# Properties

 Label 5733.2.a.br.1.3 Level $5733$ Weight $2$ Character 5733.1 Self dual yes Analytic conductor $45.778$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, 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("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5733 = 3^{2} \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5733.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: $$45.7782354788$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.4507648.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 637) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.90903$$ of defining polynomial Character $$\chi$$ $$=$$ 5733.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.264627 q^{2} -1.92997 q^{4} +1.43515 q^{5} +1.03998 q^{8} +O(q^{10})$$ $$q-0.264627 q^{2} -1.92997 q^{4} +1.43515 q^{5} +1.03998 q^{8} -0.379780 q^{10} -5.50474 q^{11} +1.00000 q^{13} +3.58474 q^{16} -4.83072 q^{17} +2.82036 q^{19} -2.76981 q^{20} +1.45670 q^{22} +5.99956 q^{23} -2.94033 q^{25} -0.264627 q^{26} -1.04188 q^{29} +9.20895 q^{31} -3.02857 q^{32} +1.27834 q^{34} +0.612497 q^{37} -0.746342 q^{38} +1.49252 q^{40} +10.6196 q^{41} -8.43685 q^{43} +10.6240 q^{44} -1.58764 q^{46} -2.40922 q^{47} +0.778091 q^{50} -1.92997 q^{52} +1.82959 q^{53} -7.90015 q^{55} +0.275709 q^{58} +0.870914 q^{59} -3.33253 q^{61} -2.43693 q^{62} -6.36804 q^{64} +1.43515 q^{65} -6.62741 q^{67} +9.32316 q^{68} +6.85856 q^{71} +3.14147 q^{73} -0.162083 q^{74} -5.44322 q^{76} -17.5723 q^{79} +5.14465 q^{80} -2.81022 q^{82} -11.4525 q^{83} -6.93283 q^{85} +2.23261 q^{86} -5.72479 q^{88} +0.995318 q^{89} -11.5790 q^{92} +0.637545 q^{94} +4.04765 q^{95} +13.5090 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{4} - 6 q^{5}+O(q^{10})$$ 6 * q + 4 * q^4 - 6 * q^5 $$6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} - 30 q^{47} - 8 q^{50} + 4 q^{52} + 14 q^{53} - 8 q^{55} - 8 q^{58} - 24 q^{59} - 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} - 28 q^{68} - 8 q^{71} - 6 q^{73} + 12 q^{74} - 16 q^{76} - 22 q^{79} + 28 q^{80} - 40 q^{82} - 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} - 26 q^{89} - 20 q^{92} - 32 q^{94} + 6 q^{95} - 14 q^{97}+O(q^{100})$$ 6 * q + 4 * q^4 - 6 * q^5 + 4 * q^10 - 4 * q^11 + 6 * q^13 - 16 * q^17 + 2 * q^19 - 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 + 6 * q^31 + 20 * q^32 - 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 - 30 * q^47 - 8 * q^50 + 4 * q^52 + 14 * q^53 - 8 * q^55 - 8 * q^58 - 24 * q^59 - 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 - 28 * q^68 - 8 * q^71 - 6 * q^73 + 12 * q^74 - 16 * q^76 - 22 * q^79 + 28 * q^80 - 40 * q^82 - 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 - 26 * q^89 - 20 * q^92 - 32 * q^94 + 6 * q^95 - 14 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.264627 −0.187119 −0.0935596 0.995614i $$-0.529825\pi$$
−0.0935596 + 0.995614i $$0.529825\pi$$
$$3$$ 0 0
$$4$$ −1.92997 −0.964986
$$5$$ 1.43515 0.641820 0.320910 0.947110i $$-0.396011\pi$$
0.320910 + 0.947110i $$0.396011\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 1.03998 0.367687
$$9$$ 0 0
$$10$$ −0.379780 −0.120097
$$11$$ −5.50474 −1.65974 −0.829871 0.557955i $$-0.811586\pi$$
−0.829871 + 0.557955i $$0.811586\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 3.58474 0.896185
$$17$$ −4.83072 −1.17162 −0.585811 0.810448i $$-0.699224\pi$$
−0.585811 + 0.810448i $$0.699224\pi$$
$$18$$ 0 0
$$19$$ 2.82036 0.647035 0.323518 0.946222i $$-0.395135\pi$$
0.323518 + 0.946222i $$0.395135\pi$$
$$20$$ −2.76981 −0.619348
$$21$$ 0 0
$$22$$ 1.45670 0.310570
$$23$$ 5.99956 1.25100 0.625498 0.780226i $$-0.284896\pi$$
0.625498 + 0.780226i $$0.284896\pi$$
$$24$$ 0 0
$$25$$ −2.94033 −0.588067
$$26$$ −0.264627 −0.0518975
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1.04188 −0.193472 −0.0967361 0.995310i $$-0.530840\pi$$
−0.0967361 + 0.995310i $$0.530840\pi$$
$$30$$ 0 0
$$31$$ 9.20895 1.65398 0.826988 0.562219i $$-0.190052\pi$$
0.826988 + 0.562219i $$0.190052\pi$$
$$32$$ −3.02857 −0.535380
$$33$$ 0 0
$$34$$ 1.27834 0.219233
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.612497 0.100694 0.0503470 0.998732i $$-0.483967\pi$$
0.0503470 + 0.998732i $$0.483967\pi$$
$$38$$ −0.746342 −0.121073
$$39$$ 0 0
$$40$$ 1.49252 0.235989
$$41$$ 10.6196 1.65850 0.829249 0.558879i $$-0.188768\pi$$
0.829249 + 0.558879i $$0.188768\pi$$
$$42$$ 0 0
$$43$$ −8.43685 −1.28661 −0.643304 0.765611i $$-0.722437\pi$$
−0.643304 + 0.765611i $$0.722437\pi$$
$$44$$ 10.6240 1.60163
$$45$$ 0 0
$$46$$ −1.58764 −0.234085
$$47$$ −2.40922 −0.351422 −0.175711 0.984442i $$-0.556222\pi$$
−0.175711 + 0.984442i $$0.556222\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0.778091 0.110039
$$51$$ 0 0
