# Properties

 Label 1638.2.a.u.1.1 Level $1638$ Weight $2$ Character 1638.1 Self dual yes Analytic conductor $13.079$ 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] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.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: $$13.0794958511$$ Analytic rank: $$1$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1638.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.56155 q^{10} +1.56155 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.68466 q^{17} -4.68466 q^{19} -3.56155 q^{20} -1.56155 q^{22} +5.56155 q^{23} +7.68466 q^{25} -1.00000 q^{26} -1.00000 q^{28} -6.68466 q^{29} +6.24621 q^{31} -1.00000 q^{32} -6.68466 q^{34} +3.56155 q^{35} -7.56155 q^{37} +4.68466 q^{38} +3.56155 q^{40} +1.12311 q^{41} -6.43845 q^{43} +1.56155 q^{44} -5.56155 q^{46} +1.00000 q^{49} -7.68466 q^{50} +1.00000 q^{52} -12.2462 q^{53} -5.56155 q^{55} +1.00000 q^{56} +6.68466 q^{58} -2.24621 q^{59} +6.68466 q^{61} -6.24621 q^{62} +1.00000 q^{64} -3.56155 q^{65} -7.12311 q^{67} +6.68466 q^{68} -3.56155 q^{70} -8.00000 q^{71} -3.56155 q^{73} +7.56155 q^{74} -4.68466 q^{76} -1.56155 q^{77} -11.1231 q^{79} -3.56155 q^{80} -1.12311 q^{82} -8.87689 q^{83} -23.8078 q^{85} +6.43845 q^{86} -1.56155 q^{88} -10.0000 q^{89} -1.00000 q^{91} +5.56155 q^{92} +16.6847 q^{95} +14.4924 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8} + 3 q^{10} - q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} + q^{17} + 3 q^{19} - 3 q^{20} + q^{22} + 7 q^{23} + 3 q^{25} - 2 q^{26} - 2 q^{28} - q^{29} - 4 q^{31} - 2 q^{32} - q^{34} + 3 q^{35} - 11 q^{37} - 3 q^{38} + 3 q^{40} - 6 q^{41} - 17 q^{43} - q^{44} - 7 q^{46} + 2 q^{49} - 3 q^{50} + 2 q^{52} - 8 q^{53} - 7 q^{55} + 2 q^{56} + q^{58} + 12 q^{59} + q^{61} + 4 q^{62} + 2 q^{64} - 3 q^{65} - 6 q^{67} + q^{68} - 3 q^{70} - 16 q^{71} - 3 q^{73} + 11 q^{74} + 3 q^{76} + q^{77} - 14 q^{79} - 3 q^{80} + 6 q^{82} - 26 q^{83} - 27 q^{85} + 17 q^{86} + q^{88} - 20 q^{89} - 2 q^{91} + 7 q^{92} + 21 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 2 * q^7 - 2 * q^8 + 3 * q^10 - q^11 + 2 * q^13 + 2 * q^14 + 2 * q^16 + q^17 + 3 * q^19 - 3 * q^20 + q^22 + 7 * q^23 + 3 * q^25 - 2 * q^26 - 2 * q^28 - q^29 - 4 * q^31 - 2 * q^32 - q^34 + 3 * q^35 - 11 * q^37 - 3 * q^38 + 3 * q^40 - 6 * q^41 - 17 * q^43 - q^44 - 7 * q^46 + 2 * q^49 - 3 * q^50 + 2 * q^52 - 8 * q^53 - 7 * q^55 + 2 * q^56 + q^58 + 12 * q^59 + q^61 + 4 * q^62 + 2 * q^64 - 3 * q^65 - 6 * q^67 + q^68 - 3 * q^70 - 16 * q^71 - 3 * q^73 + 11 * q^74 + 3 * q^76 + q^77 - 14 * q^79 - 3 * q^80 + 6 * q^82 - 26 * q^83 - 27 * q^85 + 17 * q^86 + q^88 - 20 * q^89 - 2 * q^91 + 7 * q^92 + 21 * q^95 - 4 * q^97 - 2 * q^98

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −3.56155 −1.59277 −0.796387 0.604787i $$-0.793258\pi$$
−0.796387 + 0.604787i $$0.793258\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 3.56155 1.12626
$$11$$ 1.56155 0.470826 0.235413 0.971895i $$-0.424356\pi$$
0.235413 + 0.971895i $$0.424356\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.68466 1.62127 0.810634 0.585553i $$-0.199123\pi$$
0.810634 + 0.585553i $$0.199123\pi$$
$$18$$ 0 0
$$19$$ −4.68466 −1.07473 −0.537367 0.843348i $$-0.680581\pi$$
−0.537367 + 0.843348i $$0.680581\pi$$
$$20$$ −3.56155 −0.796387
$$21$$ 0 0
$$22$$ −1.56155 −0.332924
$$23$$ 5.56155 1.15966 0.579832 0.814736i $$-0.303118\pi$$
0.579832 + 0.814736i $$0.303118\pi$$
$$24$$ 0 0
$$25$$ 7.68466 1.53693
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ −6.68466 −1.24131 −0.620655 0.784084i $$-0.713133\pi$$
−0.620655 + 0.784084i $$0.713133\pi$$
$$30$$ 0 0
$$31$$ 6.24621 1.12185 0.560926 0.827866i $$-0.310445\pi$$
0.560926 + 0.827866i $$0.310445\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −6.68466 −1.14641
$$35$$ 3.56155 0.602012
$$36$$ 0 0
$$37$$ −7.56155 −1.24311 −0.621556 0.783370i $$-0.713499\pi$$
−0.621556 + 0.783370i $$0.713499\pi$$
$$38$$ 4.68466 0.759952
$$39$$ 0 0
$$40$$ 3.56155 0.563131
$$41$$ 1.12311 0.175400 0.0876998 0.996147i $$-0.472048\pi$$
0.0876998 + 0.996147i $$0.472048\pi$$
$$42$$ 0 0
$$43$$ −6.43845 −0.981854 −0.490927 0.871201i $$-0.663342\pi$$
−0.490927 + 0.871201i $$0.663342\pi$$
$$44$$ 1.56155 0.235413
$$45$$ 0 0
$$46$$ −5.56155 −0.820006
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −7.68466 −1.08677
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ −12.2462 −1.68215 −0.841073 0.540921i $$-0.818076\pi$$
−0.841073 + 0.540921i $$0.818076\pi$$
$$54$$ 0 0
$$55$$ −5.56155 −0.749920
