# Properties

 Label 16.9.c.a.15.1 Level $16$ Weight $9$ Character 16.15 Analytic conductor $6.518$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,9,Mod(15,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("16.15");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 16.c (of order $$2$$, degree $$1$$, 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: $$6.51805776098$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 9$$ x^2 - x + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 15.1 Root $$0.500000 + 2.95804i$$ of defining polynomial Character $$\chi$$ $$=$$ 16.15 Dual form 16.9.c.a.15.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-141.986i q^{3} -510.000 q^{5} +2555.75i q^{7} -13599.0 q^{9} +O(q^{10})$$ $$q-141.986i q^{3} -510.000 q^{5} +2555.75i q^{7} -13599.0 q^{9} -19168.1i q^{11} -27710.0 q^{13} +72412.8i q^{15} +50370.0 q^{17} -108619. i q^{19} +362880. q^{21} -176347. i q^{23} -130525. q^{25} +999297. i q^{27} +54978.0 q^{29} -1.17564e6i q^{31} -2.72160e6 q^{33} -1.30343e6i q^{35} +793730. q^{37} +3.93443e6i q^{39} -75582.0 q^{41} -499648. i q^{43} +6.93549e6 q^{45} -2.86755e6i q^{47} -767039. q^{49} -7.15183e6i q^{51} +1.11662e7 q^{53} +9.77573e6i q^{55} -1.54224e7 q^{57} +2.18325e7i q^{59} -2.38266e7 q^{61} -3.47556e7i q^{63} +1.41321e7 q^{65} -7.49473e6i q^{67} -2.50387e7 q^{69} +1.00824e7i q^{71} +6.51661e6 q^{73} +1.85327e7i q^{75} +4.89888e7 q^{77} +4.87892e7i q^{79} +5.26630e7 q^{81} -7.34483e7i q^{83} -2.56887e7 q^{85} -7.80610e6i q^{87} +8.67958e7 q^{89} -7.08197e7i q^{91} -1.66925e8 q^{93} +5.53958e7i q^{95} -4.66703e7 q^{97} +2.60667e8i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1020 q^{5} - 27198 q^{9}+O(q^{10})$$ 2 * q - 1020 * q^5 - 27198 * q^9 $$2 q - 1020 q^{5} - 27198 q^{9} - 55420 q^{13} + 100740 q^{17} + 725760 q^{21} - 261050 q^{25} + 109956 q^{29} - 5443200 q^{33} + 1587460 q^{37} - 151164 q^{41} + 13870980 q^{45} - 1534078 q^{49} + 22332420 q^{53} - 30844800 q^{57} - 47653244 q^{61} + 28264200 q^{65} - 50077440 q^{69} + 13033220 q^{73} + 97977600 q^{77} + 105326082 q^{81} - 51377400 q^{85} + 173591556 q^{89} - 333849600 q^{93} - 93340540 q^{97}+O(q^{100})$$ 2 * q - 1020 * q^5 - 27198 * q^9 - 55420 * q^13 + 100740 * q^17 + 725760 * q^21 - 261050 * q^25 + 109956 * q^29 - 5443200 * q^33 + 1587460 * q^37 - 151164 * q^41 + 13870980 * q^45 - 1534078 * q^49 + 22332420 * q^53 - 30844800 * q^57 - 47653244 * q^61 + 28264200 * q^65 - 50077440 * q^69 + 13033220 * q^73 + 97977600 * q^77 + 105326082 * q^81 - 51377400 * q^85 + 173591556 * q^89 - 333849600 * q^93 - 93340540 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ − 141.986i − 1.75291i −0.481481 0.876456i $$-0.659901\pi$$
0.481481 0.876456i $$-0.340099\pi$$
$$4$$ 0 0
$$5$$ −510.000 −0.816000 −0.408000 0.912982i $$-0.633774\pi$$
−0.408000 + 0.912982i $$0.633774\pi$$
$$6$$ 0 0
$$7$$ 2555.75i 1.06445i 0.846603 + 0.532225i $$0.178644\pi$$
−0.846603 + 0.532225i $$0.821356\pi$$
$$8$$ 0 0
$$9$$ −13599.0 −2.07270
$$10$$ 0 0
$$11$$ − 19168.1i − 1.30921i −0.755972 0.654603i $$-0.772836\pi$$
0.755972 0.654603i $$-0.227164\pi$$
$$12$$ 0 0
$$13$$ −27710.0 −0.970204 −0.485102 0.874458i $$-0.661218\pi$$
−0.485102 + 0.874458i $$0.661218\pi$$
$$14$$ 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.i − 0.833474i −0.909027 0.416737i $$-0.863174\pi$$
0.909027 0.416737i $$-0.136826\pi$$
$$20$$ 0 0
$$21$$ 362880. 1.86589
$$22$$ 0 0
$$23$$ − 176347.i − 0.630167i −0.949064 0.315083i $$-0.897968\pi$$
0.949064 0.315083i $$-0.102032\pi$$
$$24$$ 0 0
$$25$$ −130525. −0.334144
$$26$$ 0 0
$$27$$ 999297.i 1.88035i
$$28$$ 0 0
$$29$$ 54978.0 0.0777315 0.0388657 0.999244i $$-0.487626\pi$$
0.0388657 + 0.999244i $$0.487626\pi$$
$$30$$ 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
$$35$$ − 1.30343e6i − 0.868592i
$$36$$ 0 0
$$37$$ 793730. 0.423512 0.211756 0.977323i $$-0.432082\pi$$
0.211756 + 0.977323i $$0.432082\pi$$
$$38$$ 0 0
$$39$$ 3.93443e6i 1.70068i
$$40$$ 0 0
$$41$$ −75582.0 −0.0267475 −0.0133737 0.999911i $$-0.504257\pi$$
−0.0133737 + 0.999911i $$0.504257\pi$$
$$42$$ 0 0
$$43$$ − 499648.i − 0.146147i −0.997327 0.0730736i $$-0.976719\pi$$
