# Properties

 Label 1520.2.a.q.1.2 Level $1520$ Weight $2$ Character 1520.1 Self dual yes Analytic conductor $12.137$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.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: $$12.1372611072$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.470683$$ of defining polynomial Character $$\chi$$ $$=$$ 1520.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.470683 q^{3} -1.00000 q^{5} -2.71982 q^{7} -2.77846 q^{9} +O(q^{10})$$ $$q+0.470683 q^{3} -1.00000 q^{5} -2.71982 q^{7} -2.77846 q^{9} +5.55691 q^{11} +2.02760 q^{13} -0.470683 q^{15} -3.77846 q^{17} -1.00000 q^{19} -1.28018 q^{21} +5.77846 q^{23} +1.00000 q^{25} -2.71982 q^{27} -5.66119 q^{29} -7.55691 q^{31} +2.61555 q^{33} +2.71982 q^{35} -3.75086 q^{37} +0.954357 q^{39} -12.6155 q^{41} +9.43965 q^{43} +2.77846 q^{45} -11.1138 q^{47} +0.397442 q^{49} -1.77846 q^{51} -8.85170 q^{53} -5.55691 q^{55} -0.470683 q^{57} -11.4526 q^{59} -10.6155 q^{61} +7.55691 q^{63} -2.02760 q^{65} +11.5845 q^{67} +2.71982 q^{69} -9.45264 q^{73} +0.470683 q^{75} -15.1138 q^{77} -8.94137 q^{79} +7.05520 q^{81} +4.94137 q^{83} +3.77846 q^{85} -2.66463 q^{87} +15.4948 q^{89} -5.51471 q^{91} -3.55691 q^{93} +1.00000 q^{95} +10.8647 q^{97} -15.4396 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 3 q^{5} + q^{7}+O(q^{10})$$ 3 * q + q^3 - 3 * q^5 + q^7 $$3 q + q^{3} - 3 q^{5} + q^{7} - 11 q^{13} - q^{15} - 3 q^{17} - 3 q^{19} - 13 q^{21} + 9 q^{23} + 3 q^{25} + q^{27} - 7 q^{29} - 6 q^{31} - 8 q^{33} - q^{35} - 20 q^{37} - 3 q^{39} - 22 q^{41} + 10 q^{43} + 12 q^{49} + 3 q^{51} - 7 q^{53} - q^{57} - 11 q^{59} - 16 q^{61} + 6 q^{63} + 11 q^{65} + q^{67} - q^{69} - 5 q^{73} + q^{75} - 12 q^{77} - 26 q^{79} - 13 q^{81} + 14 q^{83} + 3 q^{85} - 33 q^{87} - 6 q^{89} - 29 q^{91} + 6 q^{93} + 3 q^{95} + 8 q^{97} - 28 q^{99}+O(q^{100})$$ 3 * q + q^3 - 3 * q^5 + q^7 - 11 * q^13 - q^15 - 3 * q^17 - 3 * q^19 - 13 * q^21 + 9 * q^23 + 3 * q^25 + q^27 - 7 * q^29 - 6 * q^31 - 8 * q^33 - q^35 - 20 * q^37 - 3 * q^39 - 22 * q^41 + 10 * q^43 + 12 * q^49 + 3 * q^51 - 7 * q^53 - q^57 - 11 * q^59 - 16 * q^61 + 6 * q^63 + 11 * q^65 + q^67 - q^69 - 5 * q^73 + q^75 - 12 * q^77 - 26 * q^79 - 13 * q^81 + 14 * q^83 + 3 * q^85 - 33 * q^87 - 6 * q^89 - 29 * q^91 + 6 * q^93 + 3 * q^95 + 8 * q^97 - 28 * 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
$$3$$ 0.470683 0.271749 0.135875 0.990726i $$-0.456616\pi$$
0.135875 + 0.990726i $$0.456616\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.71982 −1.02800 −0.513998 0.857791i $$-0.671836\pi$$
−0.513998 + 0.857791i $$0.671836\pi$$
$$8$$ 0 0
$$9$$ −2.77846 −0.926152
$$10$$ 0 0
$$11$$ 5.55691 1.67547 0.837736 0.546075i $$-0.183879\pi$$
0.837736 + 0.546075i $$0.183879\pi$$
$$12$$ 0 0
$$13$$ 2.02760 0.562354 0.281177 0.959656i $$-0.409275\pi$$
0.281177 + 0.959656i $$0.409275\pi$$
$$14$$ 0 0
$$15$$ −0.470683 −0.121530
$$16$$ 0 0
$$17$$ −3.77846 −0.916410 −0.458205 0.888846i $$-0.651508\pi$$
−0.458205 + 0.888846i $$0.651508\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.28018 −0.279357
$$22$$ 0 0
$$23$$ 5.77846 1.20489 0.602446 0.798160i $$-0.294193\pi$$
0.602446 + 0.798160i $$0.294193\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −2.71982 −0.523430
$$28$$ 0 0
$$29$$ −5.66119 −1.05126 −0.525628 0.850714i $$-0.676170\pi$$
−0.525628 + 0.850714i $$0.676170\pi$$
$$30$$ 0 0
$$31$$ −7.55691 −1.35726 −0.678631 0.734479i $$-0.737426\pi$$
−0.678631 + 0.734479i $$0.737426\pi$$
$$32$$ 0 0
$$33$$ 2.61555 0.455308
$$34$$ 0 0
$$35$$ 2.71982 0.459734
$$36$$ 0 0
$$37$$ −3.75086 −0.616637 −0.308319 0.951283i $$-0.599766\pi$$
−0.308319 + 0.951283i $$0.599766\pi$$
$$38$$ 0 0
$$39$$ 0.954357 0.152819
$$40$$ 0 0
$$41$$ −12.6155 −1.97022 −0.985109 0.171932i $$-0.944999\pi$$
−0.985109 + 0.171932i $$0.944999\pi$$
$$42$$ 0 0
$$43$$ 9.43965 1.43953 0.719766 0.694216i $$-0.244249\pi$$
0.719766 + 0.694216i $$0.244249\pi$$
$$44$$ 0 0
$$45$$ 2.77846 0.414188
$$46$$ 0 0
$$47$$ −11.1138 −1.62112 −0.810559 0.585657i $$-0.800837\pi$$
−0.810559 + 0.585657i $$0.800837\pi$$
$$48$$ 0 0
$$49$$ 0.397442 0.0567775
$$50$$ 0 0
$$51$$ −1.77846 −0.249034
$$52$$ 0 0
$$53$$ −8.85170 −1.21587 −0.607937 0.793985i $$-0.708003\pi$$
−0.607937 + 0.793985i $$0.708003\pi$$
$$54$$ 0 0
