# Properties

 Label 7942.2.a.x.1.2 Level $7942$ Weight $2$ Character 7942.1 Self dual yes Analytic conductor $63.417$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7942 = 2 \cdot 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7942.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: $$63.4171892853$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 7942.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} +1.00000 q^{8} -0.561553 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.56155 q^{6} +3.56155 q^{7} +1.00000 q^{8} -0.561553 q^{9} +2.00000 q^{10} +1.00000 q^{11} +1.56155 q^{12} +3.56155 q^{13} +3.56155 q^{14} +3.12311 q^{15} +1.00000 q^{16} +3.56155 q^{17} -0.561553 q^{18} +2.00000 q^{20} +5.56155 q^{21} +1.00000 q^{22} +5.56155 q^{23} +1.56155 q^{24} -1.00000 q^{25} +3.56155 q^{26} -5.56155 q^{27} +3.56155 q^{28} -6.68466 q^{29} +3.12311 q^{30} -2.00000 q^{31} +1.00000 q^{32} +1.56155 q^{33} +3.56155 q^{34} +7.12311 q^{35} -0.561553 q^{36} -3.12311 q^{37} +5.56155 q^{39} +2.00000 q^{40} -2.00000 q^{41} +5.56155 q^{42} +1.00000 q^{44} -1.12311 q^{45} +5.56155 q^{46} +8.00000 q^{47} +1.56155 q^{48} +5.68466 q^{49} -1.00000 q^{50} +5.56155 q^{51} +3.56155 q^{52} +7.80776 q^{53} -5.56155 q^{54} +2.00000 q^{55} +3.56155 q^{56} -6.68466 q^{58} -4.68466 q^{59} +3.12311 q^{60} -10.2462 q^{61} -2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +7.12311 q^{65} +1.56155 q^{66} -4.68466 q^{67} +3.56155 q^{68} +8.68466 q^{69} +7.12311 q^{70} -6.00000 q^{71} -0.561553 q^{72} +2.68466 q^{73} -3.12311 q^{74} -1.56155 q^{75} +3.56155 q^{77} +5.56155 q^{78} +4.00000 q^{79} +2.00000 q^{80} -7.00000 q^{81} -2.00000 q^{82} -2.24621 q^{83} +5.56155 q^{84} +7.12311 q^{85} -10.4384 q^{87} +1.00000 q^{88} +9.12311 q^{89} -1.12311 q^{90} +12.6847 q^{91} +5.56155 q^{92} -3.12311 q^{93} +8.00000 q^{94} +1.56155 q^{96} -1.12311 q^{97} +5.68466 q^{98} -0.561553 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 4 * q^5 - q^6 + 3 * q^7 + 2 * q^8 + 3 * q^9 $$2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 3 q^{7} + 2 q^{8} + 3 q^{9} + 4 q^{10} + 2 q^{11} - q^{12} + 3 q^{13} + 3 q^{14} - 2 q^{15} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 4 q^{20} + 7 q^{21} + 2 q^{22} + 7 q^{23} - q^{24} - 2 q^{25} + 3 q^{26} - 7 q^{27} + 3 q^{28} - q^{29} - 2 q^{30} - 4 q^{31} + 2 q^{32} - q^{33} + 3 q^{34} + 6 q^{35} + 3 q^{36} + 2 q^{37} + 7 q^{39} + 4 q^{40} - 4 q^{41} + 7 q^{42} + 2 q^{44} + 6 q^{45} + 7 q^{46} + 16 q^{47} - q^{48} - q^{49} - 2 q^{50} + 7 q^{51} + 3 q^{52} - 5 q^{53} - 7 q^{54} + 4 q^{55} + 3 q^{56} - q^{58} + 3 q^{59} - 2 q^{60} - 4 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} + 6 q^{65} - q^{66} + 3 q^{67} + 3 q^{68} + 5 q^{69} + 6 q^{70} - 12 q^{71} + 3 q^{72} - 7 q^{73} + 2 q^{74} + q^{75} + 3 q^{77} + 7 q^{78} + 8 q^{79} + 4 q^{80} - 14 q^{81} - 4 q^{82} + 12 q^{83} + 7 q^{84} + 6 q^{85} - 25 q^{87} + 2 q^{88} + 10 q^{89} + 6 q^{90} + 13 q^{91} + 7 q^{92} + 2 q^{93} + 16 q^{94} - q^{96} + 6 q^{97} - q^{98} + 3 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 4 * q^5 - q^6 + 3 * q^7 + 2 * q^8 + 3 * q^9 + 4 * q^10 + 2 * q^11 - q^12 + 3 * q^13 + 3 * q^14 - 2 * q^15 + 2 * q^16 + 3 * q^17 + 3 * q^18 + 4 * q^20 + 7 * q^21 + 2 * q^22 + 7 * q^23 - q^24 - 2 * q^25 + 3 * q^26 - 7 * q^27 + 3 * q^28 - q^29 - 2 * q^30 - 4 * q^31 + 2 * q^32 - q^33 + 3 * q^34 + 6 * q^35 + 3 * q^36 + 2 * q^37 + 7 * q^39 + 4 * q^40 - 4 * q^41 + 7 * q^42 + 2 * q^44 + 6 * q^45 + 7 * q^46 + 16 * q^47 - q^48 - q^49 - 2 * q^50 + 7 * q^51 + 3 * q^52 - 5 * q^53 - 7 * q^54 + 4 * q^55 + 3 * q^56 - q^58 + 3 * q^59 - 2 * q^60 - 4 * q^61 - 4 * q^62 - 4 * q^63 + 2 * q^64 + 6 * q^65 - q^66 + 3 * q^67 + 3 * q^68 + 5 * q^69 + 6 * q^70 - 12 * q^71 + 3 * q^72 - 7 * q^73 + 2 * q^74 + q^75 + 3 * q^77 + 7 * q^78 + 8 * q^79 + 4 * q^80 - 14 * q^81 - 4 * q^82 + 12 * q^83 + 7 * q^84 + 6 * q^85 - 25 * q^87 + 2 * q^88 + 10 * q^89 + 6 * q^90 + 13 * q^91 + 7 * q^92 + 2 * q^93 + 16 * q^94 - q^96 + 6 * q^97 - q^98 + 3 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.56155 0.901563 0.450781 0.892634i $$-0.351145\pi$$
0.450781 + 0.892634i $$0.351145\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 1.56155 0.637501
$$7$$ 3.56155 1.34614 0.673070 0.739579i $$-0.264975\pi$$
0.673070 + 0.739579i $$0.264975\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −0.561553 −0.187184
$$10$$ 2.00000 0.632456
$$11$$ 1.00000 0.301511
$$12$$ 1.56155 0.450781
$$13$$ 3.56155 0.987797 0.493899 0.869520i $$-0.335571\pi$$
0.493899 + 0.869520i $$0.335571\pi$$
$$14$$ 3.56155 0.951865
$$15$$ 3.12311 0.806382
$$16$$ 1.00000 0.250000
$$17$$ 3.56155 0.863803 0.431902 0.901921i $$-0.357843\pi$$
0.431902 + 0.901921i $$0.357843\pi$$
$$18$$ −0.561553 −0.132359
$$19$$ 0 0
$$20$$ 2.00000 0.447214
$$21$$ 5.56155 1.21363
$$22$$ 1.00000 0.213201
$$23$$ 5.56155 1.15966 0.579832 0.814736i $$-0.303118\pi$$
0.579832 + 0.814736i $$0.303118\pi$$
$$24$$ 1.56155 0.318751
$$25$$ −1.00000 −0.200000
$$26$$ 3.56155 0.698478
$$27$$ −5.56155 −1.07032
$$28$$ 3.56155 0.673070
$$29$$ −6.68466 −1.24131 −0.620655 0.784084i $$-0.713133\pi$$
−0.620655 + 0.784084i $$0.713133\pi$$
