# Properties

 Label 144.9.g.g.127.2 Level $144$ Weight $9$ Character 144.127 Analytic conductor $58.663$ 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] = [144,9,Mod(127,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, 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("144.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 144.g (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: $$58.6625198488$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 127.2 Root $$0.500000 - 2.95804i$$ of defining polynomial Character $$\chi$$ $$=$$ 144.127 Dual form 144.9.g.g.127.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+510.000 q^{5} +2555.75i q^{7} +O(q^{10})$$ $$q+510.000 q^{5} +2555.75i q^{7} +19168.1i q^{11} -27710.0 q^{13} -50370.0 q^{17} -108619. i q^{19} +176347. i q^{23} -130525. q^{25} -54978.0 q^{29} -1.17564e6i q^{31} +1.30343e6i q^{35} +793730. q^{37} +75582.0 q^{41} -499648. i q^{43} +2.86755e6i q^{47} -767039. q^{49} -1.11662e7 q^{53} +9.77573e6i q^{55} -2.18325e7i q^{59} -2.38266e7 q^{61} -1.41321e7 q^{65} -7.49473e6i q^{67} -1.00824e7i q^{71} +6.51661e6 q^{73} -4.89888e7 q^{77} +4.87892e7i q^{79} +7.34483e7i q^{83} -2.56887e7 q^{85} -8.67958e7 q^{89} -7.08197e7i q^{91} -5.53958e7i q^{95} -4.66703e7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 1020 q^{5}+O(q^{10})$$ 2 * q + 1020 * q^5 $$2 q + 1020 q^{5} - 55420 q^{13} - 100740 q^{17} - 261050 q^{25} - 109956 q^{29} + 1587460 q^{37} + 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} - 47653244 q^{61} - 28264200 q^{65} + 13033220 q^{73} - 97977600 q^{77} - 51377400 q^{85} - 173591556 q^{89} - 93340540 q^{97}+O(q^{100})$$ 2 * q + 1020 * q^5 - 55420 * q^13 - 100740 * q^17 - 261050 * q^25 - 109956 * q^29 + 1587460 * q^37 + 151164 * q^41 - 1534078 * q^49 - 22332420 * q^53 - 47653244 * q^61 - 28264200 * q^65 + 13033220 * q^73 - 97977600 * q^77 - 51377400 * q^85 - 173591556 * q^89 - 93340540 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 510.000 0.816000 0.408000 0.912982i $$-0.366226\pi$$
0.408000 + 0.912982i $$0.366226\pi$$
$$6$$ 0 0
$$7$$ 2555.75i 1.06445i 0.846603 + 0.532225i $$0.178644\pi$$
−0.846603 + 0.532225i $$0.821356\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 19168.1i 1.30921i 0.755972 + 0.654603i $$0.227164\pi$$
−0.755972 + 0.654603i $$0.772836\pi$$
$$12$$ 0 0
$$13$$ −27710.0 −0.970204 −0.485102 0.874458i $$-0.661218\pi$$
−0.485102 + 0.874458i $$0.661218\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −50370.0 −0.603082 −0.301541 0.953453i $$-0.597501\pi$$
−0.301541 + 0.953453i $$0.597501\pi$$
$$18$$ 0 0
$$19$$ − 108619.i − 0.833474i −0.909027 0.416737i $$-0.863174\pi$$
0.909027 0.416737i $$-0.136826\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 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
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −54978.0 −0.0777315 −0.0388657 0.999244i $$-0.512374\pi$$
−0.0388657 + 0.999244i $$0.512374\pi$$
$$30$$ 0 0
$$31$$ − 1.17564e6i − 1.27300i −0.771276 0.636501i $$-0.780381\pi$$
0.771276 0.636501i $$-0.219619\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 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
$$39$$ 0 0
$$40$$ 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.i − 0.146147i −0.997327 0.0730736i $$-0.976719\pi$$
0.997327 0.0730736i $$-0.0232808\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.86755e6i 0.587651i 0.955859 + 0.293825i $$0.0949284\pi$$
−0.955859 + 0.293825i $$0.905072\pi$$
$$48$$ 0 0
$$49$$ −767039. −0.133056
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −1.11662e7 −1.41515 −0.707575 0.706639i $$-0.750211\pi$$
−0.707575 + 0.706639i $$0.750211\pi$$
$$54$$ 0 0
$$55$$ 9.77573e6i 1.06831i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 2.18325e7i − 1.80175i −0.434078 0.900875i $$-0.642926\pi$$
0.434078 0.900875i $$-0.357074\pi$$
$$60$$ 0 0
$$61$$ −2.38266e7 −1.72085 −0.860425 0.509577i $$-0.829802\pi$$
