Properties

 Label 256.9.d.f.127.4 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: $$\Q(i, \sqrt{35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 17x^{2} + 81$$ x^4 - 17*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{10}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 127.4 Root $$-2.95804 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 256.127 Dual form 256.9.d.f.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+141.986 q^{3} +510.000i q^{5} -2555.75i q^{7} +13599.0 q^{9} +O(q^{10})$$ $$q+141.986 q^{3} +510.000i q^{5} -2555.75i q^{7} +13599.0 q^{9} -19168.1 q^{11} -27710.0i q^{13} +72412.8i q^{15} +50370.0 q^{17} +108619. q^{19} -362880. i q^{21} +176347. i q^{23} +130525. q^{25} +999297. q^{27} +54978.0i q^{29} -1.17564e6i q^{31} -2.72160e6 q^{33} +1.30343e6 q^{35} -793730. i q^{37} -3.93443e6i q^{39} +75582.0 q^{41} -499648. q^{43} +6.93549e6i q^{45} -2.86755e6i q^{47} -767039. q^{49} +7.15183e6 q^{51} -1.11662e7i q^{53} -9.77573e6i q^{55} +1.54224e7 q^{57} +2.18325e7 q^{59} -2.38266e7i q^{61} -3.47556e7i q^{63} +1.41321e7 q^{65} +7.49473e6 q^{67} +2.50387e7i q^{69} -1.00824e7i q^{71} -6.51661e6 q^{73} +1.85327e7 q^{75} +4.89888e7i q^{77} +4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7 q^{83} +2.56887e7i q^{85} +7.80610e6i q^{87} -8.67958e7 q^{89} -7.08197e7 q^{91} -1.66925e8i q^{93} +5.53958e7i q^{95} -4.66703e7 q^{97} -2.60667e8 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 54396 q^{9}+O(q^{10})$$ 4 * q + 54396 * q^9 $$4 q + 54396 q^{9} + 201480 q^{17} + 522100 q^{25} - 10886400 q^{33} + 302328 q^{41} - 3068156 q^{49} + 61689600 q^{57} + 56528400 q^{65} - 26066440 q^{73} + 210652164 q^{81} - 347183112 q^{89} - 186681080 q^{97}+O(q^{100})$$ 4 * q + 54396 * q^9 + 201480 * q^17 + 522100 * q^25 - 10886400 * q^33 + 302328 * q^41 - 3068156 * q^49 + 61689600 * q^57 + 56528400 * q^65 - 26066440 * q^73 + 210652164 * q^81 - 347183112 * q^89 - 186681080 * 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
$$3$$ 141.986 1.75291 0.876456 0.481481i $$-0.159901\pi$$
0.876456 + 0.481481i $$0.159901\pi$$
$$4$$ 0 0
$$5$$ 510.000i 0.816000i 0.912982 + 0.408000i $$0.133774\pi$$
−0.912982 + 0.408000i $$0.866226\pi$$
$$6$$ 0 0
$$7$$ − 2555.75i − 1.06445i −0.846603 0.532225i $$-0.821356\pi$$
0.846603 0.532225i $$-0.178644\pi$$
$$8$$ 0 0
$$9$$ 13599.0 2.07270
$$10$$ 0 0
$$11$$ −19168.1 −1.30921 −0.654603 0.755972i $$-0.727164\pi$$
−0.654603 + 0.755972i $$0.727164\pi$$
$$12$$ 0 0
$$13$$ − 27710.0i − 0.970204i −0.874458 0.485102i $$-0.838782\pi$$
0.874458 0.485102i $$-0.161218\pi$$
$$14$$ 0 0
$$15$$ 72412.8i 1.43038i
$$16$$ 0 0
$$17$$ 50370.0 0.603082 0.301541 0.953453i $$-0.402499\pi$$
0.301541 + 0.953453i $$0.402499\pi$$
$$18$$ 0 0
$$19$$ 108619. 0.833474 0.416737 0.909027i $$-0.363174\pi$$
0.416737 + 0.909027i $$0.363174\pi$$
$$20$$ 0 0
$$21$$ − 362880.i − 1.86589i
$$22$$ 0 0
$$23$$ 176347.i 0.630167i 0.949064 + 0.315083i $$0.102032\pi$$
−0.949064 + 0.315083i $$0.897968\pi$$
$$24$$ 0 0
$$25$$ 130525. 0.334144
$$26$$ 0 0
$$27$$ 999297. 1.88035
$$28$$ 0 0
$$29$$ 54978.0i 0.0777315i 0.999244 + 0.0388657i $$0.0123745\pi$$
−0.999244 + 0.0388657i $$0.987626\pi$$
$$30$$ 0 0
$$31$$ − 1.17564e6i − 1.27300i −0.771276 0.636501i $$-0.780381\pi$$
0.771276 0.636501i $$-0.219619\pi$$
$$32$$ 0 0
$$33$$ −2.72160e6 −2.29493
$$34$$ 0 0
$$35$$ 1.30343e6 0.868592
$$36$$ 0 0
$$37$$ − 793730.i − 0.423512i −0.977323 0.211756i $$-0.932082\pi$$
0.977323 0.211756i $$-0.0679182\pi$$
$$38$$ 0 0
$$39$$ − 3.93443e6i − 1.70068i
$$40$$ 0 0
$$41$$ 75582.0 0.0267475 0.0133737 0.999911i $$-0.495743\pi$$
0.0133737 + 0.999911i $$0.495743\pi$$
$$42$$ 0 0
$$43$$ −499648. −0.146147 −0.0730736 0.997327i $$-0.523281\pi$$
−0.0730736 + 0.997327i $$0.523281\pi$$
$$44$$ 0 0
$$45$$ 6.93549e6i 1.69133i
$$46$$ 0 0
$$47$$ − 2.86755e6i − 0.587651i −0.955859 0.293825i $$-0.905072\pi$$
0.955859 0.293825i $$-0.0949284\pi$$
$$48$$ 0 0
$$49$$ −767039. −0.133056
$$50$$ 0 0
$$51$$ 7.15183e6 1.05715
$$52$$ 0 0
$$53$$ − 1.11662e7i − 1.41515i −0.706639 0.707575i $$-0.749789\pi$$
0.706639 0.707575i $$-0.250211\pi$$
$$54$$ 0 0
$$55$$ − 9.77573e6i − 1.06831i
$$56$$ 0 0
$$57$$ 1.54224e7 1.46101
$$58$$ 0 0
$$59$$ 2.18325e7 1.80175 0.900875 0.434078i $$-0.142926\pi$$
0.900875 + 0.434078i $$0.142926\pi$$
$$60$$ 0 0
$$61$$ − 2.38266e7i − 1.72085i −0.509577 0.860425i $$-0.670198\pi$$
0.509577 0.860425i $$-0.329802\pi$$
$$62$$ 0 0
$$63$$ − 3.47556e7i − 2.20629i
$$64$$ 0 0
$$65$$ 1.41321e7 0.791687
$$66$$ 0 0
$$67$$ 7.49473e6 0.371926 0.185963 0.982557i $$-0.440460\pi$$