$$30$$ 3.12311 0.570198
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.56155 0.271831
$$34$$ 3.56155 0.610801
$$35$$ 7.12311 1.20402
$$36$$ −0.561553 −0.0935921
$$37$$ −3.12311 −0.513435 −0.256718 0.966486i $$-0.582641\pi$$
−0.256718 + 0.966486i $$0.582641\pi$$
$$38$$ 0 0
$$39$$ 5.56155 0.890561
$$40$$ 2.00000 0.316228
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 5.56155 0.858166
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −1.12311 −0.167423
$$46$$ 5.56155 0.820006
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.56155 0.225391
$$49$$ 5.68466 0.812094
$$50$$ −1.00000 −0.141421
$$51$$ 5.56155 0.778773
$$52$$ 3.56155 0.493899
$$53$$ 7.80776 1.07248 0.536239 0.844066i $$-0.319844\pi$$
0.536239 + 0.844066i $$0.319844\pi$$
$$54$$ −5.56155 −0.756831
$$55$$ 2.00000 0.269680
$$56$$ 3.56155 0.475933
$$57$$ 0 0
$$58$$ −6.68466 −0.877739
$$59$$ −4.68466 −0.609891 −0.304945 0.952370i $$-0.598638\pi$$
−0.304945 + 0.952370i $$0.598638\pi$$
$$60$$ 3.12311 0.403191
$$61$$ −10.2462 −1.31189 −0.655946 0.754807i $$-0.727730\pi$$
−0.655946 + 0.754807i $$0.727730\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 7.12311 0.883513
$$66$$ 1.56155 0.192214
$$67$$ −4.68466 −0.572322 −0.286161 0.958182i $$-0.592379\pi$$
−0.286161 + 0.958182i $$0.592379\pi$$
$$68$$ 3.56155 0.431902
$$69$$ 8.68466 1.04551
$$70$$ 7.12311 0.851374
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ −0.561553 −0.0661796
$$73$$ 2.68466 0.314216 0.157108 0.987581i $$-0.449783\pi$$
0.157108 + 0.987581i $$0.449783\pi$$
$$74$$ −3.12311 −0.363054
$$75$$ −1.56155 −0.180313
$$76$$ 0 0
$$77$$ 3.56155 0.405877
$$78$$ 5.56155 0.629722
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 2.00000 0.223607
$$81$$ −7.00000 −0.777778
$$82$$ −2.00000 −0.220863
$$83$$ −2.24621 −0.246554 −0.123277 0.992372i $$-0.539340\pi$$
−0.123277 + 0.992372i $$0.539340\pi$$
$$84$$ 5.56155 0.606815
$$85$$ 7.12311 0.772609
$$86$$ 0 0
$$87$$ −10.4384 −1.11912
$$88$$ 1.00000 0.106600
$$89$$ 9.12311 0.967047 0.483524 0.875331i $$-0.339357\pi$$
0.483524 + 0.875331i $$0.339357\pi$$
$$90$$ −1.12311 −0.118386
$$91$$ 12.6847 1.32971
$$92$$ 5.56155 0.579832
$$93$$ −3.12311 −0.323851
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 1.56155 0.159375
$$97$$ −1.12311 −0.114034 −0.0570170 0.998373i $$-0.518159\pi$$
−0.0570170 + 0.998373i $$0.518159\pi$$
$$98$$ 5.68466 0.574237
$$99$$ −0.561553 −0.0564382
$$100$$ −1.00000 −0.100000
$$101$$ −15.1231 −1.50481 −0.752403 0.658703i $$-0.771105\pi$$
−0.752403 + 0.658703i $$0.771105\pi$$
$$102$$ 5.56155 0.550676
$$103$$ −11.3693 −1.12025 −0.560126 0.828407i $$-0.689247\pi$$
−0.560126 + 0.828407i $$0.689247\pi$$
$$104$$ 3.56155 0.349239
$$105$$ 11.1231 1.08550
$$106$$ 7.80776 0.758357
$$107$$ −18.9309 −1.83012 −0.915058 0.403322i $$-0.867855\pi$$
−0.915058 + 0.403322i $$0.867855\pi$$
$$108$$ −5.56155 −0.535161
$$109$$ 18.6847 1.78967 0.894833 0.446401i $$-0.147295\pi$$
0.894833 + 0.446401i $$0.147295\pi$$
$$110$$ 2.00000 0.190693
$$111$$ −4.87689 −0.462894
$$112$$ 3.56155 0.336535
$$113$$ −5.12311 −0.481941 −0.240971 0.970532i $$-0.577466\pi$$
−0.240971 + 0.970532i $$0.577466\pi$$
$$114$$ 0 0
$$115$$ 11.1231 1.03723
$$116$$ −6.68466 −0.620655
$$117$$ −2.00000 −0.184900
$$118$$ −4.68466 −0.431258
$$119$$ 12.6847 1.16280
$$120$$ 3.12311 0.285099
$$121$$ 1.00000 0.0909091
$$122$$ −10.2462 −0.927648
$$123$$ −3.12311 −0.281601
$$124$$ −2.00000 −0.179605
$$125$$ −12.0000 −1.07331
$$126$$ −2.00000 −0.178174
$$127$$ 14.2462 1.26415 0.632073 0.774909i $$-0.282204\pi$$
0.632073 + 0.774909i $$0.282204\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 7.12311 0.624738
$$131$$ −7.12311 −0.622349 −0.311174 0.950353i $$-0.600722\pi$$
−0.311174 + 0.950353i $$0.600722\pi$$
$$132$$ 1.56155 0.135916
$$133$$ 0 0
$$134$$ −4.68466 −0.404693
$$135$$ −11.1231 −0.957325
$$136$$ 3.56155 0.305401
$$137$$ 8.43845 0.720945 0.360473 0.932770i $$-0.382615\pi$$
0.360473 + 0.932770i $$0.382615\pi$$
$$138$$ 8.68466 0.739287
$$139$$ −17.3693 −1.47325 −0.736623 0.676303i $$-0.763581\pi$$
−0.736623 + 0.676303i $$0.763581\pi$$
$$140$$ 7.12311 0.602012
$$141$$ 12.4924 1.05205
$$142$$ −6.00000 −0.503509
$$143$$ 3.56155 0.297832
$$144$$ −0.561553 −0.0467961
$$145$$ −13.3693 −1.11026
$$146$$ 2.68466 0.222184
$$147$$ 8.87689 0.732154
$$148$$ −3.12311 −0.256718
$$149$$ −15.1231 −1.23893 −0.619467 0.785023i $$-0.712651\pi$$
−0.619467 + 0.785023i $$0.712651\pi$$
$$150$$ −1.56155 −0.127500
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 3.56155 0.286998
$$155$$ −4.00000 −0.321288
$$156$$ 5.56155 0.445281
$$157$$ 18.4924 1.47586 0.737928 0.674879i $$-0.235804\pi$$
0.737928 + 0.674879i $$0.235804\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 12.1922 0.966907
$$160$$ 2.00000 0.158114
$$161$$ 19.8078 1.56107
$$162$$ −7.00000 −0.549972
$$163$$ −13.3693 −1.04717 −0.523583 0.851975i $$-0.675405\pi$$
−0.523583 + 0.851975i $$0.675405\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 3.12311 0.243133
$$166$$ −2.24621 −0.174340
$$167$$ 25.3693 1.96314 0.981568 0.191111i $$-0.0612092\pi$$
0.981568 + 0.191111i $$0.0612092\pi$$
$$168$$ 5.56155 0.429083