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 6.68466 0.877739
$$59$$ −2.24621 −0.292432 −0.146216 0.989253i $$-0.546709\pi$$
−0.146216 + 0.989253i $$0.546709\pi$$
$$60$$ 0 0
$$61$$ 6.68466 0.855883 0.427941 0.903806i $$-0.359239\pi$$
0.427941 + 0.903806i $$0.359239\pi$$
$$62$$ −6.24621 −0.793270
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −3.56155 −0.441756
$$66$$ 0 0
$$67$$ −7.12311 −0.870226 −0.435113 0.900376i $$-0.643292\pi$$
−0.435113 + 0.900376i $$0.643292\pi$$
$$68$$ 6.68466 0.810634
$$69$$ 0 0
$$70$$ −3.56155 −0.425687
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ −3.56155 −0.416848 −0.208424 0.978039i $$-0.566833\pi$$
−0.208424 + 0.978039i $$0.566833\pi$$
$$74$$ 7.56155 0.879013
$$75$$ 0 0
$$76$$ −4.68466 −0.537367
$$77$$ −1.56155 −0.177955
$$78$$ 0 0
$$79$$ −11.1231 −1.25145 −0.625724 0.780045i $$-0.715196\pi$$
−0.625724 + 0.780045i $$0.715196\pi$$
$$80$$ −3.56155 −0.398194
$$81$$ 0 0
$$82$$ −1.12311 −0.124026
$$83$$ −8.87689 −0.974366 −0.487183 0.873300i $$-0.661975\pi$$
−0.487183 + 0.873300i $$0.661975\pi$$
$$84$$ 0 0
$$85$$ −23.8078 −2.58231
$$86$$ 6.43845 0.694276
$$87$$ 0 0
$$88$$ −1.56155 −0.166462
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 5.56155 0.579832
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 16.6847 1.71181
$$96$$ 0 0
$$97$$ 14.4924 1.47148 0.735741 0.677263i $$-0.236834\pi$$
0.735741 + 0.677263i $$0.236834\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 7.68466 0.768466
$$101$$ 5.12311 0.509768 0.254884 0.966972i $$-0.417963\pi$$
0.254884 + 0.966972i $$0.417963\pi$$
$$102$$ 0 0
$$103$$ −0.684658 −0.0674614 −0.0337307 0.999431i $$-0.510739\pi$$
−0.0337307 + 0.999431i $$0.510739\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 12.2462 1.18946
$$107$$ 8.87689 0.858162 0.429081 0.903266i $$-0.358838\pi$$
0.429081 + 0.903266i $$0.358838\pi$$
$$108$$ 0 0
$$109$$ −16.9309 −1.62168 −0.810842 0.585266i $$-0.800990\pi$$
−0.810842 + 0.585266i $$0.800990\pi$$
$$110$$ 5.56155 0.530273
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ 4.24621 0.399450 0.199725 0.979852i $$-0.435995\pi$$
0.199725 + 0.979852i $$0.435995\pi$$
$$114$$ 0 0
$$115$$ −19.8078 −1.84708
$$116$$ −6.68466 −0.620655
$$117$$ 0 0
$$118$$ 2.24621 0.206781
$$119$$ −6.68466 −0.612782
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ −6.68466 −0.605201
$$123$$ 0 0
$$124$$ 6.24621 0.560926
$$125$$ −9.56155 −0.855211
$$126$$ 0 0
$$127$$ −4.87689 −0.432754 −0.216377 0.976310i $$-0.569424\pi$$
−0.216377 + 0.976310i $$0.569424\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 3.56155 0.312369
$$131$$ 9.56155 0.835397 0.417698 0.908586i $$-0.362837\pi$$
0.417698 + 0.908586i $$0.362837\pi$$
$$132$$ 0 0
$$133$$ 4.68466 0.406211
$$134$$ 7.12311 0.615343
$$135$$ 0 0
$$136$$ −6.68466 −0.573205
$$137$$ 3.56155 0.304284 0.152142 0.988359i $$-0.451383\pi$$
0.152142 + 0.988359i $$0.451383\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 3.56155 0.301006
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ 1.56155 0.130584
$$144$$ 0 0
$$145$$ 23.8078 1.97713
$$146$$ 3.56155 0.294756
$$147$$ 0 0
$$148$$ −7.56155 −0.621556
$$149$$ 17.6155 1.44312 0.721560 0.692352i $$-0.243425\pi$$
0.721560 + 0.692352i $$0.243425\pi$$
$$150$$ 0 0
$$151$$ 11.8078 0.960902 0.480451 0.877022i $$-0.340473\pi$$
0.480451 + 0.877022i $$0.340473\pi$$
$$152$$ 4.68466 0.379976
$$153$$ 0 0
$$154$$ 1.56155 0.125834
$$155$$ −22.2462 −1.78686
$$156$$ 0 0
$$157$$ −15.5616 −1.24195 −0.620974 0.783832i $$-0.713263\pi$$
−0.620974 + 0.783832i $$0.713263\pi$$
$$158$$ 11.1231 0.884907
$$159$$ 0 0
$$160$$ 3.56155 0.281565
$$161$$ −5.56155 −0.438312
$$162$$ 0 0
$$163$$ −16.8769 −1.32190 −0.660950 0.750430i $$-0.729847\pi$$
−0.660950 + 0.750430i $$0.729847\pi$$
$$164$$ 1.12311 0.0876998
$$165$$ 0 0
$$166$$ 8.87689 0.688981
$$167$$ −22.9309 −1.77444 −0.887222 0.461343i $$-0.847368\pi$$
−0.887222 + 0.461343i $$0.847368\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 23.8078 1.82597
$$171$$ 0 0
$$172$$ −6.43845 −0.490927
$$173$$ −20.2462 −1.53929 −0.769645 0.638471i $$-0.779567\pi$$
−0.769645 + 0.638471i $$0.779567\pi$$
$$174$$ 0 0
$$175$$ −7.68466 −0.580906
$$176$$ 1.56155 0.117706
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ −16.4924 −1.23270 −0.616351 0.787472i $$-0.711390\pi$$
−0.616351 + 0.787472i $$0.711390\pi$$
$$180$$ 0 0
$$181$$ −0.246211 −0.0183007 −0.00915037 0.999958i $$-0.502913\pi$$
−0.00915037 + 0.999958i $$0.502913\pi$$
$$182$$ 1.00000 0.0741249