−0.860425 + 0.509577i $$0.829802\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 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
$$69$$ 0 0
$$70$$ 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 $$-0.463398\pi$$
0.114736 + 0.993396i $$0.463398\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 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
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 7.34483e7i 1.54764i 0.633407 + 0.773819i $$0.281656\pi$$
−0.633407 + 0.773819i $$0.718344\pi$$
$$84$$ 0 0
$$85$$ −2.56887e7 −0.492115
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −8.67958e7 −1.38337 −0.691685 0.722199i $$-0.743131\pi$$
−0.691685 + 0.722199i $$0.743131\pi$$
$$90$$ 0 0
$$91$$ − 7.08197e7i − 1.03273i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.59910e7 −0.634161 −0.317080 0.948399i $$-0.602703\pi$$
−0.317080 + 0.948399i $$0.602703\pi$$
$$102$$ 0 0
$$103$$ − 1.64884e8i − 1.46497i −0.680782 0.732486i $$-0.738360\pi$$
0.680782 0.732486i $$-0.261640\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.27326e8i 0.971364i 0.874136 + 0.485682i $$0.161429\pi$$
−0.874136 + 0.485682i $$0.838571\pi$$
$$108$$ 0 0
$$109$$ −1.56119e8 −1.10598 −0.552992 0.833186i $$-0.686514\pi$$
−0.552992 + 0.833186i $$0.686514\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.36346e8 −1.44955 −0.724776 0.688984i $$-0.758057\pi$$
−0.724776 + 0.688984i $$0.758057\pi$$
$$114$$ 0 0
$$115$$ 8.99367e7i 0.514216i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 1.28733e8i − 0.641951i
$$120$$ 0 0
$$121$$ −1.53057e8 −0.714023
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2.16350e8i 0.734636i 0.930095 + 0.367318i $$0.119724\pi$$
−0.930095 + 0.367318i $$0.880276\pi$$
$$132$$ 0 0
$$133$$ 2.77603e8 0.887193
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.86442e8 1.09699 0.548494 0.836155i $$-0.315201\pi$$
0.548494 + 0.836155i $$0.315201\pi$$
$$138$$ 0 0
$$139$$ − 3.51077e8i − 0.940465i −0.882543 0.470232i $$-0.844170\pi$$
0.882543 0.470232i $$-0.155830\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 5.31148e8i − 1.27020i
$$144$$ 0 0
$$145$$ −2.80388e7 −0.0634289
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 4.54099e8 0.921308 0.460654 0.887580i $$-0.347615\pi$$
0.460654 + 0.887580i $$0.347615\pi$$
$$150$$ 0 0
$$151$$ 6.60188e8i 1.26987i 0.772565 + 0.634936i $$0.218973\pi$$
−0.772565 + 0.634936i $$0.781027\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 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
$$159$$ 0 0
$$160$$ 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
$$165$$ 0 0
$$166$$ 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
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 1.76764e9 1.97337 0.986685 0.162644i $$-0.0520022\pi$$
0.986685 + 0.162644i $$0.0520022\pi$$
$$174$$ 0 0
$$175$$ − 3.33589e8i − 0.355680i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 7.56967e8i 0.737335i 0.929561 + 0.368668i $$0.120186\pi$$
−0.929561 + 0.368668i $$0.879814\pi$$
$$180$$ 0 0
$$181$$ 6.27094e8 0.584277 0.292138 0.956376i $$-0.405633\pi$$
0.292138 + 0.956376i $$0.405633\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.04802e8 0.345586
$$186$$ 0 0
$$187$$ − 9.65497e8i − 0.789559i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1.07924e9i 0.810933i 0.914110 + 0.405466i $$0.132891\pi$$
−0.914110 + 0.405466i $$0.867109\pi$$
$$192$$ 0 0
$$193$$ −2.96757e7 −0.0213881 −0.0106940 0.999943i $$-0.503404\pi$$
−0.0106940 + 0.999943i $$0.503404\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.12484e9 0.746837 0.373419 0.927663i $$-0.378186\pi$$
0.373419 + 0.927663i $$0.378186\pi$$
$$198$$ 0 0
$$199$$ − 1.04718e9i − 0.667742i −0.942619 0.333871i $$-0.891645\pi$$
0.942619 0.333871i $$-0.108355\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 1.40510e8i − 0.0827413i
$$204$$ 0 0
$$205$$ 3.85468e7 0.0218259
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 2.54821e8i − 0.119256i