$$169$$ −0.315342 −0.0242570
$$170$$ 7.12311 0.546317
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −10.4384 −0.791337
$$175$$ −3.56155 −0.269228
$$176$$ 1.00000 0.0753778
$$177$$ −7.31534 −0.549855
$$178$$ 9.12311 0.683806
$$179$$ −22.2462 −1.66276 −0.831380 0.555704i $$-0.812449\pi$$
−0.831380 + 0.555704i $$0.812449\pi$$
$$180$$ −1.12311 −0.0837114
$$181$$ 19.1231 1.42141 0.710705 0.703491i $$-0.248376\pi$$
0.710705 + 0.703491i $$0.248376\pi$$
$$182$$ 12.6847 0.940249
$$183$$ −16.0000 −1.18275
$$184$$ 5.56155 0.410003
$$185$$ −6.24621 −0.459231
$$186$$ −3.12311 −0.228997
$$187$$ 3.56155 0.260447
$$188$$ 8.00000 0.583460
$$189$$ −19.8078 −1.44080
$$190$$ 0 0
$$191$$ −20.6847 −1.49669 −0.748345 0.663310i $$-0.769151\pi$$
−0.748345 + 0.663310i $$0.769151\pi$$
$$192$$ 1.56155 0.112695
$$193$$ 1.12311 0.0808429 0.0404215 0.999183i $$-0.487130\pi$$
0.0404215 + 0.999183i $$0.487130\pi$$
$$194$$ −1.12311 −0.0806343
$$195$$ 11.1231 0.796542
$$196$$ 5.68466 0.406047
$$197$$ 1.75379 0.124952 0.0624761 0.998046i $$-0.480100\pi$$
0.0624761 + 0.998046i $$0.480100\pi$$
$$198$$ −0.561553 −0.0399078
$$199$$ −0.684658 −0.0485341 −0.0242671 0.999706i $$-0.507725\pi$$
−0.0242671 + 0.999706i $$0.507725\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −7.31534 −0.515984
$$202$$ −15.1231 −1.06406
$$203$$ −23.8078 −1.67098
$$204$$ 5.56155 0.389387
$$205$$ −4.00000 −0.279372
$$206$$ −11.3693 −0.792138
$$207$$ −3.12311 −0.217071
$$208$$ 3.56155 0.246949
$$209$$ 0 0
$$210$$ 11.1231 0.767567
$$211$$ 12.6847 0.873248 0.436624 0.899644i $$-0.356174\pi$$
0.436624 + 0.899644i $$0.356174\pi$$
$$212$$ 7.80776 0.536239
$$213$$ −9.36932 −0.641975
$$214$$ −18.9309 −1.29409
$$215$$ 0 0
$$216$$ −5.56155 −0.378416
$$217$$ −7.12311 −0.483548
$$218$$ 18.6847 1.26548
$$219$$ 4.19224 0.283285
$$220$$ 2.00000 0.134840
$$221$$ 12.6847 0.853262
$$222$$ −4.87689 −0.327316
$$223$$ 24.7386 1.65662 0.828311 0.560269i $$-0.189302\pi$$
0.828311 + 0.560269i $$0.189302\pi$$
$$224$$ 3.56155 0.237966
$$225$$ 0.561553 0.0374369
$$226$$ −5.12311 −0.340784
$$227$$ 17.1771 1.14008 0.570041 0.821616i $$-0.306927\pi$$
0.570041 + 0.821616i $$0.306927\pi$$
$$228$$ 0 0
$$229$$ 21.1231 1.39585 0.697927 0.716169i $$-0.254106\pi$$
0.697927 + 0.716169i $$0.254106\pi$$
$$230$$ 11.1231 0.733436
$$231$$ 5.56155 0.365923
$$232$$ −6.68466 −0.438869
$$233$$ 20.7386 1.35863 0.679317 0.733845i $$-0.262276\pi$$
0.679317 + 0.733845i $$0.262276\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 16.0000 1.04372
$$236$$ −4.68466 −0.304945
$$237$$ 6.24621 0.405735
$$238$$ 12.6847 0.822224
$$239$$ −4.93087 −0.318951 −0.159476 0.987202i $$-0.550980\pi$$
−0.159476 + 0.987202i $$0.550980\pi$$
$$240$$ 3.12311 0.201596
$$241$$ −21.6155 −1.39238 −0.696189 0.717858i $$-0.745123\pi$$
−0.696189 + 0.717858i $$0.745123\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 5.75379 0.369106
$$244$$ −10.2462 −0.655946
$$245$$ 11.3693 0.726359
$$246$$ −3.12311 −0.199122
$$247$$ 0 0
$$248$$ −2.00000 −0.127000
$$249$$ −3.50758 −0.222284
$$250$$ −12.0000 −0.758947
$$251$$ 8.87689 0.560305 0.280152 0.959956i $$-0.409615\pi$$
0.280152 + 0.959956i $$0.409615\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 5.56155 0.349652
$$254$$ 14.2462 0.893887
$$255$$ 11.1231 0.696556
$$256$$ 1.00000 0.0625000
$$257$$ 22.0000 1.37232 0.686161 0.727450i $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ 0 0
$$259$$ −11.1231 −0.691156
$$260$$ 7.12311 0.441756
$$261$$ 3.75379 0.232354
$$262$$ −7.12311 −0.440067
$$263$$ 17.1231 1.05586 0.527928 0.849289i $$-0.322969\pi$$
0.527928 + 0.849289i $$0.322969\pi$$
$$264$$ 1.56155 0.0961069
$$265$$ 15.6155 0.959254
$$266$$ 0 0
$$267$$ 14.2462 0.871854
$$268$$ −4.68466 −0.286161
$$269$$ −11.1231 −0.678188 −0.339094 0.940753i $$-0.610120\pi$$
−0.339094 + 0.940753i $$0.610120\pi$$
$$270$$ −11.1231 −0.676931
$$271$$ 13.3153 0.808849 0.404425 0.914571i $$-0.367472\pi$$
0.404425 + 0.914571i $$0.367472\pi$$
$$272$$ 3.56155 0.215951
$$273$$ 19.8078 1.19882
$$274$$ 8.43845 0.509785
$$275$$ −1.00000 −0.0603023
$$276$$ 8.68466 0.522755
$$277$$ −24.4924 −1.47161 −0.735804 0.677195i $$-0.763195\pi$$
−0.735804 + 0.677195i $$0.763195\pi$$
$$278$$ −17.3693 −1.04174
$$279$$ 1.12311 0.0672386
$$280$$ 7.12311 0.425687
$$281$$ 0.630683 0.0376234 0.0188117 0.999823i $$-0.494012\pi$$
0.0188117 + 0.999823i $$0.494012\pi$$
$$282$$ 12.4924 0.743913
$$283$$ −13.3693 −0.794723 −0.397362 0.917662i $$-0.630074\pi$$
−0.397362 + 0.917662i $$0.630074\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 3.56155 0.210599
$$287$$ −7.12311 −0.420464
$$288$$ −0.561553 −0.0330898
$$289$$ −4.31534 −0.253844
$$290$$ −13.3693 −0.785073
$$291$$ −1.75379 −0.102809
$$292$$ 2.68466 0.157108
$$293$$ 24.9309 1.45648 0.728238 0.685324i $$-0.240339\pi$$
0.728238 + 0.685324i $$0.240339\pi$$
$$294$$ 8.87689 0.517711
$$295$$ −9.36932 −0.545503
$$296$$ −3.12311 −0.181527
$$297$$ −5.56155 −0.322714
$$298$$ −15.1231 −0.876058
$$299$$ 19.8078 1.14551
$$300$$ −1.56155 −0.0901563
$$301$$ 0 0
$$302$$ −4.00000 −0.230174
$$303$$ −23.6155 −1.35668
$$304$$ 0 0
$$305$$ −20.4924 −1.17339
$$306$$ −2.00000 −0.114332