$$183$$ 0 0
$$184$$ −5.56155 −0.410003
$$185$$ 26.9309 1.98000
$$186$$ 0 0
$$187$$ 10.4384 0.763335
$$188$$ 0 0
$$189$$ 0 0
$$190$$ −16.6847 −1.21043
$$191$$ −14.9309 −1.08036 −0.540180 0.841550i $$-0.681644\pi$$
−0.540180 + 0.841550i $$0.681644\pi$$
$$192$$ 0 0
$$193$$ 19.3693 1.39423 0.697117 0.716957i $$-0.254466\pi$$
0.697117 + 0.716957i $$0.254466\pi$$
$$194$$ −14.4924 −1.04050
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −10.8769 −0.774947 −0.387473 0.921881i $$-0.626652\pi$$
−0.387473 + 0.921881i $$0.626652\pi$$
$$198$$ 0 0
$$199$$ 6.93087 0.491316 0.245658 0.969357i $$-0.420996\pi$$
0.245658 + 0.969357i $$0.420996\pi$$
$$200$$ −7.68466 −0.543387
$$201$$ 0 0
$$202$$ −5.12311 −0.360460
$$203$$ 6.68466 0.469171
$$204$$ 0 0
$$205$$ −4.00000 −0.279372
$$206$$ 0.684658 0.0477024
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −7.31534 −0.506013
$$210$$ 0 0
$$211$$ −15.8078 −1.08825 −0.544126 0.839004i $$-0.683139\pi$$
−0.544126 + 0.839004i $$0.683139\pi$$
$$212$$ −12.2462 −0.841073
$$213$$ 0 0
$$214$$ −8.87689 −0.606812
$$215$$ 22.9309 1.56387
$$216$$ 0 0
$$217$$ −6.24621 −0.424020
$$218$$ 16.9309 1.14670
$$219$$ 0 0
$$220$$ −5.56155 −0.374960
$$221$$ 6.68466 0.449659
$$222$$ 0 0
$$223$$ −23.6155 −1.58141 −0.790706 0.612196i $$-0.790287\pi$$
−0.790706 + 0.612196i $$0.790287\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −4.24621 −0.282454
$$227$$ 7.12311 0.472777 0.236389 0.971659i $$-0.424036\pi$$
0.236389 + 0.971659i $$0.424036\pi$$
$$228$$ 0 0
$$229$$ 28.2462 1.86656 0.933281 0.359147i $$-0.116932\pi$$
0.933281 + 0.359147i $$0.116932\pi$$
$$230$$ 19.8078 1.30609
$$231$$ 0 0
$$232$$ 6.68466 0.438869
$$233$$ −27.3693 −1.79302 −0.896512 0.443020i $$-0.853907\pi$$
−0.896512 + 0.443020i $$0.853907\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −2.24621 −0.146216
$$237$$ 0 0
$$238$$ 6.68466 0.433302
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 8.56155 0.550357
$$243$$ 0 0
$$244$$ 6.68466 0.427941
$$245$$ −3.56155 −0.227539
$$246$$ 0 0
$$247$$ −4.68466 −0.298078
$$248$$ −6.24621 −0.396635
$$249$$ 0 0
$$250$$ 9.56155 0.604726
$$251$$ 7.80776 0.492822 0.246411 0.969165i $$-0.420749\pi$$
0.246411 + 0.969165i $$0.420749\pi$$
$$252$$ 0 0
$$253$$ 8.68466 0.546000
$$254$$ 4.87689 0.306004
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 4.24621 0.264871 0.132436 0.991192i $$-0.457720\pi$$
0.132436 + 0.991192i $$0.457720\pi$$
$$258$$ 0 0
$$259$$ 7.56155 0.469852
$$260$$ −3.56155 −0.220878
$$261$$ 0 0
$$262$$ −9.56155 −0.590715
$$263$$ 30.2462 1.86506 0.932531 0.361091i $$-0.117596\pi$$
0.932531 + 0.361091i $$0.117596\pi$$
$$264$$ 0 0
$$265$$ 43.6155 2.67928
$$266$$ −4.68466 −0.287235
$$267$$ 0 0
$$268$$ −7.12311 −0.435113
$$269$$ −20.2462 −1.23443 −0.617217 0.786793i $$-0.711740\pi$$
−0.617217 + 0.786793i $$0.711740\pi$$
$$270$$ 0 0
$$271$$ 4.87689 0.296250 0.148125 0.988969i $$-0.452676\pi$$
0.148125 + 0.988969i $$0.452676\pi$$
$$272$$ 6.68466 0.405317
$$273$$ 0 0
$$274$$ −3.56155 −0.215161
$$275$$ 12.0000 0.723627
$$276$$ 0 0
$$277$$ −22.4924 −1.35144 −0.675719 0.737159i $$-0.736167\pi$$
−0.675719 + 0.737159i $$0.736167\pi$$
$$278$$ −12.0000 −0.719712
$$279$$ 0 0
$$280$$ −3.56155 −0.212843
$$281$$ −16.2462 −0.969168 −0.484584 0.874745i $$-0.661029\pi$$
−0.484584 + 0.874745i $$0.661029\pi$$
$$282$$ 0 0
$$283$$ 18.2462 1.08462 0.542312 0.840177i $$-0.317549\pi$$
0.542312 + 0.840177i $$0.317549\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −1.56155 −0.0923366
$$287$$ −1.12311 −0.0662948
$$288$$ 0 0
$$289$$ 27.6847 1.62851
$$290$$ −23.8078 −1.39804
$$291$$ 0 0
$$292$$ −3.56155 −0.208424
$$293$$ −24.7386 −1.44525 −0.722623 0.691242i $$-0.757064\pi$$
−0.722623 + 0.691242i $$0.757064\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 7.56155 0.439506
$$297$$ 0 0
$$298$$ −17.6155 −1.02044
$$299$$ 5.56155 0.321633
$$300$$ 0 0
$$301$$ 6.43845 0.371106
$$302$$ −11.8078 −0.679460
$$303$$ 0 0
$$304$$ −4.68466 −0.268684
$$305$$ −23.8078 −1.36323
$$306$$ 0 0
$$307$$ 26.2462 1.49795 0.748975 0.662598i $$-0.230546\pi$$
0.748975 + 0.662598i $$0.230546\pi$$
$$308$$ −1.56155 −0.0889777
$$309$$ 0 0
$$310$$ 22.2462 1.26350
$$311$$ 12.8769 0.730182 0.365091 0.930972i $$-0.381038\pi$$
0.365091 + 0.930972i $$0.381038\pi$$
$$312$$ 0 0
$$313$$ −13.6155 −0.769595 −0.384798 0.923001i $$-0.625729\pi$$
−0.384798 + 0.923001i $$0.625729\pi$$
$$314$$ 15.5616 0.878189
$$315$$ 0 0
$$316$$ −11.1231 −0.625724
$$317$$ −2.87689 −0.161582 −0.0807912 0.996731i $$-0.525745\pi$$
