# Properties

 Label 4001.2.a.b.1.3 Level $4001$ Weight $2$ Character 4001.1 Self dual yes Analytic conductor $31.948$ Analytic rank $0$ Dimension $184$ CM no Inner twists $1$

# Related objects

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 1.3 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)

 $$f(q)$$ $$=$$ $$q-2.77961 q^{2} -0.970175 q^{3} +5.72623 q^{4} +0.973772 q^{5} +2.69671 q^{6} +0.392591 q^{7} -10.3575 q^{8} -2.05876 q^{9} +O(q^{10})$$ $$q-2.77961 q^{2} -0.970175 q^{3} +5.72623 q^{4} +0.973772 q^{5} +2.69671 q^{6} +0.392591 q^{7} -10.3575 q^{8} -2.05876 q^{9} -2.70671 q^{10} -4.34690 q^{11} -5.55544 q^{12} +2.35614 q^{13} -1.09125 q^{14} -0.944729 q^{15} +17.3373 q^{16} +6.89369 q^{17} +5.72255 q^{18} +6.69926 q^{19} +5.57604 q^{20} -0.380881 q^{21} +12.0827 q^{22} -2.69910 q^{23} +10.0486 q^{24} -4.05177 q^{25} -6.54914 q^{26} +4.90788 q^{27} +2.24806 q^{28} +0.233462 q^{29} +2.62598 q^{30} +9.20330 q^{31} -27.4759 q^{32} +4.21725 q^{33} -19.1618 q^{34} +0.382294 q^{35} -11.7889 q^{36} -6.23130 q^{37} -18.6213 q^{38} -2.28586 q^{39} -10.0858 q^{40} +11.9724 q^{41} +1.05870 q^{42} +5.31449 q^{43} -24.8913 q^{44} -2.00476 q^{45} +7.50244 q^{46} -5.36754 q^{47} -16.8202 q^{48} -6.84587 q^{49} +11.2623 q^{50} -6.68808 q^{51} +13.4918 q^{52} -5.62281 q^{53} -13.6420 q^{54} -4.23289 q^{55} -4.06624 q^{56} -6.49946 q^{57} -0.648933 q^{58} -7.14186 q^{59} -5.40974 q^{60} +9.12946 q^{61} -25.5816 q^{62} -0.808250 q^{63} +41.6977 q^{64} +2.29434 q^{65} -11.7223 q^{66} -2.47276 q^{67} +39.4749 q^{68} +2.61860 q^{69} -1.06263 q^{70} -7.84018 q^{71} +21.3236 q^{72} -12.2002 q^{73} +17.3206 q^{74} +3.93092 q^{75} +38.3615 q^{76} -1.70655 q^{77} +6.35381 q^{78} -9.57153 q^{79} +16.8825 q^{80} +1.41478 q^{81} -33.2787 q^{82} +13.8667 q^{83} -2.18101 q^{84} +6.71288 q^{85} -14.7722 q^{86} -0.226499 q^{87} +45.0229 q^{88} -9.52619 q^{89} +5.57246 q^{90} +0.924997 q^{91} -15.4557 q^{92} -8.92880 q^{93} +14.9197 q^{94} +6.52355 q^{95} +26.6564 q^{96} +15.4817 q^{97} +19.0289 q^{98} +8.94923 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.77961 −1.96548 −0.982741 0.184989i $$-0.940775\pi$$
−0.982741 + 0.184989i $$0.940775\pi$$
$$3$$ −0.970175 −0.560131 −0.280065 0.959981i $$-0.590356\pi$$
−0.280065 + 0.959981i $$0.590356\pi$$
$$4$$ 5.72623 2.86312
$$5$$ 0.973772 0.435484 0.217742 0.976006i $$-0.430131\pi$$
0.217742 + 0.976006i $$0.430131\pi$$
$$6$$ 2.69671 1.10093
$$7$$ 0.392591 0.148385 0.0741926 0.997244i $$-0.476362\pi$$
0.0741926 + 0.997244i $$0.476362\pi$$
$$8$$ −10.3575 −3.66192
$$9$$ −2.05876 −0.686254
$$10$$ −2.70671 −0.855936
$$11$$ −4.34690 −1.31064 −0.655320 0.755352i $$-0.727466\pi$$
−0.655320 + 0.755352i $$0.727466\pi$$
$$12$$ −5.55544 −1.60372
$$13$$ 2.35614 0.653475 0.326737 0.945115i $$-0.394051\pi$$
0.326737 + 0.945115i $$0.394051\pi$$
$$14$$ −1.09125 −0.291648
$$15$$ −0.944729 −0.243928
$$16$$ 17.3373 4.33432
$$17$$ 6.89369 1.67196 0.835982 0.548756i $$-0.184898\pi$$
0.835982 + 0.548756i $$0.184898\pi$$
$$18$$ 5.72255 1.34882
$$19$$ 6.69926 1.53692 0.768458 0.639900i $$-0.221024\pi$$
0.768458 + 0.639900i $$0.221024\pi$$
$$20$$ 5.57604 1.24684
$$21$$ −0.380881 −0.0831151
$$22$$ 12.0827 2.57604
$$23$$ −2.69910 −0.562801 −0.281400 0.959590i $$-0.590799\pi$$
−0.281400 + 0.959590i $$0.590799\pi$$
$$24$$ 10.0486 2.05115
$$25$$ −4.05177 −0.810354
$$26$$ −6.54914 −1.28439
$$27$$ 4.90788 0.944522
$$28$$ 2.24806 0.424844
$$29$$ 0.233462 0.0433528 0.0216764 0.999765i $$-0.493100\pi$$
0.0216764 + 0.999765i $$0.493100\pi$$
$$30$$ 2.62598 0.479436
$$31$$ 9.20330 1.65296 0.826480 0.562966i $$-0.190340\pi$$
0.826480 + 0.562966i $$0.190340\pi$$
$$32$$ −27.4759 −4.85710
$$33$$ 4.21725 0.734129
$$34$$ −19.1618 −3.28622
$$35$$ 0.382294 0.0646194
$$36$$ −11.7889 −1.96482
$$37$$ −6.23130 −1.02442 −0.512210 0.858860i $$-0.671173\pi$$
−0.512210 + 0.858860i $$0.671173\pi$$
$$38$$ −18.6213 −3.02078
$$39$$ −2.28586 −0.366031
$$40$$ −10.0858 −1.59471
$$41$$ 11.9724 1.86978 0.934890 0.354937i $$-0.115498\pi$$
0.934890 + 0.354937i $$0.115498\pi$$
$$42$$ 1.05870 0.163361
$$43$$ 5.31449 0.810452 0.405226 0.914217i $$-0.367193\pi$$
0.405226 + 0.914217i $$0.367193\pi$$
$$44$$ −24.8913 −3.75251
$$45$$ −2.00476 −0.298852
$$46$$ 7.50244 1.10617
$$47$$ −5.36754 −0.782937 −0.391468 0.920192i $$-0.628033\pi$$
−0.391468 + 0.920192i $$0.628033\pi$$
$$48$$ −16.8202 −2.42778
$$49$$ −6.84587 −0.977982
$$50$$ 11.2623 1.59273
$$51$$ −6.68808 −0.936519
$$52$$ 13.4918 1.87097
$$53$$ −5.62281 −0.772352 −0.386176 0.922425i $$-0.626204\pi$$
−0.386176 + 0.922425i $$0.626204\pi$$
$$54$$ −13.6420 −1.85644
$$55$$ −4.23289 −0.570762
$$56$$ −4.06624 −0.543375
$$57$$ −6.49946 −0.860874
$$58$$ −0.648933 −0.0852091
$$59$$ −7.14186 −0.929791 −0.464896 0.885366i $$-0.653908\pi$$
−0.464896 + 0.885366i $$0.653908\pi$$
$$60$$ −5.40974 −0.698394
$$61$$ 9.12946 1.16891 0.584454 0.811427i $$-0.301309\pi$$
0.584454 + 0.811427i $$0.301309\pi$$
$$62$$ −25.5816 −3.24886
$$63$$ −0.808250 −0.101830
$$64$$ 41.6977 5.21222
$$65$$ 2.29434 0.284578
$$66$$ −11.7223 −1.44292
