# Properties

 Label 4840.2.a.y.1.3 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $1$ 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] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 3x^{2} + x + 1$$ x^4 - x^3 - 3*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 1.3 Root $$-0.477260$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.294963 q^{3} +1.00000 q^{5} -4.39026 q^{7} -2.91300 q^{9} +O(q^{10})$$ $$q+0.294963 q^{3} +1.00000 q^{5} -4.39026 q^{7} -2.91300 q^{9} +5.96281 q^{13} +0.294963 q^{15} -1.66785 q^{17} +7.69351 q^{19} -1.29496 q^{21} -0.904706 q^{23} +1.00000 q^{25} -1.74411 q^{27} -4.73899 q^{29} -5.12608 q^{31} -4.39026 q^{35} -0.184976 q^{37} +1.75881 q^{39} -2.62237 q^{41} +3.59822 q^{43} -2.91300 q^{45} -0.776180 q^{47} +12.2744 q^{49} -0.491953 q^{51} -9.59554 q^{53} +2.26930 q^{57} -11.2538 q^{59} +13.1898 q^{61} +12.7888 q^{63} +5.96281 q^{65} -7.79954 q^{67} -0.266855 q^{69} -6.97072 q^{71} +12.7541 q^{73} +0.294963 q^{75} -10.0313 q^{79} +8.22454 q^{81} +4.09529 q^{83} -1.66785 q^{85} -1.39783 q^{87} -0.466291 q^{89} -26.1783 q^{91} -1.51200 q^{93} +7.69351 q^{95} +7.09963 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 4 * q^5 - 7 * q^7 - 4 * q^9 $$4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9} + 3 q^{13} - 2 q^{15} + 11 q^{17} + 2 q^{19} - 2 q^{21} - 11 q^{23} + 4 q^{25} + q^{27} + 4 q^{29} - 17 q^{31} - 7 q^{35} - 3 q^{37} + q^{39} + 13 q^{41} - 7 q^{43} - 4 q^{45} - q^{47} - 5 q^{49} - q^{51} - 15 q^{53} + 17 q^{57} - 17 q^{59} + 4 q^{61} + 15 q^{63} + 3 q^{65} + 7 q^{67} + 4 q^{69} - 15 q^{71} + 7 q^{73} - 2 q^{75} - 12 q^{79} - 8 q^{81} + 9 q^{83} + 11 q^{85} - 23 q^{87} - 12 q^{89} - 24 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97}+O(q^{100})$$ 4 * q - 2 * q^3 + 4 * q^5 - 7 * q^7 - 4 * q^9 + 3 * q^13 - 2 * q^15 + 11 * q^17 + 2 * q^19 - 2 * q^21 - 11 * q^23 + 4 * q^25 + q^27 + 4 * q^29 - 17 * q^31 - 7 * q^35 - 3 * q^37 + q^39 + 13 * q^41 - 7 * q^43 - 4 * q^45 - q^47 - 5 * q^49 - q^51 - 15 * q^53 + 17 * q^57 - 17 * q^59 + 4 * q^61 + 15 * q^63 + 3 * q^65 + 7 * q^67 + 4 * q^69 - 15 * q^71 + 7 * q^73 - 2 * q^75 - 12 * q^79 - 8 * q^81 + 9 * q^83 + 11 * q^85 - 23 * q^87 - 12 * q^89 - 24 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97

## 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
$$3$$ 0.294963 0.170297 0.0851485 0.996368i $$-0.472864\pi$$
0.0851485 + 0.996368i $$0.472864\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.39026 −1.65936 −0.829681 0.558238i $$-0.811477\pi$$
−0.829681 + 0.558238i $$0.811477\pi$$
$$8$$ 0 0
$$9$$ −2.91300 −0.970999
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 5.96281 1.65379 0.826893 0.562359i $$-0.190106\pi$$
0.826893 + 0.562359i $$0.190106\pi$$
$$14$$ 0 0
$$15$$ 0.294963 0.0761591
$$16$$ 0 0
$$17$$ −1.66785 −0.404513 −0.202256 0.979333i $$-0.564827\pi$$
−0.202256 + 0.979333i $$0.564827\pi$$
$$18$$ 0 0
$$19$$ 7.69351 1.76501 0.882506 0.470301i $$-0.155855\pi$$
0.882506 + 0.470301i $$0.155855\pi$$
$$20$$ 0 0
$$21$$ −1.29496 −0.282584
$$22$$ 0 0
$$23$$ −0.904706 −0.188644 −0.0943221 0.995542i $$-0.530068\pi$$
−0.0943221 + 0.995542i $$0.530068\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.74411 −0.335655
$$28$$ 0 0
$$29$$ −4.73899 −0.880008 −0.440004 0.897996i $$-0.645023\pi$$
−0.440004 + 0.897996i $$0.645023\pi$$
$$30$$ 0 0
$$31$$ −5.12608 −0.920671 −0.460336 0.887745i $$-0.652271\pi$$
−0.460336 + 0.887745i $$0.652271\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.39026 −0.742089
$$36$$ 0 0
$$37$$ −0.184976 −0.0304098 −0.0152049 0.999884i $$-0.504840\pi$$
−0.0152049 + 0.999884i $$0.504840\pi$$
$$38$$ 0 0
$$39$$ 1.75881 0.281635
$$40$$ 0 0
$$41$$ −2.62237 −0.409545 −0.204773 0.978810i $$-0.565645\pi$$
−0.204773 + 0.978810i $$0.565645\pi$$
$$42$$ 0 0
$$43$$ 3.59822 0.548723 0.274361 0.961627i $$-0.411534\pi$$
0.274361 + 0.961627i $$0.411534\pi$$
$$44$$ 0 0
$$45$$ −2.91300 −0.434244
$$46$$ 0 0
$$47$$ −0.776180 −0.113217 −0.0566087 0.998396i $$-0.518029\pi$$
−0.0566087 + 0.998396i $$0.518029\pi$$
$$48$$ 0 0
$$49$$ 12.2744 1.75348
$$50$$ 0 0
$$51$$ −0.491953 −0.0688872
$$52$$ 0 0
$$53$$ −9.59554 −1.31805 −0.659024 0.752122i $$-0.729031\pi$$
−0.659024 + 0.752122i $$0.729031\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.26930 0.300576
