# Properties

 Label 4598.2.a.bo.1.1 Level $4598$ Weight $2$ Character 4598.1 Self dual yes Analytic conductor $36.715$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, 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("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4598 = 2 \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4598.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: $$36.7152148494$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 1.1 Root $$-2.14510$$ of defining polynomial Character $$\chi$$ $$=$$ 4598.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} -2.14510 q^{6} -2.89167 q^{7} +1.00000 q^{8} +1.60147 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.14510 q^{3} +1.00000 q^{4} -1.60147 q^{5} -2.14510 q^{6} -2.89167 q^{7} +1.00000 q^{8} +1.60147 q^{9} -1.60147 q^{10} -2.14510 q^{12} -4.89167 q^{13} -2.89167 q^{14} +3.43531 q^{15} +1.00000 q^{16} +5.74657 q^{17} +1.60147 q^{18} +1.00000 q^{19} -1.60147 q^{20} +6.20293 q^{21} -7.74657 q^{23} -2.14510 q^{24} -2.43531 q^{25} -4.89167 q^{26} +3.00000 q^{27} -2.89167 q^{28} -5.34803 q^{29} +3.43531 q^{30} -7.43531 q^{31} +1.00000 q^{32} +5.74657 q^{34} +4.63091 q^{35} +1.60147 q^{36} -7.49314 q^{37} +1.00000 q^{38} +10.4931 q^{39} -1.60147 q^{40} +2.05783 q^{41} +6.20293 q^{42} +10.6887 q^{43} -2.56469 q^{45} -7.74657 q^{46} -7.08727 q^{47} -2.14510 q^{48} +1.36176 q^{49} -2.43531 q^{50} -12.3270 q^{51} -4.89167 q^{52} +8.32698 q^{53} +3.00000 q^{54} -2.89167 q^{56} -2.14510 q^{57} -5.34803 q^{58} -2.54364 q^{59} +3.43531 q^{60} -13.4931 q^{61} -7.43531 q^{62} -4.63091 q^{63} +1.00000 q^{64} +7.83384 q^{65} -10.3848 q^{67} +5.74657 q^{68} +16.6172 q^{69} +4.63091 q^{70} +14.9789 q^{71} +1.60147 q^{72} +3.74657 q^{73} -7.49314 q^{74} +5.22399 q^{75} +1.00000 q^{76} +10.4931 q^{78} +8.69607 q^{79} -1.60147 q^{80} -11.2397 q^{81} +2.05783 q^{82} +6.10833 q^{83} +6.20293 q^{84} -9.20293 q^{85} +10.6887 q^{86} +11.4721 q^{87} +3.20293 q^{89} -2.56469 q^{90} +14.1451 q^{91} -7.74657 q^{92} +15.9495 q^{93} -7.08727 q^{94} -1.60147 q^{95} -2.14510 q^{96} -6.98627 q^{97} +1.36176 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 6 q^{14} - 9 q^{15} + 3 q^{16} + 9 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 15 q^{21} - 15 q^{23} + 12 q^{25} + 9 q^{27} + 6 q^{28} - 6 q^{29} - 9 q^{30} - 3 q^{31} + 3 q^{32} + 9 q^{34} + 3 q^{36} - 6 q^{37} + 3 q^{38} + 15 q^{39} - 3 q^{40} + 9 q^{41} + 15 q^{42} + 21 q^{43} - 27 q^{45} - 15 q^{46} - 12 q^{47} + 27 q^{49} + 12 q^{50} - 3 q^{51} - 9 q^{53} + 9 q^{54} + 6 q^{56} - 6 q^{58} - 3 q^{59} - 9 q^{60} - 24 q^{61} - 3 q^{62} + 3 q^{64} + 6 q^{65} + 9 q^{68} + 3 q^{69} + 21 q^{71} + 3 q^{72} + 3 q^{73} - 6 q^{74} + 36 q^{75} + 3 q^{76} + 15 q^{78} + 6 q^{79} - 3 q^{80} - 9 q^{81} + 9 q^{82} + 33 q^{83} + 15 q^{84} - 24 q^{85} + 21 q^{86} - 6 q^{87} + 6 q^{89} - 27 q^{90} + 36 q^{91} - 15 q^{92} + 36 q^{93} - 12 q^{94} - 3 q^{95} + 12 q^{97} + 27 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 6 * q^14 - 9 * q^15 + 3 * q^16 + 9 * q^17 + 3 * q^18 + 3 * q^19 - 3 * q^20 + 15 * q^21 - 15 * q^23 + 12 * q^25 + 9 * q^27 + 6 * q^28 - 6 * q^29 - 9 * q^30 - 3 * q^31 + 3 * q^32 + 9 * q^34 + 3 * q^36 - 6 * q^37 + 3 * q^38 + 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 + 21 * q^43 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 - 9 * q^53 + 9 * q^54 + 6 * q^56 - 6 * q^58 - 3 * q^59 - 9 * q^60 - 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 + 3 * q^69 + 21 * q^71 + 3 * q^72 + 3 * q^73 - 6 * q^74 + 36 * q^75 + 3 * q^76 + 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 + 33 * q^83 + 15 * q^84 - 24 * q^85 + 21 * q^86 - 6 * q^87 + 6 * q^89 - 27 * q^90 + 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * 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
$$3$$ −2.14510 −1.23848 −0.619238 0.785204i $$-0.712558\pi$$
−0.619238 + 0.785204i $$0.712558\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.60147 −0.716197 −0.358099 0.933684i $$-0.616575\pi$$
−0.358099 + 0.933684i $$0.616575\pi$$
$$6$$ −2.14510 −0.875735
$$7$$ −2.89167 −1.09295 −0.546474 0.837476i $$-0.684030\pi$$
−0.546474 + 0.837476i $$0.684030\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.60147 0.533822
$$10$$ −1.60147 −0.506428
$$11$$ 0 0
$$12$$ −2.14510 −0.619238
$$13$$ −4.89167 −1.35671 −0.678353 0.734736i $$-0.737306\pi$$
−0.678353 + 0.734736i $$0.737306\pi$$
$$14$$ −2.89167 −0.772832
$$15$$ 3.43531 0.886993
$$16$$ 1.00000 0.250000
$$17$$ 5.74657 1.39375 0.696874 0.717194i $$-0.254574\pi$$
0.696874 + 0.717194i $$0.254574\pi$$
$$18$$ 1.60147 0.377469
$$19$$ 1.00000 0.229416
$$20$$ −1.60147 −0.358099
$$21$$ 6.20293 1.35359
$$22$$ 0 0
$$23$$ −7.74657 −1.61527 −0.807636 0.589682i $$-0.799253\pi$$
−0.807636 + 0.589682i $$0.799253\pi$$
$$24$$ −2.14510 −0.437867
$$25$$ −2.43531 −0.487062
$$26$$ −4.89167 −0.959336
$$27$$ 3.00000 0.577350
$$28$$ −2.89167 −0.546474
$$29$$ −5.34803 −0.993105 −0.496552 0.868007i $$-0.665401\pi$$
−0.496552 + 0.868007i $$0.665401\pi$$
$$30$$ 3.43531 0.627199
$$31$$ −7.43531 −1.33542 −0.667710 0.744421i $$-0.732726\pi$$
−0.667710 + 0.744421i $$0.732726\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 5.74657 0.985528
