# Properties

 Label 192.8.a.q.1.1 Level $192$ Weight $8$ Character 192.1 Self dual yes Analytic conductor $59.978$ 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] = [192,8,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, 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("192.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 192.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: $$59.9779248930$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{46})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 46$$ x^2 - 46 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: no (minimal twist has level 96) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-6.78233$$ of defining polynomial Character $$\chi$$ $$=$$ 192.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-27.0000 q^{3} -423.552 q^{5} +1228.66 q^{7} +729.000 q^{9} +O(q^{10})$$ $$q-27.0000 q^{3} -423.552 q^{5} +1228.66 q^{7} +729.000 q^{9} -2781.31 q^{11} -3729.10 q^{13} +11435.9 q^{15} -11630.2 q^{17} +45178.2 q^{19} -33173.7 q^{21} +27729.3 q^{23} +101271. q^{25} -19683.0 q^{27} +210915. q^{29} +172577. q^{31} +75095.4 q^{33} -520399. q^{35} +269522. q^{37} +100686. q^{39} -379673. q^{41} -1.03330e6 q^{43} -308769. q^{45} -624148. q^{47} +686051. q^{49} +314016. q^{51} -1.84130e6 q^{53} +1.17803e6 q^{55} -1.21981e6 q^{57} -2.74738e6 q^{59} +2.70188e6 q^{61} +895690. q^{63} +1.57947e6 q^{65} -989057. q^{67} -748691. q^{69} +3.30668e6 q^{71} -1.10275e6 q^{73} -2.73432e6 q^{75} -3.41727e6 q^{77} -8.02881e6 q^{79} +531441. q^{81} +3.55645e6 q^{83} +4.92600e6 q^{85} -5.69471e6 q^{87} -5.21527e6 q^{89} -4.58178e6 q^{91} -4.65959e6 q^{93} -1.91353e7 q^{95} -7.07452e6 q^{97} -2.02758e6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9}+O(q^{10})$$ 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9 $$2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9} - 1656 q^{11} - 6156 q^{13} + 5292 q^{15} + 17108 q^{17} + 504 q^{19} - 13608 q^{21} + 51552 q^{23} + 74926 q^{25} - 39366 q^{27} + 199804 q^{29} + 257256 q^{31} + 44712 q^{33} - 685296 q^{35} + 468724 q^{37} + 166212 q^{39} - 106940 q^{41} - 1617336 q^{43} - 142884 q^{45} + 646416 q^{47} + 387634 q^{49} - 461916 q^{51} - 1469492 q^{53} + 1434096 q^{55} - 13608 q^{57} - 4541544 q^{59} + 481412 q^{61} + 367416 q^{63} + 1027224 q^{65} - 4775256 q^{67} - 1391904 q^{69} - 1094400 q^{71} - 5731884 q^{73} - 2023002 q^{75} - 4232736 q^{77} - 10402776 q^{79} + 1062882 q^{81} - 2212200 q^{83} + 11465432 q^{85} - 5394708 q^{87} - 3604364 q^{89} - 2823120 q^{91} - 6945912 q^{93} - 29300976 q^{95} - 7156188 q^{97} - 1207224 q^{99}+O(q^{100})$$ 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9 - 1656 * q^11 - 6156 * q^13 + 5292 * q^15 + 17108 * q^17 + 504 * q^19 - 13608 * q^21 + 51552 * q^23 + 74926 * q^25 - 39366 * q^27 + 199804 * q^29 + 257256 * q^31 + 44712 * q^33 - 685296 * q^35 + 468724 * q^37 + 166212 * q^39 - 106940 * q^41 - 1617336 * q^43 - 142884 * q^45 + 646416 * q^47 + 387634 * q^49 - 461916 * q^51 - 1469492 * q^53 + 1434096 * q^55 - 13608 * q^57 - 4541544 * q^59 + 481412 * q^61 + 367416 * q^63 + 1027224 * q^65 - 4775256 * q^67 - 1391904 * q^69 - 1094400 * q^71 - 5731884 * q^73 - 2023002 * q^75 - 4232736 * q^77 - 10402776 * q^79 + 1062882 * q^81 - 2212200 * q^83 + 11465432 * q^85 - 5394708 * q^87 - 3604364 * q^89 - 2823120 * q^91 - 6945912 * q^93 - 29300976 * q^95 - 7156188 * q^97 - 1207224 * 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
$$3$$ −27.0000 −0.577350
$$4$$ 0 0
$$5$$ −423.552 −1.51535 −0.757673 0.652635i $$-0.773664\pi$$
−0.757673 + 0.652635i $$0.773664\pi$$
$$6$$ 0 0
$$7$$ 1228.66 1.35390 0.676951 0.736028i $$-0.263301\pi$$
0.676951 + 0.736028i $$0.263301\pi$$
$$8$$ 0 0
$$9$$ 729.000 0.333333
$$10$$ 0 0
$$11$$ −2781.31 −0.630050 −0.315025 0.949083i $$-0.602013\pi$$
−0.315025 + 0.949083i $$0.602013\pi$$
$$12$$ 0 0
$$13$$ −3729.10 −0.470763 −0.235382 0.971903i $$-0.575634\pi$$
−0.235382 + 0.971903i $$0.575634\pi$$
$$14$$ 0 0
$$15$$ 11435.9 0.874885
$$16$$ 0 0
$$17$$ −11630.2 −0.574138 −0.287069 0.957910i $$-0.592681\pi$$
−0.287069 + 0.957910i $$0.592681\pi$$
$$18$$ 0 0
$$19$$ 45178.2 1.51109 0.755546 0.655096i $$-0.227372\pi$$
0.755546 + 0.655096i $$0.227372\pi$$
$$20$$ 0 0
$$21$$ −33173.7 −0.781675
$$22$$ 0 0
$$23$$ 27729.3 0.475217 0.237608 0.971361i $$-0.423637\pi$$
0.237608 + 0.971361i $$0.423637\pi$$
$$24$$ 0 0
$$25$$ 101271. 1.29627
$$26$$ 0 0
$$27$$ −19683.0 −0.192450
$$28$$ 0 0
$$29$$ 210915. 1.60589 0.802943 0.596056i $$-0.203266\pi$$
0.802943 + 0.596056i $$0.203266\pi$$