$$58$$ 0 0
$$59$$ −11.2538 −1.46512 −0.732561 0.680701i $$-0.761675\pi$$
−0.732561 + 0.680701i $$0.761675\pi$$
$$60$$ 0 0
$$61$$ 13.1898 1.68878 0.844390 0.535729i $$-0.179963\pi$$
0.844390 + 0.535729i $$0.179963\pi$$
$$62$$ 0 0
$$63$$ 12.7888 1.61124
$$64$$ 0 0
$$65$$ 5.96281 0.739596
$$66$$ 0 0
$$67$$ −7.79954 −0.952866 −0.476433 0.879211i $$-0.658070\pi$$
−0.476433 + 0.879211i $$0.658070\pi$$
$$68$$ 0 0
$$69$$ −0.266855 −0.0321255
$$70$$ 0 0
$$71$$ −6.97072 −0.827273 −0.413636 0.910442i $$-0.635742\pi$$
−0.413636 + 0.910442i $$0.635742\pi$$
$$72$$ 0 0
$$73$$ 12.7541 1.49275 0.746375 0.665526i $$-0.231793\pi$$
0.746375 + 0.665526i $$0.231793\pi$$
$$74$$ 0 0
$$75$$ 0.294963 0.0340594
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −10.0313 −1.12861 −0.564303 0.825568i $$-0.690855\pi$$
−0.564303 + 0.825568i $$0.690855\pi$$
$$80$$ 0 0
$$81$$ 8.22454 0.913838
$$82$$ 0 0
$$83$$ 4.09529 0.449517 0.224758 0.974415i $$-0.427841\pi$$
0.224758 + 0.974415i $$0.427841\pi$$
$$84$$ 0 0
$$85$$ −1.66785 −0.180904
$$86$$ 0 0
$$87$$ −1.39783 −0.149863
$$88$$ 0 0
$$89$$ −0.466291 −0.0494267 −0.0247134 0.999695i $$-0.507867\pi$$
−0.0247134 + 0.999695i $$0.507867\pi$$
$$90$$ 0 0
$$91$$ −26.1783 −2.74423
$$92$$ 0 0
$$93$$ −1.51200 −0.156787
$$94$$ 0 0
$$95$$ 7.69351 0.789338
$$96$$ 0 0
$$97$$ 7.09963 0.720858 0.360429 0.932787i $$-0.382630\pi$$
0.360429 + 0.932787i $$0.382630\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −16.2851 −1.62043 −0.810214 0.586135i $$-0.800649\pi$$
−0.810214 + 0.586135i $$0.800649\pi$$
$$102$$ 0 0
$$103$$ 3.19059 0.314378 0.157189 0.987569i $$-0.449757\pi$$
0.157189 + 0.987569i $$0.449757\pi$$
$$104$$ 0 0
$$105$$ −1.29496 −0.126375
$$106$$ 0 0
$$107$$ −13.1644 −1.27265 −0.636324 0.771422i $$-0.719546\pi$$
−0.636324 + 0.771422i $$0.719546\pi$$
$$108$$ 0 0
$$109$$ 1.33532 0.127900 0.0639502 0.997953i $$-0.479630\pi$$
0.0639502 + 0.997953i $$0.479630\pi$$
$$110$$ 0 0
$$111$$ −0.0545610 −0.00517870
$$112$$ 0 0
$$113$$ −1.67448 −0.157522 −0.0787611 0.996894i $$-0.525096\pi$$
−0.0787611 + 0.996894i $$0.525096\pi$$
$$114$$ 0 0
$$115$$ −0.904706 −0.0843643
$$116$$ 0 0
$$117$$ −17.3696 −1.60582
$$118$$ 0 0
$$119$$ 7.32228 0.671232
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −0.773501 −0.0697443
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 0.901470 0.0799926 0.0399963 0.999200i $$-0.487265\pi$$
0.0399963 + 0.999200i $$0.487265\pi$$
$$128$$ 0 0
$$129$$ 1.06134 0.0934458
$$130$$ 0 0
$$131$$ −14.5527 −1.27148 −0.635739 0.771904i $$-0.719305\pi$$
−0.635739 + 0.771904i $$0.719305\pi$$
$$132$$ 0 0
$$133$$ −33.7765 −2.92879
$$134$$ 0 0
$$135$$ −1.74411 −0.150109
$$136$$ 0 0
$$137$$ −20.5895 −1.75908 −0.879540 0.475824i $$-0.842150\pi$$
−0.879540 + 0.475824i $$0.842150\pi$$
$$138$$ 0 0
$$139$$ −14.9841 −1.27094 −0.635469 0.772126i $$-0.719193\pi$$
−0.635469 + 0.772126i $$0.719193\pi$$
$$140$$ 0 0
$$141$$ −0.228944 −0.0192806
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −4.73899 −0.393552
$$146$$ 0 0
$$147$$ 3.62048 0.298612
$$148$$ 0 0
$$149$$ 10.9626 0.898089 0.449045 0.893509i $$-0.351764\pi$$
0.449045 + 0.893509i $$0.351764\pi$$
$$150$$ 0 0
$$151$$ −5.91172 −0.481089 −0.240544 0.970638i $$-0.577326\pi$$
−0.240544 + 0.970638i $$0.577326\pi$$
$$152$$ 0 0
$$153$$ 4.85844 0.392781
$$154$$ 0 0
$$155$$ −5.12608 −0.411737
$$156$$ 0 0
$$157$$ −4.42628 −0.353256 −0.176628 0.984278i $$-0.556519\pi$$
−0.176628 + 0.984278i $$0.556519\pi$$
$$158$$ 0 0
$$159$$ −2.83033 −0.224460
$$160$$ 0 0
$$161$$ 3.97189 0.313029
$$162$$ 0 0
$$163$$ −7.38464 −0.578410 −0.289205 0.957267i $$-0.593391\pi$$
−0.289205 + 0.957267i $$0.593391\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −19.4876 −1.50799 −0.753996 0.656879i $$-0.771876\pi$$
−0.753996 + 0.656879i $$0.771876\pi$$
$$168$$ 0 0
$$169$$ 22.5551 1.73501
$$170$$ 0 0
$$171$$ −22.4112 −1.71383
$$172$$ 0 0
$$173$$ 9.50805 0.722883 0.361442 0.932395i $$-0.382285\pi$$
0.361442 + 0.932395i $$0.382285\pi$$
$$174$$ 0 0
$$175$$ −4.39026 −0.331872
$$176$$ 0 0
$$177$$ −3.31946 −0.249506
$$178$$ 0 0
$$179$$ 7.21253 0.539090 0.269545 0.962988i $$-0.413127\pi$$
0.269545 + 0.962988i $$0.413127\pi$$
$$180$$ 0 0
$$181$$ −23.3895 −1.73853 −0.869263 0.494350i $$-0.835406\pi$$
