# Properties

 Label 1530.2.a.s.1.1 Level $1530$ Weight $2$ Character 1530.1 Self dual yes Analytic conductor $12.217$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1530,2,Mod(1,1530)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1530, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1530.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1530.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: $$12.2171115093$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 510) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 1530.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.89898 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.89898 q^{7} +1.00000 q^{8} -1.00000 q^{10} +6.89898 q^{13} -4.89898 q^{14} +1.00000 q^{16} +1.00000 q^{17} +4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{23} +1.00000 q^{25} +6.89898 q^{26} -4.89898 q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} +4.89898 q^{35} +6.00000 q^{37} +4.00000 q^{38} -1.00000 q^{40} +2.89898 q^{41} +8.89898 q^{43} +4.00000 q^{46} -9.79796 q^{47} +17.0000 q^{49} +1.00000 q^{50} +6.89898 q^{52} +7.79796 q^{53} -4.89898 q^{56} -6.00000 q^{58} -4.89898 q^{59} +11.7980 q^{61} +4.00000 q^{62} +1.00000 q^{64} -6.89898 q^{65} +0.898979 q^{67} +1.00000 q^{68} +4.89898 q^{70} +8.89898 q^{71} -10.8990 q^{73} +6.00000 q^{74} +4.00000 q^{76} -5.79796 q^{79} -1.00000 q^{80} +2.89898 q^{82} -13.7980 q^{83} -1.00000 q^{85} +8.89898 q^{86} +7.79796 q^{89} -33.7980 q^{91} +4.00000 q^{92} -9.79796 q^{94} -4.00000 q^{95} -12.6969 q^{97} +17.0000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8} - 2 q^{10} + 4 q^{13} + 2 q^{16} + 2 q^{17} + 8 q^{19} - 2 q^{20} + 8 q^{23} + 2 q^{25} + 4 q^{26} - 12 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{34} + 12 q^{37} + 8 q^{38} - 2 q^{40} - 4 q^{41} + 8 q^{43} + 8 q^{46} + 34 q^{49} + 2 q^{50} + 4 q^{52} - 4 q^{53} - 12 q^{58} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} + 2 q^{68} + 8 q^{71} - 12 q^{73} + 12 q^{74} + 8 q^{76} + 8 q^{79} - 2 q^{80} - 4 q^{82} - 8 q^{83} - 2 q^{85} + 8 q^{86} - 4 q^{89} - 48 q^{91} + 8 q^{92} - 8 q^{95} + 4 q^{97} + 34 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^8 - 2 * q^10 + 4 * q^13 + 2 * q^16 + 2 * q^17 + 8 * q^19 - 2 * q^20 + 8 * q^23 + 2 * q^25 + 4 * q^26 - 12 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^34 + 12 * q^37 + 8 * q^38 - 2 * q^40 - 4 * q^41 + 8 * q^43 + 8 * q^46 + 34 * q^49 + 2 * q^50 + 4 * q^52 - 4 * q^53 - 12 * q^58 + 4 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 + 2 * q^68 + 8 * q^71 - 12 * q^73 + 12 * q^74 + 8 * q^76 + 8 * q^79 - 2 * q^80 - 4 * q^82 - 8 * q^83 - 2 * q^85 + 8 * q^86 - 4 * q^89 - 48 * q^91 + 8 * q^92 - 8 * q^95 + 4 * q^97 + 34 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −4.89898 −1.85164 −0.925820 0.377964i $$-0.876624\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 6.89898 1.91343 0.956716 0.291022i $$-0.0939953\pi$$
0.956716 + 0.291022i $$0.0939953\pi$$
$$14$$ −4.89898 −1.30931
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 6.89898 1.35300
$$27$$ 0 0
$$28$$ −4.89898 −0.925820
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 1.00000 0.171499
$$35$$ 4.89898 0.828079
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 2.89898 0.452745 0.226372 0.974041i $$-0.427313\pi$$
0.226372 + 0.974041i $$0.427313\pi$$
$$42$$ 0 0
$$43$$ 8.89898 1.35708 0.678541 0.734563i $$-0.262613\pi$$
0.678541 + 0.734563i $$0.262613\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ −9.79796 −1.42918 −0.714590 0.699544i $$-0.753387\pi$$
−0.714590 + 0.699544i $$0.753387\pi$$
$$48$$ 0 0
$$49$$ 17.0000 2.42857
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 6.89898 0.956716
$$53$$ 7.79796 1.07113 0.535566 0.844493i $$-0.320098\pi$$
0.535566 + 0.844493i $$0.320098\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −4.89898 −0.654654
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ −4.89898 −0.637793 −0.318896 0.947790i $$-0.603312\pi$$
−0.318896 + 0.947790i $$0.603312\pi$$
$$60$$ 0 0
$$61$$ 11.7980 1.51057 0.755287 0.655394i $$-0.227498\pi$$
0.755287 + 0.655394i $$0.227498\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −6.89898 −0.855713
$$66$$ 0 0
