# Properties

 Label 3762.2.a.y.1.2 Level $3762$ Weight $2$ Character 3762.1 Self dual yes Analytic conductor $30.040$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(1,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3762.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: $$30.0397212404$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 3762.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.56155 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.56155 q^{7} +1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -3.56155 q^{13} +3.56155 q^{14} +1.00000 q^{16} -3.56155 q^{17} -1.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -5.56155 q^{23} -1.00000 q^{25} -3.56155 q^{26} +3.56155 q^{28} -6.68466 q^{29} +2.00000 q^{31} +1.00000 q^{32} -3.56155 q^{34} -7.12311 q^{35} +3.12311 q^{37} -1.00000 q^{38} -2.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} -5.56155 q^{46} -8.00000 q^{47} +5.68466 q^{49} -1.00000 q^{50} -3.56155 q^{52} +7.80776 q^{53} +2.00000 q^{55} +3.56155 q^{56} -6.68466 q^{58} -4.68466 q^{59} -10.2462 q^{61} +2.00000 q^{62} +1.00000 q^{64} +7.12311 q^{65} +4.68466 q^{67} -3.56155 q^{68} -7.12311 q^{70} -6.00000 q^{71} +2.68466 q^{73} +3.12311 q^{74} -1.00000 q^{76} -3.56155 q^{77} -4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} +2.24621 q^{83} +7.12311 q^{85} -1.00000 q^{88} +9.12311 q^{89} -12.6847 q^{91} -5.56155 q^{92} -8.00000 q^{94} +2.00000 q^{95} +1.12311 q^{97} +5.68466 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 3 q^{7} + 2 q^{8} - 4 q^{10} - 2 q^{11} - 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - 7 q^{23} - 2 q^{25} - 3 q^{26} + 3 q^{28} - q^{29} + 4 q^{31} + 2 q^{32} - 3 q^{34} - 6 q^{35} - 2 q^{37} - 2 q^{38} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 7 q^{46} - 16 q^{47} - q^{49} - 2 q^{50} - 3 q^{52} - 5 q^{53} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 6 q^{65} - 3 q^{67} - 3 q^{68} - 6 q^{70} - 12 q^{71} - 7 q^{73} - 2 q^{74} - 2 q^{76} - 3 q^{77} - 8 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 6 q^{85} - 2 q^{88} + 10 q^{89} - 13 q^{91} - 7 q^{92} - 16 q^{94} + 4 q^{95} - 6 q^{97} - q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 3 * q^7 + 2 * q^8 - 4 * q^10 - 2 * q^11 - 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 - 2 * q^19 - 4 * q^20 - 2 * q^22 - 7 * q^23 - 2 * q^25 - 3 * q^26 + 3 * q^28 - q^29 + 4 * q^31 + 2 * q^32 - 3 * q^34 - 6 * q^35 - 2 * q^37 - 2 * q^38 - 4 * q^40 - 4 * q^41 - 2 * q^44 - 7 * q^46 - 16 * q^47 - q^49 - 2 * q^50 - 3 * q^52 - 5 * q^53 + 4 * q^55 + 3 * q^56 - q^58 + 3 * q^59 - 4 * q^61 + 4 * q^62 + 2 * q^64 + 6 * q^65 - 3 * q^67 - 3 * q^68 - 6 * q^70 - 12 * q^71 - 7 * q^73 - 2 * q^74 - 2 * q^76 - 3 * q^77 - 8 * q^79 - 4 * q^80 - 4 * q^82 - 12 * q^83 + 6 * q^85 - 2 * q^88 + 10 * q^89 - 13 * q^91 - 7 * q^92 - 16 * q^94 + 4 * q^95 - 6 * q^97 - q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ 3.56155 1.34614 0.673070 0.739579i $$-0.264975\pi$$
0.673070 + 0.739579i $$0.264975\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −2.00000 −0.632456
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −3.56155 −0.987797 −0.493899 0.869520i $$-0.664429\pi$$
−0.493899 + 0.869520i $$0.664429\pi$$
$$14$$ 3.56155 0.951865
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.56155 −0.863803 −0.431902 0.901921i $$-0.642157\pi$$
−0.431902 + 0.901921i $$0.642157\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ −5.56155 −1.15966 −0.579832 0.814736i $$-0.696882\pi$$
−0.579832 + 0.814736i $$0.696882\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ −3.56155 −0.698478
$$27$$ 0 0
$$28$$ 3.56155 0.673070
$$29$$ −6.68466 −1.24131 −0.620655 0.784084i $$-0.713133\pi$$
−0.620655 + 0.784084i $$0.713133\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −3.56155 −0.610801
$$35$$ −7.12311 −1.20402
$$36$$ 0 0
$$37$$ 3.12311 0.513435 0.256718 0.966486i $$-0.417359\pi$$
0.256718 + 0.966486i $$0.417359\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ −2.00000 −0.316228
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −5.56155 −0.820006
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 5.68466 0.812094
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −3.56155 −0.493899
$$53$$ 7.80776 1.07248 0.536239 0.844066i $$-0.319844\pi$$
0.536239 + 0.844066i $$0.319844\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 3.56155 0.475933
$$57$$ 0 0
$$58$$ −6.68466 −0.877739
$$59$$ −4.68466 −0.609891 −0.304945 0.952370i $$-0.598638\pi$$
