# Properties

 Label 592.8.a.b.1.3 Level $592$ Weight $8$ Character 592.1 Self dual yes Analytic conductor $184.932$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,8,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("592.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$592 = 2^{4} \cdot 37$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 592.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: $$184.931935087$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 405x^{2} - 2998x - 4396$$ x^4 - 2*x^3 - 405*x^2 - 2998*x - 4396 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-13.3612$$ of defining polynomial Character $$\chi$$ $$=$$ 592.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+36.0598 q^{3} -19.7362 q^{5} +50.9272 q^{7} -886.690 q^{9} +O(q^{10})$$ $$q+36.0598 q^{3} -19.7362 q^{5} +50.9272 q^{7} -886.690 q^{9} -5175.62 q^{11} +4311.51 q^{13} -711.683 q^{15} +14063.5 q^{17} -26643.5 q^{19} +1836.43 q^{21} +77473.7 q^{23} -77735.5 q^{25} -110837. q^{27} -33124.8 q^{29} +172834. q^{31} -186632. q^{33} -1005.11 q^{35} +50653.0 q^{37} +155472. q^{39} -846738. q^{41} +307387. q^{43} +17499.9 q^{45} +322473. q^{47} -820949. q^{49} +507128. q^{51} +1.83562e6 q^{53} +102147. q^{55} -960758. q^{57} +797340. q^{59} -684801. q^{61} -45156.6 q^{63} -85092.7 q^{65} +3.79463e6 q^{67} +2.79369e6 q^{69} +342906. q^{71} +2.64684e6 q^{73} -2.80313e6 q^{75} -263580. q^{77} +2.44136e6 q^{79} -2.05756e6 q^{81} -5.61596e6 q^{83} -277560. q^{85} -1.19447e6 q^{87} +8.43578e6 q^{89} +219573. q^{91} +6.23237e6 q^{93} +525840. q^{95} -3.40422e6 q^{97} +4.58917e6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9}+O(q^{10})$$ 4 * q + 53 * q^3 + 111 * q^5 + 1666 * q^7 + 4609 * q^9 $$4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9} + 4593 q^{11} + 7847 q^{13} - 18900 q^{15} + 23172 q^{17} - 23696 q^{19} + 69416 q^{21} - 24105 q^{23} - 138149 q^{25} + 433646 q^{27} - 140949 q^{29} + 664609 q^{31} - 240450 q^{33} + 248544 q^{35} + 202612 q^{37} + 2288827 q^{39} - 709737 q^{41} + 128962 q^{43} - 1755342 q^{45} + 445842 q^{47} - 1602774 q^{49} + 2883630 q^{51} - 975870 q^{53} + 644145 q^{55} + 3494630 q^{57} + 1812858 q^{59} - 2955031 q^{61} + 3362482 q^{63} + 666 q^{65} - 2737235 q^{67} - 1781673 q^{69} - 4958184 q^{71} - 931591 q^{73} - 4945810 q^{75} + 4352514 q^{77} - 5813561 q^{79} + 16394896 q^{81} - 2120460 q^{83} - 4845402 q^{85} - 7965333 q^{87} + 8833716 q^{89} + 18886274 q^{91} + 3024182 q^{93} + 3151794 q^{95} - 22666876 q^{97} + 17931894 q^{99}+O(q^{100})$$ 4 * q + 53 * q^3 + 111 * q^5 + 1666 * q^7 + 4609 * q^9 + 4593 * q^11 + 7847 * q^13 - 18900 * q^15 + 23172 * q^17 - 23696 * q^19 + 69416 * q^21 - 24105 * q^23 - 138149 * q^25 + 433646 * q^27 - 140949 * q^29 + 664609 * q^31 - 240450 * q^33 + 248544 * q^35 + 202612 * q^37 + 2288827 * q^39 - 709737 * q^41 + 128962 * q^43 - 1755342 * q^45 + 445842 * q^47 - 1602774 * q^49 + 2883630 * q^51 - 975870 * q^53 + 644145 * q^55 + 3494630 * q^57 + 1812858 * q^59 - 2955031 * q^61 + 3362482 * q^63 + 666 * q^65 - 2737235 * q^67 - 1781673 * q^69 - 4958184 * q^71 - 931591 * q^73 - 4945810 * q^75 + 4352514 * q^77 - 5813561 * q^79 + 16394896 * q^81 - 2120460 * q^83 - 4845402 * q^85 - 7965333 * q^87 + 8833716 * q^89 + 18886274 * q^91 + 3024182 * q^93 + 3151794 * q^95 - 22666876 * q^97 + 17931894 * 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
$$3$$ 36.0598 0.771079 0.385540 0.922691i $$-0.374015\pi$$
0.385540 + 0.922691i $$0.374015\pi$$
$$4$$ 0 0
$$5$$ −19.7362 −0.0706103 −0.0353051 0.999377i $$-0.511240\pi$$
−0.0353051 + 0.999377i $$0.511240\pi$$
$$6$$ 0 0
$$7$$ 50.9272 0.0561186 0.0280593 0.999606i $$-0.491067\pi$$
0.0280593 + 0.999606i $$0.491067\pi$$
$$8$$ 0 0
$$9$$ −886.690 −0.405436
$$10$$ 0 0
$$11$$ −5175.62 −1.17243 −0.586217 0.810154i $$-0.699383\pi$$
−0.586217 + 0.810154i $$0.699383\pi$$
$$12$$ 0 0
$$13$$ 4311.51 0.544287 0.272143 0.962257i $$-0.412267\pi$$
0.272143 + 0.962257i $$0.412267\pi$$
$$14$$ 0 0
$$15$$ −711.683 −0.0544461
$$16$$ 0 0
$$17$$ 14063.5 0.694262 0.347131 0.937817i $$-0.387156\pi$$
0.347131 + 0.937817i $$0.387156\pi$$
$$18$$ 0 0
$$19$$ −26643.5 −0.891154 −0.445577 0.895244i $$-0.647001\pi$$
−0.445577 + 0.895244i $$0.647001\pi$$
$$20$$ 0 0
$$21$$ 1836.43 0.0432719
$$22$$ 0 0
$$23$$ 77473.7 1.32772 0.663861 0.747856i $$-0.268917\pi$$
0.663861 + 0.747856i $$0.268917\pi$$
$$24$$ 0 0
$$25$$ −77735.5 −0.995014
$$26$$ 0 0
$$27$$ −110837. −1.08370
$$28$$ 0 0
$$29$$ −33124.8 −0.252209 −0.126104 0.992017i $$-0.540247\pi$$
