# Properties

 Label 338.10.a.a.1.1 Level $338$ Weight $10$ Character 338.1 Self dual yes Analytic conductor $174.082$ 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] = [338,10,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$174.082112623$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 338.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-16.0000 q^{2} -156.000 q^{3} +256.000 q^{4} -870.000 q^{5} +2496.00 q^{6} +952.000 q^{7} -4096.00 q^{8} +4653.00 q^{9} +O(q^{10})$$ $$q-16.0000 q^{2} -156.000 q^{3} +256.000 q^{4} -870.000 q^{5} +2496.00 q^{6} +952.000 q^{7} -4096.00 q^{8} +4653.00 q^{9} +13920.0 q^{10} +56148.0 q^{11} -39936.0 q^{12} -15232.0 q^{14} +135720. q^{15} +65536.0 q^{16} -247662. q^{17} -74448.0 q^{18} -315380. q^{19} -222720. q^{20} -148512. q^{21} -898368. q^{22} +204504. q^{23} +638976. q^{24} -1.19622e6 q^{25} +2.34468e6 q^{27} +243712. q^{28} -3.84045e6 q^{29} -2.17152e6 q^{30} +1.30941e6 q^{31} -1.04858e6 q^{32} -8.75909e6 q^{33} +3.96259e6 q^{34} -828240. q^{35} +1.19117e6 q^{36} -4.30708e6 q^{37} +5.04608e6 q^{38} +3.56352e6 q^{40} -1.51204e6 q^{41} +2.37619e6 q^{42} +3.36706e7 q^{43} +1.43739e7 q^{44} -4.04811e6 q^{45} -3.27206e6 q^{46} +1.05811e7 q^{47} -1.02236e7 q^{48} -3.94473e7 q^{49} +1.91396e7 q^{50} +3.86353e7 q^{51} +1.66162e7 q^{53} -3.75149e7 q^{54} -4.88488e7 q^{55} -3.89939e6 q^{56} +4.91993e7 q^{57} +6.14472e7 q^{58} -1.12235e8 q^{59} +3.47443e7 q^{60} -3.31972e7 q^{61} -2.09505e7 q^{62} +4.42966e6 q^{63} +1.67772e7 q^{64} +1.40145e8 q^{66} +1.21372e8 q^{67} -6.34015e7 q^{68} -3.19026e7 q^{69} +1.32518e7 q^{70} +3.87173e8 q^{71} -1.90587e7 q^{72} -2.55240e8 q^{73} +6.89132e7 q^{74} +1.86611e8 q^{75} -8.07373e7 q^{76} +5.34529e7 q^{77} +4.92102e8 q^{79} -5.70163e7 q^{80} -4.57355e8 q^{81} +2.41927e7 q^{82} +4.57420e8 q^{83} -3.80191e7 q^{84} +2.15466e8 q^{85} -5.38730e8 q^{86} +5.99110e8 q^{87} -2.29982e8 q^{88} +3.18095e7 q^{89} +6.47698e7 q^{90} +5.23530e7 q^{92} -2.04268e8 q^{93} -1.69297e8 q^{94} +2.74381e8 q^{95} +1.63578e8 q^{96} +6.73532e8 q^{97} +6.31157e8 q^{98} +2.61257e8 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −16.0000 −0.707107
$$3$$ −156.000 −1.11193 −0.555967 0.831204i $$-0.687652\pi$$
−0.555967 + 0.831204i $$0.687652\pi$$
$$4$$ 256.000 0.500000
$$5$$ −870.000 −0.622521 −0.311261 0.950325i $$-0.600751\pi$$
−0.311261 + 0.950325i $$0.600751\pi$$
$$6$$ 2496.00 0.786256
$$7$$ 952.000 0.149863 0.0749317 0.997189i $$-0.476126\pi$$
0.0749317 + 0.997189i $$0.476126\pi$$
$$8$$ −4096.00 −0.353553
$$9$$ 4653.00 0.236397
$$10$$ 13920.0 0.440189
$$11$$ 56148.0 1.15629 0.578146 0.815934i $$-0.303777\pi$$
0.578146 + 0.815934i $$0.303777\pi$$
$$12$$ −39936.0 −0.555967
$$13$$ 0 0
$$14$$ −15232.0 −0.105969
$$15$$ 135720. 0.692203
$$16$$ 65536.0 0.250000
$$17$$ −247662. −0.719183 −0.359591 0.933110i $$-0.617084\pi$$
−0.359591 + 0.933110i $$0.617084\pi$$
$$18$$ −74448.0 −0.167158
$$19$$ −315380. −0.555192 −0.277596 0.960698i $$-0.589538\pi$$
−0.277596 + 0.960698i $$0.589538\pi$$
$$20$$ −222720. −0.311261
$$21$$ −148512. −0.166638
$$22$$ −898368. −0.817621
$$23$$ 204504. 0.152380 0.0761898 0.997093i $$-0.475725\pi$$
0.0761898 + 0.997093i $$0.475725\pi$$
$$24$$ 638976. 0.393128
$$25$$ −1.19622e6 −0.612467
$$26$$ 0 0
$$27$$ 2.34468e6 0.849076
$$28$$ 243712. 0.0749317
$$29$$ −3.84045e6 −1.00830 −0.504152 0.863615i $$-0.668195\pi$$
−0.504152 + 0.863615i $$0.668195\pi$$
$$30$$ −2.17152e6 −0.489461
$$31$$ 1.30941e6 0.254652 0.127326 0.991861i $$-0.459361\pi$$
0.127326 + 0.991861i $$0.459361\pi$$
$$32$$ −1.04858e6 −0.176777
$$33$$ −8.75909e6 −1.28572
$$34$$ 3.96259e6 0.508539
$$35$$ −828240. −0.0932932
$$36$$ 1.19117e6 0.118198
$$37$$ −4.30708e6 −0.377811 −0.188906 0.981995i $$-0.560494\pi$$
−0.188906 + 0.981995i $$0.560494\pi$$
$$38$$ 5.04608e6 0.392580
$$39$$ 0 0
$$40$$ 3.56352e6 0.220095
$$41$$ −1.51204e6 −0.0835673 −0.0417837 0.999127i $$-0.513304\pi$$
−0.0417837 + 0.999127i $$0.513304\pi$$
$$42$$ 2.37619e6 0.117831
$$43$$ 3.36706e7 1.50191 0.750953 0.660355i $$-0.229594\pi$$
0.750953 + 0.660355i $$0.229594\pi$$
$$44$$ 1.43739e7 0.578146
$$45$$ −4.04811e6 −0.147162
$$46$$ −3.27206e6 −0.107749
$$47$$ 1.05811e7 0.316293 0.158146 0.987416i $$-0.449448\pi$$
0.158146 + 0.987416i $$0.449448\pi$$
$$48$$ −1.02236e7 −0.277983
$$49$$ −3.94473e7 −0.977541
$$50$$ 1.91396e7 0.433080
$$51$$ 3.86353e7 0.799684
$$52$$ 0 0
$$53$$ 1.66162e7 0.289262 0.144631 0.989486i $$-0.453801\pi$$
0.144631 + 0.989486i $$0.453801\pi$$
$$54$$ −3.75149e7 −0.600388
$$55$$ −4.88488e7 −0.719816
$$56$$ −3.89939e6 −0.0529847
$$57$$ 4.91993e7 0.617336
$$58$$ 6.14472e7 0.712978
$$59$$ −1.12235e8 −1.20585 −0.602927 0.797796i $$-0.705999\pi$$
−0.602927 + 0.797796i $$0.705999\pi$$
$$60$$ 3.47443e7 0.346101
$$61$$ −3.31972e7 −0.306985 −0.153493 0.988150i $$-0.549052\pi$$
−0.153493 + 0.988150i $$0.549052\pi$$
$$62$$ −2.09505e7 −0.180066
$$63$$ 4.42966e6 0.0354273
$$64$$ 1.67772e7 0.125000
$$65$$ 0 0
$$66$$ 1.40145e8 0.909141
$$67$$ 1.21372e8 0.735839 0.367919 0.929858i $$-0.380070\pi$$
0.367919 + 0.929858i $$0.380070\pi$$
$$68$$ −6.34015e7 −0.359591
$$69$$ −3.19026e7 −0.169436
$$70$$ 1.32518e7 0.0659683
