Properties

 Label 4001.2.a.b.1.8 Level $4001$ Weight $2$ Character 4001.1 Self dual yes Analytic conductor $31.948$ Analytic rank $0$ Dimension $184$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4001,2,Mod(1,4001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4001, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4001$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4001.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: $$31.9481458487$$ Analytic rank: $$0$$ Dimension: $$184$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.8 Character $$\chi$$ $$=$$ 4001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.68105 q^{2} -2.59933 q^{3} +5.18802 q^{4} -3.48579 q^{5} +6.96892 q^{6} -5.16960 q^{7} -8.54725 q^{8} +3.75650 q^{9} +O(q^{10})$$ $$q-2.68105 q^{2} -2.59933 q^{3} +5.18802 q^{4} -3.48579 q^{5} +6.96892 q^{6} -5.16960 q^{7} -8.54725 q^{8} +3.75650 q^{9} +9.34557 q^{10} +0.876427 q^{11} -13.4854 q^{12} +0.747353 q^{13} +13.8600 q^{14} +9.06070 q^{15} +12.5395 q^{16} +4.37103 q^{17} -10.0714 q^{18} +3.62648 q^{19} -18.0843 q^{20} +13.4375 q^{21} -2.34974 q^{22} +4.16300 q^{23} +22.2171 q^{24} +7.15071 q^{25} -2.00369 q^{26} -1.96639 q^{27} -26.8200 q^{28} +6.77142 q^{29} -24.2922 q^{30} +8.64457 q^{31} -16.5246 q^{32} -2.27812 q^{33} -11.7189 q^{34} +18.0201 q^{35} +19.4888 q^{36} +0.671547 q^{37} -9.72276 q^{38} -1.94261 q^{39} +29.7939 q^{40} -12.1738 q^{41} -36.0266 q^{42} -2.88411 q^{43} +4.54692 q^{44} -13.0944 q^{45} -11.1612 q^{46} +4.06999 q^{47} -32.5944 q^{48} +19.7248 q^{49} -19.1714 q^{50} -11.3617 q^{51} +3.87728 q^{52} -10.3253 q^{53} +5.27199 q^{54} -3.05504 q^{55} +44.1859 q^{56} -9.42640 q^{57} -18.1545 q^{58} -7.85472 q^{59} +47.0071 q^{60} +1.75285 q^{61} -23.1765 q^{62} -19.4196 q^{63} +19.2242 q^{64} -2.60511 q^{65} +6.10775 q^{66} +10.2935 q^{67} +22.6770 q^{68} -10.8210 q^{69} -48.3129 q^{70} -13.3355 q^{71} -32.1077 q^{72} +4.52167 q^{73} -1.80045 q^{74} -18.5870 q^{75} +18.8142 q^{76} -4.53078 q^{77} +5.20824 q^{78} +14.6471 q^{79} -43.7102 q^{80} -6.15821 q^{81} +32.6385 q^{82} +7.13638 q^{83} +69.7140 q^{84} -15.2365 q^{85} +7.73244 q^{86} -17.6011 q^{87} -7.49103 q^{88} +11.2472 q^{89} +35.1066 q^{90} -3.86352 q^{91} +21.5978 q^{92} -22.4701 q^{93} -10.9119 q^{94} -12.6411 q^{95} +42.9529 q^{96} +0.565715 q^{97} -52.8831 q^{98} +3.29230 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9}+O(q^{10})$$ 184 * q + 3 * q^2 + 28 * q^3 + 217 * q^4 + 15 * q^5 + 31 * q^6 + 49 * q^7 + 6 * q^8 + 210 * q^9 $$184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18} + 86 q^{19} + 26 q^{20} + 22 q^{21} + 54 q^{22} + 55 q^{23} + 72 q^{24} + 241 q^{25} + 32 q^{26} + 97 q^{27} + 75 q^{28} + 27 q^{29} - 10 q^{30} + 276 q^{31} + 20 q^{33} + 122 q^{34} + 30 q^{35} + 278 q^{36} + 42 q^{37} + 14 q^{38} + 113 q^{39} + 115 q^{40} + 39 q^{41} + 15 q^{42} + 65 q^{43} + 32 q^{44} + 54 q^{45} + 65 q^{46} + 82 q^{47} + 117 q^{48} + 297 q^{49} + 4 q^{50} + 45 q^{51} + 136 q^{52} + 21 q^{53} + 93 q^{54} + 252 q^{55} + 74 q^{56} + 14 q^{57} + 54 q^{58} + 95 q^{59} + 58 q^{60} + 131 q^{61} + 14 q^{62} + 88 q^{63} + 368 q^{64} - 9 q^{65} + 52 q^{66} + 90 q^{67} + 27 q^{68} + 101 q^{69} + 18 q^{70} + 117 q^{71} - 15 q^{72} + 72 q^{73} + 7 q^{74} + 150 q^{75} + 148 q^{76} + 7 q^{77} + 22 q^{78} + 287 q^{79} + 43 q^{80} + 244 q^{81} + 86 q^{82} + 25 q^{83} + 14 q^{84} + 41 q^{85} + 25 q^{86} + 82 q^{87} + 115 q^{88} + 48 q^{89} + 78 q^{90} + 272 q^{91} + 69 q^{92} + 44 q^{93} + 161 q^{94} + 37 q^{95} + 129 q^{96} + 106 q^{97} - 46 q^{98} + 53 q^{99}+O(q^{100})$$ 184 * q + 3 * q^2 + 28 * q^3 + 217 * q^4 + 15 * q^5 + 31 * q^6 + 49 * q^7 + 6 * q^8 + 210 * q^9 + 46 * q^10 + 25 * q^11 + 61 * q^12 + 52 * q^13 + 28 * q^14 + 59 * q^15 + 279 * q^16 + 16 * q^17 - 2 * q^18 + 86 * q^19 + 26 * q^20 + 22 * q^21 + 54 * q^22 + 55 * q^23 + 72 * q^24 + 241 * q^25 + 32 * q^26 + 97 * q^27 + 75 * q^28 + 27 * q^29 - 10 * q^30 + 276 * q^31 + 20 * q^33 + 122 * q^34 + 30 * q^35 + 278 * q^36 + 42 * q^37 + 14 * q^38 + 113 * q^39 + 115 * q^40 + 39 * q^41 + 15 * q^42 + 65 * q^43 + 32 * q^44 + 54 * q^45 + 65 * q^46 + 82 * q^47 + 117 * q^48 + 297 * q^49 + 4 * q^50 + 45 * q^51 + 136 * q^52 + 21 * q^53 + 93 * q^54 + 252 * q^55 + 74 * q^56 + 14 * q^57 + 54 * q^58 + 95 * q^59 + 58 * q^60 + 131 * q^61 + 14 * q^62 + 88 * q^63 + 368 * q^64 - 9 * q^65 + 52 * q^66 + 90 * q^67 + 27 * q^68 + 101 * q^69 + 18 * q^70 + 117 * q^71 - 15 * q^72 + 72 * q^73 + 7 * q^74 + 150 * q^75 + 148 * q^76 + 7 * q^77 + 22 * q^78 + 287 * q^79 + 43 * q^80 + 244 * q^81 + 86 * q^82 + 25 * q^83 + 14 * q^84 + 41 * q^85 + 25 * q^86 + 82 * q^87 + 115 * q^88 + 48 * q^89 + 78 * q^90 + 272 * q^91 + 69 * q^92 + 44 * q^93 + 161 * q^94 + 37 * q^95 + 129 * q^96 + 106 * q^97 - 46 * q^98 + 53 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.68105 −1.89579 −0.947894 0.318586i $$-0.896792\pi$$
−0.947894 + 0.318586i $$0.896792\pi$$
$$3$$ −2.59933 −1.50072 −0.750361 0.661028i $$-0.770120\pi$$
−0.750361 + 0.661028i $$0.770120\pi$$
$$4$$ 5.18802 2.59401
$$5$$ −3.48579 −1.55889 −0.779446 0.626470i $$-0.784499\pi$$
−0.779446 + 0.626470i $$0.784499\pi$$
$$6$$ 6.96892 2.84505
$$7$$ −5.16960 −1.95393 −0.976963 0.213408i $$-0.931544\pi$$
−0.976963 + 0.213408i $$0.931544\pi$$
$$8$$ −8.54725 −3.02191
$$9$$ 3.75650 1.25217
$$10$$ 9.34557 2.95533
$$11$$ 0.876427 0.264253 0.132126 0.991233i $$-0.457820\pi$$
