# Properties

 Label 169.2.a.c.1.2 Level $169$ Weight $2$ Character 169.1 Self dual yes Analytic conductor $1.349$ 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] = [169,2,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 169.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: $$1.34947179416$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.445042$$ of defining polynomial Character $$\chi$$ $$=$$ 169.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.554958 q^{2} +0.801938 q^{3} -1.69202 q^{4} +2.80194 q^{5} +0.445042 q^{6} +2.69202 q^{7} -2.04892 q^{8} -2.35690 q^{9} +O(q^{10})$$ $$q+0.554958 q^{2} +0.801938 q^{3} -1.69202 q^{4} +2.80194 q^{5} +0.445042 q^{6} +2.69202 q^{7} -2.04892 q^{8} -2.35690 q^{9} +1.55496 q^{10} +1.19806 q^{11} -1.35690 q^{12} +1.49396 q^{14} +2.24698 q^{15} +2.24698 q^{16} +1.13706 q^{17} -1.30798 q^{18} -1.93900 q^{19} -4.74094 q^{20} +2.15883 q^{21} +0.664874 q^{22} -4.60388 q^{23} -1.64310 q^{24} +2.85086 q^{25} -4.29590 q^{27} -4.55496 q^{28} -7.89977 q^{29} +1.24698 q^{30} -5.89977 q^{31} +5.34481 q^{32} +0.960771 q^{33} +0.631023 q^{34} +7.54288 q^{35} +3.98792 q^{36} -0.951083 q^{37} -1.07606 q^{38} -5.74094 q^{40} -3.31767 q^{41} +1.19806 q^{42} +7.15883 q^{43} -2.02715 q^{44} -6.60388 q^{45} -2.55496 q^{46} +7.69202 q^{47} +1.80194 q^{48} +0.246980 q^{49} +1.58211 q^{50} +0.911854 q^{51} +5.87263 q^{53} -2.38404 q^{54} +3.35690 q^{55} -5.51573 q^{56} -1.55496 q^{57} -4.38404 q^{58} +0.0120816 q^{59} -3.80194 q^{60} -8.03684 q^{61} -3.27413 q^{62} -6.34481 q^{63} -1.52781 q^{64} +0.533188 q^{66} +9.25667 q^{67} -1.92394 q^{68} -3.69202 q^{69} +4.18598 q^{70} +13.7409 q^{71} +4.82908 q^{72} -12.8170 q^{73} -0.527811 q^{74} +2.28621 q^{75} +3.28083 q^{76} +3.22521 q^{77} +0.807315 q^{79} +6.29590 q^{80} +3.62565 q^{81} -1.84117 q^{82} +16.3327 q^{83} -3.65279 q^{84} +3.18598 q^{85} +3.97285 q^{86} -6.33513 q^{87} -2.45473 q^{88} +14.7289 q^{89} -3.66487 q^{90} +7.78986 q^{92} -4.73125 q^{93} +4.26875 q^{94} -5.43296 q^{95} +4.28621 q^{96} -3.13169 q^{97} +0.137063 q^{98} -2.82371 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} - 2 q^{3} + 4 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 - 2 * q^3 + 4 * q^5 + q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9 $$3 q + 2 q^{2} - 2 q^{3} + 4 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} + 5 q^{10} + 8 q^{11} - 5 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 9 q^{18} + 4 q^{19} - 2 q^{21} + 3 q^{22} - 5 q^{23} - 9 q^{24} - 5 q^{25} + q^{27} - 14 q^{28} - q^{29} - q^{30} + 5 q^{31} - 7 q^{32} - 10 q^{33} - 13 q^{34} + 4 q^{35} - 7 q^{36} - 12 q^{37} + 12 q^{38} - 3 q^{40} + 7 q^{41} + 8 q^{42} + 13 q^{43} - 11 q^{45} - 8 q^{46} + 18 q^{47} + q^{48} - 4 q^{49} - q^{50} - q^{51} + q^{53} + 3 q^{54} + 6 q^{55} - 4 q^{56} - 5 q^{57} - 3 q^{58} + 19 q^{59} - 7 q^{60} + 4 q^{61} + q^{62} + 4 q^{63} - 11 q^{64} + 5 q^{66} + q^{67} - 21 q^{68} - 6 q^{69} - 2 q^{70} + 27 q^{71} + 4 q^{72} - 9 q^{73} - 8 q^{74} + 15 q^{75} + 21 q^{76} + 8 q^{77} - 5 q^{79} + 5 q^{80} - q^{81} - 14 q^{82} + 7 q^{83} + 7 q^{84} - 5 q^{85} + 18 q^{86} - 18 q^{87} + 15 q^{88} + 11 q^{89} - 12 q^{90} - 22 q^{93} + 5 q^{94} + 3 q^{95} + 21 q^{96} - 7 q^{97} - 5 q^{98} - q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 - 2 * q^3 + 4 * q^5 + q^6 + 3 * q^7 + 3 * q^8 - 3 * q^9 + 5 * q^10 + 8 * q^11 - 5 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 - 9 * q^18 + 4 * q^19 - 2 * q^21 + 3 * q^22 - 5 * q^23 - 9 * q^24 - 5 * q^25 + q^27 - 14 * q^28 - q^29 - q^30 + 5 * q^31 - 7 * q^32 - 10 * q^33 - 13 * q^34 + 4 * q^35 - 7 * q^36 - 12 * q^37 + 12 * q^38 - 3 * q^40 + 7 * q^41 + 8 * q^42 + 13 * q^43 - 11 * q^45 - 8 * q^46 + 18 * q^47 + q^48 - 4 * q^49 - q^50 - q^51 + q^53 + 3 * q^54 + 6 * q^55 - 4 * q^56 - 5 * q^57 - 3 * q^58 + 19 * q^59 - 7 * q^60 + 4 * q^61 + q^62 + 4 * q^63 - 11 * q^64 + 5 * q^66 + q^67 - 21 * q^68 - 6 * q^69 - 2 * q^70 + 27 * q^71 + 4 * q^72 - 9 * q^73 - 8 * q^74 + 15 * q^75 + 21 * q^76 + 8 * q^77 - 5 * q^79 + 5 * q^80 - q^81 - 14 * q^82 + 7 * q^83 + 7 * q^84 - 5 * q^85 + 18 * q^86 - 18 * q^87 + 15 * q^88 + 11 * q^89 - 12 * q^90 - 22 * q^93 + 5 * q^94 + 3 * q^95 + 21 * q^96 - 7 * q^97 - 5 * q^98 - q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.554958 0.392415 0.196207 0.980562i $$-0.437137\pi$$
0.196207 + 0.980562i $$0.437137\pi$$
$$3$$ 0.801938 0.462999 0.231499 0.972835i $$-0.425637\pi$$
0.231499 + 0.972835i $$0.425637\pi$$
$$4$$ −1.69202 −0.846011
$$5$$ 2.80194 1.25306 0.626532 0.779395i $$-0.284474\pi$$
0.626532 + 0.779395i $$0.284474\pi$$
$$6$$ 0.445042 0.181688
$$7$$ 2.69202 1.01749 0.508744 0.860918i $$-0.330110\pi$$
0.508744 + 0.860918i $$0.330110\pi$$
$$8$$ −2.04892 −0.724402
$$9$$ −2.35690 −0.785632
$$10$$ 1.55496 0.491721
$$11$$ 1.19806 0.361229 0.180615 0.983554i $$-0.442191\pi$$
0.180615 + 0.983554i $$0.442191\pi$$
$$12$$ −1.35690 −0.391702
$$13$$ 0 0
$$14$$ 1.49396 0.399277
$$15$$ 2.24698 0.580168
$$16$$ 2.24698 0.561745
$$17$$ 1.13706 0.275778 0.137889 0.990448i $$-0.455968\pi$$
0.137889 + 0.990448i $$0.455968\pi$$
$$18$$ −1.30798 −0.308293
$$19$$ −1.93900 −0.444837 −0.222419 0.974951i $$-0.571395\pi$$
−0.222419 + 0.974951i $$0.571395\pi$$
$$20$$ −4.74094 −1.06011
$$21$$ 2.15883 0.471096
$$22$$ 0.664874 0.141752
$$23$$ −4.60388 −0.959974 −0.479987 0.877275i $$-0.659359\pi$$
−0.479987 + 0.877275i $$0.659359\pi$$
$$24$$ −1.64310 −0.335397
