# Properties

 Label 520.2.a.f.1.2 Level $520$ Weight $2$ Character 520.1 Self dual yes Analytic conductor $4.152$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,2,Mod(1,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, 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("520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.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: $$4.15222090511$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 520.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.44949 q^{3} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q+2.44949 q^{3} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{9} +0.449490 q^{11} -1.00000 q^{13} +2.44949 q^{15} -2.89898 q^{17} -4.44949 q^{19} +4.89898 q^{21} +1.55051 q^{23} +1.00000 q^{25} +4.00000 q^{29} +0.449490 q^{31} +1.10102 q^{33} +2.00000 q^{35} -4.89898 q^{37} -2.44949 q^{39} +1.10102 q^{41} -3.34847 q^{43} +3.00000 q^{45} -2.00000 q^{47} -3.00000 q^{49} -7.10102 q^{51} +10.8990 q^{53} +0.449490 q^{55} -10.8990 q^{57} -5.34847 q^{59} +13.7980 q^{61} +6.00000 q^{63} -1.00000 q^{65} -14.8990 q^{67} +3.79796 q^{69} -8.44949 q^{71} +14.6969 q^{73} +2.44949 q^{75} +0.898979 q^{77} -4.89898 q^{79} -9.00000 q^{81} +2.00000 q^{83} -2.89898 q^{85} +9.79796 q^{87} +6.00000 q^{89} -2.00000 q^{91} +1.10102 q^{93} -4.44949 q^{95} -11.7980 q^{97} +1.34847 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 + 4 * q^7 + 6 * q^9 $$2 q + 2 q^{5} + 4 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{29} - 4 q^{31} + 12 q^{33} + 4 q^{35} + 12 q^{41} + 8 q^{43} + 6 q^{45} - 4 q^{47} - 6 q^{49} - 24 q^{51} + 12 q^{53} - 4 q^{55} - 12 q^{57} + 4 q^{59} + 8 q^{61} + 12 q^{63} - 2 q^{65} - 20 q^{67} - 12 q^{69} - 12 q^{71} - 8 q^{77} - 18 q^{81} + 4 q^{83} + 4 q^{85} + 12 q^{89} - 4 q^{91} + 12 q^{93} - 4 q^{95} - 4 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^5 + 4 * q^7 + 6 * q^9 - 4 * q^11 - 2 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 2 * q^25 + 8 * q^29 - 4 * q^31 + 12 * q^33 + 4 * q^35 + 12 * q^41 + 8 * q^43 + 6 * q^45 - 4 * q^47 - 6 * q^49 - 24 * q^51 + 12 * q^53 - 4 * q^55 - 12 * q^57 + 4 * q^59 + 8 * q^61 + 12 * q^63 - 2 * q^65 - 20 * q^67 - 12 * q^69 - 12 * q^71 - 8 * q^77 - 18 * q^81 + 4 * q^83 + 4 * q^85 + 12 * q^89 - 4 * q^91 + 12 * q^93 - 4 * q^95 - 4 * q^97 - 12 * 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
$$3$$ 2.44949 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 0.449490 0.135526 0.0677631 0.997701i $$-0.478414\pi$$
0.0677631 + 0.997701i $$0.478414\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 2.44949 0.632456
$$16$$ 0 0
$$17$$ −2.89898 −0.703106 −0.351553 0.936168i $$-0.614346\pi$$
−0.351553 + 0.936168i $$0.614346\pi$$
$$18$$ 0 0
$$19$$ −4.44949 −1.02078 −0.510391 0.859942i $$-0.670499\pi$$
−0.510391 + 0.859942i $$0.670499\pi$$
$$20$$ 0 0
$$21$$ 4.89898 1.06904
$$22$$ 0 0
$$23$$ 1.55051 0.323304 0.161652 0.986848i $$-0.448318\pi$$
0.161652 + 0.986848i $$0.448318\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ 0.449490 0.0807307 0.0403654 0.999185i $$-0.487148\pi$$
0.0403654 + 0.999185i $$0.487148\pi$$
$$32$$ 0 0
$$33$$ 1.10102 0.191663
$$34$$ 0 0
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ −4.89898 −0.805387 −0.402694 0.915335i $$-0.631926\pi$$
−0.402694 + 0.915335i $$0.631926\pi$$
$$38$$ 0 0
$$39$$ −2.44949 −0.392232
$$40$$ 0 0
$$41$$ 1.10102 0.171951 0.0859753 0.996297i $$-0.472599\pi$$
0.0859753 + 0.996297i $$0.472599\pi$$
$$42$$ 0 0
$$43$$ −3.34847 −0.510637 −0.255318 0.966857i $$-0.582180\pi$$
−0.255318 + 0.966857i $$0.582180\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −7.10102 −0.994342
$$52$$ 0 0
$$53$$ 10.8990 1.49709 0.748545 0.663084i $$-0.230753\pi$$
0.748545 + 0.663084i $$0.230753\pi$$
$$54$$ 0 0
$$55$$ 0.449490 0.0606092
$$56$$ 0 0
$$57$$ −10.8990 −1.44361
$$58$$ 0 0
$$59$$ −5.34847 −0.696311 −0.348156 0.937437i $$-0.613192\pi$$
−0.348156 + 0.937437i $$0.613192\pi$$
$$60$$ 0 0
$$61$$ 13.7980 1.76665 0.883324 0.468763i $$-0.155300\pi$$
0.883324 + 0.468763i $$0.155300\pi$$
$$62$$ 0 0
$$63$$ 6.00000 0.755929
