# Properties

 Label 927.2.a.b.1.1 Level $927$ Weight $2$ Character 927.1 Self dual yes Analytic conductor $7.402$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [927,2,Mod(1,927)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(927, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("927.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$927 = 3^{2} \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 927.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: $$7.40213226737$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 103) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 927.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.381966 q^{2} -1.85410 q^{4} +2.61803 q^{5} -1.00000 q^{7} -1.47214 q^{8} +O(q^{10})$$ $$q+0.381966 q^{2} -1.85410 q^{4} +2.61803 q^{5} -1.00000 q^{7} -1.47214 q^{8} +1.00000 q^{10} +0.381966 q^{11} +1.85410 q^{13} -0.381966 q^{14} +3.14590 q^{16} +3.38197 q^{17} -0.854102 q^{19} -4.85410 q^{20} +0.145898 q^{22} +4.47214 q^{23} +1.85410 q^{25} +0.708204 q^{26} +1.85410 q^{28} +0.763932 q^{29} +6.70820 q^{31} +4.14590 q^{32} +1.29180 q^{34} -2.61803 q^{35} -6.70820 q^{37} -0.326238 q^{38} -3.85410 q^{40} +8.94427 q^{41} +4.70820 q^{43} -0.708204 q^{44} +1.70820 q^{46} +7.09017 q^{47} -6.00000 q^{49} +0.708204 q^{50} -3.43769 q^{52} +10.0902 q^{53} +1.00000 q^{55} +1.47214 q^{56} +0.291796 q^{58} -8.61803 q^{59} +10.8541 q^{61} +2.56231 q^{62} -4.70820 q^{64} +4.85410 q^{65} -12.4164 q^{67} -6.27051 q^{68} -1.00000 q^{70} -7.09017 q^{71} -4.14590 q^{73} -2.56231 q^{74} +1.58359 q^{76} -0.381966 q^{77} +13.5623 q^{79} +8.23607 q^{80} +3.41641 q^{82} -9.32624 q^{83} +8.85410 q^{85} +1.79837 q^{86} -0.562306 q^{88} +15.7082 q^{89} -1.85410 q^{91} -8.29180 q^{92} +2.70820 q^{94} -2.23607 q^{95} +11.7082 q^{97} -2.29180 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 2 * q^7 + 6 * q^8 $$2 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8} + 2 q^{10} + 3 q^{11} - 3 q^{13} - 3 q^{14} + 13 q^{16} + 9 q^{17} + 5 q^{19} - 3 q^{20} + 7 q^{22} - 3 q^{25} - 12 q^{26} - 3 q^{28} + 6 q^{29} + 15 q^{32} + 16 q^{34} - 3 q^{35} + 15 q^{38} - q^{40} - 4 q^{43} + 12 q^{44} - 10 q^{46} + 3 q^{47} - 12 q^{49} - 12 q^{50} - 27 q^{52} + 9 q^{53} + 2 q^{55} - 6 q^{56} + 14 q^{58} - 15 q^{59} + 15 q^{61} - 15 q^{62} + 4 q^{64} + 3 q^{65} + 2 q^{67} + 21 q^{68} - 2 q^{70} - 3 q^{71} - 15 q^{73} + 15 q^{74} + 30 q^{76} - 3 q^{77} + 7 q^{79} + 12 q^{80} - 20 q^{82} - 3 q^{83} + 11 q^{85} - 21 q^{86} + 19 q^{88} + 18 q^{89} + 3 q^{91} - 30 q^{92} - 8 q^{94} + 10 q^{97} - 18 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 2 * q^7 + 6 * q^8 + 2 * q^10 + 3 * q^11 - 3 * q^13 - 3 * q^14 + 13 * q^16 + 9 * q^17 + 5 * q^19 - 3 * q^20 + 7 * q^22 - 3 * q^25 - 12 * q^26 - 3 * q^28 + 6 * q^29 + 15 * q^32 + 16 * q^34 - 3 * q^35 + 15 * q^38 - q^40 - 4 * q^43 + 12 * q^44 - 10 * q^46 + 3 * q^47 - 12 * q^49 - 12 * q^50 - 27 * q^52 + 9 * q^53 + 2 * q^55 - 6 * q^56 + 14 * q^58 - 15 * q^59 + 15 * q^61 - 15 * q^62 + 4 * q^64 + 3 * q^65 + 2 * q^67 + 21 * q^68 - 2 * q^70 - 3 * q^71 - 15 * q^73 + 15 * q^74 + 30 * q^76 - 3 * q^77 + 7 * q^79 + 12 * q^80 - 20 * q^82 - 3 * q^83 + 11 * q^85 - 21 * q^86 + 19 * q^88 + 18 * q^89 + 3 * q^91 - 30 * q^92 - 8 * q^94 + 10 * q^97 - 18 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.381966 0.270091 0.135045 0.990839i $$-0.456882\pi$$
0.135045 + 0.990839i $$0.456882\pi$$
$$3$$ 0 0
$$4$$ −1.85410 −0.927051
$$5$$ 2.61803 1.17082 0.585410 0.810737i $$-0.300933\pi$$
0.585410 + 0.810737i $$0.300933\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ −1.47214 −0.520479
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ 0.381966 0.115167 0.0575835 0.998341i $$-0.481660\pi$$
0.0575835 + 0.998341i $$0.481660\pi$$
$$12$$ 0 0
$$13$$ 1.85410 0.514235 0.257118 0.966380i $$-0.417227\pi$$
0.257118 + 0.966380i $$0.417227\pi$$
$$14$$ −0.381966 −0.102085
$$15$$ 0 0
$$16$$ 3.14590 0.786475
$$17$$ 3.38197 0.820247 0.410124 0.912030i $$-0.365486\pi$$
0.410124 + 0.912030i $$0.365486\pi$$
$$18$$ 0 0
$$19$$ −0.854102 −0.195944 −0.0979722 0.995189i $$-0.531236\pi$$
−0.0979722 + 0.995189i $$0.531236\pi$$
$$20$$ −4.85410 −1.08541
$$21$$ 0 0
$$22$$ 0.145898 0.0311056
$$23$$ 4.47214 0.932505 0.466252 0.884652i $$-0.345604\pi$$
0.466252 + 0.884652i $$0.345604\pi$$
$$24$$ 0 0
$$25$$ 1.85410 0.370820
$$26$$ 0.708204 0.138890
$$27$$ 0 0
$$28$$ 1.85410 0.350392
$$29$$ 0.763932 0.141859 0.0709293 0.997481i $$-0.477404\pi$$
0.0709293 + 0.997481i $$0.477404\pi$$
$$30$$ 0 0
$$31$$ 6.70820 1.20483 0.602414 0.798183i $$-0.294205\pi$$
0.602414 + 0.798183i $$0.294205\pi$$
$$32$$ 4.14590 0.732898
$$33$$ 0 0
$$34$$ 1.29180 0.221541