−0.126104 + 0.992017i $$0.540247\pi$$
$$30$$ 0 0
$$31$$ 172834. 1.04199 0.520995 0.853560i $$-0.325561\pi$$
0.520995 + 0.853560i $$0.325561\pi$$
$$32$$ 0 0
$$33$$ −186632. −0.904039
$$34$$ 0 0
$$35$$ −1005.11 −0.00396255
$$36$$ 0 0
$$37$$ 50653.0 0.164399
$$38$$ 0 0
$$39$$ 155472. 0.419688
$$40$$ 0 0
$$41$$ −846738. −1.91869 −0.959347 0.282229i $$-0.908926\pi$$
−0.959347 + 0.282229i $$0.908926\pi$$
$$42$$ 0 0
$$43$$ 307387. 0.589585 0.294792 0.955561i $$-0.404750\pi$$
0.294792 + 0.955561i $$0.404750\pi$$
$$44$$ 0 0
$$45$$ 17499.9 0.0286280
$$46$$ 0 0
$$47$$ 322473. 0.453056 0.226528 0.974005i $$-0.427263\pi$$
0.226528 + 0.974005i $$0.427263\pi$$
$$48$$ 0 0
$$49$$ −820949. −0.996851
$$50$$ 0 0
$$51$$ 507128. 0.535331
$$52$$ 0 0
$$53$$ 1.83562e6 1.69363 0.846814 0.531890i $$-0.178518\pi$$
0.846814 + 0.531890i $$0.178518\pi$$
$$54$$ 0 0
$$55$$ 102147. 0.0827859
$$56$$ 0 0
$$57$$ −960758. −0.687151
$$58$$ 0 0
$$59$$ 797340. 0.505430 0.252715 0.967541i $$-0.418676\pi$$
0.252715 + 0.967541i $$0.418676\pi$$
$$60$$ 0 0
$$61$$ −684801. −0.386287 −0.193144 0.981171i $$-0.561868\pi$$
−0.193144 + 0.981171i $$0.561868\pi$$
$$62$$ 0 0
$$63$$ −45156.6 −0.0227525
$$64$$ 0 0
$$65$$ −85092.7 −0.0384322
$$66$$ 0 0
$$67$$ 3.79463e6 1.54137 0.770687 0.637214i $$-0.219913\pi$$
0.770687 + 0.637214i $$0.219913\pi$$
$$68$$ 0 0
$$69$$ 2.79369e6 1.02378
$$70$$ 0 0
$$71$$ 342906. 0.113703 0.0568514 0.998383i $$-0.481894\pi$$
0.0568514 + 0.998383i $$0.481894\pi$$
$$72$$ 0 0
$$73$$ 2.64684e6 0.796337 0.398168 0.917312i $$-0.369646\pi$$
0.398168 + 0.917312i $$0.369646\pi$$
$$74$$ 0 0
$$75$$ −2.80313e6 −0.767235
$$76$$ 0 0
$$77$$ −263580. −0.0657953
$$78$$ 0 0
$$79$$ 2.44136e6 0.557104 0.278552 0.960421i $$-0.410146\pi$$
0.278552 + 0.960421i $$0.410146\pi$$
$$80$$ 0 0
$$81$$ −2.05756e6 −0.430185
$$82$$ 0 0
$$83$$ −5.61596e6 −1.07808 −0.539040 0.842280i $$-0.681213\pi$$
−0.539040 + 0.842280i $$0.681213\pi$$
$$84$$ 0 0
$$85$$ −277560. −0.0490220
$$86$$ 0 0
$$87$$ −1.19447e6 −0.194473
$$88$$ 0 0
$$89$$ 8.43578e6 1.26841 0.634206 0.773164i $$-0.281327\pi$$
0.634206 + 0.773164i $$0.281327\pi$$
$$90$$ 0 0
$$91$$ 219573. 0.0305446
$$92$$ 0 0
$$93$$ 6.23237e6 0.803457
$$94$$ 0 0
$$95$$ 525840. 0.0629246
$$96$$ 0 0
$$97$$ −3.40422e6 −0.378718 −0.189359 0.981908i $$-0.560641\pi$$
−0.189359 + 0.981908i $$0.560641\pi$$
$$98$$ 0 0
$$99$$ 4.58917e6 0.475347
$$100$$ 0 0
$$101$$ −1.62461e6 −0.156901 −0.0784504 0.996918i $$-0.524997\pi$$
−0.0784504 + 0.996918i $$0.524997\pi$$
$$102$$ 0 0
$$103$$ 7.21129e6 0.650253 0.325127 0.945671i $$-0.394593\pi$$
0.325127 + 0.945671i $$0.394593\pi$$
$$104$$ 0 0
$$105$$ −36244.0 −0.00305544
$$106$$ 0 0
$$107$$ 1.38534e7 1.09323 0.546616 0.837383i $$-0.315916\pi$$
0.546616 + 0.837383i $$0.315916\pi$$
$$108$$ 0 0
$$109$$ −5.56203e6 −0.411377 −0.205689 0.978617i $$-0.565943\pi$$
−0.205689 + 0.978617i $$0.565943\pi$$
$$110$$ 0 0
$$111$$ 1.82654e6 0.126765
$$112$$ 0 0
$$113$$ −2.13570e7 −1.39241 −0.696203 0.717845i $$-0.745129\pi$$
−0.696203 + 0.717845i $$0.745129\pi$$
$$114$$ 0 0
$$115$$ −1.52903e6 −0.0937508
$$116$$ 0 0
$$117$$ −3.82297e6 −0.220674
$$118$$ 0 0
$$119$$ 716216. 0.0389610
$$120$$ 0 0
$$121$$ 7.29990e6 0.374600
$$122$$ 0 0
$$123$$ −3.05332e7 −1.47947
$$124$$ 0 0
$$125$$ 3.07609e6 0.140869
$$126$$ 0 0
$$127$$ 2.51759e7 1.09062 0.545308 0.838236i $$-0.316413\pi$$
0.545308 + 0.838236i $$0.316413\pi$$
$$128$$ 0 0
$$129$$ 1.10843e7 0.454617
$$130$$ 0 0
$$131$$ 2.73598e7 1.06332 0.531659 0.846959i $$-0.321569\pi$$
0.531659 + 0.846959i $$0.321569\pi$$
$$132$$ 0 0
$$133$$ −1.35688e6 −0.0500103
$$134$$ 0 0
$$135$$ 2.18749e6 0.0765206
$$136$$ 0 0
$$137$$ 3.06260e7 1.01758 0.508789 0.860891i $$-0.330093\pi$$
0.508789 + 0.860891i $$0.330093\pi$$
$$138$$ 0 0
$$139$$ 3.33288e7 1.05261 0.526305 0.850296i $$-0.323577\pi$$
0.526305 + 0.850296i $$0.323577\pi$$
$$140$$ 0 0
$$141$$ 1.16283e7 0.349342
$$142$$ 0 0
$$143$$ −2.23148e7 −0.638140
$$144$$ 0 0
$$145$$ 653756. 0.0178085
$$146$$ 0 0
$$147$$ −2.96033e7 −0.768651
$$148$$ 0 0
$$149$$ 7.15248e6 0.177135 0.0885675 0.996070i $$-0.471771\pi$$
0.0885675 + 0.996070i $$0.471771\pi$$
$$150$$ 0 0
$$151$$ 4.28753e7 1.01342 0.506708 0.862118i $$-0.330862\pi$$
0.506708 + 0.862118i $$0.330862\pi$$
$$152$$ 0 0
$$153$$ −1.24700e7 −0.281479
$$154$$ 0 0
$$155$$ −3.41108e6 −0.0735752
$$156$$ 0 0
$$157$$ 6.89941e7 1.42286 0.711431 0.702756i $$-0.248047\pi$$
0.711431 + 0.702756i $$0.248047\pi$$
$$158$$ 0 0
$$159$$ 6.61922e7 1.30592
$$160$$ 0 0
