# Properties

 Label 256.9.d.e.127.2 Level $256$ Weight $9$ Character 256.127 Analytic conductor $104.289$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,9,Mod(127,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("256.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 256.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$104.288924176$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{39})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 19x^{2} + 100$$ x^4 - 19*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 4) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 127.2 Root $$3.12250 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 256.127 Dual form 256.9.d.e.127.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-99.9200 q^{3} +610.000i q^{5} +1398.88i q^{7} +3423.00 q^{9} +O(q^{10})$$ $$q-99.9200 q^{3} +610.000i q^{5} +1398.88i q^{7} +3423.00 q^{9} -18485.2 q^{11} +5470.00i q^{13} -60951.2i q^{15} +73090.0 q^{17} +19484.4 q^{19} -139776. i q^{21} -237210. i q^{23} +18525.0 q^{25} +313549. q^{27} +128222. i q^{29} +67945.6i q^{31} +1.84704e6 q^{33} -853317. q^{35} -3.47203e6i q^{37} -546562. i q^{39} -2.14688e6 q^{41} -5.92815e6 q^{43} +2.08803e6i q^{45} -7.62629e6i q^{47} +3.80794e6 q^{49} -7.30315e6 q^{51} +824290. i q^{53} -1.12760e7i q^{55} -1.94688e6 q^{57} -3.72552e6 q^{59} +1.47461e7i q^{61} +4.78836e6i q^{63} -3.33670e6 q^{65} -1.52567e7 q^{67} +2.37020e7i q^{69} -1.19604e6i q^{71} +5.72563e6 q^{73} -1.85102e6 q^{75} -2.58586e7i q^{77} -3.59132e7i q^{79} -5.37881e7 q^{81} +5.19603e7 q^{83} +4.45849e7i q^{85} -1.28119e7i q^{87} +8.33242e7 q^{89} -7.65187e6 q^{91} -6.78912e6i q^{93} +1.18855e7i q^{95} +1.20619e8 q^{97} -6.32748e7 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 13692 q^{9}+O(q^{10})$$ 4 * q + 13692 * q^9 $$4 q + 13692 q^{9} + 292360 q^{17} + 74100 q^{25} + 7388160 q^{33} - 8587528 q^{41} + 15231748 q^{49} - 7787520 q^{57} - 13346800 q^{65} + 22902520 q^{73} - 215152380 q^{81} + 333296888 q^{89} + 482476040 q^{97}+O(q^{100})$$ 4 * q + 13692 * q^9 + 292360 * q^17 + 74100 * q^25 + 7388160 * q^33 - 8587528 * q^41 + 15231748 * q^49 - 7787520 * q^57 - 13346800 * q^65 + 22902520 * q^73 - 215152380 * q^81 + 333296888 * q^89 + 482476040 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$3$$ −99.9200 −1.23358 −0.616790 0.787128i $$-0.711567\pi$$
−0.616790 + 0.787128i $$0.711567\pi$$
$$4$$ 0 0
$$5$$ 610.000i 0.976000i 0.872844 + 0.488000i $$0.162273\pi$$
−0.872844 + 0.488000i $$0.837727\pi$$
$$6$$ 0 0
$$7$$ 1398.88i 0.582624i 0.956628 + 0.291312i $$0.0940917\pi$$
−0.956628 + 0.291312i $$0.905908\pi$$
$$8$$ 0 0
$$9$$ 3423.00 0.521719
$$10$$ 0 0
$$11$$ −18485.2 −1.26256 −0.631282 0.775554i $$-0.717471\pi$$
−0.631282 + 0.775554i $$0.717471\pi$$
$$12$$ 0 0
$$13$$ 5470.00i 0.191520i 0.995404 + 0.0957600i $$0.0305281\pi$$
−0.995404 + 0.0957600i $$0.969472\pi$$
$$14$$ 0 0
$$15$$ − 60951.2i − 1.20397i
$$16$$ 0 0
$$17$$ 73090.0 0.875109 0.437555 0.899192i $$-0.355845\pi$$
0.437555 + 0.899192i $$0.355845\pi$$
$$18$$ 0 0
$$19$$ 19484.4 0.149511 0.0747554 0.997202i $$-0.476182\pi$$
0.0747554 + 0.997202i $$0.476182\pi$$
$$20$$ 0 0
$$21$$ − 139776.i − 0.718713i
$$22$$ 0 0
$$23$$ − 237210.i − 0.847660i −0.905742 0.423830i $$-0.860685\pi$$
0.905742 0.423830i $$-0.139315\pi$$
$$24$$ 0 0
$$25$$ 18525.0 0.0474240
$$26$$ 0 0
$$27$$ 313549. 0.589997
$$28$$ 0 0
$$29$$ 128222.i 0.181289i 0.995883 + 0.0906443i $$0.0288926\pi$$
−0.995883 + 0.0906443i $$0.971107\pi$$
$$30$$ 0 0
$$31$$ 67945.6i 0.0735723i 0.999323 + 0.0367862i $$0.0117120\pi$$
−0.999323 + 0.0367862i $$0.988288\pi$$
$$32$$ 0 0
$$33$$ 1.84704e6 1.55747
$$34$$ 0 0
$$35$$ −853317. −0.568641
$$36$$ 0 0
$$37$$ − 3.47203e6i − 1.85258i −0.376813 0.926289i $$-0.622980\pi$$
0.376813 0.926289i $$-0.377020\pi$$
$$38$$ 0 0
$$39$$ − 546562.i − 0.236255i
$$40$$ 0 0
$$41$$ −2.14688e6 −0.759754 −0.379877 0.925037i $$-0.624034\pi$$
−0.379877 + 0.925037i $$0.624034\pi$$
$$42$$ 0 0
$$43$$ −5.92815e6 −1.73399 −0.866993 0.498321i $$-0.833950\pi$$
−0.866993 + 0.498321i $$0.833950\pi$$
$$44$$ 0 0
$$45$$ 2.08803e6i 0.509198i
$$46$$ 0 0
$$47$$ − 7.62629e6i − 1.56287i −0.623989 0.781433i $$-0.714489\pi$$
0.623989 0.781433i $$-0.285511\pi$$
$$48$$ 0 0
$$49$$ 3.80794e6 0.660550
$$50$$ 0 0
$$51$$ −7.30315e6 −1.07952
$$52$$ 0 0
$$53$$ 824290.i 0.104466i 0.998635 + 0.0522332i $$0.0166339\pi$$
−0.998635 + 0.0522332i $$0.983366\pi$$
$$54$$ 0 0
$$55$$ − 1.12760e7i − 1.23226i
$$56$$ 0 0
$$57$$ −1.94688e6 −0.184433
$$58$$ 0 0
$$59$$ −3.72552e6 −0.307453 −0.153726 0.988113i $$-0.549127\pi$$
−0.153726 + 0.988113i $$0.549127\pi$$
