# Properties

 Label 4400.2.a.ce.1.4 Level $4400$ Weight $2$ Character 4400.1 Self dual yes Analytic conductor $35.134$ 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] = [4400,2,Mod(1,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.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: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.54764.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 9x^{2} + 3x + 2$$ x^4 - x^3 - 9*x^2 + 3*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$3.36007$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.36007 q^{3} +1.08258 q^{7} +8.29009 q^{9} +O(q^{10})$$ $$q+3.36007 q^{3} +1.08258 q^{7} +8.29009 q^{9} -1.00000 q^{11} -4.00000 q^{13} +0.107866 q^{17} +6.61228 q^{19} +3.63756 q^{21} -5.97235 q^{23} +17.7751 q^{27} +7.80273 q^{29} -1.12492 q^{31} -3.36007 q^{33} +7.05494 q^{37} -13.4403 q^{39} +5.19045 q^{41} +5.52969 q^{43} -7.69486 q^{47} -5.82801 q^{49} +0.362439 q^{51} +4.77745 q^{53} +22.2177 q^{57} +0.677809 q^{59} +0.197271 q^{61} +8.97472 q^{63} -1.41737 q^{67} -20.0675 q^{69} +6.15020 q^{71} -6.16517 q^{73} -1.08258 q^{77} +9.35562 q^{79} +34.8553 q^{81} +13.7454 q^{83} +26.2177 q^{87} -1.29009 q^{89} -4.33034 q^{91} -3.77981 q^{93} +6.60782 q^{97} -8.29009 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} + q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q + q^3 + q^7 + 7 * q^9 $$4 q + q^{3} + q^{7} + 7 q^{9} - 4 q^{11} - 16 q^{13} - 7 q^{17} + 9 q^{19} - 7 q^{21} + 6 q^{23} + 13 q^{27} + 3 q^{29} + 15 q^{31} - q^{33} - 5 q^{37} - 4 q^{39} + 10 q^{41} + 8 q^{43} - 10 q^{47} + 9 q^{49} + 23 q^{51} - 5 q^{53} + 15 q^{57} - 6 q^{59} + 29 q^{61} + 40 q^{63} + 6 q^{67} - 30 q^{69} + q^{71} - 18 q^{73} - q^{77} + 20 q^{79} + 44 q^{81} + 26 q^{83} + 31 q^{87} + 21 q^{89} - 4 q^{91} - 25 q^{93} + 4 q^{97} - 7 q^{99}+O(q^{100})$$ 4 * q + q^3 + q^7 + 7 * q^9 - 4 * q^11 - 16 * q^13 - 7 * q^17 + 9 * q^19 - 7 * q^21 + 6 * q^23 + 13 * q^27 + 3 * q^29 + 15 * q^31 - q^33 - 5 * q^37 - 4 * q^39 + 10 * q^41 + 8 * q^43 - 10 * q^47 + 9 * q^49 + 23 * q^51 - 5 * q^53 + 15 * q^57 - 6 * q^59 + 29 * q^61 + 40 * q^63 + 6 * q^67 - 30 * q^69 + q^71 - 18 * q^73 - q^77 + 20 * q^79 + 44 * q^81 + 26 * q^83 + 31 * q^87 + 21 * q^89 - 4 * q^91 - 25 * q^93 + 4 * q^97 - 7 * 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
$$3$$ 3.36007 1.93994 0.969969 0.243227i $$-0.0782060\pi$$
0.969969 + 0.243227i $$0.0782060\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.08258 0.409178 0.204589 0.978848i $$-0.434414\pi$$
0.204589 + 0.978848i $$0.434414\pi$$
$$8$$ 0 0
$$9$$ 8.29009 2.76336
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.107866 0.0261614 0.0130807 0.999914i $$-0.495836\pi$$
0.0130807 + 0.999914i $$0.495836\pi$$
$$18$$ 0 0
$$19$$ 6.61228 1.51696 0.758480 0.651696i $$-0.225942\pi$$
0.758480 + 0.651696i $$0.225942\pi$$
$$20$$ 0 0
$$21$$ 3.63756 0.793781
$$22$$ 0 0
$$23$$ −5.97235 −1.24532 −0.622661 0.782492i $$-0.713948\pi$$
−0.622661 + 0.782492i $$0.713948\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 17.7751 3.42082
$$28$$ 0 0
$$29$$ 7.80273 1.44893 0.724465 0.689311i $$-0.242087\pi$$
0.724465 + 0.689311i $$0.242087\pi$$
$$30$$ 0 0
$$31$$ −1.12492 −0.202042 −0.101021 0.994884i $$-0.532211\pi$$
−0.101021 + 0.994884i $$0.532211\pi$$
$$32$$ 0 0
$$33$$ −3.36007 −0.584914
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.05494 1.15982 0.579912 0.814679i $$-0.303087\pi$$
0.579912 + 0.814679i $$0.303087\pi$$
$$38$$ 0 0
$$39$$ −13.4403 −2.15217
$$40$$ 0 0
$$41$$ 5.19045 0.810612 0.405306 0.914181i $$-0.367165\pi$$
0.405306 + 0.914181i $$0.367165\pi$$
$$42$$ 0 0
$$43$$ 5.52969 0.843271 0.421635 0.906766i $$-0.361456\pi$$
0.421635 + 0.906766i $$0.361456\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −7.69486 −1.12241 −0.561206 0.827676i $$-0.689662\pi$$
−0.561206 + 0.827676i $$0.689662\pi$$
$$48$$ 0 0
$$49$$ −5.82801 −0.832573
$$50$$ 0 0
$$51$$ 0.362439 0.0507516
$$52$$ 0 0
$$53$$ 4.77745 0.656233 0.328116 0.944637i $$-0.393586\pi$$
0.328116 + 0.944637i $$0.393586\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 22.2177 2.94281
$$58$$ 0 0
$$59$$ 0.677809 0.0882433 0.0441216 0.999026i $$-0.485951\pi$$