$$161$$ 3.94552e6 0.0745098
$$162$$ 0 0
$$163$$ 2.76360e7 0.499826 0.249913 0.968268i $$-0.419598\pi$$
0.249913 + 0.968268i $$0.419598\pi$$
$$164$$ 0 0
$$165$$ 3.68340e6 0.0638345
$$166$$ 0 0
$$167$$ −4.33284e7 −0.719888 −0.359944 0.932974i $$-0.617204\pi$$
−0.359944 + 0.932974i $$0.617204\pi$$
$$168$$ 0 0
$$169$$ −4.41594e7 −0.703752
$$170$$ 0 0
$$171$$ 2.36245e7 0.361306
$$172$$ 0 0
$$173$$ 3.02184e7 0.443722 0.221861 0.975078i $$-0.428787\pi$$
0.221861 + 0.975078i $$0.428787\pi$$
$$174$$ 0 0
$$175$$ −3.95885e6 −0.0558388
$$176$$ 0 0
$$177$$ 2.87519e7 0.389727
$$178$$ 0 0
$$179$$ −1.27627e8 −1.66325 −0.831624 0.555340i $$-0.812588\pi$$
−0.831624 + 0.555340i $$0.812588\pi$$
$$180$$ 0 0
$$181$$ 4.76747e7 0.597603 0.298802 0.954315i $$-0.403413\pi$$
0.298802 + 0.954315i $$0.403413\pi$$
$$182$$ 0 0
$$183$$ −2.46938e7 −0.297858
$$184$$ 0 0
$$185$$ −999696. −0.0116083
$$186$$ 0 0
$$187$$ −7.27875e7 −0.813976
$$188$$ 0 0
$$189$$ −5.64460e6 −0.0608159
$$190$$ 0 0
$$191$$ −8.01658e7 −0.832477 −0.416239 0.909255i $$-0.636652\pi$$
−0.416239 + 0.909255i $$0.636652\pi$$
$$192$$ 0 0
$$193$$ 1.63480e6 0.0163687 0.00818436 0.999967i $$-0.497395\pi$$
0.00818436 + 0.999967i $$0.497395\pi$$
$$194$$ 0 0
$$195$$ −3.06843e6 −0.0296343
$$196$$ 0 0
$$197$$ −1.06595e8 −0.993356 −0.496678 0.867935i $$-0.665447\pi$$
−0.496678 + 0.867935i $$0.665447\pi$$
$$198$$ 0 0
$$199$$ −2.24394e7 −0.201848 −0.100924 0.994894i $$-0.532180\pi$$
−0.100924 + 0.994894i $$0.532180\pi$$
$$200$$ 0 0
$$201$$ 1.36834e8 1.18852
$$202$$ 0 0
$$203$$ −1.68695e6 −0.0141536
$$204$$ 0 0
$$205$$ 1.67114e7 0.135480
$$206$$ 0 0
$$207$$ −6.86951e7 −0.538307
$$208$$ 0 0
$$209$$ 1.37896e8 1.04482
$$210$$ 0 0
$$211$$ 2.68426e7 0.196714 0.0983571 0.995151i $$-0.468641\pi$$
0.0983571 + 0.995151i $$0.468641\pi$$
$$212$$ 0 0
$$213$$ 1.23651e7 0.0876739
$$214$$ 0 0
$$215$$ −6.06665e6 −0.0416307
$$216$$ 0 0
$$217$$ 8.80196e6 0.0584750
$$218$$ 0 0
$$219$$ 9.54444e7 0.614039
$$220$$ 0 0
$$221$$ 6.06351e7 0.377877
$$222$$ 0 0
$$223$$ 1.08818e7 0.0657102 0.0328551 0.999460i $$-0.489540\pi$$
0.0328551 + 0.999460i $$0.489540\pi$$
$$224$$ 0 0
$$225$$ 6.89272e7 0.403415
$$226$$ 0 0
$$227$$ 1.56786e8 0.889642 0.444821 0.895619i $$-0.353267\pi$$
0.444821 + 0.895619i $$0.353267\pi$$
$$228$$ 0 0
$$229$$ −1.95778e8 −1.07731 −0.538653 0.842528i $$-0.681067\pi$$
−0.538653 + 0.842528i $$0.681067\pi$$
$$230$$ 0 0
$$231$$ −9.50465e6 −0.0507334
$$232$$ 0 0
$$233$$ 1.45843e8 0.755337 0.377668 0.925941i $$-0.376726\pi$$
0.377668 + 0.925941i $$0.376726\pi$$
$$234$$ 0 0
$$235$$ −6.36439e6 −0.0319904
$$236$$ 0 0
$$237$$ 8.80348e7 0.429571
$$238$$ 0 0
$$239$$ 3.52653e8 1.67092 0.835458 0.549555i $$-0.185203\pi$$
0.835458 + 0.549555i $$0.185203\pi$$
$$240$$ 0 0
$$241$$ −3.09151e8 −1.42269 −0.711346 0.702842i $$-0.751914\pi$$
−0.711346 + 0.702842i $$0.751914\pi$$
$$242$$ 0 0
$$243$$ 1.68205e8 0.751997
$$244$$ 0 0
$$245$$ 1.62024e7 0.0703879
$$246$$ 0 0
$$247$$ −1.14874e8 −0.485043
$$248$$ 0 0
$$249$$ −2.02511e8 −0.831285
$$250$$ 0 0
$$251$$ 1.96899e8 0.785932 0.392966 0.919553i $$-0.371449\pi$$
0.392966 + 0.919553i $$0.371449\pi$$
$$252$$ 0 0
$$253$$ −4.00975e8 −1.55666
$$254$$ 0 0
$$255$$ −1.00088e7 −0.0377999
$$256$$ 0 0
$$257$$ 3.14559e8 1.15594 0.577972 0.816057i $$-0.303844\pi$$
0.577972 + 0.816057i $$0.303844\pi$$
$$258$$ 0 0
$$259$$ 2.57962e6 0.00922584
$$260$$ 0 0
$$261$$ 2.93714e7 0.102255
$$262$$ 0 0
$$263$$ −5.69257e7 −0.192958 −0.0964792 0.995335i $$-0.530758\pi$$
−0.0964792 + 0.995335i $$0.530758\pi$$
$$264$$ 0 0
$$265$$ −3.62282e7 −0.119587
$$266$$ 0 0
$$267$$ 3.04193e8 0.978046
$$268$$ 0 0
$$269$$ −3.56640e7 −0.111711 −0.0558557 0.998439i $$-0.517789\pi$$
−0.0558557 + 0.998439i $$0.517789\pi$$
$$270$$ 0 0
$$271$$ 5.25367e8 1.60350 0.801752 0.597657i $$-0.203901\pi$$
0.801752 + 0.597657i $$0.203901\pi$$
$$272$$ 0 0
$$273$$ 7.91777e6 0.0235523
$$274$$ 0 0
$$275$$ 4.02330e8 1.16659
$$276$$ 0 0
$$277$$ −6.45650e8 −1.82523 −0.912616 0.408818i $$-0.865941\pi$$
−0.912616 + 0.408818i $$0.865941\pi$$
$$278$$ 0 0
$$279$$ −1.53250e8 −0.422461
$$280$$ 0 0
$$281$$ 4.38689e8 1.17946 0.589731 0.807599i $$-0.299234\pi$$
0.589731 + 0.807599i $$0.299234\pi$$
$$282$$ 0 0
$$283$$ −1.91165e8 −0.501367 −0.250683 0.968069i $$-0.580655\pi$$
−0.250683 + 0.968069i $$0.580655\pi$$
$$284$$ 0 0
$$285$$ 1.89617e7 0.0485199
$$286$$ 0 0
$$287$$ −4.31220e7 −0.107674
$$288$$ 0 0
$$289$$ −2.12556e8 −0.518001
$$290$$ 0 0
$$291$$ −1.22755e8 −0.292022
$$292$$ 0 0
$$293$$ 6.78441e8 1.57571 0.787854 0.615862i $$-0.211192\pi$$
