619.97 −0.608913
$$193$$ − 1970.67i − 0.734983i −0.930027 0.367491i $$-0.880217\pi$$
0.930027 0.367491i $$-0.119783\pi$$
$$194$$ 332.943 0.123216
$$195$$ 0 0
$$196$$ 2447.14 0.891813
$$197$$ 4016.05i 1.45244i 0.687460 + 0.726222i $$0.258726\pi$$
−0.687460 + 0.726222i $$0.741274\pi$$
$$198$$ −131.970 −0.0473672
$$199$$ 4226.06 1.50541 0.752707 0.658356i $$-0.228748\pi$$
0.752707 + 0.658356i $$0.228748\pi$$
$$200$$ − 1331.53i − 0.470768i
$$201$$ − 1566.35i − 0.549662i
$$202$$ 152.885i 0.0532523i
$$203$$ − 580.245i − 0.200617i
$$204$$ 1955.85 0.671260
$$205$$ 4056.06 1.38189
$$206$$ 254.432i 0.0860541i
$$207$$ 1886.39 0.633398
$$208$$ 0 0
$$209$$ 1811.98 0.599699
$$210$$ 156.458i 0.0514126i
$$211$$ 1364.67 0.445249 0.222625 0.974904i $$-0.428538\pi$$
0.222625 + 0.974904i $$0.428538\pi$$
$$212$$ −528.244 −0.171132
$$213$$ − 563.617i − 0.181307i
$$214$$ 250.570i 0.0800401i
$$215$$ − 490.237i − 0.155506i
$$216$$ 1032.33i 0.325189i
$$217$$ −1501.86 −0.469828
$$218$$ 77.1683 0.0239748
$$219$$ − 432.093i − 0.133325i
$$220$$ 3117.71 0.955436
$$221$$ 0 0
$$222$$ −6.93332 −0.00209610
$$223$$ 1059.47i 0.318149i 0.987267 + 0.159075i $$0.0508510\pi$$
−0.987267 + 0.159075i $$0.949149\pi$$
$$224$$ −443.239 −0.132210
$$225$$ 2578.84 0.764099
$$226$$ 554.584i 0.163232i
$$227$$ − 3464.19i − 1.01289i −0.862272 0.506446i $$-0.830959\pi$$
0.862272 0.506446i $$-0.169041\pi$$
$$228$$ − 2324.75i − 0.675266i
$$229$$ − 2324.64i − 0.670815i −0.942073 0.335407i $$-0.891126\pi$$
0.942073 0.335407i $$-0.108874\pi$$
$$230$$ 1097.23 0.314563
$$231$$ −449.336 −0.127983
$$232$$ − 739.476i − 0.209263i
$$233$$ 3731.01 1.04904 0.524521 0.851398i $$-0.324245\pi$$
0.524521 + 0.851398i $$0.324245\pi$$
$$234$$ 0 0
$$235$$ −5673.86 −1.57499
$$236$$ − 2272.95i − 0.626935i
$$237$$ −746.004 −0.204465
$$238$$ 162.108 0.0441508
$$239$$ − 6044.47i − 1.63592i −0.575278 0.817958i $$-0.695106\pi$$
0.575278 0.817958i $$-0.304894\pi$$
$$240$$ − 3899.09i − 1.04869i
$$241$$ − 5173.96i − 1.38292i −0.722414 0.691461i $$-0.756967\pi$$
0.722414 0.691461i $$-0.243033\pi$$
$$242$$ 363.120i 0.0964556i
$$243$$ −3334.77 −0.880353
$$244$$ 5178.97 1.35881
$$245$$ 5581.37i 1.45543i
$$246$$ −367.967 −0.0953688
$$247$$ 0 0
$$248$$ −1914.00 −0.490076
$$249$$ 1238.62i 0.315238i
$$250$$ 524.029 0.132570
$$251$$ −5620.73 −1.41346 −0.706728 0.707486i $$-0.749829\pi$$
−0.706728 + 0.707486i $$0.749829\pi$$
$$252$$ − 569.981i − 0.142482i
$$253$$ 3151.17i 0.783054i
$$254$$ − 1141.76i − 0.282050i
$$255$$ 4460.86i 1.09549i
$$256$$ 3146.83 0.768270
$$257$$ 1674.14 0.406342 0.203171 0.979143i $$-0.434875\pi$$
0.203171 + 0.979143i $$0.434875\pi$$
$$258$$ 44.4745i 0.0107320i
$$259$$ 23.3401 0.00559954
$$260$$ 0 0
$$261$$ 1432.17 0.339653
$$262$$ 934.640i 0.220390i
$$263$$ −6309.18 −1.47924 −0.739622 0.673023i $$-0.764996\pi$$
−0.739622 + 0.673023i $$0.764996\pi$$
$$264$$ −572.644 −0.133499
$$265$$ − 1204.81i − 0.279285i
$$266$$ − 192.684i − 0.0444143i
$$267$$ − 2646.30i − 0.606558i
$$268$$ 3319.09i 0.756514i
$$269$$ −2482.73 −0.562731 −0.281366 0.959601i $$-0.590787\pi$$
−0.281366 + 0.959601i $$0.590787\pi$$
$$270$$ −1162.94 −0.262126
$$271$$ 2835.72i 0.635638i 0.948151 + 0.317819i $$0.102950\pi$$
−0.948151 + 0.317819i $$0.897050\pi$$
$$272$$ −4039.88 −0.900566
$$273$$ 0 0
$$274$$ −301.645 −0.0665075
$$275$$ 4307.88i 0.944637i
$$276$$ 4042.94 0.881725
$$277$$ 3837.51 0.832396 0.416198 0.909274i $$-0.363362\pi$$
0.416198 + 0.909274i $$0.363362\pi$$
$$278$$ − 297.960i − 0.0642822i
$$279$$ − 3706.91i − 0.795438i
$$280$$ − 671.231i − 0.143263i
$$281$$ − 9122.13i − 1.93659i −0.249819 0.968293i $$-0.580371\pi$$
0.249819 0.968293i $$-0.419629\pi$$
$$282$$ 514.735 0.108695
$$283$$ −2127.85 −0.446952 −0.223476 0.974709i $$-0.571740\pi$$
−0.223476 + 0.974709i $$0.571740\pi$$
$$284$$ 1194.30i 0.249537i
$$285$$ 5302.24 1.10203
$$286$$ 0 0
$$287$$ 1238.71 0.254769
$$288$$ − 1094.01i − 0.223838i
$$289$$ −291.061 −0.0592430
$$290$$ 833.035 0.168681
$$291$$ 2798.01i 0.563651i
$$292$$ 915.601i 0.183498i
