# Properties

 Label 6080.2.a.cg.1.4 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.78292.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 10x^{2} + 8x + 18$$ x^4 - x^3 - 10*x^2 + 8*x + 18 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3040) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.68461$$ of defining polynomial Character $$\chi$$ $$=$$ 6080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.68461 q^{3} +1.00000 q^{5} +3.20715 q^{7} +4.20715 q^{9} +O(q^{10})$$ $$q+2.68461 q^{3} +1.00000 q^{5} +3.20715 q^{7} +4.20715 q^{9} -6.44787 q^{11} -3.76325 q^{13} +2.68461 q^{15} +5.65501 q^{17} +1.00000 q^{19} +8.60995 q^{21} +7.20715 q^{23} +1.00000 q^{25} +3.24072 q^{27} +2.12851 q^{29} +10.8957 q^{31} -17.3100 q^{33} +3.20715 q^{35} +1.89176 q^{37} -10.1029 q^{39} -6.48144 q^{41} +5.49293 q^{43} +4.20715 q^{45} -2.86216 q^{47} +3.28579 q^{49} +15.1815 q^{51} +3.60597 q^{53} -6.44787 q^{55} +2.68461 q^{57} -10.7672 q^{59} +6.44787 q^{61} +13.4929 q^{63} -3.76325 q^{65} -6.62541 q^{67} +19.3484 q^{69} +4.41429 q^{71} +13.1815 q^{73} +2.68461 q^{75} -20.6793 q^{77} -9.78352 q^{79} -3.92136 q^{81} +1.49293 q^{83} +5.65501 q^{85} +5.71421 q^{87} -3.94080 q^{89} -12.0693 q^{91} +29.2508 q^{93} +1.00000 q^{95} -5.09411 q^{97} -27.1271 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} + 4 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10})$$ 4 * q + q^3 + 4 * q^5 + 5 * q^7 + 9 * q^9 $$4 q + q^{3} + 4 q^{5} + 5 q^{7} + 9 q^{9} - 6 q^{11} - 5 q^{13} + q^{15} - 5 q^{17} + 4 q^{19} + 3 q^{21} + 21 q^{23} + 4 q^{25} + q^{27} + q^{29} + 4 q^{31} - 14 q^{33} + 5 q^{35} - 10 q^{37} + 7 q^{39} - 2 q^{41} + 6 q^{43} + 9 q^{45} + 24 q^{47} + 5 q^{49} + 13 q^{51} + 5 q^{53} - 6 q^{55} + q^{57} - 11 q^{59} + 6 q^{61} + 38 q^{63} - 5 q^{65} + 19 q^{67} + 7 q^{69} + 2 q^{71} + 5 q^{73} + q^{75} - 8 q^{77} - 4 q^{79} - 16 q^{81} - 10 q^{83} - 5 q^{85} + 31 q^{87} + 20 q^{89} - 5 q^{91} + 26 q^{93} + 4 q^{95} - 6 q^{97} - 14 q^{99}+O(q^{100})$$ 4 * q + q^3 + 4 * q^5 + 5 * q^7 + 9 * q^9 - 6 * q^11 - 5 * q^13 + q^15 - 5 * q^17 + 4 * q^19 + 3 * q^21 + 21 * q^23 + 4 * q^25 + q^27 + q^29 + 4 * q^31 - 14 * q^33 + 5 * q^35 - 10 * q^37 + 7 * q^39 - 2 * q^41 + 6 * q^43 + 9 * q^45 + 24 * q^47 + 5 * q^49 + 13 * q^51 + 5 * q^53 - 6 * q^55 + q^57 - 11 * q^59 + 6 * q^61 + 38 * q^63 - 5 * q^65 + 19 * q^67 + 7 * q^69 + 2 * q^71 + 5 * q^73 + q^75 - 8 * q^77 - 4 * q^79 - 16 * q^81 - 10 * q^83 - 5 * q^85 + 31 * q^87 + 20 * q^89 - 5 * q^91 + 26 * q^93 + 4 * q^95 - 6 * q^97 - 14 * 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
$$3$$ 2.68461 1.54996 0.774981 0.631985i $$-0.217759\pi$$
0.774981 + 0.631985i $$0.217759\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 3.20715 1.21219 0.606094 0.795393i $$-0.292736\pi$$
0.606094 + 0.795393i $$0.292736\pi$$
$$8$$ 0 0
$$9$$ 4.20715 1.40238
$$10$$ 0 0
$$11$$ −6.44787 −1.94410 −0.972052 0.234764i $$-0.924568\pi$$
−0.972052 + 0.234764i $$0.924568\pi$$
$$12$$ 0 0
$$13$$ −3.76325 −1.04374 −0.521869 0.853025i $$-0.674765\pi$$
−0.521869 + 0.853025i $$0.674765\pi$$
$$14$$ 0 0
$$15$$ 2.68461 0.693164
$$16$$ 0 0
$$17$$ 5.65501 1.37154 0.685771 0.727817i $$-0.259465\pi$$
0.685771 + 0.727817i $$0.259465\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 8.60995 1.87884
$$22$$ 0 0
$$23$$ 7.20715 1.50279 0.751397 0.659850i $$-0.229380\pi$$
0.751397 + 0.659850i $$0.229380\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 3.24072 0.623677
$$28$$ 0 0
$$29$$ 2.12851 0.395254 0.197627 0.980277i $$-0.436677\pi$$
0.197627 + 0.980277i $$0.436677\pi$$
$$30$$ 0 0
$$31$$ 10.8957 1.95693 0.978466 0.206409i $$-0.0661778\pi$$
0.978466 + 0.206409i $$0.0661778\pi$$
$$32$$ 0 0
$$33$$ −17.3100 −3.01329
$$34$$ 0 0
$$35$$ 3.20715 0.542107
$$36$$ 0 0
$$37$$ 1.89176 0.311003 0.155502 0.987836i $$-0.450301\pi$$
0.155502 + 0.987836i $$0.450301\pi$$
$$38$$ 0 0
$$39$$ −10.1029 −1.61776
$$40$$ 0 0
$$41$$ −6.48144 −1.01223 −0.506116 0.862466i $$-0.668919\pi$$
−0.506116 + 0.862466i $$0.668919\pi$$
$$42$$ 0 0
$$43$$ 5.49293 0.837665 0.418832 0.908064i $$-0.362440\pi$$
0.418832 + 0.908064i $$0.362440\pi$$
$$44$$ 0 0
$$45$$ 4.20715 0.627164
$$46$$ 0 0
$$47$$ −2.86216 −0.417489 −0.208744 0.977970i $$-0.566938\pi$$
