# Properties

 Label 84.6.a.d.1.1 Level $84$ Weight $6$ Character 84.1 Self dual yes Analytic conductor $13.472$ Analytic rank $0$ 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] = [84,6,Mod(1,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("84.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 84.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: $$13.4722408643$$ Analytic rank: $$0$$ 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$11.7361$$ of defining polynomial Character $$\chi$$ $$=$$ 84.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+9.00000 q^{3} -28.4166 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$$ $$q+9.00000 q^{3} -28.4166 q^{5} +49.0000 q^{7} +81.0000 q^{9} +424.083 q^{11} +508.500 q^{13} -255.750 q^{15} -539.916 q^{17} +2603.00 q^{19} +441.000 q^{21} +261.251 q^{23} -2317.50 q^{25} +729.000 q^{27} +6879.66 q^{29} +5687.00 q^{31} +3816.75 q^{33} -1392.41 q^{35} +4909.50 q^{37} +4576.50 q^{39} -5723.42 q^{41} -1733.99 q^{43} -2301.75 q^{45} -10147.8 q^{47} +2401.00 q^{49} -4859.24 q^{51} -31181.5 q^{53} -12051.0 q^{55} +23427.0 q^{57} -38845.5 q^{59} +13651.0 q^{61} +3969.00 q^{63} -14449.8 q^{65} -30741.5 q^{67} +2351.26 q^{69} -45627.9 q^{71} +21753.5 q^{73} -20857.5 q^{75} +20780.1 q^{77} +32295.5 q^{79} +6561.00 q^{81} -46637.3 q^{83} +15342.6 q^{85} +61917.0 q^{87} -63757.4 q^{89} +24916.5 q^{91} +51183.0 q^{93} -73968.4 q^{95} +115122. q^{97} +34350.7 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 18 q^{3} + 78 q^{5} + 98 q^{7} + 162 q^{9}+O(q^{10})$$ 2 * q + 18 * q^3 + 78 * q^5 + 98 * q^7 + 162 * q^9 $$2 q + 18 q^{3} + 78 q^{5} + 98 q^{7} + 162 q^{9} + 174 q^{11} + 208 q^{13} + 702 q^{15} + 1482 q^{17} + 352 q^{19} + 882 q^{21} + 3354 q^{23} + 5882 q^{25} + 1458 q^{27} + 276 q^{29} + 6520 q^{31} + 1566 q^{33} + 3822 q^{35} + 13864 q^{37} + 1872 q^{39} - 12930 q^{41} + 12712 q^{43} + 6318 q^{45} - 28116 q^{47} + 4802 q^{49} + 13338 q^{51} - 46992 q^{53} - 38664 q^{55} + 3168 q^{57} - 65556 q^{59} - 13148 q^{61} + 7938 q^{63} - 46428 q^{65} - 75236 q^{67} + 30186 q^{69} - 66042 q^{71} + 60496 q^{73} + 52938 q^{75} + 8526 q^{77} - 34916 q^{79} + 13122 q^{81} - 82488 q^{83} + 230508 q^{85} + 2484 q^{87} + 42510 q^{89} + 10192 q^{91} + 58680 q^{93} - 313512 q^{95} + 213256 q^{97} + 14094 q^{99}+O(q^{100})$$ 2 * q + 18 * q^3 + 78 * q^5 + 98 * q^7 + 162 * q^9 + 174 * q^11 + 208 * q^13 + 702 * q^15 + 1482 * q^17 + 352 * q^19 + 882 * q^21 + 3354 * q^23 + 5882 * q^25 + 1458 * q^27 + 276 * q^29 + 6520 * q^31 + 1566 * q^33 + 3822 * q^35 + 13864 * q^37 + 1872 * q^39 - 12930 * q^41 + 12712 * q^43 + 6318 * q^45 - 28116 * q^47 + 4802 * q^49 + 13338 * q^51 - 46992 * q^53 - 38664 * q^55 + 3168 * q^57 - 65556 * q^59 - 13148 * q^61 + 7938 * q^63 - 46428 * q^65 - 75236 * q^67 + 30186 * q^69 - 66042 * q^71 + 60496 * q^73 + 52938 * q^75 + 8526 * q^77 - 34916 * q^79 + 13122 * q^81 - 82488 * q^83 + 230508 * q^85 + 2484 * q^87 + 42510 * q^89 + 10192 * q^91 + 58680 * q^93 - 313512 * q^95 + 213256 * q^97 + 14094 * 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
$$3$$ 9.00000 0.577350
$$4$$ 0 0
$$5$$ −28.4166 −0.508332 −0.254166 0.967161i $$-0.581801\pi$$
−0.254166 + 0.967161i $$0.581801\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 0 0
$$9$$ 81.0000 0.333333
$$10$$ 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
$$15$$ −255.750 −0.293486
$$16$$ 0 0
$$17$$ −539.916 −0.453110 −0.226555 0.973998i $$-0.572746\pi$$
−0.226555 + 0.973998i $$0.572746\pi$$
$$18$$ 0 0