$$30$$ 0 0
$$31$$ 172577. 1.04044 0.520221 0.854031i $$-0.325849\pi$$
0.520221 + 0.854031i $$0.325849\pi$$
$$32$$ 0 0
$$33$$ 75095.4 0.363760
$$34$$ 0 0
$$35$$ −520399. −2.05163
$$36$$ 0 0
$$37$$ 269522. 0.874757 0.437379 0.899277i $$-0.355907\pi$$
0.437379 + 0.899277i $$0.355907\pi$$
$$38$$ 0 0
$$39$$ 100686. 0.271795
$$40$$ 0 0
$$41$$ −379673. −0.860332 −0.430166 0.902750i $$-0.641545\pi$$
−0.430166 + 0.902750i $$0.641545\pi$$
$$42$$ 0 0
$$43$$ −1.03330e6 −1.98192 −0.990960 0.134154i $$-0.957168\pi$$
−0.990960 + 0.134154i $$0.957168\pi$$
$$44$$ 0 0
$$45$$ −308769. −0.505115
$$46$$ 0 0
$$47$$ −624148. −0.876890 −0.438445 0.898758i $$-0.644471\pi$$
−0.438445 + 0.898758i $$0.644471\pi$$
$$48$$ 0 0
$$49$$ 686051. 0.833049
$$50$$ 0 0
$$51$$ 314016. 0.331479
$$52$$ 0 0
$$53$$ −1.84130e6 −1.69886 −0.849431 0.527699i $$-0.823055\pi$$
−0.849431 + 0.527699i $$0.823055\pi$$
$$54$$ 0 0
$$55$$ 1.17803e6 0.954744
$$56$$ 0 0
$$57$$ −1.21981e6 −0.872429
$$58$$ 0 0
$$59$$ −2.74738e6 −1.74155 −0.870776 0.491679i $$-0.836383\pi$$
−0.870776 + 0.491679i $$0.836383\pi$$
$$60$$ 0 0
$$61$$ 2.70188e6 1.52409 0.762046 0.647523i $$-0.224195\pi$$
0.762046 + 0.647523i $$0.224195\pi$$
$$62$$ 0 0
$$63$$ 895690. 0.451300
$$64$$ 0 0
$$65$$ 1.57947e6 0.713369
$$66$$ 0 0
$$67$$ −989057. −0.401753 −0.200877 0.979617i $$-0.564379\pi$$
−0.200877 + 0.979617i $$0.564379\pi$$
$$68$$ 0 0
$$69$$ −748691. −0.274366
$$70$$ 0 0
$$71$$ 3.30668e6 1.09645 0.548224 0.836331i $$-0.315304\pi$$
0.548224 + 0.836331i $$0.315304\pi$$
$$72$$ 0 0
$$73$$ −1.10275e6 −0.331779 −0.165889 0.986144i $$-0.553049\pi$$
−0.165889 + 0.986144i $$0.553049\pi$$
$$74$$ 0 0
$$75$$ −2.73432e6 −0.748402
$$76$$ 0 0
$$77$$ −3.41727e6 −0.853026
$$78$$ 0 0
$$79$$ −8.02881e6 −1.83213 −0.916065 0.401031i $$-0.868652\pi$$
−0.916065 + 0.401031i $$0.868652\pi$$
$$80$$ 0 0
$$81$$ 531441. 0.111111
$$82$$ 0 0
$$83$$ 3.55645e6 0.682722 0.341361 0.939932i $$-0.389112\pi$$
0.341361 + 0.939932i $$0.389112\pi$$
$$84$$ 0 0
$$85$$ 4.92600e6 0.870018
$$86$$ 0 0
$$87$$ −5.69471e6 −0.927159
$$88$$ 0 0
$$89$$ −5.21527e6 −0.784172 −0.392086 0.919928i $$-0.628247\pi$$
−0.392086 + 0.919928i $$0.628247\pi$$
$$90$$ 0 0
$$91$$ −4.58178e6 −0.637367
$$92$$ 0 0
$$93$$ −4.65959e6 −0.600700
$$94$$ 0 0
$$95$$ −1.91353e7 −2.28983
$$96$$ 0 0
$$97$$ −7.07452e6 −0.787038 −0.393519 0.919316i $$-0.628743\pi$$
−0.393519 + 0.919316i $$0.628743\pi$$
$$98$$ 0 0
$$99$$ −2.02758e6 −0.210017
$$100$$ 0 0
$$101$$ 6.27236e6 0.605768 0.302884 0.953027i $$-0.402051\pi$$
0.302884 + 0.953027i $$0.402051\pi$$
$$102$$ 0 0
$$103$$ −1.22232e7 −1.10219 −0.551093 0.834444i $$-0.685789\pi$$
−0.551093 + 0.834444i $$0.685789\pi$$
$$104$$ 0 0
$$105$$ 1.40508e7 1.18451
$$106$$ 0 0
$$107$$ −8.74065e6 −0.689764 −0.344882 0.938646i $$-0.612081\pi$$
−0.344882 + 0.938646i $$0.612081\pi$$
$$108$$ 0 0
$$109$$ −5.57044e6 −0.412000 −0.206000 0.978552i $$-0.566045\pi$$
−0.206000 + 0.978552i $$0.566045\pi$$
$$110$$ 0 0
$$111$$ −7.27708e6 −0.505041
$$112$$ 0 0
$$113$$ 1.25206e7 0.816302 0.408151 0.912914i $$-0.366174\pi$$
0.408151 + 0.912914i $$0.366174\pi$$
$$114$$ 0 0
$$115$$ −1.17448e7 −0.720117
$$116$$ 0 0
$$117$$ −2.71852e6 −0.156921
$$118$$ 0 0
$$119$$ −1.42895e7 −0.777327
$$120$$ 0 0
$$121$$ −1.17515e7 −0.603037
$$122$$ 0 0
$$123$$ 1.02512e7 0.496713
$$124$$ 0 0
$$125$$ −9.80360e6 −0.448953
$$126$$ 0 0
$$127$$ −1.60536e7 −0.695438 −0.347719 0.937599i $$-0.613044\pi$$
−0.347719 + 0.937599i $$0.613044\pi$$
$$128$$ 0 0
$$129$$ 2.78991e7 1.14426
$$130$$ 0 0
$$131$$ 1.53592e7 0.596923 0.298462 0.954422i $$-0.403526\pi$$
0.298462 + 0.954422i $$0.403526\pi$$
$$132$$ 0 0
$$133$$ 5.55084e7 2.04587
$$134$$ 0 0
$$135$$ 8.33677e6 0.291628
$$136$$ 0 0
$$137$$ 2.09949e7 0.697576 0.348788 0.937202i $$-0.386593\pi$$
0.348788 + 0.937202i $$0.386593\pi$$
$$138$$ 0 0
$$139$$ 3.99151e7 1.26062 0.630312 0.776342i $$-0.282927\pi$$
0.630312 + 0.776342i $$0.282927\pi$$
$$140$$ 0 0
$$141$$ 1.68520e7 0.506273
$$142$$ 0 0
$$143$$ 1.03718e7 0.296605
$$144$$ 0 0
$$145$$ −8.93335e7 −2.43347
$$146$$ 0 0
$$147$$ −1.85234e7 −0.480961
$$148$$ 0 0
$$149$$ −5.79531e7 −1.43524 −0.717621 0.696434i $$-0.754769\pi$$
−0.717621 + 0.696434i $$0.754769\pi$$
$$150$$ 0 0
$$151$$ 1.00182e7 0.236793 0.118397 0.992966i $$-0.462225\pi$$
0.118397 + 0.992966i $$0.462225\pi$$
$$152$$ 0 0
$$153$$ −8.47843e6 −0.191379
$$154$$ 0 0
$$155$$ −7.30955e7 −1.57663
$$156$$ 0 0
$$157$$ −1.88494e7 −0.388732 −0.194366 0.980929i $$-0.562265\pi$$
−0.194366 + 0.980929i $$0.562265\pi$$
$$158$$ 0 0
$$159$$ 4.97150e7 0.980839