0.132126 + 0.991233i $$0.457820\pi$$
$$12$$ −13.4854 −3.89289
$$13$$ 0.747353 0.207278 0.103639 0.994615i $$-0.466951\pi$$
0.103639 + 0.994615i $$0.466951\pi$$
$$14$$ 13.8600 3.70423
$$15$$ 9.06070 2.33946
$$16$$ 12.5395 3.13488
$$17$$ 4.37103 1.06013 0.530065 0.847957i $$-0.322167\pi$$
0.530065 + 0.847957i $$0.322167\pi$$
$$18$$ −10.0714 −2.37384
$$19$$ 3.62648 0.831971 0.415985 0.909371i $$-0.363437\pi$$
0.415985 + 0.909371i $$0.363437\pi$$
$$20$$ −18.0843 −4.04378
$$21$$ 13.4375 2.93230
$$22$$ −2.34974 −0.500967
$$23$$ 4.16300 0.868046 0.434023 0.900902i $$-0.357094\pi$$
0.434023 + 0.900902i $$0.357094\pi$$
$$24$$ 22.2171 4.53504
$$25$$ 7.15071 1.43014
$$26$$ −2.00369 −0.392956
$$27$$ −1.96639 −0.378432
$$28$$ −26.8200 −5.06851
$$29$$ 6.77142 1.25742 0.628711 0.777639i $$-0.283583\pi$$
0.628711 + 0.777639i $$0.283583\pi$$
$$30$$ −24.2922 −4.43512
$$31$$ 8.64457 1.55261 0.776305 0.630357i $$-0.217092\pi$$
0.776305 + 0.630357i $$0.217092\pi$$
$$32$$ −16.5246 −2.92117
$$33$$ −2.27812 −0.396570
$$34$$ −11.7189 −2.00978
$$35$$ 18.0201 3.04596
$$36$$ 19.4888 3.24813
$$37$$ 0.671547 0.110402 0.0552008 0.998475i $$-0.482420\pi$$
0.0552008 + 0.998475i $$0.482420\pi$$
$$38$$ −9.72276 −1.57724
$$39$$ −1.94261 −0.311067
$$40$$ 29.7939 4.71083
$$41$$ −12.1738 −1.90122 −0.950611 0.310385i $$-0.899542\pi$$
−0.950611 + 0.310385i $$0.899542\pi$$
$$42$$ −36.0266 −5.55902
$$43$$ −2.88411 −0.439822 −0.219911 0.975520i $$-0.570577\pi$$
−0.219911 + 0.975520i $$0.570577\pi$$
$$44$$ 4.54692 0.685474
$$45$$ −13.0944 −1.95199
$$46$$ −11.1612 −1.64563
$$47$$ 4.06999 0.593670 0.296835 0.954929i $$-0.404069\pi$$
0.296835 + 0.954929i $$0.404069\pi$$
$$48$$ −32.5944 −4.70459
$$49$$ 19.7248 2.81783
$$50$$ −19.1714 −2.71125
$$51$$ −11.3617 −1.59096
$$52$$ 3.87728 0.537683
$$53$$ −10.3253 −1.41829 −0.709146 0.705062i $$-0.750919\pi$$
−0.709146 + 0.705062i $$0.750919\pi$$
$$54$$ 5.27199 0.717427
$$55$$ −3.05504 −0.411941
$$56$$ 44.1859 5.90459
$$57$$ −9.42640 −1.24856
$$58$$ −18.1545 −2.38380
$$59$$ −7.85472 −1.02260 −0.511299 0.859403i $$-0.670835\pi$$
−0.511299 + 0.859403i $$0.670835\pi$$
$$60$$ 47.0071 6.06859
$$61$$ 1.75285 0.224429 0.112214 0.993684i $$-0.464206\pi$$
0.112214 + 0.993684i $$0.464206\pi$$
$$62$$ −23.1765 −2.94342
$$63$$ −19.4196 −2.44664
$$64$$ 19.2242 2.40303
$$65$$ −2.60511 −0.323125
$$66$$ 6.10775 0.751812
$$67$$ 10.2935 1.25755 0.628776 0.777586i $$-0.283556\pi$$
0.628776 + 0.777586i $$0.283556\pi$$
$$68$$ 22.6770 2.74999
$$69$$ −10.8210 −1.30270
$$70$$ −48.3129 −5.77449
$$71$$ −13.3355 −1.58263 −0.791315 0.611409i $$-0.790603\pi$$
−0.791315 + 0.611409i $$0.790603\pi$$
$$72$$ −32.1077 −3.78393
$$73$$ 4.52167 0.529221 0.264611 0.964355i $$-0.414757\pi$$
0.264611 + 0.964355i $$0.414757\pi$$
$$74$$ −1.80045 −0.209298
$$75$$ −18.5870 −2.14625
$$76$$ 18.8142 2.15814
$$77$$ −4.53078 −0.516330
$$78$$ 5.20824 0.589718
$$79$$ 14.6471 1.64793 0.823966 0.566640i $$-0.191757\pi$$
0.823966 + 0.566640i $$0.191757\pi$$
$$80$$ −43.7102 −4.88694
$$81$$ −6.15821 −0.684245
$$82$$ 32.6385 3.60431
$$83$$ 7.13638 0.783320 0.391660 0.920110i $$-0.371901\pi$$
0.391660 + 0.920110i $$0.371901\pi$$
$$84$$ 69.7140 7.60642
$$85$$ −15.2365 −1.65263
$$86$$ 7.73244 0.833810
$$87$$ −17.6011 −1.88704
$$88$$ −7.49103 −0.798547
$$89$$ 11.2472 1.19220 0.596100 0.802910i $$-0.296716\pi$$
0.596100 + 0.802910i $$0.296716\pi$$
$$90$$ 35.1066 3.70056
$$91$$ −3.86352 −0.405007
$$92$$ 21.5978 2.25172
$$93$$ −22.4701 −2.33004
$$94$$ −10.9119 −1.12547
$$95$$ −12.6411 −1.29695
$$96$$ 42.9529 4.38386
$$97$$ 0.565715 0.0574396 0.0287198 0.999588i $$-0.490857\pi$$
0.0287198 + 0.999588i $$0.490857\pi$$
$$98$$ −52.8831 −5.34200
$$99$$ 3.29230 0.330888
$$100$$ 37.0981 3.70981
$$101$$ −0.915682 −0.0911138 −0.0455569 0.998962i $$-0.514506\pi$$
−0.0455569 + 0.998962i $$0.514506\pi$$
$$102$$ 30.4614 3.01612
$$103$$ −5.97154 −0.588393 −0.294197 0.955745i $$-0.595052\pi$$
−0.294197 + 0.955745i $$0.595052\pi$$
$$104$$ −6.38781 −0.626376
$$105$$ −46.8402 −4.57114
$$106$$ 27.6827 2.68878
$$107$$ 2.56650 0.248113 0.124057 0.992275i $$-0.460410\pi$$
0.124057 + 0.992275i $$0.460410\pi$$
$$108$$ −10.2017 −0.981657
$$109$$ 5.89465 0.564605 0.282303 0.959325i $$-0.408902\pi$$
0.282303 + 0.959325i $$0.408902\pi$$
$$110$$ 8.19070 0.780953
$$111$$ −1.74557 −0.165682
$$112$$ −64.8244 −6.12533
$$113$$ 13.6703 1.28600 0.642998 0.765868i $$-0.277690\pi$$
0.642998 + 0.765868i $$0.277690\pi$$
$$114$$ 25.2726 2.36700
$$115$$ −14.5113 −1.35319
$$116$$ 35.1303 3.26176
$$117$$ 2.80743 0.259547
$$118$$ 21.0589 1.93863
$$119$$ −22.5965 −2.07142
$$120$$ −77.4440 −7.06964
$$121$$ −10.2319 −0.930171
$$122$$ −4.69947 −0.425470
$$123$$ 31.6436 2.85321
$$124$$ 44.8482 4.02749
$$125$$ −7.49692 −0.670545
$$126$$ 52.0649 4.63831
$$127$$ 13.7879 1.22348 0.611738 0.791060i $$-0.290471\pi$$
0.611738 + 0.791060i $$0.290471\pi$$
$$128$$ −18.4919 −1.63447
$$129$$ 7.49674 0.660051
$$130$$ 6.98444 0.612576
$$131$$ 2.16019 0.188736 0.0943682 0.995537i $$-0.469917\pi$$
0.0943682 + 0.995537i $$0.469917\pi$$
$$132$$ −11.8189 −1.02871
$$133$$ −18.7474 −1.62561
$$134$$ −27.5974 −2.38405
$$135$$ 6.85442 0.589934
$$136$$ −37.3603 −3.20362
$$137$$ −3.67996 −0.314400 −0.157200 0.987567i $$-0.550247\pi$$
−0.157200 + 0.987567i $$0.550247\pi$$
$$138$$ 29.0116 2.46964
$$139$$ 21.3708 1.81264 0.906322 0.422588i $$-0.138878\pi$$
