# Properties

 Label 256.8.b.e.129.2 Level $256$ Weight $8$ Character 256.129 Analytic conductor $79.971$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,8,Mod(129,256)]

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([0, 1]))

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

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (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: $$79.9705665239$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 129.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 256.129 Dual form 256.8.b.e.129.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+84.0000i q^{3} -82.0000i q^{5} +456.000 q^{7} -4869.00 q^{9} +O(q^{10})$$ $$q+84.0000i q^{3} -82.0000i q^{5} +456.000 q^{7} -4869.00 q^{9} -2524.00i q^{11} +10778.0i q^{13} +6888.00 q^{15} -11150.0 q^{17} -4124.00i q^{19} +38304.0i q^{21} -81704.0 q^{23} +71401.0 q^{25} -225288. i q^{27} -99798.0i q^{29} -40480.0 q^{31} +212016. q^{33} -37392.0i q^{35} -419442. i q^{37} -905352. q^{39} -141402. q^{41} -690428. i q^{43} +399258. i q^{45} -682032. q^{47} -615607. q^{49} -936600. i q^{51} +1.81312e6i q^{53} -206968. q^{55} +346416. q^{57} -966028. i q^{59} -1.88767e6i q^{61} -2.22026e6 q^{63} +883796. q^{65} -2.96587e6i q^{67} -6.86314e6i q^{69} +2.54823e6 q^{71} +1.68033e6 q^{73} +5.99768e6i q^{75} -1.15094e6i q^{77} +4.03806e6 q^{79} +8.27569e6 q^{81} +5.38576e6i q^{83} +914300. i q^{85} +8.38303e6 q^{87} +6.47305e6 q^{89} +4.91477e6i q^{91} -3.40032e6i q^{93} -338168. q^{95} -6.06576e6 q^{97} +1.22894e7i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 912 q^{7} - 9738 q^{9}+O(q^{10})$$ 2 * q + 912 * q^7 - 9738 * q^9 $$2 q + 912 q^{7} - 9738 q^{9} + 13776 q^{15} - 22300 q^{17} - 163408 q^{23} + 142802 q^{25} - 80960 q^{31} + 424032 q^{33} - 1810704 q^{39} - 282804 q^{41} - 1364064 q^{47} - 1231214 q^{49} - 413936 q^{55} + 692832 q^{57} - 4440528 q^{63} + 1767592 q^{65} + 5096464 q^{71} + 3360652 q^{73} + 8076128 q^{79} + 16551378 q^{81} + 16766064 q^{87} + 12946092 q^{89} - 676336 q^{95} - 12131516 q^{97}+O(q^{100})$$ 2 * q + 912 * q^7 - 9738 * q^9 + 13776 * q^15 - 22300 * q^17 - 163408 * q^23 + 142802 * q^25 - 80960 * q^31 + 424032 * q^33 - 1810704 * q^39 - 282804 * q^41 - 1364064 * q^47 - 1231214 * q^49 - 413936 * q^55 + 692832 * q^57 - 4440528 * q^63 + 1767592 * q^65 + 5096464 * q^71 + 3360652 * q^73 + 8076128 * q^79 + 16551378 * q^81 + 16766064 * q^87 + 12946092 * q^89 - 676336 * q^95 - 12131516 * 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
$$3$$ 84.0000i 1.79620i 0.439790 + 0.898100i $$0.355053\pi$$
−0.439790 + 0.898100i $$0.644947\pi$$
$$4$$ 0 0
$$5$$ − 82.0000i − 0.293372i −0.989183 0.146686i $$-0.953139\pi$$
0.989183 0.146686i $$-0.0468607\pi$$
$$6$$ 0 0
$$7$$ 456.000 0.502483 0.251242 0.967924i $$-0.419161\pi$$
0.251242 + 0.967924i $$0.419161\pi$$
$$8$$ 0 0
$$9$$ −4869.00 −2.22634
$$10$$ 0 0
$$11$$ − 2524.00i − 0.571762i −0.958265 0.285881i $$-0.907714\pi$$
0.958265 0.285881i $$-0.0922861\pi$$
$$12$$ 0 0
$$13$$ 10778.0i 1.36062i 0.732925 + 0.680309i $$0.238155\pi$$
−0.732925 + 0.680309i $$0.761845\pi$$
$$14$$ 0 0
$$15$$ 6888.00 0.526955
$$16$$ 0 0
$$17$$ −11150.0 −0.550432 −0.275216 0.961382i $$-0.588749\pi$$
−0.275216 + 0.961382i $$0.588749\pi$$
$$18$$ 0 0
$$19$$ − 4124.00i − 0.137937i −0.997619 0.0689685i $$-0.978029\pi$$
0.997619 0.0689685i $$-0.0219708\pi$$
$$20$$ 0 0
$$21$$ 38304.0i 0.902561i
$$22$$ 0 0
$$23$$ −81704.0 −1.40022 −0.700109 0.714036i $$-0.746865\pi$$
−0.700109 + 0.714036i $$0.746865\pi$$
$$24$$ 0 0
$$25$$ 71401.0 0.913933
$$26$$ 0 0
$$27$$ − 225288.i − 2.20275i
$$28$$ 0 0
$$29$$ − 99798.0i − 0.759852i −0.925017 0.379926i $$-0.875949\pi$$
0.925017 0.379926i $$-0.124051\pi$$
$$30$$ 0 0
$$31$$ −40480.0 −0.244048 −0.122024 0.992527i $$-0.538938\pi$$
−0.122024 + 0.992527i $$0.538938\pi$$
$$32$$ 0 0
$$33$$ 212016. 1.02700
$$34$$ 0 0
$$35$$ − 37392.0i − 0.147415i
$$36$$ 0 0
$$37$$ − 419442.i − 1.36134i −0.732591 0.680669i $$-0.761689\pi$$
0.732591 0.680669i $$-0.238311\pi$$
$$38$$ 0 0
$$39$$ −905352. −2.44394
$$40$$ 0 0
$$41$$ −141402. −0.320414 −0.160207 0.987083i $$-0.551216\pi$$
−0.160207 + 0.987083i $$0.551216\pi$$
$$42$$ 0 0
$$43$$ − 690428.i − 1.32428i −0.749382 0.662138i $$-0.769649\pi$$
0.749382 0.662138i $$-0.230351\pi$$
$$44$$ 0 0
$$45$$ 399258.i 0.653145i
$$46$$ 0 0
$$47$$ −682032. −0.958213 −0.479107 0.877757i $$-0.659039\pi$$
−0.479107 + 0.877757i $$0.659039\pi$$
$$48$$ 0 0
$$49$$ −615607. −0.747510
$$50$$ 0 0
$$51$$ − 936600.i − 0.988686i
$$52$$ 0 0
$$53$$ 1.81312e6i 1.67286i 0.548071 + 0.836432i $$0.315362\pi$$
−0.548071 + 0.836432i $$0.684638\pi$$
