Properties

 Label 1008.6.a.bf.1.2 Level $1008$ Weight $6$ Character 1008.1 Self dual yes Analytic conductor $161.667$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,6,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1008.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: $$161.666890371$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{505})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 126$$ x^2 - x - 126 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$-10.7361$$ of defining polynomial Character $$\chi$$ $$=$$ 1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+28.4166 q^{5} -49.0000 q^{7} +O(q^{10})$$ $$q+28.4166 q^{5} -49.0000 q^{7} +424.083 q^{11} +508.500 q^{13} +539.916 q^{17} -2603.00 q^{19} +261.251 q^{23} -2317.50 q^{25} -6879.66 q^{29} -5687.00 q^{31} -1392.41 q^{35} +4909.50 q^{37} +5723.42 q^{41} +1733.99 q^{43} -10147.8 q^{47} +2401.00 q^{49} +31181.5 q^{53} +12051.0 q^{55} -38845.5 q^{59} +13651.0 q^{61} +14449.8 q^{65} +30741.5 q^{67} -45627.9 q^{71} +21753.5 q^{73} -20780.1 q^{77} -32295.5 q^{79} -46637.3 q^{83} +15342.6 q^{85} +63757.4 q^{89} -24916.5 q^{91} -73968.4 q^{95} +115122. q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 78 q^{5} - 98 q^{7}+O(q^{10})$$ 2 * q - 78 * q^5 - 98 * q^7 $$2 q - 78 q^{5} - 98 q^{7} + 174 q^{11} + 208 q^{13} - 1482 q^{17} - 352 q^{19} + 3354 q^{23} + 5882 q^{25} - 276 q^{29} - 6520 q^{31} + 3822 q^{35} + 13864 q^{37} + 12930 q^{41} - 12712 q^{43} - 28116 q^{47} + 4802 q^{49} + 46992 q^{53} + 38664 q^{55} - 65556 q^{59} - 13148 q^{61} + 46428 q^{65} + 75236 q^{67} - 66042 q^{71} + 60496 q^{73} - 8526 q^{77} + 34916 q^{79} - 82488 q^{83} + 230508 q^{85} - 42510 q^{89} - 10192 q^{91} - 313512 q^{95} + 213256 q^{97}+O(q^{100})$$ 2 * q - 78 * q^5 - 98 * q^7 + 174 * q^11 + 208 * q^13 - 1482 * q^17 - 352 * q^19 + 3354 * q^23 + 5882 * q^25 - 276 * q^29 - 6520 * q^31 + 3822 * q^35 + 13864 * q^37 + 12930 * q^41 - 12712 * q^43 - 28116 * q^47 + 4802 * q^49 + 46992 * q^53 + 38664 * q^55 - 65556 * q^59 - 13148 * q^61 + 46428 * q^65 + 75236 * q^67 - 66042 * q^71 + 60496 * q^73 - 8526 * q^77 + 34916 * q^79 - 82488 * q^83 + 230508 * q^85 - 42510 * q^89 - 10192 * q^91 - 313512 * q^95 + 213256 * q^97

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 0
$$4$$ 0 0
$$5$$ 28.4166 0.508332 0.254166 0.967161i $$-0.418199\pi$$
0.254166 + 0.967161i $$0.418199\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 424.083 1.05674 0.528371 0.849013i $$-0.322803\pi$$
0.528371 + 0.849013i $$0.322803\pi$$
$$12$$ 0 0
$$13$$ 508.500 0.834511 0.417256 0.908789i $$-0.362992\pi$$
0.417256 + 0.908789i $$0.362992\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 539.916 0.453110 0.226555 0.973998i $$-0.427254\pi$$
0.226555 + 0.973998i $$0.427254\pi$$
$$18$$ 0 0
$$19$$ −2603.00 −1.65421 −0.827104 0.562050i $$-0.810013\pi$$
−0.827104 + 0.562050i $$0.810013\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 261.251 0.102977 0.0514883 0.998674i $$-0.483604\pi$$
0.0514883 + 0.998674i $$0.483604\pi$$
$$24$$ 0 0
$$25$$ −2317.50 −0.741599
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6879.66 −1.51905 −0.759525 0.650478i $$-0.774569\pi$$
−0.759525 + 0.650478i $$0.774569\pi$$
$$30$$ 0 0
$$31$$ −5687.00 −1.06287 −0.531433 0.847100i $$-0.678346\pi$$
−0.531433 + 0.847100i $$0.678346\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1392.41 −0.192131
$$36$$ 0 0
$$37$$ 4909.50 0.589567 0.294783 0.955564i $$-0.404752\pi$$
0.294783 + 0.955564i $$0.404752\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5723.42 0.531736 0.265868 0.964009i $$-0.414342\pi$$
0.265868 + 0.964009i $$0.414342\pi$$
$$42$$ 0 0
$$43$$ 1733.99 0.143013 0.0715066 0.997440i $$-0.477219\pi$$
0.0715066 + 0.997440i $$0.477219\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −10147.8 −0.670083 −0.335042 0.942203i $$-0.608750\pi$$
−0.335042 + 0.942203i $$0.608750\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 31181.5 1.52478 0.762390 0.647118i $$-0.224026\pi$$
0.762390 + 0.647118i $$0.224026\pi$$
$$54$$ 0 0
$$55$$ 12051.0 0.537176
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −38845.5 −1.45282 −0.726408 0.687264i $$-0.758812\pi$$