0.0441216 + 0.999026i $$0.485951\pi$$
$$60$$ 0 0
$$61$$ 0.197271 0.0252579 0.0126290 0.999920i $$-0.495980\pi$$
0.0126290 + 0.999920i $$0.495980\pi$$
$$62$$ 0 0
$$63$$ 8.97472 1.13071
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −1.41737 −0.173160 −0.0865799 0.996245i $$-0.527594\pi$$
−0.0865799 + 0.996245i $$0.527594\pi$$
$$68$$ 0 0
$$69$$ −20.0675 −2.41585
$$70$$ 0 0
$$71$$ 6.15020 0.729895 0.364947 0.931028i $$-0.381087\pi$$
0.364947 + 0.931028i $$0.381087\pi$$
$$72$$ 0 0
$$73$$ −6.16517 −0.721578 −0.360789 0.932647i $$-0.617493\pi$$
−0.360789 + 0.932647i $$0.617493\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1.08258 −0.123372
$$78$$ 0 0
$$79$$ 9.35562 1.05259 0.526295 0.850302i $$-0.323581\pi$$
0.526295 + 0.850302i $$0.323581\pi$$
$$80$$ 0 0
$$81$$ 34.8553 3.87281
$$82$$ 0 0
$$83$$ 13.7454 1.50876 0.754378 0.656440i $$-0.227938\pi$$
0.754378 + 0.656440i $$0.227938\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 26.2177 2.81084
$$88$$ 0 0
$$89$$ −1.29009 −0.136749 −0.0683745 0.997660i $$-0.521781\pi$$
−0.0683745 + 0.997660i $$0.521781\pi$$
$$90$$ 0 0
$$91$$ −4.33034 −0.453943
$$92$$ 0 0
$$93$$ −3.77981 −0.391948
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.60782 0.670923 0.335461 0.942054i $$-0.391108\pi$$
0.335461 + 0.942054i $$0.391108\pi$$
$$98$$ 0 0
$$99$$ −8.29009 −0.833185
$$100$$ 0 0
$$101$$ −15.4403 −1.53637 −0.768183 0.640230i $$-0.778839\pi$$
−0.768183 + 0.640230i $$0.778839\pi$$
$$102$$ 0 0
$$103$$ −5.74543 −0.566114 −0.283057 0.959103i $$-0.591349\pi$$
−0.283057 + 0.959103i $$0.591349\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.30514 0.416193 0.208097 0.978108i $$-0.433273\pi$$
0.208097 + 0.978108i $$0.433273\pi$$
$$108$$ 0 0
$$109$$ −2.33034 −0.223206 −0.111603 0.993753i $$-0.535598\pi$$
−0.111603 + 0.993753i $$0.535598\pi$$
$$110$$ 0 0
$$111$$ 23.7051 2.24999
$$112$$ 0 0
$$113$$ −10.9471 −1.02981 −0.514907 0.857246i $$-0.672173\pi$$
−0.514907 + 0.857246i $$0.672173\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −33.1604 −3.06568
$$118$$ 0 0
$$119$$ 0.116774 0.0107047
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 17.4403 1.57254
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 12.5802 1.11631 0.558155 0.829737i $$-0.311509\pi$$
0.558155 + 0.829737i $$0.311509\pi$$
$$128$$ 0 0
$$129$$ 18.5802 1.63589
$$130$$ 0 0
$$131$$ −15.2430 −1.33179 −0.665894 0.746046i $$-0.731950\pi$$
−0.665894 + 0.746046i $$0.731950\pi$$
$$132$$ 0 0
$$133$$ 7.15835 0.620707
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −14.6078 −1.24803 −0.624015 0.781412i $$-0.714500\pi$$
−0.624015 + 0.781412i $$0.714500\pi$$
$$138$$ 0 0
$$139$$ 20.4150 1.73158 0.865789 0.500409i $$-0.166817\pi$$
0.865789 + 0.500409i $$0.166817\pi$$
$$140$$ 0 0
$$141$$ −25.8553 −2.17741
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −19.5825 −1.61514
$$148$$ 0 0
$$149$$ −15.8874 −1.30155 −0.650773 0.759272i $$-0.725555\pi$$
−0.650773 + 0.759272i $$0.725555\pi$$
$$150$$ 0 0
$$151$$ −6.58018 −0.535487 −0.267744 0.963490i $$-0.586278\pi$$
−0.267744 + 0.963490i $$0.586278\pi$$
$$152$$ 0 0
$$153$$ 0.894222 0.0722935
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.38535 −0.509607 −0.254803 0.966993i $$-0.582011\pi$$
−0.254803 + 0.966993i $$0.582011\pi$$
$$158$$ 0 0
$$159$$ 16.0526 1.27305
$$160$$ 0 0
$$161$$ −6.46557 −0.509559
$$162$$ 0 0
$$163$$ −18.3071 −1.43393 −0.716963 0.697111i $$-0.754468\pi$$
−0.716963 + 0.697111i $$0.754468\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0.197271 0.0152653 0.00763263 0.999971i $$-0.497570\pi$$
0.00763263 + 0.999971i $$0.497570\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 54.8164 4.19191
$$172$$ 0 0
$$173$$ −14.7201 −1.11915 −0.559576 0.828779i $$-0.689036\pi$$
−0.559576 + 0.828779i $$0.689036\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2.27749 0.171187
$$178$$ 0 0
$$179$$ −11.8518 −0.885845 −0.442923 0.896560i $$-0.646058\pi$$
−0.442923 + 0.896560i $$0.646058\pi$$
$$180$$ 0 0
$$181$$ −10.7625 −0.799969 −0.399984 0.916522i $$-0.630984\pi$$
−0.399984 + 0.916522i $$0.630984\pi$$
$$182$$ 0 0
$$183$$ 0.662843 0.0489988
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −0.107866 −0.00788797
