# Properties

 Label 192.9.g.d.127.5 Level $192$ Weight $9$ Character 192.127 Analytic conductor $78.217$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,9,Mod(127,192)]

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([1, 0, 0]))

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

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 192.g (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: $$78.2166931317$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3468738816.6 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484$$ x^8 - 2*x^7 + 2*x^6 + 18*x^5 + 77*x^4 + 8*x^2 + 88*x + 484 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{40}\cdot 3^{12}$$ Twist minimal: no (minimal twist has level 96) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 127.5 Root $$-1.27467 + 1.27467i$$ of defining polynomial Character $$\chi$$ $$=$$ 192.127 Dual form 192.9.g.d.127.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+46.7654i q^{3} -1079.08 q^{5} -3980.70i q^{7} -2187.00 q^{9} +O(q^{10})$$ $$q+46.7654i q^{3} -1079.08 q^{5} -3980.70i q^{7} -2187.00 q^{9} +20108.0i q^{11} +36104.9 q^{13} -50463.5i q^{15} +59944.2 q^{17} -126309. i q^{19} +186159. q^{21} -197109. i q^{23} +773786. q^{25} -102276. i q^{27} -485230. q^{29} +671904. i q^{31} -940357. q^{33} +4.29549e6i q^{35} +3.40344e6 q^{37} +1.68846e6i q^{39} -3.55723e6 q^{41} -736659. i q^{43} +2.35995e6 q^{45} -883461. i q^{47} -1.00811e7 q^{49} +2.80331e6i q^{51} -1.44913e7 q^{53} -2.16981e7i q^{55} +5.90690e6 q^{57} -1.08326e6i q^{59} +3.45104e6 q^{61} +8.70578e6i q^{63} -3.89601e7 q^{65} +1.43441e7i q^{67} +9.21785e6 q^{69} -3.97522e6i q^{71} +3.92004e6 q^{73} +3.61864e7i q^{75} +8.00437e7 q^{77} -5.94312e7i q^{79} +4.78297e6 q^{81} +5.42483e7i q^{83} -6.46846e7 q^{85} -2.26920e7i q^{87} -2.43253e7 q^{89} -1.43723e8i q^{91} -3.14218e7 q^{93} +1.36298e8i q^{95} -3.83933e7 q^{97} -4.39761e7i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 560 q^{5} - 17496 q^{9}+O(q^{10})$$ 8 * q - 560 * q^5 - 17496 * q^9 $$8 q - 560 q^{5} - 17496 q^{9} + 54256 q^{13} + 436176 q^{17} - 54432 q^{21} + 2456472 q^{25} - 195952 q^{29} - 1065312 q^{33} + 7023408 q^{37} - 12035120 q^{41} + 1224720 q^{45} - 9602040 q^{49} - 27342192 q^{53} + 2744928 q^{57} - 50803280 q^{61} + 34049888 q^{65} + 26687232 q^{69} + 59541648 q^{73} + 178489472 q^{77} + 38263752 q^{81} - 29428704 q^{85} - 170794992 q^{89} + 121951008 q^{93} + 272647184 q^{97}+O(q^{100})$$ 8 * q - 560 * q^5 - 17496 * q^9 + 54256 * q^13 + 436176 * q^17 - 54432 * q^21 + 2456472 * q^25 - 195952 * q^29 - 1065312 * q^33 + 7023408 * q^37 - 12035120 * q^41 + 1224720 * q^45 - 9602040 * q^49 - 27342192 * q^53 + 2744928 * q^57 - 50803280 * q^61 + 34049888 * q^65 + 26687232 * q^69 + 59541648 * q^73 + 178489472 * q^77 + 38263752 * q^81 - 29428704 * q^85 - 170794992 * q^89 + 121951008 * q^93 + 272647184 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 46.7654i 0.577350i
$$4$$ 0 0
$$5$$ −1079.08 −1.72653 −0.863263 0.504754i $$-0.831583\pi$$
−0.863263 + 0.504754i $$0.831583\pi$$
$$6$$ 0 0
$$7$$ − 3980.70i − 1.65793i −0.559298 0.828966i $$-0.688929\pi$$
0.559298 0.828966i $$-0.311071\pi$$
$$8$$ 0 0
$$9$$ −2187.00 −0.333333
$$10$$ 0 0
$$11$$ 20108.0i 1.37340i 0.726940 + 0.686701i $$0.240942\pi$$
−0.726940 + 0.686701i $$0.759058\pi$$
$$12$$ 0 0
$$13$$ 36104.9 1.26413 0.632067 0.774914i $$-0.282207\pi$$
0.632067 + 0.774914i $$0.282207\pi$$
$$14$$ 0 0
$$15$$ − 50463.5i − 0.996810i
$$16$$ 0 0
$$17$$ 59944.2 0.717715 0.358857 0.933392i $$-0.383166\pi$$
0.358857 + 0.933392i $$0.383166\pi$$
$$18$$ 0 0
$$19$$ − 126309.i − 0.969217i −0.874731 0.484608i $$-0.838962\pi$$
0.874731 0.484608i $$-0.161038\pi$$
$$20$$ 0 0
$$21$$ 186159. 0.957208
$$22$$ 0 0
$$23$$ − 197109.i − 0.704359i −0.935932 0.352179i $$-0.885441\pi$$
0.935932 0.352179i $$-0.114559\pi$$
$$24$$ 0 0
$$25$$ 773786. 1.98089
$$26$$ 0 0
$$27$$ − 102276.i − 0.192450i
$$28$$ 0 0
$$29$$ −485230. −0.686050 −0.343025 0.939326i $$-0.611452\pi$$
−0.343025 + 0.939326i $$0.611452\pi$$
$$30$$ 0 0
$$31$$ 671904.i 0.727546i 0.931488 + 0.363773i $$0.118512\pi$$
−0.931488 + 0.363773i $$0.881488\pi$$
$$32$$ 0 0
$$33$$ −940357. −0.792934
$$34$$ 0 0
$$35$$ 4.29549e6i 2.86246i
$$36$$ 0 0
$$37$$ 3.40344e6 1.81598 0.907991 0.418990i $$-0.137616\pi$$
0.907991 + 0.418990i $$0.137616\pi$$
$$38$$ 0 0
$$39$$ 1.68846e6i 0.729848i
$$40$$ 0 0
$$41$$ −3.55723e6 −1.25886 −0.629429 0.777058i $$-0.716711\pi$$
−0.629429 + 0.777058i $$0.716711\pi$$
$$42$$ 0 0
$$43$$ − 736659.i − 0.215473i −0.994179 0.107736i $$-0.965640\pi$$
0.994179 0.107736i $$-0.0343603\pi$$
