# Properties

 Label 6080.2.a.bn.1.3 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ 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] = [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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 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 1.3 Root $$-1.48119$$ 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)

 $$f(q)$$ $$=$$ $$q+0.481194 q^{3} +1.00000 q^{5} -2.15633 q^{7} -2.76845 q^{9} +O(q^{10})$$ $$q+0.481194 q^{3} +1.00000 q^{5} -2.15633 q^{7} -2.76845 q^{9} +4.15633 q^{11} +2.06300 q^{13} +0.481194 q^{15} -6.31265 q^{17} -1.00000 q^{19} -1.03761 q^{21} +3.76845 q^{23} +1.00000 q^{25} -2.77575 q^{27} -1.03761 q^{29} -3.61213 q^{31} +2.00000 q^{33} -2.15633 q^{35} +4.89938 q^{37} +0.992706 q^{39} -3.58181 q^{43} -2.76845 q^{45} +4.54420 q^{47} -2.35026 q^{49} -3.03761 q^{51} -2.45088 q^{53} +4.15633 q^{55} -0.481194 q^{57} -12.5745 q^{59} +0.156325 q^{61} +5.96968 q^{63} +2.06300 q^{65} -11.4436 q^{67} +1.81336 q^{69} -4.38787 q^{71} +9.66291 q^{73} +0.481194 q^{75} -8.96239 q^{77} -6.12601 q^{79} +6.96968 q^{81} +3.89446 q^{83} -6.31265 q^{85} -0.499293 q^{87} +5.03761 q^{89} -4.44851 q^{91} -1.73813 q^{93} -1.00000 q^{95} -1.48849 q^{97} -11.5066 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 4 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 4 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 $$3 q - 4 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} - 3 q^{19} - 14 q^{21} + 3 q^{25} - 10 q^{27} - 14 q^{29} - 10 q^{31} + 6 q^{33} + 4 q^{35} + 8 q^{37} - 10 q^{39} - 12 q^{43} + 3 q^{45} + 4 q^{47} + 3 q^{49} - 20 q^{51} - 4 q^{53} + 2 q^{55} + 4 q^{57} - 26 q^{59} - 10 q^{61} + 20 q^{63} + 2 q^{65} - 18 q^{67} + 18 q^{69} - 14 q^{71} - 2 q^{73} - 4 q^{75} - 16 q^{77} - 10 q^{79} + 23 q^{81} - 8 q^{83} + 2 q^{85} + 32 q^{87} + 26 q^{89} - 10 q^{91} + 4 q^{93} - 3 q^{95} - 12 q^{97} - 14 q^{99}+O(q^{100})$$ 3 * q - 4 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 - 3 * q^19 - 14 * q^21 + 3 * q^25 - 10 * q^27 - 14 * q^29 - 10 * q^31 + 6 * q^33 + 4 * q^35 + 8 * q^37 - 10 * q^39 - 12 * q^43 + 3 * q^45 + 4 * q^47 + 3 * q^49 - 20 * q^51 - 4 * q^53 + 2 * q^55 + 4 * q^57 - 26 * q^59 - 10 * q^61 + 20 * q^63 + 2 * q^65 - 18 * q^67 + 18 * q^69 - 14 * q^71 - 2 * q^73 - 4 * q^75 - 16 * q^77 - 10 * q^79 + 23 * q^81 - 8 * q^83 + 2 * q^85 + 32 * q^87 + 26 * q^89 - 10 * q^91 + 4 * q^93 - 3 * q^95 - 12 * 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
$$3$$ 0.481194 0.277818 0.138909 0.990305i $$-0.455641\pi$$
0.138909 + 0.990305i $$0.455641\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.15633 −0.815014 −0.407507 0.913202i $$-0.633602\pi$$
−0.407507 + 0.913202i $$0.633602\pi$$
$$8$$ 0 0
$$9$$ −2.76845 −0.922817
$$10$$ 0 0
$$11$$ 4.15633 1.25318 0.626590 0.779349i $$-0.284450\pi$$
0.626590 + 0.779349i $$0.284450\pi$$
$$12$$ 0 0
$$13$$ 2.06300 0.572174 0.286087 0.958204i $$-0.407645\pi$$
0.286087 + 0.958204i $$0.407645\pi$$
$$14$$ 0 0
$$15$$ 0.481194 0.124244
$$16$$ 0 0
$$17$$ −6.31265 −1.53104 −0.765521 0.643411i $$-0.777519\pi$$
−0.765521 + 0.643411i $$0.777519\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.03761 −0.226425
$$22$$ 0 0
$$23$$ 3.76845 0.785777 0.392888 0.919586i $$-0.371476\pi$$
0.392888 + 0.919586i $$0.371476\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −2.77575 −0.534193
$$28$$ 0 0
$$29$$ −1.03761 −0.192680 −0.0963398 0.995349i $$-0.530714\pi$$
−0.0963398 + 0.995349i $$0.530714\pi$$
$$30$$ 0 0
$$31$$ −3.61213 −0.648757 −0.324379 0.945927i $$-0.605155\pi$$
−0.324379 + 0.945927i $$0.605155\pi$$
$$32$$ 0 0
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ −2.15633 −0.364485
$$36$$ 0 0
$$37$$ 4.89938 0.805454 0.402727 0.915320i $$-0.368062\pi$$
0.402727 + 0.915320i $$0.368062\pi$$
$$38$$ 0 0
$$39$$ 0.992706 0.158960
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −3.58181 −0.546221 −0.273110 0.961983i $$-0.588052\pi$$
−0.273110 + 0.961983i $$0.588052\pi$$
$$44$$ 0 0
$$45$$ −2.76845 −0.412696
$$46$$ 0 0
$$47$$ 4.54420 0.662839 0.331420 0.943483i $$-0.392472\pi$$
0.331420 + 0.943483i $$0.392472\pi$$
$$48$$ 0 0
$$49$$ −2.35026 −0.335752
$$50$$ 0 0
$$51$$ −3.03761 −0.425351
$$52$$ 0 0
