# Properties

 Label 1881.2.a.k.1.3 Level $1881$ Weight $2$ Character 1881.1 Self dual yes Analytic conductor $15.020$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, 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("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1881 = 3^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1881.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: $$15.0198606202$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.246832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ x^5 - 2*x^4 - 5*x^3 + 6*x^2 + 7*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.15351$$ of defining polynomial Character $$\chi$$ $$=$$ 1881.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.484093 q^{2} -1.76565 q^{4} -0.637602 q^{5} +2.66942 q^{7} +1.82293 q^{8} +O(q^{10})$$ $$q-0.484093 q^{2} -1.76565 q^{4} -0.637602 q^{5} +2.66942 q^{7} +1.82293 q^{8} +0.308658 q^{10} -1.00000 q^{11} -1.20725 q^{13} -1.29225 q^{14} +2.64884 q^{16} -0.222022 q^{17} -1.00000 q^{19} +1.12578 q^{20} +0.484093 q^{22} +9.48717 q^{23} -4.59346 q^{25} +0.584420 q^{26} -4.71327 q^{28} -4.94518 q^{29} -3.05563 q^{31} -4.92814 q^{32} +0.107479 q^{34} -1.70203 q^{35} +7.18015 q^{37} +0.484093 q^{38} -1.16230 q^{40} +6.92650 q^{41} +1.53538 q^{43} +1.76565 q^{44} -4.59267 q^{46} -5.94581 q^{47} +0.125785 q^{49} +2.22366 q^{50} +2.13158 q^{52} +9.63879 q^{53} +0.637602 q^{55} +4.86615 q^{56} +2.39392 q^{58} -2.65817 q^{59} -0.809792 q^{61} +1.47921 q^{62} -2.91201 q^{64} +0.769744 q^{65} -2.22447 q^{67} +0.392015 q^{68} +0.823938 q^{70} -2.58912 q^{71} +16.6108 q^{73} -3.47586 q^{74} +1.76565 q^{76} -2.66942 q^{77} +5.69298 q^{79} -1.68891 q^{80} -3.35307 q^{82} +2.93784 q^{83} +0.141562 q^{85} -0.743267 q^{86} -1.82293 q^{88} +5.54136 q^{89} -3.22265 q^{91} -16.7511 q^{92} +2.87832 q^{94} +0.637602 q^{95} +13.7728 q^{97} -0.0608914 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10})$$ 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 $$5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29} + 11 q^{31} - 14 q^{32} - 4 q^{34} + 8 q^{35} + q^{37} + 2 q^{38} - 16 q^{40} - 2 q^{41} + 20 q^{43} - 6 q^{44} - 4 q^{46} + 20 q^{47} + 3 q^{49} + 32 q^{50} + 6 q^{52} + 14 q^{53} - 5 q^{55} + 38 q^{56} - 6 q^{58} - 3 q^{59} - 10 q^{61} + 6 q^{62} + 9 q^{67} - 24 q^{68} + 50 q^{70} - 23 q^{71} - 8 q^{74} - 6 q^{76} - 6 q^{77} + 44 q^{79} + 18 q^{80} - 30 q^{82} + 14 q^{83} - 12 q^{85} - 52 q^{86} + 6 q^{88} + 27 q^{89} + 24 q^{91} - 58 q^{92} - 8 q^{94} - 5 q^{95} + 15 q^{97} + 10 q^{98}+O(q^{100})$$ 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.484093 −0.342305 −0.171153 0.985245i $$-0.554749\pi$$
−0.171153 + 0.985245i $$0.554749\pi$$
$$3$$ 0 0
$$4$$ −1.76565 −0.882827
$$5$$ −0.637602 −0.285144 −0.142572 0.989784i $$-0.545537\pi$$
−0.142572 + 0.989784i $$0.545537\pi$$
$$6$$ 0 0
$$7$$ 2.66942 1.00894 0.504472 0.863428i $$-0.331687\pi$$
0.504472 + 0.863428i $$0.331687\pi$$
$$8$$ 1.82293 0.644502
$$9$$ 0 0
$$10$$ 0.308658 0.0976064
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −1.20725 −0.334831 −0.167415 0.985886i $$-0.553542\pi$$
−0.167415 + 0.985886i $$0.553542\pi$$
$$14$$ −1.29225 −0.345367
$$15$$ 0 0
$$16$$ 2.64884 0.662211
$$17$$ −0.222022 −0.0538483 −0.0269242 0.999637i $$-0.508571\pi$$
−0.0269242 + 0.999637i $$0.508571\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 1.12578 0.251733
$$21$$ 0 0
$$22$$ 0.484093 0.103209
$$23$$ 9.48717 1.97821 0.989106 0.147206i $$-0.0470279\pi$$
0.989106 + 0.147206i $$0.0470279\pi$$
$$24$$ 0 0
$$25$$ −4.59346 −0.918693
$$26$$ 0.584420 0.114614
$$27$$ 0 0
$$28$$ −4.71327 −0.890724
$$29$$ −4.94518 −0.918297 −0.459148 0.888360i $$-0.651845\pi$$
−0.459148 + 0.888360i $$0.651845\pi$$
$$30$$ 0 0
$$31$$ −3.05563 −0.548808 −0.274404 0.961615i $$-0.588480\pi$$
−0.274404 + 0.961615i $$0.588480\pi$$
$$32$$ −4.92814 −0.871180
$$33$$ 0 0
$$34$$ 0.107479 0.0184326
$$35$$ −1.70203 −0.287695
$$36$$ 0 0
$$37$$ 7.18015 1.18041 0.590205 0.807254i $$-0.299047\pi$$
0.590205 + 0.807254i $$0.299047\pi$$
$$38$$ 0.484093 0.0785302
$$39$$ 0 0
$$40$$ −1.16230 −0.183776
$$41$$ 6.92650 1.08174 0.540869 0.841107i $$-0.318096\pi$$
0.540869 + 0.841107i $$0.318096\pi$$
$$42$$ 0 0
$$43$$ 1.53538 0.234144 0.117072 0.993123i $$-0.462649\pi$$
0.117072 + 0.993123i $$0.462649\pi$$
$$44$$ 1.76565 0.266182
$$45$$ 0 0
$$46$$ −4.59267 −0.677152
$$47$$ −5.94581 −0.867285 −0.433642 0.901085i $$-0.642772\pi$$
−0.433642 + 0.901085i $$0.642772\pi$$
$$48$$ 0 0
$$49$$ 0.125785 0.0179692
$$50$$ 2.22366 0.314473
$$51$$ 0 0
$$52$$ 2.13158 0.295598
$$53$$ 9.63879 1.32399 0.661995 0.749509i $$-0.269710\pi$$