0.906322 + 0.422588i $$0.138878\pi$$
$$140$$ 93.4889 7.90125
$$141$$ −10.5792 −0.890933
$$142$$ 35.7531 3.00033
$$143$$ 0.655000 0.0547738
$$144$$ 47.1048 3.92540
$$145$$ −23.6037 −1.96018
$$146$$ −12.1228 −1.00329
$$147$$ −51.2712 −4.22878
$$148$$ 3.48400 0.286383
$$149$$ 2.39933 0.196561 0.0982804 0.995159i $$-0.468666\pi$$
0.0982804 + 0.995159i $$0.468666\pi$$
$$150$$ 49.8328 4.06883
$$151$$ 12.0064 0.977065 0.488533 0.872546i $$-0.337532\pi$$
0.488533 + 0.872546i $$0.337532\pi$$
$$152$$ −30.9964 −2.51414
$$153$$ 16.4198 1.32746
$$154$$ 12.1472 0.978852
$$155$$ −30.1331 −2.42035
$$156$$ −10.0783 −0.806912
$$157$$ −12.9825 −1.03612 −0.518060 0.855344i $$-0.673346\pi$$
−0.518060 + 0.855344i $$0.673346\pi$$
$$158$$ −39.2697 −3.12413
$$159$$ 26.8389 2.12846
$$160$$ 57.6013 4.55378
$$161$$ −21.5211 −1.69610
$$162$$ 16.5105 1.29718
$$163$$ −14.6077 −1.14416 −0.572082 0.820196i $$-0.693864\pi$$
−0.572082 + 0.820196i $$0.693864\pi$$
$$164$$ −63.1578 −4.93179
$$165$$ 7.94104 0.618209
$$166$$ −19.1330 −1.48501
$$167$$ 12.4688 0.964867 0.482434 0.875933i $$-0.339753\pi$$
0.482434 + 0.875933i $$0.339753\pi$$
$$168$$ −114.854 −8.86114
$$169$$ −12.4415 −0.957036
$$170$$ 40.8497 3.13303
$$171$$ 13.6229 1.04177
$$172$$ −14.9628 −1.14090
$$173$$ 14.8043 1.12555 0.562774 0.826611i $$-0.309734\pi$$
0.562774 + 0.826611i $$0.309734\pi$$
$$174$$ 47.1895 3.57743
$$175$$ −36.9663 −2.79439
$$176$$ 10.9900 0.828401
$$177$$ 20.4170 1.53463
$$178$$ −30.1543 −2.26016
$$179$$ −16.2288 −1.21299 −0.606497 0.795086i $$-0.707426\pi$$
−0.606497 + 0.795086i $$0.707426\pi$$
$$180$$ −67.9338 −5.06349
$$181$$ −5.72393 −0.425456 −0.212728 0.977111i $$-0.568235\pi$$
−0.212728 + 0.977111i $$0.568235\pi$$
$$182$$ 10.3583 0.767807
$$183$$ −4.55622 −0.336805
$$184$$ −35.5822 −2.62316
$$185$$ −2.34087 −0.172104
$$186$$ 60.2433 4.41725
$$187$$ 3.83089 0.280142
$$188$$ 21.1152 1.53999
$$189$$ 10.1655 0.739428
$$190$$ 33.8915 2.45875
$$191$$ 14.4684 1.04689 0.523447 0.852058i $$-0.324646\pi$$
0.523447 + 0.852058i $$0.324646\pi$$
$$192$$ −49.9701 −3.60628
$$193$$ −2.82574 −0.203401 −0.101701 0.994815i $$-0.532428\pi$$
−0.101701 + 0.994815i $$0.532428\pi$$
$$194$$ −1.51671 −0.108893
$$195$$ 6.77154 0.484920
$$196$$ 102.333 7.30948
$$197$$ −11.3114 −0.805904 −0.402952 0.915221i $$-0.632016\pi$$
−0.402952 + 0.915221i $$0.632016\pi$$
$$198$$ −8.82681 −0.627294
$$199$$ 25.2559 1.79034 0.895170 0.445724i $$-0.147054\pi$$
0.895170 + 0.445724i $$0.147054\pi$$
$$200$$ −61.1189 −4.32176
$$201$$ −26.7562 −1.88724
$$202$$ 2.45499 0.172732
$$203$$ −35.0055 −2.45691
$$204$$ −58.9449 −4.12697
$$205$$ 42.4351 2.96380
$$206$$ 16.0100 1.11547
$$207$$ 15.6383 1.08694
$$208$$ 9.37146 0.649794
$$209$$ 3.17834 0.219850
$$210$$ 125.581 8.66591
$$211$$ 9.06987 0.624396 0.312198 0.950017i $$-0.398935\pi$$
0.312198 + 0.950017i $$0.398935\pi$$
$$212$$ −53.5680 −3.67906
$$213$$ 34.6632 2.37509
$$214$$ −6.88092 −0.470370
$$215$$ 10.0534 0.685635
$$216$$ 16.8072 1.14359
$$217$$ −44.6890 −3.03369
$$218$$ −15.8039 −1.07037
$$219$$ −11.7533 −0.794214
$$220$$ −15.8496 −1.06858
$$221$$ 3.26670 0.219742
$$222$$ 4.67996 0.314098
$$223$$ 16.2622 1.08900 0.544499 0.838761i $$-0.316720\pi$$
0.544499 + 0.838761i $$0.316720\pi$$
$$224$$ 85.4257 5.70775
$$225$$ 26.8616 1.79078
$$226$$ −36.6508 −2.43798
$$227$$ −20.0618 −1.33155 −0.665775 0.746153i $$-0.731899\pi$$
−0.665775 + 0.746153i $$0.731899\pi$$
$$228$$ −48.9044 −3.23877
$$229$$ 21.7895 1.43989 0.719946 0.694031i $$-0.244167\pi$$
0.719946 + 0.694031i $$0.244167\pi$$
$$230$$ 38.9056 2.56536
$$231$$ 11.7770 0.774868
$$232$$ −57.8770 −3.79981
$$233$$ −21.4003 −1.40198 −0.700992 0.713169i $$-0.747259\pi$$
−0.700992 + 0.713169i $$0.747259\pi$$
$$234$$ −7.52686 −0.492046
$$235$$ −14.1871 −0.925467
$$236$$ −40.7505 −2.65263
$$237$$ −38.0727 −2.47309
$$238$$ 60.5823 3.92697
$$239$$ −25.7485 −1.66553 −0.832765 0.553627i $$-0.813243\pi$$
−0.832765 + 0.553627i $$0.813243\pi$$
$$240$$ 113.617 7.33395
$$241$$ −8.58061 −0.552726 −0.276363 0.961053i $$-0.589129\pi$$
−0.276363 + 0.961053i $$0.589129\pi$$
$$242$$ 27.4322 1.76341
$$243$$ 21.9064 1.40529
$$244$$ 9.09381 0.582171
$$245$$ −68.7564 −4.39269
$$246$$ −84.8380 −5.40907
$$247$$ 2.71026 0.172450
$$248$$ −73.8873 −4.69185
$$249$$ −18.5498 −1.17554
$$250$$ 20.0996 1.27121
$$251$$ 1.61960 0.102228 0.0511142 0.998693i $$-0.483723\pi$$
0.0511142 + 0.998693i $$0.483723\pi$$
$$252$$ −100.749 −6.34662
$$253$$ 3.64857 0.229383
$$254$$ −36.9660 −2.31945
$$255$$ 39.6046 2.48014
$$256$$ 11.1292 0.695575
$$257$$ −26.2702 −1.63869 −0.819345 0.573301i $$-0.805662\pi$$
−0.819345 + 0.573301i $$0.805662\pi$$
$$258$$ −20.0991 −1.25132
$$259$$ −3.47163 −0.215717
$$260$$ −13.5154 −0.838189
$$261$$ 25.4368 1.57450
$$262$$ −5.79157 −0.357804
$$263$$ 7.18538 0.443070 0.221535 0.975152i $$-0.428893\pi$$
0.221535 + 0.975152i $$0.428893\pi$$
$$264$$ 19.4716 1.19840
$$265$$ 35.9919 2.21096
$$266$$ 50.2628 3.08181
$$267$$ −29.2351 −1.78916
$$268$$ 53.4030 3.26211
$$269$$ 17.6646 1.07703 0.538516 0.842615i $$-0.318985\pi$$
0.538516 + 0.842615i $$0.318985\pi$$
$$270$$ −18.3770 −1.11839
$$271$$ −16.6755 −1.01297 −0.506483 0.862250i $$-0.669055\pi$$
−0.506483 + 0.862250i $$0.669055\pi$$
$$272$$ 54.8107 3.32339
$$273$$ 10.0425 0.607803
$$274$$ 9.86616 0.596036
$$275$$ 6.26707 0.377919
$$276$$ −56.1396 −3.37921
$$277$$ −2.85615 −0.171609 −0.0858047 0.996312i $$-0.527346\pi$$