$$35$$ −2.61803 −0.442529
$$36$$ 0 0
$$37$$ −6.70820 −1.10282 −0.551411 0.834234i $$-0.685910\pi$$
−0.551411 + 0.834234i $$0.685910\pi$$
$$38$$ −0.326238 −0.0529228
$$39$$ 0 0
$$40$$ −3.85410 −0.609387
$$41$$ 8.94427 1.39686 0.698430 0.715678i $$-0.253882\pi$$
0.698430 + 0.715678i $$0.253882\pi$$
$$42$$ 0 0
$$43$$ 4.70820 0.717994 0.358997 0.933339i $$-0.383119\pi$$
0.358997 + 0.933339i $$0.383119\pi$$
$$44$$ −0.708204 −0.106766
$$45$$ 0 0
$$46$$ 1.70820 0.251861
$$47$$ 7.09017 1.03421 0.517104 0.855923i $$-0.327010\pi$$
0.517104 + 0.855923i $$0.327010\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0.708204 0.100155
$$51$$ 0 0
$$52$$ −3.43769 −0.476722
$$53$$ 10.0902 1.38599 0.692996 0.720942i $$-0.256290\pi$$
0.692996 + 0.720942i $$0.256290\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ 1.47214 0.196722
$$57$$ 0 0
$$58$$ 0.291796 0.0383147
$$59$$ −8.61803 −1.12197 −0.560986 0.827825i $$-0.689578\pi$$
−0.560986 + 0.827825i $$0.689578\pi$$
$$60$$ 0 0
$$61$$ 10.8541 1.38973 0.694863 0.719142i $$-0.255465\pi$$
0.694863 + 0.719142i $$0.255465\pi$$
$$62$$ 2.56231 0.325413
$$63$$ 0 0
$$64$$ −4.70820 −0.588525
$$65$$ 4.85410 0.602077
$$66$$ 0 0
$$67$$ −12.4164 −1.51691 −0.758453 0.651728i $$-0.774044\pi$$
−0.758453 + 0.651728i $$0.774044\pi$$
$$68$$ −6.27051 −0.760411
$$69$$ 0 0
$$70$$ −1.00000 −0.119523
$$71$$ −7.09017 −0.841448 −0.420724 0.907189i $$-0.638224\pi$$
−0.420724 + 0.907189i $$0.638224\pi$$
$$72$$ 0 0
$$73$$ −4.14590 −0.485241 −0.242620 0.970121i $$-0.578007\pi$$
−0.242620 + 0.970121i $$0.578007\pi$$
$$74$$ −2.56231 −0.297862
$$75$$ 0 0
$$76$$ 1.58359 0.181650
$$77$$ −0.381966 −0.0435291
$$78$$ 0 0
$$79$$ 13.5623 1.52588 0.762939 0.646470i $$-0.223755\pi$$
0.762939 + 0.646470i $$0.223755\pi$$
$$80$$ 8.23607 0.920820
$$81$$ 0 0
$$82$$ 3.41641 0.377279
$$83$$ −9.32624 −1.02369 −0.511844 0.859079i $$-0.671037\pi$$
−0.511844 + 0.859079i $$0.671037\pi$$
$$84$$ 0 0
$$85$$ 8.85410 0.960362
$$86$$ 1.79837 0.193924
$$87$$ 0 0
$$88$$ −0.562306 −0.0599420
$$89$$ 15.7082 1.66507 0.832533 0.553975i $$-0.186890\pi$$
0.832533 + 0.553975i $$0.186890\pi$$
$$90$$ 0 0
$$91$$ −1.85410 −0.194363
$$92$$ −8.29180 −0.864479
$$93$$ 0 0
$$94$$ 2.70820 0.279330
$$95$$ −2.23607 −0.229416
$$96$$ 0 0
$$97$$ 11.7082 1.18879 0.594394 0.804174i $$-0.297392\pi$$
0.594394 + 0.804174i $$0.297392\pi$$
$$98$$ −2.29180 −0.231506
$$99$$ 0 0
$$100$$ −3.43769 −0.343769
$$101$$ −13.0902 −1.30252 −0.651260 0.758854i $$-0.725759\pi$$
−0.651260 + 0.758854i $$0.725759\pi$$
$$102$$ 0 0
$$103$$ −1.00000 −0.0985329
$$104$$ −2.72949 −0.267649
$$105$$ 0 0
$$106$$ 3.85410 0.374343
$$107$$ −4.09017 −0.395412 −0.197706 0.980261i $$-0.563349\pi$$
−0.197706 + 0.980261i $$0.563349\pi$$
$$108$$ 0 0
$$109$$ 10.5623 1.01169 0.505843 0.862626i $$-0.331182\pi$$
0.505843 + 0.862626i $$0.331182\pi$$
$$110$$ 0.381966 0.0364190
$$111$$ 0 0
$$112$$ −3.14590 −0.297259
$$113$$ 15.0000 1.41108 0.705541 0.708669i $$-0.250704\pi$$
0.705541 + 0.708669i $$0.250704\pi$$
$$114$$ 0 0
$$115$$ 11.7082 1.09180
$$116$$ −1.41641 −0.131510
$$117$$ 0 0
$$118$$ −3.29180 −0.303034
$$119$$ −3.38197 −0.310024
$$120$$ 0 0
$$121$$ −10.8541 −0.986737
$$122$$ 4.14590 0.375352
$$123$$ 0 0
$$124$$ −12.4377 −1.11694
$$125$$ −8.23607 −0.736656
$$126$$ 0 0
$$127$$ −18.2705 −1.62125 −0.810623 0.585569i $$-0.800871\pi$$
−0.810623 + 0.585569i $$0.800871\pi$$
$$128$$ −10.0902 −0.891853
$$129$$ 0 0
$$130$$ 1.85410 0.162615
$$131$$ −2.23607 −0.195366 −0.0976831 0.995218i $$-0.531143\pi$$
−0.0976831 + 0.995218i $$0.531143\pi$$
$$132$$ 0 0
$$133$$ 0.854102 0.0740600
$$134$$ −4.74265 −0.409702
$$135$$ 0 0
$$136$$ −4.97871 −0.426921
$$137$$ −0.708204 −0.0605059 −0.0302530 0.999542i $$-0.509631\pi$$
−0.0302530 + 0.999542i $$0.509631\pi$$
$$138$$ 0 0
$$139$$ −17.8541 −1.51437 −0.757183 0.653203i $$-0.773425\pi$$
−0.757183 + 0.653203i $$0.773425\pi$$
$$140$$ 4.85410 0.410246
$$141$$ 0 0
$$142$$ −2.70820 −0.227267
$$143$$ 0.708204 0.0592230
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ −1.58359 −0.131059
$$147$$ 0 0
$$148$$ 12.4377 1.02237
$$149$$ −1.47214 −0.120602 −0.0603010 0.998180i $$-0.519206\pi$$
−0.0603010 + 0.998180i $$0.519206\pi$$
$$150$$ 0 0
$$151$$ −19.0000 −1.54620 −0.773099 0.634285i $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ 1.25735 0.101985
$$153$$ 0 0
$$154$$ −0.145898 −0.0117568
$$155$$ 17.5623 1.41064
$$156$$ 0 0
$$157$$ 3.29180 0.262714 0.131357 0.991335i $$-0.458067\pi$$
0.131357 + 0.991335i $$0.458067\pi$$
$$158$$ 5.18034 0.412126
$$159$$ 0 0
$$160$$ 10.8541 0.858092
$$161$$ −4.47214 −0.352454
$$162$$ 0 0
$$163$$ −10.7082 −0.838731 −0.419366 0.907817i $$-0.637747\pi$$
