Properties

 Label 400.9.b.c.351.1 Level $400$ Weight $9$ Character 400.351 Analytic conductor $162.951$ 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] = [400,9,Mod(351,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, 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("400.351");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 400.b (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: $$162.951444024$$ 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: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 351.1 Root $$0.500000 + 2.95804i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.351 Dual form 400.9.b.c.351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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
$$3$$ − 141.986i − 1.75291i −0.481481 0.876456i $$-0.659901\pi$$
0.481481 0.876456i $$-0.340099\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 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 $$0.227164\pi$$
−0.755972 + 0.654603i $$0.772836\pi$$
$$12$$ 0 0
$$13$$ 27710.0 0.970204 0.485102 0.874458i $$-0.338782\pi$$
0.485102 + 0.874458i $$0.338782\pi$$
$$14$$ 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 $$0.136826\pi$$
−0.909027 + 0.416737i $$0.863174\pi$$
$$20$$ 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
$$25$$ 0 0
$$26$$ 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 $$0.219619\pi$$
−0.771276 + 0.636501i $$0.780381\pi$$
$$32$$ 0 0
$$33$$ 2.72160e6 2.29493
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −793730. −0.423512 −0.211756 0.977323i $$-0.567918\pi$$
−0.211756 + 0.977323i $$0.567918\pi$$
$$38$$ 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
$$45$$ 0 0
$$46$$ 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 $$-0.750211\pi$$
−0.707575 + 0.706639i $$0.750211\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.54224e7 1.46101
$$58$$ 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
$$63$$ − 3.47556e7i − 2.20629i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 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 $$-0.936431\pi$$
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
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.89888e7 −1.39359
$$78$$ 0 0
$$79$$ − 4.87892e7i − 1.25261i −0.779579 0.626304i $$-0.784567\pi$$
0.779579 0.626304i $$-0.215433\pi$$
$$80$$ 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
$$85$$ 0 0
$$86$$ 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
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.66703e7 0.527173 0.263587 0.964636i $$-0.415095\pi$$
0.263587 + 0.964636i $$0.415095\pi$$
$$98$$ 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
$$105$$ 0 0
$$106$$ 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 $$-0.758057\pi$$
−0.724776 + 0.688984i $$0.758057\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 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
$$125$$ 0 0
$$126$$ 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 $$0.119724\pi$$
−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 $$0.155830\pi$$
−0.882543 + 0.470232i $$0.844170\pi$$
$$140$$ 0 0
$$141$$ −4.07151e8 −1.03010
$$142$$ 0 0
$$143$$ 5.31148e8i 1.27020i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 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 $$-0.781027\pi$$
0.772565 0.634936i $$-0.218973\pi$$
$$152$$ 0 0
$$153$$ 6.84982e8 1.25001
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −4.35318e8 −0.716486 −0.358243 0.933628i $$-0.616624\pi$$
−0.358243 + 0.933628i $$0.616624\pi$$
$$158$$ 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
$$165$$ 0 0
$$166$$ 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 $$-0.0520022\pi$$
0.986685 + 0.162644i $$0.0520022\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −3.09990e9 −3.15831
$$178$$ 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
$$183$$ 3.38304e9i 3.01650i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 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 $$0.132891\pi$$
−0.914110 + 0.405466i $$0.867109\pi$$
$$192$$ 0 0
$$193$$ 2.96757e7 0.0213881 0.0106940 0.999943i $$-0.496596\pi$$
0.0106940 + 0.999943i $$0.496596\pi$$
$$194$$ 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 $$0.108355\pi$$
−0.942619 + 0.333871i $$0.891645\pi$$
$$200$$ 0 0
$$201$$ −1.06415e9 −0.651954
$$202$$ 0 0
$$203$$ 1.40510e8i 0.0827413i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 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 $$-0.752982\pi$$
0.713699 0.700452i $$-0.247018\pi$$
$$212$$ 0 0
$$213$$ −1.43156e9 −0.695491
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 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
$$225$$ 0 0
$$226$$ 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 $$-0.528194\pi$$
−0.0884580 + 0.996080i $$0.528194\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −6.92738e9 −2.19571
$$238$$ 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
$$243$$ − 9.21023e8i − 0.264147i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 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 $$-0.994734\pi$$
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
$$261$$ −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
$$265$$ 0 0
$$266$$ 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 $$0.000192310\pi$$
−1.00000 0.000604158i $$0.999808\pi$$
$$272$$ 0 0
$$273$$ 1.00554e10 1.81029
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.15061e9 −0.365293 −0.182647 0.983179i $$-0.558466\pi$$
−0.182647 + 0.983179i $$0.558466\pi$$
$$278$$ 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
$$285$$ 0 0
$$286$$ 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 $$-0.453670\pi$$