$$307$$ 5.75379 0.328386 0.164193 0.986428i $$-0.447498\pi$$
0.164193 + 0.986428i $$0.447498\pi$$
$$308$$ 3.56155 0.202938
$$309$$ −17.7538 −1.00998
$$310$$ −4.00000 −0.227185
$$311$$ 15.8078 0.896376 0.448188 0.893939i $$-0.352070\pi$$
0.448188 + 0.893939i $$0.352070\pi$$
$$312$$ 5.56155 0.314861
$$313$$ 3.56155 0.201311 0.100655 0.994921i $$-0.467906\pi$$
0.100655 + 0.994921i $$0.467906\pi$$
$$314$$ 18.4924 1.04359
$$315$$ −4.00000 −0.225374
$$316$$ 4.00000 0.225018
$$317$$ −30.0540 −1.68800 −0.844000 0.536344i $$-0.819805\pi$$
−0.844000 + 0.536344i $$0.819805\pi$$
$$318$$ 12.1922 0.683707
$$319$$ −6.68466 −0.374269
$$320$$ 2.00000 0.111803
$$321$$ −29.5616 −1.64996
$$322$$ 19.8078 1.10384
$$323$$ 0 0
$$324$$ −7.00000 −0.388889
$$325$$ −3.56155 −0.197559
$$326$$ −13.3693 −0.740458
$$327$$ 29.1771 1.61350
$$328$$ −2.00000 −0.110432
$$329$$ 28.4924 1.57084
$$330$$ 3.12311 0.171921
$$331$$ 10.4384 0.573749 0.286874 0.957968i $$-0.407384\pi$$
0.286874 + 0.957968i $$0.407384\pi$$
$$332$$ −2.24621 −0.123277
$$333$$ 1.75379 0.0961070
$$334$$ 25.3693 1.38815
$$335$$ −9.36932 −0.511900
$$336$$ 5.56155 0.303408
$$337$$ −2.87689 −0.156714 −0.0783572 0.996925i $$-0.524967\pi$$
−0.0783572 + 0.996925i $$0.524967\pi$$
$$338$$ −0.315342 −0.0171523
$$339$$ −8.00000 −0.434500
$$340$$ 7.12311 0.386305
$$341$$ −2.00000 −0.108306
$$342$$ 0 0
$$343$$ −4.68466 −0.252948
$$344$$ 0 0
$$345$$ 17.3693 0.935133
$$346$$ −18.0000 −0.967686
$$347$$ −31.6155 −1.69721 −0.848605 0.529027i $$-0.822557\pi$$
−0.848605 + 0.529027i $$0.822557\pi$$
$$348$$ −10.4384 −0.559560
$$349$$ −19.6155 −1.05000 −0.524998 0.851104i $$-0.675934\pi$$
−0.524998 + 0.851104i $$0.675934\pi$$
$$350$$ −3.56155 −0.190373
$$351$$ −19.8078 −1.05726
$$352$$ 1.00000 0.0533002
$$353$$ −23.1771 −1.23359 −0.616796 0.787123i $$-0.711570\pi$$
−0.616796 + 0.787123i $$0.711570\pi$$
$$354$$ −7.31534 −0.388806
$$355$$ −12.0000 −0.636894
$$356$$ 9.12311 0.483524
$$357$$ 19.8078 1.04834
$$358$$ −22.2462 −1.17575
$$359$$ 12.9309 0.682465 0.341233 0.939979i $$-0.389156\pi$$
0.341233 + 0.939979i $$0.389156\pi$$
$$360$$ −1.12311 −0.0591929
$$361$$ 0 0
$$362$$ 19.1231 1.00509
$$363$$ 1.56155 0.0819603
$$364$$ 12.6847 0.664857
$$365$$ 5.36932 0.281043
$$366$$ −16.0000 −0.836333
$$367$$ 13.7538 0.717942 0.358971 0.933349i $$-0.383128\pi$$
0.358971 + 0.933349i $$0.383128\pi$$
$$368$$ 5.56155 0.289916
$$369$$ 1.12311 0.0584665
$$370$$ −6.24621 −0.324725
$$371$$ 27.8078 1.44371
$$372$$ −3.12311 −0.161925
$$373$$ −7.56155 −0.391522 −0.195761 0.980652i $$-0.562718\pi$$
−0.195761 + 0.980652i $$0.562718\pi$$
$$374$$ 3.56155 0.184164
$$375$$ −18.7386 −0.967659
$$376$$ 8.00000 0.412568
$$377$$ −23.8078 −1.22616
$$378$$ −19.8078 −1.01880
$$379$$ −2.43845 −0.125255 −0.0626273 0.998037i $$-0.519948\pi$$
−0.0626273 + 0.998037i $$0.519948\pi$$
$$380$$ 0 0
$$381$$ 22.2462 1.13971
$$382$$ −20.6847 −1.05832
$$383$$ 2.00000 0.102195 0.0510976 0.998694i $$-0.483728\pi$$
0.0510976 + 0.998694i $$0.483728\pi$$
$$384$$ 1.56155 0.0796877
$$385$$ 7.12311 0.363027
$$386$$ 1.12311 0.0571646
$$387$$ 0 0
$$388$$ −1.12311 −0.0570170
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 11.1231 0.563240
$$391$$ 19.8078 1.00172
$$392$$ 5.68466 0.287119
$$393$$ −11.1231 −0.561086
$$394$$ 1.75379 0.0883546
$$395$$ 8.00000 0.402524
$$396$$ −0.561553 −0.0282191
$$397$$ −26.9848 −1.35433 −0.677165 0.735831i $$-0.736792\pi$$
−0.677165 + 0.735831i $$0.736792\pi$$
$$398$$ −0.684658 −0.0343188
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 24.2462 1.21080 0.605399 0.795922i $$-0.293014\pi$$
0.605399 + 0.795922i $$0.293014\pi$$
$$402$$ −7.31534 −0.364856
$$403$$ −7.12311 −0.354827
$$404$$ −15.1231 −0.752403
$$405$$ −14.0000 −0.695666
$$406$$ −23.8078 −1.18156
$$407$$ −3.12311 −0.154807
$$408$$ 5.56155 0.275338
$$409$$ 0.246211 0.0121744 0.00608718 0.999981i $$-0.498062\pi$$
0.00608718 + 0.999981i $$0.498062\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ 13.1771 0.649977
$$412$$ −11.3693 −0.560126
$$413$$ −16.6847 −0.820998
$$414$$ −3.12311 −0.153492
$$415$$ −4.49242 −0.220524
$$416$$ 3.56155 0.174619
$$417$$ −27.1231 −1.32822
$$418$$ 0 0
$$419$$ −23.1231 −1.12964 −0.564819 0.825215i $$-0.691054\pi$$
−0.564819 + 0.825215i $$0.691054\pi$$
$$420$$ 11.1231 0.542752
$$421$$ 1.06913 0.0521062 0.0260531 0.999661i $$-0.491706\pi$$
0.0260531 + 0.999661i $$0.491706\pi$$
$$422$$ 12.6847 0.617480
$$423$$ −4.49242 −0.218429
$$424$$ 7.80776 0.379179
$$425$$ −3.56155 −0.172761
$$426$$ −9.36932 −0.453945
$$427$$ −36.4924 −1.76599
$$428$$ −18.9309 −0.915058
$$429$$ 5.56155 0.268514
$$430$$ 0 0
$$431$$ 25.8617 1.24572 0.622858 0.782335i $$-0.285971\pi$$
0.622858 + 0.782335i $$0.285971\pi$$
$$432$$ −5.56155 −0.267580
$$433$$ −31.3693 −1.50751 −0.753757 0.657154i $$-0.771760\pi$$
−0.753757 + 0.657154i $$0.771760\pi$$
$$434$$ −7.12311 −0.341920
$$435$$ −20.8769 −1.00097
$$436$$ 18.6847 0.894833
$$437$$ 0 0
$$438$$ 4.19224 0.200313
$$439$$ −31.6155 −1.50893 −0.754463 0.656342i $$-0.772103\pi$$
−0.754463 + 0.656342i $$0.772103\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ −3.19224 −0.152011
$$442$$ 12.6847 0.603348
$$443$$ −17.8617 −0.848637 −0.424318 0.905513i $$-0.639486\pi$$