0.997327 0.0730736i $$-0.0232808\pi$$
$$44$$ 0 0
$$45$$ 6.93549e6 1.69133
$$46$$ 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
$$51$$ − 7.15183e6i − 1.05715i
$$52$$ 0 0
$$53$$ 1.11662e7 1.41515 0.707575 0.706639i $$-0.249789\pi$$
0.707575 + 0.706639i $$0.249789\pi$$
$$54$$ 0 0
$$55$$ 9.77573e6i 1.06831i
$$56$$ 0 0
$$57$$ −1.54224e7 −1.46101
$$58$$ 0 0
$$59$$ 2.18325e7i 1.80175i 0.434078 + 0.900875i $$0.357074\pi$$
−0.434078 + 0.900875i $$0.642926\pi$$
$$60$$ 0 0
$$61$$ −2.38266e7 −1.72085 −0.860425 0.509577i $$-0.829802\pi$$
−0.860425 + 0.509577i $$0.829802\pi$$
$$62$$ 0 0
$$63$$ − 3.47556e7i − 2.20629i
$$64$$ 0 0
$$65$$ 1.41321e7 0.791687
$$66$$ 0 0
$$67$$ − 7.49473e6i − 0.371926i −0.982557 0.185963i $$-0.940460\pi$$
0.982557 0.185963i $$-0.0595405\pi$$
$$68$$ 0 0
$$69$$ −2.50387e7 −1.10463
$$70$$ 0 0
$$71$$ 1.00824e7i 0.396763i 0.980125 + 0.198382i $$0.0635685\pi$$
−0.980125 + 0.198382i $$0.936431\pi$$
$$72$$ 0 0
$$73$$ 6.51661e6 0.229472 0.114736 0.993396i $$-0.463398\pi$$
0.114736 + 0.993396i $$0.463398\pi$$
$$74$$ 0 0
$$75$$ 1.85327e7i 0.585725i
$$76$$ 0 0
$$77$$ 4.89888e7 1.39359
$$78$$ 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
$$83$$ − 7.34483e7i − 1.54764i −0.633407 0.773819i $$-0.718344\pi$$
0.633407 0.773819i $$-0.281656\pi$$
$$84$$ 0 0
$$85$$ −2.56887e7 −0.492115
$$86$$ 0 0
$$87$$ − 7.80610e6i − 0.136256i
$$88$$ 0 0
$$89$$ 8.67958e7 1.38337 0.691685 0.722199i $$-0.256869\pi$$
0.691685 + 0.722199i $$0.256869\pi$$
$$90$$ 0 0
$$91$$ − 7.08197e7i − 1.03273i
$$92$$ 0 0
$$93$$ −1.66925e8 −2.23146
$$94$$ 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
$$99$$ 2.60667e8i 2.71360i
$$100$$ 0 0
$$101$$ 6.59910e7 0.634161 0.317080 0.948399i $$-0.397297\pi$$
0.317080 + 0.948399i $$0.397297\pi$$
$$102$$ 0 0
$$103$$ − 1.64884e8i − 1.46497i −0.680782 0.732486i $$-0.738360\pi$$
0.680782 0.732486i $$-0.261640\pi$$
$$104$$ 0 0
$$105$$ −1.85069e8 −1.52257
$$106$$ 0 0
$$107$$ − 1.27326e8i − 0.971364i −0.874136 0.485682i $$-0.838571\pi$$
0.874136 0.485682i $$-0.161429\pi$$
$$108$$ 0 0
$$109$$ −1.56119e8 −1.10598 −0.552992 0.833186i $$-0.686514\pi$$
−0.552992 + 0.833186i $$0.686514\pi$$
$$110$$ 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
$$115$$ 8.99367e7i 0.514216i
$$116$$ 0 0
$$117$$ 3.76828e8 2.01094
$$118$$ 0 0
$$119$$ 1.28733e8i 0.641951i
$$120$$ 0 0
$$121$$ −1.53057e8 −0.714023
$$122$$ 0 0
$$123$$ 1.07316e7i 0.0468860i
$$124$$ 0 0
$$125$$ 2.65786e8 1.08866
$$126$$ 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
$$131$$ − 2.16350e8i − 0.734636i −0.930095 0.367318i $$-0.880276\pi$$
0.930095 0.367318i $$-0.119724\pi$$
$$132$$ 0 0
$$133$$ 2.77603e8 0.887193
$$134$$ 0 0
$$135$$ − 5.09641e8i − 1.53437i
$$136$$ 0 0
$$137$$ −3.86442e8 −1.09699 −0.548494 0.836155i $$-0.684799\pi$$
−0.548494 + 0.836155i $$0.684799\pi$$
$$138$$ 0 0
$$139$$ − 3.51077e8i − 0.940465i −0.882543 0.470232i $$-0.844170\pi$$
0.882543 0.470232i $$-0.155830\pi$$
$$140$$ 0 0
$$141$$ −4.07151e8 −1.03010
$$142$$ 0 0
$$143$$ 5.31148e8i 1.27020i
$$144$$ 0 0
$$145$$ −2.80388e7 −0.0634289
$$146$$ 0 0
$$147$$ 1.08909e8i 0.233235i
$$148$$ 0 0
$$149$$ −4.54099e8 −0.921308 −0.460654 0.887580i $$-0.652385\pi$$
−0.460654 + 0.887580i $$0.652385\pi$$
$$150$$ 0 0
$$151$$ 6.60188e8i 1.26987i 0.772565 + 0.634936i $$0.218973\pi$$
−0.772565 + 0.634936i $$0.781027\pi$$
$$152$$ 0 0
$$153$$ −6.84982e8 −1.25001
$$154$$ 0 0
$$155$$ 5.99578e8i 1.03877i
$$156$$ 0 0
$$157$$ 4.35318e8 0.716486 0.358243 0.933628i $$-0.383376\pi$$
0.358243 + 0.933628i $$0.383376\pi$$
$$158$$ 0 0
$$159$$ − 1.58544e9i − 2.48063i
$$160$$ 0 0
$$161$$ 4.50697e8 0.670782
$$162$$ 0 0
$$163$$ − 2.44065e8i − 0.345744i −0.984944 0.172872i $$-0.944695\pi$$
0.984944 0.172872i $$-0.0553047\pi$$
$$164$$ 0 0
$$165$$ 1.38802e9 1.87266
$$166$$ 0 0
$$167$$ − 6.71351e8i − 0.863145i −0.902078 0.431573i $$-0.857959\pi$$
0.902078 0.431573i $$-0.142041\pi$$
$$168$$ 0 0
$$169$$ −4.78866e7 −0.0587040
$$170$$ 0 0
$$171$$ 1.47711e9i 1.72754i
$$172$$ 0 0
$$173$$ −1.76764e9 −1.97337 −0.986685 0.162644i $$-0.947998\pi$$
−0.986685 + 0.162644i $$0.947998\pi$$
$$174$$ 0 0