$$60$$ 0 0
$$61$$ 1.47461e7i 1.06502i 0.846424 + 0.532509i $$0.178751\pi$$
−0.846424 + 0.532509i $$0.821249\pi$$
$$62$$ 0 0
$$63$$ 4.78836e6i 0.303966i
$$64$$ 0 0
$$65$$ −3.33670e6 −0.186923
$$66$$ 0 0
$$67$$ −1.52567e7 −0.757113 −0.378557 0.925578i $$-0.623579\pi$$
−0.378557 + 0.925578i $$0.623579\pi$$
$$68$$ 0 0
$$69$$ 2.37020e7i 1.04566i
$$70$$ 0 0
$$71$$ − 1.19604e6i − 0.0470666i −0.999723 0.0235333i $$-0.992508\pi$$
0.999723 0.0235333i $$-0.00749158\pi$$
$$72$$ 0 0
$$73$$ 5.72563e6 0.201619 0.100810 0.994906i $$-0.467857\pi$$
0.100810 + 0.994906i $$0.467857\pi$$
$$74$$ 0 0
$$75$$ −1.85102e6 −0.0585013
$$76$$ 0 0
$$77$$ − 2.58586e7i − 0.735600i
$$78$$ 0 0
$$79$$ − 3.59132e7i − 0.922032i −0.887392 0.461016i $$-0.847485\pi$$
0.887392 0.461016i $$-0.152515\pi$$
$$80$$ 0 0
$$81$$ −5.37881e7 −1.24953
$$82$$ 0 0
$$83$$ 5.19603e7 1.09486 0.547431 0.836851i $$-0.315606\pi$$
0.547431 + 0.836851i $$0.315606\pi$$
$$84$$ 0 0
$$85$$ 4.45849e7i 0.854107i
$$86$$ 0 0
$$87$$ − 1.28119e7i − 0.223634i
$$88$$ 0 0
$$89$$ 8.33242e7 1.32804 0.664020 0.747715i $$-0.268849\pi$$
0.664020 + 0.747715i $$0.268849\pi$$
$$90$$ 0 0
$$91$$ −7.65187e6 −0.111584
$$92$$ 0 0
$$93$$ − 6.78912e6i − 0.0907573i
$$94$$ 0 0
$$95$$ 1.18855e7i 0.145923i
$$96$$ 0 0
$$97$$ 1.20619e8 1.36248 0.681238 0.732062i $$-0.261442\pi$$
0.681238 + 0.732062i $$0.261442\pi$$
$$98$$ 0 0
$$99$$ −6.32748e7 −0.658704
$$100$$ 0 0
$$101$$ 2.77246e7i 0.266428i 0.991087 + 0.133214i $$0.0425298\pi$$
−0.991087 + 0.133214i $$0.957470\pi$$
$$102$$ 0 0
$$103$$ 1.04501e8i 0.928477i 0.885710 + 0.464238i $$0.153672\pi$$
−0.885710 + 0.464238i $$0.846328\pi$$
$$104$$ 0 0
$$105$$ 8.52634e7 0.701464
$$106$$ 0 0
$$107$$ 1.00328e8 0.765394 0.382697 0.923874i $$-0.374995\pi$$
0.382697 + 0.923874i $$0.374995\pi$$
$$108$$ 0 0
$$109$$ 5.90716e7i 0.418478i 0.977865 + 0.209239i $$0.0670987\pi$$
−0.977865 + 0.209239i $$0.932901\pi$$
$$110$$ 0 0
$$111$$ 3.46925e8i 2.28530i
$$112$$ 0 0
$$113$$ 5.50849e7 0.337846 0.168923 0.985629i $$-0.445971\pi$$
0.168923 + 0.985629i $$0.445971\pi$$
$$114$$ 0 0
$$115$$ 1.44698e8 0.827316
$$116$$ 0 0
$$117$$ 1.87238e7i 0.0999196i
$$118$$ 0 0
$$119$$ 1.02244e8i 0.509859i
$$120$$ 0 0
$$121$$ 1.27344e8 0.594067
$$122$$ 0 0
$$123$$ 2.14516e8 0.937217
$$124$$ 0 0
$$125$$ 2.49581e8i 1.02229i
$$126$$ 0 0
$$127$$ − 2.57160e8i − 0.988529i −0.869312 0.494264i $$-0.835438\pi$$
0.869312 0.494264i $$-0.164562\pi$$
$$128$$ 0 0
$$129$$ 5.92341e8 2.13901
$$130$$ 0 0
$$131$$ 3.12175e8 1.06002 0.530009 0.847992i $$-0.322188\pi$$
0.530009 + 0.847992i $$0.322188\pi$$
$$132$$ 0 0
$$133$$ 2.72563e7i 0.0871085i
$$134$$ 0 0
$$135$$ 1.91265e8i 0.575838i
$$136$$ 0 0
$$137$$ −2.21980e8 −0.630132 −0.315066 0.949070i $$-0.602027\pi$$
−0.315066 + 0.949070i $$0.602027\pi$$
$$138$$ 0 0
$$139$$ −2.95030e8 −0.790328 −0.395164 0.918611i $$-0.629312\pi$$
−0.395164 + 0.918611i $$0.629312\pi$$
$$140$$ 0 0
$$141$$ 7.62019e8i 1.92792i
$$142$$ 0 0
$$143$$ − 1.01114e8i − 0.241806i
$$144$$ 0 0
$$145$$ −7.82154e7 −0.176938
$$146$$ 0 0
$$147$$ −3.80489e8 −0.814841
$$148$$ 0 0
$$149$$ 4.03603e8i 0.818859i 0.912342 + 0.409429i $$0.134272\pi$$
−0.912342 + 0.409429i $$0.865728\pi$$
$$150$$ 0 0
$$151$$ − 8.36985e8i − 1.60994i −0.593316 0.804970i $$-0.702181\pi$$
0.593316 0.804970i $$-0.297819\pi$$
$$152$$ 0 0
$$153$$ 2.50187e8 0.456561
$$154$$ 0 0
$$155$$ −4.14468e7 −0.0718066
$$156$$ 0 0
$$157$$ 2.71319e8i 0.446561i 0.974754 + 0.223281i $$0.0716767\pi$$
−0.974754 + 0.223281i $$0.928323\pi$$
$$158$$ 0 0
$$159$$ − 8.23630e7i − 0.128868i
$$160$$ 0 0
$$161$$ 3.31828e8 0.493867
$$162$$ 0 0
$$163$$ −5.78509e8 −0.819520 −0.409760 0.912193i $$-0.634388\pi$$
−0.409760 + 0.912193i $$0.634388\pi$$
$$164$$ 0 0
$$165$$ 1.12669e9i 1.52009i
$$166$$ 0 0
$$167$$ 4.68118e8i 0.601852i 0.953647 + 0.300926i $$0.0972958\pi$$
−0.953647 + 0.300926i $$0.902704\pi$$
$$168$$ 0 0
$$169$$ 7.85810e8 0.963320
$$170$$ 0 0
$$171$$ 6.66951e7 0.0780026
$$172$$ 0 0
$$173$$ 2.06197e8i 0.230196i 0.993354 + 0.115098i $$0.0367182\pi$$
−0.993354 + 0.115098i $$0.963282\pi$$
$$174$$ 0 0
$$175$$ 2.59142e7i 0.0276303i
$$176$$ 0 0
$$177$$ 3.72253e8 0.379268
$$178$$ 0 0
$$179$$ −1.41911e8 −0.138230 −0.0691152 0.997609i $$-0.522018\pi$$
−0.0691152 + 0.997609i $$0.522018\pi$$
$$180$$ 0 0
$$181$$ 4.82566e8i 0.449616i 0.974403 + 0.224808i $$0.0721755\pi$$
−0.974403 + 0.224808i $$0.927824\pi$$
$$182$$ 0 0
$$183$$ − 1.47343e9i − 1.31379i
$$184$$ 0 0
$$185$$ 2.11794e9 1.80812
$$186$$ 0 0
$$187$$ −1.35108e9 −1.10488
$$188$$ 0 0
$$189$$ 4.38617e8i 0.343747i
$$190$$ 0 0
$$191$$ − 9.92461e8i − 0.745727i −0.927886 0.372864i $$-0.878376\pi$$