−0.0858047 + 0.996312i $$0.527346\pi$$
$$278$$ −57.2961 −3.43639
$$279$$ 32.4733 1.94413
$$280$$ −154.023 −9.20461
$$281$$ 8.28079 0.493991 0.246995 0.969017i $$-0.420557\pi$$
0.246995 + 0.969017i $$0.420557\pi$$
$$282$$ 28.3635 1.68902
$$283$$ −20.5986 −1.22446 −0.612229 0.790680i $$-0.709727\pi$$
−0.612229 + 0.790680i $$0.709727\pi$$
$$284$$ −69.1847 −4.10536
$$285$$ 32.8584 1.94636
$$286$$ −1.75609 −0.103840
$$287$$ 62.9335 3.71485
$$288$$ −62.0748 −3.65779
$$289$$ 2.10590 0.123876
$$290$$ 63.2827 3.71609
$$291$$ −1.47048 −0.0862009
$$292$$ 23.4585 1.37281
$$293$$ −15.4501 −0.902603 −0.451301 0.892372i $$-0.649040\pi$$
−0.451301 + 0.892372i $$0.649040\pi$$
$$294$$ 137.461 8.01686
$$295$$ 27.3799 1.59412
$$296$$ −5.73988 −0.333624
$$297$$ −1.72340 −0.100002
$$298$$ −6.43272 −0.372637
$$299$$ 3.11123 0.179927
$$300$$ −96.4300 −5.56739
$$301$$ 14.9097 0.859381
$$302$$ −32.1897 −1.85231
$$303$$ 2.38016 0.136736
$$304$$ 45.4743 2.60813
$$305$$ −6.11005 −0.349860
$$306$$ −44.0222 −2.51658
$$307$$ 17.3595 0.990759 0.495380 0.868677i $$-0.335029\pi$$
0.495380 + 0.868677i $$0.335029\pi$$
$$308$$ −23.5058 −1.33937
$$309$$ 15.5220 0.883015
$$310$$ 80.7884 4.58847
$$311$$ −6.00112 −0.340292 −0.170146 0.985419i $$-0.554424\pi$$
−0.170146 + 0.985419i $$0.554424\pi$$
$$312$$ 16.6040 0.940017
$$313$$ 32.5860 1.84187 0.920936 0.389715i $$-0.127426\pi$$
0.920936 + 0.389715i $$0.127426\pi$$
$$314$$ 34.8068 1.96426
$$315$$ 67.6926 3.81405
$$316$$ 75.9897 4.27475
$$317$$ 15.3550 0.862421 0.431210 0.902251i $$-0.358087\pi$$
0.431210 + 0.902251i $$0.358087\pi$$
$$318$$ −71.9563 −4.03511
$$319$$ 5.93465 0.332277
$$320$$ −67.0116 −3.74606
$$321$$ −6.67118 −0.372349
$$322$$ 57.6991 3.21544
$$323$$ 15.8514 0.881997
$$324$$ −31.9489 −1.77494
$$325$$ 5.34411 0.296438
$$326$$ 39.1640 2.16909
$$327$$ −15.3221 −0.847316
$$328$$ 104.052 5.74532
$$329$$ −21.0403 −1.15999
$$330$$ −21.2903 −1.17199
$$331$$ −1.18210 −0.0649743 −0.0324872 0.999472i $$-0.510343\pi$$
−0.0324872 + 0.999472i $$0.510343\pi$$
$$332$$ 37.0237 2.03194
$$333$$ 2.52267 0.138241
$$334$$ −33.4296 −1.82918
$$335$$ −35.8810 −1.96039
$$336$$ 168.500 9.19242
$$337$$ 5.30255 0.288848 0.144424 0.989516i $$-0.453867\pi$$
0.144424 + 0.989516i $$0.453867\pi$$
$$338$$ 33.3562 1.81434
$$339$$ −35.5336 −1.92992
$$340$$ −79.0472 −4.28694
$$341$$ 7.57633 0.410281
$$342$$ −36.5235 −1.97497
$$343$$ −65.7821 −3.55190
$$344$$ 24.6512 1.32910
$$345$$ 37.7197 2.03076
$$346$$ −39.6910 −2.13380
$$347$$ 9.90287 0.531614 0.265807 0.964026i $$-0.414362\pi$$
0.265807 + 0.964026i $$0.414362\pi$$
$$348$$ −91.3151 −4.89500
$$349$$ 5.73195 0.306824 0.153412 0.988162i $$-0.450974\pi$$
0.153412 + 0.988162i $$0.450974\pi$$
$$350$$ 99.1086 5.29758
$$351$$ −1.46959 −0.0784408
$$352$$ −14.4826 −0.771926
$$353$$ 7.84835 0.417725 0.208863 0.977945i $$-0.433024\pi$$
0.208863 + 0.977945i $$0.433024\pi$$
$$354$$ −54.7389 −2.90934
$$355$$ 46.4846 2.46715
$$356$$ 58.3507 3.09258
$$357$$ 58.7356 3.10862
$$358$$ 43.5101 2.29958
$$359$$ 2.98498 0.157541 0.0787707 0.996893i $$-0.474901\pi$$
0.0787707 + 0.996893i $$0.474901\pi$$
$$360$$ 111.921 5.89874
$$361$$ −5.84867 −0.307825
$$362$$ 15.3461 0.806575
$$363$$ 26.5960 1.39593
$$364$$ −20.0440 −1.05059
$$365$$ −15.7616 −0.824998
$$366$$ 12.2154 0.638512
$$367$$ 21.2046 1.10687 0.553434 0.832893i $$-0.313317\pi$$
0.553434 + 0.832893i $$0.313317\pi$$
$$368$$ 52.2021 2.72122
$$369$$ −45.7307 −2.38065
$$370$$ 6.27599 0.326273
$$371$$ 53.3778 2.77124
$$372$$ −116.575 −6.04414
$$373$$ −17.1844 −0.889775 −0.444887 0.895587i $$-0.646756\pi$$
−0.444887 + 0.895587i $$0.646756\pi$$
$$374$$ −10.2708 −0.531090
$$375$$ 19.4870 1.00630
$$376$$ −34.7872 −1.79402
$$377$$ 5.06064 0.260636
$$378$$ −27.2541 −1.40180
$$379$$ 8.96136 0.460314 0.230157 0.973154i $$-0.426076\pi$$
0.230157 + 0.973154i $$0.426076\pi$$
$$380$$ −65.5824 −3.36431
$$381$$ −35.8392 −1.83610
$$382$$ −38.7904 −1.98469
$$383$$ 13.7675 0.703486 0.351743 0.936097i $$-0.385589\pi$$
0.351743 + 0.936097i $$0.385589\pi$$
$$384$$ 48.0665 2.45288
$$385$$ 15.7933 0.804902
$$386$$ 7.57594 0.385605
$$387$$ −10.8342 −0.550731
$$388$$ 2.93494 0.148999
$$389$$ −18.3355 −0.929647 −0.464823 0.885403i $$-0.653882\pi$$
−0.464823 + 0.885403i $$0.653882\pi$$
$$390$$ −18.1548 −0.919306
$$391$$ 18.1966 0.920242
$$392$$ −168.593 −8.51522
$$393$$ −5.61503 −0.283241
$$394$$ 30.3264 1.52782
$$395$$ −51.0568 −2.56895
$$396$$ 17.0805 0.858328
$$397$$ 20.4665 1.02719 0.513593 0.858034i $$-0.328314\pi$$
0.513593 + 0.858034i $$0.328314\pi$$
$$398$$ −67.7122 −3.39411
$$399$$ 48.7307 2.43959
$$400$$ 89.6666 4.48333
$$401$$ 20.4937 1.02341 0.511703 0.859163i $$-0.329015\pi$$
0.511703 + 0.859163i $$0.329015\pi$$
$$402$$ 71.7347 3.57780
$$403$$ 6.46054 0.321823
$$404$$ −4.75058 −0.236350
$$405$$ 21.4662 1.06666
$$406$$ 93.8516 4.65778
$$407$$ 0.588562 0.0291739
$$408$$ 97.1115 4.80774
$$409$$ −24.7995 −1.22626 −0.613129 0.789983i $$-0.710089\pi$$
−0.613129 + 0.789983i $$0.710089\pi$$
$$410$$ −113.771 −5.61873
$$411$$ 9.56542 0.471828
$$412$$ −30.9805 −1.52630
$$413$$ 40.6058 1.99808
$$414$$ −41.9271 −2.06060
$$415$$ −24.8759 −1.22111
$$416$$ −12.3497 −0.605495
$$417$$ −55.5496 −2.72028
$$418$$ −8.52128 −0.416790
$$419$$ 2.55337 0.124740 0.0623702 0.998053i $$-0.480134\pi$$
0.0623702 + 0.998053i $$0.480134\pi$$
$$420$$ −243.008 −11.8576