−0.304945 + 0.952370i $$0.598638\pi$$
$$60$$ 0 0
$$61$$ −10.2462 −1.31189 −0.655946 0.754807i $$-0.727730\pi$$
−0.655946 + 0.754807i $$0.727730\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 7.12311 0.883513
$$66$$ 0 0
$$67$$ 4.68466 0.572322 0.286161 0.958182i $$-0.407621\pi$$
0.286161 + 0.958182i $$0.407621\pi$$
$$68$$ −3.56155 −0.431902
$$69$$ 0 0
$$70$$ −7.12311 −0.851374
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 2.68466 0.314216 0.157108 0.987581i $$-0.449783\pi$$
0.157108 + 0.987581i $$0.449783\pi$$
$$74$$ 3.12311 0.363054
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −3.56155 −0.405877
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 0 0
$$82$$ −2.00000 −0.220863
$$83$$ 2.24621 0.246554 0.123277 0.992372i $$-0.460660\pi$$
0.123277 + 0.992372i $$0.460660\pi$$
$$84$$ 0 0
$$85$$ 7.12311 0.772609
$$86$$ 0 0
$$87$$ 0 0
$$88$$ −1.00000 −0.106600
$$89$$ 9.12311 0.967047 0.483524 0.875331i $$-0.339357\pi$$
0.483524 + 0.875331i $$0.339357\pi$$
$$90$$ 0 0
$$91$$ −12.6847 −1.32971
$$92$$ −5.56155 −0.579832
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 1.12311 0.114034 0.0570170 0.998373i $$-0.481841\pi$$
0.0570170 + 0.998373i $$0.481841\pi$$
$$98$$ 5.68466 0.574237
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 15.1231 1.50481 0.752403 0.658703i $$-0.228895\pi$$
0.752403 + 0.658703i $$0.228895\pi$$
$$102$$ 0 0
$$103$$ 11.3693 1.12025 0.560126 0.828407i $$-0.310753\pi$$
0.560126 + 0.828407i $$0.310753\pi$$
$$104$$ −3.56155 −0.349239
$$105$$ 0 0
$$106$$ 7.80776 0.758357
$$107$$ −18.9309 −1.83012 −0.915058 0.403322i $$-0.867855\pi$$
−0.915058 + 0.403322i $$0.867855\pi$$
$$108$$ 0 0
$$109$$ −18.6847 −1.78967 −0.894833 0.446401i $$-0.852705\pi$$
−0.894833 + 0.446401i $$0.852705\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 0 0
$$112$$ 3.56155 0.336535
$$113$$ −5.12311 −0.481941 −0.240971 0.970532i $$-0.577466\pi$$
−0.240971 + 0.970532i $$0.577466\pi$$
$$114$$ 0 0
$$115$$ 11.1231 1.03723
$$116$$ −6.68466 −0.620655
$$117$$ 0 0
$$118$$ −4.68466 −0.431258
$$119$$ −12.6847 −1.16280
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −10.2462 −0.927648
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ −14.2462 −1.26415 −0.632073 0.774909i $$-0.717796\pi$$
−0.632073 + 0.774909i $$0.717796\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 7.12311 0.624738
$$131$$ 7.12311 0.622349 0.311174 0.950353i $$-0.399278\pi$$
0.311174 + 0.950353i $$0.399278\pi$$
$$132$$ 0 0
$$133$$ −3.56155 −0.308826
$$134$$ 4.68466 0.404693
$$135$$ 0 0
$$136$$ −3.56155 −0.305401
$$137$$ −8.43845 −0.720945 −0.360473 0.932770i $$-0.617385\pi$$
−0.360473 + 0.932770i $$0.617385\pi$$
$$138$$ 0 0
$$139$$ −17.3693 −1.47325 −0.736623 0.676303i $$-0.763581\pi$$
−0.736623 + 0.676303i $$0.763581\pi$$
$$140$$ −7.12311 −0.602012
$$141$$ 0 0
$$142$$ −6.00000 −0.503509
$$143$$ 3.56155 0.297832
$$144$$ 0 0
$$145$$ 13.3693 1.11026
$$146$$ 2.68466 0.222184
$$147$$ 0 0
$$148$$ 3.12311 0.256718
$$149$$ 15.1231 1.23893 0.619467 0.785023i $$-0.287349\pi$$
0.619467 + 0.785023i $$0.287349\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ −3.56155 −0.286998
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 18.4924 1.47586 0.737928 0.674879i $$-0.235804\pi$$
0.737928 + 0.674879i $$0.235804\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 0 0
$$160$$ −2.00000 −0.158114
$$161$$ −19.8078 −1.56107
$$162$$ 0 0
$$163$$ −13.3693 −1.04717 −0.523583 0.851975i $$-0.675405\pi$$
−0.523583 + 0.851975i $$0.675405\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 2.24621 0.174340
$$167$$ 25.3693 1.96314 0.981568 0.191111i $$-0.0612092\pi$$
0.981568 + 0.191111i $$0.0612092\pi$$
$$168$$ 0 0
$$169$$ −0.315342 −0.0242570
$$170$$ 7.12311 0.546317
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ −3.56155 −0.269228
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 9.12311 0.683806
$$179$$ −22.2462 −1.66276 −0.831380 0.555704i $$-0.812449\pi$$
−0.831380 + 0.555704i $$0.812449\pi$$
$$180$$ 0 0
$$181$$ −19.1231 −1.42141 −0.710705 0.703491i $$-0.751624\pi$$
−0.710705 + 0.703491i $$0.751624\pi$$
$$182$$ −12.6847 −0.940249
$$183$$ 0 0
$$184$$ −5.56155 −0.410003
$$185$$ −6.24621 −0.459231
$$186$$ 0 0
$$187$$ 3.56155 0.260447
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 2.00000 0.145095
$$191$$ 20.6847 1.49669 0.748345 0.663310i $$-0.230849\pi$$
0.748345 + 0.663310i $$0.230849\pi$$
$$192$$ 0 0
$$193$$ −1.12311 −0.0808429 −0.0404215 0.999183i $$-0.512870\pi$$
−0.0404215 + 0.999183i $$0.512870\pi$$
