# Properties

 Label 2200.2.a.y.1.4 Level $2200$ Weight $2$ Character 2200.1 Self dual yes Analytic conductor $17.567$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$17.5670884447$$ 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 $$\chi$$ $$=$$ 2200.1

## $q$-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 $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −0.107866 −0.0261614 −0.0130807 0.999914i $$-0.504164\pi$$
−0.0130807 + 0.999914i $$0.504164\pi$$
$$18$$ 0 0
$$19$$ −6.61228 −1.51696 −0.758480 0.651696i $$-0.774058\pi$$
−0.758480 + 0.651696i $$0.774058\pi$$
$$20$$ 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 $$-0.467789\pi$$
0.101021 + 0.994884i $$0.467789\pi$$
$$32$$ 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 $$-0.696913\pi$$
−0.579912 + 0.814679i $$0.696913\pi$$
$$38$$ 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 $$-0.606414\pi$$
−0.328116 + 0.944637i $$0.606414\pi$$
$$54$$ 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 $$-0.514049\pi$$
−0.0441216 + 0.999026i $$0.514049\pi$$
$$60$$ 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 $$-0.618913\pi$$
−0.364947 + 0.931028i $$0.618913\pi$$
$$72$$ 0 0
$$73$$ 6.16517 0.721578 0.360789 0.932647i $$-0.382507\pi$$
0.360789 + 0.932647i $$0.382507\pi$$
$$74$$ 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 $$-0.676419\pi$$
−0.526295 + 0.850302i $$0.676419\pi$$
$$80$$ 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 $$-0.608892\pi$$
−0.335461 + 0.942054i $$0.608892\pi$$
$$98$$ 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 $$-0.327827\pi$$
0.514907 + 0.857246i $$0.327827\pi$$
$$114$$ 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 $$-0.268050\pi$$
0.665894 + 0.746046i $$0.268050\pi$$
$$132$$ 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 $$-0.285500\pi$$
0.624015 + 0.781412i $$0.285500\pi$$
$$138$$ 0 0
$$139$$ −20.4150 −1.73158 −0.865789 0.500409i $$-0.833183\pi$$
−0.865789 + 0.500409i $$0.833183\pi$$
$$140$$ 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 $$-0.413722\pi$$
0.267744 + 0.963490i $$0.413722\pi$$
$$152$$ 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 $$-0.417989\pi$$
0.254803 + 0.966993i $$0.417989\pi$$
$$158$$ 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 $$-0.310964\pi$$
0.559576 + 0.828779i $$0.310964\pi$$
$$174$$ 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 $$-0.353942\pi$$
0.442923 + 0.896560i $$0.353942\pi$$
$$180$$ 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 $$-0.480375\pi$$
0.0616157 + 0.998100i $$0.480375\pi$$
$$192$$ 0 0
$$193$$ 10.8280 0.779417 0.389709 0.920938i $$-0.372576\pi$$
0.389709 + 0.920938i $$0.372576\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.72015 −0.478791 −0.239395 0.970922i $$-0.576949\pi$$
−0.239395 + 0.970922i $$0.576949\pi$$
$$198$$ 0 0
$$199$$ −10.8280 −0.767577 −0.383789 0.923421i $$-0.625381\pi$$
−0.383789 + 0.923421i $$0.625381\pi$$
$$200$$ 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 $$-0.495101\pi$$
0.0153902 + 0.999882i $$0.495101\pi$$
$$212$$ 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 $$-0.732706\pi$$
−0.667664 + 0.744462i $$0.732706\pi$$
$$234$$ 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 $$-0.597849\pi$$
−0.302582 + 0.953123i $$0.597849\pi$$
$$240$$ 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 $$-0.612078\pi$$
−0.344874 + 0.938649i $$0.612078\pi$$
$$252$$ 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 $$-0.266849\pi$$
0.668704 + 0.743528i $$0.266849\pi$$
$$258$$ 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 $$-0.841876\pi$$
−0.879130 + 0.476582i $$0.841876\pi$$