$$25$$ 2.85086 0.570171
$$26$$ 0 0
$$27$$ −4.29590 −0.826746
$$28$$ −4.55496 −0.860806
$$29$$ −7.89977 −1.46695 −0.733475 0.679716i $$-0.762103\pi$$
−0.733475 + 0.679716i $$0.762103\pi$$
$$30$$ 1.24698 0.227666
$$31$$ −5.89977 −1.05963 −0.529815 0.848113i $$-0.677739\pi$$
−0.529815 + 0.848113i $$0.677739\pi$$
$$32$$ 5.34481 0.944839
$$33$$ 0.960771 0.167249
$$34$$ 0.631023 0.108219
$$35$$ 7.54288 1.27498
$$36$$ 3.98792 0.664653
$$37$$ −0.951083 −0.156357 −0.0781785 0.996939i $$-0.524910\pi$$
−0.0781785 + 0.996939i $$0.524910\pi$$
$$38$$ −1.07606 −0.174561
$$39$$ 0 0
$$40$$ −5.74094 −0.907722
$$41$$ −3.31767 −0.518133 −0.259066 0.965860i $$-0.583415\pi$$
−0.259066 + 0.965860i $$0.583415\pi$$
$$42$$ 1.19806 0.184865
$$43$$ 7.15883 1.09171 0.545856 0.837879i $$-0.316205\pi$$
0.545856 + 0.837879i $$0.316205\pi$$
$$44$$ −2.02715 −0.305604
$$45$$ −6.60388 −0.984448
$$46$$ −2.55496 −0.376708
$$47$$ 7.69202 1.12200 0.560998 0.827817i $$-0.310417\pi$$
0.560998 + 0.827817i $$0.310417\pi$$
$$48$$ 1.80194 0.260087
$$49$$ 0.246980 0.0352828
$$50$$ 1.58211 0.223743
$$51$$ 0.911854 0.127685
$$52$$ 0 0
$$53$$ 5.87263 0.806667 0.403334 0.915053i $$-0.367851\pi$$
0.403334 + 0.915053i $$0.367851\pi$$
$$54$$ −2.38404 −0.324427
$$55$$ 3.35690 0.452644
$$56$$ −5.51573 −0.737070
$$57$$ −1.55496 −0.205959
$$58$$ −4.38404 −0.575653
$$59$$ 0.0120816 0.00157289 0.000786444 1.00000i $$-0.499750\pi$$
0.000786444 1.00000i $$0.499750\pi$$
$$60$$ −3.80194 −0.490828
$$61$$ −8.03684 −1.02901 −0.514506 0.857487i $$-0.672025\pi$$
−0.514506 + 0.857487i $$0.672025\pi$$
$$62$$ −3.27413 −0.415815
$$63$$ −6.34481 −0.799371
$$64$$ −1.52781 −0.190976
$$65$$ 0 0
$$66$$ 0.533188 0.0656309
$$67$$ 9.25667 1.13088 0.565441 0.824789i $$-0.308706\pi$$
0.565441 + 0.824789i $$0.308706\pi$$
$$68$$ −1.92394 −0.233311
$$69$$ −3.69202 −0.444467
$$70$$ 4.18598 0.500320
$$71$$ 13.7409 1.63075 0.815375 0.578934i $$-0.196531\pi$$
0.815375 + 0.578934i $$0.196531\pi$$
$$72$$ 4.82908 0.569113
$$73$$ −12.8170 −1.50012 −0.750058 0.661372i $$-0.769975\pi$$
−0.750058 + 0.661372i $$0.769975\pi$$
$$74$$ −0.527811 −0.0613568
$$75$$ 2.28621 0.263989
$$76$$ 3.28083 0.376337
$$77$$ 3.22521 0.367547
$$78$$ 0 0
$$79$$ 0.807315 0.0908300 0.0454150 0.998968i $$-0.485539\pi$$
0.0454150 + 0.998968i $$0.485539\pi$$
$$80$$ 6.29590 0.703903
$$81$$ 3.62565 0.402850
$$82$$ −1.84117 −0.203323
$$83$$ 16.3327 1.79275 0.896375 0.443296i $$-0.146191\pi$$
0.896375 + 0.443296i $$0.146191\pi$$
$$84$$ −3.65279 −0.398552
$$85$$ 3.18598 0.345568
$$86$$ 3.97285 0.428404
$$87$$ −6.33513 −0.679197
$$88$$ −2.45473 −0.261675
$$89$$ 14.7289 1.56126 0.780628 0.624996i $$-0.214899\pi$$
0.780628 + 0.624996i $$0.214899\pi$$
$$90$$ −3.66487 −0.386312
$$91$$ 0 0
$$92$$ 7.78986 0.812149
$$93$$ −4.73125 −0.490608
$$94$$ 4.26875 0.440288
$$95$$ −5.43296 −0.557410
$$96$$ 4.28621 0.437459
$$97$$ −3.13169 −0.317975 −0.158987 0.987281i $$-0.550823\pi$$
−0.158987 + 0.987281i $$0.550823\pi$$
$$98$$ 0.137063 0.0138455
$$99$$ −2.82371 −0.283793
$$100$$ −4.82371 −0.482371
$$101$$ 5.29052 0.526426 0.263213 0.964738i $$-0.415218\pi$$
0.263213 + 0.964738i $$0.415218\pi$$
$$102$$ 0.506041 0.0501055
$$103$$ −13.5308 −1.33323 −0.666614 0.745403i $$-0.732257\pi$$
−0.666614 + 0.745403i $$0.732257\pi$$
$$104$$ 0 0
$$105$$ 6.04892 0.590314
$$106$$ 3.25906 0.316548
$$107$$ 5.63102 0.544371 0.272186 0.962245i $$-0.412253\pi$$
0.272186 + 0.962245i $$0.412253\pi$$
$$108$$ 7.26875 0.699436
$$109$$ −4.17629 −0.400016 −0.200008 0.979794i $$-0.564097\pi$$
−0.200008 + 0.979794i $$0.564097\pi$$
$$110$$ 1.86294 0.177624
$$111$$ −0.762709 −0.0723931
$$112$$ 6.04892 0.571569
$$113$$ 7.64310 0.719003 0.359501 0.933145i $$-0.382947\pi$$
0.359501 + 0.933145i $$0.382947\pi$$
$$114$$ −0.862937 −0.0808214
$$115$$ −12.8998 −1.20291
$$116$$ 13.3666 1.24106
$$117$$ 0 0
$$118$$ 0.00670477 0.000617224 0
$$119$$ 3.06100 0.280601
$$120$$ −4.60388 −0.420274
$$121$$ −9.56465 −0.869513
$$122$$ −4.46011 −0.403799
$$123$$ −2.66056 −0.239895
$$124$$ 9.98254 0.896459
$$125$$ −6.02177 −0.538604
$$126$$ −3.52111 −0.313685
$$127$$ 6.77777 0.601430 0.300715 0.953714i $$-0.402775\pi$$
0.300715 + 0.953714i $$0.402775\pi$$
$$128$$ −11.5375 −1.01978
$$129$$ 5.74094 0.505461
$$130$$ 0 0
$$131$$ −13.6799 −1.19522 −0.597611 0.801786i $$-0.703883\pi$$
−0.597611 + 0.801786i $$0.703883\pi$$
$$132$$ −1.62565 −0.141494
$$133$$ −5.21983 −0.452617
$$134$$ 5.13706 0.443775
$$135$$ −12.0368 −1.03597
$$136$$ −2.32975 −0.199774
$$137$$ −12.9879 −1.10963 −0.554816 0.831973i $$-0.687211\pi$$
−0.554816 + 0.831973i $$0.687211\pi$$
$$138$$ −2.04892 −0.174415
$$139$$ 12.0465 1.02177 0.510886 0.859648i $$-0.329317\pi$$
0.510886 + 0.859648i $$0.329317\pi$$
$$140$$ −12.7627 −1.07865
$$141$$ 6.16852 0.519483
$$142$$ 7.62565 0.639930
$$143$$ 0 0
$$144$$ −5.29590 −0.441325
$$145$$ −22.1347 −1.83818
$$146$$ −7.11290 −0.588668
$$147$$ 0.198062 0.0163359
$$148$$ 1.60925 0.132280
$$149$$ 0.740939 0.0607001 0.0303500 0.999539i $$-0.490338\pi$$
0.0303500 + 0.999539i $$0.490338\pi$$
$$150$$ 1.26875 0.103593
$$151$$ 19.0737 1.55219 0.776097 0.630614i $$-0.217197\pi$$
0.776097 + 0.630614i $$0.217197\pi$$
$$152$$ 3.97285 0.322241
$$153$$ −2.67994 −0.216660
$$154$$ 1.78986 0.144231
$$155$$ −16.5308 −1.32779
$$156$$ 0 0
$$157$$ −4.02177 −0.320972 −0.160486 0.987038i $$-0.551306\pi$$
−0.160486 + 0.987038i $$0.551306\pi$$
$$158$$ 0.448026 0.0356430
$$159$$ 4.70948 0.373486
$$160$$ 14.9758 1.18394
