# Properties

 Label 234.10.a.a.1.1 Level $234$ Weight $10$ Character 234.1 Self dual yes Analytic conductor $120.518$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,10,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 234.a (trivial)

## 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: yes Analytic conductor: $$120.518385662$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 234.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-16.0000 q^{2} +256.000 q^{4} +1979.00 q^{5} -10115.0 q^{7} -4096.00 q^{8} +O(q^{10})$$ $$q-16.0000 q^{2} +256.000 q^{4} +1979.00 q^{5} -10115.0 q^{7} -4096.00 q^{8} -31664.0 q^{10} -18850.0 q^{11} +28561.0 q^{13} +161840. q^{14} +65536.0 q^{16} +142403. q^{17} +83302.0 q^{19} +506624. q^{20} +301600. q^{22} +536544. q^{23} +1.96332e6 q^{25} -456976. q^{26} -2.58944e6 q^{28} +2.60044e6 q^{29} -2.21400e6 q^{31} -1.04858e6 q^{32} -2.27845e6 q^{34} -2.00176e7 q^{35} +1.80992e7 q^{37} -1.33283e6 q^{38} -8.10598e6 q^{40} -2.68122e7 q^{41} -4.22535e7 q^{43} -4.82560e6 q^{44} -8.58470e6 q^{46} -3.59150e7 q^{47} +6.19596e7 q^{49} -3.14131e7 q^{50} +7.31162e6 q^{52} +6.65141e7 q^{53} -3.73042e7 q^{55} +4.14310e7 q^{56} -4.16071e7 q^{58} +1.08164e8 q^{59} -2.07450e8 q^{61} +3.54241e7 q^{62} +1.67772e7 q^{64} +5.65222e7 q^{65} +1.93016e8 q^{67} +3.64552e7 q^{68} +3.20281e8 q^{70} +2.01833e8 q^{71} -1.21628e8 q^{73} -2.89588e8 q^{74} +2.13253e7 q^{76} +1.90668e8 q^{77} +1.12872e8 q^{79} +1.29696e8 q^{80} +4.28996e8 q^{82} -3.08254e8 q^{83} +2.81816e8 q^{85} +6.76056e8 q^{86} +7.72096e7 q^{88} +6.37487e6 q^{89} -2.88895e8 q^{91} +1.37355e8 q^{92} +5.74640e8 q^{94} +1.64855e8 q^{95} +8.71267e8 q^{97} -9.91354e8 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −16.0000 −0.707107
$$3$$ 0 0
$$4$$ 256.000 0.500000
$$5$$ 1979.00 1.41606 0.708029 0.706184i $$-0.249585\pi$$
0.708029 + 0.706184i $$0.249585\pi$$
$$6$$ 0 0
$$7$$ −10115.0 −1.59230 −0.796150 0.605100i $$-0.793133\pi$$
−0.796150 + 0.605100i $$0.793133\pi$$
$$8$$ −4096.00 −0.353553
$$9$$ 0 0
$$10$$ −31664.0 −1.00130
$$11$$ −18850.0 −0.388190 −0.194095 0.980983i $$-0.562177\pi$$
−0.194095 + 0.980983i $$0.562177\pi$$
$$12$$ 0 0
$$13$$ 28561.0 0.277350
$$14$$ 161840. 1.12593
$$15$$ 0 0
$$16$$ 65536.0 0.250000
$$17$$ 142403. 0.413522 0.206761 0.978391i $$-0.433708\pi$$
0.206761 + 0.978391i $$0.433708\pi$$
$$18$$ 0 0
$$19$$ 83302.0 0.146644 0.0733220 0.997308i $$-0.476640\pi$$
0.0733220 + 0.997308i $$0.476640\pi$$
$$20$$ 506624. 0.708029
$$21$$ 0 0
$$22$$ 301600. 0.274492
$$23$$ 536544. 0.399788 0.199894 0.979817i $$-0.435940\pi$$
0.199894 + 0.979817i $$0.435940\pi$$
$$24$$ 0 0
$$25$$ 1.96332e6 1.00522
$$26$$ −456976. −0.196116
$$27$$ 0 0
$$28$$ −2.58944e6 −0.796150
$$29$$ 2.60044e6 0.682741 0.341371 0.939929i $$-0.389109\pi$$
0.341371 + 0.939929i $$0.389109\pi$$
$$30$$ 0 0
$$31$$ −2.21400e6 −0.430577 −0.215288 0.976550i $$-0.569069\pi$$
−0.215288 + 0.976550i $$0.569069\pi$$
$$32$$ −1.04858e6 −0.176777
$$33$$ 0 0
$$34$$ −2.27845e6 −0.292404
$$35$$ −2.00176e7 −2.25479
$$36$$ 0 0
$$37$$ 1.80992e7 1.58764 0.793821 0.608151i $$-0.208089\pi$$
0.793821 + 0.608151i $$0.208089\pi$$
$$38$$ −1.33283e6 −0.103693
$$39$$ 0 0
$$40$$ −8.10598e6 −0.500652
$$41$$ −2.68122e7 −1.48186 −0.740928 0.671585i $$-0.765614\pi$$
−0.740928 + 0.671585i $$0.765614\pi$$
$$42$$ 0 0
$$43$$ −4.22535e7 −1.88475 −0.942376 0.334555i $$-0.891414\pi$$
−0.942376 + 0.334555i $$0.891414\pi$$
$$44$$ −4.82560e6 −0.194095
$$45$$ 0 0
$$46$$ −8.58470e6 −0.282693
$$47$$ −3.59150e7 −1.07358 −0.536791 0.843715i $$-0.680364\pi$$
−0.536791 + 0.843715i $$0.680364\pi$$
$$48$$ 0 0
$$49$$ 6.19596e7 1.53542
$$50$$ −3.14131e7 −0.710796
$$51$$ 0 0
$$52$$ 7.31162e6 0.138675
$$53$$ 6.65141e7 1.15790 0.578951 0.815362i $$-0.303462\pi$$
0.578951 + 0.815362i $$0.303462\pi$$
$$54$$ 0 0
$$55$$ −3.73042e7 −0.549699
$$56$$ 4.14310e7 0.562963
$$57$$ 0 0
$$58$$ −4.16071e7 −0.482771
$$59$$ 1.08164e8 1.16211 0.581057 0.813863i $$-0.302639\pi$$
0.581057 + 0.813863i $$0.302639\pi$$
$$60$$ 0 0
$$61$$ −2.07450e8 −1.91836 −0.959178 0.282805i $$-0.908735\pi$$
−0.959178 + 0.282805i $$0.908735\pi$$
$$62$$ 3.54241e7 0.304464
$$63$$ 0 0
$$64$$ 1.67772e7 0.125000
$$65$$ 5.65222e7 0.392744
$$66$$ 0 0
$$67$$ 1.93016e8 1.17019 0.585094 0.810966i $$-0.301058\pi$$
0.585094 + 0.810966i $$0.301058\pi$$
$$68$$ 3.64552e7 0.206761
$$69$$ 0 0
$$70$$ 3.20281e8 1.59438
$$71$$ 2.01833e8 0.942607 0.471304 0.881971i $$-0.343784\pi$$
0.471304 + 0.881971i $$0.343784\pi$$
$$72$$ 0 0
$$73$$ −1.21628e8 −0.501281 −0.250640 0.968080i $$-0.580641\pi$$
−0.250640 + 0.968080i $$0.580641\pi$$
$$74$$ −2.89588e8 −1.12263
$$75$$ 0 0
$$76$$ 2.13253e7 0.0733220
$$77$$ 1.90668e8 0.618115
$$78$$ 0 0