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −14.8990 −1.82020 −0.910100 0.414389i $$-0.863995\pi$$
−0.910100 + 0.414389i $$0.863995\pi$$
$$68$$ 0 0
$$69$$ 3.79796 0.457221
$$70$$ 0 0
$$71$$ −8.44949 −1.00277 −0.501385 0.865224i $$-0.667176\pi$$
−0.501385 + 0.865224i $$0.667176\pi$$
$$72$$ 0 0
$$73$$ 14.6969 1.72015 0.860073 0.510171i $$-0.170418\pi$$
0.860073 + 0.510171i $$0.170418\pi$$
$$74$$ 0 0
$$75$$ 2.44949 0.282843
$$76$$ 0 0
$$77$$ 0.898979 0.102448
$$78$$ 0 0
$$79$$ −4.89898 −0.551178 −0.275589 0.961276i $$-0.588873\pi$$
−0.275589 + 0.961276i $$0.588873\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ 0 0
$$83$$ 2.00000 0.219529 0.109764 0.993958i $$-0.464990\pi$$
0.109764 + 0.993958i $$0.464990\pi$$
$$84$$ 0 0
$$85$$ −2.89898 −0.314438
$$86$$ 0 0
$$87$$ 9.79796 1.05045
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 1.10102 0.114171
$$94$$ 0 0
$$95$$ −4.44949 −0.456508
$$96$$ 0 0
$$97$$ −11.7980 −1.19790 −0.598951 0.800786i $$-0.704415\pi$$
−0.598951 + 0.800786i $$0.704415\pi$$
$$98$$ 0 0
$$99$$ 1.34847 0.135526
$$100$$ 0 0
$$101$$ 3.79796 0.377911 0.188956 0.981986i $$-0.439490\pi$$
0.188956 + 0.981986i $$0.439490\pi$$
$$102$$ 0 0
$$103$$ −3.34847 −0.329934 −0.164967 0.986299i $$-0.552752\pi$$
−0.164967 + 0.986299i $$0.552752\pi$$
$$104$$ 0 0
$$105$$ 4.89898 0.478091
$$106$$ 0 0
$$107$$ 12.2474 1.18401 0.592003 0.805936i $$-0.298337\pi$$
0.592003 + 0.805936i $$0.298337\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ −12.0000 −1.13899
$$112$$ 0 0
$$113$$ 10.8990 1.02529 0.512645 0.858601i $$-0.328666\pi$$
0.512645 + 0.858601i $$0.328666\pi$$
$$114$$ 0 0
$$115$$ 1.55051 0.144586
$$116$$ 0 0
$$117$$ −3.00000 −0.277350
$$118$$ 0 0
$$119$$ −5.79796 −0.531498
$$120$$ 0 0
$$121$$ −10.7980 −0.981633
$$122$$ 0 0
$$123$$ 2.69694 0.243175
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 19.3485 1.71690 0.858450 0.512898i $$-0.171428\pi$$
0.858450 + 0.512898i $$0.171428\pi$$
$$128$$ 0 0
$$129$$ −8.20204 −0.722149
$$130$$ 0 0
$$131$$ −19.5959 −1.71210 −0.856052 0.516890i $$-0.827090\pi$$
−0.856052 + 0.516890i $$0.827090\pi$$
$$132$$ 0 0
$$133$$ −8.89898 −0.771639
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ −7.10102 −0.602301 −0.301150 0.953577i $$-0.597371\pi$$
−0.301150 + 0.953577i $$0.597371\pi$$
$$140$$ 0 0
$$141$$ −4.89898 −0.412568
$$142$$ 0 0
$$143$$ −0.449490 −0.0375882
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ 0 0
$$147$$ −7.34847 −0.606092
$$148$$ 0 0
$$149$$ −7.79796 −0.638834 −0.319417 0.947614i $$-0.603487\pi$$
−0.319417 + 0.947614i $$0.603487\pi$$
$$150$$ 0 0
$$151$$ −17.3485 −1.41180 −0.705899 0.708312i $$-0.749457\pi$$
−0.705899 + 0.708312i $$0.749457\pi$$
$$152$$ 0 0
$$153$$ −8.69694 −0.703106
$$154$$ 0 0
$$155$$ 0.449490 0.0361039
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 0 0
$$159$$ 26.6969 2.11720
$$160$$ 0 0
$$161$$ 3.10102 0.244395
$$162$$ 0 0
$$163$$ −18.8990 −1.48028 −0.740141 0.672452i $$-0.765241\pi$$
−0.740141 + 0.672452i $$0.765241\pi$$
$$164$$ 0 0
$$165$$ 1.10102 0.0857143
$$166$$ 0 0
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −13.3485 −1.02078
$$172$$ 0 0
$$173$$ −2.89898 −0.220405 −0.110203 0.993909i $$-0.535150\pi$$
−0.110203 + 0.993909i $$0.535150\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ 0 0
$$177$$ −13.1010 −0.984733
$$178$$ 0 0
$$179$$ 8.00000 0.597948 0.298974 0.954261i $$-0.403356\pi$$
0.298974 + 0.954261i $$0.403356\pi$$
$$180$$ 0 0
$$181$$ −13.7980 −1.02559 −0.512797 0.858510i $$-0.671391\pi$$
−0.512797 + 0.858510i $$0.671391\pi$$
$$182$$ 0 0
$$183$$ 33.7980 2.49842
$$184$$ 0 0
$$185$$ −4.89898 −0.360180
$$186$$ 0 0
$$187$$ −1.30306 −0.0952893
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 17.7980 1.28782 0.643908 0.765103i $$-0.277312\pi$$
0.643908 + 0.765103i $$0.277312\pi$$
$$192$$ 0 0
$$193$$ 19.7980 1.42509 0.712544 0.701627i $$-0.247543\pi$$
0.712544 + 0.701627i $$0.247543\pi$$
$$194$$ 0 0
$$195$$ −2.44949 −0.175412
$$196$$ 0 0
$$197$$ −4.20204 −0.299383 −0.149692 0.988733i $$-0.547828\pi$$
−0.149692 + 0.988733i $$0.547828\pi$$
$$198$$ 0 0
$$199$$ −13.7980 −0.978111 −0.489056 0.872253i $$-0.662659\pi$$