−0.869263 + 0.494350i $$0.835406\pi$$
$$182$$ 0 0
$$183$$ 3.89050 0.287594
$$184$$ 0 0
$$185$$ −0.184976 −0.0135997
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 7.65711 0.556973
$$190$$ 0 0
$$191$$ −4.16572 −0.301421 −0.150710 0.988578i $$-0.548156\pi$$
−0.150710 + 0.988578i $$0.548156\pi$$
$$192$$ 0 0
$$193$$ −23.1660 −1.66753 −0.833763 0.552122i $$-0.813818\pi$$
−0.833763 + 0.552122i $$0.813818\pi$$
$$194$$ 0 0
$$195$$ 1.75881 0.125951
$$196$$ 0 0
$$197$$ 10.1507 0.723209 0.361604 0.932332i $$-0.382229\pi$$
0.361604 + 0.932332i $$0.382229\pi$$
$$198$$ 0 0
$$199$$ −17.0131 −1.20603 −0.603014 0.797731i $$-0.706034\pi$$
−0.603014 + 0.797731i $$0.706034\pi$$
$$200$$ 0 0
$$201$$ −2.30058 −0.162270
$$202$$ 0 0
$$203$$ 20.8054 1.46025
$$204$$ 0 0
$$205$$ −2.62237 −0.183154
$$206$$ 0 0
$$207$$ 2.63541 0.183173
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −6.01620 −0.414173 −0.207086 0.978323i $$-0.566398\pi$$
−0.207086 + 0.978323i $$0.566398\pi$$
$$212$$ 0 0
$$213$$ −2.05611 −0.140882
$$214$$ 0 0
$$215$$ 3.59822 0.245396
$$216$$ 0 0
$$217$$ 22.5048 1.52773
$$218$$ 0 0
$$219$$ 3.76197 0.254211
$$220$$ 0 0
$$221$$ −9.94506 −0.668977
$$222$$ 0 0
$$223$$ 2.49407 0.167016 0.0835078 0.996507i $$-0.473388\pi$$
0.0835078 + 0.996507i $$0.473388\pi$$
$$224$$ 0 0
$$225$$ −2.91300 −0.194200
$$226$$ 0 0
$$227$$ −24.9953 −1.65899 −0.829497 0.558512i $$-0.811373\pi$$
−0.829497 + 0.558512i $$0.811373\pi$$
$$228$$ 0 0
$$229$$ −2.77082 −0.183101 −0.0915506 0.995800i $$-0.529182\pi$$
−0.0915506 + 0.995800i $$0.529182\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −9.60025 −0.628933 −0.314467 0.949269i $$-0.601826\pi$$
−0.314467 + 0.949269i $$0.601826\pi$$
$$234$$ 0 0
$$235$$ −0.776180 −0.0506324
$$236$$ 0 0
$$237$$ −2.95885 −0.192198
$$238$$ 0 0
$$239$$ 14.7692 0.955341 0.477671 0.878539i $$-0.341481\pi$$
0.477671 + 0.878539i $$0.341481\pi$$
$$240$$ 0 0
$$241$$ −10.4338 −0.672099 −0.336049 0.941844i $$-0.609091\pi$$
−0.336049 + 0.941844i $$0.609091\pi$$
$$242$$ 0 0
$$243$$ 7.65828 0.491279
$$244$$ 0 0
$$245$$ 12.2744 0.784180
$$246$$ 0 0
$$247$$ 45.8749 2.91895
$$248$$ 0 0
$$249$$ 1.20796 0.0765513
$$250$$ 0 0
$$251$$ −6.73394 −0.425042 −0.212521 0.977156i $$-0.568167\pi$$
−0.212521 + 0.977156i $$0.568167\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −0.491953 −0.0308073
$$256$$ 0 0
$$257$$ 7.74192 0.482928 0.241464 0.970410i $$-0.422372\pi$$
0.241464 + 0.970410i $$0.422372\pi$$
$$258$$ 0 0
$$259$$ 0.812091 0.0504609
$$260$$ 0 0
$$261$$ 13.8047 0.854487
$$262$$ 0 0
$$263$$ 18.1618 1.11990 0.559952 0.828525i $$-0.310819\pi$$
0.559952 + 0.828525i $$0.310819\pi$$
$$264$$ 0 0
$$265$$ −9.59554 −0.589449
$$266$$ 0 0
$$267$$ −0.137538 −0.00841721
$$268$$ 0 0
$$269$$ −10.7267 −0.654021 −0.327011 0.945021i $$-0.606041\pi$$
−0.327011 + 0.945021i $$0.606041\pi$$
$$270$$ 0 0
$$271$$ −18.7875 −1.14126 −0.570630 0.821207i $$-0.693301\pi$$
−0.570630 + 0.821207i $$0.693301\pi$$
$$272$$ 0 0
$$273$$ −7.72162 −0.467334
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 28.8467 1.73323 0.866614 0.498978i $$-0.166291\pi$$
0.866614 + 0.498978i $$0.166291\pi$$
$$278$$ 0 0
$$279$$ 14.9323 0.893971
$$280$$ 0 0
$$281$$ 27.0542 1.61392 0.806959 0.590608i $$-0.201112\pi$$
0.806959 + 0.590608i $$0.201112\pi$$
$$282$$ 0 0
$$283$$ 12.8731 0.765228 0.382614 0.923908i $$-0.375024\pi$$
0.382614 + 0.923908i $$0.375024\pi$$
$$284$$ 0 0
$$285$$ 2.26930 0.134422
$$286$$ 0 0
$$287$$ 11.5129 0.679583
$$288$$ 0 0
$$289$$ −14.2183 −0.836370
$$290$$ 0 0
$$291$$ 2.09413 0.122760
$$292$$ 0 0
$$293$$ 5.88873 0.344024 0.172012 0.985095i $$-0.444973\pi$$
0.172012 + 0.985095i $$0.444973\pi$$
$$294$$ 0 0
$$295$$ −11.2538 −0.655223
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −5.39459 −0.311977
$$300$$ 0 0
$$301$$ −15.7971 −0.910529
$$302$$ 0 0
$$303$$ −4.80350 −0.275954
$$304$$ 0 0
$$305$$ 13.1898 0.755246
$$306$$ 0 0
$$307$$ 6.85844 0.391432 0.195716 0.980661i $$-0.437297\pi$$
0.195716 + 0.980661i $$0.437297\pi$$
$$308$$ 0 0
$$309$$ 0.941105 0.0535376
$$310$$ 0 0
$$311$$ −14.2693 −0.809138 −0.404569 0.914508i $$-0.632578\pi$$
−0.404569 + 0.914508i $$0.632578\pi$$
$$312$$ 0 0
$$313$$ 1.59973 0.0904220 0.0452110 0.998977i $$-0.485604\pi$$