$$67$$ 0.898979 0.109828 0.0549139 0.998491i $$-0.482512\pi$$
0.0549139 + 0.998491i $$0.482512\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 0 0
$$70$$ 4.89898 0.585540
$$71$$ 8.89898 1.05611 0.528057 0.849209i $$-0.322921\pi$$
0.528057 + 0.849209i $$0.322921\pi$$
$$72$$ 0 0
$$73$$ −10.8990 −1.27563 −0.637815 0.770190i $$-0.720161\pi$$
−0.637815 + 0.770190i $$0.720161\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −5.79796 −0.652321 −0.326161 0.945314i $$-0.605755\pi$$
−0.326161 + 0.945314i $$0.605755\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 2.89898 0.320139
$$83$$ −13.7980 −1.51452 −0.757261 0.653112i $$-0.773463\pi$$
−0.757261 + 0.653112i $$0.773463\pi$$
$$84$$ 0 0
$$85$$ −1.00000 −0.108465
$$86$$ 8.89898 0.959602
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.79796 0.826582 0.413291 0.910599i $$-0.364379\pi$$
0.413291 + 0.910599i $$0.364379\pi$$
$$90$$ 0 0
$$91$$ −33.7980 −3.54299
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ −9.79796 −1.01058
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ −12.6969 −1.28918 −0.644589 0.764529i $$-0.722972\pi$$
−0.644589 + 0.764529i $$0.722972\pi$$
$$98$$ 17.0000 1.71726
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 18.8990 1.88052 0.940259 0.340459i $$-0.110582\pi$$
0.940259 + 0.340459i $$0.110582\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 6.89898 0.676501
$$105$$ 0 0
$$106$$ 7.79796 0.757405
$$107$$ −5.79796 −0.560510 −0.280255 0.959926i $$-0.590419\pi$$
−0.280255 + 0.959926i $$0.590419\pi$$
$$108$$ 0 0
$$109$$ 11.7980 1.13004 0.565020 0.825077i $$-0.308869\pi$$
0.565020 + 0.825077i $$0.308869\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −4.89898 −0.462910
$$113$$ −7.79796 −0.733570 −0.366785 0.930306i $$-0.619542\pi$$
−0.366785 + 0.930306i $$0.619542\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ −4.89898 −0.450988
$$119$$ −4.89898 −0.449089
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 11.7980 1.06814
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −6.89898 −0.605081
$$131$$ −9.79796 −0.856052 −0.428026 0.903767i $$-0.640791\pi$$
−0.428026 + 0.903767i $$0.640791\pi$$
$$132$$ 0 0
$$133$$ −19.5959 −1.69918
$$134$$ 0.898979 0.0776600
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ 13.7980 1.17033 0.585164 0.810915i $$-0.301030\pi$$
0.585164 + 0.810915i $$0.301030\pi$$
$$140$$ 4.89898 0.414039
$$141$$ 0 0
$$142$$ 8.89898 0.746786
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ −10.8990 −0.902006
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ 18.8990 1.54826 0.774132 0.633024i $$-0.218186\pi$$
0.774132 + 0.633024i $$0.218186\pi$$
$$150$$ 0 0
$$151$$ −9.79796 −0.797347 −0.398673 0.917093i $$-0.630529\pi$$
−0.398673 + 0.917093i $$0.630529\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ −1.10102 −0.0878710 −0.0439355 0.999034i $$-0.513990\pi$$
−0.0439355 + 0.999034i $$0.513990\pi$$
$$158$$ −5.79796 −0.461261
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ −19.5959 −1.54437
$$162$$ 0 0
$$163$$ −2.20204 −0.172477 −0.0862386 0.996275i $$-0.527485\pi$$
−0.0862386 + 0.996275i $$0.527485\pi$$
$$164$$ 2.89898 0.226372
$$165$$ 0 0
$$166$$ −13.7980 −1.07093
$$167$$ −2.20204 −0.170399 −0.0851995 0.996364i $$-0.527153\pi$$
−0.0851995 + 0.996364i $$0.527153\pi$$
$$168$$ 0 0
$$169$$ 34.5959 2.66122
$$170$$ −1.00000 −0.0766965
$$171$$ 0 0
$$172$$ 8.89898 0.678541
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ −4.89898 −0.370328
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 7.79796 0.584482
$$179$$ 4.89898 0.366167 0.183083 0.983097i $$-0.441392\pi$$
0.183083 + 0.983097i $$0.441392\pi$$
$$180$$ 0 0
$$181$$ −4.20204 −0.312335 −0.156168 0.987731i $$-0.549914\pi$$
−0.156168 + 0.987731i $$0.549914\pi$$
$$182$$ −33.7980 −2.50527
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −9.79796 −0.714590
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ 9.79796 0.708955 0.354478 0.935064i $$-0.384659\pi$$
0.354478 + 0.935064i $$0.384659\pi$$
$$192$$ 0 0
$$193$$ −1.10102 −0.0792532 −0.0396266 0.999215i $$-0.512617\pi$$
−0.0396266 + 0.999215i $$0.512617\pi$$
$$194$$ −12.6969 −0.911587
$$195$$ 0 0
$$196$$ 17.0000 1.21429
$$197$$ 13.5959 0.968669 0.484335 0.874883i $$-0.339062\pi$$