$$79$$ 1.12872e8 0.326035 0.163017 0.986623i $$-0.447877\pi$$
0.163017 + 0.986623i $$0.447877\pi$$
$$80$$ 1.29696e8 0.354014
$$81$$ 0 0
$$82$$ 4.28996e8 1.04783
$$83$$ −3.08254e8 −0.712948 −0.356474 0.934305i $$-0.616021\pi$$
−0.356474 + 0.934305i $$0.616021\pi$$
$$84$$ 0 0
$$85$$ 2.81816e8 0.585571
$$86$$ 6.76056e8 1.33272
$$87$$ 0 0
$$88$$ 7.72096e7 0.137246
$$89$$ 6.37487e6 0.0107700 0.00538501 0.999986i $$-0.498286\pi$$
0.00538501 + 0.999986i $$0.498286\pi$$
$$90$$ 0 0
$$91$$ −2.88895e8 −0.441624
$$92$$ 1.37355e8 0.199894
$$93$$ 0 0
$$94$$ 5.74640e8 0.759137
$$95$$ 1.64855e8 0.207656
$$96$$ 0 0
$$97$$ 8.71267e8 0.999260 0.499630 0.866239i $$-0.333469\pi$$
0.499630 + 0.866239i $$0.333469\pi$$
$$98$$ −9.91354e8 −1.08570
$$99$$ 0 0
$$100$$ 5.02609e8 0.502609
$$101$$ 8.24412e8 0.788312 0.394156 0.919044i $$-0.371037\pi$$
0.394156 + 0.919044i $$0.371037\pi$$
$$102$$ 0 0
$$103$$ −1.65896e9 −1.45234 −0.726168 0.687517i $$-0.758701\pi$$
−0.726168 + 0.687517i $$0.758701\pi$$
$$104$$ −1.16986e8 −0.0980581
$$105$$ 0 0
$$106$$ −1.06423e9 −0.818761
$$107$$ −1.15165e9 −0.849366 −0.424683 0.905342i $$-0.639614\pi$$
−0.424683 + 0.905342i $$0.639614\pi$$
$$108$$ 0 0
$$109$$ −2.78480e9 −1.88962 −0.944810 0.327620i $$-0.893753\pi$$
−0.944810 + 0.327620i $$0.893753\pi$$
$$110$$ 5.96866e8 0.388696
$$111$$ 0 0
$$112$$ −6.62897e8 −0.398075
$$113$$ −6.78547e8 −0.391496 −0.195748 0.980654i $$-0.562713\pi$$
−0.195748 + 0.980654i $$0.562713\pi$$
$$114$$ 0 0
$$115$$ 1.06182e9 0.566123
$$116$$ 6.65713e8 0.341371
$$117$$ 0 0
$$118$$ −1.73062e9 −0.821739
$$119$$ −1.44041e9 −0.658451
$$120$$ 0 0
$$121$$ −2.00263e9 −0.849309
$$122$$ 3.31920e9 1.35648
$$123$$ 0 0
$$124$$ −5.66785e8 −0.215288
$$125$$ 2.01680e7 0.00738869
$$126$$ 0 0
$$127$$ −3.48292e9 −1.18803 −0.594014 0.804455i $$-0.702458\pi$$
−0.594014 + 0.804455i $$0.702458\pi$$
$$128$$ −2.68435e8 −0.0883883
$$129$$ 0 0
$$130$$ −9.04356e8 −0.277712
$$131$$ −5.02701e9 −1.49138 −0.745691 0.666292i $$-0.767881\pi$$
−0.745691 + 0.666292i $$0.767881\pi$$
$$132$$ 0 0
$$133$$ −8.42600e8 −0.233501
$$134$$ −3.08825e9 −0.827448
$$135$$ 0 0
$$136$$ −5.83283e8 −0.146202
$$137$$ 6.38904e9 1.54950 0.774752 0.632265i $$-0.217875\pi$$
0.774752 + 0.632265i $$0.217875\pi$$
$$138$$ 0 0
$$139$$ 7.62665e9 1.73287 0.866437 0.499286i $$-0.166404\pi$$
0.866437 + 0.499286i $$0.166404\pi$$
$$140$$ −5.12450e9 −1.12739
$$141$$ 0 0
$$142$$ −3.22934e9 −0.666524
$$143$$ −5.38375e8 −0.107665
$$144$$ 0 0
$$145$$ 5.14627e9 0.966801
$$146$$ 1.94605e9 0.354459
$$147$$ 0 0
$$148$$ 4.63341e9 0.793821
$$149$$ 9.23455e9 1.53489 0.767445 0.641114i $$-0.221528\pi$$
0.767445 + 0.641114i $$0.221528\pi$$
$$150$$ 0 0
$$151$$ −3.25451e9 −0.509436 −0.254718 0.967015i $$-0.581983\pi$$
−0.254718 + 0.967015i $$0.581983\pi$$
$$152$$ −3.41205e8 −0.0518465
$$153$$ 0 0
$$154$$ −3.05068e9 −0.437073
$$155$$ −4.38151e9 −0.609722
$$156$$ 0 0
$$157$$ 1.62825e9 0.213881 0.106940 0.994265i $$-0.465895\pi$$
0.106940 + 0.994265i $$0.465895\pi$$
$$158$$ −1.80595e9 −0.230541
$$159$$ 0 0
$$160$$ −2.07513e9 −0.250326
$$161$$ −5.42714e9 −0.636583
$$162$$ 0 0
$$163$$ −1.13187e10 −1.25590 −0.627948 0.778255i $$-0.716105\pi$$
−0.627948 + 0.778255i $$0.716105\pi$$
$$164$$ −6.86393e9 −0.740928
$$165$$ 0 0
$$166$$ 4.93207e9 0.504130
$$167$$ −1.72306e9 −0.171426 −0.0857131 0.996320i $$-0.527317\pi$$
−0.0857131 + 0.996320i $$0.527317\pi$$
$$168$$ 0 0
$$169$$ 8.15731e8 0.0769231
$$170$$ −4.50905e9 −0.414061
$$171$$ 0 0
$$172$$ −1.08169e10 −0.942376
$$173$$ −2.76347e9 −0.234557 −0.117278 0.993099i $$-0.537417\pi$$
−0.117278 + 0.993099i $$0.537417\pi$$
$$174$$ 0 0
$$175$$ −1.98589e10 −1.60061
$$176$$ −1.23535e9 −0.0970475
$$177$$ 0 0
$$178$$ −1.01998e8 −0.00761555
$$179$$ −6.86682e9 −0.499939 −0.249969 0.968254i $$-0.580421\pi$$
−0.249969 + 0.968254i $$0.580421\pi$$
$$180$$ 0 0
$$181$$ −2.41534e10 −1.67273 −0.836364 0.548174i $$-0.815323\pi$$
−0.836364 + 0.548174i $$0.815323\pi$$
$$182$$ 4.62231e9 0.312276
$$183$$ 0 0
$$184$$ −2.19768e9 −0.141347
$$185$$ 3.58184e10 2.24819
$$186$$ 0 0
$$187$$ −2.68430e9 −0.160525
$$188$$ −9.19424e9 −0.536791
$$189$$ 0 0
$$190$$ −2.63767e9 −0.146835
$$191$$ −3.59983e10 −1.95719 −0.978593 0.205805i $$-0.934019\pi$$
−0.978593 + 0.205805i $$0.934019\pi$$
$$192$$ 0 0
$$193$$ −1.70031e10 −0.882107 −0.441054 0.897481i $$-0.645395\pi$$
−0.441054 + 0.897481i $$0.645395\pi$$
$$194$$ −1.39403e10 −0.706583
$$195$$ 0 0
$$196$$ 1.58617e10 0.767709
$$197$$ −3.98292e10 −1.88410 −0.942049 0.335477i $$-0.891103\pi$$
−0.942049 + 0.335477i $$0.891103\pi$$
$$198$$ 0 0
$$199$$ 1.31081e9 0.0592518 0.0296259 0.999561i $$-0.490568\pi$$
0.0296259 + 0.999561i $$0.490568\pi$$
$$200$$ −8.04174e9 −0.355398
$$201$$ 0 0
