# Properties

 Label 1045.2.a.c.1.2 Level $1045$ Weight $2$ Character 1045.1 Self dual yes Analytic conductor $8.344$ 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] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.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: $$8.34436701122$$ Analytic rank: $$0$$ 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 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1045.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.41421 q^{2} +2.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} +4.82843 q^{6} -2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.41421 q^{2} +2.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} +4.82843 q^{6} -2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} -1.00000 q^{11} +7.65685 q^{12} +6.82843 q^{13} -6.82843 q^{14} +2.00000 q^{15} +3.00000 q^{16} -4.82843 q^{17} +2.41421 q^{18} -1.00000 q^{19} +3.82843 q^{20} -5.65685 q^{21} -2.41421 q^{22} +4.00000 q^{23} +8.82843 q^{24} +1.00000 q^{25} +16.4853 q^{26} -4.00000 q^{27} -10.8284 q^{28} +3.17157 q^{29} +4.82843 q^{30} +1.17157 q^{31} -1.58579 q^{32} -2.00000 q^{33} -11.6569 q^{34} -2.82843 q^{35} +3.82843 q^{36} -1.17157 q^{37} -2.41421 q^{38} +13.6569 q^{39} +4.41421 q^{40} -8.82843 q^{41} -13.6569 q^{42} -6.82843 q^{43} -3.82843 q^{44} +1.00000 q^{45} +9.65685 q^{46} +6.00000 q^{48} +1.00000 q^{49} +2.41421 q^{50} -9.65685 q^{51} +26.1421 q^{52} -2.82843 q^{53} -9.65685 q^{54} -1.00000 q^{55} -12.4853 q^{56} -2.00000 q^{57} +7.65685 q^{58} +12.4853 q^{59} +7.65685 q^{60} -3.65685 q^{61} +2.82843 q^{62} -2.82843 q^{63} -9.82843 q^{64} +6.82843 q^{65} -4.82843 q^{66} +0.343146 q^{67} -18.4853 q^{68} +8.00000 q^{69} -6.82843 q^{70} +9.17157 q^{71} +4.41421 q^{72} -10.4853 q^{73} -2.82843 q^{74} +2.00000 q^{75} -3.82843 q^{76} +2.82843 q^{77} +32.9706 q^{78} -12.0000 q^{79} +3.00000 q^{80} -11.0000 q^{81} -21.3137 q^{82} +2.82843 q^{83} -21.6569 q^{84} -4.82843 q^{85} -16.4853 q^{86} +6.34315 q^{87} -4.41421 q^{88} -13.3137 q^{89} +2.41421 q^{90} -19.3137 q^{91} +15.3137 q^{92} +2.34315 q^{93} -1.00000 q^{95} -3.17157 q^{96} +10.1421 q^{97} +2.41421 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} + 8 q^{13} - 8 q^{14} + 4 q^{15} + 6 q^{16} - 4 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{20} - 2 q^{22} + 8 q^{23} + 12 q^{24} + 2 q^{25} + 16 q^{26} - 8 q^{27} - 16 q^{28} + 12 q^{29} + 4 q^{30} + 8 q^{31} - 6 q^{32} - 4 q^{33} - 12 q^{34} + 2 q^{36} - 8 q^{37} - 2 q^{38} + 16 q^{39} + 6 q^{40} - 12 q^{41} - 16 q^{42} - 8 q^{43} - 2 q^{44} + 2 q^{45} + 8 q^{46} + 12 q^{48} + 2 q^{49} + 2 q^{50} - 8 q^{51} + 24 q^{52} - 8 q^{54} - 2 q^{55} - 8 q^{56} - 4 q^{57} + 4 q^{58} + 8 q^{59} + 4 q^{60} + 4 q^{61} - 14 q^{64} + 8 q^{65} - 4 q^{66} + 12 q^{67} - 20 q^{68} + 16 q^{69} - 8 q^{70} + 24 q^{71} + 6 q^{72} - 4 q^{73} + 4 q^{75} - 2 q^{76} + 32 q^{78} - 24 q^{79} + 6 q^{80} - 22 q^{81} - 20 q^{82} - 32 q^{84} - 4 q^{85} - 16 q^{86} + 24 q^{87} - 6 q^{88} - 4 q^{89} + 2 q^{90} - 16 q^{91} + 8 q^{92} + 16 q^{93} - 2 q^{95} - 12 q^{96} - 8 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 + 4 * q^12 + 8 * q^13 - 8 * q^14 + 4 * q^15 + 6 * q^16 - 4 * q^17 + 2 * q^18 - 2 * q^19 + 2 * q^20 - 2 * q^22 + 8 * q^23 + 12 * q^24 + 2 * q^25 + 16 * q^26 - 8 * q^27 - 16 * q^28 + 12 * q^29 + 4 * q^30 + 8 * q^31 - 6 * q^32 - 4 * q^33 - 12 * q^34 + 2 * q^36 - 8 * q^37 - 2 * q^38 + 16 * q^39 + 6 * q^40 - 12 * q^41 - 16 * q^42 - 8 * q^43 - 2 * q^44 + 2 * q^45 + 8 * q^46 + 12 * q^48 + 2 * q^49 + 2 * q^50 - 8 * q^51 + 24 * q^52 - 8 * q^54 - 2 * q^55 - 8 * q^56 - 4 * q^57 + 4 * q^58 + 8 * q^59 + 4 * q^60 + 4 * q^61 - 14 * q^64 + 8 * q^65 - 4 * q^66 + 12 * q^67 - 20 * q^68 + 16 * q^69 - 8 * q^70 + 24 * q^71 + 6 * q^72 - 4 * q^73 + 4 * q^75 - 2 * q^76 + 32 * q^78 - 24 * q^79 + 6 * q^80 - 22 * q^81 - 20 * q^82 - 32 * q^84 - 4 * q^85 - 16 * q^86 + 24 * q^87 - 6 * q^88 - 4 * q^89 + 2 * q^90 - 16 * q^91 + 8 * q^92 + 16 * q^93 - 2 * q^95 - 12 * q^96 - 8 * q^97 + 2 * q^98 - 2 * 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 $$-0.174443\pi$$
0.853553 + 0.521005i $$0.174443\pi$$
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 3.82843 1.91421
$$5$$ 1.00000 0.447214
$$6$$ 4.82843 1.97120
$$7$$ −2.82843 −1.06904 −0.534522 0.845154i $$-0.679509\pi$$
−0.534522 + 0.845154i $$0.679509\pi$$
$$8$$ 4.41421 1.56066
$$9$$ 1.00000 0.333333
$$10$$ 2.41421 0.763441
$$11$$ −1.00000 −0.301511
$$12$$ 7.65685 2.21034
$$13$$ 6.82843 1.89386 0.946932 0.321433i $$-0.104164\pi$$
0.946932 + 0.321433i $$0.104164\pi$$
$$14$$ −6.82843 −1.82497
$$15$$ 2.00000 0.516398
$$16$$ 3.00000 0.750000
$$17$$ −4.82843 −1.17107 −0.585533 0.810649i $$-0.699115\pi$$
−0.585533 + 0.810649i $$0.699115\pi$$
$$18$$ 2.41421 0.569036
$$19$$ −1.00000 −0.229416
$$20$$ 3.82843 0.856062
$$21$$ −5.65685 −1.23443
$$22$$ −2.41421 −0.514712
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 8.82843 1.80210
$$25$$ 1.00000 0.200000
$$26$$ 16.4853 3.23303
$$27$$ −4.00000 −0.769800
$$28$$ −10.8284 −2.04638
$$29$$ 3.17157 0.588946 0.294473 0.955660i $$-0.404856\pi$$
0.294473 + 0.955660i $$0.404856\pi$$
$$30$$ 4.82843 0.881546
