# Properties

 Label 208.10.a.b.1.1 Level $208$ Weight $10$ Character 208.1 Self dual yes Analytic conductor $107.127$ 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] = [208,10,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("208.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 208.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: $$107.127453922$$ 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 $$\chi$$ $$=$$ 208.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-75.0000 q^{3} -1979.00 q^{5} +10115.0 q^{7} -14058.0 q^{9} +O(q^{10})$$ $$q-75.0000 q^{3} -1979.00 q^{5} +10115.0 q^{7} -14058.0 q^{9} -18850.0 q^{11} +28561.0 q^{13} +148425. q^{15} -142403. q^{17} -83302.0 q^{19} -758625. q^{21} +536544. q^{23} +1.96332e6 q^{25} +2.53058e6 q^{27} -2.60044e6 q^{29} +2.21400e6 q^{31} +1.41375e6 q^{33} -2.00176e7 q^{35} +1.80992e7 q^{37} -2.14208e6 q^{39} +2.68122e7 q^{41} +4.22535e7 q^{43} +2.78208e7 q^{45} -3.59150e7 q^{47} +6.19596e7 q^{49} +1.06802e7 q^{51} -6.65141e7 q^{53} +3.73042e7 q^{55} +6.24765e6 q^{57} +1.08164e8 q^{59} -2.07450e8 q^{61} -1.42197e8 q^{63} -5.65222e7 q^{65} -1.93016e8 q^{67} -4.02408e7 q^{69} +2.01833e8 q^{71} -1.21628e8 q^{73} -1.47249e8 q^{75} -1.90668e8 q^{77} -1.12872e8 q^{79} +8.69105e7 q^{81} -3.08254e8 q^{83} +2.81816e8 q^{85} +1.95033e8 q^{87} -6.37487e6 q^{89} +2.88895e8 q^{91} -1.66050e8 q^{93} +1.64855e8 q^{95} +8.71267e8 q^{97} +2.64993e8 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −75.0000 −0.534584 −0.267292 0.963616i $$-0.586129\pi$$
−0.267292 + 0.963616i $$0.586129\pi$$
$$4$$ 0 0
$$5$$ −1979.00 −1.41606 −0.708029 0.706184i $$-0.750415\pi$$
−0.708029 + 0.706184i $$0.750415\pi$$
$$6$$ 0 0
$$7$$ 10115.0 1.59230 0.796150 0.605100i $$-0.206867\pi$$
0.796150 + 0.605100i $$0.206867\pi$$
$$8$$ 0 0
$$9$$ −14058.0 −0.714220
$$10$$ 0 0
$$11$$ −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
$$14$$ 0 0
$$15$$ 148425. 0.757001
$$16$$ 0 0
$$17$$ −142403. −0.413522 −0.206761 0.978391i $$-0.566292\pi$$
−0.206761 + 0.978391i $$0.566292\pi$$
$$18$$ 0 0
$$19$$ −83302.0 −0.146644 −0.0733220 0.997308i $$-0.523360\pi$$
−0.0733220 + 0.997308i $$0.523360\pi$$
$$20$$ 0 0
$$21$$ −758625. −0.851217
$$22$$ 0 0
$$23$$ 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
$$26$$ 0 0
$$27$$ 2.53058e6 0.916394
$$28$$ 0 0
$$29$$ −2.60044e6 −0.682741 −0.341371 0.939929i $$-0.610891\pi$$
−0.341371 + 0.939929i $$0.610891\pi$$
$$30$$ 0 0
$$31$$ 2.21400e6 0.430577 0.215288 0.976550i $$-0.430931\pi$$
0.215288 + 0.976550i $$0.430931\pi$$
$$32$$ 0 0
$$33$$ 1.41375e6 0.207520
$$34$$ 0 0
$$35$$ −2.00176e7 −2.25479
$$36$$ 0 0
$$37$$ 1.80992e7 1.58764 0.793821 0.608151i $$-0.208089\pi$$
0.793821 + 0.608151i $$0.208089\pi$$
$$38$$ 0 0
$$39$$ −2.14208e6 −0.148267
$$40$$ 0 0
$$41$$ 2.68122e7 1.48186 0.740928 0.671585i $$-0.234386\pi$$
0.740928 + 0.671585i $$0.234386\pi$$
$$42$$ 0 0
$$43$$ 4.22535e7 1.88475 0.942376 0.334555i $$-0.108586\pi$$
0.942376 + 0.334555i $$0.108586\pi$$
$$44$$ 0 0
$$45$$ 2.78208e7 1.01138
$$46$$ 0 0
$$47$$ −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
$$50$$ 0 0
$$51$$ 1.06802e7 0.221062
$$52$$ 0 0
$$53$$ −6.65141e7 −1.15790 −0.578951 0.815362i $$-0.696538\pi$$
−0.578951 + 0.815362i $$0.696538\pi$$
$$54$$ 0 0
$$55$$ 3.73042e7 0.549699
$$56$$ 0 0
$$57$$ 6.24765e6 0.0783935
$$58$$ 0 0
$$59$$ 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 $$0.908735\pi$$
$$62$$ 0 0
$$63$$ −1.42197e8 −1.13725
$$64$$ 0 0
$$65$$ −5.65222e7 −0.392744
$$66$$ 0 0
$$67$$ −1.93016e8 −1.17019 −0.585094 0.810966i $$-0.698942\pi$$
−0.585094 + 0.810966i $$0.698942\pi$$
$$68$$ 0 0
$$69$$ −4.02408e7 −0.213720
$$70$$ 0 0
$$71$$ 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 $$0.580641\pi$$
$$74$$ 0 0
$$75$$ −1.47249e8 −0.537373
$$76$$ 0 0