$$55$$ −5.55691 −0.749294
$$56$$ 0 0
$$57$$ −0.470683 −0.0623435
$$58$$ 0 0
$$59$$ −11.4526 −1.49101 −0.745503 0.666502i $$-0.767791\pi$$
−0.745503 + 0.666502i $$0.767791\pi$$
$$60$$ 0 0
$$61$$ −10.6155 −1.35918 −0.679591 0.733591i $$-0.737843\pi$$
−0.679591 + 0.733591i $$0.737843\pi$$
$$62$$ 0 0
$$63$$ 7.55691 0.952082
$$64$$ 0 0
$$65$$ −2.02760 −0.251493
$$66$$ 0 0
$$67$$ 11.5845 1.41527 0.707637 0.706576i $$-0.249761\pi$$
0.707637 + 0.706576i $$0.249761\pi$$
$$68$$ 0 0
$$69$$ 2.71982 0.327428
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −9.45264 −1.10635 −0.553174 0.833066i $$-0.686583\pi$$
−0.553174 + 0.833066i $$0.686583\pi$$
$$74$$ 0 0
$$75$$ 0.470683 0.0543498
$$76$$ 0 0
$$77$$ −15.1138 −1.72238
$$78$$ 0 0
$$79$$ −8.94137 −1.00598 −0.502991 0.864292i $$-0.667767\pi$$
−0.502991 + 0.864292i $$0.667767\pi$$
$$80$$ 0 0
$$81$$ 7.05520 0.783911
$$82$$ 0 0
$$83$$ 4.94137 0.542385 0.271193 0.962525i $$-0.412582\pi$$
0.271193 + 0.962525i $$0.412582\pi$$
$$84$$ 0 0
$$85$$ 3.77846 0.409831
$$86$$ 0 0
$$87$$ −2.66463 −0.285678
$$88$$ 0 0
$$89$$ 15.4948 1.64245 0.821225 0.570604i $$-0.193291\pi$$
0.821225 + 0.570604i $$0.193291\pi$$
$$90$$ 0 0
$$91$$ −5.51471 −0.578099
$$92$$ 0 0
$$93$$ −3.55691 −0.368835
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 10.8647 1.10314 0.551571 0.834128i $$-0.314029\pi$$
0.551571 + 0.834128i $$0.314029\pi$$
$$98$$ 0 0
$$99$$ −15.4396 −1.55174
$$100$$ 0 0
$$101$$ 4.49828 0.447596 0.223798 0.974636i $$-0.428154\pi$$
0.223798 + 0.974636i $$0.428154\pi$$
$$102$$ 0 0
$$103$$ −4.36641 −0.430235 −0.215117 0.976588i $$-0.569013\pi$$
−0.215117 + 0.976588i $$0.569013\pi$$
$$104$$ 0 0
$$105$$ 1.28018 0.124932
$$106$$ 0 0
$$107$$ 1.64658 0.159181 0.0795906 0.996828i $$-0.474639\pi$$
0.0795906 + 0.996828i $$0.474639\pi$$
$$108$$ 0 0
$$109$$ −0.954357 −0.0914108 −0.0457054 0.998955i $$-0.514554\pi$$
−0.0457054 + 0.998955i $$0.514554\pi$$
$$110$$ 0 0
$$111$$ −1.76547 −0.167571
$$112$$ 0 0
$$113$$ 5.68879 0.535156 0.267578 0.963536i $$-0.413777\pi$$
0.267578 + 0.963536i $$0.413777\pi$$
$$114$$ 0 0
$$115$$ −5.77846 −0.538844
$$116$$ 0 0
$$117$$ −5.63359 −0.520826
$$118$$ 0 0
$$119$$ 10.2767 0.942067
$$120$$ 0 0
$$121$$ 19.8793 1.80721
$$122$$ 0 0
$$123$$ −5.93793 −0.535405
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 10.3043 0.914362 0.457181 0.889374i $$-0.348859\pi$$
0.457181 + 0.889374i $$0.348859\pi$$
$$128$$ 0 0
$$129$$ 4.44309 0.391192
$$130$$ 0 0
$$131$$ −3.11383 −0.272056 −0.136028 0.990705i $$-0.543434\pi$$
−0.136028 + 0.990705i $$0.543434\pi$$
$$132$$ 0 0
$$133$$ 2.71982 0.235839
$$134$$ 0 0
$$135$$ 2.71982 0.234085
$$136$$ 0 0
$$137$$ 13.4526 1.14934 0.574668 0.818386i $$-0.305131\pi$$
0.574668 + 0.818386i $$0.305131\pi$$
$$138$$ 0 0
$$139$$ −9.55691 −0.810607 −0.405303 0.914182i $$-0.632834\pi$$
−0.405303 + 0.914182i $$0.632834\pi$$
$$140$$ 0 0
$$141$$ −5.23109 −0.440538
$$142$$ 0 0
$$143$$ 11.2672 0.942209
$$144$$ 0 0
$$145$$ 5.66119 0.470136
$$146$$ 0 0
$$147$$ 0.187070 0.0154292
$$148$$ 0 0
$$149$$ 19.6121 1.60669 0.803343 0.595516i $$-0.203052\pi$$
0.803343 + 0.595516i $$0.203052\pi$$
$$150$$ 0 0
$$151$$ 17.0518 1.38765 0.693826 0.720143i $$-0.255924\pi$$
0.693826 + 0.720143i $$0.255924\pi$$
$$152$$ 0 0
$$153$$ 10.4983 0.848736
$$154$$ 0 0
$$155$$ 7.55691 0.606986
$$156$$ 0 0
$$157$$ −22.4983 −1.79556 −0.897779 0.440446i $$-0.854820\pi$$
−0.897779 + 0.440446i $$0.854820\pi$$
$$158$$ 0 0
$$159$$ −4.16635 −0.330413
$$160$$ 0 0
$$161$$ −15.7164 −1.23862
$$162$$ 0 0
$$163$$ −22.4362 −1.75734 −0.878670 0.477430i $$-0.841568\pi$$
−0.878670 + 0.477430i $$0.841568\pi$$
$$164$$ 0 0
$$165$$ −2.61555 −0.203620
$$166$$ 0 0
$$167$$ 6.69223 0.517860 0.258930 0.965896i $$-0.416630\pi$$
0.258930 + 0.965896i $$0.416630\pi$$
$$168$$ 0 0
$$169$$ −8.88885 −0.683758
$$170$$ 0 0
$$171$$ 2.77846 0.212474
$$172$$ 0 0
$$173$$ −2.86469 −0.217798 −0.108899 0.994053i $$-0.534733\pi$$
−0.108899 + 0.994053i $$0.534733\pi$$
$$174$$ 0 0
$$175$$ −2.71982 −0.205599
$$176$$ 0 0
$$177$$ −5.39057 −0.405180
$$178$$ 0 0
$$179$$ −1.38445 −0.103479 −0.0517394 0.998661i $$-0.516477\pi$$
−0.0517394 + 0.998661i $$0.516477\pi$$
$$180$$ 0 0
$$181$$ −20.8793 −1.55195 −0.775973 0.630766i $$-0.782741\pi$$
−0.775973 + 0.630766i $$0.782741\pi$$
$$182$$ 0 0