$$67$$ −2.47276 −0.302096 −0.151048 0.988526i $$-0.548265\pi$$
−0.151048 + 0.988526i $$0.548265\pi$$
$$68$$ 39.4749 4.78703
$$69$$ 2.61860 0.315242
$$70$$ −1.06263 −0.127008
$$71$$ −7.84018 −0.930458 −0.465229 0.885190i $$-0.654028\pi$$
−0.465229 + 0.885190i $$0.654028\pi$$
$$72$$ 21.3236 2.51301
$$73$$ −12.2002 −1.42792 −0.713962 0.700185i $$-0.753101\pi$$
−0.713962 + 0.700185i $$0.753101\pi$$
$$74$$ 17.3206 2.01348
$$75$$ 3.93092 0.453904
$$76$$ 38.3615 4.40037
$$77$$ −1.70655 −0.194480
$$78$$ 6.35381 0.719427
$$79$$ −9.57153 −1.07688 −0.538441 0.842663i $$-0.680986\pi$$
−0.538441 + 0.842663i $$0.680986\pi$$
$$80$$ 16.8825 1.88753
$$81$$ 1.41478 0.157198
$$82$$ −33.2787 −3.67502
$$83$$ 13.8667 1.52207 0.761037 0.648709i $$-0.224691\pi$$
0.761037 + 0.648709i $$0.224691\pi$$
$$84$$ −2.18101 −0.237968
$$85$$ 6.71288 0.728114
$$86$$ −14.7722 −1.59293
$$87$$ −0.226499 −0.0242832
$$88$$ 45.0229 4.79945
$$89$$ −9.52619 −1.00977 −0.504887 0.863185i $$-0.668466\pi$$
−0.504887 + 0.863185i $$0.668466\pi$$
$$90$$ 5.57246 0.587389
$$91$$ 0.924997 0.0969660
$$92$$ −15.4557 −1.61136
$$93$$ −8.92880 −0.925874
$$94$$ 14.9197 1.53885
$$95$$ 6.52355 0.669302
$$96$$ 26.6564 2.72061
$$97$$ 15.4817 1.57193 0.785964 0.618273i $$-0.212167\pi$$
0.785964 + 0.618273i $$0.212167\pi$$
$$98$$ 19.0289 1.92220
$$99$$ 8.94923 0.899431
$$100$$ −23.2014 −2.32014
$$101$$ 2.44716 0.243502 0.121751 0.992561i $$-0.461149\pi$$
0.121751 + 0.992561i $$0.461149\pi$$
$$102$$ 18.5903 1.84071
$$103$$ 17.2924 1.70387 0.851934 0.523650i $$-0.175430\pi$$
0.851934 + 0.523650i $$0.175430\pi$$
$$104$$ −24.4036 −2.39297
$$105$$ −0.370892 −0.0361953
$$106$$ 15.6292 1.51804
$$107$$ 6.25436 0.604632 0.302316 0.953208i $$-0.402240\pi$$
0.302316 + 0.953208i $$0.402240\pi$$
$$108$$ 28.1037 2.70428
$$109$$ −1.22278 −0.117122 −0.0585608 0.998284i $$-0.518651\pi$$
−0.0585608 + 0.998284i $$0.518651\pi$$
$$110$$ 11.7658 1.12182
$$111$$ 6.04545 0.573809
$$112$$ 6.80645 0.643149
$$113$$ −2.78485 −0.261977 −0.130988 0.991384i $$-0.541815\pi$$
−0.130988 + 0.991384i $$0.541815\pi$$
$$114$$ 18.0660 1.69203
$$115$$ −2.62830 −0.245091
$$116$$ 1.33686 0.124124
$$117$$ −4.85072 −0.448450
$$118$$ 19.8516 1.82749
$$119$$ 2.70640 0.248095
$$120$$ 9.78500 0.893244
$$121$$ 7.89552 0.717775
$$122$$ −25.3764 −2.29747
$$123$$ −11.6154 −1.04732
$$124$$ 52.7002 4.73262
$$125$$ −8.81436 −0.788380
$$126$$ 2.24662 0.200145
$$127$$ −7.66055 −0.679764 −0.339882 0.940468i $$-0.610387\pi$$
−0.339882 + 0.940468i $$0.610387\pi$$
$$128$$ −60.9517 −5.38742
$$129$$ −5.15598 −0.453959
$$130$$ −6.37737 −0.559332
$$131$$ 4.42940 0.386998 0.193499 0.981100i $$-0.438016\pi$$
0.193499 + 0.981100i $$0.438016\pi$$
$$132$$ 24.1490 2.10190
$$133$$ 2.63007 0.228056
$$134$$ 6.87331 0.593763
$$135$$ 4.77916 0.411324
$$136$$ −71.4012 −6.12260
$$137$$ −21.1506 −1.80701 −0.903507 0.428573i $$-0.859016\pi$$
−0.903507 + 0.428573i $$0.859016\pi$$
$$138$$ −7.27867 −0.619602
$$139$$ 10.6865 0.906416 0.453208 0.891405i $$-0.350279\pi$$
0.453208 + 0.891405i $$0.350279\pi$$
$$140$$ 2.18910 0.185013
$$141$$ 5.20746 0.438547
$$142$$ 21.7926 1.82880
$$143$$ −10.2419 −0.856470
$$144$$ −35.6933 −2.97444
$$145$$ 0.227339 0.0188795
$$146$$ 33.9118 2.80656
$$147$$ 6.64169 0.547798
$$148$$ −35.6819 −2.93303
$$149$$ 12.4241 1.01782 0.508911 0.860819i $$-0.330048\pi$$
0.508911 + 0.860819i $$0.330048\pi$$
$$150$$ −10.9264 −0.892140
$$151$$ −0.852439 −0.0693705 −0.0346853 0.999398i $$-0.511043\pi$$
−0.0346853 + 0.999398i $$0.511043\pi$$
$$152$$ −69.3874 −5.62806
$$153$$ −14.1925 −1.14739
$$154$$ 4.74355 0.382246
$$155$$ 8.96191 0.719838
$$156$$ −13.0894 −1.04799
$$157$$ −1.03294 −0.0824377 −0.0412188 0.999150i $$-0.513124\pi$$
−0.0412188 + 0.999150i $$0.513124\pi$$
$$158$$ 26.6051 2.11659
$$159$$ 5.45511 0.432618
$$160$$ −26.7553 −2.11519
$$161$$ −1.05964 −0.0835113
$$162$$ −3.93254 −0.308969
$$163$$ 2.54589 0.199410 0.0997048 0.995017i $$-0.468210\pi$$
0.0997048 + 0.995017i $$0.468210\pi$$
$$164$$ 68.5569 5.35340
$$165$$ 4.10664 0.319701
$$166$$ −38.5441 −2.99161
$$167$$ −5.05576 −0.391227 −0.195613 0.980681i $$-0.562670\pi$$
−0.195613 + 0.980681i $$0.562670\pi$$
$$168$$ 3.94497 0.304361
$$169$$ −7.44862 −0.572971
$$170$$ −18.6592 −1.43109
$$171$$ −13.7922 −1.05471
$$172$$ 30.4320 2.32042
$$173$$ −10.2984 −0.782974 −0.391487 0.920184i $$-0.628039\pi$$
−0.391487 + 0.920184i $$0.628039\pi$$
$$174$$ 0.629579 0.0477282
$$175$$ −1.59069 −0.120245
$$176$$ −75.3633 −5.68072
$$177$$ 6.92885 0.520804
$$178$$ 26.4791 1.98469
$$179$$ 4.01654 0.300210 0.150105 0.988670i $$-0.452039\pi$$
0.150105 + 0.988670i $$0.452039\pi$$
$$180$$ −11.4797 −0.855649
$$181$$ −8.51582 −0.632976 −0.316488 0.948597i $$-0.602504\pi$$
−0.316488 + 0.948597i $$0.602504\pi$$
$$182$$ −2.57113 −0.190585
$$183$$ −8.85718 −0.654741
$$184$$ 27.9558 2.06093
$$185$$ −6.06786 −0.446118
$$186$$ 24.8186 1.81979
$$187$$ −29.9662 −2.19134
$$188$$ −30.7358 −2.24164
$$189$$ 1.92679 0.140153
$$190$$ −18.1329 −1.31550
$$191$$ 25.2990 1.83057 0.915286 0.402805i $$-0.131965\pi$$
0.915286 + 0.402805i $$0.131965\pi$$
