Properties

 Label 1216.2.a.x.1.1 Level $1216$ Weight $2$ Character 1216.1 Self dual yes Analytic conductor $9.710$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-0.329727$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.56155 q^{3} -3.40617 q^{5} -2.50407 q^{7} -0.561553 q^{9} +O(q^{10})$$ $$q-1.56155 q^{3} -3.40617 q^{5} -2.50407 q^{7} -0.561553 q^{9} -1.40617 q^{11} -6.22101 q^{13} +5.31891 q^{15} -3.16352 q^{17} +1.00000 q^{19} +3.91023 q^{21} +7.91023 q^{23} +6.60197 q^{25} +5.56155 q^{27} -6.22101 q^{29} -8.13124 q^{31} +2.19580 q^{33} +8.52927 q^{35} +6.13124 q^{37} +9.71443 q^{39} +1.68923 q^{41} +1.71694 q^{43} +1.91274 q^{45} -9.84818 q^{47} -0.729644 q^{49} +4.94001 q^{51} -2.78713 q^{53} +4.78964 q^{55} -1.56155 q^{57} +9.25078 q^{59} -6.52927 q^{61} +1.40617 q^{63} +21.1898 q^{65} +1.87233 q^{67} -12.3522 q^{69} +13.0081 q^{71} -6.28663 q^{73} -10.3093 q^{75} +3.52114 q^{77} +11.2543 q^{79} -7.00000 q^{81} +1.80420 q^{83} +10.7755 q^{85} +9.71443 q^{87} -7.57426 q^{89} +15.5778 q^{91} +12.6974 q^{93} -3.40617 q^{95} +4.81233 q^{97} +0.789637 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - q^{5} + q^{7} + 6 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 - q^5 + q^7 + 6 * q^9 $$4 q + 2 q^{3} - q^{5} + q^{7} + 6 q^{9} + 7 q^{11} - 10 q^{13} + 8 q^{15} + 5 q^{17} + 4 q^{19} - 8 q^{21} + 8 q^{23} + 17 q^{25} + 14 q^{27} - 10 q^{29} + 6 q^{31} + 12 q^{33} + 5 q^{35} - 14 q^{37} + 12 q^{39} - 2 q^{41} + 3 q^{43} + 7 q^{45} + 3 q^{47} + 7 q^{49} - 6 q^{51} - 4 q^{53} + 35 q^{55} + 2 q^{57} + 20 q^{59} + 3 q^{61} - 7 q^{63} - 12 q^{65} + 8 q^{67} + 4 q^{69} + 30 q^{71} + 9 q^{73} + 34 q^{75} + 7 q^{77} - 10 q^{79} - 28 q^{81} + 4 q^{83} - 19 q^{85} + 12 q^{87} - 16 q^{89} + 10 q^{91} + 20 q^{93} - q^{95} - 6 q^{97} + 19 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 - q^5 + q^7 + 6 * q^9 + 7 * q^11 - 10 * q^13 + 8 * q^15 + 5 * q^17 + 4 * q^19 - 8 * q^21 + 8 * q^23 + 17 * q^25 + 14 * q^27 - 10 * q^29 + 6 * q^31 + 12 * q^33 + 5 * q^35 - 14 * q^37 + 12 * q^39 - 2 * q^41 + 3 * q^43 + 7 * q^45 + 3 * q^47 + 7 * q^49 - 6 * q^51 - 4 * q^53 + 35 * q^55 + 2 * q^57 + 20 * q^59 + 3 * q^61 - 7 * q^63 - 12 * q^65 + 8 * q^67 + 4 * q^69 + 30 * q^71 + 9 * q^73 + 34 * q^75 + 7 * q^77 - 10 * q^79 - 28 * q^81 + 4 * q^83 - 19 * q^85 + 12 * q^87 - 16 * q^89 + 10 * q^91 + 20 * q^93 - q^95 - 6 * q^97 + 19 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.56155 −0.901563 −0.450781 0.892634i $$-0.648855\pi$$
−0.450781 + 0.892634i $$0.648855\pi$$
$$4$$ 0 0
$$5$$ −3.40617 −1.52328 −0.761642 0.647998i $$-0.775606\pi$$
−0.761642 + 0.647998i $$0.775606\pi$$
$$6$$ 0 0
$$7$$ −2.50407 −0.946449 −0.473224 0.880942i $$-0.656910\pi$$
−0.473224 + 0.880942i $$0.656910\pi$$
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ −1.40617 −0.423975 −0.211988 0.977272i $$-0.567994\pi$$
−0.211988 + 0.977272i $$0.567994\pi$$
$$12$$ 0 0
$$13$$ −6.22101 −1.72540 −0.862698 0.505719i $$-0.831227\pi$$
−0.862698 + 0.505719i $$0.831227\pi$$
$$14$$ 0 0
$$15$$ 5.31891 1.37334
$$16$$ 0 0
$$17$$ −3.16352 −0.767267 −0.383633 0.923485i $$-0.625327\pi$$
−0.383633 + 0.923485i $$0.625327\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 3.91023 0.853283
$$22$$ 0 0
$$23$$ 7.91023 1.64940 0.824699 0.565572i $$-0.191345\pi$$
0.824699 + 0.565572i $$0.191345\pi$$
$$24$$ 0 0
$$25$$ 6.60197 1.32039
$$26$$ 0 0
$$27$$ 5.56155 1.07032
$$28$$ 0 0
$$29$$ −6.22101 −1.15521 −0.577606 0.816316i $$-0.696013\pi$$
−0.577606 + 0.816316i $$0.696013\pi$$
$$30$$ 0 0
$$31$$ −8.13124 −1.46041 −0.730207 0.683226i $$-0.760576\pi$$
−0.730207 + 0.683226i $$0.760576\pi$$
$$32$$ 0 0
$$33$$ 2.19580 0.382240
$$34$$ 0 0
$$35$$ 8.52927 1.44171
$$36$$ 0 0
$$37$$ 6.13124 1.00797 0.503985 0.863712i $$-0.331867\pi$$
0.503985 + 0.863712i $$0.331867\pi$$
$$38$$ 0 0
$$39$$ 9.71443 1.55555
$$40$$ 0 0
$$41$$ 1.68923 0.263813 0.131906 0.991262i $$-0.457890\pi$$
0.131906 + 0.991262i $$0.457890\pi$$
$$42$$ 0 0
$$43$$ 1.71694 0.261831 0.130915 0.991394i $$-0.458208\pi$$
0.130915 + 0.991394i $$0.458208\pi$$
$$44$$ 0 0
$$45$$ 1.91274 0.285135
$$46$$ 0 0
$$47$$ −9.84818 −1.43650 −0.718252 0.695783i $$-0.755058\pi$$
−0.718252 + 0.695783i $$0.755058\pi$$
$$48$$ 0 0
$$49$$ −0.729644 −0.104235
$$50$$ 0 0
$$51$$ 4.94001 0.691739
