# Properties

 Label 145.2.a.b.1.1 Level $145$ Weight $2$ Character 145.1 Self dual yes Analytic conductor $1.158$ Analytic rank $1$ 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] = [145,2,Mod(1,145)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(145, 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("145.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$145 = 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 145.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: $$1.15783082931$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 145.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.41421 q^{2} -2.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} +4.82843 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.41421 q^{2} -2.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} +4.82843 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} -2.41421 q^{10} -4.82843 q^{11} -7.65685 q^{12} -2.00000 q^{13} -2.00000 q^{14} -2.00000 q^{15} +3.00000 q^{16} -2.82843 q^{17} -2.41421 q^{18} +0.828427 q^{19} +3.82843 q^{20} -1.65685 q^{21} +11.6569 q^{22} -8.82843 q^{23} +8.82843 q^{24} +1.00000 q^{25} +4.82843 q^{26} +4.00000 q^{27} +3.17157 q^{28} +1.00000 q^{29} +4.82843 q^{30} -10.4853 q^{31} +1.58579 q^{32} +9.65685 q^{33} +6.82843 q^{34} +0.828427 q^{35} +3.82843 q^{36} +8.48528 q^{37} -2.00000 q^{38} +4.00000 q^{39} -4.41421 q^{40} -6.00000 q^{41} +4.00000 q^{42} -6.00000 q^{43} -18.4853 q^{44} +1.00000 q^{45} +21.3137 q^{46} -0.343146 q^{47} -6.00000 q^{48} -6.31371 q^{49} -2.41421 q^{50} +5.65685 q^{51} -7.65685 q^{52} +7.65685 q^{53} -9.65685 q^{54} -4.82843 q^{55} -3.65685 q^{56} -1.65685 q^{57} -2.41421 q^{58} -7.65685 q^{60} +7.65685 q^{61} +25.3137 q^{62} +0.828427 q^{63} -9.82843 q^{64} -2.00000 q^{65} -23.3137 q^{66} -10.4853 q^{67} -10.8284 q^{68} +17.6569 q^{69} -2.00000 q^{70} +7.31371 q^{71} -4.41421 q^{72} -8.48528 q^{73} -20.4853 q^{74} -2.00000 q^{75} +3.17157 q^{76} -4.00000 q^{77} -9.65685 q^{78} +14.4853 q^{79} +3.00000 q^{80} -11.0000 q^{81} +14.4853 q^{82} +12.8284 q^{83} -6.34315 q^{84} -2.82843 q^{85} +14.4853 q^{86} -2.00000 q^{87} +21.3137 q^{88} +3.65685 q^{89} -2.41421 q^{90} -1.65685 q^{91} -33.7990 q^{92} +20.9706 q^{93} +0.828427 q^{94} +0.828427 q^{95} -3.17157 q^{96} +4.48528 q^{97} +15.2426 q^{98} -4.82843 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 4 q^{12} - 4 q^{13} - 4 q^{14} - 4 q^{15} + 6 q^{16} - 2 q^{18} - 4 q^{19} + 2 q^{20} + 8 q^{21} + 12 q^{22} - 12 q^{23} + 12 q^{24} + 2 q^{25} + 4 q^{26} + 8 q^{27} + 12 q^{28} + 2 q^{29} + 4 q^{30} - 4 q^{31} + 6 q^{32} + 8 q^{33} + 8 q^{34} - 4 q^{35} + 2 q^{36} - 4 q^{38} + 8 q^{39} - 6 q^{40} - 12 q^{41} + 8 q^{42} - 12 q^{43} - 20 q^{44} + 2 q^{45} + 20 q^{46} - 12 q^{47} - 12 q^{48} + 10 q^{49} - 2 q^{50} - 4 q^{52} + 4 q^{53} - 8 q^{54} - 4 q^{55} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 4 q^{60} + 4 q^{61} + 28 q^{62} - 4 q^{63} - 14 q^{64} - 4 q^{65} - 24 q^{66} - 4 q^{67} - 16 q^{68} + 24 q^{69} - 4 q^{70} - 8 q^{71} - 6 q^{72} - 24 q^{74} - 4 q^{75} + 12 q^{76} - 8 q^{77} - 8 q^{78} + 12 q^{79} + 6 q^{80} - 22 q^{81} + 12 q^{82} + 20 q^{83} - 24 q^{84} + 12 q^{86} - 4 q^{87} + 20 q^{88} - 4 q^{89} - 2 q^{90} + 8 q^{91} - 28 q^{92} + 8 q^{93} - 4 q^{94} - 4 q^{95} - 12 q^{96} - 8 q^{97} + 22 q^{98} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 4 * q^12 - 4 * q^13 - 4 * q^14 - 4 * q^15 + 6 * q^16 - 2 * q^18 - 4 * q^19 + 2 * q^20 + 8 * q^21 + 12 * q^22 - 12 * q^23 + 12 * q^24 + 2 * q^25 + 4 * q^26 + 8 * q^27 + 12 * q^28 + 2 * q^29 + 4 * q^30 - 4 * q^31 + 6 * q^32 + 8 * q^33 + 8 * q^34 - 4 * q^35 + 2 * q^36 - 4 * q^38 + 8 * q^39 - 6 * q^40 - 12 * q^41 + 8 * q^42 - 12 * q^43 - 20 * q^44 + 2 * q^45 + 20 * q^46 - 12 * q^47 - 12 * q^48 + 10 * q^49 - 2 * q^50 - 4 * q^52 + 4 * q^53 - 8 * q^54 - 4 * q^55 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 4 * q^60 + 4 * q^61 + 28 * q^62 - 4 * q^63 - 14 * q^64 - 4 * q^65 - 24 * q^66 - 4 * q^67 - 16 * q^68 + 24 * q^69 - 4 * q^70 - 8 * q^71 - 6 * q^72 - 24 * q^74 - 4 * q^75 + 12 * q^76 - 8 * q^77 - 8 * q^78 + 12 * q^79 + 6 * q^80 - 22 * q^81 + 12 * q^82 + 20 * q^83 - 24 * q^84 + 12 * q^86 - 4 * q^87 + 20 * q^88 - 4 * q^89 - 2 * q^90 + 8 * q^91 - 28 * q^92 + 8 * q^93 - 4 * q^94 - 4 * q^95 - 12 * q^96 - 8 * q^97 + 22 * q^98 - 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.41421 −1.70711 −0.853553 0.521005i $$-0.825557\pi$$
−0.853553 + 0.521005i $$0.825557\pi$$
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 3.82843 1.91421
$$5$$ 1.00000 0.447214
$$6$$ 4.82843 1.97120
$$7$$ 0.828427 0.313116 0.156558 0.987669i $$-0.449960\pi$$
0.156558 + 0.987669i $$0.449960\pi$$
$$8$$ −4.41421 −1.56066
$$9$$ 1.00000 0.333333
$$10$$ −2.41421 −0.763441
$$11$$ −4.82843 −1.45583 −0.727913 0.685670i $$-0.759509\pi$$
−0.727913 + 0.685670i $$0.759509\pi$$
$$12$$ −7.65685 −2.21034
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ −2.00000 −0.516398
$$16$$ 3.00000 0.750000
$$17$$ −2.82843 −0.685994 −0.342997 0.939336i $$-0.611442\pi$$
−0.342997 + 0.939336i $$0.611442\pi$$
$$18$$ −2.41421 −0.569036
$$19$$ 0.828427 0.190054 0.0950271 0.995475i $$-0.469706\pi$$
0.0950271 + 0.995475i $$0.469706\pi$$
$$20$$ 3.82843 0.856062