0.484335 + 0.874883i $$0.339062\pi$$
$$198$$ 0 0
$$199$$ 15.5959 1.10557 0.552783 0.833325i $$-0.313566\pi$$
0.552783 + 0.833325i $$0.313566\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 18.8990 1.32973
$$203$$ 29.3939 2.06305
$$204$$ 0 0
$$205$$ −2.89898 −0.202474
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ 6.89898 0.478358
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 7.79796 0.535566
$$213$$ 0 0
$$214$$ −5.79796 −0.396340
$$215$$ −8.89898 −0.606905
$$216$$ 0 0
$$217$$ −19.5959 −1.33026
$$218$$ 11.7980 0.799059
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6.89898 0.464076
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −4.89898 −0.327327
$$225$$ 0 0
$$226$$ −7.79796 −0.518713
$$227$$ 21.7980 1.44678 0.723391 0.690439i $$-0.242583\pi$$
0.723391 + 0.690439i $$0.242583\pi$$
$$228$$ 0 0
$$229$$ −13.5959 −0.898444 −0.449222 0.893420i $$-0.648299\pi$$
−0.449222 + 0.893420i $$0.648299\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 0 0
$$235$$ 9.79796 0.639148
$$236$$ −4.89898 −0.318896
$$237$$ 0 0
$$238$$ −4.89898 −0.317554
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 13.5959 0.875790 0.437895 0.899026i $$-0.355724\pi$$
0.437895 + 0.899026i $$0.355724\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 0 0
$$244$$ 11.7980 0.755287
$$245$$ −17.0000 −1.08609
$$246$$ 0 0
$$247$$ 27.5959 1.75589
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −4.89898 −0.309221 −0.154610 0.987976i $$-0.549412\pi$$
−0.154610 + 0.987976i $$0.549412\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ −29.3939 −1.82645
$$260$$ −6.89898 −0.427857
$$261$$ 0 0
$$262$$ −9.79796 −0.605320
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ −7.79796 −0.479025
$$266$$ −19.5959 −1.20150
$$267$$ 0 0
$$268$$ 0.898979 0.0549139
$$269$$ −9.59592 −0.585073 −0.292537 0.956254i $$-0.594499\pi$$
−0.292537 + 0.956254i $$0.594499\pi$$
$$270$$ 0 0
$$271$$ −1.79796 −0.109218 −0.0546091 0.998508i $$-0.517391\pi$$
−0.0546091 + 0.998508i $$0.517391\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ 0 0
$$274$$ −14.0000 −0.845771
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7.79796 0.468534 0.234267 0.972172i $$-0.424731\pi$$
0.234267 + 0.972172i $$0.424731\pi$$
$$278$$ 13.7980 0.827547
$$279$$ 0 0
$$280$$ 4.89898 0.292770
$$281$$ −27.7980 −1.65829 −0.829144 0.559036i $$-0.811171\pi$$
−0.829144 + 0.559036i $$0.811171\pi$$
$$282$$ 0 0
$$283$$ 23.5959 1.40263 0.701316 0.712851i $$-0.252596\pi$$
0.701316 + 0.712851i $$0.252596\pi$$
$$284$$ 8.89898 0.528057
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −14.2020 −0.838320
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 6.00000 0.352332
$$291$$ 0 0
$$292$$ −10.8990 −0.637815
$$293$$ −13.5959 −0.794282 −0.397141 0.917758i $$-0.629998\pi$$
−0.397141 + 0.917758i $$0.629998\pi$$
$$294$$ 0 0
$$295$$ 4.89898 0.285230
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ 18.8990 1.09479
$$299$$ 27.5959 1.59591
$$300$$ 0 0
$$301$$ −43.5959 −2.51283
$$302$$ −9.79796 −0.563809
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ −11.7980 −0.675549
$$306$$ 0 0
$$307$$ 0.898979 0.0513075 0.0256537 0.999671i $$-0.491833\pi$$
0.0256537 + 0.999671i $$0.491833\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ 7.10102 0.402662 0.201331 0.979523i $$-0.435473\pi$$
0.201331 + 0.979523i $$0.435473\pi$$
$$312$$ 0 0
$$313$$ 6.89898 0.389953 0.194977 0.980808i $$-0.437537\pi$$
0.194977 + 0.980808i $$0.437537\pi$$
$$314$$ −1.10102 −0.0621342
$$315$$ 0 0
$$316$$ −5.79796 −0.326161
$$317$$ −14.0000 −0.786318 −0.393159 0.919470i $$-0.628618\pi$$
−0.393159 + 0.919470i $$0.628618\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ −19.5959 −1.09204
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ 6.89898 0.382687
$$326$$ −2.20204 −0.121960
$$327$$ 0 0
$$328$$ 2.89898 0.160069
$$329$$ 48.0000 2.64633
$$330$$ 0 0
$$331$$ −5.79796 −0.318685 −0.159342 0.987223i $$-0.550937\pi$$
−0.159342 + 0.987223i $$0.550937\pi$$
$$332$$ −13.7980 −0.757261
$$333$$ 0 0
$$334$$ −2.20204 −0.120490
$$335$$ −0.898979 −0.0491165
$$336$$ 0 0
$$337$$ 32.6969 1.78112 0.890558 0.454870i $$-0.150314\pi$$
0.890558 + 0.454870i $$0.150314\pi$$
$$338$$ 34.5959 1.88177
$$339$$ 0 0