$$77$$ −1.90668e8 −0.618115
$$78$$ 0 0
$$79$$ −1.12872e8 −0.326035 −0.163017 0.986623i $$-0.552123\pi$$
−0.163017 + 0.986623i $$0.552123\pi$$
$$80$$ 0 0
$$81$$ 8.69105e7 0.224331
$$82$$ 0 0
$$83$$ −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
$$86$$ 0 0
$$87$$ 1.95033e8 0.364982
$$88$$ 0 0
$$89$$ −6.37487e6 −0.0107700 −0.00538501 0.999986i $$-0.501714\pi$$
−0.00538501 + 0.999986i $$0.501714\pi$$
$$90$$ 0 0
$$91$$ 2.88895e8 0.441624
$$92$$ 0 0
$$93$$ −1.66050e8 −0.230179
$$94$$ 0 0
$$95$$ 1.64855e8 0.207656
$$96$$ 0 0
$$97$$ 8.71267e8 0.999260 0.499630 0.866239i $$-0.333469\pi$$
0.499630 + 0.866239i $$0.333469\pi$$
$$98$$ 0 0
$$99$$ 2.64993e8 0.277253
$$100$$ 0 0
$$101$$ −8.24412e8 −0.788312 −0.394156 0.919044i $$-0.628963\pi$$
−0.394156 + 0.919044i $$0.628963\pi$$
$$102$$ 0 0
$$103$$ 1.65896e9 1.45234 0.726168 0.687517i $$-0.241299\pi$$
0.726168 + 0.687517i $$0.241299\pi$$
$$104$$ 0 0
$$105$$ 1.50132e9 1.20537
$$106$$ 0 0
$$107$$ −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 $$0.893753\pi$$
$$110$$ 0 0
$$111$$ −1.35744e9 −0.848727
$$112$$ 0 0
$$113$$ 6.78547e8 0.391496 0.195748 0.980654i $$-0.437287\pi$$
0.195748 + 0.980654i $$0.437287\pi$$
$$114$$ 0 0
$$115$$ −1.06182e9 −0.566123
$$116$$ 0 0
$$117$$ −4.01511e8 −0.198089
$$118$$ 0 0
$$119$$ −1.44041e9 −0.658451
$$120$$ 0 0
$$121$$ −2.00263e9 −0.849309
$$122$$ 0 0
$$123$$ −2.01092e9 −0.792175
$$124$$ 0 0
$$125$$ −2.01680e7 −0.00738869
$$126$$ 0 0
$$127$$ 3.48292e9 1.18803 0.594014 0.804455i $$-0.297542\pi$$
0.594014 + 0.804455i $$0.297542\pi$$
$$128$$ 0 0
$$129$$ −3.16901e9 −1.00756
$$130$$ 0 0
$$131$$ −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
$$134$$ 0 0
$$135$$ −5.00801e9 −1.29767
$$136$$ 0 0
$$137$$ −6.38904e9 −1.54950 −0.774752 0.632265i $$-0.782125\pi$$
−0.774752 + 0.632265i $$0.782125\pi$$
$$138$$ 0 0
$$139$$ −7.62665e9 −1.73287 −0.866437 0.499286i $$-0.833596\pi$$
−0.866437 + 0.499286i $$0.833596\pi$$
$$140$$ 0 0
$$141$$ 2.69362e9 0.573920
$$142$$ 0 0
$$143$$ −5.38375e8 −0.107665
$$144$$ 0 0
$$145$$ 5.14627e9 0.966801
$$146$$ 0 0
$$147$$ −4.64697e9 −0.820809
$$148$$ 0 0
$$149$$ −9.23455e9 −1.53489 −0.767445 0.641114i $$-0.778472\pi$$
−0.767445 + 0.641114i $$0.778472\pi$$
$$150$$ 0 0
$$151$$ 3.25451e9 0.509436 0.254718 0.967015i $$-0.418017\pi$$
0.254718 + 0.967015i $$0.418017\pi$$
$$152$$ 0 0
$$153$$ 2.00190e9 0.295346
$$154$$ 0 0
$$155$$ −4.38151e9 −0.609722
$$156$$ 0 0
$$157$$ 1.62825e9 0.213881 0.106940 0.994265i $$-0.465895\pi$$
0.106940 + 0.994265i $$0.465895\pi$$
$$158$$ 0 0
$$159$$ 4.98855e9 0.618996
$$160$$ 0 0
$$161$$ 5.42714e9 0.636583
$$162$$ 0 0
$$163$$ 1.13187e10 1.25590 0.627948 0.778255i $$-0.283895\pi$$
0.627948 + 0.778255i $$0.283895\pi$$
$$164$$ 0 0
$$165$$ −2.79781e9 −0.293860
$$166$$ 0 0
$$167$$ −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
$$170$$ 0 0
$$171$$ 1.17106e9 0.104736
$$172$$ 0 0
$$173$$ 2.76347e9 0.234557 0.117278 0.993099i $$-0.462583\pi$$
0.117278 + 0.993099i $$0.462583\pi$$
$$174$$ 0 0
$$175$$ 1.98589e10 1.60061
$$176$$ 0 0
$$177$$ −8.11230e9 −0.621247
$$178$$ 0 0
$$179$$ −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 $$0.815323\pi$$
$$182$$ 0 0
$$183$$ 1.55587e10 1.02552
$$184$$ 0 0
$$185$$ −3.58184e10 −2.24819
$$186$$ 0 0
$$187$$ 2.68430e9 0.160525
$$188$$ 0 0
$$189$$ 2.55968e10 1.45917
$$190$$ 0 0
$$191$$ −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 $$0.645395\pi$$
$$194$$ 0 0
$$195$$ 4.23917e9 0.209954
$$196$$ 0 0
$$197$$ 3.98292e10 1.88410 0.942049 0.335477i $$-0.108897\pi$$
0.942049 + 0.335477i $$0.108897\pi$$
$$198$$ 0 0
$$199$$ −1.31081e9 −0.0592518 −0.0296259 0.999561i $$-0.509432\pi$$
−0.0296259 + 0.999561i $$0.509432\pi$$
$$200$$ 0 0
$$201$$ 1.44762e10 0.625563
$$202$$ 0 0
$$203$$ −2.63035e10 −1.08713
$$204$$ 0 0
$$205$$ −5.30614e10 −2.09839