$$53$$ −2.45088 −0.336654 −0.168327 0.985731i $$-0.553836\pi$$
−0.168327 + 0.985731i $$0.553836\pi$$
$$54$$ 0 0
$$55$$ 4.15633 0.560439
$$56$$ 0 0
$$57$$ −0.481194 −0.0637357
$$58$$ 0 0
$$59$$ −12.5745 −1.63706 −0.818531 0.574462i $$-0.805211\pi$$
−0.818531 + 0.574462i $$0.805211\pi$$
$$60$$ 0 0
$$61$$ 0.156325 0.0200154 0.0100077 0.999950i $$-0.496814\pi$$
0.0100077 + 0.999950i $$0.496814\pi$$
$$62$$ 0 0
$$63$$ 5.96968 0.752109
$$64$$ 0 0
$$65$$ 2.06300 0.255884
$$66$$ 0 0
$$67$$ −11.4436 −1.39806 −0.699028 0.715094i $$-0.746384\pi$$
−0.699028 + 0.715094i $$0.746384\pi$$
$$68$$ 0 0
$$69$$ 1.81336 0.218303
$$70$$ 0 0
$$71$$ −4.38787 −0.520745 −0.260372 0.965508i $$-0.583845\pi$$
−0.260372 + 0.965508i $$0.583845\pi$$
$$72$$ 0 0
$$73$$ 9.66291 1.13096 0.565479 0.824763i $$-0.308691\pi$$
0.565479 + 0.824763i $$0.308691\pi$$
$$74$$ 0 0
$$75$$ 0.481194 0.0555635
$$76$$ 0 0
$$77$$ −8.96239 −1.02136
$$78$$ 0 0
$$79$$ −6.12601 −0.689230 −0.344615 0.938744i $$-0.611990\pi$$
−0.344615 + 0.938744i $$0.611990\pi$$
$$80$$ 0 0
$$81$$ 6.96968 0.774409
$$82$$ 0 0
$$83$$ 3.89446 0.427473 0.213736 0.976891i $$-0.431437\pi$$
0.213736 + 0.976891i $$0.431437\pi$$
$$84$$ 0 0
$$85$$ −6.31265 −0.684703
$$86$$ 0 0
$$87$$ −0.499293 −0.0535298
$$88$$ 0 0
$$89$$ 5.03761 0.533986 0.266993 0.963699i $$-0.413970\pi$$
0.266993 + 0.963699i $$0.413970\pi$$
$$90$$ 0 0
$$91$$ −4.44851 −0.466330
$$92$$ 0 0
$$93$$ −1.73813 −0.180236
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −1.48849 −0.151133 −0.0755666 0.997141i $$-0.524077\pi$$
−0.0755666 + 0.997141i $$0.524077\pi$$
$$98$$ 0 0
$$99$$ −11.5066 −1.15646
$$100$$ 0 0
$$101$$ 12.2823 1.22214 0.611069 0.791577i $$-0.290740\pi$$
0.611069 + 0.791577i $$0.290740\pi$$
$$102$$ 0 0
$$103$$ −10.4812 −1.03274 −0.516371 0.856365i $$-0.672718\pi$$
−0.516371 + 0.856365i $$0.672718\pi$$
$$104$$ 0 0
$$105$$ −1.03761 −0.101261
$$106$$ 0 0
$$107$$ −19.6302 −1.89773 −0.948863 0.315689i $$-0.897764\pi$$
−0.948863 + 0.315689i $$0.897764\pi$$
$$108$$ 0 0
$$109$$ 13.2750 1.27152 0.635759 0.771888i $$-0.280687\pi$$
0.635759 + 0.771888i $$0.280687\pi$$
$$110$$ 0 0
$$111$$ 2.35756 0.223769
$$112$$ 0 0
$$113$$ −11.0254 −1.03718 −0.518591 0.855023i $$-0.673543\pi$$
−0.518591 + 0.855023i $$0.673543\pi$$
$$114$$ 0 0
$$115$$ 3.76845 0.351410
$$116$$ 0 0
$$117$$ −5.71133 −0.528012
$$118$$ 0 0
$$119$$ 13.6121 1.24782
$$120$$ 0 0
$$121$$ 6.27504 0.570458
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −13.5696 −1.20411 −0.602053 0.798456i $$-0.705651\pi$$
−0.602053 + 0.798456i $$0.705651\pi$$
$$128$$ 0 0
$$129$$ −1.72355 −0.151750
$$130$$ 0 0
$$131$$ −0.962389 −0.0840843 −0.0420421 0.999116i $$-0.513386\pi$$
−0.0420421 + 0.999116i $$0.513386\pi$$
$$132$$ 0 0
$$133$$ 2.15633 0.186977
$$134$$ 0 0
$$135$$ −2.77575 −0.238898
$$136$$ 0 0
$$137$$ −12.7005 −1.08508 −0.542539 0.840030i $$-0.682537\pi$$
−0.542539 + 0.840030i $$0.682537\pi$$
$$138$$ 0 0
$$139$$ −4.93207 −0.418333 −0.209166 0.977880i $$-0.567075\pi$$
−0.209166 + 0.977880i $$0.567075\pi$$
$$140$$ 0 0
$$141$$ 2.18664 0.184149
$$142$$ 0 0
$$143$$ 8.57452 0.717037
$$144$$ 0 0
$$145$$ −1.03761 −0.0861689
$$146$$ 0 0
$$147$$ −1.13093 −0.0932777
$$148$$ 0 0
$$149$$ −10.3938 −0.851489 −0.425745 0.904843i $$-0.639988\pi$$
−0.425745 + 0.904843i $$0.639988\pi$$
$$150$$ 0 0
$$151$$ 1.42548 0.116004 0.0580021 0.998316i $$-0.481527\pi$$
0.0580021 + 0.998316i $$0.481527\pi$$
$$152$$ 0 0
$$153$$ 17.4763 1.41287
$$154$$ 0 0
$$155$$ −3.61213 −0.290133
$$156$$ 0 0
$$157$$ −0.700523 −0.0559079 −0.0279539 0.999609i $$-0.508899\pi$$
−0.0279539 + 0.999609i $$0.508899\pi$$
$$158$$ 0 0
$$159$$ −1.17935 −0.0935284
$$160$$ 0 0
$$161$$ −8.12601 −0.640419
$$162$$ 0 0
$$163$$ −7.26916 −0.569365 −0.284682 0.958622i $$-0.591888\pi$$
−0.284682 + 0.958622i $$0.591888\pi$$
$$164$$ 0 0
$$165$$ 2.00000 0.155700
$$166$$ 0 0
$$167$$ 14.3453 1.11008 0.555038 0.831825i $$-0.312704\pi$$
0.555038 + 0.831825i $$0.312704\pi$$
$$168$$ 0 0
$$169$$ −8.74401 −0.672616
$$170$$ 0 0
$$171$$ 2.76845 0.211709
$$172$$ 0 0
$$173$$ −1.75035 −0.133077 −0.0665385 0.997784i $$-0.521196\pi$$
−0.0665385 + 0.997784i $$0.521196\pi$$
$$174$$ 0 0
$$175$$ −2.15633 −0.163003
$$176$$ 0 0
$$177$$ −6.05079 −0.454805
$$178$$ 0 0
$$179$$ −5.87399 −0.439043 −0.219521 0.975608i $$-0.570450\pi$$