$$188$$ 0 0
$$189$$ 19.2430 1.39972
$$190$$ 0 0
$$191$$ −1.70309 −0.123231 −0.0616157 0.998100i $$-0.519625\pi$$
−0.0616157 + 0.998100i $$0.519625\pi$$
$$192$$ 0 0
$$193$$ −10.8280 −0.779417 −0.389709 0.920938i $$-0.627424\pi$$
−0.389709 + 0.920938i $$0.627424\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.72015 0.478791 0.239395 0.970922i $$-0.423051\pi$$
0.239395 + 0.970922i $$0.423051\pi$$
$$198$$ 0 0
$$199$$ 10.8280 0.767577 0.383789 0.923421i $$-0.374619\pi$$
0.383789 + 0.923421i $$0.374619\pi$$
$$200$$ 0 0
$$201$$ −4.76248 −0.335920
$$202$$ 0 0
$$203$$ 8.44711 0.592871
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −49.5113 −3.44127
$$208$$ 0 0
$$209$$ −6.61228 −0.457381
$$210$$ 0 0
$$211$$ −0.447111 −0.0307804 −0.0153902 0.999882i $$-0.504899\pi$$
−0.0153902 + 0.999882i $$0.504899\pi$$
$$212$$ 0 0
$$213$$ 20.6651 1.41595
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1.21782 −0.0826710
$$218$$ 0 0
$$219$$ −20.7154 −1.39982
$$220$$ 0 0
$$221$$ −0.431465 −0.0290235
$$222$$ 0 0
$$223$$ −17.8577 −1.19584 −0.597922 0.801555i $$-0.704007\pi$$
−0.597922 + 0.801555i $$0.704007\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1.31396 0.0872107 0.0436054 0.999049i $$-0.486116\pi$$
0.0436054 + 0.999049i $$0.486116\pi$$
$$228$$ 0 0
$$229$$ 4.84298 0.320033 0.160016 0.987114i $$-0.448845\pi$$
0.160016 + 0.987114i $$0.448845\pi$$
$$230$$ 0 0
$$231$$ −3.63756 −0.239334
$$232$$ 0 0
$$233$$ 20.3829 1.33533 0.667664 0.744462i $$-0.267294\pi$$
0.667664 + 0.744462i $$0.267294\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 31.4356 2.04196
$$238$$ 0 0
$$239$$ 9.35562 0.605165 0.302582 0.953123i $$-0.402151\pi$$
0.302582 + 0.953123i $$0.402151\pi$$
$$240$$ 0 0
$$241$$ 22.3004 1.43650 0.718248 0.695788i $$-0.244944\pi$$
0.718248 + 0.695788i $$0.244944\pi$$
$$242$$ 0 0
$$243$$ 63.7911 4.09220
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −26.4491 −1.68292
$$248$$ 0 0
$$249$$ 46.1856 2.92690
$$250$$ 0 0
$$251$$ 10.9276 0.689747 0.344874 0.938649i $$-0.387922\pi$$
0.344874 + 0.938649i $$0.387922\pi$$
$$252$$ 0 0
$$253$$ 5.97235 0.375479
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −21.4403 −1.33741 −0.668704 0.743528i $$-0.733151\pi$$
−0.668704 + 0.743528i $$0.733151\pi$$
$$258$$ 0 0
$$259$$ 7.63756 0.474575
$$260$$ 0 0
$$261$$ 64.6853 4.00392
$$262$$ 0 0
$$263$$ −6.52287 −0.402218 −0.201109 0.979569i $$-0.564454\pi$$
−0.201109 + 0.979569i $$0.564454\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −4.33479 −0.265285
$$268$$ 0 0
$$269$$ 5.60546 0.341771 0.170885 0.985291i $$-0.445337\pi$$
0.170885 + 0.985291i $$0.445337\pi$$
$$270$$ 0 0
$$271$$ 28.9446 1.75826 0.879130 0.476582i $$-0.158124\pi$$
0.879130 + 0.476582i $$0.158124\pi$$
$$272$$ 0 0
$$273$$ −14.5502 −0.880621
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −25.3850 −1.52524 −0.762618 0.646849i $$-0.776086\pi$$
−0.762618 + 0.646849i $$0.776086\pi$$
$$278$$ 0 0
$$279$$ −9.32569 −0.558314
$$280$$ 0 0
$$281$$ −27.1604 −1.62025 −0.810125 0.586257i $$-0.800601\pi$$
−0.810125 + 0.586257i $$0.800601\pi$$
$$282$$ 0 0
$$283$$ −22.0205 −1.30898 −0.654490 0.756070i $$-0.727117\pi$$
−0.654490 + 0.756070i $$0.727117\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 5.61910 0.331685
$$288$$ 0 0
$$289$$ −16.9884 −0.999316
$$290$$ 0 0
$$291$$ 22.2028 1.30155
$$292$$ 0 0
$$293$$ −1.44029 −0.0841427 −0.0420713 0.999115i $$-0.513396\pi$$
−0.0420713 + 0.999115i $$0.513396\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −17.7751 −1.03141
$$298$$ 0 0
$$299$$ 23.8894 1.38156
$$300$$ 0 0
$$301$$ 5.98636 0.345048
$$302$$ 0 0
$$303$$ −51.8805 −2.98046
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2.80955 −0.160349 −0.0801747 0.996781i $$-0.525548\pi$$
−0.0801747 + 0.996781i $$0.525548\pi$$
$$308$$ 0 0
$$309$$ −19.3051 −1.09823
$$310$$ 0 0
$$311$$ 11.5529 0.655104 0.327552 0.944833i $$-0.393776\pi$$
0.327552 + 0.944833i $$0.393776\pi$$
$$312$$ 0 0
$$313$$ −26.1580 −1.47854 −0.739268 0.673411i $$-0.764829\pi$$
−0.739268 + 0.673411i $$0.764829\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −29.3805 −1.65018 −0.825088 0.565005i $$-0.808874\pi$$
−0.825088 + 0.565005i $$0.808874\pi$$