$$206$$ 0 0
$$207$$ −7.54274e9 −0.285537
$$208$$ 0 0
$$209$$ 1.57024e9 0.0569257
$$210$$ 0 0
$$211$$ −2.35777e10 −0.818898 −0.409449 0.912333i $$-0.634279\pi$$
−0.409449 + 0.912333i $$0.634279\pi$$
$$212$$ 0 0
$$213$$ −1.51375e10 −0.503902
$$214$$ 0 0
$$215$$ −8.36196e10 −2.66892
$$216$$ 0 0
$$217$$ 2.23947e10 0.685607
$$218$$ 0 0
$$219$$ 9.12211e9 0.267976
$$220$$ 0 0
$$221$$ −4.06717e9 −0.114690
$$222$$ 0 0
$$223$$ −2.38326e9 −0.0645356 −0.0322678 0.999479i $$-0.510273\pi$$
−0.0322678 + 0.999479i $$0.510273\pi$$
$$224$$ 0 0
$$225$$ −2.76003e10 −0.717947
$$226$$ 0 0
$$227$$ 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 $$0.397276\pi$$
$$230$$ 0 0
$$231$$ 1.43001e10 0.330434
$$232$$ 0 0
$$233$$ 2.40457e10 0.534485 0.267242 0.963629i $$-0.413888\pi$$
0.267242 + 0.963629i $$0.413888\pi$$
$$234$$ 0 0
$$235$$ 7.10758e10 1.52025
$$236$$ 0 0
$$237$$ 8.46539e9 0.174293
$$238$$ 0 0
$$239$$ −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 $$0.461725\pi$$
$$242$$ 0 0
$$243$$ −5.63276e10 −1.03632
$$244$$ 0 0
$$245$$ −1.22618e11 −2.17424
$$246$$ 0 0
$$247$$ −2.37919e9 −0.0406717
$$248$$ 0 0
$$249$$ 2.31191e10 0.381130
$$250$$ 0 0
$$251$$ −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
$$254$$ 0 0
$$255$$ −2.11362e10 −0.313037
$$256$$ 0 0
$$257$$ 2.96868e10 0.424488 0.212244 0.977217i $$-0.431923\pi$$
0.212244 + 0.977217i $$0.431923\pi$$
$$258$$ 0 0
$$259$$ 1.83074e11 2.52800
$$260$$ 0 0
$$261$$ 3.65570e10 0.487628
$$262$$ 0 0
$$263$$ 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
$$266$$ 0 0
$$267$$ 4.78115e8 0.00575747
$$268$$ 0 0
$$269$$ −2.57149e10 −0.299433 −0.149716 0.988729i $$-0.547836\pi$$
−0.149716 + 0.988729i $$0.547836\pi$$
$$270$$ 0 0
$$271$$ 8.08890e10 0.911020 0.455510 0.890231i $$-0.349457\pi$$
0.455510 + 0.890231i $$0.349457\pi$$
$$272$$ 0 0
$$273$$ −2.16671e10 −0.236085
$$274$$ 0 0
$$275$$ −3.70085e10 −0.390215
$$276$$ 0 0
$$277$$ −3.40035e10 −0.347028 −0.173514 0.984831i $$-0.555512\pi$$
−0.173514 + 0.984831i $$0.555512\pi$$
$$278$$ 0 0
$$279$$ −3.11245e10 −0.307527
$$280$$ 0 0
$$281$$ −7.14831e9 −0.0683951 −0.0341976 0.999415i $$-0.510888\pi$$
−0.0341976 + 0.999415i $$0.510888\pi$$
$$282$$ 0 0
$$283$$ 7.80508e10 0.723333 0.361667 0.932308i $$-0.382208\pi$$
0.361667 + 0.932308i $$0.382208\pi$$
$$284$$ 0 0
$$285$$ −1.23641e10 −0.111010
$$286$$ 0 0
$$287$$ 2.71206e11 2.35956
$$288$$ 0 0
$$289$$ −9.83093e10 −0.828999
$$290$$ 0 0
$$291$$ −6.53450e10 −0.534188
$$292$$ 0 0
$$293$$ −1.26662e11 −1.00402 −0.502009 0.864862i $$-0.667406\pi$$
−0.502009 + 0.864862i $$0.667406\pi$$
$$294$$ 0 0
$$295$$ −2.14057e11 −1.64562
$$296$$ 0 0
$$297$$ −4.77013e10 −0.355735
$$298$$ 0 0
$$299$$ 1.53242e10 0.110881
$$300$$ 0 0
$$301$$ 4.27394e11 3.00109
$$302$$ 0 0
$$303$$ 6.18309e10 0.421419
$$304$$ 0 0
$$305$$ 4.10543e11 2.71650
$$306$$ 0 0
$$307$$ −6.15064e10 −0.395182 −0.197591 0.980285i $$-0.563312\pi$$
−0.197591 + 0.980285i $$0.563312\pi$$
$$308$$ 0 0
$$309$$ −1.24422e11 −0.776395
$$310$$ 0 0
$$311$$ −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 $$0.243986\pi$$
$$314$$ 0 0
$$315$$ 2.81407e11 1.61041
$$316$$ 0 0
$$317$$ 7.77399e10 0.432392 0.216196 0.976350i $$-0.430635\pi$$
0.216196 + 0.976350i $$0.430635\pi$$
$$318$$ 0 0
$$319$$ 4.90183e10 0.265033
$$320$$ 0 0
$$321$$ 8.63740e10 0.454057
$$322$$ 0 0
$$323$$ 1.18625e10 0.0606406
$$324$$ 0 0
$$325$$ 5.60743e10 0.278797
$$326$$ 0 0
$$327$$ 2.08860e11 1.01016
$$328$$ 0 0
$$329$$ −3.63280e11 −1.70946
$$330$$ 0 0
$$331$$ 1.68625e11 0.772139 0.386070 0.922470i $$-0.373832\pi$$
0.386070 + 0.922470i $$0.373832\pi$$
$$332$$ 0 0
$$333$$ −2.54439e11 −1.13393
$$334$$ 0 0
$$335$$ 3.81978e11 1.65705
$$336$$ 0 0
$$337$$ −7.70797e10 −0.325541 −0.162770 0.986664i $$-0.552043\pi$$
−0.162770 + 0.986664i $$0.552043\pi$$
$$338$$ 0 0