−0.219521 + 0.975608i $$0.570450\pi$$
$$180$$ 0 0
$$181$$ 12.7005 0.944022 0.472011 0.881593i $$-0.343528\pi$$
0.472011 + 0.881593i $$0.343528\pi$$
$$182$$ 0 0
$$183$$ 0.0752228 0.00556063
$$184$$ 0 0
$$185$$ 4.89938 0.360210
$$186$$ 0 0
$$187$$ −26.2374 −1.91867
$$188$$ 0 0
$$189$$ 5.98541 0.435375
$$190$$ 0 0
$$191$$ 5.08840 0.368183 0.184092 0.982909i $$-0.441066\pi$$
0.184092 + 0.982909i $$0.441066\pi$$
$$192$$ 0 0
$$193$$ −8.71274 −0.627157 −0.313578 0.949562i $$-0.601528\pi$$
−0.313578 + 0.949562i $$0.601528\pi$$
$$194$$ 0 0
$$195$$ 0.992706 0.0710891
$$196$$ 0 0
$$197$$ −27.1246 −1.93255 −0.966274 0.257518i $$-0.917095\pi$$
−0.966274 + 0.257518i $$0.917095\pi$$
$$198$$ 0 0
$$199$$ 18.5501 1.31498 0.657490 0.753463i $$-0.271618\pi$$
0.657490 + 0.753463i $$0.271618\pi$$
$$200$$ 0 0
$$201$$ −5.50659 −0.388405
$$202$$ 0 0
$$203$$ 2.23743 0.157037
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −10.4328 −0.725128
$$208$$ 0 0
$$209$$ −4.15633 −0.287499
$$210$$ 0 0
$$211$$ 7.87399 0.542068 0.271034 0.962570i $$-0.412634\pi$$
0.271034 + 0.962570i $$0.412634\pi$$
$$212$$ 0 0
$$213$$ −2.11142 −0.144672
$$214$$ 0 0
$$215$$ −3.58181 −0.244277
$$216$$ 0 0
$$217$$ 7.78892 0.528746
$$218$$ 0 0
$$219$$ 4.64974 0.314200
$$220$$ 0 0
$$221$$ −13.0230 −0.876023
$$222$$ 0 0
$$223$$ −2.74306 −0.183689 −0.0918444 0.995773i $$-0.529276\pi$$
−0.0918444 + 0.995773i $$0.529276\pi$$
$$224$$ 0 0
$$225$$ −2.76845 −0.184563
$$226$$ 0 0
$$227$$ −8.24377 −0.547158 −0.273579 0.961850i $$-0.588208\pi$$
−0.273579 + 0.961850i $$0.588208\pi$$
$$228$$ 0 0
$$229$$ −10.7308 −0.709114 −0.354557 0.935034i $$-0.615368\pi$$
−0.354557 + 0.935034i $$0.615368\pi$$
$$230$$ 0 0
$$231$$ −4.31265 −0.283752
$$232$$ 0 0
$$233$$ 1.87399 0.122769 0.0613846 0.998114i $$-0.480448\pi$$
0.0613846 + 0.998114i $$0.480448\pi$$
$$234$$ 0 0
$$235$$ 4.54420 0.296431
$$236$$ 0 0
$$237$$ −2.94780 −0.191480
$$238$$ 0 0
$$239$$ −21.1246 −1.36644 −0.683218 0.730214i $$-0.739420\pi$$
−0.683218 + 0.730214i $$0.739420\pi$$
$$240$$ 0 0
$$241$$ 2.23743 0.144125 0.0720627 0.997400i $$-0.477042\pi$$
0.0720627 + 0.997400i $$0.477042\pi$$
$$242$$ 0 0
$$243$$ 11.6810 0.749337
$$244$$ 0 0
$$245$$ −2.35026 −0.150153
$$246$$ 0 0
$$247$$ −2.06300 −0.131266
$$248$$ 0 0
$$249$$ 1.87399 0.118759
$$250$$ 0 0
$$251$$ −25.7137 −1.62303 −0.811517 0.584329i $$-0.801358\pi$$
−0.811517 + 0.584329i $$0.801358\pi$$
$$252$$ 0 0
$$253$$ 15.6629 0.984719
$$254$$ 0 0
$$255$$ −3.03761 −0.190223
$$256$$ 0 0
$$257$$ 1.80114 0.112352 0.0561760 0.998421i $$-0.482109\pi$$
0.0561760 + 0.998421i $$0.482109\pi$$
$$258$$ 0 0
$$259$$ −10.5647 −0.656456
$$260$$ 0 0
$$261$$ 2.87258 0.177808
$$262$$ 0 0
$$263$$ 10.5296 0.649284 0.324642 0.945837i $$-0.394756\pi$$
0.324642 + 0.945837i $$0.394756\pi$$
$$264$$ 0 0
$$265$$ −2.45088 −0.150556
$$266$$ 0 0
$$267$$ 2.42407 0.148351
$$268$$ 0 0
$$269$$ 0.962389 0.0586779 0.0293389 0.999570i $$-0.490660\pi$$
0.0293389 + 0.999570i $$0.490660\pi$$
$$270$$ 0 0
$$271$$ −30.8568 −1.87442 −0.937210 0.348765i $$-0.886601\pi$$
−0.937210 + 0.348765i $$0.886601\pi$$
$$272$$ 0 0
$$273$$ −2.14060 −0.129555
$$274$$ 0 0
$$275$$ 4.15633 0.250636
$$276$$ 0 0
$$277$$ −7.40105 −0.444686 −0.222343 0.974969i $$-0.571370\pi$$
−0.222343 + 0.974969i $$0.571370\pi$$
$$278$$ 0 0
$$279$$ 10.0000 0.598684
$$280$$ 0 0
$$281$$ −2.44851 −0.146066 −0.0730329 0.997330i $$-0.523268\pi$$
−0.0730329 + 0.997330i $$0.523268\pi$$
$$282$$ 0 0
$$283$$ −26.4445 −1.57196 −0.785982 0.618249i $$-0.787842\pi$$
−0.785982 + 0.618249i $$0.787842\pi$$
$$284$$ 0 0
$$285$$ −0.481194 −0.0285035
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 22.8496 1.34409
$$290$$ 0 0
$$291$$ −0.716252 −0.0419874
$$292$$ 0 0
$$293$$ 9.86177 0.576131 0.288065 0.957611i $$-0.406988\pi$$
0.288065 + 0.957611i $$0.406988\pi$$
$$294$$ 0 0
$$295$$ −12.5745 −0.732117
$$296$$ 0 0
$$297$$ −11.5369 −0.669439
$$298$$ 0 0
$$299$$ 7.77433 0.449601
$$300$$ 0 0
$$301$$ 7.72355 0.445178
$$302$$ 0 0
$$303$$ 5.91019 0.339531
$$304$$ 0 0
$$305$$ 0.156325 0.00895115
$$306$$ 0 0
$$307$$ −0.153956 −0.00878670 −0.00439335 0.999990i $$-0.501398\pi$$
−0.00439335 + 0.999990i $$0.501398\pi$$
$$308$$ 0 0
$$309$$ −5.04349 −0.286914
$$310$$ 0 0
$$311$$ −11.6326 −0.659624 −0.329812 0.944047i $$-0.606985\pi$$