$$216$$ 0 0
$$217$$ 3.00465e9 1.35505
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 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
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 1.48863e9i − 0.560639i −0.959907 0.280319i $$-0.909560\pi$$
0.959907 0.280319i $$-0.0904404\pi$$
$$228$$ 0 0
$$229$$ 1.69447e9 0.616157 0.308079 0.951361i $$-0.400314\pi$$
0.308079 + 0.951361i $$0.400314\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −5.21423e8 −0.176916 −0.0884580 0.996080i $$-0.528194\pi$$
−0.0884580 + 0.996080i $$0.528194\pi$$
$$234$$ 0 0
$$235$$ 1.46245e9i 0.479523i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 4.40690e9i − 1.35065i −0.737522 0.675323i $$-0.764004\pi$$
0.737522 0.675323i $$-0.235996\pi$$
$$240$$ 0 0
$$241$$ −1.62148e9 −0.480666 −0.240333 0.970691i $$-0.577257\pi$$
−0.240333 + 0.970691i $$0.577257\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −3.91190e8 −0.108573
$$246$$ 0 0
$$247$$ 3.00984e9i 0.808640i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 1.31321e8i − 0.0330855i −0.999863 0.0165428i $$-0.994734\pi$$
0.999863 0.0165428i $$-0.00526597\pi$$
$$252$$ 0 0
$$253$$ −3.38023e9 −0.825019
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −5.27789e9 −1.20984 −0.604920 0.796287i $$-0.706795\pi$$
−0.604920 + 0.796287i $$0.706795\pi$$
$$258$$ 0 0
$$259$$ 2.02857e9i 0.450808i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 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
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3.46450e8 −0.0661655 −0.0330828 0.999453i $$-0.510532\pi$$
−0.0330828 + 0.999453i $$0.510532\pi$$
$$270$$ 0 0
$$271$$ − 6.51715e6i − 0.00120832i −1.00000 0.000604158i $$-0.999808\pi$$
1.00000 0.000604158i $$-0.000192310\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 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
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1.04256e10 −1.67215 −0.836074 0.548616i $$-0.815155\pi$$
−0.836074 + 0.548616i $$0.815155\pi$$
$$282$$ 0 0
$$283$$ − 1.28042e9i − 0.199622i −0.995006 0.0998108i $$-0.968176\pi$$
0.995006 0.0998108i $$-0.0318238\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1.93168e8i 0.0284714i
$$288$$ 0 0
$$289$$ −4.43862e9 −0.636292
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 2.13786e9 0.290074 0.145037 0.989426i $$-0.453670\pi$$
0.145037 + 0.989426i $$0.453670\pi$$
$$294$$ 0 0
$$295$$ − 1.11346e10i − 1.47023i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 4.88656e9i − 0.611390i
$$300$$ 0 0
$$301$$ 1.27697e9 0.155567
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 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
$$309$$ 0 0
$$310$$ 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 $$-0.504972\pi$$
−0.0156182 + 0.999878i $$0.504972\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5.31172e9 −0.526015 −0.263007 0.964794i $$-0.584714\pi$$
−0.263007 + 0.964794i $$0.584714\pi$$
$$318$$ 0 0
$$319$$ − 1.05382e9i − 0.101767i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 5.47115e9i 0.502653i
$$324$$ 0 0
$$325$$ 3.61685e9 0.324188
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2.25348e10 1.66662
$$342$$ 0 0
$$343$$ 1.27730e10i 0.922820i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 1.27822e10i 0.881636i 0.897597 + 0.440818i $$0.145312\pi$$
−0.897597 + 0.440818i $$0.854688\pi$$
$$348$$ 0 0
$$349$$ −6.39381e8 −0.0430981 −0.0215490 0.999768i $$-0.506860\pi$$
−0.0215490 + 0.999768i $$0.506860\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −2.59837e10 −1.67341 −0.836705 0.547653i $$-0.815521\pi$$
−0.836705 + 0.547653i $$0.815521\pi$$
$$354$$ 0 0
$$355$$ − 5.14203e9i − 0.323759i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 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
$$363$$ 0 0
$$364$$ 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
$$369$$ 0 0
$$370$$ 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
$$375$$ 0 0