$$339$$ −5.08910e10 −0.209287
$$340$$ 0 0
$$341$$ −4.17340e10 −0.167146
$$342$$ 0 0
$$343$$ 2.18545e11 0.852544
$$344$$ 0 0
$$345$$ 7.96365e10 0.302640
$$346$$ 0 0
$$347$$ −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 $$0.400500\pi$$
$$350$$ 0 0
$$351$$ 7.22758e10 0.254162
$$352$$ 0 0
$$353$$ 2.96506e11 1.01636 0.508180 0.861251i $$-0.330318\pi$$
0.508180 + 0.861251i $$0.330318\pi$$
$$354$$ 0 0
$$355$$ −3.99428e11 −1.33479
$$356$$ 0 0
$$357$$ 1.08030e11 0.351997
$$358$$ 0 0
$$359$$ −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
$$362$$ 0 0
$$363$$ 1.50197e11 0.454026
$$364$$ 0 0
$$365$$ 2.40702e11 0.709842
$$366$$ 0 0
$$367$$ 1.05092e11 0.302394 0.151197 0.988504i $$-0.451687\pi$$
0.151197 + 0.988504i $$0.451687\pi$$
$$368$$ 0 0
$$369$$ −3.76926e11 −1.05837
$$370$$ 0 0
$$371$$ −6.72790e11 −1.84373
$$372$$ 0 0
$$373$$ 2.19888e11 0.588182 0.294091 0.955777i $$-0.404983\pi$$
0.294091 + 0.955777i $$0.404983\pi$$
$$374$$ 0 0
$$375$$ 1.51260e9 0.00394987
$$376$$ 0 0
$$377$$ −7.42712e10 −0.189358
$$378$$ 0 0
$$379$$ 3.14748e11 0.783586 0.391793 0.920053i $$-0.371855\pi$$
0.391793 + 0.920053i $$0.371855\pi$$
$$380$$ 0 0
$$381$$ −2.61219e11 −0.635100
$$382$$ 0 0
$$383$$ 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
$$386$$ 0 0
$$387$$ −5.93999e11 −1.34613
$$388$$ 0 0
$$389$$ 4.49612e11 0.995554 0.497777 0.867305i $$-0.334150\pi$$
0.497777 + 0.867305i $$0.334150\pi$$
$$390$$ 0 0
$$391$$ −7.64055e10 −0.165321
$$392$$ 0 0
$$393$$ 3.77026e11 0.797269
$$394$$ 0 0
$$395$$ 2.23374e11 0.461684
$$396$$ 0 0
$$397$$ 2.29976e11 0.464649 0.232324 0.972638i $$-0.425367\pi$$
0.232324 + 0.972638i $$0.425367\pi$$
$$398$$ 0 0
$$399$$ 6.31950e10 0.124826
$$400$$ 0 0
$$401$$ −6.69163e11 −1.29236 −0.646178 0.763187i $$-0.723634\pi$$
−0.646178 + 0.763187i $$0.723634\pi$$
$$402$$ 0 0
$$403$$ 6.32342e10 0.119421
$$404$$ 0 0
$$405$$ −1.71996e11 −0.317666
$$406$$ 0 0
$$407$$ −3.41171e11 −0.616307
$$408$$ 0 0
$$409$$ −6.73923e11 −1.19085 −0.595423 0.803413i $$-0.703015\pi$$
−0.595423 + 0.803413i $$0.703015\pi$$
$$410$$ 0 0
$$411$$ 4.79178e11 0.828340
$$412$$ 0 0
$$413$$ 1.09408e12 1.85043
$$414$$ 0 0
$$415$$ 6.10035e11 1.00957
$$416$$ 0 0
$$417$$ 5.71999e11 0.926366
$$418$$ 0 0
$$419$$ 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 $$0.642441\pi$$
$$422$$ 0 0
$$423$$ 5.04893e11 0.766775
$$424$$ 0 0
$$425$$ −2.79582e11 −0.415680
$$426$$ 0 0
$$427$$ −2.09836e12 −3.05460
$$428$$ 0 0
$$429$$ 4.03781e10 0.0575557
$$430$$ 0 0
$$431$$ 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 $$0.820975\pi$$
$$434$$ 0 0
$$435$$ −3.85971e11 −0.516836
$$436$$ 0 0
$$437$$ −4.46952e10 −0.0586265
$$438$$ 0 0
$$439$$ 9.14852e11 1.17560 0.587801 0.809006i $$-0.299994\pi$$
0.587801 + 0.809006i $$0.299994\pi$$
$$440$$ 0 0
$$441$$ −8.71028e11 −1.09663
$$442$$ 0 0
$$443$$ 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
$$446$$ 0 0
$$447$$ 6.92591e11 0.820527
$$448$$ 0 0
$$449$$ −2.70808e10 −0.0314452 −0.0157226 0.999876i $$-0.505005\pi$$
−0.0157226 + 0.999876i $$0.505005\pi$$
$$450$$ 0 0
$$451$$ −5.05411e11 −0.575241
$$452$$ 0 0
$$453$$ −2.44088e11 −0.272336
$$454$$ 0 0
$$455$$ −5.71722e11 −0.625365
$$456$$ 0 0
$$457$$ 2.02586e11 0.217263 0.108632 0.994082i $$-0.465353\pi$$
0.108632 + 0.994082i $$0.465353\pi$$
$$458$$ 0 0
$$459$$ −3.60361e11 −0.378949
$$460$$ 0 0
$$461$$ 8.96346e11 0.924319 0.462159 0.886797i $$-0.347075\pi$$
0.462159 + 0.886797i $$0.347075\pi$$
$$462$$ 0 0
$$463$$ 5.17740e11 0.523597 0.261799 0.965123i $$-0.415684\pi$$
0.261799 + 0.965123i $$0.415684\pi$$
$$464$$ 0 0
$$465$$ 3.28614e11 0.325947
$$466$$ 0 0
$$467$$ −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
$$470$$ 0 0
$$471$$ −1.22119e11 −0.114337
$$472$$ 0 0
$$473$$ −7.96478e11 −0.731642
$$474$$ 0 0
$$475$$ −1.63548e11 −0.147409