$$52$$ −1.92997 −0.267639
$$53$$ 1.82959 0.251313 0.125657 0.992074i $$-0.459896\pi$$
0.125657 + 0.992074i $$0.459896\pi$$
$$54$$ 0 0
$$55$$ −7.90015 −1.06526
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0.275709 0.0362024
$$59$$ 0.870914 0.113383 0.0566917 0.998392i $$-0.481945\pi$$
0.0566917 + 0.998392i $$0.481945\pi$$
$$60$$ 0 0
$$61$$ −3.33253 −0.426686 −0.213343 0.976977i $$-0.568435\pi$$
−0.213343 + 0.976977i $$0.568435\pi$$
$$62$$ −2.43693 −0.309491
$$63$$ 0 0
$$64$$ −6.36804 −0.796005
$$65$$ 1.43515 0.178009
$$66$$ 0 0
$$67$$ −6.62741 −0.809667 −0.404833 0.914390i $$-0.632671\pi$$
−0.404833 + 0.914390i $$0.632671\pi$$
$$68$$ 9.32316 1.13060
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.85856 0.813961 0.406980 0.913437i $$-0.366582\pi$$
0.406980 + 0.913437i $$0.366582\pi$$
$$72$$ 0 0
$$73$$ 3.14147 0.367682 0.183841 0.982956i $$-0.441147\pi$$
0.183841 + 0.982956i $$0.441147\pi$$
$$74$$ −0.162083 −0.0188418
$$75$$ 0 0
$$76$$ −5.44322 −0.624380
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −17.5723 −1.97704 −0.988518 0.151101i $$-0.951718\pi$$
−0.988518 + 0.151101i $$0.951718\pi$$
$$80$$ 5.14465 0.575190
$$81$$ 0 0
$$82$$ −2.81022 −0.310337
$$83$$ −11.4525 −1.25708 −0.628538 0.777779i $$-0.716346\pi$$
−0.628538 + 0.777779i $$0.716346\pi$$
$$84$$ 0 0
$$85$$ −6.93283 −0.751971
$$86$$ 2.23261 0.240749
$$87$$ 0 0
$$88$$ −5.72479 −0.610265
$$89$$ 0.995318 0.105503 0.0527517 0.998608i $$-0.483201\pi$$
0.0527517 + 0.998608i $$0.483201\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −11.5790 −1.20719
$$93$$ 0 0
$$94$$ 0.637545 0.0657577
$$95$$ 4.04765 0.415280
$$96$$ 0 0
$$97$$ 13.5090 1.37163 0.685817 0.727774i $$-0.259445\pi$$
0.685817 + 0.727774i $$0.259445\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 5.67477 0.567477
$$101$$ 1.00807 0.100306 0.0501532 0.998742i $$-0.484029\pi$$
0.0501532 + 0.998742i $$0.484029\pi$$
$$102$$ 0 0
$$103$$ −12.7754 −1.25880 −0.629401 0.777081i $$-0.716700\pi$$
−0.629401 + 0.777081i $$0.716700\pi$$
$$104$$ 1.03998 0.101978
$$105$$ 0 0
$$106$$ −0.484157 −0.0470255
$$107$$ 0.685495 0.0662693 0.0331347 0.999451i $$-0.489451\pi$$
0.0331347 + 0.999451i $$0.489451\pi$$
$$108$$ 0 0
$$109$$ −2.90344 −0.278099 −0.139050 0.990285i $$-0.544405\pi$$
−0.139050 + 0.990285i $$0.544405\pi$$
$$110$$ 2.09059 0.199330
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −12.0315 −1.13183 −0.565915 0.824464i $$-0.691477\pi$$
−0.565915 + 0.824464i $$0.691477\pi$$
$$114$$ 0 0
$$115$$ 8.61029 0.802914
$$116$$ 2.01080 0.186698
$$117$$ 0 0
$$118$$ −0.230467 −0.0212162
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 19.3022 1.75474
$$122$$ 0.881875 0.0798412
$$123$$ 0 0
$$124$$ −17.7730 −1.59606
$$125$$ −11.3956 −1.01925
$$126$$ 0 0
$$127$$ 15.6659 1.39012 0.695062 0.718950i $$-0.255377\pi$$
0.695062 + 0.718950i $$0.255377\pi$$
$$128$$ 7.74229 0.684328
$$129$$ 0 0
$$130$$ −0.379780 −0.0333089
$$131$$ −12.1273 −1.05957 −0.529784 0.848132i $$-0.677727\pi$$
−0.529784 + 0.848132i $$0.677727\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 1.75379 0.151504
$$135$$ 0 0
$$136$$ −5.02383 −0.430790
$$137$$ −15.9375 −1.36163 −0.680815 0.732456i $$-0.738374\pi$$
−0.680815 + 0.732456i $$0.738374\pi$$
$$138$$ 0 0
$$139$$ −6.64088 −0.563272 −0.281636 0.959521i $$-0.590877\pi$$
−0.281636 + 0.959521i $$0.590877\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1.81496 −0.152308
$$143$$ −5.50474 −0.460330
$$144$$ 0 0
$$145$$ −1.49526 −0.124174
$$146$$ −0.831317 −0.0688003
$$147$$ 0 0
$$148$$ −1.18210 −0.0971683
$$149$$ 19.5502 1.60162 0.800809 0.598920i $$-0.204403\pi$$
0.800809 + 0.598920i $$0.204403\pi$$
$$150$$ 0 0
$$151$$ 10.6880 0.869779 0.434890 0.900484i $$-0.356787\pi$$
0.434890 + 0.900484i $$0.356787\pi$$
$$152$$ 2.93311 0.237906
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 13.2163 1.06156
$$156$$ 0 0
$$157$$ −15.0734 −1.20299 −0.601496 0.798876i $$-0.705428\pi$$
−0.601496 + 0.798876i $$0.705428\pi$$
$$158$$ 4.65009 0.369942
$$159$$ 0 0
$$160$$ −4.34646 −0.343618
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −23.8135 −1.86521 −0.932607 0.360894i $$-0.882472\pi$$
−0.932607 + 0.360894i $$0.882472\pi$$
$$164$$ −20.4955 −1.60043
$$165$$ 0 0
$$166$$ 3.03064 0.235223
$$167$$ 7.12371 0.551249 0.275625 0.961265i $$-0.411115\pi$$
0.275625 + 0.961265i $$0.411115\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 1.83461 0.140708
$$171$$ 0 0
$$172$$ 16.2829 1.24156
$$173$$ −11.2367 −0.854309 −0.427155 0.904179i $$-0.640484\pi$$
−0.427155 + 0.904179i $$0.640484\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −19.7331 −1.48744
$$177$$ 0 0
$$178$$ −0.263387 −0.0197417
$$179$$ 13.1945 0.986204 0.493102 0.869972i $$-0.335863\pi$$