$$183$$ −4.99656 −0.369357
$$184$$ 0 0
$$185$$ 3.75086 0.275769
$$186$$ 0 0
$$187$$ −20.9966 −1.53542
$$188$$ 0 0
$$189$$ 7.39744 0.538085
$$190$$ 0 0
$$191$$ −19.2733 −1.39457 −0.697284 0.716795i $$-0.745608\pi$$
−0.697284 + 0.716795i $$0.745608\pi$$
$$192$$ 0 0
$$193$$ −8.01461 −0.576904 −0.288452 0.957494i $$-0.593141\pi$$
−0.288452 + 0.957494i $$0.593141\pi$$
$$194$$ 0 0
$$195$$ −0.954357 −0.0683429
$$196$$ 0 0
$$197$$ 16.8793 1.20260 0.601300 0.799023i $$-0.294650\pi$$
0.601300 + 0.799023i $$0.294650\pi$$
$$198$$ 0 0
$$199$$ −1.28018 −0.0907493 −0.0453746 0.998970i $$-0.514448\pi$$
−0.0453746 + 0.998970i $$0.514448\pi$$
$$200$$ 0 0
$$201$$ 5.45264 0.384599
$$202$$ 0 0
$$203$$ 15.3974 1.08069
$$204$$ 0 0
$$205$$ 12.6155 0.881108
$$206$$ 0 0
$$207$$ −16.0552 −1.11591
$$208$$ 0 0
$$209$$ −5.55691 −0.384380
$$210$$ 0 0
$$211$$ 1.54392 0.106288 0.0531441 0.998587i $$-0.483076\pi$$
0.0531441 + 0.998587i $$0.483076\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −9.43965 −0.643779
$$216$$ 0 0
$$217$$ 20.5535 1.39526
$$218$$ 0 0
$$219$$ −4.44920 −0.300649
$$220$$ 0 0
$$221$$ −7.66119 −0.515347
$$222$$ 0 0
$$223$$ 3.92332 0.262725 0.131363 0.991334i $$-0.458065\pi$$
0.131363 + 0.991334i $$0.458065\pi$$
$$224$$ 0 0
$$225$$ −2.77846 −0.185230
$$226$$ 0 0
$$227$$ −10.9069 −0.723916 −0.361958 0.932194i $$-0.617892\pi$$
−0.361958 + 0.932194i $$0.617892\pi$$
$$228$$ 0 0
$$229$$ 10.1725 0.672215 0.336108 0.941824i $$-0.390889\pi$$
0.336108 + 0.941824i $$0.390889\pi$$
$$230$$ 0 0
$$231$$ −7.11383 −0.468056
$$232$$ 0 0
$$233$$ −9.11383 −0.597067 −0.298533 0.954399i $$-0.596497\pi$$
−0.298533 + 0.954399i $$0.596497\pi$$
$$234$$ 0 0
$$235$$ 11.1138 0.724986
$$236$$ 0 0
$$237$$ −4.20855 −0.273375
$$238$$ 0 0
$$239$$ −20.1595 −1.30401 −0.652004 0.758216i $$-0.726071\pi$$
−0.652004 + 0.758216i $$0.726071\pi$$
$$240$$ 0 0
$$241$$ −4.82410 −0.310748 −0.155374 0.987856i $$-0.549658\pi$$
−0.155374 + 0.987856i $$0.549658\pi$$
$$242$$ 0 0
$$243$$ 11.4802 0.736457
$$244$$ 0 0
$$245$$ −0.397442 −0.0253917
$$246$$ 0 0
$$247$$ −2.02760 −0.129013
$$248$$ 0 0
$$249$$ 2.32582 0.147393
$$250$$ 0 0
$$251$$ 18.2277 1.15052 0.575260 0.817971i $$-0.304901\pi$$
0.575260 + 0.817971i $$0.304901\pi$$
$$252$$ 0 0
$$253$$ 32.1104 2.01876
$$254$$ 0 0
$$255$$ 1.77846 0.111371
$$256$$ 0 0
$$257$$ 28.5941 1.78365 0.891824 0.452382i $$-0.149426\pi$$
0.891824 + 0.452382i $$0.149426\pi$$
$$258$$ 0 0
$$259$$ 10.2017 0.633901
$$260$$ 0 0
$$261$$ 15.7294 0.973624
$$262$$ 0 0
$$263$$ 27.3776 1.68817 0.844087 0.536206i $$-0.180143\pi$$
0.844087 + 0.536206i $$0.180143\pi$$
$$264$$ 0 0
$$265$$ 8.85170 0.543755
$$266$$ 0 0
$$267$$ 7.29317 0.446334
$$268$$ 0 0
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ −15.8337 −0.961826 −0.480913 0.876768i $$-0.659695\pi$$
−0.480913 + 0.876768i $$0.659695\pi$$
$$272$$ 0 0
$$273$$ −2.59568 −0.157098
$$274$$ 0 0
$$275$$ 5.55691 0.335095
$$276$$ 0 0
$$277$$ −6.70683 −0.402975 −0.201487 0.979491i $$-0.564577\pi$$
−0.201487 + 0.979491i $$0.564577\pi$$
$$278$$ 0 0
$$279$$ 20.9966 1.25703
$$280$$ 0 0
$$281$$ 8.38101 0.499969 0.249985 0.968250i $$-0.419574\pi$$
0.249985 + 0.968250i $$0.419574\pi$$
$$282$$ 0 0
$$283$$ 20.2897 1.20610 0.603050 0.797704i $$-0.293952\pi$$
0.603050 + 0.797704i $$0.293952\pi$$
$$284$$ 0 0
$$285$$ 0.470683 0.0278809
$$286$$ 0 0
$$287$$ 34.3121 2.02538
$$288$$ 0 0
$$289$$ −2.72326 −0.160192
$$290$$ 0 0
$$291$$ 5.11383 0.299778
$$292$$ 0 0
$$293$$ −6.76041 −0.394947 −0.197474 0.980308i $$-0.563274\pi$$
−0.197474 + 0.980308i $$0.563274\pi$$
$$294$$ 0 0
$$295$$ 11.4526 0.666798
$$296$$ 0 0
$$297$$ −15.1138 −0.876993
$$298$$ 0 0
$$299$$ 11.7164 0.677576
$$300$$ 0 0
$$301$$ −25.6742 −1.47984
$$302$$ 0 0
$$303$$ 2.11727 0.121634
$$304$$ 0 0
$$305$$ 10.6155 0.607844
$$306$$ 0 0
$$307$$ −7.74398 −0.441972 −0.220986 0.975277i $$-0.570928\pi$$
−0.220986 + 0.975277i $$0.570928\pi$$
$$308$$ 0 0
$$309$$ −2.05520 −0.116916
$$310$$ 0 0
$$311$$ 11.0456 0.626341 0.313170 0.949697i $$-0.398609\pi$$
0.313170 + 0.949697i $$0.398609\pi$$
$$312$$ 0 0
$$313$$ 1.16291 0.0657315 0.0328658 0.999460i $$-0.489537\pi$$
0.0328658 + 0.999460i $$0.489537\pi$$
$$314$$ 0 0
$$315$$ −7.55691 −0.425784
$$316$$ 0 0
$$317$$ −21.8742 −1.22858 −0.614290 0.789080i $$-0.710557\pi$$