$$44$$ 0 0
$$45$$ 2.35995e6 0.575509
$$46$$ 0 0
$$47$$ − 883461.i − 0.181049i −0.995894 0.0905245i $$-0.971146\pi$$
0.995894 0.0905245i $$-0.0288543\pi$$
$$48$$ 0 0
$$49$$ −1.00811e7 −1.74874
$$50$$ 0 0
$$51$$ 2.80331e6i 0.414373i
$$52$$ 0 0
$$53$$ −1.44913e7 −1.83655 −0.918275 0.395943i $$-0.870418\pi$$
−0.918275 + 0.395943i $$0.870418\pi$$
$$54$$ 0 0
$$55$$ − 2.16981e7i − 2.37121i
$$56$$ 0 0
$$57$$ 5.90690e6 0.559578
$$58$$ 0 0
$$59$$ − 1.08326e6i − 0.0893969i −0.999001 0.0446985i $$-0.985767\pi$$
0.999001 0.0446985i $$-0.0142327\pi$$
$$60$$ 0 0
$$61$$ 3.45104e6 0.249248 0.124624 0.992204i $$-0.460228\pi$$
0.124624 + 0.992204i $$0.460228\pi$$
$$62$$ 0 0
$$63$$ 8.70578e6i 0.552644i
$$64$$ 0 0
$$65$$ −3.89601e7 −2.18256
$$66$$ 0 0
$$67$$ 1.43441e7i 0.711828i 0.934519 + 0.355914i $$0.115830\pi$$
−0.934519 + 0.355914i $$0.884170\pi$$
$$68$$ 0 0
$$69$$ 9.21785e6 0.406662
$$70$$ 0 0
$$71$$ − 3.97522e6i − 0.156433i −0.996936 0.0782163i $$-0.975078\pi$$
0.996936 0.0782163i $$-0.0249225\pi$$
$$72$$ 0 0
$$73$$ 3.92004e6 0.138038 0.0690191 0.997615i $$-0.478013\pi$$
0.0690191 + 0.997615i $$0.478013\pi$$
$$74$$ 0 0
$$75$$ 3.61864e7i 1.14367i
$$76$$ 0 0
$$77$$ 8.00437e7 2.27701
$$78$$ 0 0
$$79$$ − 5.94312e7i − 1.52583i −0.646500 0.762914i $$-0.723768\pi$$
0.646500 0.762914i $$-0.276232\pi$$
$$80$$ 0 0
$$81$$ 4.78297e6 0.111111
$$82$$ 0 0
$$83$$ 5.42483e7i 1.14307i 0.820577 + 0.571536i $$0.193652\pi$$
−0.820577 + 0.571536i $$0.806348\pi$$
$$84$$ 0 0
$$85$$ −6.46846e7 −1.23915
$$86$$ 0 0
$$87$$ − 2.26920e7i − 0.396091i
$$88$$ 0 0
$$89$$ −2.43253e7 −0.387702 −0.193851 0.981031i $$-0.562098\pi$$
−0.193851 + 0.981031i $$0.562098\pi$$
$$90$$ 0 0
$$91$$ − 1.43723e8i − 2.09585i
$$92$$ 0 0
$$93$$ −3.14218e7 −0.420049
$$94$$ 0 0
$$95$$ 1.36298e8i 1.67338i
$$96$$ 0 0
$$97$$ −3.83933e7 −0.433679 −0.216839 0.976207i $$-0.569575\pi$$
−0.216839 + 0.976207i $$0.569575\pi$$
$$98$$ 0 0
$$99$$ − 4.39761e7i − 0.457800i
$$100$$ 0 0
$$101$$ −1.49683e8 −1.43842 −0.719212 0.694791i $$-0.755497\pi$$
−0.719212 + 0.694791i $$0.755497\pi$$
$$102$$ 0 0
$$103$$ − 7.61086e7i − 0.676215i −0.941107 0.338108i $$-0.890213\pi$$
0.941107 0.338108i $$-0.109787\pi$$
$$104$$ 0 0
$$105$$ −2.00880e8 −1.65264
$$106$$ 0 0
$$107$$ − 1.33636e8i − 1.01950i −0.860322 0.509751i $$-0.829738\pi$$
0.860322 0.509751i $$-0.170262\pi$$
$$108$$ 0 0
$$109$$ −1.88815e8 −1.33762 −0.668808 0.743436i $$-0.733195\pi$$
−0.668808 + 0.743436i $$0.733195\pi$$
$$110$$ 0 0
$$111$$ 1.59163e8i 1.04846i
$$112$$ 0 0
$$113$$ 1.75442e7 0.107602 0.0538009 0.998552i $$-0.482866\pi$$
0.0538009 + 0.998552i $$0.482866\pi$$
$$114$$ 0 0
$$115$$ 2.12696e8i 1.21609i
$$116$$ 0 0
$$117$$ −7.89615e7 −0.421378
$$118$$ 0 0
$$119$$ − 2.38620e8i − 1.18992i
$$120$$ 0 0
$$121$$ −1.89971e8 −0.886231
$$122$$ 0 0
$$123$$ − 1.66355e8i − 0.726802i
$$124$$ 0 0
$$125$$ −4.13461e8 −1.69354
$$126$$ 0 0
$$127$$ 1.91097e8i 0.734579i 0.930107 + 0.367289i $$0.119714\pi$$
−0.930107 + 0.367289i $$0.880286\pi$$
$$128$$ 0 0
$$129$$ 3.44501e7 0.124403
$$130$$ 0 0
$$131$$ 4.29123e7i 0.145712i 0.997342 + 0.0728562i $$0.0232114\pi$$
−0.997342 + 0.0728562i $$0.976789\pi$$
$$132$$ 0 0
$$133$$ −5.02799e8 −1.60690
$$134$$ 0 0
$$135$$ 1.10364e8i 0.332270i
$$136$$ 0 0
$$137$$ −2.12185e7 −0.0602328 −0.0301164 0.999546i $$-0.509588\pi$$
−0.0301164 + 0.999546i $$0.509588\pi$$
$$138$$ 0 0
$$139$$ 4.01495e8i 1.07552i 0.843096 + 0.537762i $$0.180730\pi$$
−0.843096 + 0.537762i $$0.819270\pi$$
$$140$$ 0 0
$$141$$ 4.13154e7 0.104529
$$142$$ 0 0
$$143$$ 7.25997e8i 1.73616i
$$144$$ 0 0
$$145$$ 5.23601e8 1.18448
$$146$$ 0 0
$$147$$ − 4.71449e8i − 1.00964i
$$148$$ 0 0
$$149$$ 2.84042e8 0.576285 0.288143 0.957588i $$-0.406962\pi$$
0.288143 + 0.957588i $$0.406962\pi$$
$$150$$ 0 0
$$151$$ 8.35682e8i 1.60744i 0.595011 + 0.803718i $$0.297148\pi$$
−0.595011 + 0.803718i $$0.702852\pi$$
$$152$$ 0 0
$$153$$ −1.31098e8 −0.239238
$$154$$ 0 0
$$155$$ − 7.25037e8i − 1.25613i
$$156$$ 0 0
$$157$$ −7.20894e8 −1.18651 −0.593257 0.805013i $$-0.702158\pi$$
−0.593257 + 0.805013i $$0.702158\pi$$
$$158$$ 0 0
$$159$$ − 6.77689e8i − 1.06033i
$$160$$ 0 0
$$161$$ −7.84629e8 −1.16778
$$162$$ 0 0
$$163$$ 5.61685e8i 0.795687i 0.917453 + 0.397843i $$0.130241\pi$$
−0.917453 + 0.397843i $$0.869759\pi$$
$$164$$ 0 0
$$165$$ 1.01472e9 1.36902
$$166$$ 0 0
$$167$$ − 7.29640e7i − 0.0938086i −0.998899 0.0469043i $$-0.985064\pi$$
0.998899 0.0469043i $$-0.0149356\pi$$
$$168$$ 0 0
$$169$$ 4.87835e8 0.598034
$$170$$ 0 0
$$171$$ 2.76238e8i 0.323072i
$$172$$ 0 0
$$173$$ 6.80934e8 0.760187 0.380093 0.924948i $$-0.375892\pi$$
0.380093 + 0.924948i $$0.375892\pi$$