0.493102 + 0.869972i $$0.335863\pi$$
$$180$$ 0 0
$$181$$ −13.7414 −1.02139 −0.510696 0.859761i $$-0.670612\pi$$
−0.510696 + 0.859761i $$0.670612\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 6.23939 0.459974
$$185$$ 0.879028 0.0646274
$$186$$ 0 0
$$187$$ 26.5919 1.94459
$$188$$ 4.64974 0.339117
$$189$$ 0 0
$$190$$ −1.07112 −0.0777069
$$191$$ 16.3307 1.18165 0.590824 0.806800i $$-0.298803\pi$$
0.590824 + 0.806800i $$0.298803\pi$$
$$192$$ 0 0
$$193$$ −14.0533 −1.01158 −0.505790 0.862656i $$-0.668799\pi$$
−0.505790 + 0.862656i $$0.668799\pi$$
$$194$$ −3.57485 −0.256659
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.46898 0.104660 0.0523302 0.998630i $$-0.483335\pi$$
0.0523302 + 0.998630i $$0.483335\pi$$
$$198$$ 0 0
$$199$$ −13.3772 −0.948285 −0.474142 0.880448i $$-0.657242\pi$$
−0.474142 + 0.880448i $$0.657242\pi$$
$$200$$ −3.05787 −0.216224
$$201$$ 0 0
$$202$$ −0.266761 −0.0187692
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 15.2407 1.06446
$$206$$ 3.38072 0.235546
$$207$$ 0 0
$$208$$ 3.58474 0.248557
$$209$$ −15.5254 −1.07391
$$210$$ 0 0
$$211$$ 3.47044 0.238915 0.119457 0.992839i $$-0.461885\pi$$
0.119457 + 0.992839i $$0.461885\pi$$
$$212$$ −3.53105 −0.242514
$$213$$ 0 0
$$214$$ −0.181400 −0.0124003
$$215$$ −12.1082 −0.825771
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0.768328 0.0520378
$$219$$ 0 0
$$220$$ 15.2471 1.02796
$$221$$ −4.83072 −0.324950
$$222$$ 0 0
$$223$$ −9.91318 −0.663836 −0.331918 0.943308i $$-0.607696\pi$$
−0.331918 + 0.943308i $$0.607696\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 3.18386 0.211787
$$227$$ −12.0727 −0.801292 −0.400646 0.916233i $$-0.631214\pi$$
−0.400646 + 0.916233i $$0.631214\pi$$
$$228$$ 0 0
$$229$$ 4.05171 0.267745 0.133872 0.990999i $$-0.457259\pi$$
0.133872 + 0.990999i $$0.457259\pi$$
$$230$$ −2.27851 −0.150241
$$231$$ 0 0
$$232$$ −1.08353 −0.0711372
$$233$$ −12.5450 −0.821850 −0.410925 0.911669i $$-0.634794\pi$$
−0.410925 + 0.911669i $$0.634794\pi$$
$$234$$ 0 0
$$235$$ −3.45761 −0.225549
$$236$$ −1.68084 −0.109413
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −13.3463 −0.863299 −0.431649 0.902042i $$-0.642068\pi$$
−0.431649 + 0.902042i $$0.642068\pi$$
$$240$$ 0 0
$$241$$ 20.3854 1.31314 0.656568 0.754267i $$-0.272007\pi$$
0.656568 + 0.754267i $$0.272007\pi$$
$$242$$ −5.10787 −0.328346
$$243$$ 0 0
$$244$$ 6.43169 0.411747
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.82036 0.179455
$$248$$ 9.57708 0.608145
$$249$$ 0 0
$$250$$ 3.01558 0.190722
$$251$$ −17.1921 −1.08515 −0.542577 0.840006i $$-0.682551\pi$$
−0.542577 + 0.840006i $$0.682551\pi$$
$$252$$ 0 0
$$253$$ −33.0260 −2.07633
$$254$$ −4.14561 −0.260119
$$255$$ 0 0
$$256$$ 10.6873 0.667954
$$257$$ 7.64695 0.477004 0.238502 0.971142i $$-0.423344\pi$$
0.238502 + 0.971142i $$0.423344\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2.76981 −0.171776
$$261$$ 0 0
$$262$$ 3.20921 0.198266
$$263$$ 0.101037 0.00623022 0.00311511 0.999995i $$-0.499008\pi$$
0.00311511 + 0.999995i $$0.499008\pi$$
$$264$$ 0 0
$$265$$ 2.62574 0.161298
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 12.7907 0.781318
$$269$$ 7.56852 0.461461 0.230730 0.973018i $$-0.425889\pi$$
0.230730 + 0.973018i $$0.425889\pi$$
$$270$$ 0 0
$$271$$ −13.8554 −0.841653 −0.420826 0.907141i $$-0.638260\pi$$
−0.420826 + 0.907141i $$0.638260\pi$$
$$272$$ −17.3169 −1.04999
$$273$$ 0 0
$$274$$ 4.21748 0.254787
$$275$$ 16.1858 0.976039
$$276$$ 0 0
$$277$$ 0.552935 0.0332226 0.0166113 0.999862i $$-0.494712\pi$$
0.0166113 + 0.999862i $$0.494712\pi$$
$$278$$ 1.75735 0.105399
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1.14667 −0.0684043 −0.0342022 0.999415i $$-0.510889\pi$$
−0.0342022 + 0.999415i $$0.510889\pi$$
$$282$$ 0 0
$$283$$ −4.05396 −0.240983 −0.120491 0.992714i $$-0.538447\pi$$
−0.120491 + 0.992714i $$0.538447\pi$$
$$284$$ −13.2368 −0.785461
$$285$$ 0 0
$$286$$ 1.45670 0.0861365
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 6.33588 0.372699
$$290$$ 0.395685 0.0232354
$$291$$ 0 0
$$292$$ −6.06296 −0.354808
$$293$$ −15.0649 −0.880102 −0.440051 0.897973i $$-0.645040\pi$$
−0.440051 + 0.897973i $$0.645040\pi$$
$$294$$ 0 0
$$295$$ 1.24990 0.0727718
$$296$$ 0.636982 0.0370238
$$297$$ 0 0
$$298$$ −5.17351 −0.299694
$$299$$ 5.99956 0.346964
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −2.82834 −0.162752
$$303$$ 0 0
$$304$$ 10.1103 0.579863
$$305$$ −4.78269 −0.273856
$$306$$ 0 0
$$307$$ −19.9408 −1.13808 −0.569040 0.822310i $$-0.692685\pi$$
−0.569040 + 0.822310i $$0.692685\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −3.49737 −0.198637
$$311$$ 10.8956 0.617833 0.308916 0.951089i $$-0.400034\pi$$
0.308916 + 0.951089i $$0.400034\pi$$
$$312$$ 0 0
$$313$$ −0.0519190 −0.00293464 −0.00146732 0.999999i $$-0.500467\pi$$
