# Properties

 Label 1840.2.a.r.1.3 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ 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] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$14.6924739719$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 12$$ x^3 - x^2 - 9*x + 12 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.11903$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.11903 q^{3} -1.00000 q^{5} -4.50973 q^{7} +6.72833 q^{9} +O(q^{10})$$ $$q+3.11903 q^{3} -1.00000 q^{5} -4.50973 q^{7} +6.72833 q^{9} -4.33763 q^{11} -3.72833 q^{13} -3.11903 q^{15} +1.11903 q^{17} -4.50973 q^{19} -14.0660 q^{21} +1.00000 q^{23} +1.00000 q^{25} +11.6288 q^{27} -8.23805 q^{29} -1.72833 q^{31} -13.5292 q^{33} +4.50973 q^{35} -0.781399 q^{37} -11.6288 q^{39} +3.90043 q^{41} -8.00000 q^{43} -6.72833 q^{45} +11.4567 q^{47} +13.3376 q^{49} +3.49027 q^{51} -6.00000 q^{53} +4.33763 q^{55} -14.0660 q^{57} +2.23805 q^{59} +3.55623 q^{61} -30.3429 q^{63} +3.72833 q^{65} -2.43720 q^{67} +3.11903 q^{69} -7.11903 q^{71} -9.45665 q^{73} +3.11903 q^{75} +19.5615 q^{77} +14.9133 q^{79} +16.0854 q^{81} -2.78140 q^{83} -1.11903 q^{85} -25.6947 q^{87} -7.69471 q^{89} +16.8137 q^{91} -5.39070 q^{93} +4.50973 q^{95} -0.642920 q^{97} -29.1850 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{5} - 3 q^{7} + 10 q^{9}+O(q^{10})$$ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 10 * q^9 $$3 q - q^{3} - 3 q^{5} - 3 q^{7} + 10 q^{9} - 3 q^{11} - q^{13} + q^{15} - 7 q^{17} - 3 q^{19} - 22 q^{21} + 3 q^{23} + 3 q^{25} + 14 q^{27} - 4 q^{29} + 5 q^{31} - 9 q^{33} + 3 q^{35} - 2 q^{37} - 14 q^{39} + q^{41} - 24 q^{43} - 10 q^{45} + 14 q^{47} + 30 q^{49} + 21 q^{51} - 18 q^{53} + 3 q^{55} - 22 q^{57} - 14 q^{59} + q^{61} - 8 q^{63} + q^{65} - 8 q^{67} - q^{69} - 11 q^{71} - 8 q^{73} - q^{75} - 24 q^{77} + 4 q^{79} + 7 q^{81} - 8 q^{83} + 7 q^{85} - 36 q^{87} + 18 q^{89} - q^{91} - 16 q^{93} + 3 q^{95} - 33 q^{97} - 57 q^{99}+O(q^{100})$$ 3 * q - q^3 - 3 * q^5 - 3 * q^7 + 10 * q^9 - 3 * q^11 - q^13 + q^15 - 7 * q^17 - 3 * q^19 - 22 * q^21 + 3 * q^23 + 3 * q^25 + 14 * q^27 - 4 * q^29 + 5 * q^31 - 9 * q^33 + 3 * q^35 - 2 * q^37 - 14 * q^39 + q^41 - 24 * q^43 - 10 * q^45 + 14 * q^47 + 30 * q^49 + 21 * q^51 - 18 * q^53 + 3 * q^55 - 22 * q^57 - 14 * q^59 + q^61 - 8 * q^63 + q^65 - 8 * q^67 - q^69 - 11 * q^71 - 8 * q^73 - q^75 - 24 * q^77 + 4 * q^79 + 7 * q^81 - 8 * q^83 + 7 * q^85 - 36 * q^87 + 18 * q^89 - q^91 - 16 * q^93 + 3 * q^95 - 33 * q^97 - 57 * 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
$$3$$ 3.11903 1.80077 0.900385 0.435093i $$-0.143285\pi$$
0.900385 + 0.435093i $$0.143285\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −4.50973 −1.70452 −0.852258 0.523122i $$-0.824767\pi$$
−0.852258 + 0.523122i $$0.824767\pi$$
$$8$$ 0 0
$$9$$ 6.72833 2.24278
$$10$$ 0 0
$$11$$ −4.33763 −1.30784 −0.653922 0.756562i $$-0.726878\pi$$
−0.653922 + 0.756562i $$0.726878\pi$$
$$12$$ 0 0
$$13$$ −3.72833 −1.03405 −0.517026 0.855970i $$-0.672961\pi$$
−0.517026 + 0.855970i $$0.672961\pi$$
$$14$$ 0 0
$$15$$ −3.11903 −0.805329
$$16$$ 0 0
$$17$$ 1.11903 0.271404 0.135702 0.990750i $$-0.456671\pi$$
0.135702 + 0.990750i $$0.456671\pi$$
$$18$$ 0 0
$$19$$ −4.50973 −1.03460 −0.517301 0.855803i $$-0.673063\pi$$
−0.517301 + 0.855803i $$0.673063\pi$$
$$20$$ 0 0
$$21$$ −14.0660 −3.06944
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 11.6288 2.23795
$$28$$ 0 0
$$29$$ −8.23805 −1.52977 −0.764884 0.644168i $$-0.777204\pi$$
−0.764884 + 0.644168i $$0.777204\pi$$
$$30$$ 0 0
$$31$$ −1.72833 −0.310417 −0.155208 0.987882i $$-0.549605\pi$$
−0.155208 + 0.987882i $$0.549605\pi$$
$$32$$ 0 0
$$33$$ −13.5292 −2.35513
$$34$$ 0 0
$$35$$ 4.50973 0.762283
$$36$$ 0 0
$$37$$ −0.781399 −0.128461 −0.0642306 0.997935i $$-0.520459\pi$$
−0.0642306 + 0.997935i $$0.520459\pi$$
$$38$$ 0 0
$$39$$ −11.6288 −1.86209
$$40$$ 0 0
$$41$$ 3.90043 0.609144 0.304572 0.952489i $$-0.401487\pi$$
0.304572 + 0.952489i $$0.401487\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ −6.72833 −1.00300
$$46$$ 0 0
$$47$$ 11.4567 1.67112 0.835562 0.549396i $$-0.185142\pi$$
0.835562 + 0.549396i $$0.185142\pi$$
$$48$$ 0 0
$$49$$ 13.3376 1.90538
$$50$$ 0 0
$$51$$ 3.49027 0.488736