$$194$$ 1.12311 0.0806343
$$195$$ 0 0
$$196$$ 5.68466 0.406047
$$197$$ −1.75379 −0.124952 −0.0624761 0.998046i $$-0.519900\pi$$
−0.0624761 + 0.998046i $$0.519900\pi$$
$$198$$ 0 0
$$199$$ −0.684658 −0.0485341 −0.0242671 0.999706i $$-0.507725\pi$$
−0.0242671 + 0.999706i $$0.507725\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 15.1231 1.06406
$$203$$ −23.8078 −1.67098
$$204$$ 0 0
$$205$$ 4.00000 0.279372
$$206$$ 11.3693 0.792138
$$207$$ 0 0
$$208$$ −3.56155 −0.246949
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −12.6847 −0.873248 −0.436624 0.899644i $$-0.643826\pi$$
−0.436624 + 0.899644i $$0.643826\pi$$
$$212$$ 7.80776 0.536239
$$213$$ 0 0
$$214$$ −18.9309 −1.29409
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 7.12311 0.483548
$$218$$ −18.6847 −1.26548
$$219$$ 0 0
$$220$$ 2.00000 0.134840
$$221$$ 12.6847 0.853262
$$222$$ 0 0
$$223$$ −24.7386 −1.65662 −0.828311 0.560269i $$-0.810698\pi$$
−0.828311 + 0.560269i $$0.810698\pi$$
$$224$$ 3.56155 0.237966
$$225$$ 0 0
$$226$$ −5.12311 −0.340784
$$227$$ 17.1771 1.14008 0.570041 0.821616i $$-0.306927\pi$$
0.570041 + 0.821616i $$0.306927\pi$$
$$228$$ 0 0
$$229$$ 21.1231 1.39585 0.697927 0.716169i $$-0.254106\pi$$
0.697927 + 0.716169i $$0.254106\pi$$
$$230$$ 11.1231 0.733436
$$231$$ 0 0
$$232$$ −6.68466 −0.438869
$$233$$ −20.7386 −1.35863 −0.679317 0.733845i $$-0.737724\pi$$
−0.679317 + 0.733845i $$0.737724\pi$$
$$234$$ 0 0
$$235$$ 16.0000 1.04372
$$236$$ −4.68466 −0.304945
$$237$$ 0 0
$$238$$ −12.6847 −0.822224
$$239$$ 4.93087 0.318951 0.159476 0.987202i $$-0.449020\pi$$
0.159476 + 0.987202i $$0.449020\pi$$
$$240$$ 0 0
$$241$$ 21.6155 1.39238 0.696189 0.717858i $$-0.254877\pi$$
0.696189 + 0.717858i $$0.254877\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 0 0
$$244$$ −10.2462 −0.655946
$$245$$ −11.3693 −0.726359
$$246$$ 0 0
$$247$$ 3.56155 0.226616
$$248$$ 2.00000 0.127000
$$249$$ 0 0
$$250$$ 12.0000 0.758947
$$251$$ −8.87689 −0.560305 −0.280152 0.959956i $$-0.590385\pi$$
−0.280152 + 0.959956i $$0.590385\pi$$
$$252$$ 0 0
$$253$$ 5.56155 0.349652
$$254$$ −14.2462 −0.893887
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 22.0000 1.37232 0.686161 0.727450i $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ 0 0
$$259$$ 11.1231 0.691156
$$260$$ 7.12311 0.441756
$$261$$ 0 0
$$262$$ 7.12311 0.440067
$$263$$ −17.1231 −1.05586 −0.527928 0.849289i $$-0.677031\pi$$
−0.527928 + 0.849289i $$0.677031\pi$$
$$264$$ 0 0
$$265$$ −15.6155 −0.959254
$$266$$ −3.56155 −0.218373
$$267$$ 0 0
$$268$$ 4.68466 0.286161
$$269$$ −11.1231 −0.678188 −0.339094 0.940753i $$-0.610120\pi$$
−0.339094 + 0.940753i $$0.610120\pi$$
$$270$$ 0 0
$$271$$ 13.3153 0.808849 0.404425 0.914571i $$-0.367472\pi$$
0.404425 + 0.914571i $$0.367472\pi$$
$$272$$ −3.56155 −0.215951
$$273$$ 0 0
$$274$$ −8.43845 −0.509785
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −24.4924 −1.47161 −0.735804 0.677195i $$-0.763195\pi$$
−0.735804 + 0.677195i $$0.763195\pi$$
$$278$$ −17.3693 −1.04174
$$279$$ 0 0
$$280$$ −7.12311 −0.425687
$$281$$ 0.630683 0.0376234 0.0188117 0.999823i $$-0.494012\pi$$
0.0188117 + 0.999823i $$0.494012\pi$$
$$282$$ 0 0
$$283$$ −13.3693 −0.794723 −0.397362 0.917662i $$-0.630074\pi$$
−0.397362 + 0.917662i $$0.630074\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 3.56155 0.210599
$$287$$ −7.12311 −0.420464
$$288$$ 0 0
$$289$$ −4.31534 −0.253844
$$290$$ 13.3693 0.785073
$$291$$ 0 0
$$292$$ 2.68466 0.157108
$$293$$ 24.9309 1.45648 0.728238 0.685324i $$-0.240339\pi$$
0.728238 + 0.685324i $$0.240339\pi$$
$$294$$ 0 0
$$295$$ 9.36932 0.545503
$$296$$ 3.12311 0.181527
$$297$$ 0 0
$$298$$ 15.1231 0.876058
$$299$$ 19.8078 1.14551
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 4.00000 0.230174
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 20.4924 1.17339
$$306$$ 0 0
$$307$$ −5.75379 −0.328386 −0.164193 0.986428i $$-0.552502\pi$$
−0.164193 + 0.986428i $$0.552502\pi$$
$$308$$ −3.56155 −0.202938
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ −15.8078 −0.896376 −0.448188 0.893939i $$-0.647930\pi$$
−0.448188 + 0.893939i $$0.647930\pi$$
$$312$$ 0 0
$$313$$ 3.56155 0.201311 0.100655 0.994921i $$-0.467906\pi$$
0.100655 + 0.994921i $$0.467906\pi$$
$$314$$ 18.4924 1.04359
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ −30.0540 −1.68800 −0.844000 0.536344i $$-0.819805\pi$$
−0.844000 + 0.536344i $$0.819805\pi$$
$$318$$ 0 0
$$319$$ 6.68466 0.374269
$$320$$ −2.00000 −0.111803
$$321$$ 0 0
$$322$$ −19.8078 −1.10384
$$323$$ 3.56155 0.198170