0.927886 0.372864i $$-0.121624\pi$$
$$192$$ 0 0
$$193$$ 1.17593e9 0.847526 0.423763 0.905773i $$-0.360709\pi$$
0.423763 + 0.905773i $$0.360709\pi$$
$$194$$ 0 0
$$195$$ 3.33403e8 0.230585
$$196$$ 0 0
$$197$$ 1.70538e9i 1.13229i 0.824306 + 0.566144i $$0.191565\pi$$
−0.824306 + 0.566144i $$0.808435\pi$$
$$198$$ 0 0
$$199$$ 2.49036e9i 1.58800i 0.607919 + 0.793999i $$0.292004\pi$$
−0.607919 + 0.793999i $$0.707996\pi$$
$$200$$ 0 0
$$201$$ 1.52445e9 0.933960
$$202$$ 0 0
$$203$$ −1.79367e8 −0.105623
$$204$$ 0 0
$$205$$ − 1.30960e9i − 0.741519i
$$206$$ 0 0
$$207$$ − 8.11970e8i − 0.442241i
$$208$$ 0 0
$$209$$ −3.60173e8 −0.188767
$$210$$ 0 0
$$211$$ −1.46774e9 −0.740491 −0.370245 0.928934i $$-0.620726\pi$$
−0.370245 + 0.928934i $$0.620726\pi$$
$$212$$ 0 0
$$213$$ 1.19508e8i 0.0580604i
$$214$$ 0 0
$$215$$ − 3.61617e9i − 1.69237i
$$216$$ 0 0
$$217$$ −9.50477e7 −0.0428650
$$218$$ 0 0
$$219$$ −5.72105e8 −0.248713
$$220$$ 0 0
$$221$$ 3.99802e8i 0.167601i
$$222$$ 0 0
$$223$$ 1.47920e9i 0.598147i 0.954230 + 0.299073i $$0.0966776\pi$$
−0.954230 + 0.299073i $$0.903322\pi$$
$$224$$ 0 0
$$225$$ 6.34111e7 0.0247420
$$226$$ 0 0
$$227$$ 7.50054e8 0.282481 0.141241 0.989975i $$-0.454891\pi$$
0.141241 + 0.989975i $$0.454891\pi$$
$$228$$ 0 0
$$229$$ − 2.84784e9i − 1.03556i −0.855515 0.517778i $$-0.826759\pi$$
0.855515 0.517778i $$-0.173241\pi$$
$$230$$ 0 0
$$231$$ 2.58379e9i 0.907421i
$$232$$ 0 0
$$233$$ −2.20621e8 −0.0748553 −0.0374276 0.999299i $$-0.511916\pi$$
−0.0374276 + 0.999299i $$0.511916\pi$$
$$234$$ 0 0
$$235$$ 4.65204e9 1.52536
$$236$$ 0 0
$$237$$ 3.58845e9i 1.13740i
$$238$$ 0 0
$$239$$ 4.04493e9i 1.23971i 0.784717 + 0.619855i $$0.212808\pi$$
−0.784717 + 0.619855i $$0.787192\pi$$
$$240$$ 0 0
$$241$$ 6.17983e9 1.83193 0.915964 0.401260i $$-0.131427\pi$$
0.915964 + 0.401260i $$0.131427\pi$$
$$242$$ 0 0
$$243$$ 3.31731e9 0.951395
$$244$$ 0 0
$$245$$ 2.32284e9i 0.644696i
$$246$$ 0 0
$$247$$ 1.06580e8i 0.0286343i
$$248$$ 0 0
$$249$$ −5.19187e9 −1.35060
$$250$$ 0 0
$$251$$ 5.21367e9 1.31356 0.656778 0.754084i $$-0.271919\pi$$
0.656778 + 0.754084i $$0.271919\pi$$
$$252$$ 0 0
$$253$$ 4.38487e9i 1.07022i
$$254$$ 0 0
$$255$$ − 4.45492e9i − 1.05361i
$$256$$ 0 0
$$257$$ −6.13693e9 −1.40676 −0.703378 0.710816i $$-0.748326\pi$$
−0.703378 + 0.710816i $$0.748326\pi$$
$$258$$ 0 0
$$259$$ 4.85695e9 1.07936
$$260$$ 0 0
$$261$$ 4.38904e8i 0.0945818i
$$262$$ 0 0
$$263$$ 6.96916e9i 1.45666i 0.685228 + 0.728329i $$0.259703\pi$$
−0.685228 + 0.728329i $$0.740297\pi$$
$$264$$ 0 0
$$265$$ −5.02817e8 −0.101959
$$266$$ 0 0
$$267$$ −8.32575e9 −1.63824
$$268$$ 0 0
$$269$$ − 2.70720e9i − 0.517025i −0.966008 0.258513i $$-0.916768\pi$$
0.966008 0.258513i $$-0.0832324\pi$$
$$270$$ 0 0
$$271$$ 7.99032e9i 1.48145i 0.671808 + 0.740725i $$0.265518\pi$$
−0.671808 + 0.740725i $$0.734482\pi$$
$$272$$ 0 0
$$273$$ 7.64575e8 0.137648
$$274$$ 0 0
$$275$$ −3.42438e8 −0.0598758
$$276$$ 0 0
$$277$$ − 8.22965e9i − 1.39786i −0.715192 0.698928i $$-0.753661\pi$$
0.715192 0.698928i $$-0.246339\pi$$
$$278$$ 0 0
$$279$$ 2.32578e8i 0.0383841i
$$280$$ 0 0
$$281$$ −3.08105e9 −0.494167 −0.247083 0.968994i $$-0.579472\pi$$
−0.247083 + 0.968994i $$0.579472\pi$$
$$282$$ 0 0
$$283$$ 1.17112e9 0.182582 0.0912908 0.995824i $$-0.470901\pi$$
0.0912908 + 0.995824i $$0.470901\pi$$
$$284$$ 0 0
$$285$$ − 1.18760e9i − 0.180007i
$$286$$ 0 0
$$287$$ − 3.00323e9i − 0.442650i
$$288$$ 0 0
$$289$$ −1.63361e9 −0.234184
$$290$$ 0 0
$$291$$ −1.20522e10 −1.68072
$$292$$ 0 0
$$293$$ 4.80980e9i 0.652614i 0.945264 + 0.326307i $$0.105804\pi$$
−0.945264 + 0.326307i $$0.894196\pi$$
$$294$$ 0 0
$$295$$ − 2.27256e9i − 0.300074i
$$296$$ 0 0
$$297$$ −5.79601e9 −0.744909
$$298$$ 0 0
$$299$$ 1.29754e9 0.162344
$$300$$ 0 0
$$301$$ − 8.29277e9i − 1.01026i
$$302$$ 0 0
$$303$$ − 2.77025e9i − 0.328661i
$$304$$ 0 0
$$305$$ −8.99511e9 −1.03946
$$306$$ 0 0
$$307$$ 3.49176e9 0.393089 0.196545 0.980495i $$-0.437028\pi$$
0.196545 + 0.980495i $$0.437028\pi$$
$$308$$ 0 0
$$309$$ − 1.04417e10i − 1.14535i
$$310$$ 0 0
$$311$$ − 1.29807e10i − 1.38757i −0.720182 0.693785i $$-0.755942\pi$$
0.720182 0.693785i $$-0.244058\pi$$
$$312$$ 0 0
$$313$$ 6.31165e9 0.657606 0.328803 0.944399i $$-0.393355\pi$$
0.328803 + 0.944399i $$0.393355\pi$$
$$314$$ 0 0
$$315$$ −2.92090e9 −0.296671
$$316$$ 0 0
$$317$$ − 1.65902e10i − 1.64291i −0.570273 0.821455i $$-0.693163\pi$$
0.570273 0.821455i $$-0.306837\pi$$
$$318$$ 0 0
$$319$$ − 2.37021e9i − 0.228888i
$$320$$ 0 0
$$321$$ −1.00247e10 −0.944175
$$322$$ 0 0
$$323$$ 1.42411e9 0.130838
$$324$$ 0 0
$$325$$ 1.01332e8i 0.00908264i
$$326$$ 0 0
$$327$$ − 5.90243e9i − 0.516226i