−0.424318 + 0.905513i $$0.639486\pi$$
$$444$$ −4.87689 −0.231447
$$445$$ 18.2462 0.864953
$$446$$ 24.7386 1.17141
$$447$$ −23.6155 −1.11698
$$448$$ 3.56155 0.168268
$$449$$ 17.6155 0.831328 0.415664 0.909518i $$-0.363549\pi$$
0.415664 + 0.909518i $$0.363549\pi$$
$$450$$ 0.561553 0.0264719
$$451$$ −2.00000 −0.0941763
$$452$$ −5.12311 −0.240971
$$453$$ −6.24621 −0.293473
$$454$$ 17.1771 0.806160
$$455$$ 25.3693 1.18933
$$456$$ 0 0
$$457$$ 32.4384 1.51741 0.758703 0.651436i $$-0.225833\pi$$
0.758703 + 0.651436i $$0.225833\pi$$
$$458$$ 21.1231 0.987018
$$459$$ −19.8078 −0.924547
$$460$$ 11.1231 0.518617
$$461$$ −22.7386 −1.05904 −0.529522 0.848296i $$-0.677629\pi$$
−0.529522 + 0.848296i $$0.677629\pi$$
$$462$$ 5.56155 0.258747
$$463$$ 36.4924 1.69595 0.847973 0.530039i $$-0.177823\pi$$
0.847973 + 0.530039i $$0.177823\pi$$
$$464$$ −6.68466 −0.310327
$$465$$ −6.24621 −0.289661
$$466$$ 20.7386 0.960699
$$467$$ 26.2462 1.21453 0.607265 0.794499i $$-0.292267\pi$$
0.607265 + 0.794499i $$0.292267\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ −16.6847 −0.770426
$$470$$ 16.0000 0.738025
$$471$$ 28.8769 1.33058
$$472$$ −4.68466 −0.215629
$$473$$ 0 0
$$474$$ 6.24621 0.286898
$$475$$ 0 0
$$476$$ 12.6847 0.581400
$$477$$ −4.38447 −0.200751
$$478$$ −4.93087 −0.225533
$$479$$ 11.3693 0.519477 0.259739 0.965679i $$-0.416364\pi$$
0.259739 + 0.965679i $$0.416364\pi$$
$$480$$ 3.12311 0.142550
$$481$$ −11.1231 −0.507170
$$482$$ −21.6155 −0.984560
$$483$$ 30.9309 1.40740
$$484$$ 1.00000 0.0454545
$$485$$ −2.24621 −0.101995
$$486$$ 5.75379 0.260997
$$487$$ −22.8769 −1.03665 −0.518326 0.855183i $$-0.673444\pi$$
−0.518326 + 0.855183i $$0.673444\pi$$
$$488$$ −10.2462 −0.463824
$$489$$ −20.8769 −0.944086
$$490$$ 11.3693 0.513613
$$491$$ −14.6307 −0.660273 −0.330137 0.943933i $$-0.607095\pi$$
−0.330137 + 0.943933i $$0.607095\pi$$
$$492$$ −3.12311 −0.140800
$$493$$ −23.8078 −1.07225
$$494$$ 0 0
$$495$$ −1.12311 −0.0504798
$$496$$ −2.00000 −0.0898027
$$497$$ −21.3693 −0.958545
$$498$$ −3.50758 −0.157178
$$499$$ 26.2462 1.17494 0.587471 0.809245i $$-0.300124\pi$$
0.587471 + 0.809245i $$0.300124\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 39.6155 1.76989
$$502$$ 8.87689 0.396195
$$503$$ 9.31534 0.415351 0.207675 0.978198i $$-0.433410\pi$$
0.207675 + 0.978198i $$0.433410\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ −30.2462 −1.34594
$$506$$ 5.56155 0.247241
$$507$$ −0.492423 −0.0218693
$$508$$ 14.2462 0.632073
$$509$$ 6.63068 0.293900 0.146950 0.989144i $$-0.453054\pi$$
0.146950 + 0.989144i $$0.453054\pi$$
$$510$$ 11.1231 0.492539
$$511$$ 9.56155 0.422978
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ −22.7386 −1.00198
$$516$$ 0 0
$$517$$ 8.00000 0.351840
$$518$$ −11.1231 −0.488721
$$519$$ −28.1080 −1.23380
$$520$$ 7.12311 0.312369
$$521$$ −1.12311 −0.0492042 −0.0246021 0.999697i $$-0.507832\pi$$
−0.0246021 + 0.999697i $$0.507832\pi$$
$$522$$ 3.75379 0.164299
$$523$$ 6.05398 0.264722 0.132361 0.991202i $$-0.457744\pi$$
0.132361 + 0.991202i $$0.457744\pi$$
$$524$$ −7.12311 −0.311174
$$525$$ −5.56155 −0.242726
$$526$$ 17.1231 0.746603
$$527$$ −7.12311 −0.310287
$$528$$ 1.56155 0.0679579
$$529$$ 7.93087 0.344820
$$530$$ 15.6155 0.678295
$$531$$ 2.63068 0.114162
$$532$$ 0 0
$$533$$ −7.12311 −0.308536
$$534$$ 14.2462 0.616494
$$535$$ −37.8617 −1.63691
$$536$$ −4.68466 −0.202346
$$537$$ −34.7386 −1.49908
$$538$$ −11.1231 −0.479551
$$539$$ 5.68466 0.244856
$$540$$ −11.1231 −0.478662
$$541$$ 43.2311 1.85865 0.929324 0.369265i $$-0.120391\pi$$
0.929324 + 0.369265i $$0.120391\pi$$
$$542$$ 13.3153 0.571943
$$543$$ 29.8617 1.28149
$$544$$ 3.56155 0.152700
$$545$$ 37.3693 1.60073
$$546$$ 19.8078 0.847694
$$547$$ −6.73863 −0.288123 −0.144062 0.989569i $$-0.546016\pi$$
−0.144062 + 0.989569i $$0.546016\pi$$
$$548$$ 8.43845 0.360473
$$549$$ 5.75379 0.245566
$$550$$ −1.00000 −0.0426401
$$551$$ 0 0
$$552$$ 8.68466 0.369644
$$553$$ 14.2462 0.605811
$$554$$ −24.4924 −1.04058
$$555$$ −9.75379 −0.414025
$$556$$ −17.3693 −0.736623
$$557$$ 8.87689 0.376126 0.188063 0.982157i $$-0.439779\pi$$
0.188063 + 0.982157i $$0.439779\pi$$
$$558$$ 1.12311 0.0475449
$$559$$ 0 0
$$560$$ 7.12311 0.301006
$$561$$ 5.56155 0.234809
$$562$$ 0.630683 0.0266038
$$563$$ −6.73863 −0.284000 −0.142000 0.989867i $$-0.545353\pi$$
−0.142000 + 0.989867i $$0.545353\pi$$
$$564$$ 12.4924 0.526026
$$565$$ −10.2462 −0.431061
$$566$$ −13.3693 −0.561954
$$567$$ −24.9309 −1.04700
$$568$$ −6.00000 −0.251754
$$569$$ −3.36932 −0.141249 −0.0706246 0.997503i $$-0.522499\pi$$
−0.0706246 + 0.997503i $$0.522499\pi$$
$$570$$ 0 0
$$571$$ 35.6155 1.49046 0.745232 0.666806i $$-0.232339\pi$$
0.745232 + 0.666806i $$0.232339\pi$$
$$572$$ 3.56155 0.148916
$$573$$ −32.3002 −1.34936
$$574$$ −7.12311 −0.297313
$$575$$ −5.56155 −0.231933
$$576$$ −0.561553 −0.0233980
$$577$$ 3.94602 0.164275 0.0821376 0.996621i $$-0.473825\pi$$
0.0821376 + 0.996621i $$0.473825\pi$$
$$578$$ −4.31534 −0.179495
$$579$$ 1.75379 0.0728850
$$580$$ −13.3693 −0.555131
$$581$$ −8.00000 −0.331896
$$582$$ −1.75379 −0.0726969
$$583$$ 7.80776 0.323365
$$584$$ 2.68466 0.111092
$$585$$ −4.00000 −0.165380
$$586$$ 24.9309 1.02988