$$174$$ 0 0
$$175$$ − 3.08021e9i − 3.28419i
$$176$$ 0 0
$$177$$ 5.06588e7 0.0516134
$$178$$ 0 0
$$179$$ − 4.03178e8i − 0.392721i −0.980532 0.196361i $$-0.937088\pi$$
0.980532 0.196361i $$-0.0629123\pi$$
$$180$$ 0 0
$$181$$ −1.19998e9 −1.11805 −0.559023 0.829152i $$-0.688823\pi$$
−0.559023 + 0.829152i $$0.688823\pi$$
$$182$$ 0 0
$$183$$ 1.61389e8i 0.143903i
$$184$$ 0 0
$$185$$ −3.67258e9 −3.13534
$$186$$ 0 0
$$187$$ 1.20536e9i 0.985710i
$$188$$ 0 0
$$189$$ −4.07129e8 −0.319069
$$190$$ 0 0
$$191$$ 4.09810e8i 0.307928i 0.988076 + 0.153964i $$0.0492040\pi$$
−0.988076 + 0.153964i $$0.950796\pi$$
$$192$$ 0 0
$$193$$ 8.43979e7 0.0608278 0.0304139 0.999537i $$-0.490317\pi$$
0.0304139 + 0.999537i $$0.490317\pi$$
$$194$$ 0 0
$$195$$ − 1.82198e9i − 1.26010i
$$196$$ 0 0
$$197$$ −1.18786e9 −0.788677 −0.394339 0.918965i $$-0.629026\pi$$
−0.394339 + 0.918965i $$0.629026\pi$$
$$198$$ 0 0
$$199$$ − 4.47256e8i − 0.285196i −0.989781 0.142598i $$-0.954454\pi$$
0.989781 0.142598i $$-0.0455457\pi$$
$$200$$ 0 0
$$201$$ −6.70809e8 −0.410974
$$202$$ 0 0
$$203$$ 1.93155e9i 1.13742i
$$204$$ 0 0
$$205$$ 3.83853e9 2.17345
$$206$$ 0 0
$$207$$ 4.31076e8i 0.234786i
$$208$$ 0 0
$$209$$ 2.53982e9 1.33112
$$210$$ 0 0
$$211$$ 6.71991e8i 0.339026i 0.985528 + 0.169513i $$0.0542195\pi$$
−0.985528 + 0.169513i $$0.945780\pi$$
$$212$$ 0 0
$$213$$ 1.85903e8 0.0903164
$$214$$ 0 0
$$215$$ 7.94913e8i 0.372019i
$$216$$ 0 0
$$217$$ 2.67465e9 1.20622
$$218$$ 0 0
$$219$$ 1.83322e8i 0.0796963i
$$220$$ 0 0
$$221$$ 2.16428e9 0.907287
$$222$$ 0 0
$$223$$ − 3.40869e9i − 1.37838i −0.724582 0.689188i $$-0.757967\pi$$
0.724582 0.689188i $$-0.242033\pi$$
$$224$$ 0 0
$$225$$ −1.69227e9 −0.660297
$$226$$ 0 0
$$227$$ 4.05729e9i 1.52803i 0.645198 + 0.764016i $$0.276775\pi$$
−0.645198 + 0.764016i $$0.723225\pi$$
$$228$$ 0 0
$$229$$ −3.28401e9 −1.19416 −0.597080 0.802182i $$-0.703673\pi$$
−0.597080 + 0.802182i $$0.703673\pi$$
$$230$$ 0 0
$$231$$ 3.74327e9i 1.31463i
$$232$$ 0 0
$$233$$ 4.91934e9 1.66910 0.834552 0.550929i $$-0.185727\pi$$
0.834552 + 0.550929i $$0.185727\pi$$
$$234$$ 0 0
$$235$$ 9.53325e8i 0.312586i
$$236$$ 0 0
$$237$$ 2.77932e9 0.880938
$$238$$ 0 0
$$239$$ 1.11979e9i 0.343199i 0.985167 + 0.171599i $$0.0548935\pi$$
−0.985167 + 0.171599i $$0.945107\pi$$
$$240$$ 0 0
$$241$$ −3.54686e9 −1.05142 −0.525710 0.850664i $$-0.676200\pi$$
−0.525710 + 0.850664i $$0.676200\pi$$
$$242$$ 0 0
$$243$$ 2.23677e8i 0.0641500i
$$244$$ 0 0
$$245$$ 1.08784e10 3.01925
$$246$$ 0 0
$$247$$ − 4.56039e9i − 1.22522i
$$248$$ 0 0
$$249$$ −2.53694e9 −0.659953
$$250$$ 0 0
$$251$$ − 5.42783e9i − 1.36751i −0.729710 0.683756i $$-0.760345\pi$$
0.729710 0.683756i $$-0.239655\pi$$
$$252$$ 0 0
$$253$$ 3.96345e9 0.967367
$$254$$ 0 0
$$255$$ − 3.02500e9i − 0.715425i
$$256$$ 0 0
$$257$$ 5.11925e9 1.17348 0.586738 0.809777i $$-0.300412\pi$$
0.586738 + 0.809777i $$0.300412\pi$$
$$258$$ 0 0
$$259$$ − 1.35481e10i − 3.01078i
$$260$$ 0 0
$$261$$ 1.06120e9 0.228683
$$262$$ 0 0
$$263$$ 7.63972e9i 1.59681i 0.602118 + 0.798407i $$0.294324\pi$$
−0.602118 + 0.798407i $$0.705676\pi$$
$$264$$ 0 0
$$265$$ 1.56372e10 3.17085
$$266$$ 0 0
$$267$$ − 1.13758e9i − 0.223840i
$$268$$ 0 0
$$269$$ −5.74735e9 −1.09764 −0.548818 0.835942i $$-0.684922\pi$$
−0.548818 + 0.835942i $$0.684922\pi$$
$$270$$ 0 0
$$271$$ 4.27034e9i 0.791746i 0.918305 + 0.395873i $$0.129558\pi$$
−0.918305 + 0.395873i $$0.870442\pi$$
$$272$$ 0 0
$$273$$ 6.72125e9 1.21004
$$274$$ 0 0
$$275$$ 1.55593e10i 2.72056i
$$276$$ 0 0
$$277$$ −9.64494e9 −1.63825 −0.819125 0.573614i $$-0.805541\pi$$
−0.819125 + 0.573614i $$0.805541\pi$$
$$278$$ 0 0
$$279$$ − 1.46945e9i − 0.242515i
$$280$$ 0 0
$$281$$ −2.63511e9 −0.422643 −0.211322 0.977417i $$-0.567777\pi$$
−0.211322 + 0.977417i $$0.567777\pi$$
$$282$$ 0 0
$$283$$ − 3.18282e8i − 0.0496212i −0.999692 0.0248106i $$-0.992102\pi$$
0.999692 0.0248106i $$-0.00789826\pi$$
$$284$$ 0 0
$$285$$ −6.37401e9 −0.966125
$$286$$ 0 0
$$287$$ 1.41603e10i 2.08710i
$$288$$ 0 0
$$289$$ −3.38245e9 −0.484886
$$290$$ 0 0
$$291$$ − 1.79547e9i − 0.250384i
$$292$$ 0 0
$$293$$ 1.15368e9 0.156536 0.0782682 0.996932i $$-0.475061\pi$$
0.0782682 + 0.996932i $$0.475061\pi$$
$$294$$ 0 0
$$295$$ 1.16892e9i 0.154346i
$$296$$ 0 0
$$297$$ 2.05656e9 0.264311
$$298$$ 0 0
$$299$$ − 7.11659e9i − 0.890404i
$$300$$ 0 0
$$301$$ −2.93242e9 −0.357239
$$302$$ 0 0
$$303$$ − 6.99998e9i − 0.830474i
$$304$$ 0 0
$$305$$ −3.72395e9 −0.430332
$$306$$ 0 0
$$307$$ 1.55195e10i 1.74712i 0.486715 + 0.873561i $$0.338195\pi$$
−0.486715 + 0.873561i $$0.661805\pi$$
$$308$$ 0 0
$$309$$ 3.55925e9 0.390413
$$310$$ 0 0
