Properties

 Label 6080.2.a.cl.1.2 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.387268.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2$$ x^5 - x^4 - 7*x^3 + 4*x^2 + 12*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 3040) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.81079$$ 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)

 $$f(q)$$ $$=$$ $$q-0.531822 q^{3} +1.00000 q^{5} -0.0955795 q^{7} -2.71717 q^{9} +O(q^{10})$$ $$q-0.531822 q^{3} +1.00000 q^{5} -0.0955795 q^{7} -2.71717 q^{9} -0.337570 q^{11} -4.49098 q^{13} -0.531822 q^{15} -1.68405 q^{17} -1.00000 q^{19} +0.0508313 q^{21} -2.51569 q^{23} +1.00000 q^{25} +3.04051 q^{27} +6.61268 q^{29} -6.79080 q^{31} +0.179527 q^{33} -0.0955795 q^{35} -6.28059 q^{37} +2.38840 q^{39} +1.06364 q^{41} -0.325938 q^{43} -2.71717 q^{45} +8.94884 q^{47} -6.99086 q^{49} +0.895614 q^{51} +14.4587 q^{53} -0.337570 q^{55} +0.531822 q^{57} +1.20007 q^{59} +1.21006 q^{61} +0.259705 q^{63} -4.49098 q^{65} +0.111712 q^{67} +1.33790 q^{69} +14.9003 q^{71} +2.47485 q^{73} -0.531822 q^{75} +0.0322648 q^{77} -8.33842 q^{79} +6.53448 q^{81} +17.9308 q^{83} -1.68405 q^{85} -3.51677 q^{87} -13.6749 q^{89} +0.429246 q^{91} +3.61150 q^{93} -1.00000 q^{95} -10.7213 q^{97} +0.917233 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10})$$ 5 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 7 * q^9 $$5 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 7 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{15} - 12 q^{17} - 5 q^{19} + 10 q^{21} + 8 q^{23} + 5 q^{25} + 16 q^{27} + 6 q^{29} + 10 q^{31} - 18 q^{33} + 4 q^{35} + 6 q^{37} + 18 q^{39} - 8 q^{41} + 12 q^{43} + 7 q^{45} + 16 q^{47} + 7 q^{49} - 14 q^{51} + 18 q^{53} + 2 q^{55} - 4 q^{57} + 8 q^{59} - 2 q^{61} + 36 q^{63} + 4 q^{65} + 10 q^{67} + 22 q^{69} - 18 q^{71} - 28 q^{73} + 4 q^{75} + 28 q^{77} + 14 q^{79} + 25 q^{81} + 8 q^{83} - 12 q^{85} + 24 q^{87} - 30 q^{89} + 28 q^{91} + 24 q^{93} - 5 q^{95} - 18 q^{97} - 14 q^{99}+O(q^{100})$$ 5 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 7 * q^9 + 2 * q^11 + 4 * q^13 + 4 * q^15 - 12 * q^17 - 5 * q^19 + 10 * q^21 + 8 * q^23 + 5 * q^25 + 16 * q^27 + 6 * q^29 + 10 * q^31 - 18 * q^33 + 4 * q^35 + 6 * q^37 + 18 * q^39 - 8 * q^41 + 12 * q^43 + 7 * q^45 + 16 * q^47 + 7 * q^49 - 14 * q^51 + 18 * q^53 + 2 * q^55 - 4 * q^57 + 8 * q^59 - 2 * q^61 + 36 * q^63 + 4 * q^65 + 10 * q^67 + 22 * q^69 - 18 * q^71 - 28 * q^73 + 4 * q^75 + 28 * q^77 + 14 * q^79 + 25 * q^81 + 8 * q^83 - 12 * q^85 + 24 * q^87 - 30 * q^89 + 28 * q^91 + 24 * q^93 - 5 * q^95 - 18 * 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
$$3$$ −0.531822 −0.307048 −0.153524 0.988145i $$-0.549062\pi$$
−0.153524 + 0.988145i $$0.549062\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −0.0955795 −0.0361257 −0.0180628 0.999837i $$-0.505750\pi$$
−0.0180628 + 0.999837i $$0.505750\pi$$
$$8$$ 0 0
$$9$$ −2.71717 −0.905722
$$10$$ 0 0
$$11$$ −0.337570 −0.101781 −0.0508906 0.998704i $$-0.516206\pi$$
−0.0508906 + 0.998704i $$0.516206\pi$$
$$12$$ 0 0
$$13$$ −4.49098 −1.24557 −0.622787 0.782392i $$-0.714000\pi$$
−0.622787 + 0.782392i $$0.714000\pi$$
$$14$$ 0 0
$$15$$ −0.531822 −0.137316
$$16$$ 0 0
$$17$$ −1.68405 −0.408442 −0.204221 0.978925i $$-0.565466\pi$$
−0.204221 + 0.978925i $$0.565466\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0.0508313 0.0110923
$$22$$ 0 0
$$23$$ −2.51569 −0.524558 −0.262279 0.964992i $$-0.584474\pi$$
−0.262279 + 0.964992i $$0.584474\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 3.04051 0.585147
$$28$$ 0 0
$$29$$ 6.61268 1.22794 0.613972 0.789328i $$-0.289571\pi$$
0.613972 + 0.789328i $$0.289571\pi$$
$$30$$ 0 0
$$31$$ −6.79080 −1.21966 −0.609832 0.792531i $$-0.708763\pi$$
−0.609832 + 0.792531i $$0.708763\pi$$
$$32$$ 0 0
$$33$$ 0.179527 0.0312517
$$34$$ 0 0
$$35$$ −0.0955795 −0.0161559
$$36$$ 0 0
$$37$$ −6.28059 −1.03252 −0.516262 0.856431i $$-0.672677\pi$$
−0.516262 + 0.856431i $$0.672677\pi$$
$$38$$ 0 0
$$39$$ 2.38840 0.382450
$$40$$ 0 0
$$41$$ 1.06364 0.166113 0.0830567 0.996545i $$-0.473532\pi$$
0.0830567 + 0.996545i $$0.473532\pi$$
$$42$$ 0 0
$$43$$ −0.325938 −0.0497051 −0.0248525 0.999691i $$-0.507912\pi$$
−0.0248525 + 0.999691i $$0.507912\pi$$
$$44$$ 0 0
$$45$$ −2.71717 −0.405051
$$46$$ 0 0
$$47$$ 8.94884 1.30532 0.652661 0.757650i $$-0.273653\pi$$