$$54$$ 0 0
$$55$$ −206968. −0.167739
$$56$$ 0 0
$$57$$ 346416. 0.247763
$$58$$ 0 0
$$59$$ − 966028.i − 0.612361i −0.951973 0.306181i $$-0.900949\pi$$
0.951973 0.306181i $$-0.0990511\pi$$
$$60$$ 0 0
$$61$$ − 1.88767e6i − 1.06481i −0.846490 0.532404i $$-0.821289\pi$$
0.846490 0.532404i $$-0.178711\pi$$
$$62$$ 0 0
$$63$$ −2.22026e6 −1.11870
$$64$$ 0 0
$$65$$ 883796. 0.399168
$$66$$ 0 0
$$67$$ − 2.96587e6i − 1.20473i −0.798220 0.602365i $$-0.794225\pi$$
0.798220 0.602365i $$-0.205775\pi$$
$$68$$ 0 0
$$69$$ − 6.86314e6i − 2.51507i
$$70$$ 0 0
$$71$$ 2.54823e6 0.844957 0.422479 0.906373i $$-0.361160\pi$$
0.422479 + 0.906373i $$0.361160\pi$$
$$72$$ 0 0
$$73$$ 1.68033e6 0.505549 0.252775 0.967525i $$-0.418657\pi$$
0.252775 + 0.967525i $$0.418657\pi$$
$$74$$ 0 0
$$75$$ 5.99768e6i 1.64161i
$$76$$ 0 0
$$77$$ − 1.15094e6i − 0.287301i
$$78$$ 0 0
$$79$$ 4.03806e6 0.921464 0.460732 0.887539i $$-0.347587\pi$$
0.460732 + 0.887539i $$0.347587\pi$$
$$80$$ 0 0
$$81$$ 8.27569e6 1.73024
$$82$$ 0 0
$$83$$ 5.38576e6i 1.03389i 0.856019 + 0.516945i $$0.172931\pi$$
−0.856019 + 0.516945i $$0.827069\pi$$
$$84$$ 0 0
$$85$$ 914300.i 0.161481i
$$86$$ 0 0
$$87$$ 8.38303e6 1.36485
$$88$$ 0 0
$$89$$ 6.47305e6 0.973293 0.486647 0.873599i $$-0.338220\pi$$
0.486647 + 0.873599i $$0.338220\pi$$
$$90$$ 0 0
$$91$$ 4.91477e6i 0.683688i
$$92$$ 0 0
$$93$$ − 3.40032e6i − 0.438359i
$$94$$ 0 0
$$95$$ −338168. −0.0404669
$$96$$ 0 0
$$97$$ −6.06576e6 −0.674814 −0.337407 0.941359i $$-0.609550\pi$$
−0.337407 + 0.941359i $$0.609550\pi$$
$$98$$ 0 0
$$99$$ 1.22894e7i 1.27293i
$$100$$ 0 0
$$101$$ 9.70069e6i 0.936866i 0.883499 + 0.468433i $$0.155181\pi$$
−0.883499 + 0.468433i $$0.844819\pi$$
$$102$$ 0 0
$$103$$ −4.10159e6 −0.369847 −0.184924 0.982753i $$-0.559204\pi$$
−0.184924 + 0.982753i $$0.559204\pi$$
$$104$$ 0 0
$$105$$ 3.14093e6 0.264786
$$106$$ 0 0
$$107$$ 72900.0i 0.00575287i 0.999996 + 0.00287643i $$0.000915598\pi$$
−0.999996 + 0.00287643i $$0.999084\pi$$
$$108$$ 0 0
$$109$$ − 9.55841e6i − 0.706957i −0.935443 0.353478i $$-0.884999\pi$$
0.935443 0.353478i $$-0.115001\pi$$
$$110$$ 0 0
$$111$$ 3.52331e7 2.44524
$$112$$ 0 0
$$113$$ 9.33890e6 0.608865 0.304433 0.952534i $$-0.401533\pi$$
0.304433 + 0.952534i $$0.401533\pi$$
$$114$$ 0 0
$$115$$ 6.69973e6i 0.410785i
$$116$$ 0 0
$$117$$ − 5.24781e7i − 3.02920i
$$118$$ 0 0
$$119$$ −5.08440e6 −0.276583
$$120$$ 0 0
$$121$$ 1.31166e7 0.673089
$$122$$ 0 0
$$123$$ − 1.18778e7i − 0.575529i
$$124$$ 0 0
$$125$$ − 1.22611e7i − 0.561495i
$$126$$ 0 0
$$127$$ −3.59794e7 −1.55862 −0.779311 0.626637i $$-0.784431\pi$$
−0.779311 + 0.626637i $$0.784431\pi$$
$$128$$ 0 0
$$129$$ 5.79960e7 2.37867
$$130$$ 0 0
$$131$$ 676052.i 0.0262743i 0.999914 + 0.0131371i $$0.00418180\pi$$
−0.999914 + 0.0131371i $$0.995818\pi$$
$$132$$ 0 0
$$133$$ − 1.88054e6i − 0.0693111i
$$134$$ 0 0
$$135$$ −1.84736e7 −0.646225
$$136$$ 0 0
$$137$$ 2.95841e7 0.982962 0.491481 0.870888i $$-0.336456\pi$$
0.491481 + 0.870888i $$0.336456\pi$$
$$138$$ 0 0
$$139$$ − 3.19084e7i − 1.00775i −0.863776 0.503876i $$-0.831907\pi$$
0.863776 0.503876i $$-0.168093\pi$$
$$140$$ 0 0
$$141$$ − 5.72907e7i − 1.72114i
$$142$$ 0 0
$$143$$ 2.72037e7 0.777949
$$144$$ 0 0
$$145$$ −8.18344e6 −0.222919
$$146$$ 0 0
$$147$$ − 5.17110e7i − 1.34268i
$$148$$ 0 0
$$149$$ − 1.16603e7i − 0.288773i −0.989521 0.144386i $$-0.953879\pi$$
0.989521 0.144386i $$-0.0461208\pi$$
$$150$$ 0 0
$$151$$ 1.76295e7 0.416698 0.208349 0.978055i $$-0.433191\pi$$
0.208349 + 0.978055i $$0.433191\pi$$
$$152$$ 0 0
$$153$$ 5.42894e7 1.22545
$$154$$ 0 0
$$155$$ 3.31936e6i 0.0715968i
$$156$$ 0 0
$$157$$ − 6.34658e6i − 0.130885i −0.997856 0.0654427i $$-0.979154\pi$$
0.997856 0.0654427i $$-0.0208460\pi$$
$$158$$ 0 0
$$159$$ −1.52302e8 −3.00480
$$160$$ 0 0
$$161$$ −3.72570e7 −0.703587
$$162$$ 0 0
$$163$$ − 8.04234e7i − 1.45454i −0.686351 0.727271i $$-0.740789\pi$$
0.686351 0.727271i $$-0.259211\pi$$
$$164$$ 0 0
$$165$$ − 1.73853e7i − 0.301293i
$$166$$ 0 0
$$167$$ −1.14767e8 −1.90682 −0.953411 0.301673i $$-0.902455\pi$$
−0.953411 + 0.301673i $$0.902455\pi$$
$$168$$ 0 0
$$169$$ −5.34168e7 −0.851283
$$170$$ 0 0
$$171$$ 2.00798e7i 0.307095i
$$172$$ 0 0
$$173$$ 6.33755e7i 0.930594i 0.885155 + 0.465297i $$0.154053\pi$$
−0.885155 + 0.465297i $$0.845947\pi$$
$$174$$ 0 0
$$175$$ 3.25589e7 0.459236
$$176$$ 0 0
$$177$$ 8.11464e7 1.09992
$$178$$ 0 0
$$179$$ 1.13228e7i 0.147559i 0.997275 + 0.0737796i $$0.0235061\pi$$
−0.997275 + 0.0737796i $$0.976494\pi$$
$$180$$ 0 0
$$181$$ − 5.22650e6i − 0.0655143i −0.999463 0.0327571i $$-0.989571\pi$$
0.999463 0.0327571i $$-0.0104288\pi$$
$$182$$ 0 0
$$183$$ 1.58564e8 1.91261
$$184$$ 0 0