−0.614290 + 0.789080i $$0.710557\pi$$
$$318$$ 0 0
$$319$$ −31.4588 −1.76135
$$320$$ 0 0
$$321$$ 0.775019 0.0432574
$$322$$ 0 0
$$323$$ 3.77846 0.210239
$$324$$ 0 0
$$325$$ 2.02760 0.112471
$$326$$ 0 0
$$327$$ −0.449200 −0.0248408
$$328$$ 0 0
$$329$$ 30.2277 1.66650
$$330$$ 0 0
$$331$$ −14.6646 −0.806041 −0.403020 0.915191i $$-0.632040\pi$$
−0.403020 + 0.915191i $$0.632040\pi$$
$$332$$ 0 0
$$333$$ 10.4216 0.571100
$$334$$ 0 0
$$335$$ −11.5845 −0.632929
$$336$$ 0 0
$$337$$ −20.0958 −1.09469 −0.547344 0.836908i $$-0.684361\pi$$
−0.547344 + 0.836908i $$0.684361\pi$$
$$338$$ 0 0
$$339$$ 2.67762 0.145428
$$340$$ 0 0
$$341$$ −41.9931 −2.27406
$$342$$ 0 0
$$343$$ 17.9578 0.969630
$$344$$ 0 0
$$345$$ −2.71982 −0.146430
$$346$$ 0 0
$$347$$ 11.0586 0.593659 0.296829 0.954931i $$-0.404071\pi$$
0.296829 + 0.954931i $$0.404071\pi$$
$$348$$ 0 0
$$349$$ 8.11727 0.434507 0.217254 0.976115i $$-0.430290\pi$$
0.217254 + 0.976115i $$0.430290\pi$$
$$350$$ 0 0
$$351$$ −5.51471 −0.294353
$$352$$ 0 0
$$353$$ 12.0130 0.639387 0.319693 0.947521i $$-0.396420\pi$$
0.319693 + 0.947521i $$0.396420\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4.83709 0.256006
$$358$$ 0 0
$$359$$ 17.9509 0.947413 0.473707 0.880683i $$-0.342916\pi$$
0.473707 + 0.880683i $$0.342916\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 9.35685 0.491108
$$364$$ 0 0
$$365$$ 9.45264 0.494774
$$366$$ 0 0
$$367$$ 3.11383 0.162541 0.0812703 0.996692i $$-0.474102\pi$$
0.0812703 + 0.996692i $$0.474102\pi$$
$$368$$ 0 0
$$369$$ 35.0518 1.82472
$$370$$ 0 0
$$371$$ 24.0751 1.24991
$$372$$ 0 0
$$373$$ −21.8742 −1.13261 −0.566303 0.824197i $$-0.691627\pi$$
−0.566303 + 0.824197i $$0.691627\pi$$
$$374$$ 0 0
$$375$$ −0.470683 −0.0243060
$$376$$ 0 0
$$377$$ −11.4786 −0.591179
$$378$$ 0 0
$$379$$ 5.28018 0.271224 0.135612 0.990762i $$-0.456700\pi$$
0.135612 + 0.990762i $$0.456700\pi$$
$$380$$ 0 0
$$381$$ 4.85008 0.248477
$$382$$ 0 0
$$383$$ 16.1319 0.824300 0.412150 0.911116i $$-0.364778\pi$$
0.412150 + 0.911116i $$0.364778\pi$$
$$384$$ 0 0
$$385$$ 15.1138 0.770272
$$386$$ 0 0
$$387$$ −26.2277 −1.33323
$$388$$ 0 0
$$389$$ 3.43965 0.174397 0.0871985 0.996191i $$-0.472209\pi$$
0.0871985 + 0.996191i $$0.472209\pi$$
$$390$$ 0 0
$$391$$ −21.8337 −1.10418
$$392$$ 0 0
$$393$$ −1.46563 −0.0739311
$$394$$ 0 0
$$395$$ 8.94137 0.449889
$$396$$ 0 0
$$397$$ −1.29317 −0.0649021 −0.0324511 0.999473i $$-0.510331\pi$$
−0.0324511 + 0.999473i $$0.510331\pi$$
$$398$$ 0 0
$$399$$ 1.28018 0.0640890
$$400$$ 0 0
$$401$$ −35.1070 −1.75316 −0.876579 0.481258i $$-0.840180\pi$$
−0.876579 + 0.481258i $$0.840180\pi$$
$$402$$ 0 0
$$403$$ −15.3224 −0.763262
$$404$$ 0 0
$$405$$ −7.05520 −0.350575
$$406$$ 0 0
$$407$$ −20.8432 −1.03316
$$408$$ 0 0
$$409$$ −24.8793 −1.23020 −0.615101 0.788448i $$-0.710885\pi$$
−0.615101 + 0.788448i $$0.710885\pi$$
$$410$$ 0 0
$$411$$ 6.33193 0.312331
$$412$$ 0 0
$$413$$ 31.1492 1.53275
$$414$$ 0 0
$$415$$ −4.94137 −0.242562
$$416$$ 0 0
$$417$$ −4.49828 −0.220282
$$418$$ 0 0
$$419$$ 35.4328 1.73100 0.865502 0.500905i $$-0.167000\pi$$
0.865502 + 0.500905i $$0.167000\pi$$
$$420$$ 0 0
$$421$$ 14.2147 0.692780 0.346390 0.938091i $$-0.387407\pi$$
0.346390 + 0.938091i $$0.387407\pi$$
$$422$$ 0 0
$$423$$ 30.8793 1.50140
$$424$$ 0 0
$$425$$ −3.77846 −0.183282
$$426$$ 0 0
$$427$$ 28.8724 1.39723
$$428$$ 0 0
$$429$$ 5.30328 0.256045
$$430$$ 0 0
$$431$$ 24.4983 1.18004 0.590020 0.807388i $$-0.299120\pi$$
0.590020 + 0.807388i $$0.299120\pi$$
$$432$$ 0 0
$$433$$ 9.45426 0.454343 0.227171 0.973855i $$-0.427052\pi$$
0.227171 + 0.973855i $$0.427052\pi$$
$$434$$ 0 0
$$435$$ 2.66463 0.127759
$$436$$ 0 0
$$437$$ −5.77846 −0.276421
$$438$$ 0 0
$$439$$ −9.70340 −0.463118 −0.231559 0.972821i $$-0.574383\pi$$
−0.231559 + 0.972821i $$0.574383\pi$$
$$440$$ 0 0
$$441$$ −1.10428 −0.0525846
$$442$$ 0 0
$$443$$ 6.85008 0.325457 0.162729 0.986671i $$-0.447971\pi$$
0.162729 + 0.986671i $$0.447971\pi$$
$$444$$ 0 0
$$445$$ −15.4948 −0.734526
$$446$$ 0 0
$$447$$ 9.23109 0.436616
$$448$$ 0 0
$$449$$ −25.7655 −1.21595 −0.607974 0.793957i $$-0.708017\pi$$
−0.607974 + 0.793957i $$0.708017\pi$$
$$450$$ 0 0
$$451$$ −70.1035 −3.30105
$$452$$ 0 0
$$453$$ 8.02598 0.377093
$$454$$ 0 0
$$455$$ 5.51471 0.258534
$$456$$ 0 0
$$457$$ 0.400880 0.0187524 0.00937619 0.999956i $$-0.497015\pi$$