$$376$$ 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
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 1.43419e10i − 0.666518i −0.942835 0.333259i $$-0.891852\pi$$
0.942835 0.333259i $$-0.108148\pi$$
$$384$$ 0 0
$$385$$ −2.49843e10 −1.13717
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 2.73457e10 1.19424 0.597119 0.802152i $$-0.296312\pi$$
0.597119 + 0.802152i $$0.296312\pi$$
$$390$$ 0 0
$$391$$ − 8.88257e9i − 0.380042i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 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
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2.12767e10 0.822863 0.411431 0.911441i $$-0.365029\pi$$
0.411431 + 0.911441i $$0.365029\pi$$
$$402$$ 0 0
$$403$$ 3.25771e10i 1.23507i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 5.57982e10 1.91788
$$414$$ 0 0
$$415$$ 3.74586e10i 1.26287i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 1.10009e10i 0.356922i 0.983947 + 0.178461i $$0.0571119\pi$$
−0.983947 + 0.178461i $$0.942888\pi$$
$$420$$ 0 0
$$421$$ 2.28766e10 0.728220 0.364110 0.931356i $$-0.381373\pi$$
0.364110 + 0.931356i $$0.381373\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6.57454e9 0.201516
$$426$$ 0 0
$$427$$ − 6.08948e10i − 1.83176i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 9.55108e8i − 0.0276786i −0.999904 0.0138393i $$-0.995595\pi$$
0.999904 0.0138393i $$-0.00440532\pi$$
$$432$$ 0 0
$$433$$ 3.82225e10 1.08735 0.543673 0.839297i $$-0.317033\pi$$
0.543673 + 0.839297i $$0.317033\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 7.47659e10i 1.94128i 0.240533 + 0.970641i $$0.422678\pi$$
−0.240533 + 0.970641i $$0.577322\pi$$
$$444$$ 0 0
$$445$$ −4.42658e10 −1.12883
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2.51987e10 −0.620001 −0.310000 0.950736i $$-0.600329\pi$$
−0.310000 + 0.950736i $$0.600329\pi$$
$$450$$ 0 0
$$451$$ 1.44876e9i 0.0350180i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 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
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 3.88096e10 0.859281 0.429641 0.903000i $$-0.358640\pi$$
0.429641 + 0.903000i $$0.358640\pi$$
$$462$$ 0 0
$$463$$ 3.23432e10i 0.703817i 0.936034 + 0.351908i $$0.114467\pi$$
−0.936034 + 0.351908i $$0.885533\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 2.31902e10i − 0.487570i −0.969829 0.243785i $$-0.921611\pi$$
0.969829 0.243785i $$-0.0783891\pi$$
$$468$$ 0 0
$$469$$ 1.91546e10 0.395897
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 9.57731e9 0.191337
$$474$$ 0 0
$$475$$ 1.41775e10i 0.278500i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 4.83542e9i − 0.0918528i −0.998945 0.0459264i $$-0.985376\pi$$
0.998945 0.0459264i $$-0.0146240\pi$$
$$480$$ 0 0
$$481$$ −2.19943e10 −0.410893
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 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
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 5.56483e10i − 0.957472i −0.877959 0.478736i $$-0.841095\pi$$
0.877959 0.478736i $$-0.158905\pi$$
$$492$$ 0 0
$$493$$ 2.76924e9 0.0468784
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 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
$$501$$ 0 0
$$502$$ 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
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −1.05927e10 −0.157811 −0.0789055 0.996882i $$-0.525143\pi$$
−0.0789055 + 0.996882i $$0.525143\pi$$
$$510$$ 0 0
$$511$$ 1.66548e10i 0.244262i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 8.40908e10i − 1.19542i
$$516$$ 0 0
$$517$$ −5.49654e10 −0.769356
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1.24958e11 −1.69595 −0.847973 0.530039i $$-0.822177\pi$$
−0.847973 + 0.530039i $$0.822177\pi$$
$$522$$ 0 0
$$523$$ − 2.80408e10i − 0.374786i −0.982285 0.187393i $$-0.939996\pi$$
0.982285 0.187393i $$-0.0600038\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 5.92172e10i 0.767724i
$$528$$ 0 0
$$529$$ 4.72129e10 0.602890