$$311$$ 3.12251e9i 0.333782i 0.985975 + 0.166891i $$0.0533728\pi$$
−0.985975 + 0.166891i $$0.946627\pi$$
$$312$$ 0 0
$$313$$ −9.18882e9 −0.957375 −0.478688 0.877985i $$-0.658887\pi$$
−0.478688 + 0.877985i $$0.658887\pi$$
$$314$$ 0 0
$$315$$ − 9.39423e9i − 0.954155i
$$316$$ 0 0
$$317$$ −1.03286e10 −1.02283 −0.511417 0.859333i $$-0.670879\pi$$
−0.511417 + 0.859333i $$0.670879\pi$$
$$318$$ 0 0
$$319$$ − 9.75699e9i − 0.942222i
$$320$$ 0 0
$$321$$ 6.24953e9 0.588609
$$322$$ 0 0
$$323$$ − 7.57152e9i − 0.695621i
$$324$$ 0 0
$$325$$ 2.79375e10 2.50411
$$326$$ 0 0
$$327$$ − 8.83002e9i − 0.772272i
$$328$$ 0 0
$$329$$ −3.51679e9 −0.300167
$$330$$ 0 0
$$331$$ − 2.10055e10i − 1.74993i −0.484184 0.874966i $$-0.660883\pi$$
0.484184 0.874966i $$-0.339117\pi$$
$$332$$ 0 0
$$333$$ −7.44333e9 −0.605327
$$334$$ 0 0
$$335$$ − 1.54784e10i − 1.22899i
$$336$$ 0 0
$$337$$ 6.33904e9 0.491478 0.245739 0.969336i $$-0.420969\pi$$
0.245739 + 0.969336i $$0.420969\pi$$
$$338$$ 0 0
$$339$$ 8.20461e8i 0.0621239i
$$340$$ 0 0
$$341$$ −1.35106e10 −0.999212
$$342$$ 0 0
$$343$$ 1.71821e10i 1.24136i
$$344$$ 0 0
$$345$$ −9.94679e9 −0.702112
$$346$$ 0 0
$$347$$ 9.29606e9i 0.641181i 0.947218 + 0.320591i $$0.103881\pi$$
−0.947218 + 0.320591i $$0.896119\pi$$
$$348$$ 0 0
$$349$$ 2.11040e10 1.42254 0.711269 0.702920i $$-0.248121\pi$$
0.711269 + 0.702920i $$0.248121\pi$$
$$350$$ 0 0
$$351$$ − 3.69266e9i − 0.243283i
$$352$$ 0 0
$$353$$ 1.30618e10 0.841209 0.420604 0.907244i $$-0.361818\pi$$
0.420604 + 0.907244i $$0.361818\pi$$
$$354$$ 0 0
$$355$$ 4.28957e9i 0.270085i
$$356$$ 0 0
$$357$$ 1.11591e10 0.687002
$$358$$ 0 0
$$359$$ − 1.18146e10i − 0.711279i −0.934623 0.355639i $$-0.884263\pi$$
0.934623 0.355639i $$-0.115737\pi$$
$$360$$ 0 0
$$361$$ 1.02952e9 0.0606185
$$362$$ 0 0
$$363$$ − 8.88409e9i − 0.511666i
$$364$$ 0 0
$$365$$ −4.23003e9 −0.238326
$$366$$ 0 0
$$367$$ 2.69814e10i 1.48730i 0.668567 + 0.743652i $$0.266908\pi$$
−0.668567 + 0.743652i $$0.733092\pi$$
$$368$$ 0 0
$$369$$ 7.77967e9 0.419620
$$370$$ 0 0
$$371$$ 5.76853e10i 3.04488i
$$372$$ 0 0
$$373$$ −2.07520e10 −1.07208 −0.536038 0.844194i $$-0.680080\pi$$
−0.536038 + 0.844194i $$0.680080\pi$$
$$374$$ 0 0
$$375$$ − 1.93356e10i − 0.977763i
$$376$$ 0 0
$$377$$ −1.75192e10 −0.867259
$$378$$ 0 0
$$379$$ − 3.91651e10i − 1.89820i −0.314973 0.949101i $$-0.601995\pi$$
0.314973 0.949101i $$-0.398005\pi$$
$$380$$ 0 0
$$381$$ −8.93671e9 −0.424109
$$382$$ 0 0
$$383$$ 1.04984e8i 0.00487897i 0.999997 + 0.00243949i $$0.000776513\pi$$
−0.999997 + 0.00243949i $$0.999223\pi$$
$$384$$ 0 0
$$385$$ −8.63735e10 −3.93131
$$386$$ 0 0
$$387$$ 1.61107e9i 0.0718243i
$$388$$ 0 0
$$389$$ 9.73662e9 0.425216 0.212608 0.977138i $$-0.431804\pi$$
0.212608 + 0.977138i $$0.431804\pi$$
$$390$$ 0 0
$$391$$ − 1.18155e10i − 0.505529i
$$392$$ 0 0
$$393$$ −2.00681e9 −0.0841271
$$394$$ 0 0
$$395$$ 6.41309e10i 2.63438i
$$396$$ 0 0
$$397$$ −2.07590e9 −0.0835687 −0.0417843 0.999127i $$-0.513304\pi$$
−0.0417843 + 0.999127i $$0.513304\pi$$
$$398$$ 0 0
$$399$$ − 2.35136e10i − 0.927742i
$$400$$ 0 0
$$401$$ −2.62141e10 −1.01381 −0.506905 0.862002i $$-0.669211\pi$$
−0.506905 + 0.862002i $$0.669211\pi$$
$$402$$ 0 0
$$403$$ 2.42590e10i 0.919715i
$$404$$ 0 0
$$405$$ −5.16120e9 −0.191836
$$406$$ 0 0
$$407$$ 6.84363e10i 2.49407i
$$408$$ 0 0
$$409$$ −1.51824e10 −0.542559 −0.271280 0.962501i $$-0.587447\pi$$
−0.271280 + 0.962501i $$0.587447\pi$$
$$410$$ 0 0
$$411$$ − 9.92293e8i − 0.0347754i
$$412$$ 0 0
$$413$$ −4.31211e9 −0.148214
$$414$$ 0 0
$$415$$ − 5.85382e10i − 1.97354i
$$416$$ 0 0
$$417$$ −1.87760e10 −0.620955
$$418$$ 0 0
$$419$$ − 2.04890e10i − 0.664760i −0.943145 0.332380i $$-0.892148\pi$$
0.943145 0.332380i $$-0.107852\pi$$
$$420$$ 0 0
$$421$$ −5.40434e10 −1.72034 −0.860170 0.510007i $$-0.829643\pi$$
−0.860170 + 0.510007i $$0.829643\pi$$
$$422$$ 0 0
$$423$$ 1.93213e9i 0.0603497i
$$424$$ 0 0
$$425$$ 4.63840e10 1.42172
$$426$$ 0 0
$$427$$ − 1.37376e10i − 0.413236i
$$428$$ 0 0
$$429$$ −3.39515e10 −1.00237
$$430$$ 0 0
$$431$$ − 4.04198e10i − 1.17134i −0.810548 0.585672i $$-0.800831\pi$$
0.810548 0.585672i $$-0.199169\pi$$
$$432$$ 0 0
$$433$$ −4.32070e9 −0.122914 −0.0614572 0.998110i $$-0.519575\pi$$
−0.0614572 + 0.998110i $$0.519575\pi$$
$$434$$ 0 0
$$435$$ 2.44864e10i 0.683861i
$$436$$ 0 0
$$437$$ −2.48966e10 −0.682677
$$438$$ 0 0
$$439$$ 3.97062e10i 1.06905i 0.845151 + 0.534527i $$0.179510\pi$$
−0.845151 + 0.534527i $$0.820490\pi$$
$$440$$ 0 0
$$441$$ 2.20475e10 0.582914
$$442$$ 0 0
$$443$$ − 1.97115e10i − 0.511804i −0.966703 0.255902i $$-0.917628\pi$$
0.966703 0.255902i $$-0.0823725\pi$$
$$444$$ 0 0
$$445$$ 2.62489e10 0.669378
$$446$$ 0 0