−0.419366 + 0.907817i $$0.637747\pi$$
$$164$$ −16.5836 −1.29496
$$165$$ 0 0
$$166$$ −3.56231 −0.276489
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 0 0
$$169$$ −9.56231 −0.735562
$$170$$ 3.38197 0.259385
$$171$$ 0 0
$$172$$ −8.72949 −0.665617
$$173$$ 13.0344 0.990990 0.495495 0.868611i $$-0.334987\pi$$
0.495495 + 0.868611i $$0.334987\pi$$
$$174$$ 0 0
$$175$$ −1.85410 −0.140157
$$176$$ 1.20163 0.0905760
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 1.14590 0.0856484 0.0428242 0.999083i $$-0.486364\pi$$
0.0428242 + 0.999083i $$0.486364\pi$$
$$180$$ 0 0
$$181$$ −2.85410 −0.212144 −0.106072 0.994358i $$-0.533827\pi$$
−0.106072 + 0.994358i $$0.533827\pi$$
$$182$$ −0.708204 −0.0524956
$$183$$ 0 0
$$184$$ −6.58359 −0.485349
$$185$$ −17.5623 −1.29121
$$186$$ 0 0
$$187$$ 1.29180 0.0944655
$$188$$ −13.1459 −0.958763
$$189$$ 0 0
$$190$$ −0.854102 −0.0619631
$$191$$ −3.38197 −0.244710 −0.122355 0.992486i $$-0.539045\pi$$
−0.122355 + 0.992486i $$0.539045\pi$$
$$192$$ 0 0
$$193$$ −20.1246 −1.44860 −0.724301 0.689484i $$-0.757837\pi$$
−0.724301 + 0.689484i $$0.757837\pi$$
$$194$$ 4.47214 0.321081
$$195$$ 0 0
$$196$$ 11.1246 0.794615
$$197$$ −10.4164 −0.742138 −0.371069 0.928605i $$-0.621009\pi$$
−0.371069 + 0.928605i $$0.621009\pi$$
$$198$$ 0 0
$$199$$ 3.41641 0.242183 0.121091 0.992641i $$-0.461361\pi$$
0.121091 + 0.992641i $$0.461361\pi$$
$$200$$ −2.72949 −0.193004
$$201$$ 0 0
$$202$$ −5.00000 −0.351799
$$203$$ −0.763932 −0.0536175
$$204$$ 0 0
$$205$$ 23.4164 1.63547
$$206$$ −0.381966 −0.0266128
$$207$$ 0 0
$$208$$ 5.83282 0.404433
$$209$$ −0.326238 −0.0225663
$$210$$ 0 0
$$211$$ 8.14590 0.560787 0.280393 0.959885i $$-0.409535\pi$$
0.280393 + 0.959885i $$0.409535\pi$$
$$212$$ −18.7082 −1.28488
$$213$$ 0 0
$$214$$ −1.56231 −0.106797
$$215$$ 12.3262 0.840642
$$216$$ 0 0
$$217$$ −6.70820 −0.455383
$$218$$ 4.03444 0.273247
$$219$$ 0 0
$$220$$ −1.85410 −0.125004
$$221$$ 6.27051 0.421800
$$222$$ 0 0
$$223$$ −5.70820 −0.382250 −0.191125 0.981566i $$-0.561214\pi$$
−0.191125 + 0.981566i $$0.561214\pi$$
$$224$$ −4.14590 −0.277009
$$225$$ 0 0
$$226$$ 5.72949 0.381120
$$227$$ 14.9443 0.991886 0.495943 0.868355i $$-0.334822\pi$$
0.495943 + 0.868355i $$0.334822\pi$$
$$228$$ 0 0
$$229$$ −6.70820 −0.443291 −0.221645 0.975127i $$-0.571143\pi$$
−0.221645 + 0.975127i $$0.571143\pi$$
$$230$$ 4.47214 0.294884
$$231$$ 0 0
$$232$$ −1.12461 −0.0738344
$$233$$ −8.88854 −0.582308 −0.291154 0.956676i $$-0.594039\pi$$
−0.291154 + 0.956676i $$0.594039\pi$$
$$234$$ 0 0
$$235$$ 18.5623 1.21087
$$236$$ 15.9787 1.04013
$$237$$ 0 0
$$238$$ −1.29180 −0.0837347
$$239$$ −24.3262 −1.57353 −0.786767 0.617250i $$-0.788247\pi$$
−0.786767 + 0.617250i $$0.788247\pi$$
$$240$$ 0 0
$$241$$ 25.2705 1.62782 0.813908 0.580993i $$-0.197336\pi$$
0.813908 + 0.580993i $$0.197336\pi$$
$$242$$ −4.14590 −0.266508
$$243$$ 0 0
$$244$$ −20.1246 −1.28835
$$245$$ −15.7082 −1.00356
$$246$$ 0 0
$$247$$ −1.58359 −0.100762
$$248$$ −9.87539 −0.627088
$$249$$ 0 0
$$250$$ −3.14590 −0.198964
$$251$$ −6.76393 −0.426936 −0.213468 0.976950i $$-0.568476\pi$$
−0.213468 + 0.976950i $$0.568476\pi$$
$$252$$ 0 0
$$253$$ 1.70820 0.107394
$$254$$ −6.97871 −0.437883
$$255$$ 0 0
$$256$$ 5.56231 0.347644
$$257$$ 4.52786 0.282440 0.141220 0.989978i $$-0.454897\pi$$
0.141220 + 0.989978i $$0.454897\pi$$
$$258$$ 0 0
$$259$$ 6.70820 0.416828
$$260$$ −9.00000 −0.558156
$$261$$ 0 0
$$262$$ −0.854102 −0.0527666
$$263$$ 18.3820 1.13348 0.566740 0.823896i $$-0.308204\pi$$
0.566740 + 0.823896i $$0.308204\pi$$
$$264$$ 0 0
$$265$$ 26.4164 1.62275
$$266$$ 0.326238 0.0200029
$$267$$ 0 0
$$268$$ 23.0213 1.40625
$$269$$ 3.32624 0.202804 0.101402 0.994846i $$-0.467667\pi$$
0.101402 + 0.994846i $$0.467667\pi$$
$$270$$ 0 0
$$271$$ 1.00000 0.0607457 0.0303728 0.999539i $$-0.490331\pi$$
0.0303728 + 0.999539i $$0.490331\pi$$
$$272$$ 10.6393 0.645104
$$273$$ 0 0
$$274$$ −0.270510 −0.0163421
$$275$$ 0.708204 0.0427063
$$276$$ 0 0
$$277$$ −8.70820 −0.523225 −0.261613 0.965173i $$-0.584254\pi$$
−0.261613 + 0.965173i $$0.584254\pi$$
$$278$$ −6.81966 −0.409016
$$279$$ 0 0
$$280$$ 3.85410 0.230327
$$281$$ 22.5279 1.34390 0.671950 0.740597i $$-0.265457\pi$$
0.671950 + 0.740597i $$0.265457\pi$$
$$282$$ 0 0
$$283$$ −6.29180 −0.374008 −0.187004 0.982359i $$-0.559878\pi$$
−0.187004 + 0.982359i $$0.559878\pi$$
$$284$$ 13.1459 0.780066
$$285$$ 0 0
$$286$$ 0.270510 0.0159956
$$287$$ −8.94427 −0.527964
$$288$$ 0 0
$$289$$ −5.56231 −0.327194
$$290$$ 0.763932 0.0448596
$$291$$ 0 0
$$292$$ 7.68692 0.449843
$$293$$ −9.65248 −0.563904 −0.281952 0.959429i $$-0.590982\pi$$
−0.281952 + 0.959429i $$0.590982\pi$$
$$294$$ 0 0
$$295$$ −22.5623 −1.31363
$$296$$ 9.87539 0.573995
$$297$$ 0 0