−0.726408 + 0.687264i $$0.758812\pi$$
$$60$$ 0 0
$$61$$ 13651.0 0.469720 0.234860 0.972029i $$-0.424537\pi$$
0.234860 + 0.972029i $$0.424537\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 14449.8 0.424209
$$66$$ 0 0
$$67$$ 30741.5 0.836639 0.418320 0.908300i $$-0.362619\pi$$
0.418320 + 0.908300i $$0.362619\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −45627.9 −1.07420 −0.537099 0.843519i $$-0.680480\pi$$
−0.537099 + 0.843519i $$0.680480\pi$$
$$72$$ 0 0
$$73$$ 21753.5 0.477774 0.238887 0.971047i $$-0.423218\pi$$
0.238887 + 0.971047i $$0.423218\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −20780.1 −0.399411
$$78$$ 0 0
$$79$$ −32295.5 −0.582202 −0.291101 0.956692i $$-0.594022\pi$$
−0.291101 + 0.956692i $$0.594022\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −46637.3 −0.743085 −0.371542 0.928416i $$-0.621171\pi$$
−0.371542 + 0.928416i $$0.621171\pi$$
$$84$$ 0 0
$$85$$ 15342.6 0.230330
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 63757.4 0.853209 0.426604 0.904438i $$-0.359710\pi$$
0.426604 + 0.904438i $$0.359710\pi$$
$$90$$ 0 0
$$91$$ −24916.5 −0.315416
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −73968.4 −0.840886
$$96$$ 0 0
$$97$$ 115122. 1.24231 0.621156 0.783687i $$-0.286663\pi$$
0.621156 + 0.783687i $$0.286663\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −136803. −1.33442 −0.667208 0.744872i $$-0.732511\pi$$
−0.667208 + 0.744872i $$0.732511\pi$$
$$102$$ 0 0
$$103$$ 30426.0 0.282587 0.141293 0.989968i $$-0.454874\pi$$
0.141293 + 0.989968i $$0.454874\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −134612. −1.13664 −0.568321 0.822807i $$-0.692407\pi$$
−0.568321 + 0.822807i $$0.692407\pi$$
$$108$$ 0 0
$$109$$ 7722.00 0.0622535 0.0311267 0.999515i $$-0.490090\pi$$
0.0311267 + 0.999515i $$0.490090\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −214596. −1.58098 −0.790489 0.612476i $$-0.790174\pi$$
−0.790489 + 0.612476i $$0.790174\pi$$
$$114$$ 0 0
$$115$$ 7423.87 0.0523463
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −26455.9 −0.171259
$$120$$ 0 0
$$121$$ 18795.5 0.116705
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −154657. −0.885310
$$126$$ 0 0
$$127$$ −19445.6 −0.106982 −0.0534911 0.998568i $$-0.517035\pi$$
−0.0534911 + 0.998568i $$0.517035\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −63451.1 −0.323043 −0.161522 0.986869i $$-0.551640\pi$$
−0.161522 + 0.986869i $$0.551640\pi$$
$$132$$ 0 0
$$133$$ 127547. 0.625231
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 231176. 1.05231 0.526153 0.850390i $$-0.323634\pi$$
0.526153 + 0.850390i $$0.323634\pi$$
$$138$$ 0 0
$$139$$ −419126. −1.83996 −0.919978 0.391971i $$-0.871793\pi$$
−0.919978 + 0.391971i $$0.871793\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 215646. 0.881864
$$144$$ 0 0
$$145$$ −195497. −0.772182
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 19393.6 0.0715635 0.0357818 0.999360i $$-0.488608\pi$$
0.0357818 + 0.999360i $$0.488608\pi$$
$$150$$ 0 0
$$151$$ 545294. 1.94620 0.973102 0.230376i $$-0.0739956\pi$$
0.973102 + 0.230376i $$0.0739956\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −161605. −0.540289
$$156$$ 0 0
$$157$$ 9538.03 0.0308823 0.0154411 0.999881i $$-0.495085\pi$$
0.0154411 + 0.999881i $$0.495085\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −12801.3 −0.0389215
$$162$$ 0 0
$$163$$ 566744. 1.67078 0.835388 0.549661i $$-0.185243\pi$$
0.835388 + 0.549661i $$0.185243\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 501555. 1.39164 0.695821 0.718215i $$-0.255041\pi$$
0.695821 + 0.718215i $$0.255041\pi$$
$$168$$ 0 0
$$169$$ −112721. −0.303591
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −111671. −0.283677 −0.141839 0.989890i $$-0.545301\pi$$
−0.141839 + 0.989890i $$0.545301\pi$$
$$174$$ 0 0
$$175$$ 113557. 0.280298
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −528436. −1.23271 −0.616354 0.787470i $$-0.711391\pi$$
−0.616354 + 0.787470i $$0.711391\pi$$
$$180$$ 0 0
$$181$$ −290808. −0.659797 −0.329898 0.944016i $$-0.607014\pi$$
−0.329898 + 0.944016i $$0.607014\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 139511. 0.299696