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 4.33763 0.584886
$$56$$ 0 0
$$57$$ −14.0660 −1.86308
$$58$$ 0 0
$$59$$ 2.23805 0.291370 0.145685 0.989331i $$-0.453461\pi$$
0.145685 + 0.989331i $$0.453461\pi$$
$$60$$ 0 0
$$61$$ 3.55623 0.455329 0.227664 0.973740i $$-0.426891\pi$$
0.227664 + 0.973740i $$0.426891\pi$$
$$62$$ 0 0
$$63$$ −30.3429 −3.82285
$$64$$ 0 0
$$65$$ 3.72833 0.462442
$$66$$ 0 0
$$67$$ −2.43720 −0.297752 −0.148876 0.988856i $$-0.547565\pi$$
−0.148876 + 0.988856i $$0.547565\pi$$
$$68$$ 0 0
$$69$$ 3.11903 0.375487
$$70$$ 0 0
$$71$$ −7.11903 −0.844873 −0.422437 0.906393i $$-0.638825\pi$$
−0.422437 + 0.906393i $$0.638825\pi$$
$$72$$ 0 0
$$73$$ −9.45665 −1.10682 −0.553409 0.832910i $$-0.686673\pi$$
−0.553409 + 0.832910i $$0.686673\pi$$
$$74$$ 0 0
$$75$$ 3.11903 0.360154
$$76$$ 0 0
$$77$$ 19.5615 2.22924
$$78$$ 0 0
$$79$$ 14.9133 1.67788 0.838939 0.544225i $$-0.183176\pi$$
0.838939 + 0.544225i $$0.183176\pi$$
$$80$$ 0 0
$$81$$ 16.0854 1.78727
$$82$$ 0 0
$$83$$ −2.78140 −0.305298 −0.152649 0.988280i $$-0.548780\pi$$
−0.152649 + 0.988280i $$0.548780\pi$$
$$84$$ 0 0
$$85$$ −1.11903 −0.121375
$$86$$ 0 0
$$87$$ −25.6947 −2.75476
$$88$$ 0 0
$$89$$ −7.69471 −0.815637 −0.407819 0.913063i $$-0.633710\pi$$
−0.407819 + 0.913063i $$0.633710\pi$$
$$90$$ 0 0
$$91$$ 16.8137 1.76256
$$92$$ 0 0
$$93$$ −5.39070 −0.558989
$$94$$ 0 0
$$95$$ 4.50973 0.462688
$$96$$ 0 0
$$97$$ −0.642920 −0.0652786 −0.0326393 0.999467i $$-0.510391\pi$$
−0.0326393 + 0.999467i $$0.510391\pi$$
$$98$$ 0 0
$$99$$ −29.1850 −2.93320
$$100$$ 0 0
$$101$$ −8.23805 −0.819717 −0.409858 0.912149i $$-0.634422\pi$$
−0.409858 + 0.912149i $$0.634422\pi$$
$$102$$ 0 0
$$103$$ −12.3376 −1.21566 −0.607831 0.794066i $$-0.707960\pi$$
−0.607831 + 0.794066i $$0.707960\pi$$
$$104$$ 0 0
$$105$$ 14.0660 1.37270
$$106$$ 0 0
$$107$$ 15.9328 1.54028 0.770139 0.637876i $$-0.220187\pi$$
0.770139 + 0.637876i $$0.220187\pi$$
$$108$$ 0 0
$$109$$ −1.49027 −0.142742 −0.0713712 0.997450i $$-0.522737\pi$$
−0.0713712 + 0.997450i $$0.522737\pi$$
$$110$$ 0 0
$$111$$ −2.43720 −0.231329
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ −25.0854 −2.31915
$$118$$ 0 0
$$119$$ −5.04650 −0.462612
$$120$$ 0 0
$$121$$ 7.81502 0.710456
$$122$$ 0 0
$$123$$ 12.1655 1.09693
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 0.675256 0.0599193 0.0299597 0.999551i $$-0.490462\pi$$
0.0299597 + 0.999551i $$0.490462\pi$$
$$128$$ 0 0
$$129$$ −24.9522 −2.19692
$$130$$ 0 0
$$131$$ 13.6947 1.19651 0.598256 0.801305i $$-0.295861\pi$$
0.598256 + 0.801305i $$0.295861\pi$$
$$132$$ 0 0
$$133$$ 20.3376 1.76350
$$134$$ 0 0
$$135$$ −11.6288 −1.00084
$$136$$ 0 0
$$137$$ 7.52918 0.643261 0.321631 0.946865i $$-0.395769\pi$$
0.321631 + 0.946865i $$0.395769\pi$$
$$138$$ 0 0
$$139$$ −4.67526 −0.396550 −0.198275 0.980146i $$-0.563534\pi$$
−0.198275 + 0.980146i $$0.563534\pi$$
$$140$$ 0 0
$$141$$ 35.7336 3.00931
$$142$$ 0 0
$$143$$ 16.1721 1.35238
$$144$$ 0 0
$$145$$ 8.23805 0.684133
$$146$$ 0 0
$$147$$ 41.6004 3.43114
$$148$$ 0 0
$$149$$ 7.52918 0.616814 0.308407 0.951254i $$-0.400204\pi$$
0.308407 + 0.951254i $$0.400204\pi$$
$$150$$ 0 0
$$151$$ 13.3571 1.08698 0.543492 0.839414i $$-0.317102\pi$$
0.543492 + 0.839414i $$0.317102\pi$$
$$152$$ 0 0
$$153$$ 7.52918 0.608698
$$154$$ 0 0
$$155$$ 1.72833 0.138823
$$156$$ 0 0
$$157$$ 16.2381 1.29594 0.647969 0.761667i $$-0.275619\pi$$
0.647969 + 0.761667i $$0.275619\pi$$
$$158$$ 0 0
$$159$$ −18.7142 −1.48413
$$160$$ 0 0
$$161$$ −4.50973 −0.355416
$$162$$ 0 0
$$163$$ 3.29112 0.257781 0.128890 0.991659i $$-0.458858\pi$$
0.128890 + 0.991659i $$0.458858\pi$$
$$164$$ 0 0
$$165$$ 13.5292 1.05325
$$166$$ 0 0
$$167$$ −22.9133 −1.77309 −0.886543 0.462647i $$-0.846900\pi$$
−0.886543 + 0.462647i $$0.846900\pi$$
$$168$$ 0 0
$$169$$ 0.900425 0.0692635
$$170$$ 0 0
$$171$$ −30.3429 −2.32038
$$172$$ 0 0
$$173$$ 0.575681 0.0437683 0.0218841 0.999761i $$-0.493034\pi$$
0.0218841 + 0.999761i $$0.493034\pi$$
$$174$$ 0 0
$$175$$ −4.50973 −0.340903
$$176$$ 0 0
$$177$$ 6.98055 0.524690
$$178$$ 0 0
$$179$$ −5.01945 −0.375171 −0.187586 0.982248i $$-0.560066\pi$$
−0.187586 + 0.982248i $$0.560066\pi$$
$$180$$ 0 0
$$181$$ −11.5292 −0.856957 −0.428479 0.903552i $$-0.640950\pi$$
−0.428479 + 0.903552i $$0.640950\pi$$
$$182$$ 0 0