$$71$$ 3.87173e8 1.80818 0.904091 0.427340i $$-0.140549\pi$$
0.904091 + 0.427340i $$0.140549\pi$$
$$72$$ −1.90587e7 −0.0835789
$$73$$ −2.55240e8 −1.05195 −0.525976 0.850499i $$-0.676300\pi$$
−0.525976 + 0.850499i $$0.676300\pi$$
$$74$$ 6.89132e7 0.267153
$$75$$ 1.86611e8 0.681023
$$76$$ −8.07373e7 −0.277596
$$77$$ 5.34529e7 0.173286
$$78$$ 0 0
$$79$$ 4.92102e8 1.42145 0.710727 0.703467i $$-0.248366\pi$$
0.710727 + 0.703467i $$0.248366\pi$$
$$80$$ −5.70163e7 −0.155630
$$81$$ −4.57355e8 −1.18051
$$82$$ 2.41927e7 0.0590910
$$83$$ 4.57420e8 1.05795 0.528974 0.848638i $$-0.322577\pi$$
0.528974 + 0.848638i $$0.322577\pi$$
$$84$$ −3.80191e7 −0.0833191
$$85$$ 2.15466e8 0.447707
$$86$$ −5.38730e8 −1.06201
$$87$$ 5.99110e8 1.12117
$$88$$ −2.29982e8 −0.408811
$$89$$ 3.18095e7 0.0537405 0.0268703 0.999639i $$-0.491446\pi$$
0.0268703 + 0.999639i $$0.491446\pi$$
$$90$$ 6.47698e7 0.104059
$$91$$ 0 0
$$92$$ 5.23530e7 0.0761898
$$93$$ −2.04268e8 −0.283156
$$94$$ −1.69297e8 −0.223653
$$95$$ 2.74381e8 0.345619
$$96$$ 1.63578e8 0.196564
$$97$$ 6.73532e8 0.772477 0.386238 0.922399i $$-0.373774\pi$$
0.386238 + 0.922399i $$0.373774\pi$$
$$98$$ 6.31157e8 0.691226
$$99$$ 2.61257e8 0.273344
$$100$$ −3.06234e8 −0.306234
$$101$$ 1.05772e9 1.01140 0.505701 0.862709i $$-0.331234\pi$$
0.505701 + 0.862709i $$0.331234\pi$$
$$102$$ −6.18164e8 −0.565462
$$103$$ 7.95866e8 0.696743 0.348371 0.937357i $$-0.386735\pi$$
0.348371 + 0.937357i $$0.386735\pi$$
$$104$$ 0 0
$$105$$ 1.29205e8 0.103736
$$106$$ −2.65859e8 −0.204539
$$107$$ −1.97413e9 −1.45596 −0.727981 0.685598i $$-0.759541\pi$$
−0.727981 + 0.685598i $$0.759541\pi$$
$$108$$ 6.00238e8 0.424538
$$109$$ 1.34465e9 0.912408 0.456204 0.889875i $$-0.349209\pi$$
0.456204 + 0.889875i $$0.349209\pi$$
$$110$$ 7.81580e8 0.508987
$$111$$ 6.71904e8 0.420101
$$112$$ 6.23903e7 0.0374659
$$113$$ 2.70680e9 1.56172 0.780861 0.624705i $$-0.214781\pi$$
0.780861 + 0.624705i $$0.214781\pi$$
$$114$$ −7.87188e8 −0.436523
$$115$$ −1.77918e8 −0.0948595
$$116$$ −9.83155e8 −0.504152
$$117$$ 0 0
$$118$$ 1.79576e9 0.852667
$$119$$ −2.35774e8 −0.107779
$$120$$ −5.55909e8 −0.244731
$$121$$ 7.94650e8 0.337009
$$122$$ 5.31155e8 0.217071
$$123$$ 2.35879e8 0.0929213
$$124$$ 3.35208e8 0.127326
$$125$$ 2.73993e9 1.00380
$$126$$ −7.08745e7 −0.0250509
$$127$$ 1.19960e9 0.409185 0.204593 0.978847i $$-0.434413\pi$$
0.204593 + 0.978847i $$0.434413\pi$$
$$128$$ −2.68435e8 −0.0883883
$$129$$ −5.25261e9 −1.67002
$$130$$ 0 0
$$131$$ 2.78615e9 0.826576 0.413288 0.910600i $$-0.364380\pi$$
0.413288 + 0.910600i $$0.364380\pi$$
$$132$$ −2.24233e9 −0.642860
$$133$$ −3.00242e8 −0.0832029
$$134$$ −1.94196e9 −0.520317
$$135$$ −2.03987e9 −0.528568
$$136$$ 1.01442e9 0.254269
$$137$$ −2.88233e9 −0.699039 −0.349519 0.936929i $$-0.613655\pi$$
−0.349519 + 0.936929i $$0.613655\pi$$
$$138$$ 5.10442e8 0.119809
$$139$$ 2.15641e9 0.489965 0.244982 0.969528i $$-0.421218\pi$$
0.244982 + 0.969528i $$0.421218\pi$$
$$140$$ −2.12029e8 −0.0466466
$$141$$ −1.65065e9 −0.351697
$$142$$ −6.19476e9 −1.27858
$$143$$ 0 0
$$144$$ 3.04939e8 0.0590992
$$145$$ 3.34119e9 0.627690
$$146$$ 4.08384e9 0.743843
$$147$$ 6.15378e9 1.08696
$$148$$ −1.10261e9 −0.188906
$$149$$ −7.54548e9 −1.25415 −0.627074 0.778960i $$-0.715747\pi$$
−0.627074 + 0.778960i $$0.715747\pi$$
$$150$$ −2.98578e9 −0.481556
$$151$$ 4.31308e9 0.675136 0.337568 0.941301i $$-0.390396\pi$$
0.337568 + 0.941301i $$0.390396\pi$$
$$152$$ 1.29180e9 0.196290
$$153$$ −1.15237e9 −0.170013
$$154$$ −8.55246e8 −0.122532
$$155$$ −1.13918e9 −0.158526
$$156$$ 0 0
$$157$$ −4.23157e9 −0.555845 −0.277922 0.960604i $$-0.589646\pi$$
−0.277922 + 0.960604i $$0.589646\pi$$
$$158$$ −7.87363e9 −1.00512
$$159$$ −2.59213e9 −0.321640
$$160$$ 9.12261e8 0.110047
$$161$$ 1.94688e8 0.0228361
$$162$$ 7.31768e9 0.834749
$$163$$ −8.28448e8 −0.0919223 −0.0459612 0.998943i $$-0.514635\pi$$
−0.0459612 + 0.998943i $$0.514635\pi$$
$$164$$ −3.87083e8 −0.0417837
$$165$$ 7.62041e9 0.800388
$$166$$ −7.31872e9 −0.748082
$$167$$ 2.85500e9 0.284041 0.142021 0.989864i $$-0.454640\pi$$
0.142021 + 0.989864i $$0.454640\pi$$
$$168$$ 6.08305e8 0.0589155
$$169$$ 0 0
$$170$$ −3.44746e9 −0.316576
$$171$$ −1.46746e9 −0.131246
$$172$$ 8.61967e9 0.750953
$$173$$ −1.76690e10 −1.49970 −0.749851 0.661607i $$-0.769875\pi$$
−0.749851 + 0.661607i $$0.769875\pi$$
$$174$$ −9.58576e9 −0.792784
$$175$$ −1.13881e9 −0.0917865
$$176$$ 3.67972e9 0.289073
$$177$$ 1.75087e10 1.34083
$$178$$ −5.08952e8 −0.0380003
$$179$$ −5.86732e8 −0.0427170 −0.0213585 0.999772i $$-0.506799\pi$$
−0.0213585 + 0.999772i $$0.506799\pi$$
$$180$$ −1.03632e9 −0.0735811
$$181$$ −5.43396e9 −0.376325 −0.188162 0.982138i $$-0.560253\pi$$
−0.188162 + 0.982138i $$0.560253\pi$$
$$182$$ 0 0
$$183$$ 5.17877e9 0.341347
$$184$$ −8.37648e8 −0.0538743
$$185$$ 3.74716e9 0.235196
$$186$$ 3.26828e9 0.200222
$$187$$ −1.39057e10 −0.831585
$$188$$ 2.70875e9 0.158146
$$189$$ 2.23214e9 0.127245
$$190$$ −4.39009e9 −0.244389
$$191$$ 3.23292e10 1.75770 0.878851 0.477096i $$-0.158311\pi$$
0.878851 + 0.477096i $$0.158311\pi$$
$$192$$ −2.61725e9 −0.138992
$$193$$ 1.29399e10 0.671311 0.335655 0.941985i $$-0.391042\pi$$
0.335655 + 0.941985i $$0.391042\pi$$