−0.489056 + 0.872253i $$0.662659\pi$$
$$200$$ 0 0
$$201$$ −36.4949 −2.57415
$$202$$ 0 0
$$203$$ 8.00000 0.561490
$$204$$ 0 0
$$205$$ 1.10102 0.0768986
$$206$$ 0 0
$$207$$ 4.65153 0.323304
$$208$$ 0 0
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ −6.20204 −0.426966 −0.213483 0.976947i $$-0.568481\pi$$
−0.213483 + 0.976947i $$0.568481\pi$$
$$212$$ 0 0
$$213$$ −20.6969 −1.41813
$$214$$ 0 0
$$215$$ −3.34847 −0.228364
$$216$$ 0 0
$$217$$ 0.898979 0.0610267
$$218$$ 0 0
$$219$$ 36.0000 2.43265
$$220$$ 0 0
$$221$$ 2.89898 0.195006
$$222$$ 0 0
$$223$$ 21.5959 1.44617 0.723085 0.690759i $$-0.242724\pi$$
0.723085 + 0.690759i $$0.242724\pi$$
$$224$$ 0 0
$$225$$ 3.00000 0.200000
$$226$$ 0 0
$$227$$ 13.1010 0.869545 0.434773 0.900540i $$-0.356829\pi$$
0.434773 + 0.900540i $$0.356829\pi$$
$$228$$ 0 0
$$229$$ −9.10102 −0.601412 −0.300706 0.953717i $$-0.597222\pi$$
−0.300706 + 0.953717i $$0.597222\pi$$
$$230$$ 0 0
$$231$$ 2.20204 0.144884
$$232$$ 0 0
$$233$$ 13.5959 0.890698 0.445349 0.895357i $$-0.353080\pi$$
0.445349 + 0.895357i $$0.353080\pi$$
$$234$$ 0 0
$$235$$ −2.00000 −0.130466
$$236$$ 0 0
$$237$$ −12.0000 −0.779484
$$238$$ 0 0
$$239$$ 23.1464 1.49722 0.748609 0.663012i $$-0.230722\pi$$
0.748609 + 0.663012i $$0.230722\pi$$
$$240$$ 0 0
$$241$$ 28.6969 1.84853 0.924266 0.381749i $$-0.124678\pi$$
0.924266 + 0.381749i $$0.124678\pi$$
$$242$$ 0 0
$$243$$ −22.0454 −1.41421
$$244$$ 0 0
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 4.44949 0.283114
$$248$$ 0 0
$$249$$ 4.89898 0.310460
$$250$$ 0 0
$$251$$ 14.6969 0.927663 0.463831 0.885924i $$-0.346474\pi$$
0.463831 + 0.885924i $$0.346474\pi$$
$$252$$ 0 0
$$253$$ 0.696938 0.0438161
$$254$$ 0 0
$$255$$ −7.10102 −0.444683
$$256$$ 0 0
$$257$$ 15.7980 0.985450 0.492725 0.870185i $$-0.336001\pi$$
0.492725 + 0.870185i $$0.336001\pi$$
$$258$$ 0 0
$$259$$ −9.79796 −0.608816
$$260$$ 0 0
$$261$$ 12.0000 0.742781
$$262$$ 0 0
$$263$$ 21.5505 1.32886 0.664431 0.747350i $$-0.268674\pi$$
0.664431 + 0.747350i $$0.268674\pi$$
$$264$$ 0 0
$$265$$ 10.8990 0.669519
$$266$$ 0 0
$$267$$ 14.6969 0.899438
$$268$$ 0 0
$$269$$ 26.0000 1.58525 0.792624 0.609711i $$-0.208714\pi$$
0.792624 + 0.609711i $$0.208714\pi$$
$$270$$ 0 0
$$271$$ 8.44949 0.513270 0.256635 0.966508i $$-0.417386\pi$$
0.256635 + 0.966508i $$0.417386\pi$$
$$272$$ 0 0
$$273$$ −4.89898 −0.296500
$$274$$ 0 0
$$275$$ 0.449490 0.0271053
$$276$$ 0 0
$$277$$ 26.4949 1.59192 0.795962 0.605347i $$-0.206965\pi$$
0.795962 + 0.605347i $$0.206965\pi$$
$$278$$ 0 0
$$279$$ 1.34847 0.0807307
$$280$$ 0 0
$$281$$ 2.89898 0.172939 0.0864693 0.996255i $$-0.472442\pi$$
0.0864693 + 0.996255i $$0.472442\pi$$
$$282$$ 0 0
$$283$$ −1.55051 −0.0921683 −0.0460841 0.998938i $$-0.514674\pi$$
−0.0460841 + 0.998938i $$0.514674\pi$$
$$284$$ 0 0
$$285$$ −10.8990 −0.645600
$$286$$ 0 0
$$287$$ 2.20204 0.129982
$$288$$ 0 0
$$289$$ −8.59592 −0.505642
$$290$$ 0 0
$$291$$ −28.8990 −1.69409
$$292$$ 0 0
$$293$$ −20.4949 −1.19732 −0.598662 0.801001i $$-0.704301\pi$$
−0.598662 + 0.801001i $$0.704301\pi$$
$$294$$ 0 0
$$295$$ −5.34847 −0.311400
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.55051 −0.0896683
$$300$$ 0 0
$$301$$ −6.69694 −0.386005
$$302$$ 0 0
$$303$$ 9.30306 0.534447
$$304$$ 0 0
$$305$$ 13.7980 0.790069
$$306$$ 0 0
$$307$$ −11.7980 −0.673345 −0.336673 0.941622i $$-0.609302\pi$$
−0.336673 + 0.941622i $$0.609302\pi$$
$$308$$ 0 0
$$309$$ −8.20204 −0.466598
$$310$$ 0 0
$$311$$ 12.8990 0.731434 0.365717 0.930726i $$-0.380824\pi$$
0.365717 + 0.930726i $$0.380824\pi$$
$$312$$ 0 0
$$313$$ 26.4949 1.49758 0.748790 0.662807i $$-0.230635\pi$$
0.748790 + 0.662807i $$0.230635\pi$$
$$314$$ 0 0
$$315$$ 6.00000 0.338062
$$316$$ 0 0
$$317$$ −2.69694 −0.151475 −0.0757376 0.997128i $$-0.524131\pi$$
−0.0757376 + 0.997128i $$0.524131\pi$$
$$318$$ 0 0
$$319$$ 1.79796 0.100666
$$320$$ 0 0
$$321$$ 30.0000 1.67444
$$322$$ 0 0
$$323$$ 12.8990 0.717718
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ −24.4949 −1.35457
$$328$$ 0 0
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ −8.44949 −0.464426 −0.232213 0.972665i $$-0.574597\pi$$