$$19$$ 2603.00 1.65421 0.827104 0.562050i $$-0.189987\pi$$
0.827104 + 0.562050i $$0.189987\pi$$
$$20$$ 0 0
$$21$$ 441.000 0.218218
$$22$$ 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
$$27$$ 729.000 0.192450
$$28$$ 0 0
$$29$$ 6879.66 1.51905 0.759525 0.650478i $$-0.225431\pi$$
0.759525 + 0.650478i $$0.225431\pi$$
$$30$$ 0 0
$$31$$ 5687.00 1.06287 0.531433 0.847100i $$-0.321654\pi$$
0.531433 + 0.847100i $$0.321654\pi$$
$$32$$ 0 0
$$33$$ 3816.75 0.610111
$$34$$ 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
$$39$$ 4576.50 0.481805
$$40$$ 0 0
$$41$$ −5723.42 −0.531736 −0.265868 0.964009i $$-0.585658\pi$$
−0.265868 + 0.964009i $$0.585658\pi$$
$$42$$ 0 0
$$43$$ −1733.99 −0.143013 −0.0715066 0.997440i $$-0.522781\pi$$
−0.0715066 + 0.997440i $$0.522781\pi$$
$$44$$ 0 0
$$45$$ −2301.75 −0.169444
$$46$$ 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
$$51$$ −4859.24 −0.261603
$$52$$ 0 0
$$53$$ −31181.5 −1.52478 −0.762390 0.647118i $$-0.775974\pi$$
−0.762390 + 0.647118i $$0.775974\pi$$
$$54$$ 0 0
$$55$$ −12051.0 −0.537176
$$56$$ 0 0
$$57$$ 23427.0 0.955057
$$58$$ 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
$$63$$ 3969.00 0.125988
$$64$$ 0 0
$$65$$ −14449.8 −0.424209
$$66$$ 0 0
$$67$$ −30741.5 −0.836639 −0.418320 0.908300i $$-0.637381\pi$$
−0.418320 + 0.908300i $$0.637381\pi$$
$$68$$ 0 0
$$69$$ 2351.26 0.0594536
$$70$$ 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
$$75$$ −20857.5 −0.428162
$$76$$ 0 0
$$77$$ 20780.1 0.399411
$$78$$ 0 0
$$79$$ 32295.5 0.582202 0.291101 0.956692i $$-0.405978\pi$$
0.291101 + 0.956692i $$0.405978\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 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
$$87$$ 61917.0 0.877024
$$88$$ 0 0
$$89$$ −63757.4 −0.853209 −0.426604 0.904438i $$-0.640290\pi$$
−0.426604 + 0.904438i $$0.640290\pi$$
$$90$$ 0 0
$$91$$ 24916.5 0.315416
$$92$$ 0 0
$$93$$ 51183.0 0.613646
$$94$$ 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
$$99$$ 34350.7 0.352248
$$100$$ 0 0
$$101$$ 136803. 1.33442 0.667208 0.744872i $$-0.267489\pi$$
0.667208 + 0.744872i $$0.267489\pi$$
$$102$$ 0 0
$$103$$ −30426.0 −0.282587 −0.141293 0.989968i $$-0.545126\pi$$
−0.141293 + 0.989968i $$0.545126\pi$$
$$104$$ 0 0
$$105$$ −12531.7 −0.110927
$$106$$ 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
$$111$$ 44185.5 0.340387
$$112$$ 0 0
$$113$$ 214596. 1.58098 0.790489 0.612476i $$-0.209826\pi$$
0.790489 + 0.612476i $$0.209826\pi$$
$$114$$ 0 0
$$115$$ −7423.87 −0.0523463
$$116$$ 0 0
$$117$$ 41188.5 0.278170
$$118$$ 0 0
$$119$$ −26455.9 −0.171259
$$120$$ 0 0
$$121$$ 18795.5 0.116705
$$122$$ 0 0
$$123$$ −51510.8 −0.306998
$$124$$ 0 0
$$125$$ 154657. 0.885310
$$126$$ 0 0
$$127$$ 19445.6 0.106982 0.0534911 0.998568i $$-0.482965\pi$$
0.0534911 + 0.998568i $$0.482965\pi$$
$$128$$ 0 0
$$129$$ −15605.9 −0.0825688
$$130$$ 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
$$135$$ −20715.7 −0.0978285
$$136$$ 0 0
$$137$$ −231176. −1.05231 −0.526153 0.850390i $$-0.676366\pi$$
−0.526153 + 0.850390i $$0.676366\pi$$
$$138$$ 0 0
$$139$$ 419126. 1.83996 0.919978 0.391971i $$-0.128207\pi$$
0.919978 + 0.391971i $$0.128207\pi$$
$$140$$ 0 0
$$141$$ −91330.5 −0.386873
$$142$$ 0 0
$$143$$ 215646. 0.881864
$$144$$ 0 0
$$145$$ −195497. −0.772182
$$146$$ 0 0
$$147$$ 21609.0 0.0824786
$$148$$ 0 0
$$149$$ −19393.6 −0.0715635 −0.0357818 0.999360i $$-0.511392\pi$$
−0.0357818 + 0.999360i $$0.511392\pi$$
$$150$$ 0 0
$$151$$ −545294. −1.94620 −0.973102 0.230376i $$-0.926004\pi$$