$$52$$ 0 0
$$53$$ −2.78713 −0.382842 −0.191421 0.981508i $$-0.561310\pi$$
−0.191421 + 0.981508i $$0.561310\pi$$
$$54$$ 0 0
$$55$$ 4.78964 0.645834
$$56$$ 0 0
$$57$$ −1.56155 −0.206833
$$58$$ 0 0
$$59$$ 9.25078 1.20435 0.602174 0.798365i $$-0.294301\pi$$
0.602174 + 0.798365i $$0.294301\pi$$
$$60$$ 0 0
$$61$$ −6.52927 −0.835988 −0.417994 0.908450i $$-0.637267\pi$$
−0.417994 + 0.908450i $$0.637267\pi$$
$$62$$ 0 0
$$63$$ 1.40617 0.177160
$$64$$ 0 0
$$65$$ 21.1898 2.62827
$$66$$ 0 0
$$67$$ 1.87233 0.228741 0.114370 0.993438i $$-0.463515\pi$$
0.114370 + 0.993438i $$0.463515\pi$$
$$68$$ 0 0
$$69$$ −12.3522 −1.48704
$$70$$ 0 0
$$71$$ 13.0081 1.54378 0.771891 0.635755i $$-0.219311\pi$$
0.771891 + 0.635755i $$0.219311\pi$$
$$72$$ 0 0
$$73$$ −6.28663 −0.735794 −0.367897 0.929867i $$-0.619922\pi$$
−0.367897 + 0.929867i $$0.619922\pi$$
$$74$$ 0 0
$$75$$ −10.3093 −1.19042
$$76$$ 0 0
$$77$$ 3.52114 0.401271
$$78$$ 0 0
$$79$$ 11.2543 1.26621 0.633106 0.774065i $$-0.281780\pi$$
0.633106 + 0.774065i $$0.281780\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 1.80420 0.198036 0.0990182 0.995086i $$-0.468430\pi$$
0.0990182 + 0.995086i $$0.468430\pi$$
$$84$$ 0 0
$$85$$ 10.7755 1.16877
$$86$$ 0 0
$$87$$ 9.71443 1.04150
$$88$$ 0 0
$$89$$ −7.57426 −0.802870 −0.401435 0.915888i $$-0.631488\pi$$
−0.401435 + 0.915888i $$0.631488\pi$$
$$90$$ 0 0
$$91$$ 15.5778 1.63300
$$92$$ 0 0
$$93$$ 12.6974 1.31666
$$94$$ 0 0
$$95$$ −3.40617 −0.349465
$$96$$ 0 0
$$97$$ 4.81233 0.488618 0.244309 0.969697i $$-0.421439\pi$$
0.244309 + 0.969697i $$0.421439\pi$$
$$98$$ 0 0
$$99$$ 0.789637 0.0793615
$$100$$ 0 0
$$101$$ 16.2462 1.61656 0.808279 0.588799i $$-0.200399\pi$$
0.808279 + 0.588799i $$0.200399\pi$$
$$102$$ 0 0
$$103$$ −7.68923 −0.757642 −0.378821 0.925470i $$-0.623670\pi$$
−0.378821 + 0.925470i $$0.623670\pi$$
$$104$$ 0 0
$$105$$ −13.3189 −1.29979
$$106$$ 0 0
$$107$$ −11.8886 −1.14931 −0.574657 0.818394i $$-0.694865\pi$$
−0.574657 + 0.818394i $$0.694865\pi$$
$$108$$ 0 0
$$109$$ 5.65489 0.541640 0.270820 0.962630i $$-0.412705\pi$$
0.270820 + 0.962630i $$0.412705\pi$$
$$110$$ 0 0
$$111$$ −9.57426 −0.908748
$$112$$ 0 0
$$113$$ −8.81233 −0.828995 −0.414497 0.910051i $$-0.636043\pi$$
−0.414497 + 0.910051i $$0.636043\pi$$
$$114$$ 0 0
$$115$$ −26.9436 −2.51250
$$116$$ 0 0
$$117$$ 3.49342 0.322967
$$118$$ 0 0
$$119$$ 7.92167 0.726179
$$120$$ 0 0
$$121$$ −9.02270 −0.820245
$$122$$ 0 0
$$123$$ −2.63782 −0.237844
$$124$$ 0 0
$$125$$ −5.45657 −0.488051
$$126$$ 0 0
$$127$$ −12.0504 −1.06930 −0.534650 0.845073i $$-0.679557\pi$$
−0.534650 + 0.845073i $$0.679557\pi$$
$$128$$ 0 0
$$129$$ −2.68109 −0.236057
$$130$$ 0 0
$$131$$ 0.920878 0.0804575 0.0402287 0.999190i $$-0.487191\pi$$
0.0402287 + 0.999190i $$0.487191\pi$$
$$132$$ 0 0
$$133$$ −2.50407 −0.217130
$$134$$ 0 0
$$135$$ −18.9436 −1.63040
$$136$$ 0 0
$$137$$ −0.525705 −0.0449140 −0.0224570 0.999748i $$-0.507149\pi$$
−0.0224570 + 0.999748i $$0.507149\pi$$
$$138$$ 0 0
$$139$$ −4.52927 −0.384168 −0.192084 0.981379i $$-0.561525\pi$$
−0.192084 + 0.981379i $$0.561525\pi$$
$$140$$ 0 0
$$141$$ 15.3785 1.29510
$$142$$ 0 0
$$143$$ 8.74777 0.731525
$$144$$ 0 0
$$145$$ 21.1898 1.75972
$$146$$ 0 0
$$147$$ 1.13938 0.0939743
$$148$$ 0 0
$$149$$ 18.9713 1.55419 0.777094 0.629384i $$-0.216693\pi$$
0.777094 + 0.629384i $$0.216693\pi$$
$$150$$ 0 0
$$151$$ −6.55698 −0.533600 −0.266800 0.963752i $$-0.585966\pi$$
−0.266800 + 0.963752i $$0.585966\pi$$
$$152$$ 0 0
$$153$$ 1.77648 0.143620
$$154$$ 0 0
$$155$$ 27.6964 2.22463
$$156$$ 0 0
$$157$$ −17.5651 −1.40185 −0.700925 0.713235i $$-0.747229\pi$$
−0.700925 + 0.713235i $$0.747229\pi$$
$$158$$ 0 0
$$159$$ 4.35225 0.345156
$$160$$ 0 0
$$161$$ −19.8078 −1.56107
$$162$$ 0 0
$$163$$ −22.9436 −1.79708 −0.898540 0.438892i $$-0.855371\pi$$
−0.898540 + 0.438892i $$0.855371\pi$$
$$164$$ 0 0
$$165$$ −7.47927 −0.582260
$$166$$ 0 0
$$167$$ −6.24621 −0.483346 −0.241673 0.970358i $$-0.577696\pi$$
−0.241673 + 0.970358i $$0.577696\pi$$
$$168$$ 0 0
$$169$$ 25.7009 1.97699
$$170$$ 0 0
$$171$$ −0.561553 −0.0429430
$$172$$ 0 0
$$173$$ 16.9436 1.28820 0.644098 0.764943i $$-0.277233\pi$$
0.644098 + 0.764943i $$0.277233\pi$$
$$174$$ 0 0
$$175$$ −16.5318 −1.24969
$$176$$ 0 0
$$177$$ −14.4456 −1.08580
$$178$$ 0 0
$$179$$ 18.2462 1.36379 0.681893 0.731452i $$-0.261157\pi$$