−0.232213 + 0.972665i $$0.574597\pi$$
$$332$$ 0 0
$$333$$ −14.6969 −0.805387
$$334$$ 0 0
$$335$$ −14.8990 −0.814018
$$336$$ 0 0
$$337$$ 12.6969 0.691646 0.345823 0.938300i $$-0.387600\pi$$
0.345823 + 0.938300i $$0.387600\pi$$
$$338$$ 0 0
$$339$$ 26.6969 1.44998
$$340$$ 0 0
$$341$$ 0.202041 0.0109411
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ 3.79796 0.204475
$$346$$ 0 0
$$347$$ −4.24745 −0.228015 −0.114007 0.993480i $$-0.536369\pi$$
−0.114007 + 0.993480i $$0.536369\pi$$
$$348$$ 0 0
$$349$$ −8.69694 −0.465536 −0.232768 0.972532i $$-0.574778\pi$$
−0.232768 + 0.972532i $$0.574778\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 20.4949 1.09083 0.545417 0.838165i $$-0.316371\pi$$
0.545417 + 0.838165i $$0.316371\pi$$
$$354$$ 0 0
$$355$$ −8.44949 −0.448452
$$356$$ 0 0
$$357$$ −14.2020 −0.751652
$$358$$ 0 0
$$359$$ −30.2474 −1.59640 −0.798200 0.602393i $$-0.794214\pi$$
−0.798200 + 0.602393i $$0.794214\pi$$
$$360$$ 0 0
$$361$$ 0.797959 0.0419978
$$362$$ 0 0
$$363$$ −26.4495 −1.38824
$$364$$ 0 0
$$365$$ 14.6969 0.769273
$$366$$ 0 0
$$367$$ −18.9444 −0.988889 −0.494444 0.869209i $$-0.664628\pi$$
−0.494444 + 0.869209i $$0.664628\pi$$
$$368$$ 0 0
$$369$$ 3.30306 0.171951
$$370$$ 0 0
$$371$$ 21.7980 1.13169
$$372$$ 0 0
$$373$$ −9.59592 −0.496858 −0.248429 0.968650i $$-0.579914\pi$$
−0.248429 + 0.968650i $$0.579914\pi$$
$$374$$ 0 0
$$375$$ 2.44949 0.126491
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −11.5505 −0.593310 −0.296655 0.954985i $$-0.595871\pi$$
−0.296655 + 0.954985i $$0.595871\pi$$
$$380$$ 0 0
$$381$$ 47.3939 2.42806
$$382$$ 0 0
$$383$$ 13.5959 0.694719 0.347359 0.937732i $$-0.387078\pi$$
0.347359 + 0.937732i $$0.387078\pi$$
$$384$$ 0 0
$$385$$ 0.898979 0.0458162
$$386$$ 0 0
$$387$$ −10.0454 −0.510637
$$388$$ 0 0
$$389$$ −37.5959 −1.90619 −0.953094 0.302673i $$-0.902121\pi$$
−0.953094 + 0.302673i $$0.902121\pi$$
$$390$$ 0 0
$$391$$ −4.49490 −0.227317
$$392$$ 0 0
$$393$$ −48.0000 −2.42128
$$394$$ 0 0
$$395$$ −4.89898 −0.246494
$$396$$ 0 0
$$397$$ 1.30306 0.0653988 0.0326994 0.999465i $$-0.489590\pi$$
0.0326994 + 0.999465i $$0.489590\pi$$
$$398$$ 0 0
$$399$$ −21.7980 −1.09126
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 0 0
$$403$$ −0.449490 −0.0223907
$$404$$ 0 0
$$405$$ −9.00000 −0.447214
$$406$$ 0 0
$$407$$ −2.20204 −0.109151
$$408$$ 0 0
$$409$$ 2.89898 0.143345 0.0716727 0.997428i $$-0.477166\pi$$
0.0716727 + 0.997428i $$0.477166\pi$$
$$410$$ 0 0
$$411$$ −34.2929 −1.69154
$$412$$ 0 0
$$413$$ −10.6969 −0.526362
$$414$$ 0 0
$$415$$ 2.00000 0.0981761
$$416$$ 0 0
$$417$$ −17.3939 −0.851782
$$418$$ 0 0
$$419$$ 9.30306 0.454484 0.227242 0.973838i $$-0.427029\pi$$
0.227242 + 0.973838i $$0.427029\pi$$
$$420$$ 0 0
$$421$$ 15.7980 0.769945 0.384973 0.922928i $$-0.374211\pi$$
0.384973 + 0.922928i $$0.374211\pi$$
$$422$$ 0 0
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ −2.89898 −0.140621
$$426$$ 0 0
$$427$$ 27.5959 1.33546
$$428$$ 0 0
$$429$$ −1.10102 −0.0531578
$$430$$ 0 0
$$431$$ 12.0454 0.580207 0.290103 0.956995i $$-0.406310\pi$$
0.290103 + 0.956995i $$0.406310\pi$$
$$432$$ 0 0
$$433$$ −7.79796 −0.374746 −0.187373 0.982289i $$-0.559997\pi$$
−0.187373 + 0.982289i $$0.559997\pi$$
$$434$$ 0 0
$$435$$ 9.79796 0.469776
$$436$$ 0 0
$$437$$ −6.89898 −0.330023
$$438$$ 0 0
$$439$$ 1.79796 0.0858119 0.0429059 0.999079i $$-0.486338\pi$$
0.0429059 + 0.999079i $$0.486338\pi$$
$$440$$ 0 0
$$441$$ −9.00000 −0.428571
$$442$$ 0 0
$$443$$ 22.0454 1.04741 0.523704 0.851900i $$-0.324550\pi$$
0.523704 + 0.851900i $$0.324550\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 0 0
$$447$$ −19.1010 −0.903447
$$448$$ 0 0
$$449$$ 8.69694 0.410434 0.205217 0.978717i $$-0.434210\pi$$
0.205217 + 0.978717i $$0.434210\pi$$
$$450$$ 0 0
$$451$$ 0.494897 0.0233038
$$452$$ 0 0
$$453$$ −42.4949 −1.99658
$$454$$ 0 0
$$455$$ −2.00000 −0.0937614
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3.30306 −0.153839 −0.0769195 0.997037i $$-0.524508\pi$$
−0.0769195 + 0.997037i $$0.524508\pi$$
$$462$$ 0 0
$$463$$ −13.1010 −0.608856 −0.304428 0.952535i $$-0.598465\pi$$
−0.304428 + 0.952535i $$0.598465\pi$$