$$447$$ 1.32833e10i 0.332718i
$$448$$ 0 0
$$449$$ −1.89134e10 −0.465355 −0.232677 0.972554i $$-0.574749\pi$$
−0.232677 + 0.972554i $$0.574749\pi$$
$$450$$ 0 0
$$451$$ − 7.15287e10i − 1.72892i
$$452$$ 0 0
$$453$$ −3.90810e10 −0.928053
$$454$$ 0 0
$$455$$ 1.55088e11i 3.61854i
$$456$$ 0 0
$$457$$ −5.24800e10 −1.20318 −0.601588 0.798806i $$-0.705465\pi$$
−0.601588 + 0.798806i $$0.705465\pi$$
$$458$$ 0 0
$$459$$ − 6.13085e9i − 0.138124i
$$460$$ 0 0
$$461$$ 4.67940e9 0.103606 0.0518032 0.998657i $$-0.483503\pi$$
0.0518032 + 0.998657i $$0.483503\pi$$
$$462$$ 0 0
$$463$$ − 8.06631e10i − 1.75530i −0.479304 0.877649i $$-0.659111\pi$$
0.479304 0.877649i $$-0.340889\pi$$
$$464$$ 0 0
$$465$$ 3.39066e10 0.725225
$$466$$ 0 0
$$467$$ − 2.84821e10i − 0.598832i −0.954123 0.299416i $$-0.903208\pi$$
0.954123 0.299416i $$-0.0967918\pi$$
$$468$$ 0 0
$$469$$ 5.70996e10 1.18016
$$470$$ 0 0
$$471$$ − 3.37129e10i − 0.685034i
$$472$$ 0 0
$$473$$ 1.48127e10 0.295931
$$474$$ 0 0
$$475$$ − 9.77364e10i − 1.91991i
$$476$$ 0 0
$$477$$ 3.16924e10 0.612183
$$478$$ 0 0
$$479$$ − 3.38937e10i − 0.643838i −0.946767 0.321919i $$-0.895672\pi$$
0.946767 0.321919i $$-0.104328\pi$$
$$480$$ 0 0
$$481$$ 1.22881e11 2.29564
$$482$$ 0 0
$$483$$ − 3.66935e10i − 0.674218i
$$484$$ 0 0
$$485$$ 4.14293e10 0.748757
$$486$$ 0 0
$$487$$ 1.38440e10i 0.246120i 0.992399 + 0.123060i $$0.0392708\pi$$
−0.992399 + 0.123060i $$0.960729\pi$$
$$488$$ 0 0
$$489$$ −2.62674e10 −0.459390
$$490$$ 0 0
$$491$$ 8.71026e10i 1.49867i 0.662193 + 0.749333i $$0.269626\pi$$
−0.662193 + 0.749333i $$0.730374\pi$$
$$492$$ 0 0
$$493$$ −2.90867e10 −0.492388
$$494$$ 0 0
$$495$$ 4.74537e10i 0.790404i
$$496$$ 0 0
$$497$$ −1.58241e10 −0.259355
$$498$$ 0 0
$$499$$ 5.51402e10i 0.889337i 0.895695 + 0.444668i $$0.146678\pi$$
−0.895695 + 0.444668i $$0.853322\pi$$
$$500$$ 0 0
$$501$$ 3.41219e9 0.0541604
$$502$$ 0 0
$$503$$ − 5.65908e10i − 0.884045i −0.897004 0.442022i $$-0.854261\pi$$
0.897004 0.442022i $$-0.145739\pi$$
$$504$$ 0 0
$$505$$ 1.61520e11 2.48348
$$506$$ 0 0
$$507$$ 2.28138e10i 0.345275i
$$508$$ 0 0
$$509$$ −8.63292e10 −1.28614 −0.643068 0.765809i $$-0.722339\pi$$
−0.643068 + 0.765809i $$0.722339\pi$$
$$510$$ 0 0
$$511$$ − 1.56045e10i − 0.228858i
$$512$$ 0 0
$$513$$ −1.29184e10 −0.186526
$$514$$ 0 0
$$515$$ 8.21272e10i 1.16750i
$$516$$ 0 0
$$517$$ 1.77646e10 0.248653
$$518$$ 0 0
$$519$$ 3.18441e10i 0.438894i
$$520$$ 0 0
$$521$$ −3.77681e10 −0.512595 −0.256298 0.966598i $$-0.582503\pi$$
−0.256298 + 0.966598i $$0.582503\pi$$
$$522$$ 0 0
$$523$$ − 5.86646e10i − 0.784096i −0.919945 0.392048i $$-0.871767\pi$$
0.919945 0.392048i $$-0.128233\pi$$
$$524$$ 0 0
$$525$$ 1.44047e11 1.89613
$$526$$ 0 0
$$527$$ 4.02768e10i 0.522170i
$$528$$ 0 0
$$529$$ 3.94592e10 0.503878
$$530$$ 0 0
$$531$$ 2.36908e9i 0.0297990i
$$532$$ 0 0
$$533$$ −1.28434e11 −1.59137
$$534$$ 0 0
$$535$$ 1.44204e11i 1.76020i
$$536$$ 0 0
$$537$$ 1.88548e10 0.226738
$$538$$ 0 0
$$539$$ − 2.02711e11i − 2.40172i
$$540$$ 0 0
$$541$$ −7.10071e10 −0.828920 −0.414460 0.910067i $$-0.636030\pi$$
−0.414460 + 0.910067i $$0.636030\pi$$
$$542$$ 0 0
$$543$$ − 5.61175e10i − 0.645504i
$$544$$ 0 0
$$545$$ 2.03747e11 2.30943
$$546$$ 0 0
$$547$$ 1.23143e11i 1.37550i 0.725947 + 0.687750i $$0.241402\pi$$
−0.725947 + 0.687750i $$0.758598\pi$$
$$548$$ 0 0
$$549$$ −7.54743e9 −0.0830825
$$550$$ 0 0
$$551$$ 6.12891e10i 0.664931i
$$552$$ 0 0
$$553$$ −2.36577e11 −2.52972
$$554$$ 0 0
$$555$$ − 1.71750e11i − 1.81019i
$$556$$ 0 0
$$557$$ 1.47407e11 1.53143 0.765717 0.643177i $$-0.222384\pi$$
0.765717 + 0.643177i $$0.222384\pi$$
$$558$$ 0 0
$$559$$ − 2.65970e10i − 0.272386i
$$560$$ 0 0
$$561$$ −5.63690e10 −0.569100
$$562$$ 0 0
$$563$$ 1.33296e11i 1.32673i 0.748296 + 0.663365i $$0.230872\pi$$
−0.748296 + 0.663365i $$0.769128\pi$$
$$564$$ 0 0
$$565$$ −1.89316e10 −0.185777
$$566$$ 0 0
$$567$$ − 1.90395e10i − 0.184215i
$$568$$ 0 0
$$569$$ −1.29497e11 −1.23541 −0.617703 0.786412i $$-0.711937\pi$$
−0.617703 + 0.786412i $$0.711937\pi$$
$$570$$ 0 0
$$571$$ 1.08901e11i 1.02444i 0.858854 + 0.512221i $$0.171177\pi$$
−0.858854 + 0.512221i $$0.828823\pi$$
$$572$$ 0 0
$$573$$ −1.91649e10 −0.177782
$$574$$ 0 0
$$575$$ − 1.52520e11i − 1.39526i
$$576$$ 0 0
$$577$$ 1.23772e11 1.11666 0.558330 0.829619i $$-0.311442\pi$$
0.558330 + 0.829619i $$0.311442\pi$$
$$578$$ 0 0
$$579$$ 3.94690e9i 0.0351190i
$$580$$ 0 0
$$581$$ 2.15946e11 1.89514
$$582$$ 0 0
$$583$$ − 2.91390e11i − 2.52232i
$$584$$ 0 0
$$585$$ 8.52056e10 0.727520
$$586$$ 0 0
$$587$$ − 9.78754e10i − 0.824368i −0.911101 0.412184i $$-0.864766\pi$$
0.911101 0.412184i $$-0.135234\pi$$
$$588$$ 0 0