$$194$$ −1.07765e10 −0.546224
$$195$$ 0 0
$$196$$ −1.00985e10 −0.488770
$$197$$ −8.81090e9 −0.416795 −0.208397 0.978044i $$-0.566825\pi$$
−0.208397 + 0.978044i $$0.566825\pi$$
$$198$$ −4.18011e9 −0.193283
$$199$$ −2.48534e10 −1.12343 −0.561716 0.827330i $$-0.689859\pi$$
−0.561716 + 0.827330i $$0.689859\pi$$
$$200$$ 4.89974e9 0.216540
$$201$$ −1.89341e10 −0.818204
$$202$$ −1.69235e10 −0.715170
$$203$$ −3.65611e9 −0.151108
$$204$$ 9.89063e9 0.399842
$$205$$ 1.31548e9 0.0520224
$$206$$ −1.27339e10 −0.492672
$$207$$ 9.51557e8 0.0360220
$$208$$ 0 0
$$209$$ −1.77080e10 −0.641963
$$210$$ −2.06729e9 −0.0733523
$$211$$ −4.65163e10 −1.61560 −0.807801 0.589456i $$-0.799342\pi$$
−0.807801 + 0.589456i $$0.799342\pi$$
$$212$$ 4.25375e9 0.144631
$$213$$ −6.03989e10 −2.01058
$$214$$ 3.15862e10 1.02952
$$215$$ −2.92934e10 −0.934969
$$216$$ −9.60381e9 −0.300194
$$217$$ 1.24656e9 0.0381631
$$218$$ −2.15144e10 −0.645170
$$219$$ 3.98175e10 1.16970
$$220$$ −1.25053e10 −0.359908
$$221$$ 0 0
$$222$$ −1.07505e10 −0.297056
$$223$$ −4.66347e10 −1.26281 −0.631404 0.775454i $$-0.717521\pi$$
−0.631404 + 0.775454i $$0.717521\pi$$
$$224$$ −9.98244e8 −0.0264924
$$225$$ −5.56603e9 −0.144785
$$226$$ −4.33088e10 −1.10430
$$227$$ −2.65867e10 −0.664582 −0.332291 0.943177i $$-0.607822\pi$$
−0.332291 + 0.943177i $$0.607822\pi$$
$$228$$ 1.25950e10 0.308668
$$229$$ −3.99907e10 −0.960947 −0.480474 0.877009i $$-0.659535\pi$$
−0.480474 + 0.877009i $$0.659535\pi$$
$$230$$ 2.84670e9 0.0670758
$$231$$ −8.33865e9 −0.192682
$$232$$ 1.57305e10 0.356489
$$233$$ 6.53338e9 0.145223 0.0726116 0.997360i $$-0.476867\pi$$
0.0726116 + 0.997360i $$0.476867\pi$$
$$234$$ 0 0
$$235$$ −9.20553e9 −0.196899
$$236$$ −2.87322e10 −0.602927
$$237$$ −7.67679e10 −1.58056
$$238$$ 3.77239e9 0.0762114
$$239$$ 5.66773e10 1.12362 0.561809 0.827267i $$-0.310106\pi$$
0.561809 + 0.827267i $$0.310106\pi$$
$$240$$ 8.89455e9 0.173051
$$241$$ −4.61491e9 −0.0881225 −0.0440613 0.999029i $$-0.514030\pi$$
−0.0440613 + 0.999029i $$0.514030\pi$$
$$242$$ −1.27144e10 −0.238302
$$243$$ 2.51971e10 0.463577
$$244$$ −8.49849e9 −0.153493
$$245$$ 3.43192e10 0.608540
$$246$$ −3.77406e9 −0.0657053
$$247$$ 0 0
$$248$$ −5.36334e9 −0.0900331
$$249$$ −7.13576e10 −1.17637
$$250$$ −4.38390e10 −0.709790
$$251$$ 6.80194e10 1.08169 0.540843 0.841124i $$-0.318105\pi$$
0.540843 + 0.841124i $$0.318105\pi$$
$$252$$ 1.13399e9 0.0177136
$$253$$ 1.14825e10 0.176195
$$254$$ −1.91936e10 −0.289338
$$255$$ −3.36127e10 −0.497820
$$256$$ 4.29497e9 0.0625000
$$257$$ −9.35958e10 −1.33831 −0.669156 0.743122i $$-0.733344\pi$$
−0.669156 + 0.743122i $$0.733344\pi$$
$$258$$ 8.40418e10 1.18088
$$259$$ −4.10034e9 −0.0566201
$$260$$ 0 0
$$261$$ −1.78696e10 −0.238360
$$262$$ −4.45783e10 −0.584478
$$263$$ −9.40401e10 −1.21203 −0.606013 0.795454i $$-0.707232\pi$$
−0.606013 + 0.795454i $$0.707232\pi$$
$$264$$ 3.58772e10 0.454570
$$265$$ −1.44561e10 −0.180071
$$266$$ 4.80387e9 0.0588334
$$267$$ −4.96228e9 −0.0597559
$$268$$ 3.10713e10 0.367919
$$269$$ 1.22724e11 1.42904 0.714522 0.699613i $$-0.246644\pi$$
0.714522 + 0.699613i $$0.246644\pi$$
$$270$$ 3.26379e10 0.373754
$$271$$ 1.64257e11 1.84996 0.924982 0.380012i $$-0.124080\pi$$
0.924982 + 0.380012i $$0.124080\pi$$
$$272$$ −1.62308e10 −0.179796
$$273$$ 0 0
$$274$$ 4.61173e10 0.494295
$$275$$ −6.71656e10 −0.708190
$$276$$ −8.16707e9 −0.0847180
$$277$$ 6.50134e10 0.663505 0.331752 0.943367i $$-0.392360\pi$$
0.331752 + 0.943367i $$0.392360\pi$$
$$278$$ −3.45026e10 −0.346458
$$279$$ 6.09268e9 0.0601990
$$280$$ 3.39247e9 0.0329841
$$281$$ −5.20964e10 −0.498459 −0.249230 0.968444i $$-0.580177\pi$$
−0.249230 + 0.968444i $$0.580177\pi$$
$$282$$ 2.64104e10 0.248687
$$283$$ −9.06992e10 −0.840552 −0.420276 0.907396i $$-0.638067\pi$$
−0.420276 + 0.907396i $$0.638067\pi$$
$$284$$ 9.91162e10 0.904091
$$285$$ −4.28034e10 −0.384305
$$286$$ 0 0
$$287$$ −1.43946e9 −0.0125237
$$288$$ −4.87902e9 −0.0417895
$$289$$ −5.72514e10 −0.482776
$$290$$ −5.34591e10 −0.443844
$$291$$ −1.05071e11 −0.858943
$$292$$ −6.53415e10 −0.525976
$$293$$ −7.25569e10 −0.575141 −0.287571 0.957759i $$-0.592848\pi$$
−0.287571 + 0.957759i $$0.592848\pi$$
$$294$$ −9.84605e10 −0.768597
$$295$$ 9.76445e10 0.750670
$$296$$ 1.76418e10 0.133576
$$297$$ 1.31649e11 0.981779
$$298$$ 1.20728e11 0.886816
$$299$$ 0 0
$$300$$ 4.77724e10 0.340512
$$301$$ 3.20544e10 0.225081
$$302$$ −6.90093e10 −0.477394
$$303$$ −1.65004e11 −1.12461
$$304$$ −2.06687e10 −0.138798
$$305$$ 2.88816e10 0.191105
$$306$$ 1.84379e10 0.120217
$$307$$ −1.81977e11 −1.16921 −0.584607 0.811317i $$-0.698751\pi$$
−0.584607 + 0.811317i $$0.698751\pi$$
$$308$$ 1.36839e10 0.0866429
$$309$$ −1.24155e11 −0.774732
$$310$$ 1.82270e10 0.112095
$$311$$ −8.98295e10 −0.544499 −0.272250 0.962227i $$-0.587768\pi$$
−0.272250 + 0.962227i $$0.587768\pi$$
$$312$$ 0 0
$$313$$ 5.51394e9 0.0324722 0.0162361 0.999868i $$-0.494832\pi$$
0.0162361 + 0.999868i $$0.494832\pi$$
$$314$$ 6.77052e10 0.393041
$$315$$ −3.85380e9 −0.0220542
$$316$$ 1.25978e11 0.710727
$$317$$ 1.94806e10 0.108351 0.0541757 0.998531i $$-0.482747\pi$$
0.0541757 + 0.998531i $$0.482747\pi$$
$$318$$ 4.14741e10 0.227434
$$319$$ −2.15634e11 −1.16589