−0.00146732 + 0.999999i $$0.500467\pi$$
$$314$$ 3.98883 0.225103
$$315$$ 0 0
$$316$$ 33.9140 1.90781
$$317$$ −16.1010 −0.904321 −0.452161 0.891937i $$-0.649347\pi$$
−0.452161 + 0.891937i $$0.649347\pi$$
$$318$$ 0 0
$$319$$ 5.73528 0.321114
$$320$$ −9.13912 −0.510892
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −13.6244 −0.758081
$$324$$ 0 0
$$325$$ −2.94033 −0.163100
$$326$$ 6.30167 0.349017
$$327$$ 0 0
$$328$$ 11.0441 0.609808
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 30.7862 1.69216 0.846081 0.533054i $$-0.178956\pi$$
0.846081 + 0.533054i $$0.178956\pi$$
$$332$$ 22.1030 1.21306
$$333$$ 0 0
$$334$$ −1.88512 −0.103149
$$335$$ −9.51135 −0.519661
$$336$$ 0 0
$$337$$ −2.41842 −0.131740 −0.0658700 0.997828i $$-0.520982\pi$$
−0.0658700 + 0.997828i $$0.520982\pi$$
$$338$$ −0.264627 −0.0143938
$$339$$ 0 0
$$340$$ 13.3802 0.725642
$$341$$ −50.6929 −2.74517
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −8.77411 −0.473069
$$345$$ 0 0
$$346$$ 2.97353 0.159858
$$347$$ 0.492527 0.0264403 0.0132201 0.999913i $$-0.495792\pi$$
0.0132201 + 0.999913i $$0.495792\pi$$
$$348$$ 0 0
$$349$$ −11.9442 −0.639356 −0.319678 0.947526i $$-0.603575\pi$$
−0.319678 + 0.947526i $$0.603575\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 16.6715 0.888593
$$353$$ −15.5299 −0.826575 −0.413288 0.910601i $$-0.635620\pi$$
−0.413288 + 0.910601i $$0.635620\pi$$
$$354$$ 0 0
$$355$$ 9.84308 0.522417
$$356$$ −1.92094 −0.101809
$$357$$ 0 0
$$358$$ −3.49162 −0.184538
$$359$$ 8.50709 0.448987 0.224493 0.974476i $$-0.427927\pi$$
0.224493 + 0.974476i $$0.427927\pi$$
$$360$$ 0 0
$$361$$ −11.0456 −0.581345
$$362$$ 3.63635 0.191122
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4.50850 0.235985
$$366$$ 0 0
$$367$$ 5.19084 0.270960 0.135480 0.990780i $$-0.456742\pi$$
0.135480 + 0.990780i $$0.456742\pi$$
$$368$$ 21.5069 1.12112
$$369$$ 0 0
$$370$$ −0.232614 −0.0120930
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.1427 −0.525169 −0.262585 0.964909i $$-0.584575\pi$$
−0.262585 + 0.964909i $$0.584575\pi$$
$$374$$ −7.03692 −0.363870
$$375$$ 0 0
$$376$$ −2.50553 −0.129213
$$377$$ −1.04188 −0.0536595
$$378$$ 0 0
$$379$$ −3.63670 −0.186805 −0.0934024 0.995628i $$-0.529774\pi$$
−0.0934024 + 0.995628i $$0.529774\pi$$
$$380$$ −7.81186 −0.400740
$$381$$ 0 0
$$382$$ −4.32154 −0.221109
$$383$$ −4.60281 −0.235192 −0.117596 0.993061i $$-0.537519\pi$$
−0.117596 + 0.993061i $$0.537519\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 3.71888 0.189286
$$387$$ 0 0
$$388$$ −26.0721 −1.32361
$$389$$ −19.6104 −0.994286 −0.497143 0.867669i $$-0.665618\pi$$
−0.497143 + 0.867669i $$0.665618\pi$$
$$390$$ 0 0
$$391$$ −28.9822 −1.46569
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −0.388731 −0.0195840
$$395$$ −25.2189 −1.26890
$$396$$ 0 0
$$397$$ −19.8635 −0.996919 −0.498459 0.866913i $$-0.666101\pi$$
−0.498459 + 0.866913i $$0.666101\pi$$
$$398$$ 3.53996 0.177442
$$399$$ 0 0
$$400$$ −10.5403 −0.527017
$$401$$ 15.1117 0.754644 0.377322 0.926082i $$-0.376845\pi$$
0.377322 + 0.926082i $$0.376845\pi$$
$$402$$ 0 0
$$403$$ 9.20895 0.458731
$$404$$ −1.94554 −0.0967943
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3.37164 −0.167126
$$408$$ 0 0
$$409$$ −35.2443 −1.74272 −0.871360 0.490644i $$-0.836762\pi$$
−0.871360 + 0.490644i $$0.836762\pi$$
$$410$$ −4.03310 −0.199181
$$411$$ 0 0
$$412$$ 24.6563 1.21473
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −16.4361 −0.806816
$$416$$ −3.02857 −0.148488
$$417$$ 0 0
$$418$$ 4.10842 0.200950
$$419$$ −1.50468 −0.0735084 −0.0367542 0.999324i $$-0.511702\pi$$
−0.0367542 + 0.999324i $$0.511702\pi$$
$$420$$ 0 0
$$421$$ −24.5079 −1.19444 −0.597221 0.802077i $$-0.703728\pi$$
−0.597221 + 0.802077i $$0.703728\pi$$
$$422$$ −0.918370 −0.0447056
$$423$$ 0 0
$$424$$ 1.90272 0.0924045
$$425$$ 14.2039 0.688992
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −1.32299 −0.0639490
$$429$$ 0 0
$$430$$ 3.20414 0.154518
$$431$$ −41.0655 −1.97805 −0.989027 0.147732i $$-0.952803\pi$$
−0.989027 + 0.147732i $$0.952803\pi$$
$$432$$ 0 0
$$433$$ 6.65603 0.319869 0.159934 0.987128i $$-0.448872\pi$$
0.159934 + 0.987128i $$0.448872\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 5.60357 0.268362
$$437$$ 16.9209 0.809438
$$438$$ 0 0
$$439$$ 8.22990 0.392792 0.196396 0.980525i $$-0.437076\pi$$
0.196396 + 0.980525i $$0.437076\pi$$
$$440$$ −8.21596 −0.391681
$$441$$ 0 0
$$442$$ 1.27834 0.0608043
$$443$$ −17.6856 −0.840266 −0.420133 0.907463i $$-0.638017\pi$$
−0.420133 + 0.907463i $$0.638017\pi$$
$$444$$ 0 0
$$445$$ 1.42843 0.0677143
$$446$$ 2.62329 0.124216
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 14.5250 0.685477 0.342738 0.939431i $$-0.388646\pi$$