$$293$$ 8274.77i 1.64989i 0.565215 + 0.824944i $$0.308793\pi$$
−0.565215 + 0.824944i $$0.691207\pi$$
$$294$$ − 506.344i − 0.100444i
$$295$$ 5184.10 1.02315
$$296$$ 29.7450 0.00584086
$$297$$ − 3339.86i − 0.652520i
$$298$$ −866.136 −0.168369
$$299$$ 0 0
$$300$$ 5526.99 1.06367
$$301$$ − 149.717i − 0.0286696i
$$302$$ −790.625 −0.150647
$$303$$ −1284.83 −0.243602
$$304$$ 4801.86i 0.905940i
$$305$$ 11812.1i 2.21757i
$$306$$ 400.118i 0.0747492i
$$307$$ − 3610.49i − 0.671211i −0.942003 0.335605i $$-0.891059\pi$$
0.942003 0.335605i $$-0.108941\pi$$
$$308$$ 952.140 0.176147
$$309$$ −2138.22 −0.393654
$$310$$ − 2156.16i − 0.395037i
$$311$$ −3331.06 −0.607354 −0.303677 0.952775i $$-0.598214\pi$$
−0.303677 + 0.952775i $$0.598214\pi$$
$$312$$ 0 0
$$313$$ −358.125 −0.0646724 −0.0323362 0.999477i $$-0.510295\pi$$
−0.0323362 + 0.999477i $$0.510295\pi$$
$$314$$ − 174.137i − 0.0312966i
$$315$$ 1300.00 0.232529
$$316$$ 1580.78 0.281410
$$317$$ − 3047.46i − 0.539944i −0.962868 0.269972i $$-0.912986\pi$$
0.962868 0.269972i $$-0.0870144\pi$$
$$318$$ 109.301i 0.0192744i
$$319$$ 2392.41i 0.419904i
$$320$$ 7829.23i 1.36771i
$$321$$ −2105.76 −0.366143
$$322$$ 335.093 0.0579938
$$323$$ − 5493.70i − 0.946371i
$$324$$ −1455.26 −0.249530
$$325$$ 0 0
$$326$$ −412.751 −0.0701232
$$327$$ 648.514i 0.109672i
$$328$$ 1578.64 0.265749
$$329$$ −1732.78 −0.290369
$$330$$ − 645.094i − 0.107610i
$$331$$ − 7694.77i − 1.27777i −0.769301 0.638887i $$-0.779395\pi$$
0.769301 0.638887i $$-0.220605\pi$$
$$332$$ − 2624.62i − 0.433870i
$$333$$ 57.6084i 0.00948025i
$$334$$ −1613.68 −0.264360
$$335$$ −7570.10 −1.23462
$$336$$ − 1190.77i − 0.193339i
$$337$$ −4712.21 −0.761693 −0.380846 0.924638i $$-0.624367\pi$$
−0.380846 + 0.924638i $$0.624367\pi$$
$$338$$ 0 0
$$339$$ −4660.66 −0.746703
$$340$$ − 9452.53i − 1.50775i
$$341$$ 6192.31 0.983380
$$342$$ 475.586 0.0751952
$$343$$ 3569.92i 0.561976i
$$344$$ − 190.803i − 0.0299052i
$$345$$ 9221.03i 1.43897i
$$346$$ − 623.811i − 0.0969257i
$$347$$ −5261.98 −0.814058 −0.407029 0.913415i $$-0.633435\pi$$
−0.407029 + 0.913415i $$0.633435\pi$$
$$348$$ 3069.45 0.472815
$$349$$ 50.3345i 0.00772018i 0.999993 + 0.00386009i $$0.00122871\pi$$
−0.999993 + 0.00386009i $$0.998771\pi$$
$$350$$ 458.096 0.0699608
$$351$$ 0 0
$$352$$ 1827.52 0.276725
$$353$$ − 9057.64i − 1.36569i −0.730562 0.682846i $$-0.760742\pi$$
0.730562 0.682846i $$-0.239258\pi$$
$$354$$ −470.304 −0.0706112
$$355$$ −2723.93 −0.407243
$$356$$ 5607.49i 0.834821i
$$357$$ 1362.34i 0.201968i
$$358$$ − 512.059i − 0.0755953i
$$359$$ 7177.86i 1.05525i 0.849479 + 0.527623i $$0.176917\pi$$
−0.849479 + 0.527623i $$0.823083\pi$$
$$360$$ 1656.75 0.242551
$$361$$ 329.105 0.0479815
$$362$$ 497.183i 0.0721861i
$$363$$ −3051.62 −0.441235
$$364$$ 0 0
$$365$$ −2088.28 −0.299467
$$366$$ − 1071.60i − 0.153042i
$$367$$ 4004.14 0.569522 0.284761 0.958599i $$-0.408086\pi$$
0.284761 + 0.958599i $$0.408086\pi$$
$$368$$ −8350.83 −1.18293
$$369$$ 3057.41i 0.431334i
$$370$$ 33.5084i 0.00470816i
$$371$$ − 367.945i − 0.0514899i
$$372$$ − 7944.70i − 1.10729i
$$373$$ −10014.2 −1.39012 −0.695060 0.718952i $$-0.744622\pi$$
−0.695060 + 0.718952i $$0.744622\pi$$
$$374$$ −668.388 −0.0924105
$$375$$ 4403.88i 0.606441i
$$376$$ −2208.30 −0.302883
$$377$$ 0 0
$$378$$ −355.158 −0.0483263
$$379$$ 8169.12i 1.10717i 0.832791 + 0.553587i $$0.186742\pi$$
−0.832791 + 0.553587i $$0.813258\pi$$
$$380$$ −11235.4 −1.51675
$$381$$ 9595.24 1.29023
$$382$$ 1175.97i 0.157507i
$$383$$ 7310.25i 0.975290i 0.873042 + 0.487645i $$0.162144\pi$$
−0.873042 + 0.487645i $$0.837856\pi$$
$$384$$ − 3112.70i − 0.413656i
$$385$$ 2171.62i 0.287470i
$$386$$ 864.033 0.113933
$$387$$ 369.535 0.0485388
$$388$$ − 5928.97i − 0.775767i
$$389$$ −8785.47 −1.14509 −0.572546 0.819872i $$-0.694044\pi$$
−0.572546 + 0.819872i $$0.694044\pi$$
$$390$$ 0 0
$$391$$ 9553.99 1.23572
$$392$$ 2172.30i 0.279892i
$$393$$ −7854.60 −1.00817
$$394$$ −1760.82 −0.225150
$$395$$ 3605.40i 0.459259i
$$396$$ 2350.09i 0.298224i