0.0452110 + 0.998977i $$0.485604\pi$$
$$314$$ 0 0
$$315$$ 12.7888 0.720568
$$316$$ 0 0
$$317$$ −30.8556 −1.73302 −0.866511 0.499158i $$-0.833643\pi$$
−0.866511 + 0.499158i $$0.833643\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −3.88300 −0.216728
$$322$$ 0 0
$$323$$ −12.8316 −0.713970
$$324$$ 0 0
$$325$$ 5.96281 0.330757
$$326$$ 0 0
$$327$$ 0.393870 0.0217810
$$328$$ 0 0
$$329$$ 3.40763 0.187869
$$330$$ 0 0
$$331$$ −1.16816 −0.0642079 −0.0321040 0.999485i $$-0.510221\pi$$
−0.0321040 + 0.999485i $$0.510221\pi$$
$$332$$ 0 0
$$333$$ 0.538833 0.0295279
$$334$$ 0 0
$$335$$ −7.79954 −0.426134
$$336$$ 0 0
$$337$$ 14.0291 0.764212 0.382106 0.924119i $$-0.375199\pi$$
0.382106 + 0.924119i $$0.375199\pi$$
$$338$$ 0 0
$$339$$ −0.493910 −0.0268255
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −23.1558 −1.25029
$$344$$ 0 0
$$345$$ −0.266855 −0.0143670
$$346$$ 0 0
$$347$$ 33.3514 1.79040 0.895198 0.445669i $$-0.147034\pi$$
0.895198 + 0.445669i $$0.147034\pi$$
$$348$$ 0 0
$$349$$ −35.3268 −1.89100 −0.945500 0.325623i $$-0.894426\pi$$
−0.945500 + 0.325623i $$0.894426\pi$$
$$350$$ 0 0
$$351$$ −10.3998 −0.555102
$$352$$ 0 0
$$353$$ 11.7558 0.625698 0.312849 0.949803i $$-0.398717\pi$$
0.312849 + 0.949803i $$0.398717\pi$$
$$354$$ 0 0
$$355$$ −6.97072 −0.369968
$$356$$ 0 0
$$357$$ 2.15980 0.114309
$$358$$ 0 0
$$359$$ −2.72101 −0.143609 −0.0718047 0.997419i $$-0.522876\pi$$
−0.0718047 + 0.997419i $$0.522876\pi$$
$$360$$ 0 0
$$361$$ 40.1901 2.11527
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 12.7541 0.667578
$$366$$ 0 0
$$367$$ 3.10358 0.162006 0.0810029 0.996714i $$-0.474188\pi$$
0.0810029 + 0.996714i $$0.474188\pi$$
$$368$$ 0 0
$$369$$ 7.63895 0.397668
$$370$$ 0 0
$$371$$ 42.1269 2.18712
$$372$$ 0 0
$$373$$ −27.4604 −1.42185 −0.710924 0.703269i $$-0.751723\pi$$
−0.710924 + 0.703269i $$0.751723\pi$$
$$374$$ 0 0
$$375$$ 0.294963 0.0152318
$$376$$ 0 0
$$377$$ −28.2577 −1.45535
$$378$$ 0 0
$$379$$ 13.9435 0.716229 0.358114 0.933678i $$-0.383420\pi$$
0.358114 + 0.933678i $$0.383420\pi$$
$$380$$ 0 0
$$381$$ 0.265900 0.0136225
$$382$$ 0 0
$$383$$ 8.15608 0.416756 0.208378 0.978048i $$-0.433182\pi$$
0.208378 + 0.978048i $$0.433182\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −10.4816 −0.532809
$$388$$ 0 0
$$389$$ −3.10121 −0.157237 −0.0786187 0.996905i $$-0.525051\pi$$
−0.0786187 + 0.996905i $$0.525051\pi$$
$$390$$ 0 0
$$391$$ 1.50891 0.0763090
$$392$$ 0 0
$$393$$ −4.29252 −0.216529
$$394$$ 0 0
$$395$$ −10.0313 −0.504728
$$396$$ 0 0
$$397$$ 23.1546 1.16209 0.581047 0.813870i $$-0.302643\pi$$
0.581047 + 0.813870i $$0.302643\pi$$
$$398$$ 0 0
$$399$$ −9.96281 −0.498764
$$400$$ 0 0
$$401$$ −8.24375 −0.411673 −0.205837 0.978586i $$-0.565992\pi$$
−0.205837 + 0.978586i $$0.565992\pi$$
$$402$$ 0 0
$$403$$ −30.5658 −1.52259
$$404$$ 0 0
$$405$$ 8.22454 0.408681
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 2.18649 0.108115 0.0540574 0.998538i $$-0.482785\pi$$
0.0540574 + 0.998538i $$0.482785\pi$$
$$410$$ 0 0
$$411$$ −6.07314 −0.299566
$$412$$ 0 0
$$413$$ 49.4071 2.43117
$$414$$ 0 0
$$415$$ 4.09529 0.201030
$$416$$ 0 0
$$417$$ −4.41977 −0.216437
$$418$$ 0 0
$$419$$ 10.7348 0.524429 0.262215 0.965010i $$-0.415547\pi$$
0.262215 + 0.965010i $$0.415547\pi$$
$$420$$ 0 0
$$421$$ 18.7885 0.915696 0.457848 0.889031i $$-0.348620\pi$$
0.457848 + 0.889031i $$0.348620\pi$$
$$422$$ 0 0
$$423$$ 2.26101 0.109934
$$424$$ 0 0
$$425$$ −1.66785 −0.0809025
$$426$$ 0 0
$$427$$ −57.9066 −2.80230
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −8.97878 −0.432493 −0.216246 0.976339i $$-0.569381\pi$$
−0.216246 + 0.976339i $$0.569381\pi$$
$$432$$ 0 0
$$433$$ 31.5059 1.51408 0.757039 0.653370i $$-0.226645\pi$$
0.757039 + 0.653370i $$0.226645\pi$$
$$434$$ 0 0
$$435$$ −1.39783 −0.0670206
$$436$$ 0 0
$$437$$ −6.96037 −0.332959
$$438$$ 0 0
$$439$$ 27.4750 1.31131 0.655655 0.755060i $$-0.272393\pi$$
0.655655 + 0.755060i $$0.272393\pi$$
$$440$$ 0 0
$$441$$ −35.7552 −1.70263
$$442$$ 0 0
$$443$$ −30.1960 −1.43466 −0.717328 0.696736i $$-0.754635\pi$$
−0.717328 + 0.696736i $$0.754635\pi$$
$$444$$ 0 0
$$445$$ −0.466291 −0.0221043
$$446$$ 0 0
$$447$$ 3.23355 0.152942
$$448$$ 0 0
$$449$$ −27.3820 −1.29224 −0.646118 0.763237i $$-0.723609\pi$$