$$31$$ 1.17157 0.210421 0.105210 0.994450i $$-0.466448\pi$$
0.105210 + 0.994450i $$0.466448\pi$$
$$32$$ −1.58579 −0.280330
$$33$$ −2.00000 −0.348155
$$34$$ −11.6569 −1.99913
$$35$$ −2.82843 −0.478091
$$36$$ 3.82843 0.638071
$$37$$ −1.17157 −0.192605 −0.0963027 0.995352i $$-0.530702\pi$$
−0.0963027 + 0.995352i $$0.530702\pi$$
$$38$$ −2.41421 −0.391637
$$39$$ 13.6569 2.18685
$$40$$ 4.41421 0.697948
$$41$$ −8.82843 −1.37877 −0.689384 0.724396i $$-0.742119\pi$$
−0.689384 + 0.724396i $$0.742119\pi$$
$$42$$ −13.6569 −2.10730
$$43$$ −6.82843 −1.04133 −0.520663 0.853762i $$-0.674315\pi$$
−0.520663 + 0.853762i $$0.674315\pi$$
$$44$$ −3.82843 −0.577157
$$45$$ 1.00000 0.149071
$$46$$ 9.65685 1.42383
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 6.00000 0.866025
$$49$$ 1.00000 0.142857
$$50$$ 2.41421 0.341421
$$51$$ −9.65685 −1.35223
$$52$$ 26.1421 3.62526
$$53$$ −2.82843 −0.388514 −0.194257 0.980951i $$-0.562230\pi$$
−0.194257 + 0.980951i $$0.562230\pi$$
$$54$$ −9.65685 −1.31413
$$55$$ −1.00000 −0.134840
$$56$$ −12.4853 −1.66842
$$57$$ −2.00000 −0.264906
$$58$$ 7.65685 1.00539
$$59$$ 12.4853 1.62545 0.812723 0.582651i $$-0.197984\pi$$
0.812723 + 0.582651i $$0.197984\pi$$
$$60$$ 7.65685 0.988496
$$61$$ −3.65685 −0.468212 −0.234106 0.972211i $$-0.575216\pi$$
−0.234106 + 0.972211i $$0.575216\pi$$
$$62$$ 2.82843 0.359211
$$63$$ −2.82843 −0.356348
$$64$$ −9.82843 −1.22855
$$65$$ 6.82843 0.846962
$$66$$ −4.82843 −0.594338
$$67$$ 0.343146 0.0419219 0.0209610 0.999780i $$-0.493327\pi$$
0.0209610 + 0.999780i $$0.493327\pi$$
$$68$$ −18.4853 −2.24167
$$69$$ 8.00000 0.963087
$$70$$ −6.82843 −0.816153
$$71$$ 9.17157 1.08847 0.544233 0.838934i $$-0.316821\pi$$
0.544233 + 0.838934i $$0.316821\pi$$
$$72$$ 4.41421 0.520220
$$73$$ −10.4853 −1.22721 −0.613605 0.789613i $$-0.710281\pi$$
−0.613605 + 0.789613i $$0.710281\pi$$
$$74$$ −2.82843 −0.328798
$$75$$ 2.00000 0.230940
$$76$$ −3.82843 −0.439151
$$77$$ 2.82843 0.322329
$$78$$ 32.9706 3.73318
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 3.00000 0.335410
$$81$$ −11.0000 −1.22222
$$82$$ −21.3137 −2.35371
$$83$$ 2.82843 0.310460 0.155230 0.987878i $$-0.450388\pi$$
0.155230 + 0.987878i $$0.450388\pi$$
$$84$$ −21.6569 −2.36296
$$85$$ −4.82843 −0.523716
$$86$$ −16.4853 −1.77765
$$87$$ 6.34315 0.680057
$$88$$ −4.41421 −0.470557
$$89$$ −13.3137 −1.41125 −0.705625 0.708585i $$-0.749334\pi$$
−0.705625 + 0.708585i $$0.749334\pi$$
$$90$$ 2.41421 0.254480
$$91$$ −19.3137 −2.02463
$$92$$ 15.3137 1.59656
$$93$$ 2.34315 0.242973
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ −3.17157 −0.323697
$$97$$ 10.1421 1.02978 0.514889 0.857257i $$-0.327833\pi$$
0.514889 + 0.857257i $$0.327833\pi$$
$$98$$ 2.41421 0.243872
$$99$$ −1.00000 −0.100504
$$100$$ 3.82843 0.382843
$$101$$ 7.65685 0.761885 0.380943 0.924599i $$-0.375599\pi$$
0.380943 + 0.924599i $$0.375599\pi$$
$$102$$ −23.3137 −2.30840
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 30.1421 2.95568
$$105$$ −5.65685 −0.552052
$$106$$ −6.82843 −0.663235
$$107$$ 13.3137 1.28708 0.643542 0.765410i $$-0.277464\pi$$
0.643542 + 0.765410i $$0.277464\pi$$
$$108$$ −15.3137 −1.47356
$$109$$ −0.828427 −0.0793489 −0.0396745 0.999213i $$-0.512632\pi$$
−0.0396745 + 0.999213i $$0.512632\pi$$
$$110$$ −2.41421 −0.230186
$$111$$ −2.34315 −0.222402
$$112$$ −8.48528 −0.801784
$$113$$ −12.4853 −1.17452 −0.587258 0.809400i $$-0.699793\pi$$
−0.587258 + 0.809400i $$0.699793\pi$$
$$114$$ −4.82843 −0.452224
$$115$$ 4.00000 0.373002
$$116$$ 12.1421 1.12737
$$117$$ 6.82843 0.631288
$$118$$ 30.1421 2.77481
$$119$$ 13.6569 1.25192
$$120$$ 8.82843 0.805921
$$121$$ 1.00000 0.0909091
$$122$$ −8.82843 −0.799288
$$123$$ −17.6569 −1.59206
$$124$$ 4.48528 0.402790
$$125$$ 1.00000 0.0894427
$$126$$ −6.82843 −0.608325
$$127$$ 14.9706 1.32842 0.664211 0.747545i $$-0.268768\pi$$
0.664211 + 0.747545i $$0.268768\pi$$
$$128$$ −20.5563 −1.81694
$$129$$ −13.6569 −1.20242
$$130$$ 16.4853 1.44585
$$131$$ 9.65685 0.843723 0.421862 0.906660i $$-0.361377\pi$$
0.421862 + 0.906660i $$0.361377\pi$$
$$132$$ −7.65685 −0.666444
$$133$$ 2.82843 0.245256
$$134$$ 0.828427 0.0715652
$$135$$ −4.00000 −0.344265
$$136$$ −21.3137 −1.82764
$$137$$ 9.31371 0.795724 0.397862 0.917445i $$-0.369752\pi$$
0.397862 + 0.917445i $$0.369752\pi$$
$$138$$ 19.3137 1.64409
$$139$$ 20.9706 1.77870 0.889350 0.457227i $$-0.151157\pi$$
0.889350 + 0.457227i $$0.151157\pi$$
$$140$$ −10.8284 −0.915169
$$141$$ 0 0
$$142$$ 22.1421 1.85813
$$143$$ −6.82843 −0.571022
$$144$$ 3.00000 0.250000
$$145$$ 3.17157 0.263385
$$146$$ −25.3137 −2.09498
$$147$$ 2.00000 0.164957
$$148$$ −4.48528 −0.368688
$$149$$ 7.65685 0.627274 0.313637 0.949543i $$-0.398453\pi$$
0.313637 + 0.949543i $$0.398453\pi$$
$$150$$ 4.82843 0.394239
$$151$$ 9.65685 0.785864 0.392932 0.919568i $$-0.371461\pi$$
0.392932 + 0.919568i $$0.371461\pi$$
$$152$$ −4.41421 −0.358040
$$153$$ −4.82843 −0.390355
$$154$$ 6.82843 0.550250
$$155$$ 1.17157 0.0941030
$$156$$ 52.2843 4.18609
$$157$$ −17.3137 −1.38178 −0.690892 0.722958i $$-0.742782\pi$$
−0.690892 + 0.722958i $$0.742782\pi$$
$$158$$ −28.9706 −2.30477
$$159$$ −5.65685 −0.448618
$$160$$ −1.58579 −0.125367
$$161$$ −11.3137 −0.891645