$$35$$ 4.63091 0.782767
$$36$$ 1.60147 0.266911
$$37$$ −7.49314 −1.23186 −0.615932 0.787799i $$-0.711220\pi$$
−0.615932 + 0.787799i $$0.711220\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 10.4931 1.68025
$$40$$ −1.60147 −0.253214
$$41$$ 2.05783 0.321379 0.160689 0.987005i $$-0.448628\pi$$
0.160689 + 0.987005i $$0.448628\pi$$
$$42$$ 6.20293 0.957133
$$43$$ 10.6887 1.63002 0.815009 0.579449i $$-0.196732\pi$$
0.815009 + 0.579449i $$0.196732\pi$$
$$44$$ 0 0
$$45$$ −2.56469 −0.382322
$$46$$ −7.74657 −1.14217
$$47$$ −7.08727 −1.03379 −0.516893 0.856050i $$-0.672911\pi$$
−0.516893 + 0.856050i $$0.672911\pi$$
$$48$$ −2.14510 −0.309619
$$49$$ 1.36176 0.194537
$$50$$ −2.43531 −0.344405
$$51$$ −12.3270 −1.72612
$$52$$ −4.89167 −0.678353
$$53$$ 8.32698 1.14380 0.571899 0.820324i $$-0.306207\pi$$
0.571899 + 0.820324i $$0.306207\pi$$
$$54$$ 3.00000 0.408248
$$55$$ 0 0
$$56$$ −2.89167 −0.386416
$$57$$ −2.14510 −0.284126
$$58$$ −5.34803 −0.702231
$$59$$ −2.54364 −0.331153 −0.165577 0.986197i $$-0.552949\pi$$
−0.165577 + 0.986197i $$0.552949\pi$$
$$60$$ 3.43531 0.443496
$$61$$ −13.4931 −1.72762 −0.863810 0.503818i $$-0.831928\pi$$
−0.863810 + 0.503818i $$0.831928\pi$$
$$62$$ −7.43531 −0.944285
$$63$$ −4.63091 −0.583440
$$64$$ 1.00000 0.125000
$$65$$ 7.83384 0.971669
$$66$$ 0 0
$$67$$ −10.3848 −1.26871 −0.634353 0.773043i $$-0.718733\pi$$
−0.634353 + 0.773043i $$0.718733\pi$$
$$68$$ 5.74657 0.696874
$$69$$ 16.6172 2.00047
$$70$$ 4.63091 0.553500
$$71$$ 14.9789 1.77767 0.888837 0.458224i $$-0.151514\pi$$
0.888837 + 0.458224i $$0.151514\pi$$
$$72$$ 1.60147 0.188735
$$73$$ 3.74657 0.438503 0.219251 0.975668i $$-0.429639\pi$$
0.219251 + 0.975668i $$0.429639\pi$$
$$74$$ −7.49314 −0.871059
$$75$$ 5.22399 0.603214
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 10.4931 1.18811
$$79$$ 8.69607 0.978384 0.489192 0.872176i $$-0.337292\pi$$
0.489192 + 0.872176i $$0.337292\pi$$
$$80$$ −1.60147 −0.179049
$$81$$ −11.2397 −1.24886
$$82$$ 2.05783 0.227249
$$83$$ 6.10833 0.670476 0.335238 0.942133i $$-0.391183\pi$$
0.335238 + 0.942133i $$0.391183\pi$$
$$84$$ 6.20293 0.676795
$$85$$ −9.20293 −0.998198
$$86$$ 10.6887 1.15260
$$87$$ 11.4721 1.22994
$$88$$ 0 0
$$89$$ 3.20293 0.339510 0.169755 0.985486i $$-0.445702\pi$$
0.169755 + 0.985486i $$0.445702\pi$$
$$90$$ −2.56469 −0.270342
$$91$$ 14.1451 1.48281
$$92$$ −7.74657 −0.807636
$$93$$ 15.9495 1.65389
$$94$$ −7.08727 −0.730997
$$95$$ −1.60147 −0.164307
$$96$$ −2.14510 −0.218934
$$97$$ −6.98627 −0.709349 −0.354674 0.934990i $$-0.615408\pi$$
−0.354674 + 0.934990i $$0.615408\pi$$
$$98$$ 1.36176 0.137559
$$99$$ 0 0
$$100$$ −2.43531 −0.243531
$$101$$ 1.20293 0.119696 0.0598481 0.998207i $$-0.480938\pi$$
0.0598481 + 0.998207i $$0.480938\pi$$
$$102$$ −12.3270 −1.22055
$$103$$ 16.8779 1.66303 0.831517 0.555500i $$-0.187473\pi$$
0.831517 + 0.555500i $$0.187473\pi$$
$$104$$ −4.89167 −0.479668
$$105$$ −9.93378 −0.969438
$$106$$ 8.32698 0.808788
$$107$$ 8.03677 0.776944 0.388472 0.921460i $$-0.373003\pi$$
0.388472 + 0.921460i $$0.373003\pi$$
$$108$$ 3.00000 0.288675
$$109$$ −11.5299 −1.10437 −0.552183 0.833723i $$-0.686205\pi$$
−0.552183 + 0.833723i $$0.686205\pi$$
$$110$$ 0 0
$$111$$ 16.0735 1.52563
$$112$$ −2.89167 −0.273237
$$113$$ −8.00000 −0.752577 −0.376288 0.926503i $$-0.622800\pi$$
−0.376288 + 0.926503i $$0.622800\pi$$
$$114$$ −2.14510 −0.200907
$$115$$ 12.4059 1.15685
$$116$$ −5.34803 −0.496552
$$117$$ −7.83384 −0.724239
$$118$$ −2.54364 −0.234161
$$119$$ −16.6172 −1.52329
$$120$$ 3.43531 0.313599
$$121$$ 0 0
$$122$$ −13.4931 −1.22161
$$123$$ −4.41425 −0.398020
$$124$$ −7.43531 −0.667710
$$125$$ 11.9074 1.06503
$$126$$ −4.63091 −0.412554
$$127$$ 21.2765 1.88798 0.943991 0.329971i $$-0.107039\pi$$
0.943991 + 0.329971i $$0.107039\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −22.9284 −2.01874
$$130$$ 7.83384 0.687073
$$131$$ 3.43531 0.300144 0.150072 0.988675i $$-0.452049\pi$$
0.150072 + 0.988675i $$0.452049\pi$$
$$132$$ 0 0
$$133$$ −2.89167 −0.250740
$$134$$ −10.3848 −0.897111
$$135$$ −4.80440 −0.413497
$$136$$ 5.74657 0.492764
$$137$$ 13.5877 1.16088 0.580439 0.814303i $$-0.302881\pi$$
0.580439 + 0.814303i $$0.302881\pi$$
$$138$$ 16.6172 1.41455
$$139$$ −3.19560 −0.271048 −0.135524 0.990774i $$-0.543272\pi$$
−0.135524 + 0.990774i $$0.543272\pi$$
$$140$$ 4.63091 0.391383
$$141$$ 15.2029 1.28032
$$142$$ 14.9789 1.25701
$$143$$ 0 0
$$144$$ 1.60147 0.133455
$$145$$ 8.56469 0.711259
$$146$$ 3.74657 0.310068
$$147$$ −2.92112 −0.240930
$$148$$ −7.49314 −0.615932
$$149$$ 12.0735 0.989104 0.494552 0.869148i $$-0.335332\pi$$
0.494552 + 0.869148i $$0.335332\pi$$
$$150$$ 5.22399 0.426537
$$151$$ 4.91273 0.399792 0.199896 0.979817i $$-0.435940\pi$$
0.199896 + 0.979817i $$0.435940\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 9.20293 0.744013
$$154$$ 0 0
$$155$$ 11.9074 0.956425
$$156$$ 10.4931 0.840123
$$157$$ 8.00733 0.639054 0.319527 0.947577i $$-0.396476\pi$$
0.319527 + 0.947577i $$0.396476\pi$$
$$158$$ 8.69607 0.691822
$$159$$ −17.8622 −1.41657
$$160$$ −1.60147 −0.126607
$$161$$ 22.4005 1.76541
$$162$$ −11.2397 −0.883075