−0.973102 + 0.230376i $$0.926004\pi$$
$$152$$ 0 0
$$153$$ −43733.2 −0.151037
$$154$$ 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
$$159$$ −280633. −0.880332
$$160$$ 0 0
$$161$$ 12801.3 0.0389215
$$162$$ 0 0
$$163$$ −566744. −1.67078 −0.835388 0.549661i $$-0.814757\pi$$
−0.835388 + 0.549661i $$0.814757\pi$$
$$164$$ 0 0
$$165$$ −108459. −0.310139
$$166$$ 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
$$171$$ 210843. 0.551402
$$172$$ 0 0
$$173$$ 111671. 0.283677 0.141839 0.989890i $$-0.454699\pi$$
0.141839 + 0.989890i $$0.454699\pi$$
$$174$$ 0 0
$$175$$ −113557. −0.280298
$$176$$ 0 0
$$177$$ −349609. −0.838784
$$178$$ 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
$$183$$ 122859. 0.271193
$$184$$ 0 0
$$185$$ −139511. −0.299696
$$186$$ 0 0
$$187$$ −228969. −0.478821
$$188$$ 0 0
$$189$$ 35721.0 0.0727393
$$190$$ 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
$$195$$ −130049. −0.244917
$$196$$ 0 0
$$197$$ 977503. 1.79454 0.897269 0.441485i $$-0.145548\pi$$
0.897269 + 0.441485i $$0.145548\pi$$
$$198$$ 0 0
$$199$$ 1.01780e6 1.82192 0.910962 0.412491i $$-0.135341\pi$$
0.910962 + 0.412491i $$0.135341\pi$$
$$200$$ 0 0
$$201$$ −276674. −0.483034
$$202$$ 0 0
$$203$$ 337103. 0.574147
$$204$$ 0 0
$$205$$ 162640. 0.270298
$$206$$ 0 0
$$207$$ 21161.3 0.0343255
$$208$$ 0 0
$$209$$ 1.10389e6 1.74807
$$210$$ 0 0
$$211$$ 248461. 0.384195 0.192098 0.981376i $$-0.438471\pi$$
0.192098 + 0.981376i $$0.438471\pi$$
$$212$$ 0 0
$$213$$ −410651. −0.620189
$$214$$ 0 0
$$215$$ 49274.2 0.0726982
$$216$$ 0 0
$$217$$ 278663. 0.401726
$$218$$ 0 0
$$219$$ 195782. 0.275843
$$220$$ 0 0
$$221$$ −274547. −0.378125
$$222$$ 0 0
$$223$$ −118450. −0.159505 −0.0797525 0.996815i $$-0.525413\pi$$
−0.0797525 + 0.996815i $$0.525413\pi$$
$$224$$ 0 0
$$225$$ −187717. −0.247200
$$226$$ 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
$$231$$ 187021. 0.230600
$$232$$ 0 0
$$233$$ −68576.4 −0.0827532 −0.0413766 0.999144i $$-0.513174\pi$$
−0.0413766 + 0.999144i $$0.513174\pi$$
$$234$$ 0 0
$$235$$ 288367. 0.340625
$$236$$ 0 0
$$237$$ 290659. 0.336135
$$238$$ 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
$$243$$ 59049.0 0.0641500
$$244$$ 0 0
$$245$$ −68228.3 −0.0726188
$$246$$ 0 0
$$247$$ 1.32362e6 1.38045
$$248$$ 0 0
$$249$$ −419736. −0.429020
$$250$$ 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
$$255$$ 138083. 0.132981
$$256$$ 0 0
$$257$$ 1.16584e6 1.10104 0.550522 0.834821i $$-0.314429\pi$$
0.550522 + 0.834821i $$0.314429\pi$$
$$258$$ 0 0
$$259$$ 240566. 0.222835
$$260$$ 0 0
$$261$$ 557253. 0.506350
$$262$$ 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
$$267$$ −573816. −0.492600
$$268$$ 0 0
$$269$$ −71305.8 −0.0600819 −0.0300410 0.999549i $$-0.509564\pi$$
−0.0300410 + 0.999549i $$0.509564\pi$$
$$270$$ 0 0
$$271$$ 1.20152e6 0.993821 0.496911 0.867802i $$-0.334468\pi$$
0.496911 + 0.867802i $$0.334468\pi$$
$$272$$ 0 0
$$273$$ 224248. 0.182105
$$274$$ 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
$$279$$ 460647. 0.354289
$$280$$ 0 0
$$281$$ −37163.4 −0.0280770 −0.0140385 0.999901i $$-0.504469\pi$$
−0.0140385 + 0.999901i $$0.504469\pi$$
$$282$$ 0 0
$$283$$ −19110.2 −0.0141841 −0.00709203 0.999975i $$-0.502257\pi$$
−0.00709203 + 0.999975i $$0.502257\pi$$
$$284$$ 0 0
$$285$$ −665716. −0.485486
$$286$$ 0 0
$$287$$ −280447. −0.200977
$$288$$ 0 0
$$289$$ −1.12835e6 −0.794691
$$290$$ 0 0
$$291$$ 1.03610e6 0.717249
$$292$$ 0 0
$$293$$ −1.49662e6 −1.01846 −0.509230 0.860631i $$-0.670070\pi$$
−0.509230 + 0.860631i $$0.670070\pi$$