−0.646118 + 0.763237i $$0.723609\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −1.74374 −0.0819279
$$454$$ 0 0
$$455$$ −26.1783 −1.22726
$$456$$ 0 0
$$457$$ 14.2884 0.668385 0.334193 0.942505i $$-0.391536\pi$$
0.334193 + 0.942505i $$0.391536\pi$$
$$458$$ 0 0
$$459$$ 2.90892 0.135777
$$460$$ 0 0
$$461$$ 8.78738 0.409269 0.204635 0.978838i $$-0.434399\pi$$
0.204635 + 0.978838i $$0.434399\pi$$
$$462$$ 0 0
$$463$$ −13.8394 −0.643172 −0.321586 0.946880i $$-0.604216\pi$$
−0.321586 + 0.946880i $$0.604216\pi$$
$$464$$ 0 0
$$465$$ −1.51200 −0.0701175
$$466$$ 0 0
$$467$$ 24.8893 1.15174 0.575871 0.817541i $$-0.304663\pi$$
0.575871 + 0.817541i $$0.304663\pi$$
$$468$$ 0 0
$$469$$ 34.2420 1.58115
$$470$$ 0 0
$$471$$ −1.30559 −0.0601583
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 7.69351 0.353002
$$476$$ 0 0
$$477$$ 27.9518 1.27982
$$478$$ 0 0
$$479$$ −28.5906 −1.30634 −0.653169 0.757212i $$-0.726561\pi$$
−0.653169 + 0.757212i $$0.726561\pi$$
$$480$$ 0 0
$$481$$ −1.10297 −0.0502913
$$482$$ 0 0
$$483$$ 1.17156 0.0533079
$$484$$ 0 0
$$485$$ 7.09963 0.322377
$$486$$ 0 0
$$487$$ 4.65371 0.210880 0.105440 0.994426i $$-0.466375\pi$$
0.105440 + 0.994426i $$0.466375\pi$$
$$488$$ 0 0
$$489$$ −2.17820 −0.0985014
$$490$$ 0 0
$$491$$ 24.9360 1.12535 0.562673 0.826680i $$-0.309773\pi$$
0.562673 + 0.826680i $$0.309773\pi$$
$$492$$ 0 0
$$493$$ 7.90392 0.355974
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 30.6033 1.37274
$$498$$ 0 0
$$499$$ 15.3848 0.688719 0.344359 0.938838i $$-0.388096\pi$$
0.344359 + 0.938838i $$0.388096\pi$$
$$500$$ 0 0
$$501$$ −5.74810 −0.256806
$$502$$ 0 0
$$503$$ −32.7511 −1.46030 −0.730149 0.683288i $$-0.760549\pi$$
−0.730149 + 0.683288i $$0.760549\pi$$
$$504$$ 0 0
$$505$$ −16.2851 −0.724677
$$506$$ 0 0
$$507$$ 6.65292 0.295467
$$508$$ 0 0
$$509$$ 15.7017 0.695964 0.347982 0.937501i $$-0.386867\pi$$
0.347982 + 0.937501i $$0.386867\pi$$
$$510$$ 0 0
$$511$$ −55.9936 −2.47701
$$512$$ 0 0
$$513$$ −13.4184 −0.592435
$$514$$ 0 0
$$515$$ 3.19059 0.140594
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 2.80452 0.123105
$$520$$ 0 0
$$521$$ −33.6243 −1.47311 −0.736553 0.676380i $$-0.763548\pi$$
−0.736553 + 0.676380i $$0.763548\pi$$
$$522$$ 0 0
$$523$$ −22.9574 −1.00386 −0.501928 0.864910i $$-0.667376\pi$$
−0.501928 + 0.864910i $$0.667376\pi$$
$$524$$ 0 0
$$525$$ −1.29496 −0.0565168
$$526$$ 0 0
$$527$$ 8.54952 0.372423
$$528$$ 0 0
$$529$$ −22.1815 −0.964413
$$530$$ 0 0
$$531$$ 32.7823 1.42263
$$532$$ 0 0
$$533$$ −15.6367 −0.677300
$$534$$ 0 0
$$535$$ −13.1644 −0.569145
$$536$$ 0 0
$$537$$ 2.12743 0.0918053
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.7777 −0.807314 −0.403657 0.914910i $$-0.632261\pi$$
−0.403657 + 0.914910i $$0.632261\pi$$
$$542$$ 0 0
$$543$$ −6.89902 −0.296065
$$544$$ 0 0
$$545$$ 1.33532 0.0571988
$$546$$ 0 0
$$547$$ 19.2785 0.824291 0.412146 0.911118i $$-0.364780\pi$$
0.412146 + 0.911118i $$0.364780\pi$$
$$548$$ 0 0
$$549$$ −38.4218 −1.63980
$$550$$ 0 0
$$551$$ −36.4595 −1.55323
$$552$$ 0 0
$$553$$ 44.0399 1.87277
$$554$$ 0 0
$$555$$ −0.0545610 −0.00231598
$$556$$ 0 0
$$557$$ −17.9380 −0.760057 −0.380028 0.924975i $$-0.624086\pi$$
−0.380028 + 0.924975i $$0.624086\pi$$
$$558$$ 0 0
$$559$$ 21.4555 0.907470
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −1.81288 −0.0764038 −0.0382019 0.999270i $$-0.512163\pi$$
−0.0382019 + 0.999270i $$0.512163\pi$$
$$564$$ 0 0
$$565$$ −1.67448 −0.0704460
$$566$$ 0 0
$$567$$ −36.1079 −1.51639
$$568$$ 0 0
$$569$$ 8.53549 0.357826 0.178913 0.983865i $$-0.442742\pi$$
0.178913 + 0.983865i $$0.442742\pi$$
$$570$$ 0 0
$$571$$ −32.2491 −1.34958 −0.674792 0.738008i $$-0.735767\pi$$
−0.674792 + 0.738008i $$0.735767\pi$$
$$572$$ 0 0
$$573$$ −1.22873 −0.0513310
$$574$$ 0 0
$$575$$ −0.904706 −0.0377288
$$576$$ 0 0
$$577$$ 29.9652 1.24747 0.623734 0.781637i $$-0.285615\pi$$
0.623734 + 0.781637i $$0.285615\pi$$
$$578$$ 0 0
$$579$$ −6.83312 −0.283975
$$580$$ 0 0
$$581$$ −17.9794 −0.745911
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −17.3696 −0.718147
$$586$$ 0 0
$$587$$ 9.80339 0.404629 0.202315 0.979321i $$-0.435154\pi$$
0.202315 + 0.979321i $$0.435154\pi$$
$$588$$ 0 0
$$589$$ −39.4376 −1.62500
$$590$$ 0 0
$$591$$ 2.99409 0.123160
$$592$$ 0 0
$$593$$ 15.4666 0.635136 0.317568 0.948235i $$-0.397134\pi$$