0.00937619 + 0.999956i $$0.497015\pi$$
$$458$$ 0 0
$$459$$ 10.2767 0.479677
$$460$$ 0 0
$$461$$ 23.1070 1.07620 0.538099 0.842882i $$-0.319143\pi$$
0.538099 + 0.842882i $$0.319143\pi$$
$$462$$ 0 0
$$463$$ −4.96735 −0.230852 −0.115426 0.993316i $$-0.536823\pi$$
−0.115426 + 0.993316i $$0.536823\pi$$
$$464$$ 0 0
$$465$$ 3.55691 0.164948
$$466$$ 0 0
$$467$$ −21.6190 −1.00041 −0.500204 0.865908i $$-0.666742\pi$$
−0.500204 + 0.865908i $$0.666742\pi$$
$$468$$ 0 0
$$469$$ −31.5078 −1.45490
$$470$$ 0 0
$$471$$ −10.5896 −0.487942
$$472$$ 0 0
$$473$$ 52.4553 2.41190
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 24.5941 1.12608
$$478$$ 0 0
$$479$$ −2.20855 −0.100911 −0.0504557 0.998726i $$-0.516067\pi$$
−0.0504557 + 0.998726i $$0.516067\pi$$
$$480$$ 0 0
$$481$$ −7.60523 −0.346769
$$482$$ 0 0
$$483$$ −7.39744 −0.336595
$$484$$ 0 0
$$485$$ −10.8647 −0.493340
$$486$$ 0 0
$$487$$ 11.4250 0.517718 0.258859 0.965915i $$-0.416654\pi$$
0.258859 + 0.965915i $$0.416654\pi$$
$$488$$ 0 0
$$489$$ −10.5604 −0.477556
$$490$$ 0 0
$$491$$ −23.6673 −1.06809 −0.534045 0.845456i $$-0.679329\pi$$
−0.534045 + 0.845456i $$0.679329\pi$$
$$492$$ 0 0
$$493$$ 21.3906 0.963383
$$494$$ 0 0
$$495$$ 15.4396 0.693961
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −3.99312 −0.178757 −0.0893784 0.995998i $$-0.528488\pi$$
−0.0893784 + 0.995998i $$0.528488\pi$$
$$500$$ 0 0
$$501$$ 3.14992 0.140728
$$502$$ 0 0
$$503$$ −0.338809 −0.0151068 −0.00755338 0.999971i $$-0.502404\pi$$
−0.00755338 + 0.999971i $$0.502404\pi$$
$$504$$ 0 0
$$505$$ −4.49828 −0.200171
$$506$$ 0 0
$$507$$ −4.18383 −0.185811
$$508$$ 0 0
$$509$$ −28.4914 −1.26286 −0.631430 0.775433i $$-0.717532\pi$$
−0.631430 + 0.775433i $$0.717532\pi$$
$$510$$ 0 0
$$511$$ 25.7095 1.13732
$$512$$ 0 0
$$513$$ 2.71982 0.120083
$$514$$ 0 0
$$515$$ 4.36641 0.192407
$$516$$ 0 0
$$517$$ −61.7586 −2.71614
$$518$$ 0 0
$$519$$ −1.34836 −0.0591865
$$520$$ 0 0
$$521$$ 41.4328 1.81520 0.907601 0.419833i $$-0.137911\pi$$
0.907601 + 0.419833i $$0.137911\pi$$
$$522$$ 0 0
$$523$$ 22.1741 0.969605 0.484802 0.874624i $$-0.338892\pi$$
0.484802 + 0.874624i $$0.338892\pi$$
$$524$$ 0 0
$$525$$ −1.28018 −0.0558715
$$526$$ 0 0
$$527$$ 28.5535 1.24381
$$528$$ 0 0
$$529$$ 10.3906 0.451764
$$530$$ 0 0
$$531$$ 31.8207 1.38090
$$532$$ 0 0
$$533$$ −25.5793 −1.10796
$$534$$ 0 0
$$535$$ −1.64658 −0.0711880
$$536$$ 0 0
$$537$$ −0.651639 −0.0281203
$$538$$ 0 0
$$539$$ 2.20855 0.0951291
$$540$$ 0 0
$$541$$ −7.61211 −0.327270 −0.163635 0.986521i $$-0.552322\pi$$
−0.163635 + 0.986521i $$0.552322\pi$$
$$542$$ 0 0
$$543$$ −9.82754 −0.421740
$$544$$ 0 0
$$545$$ 0.954357 0.0408801
$$546$$ 0 0
$$547$$ 18.3303 0.783748 0.391874 0.920019i $$-0.371827\pi$$
0.391874 + 0.920019i $$0.371827\pi$$
$$548$$ 0 0
$$549$$ 29.4948 1.25881
$$550$$ 0 0
$$551$$ 5.66119 0.241175
$$552$$ 0 0
$$553$$ 24.3189 1.03415
$$554$$ 0 0
$$555$$ 1.76547 0.0749399
$$556$$ 0 0
$$557$$ 36.7000 1.55503 0.777514 0.628866i $$-0.216481\pi$$
0.777514 + 0.628866i $$0.216481\pi$$
$$558$$ 0 0
$$559$$ 19.1398 0.809528
$$560$$ 0 0
$$561$$ −9.88273 −0.417249
$$562$$ 0 0
$$563$$ 16.1319 0.679877 0.339939 0.940448i $$-0.389594\pi$$
0.339939 + 0.940448i $$0.389594\pi$$
$$564$$ 0 0
$$565$$ −5.68879 −0.239329
$$566$$ 0 0
$$567$$ −19.1889 −0.805858
$$568$$ 0 0
$$569$$ −19.2051 −0.805120 −0.402560 0.915394i $$-0.631880\pi$$
−0.402560 + 0.915394i $$0.631880\pi$$
$$570$$ 0 0
$$571$$ 2.46907 0.103327 0.0516636 0.998665i $$-0.483548\pi$$
0.0516636 + 0.998665i $$0.483548\pi$$
$$572$$ 0 0
$$573$$ −9.07162 −0.378972
$$574$$ 0 0
$$575$$ 5.77846 0.240978
$$576$$ 0 0
$$577$$ 11.7233 0.488046 0.244023 0.969769i $$-0.421533\pi$$
0.244023 + 0.969769i $$0.421533\pi$$
$$578$$ 0 0
$$579$$ −3.77234 −0.156773
$$580$$ 0 0
$$581$$ −13.4396 −0.557571
$$582$$ 0 0
$$583$$ −49.1881 −2.03716
$$584$$ 0 0
$$585$$ 5.63359 0.232920
$$586$$ 0 0
$$587$$ 8.28973 0.342154 0.171077 0.985258i $$-0.445275\pi$$
0.171077 + 0.985258i $$0.445275\pi$$
$$588$$ 0 0
$$589$$ 7.55691 0.311377
$$590$$ 0 0
$$591$$ 7.94480 0.326806
$$592$$ 0 0
$$593$$ −31.5760 −1.29667 −0.648336 0.761354i $$-0.724535\pi$$
−0.648336 + 0.761354i $$0.724535\pi$$
$$594$$ 0 0
$$595$$ −10.2767 −0.421305
$$596$$ 0 0
$$597$$ −0.602558 −0.0246610
$$598$$ 0 0