$$162$$ −26.5563 −2.08646
$$163$$ 17.6569 1.38299 0.691496 0.722380i $$-0.256952\pi$$
0.691496 + 0.722380i $$0.256952\pi$$
$$164$$ −33.7990 −2.63926
$$165$$ −2.00000 −0.155700
$$166$$ 6.82843 0.529989
$$167$$ −14.0000 −1.08335 −0.541676 0.840587i $$-0.682210\pi$$
−0.541676 + 0.840587i $$0.682210\pi$$
$$168$$ −24.9706 −1.92652
$$169$$ 33.6274 2.58672
$$170$$ −11.6569 −0.894040
$$171$$ −1.00000 −0.0764719
$$172$$ −26.1421 −1.99332
$$173$$ 2.82843 0.215041 0.107521 0.994203i $$-0.465709\pi$$
0.107521 + 0.994203i $$0.465709\pi$$
$$174$$ 15.3137 1.16093
$$175$$ −2.82843 −0.213809
$$176$$ −3.00000 −0.226134
$$177$$ 24.9706 1.87690
$$178$$ −32.1421 −2.40915
$$179$$ 12.4853 0.933194 0.466597 0.884470i $$-0.345480\pi$$
0.466597 + 0.884470i $$0.345480\pi$$
$$180$$ 3.82843 0.285354
$$181$$ −5.31371 −0.394965 −0.197482 0.980306i $$-0.563277\pi$$
−0.197482 + 0.980306i $$0.563277\pi$$
$$182$$ −46.6274 −3.45625
$$183$$ −7.31371 −0.540645
$$184$$ 17.6569 1.30168
$$185$$ −1.17157 −0.0861358
$$186$$ 5.65685 0.414781
$$187$$ 4.82843 0.353090
$$188$$ 0 0
$$189$$ 11.3137 0.822951
$$190$$ −2.41421 −0.175145
$$191$$ 16.9706 1.22795 0.613973 0.789327i $$-0.289570\pi$$
0.613973 + 0.789327i $$0.289570\pi$$
$$192$$ −19.6569 −1.41861
$$193$$ 15.7990 1.13724 0.568618 0.822602i $$-0.307478\pi$$
0.568618 + 0.822602i $$0.307478\pi$$
$$194$$ 24.4853 1.75794
$$195$$ 13.6569 0.977988
$$196$$ 3.82843 0.273459
$$197$$ −13.7990 −0.983137 −0.491569 0.870839i $$-0.663576\pi$$
−0.491569 + 0.870839i $$0.663576\pi$$
$$198$$ −2.41421 −0.171571
$$199$$ 2.34315 0.166101 0.0830506 0.996545i $$-0.473534\pi$$
0.0830506 + 0.996545i $$0.473534\pi$$
$$200$$ 4.41421 0.312132
$$201$$ 0.686292 0.0484073
$$202$$ 18.4853 1.30062
$$203$$ −8.97056 −0.629610
$$204$$ −36.9706 −2.58846
$$205$$ −8.82843 −0.616604
$$206$$ 24.1421 1.68206
$$207$$ 4.00000 0.278019
$$208$$ 20.4853 1.42040
$$209$$ 1.00000 0.0691714
$$210$$ −13.6569 −0.942412
$$211$$ −24.0000 −1.65223 −0.826114 0.563503i $$-0.809453\pi$$
−0.826114 + 0.563503i $$0.809453\pi$$
$$212$$ −10.8284 −0.743699
$$213$$ 18.3431 1.25685
$$214$$ 32.1421 2.19719
$$215$$ −6.82843 −0.465695
$$216$$ −17.6569 −1.20140
$$217$$ −3.31371 −0.224949
$$218$$ −2.00000 −0.135457
$$219$$ −20.9706 −1.41706
$$220$$ −3.82843 −0.258113
$$221$$ −32.9706 −2.21784
$$222$$ −5.65685 −0.379663
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 4.48528 0.299685
$$225$$ 1.00000 0.0666667
$$226$$ −30.1421 −2.00503
$$227$$ 21.3137 1.41464 0.707320 0.706893i $$-0.249904\pi$$
0.707320 + 0.706893i $$0.249904\pi$$
$$228$$ −7.65685 −0.507088
$$229$$ 16.6274 1.09877 0.549385 0.835569i $$-0.314862\pi$$
0.549385 + 0.835569i $$0.314862\pi$$
$$230$$ 9.65685 0.636754
$$231$$ 5.65685 0.372194
$$232$$ 14.0000 0.919145
$$233$$ 10.4853 0.686914 0.343457 0.939168i $$-0.388402\pi$$
0.343457 + 0.939168i $$0.388402\pi$$
$$234$$ 16.4853 1.07768
$$235$$ 0 0
$$236$$ 47.7990 3.11145
$$237$$ −24.0000 −1.55897
$$238$$ 32.9706 2.13716
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 6.00000 0.387298
$$241$$ −16.8284 −1.08401 −0.542007 0.840374i $$-0.682335\pi$$
−0.542007 + 0.840374i $$0.682335\pi$$
$$242$$ 2.41421 0.155192
$$243$$ −10.0000 −0.641500
$$244$$ −14.0000 −0.896258
$$245$$ 1.00000 0.0638877
$$246$$ −42.6274 −2.71782
$$247$$ −6.82843 −0.434482
$$248$$ 5.17157 0.328395
$$249$$ 5.65685 0.358489
$$250$$ 2.41421 0.152688
$$251$$ 7.31371 0.461637 0.230819 0.972997i $$-0.425860\pi$$
0.230819 + 0.972997i $$0.425860\pi$$
$$252$$ −10.8284 −0.682127
$$253$$ −4.00000 −0.251478
$$254$$ 36.1421 2.26776
$$255$$ −9.65685 −0.604736
$$256$$ −29.9706 −1.87316
$$257$$ 1.85786 0.115890 0.0579452 0.998320i $$-0.481545\pi$$
0.0579452 + 0.998320i $$0.481545\pi$$
$$258$$ −32.9706 −2.05266
$$259$$ 3.31371 0.205904
$$260$$ 26.1421 1.62127
$$261$$ 3.17157 0.196315
$$262$$ 23.3137 1.44033
$$263$$ −12.4853 −0.769875 −0.384938 0.922943i $$-0.625777\pi$$
−0.384938 + 0.922943i $$0.625777\pi$$
$$264$$ −8.82843 −0.543352
$$265$$ −2.82843 −0.173749
$$266$$ 6.82843 0.418678
$$267$$ −26.6274 −1.62957
$$268$$ 1.31371 0.0802475
$$269$$ −23.6569 −1.44238 −0.721192 0.692735i $$-0.756405\pi$$
−0.721192 + 0.692735i $$0.756405\pi$$
$$270$$ −9.65685 −0.587697
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ −14.4853 −0.878299
$$273$$ −38.6274 −2.33784
$$274$$ 22.4853 1.35839
$$275$$ −1.00000 −0.0603023
$$276$$ 30.6274 1.84355
$$277$$ 8.14214 0.489214 0.244607 0.969622i $$-0.421341\pi$$
0.244607 + 0.969622i $$0.421341\pi$$
$$278$$ 50.6274 3.03643
$$279$$ 1.17157 0.0701402
$$280$$ −12.4853 −0.746138
$$281$$ 10.4853 0.625499 0.312750 0.949836i $$-0.398750\pi$$
0.312750 + 0.949836i $$0.398750\pi$$
$$282$$ 0 0
$$283$$ −16.4853 −0.979948 −0.489974 0.871737i $$-0.662994\pi$$
−0.489974 + 0.871737i $$0.662994\pi$$
$$284$$ 35.1127 2.08356
$$285$$ −2.00000 −0.118470
$$286$$ −16.4853 −0.974795
$$287$$ 24.9706 1.47397
$$288$$ −1.58579 −0.0934434
$$289$$ 6.31371 0.371395
$$290$$ 7.65685 0.449626
$$291$$ 20.2843 1.18909
$$292$$ −40.1421 −2.34914
$$293$$ −5.17157 −0.302127 −0.151063 0.988524i $$-0.548270\pi$$
−0.151063 + 0.988524i $$0.548270\pi$$
$$294$$ 4.82843 0.281600
$$295$$ 12.4853 0.726921
$$296$$ −5.17157 −0.300592