$$294$$ 0 0
$$295$$ 1.10386e6 0.738513
$$296$$ 0 0
$$297$$ 309157. 0.203370
$$298$$ 0 0
$$299$$ 132846. 0.0859351
$$300$$ 0 0
$$301$$ −84965.7 −0.0540539
$$302$$ 0 0
$$303$$ 1.23122e6 0.770425
$$304$$ 0 0
$$305$$ −387915. −0.238774
$$306$$ 0 0
$$307$$ −1.82699e6 −1.10634 −0.553172 0.833067i $$-0.686583\pi$$
−0.553172 + 0.833067i $$0.686583\pi$$
$$308$$ 0 0
$$309$$ −273834. −0.163152
$$310$$ 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
$$315$$ −112786. −0.0640438
$$316$$ 0 0
$$317$$ −1.40939e6 −0.787741 −0.393871 0.919166i $$-0.628864\pi$$
−0.393871 + 0.919166i $$0.628864\pi$$
$$318$$ 0 0
$$319$$ 2.91755e6 1.60524
$$320$$ 0 0
$$321$$ −1.21151e6 −0.656241
$$322$$ 0 0
$$323$$ −1.40540e6 −0.749538
$$324$$ 0 0
$$325$$ −1.17845e6 −0.618873
$$326$$ 0 0
$$327$$ 69498.0 0.0359421
$$328$$ 0 0
$$329$$ −497244. −0.253268
$$330$$ 0 0
$$331$$ −942703. −0.472939 −0.236469 0.971639i $$-0.575990\pi$$
−0.236469 + 0.971639i $$0.575990\pi$$
$$332$$ 0 0
$$333$$ 397670. 0.196522
$$334$$ 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
$$339$$ 1.93137e6 0.912778
$$340$$ 0 0
$$341$$ 2.41176e6 1.12318
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 0 0
$$345$$ −66814.8 −0.0302221
$$346$$ 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
$$351$$ 370696. 0.160602
$$352$$ 0 0
$$353$$ −3.72785e6 −1.59229 −0.796143 0.605108i $$-0.793130\pi$$
−0.796143 + 0.605108i $$0.793130\pi$$
$$354$$ 0 0
$$355$$ 1.29659e6 0.546049
$$356$$ 0 0
$$357$$ −238103. −0.0988767
$$358$$ 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
$$363$$ 169159. 0.0673797
$$364$$ 0 0
$$365$$ −618161. −0.242868
$$366$$ 0 0
$$367$$ 1.31628e6 0.510132 0.255066 0.966924i $$-0.417903\pi$$
0.255066 + 0.966924i $$0.417903\pi$$
$$368$$ 0 0
$$369$$ −463597. −0.177245
$$370$$ 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
$$375$$ 1.39192e6 0.511134
$$376$$ 0 0
$$377$$ 3.49831e6 1.26766
$$378$$ 0 0
$$379$$ −4.92472e6 −1.76110 −0.880549 0.473954i $$-0.842826\pi$$
−0.880549 + 0.473954i $$0.842826\pi$$
$$380$$ 0 0
$$381$$ 175010. 0.0617662
$$382$$ 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
$$387$$ −140453. −0.0476711
$$388$$ 0 0
$$389$$ 3.68937e6 1.23617 0.618085 0.786112i $$-0.287909\pi$$
0.618085 + 0.786112i $$0.287909\pi$$
$$390$$ 0 0
$$391$$ −141054. −0.0466597
$$392$$ 0 0
$$393$$ −571059. −0.186509
$$394$$ 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
$$399$$ 1.14792e6 0.360978
$$400$$ 0 0
$$401$$ 5.66295e6 1.75866 0.879331 0.476212i $$-0.157990\pi$$
0.879331 + 0.476212i $$0.157990\pi$$
$$402$$ 0 0
$$403$$ 2.89184e6 0.886975
$$404$$ 0 0
$$405$$ −186441. −0.0564813
$$406$$ 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
$$411$$ −2.08059e6 −0.607549
$$412$$ 0 0
$$413$$ −1.90343e6 −0.549113
$$414$$ 0 0
$$415$$ 1.32528e6 0.377734
$$416$$ 0 0
$$417$$ 3.77213e6 1.06230
$$418$$ 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
$$423$$ −821975. −0.223361
$$424$$ 0 0
$$425$$ 1.25125e6 0.336026
$$426$$ 0 0
$$427$$ 668898. 0.177538
$$428$$ 0 0
$$429$$ 1.94082e6 0.509144
$$430$$ 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
$$435$$ −1.75947e6 −0.445819
$$436$$ 0 0
$$437$$ 680036. 0.170345
$$438$$ 0 0
$$439$$ −3.00831e6 −0.745009 −0.372505 0.928030i $$-0.621501\pi$$
−0.372505 + 0.928030i $$0.621501\pi$$
$$440$$ 0 0
$$441$$ 194481. 0.0476190
$$442$$ 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
$$447$$ −174542. −0.0413172
$$448$$ 0 0
$$449$$ 5.41345e6 1.26724 0.633619 0.773646i $$-0.281569\pi$$
0.633619 + 0.773646i $$0.281569\pi$$