0.342738 + 0.939431i $$0.388646\pi$$
$$450$$ 0 0
$$451$$ −58.4580 −2.75268
$$452$$ 23.2205 1.09220
$$453$$ 0 0
$$454$$ 3.19475 0.149937
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.78919 0.177251 0.0886255 0.996065i $$-0.471753\pi$$
0.0886255 + 0.996065i $$0.471753\pi$$
$$458$$ −1.07219 −0.0501002
$$459$$ 0 0
$$460$$ −16.6176 −0.774801
$$461$$ −13.1107 −0.610627 −0.305314 0.952252i $$-0.598761\pi$$
−0.305314 + 0.952252i $$0.598761\pi$$
$$462$$ 0 0
$$463$$ 15.3027 0.711176 0.355588 0.934643i $$-0.384281\pi$$
0.355588 + 0.934643i $$0.384281\pi$$
$$464$$ −3.73487 −0.173387
$$465$$ 0 0
$$466$$ 3.31974 0.153784
$$467$$ −30.7738 −1.42404 −0.712022 0.702158i $$-0.752220\pi$$
−0.712022 + 0.702158i $$0.752220\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0.914975 0.0422046
$$471$$ 0 0
$$472$$ 0.905729 0.0416896
$$473$$ 46.4427 2.13544
$$474$$ 0 0
$$475$$ −8.29280 −0.380500
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 3.53178 0.161540
$$479$$ 1.71740 0.0784699 0.0392350 0.999230i $$-0.487508\pi$$
0.0392350 + 0.999230i $$0.487508\pi$$
$$480$$ 0 0
$$481$$ 0.612497 0.0279275
$$482$$ −5.39451 −0.245713
$$483$$ 0 0
$$484$$ −37.2527 −1.69330
$$485$$ 19.3875 0.880342
$$486$$ 0 0
$$487$$ 22.6805 1.02775 0.513877 0.857864i $$-0.328209\pi$$
0.513877 + 0.857864i $$0.328209\pi$$
$$488$$ −3.46575 −0.156887
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 13.1366 0.592846 0.296423 0.955057i $$-0.404206\pi$$
0.296423 + 0.955057i $$0.404206\pi$$
$$492$$ 0 0
$$493$$ 5.03303 0.226676
$$494$$ −0.746342 −0.0335795
$$495$$ 0 0
$$496$$ 33.0117 1.48227
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 14.0395 0.628495 0.314248 0.949341i $$-0.398248\pi$$
0.314248 + 0.949341i $$0.398248\pi$$
$$500$$ 21.9932 0.983566
$$501$$ 0 0
$$502$$ 4.54948 0.203053
$$503$$ −0.367865 −0.0164023 −0.00820114 0.999966i $$-0.502611\pi$$
−0.00820114 + 0.999966i $$0.502611\pi$$
$$504$$ 0 0
$$505$$ 1.44673 0.0643786
$$506$$ 8.73957 0.388521
$$507$$ 0 0
$$508$$ −30.2348 −1.34145
$$509$$ −41.2319 −1.82757 −0.913787 0.406194i $$-0.866856\pi$$
−0.913787 + 0.406194i $$0.866856\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −18.3127 −0.809315
$$513$$ 0 0
$$514$$ −2.02359 −0.0892566
$$515$$ −18.3347 −0.807925
$$516$$ 0 0
$$517$$ 13.2622 0.583269
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 1.49252 0.0654515
$$521$$ −1.04099 −0.0456065 −0.0228032 0.999740i $$-0.507259\pi$$
−0.0228032 + 0.999740i $$0.507259\pi$$
$$522$$ 0 0
$$523$$ −20.0209 −0.875451 −0.437726 0.899109i $$-0.644216\pi$$
−0.437726 + 0.899109i $$0.644216\pi$$
$$524$$ 23.4054 1.02247
$$525$$ 0 0
$$526$$ −0.0267371 −0.00116579
$$527$$ −44.4859 −1.93784
$$528$$ 0 0
$$529$$ 12.9947 0.564989
$$530$$ −0.694840 −0.0301819
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 10.6196 0.459985
$$534$$ 0 0
$$535$$ 0.983791 0.0425330
$$536$$ −6.89234 −0.297704
$$537$$ 0 0
$$538$$ −2.00283 −0.0863481
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −9.78749 −0.420797 −0.210399 0.977616i $$-0.567476\pi$$
−0.210399 + 0.977616i $$0.567476\pi$$
$$542$$ 3.66649 0.157489
$$543$$ 0 0
$$544$$ 14.6302 0.627263
$$545$$ −4.16689 −0.178490
$$546$$ 0 0
$$547$$ −2.56174 −0.109532 −0.0547660 0.998499i $$-0.517441\pi$$
−0.0547660 + 0.998499i $$0.517441\pi$$
$$548$$ 30.7589 1.31395
$$549$$ 0 0
$$550$$ −4.28319 −0.182636
$$551$$ −2.93848 −0.125183
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −0.146321 −0.00621660
$$555$$ 0 0
$$556$$ 12.8167 0.543550
$$557$$ 27.4442 1.16285 0.581424 0.813601i $$-0.302496\pi$$
0.581424 + 0.813601i $$0.302496\pi$$
$$558$$ 0 0
$$559$$ −8.43685 −0.356841
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0.303438 0.0127998
$$563$$ −0.162708 −0.00685734 −0.00342867 0.999994i $$-0.501091\pi$$
−0.00342867 + 0.999994i $$0.501091\pi$$
$$564$$ 0 0
$$565$$ −17.2671 −0.726431
$$566$$ 1.07278 0.0450925
$$567$$ 0 0
$$568$$ 7.13273 0.299283
$$569$$ 12.3901 0.519419 0.259709 0.965687i $$-0.416373\pi$$
0.259709 + 0.965687i $$0.416373\pi$$
$$570$$ 0 0
$$571$$ −21.8122 −0.912810 −0.456405 0.889772i $$-0.650863\pi$$
−0.456405 + 0.889772i $$0.650863\pi$$
$$572$$ 10.6240 0.444212
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −17.6407 −0.735669
$$576$$ 0 0
$$577$$ 6.06583 0.252524 0.126262 0.991997i $$-0.459702\pi$$
0.126262 + 0.991997i $$0.459702\pi$$
$$578$$ −1.67664 −0.0697391
$$579$$ 0 0
$$580$$ 2.88581 0.119827
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −10.0714 −0.417115
$$584$$ 3.26705 0.135192
$$585$$ 0 0
$$586$$ 3.98658 0.164684
$$587$$ −20.5820 −0.849510 −0.424755 0.905308i $$-0.639640\pi$$
−0.424755 + 0.905308i $$0.639640\pi$$
$$588$$ 0 0
$$589$$ 25.9726 1.07018
$$590$$ −0.330756 −0.0136170
$$591$$ 0 0
$$592$$ 2.19564 0.0902404