$$297$$ 4.00000 0.232104
$$298$$ 18.4853 1.07082
$$299$$ 27.3137 1.57959
$$300$$ 7.65685 0.442069
$$301$$ 19.3137 1.11322
$$302$$ 23.3137 1.34155
$$303$$ 15.3137 0.879750
$$304$$ −3.00000 −0.172062
$$305$$ −3.65685 −0.209391
$$306$$ −11.6569 −0.666378
$$307$$ −33.3137 −1.90131 −0.950657 0.310244i $$-0.899589\pi$$
−0.950657 + 0.310244i $$0.899589\pi$$
$$308$$ 10.8284 0.617007
$$309$$ 20.0000 1.13776
$$310$$ 2.82843 0.160644
$$311$$ 13.6569 0.774409 0.387205 0.921994i $$-0.373441\pi$$
0.387205 + 0.921994i $$0.373441\pi$$
$$312$$ 60.2843 3.41292
$$313$$ 26.9706 1.52447 0.762233 0.647303i $$-0.224103\pi$$
0.762233 + 0.647303i $$0.224103\pi$$
$$314$$ −41.7990 −2.35885
$$315$$ −2.82843 −0.159364
$$316$$ −45.9411 −2.58439
$$317$$ −15.7990 −0.887360 −0.443680 0.896185i $$-0.646327\pi$$
−0.443680 + 0.896185i $$0.646327\pi$$
$$318$$ −13.6569 −0.765838
$$319$$ −3.17157 −0.177574
$$320$$ −9.82843 −0.549426
$$321$$ 26.6274 1.48620
$$322$$ −27.3137 −1.52213
$$323$$ 4.82843 0.268661
$$324$$ −42.1127 −2.33959
$$325$$ 6.82843 0.378773
$$326$$ 42.6274 2.36091
$$327$$ −1.65685 −0.0916242
$$328$$ −38.9706 −2.15179
$$329$$ 0 0
$$330$$ −4.82843 −0.265796
$$331$$ −23.7990 −1.30811 −0.654055 0.756447i $$-0.726934\pi$$
−0.654055 + 0.756447i $$0.726934\pi$$
$$332$$ 10.8284 0.594287
$$333$$ −1.17157 −0.0642018
$$334$$ −33.7990 −1.84940
$$335$$ 0.343146 0.0187481
$$336$$ −16.9706 −0.925820
$$337$$ −11.5147 −0.627247 −0.313623 0.949547i $$-0.601543\pi$$
−0.313623 + 0.949547i $$0.601543\pi$$
$$338$$ 81.1838 4.41581
$$339$$ −24.9706 −1.35621
$$340$$ −18.4853 −1.00251
$$341$$ −1.17157 −0.0634442
$$342$$ −2.41421 −0.130546
$$343$$ 16.9706 0.916324
$$344$$ −30.1421 −1.62516
$$345$$ 8.00000 0.430706
$$346$$ 6.82843 0.367099
$$347$$ −31.1127 −1.67022 −0.835109 0.550085i $$-0.814595\pi$$
−0.835109 + 0.550085i $$0.814595\pi$$
$$348$$ 24.2843 1.30177
$$349$$ −24.6274 −1.31828 −0.659138 0.752022i $$-0.729079\pi$$
−0.659138 + 0.752022i $$0.729079\pi$$
$$350$$ −6.82843 −0.364995
$$351$$ −27.3137 −1.45790
$$352$$ 1.58579 0.0845227
$$353$$ −10.0000 −0.532246 −0.266123 0.963939i $$-0.585743\pi$$
−0.266123 + 0.963939i $$0.585743\pi$$
$$354$$ 60.2843 3.20407
$$355$$ 9.17157 0.486777
$$356$$ −50.9706 −2.70143
$$357$$ 27.3137 1.44559
$$358$$ 30.1421 1.59306
$$359$$ 19.3137 1.01934 0.509669 0.860370i $$-0.329768\pi$$
0.509669 + 0.860370i $$0.329768\pi$$
$$360$$ 4.41421 0.232649
$$361$$ 1.00000 0.0526316
$$362$$ −12.8284 −0.674247
$$363$$ 2.00000 0.104973
$$364$$ −73.9411 −3.87557
$$365$$ −10.4853 −0.548825
$$366$$ −17.6569 −0.922939
$$367$$ 35.3137 1.84336 0.921680 0.387950i $$-0.126817\pi$$
0.921680 + 0.387950i $$0.126817\pi$$
$$368$$ 12.0000 0.625543
$$369$$ −8.82843 −0.459590
$$370$$ −2.82843 −0.147043
$$371$$ 8.00000 0.415339
$$372$$ 8.97056 0.465102
$$373$$ −13.1716 −0.681998 −0.340999 0.940064i $$-0.610765\pi$$
−0.340999 + 0.940064i $$0.610765\pi$$
$$374$$ 11.6569 0.602762
$$375$$ 2.00000 0.103280
$$376$$ 0 0
$$377$$ 21.6569 1.11538
$$378$$ 27.3137 1.40487
$$379$$ −18.8284 −0.967151 −0.483576 0.875303i $$-0.660662\pi$$
−0.483576 + 0.875303i $$0.660662\pi$$
$$380$$ −3.82843 −0.196394
$$381$$ 29.9411 1.53393
$$382$$ 40.9706 2.09624
$$383$$ −13.3137 −0.680299 −0.340149 0.940371i $$-0.610478\pi$$
−0.340149 + 0.940371i $$0.610478\pi$$
$$384$$ −41.1127 −2.09802
$$385$$ 2.82843 0.144150
$$386$$ 38.1421 1.94138
$$387$$ −6.82843 −0.347108
$$388$$ 38.8284 1.97121
$$389$$ −16.6274 −0.843044 −0.421522 0.906818i $$-0.638504\pi$$
−0.421522 + 0.906818i $$0.638504\pi$$
$$390$$ 32.9706 1.66953
$$391$$ −19.3137 −0.976736
$$392$$ 4.41421 0.222951
$$393$$ 19.3137 0.974248
$$394$$ −33.3137 −1.67832
$$395$$ −12.0000 −0.603786
$$396$$ −3.82843 −0.192386
$$397$$ 13.3137 0.668196 0.334098 0.942538i $$-0.391568\pi$$
0.334098 + 0.942538i $$0.391568\pi$$
$$398$$ 5.65685 0.283552
$$399$$ 5.65685 0.283197
$$400$$ 3.00000 0.150000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 1.65685 0.0826364
$$403$$ 8.00000 0.398508
$$404$$ 29.3137 1.45841
$$405$$ −11.0000 −0.546594
$$406$$ −21.6569 −1.07481
$$407$$ 1.17157 0.0580727
$$408$$ −42.6274 −2.11037
$$409$$ −7.17157 −0.354611 −0.177306 0.984156i $$-0.556738\pi$$
−0.177306 + 0.984156i $$0.556738\pi$$
$$410$$ −21.3137 −1.05261
$$411$$ 18.6274 0.918823
$$412$$ 38.2843 1.88613
$$413$$ −35.3137 −1.73767
$$414$$ 9.65685 0.474608
$$415$$ 2.82843 0.138842
$$416$$ −10.8284 −0.530907
$$417$$ 41.9411 2.05387
$$418$$ 2.41421 0.118083
$$419$$ −12.9706 −0.633653 −0.316827 0.948483i $$-0.602617\pi$$
−0.316827 + 0.948483i $$0.602617\pi$$
$$420$$ −21.6569 −1.05675
$$421$$ −10.6863 −0.520818 −0.260409 0.965498i $$-0.583857\pi$$
−0.260409 + 0.965498i $$0.583857\pi$$
$$422$$ −57.9411 −2.82053
$$423$$ 0 0
$$424$$ −12.4853 −0.606339
$$425$$ −4.82843 −0.234213
$$426$$ 44.2843 2.14558
$$427$$ 10.3431 0.500540
$$428$$ 50.9706 2.46376
$$429$$ −13.6569 −0.659359
$$430$$ −16.4853 −0.794991
$$431$$ −26.6274 −1.28260 −0.641299 0.767291i $$-0.721604\pi$$
−0.641299 + 0.767291i $$0.721604\pi$$
$$432$$ −12.0000 −0.577350
$$433$$ 37.4558 1.80001 0.900006 0.435876i $$-0.143562\pi$$
0.900006 + 0.435876i $$0.143562\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 6.34315 0.304131