$$464$$ 0 0
$$465$$ 1.10102 0.0510586
$$466$$ 0 0
$$467$$ −30.9444 −1.43194 −0.715968 0.698133i $$-0.754014\pi$$
−0.715968 + 0.698133i $$0.754014\pi$$
$$468$$ 0 0
$$469$$ −29.7980 −1.37594
$$470$$ 0 0
$$471$$ 24.4949 1.12867
$$472$$ 0 0
$$473$$ −1.50510 −0.0692047
$$474$$ 0 0
$$475$$ −4.44949 −0.204157
$$476$$ 0 0
$$477$$ 32.6969 1.49709
$$478$$ 0 0
$$479$$ 5.34847 0.244378 0.122189 0.992507i $$-0.461009\pi$$
0.122189 + 0.992507i $$0.461009\pi$$
$$480$$ 0 0
$$481$$ 4.89898 0.223374
$$482$$ 0 0
$$483$$ 7.59592 0.345626
$$484$$ 0 0
$$485$$ −11.7980 −0.535718
$$486$$ 0 0
$$487$$ 2.89898 0.131365 0.0656826 0.997841i $$-0.479078\pi$$
0.0656826 + 0.997841i $$0.479078\pi$$
$$488$$ 0 0
$$489$$ −46.2929 −2.09344
$$490$$ 0 0
$$491$$ 34.2929 1.54761 0.773807 0.633421i $$-0.218350\pi$$
0.773807 + 0.633421i $$0.218350\pi$$
$$492$$ 0 0
$$493$$ −11.5959 −0.522254
$$494$$ 0 0
$$495$$ 1.34847 0.0606092
$$496$$ 0 0
$$497$$ −16.8990 −0.758023
$$498$$ 0 0
$$499$$ 21.3485 0.955689 0.477844 0.878445i $$-0.341418\pi$$
0.477844 + 0.878445i $$0.341418\pi$$
$$500$$ 0 0
$$501$$ 44.0908 1.96983
$$502$$ 0 0
$$503$$ −26.9444 −1.20139 −0.600695 0.799478i $$-0.705110\pi$$
−0.600695 + 0.799478i $$0.705110\pi$$
$$504$$ 0 0
$$505$$ 3.79796 0.169007
$$506$$ 0 0
$$507$$ 2.44949 0.108786
$$508$$ 0 0
$$509$$ 26.8990 1.19228 0.596138 0.802882i $$-0.296701\pi$$
0.596138 + 0.802882i $$0.296701\pi$$
$$510$$ 0 0
$$511$$ 29.3939 1.30031
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −3.34847 −0.147551
$$516$$ 0 0
$$517$$ −0.898979 −0.0395371
$$518$$ 0 0
$$519$$ −7.10102 −0.311700
$$520$$ 0 0
$$521$$ 6.20204 0.271716 0.135858 0.990728i $$-0.456621\pi$$
0.135858 + 0.990728i $$0.456621\pi$$
$$522$$ 0 0
$$523$$ −39.3485 −1.72059 −0.860294 0.509798i $$-0.829720\pi$$
−0.860294 + 0.509798i $$0.829720\pi$$
$$524$$ 0 0
$$525$$ 4.89898 0.213809
$$526$$ 0 0
$$527$$ −1.30306 −0.0567623
$$528$$ 0 0
$$529$$ −20.5959 −0.895475
$$530$$ 0 0
$$531$$ −16.0454 −0.696311
$$532$$ 0 0
$$533$$ −1.10102 −0.0476905
$$534$$ 0 0
$$535$$ 12.2474 0.529503
$$536$$ 0 0
$$537$$ 19.5959 0.845626
$$538$$ 0 0
$$539$$ −1.34847 −0.0580827
$$540$$ 0 0
$$541$$ 5.10102 0.219310 0.109655 0.993970i $$-0.465025\pi$$
0.109655 + 0.993970i $$0.465025\pi$$
$$542$$ 0 0
$$543$$ −33.7980 −1.45041
$$544$$ 0 0
$$545$$ −10.0000 −0.428353
$$546$$ 0 0
$$547$$ 0.651531 0.0278574 0.0139287 0.999903i $$-0.495566\pi$$
0.0139287 + 0.999903i $$0.495566\pi$$
$$548$$ 0 0
$$549$$ 41.3939 1.76665
$$550$$ 0 0
$$551$$ −17.7980 −0.758219
$$552$$ 0 0
$$553$$ −9.79796 −0.416652
$$554$$ 0 0
$$555$$ −12.0000 −0.509372
$$556$$ 0 0
$$557$$ −1.30306 −0.0552125 −0.0276062 0.999619i $$-0.508788\pi$$
−0.0276062 + 0.999619i $$0.508788\pi$$
$$558$$ 0 0
$$559$$ 3.34847 0.141625
$$560$$ 0 0
$$561$$ −3.19184 −0.134759
$$562$$ 0 0
$$563$$ −7.75255 −0.326731 −0.163366 0.986566i $$-0.552235\pi$$
−0.163366 + 0.986566i $$0.552235\pi$$
$$564$$ 0 0
$$565$$ 10.8990 0.458524
$$566$$ 0 0
$$567$$ −18.0000 −0.755929
$$568$$ 0 0
$$569$$ 10.2020 0.427692 0.213846 0.976867i $$-0.431401\pi$$
0.213846 + 0.976867i $$0.431401\pi$$
$$570$$ 0 0
$$571$$ 28.4949 1.19247 0.596237 0.802808i $$-0.296662\pi$$
0.596237 + 0.802808i $$0.296662\pi$$
$$572$$ 0 0
$$573$$ 43.5959 1.82125
$$574$$ 0 0
$$575$$ 1.55051 0.0646607
$$576$$ 0 0
$$577$$ 27.1010 1.12823 0.564115 0.825696i $$-0.309217\pi$$
0.564115 + 0.825696i $$0.309217\pi$$
$$578$$ 0 0
$$579$$ 48.4949 2.01538
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ 4.89898 0.202895
$$584$$ 0 0
$$585$$ −3.00000 −0.124035
$$586$$ 0 0
$$587$$ −10.8990 −0.449849 −0.224925 0.974376i $$-0.572214\pi$$
−0.224925 + 0.974376i $$0.572214\pi$$
$$588$$ 0 0
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ −10.2929 −0.423392
$$592$$ 0 0
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ −5.79796 −0.237693
$$596$$ 0 0
$$597$$ −33.7980 −1.38326
$$598$$ 0 0
$$599$$ −28.8990 −1.18078 −0.590390 0.807118i $$-0.701026\pi$$
−0.590390 + 0.807118i $$0.701026\pi$$
$$600$$ 0 0
$$601$$ −33.5959 −1.37041 −0.685203 0.728352i $$-0.740287\pi$$
−0.685203 + 0.728352i $$0.740287\pi$$
$$602$$ 0 0
$$603$$ −44.6969 −1.82020
$$604$$ 0 0