$$476$$ 0 0
$$477$$ 9.35055e11 0.826998
$$478$$ 0 0
$$479$$ −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
$$482$$ 0 0
$$483$$ −4.07036e11 −0.340307
$$484$$ 0 0
$$485$$ −1.72424e12 −1.41501
$$486$$ 0 0
$$487$$ −1.49591e12 −1.20511 −0.602554 0.798078i $$-0.705850\pi$$
−0.602554 + 0.798078i $$0.705850\pi$$
$$488$$ 0 0
$$489$$ −8.48906e11 −0.671382
$$490$$ 0 0
$$491$$ 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
$$494$$ 0 0
$$495$$ −5.24422e11 −0.392606
$$496$$ 0 0
$$497$$ 2.04155e12 1.50091
$$498$$ 0 0
$$499$$ 9.12174e11 0.658606 0.329303 0.944224i $$-0.393186\pi$$
0.329303 + 0.944224i $$0.393186\pi$$
$$500$$ 0 0
$$501$$ 1.29230e11 0.0916416
$$502$$ 0 0
$$503$$ 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
$$506$$ 0 0
$$507$$ −6.11798e10 −0.0411218
$$508$$ 0 0
$$509$$ −1.54192e12 −1.01820 −0.509098 0.860708i $$-0.670021\pi$$
−0.509098 + 0.860708i $$0.670021\pi$$
$$510$$ 0 0
$$511$$ −1.23027e12 −0.798189
$$512$$ 0 0
$$513$$ −2.10802e11 −0.134384
$$514$$ 0 0
$$515$$ −3.28307e12 −2.05659
$$516$$ 0 0
$$517$$ 6.76998e11 0.416754
$$518$$ 0 0
$$519$$ −2.07261e11 −0.125390
$$520$$ 0 0
$$521$$ −1.48896e12 −0.885345 −0.442672 0.896683i $$-0.645969\pi$$
−0.442672 + 0.896683i $$0.645969\pi$$
$$522$$ 0 0
$$523$$ −2.55715e12 −1.49451 −0.747256 0.664536i $$-0.768629\pi$$
−0.747256 + 0.664536i $$0.768629\pi$$
$$524$$ 0 0
$$525$$ −1.48942e12 −0.855659
$$526$$ 0 0
$$527$$ −3.15281e11 −0.178053
$$528$$ 0 0
$$529$$ −1.51327e12 −0.840169
$$530$$ 0 0
$$531$$ −1.52057e12 −0.830005
$$532$$ 0 0
$$533$$ 7.65784e11 0.410993
$$534$$ 0 0
$$535$$ 2.27912e12 1.20275
$$536$$ 0 0
$$537$$ 5.15011e11 0.267259
$$538$$ 0 0
$$539$$ −1.16794e12 −0.596033
$$540$$ 0 0
$$541$$ −2.64921e12 −1.32962 −0.664811 0.747011i $$-0.731488\pi$$
−0.664811 + 0.747011i $$0.731488\pi$$
$$542$$ 0 0
$$543$$ 1.81151e12 0.894213
$$544$$ 0 0
$$545$$ 5.51111e12 2.67581
$$546$$ 0 0
$$547$$ −2.16400e12 −1.03351 −0.516754 0.856134i $$-0.672860\pi$$
−0.516754 + 0.856134i $$0.672860\pi$$
$$548$$ 0 0
$$549$$ 2.91633e12 1.37013
$$550$$ 0 0
$$551$$ 2.16622e11 0.100120
$$552$$ 0 0
$$553$$ −1.14170e12 −0.519145
$$554$$ 0 0
$$555$$ 2.68638e12 1.20185
$$556$$ 0 0
$$557$$ −2.96364e12 −1.30460 −0.652300 0.757961i $$-0.726196\pi$$
−0.652300 + 0.757961i $$0.726196\pi$$
$$558$$ 0 0
$$559$$ 1.20680e12 0.522736
$$560$$ 0 0
$$561$$ −2.01322e11 −0.0858141
$$562$$ 0 0
$$563$$ 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
$$566$$ 0 0
$$567$$ 8.79100e11 0.357202
$$568$$ 0 0
$$569$$ −3.83703e12 −1.53458 −0.767290 0.641300i $$-0.778395\pi$$
−0.767290 + 0.641300i $$0.778395\pi$$
$$570$$ 0 0
$$571$$ −3.35374e11 −0.132028 −0.0660142 0.997819i $$-0.521028\pi$$
−0.0660142 + 0.997819i $$0.521028\pi$$
$$572$$ 0 0
$$573$$ 2.69987e12 1.04628
$$574$$ 0 0
$$575$$ 1.05341e12 0.401874
$$576$$ 0 0
$$577$$ −1.74089e11 −0.0653852 −0.0326926 0.999465i $$-0.510408\pi$$
−0.0326926 + 0.999465i $$0.510408\pi$$
$$578$$ 0 0
$$579$$ 1.27524e12 0.471560
$$580$$ 0 0
$$581$$ −3.11799e12 −1.13523
$$582$$ 0 0
$$583$$ 1.25379e12 0.449486
$$584$$ 0 0
$$585$$ 7.94589e11 0.280505
$$586$$ 0 0
$$587$$ −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
$$590$$ 0 0
$$591$$ −2.98719e12 −1.00721
$$592$$ 0 0
$$593$$ 4.16793e12 1.38412 0.692061 0.721839i $$-0.256703\pi$$
0.692061 + 0.721839i $$0.256703\pi$$
$$594$$ 0 0
$$595$$ 2.85056e12 0.932405
$$596$$ 0 0
$$597$$ 9.83110e10 0.0316751
$$598$$ 0 0
$$599$$ −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 $$0.715518\pi$$
$$602$$ 0 0
$$603$$ 2.71341e12 0.835772
$$604$$ 0 0
$$605$$ 3.96320e12 1.20267
$$606$$ 0 0
$$607$$ −1.45542e12 −0.435149 −0.217575 0.976044i $$-0.569815\pi$$
−0.217575 + 0.976044i $$0.569815\pi$$
$$608$$ 0 0
$$609$$ 1.97276e12 0.581161
$$610$$ 0 0
$$611$$ −1.02577e12 −0.297758
$$612$$ 0 0