$$185$$ −3.43942e7 −0.399379
$$186$$ 0 0
$$187$$ 2.81426e7i 0.314716i
$$188$$ 0 0
$$189$$ − 1.02731e8i − 1.10684i
$$190$$ 0 0
$$191$$ −8.50301e7 −0.882990 −0.441495 0.897264i $$-0.645552\pi$$
−0.441495 + 0.897264i $$0.645552\pi$$
$$192$$ 0 0
$$193$$ 1.15092e8 1.15237 0.576186 0.817319i $$-0.304540\pi$$
0.576186 + 0.817319i $$0.304540\pi$$
$$194$$ 0 0
$$195$$ 7.42389e7i 0.716985i
$$196$$ 0 0
$$197$$ − 1.38522e8i − 1.29088i −0.763810 0.645441i $$-0.776674\pi$$
0.763810 0.645441i $$-0.223326\pi$$
$$198$$ 0 0
$$199$$ 2.19614e7 0.197548 0.0987742 0.995110i $$-0.468508\pi$$
0.0987742 + 0.995110i $$0.468508\pi$$
$$200$$ 0 0
$$201$$ 2.49133e8 2.16394
$$202$$ 0 0
$$203$$ − 4.55079e7i − 0.381813i
$$204$$ 0 0
$$205$$ 1.15950e7i 0.0940007i
$$206$$ 0 0
$$207$$ 3.97817e8 3.11736
$$208$$ 0 0
$$209$$ −1.04090e7 −0.0788671
$$210$$ 0 0
$$211$$ 6.10208e7i 0.447187i 0.974682 + 0.223594i $$0.0717789\pi$$
−0.974682 + 0.223594i $$0.928221\pi$$
$$212$$ 0 0
$$213$$ 2.14051e8i 1.51771i
$$214$$ 0 0
$$215$$ −5.66151e7 −0.388506
$$216$$ 0 0
$$217$$ −1.84589e7 −0.122630
$$218$$ 0 0
$$219$$ 1.41147e8i 0.908068i
$$220$$ 0 0
$$221$$ − 1.20175e8i − 0.748928i
$$222$$ 0 0
$$223$$ −4.22448e7 −0.255098 −0.127549 0.991832i $$-0.540711\pi$$
−0.127549 + 0.991832i $$0.540711\pi$$
$$224$$ 0 0
$$225$$ −3.47651e8 −2.03472
$$226$$ 0 0
$$227$$ 2.39102e8i 1.35673i 0.734726 + 0.678364i $$0.237311\pi$$
−0.734726 + 0.678364i $$0.762689\pi$$
$$228$$ 0 0
$$229$$ − 4.67889e7i − 0.257465i −0.991679 0.128733i $$-0.958909\pi$$
0.991679 0.128733i $$-0.0410909\pi$$
$$230$$ 0 0
$$231$$ 9.66793e7 0.516050
$$232$$ 0 0
$$233$$ −3.45225e8 −1.78795 −0.893977 0.448113i $$-0.852096\pi$$
−0.893977 + 0.448113i $$0.852096\pi$$
$$234$$ 0 0
$$235$$ 5.59266e7i 0.281113i
$$236$$ 0 0
$$237$$ 3.39197e8i 1.65513i
$$238$$ 0 0
$$239$$ 2.34413e8 1.11068 0.555340 0.831624i $$-0.312588\pi$$
0.555340 + 0.831624i $$0.312588\pi$$
$$240$$ 0 0
$$241$$ −1.09557e8 −0.504175 −0.252087 0.967705i $$-0.581117\pi$$
−0.252087 + 0.967705i $$0.581117\pi$$
$$242$$ 0 0
$$243$$ 2.02453e8i 0.905112i
$$244$$ 0 0
$$245$$ 5.04798e7i 0.219299i
$$246$$ 0 0
$$247$$ 4.44485e7 0.187680
$$248$$ 0 0
$$249$$ −4.52404e8 −1.85707
$$250$$ 0 0
$$251$$ − 3.94031e8i − 1.57280i −0.617720 0.786398i $$-0.711943\pi$$
0.617720 0.786398i $$-0.288057\pi$$
$$252$$ 0 0
$$253$$ 2.06221e8i 0.800591i
$$254$$ 0 0
$$255$$ −7.68012e7 −0.290053
$$256$$ 0 0
$$257$$ 3.19064e8 1.17250 0.586248 0.810131i $$-0.300604\pi$$
0.586248 + 0.810131i $$0.300604\pi$$
$$258$$ 0 0
$$259$$ − 1.91266e8i − 0.684050i
$$260$$ 0 0
$$261$$ 4.85916e8i 1.69169i
$$262$$ 0 0
$$263$$ −2.19359e8 −0.743549 −0.371774 0.928323i $$-0.621250\pi$$
−0.371774 + 0.928323i $$0.621250\pi$$
$$264$$ 0 0
$$265$$ 1.48676e8 0.490772
$$266$$ 0 0
$$267$$ 5.43736e8i 1.74823i
$$268$$ 0 0
$$269$$ 1.48033e8i 0.463687i 0.972753 + 0.231844i $$0.0744757\pi$$
−0.972753 + 0.231844i $$0.925524\pi$$
$$270$$ 0 0
$$271$$ −3.69934e8 −1.12910 −0.564549 0.825399i $$-0.690950\pi$$
−0.564549 + 0.825399i $$0.690950\pi$$
$$272$$ 0 0
$$273$$ −4.12841e8 −1.22804
$$274$$ 0 0
$$275$$ − 1.80216e8i − 0.522552i
$$276$$ 0 0
$$277$$ − 3.95860e8i − 1.11908i −0.828803 0.559541i $$-0.810977\pi$$
0.828803 0.559541i $$-0.189023\pi$$
$$278$$ 0 0
$$279$$ 1.97097e8 0.543332
$$280$$ 0 0
$$281$$ 5.97760e8 1.60714 0.803572 0.595208i $$-0.202930\pi$$
0.803572 + 0.595208i $$0.202930\pi$$
$$282$$ 0 0
$$283$$ 8.05797e7i 0.211336i 0.994401 + 0.105668i $$0.0336981\pi$$
−0.994401 + 0.105668i $$0.966302\pi$$
$$284$$ 0 0
$$285$$ − 2.84061e7i − 0.0726867i
$$286$$ 0 0
$$287$$ −6.44793e7 −0.161003
$$288$$ 0 0
$$289$$ −2.86016e8 −0.697025
$$290$$ 0 0
$$291$$ − 5.09524e8i − 1.21210i
$$292$$ 0 0
$$293$$ 7.54530e8i 1.75243i 0.481924 + 0.876213i $$0.339938\pi$$
−0.481924 + 0.876213i $$0.660062\pi$$
$$294$$ 0 0
$$295$$ −7.92143e7 −0.179650
$$296$$ 0 0
$$297$$ −5.68627e8 −1.25945
$$298$$ 0 0
$$299$$ − 8.80606e8i − 1.90516i
$$300$$ 0 0
$$301$$ − 3.14835e8i − 0.665427i
$$302$$ 0 0
$$303$$ −8.14858e8 −1.68280
$$304$$ 0 0
$$305$$ −1.54789e8 −0.312385
$$306$$ 0 0
$$307$$ − 8.20472e8i − 1.61838i −0.587549 0.809188i $$-0.699907\pi$$
0.587549 0.809188i $$-0.300093\pi$$
$$308$$ 0 0
$$309$$ − 3.44534e8i − 0.664320i
$$310$$ 0 0
$$311$$ −6.53503e8 −1.23193 −0.615965 0.787773i $$-0.711234\pi$$
−0.615965 + 0.787773i $$0.711234\pi$$
$$312$$ 0 0
$$313$$ −6.63587e8 −1.22319 −0.611594 0.791172i $$-0.709471\pi$$
−0.611594 + 0.791172i $$0.709471\pi$$
$$314$$ 0 0
$$315$$ 1.82062e8i 0.328195i
$$316$$ 0 0
$$317$$ 3.54718e8i 0.625426i 0.949848 + 0.312713i $$0.101238\pi$$
−0.949848 + 0.312713i $$0.898762\pi$$
$$318$$ 0 0
$$319$$ −2.51890e8 −0.434454
$$320$$ 0 0