$$186$$ 0 0
$$187$$ 228969. 0.478821
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 51989.0 0.103116 0.0515582 0.998670i $$-0.483581\pi$$
0.0515582 + 0.998670i $$0.483581\pi$$
$$192$$ 0 0
$$193$$ −526004. −1.01647 −0.508236 0.861218i $$-0.669702\pi$$
−0.508236 + 0.861218i $$0.669702\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −977503. −1.79454 −0.897269 0.441485i $$-0.854452\pi$$
−0.897269 + 0.441485i $$0.854452\pi$$
$$198$$ 0 0
$$199$$ −1.01780e6 −1.82192 −0.910962 0.412491i $$-0.864659\pi$$
−0.910962 + 0.412491i $$0.864659\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 337103. 0.574147
$$204$$ 0 0
$$205$$ 162640. 0.270298
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1.10389e6 −1.74807
$$210$$ 0 0
$$211$$ −248461. −0.384195 −0.192098 0.981376i $$-0.561529\pi$$
−0.192098 + 0.981376i $$0.561529\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 49274.2 0.0726982
$$216$$ 0 0
$$217$$ 278663. 0.401726
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 274547. 0.378125
$$222$$ 0 0
$$223$$ 118450. 0.159505 0.0797525 0.996815i $$-0.474587\pi$$
0.0797525 + 0.996815i $$0.474587\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 717556. 0.924254 0.462127 0.886814i $$-0.347086\pi$$
0.462127 + 0.886814i $$0.347086\pi$$
$$228$$ 0 0
$$229$$ −1.33467e6 −1.68184 −0.840918 0.541162i $$-0.817984\pi$$
−0.840918 + 0.541162i $$0.817984\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 68576.4 0.0827532 0.0413766 0.999144i $$-0.486826\pi$$
0.0413766 + 0.999144i $$0.486826\pi$$
$$234$$ 0 0
$$235$$ −288367. −0.340625
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 461972. 0.523144 0.261572 0.965184i $$-0.415759\pi$$
0.261572 + 0.965184i $$0.415759\pi$$
$$240$$ 0 0
$$241$$ 142028. 0.157518 0.0787592 0.996894i $$-0.474904\pi$$
0.0787592 + 0.996894i $$0.474904\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 68228.3 0.0726188
$$246$$ 0 0
$$247$$ −1.32362e6 −1.38045
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −672493. −0.673757 −0.336879 0.941548i $$-0.609371\pi$$
−0.336879 + 0.941548i $$0.609371\pi$$
$$252$$ 0 0
$$253$$ 110792. 0.108820
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1.16584e6 −1.10104 −0.550522 0.834821i $$-0.685571\pi$$
−0.550522 + 0.834821i $$0.685571\pi$$
$$258$$ 0 0
$$259$$ −240566. −0.222835
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −1.23336e6 −1.09952 −0.549758 0.835324i $$-0.685280\pi$$
−0.549758 + 0.835324i $$0.685280\pi$$
$$264$$ 0 0
$$265$$ 886073. 0.775094
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 71305.8 0.0600819 0.0300410 0.999549i $$-0.490436\pi$$
0.0300410 + 0.999549i $$0.490436\pi$$
$$270$$ 0 0
$$271$$ −1.20152e6 −0.993821 −0.496911 0.867802i $$-0.665532\pi$$
−0.496911 + 0.867802i $$0.665532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −982811. −0.783679
$$276$$ 0 0
$$277$$ −1.64304e6 −1.28661 −0.643307 0.765608i $$-0.722438\pi$$
−0.643307 + 0.765608i $$0.722438\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 37163.4 0.0280770 0.0140385 0.999901i $$-0.495531\pi$$
0.0140385 + 0.999901i $$0.495531\pi$$
$$282$$ 0 0
$$283$$ 19110.2 0.0141841 0.00709203 0.999975i $$-0.497743\pi$$
0.00709203 + 0.999975i $$0.497743\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −280447. −0.200977
$$288$$ 0 0
$$289$$ −1.12835e6 −0.794691
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1.49662e6 1.01846 0.509230 0.860631i $$-0.329930\pi$$
0.509230 + 0.860631i $$0.329930\pi$$
$$294$$ 0 0
$$295$$ −1.10386e6 −0.738513
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 132846. 0.0859351
$$300$$ 0 0
$$301$$ −84965.7 −0.0540539
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 387915. 0.238774
$$306$$ 0 0
$$307$$ 1.82699e6 1.10634 0.553172 0.833067i $$-0.313417\pi$$
0.553172 + 0.833067i $$0.313417\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −1.65586e6 −0.970784 −0.485392 0.874297i $$-0.661323\pi$$
−0.485392 + 0.874297i $$0.661323\pi$$
$$312$$ 0 0
$$313$$ −3.03384e6 −1.75038 −0.875189 0.483781i $$-0.839263\pi$$
−0.875189 + 0.483781i $$0.839263\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.40939e6 0.787741 0.393871 0.919166i $$-0.371136\pi$$