$$436$$ −3.17157 −0.151891
$$437$$ −4.00000 −0.191346
$$438$$ −50.6274 −2.41907
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ −4.41421 −0.210439
$$441$$ 1.00000 0.0476190
$$442$$ −79.5980 −3.78609
$$443$$ 26.3431 1.25160 0.625800 0.779983i $$-0.284772\pi$$
0.625800 + 0.779983i $$0.284772\pi$$
$$444$$ −8.97056 −0.425724
$$445$$ −13.3137 −0.631130
$$446$$ 4.82843 0.228633
$$447$$ 15.3137 0.724314
$$448$$ 27.7990 1.31338
$$449$$ 23.6569 1.11644 0.558218 0.829694i $$-0.311485\pi$$
0.558218 + 0.829694i $$0.311485\pi$$
$$450$$ 2.41421 0.113807
$$451$$ 8.82843 0.415714
$$452$$ −47.7990 −2.24828
$$453$$ 19.3137 0.907437
$$454$$ 51.4558 2.41494
$$455$$ −19.3137 −0.905441
$$456$$ −8.82843 −0.413429
$$457$$ −23.4558 −1.09722 −0.548609 0.836079i $$-0.684842\pi$$
−0.548609 + 0.836079i $$0.684842\pi$$
$$458$$ 40.1421 1.87572
$$459$$ 19.3137 0.901487
$$460$$ 15.3137 0.714005
$$461$$ 38.9706 1.81504 0.907520 0.420009i $$-0.137973\pi$$
0.907520 + 0.420009i $$0.137973\pi$$
$$462$$ 13.6569 0.635374
$$463$$ 26.6274 1.23748 0.618741 0.785595i $$-0.287643\pi$$
0.618741 + 0.785595i $$0.287643\pi$$
$$464$$ 9.51472 0.441710
$$465$$ 2.34315 0.108661
$$466$$ 25.3137 1.17263
$$467$$ 24.0000 1.11059 0.555294 0.831654i $$-0.312606\pi$$
0.555294 + 0.831654i $$0.312606\pi$$
$$468$$ 26.1421 1.20842
$$469$$ −0.970563 −0.0448164
$$470$$ 0 0
$$471$$ −34.6274 −1.59555
$$472$$ 55.1127 2.53677
$$473$$ 6.82843 0.313971
$$474$$ −57.9411 −2.66132
$$475$$ −1.00000 −0.0458831
$$476$$ 52.2843 2.39645
$$477$$ −2.82843 −0.129505
$$478$$ −19.3137 −0.883388
$$479$$ −24.9706 −1.14093 −0.570467 0.821320i $$-0.693238\pi$$
−0.570467 + 0.821320i $$0.693238\pi$$
$$480$$ −3.17157 −0.144762
$$481$$ −8.00000 −0.364769
$$482$$ −40.6274 −1.85053
$$483$$ −22.6274 −1.02958
$$484$$ 3.82843 0.174019
$$485$$ 10.1421 0.460531
$$486$$ −24.1421 −1.09511
$$487$$ −9.31371 −0.422044 −0.211022 0.977481i $$-0.567679\pi$$
−0.211022 + 0.977481i $$0.567679\pi$$
$$488$$ −16.1421 −0.730720
$$489$$ 35.3137 1.59694
$$490$$ 2.41421 0.109063
$$491$$ −23.3137 −1.05213 −0.526066 0.850443i $$-0.676334\pi$$
−0.526066 + 0.850443i $$0.676334\pi$$
$$492$$ −67.5980 −3.04755
$$493$$ −15.3137 −0.689695
$$494$$ −16.4853 −0.741708
$$495$$ −1.00000 −0.0449467
$$496$$ 3.51472 0.157816
$$497$$ −25.9411 −1.16362
$$498$$ 13.6569 0.611978
$$499$$ −34.6274 −1.55014 −0.775068 0.631878i $$-0.782284\pi$$
−0.775068 + 0.631878i $$0.782284\pi$$
$$500$$ 3.82843 0.171212
$$501$$ −28.0000 −1.25095
$$502$$ 17.6569 0.788064
$$503$$ 7.79899 0.347740 0.173870 0.984769i $$-0.444373\pi$$
0.173870 + 0.984769i $$0.444373\pi$$
$$504$$ −12.4853 −0.556139
$$505$$ 7.65685 0.340726
$$506$$ −9.65685 −0.429300
$$507$$ 67.2548 2.98689
$$508$$ 57.3137 2.54288
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ −23.3137 −1.03235
$$511$$ 29.6569 1.31194
$$512$$ −31.2426 −1.38074
$$513$$ 4.00000 0.176604
$$514$$ 4.48528 0.197837
$$515$$ 10.0000 0.440653
$$516$$ −52.2843 −2.30169
$$517$$ 0 0
$$518$$ 8.00000 0.351500
$$519$$ 5.65685 0.248308
$$520$$ 30.1421 1.32182
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 7.65685 0.335131
$$523$$ 27.6569 1.20935 0.604675 0.796472i $$-0.293303\pi$$
0.604675 + 0.796472i $$0.293303\pi$$
$$524$$ 36.9706 1.61507
$$525$$ −5.65685 −0.246885
$$526$$ −30.1421 −1.31426
$$527$$ −5.65685 −0.246416
$$528$$ −6.00000 −0.261116
$$529$$ −7.00000 −0.304348
$$530$$ −6.82843 −0.296608
$$531$$ 12.4853 0.541815
$$532$$ 10.8284 0.469472
$$533$$ −60.2843 −2.61120
$$534$$ −64.2843 −2.78185
$$535$$ 13.3137 0.575602
$$536$$ 1.51472 0.0654259
$$537$$ 24.9706 1.07756
$$538$$ −57.1127 −2.46230
$$539$$ −1.00000 −0.0430730
$$540$$ −15.3137 −0.658997
$$541$$ −39.9411 −1.71720 −0.858602 0.512644i $$-0.828666\pi$$
−0.858602 + 0.512644i $$0.828666\pi$$
$$542$$ −57.9411 −2.48878
$$543$$ −10.6274 −0.456066
$$544$$ 7.65685 0.328285
$$545$$ −0.828427 −0.0354859
$$546$$ −93.2548 −3.99094
$$547$$ −34.2843 −1.46589 −0.732945 0.680288i $$-0.761855\pi$$
−0.732945 + 0.680288i $$0.761855\pi$$
$$548$$ 35.6569 1.52319
$$549$$ −3.65685 −0.156071
$$550$$ −2.41421 −0.102942
$$551$$ −3.17157 −0.135114
$$552$$ 35.3137 1.50305
$$553$$ 33.9411 1.44332
$$554$$ 19.6569 0.835140
$$555$$ −2.34315 −0.0994610
$$556$$ 80.2843 3.40481
$$557$$ 35.4558 1.50231 0.751156 0.660125i $$-0.229497\pi$$
0.751156 + 0.660125i $$0.229497\pi$$
$$558$$ 2.82843 0.119737
$$559$$ −46.6274 −1.97213
$$560$$ −8.48528 −0.358569
$$561$$ 9.65685 0.407713
$$562$$ 25.3137 1.06779
$$563$$ −12.3431 −0.520202 −0.260101 0.965581i $$-0.583756\pi$$
−0.260101 + 0.965581i $$0.583756\pi$$
$$564$$ 0 0
$$565$$ −12.4853 −0.525260
$$566$$ −39.7990 −1.67288
$$567$$ 31.1127 1.30661
$$568$$ 40.4853 1.69872
$$569$$ 20.1421 0.844402 0.422201 0.906502i $$-0.361258\pi$$
0.422201 + 0.906502i $$0.361258\pi$$
$$570$$ −4.82843 −0.202241
$$571$$ −30.3431 −1.26982 −0.634911 0.772586i $$-0.718963\pi$$
−0.634911 + 0.772586i $$0.718963\pi$$
$$572$$ −26.1421 −1.09306
$$573$$ 33.9411 1.41791
$$574$$ 60.2843 2.51622
$$575$$ 4.00000 0.166812
$$576$$ −9.82843 −0.409518
$$577$$ −17.3137 −0.720779 −0.360390 0.932802i $$-0.617356\pi$$
−0.360390 + 0.932802i $$0.617356\pi$$
$$578$$ 15.2426 0.634010
$$579$$ 31.5980 1.31317