$$183$$ 11.0920 0.819942
$$184$$ 0 0
$$185$$ 0.781399 0.0574496
$$186$$ 0 0
$$187$$ −4.85392 −0.354954
$$188$$ 0 0
$$189$$ −52.4425 −3.81463
$$190$$ 0 0
$$191$$ 18.7142 1.35411 0.677055 0.735933i $$-0.263256\pi$$
0.677055 + 0.735933i $$0.263256\pi$$
$$192$$ 0 0
$$193$$ 23.4956 1.69125 0.845624 0.533780i $$-0.179229\pi$$
0.845624 + 0.533780i $$0.179229\pi$$
$$194$$ 0 0
$$195$$ 11.6288 0.832752
$$196$$ 0 0
$$197$$ −18.1385 −1.29231 −0.646157 0.763205i $$-0.723625\pi$$
−0.646157 + 0.763205i $$0.723625\pi$$
$$198$$ 0 0
$$199$$ 23.2575 1.64868 0.824340 0.566094i $$-0.191546\pi$$
0.824340 + 0.566094i $$0.191546\pi$$
$$200$$ 0 0
$$201$$ −7.60170 −0.536183
$$202$$ 0 0
$$203$$ 37.1514 2.60751
$$204$$ 0 0
$$205$$ −3.90043 −0.272418
$$206$$ 0 0
$$207$$ 6.72833 0.467651
$$208$$ 0 0
$$209$$ 19.5615 1.35310
$$210$$ 0 0
$$211$$ −4.34420 −0.299067 −0.149533 0.988757i $$-0.547777\pi$$
−0.149533 + 0.988757i $$0.547777\pi$$
$$212$$ 0 0
$$213$$ −22.2044 −1.52142
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 7.79428 0.529110
$$218$$ 0 0
$$219$$ −29.4956 −1.99313
$$220$$ 0 0
$$221$$ −4.17210 −0.280646
$$222$$ 0 0
$$223$$ −12.4761 −0.835462 −0.417731 0.908571i $$-0.637175\pi$$
−0.417731 + 0.908571i $$0.637175\pi$$
$$224$$ 0 0
$$225$$ 6.72833 0.448555
$$226$$ 0 0
$$227$$ −15.9328 −1.05749 −0.528747 0.848779i $$-0.677338\pi$$
−0.528747 + 0.848779i $$0.677338\pi$$
$$228$$ 0 0
$$229$$ −3.56280 −0.235436 −0.117718 0.993047i $$-0.537558\pi$$
−0.117718 + 0.993047i $$0.537558\pi$$
$$230$$ 0 0
$$231$$ 61.0129 4.01435
$$232$$ 0 0
$$233$$ −27.4956 −1.80129 −0.900647 0.434552i $$-0.856907\pi$$
−0.900647 + 0.434552i $$0.856907\pi$$
$$234$$ 0 0
$$235$$ −11.4567 −0.747350
$$236$$ 0 0
$$237$$ 46.5150 3.02147
$$238$$ 0 0
$$239$$ −10.0389 −0.649363 −0.324681 0.945823i $$-0.605257\pi$$
−0.324681 + 0.945823i $$0.605257\pi$$
$$240$$ 0 0
$$241$$ −23.6947 −1.52631 −0.763155 0.646215i $$-0.776351\pi$$
−0.763155 + 0.646215i $$0.776351\pi$$
$$242$$ 0 0
$$243$$ 15.2846 0.980505
$$244$$ 0 0
$$245$$ −13.3376 −0.852110
$$246$$ 0 0
$$247$$ 16.8137 1.06983
$$248$$ 0 0
$$249$$ −8.67526 −0.549772
$$250$$ 0 0
$$251$$ −12.4425 −0.785363 −0.392681 0.919675i $$-0.628452\pi$$
−0.392681 + 0.919675i $$0.628452\pi$$
$$252$$ 0 0
$$253$$ −4.33763 −0.272704
$$254$$ 0 0
$$255$$ −3.49027 −0.218569
$$256$$ 0 0
$$257$$ 5.45665 0.340377 0.170188 0.985412i $$-0.445562\pi$$
0.170188 + 0.985412i $$0.445562\pi$$
$$258$$ 0 0
$$259$$ 3.52389 0.218964
$$260$$ 0 0
$$261$$ −55.4283 −3.43093
$$262$$ 0 0
$$263$$ −0.138479 −0.00853895 −0.00426948 0.999991i $$-0.501359\pi$$
−0.00426948 + 0.999991i $$0.501359\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ −24.0000 −1.46878
$$268$$ 0 0
$$269$$ 14.6753 0.894766 0.447383 0.894342i $$-0.352356\pi$$
0.447383 + 0.894342i $$0.352356\pi$$
$$270$$ 0 0
$$271$$ 8.31058 0.504832 0.252416 0.967619i $$-0.418775\pi$$
0.252416 + 0.967619i $$0.418775\pi$$
$$272$$ 0 0
$$273$$ 52.4425 3.17396
$$274$$ 0 0
$$275$$ −4.33763 −0.261569
$$276$$ 0 0
$$277$$ 12.9133 0.775886 0.387943 0.921683i $$-0.373186\pi$$
0.387943 + 0.921683i $$0.373186\pi$$
$$278$$ 0 0
$$279$$ −11.6288 −0.696195
$$280$$ 0 0
$$281$$ 2.67526 0.159592 0.0797962 0.996811i $$-0.474573\pi$$
0.0797962 + 0.996811i $$0.474573\pi$$
$$282$$ 0 0
$$283$$ −0.742495 −0.0441367 −0.0220684 0.999756i $$-0.507025\pi$$
−0.0220684 + 0.999756i $$0.507025\pi$$
$$284$$ 0 0
$$285$$ 14.0660 0.833195
$$286$$ 0 0
$$287$$ −17.5898 −1.03830
$$288$$ 0 0
$$289$$ −15.7478 −0.926340
$$290$$ 0 0
$$291$$ −2.00528 −0.117552
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ −2.23805 −0.130305
$$296$$ 0 0
$$297$$ −50.4412 −2.92690
$$298$$ 0 0
$$299$$ −3.72833 −0.215615
$$300$$ 0 0
$$301$$ 36.0778 2.07949
$$302$$ 0 0
$$303$$ −25.6947 −1.47612
$$304$$ 0 0
$$305$$ −3.55623 −0.203629
$$306$$ 0 0
$$307$$ −30.5084 −1.74121 −0.870604 0.491984i $$-0.836272\pi$$
−0.870604 + 0.491984i $$0.836272\pi$$
$$308$$ 0 0
$$309$$ −38.4814 −2.18913
$$310$$ 0 0
$$311$$ −5.56280 −0.315437 −0.157719 0.987484i $$-0.550414\pi$$
−0.157719 + 0.987484i $$0.550414\pi$$
$$312$$ 0 0
$$313$$ 4.07252 0.230193 0.115096 0.993354i $$-0.463282\pi$$
0.115096 + 0.993354i $$0.463282\pi$$
$$314$$ 0 0
$$315$$ 30.3429 1.70963
$$316$$ 0 0
$$317$$ −6.16553 −0.346291 −0.173145 0.984896i $$-0.555393\pi$$