0.787854 + 0.615862i $$0.211192\pi$$
$$294$$ 0 0
$$295$$ −1.57364e7 −0.0356886
$$296$$ 0 0
$$297$$ 5.73649e8 1.27057
$$298$$ 0 0
$$299$$ 3.34029e8 0.722661
$$300$$ 0 0
$$301$$ 1.56544e7 0.0330867
$$302$$ 0 0
$$303$$ −5.85833e7 −0.120983
$$304$$ 0 0
$$305$$ 1.35154e7 0.0272758
$$306$$ 0 0
$$307$$ 4.40928e8 0.869728 0.434864 0.900496i $$-0.356797\pi$$
0.434864 + 0.900496i $$0.356797\pi$$
$$308$$ 0 0
$$309$$ 2.60038e8 0.501397
$$310$$ 0 0
$$311$$ 6.01820e8 1.13450 0.567251 0.823545i $$-0.308007\pi$$
0.567251 + 0.823545i $$0.308007\pi$$
$$312$$ 0 0
$$313$$ 6.38537e8 1.17701 0.588506 0.808493i $$-0.299716\pi$$
0.588506 + 0.808493i $$0.299716\pi$$
$$314$$ 0 0
$$315$$ 891219. 0.00160656
$$316$$ 0 0
$$317$$ 6.62098e8 1.16739 0.583693 0.811974i $$-0.301607\pi$$
0.583693 + 0.811974i $$0.301607\pi$$
$$318$$ 0 0
$$319$$ 1.71441e8 0.295698
$$320$$ 0 0
$$321$$ 4.99550e8 0.842969
$$322$$ 0 0
$$323$$ −3.74701e8 −0.618694
$$324$$ 0 0
$$325$$ −3.35157e8 −0.541573
$$326$$ 0 0
$$327$$ −2.00566e8 −0.317205
$$328$$ 0 0
$$329$$ 1.64227e7 0.0254248
$$330$$ 0 0
$$331$$ 4.62478e8 0.700960 0.350480 0.936570i $$-0.386018\pi$$
0.350480 + 0.936570i $$0.386018\pi$$
$$332$$ 0 0
$$333$$ −4.49135e7 −0.0666533
$$334$$ 0 0
$$335$$ −7.48915e7 −0.108837
$$336$$ 0 0
$$337$$ 6.65983e8 0.947891 0.473946 0.880554i $$-0.342829\pi$$
0.473946 + 0.880554i $$0.342829\pi$$
$$338$$ 0 0
$$339$$ −7.70130e8 −1.07366
$$340$$ 0 0
$$341$$ −8.94524e8 −1.22166
$$342$$ 0 0
$$343$$ −8.37494e7 −0.112060
$$344$$ 0 0
$$345$$ −5.51367e7 −0.0722893
$$346$$ 0 0
$$347$$ −3.49930e8 −0.449601 −0.224801 0.974405i $$-0.572173\pi$$
−0.224801 + 0.974405i $$0.572173\pi$$
$$348$$ 0 0
$$349$$ 9.83350e8 1.23828 0.619140 0.785280i $$-0.287481\pi$$
0.619140 + 0.785280i $$0.287481\pi$$
$$350$$ 0 0
$$351$$ −4.77874e8 −0.589845
$$352$$ 0 0
$$353$$ −6.25026e8 −0.756287 −0.378144 0.925747i $$-0.623437\pi$$
−0.378144 + 0.925747i $$0.623437\pi$$
$$354$$ 0 0
$$355$$ −6.76766e6 −0.00802859
$$356$$ 0 0
$$357$$ 2.58266e7 0.0300420
$$358$$ 0 0
$$359$$ 6.83539e8 0.779709 0.389855 0.920876i $$-0.372525\pi$$
0.389855 + 0.920876i $$0.372525\pi$$
$$360$$ 0 0
$$361$$ −1.83998e8 −0.205844
$$362$$ 0 0
$$363$$ 2.63233e8 0.288847
$$364$$ 0 0
$$365$$ −5.22384e7 −0.0562296
$$366$$ 0 0
$$367$$ 5.50838e8 0.581691 0.290846 0.956770i $$-0.406063\pi$$
0.290846 + 0.956770i $$0.406063\pi$$
$$368$$ 0 0
$$369$$ 7.50794e8 0.777908
$$370$$ 0 0
$$371$$ 9.34831e7 0.0950440
$$372$$ 0 0
$$373$$ 1.03057e9 1.02824 0.514121 0.857717i $$-0.328118\pi$$
0.514121 + 0.857717i $$0.328118\pi$$
$$374$$ 0 0
$$375$$ 1.10923e8 0.108621
$$376$$ 0 0
$$377$$ −1.42818e8 −0.137274
$$378$$ 0 0
$$379$$ 1.53319e8 0.144663 0.0723317 0.997381i $$-0.476956\pi$$
0.0723317 + 0.997381i $$0.476956\pi$$
$$380$$ 0 0
$$381$$ 9.07839e8 0.840952
$$382$$ 0 0
$$383$$ 1.20085e9 1.09218 0.546089 0.837727i $$-0.316116\pi$$
0.546089 + 0.837727i $$0.316116\pi$$
$$384$$ 0 0
$$385$$ 5.20206e6 0.00464583
$$386$$ 0 0
$$387$$ −2.72557e8 −0.239039
$$388$$ 0 0
$$389$$ 9.04660e8 0.779223 0.389611 0.920979i $$-0.372609\pi$$
0.389611 + 0.920979i $$0.372609\pi$$
$$390$$ 0 0
$$391$$ 1.08955e9 0.921786
$$392$$ 0 0
$$393$$ 9.86589e8 0.819903
$$394$$ 0 0
$$395$$ −4.81830e7 −0.0393373
$$396$$ 0 0
$$397$$ 5.60100e8 0.449261 0.224630 0.974444i $$-0.427882\pi$$
0.224630 + 0.974444i $$0.427882\pi$$
$$398$$ 0 0
$$399$$ −4.89287e7 −0.0385619
$$400$$ 0 0
$$401$$ 1.75894e9 1.36222 0.681108 0.732183i $$-0.261498\pi$$
0.681108 + 0.732183i $$0.261498\pi$$
$$402$$ 0 0
$$403$$ 7.45176e8 0.567141
$$404$$ 0 0
$$405$$ 4.06084e7 0.0303755
$$406$$ 0 0
$$407$$ −2.62161e8 −0.192747
$$408$$ 0 0
$$409$$ 3.48105e8 0.251582 0.125791 0.992057i $$-0.459853\pi$$
0.125791 + 0.992057i $$0.459853\pi$$
$$410$$ 0 0
$$411$$ 1.10437e9 0.784633
$$412$$ 0 0
$$413$$ 4.06063e7 0.0283640
$$414$$ 0 0
$$415$$ 1.10838e8 0.0761235
$$416$$ 0 0
$$417$$ 1.20183e9 0.811647
$$418$$ 0 0
$$419$$ 9.10860e8 0.604927 0.302463 0.953161i $$-0.402191\pi$$
0.302463 + 0.953161i $$0.402191\pi$$
$$420$$ 0 0
$$421$$ −2.03192e9 −1.32715 −0.663574 0.748111i $$-0.730961\pi$$
−0.663574 + 0.748111i $$0.730961\pi$$
$$422$$ 0 0
$$423$$ −2.85934e8 −0.183685
$$424$$ 0 0
$$425$$ −1.09324e9 −0.690800
$$426$$ 0 0
$$427$$ −3.48750e7 −0.0216779
$$428$$ 0 0
$$429$$ −8.04666e8 −0.492057
$$430$$ 0 0
$$431$$ −1.21720e9 −0.732304 −0.366152 0.930555i $$-0.619325\pi$$
−0.366152 + 0.930555i $$0.619325\pi$$
$$432$$ 0 0
$$433$$ −1.40845e9 −0.833745 −0.416872 0.908965i $$-0.636874\pi$$
−0.416872 + 0.908965i $$0.636874\pi$$