$$397$$ 11266.8i 1.42434i 0.702006 + 0.712171i $$0.252288\pi$$
−0.702006 + 0.712171i $$0.747712\pi$$
$$398$$ 1852.90i 0.233361i
$$399$$ 1619.29 0.203173
$$400$$ −11416.2 −1.42702
$$401$$ 1576.23i 0.196293i 0.995172 + 0.0981464i $$0.0312913\pi$$
−0.995172 + 0.0981464i $$0.968709\pi$$
$$402$$ 686.763 0.0852055
$$403$$ 0 0
$$404$$ 2722.54 0.335276
$$405$$ − 3319.12i − 0.407231i
$$406$$ 254.407 0.0310985
$$407$$ −96.2335 −0.0117202
$$408$$ 1736.19i 0.210672i
$$409$$ − 6755.78i − 0.816753i −0.912814 0.408377i $$-0.866095\pi$$
0.912814 0.408377i $$-0.133905\pi$$
$$410$$ 1778.37i 0.214213i
$$411$$ − 2534.99i − 0.304238i
$$412$$ 4530.87 0.541796
$$413$$ 1583.21 0.188631
$$414$$ 827.083i 0.0981858i
$$415$$ 5986.17 0.708072
$$416$$ 0 0
$$417$$ 2504.02 0.294059
$$418$$ 794.456i 0.0929620i
$$419$$ 10756.2 1.25411 0.627057 0.778973i $$-0.284259\pi$$
0.627057 + 0.778973i $$0.284259\pi$$
$$420$$ 2786.17 0.323694
$$421$$ − 7886.03i − 0.912925i −0.889743 0.456463i $$-0.849116\pi$$
0.889743 0.456463i $$-0.150884\pi$$
$$422$$ 598.335i 0.0690201i
$$423$$ − 4276.89i − 0.491607i
$$424$$ − 468.916i − 0.0537089i
$$425$$ 13061.0 1.49071
$$426$$ 247.116 0.0281052
$$427$$ 3607.38i 0.408837i
$$428$$ 4462.08 0.503932
$$429$$ 0 0
$$430$$ 214.943 0.0241057
$$431$$ 14084.6i 1.57409i 0.616897 + 0.787044i $$0.288390\pi$$
−0.616897 + 0.787044i $$0.711610\pi$$
$$432$$ 8850.86 0.985735
$$433$$ −1864.14 −0.206894 −0.103447 0.994635i $$-0.532987\pi$$
−0.103447 + 0.994635i $$0.532987\pi$$
$$434$$ − 658.485i − 0.0728301i
$$435$$ 7000.73i 0.771630i
$$436$$ − 1374.20i − 0.150945i
$$437$$ − 11356.0i − 1.24309i
$$438$$ 189.450 0.0206673
$$439$$ −6154.49 −0.669106 −0.334553 0.942377i $$-0.608585\pi$$
−0.334553 + 0.942377i $$0.608585\pi$$
$$440$$ 2767.56i 0.299859i
$$441$$ −4207.17 −0.454289
$$442$$ 0 0
$$443$$ −14539.3 −1.55933 −0.779663 0.626200i $$-0.784609\pi$$
−0.779663 + 0.626200i $$0.784609\pi$$
$$444$$ 123.467i 0.0131970i
$$445$$ −12789.4 −1.36242
$$446$$ −464.521 −0.0493177
$$447$$ − 7278.90i − 0.770202i
$$448$$ 2391.03i 0.252155i
$$449$$ − 7043.87i − 0.740358i −0.928960 0.370179i $$-0.879296\pi$$
0.928960 0.370179i $$-0.120704\pi$$
$$450$$ 1130.68i 0.118446i
$$451$$ −5107.33 −0.533248
$$452$$ 9875.90 1.02771
$$453$$ − 6644.32i − 0.689133i
$$454$$ 1518.87 0.157013
$$455$$ 0 0
$$456$$ 2063.66 0.211929
$$457$$ − 14098.9i − 1.44314i −0.692340 0.721572i $$-0.743420\pi$$
0.692340 0.721572i $$-0.256580\pi$$
$$458$$ 1019.23 0.103986
$$459$$ −10126.1 −1.02973
$$460$$ − 19539.3i − 1.98049i
$$461$$ − 14449.7i − 1.45985i −0.683529 0.729924i $$-0.739556\pi$$
0.683529 0.729924i $$-0.260444\pi$$
$$462$$ − 197.010i − 0.0198393i
$$463$$ 15806.5i 1.58659i 0.608840 + 0.793293i $$0.291635\pi$$
−0.608840 + 0.793293i $$0.708365\pi$$
$$464$$ −6340.06 −0.634332
$$465$$ 18120.1 1.80709
$$466$$ 1635.85i 0.162617i
$$467$$ 15071.3 1.49340 0.746699 0.665162i $$-0.231638\pi$$
0.746699 + 0.665162i $$0.231638\pi$$
$$468$$ 0 0
$$469$$ −2311.89 −0.227619
$$470$$ − 2487.69i − 0.244146i
$$471$$ 1463.43 0.143166
$$472$$ 2017.68 0.196761
$$473$$ 617.299i 0.0600073i
$$474$$ − 327.083i − 0.0316950i
$$475$$ − 15524.5i − 1.49961i
$$476$$ − 2886.78i − 0.277973i
$$477$$ 908.169 0.0871744
$$478$$ 2650.18 0.253591
$$479$$ − 392.545i − 0.0374443i −0.999825 0.0187222i $$-0.994040\pi$$
0.999825 0.0187222i $$-0.00595980\pi$$
$$480$$ 5347.73 0.508519
$$481$$ 0 0
$$482$$ 2268.51 0.214373
$$483$$ 2816.08i 0.265292i
$$484$$ 6466.36 0.607284
$$485$$ 13522.7 1.26605
$$486$$ − 1462.12i − 0.136467i
$$487$$ − 9497.89i − 0.883758i −0.897075 0.441879i $$-0.854312\pi$$
0.897075 0.441879i $$-0.145688\pi$$
$$488$$ 4597.32i 0.426457i
$$489$$ − 3468.71i − 0.320778i
$$490$$ −2447.14 −0.225613
$$491$$ 1893.82 0.174067 0.0870337 0.996205i $$-0.472261\pi$$
0.0870337 + 0.996205i $$0.472261\pi$$
$$492$$ 6552.67i 0.600442i
$$493$$ 7253.52 0.662641
$$494$$ 0 0
$$495$$ −5360.04 −0.486699
$$496$$ 16410.1i 1.48555i
$$497$$ −831.881 −0.0750804
$$498$$ −543.068 −0.0488664
$$499$$ 13370.1i 1.19945i 0.800205 + 0.599727i $$0.204724\pi$$