$$318$$ 0 0
$$319$$ −7.80273 −0.436869
$$320$$ 0 0
$$321$$ 14.4656 0.807390
$$322$$ 0 0
$$323$$ 0.713242 0.0396859
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −7.83010 −0.433005
$$328$$ 0 0
$$329$$ −8.33034 −0.459266
$$330$$ 0 0
$$331$$ −12.2833 −0.675149 −0.337575 0.941299i $$-0.609607\pi$$
−0.337575 + 0.941299i $$0.609607\pi$$
$$332$$ 0 0
$$333$$ 58.4860 3.20502
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 21.3324 1.16205 0.581026 0.813885i $$-0.302652\pi$$
0.581026 + 0.813885i $$0.302652\pi$$
$$338$$ 0 0
$$339$$ −36.7829 −1.99778
$$340$$ 0 0
$$341$$ 1.12492 0.0609178
$$342$$ 0 0
$$343$$ −13.8874 −0.749849
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 10.4150 0.559107 0.279553 0.960130i $$-0.409814\pi$$
0.279553 + 0.960130i $$0.409814\pi$$
$$348$$ 0 0
$$349$$ −13.4909 −0.722149 −0.361074 0.932537i $$-0.617590\pi$$
−0.361074 + 0.932537i $$0.617590\pi$$
$$350$$ 0 0
$$351$$ −71.1003 −3.79505
$$352$$ 0 0
$$353$$ 12.7177 0.676895 0.338447 0.940985i $$-0.390098\pi$$
0.338447 + 0.940985i $$0.390098\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0.392370 0.0207664
$$358$$ 0 0
$$359$$ 19.8212 1.04612 0.523061 0.852295i $$-0.324790\pi$$
0.523061 + 0.852295i $$0.324790\pi$$
$$360$$ 0 0
$$361$$ 24.7222 1.30117
$$362$$ 0 0
$$363$$ 3.36007 0.176358
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 10.3027 0.537796 0.268898 0.963169i $$-0.413341\pi$$
0.268898 + 0.963169i $$0.413341\pi$$
$$368$$ 0 0
$$369$$ 43.0293 2.24002
$$370$$ 0 0
$$371$$ 5.17199 0.268516
$$372$$ 0 0
$$373$$ −23.3344 −1.20821 −0.604105 0.796904i $$-0.706469\pi$$
−0.604105 + 0.796904i $$0.706469\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −31.2109 −1.60744
$$378$$ 0 0
$$379$$ 21.2074 1.08935 0.544676 0.838647i $$-0.316653\pi$$
0.544676 + 0.838647i $$0.316653\pi$$
$$380$$ 0 0
$$381$$ 42.2703 2.16557
$$382$$ 0 0
$$383$$ −0.972434 −0.0496891 −0.0248445 0.999691i $$-0.507909\pi$$
−0.0248445 + 0.999691i $$0.507909\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 45.8417 2.33026
$$388$$ 0 0
$$389$$ 2.76248 0.140063 0.0700317 0.997545i $$-0.477690\pi$$
0.0700317 + 0.997545i $$0.477690\pi$$
$$390$$ 0 0
$$391$$ −0.644216 −0.0325794
$$392$$ 0 0
$$393$$ −51.2177 −2.58359
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 11.2751 0.565882 0.282941 0.959137i $$-0.408690\pi$$
0.282941 + 0.959137i $$0.408690\pi$$
$$398$$ 0 0
$$399$$ 24.0526 1.20413
$$400$$ 0 0
$$401$$ −10.4471 −0.521704 −0.260852 0.965379i $$-0.584003\pi$$
−0.260852 + 0.965379i $$0.584003\pi$$
$$402$$ 0 0
$$403$$ 4.49968 0.224145
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −7.05494 −0.349700
$$408$$ 0 0
$$409$$ −1.27512 −0.0630507 −0.0315254 0.999503i $$-0.510037\pi$$
−0.0315254 + 0.999503i $$0.510037\pi$$
$$410$$ 0 0
$$411$$ −49.0834 −2.42110
$$412$$ 0 0
$$413$$ 0.733786 0.0361072
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 68.5959 3.35916
$$418$$ 0 0
$$419$$ −18.7795 −0.917436 −0.458718 0.888582i $$-0.651691\pi$$
−0.458718 + 0.888582i $$0.651691\pi$$
$$420$$ 0 0
$$421$$ −1.27512 −0.0621457 −0.0310728 0.999517i $$-0.509892\pi$$
−0.0310728 + 0.999517i $$0.509892\pi$$
$$422$$ 0 0
$$423$$ −63.7911 −3.10163
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.213562 0.0103350
$$428$$ 0 0
$$429$$ 13.4403 0.648903
$$430$$ 0 0
$$431$$ −1.63556 −0.0787820 −0.0393910 0.999224i $$-0.512542\pi$$
−0.0393910 + 0.999224i $$0.512542\pi$$
$$432$$ 0 0
$$433$$ 26.4932 1.27318 0.636591 0.771201i $$-0.280344\pi$$
0.636591 + 0.771201i $$0.280344\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −39.4909 −1.88910
$$438$$ 0 0
$$439$$ −31.9358 −1.52421 −0.762106 0.647452i $$-0.775835\pi$$
−0.762106 + 0.647452i $$0.775835\pi$$
$$440$$ 0 0
$$441$$ −48.3147 −2.30070
$$442$$ 0 0
$$443$$ −2.02765 −0.0963365 −0.0481682 0.998839i $$-0.515338\pi$$
−0.0481682 + 0.998839i $$0.515338\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −53.3828 −2.52492
$$448$$ 0 0
$$449$$ −10.0675 −0.475116 −0.237558 0.971373i $$-0.576347\pi$$
−0.237558 + 0.971373i $$0.576347\pi$$
$$450$$ 0 0
$$451$$ −5.19045 −0.244409
$$452$$ 0 0
$$453$$ −22.1099 −1.03881
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.04839 0.282932 0.141466 0.989943i $$-0.454818\pi$$