−0.173145 + 0.984896i $$0.555393\pi$$
$$318$$ 0 0
$$319$$ 35.7336 2.00070
$$320$$ 0 0
$$321$$ 49.6947 2.77369
$$322$$ 0 0
$$323$$ −5.04650 −0.280795
$$324$$ 0 0
$$325$$ −3.72833 −0.206810
$$326$$ 0 0
$$327$$ −4.64820 −0.257046
$$328$$ 0 0
$$329$$ −51.6664 −2.84846
$$330$$ 0 0
$$331$$ −27.5886 −1.51640 −0.758202 0.652019i $$-0.773922\pi$$
−0.758202 + 0.652019i $$0.773922\pi$$
$$332$$ 0 0
$$333$$ −5.25751 −0.288110
$$334$$ 0 0
$$335$$ 2.43720 0.133159
$$336$$ 0 0
$$337$$ −17.4230 −0.949093 −0.474547 0.880230i $$-0.657388\pi$$
−0.474547 + 0.880230i $$0.657388\pi$$
$$338$$ 0 0
$$339$$ −18.7142 −1.01641
$$340$$ 0 0
$$341$$ 7.49684 0.405977
$$342$$ 0 0
$$343$$ −28.5810 −1.54323
$$344$$ 0 0
$$345$$ −3.11903 −0.167923
$$346$$ 0 0
$$347$$ −4.88097 −0.262024 −0.131012 0.991381i $$-0.541823\pi$$
−0.131012 + 0.991381i $$0.541823\pi$$
$$348$$ 0 0
$$349$$ 24.0389 1.28677 0.643387 0.765542i $$-0.277529\pi$$
0.643387 + 0.765542i $$0.277529\pi$$
$$350$$ 0 0
$$351$$ −43.3558 −2.31416
$$352$$ 0 0
$$353$$ −14.3442 −0.763464 −0.381732 0.924273i $$-0.624672\pi$$
−0.381732 + 0.924273i $$0.624672\pi$$
$$354$$ 0 0
$$355$$ 7.11903 0.377839
$$356$$ 0 0
$$357$$ −15.7402 −0.833059
$$358$$ 0 0
$$359$$ −26.7814 −1.41347 −0.706734 0.707479i $$-0.749832\pi$$
−0.706734 + 0.707479i $$0.749832\pi$$
$$360$$ 0 0
$$361$$ 1.33763 0.0704015
$$362$$ 0 0
$$363$$ 24.3752 1.27937
$$364$$ 0 0
$$365$$ 9.45665 0.494984
$$366$$ 0 0
$$367$$ 20.4761 1.06884 0.534422 0.845218i $$-0.320529\pi$$
0.534422 + 0.845218i $$0.320529\pi$$
$$368$$ 0 0
$$369$$ 26.2433 1.36617
$$370$$ 0 0
$$371$$ 27.0584 1.40480
$$372$$ 0 0
$$373$$ −3.89386 −0.201616 −0.100808 0.994906i $$-0.532143\pi$$
−0.100808 + 0.994906i $$0.532143\pi$$
$$374$$ 0 0
$$375$$ −3.11903 −0.161066
$$376$$ 0 0
$$377$$ 30.7142 1.58186
$$378$$ 0 0
$$379$$ 30.3765 1.56034 0.780169 0.625569i $$-0.215133\pi$$
0.780169 + 0.625569i $$0.215133\pi$$
$$380$$ 0 0
$$381$$ 2.10614 0.107901
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −19.5615 −0.996947
$$386$$ 0 0
$$387$$ −53.8266 −2.73616
$$388$$ 0 0
$$389$$ −18.6818 −0.947206 −0.473603 0.880738i $$-0.657047\pi$$
−0.473603 + 0.880738i $$0.657047\pi$$
$$390$$ 0 0
$$391$$ 1.11903 0.0565916
$$392$$ 0 0
$$393$$ 42.7142 2.15464
$$394$$ 0 0
$$395$$ −14.9133 −0.750370
$$396$$ 0 0
$$397$$ −28.5757 −1.43417 −0.717086 0.696985i $$-0.754525\pi$$
−0.717086 + 0.696985i $$0.754525\pi$$
$$398$$ 0 0
$$399$$ 63.4336 3.17565
$$400$$ 0 0
$$401$$ 12.1061 0.604552 0.302276 0.953220i $$-0.402254\pi$$
0.302276 + 0.953220i $$0.402254\pi$$
$$402$$ 0 0
$$403$$ 6.44377 0.320987
$$404$$ 0 0
$$405$$ −16.0854 −0.799290
$$406$$ 0 0
$$407$$ 3.38942 0.168007
$$408$$ 0 0
$$409$$ 25.2911 1.25057 0.625283 0.780398i $$-0.284984\pi$$
0.625283 + 0.780398i $$0.284984\pi$$
$$410$$ 0 0
$$411$$ 23.4837 1.15837
$$412$$ 0 0
$$413$$ −10.0930 −0.496644
$$414$$ 0 0
$$415$$ 2.78140 0.136533
$$416$$ 0 0
$$417$$ −14.5822 −0.714096
$$418$$ 0 0
$$419$$ 17.3505 0.847628 0.423814 0.905749i $$-0.360691\pi$$
0.423814 + 0.905749i $$0.360691\pi$$
$$420$$ 0 0
$$421$$ 21.4230 1.04409 0.522047 0.852916i $$-0.325168\pi$$
0.522047 + 0.852916i $$0.325168\pi$$
$$422$$ 0 0
$$423$$ 77.0841 3.74796
$$424$$ 0 0
$$425$$ 1.11903 0.0542808
$$426$$ 0 0
$$427$$ −16.0376 −0.776115
$$428$$ 0 0
$$429$$ 50.4412 2.43532
$$430$$ 0 0
$$431$$ −22.5822 −1.08775 −0.543874 0.839167i $$-0.683043\pi$$
−0.543874 + 0.839167i $$0.683043\pi$$
$$432$$ 0 0
$$433$$ 1.01417 0.0487378 0.0243689 0.999703i $$-0.492242\pi$$
0.0243689 + 0.999703i $$0.492242\pi$$
$$434$$ 0 0
$$435$$ 25.6947 1.23197
$$436$$ 0 0
$$437$$ −4.50973 −0.215729
$$438$$ 0 0
$$439$$ 26.7478 1.27660 0.638301 0.769787i $$-0.279638\pi$$
0.638301 + 0.769787i $$0.279638\pi$$
$$440$$ 0 0
$$441$$ 89.7399 4.27333
$$442$$ 0 0
$$443$$ −10.2044 −0.484827 −0.242414 0.970173i $$-0.577939\pi$$
−0.242414 + 0.970173i $$0.577939\pi$$
$$444$$ 0 0
$$445$$ 7.69471 0.364764
$$446$$ 0 0
$$447$$ 23.4837 1.11074
$$448$$ 0 0
$$449$$ 38.7867 1.83046 0.915228 0.402936i $$-0.132010\pi$$
0.915228 + 0.402936i $$0.132010\pi$$
$$450$$ 0 0
$$451$$ −16.9186 −0.796665
$$452$$ 0 0
$$453$$ 41.6611 1.95741
$$454$$ 0 0
$$455$$ −16.8137 −0.788240
$$456$$ 0 0
$$457$$ 34.9522 1.63500 0.817498 0.575932i $$-0.195361\pi$$