0.652661 + 0.757650i $$0.273653\pi$$
$$48$$ 0 0
$$49$$ −6.99086 −0.998695
$$50$$ 0 0
$$51$$ 0.895614 0.125411
$$52$$ 0 0
$$53$$ 14.4587 1.98606 0.993028 0.117875i $$-0.0376081\pi$$
0.993028 + 0.117875i $$0.0376081\pi$$
$$54$$ 0 0
$$55$$ −0.337570 −0.0455179
$$56$$ 0 0
$$57$$ 0.531822 0.0704416
$$58$$ 0 0
$$59$$ 1.20007 0.156236 0.0781178 0.996944i $$-0.475109\pi$$
0.0781178 + 0.996944i $$0.475109\pi$$
$$60$$ 0 0
$$61$$ 1.21006 0.154932 0.0774658 0.996995i $$-0.475317\pi$$
0.0774658 + 0.996995i $$0.475317\pi$$
$$62$$ 0 0
$$63$$ 0.259705 0.0327198
$$64$$ 0 0
$$65$$ −4.49098 −0.557037
$$66$$ 0 0
$$67$$ 0.111712 0.0136478 0.00682389 0.999977i $$-0.497828\pi$$
0.00682389 + 0.999977i $$0.497828\pi$$
$$68$$ 0 0
$$69$$ 1.33790 0.161064
$$70$$ 0 0
$$71$$ 14.9003 1.76834 0.884168 0.467169i $$-0.154726\pi$$
0.884168 + 0.467169i $$0.154726\pi$$
$$72$$ 0 0
$$73$$ 2.47485 0.289659 0.144829 0.989457i $$-0.453737\pi$$
0.144829 + 0.989457i $$0.453737\pi$$
$$74$$ 0 0
$$75$$ −0.531822 −0.0614095
$$76$$ 0 0
$$77$$ 0.0322648 0.00367691
$$78$$ 0 0
$$79$$ −8.33842 −0.938146 −0.469073 0.883159i $$-0.655412\pi$$
−0.469073 + 0.883159i $$0.655412\pi$$
$$80$$ 0 0
$$81$$ 6.53448 0.726054
$$82$$ 0 0
$$83$$ 17.9308 1.96816 0.984080 0.177725i $$-0.0568737\pi$$
0.984080 + 0.177725i $$0.0568737\pi$$
$$84$$ 0 0
$$85$$ −1.68405 −0.182661
$$86$$ 0 0
$$87$$ −3.51677 −0.377037
$$88$$ 0 0
$$89$$ −13.6749 −1.44954 −0.724769 0.688992i $$-0.758054\pi$$
−0.724769 + 0.688992i $$0.758054\pi$$
$$90$$ 0 0
$$91$$ 0.429246 0.0449972
$$92$$ 0 0
$$93$$ 3.61150 0.374495
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −10.7213 −1.08859 −0.544293 0.838895i $$-0.683202\pi$$
−0.544293 + 0.838895i $$0.683202\pi$$
$$98$$ 0 0
$$99$$ 0.917233 0.0921854
$$100$$ 0 0
$$101$$ 3.66243 0.364425 0.182213 0.983259i $$-0.441674\pi$$
0.182213 + 0.983259i $$0.441674\pi$$
$$102$$ 0 0
$$103$$ 18.5427 1.82706 0.913531 0.406768i $$-0.133344\pi$$
0.913531 + 0.406768i $$0.133344\pi$$
$$104$$ 0 0
$$105$$ 0.0508313 0.00496063
$$106$$ 0 0
$$107$$ 14.5024 1.40200 0.700999 0.713163i $$-0.252738\pi$$
0.700999 + 0.713163i $$0.252738\pi$$
$$108$$ 0 0
$$109$$ 10.8354 1.03785 0.518924 0.854821i $$-0.326333\pi$$
0.518924 + 0.854821i $$0.326333\pi$$
$$110$$ 0 0
$$111$$ 3.34016 0.317034
$$112$$ 0 0
$$113$$ −17.2193 −1.61986 −0.809928 0.586529i $$-0.800494\pi$$
−0.809928 + 0.586529i $$0.800494\pi$$
$$114$$ 0 0
$$115$$ −2.51569 −0.234589
$$116$$ 0 0
$$117$$ 12.2027 1.12814
$$118$$ 0 0
$$119$$ 0.160961 0.0147552
$$120$$ 0 0
$$121$$ −10.8860 −0.989641
$$122$$ 0 0
$$123$$ −0.565670 −0.0510047
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 6.78390 0.601974 0.300987 0.953628i $$-0.402684\pi$$
0.300987 + 0.953628i $$0.402684\pi$$
$$128$$ 0 0
$$129$$ 0.173341 0.0152618
$$130$$ 0 0
$$131$$ 16.7091 1.45988 0.729941 0.683510i $$-0.239548\pi$$
0.729941 + 0.683510i $$0.239548\pi$$
$$132$$ 0 0
$$133$$ 0.0955795 0.00828780
$$134$$ 0 0
$$135$$ 3.04051 0.261686
$$136$$ 0 0
$$137$$ −9.00891 −0.769683 −0.384842 0.922983i $$-0.625744\pi$$
−0.384842 + 0.922983i $$0.625744\pi$$
$$138$$ 0 0
$$139$$ 7.43282 0.630444 0.315222 0.949018i $$-0.397921\pi$$
0.315222 + 0.949018i $$0.397921\pi$$
$$140$$ 0 0
$$141$$ −4.75919 −0.400796
$$142$$ 0 0
$$143$$ 1.51602 0.126776
$$144$$ 0 0
$$145$$ 6.61268 0.549153
$$146$$ 0 0
$$147$$ 3.71790 0.306647
$$148$$ 0 0
$$149$$ −4.98045 −0.408014 −0.204007 0.978969i $$-0.565397\pi$$
−0.204007 + 0.978969i $$0.565397\pi$$
$$150$$ 0 0
$$151$$ 19.8661 1.61668 0.808339 0.588717i $$-0.200367\pi$$
0.808339 + 0.588717i $$0.200367\pi$$
$$152$$ 0 0
$$153$$ 4.57584 0.369935
$$154$$ 0 0
$$155$$ −6.79080 −0.545450
$$156$$ 0 0
$$157$$ 13.9183 1.11080 0.555401 0.831583i $$-0.312565\pi$$
0.555401 + 0.831583i $$0.312565\pi$$
$$158$$ 0 0
$$159$$ −7.68946 −0.609814
$$160$$ 0 0
$$161$$ 0.240448 0.0189500
$$162$$ 0 0
$$163$$ −7.12837 −0.558337 −0.279168 0.960242i $$-0.590059\pi$$
−0.279168 + 0.960242i $$0.590059\pi$$
$$164$$ 0 0
$$165$$ 0.179527 0.0139762
$$166$$ 0 0
$$167$$ 23.0523 1.78384 0.891919 0.452195i $$-0.149359\pi$$
0.891919 + 0.452195i $$0.149359\pi$$
$$168$$ 0 0
$$169$$ 7.16888 0.551452
$$170$$ 0 0
$$171$$ 2.71717 0.207787
$$172$$ 0 0
$$173$$ −10.4279 −0.792815 −0.396408 0.918075i $$-0.629743\pi$$
−0.396408 + 0.918075i $$0.629743\pi$$
$$174$$ 0 0
$$175$$ −0.0955795 −0.00722513