$$321$$ −6.12360e6 −0.0103333
$$322$$ 0 0
$$323$$ 4.59826e7i 0.0759250i
$$324$$ 0 0
$$325$$ 7.69560e8i 1.24351i
$$326$$ 0 0
$$327$$ 8.02906e8 1.26984
$$328$$ 0 0
$$329$$ −3.11007e8 −0.481486
$$330$$ 0 0
$$331$$ − 3.05543e8i − 0.463100i −0.972823 0.231550i $$-0.925620\pi$$
0.972823 0.231550i $$-0.0743797\pi$$
$$332$$ 0 0
$$333$$ 2.04226e9i 3.03080i
$$334$$ 0 0
$$335$$ −2.43201e8 −0.353434
$$336$$ 0 0
$$337$$ 3.54965e7 0.0505220 0.0252610 0.999681i $$-0.491958\pi$$
0.0252610 + 0.999681i $$0.491958\pi$$
$$338$$ 0 0
$$339$$ 7.84467e8i 1.09364i
$$340$$ 0 0
$$341$$ 1.02172e8i 0.139537i
$$342$$ 0 0
$$343$$ −6.56252e8 −0.878095
$$344$$ 0 0
$$345$$ −5.62777e8 −0.737853
$$346$$ 0 0
$$347$$ − 1.90594e8i − 0.244882i −0.992476 0.122441i $$-0.960928\pi$$
0.992476 0.122441i $$-0.0390723\pi$$
$$348$$ 0 0
$$349$$ − 8.60864e8i − 1.08404i −0.840366 0.542020i $$-0.817660\pi$$
0.840366 0.542020i $$-0.182340\pi$$
$$350$$ 0 0
$$351$$ 2.42815e9 2.99710
$$352$$ 0 0
$$353$$ −1.04544e9 −1.26500 −0.632498 0.774562i $$-0.717970\pi$$
−0.632498 + 0.774562i $$0.717970\pi$$
$$354$$ 0 0
$$355$$ − 2.08955e8i − 0.247887i
$$356$$ 0 0
$$357$$ − 4.27090e8i − 0.496798i
$$358$$ 0 0
$$359$$ −7.63303e8 −0.870696 −0.435348 0.900262i $$-0.643375\pi$$
−0.435348 + 0.900262i $$0.643375\pi$$
$$360$$ 0 0
$$361$$ 8.76864e8 0.980973
$$362$$ 0 0
$$363$$ 1.10179e9i 1.20900i
$$364$$ 0 0
$$365$$ − 1.37787e8i − 0.148314i
$$366$$ 0 0
$$367$$ −1.38692e9 −1.46460 −0.732302 0.680980i $$-0.761554\pi$$
−0.732302 + 0.680980i $$0.761554\pi$$
$$368$$ 0 0
$$369$$ 6.88486e8 0.713351
$$370$$ 0 0
$$371$$ 8.26782e8i 0.840586i
$$372$$ 0 0
$$373$$ 4.77105e8i 0.476029i 0.971262 + 0.238015i $$0.0764966\pi$$
−0.971262 + 0.238015i $$0.923503\pi$$
$$374$$ 0 0
$$375$$ 1.02994e9 1.00856
$$376$$ 0 0
$$377$$ 1.07562e9 1.03387
$$378$$ 0 0
$$379$$ − 3.92468e8i − 0.370311i −0.982709 0.185156i $$-0.940721\pi$$
0.982709 0.185156i $$-0.0592789\pi$$
$$380$$ 0 0
$$381$$ − 3.02227e9i − 2.79960i
$$382$$ 0 0
$$383$$ 2.10409e9 1.91368 0.956839 0.290617i $$-0.0938605\pi$$
0.956839 + 0.290617i $$0.0938605\pi$$
$$384$$ 0 0
$$385$$ −9.43774e7 −0.0842860
$$386$$ 0 0
$$387$$ 3.36169e9i 2.94829i
$$388$$ 0 0
$$389$$ − 1.26019e9i − 1.08546i −0.839907 0.542730i $$-0.817391\pi$$
0.839907 0.542730i $$-0.182609\pi$$
$$390$$ 0 0
$$391$$ 9.11000e8 0.770725
$$392$$ 0 0
$$393$$ −5.67884e7 −0.0471939
$$394$$ 0 0
$$395$$ − 3.31121e8i − 0.270332i
$$396$$ 0 0
$$397$$ 9.81298e8i 0.787107i 0.919302 + 0.393554i $$0.128754\pi$$
−0.919302 + 0.393554i $$0.871246\pi$$
$$398$$ 0 0
$$399$$ 1.57966e8 0.124497
$$400$$ 0 0
$$401$$ 9.09981e8 0.704737 0.352369 0.935861i $$-0.385376\pi$$
0.352369 + 0.935861i $$0.385376\pi$$
$$402$$ 0 0
$$403$$ − 4.36293e8i − 0.332056i
$$404$$ 0 0
$$405$$ − 6.78606e8i − 0.507604i
$$406$$ 0 0
$$407$$ −1.05867e9 −0.778361
$$408$$ 0 0
$$409$$ 3.55609e7 0.0257004 0.0128502 0.999917i $$-0.495910\pi$$
0.0128502 + 0.999917i $$0.495910\pi$$
$$410$$ 0 0
$$411$$ 2.48507e9i 1.76560i
$$412$$ 0 0
$$413$$ − 4.40509e8i − 0.307701i
$$414$$ 0 0
$$415$$ 4.41633e8 0.303314
$$416$$ 0 0
$$417$$ 2.68031e9 1.81013
$$418$$ 0 0
$$419$$ − 2.65360e9i − 1.76233i −0.472813 0.881163i $$-0.656761\pi$$
0.472813 0.881163i $$-0.343239\pi$$
$$420$$ 0 0
$$421$$ − 1.12113e9i − 0.732264i −0.930563 0.366132i $$-0.880682\pi$$
0.930563 0.366132i $$-0.119318\pi$$
$$422$$ 0 0
$$423$$ 3.32081e9 2.13331
$$424$$ 0 0
$$425$$ −7.96121e8 −0.503058
$$426$$ 0 0
$$427$$ − 8.60778e8i − 0.535049i
$$428$$ 0 0
$$429$$ 2.28511e9i 1.39735i
$$430$$ 0 0
$$431$$ −1.06344e9 −0.639799 −0.319900 0.947451i $$-0.603649\pi$$
−0.319900 + 0.947451i $$0.603649\pi$$
$$432$$ 0 0
$$433$$ −7.05962e8 −0.417901 −0.208951 0.977926i $$-0.567005\pi$$
−0.208951 + 0.977926i $$0.567005\pi$$
$$434$$ 0 0
$$435$$ − 6.87409e8i − 0.400408i
$$436$$ 0 0
$$437$$ 3.36947e8i 0.193142i
$$438$$ 0 0
$$439$$ −1.48506e9 −0.837760 −0.418880 0.908042i $$-0.637577\pi$$
−0.418880 + 0.908042i $$0.637577\pi$$
$$440$$ 0 0
$$441$$ 2.99739e9 1.66421
$$442$$ 0 0
$$443$$ − 7.22153e8i − 0.394654i −0.980338 0.197327i $$-0.936774\pi$$
0.980338 0.197327i $$-0.0632260\pi$$
$$444$$ 0 0
$$445$$ − 5.30790e8i − 0.285537i
$$446$$ 0 0
$$447$$ 9.79462e8 0.518694
$$448$$ 0 0
$$449$$ −1.22968e9 −0.641109 −0.320554 0.947230i $$-0.603869\pi$$
−0.320554 + 0.947230i $$0.603869\pi$$
$$450$$ 0 0
$$451$$ 3.56899e8i 0.183201i
$$452$$ 0 0
$$453$$ 1.48088e9i 0.748473i
$$454$$ 0 0
$$455$$ 4.03011e8 0.200575
$$456$$ 0 0
$$457$$ 8.85551e7 0.0434017 0.0217009 0.999765i $$-0.493092\pi$$
0.0217009 + 0.999765i $$0.493092\pi$$
$$458$$ 0 0
$$459$$ 2.51196e9i 1.21246i
$$460$$ 0 0
$$461$$ − 2.10937e8i − 0.100277i −0.998742 0.0501384i $$-0.984034\pi$$