$$175$$ − 3.33589e8i − 0.355680i
$$176$$ 0 0
$$177$$ 3.09990e9 3.15831
$$178$$ 0 0
$$179$$ − 7.56967e8i − 0.737335i −0.929561 0.368668i $$-0.879814\pi$$
0.929561 0.368668i $$-0.120186\pi$$
$$180$$ 0 0
$$181$$ 6.27094e8 0.584277 0.292138 0.956376i $$-0.405633\pi$$
0.292138 + 0.956376i $$0.405633\pi$$
$$182$$ 0 0
$$183$$ 3.38304e9i 3.01650i
$$184$$ 0 0
$$185$$ −4.04802e8 −0.345586
$$186$$ 0 0
$$187$$ − 9.65497e8i − 0.789559i
$$188$$ 0 0
$$189$$ −2.55395e9 −2.00154
$$190$$ 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
$$195$$ − 2.00656e9i − 1.38776i
$$196$$ 0 0
$$197$$ −1.12484e9 −0.746837 −0.373419 0.927663i $$-0.621814\pi$$
−0.373419 + 0.927663i $$0.621814\pi$$
$$198$$ 0 0
$$199$$ − 1.04718e9i − 0.667742i −0.942619 0.333871i $$-0.891645\pi$$
0.942619 0.333871i $$-0.108355\pi$$
$$200$$ 0 0
$$201$$ −1.06415e9 −0.651954
$$202$$ 0 0
$$203$$ 1.40510e8i 0.0827413i
$$204$$ 0 0
$$205$$ 3.85468e7 0.0218259
$$206$$ 0 0
$$207$$ 2.39814e9i 1.30615i
$$208$$ 0 0
$$209$$ −2.08202e9 −1.09119
$$210$$ 0 0
$$211$$ 2.77676e9i 1.40090i 0.713699 + 0.700452i $$0.247018\pi$$
−0.713699 + 0.700452i $$0.752982\pi$$
$$212$$ 0 0
$$213$$ 1.43156e9 0.695491
$$214$$ 0 0
$$215$$ 2.54821e8i 0.119256i
$$216$$ 0 0
$$217$$ 3.00465e9 1.35505
$$218$$ 0 0
$$219$$ − 9.25267e8i − 0.402245i
$$220$$ 0 0
$$221$$ −1.39575e9 −0.585113
$$222$$ 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
$$227$$ 1.48863e9i 0.560639i 0.959907 + 0.280319i $$0.0904404\pi$$
−0.959907 + 0.280319i $$0.909560\pi$$
$$228$$ 0 0
$$229$$ 1.69447e9 0.616157 0.308079 0.951361i $$-0.400314\pi$$
0.308079 + 0.951361i $$0.400314\pi$$
$$230$$ 0 0
$$231$$ − 6.95572e9i − 2.44284i
$$232$$ 0 0
$$233$$ 5.21423e8 0.176916 0.0884580 0.996080i $$-0.471806\pi$$
0.0884580 + 0.996080i $$0.471806\pi$$
$$234$$ 0 0
$$235$$ 1.46245e9i 0.479523i
$$236$$ 0 0
$$237$$ 6.92738e9 2.19571
$$238$$ 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
$$243$$ − 9.21023e8i − 0.264147i
$$244$$ 0 0
$$245$$ 3.91190e8 0.108573
$$246$$ 0 0
$$247$$ 3.00984e9i 0.808640i
$$248$$ 0 0
$$249$$ −1.04286e10 −2.71287
$$250$$ 0 0
$$251$$ 1.31321e8i 0.0330855i 0.999863 + 0.0165428i $$0.00526597\pi$$
−0.999863 + 0.0165428i $$0.994734\pi$$
$$252$$ 0 0
$$253$$ −3.38023e9 −0.825019
$$254$$ 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
$$259$$ 2.02857e9i 0.450808i
$$260$$ 0 0
$$261$$ −7.47646e8 −0.161114
$$262$$ 0 0
$$263$$ − 7.38745e9i − 1.54409i −0.635570 0.772044i $$-0.719235\pi$$
0.635570 0.772044i $$-0.280765\pi$$
$$264$$ 0 0
$$265$$ −5.69477e9 −1.15476
$$266$$ 0 0
$$267$$ − 1.23238e10i − 2.42493i
$$268$$ 0 0
$$269$$ 3.46450e8 0.0661655 0.0330828 0.999453i $$-0.489468\pi$$
0.0330828 + 0.999453i $$0.489468\pi$$
$$270$$ 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
$$275$$ 2.50192e9i 0.437464i
$$276$$ 0 0
$$277$$ 2.15061e9 0.365293 0.182647 0.983179i $$-0.441534\pi$$
0.182647 + 0.983179i $$0.441534\pi$$
$$278$$ 0 0
$$279$$ 1.59876e10i 2.63855i
$$280$$ 0 0
$$281$$ 1.04256e10 1.67215 0.836074 0.548616i $$-0.184845\pi$$
0.836074 + 0.548616i $$0.184845\pi$$
$$282$$ 0 0
$$283$$ − 1.28042e9i − 0.199622i −0.995006 0.0998108i $$-0.968176\pi$$
0.995006 0.0998108i $$-0.0318238\pi$$
$$284$$ 0 0
$$285$$ 7.86542e9 1.19218
$$286$$ 0 0
$$287$$ − 1.93168e8i − 0.0284714i
$$288$$ 0 0
$$289$$ −4.43862e9 −0.636292
$$290$$ 0 0
$$291$$ 6.62652e9i 0.924089i
$$292$$ 0 0
$$293$$ −2.13786e9 −0.290074 −0.145037 0.989426i $$-0.546330\pi$$
−0.145037 + 0.989426i $$0.546330\pi$$
$$294$$ 0 0
$$295$$ − 1.11346e10i − 1.47023i
$$296$$ 0 0
$$297$$ 1.91546e10 2.46177
$$298$$ 0 0
$$299$$ 4.88656e9i 0.611390i
$$300$$ 0 0
$$301$$ 1.27697e9 0.155567
$$302$$ 0 0
$$303$$ − 9.36980e9i − 1.11163i
$$304$$ 0 0
$$305$$ 1.21516e10 1.40421
$$306$$ 0 0
$$307$$ − 5.45140e9i − 0.613698i −0.951758 0.306849i $$-0.900725\pi$$
0.951758 0.306849i $$-0.0992747\pi$$
$$308$$ 0 0
$$309$$ −2.34112e10 −2.56797
$$310$$ 0 0
$$311$$ − 1.07550e10i − 1.14965i −0.818275 0.574827i $$-0.805069\pi$$
0.818275 0.574827i $$-0.194931\pi$$
$$312$$ 0 0
$$313$$ −2.99804e8 −0.0312364 −0.0156182 0.999878i $$-0.504972\pi$$
−0.0156182 + 0.999878i $$0.504972\pi$$
$$314$$ 0 0
$$315$$ 1.77254e10i 1.80033i