$$163$$ 1.88434 0.147593 0.0737966 0.997273i $$-0.476488\pi$$
0.0737966 + 0.997273i $$0.476488\pi$$
$$164$$ 2.05783 0.160689
$$165$$ 0 0
$$166$$ 6.10833 0.474098
$$167$$ 4.98627 0.385849 0.192925 0.981214i $$-0.438203\pi$$
0.192925 + 0.981214i $$0.438203\pi$$
$$168$$ 6.20293 0.478567
$$169$$ 10.9284 0.840650
$$170$$ −9.20293 −0.705833
$$171$$ 1.60147 0.122467
$$172$$ 10.6887 0.815009
$$173$$ −6.51419 −0.495265 −0.247632 0.968854i $$-0.579653\pi$$
−0.247632 + 0.968854i $$0.579653\pi$$
$$174$$ 11.4721 0.869696
$$175$$ 7.04211 0.532333
$$176$$ 0 0
$$177$$ 5.45636 0.410125
$$178$$ 3.20293 0.240070
$$179$$ −5.55096 −0.414899 −0.207449 0.978246i $$-0.566516\pi$$
−0.207449 + 0.978246i $$0.566516\pi$$
$$180$$ −2.56469 −0.191161
$$181$$ 1.88434 0.140062 0.0700311 0.997545i $$-0.477690\pi$$
0.0700311 + 0.997545i $$0.477690\pi$$
$$182$$ 14.1451 1.04850
$$183$$ 28.9442 2.13961
$$184$$ −7.74657 −0.571085
$$185$$ 12.0000 0.882258
$$186$$ 15.9495 1.16947
$$187$$ 0 0
$$188$$ −7.08727 −0.516893
$$189$$ −8.67501 −0.631014
$$190$$ −1.60147 −0.116183
$$191$$ 2.44264 0.176743 0.0883715 0.996088i $$-0.471834\pi$$
0.0883715 + 0.996088i $$0.471834\pi$$
$$192$$ −2.14510 −0.154809
$$193$$ 9.67501 0.696423 0.348211 0.937416i $$-0.386789\pi$$
0.348211 + 0.937416i $$0.386789\pi$$
$$194$$ −6.98627 −0.501585
$$195$$ −16.8044 −1.20339
$$196$$ 1.36176 0.0972686
$$197$$ 17.4931 1.24633 0.623167 0.782089i $$-0.285846\pi$$
0.623167 + 0.782089i $$0.285846\pi$$
$$198$$ 0 0
$$199$$ −25.2397 −1.78920 −0.894598 0.446873i $$-0.852538\pi$$
−0.894598 + 0.446873i $$0.852538\pi$$
$$200$$ −2.43531 −0.172202
$$201$$ 22.2765 1.57126
$$202$$ 1.20293 0.0846379
$$203$$ 15.4648 1.08541
$$204$$ −12.3270 −0.863061
$$205$$ −3.29554 −0.230171
$$206$$ 16.8779 1.17594
$$207$$ −12.4059 −0.862267
$$208$$ −4.89167 −0.339176
$$209$$ 0 0
$$210$$ −9.93378 −0.685496
$$211$$ 14.8338 1.02120 0.510602 0.859817i $$-0.329423\pi$$
0.510602 + 0.859817i $$0.329423\pi$$
$$212$$ 8.32698 0.571899
$$213$$ −32.1314 −2.20161
$$214$$ 8.03677 0.549383
$$215$$ −17.1176 −1.16741
$$216$$ 3.00000 0.204124
$$217$$ 21.5005 1.45955
$$218$$ −11.5299 −0.780904
$$219$$ −8.03677 −0.543075
$$220$$ 0 0
$$221$$ −28.1103 −1.89090
$$222$$ 16.0735 1.07879
$$223$$ 11.4931 0.769637 0.384819 0.922992i $$-0.374264\pi$$
0.384819 + 0.922992i $$0.374264\pi$$
$$224$$ −2.89167 −0.193208
$$225$$ −3.90006 −0.260004
$$226$$ −8.00000 −0.532152
$$227$$ −8.73284 −0.579619 −0.289810 0.957084i $$-0.593592\pi$$
−0.289810 + 0.957084i $$0.593592\pi$$
$$228$$ −2.14510 −0.142063
$$229$$ 7.84117 0.518159 0.259080 0.965856i $$-0.416581\pi$$
0.259080 + 0.965856i $$0.416581\pi$$
$$230$$ 12.4059 0.818018
$$231$$ 0 0
$$232$$ −5.34803 −0.351116
$$233$$ 12.5069 0.819352 0.409676 0.912231i $$-0.365642\pi$$
0.409676 + 0.912231i $$0.365642\pi$$
$$234$$ −7.83384 −0.512114
$$235$$ 11.3500 0.740394
$$236$$ −2.54364 −0.165577
$$237$$ −18.6540 −1.21170
$$238$$ −16.6172 −1.07713
$$239$$ −24.6245 −1.59283 −0.796414 0.604752i $$-0.793272\pi$$
−0.796414 + 0.604752i $$0.793272\pi$$
$$240$$ 3.43531 0.221748
$$241$$ −16.0809 −1.03586 −0.517930 0.855423i $$-0.673297\pi$$
−0.517930 + 0.855423i $$0.673297\pi$$
$$242$$ 0 0
$$243$$ 15.1103 0.969328
$$244$$ −13.4931 −0.863810
$$245$$ −2.18081 −0.139327
$$246$$ −4.41425 −0.281443
$$247$$ −4.89167 −0.311250
$$248$$ −7.43531 −0.472143
$$249$$ −13.1030 −0.830368
$$250$$ 11.9074 0.753089
$$251$$ 13.0873 0.826061 0.413031 0.910717i $$-0.364470\pi$$
0.413031 + 0.910717i $$0.364470\pi$$
$$252$$ −4.63091 −0.291720
$$253$$ 0 0
$$254$$ 21.2765 1.33500
$$255$$ 19.7412 1.23624
$$256$$ 1.00000 0.0625000
$$257$$ 0.723522 0.0451320 0.0225660 0.999745i $$-0.492816\pi$$
0.0225660 + 0.999745i $$0.492816\pi$$
$$258$$ −22.9284 −1.42746
$$259$$ 21.6677 1.34636
$$260$$ 7.83384 0.485834
$$261$$ −8.56469 −0.530141
$$262$$ 3.43531 0.212234
$$263$$ −13.7760 −0.849465 −0.424733 0.905319i $$-0.639632\pi$$
−0.424733 + 0.905319i $$0.639632\pi$$
$$264$$ 0 0
$$265$$ −13.3354 −0.819185
$$266$$ −2.89167 −0.177300
$$267$$ −6.87062 −0.420475
$$268$$ −10.3848 −0.634353
$$269$$ −19.9579 −1.21685 −0.608427 0.793610i $$-0.708199\pi$$
−0.608427 + 0.793610i $$0.708199\pi$$
$$270$$ −4.80440 −0.292386
$$271$$ −28.7486 −1.74635 −0.873175 0.487406i $$-0.837943\pi$$
−0.873175 + 0.487406i $$0.837943\pi$$
$$272$$ 5.74657 0.348437
$$273$$ −30.3427 −1.83642
$$274$$ 13.5877 0.820865
$$275$$ 0 0
$$276$$ 16.6172 1.00024
$$277$$ −3.08727 −0.185496 −0.0927482 0.995690i $$-0.529565\pi$$
−0.0927482 + 0.995690i $$0.529565\pi$$
$$278$$ −3.19560 −0.191660
$$279$$ −11.9074 −0.712877
$$280$$ 4.63091 0.276750
$$281$$ −9.09460 −0.542538 −0.271269 0.962504i $$-0.587443\pi$$
−0.271269 + 0.962504i $$0.587443\pi$$
$$282$$ 15.2029 0.905321
$$283$$ −13.1451 −0.781395 −0.390698 0.920519i $$-0.627766\pi$$
−0.390698 + 0.920519i $$0.627766\pi$$
$$284$$ 14.9789 0.888837
$$285$$ 3.43531 0.203490
$$286$$ 0 0
$$287$$ −5.95056 −0.351251
$$288$$ 1.60147 0.0943673
$$289$$ 16.0230 0.942532
$$290$$ 8.56469 0.502936
$$291$$ 14.9863 0.878511
$$292$$ 3.74657 0.219251
$$293$$ −3.10833 −0.181591 −0.0907953 0.995870i $$-0.528941\pi$$
−0.0907953 + 0.995870i $$0.528941\pi$$
$$294$$ −2.92112 −0.170363
$$295$$ 4.07355 0.237171