$$21$$ −1.65685 −0.361555
$$22$$ 11.6569 2.48525
$$23$$ −8.82843 −1.84085 −0.920427 0.390914i $$-0.872159\pi$$
−0.920427 + 0.390914i $$0.872159\pi$$
$$24$$ 8.82843 1.80210
$$25$$ 1.00000 0.200000
$$26$$ 4.82843 0.946932
$$27$$ 4.00000 0.769800
$$28$$ 3.17157 0.599371
$$29$$ 1.00000 0.185695
$$30$$ 4.82843 0.881546
$$31$$ −10.4853 −1.88321 −0.941606 0.336717i $$-0.890684\pi$$
−0.941606 + 0.336717i $$0.890684\pi$$
$$32$$ 1.58579 0.280330
$$33$$ 9.65685 1.68104
$$34$$ 6.82843 1.17107
$$35$$ 0.828427 0.140030
$$36$$ 3.82843 0.638071
$$37$$ 8.48528 1.39497 0.697486 0.716599i $$-0.254302\pi$$
0.697486 + 0.716599i $$0.254302\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 4.00000 0.640513
$$40$$ −4.41421 −0.697948
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 4.00000 0.617213
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ −18.4853 −2.78676
$$45$$ 1.00000 0.149071
$$46$$ 21.3137 3.14253
$$47$$ −0.343146 −0.0500530 −0.0250265 0.999687i $$-0.507967\pi$$
−0.0250265 + 0.999687i $$0.507967\pi$$
$$48$$ −6.00000 −0.866025
$$49$$ −6.31371 −0.901958
$$50$$ −2.41421 −0.341421
$$51$$ 5.65685 0.792118
$$52$$ −7.65685 −1.06181
$$53$$ 7.65685 1.05175 0.525875 0.850562i $$-0.323738\pi$$
0.525875 + 0.850562i $$0.323738\pi$$
$$54$$ −9.65685 −1.31413
$$55$$ −4.82843 −0.651065
$$56$$ −3.65685 −0.488668
$$57$$ −1.65685 −0.219456
$$58$$ −2.41421 −0.317002
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ −7.65685 −0.988496
$$61$$ 7.65685 0.980360 0.490180 0.871621i $$-0.336931\pi$$
0.490180 + 0.871621i $$0.336931\pi$$
$$62$$ 25.3137 3.21484
$$63$$ 0.828427 0.104372
$$64$$ −9.82843 −1.22855
$$65$$ −2.00000 −0.248069
$$66$$ −23.3137 −2.86972
$$67$$ −10.4853 −1.28098 −0.640490 0.767966i $$-0.721269\pi$$
−0.640490 + 0.767966i $$0.721269\pi$$
$$68$$ −10.8284 −1.31314
$$69$$ 17.6569 2.12564
$$70$$ −2.00000 −0.239046
$$71$$ 7.31371 0.867978 0.433989 0.900918i $$-0.357106\pi$$
0.433989 + 0.900918i $$0.357106\pi$$
$$72$$ −4.41421 −0.520220
$$73$$ −8.48528 −0.993127 −0.496564 0.868000i $$-0.665405\pi$$
−0.496564 + 0.868000i $$0.665405\pi$$
$$74$$ −20.4853 −2.38137
$$75$$ −2.00000 −0.230940
$$76$$ 3.17157 0.363804
$$77$$ −4.00000 −0.455842
$$78$$ −9.65685 −1.09342
$$79$$ 14.4853 1.62972 0.814861 0.579657i $$-0.196813\pi$$
0.814861 + 0.579657i $$0.196813\pi$$
$$80$$ 3.00000 0.335410
$$81$$ −11.0000 −1.22222
$$82$$ 14.4853 1.59963
$$83$$ 12.8284 1.40810 0.704051 0.710149i $$-0.251372\pi$$
0.704051 + 0.710149i $$0.251372\pi$$
$$84$$ −6.34315 −0.692094
$$85$$ −2.82843 −0.306786
$$86$$ 14.4853 1.56199
$$87$$ −2.00000 −0.214423
$$88$$ 21.3137 2.27205
$$89$$ 3.65685 0.387626 0.193813 0.981039i $$-0.437915\pi$$
0.193813 + 0.981039i $$0.437915\pi$$
$$90$$ −2.41421 −0.254480
$$91$$ −1.65685 −0.173686
$$92$$ −33.7990 −3.52379
$$93$$ 20.9706 2.17455
$$94$$ 0.828427 0.0854457
$$95$$ 0.828427 0.0849948
$$96$$ −3.17157 −0.323697
$$97$$ 4.48528 0.455411 0.227706 0.973730i $$-0.426878\pi$$
0.227706 + 0.973730i $$0.426878\pi$$
$$98$$ 15.2426 1.53974
$$99$$ −4.82843 −0.485275
$$100$$ 3.82843 0.382843
$$101$$ 4.34315 0.432159 0.216080 0.976376i $$-0.430673\pi$$
0.216080 + 0.976376i $$0.430673\pi$$
$$102$$ −13.6569 −1.35223
$$103$$ −12.1421 −1.19640 −0.598200 0.801347i $$-0.704117\pi$$
−0.598200 + 0.801347i $$0.704117\pi$$
$$104$$ 8.82843 0.865699
$$105$$ −1.65685 −0.161692
$$106$$ −18.4853 −1.79545
$$107$$ −8.14214 −0.787130 −0.393565 0.919297i $$-0.628758\pi$$
−0.393565 + 0.919297i $$0.628758\pi$$
$$108$$ 15.3137 1.47356
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 11.6569 1.11144
$$111$$ −16.9706 −1.61077
$$112$$ 2.48528 0.234837
$$113$$ 2.82843 0.266076 0.133038 0.991111i $$-0.457527\pi$$
0.133038 + 0.991111i $$0.457527\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −8.82843 −0.823255
$$116$$ 3.82843 0.355461
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ −2.34315 −0.214796
$$120$$ 8.82843 0.805921
$$121$$ 12.3137 1.11943
$$122$$ −18.4853 −1.67358
$$123$$ 12.0000 1.08200
$$124$$ −40.1421 −3.60487
$$125$$ 1.00000 0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ 20.5563 1.81694
$$129$$ 12.0000 1.05654
$$130$$ 4.82843 0.423481
$$131$$ 16.1421 1.41034 0.705172 0.709036i $$-0.250870\pi$$
0.705172 + 0.709036i $$0.250870\pi$$
$$132$$ 36.9706 3.21787
$$133$$ 0.686292 0.0595090
$$134$$ 25.3137 2.18677
$$135$$ 4.00000 0.344265
$$136$$ 12.4853 1.07060
$$137$$ −10.8284 −0.925135 −0.462567 0.886584i $$-0.653072\pi$$
−0.462567 + 0.886584i $$0.653072\pi$$
$$138$$ −42.6274 −3.62869
$$139$$ 10.3431 0.877294 0.438647 0.898659i $$-0.355458\pi$$
0.438647 + 0.898659i $$0.355458\pi$$
$$140$$ 3.17157 0.268047
$$141$$ 0.686292 0.0577962
$$142$$ −17.6569 −1.48173
$$143$$ 9.65685 0.807547
$$144$$ 3.00000 0.250000
$$145$$ 1.00000 0.0830455
$$146$$ 20.4853 1.69537
$$147$$ 12.6274 1.04149
$$148$$ 32.4853 2.67027
$$149$$ −13.3137 −1.09070 −0.545351 0.838208i $$-0.683604\pi$$
−0.545351 + 0.838208i $$0.683604\pi$$
$$150$$ 4.82843 0.394239
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ −3.65685 −0.296610
$$153$$ −2.82843 −0.228665
$$154$$ 9.65685 0.778171
$$155$$ −10.4853 −0.842198
$$156$$ 15.3137 1.22608
$$157$$ −16.4853 −1.31567 −0.657834 0.753163i $$-0.728527\pi$$
−0.657834 + 0.753163i $$0.728527\pi$$
$$158$$ −34.9706 −2.78211
$$159$$ −15.3137 −1.21446