$$192$$ −40.4541 −2.91952
$$193$$ 0.181638 0.0130746 0.00653728 0.999979i $$-0.497919\pi$$
0.00653728 + 0.999979i $$0.497919\pi$$
$$194$$ −43.0331 −3.08959
$$195$$ −2.22591 −0.159401
$$196$$ −39.2011 −2.80008
$$197$$ 13.2097 0.941149 0.470575 0.882360i $$-0.344047\pi$$
0.470575 + 0.882360i $$0.344047\pi$$
$$198$$ −24.8754 −1.76781
$$199$$ 10.3487 0.733597 0.366798 0.930300i $$-0.380454\pi$$
0.366798 + 0.930300i $$0.380454\pi$$
$$200$$ 41.9661 2.96745
$$201$$ 2.39901 0.169213
$$202$$ −6.80216 −0.478598
$$203$$ 0.0916550 0.00643292
$$204$$ −38.2975 −2.68136
$$205$$ 11.6584 0.814259
$$206$$ −48.0660 −3.34892
$$207$$ 5.55680 0.386224
$$208$$ 40.8490 2.83237
$$209$$ −29.1210 −2.01434
$$210$$ 1.03093 0.0711412
$$211$$ 7.27640 0.500928 0.250464 0.968126i $$-0.419417\pi$$
0.250464 + 0.968126i $$0.419417\pi$$
$$212$$ −32.1975 −2.21133
$$213$$ 7.60634 0.521178
$$214$$ −17.3847 −1.18839
$$215$$ 5.17510 0.352939
$$216$$ −50.8332 −3.45876
$$217$$ 3.61313 0.245275
$$218$$ 3.39886 0.230200
$$219$$ 11.8363 0.799824
$$220$$ −24.2385 −1.63416
$$221$$ 16.2425 1.09259
$$222$$ −16.8040 −1.12781
$$223$$ −10.3405 −0.692450 −0.346225 0.938151i $$-0.612537\pi$$
−0.346225 + 0.938151i $$0.612537\pi$$
$$224$$ −10.7868 −0.720722
$$225$$ 8.34162 0.556108
$$226$$ 7.74079 0.514910
$$227$$ −23.6579 −1.57023 −0.785115 0.619349i $$-0.787396\pi$$
−0.785115 + 0.619349i $$0.787396\pi$$
$$228$$ −37.2174 −2.46478
$$229$$ 1.31477 0.0868827 0.0434414 0.999056i $$-0.486168\pi$$
0.0434414 + 0.999056i $$0.486168\pi$$
$$230$$ 7.30566 0.481721
$$231$$ 1.65565 0.108934
$$232$$ −2.41808 −0.158754
$$233$$ −10.2122 −0.669026 −0.334513 0.942391i $$-0.608572\pi$$
−0.334513 + 0.942391i $$0.608572\pi$$
$$234$$ 13.4831 0.881419
$$235$$ −5.22676 −0.340956
$$236$$ −40.8960 −2.66210
$$237$$ 9.28606 0.603194
$$238$$ −7.52273 −0.487626
$$239$$ 28.3329 1.83270 0.916351 0.400376i $$-0.131121\pi$$
0.916351 + 0.400376i $$0.131121\pi$$
$$240$$ −16.3790 −1.05726
$$241$$ 13.5986 0.875961 0.437980 0.898985i $$-0.355694\pi$$
0.437980 + 0.898985i $$0.355694\pi$$
$$242$$ −21.9465 −1.41077
$$243$$ −16.0962 −1.03257
$$244$$ 52.2774 3.34672
$$245$$ −6.66632 −0.425895
$$246$$ 32.2861 2.05849
$$247$$ 15.7844 1.00434
$$248$$ −95.3229 −6.05301
$$249$$ −13.4532 −0.852560
$$250$$ 24.5005 1.54955
$$251$$ 2.90600 0.183425 0.0917124 0.995786i $$-0.470766\pi$$
0.0917124 + 0.995786i $$0.470766\pi$$
$$252$$ −4.62823 −0.291551
$$253$$ 11.7327 0.737629
$$254$$ 21.2933 1.33606
$$255$$ −6.51267 −0.407839
$$256$$ 86.0264 5.37665
$$257$$ 26.4072 1.64724 0.823618 0.567146i $$-0.191952\pi$$
0.823618 + 0.567146i $$0.191952\pi$$
$$258$$ 14.3316 0.892247
$$259$$ −2.44635 −0.152009
$$260$$ 13.1379 0.814779
$$261$$ −0.480643 −0.0297510
$$262$$ −12.3120 −0.760638
$$263$$ 3.26317 0.201215 0.100608 0.994926i $$-0.467921\pi$$
0.100608 + 0.994926i $$0.467921\pi$$
$$264$$ −43.6800 −2.68832
$$265$$ −5.47533 −0.336347
$$266$$ −7.31056 −0.448239
$$267$$ 9.24207 0.565605
$$268$$ −14.1596 −0.864935
$$269$$ 27.2067 1.65882 0.829410 0.558640i $$-0.188677\pi$$
0.829410 + 0.558640i $$0.188677\pi$$
$$270$$ −13.2842 −0.808450
$$271$$ 32.4618 1.97191 0.985956 0.167005i $$-0.0534095\pi$$
0.985956 + 0.167005i $$0.0534095\pi$$
$$272$$ 119.518 7.24683
$$273$$ −0.897409 −0.0543136
$$274$$ 58.7903 3.55165
$$275$$ 17.6126 1.06208
$$276$$ 14.9947 0.902574
$$277$$ −30.1935 −1.81415 −0.907076 0.420967i $$-0.861691\pi$$
−0.907076 + 0.420967i $$0.861691\pi$$
$$278$$ −29.7043 −1.78154
$$279$$ −18.9474 −1.13435
$$280$$ −3.95959 −0.236631
$$281$$ 6.96537 0.415519 0.207760 0.978180i $$-0.433383\pi$$
0.207760 + 0.978180i $$0.433383\pi$$
$$282$$ −14.4747 −0.861956
$$283$$ 10.3628 0.616005 0.308002 0.951386i $$-0.400340\pi$$
0.308002 + 0.951386i $$0.400340\pi$$
$$284$$ −44.8947 −2.66401
$$285$$ −6.32899 −0.374897
$$286$$ 28.4685 1.68337
$$287$$ 4.70026 0.277448
$$288$$ 56.5663 3.33320
$$289$$ 30.5229 1.79547
$$290$$ −0.631913 −0.0371072
$$291$$ −15.0199 −0.880485
$$292$$ −69.8611 −4.08831
$$293$$ 15.7984 0.922951 0.461476 0.887153i $$-0.347320\pi$$
0.461476 + 0.887153i $$0.347320\pi$$
$$294$$ −18.4613 −1.07669
$$295$$ −6.95454 −0.404909
$$296$$ 64.5405 3.75134
$$297$$ −21.3341 −1.23793
$$298$$ −34.5342 −2.00051
$$299$$ −6.35944 −0.367776
$$300$$ 22.5094 1.29958
$$301$$ 2.08642 0.120259
$$302$$ 2.36945 0.136346
$$303$$ −2.37418 −0.136393
$$304$$ 116.147 6.66148
$$305$$ 8.89002 0.509041
$$306$$ 39.4495 2.25518
$$307$$ −26.7690 −1.52779 −0.763893 0.645343i $$-0.776715\pi$$
−0.763893 + 0.645343i $$0.776715\pi$$
$$308$$ −9.77211 −0.556817
$$309$$ −16.7766 −0.954388
$$310$$ −24.9106 −1.41483
$$311$$ −4.80028 −0.272199 −0.136099 0.990695i $$-0.543457\pi$$
−0.136099 + 0.990695i $$0.543457\pi$$
$$312$$ 23.6758 1.34038
$$313$$ −12.3026 −0.695382 −0.347691 0.937609i $$-0.613034\pi$$
−0.347691 + 0.937609i $$0.613034\pi$$
$$314$$ 2.87117 0.162030
$$315$$ −0.787051 −0.0443453
$$316$$ −54.8088 −3.08324
$$317$$ 26.2909 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$318$$ −15.1631 −0.850303
$$319$$ −1.01484 −0.0568199
$$320$$ 40.6041 2.26984
$$321$$ −6.06782 −0.338673
$$322$$ 2.94539 0.164140