$$316$$ 0 0
$$317$$ 5.31172e9 0.526015 0.263007 0.964794i $$-0.415286\pi$$
0.263007 + 0.964794i $$0.415286\pi$$
$$318$$ 0 0
$$319$$ − 1.05382e9i − 0.101767i
$$320$$ 0 0
$$321$$ −1.80785e10 −1.70272
$$322$$ 0 0
$$323$$ − 5.47115e9i − 0.502653i
$$324$$ 0 0
$$325$$ 3.61685e9 0.324188
$$326$$ 0 0
$$327$$ 2.21667e10i 1.93869i
$$328$$ 0 0
$$329$$ 7.32872e9 0.625525
$$330$$ 0 0
$$331$$ 1.01004e10i 0.841446i 0.907189 + 0.420723i $$0.138224\pi$$
−0.907189 + 0.420723i $$0.861776\pi$$
$$332$$ 0 0
$$333$$ −1.07939e10 −0.877815
$$334$$ 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
$$339$$ − 3.35578e10i − 2.54094i
$$340$$ 0 0
$$341$$ −2.25348e10 −1.66662
$$342$$ 0 0
$$343$$ 1.27730e10i 0.922820i
$$344$$ 0 0
$$345$$ 1.27697e10 0.901376
$$346$$ 0 0
$$347$$ − 1.27822e10i − 0.881636i −0.897597 0.440818i $$-0.854688\pi$$
0.897597 0.440818i $$-0.145312\pi$$
$$348$$ 0 0
$$349$$ −6.39381e8 −0.0430981 −0.0215490 0.999768i $$-0.506860\pi$$
−0.0215490 + 0.999768i $$0.506860\pi$$
$$350$$ 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
$$355$$ − 5.14203e9i − 0.323759i
$$356$$ 0 0
$$357$$ 1.82783e10 1.12528
$$358$$ 0 0
$$359$$ 2.22541e10i 1.33978i 0.742461 + 0.669889i $$0.233658\pi$$
−0.742461 + 0.669889i $$0.766342\pi$$
$$360$$ 0 0
$$361$$ 5.18543e9 0.305320
$$362$$ 0 0
$$363$$ 2.17320e10i 1.25162i
$$364$$ 0 0
$$365$$ −3.32347e9 −0.187249
$$366$$ 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
$$371$$ 2.85380e10i 1.50636i
$$372$$ 0 0
$$373$$ 1.19680e10 0.618283 0.309141 0.951016i $$-0.399958\pi$$
0.309141 + 0.951016i $$0.399958\pi$$
$$374$$ 0 0
$$375$$ − 3.77379e10i − 1.90833i
$$376$$ 0 0
$$377$$ −1.52344e9 −0.0754154
$$378$$ 0 0
$$379$$ − 2.30787e10i − 1.11855i −0.828982 0.559275i $$-0.811080\pi$$
0.828982 0.559275i $$-0.188920\pi$$
$$380$$ 0 0
$$381$$ 5.22141e10 2.47792
$$382$$ 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
$$387$$ 6.79472e9i 0.302920i
$$388$$ 0 0
$$389$$ −2.73457e10 −1.19424 −0.597119 0.802152i $$-0.703688\pi$$
−0.597119 + 0.802152i $$0.703688\pi$$
$$390$$ 0 0
$$391$$ − 8.88257e9i − 0.380042i
$$392$$ 0 0
$$393$$ −3.07187e10 −1.28775
$$394$$ 0 0
$$395$$ − 2.48825e10i − 1.02213i
$$396$$ 0 0
$$397$$ 3.99456e10 1.60808 0.804039 0.594576i $$-0.202680\pi$$
0.804039 + 0.594576i $$0.202680\pi$$
$$398$$ 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
$$403$$ 3.25771e10i 1.23507i
$$404$$ 0 0
$$405$$ −2.68582e10 −0.998288
$$406$$ 0 0
$$407$$ − 1.52143e10i − 0.554465i
$$408$$ 0 0
$$409$$ 1.14283e10 0.408404 0.204202 0.978929i $$-0.434540\pi$$
0.204202 + 0.978929i $$0.434540\pi$$
$$410$$ 0 0
$$411$$ 5.48693e10i 1.92292i
$$412$$ 0 0
$$413$$ −5.57982e10 −1.91788
$$414$$ 0 0
$$415$$ 3.74586e10i 1.26287i
$$416$$ 0 0
$$417$$ −4.98479e10 −1.64855
$$418$$ 0 0
$$419$$ − 1.10009e10i − 0.356922i −0.983947 0.178461i $$-0.942888\pi$$
0.983947 0.178461i $$-0.0571119\pi$$
$$420$$ 0 0
$$421$$ 2.28766e10 0.728220 0.364110 0.931356i $$-0.381373\pi$$
0.364110 + 0.931356i $$0.381373\pi$$
$$422$$ 0 0
$$423$$ 3.89958e10i 1.21802i
$$424$$ 0 0
$$425$$ −6.57454e9 −0.201516
$$426$$ 0 0
$$427$$ − 6.08948e10i − 1.83176i
$$428$$ 0 0
$$429$$ 7.54155e10 2.22655
$$430$$ 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
$$435$$ 3.98111e9i 0.111185i
$$436$$ 0 0
$$437$$ −1.91546e10 −0.525228
$$438$$ 0 0
$$439$$ − 6.40288e10i − 1.72392i −0.506976 0.861960i $$-0.669237\pi$$
0.506976 0.861960i $$-0.330763\pi$$
$$440$$ 0 0
$$441$$ 1.04310e10 0.275785
$$442$$ 0 0
$$443$$ − 7.47659e10i − 1.94128i −0.240533 0.970641i $$-0.577322\pi$$
0.240533 0.970641i $$-0.422678\pi$$
$$444$$ 0 0
$$445$$ −4.42658e10 −1.12883
$$446$$ 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
$$451$$ 1.44876e9i 0.0350180i
$$452$$ 0 0
$$453$$ 9.37373e10 2.22597
$$454$$ 0 0
$$455$$ 3.61181e10i 0.842711i
$$456$$ 0 0
$$457$$ 4.66828e9 0.107027 0.0535133 0.998567i $$-0.482958\pi$$
0.0535133 + 0.998567i $$0.482958\pi$$
$$458$$ 0 0
$$459$$ 5.03346e10i 1.13401i
$$460$$ 0 0
$$461$$ −3.88096e10 −0.859281 −0.429641 0.903000i $$-0.641360\pi$$
−0.429641 + 0.903000i $$0.641360\pi$$
$$462$$ 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
$$467$$ 2.31902e10i 0.487570i 0.969829 + 0.243785i $$0.0783891\pi$$