$$593$$ 24.0397 0.987190 0.493595 0.869692i $$-0.335682\pi$$
0.493595 + 0.869692i $$0.335682\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −37.7314 −1.54554
$$597$$ 0 0
$$598$$ −1.58764 −0.0649236
$$599$$ −32.2523 −1.31779 −0.658896 0.752234i $$-0.728976\pi$$
−0.658896 + 0.752234i $$0.728976\pi$$
$$600$$ 0 0
$$601$$ 5.21454 0.212705 0.106353 0.994328i $$-0.466083\pi$$
0.106353 + 0.994328i $$0.466083\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −20.6276 −0.839325
$$605$$ 27.7016 1.12623
$$606$$ 0 0
$$607$$ 9.07048 0.368160 0.184080 0.982911i $$-0.441070\pi$$
0.184080 + 0.982911i $$0.441070\pi$$
$$608$$ −8.54165 −0.346410
$$609$$ 0 0
$$610$$ 1.26563 0.0512437
$$611$$ −2.40922 −0.0974668
$$612$$ 0 0
$$613$$ 20.0920 0.811507 0.405754 0.913983i $$-0.367009\pi$$
0.405754 + 0.913983i $$0.367009\pi$$
$$614$$ 5.27686 0.212957
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.9556 0.521572 0.260786 0.965397i $$-0.416018\pi$$
0.260786 + 0.965397i $$0.416018\pi$$
$$618$$ 0 0
$$619$$ −44.3644 −1.78316 −0.891578 0.452866i $$-0.850402\pi$$
−0.891578 + 0.452866i $$0.850402\pi$$
$$620$$ −25.5070 −1.02439
$$621$$ 0 0
$$622$$ −2.88327 −0.115608
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.65276 −0.0661105
$$626$$ 0.0137392 0.000549127 0
$$627$$ 0 0
$$628$$ 29.0913 1.16087
$$629$$ −2.95880 −0.117975
$$630$$ 0 0
$$631$$ 6.61717 0.263426 0.131713 0.991288i $$-0.457952\pi$$
0.131713 + 0.991288i $$0.457952\pi$$
$$632$$ −18.2747 −0.726930
$$633$$ 0 0
$$634$$ 4.26075 0.169216
$$635$$ 22.4830 0.892210
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −1.51771 −0.0600866
$$639$$ 0 0
$$640$$ 11.1114 0.439216
$$641$$ −18.9567 −0.748744 −0.374372 0.927279i $$-0.622142\pi$$
−0.374372 + 0.927279i $$0.622142\pi$$
$$642$$ 0 0
$$643$$ 13.4019 0.528517 0.264259 0.964452i $$-0.414873\pi$$
0.264259 + 0.964452i $$0.414873\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 3.60537 0.141851
$$647$$ −42.7588 −1.68102 −0.840511 0.541794i $$-0.817745\pi$$
−0.840511 + 0.541794i $$0.817745\pi$$
$$648$$ 0 0
$$649$$ −4.79416 −0.188187
$$650$$ 0.778091 0.0305192
$$651$$ 0 0
$$652$$ 45.9593 1.79991
$$653$$ −10.9852 −0.429884 −0.214942 0.976627i $$-0.568956\pi$$
−0.214942 + 0.976627i $$0.568956\pi$$
$$654$$ 0 0
$$655$$ −17.4046 −0.680052
$$656$$ 38.0684 1.48632
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 17.7614 0.691884 0.345942 0.938256i $$-0.387559\pi$$
0.345942 + 0.938256i $$0.387559\pi$$
$$660$$ 0 0
$$661$$ 8.18255 0.318264 0.159132 0.987257i $$-0.449130\pi$$
0.159132 + 0.987257i $$0.449130\pi$$
$$662$$ −8.14685 −0.316636
$$663$$ 0 0
$$664$$ −11.9103 −0.462210
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.25082 −0.242033
$$668$$ −13.7486 −0.531948
$$669$$ 0 0
$$670$$ 2.51696 0.0972385
$$671$$ 18.3447 0.708189
$$672$$ 0 0
$$673$$ 9.30129 0.358539 0.179269 0.983800i $$-0.442627\pi$$
0.179269 + 0.983800i $$0.442627\pi$$
$$674$$ 0.639979 0.0246511
$$675$$ 0 0
$$676$$ −1.92997 −0.0742297
$$677$$ 41.1552 1.58172 0.790862 0.611994i $$-0.209633\pi$$
0.790862 + 0.611994i $$0.209633\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −7.20997 −0.276490
$$681$$ 0 0
$$682$$ 13.4147 0.513675
$$683$$ −39.2842 −1.50317 −0.751583 0.659638i $$-0.770709\pi$$
−0.751583 + 0.659638i $$0.770709\pi$$
$$684$$ 0 0
$$685$$ −22.8727 −0.873921
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −30.2439 −1.15304
$$689$$ 1.82959 0.0697017
$$690$$ 0 0
$$691$$ −3.03355 −0.115402 −0.0577009 0.998334i $$-0.518377\pi$$
−0.0577009 + 0.998334i $$0.518377\pi$$
$$692$$ 21.6865 0.824397
$$693$$ 0 0
$$694$$ −0.130336 −0.00494748
$$695$$ −9.53069 −0.361520
$$696$$ 0 0
$$697$$ −51.3002 −1.94313
$$698$$ 3.16074 0.119636
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 26.2320 0.990767 0.495384 0.868674i $$-0.335028\pi$$
0.495384 + 0.868674i $$0.335028\pi$$
$$702$$ 0 0
$$703$$ 1.72746 0.0651525
$$704$$ 35.0544 1.32116
$$705$$ 0 0
$$706$$ 4.10963 0.154668
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −7.87770 −0.295853 −0.147927 0.988998i $$-0.547260\pi$$
−0.147927 + 0.988998i $$0.547260\pi$$
$$710$$ −2.60474 −0.0977542
$$711$$ 0 0
$$712$$ 1.03511 0.0387922
$$713$$ 55.2497 2.06912
$$714$$ 0 0
$$715$$ −7.90015 −0.295449
$$716$$ −25.4650 −0.951673
$$717$$ 0 0
$$718$$ −2.25120 −0.0840141
$$719$$ −45.6656 −1.70304 −0.851519 0.524323i $$-0.824318\pi$$
−0.851519 + 0.524323i $$0.824318\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 2.92295 0.108781
$$723$$ 0 0
$$724$$ 26.5206 0.985630
$$725$$ 3.06348 0.113775
$$726$$ 0 0
$$727$$ 37.5947 1.39431 0.697155 0.716921i $$-0.254449\pi$$
0.697155 + 0.716921i $$0.254449\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −1.19307 −0.0441574
$$731$$ 40.7561 1.50742
$$732$$ 0 0