$$176$$ 0 0
$$177$$ −0.638223 −0.0479718
$$178$$ 0 0
$$179$$ −13.2047 −0.986967 −0.493484 0.869755i $$-0.664277\pi$$
−0.493484 + 0.869755i $$0.664277\pi$$
$$180$$ 0 0
$$181$$ −12.3384 −0.917108 −0.458554 0.888667i $$-0.651632\pi$$
−0.458554 + 0.888667i $$0.651632\pi$$
$$182$$ 0 0
$$183$$ −0.643534 −0.0475714
$$184$$ 0 0
$$185$$ −6.28059 −0.461758
$$186$$ 0 0
$$187$$ 0.568484 0.0415717
$$188$$ 0 0
$$189$$ −0.290611 −0.0211388
$$190$$ 0 0
$$191$$ −18.3515 −1.32787 −0.663933 0.747792i $$-0.731114\pi$$
−0.663933 + 0.747792i $$0.731114\pi$$
$$192$$ 0 0
$$193$$ 5.74437 0.413489 0.206745 0.978395i $$-0.433713\pi$$
0.206745 + 0.978395i $$0.433713\pi$$
$$194$$ 0 0
$$195$$ 2.38840 0.171037
$$196$$ 0 0
$$197$$ 22.8975 1.63138 0.815688 0.578492i $$-0.196359\pi$$
0.815688 + 0.578492i $$0.196359\pi$$
$$198$$ 0 0
$$199$$ −4.38925 −0.311146 −0.155573 0.987824i $$-0.549722\pi$$
−0.155573 + 0.987824i $$0.549722\pi$$
$$200$$ 0 0
$$201$$ −0.0594109 −0.00419052
$$202$$ 0 0
$$203$$ −0.632037 −0.0443603
$$204$$ 0 0
$$205$$ 1.06364 0.0742881
$$206$$ 0 0
$$207$$ 6.83554 0.475103
$$208$$ 0 0
$$209$$ 0.337570 0.0233502
$$210$$ 0 0
$$211$$ 20.3932 1.40392 0.701961 0.712215i $$-0.252308\pi$$
0.701961 + 0.712215i $$0.252308\pi$$
$$212$$ 0 0
$$213$$ −7.92429 −0.542963
$$214$$ 0 0
$$215$$ −0.325938 −0.0222288
$$216$$ 0 0
$$217$$ 0.649061 0.0440611
$$218$$ 0 0
$$219$$ −1.31618 −0.0889390
$$220$$ 0 0
$$221$$ 7.56303 0.508744
$$222$$ 0 0
$$223$$ −27.1245 −1.81639 −0.908195 0.418548i $$-0.862539\pi$$
−0.908195 + 0.418548i $$0.862539\pi$$
$$224$$ 0 0
$$225$$ −2.71717 −0.181144
$$226$$ 0 0
$$227$$ −15.8638 −1.05292 −0.526460 0.850200i $$-0.676481\pi$$
−0.526460 + 0.850200i $$0.676481\pi$$
$$228$$ 0 0
$$229$$ 11.1600 0.737472 0.368736 0.929534i $$-0.379791\pi$$
0.368736 + 0.929534i $$0.379791\pi$$
$$230$$ 0 0
$$231$$ −0.0171591 −0.00112899
$$232$$ 0 0
$$233$$ −6.40848 −0.419833 −0.209917 0.977719i $$-0.567319\pi$$
−0.209917 + 0.977719i $$0.567319\pi$$
$$234$$ 0 0
$$235$$ 8.94884 0.583758
$$236$$ 0 0
$$237$$ 4.43456 0.288055
$$238$$ 0 0
$$239$$ 2.59204 0.167665 0.0838327 0.996480i $$-0.473284\pi$$
0.0838327 + 0.996480i $$0.473284\pi$$
$$240$$ 0 0
$$241$$ 3.86108 0.248714 0.124357 0.992238i $$-0.460313\pi$$
0.124357 + 0.992238i $$0.460313\pi$$
$$242$$ 0 0
$$243$$ −12.5967 −0.808080
$$244$$ 0 0
$$245$$ −6.99086 −0.446630
$$246$$ 0 0
$$247$$ 4.49098 0.285754
$$248$$ 0 0
$$249$$ −9.53599 −0.604319
$$250$$ 0 0
$$251$$ −13.7699 −0.869151 −0.434575 0.900635i $$-0.643102\pi$$
−0.434575 + 0.900635i $$0.643102\pi$$
$$252$$ 0 0
$$253$$ 0.849221 0.0533901
$$254$$ 0 0
$$255$$ 0.895614 0.0560856
$$256$$ 0 0
$$257$$ −10.1649 −0.634071 −0.317036 0.948414i $$-0.602687\pi$$
−0.317036 + 0.948414i $$0.602687\pi$$
$$258$$ 0 0
$$259$$ 0.600296 0.0373006
$$260$$ 0 0
$$261$$ −17.9677 −1.11217
$$262$$ 0 0
$$263$$ −9.49624 −0.585563 −0.292782 0.956179i $$-0.594581\pi$$
−0.292782 + 0.956179i $$0.594581\pi$$
$$264$$ 0 0
$$265$$ 14.4587 0.888192
$$266$$ 0 0
$$267$$ 7.27262 0.445077
$$268$$ 0 0
$$269$$ −7.83381 −0.477636 −0.238818 0.971064i $$-0.576760\pi$$
−0.238818 + 0.971064i $$0.576760\pi$$
$$270$$ 0 0
$$271$$ 24.7086 1.50094 0.750469 0.660906i $$-0.229828\pi$$
0.750469 + 0.660906i $$0.229828\pi$$
$$272$$ 0 0
$$273$$ −0.228282 −0.0138163
$$274$$ 0 0
$$275$$ −0.337570 −0.0203562
$$276$$ 0 0
$$277$$ 6.06342 0.364315 0.182158 0.983269i $$-0.441692\pi$$
0.182158 + 0.983269i $$0.441692\pi$$
$$278$$ 0 0
$$279$$ 18.4517 1.10468
$$280$$ 0 0
$$281$$ 24.7958 1.47919 0.739596 0.673051i $$-0.235016\pi$$
0.739596 + 0.673051i $$0.235016\pi$$
$$282$$ 0 0
$$283$$ −16.1626 −0.960765 −0.480382 0.877059i $$-0.659502\pi$$
−0.480382 + 0.877059i $$0.659502\pi$$
$$284$$ 0 0
$$285$$ 0.531822 0.0315024
$$286$$ 0 0
$$287$$ −0.101663 −0.00600096
$$288$$ 0 0
$$289$$ −14.1640 −0.833175
$$290$$ 0 0
$$291$$ 5.70184 0.334248
$$292$$ 0 0
$$293$$ 8.52258 0.497895 0.248947 0.968517i $$-0.419915\pi$$
0.248947 + 0.968517i $$0.419915\pi$$
$$294$$ 0 0
$$295$$ 1.20007 0.0698707
$$296$$ 0 0
$$297$$ −1.02639 −0.0595570
$$298$$ 0 0
$$299$$ 11.2979 0.653375
$$300$$ 0 0
$$301$$ 0.0311530 0.00179563
$$302$$ 0 0
$$303$$ −1.94776 −0.111896
$$304$$ 0 0
$$305$$ 1.21006 0.0692876
$$306$$ 0 0
$$307$$ 24.4232 1.39390 0.696952 0.717118i $$-0.254539\pi$$
0.696952 + 0.717118i $$0.254539\pi$$
$$308$$ 0 0