$$160$$ 0 0
$$161$$ 3.40698e7 0.643396
$$162$$ 0 0
$$163$$ −7.98770e7 −1.44466 −0.722329 0.691550i $$-0.756928\pi$$
−0.722329 + 0.691550i $$0.756928\pi$$
$$164$$ 0 0
$$165$$ −3.18068e7 −0.551221
$$166$$ 0 0
$$167$$ 2.20039e7 0.365587 0.182794 0.983151i $$-0.441486\pi$$
0.182794 + 0.983151i $$0.441486\pi$$
$$168$$ 0 0
$$169$$ −4.88423e7 −0.778382
$$170$$ 0 0
$$171$$ 3.29349e7 0.503697
$$172$$ 0 0
$$173$$ −6.83727e7 −1.00397 −0.501986 0.864876i $$-0.667397\pi$$
−0.501986 + 0.864876i $$0.667397\pi$$
$$174$$ 0 0
$$175$$ 1.24427e8 1.75502
$$176$$ 0 0
$$177$$ 7.41793e7 1.00549
$$178$$ 0 0
$$179$$ −5.95976e7 −0.776682 −0.388341 0.921516i $$-0.626952\pi$$
−0.388341 + 0.921516i $$0.626952\pi$$
$$180$$ 0 0
$$181$$ 1.51283e8 1.89634 0.948168 0.317769i $$-0.102934\pi$$
0.948168 + 0.317769i $$0.102934\pi$$
$$182$$ 0 0
$$183$$ −7.29507e7 −0.879935
$$184$$ 0 0
$$185$$ −1.14156e8 −1.32556
$$186$$ 0 0
$$187$$ 3.23472e7 0.361736
$$188$$ 0 0
$$189$$ −2.41836e7 −0.260558
$$190$$ 0 0
$$191$$ −8.66555e6 −0.0899869 −0.0449935 0.998987i $$-0.514327\pi$$
−0.0449935 + 0.998987i $$0.514327\pi$$
$$192$$ 0 0
$$193$$ 1.51312e8 1.51504 0.757520 0.652812i $$-0.226411\pi$$
0.757520 + 0.652812i $$0.226411\pi$$
$$194$$ 0 0
$$195$$ −4.26457e7 −0.411864
$$196$$ 0 0
$$197$$ −9.77091e7 −0.910549 −0.455274 0.890351i $$-0.650459\pi$$
−0.455274 + 0.890351i $$0.650459\pi$$
$$198$$ 0 0
$$199$$ 3.19522e7 0.287419 0.143709 0.989620i $$-0.454097\pi$$
0.143709 + 0.989620i $$0.454097\pi$$
$$200$$ 0 0
$$201$$ 2.67045e7 0.231952
$$202$$ 0 0
$$203$$ 2.59142e8 2.17421
$$204$$ 0 0
$$205$$ 1.60811e8 1.30370
$$206$$ 0 0
$$207$$ 2.02147e7 0.158406
$$208$$ 0 0
$$209$$ −1.25654e8 −0.952064
$$210$$ 0 0
$$211$$ 3.47789e7 0.254875 0.127437 0.991847i $$-0.459325\pi$$
0.127437 + 0.991847i $$0.459325\pi$$
$$212$$ 0 0
$$213$$ −8.92804e7 −0.633035
$$214$$ 0 0
$$215$$ 4.37656e8 3.00329
$$216$$ 0 0
$$217$$ 2.12038e8 1.40866
$$218$$ 0 0
$$219$$ 2.97743e7 0.191552
$$220$$ 0 0
$$221$$ 4.33703e7 0.270283
$$222$$ 0 0
$$223$$ 2.06077e7 0.124441 0.0622204 0.998062i $$-0.480182\pi$$
0.0622204 + 0.998062i $$0.480182\pi$$
$$224$$ 0 0
$$225$$ 7.38267e7 0.432090
$$226$$ 0 0
$$227$$ 2.38134e8 1.35124 0.675619 0.737251i $$-0.263877\pi$$
0.675619 + 0.737251i $$0.263877\pi$$
$$228$$ 0 0
$$229$$ 5.34105e7 0.293902 0.146951 0.989144i $$-0.453054\pi$$
0.146951 + 0.989144i $$0.453054\pi$$
$$230$$ 0 0
$$231$$ 9.22664e7 0.492495
$$232$$ 0 0
$$233$$ −2.32484e8 −1.20406 −0.602028 0.798475i $$-0.705640\pi$$
−0.602028 + 0.798475i $$0.705640\pi$$
$$234$$ 0 0
$$235$$ 2.64359e8 1.32879
$$236$$ 0 0
$$237$$ 2.16778e8 1.05778
$$238$$ 0 0
$$239$$ −2.35251e8 −1.11465 −0.557326 0.830294i $$-0.688173\pi$$
−0.557326 + 0.830294i $$0.688173\pi$$
$$240$$ 0 0
$$241$$ −3.44508e8 −1.58540 −0.792701 0.609610i $$-0.791326\pi$$
−0.792701 + 0.609610i $$0.791326\pi$$
$$242$$ 0 0
$$243$$ −1.43489e7 −0.0641500
$$244$$ 0 0
$$245$$ −2.90578e8 −1.26236
$$246$$ 0 0
$$247$$ −1.68474e8 −0.711367
$$248$$ 0 0
$$249$$ −9.60242e7 −0.394170
$$250$$ 0 0
$$251$$ 3.57037e7 0.142513 0.0712567 0.997458i $$-0.477299\pi$$
0.0712567 + 0.997458i $$0.477299\pi$$
$$252$$ 0 0
$$253$$ −7.71238e7 −0.299410
$$254$$ 0 0
$$255$$ −1.33002e8 −0.502305
$$256$$ 0 0
$$257$$ 1.77388e8 0.651866 0.325933 0.945393i $$-0.394322\pi$$
0.325933 + 0.945393i $$0.394322\pi$$
$$258$$ 0 0
$$259$$ 3.31149e8 1.18433
$$260$$ 0 0
$$261$$ 1.53757e8 0.535295
$$262$$ 0 0
$$263$$ −2.51237e8 −0.851604 −0.425802 0.904816i $$-0.640008\pi$$
−0.425802 + 0.904816i $$0.640008\pi$$
$$264$$ 0 0
$$265$$ 7.79885e8 2.57436
$$266$$ 0 0
$$267$$ 1.40812e8 0.452742
$$268$$ 0 0
$$269$$ −3.42878e8 −1.07400 −0.537002 0.843581i $$-0.680443\pi$$
−0.537002 + 0.843581i $$0.680443\pi$$
$$270$$ 0 0
$$271$$ −4.49518e8 −1.37200 −0.686000 0.727601i $$-0.740635\pi$$
−0.686000 + 0.727601i $$0.740635\pi$$
$$272$$ 0 0
$$273$$ 1.23708e8 0.367984
$$274$$ 0 0
$$275$$ −2.81667e8 −0.816716
$$276$$ 0 0
$$277$$ 2.29030e8 0.647461 0.323730 0.946149i $$-0.395063\pi$$
0.323730 + 0.946149i $$0.395063\pi$$
$$278$$ 0 0
$$279$$ 1.25809e8 0.346814
$$280$$ 0 0
$$281$$ −1.49209e8 −0.401165 −0.200582 0.979677i $$-0.564283\pi$$
−0.200582 + 0.979677i $$0.564283\pi$$
$$282$$ 0 0
$$283$$ −4.05047e8 −1.06231 −0.531157 0.847273i $$-0.678243\pi$$
−0.531157 + 0.847273i $$0.678243\pi$$
$$284$$ 0 0
$$285$$ 5.16653e8 1.32203
$$286$$ 0 0
$$287$$ −4.66487e8 −1.16480
$$288$$ 0 0
$$289$$ −2.75077e8 −0.670365
$$290$$ 0 0
$$291$$ 1.91012e8 0.454397
$$292$$ 0 0
$$293$$ −5.59019e7 −0.129834 −0.0649172 0.997891i $$-0.520678\pi$$
−0.0649172 + 0.997891i $$0.520678\pi$$