0.145037 + 0.989426i $$0.453670\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 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
$$305$$ 0 0
$$306$$ 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 $$0.194931\pi$$
−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
$$315$$ 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
$$321$$ −1.80785e10 −1.70272
$$322$$ 0 0
$$323$$ − 5.47115e9i − 0.502653i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 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 $$-0.861776\pi$$
0.907189 0.420723i $$-0.138224\pi$$
$$332$$ 0 0
$$333$$ 1.07939e10 0.877815
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.84359e10 1.42937 0.714684 0.699448i $$-0.246571\pi$$
0.714684 + 0.699448i $$0.246571\pi$$
$$338$$ 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
$$345$$ 0 0
$$346$$ 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 $$-0.815521\pi$$
−0.836705 + 0.547653i $$0.815521\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −1.82783e10 −1.12528
$$358$$ 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
$$363$$ 2.17320e10i 1.25162i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 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 $$-0.600042\pi$$
−0.309141 + 0.951016i $$0.600042\pi$$
$$374$$ 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 $$0.188920\pi$$
−0.828982 + 0.559275i $$0.811080\pi$$
$$380$$ 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
$$385$$ 0 0
$$386$$ 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
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −3.99456e10 −1.60808 −0.804039 0.594576i $$-0.797320\pi$$
−0.804039 + 0.594576i $$0.797320\pi$$
$$398$$ 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
$$405$$ 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
$$411$$ − 5.48693e10i − 1.92292i
$$412$$ 0 0
$$413$$ 5.57982e10 1.91788
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.98479e10 1.64855
$$418$$ 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
$$423$$ 3.89958e10i 1.21802i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 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 $$-0.995595\pi$$
0.999904 0.0138393i $$-0.00440532\pi$$
$$432$$ 0 0
$$433$$ −3.82225e10 −1.08735 −0.543673 0.839297i $$-0.682967\pi$$
−0.543673 + 0.839297i $$0.682967\pi$$
$$434$$ 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 $$0.330763\pi$$
−0.506976 + 0.861960i $$0.669237\pi$$
$$440$$ 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
$$445$$ 0 0
$$446$$ 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
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −4.66828e9 −0.107027 −0.0535133 0.998567i $$-0.517042\pi$$
−0.0535133 + 0.998567i $$0.517042\pi$$
$$458$$ 0 0
$$459$$ − 5.03346e10i − 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
$$465$$ 0 0
$$466$$ 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
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1.51849e11 2.93318
$$478$$ 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
$$483$$ − 6.39926e10i − 1.17582i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 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 $$-0.841095\pi$$
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 $$-0.780655\pi$$
0.771822 0.635838i $$-0.219345\pi$$
$$500$$ 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
$$505$$ 0 0
$$506$$ 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
$$515$$ 0 0
$$516$$ 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
$$525$$ 0 0
$$526$$ 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
$$535$$ 0 0
$$536$$ 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
$$545$$ 0 0
$$546$$ 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
$$555$$ 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
$$561$$ −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
$$565$$ 0 0
$$566$$ 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 $$-0.883727\pi$$
0.934023 0.357213i $$-0.116273\pi$$
$$572$$ 0 0
$$573$$ 1.53237e11 1.42149
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −2.13827e10 −0.192912 −0.0964560 0.995337i $$-0.530751\pi$$
−0.0964560 + 0.995337i $$0.530751\pi$$
$$578$$ 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
$$585$$ 0 0
$$586$$ 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 $$-0.655367\pi$$
−0.468949 + 0.883225i $$0.655367\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1.48685e11 1.17049
$$598$$ 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
$$603$$ 1.01921e11i 0.770892i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 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 $$-0.714788\pi$$
−0.624722 + 0.780847i $$0.714788\pi$$
$$614$$ 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 $$-0.986055\pi$$
0.999040 0.0437961i $$-0.0139452\pi$$
$$620$$ 0 0
$$621$$ 1.76223e11 1.18494
$$622$$ 0 0
$$623$$ 2.21828e11i 1.47253i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 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 $$-0.815735\pi$$
0.837073 0.547091i $$-0.184265\pi$$
$$632$$ 0 0
$$633$$ −3.94261e11 −2.45566
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 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
$$645$$ 0 0
$$646$$ 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 $$-0.505690\pi$$
−0.0178736 + 0.999840i $$0.505690\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8.86194e10 0.475628
$$658$$ 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
$$663$$ 1.98177e11i 1.02565i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 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 $$-0.573937\pi$$
−0.230196 + 0.973144i $$0.573937\pi$$
$$674$$ 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