0.998742 0.0501384i $$-0.0159662\pi$$
$$462$$ 0 0
$$463$$ −3.29775e9 −1.54413 −0.772066 0.635543i $$-0.780776\pi$$
−0.772066 + 0.635543i $$0.780776\pi$$
$$464$$ 0 0
$$465$$ −2.78826e8 −0.128602
$$466$$ 0 0
$$467$$ − 8.82873e7i − 0.0401134i −0.999799 0.0200567i $$-0.993615\pi$$
0.999799 0.0200567i $$-0.00638467\pi$$
$$468$$ 0 0
$$469$$ − 1.35244e9i − 0.605357i
$$470$$ 0 0
$$471$$ 5.33113e8 0.235096
$$472$$ 0 0
$$473$$ −1.74264e9 −0.757171
$$474$$ 0 0
$$475$$ − 2.94458e8i − 0.126065i
$$476$$ 0 0
$$477$$ − 8.82807e9i − 3.72436i
$$478$$ 0 0
$$479$$ −4.51507e9 −1.87711 −0.938557 0.345125i $$-0.887836\pi$$
−0.938557 + 0.345125i $$0.887836\pi$$
$$480$$ 0 0
$$481$$ 4.52075e9 1.85226
$$482$$ 0 0
$$483$$ − 3.12959e9i − 1.26378i
$$484$$ 0 0
$$485$$ 4.97392e8i 0.197972i
$$486$$ 0 0
$$487$$ −3.31338e9 −1.29993 −0.649964 0.759965i $$-0.725216\pi$$
−0.649964 + 0.759965i $$0.725216\pi$$
$$488$$ 0 0
$$489$$ 6.75557e9 2.61265
$$490$$ 0 0
$$491$$ − 4.01694e9i − 1.53147i −0.643154 0.765737i $$-0.722374\pi$$
0.643154 0.765737i $$-0.277626\pi$$
$$492$$ 0 0
$$493$$ 1.11275e9i 0.418247i
$$494$$ 0 0
$$495$$ 1.00773e9 0.373443
$$496$$ 0 0
$$497$$ 1.16199e9 0.424577
$$498$$ 0 0
$$499$$ − 2.70976e9i − 0.976290i −0.872763 0.488145i $$-0.837674\pi$$
0.872763 0.488145i $$-0.162326\pi$$
$$500$$ 0 0
$$501$$ − 9.64045e9i − 3.42504i
$$502$$ 0 0
$$503$$ −3.04579e8 −0.106712 −0.0533558 0.998576i $$-0.516992\pi$$
−0.0533558 + 0.998576i $$0.516992\pi$$
$$504$$ 0 0
$$505$$ 7.95456e8 0.274850
$$506$$ 0 0
$$507$$ − 4.48701e9i − 1.52908i
$$508$$ 0 0
$$509$$ 1.88202e8i 0.0632575i 0.999500 + 0.0316287i $$0.0100694\pi$$
−0.999500 + 0.0316287i $$0.989931\pi$$
$$510$$ 0 0
$$511$$ 7.66229e8 0.254030
$$512$$ 0 0
$$513$$ −9.29088e8 −0.303841
$$514$$ 0 0
$$515$$ 3.36331e8i 0.108503i
$$516$$ 0 0
$$517$$ 1.72145e9i 0.547870i
$$518$$ 0 0
$$519$$ −5.32355e9 −1.67153
$$520$$ 0 0
$$521$$ −4.14963e9 −1.28552 −0.642758 0.766069i $$-0.722210\pi$$
−0.642758 + 0.766069i $$0.722210\pi$$
$$522$$ 0 0
$$523$$ 2.51360e9i 0.768318i 0.923267 + 0.384159i $$0.125509\pi$$
−0.923267 + 0.384159i $$0.874491\pi$$
$$524$$ 0 0
$$525$$ 2.73494e9i 0.824880i
$$526$$ 0 0
$$527$$ 4.51352e8 0.134332
$$528$$ 0 0
$$529$$ 3.27072e9 0.960613
$$530$$ 0 0
$$531$$ 4.70359e9i 1.36332i
$$532$$ 0 0
$$533$$ − 1.52403e9i − 0.435962i
$$534$$ 0 0
$$535$$ 5.97780e6 0.00168773
$$536$$ 0 0
$$537$$ −9.51112e8 −0.265046
$$538$$ 0 0
$$539$$ 1.55379e9i 0.427398i
$$540$$ 0 0
$$541$$ 1.32416e9i 0.359543i 0.983708 + 0.179772i $$0.0575358\pi$$
−0.983708 + 0.179772i $$0.942464\pi$$
$$542$$ 0 0
$$543$$ 4.39026e8 0.117677
$$544$$ 0 0
$$545$$ −7.83789e8 −0.207401
$$546$$ 0 0
$$547$$ − 5.58047e8i − 0.145786i −0.997340 0.0728929i $$-0.976777\pi$$
0.997340 0.0728929i $$-0.0232231\pi$$
$$548$$ 0 0
$$549$$ 9.19107e9i 2.37062i
$$550$$ 0 0
$$551$$ −4.11567e8 −0.104812
$$552$$ 0 0
$$553$$ 1.84136e9 0.463020
$$554$$ 0 0
$$555$$ − 2.88912e9i − 0.717364i
$$556$$ 0 0
$$557$$ 3.30331e9i 0.809946i 0.914329 + 0.404973i $$0.132719\pi$$
−0.914329 + 0.404973i $$0.867281\pi$$
$$558$$ 0 0
$$559$$ 7.44143e9 1.80184
$$560$$ 0 0
$$561$$ −2.36398e9 −0.565293
$$562$$ 0 0
$$563$$ 1.22011e8i 0.0288152i 0.999896 + 0.0144076i $$0.00458623\pi$$
−0.999896 + 0.0144076i $$0.995414\pi$$
$$564$$ 0 0
$$565$$ − 7.65790e8i − 0.178624i
$$566$$ 0 0
$$567$$ 3.77371e9 0.869417
$$568$$ 0 0
$$569$$ −5.00925e8 −0.113993 −0.0569967 0.998374i $$-0.518152\pi$$
−0.0569967 + 0.998374i $$0.518152\pi$$
$$570$$ 0 0
$$571$$ − 6.98702e9i − 1.57060i −0.619116 0.785300i $$-0.712509\pi$$
0.619116 0.785300i $$-0.287491\pi$$
$$572$$ 0 0
$$573$$ − 7.14253e9i − 1.58603i
$$574$$ 0 0
$$575$$ −5.83375e9 −1.27971
$$576$$ 0 0
$$577$$ −8.16573e9 −1.76962 −0.884809 0.465954i $$-0.845711\pi$$
−0.884809 + 0.465954i $$0.845711\pi$$
$$578$$ 0 0
$$579$$ 9.66769e9i 2.06989i
$$580$$ 0 0
$$581$$ 2.45591e9i 0.519512i
$$582$$ 0 0
$$583$$ 4.57631e9 0.956479
$$584$$ 0 0
$$585$$ −4.30320e9 −0.888682
$$586$$ 0 0
$$587$$ 8.53182e9i 1.74104i 0.492135 + 0.870519i $$0.336217\pi$$
−0.492135 + 0.870519i $$0.663783\pi$$
$$588$$ 0 0
$$589$$ 1.66940e8i 0.0336632i
$$590$$ 0 0
$$591$$ 1.16358e10 2.31868
$$592$$ 0 0
$$593$$ −1.71175e9 −0.337092 −0.168546 0.985694i $$-0.553907\pi$$
−0.168546 + 0.985694i $$0.553907\pi$$
$$594$$ 0 0
$$595$$ 4.16921e8i 0.0811417i
$$596$$ 0 0
$$597$$ 1.84475e9i 0.354836i
$$598$$ 0 0
$$599$$ 4.77362e9 0.907516 0.453758 0.891125i $$-0.350083\pi$$
0.453758 + 0.891125i $$0.350083\pi$$
$$600$$ 0 0
$$601$$ −7.89998e8 −0.148445 −0.0742224 0.997242i $$-0.523647\pi$$
−0.0742224 + 0.997242i $$0.523647\pi$$
$$602$$ 0 0
$$603$$ 1.44408e10i 2.68214i
$$604$$ 0 0