$$294$$ 0 0
$$295$$ 1.16366e9 2.63905
$$296$$ 0 0
$$297$$ 5.47445e7 0.121253
$$298$$ 0 0
$$299$$ −1.03405e8 −0.223715
$$300$$ 0 0
$$301$$ −1.26957e9 −2.68333
$$302$$ 0 0
$$303$$ −1.69354e8 −0.349740
$$304$$ 0 0
$$305$$ −1.14439e9 −2.30953
$$306$$ 0 0
$$307$$ −7.13287e8 −1.40695 −0.703477 0.710718i $$-0.748370\pi$$
−0.703477 + 0.710718i $$0.748370\pi$$
$$308$$ 0 0
$$309$$ 3.30027e8 0.636347
$$310$$ 0 0
$$311$$ 2.32785e8 0.438827 0.219414 0.975632i $$-0.429586\pi$$
0.219414 + 0.975632i $$0.429586\pi$$
$$312$$ 0 0
$$313$$ 1.35376e8 0.249538 0.124769 0.992186i $$-0.460181\pi$$
0.124769 + 0.992186i $$0.460181\pi$$
$$314$$ 0 0
$$315$$ −3.79371e8 −0.683876
$$316$$ 0 0
$$317$$ 5.29613e8 0.933794 0.466897 0.884312i $$-0.345372\pi$$
0.466897 + 0.884312i $$0.345372\pi$$
$$318$$ 0 0
$$319$$ −5.86621e8 −1.01179
$$320$$ 0 0
$$321$$ 2.35997e8 0.398235
$$322$$ 0 0
$$323$$ −5.25432e8 −0.867576
$$324$$ 0 0
$$325$$ −3.77651e8 −0.610237
$$326$$ 0 0
$$327$$ 1.50402e8 0.237868
$$328$$ 0 0
$$329$$ −7.66863e8 −1.18722
$$330$$ 0 0
$$331$$ 9.24935e8 1.40189 0.700944 0.713217i $$-0.252762\pi$$
0.700944 + 0.713217i $$0.252762\pi$$
$$332$$ 0 0
$$333$$ 1.96481e8 0.291586
$$334$$ 0 0
$$335$$ 4.18917e8 0.608795
$$336$$ 0 0
$$337$$ 1.02943e9 1.46518 0.732592 0.680668i $$-0.238311\pi$$
0.732592 + 0.680668i $$0.238311\pi$$
$$338$$ 0 0
$$339$$ −3.38057e8 −0.471292
$$340$$ 0 0
$$341$$ −4.79992e8 −0.655531
$$342$$ 0 0
$$343$$ −1.68930e8 −0.226036
$$344$$ 0 0
$$345$$ 3.17110e8 0.415760
$$346$$ 0 0
$$347$$ −3.00011e8 −0.385464 −0.192732 0.981251i $$-0.561735\pi$$
−0.192732 + 0.981251i $$0.561735\pi$$
$$348$$ 0 0
$$349$$ 5.43096e8 0.683893 0.341946 0.939719i $$-0.388914\pi$$
0.341946 + 0.939719i $$0.388914\pi$$
$$350$$ 0 0
$$351$$ 7.33999e7 0.0905985
$$352$$ 0 0
$$353$$ −3.10616e8 −0.375849 −0.187924 0.982184i $$-0.560176\pi$$
−0.187924 + 0.982184i $$0.560176\pi$$
$$354$$ 0 0
$$355$$ −1.40055e9 −1.66150
$$356$$ 0 0
$$357$$ 3.85817e8 0.448790
$$358$$ 0 0
$$359$$ −2.00225e8 −0.228395 −0.114198 0.993458i $$-0.536430\pi$$
−0.114198 + 0.993458i $$0.536430\pi$$
$$360$$ 0 0
$$361$$ 1.14719e9 1.28340
$$362$$ 0 0
$$363$$ 3.17290e8 0.348163
$$364$$ 0 0
$$365$$ 4.67073e8 0.502759
$$366$$ 0 0
$$367$$ −7.09899e8 −0.749662 −0.374831 0.927093i $$-0.622299\pi$$
−0.374831 + 0.927093i $$0.622299\pi$$
$$368$$ 0 0
$$369$$ −2.76782e8 −0.286777
$$370$$ 0 0
$$371$$ −2.26232e9 −2.30009
$$372$$ 0 0
$$373$$ −1.90313e8 −0.189883 −0.0949416 0.995483i $$-0.530266\pi$$
−0.0949416 + 0.995483i $$0.530266\pi$$
$$374$$ 0 0
$$375$$ 2.64697e8 0.259203
$$376$$ 0 0
$$377$$ −7.86525e8 −0.755993
$$378$$ 0 0
$$379$$ −1.52757e9 −1.44133 −0.720665 0.693284i $$-0.756163\pi$$
−0.720665 + 0.693284i $$0.756163\pi$$
$$380$$ 0 0
$$381$$ 4.33446e8 0.401511
$$382$$ 0 0
$$383$$ 4.22576e8 0.384334 0.192167 0.981362i $$-0.438448\pi$$
0.192167 + 0.981362i $$0.438448\pi$$
$$384$$ 0 0
$$385$$ 1.44739e9 1.29263
$$386$$ 0 0
$$387$$ −7.53275e8 −0.660640
$$388$$ 0 0
$$389$$ −8.37610e8 −0.721470 −0.360735 0.932668i $$-0.617474\pi$$
−0.360735 + 0.932668i $$0.617474\pi$$
$$390$$ 0 0
$$391$$ −3.22498e8 −0.272840
$$392$$ 0 0
$$393$$ −4.14698e8 −0.344634
$$394$$ 0 0
$$395$$ 3.40062e9 2.77631
$$396$$ 0 0
$$397$$ 1.46607e9 1.17595 0.587973 0.808881i $$-0.299926\pi$$
0.587973 + 0.808881i $$0.299926\pi$$
$$398$$ 0 0
$$399$$ −1.49873e9 −1.18118
$$400$$ 0 0
$$401$$ 1.61531e9 1.25098 0.625489 0.780233i $$-0.284899\pi$$
0.625489 + 0.780233i $$0.284899\pi$$
$$402$$ 0 0
$$403$$ −6.43559e8 −0.489802
$$404$$ 0 0
$$405$$ −2.25093e8 −0.168372
$$406$$ 0 0
$$407$$ −7.49623e8 −0.551141
$$408$$ 0 0
$$409$$ −1.00624e9 −0.727230 −0.363615 0.931549i $$-0.618458\pi$$
−0.363615 + 0.931549i $$0.618458\pi$$
$$410$$ 0 0
$$411$$ −5.66862e8 −0.402746
$$412$$ 0 0
$$413$$ −3.37558e9 −2.35789
$$414$$ 0 0
$$415$$ −1.50634e9 −1.03456
$$416$$ 0 0
$$417$$ −1.07771e9 −0.727822
$$418$$ 0 0
$$419$$ −3.49577e8 −0.232164 −0.116082 0.993240i $$-0.537033\pi$$
−0.116082 + 0.993240i $$0.537033\pi$$
$$420$$ 0 0
$$421$$ 7.71712e8 0.504043 0.252022 0.967722i $$-0.418905\pi$$
0.252022 + 0.967722i $$0.418905\pi$$
$$422$$ 0 0
$$423$$ −4.55004e8 −0.292297
$$424$$ 0 0
$$425$$ −1.17781e9 −0.744239
$$426$$ 0 0
$$427$$ 3.31968e9 2.06347
$$428$$ 0 0
$$429$$ −2.80039e8 −0.171245
$$430$$ 0 0
$$431$$ −2.24587e9 −1.35118 −0.675592 0.737276i $$-0.736112\pi$$
−0.675592 + 0.737276i $$0.736112\pi$$
$$432$$ 0 0
$$433$$ −1.16044e8 −0.0686937 −0.0343468 0.999410i $$-0.510935\pi$$
−0.0343468 + 0.999410i $$0.510935\pi$$
$$434$$ 0 0
$$435$$ 2.41200e9 1.40497
$$436$$ 0 0
$$437$$ 1.25276e9 0.718096