0.681893 + 0.731452i $$0.261157\pi$$
$$180$$ 0 0
$$181$$ −10.9273 −0.812220 −0.406110 0.913824i $$-0.633115\pi$$
−0.406110 + 0.913824i $$0.633115\pi$$
$$182$$ 0 0
$$183$$ 10.1958 0.753695
$$184$$ 0 0
$$185$$ −20.8840 −1.53542
$$186$$ 0 0
$$187$$ 4.44844 0.325302
$$188$$ 0 0
$$189$$ −13.9265 −1.01300
$$190$$ 0 0
$$191$$ 15.5122 1.12242 0.561212 0.827672i $$-0.310335\pi$$
0.561212 + 0.827672i $$0.310335\pi$$
$$192$$ 0 0
$$193$$ −9.56512 −0.688512 −0.344256 0.938876i $$-0.611869\pi$$
−0.344256 + 0.938876i $$0.611869\pi$$
$$194$$ 0 0
$$195$$ −33.0890 −2.36955
$$196$$ 0 0
$$197$$ −19.1394 −1.36362 −0.681812 0.731527i $$-0.738808\pi$$
−0.681812 + 0.731527i $$0.738808\pi$$
$$198$$ 0 0
$$199$$ 0.133749 0.00948125 0.00474062 0.999989i $$-0.498491\pi$$
0.00474062 + 0.999989i $$0.498491\pi$$
$$200$$ 0 0
$$201$$ −2.92374 −0.206224
$$202$$ 0 0
$$203$$ 15.5778 1.09335
$$204$$ 0 0
$$205$$ −5.75379 −0.401862
$$206$$ 0 0
$$207$$ −4.44201 −0.308741
$$208$$ 0 0
$$209$$ −1.40617 −0.0972666
$$210$$ 0 0
$$211$$ −23.0117 −1.58419 −0.792095 0.610397i $$-0.791010\pi$$
−0.792095 + 0.610397i $$0.791010\pi$$
$$212$$ 0 0
$$213$$ −20.3129 −1.39182
$$214$$ 0 0
$$215$$ −5.84818 −0.398843
$$216$$ 0 0
$$217$$ 20.3612 1.38221
$$218$$ 0 0
$$219$$ 9.81690 0.663365
$$220$$ 0 0
$$221$$ 19.6803 1.32384
$$222$$ 0 0
$$223$$ −7.51471 −0.503222 −0.251611 0.967828i $$-0.580960\pi$$
−0.251611 + 0.967828i $$0.580960\pi$$
$$224$$ 0 0
$$225$$ −3.70735 −0.247157
$$226$$ 0 0
$$227$$ 5.81690 0.386081 0.193041 0.981191i $$-0.438165\pi$$
0.193041 + 0.981191i $$0.438165\pi$$
$$228$$ 0 0
$$229$$ 26.1702 1.72938 0.864688 0.502309i $$-0.167516\pi$$
0.864688 + 0.502309i $$0.167516\pi$$
$$230$$ 0 0
$$231$$ −5.49844 −0.361771
$$232$$ 0 0
$$233$$ 2.22352 0.145667 0.0728337 0.997344i $$-0.476796\pi$$
0.0728337 + 0.997344i $$0.476796\pi$$
$$234$$ 0 0
$$235$$ 33.5445 2.18820
$$236$$ 0 0
$$237$$ −17.5743 −1.14157
$$238$$ 0 0
$$239$$ −8.12873 −0.525804 −0.262902 0.964823i $$-0.584680\pi$$
−0.262902 + 0.964823i $$0.584680\pi$$
$$240$$ 0 0
$$241$$ 25.9517 1.67170 0.835848 0.548960i $$-0.184976\pi$$
0.835848 + 0.548960i $$0.184976\pi$$
$$242$$ 0 0
$$243$$ −5.75379 −0.369106
$$244$$ 0 0
$$245$$ 2.48529 0.158779
$$246$$ 0 0
$$247$$ −6.22101 −0.395833
$$248$$ 0 0
$$249$$ −2.81735 −0.178542
$$250$$ 0 0
$$251$$ −1.53741 −0.0970403 −0.0485202 0.998822i $$-0.515451\pi$$
−0.0485202 + 0.998822i $$0.515451\pi$$
$$252$$ 0 0
$$253$$ −11.1231 −0.699304
$$254$$ 0 0
$$255$$ −16.8265 −1.05372
$$256$$ 0 0
$$257$$ −0.370318 −0.0230998 −0.0115499 0.999933i $$-0.503677\pi$$
−0.0115499 + 0.999933i $$0.503677\pi$$
$$258$$ 0 0
$$259$$ −15.3530 −0.953992
$$260$$ 0 0
$$261$$ 3.49342 0.216238
$$262$$ 0 0
$$263$$ 18.9067 1.16584 0.582919 0.812530i $$-0.301910\pi$$
0.582919 + 0.812530i $$0.301910\pi$$
$$264$$ 0 0
$$265$$ 9.49342 0.583176
$$266$$ 0 0
$$267$$ 11.8276 0.723838
$$268$$ 0 0
$$269$$ 15.1898 0.926138 0.463069 0.886322i $$-0.346748\pi$$
0.463069 + 0.886322i $$0.346748\pi$$
$$270$$ 0 0
$$271$$ 28.4027 1.72534 0.862669 0.505768i $$-0.168791\pi$$
0.862669 + 0.505768i $$0.168791\pi$$
$$272$$ 0 0
$$273$$ −24.3256 −1.47225
$$274$$ 0 0
$$275$$ −9.28347 −0.559814
$$276$$ 0 0
$$277$$ −28.4647 −1.71028 −0.855139 0.518398i $$-0.826529\pi$$
−0.855139 + 0.518398i $$0.826529\pi$$
$$278$$ 0 0
$$279$$ 4.56612 0.273367
$$280$$ 0 0
$$281$$ −27.1394 −1.61900 −0.809500 0.587120i $$-0.800262\pi$$
−0.809500 + 0.587120i $$0.800262\pi$$
$$282$$ 0 0
$$283$$ −26.9117 −1.59974 −0.799868 0.600175i $$-0.795097\pi$$
−0.799868 + 0.600175i $$0.795097\pi$$
$$284$$ 0 0
$$285$$ 5.31891 0.315065
$$286$$ 0 0
$$287$$ −4.22994 −0.249685
$$288$$ 0 0
$$289$$ −6.99213 −0.411302
$$290$$ 0 0
$$291$$ −7.51471 −0.440520
$$292$$ 0 0
$$293$$ −10.2714 −0.600062 −0.300031 0.953929i $$-0.596997\pi$$
−0.300031 + 0.953929i $$0.596997\pi$$
$$294$$ 0 0
$$295$$ −31.5097 −1.83457
$$296$$ 0 0
$$297$$ −7.82047 −0.453790
$$298$$ 0 0
$$299$$ −49.2096 −2.84587
$$300$$ 0 0
$$301$$ −4.29933 −0.247809
$$302$$ 0 0
$$303$$ −25.3693 −1.45743
$$304$$ 0 0
$$305$$ 22.2398 1.27345
$$306$$ 0 0
$$307$$ 5.05854 0.288706 0.144353 0.989526i $$-0.453890\pi$$
0.144353 + 0.989526i $$0.453890\pi$$
$$308$$ 0 0
$$309$$ 12.0071 0.683062
$$310$$ 0 0
$$311$$ 13.4222 0.761105 0.380553 0.924759i $$-0.375734\pi$$