−0.208744 + 0.977970i $$0.566938\pi$$
$$48$$ 0 0
$$49$$ 3.28579 0.469398
$$50$$ 0 0
$$51$$ 15.1815 2.12584
$$52$$ 0 0
$$53$$ 3.60597 0.495318 0.247659 0.968847i $$-0.420339\pi$$
0.247659 + 0.968847i $$0.420339\pi$$
$$54$$ 0 0
$$55$$ −6.44787 −0.869430
$$56$$ 0 0
$$57$$ 2.68461 0.355586
$$58$$ 0 0
$$59$$ −10.7672 −1.40177 −0.700887 0.713272i $$-0.747212\pi$$
−0.700887 + 0.713272i $$0.747212\pi$$
$$60$$ 0 0
$$61$$ 6.44787 0.825565 0.412782 0.910830i $$-0.364557\pi$$
0.412782 + 0.910830i $$0.364557\pi$$
$$62$$ 0 0
$$63$$ 13.4929 1.69995
$$64$$ 0 0
$$65$$ −3.76325 −0.466774
$$66$$ 0 0
$$67$$ −6.62541 −0.809423 −0.404712 0.914444i $$-0.632628\pi$$
−0.404712 + 0.914444i $$0.632628\pi$$
$$68$$ 0 0
$$69$$ 19.3484 2.32927
$$70$$ 0 0
$$71$$ 4.41429 0.523880 0.261940 0.965084i $$-0.415638\pi$$
0.261940 + 0.965084i $$0.415638\pi$$
$$72$$ 0 0
$$73$$ 13.1815 1.54278 0.771390 0.636363i $$-0.219562\pi$$
0.771390 + 0.636363i $$0.219562\pi$$
$$74$$ 0 0
$$75$$ 2.68461 0.309992
$$76$$ 0 0
$$77$$ −20.6793 −2.35662
$$78$$ 0 0
$$79$$ −9.78352 −1.10073 −0.550366 0.834924i $$-0.685512\pi$$
−0.550366 + 0.834924i $$0.685512\pi$$
$$80$$ 0 0
$$81$$ −3.92136 −0.435707
$$82$$ 0 0
$$83$$ 1.49293 0.163871 0.0819354 0.996638i $$-0.473890\pi$$
0.0819354 + 0.996638i $$0.473890\pi$$
$$84$$ 0 0
$$85$$ 5.65501 0.613372
$$86$$ 0 0
$$87$$ 5.71421 0.612628
$$88$$ 0 0
$$89$$ −3.94080 −0.417724 −0.208862 0.977945i $$-0.566976\pi$$
−0.208862 + 0.977945i $$0.566976\pi$$
$$90$$ 0 0
$$91$$ −12.0693 −1.26521
$$92$$ 0 0
$$93$$ 29.2508 3.03317
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −5.09411 −0.517228 −0.258614 0.965981i $$-0.583266\pi$$
−0.258614 + 0.965981i $$0.583266\pi$$
$$98$$ 0 0
$$99$$ −27.1271 −2.72638
$$100$$ 0 0
$$101$$ 3.49293 0.347560 0.173780 0.984785i $$-0.444402\pi$$
0.173780 + 0.984785i $$0.444402\pi$$
$$102$$ 0 0
$$103$$ 2.07467 0.204423 0.102211 0.994763i $$-0.467408\pi$$
0.102211 + 0.994763i $$0.467408\pi$$
$$104$$ 0 0
$$105$$ 8.60995 0.840245
$$106$$ 0 0
$$107$$ 6.62541 0.640503 0.320251 0.947333i $$-0.396233\pi$$
0.320251 + 0.947333i $$0.396233\pi$$
$$108$$ 0 0
$$109$$ 17.0242 1.63063 0.815313 0.579020i $$-0.196565\pi$$
0.815313 + 0.579020i $$0.196565\pi$$
$$110$$ 0 0
$$111$$ 5.07864 0.482043
$$112$$ 0 0
$$113$$ −15.6753 −1.47461 −0.737303 0.675562i $$-0.763901\pi$$
−0.737303 + 0.675562i $$0.763901\pi$$
$$114$$ 0 0
$$115$$ 7.20715 0.672070
$$116$$ 0 0
$$117$$ −15.8326 −1.46372
$$118$$ 0 0
$$119$$ 18.1365 1.66257
$$120$$ 0 0
$$121$$ 30.5750 2.77954
$$122$$ 0 0
$$123$$ −17.4002 −1.56892
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 14.8803 1.32041 0.660205 0.751085i $$-0.270469\pi$$
0.660205 + 0.751085i $$0.270469\pi$$
$$128$$ 0 0
$$129$$ 14.7464 1.29835
$$130$$ 0 0
$$131$$ 7.85067 0.685916 0.342958 0.939351i $$-0.388571\pi$$
0.342958 + 0.939351i $$0.388571\pi$$
$$132$$ 0 0
$$133$$ 3.20715 0.278095
$$134$$ 0 0
$$135$$ 3.24072 0.278917
$$136$$ 0 0
$$137$$ 11.4977 0.982317 0.491159 0.871070i $$-0.336574\pi$$
0.491159 + 0.871070i $$0.336574\pi$$
$$138$$ 0 0
$$139$$ 7.27645 0.617181 0.308590 0.951195i $$-0.400143\pi$$
0.308590 + 0.951195i $$0.400143\pi$$
$$140$$ 0 0
$$141$$ −7.68379 −0.647092
$$142$$ 0 0
$$143$$ 24.2650 2.02914
$$144$$ 0 0
$$145$$ 2.12851 0.176763
$$146$$ 0 0
$$147$$ 8.82107 0.727549
$$148$$ 0 0
$$149$$ 12.3577 1.01238 0.506192 0.862421i $$-0.331053\pi$$
0.506192 + 0.862421i $$0.331053\pi$$
$$150$$ 0 0
$$151$$ −16.3551 −1.33096 −0.665479 0.746416i $$-0.731773\pi$$
−0.665479 + 0.746416i $$0.731773\pi$$
$$152$$ 0 0
$$153$$ 23.7915 1.92343
$$154$$ 0 0
$$155$$ 10.8957 0.875166
$$156$$ 0 0
$$157$$ −23.4769 −1.87366 −0.936830 0.349784i $$-0.886255\pi$$
−0.936830 + 0.349784i $$0.886255\pi$$
$$158$$ 0 0
$$159$$ 9.68064 0.767725
$$160$$ 0 0
$$161$$ 23.1144 1.82167
$$162$$ 0 0
$$163$$ −3.40280 −0.266528 −0.133264 0.991081i $$-0.542546\pi$$
−0.133264 + 0.991081i $$0.542546\pi$$
$$164$$ 0 0
$$165$$ −17.3100 −1.34758
$$166$$ 0 0
$$167$$ 18.0826 1.39927 0.699637 0.714498i $$-0.253345\pi$$
0.699637 + 0.714498i $$0.253345\pi$$
$$168$$ 0 0
$$169$$ 1.16208 0.0893907
$$170$$ 0 0
$$171$$ 4.20715 0.321729
$$172$$ 0 0
$$173$$ −8.14877 −0.619539 −0.309770 0.950812i $$-0.600252\pi$$
−0.309770 + 0.950812i $$0.600252\pi$$
$$174$$ 0 0
$$175$$ 3.20715 0.242437
$$176$$ 0 0