0.141466 + 0.989943i $$0.454818\pi$$
$$458$$ 0 0
$$459$$ 1.91733 0.0894934
$$460$$ 0 0
$$461$$ 31.8397 1.48292 0.741460 0.670997i $$-0.234134\pi$$
0.741460 + 0.670997i $$0.234134\pi$$
$$462$$ 0 0
$$463$$ 30.2474 1.40572 0.702858 0.711330i $$-0.251907\pi$$
0.702858 + 0.711330i $$0.251907\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −31.3417 −1.45032 −0.725160 0.688580i $$-0.758234\pi$$
−0.725160 + 0.688580i $$0.758234\pi$$
$$468$$ 0 0
$$469$$ −1.53443 −0.0708533
$$470$$ 0 0
$$471$$ −21.4553 −0.988606
$$472$$ 0 0
$$473$$ −5.52969 −0.254256
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 39.6055 1.81341
$$478$$ 0 0
$$479$$ −6.59663 −0.301408 −0.150704 0.988579i $$-0.548154\pi$$
−0.150704 + 0.988579i $$0.548154\pi$$
$$480$$ 0 0
$$481$$ −28.2197 −1.28671
$$482$$ 0 0
$$483$$ −21.7248 −0.988512
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −1.13343 −0.0513605 −0.0256802 0.999670i $$-0.508175\pi$$
−0.0256802 + 0.999670i $$0.508175\pi$$
$$488$$ 0 0
$$489$$ −61.5133 −2.78173
$$490$$ 0 0
$$491$$ −43.5569 −1.96570 −0.982848 0.184419i $$-0.940960\pi$$
−0.982848 + 0.184419i $$0.940960\pi$$
$$492$$ 0 0
$$493$$ 0.841652 0.0379061
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.65811 0.298657
$$498$$ 0 0
$$499$$ 34.5502 1.54668 0.773341 0.633991i $$-0.218584\pi$$
0.773341 + 0.633991i $$0.218584\pi$$
$$500$$ 0 0
$$501$$ 0.662843 0.0296137
$$502$$ 0 0
$$503$$ 5.92424 0.264149 0.132074 0.991240i $$-0.457836\pi$$
0.132074 + 0.991240i $$0.457836\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 10.0802 0.447678
$$508$$ 0 0
$$509$$ −42.3679 −1.87793 −0.938963 0.344018i $$-0.888212\pi$$
−0.938963 + 0.344018i $$0.888212\pi$$
$$510$$ 0 0
$$511$$ −6.67431 −0.295254
$$512$$ 0 0
$$513$$ 117.534 5.18924
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 7.69486 0.338420
$$518$$ 0 0
$$519$$ −49.4608 −2.17109
$$520$$ 0 0
$$521$$ 21.2116 0.929297 0.464648 0.885495i $$-0.346181\pi$$
0.464648 + 0.885495i $$0.346181\pi$$
$$522$$ 0 0
$$523$$ −5.19045 −0.226963 −0.113481 0.993540i $$-0.536200\pi$$
−0.113481 + 0.993540i $$0.536200\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −0.121341 −0.00528570
$$528$$ 0 0
$$529$$ 12.6690 0.550825
$$530$$ 0 0
$$531$$ 5.61910 0.243848
$$532$$ 0 0
$$533$$ −20.7618 −0.899293
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −39.8229 −1.71849
$$538$$ 0 0
$$539$$ 5.82801 0.251030
$$540$$ 0 0
$$541$$ −24.0185 −1.03263 −0.516317 0.856397i $$-0.672697\pi$$
−0.516317 + 0.856397i $$0.672697\pi$$
$$542$$ 0 0
$$543$$ −36.1627 −1.55189
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −12.8601 −0.549859 −0.274929 0.961464i $$-0.588654\pi$$
−0.274929 + 0.961464i $$0.588654\pi$$
$$548$$ 0 0
$$549$$ 1.63539 0.0697968
$$550$$ 0 0
$$551$$ 51.5938 2.19797
$$552$$ 0 0
$$553$$ 10.1282 0.430697
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 31.3344 1.32768 0.663841 0.747874i $$-0.268925\pi$$
0.663841 + 0.747874i $$0.268925\pi$$
$$558$$ 0 0
$$559$$ −22.1188 −0.935525
$$560$$ 0 0
$$561$$ −0.362439 −0.0153022
$$562$$ 0 0
$$563$$ −17.6901 −0.745550 −0.372775 0.927922i $$-0.621594\pi$$
−0.372775 + 0.927922i $$0.621594\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 37.7338 1.58467
$$568$$ 0 0
$$569$$ 26.3004 1.10257 0.551285 0.834317i $$-0.314138\pi$$
0.551285 + 0.834317i $$0.314138\pi$$
$$570$$ 0 0
$$571$$ 35.9884 1.50607 0.753033 0.657983i $$-0.228590\pi$$
0.753033 + 0.657983i $$0.228590\pi$$
$$572$$ 0 0
$$573$$ −5.72251 −0.239061
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 32.6031 1.35728 0.678642 0.734469i $$-0.262569\pi$$
0.678642 + 0.734469i $$0.262569\pi$$
$$578$$ 0 0
$$579$$ −36.3829 −1.51202
$$580$$ 0 0
$$581$$ 14.8806 0.617351
$$582$$ 0 0
$$583$$ −4.77745 −0.197862
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 17.6287 0.727612 0.363806 0.931475i $$-0.381477\pi$$
0.363806 + 0.931475i $$0.381477\pi$$
$$588$$ 0 0
$$589$$ −7.43829 −0.306489
$$590$$ 0 0
$$591$$ 22.5802 0.928824
$$592$$ 0 0
$$593$$ −13.2246 −0.543068 −0.271534 0.962429i $$-0.587531\pi$$
−0.271534 + 0.962429i $$0.587531\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 36.3829 1.48905
$$598$$ 0 0
$$599$$ −37.2771 −1.52310 −0.761551 0.648105i $$-0.775562\pi$$