$$605$$ −10.7980 −0.438999
$$606$$ 0 0
$$607$$ 21.5505 0.874708 0.437354 0.899289i $$-0.355916\pi$$
0.437354 + 0.899289i $$0.355916\pi$$
$$608$$ 0 0
$$609$$ 19.5959 0.794067
$$610$$ 0 0
$$611$$ 2.00000 0.0809113
$$612$$ 0 0
$$613$$ 45.1918 1.82528 0.912641 0.408763i $$-0.134040\pi$$
0.912641 + 0.408763i $$0.134040\pi$$
$$614$$ 0 0
$$615$$ 2.69694 0.108751
$$616$$ 0 0
$$617$$ −35.3939 −1.42490 −0.712452 0.701721i $$-0.752415\pi$$
−0.712452 + 0.701721i $$0.752415\pi$$
$$618$$ 0 0
$$619$$ −5.34847 −0.214973 −0.107487 0.994207i $$-0.534280\pi$$
−0.107487 + 0.994207i $$0.534280\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −4.89898 −0.195646
$$628$$ 0 0
$$629$$ 14.2020 0.566272
$$630$$ 0 0
$$631$$ −16.4495 −0.654844 −0.327422 0.944878i $$-0.606180\pi$$
−0.327422 + 0.944878i $$0.606180\pi$$
$$632$$ 0 0
$$633$$ −15.1918 −0.603821
$$634$$ 0 0
$$635$$ 19.3485 0.767821
$$636$$ 0 0
$$637$$ 3.00000 0.118864
$$638$$ 0 0
$$639$$ −25.3485 −1.00277
$$640$$ 0 0
$$641$$ −35.3939 −1.39797 −0.698987 0.715134i $$-0.746366\pi$$
−0.698987 + 0.715134i $$0.746366\pi$$
$$642$$ 0 0
$$643$$ −30.0000 −1.18308 −0.591542 0.806274i $$-0.701481\pi$$
−0.591542 + 0.806274i $$0.701481\pi$$
$$644$$ 0 0
$$645$$ −8.20204 −0.322955
$$646$$ 0 0
$$647$$ 14.9444 0.587524 0.293762 0.955879i $$-0.405093\pi$$
0.293762 + 0.955879i $$0.405093\pi$$
$$648$$ 0 0
$$649$$ −2.40408 −0.0943685
$$650$$ 0 0
$$651$$ 2.20204 0.0863048
$$652$$ 0 0
$$653$$ −43.7980 −1.71395 −0.856973 0.515361i $$-0.827658\pi$$
−0.856973 + 0.515361i $$0.827658\pi$$
$$654$$ 0 0
$$655$$ −19.5959 −0.765676
$$656$$ 0 0
$$657$$ 44.0908 1.72015
$$658$$ 0 0
$$659$$ 14.6969 0.572511 0.286256 0.958153i $$-0.407589\pi$$
0.286256 + 0.958153i $$0.407589\pi$$
$$660$$ 0 0
$$661$$ 39.7980 1.54796 0.773981 0.633209i $$-0.218263\pi$$
0.773981 + 0.633209i $$0.218263\pi$$
$$662$$ 0 0
$$663$$ 7.10102 0.275781
$$664$$ 0 0
$$665$$ −8.89898 −0.345088
$$666$$ 0 0
$$667$$ 6.20204 0.240144
$$668$$ 0 0
$$669$$ 52.8990 2.04519
$$670$$ 0 0
$$671$$ 6.20204 0.239427
$$672$$ 0 0
$$673$$ −1.10102 −0.0424412 −0.0212206 0.999775i $$-0.506755\pi$$
−0.0212206 + 0.999775i $$0.506755\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −40.6969 −1.56411 −0.782055 0.623209i $$-0.785829\pi$$
−0.782055 + 0.623209i $$0.785829\pi$$
$$678$$ 0 0
$$679$$ −23.5959 −0.905528
$$680$$ 0 0
$$681$$ 32.0908 1.22972
$$682$$ 0 0
$$683$$ −42.4949 −1.62602 −0.813011 0.582248i $$-0.802173\pi$$
−0.813011 + 0.582248i $$0.802173\pi$$
$$684$$ 0 0
$$685$$ −14.0000 −0.534913
$$686$$ 0 0
$$687$$ −22.2929 −0.850526
$$688$$ 0 0
$$689$$ −10.8990 −0.415218
$$690$$ 0 0
$$691$$ −14.6515 −0.557370 −0.278685 0.960382i $$-0.589899\pi$$
−0.278685 + 0.960382i $$0.589899\pi$$
$$692$$ 0 0
$$693$$ 2.69694 0.102448
$$694$$ 0 0
$$695$$ −7.10102 −0.269357
$$696$$ 0 0
$$697$$ −3.19184 −0.120899
$$698$$ 0 0
$$699$$ 33.3031 1.25964
$$700$$ 0 0
$$701$$ −37.1918 −1.40472 −0.702358 0.711824i $$-0.747869\pi$$
−0.702358 + 0.711824i $$0.747869\pi$$
$$702$$ 0 0
$$703$$ 21.7980 0.822126
$$704$$ 0 0
$$705$$ −4.89898 −0.184506
$$706$$ 0 0
$$707$$ 7.59592 0.285674
$$708$$ 0 0
$$709$$ −32.2929 −1.21278 −0.606392 0.795166i $$-0.707384\pi$$
−0.606392 + 0.795166i $$0.707384\pi$$
$$710$$ 0 0
$$711$$ −14.6969 −0.551178
$$712$$ 0 0
$$713$$ 0.696938 0.0261006
$$714$$ 0 0
$$715$$ −0.449490 −0.0168100
$$716$$ 0 0
$$717$$ 56.6969 2.11739
$$718$$ 0 0
$$719$$ −20.0000 −0.745874 −0.372937 0.927857i $$-0.621649\pi$$
−0.372937 + 0.927857i $$0.621649\pi$$
$$720$$ 0 0
$$721$$ −6.69694 −0.249407
$$722$$ 0 0
$$723$$ 70.2929 2.61422
$$724$$ 0 0
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ 18.0454 0.669267 0.334634 0.942348i $$-0.391387\pi$$
0.334634 + 0.942348i $$0.391387\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 9.70714 0.359032
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 0 0
$$735$$ −7.34847 −0.271052
$$736$$ 0 0
$$737$$ −6.69694 −0.246685
$$738$$ 0 0
$$739$$ −6.24745 −0.229816 −0.114908 0.993376i $$-0.536657\pi$$
−0.114908 + 0.993376i $$0.536657\pi$$
$$740$$ 0 0
$$741$$ 10.8990 0.400384
$$742$$ 0 0
$$743$$ −35.3939 −1.29848 −0.649238 0.760586i $$-0.724912\pi$$