$$340$$ −1.00000 −0.0542326
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −48.9898 −2.64520
$$344$$ 8.89898 0.479801
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ −21.7980 −1.17018 −0.585088 0.810970i $$-0.698940\pi$$
−0.585088 + 0.810970i $$0.698940\pi$$
$$348$$ 0 0
$$349$$ 4.20204 0.224930 0.112465 0.993656i $$-0.464125\pi$$
0.112465 + 0.993656i $$0.464125\pi$$
$$350$$ −4.89898 −0.261861
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 0 0
$$355$$ −8.89898 −0.472309
$$356$$ 7.79796 0.413291
$$357$$ 0 0
$$358$$ 4.89898 0.258919
$$359$$ −37.3939 −1.97357 −0.986787 0.162025i $$-0.948198\pi$$
−0.986787 + 0.162025i $$0.948198\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −4.20204 −0.220854
$$363$$ 0 0
$$364$$ −33.7980 −1.77149
$$365$$ 10.8990 0.570479
$$366$$ 0 0
$$367$$ −27.1010 −1.41466 −0.707331 0.706883i $$-0.750101\pi$$
−0.707331 + 0.706883i $$0.750101\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ −6.00000 −0.311925
$$371$$ −38.2020 −1.98335
$$372$$ 0 0
$$373$$ 24.6969 1.27876 0.639380 0.768891i $$-0.279191\pi$$
0.639380 + 0.768891i $$0.279191\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −9.79796 −0.505291
$$377$$ −41.3939 −2.13189
$$378$$ 0 0
$$379$$ −29.7980 −1.53062 −0.765309 0.643663i $$-0.777414\pi$$
−0.765309 + 0.643663i $$0.777414\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 0 0
$$382$$ 9.79796 0.501307
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1.10102 −0.0560405
$$387$$ 0 0
$$388$$ −12.6969 −0.644589
$$389$$ 38.4949 1.95177 0.975884 0.218288i $$-0.0700472\pi$$
0.975884 + 0.218288i $$0.0700472\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 17.0000 0.858630
$$393$$ 0 0
$$394$$ 13.5959 0.684952
$$395$$ 5.79796 0.291727
$$396$$ 0 0
$$397$$ 1.59592 0.0800968 0.0400484 0.999198i $$-0.487249\pi$$
0.0400484 + 0.999198i $$0.487249\pi$$
$$398$$ 15.5959 0.781753
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −22.8990 −1.14352 −0.571760 0.820421i $$-0.693739\pi$$
−0.571760 + 0.820421i $$0.693739\pi$$
$$402$$ 0 0
$$403$$ 27.5959 1.37465
$$404$$ 18.8990 0.940259
$$405$$ 0 0
$$406$$ 29.3939 1.45879
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −17.5959 −0.870062 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$410$$ −2.89898 −0.143170
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ 24.0000 1.18096
$$414$$ 0 0
$$415$$ 13.7980 0.677315
$$416$$ 6.89898 0.338250
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 17.7980 0.869487 0.434744 0.900554i $$-0.356839\pi$$
0.434744 + 0.900554i $$0.356839\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 0 0
$$424$$ 7.79796 0.378702
$$425$$ 1.00000 0.0485071
$$426$$ 0 0
$$427$$ −57.7980 −2.79704
$$428$$ −5.79796 −0.280255
$$429$$ 0 0
$$430$$ −8.89898 −0.429147
$$431$$ 23.1010 1.11274 0.556369 0.830936i $$-0.312194\pi$$
0.556369 + 0.830936i $$0.312194\pi$$
$$432$$ 0 0
$$433$$ −35.3939 −1.70092 −0.850461 0.526039i $$-0.823677\pi$$
−0.850461 + 0.526039i $$0.823677\pi$$
$$434$$ −19.5959 −0.940634
$$435$$ 0 0
$$436$$ 11.7980 0.565020
$$437$$ 16.0000 0.765384
$$438$$ 0 0
$$439$$ 5.79796 0.276721 0.138361 0.990382i $$-0.455817\pi$$
0.138361 + 0.990382i $$0.455817\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6.89898 0.328151
$$443$$ −37.7980 −1.79584 −0.897918 0.440164i $$-0.854920\pi$$
−0.897918 + 0.440164i $$0.854920\pi$$
$$444$$ 0 0
$$445$$ −7.79796 −0.369659
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ −4.89898 −0.231455
$$449$$ −6.89898 −0.325583 −0.162791 0.986660i $$-0.552050\pi$$
−0.162791 + 0.986660i $$0.552050\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −7.79796 −0.366785
$$453$$ 0 0
$$454$$ 21.7980 1.02303
$$455$$ 33.7980 1.58447
$$456$$ 0 0
$$457$$ 16.2020 0.757900 0.378950 0.925417i $$-0.376285\pi$$
0.378950 + 0.925417i $$0.376285\pi$$
$$458$$ −13.5959 −0.635296
$$459$$ 0 0
$$460$$ −4.00000 −0.186501
$$461$$ 28.6969 1.33655 0.668275 0.743914i $$-0.267033\pi$$
0.668275 + 0.743914i $$0.267033\pi$$
$$462$$ 0 0
$$463$$ 7.59592 0.353012 0.176506 0.984300i $$-0.443520\pi$$
0.176506 + 0.984300i $$0.443520\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ 7.59592 0.351497 0.175749 0.984435i $$-0.443765\pi$$
0.175749 + 0.984435i $$0.443765\pi$$
$$468$$ 0 0