0.393871 + 0.919166i $$0.371136\pi$$
$$318$$ 0 0
$$319$$ −2.91755e6 −1.60524
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1.40540e6 −0.749538
$$324$$ 0 0
$$325$$ −1.17845e6 −0.618873
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 497244. 0.253268
$$330$$ 0 0
$$331$$ 942703. 0.472939 0.236469 0.971639i $$-0.424010\pi$$
0.236469 + 0.971639i $$0.424010\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 873570. 0.425290
$$336$$ 0 0
$$337$$ 1.23769e6 0.593658 0.296829 0.954931i $$-0.404071\pi$$
0.296829 + 0.954931i $$0.404071\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2.41176e6 −1.12318
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −4.10799e6 −1.83150 −0.915748 0.401754i $$-0.868401\pi$$
−0.915748 + 0.401754i $$0.868401\pi$$
$$348$$ 0 0
$$349$$ 705558. 0.310076 0.155038 0.987908i $$-0.450450\pi$$
0.155038 + 0.987908i $$0.450450\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 3.72785e6 1.59229 0.796143 0.605108i $$-0.206870\pi$$
0.796143 + 0.605108i $$0.206870\pi$$
$$354$$ 0 0
$$355$$ −1.29659e6 −0.546049
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −2.87917e6 −1.17905 −0.589524 0.807751i $$-0.700685\pi$$
−0.589524 + 0.807751i $$0.700685\pi$$
$$360$$ 0 0
$$361$$ 4.29950e6 1.73640
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 618161. 0.242868
$$366$$ 0 0
$$367$$ −1.31628e6 −0.510132 −0.255066 0.966924i $$-0.582097\pi$$
−0.255066 + 0.966924i $$0.582097\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1.52789e6 −0.576313
$$372$$ 0 0
$$373$$ 4.05141e6 1.50777 0.753884 0.657008i $$-0.228178\pi$$
0.753884 + 0.657008i $$0.228178\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −3.49831e6 −1.26766
$$378$$ 0 0
$$379$$ 4.92472e6 1.76110 0.880549 0.473954i $$-0.157174\pi$$
0.880549 + 0.473954i $$0.157174\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 4.55593e6 1.58701 0.793506 0.608562i $$-0.208253\pi$$
0.793506 + 0.608562i $$0.208253\pi$$
$$384$$ 0 0
$$385$$ −590499. −0.203033
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −3.68937e6 −1.23617 −0.618085 0.786112i $$-0.712091\pi$$
−0.618085 + 0.786112i $$0.712091\pi$$
$$390$$ 0 0
$$391$$ 141054. 0.0466597
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −917728. −0.295952
$$396$$ 0 0
$$397$$ 1.54578e6 0.492233 0.246117 0.969240i $$-0.420845\pi$$
0.246117 + 0.969240i $$0.420845\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5.66295e6 −1.75866 −0.879331 0.476212i $$-0.842010\pi$$
−0.879331 + 0.476212i $$0.842010\pi$$
$$402$$ 0 0
$$403$$ −2.89184e6 −0.886975
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2.08204e6 0.623020
$$408$$ 0 0
$$409$$ 727819. 0.215137 0.107568 0.994198i $$-0.465694\pi$$
0.107568 + 0.994198i $$0.465694\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1.90343e6 0.549113
$$414$$ 0 0
$$415$$ −1.32528e6 −0.377734
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −6.46306e6 −1.79847 −0.899235 0.437465i $$-0.855876\pi$$
−0.899235 + 0.437465i $$0.855876\pi$$
$$420$$ 0 0
$$421$$ −3.50166e6 −0.962874 −0.481437 0.876481i $$-0.659885\pi$$
−0.481437 + 0.876481i $$0.659885\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −1.25125e6 −0.336026
$$426$$ 0 0
$$427$$ −668898. −0.177538
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5.35407e6 1.38832 0.694162 0.719819i $$-0.255775\pi$$
0.694162 + 0.719819i $$0.255775\pi$$
$$432$$ 0 0
$$433$$ −1.54042e6 −0.394839 −0.197420 0.980319i $$-0.563256\pi$$
−0.197420 + 0.980319i $$0.563256\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −680036. −0.170345
$$438$$ 0 0
$$439$$ 3.00831e6 0.745009 0.372505 0.928030i $$-0.378499\pi$$
0.372505 + 0.928030i $$0.378499\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −6.93545e6 −1.67906 −0.839528 0.543317i $$-0.817168\pi$$
−0.839528 + 0.543317i $$0.817168\pi$$
$$444$$ 0 0
$$445$$ 1.81177e6 0.433713
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −5.41345e6 −1.26724 −0.633619 0.773646i $$-0.718431\pi$$
−0.633619 + 0.773646i $$0.718431\pi$$
$$450$$ 0 0
$$451$$ 2.42720e6 0.561908
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −708042. −0.160336
$$456$$ 0 0
$$457$$ 551975. 0.123632 0.0618158 0.998088i $$-0.480311\pi$$