$$296$$ −7.49314 −0.435530
$$297$$ 0 0
$$298$$ 12.0735 0.699402
$$299$$ 37.8937 2.19145
$$300$$ 5.22399 0.301607
$$301$$ −30.9083 −1.78153
$$302$$ 4.91273 0.282696
$$303$$ −2.58041 −0.148241
$$304$$ 1.00000 0.0573539
$$305$$ 21.6088 1.23732
$$306$$ 9.20293 0.526097
$$307$$ −2.18920 −0.124944 −0.0624722 0.998047i $$-0.519898\pi$$
−0.0624722 + 0.998047i $$0.519898\pi$$
$$308$$ 0 0
$$309$$ −36.2049 −2.05963
$$310$$ 11.9074 0.676294
$$311$$ 3.92112 0.222346 0.111173 0.993801i $$-0.464539\pi$$
0.111173 + 0.993801i $$0.464539\pi$$
$$312$$ 10.4931 0.594057
$$313$$ −21.7118 −1.22722 −0.613611 0.789608i $$-0.710284\pi$$
−0.613611 + 0.789608i $$0.710284\pi$$
$$314$$ 8.00733 0.451880
$$315$$ 7.41625 0.417858
$$316$$ 8.69607 0.489192
$$317$$ 6.90739 0.387958 0.193979 0.981006i $$-0.437861\pi$$
0.193979 + 0.981006i $$0.437861\pi$$
$$318$$ −17.8622 −1.00166
$$319$$ 0 0
$$320$$ −1.60147 −0.0895246
$$321$$ −17.2397 −0.962226
$$322$$ 22.4005 1.24833
$$323$$ 5.74657 0.319748
$$324$$ −11.2397 −0.624428
$$325$$ 11.9127 0.660799
$$326$$ 1.88434 0.104364
$$327$$ 24.7328 1.36773
$$328$$ 2.05783 0.113625
$$329$$ 20.4941 1.12987
$$330$$ 0 0
$$331$$ 32.1030 1.76454 0.882270 0.470744i $$-0.156014\pi$$
0.882270 + 0.470744i $$0.156014\pi$$
$$332$$ 6.10833 0.335238
$$333$$ −12.0000 −0.657596
$$334$$ 4.98627 0.272837
$$335$$ 16.6309 0.908644
$$336$$ 6.20293 0.338398
$$337$$ 1.55096 0.0844864 0.0422432 0.999107i $$-0.486550\pi$$
0.0422432 + 0.999107i $$0.486550\pi$$
$$338$$ 10.9284 0.594429
$$339$$ 17.1608 0.932048
$$340$$ −9.20293 −0.499099
$$341$$ 0 0
$$342$$ 1.60147 0.0865973
$$343$$ 16.3039 0.880330
$$344$$ 10.6887 0.576298
$$345$$ −26.6118 −1.43273
$$346$$ −6.51419 −0.350205
$$347$$ 0.506864 0.0272099 0.0136049 0.999907i $$-0.495669\pi$$
0.0136049 + 0.999907i $$0.495669\pi$$
$$348$$ 11.4721 0.614968
$$349$$ −10.8117 −0.578738 −0.289369 0.957218i $$-0.593446\pi$$
−0.289369 + 0.957218i $$0.593446\pi$$
$$350$$ 7.04211 0.376417
$$351$$ −14.6750 −0.783294
$$352$$ 0 0
$$353$$ 27.0966 1.44221 0.721103 0.692828i $$-0.243635\pi$$
0.721103 + 0.692828i $$0.243635\pi$$
$$354$$ 5.45636 0.290002
$$355$$ −23.9883 −1.27316
$$356$$ 3.20293 0.169755
$$357$$ 35.6456 1.88656
$$358$$ −5.55096 −0.293378
$$359$$ 20.7486 1.09507 0.547534 0.836784i $$-0.315567\pi$$
0.547534 + 0.836784i $$0.315567\pi$$
$$360$$ −2.56469 −0.135171
$$361$$ 1.00000 0.0526316
$$362$$ 1.88434 0.0990389
$$363$$ 0 0
$$364$$ 14.1451 0.741405
$$365$$ −6.00000 −0.314054
$$366$$ 28.9442 1.51294
$$367$$ 25.6402 1.33841 0.669205 0.743078i $$-0.266635\pi$$
0.669205 + 0.743078i $$0.266635\pi$$
$$368$$ −7.74657 −0.403818
$$369$$ 3.29554 0.171559
$$370$$ 12.0000 0.623850
$$371$$ −24.0789 −1.25011
$$372$$ 15.9495 0.826943
$$373$$ −25.2049 −1.30506 −0.652531 0.757762i $$-0.726293\pi$$
−0.652531 + 0.757762i $$0.726293\pi$$
$$374$$ 0 0
$$375$$ −25.5426 −1.31901
$$376$$ −7.08727 −0.365498
$$377$$ 26.1608 1.34735
$$378$$ −8.67501 −0.446195
$$379$$ 3.47208 0.178349 0.0891744 0.996016i $$-0.471577\pi$$
0.0891744 + 0.996016i $$0.471577\pi$$
$$380$$ −1.60147 −0.0821534
$$381$$ −45.6402 −2.33822
$$382$$ 2.44264 0.124976
$$383$$ 23.1176 1.18126 0.590628 0.806944i $$-0.298880\pi$$
0.590628 + 0.806944i $$0.298880\pi$$
$$384$$ −2.14510 −0.109467
$$385$$ 0 0
$$386$$ 9.67501 0.492445
$$387$$ 17.1176 0.870139
$$388$$ −6.98627 −0.354674
$$389$$ −26.7623 −1.35690 −0.678451 0.734646i $$-0.737348\pi$$
−0.678451 + 0.734646i $$0.737348\pi$$
$$390$$ −16.8044 −0.850924
$$391$$ −44.5162 −2.25128
$$392$$ 1.36176 0.0687793
$$393$$ −7.36909 −0.371721
$$394$$ 17.4931 0.881291
$$395$$ −13.9265 −0.700716
$$396$$ 0 0
$$397$$ −31.1030 −1.56101 −0.780507 0.625147i $$-0.785039\pi$$
−0.780507 + 0.625147i $$0.785039\pi$$
$$398$$ −25.2397 −1.26515
$$399$$ 6.20293 0.310535
$$400$$ −2.43531 −0.121765
$$401$$ 29.3500 1.46567 0.732835 0.680406i $$-0.238197\pi$$
0.732835 + 0.680406i $$0.238197\pi$$
$$402$$ 22.2765 1.11105
$$403$$ 36.3711 1.81177
$$404$$ 1.20293 0.0598481
$$405$$ 18.0000 0.894427
$$406$$ 15.4648 0.767503
$$407$$ 0 0
$$408$$ −12.3270 −0.610276
$$409$$ −8.58774 −0.424636 −0.212318 0.977201i $$-0.568101\pi$$
−0.212318 + 0.977201i $$0.568101\pi$$
$$410$$ −3.29554 −0.162755
$$411$$ −29.1471 −1.43772
$$412$$ 16.8779 0.831517
$$413$$ 7.35536 0.361934
$$414$$ −12.4059 −0.609715
$$415$$ −9.78228 −0.480193
$$416$$ −4.89167 −0.239834
$$417$$ 6.85490 0.335686
$$418$$ 0 0
$$419$$ 7.59414 0.370998 0.185499 0.982644i $$-0.440610\pi$$
0.185499 + 0.982644i $$0.440610\pi$$
$$420$$ −9.93378 −0.484719
$$421$$ 8.76030 0.426951 0.213475 0.976948i $$-0.431522\pi$$
0.213475 + 0.976948i $$0.431522\pi$$
$$422$$ 14.8338 0.722100
$$423$$ −11.3500 −0.551857
$$424$$ 8.32698 0.404394
$$425$$ −13.9947 −0.678841
$$426$$ −32.1314 −1.55677
$$427$$ 39.0177 1.88820
$$428$$ 8.03677 0.388472
$$429$$ 0 0
$$430$$ −17.1176 −0.825486
$$431$$ 33.6677 1.62172 0.810858 0.585243i $$-0.199001\pi$$
0.810858 + 0.585243i $$0.199001\pi$$
$$432$$ 3.00000 0.144338
$$433$$ 30.5383 1.46758 0.733789 0.679378i $$-0.237750\pi$$
0.733789 + 0.679378i $$0.237750\pi$$
$$434$$ 21.5005 1.03206
$$435$$ −18.3721 −0.880877
$$436$$ −11.5299 −0.552183
$$437$$ −7.74657 −0.370569
$$438$$ −8.03677 −0.384012