$$587$$ −7.50758 −0.309871 −0.154935 0.987925i $$-0.549517\pi$$
−0.154935 + 0.987925i $$0.549517\pi$$
$$588$$ 8.87689 0.366077
$$589$$ 0 0
$$590$$ −9.36932 −0.385729
$$591$$ 2.73863 0.112652
$$592$$ −3.12311 −0.128359
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ −5.56155 −0.228193
$$595$$ 25.3693 1.04004
$$596$$ −15.1231 −0.619467
$$597$$ −1.06913 −0.0437566
$$598$$ 19.8078 0.810000
$$599$$ −39.8617 −1.62871 −0.814353 0.580370i $$-0.802908\pi$$
−0.814353 + 0.580370i $$0.802908\pi$$
$$600$$ −1.56155 −0.0637501
$$601$$ 3.36932 0.137437 0.0687187 0.997636i $$-0.478109\pi$$
0.0687187 + 0.997636i $$0.478109\pi$$
$$602$$ 0 0
$$603$$ 2.63068 0.107130
$$604$$ −4.00000 −0.162758
$$605$$ 2.00000 0.0813116
$$606$$ −23.6155 −0.959315
$$607$$ −27.6155 −1.12088 −0.560440 0.828195i $$-0.689368\pi$$
−0.560440 + 0.828195i $$0.689368\pi$$
$$608$$ 0 0
$$609$$ −37.1771 −1.50649
$$610$$ −20.4924 −0.829714
$$611$$ 28.4924 1.15268
$$612$$ −2.00000 −0.0808452
$$613$$ 1.75379 0.0708349 0.0354174 0.999373i $$-0.488724\pi$$
0.0354174 + 0.999373i $$0.488724\pi$$
$$614$$ 5.75379 0.232204
$$615$$ −6.24621 −0.251872
$$616$$ 3.56155 0.143499
$$617$$ −4.73863 −0.190770 −0.0953851 0.995440i $$-0.530408\pi$$
−0.0953851 + 0.995440i $$0.530408\pi$$
$$618$$ −17.7538 −0.714162
$$619$$ −26.2462 −1.05492 −0.527462 0.849579i $$-0.676856\pi$$
−0.527462 + 0.849579i $$0.676856\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ −30.9309 −1.24121
$$622$$ 15.8078 0.633834
$$623$$ 32.4924 1.30178
$$624$$ 5.56155 0.222640
$$625$$ −19.0000 −0.760000
$$626$$ 3.56155 0.142348
$$627$$ 0 0
$$628$$ 18.4924 0.737928
$$629$$ −11.1231 −0.443507
$$630$$ −4.00000 −0.159364
$$631$$ 36.4924 1.45274 0.726370 0.687304i $$-0.241206\pi$$
0.726370 + 0.687304i $$0.241206\pi$$
$$632$$ 4.00000 0.159111
$$633$$ 19.8078 0.787288
$$634$$ −30.0540 −1.19360
$$635$$ 28.4924 1.13069
$$636$$ 12.1922 0.483454
$$637$$ 20.2462 0.802184
$$638$$ −6.68466 −0.264648
$$639$$ 3.36932 0.133288
$$640$$ 2.00000 0.0790569
$$641$$ 9.61553 0.379791 0.189895 0.981804i $$-0.439185\pi$$
0.189895 + 0.981804i $$0.439185\pi$$
$$642$$ −29.5616 −1.16670
$$643$$ −37.3693 −1.47370 −0.736851 0.676055i $$-0.763688\pi$$
−0.736851 + 0.676055i $$0.763688\pi$$
$$644$$ 19.8078 0.780535
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 22.5464 0.886390 0.443195 0.896425i $$-0.353845\pi$$
0.443195 + 0.896425i $$0.353845\pi$$
$$648$$ −7.00000 −0.274986
$$649$$ −4.68466 −0.183889
$$650$$ −3.56155 −0.139696
$$651$$ −11.1231 −0.435949
$$652$$ −13.3693 −0.523583
$$653$$ 25.6155 1.00241 0.501207 0.865328i $$-0.332890\pi$$
0.501207 + 0.865328i $$0.332890\pi$$
$$654$$ 29.1771 1.14091
$$655$$ −14.2462 −0.556646
$$656$$ −2.00000 −0.0780869
$$657$$ −1.50758 −0.0588162
$$658$$ 28.4924 1.11075
$$659$$ −47.4233 −1.84735 −0.923675 0.383178i $$-0.874830\pi$$
−0.923675 + 0.383178i $$0.874830\pi$$
$$660$$ 3.12311 0.121567
$$661$$ 22.4384 0.872754 0.436377 0.899764i $$-0.356261\pi$$
0.436377 + 0.899764i $$0.356261\pi$$
$$662$$ 10.4384 0.405702
$$663$$ 19.8078 0.769270
$$664$$ −2.24621 −0.0871699
$$665$$ 0 0
$$666$$ 1.75379 0.0679579
$$667$$ −37.1771 −1.43950
$$668$$ 25.3693 0.981568
$$669$$ 38.6307 1.49355
$$670$$ −9.36932 −0.361968
$$671$$ −10.2462 −0.395551
$$672$$ 5.56155 0.214542
$$673$$ 18.8769 0.727651 0.363825 0.931467i $$-0.381470\pi$$
0.363825 + 0.931467i $$0.381470\pi$$
$$674$$ −2.87689 −0.110814
$$675$$ 5.56155 0.214064
$$676$$ −0.315342 −0.0121285
$$677$$ 39.1771 1.50570 0.752849 0.658194i $$-0.228679\pi$$
0.752849 + 0.658194i $$0.228679\pi$$
$$678$$ −8.00000 −0.307238
$$679$$ −4.00000 −0.153506
$$680$$ 7.12311 0.273159
$$681$$ 26.8229 1.02786
$$682$$ −2.00000 −0.0765840
$$683$$ 4.49242 0.171898 0.0859489 0.996300i $$-0.472608\pi$$
0.0859489 + 0.996300i $$0.472608\pi$$
$$684$$ 0 0
$$685$$ 16.8769 0.644833
$$686$$ −4.68466 −0.178861
$$687$$ 32.9848 1.25845
$$688$$ 0 0
$$689$$ 27.8078 1.05939
$$690$$ 17.3693 0.661239
$$691$$ −43.6155 −1.65921 −0.829606 0.558349i $$-0.811435\pi$$
−0.829606 + 0.558349i $$0.811435\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ −2.00000 −0.0759737
$$694$$ −31.6155 −1.20011
$$695$$ −34.7386 −1.31771
$$696$$ −10.4384 −0.395668
$$697$$ −7.12311 −0.269807
$$698$$ −19.6155 −0.742459
$$699$$ 32.3845 1.22489
$$700$$ −3.56155 −0.134614
$$701$$ 24.9848 0.943665 0.471832 0.881688i $$-0.343593\pi$$
0.471832 + 0.881688i $$0.343593\pi$$
$$702$$ −19.8078 −0.747596
$$703$$ 0 0
$$704$$ 1.00000 0.0376889
$$705$$ 24.9848 0.940984
$$706$$ −23.1771 −0.872281
$$707$$ −53.8617 −2.02568
$$708$$ −7.31534 −0.274927
$$709$$ −9.50758 −0.357065 −0.178532 0.983934i $$-0.557135\pi$$
−0.178532 + 0.983934i $$0.557135\pi$$
$$710$$ −12.0000 −0.450352
$$711$$ −2.24621 −0.0842395
$$712$$ 9.12311 0.341903
$$713$$ −11.1231 −0.416564
$$714$$ 19.8078 0.741287
$$715$$ 7.12311 0.266389
$$716$$ −22.2462 −0.831380
$$717$$ −7.69981 −0.287555
$$718$$ 12.9309 0.482576
$$719$$ −6.43845 −0.240114 −0.120057 0.992767i $$-0.538308\pi$$
−0.120057 + 0.992767i $$0.538308\pi$$
$$720$$ −1.12311 −0.0418557
$$721$$ −40.4924 −1.50802
$$722$$ 0 0
$$723$$ −33.7538 −1.25532
$$724$$ 19.1231 0.710705
$$725$$ 6.68466 0.248262