−0.329812 + 0.944047i $$0.606985\pi$$
$$312$$ 0 0
$$313$$ 0.826531 0.0467183 0.0233592 0.999727i $$-0.492564\pi$$
0.0233592 + 0.999727i $$0.492564\pi$$
$$314$$ 0 0
$$315$$ 5.96968 0.336354
$$316$$ 0 0
$$317$$ 18.1236 1.01792 0.508962 0.860789i $$-0.330029\pi$$
0.508962 + 0.860789i $$0.330029\pi$$
$$318$$ 0 0
$$319$$ −4.31265 −0.241462
$$320$$ 0 0
$$321$$ −9.44595 −0.527222
$$322$$ 0 0
$$323$$ 6.31265 0.351245
$$324$$ 0 0
$$325$$ 2.06300 0.114435
$$326$$ 0 0
$$327$$ 6.38787 0.353250
$$328$$ 0 0
$$329$$ −9.79877 −0.540224
$$330$$ 0 0
$$331$$ 11.8134 0.649321 0.324660 0.945831i $$-0.394750\pi$$
0.324660 + 0.945831i $$0.394750\pi$$
$$332$$ 0 0
$$333$$ −13.5637 −0.743287
$$334$$ 0 0
$$335$$ −11.4436 −0.625230
$$336$$ 0 0
$$337$$ 35.1632 1.91546 0.957730 0.287670i $$-0.0928805\pi$$
0.957730 + 0.287670i $$0.0928805\pi$$
$$338$$ 0 0
$$339$$ −5.30536 −0.288147
$$340$$ 0 0
$$341$$ −15.0132 −0.813009
$$342$$ 0 0
$$343$$ 20.1622 1.08866
$$344$$ 0 0
$$345$$ 1.81336 0.0976279
$$346$$ 0 0
$$347$$ −30.8423 −1.65570 −0.827850 0.560950i $$-0.810436\pi$$
−0.827850 + 0.560950i $$0.810436\pi$$
$$348$$ 0 0
$$349$$ −9.87399 −0.528543 −0.264271 0.964448i $$-0.585131\pi$$
−0.264271 + 0.964448i $$0.585131\pi$$
$$350$$ 0 0
$$351$$ −5.72638 −0.305651
$$352$$ 0 0
$$353$$ −13.3865 −0.712489 −0.356245 0.934393i $$-0.615943\pi$$
−0.356245 + 0.934393i $$0.615943\pi$$
$$354$$ 0 0
$$355$$ −4.38787 −0.232884
$$356$$ 0 0
$$357$$ 6.55008 0.346667
$$358$$ 0 0
$$359$$ 22.3331 1.17870 0.589348 0.807879i $$-0.299385\pi$$
0.589348 + 0.807879i $$0.299385\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 3.01951 0.158483
$$364$$ 0 0
$$365$$ 9.66291 0.505780
$$366$$ 0 0
$$367$$ 2.86670 0.149640 0.0748202 0.997197i $$-0.476162\pi$$
0.0748202 + 0.997197i $$0.476162\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 5.28489 0.274378
$$372$$ 0 0
$$373$$ −31.2873 −1.61999 −0.809996 0.586435i $$-0.800531\pi$$
−0.809996 + 0.586435i $$0.800531\pi$$
$$374$$ 0 0
$$375$$ 0.481194 0.0248488
$$376$$ 0 0
$$377$$ −2.14060 −0.110246
$$378$$ 0 0
$$379$$ 31.9511 1.64122 0.820610 0.571489i $$-0.193634\pi$$
0.820610 + 0.571489i $$0.193634\pi$$
$$380$$ 0 0
$$381$$ −6.52961 −0.334522
$$382$$ 0 0
$$383$$ −17.3684 −0.887482 −0.443741 0.896155i $$-0.646349\pi$$
−0.443741 + 0.896155i $$0.646349\pi$$
$$384$$ 0 0
$$385$$ −8.96239 −0.456766
$$386$$ 0 0
$$387$$ 9.91607 0.504062
$$388$$ 0 0
$$389$$ 12.7250 0.645181 0.322591 0.946539i $$-0.395446\pi$$
0.322591 + 0.946539i $$0.395446\pi$$
$$390$$ 0 0
$$391$$ −23.7889 −1.20306
$$392$$ 0 0
$$393$$ −0.463096 −0.0233601
$$394$$ 0 0
$$395$$ −6.12601 −0.308233
$$396$$ 0 0
$$397$$ −15.2750 −0.766632 −0.383316 0.923617i $$-0.625218\pi$$
−0.383316 + 0.923617i $$0.625218\pi$$
$$398$$ 0 0
$$399$$ 1.03761 0.0519455
$$400$$ 0 0
$$401$$ 26.7875 1.33770 0.668852 0.743396i $$-0.266786\pi$$
0.668852 + 0.743396i $$0.266786\pi$$
$$402$$ 0 0
$$403$$ −7.45183 −0.371202
$$404$$ 0 0
$$405$$ 6.96968 0.346326
$$406$$ 0 0
$$407$$ 20.3634 1.00938
$$408$$ 0 0
$$409$$ −22.8872 −1.13170 −0.565849 0.824509i $$-0.691451\pi$$
−0.565849 + 0.824509i $$0.691451\pi$$
$$410$$ 0 0
$$411$$ −6.11142 −0.301454
$$412$$ 0 0
$$413$$ 27.1147 1.33423
$$414$$ 0 0
$$415$$ 3.89446 0.191172
$$416$$ 0 0
$$417$$ −2.37328 −0.116220
$$418$$ 0 0
$$419$$ 12.1260 0.592394 0.296197 0.955127i $$-0.404281\pi$$
0.296197 + 0.955127i $$0.404281\pi$$
$$420$$ 0 0
$$421$$ 21.7137 1.05826 0.529130 0.848541i $$-0.322518\pi$$
0.529130 + 0.848541i $$0.322518\pi$$
$$422$$ 0 0
$$423$$ −12.5804 −0.611680
$$424$$ 0 0
$$425$$ −6.31265 −0.306209
$$426$$ 0 0
$$427$$ −0.337088 −0.0163128
$$428$$ 0 0
$$429$$ 4.12601 0.199206
$$430$$ 0 0
$$431$$ −34.7367 −1.67321 −0.836604 0.547807i $$-0.815463\pi$$
−0.836604 + 0.547807i $$0.815463\pi$$
$$432$$ 0 0
$$433$$ −6.77338 −0.325508 −0.162754 0.986667i $$-0.552038\pi$$
−0.162754 + 0.986667i $$0.552038\pi$$
$$434$$ 0 0
$$435$$ −0.499293 −0.0239393
$$436$$ 0 0
$$437$$ −3.76845 −0.180270
$$438$$ 0 0
$$439$$ 21.9149 1.04594 0.522971 0.852350i $$-0.324824\pi$$
0.522971 + 0.852350i $$0.324824\pi$$
$$440$$ 0 0
$$441$$ 6.50659 0.309837
$$442$$ 0 0
$$443$$ −20.3938 −0.968936 −0.484468 0.874809i $$-0.660987\pi$$
−0.484468 + 0.874809i $$0.660987\pi$$
$$444$$ 0 0
$$445$$ 5.03761 0.238806