−0.969829 + 0.243785i $$0.921611\pi$$
$$468$$ 0 0
$$469$$ 1.91546e10 0.395897
$$470$$ 0 0
$$471$$ − 6.18090e10i − 1.25594i
$$472$$ 0 0
$$473$$ −9.57731e9 −0.191337
$$474$$ 0 0
$$475$$ 1.41775e10i 0.278500i
$$476$$ 0 0
$$477$$ −1.51849e11 −2.93318
$$478$$ 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
$$483$$ − 6.39926e10i − 1.17582i
$$484$$ 0 0
$$485$$ 2.38018e10 0.430173
$$486$$ 0 0
$$487$$ − 3.03878e10i − 0.540236i −0.962827 0.270118i $$-0.912937\pi$$
0.962827 0.270118i $$-0.0870628\pi$$
$$488$$ 0 0
$$489$$ −3.46538e10 −0.606059
$$490$$ 0 0
$$491$$ 5.56483e10i 0.957472i 0.877959 + 0.478736i $$0.158905\pi$$
−0.877959 + 0.478736i $$0.841095\pi$$
$$492$$ 0 0
$$493$$ 2.76924e9 0.0468784
$$494$$ 0 0
$$495$$ − 1.32940e11i − 2.21429i
$$496$$ 0 0
$$497$$ −2.57681e10 −0.422335
$$498$$ 0 0
$$499$$ 7.88458e10i 1.27168i 0.771822 + 0.635838i $$0.219345\pi$$
−0.771822 + 0.635838i $$0.780655\pi$$
$$500$$ 0 0
$$501$$ −9.53224e10 −1.51302
$$502$$ 0 0
$$503$$ − 4.41092e10i − 0.689061i −0.938775 0.344530i $$-0.888038\pi$$
0.938775 0.344530i $$-0.111962\pi$$
$$504$$ 0 0
$$505$$ −3.36554e10 −0.517475
$$506$$ 0 0
$$507$$ 6.79923e9i 0.102903i
$$508$$ 0 0
$$509$$ 1.05927e10 0.157811 0.0789055 0.996882i $$-0.474857\pi$$
0.0789055 + 0.996882i $$0.474857\pi$$
$$510$$ 0 0
$$511$$ 1.66548e10i 0.244262i
$$512$$ 0 0
$$513$$ 1.08543e11 1.56723
$$514$$ 0 0
$$515$$ 8.40908e10i 1.19542i
$$516$$ 0 0
$$517$$ −5.49654e10 −0.769356
$$518$$ 0 0
$$519$$ 2.50979e11i 3.45914i
$$520$$ 0 0
$$521$$ 1.24958e11 1.69595 0.847973 0.530039i $$-0.177823\pi$$
0.847973 + 0.530039i $$0.177823\pi$$
$$522$$ 0 0
$$523$$ − 2.80408e10i − 0.374786i −0.982285 0.187393i $$-0.939996\pi$$
0.982285 0.187393i $$-0.0600038\pi$$
$$524$$ 0 0
$$525$$ −4.73649e10 −0.623476
$$526$$ 0 0
$$527$$ − 5.92172e10i − 0.767724i
$$528$$ 0 0
$$529$$ 4.72129e10 0.602890
$$530$$ 0 0
$$531$$ − 2.96900e11i − 3.73449i
$$532$$ 0 0
$$533$$ 2.09438e9 0.0259505
$$534$$ 0 0
$$535$$ 6.49363e10i 0.792633i
$$536$$ 0 0
$$537$$ −1.07479e11 −1.29248
$$538$$ 0 0
$$539$$ 1.47027e10i 0.174197i
$$540$$ 0 0
$$541$$ 1.44659e11 1.68871 0.844356 0.535782i $$-0.179983\pi$$
0.844356 + 0.535782i $$0.179983\pi$$
$$542$$ 0 0
$$543$$ − 8.90386e10i − 1.02419i
$$544$$ 0 0
$$545$$ 7.96205e10 0.902483
$$546$$ 0 0
$$547$$ − 1.03774e10i − 0.115915i −0.998319 0.0579573i $$-0.981541\pi$$
0.998319 0.0579573i $$-0.0184587\pi$$
$$548$$ 0 0
$$549$$ 3.24018e11 3.56681
$$550$$ 0 0
$$551$$ − 5.97167e9i − 0.0647872i
$$552$$ 0 0
$$553$$ −1.24693e11 −1.33334
$$554$$ 0 0
$$555$$ 5.74762e10i 0.605782i
$$556$$ 0 0
$$557$$ −5.47312e9 −0.0568610 −0.0284305 0.999596i $$-0.509051\pi$$
−0.0284305 + 0.999596i $$0.509051\pi$$
$$558$$ 0 0
$$559$$ 1.38453e10i 0.141793i
$$560$$ 0 0
$$561$$ −1.37087e11 −1.38403
$$562$$ 0 0
$$563$$ 4.36118e10i 0.434081i 0.976163 + 0.217040i $$0.0696403\pi$$
−0.976163 + 0.217040i $$0.930360\pi$$
$$564$$ 0 0
$$565$$ −1.20536e11 −1.18283
$$566$$ 0 0
$$567$$ 1.34593e11i 1.30224i
$$568$$ 0 0
$$569$$ 1.27822e10 0.121943 0.0609716 0.998140i $$-0.480580\pi$$
0.0609716 + 0.998140i $$0.480580\pi$$
$$570$$ 0 0
$$571$$ 7.59455e10i 0.714427i 0.934023 + 0.357213i $$0.116273\pi$$
−0.934023 + 0.357213i $$0.883727\pi$$
$$572$$ 0 0
$$573$$ −1.53237e11 −1.42149
$$574$$ 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
$$579$$ 4.21353e9i 0.0374914i
$$580$$ 0 0
$$581$$ 1.87715e11 1.64738
$$582$$ 0 0
$$583$$ − 2.14035e11i − 1.85272i
$$584$$ 0 0
$$585$$ −1.92182e11 −1.64093
$$586$$ 0 0
$$587$$ − 6.07298e10i − 0.511504i −0.966742 0.255752i $$-0.917677\pi$$
0.966742 0.255752i $$-0.0823231\pi$$
$$588$$ 0 0
$$589$$ −1.27697e11 −1.06101
$$590$$ 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
$$595$$ − 6.56538e10i − 0.523832i
$$596$$ 0 0
$$597$$ −1.48685e11 −1.17049
$$598$$ 0 0
$$599$$ − 2.40647e11i − 1.86927i −0.355607 0.934636i $$-0.615726\pi$$
0.355607 0.934636i $$-0.384274\pi$$
$$600$$ 0 0
$$601$$ −1.92942e11 −1.47887 −0.739434 0.673229i $$-0.764907\pi$$
−0.739434 + 0.673229i $$0.764907\pi$$
$$602$$ 0 0
$$603$$ 1.01921e11i 0.770892i
$$604$$ 0 0
$$605$$ 7.80591e10 0.582643
$$606$$ 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