0.661995 + 0.749509i $$0.269710\pi$$
$$54$$ 0 0
$$55$$ 0.637602 0.0859742
$$56$$ 4.86615 0.650266
$$57$$ 0 0
$$58$$ 2.39392 0.314338
$$59$$ −2.65817 −0.346065 −0.173032 0.984916i $$-0.555357\pi$$
−0.173032 + 0.984916i $$0.555357\pi$$
$$60$$ 0 0
$$61$$ −0.809792 −0.103683 −0.0518416 0.998655i $$-0.516509\pi$$
−0.0518416 + 0.998655i $$0.516509\pi$$
$$62$$ 1.47921 0.187860
$$63$$ 0 0
$$64$$ −2.91201 −0.364001
$$65$$ 0.769744 0.0954750
$$66$$ 0 0
$$67$$ −2.22447 −0.271763 −0.135881 0.990725i $$-0.543387\pi$$
−0.135881 + 0.990725i $$0.543387\pi$$
$$68$$ 0.392015 0.0475387
$$69$$ 0 0
$$70$$ 0.823938 0.0984794
$$71$$ −2.58912 −0.307272 −0.153636 0.988127i $$-0.549098\pi$$
−0.153636 + 0.988127i $$0.549098\pi$$
$$72$$ 0 0
$$73$$ 16.6108 1.94414 0.972071 0.234687i $$-0.0754065\pi$$
0.972071 + 0.234687i $$0.0754065\pi$$
$$74$$ −3.47586 −0.404060
$$75$$ 0 0
$$76$$ 1.76565 0.202534
$$77$$ −2.66942 −0.304208
$$78$$ 0 0
$$79$$ 5.69298 0.640510 0.320255 0.947331i $$-0.396231\pi$$
0.320255 + 0.947331i $$0.396231\pi$$
$$80$$ −1.68891 −0.188826
$$81$$ 0 0
$$82$$ −3.35307 −0.370284
$$83$$ 2.93784 0.322470 0.161235 0.986916i $$-0.448452\pi$$
0.161235 + 0.986916i $$0.448452\pi$$
$$84$$ 0 0
$$85$$ 0.141562 0.0153545
$$86$$ −0.743267 −0.0801486
$$87$$ 0 0
$$88$$ −1.82293 −0.194325
$$89$$ 5.54136 0.587383 0.293692 0.955900i $$-0.405116\pi$$
0.293692 + 0.955900i $$0.405116\pi$$
$$90$$ 0 0
$$91$$ −3.22265 −0.337826
$$92$$ −16.7511 −1.74642
$$93$$ 0 0
$$94$$ 2.87832 0.296876
$$95$$ 0.637602 0.0654166
$$96$$ 0 0
$$97$$ 13.7728 1.39842 0.699209 0.714917i $$-0.253536\pi$$
0.699209 + 0.714917i $$0.253536\pi$$
$$98$$ −0.0608914 −0.00615096
$$99$$ 0 0
$$100$$ 8.11047 0.811047
$$101$$ −12.0308 −1.19711 −0.598555 0.801082i $$-0.704258\pi$$
−0.598555 + 0.801082i $$0.704258\pi$$
$$102$$ 0 0
$$103$$ 8.53068 0.840553 0.420276 0.907396i $$-0.361933\pi$$
0.420276 + 0.907396i $$0.361933\pi$$
$$104$$ −2.20073 −0.215799
$$105$$ 0 0
$$106$$ −4.66607 −0.453208
$$107$$ −1.64585 −0.159110 −0.0795552 0.996830i $$-0.525350\pi$$
−0.0795552 + 0.996830i $$0.525350\pi$$
$$108$$ 0 0
$$109$$ 16.8065 1.60977 0.804886 0.593430i $$-0.202226\pi$$
0.804886 + 0.593430i $$0.202226\pi$$
$$110$$ −0.308658 −0.0294294
$$111$$ 0 0
$$112$$ 7.07087 0.668134
$$113$$ 11.2947 1.06252 0.531258 0.847210i $$-0.321720\pi$$
0.531258 + 0.847210i $$0.321720\pi$$
$$114$$ 0 0
$$115$$ −6.04904 −0.564076
$$116$$ 8.73148 0.810697
$$117$$ 0 0
$$118$$ 1.28680 0.118460
$$119$$ −0.592670 −0.0543300
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0.392015 0.0354913
$$123$$ 0 0
$$124$$ 5.39519 0.484502
$$125$$ 6.11681 0.547104
$$126$$ 0 0
$$127$$ 18.5894 1.64954 0.824771 0.565467i $$-0.191304\pi$$
0.824771 + 0.565467i $$0.191304\pi$$
$$128$$ 11.2660 0.995779
$$129$$ 0 0
$$130$$ −0.372628 −0.0326816
$$131$$ 3.09098 0.270060 0.135030 0.990842i $$-0.456887\pi$$
0.135030 + 0.990842i $$0.456887\pi$$
$$132$$ 0 0
$$133$$ −2.66942 −0.231468
$$134$$ 1.07685 0.0930257
$$135$$ 0 0
$$136$$ −0.404730 −0.0347053
$$137$$ 8.67619 0.741257 0.370629 0.928781i $$-0.379142\pi$$
0.370629 + 0.928781i $$0.379142\pi$$
$$138$$ 0 0
$$139$$ 14.5426 1.23348 0.616742 0.787166i $$-0.288452\pi$$
0.616742 + 0.787166i $$0.288452\pi$$
$$140$$ 3.00519 0.253985
$$141$$ 0 0
$$142$$ 1.25338 0.105181
$$143$$ 1.20725 0.100955
$$144$$ 0 0
$$145$$ 3.15306 0.261847
$$146$$ −8.04115 −0.665490
$$147$$ 0 0
$$148$$ −12.6777 −1.04210
$$149$$ 18.4716 1.51325 0.756625 0.653849i $$-0.226847\pi$$
0.756625 + 0.653849i $$0.226847\pi$$
$$150$$ 0 0
$$151$$ −0.913992 −0.0743796 −0.0371898 0.999308i $$-0.511841\pi$$
−0.0371898 + 0.999308i $$0.511841\pi$$
$$152$$ −1.82293 −0.147859
$$153$$ 0 0
$$154$$ 1.29225 0.104132
$$155$$ 1.94828 0.156489
$$156$$ 0 0
$$157$$ −2.39421 −0.191079 −0.0955395 0.995426i $$-0.530458\pi$$
−0.0955395 + 0.995426i $$0.530458\pi$$
$$158$$ −2.75593 −0.219250
$$159$$ 0 0
$$160$$ 3.14219 0.248412
$$161$$ 25.3252 1.99591
$$162$$ 0 0
$$163$$ 14.1986 1.11212 0.556061 0.831141i $$-0.312312\pi$$
0.556061 + 0.831141i $$0.312312\pi$$
$$164$$ −12.2298 −0.954987
$$165$$ 0 0
$$166$$ −1.42219 −0.110383
$$167$$ −9.25056 −0.715830 −0.357915 0.933754i $$-0.616512\pi$$
−0.357915 + 0.933754i $$0.616512\pi$$
$$168$$ 0 0
$$169$$ −11.5426 −0.887888
$$170$$ −0.0685290 −0.00525594
$$171$$ 0 0
$$172$$ −2.71095 −0.206708
$$173$$ −5.39376 −0.410080 −0.205040 0.978754i $$-0.565732\pi$$
−0.205040 + 0.978754i $$0.565732\pi$$
$$174$$ 0 0
$$175$$ −12.2619 −0.926910
$$176$$ −2.64884 −0.199664
$$177$$ 0 0
$$178$$ −2.68253 −0.201064
$$179$$ −1.30720 −0.0977048 −0.0488524 0.998806i $$-0.515556\pi$$
−0.0488524 + 0.998806i $$0.515556\pi$$