$$469$$ −4.40408 −0.203362
$$470$$ 9.79796 0.451946
$$471$$ 0 0
$$472$$ −4.89898 −0.225494
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ −4.89898 −0.224544
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 32.8990 1.50319 0.751596 0.659623i $$-0.229284\pi$$
0.751596 + 0.659623i $$0.229284\pi$$
$$480$$ 0 0
$$481$$ 41.3939 1.88740
$$482$$ 13.5959 0.619277
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 12.6969 0.576538
$$486$$ 0 0
$$487$$ −20.8990 −0.947023 −0.473512 0.880788i $$-0.657014\pi$$
−0.473512 + 0.880788i $$0.657014\pi$$
$$488$$ 11.7980 0.534069
$$489$$ 0 0
$$490$$ −17.0000 −0.767982
$$491$$ 1.30306 0.0588063 0.0294032 0.999568i $$-0.490639\pi$$
0.0294032 + 0.999568i $$0.490639\pi$$
$$492$$ 0 0
$$493$$ −6.00000 −0.270226
$$494$$ 27.5959 1.24160
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ −43.5959 −1.95554
$$498$$ 0 0
$$499$$ −39.5959 −1.77256 −0.886278 0.463153i $$-0.846718\pi$$
−0.886278 + 0.463153i $$0.846718\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ −4.89898 −0.218652
$$503$$ −4.00000 −0.178351 −0.0891756 0.996016i $$-0.528423\pi$$
−0.0891756 + 0.996016i $$0.528423\pi$$
$$504$$ 0 0
$$505$$ −18.8990 −0.840994
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −12.0000 −0.532414
$$509$$ 15.3031 0.678296 0.339148 0.940733i $$-0.389861\pi$$
0.339148 + 0.940733i $$0.389861\pi$$
$$510$$ 0 0
$$511$$ 53.3939 2.36201
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 2.00000 0.0882162
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −29.3939 −1.29149
$$519$$ 0 0
$$520$$ −6.89898 −0.302540
$$521$$ 40.2929 1.76526 0.882631 0.470066i $$-0.155770\pi$$
0.882631 + 0.470066i $$0.155770\pi$$
$$522$$ 0 0
$$523$$ 18.6969 0.817560 0.408780 0.912633i $$-0.365954\pi$$
0.408780 + 0.912633i $$0.365954\pi$$
$$524$$ −9.79796 −0.428026
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −7.79796 −0.338722
$$531$$ 0 0
$$532$$ −19.5959 −0.849591
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ 5.79796 0.250668
$$536$$ 0.898979 0.0388300
$$537$$ 0 0
$$538$$ −9.59592 −0.413709
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −7.79796 −0.335260 −0.167630 0.985850i $$-0.553611\pi$$
−0.167630 + 0.985850i $$0.553611\pi$$
$$542$$ −1.79796 −0.0772290
$$543$$ 0 0
$$544$$ 1.00000 0.0428746
$$545$$ −11.7980 −0.505369
$$546$$ 0 0
$$547$$ 0.404082 0.0172773 0.00863865 0.999963i $$-0.497250\pi$$
0.00863865 + 0.999963i $$0.497250\pi$$
$$548$$ −14.0000 −0.598050
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 28.4041 1.20786
$$554$$ 7.79796 0.331304
$$555$$ 0 0
$$556$$ 13.7980 0.585164
$$557$$ −19.7980 −0.838866 −0.419433 0.907786i $$-0.637771\pi$$
−0.419433 + 0.907786i $$0.637771\pi$$
$$558$$ 0 0
$$559$$ 61.3939 2.59668
$$560$$ 4.89898 0.207020
$$561$$ 0 0
$$562$$ −27.7980 −1.17259
$$563$$ −21.7980 −0.918674 −0.459337 0.888262i $$-0.651913\pi$$
−0.459337 + 0.888262i $$0.651913\pi$$
$$564$$ 0 0
$$565$$ 7.79796 0.328063
$$566$$ 23.5959 0.991810
$$567$$ 0 0
$$568$$ 8.89898 0.373393
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ 29.7980 1.24701 0.623503 0.781821i $$-0.285709\pi$$
0.623503 + 0.781821i $$0.285709\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −14.2020 −0.592782
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ −27.3939 −1.14042 −0.570211 0.821498i $$-0.693139\pi$$
−0.570211 + 0.821498i $$0.693139\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ 6.00000 0.249136
$$581$$ 67.5959 2.80435
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −10.8990 −0.451003
$$585$$ 0 0
$$586$$ −13.5959 −0.561642
$$587$$ 41.3939 1.70851 0.854254 0.519856i $$-0.174014\pi$$
0.854254 + 0.519856i $$0.174014\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 4.89898 0.201688
$$591$$ 0 0
$$592$$ 6.00000 0.246598
$$593$$ −1.59592 −0.0655365 −0.0327682 0.999463i $$-0.510432\pi$$
−0.0327682 + 0.999463i $$0.510432\pi$$
$$594$$ 0 0
$$595$$ 4.89898 0.200839
$$596$$ 18.8990 0.774132
$$597$$ 0 0
$$598$$ 27.5959 1.12848
$$599$$ −21.3939 −0.874130 −0.437065 0.899430i $$-0.643982\pi$$
−0.437065 + 0.899430i $$0.643982\pi$$
$$600$$ 0 0
$$601$$ 16.2020 0.660895 0.330448 0.943824i $$-0.392800\pi$$
0.330448 + 0.943824i $$0.392800\pi$$
$$602$$ −43.5959 −1.77684
$$603$$ 0 0
$$604$$ −9.79796 −0.398673