$$202$$ −1.31906e10 −0.557421
$$203$$ −2.63035e10 −1.08713
$$204$$ 0 0
$$205$$ −5.30614e10 −2.09839
$$206$$ 2.65433e10 1.02696
$$207$$ 0 0
$$208$$ 1.87177e9 0.0693375
$$209$$ −1.57024e9 −0.0569257
$$210$$ 0 0
$$211$$ 2.35777e10 0.818898 0.409449 0.912333i $$-0.365721\pi$$
0.409449 + 0.912333i $$0.365721\pi$$
$$212$$ 1.70276e10 0.578951
$$213$$ 0 0
$$214$$ 1.84265e10 0.600593
$$215$$ −8.36196e10 −2.66892
$$216$$ 0 0
$$217$$ 2.23947e10 0.685607
$$218$$ 4.45568e10 1.33616
$$219$$ 0 0
$$220$$ −9.54986e9 −0.274850
$$221$$ 4.06717e9 0.114690
$$222$$ 0 0
$$223$$ 2.38326e9 0.0645356 0.0322678 0.999479i $$-0.489727\pi$$
0.0322678 + 0.999479i $$0.489727\pi$$
$$224$$ 1.06063e10 0.281481
$$225$$ 0 0
$$226$$ 1.08568e10 0.276829
$$227$$ 7.46548e9 0.186613 0.0933064 0.995637i $$-0.470256\pi$$
0.0933064 + 0.995637i $$0.470256\pi$$
$$228$$ 0 0
$$229$$ 2.63966e10 0.634292 0.317146 0.948377i $$-0.397276\pi$$
0.317146 + 0.948377i $$0.397276\pi$$
$$230$$ −1.69891e10 −0.400309
$$231$$ 0 0
$$232$$ −1.06514e10 −0.241386
$$233$$ −2.40457e10 −0.534485 −0.267242 0.963629i $$-0.586112\pi$$
−0.267242 + 0.963629i $$0.586112\pi$$
$$234$$ 0 0
$$235$$ −7.10758e10 −1.52025
$$236$$ 2.76900e10 0.581057
$$237$$ 0 0
$$238$$ 2.30465e10 0.465595
$$239$$ −5.96318e10 −1.18219 −0.591095 0.806602i $$-0.701304\pi$$
−0.591095 + 0.806602i $$0.701304\pi$$
$$240$$ 0 0
$$241$$ 1.25639e10 0.239910 0.119955 0.992779i $$-0.461725\pi$$
0.119955 + 0.992779i $$0.461725\pi$$
$$242$$ 3.20420e10 0.600552
$$243$$ 0 0
$$244$$ −5.31072e10 −0.959178
$$245$$ 1.22618e11 2.17424
$$246$$ 0 0
$$247$$ 2.37919e9 0.0406717
$$248$$ 9.06856e9 0.152232
$$249$$ 0 0
$$250$$ −3.22688e8 −0.00522459
$$251$$ −2.41771e10 −0.384479 −0.192240 0.981348i $$-0.561575\pi$$
−0.192240 + 0.981348i $$0.561575\pi$$
$$252$$ 0 0
$$253$$ −1.01139e10 −0.155194
$$254$$ 5.57267e10 0.840062
$$255$$ 0 0
$$256$$ 4.29497e9 0.0625000
$$257$$ −2.96868e10 −0.424488 −0.212244 0.977217i $$-0.568077\pi$$
−0.212244 + 0.977217i $$0.568077\pi$$
$$258$$ 0 0
$$259$$ −1.83074e11 −2.52800
$$260$$ 1.44697e10 0.196372
$$261$$ 0 0
$$262$$ 8.04322e10 1.05457
$$263$$ 7.59146e10 0.978418 0.489209 0.872167i $$-0.337286\pi$$
0.489209 + 0.872167i $$0.337286\pi$$
$$264$$ 0 0
$$265$$ 1.31631e11 1.63966
$$266$$ 1.34816e10 0.165110
$$267$$ 0 0
$$268$$ 4.94120e10 0.585094
$$269$$ 2.57149e10 0.299433 0.149716 0.988729i $$-0.452164\pi$$
0.149716 + 0.988729i $$0.452164\pi$$
$$270$$ 0 0
$$271$$ −8.08890e10 −0.911020 −0.455510 0.890231i $$-0.650543\pi$$
−0.455510 + 0.890231i $$0.650543\pi$$
$$272$$ 9.33252e9 0.103381
$$273$$ 0 0
$$274$$ −1.02225e11 −1.09567
$$275$$ −3.70085e10 −0.390215
$$276$$ 0 0
$$277$$ −3.40035e10 −0.347028 −0.173514 0.984831i $$-0.555512\pi$$
−0.173514 + 0.984831i $$0.555512\pi$$
$$278$$ −1.22026e11 −1.22533
$$279$$ 0 0
$$280$$ 8.19920e10 0.797188
$$281$$ 7.14831e9 0.0683951 0.0341976 0.999415i $$-0.489112\pi$$
0.0341976 + 0.999415i $$0.489112\pi$$
$$282$$ 0 0
$$283$$ −7.80508e10 −0.723333 −0.361667 0.932308i $$-0.617792\pi$$
−0.361667 + 0.932308i $$0.617792\pi$$
$$284$$ 5.16694e10 0.471304
$$285$$ 0 0
$$286$$ 8.61400e9 0.0761303
$$287$$ 2.71206e11 2.35956
$$288$$ 0 0
$$289$$ −9.83093e10 −0.828999
$$290$$ −8.23404e10 −0.683631
$$291$$ 0 0
$$292$$ −3.11368e10 −0.250640
$$293$$ 1.26662e11 1.00402 0.502009 0.864862i $$-0.332594\pi$$
0.502009 + 0.864862i $$0.332594\pi$$
$$294$$ 0 0
$$295$$ 2.14057e11 1.64562
$$296$$ −7.41345e10 −0.561316
$$297$$ 0 0
$$298$$ −1.47753e11 −1.08533
$$299$$ 1.53242e10 0.110881
$$300$$ 0 0
$$301$$ 4.27394e11 3.00109
$$302$$ 5.20722e10 0.360226
$$303$$ 0 0
$$304$$ 5.45928e9 0.0366610
$$305$$ −4.10543e11 −2.71650
$$306$$ 0 0
$$307$$ 6.15064e10 0.395182 0.197591 0.980285i $$-0.436688\pi$$
0.197591 + 0.980285i $$0.436688\pi$$
$$308$$ 4.88109e10 0.309057
$$309$$ 0 0
$$310$$ 7.01042e10 0.431138
$$311$$ −2.16398e11 −1.31169 −0.655846 0.754894i $$-0.727688\pi$$
−0.655846 + 0.754894i $$0.727688\pi$$
$$312$$ 0 0
$$313$$ 2.44634e11 1.44068 0.720340 0.693621i $$-0.243986\pi$$
0.720340 + 0.693621i $$0.243986\pi$$
$$314$$ −2.60519e10 −0.151237
$$315$$ 0 0
$$316$$ 2.88952e10 0.163017
$$317$$ −7.77399e10 −0.432392 −0.216196 0.976350i $$-0.569365\pi$$
−0.216196 + 0.976350i $$0.569365\pi$$
$$318$$ 0 0
$$319$$ −4.90183e10 −0.265033
$$320$$ 3.32021e10 0.177007
$$321$$ 0 0
$$322$$ 8.68343e10 0.450132
$$323$$ 1.18625e10 0.0606406
$$324$$ 0 0
$$325$$ 5.60743e10 0.278797
$$326$$ 1.81100e11 0.888053
$$327$$ 0 0
$$328$$ 1.09823e11 0.523915
$$329$$ 3.63280e11 1.70946
$$330$$ 0 0
$$331$$ −1.68625e11 −0.772139 −0.386070 0.922470i $$-0.626168\pi$$
−0.386070 + 0.922470i $$0.626168\pi$$
$$332$$ −7.89131e10 −0.356474
$$333$$ 0 0
$$334$$ 2.75690e10 0.121217
$$335$$ 3.81978e11 1.65705