$$446$$ 0 0
$$447$$ −5.00141 −0.236559
$$448$$ 0 0
$$449$$ 31.1998 1.47241 0.736205 0.676758i $$-0.236616\pi$$
0.736205 + 0.676758i $$0.236616\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0.685935 0.0322280
$$454$$ 0 0
$$455$$ −4.44851 −0.208549
$$456$$ 0 0
$$457$$ 26.9986 1.26294 0.631470 0.775400i $$-0.282452\pi$$
0.631470 + 0.775400i $$0.282452\pi$$
$$458$$ 0 0
$$459$$ 17.5223 0.817872
$$460$$ 0 0
$$461$$ 19.5223 0.909245 0.454622 0.890684i $$-0.349774\pi$$
0.454622 + 0.890684i $$0.349774\pi$$
$$462$$ 0 0
$$463$$ 1.84367 0.0856828 0.0428414 0.999082i $$-0.486359\pi$$
0.0428414 + 0.999082i $$0.486359\pi$$
$$464$$ 0 0
$$465$$ −1.73813 −0.0806041
$$466$$ 0 0
$$467$$ −9.84367 −0.455511 −0.227755 0.973718i $$-0.573139\pi$$
−0.227755 + 0.973718i $$0.573139\pi$$
$$468$$ 0 0
$$469$$ 24.6761 1.13944
$$470$$ 0 0
$$471$$ −0.337088 −0.0155322
$$472$$ 0 0
$$473$$ −14.8872 −0.684513
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 6.78514 0.310670
$$478$$ 0 0
$$479$$ −8.31853 −0.380083 −0.190042 0.981776i $$-0.560862\pi$$
−0.190042 + 0.981776i $$0.560862\pi$$
$$480$$ 0 0
$$481$$ 10.1075 0.460860
$$482$$ 0 0
$$483$$ −3.91019 −0.177920
$$484$$ 0 0
$$485$$ −1.48849 −0.0675888
$$486$$ 0 0
$$487$$ −34.1197 −1.54611 −0.773055 0.634339i $$-0.781272\pi$$
−0.773055 + 0.634339i $$0.781272\pi$$
$$488$$ 0 0
$$489$$ −3.49788 −0.158180
$$490$$ 0 0
$$491$$ 38.0870 1.71884 0.859421 0.511269i $$-0.170824\pi$$
0.859421 + 0.511269i $$0.170824\pi$$
$$492$$ 0 0
$$493$$ 6.55008 0.295001
$$494$$ 0 0
$$495$$ −11.5066 −0.517183
$$496$$ 0 0
$$497$$ 9.46168 0.424414
$$498$$ 0 0
$$499$$ 25.0336 1.12066 0.560330 0.828270i $$-0.310674\pi$$
0.560330 + 0.828270i $$0.310674\pi$$
$$500$$ 0 0
$$501$$ 6.90289 0.308399
$$502$$ 0 0
$$503$$ 14.7210 0.656377 0.328188 0.944612i $$-0.393562\pi$$
0.328188 + 0.944612i $$0.393562\pi$$
$$504$$ 0 0
$$505$$ 12.2823 0.546557
$$506$$ 0 0
$$507$$ −4.20757 −0.186865
$$508$$ 0 0
$$509$$ −37.1998 −1.64885 −0.824426 0.565969i $$-0.808502\pi$$
−0.824426 + 0.565969i $$0.808502\pi$$
$$510$$ 0 0
$$511$$ −20.8364 −0.921747
$$512$$ 0 0
$$513$$ 2.77575 0.122552
$$514$$ 0 0
$$515$$ −10.4812 −0.461857
$$516$$ 0 0
$$517$$ 18.8872 0.830657
$$518$$ 0 0
$$519$$ −0.842260 −0.0369711
$$520$$ 0 0
$$521$$ −25.0132 −1.09585 −0.547924 0.836528i $$-0.684582\pi$$
−0.547924 + 0.836528i $$0.684582\pi$$
$$522$$ 0 0
$$523$$ −5.78067 −0.252771 −0.126386 0.991981i $$-0.540338\pi$$
−0.126386 + 0.991981i $$0.540338\pi$$
$$524$$ 0 0
$$525$$ −1.03761 −0.0452851
$$526$$ 0 0
$$527$$ 22.8021 0.993275
$$528$$ 0 0
$$529$$ −8.79877 −0.382555
$$530$$ 0 0
$$531$$ 34.8119 1.51071
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −19.6302 −0.848689
$$536$$ 0 0
$$537$$ −2.82653 −0.121974
$$538$$ 0 0
$$539$$ −9.76845 −0.420757
$$540$$ 0 0
$$541$$ −18.5804 −0.798834 −0.399417 0.916769i $$-0.630787\pi$$
−0.399417 + 0.916769i $$0.630787\pi$$
$$542$$ 0 0
$$543$$ 6.11142 0.262266
$$544$$ 0 0
$$545$$ 13.2750 0.568640
$$546$$ 0 0
$$547$$ −4.54657 −0.194397 −0.0971986 0.995265i $$-0.530988\pi$$
−0.0971986 + 0.995265i $$0.530988\pi$$
$$548$$ 0 0
$$549$$ −0.432779 −0.0184705
$$550$$ 0 0
$$551$$ 1.03761 0.0442037
$$552$$ 0 0
$$553$$ 13.2097 0.561732
$$554$$ 0 0
$$555$$ 2.35756 0.100073
$$556$$ 0 0
$$557$$ 44.3996 1.88127 0.940636 0.339416i $$-0.110229\pi$$
0.940636 + 0.339416i $$0.110229\pi$$
$$558$$ 0 0
$$559$$ −7.38929 −0.312534
$$560$$ 0 0
$$561$$ −12.6253 −0.533041
$$562$$ 0 0
$$563$$ 9.59403 0.404340 0.202170 0.979350i $$-0.435201\pi$$
0.202170 + 0.979350i $$0.435201\pi$$
$$564$$ 0 0
$$565$$ −11.0254 −0.463842
$$566$$ 0 0
$$567$$ −15.0289 −0.631155
$$568$$ 0 0
$$569$$ −9.36485 −0.392595 −0.196297 0.980544i $$-0.562892\pi$$
−0.196297 + 0.980544i $$0.562892\pi$$
$$570$$ 0 0
$$571$$ 16.4083 0.686668 0.343334 0.939213i $$-0.388444\pi$$
0.343334 + 0.939213i $$0.388444\pi$$
$$572$$ 0 0
$$573$$ 2.44851 0.102288
$$574$$ 0 0
$$575$$ 3.76845 0.157155
$$576$$ 0 0
$$577$$ 28.2736 1.17705 0.588523 0.808480i $$-0.299710\pi$$
0.588523 + 0.808480i $$0.299710\pi$$
$$578$$ 0 0
$$579$$ −4.19252 −0.174235
$$580$$ 0 0
$$581$$ −8.39772 −0.348396
$$582$$ 0 0
$$583$$ −10.1866 −0.421888
$$584$$ 0 0
$$585$$ −5.71133 −0.236134
$$586$$ 0 0
$$587$$ −35.2203 −1.45370 −0.726848 0.686798i $$-0.759016\pi$$