$$605$$ 11.0000 0.447214
$$606$$ 0 0
$$607$$ −32.4949 −1.31893 −0.659464 0.751736i $$-0.729217\pi$$
−0.659464 + 0.751736i $$0.729217\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ −11.7980 −0.477685
$$611$$ −67.5959 −2.73464
$$612$$ 0 0
$$613$$ −14.4949 −0.585443 −0.292722 0.956198i $$-0.594561\pi$$
−0.292722 + 0.956198i $$0.594561\pi$$
$$614$$ 0.898979 0.0362799
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 37.5959 1.51355 0.756777 0.653673i $$-0.226773\pi$$
0.756777 + 0.653673i $$0.226773\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ 7.10102 0.284725
$$623$$ −38.2020 −1.53053
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 6.89898 0.275739
$$627$$ 0 0
$$628$$ −1.10102 −0.0439355
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −5.79796 −0.230630
$$633$$ 0 0
$$634$$ −14.0000 −0.556011
$$635$$ 12.0000 0.476205
$$636$$ 0 0
$$637$$ 117.283 4.64691
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 20.6969 0.817480 0.408740 0.912651i $$-0.365968\pi$$
0.408740 + 0.912651i $$0.365968\pi$$
$$642$$ 0 0
$$643$$ 29.7980 1.17512 0.587558 0.809182i $$-0.300089\pi$$
0.587558 + 0.809182i $$0.300089\pi$$
$$644$$ −19.5959 −0.772187
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 6.89898 0.270600
$$651$$ 0 0
$$652$$ −2.20204 −0.0862386
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ 9.79796 0.382838
$$656$$ 2.89898 0.113186
$$657$$ 0 0
$$658$$ 48.0000 1.87123
$$659$$ −30.6969 −1.19578 −0.597891 0.801577i $$-0.703995\pi$$
−0.597891 + 0.801577i $$0.703995\pi$$
$$660$$ 0 0
$$661$$ −19.7980 −0.770051 −0.385026 0.922906i $$-0.625807\pi$$
−0.385026 + 0.922906i $$0.625807\pi$$
$$662$$ −5.79796 −0.225344
$$663$$ 0 0
$$664$$ −13.7980 −0.535465
$$665$$ 19.5959 0.759897
$$666$$ 0 0
$$667$$ −24.0000 −0.929284
$$668$$ −2.20204 −0.0851995
$$669$$ 0 0
$$670$$ −0.898979 −0.0347306
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −33.1010 −1.27595 −0.637975 0.770057i $$-0.720228\pi$$
−0.637975 + 0.770057i $$0.720228\pi$$
$$674$$ 32.6969 1.25944
$$675$$ 0 0
$$676$$ 34.5959 1.33061
$$677$$ 13.5959 0.522534 0.261267 0.965267i $$-0.415860\pi$$
0.261267 + 0.965267i $$0.415860\pi$$
$$678$$ 0 0
$$679$$ 62.2020 2.38710
$$680$$ −1.00000 −0.0383482
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 14.0000 0.534913
$$686$$ −48.9898 −1.87044
$$687$$ 0 0
$$688$$ 8.89898 0.339270
$$689$$ 53.7980 2.04954
$$690$$ 0 0
$$691$$ −27.1918 −1.03443 −0.517213 0.855857i $$-0.673031\pi$$
−0.517213 + 0.855857i $$0.673031\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ −21.7980 −0.827439
$$695$$ −13.7980 −0.523386
$$696$$ 0 0
$$697$$ 2.89898 0.109807
$$698$$ 4.20204 0.159050
$$699$$ 0 0
$$700$$ −4.89898 −0.185164
$$701$$ −14.8990 −0.562727 −0.281363 0.959601i $$-0.590787\pi$$
−0.281363 + 0.959601i $$0.590787\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ −92.5857 −3.48204
$$708$$ 0 0
$$709$$ −31.7980 −1.19420 −0.597099 0.802168i $$-0.703680\pi$$
−0.597099 + 0.802168i $$0.703680\pi$$
$$710$$ −8.89898 −0.333973
$$711$$ 0 0
$$712$$ 7.79796 0.292241
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.89898 0.183083
$$717$$ 0 0
$$718$$ −37.3939 −1.39553
$$719$$ 12.4949 0.465981 0.232991 0.972479i $$-0.425149\pi$$
0.232991 + 0.972479i $$0.425149\pi$$
$$720$$ 0 0
$$721$$ −19.5959 −0.729790
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ −4.20204 −0.156168
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ 0.404082 0.0149866 0.00749329 0.999972i $$-0.497615\pi$$
0.00749329 + 0.999972i $$0.497615\pi$$
$$728$$ −33.7980 −1.25264
$$729$$ 0 0
$$730$$ 10.8990 0.403389
$$731$$ 8.89898 0.329141
$$732$$ 0 0
$$733$$ −46.4949 −1.71733 −0.858664 0.512539i $$-0.828705\pi$$
−0.858664 + 0.512539i $$0.828705\pi$$
$$734$$ −27.1010 −1.00032
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −9.39388 −0.345559 −0.172780 0.984960i $$-0.555275\pi$$
−0.172780 + 0.984960i $$0.555275\pi$$
$$740$$ −6.00000 −0.220564
$$741$$ 0 0
$$742$$ −38.2020 −1.40244
$$743$$ 9.39388 0.344628 0.172314 0.985042i $$-0.444876\pi$$
0.172314 + 0.985042i $$0.444876\pi$$
$$744$$ 0 0
$$745$$ −18.8990 −0.692405
$$746$$ 24.6969 0.904219
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 28.4041 1.03786
$$750$$ 0 0
$$751$$ 21.7980 0.795419 0.397709 0.917511i $$-0.369805\pi$$