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −2.09438e9 −0.0259505
$$534$$ 0 0
$$535$$ 6.49363e10i 0.792633i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5.97167e9i 0.0647872i
$$552$$ 0 0
$$553$$ −1.24693e11 −1.33334
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 5.47312e9 0.0568610 0.0284305 0.999596i $$-0.490949\pi$$
0.0284305 + 0.999596i $$0.490949\pi$$
$$558$$ 0 0
$$559$$ 1.38453e10i 0.141793i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 4.36118e10i − 0.434081i −0.976163 0.217040i $$-0.930360\pi$$
0.976163 0.217040i $$-0.0696403\pi$$
$$564$$ 0 0
$$565$$ −1.20536e11 −1.18283
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −1.27822e10 −0.121943 −0.0609716 0.998140i $$-0.519420\pi$$
−0.0609716 + 0.998140i $$0.519420\pi$$
$$570$$ 0 0
$$571$$ 7.59455e10i 0.714427i 0.934023 + 0.357213i $$0.116273\pi$$
−0.934023 + 0.357213i $$0.883727\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 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
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1.87715e11 −1.64738
$$582$$ 0 0
$$583$$ − 2.14035e11i − 1.85272i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 6.07298e10i 0.511504i 0.966742 + 0.255752i $$0.0823231\pi$$
−0.966742 + 0.255752i $$0.917677\pi$$
$$588$$ 0 0
$$589$$ −1.27697e11 −1.06101
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −1.15978e11 −0.937899 −0.468949 0.883225i $$-0.655367\pi$$
−0.468949 + 0.883225i $$0.655367\pi$$
$$594$$ 0 0
$$595$$ − 6.56538e10i − 0.523832i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 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 $$-0.764907\pi$$
−0.739434 + 0.673229i $$0.764907\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 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
$$609$$ 0 0
$$610$$ 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
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9.84986e10 0.679656 0.339828 0.940488i $$-0.389631\pi$$
0.339828 + 0.940488i $$0.389631\pi$$
$$618$$ 0 0
$$619$$ 1.28596e10i 0.0875923i 0.999040 + 0.0437961i $$0.0139452\pi$$
−0.999040 + 0.0437961i $$0.986055\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 2.21828e11i − 1.47253i
$$624$$ 0 0
$$625$$ −8.45648e10 −0.554204
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 1.87548e11i 1.15350i
$$636$$ 0 0
$$637$$ 2.12547e10 0.129091
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.13903e11 0.674690 0.337345 0.941381i $$-0.390471\pi$$
0.337345 + 0.941381i $$0.390471\pi$$
$$642$$ 0 0
$$643$$ 7.04067e10i 0.411879i 0.978565 + 0.205940i $$0.0660250\pi$$
−0.978565 + 0.205940i $$0.933975\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 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
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −6.49972e9 −0.0357472 −0.0178736 0.999840i $$-0.505690\pi$$
−0.0178736 + 0.999840i $$0.505690\pi$$
$$654$$ 0 0
$$655$$ 1.10339e11i 0.599463i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 2.20982e11i 1.17170i 0.810421 + 0.585848i $$0.199239\pi$$
−0.810421 + 0.585848i $$0.800761\pi$$
$$660$$ 0 0
$$661$$ −2.69549e11 −1.41199 −0.705995 0.708217i $$-0.749500\pi$$
−0.705995 + 0.708217i $$0.749500\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1.41578e11 0.723949
$$666$$ 0 0
$$667$$ − 9.69518e9i − 0.0489838i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −8.02735e10 −0.382136 −0.191068 0.981577i $$-0.561195\pi$$
−0.191068 + 0.981577i $$0.561195\pi$$
$$678$$ 0 0
$$679$$ − 1.19277e11i − 0.561150i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 3.00783e11i − 1.38220i −0.722760 0.691099i $$-0.757127\pi$$
0.722760 0.691099i $$-0.242873\pi$$
$$684$$ 0 0
$$685$$ 1.97085e11 0.895142
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 1.79049e11i − 0.767419i
$$696$$ 0 0
$$697$$ −3.80707e9 −0.0161309
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.73603e11 1.13305 0.566524 0.824045i $$-0.308288\pi$$
0.566524 + 0.824045i $$0.308288\pi$$
$$702$$ 0 0