$$324$$ 0 0
$$325$$ 3.56155 0.197559
$$326$$ −13.3693 −0.740458
$$327$$ 0 0
$$328$$ −2.00000 −0.110432
$$329$$ −28.4924 −1.57084
$$330$$ 0 0
$$331$$ −10.4384 −0.573749 −0.286874 0.957968i $$-0.592616\pi$$
−0.286874 + 0.957968i $$0.592616\pi$$
$$332$$ 2.24621 0.123277
$$333$$ 0 0
$$334$$ 25.3693 1.38815
$$335$$ −9.36932 −0.511900
$$336$$ 0 0
$$337$$ 2.87689 0.156714 0.0783572 0.996925i $$-0.475033\pi$$
0.0783572 + 0.996925i $$0.475033\pi$$
$$338$$ −0.315342 −0.0171523
$$339$$ 0 0
$$340$$ 7.12311 0.386305
$$341$$ −2.00000 −0.108306
$$342$$ 0 0
$$343$$ −4.68466 −0.252948
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 31.6155 1.69721 0.848605 0.529027i $$-0.177443\pi$$
0.848605 + 0.529027i $$0.177443\pi$$
$$348$$ 0 0
$$349$$ −19.6155 −1.05000 −0.524998 0.851104i $$-0.675934\pi$$
−0.524998 + 0.851104i $$0.675934\pi$$
$$350$$ −3.56155 −0.190373
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ 23.1771 1.23359 0.616796 0.787123i $$-0.288430\pi$$
0.616796 + 0.787123i $$0.288430\pi$$
$$354$$ 0 0
$$355$$ 12.0000 0.636894
$$356$$ 9.12311 0.483524
$$357$$ 0 0
$$358$$ −22.2462 −1.17575
$$359$$ −12.9309 −0.682465 −0.341233 0.939979i $$-0.610844\pi$$
−0.341233 + 0.939979i $$0.610844\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −19.1231 −1.00509
$$363$$ 0 0
$$364$$ −12.6847 −0.664857
$$365$$ −5.36932 −0.281043
$$366$$ 0 0
$$367$$ 13.7538 0.717942 0.358971 0.933349i $$-0.383128\pi$$
0.358971 + 0.933349i $$0.383128\pi$$
$$368$$ −5.56155 −0.289916
$$369$$ 0 0
$$370$$ −6.24621 −0.324725
$$371$$ 27.8078 1.44371
$$372$$ 0 0
$$373$$ 7.56155 0.391522 0.195761 0.980652i $$-0.437282\pi$$
0.195761 + 0.980652i $$0.437282\pi$$
$$374$$ 3.56155 0.184164
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ 23.8078 1.22616
$$378$$ 0 0
$$379$$ 2.43845 0.125255 0.0626273 0.998037i $$-0.480052\pi$$
0.0626273 + 0.998037i $$0.480052\pi$$
$$380$$ 2.00000 0.102598
$$381$$ 0 0
$$382$$ 20.6847 1.05832
$$383$$ 2.00000 0.102195 0.0510976 0.998694i $$-0.483728\pi$$
0.0510976 + 0.998694i $$0.483728\pi$$
$$384$$ 0 0
$$385$$ 7.12311 0.363027
$$386$$ −1.12311 −0.0571646
$$387$$ 0 0
$$388$$ 1.12311 0.0570170
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 19.8078 1.00172
$$392$$ 5.68466 0.287119
$$393$$ 0 0
$$394$$ −1.75379 −0.0883546
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ −26.9848 −1.35433 −0.677165 0.735831i $$-0.736792\pi$$
−0.677165 + 0.735831i $$0.736792\pi$$
$$398$$ −0.684658 −0.0343188
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 24.2462 1.21080 0.605399 0.795922i $$-0.293014\pi$$
0.605399 + 0.795922i $$0.293014\pi$$
$$402$$ 0 0
$$403$$ −7.12311 −0.354827
$$404$$ 15.1231 0.752403
$$405$$ 0 0
$$406$$ −23.8078 −1.18156
$$407$$ −3.12311 −0.154807
$$408$$ 0 0
$$409$$ −0.246211 −0.0121744 −0.00608718 0.999981i $$-0.501938\pi$$
−0.00608718 + 0.999981i $$0.501938\pi$$
$$410$$ 4.00000 0.197546
$$411$$ 0 0
$$412$$ 11.3693 0.560126
$$413$$ −16.6847 −0.820998
$$414$$ 0 0
$$415$$ −4.49242 −0.220524
$$416$$ −3.56155 −0.174619
$$417$$ 0 0
$$418$$ 1.00000 0.0489116
$$419$$ 23.1231 1.12964 0.564819 0.825215i $$-0.308946\pi$$
0.564819 + 0.825215i $$0.308946\pi$$
$$420$$ 0 0
$$421$$ −1.06913 −0.0521062 −0.0260531 0.999661i $$-0.508294\pi$$
−0.0260531 + 0.999661i $$0.508294\pi$$
$$422$$ −12.6847 −0.617480
$$423$$ 0 0
$$424$$ 7.80776 0.379179
$$425$$ 3.56155 0.172761
$$426$$ 0 0
$$427$$ −36.4924 −1.76599
$$428$$ −18.9309 −0.915058
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 25.8617 1.24572 0.622858 0.782335i $$-0.285971\pi$$
0.622858 + 0.782335i $$0.285971\pi$$
$$432$$ 0 0
$$433$$ 31.3693 1.50751 0.753757 0.657154i $$-0.228240\pi$$
0.753757 + 0.657154i $$0.228240\pi$$
$$434$$ 7.12311 0.341920
$$435$$ 0 0
$$436$$ −18.6847 −0.894833
$$437$$ 5.56155 0.266045
$$438$$ 0 0
$$439$$ 31.6155 1.50893 0.754463 0.656342i $$-0.227897\pi$$
0.754463 + 0.656342i $$0.227897\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ 0 0
$$442$$ 12.6847 0.603348
$$443$$ 17.8617 0.848637 0.424318 0.905513i $$-0.360514\pi$$
0.424318 + 0.905513i $$0.360514\pi$$
$$444$$ 0 0
$$445$$ −18.2462 −0.864953
$$446$$ −24.7386 −1.17141
$$447$$ 0 0
$$448$$ 3.56155 0.168268
$$449$$ 17.6155 0.831328 0.415664 0.909518i $$-0.363549\pi$$
0.415664 + 0.909518i $$0.363549\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ −5.12311 −0.240971
$$453$$ 0 0
$$454$$ 17.1771 0.806160
$$455$$ 25.3693 1.18933
$$456$$ 0 0
$$457$$ 32.4384 1.51741 0.758703 0.651436i $$-0.225833\pi$$
0.758703 + 0.651436i $$0.225833\pi$$
$$458$$ 21.1231 0.987018