$$320$$ −1.45962e10 −0.0778152
$$321$$ 3.07965e11 1.61893
$$322$$ −3.11500e9 −0.0161476
$$323$$ 7.81076e10 0.399284
$$324$$ −1.17083e11 −0.590257
$$325$$ 0 0
$$326$$ 1.32552e10 0.0649989
$$327$$ −2.09765e11 −1.01454
$$328$$ 6.19332e9 0.0295455
$$329$$ 1.00732e10 0.0474007
$$330$$ −1.21927e11 −0.565960
$$331$$ 1.06801e11 0.489046 0.244523 0.969644i $$-0.421369\pi$$
0.244523 + 0.969644i $$0.421369\pi$$
$$332$$ 1.17100e11 0.528974
$$333$$ −2.00408e10 −0.0893134
$$334$$ −4.56800e10 −0.200848
$$335$$ −1.05594e11 −0.458075
$$336$$ −9.73288e9 −0.0416596
$$337$$ −1.75776e11 −0.742380 −0.371190 0.928557i $$-0.621050\pi$$
−0.371190 + 0.928557i $$0.621050\pi$$
$$338$$ 0 0
$$339$$ −4.22261e11 −1.73653
$$340$$ 5.51593e10 0.223853
$$341$$ 7.35206e10 0.294452
$$342$$ 2.34794e10 0.0928046
$$343$$ −7.59705e10 −0.296361
$$344$$ −1.37915e11 −0.531004
$$345$$ 2.77553e10 0.105477
$$346$$ 2.82704e11 1.06045
$$347$$ 8.33026e10 0.308444 0.154222 0.988036i $$-0.450713\pi$$
0.154222 + 0.988036i $$0.450713\pi$$
$$348$$ 1.53372e11 0.560583
$$349$$ −3.21368e11 −1.15955 −0.579773 0.814778i $$-0.696859\pi$$
−0.579773 + 0.814778i $$0.696859\pi$$
$$350$$ 1.82209e10 0.0649028
$$351$$ 0 0
$$352$$ −5.88754e10 −0.204405
$$353$$ −4.07688e11 −1.39747 −0.698735 0.715381i $$-0.746253\pi$$
−0.698735 + 0.715381i $$0.746253\pi$$
$$354$$ −2.80139e11 −0.948110
$$355$$ −3.36840e11 −1.12563
$$356$$ 8.14323e9 0.0268703
$$357$$ 3.67808e10 0.119843
$$358$$ 9.38771e9 0.0302055
$$359$$ 5.60079e11 1.77961 0.889804 0.456343i $$-0.150841\pi$$
0.889804 + 0.456343i $$0.150841\pi$$
$$360$$ 1.65811e10 0.0520297
$$361$$ −2.23223e11 −0.691762
$$362$$ 8.69433e10 0.266102
$$363$$ −1.23965e11 −0.374732
$$364$$ 0 0
$$365$$ 2.22059e11 0.654863
$$366$$ −8.28603e10 −0.241369
$$367$$ 3.76056e11 1.08207 0.541036 0.841000i $$-0.318032\pi$$
0.541036 + 0.841000i $$0.318032\pi$$
$$368$$ 1.34024e10 0.0380949
$$369$$ −7.03553e9 −0.0197551
$$370$$ −5.99545e10 −0.166308
$$371$$ 1.58186e10 0.0433497
$$372$$ −5.22925e10 −0.141578
$$373$$ −8.12245e10 −0.217269 −0.108634 0.994082i $$-0.534648\pi$$
−0.108634 + 0.994082i $$0.534648\pi$$
$$374$$ 2.22492e11 0.588019
$$375$$ −4.27430e11 −1.11615
$$376$$ −4.33401e10 −0.111826
$$377$$ 0 0
$$378$$ −3.57142e10 −0.0899762
$$379$$ −2.02729e11 −0.504708 −0.252354 0.967635i $$-0.581205\pi$$
−0.252354 + 0.967635i $$0.581205\pi$$
$$380$$ 7.02414e10 0.172809
$$381$$ −1.87138e11 −0.454987
$$382$$ −5.17268e11 −1.24288
$$383$$ 4.76816e10 0.113229 0.0566143 0.998396i $$-0.481969\pi$$
0.0566143 + 0.998396i $$0.481969\pi$$
$$384$$ 4.18759e10 0.0982820
$$385$$ −4.65040e10 −0.107874
$$386$$ −2.07039e11 −0.474688
$$387$$ 1.56669e11 0.355046
$$388$$ 1.72424e11 0.386238
$$389$$ −1.89795e11 −0.420253 −0.210126 0.977674i $$-0.567388\pi$$
−0.210126 + 0.977674i $$0.567388\pi$$
$$390$$ 0 0
$$391$$ −5.06479e10 −0.109589
$$392$$ 1.61576e11 0.345613
$$393$$ −4.34639e11 −0.919098
$$394$$ 1.40974e11 0.294718
$$395$$ −4.28129e11 −0.884886
$$396$$ 6.68817e10 0.136672
$$397$$ −4.00237e11 −0.808649 −0.404325 0.914616i $$-0.632493\pi$$
−0.404325 + 0.914616i $$0.632493\pi$$
$$398$$ 3.97654e11 0.794387
$$399$$ 4.68377e10 0.0925162
$$400$$ −7.83958e10 −0.153117
$$401$$ −8.76186e11 −1.69218 −0.846090 0.533040i $$-0.821049\pi$$
−0.846090 + 0.533040i $$0.821049\pi$$
$$402$$ 3.02945e11 0.578558
$$403$$ 0 0
$$404$$ 2.70776e11 0.505701
$$405$$ 3.97899e11 0.734895
$$406$$ 5.84977e10 0.106849
$$407$$ −2.41834e11 −0.436860
$$408$$ −1.58250e11 −0.282731
$$409$$ −5.72300e11 −1.01127 −0.505637 0.862746i $$-0.668742\pi$$
−0.505637 + 0.862746i $$0.668742\pi$$
$$410$$ −2.10476e10 −0.0367854
$$411$$ 4.49643e11 0.777285
$$412$$ 2.03742e11 0.348371
$$413$$ −1.06848e11 −0.180713
$$414$$ −1.52249e10 −0.0254714
$$415$$ −3.97956e11 −0.658595
$$416$$ 0 0
$$417$$ −3.36400e11 −0.544809
$$418$$ 2.83327e11 0.453937
$$419$$ 4.16693e11 0.660469 0.330235 0.943899i $$-0.392872\pi$$
0.330235 + 0.943899i $$0.392872\pi$$
$$420$$ 3.30766e10 0.0518679
$$421$$ 1.19043e12 1.84687 0.923434 0.383757i $$-0.125370\pi$$
0.923434 + 0.383757i $$0.125370\pi$$
$$422$$ 7.44261e11 1.14240
$$423$$ 4.92337e10 0.0747706
$$424$$ −6.80600e10 −0.102269
$$425$$ 2.96259e11 0.440476
$$426$$ 9.66383e11 1.42169
$$427$$ −3.16038e10 −0.0460059
$$428$$ −5.05379e11 −0.727981
$$429$$ 0 0
$$430$$ 4.68695e11 0.661123
$$431$$ 4.36455e11 0.609245 0.304622 0.952473i $$-0.401470\pi$$
0.304622 + 0.952473i $$0.401470\pi$$
$$432$$ 1.53661e11 0.212269
$$433$$ 4.48430e11 0.613055 0.306527 0.951862i $$-0.400833\pi$$
0.306527 + 0.951862i $$0.400833\pi$$
$$434$$ −1.99449e10 −0.0269854
$$435$$ −5.21226e11 −0.697950
$$436$$ 3.44230e11 0.456204
$$437$$ −6.44965e10 −0.0845998
$$438$$ −6.37079e11 −0.827104
$$439$$ −6.59703e11 −0.847731 −0.423866 0.905725i $$-0.639327\pi$$
−0.423866 + 0.905725i $$0.639327\pi$$
$$440$$ 2.00085e11 0.254493
$$441$$ −1.83548e11 −0.231088
$$442$$ 0 0
$$443$$ 9.48507e11 1.17010 0.585051 0.810997i $$-0.301075\pi$$
0.585051 + 0.810997i $$0.301075\pi$$
$$444$$ 1.72007e11 0.210051
$$445$$ −2.76743e10 −0.0334546
$$446$$ 7.46156e11 0.892941
$$447$$ 1.17709e12 1.39453
$$448$$ 1.59719e10 0.0187329
$$449$$ 6.11763e11 0.710354 0.355177 0.934799i $$-0.384421\pi$$