−0.0807912 + 0.996731i $$0.525745\pi$$
$$318$$ 0 0
$$319$$ −10.4384 −0.584441
$$320$$ −3.56155 −0.199097
$$321$$ 0 0
$$322$$ 5.56155 0.309933
$$323$$ −31.3153 −1.74243
$$324$$ 0 0
$$325$$ 7.68466 0.426268
$$326$$ 16.8769 0.934725
$$327$$ 0 0
$$328$$ −1.12311 −0.0620131
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 13.3693 0.734844 0.367422 0.930054i $$-0.380240\pi$$
0.367422 + 0.930054i $$0.380240\pi$$
$$332$$ −8.87689 −0.487183
$$333$$ 0 0
$$334$$ 22.9309 1.25472
$$335$$ 25.3693 1.38607
$$336$$ 0 0
$$337$$ 4.43845 0.241778 0.120889 0.992666i $$-0.461426\pi$$
0.120889 + 0.992666i $$0.461426\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ −23.8078 −1.29116
$$341$$ 9.75379 0.528197
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 6.43845 0.347138
$$345$$ 0 0
$$346$$ 20.2462 1.08844
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 0 0
$$349$$ 7.75379 0.415051 0.207525 0.978230i $$-0.433459\pi$$
0.207525 + 0.978230i $$0.433459\pi$$
$$350$$ 7.68466 0.410762
$$351$$ 0 0
$$352$$ −1.56155 −0.0832310
$$353$$ −30.4924 −1.62295 −0.811474 0.584389i $$-0.801334\pi$$
−0.811474 + 0.584389i $$0.801334\pi$$
$$354$$ 0 0
$$355$$ 28.4924 1.51222
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 16.4924 0.871652
$$359$$ 26.7386 1.41121 0.705606 0.708605i $$-0.250675\pi$$
0.705606 + 0.708605i $$0.250675\pi$$
$$360$$ 0 0
$$361$$ 2.94602 0.155054
$$362$$ 0.246211 0.0129406
$$363$$ 0 0
$$364$$ −1.00000 −0.0524142
$$365$$ 12.6847 0.663945
$$366$$ 0 0
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ 5.56155 0.289916
$$369$$ 0 0
$$370$$ −26.9309 −1.40007
$$371$$ 12.2462 0.635792
$$372$$ 0 0
$$373$$ −17.6155 −0.912097 −0.456049 0.889955i $$-0.650736\pi$$
−0.456049 + 0.889955i $$0.650736\pi$$
$$374$$ −10.4384 −0.539759
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.68466 −0.344277
$$378$$ 0 0
$$379$$ 34.2462 1.75911 0.879555 0.475797i $$-0.157840\pi$$
0.879555 + 0.475797i $$0.157840\pi$$
$$380$$ 16.6847 0.855905
$$381$$ 0 0
$$382$$ 14.9309 0.763930
$$383$$ −27.4233 −1.40126 −0.700632 0.713522i $$-0.747099\pi$$
−0.700632 + 0.713522i $$0.747099\pi$$
$$384$$ 0 0
$$385$$ 5.56155 0.283443
$$386$$ −19.3693 −0.985872
$$387$$ 0 0
$$388$$ 14.4924 0.735741
$$389$$ 28.7386 1.45711 0.728553 0.684989i $$-0.240193\pi$$
0.728553 + 0.684989i $$0.240193\pi$$
$$390$$ 0 0
$$391$$ 37.1771 1.88013
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ 10.8769 0.547970
$$395$$ 39.6155 1.99327
$$396$$ 0 0
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ −6.93087 −0.347413
$$399$$ 0 0
$$400$$ 7.68466 0.384233
$$401$$ −8.24621 −0.411796 −0.205898 0.978573i $$-0.566012\pi$$
−0.205898 + 0.978573i $$0.566012\pi$$
$$402$$ 0 0
$$403$$ 6.24621 0.311146
$$404$$ 5.12311 0.254884
$$405$$ 0 0
$$406$$ −6.68466 −0.331754
$$407$$ −11.8078 −0.585289
$$408$$ 0 0
$$409$$ −4.93087 −0.243816 −0.121908 0.992541i $$-0.538901\pi$$
−0.121908 + 0.992541i $$0.538901\pi$$
$$410$$ 4.00000 0.197546
$$411$$ 0 0
$$412$$ −0.684658 −0.0337307
$$413$$ 2.24621 0.110529
$$414$$ 0 0
$$415$$ 31.6155 1.55195
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ 7.31534 0.357805
$$419$$ 36.6847 1.79216 0.896081 0.443890i $$-0.146402\pi$$
0.896081 + 0.443890i $$0.146402\pi$$
$$420$$ 0 0
$$421$$ −3.75379 −0.182948 −0.0914742 0.995807i $$-0.529158\pi$$
−0.0914742 + 0.995807i $$0.529158\pi$$
$$422$$ 15.8078 0.769510
$$423$$ 0 0
$$424$$ 12.2462 0.594729
$$425$$ 51.3693 2.49178
$$426$$ 0 0
$$427$$ −6.68466 −0.323493
$$428$$ 8.87689 0.429081
$$429$$ 0 0
$$430$$ −22.9309 −1.10582
$$431$$ 33.3693 1.60734 0.803672 0.595073i $$-0.202877\pi$$
0.803672 + 0.595073i $$0.202877\pi$$
$$432$$ 0 0
$$433$$ −31.3693 −1.50751 −0.753757 0.657154i $$-0.771760\pi$$
−0.753757 + 0.657154i $$0.771760\pi$$
$$434$$ 6.24621 0.299828
$$435$$ 0 0
$$436$$ −16.9309 −0.810842
$$437$$ −26.0540 −1.24633
$$438$$ 0 0
$$439$$ −6.93087 −0.330792 −0.165396 0.986227i $$-0.552890\pi$$
−0.165396 + 0.986227i $$0.552890\pi$$
$$440$$ 5.56155 0.265137
$$441$$ 0 0
$$442$$ −6.68466 −0.317957
$$443$$ 5.36932 0.255104 0.127552 0.991832i $$-0.459288\pi$$
0.127552 + 0.991832i $$0.459288\pi$$
$$444$$ 0 0
$$445$$ 35.6155 1.68834
$$446$$ 23.6155 1.11823
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ −10.6847 −0.504240 −0.252120 0.967696i $$-0.581128\pi$$
−0.252120 + 0.967696i $$0.581128\pi$$
$$450$$ 0 0
$$451$$ 1.75379 0.0825827
$$452$$ 4.24621 0.199725
$$453$$ 0 0
$$454$$ −7.12311 −0.334304
$$455$$ 3.56155 0.166968
$$456$$ 0 0
$$457$$ 27.3693 1.28028 0.640141 0.768257i $$-0.278876\pi$$