$$180$$ 0 0
$$181$$ −10.7624 −0.799961 −0.399980 0.916524i $$-0.630983\pi$$
−0.399980 + 0.916524i $$0.630983\pi$$
$$182$$ 1.56006 0.115639
$$183$$ 0 0
$$184$$ 17.2944 1.27496
$$185$$ −4.57808 −0.336587
$$186$$ 0 0
$$187$$ 0.222022 0.0162359
$$188$$ 10.4982 0.765663
$$189$$ 0 0
$$190$$ −0.308658 −0.0223924
$$191$$ −15.3041 −1.10737 −0.553684 0.832727i $$-0.686778\pi$$
−0.553684 + 0.832727i $$0.686778\pi$$
$$192$$ 0 0
$$193$$ −18.3032 −1.31749 −0.658747 0.752364i $$-0.728913\pi$$
−0.658747 + 0.752364i $$0.728913\pi$$
$$194$$ −6.66732 −0.478686
$$195$$ 0 0
$$196$$ −0.222092 −0.0158637
$$197$$ 0.576171 0.0410505 0.0205252 0.999789i $$-0.493466\pi$$
0.0205252 + 0.999789i $$0.493466\pi$$
$$198$$ 0 0
$$199$$ −10.1678 −0.720778 −0.360389 0.932802i $$-0.617356\pi$$
−0.360389 + 0.932802i $$0.617356\pi$$
$$200$$ −8.37354 −0.592099
$$201$$ 0 0
$$202$$ 5.82402 0.409777
$$203$$ −13.2007 −0.926510
$$204$$ 0 0
$$205$$ −4.41635 −0.308451
$$206$$ −4.12964 −0.287726
$$207$$ 0 0
$$208$$ −3.19781 −0.221728
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −7.25837 −0.499687 −0.249843 0.968286i $$-0.580379\pi$$
−0.249843 + 0.968286i $$0.580379\pi$$
$$212$$ −17.0188 −1.16885
$$213$$ 0 0
$$214$$ 0.796745 0.0544643
$$215$$ −0.978963 −0.0667647
$$216$$ 0 0
$$217$$ −8.15675 −0.553716
$$218$$ −8.13591 −0.551033
$$219$$ 0 0
$$220$$ −1.12578 −0.0759004
$$221$$ 0.268036 0.0180301
$$222$$ 0 0
$$223$$ 16.2772 1.09000 0.545000 0.838436i $$-0.316529\pi$$
0.545000 + 0.838436i $$0.316529\pi$$
$$224$$ −13.1553 −0.878972
$$225$$ 0 0
$$226$$ −5.46768 −0.363705
$$227$$ 20.9260 1.38890 0.694452 0.719539i $$-0.255647\pi$$
0.694452 + 0.719539i $$0.255647\pi$$
$$228$$ 0 0
$$229$$ −10.8083 −0.714234 −0.357117 0.934060i $$-0.616240\pi$$
−0.357117 + 0.934060i $$0.616240\pi$$
$$230$$ 2.92830 0.193086
$$231$$ 0 0
$$232$$ −9.01469 −0.591844
$$233$$ 25.4173 1.66514 0.832570 0.553919i $$-0.186868\pi$$
0.832570 + 0.553919i $$0.186868\pi$$
$$234$$ 0 0
$$235$$ 3.79106 0.247301
$$236$$ 4.69342 0.305515
$$237$$ 0 0
$$238$$ 0.286907 0.0185974
$$239$$ −0.352238 −0.0227844 −0.0113922 0.999935i $$-0.503626\pi$$
−0.0113922 + 0.999935i $$0.503626\pi$$
$$240$$ 0 0
$$241$$ −15.6832 −1.01024 −0.505122 0.863048i $$-0.668552\pi$$
−0.505122 + 0.863048i $$0.668552\pi$$
$$242$$ −0.484093 −0.0311187
$$243$$ 0 0
$$244$$ 1.42981 0.0915344
$$245$$ −0.0802004 −0.00512382
$$246$$ 0 0
$$247$$ 1.20725 0.0768154
$$248$$ −5.57019 −0.353707
$$249$$ 0 0
$$250$$ −2.96110 −0.187277
$$251$$ −1.16332 −0.0734282 −0.0367141 0.999326i $$-0.511689\pi$$
−0.0367141 + 0.999326i $$0.511689\pi$$
$$252$$ 0 0
$$253$$ −9.48717 −0.596453
$$254$$ −8.99899 −0.564647
$$255$$ 0 0
$$256$$ 0.370256 0.0231410
$$257$$ 15.0510 0.938857 0.469428 0.882971i $$-0.344460\pi$$
0.469428 + 0.882971i $$0.344460\pi$$
$$258$$ 0 0
$$259$$ 19.1668 1.19097
$$260$$ −1.35910 −0.0842879
$$261$$ 0 0
$$262$$ −1.49632 −0.0924429
$$263$$ 20.6609 1.27400 0.637002 0.770862i $$-0.280174\pi$$
0.637002 + 0.770862i $$0.280174\pi$$
$$264$$ 0 0
$$265$$ −6.14571 −0.377528
$$266$$ 1.29225 0.0792326
$$267$$ 0 0
$$268$$ 3.92765 0.239919
$$269$$ 2.82662 0.172342 0.0861711 0.996280i $$-0.472537\pi$$
0.0861711 + 0.996280i $$0.472537\pi$$
$$270$$ 0 0
$$271$$ 23.5970 1.43341 0.716707 0.697375i $$-0.245649\pi$$
0.716707 + 0.697375i $$0.245649\pi$$
$$272$$ −0.588102 −0.0356589
$$273$$ 0 0
$$274$$ −4.20008 −0.253736
$$275$$ 4.59346 0.276996
$$276$$ 0 0
$$277$$ −18.7773 −1.12822 −0.564110 0.825700i $$-0.690781\pi$$
−0.564110 + 0.825700i $$0.690781\pi$$
$$278$$ −7.03994 −0.422228
$$279$$ 0 0
$$280$$ −3.10267 −0.185420
$$281$$ −27.2632 −1.62639 −0.813194 0.581993i $$-0.802273\pi$$
−0.813194 + 0.581993i $$0.802273\pi$$
$$282$$ 0 0
$$283$$ −27.2601 −1.62044 −0.810221 0.586124i $$-0.800653\pi$$
−0.810221 + 0.586124i $$0.800653\pi$$
$$284$$ 4.57149 0.271268
$$285$$ 0 0
$$286$$ −0.584420 −0.0345575
$$287$$ 18.4897 1.09141
$$288$$ 0 0
$$289$$ −16.9507 −0.997100
$$290$$ −1.52637 −0.0896316
$$291$$ 0 0
$$292$$ −29.3289 −1.71634
$$293$$ −30.9996 −1.81101 −0.905507 0.424332i $$-0.860509\pi$$
−0.905507 + 0.424332i $$0.860509\pi$$
$$294$$ 0 0
$$295$$ 1.69486 0.0986784
$$296$$ 13.0889 0.760776
$$297$$ 0 0
$$298$$ −8.94195 −0.517993
$$299$$ −11.4534 −0.662366
$$300$$ 0 0
$$301$$ 4.09858 0.236238
$$302$$ 0.442457 0.0254605
$$303$$ 0 0
$$304$$ −2.64884 −0.151922
$$305$$ 0.516325 0.0295647
$$306$$ 0 0
$$307$$ −18.2495 −1.04156 −0.520778 0.853692i $$-0.674358\pi$$
−0.520778 + 0.853692i $$0.674358\pi$$
$$308$$ 4.71327 0.268563
$$309$$ 0 0
$$310$$ −0.943146 −0.0535671
$$311$$ −30.8931 −1.75179 −0.875894 0.482503i $$-0.839728\pi$$
−0.875894 + 0.482503i $$0.839728\pi$$