$$726$$ 1.56155 0.0579547
$$727$$ −39.4233 −1.46213 −0.731064 0.682308i $$-0.760976\pi$$
−0.731064 + 0.682308i $$0.760976\pi$$
$$728$$ 12.6847 0.470125
$$729$$ 29.9848 1.11055
$$730$$ 5.36932 0.198727
$$731$$ 0 0
$$732$$ −16.0000 −0.591377
$$733$$ −48.1080 −1.77691 −0.888454 0.458966i $$-0.848220\pi$$
−0.888454 + 0.458966i $$0.848220\pi$$
$$734$$ 13.7538 0.507662
$$735$$ 17.7538 0.654858
$$736$$ 5.56155 0.205002
$$737$$ −4.68466 −0.172562
$$738$$ 1.12311 0.0413421
$$739$$ 20.9848 0.771940 0.385970 0.922511i $$-0.373867\pi$$
0.385970 + 0.922511i $$0.373867\pi$$
$$740$$ −6.24621 −0.229615
$$741$$ 0 0
$$742$$ 27.8078 1.02086
$$743$$ 25.7538 0.944815 0.472407 0.881380i $$-0.343385\pi$$
0.472407 + 0.881380i $$0.343385\pi$$
$$744$$ −3.12311 −0.114499
$$745$$ −30.2462 −1.10814
$$746$$ −7.56155 −0.276848
$$747$$ 1.26137 0.0461510
$$748$$ 3.56155 0.130223
$$749$$ −67.4233 −2.46359
$$750$$ −18.7386 −0.684238
$$751$$ 41.2311 1.50454 0.752271 0.658853i $$-0.228958\pi$$
0.752271 + 0.658853i $$0.228958\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 13.8617 0.505150
$$754$$ −23.8078 −0.867028
$$755$$ −8.00000 −0.291150
$$756$$ −19.8078 −0.720401
$$757$$ −9.50758 −0.345559 −0.172779 0.984961i $$-0.555275\pi$$
−0.172779 + 0.984961i $$0.555275\pi$$
$$758$$ −2.43845 −0.0885684
$$759$$ 8.68466 0.315233
$$760$$ 0 0
$$761$$ −22.1922 −0.804468 −0.402234 0.915537i $$-0.631766\pi$$
−0.402234 + 0.915537i $$0.631766\pi$$
$$762$$ 22.2462 0.805895
$$763$$ 66.5464 2.40914
$$764$$ −20.6847 −0.748345
$$765$$ −4.00000 −0.144620
$$766$$ 2.00000 0.0722629
$$767$$ −16.6847 −0.602448
$$768$$ 1.56155 0.0563477
$$769$$ −1.31534 −0.0474324 −0.0237162 0.999719i $$-0.507550\pi$$
−0.0237162 + 0.999719i $$0.507550\pi$$
$$770$$ 7.12311 0.256699
$$771$$ 34.3542 1.23723
$$772$$ 1.12311 0.0404215
$$773$$ −9.06913 −0.326194 −0.163097 0.986610i $$-0.552148\pi$$
−0.163097 + 0.986610i $$0.552148\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ −1.12311 −0.0403171
$$777$$ −17.3693 −0.623121
$$778$$ 18.0000 0.645331
$$779$$ 0 0
$$780$$ 11.1231 0.398271
$$781$$ −6.00000 −0.214697
$$782$$ 19.8078 0.708324
$$783$$ 37.1771 1.32860
$$784$$ 5.68466 0.203024
$$785$$ 36.9848 1.32005
$$786$$ −11.1231 −0.396748
$$787$$ −3.31534 −0.118179 −0.0590896 0.998253i $$-0.518820\pi$$
−0.0590896 + 0.998253i $$0.518820\pi$$
$$788$$ 1.75379 0.0624761
$$789$$ 26.7386 0.951921
$$790$$ 8.00000 0.284627
$$791$$ −18.2462 −0.648761
$$792$$ −0.561553 −0.0199539
$$793$$ −36.4924 −1.29588
$$794$$ −26.9848 −0.957656
$$795$$ 24.3845 0.864828
$$796$$ −0.684658 −0.0242671
$$797$$ 35.8078 1.26838 0.634188 0.773179i $$-0.281335\pi$$
0.634188 + 0.773179i $$0.281335\pi$$
$$798$$ 0 0
$$799$$ 28.4924 1.00799
$$800$$ −1.00000 −0.0353553
$$801$$ −5.12311 −0.181016
$$802$$ 24.2462 0.856163
$$803$$ 2.68466 0.0947395
$$804$$ −7.31534 −0.257992
$$805$$ 39.6155 1.39626
$$806$$ −7.12311 −0.250901
$$807$$ −17.3693 −0.611429
$$808$$ −15.1231 −0.532029
$$809$$ 36.9309 1.29842 0.649210 0.760609i $$-0.275100\pi$$
0.649210 + 0.760609i $$0.275100\pi$$
$$810$$ −14.0000 −0.491910
$$811$$ −20.6847 −0.726337 −0.363168 0.931724i $$-0.618305\pi$$
−0.363168 + 0.931724i $$0.618305\pi$$
$$812$$ −23.8078 −0.835489
$$813$$ 20.7926 0.729229
$$814$$ −3.12311 −0.109465
$$815$$ −26.7386 −0.936613
$$816$$ 5.56155 0.194693
$$817$$ 0 0
$$818$$ 0.246211 0.00860857
$$819$$ −7.12311 −0.248901
$$820$$ −4.00000 −0.139686
$$821$$ 44.9848 1.56998 0.784991 0.619507i $$-0.212668\pi$$
0.784991 + 0.619507i $$0.212668\pi$$
$$822$$ 13.1771 0.459603
$$823$$ 14.9309 0.520457 0.260229 0.965547i $$-0.416202\pi$$
0.260229 + 0.965547i $$0.416202\pi$$
$$824$$ −11.3693 −0.396069
$$825$$ −1.56155 −0.0543663
$$826$$ −16.6847 −0.580534
$$827$$ −21.0691 −0.732645 −0.366323 0.930488i $$-0.619383\pi$$
−0.366323 + 0.930488i $$0.619383\pi$$
$$828$$ −3.12311 −0.108535
$$829$$ 33.1771 1.15229 0.576144 0.817348i $$-0.304557\pi$$
0.576144 + 0.817348i $$0.304557\pi$$
$$830$$ −4.49242 −0.155934
$$831$$ −38.2462 −1.32675
$$832$$ 3.56155 0.123475
$$833$$ 20.2462 0.701490
$$834$$ −27.1231 −0.939196
$$835$$ 50.7386 1.75588
$$836$$ 0 0
$$837$$ 11.1231 0.384471
$$838$$ −23.1231 −0.798774
$$839$$ −1.12311 −0.0387739 −0.0193870 0.999812i $$-0.506171\pi$$
−0.0193870 + 0.999812i $$0.506171\pi$$
$$840$$ 11.1231 0.383784
$$841$$ 15.6847 0.540850
$$842$$ 1.06913 0.0368447
$$843$$ 0.984845 0.0339199
$$844$$ 12.6847 0.436624
$$845$$ −0.630683 −0.0216962
$$846$$ −4.49242 −0.154453
$$847$$ 3.56155 0.122376
$$848$$ 7.80776 0.268120
$$849$$ −20.8769 −0.716493
$$850$$ −3.56155 −0.122160
$$851$$ −17.3693 −0.595413
$$852$$ −9.36932 −0.320988
$$853$$ 38.3542 1.31322 0.656611 0.754230i $$-0.271989\pi$$
0.656611 + 0.754230i $$0.271989\pi$$
$$854$$ −36.4924 −1.24874
$$855$$ 0 0
$$856$$ −18.9309 −0.647044
$$857$$ 10.0000 0.341593 0.170797 0.985306i $$-0.445366\pi$$
0.170797 + 0.985306i $$0.445366\pi$$
$$858$$ 5.56155 0.189868
$$859$$ −41.8617 −1.42830 −0.714152 0.699991i $$-0.753188\pi$$
−0.714152 + 0.699991i $$0.753188\pi$$
$$860$$ 0 0
$$861$$ −11.1231 −0.379074
$$862$$ 25.8617 0.880854
$$863$$ 16.2462 0.553027 0.276514 0.961010i $$-0.410821\pi$$
0.276514 + 0.961010i $$0.410821\pi$$
$$864$$ −5.56155 −0.189208