$$298$$ −0.562306 −0.0325735
$$299$$ 8.29180 0.479527
$$300$$ 0 0
$$301$$ −4.70820 −0.271376
$$302$$ −7.25735 −0.417614
$$303$$ 0 0
$$304$$ −2.68692 −0.154105
$$305$$ 28.4164 1.62712
$$306$$ 0 0
$$307$$ 3.85410 0.219965 0.109983 0.993934i $$-0.464920\pi$$
0.109983 + 0.993934i $$0.464920\pi$$
$$308$$ 0.708204 0.0403537
$$309$$ 0 0
$$310$$ 6.70820 0.381000
$$311$$ 32.8885 1.86494 0.932469 0.361250i $$-0.117650\pi$$
0.932469 + 0.361250i $$0.117650\pi$$
$$312$$ 0 0
$$313$$ 29.7082 1.67921 0.839603 0.543200i $$-0.182787\pi$$
0.839603 + 0.543200i $$0.182787\pi$$
$$314$$ 1.25735 0.0709566
$$315$$ 0 0
$$316$$ −25.1459 −1.41457
$$317$$ −1.58359 −0.0889434 −0.0444717 0.999011i $$-0.514160\pi$$
−0.0444717 + 0.999011i $$0.514160\pi$$
$$318$$ 0 0
$$319$$ 0.291796 0.0163374
$$320$$ −12.3262 −0.689058
$$321$$ 0 0
$$322$$ −1.70820 −0.0951945
$$323$$ −2.88854 −0.160723
$$324$$ 0 0
$$325$$ 3.43769 0.190689
$$326$$ −4.09017 −0.226534
$$327$$ 0 0
$$328$$ −13.1672 −0.727036
$$329$$ −7.09017 −0.390894
$$330$$ 0 0
$$331$$ −22.8541 −1.25618 −0.628088 0.778143i $$-0.716162\pi$$
−0.628088 + 0.778143i $$0.716162\pi$$
$$332$$ 17.2918 0.949011
$$333$$ 0 0
$$334$$ −3.43769 −0.188102
$$335$$ −32.5066 −1.77602
$$336$$ 0 0
$$337$$ −22.5623 −1.22905 −0.614524 0.788898i $$-0.710652\pi$$
−0.614524 + 0.788898i $$0.710652\pi$$
$$338$$ −3.65248 −0.198668
$$339$$ 0 0
$$340$$ −16.4164 −0.890305
$$341$$ 2.56231 0.138757
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ −6.93112 −0.373701
$$345$$ 0 0
$$346$$ 4.97871 0.267657
$$347$$ −7.47214 −0.401125 −0.200563 0.979681i $$-0.564277\pi$$
−0.200563 + 0.979681i $$0.564277\pi$$
$$348$$ 0 0
$$349$$ −11.4164 −0.611106 −0.305553 0.952175i $$-0.598841\pi$$
−0.305553 + 0.952175i $$0.598841\pi$$
$$350$$ −0.708204 −0.0378551
$$351$$ 0 0
$$352$$ 1.58359 0.0844057
$$353$$ −4.03444 −0.214732 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$354$$ 0 0
$$355$$ −18.5623 −0.985185
$$356$$ −29.1246 −1.54360
$$357$$ 0 0
$$358$$ 0.437694 0.0231329
$$359$$ −30.3262 −1.60056 −0.800279 0.599628i $$-0.795315\pi$$
−0.800279 + 0.599628i $$0.795315\pi$$
$$360$$ 0 0
$$361$$ −18.2705 −0.961606
$$362$$ −1.09017 −0.0572981
$$363$$ 0 0
$$364$$ 3.43769 0.180184
$$365$$ −10.8541 −0.568130
$$366$$ 0 0
$$367$$ 36.5623 1.90854 0.954268 0.298951i $$-0.0966368\pi$$
0.954268 + 0.298951i $$0.0966368\pi$$
$$368$$ 14.0689 0.733391
$$369$$ 0 0
$$370$$ −6.70820 −0.348743
$$371$$ −10.0902 −0.523856
$$372$$ 0 0
$$373$$ 37.6869 1.95135 0.975677 0.219212i $$-0.0703485\pi$$
0.975677 + 0.219212i $$0.0703485\pi$$
$$374$$ 0.493422 0.0255143
$$375$$ 0 0
$$376$$ −10.4377 −0.538283
$$377$$ 1.41641 0.0729487
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 4.14590 0.212680
$$381$$ 0 0
$$382$$ −1.29180 −0.0660940
$$383$$ −23.1803 −1.18446 −0.592230 0.805769i $$-0.701752\pi$$
−0.592230 + 0.805769i $$0.701752\pi$$
$$384$$ 0 0
$$385$$ −1.00000 −0.0509647
$$386$$ −7.68692 −0.391254
$$387$$ 0 0
$$388$$ −21.7082 −1.10207
$$389$$ 19.4164 0.984451 0.492225 0.870468i $$-0.336184\pi$$
0.492225 + 0.870468i $$0.336184\pi$$
$$390$$ 0 0
$$391$$ 15.1246 0.764884
$$392$$ 8.83282 0.446125
$$393$$ 0 0
$$394$$ −3.97871 −0.200445
$$395$$ 35.5066 1.78653
$$396$$ 0 0
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ 1.30495 0.0654113
$$399$$ 0 0
$$400$$ 5.83282 0.291641
$$401$$ −11.8885 −0.593686 −0.296843 0.954926i $$-0.595934\pi$$
−0.296843 + 0.954926i $$0.595934\pi$$
$$402$$ 0 0
$$403$$ 12.4377 0.619566
$$404$$ 24.2705 1.20750
$$405$$ 0 0
$$406$$ −0.291796 −0.0144816
$$407$$ −2.56231 −0.127009
$$408$$ 0 0
$$409$$ 23.2918 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$410$$ 8.94427 0.441726
$$411$$ 0 0
$$412$$ 1.85410 0.0913450
$$413$$ 8.61803 0.424066
$$414$$ 0 0
$$415$$ −24.4164 −1.19855
$$416$$ 7.68692 0.376882
$$417$$ 0 0
$$418$$ −0.124612 −0.00609496
$$419$$ −7.09017 −0.346377 −0.173189 0.984889i $$-0.555407\pi$$
−0.173189 + 0.984889i $$0.555407\pi$$
$$420$$ 0 0
$$421$$ 3.00000 0.146211 0.0731055 0.997324i $$-0.476709\pi$$
0.0731055 + 0.997324i $$0.476709\pi$$
$$422$$ 3.11146 0.151463
$$423$$ 0 0
$$424$$ −14.8541 −0.721379
$$425$$ 6.27051 0.304164
$$426$$ 0 0
$$427$$ −10.8541 −0.525267
$$428$$ 7.58359 0.366567
$$429$$ 0 0
$$430$$ 4.70820 0.227050
$$431$$ 10.3607 0.499056 0.249528 0.968368i $$-0.419724\pi$$
0.249528 + 0.968368i $$0.419724\pi$$
$$432$$ 0 0
$$433$$ −12.4164 −0.596694 −0.298347 0.954457i $$-0.596435\pi$$
−0.298347 + 0.954457i $$0.596435\pi$$
$$434$$ −2.56231 −0.122995
$$435$$ 0 0
$$436$$ −19.5836 −0.937884
$$437$$ −3.81966 −0.182719
$$438$$ 0 0
$$439$$ 9.43769 0.450437 0.225218 0.974308i $$-0.427690\pi$$
0.225218 + 0.974308i $$0.427690\pi$$
$$440$$ −1.47214 −0.0701813