$$434$$ 0 0
$$435$$ 2.35743e7 0.0137318
$$436$$ 0 0
$$437$$ −2.06417e9 −1.18320
$$438$$ 0 0
$$439$$ −1.96037e9 −1.10589 −0.552946 0.833217i $$-0.686496\pi$$
−0.552946 + 0.833217i $$0.686496\pi$$
$$440$$ 0 0
$$441$$ 7.27927e8 0.404160
$$442$$ 0 0
$$443$$ −2.51266e9 −1.37316 −0.686578 0.727056i $$-0.740888\pi$$
−0.686578 + 0.727056i $$0.740888\pi$$
$$444$$ 0 0
$$445$$ −1.66490e8 −0.0895629
$$446$$ 0 0
$$447$$ 2.57917e8 0.136585
$$448$$ 0 0
$$449$$ −1.67892e9 −0.875324 −0.437662 0.899140i $$-0.644193\pi$$
−0.437662 + 0.899140i $$0.644193\pi$$
$$450$$ 0 0
$$451$$ 4.38240e9 2.24954
$$452$$ 0 0
$$453$$ 1.54607e9 0.781424
$$454$$ 0 0
$$455$$ −4.33354e6 −0.00215676
$$456$$ 0 0
$$457$$ −6.46678e8 −0.316943 −0.158472 0.987364i $$-0.550657\pi$$
−0.158472 + 0.987364i $$0.550657\pi$$
$$458$$ 0 0
$$459$$ −1.55876e9 −0.752373
$$460$$ 0 0
$$461$$ 1.32962e9 0.632083 0.316041 0.948745i $$-0.397646\pi$$
0.316041 + 0.948745i $$0.397646\pi$$
$$462$$ 0 0
$$463$$ −3.34991e9 −1.56856 −0.784278 0.620409i $$-0.786967\pi$$
−0.784278 + 0.620409i $$0.786967\pi$$
$$464$$ 0 0
$$465$$ −1.23003e8 −0.0567323
$$466$$ 0 0
$$467$$ −6.13048e8 −0.278539 −0.139269 0.990255i $$-0.544475\pi$$
−0.139269 + 0.990255i $$0.544475\pi$$
$$468$$ 0 0
$$469$$ 1.93250e8 0.0864997
$$470$$ 0 0
$$471$$ 2.48791e9 1.09714
$$472$$ 0 0
$$473$$ −1.59092e9 −0.691249
$$474$$ 0 0
$$475$$ 2.07114e9 0.886711
$$476$$ 0 0
$$477$$ −1.62763e9 −0.686658
$$478$$ 0 0
$$479$$ 3.20691e9 1.33325 0.666627 0.745392i $$-0.267738\pi$$
0.666627 + 0.745392i $$0.267738\pi$$
$$480$$ 0 0
$$481$$ 2.18391e8 0.0894802
$$482$$ 0 0
$$483$$ 1.42275e8 0.0574530
$$484$$ 0 0
$$485$$ 6.71862e7 0.0267414
$$486$$ 0 0
$$487$$ −2.67973e9 −1.05133 −0.525665 0.850692i $$-0.676183\pi$$
−0.525665 + 0.850692i $$0.676183\pi$$
$$488$$ 0 0
$$489$$ 9.96550e8 0.385406
$$490$$ 0 0
$$491$$ −1.74917e9 −0.666879 −0.333439 0.942772i $$-0.608209\pi$$
−0.333439 + 0.942772i $$0.608209\pi$$
$$492$$ 0 0
$$493$$ −4.65851e8 −0.175099
$$494$$ 0 0
$$495$$ −9.05727e7 −0.0335644
$$496$$ 0 0
$$497$$ 1.74633e7 0.00638084
$$498$$ 0 0
$$499$$ 2.21759e9 0.798967 0.399484 0.916740i $$-0.369189\pi$$
0.399484 + 0.916740i $$0.369189\pi$$
$$500$$ 0 0
$$501$$ −1.56241e9 −0.555091
$$502$$ 0 0
$$503$$ −4.21833e9 −1.47792 −0.738962 0.673747i $$-0.764684\pi$$
−0.738962 + 0.673747i $$0.764684\pi$$
$$504$$ 0 0
$$505$$ 3.20637e7 0.0110788
$$506$$ 0 0
$$507$$ −1.59238e9 −0.542649
$$508$$ 0 0
$$509$$ −4.92334e9 −1.65481 −0.827404 0.561607i $$-0.810183\pi$$
−0.827404 + 0.561607i $$0.810183\pi$$
$$510$$ 0 0
$$511$$ 1.34796e8 0.0446893
$$512$$ 0 0
$$513$$ 2.95307e9 0.965747
$$514$$ 0 0
$$515$$ −1.42323e8 −0.0459146
$$516$$ 0 0
$$517$$ −1.66900e9 −0.531178
$$518$$ 0 0
$$519$$ 1.08967e9 0.342145
$$520$$ 0 0
$$521$$ 3.66580e9 1.13563 0.567814 0.823157i $$-0.307789\pi$$
0.567814 + 0.823157i $$0.307789\pi$$
$$522$$ 0 0
$$523$$ −5.07711e9 −1.55189 −0.775945 0.630800i $$-0.782727\pi$$
−0.775945 + 0.630800i $$0.782727\pi$$
$$524$$ 0 0
$$525$$ −1.42755e8 −0.0430561
$$526$$ 0 0
$$527$$ 2.43066e9 0.723414
$$528$$ 0 0
$$529$$ 2.59735e9 0.762843
$$530$$ 0 0
$$531$$ −7.06993e8 −0.204920
$$532$$ 0 0
$$533$$ −3.65072e9 −1.04432
$$534$$ 0 0
$$535$$ −2.73413e8 −0.0771934
$$536$$ 0 0
$$537$$ −4.60221e9 −1.28250
$$538$$ 0 0
$$539$$ 4.24892e9 1.16874
$$540$$ 0 0
$$541$$ 5.97271e9 1.62174 0.810870 0.585226i $$-0.198994\pi$$
0.810870 + 0.585226i $$0.198994\pi$$
$$542$$ 0 0
$$543$$ 1.71914e9 0.460800
$$544$$ 0 0
$$545$$ 1.09773e8 0.0290475
$$546$$ 0 0
$$547$$ 4.63928e9 1.21198 0.605990 0.795473i $$-0.292777\pi$$
0.605990 + 0.795473i $$0.292777\pi$$
$$548$$ 0 0
$$549$$ 6.07206e8 0.156615
$$550$$ 0 0
$$551$$ 8.82558e8 0.224757
$$552$$ 0 0
$$553$$ 1.24331e8 0.0312639
$$554$$ 0 0
$$555$$ −3.60489e7 −0.00895089
$$556$$ 0 0
$$557$$ −2.87674e9 −0.705354 −0.352677 0.935745i $$-0.614729\pi$$
−0.352677 + 0.935745i $$0.614729\pi$$
$$558$$ 0 0
$$559$$ 1.32530e9 0.320903
$$560$$ 0 0
$$561$$ −2.62471e9 −0.627640
$$562$$ 0 0
$$563$$ −5.48798e9 −1.29608 −0.648042 0.761604i $$-0.724412\pi$$
−0.648042 + 0.761604i $$0.724412\pi$$
$$564$$ 0 0
$$565$$ 4.21505e8 0.0983181
$$566$$ 0 0
$$567$$ −1.04786e8 −0.0241414
$$568$$ 0 0
$$569$$ 5.47367e9 1.24562 0.622810 0.782373i $$-0.285991\pi$$
0.622810 + 0.782373i $$0.285991\pi$$
$$570$$ 0 0
$$571$$ −1.73348e9 −0.389665 −0.194833 0.980837i $$-0.562416\pi$$
−0.194833 + 0.980837i $$0.562416\pi$$
$$572$$ 0 0
$$573$$ −2.89076e9 −0.641906
$$574$$ 0 0
$$575$$ −6.02246e9 −1.32110
$$576$$ 0 0
$$577$$ −2.24479e8 −0.0486475 −0.0243238 0.999704i $$-0.507743\pi$$