0.355177 + 0.934799i $$0.384421\pi$$
$$450$$ 8.90566e10 0.102379
$$451$$ −8.48981e10 −0.0966282
$$452$$ 6.92941e11 0.780861
$$453$$ −6.72841e11 −0.750707
$$454$$ 4.25388e11 0.469930
$$455$$ 0 0
$$456$$ −2.01520e11 −0.218261
$$457$$ 3.79033e11 0.406494 0.203247 0.979128i $$-0.434851\pi$$
0.203247 + 0.979128i $$0.434851\pi$$
$$458$$ 6.39852e11 0.679492
$$459$$ −5.80688e11 −0.610641
$$460$$ −4.55471e10 −0.0474297
$$461$$ 8.90062e11 0.917839 0.458919 0.888478i $$-0.348237\pi$$
0.458919 + 0.888478i $$0.348237\pi$$
$$462$$ 1.33418e11 0.136247
$$463$$ −1.32852e12 −1.34355 −0.671776 0.740755i $$-0.734468\pi$$
−0.671776 + 0.740755i $$0.734468\pi$$
$$464$$ −2.51688e11 −0.252076
$$465$$ 1.77713e11 0.176271
$$466$$ −1.04534e11 −0.102688
$$467$$ 1.65638e12 1.61151 0.805755 0.592249i $$-0.201760\pi$$
0.805755 + 0.592249i $$0.201760\pi$$
$$468$$ 0 0
$$469$$ 1.15546e11 0.110275
$$470$$ 1.47289e11 0.139229
$$471$$ 6.60125e11 0.618062
$$472$$ 4.59715e11 0.426334
$$473$$ 1.89054e12 1.73664
$$474$$ 1.22829e12 1.11763
$$475$$ 3.77265e11 0.340037
$$476$$ −6.03582e10 −0.0538896
$$477$$ 7.73152e10 0.0683805
$$478$$ −9.06837e11 −0.794518
$$479$$ 1.07973e12 0.937143 0.468571 0.883426i $$-0.344769\pi$$
0.468571 + 0.883426i $$0.344769\pi$$
$$480$$ −1.42313e11 −0.122365
$$481$$ 0 0
$$482$$ 7.38386e10 0.0623120
$$483$$ −3.03713e10 −0.0253923
$$484$$ 2.03430e11 0.168505
$$485$$ −5.85973e11 −0.480883
$$486$$ −4.03153e11 −0.327798
$$487$$ −1.60549e12 −1.29338 −0.646690 0.762753i $$-0.723847\pi$$
−0.646690 + 0.762753i $$0.723847\pi$$
$$488$$ 1.35976e11 0.108536
$$489$$ 1.29238e11 0.102212
$$490$$ −5.49106e11 −0.430303
$$491$$ 7.93629e11 0.616242 0.308121 0.951347i $$-0.400300\pi$$
0.308121 + 0.951347i $$0.400300\pi$$
$$492$$ 6.03849e10 0.0464607
$$493$$ 9.51134e11 0.725154
$$494$$ 0 0
$$495$$ −2.27293e11 −0.170162
$$496$$ 8.58134e10 0.0636630
$$497$$ 3.68588e11 0.270980
$$498$$ 1.14172e12 0.831817
$$499$$ 1.96951e12 1.42202 0.711010 0.703182i $$-0.248238\pi$$
0.711010 + 0.703182i $$0.248238\pi$$
$$500$$ 7.01423e11 0.501898
$$501$$ −4.45380e11 −0.315835
$$502$$ −1.08831e12 −0.764867
$$503$$ −5.42230e11 −0.377683 −0.188842 0.982008i $$-0.560473\pi$$
−0.188842 + 0.982008i $$0.560473\pi$$
$$504$$ −1.81439e10 −0.0125254
$$505$$ −9.20216e11 −0.629620
$$506$$ −1.83720e11 −0.124589
$$507$$ 0 0
$$508$$ 3.07098e11 0.204593
$$509$$ 1.69215e12 1.11740 0.558699 0.829370i $$-0.311301\pi$$
0.558699 + 0.829370i $$0.311301\pi$$
$$510$$ 5.37803e11 0.352012
$$511$$ −2.42989e11 −0.157649
$$512$$ −6.87195e10 −0.0441942
$$513$$ −7.39465e11 −0.471400
$$514$$ 1.49753e12 0.946329
$$515$$ −6.92403e11 −0.433737
$$516$$ −1.34467e12 −0.835010
$$517$$ 5.94106e11 0.365727
$$518$$ 6.56054e10 0.0400365
$$519$$ 2.75637e12 1.66757
$$520$$ 0 0
$$521$$ 2.97596e12 1.76953 0.884764 0.466040i $$-0.154320\pi$$
0.884764 + 0.466040i $$0.154320\pi$$
$$522$$ 2.85914e11 0.168546
$$523$$ −1.07989e12 −0.631137 −0.315568 0.948903i $$-0.602195\pi$$
−0.315568 + 0.948903i $$0.602195\pi$$
$$524$$ 7.13253e11 0.413288
$$525$$ 1.77654e11 0.102060
$$526$$ 1.50464e12 0.857032
$$527$$ −3.24291e11 −0.183141
$$528$$ −5.74036e11 −0.321430
$$529$$ −1.75933e12 −0.976780
$$530$$ 2.31298e11 0.127330
$$531$$ −5.22230e11 −0.285060
$$532$$ −7.68619e10 −0.0416015
$$533$$ 0 0
$$534$$ 7.93965e10 0.0422538
$$535$$ 1.71750e12 0.906367
$$536$$ −4.97141e11 −0.260158
$$537$$ 9.15302e10 0.0474985
$$538$$ −1.96359e12 −1.01049
$$539$$ −2.21489e12 −1.13032
$$540$$ −5.22207e11 −0.264284
$$541$$ 2.02167e12 1.01467 0.507333 0.861750i $$-0.330631\pi$$
0.507333 + 0.861750i $$0.330631\pi$$
$$542$$ −2.62812e12 −1.30812
$$543$$ 8.47697e11 0.418448
$$544$$ 2.59692e11 0.127135
$$545$$ −1.16984e12 −0.567994
$$546$$ 0 0
$$547$$ −2.62612e12 −1.25421 −0.627107 0.778933i $$-0.715761\pi$$
−0.627107 + 0.778933i $$0.715761\pi$$
$$548$$ −7.37876e11 −0.349519
$$549$$ −1.54467e11 −0.0725703
$$550$$ 1.07465e12 0.500766
$$551$$ 1.21120e12 0.559802
$$552$$ 1.30673e11 0.0599046
$$553$$ 4.68481e11 0.213024
$$554$$ −1.04021e12 −0.469169
$$555$$ −5.84557e11 −0.261522
$$556$$ 5.52041e11 0.244982
$$557$$ −3.48482e12 −1.53402 −0.767012 0.641633i $$-0.778257\pi$$
−0.767012 + 0.641633i $$0.778257\pi$$
$$558$$ −9.74828e10 −0.0425671
$$559$$ 0 0
$$560$$ −5.42795e10 −0.0233233
$$561$$ 2.16929e12 0.924667
$$562$$ 8.33543e11 0.352464
$$563$$ −2.77091e12 −1.16235 −0.581173 0.813780i $$-0.697406\pi$$
−0.581173 + 0.813780i $$0.697406\pi$$
$$564$$ −4.22566e11 −0.175848
$$565$$ −2.35492e12 −0.972205
$$566$$ 1.45119e12 0.594360
$$567$$ −4.35402e11 −0.176916
$$568$$ −1.58586e12 −0.639289
$$569$$ −6.70382e11 −0.268113 −0.134056 0.990974i $$-0.542800\pi$$
−0.134056 + 0.990974i $$0.542800\pi$$
$$570$$ 6.84854e11 0.271745
$$571$$ 2.67123e12 1.05160 0.525798 0.850609i $$-0.323767\pi$$
0.525798 + 0.850609i $$0.323767\pi$$
$$572$$ 0 0
$$573$$ −5.04336e12 −1.95445
$$574$$ 2.30314e10 0.00885559
$$575$$ −2.44633e11 −0.0933274
$$576$$ 7.80644e10 0.0295496
$$577$$ 6.59284e11 0.247618 0.123809 0.992306i $$-0.460489\pi$$
0.123809 + 0.992306i $$0.460489\pi$$
$$578$$ 9.16023e11 0.341374
$$579$$ −2.01863e12 −0.746453
$$580$$ 8.55345e11 0.313845
$$581$$ 4.35464e11 0.158548
$$582$$ 1.68114e12 0.607365