$$177$$ −28.9058 −2.17270
$$178$$ 0 0
$$179$$ 8.26496 0.617752 0.308876 0.951102i $$-0.400047\pi$$
0.308876 + 0.951102i $$0.400047\pi$$
$$180$$ 0 0
$$181$$ 0.0901336 0.00669958 0.00334979 0.999994i $$-0.498934\pi$$
0.00334979 + 0.999994i $$0.498934\pi$$
$$182$$ 0 0
$$183$$ 17.3100 1.27959
$$184$$ 0 0
$$185$$ 1.89176 0.139085
$$186$$ 0 0
$$187$$ −36.4628 −2.66642
$$188$$ 0 0
$$189$$ 10.3935 0.756013
$$190$$ 0 0
$$191$$ 0.916561 0.0663201 0.0331600 0.999450i $$-0.489443\pi$$
0.0331600 + 0.999450i $$0.489443\pi$$
$$192$$ 0 0
$$193$$ −17.7424 −1.27713 −0.638564 0.769569i $$-0.720471\pi$$
−0.638564 + 0.769569i $$0.720471\pi$$
$$194$$ 0 0
$$195$$ −10.1029 −0.723482
$$196$$ 0 0
$$197$$ −26.6121 −1.89603 −0.948017 0.318220i $$-0.896915\pi$$
−0.948017 + 0.318220i $$0.896915\pi$$
$$198$$ 0 0
$$199$$ 3.58786 0.254337 0.127168 0.991881i $$-0.459411\pi$$
0.127168 + 0.991881i $$0.459411\pi$$
$$200$$ 0 0
$$201$$ −17.7867 −1.25457
$$202$$ 0 0
$$203$$ 6.82643 0.479121
$$204$$ 0 0
$$205$$ −6.48144 −0.452683
$$206$$ 0 0
$$207$$ 30.3215 2.10749
$$208$$ 0 0
$$209$$ −6.44787 −0.446008
$$210$$ 0 0
$$211$$ 8.45266 0.581905 0.290953 0.956737i $$-0.406028\pi$$
0.290953 + 0.956737i $$0.406028\pi$$
$$212$$ 0 0
$$213$$ 11.8507 0.811994
$$214$$ 0 0
$$215$$ 5.49293 0.374615
$$216$$ 0 0
$$217$$ 34.9442 2.37217
$$218$$ 0 0
$$219$$ 35.3873 2.39125
$$220$$ 0 0
$$221$$ −21.2812 −1.43153
$$222$$ 0 0
$$223$$ −28.9191 −1.93657 −0.968285 0.249849i $$-0.919619\pi$$
−0.968285 + 0.249849i $$0.919619\pi$$
$$224$$ 0 0
$$225$$ 4.20715 0.280476
$$226$$ 0 0
$$227$$ −12.7518 −0.846364 −0.423182 0.906045i $$-0.639087\pi$$
−0.423182 + 0.906045i $$0.639087\pi$$
$$228$$ 0 0
$$229$$ 6.98851 0.461814 0.230907 0.972976i $$-0.425831\pi$$
0.230907 + 0.972976i $$0.425831\pi$$
$$230$$ 0 0
$$231$$ −55.5158 −3.65267
$$232$$ 0 0
$$233$$ −7.04507 −0.461538 −0.230769 0.973009i $$-0.574124\pi$$
−0.230769 + 0.973009i $$0.574124\pi$$
$$234$$ 0 0
$$235$$ −2.86216 −0.186707
$$236$$ 0 0
$$237$$ −26.2650 −1.70609
$$238$$ 0 0
$$239$$ −13.3388 −0.862815 −0.431408 0.902157i $$-0.641983\pi$$
−0.431408 + 0.902157i $$0.641983\pi$$
$$240$$ 0 0
$$241$$ −11.6342 −0.749424 −0.374712 0.927141i $$-0.622258\pi$$
−0.374712 + 0.927141i $$0.622258\pi$$
$$242$$ 0 0
$$243$$ −20.2495 −1.29901
$$244$$ 0 0
$$245$$ 3.28579 0.209921
$$246$$ 0 0
$$247$$ −3.76325 −0.239450
$$248$$ 0 0
$$249$$ 4.00795 0.253993
$$250$$ 0 0
$$251$$ −4.58117 −0.289161 −0.144580 0.989493i $$-0.546183\pi$$
−0.144580 + 0.989493i $$0.546183\pi$$
$$252$$ 0 0
$$253$$ −46.4707 −2.92159
$$254$$ 0 0
$$255$$ 15.1815 0.950704
$$256$$ 0 0
$$257$$ 5.67528 0.354014 0.177007 0.984210i $$-0.443358\pi$$
0.177007 + 0.984210i $$0.443358\pi$$
$$258$$ 0 0
$$259$$ 6.06715 0.376994
$$260$$ 0 0
$$261$$ 8.95493 0.554296
$$262$$ 0 0
$$263$$ −12.7049 −0.783416 −0.391708 0.920090i $$-0.628116\pi$$
−0.391708 + 0.920090i $$0.628116\pi$$
$$264$$ 0 0
$$265$$ 3.60597 0.221513
$$266$$ 0 0
$$267$$ −10.5795 −0.647456
$$268$$ 0 0
$$269$$ −4.54064 −0.276848 −0.138424 0.990373i $$-0.544204\pi$$
−0.138424 + 0.990373i $$0.544204\pi$$
$$270$$ 0 0
$$271$$ −3.62144 −0.219987 −0.109993 0.993932i $$-0.535083\pi$$
−0.109993 + 0.993932i $$0.535083\pi$$
$$272$$ 0 0
$$273$$ −32.4014 −1.96102
$$274$$ 0 0
$$275$$ −6.44787 −0.388821
$$276$$ 0 0
$$277$$ 18.5716 1.11586 0.557929 0.829889i $$-0.311596\pi$$
0.557929 + 0.829889i $$0.311596\pi$$
$$278$$ 0 0
$$279$$ 45.8399 2.74437
$$280$$ 0 0
$$281$$ −23.3100 −1.39056 −0.695280 0.718739i $$-0.744720\pi$$
−0.695280 + 0.718739i $$0.744720\pi$$
$$282$$ 0 0
$$283$$ 26.0741 1.54994 0.774972 0.631995i $$-0.217764\pi$$
0.774972 + 0.631995i $$0.217764\pi$$
$$284$$ 0 0
$$285$$ 2.68461 0.159023
$$286$$ 0 0
$$287$$ −20.7869 −1.22701
$$288$$ 0 0
$$289$$ 14.9792 0.881128
$$290$$ 0 0
$$291$$ −13.6757 −0.801684
$$292$$ 0 0
$$293$$ −14.6281 −0.854580 −0.427290 0.904115i $$-0.640532\pi$$
−0.427290 + 0.904115i $$0.640532\pi$$
$$294$$ 0 0
$$295$$ −10.7672 −0.626892
$$296$$ 0 0
$$297$$ −20.8957 −1.21249
$$298$$ 0 0
$$299$$ −27.1223 −1.56852
$$300$$ 0 0
$$301$$ 17.6166 1.01541
$$302$$ 0 0
$$303$$ 9.37717 0.538705
$$304$$ 0 0
$$305$$ 6.44787 0.369204
$$306$$ 0 0
$$307$$ 32.3802 1.84803 0.924017 0.382353i $$-0.124886\pi$$
0.924017 + 0.382353i $$0.124886\pi$$
$$308$$ 0 0
$$309$$ 5.56968 0.316848