$$438$$ 0 0
$$439$$ 2.46503e9 1.39058 0.695290 0.718729i $$-0.255276\pi$$
0.695290 + 0.718729i $$0.255276\pi$$
$$440$$ 0 0
$$441$$ 5.00131e8 0.277683
$$442$$ 0 0
$$443$$ 2.71781e9 1.48527 0.742635 0.669696i $$-0.233576\pi$$
0.742635 + 0.669696i $$0.233576\pi$$
$$444$$ 0 0
$$445$$ 2.20894e9 1.18829
$$446$$ 0 0
$$447$$ 1.56474e9 0.828637
$$448$$ 0 0
$$449$$ −3.31427e9 −1.72793 −0.863964 0.503553i $$-0.832026\pi$$
−0.863964 + 0.503553i $$0.832026\pi$$
$$450$$ 0 0
$$451$$ 1.05599e9 0.542052
$$452$$ 0 0
$$453$$ −2.70491e8 −0.136713
$$454$$ 0 0
$$455$$ 1.94062e9 0.965831
$$456$$ 0 0
$$457$$ −1.13520e9 −0.556374 −0.278187 0.960527i $$-0.589733\pi$$
−0.278187 + 0.960527i $$0.589733\pi$$
$$458$$ 0 0
$$459$$ 2.28918e8 0.110493
$$460$$ 0 0
$$461$$ −1.89698e9 −0.901799 −0.450900 0.892575i $$-0.648897\pi$$
−0.450900 + 0.892575i $$0.648897\pi$$
$$462$$ 0 0
$$463$$ 1.13818e9 0.532940 0.266470 0.963843i $$-0.414143\pi$$
0.266470 + 0.963843i $$0.414143\pi$$
$$464$$ 0 0
$$465$$ 1.97358e9 0.910268
$$466$$ 0 0
$$467$$ −2.10626e9 −0.956978 −0.478489 0.878094i $$-0.658815\pi$$
−0.478489 + 0.878094i $$0.658815\pi$$
$$468$$ 0 0
$$469$$ −1.21521e9 −0.543935
$$470$$ 0 0
$$471$$ 5.08935e8 0.224434
$$472$$ 0 0
$$473$$ 2.87393e9 1.24871
$$474$$ 0 0
$$475$$ 4.57524e9 1.95878
$$476$$ 0 0
$$477$$ −1.34231e9 −0.566288
$$478$$ 0 0
$$479$$ −1.63874e8 −0.0681295 −0.0340648 0.999420i $$-0.510845\pi$$
−0.0340648 + 0.999420i $$0.510845\pi$$
$$480$$ 0 0
$$481$$ −1.00507e9 −0.411804
$$482$$ 0 0
$$483$$ −9.19884e8 −0.371465
$$484$$ 0 0
$$485$$ 2.99643e9 1.19263
$$486$$ 0 0
$$487$$ −4.01594e8 −0.157556 −0.0787781 0.996892i $$-0.525102\pi$$
−0.0787781 + 0.996892i $$0.525102\pi$$
$$488$$ 0 0
$$489$$ 2.15668e9 0.834073
$$490$$ 0 0
$$491$$ −2.05766e9 −0.784490 −0.392245 0.919861i $$-0.628301\pi$$
−0.392245 + 0.919861i $$0.628301\pi$$
$$492$$ 0 0
$$493$$ −2.45299e9 −0.922001
$$494$$ 0 0
$$495$$ 8.58783e8 0.318248
$$496$$ 0 0
$$497$$ 4.06277e9 1.48448
$$498$$ 0 0
$$499$$ 8.05327e8 0.290148 0.145074 0.989421i $$-0.453658\pi$$
0.145074 + 0.989421i $$0.453658\pi$$
$$500$$ 0 0
$$501$$ −5.94104e8 −0.211072
$$502$$ 0 0
$$503$$ 1.32490e9 0.464189 0.232094 0.972693i $$-0.425442\pi$$
0.232094 + 0.972693i $$0.425442\pi$$
$$504$$ 0 0
$$505$$ −2.65667e9 −0.917948
$$506$$ 0 0
$$507$$ 1.31874e9 0.449399
$$508$$ 0 0
$$509$$ 2.83357e9 0.952406 0.476203 0.879335i $$-0.342013\pi$$
0.476203 + 0.879335i $$0.342013\pi$$
$$510$$ 0 0
$$511$$ −1.35490e9 −0.449195
$$512$$ 0 0
$$513$$ −8.89242e8 −0.290810
$$514$$ 0 0
$$515$$ 5.17716e9 1.67019
$$516$$ 0 0
$$517$$ 1.73595e9 0.552485
$$518$$ 0 0
$$519$$ 1.84606e9 0.579643
$$520$$ 0 0
$$521$$ −1.08648e8 −0.0336580 −0.0168290 0.999858i $$-0.505357\pi$$
−0.0168290 + 0.999858i $$0.505357\pi$$
$$522$$ 0 0
$$523$$ −5.09356e9 −1.55692 −0.778458 0.627696i $$-0.783998\pi$$
−0.778458 + 0.627696i $$0.783998\pi$$
$$524$$ 0 0
$$525$$ −3.35954e9 −1.01326
$$526$$ 0 0
$$527$$ −2.00711e9 −0.597358
$$528$$ 0 0
$$529$$ −2.63591e9 −0.774169
$$530$$ 0 0
$$531$$ −2.00284e9 −0.580518
$$532$$ 0 0
$$533$$ 1.41584e9 0.405013
$$534$$ 0 0
$$535$$ 3.70212e9 1.04523
$$536$$ 0 0
$$537$$ 1.60914e9 0.448418
$$538$$ 0 0
$$539$$ −1.90812e9 −0.524862
$$540$$ 0 0
$$541$$ 3.98616e9 1.08234 0.541171 0.840913i $$-0.317981\pi$$
0.541171 + 0.840913i $$0.317981\pi$$
$$542$$ 0 0
$$543$$ −4.08464e9 −1.09485
$$544$$ 0 0
$$545$$ 2.35937e9 0.624322
$$546$$ 0 0
$$547$$ −4.49694e9 −1.17479 −0.587397 0.809299i $$-0.699847\pi$$
−0.587397 + 0.809299i $$0.699847\pi$$
$$548$$ 0 0
$$549$$ 1.96967e9 0.508031
$$550$$ 0 0
$$551$$ 9.52876e9 2.42664
$$552$$ 0 0
$$553$$ −9.86464e9 −2.48052
$$554$$ 0 0
$$555$$ 3.08222e9 0.765312
$$556$$ 0 0
$$557$$ −6.10617e9 −1.49719 −0.748593 0.663030i $$-0.769270\pi$$
−0.748593 + 0.663030i $$0.769270\pi$$
$$558$$ 0 0
$$559$$ 3.85328e9 0.933016
$$560$$ 0 0
$$561$$ −8.73376e8 −0.208848
$$562$$ 0 0
$$563$$ 1.38158e9 0.326284 0.163142 0.986603i $$-0.447837\pi$$
0.163142 + 0.986603i $$0.447837\pi$$
$$564$$ 0 0
$$565$$ −5.30313e9 −1.23698
$$566$$ 0 0
$$567$$ 6.52958e8 0.150433
$$568$$ 0 0
$$569$$ 8.73612e9 1.98804 0.994021 0.109188i $$-0.0348251\pi$$
0.994021 + 0.109188i $$0.0348251\pi$$
$$570$$ 0 0
$$571$$ −1.33535e9 −0.300170 −0.150085 0.988673i $$-0.547955\pi$$
−0.150085 + 0.988673i $$0.547955\pi$$
$$572$$ 0 0
$$573$$ 2.33970e8 0.0519540
$$574$$ 0 0
$$575$$ 2.80818e9 0.616009
$$576$$ 0 0
$$577$$ 5.50143e8 0.119223 0.0596115 0.998222i $$-0.481014\pi$$
0.0596115 + 0.998222i $$0.481014\pi$$
$$578$$ 0 0
$$579$$ −4.08544e9 −0.874709
$$580$$ 0 0
$$581$$ 4.36966e9 0.924338
$$582$$ 0 0