$$160$$ 1.58579 0.125367
$$161$$ −7.31371 −0.576401
$$162$$ 26.5563 2.08646
$$163$$ −19.6569 −1.53964 −0.769822 0.638259i $$-0.779655\pi$$
−0.769822 + 0.638259i $$0.779655\pi$$
$$164$$ −22.9706 −1.79370
$$165$$ 9.65685 0.751785
$$166$$ −30.9706 −2.40378
$$167$$ 14.4853 1.12090 0.560452 0.828187i $$-0.310627\pi$$
0.560452 + 0.828187i $$0.310627\pi$$
$$168$$ 7.31371 0.564265
$$169$$ −9.00000 −0.692308
$$170$$ 6.82843 0.523716
$$171$$ 0.828427 0.0633514
$$172$$ −22.9706 −1.75149
$$173$$ −5.31371 −0.403994 −0.201997 0.979386i $$-0.564743\pi$$
−0.201997 + 0.979386i $$0.564743\pi$$
$$174$$ 4.82843 0.366042
$$175$$ 0.828427 0.0626232
$$176$$ −14.4853 −1.09187
$$177$$ 0 0
$$178$$ −8.82843 −0.661719
$$179$$ −0.686292 −0.0512958 −0.0256479 0.999671i $$-0.508165\pi$$
−0.0256479 + 0.999671i $$0.508165\pi$$
$$180$$ 3.82843 0.285354
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 4.00000 0.296500
$$183$$ −15.3137 −1.13202
$$184$$ 38.9706 2.87295
$$185$$ 8.48528 0.623850
$$186$$ −50.6274 −3.71218
$$187$$ 13.6569 0.998688
$$188$$ −1.31371 −0.0958120
$$189$$ 3.31371 0.241037
$$190$$ −2.00000 −0.145095
$$191$$ −15.1716 −1.09778 −0.548888 0.835896i $$-0.684949\pi$$
−0.548888 + 0.835896i $$0.684949\pi$$
$$192$$ 19.6569 1.41861
$$193$$ −12.4853 −0.898710 −0.449355 0.893353i $$-0.648346\pi$$
−0.449355 + 0.893353i $$0.648346\pi$$
$$194$$ −10.8284 −0.777436
$$195$$ 4.00000 0.286446
$$196$$ −24.1716 −1.72654
$$197$$ −8.34315 −0.594425 −0.297212 0.954811i $$-0.596057\pi$$
−0.297212 + 0.954811i $$0.596057\pi$$
$$198$$ 11.6569 0.828417
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ −4.41421 −0.312132
$$201$$ 20.9706 1.47915
$$202$$ −10.4853 −0.737742
$$203$$ 0.828427 0.0581442
$$204$$ 21.6569 1.51628
$$205$$ −6.00000 −0.419058
$$206$$ 29.3137 2.04238
$$207$$ −8.82843 −0.613618
$$208$$ −6.00000 −0.416025
$$209$$ −4.00000 −0.276686
$$210$$ 4.00000 0.276026
$$211$$ 4.82843 0.332403 0.166201 0.986092i $$-0.446850\pi$$
0.166201 + 0.986092i $$0.446850\pi$$
$$212$$ 29.3137 2.01327
$$213$$ −14.6274 −1.00225
$$214$$ 19.6569 1.34371
$$215$$ −6.00000 −0.409197
$$216$$ −17.6569 −1.20140
$$217$$ −8.68629 −0.589664
$$218$$ −4.82843 −0.327022
$$219$$ 16.9706 1.14676
$$220$$ −18.4853 −1.24628
$$221$$ 5.65685 0.380521
$$222$$ 40.9706 2.74976
$$223$$ 21.7990 1.45977 0.729884 0.683571i $$-0.239574\pi$$
0.729884 + 0.683571i $$0.239574\pi$$
$$224$$ 1.31371 0.0877758
$$225$$ 1.00000 0.0666667
$$226$$ −6.82843 −0.454220
$$227$$ −8.14214 −0.540413 −0.270206 0.962802i $$-0.587092\pi$$
−0.270206 + 0.962802i $$0.587092\pi$$
$$228$$ −6.34315 −0.420085
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 21.3137 1.40538
$$231$$ 8.00000 0.526361
$$232$$ −4.41421 −0.289807
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 4.82843 0.315644
$$235$$ −0.343146 −0.0223844
$$236$$ 0 0
$$237$$ −28.9706 −1.88184
$$238$$ 5.65685 0.366679
$$239$$ −23.3137 −1.50804 −0.754019 0.656852i $$-0.771887\pi$$
−0.754019 + 0.656852i $$0.771887\pi$$
$$240$$ −6.00000 −0.387298
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −29.7279 −1.91098
$$243$$ 10.0000 0.641500
$$244$$ 29.3137 1.87662
$$245$$ −6.31371 −0.403368
$$246$$ −28.9706 −1.84710
$$247$$ −1.65685 −0.105423
$$248$$ 46.2843 2.93905
$$249$$ −25.6569 −1.62594
$$250$$ −2.41421 −0.152688
$$251$$ 3.17157 0.200188 0.100094 0.994978i $$-0.468086\pi$$
0.100094 + 0.994978i $$0.468086\pi$$
$$252$$ 3.17157 0.199790
$$253$$ 42.6274 2.67996
$$254$$ −14.4853 −0.908887
$$255$$ 5.65685 0.354246
$$256$$ −29.9706 −1.87316
$$257$$ 29.3137 1.82854 0.914269 0.405107i $$-0.132766\pi$$
0.914269 + 0.405107i $$0.132766\pi$$
$$258$$ −28.9706 −1.80363
$$259$$ 7.02944 0.436788
$$260$$ −7.65685 −0.474858
$$261$$ 1.00000 0.0618984
$$262$$ −38.9706 −2.40761
$$263$$ −8.34315 −0.514460 −0.257230 0.966350i $$-0.582810\pi$$
−0.257230 + 0.966350i $$0.582810\pi$$
$$264$$ −42.6274 −2.62354
$$265$$ 7.65685 0.470357
$$266$$ −1.65685 −0.101588
$$267$$ −7.31371 −0.447592
$$268$$ −40.1421 −2.45207
$$269$$ 1.31371 0.0800982 0.0400491 0.999198i $$-0.487249\pi$$
0.0400491 + 0.999198i $$0.487249\pi$$
$$270$$ −9.65685 −0.587697
$$271$$ 29.7990 1.81016 0.905080 0.425242i $$-0.139811\pi$$
0.905080 + 0.425242i $$0.139811\pi$$
$$272$$ −8.48528 −0.514496
$$273$$ 3.31371 0.200555
$$274$$ 26.1421 1.57930
$$275$$ −4.82843 −0.291165
$$276$$ 67.5980 4.06892
$$277$$ 7.65685 0.460056 0.230028 0.973184i $$-0.426118\pi$$
0.230028 + 0.973184i $$0.426118\pi$$
$$278$$ −24.9706 −1.49763
$$279$$ −10.4853 −0.627737
$$280$$ −3.65685 −0.218539
$$281$$ −6.68629 −0.398871 −0.199435 0.979911i $$-0.563911\pi$$
−0.199435 + 0.979911i $$0.563911\pi$$
$$282$$ −1.65685 −0.0986642
$$283$$ −0.828427 −0.0492449 −0.0246224 0.999697i $$-0.507838\pi$$
−0.0246224 + 0.999697i $$0.507838\pi$$
$$284$$ 28.0000 1.66149
$$285$$ −1.65685 −0.0981436
$$286$$ −23.3137 −1.37857
$$287$$ −4.97056 −0.293403
$$288$$ 1.58579 0.0934434
$$289$$ −9.00000 −0.529412
$$290$$ −2.41421 −0.141768
$$291$$ −8.97056 −0.525864
$$292$$ −32.4853 −1.90106
$$293$$ −8.48528 −0.495715 −0.247858 0.968796i $$-0.579727\pi$$
−0.247858 + 0.968796i $$0.579727\pi$$
$$294$$ −30.4853 −1.77794
$$295$$ 0 0
$$296$$ −37.4558 −2.17708
$$297$$ −19.3137 −1.12070
$$298$$ 32.1421 1.86194
$$299$$ 17.6569 1.02112
$$300$$ −7.65685 −0.442069