0.185963 + 0.982557i $$0.440460\pi$$
$$68$$ 0 0
$$69$$ 2.50387e7i 1.10463i
$$70$$ 0 0
$$71$$ − 1.00824e7i − 0.396763i −0.980125 0.198382i $$-0.936431\pi$$
0.980125 0.198382i $$-0.0635685\pi$$
$$72$$ 0 0
$$73$$ −6.51661e6 −0.229472 −0.114736 0.993396i $$-0.536602\pi$$
−0.114736 + 0.993396i $$0.536602\pi$$
$$74$$ 0 0
$$75$$ 1.85327e7 0.585725
$$76$$ 0 0
$$77$$ 4.89888e7i 1.39359i
$$78$$ 0 0
$$79$$ 4.87892e7i 1.25261i 0.779579 + 0.626304i $$0.215433\pi$$
−0.779579 + 0.626304i $$0.784567\pi$$
$$80$$ 0 0
$$81$$ 5.26630e7 1.22339
$$82$$ 0 0
$$83$$ 7.34483e7 1.54764 0.773819 0.633407i $$-0.218344\pi$$
0.773819 + 0.633407i $$0.218344\pi$$
$$84$$ 0 0
$$85$$ 2.56887e7i 0.492115i
$$86$$ 0 0
$$87$$ 7.80610e6i 0.136256i
$$88$$ 0 0
$$89$$ −8.67958e7 −1.38337 −0.691685 0.722199i $$-0.743131\pi$$
−0.691685 + 0.722199i $$0.743131\pi$$
$$90$$ 0 0
$$91$$ −7.08197e7 −1.03273
$$92$$ 0 0
$$93$$ − 1.66925e8i − 2.23146i
$$94$$ 0 0
$$95$$ 5.53958e7i 0.680115i
$$96$$ 0 0
$$97$$ −4.66703e7 −0.527173 −0.263587 0.964636i $$-0.584905\pi$$
−0.263587 + 0.964636i $$0.584905\pi$$
$$98$$ 0 0
$$99$$ −2.60667e8 −2.71360
$$100$$ 0 0
$$101$$ − 6.59910e7i − 0.634161i −0.948399 0.317080i $$-0.897297\pi$$
0.948399 0.317080i $$-0.102703\pi$$
$$102$$ 0 0
$$103$$ 1.64884e8i 1.46497i 0.680782 + 0.732486i $$0.261640\pi$$
−0.680782 + 0.732486i $$0.738360\pi$$
$$104$$ 0 0
$$105$$ 1.85069e8 1.52257
$$106$$ 0 0
$$107$$ −1.27326e8 −0.971364 −0.485682 0.874136i $$-0.661429\pi$$
−0.485682 + 0.874136i $$0.661429\pi$$
$$108$$ 0 0
$$109$$ − 1.56119e8i − 1.10598i −0.833186 0.552992i $$-0.813486\pi$$
0.833186 0.552992i $$-0.186514\pi$$
$$110$$ 0 0
$$111$$ − 1.12698e8i − 0.742380i
$$112$$ 0 0
$$113$$ 2.36346e8 1.44955 0.724776 0.688984i $$-0.241943\pi$$
0.724776 + 0.688984i $$0.241943\pi$$
$$114$$ 0 0
$$115$$ −8.99367e7 −0.514216
$$116$$ 0 0
$$117$$ − 3.76828e8i − 2.01094i
$$118$$ 0 0
$$119$$ − 1.28733e8i − 0.641951i
$$120$$ 0 0
$$121$$ 1.53057e8 0.714023
$$122$$ 0 0
$$123$$ 1.07316e7 0.0468860
$$124$$ 0 0
$$125$$ 2.65786e8i 1.08866i
$$126$$ 0 0
$$127$$ 3.67741e8i 1.41360i 0.707412 + 0.706802i $$0.249863\pi$$
−0.707412 + 0.706802i $$0.750137\pi$$
$$128$$ 0 0
$$129$$ −7.09430e7 −0.256183
$$130$$ 0 0
$$131$$ 2.16350e8 0.734636 0.367318 0.930095i $$-0.380276\pi$$
0.367318 + 0.930095i $$0.380276\pi$$
$$132$$ 0 0
$$133$$ − 2.77603e8i − 0.887193i
$$134$$ 0 0
$$135$$ 5.09641e8i 1.53437i
$$136$$ 0 0
$$137$$ 3.86442e8 1.09699 0.548494 0.836155i $$-0.315201\pi$$
0.548494 + 0.836155i $$0.315201\pi$$
$$138$$ 0 0
$$139$$ −3.51077e8 −0.940465 −0.470232 0.882543i $$-0.655830\pi$$
−0.470232 + 0.882543i $$0.655830\pi$$
$$140$$ 0 0
$$141$$ − 4.07151e8i − 1.03010i
$$142$$ 0 0
$$143$$ 5.31148e8i 1.27020i
$$144$$ 0 0
$$145$$ −2.80388e7 −0.0634289
$$146$$ 0 0
$$147$$ −1.08909e8 −0.233235
$$148$$ 0 0
$$149$$ 4.54099e8i 0.921308i 0.887580 + 0.460654i $$0.152385\pi$$
−0.887580 + 0.460654i $$0.847615\pi$$
$$150$$ 0 0
$$151$$ − 6.60188e8i − 1.26987i −0.772565 0.634936i $$-0.781027\pi$$
0.772565 0.634936i $$-0.218973\pi$$
$$152$$ 0 0
$$153$$ 6.84982e8 1.25001
$$154$$ 0 0
$$155$$ 5.99578e8 1.03877
$$156$$ 0 0
$$157$$ 4.35318e8i 0.716486i 0.933628 + 0.358243i $$0.116624\pi$$
−0.933628 + 0.358243i $$0.883376\pi$$
$$158$$ 0 0
$$159$$ − 1.58544e9i − 2.48063i
$$160$$ 0 0
$$161$$ 4.50697e8 0.670782
$$162$$ 0 0
$$163$$ 2.44065e8 0.345744 0.172872 0.984944i $$-0.444695\pi$$
0.172872 + 0.984944i $$0.444695\pi$$
$$164$$ 0 0
$$165$$ − 1.38802e9i − 1.87266i
$$166$$ 0 0
$$167$$ 6.71351e8i 0.863145i 0.902078 + 0.431573i $$0.142041\pi$$
−0.902078 + 0.431573i $$0.857959\pi$$
$$168$$ 0 0
$$169$$ 4.78866e7 0.0587040
$$170$$ 0 0
$$171$$ 1.47711e9 1.72754
$$172$$ 0 0
$$173$$ − 1.76764e9i − 1.97337i −0.162644 0.986685i $$-0.552002\pi$$
0.162644 0.986685i $$-0.447998\pi$$
$$174$$ 0 0
$$175$$ − 3.33589e8i − 0.355680i
$$176$$ 0 0
$$177$$ 3.09990e9 3.15831
$$178$$ 0 0
$$179$$ 7.56967e8 0.737335 0.368668 0.929561i $$-0.379814\pi$$
0.368668 + 0.929561i $$0.379814\pi$$
$$180$$ 0 0
$$181$$ − 6.27094e8i − 0.584277i −0.956376 0.292138i $$-0.905633\pi$$
0.956376 0.292138i $$-0.0943668\pi$$
$$182$$ 0 0
$$183$$ − 3.38304e9i − 3.01650i
$$184$$ 0 0
$$185$$ 4.04802e8 0.345586
$$186$$ 0 0
$$187$$ −9.65497e8 −0.789559
$$188$$ 0 0
$$189$$ − 2.55395e9i − 2.00154i
$$190$$ 0 0
$$191$$ − 1.07924e9i − 0.810933i −0.914110 0.405466i $$-0.867109\pi$$
0.914110 0.405466i $$-0.132891\pi$$
$$192$$ 0 0
$$193$$ −2.96757e7 −0.0213881 −0.0106940 0.999943i $$-0.503404\pi$$
−0.0106940 + 0.999943i $$0.503404\pi$$
$$194$$ 0 0
$$195$$ 2.00656e9 1.38776
$$196$$ 0 0