−0.761551 + 0.648105i $$0.775562\pi$$
$$600$$ 0 0
$$601$$ 17.0594 0.695867 0.347934 0.937519i $$-0.386883\pi$$
0.347934 + 0.937519i $$0.386883\pi$$
$$602$$ 0 0
$$603$$ −11.7502 −0.478503
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7.92624 0.321716 0.160858 0.986978i $$-0.448574\pi$$
0.160858 + 0.986978i $$0.448574\pi$$
$$608$$ 0 0
$$609$$ 28.3829 1.15013
$$610$$ 0 0
$$611$$ 30.7795 1.24520
$$612$$ 0 0
$$613$$ −9.93596 −0.401310 −0.200655 0.979662i $$-0.564307\pi$$
−0.200655 + 0.979662i $$0.564307\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −38.1010 −1.53389 −0.766944 0.641715i $$-0.778223\pi$$
−0.766944 + 0.641715i $$0.778223\pi$$
$$618$$ 0 0
$$619$$ −5.70309 −0.229227 −0.114613 0.993410i $$-0.536563\pi$$
−0.114613 + 0.993410i $$0.536563\pi$$
$$620$$ 0 0
$$621$$ −106.159 −4.26002
$$622$$ 0 0
$$623$$ −1.39663 −0.0559548
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −22.2177 −0.887291
$$628$$ 0 0
$$629$$ 0.760990 0.0303427
$$630$$ 0 0
$$631$$ −17.3242 −0.689665 −0.344833 0.938664i $$-0.612064\pi$$
−0.344833 + 0.938664i $$0.612064\pi$$
$$632$$ 0 0
$$633$$ −1.50232 −0.0597120
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 23.3120 0.923657
$$638$$ 0 0
$$639$$ 50.9857 2.01696
$$640$$ 0 0
$$641$$ −22.3523 −0.882863 −0.441431 0.897295i $$-0.645529\pi$$
−0.441431 + 0.897295i $$0.645529\pi$$
$$642$$ 0 0
$$643$$ −25.2911 −0.997385 −0.498693 0.866779i $$-0.666186\pi$$
−0.498693 + 0.866779i $$0.666186\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 22.4262 0.881665 0.440832 0.897589i $$-0.354683\pi$$
0.440832 + 0.897589i $$0.354683\pi$$
$$648$$ 0 0
$$649$$ −0.677809 −0.0266063
$$650$$ 0 0
$$651$$ −4.09197 −0.160377
$$652$$ 0 0
$$653$$ −11.8391 −0.463301 −0.231650 0.972799i $$-0.574413\pi$$
−0.231650 + 0.972799i $$0.574413\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −51.1098 −1.99398
$$658$$ 0 0
$$659$$ 15.7522 0.613617 0.306809 0.951771i $$-0.400739\pi$$
0.306809 + 0.951771i $$0.400739\pi$$
$$660$$ 0 0
$$661$$ −7.15702 −0.278376 −0.139188 0.990266i $$-0.544449\pi$$
−0.139188 + 0.990266i $$0.544449\pi$$
$$662$$ 0 0
$$663$$ −1.44976 −0.0563038
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −46.6006 −1.80438
$$668$$ 0 0
$$669$$ −60.0033 −2.31986
$$670$$ 0 0
$$671$$ −0.197271 −0.00761555
$$672$$ 0 0
$$673$$ 11.4976 0.443200 0.221600 0.975138i $$-0.428872\pi$$
0.221600 + 0.975138i $$0.428872\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 21.9953 0.845347 0.422673 0.906282i $$-0.361092\pi$$
0.422673 + 0.906282i $$0.361092\pi$$
$$678$$ 0 0
$$679$$ 7.15353 0.274527
$$680$$ 0 0
$$681$$ 4.41501 0.169183
$$682$$ 0 0
$$683$$ 0.468300 0.0179190 0.00895950 0.999960i $$-0.497148\pi$$
0.00895950 + 0.999960i $$0.497148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 16.2728 0.620844
$$688$$ 0 0
$$689$$ −19.1098 −0.728025
$$690$$ 0 0
$$691$$ 36.9140 1.40428 0.702138 0.712041i $$-0.252229\pi$$
0.702138 + 0.712041i $$0.252229\pi$$
$$692$$ 0 0
$$693$$ −8.97472 −0.340921
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0.559875 0.0212068
$$698$$ 0 0
$$699$$ 68.4880 2.59046
$$700$$ 0 0
$$701$$ 18.6628 0.704886 0.352443 0.935833i $$-0.385351\pi$$
0.352443 + 0.935833i $$0.385351\pi$$
$$702$$ 0 0
$$703$$ 46.6492 1.75941
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −16.7154 −0.628648
$$708$$ 0 0
$$709$$ 6.66135 0.250172 0.125086 0.992146i $$-0.460079\pi$$
0.125086 + 0.992146i $$0.460079\pi$$
$$710$$ 0 0
$$711$$ 77.5589 2.90869
$$712$$ 0 0
$$713$$ 6.71842 0.251607
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 31.4356 1.17398
$$718$$ 0 0
$$719$$ 3.11128 0.116031 0.0580156 0.998316i $$-0.481523\pi$$
0.0580156 + 0.998316i $$0.481523\pi$$
$$720$$ 0 0
$$721$$ −6.21991 −0.231641
$$722$$ 0 0
$$723$$ 74.9310 2.78671
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 33.9540 1.25928 0.629642 0.776886i $$-0.283202\pi$$
0.629642 + 0.776886i $$0.283202\pi$$
$$728$$ 0 0
$$729$$ 109.777 4.06581
$$730$$ 0 0
$$731$$ 0.596468 0.0220612
$$732$$ 0 0
$$733$$ −21.0457 −0.777342 −0.388671 0.921377i $$-0.627066\pi$$
−0.388671 + 0.921377i $$0.627066\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.41737 0.0522097
$$738$$ 0 0