$$450$$ 0 0
$$451$$ −2.42720e6 −0.561908
$$452$$ 0 0
$$453$$ −4.90764e6 −1.12364
$$454$$ 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
$$459$$ −393599. −0.0872011
$$460$$ 0 0
$$461$$ 4.44254e6 0.973597 0.486799 0.873514i $$-0.338165\pi$$
0.486799 + 0.873514i $$0.338165\pi$$
$$462$$ 0 0
$$463$$ −4.62590e6 −1.00287 −0.501434 0.865196i $$-0.667194\pi$$
−0.501434 + 0.865196i $$0.667194\pi$$
$$464$$ 0 0
$$465$$ −1.45445e6 −0.311936
$$466$$ 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
$$471$$ 85842.2 0.0178299
$$472$$ 0 0
$$473$$ −735357. −0.151128
$$474$$ 0 0
$$475$$ −6.03244e6 −1.22676
$$476$$ 0 0
$$477$$ −2.52570e6 −0.508260
$$478$$ 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
$$483$$ 115212. 0.0224713
$$484$$ 0 0
$$485$$ −3.27139e6 −0.631507
$$486$$ 0 0
$$487$$ 3.74040e6 0.714653 0.357327 0.933979i $$-0.383688\pi$$
0.357327 + 0.933979i $$0.383688\pi$$
$$488$$ 0 0
$$489$$ −5.10070e6 −0.964623
$$490$$ 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
$$495$$ −976131. −0.179059
$$496$$ 0 0
$$497$$ −2.23577e6 −0.406009
$$498$$ 0 0
$$499$$ 2.58167e6 0.464141 0.232070 0.972699i $$-0.425450\pi$$
0.232070 + 0.972699i $$0.425450\pi$$
$$500$$ 0 0
$$501$$ 4.51400e6 0.803465
$$502$$ 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
$$507$$ −1.01449e6 −0.175278
$$508$$ 0 0
$$509$$ 1.13802e7 1.94696 0.973479 0.228776i $$-0.0734723\pi$$
0.973479 + 0.228776i $$0.0734723\pi$$
$$510$$ 0 0
$$511$$ 1.06592e6 0.180581
$$512$$ 0 0
$$513$$ 1.89759e6 0.318352
$$514$$ 0 0
$$515$$ 864604. 0.143648
$$516$$ 0 0
$$517$$ −4.30353e6 −0.708106
$$518$$ 0 0
$$519$$ 1.00504e6 0.163781
$$520$$ 0 0
$$521$$ −4.72031e6 −0.761862 −0.380931 0.924603i $$-0.624396\pi$$
−0.380931 + 0.924603i $$0.624396\pi$$
$$522$$ 0 0
$$523$$ −8.46281e6 −1.35288 −0.676441 0.736496i $$-0.736479\pi$$
−0.676441 + 0.736496i $$0.736479\pi$$
$$524$$ 0 0
$$525$$ −1.02202e6 −0.161830
$$526$$ 0 0
$$527$$ −3.07050e6 −0.481596
$$528$$ 0 0
$$529$$ −6.36809e6 −0.989396
$$530$$ 0 0
$$531$$ −3.14649e6 −0.484272
$$532$$ 0 0
$$533$$ −2.91036e6 −0.443739
$$534$$ 0 0
$$535$$ 3.82521e6 0.577792
$$536$$ 0 0
$$537$$ −4.75592e6 −0.711704
$$538$$ 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
$$543$$ −2.61727e6 −0.380934
$$544$$ 0 0
$$545$$ −219433. −0.0316454
$$546$$ 0 0
$$547$$ −5.04856e6 −0.721438 −0.360719 0.932675i $$-0.617469\pi$$
−0.360719 + 0.932675i $$0.617469\pi$$
$$548$$ 0 0
$$549$$ 1.10573e6 0.156573
$$550$$ 0 0
$$551$$ 1.79077e7 2.51282
$$552$$ 0 0
$$553$$ 1.58248e6 0.220052
$$554$$ 0 0
$$555$$ −1.25560e6 −0.173029
$$556$$ 0 0
$$557$$ −1.09175e7 −1.49103 −0.745516 0.666488i $$-0.767797\pi$$
−0.745516 + 0.666488i $$0.767797\pi$$
$$558$$ 0 0
$$559$$ −881735. −0.119346
$$560$$ 0 0
$$561$$ −2.06072e6 −0.276447
$$562$$ 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
$$567$$ 321489. 0.0419961
$$568$$ 0 0
$$569$$ 9.80819e6 1.27001 0.635007 0.772507i $$-0.280997\pi$$
0.635007 + 0.772507i $$0.280997\pi$$
$$570$$ 0 0
$$571$$ 1.54139e7 1.97844 0.989221 0.146430i $$-0.0467783\pi$$
0.989221 + 0.146430i $$0.0467783\pi$$
$$572$$ 0 0
$$573$$ 467901. 0.0595343
$$574$$ 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
$$579$$ −4.73403e6 −0.586860
$$580$$ 0 0
$$581$$ −2.28523e6 −0.280860
$$582$$ 0 0
$$583$$ −1.32235e7 −1.61130
$$584$$ 0 0
$$585$$ −1.17044e6 −0.141403
$$586$$ 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
$$591$$ 8.79753e6 1.03608
$$592$$ 0 0
$$593$$ 6.01144e6 0.702007 0.351004 0.936374i $$-0.385840\pi$$