$$161$$ −12.3937 −0.976763
$$162$$ 2.01208 0.158084
$$163$$ −15.1371 −1.18563 −0.592813 0.805340i $$-0.701983\pi$$
−0.592813 + 0.805340i $$0.701983\pi$$
$$164$$ 5.61356 0.438346
$$165$$ 2.69202 0.209574
$$166$$ 9.06398 0.703502
$$167$$ 6.26337 0.484674 0.242337 0.970192i $$-0.422086\pi$$
0.242337 + 0.970192i $$0.422086\pi$$
$$168$$ −4.42327 −0.341263
$$169$$ 0 0
$$170$$ 1.76809 0.135606
$$171$$ 4.57002 0.349478
$$172$$ −12.1129 −0.923600
$$173$$ 16.3913 1.24621 0.623105 0.782138i $$-0.285871\pi$$
0.623105 + 0.782138i $$0.285871\pi$$
$$174$$ −3.51573 −0.266527
$$175$$ 7.67456 0.580142
$$176$$ 2.69202 0.202919
$$177$$ 0.00968868 0.000728246 0
$$178$$ 8.17390 0.612660
$$179$$ −2.45473 −0.183475 −0.0917376 0.995783i $$-0.529242\pi$$
−0.0917376 + 0.995783i $$0.529242\pi$$
$$180$$ 11.1739 0.832853
$$181$$ 11.8073 0.877631 0.438815 0.898577i $$-0.355398\pi$$
0.438815 + 0.898577i $$0.355398\pi$$
$$182$$ 0 0
$$183$$ −6.44504 −0.476431
$$184$$ 9.43296 0.695407
$$185$$ −2.66487 −0.195925
$$186$$ −2.62565 −0.192522
$$187$$ 1.36227 0.0996192
$$188$$ −13.0151 −0.949221
$$189$$ −11.5646 −0.841204
$$190$$ −3.01507 −0.218736
$$191$$ −8.99330 −0.650732 −0.325366 0.945588i $$-0.605488\pi$$
−0.325366 + 0.945588i $$0.605488\pi$$
$$192$$ −1.22521 −0.0884219
$$193$$ −13.5254 −0.973581 −0.486790 0.873519i $$-0.661832\pi$$
−0.486790 + 0.873519i $$0.661832\pi$$
$$194$$ −1.73795 −0.124778
$$195$$ 0 0
$$196$$ −0.417895 −0.0298496
$$197$$ −12.9758 −0.924490 −0.462245 0.886752i $$-0.652956\pi$$
−0.462245 + 0.886752i $$0.652956\pi$$
$$198$$ −1.56704 −0.111365
$$199$$ −13.5864 −0.963116 −0.481558 0.876414i $$-0.659929\pi$$
−0.481558 + 0.876414i $$0.659929\pi$$
$$200$$ −5.84117 −0.413033
$$201$$ 7.42327 0.523597
$$202$$ 2.93602 0.206577
$$203$$ −21.2664 −1.49261
$$204$$ −1.54288 −0.108023
$$205$$ −9.29590 −0.649254
$$206$$ −7.50902 −0.523179
$$207$$ 10.8509 0.754187
$$208$$ 0 0
$$209$$ −2.32304 −0.160688
$$210$$ 3.35690 0.231648
$$211$$ 10.4601 0.720103 0.360052 0.932932i $$-0.382759\pi$$
0.360052 + 0.932932i $$0.382759\pi$$
$$212$$ −9.93661 −0.682449
$$213$$ 11.0194 0.755035
$$214$$ 3.12498 0.213619
$$215$$ 20.0586 1.36799
$$216$$ 8.80194 0.598896
$$217$$ −15.8823 −1.07816
$$218$$ −2.31767 −0.156972
$$219$$ −10.2784 −0.694553
$$220$$ −5.67994 −0.382941
$$221$$ 0 0
$$222$$ −0.423272 −0.0284081
$$223$$ 11.4058 0.763790 0.381895 0.924206i $$-0.375272\pi$$
0.381895 + 0.924206i $$0.375272\pi$$
$$224$$ 14.3884 0.961362
$$225$$ −6.71917 −0.447945
$$226$$ 4.24160 0.282147
$$227$$ −10.6407 −0.706249 −0.353124 0.935576i $$-0.614881\pi$$
−0.353124 + 0.935576i $$0.614881\pi$$
$$228$$ 2.63102 0.174244
$$229$$ 1.13946 0.0752974 0.0376487 0.999291i $$-0.488013\pi$$
0.0376487 + 0.999291i $$0.488013\pi$$
$$230$$ −7.15883 −0.472040
$$231$$ 2.58642 0.170174
$$232$$ 16.1860 1.06266
$$233$$ −10.8509 −0.710863 −0.355432 0.934702i $$-0.615666\pi$$
−0.355432 + 0.934702i $$0.615666\pi$$
$$234$$ 0 0
$$235$$ 21.5526 1.40593
$$236$$ −0.0204423 −0.00133068
$$237$$ 0.647416 0.0420542
$$238$$ 1.69873 0.110112
$$239$$ 11.9293 0.771643 0.385822 0.922573i $$-0.373918\pi$$
0.385822 + 0.922573i $$0.373918\pi$$
$$240$$ 5.04892 0.325906
$$241$$ 3.64848 0.235019 0.117510 0.993072i $$-0.462509\pi$$
0.117510 + 0.993072i $$0.462509\pi$$
$$242$$ −5.30798 −0.341210
$$243$$ 15.7952 1.01326
$$244$$ 13.5985 0.870555
$$245$$ 0.692021 0.0442116
$$246$$ −1.47650 −0.0941383
$$247$$ 0 0
$$248$$ 12.0881 0.767598
$$249$$ 13.0978 0.830042
$$250$$ −3.34183 −0.211356
$$251$$ 1.37329 0.0866813 0.0433406 0.999060i $$-0.486200\pi$$
0.0433406 + 0.999060i $$0.486200\pi$$
$$252$$ 10.7356 0.676277
$$253$$ −5.51573 −0.346771
$$254$$ 3.76138 0.236010
$$255$$ 2.55496 0.159998
$$256$$ −3.34721 −0.209200
$$257$$ 29.4359 1.83616 0.918082 0.396391i $$-0.129737\pi$$
0.918082 + 0.396391i $$0.129737\pi$$
$$258$$ 3.18598 0.198350
$$259$$ −2.56033 −0.159091
$$260$$ 0 0
$$261$$ 18.6189 1.15248
$$262$$ −7.59179 −0.469023
$$263$$ 10.6963 0.659564 0.329782 0.944057i $$-0.393025\pi$$
0.329782 + 0.944057i $$0.393025\pi$$
$$264$$ −1.96854 −0.121155
$$265$$ 16.4547 1.01081
$$266$$ −2.89679 −0.177613
$$267$$ 11.8116 0.722860
$$268$$ −15.6625 −0.956738
$$269$$ −10.1860 −0.621050 −0.310525 0.950565i $$-0.600505\pi$$
−0.310525 + 0.950565i $$0.600505\pi$$
$$270$$ −6.67994 −0.406528
$$271$$ 29.4523 1.78910 0.894551 0.446966i $$-0.147495\pi$$
0.894551 + 0.446966i $$0.147495\pi$$
$$272$$ 2.55496 0.154917
$$273$$ 0 0
$$274$$ −7.20775 −0.435436
$$275$$ 3.41550 0.205963
$$276$$ 6.24698 0.376024
$$277$$ −10.2446 −0.615538 −0.307769 0.951461i $$-0.599582\pi$$
−0.307769 + 0.951461i $$0.599582\pi$$
$$278$$ 6.68532 0.400959
$$279$$ 13.9051 0.832480
$$280$$ −15.4547 −0.923597
$$281$$ 11.5646 0.689889 0.344944 0.938623i $$-0.387898\pi$$
0.344944 + 0.938623i $$0.387898\pi$$
$$282$$ 3.42327 0.203853
$$283$$ −30.7090 −1.82546 −0.912730 0.408562i $$-0.866030\pi$$
−0.912730 + 0.408562i $$0.866030\pi$$
$$284$$ −23.2500 −1.37963
$$285$$ −4.35690 −0.258080
$$286$$ 0 0
$$287$$ −8.93123 −0.527194
$$288$$ −12.5972 −0.742295
$$289$$ −15.7071 −0.923946
$$290$$ −12.2838 −0.721330
$$291$$ −2.51142 −0.147222
$$292$$ 21.6866 1.26911
$$293$$ 18.6082 1.08710 0.543551 0.839376i $$-0.317079\pi$$
0.543551 + 0.839376i $$0.317079\pi$$
$$294$$ 0.109916 0.00641045
$$295$$ 0.0338518 0.00197093
$$296$$ 1.94869 0.113265
$$297$$ −5.14675 −0.298645
$$298$$ 0.411190 0.0238196
$$299$$ 0 0
$$300$$ −3.86831 −0.223337
$$301$$ 19.2717 1.11080