$$421$$ 3.17201 0.154594 0.0772972 0.997008i $$-0.475371\pi$$
0.0772972 + 0.997008i $$0.475371\pi$$
$$422$$ −24.3168 −1.18372
$$423$$ 15.2889 0.743373
$$424$$ 88.2530 4.28595
$$425$$ 31.2560 1.51614
$$426$$ −92.9339 −4.50266
$$427$$ −9.06152 −0.438518
$$428$$ 13.3151 0.643609
$$429$$ −1.70256 −0.0822003
$$430$$ −26.9536 −1.29982
$$431$$ −5.48399 −0.264154 −0.132077 0.991239i $$-0.542165\pi$$
−0.132077 + 0.991239i $$0.542165\pi$$
$$432$$ −24.6576 −1.18634
$$433$$ 1.03307 0.0496462 0.0248231 0.999692i $$-0.492098\pi$$
0.0248231 + 0.999692i $$0.492098\pi$$
$$434$$ 119.813 5.75122
$$435$$ 61.3538 2.94169
$$436$$ 30.5816 1.46459
$$437$$ 15.0970 0.722189
$$438$$ 31.5111 1.50566
$$439$$ −27.3336 −1.30456 −0.652280 0.757978i $$-0.726187\pi$$
−0.652280 + 0.757978i $$0.726187\pi$$
$$440$$ 26.1121 1.24485
$$441$$ 74.0962 3.52839
$$442$$ −8.75819 −0.416584
$$443$$ 23.2589 1.10507 0.552533 0.833491i $$-0.313662\pi$$
0.552533 + 0.833491i $$0.313662\pi$$
$$444$$ −9.05606 −0.429782
$$445$$ −39.2053 −1.85851
$$446$$ −43.5998 −2.06451
$$447$$ −6.23664 −0.294983
$$448$$ −99.3817 −4.69535
$$449$$ −21.4726 −1.01336 −0.506678 0.862135i $$-0.669127\pi$$
−0.506678 + 0.862135i $$0.669127\pi$$
$$450$$ −72.0174 −3.39493
$$451$$ −10.6694 −0.502403
$$452$$ 70.9220 3.33589
$$453$$ −31.2085 −1.46630
$$454$$ 53.7867 2.52434
$$455$$ 13.4674 0.631361
$$456$$ 80.5697 3.77302
$$457$$ −30.9583 −1.44817 −0.724084 0.689711i $$-0.757737\pi$$
−0.724084 + 0.689711i $$0.757737\pi$$
$$458$$ −58.4187 −2.72973
$$459$$ −8.59515 −0.401187
$$460$$ −75.2852 −3.51019
$$461$$ 8.38181 0.390380 0.195190 0.980765i $$-0.437468\pi$$
0.195190 + 0.980765i $$0.437468\pi$$
$$462$$ −31.5746 −1.46898
$$463$$ 37.7697 1.75531 0.877653 0.479297i $$-0.159108\pi$$
0.877653 + 0.479297i $$0.159108\pi$$
$$464$$ 84.9105 3.94187
$$465$$ 78.3258 3.63227
$$466$$ 57.3754 2.65786
$$467$$ 15.1792 0.702411 0.351205 0.936298i $$-0.385772\pi$$
0.351205 + 0.936298i $$0.385772\pi$$
$$468$$ 14.5650 0.673268
$$469$$ −53.2134 −2.45717
$$470$$ 38.0364 1.75449
$$471$$ 33.7459 1.55493
$$472$$ 67.1362 3.09020
$$473$$ −2.52771 −0.116224
$$474$$ 102.075 4.68845
$$475$$ 25.9319 1.18984
$$476$$ −117.231 −5.37328
$$477$$ −38.7871 −1.77594
$$478$$ 69.0329 3.15749
$$479$$ 24.6932 1.12826 0.564131 0.825685i $$-0.309211\pi$$
0.564131 + 0.825685i $$0.309211\pi$$
$$480$$ −149.725 −6.83396
$$481$$ 0.501883 0.0228839
$$482$$ 23.0050 1.04785
$$483$$ 55.9403 2.54537
$$484$$ −53.0832 −2.41287
$$485$$ −1.97196 −0.0895422
$$486$$ −58.7320 −2.66414
$$487$$ 23.2653 1.05425 0.527125 0.849787i $$-0.323270\pi$$
0.527125 + 0.849787i $$0.323270\pi$$
$$488$$ −14.9820 −0.678204
$$489$$ 37.9702 1.71707
$$490$$ 184.339 8.32760
$$491$$ −24.0510 −1.08541 −0.542704 0.839924i $$-0.682600\pi$$
−0.542704 + 0.839924i $$0.682600\pi$$
$$492$$ 164.168 7.40125
$$493$$ 29.5981 1.33303
$$494$$ −7.26633 −0.326928
$$495$$ −11.4762 −0.515819
$$496$$ 108.399 4.86725
$$497$$ 68.9391 3.09234
$$498$$ 49.7329 2.22858
$$499$$ −32.8536 −1.47073 −0.735365 0.677671i $$-0.762989\pi$$
−0.735365 + 0.677671i $$0.762989\pi$$
$$500$$ −38.8942 −1.73940
$$501$$ −32.4106 −1.44800
$$502$$ −4.34224 −0.193804
$$503$$ 37.9490 1.69206 0.846030 0.533135i $$-0.178986\pi$$
0.846030 + 0.533135i $$0.178986\pi$$
$$504$$ 165.984 7.39352
$$505$$ 3.19187 0.142036
$$506$$ −9.78199 −0.434862
$$507$$ 32.3394 1.43624
$$508$$ 71.5318 3.17371
$$509$$ −17.0253 −0.754635 −0.377317 0.926084i $$-0.623153\pi$$
−0.377317 + 0.926084i $$0.623153\pi$$
$$510$$ −106.182 −4.70181
$$511$$ −23.3752 −1.03406
$$512$$ 7.14588 0.315806
$$513$$ −7.13107 −0.314844
$$514$$ 70.4317 3.10661
$$515$$ 20.8155 0.917241
$$516$$ 38.8933 1.71218
$$517$$ 3.56705 0.156879
$$518$$ 9.30761 0.408953
$$519$$ −38.4811 −1.68913
$$520$$ 22.2665 0.976453
$$521$$ −16.4070 −0.718804 −0.359402 0.933183i $$-0.617019\pi$$
−0.359402 + 0.933183i $$0.617019\pi$$
$$522$$ −68.1974 −2.98492
$$523$$ −38.4907 −1.68308 −0.841540 0.540194i $$-0.818351\pi$$
−0.841540 + 0.540194i $$0.818351\pi$$
$$524$$ 11.2071 0.489584
$$525$$ 96.0876 4.19361
$$526$$ −19.2644 −0.839966
$$527$$ 37.7857 1.64597
$$528$$ −28.5666 −1.24320
$$529$$ −5.66940 −0.246496
$$530$$ −96.4959 −4.19152
$$531$$ −29.5063 −1.28046
$$532$$ −97.2622 −4.21685
$$533$$ −9.09810 −0.394082
$$534$$ 78.3808 3.39187
$$535$$ −8.94629 −0.386782
$$536$$ −87.9812 −3.80021
$$537$$ 42.1839 1.82037
$$538$$ −47.3598 −2.04183
$$539$$ 17.2873 0.744618
$$540$$ 35.5609 1.53030
$$541$$ −29.8843 −1.28483 −0.642414 0.766358i $$-0.722067\pi$$
−0.642414 + 0.766358i $$0.722067\pi$$
$$542$$ 44.7079 1.92037
$$543$$ 14.8784 0.638492
$$544$$ −72.2296 −3.09682
$$545$$ −20.5475 −0.880159
$$546$$ −26.9246 −1.15226
$$547$$ 20.3017 0.868038 0.434019 0.900904i $$-0.357095\pi$$
0.434019 + 0.900904i $$0.357095\pi$$
$$548$$ −19.0917 −0.815558
$$549$$ 6.58457 0.281022
$$550$$ −16.8023 −0.716454
$$551$$ 24.5564 1.04614
$$552$$ 92.4898 3.93663
$$553$$ −75.7199 −3.21994
$$554$$ 7.65748 0.325335
$$555$$ 6.08469 0.258281
$$556$$ 110.872 4.70202
$$557$$ 5.33868 0.226207 0.113104 0.993583i $$-0.463921\pi$$
0.113104 + 0.993583i $$0.463921\pi$$
$$558$$ −87.0626 −3.68565
$$559$$ −2.15545 −0.0911657
$$560$$ 225.964 9.54873
$$561$$ −9.95772 −0.420415
$$562$$ −22.2012 −0.936502
$$563$$ −1.04621 −0.0440925 −0.0220463 0.999757i $$-0.507018\pi$$
−0.0220463 + 0.999757i $$0.507018\pi$$
$$564$$ −54.8854 −2.31109
$$565$$ −47.6519 −2.00473
$$566$$ 55.2258 2.32131
$$567$$ 31.8355 1.33696