$$197$$ 1.12484e9i 0.746837i 0.927663 + 0.373419i $$0.121814\pi$$
−0.927663 + 0.373419i $$0.878186\pi$$
$$198$$ 0 0
$$199$$ 1.04718e9i 0.667742i 0.942619 + 0.333871i $$0.108355\pi$$
−0.942619 + 0.333871i $$0.891645\pi$$
$$200$$ 0 0
$$201$$ 1.06415e9 0.651954
$$202$$ 0 0
$$203$$ 1.40510e8 0.0827413
$$204$$ 0 0
$$205$$ 3.85468e7i 0.0218259i
$$206$$ 0 0
$$207$$ 2.39814e9i 1.30615i
$$208$$ 0 0
$$209$$ −2.08202e9 −1.09119
$$210$$ 0 0
$$211$$ −2.77676e9 −1.40090 −0.700452 0.713699i $$-0.747018\pi$$
−0.700452 + 0.713699i $$0.747018\pi$$
$$212$$ 0 0
$$213$$ − 1.43156e9i − 0.695491i
$$214$$ 0 0
$$215$$ − 2.54821e8i − 0.119256i
$$216$$ 0 0
$$217$$ −3.00465e9 −1.35505
$$218$$ 0 0
$$219$$ −9.25267e8 −0.402245
$$220$$ 0 0
$$221$$ − 1.39575e9i − 0.585113i
$$222$$ 0 0
$$223$$ − 2.27822e9i − 0.921248i −0.887595 0.460624i $$-0.847626\pi$$
0.887595 0.460624i $$-0.152374\pi$$
$$224$$ 0 0
$$225$$ 1.77501e9 0.692581
$$226$$ 0 0
$$227$$ −1.48863e9 −0.560639 −0.280319 0.959907i $$-0.590440\pi$$
−0.280319 + 0.959907i $$0.590440\pi$$
$$228$$ 0 0
$$229$$ − 1.69447e9i − 0.616157i −0.951361 0.308079i $$-0.900314\pi$$
0.951361 0.308079i $$-0.0996860\pi$$
$$230$$ 0 0
$$231$$ 6.95572e9i 2.44284i
$$232$$ 0 0
$$233$$ −5.21423e8 −0.176916 −0.0884580 0.996080i $$-0.528194\pi$$
−0.0884580 + 0.996080i $$0.528194\pi$$
$$234$$ 0 0
$$235$$ 1.46245e9 0.479523
$$236$$ 0 0
$$237$$ 6.92738e9i 2.19571i
$$238$$ 0 0
$$239$$ 4.40690e9i 1.35065i 0.737522 + 0.675323i $$0.235996\pi$$
−0.737522 + 0.675323i $$0.764004\pi$$
$$240$$ 0 0
$$241$$ −1.62148e9 −0.480666 −0.240333 0.970691i $$-0.577257\pi$$
−0.240333 + 0.970691i $$0.577257\pi$$
$$242$$ 0 0
$$243$$ 9.21023e8 0.264147
$$244$$ 0 0
$$245$$ − 3.91190e8i − 0.108573i
$$246$$ 0 0
$$247$$ − 3.00984e9i − 0.808640i
$$248$$ 0 0
$$249$$ 1.04286e10 2.71287
$$250$$ 0 0
$$251$$ 1.31321e8 0.0330855 0.0165428 0.999863i $$-0.494734\pi$$
0.0165428 + 0.999863i $$0.494734\pi$$
$$252$$ 0 0
$$253$$ − 3.38023e9i − 0.825019i
$$254$$ 0 0
$$255$$ 3.64743e9i 0.862634i
$$256$$ 0 0
$$257$$ 5.27789e9 1.20984 0.604920 0.796287i $$-0.293205\pi$$
0.604920 + 0.796287i $$0.293205\pi$$
$$258$$ 0 0
$$259$$ −2.02857e9 −0.450808
$$260$$ 0 0
$$261$$ 7.47646e8i 0.161114i
$$262$$ 0 0
$$263$$ 7.38745e9i 1.54409i 0.635570 + 0.772044i $$0.280765\pi$$
−0.635570 + 0.772044i $$0.719235\pi$$
$$264$$ 0 0
$$265$$ 5.69477e9 1.15476
$$266$$ 0 0
$$267$$ −1.23238e10 −2.42493
$$268$$ 0 0
$$269$$ 3.46450e8i 0.0661655i 0.999453 + 0.0330828i $$0.0105325\pi$$
−0.999453 + 0.0330828i $$0.989468\pi$$
$$270$$ 0 0
$$271$$ − 6.51715e6i − 0.00120832i −1.00000 0.000604158i $$-0.999808\pi$$
1.00000 0.000604158i $$-0.000192310\pi$$
$$272$$ 0 0
$$273$$ −1.00554e10 −1.81029
$$274$$ 0 0
$$275$$ −2.50192e9 −0.437464
$$276$$ 0 0
$$277$$ − 2.15061e9i − 0.365293i −0.983179 0.182647i $$-0.941534\pi$$
0.983179 0.182647i $$-0.0584664\pi$$
$$278$$ 0 0
$$279$$ − 1.59876e10i − 2.63855i
$$280$$ 0 0
$$281$$ −1.04256e10 −1.67215 −0.836074 0.548616i $$-0.815155\pi$$
−0.836074 + 0.548616i $$0.815155\pi$$
$$282$$ 0 0
$$283$$ −1.28042e9 −0.199622 −0.0998108 0.995006i $$-0.531824\pi$$
−0.0998108 + 0.995006i $$0.531824\pi$$
$$284$$ 0 0
$$285$$ 7.86542e9i 1.19218i
$$286$$ 0 0
$$287$$ − 1.93168e8i − 0.0284714i
$$288$$ 0 0
$$289$$ −4.43862e9 −0.636292
$$290$$ 0 0
$$291$$ −6.62652e9 −0.924089
$$292$$ 0 0
$$293$$ 2.13786e9i 0.290074i 0.989426 + 0.145037i $$0.0463301\pi$$
−0.989426 + 0.145037i $$0.953670\pi$$
$$294$$ 0 0
$$295$$ 1.11346e10i 1.47023i
$$296$$ 0 0
$$297$$ −1.91546e10 −2.46177
$$298$$ 0 0
$$299$$ 4.88656e9 0.611390
$$300$$ 0 0
$$301$$ 1.27697e9i 0.155567i
$$302$$ 0 0
$$303$$ − 9.36980e9i − 1.11163i
$$304$$ 0 0
$$305$$ 1.21516e10 1.40421
$$306$$ 0 0
$$307$$ 5.45140e9 0.613698 0.306849 0.951758i $$-0.400725\pi$$
0.306849 + 0.951758i $$0.400725\pi$$
$$308$$ 0 0
$$309$$ 2.34112e10i 2.56797i
$$310$$ 0 0
$$311$$ 1.07550e10i 1.14965i 0.818275 + 0.574827i $$0.194931\pi$$
−0.818275 + 0.574827i $$0.805069\pi$$
$$312$$ 0 0
$$313$$ 2.99804e8 0.0312364 0.0156182 0.999878i $$-0.495028\pi$$
0.0156182 + 0.999878i $$0.495028\pi$$
$$314$$ 0 0
$$315$$ 1.77254e10 1.80033
$$316$$ 0 0
$$317$$ 5.31172e9i 0.526015i 0.964794 + 0.263007i $$0.0847144\pi$$
−0.964794 + 0.263007i $$0.915286\pi$$
$$318$$ 0 0
$$319$$ − 1.05382e9i − 0.101767i
$$320$$ 0 0
$$321$$ −1.80785e10 −1.70272
$$322$$ 0 0
$$323$$ 5.47115e9 0.502653
$$324$$ 0 0
$$325$$ − 3.61685e9i − 0.324188i
$$326$$ 0 0
$$327$$ − 2.21667e10i − 1.93869i
$$328$$ 0 0
$$329$$ −7.32872e9 −0.625525
$$330$$ 0 0
$$331$$ 1.01004e10 0.841446 0.420723 0.907189i $$-0.361776\pi$$
0.420723 + 0.907189i $$0.361776\pi$$
$$332$$ 0 0