$$310$$ 0 0
$$311$$ 14.6840 0.832656 0.416328 0.909214i $$-0.363317\pi$$
0.416328 + 0.909214i $$0.363317\pi$$
$$312$$ 0 0
$$313$$ −5.50568 −0.311199 −0.155600 0.987820i $$-0.549731\pi$$
−0.155600 + 0.987820i $$0.549731\pi$$
$$314$$ 0 0
$$315$$ 13.4929 0.760241
$$316$$ 0 0
$$317$$ −12.6014 −0.707767 −0.353884 0.935290i $$-0.615139\pi$$
−0.353884 + 0.935290i $$0.615139\pi$$
$$318$$ 0 0
$$319$$ −13.7243 −0.768414
$$320$$ 0 0
$$321$$ 17.7867 0.992755
$$322$$ 0 0
$$323$$ 5.65501 0.314653
$$324$$ 0 0
$$325$$ −3.76325 −0.208748
$$326$$ 0 0
$$327$$ 45.7035 2.52741
$$328$$ 0 0
$$329$$ −9.17936 −0.506075
$$330$$ 0 0
$$331$$ 28.1285 1.54608 0.773041 0.634356i $$-0.218735\pi$$
0.773041 + 0.634356i $$0.218735\pi$$
$$332$$ 0 0
$$333$$ 7.95891 0.436145
$$334$$ 0 0
$$335$$ −6.62541 −0.361985
$$336$$ 0 0
$$337$$ −12.2981 −0.669920 −0.334960 0.942232i $$-0.608723\pi$$
−0.334960 + 0.942232i $$0.608723\pi$$
$$338$$ 0 0
$$339$$ −42.0821 −2.28558
$$340$$ 0 0
$$341$$ −70.2542 −3.80448
$$342$$ 0 0
$$343$$ −11.9120 −0.643189
$$344$$ 0 0
$$345$$ 19.3484 1.04168
$$346$$ 0 0
$$347$$ −20.0149 −1.07446 −0.537228 0.843437i $$-0.680529\pi$$
−0.537228 + 0.843437i $$0.680529\pi$$
$$348$$ 0 0
$$349$$ −26.6121 −1.42451 −0.712257 0.701919i $$-0.752327\pi$$
−0.712257 + 0.701919i $$0.752327\pi$$
$$350$$ 0 0
$$351$$ −12.1957 −0.650956
$$352$$ 0 0
$$353$$ 20.4510 1.08850 0.544249 0.838924i $$-0.316815\pi$$
0.544249 + 0.838924i $$0.316815\pi$$
$$354$$ 0 0
$$355$$ 4.41429 0.234286
$$356$$ 0 0
$$357$$ 48.6894 2.57691
$$358$$ 0 0
$$359$$ 4.88299 0.257714 0.128857 0.991663i $$-0.458869\pi$$
0.128857 + 0.991663i $$0.458869\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 82.0820 4.30819
$$364$$ 0 0
$$365$$ 13.1815 0.689952
$$366$$ 0 0
$$367$$ −19.6086 −1.02356 −0.511779 0.859117i $$-0.671013\pi$$
−0.511779 + 0.859117i $$0.671013\pi$$
$$368$$ 0 0
$$369$$ −27.2684 −1.41953
$$370$$ 0 0
$$371$$ 11.5649 0.600419
$$372$$ 0 0
$$373$$ −4.71819 −0.244298 −0.122149 0.992512i $$-0.538979\pi$$
−0.122149 + 0.992512i $$0.538979\pi$$
$$374$$ 0 0
$$375$$ 2.68461 0.138633
$$376$$ 0 0
$$377$$ −8.01011 −0.412541
$$378$$ 0 0
$$379$$ 6.89358 0.354099 0.177050 0.984202i $$-0.443345\pi$$
0.177050 + 0.984202i $$0.443345\pi$$
$$380$$ 0 0
$$381$$ 39.9478 2.04659
$$382$$ 0 0
$$383$$ 18.2399 0.932015 0.466008 0.884781i $$-0.345692\pi$$
0.466008 + 0.884781i $$0.345692\pi$$
$$384$$ 0 0
$$385$$ −20.6793 −1.05391
$$386$$ 0 0
$$387$$ 23.1096 1.17473
$$388$$ 0 0
$$389$$ 6.04053 0.306267 0.153133 0.988206i $$-0.451064\pi$$
0.153133 + 0.988206i $$0.451064\pi$$
$$390$$ 0 0
$$391$$ 40.7565 2.06115
$$392$$ 0 0
$$393$$ 21.0760 1.06314
$$394$$ 0 0
$$395$$ −9.78352 −0.492262
$$396$$ 0 0
$$397$$ 11.6166 0.583023 0.291511 0.956567i $$-0.405842\pi$$
0.291511 + 0.956567i $$0.405842\pi$$
$$398$$ 0 0
$$399$$ 8.60995 0.431036
$$400$$ 0 0
$$401$$ 7.21030 0.360065 0.180033 0.983661i $$-0.442380\pi$$
0.180033 + 0.983661i $$0.442380\pi$$
$$402$$ 0 0
$$403$$ −41.0034 −2.04253
$$404$$ 0 0
$$405$$ −3.92136 −0.194854
$$406$$ 0 0
$$407$$ −12.1978 −0.604623
$$408$$ 0 0
$$409$$ −27.9893 −1.38398 −0.691990 0.721907i $$-0.743266\pi$$
−0.691990 + 0.721907i $$0.743266\pi$$
$$410$$ 0 0
$$411$$ 30.8670 1.52255
$$412$$ 0 0
$$413$$ −34.5321 −1.69921
$$414$$ 0 0
$$415$$ 1.49293 0.0732852
$$416$$ 0 0
$$417$$ 19.5345 0.956606
$$418$$ 0 0
$$419$$ 23.2278 1.13475 0.567377 0.823458i $$-0.307958\pi$$
0.567377 + 0.823458i $$0.307958\pi$$
$$420$$ 0 0
$$421$$ 9.74515 0.474949 0.237475 0.971394i $$-0.423680\pi$$
0.237475 + 0.971394i $$0.423680\pi$$
$$422$$ 0 0
$$423$$ −12.0415 −0.585479
$$424$$ 0 0
$$425$$ 5.65501 0.274308
$$426$$ 0 0
$$427$$ 20.6793 1.00074
$$428$$ 0 0
$$429$$ 65.1420 3.14509
$$430$$ 0 0
$$431$$ 7.53610 0.363001 0.181501 0.983391i $$-0.441905\pi$$
0.181501 + 0.983391i $$0.441905\pi$$
$$432$$ 0 0
$$433$$ −38.5480 −1.85250 −0.926250 0.376910i $$-0.876987\pi$$
−0.926250 + 0.376910i $$0.876987\pi$$
$$434$$ 0 0
$$435$$ 5.71421 0.273976
$$436$$ 0 0
$$437$$ 7.20715 0.344765
$$438$$ 0 0
$$439$$ 12.2554 0.584917 0.292458 0.956278i $$-0.405527\pi$$
0.292458 + 0.956278i $$0.405527\pi$$
$$440$$ 0 0
$$441$$ 13.8238 0.658276
$$442$$ 0 0
$$443$$ −17.9823 −0.854366 −0.427183 0.904165i $$-0.640494\pi$$
−0.427183 + 0.904165i $$0.640494\pi$$
$$444$$ 0 0