$$580$$ 12.1421 0.504175
$$581$$ −8.00000 −0.331896
$$582$$ 48.9706 2.02990
$$583$$ 2.82843 0.117141
$$584$$ −46.2843 −1.91526
$$585$$ 6.82843 0.282321
$$586$$ −12.4853 −0.515762
$$587$$ 32.0000 1.32078 0.660391 0.750922i $$-0.270391\pi$$
0.660391 + 0.750922i $$0.270391\pi$$
$$588$$ 7.65685 0.315763
$$589$$ −1.17157 −0.0482738
$$590$$ 30.1421 1.24093
$$591$$ −27.5980 −1.13523
$$592$$ −3.51472 −0.144454
$$593$$ 36.8284 1.51236 0.756181 0.654362i $$-0.227063\pi$$
0.756181 + 0.654362i $$0.227063\pi$$
$$594$$ 9.65685 0.396226
$$595$$ 13.6569 0.559876
$$596$$ 29.3137 1.20074
$$597$$ 4.68629 0.191797
$$598$$ 65.9411 2.69653
$$599$$ −27.7990 −1.13584 −0.567918 0.823085i $$-0.692251\pi$$
−0.567918 + 0.823085i $$0.692251\pi$$
$$600$$ 8.82843 0.360419
$$601$$ 36.8284 1.50226 0.751131 0.660153i $$-0.229508\pi$$
0.751131 + 0.660153i $$0.229508\pi$$
$$602$$ 46.6274 1.90039
$$603$$ 0.343146 0.0139740
$$604$$ 36.9706 1.50431
$$605$$ 1.00000 0.0406558
$$606$$ 36.9706 1.50183
$$607$$ −6.97056 −0.282926 −0.141463 0.989944i $$-0.545181\pi$$
−0.141463 + 0.989944i $$0.545181\pi$$
$$608$$ 1.58579 0.0643121
$$609$$ −17.9411 −0.727011
$$610$$ −8.82843 −0.357453
$$611$$ 0 0
$$612$$ −18.4853 −0.747223
$$613$$ −28.1421 −1.13665 −0.568325 0.822804i $$-0.692408\pi$$
−0.568325 + 0.822804i $$0.692408\pi$$
$$614$$ −80.4264 −3.24575
$$615$$ −17.6569 −0.711993
$$616$$ 12.4853 0.503046
$$617$$ 22.9706 0.924760 0.462380 0.886682i $$-0.346996\pi$$
0.462380 + 0.886682i $$0.346996\pi$$
$$618$$ 48.2843 1.94228
$$619$$ 47.3137 1.90170 0.950849 0.309654i $$-0.100213\pi$$
0.950849 + 0.309654i $$0.100213\pi$$
$$620$$ 4.48528 0.180133
$$621$$ −16.0000 −0.642058
$$622$$ 32.9706 1.32200
$$623$$ 37.6569 1.50869
$$624$$ 40.9706 1.64014
$$625$$ 1.00000 0.0400000
$$626$$ 65.1127 2.60243
$$627$$ 2.00000 0.0798723
$$628$$ −66.2843 −2.64503
$$629$$ 5.65685 0.225554
$$630$$ −6.82843 −0.272051
$$631$$ −5.65685 −0.225196 −0.112598 0.993641i $$-0.535917\pi$$
−0.112598 + 0.993641i $$0.535917\pi$$
$$632$$ −52.9706 −2.10706
$$633$$ −48.0000 −1.90783
$$634$$ −38.1421 −1.51482
$$635$$ 14.9706 0.594089
$$636$$ −21.6569 −0.858750
$$637$$ 6.82843 0.270552
$$638$$ −7.65685 −0.303138
$$639$$ 9.17157 0.362822
$$640$$ −20.5563 −0.812561
$$641$$ −19.9411 −0.787627 −0.393814 0.919190i $$-0.628844\pi$$
−0.393814 + 0.919190i $$0.628844\pi$$
$$642$$ 64.2843 2.53710
$$643$$ −21.6569 −0.854063 −0.427031 0.904237i $$-0.640441\pi$$
−0.427031 + 0.904237i $$0.640441\pi$$
$$644$$ −43.3137 −1.70680
$$645$$ −13.6569 −0.537738
$$646$$ 11.6569 0.458633
$$647$$ 20.9706 0.824438 0.412219 0.911085i $$-0.364754\pi$$
0.412219 + 0.911085i $$0.364754\pi$$
$$648$$ −48.5563 −1.90747
$$649$$ −12.4853 −0.490090
$$650$$ 16.4853 0.646606
$$651$$ −6.62742 −0.259749
$$652$$ 67.5980 2.64734
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ 9.65685 0.377325
$$656$$ −26.4853 −1.03408
$$657$$ −10.4853 −0.409070
$$658$$ 0 0
$$659$$ 7.02944 0.273828 0.136914 0.990583i $$-0.456282\pi$$
0.136914 + 0.990583i $$0.456282\pi$$
$$660$$ −7.65685 −0.298043
$$661$$ 42.9706 1.67136 0.835681 0.549216i $$-0.185073\pi$$
0.835681 + 0.549216i $$0.185073\pi$$
$$662$$ −57.4558 −2.23308
$$663$$ −65.9411 −2.56094
$$664$$ 12.4853 0.484523
$$665$$ 2.82843 0.109682
$$666$$ −2.82843 −0.109599
$$667$$ 12.6863 0.491215
$$668$$ −53.5980 −2.07377
$$669$$ 4.00000 0.154649
$$670$$ 0.828427 0.0320049
$$671$$ 3.65685 0.141171
$$672$$ 8.97056 0.346047
$$673$$ −33.4558 −1.28963 −0.644814 0.764340i $$-0.723065\pi$$
−0.644814 + 0.764340i $$0.723065\pi$$
$$674$$ −27.7990 −1.07078
$$675$$ −4.00000 −0.153960
$$676$$ 128.740 4.95154
$$677$$ −1.85786 −0.0714035 −0.0357018 0.999362i $$-0.511367\pi$$
−0.0357018 + 0.999362i $$0.511367\pi$$
$$678$$ −60.2843 −2.31520
$$679$$ −28.6863 −1.10088
$$680$$ −21.3137 −0.817343
$$681$$ 42.6274 1.63349
$$682$$ −2.82843 −0.108306
$$683$$ −5.31371 −0.203323 −0.101662 0.994819i $$-0.532416\pi$$
−0.101662 + 0.994819i $$0.532416\pi$$
$$684$$ −3.82843 −0.146384
$$685$$ 9.31371 0.355859
$$686$$ 40.9706 1.56426
$$687$$ 33.2548 1.26875
$$688$$ −20.4853 −0.780994
$$689$$ −19.3137 −0.735794
$$690$$ 19.3137 0.735260
$$691$$ 10.6274 0.404286 0.202143 0.979356i $$-0.435209\pi$$
0.202143 + 0.979356i $$0.435209\pi$$
$$692$$ 10.8284 0.411635
$$693$$ 2.82843 0.107443
$$694$$ −75.1127 −2.85124
$$695$$ 20.9706 0.795459
$$696$$ 28.0000 1.06134
$$697$$ 42.6274 1.61463
$$698$$ −59.4558 −2.25044
$$699$$ 20.9706 0.793180
$$700$$ −10.8284 −0.409276
$$701$$ −39.9411 −1.50856 −0.754278 0.656555i $$-0.772013\pi$$
−0.754278 + 0.656555i $$0.772013\pi$$
$$702$$ −65.9411 −2.48879
$$703$$ 1.17157 0.0441867
$$704$$ 9.82843 0.370423
$$705$$ 0 0
$$706$$ −24.1421 −0.908601
$$707$$ −21.6569 −0.814490
$$708$$ 95.5980 3.59279
$$709$$ −42.0000 −1.57734 −0.788672 0.614815i $$-0.789231\pi$$
−0.788672 + 0.614815i $$0.789231\pi$$
$$710$$ 22.1421 0.830980
$$711$$ −12.0000 −0.450035
$$712$$ −58.7696 −2.20248
$$713$$ 4.68629 0.175503
$$714$$ 65.9411 2.46778
$$715$$ −6.82843 −0.255369
$$716$$ 47.7990 1.78633
$$717$$ −16.0000 −0.597531
$$718$$ 46.6274 1.74012
$$719$$ −12.6863 −0.473119 −0.236559 0.971617i $$-0.576020\pi$$
−0.236559 + 0.971617i $$0.576020\pi$$