$$312$$ 0 0
$$313$$ 28.7973 1.62772 0.813859 0.581062i $$-0.197363\pi$$
0.813859 + 0.581062i $$0.197363\pi$$
$$314$$ 1.15902 0.0654073
$$315$$ 0 0
$$316$$ −10.0518 −0.565460
$$317$$ −5.96040 −0.334769 −0.167385 0.985892i $$-0.553532\pi$$
−0.167385 + 0.985892i $$0.553532\pi$$
$$318$$ 0 0
$$319$$ 4.94518 0.276877
$$320$$ 1.85670 0.103793
$$321$$ 0 0
$$322$$ −12.2597 −0.683209
$$323$$ 0.222022 0.0123536
$$324$$ 0 0
$$325$$ 5.54545 0.307606
$$326$$ −6.87345 −0.380685
$$327$$ 0 0
$$328$$ 12.6265 0.697181
$$329$$ −15.8718 −0.875043
$$330$$ 0 0
$$331$$ 33.4010 1.83589 0.917944 0.396711i $$-0.129849\pi$$
0.917944 + 0.396711i $$0.129849\pi$$
$$332$$ −5.18722 −0.284686
$$333$$ 0 0
$$334$$ 4.47813 0.245032
$$335$$ 1.41833 0.0774915
$$336$$ 0 0
$$337$$ 24.9988 1.36177 0.680885 0.732390i $$-0.261595\pi$$
0.680885 + 0.732390i $$0.261595\pi$$
$$338$$ 5.58766 0.303929
$$339$$ 0 0
$$340$$ −0.249949 −0.0135554
$$341$$ 3.05563 0.165472
$$342$$ 0 0
$$343$$ −18.3501 −0.990815
$$344$$ 2.79889 0.150906
$$345$$ 0 0
$$346$$ 2.61108 0.140372
$$347$$ 8.35577 0.448561 0.224281 0.974525i $$-0.427997\pi$$
0.224281 + 0.974525i $$0.427997\pi$$
$$348$$ 0 0
$$349$$ −28.3911 −1.51974 −0.759871 0.650074i $$-0.774738\pi$$
−0.759871 + 0.650074i $$0.774738\pi$$
$$350$$ 5.93588 0.317286
$$351$$ 0 0
$$352$$ 4.92814 0.262671
$$353$$ 12.4082 0.660423 0.330212 0.943907i $$-0.392880\pi$$
0.330212 + 0.943907i $$0.392880\pi$$
$$354$$ 0 0
$$355$$ 1.65083 0.0876169
$$356$$ −9.78413 −0.518558
$$357$$ 0 0
$$358$$ 0.632806 0.0334448
$$359$$ −12.1134 −0.639319 −0.319659 0.947533i $$-0.603568\pi$$
−0.319659 + 0.947533i $$0.603568\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 5.20999 0.273831
$$363$$ 0 0
$$364$$ 5.69009 0.298242
$$365$$ −10.5911 −0.554361
$$366$$ 0 0
$$367$$ −4.06064 −0.211964 −0.105982 0.994368i $$-0.533799\pi$$
−0.105982 + 0.994368i $$0.533799\pi$$
$$368$$ 25.1300 1.30999
$$369$$ 0 0
$$370$$ 2.21621 0.115216
$$371$$ 25.7299 1.33583
$$372$$ 0 0
$$373$$ 35.4898 1.83759 0.918796 0.394732i $$-0.129163\pi$$
0.918796 + 0.394732i $$0.129163\pi$$
$$374$$ −0.107479 −0.00555763
$$375$$ 0 0
$$376$$ −10.8388 −0.558967
$$377$$ 5.97006 0.307474
$$378$$ 0 0
$$379$$ 13.5584 0.696449 0.348224 0.937411i $$-0.386785\pi$$
0.348224 + 0.937411i $$0.386785\pi$$
$$380$$ −1.12578 −0.0577515
$$381$$ 0 0
$$382$$ 7.40862 0.379058
$$383$$ 23.8326 1.21779 0.608894 0.793251i $$-0.291613\pi$$
0.608894 + 0.793251i $$0.291613\pi$$
$$384$$ 0 0
$$385$$ 1.70203 0.0867432
$$386$$ 8.86045 0.450985
$$387$$ 0 0
$$388$$ −24.3180 −1.23456
$$389$$ 12.7567 0.646789 0.323395 0.946264i $$-0.395176\pi$$
0.323395 + 0.946264i $$0.395176\pi$$
$$390$$ 0 0
$$391$$ −2.10636 −0.106523
$$392$$ 0.229296 0.0115812
$$393$$ 0 0
$$394$$ −0.278920 −0.0140518
$$395$$ −3.62986 −0.182638
$$396$$ 0 0
$$397$$ −30.3169 −1.52156 −0.760780 0.649009i $$-0.775184\pi$$
−0.760780 + 0.649009i $$0.775184\pi$$
$$398$$ 4.92217 0.246726
$$399$$ 0 0
$$400$$ −12.1674 −0.608368
$$401$$ −32.6370 −1.62982 −0.814908 0.579591i $$-0.803212\pi$$
−0.814908 + 0.579591i $$0.803212\pi$$
$$402$$ 0 0
$$403$$ 3.68891 0.183758
$$404$$ 21.2422 1.05684
$$405$$ 0 0
$$406$$ 6.39038 0.317149
$$407$$ −7.18015 −0.355907
$$408$$ 0 0
$$409$$ −23.1587 −1.14512 −0.572562 0.819862i $$-0.694050\pi$$
−0.572562 + 0.819862i $$0.694050\pi$$
$$410$$ 2.13792 0.105584
$$411$$ 0 0
$$412$$ −15.0622 −0.742063
$$413$$ −7.09578 −0.349160
$$414$$ 0 0
$$415$$ −1.87318 −0.0919506
$$416$$ 5.94949 0.291698
$$417$$ 0 0
$$418$$ −0.484093 −0.0236777
$$419$$ 14.2213 0.694755 0.347377 0.937725i $$-0.387072\pi$$
0.347377 + 0.937725i $$0.387072\pi$$
$$420$$ 0 0
$$421$$ 15.9605 0.777866 0.388933 0.921266i $$-0.372844\pi$$
0.388933 + 0.921266i $$0.372844\pi$$
$$422$$ 3.51373 0.171045
$$423$$ 0 0
$$424$$ 17.5708 0.853313
$$425$$ 1.01985 0.0494701
$$426$$ 0 0
$$427$$ −2.16167 −0.104611
$$428$$ 2.90600 0.140467
$$429$$ 0 0
$$430$$ 0.473909 0.0228539
$$431$$ −24.5100 −1.18061 −0.590304 0.807181i $$-0.700992\pi$$
−0.590304 + 0.807181i $$0.700992\pi$$
$$432$$ 0 0
$$433$$ −19.1839 −0.921921 −0.460961 0.887421i $$-0.652495\pi$$
−0.460961 + 0.887421i $$0.652495\pi$$
$$434$$ 3.94862 0.189540
$$435$$ 0 0
$$436$$ −29.6745 −1.42115
$$437$$ −9.48717 −0.453833
$$438$$ 0 0
$$439$$ 13.7961 0.658450 0.329225 0.944251i $$-0.393212\pi$$
0.329225 + 0.944251i $$0.393212\pi$$
$$440$$ 1.16230 0.0554105
$$441$$ 0 0
$$442$$ −0.129754 −0.00617178
$$443$$ −14.1259 −0.671140 −0.335570 0.942015i $$-0.608929\pi$$
−0.335570 + 0.942015i $$0.608929\pi$$
$$444$$ 0 0
$$445$$ −3.53318 −0.167489
$$446$$ −7.87967 −0.373113
$$447$$ 0 0
$$448$$ −7.77337 −0.367257
$$449$$ 21.0016 0.991129 0.495565 0.868571i $$-0.334961\pi$$