$$309$$ −9.86140 −0.560995
$$310$$ 0 0
$$311$$ 7.67265 0.435076 0.217538 0.976052i $$-0.430197\pi$$
0.217538 + 0.976052i $$0.430197\pi$$
$$312$$ 0 0
$$313$$ 2.02932 0.114704 0.0573518 0.998354i $$-0.481734\pi$$
0.0573518 + 0.998354i $$0.481734\pi$$
$$314$$ 0 0
$$315$$ 0.259705 0.0146327
$$316$$ 0 0
$$317$$ 5.44515 0.305830 0.152915 0.988239i $$-0.451134\pi$$
0.152915 + 0.988239i $$0.451134\pi$$
$$318$$ 0 0
$$319$$ −2.23224 −0.124982
$$320$$ 0 0
$$321$$ −7.71268 −0.430480
$$322$$ 0 0
$$323$$ 1.68405 0.0937030
$$324$$ 0 0
$$325$$ −4.49098 −0.249115
$$326$$ 0 0
$$327$$ −5.76253 −0.318668
$$328$$ 0 0
$$329$$ −0.855326 −0.0471556
$$330$$ 0 0
$$331$$ −6.87780 −0.378038 −0.189019 0.981973i $$-0.560531\pi$$
−0.189019 + 0.981973i $$0.560531\pi$$
$$332$$ 0 0
$$333$$ 17.0654 0.935179
$$334$$ 0 0
$$335$$ 0.111712 0.00610348
$$336$$ 0 0
$$337$$ −4.21120 −0.229398 −0.114699 0.993400i $$-0.536590\pi$$
−0.114699 + 0.993400i $$0.536590\pi$$
$$338$$ 0 0
$$339$$ 9.15761 0.497373
$$340$$ 0 0
$$341$$ 2.29237 0.124139
$$342$$ 0 0
$$343$$ 1.33724 0.0722042
$$344$$ 0 0
$$345$$ 1.33790 0.0720301
$$346$$ 0 0
$$347$$ −3.64180 −0.195502 −0.0977510 0.995211i $$-0.531165\pi$$
−0.0977510 + 0.995211i $$0.531165\pi$$
$$348$$ 0 0
$$349$$ −28.5835 −1.53004 −0.765020 0.644006i $$-0.777271\pi$$
−0.765020 + 0.644006i $$0.777271\pi$$
$$350$$ 0 0
$$351$$ −13.6549 −0.728844
$$352$$ 0 0
$$353$$ 7.26564 0.386711 0.193356 0.981129i $$-0.438063\pi$$
0.193356 + 0.981129i $$0.438063\pi$$
$$354$$ 0 0
$$355$$ 14.9003 0.790824
$$356$$ 0 0
$$357$$ −0.0856024 −0.00453056
$$358$$ 0 0
$$359$$ 4.80851 0.253784 0.126892 0.991917i $$-0.459500\pi$$
0.126892 + 0.991917i $$0.459500\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 5.78944 0.303867
$$364$$ 0 0
$$365$$ 2.47485 0.129539
$$366$$ 0 0
$$367$$ 27.9947 1.46131 0.730655 0.682747i $$-0.239215\pi$$
0.730655 + 0.682747i $$0.239215\pi$$
$$368$$ 0 0
$$369$$ −2.89010 −0.150452
$$370$$ 0 0
$$371$$ −1.38196 −0.0717476
$$372$$ 0 0
$$373$$ 30.9677 1.60344 0.801722 0.597697i $$-0.203917\pi$$
0.801722 + 0.597697i $$0.203917\pi$$
$$374$$ 0 0
$$375$$ −0.531822 −0.0274632
$$376$$ 0 0
$$377$$ −29.6974 −1.52949
$$378$$ 0 0
$$379$$ −20.3238 −1.04396 −0.521981 0.852957i $$-0.674807\pi$$
−0.521981 + 0.852957i $$0.674807\pi$$
$$380$$ 0 0
$$381$$ −3.60783 −0.184835
$$382$$ 0 0
$$383$$ 19.3567 0.989082 0.494541 0.869154i $$-0.335336\pi$$
0.494541 + 0.869154i $$0.335336\pi$$
$$384$$ 0 0
$$385$$ 0.0322648 0.00164437
$$386$$ 0 0
$$387$$ 0.885627 0.0450190
$$388$$ 0 0
$$389$$ 24.5206 1.24324 0.621621 0.783318i $$-0.286474\pi$$
0.621621 + 0.783318i $$0.286474\pi$$
$$390$$ 0 0
$$391$$ 4.23654 0.214251
$$392$$ 0 0
$$393$$ −8.88627 −0.448253
$$394$$ 0 0
$$395$$ −8.33842 −0.419552
$$396$$ 0 0
$$397$$ 9.23699 0.463591 0.231796 0.972765i $$-0.425540\pi$$
0.231796 + 0.972765i $$0.425540\pi$$
$$398$$ 0 0
$$399$$ −0.0508313 −0.00254475
$$400$$ 0 0
$$401$$ 21.0570 1.05154 0.525768 0.850628i $$-0.323778\pi$$
0.525768 + 0.850628i $$0.323778\pi$$
$$402$$ 0 0
$$403$$ 30.4973 1.51918
$$404$$ 0 0
$$405$$ 6.53448 0.324701
$$406$$ 0 0
$$407$$ 2.12014 0.105091
$$408$$ 0 0
$$409$$ 29.4567 1.45654 0.728269 0.685291i $$-0.240325\pi$$
0.728269 + 0.685291i $$0.240325\pi$$
$$410$$ 0 0
$$411$$ 4.79114 0.236329
$$412$$ 0 0
$$413$$ −0.114702 −0.00564411
$$414$$ 0 0
$$415$$ 17.9308 0.880188
$$416$$ 0 0
$$417$$ −3.95294 −0.193576
$$418$$ 0 0
$$419$$ 26.4973 1.29448 0.647239 0.762287i $$-0.275923\pi$$
0.647239 + 0.762287i $$0.275923\pi$$
$$420$$ 0 0
$$421$$ −20.0442 −0.976894 −0.488447 0.872594i $$-0.662436\pi$$
−0.488447 + 0.872594i $$0.662436\pi$$
$$422$$ 0 0
$$423$$ −24.3155 −1.18226
$$424$$ 0 0
$$425$$ −1.68405 −0.0816884
$$426$$ 0 0
$$427$$ −0.115657 −0.00559701
$$428$$ 0 0
$$429$$ −0.806253 −0.0389262
$$430$$ 0 0
$$431$$ −17.6186 −0.848660 −0.424330 0.905508i $$-0.639490\pi$$
−0.424330 + 0.905508i $$0.639490\pi$$
$$432$$ 0 0
$$433$$ 8.99700 0.432368 0.216184 0.976353i $$-0.430639\pi$$
0.216184 + 0.976353i $$0.430639\pi$$
$$434$$ 0 0
$$435$$ −3.51677 −0.168616
$$436$$ 0 0
$$437$$ 2.51569 0.120342
$$438$$ 0 0
$$439$$ −16.0456 −0.765815 −0.382907 0.923787i $$-0.625077\pi$$
−0.382907 + 0.923787i $$0.625077\pi$$
$$440$$ 0 0
$$441$$ 18.9953 0.904540
$$442$$ 0 0
$$443$$ −17.7087 −0.841365 −0.420683 0.907208i $$-0.638209\pi$$
−0.420683 + 0.907208i $$0.638209\pi$$