$$439$$ 12.5069 0.596920 0.298460 0.954422i $$-0.403527\pi$$
0.298460 + 0.954422i $$0.403527\pi$$
$$440$$ 0 0
$$441$$ 2.18081 0.103848
$$442$$ −28.1103 −1.33707
$$443$$ −17.6677 −0.839417 −0.419709 0.907659i $$-0.637868\pi$$
−0.419709 + 0.907659i $$0.637868\pi$$
$$444$$ 16.0735 0.762817
$$445$$ −5.12938 −0.243156
$$446$$ 11.4931 0.544216
$$447$$ −25.8990 −1.22498
$$448$$ −2.89167 −0.136619
$$449$$ −17.7098 −0.835777 −0.417888 0.908498i $$-0.637230\pi$$
−0.417888 + 0.908498i $$0.637230\pi$$
$$450$$ −3.90006 −0.183851
$$451$$ 0 0
$$452$$ −8.00000 −0.376288
$$453$$ −10.5383 −0.495133
$$454$$ −8.73284 −0.409853
$$455$$ −22.6529 −1.06198
$$456$$ −2.14510 −0.100454
$$457$$ 24.8064 1.16039 0.580197 0.814476i $$-0.302976\pi$$
0.580197 + 0.814476i $$0.302976\pi$$
$$458$$ 7.84117 0.366394
$$459$$ 17.2397 0.804681
$$460$$ 12.4059 0.578426
$$461$$ 2.98627 0.139085 0.0695423 0.997579i $$-0.477846\pi$$
0.0695423 + 0.997579i $$0.477846\pi$$
$$462$$ 0 0
$$463$$ −39.5667 −1.83882 −0.919410 0.393301i $$-0.871333\pi$$
−0.919410 + 0.393301i $$0.871333\pi$$
$$464$$ −5.34803 −0.248276
$$465$$ −25.5426 −1.18451
$$466$$ 12.5069 0.579369
$$467$$ 4.46475 0.206604 0.103302 0.994650i $$-0.467059\pi$$
0.103302 + 0.994650i $$0.467059\pi$$
$$468$$ −7.83384 −0.362119
$$469$$ 30.0294 1.38663
$$470$$ 11.3500 0.523538
$$471$$ −17.1765 −0.791453
$$472$$ −2.54364 −0.117080
$$473$$ 0 0
$$474$$ −18.6540 −0.856805
$$475$$ −2.43531 −0.111740
$$476$$ −16.6172 −0.761647
$$477$$ 13.3354 0.610585
$$478$$ −24.6245 −1.12630
$$479$$ 20.8863 0.954321 0.477161 0.878816i $$-0.341666\pi$$
0.477161 + 0.878816i $$0.341666\pi$$
$$480$$ 3.43531 0.156800
$$481$$ 36.6540 1.67128
$$482$$ −16.0809 −0.732464
$$483$$ −48.0514 −2.18642
$$484$$ 0 0
$$485$$ 11.1883 0.508033
$$486$$ 15.1103 0.685418
$$487$$ 15.2838 0.692575 0.346288 0.938128i $$-0.387442\pi$$
0.346288 + 0.938128i $$0.387442\pi$$
$$488$$ −13.4931 −0.610806
$$489$$ −4.04211 −0.182791
$$490$$ −2.18081 −0.0985191
$$491$$ −3.31965 −0.149814 −0.0749069 0.997191i $$-0.523866\pi$$
−0.0749069 + 0.997191i $$0.523866\pi$$
$$492$$ −4.41425 −0.199010
$$493$$ −30.7328 −1.38414
$$494$$ −4.89167 −0.220087
$$495$$ 0 0
$$496$$ −7.43531 −0.333855
$$497$$ −43.3142 −1.94291
$$498$$ −13.1030 −0.587159
$$499$$ −2.07355 −0.0928247 −0.0464124 0.998922i $$-0.514779\pi$$
−0.0464124 + 0.998922i $$0.514779\pi$$
$$500$$ 11.9074 0.532515
$$501$$ −10.6961 −0.477865
$$502$$ 13.0873 0.584114
$$503$$ 1.11672 0.0497921 0.0248960 0.999690i $$-0.492075\pi$$
0.0248960 + 0.999690i $$0.492075\pi$$
$$504$$ −4.63091 −0.206277
$$505$$ −1.92645 −0.0857260
$$506$$ 0 0
$$507$$ −23.4426 −1.04112
$$508$$ 21.2765 0.943991
$$509$$ −38.2481 −1.69532 −0.847659 0.530542i $$-0.821988\pi$$
−0.847659 + 0.530542i $$0.821988\pi$$
$$510$$ 19.7412 0.874156
$$511$$ −10.8338 −0.479261
$$512$$ 1.00000 0.0441942
$$513$$ 3.00000 0.132453
$$514$$ 0.723522 0.0319132
$$515$$ −27.0294 −1.19106
$$516$$ −22.9284 −1.00937
$$517$$ 0 0
$$518$$ 21.6677 0.952023
$$519$$ 13.9736 0.613373
$$520$$ 7.83384 0.343537
$$521$$ 10.8853 0.476892 0.238446 0.971156i $$-0.423362\pi$$
0.238446 + 0.971156i $$0.423362\pi$$
$$522$$ −8.56469 −0.374866
$$523$$ 15.7191 0.687349 0.343674 0.939089i $$-0.388328\pi$$
0.343674 + 0.939089i $$0.388328\pi$$
$$524$$ 3.43531 0.150072
$$525$$ −15.1060 −0.659282
$$526$$ −13.7760 −0.600663
$$527$$ −42.7275 −1.86124
$$528$$ 0 0
$$529$$ 37.0093 1.60910
$$530$$ −13.3354 −0.579251
$$531$$ −4.07355 −0.176777
$$532$$ −2.89167 −0.125370
$$533$$ −10.0662 −0.436016
$$534$$ −6.87062 −0.297321
$$535$$ −12.8706 −0.556445
$$536$$ −10.3848 −0.448555
$$537$$ 11.9074 0.513842
$$538$$ −19.9579 −0.860446
$$539$$ 0 0
$$540$$ −4.80440 −0.206748
$$541$$ 26.4373 1.13663 0.568314 0.822812i $$-0.307596\pi$$
0.568314 + 0.822812i $$0.307596\pi$$
$$542$$ −28.7486 −1.23486
$$543$$ −4.04211 −0.173464
$$544$$ 5.74657 0.246382
$$545$$ 18.4648 0.790943
$$546$$ −30.3427 −1.29855
$$547$$ 24.9588 1.06716 0.533581 0.845749i $$-0.320846\pi$$
0.533581 + 0.845749i $$0.320846\pi$$
$$548$$ 13.5877 0.580439
$$549$$ −21.6088 −0.922241
$$550$$ 0 0
$$551$$ −5.34803 −0.227834
$$552$$ 16.6172 0.707274
$$553$$ −25.1462 −1.06932
$$554$$ −3.08727 −0.131166
$$555$$ −25.7412 −1.09265
$$556$$ −3.19560 −0.135524
$$557$$ −8.30486 −0.351888 −0.175944 0.984400i $$-0.556298\pi$$
−0.175944 + 0.984400i $$0.556298\pi$$
$$558$$ −11.9074 −0.504080
$$559$$ −52.2858 −2.21145
$$560$$ 4.63091 0.195692
$$561$$ 0 0
$$562$$ −9.09460 −0.383633
$$563$$ −41.6402 −1.75493 −0.877463 0.479644i $$-0.840766\pi$$
−0.877463 + 0.479644i $$0.840766\pi$$
$$564$$ 15.2029 0.640159
$$565$$ 12.8117 0.538993
$$566$$ −13.1451 −0.552530
$$567$$ 32.5015 1.36494
$$568$$ 14.9789 0.628503
$$569$$ 14.9138 0.625219 0.312609 0.949882i $$-0.398797\pi$$
0.312609 + 0.949882i $$0.398797\pi$$
$$570$$ 3.43531 0.143889
$$571$$ 23.0020 0.962603 0.481302 0.876555i $$-0.340164\pi$$
0.481302 + 0.876555i $$0.340164\pi$$
$$572$$ 0 0
$$573$$ −5.23970 −0.218892
$$574$$ −5.95056 −0.248372
$$575$$ 18.8653 0.786737
$$576$$ 1.60147 0.0667277
$$577$$ −33.6613 −1.40134 −0.700669 0.713487i $$-0.747115\pi$$
−0.700669 + 0.713487i $$0.747115\pi$$
$$578$$ 16.0230 0.666471
$$579$$ −20.7539 −0.862502