−0.800205 + 0.599727i $$0.795276\pi$$
$$500$$ − 9331.79i − 0.834661i
$$501$$ − 13561.1i − 1.20932i
$$502$$ − 2464.39i − 0.219106i
$$503$$ 5554.71 0.492391 0.246195 0.969220i $$-0.420820\pi$$
0.246195 + 0.969220i $$0.420820\pi$$
$$504$$ 505.966 0.0447173
$$505$$ 6209.51i 0.547168i
$$506$$ −1381.62 −0.121385
$$507$$ 0 0
$$508$$ −20332.3 −1.77578
$$509$$ − 2197.55i − 0.191365i −0.995412 0.0956824i $$-0.969497\pi$$
0.995412 0.0956824i $$-0.0305033\pi$$
$$510$$ −1955.85 −0.169817
$$511$$ −637.756 −0.0552107
$$512$$ 8137.89i 0.702437i
$$513$$ 12036.0i 1.03587i
$$514$$ 734.022i 0.0629890i
$$515$$ 10333.9i 0.884206i
$$516$$ 791.991 0.0675687
$$517$$ 7144.45 0.607761
$$518$$ 10.2334i 0 0.000868010i
$$519$$ 5242.44 0.443386
$$520$$ 0 0
$$521$$ 17005.2 1.42997 0.714983 0.699142i $$-0.246435\pi$$
0.714983 + 0.699142i $$0.246435\pi$$
$$522$$ 627.932i 0.0526511i
$$523$$ −14486.2 −1.21116 −0.605581 0.795783i $$-0.707059\pi$$
−0.605581 + 0.795783i $$0.707059\pi$$
$$524$$ 16643.8 1.38758
$$525$$ 3849.79i 0.320035i
$$526$$ − 2766.24i − 0.229304i
$$527$$ − 18774.4i − 1.55185i
$$528$$ 4909.68i 0.404671i
$$529$$ 7582.03 0.623163
$$530$$ 528.244 0.0432933
$$531$$ 3907.72i 0.319361i
$$532$$ −3431.27 −0.279632
$$533$$ 0 0
$$534$$ 1160.26 0.0940252
$$535$$ 10177.0i 0.822413i
$$536$$ −2946.32 −0.237429
$$537$$ 4303.28 0.345810
$$538$$ − 1088.55i − 0.0872315i
$$539$$ − 7027.98i − 0.561626i
$$540$$ 20709.3i 1.65034i
$$541$$ − 15266.7i − 1.21325i −0.794990 0.606623i $$-0.792524\pi$$
0.794990 0.606623i $$-0.207476\pi$$
$$542$$ −1243.31 −0.0985330
$$543$$ −4178.27 −0.330215
$$544$$ − 5540.83i − 0.436693i
$$545$$ 3134.23 0.246341
$$546$$ 0 0
$$547$$ 15260.5 1.19286 0.596430 0.802665i $$-0.296586\pi$$
0.596430 + 0.802665i $$0.296586\pi$$
$$548$$ 5371.62i 0.418731i
$$549$$ −8903.81 −0.692177
$$550$$ −1888.78 −0.146432
$$551$$ − 8621.64i − 0.666595i
$$552$$ 3588.87i 0.276726i
$$553$$ 1101.08i 0.0846703i
$$554$$ 1682.55i 0.129033i
$$555$$ −281.601 −0.0215374
$$556$$ −5306.00 −0.404721
$$557$$ − 10442.1i − 0.794337i −0.917746 0.397169i $$-0.869993\pi$$
0.917746 0.397169i $$-0.130007\pi$$
$$558$$ 1625.29 0.123304
$$559$$ 0 0
$$560$$ −5754.94 −0.434269
$$561$$ − 5617.06i − 0.422731i
$$562$$ 3999.57 0.300199
$$563$$ −7145.26 −0.534879 −0.267440 0.963575i $$-0.586178\pi$$
−0.267440 + 0.963575i $$0.586178\pi$$
$$564$$ − 9166.29i − 0.684344i
$$565$$ 22524.7i 1.67721i
$$566$$ − 932.950i − 0.0692841i
$$567$$ − 1013.65i − 0.0750783i
$$568$$ −1060.17 −0.0783162
$$569$$ −4438.86 −0.327042 −0.163521 0.986540i $$-0.552285\pi$$
−0.163521 + 0.986540i $$0.552285\pi$$
$$570$$ 2324.75i 0.170830i
$$571$$ −10117.3 −0.741497 −0.370748 0.928733i $$-0.620899\pi$$
−0.370748 + 0.928733i $$0.620899\pi$$
$$572$$ 0 0
$$573$$ −9882.70 −0.720516
$$574$$ 543.109i 0.0394929i
$$575$$ 26998.4 1.95810
$$576$$ −5901.58 −0.426909
$$577$$ − 3105.60i − 0.224069i −0.993704 0.112035i $$-0.964263\pi$$
0.993704 0.112035i $$-0.0357368\pi$$
$$578$$ − 127.615i − 0.00918352i
$$579$$ 7261.23i 0.521186i
$$580$$ − 14834.5i − 1.06202i
$$581$$ 1828.16 0.130542
$$582$$ −1226.78 −0.0873741
$$583$$ 1517.08i 0.107772i
$$584$$ −812.769 −0.0575901
$$585$$ 0 0
$$586$$ −3628.05 −0.255757
$$587$$ − 19662.3i − 1.38254i −0.722597 0.691270i $$-0.757052\pi$$
0.722597 0.691270i $$-0.242948\pi$$
$$588$$ −9016.86 −0.632396
$$589$$ −22315.5 −1.56111
$$590$$ 2272.95i 0.158603i
$$591$$ − 14797.8i − 1.02995i
$$592$$ − 255.026i − 0.0177052i
$$593$$ 6395.51i 0.442888i 0.975173 + 0.221444i $$0.0710769\pi$$
−0.975173 + 0.221444i $$0.928923\pi$$
$$594$$ 1464.35 0.101150
$$595$$ 6584.10 0.453650
$$596$$ 15423.9i 1.06005i
$$597$$ −15571.6 −1.06751
$$598$$ 0 0
$$599$$ 8878.48 0.605618 0.302809 0.953051i $$-0.402076\pi$$
0.302809 + 0.953051i $$0.402076\pi$$
$$600$$ 4906.25i 0.333828i
$$601$$ 19100.6 1.29639 0.648194 0.761475i $$-0.275525\pi$$
0.648194 + 0.761475i $$0.275525\pi$$
$$602$$ 65.6430 0.00444420
$$603$$ − 5706.26i − 0.385368i
$$604$$ 14079.3i 0.948472i
$$605$$ 14748.3i 0.991082i