$$333$$ − 1.07939e10i − 0.877815i
$$334$$ 0 0
$$335$$ 3.82231e9i 0.303492i
$$336$$ 0 0
$$337$$ −1.84359e10 −1.42937 −0.714684 0.699448i $$-0.753429\pi$$
−0.714684 + 0.699448i $$0.753429\pi$$
$$338$$ 0 0
$$339$$ 3.35578e10 2.54094
$$340$$ 0 0
$$341$$ 2.25348e10i 1.66662i
$$342$$ 0 0
$$343$$ − 1.27730e10i − 0.922820i
$$344$$ 0 0
$$345$$ −1.27697e10 −0.901376
$$346$$ 0 0
$$347$$ −1.27822e10 −0.881636 −0.440818 0.897597i $$-0.645312\pi$$
−0.440818 + 0.897597i $$0.645312\pi$$
$$348$$ 0 0
$$349$$ − 6.39381e8i − 0.0430981i −0.999768 0.0215490i $$-0.993140\pi$$
0.999768 0.0215490i $$-0.00685980\pi$$
$$350$$ 0 0
$$351$$ − 2.76905e10i − 1.82433i
$$352$$ 0 0
$$353$$ 2.59837e10 1.67341 0.836705 0.547653i $$-0.184479\pi$$
0.836705 + 0.547653i $$0.184479\pi$$
$$354$$ 0 0
$$355$$ 5.14203e9 0.323759
$$356$$ 0 0
$$357$$ − 1.82783e10i − 1.12528i
$$358$$ 0 0
$$359$$ − 2.22541e10i − 1.33978i −0.742461 0.669889i $$-0.766342\pi$$
0.742461 0.669889i $$-0.233658\pi$$
$$360$$ 0 0
$$361$$ −5.18543e9 −0.305320
$$362$$ 0 0
$$363$$ 2.17320e10 1.25162
$$364$$ 0 0
$$365$$ − 3.32347e9i − 0.187249i
$$366$$ 0 0
$$367$$ 1.86229e10i 1.02656i 0.858221 + 0.513280i $$0.171570\pi$$
−0.858221 + 0.513280i $$0.828430\pi$$
$$368$$ 0 0
$$369$$ 1.02784e9 0.0554396
$$370$$ 0 0
$$371$$ −2.85380e10 −1.50636
$$372$$ 0 0
$$373$$ − 1.19680e10i − 0.618283i −0.951016 0.309141i $$-0.899958\pi$$
0.951016 0.309141i $$-0.100042\pi$$
$$374$$ 0 0
$$375$$ 3.77379e10i 1.90833i
$$376$$ 0 0
$$377$$ 1.52344e9 0.0754154
$$378$$ 0 0
$$379$$ −2.30787e10 −1.11855 −0.559275 0.828982i $$-0.688920\pi$$
−0.559275 + 0.828982i $$0.688920\pi$$
$$380$$ 0 0
$$381$$ 5.22141e10i 2.47792i
$$382$$ 0 0
$$383$$ 1.43419e10i 0.666518i 0.942835 + 0.333259i $$0.108148\pi$$
−0.942835 + 0.333259i $$0.891852\pi$$
$$384$$ 0 0
$$385$$ −2.49843e10 −1.13717
$$386$$ 0 0
$$387$$ −6.79472e9 −0.302920
$$388$$ 0 0
$$389$$ 2.73457e10i 1.19424i 0.802152 + 0.597119i $$0.203688\pi$$
−0.802152 + 0.597119i $$0.796312\pi$$
$$390$$ 0 0
$$391$$ 8.88257e9i 0.380042i
$$392$$ 0 0
$$393$$ 3.07187e10 1.28775
$$394$$ 0 0
$$395$$ −2.48825e10 −1.02213
$$396$$ 0 0
$$397$$ 3.99456e10i 1.60808i 0.594576 + 0.804039i $$0.297320\pi$$
−0.594576 + 0.804039i $$0.702680\pi$$
$$398$$ 0 0
$$399$$ − 3.94157e10i − 1.55517i
$$400$$ 0 0
$$401$$ −2.12767e10 −0.822863 −0.411431 0.911441i $$-0.634971\pi$$
−0.411431 + 0.911441i $$0.634971\pi$$
$$402$$ 0 0
$$403$$ −3.25771e10 −1.23507
$$404$$ 0 0
$$405$$ 2.68582e10i 0.998288i
$$406$$ 0 0
$$407$$ 1.52143e10i 0.554465i
$$408$$ 0 0
$$409$$ −1.14283e10 −0.408404 −0.204202 0.978929i $$-0.565460\pi$$
−0.204202 + 0.978929i $$0.565460\pi$$
$$410$$ 0 0
$$411$$ 5.48693e10 1.92292
$$412$$ 0 0
$$413$$ − 5.57982e10i − 1.91788i
$$414$$ 0 0
$$415$$ 3.74586e10i 1.26287i
$$416$$ 0 0
$$417$$ −4.98479e10 −1.64855
$$418$$ 0 0
$$419$$ 1.10009e10 0.356922 0.178461 0.983947i $$-0.442888\pi$$
0.178461 + 0.983947i $$0.442888\pi$$
$$420$$ 0 0
$$421$$ − 2.28766e10i − 0.728220i −0.931356 0.364110i $$-0.881373\pi$$
0.931356 0.364110i $$-0.118627\pi$$
$$422$$ 0 0
$$423$$ − 3.89958e10i − 1.21802i
$$424$$ 0 0
$$425$$ 6.57454e9 0.201516
$$426$$ 0 0
$$427$$ −6.08948e10 −1.83176
$$428$$ 0 0
$$429$$ 7.54155e10i 2.22655i
$$430$$ 0 0
$$431$$ 9.55108e8i 0.0276786i 0.999904 + 0.0138393i $$0.00440532\pi$$
−0.999904 + 0.0138393i $$0.995595\pi$$
$$432$$ 0 0
$$433$$ 3.82225e10 1.08735 0.543673 0.839297i $$-0.317033\pi$$
0.543673 + 0.839297i $$0.317033\pi$$
$$434$$ 0 0
$$435$$ −3.98111e9 −0.111185
$$436$$ 0 0
$$437$$ 1.91546e10i 0.525228i
$$438$$ 0 0
$$439$$ 6.40288e10i 1.72392i 0.506976 + 0.861960i $$0.330763\pi$$
−0.506976 + 0.861960i $$0.669237\pi$$
$$440$$ 0 0
$$441$$ −1.04310e10 −0.275785
$$442$$ 0 0
$$443$$ −7.47659e10 −1.94128 −0.970641 0.240533i $$-0.922678\pi$$
−0.970641 + 0.240533i $$0.922678\pi$$
$$444$$ 0 0
$$445$$ − 4.42658e10i − 1.12883i
$$446$$ 0 0
$$447$$ 6.44756e10i 1.61497i
$$448$$ 0 0
$$449$$ 2.51987e10 0.620001 0.310000 0.950736i $$-0.399671\pi$$
0.310000 + 0.950736i $$0.399671\pi$$
$$450$$ 0 0
$$451$$ −1.44876e9 −0.0350180
$$452$$ 0 0
$$453$$ − 9.37373e10i − 2.22597i
$$454$$ 0 0
$$455$$ − 3.61181e10i − 0.842711i
$$456$$ 0 0
$$457$$ −4.66828e9 −0.107027 −0.0535133 0.998567i $$-0.517042\pi$$
−0.0535133 + 0.998567i $$0.517042\pi$$
$$458$$ 0 0
$$459$$ 5.03346e10 1.13401
$$460$$ 0 0
$$461$$ − 3.88096e10i − 0.859281i −0.903000 0.429641i $$-0.858640\pi$$
0.903000 0.429641i $$-0.141360\pi$$
$$462$$ 0 0
$$463$$ 3.23432e10i 0.703817i 0.936034 + 0.351908i $$0.114467\pi$$
−0.936034 + 0.351908i $$0.885533\pi$$
$$464$$ 0 0
$$465$$ 8.51316e10 1.82087
$$466$$ 0 0