$$739$$ 17.0253 0.626285 0.313143 0.949706i $$-0.398618\pi$$
0.313143 + 0.949706i $$0.398618\pi$$
$$740$$ 0 0
$$741$$ −88.8710 −3.26476
$$742$$ 0 0
$$743$$ 23.7563 0.871536 0.435768 0.900059i $$-0.356477\pi$$
0.435768 + 0.900059i $$0.356477\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 113.951 4.16924
$$748$$ 0 0
$$749$$ 4.66067 0.170297
$$750$$ 0 0
$$751$$ 44.8654 1.63716 0.818582 0.574390i $$-0.194761\pi$$
0.818582 + 0.574390i $$0.194761\pi$$
$$752$$ 0 0
$$753$$ 36.7177 1.33807
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −41.3761 −1.50384 −0.751920 0.659255i $$-0.770872\pi$$
−0.751920 + 0.659255i $$0.770872\pi$$
$$758$$ 0 0
$$759$$ 20.0675 0.728405
$$760$$ 0 0
$$761$$ 16.3002 0.590883 0.295442 0.955361i $$-0.404533\pi$$
0.295442 + 0.955361i $$0.404533\pi$$
$$762$$ 0 0
$$763$$ −2.52279 −0.0913310
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −2.71124 −0.0978971
$$768$$ 0 0
$$769$$ 41.1262 1.48305 0.741525 0.670925i $$-0.234103\pi$$
0.741525 + 0.670925i $$0.234103\pi$$
$$770$$ 0 0
$$771$$ −72.0409 −2.59449
$$772$$ 0 0
$$773$$ −9.77280 −0.351503 −0.175752 0.984435i $$-0.556236\pi$$
−0.175752 + 0.984435i $$0.556236\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 25.6628 0.920646
$$778$$ 0 0
$$779$$ 34.3207 1.22967
$$780$$ 0 0
$$781$$ −6.15020 −0.220072
$$782$$ 0 0
$$783$$ 138.694 4.95652
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 5.25466 0.187308 0.0936541 0.995605i $$-0.470145\pi$$
0.0936541 + 0.995605i $$0.470145\pi$$
$$788$$ 0 0
$$789$$ −21.9173 −0.780278
$$790$$ 0 0
$$791$$ −11.8511 −0.421377
$$792$$ 0 0
$$793$$ −0.789082 −0.0280211
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 14.2133 0.503460 0.251730 0.967797i $$-0.419000\pi$$
0.251730 + 0.967797i $$0.419000\pi$$
$$798$$ 0 0
$$799$$ −0.830017 −0.0293639
$$800$$ 0 0
$$801$$ −10.6949 −0.377887
$$802$$ 0 0
$$803$$ 6.16517 0.217564
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18.8347 0.663015
$$808$$ 0 0
$$809$$ −12.8641 −0.452279 −0.226139 0.974095i $$-0.572610\pi$$
−0.226139 + 0.974095i $$0.572610\pi$$
$$810$$ 0 0
$$811$$ −4.31605 −0.151557 −0.0757785 0.997125i $$-0.524144\pi$$
−0.0757785 + 0.997125i $$0.524144\pi$$
$$812$$ 0 0
$$813$$ 97.2560 3.41092
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 36.5639 1.27921
$$818$$ 0 0
$$819$$ −35.8989 −1.25441
$$820$$ 0 0
$$821$$ −42.9994 −1.50069 −0.750344 0.661048i $$-0.770112\pi$$
−0.750344 + 0.661048i $$0.770112\pi$$
$$822$$ 0 0
$$823$$ −48.1834 −1.67957 −0.839783 0.542922i $$-0.817318\pi$$
−0.839783 + 0.542922i $$0.817318\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 32.2868 1.12272 0.561360 0.827571i $$-0.310278\pi$$
0.561360 + 0.827571i $$0.310278\pi$$
$$828$$ 0 0
$$829$$ 54.9345 1.90795 0.953977 0.299881i $$-0.0969470\pi$$
0.953977 + 0.299881i $$0.0969470\pi$$
$$830$$ 0 0
$$831$$ −85.2954 −2.95887
$$832$$ 0 0
$$833$$ −0.628646 −0.0217813
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −19.9955 −0.691147
$$838$$ 0 0
$$839$$ −11.2281 −0.387635 −0.193818 0.981038i $$-0.562087\pi$$
−0.193818 + 0.981038i $$0.562087\pi$$
$$840$$ 0 0
$$841$$ 31.8826 1.09940
$$842$$ 0 0
$$843$$ −91.2608 −3.14319
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1.08258 0.0371980
$$848$$ 0 0
$$849$$ −73.9904 −2.53934
$$850$$ 0 0
$$851$$ −42.1346 −1.44435
$$852$$ 0 0
$$853$$ −2.40328 −0.0822869 −0.0411434 0.999153i $$-0.513100\pi$$
−0.0411434 + 0.999153i $$0.513100\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −34.8280 −1.18970 −0.594851 0.803836i $$-0.702789\pi$$
−0.594851 + 0.803836i $$0.702789\pi$$
$$858$$ 0 0
$$859$$ −27.0627 −0.923368 −0.461684 0.887044i $$-0.652755\pi$$
−0.461684 + 0.887044i $$0.652755\pi$$
$$860$$ 0 0
$$861$$ 18.8806 0.643448
$$862$$ 0 0
$$863$$ 40.0157 1.36215 0.681076 0.732213i $$-0.261512\pi$$
0.681076 + 0.732213i $$0.261512\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −57.0821 −1.93861
$$868$$ 0 0
$$869$$ −9.35562 −0.317368
$$870$$ 0 0
$$871$$ 5.66950 0.192104
$$872$$ 0 0
$$873$$ 54.7795 1.85400
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 18.3167 0.618511 0.309255 0.950979i $$-0.399920\pi$$
0.309255 + 0.950979i $$0.399920\pi$$
$$878$$ 0 0
$$879$$ −4.83948 −0.163232