−0.649238 + 0.760586i $$0.724912\pi$$
$$744$$ 0 0
$$745$$ −7.79796 −0.285695
$$746$$ 0 0
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ 24.4949 0.895024
$$750$$ 0 0
$$751$$ 31.1010 1.13489 0.567446 0.823410i $$-0.307931\pi$$
0.567446 + 0.823410i $$0.307931\pi$$
$$752$$ 0 0
$$753$$ 36.0000 1.31191
$$754$$ 0 0
$$755$$ −17.3485 −0.631375
$$756$$ 0 0
$$757$$ 26.4949 0.962973 0.481487 0.876453i $$-0.340097\pi$$
0.481487 + 0.876453i $$0.340097\pi$$
$$758$$ 0 0
$$759$$ 1.70714 0.0619654
$$760$$ 0 0
$$761$$ −10.4041 −0.377148 −0.188574 0.982059i $$-0.560386\pi$$
−0.188574 + 0.982059i $$0.560386\pi$$
$$762$$ 0 0
$$763$$ −20.0000 −0.724049
$$764$$ 0 0
$$765$$ −8.69694 −0.314438
$$766$$ 0 0
$$767$$ 5.34847 0.193122
$$768$$ 0 0
$$769$$ 39.3939 1.42058 0.710290 0.703909i $$-0.248564\pi$$
0.710290 + 0.703909i $$0.248564\pi$$
$$770$$ 0 0
$$771$$ 38.6969 1.39364
$$772$$ 0 0
$$773$$ 44.0908 1.58584 0.792918 0.609328i $$-0.208561\pi$$
0.792918 + 0.609328i $$0.208561\pi$$
$$774$$ 0 0
$$775$$ 0.449490 0.0161461
$$776$$ 0 0
$$777$$ −24.0000 −0.860995
$$778$$ 0 0
$$779$$ −4.89898 −0.175524
$$780$$ 0 0
$$781$$ −3.79796 −0.135902
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ 38.0000 1.35455 0.677277 0.735728i $$-0.263160\pi$$
0.677277 + 0.735728i $$0.263160\pi$$
$$788$$ 0 0
$$789$$ 52.7878 1.87929
$$790$$ 0 0
$$791$$ 21.7980 0.775046
$$792$$ 0 0
$$793$$ −13.7980 −0.489980
$$794$$ 0 0
$$795$$ 26.6969 0.946843
$$796$$ 0 0
$$797$$ 27.7980 0.984654 0.492327 0.870410i $$-0.336146\pi$$
0.492327 + 0.870410i $$0.336146\pi$$
$$798$$ 0 0
$$799$$ 5.79796 0.205117
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 0 0
$$803$$ 6.60612 0.233125
$$804$$ 0 0
$$805$$ 3.10102 0.109297
$$806$$ 0 0
$$807$$ 63.6867 2.24188
$$808$$ 0 0
$$809$$ 12.0000 0.421898 0.210949 0.977497i $$-0.432345\pi$$
0.210949 + 0.977497i $$0.432345\pi$$
$$810$$ 0 0
$$811$$ −53.8434 −1.89070 −0.945348 0.326063i $$-0.894278\pi$$
−0.945348 + 0.326063i $$0.894278\pi$$
$$812$$ 0 0
$$813$$ 20.6969 0.725873
$$814$$ 0 0
$$815$$ −18.8990 −0.662002
$$816$$ 0 0
$$817$$ 14.8990 0.521249
$$818$$ 0 0
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ −53.1918 −1.85641 −0.928204 0.372072i $$-0.878648\pi$$
−0.928204 + 0.372072i $$0.878648\pi$$
$$822$$ 0 0
$$823$$ 30.4495 1.06140 0.530701 0.847559i $$-0.321929\pi$$
0.530701 + 0.847559i $$0.321929\pi$$
$$824$$ 0 0
$$825$$ 1.10102 0.0383326
$$826$$ 0 0
$$827$$ 57.1918 1.98875 0.994377 0.105893i $$-0.0337702\pi$$
0.994377 + 0.105893i $$0.0337702\pi$$
$$828$$ 0 0
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ 0 0
$$831$$ 64.8990 2.25132
$$832$$ 0 0
$$833$$ 8.69694 0.301331
$$834$$ 0 0
$$835$$ 18.0000 0.622916
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −19.5505 −0.674959 −0.337479 0.941333i $$-0.609574\pi$$
−0.337479 + 0.941333i $$0.609574\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 0 0
$$843$$ 7.10102 0.244572
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ −21.5959 −0.742045
$$848$$ 0 0
$$849$$ −3.79796 −0.130346
$$850$$ 0 0
$$851$$ −7.59592 −0.260385
$$852$$ 0 0
$$853$$ −18.6969 −0.640171 −0.320085 0.947389i $$-0.603712\pi$$
−0.320085 + 0.947389i $$0.603712\pi$$
$$854$$ 0 0
$$855$$ −13.3485 −0.456508
$$856$$ 0 0
$$857$$ 35.3939 1.20903 0.604516 0.796593i $$-0.293367\pi$$
0.604516 + 0.796593i $$0.293367\pi$$
$$858$$ 0 0
$$859$$ −30.6969 −1.04737 −0.523683 0.851913i $$-0.675442\pi$$
−0.523683 + 0.851913i $$0.675442\pi$$
$$860$$ 0 0
$$861$$ 5.39388 0.183823
$$862$$ 0 0
$$863$$ 23.3939 0.796337 0.398168 0.917312i $$-0.369646\pi$$
0.398168 + 0.917312i $$0.369646\pi$$
$$864$$ 0 0
$$865$$ −2.89898 −0.0985683
$$866$$ 0 0
$$867$$ −21.0556 −0.715086
$$868$$ 0 0
$$869$$ −2.20204 −0.0746991
$$870$$ 0 0
$$871$$ 14.8990 0.504833
$$872$$ 0 0
$$873$$ −35.3939 −1.19790
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ 17.5959 0.594172 0.297086 0.954851i $$-0.403985\pi$$
0.297086 + 0.954851i $$0.403985\pi$$
$$878$$ 0 0
$$879$$ −50.2020 −1.69327
$$880$$ 0 0
$$881$$ 9.79796 0.330102 0.165051 0.986285i $$-0.447221\pi$$
0.165051 + 0.986285i $$0.447221\pi$$
$$882$$ 0 0