$$599$$ −1.28629 −0.0525564 −0.0262782 0.999655i $$-0.508366\pi$$
−0.0262782 + 0.999655i $$0.508366\pi$$
$$600$$ 0 0
$$601$$ 38.8432 1.58445 0.792224 0.610231i $$-0.208923\pi$$
0.792224 + 0.610231i $$0.208923\pi$$
$$602$$ 0 0
$$603$$ −32.1871 −1.31076
$$604$$ 0 0
$$605$$ −19.8793 −0.808208
$$606$$ 0 0
$$607$$ −43.6888 −1.77327 −0.886637 0.462467i $$-0.846964\pi$$
−0.886637 + 0.462467i $$0.846964\pi$$
$$608$$ 0 0
$$609$$ 7.24732 0.293676
$$610$$ 0 0
$$611$$ −22.5344 −0.911643
$$612$$ 0 0
$$613$$ −30.8172 −1.24470 −0.622348 0.782741i $$-0.713821\pi$$
−0.622348 + 0.782741i $$0.713821\pi$$
$$614$$ 0 0
$$615$$ 5.93793 0.239440
$$616$$ 0 0
$$617$$ 10.4983 0.422645 0.211322 0.977416i $$-0.432223\pi$$
0.211322 + 0.977416i $$0.432223\pi$$
$$618$$ 0 0
$$619$$ −13.3224 −0.535472 −0.267736 0.963492i $$-0.586275\pi$$
−0.267736 + 0.963492i $$0.586275\pi$$
$$620$$ 0 0
$$621$$ −15.7164 −0.630677
$$622$$ 0 0
$$623$$ −42.1432 −1.68843
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −2.61555 −0.104455
$$628$$ 0 0
$$629$$ 14.1725 0.565093
$$630$$ 0 0
$$631$$ −1.21199 −0.0482486 −0.0241243 0.999709i $$-0.507680\pi$$
−0.0241243 + 0.999709i $$0.507680\pi$$
$$632$$ 0 0
$$633$$ 0.726700 0.0288837
$$634$$ 0 0
$$635$$ −10.3043 −0.408915
$$636$$ 0 0
$$637$$ 0.805853 0.0319291
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −49.6965 −1.96289 −0.981447 0.191732i $$-0.938589\pi$$
−0.981447 + 0.191732i $$0.938589\pi$$
$$642$$ 0 0
$$643$$ 39.0449 1.53978 0.769890 0.638177i $$-0.220311\pi$$
0.769890 + 0.638177i $$0.220311\pi$$
$$644$$ 0 0
$$645$$ −4.44309 −0.174946
$$646$$ 0 0
$$647$$ 28.4232 1.11743 0.558716 0.829359i $$-0.311294\pi$$
0.558716 + 0.829359i $$0.311294\pi$$
$$648$$ 0 0
$$649$$ −63.6413 −2.49814
$$650$$ 0 0
$$651$$ 9.67418 0.379161
$$652$$ 0 0
$$653$$ 29.0586 1.13715 0.568576 0.822631i $$-0.307494\pi$$
0.568576 + 0.822631i $$0.307494\pi$$
$$654$$ 0 0
$$655$$ 3.11383 0.121667
$$656$$ 0 0
$$657$$ 26.2637 1.02465
$$658$$ 0 0
$$659$$ −11.8957 −0.463392 −0.231696 0.972788i $$-0.574427\pi$$
−0.231696 + 0.972788i $$0.574427\pi$$
$$660$$ 0 0
$$661$$ 30.8923 1.20157 0.600785 0.799410i $$-0.294855\pi$$
0.600785 + 0.799410i $$0.294855\pi$$
$$662$$ 0 0
$$663$$ −3.60600 −0.140045
$$664$$ 0 0
$$665$$ −2.71982 −0.105470
$$666$$ 0 0
$$667$$ −32.7129 −1.26665
$$668$$ 0 0
$$669$$ 1.84664 0.0713953
$$670$$ 0 0
$$671$$ −58.9897 −2.27727
$$672$$ 0 0
$$673$$ 37.5354 1.44688 0.723442 0.690385i $$-0.242559\pi$$
0.723442 + 0.690385i $$0.242559\pi$$
$$674$$ 0 0
$$675$$ −2.71982 −0.104686
$$676$$ 0 0
$$677$$ −31.6466 −1.21628 −0.608138 0.793831i $$-0.708083\pi$$
−0.608138 + 0.793831i $$0.708083\pi$$
$$678$$ 0 0
$$679$$ −29.5500 −1.13403
$$680$$ 0 0
$$681$$ −5.13369 −0.196724
$$682$$ 0 0
$$683$$ −22.6233 −0.865656 −0.432828 0.901477i $$-0.642484\pi$$
−0.432828 + 0.901477i $$0.642484\pi$$
$$684$$ 0 0
$$685$$ −13.4526 −0.513999
$$686$$ 0 0
$$687$$ 4.78801 0.182674
$$688$$ 0 0
$$689$$ −17.9477 −0.683752
$$690$$ 0 0
$$691$$ −17.7655 −0.675830 −0.337915 0.941177i $$-0.609722\pi$$
−0.337915 + 0.941177i $$0.609722\pi$$
$$692$$ 0 0
$$693$$ 41.9931 1.59519
$$694$$ 0 0
$$695$$ 9.55691 0.362514
$$696$$ 0 0
$$697$$ 47.6673 1.80553
$$698$$ 0 0
$$699$$ −4.28973 −0.162252
$$700$$ 0 0
$$701$$ 25.6052 0.967096 0.483548 0.875318i $$-0.339348\pi$$
0.483548 + 0.875318i $$0.339348\pi$$
$$702$$ 0 0
$$703$$ 3.75086 0.141466
$$704$$ 0 0
$$705$$ 5.23109 0.197014
$$706$$ 0 0
$$707$$ −12.2345 −0.460127
$$708$$ 0 0
$$709$$ 23.2311 0.872462 0.436231 0.899835i $$-0.356313\pi$$
0.436231 + 0.899835i $$0.356313\pi$$
$$710$$ 0 0
$$711$$ 24.8432 0.931693
$$712$$ 0 0
$$713$$ −43.6673 −1.63535
$$714$$ 0 0
$$715$$ −11.2672 −0.421369
$$716$$ 0 0
$$717$$ −9.48873 −0.354363
$$718$$ 0 0
$$719$$ −24.9544 −0.930640 −0.465320 0.885142i $$-0.654061\pi$$
−0.465320 + 0.885142i $$0.654061\pi$$
$$720$$ 0 0
$$721$$ 11.8759 0.442280
$$722$$ 0 0
$$723$$ −2.27062 −0.0844454
$$724$$ 0 0
$$725$$ −5.66119 −0.210251
$$726$$ 0 0
$$727$$ 3.39744 0.126004 0.0630021 0.998013i $$-0.479933\pi$$
0.0630021 + 0.998013i $$0.479933\pi$$
$$728$$ 0 0
$$729$$ −15.7620 −0.583779
$$730$$ 0 0
$$731$$ −35.6673 −1.31920
$$732$$ 0 0
$$733$$ −9.66730 −0.357070 −0.178535 0.983934i $$-0.557136\pi$$
−0.178535 + 0.983934i $$0.557136\pi$$
$$734$$ 0 0
$$735$$ −0.187070 −0.00690016
$$736$$ 0 0