$$467$$ −2.31902e10 −0.487570 −0.243785 0.969829i $$-0.578389\pi$$
−0.243785 + 0.969829i $$0.578389\pi$$
$$468$$ 0 0
$$469$$ − 1.91546e10i − 0.395897i
$$470$$ 0 0
$$471$$ 6.18090e10i 1.25594i
$$472$$ 0 0
$$473$$ 9.57731e9 0.191337
$$474$$ 0 0
$$475$$ 1.41775e10 0.278500
$$476$$ 0 0
$$477$$ − 1.51849e11i − 2.93318i
$$478$$ 0 0
$$479$$ 4.83542e9i 0.0918528i 0.998945 + 0.0459264i $$0.0146240\pi$$
−0.998945 + 0.0459264i $$0.985376\pi$$
$$480$$ 0 0
$$481$$ −2.19943e10 −0.410893
$$482$$ 0 0
$$483$$ 6.39926e10 1.17582
$$484$$ 0 0
$$485$$ − 2.38018e10i − 0.430173i
$$486$$ 0 0
$$487$$ 3.03878e10i 0.540236i 0.962827 + 0.270118i $$0.0870628\pi$$
−0.962827 + 0.270118i $$0.912937\pi$$
$$488$$ 0 0
$$489$$ 3.46538e10 0.606059
$$490$$ 0 0
$$491$$ 5.56483e10 0.957472 0.478736 0.877959i $$-0.341095\pi$$
0.478736 + 0.877959i $$0.341095\pi$$
$$492$$ 0 0
$$493$$ 2.76924e9i 0.0468784i
$$494$$ 0 0
$$495$$ − 1.32940e11i − 2.21429i
$$496$$ 0 0
$$497$$ −2.57681e10 −0.422335
$$498$$ 0 0
$$499$$ −7.88458e10 −1.27168 −0.635838 0.771822i $$-0.719345\pi$$
−0.635838 + 0.771822i $$0.719345\pi$$
$$500$$ 0 0
$$501$$ 9.53224e10i 1.51302i
$$502$$ 0 0
$$503$$ 4.41092e10i 0.689061i 0.938775 + 0.344530i $$0.111962\pi$$
−0.938775 + 0.344530i $$0.888038\pi$$
$$504$$ 0 0
$$505$$ 3.36554e10 0.517475
$$506$$ 0 0
$$507$$ 6.79923e9 0.102903
$$508$$ 0 0
$$509$$ 1.05927e10i 0.157811i 0.996882 + 0.0789055i $$0.0251425\pi$$
−0.996882 + 0.0789055i $$0.974857\pi$$
$$510$$ 0 0
$$511$$ 1.66548e10i 0.244262i
$$512$$ 0 0
$$513$$ 1.08543e11 1.56723
$$514$$ 0 0
$$515$$ −8.40908e10 −1.19542
$$516$$ 0 0
$$517$$ 5.49654e10i 0.769356i
$$518$$ 0 0
$$519$$ − 2.50979e11i − 3.45914i
$$520$$ 0 0
$$521$$ −1.24958e11 −1.69595 −0.847973 0.530039i $$-0.822177\pi$$
−0.847973 + 0.530039i $$0.822177\pi$$
$$522$$ 0 0
$$523$$ −2.80408e10 −0.374786 −0.187393 0.982285i $$-0.560004\pi$$
−0.187393 + 0.982285i $$0.560004\pi$$
$$524$$ 0 0
$$525$$ − 4.73649e10i − 0.623476i
$$526$$ 0 0
$$527$$ − 5.92172e10i − 0.767724i
$$528$$ 0 0
$$529$$ 4.72129e10 0.602890
$$530$$ 0 0
$$531$$ 2.96900e11 3.73449
$$532$$ 0 0
$$533$$ − 2.09438e9i − 0.0259505i
$$534$$ 0 0
$$535$$ − 6.49363e10i − 0.792633i
$$536$$ 0 0
$$537$$ 1.07479e11 1.29248
$$538$$ 0 0
$$539$$ 1.47027e10 0.174197
$$540$$ 0 0
$$541$$ 1.44659e11i 1.68871i 0.535782 + 0.844356i $$0.320017\pi$$
−0.535782 + 0.844356i $$0.679983\pi$$
$$542$$ 0 0
$$543$$ − 8.90386e10i − 1.02419i
$$544$$ 0 0
$$545$$ 7.96205e10 0.902483
$$546$$ 0 0
$$547$$ 1.03774e10 0.115915 0.0579573 0.998319i $$-0.481541\pi$$
0.0579573 + 0.998319i $$0.481541\pi$$
$$548$$ 0 0
$$549$$ − 3.24018e11i − 3.56681i
$$550$$ 0 0
$$551$$ 5.97167e9i 0.0647872i
$$552$$ 0 0
$$553$$ 1.24693e11 1.33334
$$554$$ 0 0
$$555$$ 5.74762e10 0.605782
$$556$$ 0 0
$$557$$ − 5.47312e9i − 0.0568610i −0.999596 0.0284305i $$-0.990949\pi$$
0.999596 0.0284305i $$-0.00905093\pi$$
$$558$$ 0 0
$$559$$ 1.38453e10i 0.141793i
$$560$$ 0 0
$$561$$ −1.37087e11 −1.38403
$$562$$ 0 0
$$563$$ −4.36118e10 −0.434081 −0.217040 0.976163i $$-0.569640\pi$$
−0.217040 + 0.976163i $$0.569640\pi$$
$$564$$ 0 0
$$565$$ 1.20536e11i 1.18283i
$$566$$ 0 0
$$567$$ − 1.34593e11i − 1.30224i
$$568$$ 0 0
$$569$$ −1.27822e10 −0.121943 −0.0609716 0.998140i $$-0.519420\pi$$
−0.0609716 + 0.998140i $$0.519420\pi$$
$$570$$ 0 0
$$571$$ 7.59455e10 0.714427 0.357213 0.934023i $$-0.383727\pi$$
0.357213 + 0.934023i $$0.383727\pi$$
$$572$$ 0 0
$$573$$ − 1.53237e11i − 1.42149i
$$574$$ 0 0
$$575$$ 2.30176e10i 0.210566i
$$576$$ 0 0
$$577$$ 2.13827e10 0.192912 0.0964560 0.995337i $$-0.469249\pi$$
0.0964560 + 0.995337i $$0.469249\pi$$
$$578$$ 0 0
$$579$$ −4.21353e9 −0.0374914
$$580$$ 0 0
$$581$$ − 1.87715e11i − 1.64738i
$$582$$ 0 0
$$583$$ 2.14035e11i 1.85272i
$$584$$ 0 0
$$585$$ 1.92182e11 1.64093
$$586$$ 0 0
$$587$$ −6.07298e10 −0.511504 −0.255752 0.966742i $$-0.582323\pi$$
−0.255752 + 0.966742i $$0.582323\pi$$
$$588$$ 0 0
$$589$$ − 1.27697e11i − 1.06101i
$$590$$ 0 0
$$591$$ 1.59711e11i 1.30914i
$$592$$ 0 0
$$593$$ 1.15978e11 0.937899 0.468949 0.883225i $$-0.344633\pi$$
0.468949 + 0.883225i $$0.344633\pi$$
$$594$$ 0 0
$$595$$ 6.56538e10 0.523832
$$596$$ 0 0
$$597$$ 1.48685e11i 1.17049i
$$598$$ 0 0
$$599$$ 2.40647e11i 1.86927i 0.355607 + 0.934636i $$0.384274\pi$$
−0.355607 + 0.934636i $$0.615726\pi$$
$$600$$ 0 0
$$601$$ 1.92942e11 1.47887 0.739434 0.673229i $$-0.235093\pi$$
0.739434 + 0.673229i $$0.235093\pi$$
$$602$$ 0 0
$$603$$ 1.01921e11 0.770892
$$604$$ 0 0
$$605$$ 7.80591e10i 0.582643i
$$606$$ 0 0
$$607$$ − 1.62042e11i − 1.19364i −0.802376 0.596819i $$-0.796431\pi$$
0.802376 0.596819i $$-0.203569\pi$$