0.351004 + 0.936374i $$0.385840\pi$$
$$594$$ 0 0
$$595$$ 751786. 0.0870567
$$596$$ 0 0
$$597$$ 9.16021e6 1.05189
$$598$$ 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
$$603$$ −2.49006e6 −0.278880
$$604$$ 0 0
$$605$$ −534103. −0.0593249
$$606$$ 0 0
$$607$$ 1.29052e7 1.42165 0.710823 0.703371i $$-0.248323\pi$$
0.710823 + 0.703371i $$0.248323\pi$$
$$608$$ 0 0
$$609$$ 3.03393e6 0.331484
$$610$$ 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
$$615$$ 1.46376e6 0.156057
$$616$$ 0 0
$$617$$ 1.31134e7 1.38676 0.693381 0.720571i $$-0.256120\pi$$
0.693381 + 0.720571i $$0.256120\pi$$
$$618$$ 0 0
$$619$$ −2.56630e6 −0.269203 −0.134602 0.990900i $$-0.542975\pi$$
−0.134602 + 0.990900i $$0.542975\pi$$
$$620$$ 0 0
$$621$$ 190452. 0.0198179
$$622$$ 0 0
$$623$$ −3.12411e6 −0.322483
$$624$$ 0 0
$$625$$ 2.84734e6 0.291567
$$626$$ 0 0
$$627$$ 9.93499e6 1.00925
$$628$$ 0 0
$$629$$ −2.65072e6 −0.267139
$$630$$ 0 0
$$631$$ 1.08637e7 1.08618 0.543092 0.839673i $$-0.317253\pi$$
0.543092 + 0.839673i $$0.317253\pi$$
$$632$$ 0 0
$$633$$ 2.23615e6 0.221815
$$634$$ 0 0
$$635$$ −552577. −0.0543824
$$636$$ 0 0
$$637$$ 1.22091e6 0.119216
$$638$$ 0 0
$$639$$ −3.69586e6 −0.358066
$$640$$ 0 0
$$641$$ −1.55104e7 −1.49100 −0.745500 0.666506i $$-0.767789\pi$$
−0.745500 + 0.666506i $$0.767789\pi$$
$$642$$ 0 0
$$643$$ 3.47784e6 0.331728 0.165864 0.986149i $$-0.446959\pi$$
0.165864 + 0.986149i $$0.446959\pi$$
$$644$$ 0 0
$$645$$ 443468. 0.0419723
$$646$$ 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
$$651$$ 2.50797e6 0.231937
$$652$$ 0 0
$$653$$ −1.27403e7 −1.16922 −0.584610 0.811315i $$-0.698752\pi$$
−0.584610 + 0.811315i $$0.698752\pi$$
$$654$$ 0 0
$$655$$ 1.80306e6 0.164213
$$656$$ 0 0
$$657$$ 1.76203e6 0.159258
$$658$$ 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
$$663$$ −2.47092e6 −0.218311
$$664$$ 0 0
$$665$$ −3.62445e6 −0.317825
$$666$$ 0 0
$$667$$ 1.79732e6 0.156427
$$668$$ 0 0
$$669$$ −1.06605e6 −0.0920903
$$670$$ 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
$$675$$ −1.68945e6 −0.142721
$$676$$ 0 0
$$677$$ −2.35417e7 −1.97408 −0.987042 0.160464i $$-0.948701\pi$$
−0.987042 + 0.160464i $$0.948701\pi$$
$$678$$ 0 0
$$679$$ 5.64100e6 0.469550
$$680$$ 0 0
$$681$$ 6.45801e6 0.533618
$$682$$ 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
$$687$$ −1.20120e7 −0.971009
$$688$$ 0 0
$$689$$ −1.58558e7 −1.27245
$$690$$ 0 0
$$691$$ −9.74172e6 −0.776140 −0.388070 0.921630i $$-0.626858\pi$$
−0.388070 + 0.921630i $$0.626858\pi$$
$$692$$ 0 0
$$693$$ 1.68319e6 0.133137
$$694$$ 0 0
$$695$$ −1.19101e7 −0.935308
$$696$$ 0 0
$$697$$ 3.09016e6 0.240935
$$698$$ 0 0
$$699$$ −617187. −0.0477776
$$700$$ 0 0
$$701$$ −1.23931e7 −0.952542 −0.476271 0.879299i $$-0.658012\pi$$
−0.476271 + 0.879299i $$0.658012\pi$$
$$702$$ 0 0
$$703$$ 1.27794e7 0.975266
$$704$$ 0 0
$$705$$ 2.59530e6 0.196660
$$706$$ 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
$$711$$ 2.61593e6 0.194067
$$712$$ 0 0
$$713$$ 1.48573e6 0.109450
$$714$$ 0 0
$$715$$ −6.12793e6 −0.448279
$$716$$ 0 0
$$717$$ 4.15775e6 0.302037
$$718$$ 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
$$723$$ 1.27825e6 0.0909433
$$724$$ 0 0
$$725$$ −1.59436e7 −1.12653
$$726$$ 0 0
$$727$$ −9.34544e6 −0.655788 −0.327894 0.944714i $$-0.606339\pi$$
−0.327894 + 0.944714i $$0.606339\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ 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
$$735$$ −614055. −0.0419265
$$736$$ 0 0
$$737$$ −1.30370e7 −0.884112
$$738$$ 0 0
$$739$$ −1.75852e7 −1.18451 −0.592253 0.805752i $$-0.701761\pi$$