$$301$$ −4.97056 −0.286498
$$302$$ 28.9706 1.66707
$$303$$ −8.68629 −0.499014
$$304$$ 2.48528 0.142541
$$305$$ 7.65685 0.438430
$$306$$ 6.82843 0.390355
$$307$$ 10.9706 0.626123 0.313062 0.949733i $$-0.398645\pi$$
0.313062 + 0.949733i $$0.398645\pi$$
$$308$$ −15.3137 −0.872580
$$309$$ 24.2843 1.38148
$$310$$ 25.3137 1.43772
$$311$$ −2.48528 −0.140927 −0.0704637 0.997514i $$-0.522448\pi$$
−0.0704637 + 0.997514i $$0.522448\pi$$
$$312$$ −17.6569 −0.999623
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 39.7990 2.24599
$$315$$ 0.828427 0.0466766
$$316$$ 55.4558 3.11963
$$317$$ −2.82843 −0.158860 −0.0794301 0.996840i $$-0.525310\pi$$
−0.0794301 + 0.996840i $$0.525310\pi$$
$$318$$ 36.9706 2.07321
$$319$$ −4.82843 −0.270340
$$320$$ −9.82843 −0.549426
$$321$$ 16.2843 0.908899
$$322$$ 17.6569 0.983978
$$323$$ −2.34315 −0.130376
$$324$$ −42.1127 −2.33959
$$325$$ −2.00000 −0.110940
$$326$$ 47.4558 2.62834
$$327$$ −4.00000 −0.221201
$$328$$ 26.4853 1.46241
$$329$$ −0.284271 −0.0156724
$$330$$ −23.3137 −1.28338
$$331$$ −17.7990 −0.978321 −0.489160 0.872194i $$-0.662697\pi$$
−0.489160 + 0.872194i $$0.662697\pi$$
$$332$$ 49.1127 2.69541
$$333$$ 8.48528 0.464991
$$334$$ −34.9706 −1.91350
$$335$$ −10.4853 −0.572872
$$336$$ −4.97056 −0.271166
$$337$$ −6.82843 −0.371968 −0.185984 0.982553i $$-0.559547\pi$$
−0.185984 + 0.982553i $$0.559547\pi$$
$$338$$ 21.7279 1.18184
$$339$$ −5.65685 −0.307238
$$340$$ −10.8284 −0.587254
$$341$$ 50.6274 2.74163
$$342$$ −2.00000 −0.108148
$$343$$ −11.0294 −0.595534
$$344$$ 26.4853 1.42799
$$345$$ 17.6569 0.950613
$$346$$ 12.8284 0.689661
$$347$$ 20.1421 1.08129 0.540643 0.841252i $$-0.318181\pi$$
0.540643 + 0.841252i $$0.318181\pi$$
$$348$$ −7.65685 −0.410450
$$349$$ −24.6274 −1.31828 −0.659138 0.752022i $$-0.729079\pi$$
−0.659138 + 0.752022i $$0.729079\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ −8.00000 −0.427008
$$352$$ −7.65685 −0.408112
$$353$$ −15.6569 −0.833330 −0.416665 0.909060i $$-0.636801\pi$$
−0.416665 + 0.909060i $$0.636801\pi$$
$$354$$ 0 0
$$355$$ 7.31371 0.388171
$$356$$ 14.0000 0.741999
$$357$$ 4.68629 0.248025
$$358$$ 1.65685 0.0875675
$$359$$ −32.1421 −1.69640 −0.848199 0.529678i $$-0.822313\pi$$
−0.848199 + 0.529678i $$0.822313\pi$$
$$360$$ −4.41421 −0.232649
$$361$$ −18.3137 −0.963879
$$362$$ 14.4853 0.761329
$$363$$ −24.6274 −1.29260
$$364$$ −6.34315 −0.332471
$$365$$ −8.48528 −0.444140
$$366$$ 36.9706 1.93248
$$367$$ 18.0000 0.939592 0.469796 0.882775i $$-0.344327\pi$$
0.469796 + 0.882775i $$0.344327\pi$$
$$368$$ −26.4853 −1.38064
$$369$$ −6.00000 −0.312348
$$370$$ −20.4853 −1.06498
$$371$$ 6.34315 0.329320
$$372$$ 80.2843 4.16255
$$373$$ 26.9706 1.39648 0.698241 0.715862i $$-0.253966\pi$$
0.698241 + 0.715862i $$0.253966\pi$$
$$374$$ −32.9706 −1.70487
$$375$$ −2.00000 −0.103280
$$376$$ 1.51472 0.0781156
$$377$$ −2.00000 −0.103005
$$378$$ −8.00000 −0.411476
$$379$$ 5.51472 0.283272 0.141636 0.989919i $$-0.454764\pi$$
0.141636 + 0.989919i $$0.454764\pi$$
$$380$$ 3.17157 0.162698
$$381$$ −12.0000 −0.614779
$$382$$ 36.6274 1.87402
$$383$$ −14.4853 −0.740163 −0.370082 0.928999i $$-0.620670\pi$$
−0.370082 + 0.928999i $$0.620670\pi$$
$$384$$ −41.1127 −2.09802
$$385$$ −4.00000 −0.203859
$$386$$ 30.1421 1.53419
$$387$$ −6.00000 −0.304997
$$388$$ 17.1716 0.871755
$$389$$ −6.68629 −0.339008 −0.169504 0.985529i $$-0.554217\pi$$
−0.169504 + 0.985529i $$0.554217\pi$$
$$390$$ −9.65685 −0.488994
$$391$$ 24.9706 1.26282
$$392$$ 27.8701 1.40765
$$393$$ −32.2843 −1.62853
$$394$$ 20.1421 1.01475
$$395$$ 14.4853 0.728834
$$396$$ −18.4853 −0.928920
$$397$$ −8.34315 −0.418730 −0.209365 0.977838i $$-0.567140\pi$$
−0.209365 + 0.977838i $$0.567140\pi$$
$$398$$ −28.9706 −1.45216
$$399$$ −1.37258 −0.0687151
$$400$$ 3.00000 0.150000
$$401$$ −29.3137 −1.46386 −0.731928 0.681382i $$-0.761379\pi$$
−0.731928 + 0.681382i $$0.761379\pi$$
$$402$$ −50.6274 −2.52507
$$403$$ 20.9706 1.04462
$$404$$ 16.6274 0.827245
$$405$$ −11.0000 −0.546594
$$406$$ −2.00000 −0.0992583
$$407$$ −40.9706 −2.03084
$$408$$ −24.9706 −1.23623
$$409$$ 30.9706 1.53140 0.765698 0.643200i $$-0.222394\pi$$
0.765698 + 0.643200i $$0.222394\pi$$
$$410$$ 14.4853 0.715377
$$411$$ 21.6569 1.06825
$$412$$ −46.4853 −2.29017
$$413$$ 0 0
$$414$$ 21.3137 1.04751
$$415$$ 12.8284 0.629723
$$416$$ −3.17157 −0.155499
$$417$$ −20.6863 −1.01301
$$418$$ 9.65685 0.472332
$$419$$ −4.97056 −0.242828 −0.121414 0.992602i $$-0.538743\pi$$
−0.121414 + 0.992602i $$0.538743\pi$$
$$420$$ −6.34315 −0.309514
$$421$$ −14.9706 −0.729621 −0.364810 0.931082i $$-0.618866\pi$$
−0.364810 + 0.931082i $$0.618866\pi$$
$$422$$ −11.6569 −0.567447
$$423$$ −0.343146 −0.0166843
$$424$$ −33.7990 −1.64142
$$425$$ −2.82843 −0.137199
$$426$$ 35.3137 1.71095
$$427$$ 6.34315 0.306966
$$428$$ −31.1716 −1.50673
$$429$$ −19.3137 −0.932475
$$430$$ 14.4853 0.698542
$$431$$ −19.3137 −0.930309 −0.465154 0.885230i $$-0.654001\pi$$
−0.465154 + 0.885230i $$0.654001\pi$$
$$432$$ 12.0000 0.577350
$$433$$ −34.8284 −1.67375 −0.836874 0.547396i $$-0.815619\pi$$
−0.836874 + 0.547396i $$0.815619\pi$$
$$434$$ 20.9706 1.00662
$$435$$ −2.00000 −0.0958927
$$436$$ 7.65685 0.366697
$$437$$ −7.31371 −0.349862
$$438$$ −40.9706 −1.95765
$$439$$ −21.6569 −1.03363 −0.516813 0.856099i $$-0.672882\pi$$