0.0618158 + 0.998088i $$0.480311\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −4.44254e6 −0.973597 −0.486799 0.873514i $$-0.661835\pi$$
−0.486799 + 0.873514i $$0.661835\pi$$
$$462$$ 0 0
$$463$$ 4.62590e6 1.00287 0.501434 0.865196i $$-0.332806\pi$$
0.501434 + 0.865196i $$0.332806\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 5.50087e6 1.16718 0.583592 0.812047i $$-0.301647\pi$$
0.583592 + 0.812047i $$0.301647\pi$$
$$468$$ 0 0
$$469$$ −1.50633e6 −0.316220
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 735357. 0.151128
$$474$$ 0 0
$$475$$ 6.03244e6 1.22676
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −3.56375e6 −0.709690 −0.354845 0.934925i $$-0.615466\pi$$
−0.354845 + 0.934925i $$0.615466\pi$$
$$480$$ 0 0
$$481$$ 2.49648e6 0.492000
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3.27139e6 0.631507
$$486$$ 0 0
$$487$$ −3.74040e6 −0.714653 −0.357327 0.933979i $$-0.616312\pi$$
−0.357327 + 0.933979i $$0.616312\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 5.00459e6 0.936838 0.468419 0.883506i $$-0.344824\pi$$
0.468419 + 0.883506i $$0.344824\pi$$
$$492$$ 0 0
$$493$$ −3.71444e6 −0.688297
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.23577e6 0.406009
$$498$$ 0 0
$$499$$ −2.58167e6 −0.464141 −0.232070 0.972699i $$-0.574550\pi$$
−0.232070 + 0.972699i $$0.574550\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −4.02089e6 −0.708601 −0.354301 0.935132i $$-0.615281\pi$$
−0.354301 + 0.935132i $$0.615281\pi$$
$$504$$ 0 0
$$505$$ −3.88747e6 −0.678326
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −1.13802e7 −1.94696 −0.973479 0.228776i $$-0.926528\pi$$
−0.973479 + 0.228776i $$0.926528\pi$$
$$510$$ 0 0
$$511$$ −1.06592e6 −0.180581
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 864604. 0.143648
$$516$$ 0 0
$$517$$ −4.30353e6 −0.708106
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 4.72031e6 0.761862 0.380931 0.924603i $$-0.375604\pi$$
0.380931 + 0.924603i $$0.375604\pi$$
$$522$$ 0 0
$$523$$ 8.46281e6 1.35288 0.676441 0.736496i $$-0.263521\pi$$
0.676441 + 0.736496i $$0.263521\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −3.07050e6 −0.481596
$$528$$ 0 0
$$529$$ −6.36809e6 −0.989396
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 2.91036e6 0.443739
$$534$$ 0 0
$$535$$ −3.82521e6 −0.577792
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 1.01822e6 0.150963
$$540$$ 0 0
$$541$$ 1.09051e7 1.60190 0.800952 0.598728i $$-0.204327\pi$$
0.800952 + 0.598728i $$0.204327\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 219433. 0.0316454
$$546$$ 0 0
$$547$$ 5.04856e6 0.721438 0.360719 0.932675i $$-0.382531\pi$$
0.360719 + 0.932675i $$0.382531\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1.79077e7 2.51282
$$552$$ 0 0
$$553$$ 1.58248e6 0.220052
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1.09175e7 1.49103 0.745516 0.666488i $$-0.232203\pi$$
0.745516 + 0.666488i $$0.232203\pi$$
$$558$$ 0 0
$$559$$ 881735. 0.119346
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −2.16432e6 −0.287774 −0.143887 0.989594i $$-0.545960\pi$$
−0.143887 + 0.989594i $$0.545960\pi$$
$$564$$ 0 0
$$565$$ −6.09810e6 −0.803662
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −9.80819e6 −1.27001 −0.635007 0.772507i $$-0.719003\pi$$
−0.635007 + 0.772507i $$0.719003\pi$$
$$570$$ 0 0
$$571$$ −1.54139e7 −1.97844 −0.989221 0.146430i $$-0.953222\pi$$
−0.989221 + 0.146430i $$0.953222\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −605448. −0.0763673
$$576$$ 0 0
$$577$$ −7.40379e6 −0.925794 −0.462897 0.886412i $$-0.653190\pi$$
−0.462897 + 0.886412i $$0.653190\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2.28523e6 0.280860
$$582$$ 0 0
$$583$$ 1.32235e7 1.61130
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 9.83732e6 1.17837 0.589185 0.807998i $$-0.299449\pi$$
0.589185 + 0.807998i $$0.299449\pi$$
$$588$$ 0 0
$$589$$ 1.48032e7 1.75820
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −6.01144e6 −0.702007 −0.351004 0.936374i $$-0.614160\pi$$
−0.351004 + 0.936374i $$0.614160\pi$$
$$594$$ 0 0
$$595$$ −751786. −0.0870567
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −604201. −0.0688041 −0.0344020 0.999408i $$-0.510953\pi$$