$$302$$ 10.5851 0.609103
$$303$$ 4.24267 0.243735
$$304$$ −4.35690 −0.249885
$$305$$ −22.5187 −1.28942
$$306$$ −1.48725 −0.0850207
$$307$$ −8.94438 −0.510483 −0.255241 0.966877i $$-0.582155\pi$$
−0.255241 + 0.966877i $$0.582155\pi$$
$$308$$ −5.45712 −0.310948
$$309$$ −10.8509 −0.617284
$$310$$ −9.17390 −0.521042
$$311$$ 21.0398 1.19306 0.596529 0.802591i $$-0.296546\pi$$
0.596529 + 0.802591i $$0.296546\pi$$
$$312$$ 0 0
$$313$$ −7.12737 −0.402863 −0.201432 0.979503i $$-0.564559\pi$$
−0.201432 + 0.979503i $$0.564559\pi$$
$$314$$ −2.23191 −0.125954
$$315$$ −17.7778 −1.00166
$$316$$ −1.36599 −0.0768431
$$317$$ −23.9651 −1.34601 −0.673007 0.739636i $$-0.734997\pi$$
−0.673007 + 0.739636i $$0.734997\pi$$
$$318$$ 2.61356 0.146561
$$319$$ −9.46442 −0.529906
$$320$$ −4.28083 −0.239306
$$321$$ 4.51573 0.252043
$$322$$ −6.87800 −0.383296
$$323$$ −2.20477 −0.122677
$$324$$ −6.13467 −0.340815
$$325$$ 0 0
$$326$$ −8.40044 −0.465257
$$327$$ −3.34913 −0.185207
$$328$$ 6.79763 0.375336
$$329$$ 20.7071 1.14162
$$330$$ 1.49396 0.0822397
$$331$$ −2.89546 −0.159149 −0.0795745 0.996829i $$-0.525356\pi$$
−0.0795745 + 0.996829i $$0.525356\pi$$
$$332$$ −27.6353 −1.51669
$$333$$ 2.24160 0.122839
$$334$$ 3.47591 0.190193
$$335$$ 25.9366 1.41707
$$336$$ 4.85086 0.264636
$$337$$ −3.10560 −0.169173 −0.0845865 0.996416i $$-0.526957\pi$$
−0.0845865 + 0.996416i $$0.526957\pi$$
$$338$$ 0 0
$$339$$ 6.12929 0.332898
$$340$$ −5.39075 −0.292354
$$341$$ −7.06829 −0.382770
$$342$$ 2.53617 0.137140
$$343$$ −18.1793 −0.981589
$$344$$ −14.6679 −0.790838
$$345$$ −10.3448 −0.556946
$$346$$ 9.09651 0.489031
$$347$$ −11.3787 −0.610839 −0.305419 0.952218i $$-0.598797\pi$$
−0.305419 + 0.952218i $$0.598797\pi$$
$$348$$ 10.7192 0.574608
$$349$$ 3.34721 0.179172 0.0895859 0.995979i $$-0.471446\pi$$
0.0895859 + 0.995979i $$0.471446\pi$$
$$350$$ 4.25906 0.227656
$$351$$ 0 0
$$352$$ 6.40342 0.341303
$$353$$ −0.637727 −0.0339428 −0.0169714 0.999856i $$-0.505402\pi$$
−0.0169714 + 0.999856i $$0.505402\pi$$
$$354$$ 0.00537681 0.000285774 0
$$355$$ 38.5013 2.04343
$$356$$ −24.9215 −1.32084
$$357$$ 2.45473 0.129918
$$358$$ −1.36227 −0.0719983
$$359$$ 21.4590 1.13256 0.566282 0.824211i $$-0.308381\pi$$
0.566282 + 0.824211i $$0.308381\pi$$
$$360$$ 13.5308 0.713136
$$361$$ −15.2403 −0.802120
$$362$$ 6.55257 0.344395
$$363$$ −7.67025 −0.402584
$$364$$ 0 0
$$365$$ −35.9124 −1.87974
$$366$$ −3.57673 −0.186959
$$367$$ −9.38703 −0.489999 −0.244999 0.969523i $$-0.578788\pi$$
−0.244999 + 0.969523i $$0.578788\pi$$
$$368$$ −10.3448 −0.539261
$$369$$ 7.81940 0.407062
$$370$$ −1.47889 −0.0768840
$$371$$ 15.8092 0.820775
$$372$$ 8.00538 0.415059
$$373$$ 27.7265 1.43562 0.717811 0.696238i $$-0.245144\pi$$
0.717811 + 0.696238i $$0.245144\pi$$
$$374$$ 0.756004 0.0390921
$$375$$ −4.82908 −0.249373
$$376$$ −15.7603 −0.812776
$$377$$ 0 0
$$378$$ −6.41789 −0.330101
$$379$$ −35.8702 −1.84253 −0.921265 0.388935i $$-0.872843\pi$$
−0.921265 + 0.388935i $$0.872843\pi$$
$$380$$ 9.19269 0.471575
$$381$$ 5.43535 0.278462
$$382$$ −4.99090 −0.255357
$$383$$ −4.85517 −0.248087 −0.124044 0.992277i $$-0.539586\pi$$
−0.124044 + 0.992277i $$0.539586\pi$$
$$384$$ −9.25236 −0.472157
$$385$$ 9.03684 0.460560
$$386$$ −7.50604 −0.382047
$$387$$ −16.8726 −0.857684
$$388$$ 5.29888 0.269010
$$389$$ 2.38537 0.120943 0.0604716 0.998170i $$-0.480740\pi$$
0.0604716 + 0.998170i $$0.480740\pi$$
$$390$$ 0 0
$$391$$ −5.23490 −0.264740
$$392$$ −0.506041 −0.0255589
$$393$$ −10.9705 −0.553387
$$394$$ −7.20105 −0.362783
$$395$$ 2.26205 0.113816
$$396$$ 4.77777 0.240092
$$397$$ 15.2664 0.766196 0.383098 0.923708i $$-0.374857\pi$$
0.383098 + 0.923708i $$0.374857\pi$$
$$398$$ −7.53989 −0.377941
$$399$$ −4.18598 −0.209561
$$400$$ 6.40581 0.320291
$$401$$ −12.7584 −0.637124 −0.318562 0.947902i $$-0.603200\pi$$
−0.318562 + 0.947902i $$0.603200\pi$$
$$402$$ 4.11960 0.205467
$$403$$ 0 0
$$404$$ −8.95167 −0.445362
$$405$$ 10.1588 0.504797
$$406$$ −11.8019 −0.585720
$$407$$ −1.13946 −0.0564807
$$408$$ −1.86831 −0.0924953
$$409$$ 25.3588 1.25391 0.626956 0.779054i $$-0.284300\pi$$
0.626956 + 0.779054i $$0.284300\pi$$
$$410$$ −5.15883 −0.254777
$$411$$ −10.4155 −0.513759
$$412$$ 22.8944 1.12793
$$413$$ 0.0325239 0.00160040
$$414$$ 6.02177 0.295954
$$415$$ 45.7633 2.24643
$$416$$ 0 0
$$417$$ 9.66056 0.473080
$$418$$ −1.28919 −0.0630565
$$419$$ −11.6673 −0.569983 −0.284992 0.958530i $$-0.591991\pi$$
−0.284992 + 0.958530i $$0.591991\pi$$
$$420$$ −10.2349 −0.499412
$$421$$ 8.29291 0.404172 0.202086 0.979368i $$-0.435228\pi$$
0.202086 + 0.979368i $$0.435228\pi$$
$$422$$ 5.80492 0.282579
$$423$$ −18.1293 −0.881476
$$424$$ −12.0325 −0.584351
$$425$$ 3.24160 0.157241
$$426$$ 6.11529 0.296287
$$427$$ −21.6353 −1.04701
$$428$$ −9.52781 −0.460544
$$429$$ 0 0
$$430$$ 11.1317 0.536818
$$431$$ −0.932296 −0.0449071 −0.0224536 0.999748i $$-0.507148\pi$$
−0.0224536 + 0.999748i $$0.507148\pi$$
$$432$$ −9.65279 −0.464420
$$433$$ −13.3502 −0.641569 −0.320785 0.947152i $$-0.603947\pi$$
−0.320785 + 0.947152i $$0.603947\pi$$
$$434$$ −8.81402 −0.423086
$$435$$ −17.7506 −0.851077
$$436$$ 7.06638 0.338418
$$437$$ 8.92692 0.427032
$$438$$ −5.70410 −0.272553
$$439$$ 13.9922 0.667813 0.333906 0.942606i $$-0.391633\pi$$
0.333906 + 0.942606i $$0.391633\pi$$
$$440$$ −6.87800 −0.327896
$$441$$ −0.582105 −0.0277193
$$442$$ 0 0
$$443$$ 23.7017 1.12610 0.563051 0.826422i $$-0.309627\pi$$
0.563051 + 0.826422i $$0.309627\pi$$
$$444$$ 1.29052 0.0612454