−0.516813 + 0.856099i $$0.672882\pi$$
$$440$$ 21.3137 1.01609
$$441$$ −6.31371 −0.300653
$$442$$ −13.6569 −0.649590
$$443$$ 3.65685 0.173742 0.0868712 0.996220i $$-0.472313\pi$$
0.0868712 + 0.996220i $$0.472313\pi$$
$$444$$ −64.9706 −3.08337
$$445$$ 3.65685 0.173352
$$446$$ −52.6274 −2.49198
$$447$$ 26.6274 1.25943
$$448$$ −8.14214 −0.384680
$$449$$ 0.343146 0.0161940 0.00809702 0.999967i $$-0.497423\pi$$
0.00809702 + 0.999967i $$0.497423\pi$$
$$450$$ −2.41421 −0.113807
$$451$$ 28.9706 1.36417
$$452$$ 10.8284 0.509326
$$453$$ 24.0000 1.12762
$$454$$ 19.6569 0.922542
$$455$$ −1.65685 −0.0776745
$$456$$ 7.31371 0.342496
$$457$$ 8.34315 0.390276 0.195138 0.980776i $$-0.437485\pi$$
0.195138 + 0.980776i $$0.437485\pi$$
$$458$$ 4.82843 0.225618
$$459$$ −11.3137 −0.528079
$$460$$ −33.7990 −1.57589
$$461$$ −24.3431 −1.13377 −0.566887 0.823796i $$-0.691852\pi$$
−0.566887 + 0.823796i $$0.691852\pi$$
$$462$$ −19.3137 −0.898555
$$463$$ −17.7990 −0.827189 −0.413595 0.910461i $$-0.635727\pi$$
−0.413595 + 0.910461i $$0.635727\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 20.9706 0.972487
$$466$$ −43.4558 −2.01305
$$467$$ −22.9706 −1.06295 −0.531475 0.847074i $$-0.678362\pi$$
−0.531475 + 0.847074i $$0.678362\pi$$
$$468$$ −7.65685 −0.353938
$$469$$ −8.68629 −0.401096
$$470$$ 0.828427 0.0382125
$$471$$ 32.9706 1.51920
$$472$$ 0 0
$$473$$ 28.9706 1.33207
$$474$$ 69.9411 3.21250
$$475$$ 0.828427 0.0380108
$$476$$ −8.97056 −0.411165
$$477$$ 7.65685 0.350583
$$478$$ 56.2843 2.57438
$$479$$ 12.8284 0.586146 0.293073 0.956090i $$-0.405322\pi$$
0.293073 + 0.956090i $$0.405322\pi$$
$$480$$ −3.17157 −0.144762
$$481$$ −16.9706 −0.773791
$$482$$ −24.1421 −1.09964
$$483$$ 14.6274 0.665571
$$484$$ 47.1421 2.14282
$$485$$ 4.48528 0.203666
$$486$$ −24.1421 −1.09511
$$487$$ −29.7990 −1.35032 −0.675161 0.737671i $$-0.735926\pi$$
−0.675161 + 0.737671i $$0.735926\pi$$
$$488$$ −33.7990 −1.53001
$$489$$ 39.3137 1.77783
$$490$$ 15.2426 0.688592
$$491$$ 43.4558 1.96113 0.980567 0.196183i $$-0.0628545\pi$$
0.980567 + 0.196183i $$0.0628545\pi$$
$$492$$ 45.9411 2.07119
$$493$$ −2.82843 −0.127386
$$494$$ 4.00000 0.179969
$$495$$ −4.82843 −0.217022
$$496$$ −31.4558 −1.41241
$$497$$ 6.05887 0.271778
$$498$$ 61.9411 2.77565
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 3.82843 0.171212
$$501$$ −28.9706 −1.29431
$$502$$ −7.65685 −0.341742
$$503$$ −30.0000 −1.33763 −0.668817 0.743427i $$-0.733199\pi$$
−0.668817 + 0.743427i $$0.733199\pi$$
$$504$$ −3.65685 −0.162889
$$505$$ 4.34315 0.193267
$$506$$ −102.912 −4.57498
$$507$$ 18.0000 0.799408
$$508$$ 22.9706 1.01915
$$509$$ −44.6274 −1.97808 −0.989038 0.147663i $$-0.952825\pi$$
−0.989038 + 0.147663i $$0.952825\pi$$
$$510$$ −13.6569 −0.604736
$$511$$ −7.02944 −0.310964
$$512$$ 31.2426 1.38074
$$513$$ 3.31371 0.146304
$$514$$ −70.7696 −3.12151
$$515$$ −12.1421 −0.535046
$$516$$ 45.9411 2.02245
$$517$$ 1.65685 0.0728684
$$518$$ −16.9706 −0.745644
$$519$$ 10.6274 0.466492
$$520$$ 8.82843 0.387152
$$521$$ 1.31371 0.0575546 0.0287773 0.999586i $$-0.490839\pi$$
0.0287773 + 0.999586i $$0.490839\pi$$
$$522$$ −2.41421 −0.105667
$$523$$ −14.4853 −0.633397 −0.316699 0.948526i $$-0.602574\pi$$
−0.316699 + 0.948526i $$0.602574\pi$$
$$524$$ 61.7990 2.69970
$$525$$ −1.65685 −0.0723110
$$526$$ 20.1421 0.878239
$$527$$ 29.6569 1.29187
$$528$$ 28.9706 1.26078
$$529$$ 54.9411 2.38874
$$530$$ −18.4853 −0.802949
$$531$$ 0 0
$$532$$ 2.62742 0.113913
$$533$$ 12.0000 0.519778
$$534$$ 17.6569 0.764087
$$535$$ −8.14214 −0.352015
$$536$$ 46.2843 1.99918
$$537$$ 1.37258 0.0592313
$$538$$ −3.17157 −0.136736
$$539$$ 30.4853 1.31309
$$540$$ 15.3137 0.658997
$$541$$ −38.9706 −1.67548 −0.837738 0.546073i $$-0.816122\pi$$
−0.837738 + 0.546073i $$0.816122\pi$$
$$542$$ −71.9411 −3.09014
$$543$$ 12.0000 0.514969
$$544$$ −4.48528 −0.192305
$$545$$ 2.00000 0.0856706
$$546$$ −8.00000 −0.342368
$$547$$ 14.4853 0.619346 0.309673 0.950843i $$-0.399780\pi$$
0.309673 + 0.950843i $$0.399780\pi$$
$$548$$ −41.4558 −1.77091
$$549$$ 7.65685 0.326787
$$550$$ 11.6569 0.497050
$$551$$ 0.828427 0.0352922
$$552$$ −77.9411 −3.31739
$$553$$ 12.0000 0.510292
$$554$$ −18.4853 −0.785364
$$555$$ −16.9706 −0.720360
$$556$$ 39.5980 1.67933
$$557$$ 39.9411 1.69236 0.846180 0.532897i $$-0.178897\pi$$
0.846180 + 0.532897i $$0.178897\pi$$
$$558$$ 25.3137 1.07161
$$559$$ 12.0000 0.507546
$$560$$ 2.48528 0.105022
$$561$$ −27.3137 −1.15319
$$562$$ 16.1421 0.680915
$$563$$ −3.65685 −0.154118 −0.0770590 0.997027i $$-0.524553\pi$$
−0.0770590 + 0.997027i $$0.524553\pi$$
$$564$$ 2.62742 0.110634
$$565$$ 2.82843 0.118993
$$566$$ 2.00000 0.0840663
$$567$$ −9.11270 −0.382697
$$568$$ −32.2843 −1.35462
$$569$$ 16.3431 0.685140 0.342570 0.939492i $$-0.388703\pi$$
0.342570 + 0.939492i $$0.388703\pi$$
$$570$$ 4.00000 0.167542
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 36.9706 1.54582
$$573$$ 30.3431 1.26760
$$574$$ 12.0000 0.500870
$$575$$ −8.82843 −0.368171
$$576$$ −9.82843 −0.409518
$$577$$ −15.7990 −0.657721 −0.328860 0.944379i $$-0.606665\pi$$
−0.328860 + 0.944379i $$0.606665\pi$$
$$578$$ 21.7279 0.903762
$$579$$ 24.9706 1.03774
$$580$$ 3.82843 0.158967
$$581$$ 10.6274 0.440900
$$582$$ 21.6569 0.897705
$$583$$ −36.9706 −1.53116
$$584$$ 37.4558 1.54993