$$444$$ 0 0
$$445$$ −13.6749 −0.648253
$$446$$ 0 0
$$447$$ 2.64871 0.125280
$$448$$ 0 0
$$449$$ 29.2271 1.37931 0.689655 0.724138i $$-0.257762\pi$$
0.689655 + 0.724138i $$0.257762\pi$$
$$450$$ 0 0
$$451$$ −0.359054 −0.0169072
$$452$$ 0 0
$$453$$ −10.5652 −0.496397
$$454$$ 0 0
$$455$$ 0.429246 0.0201233
$$456$$ 0 0
$$457$$ −6.29269 −0.294359 −0.147180 0.989110i $$-0.547020\pi$$
−0.147180 + 0.989110i $$0.547020\pi$$
$$458$$ 0 0
$$459$$ −5.12038 −0.238999
$$460$$ 0 0
$$461$$ −8.76201 −0.408087 −0.204044 0.978962i $$-0.565408\pi$$
−0.204044 + 0.978962i $$0.565408\pi$$
$$462$$ 0 0
$$463$$ −3.21624 −0.149471 −0.0747357 0.997203i $$-0.523811\pi$$
−0.0747357 + 0.997203i $$0.523811\pi$$
$$464$$ 0 0
$$465$$ 3.61150 0.167479
$$466$$ 0 0
$$467$$ 31.4584 1.45572 0.727862 0.685724i $$-0.240514\pi$$
0.727862 + 0.685724i $$0.240514\pi$$
$$468$$ 0 0
$$469$$ −0.0106774 −0.000493035 0
$$470$$ 0 0
$$471$$ −7.40207 −0.341069
$$472$$ 0 0
$$473$$ 0.110027 0.00505904
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ −39.2867 −1.79881
$$478$$ 0 0
$$479$$ −13.7313 −0.627399 −0.313699 0.949522i $$-0.601568\pi$$
−0.313699 + 0.949522i $$0.601568\pi$$
$$480$$ 0 0
$$481$$ 28.2060 1.28608
$$482$$ 0 0
$$483$$ −0.127876 −0.00581855
$$484$$ 0 0
$$485$$ −10.7213 −0.486831
$$486$$ 0 0
$$487$$ −6.70766 −0.303953 −0.151977 0.988384i $$-0.548564\pi$$
−0.151977 + 0.988384i $$0.548564\pi$$
$$488$$ 0 0
$$489$$ 3.79102 0.171436
$$490$$ 0 0
$$491$$ −16.7613 −0.756429 −0.378214 0.925718i $$-0.623462\pi$$
−0.378214 + 0.925718i $$0.623462\pi$$
$$492$$ 0 0
$$493$$ −11.1361 −0.501543
$$494$$ 0 0
$$495$$ 0.917233 0.0412266
$$496$$ 0 0
$$497$$ −1.42416 −0.0638823
$$498$$ 0 0
$$499$$ −17.3105 −0.774926 −0.387463 0.921885i $$-0.626648\pi$$
−0.387463 + 0.921885i $$0.626648\pi$$
$$500$$ 0 0
$$501$$ −12.2597 −0.547723
$$502$$ 0 0
$$503$$ 38.7748 1.72888 0.864441 0.502734i $$-0.167673\pi$$
0.864441 + 0.502734i $$0.167673\pi$$
$$504$$ 0 0
$$505$$ 3.66243 0.162976
$$506$$ 0 0
$$507$$ −3.81257 −0.169322
$$508$$ 0 0
$$509$$ 3.39492 0.150477 0.0752385 0.997166i $$-0.476028\pi$$
0.0752385 + 0.997166i $$0.476028\pi$$
$$510$$ 0 0
$$511$$ −0.236545 −0.0104641
$$512$$ 0 0
$$513$$ −3.04051 −0.134242
$$514$$ 0 0
$$515$$ 18.5427 0.817087
$$516$$ 0 0
$$517$$ −3.02086 −0.132857
$$518$$ 0 0
$$519$$ 5.54576 0.243432
$$520$$ 0 0
$$521$$ 5.32985 0.233505 0.116753 0.993161i $$-0.462752\pi$$
0.116753 + 0.993161i $$0.462752\pi$$
$$522$$ 0 0
$$523$$ −7.55465 −0.330342 −0.165171 0.986265i $$-0.552818\pi$$
−0.165171 + 0.986265i $$0.552818\pi$$
$$524$$ 0 0
$$525$$ 0.0508313 0.00221846
$$526$$ 0 0
$$527$$ 11.4360 0.498161
$$528$$ 0 0
$$529$$ −16.6713 −0.724839
$$530$$ 0 0
$$531$$ −3.26078 −0.141506
$$532$$ 0 0
$$533$$ −4.77680 −0.206906
$$534$$ 0 0
$$535$$ 14.5024 0.626992
$$536$$ 0 0
$$537$$ 7.02256 0.303046
$$538$$ 0 0
$$539$$ 2.35991 0.101648
$$540$$ 0 0
$$541$$ −28.8850 −1.24186 −0.620931 0.783865i $$-0.713245\pi$$
−0.620931 + 0.783865i $$0.713245\pi$$
$$542$$ 0 0
$$543$$ 6.56185 0.281596
$$544$$ 0 0
$$545$$ 10.8354 0.464139
$$546$$ 0 0
$$547$$ 28.7106 1.22757 0.613787 0.789471i $$-0.289645\pi$$
0.613787 + 0.789471i $$0.289645\pi$$
$$548$$ 0 0
$$549$$ −3.28792 −0.140325
$$550$$ 0 0
$$551$$ −6.61268 −0.281709
$$552$$ 0 0
$$553$$ 0.796983 0.0338911
$$554$$ 0 0
$$555$$ 3.34016 0.141782
$$556$$ 0 0
$$557$$ 5.93658 0.251541 0.125771 0.992059i $$-0.459860\pi$$
0.125771 + 0.992059i $$0.459860\pi$$
$$558$$ 0 0
$$559$$ 1.46378 0.0619113
$$560$$ 0 0
$$561$$ −0.302333 −0.0127645
$$562$$ 0 0
$$563$$ 15.9031 0.670235 0.335117 0.942176i $$-0.391224\pi$$
0.335117 + 0.942176i $$0.391224\pi$$
$$564$$ 0 0
$$565$$ −17.2193 −0.724422
$$566$$ 0 0
$$567$$ −0.624563 −0.0262292
$$568$$ 0 0
$$569$$ −18.4451 −0.773258 −0.386629 0.922235i $$-0.626361\pi$$
−0.386629 + 0.922235i $$0.626361\pi$$
$$570$$ 0 0
$$571$$ 34.2148 1.43184 0.715922 0.698180i $$-0.246006\pi$$
0.715922 + 0.698180i $$0.246006\pi$$
$$572$$ 0 0
$$573$$ 9.75971 0.407718
$$574$$ 0 0
$$575$$ −2.51569 −0.104912
$$576$$ 0 0
$$577$$ 7.51233 0.312742 0.156371 0.987698i $$-0.450020\pi$$
0.156371 + 0.987698i $$0.450020\pi$$
$$578$$ 0 0
$$579$$ −3.05498 −0.126961
$$580$$ 0 0
$$581$$ −1.71382 −0.0711011
$$582$$ 0 0
$$583$$ −4.88083 −0.202143
$$584$$ 0 0
$$585$$ 12.2027 0.504521
$$586$$ 0 0
$$587$$ 18.0757 0.746064 0.373032 0.927819i $$-0.378318\pi$$