$$720$$ 3.00000 0.111803
$$721$$ −28.2843 −1.05336
$$722$$ 2.41421 0.0898477
$$723$$ −33.6569 −1.25171
$$724$$ −20.3431 −0.756047
$$725$$ 3.17157 0.117789
$$726$$ 4.82843 0.179200
$$727$$ 24.9706 0.926107 0.463053 0.886330i $$-0.346754\pi$$
0.463053 + 0.886330i $$0.346754\pi$$
$$728$$ −85.2548 −3.15975
$$729$$ 13.0000 0.481481
$$730$$ −25.3137 −0.936902
$$731$$ 32.9706 1.21946
$$732$$ −28.0000 −1.03491
$$733$$ 6.48528 0.239539 0.119770 0.992802i $$-0.461784\pi$$
0.119770 + 0.992802i $$0.461784\pi$$
$$734$$ 85.2548 3.14681
$$735$$ 2.00000 0.0737711
$$736$$ −6.34315 −0.233811
$$737$$ −0.343146 −0.0126399
$$738$$ −21.3137 −0.784568
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ −4.48528 −0.164882
$$741$$ −13.6569 −0.501697
$$742$$ 19.3137 0.709029
$$743$$ 27.6569 1.01463 0.507316 0.861760i $$-0.330638\pi$$
0.507316 + 0.861760i $$0.330638\pi$$
$$744$$ 10.3431 0.379198
$$745$$ 7.65685 0.280525
$$746$$ −31.7990 −1.16424
$$747$$ 2.82843 0.103487
$$748$$ 18.4853 0.675889
$$749$$ −37.6569 −1.37595
$$750$$ 4.82843 0.176309
$$751$$ −51.7990 −1.89017 −0.945086 0.326822i $$-0.894022\pi$$
−0.945086 + 0.326822i $$0.894022\pi$$
$$752$$ 0 0
$$753$$ 14.6274 0.533053
$$754$$ 52.2843 1.90408
$$755$$ 9.65685 0.351449
$$756$$ 43.3137 1.57530
$$757$$ −6.68629 −0.243017 −0.121509 0.992590i $$-0.538773\pi$$
−0.121509 + 0.992590i $$0.538773\pi$$
$$758$$ −45.4558 −1.65103
$$759$$ −8.00000 −0.290382
$$760$$ −4.41421 −0.160120
$$761$$ 48.9117 1.77305 0.886524 0.462683i $$-0.153113\pi$$
0.886524 + 0.462683i $$0.153113\pi$$
$$762$$ 72.2843 2.61858
$$763$$ 2.34315 0.0848276
$$764$$ 64.9706 2.35055
$$765$$ −4.82843 −0.174572
$$766$$ −32.1421 −1.16134
$$767$$ 85.2548 3.07837
$$768$$ −59.9411 −2.16294
$$769$$ −1.02944 −0.0371225 −0.0185612 0.999828i $$-0.505909\pi$$
−0.0185612 + 0.999828i $$0.505909\pi$$
$$770$$ 6.82843 0.246079
$$771$$ 3.71573 0.133819
$$772$$ 60.4853 2.17691
$$773$$ −31.7990 −1.14373 −0.571865 0.820348i $$-0.693780\pi$$
−0.571865 + 0.820348i $$0.693780\pi$$
$$774$$ −16.4853 −0.592551
$$775$$ 1.17157 0.0420841
$$776$$ 44.7696 1.60713
$$777$$ 6.62742 0.237757
$$778$$ −40.1421 −1.43917
$$779$$ 8.82843 0.316311
$$780$$ 52.2843 1.87208
$$781$$ −9.17157 −0.328185
$$782$$ −46.6274 −1.66739
$$783$$ −12.6863 −0.453371
$$784$$ 3.00000 0.107143
$$785$$ −17.3137 −0.617953
$$786$$ 46.6274 1.66314
$$787$$ −14.9706 −0.533643 −0.266821 0.963746i $$-0.585973\pi$$
−0.266821 + 0.963746i $$0.585973\pi$$
$$788$$ −52.8284 −1.88193
$$789$$ −24.9706 −0.888976
$$790$$ −28.9706 −1.03073
$$791$$ 35.3137 1.25561
$$792$$ −4.41421 −0.156852
$$793$$ −24.9706 −0.886731
$$794$$ 32.1421 1.14068
$$795$$ −5.65685 −0.200628
$$796$$ 8.97056 0.317953
$$797$$ −10.8284 −0.383563 −0.191781 0.981438i $$-0.561426\pi$$
−0.191781 + 0.981438i $$0.561426\pi$$
$$798$$ 13.6569 0.483447
$$799$$ 0 0
$$800$$ −1.58579 −0.0560660
$$801$$ −13.3137 −0.470417
$$802$$ 43.4558 1.53448
$$803$$ 10.4853 0.370018
$$804$$ 2.62742 0.0926619
$$805$$ −11.3137 −0.398756
$$806$$ 19.3137 0.680296
$$807$$ −47.3137 −1.66552
$$808$$ 33.7990 1.18904
$$809$$ −43.6569 −1.53489 −0.767447 0.641113i $$-0.778473\pi$$
−0.767447 + 0.641113i $$0.778473\pi$$
$$810$$ −26.5563 −0.933095
$$811$$ −7.31371 −0.256819 −0.128410 0.991721i $$-0.540987\pi$$
−0.128410 + 0.991721i $$0.540987\pi$$
$$812$$ −34.3431 −1.20521
$$813$$ −48.0000 −1.68343
$$814$$ 2.82843 0.0991363
$$815$$ 17.6569 0.618493
$$816$$ −28.9706 −1.01417
$$817$$ 6.82843 0.238896
$$818$$ −17.3137 −0.605360
$$819$$ −19.3137 −0.674876
$$820$$ −33.7990 −1.18031
$$821$$ 44.6274 1.55751 0.778754 0.627330i $$-0.215852\pi$$
0.778754 + 0.627330i $$0.215852\pi$$
$$822$$ 44.9706 1.56853
$$823$$ 5.65685 0.197186 0.0985928 0.995128i $$-0.468566\pi$$
0.0985928 + 0.995128i $$0.468566\pi$$
$$824$$ 44.1421 1.53776
$$825$$ −2.00000 −0.0696311
$$826$$ −85.2548 −2.96640
$$827$$ 35.6569 1.23991 0.619955 0.784637i $$-0.287151\pi$$
0.619955 + 0.784637i $$0.287151\pi$$
$$828$$ 15.3137 0.532188
$$829$$ −26.9706 −0.936726 −0.468363 0.883536i $$-0.655156\pi$$
−0.468363 + 0.883536i $$0.655156\pi$$
$$830$$ 6.82843 0.237018
$$831$$ 16.2843 0.564895
$$832$$ −67.1127 −2.32671
$$833$$ −4.82843 −0.167295
$$834$$ 101.255 3.50617
$$835$$ −14.0000 −0.484490
$$836$$ 3.82843 0.132409
$$837$$ −4.68629 −0.161982
$$838$$ −31.3137 −1.08171
$$839$$ −7.51472 −0.259437 −0.129718 0.991551i $$-0.541407\pi$$
−0.129718 + 0.991551i $$0.541407\pi$$
$$840$$ −24.9706 −0.861566
$$841$$ −18.9411 −0.653142
$$842$$ −25.7990 −0.889092
$$843$$ 20.9706 0.722265
$$844$$ −91.8823 −3.16272
$$845$$ 33.6274 1.15682
$$846$$ 0 0
$$847$$ −2.82843 −0.0971859
$$848$$ −8.48528 −0.291386
$$849$$ −32.9706 −1.13155
$$850$$ −11.6569 −0.399827
$$851$$ −4.68629 −0.160644
$$852$$ 70.2254 2.40588
$$853$$ 42.7696 1.46440 0.732201 0.681089i $$-0.238493\pi$$
0.732201 + 0.681089i $$0.238493\pi$$
$$854$$ 24.9706 0.854475
$$855$$ −1.00000 −0.0341993
$$856$$ 58.7696 2.00870
$$857$$ −9.45584 −0.323005 −0.161503 0.986872i $$-0.551634\pi$$
−0.161503 + 0.986872i $$0.551634\pi$$
$$858$$ −32.9706 −1.12560
$$859$$ 31.3137 1.06841 0.534205 0.845355i $$-0.320611\pi$$
0.534205 + 0.845355i $$0.320611\pi$$
$$860$$ −26.1421 −0.891439
$$861$$ 49.9411 1.70199