$$606$$ − 563.330i − 0.0377619i
$$607$$ 16595.8 1.10972 0.554861 0.831943i $$-0.312771\pi$$
0.554861 + 0.831943i $$0.312771\pi$$
$$608$$ −6585.91 −0.439299
$$609$$ 2138.01i 0.142260i
$$610$$ −5178.97 −0.343755
$$611$$ 0 0
$$612$$ 7125.21 0.470620
$$613$$ − 16469.2i − 1.08513i −0.840015 0.542564i $$-0.817454\pi$$
0.840015 0.542564i $$-0.182546\pi$$
$$614$$ 1583.01 0.104047
$$615$$ −14945.2 −0.979915
$$616$$ 845.205i 0.0552829i
$$617$$ − 10116.0i − 0.660055i −0.943971 0.330027i $$-0.892942\pi$$
0.943971 0.330027i $$-0.107058\pi$$
$$618$$ − 937.496i − 0.0610220i
$$619$$ 18854.8i 1.22430i 0.790743 + 0.612148i $$0.209694\pi$$
−0.790743 + 0.612148i $$0.790306\pi$$
$$620$$ −38396.3 −2.48715
$$621$$ −20931.6 −1.35258
$$622$$ − 1460.49i − 0.0941487i
$$623$$ −3905.86 −0.251180
$$624$$ 0 0
$$625$$ −2730.82 −0.174773
$$626$$ − 157.019i − 0.0100252i
$$627$$ −6676.51 −0.425254
$$628$$ −3100.99 −0.197043
$$629$$ 291.769i 0.0184954i
$$630$$ 569.981i 0.0360454i
$$631$$ 18946.2i 1.19531i 0.801755 + 0.597653i $$0.203900\pi$$
−0.801755 + 0.597653i $$0.796100\pi$$
$$632$$ 1403.24i 0.0883194i
$$633$$ −5028.33 −0.315732
$$634$$ 1336.15 0.0836991
$$635$$ − 46373.3i − 2.89806i
$$636$$ 1946.40 0.121352
$$637$$ 0 0
$$638$$ −1048.95 −0.0650912
$$639$$ − 2053.27i − 0.127114i
$$640$$ −15043.5 −0.929135
$$641$$ 23586.9 1.45340 0.726698 0.686957i $$-0.241054\pi$$
0.726698 + 0.686957i $$0.241054\pi$$
$$642$$ − 923.263i − 0.0567575i
$$643$$ 27153.0i 1.66534i 0.553772 + 0.832669i $$0.313188\pi$$
−0.553772 + 0.832669i $$0.686812\pi$$
$$644$$ − 5967.25i − 0.365128i
$$645$$ 1806.35i 0.110272i
$$646$$ 2408.70 0.146701
$$647$$ −6856.72 −0.416639 −0.208319 0.978061i $$-0.566799\pi$$
−0.208319 + 0.978061i $$0.566799\pi$$
$$648$$ − 1291.82i − 0.0783140i
$$649$$ −6527.75 −0.394817
$$650$$ 0 0
$$651$$ 5533.83 0.333161
$$652$$ 7350.17i 0.441495i
$$653$$ −8073.89 −0.483853 −0.241926 0.970295i $$-0.577779\pi$$
−0.241926 + 0.970295i $$0.577779\pi$$
$$654$$ −284.339 −0.0170008
$$655$$ 37960.9i 2.26451i
$$656$$ − 13534.8i − 0.805555i
$$657$$ − 1574.12i − 0.0934739i
$$658$$ − 759.734i − 0.0450114i
$$659$$ 5305.73 0.313629 0.156815 0.987628i $$-0.449877\pi$$
0.156815 + 0.987628i $$0.449877\pi$$
$$660$$ −11487.7 −0.677512
$$661$$ 25848.3i 1.52100i 0.649336 + 0.760502i $$0.275047\pi$$
−0.649336 + 0.760502i $$0.724953\pi$$
$$662$$ 3373.75 0.198073
$$663$$ 0 0
$$664$$ 2329.85 0.136168
$$665$$ − 7825.96i − 0.456357i
$$666$$ −25.2583 −0.00146958
$$667$$ 14993.7 0.870404
$$668$$ 28735.9i 1.66441i
$$669$$ − 3903.78i − 0.225604i
$$670$$ − 3319.09i − 0.191385i
$$671$$ − 14873.6i − 0.855722i
$$672$$ 1633.18 0.0937521
$$673$$ 14529.1 0.832177 0.416089 0.909324i $$-0.363401\pi$$
0.416089 + 0.909324i $$0.363401\pi$$
$$674$$ − 2066.06i − 0.118073i
$$675$$ −28615.0 −1.63169
$$676$$ 0 0
$$677$$ 12058.1 0.684535 0.342267 0.939603i $$-0.388805\pi$$
0.342267 + 0.939603i $$0.388805\pi$$
$$678$$ − 2043.45i − 0.115750i
$$679$$ 4129.78 0.233412
$$680$$ 8390.91 0.473201
$$681$$ 12764.4i 0.718255i
$$682$$ 2715.00i 0.152438i
$$683$$ 30028.8i 1.68231i 0.540792 + 0.841156i $$0.318125\pi$$
−0.540792 + 0.841156i $$0.681875\pi$$
$$684$$ − 8469.13i − 0.473429i
$$685$$ −12251.5 −0.683364
$$686$$ −1565.22 −0.0871144
$$687$$ 8565.50i 0.475683i
$$688$$ −1635.89 −0.0906505
$$689$$ 0 0
$$690$$ −4042.94 −0.223061
$$691$$ − 449.696i − 0.0247572i −0.999923 0.0123786i $$-0.996060\pi$$
0.999923 0.0123786i $$-0.00394034\pi$$
$$692$$ −11108.7 −0.610244
$$693$$ −1636.94 −0.0897291
$$694$$ − 2307.10i − 0.126191i
$$695$$ − 12101.8i − 0.660500i
$$696$$ 2724.72i 0.148391i
$$697$$ 15484.8i 0.841506i
$$698$$ −22.0690 −0.00119674
$$699$$ −13747.5 −0.743889
$$700$$ − 8157.67i − 0.440473i
$$701$$ 26986.0 1.45399 0.726994 0.686644i $$-0.240917\pi$$
0.726994 + 0.686644i $$0.240917\pi$$
$$702$$ 0 0
$$703$$ 346.801 0.0186057
$$704$$ − 9858.46i − 0.527776i
$$705$$ 20906.2 1.11684
$$706$$ 3971.29 0.211702
$$707$$ 1896.37i 0.100877i
$$708$$ 8375.06i 0.444568i
$$709$$ − 9098.87i − 0.481968i −0.970529 0.240984i $$-0.922530\pi$$