$$445$$ −3.94080 −0.186812
$$446$$ 0 0
$$447$$ 33.1757 1.56916
$$448$$ 0 0
$$449$$ −15.5265 −0.732741 −0.366371 0.930469i $$-0.619400\pi$$
−0.366371 + 0.930469i $$0.619400\pi$$
$$450$$ 0 0
$$451$$ 41.7915 1.96788
$$452$$ 0 0
$$453$$ −43.9071 −2.06294
$$454$$ 0 0
$$455$$ −12.0693 −0.565818
$$456$$ 0 0
$$457$$ −35.5136 −1.66126 −0.830629 0.556827i $$-0.812019\pi$$
−0.830629 + 0.556827i $$0.812019\pi$$
$$458$$ 0 0
$$459$$ 18.3263 0.855399
$$460$$ 0 0
$$461$$ −40.7543 −1.89812 −0.949060 0.315097i $$-0.897963\pi$$
−0.949060 + 0.315097i $$0.897963\pi$$
$$462$$ 0 0
$$463$$ 11.4433 0.531817 0.265908 0.963998i $$-0.414328\pi$$
0.265908 + 0.963998i $$0.414328\pi$$
$$464$$ 0 0
$$465$$ 29.2508 1.35647
$$466$$ 0 0
$$467$$ 37.6165 1.74068 0.870342 0.492447i $$-0.163898\pi$$
0.870342 + 0.492447i $$0.163898\pi$$
$$468$$ 0 0
$$469$$ −21.2487 −0.981172
$$470$$ 0 0
$$471$$ −63.0264 −2.90410
$$472$$ 0 0
$$473$$ −35.4177 −1.62851
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 15.1709 0.694626
$$478$$ 0 0
$$479$$ 27.3436 1.24936 0.624680 0.780880i $$-0.285229\pi$$
0.624680 + 0.780880i $$0.285229\pi$$
$$480$$ 0 0
$$481$$ −7.11917 −0.324606
$$482$$ 0 0
$$483$$ 62.0531 2.82352
$$484$$ 0 0
$$485$$ −5.09411 −0.231312
$$486$$ 0 0
$$487$$ −1.28661 −0.0583019 −0.0291509 0.999575i $$-0.509280\pi$$
−0.0291509 + 0.999575i $$0.509280\pi$$
$$488$$ 0 0
$$489$$ −9.13520 −0.413108
$$490$$ 0 0
$$491$$ −26.6296 −1.20178 −0.600890 0.799332i $$-0.705187\pi$$
−0.600890 + 0.799332i $$0.705187\pi$$
$$492$$ 0 0
$$493$$ 12.0367 0.542107
$$494$$ 0 0
$$495$$ −27.1271 −1.21927
$$496$$ 0 0
$$497$$ 14.1573 0.635041
$$498$$ 0 0
$$499$$ −18.7049 −0.837345 −0.418673 0.908137i $$-0.637505\pi$$
−0.418673 + 0.908137i $$0.637505\pi$$
$$500$$ 0 0
$$501$$ 48.5448 2.16882
$$502$$ 0 0
$$503$$ −18.8989 −0.842660 −0.421330 0.906907i $$-0.638437\pi$$
−0.421330 + 0.906907i $$0.638437\pi$$
$$504$$ 0 0
$$505$$ 3.49293 0.155434
$$506$$ 0 0
$$507$$ 3.11973 0.138552
$$508$$ 0 0
$$509$$ −1.91781 −0.0850057 −0.0425028 0.999096i $$-0.513533\pi$$
−0.0425028 + 0.999096i $$0.513533\pi$$
$$510$$ 0 0
$$511$$ 42.2751 1.87014
$$512$$ 0 0
$$513$$ 3.24072 0.143081
$$514$$ 0 0
$$515$$ 2.07467 0.0914207
$$516$$ 0 0
$$517$$ 18.4548 0.811642
$$518$$ 0 0
$$519$$ −21.8763 −0.960263
$$520$$ 0 0
$$521$$ −11.3330 −0.496508 −0.248254 0.968695i $$-0.579857\pi$$
−0.248254 + 0.968695i $$0.579857\pi$$
$$522$$ 0 0
$$523$$ −40.0431 −1.75096 −0.875482 0.483251i $$-0.839456\pi$$
−0.875482 + 0.483251i $$0.839456\pi$$
$$524$$ 0 0
$$525$$ 8.60995 0.375769
$$526$$ 0 0
$$527$$ 61.6155 2.68401
$$528$$ 0 0
$$529$$ 28.9430 1.25839
$$530$$ 0 0
$$531$$ −45.2993 −1.96582
$$532$$ 0 0
$$533$$ 24.3913 1.05650
$$534$$ 0 0
$$535$$ 6.62541 0.286442
$$536$$ 0 0
$$537$$ 22.1882 0.957492
$$538$$ 0 0
$$539$$ −21.1863 −0.912559
$$540$$ 0 0
$$541$$ 14.6131 0.628266 0.314133 0.949379i $$-0.398286\pi$$
0.314133 + 0.949379i $$0.398286\pi$$
$$542$$ 0 0
$$543$$ 0.241974 0.0103841
$$544$$ 0 0
$$545$$ 17.0242 0.729238
$$546$$ 0 0
$$547$$ −16.8616 −0.720950 −0.360475 0.932769i $$-0.617385\pi$$
−0.360475 + 0.932769i $$0.617385\pi$$
$$548$$ 0 0
$$549$$ 27.1271 1.15776
$$550$$ 0 0
$$551$$ 2.12851 0.0906774
$$552$$ 0 0
$$553$$ −31.3772 −1.33429
$$554$$ 0 0
$$555$$ 5.07864 0.215576
$$556$$ 0 0
$$557$$ −15.5186 −0.657542 −0.328771 0.944410i $$-0.606635\pi$$
−0.328771 + 0.944410i $$0.606635\pi$$
$$558$$ 0 0
$$559$$ −20.6713 −0.874303
$$560$$ 0 0
$$561$$ −97.8884 −4.13285
$$562$$ 0 0
$$563$$ −39.2354 −1.65357 −0.826787 0.562516i $$-0.809834\pi$$
−0.826787 + 0.562516i $$0.809834\pi$$
$$564$$ 0 0
$$565$$ −15.6753 −0.659464
$$566$$ 0 0
$$567$$ −12.5764 −0.528158
$$568$$ 0 0
$$569$$ 11.7323 0.491842 0.245921 0.969290i $$-0.420910\pi$$
0.245921 + 0.969290i $$0.420910\pi$$
$$570$$ 0 0
$$571$$ 10.1237 0.423664 0.211832 0.977306i $$-0.432057\pi$$
0.211832 + 0.977306i $$0.432057\pi$$
$$572$$ 0 0
$$573$$ 2.46061 0.102794
$$574$$ 0 0
$$575$$ 7.20715 0.300559
$$576$$ 0 0
$$577$$ 31.1628 1.29733 0.648663 0.761076i $$-0.275328\pi$$
0.648663 + 0.761076i $$0.275328\pi$$
$$578$$ 0 0
$$579$$ −47.6315 −1.97950
$$580$$ 0 0
$$581$$ 4.78806 0.198642
$$582$$ 0 0
$$583$$ −23.2508 −0.962951
$$584$$ 0 0
$$585$$ −15.8326 −0.654596
$$586$$ 0 0
$$587$$ 14.2084 0.586443 0.293222 0.956044i $$-0.405273\pi$$