$$272$$ 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 $$-0.223914\pi$$
0.762618 + 0.646849i $$0.223914\pi$$
$$278$$ 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 $$-0.486604\pi$$
0.0420713 + 0.999115i $$0.486604\pi$$
$$294$$ 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 $$-0.606224\pi$$
−0.327552 + 0.944833i $$0.606224\pi$$
$$312$$ 0 0
$$313$$ 26.1580 1.47854 0.739268 0.673411i $$-0.235171\pi$$
0.739268 + 0.673411i $$0.235171\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 29.3805 1.65018 0.825088 0.565005i $$-0.191126\pi$$
0.825088 + 0.565005i $$0.191126\pi$$
$$318$$ 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 $$-0.390393\pi$$
0.337575 + 0.941299i $$0.390393\pi$$
$$332$$ 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 $$-0.697348\pi$$
−0.581026 + 0.813885i $$0.697348\pi$$
$$338$$ 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 $$-0.609902\pi$$
−0.338447 + 0.940985i $$0.609902\pi$$
$$354$$ 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 $$-0.675210\pi$$
−0.523061 + 0.852295i $$0.675210\pi$$
$$360$$ 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 $$-0.293531\pi$$
0.604105 + 0.796904i $$0.293531\pi$$
$$374$$ 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 $$-0.683347\pi$$
−0.544676 + 0.838647i $$0.683347\pi$$
$$380$$ 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 $$-0.591310\pi$$
−0.282941 + 0.959137i $$0.591310\pi$$
$$398$$ 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 $$-0.348309\pi$$
0.458718 + 0.888582i $$0.348309\pi$$
$$420$$ 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 $$-0.487458\pi$$
0.0393910 + 0.999224i $$0.487458\pi$$
$$432$$ 0 0
$$433$$ −26.4932 −1.27318 −0.636591 0.771201i $$-0.719656\pi$$
−0.636591 + 0.771201i $$0.719656\pi$$
$$434$$ 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 $$-0.224165\pi$$
0.762106 + 0.647452i $$0.224165\pi$$
$$440$$ 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 $$-0.545182\pi$$
−0.141466 + 0.989943i $$0.545182\pi$$
$$458$$ 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 $$-0.451846\pi$$
0.150704 + 0.988579i $$0.451846\pi$$
$$480$$ 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 $$-0.0590404\pi$$
0.982848 + 0.184419i $$0.0590404\pi$$
$$492$$ 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 $$-0.781416\pi$$
−0.773341 + 0.633991i $$0.781416\pi$$
$$500$$ 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 $$-0.731075\pi$$
−0.663841 + 0.747874i $$0.731075\pi$$
$$558$$ 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 $$-0.771410\pi$$
−0.753033 + 0.657983i $$0.771410\pi$$
$$572$$ 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 $$-0.737431\pi$$
−0.678642 + 0.734469i $$0.737431\pi$$
$$578$$ 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 $$-0.412469\pi$$
0.271534 + 0.962429i $$0.412469\pi$$
$$594$$ 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 $$-0.224438\pi$$
0.761551 + 0.648105i $$0.224438\pi$$
$$600$$ 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 $$-0.435693\pi$$
0.200655 + 0.979662i $$0.435693\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 38.1010 1.53389 0.766944 0.641715i $$-0.221777\pi$$
0.766944 + 0.641715i $$0.221777\pi$$
$$618$$ 0 0
$$619$$ 5.70309 0.229227 0.114613 0.993410i $$-0.463437\pi$$
0.114613 + 0.993410i $$0.463437\pi$$
$$620$$ 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 $$-0.387936\pi$$
0.344833 + 0.938664i $$0.387936\pi$$
$$632$$ 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 $$-0.425587\pi$$
0.231650 + 0.972799i $$0.425587\pi$$
$$654$$ 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 $$-0.599261\pi$$