−0.726848 + 0.686798i $$0.759016\pi$$
$$588$$ 0 0
$$589$$ 3.61213 0.148835
$$590$$ 0 0
$$591$$ −13.0522 −0.536896
$$592$$ 0 0
$$593$$ 17.0738 0.701137 0.350569 0.936537i $$-0.385988\pi$$
0.350569 + 0.936537i $$0.385988\pi$$
$$594$$ 0 0
$$595$$ 13.6121 0.558043
$$596$$ 0 0
$$597$$ 8.92619 0.365325
$$598$$ 0 0
$$599$$ −18.2882 −0.747236 −0.373618 0.927583i $$-0.621883\pi$$
−0.373618 + 0.927583i $$0.621883\pi$$
$$600$$ 0 0
$$601$$ 22.3879 0.913220 0.456610 0.889667i $$-0.349063\pi$$
0.456610 + 0.889667i $$0.349063\pi$$
$$602$$ 0 0
$$603$$ 31.6810 1.29015
$$604$$ 0 0
$$605$$ 6.27504 0.255117
$$606$$ 0 0
$$607$$ 0.471345 0.0191313 0.00956566 0.999954i $$-0.496955\pi$$
0.00956566 + 0.999954i $$0.496955\pi$$
$$608$$ 0 0
$$609$$ 1.07664 0.0436275
$$610$$ 0 0
$$611$$ 9.37470 0.379260
$$612$$ 0 0
$$613$$ 22.9135 0.925468 0.462734 0.886497i $$-0.346868\pi$$
0.462734 + 0.886497i $$0.346868\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.06537 0.244183 0.122091 0.992519i $$-0.461040\pi$$
0.122091 + 0.992519i $$0.461040\pi$$
$$618$$ 0 0
$$619$$ −3.53102 −0.141924 −0.0709619 0.997479i $$-0.522607\pi$$
−0.0709619 + 0.997479i $$0.522607\pi$$
$$620$$ 0 0
$$621$$ −10.4603 −0.419756
$$622$$ 0 0
$$623$$ −10.8627 −0.435206
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −2.00000 −0.0798723
$$628$$ 0 0
$$629$$ −30.9281 −1.23318
$$630$$ 0 0
$$631$$ −31.0797 −1.23726 −0.618631 0.785681i $$-0.712313\pi$$
−0.618631 + 0.785681i $$0.712313\pi$$
$$632$$ 0 0
$$633$$ 3.78892 0.150596
$$634$$ 0 0
$$635$$ −13.5696 −0.538493
$$636$$ 0 0
$$637$$ −4.84860 −0.192109
$$638$$ 0 0
$$639$$ 12.1476 0.480552
$$640$$ 0 0
$$641$$ 42.0870 1.66234 0.831168 0.556021i $$-0.187673\pi$$
0.831168 + 0.556021i $$0.187673\pi$$
$$642$$ 0 0
$$643$$ 14.6194 0.576534 0.288267 0.957550i $$-0.406921\pi$$
0.288267 + 0.957550i $$0.406921\pi$$
$$644$$ 0 0
$$645$$ −1.72355 −0.0678646
$$646$$ 0 0
$$647$$ 8.58040 0.337330 0.168665 0.985673i $$-0.446054\pi$$
0.168665 + 0.985673i $$0.446054\pi$$
$$648$$ 0 0
$$649$$ −52.2638 −2.05153
$$650$$ 0 0
$$651$$ 3.74798 0.146895
$$652$$ 0 0
$$653$$ 35.5877 1.39265 0.696327 0.717725i $$-0.254816\pi$$
0.696327 + 0.717725i $$0.254816\pi$$
$$654$$ 0 0
$$655$$ −0.962389 −0.0376036
$$656$$ 0 0
$$657$$ −26.7513 −1.04367
$$658$$ 0 0
$$659$$ −25.8251 −1.00600 −0.503002 0.864285i $$-0.667771\pi$$
−0.503002 + 0.864285i $$0.667771\pi$$
$$660$$ 0 0
$$661$$ −0.584365 −0.0227291 −0.0113646 0.999935i $$-0.503618\pi$$
−0.0113646 + 0.999935i $$0.503618\pi$$
$$662$$ 0 0
$$663$$ −6.26660 −0.243375
$$664$$ 0 0
$$665$$ 2.15633 0.0836187
$$666$$ 0 0
$$667$$ −3.91019 −0.151403
$$668$$ 0 0
$$669$$ −1.31994 −0.0510320
$$670$$ 0 0
$$671$$ 0.649738 0.0250829
$$672$$ 0 0
$$673$$ −11.3136 −0.436107 −0.218054 0.975937i $$-0.569971\pi$$
−0.218054 + 0.975937i $$0.569971\pi$$
$$674$$ 0 0
$$675$$ −2.77575 −0.106839
$$676$$ 0 0
$$677$$ −8.58673 −0.330015 −0.165008 0.986292i $$-0.552765\pi$$
−0.165008 + 0.986292i $$0.552765\pi$$
$$678$$ 0 0
$$679$$ 3.20967 0.123176
$$680$$ 0 0
$$681$$ −3.96685 −0.152010
$$682$$ 0 0
$$683$$ −30.5564 −1.16921 −0.584604 0.811318i $$-0.698750\pi$$
−0.584604 + 0.811318i $$0.698750\pi$$
$$684$$ 0 0
$$685$$ −12.7005 −0.485262
$$686$$ 0 0
$$687$$ −5.16362 −0.197004
$$688$$ 0 0
$$689$$ −5.05617 −0.192625
$$690$$ 0 0
$$691$$ 3.38058 0.128603 0.0643016 0.997931i $$-0.479518\pi$$
0.0643016 + 0.997931i $$0.479518\pi$$
$$692$$ 0 0
$$693$$ 24.8119 0.942528
$$694$$ 0 0
$$695$$ −4.93207 −0.187084
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0.901754 0.0341075
$$700$$ 0 0
$$701$$ −16.2579 −0.614052 −0.307026 0.951701i $$-0.599334\pi$$
−0.307026 + 0.951701i $$0.599334\pi$$
$$702$$ 0 0
$$703$$ −4.89938 −0.184784
$$704$$ 0 0
$$705$$ 2.18664 0.0823537
$$706$$ 0 0
$$707$$ −26.4847 −0.996060
$$708$$ 0 0
$$709$$ −35.2262 −1.32295 −0.661473 0.749969i $$-0.730068\pi$$
−0.661473 + 0.749969i $$0.730068\pi$$
$$710$$ 0 0
$$711$$ 16.9596 0.636033
$$712$$ 0 0
$$713$$ −13.6121 −0.509778
$$714$$ 0 0
$$715$$ 8.57452 0.320669
$$716$$ 0 0
$$717$$ −10.1650 −0.379620
$$718$$ 0 0
$$719$$ 28.6194 1.06732 0.533662 0.845698i $$-0.320815\pi$$
0.533662 + 0.845698i $$0.320815\pi$$
$$720$$ 0 0
$$721$$ 22.6009 0.841700
$$722$$ 0 0
$$723$$ 1.07664 0.0400406
$$724$$ 0 0
$$725$$ −1.03761 −0.0385359
$$726$$ 0 0