$$611$$ 7.94597e10i 0.570141i
$$612$$ 0 0
$$613$$ 1.76424e11 1.24944 0.624722 0.780847i $$-0.285212\pi$$
0.624722 + 0.780847i $$0.285212\pi$$
$$614$$ 0 0
$$615$$ − 5.47311e9i − 0.0382590i
$$616$$ 0 0
$$617$$ −9.84986e10 −0.679656 −0.339828 0.940488i $$-0.610369\pi$$
−0.339828 + 0.940488i $$0.610369\pi$$
$$618$$ 0 0
$$619$$ 1.28596e10i 0.0875923i 0.999040 + 0.0437961i $$0.0139452\pi$$
−0.999040 + 0.0437961i $$0.986055\pi$$
$$620$$ 0 0
$$621$$ 1.76223e11 1.18494
$$622$$ 0 0
$$623$$ 2.21828e11i 1.47253i
$$624$$ 0 0
$$625$$ −8.45648e10 −0.554204
$$626$$ 0 0
$$627$$ 2.95618e11i 1.91276i
$$628$$ 0 0
$$629$$ 3.99802e10 0.255413
$$630$$ 0 0
$$631$$ 1.73463e11i 1.09418i 0.837073 + 0.547091i $$0.184265\pi$$
−0.837073 + 0.547091i $$0.815735\pi$$
$$632$$ 0 0
$$633$$ 3.94261e11 2.45566
$$634$$ 0 0
$$635$$ − 1.87548e11i − 1.15350i
$$636$$ 0 0
$$637$$ 2.12547e10 0.129091
$$638$$ 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
$$643$$ 7.04067e10i 0.411879i 0.978565 + 0.205940i $$0.0660250\pi$$
−0.978565 + 0.205940i $$0.933975\pi$$
$$644$$ 0 0
$$645$$ 3.61810e10 0.209046
$$646$$ 0 0
$$647$$ − 1.99175e11i − 1.13663i −0.822812 0.568314i $$-0.807596\pi$$
0.822812 0.568314i $$-0.192404\pi$$
$$648$$ 0 0
$$649$$ 4.18487e11 2.35886
$$650$$ 0 0
$$651$$ − 4.26617e11i − 2.37528i
$$652$$ 0 0
$$653$$ 6.49972e9 0.0357472 0.0178736 0.999840i $$-0.494310\pi$$
0.0178736 + 0.999840i $$0.494310\pi$$
$$654$$ 0 0
$$655$$ 1.10339e11i 0.599463i
$$656$$ 0 0
$$657$$ −8.86194e10 −0.475628
$$658$$ 0 0
$$659$$ − 2.20982e11i − 1.17170i −0.810421 0.585848i $$-0.800761\pi$$
0.810421 0.585848i $$-0.199239\pi$$
$$660$$ 0 0
$$661$$ −2.69549e11 −1.41199 −0.705995 0.708217i $$-0.749500\pi$$
−0.705995 + 0.708217i $$0.749500\pi$$
$$662$$ 0 0
$$663$$ 1.98177e11i 1.02565i
$$664$$ 0 0
$$665$$ −1.41578e11 −0.723949
$$666$$ 0 0
$$667$$ − 9.69518e9i − 0.0489838i
$$668$$ 0 0
$$669$$ −3.23476e11 −1.61487
$$670$$ 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
$$675$$ − 1.30433e11i − 0.628309i
$$676$$ 0 0
$$677$$ 8.02735e10 0.382136 0.191068 0.981577i $$-0.438805\pi$$
0.191068 + 0.981577i $$0.438805\pi$$
$$678$$ 0 0
$$679$$ − 1.19277e11i − 0.561150i
$$680$$ 0 0
$$681$$ 2.11364e11 0.982751
$$682$$ 0 0
$$683$$ 3.00783e11i 1.38220i 0.722760 + 0.691099i $$0.242873\pi$$
−0.722760 + 0.691099i $$0.757127\pi$$
$$684$$ 0 0
$$685$$ 1.97085e11 0.895142
$$686$$ 0 0
$$687$$ − 2.40591e11i − 1.08007i
$$688$$ 0 0
$$689$$ −3.09416e11 −1.37298
$$690$$ 0 0
$$691$$ 3.06208e11i 1.34309i 0.740964 + 0.671544i $$0.234369\pi$$
−0.740964 + 0.671544i $$0.765631\pi$$
$$692$$ 0 0
$$693$$ −6.66199e11 −2.88849
$$694$$ 0 0
$$695$$ 1.79049e11i 0.767419i
$$696$$ 0 0
$$697$$ −3.80707e9 −0.0161309
$$698$$ 0 0
$$699$$ − 7.40348e10i − 0.310118i
$$700$$ 0 0
$$701$$ −2.73603e11 −1.13305 −0.566524 0.824045i $$-0.691712\pi$$
−0.566524 + 0.824045i $$0.691712\pi$$
$$702$$ 0 0
$$703$$ − 8.62143e10i − 0.352987i
$$704$$ 0 0
$$705$$ 2.07647e11 0.840562
$$706$$ 0 0
$$707$$ 1.68656e11i 0.675033i
$$708$$ 0 0
$$709$$ −1.76662e11 −0.699129 −0.349564 0.936912i $$-0.613670\pi$$
−0.349564 + 0.936912i $$0.613670\pi$$
$$710$$ 0 0
$$711$$ − 6.63484e11i − 2.59628i
$$712$$ 0 0
$$713$$ −2.07321e11 −0.802203
$$714$$ 0 0
$$715$$ − 2.70885e11i − 1.03648i
$$716$$ 0 0
$$717$$ 6.25718e11 2.36756
$$718$$ 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
$$723$$ 2.30227e11i 0.842565i
$$724$$ 0 0
$$725$$ −7.17600e9 −0.0259735
$$726$$ 0 0
$$727$$ 2.87080e11i 1.02770i 0.857881 + 0.513849i $$0.171781\pi$$
−0.857881 + 0.513849i $$0.828219\pi$$
$$728$$ 0 0
$$729$$ 2.14750e11 0.760366
$$730$$ 0 0
$$731$$ − 2.51673e10i − 0.0881388i
$$732$$ 0 0
$$733$$ −2.94176e11 −1.01904 −0.509520 0.860459i $$-0.670177\pi$$
−0.509520 + 0.860459i $$0.670177\pi$$
$$734$$ 0 0
$$735$$ − 5.55435e10i − 0.190320i
$$736$$ 0 0
$$737$$ −1.43660e11 −0.486928
$$738$$ 0 0
$$739$$ − 9.69888e10i − 0.325195i −0.986692 0.162598i $$-0.948013\pi$$
0.986692 0.162598i $$-0.0519872\pi$$
$$740$$ 0 0
$$741$$ 4.27355e11 1.41748
$$742$$ 0 0
$$743$$ − 1.01567e11i − 0.333271i −0.986019 0.166635i $$-0.946710\pi$$
0.986019 0.166635i $$-0.0532902\pi$$
$$744$$ 0 0