$$589$$ 8.48677e10 0.705150
$$590$$ 0 0
$$591$$ − 5.55506e10i − 0.455343i
$$592$$ 0 0
$$593$$ −6.13187e10 −0.495877 −0.247938 0.968776i $$-0.579753\pi$$
−0.247938 + 0.968776i $$0.579753\pi$$
$$594$$ 0 0
$$595$$ 2.57490e11i 2.05443i
$$596$$ 0 0
$$597$$ 2.09161e10 0.164658
$$598$$ 0 0
$$599$$ 4.28422e10i 0.332786i 0.986060 + 0.166393i $$0.0532120\pi$$
−0.986060 + 0.166393i $$0.946788\pi$$
$$600$$ 0 0
$$601$$ 9.04002e10 0.692901 0.346451 0.938068i $$-0.387387\pi$$
0.346451 + 0.938068i $$0.387387\pi$$
$$602$$ 0 0
$$603$$ − 3.13706e10i − 0.237276i
$$604$$ 0 0
$$605$$ 2.04994e11 1.53010
$$606$$ 0 0
$$607$$ 1.05084e11i 0.774069i 0.922065 + 0.387035i $$0.126501\pi$$
−0.922065 + 0.387035i $$0.873499\pi$$
$$608$$ 0 0
$$609$$ −9.03298e10 −0.656692
$$610$$ 0 0
$$611$$ − 3.18973e10i − 0.228870i
$$612$$ 0 0
$$613$$ 2.50049e11 1.77085 0.885427 0.464778i $$-0.153866\pi$$
0.885427 + 0.464778i $$0.153866\pi$$
$$614$$ 0 0
$$615$$ 1.79511e11i 1.25484i
$$616$$ 0 0
$$617$$ 5.63479e10 0.388810 0.194405 0.980921i $$-0.437722\pi$$
0.194405 + 0.980921i $$0.437722\pi$$
$$618$$ 0 0
$$619$$ 2.37696e11i 1.61904i 0.587090 + 0.809522i $$0.300274\pi$$
−0.587090 + 0.809522i $$0.699726\pi$$
$$620$$ 0 0
$$621$$ −2.01594e10 −0.135554
$$622$$ 0 0
$$623$$ 9.68316e10i 0.642784i
$$624$$ 0 0
$$625$$ 1.43897e11 0.943041
$$626$$ 0 0
$$627$$ 1.18776e11i 0.768525i
$$628$$ 0 0
$$629$$ 2.04017e11 1.30336
$$630$$ 0 0
$$631$$ 3.96989e10i 0.250416i 0.992131 + 0.125208i $$0.0399598\pi$$
−0.992131 + 0.125208i $$0.960040\pi$$
$$632$$ 0 0
$$633$$ −3.14259e10 −0.195737
$$634$$ 0 0
$$635$$ − 2.06208e11i − 1.26827i
$$636$$ 0 0
$$637$$ −3.63979e11 −2.21064
$$638$$ 0 0
$$639$$ 8.69380e9i 0.0521442i
$$640$$ 0 0
$$641$$ −2.16918e11 −1.28488 −0.642442 0.766334i $$-0.722079\pi$$
−0.642442 + 0.766334i $$0.722079\pi$$
$$642$$ 0 0
$$643$$ 3.00637e10i 0.175873i 0.996126 + 0.0879365i $$0.0280273\pi$$
−0.996126 + 0.0879365i $$0.971973\pi$$
$$644$$ 0 0
$$645$$ −3.71744e10 −0.214786
$$646$$ 0 0
$$647$$ − 2.17396e11i − 1.24061i −0.784361 0.620305i $$-0.787009\pi$$
0.784361 0.620305i $$-0.212991\pi$$
$$648$$ 0 0
$$649$$ 2.17821e10 0.122778
$$650$$ 0 0
$$651$$ 1.25081e11i 0.696413i
$$652$$ 0 0
$$653$$ 9.82145e10 0.540161 0.270080 0.962838i $$-0.412950\pi$$
0.270080 + 0.962838i $$0.412950\pi$$
$$654$$ 0 0
$$655$$ − 4.63058e10i − 0.251576i
$$656$$ 0 0
$$657$$ −8.57313e9 −0.0460127
$$658$$ 0 0
$$659$$ − 8.83128e10i − 0.468255i −0.972206 0.234127i $$-0.924777\pi$$
0.972206 0.234127i $$-0.0752233\pi$$
$$660$$ 0 0
$$661$$ −9.04522e10 −0.473820 −0.236910 0.971532i $$-0.576135\pi$$
−0.236910 + 0.971532i $$0.576135\pi$$
$$662$$ 0 0
$$663$$ 1.01213e11i 0.523823i
$$664$$ 0 0
$$665$$ 5.42560e11 2.77435
$$666$$ 0 0
$$667$$ 9.56429e10i 0.483225i
$$668$$ 0 0
$$669$$ 1.59409e11 0.795806
$$670$$ 0 0
$$671$$ 6.93934e10i 0.342317i
$$672$$ 0 0
$$673$$ 2.22996e11 1.08702 0.543510 0.839403i $$-0.317095\pi$$
0.543510 + 0.839403i $$0.317095\pi$$
$$674$$ 0 0
$$675$$ − 7.91396e10i − 0.381223i
$$676$$ 0 0
$$677$$ −3.19359e11 −1.52028 −0.760142 0.649757i $$-0.774871\pi$$
−0.760142 + 0.649757i $$0.774871\pi$$
$$678$$ 0 0
$$679$$ 1.52832e11i 0.719010i
$$680$$ 0 0
$$681$$ −1.89741e11 −0.882209
$$682$$ 0 0
$$683$$ − 3.69050e11i − 1.69591i −0.530070 0.847954i $$-0.677834\pi$$
0.530070 0.847954i $$-0.322166\pi$$
$$684$$ 0 0
$$685$$ 2.28965e10 0.103994
$$686$$ 0 0
$$687$$ − 1.53578e11i − 0.689449i
$$688$$ 0 0
$$689$$ −5.23206e11 −2.32164
$$690$$ 0 0
$$691$$ − 3.14143e11i − 1.37789i −0.724813 0.688946i $$-0.758074\pi$$
0.724813 0.688946i $$-0.241926\pi$$
$$692$$ 0 0
$$693$$ −1.75056e11 −0.759002
$$694$$ 0 0
$$695$$ − 4.33244e11i − 1.85692i
$$696$$ 0 0
$$697$$ −2.13236e11 −0.903501
$$698$$ 0 0
$$699$$ 2.30055e11i 0.963657i
$$700$$ 0 0
$$701$$ −2.36655e11 −0.980040 −0.490020 0.871711i $$-0.663011\pi$$
−0.490020 + 0.871711i $$0.663011\pi$$
$$702$$ 0 0
$$703$$ − 4.29887e11i − 1.76008i
$$704$$ 0 0
$$705$$ −4.45826e10 −0.180472
$$706$$ 0 0
$$707$$ 5.95843e11i 2.38481i
$$708$$ 0 0
$$709$$ −1.91093e11 −0.756240 −0.378120 0.925757i $$-0.623429\pi$$
−0.378120 + 0.925757i $$0.623429\pi$$
$$710$$ 0 0
$$711$$ 1.29976e11i 0.508610i
$$712$$ 0 0
$$713$$ 1.32438e11 0.512453
$$714$$ 0 0
$$715$$ − 7.83408e11i − 2.99753i
$$716$$ 0 0
$$717$$ −5.23675e10 −0.198146
$$718$$ 0 0
$$719$$ − 1.08030e11i − 0.404230i −0.979362 0.202115i $$-0.935219\pi$$
0.979362 0.202115i $$-0.0647814\pi$$
$$720$$ 0 0
$$721$$ −3.02965e11 −1.12112
$$722$$ 0 0
$$723$$ − 1.65870e11i − 0.607038i
$$724$$ 0 0
$$725$$ −3.75464e11 −1.35899
$$726$$ 0 0
$$727$$ 2.11129e11i 0.755804i 0.925846 + 0.377902i $$0.123354\pi$$
−0.925846 + 0.377902i $$0.876646\pi$$