0.293222 + 0.956044i $$0.405273\pi$$
$$588$$ 0 0
$$589$$ 10.8957 0.448951
$$590$$ 0 0
$$591$$ −71.4432 −2.93878
$$592$$ 0 0
$$593$$ −40.3055 −1.65515 −0.827574 0.561357i $$-0.810280\pi$$
−0.827574 + 0.561357i $$0.810280\pi$$
$$594$$ 0 0
$$595$$ 18.1365 0.743522
$$596$$ 0 0
$$597$$ 9.63203 0.394213
$$598$$ 0 0
$$599$$ 12.7694 0.521743 0.260871 0.965374i $$-0.415990\pi$$
0.260871 + 0.965374i $$0.415990\pi$$
$$600$$ 0 0
$$601$$ 44.6014 1.81933 0.909664 0.415345i $$-0.136339\pi$$
0.909664 + 0.415345i $$0.136339\pi$$
$$602$$ 0 0
$$603$$ −27.8741 −1.13512
$$604$$ 0 0
$$605$$ 30.5750 1.24305
$$606$$ 0 0
$$607$$ 33.2044 1.34773 0.673863 0.738856i $$-0.264634\pi$$
0.673863 + 0.738856i $$0.264634\pi$$
$$608$$ 0 0
$$609$$ 18.3263 0.742620
$$610$$ 0 0
$$611$$ 10.7710 0.435749
$$612$$ 0 0
$$613$$ 30.3038 1.22396 0.611980 0.790873i $$-0.290373\pi$$
0.611980 + 0.790873i $$0.290373\pi$$
$$614$$ 0 0
$$615$$ −17.4002 −0.701642
$$616$$ 0 0
$$617$$ −23.5078 −0.946390 −0.473195 0.880958i $$-0.656899\pi$$
−0.473195 + 0.880958i $$0.656899\pi$$
$$618$$ 0 0
$$619$$ 44.9336 1.80603 0.903017 0.429604i $$-0.141347\pi$$
0.903017 + 0.429604i $$0.141347\pi$$
$$620$$ 0 0
$$621$$ 23.3563 0.937258
$$622$$ 0 0
$$623$$ −12.6387 −0.506360
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −17.3100 −0.691296
$$628$$ 0 0
$$629$$ 10.6979 0.426554
$$630$$ 0 0
$$631$$ 12.4196 0.494417 0.247208 0.968962i $$-0.420487\pi$$
0.247208 + 0.968962i $$0.420487\pi$$
$$632$$ 0 0
$$633$$ 22.6921 0.901931
$$634$$ 0 0
$$635$$ 14.8803 0.590505
$$636$$ 0 0
$$637$$ −12.3653 −0.489929
$$638$$ 0 0
$$639$$ 18.5716 0.734680
$$640$$ 0 0
$$641$$ −50.4345 −1.99204 −0.996022 0.0891124i $$-0.971597\pi$$
−0.996022 + 0.0891124i $$0.971597\pi$$
$$642$$ 0 0
$$643$$ 3.61928 0.142731 0.0713653 0.997450i $$-0.477264\pi$$
0.0713653 + 0.997450i $$0.477264\pi$$
$$644$$ 0 0
$$645$$ 14.7464 0.580639
$$646$$ 0 0
$$647$$ −14.4810 −0.569305 −0.284653 0.958631i $$-0.591878\pi$$
−0.284653 + 0.958631i $$0.591878\pi$$
$$648$$ 0 0
$$649$$ 69.4257 2.72520
$$650$$ 0 0
$$651$$ 93.8117 3.67677
$$652$$ 0 0
$$653$$ −1.57776 −0.0617425 −0.0308712 0.999523i $$-0.509828\pi$$
−0.0308712 + 0.999523i $$0.509828\pi$$
$$654$$ 0 0
$$655$$ 7.85067 0.306751
$$656$$ 0 0
$$657$$ 55.4566 2.16357
$$658$$ 0 0
$$659$$ −36.9341 −1.43875 −0.719374 0.694623i $$-0.755571\pi$$
−0.719374 + 0.694623i $$0.755571\pi$$
$$660$$ 0 0
$$661$$ −10.5737 −0.411270 −0.205635 0.978629i $$-0.565926\pi$$
−0.205635 + 0.978629i $$0.565926\pi$$
$$662$$ 0 0
$$663$$ −57.1319 −2.21882
$$664$$ 0 0
$$665$$ 3.20715 0.124368
$$666$$ 0 0
$$667$$ 15.3404 0.593985
$$668$$ 0 0
$$669$$ −77.6367 −3.00161
$$670$$ 0 0
$$671$$ −41.5750 −1.60498
$$672$$ 0 0
$$673$$ −2.69736 −0.103976 −0.0519878 0.998648i $$-0.516556\pi$$
−0.0519878 + 0.998648i $$0.516556\pi$$
$$674$$ 0 0
$$675$$ 3.24072 0.124735
$$676$$ 0 0
$$677$$ −16.7341 −0.643143 −0.321572 0.946885i $$-0.604211\pi$$
−0.321572 + 0.946885i $$0.604211\pi$$
$$678$$ 0 0
$$679$$ −16.3375 −0.626978
$$680$$ 0 0
$$681$$ −34.2335 −1.31183
$$682$$ 0 0
$$683$$ 11.3847 0.435623 0.217812 0.975991i $$-0.430108\pi$$
0.217812 + 0.975991i $$0.430108\pi$$
$$684$$ 0 0
$$685$$ 11.4977 0.439306
$$686$$ 0 0
$$687$$ 18.7614 0.715793
$$688$$ 0 0
$$689$$ −13.5702 −0.516983
$$690$$ 0 0
$$691$$ −9.75789 −0.371208 −0.185604 0.982625i $$-0.559424\pi$$
−0.185604 + 0.982625i $$0.559424\pi$$
$$692$$ 0 0
$$693$$ −87.0006 −3.30488
$$694$$ 0 0
$$695$$ 7.27645 0.276012
$$696$$ 0 0
$$697$$ −36.6526 −1.38832
$$698$$ 0 0
$$699$$ −18.9133 −0.715366
$$700$$ 0 0
$$701$$ −13.0520 −0.492968 −0.246484 0.969147i $$-0.579275\pi$$
−0.246484 + 0.969147i $$0.579275\pi$$
$$702$$ 0 0
$$703$$ 1.89176 0.0713490
$$704$$ 0 0
$$705$$ −7.68379 −0.289388
$$706$$ 0 0
$$707$$ 11.2023 0.421308
$$708$$ 0 0
$$709$$ 9.46895 0.355614 0.177807 0.984065i $$-0.443100\pi$$
0.177807 + 0.984065i $$0.443100\pi$$
$$710$$ 0 0
$$711$$ −41.1607 −1.54365
$$712$$ 0 0
$$713$$ 78.5271 2.94086
$$714$$ 0 0
$$715$$ 24.2650 0.907458
$$716$$ 0 0
$$717$$ −35.8095 −1.33733
$$718$$ 0 0
$$719$$ −27.3228 −1.01897 −0.509484 0.860480i $$-0.670164\pi$$
−0.509484 + 0.860480i $$0.670164\pi$$
$$720$$ 0 0
$$721$$ 6.65376 0.247799
$$722$$ 0 0
$$723$$ −31.2333 −1.16158
$$724$$ 0 0
$$725$$ 2.12851 0.0790507
$$726$$ 0 0