0.397709 + 0.917511i $$0.369805\pi$$
$$752$$ −9.79796 −0.357295
$$753$$ 0 0
$$754$$ −41.3939 −1.50748
$$755$$ 9.79796 0.356584
$$756$$ 0 0
$$757$$ −4.69694 −0.170713 −0.0853566 0.996350i $$-0.527203\pi$$
−0.0853566 + 0.996350i $$0.527203\pi$$
$$758$$ −29.7980 −1.08231
$$759$$ 0 0
$$760$$ −4.00000 −0.145095
$$761$$ 1.59592 0.0578520 0.0289260 0.999582i $$-0.490791\pi$$
0.0289260 + 0.999582i $$0.490791\pi$$
$$762$$ 0 0
$$763$$ −57.7980 −2.09243
$$764$$ 9.79796 0.354478
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ −33.7980 −1.22037
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −1.10102 −0.0396266
$$773$$ 35.3939 1.27303 0.636515 0.771265i $$-0.280375\pi$$
0.636515 + 0.771265i $$0.280375\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ −12.6969 −0.455794
$$777$$ 0 0
$$778$$ 38.4949 1.38011
$$779$$ 11.5959 0.415467
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 4.00000 0.143040
$$783$$ 0 0
$$784$$ 17.0000 0.607143
$$785$$ 1.10102 0.0392971
$$786$$ 0 0
$$787$$ −45.7980 −1.63252 −0.816260 0.577684i $$-0.803957\pi$$
−0.816260 + 0.577684i $$0.803957\pi$$
$$788$$ 13.5959 0.484335
$$789$$ 0 0
$$790$$ 5.79796 0.206282
$$791$$ 38.2020 1.35831
$$792$$ 0 0
$$793$$ 81.3939 2.89038
$$794$$ 1.59592 0.0566370
$$795$$ 0 0
$$796$$ 15.5959 0.552783
$$797$$ −26.0000 −0.920967 −0.460484 0.887668i $$-0.652324\pi$$
−0.460484 + 0.887668i $$0.652324\pi$$
$$798$$ 0 0
$$799$$ −9.79796 −0.346627
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ −22.8990 −0.808591
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 19.5959 0.690665
$$806$$ 27.5959 0.972025
$$807$$ 0 0
$$808$$ 18.8990 0.664864
$$809$$ −32.6969 −1.14956 −0.574782 0.818307i $$-0.694913\pi$$
−0.574782 + 0.818307i $$0.694913\pi$$
$$810$$ 0 0
$$811$$ 25.3939 0.891700 0.445850 0.895108i $$-0.352902\pi$$
0.445850 + 0.895108i $$0.352902\pi$$
$$812$$ 29.3939 1.03152
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 2.20204 0.0771341
$$816$$ 0 0
$$817$$ 35.5959 1.24534
$$818$$ −17.5959 −0.615227
$$819$$ 0 0
$$820$$ −2.89898 −0.101237
$$821$$ −15.7980 −0.551353 −0.275676 0.961251i $$-0.588902\pi$$
−0.275676 + 0.961251i $$0.588902\pi$$
$$822$$ 0 0
$$823$$ 20.8990 0.728493 0.364246 0.931303i $$-0.381327\pi$$
0.364246 + 0.931303i $$0.381327\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 4.00000 0.139094 0.0695468 0.997579i $$-0.477845\pi$$
0.0695468 + 0.997579i $$0.477845\pi$$
$$828$$ 0 0
$$829$$ 3.39388 0.117874 0.0589371 0.998262i $$-0.481229\pi$$
0.0589371 + 0.998262i $$0.481229\pi$$
$$830$$ 13.7980 0.478934
$$831$$ 0 0
$$832$$ 6.89898 0.239179
$$833$$ 17.0000 0.589015
$$834$$ 0 0
$$835$$ 2.20204 0.0762048
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 17.7980 0.614820
$$839$$ 7.10102 0.245154 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 14.0000 0.482472
$$843$$ 0 0
$$844$$ −12.0000 −0.413057
$$845$$ −34.5959 −1.19014
$$846$$ 0 0
$$847$$ 53.8888 1.85164
$$848$$ 7.79796 0.267783
$$849$$ 0 0
$$850$$ 1.00000 0.0342997
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ 11.3939 0.390119 0.195059 0.980791i $$-0.437510\pi$$
0.195059 + 0.980791i $$0.437510\pi$$
$$854$$ −57.7980 −1.97781
$$855$$ 0 0
$$856$$ −5.79796 −0.198170
$$857$$ 10.0000 0.341593 0.170797 0.985306i $$-0.445366\pi$$
0.170797 + 0.985306i $$0.445366\pi$$
$$858$$ 0 0
$$859$$ 5.79796 0.197824 0.0989119 0.995096i $$-0.468464\pi$$
0.0989119 + 0.995096i $$0.468464\pi$$
$$860$$ −8.89898 −0.303453
$$861$$ 0 0
$$862$$ 23.1010 0.786824
$$863$$ 45.3939 1.54523 0.772613 0.634878i $$-0.218949\pi$$
0.772613 + 0.634878i $$0.218949\pi$$
$$864$$ 0 0
$$865$$ 6.00000 0.204006
$$866$$ −35.3939 −1.20273
$$867$$ 0 0
$$868$$ −19.5959 −0.665129
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 6.20204 0.210148
$$872$$ 11.7980 0.399529
$$873$$ 0 0
$$874$$ 16.0000 0.541208
$$875$$ 4.89898 0.165616
$$876$$ 0 0
$$877$$ −31.3939 −1.06010 −0.530048 0.847968i $$-0.677826\pi$$
−0.530048 + 0.847968i $$0.677826\pi$$
$$878$$ 5.79796 0.195672
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −34.4949 −1.16216 −0.581081 0.813846i $$-0.697370\pi$$
−0.581081 + 0.813846i $$0.697370\pi$$
$$882$$ 0 0
$$883$$ 26.6969 0.898424 0.449212 0.893425i $$-0.351705\pi$$