$$328$$ 0 0
$$329$$ 1.06683e10 0.910563
$$330$$ 0 0
$$331$$ 5.48640e9 0.457062 0.228531 0.973537i $$-0.426608\pi$$
0.228531 + 0.973537i $$0.426608\pi$$
$$332$$ 0 0
$$333$$ − 1.18848e10i − 0.966526i
$$334$$ 0 0
$$335$$ − 9.30657e9i − 0.738942i
$$336$$ 0 0
$$337$$ −3.56226e8 −0.0276189 −0.0138095 0.999905i $$-0.504396\pi$$
−0.0138095 + 0.999905i $$0.504396\pi$$
$$338$$ 0 0
$$339$$ −5.50408e9 −0.416760
$$340$$ 0 0
$$341$$ − 1.25599e9i − 0.0928897i
$$342$$ 0 0
$$343$$ 1.33911e10i 0.967476i
$$344$$ 0 0
$$345$$ −1.44582e10 −1.02056
$$346$$ 0 0
$$347$$ −1.59731e10 −1.10172 −0.550859 0.834599i $$-0.685700\pi$$
−0.550859 + 0.834599i $$0.685700\pi$$
$$348$$ 0 0
$$349$$ − 1.03634e10i − 0.698553i −0.937020 0.349277i $$-0.886427\pi$$
0.937020 0.349277i $$-0.113573\pi$$
$$350$$ 0 0
$$351$$ 1.71511e9i 0.112996i
$$352$$ 0 0
$$353$$ −1.30979e10 −0.843536 −0.421768 0.906704i $$-0.638590\pi$$
−0.421768 + 0.906704i $$0.638590\pi$$
$$354$$ 0 0
$$355$$ 7.29586e8 0.0459370
$$356$$ 0 0
$$357$$ − 1.02162e10i − 0.628952i
$$358$$ 0 0
$$359$$ 3.31454e9i 0.199547i 0.995010 + 0.0997737i $$0.0318119\pi$$
−0.995010 + 0.0997737i $$0.968188\pi$$
$$360$$ 0 0
$$361$$ −1.66039e10 −0.977647
$$362$$ 0 0
$$363$$ −1.27242e10 −0.732829
$$364$$ 0 0
$$365$$ 3.49263e9i 0.196780i
$$366$$ 0 0
$$367$$ − 1.96628e10i − 1.08388i −0.840418 0.541939i $$-0.817691\pi$$
0.840418 0.541939i $$-0.182309\pi$$
$$368$$ 0 0
$$369$$ −7.34878e9 −0.396378
$$370$$ 0 0
$$371$$ −1.15308e9 −0.0608646
$$372$$ 0 0
$$373$$ − 2.10063e10i − 1.08521i −0.839987 0.542606i $$-0.817438\pi$$
0.839987 0.542606i $$-0.182562\pi$$
$$374$$ 0 0
$$375$$ − 2.49382e10i − 1.26107i
$$376$$ 0 0
$$377$$ −7.01374e8 −0.0347204
$$378$$ 0 0
$$379$$ 3.04816e9 0.147734 0.0738670 0.997268i $$-0.476466\pi$$
0.0738670 + 0.997268i $$0.476466\pi$$
$$380$$ 0 0
$$381$$ 2.56955e10i 1.21943i
$$382$$ 0 0
$$383$$ 2.23357e10i 1.03802i 0.854770 + 0.519008i $$0.173698\pi$$
−0.854770 + 0.519008i $$0.826302\pi$$
$$384$$ 0 0
$$385$$ 1.57737e10 0.717945
$$386$$ 0 0
$$387$$ −2.02921e10 −0.904654
$$388$$ 0 0
$$389$$ 3.13680e10i 1.36990i 0.728592 + 0.684948i $$0.240175\pi$$
−0.728592 + 0.684948i $$0.759825\pi$$
$$390$$ 0 0
$$391$$ − 1.73377e10i − 0.741795i
$$392$$ 0 0
$$393$$ −3.11926e10 −1.30762
$$394$$ 0 0
$$395$$ 2.19071e10 0.899904
$$396$$ 0 0
$$397$$ − 7.65788e9i − 0.308281i −0.988049 0.154140i $$-0.950739\pi$$
0.988049 0.154140i $$-0.0492608\pi$$
$$398$$ 0 0
$$399$$ − 2.72345e9i − 0.107455i
$$400$$ 0 0
$$401$$ −3.26120e10 −1.26125 −0.630623 0.776089i $$-0.717201\pi$$
−0.630623 + 0.776089i $$0.717201\pi$$
$$402$$ 0 0
$$403$$ −3.71662e8 −0.0140906
$$404$$ 0 0
$$405$$ − 3.28107e10i − 1.21954i
$$406$$ 0 0
$$407$$ 6.41811e10i 2.33900i
$$408$$ 0 0
$$409$$ −2.26168e10 −0.808236 −0.404118 0.914707i $$-0.632422\pi$$
−0.404118 + 0.914707i $$0.632422\pi$$
$$410$$ 0 0
$$411$$ 2.21802e10 0.777318
$$412$$ 0 0
$$413$$ − 5.21155e9i − 0.179129i
$$414$$ 0 0
$$415$$ 3.16958e10i 1.06858i
$$416$$ 0 0
$$417$$ 2.94794e10 0.974932
$$418$$ 0 0
$$419$$ 4.94503e10 1.60440 0.802201 0.597054i $$-0.203662\pi$$
0.802201 + 0.597054i $$0.203662\pi$$
$$420$$ 0 0
$$421$$ − 3.34077e10i − 1.06345i −0.846916 0.531726i $$-0.821543\pi$$
0.846916 0.531726i $$-0.178457\pi$$
$$422$$ 0 0
$$423$$ − 2.61048e10i − 0.815378i
$$424$$ 0 0
$$425$$ 1.35399e9 0.0415012
$$426$$ 0 0
$$427$$ −2.06280e10 −0.620505
$$428$$ 0 0
$$429$$ 1.01033e10i 0.298287i
$$430$$ 0 0
$$431$$ 3.06956e10i 0.889544i 0.895644 + 0.444772i $$0.146715\pi$$
−0.895644 + 0.444772i $$0.853285\pi$$
$$432$$ 0 0
$$433$$ 2.88433e9 0.0820529 0.0410265 0.999158i $$-0.486937\pi$$
0.0410265 + 0.999158i $$0.486937\pi$$
$$434$$ 0 0
$$435$$ 7.81528e9 0.218267
$$436$$ 0 0
$$437$$ − 4.62189e9i − 0.126734i
$$438$$ 0 0
$$439$$ 6.92422e10i 1.86429i 0.362088 + 0.932144i $$0.382064\pi$$
−0.362088 + 0.932144i $$0.617936\pi$$
$$440$$ 0 0
$$441$$ 1.30346e10 0.344621
$$442$$ 0 0
$$443$$ 2.06609e10 0.536455 0.268228 0.963356i $$-0.413562\pi$$
0.268228 + 0.963356i $$0.413562\pi$$
$$444$$ 0 0
$$445$$ 5.08278e10i 1.29617i
$$446$$ 0 0
$$447$$ − 4.03280e10i − 1.01013i
$$448$$ 0 0
$$449$$ 2.11092e10 0.519382 0.259691 0.965692i $$-0.416379\pi$$
0.259691 + 0.965692i $$0.416379\pi$$
$$450$$ 0 0
$$451$$ 3.96855e10 0.959237
$$452$$ 0 0
$$453$$ 8.36315e10i 1.98599i
$$454$$ 0 0
$$455$$ − 4.66764e9i − 0.108906i
$$456$$ 0 0
$$457$$ 2.06831e10 0.474188 0.237094 0.971487i $$-0.423805\pi$$
0.237094 + 0.971487i $$0.423805\pi$$
$$458$$ 0 0
$$459$$ 2.29173e10 0.516312
$$460$$ 0 0
$$461$$ − 7.65072e10i − 1.69394i −0.531640 0.846971i $$-0.678424\pi$$
0.531640 0.846971i $$-0.321576\pi$$
$$462$$ 0 0