$$745$$ 2.31590e11 0.751788
$$746$$ 0 0
$$747$$ 9.98824e11i 3.20779i
$$748$$ 0 0
$$749$$ 3.25413e11 1.03397
$$750$$ 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
$$755$$ − 3.36696e11i − 1.03621i
$$756$$ 0 0
$$757$$ 1.82006e11 0.554244 0.277122 0.960835i $$-0.410619\pi$$
0.277122 + 0.960835i $$0.410619\pi$$
$$758$$ 0 0
$$759$$ 4.79945e11i 1.44619i
$$760$$ 0 0
$$761$$ −4.27419e11 −1.27443 −0.637213 0.770687i $$-0.719913\pi$$
−0.637213 + 0.770687i $$0.719913\pi$$
$$762$$ 0 0
$$763$$ − 3.99000e11i − 1.17727i
$$764$$ 0 0
$$765$$ 3.49341e11 1.02001
$$766$$ 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
$$771$$ − 7.49386e11i − 2.12074i
$$772$$ 0 0
$$773$$ −1.49408e11 −0.418462 −0.209231 0.977866i $$-0.567096\pi$$
−0.209231 + 0.977866i $$0.567096\pi$$
$$774$$ 0 0
$$775$$ 1.53451e11i 0.425366i
$$776$$ 0 0
$$777$$ 2.88029e11 0.790227
$$778$$ 0 0
$$779$$ 8.20966e9i 0.0222933i
$$780$$ 0 0
$$781$$ 1.93261e11 0.519445
$$782$$ 0 0
$$783$$ 5.49393e10i 0.146163i
$$784$$ 0 0
$$785$$ −2.22012e11 −0.584652
$$786$$ 0 0
$$787$$ − 7.33252e11i − 1.91141i −0.294323 0.955706i $$-0.595094\pi$$
0.294323 0.955706i $$-0.404906\pi$$
$$788$$ 0 0
$$789$$ −1.04891e12 −2.70665
$$790$$ 0 0
$$791$$ 6.04040e11i 1.54298i
$$792$$ 0 0
$$793$$ 6.60236e11 1.66958
$$794$$ 0 0
$$795$$ 8.08577e11i 2.02420i
$$796$$ 0 0
$$797$$ 3.02703e11 0.750212 0.375106 0.926982i $$-0.377606\pi$$
0.375106 + 0.926982i $$0.377606\pi$$
$$798$$ 0 0
$$799$$ − 1.44438e11i − 0.354401i
$$800$$ 0 0
$$801$$ −1.18034e12 −2.86732
$$802$$ 0 0
$$803$$ − 1.24911e11i − 0.300427i
$$804$$ 0 0
$$805$$ −2.29855e11 −0.547358
$$806$$ 0 0
$$807$$ − 4.91911e10i − 0.115982i
$$808$$ 0 0
$$809$$ −5.84316e11 −1.36412 −0.682062 0.731295i $$-0.738916\pi$$
−0.682062 + 0.731295i $$0.738916\pi$$
$$810$$ 0 0
$$811$$ 1.21470e11i 0.280793i 0.990095 + 0.140396i $$0.0448377\pi$$
−0.990095 + 0.140396i $$0.955162\pi$$
$$812$$ 0 0
$$813$$ −9.25344e8 −0.00211807
$$814$$ 0 0
$$815$$ 1.24473e11i 0.282127i
$$816$$ 0 0
$$817$$ −5.42714e10 −0.121810
$$818$$ 0 0
$$819$$ 9.63078e11i 2.14055i
$$820$$ 0 0
$$821$$ 4.52470e11 0.995903 0.497952 0.867205i $$-0.334086\pi$$
0.497952 + 0.867205i $$0.334086\pi$$
$$822$$ 0 0
$$823$$ − 3.06704e11i − 0.668528i −0.942479 0.334264i $$-0.891512\pi$$
0.942479 0.334264i $$-0.108488\pi$$
$$824$$ 0 0
$$825$$ 3.55237e11 0.766835
$$826$$ 0 0
$$827$$ − 2.93276e11i − 0.626982i −0.949591 0.313491i $$-0.898501\pi$$
0.949591 0.313491i $$-0.101499\pi$$
$$828$$ 0 0
$$829$$ 3.35532e11 0.710421 0.355210 0.934786i $$-0.384409\pi$$
0.355210 + 0.934786i $$0.384409\pi$$
$$830$$ 0 0
$$831$$ − 3.05356e11i − 0.640327i
$$832$$ 0 0
$$833$$ −3.86358e10 −0.0802434
$$834$$ 0 0
$$835$$ 3.42389e11i 0.704326i
$$836$$ 0 0
$$837$$ 1.17482e12 2.39369
$$838$$ 0 0
$$839$$ 3.42844e11i 0.691908i 0.938252 + 0.345954i $$0.112445\pi$$
−0.938252 + 0.345954i $$0.887555\pi$$
$$840$$ 0 0
$$841$$ −4.97224e11 −0.993958
$$842$$ 0 0
$$843$$ − 1.48029e12i − 2.93113i
$$844$$ 0 0
$$845$$ 2.44222e10 0.0479024
$$846$$ 0 0
$$847$$ − 3.91175e11i − 0.760042i
$$848$$ 0 0
$$849$$ −1.81802e11 −0.349919
$$850$$ 0 0
$$851$$ − 1.39972e11i − 0.266883i
$$852$$ 0 0
$$853$$ −5.08662e11 −0.960801 −0.480400 0.877049i $$-0.659509\pi$$
−0.480400 + 0.877049i $$0.659509\pi$$
$$854$$ 0 0
$$855$$ − 7.53328e11i − 1.40968i
$$856$$ 0 0
$$857$$ 6.06764e11 1.12486 0.562428 0.826846i $$-0.309867\pi$$
0.562428 + 0.826846i $$0.309867\pi$$
$$858$$ 0 0
$$859$$ 9.49431e11i 1.74378i 0.489705 + 0.871888i $$0.337105\pi$$
−0.489705 + 0.871888i $$0.662895\pi$$
$$860$$ 0 0
$$861$$ −2.74272e10 −0.0499078
$$862$$ 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
$$867$$ 6.30222e11i 1.11536i
$$868$$ 0 0
$$869$$ 9.35196e11 1.63992
$$870$$ 0 0
$$871$$ 2.07679e11i 0.360844i
$$872$$ 0 0
$$873$$ 6.34669e11 1.09267
$$874$$ 0 0
$$875$$ 6.79283e11i 1.15883i
$$876$$ 0 0
$$877$$ −8.80195e11 −1.48792 −0.743961 0.668223i $$-0.767055\pi$$
−0.743961 + 0.668223i $$0.767055\pi$$
$$878$$ 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
$$883$$ 7.60446e11i 1.25091i 0.780261 + 0.625454i $$0.215086\pi$$
−0.780261 + 0.625454i $$0.784914\pi$$
$$884$$ 0 0
$$885$$ −1.58095e12 −2.57718