$$605$$ − 1.07556e9i − 0.197465i
$$606$$ 0 0
$$607$$ −1.82652e9 −0.331485 −0.165743 0.986169i $$-0.553002\pi$$
−0.165743 + 0.986169i $$0.553002\pi$$
$$608$$ 0 0
$$609$$ 3.82266e9 0.685813
$$610$$ 0 0
$$611$$ − 7.35094e9i − 1.30376i
$$612$$ 0 0
$$613$$ − 6.90339e9i − 1.21046i −0.796050 0.605231i $$-0.793081\pi$$
0.796050 0.605231i $$-0.206919\pi$$
$$614$$ 0 0
$$615$$ −9.73977e8 −0.168844
$$616$$ 0 0
$$617$$ 5.69235e9 0.975649 0.487825 0.872942i $$-0.337791\pi$$
0.487825 + 0.872942i $$0.337791\pi$$
$$618$$ 0 0
$$619$$ − 4.28594e9i − 0.726321i −0.931727 0.363161i $$-0.881698\pi$$
0.931727 0.363161i $$-0.118302\pi$$
$$620$$ 0 0
$$621$$ 1.84069e10i 3.08433i
$$622$$ 0 0
$$623$$ 2.95171e9 0.489064
$$624$$ 0 0
$$625$$ 4.57279e9 0.749206
$$626$$ 0 0
$$627$$ − 8.74354e8i − 0.141661i
$$628$$ 0 0
$$629$$ 4.67678e9i 0.749324i
$$630$$ 0 0
$$631$$ 5.61602e8 0.0889869 0.0444935 0.999010i $$-0.485833\pi$$
0.0444935 + 0.999010i $$0.485833\pi$$
$$632$$ 0 0
$$633$$ −5.12575e9 −0.803238
$$634$$ 0 0
$$635$$ 2.95031e9i 0.457256i
$$636$$ 0 0
$$637$$ − 6.63501e9i − 1.01708i
$$638$$ 0 0
$$639$$ −1.24073e10 −1.88116
$$640$$ 0 0
$$641$$ 5.17445e9 0.775998 0.387999 0.921660i $$-0.373166\pi$$
0.387999 + 0.921660i $$0.373166\pi$$
$$642$$ 0 0
$$643$$ 1.04374e10i 1.54830i 0.633004 + 0.774148i $$0.281822\pi$$
−0.633004 + 0.774148i $$0.718178\pi$$
$$644$$ 0 0
$$645$$ − 4.75567e9i − 0.697835i
$$646$$ 0 0
$$647$$ 9.71623e8 0.141037 0.0705185 0.997510i $$-0.477535\pi$$
0.0705185 + 0.997510i $$0.477535\pi$$
$$648$$ 0 0
$$649$$ −2.43825e9 −0.350125
$$650$$ 0 0
$$651$$ − 1.55055e9i − 0.220268i
$$652$$ 0 0
$$653$$ 7.25223e9i 1.01924i 0.860400 + 0.509619i $$0.170214\pi$$
−0.860400 + 0.509619i $$0.829786\pi$$
$$654$$ 0 0
$$655$$ 5.54363e7 0.00770814
$$656$$ 0 0
$$657$$ −8.18151e9 −1.12552
$$658$$ 0 0
$$659$$ − 3.81924e9i − 0.519851i −0.965629 0.259925i $$-0.916302\pi$$
0.965629 0.259925i $$-0.0836979\pi$$
$$660$$ 0 0
$$661$$ 1.07881e10i 1.45292i 0.687210 + 0.726459i $$0.258835\pi$$
−0.687210 + 0.726459i $$0.741165\pi$$
$$662$$ 0 0
$$663$$ 1.00947e10 1.34523
$$664$$ 0 0
$$665$$ −1.54205e8 −0.0203339
$$666$$ 0 0
$$667$$ 8.15390e9i 1.06396i
$$668$$ 0 0
$$669$$ − 3.54857e9i − 0.458207i
$$670$$ 0 0
$$671$$ −4.76448e9 −0.608817
$$672$$ 0 0
$$673$$ −6.34833e9 −0.802798 −0.401399 0.915903i $$-0.631476\pi$$
−0.401399 + 0.915903i $$0.631476\pi$$
$$674$$ 0 0
$$675$$ − 1.60858e10i − 2.01316i
$$676$$ 0 0
$$677$$ 8.82566e9i 1.09317i 0.837404 + 0.546584i $$0.184072\pi$$
−0.837404 + 0.546584i $$0.815928\pi$$
$$678$$ 0 0
$$679$$ −2.76599e9 −0.339083
$$680$$ 0 0
$$681$$ −2.00846e10 −2.43696
$$682$$ 0 0
$$683$$ 4.92331e9i 0.591268i 0.955301 + 0.295634i $$0.0955309\pi$$
−0.955301 + 0.295634i $$0.904469\pi$$
$$684$$ 0 0
$$685$$ − 2.42590e9i − 0.288374i
$$686$$ 0 0
$$687$$ 3.93027e9 0.462459
$$688$$ 0 0
$$689$$ −1.95418e10 −2.27613
$$690$$ 0 0
$$691$$ 5.68449e9i 0.655418i 0.944779 + 0.327709i $$0.106277\pi$$
−0.944779 + 0.327709i $$0.893723\pi$$
$$692$$ 0 0
$$693$$ 5.60395e9i 0.639628i
$$694$$ 0 0
$$695$$ −2.61649e9 −0.295646
$$696$$ 0 0
$$697$$ 1.57663e9 0.176366
$$698$$ 0 0
$$699$$ − 2.89989e10i − 3.21152i
$$700$$ 0 0
$$701$$ 1.70567e9i 0.187017i 0.995618 + 0.0935085i $$0.0298082\pi$$
−0.995618 + 0.0935085i $$0.970192\pi$$
$$702$$ 0 0
$$703$$ −1.72978e9 −0.187779
$$704$$ 0 0
$$705$$ −4.69784e9 −0.504936
$$706$$ 0 0
$$707$$ 4.42351e9i 0.470760i
$$708$$ 0 0
$$709$$ 4.52189e9i 0.476495i 0.971204 + 0.238248i $$0.0765730\pi$$
−0.971204 + 0.238248i $$0.923427\pi$$
$$710$$ 0 0
$$711$$ −1.96613e10 −2.05149
$$712$$ 0 0
$$713$$ 3.30738e9 0.341720
$$714$$ 0 0
$$715$$ − 2.23070e9i − 0.228229i
$$716$$ 0 0
$$717$$ 1.96907e10i 1.99500i
$$718$$ 0 0
$$719$$ −3.09206e9 −0.310239 −0.155120 0.987896i $$-0.549576\pi$$
−0.155120 + 0.987896i $$0.549576\pi$$
$$720$$ 0 0
$$721$$ −1.87033e9 −0.185842
$$722$$ 0 0
$$723$$ − 9.20280e9i − 0.905599i
$$724$$ 0 0
$$725$$ − 7.12568e9i − 0.694453i
$$726$$ 0 0
$$727$$ −1.44622e10 −1.39593 −0.697965 0.716132i $$-0.745911\pi$$
−0.697965 + 0.716132i $$0.745911\pi$$
$$728$$ 0 0
$$729$$ 1.09288e9 0.104478
$$730$$ 0 0
$$731$$ 7.69827e9i 0.728924i
$$732$$ 0 0
$$733$$ − 3.15415e9i − 0.295814i −0.989001 0.147907i $$-0.952746\pi$$
0.989001 0.147907i $$-0.0472536\pi$$
$$734$$ 0 0
$$735$$ −4.24030e9 −0.393905
$$736$$ 0 0
$$737$$ −7.48585e9 −0.688819
$$738$$ 0 0
$$739$$ − 1.54236e10i − 1.40582i −0.711277 0.702912i $$-0.751883\pi$$
0.711277 0.702912i $$-0.248117\pi$$
$$740$$ 0 0
$$741$$ 3.73367e9i 0.337111i
$$742$$ 0 0
$$743$$ 1.59520e10 1.42677 0.713385 0.700772i $$-0.247161\pi$$
0.713385 + 0.700772i $$0.247161\pi$$
$$744$$ 0 0
$$745$$ −9.56141e8 −0.0847179
$$746$$ 0 0