$$463$$ − 3.41303e9i − 0.0742704i −0.999310 0.0371352i $$-0.988177\pi$$
0.999310 0.0371352i $$-0.0118232\pi$$
$$464$$ 0 0
$$465$$ 4.14136e9 0.0885791
$$466$$ 0 0
$$467$$ −1.92903e10 −0.405576 −0.202788 0.979223i $$-0.565000\pi$$
−0.202788 + 0.979223i $$0.565000\pi$$
$$468$$ 0 0
$$469$$ − 2.13423e10i − 0.441112i
$$470$$ 0 0
$$471$$ − 2.71102e10i − 0.550869i
$$472$$ 0 0
$$473$$ 1.09583e11 2.18927
$$474$$ 0 0
$$475$$ 3.60948e8 0.00709040
$$476$$ 0 0
$$477$$ 2.82154e9i 0.0545021i
$$478$$ 0 0
$$479$$ 2.43887e10i 0.463282i 0.972801 + 0.231641i $$0.0744095\pi$$
−0.972801 + 0.231641i $$0.925590\pi$$
$$480$$ 0 0
$$481$$ 1.89920e10 0.354806
$$482$$ 0 0
$$483$$ −3.31563e10 −0.609224
$$484$$ 0 0
$$485$$ 7.35776e10i 1.32978i
$$486$$ 0 0
$$487$$ − 9.30801e10i − 1.65478i −0.561626 0.827391i $$-0.689824\pi$$
0.561626 0.827391i $$-0.310176\pi$$
$$488$$ 0 0
$$489$$ 5.78046e10 1.01094
$$490$$ 0 0
$$491$$ −2.12850e9 −0.0366225 −0.0183113 0.999832i $$-0.505829\pi$$
−0.0183113 + 0.999832i $$0.505829\pi$$
$$492$$ 0 0
$$493$$ 9.37175e9i 0.158647i
$$494$$ 0 0
$$495$$ − 3.85976e10i − 0.642895i
$$496$$ 0 0
$$497$$ 1.67312e9 0.0274221
$$498$$ 0 0
$$499$$ −1.04101e10 −0.167901 −0.0839503 0.996470i $$-0.526754\pi$$
−0.0839503 + 0.996470i $$0.526754\pi$$
$$500$$ 0 0
$$501$$ − 4.67744e10i − 0.742433i
$$502$$ 0 0
$$503$$ 3.93019e10i 0.613962i 0.951716 + 0.306981i $$0.0993188\pi$$
−0.951716 + 0.306981i $$0.900681\pi$$
$$504$$ 0 0
$$505$$ −1.69120e10 −0.260034
$$506$$ 0 0
$$507$$ −7.85181e10 −1.18833
$$508$$ 0 0
$$509$$ 3.25113e10i 0.484354i 0.970232 + 0.242177i $$0.0778615\pi$$
−0.970232 + 0.242177i $$0.922139\pi$$
$$510$$ 0 0
$$511$$ 8.00947e9i 0.117468i
$$512$$ 0 0
$$513$$ 6.10931e9 0.0882110
$$514$$ 0 0
$$515$$ −6.37455e10 −0.906194
$$516$$ 0 0
$$517$$ 1.40973e11i 1.97322i
$$518$$ 0 0
$$519$$ − 2.06032e10i − 0.283965i
$$520$$ 0 0
$$521$$ −1.84550e9 −0.0250475 −0.0125237 0.999922i $$-0.503987\pi$$
−0.0125237 + 0.999922i $$0.503987\pi$$
$$522$$ 0 0
$$523$$ 6.23770e10 0.833715 0.416858 0.908972i $$-0.363131\pi$$
0.416858 + 0.908972i $$0.363131\pi$$
$$524$$ 0 0
$$525$$ − 2.58935e9i − 0.0340842i
$$526$$ 0 0
$$527$$ 4.96614e9i 0.0643838i
$$528$$ 0 0
$$529$$ 2.20424e10 0.281473
$$530$$ 0 0
$$531$$ −1.27524e10 −0.160404
$$532$$ 0 0
$$533$$ − 1.17434e10i − 0.145508i
$$534$$ 0 0
$$535$$ 6.11998e10i 0.747025i
$$536$$ 0 0
$$537$$ 1.41797e10 0.170518
$$538$$ 0 0
$$539$$ −7.03905e10 −0.833986
$$540$$ 0 0
$$541$$ 7.45917e10i 0.870766i 0.900245 + 0.435383i $$0.143387\pi$$
−0.900245 + 0.435383i $$0.856613\pi$$
$$542$$ 0 0
$$543$$ − 4.82179e10i − 0.554638i
$$544$$ 0 0
$$545$$ −3.60337e10 −0.408435
$$546$$ 0 0
$$547$$ −1.41531e9 −0.0158089 −0.00790445 0.999969i $$-0.502516\pi$$
−0.00790445 + 0.999969i $$0.502516\pi$$
$$548$$ 0 0
$$549$$ 5.04758e10i 0.555641i
$$550$$ 0 0
$$551$$ 2.49833e9i 0.0271046i
$$552$$ 0 0
$$553$$ 5.02383e10 0.537198
$$554$$ 0 0
$$555$$ −2.11624e11 −2.23046
$$556$$ 0 0
$$557$$ 1.37543e11i 1.42895i 0.699661 + 0.714475i $$0.253334\pi$$
−0.699661 + 0.714475i $$0.746666\pi$$
$$558$$ 0 0
$$559$$ − 3.24270e10i − 0.332093i
$$560$$ 0 0
$$561$$ 1.35000e11 1.36296
$$562$$ 0 0
$$563$$ 1.06415e11 1.05918 0.529589 0.848255i $$-0.322346\pi$$
0.529589 + 0.848255i $$0.322346\pi$$
$$564$$ 0 0
$$565$$ 3.36018e10i 0.329738i
$$566$$ 0 0
$$567$$ − 7.52431e10i − 0.728005i
$$568$$ 0 0
$$569$$ −4.02429e10 −0.383919 −0.191960 0.981403i $$-0.561484\pi$$
−0.191960 + 0.981403i $$0.561484\pi$$
$$570$$ 0 0
$$571$$ 1.50341e11 1.41427 0.707137 0.707077i $$-0.249986\pi$$
0.707137 + 0.707077i $$0.249986\pi$$
$$572$$ 0 0
$$573$$ 9.91667e10i 0.919914i
$$574$$ 0 0
$$575$$ − 4.39432e9i − 0.0401994i
$$576$$ 0 0
$$577$$ 4.96477e9 0.0447915 0.0223958 0.999749i $$-0.492871\pi$$
0.0223958 + 0.999749i $$0.492871\pi$$
$$578$$ 0 0
$$579$$ −1.17499e11 −1.04549
$$580$$ 0 0
$$581$$ 7.26862e10i 0.637892i
$$582$$ 0 0
$$583$$ − 1.52372e10i − 0.131895i
$$584$$ 0 0
$$585$$ −1.14215e10 −0.0975216
$$586$$ 0 0
$$587$$ 1.53440e11 1.29237 0.646185 0.763181i $$-0.276363\pi$$
0.646185 + 0.763181i $$0.276363\pi$$
$$588$$ 0 0
$$589$$ 1.32388e9i 0.0109999i
$$590$$ 0 0
$$591$$ − 1.70402e11i − 1.39677i
$$592$$ 0 0
$$593$$ 2.06036e11 1.66619 0.833094 0.553131i $$-0.186567\pi$$
0.833094 + 0.553131i $$0.186567\pi$$
$$594$$ 0 0
$$595$$ −6.23689e10 −0.497623
$$596$$ 0 0
$$597$$ − 2.48837e11i − 1.95892i
$$598$$ 0 0
$$599$$ − 2.30634e11i − 1.79150i −0.444558 0.895750i $$-0.646639\pi$$
0.444558 0.895750i $$-0.353361\pi$$
$$600$$ 0 0
$$601$$ −1.01422e11 −0.777382 −0.388691 0.921368i $$-0.627073\pi$$
−0.388691 + 0.921368i $$0.627073\pi$$
$$602$$ 0 0
$$603$$ −5.22236e10 −0.395001
$$604$$ 0 0