$$585$$ −2.00000 −0.0826898
$$586$$ 20.4853 0.846239
$$587$$ 9.79899 0.404448 0.202224 0.979339i $$-0.435183\pi$$
0.202224 + 0.979339i $$0.435183\pi$$
$$588$$ 48.3431 1.99364
$$589$$ −8.68629 −0.357912
$$590$$ 0 0
$$591$$ 16.6863 0.686382
$$592$$ 25.4558 1.04623
$$593$$ 3.65685 0.150169 0.0750845 0.997177i $$-0.476077\pi$$
0.0750845 + 0.997177i $$0.476077\pi$$
$$594$$ 46.6274 1.91315
$$595$$ −2.34315 −0.0960596
$$596$$ −50.9706 −2.08784
$$597$$ −24.0000 −0.982255
$$598$$ −42.6274 −1.74316
$$599$$ 1.79899 0.0735047 0.0367524 0.999324i $$-0.488299\pi$$
0.0367524 + 0.999324i $$0.488299\pi$$
$$600$$ 8.82843 0.360419
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 12.0000 0.489083
$$603$$ −10.4853 −0.426994
$$604$$ −45.9411 −1.86932
$$605$$ 12.3137 0.500623
$$606$$ 20.9706 0.851871
$$607$$ 42.9706 1.74412 0.872061 0.489398i $$-0.162783\pi$$
0.872061 + 0.489398i $$0.162783\pi$$
$$608$$ 1.31371 0.0532779
$$609$$ −1.65685 −0.0671391
$$610$$ −18.4853 −0.748447
$$611$$ 0.686292 0.0277644
$$612$$ −10.8284 −0.437713
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −26.4853 −1.06886
$$615$$ 12.0000 0.483887
$$616$$ 17.6569 0.711415
$$617$$ −14.8284 −0.596970 −0.298485 0.954414i $$-0.596481\pi$$
−0.298485 + 0.954414i $$0.596481\pi$$
$$618$$ −58.6274 −2.35834
$$619$$ 29.7990 1.19772 0.598861 0.800853i $$-0.295620\pi$$
0.598861 + 0.800853i $$0.295620\pi$$
$$620$$ −40.1421 −1.61215
$$621$$ −35.3137 −1.41709
$$622$$ 6.00000 0.240578
$$623$$ 3.02944 0.121372
$$624$$ 12.0000 0.480384
$$625$$ 1.00000 0.0400000
$$626$$ 14.4853 0.578948
$$627$$ 8.00000 0.319489
$$628$$ −63.1127 −2.51847
$$629$$ −24.0000 −0.956943
$$630$$ −2.00000 −0.0796819
$$631$$ 3.02944 0.120600 0.0603000 0.998180i $$-0.480794\pi$$
0.0603000 + 0.998180i $$0.480794\pi$$
$$632$$ −63.9411 −2.54344
$$633$$ −9.65685 −0.383825
$$634$$ 6.82843 0.271191
$$635$$ 6.00000 0.238103
$$636$$ −58.6274 −2.32473
$$637$$ 12.6274 0.500316
$$638$$ 11.6569 0.461499
$$639$$ 7.31371 0.289326
$$640$$ 20.5563 0.812561
$$641$$ −44.6274 −1.76268 −0.881338 0.472485i $$-0.843357\pi$$
−0.881338 + 0.472485i $$0.843357\pi$$
$$642$$ −39.3137 −1.55159
$$643$$ −31.4558 −1.24050 −0.620249 0.784405i $$-0.712968\pi$$
−0.620249 + 0.784405i $$0.712968\pi$$
$$644$$ −28.0000 −1.10335
$$645$$ 12.0000 0.472500
$$646$$ 5.65685 0.222566
$$647$$ 21.1127 0.830026 0.415013 0.909816i $$-0.363777\pi$$
0.415013 + 0.909816i $$0.363777\pi$$
$$648$$ 48.5563 1.90747
$$649$$ 0 0
$$650$$ 4.82843 0.189386
$$651$$ 17.3726 0.680885
$$652$$ −75.2548 −2.94721
$$653$$ −22.8284 −0.893345 −0.446673 0.894697i $$-0.647391\pi$$
−0.446673 + 0.894697i $$0.647391\pi$$
$$654$$ 9.65685 0.377613
$$655$$ 16.1421 0.630725
$$656$$ −18.0000 −0.702782
$$657$$ −8.48528 −0.331042
$$658$$ 0.686292 0.0267544
$$659$$ −37.7990 −1.47244 −0.736220 0.676743i $$-0.763391\pi$$
−0.736220 + 0.676743i $$0.763391\pi$$
$$660$$ 36.9706 1.43908
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ 42.9706 1.67010
$$663$$ −11.3137 −0.439388
$$664$$ −56.6274 −2.19757
$$665$$ 0.686292 0.0266132
$$666$$ −20.4853 −0.793789
$$667$$ −8.82843 −0.341838
$$668$$ 55.4558 2.14565
$$669$$ −43.5980 −1.68560
$$670$$ 25.3137 0.977954
$$671$$ −36.9706 −1.42723
$$672$$ −2.62742 −0.101355
$$673$$ −10.9706 −0.422884 −0.211442 0.977391i $$-0.567816\pi$$
−0.211442 + 0.977391i $$0.567816\pi$$
$$674$$ 16.4853 0.634989
$$675$$ 4.00000 0.153960
$$676$$ −34.4558 −1.32522
$$677$$ 36.7696 1.41317 0.706584 0.707629i $$-0.250235\pi$$
0.706584 + 0.707629i $$0.250235\pi$$
$$678$$ 13.6569 0.524488
$$679$$ 3.71573 0.142597
$$680$$ 12.4853 0.478789
$$681$$ 16.2843 0.624015
$$682$$ −122.225 −4.68025
$$683$$ 40.1421 1.53600 0.767998 0.640452i $$-0.221253\pi$$
0.767998 + 0.640452i $$0.221253\pi$$
$$684$$ 3.17157 0.121268
$$685$$ −10.8284 −0.413733
$$686$$ 26.6274 1.01664
$$687$$ 4.00000 0.152610
$$688$$ −18.0000 −0.686244
$$689$$ −15.3137 −0.583406
$$690$$ −42.6274 −1.62280
$$691$$ −11.0294 −0.419580 −0.209790 0.977747i $$-0.567278\pi$$
−0.209790 + 0.977747i $$0.567278\pi$$
$$692$$ −20.3431 −0.773330
$$693$$ −4.00000 −0.151947
$$694$$ −48.6274 −1.84587
$$695$$ 10.3431 0.392338
$$696$$ 8.82843 0.334641
$$697$$ 16.9706 0.642806
$$698$$ 59.4558 2.25044
$$699$$ −36.0000 −1.36165
$$700$$ 3.17157 0.119874
$$701$$ 29.3137 1.10716 0.553582 0.832795i $$-0.313261\pi$$
0.553582 + 0.832795i $$0.313261\pi$$
$$702$$ 19.3137 0.728949
$$703$$ 7.02944 0.265120
$$704$$ 47.4558 1.78856
$$705$$ 0.686292 0.0258472
$$706$$ 37.7990 1.42258
$$707$$ 3.59798 0.135316
$$708$$ 0 0
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ −17.6569 −0.662650
$$711$$ 14.4853 0.543240
$$712$$ −16.1421 −0.604952
$$713$$ 92.5685 3.46672
$$714$$ −11.3137 −0.423405
$$715$$ 9.65685 0.361146
$$716$$ −2.62742 −0.0981912
$$717$$ 46.6274 1.74133
$$718$$ 77.5980 2.89593
$$719$$ 10.6274 0.396336 0.198168 0.980168i $$-0.436501\pi$$
0.198168 + 0.980168i $$0.436501\pi$$
$$720$$ 3.00000 0.111803
$$721$$ −10.0589 −0.374612
$$722$$ 44.2132 1.64545
$$723$$ −20.0000 −0.743808
$$724$$ −22.9706 −0.853694
$$725$$ 1.00000 0.0371391
$$726$$ 59.4558 2.20661
$$727$$ 43.9411 1.62969 0.814843 0.579682i $$-0.196823\pi$$
0.814843 + 0.579682i $$0.196823\pi$$
$$728$$ 7.31371 0.271064
$$729$$ 13.0000 0.481481
$$730$$ 20.4853 0.758194
$$731$$ 16.9706 0.627679