$$323$$ 46.1826 2.56967
$$324$$ 8.10136 0.450076
$$325$$ −9.54652 −0.529546
$$326$$ −7.07659 −0.391936
$$327$$ 1.18631 0.0656034
$$328$$ −124.004 −6.84698
$$329$$ −2.10725 −0.116176
$$330$$ −11.4149 −0.628367
$$331$$ 32.6548 1.79487 0.897435 0.441147i $$-0.145428\pi$$
0.897435 + 0.441147i $$0.145428\pi$$
$$332$$ 79.4042 4.35787
$$333$$ 12.8288 0.703012
$$334$$ 14.0530 0.768949
$$335$$ −2.40790 −0.131558
$$336$$ −6.60344 −0.360247
$$337$$ 2.86894 0.156281 0.0781407 0.996942i $$-0.475102\pi$$
0.0781407 + 0.996942i $$0.475102\pi$$
$$338$$ 20.7043 1.12616
$$339$$ 2.70179 0.146741
$$340$$ 38.4395 2.08467
$$341$$ −40.0058 −2.16644
$$342$$ 38.3369 2.07302
$$343$$ −5.43576 −0.293503
$$344$$ −55.0446 −2.96781
$$345$$ 2.54991 0.137283
$$346$$ 28.6256 1.53892
$$347$$ −2.06265 −0.110729 −0.0553644 0.998466i $$-0.517632\pi$$
−0.0553644 + 0.998466i $$0.517632\pi$$
$$348$$ −1.29699 −0.0695257
$$349$$ −8.40748 −0.450042 −0.225021 0.974354i $$-0.572245\pi$$
−0.225021 + 0.974354i $$0.572245\pi$$
$$350$$ 4.42149 0.236338
$$351$$ 11.5636 0.617222
$$352$$ 119.435 6.36590
$$353$$ 4.46462 0.237628 0.118814 0.992917i $$-0.462091\pi$$
0.118814 + 0.992917i $$0.462091\pi$$
$$354$$ −19.2595 −1.02363
$$355$$ −7.63454 −0.405200
$$356$$ −54.5492 −2.89110
$$357$$ −2.62568 −0.138966
$$358$$ −11.1644 −0.590057
$$359$$ 0.452733 0.0238943 0.0119472 0.999929i $$-0.496197\pi$$
0.0119472 + 0.999929i $$0.496197\pi$$
$$360$$ 20.7643 1.09437
$$361$$ 25.8801 1.36211
$$362$$ 23.6707 1.24410
$$363$$ −7.66004 −0.402048
$$364$$ 5.29675 0.277625
$$365$$ −11.8802 −0.621838
$$366$$ 24.6195 1.28688
$$367$$ 17.8679 0.932697 0.466348 0.884601i $$-0.345569\pi$$
0.466348 + 0.884601i $$0.345569\pi$$
$$368$$ −46.7950 −2.43936
$$369$$ −24.6484 −1.28314
$$370$$ 16.8663 0.876837
$$371$$ −2.20746 −0.114606
$$372$$ −51.1284 −2.65088
$$373$$ −8.93177 −0.462469 −0.231235 0.972898i $$-0.574277\pi$$
−0.231235 + 0.972898i $$0.574277\pi$$
$$374$$ 83.2942 4.30704
$$375$$ 8.55147 0.441596
$$376$$ 55.5942 2.86705
$$377$$ 0.550069 0.0283300
$$378$$ −5.35572 −0.275468
$$379$$ −13.6225 −0.699741 −0.349870 0.936798i $$-0.613774\pi$$
−0.349870 + 0.936798i $$0.613774\pi$$
$$380$$ 37.3554 1.91629
$$381$$ 7.43207 0.380757
$$382$$ −70.3214 −3.59795
$$383$$ −12.3858 −0.632887 −0.316444 0.948611i $$-0.602489\pi$$
−0.316444 + 0.948611i $$0.602489\pi$$
$$384$$ 59.1338 3.01766
$$385$$ −1.66179 −0.0846927
$$386$$ −0.504882 −0.0256978
$$387$$ −10.9413 −0.556175
$$388$$ 88.6518 4.50061
$$389$$ −3.63343 −0.184222 −0.0921111 0.995749i $$-0.529362\pi$$
−0.0921111 + 0.995749i $$0.529362\pi$$
$$390$$ 6.18716 0.313299
$$391$$ −18.6067 −0.940983
$$392$$ 70.9059 3.58129
$$393$$ −4.29729 −0.216770
$$394$$ −36.7177 −1.84981
$$395$$ −9.32049 −0.468965
$$396$$ 51.2453 2.57518
$$397$$ 13.9378 0.699517 0.349758 0.936840i $$-0.386264\pi$$
0.349758 + 0.936840i $$0.386264\pi$$
$$398$$ −28.7652 −1.44187
$$399$$ −2.55162 −0.127741
$$400$$ −70.2466 −3.51233
$$401$$ 4.09635 0.204562 0.102281 0.994756i $$-0.467386\pi$$
0.102281 + 0.994756i $$0.467386\pi$$
$$402$$ −6.66831 −0.332585
$$403$$ 21.6842 1.08017
$$404$$ 14.0130 0.697174
$$405$$ 1.37767 0.0684572
$$406$$ −0.254765 −0.0126438
$$407$$ 27.0868 1.34264
$$408$$ 69.2716 3.42946
$$409$$ 39.6504 1.96059 0.980294 0.197546i $$-0.0632972\pi$$
0.980294 + 0.197546i $$0.0632972\pi$$
$$410$$ −32.4059 −1.60041
$$411$$ 20.5197 1.01216
$$412$$ 99.0201 4.87837
$$413$$ −2.80383 −0.137967
$$414$$ −15.4457 −0.759116
$$415$$ 13.5030 0.662839
$$416$$ −64.7370 −3.17399
$$417$$ −10.3678 −0.507711
$$418$$ 80.9451 3.95915
$$419$$ 27.1912 1.32838 0.664190 0.747564i $$-0.268777\pi$$
0.664190 + 0.747564i $$0.268777\pi$$
$$420$$ −2.12381 −0.103631
$$421$$ 6.93651 0.338065 0.169032 0.985611i $$-0.445936\pi$$
0.169032 + 0.985611i $$0.445936\pi$$
$$422$$ −20.2256 −0.984565
$$423$$ 11.0505 0.537293
$$424$$ 58.2381 2.82829
$$425$$ −27.9316 −1.35488
$$426$$ −21.1427 −1.02437
$$427$$ 3.58414 0.173449
$$428$$ 35.8139 1.73113
$$429$$ 9.93642 0.479735
$$430$$ −14.3847 −0.693694
$$431$$ 14.5160 0.699210 0.349605 0.936897i $$-0.386316\pi$$
0.349605 + 0.936897i $$0.386316\pi$$
$$432$$ 85.0893 4.09386
$$433$$ −14.6779 −0.705373 −0.352686 0.935742i $$-0.614732\pi$$
−0.352686 + 0.935742i $$0.614732\pi$$
$$434$$ −10.0431 −0.482083
$$435$$ −0.220558 −0.0105750
$$436$$ −7.00195 −0.335333
$$437$$ −18.0820 −0.864978
$$438$$ −32.9003 −1.57204
$$439$$ 30.1764 1.44024 0.720121 0.693849i $$-0.244086\pi$$
0.720121 + 0.693849i $$0.244086\pi$$
$$440$$ 43.8420 2.09009
$$441$$ 14.0940 0.671144
$$442$$ −45.1477 −2.14746
$$443$$ 26.7445 1.27067 0.635335 0.772237i $$-0.280862\pi$$
0.635335 + 0.772237i $$0.280862\pi$$
$$444$$ 34.6176 1.64288
$$445$$ −9.27633 −0.439740
$$446$$ 28.7425 1.36100
$$447$$ −12.0536 −0.570113
$$448$$ 16.3701 0.773416
$$449$$ −17.2811 −0.815546 −0.407773 0.913083i $$-0.633695\pi$$
−0.407773 + 0.913083i $$0.633695\pi$$
$$450$$ −23.1865 −1.09302
$$451$$ −52.0430 −2.45061
$$452$$ −15.9467 −0.750069
$$453$$ 0.827015 0.0388565
$$454$$ 65.7598 3.08626
$$455$$ 0.900736 0.0422272
$$456$$ 67.3179 3.15245