$$605$$ 7.76795e10i 0.579809i
$$606$$ 0 0
$$607$$ 1.97883e11i 1.45765i 0.684700 + 0.728825i $$0.259933\pi$$
−0.684700 + 0.728825i $$0.740067\pi$$
$$608$$ 0 0
$$609$$ 1.79224e10 0.130294
$$610$$ 0 0
$$611$$ 4.17158e10 0.299320
$$612$$ 0 0
$$613$$ 1.27158e11i 0.900538i 0.892893 + 0.450269i $$0.148672\pi$$
−0.892893 + 0.450269i $$0.851328\pi$$
$$614$$ 0 0
$$615$$ 1.30855e11i 0.914723i
$$616$$ 0 0
$$617$$ 5.06702e10 0.349632 0.174816 0.984601i $$-0.444067\pi$$
0.174816 + 0.984601i $$0.444067\pi$$
$$618$$ 0 0
$$619$$ −7.06748e10 −0.481395 −0.240698 0.970600i $$-0.577376\pi$$
−0.240698 + 0.970600i $$0.577376\pi$$
$$620$$ 0 0
$$621$$ − 7.43769e10i − 0.500117i
$$622$$ 0 0
$$623$$ 1.16561e11i 0.773748i
$$624$$ 0 0
$$625$$ −1.45008e11 −0.950327
$$626$$ 0 0
$$627$$ 3.59885e10 0.232859
$$628$$ 0 0
$$629$$ − 2.53771e11i − 1.62121i
$$630$$ 0 0
$$631$$ 1.65273e11i 1.04252i 0.853399 + 0.521259i $$0.174537\pi$$
−0.853399 + 0.521259i $$0.825463\pi$$
$$632$$ 0 0
$$633$$ 1.46657e11 0.913454
$$634$$ 0 0
$$635$$ 1.56868e11 0.964804
$$636$$ 0 0
$$637$$ 2.08294e10i 0.126508i
$$638$$ 0 0
$$639$$ − 4.09405e9i − 0.0245556i
$$640$$ 0 0
$$641$$ 1.12013e11 0.663490 0.331745 0.943369i $$-0.392363\pi$$
0.331745 + 0.943369i $$0.392363\pi$$
$$642$$ 0 0
$$643$$ 2.65913e11 1.55559 0.777795 0.628518i $$-0.216338\pi$$
0.777795 + 0.628518i $$0.216338\pi$$
$$644$$ 0 0
$$645$$ 3.61328e11i 2.08767i
$$646$$ 0 0
$$647$$ 2.71996e11i 1.55219i 0.630614 + 0.776097i $$0.282803\pi$$
−0.630614 + 0.776097i $$0.717197\pi$$
$$648$$ 0 0
$$649$$ 6.88669e10 0.388179
$$650$$ 0 0
$$651$$ 9.49716e9 0.0528774
$$652$$ 0 0
$$653$$ − 3.03789e11i − 1.67078i −0.549656 0.835391i $$-0.685241\pi$$
0.549656 0.835391i $$-0.314759\pi$$
$$654$$ 0 0
$$655$$ 1.90427e11i 1.03458i
$$656$$ 0 0
$$657$$ 1.95988e10 0.105189
$$658$$ 0 0
$$659$$ −4.18575e10 −0.221938 −0.110969 0.993824i $$-0.535395\pi$$
−0.110969 + 0.993824i $$0.535395\pi$$
$$660$$ 0 0
$$661$$ − 2.46529e11i − 1.29141i −0.763589 0.645703i $$-0.776565\pi$$
0.763589 0.645703i $$-0.223435\pi$$
$$662$$ 0 0
$$663$$ − 3.99482e10i − 0.206749i
$$664$$ 0 0
$$665$$ −1.66264e10 −0.0850179
$$666$$ 0 0
$$667$$ 3.04155e10 0.153671
$$668$$ 0 0
$$669$$ − 1.47802e11i − 0.737862i
$$670$$ 0 0
$$671$$ − 2.72584e11i − 1.34465i
$$672$$ 0 0
$$673$$ −3.15336e11 −1.53714 −0.768569 0.639767i $$-0.779031\pi$$
−0.768569 + 0.639767i $$0.779031\pi$$
$$674$$ 0 0
$$675$$ 5.80849e9 0.0279800
$$676$$ 0 0
$$677$$ − 2.47236e10i − 0.117695i −0.998267 0.0588475i $$-0.981257\pi$$
0.998267 0.0588475i $$-0.0187426\pi$$
$$678$$ 0 0
$$679$$ 1.68731e11i 0.793811i
$$680$$ 0 0
$$681$$ −7.49454e10 −0.348463
$$682$$ 0 0
$$683$$ 7.20843e10 0.331251 0.165626 0.986189i $$-0.447036\pi$$
0.165626 + 0.986189i $$0.447036\pi$$
$$684$$ 0 0
$$685$$ − 1.35408e11i − 0.615009i
$$686$$ 0 0
$$687$$ 2.84556e11i 1.27744i
$$688$$ 0 0
$$689$$ −4.50887e9 −0.0200074
$$690$$ 0 0
$$691$$ 2.95424e11 1.29578 0.647892 0.761732i $$-0.275651\pi$$
0.647892 + 0.761732i $$0.275651\pi$$
$$692$$ 0 0
$$693$$ − 8.85139e10i − 0.383776i
$$694$$ 0 0
$$695$$ − 1.79968e11i − 0.771360i
$$696$$ 0 0
$$697$$ −1.56916e11 −0.664867
$$698$$ 0 0
$$699$$ 2.20444e10 0.0923400
$$700$$ 0 0
$$701$$ 2.87925e11i 1.19236i 0.802851 + 0.596180i $$0.203315\pi$$
−0.802851 + 0.596180i $$0.796685\pi$$
$$702$$ 0 0
$$703$$ − 6.76504e10i − 0.276980i
$$704$$ 0 0
$$705$$ −4.64831e11 −1.88165
$$706$$ 0 0
$$707$$ −3.87834e10 −0.155227
$$708$$ 0 0
$$709$$ 2.51685e11i 0.996030i 0.867168 + 0.498015i $$0.165938\pi$$
−0.867168 + 0.498015i $$0.834062\pi$$
$$710$$ 0 0
$$711$$ − 1.22931e11i − 0.481042i
$$712$$ 0 0
$$713$$ 1.61174e10 0.0623643
$$714$$ 0 0
$$715$$ 6.16795e10 0.236003
$$716$$ 0 0
$$717$$ − 4.04170e11i − 1.52928i
$$718$$ 0 0
$$719$$ 1.38856e11i 0.519574i 0.965666 + 0.259787i $$0.0836524\pi$$
−0.965666 + 0.259787i $$0.916348\pi$$
$$720$$ 0 0
$$721$$ −1.46184e11 −0.540953
$$722$$ 0 0
$$723$$ −6.17489e11 −2.25983
$$724$$ 0 0
$$725$$ 2.37531e9i 0.00859743i
$$726$$ 0 0
$$727$$ − 1.79083e11i − 0.641088i −0.947234 0.320544i $$-0.896134\pi$$
0.947234 0.320544i $$-0.103866\pi$$
$$728$$ 0 0
$$729$$ 2.14381e10 0.0759061
$$730$$ 0 0
$$731$$ −4.33289e11 −1.51743
$$732$$ 0 0
$$733$$ − 2.17618e11i − 0.753839i −0.926246 0.376920i $$-0.876983\pi$$
0.926246 0.376920i $$-0.123017\pi$$
$$734$$ 0 0
$$735$$ − 2.32098e11i − 0.795285i
$$736$$ 0 0
$$737$$ 2.82023e11 0.955904
$$738$$ 0 0
$$739$$ −4.84950e11 −1.62599 −0.812997 0.582268i $$-0.802166\pi$$
−0.812997 + 0.582268i $$0.802166\pi$$
$$740$$ 0 0
$$741$$ − 1.06494e10i − 0.0353227i
$$742$$ 0 0
$$743$$ 2.03509e11i 0.667771i 0.942614 + 0.333886i $$0.108360\pi$$
−0.942614 + 0.333886i $$0.891640\pi$$