$$727$$ −21.1041 −0.782709 −0.391354 0.920240i $$-0.627993\pi$$
−0.391354 + 0.920240i $$0.627993\pi$$
$$728$$ 0 0
$$729$$ −15.2882 −0.566230
$$730$$ 0 0
$$731$$ 22.6107 0.836287
$$732$$ 0 0
$$733$$ −47.1655 −1.74210 −0.871049 0.491196i $$-0.836560\pi$$
−0.871049 + 0.491196i $$0.836560\pi$$
$$734$$ 0 0
$$735$$ −1.13093 −0.0417151
$$736$$ 0 0
$$737$$ −47.5633 −1.75201
$$738$$ 0 0
$$739$$ 12.9262 0.475498 0.237749 0.971327i $$-0.423590\pi$$
0.237749 + 0.971327i $$0.423590\pi$$
$$740$$ 0 0
$$741$$ −0.992706 −0.0364680
$$742$$ 0 0
$$743$$ −40.3717 −1.48109 −0.740547 0.672005i $$-0.765433\pi$$
−0.740547 + 0.672005i $$0.765433\pi$$
$$744$$ 0 0
$$745$$ −10.3938 −0.380798
$$746$$ 0 0
$$747$$ −10.7816 −0.394479
$$748$$ 0 0
$$749$$ 42.3291 1.54667
$$750$$ 0 0
$$751$$ 14.2130 0.518639 0.259320 0.965792i $$-0.416502\pi$$
0.259320 + 0.965792i $$0.416502\pi$$
$$752$$ 0 0
$$753$$ −12.3733 −0.450908
$$754$$ 0 0
$$755$$ 1.42548 0.0518787
$$756$$ 0 0
$$757$$ 24.3028 0.883300 0.441650 0.897187i $$-0.354393\pi$$
0.441650 + 0.897187i $$0.354393\pi$$
$$758$$ 0 0
$$759$$ 7.53690 0.273572
$$760$$ 0 0
$$761$$ −5.05808 −0.183355 −0.0916776 0.995789i $$-0.529223\pi$$
−0.0916776 + 0.995789i $$0.529223\pi$$
$$762$$ 0 0
$$763$$ −28.6253 −1.03631
$$764$$ 0 0
$$765$$ 17.4763 0.631856
$$766$$ 0 0
$$767$$ −25.9413 −0.936685
$$768$$ 0 0
$$769$$ −42.8324 −1.54458 −0.772288 0.635272i $$-0.780888\pi$$
−0.772288 + 0.635272i $$0.780888\pi$$
$$770$$ 0 0
$$771$$ 0.866698 0.0312134
$$772$$ 0 0
$$773$$ −21.5755 −0.776016 −0.388008 0.921656i $$-0.626837\pi$$
−0.388008 + 0.921656i $$0.626837\pi$$
$$774$$ 0 0
$$775$$ −3.61213 −0.129751
$$776$$ 0 0
$$777$$ −5.08366 −0.182375
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −18.2374 −0.652586
$$782$$ 0 0
$$783$$ 2.88015 0.102928
$$784$$ 0 0
$$785$$ −0.700523 −0.0250028
$$786$$ 0 0
$$787$$ −24.3961 −0.869628 −0.434814 0.900520i $$-0.643186\pi$$
−0.434814 + 0.900520i $$0.643186\pi$$
$$788$$ 0 0
$$789$$ 5.06679 0.180382
$$790$$ 0 0
$$791$$ 23.7743 0.845318
$$792$$ 0 0
$$793$$ 0.322499 0.0114523
$$794$$ 0 0
$$795$$ −1.17935 −0.0418272
$$796$$ 0 0
$$797$$ −3.22662 −0.114293 −0.0571464 0.998366i $$-0.518200\pi$$
−0.0571464 + 0.998366i $$0.518200\pi$$
$$798$$ 0 0
$$799$$ −28.6859 −1.01484
$$800$$ 0 0
$$801$$ −13.9464 −0.492771
$$802$$ 0 0
$$803$$ 40.1622 1.41729
$$804$$ 0 0
$$805$$ −8.12601 −0.286404
$$806$$ 0 0
$$807$$ 0.463096 0.0163017
$$808$$ 0 0
$$809$$ 29.9756 1.05388 0.526942 0.849901i $$-0.323338\pi$$
0.526942 + 0.849901i $$0.323338\pi$$
$$810$$ 0 0
$$811$$ 40.0870 1.40764 0.703822 0.710376i $$-0.251475\pi$$
0.703822 + 0.710376i $$0.251475\pi$$
$$812$$ 0 0
$$813$$ −14.8481 −0.520747
$$814$$ 0 0
$$815$$ −7.26916 −0.254628
$$816$$ 0 0
$$817$$ 3.58181 0.125312
$$818$$ 0 0
$$819$$ 12.3155 0.430338
$$820$$ 0 0
$$821$$ −11.3766 −0.397046 −0.198523 0.980096i $$-0.563615\pi$$
−0.198523 + 0.980096i $$0.563615\pi$$
$$822$$ 0 0
$$823$$ −44.7426 −1.55963 −0.779814 0.626011i $$-0.784687\pi$$
−0.779814 + 0.626011i $$0.784687\pi$$
$$824$$ 0 0
$$825$$ 2.00000 0.0696311
$$826$$ 0 0
$$827$$ −31.8070 −1.10604 −0.553019 0.833169i $$-0.686524\pi$$
−0.553019 + 0.833169i $$0.686524\pi$$
$$828$$ 0 0
$$829$$ 5.02302 0.174457 0.0872284 0.996188i $$-0.472199\pi$$
0.0872284 + 0.996188i $$0.472199\pi$$
$$830$$ 0 0
$$831$$ −3.56134 −0.123542
$$832$$ 0 0
$$833$$ 14.8364 0.514050
$$834$$ 0 0
$$835$$ 14.3453 0.496441
$$836$$ 0 0
$$837$$ 10.0263 0.346561
$$838$$ 0 0
$$839$$ 49.8858 1.72225 0.861124 0.508395i $$-0.169761\pi$$
0.861124 + 0.508395i $$0.169761\pi$$
$$840$$ 0 0
$$841$$ −27.9234 −0.962875
$$842$$ 0 0
$$843$$ −1.17821 −0.0405796
$$844$$ 0 0
$$845$$ −8.74401 −0.300803
$$846$$ 0 0
$$847$$ −13.5310 −0.464932
$$848$$ 0 0
$$849$$ −12.7250 −0.436720
$$850$$ 0 0
$$851$$ 18.4631 0.632907
$$852$$ 0 0
$$853$$ 7.61213 0.260634 0.130317 0.991472i $$-0.458400\pi$$
0.130317 + 0.991472i $$0.458400\pi$$
$$854$$ 0 0
$$855$$ 2.76845 0.0946791
$$856$$ 0 0
$$857$$ 13.1758 0.450078 0.225039 0.974350i $$-0.427749\pi$$
0.225039 + 0.974350i $$0.427749\pi$$
$$858$$ 0 0
$$859$$ 37.9003 1.29314 0.646571 0.762853i $$-0.276202\pi$$
0.646571 + 0.762853i $$0.276202\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −16.2292 −0.552448 −0.276224 0.961093i $$-0.589083\pi$$
−0.276224 + 0.961093i $$0.589083\pi$$