$$457$$ −14.4572 −0.676278 −0.338139 0.941096i $$-0.609797\pi$$
−0.338139 + 0.941096i $$0.609797\pi$$
$$458$$ −3.65456 −0.170766
$$459$$ 33.8334 1.57921
$$460$$ −15.0503 −0.701723
$$461$$ −38.8363 −1.80879 −0.904394 0.426698i $$-0.859677\pi$$
−0.904394 + 0.426698i $$0.859677\pi$$
$$462$$ −4.60207 −0.214108
$$463$$ 6.46308 0.300365 0.150182 0.988658i $$-0.452014\pi$$
0.150182 + 0.988658i $$0.452014\pi$$
$$464$$ 4.04759 0.187905
$$465$$ −8.69462 −0.403203
$$466$$ 28.3860 1.31496
$$467$$ −17.3404 −0.802420 −0.401210 0.915986i $$-0.631410\pi$$
−0.401210 + 0.915986i $$0.631410\pi$$
$$468$$ −27.7764 −1.28396
$$469$$ −0.970782 −0.0448265
$$470$$ 14.5284 0.670143
$$471$$ 1.00213 0.0461759
$$472$$ 73.9716 3.40482
$$473$$ −23.1015 −1.06221
$$474$$ −25.8116 −1.18557
$$475$$ −27.1439 −1.24545
$$476$$ 15.4975 0.710325
$$477$$ 11.5760 0.530029
$$478$$ −78.7544 −3.60214
$$479$$ 1.84876 0.0844720 0.0422360 0.999108i $$-0.486552\pi$$
0.0422360 + 0.999108i $$0.486552\pi$$
$$480$$ 25.9573 1.18478
$$481$$ −14.6818 −0.669432
$$482$$ −37.7987 −1.72168
$$483$$ 1.02804 0.0467772
$$484$$ 45.2116 2.05507
$$485$$ 15.0756 0.684549
$$486$$ 44.7412 2.02950
$$487$$ −6.00067 −0.271916 −0.135958 0.990715i $$-0.543411\pi$$
−0.135958 + 0.990715i $$0.543411\pi$$
$$488$$ −94.5582 −4.28045
$$489$$ −2.46996 −0.111695
$$490$$ 18.5298 0.837089
$$491$$ −30.6878 −1.38492 −0.692461 0.721456i $$-0.743473\pi$$
−0.692461 + 0.721456i $$0.743473\pi$$
$$492$$ −66.5122 −2.99860
$$493$$ 1.60941 0.0724844
$$494$$ −43.8744 −1.97400
$$495$$ 8.71450 0.391688
$$496$$ 159.560 7.16446
$$497$$ −3.07798 −0.138066
$$498$$ 37.3946 1.67569
$$499$$ −0.291563 −0.0130521 −0.00652607 0.999979i $$-0.502077\pi$$
−0.00652607 + 0.999979i $$0.502077\pi$$
$$500$$ −50.4730 −2.25722
$$501$$ 4.90497 0.219138
$$502$$ −8.07754 −0.360518
$$503$$ −9.25364 −0.412599 −0.206300 0.978489i $$-0.566142\pi$$
−0.206300 + 0.978489i $$0.566142\pi$$
$$504$$ 8.37143 0.372893
$$505$$ 2.38298 0.106041
$$506$$ −32.6123 −1.44979
$$507$$ 7.22646 0.320938
$$508$$ −43.8661 −1.94624
$$509$$ 15.4761 0.685966 0.342983 0.939342i $$-0.388563\pi$$
0.342983 + 0.939342i $$0.388563\pi$$
$$510$$ 18.1027 0.801600
$$511$$ −4.78968 −0.211883
$$512$$ −117.216 −5.18029
$$513$$ 32.8792 1.45165
$$514$$ −73.4017 −3.23761
$$515$$ 16.8388 0.742007
$$516$$ −29.5243 −1.29974
$$517$$ 23.3322 1.02615
$$518$$ 6.79990 0.298770
$$519$$ 9.99126 0.438568
$$520$$ −23.7636 −1.04210
$$521$$ −6.10225 −0.267344 −0.133672 0.991026i $$-0.542677\pi$$
−0.133672 + 0.991026i $$0.542677\pi$$
$$522$$ 1.33600 0.0584751
$$523$$ 17.3036 0.756633 0.378317 0.925676i $$-0.376503\pi$$
0.378317 + 0.925676i $$0.376503\pi$$
$$524$$ 25.3638 1.10802
$$525$$ 1.54324 0.0673526
$$526$$ −9.07033 −0.395485
$$527$$ 63.4447 2.76369
$$528$$ 73.1156 3.18195
$$529$$ −15.7149 −0.683255
$$530$$ 15.2193 0.661084
$$531$$ 14.7034 0.638073
$$532$$ 15.0604 0.652950
$$533$$ 28.2087 1.22185
$$534$$ −25.6893 −1.11169
$$535$$ 6.09032 0.263308
$$536$$ 25.6115 1.10625
$$537$$ −3.89674 −0.168157
$$538$$ −75.6239 −3.26038
$$539$$ 29.7583 1.28178
$$540$$ 27.3666 1.17767
$$541$$ 21.6632 0.931373 0.465687 0.884950i $$-0.345807\pi$$
0.465687 + 0.884950i $$0.345807\pi$$
$$542$$ −90.2311 −3.87576
$$543$$ 8.26183 0.354549
$$544$$ −189.410 −8.12090
$$545$$ −1.19071 −0.0510046
$$546$$ 2.49445 0.106752
$$547$$ 14.8153 0.633456 0.316728 0.948516i $$-0.397416\pi$$
0.316728 + 0.948516i $$0.397416\pi$$
$$548$$ −121.113 −5.17369
$$549$$ −18.7954 −0.802168
$$550$$ −48.9562 −2.08750
$$551$$ 1.56402 0.0666296
$$552$$ −27.1220 −1.15439
$$553$$ −3.75769 −0.159793
$$554$$ 83.9262 3.56568
$$555$$ 5.88689 0.249884
$$556$$ 61.1933 2.59517
$$557$$ −33.5218 −1.42036 −0.710181 0.704019i $$-0.751387\pi$$
−0.710181 + 0.704019i $$0.751387\pi$$
$$558$$ 52.6663 2.22954
$$559$$ 12.5217 0.529610
$$560$$ 6.62793 0.280081
$$561$$ 29.0724 1.22744
$$562$$ −19.3610 −0.816695
$$563$$ −27.3604 −1.15310 −0.576552 0.817061i $$-0.695602\pi$$
−0.576552 + 0.817061i $$0.695602\pi$$
$$564$$ 29.8191 1.25561
$$565$$ −2.71181 −0.114087
$$566$$ −28.8045 −1.21075
$$567$$ 0.555430 0.0233258
$$568$$ 81.2044 3.40726
$$569$$ 14.8151 0.621081 0.310541 0.950560i $$-0.399490\pi$$
0.310541 + 0.950560i $$0.399490\pi$$
$$570$$ 17.5921 0.736853
$$571$$ 0.893242 0.0373810 0.0186905 0.999825i $$-0.494050\pi$$
0.0186905 + 0.999825i $$0.494050\pi$$
$$572$$ −58.6474 −2.45217
$$573$$ −24.5445 −1.02536
$$574$$ −13.0649 −0.545318
$$575$$ 10.9361 0.456068
$$576$$ −85.8457 −3.57690
$$577$$ −29.1151 −1.21208 −0.606039 0.795435i $$-0.707243\pi$$
−0.606039 + 0.795435i $$0.707243\pi$$
$$578$$ −84.8419 −3.52896
$$579$$ −0.176220 −0.00732346
$$580$$ 1.30179 0.0540541
$$581$$ 5.44395 0.225853
$$582$$ 41.7496 1.73058
$$583$$ 24.4418 1.01227
$$584$$ 126.363 5.22894
$$585$$ −4.72350 −0.195293
$$586$$ −43.9133 −1.81404
$$587$$ −0.453052 −0.0186995 −0.00934974 0.999956i $$-0.502976\pi$$
−0.00934974 + 0.999956i $$0.502976\pi$$
$$588$$ 38.0319 1.56841
$$589$$ 61.6553 2.54046
$$590$$ 19.3309 0.795841
$$591$$ −12.8157 −0.527166
$$592$$ −108.034 −4.44016
$$593$$ 36.1971 1.48644 0.743219 0.669048i $$-0.233298\pi$$