−0.592253 + 0.805752i $$0.701761\pi$$
$$740$$ 0 0
$$741$$ 1.19126e7 0.797006
$$742$$ 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
$$747$$ −3.77762e6 −0.247695
$$748$$ 0 0
$$749$$ −6.59598e6 −0.429610
$$750$$ 0 0
$$751$$ −1.73284e7 −1.12114 −0.560570 0.828107i $$-0.689418\pi$$
−0.560570 + 0.828107i $$0.689418\pi$$
$$752$$ 0 0
$$753$$ −6.05244e6 −0.388994
$$754$$ 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
$$759$$ 997129. 0.0628271
$$760$$ 0 0
$$761$$ −615867. −0.0385501 −0.0192751 0.999814i $$-0.506136\pi$$
−0.0192751 + 0.999814i $$0.506136\pi$$
$$762$$ 0 0
$$763$$ 378378. 0.0235296
$$764$$ 0 0
$$765$$ 1.24275e6 0.0767767
$$766$$ 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
$$771$$ 1.04925e7 0.635688
$$772$$ 0 0
$$773$$ −1.80327e7 −1.08546 −0.542729 0.839908i $$-0.682609\pi$$
−0.542729 + 0.839908i $$0.682609\pi$$
$$774$$ 0 0
$$775$$ −1.31796e7 −0.788221
$$776$$ 0 0
$$777$$ 2.16509e6 0.128654
$$778$$ 0 0
$$779$$ −1.48980e7 −0.879601
$$780$$ 0 0
$$781$$ −1.93500e7 −1.13515
$$782$$ 0 0
$$783$$ 5.01527e6 0.292341
$$784$$ 0 0
$$785$$ −271038. −0.0156984
$$786$$ 0 0
$$787$$ 3.13051e7 1.80168 0.900841 0.434149i $$-0.142951\pi$$
0.900841 + 0.434149i $$0.142951\pi$$
$$788$$ 0 0
$$789$$ −1.11003e7 −0.634806
$$790$$ 0 0
$$791$$ 1.05152e7 0.597554
$$792$$ 0 0
$$793$$ 6.94152e6 0.391987
$$794$$ 0 0
$$795$$ 7.97465e6 0.447501
$$796$$ 0 0
$$797$$ 1.38176e7 0.770525 0.385262 0.922807i $$-0.374111\pi$$
0.385262 + 0.922807i $$0.374111\pi$$
$$798$$ 0 0
$$799$$ 5.47898e6 0.303621
$$800$$ 0 0
$$801$$ −5.16435e6 −0.284403
$$802$$ 0 0
$$803$$ 9.22529e6 0.504884
$$804$$ 0 0
$$805$$ −363770. −0.0197850
$$806$$ 0 0
$$807$$ −641752. −0.0346883
$$808$$ 0 0
$$809$$ 8.02702e6 0.431204 0.215602 0.976481i $$-0.430829\pi$$
0.215602 + 0.976481i $$0.430829\pi$$
$$810$$ 0 0
$$811$$ 5.68487e6 0.303507 0.151753 0.988418i $$-0.451508\pi$$
0.151753 + 0.988418i $$0.451508\pi$$
$$812$$ 0 0
$$813$$ 1.08137e7 0.573783
$$814$$ 0 0
$$815$$ 1.61050e7 0.849308
$$816$$ 0 0
$$817$$ −4.51358e6 −0.236574
$$818$$ 0 0
$$819$$ 2.01824e6 0.105139
$$820$$ 0 0
$$821$$ 1.26582e7 0.655412 0.327706 0.944780i $$-0.393724\pi$$
0.327706 + 0.944780i $$0.393724\pi$$
$$822$$ 0 0
$$823$$ −6.69435e6 −0.344516 −0.172258 0.985052i $$-0.555106\pi$$
−0.172258 + 0.985052i $$0.555106\pi$$
$$824$$ 0 0
$$825$$ −8.84530e6 −0.452457
$$826$$ 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
$$831$$ −1.47873e7 −0.742827
$$832$$ 0 0
$$833$$ −1.29634e6 −0.0647300
$$834$$ 0 0
$$835$$ −1.42525e7 −0.707416
$$836$$ 0 0
$$837$$ 4.14582e6 0.204549
$$838$$ 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
$$843$$ −334471. −0.0162102
$$844$$ 0 0
$$845$$ 3.20315e6 0.154325
$$846$$ 0 0
$$847$$ 920977. 0.0441103
$$848$$ 0 0
$$849$$ −171992. −0.00818917
$$850$$ 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
$$855$$ −5.99144e6 −0.280295
$$856$$ 0 0
$$857$$ 2.15656e7 1.00302 0.501509 0.865152i $$-0.332778\pi$$
0.501509 + 0.865152i $$0.332778\pi$$
$$858$$ 0 0
$$859$$ 1.68364e7 0.778514 0.389257 0.921129i $$-0.372732\pi$$
0.389257 + 0.921129i $$0.372732\pi$$
$$860$$ 0 0
$$861$$ −2.52403e6 −0.116034
$$862$$ 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
$$867$$ −1.01551e7 −0.458815
$$868$$ 0 0
$$869$$ 1.36960e7 0.615238
$$870$$ 0 0
$$871$$ −1.56320e7 −0.698185
$$872$$ 0 0
$$873$$ 9.32492e6 0.414104
$$874$$ 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
$$879$$ −1.34696e7 −0.588008
$$880$$ 0 0
$$881$$ 2.59165e7 1.12496 0.562480 0.826811i $$-0.309847\pi$$