$$681$$ 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
$$685$$ 0 0
$$686$$ 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 $$-0.765631\pi$$
0.740964 0.671544i $$-0.234369\pi$$
$$692$$ 0 0
$$693$$ 6.66199e11 2.88849
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 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
$$705$$ 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
$$711$$ 6.63484e11i 2.59628i
$$712$$ 0 0
$$713$$ 2.07321e11 0.802203
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −6.25718e11 −2.36756
$$718$$ 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
$$723$$ 2.30227e11i 0.842565i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 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 $$-0.329823\pi$$
0.509520 + 0.860459i $$0.329823\pi$$
$$734$$ 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 $$0.0519872\pi$$
−0.986692 + 0.162598i $$0.948013\pi$$
$$740$$ 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
$$745$$ 0 0
$$746$$ 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 $$-0.771879\pi$$
0.754001 0.656873i $$-0.228121\pi$$
$$752$$ 0 0
$$753$$ −1.86457e10 −0.0579960
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −1.82006e11 −0.554244 −0.277122 0.960835i $$-0.589381\pi$$
−0.277122 + 0.960835i $$0.589381\pi$$
$$758$$ 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
$$765$$ 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
$$771$$ 7.49386e11i 2.12074i
$$772$$ 0 0
$$773$$ 1.49408e11 0.418462 0.209231 0.977866i $$-0.432904\pi$$
0.209231 + 0.977866i $$0.432904\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 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
$$785$$ 0 0
$$786$$ 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
$$795$$ 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
$$801$$ −1.18034e12 −2.86732
$$802$$ 0 0
$$803$$ − 1.24911e11i − 0.300427i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 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 $$-0.955162\pi$$
0.990095 0.140396i $$-0.0448377\pi$$
$$812$$ 0 0
$$813$$ 9.25344e8 0.00211807
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 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
$$825$$ 0 0
$$826$$ 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
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1.17482e12 −2.39369
$$838$$ 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
$$843$$ − 1.48029e12i − 2.93113i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 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 $$-0.340491\pi$$
0.480400 + 0.877049i $$0.340491\pi$$
$$854$$ 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 $$-0.662895\pi$$
0.489705 0.871888i $$-0.337105\pi$$
$$860$$ 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
$$865$$ 0 0
$$866$$ 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
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.80195e11 1.48792 0.743961 0.668223i $$-0.232945\pi$$
0.743961 + 0.668223i $$0.232945\pi$$
$$878$$ 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
$$885$$ 0 0
$$886$$ 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
$$895$$ 0 0
$$896$$ 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
$$905$$ 0 0
$$906$$ 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 $$-0.910188\pi$$
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 $$0.158861\pi$$
−0.878025 + 0.478615i $$0.841139\pi$$
$$920$$ 0 0
$$921$$ −7.74022e11 −1.07576
$$922$$ 0 0
$$923$$ − 2.79384e11i − 0.384941i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 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
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1.03941e12 −1.34843 −0.674217 0.738533i $$-0.735519\pi$$
−0.674217 + 0.738533i $$0.735519\pi$$
$$938$$ 0 0
$$939$$ − 4.25680e10i − 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
$$945$$ 0 0
$$946$$ 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 $$-0.630643\pi$$
−0.399001 + 0.916951i $$0.630643\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 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
$$965$$ 0 0
$$966$$ 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 $$-0.936015\pi$$
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
$$981$$ 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
$$985$$ 0 0
$$986$$ 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 $$0.175567\pi$$
−0.851709 + 0.524015i $$0.824433\pi$$
$$992$$ 0 0
$$993$$ −1.43411e12 −1.47498
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 3.28556e11 0.332528 0.166264 0.986081i $$-0.446830\pi$$
0.166264 + 0.986081i $$0.446830\pi$$
$$998$$ 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 400.9.b.c.351.1 2
4.3 odd 2 inner 400.9.b.c.351.2 2
5.2 odd 4 400.9.h.b.399.2 4
5.3 odd 4 400.9.h.b.399.4 4
5.4 even 2 16.9.c.a.15.2 yes 2
15.14 odd 2 144.9.g.g.127.1 2
20.3 even 4 400.9.h.b.399.1 4
20.7 even 4 400.9.h.b.399.3 4
20.19 odd 2 16.9.c.a.15.1 2
40.19 odd 2 64.9.c.d.63.2 2
40.29 even 2 64.9.c.d.63.1 2
60.59 even 2 144.9.g.g.127.2 2
80.19 odd 4 256.9.d.f.127.4 4
80.29 even 4 256.9.d.f.127.2 4
80.59 odd 4 256.9.d.f.127.1 4
80.69 even 4 256.9.d.f.127.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
16.9.c.a.15.1 2 20.19 odd 2
16.9.c.a.15.2 yes 2 5.4 even 2
64.9.c.d.63.1 2 40.29 even 2
64.9.c.d.63.2 2 40.19 odd 2
144.9.g.g.127.1 2 15.14 odd 2
144.9.g.g.127.2 2 60.59 even 2
256.9.d.f.127.1 4 80.59 odd 4
256.9.d.f.127.2 4 80.29 even 4
256.9.d.f.127.3 4 80.69 even 4
256.9.d.f.127.4 4 80.19 odd 4
400.9.b.c.351.1 2 1.1 even 1 trivial
400.9.b.c.351.2 2 4.3 odd 2 inner
400.9.h.b.399.1 4 20.3 even 4
400.9.h.b.399.2 4 5.2 odd 4
400.9.h.b.399.3 4 20.7 even 4
400.9.h.b.399.4 4 5.3 odd 4