$$728$$ 0 0
$$729$$ −1.04604e10 −0.0370370
$$730$$ 0 0
$$731$$ − 4.41584e10i − 0.154648i
$$732$$ 0 0
$$733$$ 1.48237e10 0.0513500 0.0256750 0.999670i $$-0.491826\pi$$
0.0256750 + 0.999670i $$0.491826\pi$$
$$734$$ 0 0
$$735$$ 5.08730e11i 1.74316i
$$736$$ 0 0
$$737$$ −2.88431e11 −0.977625
$$738$$ 0 0
$$739$$ 2.87737e10i 0.0964757i 0.998836 + 0.0482378i $$0.0153605\pi$$
−0.998836 + 0.0482378i $$0.984639\pi$$
$$740$$ 0 0
$$741$$ 2.13268e11 0.707381
$$742$$ 0 0
$$743$$ 4.75481e11i 1.56019i 0.625660 + 0.780096i $$0.284830\pi$$
−0.625660 + 0.780096i $$0.715170\pi$$
$$744$$ 0 0
$$745$$ −3.06504e11 −0.994971
$$746$$ 0 0
$$747$$ − 1.18641e11i − 0.381024i
$$748$$ 0 0
$$749$$ −5.31964e11 −1.69026
$$750$$ 0 0
$$751$$ 2.15430e11i 0.677245i 0.940922 + 0.338623i $$0.109961\pi$$
−0.940922 + 0.338623i $$0.890039\pi$$
$$752$$ 0 0
$$753$$ 2.53835e11 0.789534
$$754$$ 0 0
$$755$$ − 9.01767e11i − 2.77528i
$$756$$ 0 0
$$757$$ −3.87925e11 −1.18131 −0.590656 0.806924i $$-0.701131\pi$$
−0.590656 + 0.806924i $$0.701131\pi$$
$$758$$ 0 0
$$759$$ 1.85352e11i 0.558510i
$$760$$ 0 0
$$761$$ −1.81201e11 −0.540284 −0.270142 0.962821i $$-0.587071\pi$$
−0.270142 + 0.962821i $$0.587071\pi$$
$$762$$ 0 0
$$763$$ 7.51616e11i 2.21768i
$$764$$ 0 0
$$765$$ 1.41465e11 0.413051
$$766$$ 0 0
$$767$$ − 3.91108e10i − 0.113010i
$$768$$ 0 0
$$769$$ 1.65080e10 0.0472052 0.0236026 0.999721i $$-0.492486\pi$$
0.0236026 + 0.999721i $$0.492486\pi$$
$$770$$ 0 0
$$771$$ 2.39404e11i 0.677506i
$$772$$ 0 0
$$773$$ −1.35215e11 −0.378710 −0.189355 0.981909i $$-0.560640\pi$$
−0.189355 + 0.981909i $$0.560640\pi$$
$$774$$ 0 0
$$775$$ 5.19910e11i 1.44119i
$$776$$ 0 0
$$777$$ 6.33581e11 1.73827
$$778$$ 0 0
$$779$$ 4.49312e11i 1.22011i
$$780$$ 0 0
$$781$$ 7.99335e10 0.214845
$$782$$ 0 0
$$783$$ 4.96273e10i 0.132030i
$$784$$ 0 0
$$785$$ 7.77901e11 2.04855
$$786$$ 0 0
$$787$$ − 3.83876e11i − 1.00067i −0.865831 0.500337i $$-0.833209\pi$$
0.865831 0.500337i $$-0.166791\pi$$
$$788$$ 0 0
$$789$$ −3.57274e11 −0.921921
$$790$$ 0 0
$$791$$ − 6.98381e10i − 0.178397i
$$792$$ 0 0
$$793$$ 1.24600e11 0.315082
$$794$$ 0 0
$$795$$ 7.31280e11i 1.83069i
$$796$$ 0 0
$$797$$ −2.54150e10 −0.0629878 −0.0314939 0.999504i $$-0.510026\pi$$
−0.0314939 + 0.999504i $$0.510026\pi$$
$$798$$ 0 0
$$799$$ − 5.29584e10i − 0.129942i
$$800$$ 0 0
$$801$$ 5.31994e10 0.129234
$$802$$ 0 0
$$803$$ 7.88240e10i 0.189582i
$$804$$ 0 0
$$805$$ 8.46677e11 2.01620
$$806$$ 0 0
$$807$$ − 2.68777e11i − 0.633720i
$$808$$ 0 0
$$809$$ −1.93064e11 −0.450721 −0.225361 0.974275i $$-0.572356\pi$$
−0.225361 + 0.974275i $$0.572356\pi$$
$$810$$ 0 0
$$811$$ 1.36589e11i 0.315741i 0.987460 + 0.157871i $$0.0504629\pi$$
−0.987460 + 0.157871i $$0.949537\pi$$
$$812$$ 0 0
$$813$$ −1.99704e11 −0.457115
$$814$$ 0 0
$$815$$ − 6.06102e11i − 1.37377i
$$816$$ 0 0
$$817$$ −9.30469e10 −0.208840
$$818$$ 0 0
$$819$$ 3.14322e11i 0.698616i
$$820$$ 0 0
$$821$$ 6.34575e11 1.39672 0.698362 0.715745i $$-0.253913\pi$$
0.698362 + 0.715745i $$0.253913\pi$$
$$822$$ 0 0
$$823$$ − 6.86712e10i − 0.149684i −0.997195 0.0748419i $$-0.976155\pi$$
0.997195 0.0748419i $$-0.0238452\pi$$
$$824$$ 0 0
$$825$$ −7.27635e11 −1.57072
$$826$$ 0 0
$$827$$ − 9.21952e11i − 1.97100i −0.169679 0.985499i $$-0.554273\pi$$
0.169679 0.985499i $$-0.445727\pi$$
$$828$$ 0 0
$$829$$ −7.03277e10 −0.148905 −0.0744523 0.997225i $$-0.523721\pi$$
−0.0744523 + 0.997225i $$0.523721\pi$$
$$830$$ 0 0
$$831$$ − 4.51049e11i − 0.945845i
$$832$$ 0 0
$$833$$ −6.04307e11 −1.25510
$$834$$ 0 0
$$835$$ 7.87339e10i 0.161963i
$$836$$ 0 0
$$837$$ 6.87195e10 0.140016
$$838$$ 0 0
$$839$$ − 8.07774e11i − 1.63020i −0.579317 0.815102i $$-0.696681\pi$$
0.579317 0.815102i $$-0.303319\pi$$
$$840$$ 0 0
$$841$$ −2.64798e11 −0.529336
$$842$$ 0 0
$$843$$ − 1.23232e11i − 0.244013i
$$844$$ 0 0
$$845$$ −5.26412e11 −1.03252
$$846$$ 0 0
$$847$$ 7.56219e11i 1.46931i
$$848$$ 0 0
$$849$$ 1.48846e10 0.0286488
$$850$$ 0 0
$$851$$ − 6.70847e11i − 1.27910i
$$852$$ 0 0
$$853$$ 1.50102e11 0.283525 0.141762 0.989901i $$-0.454723\pi$$
0.141762 + 0.989901i $$0.454723\pi$$
$$854$$ 0 0
$$855$$ − 2.98083e11i − 0.557793i
$$856$$ 0 0
$$857$$ −7.35461e11 −1.36344 −0.681720 0.731613i $$-0.738768\pi$$
−0.681720 + 0.731613i $$0.738768\pi$$
$$858$$ 0 0
$$859$$ 5.72116e11i 1.05078i 0.850862 + 0.525389i $$0.176080\pi$$
−0.850862 + 0.525389i $$0.823920\pi$$
$$860$$ 0 0
$$861$$ −6.62210e11 −1.20499
$$862$$ 0 0
$$863$$ − 3.82637e11i − 0.689832i −0.938634 0.344916i $$-0.887907\pi$$
0.938634 0.344916i $$-0.112093\pi$$
$$864$$ 0 0
$$865$$ −7.34781e11 −1.31248
$$866$$ 0 0
$$867$$ − 1.58181e11i − 0.279949i
$$868$$ 0 0