0.373032 + 0.927819i $$0.378318\pi$$
$$588$$ 0 0
$$589$$ 6.79080 0.279810
$$590$$ 0 0
$$591$$ −12.1774 −0.500910
$$592$$ 0 0
$$593$$ −2.52243 −0.103584 −0.0517919 0.998658i $$-0.516493\pi$$
−0.0517919 + 0.998658i $$0.516493\pi$$
$$594$$ 0 0
$$595$$ 0.160961 0.00659874
$$596$$ 0 0
$$597$$ 2.33430 0.0955366
$$598$$ 0 0
$$599$$ 34.6184 1.41447 0.707235 0.706978i $$-0.249942\pi$$
0.707235 + 0.706978i $$0.249942\pi$$
$$600$$ 0 0
$$601$$ 38.0768 1.55318 0.776592 0.630004i $$-0.216947\pi$$
0.776592 + 0.630004i $$0.216947\pi$$
$$602$$ 0 0
$$603$$ −0.303540 −0.0123611
$$604$$ 0 0
$$605$$ −10.8860 −0.442581
$$606$$ 0 0
$$607$$ 24.4616 0.992867 0.496433 0.868075i $$-0.334643\pi$$
0.496433 + 0.868075i $$0.334643\pi$$
$$608$$ 0 0
$$609$$ 0.336131 0.0136207
$$610$$ 0 0
$$611$$ −40.1890 −1.62587
$$612$$ 0 0
$$613$$ −16.6663 −0.673146 −0.336573 0.941657i $$-0.609268\pi$$
−0.336573 + 0.941657i $$0.609268\pi$$
$$614$$ 0 0
$$615$$ −0.565670 −0.0228100
$$616$$ 0 0
$$617$$ 40.2955 1.62224 0.811118 0.584883i $$-0.198860\pi$$
0.811118 + 0.584883i $$0.198860\pi$$
$$618$$ 0 0
$$619$$ 20.4210 0.820788 0.410394 0.911908i $$-0.365391\pi$$
0.410394 + 0.911908i $$0.365391\pi$$
$$620$$ 0 0
$$621$$ −7.64899 −0.306943
$$622$$ 0 0
$$623$$ 1.30704 0.0523655
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −0.179527 −0.00716962
$$628$$ 0 0
$$629$$ 10.5768 0.421726
$$630$$ 0 0
$$631$$ −3.64134 −0.144960 −0.0724798 0.997370i $$-0.523091\pi$$
−0.0724798 + 0.997370i $$0.523091\pi$$
$$632$$ 0 0
$$633$$ −10.8455 −0.431071
$$634$$ 0 0
$$635$$ 6.78390 0.269211
$$636$$ 0 0
$$637$$ 31.3958 1.24395
$$638$$ 0 0
$$639$$ −40.4865 −1.60162
$$640$$ 0 0
$$641$$ −30.6730 −1.21151 −0.605756 0.795651i $$-0.707129\pi$$
−0.605756 + 0.795651i $$0.707129\pi$$
$$642$$ 0 0
$$643$$ −10.9021 −0.429938 −0.214969 0.976621i $$-0.568965\pi$$
−0.214969 + 0.976621i $$0.568965\pi$$
$$644$$ 0 0
$$645$$ 0.173341 0.00682530
$$646$$ 0 0
$$647$$ −5.79113 −0.227673 −0.113836 0.993500i $$-0.536314\pi$$
−0.113836 + 0.993500i $$0.536314\pi$$
$$648$$ 0 0
$$649$$ −0.405107 −0.0159018
$$650$$ 0 0
$$651$$ −0.345185 −0.0135289
$$652$$ 0 0
$$653$$ 27.5926 1.07978 0.539890 0.841735i $$-0.318466\pi$$
0.539890 + 0.841735i $$0.318466\pi$$
$$654$$ 0 0
$$655$$ 16.7091 0.652879
$$656$$ 0 0
$$657$$ −6.72456 −0.262350
$$658$$ 0 0
$$659$$ −23.8541 −0.929224 −0.464612 0.885514i $$-0.653806\pi$$
−0.464612 + 0.885514i $$0.653806\pi$$
$$660$$ 0 0
$$661$$ 33.2311 1.29254 0.646270 0.763109i $$-0.276328\pi$$
0.646270 + 0.763109i $$0.276328\pi$$
$$662$$ 0 0
$$663$$ −4.02218 −0.156209
$$664$$ 0 0
$$665$$ 0.0955795 0.00370642
$$666$$ 0 0
$$667$$ −16.6354 −0.644127
$$668$$ 0 0
$$669$$ 14.4254 0.557718
$$670$$ 0 0
$$671$$ −0.408478 −0.0157691
$$672$$ 0 0
$$673$$ 25.6622 0.989206 0.494603 0.869119i $$-0.335313\pi$$
0.494603 + 0.869119i $$0.335313\pi$$
$$674$$ 0 0
$$675$$ 3.04051 0.117029
$$676$$ 0 0
$$677$$ 9.59071 0.368601 0.184300 0.982870i $$-0.440998\pi$$
0.184300 + 0.982870i $$0.440998\pi$$
$$678$$ 0 0
$$679$$ 1.02474 0.0393259
$$680$$ 0 0
$$681$$ 8.43674 0.323296
$$682$$ 0 0
$$683$$ 36.4798 1.39586 0.697929 0.716166i $$-0.254105\pi$$
0.697929 + 0.716166i $$0.254105\pi$$
$$684$$ 0 0
$$685$$ −9.00891 −0.344213
$$686$$ 0 0
$$687$$ −5.93512 −0.226439
$$688$$ 0 0
$$689$$ −64.9338 −2.47378
$$690$$ 0 0
$$691$$ −20.3671 −0.774799 −0.387400 0.921912i $$-0.626627\pi$$
−0.387400 + 0.921912i $$0.626627\pi$$
$$692$$ 0 0
$$693$$ −0.0876687 −0.00333026
$$694$$ 0 0
$$695$$ 7.43282 0.281943
$$696$$ 0 0
$$697$$ −1.79123 −0.0678476
$$698$$ 0 0
$$699$$ 3.40817 0.128909
$$700$$ 0 0
$$701$$ −37.7648 −1.42636 −0.713178 0.700983i $$-0.752745\pi$$
−0.713178 + 0.700983i $$0.752745\pi$$
$$702$$ 0 0
$$703$$ 6.28059 0.236877
$$704$$ 0 0
$$705$$ −4.75919 −0.179241
$$706$$ 0 0
$$707$$ −0.350053 −0.0131651
$$708$$ 0 0
$$709$$ −27.8421 −1.04563 −0.522817 0.852445i $$-0.675119\pi$$
−0.522817 + 0.852445i $$0.675119\pi$$
$$710$$ 0 0
$$711$$ 22.6569 0.849699
$$712$$ 0 0
$$713$$ 17.0835 0.639783
$$714$$ 0 0
$$715$$ 1.51602 0.0566959
$$716$$ 0 0
$$717$$ −1.37851 −0.0514813
$$718$$ 0 0
$$719$$ −30.1635 −1.12491 −0.562455 0.826828i $$-0.690143\pi$$
−0.562455 + 0.826828i $$0.690143\pi$$
$$720$$ 0 0
$$721$$ −1.77230 −0.0660039
$$722$$ 0 0
$$723$$ −2.05341 −0.0763670
$$724$$ 0 0
$$725$$ 6.61268 0.245589
$$726$$ 0 0