0.640141 + 0.768257i $$0.278876\pi$$
$$458$$ −28.2462 −1.31986
$$459$$ 0 0
$$460$$ −19.8078 −0.923542
$$461$$ 28.0540 1.30660 0.653302 0.757097i $$-0.273383\pi$$
0.653302 + 0.757097i $$0.273383\pi$$
$$462$$ 0 0
$$463$$ −8.68466 −0.403610 −0.201805 0.979426i $$-0.564681\pi$$
−0.201805 + 0.979426i $$0.564681\pi$$
$$464$$ −6.68466 −0.310327
$$465$$ 0 0
$$466$$ 27.3693 1.26786
$$467$$ 3.31534 0.153416 0.0767079 0.997054i $$-0.475559\pi$$
0.0767079 + 0.997054i $$0.475559\pi$$
$$468$$ 0 0
$$469$$ 7.12311 0.328914
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 2.24621 0.103390
$$473$$ −10.0540 −0.462282
$$474$$ 0 0
$$475$$ −36.0000 −1.65179
$$476$$ −6.68466 −0.306391
$$477$$ 0 0
$$478$$ 16.0000 0.731823
$$479$$ −32.3002 −1.47583 −0.737917 0.674892i $$-0.764190\pi$$
−0.737917 + 0.674892i $$0.764190\pi$$
$$480$$ 0 0
$$481$$ −7.56155 −0.344777
$$482$$ 14.0000 0.637683
$$483$$ 0 0
$$484$$ −8.56155 −0.389161
$$485$$ −51.6155 −2.34374
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ −6.68466 −0.302600
$$489$$ 0 0
$$490$$ 3.56155 0.160895
$$491$$ 8.87689 0.400609 0.200304 0.979734i $$-0.435807\pi$$
0.200304 + 0.979734i $$0.435807\pi$$
$$492$$ 0 0
$$493$$ −44.6847 −2.01250
$$494$$ 4.68466 0.210773
$$495$$ 0 0
$$496$$ 6.24621 0.280463
$$497$$ 8.00000 0.358849
$$498$$ 0 0
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ −9.56155 −0.427606
$$501$$ 0 0
$$502$$ −7.80776 −0.348478
$$503$$ 3.12311 0.139252 0.0696262 0.997573i $$-0.477819\pi$$
0.0696262 + 0.997573i $$0.477819\pi$$
$$504$$ 0 0
$$505$$ −18.2462 −0.811946
$$506$$ −8.68466 −0.386080
$$507$$ 0 0
$$508$$ −4.87689 −0.216377
$$509$$ 12.0540 0.534283 0.267142 0.963657i $$-0.413921\pi$$
0.267142 + 0.963657i $$0.413921\pi$$
$$510$$ 0 0
$$511$$ 3.56155 0.157554
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −4.24621 −0.187292
$$515$$ 2.43845 0.107451
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −7.56155 −0.332236
$$519$$ 0 0
$$520$$ 3.56155 0.156184
$$521$$ −2.68466 −0.117617 −0.0588085 0.998269i $$-0.518730\pi$$
−0.0588085 + 0.998269i $$0.518730\pi$$
$$522$$ 0 0
$$523$$ 21.7538 0.951227 0.475613 0.879654i $$-0.342226\pi$$
0.475613 + 0.879654i $$0.342226\pi$$
$$524$$ 9.56155 0.417698
$$525$$ 0 0
$$526$$ −30.2462 −1.31880
$$527$$ 41.7538 1.81882
$$528$$ 0 0
$$529$$ 7.93087 0.344820
$$530$$ −43.6155 −1.89454
$$531$$ 0 0
$$532$$ 4.68466 0.203106
$$533$$ 1.12311 0.0486471
$$534$$ 0 0
$$535$$ −31.6155 −1.36686
$$536$$ 7.12311 0.307671
$$537$$ 0 0
$$538$$ 20.2462 0.872876
$$539$$ 1.56155 0.0672608
$$540$$ 0 0
$$541$$ −32.9309 −1.41581 −0.707904 0.706308i $$-0.750359\pi$$
−0.707904 + 0.706308i $$0.750359\pi$$
$$542$$ −4.87689 −0.209481
$$543$$ 0 0
$$544$$ −6.68466 −0.286602
$$545$$ 60.3002 2.58298
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 3.56155 0.152142
$$549$$ 0 0
$$550$$ −12.0000 −0.511682
$$551$$ 31.3153 1.33408
$$552$$ 0 0
$$553$$ 11.1231 0.473003
$$554$$ 22.4924 0.955611
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ 3.36932 0.142763 0.0713813 0.997449i $$-0.477259\pi$$
0.0713813 + 0.997449i $$0.477259\pi$$
$$558$$ 0 0
$$559$$ −6.43845 −0.272317
$$560$$ 3.56155 0.150503
$$561$$ 0 0
$$562$$ 16.2462 0.685305
$$563$$ 6.05398 0.255145 0.127572 0.991829i $$-0.459282\pi$$
0.127572 + 0.991829i $$0.459282\pi$$
$$564$$ 0 0
$$565$$ −15.1231 −0.636234
$$566$$ −18.2462 −0.766945
$$567$$ 0 0
$$568$$ 8.00000 0.335673
$$569$$ −41.2311 −1.72850 −0.864248 0.503066i $$-0.832205\pi$$
−0.864248 + 0.503066i $$0.832205\pi$$
$$570$$ 0 0
$$571$$ 18.2462 0.763580 0.381790 0.924249i $$-0.375308\pi$$
0.381790 + 0.924249i $$0.375308\pi$$
$$572$$ 1.56155 0.0652918
$$573$$ 0 0
$$574$$ 1.12311 0.0468775
$$575$$ 42.7386 1.78232
$$576$$ 0 0
$$577$$ −30.0000 −1.24892 −0.624458 0.781058i $$-0.714680\pi$$
−0.624458 + 0.781058i $$0.714680\pi$$
$$578$$ −27.6847 −1.15153
$$579$$ 0 0
$$580$$ 23.8078 0.988564
$$581$$ 8.87689 0.368276
$$582$$ 0 0
$$583$$ −19.1231 −0.791998
$$584$$ 3.56155 0.147378
$$585$$ 0 0
$$586$$ 24.7386 1.02194
$$587$$ −13.3693 −0.551811 −0.275905 0.961185i $$-0.588978\pi$$
−0.275905 + 0.961185i $$0.588978\pi$$
$$588$$ 0 0
$$589$$ −29.2614 −1.20569
$$590$$ −8.00000 −0.329355
$$591$$ 0 0
$$592$$ −7.56155 −0.310778
$$593$$ 20.2462 0.831412 0.415706 0.909499i $$-0.363534\pi$$
0.415706 + 0.909499i $$0.363534\pi$$
$$594$$ 0 0
$$595$$ 23.8078 0.976023
$$596$$ 17.6155 0.721560
$$597$$ 0 0
$$598$$ −5.56155 −0.227429
$$599$$ −21.5616 −0.880981 −0.440491 0.897757i $$-0.645195\pi$$