$$459$$ 0 0
$$460$$ 11.1231 0.518617
$$461$$ 22.7386 1.05904 0.529522 0.848296i $$-0.322371\pi$$
0.529522 + 0.848296i $$0.322371\pi$$
$$462$$ 0 0
$$463$$ 36.4924 1.69595 0.847973 0.530039i $$-0.177823\pi$$
0.847973 + 0.530039i $$0.177823\pi$$
$$464$$ −6.68466 −0.310327
$$465$$ 0 0
$$466$$ −20.7386 −0.960699
$$467$$ −26.2462 −1.21453 −0.607265 0.794499i $$-0.707733\pi$$
−0.607265 + 0.794499i $$0.707733\pi$$
$$468$$ 0 0
$$469$$ 16.6847 0.770426
$$470$$ 16.0000 0.738025
$$471$$ 0 0
$$472$$ −4.68466 −0.215629
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ −12.6847 −0.581400
$$477$$ 0 0
$$478$$ 4.93087 0.225533
$$479$$ −11.3693 −0.519477 −0.259739 0.965679i $$-0.583636\pi$$
−0.259739 + 0.965679i $$0.583636\pi$$
$$480$$ 0 0
$$481$$ −11.1231 −0.507170
$$482$$ 21.6155 0.984560
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −2.24621 −0.101995
$$486$$ 0 0
$$487$$ 22.8769 1.03665 0.518326 0.855183i $$-0.326556\pi$$
0.518326 + 0.855183i $$0.326556\pi$$
$$488$$ −10.2462 −0.463824
$$489$$ 0 0
$$490$$ −11.3693 −0.513613
$$491$$ 14.6307 0.660273 0.330137 0.943933i $$-0.392905\pi$$
0.330137 + 0.943933i $$0.392905\pi$$
$$492$$ 0 0
$$493$$ 23.8078 1.07225
$$494$$ 3.56155 0.160242
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ −21.3693 −0.958545
$$498$$ 0 0
$$499$$ 26.2462 1.17494 0.587471 0.809245i $$-0.300124\pi$$
0.587471 + 0.809245i $$0.300124\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 0 0
$$502$$ −8.87689 −0.396195
$$503$$ −9.31534 −0.415351 −0.207675 0.978198i $$-0.566590\pi$$
−0.207675 + 0.978198i $$0.566590\pi$$
$$504$$ 0 0
$$505$$ −30.2462 −1.34594
$$506$$ 5.56155 0.247241
$$507$$ 0 0
$$508$$ −14.2462 −0.632073
$$509$$ 6.63068 0.293900 0.146950 0.989144i $$-0.453054\pi$$
0.146950 + 0.989144i $$0.453054\pi$$
$$510$$ 0 0
$$511$$ 9.56155 0.422978
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ −22.7386 −1.00198
$$516$$ 0 0
$$517$$ 8.00000 0.351840
$$518$$ 11.1231 0.488721
$$519$$ 0 0
$$520$$ 7.12311 0.312369
$$521$$ −1.12311 −0.0492042 −0.0246021 0.999697i $$-0.507832\pi$$
−0.0246021 + 0.999697i $$0.507832\pi$$
$$522$$ 0 0
$$523$$ −6.05398 −0.264722 −0.132361 0.991202i $$-0.542256\pi$$
−0.132361 + 0.991202i $$0.542256\pi$$
$$524$$ 7.12311 0.311174
$$525$$ 0 0
$$526$$ −17.1231 −0.746603
$$527$$ −7.12311 −0.310287
$$528$$ 0 0
$$529$$ 7.93087 0.344820
$$530$$ −15.6155 −0.678295
$$531$$ 0 0
$$532$$ −3.56155 −0.154413
$$533$$ 7.12311 0.308536
$$534$$ 0 0
$$535$$ 37.8617 1.63691
$$536$$ 4.68466 0.202346
$$537$$ 0 0
$$538$$ −11.1231 −0.479551
$$539$$ −5.68466 −0.244856
$$540$$ 0 0
$$541$$ 43.2311 1.85865 0.929324 0.369265i $$-0.120391\pi$$
0.929324 + 0.369265i $$0.120391\pi$$
$$542$$ 13.3153 0.571943
$$543$$ 0 0
$$544$$ −3.56155 −0.152700
$$545$$ 37.3693 1.60073
$$546$$ 0 0
$$547$$ 6.73863 0.288123 0.144062 0.989569i $$-0.453984\pi$$
0.144062 + 0.989569i $$0.453984\pi$$
$$548$$ −8.43845 −0.360473
$$549$$ 0 0
$$550$$ 1.00000 0.0426401
$$551$$ 6.68466 0.284776
$$552$$ 0 0
$$553$$ −14.2462 −0.605811
$$554$$ −24.4924 −1.04058
$$555$$ 0 0
$$556$$ −17.3693 −0.736623
$$557$$ −8.87689 −0.376126 −0.188063 0.982157i $$-0.560221\pi$$
−0.188063 + 0.982157i $$0.560221\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −7.12311 −0.301006
$$561$$ 0 0
$$562$$ 0.630683 0.0266038
$$563$$ −6.73863 −0.284000 −0.142000 0.989867i $$-0.545353\pi$$
−0.142000 + 0.989867i $$0.545353\pi$$
$$564$$ 0 0
$$565$$ 10.2462 0.431061
$$566$$ −13.3693 −0.561954
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ −3.36932 −0.141249 −0.0706246 0.997503i $$-0.522499\pi$$
−0.0706246 + 0.997503i $$0.522499\pi$$
$$570$$ 0 0
$$571$$ 35.6155 1.49046 0.745232 0.666806i $$-0.232339\pi$$
0.745232 + 0.666806i $$0.232339\pi$$
$$572$$ 3.56155 0.148916
$$573$$ 0 0
$$574$$ −7.12311 −0.297313
$$575$$ 5.56155 0.231933
$$576$$ 0 0
$$577$$ 3.94602 0.164275 0.0821376 0.996621i $$-0.473825\pi$$
0.0821376 + 0.996621i $$0.473825\pi$$
$$578$$ −4.31534 −0.179495
$$579$$ 0 0
$$580$$ 13.3693 0.555131
$$581$$ 8.00000 0.331896
$$582$$ 0 0
$$583$$ −7.80776 −0.323365
$$584$$ 2.68466 0.111092
$$585$$ 0 0
$$586$$ 24.9309 1.02988
$$587$$ 7.50758 0.309871 0.154935 0.987925i $$-0.450483\pi$$
0.154935 + 0.987925i $$0.450483\pi$$
$$588$$ 0 0
$$589$$ −2.00000 −0.0824086
$$590$$ 9.36932 0.385729
$$591$$ 0 0
$$592$$ 3.12311 0.128359
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 0 0
$$595$$ 25.3693 1.04004
$$596$$ 15.1231 0.619467
$$597$$ 0 0
$$598$$ 19.8078 0.810000
$$599$$ −39.8617 −1.62871 −0.814353 0.580370i $$-0.802908\pi$$