0.743219 + 0.669048i $$0.233298\pi$$
$$594$$ 59.3004 2.43312
$$595$$ 2.63541 0.108041
$$596$$ 71.1433 2.91414
$$597$$ −10.0400 −0.410910
$$598$$ 17.6768 0.722857
$$599$$ 41.1469 1.68122 0.840609 0.541642i $$-0.182197\pi$$
0.840609 + 0.541642i $$0.182197\pi$$
$$600$$ −40.7144 −1.66216
$$601$$ −5.59294 −0.228141 −0.114070 0.993473i $$-0.536389\pi$$
−0.114070 + 0.993473i $$0.536389\pi$$
$$602$$ −5.79942 −0.236367
$$603$$ 5.09082 0.207314
$$604$$ −4.88126 −0.198616
$$605$$ 7.68844 0.312579
$$606$$ 6.59928 0.268078
$$607$$ 22.5401 0.914874 0.457437 0.889242i $$-0.348768\pi$$
0.457437 + 0.889242i $$0.348768\pi$$
$$608$$ −184.068 −7.46495
$$609$$ −0.0889213 −0.00360327
$$610$$ −24.7108 −1.00051
$$611$$ −12.6467 −0.511629
$$612$$ −81.2693 −3.28512
$$613$$ 1.19664 0.0483320 0.0241660 0.999708i $$-0.492307\pi$$
0.0241660 + 0.999708i $$0.492307\pi$$
$$614$$ 74.4073 3.00283
$$615$$ −11.3107 −0.456092
$$616$$ 17.6756 0.712168
$$617$$ −27.6002 −1.11114 −0.555571 0.831469i $$-0.687500\pi$$
−0.555571 + 0.831469i $$0.687500\pi$$
$$618$$ 46.6324 1.87583
$$619$$ −16.6919 −0.670905 −0.335453 0.942057i $$-0.608889\pi$$
−0.335453 + 0.942057i $$0.608889\pi$$
$$620$$ 51.3180 2.06098
$$621$$ −13.2468 −0.531578
$$622$$ 13.3429 0.535002
$$623$$ −3.73989 −0.149836
$$624$$ −39.6306 −1.58650
$$625$$ 11.6757 0.467027
$$626$$ 34.1963 1.36676
$$627$$ 28.2525 1.12830
$$628$$ −5.91486 −0.236029
$$629$$ −42.9566 −1.71279
$$630$$ 2.18770 0.0871599
$$631$$ −1.49618 −0.0595620 −0.0297810 0.999556i $$-0.509481\pi$$
−0.0297810 + 0.999556i $$0.509481\pi$$
$$632$$ 99.1369 3.94345
$$633$$ −7.05938 −0.280585
$$634$$ −73.0783 −2.90231
$$635$$ −7.45963 −0.296026
$$636$$ 31.2372 1.23864
$$637$$ −16.1298 −0.639086
$$638$$ 2.82085 0.111678
$$639$$ 16.1411 0.638530
$$640$$ −59.3530 −2.34613
$$641$$ 18.1503 0.716893 0.358447 0.933550i $$-0.383306\pi$$
0.358447 + 0.933550i $$0.383306\pi$$
$$642$$ 16.8662 0.665655
$$643$$ 49.5946 1.95582 0.977909 0.209030i $$-0.0670305\pi$$
0.977909 + 0.209030i $$0.0670305\pi$$
$$644$$ −6.06774 −0.239103
$$645$$ −5.02075 −0.197692
$$646$$ −128.370 −5.05064
$$647$$ 34.1541 1.34274 0.671368 0.741124i $$-0.265707\pi$$
0.671368 + 0.741124i $$0.265707\pi$$
$$648$$ −14.6536 −0.575646
$$649$$ 31.0449 1.21862
$$650$$ 26.5356 1.04081
$$651$$ −3.50536 −0.137386
$$652$$ 14.5784 0.570933
$$653$$ −35.1164 −1.37421 −0.687105 0.726558i $$-0.741119\pi$$
−0.687105 + 0.726558i $$0.741119\pi$$
$$654$$ −3.29749 −0.128942
$$655$$ 4.31322 0.168532
$$656$$ 207.569 8.10422
$$657$$ 25.1173 0.979918
$$658$$ 5.85732 0.228342
$$659$$ 11.6728 0.454707 0.227354 0.973812i $$-0.426993\pi$$
0.227354 + 0.973812i $$0.426993\pi$$
$$660$$ 23.5156 0.915342
$$661$$ −11.0462 −0.429645 −0.214823 0.976653i $$-0.568917\pi$$
−0.214823 + 0.976653i $$0.568917\pi$$
$$662$$ −90.7676 −3.52778
$$663$$ −15.7580 −0.611991
$$664$$ −143.624 −5.57371
$$665$$ 2.56109 0.0993146
$$666$$ −35.6589 −1.38176
$$667$$ −0.630137 −0.0243990
$$668$$ −28.9505 −1.12013
$$669$$ 10.0321 0.387863
$$670$$ 6.69303 0.258574
$$671$$ −39.6849 −1.53202
$$672$$ 10.4651 0.403698
$$673$$ 45.7913 1.76512 0.882562 0.470196i $$-0.155817\pi$$
0.882562 + 0.470196i $$0.155817\pi$$
$$674$$ −7.97455 −0.307168
$$675$$ −19.8856 −0.765397
$$676$$ −42.6525 −1.64048
$$677$$ −26.8492 −1.03190 −0.515949 0.856619i $$-0.672561\pi$$
−0.515949 + 0.856619i $$0.672561\pi$$
$$678$$ −7.50992 −0.288417
$$679$$ 6.07797 0.233251
$$680$$ −69.5284 −2.66629
$$681$$ 22.9523 0.879534
$$682$$ 111.200 4.25809
$$683$$ 38.5149 1.47373 0.736867 0.676038i $$-0.236304\pi$$
0.736867 + 0.676038i $$0.236304\pi$$
$$684$$ −78.9772 −3.01977
$$685$$ −20.5958 −0.786926
$$686$$ 15.1093 0.576875
$$687$$ −1.27556 −0.0486657
$$688$$ 92.1387 3.51275
$$689$$ −13.2481 −0.504713
$$690$$ −7.08777 −0.269827
$$691$$ 37.1327 1.41259 0.706297 0.707916i $$-0.250364\pi$$
0.706297 + 0.707916i $$0.250364\pi$$
$$692$$ −58.9711 −2.24175
$$693$$ 3.51338 0.133462
$$694$$ 5.73336 0.217636
$$695$$ 10.4062 0.394730
$$696$$ 2.34596 0.0889232
$$697$$ 82.5342 3.12621
$$698$$ 23.3695 0.884550
$$699$$ 9.90765 0.374742
$$700$$ −9.10864 −0.344274
$$701$$ −1.35551 −0.0511969 −0.0255984 0.999672i $$-0.508149\pi$$
−0.0255984 + 0.999672i $$0.508149\pi$$
$$702$$ −32.1424 −1.21314
$$703$$ −41.7451 −1.57445
$$704$$ −181.256 −6.83134
$$705$$ 5.07087 0.190980
$$706$$ −12.4099 −0.467053
$$707$$ 0.960733 0.0361321
$$708$$ 39.6762 1.49112
$$709$$ −47.2669 −1.77515 −0.887573 0.460667i $$-0.847610\pi$$
−0.887573 + 0.460667i $$0.847610\pi$$
$$710$$ 21.2211 0.796412
$$711$$ 19.7055 0.739014
$$712$$ 98.6672 3.69771
$$713$$ −24.8406 −0.930287
$$714$$ 7.29836 0.273134
$$715$$ −9.97326 −0.372979
$$716$$ 22.9996 0.859536
$$717$$ −27.4878 −1.02655
$$718$$ −1.25842 −0.0469638
$$719$$ 17.9037 0.667695 0.333847 0.942627i $$-0.391653\pi$$
0.333847 + 0.942627i $$0.391653\pi$$
$$720$$ −34.7571 −1.29532
$$721$$ 6.78882 0.252829
$$722$$ −71.9367 −2.67721
$$723$$ −13.1930 −0.490652
$$724$$ −48.7636 −1.81228
$$725$$ −0.945934 −0.0351311
$$726$$ 21.2919 0.790217
$$727$$ −22.0310 −0.817085 −0.408542 0.912739i $$-0.633963\pi$$
−0.408542 + 0.912739i $$0.633963\pi$$