$$744$$ 0 0
$$745$$ −2.46198e11 −0.799206
$$746$$ 0 0
$$747$$ 1.77860e11 0.571210
$$748$$ 0 0
$$749$$ 1.40346e11i 0.445937i
$$750$$ 0 0
$$751$$ − 2.34693e11i − 0.737804i −0.929468 0.368902i $$-0.879734\pi$$
0.929468 0.368902i $$-0.120266\pi$$
$$752$$ 0 0
$$753$$ −5.20950e11 −1.62038
$$754$$ 0 0
$$755$$ 5.10561e11 1.57130
$$756$$ 0 0
$$757$$ − 3.84882e11i − 1.17204i −0.810295 0.586022i $$-0.800693\pi$$
0.810295 0.586022i $$-0.199307\pi$$
$$758$$ 0 0
$$759$$ − 4.38136e11i − 1.32021i
$$760$$ 0 0
$$761$$ −2.39209e11 −0.713244 −0.356622 0.934249i $$-0.616072\pi$$
−0.356622 + 0.934249i $$0.616072\pi$$
$$762$$ 0 0
$$763$$ −8.26340e10 −0.243815
$$764$$ 0 0
$$765$$ 1.52614e11i 0.445604i
$$766$$ 0 0
$$767$$ − 2.03786e10i − 0.0588833i
$$768$$ 0 0
$$769$$ 2.08457e11 0.596089 0.298045 0.954552i $$-0.403666\pi$$
0.298045 + 0.954552i $$0.403666\pi$$
$$770$$ 0 0
$$771$$ 6.13202e11 1.73535
$$772$$ 0 0
$$773$$ − 5.54469e10i − 0.155296i −0.996981 0.0776478i $$-0.975259\pi$$
0.996981 0.0776478i $$-0.0247410\pi$$
$$774$$ 0 0
$$775$$ 1.25869e9i 0.00348909i
$$776$$ 0 0
$$777$$ −4.85306e11 −1.33147
$$778$$ 0 0
$$779$$ −4.18307e10 −0.113591
$$780$$ 0 0
$$781$$ 2.21091e10i 0.0594246i
$$782$$ 0 0
$$783$$ 4.02039e10i 0.106960i
$$784$$ 0 0
$$785$$ −1.65504e11 −0.435844
$$786$$ 0 0
$$787$$ −4.05908e11 −1.05811 −0.529053 0.848589i $$-0.677453\pi$$
−0.529053 + 0.848589i $$0.677453\pi$$
$$788$$ 0 0
$$789$$ − 6.96358e11i − 1.79690i
$$790$$ 0 0
$$791$$ 7.70572e10i 0.196837i
$$792$$ 0 0
$$793$$ −8.06610e10 −0.203972
$$794$$ 0 0
$$795$$ 5.02414e10 0.125775
$$796$$ 0 0
$$797$$ 3.09015e11i 0.765855i 0.923778 + 0.382927i $$0.125084\pi$$
−0.923778 + 0.382927i $$0.874916\pi$$
$$798$$ 0 0
$$799$$ − 5.57406e11i − 1.36768i
$$800$$ 0 0
$$801$$ 2.85219e11 0.692864
$$802$$ 0 0
$$803$$ −1.05839e11 −0.254557
$$804$$ 0 0
$$805$$ 2.02415e11i 0.482014i
$$806$$ 0 0
$$807$$ 2.70504e11i 0.637792i
$$808$$ 0 0
$$809$$ −4.77958e11 −1.11582 −0.557912 0.829900i $$-0.688397\pi$$
−0.557912 + 0.829900i $$0.688397\pi$$
$$810$$ 0 0
$$811$$ −6.37503e11 −1.47366 −0.736832 0.676075i $$-0.763679\pi$$
−0.736832 + 0.676075i $$0.763679\pi$$
$$812$$ 0 0
$$813$$ − 7.98393e11i − 1.82749i
$$814$$ 0 0
$$815$$ − 3.52890e11i − 0.799851i
$$816$$ 0 0
$$817$$ −1.15506e11 −0.259250
$$818$$ 0 0
$$819$$ −2.61924e10 −0.0582155
$$820$$ 0 0
$$821$$ − 8.43824e11i − 1.85729i −0.370973 0.928644i $$-0.620976\pi$$
0.370973 0.928644i $$-0.379024\pi$$
$$822$$ 0 0
$$823$$ − 2.60916e11i − 0.568723i −0.958717 0.284362i $$-0.908218\pi$$
0.958717 0.284362i $$-0.0917817\pi$$
$$824$$ 0 0
$$825$$ 3.42164e10 0.0738616
$$826$$ 0 0
$$827$$ 2.78675e11 0.595765 0.297883 0.954602i $$-0.403720\pi$$
0.297883 + 0.954602i $$0.403720\pi$$
$$828$$ 0 0
$$829$$ − 4.75156e11i − 1.00605i −0.864273 0.503023i $$-0.832221\pi$$
0.864273 0.503023i $$-0.167779\pi$$
$$830$$ 0 0
$$831$$ 8.22306e11i 1.72437i
$$832$$ 0 0
$$833$$ 2.78322e11 0.578053
$$834$$ 0 0
$$835$$ −2.85552e11 −0.587408
$$836$$ 0 0
$$837$$ 2.13043e10i 0.0434075i
$$838$$ 0 0
$$839$$ − 3.35440e11i − 0.676966i −0.940973 0.338483i $$-0.890086\pi$$
0.940973 0.338483i $$-0.109914\pi$$
$$840$$ 0 0
$$841$$ 4.83806e11 0.967134
$$842$$ 0 0
$$843$$ 3.07858e11 0.609594
$$844$$ 0 0
$$845$$ 4.79344e11i 0.940200i
$$846$$ 0 0
$$847$$ 1.78138e11i 0.346117i
$$848$$ 0 0
$$849$$ −1.17019e11 −0.225229
$$850$$ 0 0
$$851$$ −8.23600e11 −1.57036
$$852$$ 0 0
$$853$$ − 9.75408e10i − 0.184243i −0.995748 0.0921213i $$-0.970635\pi$$
0.995748 0.0921213i $$-0.0293648\pi$$
$$854$$ 0 0
$$855$$ 4.06840e10i 0.0761306i
$$856$$ 0 0
$$857$$ 7.94769e10 0.147339 0.0736695 0.997283i $$-0.476529\pi$$
0.0736695 + 0.997283i $$0.476529\pi$$
$$858$$ 0 0
$$859$$ −4.15618e11 −0.763347 −0.381673 0.924297i $$-0.624652\pi$$
−0.381673 + 0.924297i $$0.624652\pi$$
$$860$$ 0 0
$$861$$ 3.00083e11i 0.546045i
$$862$$ 0 0
$$863$$ 4.80012e11i 0.865383i 0.901542 + 0.432692i $$0.142436\pi$$
−0.901542 + 0.432692i $$0.857564\pi$$
$$864$$ 0 0
$$865$$ −1.25780e11 −0.224671
$$866$$ 0 0
$$867$$ 1.63230e11 0.288884
$$868$$ 0 0
$$869$$ 6.63863e11i 1.16412i
$$870$$ 0 0
$$871$$ − 8.34540e10i − 0.145002i
$$872$$ 0 0
$$873$$ 4.12879e11 0.710830
$$874$$ 0 0
$$875$$ −3.49134e11 −0.595608
$$876$$ 0 0
$$877$$ − 2.74155e11i − 0.463444i −0.972782 0.231722i $$-0.925564\pi$$
0.972782 0.231722i $$-0.0744360\pi$$
$$878$$ 0 0
$$879$$ − 4.80595e11i − 0.805051i
$$880$$ 0 0
$$881$$ 8.01838e11 1.33101 0.665507 0.746391i $$-0.268215\pi$$
0.665507 + 0.746391i $$0.268215\pi$$
$$882$$ 0 0
$$883$$ 9.95008e11 1.63676 0.818378 0.574681i $$-0.194874\pi$$
0.818378 + 0.574681i $$0.194874\pi$$
$$884$$ 0 0
$$885$$ 2.27075e11i 0.370165i