−0.814353 + 0.580370i $$0.802908\pi$$
$$600$$ 0 0
$$601$$ −3.36932 −0.137437 −0.0687187 0.997636i $$-0.521891\pi$$
−0.0687187 + 0.997636i $$0.521891\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 4.00000 0.162758
$$605$$ −2.00000 −0.0813116
$$606$$ 0 0
$$607$$ 27.6155 1.12088 0.560440 0.828195i $$-0.310632\pi$$
0.560440 + 0.828195i $$0.310632\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 20.4924 0.829714
$$611$$ 28.4924 1.15268
$$612$$ 0 0
$$613$$ 1.75379 0.0708349 0.0354174 0.999373i $$-0.488724\pi$$
0.0354174 + 0.999373i $$0.488724\pi$$
$$614$$ −5.75379 −0.232204
$$615$$ 0 0
$$616$$ −3.56155 −0.143499
$$617$$ 4.73863 0.190770 0.0953851 0.995440i $$-0.469592\pi$$
0.0953851 + 0.995440i $$0.469592\pi$$
$$618$$ 0 0
$$619$$ −26.2462 −1.05492 −0.527462 0.849579i $$-0.676856\pi$$
−0.527462 + 0.849579i $$0.676856\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ −15.8078 −0.633834
$$623$$ 32.4924 1.30178
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 3.56155 0.142348
$$627$$ 0 0
$$628$$ 18.4924 0.737928
$$629$$ −11.1231 −0.443507
$$630$$ 0 0
$$631$$ 36.4924 1.45274 0.726370 0.687304i $$-0.241206\pi$$
0.726370 + 0.687304i $$0.241206\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ 0 0
$$634$$ −30.0540 −1.19360
$$635$$ 28.4924 1.13069
$$636$$ 0 0
$$637$$ −20.2462 −0.802184
$$638$$ 6.68466 0.264648
$$639$$ 0 0
$$640$$ −2.00000 −0.0790569
$$641$$ 9.61553 0.379791 0.189895 0.981804i $$-0.439185\pi$$
0.189895 + 0.981804i $$0.439185\pi$$
$$642$$ 0 0
$$643$$ −37.3693 −1.47370 −0.736851 0.676055i $$-0.763688\pi$$
−0.736851 + 0.676055i $$0.763688\pi$$
$$644$$ −19.8078 −0.780535
$$645$$ 0 0
$$646$$ 3.56155 0.140127
$$647$$ −22.5464 −0.886390 −0.443195 0.896425i $$-0.646155\pi$$
−0.443195 + 0.896425i $$0.646155\pi$$
$$648$$ 0 0
$$649$$ 4.68466 0.183889
$$650$$ 3.56155 0.139696
$$651$$ 0 0
$$652$$ −13.3693 −0.523583
$$653$$ −25.6155 −1.00241 −0.501207 0.865328i $$-0.667110\pi$$
−0.501207 + 0.865328i $$0.667110\pi$$
$$654$$ 0 0
$$655$$ −14.2462 −0.556646
$$656$$ −2.00000 −0.0780869
$$657$$ 0 0
$$658$$ −28.4924 −1.11075
$$659$$ −47.4233 −1.84735 −0.923675 0.383178i $$-0.874830\pi$$
−0.923675 + 0.383178i $$0.874830\pi$$
$$660$$ 0 0
$$661$$ −22.4384 −0.872754 −0.436377 0.899764i $$-0.643739\pi$$
−0.436377 + 0.899764i $$0.643739\pi$$
$$662$$ −10.4384 −0.405702
$$663$$ 0 0
$$664$$ 2.24621 0.0871699
$$665$$ 7.12311 0.276222
$$666$$ 0 0
$$667$$ 37.1771 1.43950
$$668$$ 25.3693 0.981568
$$669$$ 0 0
$$670$$ −9.36932 −0.361968
$$671$$ 10.2462 0.395551
$$672$$ 0 0
$$673$$ −18.8769 −0.727651 −0.363825 0.931467i $$-0.618530\pi$$
−0.363825 + 0.931467i $$0.618530\pi$$
$$674$$ 2.87689 0.110814
$$675$$ 0 0
$$676$$ −0.315342 −0.0121285
$$677$$ 39.1771 1.50570 0.752849 0.658194i $$-0.228679\pi$$
0.752849 + 0.658194i $$0.228679\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ 7.12311 0.273159
$$681$$ 0 0
$$682$$ −2.00000 −0.0765840
$$683$$ 4.49242 0.171898 0.0859489 0.996300i $$-0.472608\pi$$
0.0859489 + 0.996300i $$0.472608\pi$$
$$684$$ 0 0
$$685$$ 16.8769 0.644833
$$686$$ −4.68466 −0.178861
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −27.8078 −1.05939
$$690$$ 0 0
$$691$$ −43.6155 −1.65921 −0.829606 0.558349i $$-0.811435\pi$$
−0.829606 + 0.558349i $$0.811435\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ 31.6155 1.20011
$$695$$ 34.7386 1.31771
$$696$$ 0 0
$$697$$ 7.12311 0.269807
$$698$$ −19.6155 −0.742459
$$699$$ 0 0
$$700$$ −3.56155 −0.134614
$$701$$ −24.9848 −0.943665 −0.471832 0.881688i $$-0.656407\pi$$
−0.471832 + 0.881688i $$0.656407\pi$$
$$702$$ 0 0
$$703$$ −3.12311 −0.117790
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 23.1771 0.872281
$$707$$ 53.8617 2.02568
$$708$$ 0 0
$$709$$ −9.50758 −0.357065 −0.178532 0.983934i $$-0.557135\pi$$
−0.178532 + 0.983934i $$0.557135\pi$$
$$710$$ 12.0000 0.450352
$$711$$ 0 0
$$712$$ 9.12311 0.341903
$$713$$ −11.1231 −0.416564
$$714$$ 0 0
$$715$$ −7.12311 −0.266389
$$716$$ −22.2462 −0.831380
$$717$$ 0 0
$$718$$ −12.9309 −0.482576
$$719$$ 6.43845 0.240114 0.120057 0.992767i $$-0.461692\pi$$
0.120057 + 0.992767i $$0.461692\pi$$
$$720$$ 0 0
$$721$$ 40.4924 1.50802
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ −19.1231 −0.710705
$$725$$ 6.68466 0.248262
$$726$$ 0 0
$$727$$ −39.4233 −1.46213 −0.731064 0.682308i $$-0.760976\pi$$
−0.731064 + 0.682308i $$0.760976\pi$$
$$728$$ −12.6847 −0.470125
$$729$$ 0 0
$$730$$ −5.36932 −0.198727
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −48.1080 −1.77691 −0.888454 0.458966i $$-0.848220\pi$$