$$737$$ 64.3741 2.37125
$$738$$ 0 0
$$739$$ −17.0225 −0.626184 −0.313092 0.949723i $$-0.601365\pi$$
−0.313092 + 0.949723i $$0.601365\pi$$
$$740$$ 0 0
$$741$$ −0.954357 −0.0350592
$$742$$ 0 0
$$743$$ −8.57496 −0.314585 −0.157292 0.987552i $$-0.550277\pi$$
−0.157292 + 0.987552i $$0.550277\pi$$
$$744$$ 0 0
$$745$$ −19.6121 −0.718532
$$746$$ 0 0
$$747$$ −13.7294 −0.502332
$$748$$ 0 0
$$749$$ −4.47842 −0.163638
$$750$$ 0 0
$$751$$ −12.8310 −0.468209 −0.234104 0.972211i $$-0.575216\pi$$
−0.234104 + 0.972211i $$0.575216\pi$$
$$752$$ 0 0
$$753$$ 8.57946 0.312653
$$754$$ 0 0
$$755$$ −17.0518 −0.620577
$$756$$ 0 0
$$757$$ −35.8827 −1.30418 −0.652090 0.758142i $$-0.726108\pi$$
−0.652090 + 0.758142i $$0.726108\pi$$
$$758$$ 0 0
$$759$$ 15.1138 0.548597
$$760$$ 0 0
$$761$$ −30.1234 −1.09197 −0.545986 0.837794i $$-0.683845\pi$$
−0.545986 + 0.837794i $$0.683845\pi$$
$$762$$ 0 0
$$763$$ 2.59568 0.0939700
$$764$$ 0 0
$$765$$ −10.4983 −0.379566
$$766$$ 0 0
$$767$$ −23.2213 −0.838474
$$768$$ 0 0
$$769$$ −6.22154 −0.224355 −0.112177 0.993688i $$-0.535782\pi$$
−0.112177 + 0.993688i $$0.535782\pi$$
$$770$$ 0 0
$$771$$ 13.4588 0.484705
$$772$$ 0 0
$$773$$ 10.2362 0.368169 0.184084 0.982910i $$-0.441068\pi$$
0.184084 + 0.982910i $$0.441068\pi$$
$$774$$ 0 0
$$775$$ −7.55691 −0.271452
$$776$$ 0 0
$$777$$ 4.80176 0.172262
$$778$$ 0 0
$$779$$ 12.6155 0.451999
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 15.3974 0.550260
$$784$$ 0 0
$$785$$ 22.4983 0.802998
$$786$$ 0 0
$$787$$ 4.34654 0.154937 0.0774687 0.996995i $$-0.475316\pi$$
0.0774687 + 0.996995i $$0.475316\pi$$
$$788$$ 0 0
$$789$$ 12.8862 0.458760
$$790$$ 0 0
$$791$$ −15.4725 −0.550139
$$792$$ 0 0
$$793$$ −21.5241 −0.764342
$$794$$ 0 0
$$795$$ 4.16635 0.147765
$$796$$ 0 0
$$797$$ −30.1932 −1.06950 −0.534749 0.845011i $$-0.679594\pi$$
−0.534749 + 0.845011i $$0.679594\pi$$
$$798$$ 0 0
$$799$$ 41.9931 1.48561
$$800$$ 0 0
$$801$$ −43.0518 −1.52116
$$802$$ 0 0
$$803$$ −52.5275 −1.85366
$$804$$ 0 0
$$805$$ 15.7164 0.553930
$$806$$ 0 0
$$807$$ −6.58957 −0.231964
$$808$$ 0 0
$$809$$ −11.8077 −0.415136 −0.207568 0.978221i $$-0.566555\pi$$
−0.207568 + 0.978221i $$0.566555\pi$$
$$810$$ 0 0
$$811$$ −0.811111 −0.0284820 −0.0142410 0.999899i $$-0.504533\pi$$
−0.0142410 + 0.999899i $$0.504533\pi$$
$$812$$ 0 0
$$813$$ −7.45264 −0.261375
$$814$$ 0 0
$$815$$ 22.4362 0.785906
$$816$$ 0 0
$$817$$ −9.43965 −0.330251
$$818$$ 0 0
$$819$$ 15.3224 0.535407
$$820$$ 0 0
$$821$$ 40.6707 1.41942 0.709709 0.704495i $$-0.248826\pi$$
0.709709 + 0.704495i $$0.248826\pi$$
$$822$$ 0 0
$$823$$ 7.48185 0.260801 0.130401 0.991461i $$-0.458374\pi$$
0.130401 + 0.991461i $$0.458374\pi$$
$$824$$ 0 0
$$825$$ 2.61555 0.0910617
$$826$$ 0 0
$$827$$ 46.4346 1.61469 0.807344 0.590080i $$-0.200904\pi$$
0.807344 + 0.590080i $$0.200904\pi$$
$$828$$ 0 0
$$829$$ −10.7267 −0.372554 −0.186277 0.982497i $$-0.559642\pi$$
−0.186277 + 0.982497i $$0.559642\pi$$
$$830$$ 0 0
$$831$$ −3.15680 −0.109508
$$832$$ 0 0
$$833$$ −1.50172 −0.0520315
$$834$$ 0 0
$$835$$ −6.69223 −0.231594
$$836$$ 0 0
$$837$$ 20.5535 0.710432
$$838$$ 0 0
$$839$$ −10.1465 −0.350295 −0.175148 0.984542i $$-0.556040\pi$$
−0.175148 + 0.984542i $$0.556040\pi$$
$$840$$ 0 0
$$841$$ 3.04908 0.105141
$$842$$ 0 0
$$843$$ 3.94480 0.135866
$$844$$ 0 0
$$845$$ 8.88885 0.305786
$$846$$ 0 0
$$847$$ −54.0682 −1.85780
$$848$$ 0 0
$$849$$ 9.55004 0.327756
$$850$$ 0 0
$$851$$ −21.6742 −0.742981
$$852$$ 0 0
$$853$$ −26.1104 −0.894003 −0.447001 0.894533i $$-0.647508\pi$$
−0.447001 + 0.894533i $$0.647508\pi$$
$$854$$ 0 0
$$855$$ −2.77846 −0.0950212
$$856$$ 0 0
$$857$$ 20.6922 0.706833 0.353416 0.935466i $$-0.385020\pi$$
0.353416 + 0.935466i $$0.385020\pi$$
$$858$$ 0 0
$$859$$ −3.53093 −0.120474 −0.0602370 0.998184i $$-0.519186\pi$$
−0.0602370 + 0.998184i $$0.519186\pi$$
$$860$$ 0 0
$$861$$ 16.1501 0.550395
$$862$$ 0 0
$$863$$ 3.39906 0.115705 0.0578527 0.998325i $$-0.481575\pi$$
0.0578527 + 0.998325i $$0.481575\pi$$
$$864$$ 0 0
$$865$$ 2.86469 0.0974023
$$866$$ 0 0
$$867$$ −1.28179 −0.0435320
$$868$$ 0 0
$$869$$ −49.6864 −1.68550
$$870$$ 0 0
$$871$$ 23.4887 0.795885
$$872$$ 0 0
$$873$$ −30.1871 −1.02168
$$874$$ 0 0
$$875$$ 2.71982 0.0919468
$$876$$ 0 0
$$877$$ −0.422364 −0.0142622 −0.00713111 0.999975i $$-0.502270\pi$$