$$336$$ 0 0
$$337$$ −7.70797e10 −0.325541 −0.162770 0.986664i $$-0.552043\pi$$
−0.162770 + 0.986664i $$0.552043\pi$$
$$338$$ −1.30517e10 −0.0543928
$$339$$ 0 0
$$340$$ 7.21448e10 0.292786
$$341$$ 4.17340e10 0.167146
$$342$$ 0 0
$$343$$ −2.18545e11 −0.852544
$$344$$ 1.73070e11 0.666361
$$345$$ 0 0
$$346$$ 4.42156e10 0.165857
$$347$$ −3.54609e11 −1.31301 −0.656504 0.754322i $$-0.727966\pi$$
−0.656504 + 0.754322i $$0.727966\pi$$
$$348$$ 0 0
$$349$$ 1.70460e11 0.615048 0.307524 0.951540i $$-0.400500\pi$$
0.307524 + 0.951540i $$0.400500\pi$$
$$350$$ 3.17743e11 1.13180
$$351$$ 0 0
$$352$$ 1.97657e10 0.0686229
$$353$$ −2.96506e11 −1.01636 −0.508180 0.861251i $$-0.669682\pi$$
−0.508180 + 0.861251i $$0.669682\pi$$
$$354$$ 0 0
$$355$$ 3.99428e11 1.33479
$$356$$ 1.63197e9 0.00538501
$$357$$ 0 0
$$358$$ 1.09869e11 0.353510
$$359$$ −7.20144e10 −0.228820 −0.114410 0.993434i $$-0.536498\pi$$
−0.114410 + 0.993434i $$0.536498\pi$$
$$360$$ 0 0
$$361$$ −3.15748e11 −0.978496
$$362$$ 3.86455e11 1.18280
$$363$$ 0 0
$$364$$ −7.39570e10 −0.220812
$$365$$ −2.40702e11 −0.709842
$$366$$ 0 0
$$367$$ −1.05092e11 −0.302394 −0.151197 0.988504i $$-0.548313\pi$$
−0.151197 + 0.988504i $$0.548313\pi$$
$$368$$ 3.51629e10 0.0999471
$$369$$ 0 0
$$370$$ −5.73094e11 −1.58971
$$371$$ −6.72790e11 −1.84373
$$372$$ 0 0
$$373$$ 2.19888e11 0.588182 0.294091 0.955777i $$-0.404983\pi$$
0.294091 + 0.955777i $$0.404983\pi$$
$$374$$ 4.29487e10 0.113508
$$375$$ 0 0
$$376$$ 1.47108e11 0.379569
$$377$$ 7.42712e10 0.189358
$$378$$ 0 0
$$379$$ −3.14748e11 −0.783586 −0.391793 0.920053i $$-0.628145\pi$$
−0.391793 + 0.920053i $$0.628145\pi$$
$$380$$ 4.22028e10 0.103828
$$381$$ 0 0
$$382$$ 5.75973e11 1.38394
$$383$$ 3.41027e10 0.0809831 0.0404915 0.999180i $$-0.487108\pi$$
0.0404915 + 0.999180i $$0.487108\pi$$
$$384$$ 0 0
$$385$$ 3.77331e11 0.875286
$$386$$ 2.72050e11 0.623744
$$387$$ 0 0
$$388$$ 2.23044e11 0.499630
$$389$$ −4.49612e11 −0.995554 −0.497777 0.867305i $$-0.665850\pi$$
−0.497777 + 0.867305i $$0.665850\pi$$
$$390$$ 0 0
$$391$$ 7.64055e10 0.165321
$$392$$ −2.53787e11 −0.542852
$$393$$ 0 0
$$394$$ 6.37267e11 1.33226
$$395$$ 2.23374e11 0.461684
$$396$$ 0 0
$$397$$ 2.29976e11 0.464649 0.232324 0.972638i $$-0.425367\pi$$
0.232324 + 0.972638i $$0.425367\pi$$
$$398$$ −2.09730e10 −0.0418974
$$399$$ 0 0
$$400$$ 1.28668e11 0.251304
$$401$$ 6.69163e11 1.29236 0.646178 0.763187i $$-0.276366\pi$$
0.646178 + 0.763187i $$0.276366\pi$$
$$402$$ 0 0
$$403$$ −6.32342e10 −0.119421
$$404$$ 2.11049e11 0.394156
$$405$$ 0 0
$$406$$ 4.20856e11 0.768716
$$407$$ −3.41171e11 −0.616307
$$408$$ 0 0
$$409$$ −6.73923e11 −1.19085 −0.595423 0.803413i $$-0.703015\pi$$
−0.595423 + 0.803413i $$0.703015\pi$$
$$410$$ 8.48983e11 1.48379
$$411$$ 0 0
$$412$$ −4.24693e11 −0.726168
$$413$$ −1.09408e12 −1.85043
$$414$$ 0 0
$$415$$ −6.10035e11 −1.00957
$$416$$ −2.99484e10 −0.0490290
$$417$$ 0 0
$$418$$ 2.51239e10 0.0402526
$$419$$ 3.45053e11 0.546919 0.273460 0.961883i $$-0.411832\pi$$
0.273460 + 0.961883i $$0.411832\pi$$
$$420$$ 0 0
$$421$$ −5.57817e11 −0.865411 −0.432706 0.901535i $$-0.642441\pi$$
−0.432706 + 0.901535i $$0.642441\pi$$
$$422$$ −3.77243e11 −0.579049
$$423$$ 0 0
$$424$$ −2.72442e11 −0.409380
$$425$$ 2.79582e11 0.415680
$$426$$ 0 0
$$427$$ 2.09836e12 3.05460
$$428$$ −2.94823e11 −0.424683
$$429$$ 0 0
$$430$$ 1.33791e12 1.88721
$$431$$ 6.39243e11 0.892315 0.446158 0.894954i $$-0.352792\pi$$
0.446158 + 0.894954i $$0.352792\pi$$
$$432$$ 0 0
$$433$$ −1.23759e12 −1.69193 −0.845965 0.533238i $$-0.820975\pi$$
−0.845965 + 0.533238i $$0.820975\pi$$
$$434$$ −3.58314e11 −0.484798
$$435$$ 0 0
$$436$$ −7.12908e11 −0.944810
$$437$$ 4.46952e10 0.0586265
$$438$$ 0 0
$$439$$ −9.14852e11 −1.17560 −0.587801 0.809006i $$-0.700006\pi$$
−0.587801 + 0.809006i $$0.700006\pi$$
$$440$$ 1.52798e11 0.194348
$$441$$ 0 0
$$442$$ −6.50748e10 −0.0810984
$$443$$ 1.18651e12 1.46370 0.731852 0.681464i $$-0.238656\pi$$
0.731852 + 0.681464i $$0.238656\pi$$
$$444$$ 0 0
$$445$$ 1.26159e10 0.0152510
$$446$$ −3.81322e10 −0.0456336
$$447$$ 0 0
$$448$$ −1.69702e11 −0.199037
$$449$$ 2.70808e10 0.0314452 0.0157226 0.999876i $$-0.494995\pi$$
0.0157226 + 0.999876i $$0.494995\pi$$
$$450$$ 0 0
$$451$$ 5.05411e11 0.575241
$$452$$ −1.73708e11 −0.195748
$$453$$ 0 0
$$454$$ −1.19448e11 −0.131955
$$455$$ −5.71722e11 −0.625365
$$456$$ 0 0
$$457$$ 2.02586e11 0.217263 0.108632 0.994082i $$-0.465353\pi$$
0.108632 + 0.994082i $$0.465353\pi$$
$$458$$ −4.22346e11 −0.448512
$$459$$ 0 0
$$460$$ 2.71826e11 0.283062
$$461$$ −8.96346e11 −0.924319 −0.462159 0.886797i $$-0.652925\pi$$
−0.462159 + 0.886797i $$0.652925\pi$$
$$462$$ 0 0
$$463$$ −5.17740e11 −0.523597 −0.261799 0.965123i $$-0.584316\pi$$