−0.0344020 + 0.999408i $$0.510953\pi$$
$$600$$ 0 0
$$601$$ 2.52672e6 0.285346 0.142673 0.989770i $$-0.454430\pi$$
0.142673 + 0.989770i $$0.454430\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 534103. 0.0593249
$$606$$ 0 0
$$607$$ −1.29052e7 −1.42165 −0.710823 0.703371i $$-0.751677\pi$$
−0.710823 + 0.703371i $$0.751677\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5.16017e6 −0.559192
$$612$$ 0 0
$$613$$ −3.27264e6 −0.351760 −0.175880 0.984412i $$-0.556277\pi$$
−0.175880 + 0.984412i $$0.556277\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1.31134e7 −1.38676 −0.693381 0.720571i $$-0.743880\pi$$
−0.693381 + 0.720571i $$0.743880\pi$$
$$618$$ 0 0
$$619$$ 2.56630e6 0.269203 0.134602 0.990900i $$-0.457025\pi$$
0.134602 + 0.990900i $$0.457025\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −3.12411e6 −0.322483
$$624$$ 0 0
$$625$$ 2.84734e6 0.291567
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 2.65072e6 0.267139
$$630$$ 0 0
$$631$$ −1.08637e7 −1.08618 −0.543092 0.839673i $$-0.682747\pi$$
−0.543092 + 0.839673i $$0.682747\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −552577. −0.0543824
$$636$$ 0 0
$$637$$ 1.22091e6 0.119216
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.55104e7 1.49100 0.745500 0.666506i $$-0.232211\pi$$
0.745500 + 0.666506i $$0.232211\pi$$
$$642$$ 0 0
$$643$$ −3.47784e6 −0.331728 −0.165864 0.986149i $$-0.553041\pi$$
−0.165864 + 0.986149i $$0.553041\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.45505e6 0.887980 0.443990 0.896032i $$-0.353563\pi$$
0.443990 + 0.896032i $$0.353563\pi$$
$$648$$ 0 0
$$649$$ −1.64737e7 −1.53525
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1.27403e7 1.16922 0.584610 0.811315i $$-0.301248\pi$$
0.584610 + 0.811315i $$0.301248\pi$$
$$654$$ 0 0
$$655$$ −1.80306e6 −0.164213
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 8.02252e6 0.719610 0.359805 0.933027i $$-0.382843\pi$$
0.359805 + 0.933027i $$0.382843\pi$$
$$660$$ 0 0
$$661$$ 2.12008e7 1.88734 0.943669 0.330892i $$-0.107350\pi$$
0.943669 + 0.330892i $$0.107350\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 3.62445e6 0.317825
$$666$$ 0 0
$$667$$ −1.79732e6 −0.156427
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 5.78915e6 0.496374
$$672$$ 0 0
$$673$$ −1.73938e7 −1.48032 −0.740162 0.672428i $$-0.765251\pi$$
−0.740162 + 0.672428i $$0.765251\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 2.35417e7 1.97408 0.987042 0.160464i $$-0.0512991\pi$$
0.987042 + 0.160464i $$0.0512991\pi$$
$$678$$ 0 0
$$679$$ −5.64100e6 −0.469550
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1.31970e7 −1.08249 −0.541243 0.840866i $$-0.682046\pi$$
−0.541243 + 0.840866i $$0.682046\pi$$
$$684$$ 0 0
$$685$$ 6.56925e6 0.534921
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 1.58558e7 1.27245
$$690$$ 0 0
$$691$$ 9.74172e6 0.776140 0.388070 0.921630i $$-0.373142\pi$$
0.388070 + 0.921630i $$0.373142\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1.19101e7 −0.935308
$$696$$ 0 0
$$697$$ 3.09016e6 0.240935
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.23931e7 0.952542 0.476271 0.879299i $$-0.341988\pi$$
0.476271 + 0.879299i $$0.341988\pi$$
$$702$$ 0 0
$$703$$ −1.27794e7 −0.975266
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6.70333e6 0.504362
$$708$$ 0 0
$$709$$ 1.18843e7 0.887884 0.443942 0.896055i $$-0.353580\pi$$
0.443942 + 0.896055i $$0.353580\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −1.48573e6 −0.109450
$$714$$ 0 0
$$715$$ 6.12793e6 0.448279
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 1.57469e7 1.13599 0.567993 0.823034i $$-0.307720\pi$$
0.567993 + 0.823034i $$0.307720\pi$$
$$720$$ 0 0
$$721$$ −1.49087e6 −0.106808
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1.59436e7 1.12653
$$726$$ 0 0
$$727$$ 9.34544e6 0.655788 0.327894 0.944714i $$-0.393661\pi$$
0.327894 + 0.944714i $$0.393661\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 936210. 0.0648008
$$732$$ 0 0
$$733$$ −1.42667e7 −0.980763 −0.490381 0.871508i $$-0.663143\pi$$
−0.490381 + 0.871508i $$0.663143\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.30370e7 0.884112