$$613$$ −2.55645e12 −0.731248 −0.365624 0.930763i $$-0.619144\pi$$
−0.365624 + 0.930763i $$0.619144\pi$$
$$614$$ 0 0
$$615$$ 3.97961e12 1.12177
$$616$$ 0 0
$$617$$ 2.69727e11 0.0749276 0.0374638 0.999298i $$-0.488072\pi$$
0.0374638 + 0.999298i $$0.488072\pi$$
$$618$$ 0 0
$$619$$ −4.28111e11 −0.117206 −0.0586028 0.998281i $$-0.518665\pi$$
−0.0586028 + 0.998281i $$0.518665\pi$$
$$620$$ 0 0
$$621$$ 1.35776e12 0.366364
$$622$$ 0 0
$$623$$ −6.44818e10 −0.0171491
$$624$$ 0 0
$$625$$ −3.79469e12 −0.994755
$$626$$ 0 0
$$627$$ −1.17768e11 −0.0304316
$$628$$ 0 0
$$629$$ −2.57739e12 −0.656525
$$630$$ 0 0
$$631$$ 2.98421e11 0.0749372 0.0374686 0.999298i $$-0.488071\pi$$
0.0374686 + 0.999298i $$0.488071\pi$$
$$632$$ 0 0
$$633$$ 1.76833e12 0.437770
$$634$$ 0 0
$$635$$ −6.89269e12 −1.68231
$$636$$ 0 0
$$637$$ 1.76963e12 0.425848
$$638$$ 0 0
$$639$$ −2.83738e12 −0.673229
$$640$$ 0 0
$$641$$ −2.34144e12 −0.547799 −0.273899 0.961758i $$-0.588314\pi$$
−0.273899 + 0.961758i $$0.588314\pi$$
$$642$$ 0 0
$$643$$ −7.75186e12 −1.78837 −0.894183 0.447701i $$-0.852243\pi$$
−0.894183 + 0.447701i $$0.852243\pi$$
$$644$$ 0 0
$$645$$ 6.27147e12 1.42676
$$646$$ 0 0
$$647$$ −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
$$650$$ 0 0
$$651$$ −1.67960e12 −0.366514
$$652$$ 0 0
$$653$$ 2.82022e12 0.606979 0.303490 0.952835i $$-0.401848\pi$$
0.303490 + 0.952835i $$0.401848\pi$$
$$654$$ 0 0
$$655$$ 9.94846e12 2.11188
$$656$$ 0 0
$$657$$ 1.70985e12 0.358025
$$658$$ 0 0
$$659$$ 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 $$0.510345\pi$$
$$662$$ 0 0
$$663$$ 3.05038e11 0.0613116
$$664$$ 0 0
$$665$$ 1.66750e12 0.330651
$$666$$ 0 0
$$667$$ −1.39525e12 −0.272952
$$668$$ 0 0
$$669$$ 1.78745e11 0.0344997
$$670$$ 0 0
$$671$$ 3.91043e12 0.744686
$$672$$ 0 0
$$673$$ 4.82897e12 0.907375 0.453687 0.891161i $$-0.350108\pi$$
0.453687 + 0.891161i $$0.350108\pi$$
$$674$$ 0 0
$$675$$ 4.96832e12 0.921176
$$676$$ 0 0
$$677$$ −7.47095e12 −1.36687 −0.683434 0.730012i $$-0.739514\pi$$
−0.683434 + 0.730012i $$0.739514\pi$$
$$678$$ 0 0
$$679$$ 8.81286e12 1.59112
$$680$$ 0 0
$$681$$ −5.59911e11 −0.0997601
$$682$$ 0 0
$$683$$ 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
$$686$$ 0 0
$$687$$ −1.97975e12 −0.339082
$$688$$ 0 0
$$689$$ −1.89971e12 −0.321144
$$690$$ 0 0
$$691$$ −4.50776e12 −0.752159 −0.376079 0.926587i $$-0.622728\pi$$
−0.376079 + 0.926587i $$0.622728\pi$$
$$692$$ 0 0
$$693$$ 2.68041e12 0.441470
$$694$$ 0 0
$$695$$ 1.50931e13 2.45385
$$696$$ 0 0
$$697$$ −3.81814e12 −0.612780
$$698$$ 0 0
$$699$$ −1.80343e12 −0.285727
$$700$$ 0 0
$$701$$ −1.38907e12 −0.217267 −0.108633 0.994082i $$-0.534647\pi$$
−0.108633 + 0.994082i $$0.534647\pi$$
$$702$$ 0 0
$$703$$ −1.50770e12 −0.232818
$$704$$ 0 0
$$705$$ −5.33068e12 −0.812703
$$706$$ 0 0
$$707$$ −8.33893e12 −1.25523
$$708$$ 0 0
$$709$$ 5.59251e12 0.831187 0.415593 0.909551i $$-0.363574\pi$$
0.415593 + 0.909551i $$0.363574\pi$$
$$710$$ 0 0
$$711$$ 1.58675e12 0.232861
$$712$$ 0 0
$$713$$ 1.18791e12 0.172140
$$714$$ 0 0
$$715$$ 1.06544e12 0.152459
$$716$$ 0 0
$$717$$ 4.47238e12 0.631979
$$718$$ 0 0
$$719$$ 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
$$722$$ 0 0
$$723$$ −9.42295e11 −0.128252
$$724$$ 0 0
$$725$$ −5.10549e12 −0.686304
$$726$$ 0 0
$$727$$ 5.13697e12 0.682028 0.341014 0.940058i $$-0.389230\pi$$
0.341014 + 0.940058i $$0.389230\pi$$
$$728$$ 0 0
$$729$$ 2.51391e12 0.329667
$$730$$ 0 0
$$731$$ −6.01702e12 −0.779387
$$732$$ 0 0
$$733$$ 9.72563e12 1.24437 0.622185 0.782870i $$-0.286245\pi$$
0.622185 + 0.782870i $$0.286245\pi$$
$$734$$ 0 0
$$735$$ 9.19636e12 1.16231
$$736$$ 0 0
$$737$$ 3.63834e12 0.454255
$$738$$ 0 0
$$739$$ 1.15533e13 1.42498 0.712488 0.701684i $$-0.247568\pi$$
0.712488 + 0.701684i $$0.247568\pi$$
$$740$$ 0 0
$$741$$ 1.78439e11 0.0217424
$$742$$ 0 0