−0.888454 + 0.458966i $$0.848220\pi$$
$$734$$ 13.7538 0.507662
$$735$$ 0 0
$$736$$ −5.56155 −0.205002
$$737$$ −4.68466 −0.172562
$$738$$ 0 0
$$739$$ 20.9848 0.771940 0.385970 0.922511i $$-0.373867\pi$$
0.385970 + 0.922511i $$0.373867\pi$$
$$740$$ −6.24621 −0.229615
$$741$$ 0 0
$$742$$ 27.8078 1.02086
$$743$$ 25.7538 0.944815 0.472407 0.881380i $$-0.343385\pi$$
0.472407 + 0.881380i $$0.343385\pi$$
$$744$$ 0 0
$$745$$ −30.2462 −1.10814
$$746$$ 7.56155 0.276848
$$747$$ 0 0
$$748$$ 3.56155 0.130223
$$749$$ −67.4233 −2.46359
$$750$$ 0 0
$$751$$ −41.2311 −1.50454 −0.752271 0.658853i $$-0.771042\pi$$
−0.752271 + 0.658853i $$0.771042\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 23.8078 0.867028
$$755$$ −8.00000 −0.291150
$$756$$ 0 0
$$757$$ −9.50758 −0.345559 −0.172779 0.984961i $$-0.555275\pi$$
−0.172779 + 0.984961i $$0.555275\pi$$
$$758$$ 2.43845 0.0885684
$$759$$ 0 0
$$760$$ 2.00000 0.0725476
$$761$$ 22.1922 0.804468 0.402234 0.915537i $$-0.368234\pi$$
0.402234 + 0.915537i $$0.368234\pi$$
$$762$$ 0 0
$$763$$ −66.5464 −2.40914
$$764$$ 20.6847 0.748345
$$765$$ 0 0
$$766$$ 2.00000 0.0722629
$$767$$ 16.6847 0.602448
$$768$$ 0 0
$$769$$ −1.31534 −0.0474324 −0.0237162 0.999719i $$-0.507550\pi$$
−0.0237162 + 0.999719i $$0.507550\pi$$
$$770$$ 7.12311 0.256699
$$771$$ 0 0
$$772$$ −1.12311 −0.0404215
$$773$$ −9.06913 −0.326194 −0.163097 0.986610i $$-0.552148\pi$$
−0.163097 + 0.986610i $$0.552148\pi$$
$$774$$ 0 0
$$775$$ −2.00000 −0.0718421
$$776$$ 1.12311 0.0403171
$$777$$ 0 0
$$778$$ −18.0000 −0.645331
$$779$$ 2.00000 0.0716574
$$780$$ 0 0
$$781$$ 6.00000 0.214697
$$782$$ 19.8078 0.708324
$$783$$ 0 0
$$784$$ 5.68466 0.203024
$$785$$ −36.9848 −1.32005
$$786$$ 0 0
$$787$$ 3.31534 0.118179 0.0590896 0.998253i $$-0.481180\pi$$
0.0590896 + 0.998253i $$0.481180\pi$$
$$788$$ −1.75379 −0.0624761
$$789$$ 0 0
$$790$$ 8.00000 0.284627
$$791$$ −18.2462 −0.648761
$$792$$ 0 0
$$793$$ 36.4924 1.29588
$$794$$ −26.9848 −0.957656
$$795$$ 0 0
$$796$$ −0.684658 −0.0242671
$$797$$ 35.8078 1.26838 0.634188 0.773179i $$-0.281335\pi$$
0.634188 + 0.773179i $$0.281335\pi$$
$$798$$ 0 0
$$799$$ 28.4924 1.00799
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 24.2462 0.856163
$$803$$ −2.68466 −0.0947395
$$804$$ 0 0
$$805$$ 39.6155 1.39626
$$806$$ −7.12311 −0.250901
$$807$$ 0 0
$$808$$ 15.1231 0.532029
$$809$$ −36.9309 −1.29842 −0.649210 0.760609i $$-0.724900\pi$$
−0.649210 + 0.760609i $$0.724900\pi$$
$$810$$ 0 0
$$811$$ 20.6847 0.726337 0.363168 0.931724i $$-0.381695\pi$$
0.363168 + 0.931724i $$0.381695\pi$$
$$812$$ −23.8078 −0.835489
$$813$$ 0 0
$$814$$ −3.12311 −0.109465
$$815$$ 26.7386 0.936613
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −0.246211 −0.00860857
$$819$$ 0 0
$$820$$ 4.00000 0.139686
$$821$$ −44.9848 −1.56998 −0.784991 0.619507i $$-0.787332\pi$$
−0.784991 + 0.619507i $$0.787332\pi$$
$$822$$ 0 0
$$823$$ 14.9309 0.520457 0.260229 0.965547i $$-0.416202\pi$$
0.260229 + 0.965547i $$0.416202\pi$$
$$824$$ 11.3693 0.396069
$$825$$ 0 0
$$826$$ −16.6847 −0.580534
$$827$$ −21.0691 −0.732645 −0.366323 0.930488i $$-0.619383\pi$$
−0.366323 + 0.930488i $$0.619383\pi$$
$$828$$ 0 0
$$829$$ −33.1771 −1.15229 −0.576144 0.817348i $$-0.695443\pi$$
−0.576144 + 0.817348i $$0.695443\pi$$
$$830$$ −4.49242 −0.155934
$$831$$ 0 0
$$832$$ −3.56155 −0.123475
$$833$$ −20.2462 −0.701490
$$834$$ 0 0
$$835$$ −50.7386 −1.75588
$$836$$ 1.00000 0.0345857
$$837$$ 0 0
$$838$$ 23.1231 0.798774
$$839$$ −1.12311 −0.0387739 −0.0193870 0.999812i $$-0.506171\pi$$
−0.0193870 + 0.999812i $$0.506171\pi$$
$$840$$ 0 0
$$841$$ 15.6847 0.540850
$$842$$ −1.06913 −0.0368447
$$843$$ 0 0
$$844$$ −12.6847 −0.436624
$$845$$ 0.630683 0.0216962
$$846$$ 0 0
$$847$$ 3.56155 0.122376
$$848$$ 7.80776 0.268120
$$849$$ 0 0
$$850$$ 3.56155 0.122160
$$851$$ −17.3693 −0.595413
$$852$$ 0 0
$$853$$ 38.3542 1.31322 0.656611 0.754230i $$-0.271989\pi$$
0.656611 + 0.754230i $$0.271989\pi$$
$$854$$ −36.4924 −1.24874
$$855$$ 0 0
$$856$$ −18.9309 −0.647044
$$857$$ 10.0000 0.341593 0.170797 0.985306i $$-0.445366\pi$$
0.170797 + 0.985306i $$0.445366\pi$$
$$858$$ 0 0
$$859$$ −41.8617 −1.42830 −0.714152 0.699991i $$-0.753188\pi$$
−0.714152 + 0.699991i $$0.753188\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 25.8617 0.880854
$$863$$ 16.2462 0.553027 0.276514 0.961010i $$-0.410821\pi$$
0.276514 + 0.961010i $$0.410821\pi$$
$$864$$ 0 0
$$865$$ 36.0000 1.22404
$$866$$ 31.3693 1.06597
$$867$$ 0 0
$$868$$ 7.12311 0.241774
$$869$$ 4.00000 0.135691
$$870$$ 0 0