−0.261799 + 0.965123i $$0.584316\pi$$
$$464$$ 1.70423e11 0.170685
$$465$$ 0 0
$$466$$ 3.84731e11 0.377938
$$467$$ −9.20785e11 −0.895843 −0.447922 0.894073i $$-0.647836\pi$$
−0.447922 + 0.894073i $$0.647836\pi$$
$$468$$ 0 0
$$469$$ −1.95235e12 −1.86329
$$470$$ 1.13721e12 1.07498
$$471$$ 0 0
$$472$$ −4.43040e11 −0.410869
$$473$$ 7.96478e11 0.731642
$$474$$ 0 0
$$475$$ 1.63548e11 0.147409
$$476$$ −3.68744e11 −0.329226
$$477$$ 0 0
$$478$$ 9.54108e11 0.835934
$$479$$ −1.73253e12 −1.50373 −0.751867 0.659315i $$-0.770846\pi$$
−0.751867 + 0.659315i $$0.770846\pi$$
$$480$$ 0 0
$$481$$ 5.16932e11 0.440333
$$482$$ −2.01023e11 −0.169642
$$483$$ 0 0
$$484$$ −5.12672e11 −0.424654
$$485$$ 1.72424e12 1.41501
$$486$$ 0 0
$$487$$ 1.49591e12 1.20511 0.602554 0.798078i $$-0.294150\pi$$
0.602554 + 0.798078i $$0.294150\pi$$
$$488$$ 8.49715e11 0.678241
$$489$$ 0 0
$$490$$ −1.96189e12 −1.53742
$$491$$ 4.28954e11 0.333076 0.166538 0.986035i $$-0.446741\pi$$
0.166538 + 0.986035i $$0.446741\pi$$
$$492$$ 0 0
$$493$$ 3.70311e11 0.282329
$$494$$ −3.80670e10 −0.0287592
$$495$$ 0 0
$$496$$ −1.45097e11 −0.107644
$$497$$ −2.04155e12 −1.50091
$$498$$ 0 0
$$499$$ −9.12174e11 −0.658606 −0.329303 0.944224i $$-0.606814\pi$$
−0.329303 + 0.944224i $$0.606814\pi$$
$$500$$ 5.16301e9 0.00369435
$$501$$ 0 0
$$502$$ 3.86834e11 0.271868
$$503$$ 1.26835e12 0.883456 0.441728 0.897149i $$-0.354366\pi$$
0.441728 + 0.897149i $$0.354366\pi$$
$$504$$ 0 0
$$505$$ 1.63151e12 1.11629
$$506$$ 1.61822e11 0.109739
$$507$$ 0 0
$$508$$ −8.91627e11 −0.594014
$$509$$ 1.54192e12 1.01820 0.509098 0.860708i $$-0.329979\pi$$
0.509098 + 0.860708i $$0.329979\pi$$
$$510$$ 0 0
$$511$$ 1.23027e12 0.798189
$$512$$ −6.87195e10 −0.0441942
$$513$$ 0 0
$$514$$ 4.74990e11 0.300158
$$515$$ −3.28307e12 −2.05659
$$516$$ 0 0
$$517$$ 6.76998e11 0.416754
$$518$$ 2.92918e12 1.78757
$$519$$ 0 0
$$520$$ −2.31515e11 −0.138856
$$521$$ 1.48896e12 0.885345 0.442672 0.896683i $$-0.354031\pi$$
0.442672 + 0.896683i $$0.354031\pi$$
$$522$$ 0 0
$$523$$ 2.55715e12 1.49451 0.747256 0.664536i $$-0.231371\pi$$
0.747256 + 0.664536i $$0.231371\pi$$
$$524$$ −1.28691e12 −0.745691
$$525$$ 0 0
$$526$$ −1.21463e12 −0.691846
$$527$$ −3.15281e11 −0.178053
$$528$$ 0 0
$$529$$ −1.51327e12 −0.840169
$$530$$ −2.10610e12 −1.15941
$$531$$ 0 0
$$532$$ −2.15706e11 −0.116751
$$533$$ −7.65784e11 −0.410993
$$534$$ 0 0
$$535$$ −2.27912e12 −1.20275
$$536$$ −7.90592e11 −0.413724
$$537$$ 0 0
$$538$$ −4.11438e11 −0.211731
$$539$$ −1.16794e12 −0.596033
$$540$$ 0 0
$$541$$ −2.64921e12 −1.32962 −0.664811 0.747011i $$-0.731488\pi$$
−0.664811 + 0.747011i $$0.731488\pi$$
$$542$$ 1.29422e12 0.644188
$$543$$ 0 0
$$544$$ −1.49320e11 −0.0731011
$$545$$ −5.51111e12 −2.67581
$$546$$ 0 0
$$547$$ 2.16400e12 1.03351 0.516754 0.856134i $$-0.327140\pi$$
0.516754 + 0.856134i $$0.327140\pi$$
$$548$$ 1.63559e12 0.774752
$$549$$ 0 0
$$550$$ 5.92136e11 0.275924
$$551$$ 2.16622e11 0.100120
$$552$$ 0 0
$$553$$ −1.14170e12 −0.519145
$$554$$ 5.44056e11 0.245386
$$555$$ 0 0
$$556$$ 1.95242e12 0.866437
$$557$$ 2.96364e12 1.30460 0.652300 0.757961i $$-0.273804\pi$$
0.652300 + 0.757961i $$0.273804\pi$$
$$558$$ 0 0
$$559$$ −1.20680e12 −0.522736
$$560$$ −1.31187e12 −0.563697
$$561$$ 0 0
$$562$$ −1.14373e11 −0.0483627
$$563$$ 3.46859e12 1.45501 0.727504 0.686104i $$-0.240680\pi$$
0.727504 + 0.686104i $$0.240680\pi$$
$$564$$ 0 0
$$565$$ −1.34284e12 −0.554380
$$566$$ 1.24881e12 0.511474
$$567$$ 0 0
$$568$$ −8.26710e11 −0.333262
$$569$$ 3.83703e12 1.53458 0.767290 0.641300i $$-0.221605\pi$$
0.767290 + 0.641300i $$0.221605\pi$$
$$570$$ 0 0
$$571$$ 3.35374e11 0.132028 0.0660142 0.997819i $$-0.478972\pi$$
0.0660142 + 0.997819i $$0.478972\pi$$
$$572$$ −1.37824e11 −0.0538323
$$573$$ 0 0
$$574$$ −4.33929e12 −1.66846
$$575$$ 1.05341e12 0.401874
$$576$$ 0 0
$$577$$ −1.74089e11 −0.0653852 −0.0326926 0.999465i $$-0.510408\pi$$
−0.0326926 + 0.999465i $$0.510408\pi$$
$$578$$ 1.57295e12 0.586191
$$579$$ 0 0
$$580$$ 1.31745e12 0.483400
$$581$$ 3.11799e12 1.13523
$$582$$ 0 0
$$583$$ −1.25379e12 −0.449486
$$584$$ 4.98189e11 0.177230
$$585$$ 0 0
$$586$$ −2.02659e12 −0.709948
$$587$$ −1.87089e12 −0.650395 −0.325198 0.945646i $$-0.605431\pi$$
−0.325198 + 0.945646i $$0.605431\pi$$
$$588$$ 0 0
$$589$$ −1.84431e11 −0.0631415
$$590$$ −3.42490e12 −1.16363
$$591$$ 0 0
$$592$$ 1.18615e12 0.396911
$$593$$ −4.16793e12 −1.38412 −0.692061 0.721839i $$-0.743297\pi$$
−0.692061 + 0.721839i $$0.743297\pi$$
$$594$$ 0 0
$$595$$ −2.85056e12 −0.932405
$$596$$ 2.36404e12 0.767445
$$597$$ 0 0
$$598$$ −2.45188e11 −0.0784049
$$599$$ −8.16635e10 −0.0259183 −0.0129592 0.999916i $$-0.504125\pi$$
−0.0129592 + 0.999916i $$0.504125\pi$$
$$600$$ 0 0
$$601$$ −4.00769e12 −1.25302 −0.626511 0.779412i $$-0.715518\pi$$