$$583$$ 9.32967e11 0.334471
$$584$$ 1.04546e12 0.371921
$$585$$ 0 0
$$586$$ 1.16091e12 0.406686
$$587$$ −1.04947e12 −0.364835 −0.182418 0.983221i $$-0.558392\pi$$
−0.182418 + 0.983221i $$0.558392\pi$$
$$588$$ 1.57537e12 0.543480
$$589$$ −4.12961e11 −0.141381
$$590$$ −1.56231e12 −0.530804
$$591$$ 1.37450e12 0.463448
$$592$$ −2.82269e11 −0.0944528
$$593$$ 1.31188e12 0.435662 0.217831 0.975987i $$-0.430102\pi$$
0.217831 + 0.975987i $$0.430102\pi$$
$$594$$ −2.10639e12 −0.694223
$$595$$ 2.05124e11 0.0670949
$$596$$ −1.93164e12 −0.627074
$$597$$ 3.87713e12 1.24918
$$598$$ 0 0
$$599$$ −3.37603e12 −1.07148 −0.535742 0.844382i $$-0.679968\pi$$
−0.535742 + 0.844382i $$0.679968\pi$$
$$600$$ −7.64359e11 −0.240778
$$601$$ 2.63880e12 0.825034 0.412517 0.910950i $$-0.364650\pi$$
0.412517 + 0.910950i $$0.364650\pi$$
$$602$$ −5.12871e11 −0.159156
$$603$$ 5.64745e11 0.173950
$$604$$ 1.10415e12 0.337568
$$605$$ −6.91346e11 −0.209795
$$606$$ 2.64007e12 0.795221
$$607$$ −5.15939e12 −1.54259 −0.771293 0.636481i $$-0.780390\pi$$
−0.771293 + 0.636481i $$0.780390\pi$$
$$608$$ 3.30700e11 0.0981449
$$609$$ 5.70353e11 0.168022
$$610$$ −4.62105e11 −0.135132
$$611$$ 0 0
$$612$$ −2.95007e11 −0.0850063
$$613$$ 5.37354e11 0.153705 0.0768525 0.997042i $$-0.475513\pi$$
0.0768525 + 0.997042i $$0.475513\pi$$
$$614$$ 2.91163e12 0.826759
$$615$$ −2.05214e11 −0.0578455
$$616$$ −2.18943e11 −0.0612658
$$617$$ −4.63358e12 −1.28716 −0.643582 0.765378i $$-0.722552\pi$$
−0.643582 + 0.765378i $$0.722552\pi$$
$$618$$ 1.98648e12 0.547818
$$619$$ −3.06267e12 −0.838480 −0.419240 0.907876i $$-0.637703\pi$$
−0.419240 + 0.907876i $$0.637703\pi$$
$$620$$ −2.91631e11 −0.0792632
$$621$$ 4.79496e11 0.129382
$$622$$ 1.43727e12 0.385019
$$623$$ 3.02827e10 0.00805374
$$624$$ 0 0
$$625$$ −4.73661e10 −0.0124167
$$626$$ −8.82230e10 −0.0229613
$$627$$ 2.76244e12 0.713821
$$628$$ −1.08328e12 −0.277922
$$629$$ 1.06670e12 0.271715
$$630$$ 6.16608e10 0.0155947
$$631$$ 5.46928e10 0.0137340 0.00686702 0.999976i $$-0.497814\pi$$
0.00686702 + 0.999976i $$0.497814\pi$$
$$632$$ −2.01565e12 −0.502560
$$633$$ 7.25654e12 1.79644
$$634$$ −3.11689e11 −0.0766160
$$635$$ −1.04365e12 −0.254727
$$636$$ −6.63585e11 −0.160820
$$637$$ 0 0
$$638$$ 3.45014e12 0.824410
$$639$$ 1.80151e12 0.427449
$$640$$ 2.33539e11 0.0550236
$$641$$ −4.78068e12 −1.11848 −0.559240 0.829006i $$-0.688907\pi$$
−0.559240 + 0.829006i $$0.688907\pi$$
$$642$$ −4.92744e12 −1.14476
$$643$$ −4.10484e12 −0.946994 −0.473497 0.880795i $$-0.657008\pi$$
−0.473497 + 0.880795i $$0.657008\pi$$
$$644$$ 4.98401e10 0.0114181
$$645$$ 4.56977e12 1.03962
$$646$$ −1.24972e12 −0.282337
$$647$$ −5.49263e12 −1.23228 −0.616142 0.787635i $$-0.711305\pi$$
−0.616142 + 0.787635i $$0.711305\pi$$
$$648$$ 1.87333e12 0.417375
$$649$$ −6.30178e12 −1.39432
$$650$$ 0 0
$$651$$ −1.94463e11 −0.0424348
$$652$$ −2.12083e11 −0.0459612
$$653$$ 4.15994e12 0.895320 0.447660 0.894204i $$-0.352258\pi$$
0.447660 + 0.894204i $$0.352258\pi$$
$$654$$ 3.35624e12 0.717386
$$655$$ −2.42395e12 −0.514561
$$656$$ −9.90932e10 −0.0208918
$$657$$ −1.18763e12 −0.248678
$$658$$ −1.61171e11 −0.0335174
$$659$$ 2.15295e12 0.444681 0.222341 0.974969i $$-0.428630\pi$$
0.222341 + 0.974969i $$0.428630\pi$$
$$660$$ 1.95082e12 0.400194
$$661$$ −8.63978e12 −1.76034 −0.880169 0.474660i $$-0.842571\pi$$
−0.880169 + 0.474660i $$0.842571\pi$$
$$662$$ −1.70882e12 −0.345807
$$663$$ 0 0
$$664$$ −1.87359e12 −0.374041
$$665$$ 2.61210e11 0.0517956
$$666$$ 3.20653e11 0.0631541
$$667$$ −7.85387e11 −0.153645
$$668$$ 7.30880e11 0.142021
$$669$$ 7.27502e12 1.40416
$$670$$ 1.68950e12 0.323908
$$671$$ −1.86396e12 −0.354964
$$672$$ 1.55726e11 0.0294578
$$673$$ −2.90788e12 −0.546398 −0.273199 0.961957i $$-0.588082\pi$$
−0.273199 + 0.961957i $$0.588082\pi$$
$$674$$ 2.81242e12 0.524942
$$675$$ −2.80476e12 −0.520031
$$676$$ 0 0
$$677$$ 4.26822e12 0.780904 0.390452 0.920623i $$-0.372319\pi$$
0.390452 + 0.920623i $$0.372319\pi$$
$$678$$ 6.75618e12 1.22791
$$679$$ 6.41203e11 0.115766
$$680$$ −8.82548e11 −0.158288
$$681$$ 4.14753e12 0.738971
$$682$$ −1.17633e12 −0.208209
$$683$$ 7.69165e11 0.135247 0.0676233 0.997711i $$-0.478458\pi$$
0.0676233 + 0.997711i $$0.478458\pi$$
$$684$$ −3.75671e11 −0.0656228
$$685$$ 2.50763e12 0.435166
$$686$$ 1.21553e12 0.209559
$$687$$ 6.23855e12 1.06851
$$688$$ 2.20664e12 0.375477
$$689$$ 0 0
$$690$$ −4.44085e11 −0.0745838
$$691$$ −1.38648e12 −0.231347 −0.115673 0.993287i $$-0.536903\pi$$
−0.115673 + 0.993287i $$0.536903\pi$$
$$692$$ −4.52327e12 −0.749851
$$693$$ 2.48716e11 0.0409642
$$694$$ −1.33284e12 −0.218103
$$695$$ −1.87608e12 −0.305014
$$696$$ −2.45396e12 −0.396392
$$697$$ 3.74475e11 0.0601002
$$698$$ 5.14189e12 0.819923
$$699$$ −1.01921e12 −0.161479
$$700$$ −2.91534e11 −0.0458932
$$701$$ −5.51186e12 −0.862119 −0.431059 0.902324i $$-0.641860\pi$$
−0.431059 + 0.902324i $$0.641860\pi$$
$$702$$ 0 0
$$703$$ 1.35837e12 0.209758
$$704$$ 9.42007e11 0.144536
$$705$$ 1.43606e12 0.218939
$$706$$ 6.52302e12 0.988160
$$707$$ 1.00695e12 0.151572
$$708$$ 4.48222e12 0.670415
$$709$$ −2.43152e12 −0.361385 −0.180693 0.983540i $$-0.557834\pi$$
−0.180693 + 0.983540i $$0.557834\pi$$
$$710$$ 5.38944e12 0.795942