$$445$$ 41.2693 1.95635
$$446$$ 6.32975 0.299722
$$447$$ 0.594187 0.0281041
$$448$$ −4.11290 −0.194316
$$449$$ 12.5864 0.593990 0.296995 0.954879i $$-0.404016\pi$$
0.296995 + 0.954879i $$0.404016\pi$$
$$450$$ −3.72886 −0.175780
$$451$$ −3.97477 −0.187165
$$452$$ −12.9323 −0.608284
$$453$$ 15.2959 0.718664
$$454$$ −5.90515 −0.277142
$$455$$ 0 0
$$456$$ 3.18598 0.149197
$$457$$ 33.6383 1.57353 0.786767 0.617250i $$-0.211753\pi$$
0.786767 + 0.617250i $$0.211753\pi$$
$$458$$ 0.632351 0.0295478
$$459$$ −4.88471 −0.227999
$$460$$ 21.8267 1.01767
$$461$$ 1.40283 0.0653363 0.0326681 0.999466i $$-0.489600\pi$$
0.0326681 + 0.999466i $$0.489600\pi$$
$$462$$ 1.43535 0.0667787
$$463$$ 15.2010 0.706453 0.353226 0.935538i $$-0.385085\pi$$
0.353226 + 0.935538i $$0.385085\pi$$
$$464$$ −17.7506 −0.824052
$$465$$ −13.2567 −0.614763
$$466$$ −6.02177 −0.278953
$$467$$ −39.3414 −1.82050 −0.910250 0.414058i $$-0.864111\pi$$
−0.910250 + 0.414058i $$0.864111\pi$$
$$468$$ 0 0
$$469$$ 24.9191 1.15066
$$470$$ 11.9608 0.551709
$$471$$ −3.22521 −0.148610
$$472$$ −0.0247542 −0.00113940
$$473$$ 8.57673 0.394358
$$474$$ 0.359289 0.0165027
$$475$$ −5.52781 −0.253633
$$476$$ −5.17928 −0.237392
$$477$$ −13.8412 −0.633743
$$478$$ 6.62027 0.302804
$$479$$ 22.3690 1.02206 0.511032 0.859561i $$-0.329263\pi$$
0.511032 + 0.859561i $$0.329263\pi$$
$$480$$ 12.0097 0.548165
$$481$$ 0 0
$$482$$ 2.02475 0.0922250
$$483$$ −9.93900 −0.452240
$$484$$ 16.1836 0.735618
$$485$$ −8.77479 −0.398443
$$486$$ 8.76569 0.397620
$$487$$ −22.9205 −1.03863 −0.519313 0.854584i $$-0.673812\pi$$
−0.519313 + 0.854584i $$0.673812\pi$$
$$488$$ 16.4668 0.745418
$$489$$ −12.1390 −0.548944
$$490$$ 0.384043 0.0173493
$$491$$ 1.84356 0.0831987 0.0415993 0.999134i $$-0.486755\pi$$
0.0415993 + 0.999134i $$0.486755\pi$$
$$492$$ 4.50173 0.202954
$$493$$ −8.98254 −0.404553
$$494$$ 0 0
$$495$$ −7.91185 −0.355611
$$496$$ −13.2567 −0.595242
$$497$$ 36.9909 1.65927
$$498$$ 7.26875 0.325720
$$499$$ 12.0344 0.538736 0.269368 0.963037i $$-0.413185\pi$$
0.269368 + 0.963037i $$0.413185\pi$$
$$500$$ 10.1890 0.455664
$$501$$ 5.02284 0.224404
$$502$$ 0.762118 0.0340150
$$503$$ −30.5056 −1.36018 −0.680088 0.733130i $$-0.738058\pi$$
−0.680088 + 0.733130i $$0.738058\pi$$
$$504$$ 13.0000 0.579066
$$505$$ 14.8237 0.659646
$$506$$ −3.06100 −0.136078
$$507$$ 0 0
$$508$$ −11.4681 −0.508816
$$509$$ 1.51142 0.0669924 0.0334962 0.999439i $$-0.489336\pi$$
0.0334962 + 0.999439i $$0.489336\pi$$
$$510$$ 1.41789 0.0627854
$$511$$ −34.5036 −1.52635
$$512$$ 21.2174 0.937687
$$513$$ 8.32975 0.367767
$$514$$ 16.3357 0.720538
$$515$$ −37.9124 −1.67062
$$516$$ −9.71379 −0.427626
$$517$$ 9.21552 0.405298
$$518$$ −1.42088 −0.0624298
$$519$$ 13.1448 0.576994
$$520$$ 0 0
$$521$$ −5.64012 −0.247098 −0.123549 0.992338i $$-0.539428\pi$$
−0.123549 + 0.992338i $$0.539428\pi$$
$$522$$ 10.3327 0.452251
$$523$$ −31.7506 −1.38836 −0.694179 0.719802i $$-0.744232\pi$$
−0.694179 + 0.719802i $$0.744232\pi$$
$$524$$ 23.1468 1.01117
$$525$$ 6.15452 0.268605
$$526$$ 5.93602 0.258823
$$527$$ −6.70841 −0.292223
$$528$$ 2.15883 0.0939512
$$529$$ −1.80433 −0.0784492
$$530$$ 9.13169 0.396655
$$531$$ −0.0284750 −0.00123571
$$532$$ 8.83207 0.382919
$$533$$ 0 0
$$534$$ 6.55496 0.283661
$$535$$ 15.7778 0.682133
$$536$$ −18.9661 −0.819213
$$537$$ −1.96854 −0.0849488
$$538$$ −5.65279 −0.243709
$$539$$ 0.295897 0.0127452
$$540$$ 20.3666 0.876438
$$541$$ −24.3297 −1.04602 −0.523009 0.852327i $$-0.675191\pi$$
−0.523009 + 0.852327i $$0.675191\pi$$
$$542$$ 16.3448 0.702070
$$543$$ 9.46873 0.406342
$$544$$ 6.07739 0.260566
$$545$$ −11.7017 −0.501246
$$546$$ 0 0
$$547$$ −8.18896 −0.350135 −0.175067 0.984556i $$-0.556014\pi$$
−0.175067 + 0.984556i $$0.556014\pi$$
$$548$$ 21.9758 0.938761
$$549$$ 18.9420 0.808424
$$550$$ 1.89546 0.0808227
$$551$$ 15.3177 0.652555
$$552$$ 7.56465 0.321973
$$553$$ 2.17331 0.0924185
$$554$$ −5.68532 −0.241546
$$555$$ −2.13706 −0.0907133
$$556$$ −20.3830 −0.864431
$$557$$ −25.3327 −1.07338 −0.536691 0.843779i $$-0.680326\pi$$
−0.536691 + 0.843779i $$0.680326\pi$$
$$558$$ 7.71678 0.326677
$$559$$ 0 0
$$560$$ 16.9487 0.716213
$$561$$ 1.09246 0.0461236
$$562$$ 6.41789 0.270723
$$563$$ −25.3937 −1.07022 −0.535109 0.844783i $$-0.679730\pi$$
−0.535109 + 0.844783i $$0.679730\pi$$
$$564$$ −10.4373 −0.439488
$$565$$ 21.4155 0.900957
$$566$$ −17.0422 −0.716338
$$567$$ 9.76032 0.409895
$$568$$ −28.1540 −1.18132
$$569$$ −31.1347 −1.30523 −0.652617 0.757688i $$-0.726329\pi$$
−0.652617 + 0.757688i $$0.726329\pi$$
$$570$$ −2.41789 −0.101274
$$571$$ −20.5090 −0.858276 −0.429138 0.903239i $$-0.641183\pi$$
−0.429138 + 0.903239i $$0.641183\pi$$
$$572$$ 0 0
$$573$$ −7.21206 −0.301288
$$574$$ −4.95646 −0.206879
$$575$$ −13.1250 −0.547350
$$576$$ 3.60089 0.150037
$$577$$ −15.6890 −0.653143 −0.326572 0.945172i $$-0.605893\pi$$
−0.326572 + 0.945172i $$0.605893\pi$$
$$578$$ −8.71678 −0.362570
$$579$$ −10.8465 −0.450767
$$580$$ 37.4523 1.55512
$$581$$ 43.9681 1.82410
$$582$$ −1.39373 −0.0577720
$$583$$ 7.03577 0.291392
$$584$$ 26.2610 1.08669
$$585$$ 0 0
$$586$$ 10.3268 0.426595
$$587$$ −30.5687 −1.26171 −0.630853 0.775903i $$-0.717295\pi$$
−0.630853 + 0.775903i $$0.717295\pi$$
$$588$$ −0.335126 −0.0138203
$$589$$ 11.4397 0.471363
$$590$$ 0.0187864 0.000773422 0
$$591$$ −10.4058 −0.428038
$$592$$ −2.13706 −0.0878328
$$593$$ 29.6883 1.21915 0.609576 0.792727i $$-0.291340\pi$$
0.609576 + 0.792727i $$0.291340\pi$$