$$732$$ −58.6274 −2.16693
$$733$$ −17.1716 −0.634247 −0.317123 0.948384i $$-0.602717\pi$$
−0.317123 + 0.948384i $$0.602717\pi$$
$$734$$ −43.4558 −1.60398
$$735$$ 12.6274 0.465769
$$736$$ −14.0000 −0.516047
$$737$$ 50.6274 1.86488
$$738$$ 14.4853 0.533211
$$739$$ −2.48528 −0.0914226 −0.0457113 0.998955i $$-0.514555\pi$$
−0.0457113 + 0.998955i $$0.514555\pi$$
$$740$$ 32.4853 1.19418
$$741$$ 3.31371 0.121732
$$742$$ −15.3137 −0.562184
$$743$$ −7.37258 −0.270474 −0.135237 0.990813i $$-0.543180\pi$$
−0.135237 + 0.990813i $$0.543180\pi$$
$$744$$ −92.5685 −3.39373
$$745$$ −13.3137 −0.487777
$$746$$ −65.1127 −2.38395
$$747$$ 12.8284 0.469368
$$748$$ 52.2843 1.91170
$$749$$ −6.74517 −0.246463
$$750$$ 4.82843 0.176309
$$751$$ −12.1421 −0.443073 −0.221536 0.975152i $$-0.571107\pi$$
−0.221536 + 0.975152i $$0.571107\pi$$
$$752$$ −1.02944 −0.0375397
$$753$$ −6.34315 −0.231157
$$754$$ 4.82843 0.175841
$$755$$ −12.0000 −0.436725
$$756$$ 12.6863 0.461396
$$757$$ 36.4853 1.32608 0.663040 0.748584i $$-0.269266\pi$$
0.663040 + 0.748584i $$0.269266\pi$$
$$758$$ −13.3137 −0.483576
$$759$$ −85.2548 −3.09455
$$760$$ −3.65685 −0.132648
$$761$$ −36.6274 −1.32774 −0.663871 0.747847i $$-0.731088\pi$$
−0.663871 + 0.747847i $$0.731088\pi$$
$$762$$ 28.9706 1.04949
$$763$$ 1.65685 0.0599822
$$764$$ −58.0833 −2.10138
$$765$$ −2.82843 −0.102262
$$766$$ 34.9706 1.26354
$$767$$ 0 0
$$768$$ 59.9411 2.16294
$$769$$ −4.34315 −0.156618 −0.0783089 0.996929i $$-0.524952\pi$$
−0.0783089 + 0.996929i $$0.524952\pi$$
$$770$$ 9.65685 0.348009
$$771$$ −58.6274 −2.11141
$$772$$ −47.7990 −1.72032
$$773$$ 8.48528 0.305194 0.152597 0.988288i $$-0.451236\pi$$
0.152597 + 0.988288i $$0.451236\pi$$
$$774$$ 14.4853 0.520663
$$775$$ −10.4853 −0.376642
$$776$$ −19.7990 −0.710742
$$777$$ −14.0589 −0.504359
$$778$$ 16.1421 0.578724
$$779$$ −4.97056 −0.178089
$$780$$ 15.3137 0.548319
$$781$$ −35.3137 −1.26362
$$782$$ −60.2843 −2.15576
$$783$$ 4.00000 0.142948
$$784$$ −18.9411 −0.676469
$$785$$ −16.4853 −0.588385
$$786$$ 77.9411 2.78007
$$787$$ −21.7990 −0.777050 −0.388525 0.921438i $$-0.627015\pi$$
−0.388525 + 0.921438i $$0.627015\pi$$
$$788$$ −31.9411 −1.13786
$$789$$ 16.6863 0.594048
$$790$$ −34.9706 −1.24420
$$791$$ 2.34315 0.0833127
$$792$$ 21.3137 0.757350
$$793$$ −15.3137 −0.543806
$$794$$ 20.1421 0.714818
$$795$$ −15.3137 −0.543121
$$796$$ 45.9411 1.62834
$$797$$ −34.1421 −1.20938 −0.604688 0.796462i $$-0.706702\pi$$
−0.604688 + 0.796462i $$0.706702\pi$$
$$798$$ 3.31371 0.117304
$$799$$ 0.970563 0.0343360
$$800$$ 1.58579 0.0560660
$$801$$ 3.65685 0.129209
$$802$$ 70.7696 2.49896
$$803$$ 40.9706 1.44582
$$804$$ 80.2843 2.83141
$$805$$ −7.31371 −0.257774
$$806$$ −50.6274 −1.78327
$$807$$ −2.62742 −0.0924895
$$808$$ −19.1716 −0.674454
$$809$$ −14.2843 −0.502208 −0.251104 0.967960i $$-0.580794\pi$$
−0.251104 + 0.967960i $$0.580794\pi$$
$$810$$ 26.5563 0.933095
$$811$$ 26.3431 0.925033 0.462516 0.886611i $$-0.346947\pi$$
0.462516 + 0.886611i $$0.346947\pi$$
$$812$$ 3.17157 0.111300
$$813$$ −59.5980 −2.09019
$$814$$ 98.9117 3.46685
$$815$$ −19.6569 −0.688550
$$816$$ 16.9706 0.594089
$$817$$ −4.97056 −0.173898
$$818$$ −74.7696 −2.61426
$$819$$ −1.65685 −0.0578952
$$820$$ −22.9706 −0.802167
$$821$$ −45.3137 −1.58146 −0.790730 0.612165i $$-0.790299\pi$$
−0.790730 + 0.612165i $$0.790299\pi$$
$$822$$ −52.2843 −1.82362
$$823$$ 2.97056 0.103547 0.0517737 0.998659i $$-0.483513\pi$$
0.0517737 + 0.998659i $$0.483513\pi$$
$$824$$ 53.5980 1.86717
$$825$$ 9.65685 0.336209
$$826$$ 0 0
$$827$$ −5.31371 −0.184776 −0.0923879 0.995723i $$-0.529450\pi$$
−0.0923879 + 0.995723i $$0.529450\pi$$
$$828$$ −33.7990 −1.17460
$$829$$ −24.6274 −0.855346 −0.427673 0.903934i $$-0.640666\pi$$
−0.427673 + 0.903934i $$0.640666\pi$$
$$830$$ −30.9706 −1.07500
$$831$$ −15.3137 −0.531227
$$832$$ 19.6569 0.681479
$$833$$ 17.8579 0.618738
$$834$$ 49.9411 1.72932
$$835$$ 14.4853 0.501284
$$836$$ −15.3137 −0.529636
$$837$$ −41.9411 −1.44970
$$838$$ 12.0000 0.414533
$$839$$ −14.4853 −0.500087 −0.250044 0.968235i $$-0.580445\pi$$
−0.250044 + 0.968235i $$0.580445\pi$$
$$840$$ 7.31371 0.252347
$$841$$ 1.00000 0.0344828
$$842$$ 36.1421 1.24554
$$843$$ 13.3726 0.460576
$$844$$ 18.4853 0.636290
$$845$$ −9.00000 −0.309609
$$846$$ 0.828427 0.0284819
$$847$$ 10.2010 0.350511
$$848$$ 22.9706 0.788812
$$849$$ 1.65685 0.0568631
$$850$$ 6.82843 0.234213
$$851$$ −74.9117 −2.56794
$$852$$ −56.0000 −1.91853
$$853$$ −11.1127 −0.380492 −0.190246 0.981736i $$-0.560928\pi$$
−0.190246 + 0.981736i $$0.560928\pi$$
$$854$$ −15.3137 −0.524024
$$855$$ 0.828427 0.0283316
$$856$$ 35.9411 1.22844
$$857$$ 48.6274 1.66108 0.830540 0.556958i $$-0.188032\pi$$
0.830540 + 0.556958i $$0.188032\pi$$
$$858$$ 46.6274 1.59183
$$859$$ 28.4264 0.969896 0.484948 0.874543i $$-0.338838\pi$$
0.484948 + 0.874543i $$0.338838\pi$$
$$860$$ −22.9706 −0.783290
$$861$$ 9.94113 0.338793
$$862$$ 46.6274 1.58814
$$863$$ −7.85786 −0.267485 −0.133742 0.991016i $$-0.542699\pi$$
−0.133742 + 0.991016i $$0.542699\pi$$
$$864$$ 6.34315 0.215798
$$865$$ −5.31371 −0.180672
$$866$$ 84.0833 2.85727
$$867$$ 18.0000 0.611312
$$868$$ −33.2548 −1.12874
$$869$$ −69.9411 −2.37259
$$870$$ 4.82843 0.163699
$$871$$ 20.9706 0.710560
$$872$$ −8.82843 −0.298968
$$873$$ 4.48528 0.151804