$$568$$ 113.982 4.78256
$$569$$ −27.6822 −1.16050 −0.580249 0.814439i $$-0.697045\pi$$
−0.580249 + 0.814439i $$0.697045\pi$$
$$570$$ −88.0950 −3.68989
$$571$$ −14.0987 −0.590012 −0.295006 0.955495i $$-0.595322\pi$$
−0.295006 + 0.955495i $$0.595322\pi$$
$$572$$ 3.39815 0.142084
$$573$$ −37.6080 −1.57110
$$574$$ −168.728 −7.04256
$$575$$ 29.7684 1.24143
$$576$$ 72.2159 3.00900
$$577$$ 7.17715 0.298789 0.149394 0.988778i $$-0.452268\pi$$
0.149394 + 0.988778i $$0.452268\pi$$
$$578$$ −5.64601 −0.234843
$$579$$ 7.34502 0.305249
$$580$$ −122.457 −5.08474
$$581$$ −36.8923 −1.53055
$$582$$ 3.94242 0.163419
$$583$$ −9.04938 −0.374787
$$584$$ −38.6478 −1.59926
$$585$$ −9.78611 −0.404606
$$586$$ 41.4224 1.71114
$$587$$ −19.8595 −0.819690 −0.409845 0.912155i $$-0.634417\pi$$
−0.409845 + 0.912155i $$0.634417\pi$$
$$588$$ −265.996 −10.9695
$$589$$ 31.3493 1.29173
$$590$$ −73.4068 −3.02211
$$591$$ 29.4020 1.20944
$$592$$ 8.42089 0.346096
$$593$$ 10.8390 0.445105 0.222553 0.974921i $$-0.428561\pi$$
0.222553 + 0.974921i $$0.428561\pi$$
$$594$$ 4.62051 0.189582
$$595$$ 78.7665 3.22911
$$596$$ 12.4478 0.509881
$$597$$ −65.6482 −2.68680
$$598$$ −8.34137 −0.341104
$$599$$ 3.82319 0.156211 0.0781056 0.996945i $$-0.475113\pi$$
0.0781056 + 0.996945i $$0.475113\pi$$
$$600$$ 158.868 6.48576
$$601$$ 46.1006 1.88048 0.940241 0.340511i $$-0.110600\pi$$
0.940241 + 0.340511i $$0.110600\pi$$
$$602$$ −39.9736 −1.62920
$$603$$ 38.6676 1.57467
$$604$$ 62.2894 2.53452
$$605$$ 35.6661 1.45003
$$606$$ −6.38132 −0.259223
$$607$$ 8.72839 0.354274 0.177137 0.984186i $$-0.443316\pi$$
0.177137 + 0.984186i $$0.443316\pi$$
$$608$$ −59.9262 −2.43033
$$609$$ 90.9909 3.68714
$$610$$ 16.3813 0.663261
$$611$$ 3.04172 0.123055
$$612$$ 85.1862 3.44345
$$613$$ −23.1545 −0.935200 −0.467600 0.883940i $$-0.654881\pi$$
−0.467600 + 0.883940i $$0.654881\pi$$
$$614$$ −46.5417 −1.87827
$$615$$ −110.303 −4.44784
$$616$$ 38.7257 1.56030
$$617$$ −22.5030 −0.905938 −0.452969 0.891526i $$-0.649635\pi$$
−0.452969 + 0.891526i $$0.649635\pi$$
$$618$$ −41.6152 −1.67401
$$619$$ 16.3434 0.656897 0.328448 0.944522i $$-0.393474\pi$$
0.328448 + 0.944522i $$0.393474\pi$$
$$620$$ −156.331 −6.27842
$$621$$ −8.18609 −0.328497
$$622$$ 16.0893 0.645122
$$623$$ −58.1435 −2.32947
$$624$$ −24.3595 −0.975160
$$625$$ −9.62088 −0.384835
$$626$$ −87.3647 −3.49180
$$627$$ −8.26154 −0.329934
$$628$$ −67.3537 −2.68771
$$629$$ 2.93535 0.117040
$$630$$ −181.487 −7.23063
$$631$$ 19.9320 0.793482 0.396741 0.917931i $$-0.370141\pi$$
0.396741 + 0.917931i $$0.370141\pi$$
$$632$$ −125.193 −4.97990
$$633$$ −23.5756 −0.937044
$$634$$ −41.1674 −1.63497
$$635$$ −48.0616 −1.90727
$$636$$ 139.241 5.52125
$$637$$ 14.7414 0.584075
$$638$$ −15.9111 −0.629926
$$639$$ −50.0947 −1.98172
$$640$$ 64.4588 2.54796
$$641$$ 2.36150 0.0932737 0.0466368 0.998912i $$-0.485150\pi$$
0.0466368 + 0.998912i $$0.485150\pi$$
$$642$$ 17.8858 0.705895
$$643$$ 15.3925 0.607019 0.303510 0.952828i $$-0.401842\pi$$
0.303510 + 0.952828i $$0.401842\pi$$
$$644$$ −111.652 −4.39970
$$645$$ −26.1320 −1.02895
$$646$$ −42.4985 −1.67208
$$647$$ 18.7879 0.738627 0.369313 0.929305i $$-0.379593\pi$$
0.369313 + 0.929305i $$0.379593\pi$$
$$648$$ 52.6357 2.06773
$$649$$ −6.88409 −0.270224
$$650$$ −14.3278 −0.561983
$$651$$ 116.161 4.55272
$$652$$ −75.7852 −2.96798
$$653$$ 14.6954 0.575075 0.287538 0.957769i $$-0.407163\pi$$
0.287538 + 0.957769i $$0.407163\pi$$
$$654$$ 41.0794 1.60633
$$655$$ −7.52995 −0.294220
$$656$$ −152.653 −5.96011
$$657$$ 16.9856 0.662673
$$658$$ 56.4100 2.19909
$$659$$ −8.89957 −0.346678 −0.173339 0.984862i $$-0.555456\pi$$
−0.173339 + 0.984862i $$0.555456\pi$$
$$660$$ 41.1983 1.60364
$$661$$ −49.6246 −1.93017 −0.965086 0.261933i $$-0.915640\pi$$
−0.965086 + 0.261933i $$0.915640\pi$$
$$662$$ 3.16928 0.123178
$$663$$ −8.49122 −0.329772
$$664$$ −60.9964 −2.36712
$$665$$ 65.3496 2.53415
$$666$$ −6.76339 −0.262076
$$667$$ 28.1894 1.09150
$$668$$ 64.6886 2.50288
$$669$$ −42.2708 −1.63428
$$670$$ 96.1987 3.71648
$$671$$ 1.53624 0.0593059
$$672$$ −222.049 −8.56574
$$673$$ 8.50990 0.328033 0.164016 0.986458i $$-0.447555\pi$$
0.164016 + 0.986458i $$0.447555\pi$$
$$674$$ −14.2164 −0.547595
$$675$$ −14.0611 −0.541212
$$676$$ −64.5466 −2.48256
$$677$$ 12.5579 0.482639 0.241319 0.970446i $$-0.422420\pi$$
0.241319 + 0.970446i $$0.422420\pi$$
$$678$$ 95.2675 3.65872
$$679$$ −2.92452 −0.112233
$$680$$ 130.230 4.99409
$$681$$ 52.1472 1.99829
$$682$$ −20.3125 −0.777806
$$683$$ −28.8857 −1.10528 −0.552639 0.833420i $$-0.686379\pi$$
−0.552639 + 0.833420i $$0.686379\pi$$
$$684$$ 70.6757 2.70235
$$685$$ 12.8276 0.490116
$$686$$ 176.365 6.73365
$$687$$ −56.6380 −2.16088
$$688$$ −36.1654 −1.37879
$$689$$ −7.71666 −0.293981
$$690$$ −101.128 −3.84989
$$691$$ −21.6329 −0.822953 −0.411476 0.911420i $$-0.634987\pi$$
−0.411476 + 0.911420i $$0.634987\pi$$
$$692$$ 76.8049 2.91968
$$693$$ −17.0199 −0.646531
$$694$$ −26.5501 −1.00783
$$695$$ −74.4939 −2.82572
$$696$$ 150.441 5.70246
$$697$$ −53.2119 −2.01554
$$698$$ −15.3676 −0.581674
$$699$$ 55.6265 2.10399
$$700$$ −191.782 −7.24869
$$701$$ 16.4987 0.623148 0.311574 0.950222i $$-0.399144\pi$$
0.311574 + 0.950222i $$0.399144\pi$$
$$702$$ 3.94004 0.148707
$$703$$ 2.43535 0.0918509
$$704$$ 16.8486 0.635007
$$705$$ 36.8770 1.38887
$$706$$ −21.0418 −0.791919
$$707$$ 4.73371 0.178030
$$708$$ 105.924 3.98086