$$743$$ 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
$$746$$ 0 0
$$747$$ 4.33344e12 0.509202
$$748$$ 0 0
$$749$$ −1.16490e13 −1.35245
$$750$$ 0 0
$$751$$ −1.92230e12 −0.220516 −0.110258 0.993903i $$-0.535168\pi$$
−0.110258 + 0.993903i $$0.535168\pi$$
$$752$$ 0 0
$$753$$ 1.81328e12 0.205536
$$754$$ 0 0
$$755$$ −6.44068e12 −0.721390
$$756$$ 0 0
$$757$$ −1.27412e12 −0.141019 −0.0705096 0.997511i $$-0.522463\pi$$
−0.0705096 + 0.997511i $$0.522463\pi$$
$$758$$ 0 0
$$759$$ 7.58539e11 0.0829641
$$760$$ 0 0
$$761$$ 1.21619e13 1.31453 0.657267 0.753658i $$-0.271712\pi$$
0.657267 + 0.753658i $$0.271712\pi$$
$$762$$ 0 0
$$763$$ −2.81682e13 −3.00884
$$764$$ 0 0
$$765$$ −3.96176e12 −0.418227
$$766$$ 0 0
$$767$$ 3.08927e12 0.322312
$$768$$ 0 0
$$769$$ −5.98877e12 −0.617546 −0.308773 0.951136i $$-0.599918\pi$$
−0.308773 + 0.951136i $$0.599918\pi$$
$$770$$ 0 0
$$771$$ −2.22651e12 −0.226924
$$772$$ 0 0
$$773$$ −9.64838e12 −0.971956 −0.485978 0.873971i $$-0.661536\pi$$
−0.485978 + 0.873971i $$0.661536\pi$$
$$774$$ 0 0
$$775$$ 4.34679e12 0.432824
$$776$$ 0 0
$$777$$ −1.37305e13 −1.35143
$$778$$ 0 0
$$779$$ −2.23351e12 −0.217305
$$780$$ 0 0
$$781$$ −3.80456e12 −0.365911
$$782$$ 0 0
$$783$$ −6.58061e12 −0.625660
$$784$$ 0 0
$$785$$ −3.22230e12 −0.302867
$$786$$ 0 0
$$787$$ −1.40829e13 −1.30860 −0.654299 0.756236i $$-0.727036\pi$$
−0.654299 + 0.756236i $$0.727036\pi$$
$$788$$ 0 0
$$789$$ −5.69360e12 −0.523046
$$790$$ 0 0
$$791$$ 6.86350e12 0.623378
$$792$$ 0 0
$$793$$ −5.92498e12 −0.532056
$$794$$ 0 0
$$795$$ −9.87235e12 −0.876533
$$796$$ 0 0
$$797$$ −8.45863e12 −0.742570 −0.371285 0.928519i $$-0.621083\pi$$
−0.371285 + 0.928519i $$0.621083\pi$$
$$798$$ 0 0
$$799$$ 5.11440e12 0.443950
$$800$$ 0 0
$$801$$ 8.96179e10 0.00769216
$$802$$ 0 0
$$803$$ 2.29269e12 0.194592
$$804$$ 0 0
$$805$$ −1.07403e13 −0.901437
$$806$$ 0 0
$$807$$ 1.92862e12 0.160072
$$808$$ 0 0
$$809$$ −9.95988e12 −0.817496 −0.408748 0.912647i $$-0.634035\pi$$
−0.408748 + 0.912647i $$0.634035\pi$$
$$810$$ 0 0
$$811$$ −9.83262e12 −0.798133 −0.399067 0.916922i $$-0.630666\pi$$
−0.399067 + 0.916922i $$0.630666\pi$$
$$812$$ 0 0
$$813$$ −6.06668e12 −0.487016
$$814$$ 0 0
$$815$$ −2.23998e13 −1.77842
$$816$$ 0 0
$$817$$ −3.51980e12 −0.276388
$$818$$ 0 0
$$819$$ −4.06128e12 −0.315417
$$820$$ 0 0
$$821$$ 8.67049e12 0.666039 0.333019 0.942920i $$-0.391933\pi$$
0.333019 + 0.942920i $$0.391933\pi$$
$$822$$ 0 0
$$823$$ 1.26458e13 0.960834 0.480417 0.877040i $$-0.340485\pi$$
0.480417 + 0.877040i $$0.340485\pi$$
$$824$$ 0 0
$$825$$ 2.77564e12 0.208603
$$826$$ 0 0
$$827$$ 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 $$0.621033\pi$$
$$830$$ 0 0
$$831$$ 2.55026e12 0.185516
$$832$$ 0 0
$$833$$ −8.82324e12 −0.634929
$$834$$ 0 0
$$835$$ 3.40994e12 0.242749
$$836$$ 0 0
$$837$$ 5.60270e12 0.394578
$$838$$ 0 0
$$839$$ −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
$$842$$ 0 0
$$843$$ 5.36123e11 0.0365629
$$844$$ 0 0
$$845$$ −1.61433e12 −0.108927
$$846$$ 0 0
$$847$$ −2.02566e13 −1.35235
$$848$$ 0 0
$$849$$ −5.85381e12 −0.386682
$$850$$ 0 0
$$851$$ 9.71104e12 0.634721
$$852$$ 0 0
$$853$$ 2.56415e13 1.65834 0.829170 0.558997i $$-0.188814\pi$$
0.829170 + 0.558997i $$0.188814\pi$$
$$854$$ 0 0
$$855$$ −2.31753e12 −0.148312
$$856$$ 0 0
$$857$$ −1.44941e11 −0.00917862 −0.00458931 0.999989i $$-0.501461\pi$$
−0.00458931 + 0.999989i $$0.501461\pi$$
$$858$$ 0 0
$$859$$ 9.47728e12 0.593902 0.296951 0.954893i $$-0.404030\pi$$
0.296951 + 0.954893i $$0.404030\pi$$
$$860$$ 0 0
$$861$$ −2.03404e13 −1.26138
$$862$$ 0 0
$$863$$ 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
$$866$$ 0 0
$$867$$ 7.37319e12 0.443169
$$868$$ 0 0
$$869$$ 2.12764e12 0.126563
$$870$$ 0 0
$$871$$ −5.51272e12 −0.324552
$$872$$ 0 0
$$873$$ −1.22483e13 −0.713692
$$874$$ 0 0
$$875$$ −2.03999e11 −0.0117650