$$727$$ 17.8911 0.663544 0.331772 0.943360i $$-0.392354\pi$$
0.331772 + 0.943360i $$0.392354\pi$$
$$728$$ 0 0
$$729$$ −12.9042 −0.477934
$$730$$ 0 0
$$731$$ 0.548896 0.0203016
$$732$$ 0 0
$$733$$ −12.7778 −0.471958 −0.235979 0.971758i $$-0.575830\pi$$
−0.235979 + 0.971758i $$0.575830\pi$$
$$734$$ 0 0
$$735$$ 3.71790 0.137137
$$736$$ 0 0
$$737$$ −0.0377106 −0.00138909
$$738$$ 0 0
$$739$$ −19.5006 −0.717340 −0.358670 0.933464i $$-0.616770\pi$$
−0.358670 + 0.933464i $$0.616770\pi$$
$$740$$ 0 0
$$741$$ −2.38840 −0.0877401
$$742$$ 0 0
$$743$$ −41.0316 −1.50530 −0.752652 0.658418i $$-0.771226\pi$$
−0.752652 + 0.658418i $$0.771226\pi$$
$$744$$ 0 0
$$745$$ −4.98045 −0.182469
$$746$$ 0 0
$$747$$ −48.7209 −1.78261
$$748$$ 0 0
$$749$$ −1.38613 −0.0506481
$$750$$ 0 0
$$751$$ −23.6623 −0.863449 −0.431724 0.902006i $$-0.642095\pi$$
−0.431724 + 0.902006i $$0.642095\pi$$
$$752$$ 0 0
$$753$$ 7.32316 0.266871
$$754$$ 0 0
$$755$$ 19.8661 0.723000
$$756$$ 0 0
$$757$$ −7.29305 −0.265070 −0.132535 0.991178i $$-0.542312\pi$$
−0.132535 + 0.991178i $$0.542312\pi$$
$$758$$ 0 0
$$759$$ −0.451635 −0.0163933
$$760$$ 0 0
$$761$$ 34.8131 1.26197 0.630987 0.775793i $$-0.282650\pi$$
0.630987 + 0.775793i $$0.282650\pi$$
$$762$$ 0 0
$$763$$ −1.03565 −0.0374929
$$764$$ 0 0
$$765$$ 4.57584 0.165440
$$766$$ 0 0
$$767$$ −5.38948 −0.194603
$$768$$ 0 0
$$769$$ 12.5028 0.450862 0.225431 0.974259i $$-0.427621\pi$$
0.225431 + 0.974259i $$0.427621\pi$$
$$770$$ 0 0
$$771$$ 5.40594 0.194690
$$772$$ 0 0
$$773$$ −0.0576140 −0.00207223 −0.00103612 0.999999i $$-0.500330\pi$$
−0.00103612 + 0.999999i $$0.500330\pi$$
$$774$$ 0 0
$$775$$ −6.79080 −0.243933
$$776$$ 0 0
$$777$$ −0.319251 −0.0114531
$$778$$ 0 0
$$779$$ −1.06364 −0.0381090
$$780$$ 0 0
$$781$$ −5.02988 −0.179983
$$782$$ 0 0
$$783$$ 20.1059 0.718528
$$784$$ 0 0
$$785$$ 13.9183 0.496766
$$786$$ 0 0
$$787$$ 6.52348 0.232537 0.116268 0.993218i $$-0.462907\pi$$
0.116268 + 0.993218i $$0.462907\pi$$
$$788$$ 0 0
$$789$$ 5.05031 0.179796
$$790$$ 0 0
$$791$$ 1.64581 0.0585184
$$792$$ 0 0
$$793$$ −5.43433 −0.192979
$$794$$ 0 0
$$795$$ −7.68946 −0.272717
$$796$$ 0 0
$$797$$ 17.9775 0.636797 0.318399 0.947957i $$-0.396855\pi$$
0.318399 + 0.947957i $$0.396855\pi$$
$$798$$ 0 0
$$799$$ −15.0703 −0.533148
$$800$$ 0 0
$$801$$ 37.1570 1.31288
$$802$$ 0 0
$$803$$ −0.835433 −0.0294818
$$804$$ 0 0
$$805$$ 0.240448 0.00847470
$$806$$ 0 0
$$807$$ 4.16619 0.146657
$$808$$ 0 0
$$809$$ −9.96787 −0.350452 −0.175226 0.984528i $$-0.556066\pi$$
−0.175226 + 0.984528i $$0.556066\pi$$
$$810$$ 0 0
$$811$$ −7.56773 −0.265739 −0.132870 0.991134i $$-0.542419\pi$$
−0.132870 + 0.991134i $$0.542419\pi$$
$$812$$ 0 0
$$813$$ −13.1406 −0.460859
$$814$$ 0 0
$$815$$ −7.12837 −0.249696
$$816$$ 0 0
$$817$$ 0.325938 0.0114031
$$818$$ 0 0
$$819$$ −1.16633 −0.0407549
$$820$$ 0 0
$$821$$ −38.1434 −1.33122 −0.665608 0.746302i $$-0.731828\pi$$
−0.665608 + 0.746302i $$0.731828\pi$$
$$822$$ 0 0
$$823$$ 12.9509 0.451440 0.225720 0.974192i $$-0.427527\pi$$
0.225720 + 0.974192i $$0.427527\pi$$
$$824$$ 0 0
$$825$$ 0.179527 0.00625033
$$826$$ 0 0
$$827$$ 13.3538 0.464356 0.232178 0.972673i $$-0.425415\pi$$
0.232178 + 0.972673i $$0.425415\pi$$
$$828$$ 0 0
$$829$$ 15.5768 0.541005 0.270503 0.962719i $$-0.412810\pi$$
0.270503 + 0.962719i $$0.412810\pi$$
$$830$$ 0 0
$$831$$ −3.22466 −0.111862
$$832$$ 0 0
$$833$$ 11.7730 0.407909
$$834$$ 0 0
$$835$$ 23.0523 0.797757
$$836$$ 0 0
$$837$$ −20.6475 −0.713683
$$838$$ 0 0
$$839$$ 28.7351 0.992045 0.496023 0.868310i $$-0.334793\pi$$
0.496023 + 0.868310i $$0.334793\pi$$
$$840$$ 0 0
$$841$$ 14.7275 0.507845
$$842$$ 0 0
$$843$$ −13.1869 −0.454183
$$844$$ 0 0
$$845$$ 7.16888 0.246617
$$846$$ 0 0
$$847$$ 1.04048 0.0357514
$$848$$ 0 0
$$849$$ 8.59561 0.295000
$$850$$ 0 0
$$851$$ 15.8000 0.541618
$$852$$ 0 0
$$853$$ −52.3067 −1.79095 −0.895473 0.445116i $$-0.853163\pi$$
−0.895473 + 0.445116i $$0.853163\pi$$
$$854$$ 0 0
$$855$$ 2.71717 0.0929251
$$856$$ 0 0
$$857$$ −53.7133 −1.83481 −0.917405 0.397955i $$-0.869720\pi$$
−0.917405 + 0.397955i $$0.869720\pi$$
$$858$$ 0 0
$$859$$ −28.2653 −0.964399 −0.482199 0.876061i $$-0.660162\pi$$
−0.482199 + 0.876061i $$0.660162\pi$$
$$860$$ 0 0
$$861$$ 0.0540664 0.00184258
$$862$$ 0 0
$$863$$ −21.4628 −0.730603 −0.365301 0.930889i $$-0.619034\pi$$
−0.365301 + 0.930889i $$0.619034\pi$$