0.562480 + 0.826811i $$0.309847\pi$$
$$882$$ 0 0
$$883$$ −8.24975e6 −0.356073 −0.178036 0.984024i $$-0.556974\pi$$
−0.178036 + 0.984024i $$0.556974\pi$$
$$884$$ 0 0
$$885$$ 9.93472e6 0.426381
$$886$$ 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
$$891$$ 2.78241e6 0.117416
$$892$$ 0 0
$$893$$ −2.64148e7 −1.10846
$$894$$ 0 0
$$895$$ 1.50164e7 0.626624
$$896$$ 0 0
$$897$$ 1.19561e6 0.0496147
$$898$$ 0 0
$$899$$ 3.91246e7 1.61455
$$900$$ 0 0
$$901$$ 1.68354e7 0.690893
$$902$$ 0 0
$$903$$ −764691. −0.0312081
$$904$$ 0 0
$$905$$ 8.26378e6 0.335396
$$906$$ 0 0
$$907$$ 3.52189e6 0.142153 0.0710767 0.997471i $$-0.477356\pi$$
0.0710767 + 0.997471i $$0.477356\pi$$
$$908$$ 0 0
$$909$$ 1.10810e7 0.444805
$$910$$ 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
$$915$$ −3.49123e6 −0.137856
$$916$$ 0 0
$$917$$ −3.10910e6 −0.122099
$$918$$ 0 0
$$919$$ −4.21717e7 −1.64715 −0.823574 0.567209i $$-0.808023\pi$$
−0.823574 + 0.567209i $$0.808023\pi$$
$$920$$ 0 0
$$921$$ −1.64429e7 −0.638748
$$922$$ 0 0
$$923$$ −2.32018e7 −0.896431
$$924$$ 0 0
$$925$$ −1.13778e7 −0.437222
$$926$$ 0 0
$$927$$ −2.46451e6 −0.0941956
$$928$$ 0 0
$$929$$ 4.24746e6 0.161469 0.0807347 0.996736i $$-0.474273\pi$$
0.0807347 + 0.996736i $$0.474273\pi$$
$$930$$ 0 0
$$931$$ 6.24980e6 0.236315
$$932$$ 0 0
$$933$$ −1.49027e7 −0.560483
$$934$$ 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
$$939$$ −2.73046e7 −1.01058
$$940$$ 0 0
$$941$$ −924321. −0.0340290 −0.0170145 0.999855i $$-0.505416\pi$$
−0.0170145 + 0.999855i $$0.505416\pi$$
$$942$$ 0 0
$$943$$ −1.49525e6 −0.0547563
$$944$$ 0 0
$$945$$ −1.01507e6 −0.0369757
$$946$$ 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
$$951$$ −1.26845e7 −0.454803
$$952$$ 0 0
$$953$$ 2.28022e7 0.813289 0.406644 0.913587i $$-0.366699\pi$$
0.406644 + 0.913587i $$0.366699\pi$$
$$954$$ 0 0
$$955$$ −1.47735e6 −0.0524174
$$956$$ 0 0
$$957$$ 2.62579e7 0.926789
$$958$$ 0 0
$$959$$ −1.13276e7 −0.397734
$$960$$ 0 0
$$961$$ 3.71280e6 0.129686
$$962$$ 0 0
$$963$$ −1.09036e7 −0.378881
$$964$$ 0 0
$$965$$ 1.49472e7 0.516705
$$966$$ 0 0
$$967$$ −3.66797e7 −1.26142 −0.630709 0.776019i $$-0.717236\pi$$
−0.630709 + 0.776019i $$0.717236\pi$$
$$968$$ 0 0
$$969$$ −1.26486e7 −0.432746
$$970$$ 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
$$975$$ −1.06060e7 −0.357306
$$976$$ 0 0
$$977$$ 2.05752e7 0.689616 0.344808 0.938673i $$-0.387944\pi$$
0.344808 + 0.938673i $$0.387944\pi$$
$$978$$ 0 0
$$979$$ −2.70384e7 −0.901622
$$980$$ 0 0
$$981$$ 625482. 0.0207512
$$982$$ 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
$$987$$ −4.47520e6 −0.146224
$$988$$ 0 0
$$989$$ −453008. −0.0147270
$$990$$ 0 0
$$991$$ −3.65391e7 −1.18188 −0.590940 0.806715i $$-0.701243\pi$$
−0.590940 + 0.806715i $$0.701243\pi$$
$$992$$ 0 0
$$993$$ −8.48432e6 −0.273051
$$994$$ 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
$$999$$ 3.57903e6 0.113462
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 84.6.a.d.1.1 2
3.2 odd 2 252.6.a.e.1.2 2
4.3 odd 2 336.6.a.u.1.1 2
7.2 even 3 588.6.i.h.361.2 4
7.3 odd 6 588.6.i.n.373.1 4
7.4 even 3 588.6.i.h.373.2 4
7.5 odd 6 588.6.i.n.361.1 4
7.6 odd 2 588.6.a.g.1.2 2
12.11 even 2 1008.6.a.bf.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.a.d.1.1 2 1.1 even 1 trivial
252.6.a.e.1.2 2 3.2 odd 2
336.6.a.u.1.1 2 4.3 odd 2
588.6.a.g.1.2 2 7.6 odd 2
588.6.i.h.361.2 4 7.2 even 3
588.6.i.h.373.2 4 7.4 even 3
588.6.i.n.361.1 4 7.5 odd 6
588.6.i.n.373.1 4 7.3 odd 6
1008.6.a.bf.1.2 2 12.11 even 2