$$727$$ 6.16208 0.228539 0.114269 0.993450i $$-0.463547\pi$$
0.114269 + 0.993450i $$0.463547\pi$$
$$728$$ 0 0
$$729$$ −42.5980 −1.57770
$$730$$ 0 0
$$731$$ 31.0626 1.14889
$$732$$ 0 0
$$733$$ −40.9804 −1.51365 −0.756823 0.653620i $$-0.773250\pi$$
−0.756823 + 0.653620i $$0.773250\pi$$
$$734$$ 0 0
$$735$$ 8.82107 0.325370
$$736$$ 0 0
$$737$$ 42.7198 1.57360
$$738$$ 0 0
$$739$$ 14.8612 0.546677 0.273338 0.961918i $$-0.411872\pi$$
0.273338 + 0.961918i $$0.411872\pi$$
$$740$$ 0 0
$$741$$ −10.1029 −0.371139
$$742$$ 0 0
$$743$$ −4.80517 −0.176285 −0.0881423 0.996108i $$-0.528093\pi$$
−0.0881423 + 0.996108i $$0.528093\pi$$
$$744$$ 0 0
$$745$$ 12.3577 0.452752
$$746$$ 0 0
$$747$$ 6.28099 0.229809
$$748$$ 0 0
$$749$$ 21.2487 0.776409
$$750$$ 0 0
$$751$$ −24.9363 −0.909937 −0.454969 0.890507i $$-0.650350\pi$$
−0.454969 + 0.890507i $$0.650350\pi$$
$$752$$ 0 0
$$753$$ −12.2987 −0.448188
$$754$$ 0 0
$$755$$ −16.3551 −0.595223
$$756$$ 0 0
$$757$$ 22.3134 0.810996 0.405498 0.914096i $$-0.367098\pi$$
0.405498 + 0.914096i $$0.367098\pi$$
$$758$$ 0 0
$$759$$ −124.756 −4.52835
$$760$$ 0 0
$$761$$ −43.6064 −1.58073 −0.790365 0.612636i $$-0.790109\pi$$
−0.790365 + 0.612636i $$0.790109\pi$$
$$762$$ 0 0
$$763$$ 54.5992 1.97662
$$764$$ 0 0
$$765$$ 23.7915 0.860182
$$766$$ 0 0
$$767$$ 40.5198 1.46309
$$768$$ 0 0
$$769$$ 7.09039 0.255686 0.127843 0.991794i $$-0.459195\pi$$
0.127843 + 0.991794i $$0.459195\pi$$
$$770$$ 0 0
$$771$$ 15.2359 0.548708
$$772$$ 0 0
$$773$$ 9.38949 0.337716 0.168858 0.985640i $$-0.445992\pi$$
0.168858 + 0.985640i $$0.445992\pi$$
$$774$$ 0 0
$$775$$ 10.8957 0.391386
$$776$$ 0 0
$$777$$ 16.2879 0.584327
$$778$$ 0 0
$$779$$ −6.48144 −0.232222
$$780$$ 0 0
$$781$$ −28.4628 −1.01848
$$782$$ 0 0
$$783$$ 6.89789 0.246511
$$784$$ 0 0
$$785$$ −23.4769 −0.837927
$$786$$ 0 0
$$787$$ −8.54867 −0.304727 −0.152364 0.988325i $$-0.548688\pi$$
−0.152364 + 0.988325i $$0.548688\pi$$
$$788$$ 0 0
$$789$$ −34.1077 −1.21427
$$790$$ 0 0
$$791$$ −50.2729 −1.78750
$$792$$ 0 0
$$793$$ −24.2650 −0.861674
$$794$$ 0 0
$$795$$ 9.68064 0.343337
$$796$$ 0 0
$$797$$ 37.5777 1.33107 0.665535 0.746366i $$-0.268203\pi$$
0.665535 + 0.746366i $$0.268203\pi$$
$$798$$ 0 0
$$799$$ −16.1855 −0.572604
$$800$$ 0 0
$$801$$ −16.5795 −0.585809
$$802$$ 0 0
$$803$$ −84.9927 −2.99933
$$804$$ 0 0
$$805$$ 23.1144 0.814675
$$806$$ 0 0
$$807$$ −12.1899 −0.429103
$$808$$ 0 0
$$809$$ −18.0773 −0.635562 −0.317781 0.948164i $$-0.602938\pi$$
−0.317781 + 0.948164i $$0.602938\pi$$
$$810$$ 0 0
$$811$$ 6.73629 0.236543 0.118272 0.992981i $$-0.462265\pi$$
0.118272 + 0.992981i $$0.462265\pi$$
$$812$$ 0 0
$$813$$ −9.72216 −0.340971
$$814$$ 0 0
$$815$$ −3.40280 −0.119195
$$816$$ 0 0
$$817$$ 5.49293 0.192173
$$818$$ 0 0
$$819$$ −50.7773 −1.77430
$$820$$ 0 0
$$821$$ −40.4803 −1.41277 −0.706386 0.707826i $$-0.749676\pi$$
−0.706386 + 0.707826i $$0.749676\pi$$
$$822$$ 0 0
$$823$$ −23.6390 −0.824003 −0.412002 0.911183i $$-0.635170\pi$$
−0.412002 + 0.911183i $$0.635170\pi$$
$$824$$ 0 0
$$825$$ −17.3100 −0.602658
$$826$$ 0 0
$$827$$ −10.4276 −0.362603 −0.181302 0.983428i $$-0.558031\pi$$
−0.181302 + 0.983428i $$0.558031\pi$$
$$828$$ 0 0
$$829$$ −50.1257 −1.74094 −0.870469 0.492223i $$-0.836185\pi$$
−0.870469 + 0.492223i $$0.836185\pi$$
$$830$$ 0 0
$$831$$ 49.8575 1.72954
$$832$$ 0 0
$$833$$ 18.5812 0.643799
$$834$$ 0 0
$$835$$ 18.0826 0.625775
$$836$$ 0 0
$$837$$ 35.3100 1.22049
$$838$$ 0 0
$$839$$ −12.3321 −0.425752 −0.212876 0.977079i $$-0.568283\pi$$
−0.212876 + 0.977079i $$0.568283\pi$$
$$840$$ 0 0
$$841$$ −24.4695 −0.843775
$$842$$ 0 0
$$843$$ −62.5784 −2.15531
$$844$$ 0 0
$$845$$ 1.16208 0.0399767
$$846$$ 0 0
$$847$$ 98.0584 3.36933
$$848$$ 0 0
$$849$$ 69.9989 2.40236
$$850$$ 0 0
$$851$$ 13.6342 0.467374
$$852$$ 0 0
$$853$$ −43.9205 −1.50381 −0.751904 0.659272i $$-0.770864\pi$$
−0.751904 + 0.659272i $$0.770864\pi$$
$$854$$ 0 0
$$855$$ 4.20715 0.143881
$$856$$ 0 0
$$857$$ −30.1488 −1.02986 −0.514931 0.857232i $$-0.672183\pi$$
−0.514931 + 0.857232i $$0.672183\pi$$
$$858$$ 0 0
$$859$$ 52.1140 1.77811 0.889053 0.457804i $$-0.151364\pi$$
0.889053 + 0.457804i $$0.151364\pi$$
$$860$$ 0 0
$$861$$ −55.8049 −1.90182
$$862$$ 0 0
$$863$$ −0.166447 −0.00566592 −0.00283296 0.999996i $$-0.500902\pi$$
−0.00283296 + 0.999996i $$0.500902\pi$$