$$580$$ 8.56469 0.355629
$$581$$ −17.6633 −0.732796
$$582$$ 14.9863 0.621201
$$583$$ 0 0
$$584$$ 3.74657 0.155034
$$585$$ 12.5456 0.518698
$$586$$ −3.10833 −0.128404
$$587$$ 4.00000 0.165098 0.0825488 0.996587i $$-0.473694\pi$$
0.0825488 + 0.996587i $$0.473694\pi$$
$$588$$ −2.92112 −0.120465
$$589$$ −7.43531 −0.306367
$$590$$ 4.07355 0.167705
$$591$$ −37.5246 −1.54355
$$592$$ −7.49314 −0.307966
$$593$$ 30.9588 1.27133 0.635663 0.771967i $$-0.280727\pi$$
0.635663 + 0.771967i $$0.280727\pi$$
$$594$$ 0 0
$$595$$ 26.6118 1.09098
$$596$$ 12.0735 0.494552
$$597$$ 54.1418 2.21587
$$598$$ 37.8937 1.54959
$$599$$ 3.93378 0.160730 0.0803650 0.996766i $$-0.474391\pi$$
0.0803650 + 0.996766i $$0.474391\pi$$
$$600$$ 5.22399 0.213268
$$601$$ −23.1451 −0.944108 −0.472054 0.881570i $$-0.656487\pi$$
−0.472054 + 0.881570i $$0.656487\pi$$
$$602$$ −30.9083 −1.25973
$$603$$ −16.6309 −0.677263
$$604$$ 4.91273 0.199896
$$605$$ 0 0
$$606$$ −2.58041 −0.104822
$$607$$ 1.78334 0.0723836 0.0361918 0.999345i $$-0.488477\pi$$
0.0361918 + 0.999345i $$0.488477\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ −33.1735 −1.34426
$$610$$ 21.6088 0.874914
$$611$$ 34.6686 1.40254
$$612$$ 9.20293 0.372006
$$613$$ 33.0598 1.33527 0.667637 0.744487i $$-0.267306\pi$$
0.667637 + 0.744487i $$0.267306\pi$$
$$614$$ −2.18920 −0.0883491
$$615$$ 7.06927 0.285061
$$616$$ 0 0
$$617$$ −21.8412 −0.879292 −0.439646 0.898171i $$-0.644896\pi$$
−0.439646 + 0.898171i $$0.644896\pi$$
$$618$$ −36.2049 −1.45638
$$619$$ −46.2206 −1.85776 −0.928882 0.370375i $$-0.879229\pi$$
−0.928882 + 0.370375i $$0.879229\pi$$
$$620$$ 11.9074 0.478212
$$621$$ −23.2397 −0.932577
$$622$$ 3.92112 0.157222
$$623$$ −9.26182 −0.371067
$$624$$ 10.4931 0.420062
$$625$$ −6.89273 −0.275709
$$626$$ −21.7118 −0.867778
$$627$$ 0 0
$$628$$ 8.00733 0.319527
$$629$$ −43.0598 −1.71691
$$630$$ 7.41625 0.295470
$$631$$ 0.317659 0.0126458 0.00632291 0.999980i $$-0.497987\pi$$
0.00632291 + 0.999980i $$0.497987\pi$$
$$632$$ 8.69607 0.345911
$$633$$ −31.8201 −1.26474
$$634$$ 6.90739 0.274327
$$635$$ −34.0735 −1.35217
$$636$$ −17.8622 −0.708283
$$637$$ −6.66129 −0.263930
$$638$$ 0 0
$$639$$ 23.9883 0.948961
$$640$$ −1.60147 −0.0633035
$$641$$ −41.5246 −1.64012 −0.820061 0.572276i $$-0.806061\pi$$
−0.820061 + 0.572276i $$0.806061\pi$$
$$642$$ −17.2397 −0.680397
$$643$$ 7.26182 0.286378 0.143189 0.989695i $$-0.454264\pi$$
0.143189 + 0.989695i $$0.454264\pi$$
$$644$$ 22.4005 0.882704
$$645$$ 36.7191 1.44581
$$646$$ 5.74657 0.226096
$$647$$ 29.0819 1.14333 0.571664 0.820487i $$-0.306298\pi$$
0.571664 + 0.820487i $$0.306298\pi$$
$$648$$ −11.2397 −0.441537
$$649$$ 0 0
$$650$$ 11.9127 0.467256
$$651$$ −46.1207 −1.80761
$$652$$ 1.88434 0.0737966
$$653$$ 12.6529 0.495146 0.247573 0.968869i $$-0.420367\pi$$
0.247573 + 0.968869i $$0.420367\pi$$
$$654$$ 24.7328 0.967131
$$655$$ −5.50153 −0.214962
$$656$$ 2.05783 0.0803447
$$657$$ 6.00000 0.234082
$$658$$ 20.4941 0.798942
$$659$$ −49.4878 −1.92777 −0.963886 0.266317i $$-0.914193\pi$$
−0.963886 + 0.266317i $$0.914193\pi$$
$$660$$ 0 0
$$661$$ 4.94950 0.192513 0.0962566 0.995357i $$-0.469313\pi$$
0.0962566 + 0.995357i $$0.469313\pi$$
$$662$$ 32.1030 1.24772
$$663$$ 60.2995 2.34184
$$664$$ 6.10833 0.237049
$$665$$ 4.63091 0.179579
$$666$$ −12.0000 −0.464991
$$667$$ 41.4289 1.60413
$$668$$ 4.98627 0.192925
$$669$$ −24.6540 −0.953177
$$670$$ 16.6309 0.642508
$$671$$ 0 0
$$672$$ 6.20293 0.239283
$$673$$ −21.9001 −0.844185 −0.422093 0.906553i $$-0.638704\pi$$
−0.422093 + 0.906553i $$0.638704\pi$$
$$674$$ 1.55096 0.0597409
$$675$$ −7.30592 −0.281205
$$676$$ 10.9284 0.420325
$$677$$ −26.4353 −1.01599 −0.507996 0.861360i $$-0.669613\pi$$
−0.507996 + 0.861360i $$0.669613\pi$$
$$678$$ 17.1608 0.659057
$$679$$ 20.2020 0.775282
$$680$$ −9.20293 −0.352916
$$681$$ 18.7328 0.717844
$$682$$ 0 0
$$683$$ 9.82545 0.375960 0.187980 0.982173i $$-0.439806\pi$$
0.187980 + 0.982173i $$0.439806\pi$$
$$684$$ 1.60147 0.0612336
$$685$$ −21.7603 −0.831418
$$686$$ 16.3039 0.622487
$$687$$ −16.8201 −0.641727
$$688$$ 10.6887 0.407504
$$689$$ −40.7328 −1.55180
$$690$$ −26.6118 −1.01310
$$691$$ −19.8255 −0.754196 −0.377098 0.926173i $$-0.623078\pi$$
−0.377098 + 0.926173i $$0.623078\pi$$
$$692$$ −6.51419 −0.247632
$$693$$ 0 0
$$694$$ 0.506864 0.0192403
$$695$$ 5.11765 0.194123
$$696$$ 11.4721 0.434848
$$697$$ 11.8255 0.447921
$$698$$ −10.8117 −0.409230
$$699$$ −26.8285 −1.01475
$$700$$ 7.04211 0.266167
$$701$$ 10.7550 0.406209 0.203105 0.979157i $$-0.434897\pi$$
0.203105 + 0.979157i $$0.434897\pi$$
$$702$$ −14.6750 −0.553873
$$703$$ −7.49314 −0.282609
$$704$$ 0 0
$$705$$ −24.3470 −0.916960
$$706$$ 27.0966 1.01979
$$707$$ −3.47848 −0.130822
$$708$$ 5.45636 0.205063
$$709$$ −1.82651 −0.0685962 −0.0342981 0.999412i $$-0.510920\pi$$
−0.0342981 + 0.999412i $$0.510920\pi$$
$$710$$ −23.9883 −0.900264
$$711$$ 13.9265 0.522283
$$712$$ 3.20293 0.120035
$$713$$ 57.5981 2.15707
$$714$$ 35.6456 1.33400
$$715$$ 0 0
$$716$$ −5.55096 −0.207449
$$717$$ 52.8221 1.97268
$$718$$ 20.7486 0.774329
$$719$$ −4.65929 −0.173762 −0.0868812 0.996219i $$-0.527690\pi$$
−0.0868812 + 0.996219i $$0.527690\pi$$
$$720$$ −2.56469 −0.0955804
$$721$$ −48.8055 −1.81761