0.317568 + 0.948235i $$0.397134\pi$$
$$594$$ 0 0
$$595$$ 7.32228 0.300184
$$596$$ 0 0
$$597$$ −5.01824 −0.205383
$$598$$ 0 0
$$599$$ −0.278191 −0.0113666 −0.00568328 0.999984i $$-0.501809\pi$$
−0.00568328 + 0.999984i $$0.501809\pi$$
$$600$$ 0 0
$$601$$ 30.0055 1.22395 0.611974 0.790878i $$-0.290376\pi$$
0.611974 + 0.790878i $$0.290376\pi$$
$$602$$ 0 0
$$603$$ 22.7200 0.925232
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −22.2971 −0.905011 −0.452505 0.891762i $$-0.649470\pi$$
−0.452505 + 0.891762i $$0.649470\pi$$
$$608$$ 0 0
$$609$$ 6.13682 0.248676
$$610$$ 0 0
$$611$$ −4.62821 −0.187237
$$612$$ 0 0
$$613$$ −28.3748 −1.14605 −0.573024 0.819539i $$-0.694230\pi$$
−0.573024 + 0.819539i $$0.694230\pi$$
$$614$$ 0 0
$$615$$ −0.773501 −0.0311906
$$616$$ 0 0
$$617$$ −15.1603 −0.610332 −0.305166 0.952299i $$-0.598712\pi$$
−0.305166 + 0.952299i $$0.598712\pi$$
$$618$$ 0 0
$$619$$ −29.5238 −1.18666 −0.593331 0.804959i $$-0.702187\pi$$
−0.593331 + 0.804959i $$0.702187\pi$$
$$620$$ 0 0
$$621$$ 1.57791 0.0633194
$$622$$ 0 0
$$623$$ 2.04714 0.0820167
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0.308511 0.0123011
$$630$$ 0 0
$$631$$ −45.6894 −1.81887 −0.909433 0.415849i $$-0.863484\pi$$
−0.909433 + 0.415849i $$0.863484\pi$$
$$632$$ 0 0
$$633$$ −1.77456 −0.0705323
$$634$$ 0 0
$$635$$ 0.901470 0.0357738
$$636$$ 0 0
$$637$$ 73.1897 2.89988
$$638$$ 0 0
$$639$$ 20.3057 0.803281
$$640$$ 0 0
$$641$$ 36.1151 1.42646 0.713231 0.700929i $$-0.247231\pi$$
0.713231 + 0.700929i $$0.247231\pi$$
$$642$$ 0 0
$$643$$ −33.4972 −1.32100 −0.660500 0.750826i $$-0.729656\pi$$
−0.660500 + 0.750826i $$0.729656\pi$$
$$644$$ 0 0
$$645$$ 1.06134 0.0417902
$$646$$ 0 0
$$647$$ 39.2066 1.54137 0.770686 0.637215i $$-0.219914\pi$$
0.770686 + 0.637215i $$0.219914\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 6.63808 0.260167
$$652$$ 0 0
$$653$$ 24.7595 0.968913 0.484456 0.874815i $$-0.339017\pi$$
0.484456 + 0.874815i $$0.339017\pi$$
$$654$$ 0 0
$$655$$ −14.5527 −0.568622
$$656$$ 0 0
$$657$$ −37.1525 −1.44946
$$658$$ 0 0
$$659$$ 24.8813 0.969238 0.484619 0.874725i $$-0.338958\pi$$
0.484619 + 0.874725i $$0.338958\pi$$
$$660$$ 0 0
$$661$$ −5.94396 −0.231193 −0.115597 0.993296i $$-0.536878\pi$$
−0.115597 + 0.993296i $$0.536878\pi$$
$$662$$ 0 0
$$663$$ −2.93342 −0.113925
$$664$$ 0 0
$$665$$ −33.7765 −1.30980
$$666$$ 0 0
$$667$$ 4.28739 0.166009
$$668$$ 0 0
$$669$$ 0.735660 0.0284422
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 37.8500 1.45901 0.729505 0.683975i $$-0.239750\pi$$
0.729505 + 0.683975i $$0.239750\pi$$
$$674$$ 0 0
$$675$$ −1.74411 −0.0671310
$$676$$ 0 0
$$677$$ 0.190100 0.00730612 0.00365306 0.999993i $$-0.498837\pi$$
0.00365306 + 0.999993i $$0.498837\pi$$
$$678$$ 0 0
$$679$$ −31.1692 −1.19616
$$680$$ 0 0
$$681$$ −7.37267 −0.282521
$$682$$ 0 0
$$683$$ 23.0773 0.883030 0.441515 0.897254i $$-0.354441\pi$$
0.441515 + 0.897254i $$0.354441\pi$$
$$684$$ 0 0
$$685$$ −20.5895 −0.786685
$$686$$ 0 0
$$687$$ −0.817290 −0.0311816
$$688$$ 0 0
$$689$$ −57.2164 −2.17977
$$690$$ 0 0
$$691$$ 46.7082 1.77686 0.888432 0.459008i $$-0.151795\pi$$
0.888432 + 0.459008i $$0.151795\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −14.9841 −0.568381
$$696$$ 0 0
$$697$$ 4.37371 0.165666
$$698$$ 0 0
$$699$$ −2.83172 −0.107105
$$700$$ 0 0
$$701$$ 4.17239 0.157589 0.0787946 0.996891i $$-0.474893\pi$$
0.0787946 + 0.996891i $$0.474893\pi$$
$$702$$ 0 0
$$703$$ −1.42311 −0.0536737
$$704$$ 0 0
$$705$$ −0.228944 −0.00862254
$$706$$ 0 0
$$707$$ 71.4957 2.68887
$$708$$ 0 0
$$709$$ −8.74277 −0.328342 −0.164171 0.986432i $$-0.552495\pi$$
−0.164171 + 0.986432i $$0.552495\pi$$
$$710$$ 0 0
$$711$$ 29.2211 1.09588
$$712$$ 0 0
$$713$$ 4.63760 0.173679
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 4.35637 0.162692
$$718$$ 0 0
$$719$$ −7.40160 −0.276033 −0.138017 0.990430i $$-0.544073\pi$$
−0.138017 + 0.990430i $$0.544073\pi$$
$$720$$ 0 0
$$721$$ −14.0075 −0.521667
$$722$$ 0 0
$$723$$ −3.07758 −0.114456
$$724$$ 0 0
$$725$$ −4.73899 −0.176002
$$726$$ 0 0
$$727$$ 18.0187 0.668275 0.334137 0.942524i $$-0.391555\pi$$
0.334137 + 0.942524i $$0.391555\pi$$
$$728$$ 0 0
$$729$$ −22.4147 −0.830175
$$730$$ 0 0
$$731$$ −6.00128 −0.221965
$$732$$ 0 0
$$733$$ −36.3939 −1.34424 −0.672120 0.740442i $$-0.734616\pi$$