$$876$$ 0 0
$$877$$ 1.42035e13 0.810768 0.405384 0.914147i $$-0.367138\pi$$
0.405384 + 0.914147i $$0.367138\pi$$
$$878$$ 0 0
$$879$$ 9.49964e12 0.536732
$$880$$ 0 0
$$881$$ −1.12878e13 −0.631275 −0.315638 0.948880i $$-0.602218\pi$$
−0.315638 + 0.948880i $$0.602218\pi$$
$$882$$ 0 0
$$883$$ −2.25433e12 −0.124794 −0.0623972 0.998051i $$-0.519875\pi$$
−0.0623972 + 0.998051i $$0.519875\pi$$
$$884$$ 0 0
$$885$$ 1.60542e13 0.879721
$$886$$ 0 0
$$887$$ 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
$$890$$ 0 0
$$891$$ −1.63826e12 −0.0870831
$$892$$ 0 0
$$893$$ 2.99179e12 0.157434
$$894$$ 0 0
$$895$$ 1.35894e13 0.707942
$$896$$ 0 0
$$897$$ −1.14932e12 −0.0592753
$$898$$ 0 0
$$899$$ −5.75739e12 −0.293973
$$900$$ 0 0
$$901$$ 9.47180e12 0.478819
$$902$$ 0 0
$$903$$ −3.20545e13 −1.60433
$$904$$ 0 0
$$905$$ 4.77997e13 2.36868
$$906$$ 0 0
$$907$$ −1.77171e13 −0.869280 −0.434640 0.900604i $$-0.643124\pi$$
−0.434640 + 0.900604i $$0.643124\pi$$
$$908$$ 0 0
$$909$$ 1.15896e13 0.563028
$$910$$ 0 0
$$911$$ −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
$$914$$ 0 0
$$915$$ −3.07908e13 −1.45220
$$916$$ 0 0
$$917$$ −5.08482e13 −2.37473
$$918$$ 0 0
$$919$$ 1.66013e13 0.767754 0.383877 0.923384i $$-0.374589\pi$$
0.383877 + 0.923384i $$0.374589\pi$$
$$920$$ 0 0
$$921$$ 4.61298e12 0.211258
$$922$$ 0 0
$$923$$ 5.76457e12 0.261432
$$924$$ 0 0
$$925$$ 3.55345e13 1.59593
$$926$$ 0 0
$$927$$ −2.33216e13 −1.03729
$$928$$ 0 0
$$929$$ −1.45567e13 −0.641197 −0.320598 0.947215i $$-0.603884\pi$$
−0.320598 + 0.947215i $$0.603884\pi$$
$$930$$ 0 0
$$931$$ −5.16136e12 −0.225160
$$932$$ 0 0
$$933$$ 1.62299e13 0.701210
$$934$$ 0 0
$$935$$ −5.31222e12 −0.227313
$$936$$ 0 0
$$937$$ −1.38682e13 −0.587751 −0.293875 0.955844i $$-0.594945\pi$$
−0.293875 + 0.955844i $$0.594945\pi$$
$$938$$ 0 0
$$939$$ −1.83476e13 −0.770164
$$940$$ 0 0
$$941$$ −2.15767e13 −0.897081 −0.448540 0.893763i $$-0.648056\pi$$
−0.448540 + 0.893763i $$0.648056\pi$$
$$942$$ 0 0
$$943$$ 1.43859e13 0.592428
$$944$$ 0 0
$$945$$ −5.06560e13 −2.06627
$$946$$ 0 0
$$947$$ −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
$$950$$ 0 0
$$951$$ −5.83049e12 −0.231149
$$952$$ 0 0
$$953$$ 2.71758e13 1.06724 0.533622 0.845723i $$-0.320830\pi$$
0.533622 + 0.845723i $$0.320830\pi$$
$$954$$ 0 0
$$955$$ 7.12407e13 2.77149
$$956$$ 0 0
$$957$$ −3.67637e12 −0.141682
$$958$$ 0 0
$$959$$ −6.46251e13 −2.46727
$$960$$ 0 0
$$961$$ −2.15378e13 −0.814603
$$962$$ 0 0
$$963$$ 1.61899e13 0.606635
$$964$$ 0 0
$$965$$ 3.36492e13 1.24911
$$966$$ 0 0
$$967$$ −1.20281e13 −0.442364 −0.221182 0.975233i $$-0.570991\pi$$
−0.221182 + 0.975233i $$0.570991\pi$$
$$968$$ 0 0
$$969$$ −8.89684e11 −0.0324174
$$970$$ 0 0
$$971$$ 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
$$974$$ 0 0
$$975$$ −4.20557e12 −0.149040
$$976$$ 0 0
$$977$$ −4.02945e13 −1.41488 −0.707441 0.706773i $$-0.750150\pi$$
−0.707441 + 0.706773i $$0.750150\pi$$
$$978$$ 0 0
$$979$$ 1.20166e11 0.00418081
$$980$$ 0 0
$$981$$ 3.91487e13 1.34960
$$982$$ 0 0
$$983$$ −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
$$986$$ 0 0
$$987$$ 2.72460e13 0.913852
$$988$$ 0 0
$$989$$ 2.26708e13 0.753502
$$990$$ 0 0
$$991$$ 2.37023e13 0.780653 0.390327 0.920676i $$-0.372362\pi$$
0.390327 + 0.920676i $$0.372362\pi$$
$$992$$ 0 0
$$993$$ −1.26469e13 −0.412773
$$994$$ 0 0
$$995$$ 2.59410e12 0.0839040
$$996$$ 0 0
$$997$$ −3.80094e13 −1.21832 −0.609162 0.793045i $$-0.708494\pi$$
−0.609162 + 0.793045i $$0.708494\pi$$
$$998$$ 0 0
$$999$$ 4.58015e13 1.45491
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 208.10.a.b.1.1 1
4.3 odd 2 26.10.a.c.1.1 1
12.11 even 2 234.10.a.a.1.1 1
52.51 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 4.3 odd 2
208.10.a.b.1.1 1 1.1 even 1 trivial
234.10.a.a.1.1 1 12.11 even 2
338.10.a.b.1.1 1 52.51 odd 2