−0.440491 + 0.897757i $$0.645195\pi$$
$$600$$ 0 0
$$601$$ −10.8769 −0.443678 −0.221839 0.975083i $$-0.571206\pi$$
−0.221839 + 0.975083i $$0.571206\pi$$
$$602$$ −6.43845 −0.262412
$$603$$ 0 0
$$604$$ 11.8078 0.480451
$$605$$ 30.4924 1.23969
$$606$$ 0 0
$$607$$ −18.4384 −0.748393 −0.374197 0.927349i $$-0.622082\pi$$
−0.374197 + 0.927349i $$0.622082\pi$$
$$608$$ 4.68466 0.189988
$$609$$ 0 0
$$610$$ 23.8078 0.963948
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 44.5464 1.79921 0.899606 0.436702i $$-0.143854\pi$$
0.899606 + 0.436702i $$0.143854\pi$$
$$614$$ −26.2462 −1.05921
$$615$$ 0 0
$$616$$ 1.56155 0.0629168
$$617$$ 33.4233 1.34557 0.672786 0.739838i $$-0.265098\pi$$
0.672786 + 0.739838i $$0.265098\pi$$
$$618$$ 0 0
$$619$$ 19.3153 0.776349 0.388175 0.921586i $$-0.373106\pi$$
0.388175 + 0.921586i $$0.373106\pi$$
$$620$$ −22.2462 −0.893429
$$621$$ 0 0
$$622$$ −12.8769 −0.516316
$$623$$ 10.0000 0.400642
$$624$$ 0 0
$$625$$ −4.36932 −0.174773
$$626$$ 13.6155 0.544186
$$627$$ 0 0
$$628$$ −15.5616 −0.620974
$$629$$ −50.5464 −2.01542
$$630$$ 0 0
$$631$$ −22.9309 −0.912864 −0.456432 0.889758i $$-0.650873\pi$$
−0.456432 + 0.889758i $$0.650873\pi$$
$$632$$ 11.1231 0.442453
$$633$$ 0 0
$$634$$ 2.87689 0.114256
$$635$$ 17.3693 0.689280
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 10.4384 0.413262
$$639$$ 0 0
$$640$$ 3.56155 0.140783
$$641$$ 20.2462 0.799677 0.399839 0.916586i $$-0.369066\pi$$
0.399839 + 0.916586i $$0.369066\pi$$
$$642$$ 0 0
$$643$$ 16.1922 0.638559 0.319280 0.947661i $$-0.396559\pi$$
0.319280 + 0.947661i $$0.396559\pi$$
$$644$$ −5.56155 −0.219156
$$645$$ 0 0
$$646$$ 31.3153 1.23209
$$647$$ 14.2462 0.560076 0.280038 0.959989i $$-0.409653\pi$$
0.280038 + 0.959989i $$0.409653\pi$$
$$648$$ 0 0
$$649$$ −3.50758 −0.137684
$$650$$ −7.68466 −0.301417
$$651$$ 0 0
$$652$$ −16.8769 −0.660950
$$653$$ 16.9309 0.662556 0.331278 0.943533i $$-0.392520\pi$$
0.331278 + 0.943533i $$0.392520\pi$$
$$654$$ 0 0
$$655$$ −34.0540 −1.33060
$$656$$ 1.12311 0.0438499
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −21.3693 −0.832430 −0.416215 0.909266i $$-0.636644\pi$$
−0.416215 + 0.909266i $$0.636644\pi$$
$$660$$ 0 0
$$661$$ −0.246211 −0.00957651 −0.00478825 0.999989i $$-0.501524\pi$$
−0.00478825 + 0.999989i $$0.501524\pi$$
$$662$$ −13.3693 −0.519613
$$663$$ 0 0
$$664$$ 8.87689 0.344490
$$665$$ −16.6847 −0.647003
$$666$$ 0 0
$$667$$ −37.1771 −1.43950
$$668$$ −22.9309 −0.887222
$$669$$ 0 0
$$670$$ −25.3693 −0.980102
$$671$$ 10.4384 0.402972
$$672$$ 0 0
$$673$$ −33.8078 −1.30319 −0.651597 0.758566i $$-0.725901\pi$$
−0.651597 + 0.758566i $$0.725901\pi$$
$$674$$ −4.43845 −0.170963
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −2.49242 −0.0957916 −0.0478958 0.998852i $$-0.515252\pi$$
−0.0478958 + 0.998852i $$0.515252\pi$$
$$678$$ 0 0
$$679$$ −14.4924 −0.556168
$$680$$ 23.8078 0.912986
$$681$$ 0 0
$$682$$ −9.75379 −0.373492
$$683$$ −35.3153 −1.35130 −0.675652 0.737221i $$-0.736138\pi$$
−0.675652 + 0.737221i $$0.736138\pi$$
$$684$$ 0 0
$$685$$ −12.6847 −0.484656
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ −6.43845 −0.245463
$$689$$ −12.2462 −0.466543
$$690$$ 0 0
$$691$$ 16.4924 0.627401 0.313701 0.949522i $$-0.398431\pi$$
0.313701 + 0.949522i $$0.398431\pi$$
$$692$$ −20.2462 −0.769645
$$693$$ 0 0
$$694$$ −20.0000 −0.759190
$$695$$ −42.7386 −1.62117
$$696$$ 0 0
$$697$$ 7.50758 0.284370
$$698$$ −7.75379 −0.293485
$$699$$ 0 0
$$700$$ −7.68466 −0.290453
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ 35.4233 1.33601
$$704$$ 1.56155 0.0588532
$$705$$ 0 0
$$706$$ 30.4924 1.14760
$$707$$ −5.12311 −0.192674
$$708$$ 0 0
$$709$$ −16.2462 −0.610139 −0.305070 0.952330i $$-0.598680\pi$$
−0.305070 + 0.952330i $$0.598680\pi$$
$$710$$ −28.4924 −1.06930
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ 34.7386 1.30097
$$714$$ 0 0
$$715$$ −5.56155 −0.207990
$$716$$ −16.4924 −0.616351
$$717$$ 0 0
$$718$$ −26.7386 −0.997877
$$719$$ −43.1231 −1.60822 −0.804110 0.594480i $$-0.797358\pi$$
−0.804110 + 0.594480i $$0.797358\pi$$
$$720$$ 0 0
$$721$$ 0.684658 0.0254980
$$722$$ −2.94602 −0.109640
$$723$$ 0 0
$$724$$ −0.246211 −0.00915037
$$725$$ −51.3693 −1.90781
$$726$$ 0 0
$$727$$ −6.93087 −0.257052 −0.128526 0.991706i $$-0.541025\pi$$
−0.128526 + 0.991706i $$0.541025\pi$$
$$728$$ 1.00000 0.0370625
$$729$$ 0 0
$$730$$ −12.6847 −0.469480
$$731$$ −43.0388 −1.59185
$$732$$ 0 0
$$733$$ −41.6155 −1.53710 −0.768552 0.639787i $$-0.779023\pi$$
−0.768552 + 0.639787i $$0.779023\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ −5.56155 −0.205002