0.495565 + 0.868571i $$0.334961\pi$$
$$450$$ 0 0
$$451$$ −6.92650 −0.326156
$$452$$ −19.9425 −0.938018
$$453$$ 0 0
$$454$$ −10.1301 −0.475429
$$455$$ 2.05477 0.0963290
$$456$$ 0 0
$$457$$ −22.4621 −1.05073 −0.525367 0.850876i $$-0.676072\pi$$
−0.525367 + 0.850876i $$0.676072\pi$$
$$458$$ 5.23223 0.244486
$$459$$ 0 0
$$460$$ 10.6805 0.497981
$$461$$ 31.1213 1.44946 0.724732 0.689031i $$-0.241964\pi$$
0.724732 + 0.689031i $$0.241964\pi$$
$$462$$ 0 0
$$463$$ 31.6932 1.47291 0.736455 0.676487i $$-0.236498\pi$$
0.736455 + 0.676487i $$0.236498\pi$$
$$464$$ −13.0990 −0.608106
$$465$$ 0 0
$$466$$ −12.3043 −0.569986
$$467$$ 1.75474 0.0811995 0.0405997 0.999175i $$-0.487073\pi$$
0.0405997 + 0.999175i $$0.487073\pi$$
$$468$$ 0 0
$$469$$ −5.93804 −0.274193
$$470$$ −1.83522 −0.0846525
$$471$$ 0 0
$$472$$ −4.84566 −0.223039
$$473$$ −1.53538 −0.0705970
$$474$$ 0 0
$$475$$ 4.59346 0.210763
$$476$$ 1.04645 0.0479640
$$477$$ 0 0
$$478$$ 0.170516 0.00779922
$$479$$ 11.1294 0.508515 0.254257 0.967137i $$-0.418169\pi$$
0.254257 + 0.967137i $$0.418169\pi$$
$$480$$ 0 0
$$481$$ −8.66823 −0.395237
$$482$$ 7.59212 0.345812
$$483$$ 0 0
$$484$$ −1.76565 −0.0802570
$$485$$ −8.78158 −0.398751
$$486$$ 0 0
$$487$$ 3.27007 0.148181 0.0740905 0.997252i $$-0.476395\pi$$
0.0740905 + 0.997252i $$0.476395\pi$$
$$488$$ −1.47619 −0.0668240
$$489$$ 0 0
$$490$$ 0.0388244 0.00175391
$$491$$ −29.2686 −1.32087 −0.660437 0.750882i $$-0.729629\pi$$
−0.660437 + 0.750882i $$0.729629\pi$$
$$492$$ 0 0
$$493$$ 1.09794 0.0494487
$$494$$ −0.584420 −0.0262943
$$495$$ 0 0
$$496$$ −8.09389 −0.363426
$$497$$ −6.91145 −0.310021
$$498$$ 0 0
$$499$$ −1.04828 −0.0469275 −0.0234638 0.999725i $$-0.507469\pi$$
−0.0234638 + 0.999725i $$0.507469\pi$$
$$500$$ −10.8002 −0.482998
$$501$$ 0 0
$$502$$ 0.563155 0.0251348
$$503$$ −28.4724 −1.26952 −0.634761 0.772709i $$-0.718901\pi$$
−0.634761 + 0.772709i $$0.718901\pi$$
$$504$$ 0 0
$$505$$ 7.67086 0.341349
$$506$$ 4.59267 0.204169
$$507$$ 0 0
$$508$$ −32.8224 −1.45626
$$509$$ −39.6801 −1.75879 −0.879396 0.476091i $$-0.842053\pi$$
−0.879396 + 0.476091i $$0.842053\pi$$
$$510$$ 0 0
$$511$$ 44.3410 1.96153
$$512$$ −22.7112 −1.00370
$$513$$ 0 0
$$514$$ −7.28609 −0.321376
$$515$$ −5.43918 −0.239679
$$516$$ 0 0
$$517$$ 5.94581 0.261496
$$518$$ −9.27852 −0.407675
$$519$$ 0 0
$$520$$ 1.40319 0.0615338
$$521$$ 15.3168 0.671043 0.335522 0.942033i $$-0.391087\pi$$
0.335522 + 0.942033i $$0.391087\pi$$
$$522$$ 0 0
$$523$$ −4.17850 −0.182713 −0.0913566 0.995818i $$-0.529120\pi$$
−0.0913566 + 0.995818i $$0.529120\pi$$
$$524$$ −5.45760 −0.238416
$$525$$ 0 0
$$526$$ −10.0018 −0.436098
$$527$$ 0.678418 0.0295524
$$528$$ 0 0
$$529$$ 67.0064 2.91332
$$530$$ 2.97509 0.129230
$$531$$ 0 0
$$532$$ 4.71327 0.204346
$$533$$ −8.36201 −0.362199
$$534$$ 0 0
$$535$$ 1.04940 0.0453694
$$536$$ −4.05505 −0.175151
$$537$$ 0 0
$$538$$ −1.36835 −0.0589937
$$539$$ −0.125785 −0.00541792
$$540$$ 0 0
$$541$$ −14.6201 −0.628565 −0.314283 0.949329i $$-0.601764\pi$$
−0.314283 + 0.949329i $$0.601764\pi$$
$$542$$ −11.4231 −0.490665
$$543$$ 0 0
$$544$$ 1.09416 0.0469116
$$545$$ −10.7159 −0.459017
$$546$$ 0 0
$$547$$ −34.7220 −1.48461 −0.742303 0.670064i $$-0.766267\pi$$
−0.742303 + 0.670064i $$0.766267\pi$$
$$548$$ −15.3192 −0.654402
$$549$$ 0 0
$$550$$ −2.22366 −0.0948173
$$551$$ 4.94518 0.210672
$$552$$ 0 0
$$553$$ 15.1969 0.646240
$$554$$ 9.08997 0.386196
$$555$$ 0 0
$$556$$ −25.6771 −1.08895
$$557$$ 7.07603 0.299821 0.149910 0.988700i $$-0.452101\pi$$
0.149910 + 0.988700i $$0.452101\pi$$
$$558$$ 0 0
$$559$$ −1.85359 −0.0783985
$$560$$ −4.50840 −0.190515
$$561$$ 0 0
$$562$$ 13.1979 0.556721
$$563$$ 32.5712 1.37271 0.686357 0.727265i $$-0.259209\pi$$
0.686357 + 0.727265i $$0.259209\pi$$
$$564$$ 0 0
$$565$$ −7.20152 −0.302970
$$566$$ 13.1964 0.554686
$$567$$ 0 0
$$568$$ −4.71978 −0.198037
$$569$$ −32.6457 −1.36858 −0.684290 0.729210i $$-0.739888\pi$$
−0.684290 + 0.729210i $$0.739888\pi$$
$$570$$ 0 0
$$571$$ −35.8059 −1.49843 −0.749216 0.662326i $$-0.769569\pi$$
−0.749216 + 0.662326i $$0.769569\pi$$
$$572$$ −2.13158 −0.0891260
$$573$$ 0 0
$$574$$ −8.95073 −0.373596
$$575$$ −43.5790 −1.81737
$$576$$ 0 0
$$577$$ 15.7218 0.654509 0.327254 0.944936i $$-0.393877\pi$$
0.327254 + 0.944936i $$0.393877\pi$$
$$578$$ 8.20571 0.341313
$$579$$ 0 0
$$580$$ −5.56721 −0.231166
$$581$$ 7.84233 0.325355
$$582$$ 0 0
$$583$$ −9.63879 −0.399198
$$584$$ 30.2802 1.25300
$$585$$ 0 0
$$586$$ 15.0067 0.619919
$$587$$ −24.4810 −1.01044 −0.505220 0.862991i $$-0.668588\pi$$
−0.505220 + 0.862991i $$0.668588\pi$$
$$588$$ 0 0
$$589$$ 3.05563 0.125905
$$590$$ −0.820468 −0.0337781