$$594$$ −2.85623 −0.117193
$$595$$ 8.57673 0.351612
$$596$$ −1.25368 −0.0513529
$$597$$ −10.8955 −0.445922
$$598$$ 0 0
$$599$$ 24.2325 0.990113 0.495057 0.868861i $$-0.335147\pi$$
0.495057 + 0.868861i $$0.335147\pi$$
$$600$$ −4.68425 −0.191234
$$601$$ 16.4819 0.672310 0.336155 0.941807i $$-0.390873\pi$$
0.336155 + 0.941807i $$0.390873\pi$$
$$602$$ 10.6950 0.435896
$$603$$ −21.8170 −0.888457
$$604$$ −32.2731 −1.31317
$$605$$ −26.7995 −1.08956
$$606$$ 2.35450 0.0956451
$$607$$ 1.43190 0.0581188 0.0290594 0.999578i $$-0.490749\pi$$
0.0290594 + 0.999578i $$0.490749\pi$$
$$608$$ −10.3636 −0.420300
$$609$$ −17.0543 −0.691075
$$610$$ −12.4969 −0.505986
$$611$$ 0 0
$$612$$ 4.53452 0.183297
$$613$$ −3.84846 −0.155438 −0.0777190 0.996975i $$-0.524764\pi$$
−0.0777190 + 0.996975i $$0.524764\pi$$
$$614$$ −4.96376 −0.200321
$$615$$ −7.45473 −0.300604
$$616$$ −6.60819 −0.266251
$$617$$ 15.0388 0.605437 0.302719 0.953080i $$-0.402106\pi$$
0.302719 + 0.953080i $$0.402106\pi$$
$$618$$ −6.02177 −0.242231
$$619$$ −12.8170 −0.515159 −0.257579 0.966257i $$-0.582925\pi$$
−0.257579 + 0.966257i $$0.582925\pi$$
$$620$$ 27.9705 1.12332
$$621$$ 19.7778 0.793655
$$622$$ 11.6762 0.468174
$$623$$ 39.6504 1.58856
$$624$$ 0 0
$$625$$ −31.1269 −1.24508
$$626$$ −3.95539 −0.158089
$$627$$ −1.86294 −0.0743985
$$628$$ 6.80492 0.271546
$$629$$ −1.08144 −0.0431199
$$630$$ −9.86592 −0.393068
$$631$$ −25.7517 −1.02516 −0.512579 0.858640i $$-0.671310\pi$$
−0.512579 + 0.858640i $$0.671310\pi$$
$$632$$ −1.65412 −0.0657974
$$633$$ 8.38835 0.333407
$$634$$ −13.2996 −0.528195
$$635$$ 18.9909 0.753631
$$636$$ −7.96854 −0.315973
$$637$$ 0 0
$$638$$ −5.25236 −0.207943
$$639$$ −32.3860 −1.28117
$$640$$ −32.3274 −1.27785
$$641$$ 24.4571 0.965998 0.482999 0.875621i $$-0.339547\pi$$
0.482999 + 0.875621i $$0.339547\pi$$
$$642$$ 2.50604 0.0989055
$$643$$ 9.97344 0.393314 0.196657 0.980472i $$-0.436991\pi$$
0.196657 + 0.980472i $$0.436991\pi$$
$$644$$ 20.9705 0.826352
$$645$$ 16.0858 0.633376
$$646$$ −1.22355 −0.0481401
$$647$$ 11.8431 0.465600 0.232800 0.972525i $$-0.425211\pi$$
0.232800 + 0.972525i $$0.425211\pi$$
$$648$$ −7.42865 −0.291825
$$649$$ 0.0144745 0.000568173 0
$$650$$ 0 0
$$651$$ −12.7366 −0.499188
$$652$$ 25.6122 1.00305
$$653$$ −7.47411 −0.292484 −0.146242 0.989249i $$-0.546718\pi$$
−0.146242 + 0.989249i $$0.546718\pi$$
$$654$$ −1.85862 −0.0726780
$$655$$ −38.3303 −1.49769
$$656$$ −7.45473 −0.291058
$$657$$ 30.2083 1.17854
$$658$$ 11.4916 0.447988
$$659$$ 34.1739 1.33123 0.665613 0.746297i $$-0.268170\pi$$
0.665613 + 0.746297i $$0.268170\pi$$
$$660$$ −4.55496 −0.177302
$$661$$ −33.6088 −1.30723 −0.653615 0.756827i $$-0.726748\pi$$
−0.653615 + 0.756827i $$0.726748\pi$$
$$662$$ −1.60686 −0.0624524
$$663$$ 0 0
$$664$$ −33.4644 −1.29867
$$665$$ −14.6256 −0.567158
$$666$$ 1.24400 0.0482039
$$667$$ 36.3696 1.40824
$$668$$ −10.5978 −0.410040
$$669$$ 9.14675 0.353634
$$670$$ 14.3937 0.556078
$$671$$ −9.62863 −0.371709
$$672$$ 11.5386 0.445110
$$673$$ 48.0320 1.85150 0.925750 0.378137i $$-0.123435\pi$$
0.925750 + 0.378137i $$0.123435\pi$$
$$674$$ −1.72348 −0.0663860
$$675$$ −12.2470 −0.471386
$$676$$ 0 0
$$677$$ −33.6582 −1.29359 −0.646794 0.762665i $$-0.723891\pi$$
−0.646794 + 0.762665i $$0.723891\pi$$
$$678$$ 3.40150 0.130634
$$679$$ −8.43057 −0.323535
$$680$$ −6.52781 −0.250330
$$681$$ −8.53319 −0.326992
$$682$$ −3.92261 −0.150204
$$683$$ −15.9041 −0.608553 −0.304276 0.952584i $$-0.598415\pi$$
−0.304276 + 0.952584i $$0.598415\pi$$
$$684$$ −7.73258 −0.295663
$$685$$ −36.3913 −1.39044
$$686$$ −10.0887 −0.385190
$$687$$ 0.913773 0.0348626
$$688$$ 16.0858 0.613264
$$689$$ 0 0
$$690$$ −5.74094 −0.218554
$$691$$ −33.1903 −1.26262 −0.631309 0.775531i $$-0.717482\pi$$
−0.631309 + 0.775531i $$0.717482\pi$$
$$692$$ −27.7345 −1.05431
$$693$$ −7.60148 −0.288756
$$694$$ −6.31468 −0.239702
$$695$$ 33.7536 1.28035
$$696$$ 12.9801 0.492011
$$697$$ −3.77240 −0.142890
$$698$$ 1.85756 0.0703097
$$699$$ −8.70171 −0.329129
$$700$$ −12.9855 −0.490807
$$701$$ −14.9129 −0.563253 −0.281627 0.959524i $$-0.590874\pi$$
−0.281627 + 0.959524i $$0.590874\pi$$
$$702$$ 0 0
$$703$$ 1.84415 0.0695534
$$704$$ −1.83041 −0.0689863
$$705$$ 17.2838 0.650946
$$706$$ −0.353912 −0.0133197
$$707$$ 14.2422 0.535633
$$708$$ −0.0163935 −0.000616104 0
$$709$$ −38.4312 −1.44331 −0.721656 0.692252i $$-0.756619\pi$$
−0.721656 + 0.692252i $$0.756619\pi$$
$$710$$ 21.3666 0.801874
$$711$$ −1.90276 −0.0713589
$$712$$ −30.1782 −1.13098
$$713$$ 27.1618 1.01722
$$714$$ 1.36227 0.0509818
$$715$$ 0 0
$$716$$ 4.15346 0.155222
$$717$$ 9.56657 0.357270
$$718$$ 11.9089 0.444435
$$719$$ −11.4373 −0.426538 −0.213269 0.976993i $$-0.568411\pi$$
−0.213269 + 0.976993i $$0.568411\pi$$
$$720$$ −14.8388 −0.553008
$$721$$ −36.4252 −1.35654
$$722$$ −8.45771 −0.314764
$$723$$ 2.92585 0.108814
$$724$$ −19.9782 −0.742485
$$725$$ −22.5211 −0.836413
$$726$$ −4.25667 −0.157980
$$727$$ 3.63640 0.134867 0.0674333 0.997724i $$-0.478519\pi$$
0.0674333 + 0.997724i $$0.478519\pi$$
$$728$$ 0 0
$$729$$ 1.78986 0.0662910
$$730$$ −19.9299 −0.737639
$$731$$ 8.14005 0.301071
$$732$$ 10.9051 0.403066
$$733$$ 3.52217 0.130094 0.0650472 0.997882i $$-0.479280\pi$$
0.0650472 + 0.997882i $$0.479280\pi$$
$$734$$ −5.20941 −0.192283
$$735$$ 0.554958 0.0204699
$$736$$ −24.6069 −0.907021
$$737$$ 11.0901 0.408508
$$738$$ 4.33944 0.159737
$$739$$ −0.420288 −0.0154605 −0.00773027 0.999970i $$-0.502461\pi$$
−0.00773027 + 0.999970i $$0.502461\pi$$