−0.00713111 + 0.999975i $$0.502270\pi$$
$$878$$ 0 0
$$879$$ −3.18201 −0.107327
$$880$$ 0 0
$$881$$ 16.5243 0.556716 0.278358 0.960477i $$-0.410210\pi$$
0.278358 + 0.960477i $$0.410210\pi$$
$$882$$ 0 0
$$883$$ −24.5535 −0.826290 −0.413145 0.910665i $$-0.635570\pi$$
−0.413145 + 0.910665i $$0.635570\pi$$
$$884$$ 0 0
$$885$$ 5.39057 0.181202
$$886$$ 0 0
$$887$$ 41.5975 1.39671 0.698354 0.715753i $$-0.253916\pi$$
0.698354 + 0.715753i $$0.253916\pi$$
$$888$$ 0 0
$$889$$ −28.0260 −0.939961
$$890$$ 0 0
$$891$$ 39.2051 1.31342
$$892$$ 0 0
$$893$$ 11.1138 0.371910
$$894$$ 0 0
$$895$$ 1.38445 0.0462771
$$896$$ 0 0
$$897$$ 5.51471 0.184131
$$898$$ 0 0
$$899$$ 42.7811 1.42683
$$900$$ 0 0
$$901$$ 33.4458 1.11424
$$902$$ 0 0
$$903$$ −12.0844 −0.402144
$$904$$ 0 0
$$905$$ 20.8793 0.694051
$$906$$ 0 0
$$907$$ −31.9294 −1.06020 −0.530100 0.847935i $$-0.677846\pi$$
−0.530100 + 0.847935i $$0.677846\pi$$
$$908$$ 0 0
$$909$$ −12.4983 −0.414542
$$910$$ 0 0
$$911$$ −26.9605 −0.893240 −0.446620 0.894724i $$-0.647372\pi$$
−0.446620 + 0.894724i $$0.647372\pi$$
$$912$$ 0 0
$$913$$ 27.4588 0.908752
$$914$$ 0 0
$$915$$ 4.99656 0.165181
$$916$$ 0 0
$$917$$ 8.46907 0.279673
$$918$$ 0 0
$$919$$ 0.394005 0.0129970 0.00649850 0.999979i $$-0.497931\pi$$
0.00649850 + 0.999979i $$0.497931\pi$$
$$920$$ 0 0
$$921$$ −3.64496 −0.120106
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −3.75086 −0.123327
$$926$$ 0 0
$$927$$ 12.1319 0.398463
$$928$$ 0 0
$$929$$ −32.9284 −1.08035 −0.540173 0.841554i $$-0.681641\pi$$
−0.540173 + 0.841554i $$0.681641\pi$$
$$930$$ 0 0
$$931$$ −0.397442 −0.0130256
$$932$$ 0 0
$$933$$ 5.19900 0.170208
$$934$$ 0 0
$$935$$ 20.9966 0.686661
$$936$$ 0 0
$$937$$ −26.3388 −0.860451 −0.430226 0.902721i $$-0.641566\pi$$
−0.430226 + 0.902721i $$0.641566\pi$$
$$938$$ 0 0
$$939$$ 0.547362 0.0178625
$$940$$ 0 0
$$941$$ 4.71982 0.153862 0.0769309 0.997036i $$-0.475488\pi$$
0.0769309 + 0.997036i $$0.475488\pi$$
$$942$$ 0 0
$$943$$ −72.8984 −2.37390
$$944$$ 0 0
$$945$$ −7.39744 −0.240639
$$946$$ 0 0
$$947$$ −53.9639 −1.75359 −0.876796 0.480863i $$-0.840323\pi$$
−0.876796 + 0.480863i $$0.840323\pi$$
$$948$$ 0 0
$$949$$ −19.1661 −0.622159
$$950$$ 0 0
$$951$$ −10.2958 −0.333866
$$952$$ 0 0
$$953$$ −13.0801 −0.423707 −0.211853 0.977301i $$-0.567950\pi$$
−0.211853 + 0.977301i $$0.567950\pi$$
$$954$$ 0 0
$$955$$ 19.2733 0.623669
$$956$$ 0 0
$$957$$ −14.8071 −0.478646
$$958$$ 0 0
$$959$$ −36.5888 −1.18151
$$960$$ 0 0
$$961$$ 26.1070 0.842160
$$962$$ 0 0
$$963$$ −4.57496 −0.147426
$$964$$ 0 0
$$965$$ 8.01461 0.257999
$$966$$ 0 0
$$967$$ −10.4914 −0.337381 −0.168690 0.985669i $$-0.553954\pi$$
−0.168690 + 0.985669i $$0.553954\pi$$
$$968$$ 0 0
$$969$$ 1.77846 0.0571323
$$970$$ 0 0
$$971$$ 12.4691 0.400151 0.200076 0.979780i $$-0.435881\pi$$
0.200076 + 0.979780i $$0.435881\pi$$
$$972$$ 0 0
$$973$$ 25.9931 0.833301
$$974$$ 0 0
$$975$$ 0.954357 0.0305639
$$976$$ 0 0
$$977$$ −52.4699 −1.67866 −0.839331 0.543621i $$-0.817053\pi$$
−0.839331 + 0.543621i $$0.817053\pi$$
$$978$$ 0 0
$$979$$ 86.1035 2.75188
$$980$$ 0 0
$$981$$ 2.65164 0.0846603
$$982$$ 0 0
$$983$$ −14.1939 −0.452717 −0.226358 0.974044i $$-0.572682\pi$$
−0.226358 + 0.974044i $$0.572682\pi$$
$$984$$ 0 0
$$985$$ −16.8793 −0.537819
$$986$$ 0 0
$$987$$ 14.2277 0.452871
$$988$$ 0 0
$$989$$ 54.5466 1.73448
$$990$$ 0 0
$$991$$ −19.4036 −0.616374 −0.308187 0.951326i $$-0.599722\pi$$
−0.308187 + 0.951326i $$0.599722\pi$$
$$992$$ 0 0
$$993$$ −6.90240 −0.219041
$$994$$ 0 0
$$995$$ 1.28018 0.0405843
$$996$$ 0 0
$$997$$ −8.14648 −0.258002 −0.129001 0.991644i $$-0.541177\pi$$
−0.129001 + 0.991644i $$0.541177\pi$$
$$998$$ 0 0
$$999$$ 10.2017 0.322767
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 1520.2.a.q.1.2 3
4.3 odd 2 760.2.a.i.1.2 3
5.4 even 2 7600.2.a.bp.1.2 3
8.3 odd 2 6080.2.a.bx.1.2 3
8.5 even 2 6080.2.a.br.1.2 3
12.11 even 2 6840.2.a.bm.1.3 3
20.3 even 4 3800.2.d.n.3649.3 6
20.7 even 4 3800.2.d.n.3649.4 6
20.19 odd 2 3800.2.a.w.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.2 3 4.3 odd 2
1520.2.a.q.1.2 3 1.1 even 1 trivial
3800.2.a.w.1.2 3 20.19 odd 2
3800.2.d.n.3649.3 6 20.3 even 4
3800.2.d.n.3649.4 6 20.7 even 4
6080.2.a.br.1.2 3 8.5 even 2
6080.2.a.bx.1.2 3 8.3 odd 2
6840.2.a.bm.1.3 3 12.11 even 2
7600.2.a.bp.1.2 3 5.4 even 2