0.970529 0.240984i $$-0.0774701\pi$$
$$710$$ − 1194.30i − 0.0631285i
$$711$$ −2717.71 −0.143350
$$712$$ −4977.71 −0.262005
$$713$$ − 38808.4i − 2.03841i
$$714$$ −597.312 −0.0313079
$$715$$ 0 0
$$716$$ −9118.62 −0.475948
$$717$$ 22271.8i 1.16005i
$$718$$ −3147.11 −0.163578
$$719$$ 6293.55 0.326439 0.163220 0.986590i $$-0.447812\pi$$
0.163220 + 0.986590i $$0.447812\pi$$
$$720$$ − 14204.5i − 0.735235i
$$721$$ 3155.95i 0.163015i
$$722$$ 144.295i 0.00743783i
$$723$$ 19064.3i 0.980648i
$$724$$ 8853.72 0.454483
$$725$$ 20497.5 1.05001
$$726$$ − 1337.97i − 0.0683979i
$$727$$ −18070.7 −0.921878 −0.460939 0.887432i $$-0.652487\pi$$
−0.460939 + 0.887432i $$0.652487\pi$$
$$728$$ 0 0
$$729$$ 17319.9 0.879944
$$730$$ − 915.601i − 0.0464218i
$$731$$ 1871.58 0.0946962
$$732$$ −19082.7 −0.963550
$$733$$ − 34771.5i − 1.75214i −0.482188 0.876068i $$-0.660158\pi$$
0.482188 0.876068i $$-0.339842\pi$$
$$734$$ 1755.61i 0.0882842i
$$735$$ − 20565.4i − 1.03206i
$$736$$ − 11453.4i − 0.573613i
$$737$$ 9532.17 0.476421
$$738$$ −1340.51 −0.0668631
$$739$$ 23631.5i 1.17632i 0.808746 + 0.588158i $$0.200147\pi$$
−0.808746 + 0.588158i $$0.799853\pi$$
$$740$$ 596.710 0.0296425
$$741$$ 0 0
$$742$$ 161.324 0.00798167
$$743$$ − 32502.8i − 1.60486i −0.596745 0.802431i $$-0.703540\pi$$
0.596745 0.802431i $$-0.296460\pi$$
$$744$$ 7052.42 0.347519
$$745$$ −35178.6 −1.72999
$$746$$ − 4390.69i − 0.215489i
$$747$$ 4512.31i 0.221013i
$$748$$ 11902.5i 0.581816i
$$749$$ 3108.04i 0.151622i
$$750$$ −1930.87 −0.0940072
$$751$$ −2020.86 −0.0981920 −0.0490960 0.998794i $$-0.515634\pi$$
−0.0490960 + 0.998794i $$0.515634\pi$$
$$752$$ 18933.3i 0.918120i
$$753$$ 20710.5 1.00230
$$754$$ 0 0
$$755$$ −32111.6 −1.54790
$$756$$ 6324.56i 0.304262i
$$757$$ 12568.2 0.603434 0.301717 0.953398i $$-0.402440\pi$$
0.301717 + 0.953398i $$0.402440\pi$$
$$758$$ −3581.73 −0.171628
$$759$$ − 11611.0i − 0.555273i
$$760$$ − 9973.56i − 0.476025i
$$761$$ − 8704.81i − 0.414651i −0.978272 0.207325i $$-0.933524\pi$$
0.978272 0.207325i $$-0.0664758\pi$$
$$762$$ 4207.01i 0.200005i
$$763$$ 957.187 0.0454161
$$764$$ 20941.4 0.991665
$$765$$ 16251.0i 0.768048i
$$766$$ −3205.16 −0.151184
$$767$$ 0 0
$$768$$ −11595.0 −0.544790
$$769$$ 21915.9i 1.02771i 0.857878 + 0.513853i $$0.171782\pi$$
−0.857878 + 0.513853i $$0.828218\pi$$
$$770$$ −952.140 −0.0445620
$$771$$ −6168.63 −0.288143
$$772$$ − 15386.5i − 0.717322i
$$773$$ 23077.5i 1.07379i 0.843649 + 0.536896i $$0.180403\pi$$
−0.843649 + 0.536896i $$0.819597\pi$$
$$774$$ 162.022i 0.00752422i
$$775$$ − 53054.0i − 2.45904i
$$776$$ 5263.08 0.243471
$$777$$ −86.0001 −0.00397070
$$778$$ − 3851.96i − 0.177506i
$$779$$ 18405.5 0.846528
$$780$$ 0 0
$$781$$ 3429.94 0.157148
$$782$$ 4188.92i 0.191554i
$$783$$ −15891.5 −0.725309
$$784$$ 18624.6 0.848426
$$785$$ − 7072.67i − 0.321572i
$$786$$ − 3443.83i − 0.156282i
$$787$$ − 16522.4i − 0.748362i −0.927356 0.374181i $$-0.877924\pi$$
0.927356 0.374181i $$-0.122076\pi$$
$$788$$ 31356.4i 1.41754i
$$789$$ 23247.2 1.04895
$$790$$ −1580.78 −0.0711918
$$791$$ 6879.00i 0.309215i
$$792$$ −2086.15 −0.0935962
$$793$$ 0 0
$$794$$ −4939.89 −0.220794
$$795$$ 4439.30i 0.198045i
$$796$$ 32996.1 1.46924
$$797$$ 11719.4 0.520855 0.260427 0.965493i $$-0.416137\pi$$
0.260427 + 0.965493i $$0.416137\pi$$
$$798$$ 709.974i 0.0314948i
$$799$$ − 21661.2i − 0.959095i
$$800$$ − 15657.7i − 0.691978i
$$801$$ − 9640.53i − 0.425258i
$$802$$ −691.096 −0.0304282
$$803$$ 2629.53 0.115559
$$804$$ − 12229.7i − 0.536454i
$$805$$ 13610.0 0.595886
$$806$$ 0 0
$$807$$ 9148.01 0.399040
$$808$$ 2416.77i 0.105225i
$$809$$ −24096.0 −1.04718 −0.523592 0.851969i $$-0.675408\pi$$
−0.523592 + 0.851969i $$0.675408\pi$$
$$810$$ 1455.26 0.0631267
$$811$$ − 16622.6i − 0.719729i −0.933005 0.359864i $$-0.882823\pi$$
0.933005 0.359864i $$-0.117177\pi$$
$$812$$ − 4530.42i − 0.195796i
$$813$$ − 10448.7i − 0.450739i
$$814$$ − 42.1933i − 0.00181680i
$$815$$ −16764.1 −0.720516
$$816$$ 14885.6 0.638603
$$817$$ − 2224.59i − 0.0952613i
$$818$$ 2962.05 0.126609