$$728$$ −9.58063 −0.355082
$$729$$ 11.3718 0.421178
$$730$$ 33.0223 1.22221
$$731$$ 36.6364 1.35505
$$732$$ −50.7182 −1.87460
$$733$$ 40.7096 1.50364 0.751821 0.659367i $$-0.229176\pi$$
0.751821 + 0.659367i $$0.229176\pi$$
$$734$$ −49.6658 −1.83320
$$735$$ 6.46749 0.238557
$$736$$ 74.1601 2.73358
$$737$$ 10.7488 0.395938
$$738$$ 68.5129 2.52199
$$739$$ −0.0849355 −0.00312440 −0.00156220 0.999999i $$-0.500497\pi$$
−0.00156220 + 0.999999i $$0.500497\pi$$
$$740$$ −34.7460 −1.27729
$$741$$ −15.3136 −0.562559
$$742$$ 6.13588 0.225255
$$743$$ 16.1883 0.593891 0.296946 0.954894i $$-0.404032\pi$$
0.296946 + 0.954894i $$0.404032\pi$$
$$744$$ 92.4798 3.39048
$$745$$ 12.0982 0.443245
$$746$$ 24.8268 0.908975
$$747$$ −28.5483 −1.04453
$$748$$ −171.593 −6.27407
$$749$$ 2.45540 0.0897185
$$750$$ −23.7697 −0.867948
$$751$$ 7.48195 0.273020 0.136510 0.990639i $$-0.456411\pi$$
0.136510 + 0.990639i $$0.456411\pi$$
$$752$$ −93.0585 −3.39350
$$753$$ −2.81932 −0.102742
$$754$$ −1.52898 −0.0556820
$$755$$ −0.830081 −0.0302097
$$756$$ 11.0332 0.401275
$$757$$ 25.0984 0.912216 0.456108 0.889924i $$-0.349243\pi$$
0.456108 + 0.889924i $$0.349243\pi$$
$$758$$ 37.8652 1.37533
$$759$$ −11.3828 −0.413168
$$760$$ −67.5675 −2.45093
$$761$$ 12.8860 0.467117 0.233558 0.972343i $$-0.424963\pi$$
0.233558 + 0.972343i $$0.424963\pi$$
$$762$$ −20.6583 −0.748370
$$763$$ −0.480054 −0.0173791
$$764$$ 144.868 5.24114
$$765$$ −13.8202 −0.499671
$$766$$ 34.4278 1.24393
$$767$$ −16.8272 −0.607595
$$768$$ −83.4606 −3.01163
$$769$$ −7.68380 −0.277085 −0.138542 0.990357i $$-0.544242\pi$$
−0.138542 + 0.990357i $$0.544242\pi$$
$$770$$ 4.61913 0.166462
$$771$$ −25.6196 −0.922667
$$772$$ 1.04010 0.0374340
$$773$$ 9.86788 0.354923 0.177461 0.984128i $$-0.443211\pi$$
0.177461 + 0.984128i $$0.443211\pi$$
$$774$$ 30.4124 1.09315
$$775$$ −37.2896 −1.33948
$$776$$ −160.351 −5.75627
$$777$$ 2.37339 0.0851447
$$778$$ 10.0995 0.362085
$$779$$ 80.2065 2.87370
$$780$$ −12.7461 −0.456383
$$781$$ 34.0805 1.21949
$$782$$ 51.7195 1.84948
$$783$$ 1.14580 0.0409477
$$784$$ −118.689 −4.23888
$$785$$ −1.00585 −0.0359003
$$786$$ 11.9448 0.426057
$$787$$ 52.9736 1.88830 0.944152 0.329510i $$-0.106884\pi$$
0.944152 + 0.329510i $$0.106884\pi$$
$$788$$ 75.6416 2.69462
$$789$$ −3.16584 −0.112707
$$790$$ 25.9073 0.921741
$$791$$ −1.09331 −0.0388735
$$792$$ −92.6913 −3.29364
$$793$$ 21.5103 0.763852
$$794$$ −38.7416 −1.37489
$$795$$ 5.31203 0.188398
$$796$$ 59.2588 2.10037
$$797$$ 37.0332 1.31178 0.655891 0.754856i $$-0.272293\pi$$
0.655891 + 0.754856i $$0.272293\pi$$
$$798$$ 7.09252 0.251073
$$799$$ −37.0022 −1.30904
$$800$$ 111.326 3.93597
$$801$$ 19.6121 0.692961
$$802$$ −11.3862 −0.402062
$$803$$ 53.0330 1.87149
$$804$$ 13.7373 0.484477
$$805$$ −1.03185 −0.0363678
$$806$$ −60.2737 −2.12305
$$807$$ −26.3952 −0.929156
$$808$$ −25.3464 −0.891684
$$809$$ 2.80512 0.0986228 0.0493114 0.998783i $$-0.484297\pi$$
0.0493114 + 0.998783i $$0.484297\pi$$
$$810$$ −3.82940 −0.134551
$$811$$ 6.48967 0.227883 0.113942 0.993487i $$-0.463652\pi$$
0.113942 + 0.993487i $$0.463652\pi$$
$$812$$ 0.524838 0.0184182
$$813$$ −31.4936 −1.10453
$$814$$ −75.2908 −2.63894
$$815$$ 2.47912 0.0868397
$$816$$ −115.953 −4.05917
$$817$$ 35.6031 1.24560
$$818$$ −110.213 −3.85350
$$819$$ −1.90435 −0.0665433
$$820$$ 66.7588 2.33132
$$821$$ −6.85515 −0.239246 −0.119623 0.992819i $$-0.538169\pi$$
−0.119623 + 0.992819i $$0.538169\pi$$
$$822$$ −57.0369 −1.98939
$$823$$ −11.3055 −0.394084 −0.197042 0.980395i $$-0.563133\pi$$
−0.197042 + 0.980395i $$0.563133\pi$$
$$824$$ −179.105 −6.23942
$$825$$ −17.0873 −0.594904
$$826$$ 7.79355 0.271172
$$827$$ 30.4961 1.06045 0.530226 0.847856i $$-0.322107\pi$$
0.530226 + 0.847856i $$0.322107\pi$$
$$828$$ 31.8195 1.10580
$$829$$ 46.5293 1.61603 0.808015 0.589162i $$-0.200542\pi$$
0.808015 + 0.589162i $$0.200542\pi$$
$$830$$ −37.5332 −1.30280
$$831$$ 29.2930 1.01616
$$832$$ 98.2456 3.40605
$$833$$ −47.1933 −1.63515
$$834$$ 28.8183 0.997897
$$835$$ −4.92316 −0.170373
$$836$$ −166.754 −5.76730
$$837$$ 45.1687 1.56126
$$838$$ −75.5811 −2.61090
$$839$$ −33.6644 −1.16222 −0.581112 0.813824i $$-0.697382\pi$$
−0.581112 + 0.813824i $$0.697382\pi$$
$$840$$ 3.84150 0.132544
$$841$$ −28.9455 −0.998121
$$842$$ −19.2808 −0.664460
$$843$$ −6.75763 −0.232745
$$844$$ 41.6664 1.43422
$$845$$ −7.25326 −0.249520
$$846$$ −30.7161 −1.05604
$$847$$ 3.09971 0.106507
$$848$$ −97.4841 −3.34762
$$849$$ −10.0537 −0.345043
$$850$$ 77.6390 2.66300
$$851$$ 16.8189 0.576544
$$852$$ 43.5557 1.49219
$$853$$ −24.3110 −0.832393 −0.416197 0.909275i $$-0.636637\pi$$
−0.416197 + 0.909275i $$0.636637\pi$$
$$854$$ −9.96251 −0.340910
$$855$$ −13.4304 −0.459311
$$856$$ −64.7794 −2.21411
$$857$$ −6.58629 −0.224983 −0.112492 0.993653i $$-0.535883\pi$$
−0.112492 + 0.993653i $$0.535883\pi$$
$$858$$ −27.6194 −0.942910
$$859$$ −16.8816 −0.575994 −0.287997 0.957631i $$-0.592989\pi$$
−0.287997 + 0.957631i $$0.592989\pi$$
$$860$$ 29.6338 1.01050
$$861$$ −4.56008 −0.155407
$$862$$ −40.3487 −1.37428
$$863$$ 51.6720 1.75894 0.879468 0.475957i $$-0.157898\pi$$