$$738$$ 0 0
$$739$$ 1.75852e7 1.18451 0.592253 0.805752i $$-0.298239\pi$$
0.592253 + 0.805752i $$0.298239\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 36857.8 0.00244939 0.00122469 0.999999i $$-0.499610\pi$$
0.00122469 + 0.999999i $$0.499610\pi$$
$$744$$ 0 0
$$745$$ 551099. 0.0363780
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 6.59598e6 0.429610
$$750$$ 0 0
$$751$$ 1.73284e7 1.12114 0.560570 0.828107i $$-0.310582\pi$$
0.560570 + 0.828107i $$0.310582\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1.54954e7 0.989317
$$756$$ 0 0
$$757$$ 1.04739e7 0.664308 0.332154 0.943225i $$-0.392225\pi$$
0.332154 + 0.943225i $$0.392225\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 615867. 0.0385501 0.0192751 0.999814i $$-0.493864\pi$$
0.0192751 + 0.999814i $$0.493864\pi$$
$$762$$ 0 0
$$763$$ −378378. −0.0235296
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −1.97529e7 −1.21239
$$768$$ 0 0
$$769$$ 2.82564e7 1.72306 0.861531 0.507705i $$-0.169506\pi$$
0.861531 + 0.507705i $$0.169506\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1.80327e7 1.08546 0.542729 0.839908i $$-0.317391\pi$$
0.542729 + 0.839908i $$0.317391\pi$$
$$774$$ 0 0
$$775$$ 1.31796e7 0.788221
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.48980e7 −0.879601
$$780$$ 0 0
$$781$$ −1.93500e7 −1.13515
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 271038. 0.0156984
$$786$$ 0 0
$$787$$ −3.13051e7 −1.80168 −0.900841 0.434149i $$-0.857049\pi$$
−0.900841 + 0.434149i $$0.857049\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1.05152e7 0.597554
$$792$$ 0 0
$$793$$ 6.94152e6 0.391987
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −1.38176e7 −0.770525 −0.385262 0.922807i $$-0.625889\pi$$
−0.385262 + 0.922807i $$0.625889\pi$$
$$798$$ 0 0
$$799$$ −5.47898e6 −0.303621
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 9.22529e6 0.504884
$$804$$ 0 0
$$805$$ −363770. −0.0197850
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −8.02702e6 −0.431204 −0.215602 0.976481i $$-0.569171\pi$$
−0.215602 + 0.976481i $$0.569171\pi$$
$$810$$ 0 0
$$811$$ −5.68487e6 −0.303507 −0.151753 0.988418i $$-0.548492\pi$$
−0.151753 + 0.988418i $$0.548492\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 1.61050e7 0.849308
$$816$$ 0 0
$$817$$ −4.51358e6 −0.236574
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1.26582e7 −0.655412 −0.327706 0.944780i $$-0.606276\pi$$
−0.327706 + 0.944780i $$0.606276\pi$$
$$822$$ 0 0
$$823$$ 6.69435e6 0.344516 0.172258 0.985052i $$-0.444894\pi$$
0.172258 + 0.985052i $$0.444894\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 6.41484e6 0.326153 0.163077 0.986613i $$-0.447858\pi$$
0.163077 + 0.986613i $$0.447858\pi$$
$$828$$ 0 0
$$829$$ −4.66483e6 −0.235749 −0.117874 0.993029i $$-0.537608\pi$$
−0.117874 + 0.993029i $$0.537608\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1.29634e6 0.0647300
$$834$$ 0 0
$$835$$ 1.42525e7 0.707416
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −7.16720e6 −0.351516 −0.175758 0.984433i $$-0.556238\pi$$
−0.175758 + 0.984433i $$0.556238\pi$$
$$840$$ 0 0
$$841$$ 2.68186e7 1.30751
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −3.20315e6 −0.154325
$$846$$ 0 0
$$847$$ −920977. −0.0441103
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 1.28261e6 0.0607116
$$852$$ 0 0
$$853$$ −2.18869e7 −1.02994 −0.514970 0.857208i $$-0.672197\pi$$
−0.514970 + 0.857208i $$0.672197\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −2.15656e7 −1.00302 −0.501509 0.865152i $$-0.667222\pi$$
−0.501509 + 0.865152i $$0.667222\pi$$
$$858$$ 0 0
$$859$$ −1.68364e7 −0.778514 −0.389257 0.921129i $$-0.627268\pi$$
−0.389257 + 0.921129i $$0.627268\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −5.28356e6 −0.241490 −0.120745 0.992684i $$-0.538528\pi$$
−0.120745 + 0.992684i $$0.538528\pi$$
$$864$$ 0 0
$$865$$ −3.17331e6 −0.144202
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −1.36960e7 −0.615238
$$870$$ 0 0
$$871$$ 1.56320e7 0.698185
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 7.57821e6 0.334616
$$876$$ 0 0
$$877$$ 3.77000e7 1.65517 0.827584 0.561342i $$-0.189715\pi$$