$$886$$ 0 0
$$887$$ 5.46038e11i 0.882122i 0.897477 + 0.441061i $$0.145398\pi$$
−0.897477 + 0.441061i $$0.854602\pi$$
$$888$$ 0 0
$$889$$ 3.59736e11 0.575940
$$890$$ 0 0
$$891$$ 9.94283e11 1.57761
$$892$$ 0 0
$$893$$ − 1.48594e11i − 0.233665i
$$894$$ 0 0
$$895$$ − 8.65656e10i − 0.134913i
$$896$$ 0 0
$$897$$ −1.29650e11 −0.200264
$$898$$ 0 0
$$899$$ −8.71212e9 −0.0133378
$$900$$ 0 0
$$901$$ 6.02474e10i 0.0914195i
$$902$$ 0 0
$$903$$ 8.28613e11i 1.24624i
$$904$$ 0 0
$$905$$ −2.94365e11 −0.438826
$$906$$ 0 0
$$907$$ 6.55018e11 0.967886 0.483943 0.875100i $$-0.339204\pi$$
0.483943 + 0.875100i $$0.339204\pi$$
$$908$$ 0 0
$$909$$ 9.49014e10i 0.139001i
$$910$$ 0 0
$$911$$ 1.19425e11i 0.173389i 0.996235 + 0.0866943i $$0.0276304\pi$$
−0.996235 + 0.0866943i $$0.972370\pi$$
$$912$$ 0 0
$$913$$ −9.60496e11 −1.38233
$$914$$ 0 0
$$915$$ 8.98791e11 1.28225
$$916$$ 0 0
$$917$$ 4.36696e11i 0.617592i
$$918$$ 0 0
$$919$$ 5.33989e10i 0.0748635i 0.999299 + 0.0374318i $$0.0119177\pi$$
−0.999299 + 0.0374318i $$0.988082\pi$$
$$920$$ 0 0
$$921$$ −3.48897e11 −0.484907
$$922$$ 0 0
$$923$$ 6.54235e9 0.00901420
$$924$$ 0 0
$$925$$ − 6.43194e10i − 0.0878567i
$$926$$ 0 0
$$927$$ 3.57707e11i 0.484404i
$$928$$ 0 0
$$929$$ −6.30991e11 −0.847150 −0.423575 0.905861i $$-0.639225\pi$$
−0.423575 + 0.905861i $$0.639225\pi$$
$$930$$ 0 0
$$931$$ 7.41953e10 0.0987593
$$932$$ 0 0
$$933$$ 1.29703e12i 1.71168i
$$934$$ 0 0
$$935$$ − 8.24161e11i − 1.07836i
$$936$$ 0 0
$$937$$ 8.41436e11 1.09160 0.545799 0.837916i $$-0.316226\pi$$
0.545799 + 0.837916i $$0.316226\pi$$
$$938$$ 0 0
$$939$$ −6.30660e11 −0.811209
$$940$$ 0 0
$$941$$ 4.52935e11i 0.577666i 0.957379 + 0.288833i $$0.0932673\pi$$
−0.957379 + 0.288833i $$0.906733\pi$$
$$942$$ 0 0
$$943$$ 5.09262e11i 0.644013i
$$944$$ 0 0
$$945$$ −2.67556e11 −0.335497
$$946$$ 0 0
$$947$$ 1.00309e12 1.24721 0.623605 0.781739i $$-0.285667\pi$$
0.623605 + 0.781739i $$0.285667\pi$$
$$948$$ 0 0
$$949$$ 3.13192e10i 0.0386141i
$$950$$ 0 0
$$951$$ 1.65769e12i 2.02666i
$$952$$ 0 0
$$953$$ 1.39040e12 1.68565 0.842826 0.538186i $$-0.180890\pi$$
0.842826 + 0.538186i $$0.180890\pi$$
$$954$$ 0 0
$$955$$ 6.05401e11 0.727830
$$956$$ 0 0
$$957$$ 2.36831e11i 0.282352i
$$958$$ 0 0
$$959$$ − 3.10523e11i − 0.367130i
$$960$$ 0 0
$$961$$ 8.48274e11 0.994587
$$962$$ 0 0
$$963$$ 3.43421e11 0.399321
$$964$$ 0 0
$$965$$ 7.17318e11i 0.827185i
$$966$$ 0 0
$$967$$ − 9.01843e11i − 1.03140i −0.856771 0.515698i $$-0.827533\pi$$
0.856771 0.515698i $$-0.172467\pi$$
$$968$$ 0 0
$$969$$ −1.42297e11 −0.161399
$$970$$ 0 0
$$971$$ 1.28411e12 1.44452 0.722261 0.691621i $$-0.243103\pi$$
0.722261 + 0.691621i $$0.243103\pi$$
$$972$$ 0 0
$$973$$ − 4.12712e11i − 0.460464i
$$974$$ 0 0
$$975$$ − 1.01251e10i − 0.0112042i
$$976$$ 0 0
$$977$$ 1.20120e12 1.31837 0.659183 0.751983i $$-0.270902\pi$$
0.659183 + 0.751983i $$0.270902\pi$$
$$978$$ 0 0
$$979$$ −1.54026e12 −1.67674
$$980$$ 0 0
$$981$$ 2.02202e11i 0.218328i
$$982$$ 0 0
$$983$$ 3.77388e11i 0.404179i 0.979367 + 0.202090i $$0.0647733\pi$$
−0.979367 + 0.202090i $$0.935227\pi$$
$$984$$ 0 0
$$985$$ −1.04028e12 −1.10511
$$986$$ 0 0
$$987$$ −1.06597e12 −1.12325
$$988$$ 0 0
$$989$$ 1.40622e12i 1.46983i
$$990$$ 0 0
$$991$$ 3.35563e11i 0.347920i 0.984753 + 0.173960i $$0.0556563\pi$$
−0.984753 + 0.173960i $$0.944344\pi$$
$$992$$ 0 0
$$993$$ −5.48200e11 −0.563822
$$994$$ 0 0
$$995$$ −1.51912e12 −1.54989
$$996$$ 0 0
$$997$$ − 1.03163e12i − 1.04410i −0.852915 0.522050i $$-0.825167\pi$$
0.852915 0.522050i $$-0.174833\pi$$
$$998$$ 0 0
$$999$$ − 1.08865e12i − 1.09302i
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 256.9.d.e.127.2 4
4.3 odd 2 inner 256.9.d.e.127.4 4
8.3 odd 2 inner 256.9.d.e.127.1 4
8.5 even 2 inner 256.9.d.e.127.3 4
16.3 odd 4 4.9.b.b.3.2 yes 2
16.5 even 4 64.9.c.b.63.2 2
16.11 odd 4 64.9.c.b.63.1 2
16.13 even 4 4.9.b.b.3.1 2
48.29 odd 4 36.9.d.b.19.2 2
48.35 even 4 36.9.d.b.19.1 2
80.3 even 4 100.9.d.b.99.4 4
80.13 odd 4 100.9.d.b.99.2 4
80.19 odd 4 100.9.b.c.51.1 2
80.29 even 4 100.9.b.c.51.2 2
80.67 even 4 100.9.d.b.99.1 4
80.77 odd 4 100.9.d.b.99.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4.9.b.b.3.1 2 16.13 even 4
4.9.b.b.3.2 yes 2 16.3 odd 4
36.9.d.b.19.1 2 48.35 even 4
36.9.d.b.19.2 2 48.29 odd 4
64.9.c.b.63.1 2 16.11 odd 4
64.9.c.b.63.2 2 16.5 even 4
100.9.b.c.51.1 2 80.19 odd 4
100.9.b.c.51.2 2 80.29 even 4
100.9.d.b.99.1 4 80.67 even 4
100.9.d.b.99.2 4 80.13 odd 4
100.9.d.b.99.3 4 80.77 odd 4
100.9.d.b.99.4 4 80.3 even 4
256.9.d.e.127.1 4 8.3 odd 2 inner
256.9.d.e.127.2 4 1.1 even 1 trivial
256.9.d.e.127.3 4 8.5 even 2 inner
256.9.d.e.127.4 4 4.3 odd 2 inner