$$703$$ − 8.62143e10i − 0.352987i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 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
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 2.07321e11 0.802203
$$714$$ 0 0
$$715$$ − 2.70885e11i − 1.03648i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2.25510e11i 0.843821i 0.906637 + 0.421911i $$0.138640\pi$$
−0.906637 + 0.421911i $$0.861360\pi$$
$$720$$ 0 0
$$721$$ 4.21402e11 1.55939
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 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
$$729$$ 0 0
$$730$$ 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
$$735$$ 0 0
$$736$$ 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
$$741$$ 0 0
$$742$$ 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
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ 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
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 4.27419e11 1.27443 0.637213 0.770687i $$-0.280087\pi$$
0.637213 + 0.770687i $$0.280087\pi$$
$$762$$ 0 0
$$763$$ − 3.99000e11i − 1.17727i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 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
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1.49408e11 0.418462 0.209231 0.977866i $$-0.432904\pi$$
0.209231 + 0.977866i $$0.432904\pi$$
$$774$$ 0 0
$$775$$ 1.53451e11i 0.425366i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 8.20966e9i − 0.0222933i
$$780$$ 0 0
$$781$$ 1.93261e11 0.519445
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 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
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 6.04040e11i − 1.54298i
$$792$$ 0 0
$$793$$ 6.60236e11 1.66958
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −3.02703e11 −0.750212 −0.375106 0.926982i $$-0.622394\pi$$
−0.375106 + 0.926982i $$0.622394\pi$$
$$798$$ 0 0
$$799$$ − 1.44438e11i − 0.354401i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 1.24911e11i 0.300427i
$$804$$ 0 0
$$805$$ −2.29855e11 −0.547358
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 5.84316e11 1.36412 0.682062 0.731295i $$-0.261084\pi$$
0.682062 + 0.731295i $$0.261084\pi$$
$$810$$ 0 0
$$811$$ 1.21470e11i 0.280793i 0.990095 + 0.140396i $$0.0448377\pi$$
−0.990095 + 0.140396i $$0.955162\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 1.24473e11i − 0.282127i
$$816$$ 0 0
$$817$$ −5.42714e10 −0.121810
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −4.52470e11 −0.995903 −0.497952 0.867205i $$-0.665914\pi$$
−0.497952 + 0.867205i $$0.665914\pi$$
$$822$$ 0 0
$$823$$ − 3.06704e11i − 0.668528i −0.942479 0.334264i $$-0.891512\pi$$
0.942479 0.334264i $$-0.108488\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.93276e11i 0.626982i 0.949591 + 0.313491i $$0.101499\pi$$
−0.949591 + 0.313491i $$0.898501\pi$$
$$828$$ 0 0
$$829$$ 3.35532e11 0.710421 0.355210 0.934786i $$-0.384409\pi$$
0.355210 + 0.934786i $$0.384409\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 3.86358e10 0.0802434
$$834$$ 0 0
$$835$$ 3.42389e11i 0.704326i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 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
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −2.44222e10 −0.0479024
$$846$$ 0 0
$$847$$ − 3.91175e11i − 0.760042i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 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
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −6.06764e11 −1.12486 −0.562428 0.826846i $$-0.690133\pi$$
−0.562428 + 0.826846i $$0.690133\pi$$
$$858$$ 0 0
$$859$$ 9.49431e11i 1.74378i 0.489705 + 0.871888i $$0.337105\pi$$
−0.489705 + 0.871888i $$0.662895\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 2.99836e10i − 0.0540556i −0.999635 0.0270278i $$-0.991396\pi$$
0.999635 0.0270278i $$-0.00860426\pi$$
$$864$$ 0 0
$$865$$ 9.01494e11 1.61027
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −9.35196e11 −1.63992