$$608$$ 0 0
$$609$$ 1.99504e10 0.145038
$$610$$ 0 0
$$611$$ −7.94597e10 −0.570141
$$612$$ 0 0
$$613$$ − 1.76424e11i − 1.24944i −0.780847 0.624722i $$-0.785212\pi$$
0.780847 0.624722i $$-0.214788\pi$$
$$614$$ 0 0
$$615$$ 5.47311e9i 0.0382590i
$$616$$ 0 0
$$617$$ 9.84986e10 0.679656 0.339828 0.940488i $$-0.389631\pi$$
0.339828 + 0.940488i $$0.389631\pi$$
$$618$$ 0 0
$$619$$ 1.28596e10 0.0875923 0.0437961 0.999040i $$-0.486055\pi$$
0.0437961 + 0.999040i $$0.486055\pi$$
$$620$$ 0 0
$$621$$ 1.76223e11i 1.18494i
$$622$$ 0 0
$$623$$ 2.21828e11i 1.47253i
$$624$$ 0 0
$$625$$ −8.45648e10 −0.554204
$$626$$ 0 0
$$627$$ −2.95618e11 −1.91276
$$628$$ 0 0
$$629$$ − 3.99802e10i − 0.255413i
$$630$$ 0 0
$$631$$ − 1.73463e11i − 1.09418i −0.837073 0.547091i $$-0.815735\pi$$
0.837073 0.547091i $$-0.184265\pi$$
$$632$$ 0 0
$$633$$ −3.94261e11 −2.45566
$$634$$ 0 0
$$635$$ −1.87548e11 −1.15350
$$636$$ 0 0
$$637$$ 2.12547e10i 0.129091i
$$638$$ 0 0
$$639$$ − 1.37111e11i − 0.822372i
$$640$$ 0 0
$$641$$ −1.13903e11 −0.674690 −0.337345 0.941381i $$-0.609529\pi$$
−0.337345 + 0.941381i $$0.609529\pi$$
$$642$$ 0 0
$$643$$ −7.04067e10 −0.411879 −0.205940 0.978565i $$-0.566025\pi$$
−0.205940 + 0.978565i $$0.566025\pi$$
$$644$$ 0 0
$$645$$ − 3.61810e10i − 0.209046i
$$646$$ 0 0
$$647$$ 1.99175e11i 1.13663i 0.822812 + 0.568314i $$0.192404\pi$$
−0.822812 + 0.568314i $$0.807596\pi$$
$$648$$ 0 0
$$649$$ −4.18487e11 −2.35886
$$650$$ 0 0
$$651$$ −4.26617e11 −2.37528
$$652$$ 0 0
$$653$$ 6.49972e9i 0.0357472i 0.999840 + 0.0178736i $$0.00568964\pi$$
−0.999840 + 0.0178736i $$0.994310\pi$$
$$654$$ 0 0
$$655$$ 1.10339e11i 0.599463i
$$656$$ 0 0
$$657$$ −8.86194e10 −0.475628
$$658$$ 0 0
$$659$$ 2.20982e11 1.17170 0.585848 0.810421i $$-0.300761\pi$$
0.585848 + 0.810421i $$0.300761\pi$$
$$660$$ 0 0
$$661$$ 2.69549e11i 1.41199i 0.708217 + 0.705995i $$0.249500\pi$$
−0.708217 + 0.705995i $$0.750500\pi$$
$$662$$ 0 0
$$663$$ − 1.98177e11i − 1.02565i
$$664$$ 0 0
$$665$$ 1.41578e11 0.723949
$$666$$ 0 0
$$667$$ −9.69518e9 −0.0489838
$$668$$ 0 0
$$669$$ − 3.23476e11i − 1.61487i
$$670$$ 0 0
$$671$$ 4.56711e11i 2.25295i
$$672$$ 0 0
$$673$$ 9.44470e10 0.460392 0.230196 0.973144i $$-0.426063\pi$$
0.230196 + 0.973144i $$0.426063\pi$$
$$674$$ 0 0
$$675$$ 1.30433e11 0.628309
$$676$$ 0 0
$$677$$ − 8.02735e10i − 0.382136i −0.981577 0.191068i $$-0.938805\pi$$
0.981577 0.191068i $$-0.0611950\pi$$
$$678$$ 0 0
$$679$$ 1.19277e11i 0.561150i
$$680$$ 0 0
$$681$$ −2.11364e11 −0.982751
$$682$$ 0 0
$$683$$ 3.00783e11 1.38220 0.691099 0.722760i $$-0.257127\pi$$
0.691099 + 0.722760i $$0.257127\pi$$
$$684$$ 0 0
$$685$$ 1.97085e11i 0.895142i
$$686$$ 0 0
$$687$$ − 2.40591e11i − 1.08007i
$$688$$ 0 0
$$689$$ −3.09416e11 −1.37298
$$690$$ 0 0
$$691$$ −3.06208e11 −1.34309 −0.671544 0.740964i $$-0.734369\pi$$
−0.671544 + 0.740964i $$0.734369\pi$$
$$692$$ 0 0
$$693$$ 6.66199e11i 2.88849i
$$694$$ 0 0
$$695$$ − 1.79049e11i − 0.767419i
$$696$$ 0 0
$$697$$ 3.80707e9 0.0161309
$$698$$ 0 0
$$699$$ −7.40348e10 −0.310118
$$700$$ 0 0
$$701$$ − 2.73603e11i − 1.13305i −0.824045 0.566524i $$-0.808288\pi$$
0.824045 0.566524i $$-0.191712\pi$$
$$702$$ 0 0
$$703$$ − 8.62143e10i − 0.352987i
$$704$$ 0 0
$$705$$ 2.07647e11 0.840562
$$706$$ 0 0
$$707$$ −1.68656e11 −0.675033
$$708$$ 0 0
$$709$$ 1.76662e11i 0.699129i 0.936912 + 0.349564i $$0.113670\pi$$
−0.936912 + 0.349564i $$0.886330\pi$$
$$710$$ 0 0
$$711$$ 6.63484e11i 2.59628i
$$712$$ 0 0
$$713$$ 2.07321e11 0.802203
$$714$$ 0 0
$$715$$ −2.70885e11 −1.03648
$$716$$ 0 0
$$717$$ 6.25718e11i 2.36756i
$$718$$ 0 0
$$719$$ − 2.25510e11i − 0.843821i −0.906637 0.421911i $$-0.861360\pi$$
0.906637 0.421911i $$-0.138640\pi$$
$$720$$ 0 0
$$721$$ 4.21402e11 1.55939
$$722$$ 0 0
$$723$$ −2.30227e11 −0.842565
$$724$$ 0 0
$$725$$ 7.17600e9i 0.0259735i
$$726$$ 0 0
$$727$$ − 2.87080e11i − 1.02770i −0.857881 0.513849i $$-0.828219\pi$$
0.857881 0.513849i $$-0.171781\pi$$
$$728$$ 0 0
$$729$$ −2.14750e11 −0.760366
$$730$$ 0 0
$$731$$ −2.51673e10 −0.0881388
$$732$$ 0 0
$$733$$ − 2.94176e11i − 1.01904i −0.860459 0.509520i $$-0.829823\pi$$
0.860459 0.509520i $$-0.170177\pi$$
$$734$$ 0 0
$$735$$ − 5.55435e10i − 0.190320i
$$736$$ 0 0
$$737$$ −1.43660e11 −0.486928
$$738$$ 0 0
$$739$$ 9.69888e10 0.325195 0.162598 0.986692i $$-0.448013\pi$$
0.162598 + 0.986692i $$0.448013\pi$$
$$740$$ 0 0
$$741$$ − 4.27355e11i − 1.41748i
$$742$$ 0 0
$$743$$ 1.01567e11i 0.333271i 0.986019 + 0.166635i $$0.0532902\pi$$
−0.986019 + 0.166635i $$0.946710\pi$$
$$744$$ 0 0
$$745$$ −2.31590e11 −0.751788
$$746$$ 0 0
$$747$$ 9.98824e11 3.20779
$$748$$ 0 0