$$709$$ 17.8019 0.668565 0.334282 0.942473i $$-0.391506\pi$$
0.334282 + 0.942473i $$0.391506\pi$$
$$710$$ −124.628 −4.67719
$$711$$ 55.0219 2.06348
$$712$$ −96.1326 −3.60272
$$713$$ 35.9874 1.34774
$$714$$ −157.473 −5.89328
$$715$$ −2.28319 −0.0853865
$$716$$ −84.1952 −3.14652
$$717$$ 66.9287 2.49950
$$718$$ −8.00288 −0.298665
$$719$$ 5.51006 0.205491 0.102745 0.994708i $$-0.467237\pi$$
0.102745 + 0.994708i $$0.467237\pi$$
$$720$$ −164.197 −6.11927
$$721$$ 30.8705 1.14968
$$722$$ 15.6806 0.583570
$$723$$ 22.3038 0.829488
$$724$$ −29.6959 −1.10364
$$725$$ 48.4205 1.79829
$$726$$ −71.3052 −2.64638
$$727$$ −50.0027 −1.85450 −0.927248 0.374447i $$-0.877833\pi$$
−0.927248 + 0.374447i $$0.877833\pi$$
$$728$$ 33.0224 1.22389
$$729$$ −38.4672 −1.42471
$$730$$ 42.2575 1.56402
$$731$$ −12.6065 −0.466269
$$732$$ −23.6378 −0.873677
$$733$$ −17.1710 −0.634224 −0.317112 0.948388i $$-0.602713\pi$$
−0.317112 + 0.948388i $$0.602713\pi$$
$$734$$ −56.8504 −2.09839
$$735$$ 178.720 6.59220
$$736$$ −68.7921 −2.53571
$$737$$ 9.02151 0.332311
$$738$$ 122.606 4.51320
$$739$$ −24.8717 −0.914920 −0.457460 0.889230i $$-0.651241\pi$$
−0.457460 + 0.889230i $$0.651241\pi$$
$$740$$ −12.1445 −0.446440
$$741$$ −7.04485 −0.258799
$$742$$ −143.108 −5.25368
$$743$$ 6.44568 0.236469 0.118234 0.992986i $$-0.462277\pi$$
0.118234 + 0.992986i $$0.462277\pi$$
$$744$$ 192.057 7.04116
$$745$$ −8.36355 −0.306417
$$746$$ 46.0722 1.68682
$$747$$ 26.8078 0.980847
$$748$$ 19.8747 0.726692
$$749$$ −13.2678 −0.484795
$$750$$ −52.2455 −1.90773
$$751$$ 43.9422 1.60348 0.801738 0.597676i $$-0.203909\pi$$
0.801738 + 0.597676i $$0.203909\pi$$
$$752$$ 51.0359 1.86109
$$753$$ −4.20988 −0.153417
$$754$$ −13.5678 −0.494111
$$755$$ −41.8517 −1.52314
$$756$$ 52.7386 1.91809
$$757$$ −11.0556 −0.401823 −0.200912 0.979609i $$-0.564390\pi$$
−0.200912 + 0.979609i $$0.564390\pi$$
$$758$$ −24.0258 −0.872658
$$759$$ −9.48382 −0.344241
$$760$$ 108.047 3.91927
$$761$$ 17.5540 0.636332 0.318166 0.948035i $$-0.396933\pi$$
0.318166 + 0.948035i $$0.396933\pi$$
$$762$$ 96.0866 3.48085
$$763$$ −30.4730 −1.10320
$$764$$ 75.0622 2.71565
$$765$$ −57.2358 −2.06937
$$766$$ −36.9113 −1.33366
$$767$$ −5.87025 −0.211962
$$768$$ −28.9284 −1.04386
$$769$$ −25.6853 −0.926236 −0.463118 0.886297i $$-0.653269\pi$$
−0.463118 + 0.886297i $$0.653269\pi$$
$$770$$ −42.3427 −1.52592
$$771$$ 68.2848 2.45922
$$772$$ −14.6600 −0.527625
$$773$$ −8.40534 −0.302319 −0.151160 0.988509i $$-0.548301\pi$$
−0.151160 + 0.988509i $$0.548301\pi$$
$$774$$ 29.0469 1.04407
$$775$$ 61.8148 2.22045
$$776$$ −4.83530 −0.173577
$$777$$ 9.02390 0.323731
$$778$$ 49.1584 1.76241
$$779$$ −44.1479 −1.58176
$$780$$ 35.1309 1.25789
$$781$$ −11.6876 −0.418214
$$782$$ −48.7860 −1.74458
$$783$$ −13.3153 −0.475848
$$784$$ 247.340 8.83357
$$785$$ 45.2544 1.61520
$$786$$ 15.0542 0.536965
$$787$$ −42.4076 −1.51167 −0.755834 0.654764i $$-0.772768\pi$$
−0.755834 + 0.654764i $$0.772768\pi$$
$$788$$ −58.6838 −2.09052
$$789$$ −18.6772 −0.664924
$$790$$ 136.886 4.87018
$$791$$ −70.6702 −2.51274
$$792$$ −28.1401 −0.999914
$$793$$ 1.30999 0.0465193
$$794$$ −54.8718 −1.94733
$$795$$ −93.5546 −3.31804
$$796$$ 131.028 4.64416
$$797$$ 36.6030 1.29655 0.648273 0.761408i $$-0.275492\pi$$
0.648273 + 0.761408i $$0.275492\pi$$
$$798$$ −130.649 −4.62494
$$799$$ 17.7901 0.629367
$$800$$ −118.163 −4.17769
$$801$$ 42.2501 1.49283
$$802$$ −54.9445 −1.94016
$$803$$ 3.96291 0.139848
$$804$$ −138.812 −4.89551
$$805$$ 75.0179 2.64403
$$806$$ −17.3210 −0.610107
$$807$$ −45.9162 −1.61633
$$808$$ 7.82656 0.275337
$$809$$ 40.1918 1.41307 0.706535 0.707678i $$-0.250257\pi$$
0.706535 + 0.707678i $$0.250257\pi$$
$$810$$ −57.5519 −2.02217
$$811$$ −35.0857 −1.23203 −0.616013 0.787736i $$-0.711253\pi$$
−0.616013 + 0.787736i $$0.711253\pi$$
$$812$$ −181.610 −6.37325
$$813$$ 43.3452 1.52018
$$814$$ −1.57796 −0.0553076
$$815$$ 50.9194 1.78363
$$816$$ −142.471 −4.98748
$$817$$ −10.4592 −0.365919
$$818$$ 66.4888 2.32472
$$819$$ −14.5133 −0.507136
$$820$$ 220.155 7.68813
$$821$$ 32.3118 1.12769 0.563844 0.825881i $$-0.309322\pi$$
0.563844 + 0.825881i $$0.309322\pi$$
$$822$$ −25.6454 −0.894485
$$823$$ −17.0148 −0.593100 −0.296550 0.955017i $$-0.595836\pi$$
−0.296550 + 0.955017i $$0.595836\pi$$
$$824$$ 51.0402 1.77807
$$825$$ −16.2902 −0.567151
$$826$$ −108.866 −3.78794
$$827$$ 1.15823 0.0402756 0.0201378 0.999797i $$-0.493590\pi$$
0.0201378 + 0.999797i $$0.493590\pi$$
$$828$$ 81.1320 2.81953
$$829$$ 7.70413 0.267576 0.133788 0.991010i $$-0.457286\pi$$
0.133788 + 0.991010i $$0.457286\pi$$
$$830$$ 66.6935 2.31497
$$831$$ 7.42407 0.257538
$$832$$ 14.3673 0.498096
$$833$$ 86.2176 2.98726
$$834$$ 148.931 5.15706
$$835$$ −43.4637 −1.50412
$$836$$ 16.4893 0.570294
$$837$$ −16.9986 −0.587557
$$838$$ −6.84572 −0.236482
$$839$$ −45.8973 −1.58455 −0.792276 0.610163i $$-0.791104\pi$$
−0.792276 + 0.610163i $$0.791104\pi$$
$$840$$ 400.355 13.8136
$$841$$ 16.8521 0.581108
$$842$$ −8.50432 −0.293078
$$843$$ −21.5245 −0.741343
$$844$$ 47.0547 1.61969
$$845$$ 43.3683 1.49191
$$846$$ −40.9904 −1.40928
$$847$$ 52.8947 1.81748
$$848$$ −129.475 −4.44618
$$849$$ 53.5424 1.83757
$$850$$ −83.7988 −2.87427
$$851$$ 2.79565 0.0958337
$$852$$ 179.834 6.16100
$$853$$ 15.9270 0.545328 0.272664 0.962109i $$-0.412095\pi$$
0.272664 + 0.962109i $$0.412095\pi$$
$$854$$ 24.2944 0.831336
$$855$$ −47.4864 −1.62400
$$856$$ −21.9365 −0.749776