−0.0243238 + 0.999704i $$0.507743\pi$$
$$578$$ 0 0
$$579$$ 5.89507e7 0.0126216
$$580$$ 0 0
$$581$$ −2.86005e8 −0.0605003
$$582$$ 0 0
$$583$$ −9.50049e9 −1.98567
$$584$$ 0 0
$$585$$ 7.54508e7 0.0155818
$$586$$ 0 0
$$587$$ −2.44680e9 −0.499305 −0.249652 0.968336i $$-0.580316\pi$$
−0.249652 + 0.968336i $$0.580316\pi$$
$$588$$ 0 0
$$589$$ −4.60490e9 −0.928574
$$590$$ 0 0
$$591$$ −3.84380e9 −0.765956
$$592$$ 0 0
$$593$$ −1.85980e9 −0.366248 −0.183124 0.983090i $$-0.558621\pi$$
−0.183124 + 0.983090i $$0.558621\pi$$
$$594$$ 0 0
$$595$$ −1.41354e7 −0.00275105
$$596$$ 0 0
$$597$$ −8.09161e8 −0.155641
$$598$$ 0 0
$$599$$ 1.26622e8 0.0240722 0.0120361 0.999928i $$-0.496169\pi$$
0.0120361 + 0.999928i $$0.496169\pi$$
$$600$$ 0 0
$$601$$ −1.07228e9 −0.201488 −0.100744 0.994912i $$-0.532122\pi$$
−0.100744 + 0.994912i $$0.532122\pi$$
$$602$$ 0 0
$$603$$ −3.36466e9 −0.624929
$$604$$ 0 0
$$605$$ −1.44072e8 −0.0264506
$$606$$ 0 0
$$607$$ 5.73945e9 1.04162 0.520811 0.853672i $$-0.325630\pi$$
0.520811 + 0.853672i $$0.325630\pi$$
$$608$$ 0 0
$$609$$ −6.08312e7 −0.0109135
$$610$$ 0 0
$$611$$ 1.39035e9 0.246592
$$612$$ 0 0
$$613$$ 8.64474e8 0.151579 0.0757897 0.997124i $$-0.475852\pi$$
0.0757897 + 0.997124i $$0.475852\pi$$
$$614$$ 0 0
$$615$$ 6.02609e8 0.104465
$$616$$ 0 0
$$617$$ −8.52653e9 −1.46142 −0.730708 0.682690i $$-0.760810\pi$$
−0.730708 + 0.682690i $$0.760810\pi$$
$$618$$ 0 0
$$619$$ −3.14258e9 −0.532561 −0.266281 0.963896i $$-0.585795\pi$$
−0.266281 + 0.963896i $$0.585795\pi$$
$$620$$ 0 0
$$621$$ −8.58693e9 −1.43886
$$622$$ 0 0
$$623$$ 4.29611e8 0.0711815
$$624$$ 0 0
$$625$$ 6.01237e9 0.985067
$$626$$ 0 0
$$627$$ 4.97252e9 0.805639
$$628$$ 0 0
$$629$$ 7.12360e8 0.114136
$$630$$ 0 0
$$631$$ −4.61616e9 −0.731439 −0.365719 0.930725i $$-0.619177\pi$$
−0.365719 + 0.930725i $$0.619177\pi$$
$$632$$ 0 0
$$633$$ 9.67939e8 0.151682
$$634$$ 0 0
$$635$$ −4.96876e8 −0.0770087
$$636$$ 0 0
$$637$$ −3.53953e9 −0.542573
$$638$$ 0 0
$$639$$ −3.04051e8 −0.0460993
$$640$$ 0 0
$$641$$ −1.00510e10 −1.50732 −0.753658 0.657266i $$-0.771713\pi$$
−0.753658 + 0.657266i $$0.771713\pi$$
$$642$$ 0 0
$$643$$ 3.42743e9 0.508429 0.254214 0.967148i $$-0.418183\pi$$
0.254214 + 0.967148i $$0.418183\pi$$
$$644$$ 0 0
$$645$$ −2.18762e8 −0.0321006
$$646$$ 0 0
$$647$$ 1.28014e9 0.185821 0.0929103 0.995674i $$-0.470383\pi$$
0.0929103 + 0.995674i $$0.470383\pi$$
$$648$$ 0 0
$$649$$ −4.12673e9 −0.592584
$$650$$ 0 0
$$651$$ 3.17397e8 0.0450889
$$652$$ 0 0
$$653$$ 6.73100e9 0.945983 0.472992 0.881067i $$-0.343174\pi$$
0.472992 + 0.881067i $$0.343174\pi$$
$$654$$ 0 0
$$655$$ −5.39977e8 −0.0750812
$$656$$ 0 0
$$657$$ −2.34692e9 −0.322864
$$658$$ 0 0
$$659$$ 8.07982e9 1.09977 0.549886 0.835240i $$-0.314671\pi$$
0.549886 + 0.835240i $$0.314671\pi$$
$$660$$ 0 0
$$661$$ −1.58791e8 −0.0213856 −0.0106928 0.999943i $$-0.503404\pi$$
−0.0106928 + 0.999943i $$0.503404\pi$$
$$662$$ 0 0
$$663$$ 2.18649e9 0.291374
$$664$$ 0 0
$$665$$ 2.67796e7 0.00353124
$$666$$ 0 0
$$667$$ −2.56630e9 −0.334863
$$668$$ 0 0
$$669$$ 3.92395e8 0.0506678
$$670$$ 0 0
$$671$$ 3.54427e9 0.452896
$$672$$ 0 0
$$673$$ −7.20617e8 −0.0911280 −0.0455640 0.998961i $$-0.514509\pi$$
−0.0455640 + 0.998961i $$0.514509\pi$$
$$674$$ 0 0
$$675$$ 8.61594e9 1.07830
$$676$$ 0 0
$$677$$ −8.42691e9 −1.04378 −0.521889 0.853013i $$-0.674773\pi$$
−0.521889 + 0.853013i $$0.674773\pi$$
$$678$$ 0 0
$$679$$ −1.73367e8 −0.0212531
$$680$$ 0 0
$$681$$ 5.65366e9 0.685985
$$682$$ 0 0
$$683$$ −5.36376e9 −0.644164 −0.322082 0.946712i $$-0.604383\pi$$
−0.322082 + 0.946712i $$0.604383\pi$$
$$684$$ 0 0
$$685$$ −6.04439e8 −0.0718515
$$686$$ 0 0
$$687$$ −7.05971e9 −0.830688
$$688$$ 0 0
$$689$$ 7.91431e9 0.921819
$$690$$ 0 0
$$691$$ −1.48364e10 −1.71063 −0.855313 0.518112i $$-0.826635\pi$$
−0.855313 + 0.518112i $$0.826635\pi$$
$$692$$ 0 0
$$693$$ 2.33714e8 0.0266758
$$694$$ 0 0
$$695$$ −6.57783e8 −0.0743252
$$696$$ 0 0
$$697$$ −1.19081e10 −1.33208
$$698$$ 0 0
$$699$$ 5.25908e9 0.582425
$$700$$ 0 0
$$701$$ 6.52624e9 0.715566 0.357783 0.933805i $$-0.383533\pi$$
0.357783 + 0.933805i $$0.383533\pi$$
$$702$$ 0 0
$$703$$ −1.34957e9 −0.146505
$$704$$ 0 0
$$705$$ −2.29499e8 −0.0246671
$$706$$ 0 0
$$707$$ −8.27371e7 −0.00880506
$$708$$ 0 0
$$709$$ 1.69533e10 1.78646 0.893228 0.449603i $$-0.148435\pi$$
0.893228 + 0.449603i $$0.148435\pi$$
$$710$$ 0 0
$$711$$ −2.16472e9 −0.225870
$$712$$ 0 0
$$713$$ 1.33901e10 1.38347
$$714$$ 0 0
$$715$$ 4.40408e8 0.0450592
$$716$$ 0 0
$$717$$ 1.27166e10 1.28841
$$718$$ 0 0