0.380553 + 0.924759i $$0.375734\pi$$
$$312$$ 0 0
$$313$$ 6.94001 0.392272 0.196136 0.980577i $$-0.437161\pi$$
0.196136 + 0.980577i $$0.437161\pi$$
$$314$$ 0 0
$$315$$ −4.78964 −0.269865
$$316$$ 0 0
$$317$$ −5.60448 −0.314779 −0.157389 0.987537i $$-0.550308\pi$$
−0.157389 + 0.987537i $$0.550308\pi$$
$$318$$ 0 0
$$319$$ 8.74777 0.489781
$$320$$ 0 0
$$321$$ 18.5647 1.03618
$$322$$ 0 0
$$323$$ −3.16352 −0.176023
$$324$$ 0 0
$$325$$ −41.0709 −2.27820
$$326$$ 0 0
$$327$$ −8.83040 −0.488322
$$328$$ 0 0
$$329$$ 24.6605 1.35958
$$330$$ 0 0
$$331$$ 4.11140 0.225983 0.112992 0.993596i $$-0.463957\pi$$
0.112992 + 0.993596i $$0.463957\pi$$
$$332$$ 0 0
$$333$$ −3.44302 −0.188676
$$334$$ 0 0
$$335$$ −6.37745 −0.348437
$$336$$ 0 0
$$337$$ −0.632801 −0.0344709 −0.0172354 0.999851i $$-0.505486\pi$$
−0.0172354 + 0.999851i $$0.505486\pi$$
$$338$$ 0 0
$$339$$ 13.7609 0.747391
$$340$$ 0 0
$$341$$ 11.4339 0.619179
$$342$$ 0 0
$$343$$ 19.3556 1.04510
$$344$$ 0 0
$$345$$ 42.0738 2.26518
$$346$$ 0 0
$$347$$ −3.02871 −0.162590 −0.0812949 0.996690i $$-0.525906\pi$$
−0.0812949 + 0.996690i $$0.525906\pi$$
$$348$$ 0 0
$$349$$ −19.0954 −1.02215 −0.511076 0.859535i $$-0.670753\pi$$
−0.511076 + 0.859535i $$0.670753\pi$$
$$350$$ 0 0
$$351$$ −34.5985 −1.84673
$$352$$ 0 0
$$353$$ 15.2025 0.809147 0.404573 0.914506i $$-0.367420\pi$$
0.404573 + 0.914506i $$0.367420\pi$$
$$354$$ 0 0
$$355$$ −44.3079 −2.35162
$$356$$ 0 0
$$357$$ −12.3701 −0.654696
$$358$$ 0 0
$$359$$ 34.9411 1.84412 0.922059 0.387048i $$-0.126505\pi$$
0.922059 + 0.387048i $$0.126505\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 14.0894 0.739503
$$364$$ 0 0
$$365$$ 21.4133 1.12082
$$366$$ 0 0
$$367$$ −7.60839 −0.397155 −0.198577 0.980085i $$-0.563632\pi$$
−0.198577 + 0.980085i $$0.563632\pi$$
$$368$$ 0 0
$$369$$ −0.948590 −0.0493816
$$370$$ 0 0
$$371$$ 6.97916 0.362340
$$372$$ 0 0
$$373$$ −9.38739 −0.486060 −0.243030 0.970019i $$-0.578141\pi$$
−0.243030 + 0.970019i $$0.578141\pi$$
$$374$$ 0 0
$$375$$ 8.52073 0.440009
$$376$$ 0 0
$$377$$ 38.7009 1.99320
$$378$$ 0 0
$$379$$ 11.0046 0.565267 0.282633 0.959228i $$-0.408792\pi$$
0.282633 + 0.959228i $$0.408792\pi$$
$$380$$ 0 0
$$381$$ 18.8173 0.964042
$$382$$ 0 0
$$383$$ −20.6923 −1.05733 −0.528665 0.848831i $$-0.677307\pi$$
−0.528665 + 0.848831i $$0.677307\pi$$
$$384$$ 0 0
$$385$$ −11.9936 −0.611249
$$386$$ 0 0
$$387$$ −0.964152 −0.0490106
$$388$$ 0 0
$$389$$ 14.4306 0.731659 0.365830 0.930682i $$-0.380785\pi$$
0.365830 + 0.930682i $$0.380785\pi$$
$$390$$ 0 0
$$391$$ −25.0242 −1.26553
$$392$$ 0 0
$$393$$ −1.43800 −0.0725375
$$394$$ 0 0
$$395$$ −38.3342 −1.92880
$$396$$ 0 0
$$397$$ 14.2739 0.716388 0.358194 0.933647i $$-0.383393\pi$$
0.358194 + 0.933647i $$0.383393\pi$$
$$398$$ 0 0
$$399$$ 3.91023 0.195757
$$400$$ 0 0
$$401$$ 5.38559 0.268943 0.134472 0.990917i $$-0.457066\pi$$
0.134472 + 0.990917i $$0.457066\pi$$
$$402$$ 0 0
$$403$$ 50.5845 2.51979
$$404$$ 0 0
$$405$$ 23.8432 1.18478
$$406$$ 0 0
$$407$$ −8.62155 −0.427354
$$408$$ 0 0
$$409$$ 5.07983 0.251182 0.125591 0.992082i $$-0.459917\pi$$
0.125591 + 0.992082i $$0.459917\pi$$
$$410$$ 0 0
$$411$$ 0.820916 0.0404928
$$412$$ 0 0
$$413$$ −23.1646 −1.13985
$$414$$ 0 0
$$415$$ −6.14539 −0.301666
$$416$$ 0 0
$$417$$ 7.07270 0.346351
$$418$$ 0 0
$$419$$ 4.30576 0.210350 0.105175 0.994454i $$-0.466460\pi$$
0.105175 + 0.994454i $$0.466460\pi$$
$$420$$ 0 0
$$421$$ 17.8670 0.870782 0.435391 0.900241i $$-0.356610\pi$$
0.435391 + 0.900241i $$0.356610\pi$$
$$422$$ 0 0
$$423$$ 5.53027 0.268891
$$424$$ 0 0
$$425$$ −20.8855 −1.01309
$$426$$ 0 0
$$427$$ 16.3497 0.791219
$$428$$ 0 0
$$429$$ −13.6601 −0.659516
$$430$$ 0 0
$$431$$ 18.1958 0.876461 0.438230 0.898863i $$-0.355605\pi$$
0.438230 + 0.898863i $$0.355605\pi$$
$$432$$ 0 0
$$433$$ −16.6074 −0.798100 −0.399050 0.916929i $$-0.630660\pi$$
−0.399050 + 0.916929i $$0.630660\pi$$
$$434$$ 0 0
$$435$$ −33.0890 −1.58649
$$436$$ 0 0
$$437$$ 7.91023 0.378398
$$438$$ 0 0
$$439$$ 11.6084 0.554038 0.277019 0.960864i $$-0.410653\pi$$
0.277019 + 0.960864i $$0.410653\pi$$
$$440$$ 0 0
$$441$$ 0.409734 0.0195111
$$442$$ 0 0
$$443$$ 23.8140 1.13144 0.565720 0.824598i $$-0.308598\pi$$
0.565720 + 0.824598i $$0.308598\pi$$
$$444$$ 0 0
$$445$$ 25.7992 1.22300
$$446$$ 0 0
$$447$$ −29.6247 −1.40120
$$448$$ 0 0