$$869$$ 1.19504e12 2.09558
$$870$$ 0 0
$$871$$ 5.17894e11i 0.899846i
$$872$$ 0 0
$$873$$ 8.39660e10 0.144560
$$874$$ 0 0
$$875$$ 1.64586e12i 2.80777i
$$876$$ 0 0
$$877$$ −1.84222e11 −0.311418 −0.155709 0.987803i $$-0.549766\pi$$
−0.155709 + 0.987803i $$0.549766\pi$$
$$878$$ 0 0
$$879$$ 5.39523e10i 0.0903763i
$$880$$ 0 0
$$881$$ 5.11578e11 0.849196 0.424598 0.905382i $$-0.360415\pi$$
0.424598 + 0.905382i $$0.360415\pi$$
$$882$$ 0 0
$$883$$ 1.58004e11i 0.259912i 0.991520 + 0.129956i $$0.0414836\pi$$
−0.991520 + 0.129956i $$0.958516\pi$$
$$884$$ 0 0
$$885$$ −5.46649e10 −0.0891118
$$886$$ 0 0
$$887$$ − 4.26843e11i − 0.689562i −0.938683 0.344781i $$-0.887953\pi$$
0.938683 0.344781i $$-0.112047\pi$$
$$888$$ 0 0
$$889$$ 7.60698e11 1.21788
$$890$$ 0 0
$$891$$ 9.61758e10i 0.152600i
$$892$$ 0 0
$$893$$ −1.11589e11 −0.175476
$$894$$ 0 0
$$895$$ 4.35061e11i 0.678043i
$$896$$ 0 0
$$897$$ 3.32810e11 0.514075
$$898$$ 0 0
$$899$$ − 3.26028e11i − 0.499133i
$$900$$ 0 0
$$901$$ −8.68668e11 −1.31812
$$902$$ 0 0
$$903$$ − 1.37135e11i − 0.206252i
$$904$$ 0 0
$$905$$ 1.29487e12 1.93033
$$906$$ 0 0
$$907$$ − 9.37783e11i − 1.38571i −0.721075 0.692857i $$-0.756352\pi$$
0.721075 0.692857i $$-0.243648\pi$$
$$908$$ 0 0
$$909$$ 3.27357e11 0.479475
$$910$$ 0 0
$$911$$ 3.95324e11i 0.573958i 0.957937 + 0.286979i $$0.0926509\pi$$
−0.957937 + 0.286979i $$0.907349\pi$$
$$912$$ 0 0
$$913$$ −1.09082e12 −1.56990
$$914$$ 0 0
$$915$$ − 1.74152e11i − 0.248453i
$$916$$ 0 0
$$917$$ 1.70821e11 0.241582
$$918$$ 0 0
$$919$$ − 1.86250e11i − 0.261116i −0.991441 0.130558i $$-0.958323\pi$$
0.991441 0.130558i $$-0.0416769\pi$$
$$920$$ 0 0
$$921$$ −7.25773e11 −1.00870
$$922$$ 0 0
$$923$$ − 1.43525e11i − 0.197752i
$$924$$ 0 0
$$925$$ 2.63354e12 3.59726
$$926$$ 0 0
$$927$$ 1.66450e11i 0.225405i
$$928$$ 0 0
$$929$$ 4.97016e11 0.667279 0.333639 0.942701i $$-0.391723\pi$$
0.333639 + 0.942701i $$0.391723\pi$$
$$930$$ 0 0
$$931$$ 1.27334e12i 1.69491i
$$932$$ 0 0
$$933$$ −1.46025e11 −0.192709
$$934$$ 0 0
$$935$$ − 1.30067e12i − 1.70185i
$$936$$ 0 0
$$937$$ −9.17550e11 −1.19034 −0.595171 0.803599i $$-0.702915\pi$$
−0.595171 + 0.803599i $$0.702915\pi$$
$$938$$ 0 0
$$939$$ − 4.29718e11i − 0.552741i
$$940$$ 0 0
$$941$$ −1.67021e11 −0.213016 −0.106508 0.994312i $$-0.533967\pi$$
−0.106508 + 0.994312i $$0.533967\pi$$
$$942$$ 0 0
$$943$$ 7.01161e11i 0.886688i
$$944$$ 0 0
$$945$$ 4.39324e11 0.550881
$$946$$ 0 0
$$947$$ − 6.85774e11i − 0.852670i −0.904565 0.426335i $$-0.859805\pi$$
0.904565 0.426335i $$-0.140195\pi$$
$$948$$ 0 0
$$949$$ 1.41533e11 0.174499
$$950$$ 0 0
$$951$$ − 4.83022e11i − 0.590534i
$$952$$ 0 0
$$953$$ −5.38436e11 −0.652774 −0.326387 0.945236i $$-0.605831\pi$$
−0.326387 + 0.945236i $$0.605831\pi$$
$$954$$ 0 0
$$955$$ − 4.42218e11i − 0.531646i
$$956$$ 0 0
$$957$$ 4.56289e11 0.543992
$$958$$ 0 0
$$959$$ 8.44645e10i 0.0998619i
$$960$$ 0 0
$$961$$ 4.01436e11 0.470677
$$962$$ 0 0
$$963$$ 2.92262e11i 0.339834i
$$964$$ 0 0
$$965$$ −9.10719e10 −0.105021
$$966$$ 0 0
$$967$$ − 1.61685e12i − 1.84912i −0.381038 0.924559i $$-0.624433\pi$$
0.381038 0.924559i $$-0.375567\pi$$
$$968$$ 0 0
$$969$$ 3.54085e11 0.401617
$$970$$ 0 0
$$971$$ 1.35680e12i 1.52630i 0.646223 + 0.763148i $$0.276347\pi$$
−0.646223 + 0.763148i $$0.723653\pi$$
$$972$$ 0 0
$$973$$ 1.59823e12 1.78315
$$974$$ 0 0
$$975$$ 1.30651e12i 1.44575i
$$976$$ 0 0
$$977$$ 1.55870e12 1.71074 0.855372 0.518014i $$-0.173328\pi$$
0.855372 + 0.518014i $$0.173328\pi$$
$$978$$ 0 0
$$979$$ − 4.89132e11i − 0.532470i
$$980$$ 0 0
$$981$$ 4.12939e11 0.445872
$$982$$ 0 0
$$983$$ 3.31667e11i 0.355213i 0.984102 + 0.177606i $$0.0568354\pi$$
−0.984102 + 0.177606i $$0.943165\pi$$
$$984$$ 0 0
$$985$$ 1.28179e12 1.36167
$$986$$ 0 0
$$987$$ − 1.64464e11i − 0.173302i
$$988$$ 0 0
$$989$$ −1.45202e11 −0.151770
$$990$$ 0 0
$$991$$ 9.22831e11i 0.956814i 0.878138 + 0.478407i $$0.158786\pi$$
−0.878138 + 0.478407i $$0.841214\pi$$
$$992$$ 0 0
$$993$$ 9.82331e11 1.01032
$$994$$ 0 0
$$995$$ 4.82625e11i 0.492399i
$$996$$ 0 0
$$997$$ −1.00763e12 −1.01981 −0.509905 0.860231i $$-0.670320\pi$$
−0.509905 + 0.860231i $$0.670320\pi$$
$$998$$ 0 0
$$999$$ − 3.48090e11i − 0.349486i
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 192.9.g.d.127.5 8
4.3 odd 2 inner 192.9.g.d.127.1 8
8.3 odd 2 96.9.g.b.31.8 yes 8
8.5 even 2 96.9.g.b.31.4 8
24.5 odd 2 288.9.g.c.127.1 8
24.11 even 2 288.9.g.c.127.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
96.9.g.b.31.4 8 8.5 even 2
96.9.g.b.31.8 yes 8 8.3 odd 2
192.9.g.d.127.1 8 4.3 odd 2 inner
192.9.g.d.127.5 8 1.1 even 1 trivial
288.9.g.c.127.1 8 24.5 odd 2
288.9.g.c.127.2 8 24.11 even 2