$$864$$ 0 0
$$865$$ −8.14877 −0.277066
$$866$$ 0 0
$$867$$ 40.2133 1.36571
$$868$$ 0 0
$$869$$ 63.0828 2.13994
$$870$$ 0 0
$$871$$ 24.9331 0.844826
$$872$$ 0 0
$$873$$ −21.4317 −0.725352
$$874$$ 0 0
$$875$$ 3.20715 0.108421
$$876$$ 0 0
$$877$$ −1.66517 −0.0562289 −0.0281144 0.999605i $$-0.508950\pi$$
−0.0281144 + 0.999605i $$0.508950\pi$$
$$878$$ 0 0
$$879$$ −39.2707 −1.32457
$$880$$ 0 0
$$881$$ −37.2419 −1.25471 −0.627355 0.778733i $$-0.715863\pi$$
−0.627355 + 0.778733i $$0.715863\pi$$
$$882$$ 0 0
$$883$$ −48.5961 −1.63539 −0.817694 0.575653i $$-0.804748\pi$$
−0.817694 + 0.575653i $$0.804748\pi$$
$$884$$ 0 0
$$885$$ −28.9058 −0.971659
$$886$$ 0 0
$$887$$ −12.7619 −0.428502 −0.214251 0.976779i $$-0.568731\pi$$
−0.214251 + 0.976779i $$0.568731\pi$$
$$888$$ 0 0
$$889$$ 47.7232 1.60058
$$890$$ 0 0
$$891$$ 25.2844 0.847059
$$892$$ 0 0
$$893$$ −2.86216 −0.0957785
$$894$$ 0 0
$$895$$ 8.26496 0.276267
$$896$$ 0 0
$$897$$ −72.8129 −2.43115
$$898$$ 0 0
$$899$$ 23.1916 0.773484
$$900$$ 0 0
$$901$$ 20.3918 0.679350
$$902$$ 0 0
$$903$$ 47.2939 1.57384
$$904$$ 0 0
$$905$$ 0.0901336 0.00299614
$$906$$ 0 0
$$907$$ 14.7438 0.489560 0.244780 0.969579i $$-0.421284\pi$$
0.244780 + 0.969579i $$0.421284\pi$$
$$908$$ 0 0
$$909$$ 14.6953 0.487412
$$910$$ 0 0
$$911$$ 27.4177 0.908389 0.454195 0.890903i $$-0.349927\pi$$
0.454195 + 0.890903i $$0.349927\pi$$
$$912$$ 0 0
$$913$$ −9.62624 −0.318582
$$914$$ 0 0
$$915$$ 17.3100 0.572252
$$916$$ 0 0
$$917$$ 25.1782 0.831459
$$918$$ 0 0
$$919$$ 44.8066 1.47803 0.739017 0.673687i $$-0.235290\pi$$
0.739017 + 0.673687i $$0.235290\pi$$
$$920$$ 0 0
$$921$$ 86.9282 2.86438
$$922$$ 0 0
$$923$$ −16.6121 −0.546794
$$924$$ 0 0
$$925$$ 1.89176 0.0622007
$$926$$ 0 0
$$927$$ 8.72843 0.286679
$$928$$ 0 0
$$929$$ 41.9375 1.37593 0.687963 0.725746i $$-0.258505\pi$$
0.687963 + 0.725746i $$0.258505\pi$$
$$930$$ 0 0
$$931$$ 3.28579 0.107687
$$932$$ 0 0
$$933$$ 39.4210 1.29059
$$934$$ 0 0
$$935$$ −36.4628 −1.19246
$$936$$ 0 0
$$937$$ 51.1073 1.66960 0.834801 0.550552i $$-0.185583\pi$$
0.834801 + 0.550552i $$0.185583\pi$$
$$938$$ 0 0
$$939$$ −14.7806 −0.482347
$$940$$ 0 0
$$941$$ −12.8264 −0.418130 −0.209065 0.977902i $$-0.567042\pi$$
−0.209065 + 0.977902i $$0.567042\pi$$
$$942$$ 0 0
$$943$$ −46.7127 −1.52117
$$944$$ 0 0
$$945$$ 10.3935 0.338099
$$946$$ 0 0
$$947$$ −24.4575 −0.794761 −0.397380 0.917654i $$-0.630081\pi$$
−0.397380 + 0.917654i $$0.630081\pi$$
$$948$$ 0 0
$$949$$ −49.6054 −1.61026
$$950$$ 0 0
$$951$$ −33.8300 −1.09701
$$952$$ 0 0
$$953$$ 21.4854 0.695981 0.347990 0.937498i $$-0.386864\pi$$
0.347990 + 0.937498i $$0.386864\pi$$
$$954$$ 0 0
$$955$$ 0.916561 0.0296592
$$956$$ 0 0
$$957$$ −36.8445 −1.19101
$$958$$ 0 0
$$959$$ 36.8749 1.19075
$$960$$ 0 0
$$961$$ 87.7170 2.82958
$$962$$ 0 0
$$963$$ 27.8741 0.898230
$$964$$ 0 0
$$965$$ −17.7424 −0.571149
$$966$$ 0 0
$$967$$ −41.4000 −1.33134 −0.665668 0.746248i $$-0.731853\pi$$
−0.665668 + 0.746248i $$0.731853\pi$$
$$968$$ 0 0
$$969$$ 15.1815 0.487701
$$970$$ 0 0
$$971$$ 21.5095 0.690272 0.345136 0.938553i $$-0.387833\pi$$
0.345136 + 0.938553i $$0.387833\pi$$
$$972$$ 0 0
$$973$$ 23.3366 0.748139
$$974$$ 0 0
$$975$$ −10.1029 −0.323551
$$976$$ 0 0
$$977$$ 42.0896 1.34656 0.673282 0.739385i $$-0.264884\pi$$
0.673282 + 0.739385i $$0.264884\pi$$
$$978$$ 0 0
$$979$$ 25.4098 0.812099
$$980$$ 0 0
$$981$$ 71.6235 2.28676
$$982$$ 0 0
$$983$$ 57.9968 1.84981 0.924905 0.380198i $$-0.124144\pi$$
0.924905 + 0.380198i $$0.124144\pi$$
$$984$$ 0 0
$$985$$ −26.6121 −0.847932
$$986$$ 0 0
$$987$$ −24.6430 −0.784397
$$988$$ 0 0
$$989$$ 39.5884 1.25884
$$990$$ 0 0
$$991$$ −17.1218 −0.543892 −0.271946 0.962312i $$-0.587667\pi$$
−0.271946 + 0.962312i $$0.587667\pi$$
$$992$$ 0 0
$$993$$ 75.5141 2.39637
$$994$$ 0 0
$$995$$ 3.58786 0.113743
$$996$$ 0 0
$$997$$ 21.7526 0.688911 0.344456 0.938803i $$-0.388064\pi$$
0.344456 + 0.938803i $$0.388064\pi$$
$$998$$ 0 0
$$999$$ 6.13066 0.193966
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 6080.2.a.cg.1.4 4
4.3 odd 2 6080.2.a.ce.1.1 4
8.3 odd 2 3040.2.a.s.1.4 yes 4
8.5 even 2 3040.2.a.q.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.q.1.1 4 8.5 even 2
3040.2.a.s.1.4 yes 4 8.3 odd 2
6080.2.a.ce.1.1 4 4.3 odd 2
6080.2.a.cg.1.4 4 1.1 even 1 trivial