$$857$$ −16.3697 −0.559179 −0.279590 0.960120i $$-0.590198\pi$$
−0.279590 + 0.960120i $$0.590198\pi$$
$$858$$ 4.56464 0.155834
$$859$$ 35.9599 1.22694 0.613468 0.789720i $$-0.289774\pi$$
0.613468 + 0.789720i $$0.289774\pi$$
$$860$$ 52.1572 1.77855
$$861$$ −163.585 −5.57495
$$862$$ 14.7028 0.500781
$$863$$ −23.3613 −0.795228 −0.397614 0.917553i $$-0.630162\pi$$
−0.397614 + 0.917553i $$0.630162\pi$$
$$864$$ 32.4939 1.10546
$$865$$ −51.6045 −1.75461
$$866$$ −2.76971 −0.0941186
$$867$$ −5.47391 −0.185904
$$868$$ −231.847 −7.86942
$$869$$ 12.8371 0.435470
$$870$$ −164.493 −5.57682
$$871$$ 7.69289 0.260664
$$872$$ −50.3831 −1.70619
$$873$$ 2.12511 0.0719240
$$874$$ −40.4759 −1.36912
$$875$$ 38.7561 1.31020
$$876$$ −60.9763 −2.06020
$$877$$ −11.0060 −0.371646 −0.185823 0.982583i $$-0.559495\pi$$
−0.185823 + 0.982583i $$0.559495\pi$$
$$878$$ 73.2826 2.47317
$$879$$ 40.1598 1.35456
$$880$$ −38.3087 −1.29139
$$881$$ 30.4744 1.02671 0.513354 0.858177i $$-0.328403\pi$$
0.513354 + 0.858177i $$0.328403\pi$$
$$882$$ −198.656 −6.68908
$$883$$ 2.53253 0.0852266 0.0426133 0.999092i $$-0.486432\pi$$
0.0426133 + 0.999092i $$0.486432\pi$$
$$884$$ 16.9477 0.570014
$$885$$ −71.1693 −2.39233
$$886$$ −62.3583 −2.09497
$$887$$ 38.8851 1.30563 0.652817 0.757516i $$-0.273587\pi$$
0.652817 + 0.757516i $$0.273587\pi$$
$$888$$ 14.9198 0.500676
$$889$$ −71.2778 −2.39058
$$890$$ 105.111 3.52334
$$891$$ −5.39722 −0.180814
$$892$$ 84.3688 2.82488
$$893$$ 14.7597 0.493916
$$894$$ 16.7207 0.559225
$$895$$ 56.5700 1.89093
$$896$$ 95.5958 3.19363
$$897$$ −8.08711 −0.270021
$$898$$ 57.5692 1.92111
$$899$$ 58.5360 1.95228
$$900$$ 139.359 4.64530
$$901$$ −45.1323 −1.50357
$$902$$ 28.6052 0.952449
$$903$$ −38.7552 −1.28969
$$904$$ −116.844 −3.88616
$$905$$ 19.9524 0.663240
$$906$$ 83.6715 2.77980
$$907$$ −20.5975 −0.683930 −0.341965 0.939713i $$-0.611092\pi$$
−0.341965 + 0.939713i $$0.611092\pi$$
$$908$$ −104.081 −3.45405
$$909$$ −3.43976 −0.114090
$$910$$ −36.1068 −1.19693
$$911$$ −47.3267 −1.56800 −0.784001 0.620759i $$-0.786824\pi$$
−0.784001 + 0.620759i $$0.786824\pi$$
$$912$$ −118.203 −3.91408
$$913$$ 6.25451 0.206994
$$914$$ 83.0007 2.74542
$$915$$ 15.8820 0.525043
$$916$$ 113.044 3.73509
$$917$$ −11.1673 −0.368777
$$918$$ 23.0440 0.760566
$$919$$ −44.2802 −1.46067 −0.730335 0.683089i $$-0.760636\pi$$
−0.730335 + 0.683089i $$0.760636\pi$$
$$920$$ 124.032 4.08921
$$921$$ −45.1230 −1.48685
$$922$$ −22.4720 −0.740077
$$923$$ −9.96630 −0.328045
$$924$$ 61.0992 2.01002
$$925$$ 4.80204 0.157890
$$926$$ −101.262 −3.32769
$$927$$ −22.4321 −0.736766
$$928$$ −111.895 −3.67314
$$929$$ −42.2540 −1.38631 −0.693155 0.720788i $$-0.743780\pi$$
−0.693155 + 0.720788i $$0.743780\pi$$
$$930$$ −209.995 −6.88602
$$931$$ 71.5315 2.34435
$$932$$ −111.026 −3.63676
$$933$$ 15.5989 0.510684
$$934$$ −40.6962 −1.33162
$$935$$ −13.3537 −0.436711
$$936$$ −23.9958 −0.784327
$$937$$ −19.0600 −0.622665 −0.311332 0.950301i $$-0.600775\pi$$
−0.311332 + 0.950301i $$0.600775\pi$$
$$938$$ 142.668 4.65826
$$939$$ −84.7017 −2.76414
$$940$$ −73.6032 −2.40067
$$941$$ −28.9026 −0.942197 −0.471098 0.882081i $$-0.656142\pi$$
−0.471098 + 0.882081i $$0.656142\pi$$
$$942$$ −90.4743 −2.94781
$$943$$ −50.6794 −1.65035
$$944$$ −98.4946 −3.20573
$$945$$ −35.4346 −1.15269
$$946$$ 6.77691 0.220336
$$947$$ 48.2970 1.56944 0.784721 0.619849i $$-0.212806\pi$$
0.784721 + 0.619849i $$0.212806\pi$$
$$948$$ −197.522 −6.41522
$$949$$ 3.37928 0.109696
$$950$$ −69.5247 −2.25568
$$951$$ −39.9126 −1.29425
$$952$$ 193.138 6.25963
$$953$$ 38.9193 1.26072 0.630360 0.776303i $$-0.282907\pi$$
0.630360 + 0.776303i $$0.282907\pi$$
$$954$$ 103.990 3.36680
$$955$$ −50.4336 −1.63199
$$956$$ −133.584 −4.32040
$$957$$ −15.4261 −0.498655
$$958$$ −66.2038 −2.13895
$$959$$ 19.0239 0.614315
$$960$$ 174.185 5.62180
$$961$$ 43.7286 1.41060
$$962$$ −1.34557 −0.0433830
$$963$$ 9.64108 0.310679
$$964$$ −44.5164 −1.43378
$$965$$ 9.84992 0.317080
$$966$$ −149.979 −4.82549
$$967$$ 17.9995 0.578823 0.289412 0.957205i $$-0.406540\pi$$
0.289412 + 0.957205i $$0.406540\pi$$
$$968$$ 87.4544 2.81089
$$969$$ −41.2031 −1.32363
$$970$$ 5.28693 0.169753
$$971$$ 51.1998 1.64308 0.821540 0.570150i $$-0.193115\pi$$
0.821540 + 0.570150i $$0.193115\pi$$
$$972$$ 113.651 3.64535
$$973$$ −110.478 −3.54177
$$974$$ −62.3754 −1.99864
$$975$$ −13.8911 −0.444870
$$976$$ 21.9799 0.703559
$$977$$ −29.4243 −0.941366 −0.470683 0.882302i $$-0.655992\pi$$
−0.470683 + 0.882302i $$0.655992\pi$$
$$978$$ −101.800 −3.25521
$$979$$ 9.85734 0.315042
$$980$$ −356.710 −11.3947
$$981$$ 22.1433 0.706980
$$982$$ 64.4820 2.05770
$$983$$ 7.77916 0.248117 0.124058 0.992275i $$-0.460409\pi$$
0.124058 + 0.992275i $$0.460409\pi$$
$$984$$ −270.466 −8.62213
$$985$$ 39.4291 1.25632
$$986$$ −79.3539 −2.52714
$$987$$ 54.6905 1.74082
$$988$$ 14.0609 0.447336
$$989$$ −12.0066 −0.381786
$$990$$ 30.7684 0.977883
$$991$$ 49.1705 1.56195 0.780976 0.624561i $$-0.214722\pi$$
0.780976 + 0.624561i $$0.214722\pi$$
$$992$$ −142.848 −4.53544
$$993$$ 3.07268 0.0975084
$$994$$ −184.829 −5.86242
$$995$$ −88.0366 −2.79095
$$996$$ −96.2367 −3.04938
$$997$$ −22.6434 −0.717124 −0.358562 0.933506i $$-0.616733\pi$$
−0.358562 + 0.933506i $$0.616733\pi$$
$$998$$ 88.0821 2.78819
$$999$$ −1.32052 −0.0417795
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 4001.2.a.b.1.8 184

By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.8 184 1.1 even 1 trivial