$$864$$ 0 0
$$865$$ −1.75035 −0.0595138
$$866$$ 0 0
$$867$$ 10.9951 0.373412
$$868$$ 0 0
$$869$$ −25.4617 −0.863728
$$870$$ 0 0
$$871$$ −23.6082 −0.799932
$$872$$ 0 0
$$873$$ 4.12081 0.139468
$$874$$ 0 0
$$875$$ −2.15633 −0.0728971
$$876$$ 0 0
$$877$$ −15.3888 −0.519644 −0.259822 0.965657i $$-0.583664\pi$$
−0.259822 + 0.965657i $$0.583664\pi$$
$$878$$ 0 0
$$879$$ 4.74543 0.160059
$$880$$ 0 0
$$881$$ −51.5428 −1.73652 −0.868260 0.496109i $$-0.834762\pi$$
−0.868260 + 0.496109i $$0.834762\pi$$
$$882$$ 0 0
$$883$$ 26.3185 0.885689 0.442845 0.896598i $$-0.353969\pi$$
0.442845 + 0.896598i $$0.353969\pi$$
$$884$$ 0 0
$$885$$ −6.05079 −0.203395
$$886$$ 0 0
$$887$$ −6.30631 −0.211745 −0.105873 0.994380i $$-0.533764\pi$$
−0.105873 + 0.994380i $$0.533764\pi$$
$$888$$ 0 0
$$889$$ 29.2605 0.981364
$$890$$ 0 0
$$891$$ 28.9683 0.970473
$$892$$ 0 0
$$893$$ −4.54420 −0.152066
$$894$$ 0 0
$$895$$ −5.87399 −0.196346
$$896$$ 0 0
$$897$$ 3.74096 0.124907
$$898$$ 0 0
$$899$$ 3.74798 0.125002
$$900$$ 0 0
$$901$$ 15.4715 0.515431
$$902$$ 0 0
$$903$$ 3.71653 0.123678
$$904$$ 0 0
$$905$$ 12.7005 0.422180
$$906$$ 0 0
$$907$$ −21.7054 −0.720718 −0.360359 0.932814i $$-0.617346\pi$$
−0.360359 + 0.932814i $$0.617346\pi$$
$$908$$ 0 0
$$909$$ −34.0031 −1.12781
$$910$$ 0 0
$$911$$ −10.3634 −0.343356 −0.171678 0.985153i $$-0.554919\pi$$
−0.171678 + 0.985153i $$0.554919\pi$$
$$912$$ 0 0
$$913$$ 16.1866 0.535700
$$914$$ 0 0
$$915$$ 0.0752228 0.00248679
$$916$$ 0 0
$$917$$ 2.07522 0.0685299
$$918$$ 0 0
$$919$$ 42.3752 1.39783 0.698914 0.715205i $$-0.253667\pi$$
0.698914 + 0.715205i $$0.253667\pi$$
$$920$$ 0 0
$$921$$ −0.0740825 −0.00244110
$$922$$ 0 0
$$923$$ −9.05220 −0.297957
$$924$$ 0 0
$$925$$ 4.89938 0.161091
$$926$$ 0 0
$$927$$ 29.0167 0.953033
$$928$$ 0 0
$$929$$ 27.1002 0.889127 0.444564 0.895747i $$-0.353359\pi$$
0.444564 + 0.895747i $$0.353359\pi$$
$$930$$ 0 0
$$931$$ 2.35026 0.0770267
$$932$$ 0 0
$$933$$ −5.59754 −0.183255
$$934$$ 0 0
$$935$$ −26.2374 −0.858056
$$936$$ 0 0
$$937$$ −38.1866 −1.24750 −0.623752 0.781623i $$-0.714392\pi$$
−0.623752 + 0.781623i $$0.714392\pi$$
$$938$$ 0 0
$$939$$ 0.397722 0.0129792
$$940$$ 0 0
$$941$$ −60.5910 −1.97521 −0.987605 0.156958i $$-0.949831\pi$$
−0.987605 + 0.156958i $$0.949831\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 5.98541 0.194705
$$946$$ 0 0
$$947$$ 32.2433 1.04777 0.523883 0.851790i $$-0.324483\pi$$
0.523883 + 0.851790i $$0.324483\pi$$
$$948$$ 0 0
$$949$$ 19.9346 0.647105
$$950$$ 0 0
$$951$$ 8.72099 0.282798
$$952$$ 0 0
$$953$$ 7.88812 0.255521 0.127761 0.991805i $$-0.459221\pi$$
0.127761 + 0.991805i $$0.459221\pi$$
$$954$$ 0 0
$$955$$ 5.08840 0.164657
$$956$$ 0 0
$$957$$ −2.07522 −0.0670824
$$958$$ 0 0
$$959$$ 27.3865 0.884355
$$960$$ 0 0
$$961$$ −17.9525 −0.579114
$$962$$ 0 0
$$963$$ 54.3453 1.75125
$$964$$ 0 0
$$965$$ −8.71274 −0.280473
$$966$$ 0 0
$$967$$ 40.3693 1.29819 0.649095 0.760708i $$-0.275148\pi$$
0.649095 + 0.760708i $$0.275148\pi$$
$$968$$ 0 0
$$969$$ 3.03761 0.0975821
$$970$$ 0 0
$$971$$ 19.2995 0.619350 0.309675 0.950843i $$-0.399780\pi$$
0.309675 + 0.950843i $$0.399780\pi$$
$$972$$ 0 0
$$973$$ 10.6351 0.340947
$$974$$ 0 0
$$975$$ 0.992706 0.0317920
$$976$$ 0 0
$$977$$ 22.5623 0.721832 0.360916 0.932598i $$-0.382464\pi$$
0.360916 + 0.932598i $$0.382464\pi$$
$$978$$ 0 0
$$979$$ 20.9380 0.669180
$$980$$ 0 0
$$981$$ −36.7513 −1.17338
$$982$$ 0 0
$$983$$ 44.4666 1.41826 0.709132 0.705076i $$-0.249087\pi$$
0.709132 + 0.705076i $$0.249087\pi$$
$$984$$ 0 0
$$985$$ −27.1246 −0.864261
$$986$$ 0 0
$$987$$ −4.71511 −0.150084
$$988$$ 0 0
$$989$$ −13.4979 −0.429208
$$990$$ 0 0
$$991$$ 3.82512 0.121509 0.0607544 0.998153i $$-0.480649\pi$$
0.0607544 + 0.998153i $$0.480649\pi$$
$$992$$ 0 0
$$993$$ 5.68452 0.180393
$$994$$ 0 0
$$995$$ 18.5501 0.588077
$$996$$ 0 0
$$997$$ 52.8627 1.67418 0.837090 0.547066i $$-0.184255\pi$$
0.837090 + 0.547066i $$0.184255\pi$$
$$998$$ 0 0
$$999$$ −13.5994 −0.430268
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.bn.1.3 3
4.3 odd 2 6080.2.a.cb.1.1 3
8.3 odd 2 3040.2.a.i.1.3 3
8.5 even 2 3040.2.a.o.1.1 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.i.1.3 3 8.3 odd 2
3040.2.a.o.1.1 yes 3 8.5 even 2
6080.2.a.bn.1.3 3 1.1 even 1 trivial
6080.2.a.cb.1.1 3 4.3 odd 2