$$880$$ 0 0
$$881$$ 11.8560 0.399438 0.199719 0.979853i $$-0.435997\pi$$
0.199719 + 0.979853i $$0.435997\pi$$
$$882$$ 0 0
$$883$$ 15.2478 0.513128 0.256564 0.966527i $$-0.417410\pi$$
0.256564 + 0.966527i $$0.417410\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 3.29631 0.110679 0.0553397 0.998468i $$-0.482376\pi$$
0.0553397 + 0.998468i $$0.482376\pi$$
$$888$$ 0 0
$$889$$ 13.6191 0.456770
$$890$$ 0 0
$$891$$ −34.8553 −1.16770
$$892$$ 0 0
$$893$$ −50.8806 −1.70265
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 80.2701 2.68014
$$898$$ 0 0
$$899$$ −8.77745 −0.292744
$$900$$ 0 0
$$901$$ 0.515326 0.0171680
$$902$$ 0 0
$$903$$ 20.1146 0.669372
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −16.3577 −0.543149 −0.271574 0.962417i $$-0.587544\pi$$
−0.271574 + 0.962417i $$0.587544\pi$$
$$908$$ 0 0
$$909$$ −128.001 −4.24554
$$910$$ 0 0
$$911$$ 8.34598 0.276515 0.138257 0.990396i $$-0.455850\pi$$
0.138257 + 0.990396i $$0.455850\pi$$
$$912$$ 0 0
$$913$$ −13.7454 −0.454907
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −16.5019 −0.544939
$$918$$ 0 0
$$919$$ 12.9611 0.427546 0.213773 0.976883i $$-0.431425\pi$$
0.213773 + 0.976883i $$0.431425\pi$$
$$920$$ 0 0
$$921$$ −9.44029 −0.311068
$$922$$ 0 0
$$923$$ −24.6008 −0.809746
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −47.6301 −1.56438
$$928$$ 0 0
$$929$$ 37.2635 1.22258 0.611288 0.791408i $$-0.290652\pi$$
0.611288 + 0.791408i $$0.290652\pi$$
$$930$$ 0 0
$$931$$ −38.5364 −1.26298
$$932$$ 0 0
$$933$$ 38.8185 1.27086
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 4.28395 0.139950 0.0699752 0.997549i $$-0.477708\pi$$
0.0699752 + 0.997549i $$0.477708\pi$$
$$938$$ 0 0
$$939$$ −87.8927 −2.86827
$$940$$ 0 0
$$941$$ 23.2089 0.756589 0.378294 0.925685i $$-0.376511\pi$$
0.378294 + 0.925685i $$0.376511\pi$$
$$942$$ 0 0
$$943$$ −30.9992 −1.00947
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 20.8646 0.678007 0.339004 0.940785i $$-0.389910\pi$$
0.339004 + 0.940785i $$0.389910\pi$$
$$948$$ 0 0
$$949$$ 24.6607 0.800519
$$950$$ 0 0
$$951$$ −98.7207 −3.20124
$$952$$ 0 0
$$953$$ −45.0061 −1.45789 −0.728945 0.684572i $$-0.759989\pi$$
−0.728945 + 0.684572i $$0.759989\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −26.2177 −0.847499
$$958$$ 0 0
$$959$$ −15.8142 −0.510667
$$960$$ 0 0
$$961$$ −29.7346 −0.959179
$$962$$ 0 0
$$963$$ 35.6900 1.15009
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −21.9679 −0.706440 −0.353220 0.935540i $$-0.614913\pi$$
−0.353220 + 0.935540i $$0.614913\pi$$
$$968$$ 0 0
$$969$$ 2.39655 0.0769882
$$970$$ 0 0
$$971$$ 16.5837 0.532195 0.266098 0.963946i $$-0.414266\pi$$
0.266098 + 0.963946i $$0.414266\pi$$
$$972$$ 0 0
$$973$$ 22.1010 0.708524
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −19.8189 −0.634063 −0.317032 0.948415i $$-0.602686\pi$$
−0.317032 + 0.948415i $$0.602686\pi$$
$$978$$ 0 0
$$979$$ 1.29009 0.0412314
$$980$$ 0 0
$$981$$ −19.3187 −0.616798
$$982$$ 0 0
$$983$$ −13.9634 −0.445365 −0.222682 0.974891i $$-0.571481\pi$$
−0.222682 + 0.974891i $$0.571481\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −27.9905 −0.890949
$$988$$ 0 0
$$989$$ −33.0253 −1.05014
$$990$$ 0 0
$$991$$ −4.10113 −0.130277 −0.0651383 0.997876i $$-0.520749\pi$$
−0.0651383 + 0.997876i $$0.520749\pi$$
$$992$$ 0 0
$$993$$ −41.2727 −1.30975
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −33.3208 −1.05528 −0.527640 0.849468i $$-0.676923\pi$$
−0.527640 + 0.849468i $$0.676923\pi$$
$$998$$ 0 0
$$999$$ 125.402 3.96755
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 4400.2.a.ce.1.4 4
4.3 odd 2 2200.2.a.x.1.1 4
5.2 odd 4 880.2.b.j.529.1 8
5.3 odd 4 880.2.b.j.529.8 8
5.4 even 2 4400.2.a.cb.1.1 4
20.3 even 4 440.2.b.d.89.1 8
20.7 even 4 440.2.b.d.89.8 yes 8
20.19 odd 2 2200.2.a.y.1.4 4
60.23 odd 4 3960.2.d.f.3169.6 8
60.47 odd 4 3960.2.d.f.3169.5 8

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.b.d.89.1 8 20.3 even 4
440.2.b.d.89.8 yes 8 20.7 even 4
880.2.b.j.529.1 8 5.2 odd 4
880.2.b.j.529.8 8 5.3 odd 4
2200.2.a.x.1.1 4 4.3 odd 2
2200.2.a.y.1.4 4 20.19 odd 2
3960.2.d.f.3169.5 8 60.47 odd 4
3960.2.d.f.3169.6 8 60.23 odd 4
4400.2.a.cb.1.1 4 5.4 even 2
4400.2.a.ce.1.4 4 1.1 even 1 trivial