$$449$$ 3.72336 0.175716 0.0878582 0.996133i $$-0.471998\pi$$
0.0878582 + 0.996133i $$0.471998\pi$$
$$450$$ 0 0
$$451$$ −2.37533 −0.111850
$$452$$ 0 0
$$453$$ 10.2391 0.481074
$$454$$ 0 0
$$455$$ −53.0607 −2.48752
$$456$$ 0 0
$$457$$ 26.7719 1.25234 0.626169 0.779688i $$-0.284622\pi$$
0.626169 + 0.779688i $$0.284622\pi$$
$$458$$ 0 0
$$459$$ −17.5941 −0.821222
$$460$$ 0 0
$$461$$ −7.76735 −0.361761 −0.180881 0.983505i $$-0.557895\pi$$
−0.180881 + 0.983505i $$0.557895\pi$$
$$462$$ 0 0
$$463$$ 13.7928 0.641004 0.320502 0.947248i $$-0.396148\pi$$
0.320502 + 0.947248i $$0.396148\pi$$
$$464$$ 0 0
$$465$$ −43.2493 −2.00564
$$466$$ 0 0
$$467$$ 27.7282 1.28311 0.641554 0.767078i $$-0.278290\pi$$
0.641554 + 0.767078i $$0.278290\pi$$
$$468$$ 0 0
$$469$$ −4.68843 −0.216492
$$470$$ 0 0
$$471$$ 27.4289 1.26386
$$472$$ 0 0
$$473$$ −2.41430 −0.111010
$$474$$ 0 0
$$475$$ 6.60197 0.302919
$$476$$ 0 0
$$477$$ 1.56512 0.0716619
$$478$$ 0 0
$$479$$ −14.0163 −0.640420 −0.320210 0.947347i $$-0.603753\pi$$
−0.320210 + 0.947347i $$0.603753\pi$$
$$480$$ 0 0
$$481$$ −38.1425 −1.73915
$$482$$ 0 0
$$483$$ 30.9309 1.40740
$$484$$ 0 0
$$485$$ −16.3916 −0.744304
$$486$$ 0 0
$$487$$ 37.5764 1.70275 0.851374 0.524559i $$-0.175770\pi$$
0.851374 + 0.524559i $$0.175770\pi$$
$$488$$ 0 0
$$489$$ 35.8276 1.62018
$$490$$ 0 0
$$491$$ 5.24309 0.236617 0.118309 0.992977i $$-0.462253\pi$$
0.118309 + 0.992977i $$0.462253\pi$$
$$492$$ 0 0
$$493$$ 19.6803 0.886356
$$494$$ 0 0
$$495$$ −2.68963 −0.120890
$$496$$ 0 0
$$497$$ −32.5733 −1.46111
$$498$$ 0 0
$$499$$ 29.0542 1.30065 0.650323 0.759658i $$-0.274633\pi$$
0.650323 + 0.759658i $$0.274633\pi$$
$$500$$ 0 0
$$501$$ 9.75379 0.435767
$$502$$ 0 0
$$503$$ 31.5512 1.40680 0.703399 0.710796i $$-0.251665\pi$$
0.703399 + 0.710796i $$0.251665\pi$$
$$504$$ 0 0
$$505$$ −55.3373 −2.46248
$$506$$ 0 0
$$507$$ −40.1334 −1.78239
$$508$$ 0 0
$$509$$ −17.5097 −0.776104 −0.388052 0.921638i $$-0.626852\pi$$
−0.388052 + 0.921638i $$0.626852\pi$$
$$510$$ 0 0
$$511$$ 15.7421 0.696391
$$512$$ 0 0
$$513$$ 5.56155 0.245549
$$514$$ 0 0
$$515$$ 26.1908 1.15410
$$516$$ 0 0
$$517$$ 13.8482 0.609042
$$518$$ 0 0
$$519$$ −26.4583 −1.16139
$$520$$ 0 0
$$521$$ −37.8780 −1.65947 −0.829733 0.558161i $$-0.811507\pi$$
−0.829733 + 0.558161i $$0.811507\pi$$
$$522$$ 0 0
$$523$$ 28.3739 1.24070 0.620352 0.784324i $$-0.286990\pi$$
0.620352 + 0.784324i $$0.286990\pi$$
$$524$$ 0 0
$$525$$ 25.8152 1.12667
$$526$$ 0 0
$$527$$ 25.7234 1.12053
$$528$$ 0 0
$$529$$ 39.5718 1.72051
$$530$$ 0 0
$$531$$ −5.19480 −0.225435
$$532$$ 0 0
$$533$$ −10.5087 −0.455182
$$534$$ 0 0
$$535$$ 40.4945 1.75073
$$536$$ 0 0
$$537$$ −28.4924 −1.22954
$$538$$ 0 0
$$539$$ 1.02600 0.0441930
$$540$$ 0 0
$$541$$ −38.5868 −1.65898 −0.829488 0.558524i $$-0.811368\pi$$
−0.829488 + 0.558524i $$0.811368\pi$$
$$542$$ 0 0
$$543$$ 17.0636 0.732267
$$544$$ 0 0
$$545$$ −19.2615 −0.825071
$$546$$ 0 0
$$547$$ −13.3947 −0.572717 −0.286359 0.958123i $$-0.592445\pi$$
−0.286359 + 0.958123i $$0.592445\pi$$
$$548$$ 0 0
$$549$$ 3.66653 0.156484
$$550$$ 0 0
$$551$$ −6.22101 −0.265024
$$552$$ 0 0
$$553$$ −28.1816 −1.19841
$$554$$ 0 0
$$555$$ 32.6115 1.38428
$$556$$ 0 0
$$557$$ −4.46471 −0.189176 −0.0945879 0.995517i $$-0.530153\pi$$
−0.0945879 + 0.995517i $$0.530153\pi$$
$$558$$ 0 0
$$559$$ −10.6811 −0.451762
$$560$$ 0 0
$$561$$ −6.94647 −0.293280
$$562$$ 0 0
$$563$$ −22.1171 −0.932124 −0.466062 0.884752i $$-0.654328\pi$$
−0.466062 + 0.884752i $$0.654328\pi$$
$$564$$ 0 0
$$565$$ 30.0163 1.26279
$$566$$ 0 0
$$567$$ 17.5285 0.736127
$$568$$ 0 0
$$569$$ 6.70238 0.280978 0.140489 0.990082i $$-0.455132\pi$$
0.140489 + 0.990082i $$0.455132\pi$$
$$570$$ 0 0
$$571$$ −6.37533 −0.266799 −0.133400 0.991062i $$-0.542589\pi$$
−0.133400 + 0.991062i $$0.542589\pi$$
$$572$$ 0 0
$$573$$ −24.2231 −1.01194
$$574$$ 0 0
$$575$$ 52.2231 2.17785
$$576$$ 0 0
$$577$$ 44.5654 1.85528 0.927641 0.373474i $$-0.121834\pi$$
0.927641 + 0.373474i $$0.121834\pi$$
$$578$$ 0 0
$$579$$ 14.9364 0.620737
$$580$$ 0 0
$$581$$ −4.51783 −0.187431
$$582$$ 0 0
$$583$$ 3.91917 0.162315
$$584$$ 0 0
$$585$$ −11.8992 −0.491971
$$586$$ 0 0
$$587$$ −29.6311 −1.22301 −0.611503 0.791242i $$-0.709435\pi$$
−0.611503 + 0.791242i $$0.709435\pi$$
$$588$$ 0 0
$$589$$ −8.13124 −0.335042
$$590$$ 0 0