0.817498 + 0.575932i $$0.195361\pi$$
$$458$$ 0 0
$$459$$ 13.0129 0.607389
$$460$$ 0 0
$$461$$ 16.3700 0.762425 0.381213 0.924487i $$-0.375507\pi$$
0.381213 + 0.924487i $$0.375507\pi$$
$$462$$ 0 0
$$463$$ −29.2186 −1.35790 −0.678952 0.734183i $$-0.737565\pi$$
−0.678952 + 0.734183i $$0.737565\pi$$
$$464$$ 0 0
$$465$$ 5.39070 0.249988
$$466$$ 0 0
$$467$$ −24.2770 −1.12340 −0.561702 0.827340i $$-0.689853\pi$$
−0.561702 + 0.827340i $$0.689853\pi$$
$$468$$ 0 0
$$469$$ 10.9911 0.507523
$$470$$ 0 0
$$471$$ 50.6469 2.33369
$$472$$ 0 0
$$473$$ 34.7010 1.59555
$$474$$ 0 0
$$475$$ −4.50973 −0.206920
$$476$$ 0 0
$$477$$ −40.3700 −1.84841
$$478$$ 0 0
$$479$$ −24.6080 −1.12437 −0.562185 0.827012i $$-0.690039\pi$$
−0.562185 + 0.827012i $$0.690039\pi$$
$$480$$ 0 0
$$481$$ 2.91331 0.132835
$$482$$ 0 0
$$483$$ −14.0660 −0.640023
$$484$$ 0 0
$$485$$ 0.642920 0.0291935
$$486$$ 0 0
$$487$$ 30.2381 1.37022 0.685108 0.728441i $$-0.259755\pi$$
0.685108 + 0.728441i $$0.259755\pi$$
$$488$$ 0 0
$$489$$ 10.2651 0.464204
$$490$$ 0 0
$$491$$ −12.3311 −0.556493 −0.278246 0.960510i $$-0.589753\pi$$
−0.278246 + 0.960510i $$0.589753\pi$$
$$492$$ 0 0
$$493$$ −9.21860 −0.415185
$$494$$ 0 0
$$495$$ 29.1850 1.31177
$$496$$ 0 0
$$497$$ 32.1049 1.44010
$$498$$ 0 0
$$499$$ 26.9133 1.20481 0.602403 0.798192i $$-0.294210\pi$$
0.602403 + 0.798192i $$0.294210\pi$$
$$500$$ 0 0
$$501$$ −71.4672 −3.19292
$$502$$ 0 0
$$503$$ −20.5097 −0.914483 −0.457242 0.889342i $$-0.651163\pi$$
−0.457242 + 0.889342i $$0.651163\pi$$
$$504$$ 0 0
$$505$$ 8.23805 0.366589
$$506$$ 0 0
$$507$$ 2.80845 0.124728
$$508$$ 0 0
$$509$$ −36.7142 −1.62733 −0.813663 0.581336i $$-0.802530\pi$$
−0.813663 + 0.581336i $$0.802530\pi$$
$$510$$ 0 0
$$511$$ 42.6469 1.88659
$$512$$ 0 0
$$513$$ −52.4425 −2.31539
$$514$$ 0 0
$$515$$ 12.3376 0.543661
$$516$$ 0 0
$$517$$ −49.6947 −2.18557
$$518$$ 0 0
$$519$$ 1.79557 0.0788166
$$520$$ 0 0
$$521$$ −4.91331 −0.215256 −0.107628 0.994191i $$-0.534326\pi$$
−0.107628 + 0.994191i $$0.534326\pi$$
$$522$$ 0 0
$$523$$ 0.344196 0.0150506 0.00752531 0.999972i $$-0.497605\pi$$
0.00752531 + 0.999972i $$0.497605\pi$$
$$524$$ 0 0
$$525$$ −14.0660 −0.613889
$$526$$ 0 0
$$527$$ −1.93404 −0.0842483
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 15.0584 0.653477
$$532$$ 0 0
$$533$$ −14.5421 −0.629887
$$534$$ 0 0
$$535$$ −15.9328 −0.688833
$$536$$ 0 0
$$537$$ −15.6558 −0.675598
$$538$$ 0 0
$$539$$ −57.8537 −2.49193
$$540$$ 0 0
$$541$$ −6.13191 −0.263631 −0.131816 0.991274i $$-0.542081\pi$$
−0.131816 + 0.991274i $$0.542081\pi$$
$$542$$ 0 0
$$543$$ −35.9598 −1.54318
$$544$$ 0 0
$$545$$ 1.49027 0.0638363
$$546$$ 0 0
$$547$$ 9.18498 0.392721 0.196361 0.980532i $$-0.437088\pi$$
0.196361 + 0.980532i $$0.437088\pi$$
$$548$$ 0 0
$$549$$ 23.9275 1.02120
$$550$$ 0 0
$$551$$ 37.1514 1.58270
$$552$$ 0 0
$$553$$ −67.2549 −2.85997
$$554$$ 0 0
$$555$$ 2.43720 0.103454
$$556$$ 0 0
$$557$$ −4.30529 −0.182421 −0.0912105 0.995832i $$-0.529074\pi$$
−0.0912105 + 0.995832i $$0.529074\pi$$
$$558$$ 0 0
$$559$$ 29.8266 1.26153
$$560$$ 0 0
$$561$$ −15.1395 −0.639191
$$562$$ 0 0
$$563$$ −11.1256 −0.468888 −0.234444 0.972130i $$-0.575327\pi$$
−0.234444 + 0.972130i $$0.575327\pi$$
$$564$$ 0 0
$$565$$ 6.00000 0.252422
$$566$$ 0 0
$$567$$ −72.5408 −3.04643
$$568$$ 0 0
$$569$$ −16.0389 −0.672386 −0.336193 0.941793i $$-0.609139\pi$$
−0.336193 + 0.941793i $$0.609139\pi$$
$$570$$ 0 0
$$571$$ −17.9004 −0.749109 −0.374555 0.927205i $$-0.622204\pi$$
−0.374555 + 0.927205i $$0.622204\pi$$
$$572$$ 0 0
$$573$$ 58.3700 2.43844
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −9.12559 −0.379903 −0.189952 0.981793i $$-0.560833\pi$$
−0.189952 + 0.981793i $$0.560833\pi$$
$$578$$ 0 0
$$579$$ 73.2833 3.04555
$$580$$ 0 0
$$581$$ 12.5433 0.520386
$$582$$ 0 0
$$583$$ 26.0258 1.07788
$$584$$ 0 0
$$585$$ 25.0854 1.03715
$$586$$ 0 0
$$587$$ 33.6340 1.38823 0.694113 0.719866i $$-0.255797\pi$$
0.694113 + 0.719866i $$0.255797\pi$$
$$588$$ 0 0
$$589$$ 7.79428 0.321158
$$590$$ 0 0
$$591$$ −56.5744 −2.32716
$$592$$ 0 0
$$593$$ −17.4567 −0.716859 −0.358429 0.933557i $$-0.616688\pi$$
−0.358429 + 0.933557i $$0.616688\pi$$
$$594$$ 0 0
$$595$$ 5.04650 0.206886
$$596$$ 0 0
$$597$$ 72.5408 2.96890
$$598$$ 0 0
$$599$$ 11.5951 0.473764 0.236882 0.971538i $$-0.423874\pi$$