−0.626511 + 0.779412i $$0.715518\pi$$
$$602$$ −6.83830e12 −2.12209
$$603$$ 0 0
$$604$$ −8.33155e11 −0.254718
$$605$$ −3.96320e12 −1.20267
$$606$$ 0 0
$$607$$ 1.45542e12 0.435149 0.217575 0.976044i $$-0.430185\pi$$
0.217575 + 0.976044i $$0.430185\pi$$
$$608$$ −8.73485e10 −0.0259232
$$609$$ 0 0
$$610$$ 6.56869e12 1.92086
$$611$$ −1.02577e12 −0.297758
$$612$$ 0 0
$$613$$ −2.55645e12 −0.731248 −0.365624 0.930763i $$-0.619144\pi$$
−0.365624 + 0.930763i $$0.619144\pi$$
$$614$$ −9.84102e11 −0.279436
$$615$$ 0 0
$$616$$ −7.80975e11 −0.218537
$$617$$ −2.69727e11 −0.0749276 −0.0374638 0.999298i $$-0.511928\pi$$
−0.0374638 + 0.999298i $$0.511928\pi$$
$$618$$ 0 0
$$619$$ 4.28111e11 0.117206 0.0586028 0.998281i $$-0.481335\pi$$
0.0586028 + 0.998281i $$0.481335\pi$$
$$620$$ −1.12167e12 −0.304861
$$621$$ 0 0
$$622$$ 3.46237e12 0.927507
$$623$$ −6.44818e10 −0.0171491
$$624$$ 0 0
$$625$$ −3.79469e12 −0.994755
$$626$$ −3.91414e12 −1.01871
$$627$$ 0 0
$$628$$ 4.16831e11 0.106940
$$629$$ 2.57739e12 0.656525
$$630$$ 0 0
$$631$$ −2.98421e11 −0.0749372 −0.0374686 0.999298i $$-0.511929\pi$$
−0.0374686 + 0.999298i $$0.511929\pi$$
$$632$$ −4.62323e11 −0.115271
$$633$$ 0 0
$$634$$ 1.24384e12 0.305747
$$635$$ −6.89269e12 −1.68231
$$636$$ 0 0
$$637$$ 1.76963e12 0.425848
$$638$$ 7.84293e11 0.187407
$$639$$ 0 0
$$640$$ −5.31234e11 −0.125163
$$641$$ 2.34144e12 0.547799 0.273899 0.961758i $$-0.411686\pi$$
0.273899 + 0.961758i $$0.411686\pi$$
$$642$$ 0 0
$$643$$ 7.75186e12 1.78837 0.894183 0.447701i $$-0.147757\pi$$
0.894183 + 0.447701i $$0.147757\pi$$
$$644$$ −1.38935e12 −0.318291
$$645$$ 0 0
$$646$$ −1.89799e11 −0.0428794
$$647$$ −3.74980e12 −0.841278 −0.420639 0.907228i $$-0.638194\pi$$
−0.420639 + 0.907228i $$0.638194\pi$$
$$648$$ 0 0
$$649$$ −2.03889e12 −0.451121
$$650$$ −8.97188e11 −0.197139
$$651$$ 0 0
$$652$$ −2.89760e12 −0.627948
$$653$$ −2.82022e12 −0.606979 −0.303490 0.952835i $$-0.598152\pi$$
−0.303490 + 0.952835i $$0.598152\pi$$
$$654$$ 0 0
$$655$$ −9.94846e12 −2.11188
$$656$$ −1.75717e12 −0.370464
$$657$$ 0 0
$$658$$ −5.81248e12 −1.20877
$$659$$ 1.16074e12 0.239746 0.119873 0.992789i $$-0.461751\pi$$
0.119873 + 0.992789i $$0.461751\pi$$
$$660$$ 0 0
$$661$$ −3.18976e11 −0.0649907 −0.0324954 0.999472i $$-0.510345\pi$$
−0.0324954 + 0.999472i $$0.510345\pi$$
$$662$$ 2.69800e12 0.545985
$$663$$ 0 0
$$664$$ 1.26261e12 0.252065
$$665$$ −1.66750e12 −0.330651
$$666$$ 0 0
$$667$$ 1.39525e12 0.272952
$$668$$ −4.41104e11 −0.0857131
$$669$$ 0 0
$$670$$ −6.11164e12 −1.17171
$$671$$ 3.91043e12 0.744686
$$672$$ 0 0
$$673$$ 4.82897e12 0.907375 0.453687 0.891161i $$-0.350108\pi$$
0.453687 + 0.891161i $$0.350108\pi$$
$$674$$ 1.23328e12 0.230192
$$675$$ 0 0
$$676$$ 2.08827e11 0.0384615
$$677$$ 7.47095e12 1.36687 0.683434 0.730012i $$-0.260486\pi$$
0.683434 + 0.730012i $$0.260486\pi$$
$$678$$ 0 0
$$679$$ −8.81286e12 −1.59112
$$680$$ −1.15432e12 −0.207031
$$681$$ 0 0
$$682$$ −6.67744e11 −0.118190
$$683$$ 4.13060e12 0.726306 0.363153 0.931730i $$-0.381700\pi$$
0.363153 + 0.931730i $$0.381700\pi$$
$$684$$ 0 0
$$685$$ 1.26439e13 2.19419
$$686$$ 3.49672e12 0.602840
$$687$$ 0 0
$$688$$ −2.76912e12 −0.471188
$$689$$ 1.89971e12 0.321144
$$690$$ 0 0
$$691$$ 4.50776e12 0.752159 0.376079 0.926587i $$-0.377272\pi$$
0.376079 + 0.926587i $$0.377272\pi$$
$$692$$ −7.07449e11 −0.117278
$$693$$ 0 0
$$694$$ 5.67375e12 0.928437
$$695$$ 1.50931e13 2.45385
$$696$$ 0 0
$$697$$ −3.81814e12 −0.612780
$$698$$ −2.72737e12 −0.434905
$$699$$ 0 0
$$700$$ −5.08389e12 −0.800304
$$701$$ 1.38907e12 0.217267 0.108633 0.994082i $$-0.465353\pi$$
0.108633 + 0.994082i $$0.465353\pi$$
$$702$$ 0 0
$$703$$ 1.50770e12 0.232818
$$704$$ −3.16251e11 −0.0485237
$$705$$ 0 0
$$706$$ 4.74410e12 0.718675
$$707$$ −8.33893e12 −1.25523
$$708$$ 0 0
$$709$$ 5.59251e12 0.831187 0.415593 0.909551i $$-0.363574\pi$$
0.415593 + 0.909551i $$0.363574\pi$$
$$710$$ −6.39086e12 −0.943836
$$711$$ 0 0
$$712$$ −2.61115e10 −0.00380778
$$713$$ −1.18791e12 −0.172140
$$714$$ 0 0
$$715$$ −1.06544e12 −0.152459
$$716$$ −1.75791e12 −0.249969
$$717$$ 0 0
$$718$$ 1.15223e12 0.161800
$$719$$ 8.33742e12 1.16346 0.581730 0.813382i $$-0.302376\pi$$
0.581730 + 0.813382i $$0.302376\pi$$
$$720$$ 0 0
$$721$$ 1.67803e13 2.31255
$$722$$ 5.05198e12 0.691901
$$723$$ 0 0
$$724$$ −6.18328e12 −0.836364
$$725$$ 5.10549e12 0.686304
$$726$$ 0 0
$$727$$ −5.13697e12 −0.682028 −0.341014 0.940058i $$-0.610770\pi$$
−0.341014 + 0.940058i $$0.610770\pi$$
$$728$$ 1.18331e12 0.156138
$$729$$ 0 0
$$730$$ 3.85123e12 0.501934
$$731$$ −6.01702e12 −0.779387
$$732$$ 0 0
$$733$$ 9.72563e12 1.24437 0.622185 0.782870i $$-0.286245\pi$$
0.622185 + 0.782870i $$0.286245\pi$$
$$734$$ 1.68147e12 0.213825