$$591$$ 29.8871 1.22939
$$592$$ 0 0
$$593$$ 43.6572 1.79279 0.896393 0.443259i $$-0.146178\pi$$
0.896393 + 0.443259i $$0.146178\pi$$
$$594$$ 0 0
$$595$$ −26.9825 −1.10618
$$596$$ 0 0
$$597$$ −0.208857 −0.00854794
$$598$$ 0 0
$$599$$ 12.5661 0.513438 0.256719 0.966486i $$-0.417359\pi$$
0.256719 + 0.966486i $$0.417359\pi$$
$$600$$ 0 0
$$601$$ −37.9213 −1.54684 −0.773421 0.633893i $$-0.781456\pi$$
−0.773421 + 0.633893i $$0.781456\pi$$
$$602$$ 0 0
$$603$$ −1.05141 −0.0428167
$$604$$ 0 0
$$605$$ 30.7328 1.24947
$$606$$ 0 0
$$607$$ −1.93332 −0.0784710 −0.0392355 0.999230i $$-0.512492\pi$$
−0.0392355 + 0.999230i $$0.512492\pi$$
$$608$$ 0 0
$$609$$ −24.3256 −0.985723
$$610$$ 0 0
$$611$$ 61.2656 2.47854
$$612$$ 0 0
$$613$$ −18.4306 −0.744404 −0.372202 0.928152i $$-0.621397\pi$$
−0.372202 + 0.928152i $$0.621397\pi$$
$$614$$ 0 0
$$615$$ 8.98485 0.362304
$$616$$ 0 0
$$617$$ 30.5314 1.22915 0.614574 0.788859i $$-0.289328\pi$$
0.614574 + 0.788859i $$0.289328\pi$$
$$618$$ 0 0
$$619$$ 28.9094 1.16197 0.580984 0.813915i $$-0.302668\pi$$
0.580984 + 0.813915i $$0.302668\pi$$
$$620$$ 0 0
$$621$$ 43.9932 1.76539
$$622$$ 0 0
$$623$$ 18.9665 0.759875
$$624$$ 0 0
$$625$$ −14.4238 −0.576954
$$626$$ 0 0
$$627$$ 2.19580 0.0876919
$$628$$ 0 0
$$629$$ −19.3963 −0.773382
$$630$$ 0 0
$$631$$ −32.5314 −1.29505 −0.647527 0.762042i $$-0.724197\pi$$
−0.647527 + 0.762042i $$0.724197\pi$$
$$632$$ 0 0
$$633$$ 35.9340 1.42825
$$634$$ 0 0
$$635$$ 41.0457 1.62885
$$636$$ 0 0
$$637$$ 4.53912 0.179846
$$638$$ 0 0
$$639$$ −7.30476 −0.288972
$$640$$ 0 0
$$641$$ −30.1312 −1.19011 −0.595056 0.803684i $$-0.702870\pi$$
−0.595056 + 0.803684i $$0.702870\pi$$
$$642$$ 0 0
$$643$$ 20.1752 0.795633 0.397817 0.917465i $$-0.369768\pi$$
0.397817 + 0.917465i $$0.369768\pi$$
$$644$$ 0 0
$$645$$ 9.13224 0.359582
$$646$$ 0 0
$$647$$ −14.2741 −0.561174 −0.280587 0.959829i $$-0.590529\pi$$
−0.280587 + 0.959829i $$0.590529\pi$$
$$648$$ 0 0
$$649$$ −13.0081 −0.510614
$$650$$ 0 0
$$651$$ −31.7951 −1.24615
$$652$$ 0 0
$$653$$ −9.21538 −0.360626 −0.180313 0.983609i $$-0.557711\pi$$
−0.180313 + 0.983609i $$0.557711\pi$$
$$654$$ 0 0
$$655$$ −3.13666 −0.122560
$$656$$ 0 0
$$657$$ 3.53027 0.137729
$$658$$ 0 0
$$659$$ −12.1511 −0.473339 −0.236669 0.971590i $$-0.576056\pi$$
−0.236669 + 0.971590i $$0.576056\pi$$
$$660$$ 0 0
$$661$$ −13.6924 −0.532574 −0.266287 0.963894i $$-0.585797\pi$$
−0.266287 + 0.963894i $$0.585797\pi$$
$$662$$ 0 0
$$663$$ −30.7318 −1.19352
$$664$$ 0 0
$$665$$ 8.52927 0.330751
$$666$$ 0 0
$$667$$ −49.2096 −1.90540
$$668$$ 0 0
$$669$$ 11.7346 0.453687
$$670$$ 0 0
$$671$$ 9.18124 0.354438
$$672$$ 0 0
$$673$$ 40.5341 1.56247 0.781237 0.624234i $$-0.214589\pi$$
0.781237 + 0.624234i $$0.214589\pi$$
$$674$$ 0 0
$$675$$ 36.7172 1.41325
$$676$$ 0 0
$$677$$ −11.5653 −0.444492 −0.222246 0.974991i $$-0.571339\pi$$
−0.222246 + 0.974991i $$0.571339\pi$$
$$678$$ 0 0
$$679$$ −12.0504 −0.462452
$$680$$ 0 0
$$681$$ −9.08340 −0.348077
$$682$$ 0 0
$$683$$ 44.2625 1.69366 0.846828 0.531866i $$-0.178509\pi$$
0.846828 + 0.531866i $$0.178509\pi$$
$$684$$ 0 0
$$685$$ 1.79064 0.0684168
$$686$$ 0 0
$$687$$ −40.8662 −1.55914
$$688$$ 0 0
$$689$$ 17.3387 0.660554
$$690$$ 0 0
$$691$$ −32.6676 −1.24274 −0.621368 0.783519i $$-0.713423\pi$$
−0.621368 + 0.783519i $$0.713423\pi$$
$$692$$ 0 0
$$693$$ −1.97730 −0.0751116
$$694$$ 0 0
$$695$$ 15.4275 0.585197
$$696$$ 0 0
$$697$$ −5.34391 −0.202415
$$698$$ 0 0
$$699$$ −3.47214 −0.131328
$$700$$ 0 0
$$701$$ 8.73150 0.329784 0.164892 0.986312i $$-0.447272\pi$$
0.164892 + 0.986312i $$0.447272\pi$$
$$702$$ 0 0
$$703$$ 6.13124 0.231244
$$704$$ 0 0
$$705$$ −52.3816 −1.97280
$$706$$ 0 0
$$707$$ −40.6816 −1.52999
$$708$$ 0 0
$$709$$ −14.3058 −0.537264 −0.268632 0.963243i $$-0.586572\pi$$
−0.268632 + 0.963243i $$0.586572\pi$$
$$710$$ 0 0
$$711$$ −6.31991 −0.237015
$$712$$ 0 0
$$713$$ −64.3200 −2.40880
$$714$$ 0 0
$$715$$ −29.7964 −1.11432
$$716$$ 0 0
$$717$$ 12.6934 0.474045
$$718$$ 0 0
$$719$$ 21.4959 0.801663 0.400831 0.916152i $$-0.368721\pi$$
0.400831 + 0.916152i $$0.368721\pi$$
$$720$$ 0 0
$$721$$ 19.2543 0.717069
$$722$$ 0 0
$$723$$ −40.5250 −1.50714
$$724$$ 0 0
$$725$$ −41.0709 −1.52533
$$726$$ 0 0
$$727$$ 47.6002 1.76539 0.882696 0.469944i $$-0.155726\pi$$