0.449212 + 0.893425i $$0.351705\pi$$
$$884$$ 6.89898 0.232038
$$885$$ 0 0
$$886$$ −37.7980 −1.26985
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 58.7878 1.97168
$$890$$ −7.79796 −0.261388
$$891$$ 0 0
$$892$$ −4.00000 −0.133930
$$893$$ −39.1918 −1.31150
$$894$$ 0 0
$$895$$ −4.89898 −0.163755
$$896$$ −4.89898 −0.163663
$$897$$ 0 0
$$898$$ −6.89898 −0.230222
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 7.79796 0.259788
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −7.79796 −0.259356
$$905$$ 4.20204 0.139681
$$906$$ 0 0
$$907$$ −33.3939 −1.10883 −0.554413 0.832242i $$-0.687057\pi$$
−0.554413 + 0.832242i $$0.687057\pi$$
$$908$$ 21.7980 0.723391
$$909$$ 0 0
$$910$$ 33.7980 1.12039
$$911$$ −23.1010 −0.765371 −0.382685 0.923879i $$-0.625001\pi$$
−0.382685 + 0.923879i $$0.625001\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 16.2020 0.535916
$$915$$ 0 0
$$916$$ −13.5959 −0.449222
$$917$$ 48.0000 1.58510
$$918$$ 0 0
$$919$$ 1.79796 0.0593092 0.0296546 0.999560i $$-0.490559\pi$$
0.0296546 + 0.999560i $$0.490559\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 0 0
$$922$$ 28.6969 0.945083
$$923$$ 61.3939 2.02080
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ 7.59592 0.249617
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ −36.2929 −1.19073 −0.595365 0.803455i $$-0.702993\pi$$
−0.595365 + 0.803455i $$0.702993\pi$$
$$930$$ 0 0
$$931$$ 68.0000 2.22861
$$932$$ −14.0000 −0.458585
$$933$$ 0 0
$$934$$ 7.59592 0.248546
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 39.3939 1.28694 0.643471 0.765471i $$-0.277494\pi$$
0.643471 + 0.765471i $$0.277494\pi$$
$$938$$ −4.40408 −0.143798
$$939$$ 0 0
$$940$$ 9.79796 0.319574
$$941$$ 16.2020 0.528171 0.264086 0.964499i $$-0.414930\pi$$
0.264086 + 0.964499i $$0.414930\pi$$
$$942$$ 0 0
$$943$$ 11.5959 0.377615
$$944$$ −4.89898 −0.159448
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −21.7980 −0.708338 −0.354169 0.935181i $$-0.615236\pi$$
−0.354169 + 0.935181i $$0.615236\pi$$
$$948$$ 0 0
$$949$$ −75.1918 −2.44083
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ −4.89898 −0.158777
$$953$$ 41.1918 1.33433 0.667167 0.744908i $$-0.267507\pi$$
0.667167 + 0.744908i $$0.267507\pi$$
$$954$$ 0 0
$$955$$ −9.79796 −0.317055
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 32.8990 1.06292
$$959$$ 68.5857 2.21475
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 41.3939 1.33459
$$963$$ 0 0
$$964$$ 13.5959 0.437895
$$965$$ 1.10102 0.0354431
$$966$$ 0 0
$$967$$ −41.3939 −1.33114 −0.665569 0.746337i $$-0.731811\pi$$
−0.665569 + 0.746337i $$0.731811\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ 0 0
$$970$$ 12.6969 0.407674
$$971$$ −38.6969 −1.24184 −0.620922 0.783872i $$-0.713242\pi$$
−0.620922 + 0.783872i $$0.713242\pi$$
$$972$$ 0 0
$$973$$ −67.5959 −2.16703
$$974$$ −20.8990 −0.669646
$$975$$ 0 0
$$976$$ 11.7980 0.377643
$$977$$ 29.5959 0.946857 0.473429 0.880832i $$-0.343016\pi$$
0.473429 + 0.880832i $$0.343016\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −17.0000 −0.543045
$$981$$ 0 0
$$982$$ 1.30306 0.0415824
$$983$$ −9.39388 −0.299618 −0.149809 0.988715i $$-0.547866\pi$$
−0.149809 + 0.988715i $$0.547866\pi$$
$$984$$ 0 0
$$985$$ −13.5959 −0.433202
$$986$$ −6.00000 −0.191079
$$987$$ 0 0
$$988$$ 27.5959 0.877943
$$989$$ 35.5959 1.13188
$$990$$ 0 0
$$991$$ −15.5959 −0.495421 −0.247710 0.968834i $$-0.579678\pi$$
−0.247710 + 0.968834i $$0.579678\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ −43.5959 −1.38278
$$995$$ −15.5959 −0.494424
$$996$$ 0 0
$$997$$ −21.5959 −0.683950 −0.341975 0.939709i $$-0.611096\pi$$
−0.341975 + 0.939709i $$0.611096\pi$$
$$998$$ −39.5959 −1.25339
$$999$$ 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 1530.2.a.s.1.1 2
3.2 odd 2 510.2.a.h.1.1 2
5.4 even 2 7650.2.a.cu.1.2 2
12.11 even 2 4080.2.a.bq.1.2 2
15.2 even 4 2550.2.d.u.2449.1 4
15.8 even 4 2550.2.d.u.2449.4 4
15.14 odd 2 2550.2.a.bl.1.2 2
51.50 odd 2 8670.2.a.be.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.h.1.1 2 3.2 odd 2
1530.2.a.s.1.1 2 1.1 even 1 trivial
2550.2.a.bl.1.2 2 15.14 odd 2
2550.2.d.u.2449.1 4 15.2 even 4
2550.2.d.u.2449.4 4 15.8 even 4
4080.2.a.bq.1.2 2 12.11 even 2
7650.2.a.cu.1.2 2 5.4 even 2
8670.2.a.be.1.2 2 51.50 odd 2