$$441$$ 0 0
$$442$$ 2.39512 0.113924
$$443$$ 35.5623 1.68962 0.844808 0.535069i $$-0.179715\pi$$
0.844808 + 0.535069i $$0.179715\pi$$
$$444$$ 0 0
$$445$$ 41.1246 1.94949
$$446$$ −2.18034 −0.103242
$$447$$ 0 0
$$448$$ 4.70820 0.222442
$$449$$ −31.3607 −1.48000 −0.740001 0.672606i $$-0.765175\pi$$
−0.740001 + 0.672606i $$0.765175\pi$$
$$450$$ 0 0
$$451$$ 3.41641 0.160872
$$452$$ −27.8115 −1.30814
$$453$$ 0 0
$$454$$ 5.70820 0.267899
$$455$$ −4.85410 −0.227564
$$456$$ 0 0
$$457$$ −3.14590 −0.147159 −0.0735795 0.997289i $$-0.523442\pi$$
−0.0735795 + 0.997289i $$0.523442\pi$$
$$458$$ −2.56231 −0.119729
$$459$$ 0 0
$$460$$ −21.7082 −1.01215
$$461$$ 39.2148 1.82641 0.913207 0.407495i $$-0.133598\pi$$
0.913207 + 0.407495i $$0.133598\pi$$
$$462$$ 0 0
$$463$$ 5.41641 0.251722 0.125861 0.992048i $$-0.459831\pi$$
0.125861 + 0.992048i $$0.459831\pi$$
$$464$$ 2.40325 0.111568
$$465$$ 0 0
$$466$$ −3.39512 −0.157276
$$467$$ −27.6525 −1.27960 −0.639802 0.768540i $$-0.720984\pi$$
−0.639802 + 0.768540i $$0.720984\pi$$
$$468$$ 0 0
$$469$$ 12.4164 0.573336
$$470$$ 7.09017 0.327045
$$471$$ 0 0
$$472$$ 12.6869 0.583963
$$473$$ 1.79837 0.0826893
$$474$$ 0 0
$$475$$ −1.58359 −0.0726602
$$476$$ 6.27051 0.287408
$$477$$ 0 0
$$478$$ −9.29180 −0.424997
$$479$$ 14.1803 0.647916 0.323958 0.946071i $$-0.394986\pi$$
0.323958 + 0.946071i $$0.394986\pi$$
$$480$$ 0 0
$$481$$ −12.4377 −0.567110
$$482$$ 9.65248 0.439658
$$483$$ 0 0
$$484$$ 20.1246 0.914755
$$485$$ 30.6525 1.39186
$$486$$ 0 0
$$487$$ −23.0000 −1.04223 −0.521115 0.853487i $$-0.674484\pi$$
−0.521115 + 0.853487i $$0.674484\pi$$
$$488$$ −15.9787 −0.723322
$$489$$ 0 0
$$490$$ −6.00000 −0.271052
$$491$$ 35.7771 1.61460 0.807299 0.590143i $$-0.200929\pi$$
0.807299 + 0.590143i $$0.200929\pi$$
$$492$$ 0 0
$$493$$ 2.58359 0.116359
$$494$$ −0.604878 −0.0272148
$$495$$ 0 0
$$496$$ 21.1033 0.947567
$$497$$ 7.09017 0.318038
$$498$$ 0 0
$$499$$ −20.2705 −0.907433 −0.453716 0.891146i $$-0.649902\pi$$
−0.453716 + 0.891146i $$0.649902\pi$$
$$500$$ 15.2705 0.682918
$$501$$ 0 0
$$502$$ −2.58359 −0.115311
$$503$$ 31.3607 1.39830 0.699152 0.714973i $$-0.253561\pi$$
0.699152 + 0.714973i $$0.253561\pi$$
$$504$$ 0 0
$$505$$ −34.2705 −1.52502
$$506$$ 0.652476 0.0290061
$$507$$ 0 0
$$508$$ 33.8754 1.50298
$$509$$ −24.3820 −1.08071 −0.540356 0.841437i $$-0.681710\pi$$
−0.540356 + 0.841437i $$0.681710\pi$$
$$510$$ 0 0
$$511$$ 4.14590 0.183404
$$512$$ 22.3050 0.985749
$$513$$ 0 0
$$514$$ 1.72949 0.0762845
$$515$$ −2.61803 −0.115364
$$516$$ 0 0
$$517$$ 2.70820 0.119107
$$518$$ 2.56231 0.112581
$$519$$ 0 0
$$520$$ −7.14590 −0.313368
$$521$$ 5.18034 0.226955 0.113477 0.993541i $$-0.463801\pi$$
0.113477 + 0.993541i $$0.463801\pi$$
$$522$$ 0 0
$$523$$ −37.4164 −1.63611 −0.818053 0.575143i $$-0.804946\pi$$
−0.818053 + 0.575143i $$0.804946\pi$$
$$524$$ 4.14590 0.181114
$$525$$ 0 0
$$526$$ 7.02129 0.306143
$$527$$ 22.6869 0.988258
$$528$$ 0 0
$$529$$ −3.00000 −0.130435
$$530$$ 10.0902 0.438289
$$531$$ 0 0
$$532$$ −1.58359 −0.0686574
$$533$$ 16.5836 0.718315
$$534$$ 0 0
$$535$$ −10.7082 −0.462956
$$536$$ 18.2786 0.789517
$$537$$ 0 0
$$538$$ 1.27051 0.0547756
$$539$$ −2.29180 −0.0987146
$$540$$ 0 0
$$541$$ 15.8541 0.681621 0.340811 0.940132i $$-0.389299\pi$$
0.340811 + 0.940132i $$0.389299\pi$$
$$542$$ 0.381966 0.0164068
$$543$$ 0 0
$$544$$ 14.0213 0.601158
$$545$$ 27.6525 1.18450
$$546$$ 0 0
$$547$$ 31.2705 1.33703 0.668515 0.743698i $$-0.266930\pi$$
0.668515 + 0.743698i $$0.266930\pi$$
$$548$$ 1.31308 0.0560921
$$549$$ 0 0
$$550$$ 0.270510 0.0115346
$$551$$ −0.652476 −0.0277964
$$552$$ 0 0
$$553$$ −13.5623 −0.576728
$$554$$ −3.32624 −0.141318
$$555$$ 0 0
$$556$$ 33.1033 1.40389
$$557$$ −28.6869 −1.21550 −0.607752 0.794127i $$-0.707928\pi$$
−0.607752 + 0.794127i $$0.707928\pi$$
$$558$$ 0 0
$$559$$ 8.72949 0.369218
$$560$$ −8.23607 −0.348037
$$561$$ 0 0
$$562$$ 8.60488 0.362975
$$563$$ 25.7984 1.08727 0.543636 0.839321i $$-0.317047\pi$$
0.543636 + 0.839321i $$0.317047\pi$$
$$564$$ 0 0
$$565$$ 39.2705 1.65212
$$566$$ −2.40325 −0.101016
$$567$$ 0 0
$$568$$ 10.4377 0.437956
$$569$$ −42.1591 −1.76740 −0.883700 0.468054i $$-0.844955\pi$$
−0.883700 + 0.468054i $$0.844955\pi$$
$$570$$ 0 0
$$571$$ −2.43769 −0.102014 −0.0510072 0.998698i $$-0.516243\pi$$
−0.0510072 + 0.998698i $$0.516243\pi$$
$$572$$ −1.31308 −0.0549027
$$573$$ 0 0
$$574$$ −3.41641 −0.142598
$$575$$ 8.29180 0.345792
$$576$$ 0 0
$$577$$ 16.8328 0.700759 0.350380 0.936608i $$-0.386053\pi$$
0.350380 + 0.936608i $$0.386053\pi$$
$$578$$ −2.12461 −0.0883722
$$579$$ 0 0
$$580$$ −3.70820 −0.153975
$$581$$ 9.32624 0.386918
$$582$$ 0 0
$$583$$ 3.85410 0.159621
$$584$$ 6.10333 0.252557
$$585$$ 0 0