$$737$$ −11.1231 −0.409725
$$738$$ 0 0
$$739$$ −32.1080 −1.18111 −0.590555 0.806997i $$-0.701091\pi$$
−0.590555 + 0.806997i $$0.701091\pi$$
$$740$$ 26.9309 0.989998
$$741$$ 0 0
$$742$$ −12.2462 −0.449573
$$743$$ −28.1080 −1.03118 −0.515590 0.856835i $$-0.672427\pi$$
−0.515590 + 0.856835i $$0.672427\pi$$
$$744$$ 0 0
$$745$$ −62.7386 −2.29857
$$746$$ 17.6155 0.644950
$$747$$ 0 0
$$748$$ 10.4384 0.381667
$$749$$ −8.87689 −0.324355
$$750$$ 0 0
$$751$$ −6.24621 −0.227927 −0.113964 0.993485i $$-0.536355\pi$$
−0.113964 + 0.993485i $$0.536355\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 6.68466 0.243441
$$755$$ −42.0540 −1.53050
$$756$$ 0 0
$$757$$ 34.4924 1.25365 0.626824 0.779161i $$-0.284354\pi$$
0.626824 + 0.779161i $$0.284354\pi$$
$$758$$ −34.2462 −1.24388
$$759$$ 0 0
$$760$$ −16.6847 −0.605216
$$761$$ −5.12311 −0.185712 −0.0928562 0.995680i $$-0.529600\pi$$
−0.0928562 + 0.995680i $$0.529600\pi$$
$$762$$ 0 0
$$763$$ 16.9309 0.612939
$$764$$ −14.9309 −0.540180
$$765$$ 0 0
$$766$$ 27.4233 0.990844
$$767$$ −2.24621 −0.0811060
$$768$$ 0 0
$$769$$ −16.4384 −0.592786 −0.296393 0.955066i $$-0.595784\pi$$
−0.296393 + 0.955066i $$0.595784\pi$$
$$770$$ −5.56155 −0.200424
$$771$$ 0 0
$$772$$ 19.3693 0.697117
$$773$$ −52.9309 −1.90379 −0.951896 0.306423i $$-0.900868\pi$$
−0.951896 + 0.306423i $$0.900868\pi$$
$$774$$ 0 0
$$775$$ 48.0000 1.72421
$$776$$ −14.4924 −0.520248
$$777$$ 0 0
$$778$$ −28.7386 −1.03033
$$779$$ −5.26137 −0.188508
$$780$$ 0 0
$$781$$ −12.4924 −0.447014
$$782$$ −37.1771 −1.32945
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 55.4233 1.97814
$$786$$ 0 0
$$787$$ 20.3002 0.723624 0.361812 0.932251i $$-0.382158\pi$$
0.361812 + 0.932251i $$0.382158\pi$$
$$788$$ −10.8769 −0.387473
$$789$$ 0 0
$$790$$ −39.6155 −1.40946
$$791$$ −4.24621 −0.150978
$$792$$ 0 0
$$793$$ 6.68466 0.237379
$$794$$ 18.0000 0.638796
$$795$$ 0 0
$$796$$ 6.93087 0.245658
$$797$$ −31.3693 −1.11116 −0.555579 0.831464i $$-0.687503\pi$$
−0.555579 + 0.831464i $$0.687503\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −7.68466 −0.271694
$$801$$ 0 0
$$802$$ 8.24621 0.291184
$$803$$ −5.56155 −0.196263
$$804$$ 0 0
$$805$$ 19.8078 0.698132
$$806$$ −6.24621 −0.220013
$$807$$ 0 0
$$808$$ −5.12311 −0.180230
$$809$$ 2.49242 0.0876289 0.0438145 0.999040i $$-0.486049\pi$$
0.0438145 + 0.999040i $$0.486049\pi$$
$$810$$ 0 0
$$811$$ 41.5616 1.45942 0.729712 0.683755i $$-0.239654\pi$$
0.729712 + 0.683755i $$0.239654\pi$$
$$812$$ 6.68466 0.234586
$$813$$ 0 0
$$814$$ 11.8078 0.413862
$$815$$ 60.1080 2.10549
$$816$$ 0 0
$$817$$ 30.1619 1.05523
$$818$$ 4.93087 0.172404
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ −14.3845 −0.502022 −0.251011 0.967984i $$-0.580763\pi$$
−0.251011 + 0.967984i $$0.580763\pi$$
$$822$$ 0 0
$$823$$ 4.49242 0.156596 0.0782980 0.996930i $$-0.475051\pi$$
0.0782980 + 0.996930i $$0.475051\pi$$
$$824$$ 0.684658 0.0238512
$$825$$ 0 0
$$826$$ −2.24621 −0.0781557
$$827$$ −26.9309 −0.936478 −0.468239 0.883602i $$-0.655111\pi$$
−0.468239 + 0.883602i $$0.655111\pi$$
$$828$$ 0 0
$$829$$ 38.6847 1.34357 0.671787 0.740744i $$-0.265527\pi$$
0.671787 + 0.740744i $$0.265527\pi$$
$$830$$ −31.6155 −1.09739
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 6.68466 0.231610
$$834$$ 0 0
$$835$$ 81.6695 2.82629
$$836$$ −7.31534 −0.253006
$$837$$ 0 0
$$838$$ −36.6847 −1.26725
$$839$$ 10.7386 0.370739 0.185369 0.982669i $$-0.440652\pi$$
0.185369 + 0.982669i $$0.440652\pi$$
$$840$$ 0 0
$$841$$ 15.6847 0.540850
$$842$$ 3.75379 0.129364
$$843$$ 0 0
$$844$$ −15.8078 −0.544126
$$845$$ −3.56155 −0.122521
$$846$$ 0 0
$$847$$ 8.56155 0.294178
$$848$$ −12.2462 −0.420537
$$849$$ 0 0
$$850$$ −51.3693 −1.76195
$$851$$ −42.0540 −1.44159
$$852$$ 0 0
$$853$$ −27.3693 −0.937108 −0.468554 0.883435i $$-0.655225\pi$$
−0.468554 + 0.883435i $$0.655225\pi$$
$$854$$ 6.68466 0.228744
$$855$$ 0 0
$$856$$ −8.87689 −0.303406
$$857$$ 34.4924 1.17824 0.589119 0.808046i $$-0.299475\pi$$
0.589119 + 0.808046i $$0.299475\pi$$
$$858$$ 0 0
$$859$$ 5.75379 0.196317 0.0981584 0.995171i $$-0.468705\pi$$
0.0981584 + 0.995171i $$0.468705\pi$$
$$860$$ 22.9309 0.781936
$$861$$ 0 0
$$862$$ −33.3693 −1.13656
$$863$$ 17.3693 0.591258 0.295629 0.955303i $$-0.404471\pi$$
0.295629 + 0.955303i $$0.404471\pi$$
$$864$$ 0 0
$$865$$ 72.1080 2.45174
$$866$$ 31.3693 1.06597
$$867$$ 0 0
$$868$$ −6.24621 −0.212010
$$869$$ −17.3693 −0.589214
$$870$$ 0 0
$$871$$ −7.12311 −0.241357
$$872$$ 16.9309 0.573352