$$870$$ 0 0
$$871$$ 2.07679e11i 0.360844i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 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
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −1.04085e12 −1.72776 −0.863879 0.503699i $$-0.831972\pi$$
−0.863879 + 0.503699i $$0.831972\pi$$
$$882$$ 0 0
$$883$$ 7.60446e11i 1.25091i 0.780261 + 0.625454i $$0.215086\pi$$
−0.780261 + 0.625454i $$0.784914\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 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
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 3.11471e11 0.489792
$$894$$ 0 0
$$895$$ 3.86053e11i 0.601666i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 6.46345e10i 0.0989523i
$$900$$ 0 0
$$901$$ 5.62442e11 0.853451
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 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
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 3.83541e11i − 0.556851i −0.960458 0.278425i $$-0.910188\pi$$
0.960458 0.278425i $$-0.0898125\pi$$
$$912$$ 0 0
$$913$$ −1.40786e12 −2.02618
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 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
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 2.79384e11i 0.384941i
$$924$$ 0 0
$$925$$ −1.03602e11 −0.141514
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −2.94973e11 −0.396021 −0.198011 0.980200i $$-0.563448\pi$$
−0.198011 + 0.980200i $$0.563448\pi$$
$$930$$ 0 0
$$931$$ 8.33152e10i 0.110898i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 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
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1.26891e11 −0.161836 −0.0809178 0.996721i $$-0.525785\pi$$
−0.0809178 + 0.996721i $$0.525785\pi$$
$$942$$ 0 0
$$943$$ 1.33286e10i 0.0168554i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 6.13064e11i − 0.762265i −0.924520 0.381133i $$-0.875534\pi$$
0.924520 0.381133i $$-0.124466\pi$$
$$948$$ 0 0
$$949$$ −1.80575e11 −0.222635
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −6.58227e11 −0.798002 −0.399001 0.916951i $$-0.630643\pi$$
−0.399001 + 0.916951i $$0.630643\pi$$
$$954$$ 0 0
$$955$$ 5.50413e11i 0.661721i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 9.87647e11i 1.16769i
$$960$$ 0 0
$$961$$ −5.29246e11 −0.620532
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 3.54981e11i − 0.399327i −0.979865 0.199663i $$-0.936015\pi$$
0.979865 0.199663i $$-0.0639849\pi$$
$$972$$ 0 0
$$973$$ 8.97263e11 1.00108
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −6.02238e11 −0.660982 −0.330491 0.943809i $$-0.607214\pi$$
−0.330491 + 0.943809i $$0.607214\pi$$
$$978$$ 0 0
$$979$$ − 1.66371e12i − 1.81112i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 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
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 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
$$999$$ 0 0
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 144.9.g.g.127.2 2
3.2 odd 2 16.9.c.a.15.1 2
4.3 odd 2 inner 144.9.g.g.127.1 2
12.11 even 2 16.9.c.a.15.2 yes 2
15.2 even 4 400.9.h.b.399.1 4
15.8 even 4 400.9.h.b.399.3 4
15.14 odd 2 400.9.b.c.351.2 2
24.5 odd 2 64.9.c.d.63.2 2
24.11 even 2 64.9.c.d.63.1 2
48.5 odd 4 256.9.d.f.127.1 4
48.11 even 4 256.9.d.f.127.3 4
48.29 odd 4 256.9.d.f.127.4 4
48.35 even 4 256.9.d.f.127.2 4
60.23 odd 4 400.9.h.b.399.2 4
60.47 odd 4 400.9.h.b.399.4 4
60.59 even 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 3.2 odd 2
16.9.c.a.15.2 yes 2 12.11 even 2
64.9.c.d.63.1 2 24.11 even 2
64.9.c.d.63.2 2 24.5 odd 2
144.9.g.g.127.1 2 4.3 odd 2 inner
144.9.g.g.127.2 2 1.1 even 1 trivial
256.9.d.f.127.1 4 48.5 odd 4
256.9.d.f.127.2 4 48.35 even 4
256.9.d.f.127.3 4 48.11 even 4
256.9.d.f.127.4 4 48.29 odd 4
400.9.b.c.351.1 2 60.59 even 2
400.9.b.c.351.2 2 15.14 odd 2
400.9.h.b.399.1 4 15.2 even 4
400.9.h.b.399.2 4 60.23 odd 4
400.9.h.b.399.3 4 15.8 even 4
400.9.h.b.399.4 4 60.47 odd 4