$$586$$ −3.68692 −0.152305
$$587$$ −11.0689 −0.456862 −0.228431 0.973560i $$-0.573359\pi$$
−0.228431 + 0.973560i $$0.573359\pi$$
$$588$$ 0 0
$$589$$ −5.72949 −0.236080
$$590$$ −8.61803 −0.354799
$$591$$ 0 0
$$592$$ −21.1033 −0.867341
$$593$$ 20.1803 0.828707 0.414354 0.910116i $$-0.364008\pi$$
0.414354 + 0.910116i $$0.364008\pi$$
$$594$$ 0 0
$$595$$ −8.85410 −0.362983
$$596$$ 2.72949 0.111804
$$597$$ 0 0
$$598$$ 3.16718 0.129516
$$599$$ −20.4508 −0.835599 −0.417800 0.908539i $$-0.637199\pi$$
−0.417800 + 0.908539i $$0.637199\pi$$
$$600$$ 0 0
$$601$$ −16.5623 −0.675591 −0.337795 0.941220i $$-0.609681\pi$$
−0.337795 + 0.941220i $$0.609681\pi$$
$$602$$ −1.79837 −0.0732962
$$603$$ 0 0
$$604$$ 35.2279 1.43340
$$605$$ −28.4164 −1.15529
$$606$$ 0 0
$$607$$ −7.70820 −0.312866 −0.156433 0.987689i $$-0.550000\pi$$
−0.156433 + 0.987689i $$0.550000\pi$$
$$608$$ −3.54102 −0.143607
$$609$$ 0 0
$$610$$ 10.8541 0.439470
$$611$$ 13.1459 0.531826
$$612$$ 0 0
$$613$$ −2.41641 −0.0975978 −0.0487989 0.998809i $$-0.515539\pi$$
−0.0487989 + 0.998809i $$0.515539\pi$$
$$614$$ 1.47214 0.0594106
$$615$$ 0 0
$$616$$ 0.562306 0.0226560
$$617$$ 30.2705 1.21864 0.609322 0.792923i $$-0.291442\pi$$
0.609322 + 0.792923i $$0.291442\pi$$
$$618$$ 0 0
$$619$$ 31.6869 1.27360 0.636802 0.771027i $$-0.280257\pi$$
0.636802 + 0.771027i $$0.280257\pi$$
$$620$$ −32.5623 −1.30773
$$621$$ 0 0
$$622$$ 12.5623 0.503703
$$623$$ −15.7082 −0.629336
$$624$$ 0 0
$$625$$ −30.8328 −1.23331
$$626$$ 11.3475 0.453538
$$627$$ 0 0
$$628$$ −6.10333 −0.243549
$$629$$ −22.6869 −0.904587
$$630$$ 0 0
$$631$$ 8.72949 0.347516 0.173758 0.984788i $$-0.444409\pi$$
0.173758 + 0.984788i $$0.444409\pi$$
$$632$$ −19.9656 −0.794187
$$633$$ 0 0
$$634$$ −0.604878 −0.0240228
$$635$$ −47.8328 −1.89819
$$636$$ 0 0
$$637$$ −11.1246 −0.440773
$$638$$ 0.111456 0.00441259
$$639$$ 0 0
$$640$$ −26.4164 −1.04420
$$641$$ 15.0000 0.592464 0.296232 0.955116i $$-0.404270\pi$$
0.296232 + 0.955116i $$0.404270\pi$$
$$642$$ 0 0
$$643$$ 5.00000 0.197181 0.0985904 0.995128i $$-0.468567\pi$$
0.0985904 + 0.995128i $$0.468567\pi$$
$$644$$ 8.29180 0.326743
$$645$$ 0 0
$$646$$ −1.10333 −0.0434098
$$647$$ −43.7426 −1.71970 −0.859850 0.510546i $$-0.829443\pi$$
−0.859850 + 0.510546i $$0.829443\pi$$
$$648$$ 0 0
$$649$$ −3.29180 −0.129214
$$650$$ 1.31308 0.0515033
$$651$$ 0 0
$$652$$ 19.8541 0.777547
$$653$$ 0.763932 0.0298950 0.0149475 0.999888i $$-0.495242\pi$$
0.0149475 + 0.999888i $$0.495242\pi$$
$$654$$ 0 0
$$655$$ −5.85410 −0.228739
$$656$$ 28.1378 1.09860
$$657$$ 0 0
$$658$$ −2.70820 −0.105577
$$659$$ 12.5967 0.490700 0.245350 0.969435i $$-0.421097\pi$$
0.245350 + 0.969435i $$0.421097\pi$$
$$660$$ 0 0
$$661$$ −11.4377 −0.444875 −0.222437 0.974947i $$-0.571401\pi$$
−0.222437 + 0.974947i $$0.571401\pi$$
$$662$$ −8.72949 −0.339281
$$663$$ 0 0
$$664$$ 13.7295 0.532808
$$665$$ 2.23607 0.0867110
$$666$$ 0 0
$$667$$ 3.41641 0.132284
$$668$$ 16.6869 0.645636
$$669$$ 0 0
$$670$$ −12.4164 −0.479688
$$671$$ 4.14590 0.160051
$$672$$ 0 0
$$673$$ −37.7082 −1.45354 −0.726772 0.686879i $$-0.758980\pi$$
−0.726772 + 0.686879i $$0.758980\pi$$
$$674$$ −8.61803 −0.331954
$$675$$ 0 0
$$676$$ 17.7295 0.681903
$$677$$ −10.9656 −0.421441 −0.210720 0.977546i $$-0.567581\pi$$
−0.210720 + 0.977546i $$0.567581\pi$$
$$678$$ 0 0
$$679$$ −11.7082 −0.449320
$$680$$ −13.0344 −0.499848
$$681$$ 0 0
$$682$$ 0.978714 0.0374769
$$683$$ −42.7639 −1.63632 −0.818158 0.574993i $$-0.805005\pi$$
−0.818158 + 0.574993i $$0.805005\pi$$
$$684$$ 0 0
$$685$$ −1.85410 −0.0708416
$$686$$ 4.96556 0.189586
$$687$$ 0 0
$$688$$ 14.8115 0.564684
$$689$$ 18.7082 0.712726
$$690$$ 0 0
$$691$$ 1.14590 0.0435920 0.0217960 0.999762i $$-0.493062\pi$$
0.0217960 + 0.999762i $$0.493062\pi$$
$$692$$ −24.1672 −0.918698
$$693$$ 0 0
$$694$$ −2.85410 −0.108340
$$695$$ −46.7426 −1.77305
$$696$$ 0 0
$$697$$ 30.2492 1.14577
$$698$$ −4.36068 −0.165054
$$699$$ 0 0
$$700$$ 3.43769 0.129933
$$701$$ 1.20163 0.0453848 0.0226924 0.999742i $$-0.492776\pi$$
0.0226924 + 0.999742i $$0.492776\pi$$
$$702$$ 0 0
$$703$$ 5.72949 0.216092
$$704$$ −1.79837 −0.0677788
$$705$$ 0 0
$$706$$ −1.54102 −0.0579970
$$707$$ 13.0902 0.492307
$$708$$ 0 0
$$709$$ −47.9787 −1.80188 −0.900939 0.433945i $$-0.857121\pi$$
−0.900939 + 0.433945i $$0.857121\pi$$
$$710$$ −7.09017 −0.266089
$$711$$ 0 0
$$712$$ −23.1246 −0.866631
$$713$$ 30.0000 1.12351
$$714$$ 0 0
$$715$$ 1.85410 0.0693395
$$716$$ −2.12461 −0.0794005
$$717$$ 0 0
$$718$$ −11.5836 −0.432296
$$719$$ −23.6738 −0.882882 −0.441441 0.897290i $$-0.645533\pi$$
−0.441441 + 0.897290i $$0.645533\pi$$
$$720$$ 0 0
$$721$$ 1.00000 0.0372419
$$722$$ −6.97871 −0.259721
$$723$$ 0 0
$$724$$ 5.29180 0.196668