$$733$$ 53.1810 1.96429 0.982143 0.188138i $$-0.0602451\pi$$
0.982143 + 0.188138i $$0.0602451\pi$$
$$734$$ −1.37363 −0.0507018
$$735$$ 0 0
$$736$$ −18.1701 −0.669758
$$737$$ 36.4822 1.34384
$$738$$ 0 0
$$739$$ 41.9633 1.54364 0.771822 0.635839i $$-0.219346\pi$$
0.771822 + 0.635839i $$0.219346\pi$$
$$740$$ −1.69650 −0.0623646
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 38.5424 1.41398 0.706991 0.707222i $$-0.250052\pi$$
0.706991 + 0.707222i $$0.250052\pi$$
$$744$$ 0 0
$$745$$ 28.0576 1.02795
$$746$$ 2.68403 0.0982693
$$747$$ 0 0
$$748$$ −51.3216 −1.87650
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 36.2434 1.32254 0.661270 0.750148i $$-0.270018\pi$$
0.661270 + 0.750148i $$0.270018\pi$$
$$752$$ −8.63644 −0.314939
$$753$$ 0 0
$$754$$ 0.275709 0.0100407
$$755$$ 15.3390 0.558242
$$756$$ 0 0
$$757$$ 19.4752 0.707837 0.353919 0.935276i $$-0.384849\pi$$
0.353919 + 0.935276i $$0.384849\pi$$
$$758$$ 0.962368 0.0349548
$$759$$ 0 0
$$760$$ 4.20946 0.152693
$$761$$ 51.9059 1.88159 0.940793 0.338981i $$-0.110082\pi$$
0.940793 + 0.338981i $$0.110082\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −31.5178 −1.14028
$$765$$ 0 0
$$766$$ 1.21803 0.0440090
$$767$$ 0.870914 0.0314469
$$768$$ 0 0
$$769$$ −7.31376 −0.263741 −0.131870 0.991267i $$-0.542098\pi$$
−0.131870 + 0.991267i $$0.542098\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 27.1225 0.976161
$$773$$ 14.1844 0.510178 0.255089 0.966918i $$-0.417895\pi$$
0.255089 + 0.966918i $$0.417895\pi$$
$$774$$ 0 0
$$775$$ −27.0774 −0.972649
$$776$$ 14.0491 0.504332
$$777$$ 0 0
$$778$$ 5.18943 0.186050
$$779$$ 29.9510 1.07311
$$780$$ 0 0
$$781$$ −37.7546 −1.35097
$$782$$ 7.66946 0.274259
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −21.6327 −0.772104
$$786$$ 0 0
$$787$$ 31.2777 1.11493 0.557465 0.830201i $$-0.311774\pi$$
0.557465 + 0.830201i $$0.311774\pi$$
$$788$$ −2.83509 −0.100996
$$789$$ 0 0
$$790$$ 6.67360 0.237436
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −3.33253 −0.118342
$$794$$ 5.25640 0.186543
$$795$$ 0 0
$$796$$ 25.8176 0.915082
$$797$$ −20.2422 −0.717017 −0.358509 0.933526i $$-0.616715\pi$$
−0.358509 + 0.933526i $$0.616715\pi$$
$$798$$ 0 0
$$799$$ 11.6383 0.411733
$$800$$ 8.90500 0.314839
$$801$$ 0 0
$$802$$ −3.99897 −0.141208
$$803$$ −17.2930 −0.610257
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −2.43693 −0.0858373
$$807$$ 0 0
$$808$$ 1.04836 0.0368813
$$809$$ −7.88265 −0.277139 −0.138570 0.990353i $$-0.544250\pi$$
−0.138570 + 0.990353i $$0.544250\pi$$
$$810$$ 0 0
$$811$$ −5.99962 −0.210675 −0.105338 0.994437i $$-0.533592\pi$$
−0.105338 + 0.994437i $$0.533592\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0.892226 0.0312725
$$815$$ −34.1760 −1.19713
$$816$$ 0 0
$$817$$ −23.7950 −0.832480
$$818$$ 9.32659 0.326096
$$819$$ 0 0
$$820$$ −29.4142 −1.02719
$$821$$ 19.1692 0.669011 0.334505 0.942394i $$-0.391431\pi$$
0.334505 + 0.942394i $$0.391431\pi$$
$$822$$ 0 0
$$823$$ −30.3735 −1.05875 −0.529376 0.848387i $$-0.677574\pi$$
−0.529376 + 0.848387i $$0.677574\pi$$
$$824$$ −13.2861 −0.462845
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 14.6870 0.510717 0.255359 0.966846i $$-0.417807\pi$$
0.255359 + 0.966846i $$0.417807\pi$$
$$828$$ 0 0
$$829$$ −34.9985 −1.21555 −0.607774 0.794110i $$-0.707938\pi$$
−0.607774 + 0.794110i $$0.707938\pi$$
$$830$$ 4.34943 0.150971
$$831$$ 0 0
$$832$$ −6.36804 −0.220772
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 10.2236 0.353803
$$836$$ 29.9635 1.03631
$$837$$ 0 0
$$838$$ 0.398178 0.0137548
$$839$$ −27.6333 −0.954008 −0.477004 0.878901i $$-0.658277\pi$$
−0.477004 + 0.878901i $$0.658277\pi$$
$$840$$ 0 0
$$841$$ −27.9145 −0.962568
$$842$$ 6.48544 0.223503
$$843$$ 0 0
$$844$$ −6.69785 −0.230550
$$845$$ 1.43515 0.0493708
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6.55859 0.225223
$$849$$ 0 0
$$850$$ −3.75874 −0.128924
$$851$$ 3.67472 0.125968
$$852$$ 0 0
$$853$$ 32.6336 1.11735 0.558676 0.829386i $$-0.311310\pi$$
0.558676 + 0.829386i $$0.311310\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0.712898 0.0243664
$$857$$ 18.8742 0.644730 0.322365 0.946616i $$-0.395522\pi$$
0.322365 + 0.946616i $$0.395522\pi$$
$$858$$ 0 0
$$859$$ 15.8242 0.539915 0.269957 0.962872i $$-0.412990\pi$$
0.269957 + 0.962872i $$0.412990\pi$$
$$860$$ 23.3684 0.796857
$$861$$ 0 0
$$862$$ 10.8670 0.370132
$$863$$ −52.3212 −1.78104 −0.890518 0.454948i $$-0.849658\pi$$
−0.890518 + 0.454948i $$0.849658\pi$$
$$864$$ 0 0
$$865$$ −16.1264 −0.548313
$$866$$ −1.76136 −0.0598536
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 96.7309 3.28137
$$870$$ 0 0
$$871$$ −6.62741 −0.224561
$$872$$ −3.01951 −0.102253
$$873$$ 0 0
$$874$$ −4.47773 −0.151461
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −54.0162 −1.82400 −0.911999 0.410193i $$-0.865461\pi$$