$$886$$ 0 0
$$887$$ − 3.34097e11i − 0.539732i −0.962898 0.269866i $$-0.913021\pi$$
0.962898 0.269866i $$-0.0869794\pi$$
$$888$$ 0 0
$$889$$ −9.39853e11 −1.50471
$$890$$ 0 0
$$891$$ − 1.00945e12i − 1.60167i
$$892$$ 0 0
$$893$$ −3.11471e11 −0.489792
$$894$$ 0 0
$$895$$ 3.86053e11i 0.601666i
$$896$$ 0 0
$$897$$ 6.93823e11 1.07171
$$898$$ 0 0
$$899$$ − 6.46345e10i − 0.0989523i
$$900$$ 0 0
$$901$$ 5.62442e11 0.853451
$$902$$ 0 0
$$903$$ − 1.81312e11i − 0.272695i
$$904$$ 0 0
$$905$$ −3.19818e11 −0.476770
$$906$$ 0 0
$$907$$ − 7.65213e11i − 1.13071i −0.824846 0.565357i $$-0.808738\pi$$
0.824846 0.565357i $$-0.191262\pi$$
$$908$$ 0 0
$$909$$ −8.97412e11 −1.31443
$$910$$ 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
$$915$$ − 1.72535e12i − 2.46146i
$$916$$ 0 0
$$917$$ 5.52937e11 0.781984
$$918$$ 0 0
$$919$$ − 6.82775e11i − 0.957229i −0.878025 0.478615i $$-0.841139\pi$$
0.878025 0.478615i $$-0.158861\pi$$
$$920$$ 0 0
$$921$$ −7.74022e11 −1.07576
$$922$$ 0 0
$$923$$ − 2.79384e11i − 0.384941i
$$924$$ 0 0
$$925$$ −1.03602e11 −0.141514
$$926$$ 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
$$931$$ 8.33152e10i 0.110898i
$$932$$ 0 0
$$933$$ −1.52705e12 −2.01524
$$934$$ 0 0
$$935$$ 4.92404e11i 0.644280i
$$936$$ 0 0
$$937$$ 1.03941e12 1.34843 0.674217 0.738533i $$-0.264481\pi$$
0.674217 + 0.738533i $$0.264481\pi$$
$$938$$ 0 0
$$939$$ 4.25680e10i 0.0547546i
$$940$$ 0 0
$$941$$ 1.26891e11 0.161836 0.0809178 0.996721i $$-0.474215\pi$$
0.0809178 + 0.996721i $$0.474215\pi$$
$$942$$ 0 0
$$943$$ 1.33286e10i 0.0168554i
$$944$$ 0 0
$$945$$ 1.30251e12 1.63326
$$946$$ 0 0
$$947$$ 6.13064e11i 0.762265i 0.924520 + 0.381133i $$0.124466\pi$$
−0.924520 + 0.381133i $$0.875534\pi$$
$$948$$ 0 0
$$949$$ −1.80575e11 −0.222635
$$950$$ 0 0
$$951$$ − 7.54189e11i − 0.922058i
$$952$$ 0 0
$$953$$ 6.58227e11 0.798002 0.399001 0.916951i $$-0.369357\pi$$
0.399001 + 0.916951i $$0.369357\pi$$
$$954$$ 0 0
$$955$$ 5.50413e11i 0.661721i
$$956$$ 0 0
$$957$$ −1.49628e11 −0.178388
$$958$$ 0 0
$$959$$ − 9.87647e11i − 1.16769i
$$960$$ 0 0
$$961$$ −5.29246e11 −0.620532
$$962$$ 0 0
$$963$$ 1.73151e12i 2.01335i
$$964$$ 0 0
$$965$$ 1.51346e10 0.0174527
$$966$$ 0 0
$$967$$ 5.41485e11i 0.619271i 0.950855 + 0.309635i $$0.100207\pi$$
−0.950855 + 0.309635i $$0.899793\pi$$
$$968$$ 0 0
$$969$$ −7.76826e11 −0.881107
$$970$$ 0 0
$$971$$ 3.54981e11i 0.399327i 0.979865 + 0.199663i $$0.0639849\pi$$
−0.979865 + 0.199663i $$0.936015\pi$$
$$972$$ 0 0
$$973$$ 8.97263e11 1.00108
$$974$$ 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
$$979$$ − 1.66371e12i − 1.81112i
$$980$$ 0 0
$$981$$ 2.12306e12 2.29238
$$982$$ 0 0
$$983$$ − 1.37465e12i − 1.47223i −0.676854 0.736117i $$-0.736657\pi$$
0.676854 0.736117i $$-0.263343\pi$$
$$984$$ 0 0
$$985$$ 5.73668e11 0.609419
$$986$$ 0 0
$$987$$ − 1.04058e12i − 1.09649i
$$988$$ 0 0
$$989$$ −8.81113e10 −0.0920972
$$990$$ 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
$$995$$ 5.34061e11i 0.544877i
$$996$$ 0 0
$$997$$ −3.28556e11 −0.332528 −0.166264 0.986081i $$-0.553170\pi$$
−0.166264 + 0.986081i $$0.553170\pi$$
$$998$$ 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 16.9.c.a.15.1 2
3.2 odd 2 144.9.g.g.127.2 2
4.3 odd 2 inner 16.9.c.a.15.2 yes 2
5.2 odd 4 400.9.h.b.399.1 4
5.3 odd 4 400.9.h.b.399.3 4
5.4 even 2 400.9.b.c.351.2 2
8.3 odd 2 64.9.c.d.63.1 2
8.5 even 2 64.9.c.d.63.2 2
12.11 even 2 144.9.g.g.127.1 2
16.3 odd 4 256.9.d.f.127.2 4
16.5 even 4 256.9.d.f.127.1 4
16.11 odd 4 256.9.d.f.127.3 4
16.13 even 4 256.9.d.f.127.4 4
20.3 even 4 400.9.h.b.399.2 4
20.7 even 4 400.9.h.b.399.4 4
20.19 odd 2 400.9.b.c.351.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
16.9.c.a.15.1 2 1.1 even 1 trivial
16.9.c.a.15.2 yes 2 4.3 odd 2 inner
64.9.c.d.63.1 2 8.3 odd 2
64.9.c.d.63.2 2 8.5 even 2
144.9.g.g.127.1 2 12.11 even 2
144.9.g.g.127.2 2 3.2 odd 2
256.9.d.f.127.1 4 16.5 even 4
256.9.d.f.127.2 4 16.3 odd 4
256.9.d.f.127.3 4 16.11 odd 4
256.9.d.f.127.4 4 16.13 even 4
400.9.b.c.351.1 2 20.19 odd 2
400.9.b.c.351.2 2 5.4 even 2
400.9.h.b.399.1 4 5.2 odd 4
400.9.h.b.399.2 4 20.3 even 4
400.9.h.b.399.3 4 5.3 odd 4
400.9.h.b.399.4 4 20.7 even 4