$$749$$ 3.25413e11i 1.03397i
$$750$$ 0 0
$$751$$ 4.17899e11i 1.31375i 0.754001 + 0.656873i $$0.228121\pi$$
−0.754001 + 0.656873i $$0.771879\pi$$
$$752$$ 0 0
$$753$$ 1.86457e10 0.0579960
$$754$$ 0 0
$$755$$ 3.36696e11 1.03621
$$756$$ 0 0
$$757$$ − 1.82006e11i − 0.554244i −0.960835 0.277122i $$-0.910619\pi$$
0.960835 0.277122i $$-0.0893806\pi$$
$$758$$ 0 0
$$759$$ − 4.79945e11i − 1.44619i
$$760$$ 0 0
$$761$$ 4.27419e11 1.27443 0.637213 0.770687i $$-0.280087\pi$$
0.637213 + 0.770687i $$0.280087\pi$$
$$762$$ 0 0
$$763$$ −3.99000e11 −1.17727
$$764$$ 0 0
$$765$$ 3.49341e11i 1.02001i
$$766$$ 0 0
$$767$$ − 6.04978e11i − 1.74807i
$$768$$ 0 0
$$769$$ −5.09969e11 −1.45827 −0.729136 0.684368i $$-0.760078\pi$$
−0.729136 + 0.684368i $$0.760078\pi$$
$$770$$ 0 0
$$771$$ 7.49386e11 2.12074
$$772$$ 0 0
$$773$$ 1.49408e11i 0.418462i 0.977866 + 0.209231i $$0.0670961\pi$$
−0.977866 + 0.209231i $$0.932904\pi$$
$$774$$ 0 0
$$775$$ − 1.53451e11i − 0.425366i
$$776$$ 0 0
$$777$$ −2.88029e11 −0.790227
$$778$$ 0 0
$$779$$ 8.20966e9 0.0222933
$$780$$ 0 0
$$781$$ 1.93261e11i 0.519445i
$$782$$ 0 0
$$783$$ 5.49393e10i 0.146163i
$$784$$ 0 0
$$785$$ −2.22012e11 −0.584652
$$786$$ 0 0
$$787$$ 7.33252e11 1.91141 0.955706 0.294323i $$-0.0950943\pi$$
0.955706 + 0.294323i $$0.0950943\pi$$
$$788$$ 0 0
$$789$$ 1.04891e12i 2.70665i
$$790$$ 0 0
$$791$$ − 6.04040e11i − 1.54298i
$$792$$ 0 0
$$793$$ −6.60236e11 −1.66958
$$794$$ 0 0
$$795$$ 8.08577e11 2.02420
$$796$$ 0 0
$$797$$ 3.02703e11i 0.750212i 0.926982 + 0.375106i $$0.122394\pi$$
−0.926982 + 0.375106i $$0.877606\pi$$
$$798$$ 0 0
$$799$$ − 1.44438e11i − 0.354401i
$$800$$ 0 0
$$801$$ −1.18034e12 −2.86732
$$802$$ 0 0
$$803$$ 1.24911e11 0.300427
$$804$$ 0 0
$$805$$ 2.29855e11i 0.547358i
$$806$$ 0 0
$$807$$ 4.91911e10i 0.115982i
$$808$$ 0 0
$$809$$ 5.84316e11 1.36412 0.682062 0.731295i $$-0.261084\pi$$
0.682062 + 0.731295i $$0.261084\pi$$
$$810$$ 0 0
$$811$$ 1.21470e11 0.280793 0.140396 0.990095i $$-0.455162\pi$$
0.140396 + 0.990095i $$0.455162\pi$$
$$812$$ 0 0
$$813$$ − 9.25344e8i − 0.00211807i
$$814$$ 0 0
$$815$$ 1.24473e11i 0.282127i
$$816$$ 0 0
$$817$$ −5.42714e10 −0.121810
$$818$$ 0 0
$$819$$ −9.63078e11 −2.14055
$$820$$ 0 0
$$821$$ − 4.52470e11i − 0.995903i −0.867205 0.497952i $$-0.834086\pi$$
0.867205 0.497952i $$-0.165914\pi$$
$$822$$ 0 0
$$823$$ 3.06704e11i 0.668528i 0.942479 + 0.334264i $$0.108488\pi$$
−0.942479 + 0.334264i $$0.891512\pi$$
$$824$$ 0 0
$$825$$ −3.55237e11 −0.766835
$$826$$ 0 0
$$827$$ −2.93276e11 −0.626982 −0.313491 0.949591i $$-0.601499\pi$$
−0.313491 + 0.949591i $$0.601499\pi$$
$$828$$ 0 0
$$829$$ 3.35532e11i 0.710421i 0.934786 + 0.355210i $$0.115591\pi$$
−0.934786 + 0.355210i $$0.884409\pi$$
$$830$$ 0 0
$$831$$ − 3.05356e11i − 0.640327i
$$832$$ 0 0
$$833$$ −3.86358e10 −0.0802434
$$834$$ 0 0
$$835$$ −3.42389e11 −0.704326
$$836$$ 0 0
$$837$$ − 1.17482e12i − 2.39369i
$$838$$ 0 0
$$839$$ − 3.42844e11i − 0.691908i −0.938252 0.345954i $$-0.887555\pi$$
0.938252 0.345954i $$-0.112445\pi$$
$$840$$ 0 0
$$841$$ 4.97224e11 0.993958
$$842$$ 0 0
$$843$$ −1.48029e12 −2.93113
$$844$$ 0 0
$$845$$ 2.44222e10i 0.0479024i
$$846$$ 0 0
$$847$$ − 3.91175e11i − 0.760042i
$$848$$ 0 0
$$849$$ −1.81802e11 −0.349919
$$850$$ 0 0
$$851$$ 1.39972e11 0.266883
$$852$$ 0 0
$$853$$ 5.08662e11i 0.960801i 0.877049 + 0.480400i $$0.159509\pi$$
−0.877049 + 0.480400i $$0.840491\pi$$
$$854$$ 0 0
$$855$$ 7.53328e11i 1.40968i
$$856$$ 0 0
$$857$$ −6.06764e11 −1.12486 −0.562428 0.826846i $$-0.690133\pi$$
−0.562428 + 0.826846i $$0.690133\pi$$
$$858$$ 0 0
$$859$$ 9.49431e11 1.74378 0.871888 0.489705i $$-0.162895\pi$$
0.871888 + 0.489705i $$0.162895\pi$$
$$860$$ 0 0
$$861$$ − 2.74272e10i − 0.0499078i
$$862$$ 0 0
$$863$$ 2.99836e10i 0.0540556i 0.999635 + 0.0270278i $$0.00860426\pi$$
−0.999635 + 0.0270278i $$0.991396\pi$$
$$864$$ 0 0
$$865$$ 9.01494e11 1.61027
$$866$$ 0 0
$$867$$ −6.30222e11 −1.11536
$$868$$ 0 0
$$869$$ − 9.35196e11i − 1.63992i
$$870$$ 0 0
$$871$$ − 2.07679e11i − 0.360844i
$$872$$ 0 0
$$873$$ −6.34669e11 −1.09267
$$874$$ 0 0
$$875$$ 6.79283e11 1.15883
$$876$$ 0 0
$$877$$ − 8.80195e11i − 1.48792i −0.668223 0.743961i $$-0.732945\pi$$
0.668223 0.743961i $$-0.267055\pi$$
$$878$$ 0 0
$$879$$ 3.03546e11i 0.508474i
$$880$$ 0 0
$$881$$ 1.04085e12 1.72776 0.863879 0.503699i $$-0.168028\pi$$
0.863879 + 0.503699i $$0.168028\pi$$
$$882$$ 0 0
$$883$$ −7.60446e11 −1.25091 −0.625454 0.780261i $$-0.715086\pi$$
−0.625454 + 0.780261i $$0.715086\pi$$
$$884$$ 0 0
$$885$$ 1.58095e12i 2.57718i
$$886$$ 0 0
$$887$$ 3.34097e11i 0.539732i 0.962898 + 0.269866i $$0.0869794\pi$$