$$719$$ 4.30200e9 0.431637 0.215819 0.976433i $$-0.430758\pi$$
0.215819 + 0.976433i $$0.430758\pi$$
$$720$$ 0 0
$$721$$ 3.67251e8 0.0364913
$$722$$ 0 0
$$723$$ −1.11479e10 −1.09701
$$724$$ 0 0
$$725$$ 2.57497e9 0.250951
$$726$$ 0 0
$$727$$ −1.60836e10 −1.55243 −0.776217 0.630465i $$-0.782864\pi$$
−0.776217 + 0.630465i $$0.782864\pi$$
$$728$$ 0 0
$$729$$ 1.05653e10 1.01003
$$730$$ 0 0
$$731$$ 4.32295e9 0.409326
$$732$$ 0 0
$$733$$ 1.61253e9 0.151232 0.0756162 0.997137i $$-0.475908\pi$$
0.0756162 + 0.997137i $$0.475908\pi$$
$$734$$ 0 0
$$735$$ 5.84256e8 0.0542747
$$736$$ 0 0
$$737$$ −1.96396e10 −1.80716
$$738$$ 0 0
$$739$$ −5.21771e9 −0.475581 −0.237790 0.971317i $$-0.576423\pi$$
−0.237790 + 0.971317i $$0.576423\pi$$
$$740$$ 0 0
$$741$$ −4.14232e9 −0.374007
$$742$$ 0 0
$$743$$ −3.55715e9 −0.318157 −0.159078 0.987266i $$-0.550852\pi$$
−0.159078 + 0.987266i $$0.550852\pi$$
$$744$$ 0 0
$$745$$ −1.41163e8 −0.0125076
$$746$$ 0 0
$$747$$ 4.97961e9 0.437093
$$748$$ 0 0
$$749$$ 7.05514e8 0.0613507
$$750$$ 0 0
$$751$$ −2.02483e10 −1.74441 −0.872205 0.489140i $$-0.837311\pi$$
−0.872205 + 0.489140i $$0.837311\pi$$
$$752$$ 0 0
$$753$$ 7.10014e9 0.606016
$$754$$ 0 0
$$755$$ −8.46194e8 −0.0715576
$$756$$ 0 0
$$757$$ 1.87776e10 1.57327 0.786636 0.617417i $$-0.211821\pi$$
0.786636 + 0.617417i $$0.211821\pi$$
$$758$$ 0 0
$$759$$ −1.44591e10 −1.20031
$$760$$ 0 0
$$761$$ −9.48144e9 −0.779880 −0.389940 0.920840i $$-0.627504\pi$$
−0.389940 + 0.920840i $$0.627504\pi$$
$$762$$ 0 0
$$763$$ −2.83259e8 −0.0230859
$$764$$ 0 0
$$765$$ 2.46110e8 0.0198753
$$766$$ 0 0
$$767$$ 3.43774e9 0.275099
$$768$$ 0 0
$$769$$ −7.67008e9 −0.608216 −0.304108 0.952638i $$-0.598358\pi$$
−0.304108 + 0.952638i $$0.598358\pi$$
$$770$$ 0 0
$$771$$ 1.13430e10 0.891324
$$772$$ 0 0
$$773$$ −7.33496e9 −0.571175 −0.285588 0.958353i $$-0.592189\pi$$
−0.285588 + 0.958353i $$0.592189\pi$$
$$774$$ 0 0
$$775$$ −1.34353e10 −1.03680
$$776$$ 0 0
$$777$$ 9.30205e7 0.00711386
$$778$$ 0 0
$$779$$ 2.25600e10 1.70985
$$780$$ 0 0
$$781$$ −1.77475e9 −0.133309
$$782$$ 0 0
$$783$$ 3.67144e9 0.273319
$$784$$ 0 0
$$785$$ −1.36168e9 −0.100469
$$786$$ 0 0
$$787$$ 3.05525e9 0.223427 0.111713 0.993740i $$-0.464366\pi$$
0.111713 + 0.993740i $$0.464366\pi$$
$$788$$ 0 0
$$789$$ −2.05273e9 −0.148786
$$790$$ 0 0
$$791$$ −1.08765e9 −0.0781398
$$792$$ 0 0
$$793$$ −2.95253e9 −0.210251
$$794$$ 0 0
$$795$$ −1.30638e9 −0.0922115
$$796$$ 0 0
$$797$$ −1.30056e10 −0.909971 −0.454985 0.890499i $$-0.650355\pi$$
−0.454985 + 0.890499i $$0.650355\pi$$
$$798$$ 0 0
$$799$$ 4.53512e9 0.314539
$$800$$ 0 0
$$801$$ −7.47992e9 −0.514260
$$802$$ 0 0
$$803$$ −1.36990e10 −0.933652
$$804$$ 0 0
$$805$$ −7.78695e7 −0.00526116
$$806$$ 0 0
$$807$$ −1.28604e9 −0.0861383
$$808$$ 0 0
$$809$$ −2.37212e10 −1.57513 −0.787565 0.616231i $$-0.788659\pi$$
−0.787565 + 0.616231i $$0.788659\pi$$
$$810$$ 0 0
$$811$$ 1.28296e10 0.844577 0.422288 0.906462i $$-0.361227\pi$$
0.422288 + 0.906462i $$0.361227\pi$$
$$812$$ 0 0
$$813$$ 1.89446e10 1.23643
$$814$$ 0 0
$$815$$ −5.45429e8 −0.0352929
$$816$$ 0 0
$$817$$ −8.18986e9 −0.525411
$$818$$ 0 0
$$819$$ −1.94693e8 −0.0123839
$$820$$ 0 0
$$821$$ −7.10512e9 −0.448096 −0.224048 0.974578i $$-0.571927\pi$$
−0.224048 + 0.974578i $$0.571927\pi$$
$$822$$ 0 0
$$823$$ −3.31606e9 −0.207359 −0.103680 0.994611i $$-0.533062\pi$$
−0.103680 + 0.994611i $$0.533062\pi$$
$$824$$ 0 0
$$825$$ 1.45079e10 0.899532
$$826$$ 0 0
$$827$$ 2.02601e10 1.24558 0.622791 0.782388i $$-0.285999\pi$$
0.622791 + 0.782388i $$0.285999\pi$$
$$828$$ 0 0
$$829$$ 3.28928e9 0.200521 0.100261 0.994961i $$-0.468032\pi$$
0.100261 + 0.994961i $$0.468032\pi$$
$$830$$ 0 0
$$831$$ −2.32820e10 −1.40740
$$832$$ 0 0
$$833$$ −1.15454e10 −0.692075
$$834$$ 0 0
$$835$$ 8.55136e8 0.0508315
$$836$$ 0 0
$$837$$ −1.91564e10 −1.12921
$$838$$ 0 0
$$839$$ 1.33201e10 0.778646 0.389323 0.921101i $$-0.372709\pi$$
0.389323 + 0.921101i $$0.372709\pi$$
$$840$$ 0 0
$$841$$ −1.61526e10 −0.936391
$$842$$ 0 0
$$843$$ 1.58190e10 0.909460
$$844$$ 0 0
$$845$$ 8.71537e8 0.0496921
$$846$$ 0 0
$$847$$ 3.71764e8 0.0210220
$$848$$ 0 0
$$849$$ −6.89337e9 −0.386593
$$850$$ 0 0
$$851$$ 3.92427e9 0.218276
$$852$$ 0 0
$$853$$ −2.73074e10 −1.50646 −0.753231 0.657756i $$-0.771506\pi$$
−0.753231 + 0.657756i $$0.771506\pi$$
$$854$$ 0 0
$$855$$ −4.66257e8 −0.0255119
$$856$$ 0 0
$$857$$ 1.77629e10 0.964008 0.482004 0.876169i $$-0.339909\pi$$
0.482004 + 0.876169i $$0.339909\pi$$
$$858$$ 0 0
$$859$$ −3.55499e10 −1.91365 −0.956825 0.290664i $$-0.906124\pi$$