$$747$$ − 2.62233e10i − 2.30179i
$$748$$ 0 0
$$749$$ 3.32424e7i 0.00289072i
$$750$$ 0 0
$$751$$ 6.13964e9 0.528936 0.264468 0.964395i $$-0.414804\pi$$
0.264468 + 0.964395i $$0.414804\pi$$
$$752$$ 0 0
$$753$$ 3.30986e10 2.82506
$$754$$ 0 0
$$755$$ − 1.44562e9i − 0.122248i
$$756$$ 0 0
$$757$$ − 1.42818e10i − 1.19660i −0.801273 0.598299i $$-0.795843\pi$$
0.801273 0.598299i $$-0.204157\pi$$
$$758$$ 0 0
$$759$$ −1.73226e10 −1.43802
$$760$$ 0 0
$$761$$ −1.47536e10 −1.21353 −0.606767 0.794880i $$-0.707534\pi$$
−0.606767 + 0.794880i $$0.707534\pi$$
$$762$$ 0 0
$$763$$ − 4.35863e9i − 0.355234i
$$764$$ 0 0
$$765$$ − 4.45173e9i − 0.359512i
$$766$$ 0 0
$$767$$ 1.04118e10 0.833190
$$768$$ 0 0
$$769$$ 1.97592e10 1.56685 0.783424 0.621487i $$-0.213471\pi$$
0.783424 + 0.621487i $$0.213471\pi$$
$$770$$ 0 0
$$771$$ 2.68014e10i 2.10604i
$$772$$ 0 0
$$773$$ 1.01370e10i 0.789374i 0.918816 + 0.394687i $$0.129147\pi$$
−0.918816 + 0.394687i $$0.870853\pi$$
$$774$$ 0 0
$$775$$ −2.89031e9 −0.223043
$$776$$ 0 0
$$777$$ 1.60663e10 1.22869
$$778$$ 0 0
$$779$$ 5.83142e8i 0.0441970i
$$780$$ 0 0
$$781$$ − 6.43174e9i − 0.483114i
$$782$$ 0 0
$$783$$ −2.24833e10 −1.67376
$$784$$ 0 0
$$785$$ −5.20420e8 −0.0383981
$$786$$ 0 0
$$787$$ 1.27882e10i 0.935188i 0.883943 + 0.467594i $$0.154879\pi$$
−0.883943 + 0.467594i $$0.845121\pi$$
$$788$$ 0 0
$$789$$ − 1.84261e10i − 1.33556i
$$790$$ 0 0
$$791$$ 4.25854e9 0.305945
$$792$$ 0 0
$$793$$ 2.03453e10 1.44880
$$794$$ 0 0
$$795$$ 1.24888e10i 0.881524i
$$796$$ 0 0
$$797$$ 7.38617e9i 0.516791i 0.966039 + 0.258396i $$0.0831938\pi$$
−0.966039 + 0.258396i $$0.916806\pi$$
$$798$$ 0 0
$$799$$ 7.60466e9 0.527431
$$800$$ 0 0
$$801$$ −3.15173e10 −2.16688
$$802$$ 0 0
$$803$$ − 4.24114e9i − 0.289054i
$$804$$ 0 0
$$805$$ 3.05508e9i 0.206413i
$$806$$ 0 0
$$807$$ −1.24348e10 −0.832875
$$808$$ 0 0
$$809$$ −1.53742e10 −1.02087 −0.510437 0.859915i $$-0.670516\pi$$
−0.510437 + 0.859915i $$0.670516\pi$$
$$810$$ 0 0
$$811$$ 9.77882e9i 0.643744i 0.946783 + 0.321872i $$0.104312\pi$$
−0.946783 + 0.321872i $$0.895688\pi$$
$$812$$ 0 0
$$813$$ − 3.10745e10i − 2.02809i
$$814$$ 0 0
$$815$$ −6.59472e9 −0.426722
$$816$$ 0 0
$$817$$ −2.84733e9 −0.182667
$$818$$ 0 0
$$819$$ − 2.39300e10i − 1.52212i
$$820$$ 0 0
$$821$$ 1.83470e10i 1.15708i 0.815654 + 0.578540i $$0.196377\pi$$
−0.815654 + 0.578540i $$0.803623\pi$$
$$822$$ 0 0
$$823$$ 3.16960e10 1.98201 0.991004 0.133829i $$-0.0427272\pi$$
0.991004 + 0.133829i $$0.0427272\pi$$
$$824$$ 0 0
$$825$$ 1.51382e10 0.938608
$$826$$ 0 0
$$827$$ − 6.12845e9i − 0.376774i −0.982095 0.188387i $$-0.939674\pi$$
0.982095 0.188387i $$-0.0603260\pi$$
$$828$$ 0 0
$$829$$ 1.24652e10i 0.759904i 0.925006 + 0.379952i $$0.124060\pi$$
−0.925006 + 0.379952i $$0.875940\pi$$
$$830$$ 0 0
$$831$$ 3.32522e10 2.01010
$$832$$ 0 0
$$833$$ 6.86402e9 0.411454
$$834$$ 0 0
$$835$$ 9.41091e9i 0.559409i
$$836$$ 0 0
$$837$$ 9.11966e9i 0.537575i
$$838$$ 0 0
$$839$$ 1.82237e10 1.06530 0.532648 0.846337i $$-0.321197\pi$$
0.532648 + 0.846337i $$0.321197\pi$$
$$840$$ 0 0
$$841$$ 7.29024e9 0.422625
$$842$$ 0 0
$$843$$ 5.02118e10i 2.88675i
$$844$$ 0 0
$$845$$ 4.38017e9i 0.249743i
$$846$$ 0 0
$$847$$ 5.98117e9 0.338216
$$848$$ 0 0
$$849$$ −6.76870e9 −0.379602
$$850$$ 0 0
$$851$$ 3.42701e10i 1.90617i
$$852$$ 0 0
$$853$$ − 2.48619e10i − 1.37155i −0.727812 0.685777i $$-0.759463\pi$$
0.727812 0.685777i $$-0.240537\pi$$
$$854$$ 0 0
$$855$$ 1.64654e9 0.0900930
$$856$$ 0 0
$$857$$ −2.96761e10 −1.61055 −0.805275 0.592902i $$-0.797982\pi$$
−0.805275 + 0.592902i $$0.797982\pi$$
$$858$$ 0 0
$$859$$ 1.14772e10i 0.617819i 0.951091 + 0.308910i $$0.0999640\pi$$
−0.951091 + 0.308910i $$0.900036\pi$$
$$860$$ 0 0
$$861$$ − 5.41626e9i − 0.289194i
$$862$$ 0 0
$$863$$ 2.13485e10 1.13066 0.565328 0.824866i $$-0.308750\pi$$
0.565328 + 0.824866i $$0.308750\pi$$
$$864$$ 0 0
$$865$$ 5.19679e9 0.273010
$$866$$ 0 0
$$867$$ − 2.40254e10i − 1.25200i
$$868$$ 0 0
$$869$$ − 1.01921e10i − 0.526858i
$$870$$ 0 0
$$871$$ 3.19661e10 1.63918
$$872$$ 0 0
$$873$$ 2.95342e10 1.50236
$$874$$ 0 0
$$875$$ − 5.59108e9i − 0.282142i
$$876$$ 0 0
$$877$$ 7.92753e9i 0.396862i 0.980115 + 0.198431i $$0.0635846\pi$$
−0.980115 + 0.198431i $$0.936415\pi$$
$$878$$ 0 0
$$879$$ −6.33805e10 −3.14771
$$880$$ 0 0
$$881$$ 7.32045e9 0.360680 0.180340 0.983604i $$-0.442280\pi$$
0.180340 + 0.983604i $$0.442280\pi$$
$$882$$ 0 0
$$883$$ 3.54988e9i 0.173521i 0.996229 + 0.0867604i $$0.0276514\pi$$
−0.996229 + 0.0867604i $$0.972349\pi$$
$$884$$ 0 0
$$885$$ − 6.65400e9i − 0.322687i
$$886$$ 0 0
$$887$$ −5.80634e9 −0.279364 −0.139682 0.990196i $$-0.544608\pi$$
−0.139682 + 0.990196i $$0.544608\pi$$