0.236882 + 0.971538i $$0.423874\pi$$
$$600$$ 0 0
$$601$$ −31.6611 −1.29148 −0.645741 0.763556i $$-0.723452\pi$$
−0.645741 + 0.763556i $$0.723452\pi$$
$$602$$ 0 0
$$603$$ −16.3983 −0.667790
$$604$$ 0 0
$$605$$ −7.81502 −0.317726
$$606$$ 0 0
$$607$$ 36.0778 1.46435 0.732177 0.681115i $$-0.238505\pi$$
0.732177 + 0.681115i $$0.238505\pi$$
$$608$$ 0 0
$$609$$ 115.876 4.69554
$$610$$ 0 0
$$611$$ −42.7142 −1.72803
$$612$$ 0 0
$$613$$ −32.0389 −1.29404 −0.647020 0.762473i $$-0.723985\pi$$
−0.647020 + 0.762473i $$0.723985\pi$$
$$614$$ 0 0
$$615$$ −12.1655 −0.490562
$$616$$ 0 0
$$617$$ −13.3960 −0.539302 −0.269651 0.962958i $$-0.586908\pi$$
−0.269651 + 0.962958i $$0.586908\pi$$
$$618$$ 0 0
$$619$$ −37.1309 −1.49242 −0.746208 0.665713i $$-0.768128\pi$$
−0.746208 + 0.665713i $$0.768128\pi$$
$$620$$ 0 0
$$621$$ 11.6288 0.466646
$$622$$ 0 0
$$623$$ 34.7010 1.39027
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 61.0129 2.43662
$$628$$ 0 0
$$629$$ −0.874406 −0.0348648
$$630$$ 0 0
$$631$$ −11.1125 −0.442380 −0.221190 0.975231i $$-0.570994\pi$$
−0.221190 + 0.975231i $$0.570994\pi$$
$$632$$ 0 0
$$633$$ −13.5497 −0.538551
$$634$$ 0 0
$$635$$ −0.675256 −0.0267967
$$636$$ 0 0
$$637$$ −49.7270 −1.97026
$$638$$ 0 0
$$639$$ −47.8991 −1.89486
$$640$$ 0 0
$$641$$ 12.3831 0.489103 0.244552 0.969636i $$-0.421359\pi$$
0.244552 + 0.969636i $$0.421359\pi$$
$$642$$ 0 0
$$643$$ 37.4956 1.47868 0.739340 0.673332i $$-0.235138\pi$$
0.739340 + 0.673332i $$0.235138\pi$$
$$644$$ 0 0
$$645$$ 24.9522 0.982492
$$646$$ 0 0
$$647$$ −14.5691 −0.572771 −0.286385 0.958114i $$-0.592454\pi$$
−0.286385 + 0.958114i $$0.592454\pi$$
$$648$$ 0 0
$$649$$ −9.70784 −0.381066
$$650$$ 0 0
$$651$$ 24.3106 0.952807
$$652$$ 0 0
$$653$$ −4.41672 −0.172840 −0.0864198 0.996259i $$-0.527543\pi$$
−0.0864198 + 0.996259i $$0.527543\pi$$
$$654$$ 0 0
$$655$$ −13.6947 −0.535097
$$656$$ 0 0
$$657$$ −63.6275 −2.48234
$$658$$ 0 0
$$659$$ 31.8655 1.24130 0.620652 0.784086i $$-0.286868\pi$$
0.620652 + 0.784086i $$0.286868\pi$$
$$660$$ 0 0
$$661$$ 33.1190 1.28818 0.644090 0.764949i $$-0.277236\pi$$
0.644090 + 0.764949i $$0.277236\pi$$
$$662$$ 0 0
$$663$$ −13.0129 −0.505379
$$664$$ 0 0
$$665$$ −20.3376 −0.788659
$$666$$ 0 0
$$667$$ −8.23805 −0.318979
$$668$$ 0 0
$$669$$ −38.9133 −1.50448
$$670$$ 0 0
$$671$$ −15.4256 −0.595499
$$672$$ 0 0
$$673$$ 19.3505 0.745907 0.372954 0.927850i $$-0.378345\pi$$
0.372954 + 0.927850i $$0.378345\pi$$
$$674$$ 0 0
$$675$$ 11.6288 0.447591
$$676$$ 0 0
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ 2.89939 0.111268
$$680$$ 0 0
$$681$$ −49.6947 −1.90431
$$682$$ 0 0
$$683$$ −38.3495 −1.46740 −0.733701 0.679472i $$-0.762209\pi$$
−0.733701 + 0.679472i $$0.762209\pi$$
$$684$$ 0 0
$$685$$ −7.52918 −0.287675
$$686$$ 0 0
$$687$$ −11.1125 −0.423967
$$688$$ 0 0
$$689$$ 22.3700 0.852228
$$690$$ 0 0
$$691$$ 21.0195 0.799618 0.399809 0.916599i $$-0.369077\pi$$
0.399809 + 0.916599i $$0.369077\pi$$
$$692$$ 0 0
$$693$$ 131.616 4.99969
$$694$$ 0 0
$$695$$ 4.67526 0.177343
$$696$$ 0 0
$$697$$ 4.36468 0.165324
$$698$$ 0 0
$$699$$ −85.7594 −3.24372
$$700$$ 0 0
$$701$$ 19.3169 0.729589 0.364794 0.931088i $$-0.381139\pi$$
0.364794 + 0.931088i $$0.381139\pi$$
$$702$$ 0 0
$$703$$ 3.52389 0.132906
$$704$$ 0 0
$$705$$ −35.7336 −1.34581
$$706$$ 0 0
$$707$$ 37.1514 1.39722
$$708$$ 0 0
$$709$$ 12.2315 0.459363 0.229682 0.973266i $$-0.426232\pi$$
0.229682 + 0.973266i $$0.426232\pi$$
$$710$$ 0 0
$$711$$ 100.342 3.76311
$$712$$ 0 0
$$713$$ −1.72833 −0.0647264
$$714$$ 0 0
$$715$$ −16.1721 −0.604802
$$716$$ 0 0
$$717$$ −31.3116 −1.16935
$$718$$ 0 0
$$719$$ 40.6416 1.51568 0.757839 0.652442i $$-0.226255\pi$$
0.757839 + 0.652442i $$0.226255\pi$$
$$720$$ 0 0
$$721$$ 55.6393 2.07212
$$722$$ 0 0
$$723$$ −73.9044 −2.74854
$$724$$ 0 0
$$725$$ −8.23805 −0.305954
$$726$$ 0 0
$$727$$ 23.4501 0.869716 0.434858 0.900499i $$-0.356799\pi$$
0.434858 + 0.900499i $$0.356799\pi$$
$$728$$ 0 0
$$729$$ −0.583281 −0.0216030
$$730$$ 0 0
$$731$$ −8.95221 −0.331110
$$732$$ 0 0
$$733$$ −12.5150 −0.462252 −0.231126 0.972924i $$-0.574241\pi$$
−0.231126 + 0.972924i $$0.574241\pi$$
$$734$$ 0 0
$$735$$ −41.6004 −1.53445
$$736$$ 0 0
$$737$$ 10.5717 0.389413
$$738$$ 0 0
$$739$$ 21.3505 0.785391 0.392696 0.919668i $$-0.371543\pi$$