−0.956825 + 0.290664i $$0.906124\pi$$
$$860$$ 0 0
$$861$$ −1.55497e9 −0.0830255
$$862$$ 0 0
$$863$$ 1.54322e10 0.817317 0.408658 0.912687i $$-0.365997\pi$$
0.408658 + 0.912687i $$0.365997\pi$$
$$864$$ 0 0
$$865$$ −5.96396e8 −0.0313313
$$866$$ 0 0
$$867$$ −7.66472e9 −0.399420
$$868$$ 0 0
$$869$$ −1.26355e10 −0.653167
$$870$$ 0 0
$$871$$ 1.63606e10 0.838950
$$872$$ 0 0
$$873$$ 3.01848e9 0.153546
$$874$$ 0 0
$$875$$ 1.56657e8 0.00790534
$$876$$ 0 0
$$877$$ 2.45939e10 1.23120 0.615599 0.788059i $$-0.288914\pi$$
0.615599 + 0.788059i $$0.288914\pi$$
$$878$$ 0 0
$$879$$ 2.44645e10 1.21500
$$880$$ 0 0
$$881$$ −3.24805e10 −1.60032 −0.800159 0.599788i $$-0.795252\pi$$
−0.800159 + 0.599788i $$0.795252\pi$$
$$882$$ 0 0
$$883$$ 1.24274e10 0.607460 0.303730 0.952758i $$-0.401768\pi$$
0.303730 + 0.952758i $$0.401768\pi$$
$$884$$ 0 0
$$885$$ −5.67453e8 −0.0275187
$$886$$ 0 0
$$887$$ 2.06132e10 0.991774 0.495887 0.868387i $$-0.334843\pi$$
0.495887 + 0.868387i $$0.334843\pi$$
$$888$$ 0 0
$$889$$ 1.28214e9 0.0612039
$$890$$ 0 0
$$891$$ 1.06492e10 0.504363
$$892$$ 0 0
$$893$$ −8.59181e9 −0.403742
$$894$$ 0 0
$$895$$ 2.51887e9 0.117442
$$896$$ 0 0
$$897$$ 1.20450e10 0.557229
$$898$$ 0 0
$$899$$ −5.72509e9 −0.262799
$$900$$ 0 0
$$901$$ 2.58153e10 1.17582
$$902$$ 0 0
$$903$$ 5.64494e8 0.0255125
$$904$$ 0 0
$$905$$ −9.40916e8 −0.0421969
$$906$$ 0 0
$$907$$ 2.15584e10 0.959380 0.479690 0.877438i $$-0.340749\pi$$
0.479690 + 0.877438i $$0.340749\pi$$
$$908$$ 0 0
$$909$$ 1.44053e9 0.0636133
$$910$$ 0 0
$$911$$ −2.47542e10 −1.08476 −0.542381 0.840132i $$-0.682477\pi$$
−0.542381 + 0.840132i $$0.682477\pi$$
$$912$$ 0 0
$$913$$ 2.90661e10 1.26398
$$914$$ 0 0
$$915$$ 4.87361e8 0.0210318
$$916$$ 0 0
$$917$$ 1.39336e9 0.0596719
$$918$$ 0 0
$$919$$ −3.82049e10 −1.62373 −0.811866 0.583843i $$-0.801548\pi$$
−0.811866 + 0.583843i $$0.801548\pi$$
$$920$$ 0 0
$$921$$ 1.58998e10 0.670629
$$922$$ 0 0
$$923$$ 1.47844e9 0.0618869
$$924$$ 0 0
$$925$$ −3.93754e9 −0.163579
$$926$$ 0 0
$$927$$ −6.39417e9 −0.263636
$$928$$ 0 0
$$929$$ −1.51912e10 −0.621636 −0.310818 0.950469i $$-0.600603\pi$$
−0.310818 + 0.950469i $$0.600603\pi$$
$$930$$ 0 0
$$931$$ 2.18729e10 0.888348
$$932$$ 0 0
$$933$$ 2.17015e10 0.874791
$$934$$ 0 0
$$935$$ 1.43655e9 0.0574750
$$936$$ 0 0
$$937$$ −2.93618e10 −1.16599 −0.582995 0.812476i $$-0.698119\pi$$
−0.582995 + 0.812476i $$0.698119\pi$$
$$938$$ 0 0
$$939$$ 2.30255e10 0.907570
$$940$$ 0 0
$$941$$ −1.88409e10 −0.737119 −0.368559 0.929604i $$-0.620149\pi$$
−0.368559 + 0.929604i $$0.620149\pi$$
$$942$$ 0 0
$$943$$ −6.55999e10 −2.54749
$$944$$ 0 0
$$945$$ 1.11403e8 0.00429423
$$946$$ 0 0
$$947$$ 1.73098e10 0.662319 0.331159 0.943575i $$-0.392560\pi$$
0.331159 + 0.943575i $$0.392560\pi$$
$$948$$ 0 0
$$949$$ 1.14119e10 0.433436
$$950$$ 0 0
$$951$$ 2.38751e10 0.900148
$$952$$ 0 0
$$953$$ 1.22640e10 0.458993 0.229497 0.973309i $$-0.426292\pi$$
0.229497 + 0.973309i $$0.426292\pi$$
$$954$$ 0 0
$$955$$ 1.58217e9 0.0587815
$$956$$ 0 0
$$957$$ 6.18214e9 0.228007
$$958$$ 0 0
$$959$$ 1.55969e9 0.0571050
$$960$$ 0 0
$$961$$ 2.35903e9 0.0857436
$$962$$ 0 0
$$963$$ −1.22836e10 −0.443236
$$964$$ 0 0
$$965$$ −3.22647e7 −0.00115580
$$966$$ 0 0
$$967$$ 2.65054e10 0.942632 0.471316 0.881964i $$-0.343779\pi$$
0.471316 + 0.881964i $$0.343779\pi$$
$$968$$ 0 0
$$969$$ −1.35116e10 −0.477062
$$970$$ 0 0
$$971$$ 4.44819e10 1.55925 0.779626 0.626246i $$-0.215409\pi$$
0.779626 + 0.626246i $$0.215409\pi$$
$$972$$ 0 0
$$973$$ 1.69734e9 0.0590710
$$974$$ 0 0
$$975$$ −1.20857e10 −0.417596
$$976$$ 0 0
$$977$$ 2.66074e10 0.912792 0.456396 0.889777i $$-0.349140\pi$$
0.456396 + 0.889777i $$0.349140\pi$$
$$978$$ 0 0
$$979$$ −4.36604e10 −1.48713
$$980$$ 0 0
$$981$$ 4.93179e9 0.166787
$$982$$ 0 0
$$983$$ −3.98658e10 −1.33864 −0.669319 0.742975i $$-0.733414\pi$$
−0.669319 + 0.742975i $$0.733414\pi$$
$$984$$ 0 0
$$985$$ 2.10378e9 0.0701411
$$986$$ 0 0
$$987$$ 5.92199e8 0.0196046
$$988$$ 0 0
$$989$$ 2.38144e10 0.782804
$$990$$ 0 0
$$991$$ 1.96883e10 0.642613 0.321306 0.946975i $$-0.395878\pi$$
0.321306 + 0.946975i $$0.395878\pi$$
$$992$$ 0 0
$$993$$ 1.66769e10 0.540496
$$994$$ 0 0
$$995$$ 4.42868e8 0.0142526
$$996$$ 0 0
$$997$$ 7.78796e9 0.248880 0.124440 0.992227i $$-0.460287\pi$$
0.124440 + 0.992227i $$0.460287\pi$$
$$998$$ 0 0
$$999$$ −5.61421e9 −0.178160
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 592.8.a.b.1.3 4
4.3 odd 2 74.8.a.a.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
74.8.a.a.1.2 4 4.3 odd 2
592.8.a.b.1.3 4 1.1 even 1 trivial