−0.672120 + 0.740442i $$0.734616\pi$$
$$734$$ 0 0
$$735$$ 3.62048 0.133543
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −12.4319 −0.457313 −0.228657 0.973507i $$-0.573433\pi$$
−0.228657 + 0.973507i $$0.573433\pi$$
$$740$$ 0 0
$$741$$ 13.5314 0.497089
$$742$$ 0 0
$$743$$ 37.3696 1.37096 0.685479 0.728092i $$-0.259593\pi$$
0.685479 + 0.728092i $$0.259593\pi$$
$$744$$ 0 0
$$745$$ 10.9626 0.401638
$$746$$ 0 0
$$747$$ −11.9296 −0.436480
$$748$$ 0 0
$$749$$ 57.7950 2.11178
$$750$$ 0 0
$$751$$ −7.89958 −0.288260 −0.144130 0.989559i $$-0.546038\pi$$
−0.144130 + 0.989559i $$0.546038\pi$$
$$752$$ 0 0
$$753$$ −1.98626 −0.0723834
$$754$$ 0 0
$$755$$ −5.91172 −0.215149
$$756$$ 0 0
$$757$$ −3.17036 −0.115229 −0.0576143 0.998339i $$-0.518349\pi$$
−0.0576143 + 0.998339i $$0.518349\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 46.7689 1.69537 0.847685 0.530499i $$-0.177996\pi$$
0.847685 + 0.530499i $$0.177996\pi$$
$$762$$ 0 0
$$763$$ −5.86239 −0.212233
$$764$$ 0 0
$$765$$ 4.85844 0.175657
$$766$$ 0 0
$$767$$ −67.1044 −2.42300
$$768$$ 0 0
$$769$$ −1.95610 −0.0705388 −0.0352694 0.999378i $$-0.511229\pi$$
−0.0352694 + 0.999378i $$0.511229\pi$$
$$770$$ 0 0
$$771$$ 2.28358 0.0822411
$$772$$ 0 0
$$773$$ −20.4428 −0.735277 −0.367639 0.929969i $$-0.619834\pi$$
−0.367639 + 0.929969i $$0.619834\pi$$
$$774$$ 0 0
$$775$$ −5.12608 −0.184134
$$776$$ 0 0
$$777$$ 0.239537 0.00859333
$$778$$ 0 0
$$779$$ −20.1752 −0.722852
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 8.26534 0.295379
$$784$$ 0 0
$$785$$ −4.42628 −0.157981
$$786$$ 0 0
$$787$$ −2.59196 −0.0923933 −0.0461967 0.998932i $$-0.514710\pi$$
−0.0461967 + 0.998932i $$0.514710\pi$$
$$788$$ 0 0
$$789$$ 5.35706 0.190716
$$790$$ 0 0
$$791$$ 7.35141 0.261386
$$792$$ 0 0
$$793$$ 78.6483 2.79288
$$794$$ 0 0
$$795$$ −2.83033 −0.100381
$$796$$ 0 0
$$797$$ 22.3867 0.792977 0.396488 0.918040i $$-0.370229\pi$$
0.396488 + 0.918040i $$0.370229\pi$$
$$798$$ 0 0
$$799$$ 1.29455 0.0457979
$$800$$ 0 0
$$801$$ 1.35830 0.0479933
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 3.97189 0.139991
$$806$$ 0 0
$$807$$ −3.16399 −0.111378
$$808$$ 0 0
$$809$$ 11.9561 0.420354 0.210177 0.977663i $$-0.432596\pi$$
0.210177 + 0.977663i $$0.432596\pi$$
$$810$$ 0 0
$$811$$ 18.4696 0.648554 0.324277 0.945962i $$-0.394879\pi$$
0.324277 + 0.945962i $$0.394879\pi$$
$$812$$ 0 0
$$813$$ −5.54162 −0.194353
$$814$$ 0 0
$$815$$ −7.38464 −0.258673
$$816$$ 0 0
$$817$$ 27.6829 0.968503
$$818$$ 0 0
$$819$$ 76.2572 2.66464
$$820$$ 0 0
$$821$$ 55.4734 1.93604 0.968018 0.250881i $$-0.0807201\pi$$
0.968018 + 0.250881i $$0.0807201\pi$$
$$822$$ 0 0
$$823$$ 43.7133 1.52375 0.761874 0.647725i $$-0.224279\pi$$
0.761874 + 0.647725i $$0.224279\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 31.7396 1.10369 0.551846 0.833946i $$-0.313924\pi$$
0.551846 + 0.833946i $$0.313924\pi$$
$$828$$ 0 0
$$829$$ −35.0302 −1.21665 −0.608325 0.793688i $$-0.708158\pi$$
−0.608325 + 0.793688i $$0.708158\pi$$
$$830$$ 0 0
$$831$$ 8.50870 0.295164
$$832$$ 0 0
$$833$$ −20.4718 −0.709304
$$834$$ 0 0
$$835$$ −19.4876 −0.674394
$$836$$ 0 0
$$837$$ 8.94047 0.309028
$$838$$ 0 0
$$839$$ −15.8542 −0.547349 −0.273674 0.961822i $$-0.588239\pi$$
−0.273674 + 0.961822i $$0.588239\pi$$
$$840$$ 0 0
$$841$$ −6.54197 −0.225585
$$842$$ 0 0
$$843$$ 7.97998 0.274845
$$844$$ 0 0
$$845$$ 22.5551 0.775919
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 3.79710 0.130316
$$850$$ 0 0
$$851$$ 0.167349 0.00573664
$$852$$ 0 0
$$853$$ −22.5865 −0.773346 −0.386673 0.922217i $$-0.626376\pi$$
−0.386673 + 0.922217i $$0.626376\pi$$
$$854$$ 0 0
$$855$$ −22.4112 −0.766446
$$856$$ 0 0
$$857$$ 40.9108 1.39749 0.698744 0.715372i $$-0.253743\pi$$
0.698744 + 0.715372i $$0.253743\pi$$
$$858$$ 0 0
$$859$$ 15.9523 0.544284 0.272142 0.962257i $$-0.412268\pi$$
0.272142 + 0.962257i $$0.412268\pi$$
$$860$$ 0 0
$$861$$ 3.39587 0.115731
$$862$$ 0 0
$$863$$ 7.28894 0.248118 0.124059 0.992275i $$-0.460409\pi$$
0.124059 + 0.992275i $$0.460409\pi$$
$$864$$ 0 0
$$865$$ 9.50805 0.323283
$$866$$ 0 0
$$867$$ −4.19387 −0.142431
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −46.5072 −1.57584
$$872$$ 0 0
$$873$$ −20.6812 −0.699952
$$874$$ 0 0
$$875$$ −4.39026 −0.148418
$$876$$ 0 0