$$865$$ −36.0000 −1.22404
$$866$$ −31.3693 −1.06597
$$867$$ −6.73863 −0.228856
$$868$$ −7.12311 −0.241774
$$869$$ 4.00000 0.135691
$$870$$ −20.8769 −0.707793
$$871$$ −16.6847 −0.565338
$$872$$ 18.6847 0.632742
$$873$$ 0.630683 0.0213454
$$874$$ 0 0
$$875$$ −42.7386 −1.44483
$$876$$ 4.19224 0.141643
$$877$$ 40.4384 1.36551 0.682755 0.730648i $$-0.260782\pi$$
0.682755 + 0.730648i $$0.260782\pi$$
$$878$$ −31.6155 −1.06697
$$879$$ 38.9309 1.31311
$$880$$ 2.00000 0.0674200
$$881$$ −28.7386 −0.968229 −0.484115 0.875005i $$-0.660858\pi$$
−0.484115 + 0.875005i $$0.660858\pi$$
$$882$$ −3.19224 −0.107488
$$883$$ −54.7386 −1.84210 −0.921051 0.389442i $$-0.872668\pi$$
−0.921051 + 0.389442i $$0.872668\pi$$
$$884$$ 12.6847 0.426631
$$885$$ −14.6307 −0.491805
$$886$$ −17.8617 −0.600077
$$887$$ 32.0000 1.07445 0.537227 0.843437i $$-0.319472\pi$$
0.537227 + 0.843437i $$0.319472\pi$$
$$888$$ −4.87689 −0.163658
$$889$$ 50.7386 1.70172
$$890$$ 18.2462 0.611614
$$891$$ −7.00000 −0.234509
$$892$$ 24.7386 0.828311
$$893$$ 0 0
$$894$$ −23.6155 −0.789821
$$895$$ −44.4924 −1.48722
$$896$$ 3.56155 0.118983
$$897$$ 30.9309 1.03275
$$898$$ 17.6155 0.587838
$$899$$ 13.3693 0.445892
$$900$$ 0.561553 0.0187184
$$901$$ 27.8078 0.926411
$$902$$ −2.00000 −0.0665927
$$903$$ 0 0
$$904$$ −5.12311 −0.170392
$$905$$ 38.2462 1.27135
$$906$$ −6.24621 −0.207516
$$907$$ −56.7926 −1.88577 −0.942884 0.333122i $$-0.891898\pi$$
−0.942884 + 0.333122i $$0.891898\pi$$
$$908$$ 17.1771 0.570041
$$909$$ 8.49242 0.281676
$$910$$ 25.3693 0.840985
$$911$$ 46.9848 1.55668 0.778339 0.627845i $$-0.216063\pi$$
0.778339 + 0.627845i $$0.216063\pi$$
$$912$$ 0 0
$$913$$ −2.24621 −0.0743387
$$914$$ 32.4384 1.07297
$$915$$ −32.0000 −1.05789
$$916$$ 21.1231 0.697927
$$917$$ −25.3693 −0.837769
$$918$$ −19.8078 −0.653754
$$919$$ 28.4384 0.938098 0.469049 0.883172i $$-0.344597\pi$$
0.469049 + 0.883172i $$0.344597\pi$$
$$920$$ 11.1231 0.366718
$$921$$ 8.98485 0.296061
$$922$$ −22.7386 −0.748857
$$923$$ −21.3693 −0.703380
$$924$$ 5.56155 0.182962
$$925$$ 3.12311 0.102687
$$926$$ 36.4924 1.19922
$$927$$ 6.38447 0.209694
$$928$$ −6.68466 −0.219435
$$929$$ −16.9309 −0.555484 −0.277742 0.960656i $$-0.589586\pi$$
−0.277742 + 0.960656i $$0.589586\pi$$
$$930$$ −6.24621 −0.204821
$$931$$ 0 0
$$932$$ 20.7386 0.679317
$$933$$ 24.6847 0.808139
$$934$$ 26.2462 0.858802
$$935$$ 7.12311 0.232950
$$936$$ −2.00000 −0.0653720
$$937$$ −54.3002 −1.77391 −0.886955 0.461856i $$-0.847184\pi$$
−0.886955 + 0.461856i $$0.847184\pi$$
$$938$$ −16.6847 −0.544773
$$939$$ 5.56155 0.181494
$$940$$ 16.0000 0.521862
$$941$$ −20.4384 −0.666274 −0.333137 0.942878i $$-0.608107\pi$$
−0.333137 + 0.942878i $$0.608107\pi$$
$$942$$ 28.8769 0.940860
$$943$$ −11.1231 −0.362218
$$944$$ −4.68466 −0.152473
$$945$$ −39.6155 −1.28869
$$946$$ 0 0
$$947$$ 28.9848 0.941881 0.470940 0.882165i $$-0.343915\pi$$
0.470940 + 0.882165i $$0.343915\pi$$
$$948$$ 6.24621 0.202868
$$949$$ 9.56155 0.310381
$$950$$ 0 0
$$951$$ −46.9309 −1.52184
$$952$$ 12.6847 0.411112
$$953$$ 9.12311 0.295526 0.147763 0.989023i $$-0.452793\pi$$
0.147763 + 0.989023i $$0.452793\pi$$
$$954$$ −4.38447 −0.141953
$$955$$ −41.3693 −1.33868
$$956$$ −4.93087 −0.159476
$$957$$ −10.4384 −0.337427
$$958$$ 11.3693 0.367326
$$959$$ 30.0540 0.970493
$$960$$ 3.12311 0.100798
$$961$$ −27.0000 −0.870968
$$962$$ −11.1231 −0.358623
$$963$$ 10.6307 0.342569
$$964$$ −21.6155 −0.696189
$$965$$ 2.24621 0.0723081
$$966$$ 30.9309 0.995184
$$967$$ −3.86174 −0.124185 −0.0620926 0.998070i $$-0.519777\pi$$
−0.0620926 + 0.998070i $$0.519777\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ −2.24621 −0.0721215
$$971$$ 14.2462 0.457183 0.228591 0.973522i $$-0.426588\pi$$
0.228591 + 0.973522i $$0.426588\pi$$
$$972$$ 5.75379 0.184553
$$973$$ −61.8617 −1.98320
$$974$$ −22.8769 −0.733023
$$975$$ −5.56155 −0.178112
$$976$$ −10.2462 −0.327973
$$977$$ −55.3693 −1.77142 −0.885711 0.464238i $$-0.846328\pi$$
−0.885711 + 0.464238i $$0.846328\pi$$
$$978$$ −20.8769 −0.667569
$$979$$ 9.12311 0.291576
$$980$$ 11.3693 0.363180
$$981$$ −10.4924 −0.334997
$$982$$ −14.6307 −0.466884
$$983$$ 48.2462 1.53882 0.769408 0.638758i $$-0.220552\pi$$
0.769408 + 0.638758i $$0.220552\pi$$
$$984$$ −3.12311 −0.0995610
$$985$$ 3.50758 0.111761
$$986$$ −23.8078 −0.758194
$$987$$ 44.4924 1.41621
$$988$$ 0 0
$$989$$ 0 0
$$990$$ −1.12311 −0.0356946
$$991$$ −21.1231 −0.670998 −0.335499 0.942041i $$-0.608905\pi$$
−0.335499 + 0.942041i $$0.608905\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 16.3002 0.517271
$$994$$ −21.3693 −0.677794
$$995$$ −1.36932 −0.0434103
$$996$$ −3.50758 −0.111142
$$997$$ −53.3693 −1.69022 −0.845112 0.534590i $$-0.820466\pi$$
−0.845112 + 0.534590i $$0.820466\pi$$
$$998$$ 26.2462 0.830809
$$999$$ 17.3693 0.549541
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 7942.2.a.x.1.2 2
19.18 odd 2 418.2.a.e.1.1 2
57.56 even 2 3762.2.a.y.1.2 2
76.75 even 2 3344.2.a.k.1.2 2
209.208 even 2 4598.2.a.bj.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.1 2 19.18 odd 2
3344.2.a.k.1.2 2 76.75 even 2
3762.2.a.y.1.2 2 57.56 even 2
4598.2.a.bj.1.1 2 209.208 even 2
7942.2.a.x.1.2 2 1.1 even 1 trivial