$$883$$ −11.3485 −0.381906 −0.190953 0.981599i $$-0.561158\pi$$
−0.190953 + 0.981599i $$0.561158\pi$$
$$884$$ 0 0
$$885$$ −13.1010 −0.440386
$$886$$ 0 0
$$887$$ 4.24745 0.142615 0.0713077 0.997454i $$-0.477283\pi$$
0.0713077 + 0.997454i $$0.477283\pi$$
$$888$$ 0 0
$$889$$ 38.6969 1.29785
$$890$$ 0 0
$$891$$ −4.04541 −0.135526
$$892$$ 0 0
$$893$$ 8.89898 0.297793
$$894$$ 0 0
$$895$$ 8.00000 0.267411
$$896$$ 0 0
$$897$$ −3.79796 −0.126810
$$898$$ 0 0
$$899$$ 1.79796 0.0599653
$$900$$ 0 0
$$901$$ −31.5959 −1.05261
$$902$$ 0 0
$$903$$ −16.4041 −0.545894
$$904$$ 0 0
$$905$$ −13.7980 −0.458660
$$906$$ 0 0
$$907$$ 43.8434 1.45580 0.727898 0.685686i $$-0.240498\pi$$
0.727898 + 0.685686i $$0.240498\pi$$
$$908$$ 0 0
$$909$$ 11.3939 0.377911
$$910$$ 0 0
$$911$$ −49.3939 −1.63649 −0.818246 0.574868i $$-0.805053\pi$$
−0.818246 + 0.574868i $$0.805053\pi$$
$$912$$ 0 0
$$913$$ 0.898979 0.0297519
$$914$$ 0 0
$$915$$ 33.7980 1.11733
$$916$$ 0 0
$$917$$ −39.1918 −1.29423
$$918$$ 0 0
$$919$$ −11.1010 −0.366189 −0.183094 0.983095i $$-0.558611\pi$$
−0.183094 + 0.983095i $$0.558611\pi$$
$$920$$ 0 0
$$921$$ −28.8990 −0.952254
$$922$$ 0 0
$$923$$ 8.44949 0.278118
$$924$$ 0 0
$$925$$ −4.89898 −0.161077
$$926$$ 0 0
$$927$$ −10.0454 −0.329934
$$928$$ 0 0
$$929$$ 24.6969 0.810280 0.405140 0.914255i $$-0.367223\pi$$
0.405140 + 0.914255i $$0.367223\pi$$
$$930$$ 0 0
$$931$$ 13.3485 0.437478
$$932$$ 0 0
$$933$$ 31.5959 1.03440
$$934$$ 0 0
$$935$$ −1.30306 −0.0426147
$$936$$ 0 0
$$937$$ −19.3939 −0.633570 −0.316785 0.948497i $$-0.602603\pi$$
−0.316785 + 0.948497i $$0.602603\pi$$
$$938$$ 0 0
$$939$$ 64.8990 2.11790
$$940$$ 0 0
$$941$$ −23.3031 −0.759658 −0.379829 0.925057i $$-0.624017\pi$$
−0.379829 + 0.925057i $$0.624017\pi$$
$$942$$ 0 0
$$943$$ 1.70714 0.0555922
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −47.7980 −1.55322 −0.776612 0.629979i $$-0.783064\pi$$
−0.776612 + 0.629979i $$0.783064\pi$$
$$948$$ 0 0
$$949$$ −14.6969 −0.477083
$$950$$ 0 0
$$951$$ −6.60612 −0.214218
$$952$$ 0 0
$$953$$ −57.1918 −1.85263 −0.926313 0.376756i $$-0.877040\pi$$
−0.926313 + 0.376756i $$0.877040\pi$$
$$954$$ 0 0
$$955$$ 17.7980 0.575928
$$956$$ 0 0
$$957$$ 4.40408 0.142364
$$958$$ 0 0
$$959$$ −28.0000 −0.904167
$$960$$ 0 0
$$961$$ −30.7980 −0.993483
$$962$$ 0 0
$$963$$ 36.7423 1.18401
$$964$$ 0 0
$$965$$ 19.7980 0.637319
$$966$$ 0 0
$$967$$ 25.1010 0.807194 0.403597 0.914937i $$-0.367760\pi$$
0.403597 + 0.914937i $$0.367760\pi$$
$$968$$ 0 0
$$969$$ 31.5959 1.01501
$$970$$ 0 0
$$971$$ 27.5959 0.885595 0.442798 0.896622i $$-0.353986\pi$$
0.442798 + 0.896622i $$0.353986\pi$$
$$972$$ 0 0
$$973$$ −14.2020 −0.455297
$$974$$ 0 0
$$975$$ −2.44949 −0.0784465
$$976$$ 0 0
$$977$$ −1.30306 −0.0416886 −0.0208443 0.999783i $$-0.506635\pi$$
−0.0208443 + 0.999783i $$0.506635\pi$$
$$978$$ 0 0
$$979$$ 2.69694 0.0861945
$$980$$ 0 0
$$981$$ −30.0000 −0.957826
$$982$$ 0 0
$$983$$ 58.8990 1.87859 0.939293 0.343117i $$-0.111483\pi$$
0.939293 + 0.343117i $$0.111483\pi$$
$$984$$ 0 0
$$985$$ −4.20204 −0.133888
$$986$$ 0 0
$$987$$ −9.79796 −0.311872
$$988$$ 0 0
$$989$$ −5.19184 −0.165091
$$990$$ 0 0
$$991$$ −30.2020 −0.959399 −0.479700 0.877433i $$-0.659254\pi$$
−0.479700 + 0.877433i $$0.659254\pi$$
$$992$$ 0 0
$$993$$ −20.6969 −0.656797
$$994$$ 0 0
$$995$$ −13.7980 −0.437425
$$996$$ 0 0
$$997$$ 25.1010 0.794957 0.397479 0.917611i $$-0.369885\pi$$
0.397479 + 0.917611i $$0.369885\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 520.2.a.f.1.2 2
3.2 odd 2 4680.2.a.z.1.1 2
4.3 odd 2 1040.2.a.k.1.1 2
5.2 odd 4 2600.2.d.h.1249.2 4
5.3 odd 4 2600.2.d.h.1249.3 4
5.4 even 2 2600.2.a.q.1.1 2
8.3 odd 2 4160.2.a.ba.1.2 2
8.5 even 2 4160.2.a.bg.1.1 2
12.11 even 2 9360.2.a.cc.1.2 2
13.12 even 2 6760.2.a.s.1.2 2
20.19 odd 2 5200.2.a.bv.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.a.f.1.2 2 1.1 even 1 trivial
1040.2.a.k.1.1 2 4.3 odd 2
2600.2.a.q.1.1 2 5.4 even 2
2600.2.d.h.1249.2 4 5.2 odd 4
2600.2.d.h.1249.3 4 5.3 odd 4
4160.2.a.ba.1.2 2 8.3 odd 2
4160.2.a.bg.1.1 2 8.5 even 2
4680.2.a.z.1.1 2 3.2 odd 2
5200.2.a.bv.1.2 2 20.19 odd 2
6760.2.a.s.1.2 2 13.12 even 2
9360.2.a.cc.1.2 2 12.11 even 2