0.827584 + 0.561342i $$0.189715\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −2.59165e7 −1.12496 −0.562480 0.826811i $$-0.690153\pi$$
−0.562480 + 0.826811i $$0.690153\pi$$
$$882$$ 0 0
$$883$$ 8.24975e6 0.356073 0.178036 0.984024i $$-0.443026\pi$$
0.178036 + 0.984024i $$0.443026\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 8.84834e6 0.377618 0.188809 0.982014i $$-0.439537\pi$$
0.188809 + 0.982014i $$0.439537\pi$$
$$888$$ 0 0
$$889$$ 952833. 0.0404355
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 2.64148e7 1.10846
$$894$$ 0 0
$$895$$ −1.50164e7 −0.626624
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 3.91246e7 1.61455
$$900$$ 0 0
$$901$$ 1.68354e7 0.690893
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −8.26378e6 −0.335396
$$906$$ 0 0
$$907$$ −3.52189e6 −0.142153 −0.0710767 0.997471i $$-0.522644\pi$$
−0.0710767 + 0.997471i $$0.522644\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 2.67890e7 1.06945 0.534726 0.845026i $$-0.320415\pi$$
0.534726 + 0.845026i $$0.320415\pi$$
$$912$$ 0 0
$$913$$ −1.97781e7 −0.785250
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 3.10910e6 0.122099
$$918$$ 0 0
$$919$$ 4.21717e7 1.64715 0.823574 0.567209i $$-0.191977\pi$$
0.823574 + 0.567209i $$0.191977\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −2.32018e7 −0.896431
$$924$$ 0 0
$$925$$ −1.13778e7 −0.437222
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −4.24746e6 −0.161469 −0.0807347 0.996736i $$-0.525727\pi$$
−0.0807347 + 0.996736i $$0.525727\pi$$
$$930$$ 0 0
$$931$$ −6.24980e6 −0.236315
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 6.50653e6 0.243400
$$936$$ 0 0
$$937$$ 4.39330e6 0.163471 0.0817357 0.996654i $$-0.473954\pi$$
0.0817357 + 0.996654i $$0.473954\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 924321. 0.0340290 0.0170145 0.999855i $$-0.494584\pi$$
0.0170145 + 0.999855i $$0.494584\pi$$
$$942$$ 0 0
$$943$$ 1.49525e6 0.0547563
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 3.87692e7 1.40479 0.702395 0.711787i $$-0.252114\pi$$
0.702395 + 0.711787i $$0.252114\pi$$
$$948$$ 0 0
$$949$$ 1.10617e7 0.398708
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2.28022e7 −0.813289 −0.406644 0.913587i $$-0.633301\pi$$
−0.406644 + 0.913587i $$0.633301\pi$$
$$954$$ 0 0
$$955$$ 1.47735e6 0.0524174
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −1.13276e7 −0.397734
$$960$$ 0 0
$$961$$ 3.71280e6 0.129686
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −1.49472e7 −0.516705
$$966$$ 0 0
$$967$$ 3.66797e7 1.26142 0.630709 0.776019i $$-0.282764\pi$$
0.630709 + 0.776019i $$0.282764\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −1.38200e7 −0.470391 −0.235195 0.971948i $$-0.575573\pi$$
−0.235195 + 0.971948i $$0.575573\pi$$
$$972$$ 0 0
$$973$$ 2.05372e7 0.695438
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −2.05752e7 −0.689616 −0.344808 0.938673i $$-0.612056\pi$$
−0.344808 + 0.938673i $$0.612056\pi$$
$$978$$ 0 0
$$979$$ 2.70384e7 0.901622
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −5.77676e7 −1.90678 −0.953390 0.301742i $$-0.902432\pi$$
−0.953390 + 0.301742i $$0.902432\pi$$
$$984$$ 0 0
$$985$$ −2.77773e7 −0.912221
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 453008. 0.0147270
$$990$$ 0 0
$$991$$ 3.65391e7 1.18188 0.590940 0.806715i $$-0.298757\pi$$
0.590940 + 0.806715i $$0.298757\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −2.89224e7 −0.926142
$$996$$ 0 0
$$997$$ −3.57190e7 −1.13805 −0.569024 0.822321i $$-0.692679\pi$$
−0.569024 + 0.822321i $$0.692679\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 1008.6.a.bf.1.2 2
3.2 odd 2 336.6.a.u.1.1 2
4.3 odd 2 252.6.a.e.1.2 2
12.11 even 2 84.6.a.d.1.1 2
84.11 even 6 588.6.i.h.373.2 4
84.23 even 6 588.6.i.h.361.2 4
84.47 odd 6 588.6.i.n.361.1 4
84.59 odd 6 588.6.i.n.373.1 4
84.83 odd 2 588.6.a.g.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.a.d.1.1 2 12.11 even 2
252.6.a.e.1.2 2 4.3 odd 2
336.6.a.u.1.1 2 3.2 odd 2
588.6.a.g.1.2 2 84.83 odd 2
588.6.i.h.361.2 4 84.23 even 6
588.6.i.h.373.2 4 84.11 even 6
588.6.i.n.361.1 4 84.47 odd 6
588.6.i.n.373.1 4 84.59 odd 6
1008.6.a.bf.1.2 2 1.1 even 1 trivial