$$874$$ 17.6569 0.597252
$$875$$ 0.828427 0.0280059
$$876$$ 64.9706 2.19515
$$877$$ −18.2843 −0.617416 −0.308708 0.951157i $$-0.599897\pi$$
−0.308708 + 0.951157i $$0.599897\pi$$
$$878$$ 52.2843 1.76451
$$879$$ 16.9706 0.572403
$$880$$ −14.4853 −0.488299
$$881$$ 6.68629 0.225267 0.112633 0.993637i $$-0.464071\pi$$
0.112633 + 0.993637i $$0.464071\pi$$
$$882$$ 15.2426 0.513246
$$883$$ 2.48528 0.0836364 0.0418182 0.999125i $$-0.486685\pi$$
0.0418182 + 0.999125i $$0.486685\pi$$
$$884$$ 21.6569 0.728399
$$885$$ 0 0
$$886$$ −8.82843 −0.296597
$$887$$ −29.3137 −0.984258 −0.492129 0.870522i $$-0.663781\pi$$
−0.492129 + 0.870522i $$0.663781\pi$$
$$888$$ 74.9117 2.51387
$$889$$ 4.97056 0.166707
$$890$$ −8.82843 −0.295930
$$891$$ 53.1127 1.77934
$$892$$ 83.4558 2.79431
$$893$$ −0.284271 −0.00951277
$$894$$ −64.2843 −2.14999
$$895$$ −0.686292 −0.0229402
$$896$$ 17.0294 0.568914
$$897$$ −35.3137 −1.17909
$$898$$ −0.828427 −0.0276450
$$899$$ −10.4853 −0.349704
$$900$$ 3.82843 0.127614
$$901$$ −21.6569 −0.721494
$$902$$ −69.9411 −2.32878
$$903$$ 9.94113 0.330820
$$904$$ −12.4853 −0.415254
$$905$$ −6.00000 −0.199447
$$906$$ −57.9411 −1.92496
$$907$$ −10.0000 −0.332045 −0.166022 0.986122i $$-0.553092\pi$$
−0.166022 + 0.986122i $$0.553092\pi$$
$$908$$ −31.1716 −1.03446
$$909$$ 4.34315 0.144053
$$910$$ 4.00000 0.132599
$$911$$ 3.85786 0.127817 0.0639084 0.997956i $$-0.479643\pi$$
0.0639084 + 0.997956i $$0.479643\pi$$
$$912$$ −4.97056 −0.164592
$$913$$ −61.9411 −2.04995
$$914$$ −20.1421 −0.666243
$$915$$ −15.3137 −0.506256
$$916$$ −7.65685 −0.252990
$$917$$ 13.3726 0.441602
$$918$$ 27.3137 0.901487
$$919$$ −36.0000 −1.18753 −0.593765 0.804638i $$-0.702359\pi$$
−0.593765 + 0.804638i $$0.702359\pi$$
$$920$$ 38.9706 1.28482
$$921$$ −21.9411 −0.722985
$$922$$ 58.7696 1.93547
$$923$$ −14.6274 −0.481467
$$924$$ 30.6274 1.00757
$$925$$ 8.48528 0.278994
$$926$$ 42.9706 1.41210
$$927$$ −12.1421 −0.398800
$$928$$ 1.58579 0.0520560
$$929$$ 40.6274 1.33294 0.666471 0.745531i $$-0.267804\pi$$
0.666471 + 0.745531i $$0.267804\pi$$
$$930$$ −50.6274 −1.66014
$$931$$ −5.23045 −0.171421
$$932$$ 68.9117 2.25728
$$933$$ 4.97056 0.162729
$$934$$ 55.4558 1.81457
$$935$$ 13.6569 0.446627
$$936$$ 8.82843 0.288566
$$937$$ 8.34315 0.272559 0.136279 0.990670i $$-0.456486\pi$$
0.136279 + 0.990670i $$0.456486\pi$$
$$938$$ 20.9706 0.684713
$$939$$ 12.0000 0.391605
$$940$$ −1.31371 −0.0428484
$$941$$ 39.9411 1.30204 0.651022 0.759059i $$-0.274341\pi$$
0.651022 + 0.759059i $$0.274341\pi$$
$$942$$ −79.5980 −2.59344
$$943$$ 52.9706 1.72496
$$944$$ 0 0
$$945$$ 3.31371 0.107795
$$946$$ −69.9411 −2.27398
$$947$$ −56.9117 −1.84938 −0.924691 0.380719i $$-0.875676\pi$$
−0.924691 + 0.380719i $$0.875676\pi$$
$$948$$ −110.912 −3.60224
$$949$$ 16.9706 0.550888
$$950$$ −2.00000 −0.0648886
$$951$$ 5.65685 0.183436
$$952$$ 10.3431 0.335223
$$953$$ 6.68629 0.216590 0.108295 0.994119i $$-0.465461\pi$$
0.108295 + 0.994119i $$0.465461\pi$$
$$954$$ −18.4853 −0.598483
$$955$$ −15.1716 −0.490941
$$956$$ −89.2548 −2.88671
$$957$$ 9.65685 0.312162
$$958$$ −30.9706 −1.00061
$$959$$ −8.97056 −0.289675
$$960$$ 19.6569 0.634422
$$961$$ 78.9411 2.54649
$$962$$ 40.9706 1.32094
$$963$$ −8.14214 −0.262377
$$964$$ 38.2843 1.23305
$$965$$ −12.4853 −0.401915
$$966$$ −35.3137 −1.13620
$$967$$ −18.9706 −0.610052 −0.305026 0.952344i $$-0.598665\pi$$
−0.305026 + 0.952344i $$0.598665\pi$$
$$968$$ −54.3553 −1.74705
$$969$$ 4.68629 0.150545
$$970$$ −10.8284 −0.347680
$$971$$ −0.142136 −0.00456135 −0.00228067 0.999997i $$-0.500726\pi$$
−0.00228067 + 0.999997i $$0.500726\pi$$
$$972$$ 38.2843 1.22797
$$973$$ 8.56854 0.274695
$$974$$ 71.9411 2.30514
$$975$$ 4.00000 0.128103
$$976$$ 22.9706 0.735270
$$977$$ −25.3137 −0.809857 −0.404929 0.914348i $$-0.632704\pi$$
−0.404929 + 0.914348i $$0.632704\pi$$
$$978$$ −94.9117 −3.03494
$$979$$ −17.6569 −0.564316
$$980$$ −24.1716 −0.772133
$$981$$ 2.00000 0.0638551
$$982$$ −104.912 −3.34787
$$983$$ 13.3137 0.424641 0.212321 0.977200i $$-0.431898\pi$$
0.212321 + 0.977200i $$0.431898\pi$$
$$984$$ −52.9706 −1.68864
$$985$$ −8.34315 −0.265835
$$986$$ 6.82843 0.217461
$$987$$ 0.568542 0.0180969
$$988$$ −6.34315 −0.201802
$$989$$ 52.9706 1.68437
$$990$$ 11.6569 0.370479
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ −16.6274 −0.527921
$$993$$ 35.5980 1.12967
$$994$$ −14.6274 −0.463953
$$995$$ 12.0000 0.380426
$$996$$ −98.2254 −3.11239
$$997$$ 1.17157 0.0371041 0.0185520 0.999828i $$-0.494094\pi$$
0.0185520 + 0.999828i $$0.494094\pi$$
$$998$$ −86.9117 −2.75114
$$999$$ 33.9411 1.07385
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 145.2.a.b.1.1 2
3.2 odd 2 1305.2.a.n.1.2 2
4.3 odd 2 2320.2.a.k.1.1 2
5.2 odd 4 725.2.b.c.349.1 4
5.3 odd 4 725.2.b.c.349.4 4
5.4 even 2 725.2.a.c.1.2 2
7.6 odd 2 7105.2.a.e.1.1 2
8.3 odd 2 9280.2.a.w.1.1 2
8.5 even 2 9280.2.a.be.1.2 2
15.14 odd 2 6525.2.a.p.1.1 2
29.28 even 2 4205.2.a.d.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.1 2 1.1 even 1 trivial
725.2.a.c.1.2 2 5.4 even 2
725.2.b.c.349.1 4 5.2 odd 4
725.2.b.c.349.4 4 5.3 odd 4
1305.2.a.n.1.2 2 3.2 odd 2
2320.2.a.k.1.1 2 4.3 odd 2
4205.2.a.d.1.2 2 29.28 even 2
6525.2.a.p.1.1 2 15.14 odd 2
7105.2.a.e.1.1 2 7.6 odd 2
9280.2.a.w.1.1 2 8.3 odd 2
9280.2.a.be.1.2 2 8.5 even 2