$$583$$ 5.12122e9 1.07037
$$584$$ 0 0
$$585$$ 1.15143e9 0.237790
$$586$$ 0 0
$$587$$ −3.96942e9 −0.810017 −0.405008 0.914313i $$-0.632731\pi$$
−0.405008 + 0.914313i $$0.632731\pi$$
$$588$$ 0 0
$$589$$ 7.79673e9 1.57220
$$590$$ 0 0
$$591$$ 2.63815e9 0.525706
$$592$$ 0 0
$$593$$ 2.66508e9 0.524830 0.262415 0.964955i $$-0.415481\pi$$
0.262415 + 0.964955i $$0.415481\pi$$
$$594$$ 0 0
$$595$$ 6.05236e9 1.17792
$$596$$ 0 0
$$597$$ −8.62709e8 −0.165941
$$598$$ 0 0
$$599$$ 1.20668e7 0.00229403 0.00114701 0.999999i $$-0.499635\pi$$
0.00114701 + 0.999999i $$0.499635\pi$$
$$600$$ 0 0
$$601$$ 1.12678e9 0.211728 0.105864 0.994381i $$-0.466239\pi$$
0.105864 + 0.994381i $$0.466239\pi$$
$$602$$ 0 0
$$603$$ −7.21023e8 −0.133918
$$604$$ 0 0
$$605$$ 4.97736e9 0.913809
$$606$$ 0 0
$$607$$ −3.33232e9 −0.604765 −0.302382 0.953187i $$-0.597782\pi$$
−0.302382 + 0.953187i $$0.597782\pi$$
$$608$$ 0 0
$$609$$ −6.99684e9 −1.25528
$$610$$ 0 0
$$611$$ 2.32751e9 0.412808
$$612$$ 0 0
$$613$$ −4.28653e9 −0.751612 −0.375806 0.926698i $$-0.622634\pi$$
−0.375806 + 0.926698i $$0.622634\pi$$
$$614$$ 0 0
$$615$$ −4.34190e9 −0.752692
$$616$$ 0 0
$$617$$ 3.82726e9 0.655978 0.327989 0.944681i $$-0.393629\pi$$
0.327989 + 0.944681i $$0.393629\pi$$
$$618$$ 0 0
$$619$$ 1.89529e9 0.321187 0.160594 0.987021i $$-0.448659\pi$$
0.160594 + 0.987021i $$0.448659\pi$$
$$620$$ 0 0
$$621$$ −5.45796e8 −0.0914555
$$622$$ 0 0
$$623$$ −6.40777e9 −1.06169
$$624$$ 0 0
$$625$$ −3.75948e9 −0.615953
$$626$$ 0 0
$$627$$ 3.39267e9 0.549674
$$628$$ 0 0
$$629$$ −3.13459e9 −0.502232
$$630$$ 0 0
$$631$$ −1.20090e10 −1.90285 −0.951424 0.307882i $$-0.900380\pi$$
−0.951424 + 0.307882i $$0.900380\pi$$
$$632$$ 0 0
$$633$$ −9.39029e8 −0.147152
$$634$$ 0 0
$$635$$ 6.79951e9 1.05383
$$636$$ 0 0
$$637$$ −2.55836e9 −0.392169
$$638$$ 0 0
$$639$$ 2.41057e9 0.365483
$$640$$ 0 0
$$641$$ 3.33728e9 0.500484 0.250242 0.968183i $$-0.419490\pi$$
0.250242 + 0.968183i $$0.419490\pi$$
$$642$$ 0 0
$$643$$ −7.80682e9 −1.15807 −0.579037 0.815302i $$-0.696571\pi$$
−0.579037 + 0.815302i $$0.696571\pi$$
$$644$$ 0 0
$$645$$ −1.18167e10 −1.73395
$$646$$ 0 0
$$647$$ −4.81228e9 −0.698531 −0.349265 0.937024i $$-0.613569\pi$$
−0.349265 + 0.937024i $$0.613569\pi$$
$$648$$ 0 0
$$649$$ 7.64132e9 1.09727
$$650$$ 0 0
$$651$$ −5.72503e9 −0.813288
$$652$$ 0 0
$$653$$ 4.16908e9 0.585927 0.292964 0.956124i $$-0.405359\pi$$
0.292964 + 0.956124i $$0.405359\pi$$
$$654$$ 0 0
$$655$$ −6.50541e9 −0.904545
$$656$$ 0 0
$$657$$ −8.03907e8 −0.110593
$$658$$ 0 0
$$659$$ −1.02165e10 −1.39060 −0.695299 0.718721i $$-0.744728\pi$$
−0.695299 + 0.718721i $$0.744728\pi$$
$$660$$ 0 0
$$661$$ −1.41924e9 −0.191139 −0.0955697 0.995423i $$-0.530467\pi$$
−0.0955697 + 0.995423i $$0.530467\pi$$
$$662$$ 0 0
$$663$$ −1.17100e9 −0.156048
$$664$$ 0 0
$$665$$ −2.35107e10 −3.10020
$$666$$ 0 0
$$667$$ 5.84853e9 0.763144
$$668$$ 0 0
$$669$$ −5.56408e8 −0.0718459
$$670$$ 0 0
$$671$$ −7.51476e9 −0.960255
$$672$$ 0 0
$$673$$ 8.90621e9 1.12626 0.563132 0.826367i $$-0.309596\pi$$
0.563132 + 0.826367i $$0.309596\pi$$
$$674$$ 0 0
$$675$$ −1.99332e9 −0.249467
$$676$$ 0 0
$$677$$ 3.21191e9 0.397834 0.198917 0.980016i $$-0.436258\pi$$
0.198917 + 0.980016i $$0.436258\pi$$
$$678$$ 0 0
$$679$$ −8.69215e9 −1.06557
$$680$$ 0 0
$$681$$ −6.42963e9 −0.780137
$$682$$ 0 0
$$683$$ −1.49063e10 −1.79018 −0.895091 0.445883i $$-0.852890\pi$$
−0.895091 + 0.445883i $$0.852890\pi$$
$$684$$ 0 0
$$685$$ −8.89242e9 −1.05707
$$686$$ 0 0
$$687$$ −1.44208e9 −0.169684
$$688$$ 0 0
$$689$$ 6.86639e9 0.799762
$$690$$ 0 0
$$691$$ −3.38129e9 −0.389860 −0.194930 0.980817i $$-0.562448\pi$$
−0.194930 + 0.980817i $$0.562448\pi$$
$$692$$ 0 0
$$693$$ −2.49119e9 −0.284342
$$694$$ 0 0
$$695$$ −1.69061e10 −1.91028
$$696$$ 0 0
$$697$$ 4.41568e9 0.493950
$$698$$ 0 0
$$699$$ 6.27706e9 0.695162
$$700$$ 0 0
$$701$$ 1.19330e10 1.30839 0.654195 0.756326i $$-0.273008\pi$$
0.654195 + 0.756326i $$0.273008\pi$$
$$702$$ 0 0
$$703$$ 1.21765e10 1.32184
$$704$$ 0 0
$$705$$ −7.13769e9 −0.767178
$$706$$ 0 0
$$707$$ 7.70657e9 0.820150
$$708$$ 0 0
$$709$$ −1.92709e9 −0.203068 −0.101534 0.994832i $$-0.532375\pi$$
−0.101534 + 0.994832i $$0.532375\pi$$
$$710$$ 0 0
$$711$$ −5.85300e9 −0.610710
$$712$$ 0 0
$$713$$ 4.78546e9 0.494436
$$714$$ 0 0
$$715$$ −4.39299e9 −0.449458
$$716$$ 0 0
$$717$$ 6.35179e9 0.643545
$$718$$ 0 0
$$719$$ 5.16235e9 0.517960 0.258980 0.965883i $$-0.416614\pi$$
0.258980 + 0.965883i $$0.416614\pi$$
$$720$$ 0 0
$$721$$ −1.50181e10 −1.49225
$$722$$ 0 0
$$723$$ 9.30171e9 0.915333
$$724$$ 0 0
$$725$$ 2.13596e10 2.08166
$$726$$ 0 0