0.392696 + 0.919668i $$0.371543\pi$$
$$740$$ 0 0
$$741$$ 52.4425 1.92652
$$742$$ 0 0
$$743$$ −24.9858 −0.916641 −0.458321 0.888787i $$-0.651549\pi$$
−0.458321 + 0.888787i $$0.651549\pi$$
$$744$$ 0 0
$$745$$ −7.52918 −0.275848
$$746$$ 0 0
$$747$$ −18.7142 −0.684715
$$748$$ 0 0
$$749$$ −71.8524 −2.62543
$$750$$ 0 0
$$751$$ 33.6275 1.22708 0.613542 0.789662i $$-0.289744\pi$$
0.613542 + 0.789662i $$0.289744\pi$$
$$752$$ 0 0
$$753$$ −38.8085 −1.41426
$$754$$ 0 0
$$755$$ −13.3571 −0.486114
$$756$$ 0 0
$$757$$ −37.1230 −1.34926 −0.674630 0.738156i $$-0.735697\pi$$
−0.674630 + 0.738156i $$0.735697\pi$$
$$758$$ 0 0
$$759$$ −13.5292 −0.491078
$$760$$ 0 0
$$761$$ −3.87337 −0.140410 −0.0702048 0.997533i $$-0.522365\pi$$
−0.0702048 + 0.997533i $$0.522365\pi$$
$$762$$ 0 0
$$763$$ 6.72073 0.243307
$$764$$ 0 0
$$765$$ −7.52918 −0.272218
$$766$$ 0 0
$$767$$ −8.34420 −0.301291
$$768$$ 0 0
$$769$$ 23.1645 0.835333 0.417667 0.908600i $$-0.362848\pi$$
0.417667 + 0.908600i $$0.362848\pi$$
$$770$$ 0 0
$$771$$ 17.0195 0.612941
$$772$$ 0 0
$$773$$ −8.78140 −0.315845 −0.157922 0.987452i $$-0.550480\pi$$
−0.157922 + 0.987452i $$0.550480\pi$$
$$774$$ 0 0
$$775$$ −1.72833 −0.0620834
$$776$$ 0 0
$$777$$ 10.9911 0.394304
$$778$$ 0 0
$$779$$ −17.5898 −0.630222
$$780$$ 0 0
$$781$$ 30.8797 1.10496
$$782$$ 0 0
$$783$$ −95.7983 −3.42355
$$784$$ 0 0
$$785$$ −16.2381 −0.579561
$$786$$ 0 0
$$787$$ −49.6275 −1.76903 −0.884514 0.466513i $$-0.845510\pi$$
−0.884514 + 0.466513i $$0.845510\pi$$
$$788$$ 0 0
$$789$$ −0.431918 −0.0153767
$$790$$ 0 0
$$791$$ 27.0584 0.962084
$$792$$ 0 0
$$793$$ −13.2588 −0.470833
$$794$$ 0 0
$$795$$ 18.7142 0.663723
$$796$$ 0 0
$$797$$ 18.3311 0.649319 0.324660 0.945831i $$-0.394750\pi$$
0.324660 + 0.945831i $$0.394750\pi$$
$$798$$ 0 0
$$799$$ 12.8203 0.453550
$$800$$ 0 0
$$801$$ −51.7725 −1.82929
$$802$$ 0 0
$$803$$ 41.0195 1.44755
$$804$$ 0 0
$$805$$ 4.50973 0.158947
$$806$$ 0 0
$$807$$ 45.7725 1.61127
$$808$$ 0 0
$$809$$ −1.93933 −0.0681832 −0.0340916 0.999419i $$-0.510854\pi$$
−0.0340916 + 0.999419i $$0.510854\pi$$
$$810$$ 0 0
$$811$$ 5.41775 0.190243 0.0951215 0.995466i $$-0.469676\pi$$
0.0951215 + 0.995466i $$0.469676\pi$$
$$812$$ 0 0
$$813$$ 25.9209 0.909086
$$814$$ 0 0
$$815$$ −3.29112 −0.115283
$$816$$ 0 0
$$817$$ 36.0778 1.26220
$$818$$ 0 0
$$819$$ 113.128 3.95302
$$820$$ 0 0
$$821$$ 37.8655 1.32152 0.660758 0.750599i $$-0.270235\pi$$
0.660758 + 0.750599i $$0.270235\pi$$
$$822$$ 0 0
$$823$$ −43.7336 −1.52446 −0.762229 0.647308i $$-0.775895\pi$$
−0.762229 + 0.647308i $$0.775895\pi$$
$$824$$ 0 0
$$825$$ −13.5292 −0.471026
$$826$$ 0 0
$$827$$ −28.1991 −0.980581 −0.490290 0.871559i $$-0.663109\pi$$
−0.490290 + 0.871559i $$0.663109\pi$$
$$828$$ 0 0
$$829$$ 1.12559 0.0390935 0.0195468 0.999809i $$-0.493778\pi$$
0.0195468 + 0.999809i $$0.493778\pi$$
$$830$$ 0 0
$$831$$ 40.2770 1.39719
$$832$$ 0 0
$$833$$ 14.9252 0.517126
$$834$$ 0 0
$$835$$ 22.9133 0.792948
$$836$$ 0 0
$$837$$ −20.0983 −0.694699
$$838$$ 0 0
$$839$$ −39.9328 −1.37863 −0.689316 0.724461i $$-0.742089\pi$$
−0.689316 + 0.724461i $$0.742089\pi$$
$$840$$ 0 0
$$841$$ 38.8655 1.34019
$$842$$ 0 0
$$843$$ 8.34420 0.287389
$$844$$ 0 0
$$845$$ −0.900425 −0.0309756
$$846$$ 0 0
$$847$$ −35.2436 −1.21098
$$848$$ 0 0
$$849$$ −2.31586 −0.0794801
$$850$$ 0 0
$$851$$ −0.781399 −0.0267860
$$852$$ 0 0
$$853$$ 47.9921 1.64322 0.821610 0.570050i $$-0.193076\pi$$
0.821610 + 0.570050i $$0.193076\pi$$
$$854$$ 0 0
$$855$$ 30.3429 1.03771
$$856$$ 0 0
$$857$$ −43.4283 −1.48348 −0.741742 0.670686i $$-0.766000\pi$$
−0.741742 + 0.670686i $$0.766000\pi$$
$$858$$ 0 0
$$859$$ −32.5433 −1.11036 −0.555182 0.831729i $$-0.687352\pi$$
−0.555182 + 0.831729i $$0.687352\pi$$
$$860$$ 0 0
$$861$$ −54.8632 −1.86973
$$862$$ 0 0
$$863$$ −46.1036 −1.56938 −0.784692 0.619886i $$-0.787179\pi$$
−0.784692 + 0.619886i $$0.787179\pi$$
$$864$$ 0 0
$$865$$ −0.575681 −0.0195738
$$866$$ 0 0
$$867$$ −49.1177 −1.66813
$$868$$ 0 0
$$869$$ −64.6884 −2.19440
$$870$$ 0 0
$$871$$ 9.08669 0.307891
$$872$$ 0 0
$$873$$ −4.32578 −0.146405
$$874$$ 0 0
$$875$$ 4.50973 0.152457
$$876$$ 0 0
$$877$$ −24.0996 −0.813785 −0.406892 0.913476i $$-0.633388\pi$$
−0.406892 + 0.913476i $$0.633388\pi$$
$$878$$ 0 0
$$879$$ −18.7142 −0.631213