$$722$$ 1.00000 0.0372161
$$723$$ 34.4951 1.28289
$$724$$ 1.88434 0.0700311
$$725$$ 13.0241 0.483703
$$726$$ 0 0
$$727$$ 19.9358 0.739377 0.369688 0.929156i $$-0.379464\pi$$
0.369688 + 0.929156i $$0.379464\pi$$
$$728$$ 14.1451 0.524252
$$729$$ 1.30592 0.0483676
$$730$$ −6.00000 −0.222070
$$731$$ 61.4236 2.27183
$$732$$ 28.9442 1.06981
$$733$$ −13.7687 −0.508558 −0.254279 0.967131i $$-0.581838\pi$$
−0.254279 + 0.967131i $$0.581838\pi$$
$$734$$ 25.6402 0.946398
$$735$$ 4.67807 0.172553
$$736$$ −7.74657 −0.285542
$$737$$ 0 0
$$738$$ 3.29554 0.121311
$$739$$ −22.4867 −0.827188 −0.413594 0.910461i $$-0.635727\pi$$
−0.413594 + 0.910461i $$0.635727\pi$$
$$740$$ 12.0000 0.441129
$$741$$ 10.4931 0.385475
$$742$$ −24.0789 −0.883964
$$743$$ 50.6540 1.85831 0.929157 0.369686i $$-0.120535\pi$$
0.929157 + 0.369686i $$0.120535\pi$$
$$744$$ 15.9495 0.584737
$$745$$ −19.3354 −0.708393
$$746$$ −25.2049 −0.922818
$$747$$ 9.78228 0.357915
$$748$$ 0 0
$$749$$ −23.2397 −0.849160
$$750$$ −25.5426 −0.932683
$$751$$ −38.9127 −1.41995 −0.709973 0.704229i $$-0.751293\pi$$
−0.709973 + 0.704229i $$0.751293\pi$$
$$752$$ −7.08727 −0.258446
$$753$$ −28.0735 −1.02306
$$754$$ 26.1608 0.952721
$$755$$ −7.86756 −0.286330
$$756$$ −8.67501 −0.315507
$$757$$ −20.5877 −0.748274 −0.374137 0.927373i $$-0.622061\pi$$
−0.374137 + 0.927373i $$0.622061\pi$$
$$758$$ 3.47208 0.126112
$$759$$ 0 0
$$760$$ −1.60147 −0.0580913
$$761$$ −10.6025 −0.384341 −0.192171 0.981362i $$-0.561553\pi$$
−0.192171 + 0.981362i $$0.561553\pi$$
$$762$$ −45.6402 −1.65337
$$763$$ 33.3407 1.20701
$$764$$ 2.44264 0.0883715
$$765$$ −14.7382 −0.532860
$$766$$ 23.1176 0.835275
$$767$$ 12.4426 0.449278
$$768$$ −2.14510 −0.0774047
$$769$$ 8.77495 0.316433 0.158216 0.987404i $$-0.449426\pi$$
0.158216 + 0.987404i $$0.449426\pi$$
$$770$$ 0 0
$$771$$ −1.55203 −0.0558949
$$772$$ 9.67501 0.348211
$$773$$ −46.8653 −1.68563 −0.842813 0.538206i $$-0.819102\pi$$
−0.842813 + 0.538206i $$0.819102\pi$$
$$774$$ 17.1176 0.615281
$$775$$ 18.1073 0.650432
$$776$$ −6.98627 −0.250793
$$777$$ −46.4794 −1.66744
$$778$$ −26.7623 −0.959474
$$779$$ 2.05783 0.0737294
$$780$$ −16.8044 −0.601694
$$781$$ 0 0
$$782$$ −44.5162 −1.59190
$$783$$ −16.0441 −0.573369
$$784$$ 1.36176 0.0486343
$$785$$ −12.8235 −0.457689
$$786$$ −7.36909 −0.262847
$$787$$ 27.4143 0.977213 0.488606 0.872504i $$-0.337505\pi$$
0.488606 + 0.872504i $$0.337505\pi$$
$$788$$ 17.4931 0.623167
$$789$$ 29.5510 1.05204
$$790$$ −13.9265 −0.495481
$$791$$ 23.1334 0.822528
$$792$$ 0 0
$$793$$ 66.0040 2.34387
$$794$$ −31.1030 −1.10380
$$795$$ 28.6057 1.01454
$$796$$ −25.2397 −0.894598
$$797$$ −9.16616 −0.324682 −0.162341 0.986735i $$-0.551904\pi$$
−0.162341 + 0.986735i $$0.551904\pi$$
$$798$$ 6.20293 0.219581
$$799$$ −40.7275 −1.44084
$$800$$ −2.43531 −0.0861011
$$801$$ 5.12938 0.181238
$$802$$ 29.3500 1.03639
$$803$$ 0 0
$$804$$ 22.2765 0.785631
$$805$$ −35.8737 −1.26438
$$806$$ 36.3711 1.28112
$$807$$ 42.8117 1.50704
$$808$$ 1.20293 0.0423190
$$809$$ −48.0829 −1.69050 −0.845252 0.534368i $$-0.820550\pi$$
−0.845252 + 0.534368i $$0.820550\pi$$
$$810$$ 18.0000 0.632456
$$811$$ 2.67395 0.0938951 0.0469475 0.998897i $$-0.485051\pi$$
0.0469475 + 0.998897i $$0.485051\pi$$
$$812$$ 15.4648 0.542706
$$813$$ 61.6686 2.16281
$$814$$ 0 0
$$815$$ −3.01771 −0.105706
$$816$$ −12.3270 −0.431531
$$817$$ 10.6887 0.373952
$$818$$ −8.58774 −0.300263
$$819$$ 22.6529 0.791556
$$820$$ −3.29554 −0.115085
$$821$$ −29.5078 −1.02983 −0.514915 0.857242i $$-0.672176\pi$$
−0.514915 + 0.857242i $$0.672176\pi$$
$$822$$ −29.1471 −1.01662
$$823$$ −2.39654 −0.0835382 −0.0417691 0.999127i $$-0.513299\pi$$
−0.0417691 + 0.999127i $$0.513299\pi$$
$$824$$ 16.8779 0.587971
$$825$$ 0 0
$$826$$ 7.35536 0.255926
$$827$$ −7.78868 −0.270839 −0.135419 0.990788i $$-0.543238\pi$$
−0.135419 + 0.990788i $$0.543238\pi$$
$$828$$ −12.4059 −0.431134
$$829$$ 13.2818 0.461296 0.230648 0.973037i $$-0.425915\pi$$
0.230648 + 0.973037i $$0.425915\pi$$
$$830$$ −9.78228 −0.339548
$$831$$ 6.62252 0.229733
$$832$$ −4.89167 −0.169588
$$833$$ 7.82545 0.271136
$$834$$ 6.85490 0.237366
$$835$$ −7.98534 −0.276344
$$836$$ 0 0
$$837$$ −22.3059 −0.771006
$$838$$ 7.59414 0.262335
$$839$$ −33.9063 −1.17058 −0.585288 0.810825i $$-0.699019\pi$$
−0.585288 + 0.810825i $$0.699019\pi$$
$$840$$ −9.93378 −0.342748
$$841$$ −0.398534 −0.0137426
$$842$$ 8.76030 0.301900
$$843$$ 19.5089 0.671921
$$844$$ 14.8338 0.510602
$$845$$ −17.5015 −0.602071
$$846$$ −11.3500 −0.390222
$$847$$ 0 0
$$848$$ 8.32698 0.285950
$$849$$ 28.1976 0.967739
$$850$$ −13.9947 −0.480013
$$851$$ 58.0461 1.98979
$$852$$ −32.1314 −1.10080
$$853$$ −40.0735 −1.37209 −0.686046 0.727558i $$-0.740655\pi$$
−0.686046 + 0.727558i $$0.740655\pi$$
$$854$$ 39.0177 1.33516
$$855$$ −2.56469 −0.0877106
$$856$$ 8.03677 0.274691
$$857$$ −0.665693 −0.0227396 −0.0113698 0.999935i $$-0.503619\pi$$
−0.0113698 + 0.999935i $$0.503619\pi$$
$$858$$ 0 0
$$859$$ 13.8990 0.474228 0.237114 0.971482i $$-0.423799\pi$$
0.237114 + 0.971482i $$0.423799\pi$$
$$860$$ −17.1176 −0.583707
$$861$$ 12.7646 0.435015
$$862$$ 33.6677 1.14673
$$863$$ −40.3941 −1.37503 −0.687516 0.726169i $$-0.741299\pi$$