$$735$$ 0 0
$$736$$ −5.62607e11 −0.0706733
$$737$$ −3.63834e12 −0.454255
$$738$$ 0 0
$$739$$ −1.15533e13 −1.42498 −0.712488 0.701684i $$-0.752432\pi$$
−0.712488 + 0.701684i $$0.752432\pi$$
$$740$$ 9.16951e12 1.12410
$$741$$ 0 0
$$742$$ 1.07646e13 1.30371
$$743$$ 1.04495e13 1.25790 0.628948 0.777447i $$-0.283486\pi$$
0.628948 + 0.777447i $$0.283486\pi$$
$$744$$ 0 0
$$745$$ 1.82752e13 2.17349
$$746$$ −3.51821e12 −0.415907
$$747$$ 0 0
$$748$$ −6.87180e11 −0.0802626
$$749$$ 1.16490e13 1.35245
$$750$$ 0 0
$$751$$ 1.92230e12 0.220516 0.110258 0.993903i $$-0.464832\pi$$
0.110258 + 0.993903i $$0.464832\pi$$
$$752$$ −2.35372e12 −0.268396
$$753$$ 0 0
$$754$$ −1.18834e12 −0.133897
$$755$$ −6.44068e12 −0.721390
$$756$$ 0 0
$$757$$ −1.27412e12 −0.141019 −0.0705096 0.997511i $$-0.522463\pi$$
−0.0705096 + 0.997511i $$0.522463\pi$$
$$758$$ 5.03597e12 0.554079
$$759$$ 0 0
$$760$$ −6.75245e11 −0.0734176
$$761$$ −1.21619e13 −1.31453 −0.657267 0.753658i $$-0.728288\pi$$
−0.657267 + 0.753658i $$0.728288\pi$$
$$762$$ 0 0
$$763$$ 2.81682e13 3.00884
$$764$$ −9.21557e12 −0.978593
$$765$$ 0 0
$$766$$ −5.45643e11 −0.0572637
$$767$$ 3.08927e12 0.322312
$$768$$ 0 0
$$769$$ −5.98877e12 −0.617546 −0.308773 0.951136i $$-0.599918\pi$$
−0.308773 + 0.951136i $$0.599918\pi$$
$$770$$ −6.03730e12 −0.618920
$$771$$ 0 0
$$772$$ −4.35281e12 −0.441054
$$773$$ 9.64838e12 0.971956 0.485978 0.873971i $$-0.338464\pi$$
0.485978 + 0.873971i $$0.338464\pi$$
$$774$$ 0 0
$$775$$ −4.34679e12 −0.432824
$$776$$ −3.56871e12 −0.353292
$$777$$ 0 0
$$778$$ 7.19379e12 0.703963
$$779$$ −2.23351e12 −0.217305
$$780$$ 0 0
$$781$$ −3.80456e12 −0.365911
$$782$$ −1.22249e12 −0.116900
$$783$$ 0 0
$$784$$ 4.06059e12 0.383854
$$785$$ 3.22230e12 0.302867
$$786$$ 0 0
$$787$$ 1.40829e13 1.30860 0.654299 0.756236i $$-0.272964\pi$$
0.654299 + 0.756236i $$0.272964\pi$$
$$788$$ −1.01963e13 −0.942049
$$789$$ 0 0
$$790$$ −3.57398e12 −0.326460
$$791$$ 6.86350e12 0.623378
$$792$$ 0 0
$$793$$ −5.92498e12 −0.532056
$$794$$ −3.67961e12 −0.328556
$$795$$ 0 0
$$796$$ 3.35568e11 0.0296259
$$797$$ 8.45863e12 0.742570 0.371285 0.928519i $$-0.378917\pi$$
0.371285 + 0.928519i $$0.378917\pi$$
$$798$$ 0 0
$$799$$ −5.11440e12 −0.443950
$$800$$ −2.05869e12 −0.177699
$$801$$ 0 0
$$802$$ −1.07066e13 −0.913834
$$803$$ 2.29269e12 0.194592
$$804$$ 0 0
$$805$$ −1.07403e13 −0.901437
$$806$$ 1.01175e12 0.0844431
$$807$$ 0 0
$$808$$ −3.37679e12 −0.278710
$$809$$ 9.95988e12 0.817496 0.408748 0.912647i $$-0.365965\pi$$
0.408748 + 0.912647i $$0.365965\pi$$
$$810$$ 0 0
$$811$$ 9.83262e12 0.798133 0.399067 0.916922i $$-0.369334\pi$$
0.399067 + 0.916922i $$0.369334\pi$$
$$812$$ −6.73369e12 −0.543564
$$813$$ 0 0
$$814$$ 5.45873e12 0.435795
$$815$$ −2.23998e13 −1.77842
$$816$$ 0 0
$$817$$ −3.51980e12 −0.276388
$$818$$ 1.07828e13 0.842055
$$819$$ 0 0
$$820$$ −1.35837e13 −1.04920
$$821$$ −8.67049e12 −0.666039 −0.333019 0.942920i $$-0.608067\pi$$
−0.333019 + 0.942920i $$0.608067\pi$$
$$822$$ 0 0
$$823$$ −1.26458e13 −0.960834 −0.480417 0.877040i $$-0.659515\pi$$
−0.480417 + 0.877040i $$0.659515\pi$$
$$824$$ 6.79508e12 0.513479
$$825$$ 0 0
$$826$$ 1.75053e13 1.30845
$$827$$ 2.04741e13 1.52205 0.761027 0.648720i $$-0.224695\pi$$
0.761027 + 0.648720i $$0.224695\pi$$
$$828$$ 0 0
$$829$$ −1.00940e13 −0.742283 −0.371141 0.928576i $$-0.621033\pi$$
−0.371141 + 0.928576i $$0.621033\pi$$
$$830$$ 9.76056e12 0.713877
$$831$$ 0 0
$$832$$ 4.79174e11 0.0346688
$$833$$ 8.82324e12 0.634929
$$834$$ 0 0
$$835$$ −3.40994e12 −0.242749
$$836$$ −4.01982e11 −0.0284629
$$837$$ 0 0
$$838$$ −5.52085e12 −0.386730
$$839$$ −1.54505e13 −1.07650 −0.538249 0.842786i $$-0.680914\pi$$
−0.538249 + 0.842786i $$0.680914\pi$$
$$840$$ 0 0
$$841$$ −7.74485e12 −0.533864
$$842$$ 8.92508e12 0.611938
$$843$$ 0 0
$$844$$ 6.03589e12 0.409449
$$845$$ 1.61433e12 0.108927
$$846$$ 0 0
$$847$$ 2.02566e13 1.35235
$$848$$ 4.35907e12 0.289476
$$849$$ 0 0
$$850$$ −4.47331e12 −0.293930
$$851$$ 9.71104e12 0.634721
$$852$$ 0 0
$$853$$ 2.56415e13 1.65834 0.829170 0.558997i $$-0.188814\pi$$
0.829170 + 0.558997i $$0.188814\pi$$
$$854$$ −3.35737e13 −2.15993
$$855$$ 0 0
$$856$$ 4.71717e12 0.300296
$$857$$ 1.44941e11 0.00917862 0.00458931 0.999989i $$-0.498539\pi$$
0.00458931 + 0.999989i $$0.498539\pi$$
$$858$$ 0 0
$$859$$ −9.47728e12 −0.593902 −0.296951 0.954893i $$-0.595970\pi$$
−0.296951 + 0.954893i $$0.595970\pi$$
$$860$$ −2.14066e13 −1.33446
$$861$$ 0 0
$$862$$ −1.02279e13 −0.630962
$$863$$ 4.25726e12 0.261265 0.130633 0.991431i $$-0.458299\pi$$
0.130633 + 0.991431i $$0.458299\pi$$
$$864$$ 0 0
$$865$$ −5.46892e12 −0.332146
$$866$$ 1.98015e13 1.19638
$$867$$ 0 0
$$868$$ 5.73303e12 0.342804
$$869$$ −2.12764e12 −0.126563
$$870$$ 0 0