$$888$$ 0 0
$$889$$ −1.64066e10 −0.783182
$$890$$ 0 0
$$891$$ − 2.08878e10i − 0.989285i
$$892$$ 0 0
$$893$$ 2.81270e9i 0.132173i
$$894$$ 0 0
$$895$$ 9.28466e8 0.0432898
$$896$$ 0 0
$$897$$ 7.39709e10 3.42206
$$898$$ 0 0
$$899$$ 4.03982e9i 0.185440i
$$900$$ 0 0
$$901$$ − 2.02163e10i − 0.920798i
$$902$$ 0 0
$$903$$ 2.64462e10 1.19524
$$904$$ 0 0
$$905$$ −4.28573e8 −0.0192201
$$906$$ 0 0
$$907$$ 1.78240e10i 0.793196i 0.917992 + 0.396598i $$0.129809\pi$$
−0.917992 + 0.396598i $$0.870191\pi$$
$$908$$ 0 0
$$909$$ − 4.72326e10i − 2.08578i
$$910$$ 0 0
$$911$$ 1.87703e10 0.822538 0.411269 0.911514i $$-0.365086\pi$$
0.411269 + 0.911514i $$0.365086\pi$$
$$912$$ 0 0
$$913$$ 1.35937e10 0.591138
$$914$$ 0 0
$$915$$ − 1.30023e10i − 0.561107i
$$916$$ 0 0
$$917$$ 3.08280e8i 0.0132024i
$$918$$ 0 0
$$919$$ −3.75844e10 −1.59736 −0.798681 0.601754i $$-0.794469\pi$$
−0.798681 + 0.601754i $$0.794469\pi$$
$$920$$ 0 0
$$921$$ 6.89197e10 2.90693
$$922$$ 0 0
$$923$$ 2.74648e10i 1.14966i
$$924$$ 0 0
$$925$$ − 2.99486e10i − 1.24417i
$$926$$ 0 0
$$927$$ 1.99707e10 0.823404
$$928$$ 0 0
$$929$$ −1.92372e10 −0.787205 −0.393602 0.919281i $$-0.628771\pi$$
−0.393602 + 0.919281i $$0.628771\pi$$
$$930$$ 0 0
$$931$$ 2.53876e9i 0.103109i
$$932$$ 0 0
$$933$$ − 5.48943e10i − 2.21279i
$$934$$ 0 0
$$935$$ 2.30769e9 0.0923289
$$936$$ 0 0
$$937$$ −1.04732e9 −0.0415900 −0.0207950 0.999784i $$-0.506620\pi$$
−0.0207950 + 0.999784i $$0.506620\pi$$
$$938$$ 0 0
$$939$$ − 5.57413e10i − 2.19709i
$$940$$ 0 0
$$941$$ 7.97861e9i 0.312150i 0.987745 + 0.156075i $$0.0498842\pi$$
−0.987745 + 0.156075i $$0.950116\pi$$
$$942$$ 0 0
$$943$$ 1.15531e10 0.448650
$$944$$ 0 0
$$945$$ −8.42397e9 −0.324717
$$946$$ 0 0
$$947$$ 4.26943e9i 0.163360i 0.996659 + 0.0816799i $$0.0260285\pi$$
−0.996659 + 0.0816799i $$0.973971\pi$$
$$948$$ 0 0
$$949$$ 1.81106e10i 0.687860i
$$950$$ 0 0
$$951$$ −2.97963e10 −1.12339
$$952$$ 0 0
$$953$$ 1.06048e10 0.396897 0.198449 0.980111i $$-0.436410\pi$$
0.198449 + 0.980111i $$0.436410\pi$$
$$954$$ 0 0
$$955$$ 6.97247e9i 0.259045i
$$956$$ 0 0
$$957$$ − 2.11588e10i − 0.780367i
$$958$$ 0 0
$$959$$ 1.34904e10 0.493922
$$960$$ 0 0
$$961$$ −2.58740e10 −0.940441
$$962$$ 0 0
$$963$$ − 3.54950e8i − 0.0128078i
$$964$$ 0 0
$$965$$ − 9.43750e9i − 0.338074i
$$966$$ 0 0
$$967$$ 1.65090e10 0.587123 0.293562 0.955940i $$-0.405159\pi$$
0.293562 + 0.955940i $$0.405159\pi$$
$$968$$ 0 0
$$969$$ −3.86254e9 −0.136377
$$970$$ 0 0
$$971$$ − 2.46094e10i − 0.862649i −0.902197 0.431324i $$-0.858046\pi$$
0.902197 0.431324i $$-0.141954\pi$$
$$972$$ 0 0
$$973$$ − 1.45503e10i − 0.506379i
$$974$$ 0 0
$$975$$ −6.46430e10 −2.23360
$$976$$ 0 0
$$977$$ −2.53886e10 −0.870979 −0.435489 0.900194i $$-0.643425\pi$$
−0.435489 + 0.900194i $$0.643425\pi$$
$$978$$ 0 0
$$979$$ − 1.63380e10i − 0.556492i
$$980$$ 0 0
$$981$$ 4.65399e10i 1.57392i
$$982$$ 0 0
$$983$$ −1.87585e10 −0.629884 −0.314942 0.949111i $$-0.601985\pi$$
−0.314942 + 0.949111i $$0.601985\pi$$
$$984$$ 0 0
$$985$$ −1.13588e10 −0.378709
$$986$$ 0 0
$$987$$ − 2.61246e10i − 0.864846i
$$988$$ 0 0
$$989$$ 5.64107e10i 1.85428i
$$990$$ 0 0
$$991$$ −3.59792e9 −0.117434 −0.0587170 0.998275i $$-0.518701\pi$$
−0.0587170 + 0.998275i $$0.518701\pi$$
$$992$$ 0 0
$$993$$ 2.56656e10 0.831821
$$994$$ 0 0
$$995$$ − 1.80083e9i − 0.0579552i
$$996$$ 0 0
$$997$$ − 1.34287e10i − 0.429143i −0.976708 0.214571i $$-0.931165\pi$$
0.976708 0.214571i $$-0.0688354\pi$$
$$998$$ 0 0
$$999$$ −9.44952e10 −2.99868
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 256.8.b.e.129.2 2
4.3 odd 2 256.8.b.c.129.1 2
8.3 odd 2 256.8.b.c.129.2 2
8.5 even 2 inner 256.8.b.e.129.1 2
16.3 odd 4 16.8.a.c.1.1 1
16.5 even 4 64.8.a.g.1.1 1
16.11 odd 4 64.8.a.a.1.1 1
16.13 even 4 8.8.a.a.1.1 1
48.5 odd 4 576.8.a.j.1.1 1
48.11 even 4 576.8.a.k.1.1 1
48.29 odd 4 72.8.a.d.1.1 1
48.35 even 4 144.8.a.g.1.1 1
80.3 even 4 400.8.c.b.49.2 2
80.13 odd 4 200.8.c.a.49.1 2
80.19 odd 4 400.8.a.b.1.1 1
80.29 even 4 200.8.a.i.1.1 1
80.67 even 4 400.8.c.b.49.1 2
80.77 odd 4 200.8.c.a.49.2 2
112.13 odd 4 392.8.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.a.a.1.1 1 16.13 even 4
16.8.a.c.1.1 1 16.3 odd 4
64.8.a.a.1.1 1 16.11 odd 4
64.8.a.g.1.1 1 16.5 even 4
72.8.a.d.1.1 1 48.29 odd 4
144.8.a.g.1.1 1 48.35 even 4
200.8.a.i.1.1 1 80.29 even 4
200.8.c.a.49.1 2 80.13 odd 4
200.8.c.a.49.2 2 80.77 odd 4
256.8.b.c.129.1 2 4.3 odd 2
256.8.b.c.129.2 2 8.3 odd 2
256.8.b.e.129.1 2 8.5 even 2 inner
256.8.b.e.129.2 2 1.1 even 1 trivial
392.8.a.d.1.1 1 112.13 odd 4
400.8.a.b.1.1 1 80.19 odd 4
400.8.c.b.49.1 2 80.67 even 4
400.8.c.b.49.2 2 80.3 even 4
576.8.a.j.1.1 1 48.5 odd 4
576.8.a.k.1.1 1 48.11 even 4