−0.687516 + 0.726169i $$0.741299\pi$$
$$864$$ 3.00000 0.102062
$$865$$ 10.4323 0.354707
$$866$$ 30.5383 1.03773
$$867$$ −34.3711 −1.16730
$$868$$ 21.5005 0.729773
$$869$$ 0 0
$$870$$ −18.3721 −0.622874
$$871$$ 50.7991 1.72126
$$872$$ −11.5299 −0.390452
$$873$$ −11.1883 −0.378666
$$874$$ −7.74657 −0.262032
$$875$$ −34.4323 −1.16402
$$876$$ −8.03677 −0.271537
$$877$$ −15.6108 −0.527139 −0.263569 0.964640i $$-0.584900\pi$$
−0.263569 + 0.964640i $$0.584900\pi$$
$$878$$ 12.5069 0.422086
$$879$$ 6.66769 0.224895
$$880$$ 0 0
$$881$$ 19.9843 0.673288 0.336644 0.941632i $$-0.390708\pi$$
0.336644 + 0.941632i $$0.390708\pi$$
$$882$$ 2.18081 0.0734318
$$883$$ 20.2167 0.680345 0.340172 0.940363i $$-0.389515\pi$$
0.340172 + 0.940363i $$0.389515\pi$$
$$884$$ −28.1103 −0.945452
$$885$$ −8.73818 −0.293731
$$886$$ −17.6677 −0.593557
$$887$$ −20.1471 −0.676473 −0.338237 0.941061i $$-0.609830\pi$$
−0.338237 + 0.941061i $$0.609830\pi$$
$$888$$ 16.0735 0.539393
$$889$$ −61.5246 −2.06347
$$890$$ −5.12938 −0.171937
$$891$$ 0 0
$$892$$ 11.4931 0.384819
$$893$$ −7.08727 −0.237167
$$894$$ −25.8990 −0.866192
$$895$$ 8.88968 0.297149
$$896$$ −2.89167 −0.0966039
$$897$$ −81.2858 −2.71405
$$898$$ −17.7098 −0.590984
$$899$$ 39.7643 1.32621
$$900$$ −3.90006 −0.130002
$$901$$ 47.8516 1.59417
$$902$$ 0 0
$$903$$ 66.3015 2.20638
$$904$$ −8.00000 −0.266076
$$905$$ −3.01771 −0.100312
$$906$$ −10.5383 −0.350112
$$907$$ 4.57109 0.151781 0.0758903 0.997116i $$-0.475820\pi$$
0.0758903 + 0.997116i $$0.475820\pi$$
$$908$$ −8.73284 −0.289810
$$909$$ 1.92645 0.0638964
$$910$$ −22.6529 −0.750936
$$911$$ 51.6402 1.71092 0.855459 0.517871i $$-0.173275\pi$$
0.855459 + 0.517871i $$0.173275\pi$$
$$912$$ −2.14510 −0.0710314
$$913$$ 0 0
$$914$$ 24.8064 0.820522
$$915$$ −46.3531 −1.53239
$$916$$ 7.84117 0.259080
$$917$$ −9.93378 −0.328042
$$918$$ 17.2397 0.568995
$$919$$ 48.8412 1.61112 0.805561 0.592513i $$-0.201864\pi$$
0.805561 + 0.592513i $$0.201864\pi$$
$$920$$ 12.4059 0.409009
$$921$$ 4.69607 0.154741
$$922$$ 2.98627 0.0983477
$$923$$ −73.2721 −2.41178
$$924$$ 0 0
$$925$$ 18.2481 0.599994
$$926$$ −39.5667 −1.30024
$$927$$ 27.0294 0.887763
$$928$$ −5.34803 −0.175558
$$929$$ −57.8819 −1.89904 −0.949522 0.313700i $$-0.898431\pi$$
−0.949522 + 0.313700i $$0.898431\pi$$
$$930$$ −25.5426 −0.837574
$$931$$ 1.36176 0.0446299
$$932$$ 12.5069 0.409676
$$933$$ −8.41120 −0.275370
$$934$$ 4.46475 0.146091
$$935$$ 0 0
$$936$$ −7.83384 −0.256057
$$937$$ 57.6496 1.88333 0.941664 0.336553i $$-0.109261\pi$$
0.941664 + 0.336553i $$0.109261\pi$$
$$938$$ 30.0294 0.980496
$$939$$ 46.5740 1.51989
$$940$$ 11.3500 0.370197
$$941$$ 15.0966 0.492135 0.246067 0.969253i $$-0.420862\pi$$
0.246067 + 0.969253i $$0.420862\pi$$
$$942$$ −17.1765 −0.559642
$$943$$ −15.9411 −0.519114
$$944$$ −2.54364 −0.0827883
$$945$$ 13.8927 0.451931
$$946$$ 0 0
$$947$$ 33.3775 1.08462 0.542311 0.840178i $$-0.317549\pi$$
0.542311 + 0.840178i $$0.317549\pi$$
$$948$$ −18.6540 −0.605852
$$949$$ −18.3270 −0.594919
$$950$$ −2.43531 −0.0790118
$$951$$ −14.8171 −0.480476
$$952$$ −16.6172 −0.538566
$$953$$ 12.4794 0.404248 0.202124 0.979360i $$-0.435216\pi$$
0.202124 + 0.979360i $$0.435216\pi$$
$$954$$ 13.3354 0.431749
$$955$$ −3.91180 −0.126583
$$956$$ −24.6245 −0.796414
$$957$$ 0 0
$$958$$ 20.8863 0.674807
$$959$$ −39.2913 −1.26878
$$960$$ 3.43531 0.110874
$$961$$ 24.2838 0.783349
$$962$$ 36.6540 1.18177
$$963$$ 12.8706 0.414750
$$964$$ −16.0809 −0.517930
$$965$$ −15.4942 −0.498776
$$966$$ −48.0514 −1.54603
$$967$$ 12.5804 0.404559 0.202279 0.979328i $$-0.435165\pi$$
0.202279 + 0.979328i $$0.435165\pi$$
$$968$$ 0 0
$$969$$ −12.3270 −0.396000
$$970$$ 11.1883 0.359234
$$971$$ −1.79067 −0.0574653 −0.0287327 0.999587i $$-0.509147\pi$$
−0.0287327 + 0.999587i $$0.509147\pi$$
$$972$$ 15.1103 0.484664
$$973$$ 9.24063 0.296241
$$974$$ 15.2838 0.489725
$$975$$ −25.5540 −0.818384
$$976$$ −13.4931 −0.431905
$$977$$ 6.40586 0.204942 0.102471 0.994736i $$-0.467325\pi$$
0.102471 + 0.994736i $$0.467325\pi$$
$$978$$ −4.04211 −0.129252
$$979$$ 0 0
$$980$$ −2.18081 −0.0696635
$$981$$ −18.4648 −0.589534
$$982$$ −3.31965 −0.105934
$$983$$ −20.5647 −0.655912 −0.327956 0.944693i $$-0.606360\pi$$
−0.327956 + 0.944693i $$0.606360\pi$$
$$984$$ −4.41425 −0.140721
$$985$$ −28.0147 −0.892621
$$986$$ −30.7328 −0.978733
$$987$$ −43.9619 −1.39932
$$988$$ −4.89167 −0.155625
$$989$$ −82.8011 −2.63292
$$990$$ 0 0
$$991$$ −29.0166 −0.921744 −0.460872 0.887467i $$-0.652463\pi$$
−0.460872 + 0.887467i $$0.652463\pi$$
$$992$$ −7.43531 −0.236071
$$993$$ −68.8642 −2.18534
$$994$$ −43.3142 −1.37384
$$995$$ 40.4205 1.28142
$$996$$ −13.1030 −0.415184
$$997$$ −42.3638 −1.34167 −0.670837 0.741605i $$-0.734065\pi$$
−0.670837 + 0.741605i $$0.734065\pi$$
$$998$$ −2.07355 −0.0656370
$$999$$ −22.4794 −0.711217
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 4598.2.a.bo.1.1 3
11.10 odd 2 418.2.a.g.1.1 3
33.32 even 2 3762.2.a.bg.1.2 3
44.43 even 2 3344.2.a.q.1.3 3
209.208 even 2 7942.2.a.bi.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.1 3 11.10 odd 2
3344.2.a.q.1.3 3 44.43 even 2
3762.2.a.bg.1.2 3 33.32 even 2
4598.2.a.bo.1.1 3 1.1 even 1 trivial
7942.2.a.bi.1.3 3 209.208 even 2