$$725$$ 1.41641 0.0526041
$$726$$ 0 0
$$727$$ −38.2705 −1.41937 −0.709687 0.704517i $$-0.751164\pi$$
−0.709687 + 0.704517i $$0.751164\pi$$
$$728$$ 2.72949 0.101162
$$729$$ 0 0
$$730$$ −4.14590 −0.153447
$$731$$ 15.9230 0.588933
$$732$$ 0 0
$$733$$ 1.29180 0.0477136 0.0238568 0.999715i $$-0.492405\pi$$
0.0238568 + 0.999715i $$0.492405\pi$$
$$734$$ 13.9656 0.515478
$$735$$ 0 0
$$736$$ 18.5410 0.683431
$$737$$ −4.74265 −0.174698
$$738$$ 0 0
$$739$$ 36.8328 1.35492 0.677459 0.735561i $$-0.263081\pi$$
0.677459 + 0.735561i $$0.263081\pi$$
$$740$$ 32.5623 1.19701
$$741$$ 0 0
$$742$$ −3.85410 −0.141489
$$743$$ −17.2918 −0.634374 −0.317187 0.948363i $$-0.602738\pi$$
−0.317187 + 0.948363i $$0.602738\pi$$
$$744$$ 0 0
$$745$$ −3.85410 −0.141203
$$746$$ 14.3951 0.527043
$$747$$ 0 0
$$748$$ −2.39512 −0.0875743
$$749$$ 4.09017 0.149452
$$750$$ 0 0
$$751$$ −3.87539 −0.141415 −0.0707075 0.997497i $$-0.522526\pi$$
−0.0707075 + 0.997497i $$0.522526\pi$$
$$752$$ 22.3050 0.813378
$$753$$ 0 0
$$754$$ 0.541020 0.0197028
$$755$$ −49.7426 −1.81032
$$756$$ 0 0
$$757$$ −34.7082 −1.26149 −0.630746 0.775990i $$-0.717251\pi$$
−0.630746 + 0.775990i $$0.717251\pi$$
$$758$$ 1.90983 0.0693682
$$759$$ 0 0
$$760$$ 3.29180 0.119406
$$761$$ −1.47214 −0.0533649 −0.0266824 0.999644i $$-0.508494\pi$$
−0.0266824 + 0.999644i $$0.508494\pi$$
$$762$$ 0 0
$$763$$ −10.5623 −0.382381
$$764$$ 6.27051 0.226859
$$765$$ 0 0
$$766$$ −8.85410 −0.319912
$$767$$ −15.9787 −0.576958
$$768$$ 0 0
$$769$$ −42.3951 −1.52881 −0.764404 0.644738i $$-0.776966\pi$$
−0.764404 + 0.644738i $$0.776966\pi$$
$$770$$ −0.381966 −0.0137651
$$771$$ 0 0
$$772$$ 37.3131 1.34293
$$773$$ 25.4721 0.916169 0.458085 0.888909i $$-0.348536\pi$$
0.458085 + 0.888909i $$0.348536\pi$$
$$774$$ 0 0
$$775$$ 12.4377 0.446775
$$776$$ −17.2361 −0.618739
$$777$$ 0 0
$$778$$ 7.41641 0.265891
$$779$$ −7.63932 −0.273707
$$780$$ 0 0
$$781$$ −2.70820 −0.0969072
$$782$$ 5.77709 0.206588
$$783$$ 0 0
$$784$$ −18.8754 −0.674121
$$785$$ 8.61803 0.307591
$$786$$ 0 0
$$787$$ 16.5836 0.591141 0.295571 0.955321i $$-0.404490\pi$$
0.295571 + 0.955321i $$0.404490\pi$$
$$788$$ 19.3131 0.688000
$$789$$ 0 0
$$790$$ 13.5623 0.482525
$$791$$ −15.0000 −0.533339
$$792$$ 0 0
$$793$$ 20.1246 0.714646
$$794$$ 7.63932 0.271109
$$795$$ 0 0
$$796$$ −6.33437 −0.224516
$$797$$ 9.87539 0.349804 0.174902 0.984586i $$-0.444039\pi$$
0.174902 + 0.984586i $$0.444039\pi$$
$$798$$ 0 0
$$799$$ 23.9787 0.848306
$$800$$ 7.68692 0.271774
$$801$$ 0 0
$$802$$ −4.54102 −0.160349
$$803$$ −1.58359 −0.0558838
$$804$$ 0 0
$$805$$ −11.7082 −0.412660
$$806$$ 4.75078 0.167339
$$807$$ 0 0
$$808$$ 19.2705 0.677934
$$809$$ 14.9443 0.525413 0.262706 0.964876i $$-0.415385\pi$$
0.262706 + 0.964876i $$0.415385\pi$$
$$810$$ 0 0
$$811$$ −45.5410 −1.59916 −0.799581 0.600559i $$-0.794945\pi$$
−0.799581 + 0.600559i $$0.794945\pi$$
$$812$$ 1.41641 0.0497062
$$813$$ 0 0
$$814$$ −0.978714 −0.0343039
$$815$$ −28.0344 −0.982004
$$816$$ 0 0
$$817$$ −4.02129 −0.140687
$$818$$ 8.89667 0.311065
$$819$$ 0 0
$$820$$ −43.4164 −1.51617
$$821$$ −33.0000 −1.15171 −0.575854 0.817553i $$-0.695330\pi$$
−0.575854 + 0.817553i $$0.695330\pi$$
$$822$$ 0 0
$$823$$ 23.5623 0.821330 0.410665 0.911786i $$-0.365297\pi$$
0.410665 + 0.911786i $$0.365297\pi$$
$$824$$ 1.47214 0.0512843
$$825$$ 0 0
$$826$$ 3.29180 0.114536
$$827$$ 6.70820 0.233267 0.116634 0.993175i $$-0.462790\pi$$
0.116634 + 0.993175i $$0.462790\pi$$
$$828$$ 0 0
$$829$$ −19.2705 −0.669292 −0.334646 0.942344i $$-0.608617\pi$$
−0.334646 + 0.942344i $$0.608617\pi$$
$$830$$ −9.32624 −0.323718
$$831$$ 0 0
$$832$$ −8.72949 −0.302641
$$833$$ −20.2918 −0.703069
$$834$$ 0 0
$$835$$ −23.5623 −0.815407
$$836$$ 0.604878 0.0209202
$$837$$ 0 0
$$838$$ −2.70820 −0.0935534
$$839$$ −21.3820 −0.738187 −0.369094 0.929392i $$-0.620332\pi$$
−0.369094 + 0.929392i $$0.620332\pi$$
$$840$$ 0 0
$$841$$ −28.4164 −0.979876
$$842$$ 1.14590 0.0394903
$$843$$ 0 0
$$844$$ −15.1033 −0.519878
$$845$$ −25.0344 −0.861211
$$846$$ 0 0
$$847$$ 10.8541 0.372951
$$848$$ 31.7426 1.09005
$$849$$ 0 0
$$850$$ 2.39512 0.0821520
$$851$$ −30.0000 −1.02839
$$852$$ 0 0
$$853$$ −10.7295 −0.367371 −0.183685 0.982985i $$-0.558803\pi$$
−0.183685 + 0.982985i $$0.558803\pi$$
$$854$$ −4.14590 −0.141870
$$855$$ 0 0
$$856$$ 6.02129 0.205803
$$857$$ −3.76393 −0.128573 −0.0642867 0.997931i $$-0.520477\pi$$
−0.0642867 + 0.997931i $$0.520477\pi$$
$$858$$ 0 0
$$859$$ −9.56231 −0.326262 −0.163131 0.986604i $$-0.552159\pi$$
−0.163131 + 0.986604i $$0.552159\pi$$
$$860$$ −22.8541 −0.779318
$$861$$ 0 0
$$862$$ 3.95743 0.134791
$$863$$ 8.29180 0.282256 0.141128 0.989991i $$-0.454927\pi$$
0.141128 + 0.989991i $$0.454927\pi$$
$$864$$ 0 0