$$740$$ 4.50902 0.165755
$$741$$ 0 0
$$742$$ 8.77346 0.322084
$$743$$ 25.3623 0.930452 0.465226 0.885192i $$-0.345973\pi$$
0.465226 + 0.885192i $$0.345973\pi$$
$$744$$ 9.69394 0.355397
$$745$$ 2.07606 0.0760611
$$746$$ 15.3870 0.563359
$$747$$ −38.4946 −1.40844
$$748$$ −2.30499 −0.0842790
$$749$$ 15.1588 0.553892
$$750$$ −2.67994 −0.0978576
$$751$$ 0.650874 0.0237507 0.0118754 0.999929i $$-0.496220\pi$$
0.0118754 + 0.999929i $$0.496220\pi$$
$$752$$ 17.2838 0.630276
$$753$$ 1.10129 0.0401333
$$754$$ 0 0
$$755$$ 53.4432 1.94500
$$756$$ 19.5676 0.711668
$$757$$ −16.7909 −0.610276 −0.305138 0.952308i $$-0.598703\pi$$
−0.305138 + 0.952308i $$0.598703\pi$$
$$758$$ −19.9065 −0.723036
$$759$$ −4.42327 −0.160555
$$760$$ 11.1317 0.403789
$$761$$ −30.9221 −1.12093 −0.560463 0.828179i $$-0.689377\pi$$
−0.560463 + 0.828179i $$0.689377\pi$$
$$762$$ 3.01639 0.109272
$$763$$ −11.2427 −0.407012
$$764$$ 15.2168 0.550526
$$765$$ −7.50902 −0.271489
$$766$$ −2.69441 −0.0973531
$$767$$ 0 0
$$768$$ −2.68425 −0.0968596
$$769$$ 43.7689 1.57835 0.789174 0.614169i $$-0.210509\pi$$
0.789174 + 0.614169i $$0.210509\pi$$
$$770$$ 5.01507 0.180730
$$771$$ 23.6058 0.850142
$$772$$ 22.8853 0.823660
$$773$$ 42.4209 1.52577 0.762886 0.646532i $$-0.223782\pi$$
0.762886 + 0.646532i $$0.223782\pi$$
$$774$$ −9.36360 −0.336568
$$775$$ −16.8194 −0.604171
$$776$$ 6.41657 0.230341
$$777$$ −2.05323 −0.0736592
$$778$$ 1.32378 0.0474598
$$779$$ 6.43296 0.230485
$$780$$ 0 0
$$781$$ 16.4625 0.589075
$$782$$ −2.90515 −0.103888
$$783$$ 33.9366 1.21280
$$784$$ 0.554958 0.0198199
$$785$$ −11.2687 −0.402199
$$786$$ −6.08815 −0.217157
$$787$$ 36.0116 1.28368 0.641838 0.766841i $$-0.278172\pi$$
0.641838 + 0.766841i $$0.278172\pi$$
$$788$$ 21.9554 0.782129
$$789$$ 8.57779 0.305378
$$790$$ 1.25534 0.0446630
$$791$$ 20.5754 0.731577
$$792$$ 5.78554 0.205580
$$793$$ 0 0
$$794$$ 8.47219 0.300667
$$795$$ 13.1957 0.468002
$$796$$ 22.9885 0.814806
$$797$$ −31.7101 −1.12323 −0.561614 0.827399i $$-0.689819\pi$$
−0.561614 + 0.827399i $$0.689819\pi$$
$$798$$ −2.32304 −0.0822349
$$799$$ 8.74632 0.309422
$$800$$ 15.2373 0.538720
$$801$$ −34.7144 −1.22657
$$802$$ −7.08038 −0.250017
$$803$$ −15.3556 −0.541886
$$804$$ −12.5603 −0.442969
$$805$$ −34.7265 −1.22395
$$806$$ 0 0
$$807$$ −8.16852 −0.287546
$$808$$ −10.8398 −0.381344
$$809$$ −45.2814 −1.59201 −0.796005 0.605290i $$-0.793057\pi$$
−0.796005 + 0.605290i $$0.793057\pi$$
$$810$$ 5.63773 0.198090
$$811$$ 42.8635 1.50514 0.752571 0.658511i $$-0.228813\pi$$
0.752571 + 0.658511i $$0.228813\pi$$
$$812$$ 35.9831 1.26276
$$813$$ 23.6189 0.828352
$$814$$ −0.632351 −0.0221639
$$815$$ −42.4131 −1.48567
$$816$$ 2.04892 0.0717265
$$817$$ −13.8810 −0.485634
$$818$$ 14.0731 0.492054
$$819$$ 0 0
$$820$$ 15.7289 0.549276
$$821$$ 7.82776 0.273191 0.136595 0.990627i $$-0.456384\pi$$
0.136595 + 0.990627i $$0.456384\pi$$
$$822$$ −5.78017 −0.201606
$$823$$ −36.7754 −1.28191 −0.640955 0.767579i $$-0.721461\pi$$
−0.640955 + 0.767579i $$0.721461\pi$$
$$824$$ 27.7235 0.965793
$$825$$ 2.73902 0.0953604
$$826$$ 0.0180494 0.000628019 0
$$827$$ −47.3293 −1.64580 −0.822900 0.568186i $$-0.807645\pi$$
−0.822900 + 0.568186i $$0.807645\pi$$
$$828$$ −18.3599 −0.638050
$$829$$ 25.2687 0.877620 0.438810 0.898580i $$-0.355400\pi$$
0.438810 + 0.898580i $$0.355400\pi$$
$$830$$ 25.3967 0.881533
$$831$$ −8.21552 −0.284993
$$832$$ 0 0
$$833$$ 0.280831 0.00973023
$$834$$ 5.36121 0.185643
$$835$$ 17.5496 0.607328
$$836$$ 3.93064 0.135944
$$837$$ 25.3448 0.876045
$$838$$ −6.47484 −0.223670
$$839$$ 37.6883 1.30114 0.650572 0.759444i $$-0.274529\pi$$
0.650572 + 0.759444i $$0.274529\pi$$
$$840$$ −12.3937 −0.427624
$$841$$ 33.4064 1.15194
$$842$$ 4.60222 0.158603
$$843$$ 9.27413 0.319418
$$844$$ −17.6987 −0.609215
$$845$$ 0 0
$$846$$ −10.0610 −0.345904
$$847$$ −25.7482 −0.884720
$$848$$ 13.1957 0.453141
$$849$$ −24.6267 −0.845187
$$850$$ 1.79895 0.0617036
$$851$$ 4.37867 0.150099
$$852$$ −18.6450 −0.638768
$$853$$ 31.0121 1.06183 0.530917 0.847424i $$-0.321848\pi$$
0.530917 + 0.847424i $$0.321848\pi$$
$$854$$ −12.0067 −0.410861
$$855$$ 12.8049 0.437919
$$856$$ −11.5375 −0.394344
$$857$$ 12.4692 0.425940 0.212970 0.977059i $$-0.431686\pi$$
0.212970 + 0.977059i $$0.431686\pi$$
$$858$$ 0 0
$$859$$ −17.3163 −0.590826 −0.295413 0.955370i $$-0.595457\pi$$
−0.295413 + 0.955370i $$0.595457\pi$$
$$860$$ −33.9396 −1.15733
$$861$$ −7.16229 −0.244090
$$862$$ −0.517385 −0.0176222
$$863$$ 3.46383 0.117910 0.0589550 0.998261i $$-0.481223\pi$$
0.0589550 + 0.998261i $$0.481223\pi$$
$$864$$ −22.9608 −0.781141
$$865$$ 45.9275 1.56158
$$866$$ −7.40880 −0.251761
$$867$$ −12.5961 −0.427786
$$868$$ 26.8732 0.912136
$$869$$ 0.967213 0.0328105
$$870$$ −9.85086 −0.333975
$$871$$ 0 0
$$872$$ 8.55688 0.289772
$$873$$ 7.38106 0.249811
$$874$$ 4.95407 0.167574
$$875$$ −16.2107 −0.548023
$$876$$ 17.3913 0.587599
$$877$$ −57.2549 −1.93336 −0.966680 0.255989i $$-0.917599\pi$$
−0.966680 + 0.255989i $$0.917599\pi$$
$$878$$ 7.76510 0.262059
$$879$$ 14.9226 0.503327
$$880$$ 7.54288 0.254270
$$881$$ −43.1782 −1.45471 −0.727355 0.686261i $$-0.759251\pi$$
−0.727355 + 0.686261i $$0.759251\pi$$
$$882$$ −0.323044 −0.0108775
$$883$$ 49.9560 1.68115 0.840576 0.541693i $$-0.182216\pi$$
0.840576 + 0.541693i $$0.182216\pi$$
$$884$$ 0 0
$$885$$ 0.0271471 0.000912539 0
$$886$$ 13.1535 0.441899
$$887$$ 17.6746 0.593454 0.296727 0.954962i $$-0.404105\pi$$
0.296727 + 0.954962i $$0.404105\pi$$