$$873$$ 0 0
$$874$$ 26.0540 0.881289
$$875$$ 9.56155 0.323239
$$876$$ 0 0
$$877$$ −40.2462 −1.35902 −0.679509 0.733667i $$-0.737807\pi$$
−0.679509 + 0.733667i $$0.737807\pi$$
$$878$$ 6.93087 0.233906
$$879$$ 0 0
$$880$$ −5.56155 −0.187480
$$881$$ 19.1771 0.646092 0.323046 0.946383i $$-0.395293\pi$$
0.323046 + 0.946383i $$0.395293\pi$$
$$882$$ 0 0
$$883$$ 1.56155 0.0525504 0.0262752 0.999655i $$-0.491635\pi$$
0.0262752 + 0.999655i $$0.491635\pi$$
$$884$$ 6.68466 0.224829
$$885$$ 0 0
$$886$$ −5.36932 −0.180386
$$887$$ 52.4924 1.76252 0.881262 0.472629i $$-0.156695\pi$$
0.881262 + 0.472629i $$0.156695\pi$$
$$888$$ 0 0
$$889$$ 4.87689 0.163566
$$890$$ −35.6155 −1.19384
$$891$$ 0 0
$$892$$ −23.6155 −0.790706
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 58.7386 1.96342
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 10.6847 0.356552
$$899$$ −41.7538 −1.39257
$$900$$ 0 0
$$901$$ −81.8617 −2.72721
$$902$$ −1.75379 −0.0583948
$$903$$ 0 0
$$904$$ −4.24621 −0.141227
$$905$$ 0.876894 0.0291490
$$906$$ 0 0
$$907$$ −7.50758 −0.249285 −0.124643 0.992202i $$-0.539778\pi$$
−0.124643 + 0.992202i $$0.539778\pi$$
$$908$$ 7.12311 0.236389
$$909$$ 0 0
$$910$$ −3.56155 −0.118064
$$911$$ 11.4233 0.378471 0.189235 0.981932i $$-0.439399\pi$$
0.189235 + 0.981932i $$0.439399\pi$$
$$912$$ 0 0
$$913$$ −13.8617 −0.458757
$$914$$ −27.3693 −0.905297
$$915$$ 0 0
$$916$$ 28.2462 0.933281
$$917$$ −9.56155 −0.315750
$$918$$ 0 0
$$919$$ 19.1231 0.630813 0.315407 0.948957i $$-0.397859\pi$$
0.315407 + 0.948957i $$0.397859\pi$$
$$920$$ 19.8078 0.653043
$$921$$ 0 0
$$922$$ −28.0540 −0.923908
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ −58.1080 −1.91058
$$926$$ 8.68466 0.285396
$$927$$ 0 0
$$928$$ 6.68466 0.219435
$$929$$ −25.6155 −0.840418 −0.420209 0.907427i $$-0.638043\pi$$
−0.420209 + 0.907427i $$0.638043\pi$$
$$930$$ 0 0
$$931$$ −4.68466 −0.153533
$$932$$ −27.3693 −0.896512
$$933$$ 0 0
$$934$$ −3.31534 −0.108481
$$935$$ −37.1771 −1.21582
$$936$$ 0 0
$$937$$ −46.9848 −1.53493 −0.767464 0.641092i $$-0.778482\pi$$
−0.767464 + 0.641092i $$0.778482\pi$$
$$938$$ −7.12311 −0.232578
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 34.0000 1.10837 0.554184 0.832394i $$-0.313030\pi$$
0.554184 + 0.832394i $$0.313030\pi$$
$$942$$ 0 0
$$943$$ 6.24621 0.203405
$$944$$ −2.24621 −0.0731079
$$945$$ 0 0
$$946$$ 10.0540 0.326883
$$947$$ 40.7926 1.32558 0.662791 0.748805i $$-0.269372\pi$$
0.662791 + 0.748805i $$0.269372\pi$$
$$948$$ 0 0
$$949$$ −3.56155 −0.115613
$$950$$ 36.0000 1.16799
$$951$$ 0 0
$$952$$ 6.68466 0.216651
$$953$$ −11.3693 −0.368288 −0.184144 0.982899i $$-0.558951\pi$$
−0.184144 + 0.982899i $$0.558951\pi$$
$$954$$ 0 0
$$955$$ 53.1771 1.72077
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 32.3002 1.04357
$$959$$ −3.56155 −0.115009
$$960$$ 0 0
$$961$$ 8.01515 0.258553
$$962$$ 7.56155 0.243794
$$963$$ 0 0
$$964$$ −14.0000 −0.450910
$$965$$ −68.9848 −2.22070
$$966$$ 0 0
$$967$$ −21.5616 −0.693373 −0.346686 0.937981i $$-0.612693\pi$$
−0.346686 + 0.937981i $$0.612693\pi$$
$$968$$ 8.56155 0.275179
$$969$$ 0 0
$$970$$ 51.6155 1.65727
$$971$$ −2.24621 −0.0720843 −0.0360422 0.999350i $$-0.511475\pi$$
−0.0360422 + 0.999350i $$0.511475\pi$$
$$972$$ 0 0
$$973$$ −12.0000 −0.384702
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ 6.68466 0.213971
$$977$$ 22.6847 0.725747 0.362873 0.931839i $$-0.381796\pi$$
0.362873 + 0.931839i $$0.381796\pi$$
$$978$$ 0 0
$$979$$ −15.6155 −0.499074
$$980$$ −3.56155 −0.113770
$$981$$ 0 0
$$982$$ −8.87689 −0.283273
$$983$$ 13.1771 0.420284 0.210142 0.977671i $$-0.432607\pi$$
0.210142 + 0.977671i $$0.432607\pi$$
$$984$$ 0 0
$$985$$ 38.7386 1.23432
$$986$$ 44.6847 1.42305
$$987$$ 0 0
$$988$$ −4.68466 −0.149039
$$989$$ −35.8078 −1.13862
$$990$$ 0 0
$$991$$ −7.61553 −0.241915 −0.120958 0.992658i $$-0.538597\pi$$
−0.120958 + 0.992658i $$0.538597\pi$$
$$992$$ −6.24621 −0.198317
$$993$$ 0 0
$$994$$ −8.00000 −0.253745
$$995$$ −24.6847 −0.782556
$$996$$ 0 0
$$997$$ 40.7386 1.29021 0.645103 0.764096i $$-0.276815\pi$$
0.645103 + 0.764096i $$0.276815\pi$$
$$998$$ −36.0000 −1.13956
$$999$$ 0 0
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 1638.2.a.u.1.1 2
3.2 odd 2 546.2.a.j.1.2 2
12.11 even 2 4368.2.a.be.1.2 2
21.20 even 2 3822.2.a.bo.1.1 2
39.38 odd 2 7098.2.a.bl.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.j.1.2 2 3.2 odd 2
1638.2.a.u.1.1 2 1.1 even 1 trivial
3822.2.a.bo.1.1 2 21.20 even 2
4368.2.a.be.1.2 2 12.11 even 2
7098.2.a.bl.1.1 2 39.38 odd 2