−0.962898 + 0.269866i $$0.913021\pi$$
$$888$$ 0 0
$$889$$ 9.39853e11 1.50471
$$890$$ 0 0
$$891$$ −1.00945e12 −1.60167
$$892$$ 0 0
$$893$$ − 3.11471e11i − 0.489792i
$$894$$ 0 0
$$895$$ 3.86053e11i 0.601666i
$$896$$ 0 0
$$897$$ 6.93823e11 1.07171
$$898$$ 0 0
$$899$$ 6.46345e10 0.0989523
$$900$$ 0 0
$$901$$ − 5.62442e11i − 0.853451i
$$902$$ 0 0
$$903$$ 1.81312e11i 0.272695i
$$904$$ 0 0
$$905$$ 3.19818e11 0.476770
$$906$$ 0 0
$$907$$ −7.65213e11 −1.13071 −0.565357 0.824846i $$-0.691262\pi$$
−0.565357 + 0.824846i $$0.691262\pi$$
$$908$$ 0 0
$$909$$ − 8.97412e11i − 1.31443i
$$910$$ 0 0
$$911$$ 3.83541e11i 0.556851i 0.960458 + 0.278425i $$0.0898125\pi$$
−0.960458 + 0.278425i $$0.910188\pi$$
$$912$$ 0 0
$$913$$ −1.40786e12 −2.02618
$$914$$ 0 0
$$915$$ 1.72535e12 2.46146
$$916$$ 0 0
$$917$$ − 5.52937e11i − 0.781984i
$$918$$ 0 0
$$919$$ 6.82775e11i 0.957229i 0.878025 + 0.478615i $$0.158861\pi$$
−0.878025 + 0.478615i $$0.841139\pi$$
$$920$$ 0 0
$$921$$ 7.74022e11 1.07576
$$922$$ 0 0
$$923$$ −2.79384e11 −0.384941
$$924$$ 0 0
$$925$$ − 1.03602e11i − 0.141514i
$$926$$ 0 0
$$927$$ 2.24226e12i 3.03645i
$$928$$ 0 0
$$929$$ 2.94973e11 0.396021 0.198011 0.980200i $$-0.436552\pi$$
0.198011 + 0.980200i $$0.436552\pi$$
$$930$$ 0 0
$$931$$ −8.33152e10 −0.110898
$$932$$ 0 0
$$933$$ 1.52705e12i 2.01524i
$$934$$ 0 0
$$935$$ − 4.92404e11i − 0.644280i
$$936$$ 0 0
$$937$$ −1.03941e12 −1.34843 −0.674217 0.738533i $$-0.735519\pi$$
−0.674217 + 0.738533i $$0.735519\pi$$
$$938$$ 0 0
$$939$$ 4.25680e10 0.0547546
$$940$$ 0 0
$$941$$ 1.26891e11i 0.161836i 0.996721 + 0.0809178i $$0.0257851\pi$$
−0.996721 + 0.0809178i $$0.974215\pi$$
$$942$$ 0 0
$$943$$ 1.33286e10i 0.0168554i
$$944$$ 0 0
$$945$$ 1.30251e12 1.63326
$$946$$ 0 0
$$947$$ −6.13064e11 −0.762265 −0.381133 0.924520i $$-0.624466\pi$$
−0.381133 + 0.924520i $$0.624466\pi$$
$$948$$ 0 0
$$949$$ 1.80575e11i 0.222635i
$$950$$ 0 0
$$951$$ 7.54189e11i 0.922058i
$$952$$ 0 0
$$953$$ −6.58227e11 −0.798002 −0.399001 0.916951i $$-0.630643\pi$$
−0.399001 + 0.916951i $$0.630643\pi$$
$$954$$ 0 0
$$955$$ 5.50413e11 0.661721
$$956$$ 0 0
$$957$$ − 1.49628e11i − 0.178388i
$$958$$ 0 0
$$959$$ − 9.87647e11i − 1.16769i
$$960$$ 0 0
$$961$$ −5.29246e11 −0.620532
$$962$$ 0 0
$$963$$ −1.73151e12 −2.01335
$$964$$ 0 0
$$965$$ − 1.51346e10i − 0.0174527i
$$966$$ 0 0
$$967$$ − 5.41485e11i − 0.619271i −0.950855 0.309635i $$-0.899793\pi$$
0.950855 0.309635i $$-0.100207\pi$$
$$968$$ 0 0
$$969$$ 7.76826e11 0.881107
$$970$$ 0 0
$$971$$ 3.54981e11 0.399327 0.199663 0.979865i $$-0.436015\pi$$
0.199663 + 0.979865i $$0.436015\pi$$
$$972$$ 0 0
$$973$$ 8.97263e11i 1.00108i
$$974$$ 0 0
$$975$$ − 5.13541e11i − 0.568273i
$$976$$ 0 0
$$977$$ 6.02238e11 0.660982 0.330491 0.943809i $$-0.392786\pi$$
0.330491 + 0.943809i $$0.392786\pi$$
$$978$$ 0 0
$$979$$ 1.66371e12 1.81112
$$980$$ 0 0
$$981$$ − 2.12306e12i − 2.29238i
$$982$$ 0 0
$$983$$ 1.37465e12i 1.47223i 0.676854 + 0.736117i $$0.263343\pi$$
−0.676854 + 0.736117i $$0.736657\pi$$
$$984$$ 0 0
$$985$$ −5.73668e11 −0.609419
$$986$$ 0 0
$$987$$ −1.04058e12 −1.09649
$$988$$ 0 0
$$989$$ − 8.81113e10i − 0.0920972i
$$990$$ 0 0
$$991$$ − 1.01081e12i − 1.04803i −0.851709 0.524015i $$-0.824433\pi$$
0.851709 0.524015i $$-0.175567\pi$$
$$992$$ 0 0
$$993$$ 1.43411e12 1.47498
$$994$$ 0 0
$$995$$ −5.34061e11 −0.544877
$$996$$ 0 0
$$997$$ 3.28556e11i 0.332528i 0.986081 + 0.166264i $$0.0531704\pi$$
−0.986081 + 0.166264i $$0.946830\pi$$
$$998$$ 0 0
$$999$$ − 7.93172e11i − 0.796353i
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.f.127.4 4
4.3 odd 2 inner 256.9.d.f.127.2 4
8.3 odd 2 inner 256.9.d.f.127.3 4
8.5 even 2 inner 256.9.d.f.127.1 4
16.3 odd 4 64.9.c.d.63.1 2
16.5 even 4 16.9.c.a.15.1 2
16.11 odd 4 16.9.c.a.15.2 yes 2
16.13 even 4 64.9.c.d.63.2 2
48.5 odd 4 144.9.g.g.127.2 2
48.11 even 4 144.9.g.g.127.1 2
80.27 even 4 400.9.h.b.399.4 4
80.37 odd 4 400.9.h.b.399.1 4
80.43 even 4 400.9.h.b.399.2 4
80.53 odd 4 400.9.h.b.399.3 4
80.59 odd 4 400.9.b.c.351.1 2
80.69 even 4 400.9.b.c.351.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
16.9.c.a.15.1 2 16.5 even 4
16.9.c.a.15.2 yes 2 16.11 odd 4
64.9.c.d.63.1 2 16.3 odd 4
64.9.c.d.63.2 2 16.13 even 4
144.9.g.g.127.1 2 48.11 even 4
144.9.g.g.127.2 2 48.5 odd 4
256.9.d.f.127.1 4 8.5 even 2 inner
256.9.d.f.127.2 4 4.3 odd 2 inner
256.9.d.f.127.3 4 8.3 odd 2 inner
256.9.d.f.127.4 4 1.1 even 1 trivial
400.9.b.c.351.1 2 80.59 odd 4
400.9.b.c.351.2 2 80.69 even 4
400.9.h.b.399.1 4 80.37 odd 4
400.9.h.b.399.2 4 80.43 even 4
400.9.h.b.399.3 4 80.53 odd 4
400.9.h.b.399.4 4 80.27 even 4