$$819$$ 0 0
$$820$$ 31668.7 1.34868
$$821$$ 38005.5i 1.61559i 0.589461 + 0.807797i $$0.299340\pi$$
−0.589461 + 0.807797i $$0.700660\pi$$
$$822$$ 1111.46 0.0471613
$$823$$ 15859.5 0.671722 0.335861 0.941912i $$-0.390973\pi$$
0.335861 + 0.941912i $$0.390973\pi$$
$$824$$ 4022.01i 0.170040i
$$825$$ − 15873.1i − 0.669854i
$$826$$ 694.155i 0.0292406i
$$827$$ 12201.0i 0.513023i 0.966541 + 0.256512i $$0.0825732\pi$$
−0.966541 + 0.256512i $$0.917427\pi$$
$$828$$ 14728.5 0.618177
$$829$$ 5431.41 0.227552 0.113776 0.993506i $$-0.463705\pi$$
0.113776 + 0.993506i $$0.463705\pi$$
$$830$$ 2624.62i 0.109761i
$$831$$ −14139.9 −0.590263
$$832$$ 0 0
$$833$$ −21308.0 −0.886290
$$834$$ 1097.88i 0.0455834i
$$835$$ −65540.3 −2.71630
$$836$$ 14147.5 0.585288
$$837$$ 41132.2i 1.69861i
$$838$$ 4716.02i 0.194406i
$$839$$ − 7960.90i − 0.327582i −0.986495 0.163791i $$-0.947628\pi$$
0.986495 0.163791i $$-0.0523722\pi$$
$$840$$ 2473.26i 0.101590i
$$841$$ −13005.6 −0.533255
$$842$$ 3457.61 0.141517
$$843$$ 33611.9i 1.37326i
$$844$$ 10655.0 0.434550
$$845$$ 0 0
$$846$$ 1875.19 0.0762062
$$847$$ 4504.10i 0.182719i
$$848$$ −4020.35 −0.162806
$$849$$ 7840.40 0.316940
$$850$$ 5726.56i 0.231082i
$$851$$ 603.115i 0.0242944i
$$852$$ − 4400.59i − 0.176950i
$$853$$ 13576.7i 0.544969i 0.962160 + 0.272485i $$0.0878454\pi$$
−0.962160 + 0.272485i $$0.912155\pi$$
$$854$$ −1581.65 −0.0633756
$$855$$ 19316.2 0.772631
$$856$$ 3960.95i 0.158157i
$$857$$ 31223.9 1.24456 0.622281 0.782794i $$-0.286206\pi$$
0.622281 + 0.782794i $$0.286206\pi$$
$$858$$ 0 0
$$859$$ −11815.8 −0.469323 −0.234661 0.972077i $$-0.575398\pi$$
−0.234661 + 0.972077i $$0.575398\pi$$
$$860$$ − 3827.65i − 0.151770i
$$861$$ −4564.22 −0.180660
$$862$$ −6175.36 −0.244006
$$863$$ 1790.84i 0.0706384i 0.999376 + 0.0353192i $$0.0112448\pi$$
−0.999376 + 0.0353192i $$0.988755\pi$$
$$864$$ 12139.2i 0.477992i
$$865$$ − 25336.4i − 0.995912i
$$866$$ − 817.328i − 0.0320715i
$$867$$ 1072.46 0.0420100
$$868$$ −11726.1 −0.458538
$$869$$ − 4539.87i − 0.177220i
$$870$$ −3069.45 −0.119614
$$871$$ 0 0
$$872$$ 1219.86 0.0473734
$$873$$ 10193.2i 0.395176i
$$874$$ 4979.01 0.192698
$$875$$ 6500.00 0.251132
$$876$$ − 3373.68i − 0.130121i
$$877$$ − 43542.5i − 1.67654i −0.545255 0.838270i $$-0.683567\pi$$
0.545255 0.838270i $$-0.316433\pi$$
$$878$$ − 2698.42i − 0.103721i
$$879$$ − 30489.7i − 1.16996i
$$880$$ 23728.2 0.908953
$$881$$ 1020.04 0.0390080 0.0195040 0.999810i $$-0.493791\pi$$
0.0195040 + 0.999810i $$0.493791\pi$$
$$882$$ − 1844.62i − 0.0704214i
$$883$$ 34781.9 1.32560 0.662800 0.748797i $$-0.269368\pi$$
0.662800 + 0.748797i $$0.269368\pi$$
$$884$$ 0 0
$$885$$ −19101.6 −0.725531
$$886$$ − 6374.70i − 0.241718i
$$887$$ 49785.1 1.88458 0.942288 0.334802i $$-0.108670\pi$$
0.942288 + 0.334802i $$0.108670\pi$$
$$888$$ −109.600 −0.00414183
$$889$$ − 14162.3i − 0.534295i
$$890$$ − 5607.49i − 0.211195i
$$891$$ 4179.40i 0.157144i
$$892$$ 8272.08i 0.310504i
$$893$$ −25746.8 −0.964818
$$894$$ 3191.41 0.119392
$$895$$ − 20797.5i − 0.776743i
$$896$$ −4594.25 −0.171298
$$897$$ 0 0
$$898$$ 3088.36 0.114766
$$899$$ − 29463.9i − 1.09308i
$$900$$ 20134.9 0.745738
$$901$$ 4599.60 0.170072
$$902$$ − 2239.29i − 0.0826611i
$$903$$ 551.656i 0.0203300i
$$904$$ 8766.74i 0.322541i
$$905$$ 20193.3i 0.741712i
$$906$$ 2913.18 0.106826
$$907$$ −17389.9 −0.636627 −0.318314 0.947985i $$-0.603116\pi$$
−0.318314 + 0.947985i $$0.603116\pi$$
$$908$$ − 27047.6i − 0.988553i
$$909$$ −4680.66 −0.170790
$$910$$ 0 0
$$911$$ 20419.5 0.742621 0.371311 0.928509i $$-0.378909\pi$$
0.371311 + 0.928509i $$0.378909\pi$$
$$912$$ − 17693.2i − 0.642414i
$$913$$ −7537.71 −0.273233
$$914$$ 6181.60 0.223708
$$915$$ − 43523.5i − 1.57250i
$$916$$ − 18150.2i − 0.654695i
$$917$$ 11593.2i 0.417492i
$$918$$ − 4439.75i − 0.159623i
$$919$$ −33231.8 −1.19283 −0.596417 0.802674i $$-0.703410\pi$$
−0.596417 + 0.802674i $$0.703410\pi$$
$$920$$ 17344.8 0.621567
$$921$$ 13303.4i 0.475964i
$$922$$ 6335.43 0.226297
$$923$$ 0 0
$$924$$ −3508.31 −0.124908