$$727$$ −1.38758e10 −1.33933 −0.669667 0.742662i $$-0.733563\pi$$
−0.669667 + 0.742662i $$0.733563\pi$$
$$728$$ 0 0
$$729$$ 3.87420e8 0.0370370
$$730$$ 0 0
$$731$$ 1.20175e10 1.13790
$$732$$ 0 0
$$733$$ −3.26388e9 −0.306104 −0.153052 0.988218i $$-0.548910\pi$$
−0.153052 + 0.988218i $$0.548910\pi$$
$$734$$ 0 0
$$735$$ 7.84561e9 0.728822
$$736$$ 0 0
$$737$$ 2.75088e9 0.253125
$$738$$ 0 0
$$739$$ 5.49185e8 0.0500568 0.0250284 0.999687i $$-0.492032\pi$$
0.0250284 + 0.999687i $$0.492032\pi$$
$$740$$ 0 0
$$741$$ 4.54880e9 0.410708
$$742$$ 0 0
$$743$$ −1.11208e10 −0.994663 −0.497332 0.867561i $$-0.665687\pi$$
−0.497332 + 0.867561i $$0.665687\pi$$
$$744$$ 0 0
$$745$$ 2.45462e10 2.17489
$$746$$ 0 0
$$747$$ 2.59265e9 0.227574
$$748$$ 0 0
$$749$$ −1.07392e10 −0.933872
$$750$$ 0 0
$$751$$ 5.06853e9 0.436659 0.218329 0.975875i $$-0.429939\pi$$
0.218329 + 0.975875i $$0.429939\pi$$
$$752$$ 0 0
$$753$$ −9.64001e8 −0.0822801
$$754$$ 0 0
$$755$$ −4.24322e9 −0.358824
$$756$$ 0 0
$$757$$ −5.64424e9 −0.472901 −0.236450 0.971644i $$-0.575984\pi$$
−0.236450 + 0.971644i $$0.575984\pi$$
$$758$$ 0 0
$$759$$ 2.08234e9 0.172865
$$760$$ 0 0
$$761$$ 1.57769e10 1.29770 0.648850 0.760916i $$-0.275250\pi$$
0.648850 + 0.760916i $$0.275250\pi$$
$$762$$ 0 0
$$763$$ −6.84416e9 −0.557807
$$764$$ 0 0
$$765$$ 3.59105e9 0.290006
$$766$$ 0 0
$$767$$ 1.02453e10 0.819859
$$768$$ 0 0
$$769$$ 1.05126e10 0.833616 0.416808 0.908994i $$-0.363149\pi$$
0.416808 + 0.908994i $$0.363149\pi$$
$$770$$ 0 0
$$771$$ −4.78947e9 −0.376355
$$772$$ 0 0
$$773$$ 4.70533e9 0.366405 0.183203 0.983075i $$-0.441354\pi$$
0.183203 + 0.983075i $$0.441354\pi$$
$$774$$ 0 0
$$775$$ 1.74771e10 1.34870
$$776$$ 0 0
$$777$$ −8.94103e9 −0.683776
$$778$$ 0 0
$$779$$ −1.71529e10 −1.30004
$$780$$ 0 0
$$781$$ −9.19691e9 −0.690818
$$782$$ 0 0
$$783$$ −4.15144e9 −0.309053
$$784$$ 0 0
$$785$$ 7.98372e9 0.589063
$$786$$ 0 0
$$787$$ 2.07916e10 1.52046 0.760232 0.649652i $$-0.225085\pi$$
0.760232 + 0.649652i $$0.225085\pi$$
$$788$$ 0 0
$$789$$ 6.78339e9 0.491674
$$790$$ 0 0
$$791$$ 1.53835e10 1.10519
$$792$$ 0 0
$$793$$ −1.00756e10 −0.717487
$$794$$ 0 0
$$795$$ −2.10569e10 −1.48631
$$796$$ 0 0
$$797$$ −1.09035e10 −0.762890 −0.381445 0.924392i $$-0.624573\pi$$
−0.381445 + 0.924392i $$0.624573\pi$$
$$798$$ 0 0
$$799$$ 7.25897e9 0.503456
$$800$$ 0 0
$$801$$ −3.80193e9 −0.261391
$$802$$ 0 0
$$803$$ 3.06710e9 0.209037
$$804$$ 0 0
$$805$$ −1.44303e10 −0.974968
$$806$$ 0 0
$$807$$ 9.25769e9 0.620077
$$808$$ 0 0
$$809$$ −1.20873e10 −0.802620 −0.401310 0.915942i $$-0.631445\pi$$
−0.401310 + 0.915942i $$0.631445\pi$$
$$810$$ 0 0
$$811$$ −1.39636e10 −0.919229 −0.459615 0.888118i $$-0.652012\pi$$
−0.459615 + 0.888118i $$0.652012\pi$$
$$812$$ 0 0
$$813$$ 1.21370e10 0.792125
$$814$$ 0 0
$$815$$ 3.38320e10 2.18915
$$816$$ 0 0
$$817$$ −4.66825e10 −2.99486
$$818$$ 0 0
$$819$$ −3.34012e9 −0.212456
$$820$$ 0 0
$$821$$ −6.90599e9 −0.435537 −0.217769 0.976000i $$-0.569878\pi$$
−0.217769 + 0.976000i $$0.569878\pi$$
$$822$$ 0 0
$$823$$ −1.31839e9 −0.0824413 −0.0412206 0.999150i $$-0.513125\pi$$
−0.0412206 + 0.999150i $$0.513125\pi$$
$$824$$ 0 0
$$825$$ 7.60500e9 0.471531
$$826$$ 0 0
$$827$$ 1.16284e10 0.714907 0.357454 0.933931i $$-0.383645\pi$$
0.357454 + 0.933931i $$0.383645\pi$$
$$828$$ 0 0
$$829$$ −1.53849e10 −0.937893 −0.468947 0.883226i $$-0.655366\pi$$
−0.468947 + 0.883226i $$0.655366\pi$$
$$830$$ 0 0
$$831$$ −6.18382e9 −0.373812
$$832$$ 0 0
$$833$$ −7.97892e9 −0.478285
$$834$$ 0 0
$$835$$ −9.31978e9 −0.553991
$$836$$ 0 0
$$837$$ −3.39684e9 −0.200233
$$838$$ 0 0
$$839$$ 4.92329e9 0.287799 0.143899 0.989592i $$-0.454036\pi$$
0.143899 + 0.989592i $$0.454036\pi$$
$$840$$ 0 0
$$841$$ 2.72353e10 1.57887
$$842$$ 0 0
$$843$$ 4.02864e9 0.231613
$$844$$ 0 0
$$845$$ 2.06872e10 1.17952
$$846$$ 0 0
$$847$$ −1.44385e10 −0.816452
$$848$$ 0 0
$$849$$ 1.09363e10 0.613327
$$850$$ 0 0
$$851$$ 7.47365e9 0.415699
$$852$$ 0 0
$$853$$ 7.54820e9 0.416411 0.208205 0.978085i $$-0.433238\pi$$
0.208205 + 0.978085i $$0.433238\pi$$
$$854$$ 0 0
$$855$$ −1.39496e10 −0.763275
$$856$$ 0 0
$$857$$ −1.27814e10 −0.693656 −0.346828 0.937929i $$-0.612741\pi$$
−0.346828 + 0.937929i $$0.612741\pi$$
$$858$$ 0 0
$$859$$ −1.33691e10 −0.719657 −0.359829 0.933018i $$-0.617165\pi$$
−0.359829 + 0.933018i $$0.617165\pi$$
$$860$$ 0 0
$$861$$ 1.25952e10 0.672500
$$862$$ 0 0
$$863$$ 9.25958e9 0.490404 0.245202 0.969472i $$-0.421146\pi$$
0.245202 + 0.969472i $$0.421146\pi$$
$$864$$ 0 0
$$865$$ 2.89594e10 1.52136
$$866$$ 0 0
$$867$$ 7.42707e9 0.387036
$$868$$ 0 0
$$869$$ 2.23306e10 1.15433