−0.911999 + 0.410193i $$0.865461\pi$$
$$878$$ −2.17785 −0.0734989
$$879$$ 0 0
$$880$$ −28.3200 −0.954667
$$881$$ 42.0823 1.41779 0.708895 0.705314i $$-0.249194\pi$$
0.708895 + 0.705314i $$0.249194\pi$$
$$882$$ 0 0
$$883$$ 36.5314 1.22938 0.614689 0.788769i $$-0.289281\pi$$
0.614689 + 0.788769i $$0.289281\pi$$
$$884$$ 9.32316 0.313572
$$885$$ 0 0
$$886$$ 4.68007 0.157230
$$887$$ 11.4648 0.384950 0.192475 0.981302i $$-0.438349\pi$$
0.192475 + 0.981302i $$0.438349\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −0.378001 −0.0126706
$$891$$ 0 0
$$892$$ 19.1322 0.640592
$$893$$ −6.79488 −0.227382
$$894$$ 0 0
$$895$$ 18.9361 0.632965
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −3.84370 −0.128266
$$899$$ −9.59462 −0.319999
$$900$$ 0 0
$$901$$ −8.83823 −0.294444
$$902$$ 15.4695 0.515079
$$903$$ 0 0
$$904$$ −12.5125 −0.416159
$$905$$ −19.7211 −0.655550
$$906$$ 0 0
$$907$$ −5.04665 −0.167571 −0.0837856 0.996484i $$-0.526701\pi$$
−0.0837856 + 0.996484i $$0.526701\pi$$
$$908$$ 23.2999 0.773236
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −47.5236 −1.57453 −0.787263 0.616618i $$-0.788503\pi$$
−0.787263 + 0.616618i $$0.788503\pi$$
$$912$$ 0 0
$$913$$ 63.0431 2.08642
$$914$$ −1.00272 −0.0331671
$$915$$ 0 0
$$916$$ −7.81969 −0.258370
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −45.6698 −1.50651 −0.753254 0.657730i $$-0.771517\pi$$
−0.753254 + 0.657730i $$0.771517\pi$$
$$920$$ 8.95449 0.295221
$$921$$ 0 0
$$922$$ 3.46945 0.114260
$$923$$ 6.85856 0.225752
$$924$$ 0 0
$$925$$ −1.80095 −0.0592148
$$926$$ −4.04950 −0.133075
$$927$$ 0 0
$$928$$ 3.15540 0.103581
$$929$$ 44.4449 1.45819 0.729095 0.684412i $$-0.239941\pi$$
0.729095 + 0.684412i $$0.239941\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 24.2115 0.793074
$$933$$ 0 0
$$934$$ 8.14357 0.266466
$$935$$ 38.1634 1.24808
$$936$$ 0 0
$$937$$ 46.9796 1.53476 0.767379 0.641194i $$-0.221561\pi$$
0.767379 + 0.641194i $$0.221561\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 6.67309 0.217652
$$941$$ −35.5654 −1.15940 −0.579699 0.814830i $$-0.696830\pi$$
−0.579699 + 0.814830i $$0.696830\pi$$
$$942$$ 0 0
$$943$$ 63.7128 2.07477
$$944$$ 3.12200 0.101613
$$945$$ 0 0
$$946$$ −12.2900 −0.399581
$$947$$ 22.3592 0.726576 0.363288 0.931677i $$-0.381654\pi$$
0.363288 + 0.931677i $$0.381654\pi$$
$$948$$ 0 0
$$949$$ 3.14147 0.101977
$$950$$ 2.19450 0.0711989
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 46.7684 1.51498 0.757488 0.652849i $$-0.226426\pi$$
0.757488 + 0.652849i $$0.226426\pi$$
$$954$$ 0 0
$$955$$ 23.4371 0.758406
$$956$$ 25.7579 0.833071
$$957$$ 0 0
$$958$$ −0.454469 −0.0146832
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 53.8048 1.73564
$$962$$ −0.162083 −0.00522577
$$963$$ 0 0
$$964$$ −39.3432 −1.26716
$$965$$ −20.1687 −0.649253
$$966$$ 0 0
$$967$$ 8.22976 0.264651 0.132326 0.991206i $$-0.457756\pi$$
0.132326 + 0.991206i $$0.457756\pi$$
$$968$$ 20.0738 0.645196
$$969$$ 0 0
$$970$$ −5.13046 −0.164729
$$971$$ 23.6326 0.758407 0.379204 0.925313i $$-0.376198\pi$$
0.379204 + 0.925313i $$0.376198\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −6.00187 −0.192312
$$975$$ 0 0
$$976$$ −11.9462 −0.382390
$$977$$ 26.6428 0.852379 0.426190 0.904634i $$-0.359856\pi$$
0.426190 + 0.904634i $$0.359856\pi$$
$$978$$ 0 0
$$979$$ −5.47897 −0.175109
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −3.47629 −0.110933
$$983$$ −43.6302 −1.39159 −0.695793 0.718242i $$-0.744947\pi$$
−0.695793 + 0.718242i $$0.744947\pi$$
$$984$$ 0 0
$$985$$ 2.10821 0.0671732
$$986$$ −1.33187 −0.0424155
$$987$$ 0 0
$$988$$ −5.44322 −0.173172
$$989$$ −50.6174 −1.60954
$$990$$ 0 0
$$991$$ −7.60816 −0.241681 −0.120841 0.992672i $$-0.538559\pi$$
−0.120841 + 0.992672i $$0.538559\pi$$
$$992$$ −27.8899 −0.885506
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −19.1983 −0.608628
$$996$$ 0 0
$$997$$ 19.4356 0.615532 0.307766 0.951462i $$-0.400419\pi$$
0.307766 + 0.951462i $$0.400419\pi$$
$$998$$ −3.71523 −0.117604
$$999$$ 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 5733.2.a.br.1.3 6
3.2 odd 2 637.2.a.n.1.4 yes 6
7.6 odd 2 5733.2.a.bu.1.3 6
21.2 odd 6 637.2.e.n.508.3 12
21.5 even 6 637.2.e.o.508.3 12
21.11 odd 6 637.2.e.n.79.3 12
21.17 even 6 637.2.e.o.79.3 12
21.20 even 2 637.2.a.m.1.4 6
39.38 odd 2 8281.2.a.cd.1.3 6
273.272 even 2 8281.2.a.cc.1.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.4 6 21.20 even 2
637.2.a.n.1.4 yes 6 3.2 odd 2
637.2.e.n.79.3 12 21.11 odd 6
637.2.e.n.508.3 12 21.2 odd 6
637.2.e.o.79.3 12 21.17 even 6
637.2.e.o.508.3 12 21.5 even 6
5733.2.a.br.1.3 6 1.1 even 1 trivial
5733.2.a.bu.1.3 6 7.6 odd 2
8281.2.a.cc.1.3 6 273.272 even 2
8281.2.a.cd.1.3 6 39.38 odd 2