$$591$$ 0 0
$$592$$ 19.0191 0.781680
$$593$$ 21.2995 0.874666 0.437333 0.899300i $$-0.355923\pi$$
0.437333 + 0.899300i $$0.355923\pi$$
$$594$$ 0 0
$$595$$ 0.377887 0.0154919
$$596$$ −32.6144 −1.33594
$$597$$ 0 0
$$598$$ 5.54450 0.226731
$$599$$ 8.90839 0.363987 0.181994 0.983300i $$-0.441745\pi$$
0.181994 + 0.983300i $$0.441745\pi$$
$$600$$ 0 0
$$601$$ −8.36794 −0.341335 −0.170668 0.985329i $$-0.554592\pi$$
−0.170668 + 0.985329i $$0.554592\pi$$
$$602$$ −1.98409 −0.0808655
$$603$$ 0 0
$$604$$ 1.61379 0.0656643
$$605$$ −0.637602 −0.0259222
$$606$$ 0 0
$$607$$ 43.2321 1.75474 0.877368 0.479819i $$-0.159298\pi$$
0.877368 + 0.479819i $$0.159298\pi$$
$$608$$ 4.92814 0.199862
$$609$$ 0 0
$$610$$ −0.249949 −0.0101201
$$611$$ 7.17807 0.290394
$$612$$ 0 0
$$613$$ −12.4893 −0.504438 −0.252219 0.967670i $$-0.581160\pi$$
−0.252219 + 0.967670i $$0.581160\pi$$
$$614$$ 8.83447 0.356530
$$615$$ 0 0
$$616$$ −4.86615 −0.196063
$$617$$ 18.4079 0.741073 0.370536 0.928818i $$-0.379174\pi$$
0.370536 + 0.928818i $$0.379174\pi$$
$$618$$ 0 0
$$619$$ −7.30900 −0.293773 −0.146887 0.989153i $$-0.546925\pi$$
−0.146887 + 0.989153i $$0.546925\pi$$
$$620$$ −3.43998 −0.138153
$$621$$ 0 0
$$622$$ 14.9551 0.599646
$$623$$ 14.7922 0.592637
$$624$$ 0 0
$$625$$ 19.0672 0.762689
$$626$$ −13.9406 −0.557177
$$627$$ 0 0
$$628$$ 4.22735 0.168690
$$629$$ −1.59415 −0.0635631
$$630$$ 0 0
$$631$$ −11.8397 −0.471329 −0.235665 0.971834i $$-0.575727\pi$$
−0.235665 + 0.971834i $$0.575727\pi$$
$$632$$ 10.3779 0.412810
$$633$$ 0 0
$$634$$ 2.88538 0.114593
$$635$$ −11.8526 −0.470357
$$636$$ 0 0
$$637$$ −0.151853 −0.00601664
$$638$$ −2.39392 −0.0947764
$$639$$ 0 0
$$640$$ −7.18320 −0.283941
$$641$$ 36.5974 1.44551 0.722756 0.691104i $$-0.242875\pi$$
0.722756 + 0.691104i $$0.242875\pi$$
$$642$$ 0 0
$$643$$ −14.0452 −0.553888 −0.276944 0.960886i $$-0.589322\pi$$
−0.276944 + 0.960886i $$0.589322\pi$$
$$644$$ −44.7156 −1.76204
$$645$$ 0 0
$$646$$ −0.107479 −0.00422872
$$647$$ 43.5266 1.71121 0.855604 0.517631i $$-0.173186\pi$$
0.855604 + 0.517631i $$0.173186\pi$$
$$648$$ 0 0
$$649$$ 2.65817 0.104342
$$650$$ −2.68451 −0.105295
$$651$$ 0 0
$$652$$ −25.0699 −0.981812
$$653$$ −44.0328 −1.72314 −0.861568 0.507642i $$-0.830517\pi$$
−0.861568 + 0.507642i $$0.830517\pi$$
$$654$$ 0 0
$$655$$ −1.97081 −0.0770060
$$656$$ 18.3472 0.716338
$$657$$ 0 0
$$658$$ 7.68344 0.299532
$$659$$ 3.86784 0.150670 0.0753349 0.997158i $$-0.475997\pi$$
0.0753349 + 0.997158i $$0.475997\pi$$
$$660$$ 0 0
$$661$$ 7.96999 0.309997 0.154998 0.987915i $$-0.450463\pi$$
0.154998 + 0.987915i $$0.450463\pi$$
$$662$$ −16.1692 −0.628434
$$663$$ 0 0
$$664$$ 5.35547 0.207833
$$665$$ 1.70203 0.0660017
$$666$$ 0 0
$$667$$ −46.9158 −1.81659
$$668$$ 16.3333 0.631954
$$669$$ 0 0
$$670$$ −0.686602 −0.0265258
$$671$$ 0.809792 0.0312617
$$672$$ 0 0
$$673$$ 21.7872 0.839835 0.419917 0.907562i $$-0.362059\pi$$
0.419917 + 0.907562i $$0.362059\pi$$
$$674$$ −12.1017 −0.466141
$$675$$ 0 0
$$676$$ 20.3802 0.783852
$$677$$ −33.2004 −1.27599 −0.637997 0.770039i $$-0.720237\pi$$
−0.637997 + 0.770039i $$0.720237\pi$$
$$678$$ 0 0
$$679$$ 36.7654 1.41093
$$680$$ 0.258057 0.00989602
$$681$$ 0 0
$$682$$ −1.47921 −0.0566418
$$683$$ 34.9595 1.33769 0.668843 0.743403i $$-0.266790\pi$$
0.668843 + 0.743403i $$0.266790\pi$$
$$684$$ 0 0
$$685$$ −5.53196 −0.211365
$$686$$ 8.88317 0.339161
$$687$$ 0 0
$$688$$ 4.06699 0.155052
$$689$$ −11.6364 −0.443312
$$690$$ 0 0
$$691$$ −30.4626 −1.15885 −0.579426 0.815025i $$-0.696723\pi$$
−0.579426 + 0.815025i $$0.696723\pi$$
$$692$$ 9.52351 0.362030
$$693$$ 0 0
$$694$$ −4.04497 −0.153545
$$695$$ −9.27236 −0.351721
$$696$$ 0 0
$$697$$ −1.53784 −0.0582497
$$698$$ 13.7439 0.520215
$$699$$ 0 0
$$700$$ 21.6502 0.818301
$$701$$ −10.3329 −0.390269 −0.195135 0.980776i $$-0.562514\pi$$
−0.195135 + 0.980776i $$0.562514\pi$$
$$702$$ 0 0
$$703$$ −7.18015 −0.270805
$$704$$ 2.91201 0.109751
$$705$$ 0 0
$$706$$ −6.00673 −0.226066
$$707$$ −32.1152 −1.20782
$$708$$ 0 0
$$709$$ −45.4499 −1.70691 −0.853454 0.521169i $$-0.825496\pi$$
−0.853454 + 0.521169i $$0.825496\pi$$
$$710$$ −0.799154 −0.0299917
$$711$$ 0 0
$$712$$ 10.1015 0.378570
$$713$$ −28.9893 −1.08566
$$714$$ 0 0
$$715$$ −0.769744 −0.0287868
$$716$$ 2.30806 0.0862564
$$717$$ 0 0
$$718$$ 5.86399 0.218842
$$719$$ 7.51239 0.280165 0.140082 0.990140i $$-0.455263\pi$$
0.140082 + 0.990140i $$0.455263\pi$$
$$720$$ 0 0
$$721$$ 22.7719 0.848071
$$722$$ −0.484093 −0.0180161
$$723$$ 0 0
$$724$$ 19.0026 0.706227
$$725$$ 22.7155 0.843632
$$726$$ 0 0
$$727$$ 4.79898 0.177984 0.0889922 0.996032i $$-0.471635\pi$$
0.0889922 + 0.996032i $$0.471635\pi$$