$$711$$ 2.28975e12 0.336028
$$712$$ −1.30292e11 −0.0190001
$$713$$ 2.67779e11 0.0388038
$$714$$ −5.88492e11 −0.0847420
$$715$$ 0 0
$$716$$ −1.50203e11 −0.0213585
$$717$$ −8.84166e12 −1.24939
$$718$$ −8.96126e12 −1.25837
$$719$$ −2.70890e12 −0.378018 −0.189009 0.981975i $$-0.560528\pi$$
−0.189009 + 0.981975i $$0.560528\pi$$
$$720$$ −2.65297e11 −0.0367905
$$721$$ 7.57664e11 0.104416
$$722$$ 3.57157e12 0.489150
$$723$$ 7.19927e11 0.0979864
$$724$$ −1.39109e12 −0.188162
$$725$$ 4.59404e12 0.617553
$$726$$ 1.98345e12 0.264976
$$727$$ 5.26647e11 0.0699221 0.0349611 0.999389i $$-0.488869\pi$$
0.0349611 + 0.999389i $$0.488869\pi$$
$$728$$ 0 0
$$729$$ 5.07138e12 0.665047
$$730$$ −3.55294e12 −0.463058
$$731$$ −8.33893e12 −1.08015
$$732$$ 1.32576e12 0.170674
$$733$$ 2.78009e12 0.355706 0.177853 0.984057i $$-0.443085\pi$$
0.177853 + 0.984057i $$0.443085\pi$$
$$734$$ −6.01690e12 −0.765140
$$735$$ −5.35379e12 −0.676656
$$736$$ −2.14438e11 −0.0269371
$$737$$ 6.81481e12 0.850844
$$738$$ 1.12569e11 0.0139689
$$739$$ 2.36558e12 0.291768 0.145884 0.989302i $$-0.453397\pi$$
0.145884 + 0.989302i $$0.453397\pi$$
$$740$$ 9.59272e11 0.117598
$$741$$ 0 0
$$742$$ −2.53098e11 −0.0306529
$$743$$ −1.31398e13 −1.58175 −0.790876 0.611977i $$-0.790375\pi$$
−0.790876 + 0.611977i $$0.790375\pi$$
$$744$$ 8.36680e11 0.100111
$$745$$ 6.56457e12 0.780733
$$746$$ 1.29959e12 0.153632
$$747$$ 2.12838e12 0.250095
$$748$$ −3.55987e12 −0.415792
$$749$$ −1.87938e12 −0.218195
$$750$$ 6.83888e12 0.789240
$$751$$ −7.29436e12 −0.836773 −0.418387 0.908269i $$-0.637404\pi$$
−0.418387 + 0.908269i $$0.637404\pi$$
$$752$$ 6.93441e11 0.0790732
$$753$$ −1.06110e13 −1.20276
$$754$$ 0 0
$$755$$ −3.75238e12 −0.420287
$$756$$ 5.71427e11 0.0636227
$$757$$ −1.63020e13 −1.80430 −0.902150 0.431423i $$-0.858012\pi$$
−0.902150 + 0.431423i $$0.858012\pi$$
$$758$$ 3.24367e12 0.356882
$$759$$ −1.79127e12 −0.195917
$$760$$ −1.12386e12 −0.122195
$$761$$ 9.68945e12 1.04729 0.523646 0.851936i $$-0.324571\pi$$
0.523646 + 0.851936i $$0.324571\pi$$
$$762$$ 2.99420e12 0.321724
$$763$$ 1.28010e12 0.136737
$$764$$ 8.27629e12 0.878851
$$765$$ 1.00256e12 0.105836
$$766$$ −7.62906e11 −0.0800648
$$767$$ 0 0
$$768$$ −6.70015e11 −0.0694959
$$769$$ −1.21329e13 −1.25111 −0.625554 0.780180i $$-0.715127\pi$$
−0.625554 + 0.780180i $$0.715127\pi$$
$$770$$ 7.44064e11 0.0762785
$$771$$ 1.46009e13 1.48811
$$772$$ 3.31262e12 0.335655
$$773$$ 1.68647e13 1.69891 0.849454 0.527663i $$-0.176932\pi$$
0.849454 + 0.527663i $$0.176932\pi$$
$$774$$ −2.50671e12 −0.251055
$$775$$ −1.56635e12 −0.155966
$$776$$ −2.75879e12 −0.273112
$$777$$ 6.39653e11 0.0629578
$$778$$ 3.03671e12 0.297164
$$779$$ 4.76868e11 0.0463959
$$780$$ 0 0
$$781$$ 2.17390e13 2.09079
$$782$$ 8.10366e11 0.0774909
$$783$$ −9.00463e12 −0.856126
$$784$$ −2.58522e12 −0.244385
$$785$$ 3.68147e12 0.346025
$$786$$ 6.95422e12 0.649901
$$787$$ 6.00116e12 0.557634 0.278817 0.960344i $$-0.410058\pi$$
0.278817 + 0.960344i $$0.410058\pi$$
$$788$$ −2.25559e12 −0.208397
$$789$$ 1.46703e13 1.34769
$$790$$ 6.85006e12 0.625709
$$791$$ 2.57688e12 0.234045
$$792$$ −1.07011e12 −0.0966416
$$793$$ 0 0
$$794$$ 6.40380e12 0.571801
$$795$$ 2.25515e12 0.200228
$$796$$ −6.36247e12 −0.561716
$$797$$ −1.43155e13 −1.25674 −0.628368 0.777916i $$-0.716277\pi$$
−0.628368 + 0.777916i $$0.716277\pi$$
$$798$$ −7.49403e11 −0.0654188
$$799$$ −2.62053e12 −0.227472
$$800$$ 1.25433e12 0.108270
$$801$$ 1.48010e11 0.0127041
$$802$$ 1.40190e13 1.19655
$$803$$ −1.43312e13 −1.21636
$$804$$ −4.84712e12 −0.409102
$$805$$ −1.69378e11 −0.0142160
$$806$$ 0 0
$$807$$ −1.91450e13 −1.58900
$$808$$ −4.33242e12 −0.357585
$$809$$ −1.09893e13 −0.901992 −0.450996 0.892526i $$-0.648931\pi$$
−0.450996 + 0.892526i $$0.648931\pi$$
$$810$$ −6.36638e12 −0.519649
$$811$$ −2.15444e13 −1.74880 −0.874399 0.485207i $$-0.838744\pi$$
−0.874399 + 0.485207i $$0.838744\pi$$
$$812$$ −9.35964e11 −0.0755539
$$813$$ −2.56242e13 −2.05704
$$814$$ 3.86934e12 0.308907
$$815$$ 7.20750e11 0.0572236
$$816$$ 2.53200e12 0.199921
$$817$$ −1.06190e13 −0.833846
$$818$$ 9.15680e12 0.715079
$$819$$ 0 0
$$820$$ 3.36762e11 0.0260112
$$821$$ −5.71748e12 −0.439198 −0.219599 0.975590i $$-0.570475\pi$$
−0.219599 + 0.975590i $$0.570475\pi$$
$$822$$ −7.19430e12 −0.549623
$$823$$ −1.00524e13 −0.763787 −0.381893 0.924206i $$-0.624728\pi$$
−0.381893 + 0.924206i $$0.624728\pi$$
$$824$$ −3.25987e12 −0.246336
$$825$$ 1.04778e13 0.787461
$$826$$ 1.70957e12 0.127784
$$827$$ −2.29581e13 −1.70672 −0.853359 0.521324i $$-0.825438\pi$$
−0.853359 + 0.521324i $$0.825438\pi$$
$$828$$ 2.43599e11 0.0180110
$$829$$ −1.57277e13 −1.15657 −0.578283 0.815836i $$-0.696277\pi$$
−0.578283 + 0.815836i $$0.696277\pi$$
$$830$$ 6.36729e12 0.465697
$$831$$ −1.01421e13 −0.737773
$$832$$ 0 0
$$833$$ 9.76960e12 0.703031
$$834$$ 5.38240e12 0.385238
$$835$$ −2.48385e12 −0.176822
$$836$$ −4.53324e12 −0.320982
$$837$$ 3.07014e12 0.216219
$$838$$ −6.66708e12 −0.467022
$$839$$ 8.52168e12 0.593740 0.296870 0.954918i $$-0.404057\pi$$
0.296870 + 0.954918i $$0.404057\pi$$
$$840$$ −5.29225e11 −0.0366762
$$841$$ 2.41910e11 0.0166752
$$842$$ −1.90469e13 −1.30593
$$843$$ 8.12705e12 0.554254