$$877$$ 7.14718 0.241343 0.120671 0.992692i $$-0.461495\pi$$
0.120671 + 0.992692i $$0.461495\pi$$
$$878$$ 0 0
$$879$$ 1.73696 0.0585861
$$880$$ 0 0
$$881$$ −17.5763 −0.592160 −0.296080 0.955163i $$-0.595680\pi$$
−0.296080 + 0.955163i $$0.595680\pi$$
$$882$$ 0 0
$$883$$ −6.26855 −0.210954 −0.105477 0.994422i $$-0.533637\pi$$
−0.105477 + 0.994422i $$0.533637\pi$$
$$884$$ 0 0
$$885$$ −3.31946 −0.111582
$$886$$ 0 0
$$887$$ −24.6894 −0.828989 −0.414495 0.910052i $$-0.636042\pi$$
−0.414495 + 0.910052i $$0.636042\pi$$
$$888$$ 0 0
$$889$$ −3.95769 −0.132737
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −5.97155 −0.199830
$$894$$ 0 0
$$895$$ 7.21253 0.241088
$$896$$ 0 0
$$897$$ −1.59120 −0.0531288
$$898$$ 0 0
$$899$$ 24.2924 0.810199
$$900$$ 0 0
$$901$$ 16.0039 0.533167
$$902$$ 0 0
$$903$$ −4.65956 −0.155060
$$904$$ 0 0
$$905$$ −23.3895 −0.777492
$$906$$ 0 0
$$907$$ 25.4380 0.844655 0.422327 0.906443i $$-0.361213\pi$$
0.422327 + 0.906443i $$0.361213\pi$$
$$908$$ 0 0
$$909$$ 47.4384 1.57343
$$910$$ 0 0
$$911$$ −24.5296 −0.812701 −0.406350 0.913717i $$-0.633199\pi$$
−0.406350 + 0.913717i $$0.633199\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 3.89050 0.128616
$$916$$ 0 0
$$917$$ 63.8903 2.10984
$$918$$ 0 0
$$919$$ 21.2250 0.700148 0.350074 0.936722i $$-0.386156\pi$$
0.350074 + 0.936722i $$0.386156\pi$$
$$920$$ 0 0
$$921$$ 2.02298 0.0666596
$$922$$ 0 0
$$923$$ −41.5651 −1.36813
$$924$$ 0 0
$$925$$ −0.184976 −0.00608196
$$926$$ 0 0
$$927$$ −9.29417 −0.305261
$$928$$ 0 0
$$929$$ 34.2535 1.12382 0.561910 0.827198i $$-0.310067\pi$$
0.561910 + 0.827198i $$0.310067\pi$$
$$930$$ 0 0
$$931$$ 94.4329 3.09491
$$932$$ 0 0
$$933$$ −4.20891 −0.137794
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 34.9322 1.14119 0.570593 0.821233i $$-0.306713\pi$$
0.570593 + 0.821233i $$0.306713\pi$$
$$938$$ 0 0
$$939$$ 0.471860 0.0153986
$$940$$ 0 0
$$941$$ −18.7014 −0.609649 −0.304825 0.952409i $$-0.598598\pi$$
−0.304825 + 0.952409i $$0.598598\pi$$
$$942$$ 0 0
$$943$$ 2.37247 0.0772583
$$944$$ 0 0
$$945$$ 7.65711 0.249086
$$946$$ 0 0
$$947$$ 12.7418 0.414052 0.207026 0.978335i $$-0.433621\pi$$
0.207026 + 0.978335i $$0.433621\pi$$
$$948$$ 0 0
$$949$$ 76.0501 2.46869
$$950$$ 0 0
$$951$$ −9.10125 −0.295128
$$952$$ 0 0
$$953$$ 5.77686 0.187131 0.0935654 0.995613i $$-0.470174\pi$$
0.0935654 + 0.995613i $$0.470174\pi$$
$$954$$ 0 0
$$955$$ −4.16572 −0.134799
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 90.3933 2.91895
$$960$$ 0 0
$$961$$ −4.72330 −0.152364
$$962$$ 0 0
$$963$$ 38.3478 1.23574
$$964$$ 0 0
$$965$$ −23.1660 −0.745741
$$966$$ 0 0
$$967$$ −36.2900 −1.16701 −0.583504 0.812110i $$-0.698319\pi$$
−0.583504 + 0.812110i $$0.698319\pi$$
$$968$$ 0 0
$$969$$ −3.78485 −0.121587
$$970$$ 0 0
$$971$$ −6.98883 −0.224282 −0.112141 0.993692i $$-0.535771\pi$$
−0.112141 + 0.993692i $$0.535771\pi$$
$$972$$ 0 0
$$973$$ 65.7842 2.10895
$$974$$ 0 0
$$975$$ 1.75881 0.0563269
$$976$$ 0 0
$$977$$ 25.9994 0.831795 0.415897 0.909412i $$-0.363468\pi$$
0.415897 + 0.909412i $$0.363468\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −3.88978 −0.124191
$$982$$ 0 0
$$983$$ 28.6461 0.913669 0.456834 0.889552i $$-0.348983\pi$$
0.456834 + 0.889552i $$0.348983\pi$$
$$984$$ 0 0
$$985$$ 10.1507 0.323429
$$986$$ 0 0
$$987$$ 1.00512 0.0319935
$$988$$ 0 0
$$989$$ −3.25533 −0.103513
$$990$$ 0 0
$$991$$ 27.9673 0.888409 0.444205 0.895925i $$-0.353486\pi$$
0.444205 + 0.895925i $$0.353486\pi$$
$$992$$ 0 0
$$993$$ −0.344564 −0.0109344
$$994$$ 0 0
$$995$$ −17.0131 −0.539352
$$996$$ 0 0
$$997$$ 13.0579 0.413547 0.206773 0.978389i $$-0.433704\pi$$
0.206773 + 0.978389i $$0.433704\pi$$
$$998$$ 0 0
$$999$$ 0.322619 0.0102072
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 4840.2.a.y.1.3 4
4.3 odd 2 9680.2.a.cu.1.2 4
11.5 even 5 440.2.y.a.201.2 yes 8
11.9 even 5 440.2.y.a.81.2 8
11.10 odd 2 4840.2.a.z.1.3 4
44.27 odd 10 880.2.bo.d.641.1 8
44.31 odd 10 880.2.bo.d.81.1 8
44.43 even 2 9680.2.a.ct.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.81.2 8 11.9 even 5
440.2.y.a.201.2 yes 8 11.5 even 5
880.2.bo.d.81.1 8 44.31 odd 10
880.2.bo.d.641.1 8 44.27 odd 10
4840.2.a.y.1.3 4 1.1 even 1 trivial
4840.2.a.z.1.3 4 11.10 odd 2
9680.2.a.ct.1.2 4 44.43 even 2
9680.2.a.cu.1.2 4 4.3 odd 2