$$728$$ −5.87465 −0.217729
$$729$$ 0 0
$$730$$ 5.12705 0.189761
$$731$$ −0.340889 −0.0126082
$$732$$ 0 0
$$733$$ −8.55077 −0.315830 −0.157915 0.987453i $$-0.550477\pi$$
−0.157915 + 0.987453i $$0.550477\pi$$
$$734$$ 1.96572 0.0725562
$$735$$ 0 0
$$736$$ −46.7541 −1.72338
$$737$$ 2.22447 0.0819395
$$738$$ 0 0
$$739$$ 38.8820 1.43030 0.715149 0.698972i $$-0.246359\pi$$
0.715149 + 0.698972i $$0.246359\pi$$
$$740$$ 8.08330 0.297148
$$741$$ 0 0
$$742$$ −12.4557 −0.457262
$$743$$ −28.4350 −1.04318 −0.521589 0.853197i $$-0.674660\pi$$
−0.521589 + 0.853197i $$0.674660\pi$$
$$744$$ 0 0
$$745$$ −11.7775 −0.431494
$$746$$ −17.1804 −0.629018
$$747$$ 0 0
$$748$$ −0.392015 −0.0143335
$$749$$ −4.39346 −0.160534
$$750$$ 0 0
$$751$$ 18.2732 0.666799 0.333400 0.942786i $$-0.391804\pi$$
0.333400 + 0.942786i $$0.391804\pi$$
$$752$$ −15.7495 −0.574326
$$753$$ 0 0
$$754$$ −2.89006 −0.105250
$$755$$ 0.582763 0.0212089
$$756$$ 0 0
$$757$$ 1.10637 0.0402117 0.0201058 0.999798i $$-0.493600\pi$$
0.0201058 + 0.999798i $$0.493600\pi$$
$$758$$ −6.56352 −0.238398
$$759$$ 0 0
$$760$$ 1.16230 0.0421611
$$761$$ −1.10110 −0.0399147 −0.0199573 0.999801i $$-0.506353\pi$$
−0.0199573 + 0.999801i $$0.506353\pi$$
$$762$$ 0 0
$$763$$ 44.8636 1.62417
$$764$$ 27.0218 0.977615
$$765$$ 0 0
$$766$$ −11.5372 −0.416855
$$767$$ 3.20908 0.115873
$$768$$ 0 0
$$769$$ 30.2914 1.09234 0.546169 0.837675i $$-0.316086\pi$$
0.546169 + 0.837675i $$0.316086\pi$$
$$770$$ −0.823938 −0.0296927
$$771$$ 0 0
$$772$$ 32.3171 1.16312
$$773$$ 43.9677 1.58141 0.790704 0.612199i $$-0.209715\pi$$
0.790704 + 0.612199i $$0.209715\pi$$
$$774$$ 0 0
$$775$$ 14.0359 0.504186
$$776$$ 25.1068 0.901283
$$777$$ 0 0
$$778$$ −6.17542 −0.221399
$$779$$ −6.92650 −0.248168
$$780$$ 0 0
$$781$$ 2.58912 0.0926461
$$782$$ 1.01967 0.0364635
$$783$$ 0 0
$$784$$ 0.333183 0.0118994
$$785$$ 1.52655 0.0544851
$$786$$ 0 0
$$787$$ −37.8350 −1.34867 −0.674337 0.738424i $$-0.735570\pi$$
−0.674337 + 0.738424i $$0.735570\pi$$
$$788$$ −1.01732 −0.0362405
$$789$$ 0 0
$$790$$ 1.75719 0.0625179
$$791$$ 30.1502 1.07202
$$792$$ 0 0
$$793$$ 0.977621 0.0347163
$$794$$ 14.6762 0.520838
$$795$$ 0 0
$$796$$ 17.9529 0.636323
$$797$$ −11.2902 −0.399918 −0.199959 0.979804i $$-0.564081\pi$$
−0.199959 + 0.979804i $$0.564081\pi$$
$$798$$ 0 0
$$799$$ 1.32010 0.0467018
$$800$$ 22.6372 0.800347
$$801$$ 0 0
$$802$$ 15.7993 0.557894
$$803$$ −16.6108 −0.586181
$$804$$ 0 0
$$805$$ −16.1474 −0.569121
$$806$$ −1.78577 −0.0629012
$$807$$ 0 0
$$808$$ −21.9313 −0.771539
$$809$$ −17.8104 −0.626180 −0.313090 0.949723i $$-0.601364\pi$$
−0.313090 + 0.949723i $$0.601364\pi$$
$$810$$ 0 0
$$811$$ 28.1638 0.988965 0.494482 0.869188i $$-0.335358\pi$$
0.494482 + 0.869188i $$0.335358\pi$$
$$812$$ 23.3079 0.817949
$$813$$ 0 0
$$814$$ 3.47586 0.121829
$$815$$ −9.05307 −0.317115
$$816$$ 0 0
$$817$$ −1.53538 −0.0537162
$$818$$ 11.2110 0.391982
$$819$$ 0 0
$$820$$ 7.79774 0.272309
$$821$$ 44.2644 1.54484 0.772418 0.635114i $$-0.219047\pi$$
0.772418 + 0.635114i $$0.219047\pi$$
$$822$$ 0 0
$$823$$ −22.4258 −0.781716 −0.390858 0.920451i $$-0.627822\pi$$
−0.390858 + 0.920451i $$0.627822\pi$$
$$824$$ 15.5508 0.541738
$$825$$ 0 0
$$826$$ 3.43501 0.119519
$$827$$ −19.7303 −0.686090 −0.343045 0.939319i $$-0.611458\pi$$
−0.343045 + 0.939319i $$0.611458\pi$$
$$828$$ 0 0
$$829$$ 5.70836 0.198259 0.0991297 0.995075i $$-0.468394\pi$$
0.0991297 + 0.995075i $$0.468394\pi$$
$$830$$ 0.906791 0.0314752
$$831$$ 0 0
$$832$$ 3.51552 0.121879
$$833$$ −0.0279270 −0.000967612 0
$$834$$ 0 0
$$835$$ 5.89817 0.204115
$$836$$ −1.76565 −0.0610664
$$837$$ 0 0
$$838$$ −6.88441 −0.237818
$$839$$ 0.0666011 0.00229932 0.00114966 0.999999i $$-0.499634\pi$$
0.00114966 + 0.999999i $$0.499634\pi$$
$$840$$ 0 0
$$841$$ −4.54521 −0.156731
$$842$$ −7.72635 −0.266268
$$843$$ 0 0
$$844$$ 12.8158 0.441137
$$845$$ 7.35955 0.253176
$$846$$ 0 0
$$847$$ 2.66942 0.0917222
$$848$$ 25.5316 0.876760
$$849$$ 0 0
$$850$$ −0.493703 −0.0169339
$$851$$ 68.1193 2.33510
$$852$$ 0 0
$$853$$ −14.8615 −0.508848 −0.254424 0.967093i $$-0.581886\pi$$
−0.254424 + 0.967093i $$0.581886\pi$$
$$854$$ 1.04645 0.0358088
$$855$$ 0 0
$$856$$ −3.00026 −0.102547
$$857$$ −12.5169 −0.427569 −0.213785 0.976881i $$-0.568579\pi$$
−0.213785 + 0.976881i $$0.568579\pi$$
$$858$$ 0 0
$$859$$ −21.2738 −0.725854 −0.362927 0.931818i $$-0.618223\pi$$
−0.362927 + 0.931818i $$0.618223\pi$$
$$860$$ 1.72851 0.0589417
$$861$$ 0 0
$$862$$ 11.8651 0.404128
$$863$$ 34.4402 1.17236 0.586179 0.810181i $$-0.300631\pi$$
0.586179 + 0.810181i $$0.300631\pi$$
$$864$$ 0 0
$$865$$ 3.43907 0.116932
$$866$$ 9.28680 0.315578
$$867$$ 0 0
$$868$$ 14.4020 0.488836