$$864$$ 0 0
$$865$$ −10.4279 −0.354558
$$866$$ 0 0
$$867$$ 7.53272 0.255824
$$868$$ 0 0
$$869$$ 2.81480 0.0954856
$$870$$ 0 0
$$871$$ −0.501696 −0.0169993
$$872$$ 0 0
$$873$$ 29.1316 0.985957
$$874$$ 0 0
$$875$$ −0.0955795 −0.00323118
$$876$$ 0 0
$$877$$ −7.69073 −0.259697 −0.129849 0.991534i $$-0.541449\pi$$
−0.129849 + 0.991534i $$0.541449\pi$$
$$878$$ 0 0
$$879$$ −4.53250 −0.152877
$$880$$ 0 0
$$881$$ 41.6429 1.40299 0.701493 0.712676i $$-0.252517\pi$$
0.701493 + 0.712676i $$0.252517\pi$$
$$882$$ 0 0
$$883$$ −9.41566 −0.316862 −0.158431 0.987370i $$-0.550644\pi$$
−0.158431 + 0.987370i $$0.550644\pi$$
$$884$$ 0 0
$$885$$ −0.638223 −0.0214536
$$886$$ 0 0
$$887$$ 24.2210 0.813263 0.406632 0.913592i $$-0.366703\pi$$
0.406632 + 0.913592i $$0.366703\pi$$
$$888$$ 0 0
$$889$$ −0.648402 −0.0217467
$$890$$ 0 0
$$891$$ −2.20585 −0.0738986
$$892$$ 0 0
$$893$$ −8.94884 −0.299461
$$894$$ 0 0
$$895$$ −13.2047 −0.441385
$$896$$ 0 0
$$897$$ −6.00848 −0.200617
$$898$$ 0 0
$$899$$ −44.9053 −1.49768
$$900$$ 0 0
$$901$$ −24.3492 −0.811189
$$902$$ 0 0
$$903$$ −0.0165679 −0.000551344 0
$$904$$ 0 0
$$905$$ −12.3384 −0.410143
$$906$$ 0 0
$$907$$ 35.2614 1.17083 0.585417 0.810732i $$-0.300931\pi$$
0.585417 + 0.810732i $$0.300931\pi$$
$$908$$ 0 0
$$909$$ −9.95143 −0.330068
$$910$$ 0 0
$$911$$ −58.6348 −1.94266 −0.971328 0.237742i $$-0.923593\pi$$
−0.971328 + 0.237742i $$0.923593\pi$$
$$912$$ 0 0
$$913$$ −6.05290 −0.200322
$$914$$ 0 0
$$915$$ −0.643534 −0.0212746
$$916$$ 0 0
$$917$$ −1.59705 −0.0527392
$$918$$ 0 0
$$919$$ −20.0824 −0.662458 −0.331229 0.943550i $$-0.607463\pi$$
−0.331229 + 0.943550i $$0.607463\pi$$
$$920$$ 0 0
$$921$$ −12.9888 −0.427995
$$922$$ 0 0
$$923$$ −66.9168 −2.20259
$$924$$ 0 0
$$925$$ −6.28059 −0.206505
$$926$$ 0 0
$$927$$ −50.3835 −1.65481
$$928$$ 0 0
$$929$$ −17.7731 −0.583116 −0.291558 0.956553i $$-0.594174\pi$$
−0.291558 + 0.956553i $$0.594174\pi$$
$$930$$ 0 0
$$931$$ 6.99086 0.229116
$$932$$ 0 0
$$933$$ −4.08049 −0.133589
$$934$$ 0 0
$$935$$ 0.568484 0.0185914
$$936$$ 0 0
$$937$$ −28.1246 −0.918792 −0.459396 0.888232i $$-0.651934\pi$$
−0.459396 + 0.888232i $$0.651934\pi$$
$$938$$ 0 0
$$939$$ −1.07923 −0.0352195
$$940$$ 0 0
$$941$$ 22.0914 0.720160 0.360080 0.932921i $$-0.382749\pi$$
0.360080 + 0.932921i $$0.382749\pi$$
$$942$$ 0 0
$$943$$ −2.67580 −0.0871360
$$944$$ 0 0
$$945$$ −0.290611 −0.00945358
$$946$$ 0 0
$$947$$ 38.3266 1.24545 0.622724 0.782442i $$-0.286026\pi$$
0.622724 + 0.782442i $$0.286026\pi$$
$$948$$ 0 0
$$949$$ −11.1145 −0.360791
$$950$$ 0 0
$$951$$ −2.89585 −0.0939044
$$952$$ 0 0
$$953$$ 14.8386 0.480670 0.240335 0.970690i $$-0.422743\pi$$
0.240335 + 0.970690i $$0.422743\pi$$
$$954$$ 0 0
$$955$$ −18.3515 −0.593839
$$956$$ 0 0
$$957$$ 1.18716 0.0383753
$$958$$ 0 0
$$959$$ 0.861067 0.0278053
$$960$$ 0 0
$$961$$ 15.1149 0.487578
$$962$$ 0 0
$$963$$ −39.4053 −1.26982
$$964$$ 0 0
$$965$$ 5.74437 0.184918
$$966$$ 0 0
$$967$$ 28.2875 0.909664 0.454832 0.890577i $$-0.349699\pi$$
0.454832 + 0.890577i $$0.349699\pi$$
$$968$$ 0 0
$$969$$ −0.895614 −0.0287713
$$970$$ 0 0
$$971$$ −41.8695 −1.34366 −0.671828 0.740707i $$-0.734490\pi$$
−0.671828 + 0.740707i $$0.734490\pi$$
$$972$$ 0 0
$$973$$ −0.710426 −0.0227752
$$974$$ 0 0
$$975$$ 2.38840 0.0764901
$$976$$ 0 0
$$977$$ 7.36966 0.235776 0.117888 0.993027i $$-0.462388\pi$$
0.117888 + 0.993027i $$0.462388\pi$$
$$978$$ 0 0
$$979$$ 4.61624 0.147536
$$980$$ 0 0
$$981$$ −29.4417 −0.940001
$$982$$ 0 0
$$983$$ −18.5640 −0.592098 −0.296049 0.955173i $$-0.595669\pi$$
−0.296049 + 0.955173i $$0.595669\pi$$
$$984$$ 0 0
$$985$$ 22.8975 0.729573
$$986$$ 0 0
$$987$$ 0.454881 0.0144790
$$988$$ 0 0
$$989$$ 0.819959 0.0260732
$$990$$ 0 0
$$991$$ 42.5301 1.35101 0.675506 0.737354i $$-0.263925\pi$$
0.675506 + 0.737354i $$0.263925\pi$$
$$992$$ 0 0
$$993$$ 3.65776 0.116076
$$994$$ 0 0
$$995$$ −4.38925 −0.139149
$$996$$ 0 0
$$997$$ 52.5775 1.66515 0.832574 0.553914i $$-0.186866\pi$$
0.832574 + 0.553914i $$0.186866\pi$$
$$998$$ 0 0
$$999$$ −19.0962 −0.604178
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.cl.1.2 5
4.3 odd 2 6080.2.a.ci.1.4 5
8.3 odd 2 3040.2.a.x.1.2 yes 5
8.5 even 2 3040.2.a.u.1.4 5

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.u.1.4 5 8.5 even 2
3040.2.a.x.1.2 yes 5 8.3 odd 2
6080.2.a.ci.1.4 5 4.3 odd 2
6080.2.a.cl.1.2 5 1.1 even 1 trivial