$$880$$ 0 0
$$881$$ 2.34420 0.0789780 0.0394890 0.999220i $$-0.487427\pi$$
0.0394890 + 0.999220i $$0.487427\pi$$
$$882$$ 0 0
$$883$$ 41.0505 1.38146 0.690730 0.723113i $$-0.257289\pi$$
0.690730 + 0.723113i $$0.257289\pi$$
$$884$$ 0 0
$$885$$ −6.98055 −0.234649
$$886$$ 0 0
$$887$$ −54.7788 −1.83929 −0.919647 0.392747i $$-0.871525\pi$$
−0.919647 + 0.392747i $$0.871525\pi$$
$$888$$ 0 0
$$889$$ −3.04522 −0.102133
$$890$$ 0 0
$$891$$ −69.7725 −2.33747
$$892$$ 0 0
$$893$$ −51.6664 −1.72895
$$894$$ 0 0
$$895$$ 5.01945 0.167782
$$896$$ 0 0
$$897$$ −11.6288 −0.388273
$$898$$ 0 0
$$899$$ 14.2381 0.474866
$$900$$ 0 0
$$901$$ −6.71416 −0.223681
$$902$$ 0 0
$$903$$ 112.528 3.74469
$$904$$ 0 0
$$905$$ 11.5292 0.383243
$$906$$ 0 0
$$907$$ 10.1061 0.335569 0.167784 0.985824i $$-0.446339\pi$$
0.167784 + 0.985824i $$0.446339\pi$$
$$908$$ 0 0
$$909$$ −55.4283 −1.83844
$$910$$ 0 0
$$911$$ 25.4178 0.842128 0.421064 0.907031i $$-0.361657\pi$$
0.421064 + 0.907031i $$0.361657\pi$$
$$912$$ 0 0
$$913$$ 12.0647 0.399282
$$914$$ 0 0
$$915$$ −11.0920 −0.366689
$$916$$ 0 0
$$917$$ −61.7594 −2.03947
$$918$$ 0 0
$$919$$ −23.6017 −0.778548 −0.389274 0.921122i $$-0.627274\pi$$
−0.389274 + 0.921122i $$0.627274\pi$$
$$920$$ 0 0
$$921$$ −95.1566 −3.13552
$$922$$ 0 0
$$923$$ 26.5421 0.873643
$$924$$ 0 0
$$925$$ −0.781399 −0.0256922
$$926$$ 0 0
$$927$$ −83.0116 −2.72646
$$928$$ 0 0
$$929$$ −7.08669 −0.232507 −0.116253 0.993220i $$-0.537088\pi$$
−0.116253 + 0.993220i $$0.537088\pi$$
$$930$$ 0 0
$$931$$ −60.1490 −1.97131
$$932$$ 0 0
$$933$$ −17.3505 −0.568030
$$934$$ 0 0
$$935$$ 4.85392 0.158740
$$936$$ 0 0
$$937$$ 27.3169 0.892404 0.446202 0.894932i $$-0.352776\pi$$
0.446202 + 0.894932i $$0.352776\pi$$
$$938$$ 0 0
$$939$$ 12.7023 0.414524
$$940$$ 0 0
$$941$$ 55.8979 1.82222 0.911109 0.412165i $$-0.135227\pi$$
0.911109 + 0.412165i $$0.135227\pi$$
$$942$$ 0 0
$$943$$ 3.90043 0.127015
$$944$$ 0 0
$$945$$ 52.4425 1.70595
$$946$$ 0 0
$$947$$ −37.5939 −1.22164 −0.610818 0.791771i $$-0.709159\pi$$
−0.610818 + 0.791771i $$0.709159\pi$$
$$948$$ 0 0
$$949$$ 35.2575 1.14451
$$950$$ 0 0
$$951$$ −19.2305 −0.623590
$$952$$ 0 0
$$953$$ −29.3828 −0.951804 −0.475902 0.879498i $$-0.657878\pi$$
−0.475902 + 0.879498i $$0.657878\pi$$
$$954$$ 0 0
$$955$$ −18.7142 −0.605576
$$956$$ 0 0
$$957$$ 111.454 3.60280
$$958$$ 0 0
$$959$$ −33.9545 −1.09645
$$960$$ 0 0
$$961$$ −28.0129 −0.903641
$$962$$ 0 0
$$963$$ 107.201 3.45450
$$964$$ 0 0
$$965$$ −23.4956 −0.756349
$$966$$ 0 0
$$967$$ 49.2292 1.58310 0.791552 0.611102i $$-0.209274\pi$$
0.791552 + 0.611102i $$0.209274\pi$$
$$968$$ 0 0
$$969$$ −15.7402 −0.505647
$$970$$ 0 0
$$971$$ −9.62347 −0.308832 −0.154416 0.988006i $$-0.549350\pi$$
−0.154416 + 0.988006i $$0.549350\pi$$
$$972$$ 0 0
$$973$$ 21.0841 0.675926
$$974$$ 0 0
$$975$$ −11.6288 −0.372418
$$976$$ 0 0
$$977$$ 18.9858 0.607411 0.303705 0.952766i $$-0.401776\pi$$
0.303705 + 0.952766i $$0.401776\pi$$
$$978$$ 0 0
$$979$$ 33.3768 1.06673
$$980$$ 0 0
$$981$$ −10.0271 −0.320139
$$982$$ 0 0
$$983$$ −4.33763 −0.138349 −0.0691744 0.997605i $$-0.522037\pi$$
−0.0691744 + 0.997605i $$0.522037\pi$$
$$984$$ 0 0
$$985$$ 18.1385 0.577940
$$986$$ 0 0
$$987$$ −161.149 −5.12942
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ −1.96766 −0.0625049 −0.0312524 0.999512i $$-0.509950\pi$$
−0.0312524 + 0.999512i $$0.509950\pi$$
$$992$$ 0 0
$$993$$ −86.0495 −2.73070
$$994$$ 0 0
$$995$$ −23.2575 −0.737312
$$996$$ 0 0
$$997$$ 3.96110 0.125449 0.0627246 0.998031i $$-0.480021\pi$$
0.0627246 + 0.998031i $$0.480021\pi$$
$$998$$ 0 0
$$999$$ −9.08669 −0.287490
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 1840.2.a.r.1.3 3
4.3 odd 2 230.2.a.d.1.1 3
5.4 even 2 9200.2.a.cf.1.1 3
8.3 odd 2 7360.2.a.bz.1.3 3
8.5 even 2 7360.2.a.ce.1.1 3
12.11 even 2 2070.2.a.z.1.3 3
20.3 even 4 1150.2.b.j.599.1 6
20.7 even 4 1150.2.b.j.599.6 6
20.19 odd 2 1150.2.a.q.1.3 3
92.91 even 2 5290.2.a.r.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 3 4.3 odd 2
1150.2.a.q.1.3 3 20.19 odd 2
1150.2.b.j.599.1 6 20.3 even 4
1150.2.b.j.599.6 6 20.7 even 4
1840.2.a.r.1.3 3 1.1 even 1 trivial
2070.2.a.z.1.3 3 12.11 even 2
5290.2.a.r.1.1 3 92.91 even 2
7360.2.a.bz.1.3 3 8.3 odd 2
7360.2.a.ce.1.1 3 8.5 even 2
9200.2.a.cf.1.1 3 5.4 even 2