$$865$$ 34.1246 1.16027
$$866$$ −4.74265 −0.161162
$$867$$ 0 0
$$868$$ 12.4377 0.422163
$$869$$ 5.18034 0.175731
$$870$$ 0 0
$$871$$ −23.0213 −0.780047
$$872$$ −15.5492 −0.526561
$$873$$ 0 0
$$874$$ −1.45898 −0.0493507
$$875$$ 8.23607 0.278430
$$876$$ 0 0
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ 3.60488 0.121659
$$879$$ 0 0
$$880$$ 3.14590 0.106048
$$881$$ 5.88854 0.198390 0.0991950 0.995068i $$-0.468373\pi$$
0.0991950 + 0.995068i $$0.468373\pi$$
$$882$$ 0 0
$$883$$ 46.1246 1.55222 0.776108 0.630600i $$-0.217191\pi$$
0.776108 + 0.630600i $$0.217191\pi$$
$$884$$ −11.6262 −0.391030
$$885$$ 0 0
$$886$$ 13.5836 0.456350
$$887$$ 58.1935 1.95395 0.976973 0.213362i $$-0.0684414\pi$$
0.976973 + 0.213362i $$0.0684414\pi$$
$$888$$ 0 0
$$889$$ 18.2705 0.612773
$$890$$ 15.7082 0.526540
$$891$$ 0 0
$$892$$ 10.5836 0.354365
$$893$$ −6.05573 −0.202647
$$894$$ 0 0
$$895$$ 3.00000 0.100279
$$896$$ 10.0902 0.337089
$$897$$ 0 0
$$898$$ −11.9787 −0.399735
$$899$$ 5.12461 0.170915
$$900$$ 0 0
$$901$$ 34.1246 1.13686
$$902$$ 1.30495 0.0434501
$$903$$ 0 0
$$904$$ −22.0820 −0.734438
$$905$$ −7.47214 −0.248382
$$906$$ 0 0
$$907$$ 33.1246 1.09988 0.549942 0.835203i $$-0.314650\pi$$
0.549942 + 0.835203i $$0.314650\pi$$
$$908$$ −27.7082 −0.919529
$$909$$ 0 0
$$910$$ −1.85410 −0.0614629
$$911$$ −19.0344 −0.630639 −0.315320 0.948986i $$-0.602112\pi$$
−0.315320 + 0.948986i $$0.602112\pi$$
$$912$$ 0 0
$$913$$ −3.56231 −0.117895
$$914$$ −1.20163 −0.0397463
$$915$$ 0 0
$$916$$ 12.4377 0.410953
$$917$$ 2.23607 0.0738415
$$918$$ 0 0
$$919$$ 18.9787 0.626050 0.313025 0.949745i $$-0.398658\pi$$
0.313025 + 0.949745i $$0.398658\pi$$
$$920$$ −17.2361 −0.568256
$$921$$ 0 0
$$922$$ 14.9787 0.493298
$$923$$ −13.1459 −0.432703
$$924$$ 0 0
$$925$$ −12.4377 −0.408949
$$926$$ 2.06888 0.0679877
$$927$$ 0 0
$$928$$ 3.16718 0.103968
$$929$$ 2.94427 0.0965984 0.0482992 0.998833i $$-0.484620\pi$$
0.0482992 + 0.998833i $$0.484620\pi$$
$$930$$ 0 0
$$931$$ 5.12461 0.167952
$$932$$ 16.4803 0.539829
$$933$$ 0 0
$$934$$ −10.5623 −0.345609
$$935$$ 3.38197 0.110602
$$936$$ 0 0
$$937$$ −11.0000 −0.359354 −0.179677 0.983726i $$-0.557505\pi$$
−0.179677 + 0.983726i $$0.557505\pi$$
$$938$$ 4.74265 0.154853
$$939$$ 0 0
$$940$$ −34.4164 −1.12254
$$941$$ −50.3951 −1.64283 −0.821417 0.570328i $$-0.806816\pi$$
−0.821417 + 0.570328i $$0.806816\pi$$
$$942$$ 0 0
$$943$$ 40.0000 1.30258
$$944$$ −27.1115 −0.882403
$$945$$ 0 0
$$946$$ 0.686918 0.0223336
$$947$$ −35.0132 −1.13777 −0.568887 0.822415i $$-0.692626\pi$$
−0.568887 + 0.822415i $$0.692626\pi$$
$$948$$ 0 0
$$949$$ −7.68692 −0.249528
$$950$$ −0.604878 −0.0196248
$$951$$ 0 0
$$952$$ 4.97871 0.161361
$$953$$ 31.3607 1.01587 0.507936 0.861395i $$-0.330409\pi$$
0.507936 + 0.861395i $$0.330409\pi$$
$$954$$ 0 0
$$955$$ −8.85410 −0.286512
$$956$$ 45.1033 1.45875
$$957$$ 0 0
$$958$$ 5.41641 0.174996
$$959$$ 0.708204 0.0228691
$$960$$ 0 0
$$961$$ 14.0000 0.451613
$$962$$ −4.75078 −0.153171
$$963$$ 0 0
$$964$$ −46.8541 −1.50907
$$965$$ −52.6869 −1.69605
$$966$$ 0 0
$$967$$ 12.4164 0.399285 0.199642 0.979869i $$-0.436022\pi$$
0.199642 + 0.979869i $$0.436022\pi$$
$$968$$ 15.9787 0.513575
$$969$$ 0 0
$$970$$ 11.7082 0.375928
$$971$$ −23.0132 −0.738527 −0.369264 0.929325i $$-0.620390\pi$$
−0.369264 + 0.929325i $$0.620390\pi$$
$$972$$ 0 0
$$973$$ 17.8541 0.572376
$$974$$ −8.78522 −0.281497
$$975$$ 0 0
$$976$$ 34.1459 1.09298
$$977$$ −4.25735 −0.136205 −0.0681024 0.997678i $$-0.521694\pi$$
−0.0681024 + 0.997678i $$0.521694\pi$$
$$978$$ 0 0
$$979$$ 6.00000 0.191761
$$980$$ 29.1246 0.930352
$$981$$ 0 0
$$982$$ 13.6656 0.436088
$$983$$ 12.6525 0.403551 0.201776 0.979432i $$-0.435329\pi$$
0.201776 + 0.979432i $$0.435329\pi$$
$$984$$ 0 0
$$985$$ −27.2705 −0.868911
$$986$$ 0.986844 0.0314275
$$987$$ 0 0
$$988$$ 2.93614 0.0934111
$$989$$ 21.0557 0.669533
$$990$$ 0 0
$$991$$ −9.27051 −0.294487 −0.147244 0.989100i $$-0.547040\pi$$
−0.147244 + 0.989100i $$0.547040\pi$$
$$992$$ 27.8115 0.883017
$$993$$ 0 0
$$994$$ 2.70820 0.0858990
$$995$$ 8.94427 0.283552
$$996$$ 0 0
$$997$$ −52.7082 −1.66929 −0.834643 0.550792i $$-0.814326\pi$$
−0.834643 + 0.550792i $$0.814326\pi$$
$$998$$ −7.74265 −0.245089
$$999$$ 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 927.2.a.b.1.1 2
3.2 odd 2 103.2.a.a.1.2 2
12.11 even 2 1648.2.a.f.1.1 2
15.14 odd 2 2575.2.a.g.1.1 2
21.20 even 2 5047.2.a.a.1.2 2
24.5 odd 2 6592.2.a.t.1.2 2
24.11 even 2 6592.2.a.h.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.2 2 3.2 odd 2
927.2.a.b.1.1 2 1.1 even 1 trivial
1648.2.a.f.1.1 2 12.11 even 2
2575.2.a.g.1.1 2 15.14 odd 2
5047.2.a.a.1.2 2 21.20 even 2
6592.2.a.h.1.2 2 24.11 even 2
6592.2.a.t.1.2 2 24.5 odd 2