0.882696 + 0.469944i $$0.155726\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ −5.43158 −0.200894
$$732$$ 0 0
$$733$$ 49.7239 1.83659 0.918297 0.395892i $$-0.129565\pi$$
0.918297 + 0.395892i $$0.129565\pi$$
$$734$$ 0 0
$$735$$ −3.88091 −0.143149
$$736$$ 0 0
$$737$$ −2.63280 −0.0969805
$$738$$ 0 0
$$739$$ 35.5283 1.30693 0.653464 0.756957i $$-0.273315\pi$$
0.653464 + 0.756957i $$0.273315\pi$$
$$740$$ 0 0
$$741$$ 9.71443 0.356869
$$742$$ 0 0
$$743$$ −20.3612 −0.746979 −0.373490 0.927634i $$-0.621839\pi$$
−0.373490 + 0.927634i $$0.621839\pi$$
$$744$$ 0 0
$$745$$ −64.6194 −2.36747
$$746$$ 0 0
$$747$$ −1.01315 −0.0370693
$$748$$ 0 0
$$749$$ 29.7699 1.08777
$$750$$ 0 0
$$751$$ −18.9868 −0.692840 −0.346420 0.938080i $$-0.612603\pi$$
−0.346420 + 0.938080i $$0.612603\pi$$
$$752$$ 0 0
$$753$$ 2.40074 0.0874880
$$754$$ 0 0
$$755$$ 22.3342 0.812824
$$756$$ 0 0
$$757$$ 22.4738 0.816826 0.408413 0.912797i $$-0.366082\pi$$
0.408413 + 0.912797i $$0.366082\pi$$
$$758$$ 0 0
$$759$$ 17.3693 0.630466
$$760$$ 0 0
$$761$$ −1.40973 −0.0511028 −0.0255514 0.999674i $$-0.508134\pi$$
−0.0255514 + 0.999674i $$0.508134\pi$$
$$762$$ 0 0
$$763$$ −14.1602 −0.512634
$$764$$ 0 0
$$765$$ −6.05100 −0.218774
$$766$$ 0 0
$$767$$ −57.5492 −2.07798
$$768$$ 0 0
$$769$$ −0.114116 −0.00411512 −0.00205756 0.999998i $$-0.500655\pi$$
−0.00205756 + 0.999998i $$0.500655\pi$$
$$770$$ 0 0
$$771$$ 0.578272 0.0208260
$$772$$ 0 0
$$773$$ 46.1799 1.66097 0.830487 0.557038i $$-0.188062\pi$$
0.830487 + 0.557038i $$0.188062\pi$$
$$774$$ 0 0
$$775$$ −53.6822 −1.92832
$$776$$ 0 0
$$777$$ 23.9746 0.860084
$$778$$ 0 0
$$779$$ 1.68923 0.0605228
$$780$$ 0 0
$$781$$ −18.2916 −0.654525
$$782$$ 0 0
$$783$$ −34.5985 −1.23645
$$784$$ 0 0
$$785$$ 59.8297 2.13541
$$786$$ 0 0
$$787$$ 12.6847 0.452159 0.226080 0.974109i $$-0.427409\pi$$
0.226080 + 0.974109i $$0.427409\pi$$
$$788$$ 0 0
$$789$$ −29.5238 −1.05108
$$790$$ 0 0
$$791$$ 22.0667 0.784601
$$792$$ 0 0
$$793$$ 40.6186 1.44241
$$794$$ 0 0
$$795$$ −14.8245 −0.525770
$$796$$ 0 0
$$797$$ −39.2796 −1.39135 −0.695677 0.718355i $$-0.744895\pi$$
−0.695677 + 0.718355i $$0.744895\pi$$
$$798$$ 0 0
$$799$$ 31.1549 1.10218
$$800$$ 0 0
$$801$$ 4.25335 0.150285
$$802$$ 0 0
$$803$$ 8.84004 0.311958
$$804$$ 0 0
$$805$$ 67.4685 2.37795
$$806$$ 0 0
$$807$$ −23.7197 −0.834971
$$808$$ 0 0
$$809$$ −1.09896 −0.0386374 −0.0193187 0.999813i $$-0.506150\pi$$
−0.0193187 + 0.999813i $$0.506150\pi$$
$$810$$ 0 0
$$811$$ 36.5627 1.28389 0.641944 0.766751i $$-0.278128\pi$$
0.641944 + 0.766751i $$0.278128\pi$$
$$812$$ 0 0
$$813$$ −44.3522 −1.55550
$$814$$ 0 0
$$815$$ 78.1496 2.73746
$$816$$ 0 0
$$817$$ 1.71694 0.0600681
$$818$$ 0 0
$$819$$ −8.74777 −0.305672
$$820$$ 0 0
$$821$$ 17.8823 0.624097 0.312049 0.950066i $$-0.398985\pi$$
0.312049 + 0.950066i $$0.398985\pi$$
$$822$$ 0 0
$$823$$ 8.31328 0.289783 0.144891 0.989448i $$-0.453717\pi$$
0.144891 + 0.989448i $$0.453717\pi$$
$$824$$ 0 0
$$825$$ 14.4966 0.504708
$$826$$ 0 0
$$827$$ 0.280204 0.00974363 0.00487182 0.999988i $$-0.498449\pi$$
0.00487182 + 0.999988i $$0.498449\pi$$
$$828$$ 0 0
$$829$$ −6.04148 −0.209829 −0.104915 0.994481i $$-0.533457\pi$$
−0.104915 + 0.994481i $$0.533457\pi$$
$$830$$ 0 0
$$831$$ 44.4491 1.54192
$$832$$ 0 0
$$833$$ 2.30824 0.0799759
$$834$$ 0 0
$$835$$ 21.2756 0.736274
$$836$$ 0 0
$$837$$ −45.2223 −1.56311
$$838$$ 0 0
$$839$$ −29.5942 −1.02171 −0.510853 0.859668i $$-0.670670\pi$$
−0.510853 + 0.859668i $$0.670670\pi$$
$$840$$ 0 0
$$841$$ 9.70093 0.334515
$$842$$ 0 0
$$843$$ 42.3796 1.45963
$$844$$ 0 0
$$845$$ −87.5416 −3.01152
$$846$$ 0 0
$$847$$ 22.5934 0.776320
$$848$$ 0 0
$$849$$ 42.0241 1.44226
$$850$$ 0 0
$$851$$ 48.4996 1.66254
$$852$$ 0 0
$$853$$ −15.2331 −0.521570 −0.260785 0.965397i $$-0.583981\pi$$
−0.260785 + 0.965397i $$0.583981\pi$$
$$854$$ 0 0
$$855$$ 1.91274 0.0654144
$$856$$ 0 0
$$857$$ −2.25535 −0.0770412 −0.0385206 0.999258i $$-0.512265\pi$$
−0.0385206 + 0.999258i $$0.512265\pi$$
$$858$$ 0 0
$$859$$ −1.91686 −0.0654025 −0.0327013 0.999465i $$-0.510411\pi$$
−0.0327013 + 0.999465i $$0.510411\pi$$
$$860$$ 0 0
$$861$$ 6.60527 0.225107
$$862$$ 0 0
$$863$$ 35.8530 1.22045 0.610225 0.792228i $$-0.291079\pi$$
0.610225 + 0.792228i $$0.291079\pi$$
$$864$$ 0 0
$$865$$ −57.7126 −1.96229
$$866$$ 0 0
$$867$$ 10.9186 0.370814
$$868$$ 0 0
$$869$$ −15.8255 −0.536843