$$888$$ 1.56273 0.0524417
$$889$$ 18.2459 0.611948
$$890$$ 22.9028 0.767702
$$891$$ 4.34375 0.145521
$$892$$ −19.2989 −0.646174
$$893$$ −14.9148 −0.499106
$$894$$ 0.329749 0.0110284
$$895$$ −6.87800 −0.229906
$$896$$ −31.0592 −1.03761
$$897$$ 0 0
$$898$$ 6.98493 0.233090
$$899$$ 46.6069 1.55443
$$900$$ 11.3690 0.378966
$$901$$ 6.67755 0.222461
$$902$$ −2.20583 −0.0734462
$$903$$ 15.4547 0.514301
$$904$$ −15.6601 −0.520847
$$905$$ 33.0834 1.09973
$$906$$ 8.48858 0.282014
$$907$$ 7.73423 0.256811 0.128406 0.991722i $$-0.459014\pi$$
0.128406 + 0.991722i $$0.459014\pi$$
$$908$$ 18.0043 0.597494
$$909$$ −12.4692 −0.413577
$$910$$ 0 0
$$911$$ 39.6179 1.31260 0.656299 0.754501i $$-0.272121\pi$$
0.656299 + 0.754501i $$0.272121\pi$$
$$912$$ −3.49396 −0.115697
$$913$$ 19.5676 0.647594
$$914$$ 18.6679 0.617478
$$915$$ −18.0586 −0.596999
$$916$$ −1.92798 −0.0637024
$$917$$ −36.8267 −1.21612
$$918$$ −2.71081 −0.0894700
$$919$$ 14.6213 0.482313 0.241157 0.970486i $$-0.422473\pi$$
0.241157 + 0.970486i $$0.422473\pi$$
$$920$$ 26.4306 0.871390
$$921$$ −7.17283 −0.236353
$$922$$ 0.778512 0.0256389
$$923$$ 0 0
$$924$$ −4.37627 −0.143969
$$925$$ −2.71140 −0.0891502
$$926$$ 8.43594 0.277222
$$927$$ 31.8907 1.04743
$$928$$ −42.2228 −1.38603
$$929$$ −3.55735 −0.116713 −0.0583565 0.998296i $$-0.518586\pi$$
−0.0583565 + 0.998296i $$0.518586\pi$$
$$930$$ −7.35690 −0.241242
$$931$$ −0.478894 −0.0156951
$$932$$ 18.3599 0.601398
$$933$$ 16.8726 0.552385
$$934$$ −21.8328 −0.714391
$$935$$ 3.81700 0.124829
$$936$$ 0 0
$$937$$ 34.5526 1.12878 0.564392 0.825507i $$-0.309111\pi$$
0.564392 + 0.825507i $$0.309111\pi$$
$$938$$ 13.8291 0.451536
$$939$$ −5.71571 −0.186525
$$940$$ −36.4674 −1.18944
$$941$$ −20.6233 −0.672299 −0.336149 0.941809i $$-0.609125\pi$$
−0.336149 + 0.941809i $$0.609125\pi$$
$$942$$ −1.78986 −0.0583167
$$943$$ 15.2741 0.497394
$$944$$ 0.0271471 0.000883562 0
$$945$$ −32.4034 −1.05408
$$946$$ 4.75973 0.154752
$$947$$ −29.4999 −0.958619 −0.479309 0.877646i $$-0.659113\pi$$
−0.479309 + 0.877646i $$0.659113\pi$$
$$948$$ −1.09544 −0.0355783
$$949$$ 0 0
$$950$$ −3.06770 −0.0995295
$$951$$ −19.2185 −0.623203
$$952$$ −6.27173 −0.203268
$$953$$ 26.2389 0.849963 0.424981 0.905202i $$-0.360281\pi$$
0.424981 + 0.905202i $$0.360281\pi$$
$$954$$ −7.68127 −0.248690
$$955$$ −25.1987 −0.815409
$$956$$ −20.1847 −0.652818
$$957$$ −7.58987 −0.245346
$$958$$ 12.4138 0.401073
$$959$$ −34.9638 −1.12904
$$960$$ −3.43296 −0.110798
$$961$$ 3.80731 0.122817
$$962$$ 0 0
$$963$$ −13.2717 −0.427676
$$964$$ −6.17331 −0.198829
$$965$$ −37.8974 −1.21996
$$966$$ −5.51573 −0.177466
$$967$$ −17.5176 −0.563330 −0.281665 0.959513i $$-0.590887\pi$$
−0.281665 + 0.959513i $$0.590887\pi$$
$$968$$ 19.5972 0.629877
$$969$$ −1.76809 −0.0567991
$$970$$ −4.86964 −0.156355
$$971$$ 20.5120 0.658262 0.329131 0.944284i $$-0.393244\pi$$
0.329131 + 0.944284i $$0.393244\pi$$
$$972$$ −26.7259 −0.857233
$$973$$ 32.4295 1.03964
$$974$$ −12.7199 −0.407572
$$975$$ 0 0
$$976$$ −18.0586 −0.578042
$$977$$ −25.4450 −0.814059 −0.407030 0.913415i $$-0.633435\pi$$
−0.407030 + 0.913415i $$0.633435\pi$$
$$978$$ −6.73663 −0.215414
$$979$$ 17.6461 0.563971
$$980$$ −1.17092 −0.0374035
$$981$$ 9.84309 0.314266
$$982$$ 1.02310 0.0326484
$$983$$ 39.5244 1.26063 0.630316 0.776339i $$-0.282926\pi$$
0.630316 + 0.776339i $$0.282926\pi$$
$$984$$ 5.45127 0.173780
$$985$$ −36.3575 −1.15845
$$986$$ −4.98493 −0.158753
$$987$$ 16.6058 0.528568
$$988$$ 0 0
$$989$$ −32.9584 −1.04802
$$990$$ −4.39075 −0.139547
$$991$$ −29.8377 −0.947826 −0.473913 0.880572i $$-0.657159\pi$$
−0.473913 + 0.880572i $$0.657159\pi$$
$$992$$ −31.5332 −1.00118
$$993$$ −2.32198 −0.0736858
$$994$$ 20.5284 0.651121
$$995$$ −38.0683 −1.20685
$$996$$ −22.1618 −0.702224
$$997$$ −4.93123 −0.156174 −0.0780868 0.996947i $$-0.524881\pi$$
−0.0780868 + 0.996947i $$0.524881\pi$$
$$998$$ 6.67861 0.211408
$$999$$ 4.08575 0.129268
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 169.2.a.c.1.2 yes 3
3.2 odd 2 1521.2.a.o.1.2 3
4.3 odd 2 2704.2.a.ba.1.1 3
5.4 even 2 4225.2.a.bb.1.2 3
7.6 odd 2 8281.2.a.bj.1.2 3
13.2 odd 12 169.2.e.b.147.4 12
13.3 even 3 169.2.c.b.22.2 6
13.4 even 6 169.2.c.c.146.2 6
13.5 odd 4 169.2.b.b.168.3 6
13.6 odd 12 169.2.e.b.23.3 12
13.7 odd 12 169.2.e.b.23.4 12
13.8 odd 4 169.2.b.b.168.4 6
13.9 even 3 169.2.c.b.146.2 6
13.10 even 6 169.2.c.c.22.2 6
13.11 odd 12 169.2.e.b.147.3 12
13.12 even 2 169.2.a.b.1.2 3
39.5 even 4 1521.2.b.l.1351.4 6
39.8 even 4 1521.2.b.l.1351.3 6
39.38 odd 2 1521.2.a.r.1.2 3
52.31 even 4 2704.2.f.o.337.1 6
52.47 even 4 2704.2.f.o.337.2 6
52.51 odd 2 2704.2.a.z.1.1 3
65.64 even 2 4225.2.a.bg.1.2 3
91.90 odd 2 8281.2.a.bf.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.2 3 13.12 even 2
169.2.a.c.1.2 yes 3 1.1 even 1 trivial
169.2.b.b.168.3 6 13.5 odd 4
169.2.b.b.168.4 6 13.8 odd 4
169.2.c.b.22.2 6 13.3 even 3
169.2.c.b.146.2 6 13.9 even 3
169.2.c.c.22.2 6 13.10 even 6
169.2.c.c.146.2 6 13.4 even 6
169.2.e.b.23.3 12 13.6 odd 12
169.2.e.b.23.4 12 13.7 odd 12
169.2.e.b.147.3 12 13.11 odd 12
169.2.e.b.147.4 12 13.2 odd 12
1521.2.a.o.1.2 3 3.2 odd 2
1521.2.a.r.1.2 3 39.38 odd 2
1521.2.b.l.1351.3 6 39.8 even 4
1521.2.b.l.1351.4 6 39.5 even 4
2704.2.a.z.1.1 3 52.51 odd 2
2704.2.a.ba.1.1 3 4.3 odd 2
2704.2.f.o.337.1 6 52.31 even 4
2704.2.f.o.337.2 6 52.47 even 4
4225.2.a.bb.1.2 3 5.4 even 2
4225.2.a.bg.1.2 3 65.64 even 2
8281.2.a.bf.1.2 3 91.90 odd 2
8281.2.a.bj.1.2 3 7.6 odd 2