$$871$$ −16.6847 −0.565338
$$872$$ −18.6847 −0.632742
$$873$$ 0 0
$$874$$ 5.56155 0.188122
$$875$$ 42.7386 1.44483
$$876$$ 0 0
$$877$$ −40.4384 −1.36551 −0.682755 0.730648i $$-0.739218\pi$$
−0.682755 + 0.730648i $$0.739218\pi$$
$$878$$ 31.6155 1.06697
$$879$$ 0 0
$$880$$ 2.00000 0.0674200
$$881$$ 28.7386 0.968229 0.484115 0.875005i $$-0.339142\pi$$
0.484115 + 0.875005i $$0.339142\pi$$
$$882$$ 0 0
$$883$$ −54.7386 −1.84210 −0.921051 0.389442i $$-0.872668\pi$$
−0.921051 + 0.389442i $$0.872668\pi$$
$$884$$ 12.6847 0.426631
$$885$$ 0 0
$$886$$ 17.8617 0.600077
$$887$$ 32.0000 1.07445 0.537227 0.843437i $$-0.319472\pi$$
0.537227 + 0.843437i $$0.319472\pi$$
$$888$$ 0 0
$$889$$ −50.7386 −1.70172
$$890$$ −18.2462 −0.611614
$$891$$ 0 0
$$892$$ −24.7386 −0.828311
$$893$$ 8.00000 0.267710
$$894$$ 0 0
$$895$$ 44.4924 1.48722
$$896$$ 3.56155 0.118983
$$897$$ 0 0
$$898$$ 17.6155 0.587838
$$899$$ −13.3693 −0.445892
$$900$$ 0 0
$$901$$ −27.8078 −0.926411
$$902$$ 2.00000 0.0665927
$$903$$ 0 0
$$904$$ −5.12311 −0.170392
$$905$$ 38.2462 1.27135
$$906$$ 0 0
$$907$$ 56.7926 1.88577 0.942884 0.333122i $$-0.108102\pi$$
0.942884 + 0.333122i $$0.108102\pi$$
$$908$$ 17.1771 0.570041
$$909$$ 0 0
$$910$$ 25.3693 0.840985
$$911$$ 46.9848 1.55668 0.778339 0.627845i $$-0.216063\pi$$
0.778339 + 0.627845i $$0.216063\pi$$
$$912$$ 0 0
$$913$$ −2.24621 −0.0743387
$$914$$ 32.4384 1.07297
$$915$$ 0 0
$$916$$ 21.1231 0.697927
$$917$$ 25.3693 0.837769
$$918$$ 0 0
$$919$$ 28.4384 0.938098 0.469049 0.883172i $$-0.344597\pi$$
0.469049 + 0.883172i $$0.344597\pi$$
$$920$$ 11.1231 0.366718
$$921$$ 0 0
$$922$$ 22.7386 0.748857
$$923$$ 21.3693 0.703380
$$924$$ 0 0
$$925$$ −3.12311 −0.102687
$$926$$ 36.4924 1.19922
$$927$$ 0 0
$$928$$ −6.68466 −0.219435
$$929$$ 16.9309 0.555484 0.277742 0.960656i $$-0.410414\pi$$
0.277742 + 0.960656i $$0.410414\pi$$
$$930$$ 0 0
$$931$$ −5.68466 −0.186307
$$932$$ −20.7386 −0.679317
$$933$$ 0 0
$$934$$ −26.2462 −0.858802
$$935$$ −7.12311 −0.232950
$$936$$ 0 0
$$937$$ −54.3002 −1.77391 −0.886955 0.461856i $$-0.847184\pi$$
−0.886955 + 0.461856i $$0.847184\pi$$
$$938$$ 16.6847 0.544773
$$939$$ 0 0
$$940$$ 16.0000 0.521862
$$941$$ −20.4384 −0.666274 −0.333137 0.942878i $$-0.608107\pi$$
−0.333137 + 0.942878i $$0.608107\pi$$
$$942$$ 0 0
$$943$$ 11.1231 0.362218
$$944$$ −4.68466 −0.152473
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −28.9848 −0.941881 −0.470940 0.882165i $$-0.656085\pi$$
−0.470940 + 0.882165i $$0.656085\pi$$
$$948$$ 0 0
$$949$$ −9.56155 −0.310381
$$950$$ 1.00000 0.0324443
$$951$$ 0 0
$$952$$ −12.6847 −0.411112
$$953$$ 9.12311 0.295526 0.147763 0.989023i $$-0.452793\pi$$
0.147763 + 0.989023i $$0.452793\pi$$
$$954$$ 0 0
$$955$$ −41.3693 −1.33868
$$956$$ 4.93087 0.159476
$$957$$ 0 0
$$958$$ −11.3693 −0.367326
$$959$$ −30.0540 −0.970493
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −11.1231 −0.358623
$$963$$ 0 0
$$964$$ 21.6155 0.696189
$$965$$ 2.24621 0.0723081
$$966$$ 0 0
$$967$$ −3.86174 −0.124185 −0.0620926 0.998070i $$-0.519777\pi$$
−0.0620926 + 0.998070i $$0.519777\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ −2.24621 −0.0721215
$$971$$ 14.2462 0.457183 0.228591 0.973522i $$-0.426588\pi$$
0.228591 + 0.973522i $$0.426588\pi$$
$$972$$ 0 0
$$973$$ −61.8617 −1.98320
$$974$$ 22.8769 0.733023
$$975$$ 0 0
$$976$$ −10.2462 −0.327973
$$977$$ −55.3693 −1.77142 −0.885711 0.464238i $$-0.846328\pi$$
−0.885711 + 0.464238i $$0.846328\pi$$
$$978$$ 0 0
$$979$$ −9.12311 −0.291576
$$980$$ −11.3693 −0.363180
$$981$$ 0 0
$$982$$ 14.6307 0.466884
$$983$$ 48.2462 1.53882 0.769408 0.638758i $$-0.220552\pi$$
0.769408 + 0.638758i $$0.220552\pi$$
$$984$$ 0 0
$$985$$ 3.50758 0.111761
$$986$$ 23.8078 0.758194
$$987$$ 0 0
$$988$$ 3.56155 0.113308
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 21.1231 0.670998 0.335499 0.942041i $$-0.391095\pi$$
0.335499 + 0.942041i $$0.391095\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ 0 0
$$994$$ −21.3693 −0.677794
$$995$$ 1.36932 0.0434103
$$996$$ 0 0
$$997$$ −53.3693 −1.69022 −0.845112 0.534590i $$-0.820466\pi$$
−0.845112 + 0.534590i $$0.820466\pi$$
$$998$$ 26.2462 0.830809
$$999$$ 0 0
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 3762.2.a.y.1.2 2
3.2 odd 2 418.2.a.e.1.1 2
12.11 even 2 3344.2.a.k.1.2 2
33.32 even 2 4598.2.a.bj.1.1 2
57.56 even 2 7942.2.a.x.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.1 2 3.2 odd 2
3344.2.a.k.1.2 2 12.11 even 2
3762.2.a.y.1.2 2 1.1 even 1 trivial
4598.2.a.bj.1.1 2 33.32 even 2
7942.2.a.x.1.2 2 57.56 even 2