$$925$$ 824.502i 0.0293075i
$$926$$ −6930.31 −0.245944
$$927$$ −7789.58 −0.275991
$$928$$ − 8695.59i − 0.307594i
$$929$$ 25222.8i 0.890780i 0.895337 + 0.445390i $$0.146935\pi$$
−0.895337 + 0.445390i $$0.853065\pi$$
$$930$$ 7944.70i 0.280126i
$$931$$ 25327.0i 0.891579i
$$932$$ 29130.9 1.02383
$$933$$ 12273.8 0.430683
$$934$$ 6607.97i 0.231498i
$$935$$ −27146.9 −0.949519
$$936$$ 0 0
$$937$$ −26979.4 −0.940639 −0.470319 0.882496i $$-0.655861\pi$$
−0.470319 + 0.882496i $$0.655861\pi$$
$$938$$ − 1013.64i − 0.0352842i
$$939$$ 1319.57 0.0458600
$$940$$ −44300.2 −1.53714
$$941$$ − 7641.67i − 0.264730i −0.991201 0.132365i $$-0.957743\pi$$
0.991201 0.132365i $$-0.0422572\pi$$
$$942$$ 641.635i 0.0221928i
$$943$$ 32008.7i 1.10535i
$$944$$ − 17299.0i − 0.596434i
$$945$$ −14424.9 −0.496553
$$946$$ −270.653 −0.00930200
$$947$$ − 2869.32i − 0.0984587i −0.998788 0.0492293i $$-0.984323\pi$$
0.998788 0.0492293i $$-0.0156765\pi$$
$$948$$ −5824.62 −0.199552
$$949$$ 0 0
$$950$$ 6806.67 0.232461
$$951$$ 11228.8i 0.382881i
$$952$$ 2562.56 0.0872407
$$953$$ −12313.6 −0.418548 −0.209274 0.977857i $$-0.567110\pi$$
−0.209274 + 0.977857i $$0.567110\pi$$
$$954$$ 398.184i 0.0135133i
$$955$$ 47762.6i 1.61839i
$$956$$ − 47193.8i − 1.59661i
$$957$$ − 8815.22i − 0.297759i
$$958$$ 172.110 0.00580442
$$959$$ −3741.57 −0.125987
$$960$$ − 28848.0i − 0.969861i
$$961$$ −46470.7 −1.55989
$$962$$ 0 0
$$963$$ −7671.32 −0.256703
$$964$$ − 40397.1i − 1.34969i
$$965$$ 35093.2 1.17066
$$966$$ −1234.70 −0.0411241
$$967$$ 17838.0i 0.593207i 0.955001 + 0.296603i $$0.0958540\pi$$
−0.955001 + 0.296603i $$0.904146\pi$$
$$968$$ 5740.12i 0.190593i
$$969$$ 20242.4i 0.671084i
$$970$$ 5928.97i 0.196255i
$$971$$ 41525.3 1.37241 0.686206 0.727408i $$-0.259275\pi$$
0.686206 + 0.727408i $$0.259275\pi$$
$$972$$ −26037.1 −0.859199
$$973$$ − 3695.86i − 0.121772i
$$974$$ 4164.32 0.136995
$$975$$ 0 0
$$976$$ 39416.1 1.29270
$$977$$ 31654.4i 1.03655i 0.855213 + 0.518277i $$0.173426\pi$$
−0.855213 + 0.518277i $$0.826574\pi$$
$$978$$ 1520.85 0.0497253
$$979$$ 16104.3 0.525735
$$980$$ 43578.0i 1.42046i
$$981$$ 2362.55i 0.0768913i
$$982$$ 830.342i 0.0269830i
$$983$$ − 39913.2i − 1.29505i −0.762045 0.647525i $$-0.775804\pi$$
0.762045 0.647525i $$-0.224196\pi$$
$$984$$ −5816.74 −0.188446
$$985$$ −71516.8 −2.31342
$$986$$ 3180.28i 0.102719i
$$987$$ 6384.72 0.205905
$$988$$ 0 0
$$989$$ 3868.74 0.124387
$$990$$ − 2350.09i − 0.0754453i
$$991$$ −2700.94 −0.0865773 −0.0432887 0.999063i $$-0.513784\pi$$
−0.0432887 + 0.999063i $$0.513784\pi$$
$$992$$ −22506.9 −0.720358
$$993$$ 28352.6i 0.906085i
$$994$$ − 364.736i − 0.0116386i
$$995$$ 75256.6i 2.39778i
$$996$$ 9670.83i 0.307663i
$$997$$ −9729.08 −0.309050 −0.154525 0.987989i $$-0.549385\pi$$
−0.154525 + 0.987989i $$0.549385\pi$$
$$998$$ −5862.08 −0.185933
$$999$$ − 639.228i − 0.0202445i
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.b.e.168.3 4
13.2 odd 12 169.4.c.f.22.2 4
13.3 even 3 169.4.e.g.147.2 8
13.4 even 6 169.4.e.g.23.2 8
13.5 odd 4 169.4.a.j.1.1 2
13.6 odd 12 169.4.c.f.146.2 4
13.7 odd 12 13.4.c.b.3.1 4
13.8 odd 4 169.4.a.f.1.2 2
13.9 even 3 169.4.e.g.23.3 8
13.10 even 6 169.4.e.g.147.3 8
13.11 odd 12 13.4.c.b.9.1 yes 4
13.12 even 2 inner 169.4.b.e.168.2 4
39.5 even 4 1521.4.a.l.1.2 2
39.8 even 4 1521.4.a.t.1.1 2
39.11 even 12 117.4.g.d.100.2 4
39.20 even 12 117.4.g.d.55.2 4
52.7 even 12 208.4.i.e.81.1 4
52.11 even 12 208.4.i.e.113.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.b.3.1 4 13.7 odd 12
13.4.c.b.9.1 yes 4 13.11 odd 12
117.4.g.d.55.2 4 39.20 even 12
117.4.g.d.100.2 4 39.11 even 12
169.4.a.f.1.2 2 13.8 odd 4
169.4.a.j.1.1 2 13.5 odd 4
169.4.b.e.168.2 4 13.12 even 2 inner
169.4.b.e.168.3 4 1.1 even 1 trivial
169.4.c.f.22.2 4 13.2 odd 12
169.4.c.f.146.2 4 13.6 odd 12
169.4.e.g.23.2 8 13.4 even 6
169.4.e.g.23.3 8 13.9 even 3
169.4.e.g.147.2 8 13.3 even 3
169.4.e.g.147.3 8 13.10 even 6
208.4.i.e.81.1 4 52.7 even 12
208.4.i.e.113.1 4 52.11 even 12
1521.4.a.l.1.2 2 39.5 even 4
1521.4.a.t.1.1 2 39.8 even 4