$$869$$ −5.69298 −0.193121
$$870$$ 0 0
$$871$$ 2.68549 0.0909944
$$872$$ 30.6370 1.03750
$$873$$ 0 0
$$874$$ 4.59267 0.155349
$$875$$ 16.3283 0.551998
$$876$$ 0 0
$$877$$ −0.910260 −0.0307373 −0.0153687 0.999882i $$-0.504892\pi$$
−0.0153687 + 0.999882i $$0.504892\pi$$
$$878$$ −6.67857 −0.225391
$$879$$ 0 0
$$880$$ 1.68891 0.0569331
$$881$$ −33.7502 −1.13707 −0.568537 0.822658i $$-0.692490\pi$$
−0.568537 + 0.822658i $$0.692490\pi$$
$$882$$ 0 0
$$883$$ 39.8800 1.34207 0.671035 0.741426i $$-0.265850\pi$$
0.671035 + 0.741426i $$0.265850\pi$$
$$884$$ −0.473259 −0.0159174
$$885$$ 0 0
$$886$$ 6.83822 0.229735
$$887$$ 16.1868 0.543500 0.271750 0.962368i $$-0.412398\pi$$
0.271750 + 0.962368i $$0.412398\pi$$
$$888$$ 0 0
$$889$$ 49.6228 1.66430
$$890$$ 1.71039 0.0573324
$$891$$ 0 0
$$892$$ −28.7399 −0.962282
$$893$$ 5.94581 0.198969
$$894$$ 0 0
$$895$$ 0.833474 0.0278599
$$896$$ 30.0735 1.00469
$$897$$ 0 0
$$898$$ −10.1667 −0.339269
$$899$$ 15.1106 0.503968
$$900$$ 0 0
$$901$$ −2.14003 −0.0712946
$$902$$ 3.35307 0.111645
$$903$$ 0 0
$$904$$ 20.5894 0.684793
$$905$$ 6.86211 0.228104
$$906$$ 0 0
$$907$$ −14.3005 −0.474839 −0.237419 0.971407i $$-0.576302\pi$$
−0.237419 + 0.971407i $$0.576302\pi$$
$$908$$ −36.9480 −1.22616
$$909$$ 0 0
$$910$$ −0.994698 −0.0329739
$$911$$ 8.50108 0.281653 0.140827 0.990034i $$-0.455024\pi$$
0.140827 + 0.990034i $$0.455024\pi$$
$$912$$ 0 0
$$913$$ −2.93784 −0.0972285
$$914$$ 10.8738 0.359672
$$915$$ 0 0
$$916$$ 19.0838 0.630545
$$917$$ 8.25111 0.272476
$$918$$ 0 0
$$919$$ −26.0156 −0.858177 −0.429088 0.903263i $$-0.641165\pi$$
−0.429088 + 0.903263i $$0.641165\pi$$
$$920$$ −11.0269 −0.363548
$$921$$ 0 0
$$922$$ −15.0656 −0.496159
$$923$$ 3.12571 0.102884
$$924$$ 0 0
$$925$$ −32.9818 −1.08443
$$926$$ −15.3425 −0.504185
$$927$$ 0 0
$$928$$ 24.3705 0.800001
$$929$$ −5.99773 −0.196779 −0.0983896 0.995148i $$-0.531369\pi$$
−0.0983896 + 0.995148i $$0.531369\pi$$
$$930$$ 0 0
$$931$$ −0.125785 −0.00412242
$$932$$ −44.8781 −1.47003
$$933$$ 0 0
$$934$$ −0.849454 −0.0277950
$$935$$ −0.141562 −0.00462957
$$936$$ 0 0
$$937$$ −50.5847 −1.65253 −0.826265 0.563282i $$-0.809539\pi$$
−0.826265 + 0.563282i $$0.809539\pi$$
$$938$$ 2.87456 0.0938578
$$939$$ 0 0
$$940$$ −6.69370 −0.218324
$$941$$ 21.8640 0.712747 0.356373 0.934344i $$-0.384013\pi$$
0.356373 + 0.934344i $$0.384013\pi$$
$$942$$ 0 0
$$943$$ 65.7129 2.13991
$$944$$ −7.04109 −0.229168
$$945$$ 0 0
$$946$$ 0.743267 0.0241657
$$947$$ 33.1722 1.07795 0.538976 0.842321i $$-0.318812\pi$$
0.538976 + 0.842321i $$0.318812\pi$$
$$948$$ 0 0
$$949$$ −20.0533 −0.650958
$$950$$ −2.22366 −0.0721451
$$951$$ 0 0
$$952$$ −1.08039 −0.0350157
$$953$$ −22.4494 −0.727209 −0.363604 0.931553i $$-0.618454\pi$$
−0.363604 + 0.931553i $$0.618454\pi$$
$$954$$ 0 0
$$955$$ 9.75794 0.315760
$$956$$ 0.621931 0.0201147
$$957$$ 0 0
$$958$$ −5.38765 −0.174067
$$959$$ 23.1604 0.747887
$$960$$ 0 0
$$961$$ −21.6631 −0.698810
$$962$$ 4.19623 0.135292
$$963$$ 0 0
$$964$$ 27.6911 0.891870
$$965$$ 11.6702 0.375676
$$966$$ 0 0
$$967$$ −0.790013 −0.0254051 −0.0127026 0.999919i $$-0.504043\pi$$
−0.0127026 + 0.999919i $$0.504043\pi$$
$$968$$ 1.82293 0.0585911
$$969$$ 0 0
$$970$$ 4.25110 0.136495
$$971$$ −51.6970 −1.65904 −0.829518 0.558479i $$-0.811385\pi$$
−0.829518 + 0.558479i $$0.811385\pi$$
$$972$$ 0 0
$$973$$ 38.8201 1.24452
$$974$$ −1.58302 −0.0507231
$$975$$ 0 0
$$976$$ −2.14501 −0.0686602
$$977$$ −21.4536 −0.686360 −0.343180 0.939270i $$-0.611504\pi$$
−0.343180 + 0.939270i $$0.611504\pi$$
$$978$$ 0 0
$$979$$ −5.54136 −0.177103
$$980$$ 0.141606 0.00452345
$$981$$ 0 0
$$982$$ 14.1687 0.452142
$$983$$ 42.9916 1.37122 0.685610 0.727969i $$-0.259536\pi$$
0.685610 + 0.727969i $$0.259536\pi$$
$$984$$ 0 0
$$985$$ −0.367368 −0.0117053
$$986$$ −0.531505 −0.0169266
$$987$$ 0 0
$$988$$ −2.13158 −0.0678147
$$989$$ 14.5664 0.463186
$$990$$ 0 0
$$991$$ 57.7319 1.83392 0.916958 0.398985i $$-0.130637\pi$$
0.916958 + 0.398985i $$0.130637\pi$$
$$992$$ 15.0586 0.478110
$$993$$ 0 0
$$994$$ 3.34578 0.106122
$$995$$ 6.48303 0.205526
$$996$$ 0 0
$$997$$ −30.3487 −0.961152 −0.480576 0.876953i $$-0.659572\pi$$
−0.480576 + 0.876953i $$0.659572\pi$$
$$998$$ 0.507466 0.0160635
$$999$$ 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 1881.2.a.k.1.3 5
3.2 odd 2 209.2.a.c.1.3 5
12.11 even 2 3344.2.a.t.1.1 5
15.14 odd 2 5225.2.a.h.1.3 5
33.32 even 2 2299.2.a.n.1.3 5
57.56 even 2 3971.2.a.h.1.3 5

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.3 5 3.2 odd 2
1881.2.a.k.1.3 5 1.1 even 1 trivial
2299.2.a.n.1.3 5 33.32 even 2
3344.2.a.t.1.1 5 12.11 even 2
3971.2.a.h.1.3 5 57.56 even 2
5225.2.a.h.1.3 5 15.14 odd 2