$$870$$ 0 0
$$871$$ 3.68830e9 0.189131
$$872$$ 0 0
$$873$$ −5.15733e9 −0.262346
$$874$$ 0 0
$$875$$ −1.20452e10 −0.607837
$$876$$ 0 0
$$877$$ −1.40971e10 −0.705717 −0.352859 0.935677i $$-0.614790\pi$$
−0.352859 + 0.935677i $$0.614790\pi$$
$$878$$ 0 0
$$879$$ 1.50935e9 0.0749600
$$880$$ 0 0
$$881$$ 1.97871e10 0.974914 0.487457 0.873147i $$-0.337925\pi$$
0.487457 + 0.873147i $$0.337925\pi$$
$$882$$ 0 0
$$883$$ 3.39322e10 1.65863 0.829314 0.558782i $$-0.188731\pi$$
0.829314 + 0.558782i $$0.188731\pi$$
$$884$$ 0 0
$$885$$ −3.14188e10 −1.52366
$$886$$ 0 0
$$887$$ −3.05047e10 −1.46769 −0.733846 0.679316i $$-0.762276\pi$$
−0.733846 + 0.679316i $$0.762276\pi$$
$$888$$ 0 0
$$889$$ −1.97243e10 −0.941554
$$890$$ 0 0
$$891$$ −1.47810e9 −0.0700056
$$892$$ 0 0
$$893$$ −2.81978e10 −1.32506
$$894$$ 0 0
$$895$$ 2.52427e10 1.17694
$$896$$ 0 0
$$897$$ 2.79195e9 0.129162
$$898$$ 0 0
$$899$$ 3.63992e10 1.67083
$$900$$ 0 0
$$901$$ 2.14147e10 0.975382
$$902$$ 0 0
$$903$$ 3.42783e10 1.54922
$$904$$ 0 0
$$905$$ −6.40762e10 −2.87360
$$906$$ 0 0
$$907$$ −6.76417e9 −0.301016 −0.150508 0.988609i $$-0.548091\pi$$
−0.150508 + 0.988609i $$0.548091\pi$$
$$908$$ 0 0
$$909$$ 4.57255e9 0.201923
$$910$$ 0 0
$$911$$ 1.14881e10 0.503424 0.251712 0.967802i $$-0.419006\pi$$
0.251712 + 0.967802i $$0.419006\pi$$
$$912$$ 0 0
$$913$$ −9.89160e9 −0.430149
$$914$$ 0 0
$$915$$ 3.08984e10 1.33341
$$916$$ 0 0
$$917$$ 1.88711e10 0.808175
$$918$$ 0 0
$$919$$ 3.87986e10 1.64897 0.824483 0.565887i $$-0.191466\pi$$
0.824483 + 0.565887i $$0.191466\pi$$
$$920$$ 0 0
$$921$$ 1.92587e10 0.812305
$$922$$ 0 0
$$923$$ −1.23310e10 −0.516168
$$924$$ 0 0
$$925$$ 2.72948e10 1.13392
$$926$$ 0 0
$$927$$ −8.91072e9 −0.367395
$$928$$ 0 0
$$929$$ 4.14694e10 1.69696 0.848482 0.529224i $$-0.177517\pi$$
0.848482 + 0.529224i $$0.177517\pi$$
$$930$$ 0 0
$$931$$ 3.09945e10 1.25881
$$932$$ 0 0
$$933$$ −6.28519e9 −0.253357
$$934$$ 0 0
$$935$$ −1.37007e10 −0.548155
$$936$$ 0 0
$$937$$ 1.25279e10 0.497497 0.248748 0.968568i $$-0.419981\pi$$
0.248748 + 0.968568i $$0.419981\pi$$
$$938$$ 0 0
$$939$$ −3.65515e9 −0.144071
$$940$$ 0 0
$$941$$ −3.90647e10 −1.52834 −0.764171 0.645014i $$-0.776852\pi$$
−0.764171 + 0.645014i $$0.776852\pi$$
$$942$$ 0 0
$$943$$ −1.05281e10 −0.408844
$$944$$ 0 0
$$945$$ 1.02430e10 0.394836
$$946$$ 0 0
$$947$$ 1.55171e10 0.593726 0.296863 0.954920i $$-0.404059\pi$$
0.296863 + 0.954920i $$0.404059\pi$$
$$948$$ 0 0
$$949$$ 4.11228e9 0.156189
$$950$$ 0 0
$$951$$ −1.42995e10 −0.539126
$$952$$ 0 0
$$953$$ −3.71024e10 −1.38860 −0.694299 0.719687i $$-0.744285\pi$$
−0.694299 + 0.719687i $$0.744285\pi$$
$$954$$ 0 0
$$955$$ 3.67031e9 0.136361
$$956$$ 0 0
$$957$$ 1.58388e10 0.584157
$$958$$ 0 0
$$959$$ 2.57955e10 0.944449
$$960$$ 0 0
$$961$$ 2.27038e9 0.0825214
$$962$$ 0 0
$$963$$ −6.37193e9 −0.229921
$$964$$ 0 0
$$965$$ −6.40887e10 −2.29581
$$966$$ 0 0
$$967$$ −2.56321e10 −0.911574 −0.455787 0.890089i $$-0.650642\pi$$
−0.455787 + 0.890089i $$0.650642\pi$$
$$968$$ 0 0
$$969$$ 1.41867e10 0.500895
$$970$$ 0 0
$$971$$ 5.08869e10 1.78377 0.891884 0.452264i $$-0.149384\pi$$
0.891884 + 0.452264i $$0.149384\pi$$
$$972$$ 0 0
$$973$$ 4.90420e10 1.70676
$$974$$ 0 0
$$975$$ 1.01966e10 0.352320
$$976$$ 0 0
$$977$$ 1.68684e10 0.578686 0.289343 0.957225i $$-0.406563\pi$$
0.289343 + 0.957225i $$0.406563\pi$$
$$978$$ 0 0
$$979$$ 1.45053e10 0.494068
$$980$$ 0 0
$$981$$ −4.06085e9 −0.137333
$$982$$ 0 0
$$983$$ −1.61062e10 −0.540825 −0.270412 0.962745i $$-0.587160\pi$$
−0.270412 + 0.962745i $$0.587160\pi$$
$$984$$ 0 0
$$985$$ 4.13849e10 1.37980
$$986$$ 0 0
$$987$$ 2.07053e10 0.685443
$$988$$ 0 0
$$989$$ −2.86527e10 −0.941842
$$990$$ 0 0
$$991$$ −3.98303e9 −0.130004 −0.0650019 0.997885i $$-0.520705\pi$$
−0.0650019 + 0.997885i $$0.520705\pi$$
$$992$$ 0 0
$$993$$ −2.49732e10 −0.809380
$$994$$ 0 0
$$995$$ −1.35334e10 −0.435538
$$996$$ 0 0
$$997$$ 4.19886e10 1.34183 0.670917 0.741533i $$-0.265901\pi$$
0.670917 + 0.741533i $$0.265901\pi$$
$$998$$ 0 0
$$999$$ −5.30499e9 −0.168347
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 192.8.a.q.1.1 2
3.2 odd 2 576.8.a.bo.1.2 2
4.3 odd 2 192.8.a.t.1.1 2
8.3 odd 2 96.8.a.e.1.2 2
8.5 even 2 96.8.a.h.1.2 yes 2
12.11 even 2 576.8.a.bn.1.2 2
24.5 odd 2 288.8.a.h.1.1 2
24.11 even 2 288.8.a.g.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
96.8.a.e.1.2 2 8.3 odd 2
96.8.a.h.1.2 yes 2 8.5 even 2
192.8.a.q.1.1 2 1.1 even 1 trivial
192.8.a.t.1.1 2 4.3 odd 2
288.8.a.g.1.1 2 24.11 even 2
288.8.a.h.1.1 2 24.5 odd 2
576.8.a.bn.1.2 2 12.11 even 2
576.8.a.bo.1.2 2 3.2 odd 2