0.879468 + 0.475957i $$0.157898\pi$$
$$864$$ −134.848 −4.58764
$$865$$ −10.0283 −0.340973
$$866$$ 40.7987 1.38640
$$867$$ −29.6126 −1.00570
$$868$$ 20.6896 0.702251
$$869$$ 41.6065 1.41140
$$870$$ 0.613066 0.0207849
$$871$$ −5.82616 −0.197412
$$872$$ 12.6650 0.428890
$$873$$ −31.8731 −1.07874
$$874$$ 50.2608 1.70010
$$875$$ −3.46043 −0.116984
$$876$$ 67.7775 2.28999
$$877$$ −3.24977 −0.109737 −0.0548685 0.998494i $$-0.517474\pi$$
−0.0548685 + 0.998494i $$0.517474\pi$$
$$878$$ −83.8786 −2.83077
$$879$$ −15.3272 −0.516973
$$880$$ −73.3867 −2.47386
$$881$$ 36.0616 1.21495 0.607473 0.794341i $$-0.292183\pi$$
0.607473 + 0.794341i $$0.292183\pi$$
$$882$$ −39.1759 −1.31912
$$883$$ −49.1387 −1.65365 −0.826825 0.562459i $$-0.809855\pi$$
−0.826825 + 0.562459i $$0.809855\pi$$
$$884$$ 93.0082 3.12820
$$885$$ 6.74712 0.226802
$$886$$ −74.3393 −2.49748
$$887$$ 40.7917 1.36965 0.684825 0.728707i $$-0.259878\pi$$
0.684825 + 0.728707i $$0.259878\pi$$
$$888$$ −62.6156 −2.10124
$$889$$ −3.00746 −0.100867
$$890$$ 25.7846 0.864301
$$891$$ −6.14991 −0.206030
$$892$$ −59.2120 −1.98257
$$893$$ −35.9586 −1.20331
$$894$$ 33.5042 1.12055
$$895$$ 3.91119 0.130737
$$896$$ −23.9290 −0.799413
$$897$$ 6.16977 0.206003
$$898$$ 48.0347 1.60294
$$899$$ 2.14862 0.0716605
$$900$$ 47.7661 1.59220
$$901$$ −38.7619 −1.29135
$$902$$ 144.659 4.81662
$$903$$ −2.02419 −0.0673608
$$904$$ 28.8440 0.959337
$$905$$ −8.29246 −0.275651
$$906$$ −2.29878 −0.0763718
$$907$$ −11.2754 −0.374393 −0.187196 0.982323i $$-0.559940\pi$$
−0.187196 + 0.982323i $$0.559940\pi$$
$$908$$ −135.471 −4.49575
$$909$$ −5.03813 −0.167104
$$910$$ −2.50369 −0.0829967
$$911$$ 16.8216 0.557324 0.278662 0.960389i $$-0.410109\pi$$
0.278662 + 0.960389i $$0.410109\pi$$
$$912$$ −112.683 −3.73130
$$913$$ −60.2773 −1.99489
$$914$$ 40.1853 1.32921
$$915$$ −8.62487 −0.285129
$$916$$ 7.52870 0.248755
$$917$$ 1.73894 0.0574248
$$918$$ −94.0437 −3.10390
$$919$$ 10.9219 0.360281 0.180141 0.983641i $$-0.442345\pi$$
0.180141 + 0.983641i $$0.442345\pi$$
$$920$$ 27.2226 0.897502
$$921$$ 25.9706 0.855760
$$922$$ 107.950 3.55514
$$923$$ −18.4725 −0.608031
$$924$$ 9.48065 0.311890
$$925$$ 25.2478 0.830142
$$926$$ −17.9649 −0.590362
$$927$$ −35.6008 −1.16929
$$928$$ −6.41458 −0.210569
$$929$$ 2.42455 0.0795468 0.0397734 0.999209i $$-0.487336\pi$$
0.0397734 + 0.999209i $$0.487336\pi$$
$$930$$ 24.1676 0.792488
$$931$$ −45.8623 −1.50308
$$932$$ −58.4776 −1.91550
$$933$$ 4.65711 0.152467
$$934$$ 48.1996 1.57714
$$935$$ −29.1802 −0.954295
$$936$$ 50.2412 1.64219
$$937$$ −20.3532 −0.664910 −0.332455 0.943119i $$-0.607877\pi$$
−0.332455 + 0.943119i $$0.607877\pi$$
$$938$$ 2.69840 0.0881057
$$939$$ 11.9356 0.389505
$$940$$ −29.9297 −0.976198
$$941$$ −43.4342 −1.41592 −0.707958 0.706255i $$-0.750383\pi$$
−0.707958 + 0.706255i $$0.750383\pi$$
$$942$$ −2.78554 −0.0907578
$$943$$ −32.3148 −1.05231
$$944$$ −123.820 −4.03001
$$945$$ 1.87625 0.0610345
$$946$$ 64.2132 2.08775
$$947$$ −44.1484 −1.43463 −0.717315 0.696749i $$-0.754629\pi$$
−0.717315 + 0.696749i $$0.754629\pi$$
$$948$$ 53.1741 1.72701
$$949$$ −28.7453 −0.933112
$$950$$ 75.4494 2.44790
$$951$$ −25.5067 −0.827112
$$952$$ −28.0314 −0.908504
$$953$$ 21.3291 0.690917 0.345458 0.938434i $$-0.387723\pi$$
0.345458 + 0.938434i $$0.387723\pi$$
$$954$$ −32.1768 −1.04176
$$955$$ 24.6355 0.797185
$$956$$ 162.241 5.24724
$$957$$ 0.984568 0.0318266
$$958$$ −5.13883 −0.166028
$$959$$ −8.30351 −0.268134
$$960$$ −39.3931 −1.27141
$$961$$ 53.7007 1.73228
$$962$$ 40.8097 1.31576
$$963$$ −12.8762 −0.414931
$$964$$ 77.8685 2.50798
$$965$$ 0.176874 0.00569376
$$966$$ −2.85754 −0.0919398
$$967$$ −38.9584 −1.25282 −0.626409 0.779495i $$-0.715476\pi$$
−0.626409 + 0.779495i $$0.715476\pi$$
$$968$$ −81.7777 −2.62843
$$969$$ −44.8052 −1.43935
$$970$$ −41.9044 −1.34547
$$971$$ −45.1226 −1.44805 −0.724027 0.689771i $$-0.757711\pi$$
−0.724027 + 0.689771i $$0.757711\pi$$
$$972$$ −92.1707 −2.95638
$$973$$ 4.19541 0.134499
$$974$$ 16.6795 0.534446
$$975$$ 9.26179 0.296615
$$976$$ 158.280 5.06642
$$977$$ −49.0844 −1.57035 −0.785174 0.619275i $$-0.787426\pi$$
−0.785174 + 0.619275i $$0.787426\pi$$
$$978$$ 6.86552 0.219535
$$979$$ 41.4094 1.32345
$$980$$ −38.1729 −1.21939
$$981$$ 2.51742 0.0803751
$$982$$ 85.3001 2.72204
$$983$$ 11.7156 0.373669 0.186834 0.982391i $$-0.440177\pi$$
0.186834 + 0.982391i $$0.440177\pi$$
$$984$$ 120.306 3.83520
$$985$$ 12.8632 0.409855
$$986$$ −4.47354 −0.142467
$$987$$ 2.04440 0.0650739
$$988$$ 90.3850 2.87553
$$989$$ −14.3443 −0.456123
$$990$$ −24.2229 −0.769855
$$991$$ −13.7963 −0.438253 −0.219127 0.975696i $$-0.570321\pi$$
−0.219127 + 0.975696i $$0.570321\pi$$
$$992$$ −252.869 −8.02859
$$993$$ −31.6809 −1.00536
$$994$$ 8.55558 0.271367
$$995$$ 10.0772 0.319470
$$996$$ −77.0360 −2.44098
$$997$$ −14.6007 −0.462409 −0.231204 0.972905i $$-0.574267\pi$$
−0.231204 + 0.972905i $$0.574267\pi$$
$$998$$ 0.810431 0.0256537
$$999$$ −30.5825 −0.967587
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.1.3 184

By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.3 184 1.1 even 1 trivial