$$862$$ −64.2843 −2.18953
$$863$$ 13.0294 0.443527 0.221764 0.975100i $$-0.428819\pi$$
0.221764 + 0.975100i $$0.428819\pi$$
$$864$$ 6.34315 0.215798
$$865$$ 2.82843 0.0961694
$$866$$ 90.4264 3.07281
$$867$$ 12.6274 0.428850
$$868$$ −12.6863 −0.430601
$$869$$ 12.0000 0.407072
$$870$$ 15.3137 0.519183
$$871$$ 2.34315 0.0793945
$$872$$ −3.65685 −0.123837
$$873$$ 10.1421 0.343259
$$874$$ −9.65685 −0.326648
$$875$$ −2.82843 −0.0956183
$$876$$ −80.2843 −2.71255
$$877$$ 15.5147 0.523895 0.261947 0.965082i $$-0.415635\pi$$
0.261947 + 0.965082i $$0.415635\pi$$
$$878$$ 48.2843 1.62952
$$879$$ −10.3431 −0.348866
$$880$$ −3.00000 −0.101130
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ 2.41421 0.0812908
$$883$$ 48.2843 1.62490 0.812448 0.583034i $$-0.198135\pi$$
0.812448 + 0.583034i $$0.198135\pi$$
$$884$$ −126.225 −4.24542
$$885$$ 24.9706 0.839376
$$886$$ 63.5980 2.13662
$$887$$ 25.3137 0.849951 0.424976 0.905205i $$-0.360283\pi$$
0.424976 + 0.905205i $$0.360283\pi$$
$$888$$ −10.3431 −0.347093
$$889$$ −42.3431 −1.42014
$$890$$ −32.1421 −1.07741
$$891$$ 11.0000 0.368514
$$892$$ 7.65685 0.256370
$$893$$ 0 0
$$894$$ 36.9706 1.23648
$$895$$ 12.4853 0.417337
$$896$$ 58.1421 1.94239
$$897$$ 54.6274 1.82396
$$898$$ 57.1127 1.90588
$$899$$ 3.71573 0.123926
$$900$$ 3.82843 0.127614
$$901$$ 13.6569 0.454976
$$902$$ 21.3137 0.709669
$$903$$ 38.6274 1.28544
$$904$$ −55.1127 −1.83302
$$905$$ −5.31371 −0.176634
$$906$$ 46.6274 1.54909
$$907$$ −3.65685 −0.121424 −0.0607119 0.998155i $$-0.519337\pi$$
−0.0607119 + 0.998155i $$0.519337\pi$$
$$908$$ 81.5980 2.70792
$$909$$ 7.65685 0.253962
$$910$$ −46.6274 −1.54568
$$911$$ 17.4558 0.578338 0.289169 0.957278i $$-0.406621\pi$$
0.289169 + 0.957278i $$0.406621\pi$$
$$912$$ −6.00000 −0.198680
$$913$$ −2.82843 −0.0936073
$$914$$ −56.6274 −1.87307
$$915$$ −7.31371 −0.241784
$$916$$ 63.6569 2.10328
$$917$$ −27.3137 −0.901978
$$918$$ 46.6274 1.53893
$$919$$ −0.970563 −0.0320159 −0.0160080 0.999872i $$-0.505096\pi$$
−0.0160080 + 0.999872i $$0.505096\pi$$
$$920$$ 17.6569 0.582129
$$921$$ −66.6274 −2.19545
$$922$$ 94.0833 3.09847
$$923$$ 62.6274 2.06141
$$924$$ 21.6569 0.712458
$$925$$ −1.17157 −0.0385211
$$926$$ 64.2843 2.11251
$$927$$ 10.0000 0.328443
$$928$$ −5.02944 −0.165099
$$929$$ −46.0000 −1.50921 −0.754606 0.656179i $$-0.772172\pi$$
−0.754606 + 0.656179i $$0.772172\pi$$
$$930$$ 5.65685 0.185496
$$931$$ −1.00000 −0.0327737
$$932$$ 40.1421 1.31490
$$933$$ 27.3137 0.894211
$$934$$ 57.9411 1.89589
$$935$$ 4.82843 0.157906
$$936$$ 30.1421 0.985227
$$937$$ −17.5147 −0.572181 −0.286090 0.958203i $$-0.592356\pi$$
−0.286090 + 0.958203i $$0.592356\pi$$
$$938$$ −2.34315 −0.0765064
$$939$$ 53.9411 1.76030
$$940$$ 0 0
$$941$$ −4.14214 −0.135030 −0.0675149 0.997718i $$-0.521507\pi$$
−0.0675149 + 0.997718i $$0.521507\pi$$
$$942$$ −83.5980 −2.72377
$$943$$ −35.3137 −1.14997
$$944$$ 37.4558 1.21908
$$945$$ 11.3137 0.368035
$$946$$ 16.4853 0.535983
$$947$$ 19.3137 0.627611 0.313806 0.949487i $$-0.398396\pi$$
0.313806 + 0.949487i $$0.398396\pi$$
$$948$$ −91.8823 −2.98420
$$949$$ −71.5980 −2.32417
$$950$$ −2.41421 −0.0783274
$$951$$ −31.5980 −1.02463
$$952$$ 60.2843 1.95382
$$953$$ −38.1421 −1.23554 −0.617772 0.786357i $$-0.711965\pi$$
−0.617772 + 0.786357i $$0.711965\pi$$
$$954$$ −6.82843 −0.221078
$$955$$ 16.9706 0.549155
$$956$$ −30.6274 −0.990561
$$957$$ −6.34315 −0.205045
$$958$$ −60.2843 −1.94770
$$959$$ −26.3431 −0.850665
$$960$$ −19.6569 −0.634422
$$961$$ −29.6274 −0.955723
$$962$$ −19.3137 −0.622699
$$963$$ 13.3137 0.429028
$$964$$ −64.4264 −2.07503
$$965$$ 15.7990 0.508587
$$966$$ −54.6274 −1.75761
$$967$$ −24.7696 −0.796535 −0.398268 0.917269i $$-0.630388\pi$$
−0.398268 + 0.917269i $$0.630388\pi$$
$$968$$ 4.41421 0.141878
$$969$$ 9.65685 0.310223
$$970$$ 24.4853 0.786175
$$971$$ 7.51472 0.241159 0.120579 0.992704i $$-0.461525\pi$$
0.120579 + 0.992704i $$0.461525\pi$$
$$972$$ −38.2843 −1.22797
$$973$$ −59.3137 −1.90151
$$974$$ −22.4853 −0.720475
$$975$$ 13.6569 0.437369
$$976$$ −10.9706 −0.351159
$$977$$ 21.1716 0.677339 0.338669 0.940905i $$-0.390023\pi$$
0.338669 + 0.940905i $$0.390023\pi$$
$$978$$ 85.2548 2.72615
$$979$$ 13.3137 0.425508
$$980$$ 3.82843 0.122295
$$981$$ −0.828427 −0.0264496
$$982$$ −56.2843 −1.79610
$$983$$ 51.6569 1.64760 0.823799 0.566882i $$-0.191851\pi$$
0.823799 + 0.566882i $$0.191851\pi$$
$$984$$ −77.9411 −2.48467
$$985$$ −13.7990 −0.439672
$$986$$ −36.9706 −1.17738
$$987$$ 0 0
$$988$$ −26.1421 −0.831692
$$989$$ −27.3137 −0.868525
$$990$$ −2.41421 −0.0767287
$$991$$ −40.0833 −1.27329 −0.636643 0.771158i $$-0.719678\pi$$
−0.636643 + 0.771158i $$0.719678\pi$$
$$992$$ −1.85786 −0.0589873
$$993$$ −47.5980 −1.51048
$$994$$ −62.6274 −1.98642
$$995$$ 2.34315 0.0742827
$$996$$ 21.6569 0.686224
$$997$$ −4.14214 −0.131183 −0.0655914 0.997847i $$-0.520893\pi$$
−0.0655914 + 0.997847i $$0.520893\pi$$
$$998$$ −83.5980 −2.64625
$$999$$ 4.68629 0.148268
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 1045.2.a.c.1.2 2
3.2 odd 2 9405.2.a.o.1.1 2
5.4 even 2 5225.2.a.e.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.c.1.2 2 1.1 even 1 trivial
5225.2.a.e.1.1 2 5.4 even 2
9405.2.a.o.1.1 2 3.2 odd 2