$$871$$ 5.51272e12 0.324552
$$872$$ 1.14065e13 0.668081
$$873$$ 0 0
$$874$$ −7.15123e11 −0.0414552
$$875$$ −2.03999e11 −0.0117650
$$876$$ 0 0
$$877$$ 1.42035e13 0.810768 0.405384 0.914147i $$-0.367138\pi$$
0.405384 + 0.914147i $$0.367138\pi$$
$$878$$ 1.46376e13 0.831276
$$879$$ 0 0
$$880$$ −2.44476e12 −0.137425
$$881$$ 1.12878e13 0.631275 0.315638 0.948880i $$-0.397782\pi$$
0.315638 + 0.948880i $$0.397782\pi$$
$$882$$ 0 0
$$883$$ 2.25433e12 0.124794 0.0623972 0.998051i $$-0.480125\pi$$
0.0623972 + 0.998051i $$0.480125\pi$$
$$884$$ 1.04120e12 0.0573452
$$885$$ 0 0
$$886$$ −1.89841e13 −1.03499
$$887$$ 3.13450e13 1.70024 0.850122 0.526585i $$-0.176528\pi$$
0.850122 + 0.526585i $$0.176528\pi$$
$$888$$ 0 0
$$889$$ 3.52297e13 1.89170
$$890$$ −2.01854e11 −0.0107841
$$891$$ 0 0
$$892$$ 6.10115e11 0.0322678
$$893$$ −2.99179e12 −0.157434
$$894$$ 0 0
$$895$$ −1.35894e13 −0.707942
$$896$$ 2.71522e12 0.140741
$$897$$ 0 0
$$898$$ −4.33294e11 −0.0222351
$$899$$ −5.75739e12 −0.293973
$$900$$ 0 0
$$901$$ 9.47180e12 0.478819
$$902$$ −8.08657e12 −0.406757
$$903$$ 0 0
$$904$$ 2.77933e12 0.138415
$$905$$ −4.77997e13 −2.36868
$$906$$ 0 0
$$907$$ 1.77171e13 0.869280 0.434640 0.900604i $$-0.356876\pi$$
0.434640 + 0.900604i $$0.356876\pi$$
$$908$$ 1.91116e12 0.0933064
$$909$$ 0 0
$$910$$ 9.14756e12 0.442200
$$911$$ −2.93419e13 −1.41142 −0.705710 0.708501i $$-0.749372\pi$$
−0.705710 + 0.708501i $$0.749372\pi$$
$$912$$ 0 0
$$913$$ 5.81059e12 0.276759
$$914$$ −3.24138e12 −0.153628
$$915$$ 0 0
$$916$$ 6.75754e12 0.317146
$$917$$ 5.08482e13 2.37473
$$918$$ 0 0
$$919$$ −1.66013e13 −0.767754 −0.383877 0.923384i $$-0.625411\pi$$
−0.383877 + 0.923384i $$0.625411\pi$$
$$920$$ −4.34922e12 −0.200155
$$921$$ 0 0
$$922$$ 1.43415e13 0.653592
$$923$$ 5.76457e12 0.261432
$$924$$ 0 0
$$925$$ 3.55345e13 1.59593
$$926$$ 8.28384e12 0.370239
$$927$$ 0 0
$$928$$ −2.72676e12 −0.120693
$$929$$ 1.45567e13 0.641197 0.320598 0.947215i $$-0.396116\pi$$
0.320598 + 0.947215i $$0.396116\pi$$
$$930$$ 0 0
$$931$$ 5.16136e12 0.225160
$$932$$ −6.15569e12 −0.267242
$$933$$ 0 0
$$934$$ 1.47326e13 0.633457
$$935$$ −5.31222e12 −0.227313
$$936$$ 0 0
$$937$$ −1.38682e13 −0.587751 −0.293875 0.955844i $$-0.594945\pi$$
−0.293875 + 0.955844i $$0.594945\pi$$
$$938$$ 3.12376e13 1.31754
$$939$$ 0 0
$$940$$ −1.81954e13 −0.760127
$$941$$ 2.15767e13 0.897081 0.448540 0.893763i $$-0.351944\pi$$
0.448540 + 0.893763i $$0.351944\pi$$
$$942$$ 0 0
$$943$$ −1.43859e13 −0.592428
$$944$$ 7.08864e12 0.290528
$$945$$ 0 0
$$946$$ −1.27436e13 −0.517349
$$947$$ −7.20814e12 −0.291238 −0.145619 0.989341i $$-0.546517\pi$$
−0.145619 + 0.989341i $$0.546517\pi$$
$$948$$ 0 0
$$949$$ −3.47382e12 −0.139030
$$950$$ −2.61677e12 −0.104234
$$951$$ 0 0
$$952$$ 5.89990e12 0.232798
$$953$$ −2.71758e13 −1.06724 −0.533622 0.845723i $$-0.679170\pi$$
−0.533622 + 0.845723i $$0.679170\pi$$
$$954$$ 0 0
$$955$$ −7.12407e13 −2.77149
$$956$$ −1.52657e13 −0.591095
$$957$$ 0 0
$$958$$ 2.77205e13 1.06330
$$959$$ −6.46251e13 −2.46727
$$960$$ 0 0
$$961$$ −2.15378e13 −0.814603
$$962$$ −8.27092e12 −0.311362
$$963$$ 0 0
$$964$$ 3.21637e12 0.119955
$$965$$ −3.36492e13 −1.24911
$$966$$ 0 0
$$967$$ 1.20281e13 0.442364 0.221182 0.975233i $$-0.429009\pi$$
0.221182 + 0.975233i $$0.429009\pi$$
$$968$$ 8.20275e12 0.300276
$$969$$ 0 0
$$970$$ −2.75878e13 −1.00056
$$971$$ 1.10321e12 0.0398265 0.0199132 0.999802i $$-0.493661\pi$$
0.0199132 + 0.999802i $$0.493661\pi$$
$$972$$ 0 0
$$973$$ −7.71436e13 −2.75926
$$974$$ −2.39346e13 −0.852140
$$975$$ 0 0
$$976$$ −1.35954e13 −0.479589
$$977$$ 4.02945e13 1.41488 0.707441 0.706773i $$-0.249850\pi$$
0.707441 + 0.706773i $$0.249850\pi$$
$$978$$ 0 0
$$979$$ −1.20166e11 −0.00418081
$$980$$ 3.13902e13 1.08712
$$981$$ 0 0
$$982$$ −6.86326e12 −0.235520
$$983$$ −5.04270e13 −1.72255 −0.861276 0.508137i $$-0.830334\pi$$
−0.861276 + 0.508137i $$0.830334\pi$$
$$984$$ 0 0
$$985$$ −7.88219e13 −2.66799
$$986$$ −5.92497e12 −0.199637
$$987$$ 0 0
$$988$$ 6.09072e11 0.0203359
$$989$$ −2.26708e13 −0.753502
$$990$$ 0 0
$$991$$ −2.37023e13 −0.780653 −0.390327 0.920676i $$-0.627638\pi$$
−0.390327 + 0.920676i $$0.627638\pi$$
$$992$$ 2.32155e12 0.0761160
$$993$$ 0 0
$$994$$ 3.26647e13 1.06131
$$995$$ 2.59410e12 0.0839040
$$996$$ 0 0
$$997$$ −3.80094e13 −1.21832 −0.609162 0.793045i $$-0.708494\pi$$
−0.609162 + 0.793045i $$0.708494\pi$$
$$998$$ 1.45948e13 0.465704
$$999$$ 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 234.10.a.a.1.1 1
3.2 odd 2 26.10.a.c.1.1 1
12.11 even 2 208.10.a.b.1.1 1
39.38 odd 2 338.10.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.10.a.c.1.1 1 3.2 odd 2
208.10.a.b.1.1 1 12.11 even 2
234.10.a.a.1.1 1 1.1 even 1 trivial
338.10.a.b.1.1 1 39.38 odd 2