$$870$$ 0 0
$$871$$ −11.6478 −0.394669
$$872$$ 0 0
$$873$$ −2.70238 −0.0914617
$$874$$ 0 0
$$875$$ 13.6636 0.461915
$$876$$ 0 0
$$877$$ −48.2323 −1.62869 −0.814344 0.580383i $$-0.802903\pi$$
−0.814344 + 0.580383i $$0.802903\pi$$
$$878$$ 0 0
$$879$$ 16.0394 0.540994
$$880$$ 0 0
$$881$$ 4.77336 0.160819 0.0804094 0.996762i $$-0.474377\pi$$
0.0804094 + 0.996762i $$0.474377\pi$$
$$882$$ 0 0
$$883$$ −22.2144 −0.747573 −0.373787 0.927515i $$-0.621941\pi$$
−0.373787 + 0.927515i $$0.621941\pi$$
$$884$$ 0 0
$$885$$ 49.2041 1.65398
$$886$$ 0 0
$$887$$ −42.1729 −1.41603 −0.708014 0.706198i $$-0.750409\pi$$
−0.708014 + 0.706198i $$0.750409\pi$$
$$888$$ 0 0
$$889$$ 30.1750 1.01204
$$890$$ 0 0
$$891$$ 9.84316 0.329758
$$892$$ 0 0
$$893$$ −9.84818 −0.329557
$$894$$ 0 0
$$895$$ −62.1496 −2.07743
$$896$$ 0 0
$$897$$ 76.8434 2.56573
$$898$$ 0 0
$$899$$ 50.5845 1.68709
$$900$$ 0 0
$$901$$ 8.81714 0.293742
$$902$$ 0 0
$$903$$ 6.71363 0.223416
$$904$$ 0 0
$$905$$ 37.2202 1.23724
$$906$$ 0 0
$$907$$ −26.7564 −0.888430 −0.444215 0.895920i $$-0.646517\pi$$
−0.444215 + 0.895920i $$0.646517\pi$$
$$908$$ 0 0
$$909$$ −9.12311 −0.302594
$$910$$ 0 0
$$911$$ 11.1281 0.368691 0.184346 0.982861i $$-0.440983\pi$$
0.184346 + 0.982861i $$0.440983\pi$$
$$912$$ 0 0
$$913$$ −2.53700 −0.0839625
$$914$$ 0 0
$$915$$ −34.7286 −1.14809
$$916$$ 0 0
$$917$$ −2.30594 −0.0761489
$$918$$ 0 0
$$919$$ 5.45195 0.179843 0.0899216 0.995949i $$-0.471338\pi$$
0.0899216 + 0.995949i $$0.471338\pi$$
$$920$$ 0 0
$$921$$ −7.89918 −0.260287
$$922$$ 0 0
$$923$$ −80.9237 −2.66364
$$924$$ 0 0
$$925$$ 40.4783 1.33092
$$926$$ 0 0
$$927$$ 4.31791 0.141819
$$928$$ 0 0
$$929$$ −1.94402 −0.0637813 −0.0318906 0.999491i $$-0.510153\pi$$
−0.0318906 + 0.999491i $$0.510153\pi$$
$$930$$ 0 0
$$931$$ −0.729644 −0.0239131
$$932$$ 0 0
$$933$$ −20.9595 −0.686184
$$934$$ 0 0
$$935$$ −15.1521 −0.495527
$$936$$ 0 0
$$937$$ −11.2897 −0.368820 −0.184410 0.982849i $$-0.559037\pi$$
−0.184410 + 0.982849i $$0.559037\pi$$
$$938$$ 0 0
$$939$$ −10.8372 −0.353658
$$940$$ 0 0
$$941$$ −51.4229 −1.67634 −0.838170 0.545409i $$-0.816374\pi$$
−0.838170 + 0.545409i $$0.816374\pi$$
$$942$$ 0 0
$$943$$ 13.3622 0.435133
$$944$$ 0 0
$$945$$ 47.4360 1.54309
$$946$$ 0 0
$$947$$ −1.94045 −0.0630563 −0.0315281 0.999503i $$-0.510037\pi$$
−0.0315281 + 0.999503i $$0.510037\pi$$
$$948$$ 0 0
$$949$$ 39.1092 1.26954
$$950$$ 0 0
$$951$$ 8.75169 0.283793
$$952$$ 0 0
$$953$$ −23.4523 −0.759693 −0.379847 0.925049i $$-0.624023\pi$$
−0.379847 + 0.925049i $$0.624023\pi$$
$$954$$ 0 0
$$955$$ −52.8371 −1.70977
$$956$$ 0 0
$$957$$ −13.6601 −0.441569
$$958$$ 0 0
$$959$$ 1.31640 0.0425088
$$960$$ 0 0
$$961$$ 35.1171 1.13281
$$962$$ 0 0
$$963$$ 6.67608 0.215134
$$964$$ 0 0
$$965$$ 32.5804 1.04880
$$966$$ 0 0
$$967$$ −59.0194 −1.89794 −0.948968 0.315373i $$-0.897870\pi$$
−0.948968 + 0.315373i $$0.897870\pi$$
$$968$$ 0 0
$$969$$ 4.94001 0.158696
$$970$$ 0 0
$$971$$ 39.7743 1.27642 0.638209 0.769863i $$-0.279676\pi$$
0.638209 + 0.769863i $$0.279676\pi$$
$$972$$ 0 0
$$973$$ 11.3416 0.363595
$$974$$ 0 0
$$975$$ 64.1344 2.05394
$$976$$ 0 0
$$977$$ −41.9538 −1.34222 −0.671111 0.741357i $$-0.734183\pi$$
−0.671111 + 0.741357i $$0.734183\pi$$
$$978$$ 0 0
$$979$$ 10.6507 0.340397
$$980$$ 0 0
$$981$$ −3.17552 −0.101386
$$982$$ 0 0
$$983$$ −7.93132 −0.252970 −0.126485 0.991969i $$-0.540370\pi$$
−0.126485 + 0.991969i $$0.540370\pi$$
$$984$$ 0 0
$$985$$ 65.1919 2.07719
$$986$$ 0 0
$$987$$ −38.5087 −1.22575
$$988$$ 0 0
$$989$$ 13.5814 0.431863
$$990$$ 0 0
$$991$$ −24.8408 −0.789093 −0.394546 0.918876i $$-0.629098\pi$$
−0.394546 + 0.918876i $$0.629098\pi$$
$$992$$ 0 0
$$993$$ −6.42017 −0.203738
$$994$$ 0 0
$$995$$ −0.455573 −0.0144426
$$996$$ 0 0
$$997$$ −28.1157 −0.890433 −0.445216 0.895423i $$-0.646873\pi$$
−0.445216 + 0.895423i $$0.646873\pi$$
$$998$$ 0 0
$$999$$ 34.0992 1.07885
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 1216.2.a.x.1.1 4
4.3 odd 2 1216.2.a.w.1.3 4
8.3 odd 2 608.2.a.j.1.2 yes 4
8.5 even 2 608.2.a.i.1.4 4
24.5 odd 2 5472.2.a.bt.1.1 4
24.11 even 2 5472.2.a.bs.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.4 4 8.5 even 2
608.2.a.j.1.2 yes 4 8.3 odd 2
1216.2.a.w.1.3 4 4.3 odd 2
1216.2.a.x.1.1 4 1.1 even 1 trivial
5472.2.a.bs.1.1 4 24.11 even 2
5472.2.a.bt.1.1 4 24.5 odd 2