−0.306809 + 0.951771i $$0.599261\pi$$
$$660$$ 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 $$-0.571128\pi$$
−0.221600 + 0.975138i $$0.571128\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −21.9953 −0.845347 −0.422673 0.906282i $$-0.638908\pi$$
−0.422673 + 0.906282i $$0.638908\pi$$
$$678$$ 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 $$-0.747771\pi$$
−0.702138 + 0.712041i $$0.747771\pi$$
$$692$$ 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 $$-0.518477\pi$$
−0.0580156 + 0.998316i $$0.518477\pi$$
$$720$$ 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 $$-0.372934\pi$$
0.388671 + 0.921377i $$0.372934\pi$$
$$734$$ 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 $$-0.601382\pi$$
−0.313143 + 0.949706i $$0.601382\pi$$
$$740$$ 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 $$-0.805239\pi$$
−0.818582 + 0.574390i $$0.805239\pi$$
$$752$$ 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 $$-0.229128\pi$$
0.751920 + 0.659255i $$0.229128\pi$$
$$758$$ 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 $$-0.443764\pi$$
0.175752 + 0.984435i $$0.443764\pi$$
$$774$$ 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 $$-0.581000\pi$$
−0.251730 + 0.967797i $$0.581000\pi$$
$$798$$ 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 $$-0.475856\pi$$
0.0757785 + 0.997125i $$0.475856\pi$$
$$812$$ 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 $$-0.437913\pi$$
0.193818 + 0.981038i $$0.437913\pi$$
$$840$$ 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 $$-0.486900\pi$$
0.0411434 + 0.999153i $$0.486900\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 34.8280 1.18970 0.594851 0.803836i $$-0.297211\pi$$
0.594851 + 0.803836i $$0.297211\pi$$
$$858$$ 0 0
$$859$$ 27.0627 0.923368 0.461684 0.887044i $$-0.347245\pi$$
0.461684 + 0.887044i $$0.347245\pi$$
$$860$$ 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 $$-0.600080\pi$$
−0.309255 + 0.950979i $$0.600080\pi$$
$$878$$ 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 $$-0.544150\pi$$
−0.138257 + 0.990396i $$0.544150\pi$$
$$912$$ 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 $$-0.568575\pi$$
−0.213773 + 0.976883i $$0.568575\pi$$
$$920$$ 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 $$-0.522292\pi$$
−0.0699752 + 0.997549i $$0.522292\pi$$
$$938$$ 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 $$-0.240011\pi$$
0.728945 + 0.684572i $$0.240011\pi$$
$$954$$ 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 $$-0.585734\pi$$
−0.266098 + 0.963946i $$0.585734\pi$$
$$972$$ 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 $$-0.397314\pi$$
0.317032 + 0.948415i $$0.397314\pi$$
$$978$$ 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 $$-0.479251\pi$$
0.0651383 + 0.997876i $$0.479251\pi$$
$$992$$ 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 $$-0.323077\pi$$
0.527640 + 0.849468i $$0.323077\pi$$
$$998$$ 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 2200.2.a.y.1.4 4
4.3 odd 2 4400.2.a.cb.1.1 4
5.2 odd 4 440.2.b.d.89.1 8
5.3 odd 4 440.2.b.d.89.8 yes 8
5.4 even 2 2200.2.a.x.1.1 4
15.2 even 4 3960.2.d.f.3169.6 8
15.8 even 4 3960.2.d.f.3169.5 8
20.3 even 4 880.2.b.j.529.1 8
20.7 even 4 880.2.b.j.529.8 8
20.19 odd 2 4400.2.a.ce.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.b.d.89.1 8 5.2 odd 4
440.2.b.d.89.8 yes 8 5.3 odd 4
880.2.b.j.529.1 8 20.3 even 4
880.2.b.j.529.8 8 20.7 even 4
2200.2.a.x.1.1 4 5.4 even 2
2200.2.a.y.1.4 4 1.1 even 1 trivial
3960.2.d.f.3169.5 8 15.8 even 4
3960.2.d.f.3169.6 8 15.2 even 4
4400.2.a.cb.1.1 4 4.3 odd 2
4400.2.a.ce.1.4 4 20.19 odd 2