$$844$$ −1.19082e13 −0.807801
$$845$$ 0 0
$$846$$ −7.87740e11 −0.0528708
$$847$$ 7.56507e11 0.0505054
$$848$$ 1.08896e12 0.0723154
$$849$$ 1.41491e13 0.934638
$$850$$ −4.74015e12 −0.311463
$$851$$ −8.80815e11 −0.0575707
$$852$$ −1.54621e13 −1.00529
$$853$$ 1.83160e12 0.118457 0.0592285 0.998244i $$-0.481136\pi$$
0.0592285 + 0.998244i $$0.481136\pi$$
$$854$$ 5.05660e11 0.0325311
$$855$$ 1.27669e12 0.0817032
$$856$$ 8.08606e12 0.514760
$$857$$ −6.20072e11 −0.0392671 −0.0196335 0.999807i $$-0.506250\pi$$
−0.0196335 + 0.999807i $$0.506250\pi$$
$$858$$ 0 0
$$859$$ −5.71581e12 −0.358186 −0.179093 0.983832i $$-0.557316\pi$$
−0.179093 + 0.983832i $$0.557316\pi$$
$$860$$ −7.49912e12 −0.467484
$$861$$ 2.24556e11 0.0139255
$$862$$ −6.98328e12 −0.430801
$$863$$ −2.02596e13 −1.24332 −0.621658 0.783289i $$-0.713540\pi$$
−0.621658 + 0.783289i $$0.713540\pi$$
$$864$$ −2.45858e12 −0.150097
$$865$$ 1.53720e13 0.933596
$$866$$ −7.17488e12 −0.433495
$$867$$ 8.93122e12 0.536815
$$868$$ 3.19118e11 0.0190815
$$869$$ 2.76305e13 1.64362
$$870$$ 8.33961e12 0.493525
$$871$$ 0 0
$$872$$ −5.50768e12 −0.322585
$$873$$ 3.13394e12 0.182611
$$874$$ 1.03194e12 0.0598211
$$875$$ 2.60842e12 0.150432
$$876$$ 1.01933e13 0.584851
$$877$$ 9.14573e10 0.00522059 0.00261030 0.999997i $$-0.499169\pi$$
0.00261030 + 0.999997i $$0.499169\pi$$
$$878$$ 1.05552e13 0.599436
$$879$$ 1.13189e13 0.639519
$$880$$ −3.20135e12 −0.179954
$$881$$ 1.73150e13 0.968347 0.484174 0.874972i $$-0.339120\pi$$
0.484174 + 0.874972i $$0.339120\pi$$
$$882$$ 2.93677e12 0.163404
$$883$$ 1.38781e13 0.768259 0.384130 0.923279i $$-0.374502\pi$$
0.384130 + 0.923279i $$0.374502\pi$$
$$884$$ 0 0
$$885$$ −1.52325e13 −0.834695
$$886$$ −1.51761e13 −0.827387
$$887$$ 1.76586e13 0.957853 0.478926 0.877855i $$-0.341026\pi$$
0.478926 + 0.877855i $$0.341026\pi$$
$$888$$ −2.75212e12 −0.148528
$$889$$ 1.14202e12 0.0613219
$$890$$ 4.42788e11 0.0236560
$$891$$ −2.56796e13 −1.36502
$$892$$ −1.19385e13 −0.631404
$$893$$ −3.33706e12 −0.175603
$$894$$ −1.88335e13 −0.986081
$$895$$ 5.10457e11 0.0265923
$$896$$ −2.55551e11 −0.0132462
$$897$$ 0 0
$$898$$ −9.78821e12 −0.502296
$$899$$ −5.02872e12 −0.256767
$$900$$ −1.42490e12 −0.0723927
$$901$$ −4.11520e12 −0.208032
$$902$$ 1.35837e12 0.0683264
$$903$$ −5.00049e12 −0.250275
$$904$$ −1.10871e13 −0.552152
$$905$$ 4.72754e12 0.234270
$$906$$ 1.07655e13 0.530830
$$907$$ 1.23924e13 0.608025 0.304013 0.952668i $$-0.401673\pi$$
0.304013 + 0.952668i $$0.401673\pi$$
$$908$$ −6.80620e12 −0.332291
$$909$$ 4.92157e12 0.239092
$$910$$ 0 0
$$911$$ −1.38104e13 −0.664313 −0.332157 0.943224i $$-0.607776\pi$$
−0.332157 + 0.943224i $$0.607776\pi$$
$$912$$ 3.22432e12 0.154334
$$913$$ 2.56832e13 1.22329
$$914$$ −6.06452e12 −0.287434
$$915$$ −4.50553e12 −0.212496
$$916$$ −1.02376e13 −0.480474
$$917$$ 2.65241e12 0.123874
$$918$$ 9.29101e12 0.431788
$$919$$ 8.56606e11 0.0396152 0.0198076 0.999804i $$-0.493695\pi$$
0.0198076 + 0.999804i $$0.493695\pi$$
$$920$$ 7.28754e11 0.0335379
$$921$$ 2.83884e13 1.30009
$$922$$ −1.42410e13 −0.649010
$$923$$ 0 0
$$924$$ −2.13469e12 −0.0963412
$$925$$ 5.15223e12 0.231397
$$926$$ 2.12563e13 0.950034
$$927$$ 3.70316e12 0.164708
$$928$$ 4.02700e12 0.178244
$$929$$ −1.29169e13 −0.568968 −0.284484 0.958681i $$-0.591822\pi$$
−0.284484 + 0.958681i $$0.591822\pi$$
$$930$$ −2.84341e12 −0.124642
$$931$$ 1.24409e13 0.542723
$$932$$ 1.67254e12 0.0726116
$$933$$ 1.40134e13 0.605447
$$934$$ −2.65020e13 −1.13951
$$935$$ 1.20980e13 0.517679
$$936$$ 0 0
$$937$$ −3.67867e12 −0.155906 −0.0779530 0.996957i $$-0.524838\pi$$
−0.0779530 + 0.996957i $$0.524838\pi$$
$$938$$ −1.84874e12 −0.0779765
$$939$$ −8.60174e11 −0.0361070
$$940$$ −2.35662e12 −0.0984495
$$941$$ −4.45145e13 −1.85075 −0.925376 0.379051i $$-0.876251\pi$$
−0.925376 + 0.379051i $$0.876251\pi$$
$$942$$ −1.05620e13 −0.437036
$$943$$ −3.09219e11 −0.0127339
$$944$$ −7.35544e12 −0.301463
$$945$$ −1.94196e12 −0.0792130
$$946$$ −3.02486e13 −1.22799
$$947$$ 1.99543e13 0.806236 0.403118 0.915148i $$-0.367926\pi$$
0.403118 + 0.915148i $$0.367926\pi$$
$$948$$ −1.96526e13 −0.790282
$$949$$ 0 0
$$950$$ −6.03625e12 −0.240442
$$951$$ −3.03897e12 −0.120480
$$952$$ 9.65731e11 0.0381057
$$953$$ 7.13202e12 0.280088 0.140044 0.990145i $$-0.455276\pi$$
0.140044 + 0.990145i $$0.455276\pi$$
$$954$$ −1.23704e12 −0.0483523
$$955$$ −2.81264e13 −1.09421
$$956$$ 1.45094e13 0.561809
$$957$$ 3.36388e13 1.29639
$$958$$ −1.72757e13 −0.662660
$$959$$ −2.74398e12 −0.104760
$$960$$ 2.27700e12 0.0865253
$$961$$ −2.47251e13 −0.935152
$$962$$ 0 0
$$963$$ −9.18565e12 −0.344185
$$964$$ −1.18142e12 −0.0440613
$$965$$ −1.12577e13 −0.417905
$$966$$ 4.85941e11 0.0179550
$$967$$ −2.84176e12 −0.104512 −0.0522562 0.998634i $$-0.516641\pi$$
−0.0522562 + 0.998634i $$0.516641\pi$$
$$968$$ −3.25489e12 −0.119151
$$969$$ −1.21848e13 −0.443978
$$970$$ 9.37557e12 0.340036
$$971$$ −3.90309e13 −1.40903 −0.704517 0.709687i $$-0.748836\pi$$
−0.704517 + 0.709687i $$0.748836\pi$$
$$972$$ 6.45045e12 0.231788
$$973$$ 2.05290e12 0.0734278
$$974$$ 2.56878e13 0.914558
$$975$$ 0 0
$$976$$ −2.17561e12 −0.0767463
$$977$$ 2.17556e13 0.763915 0.381958 0.924180i $$-0.375250\pi$$
0.381958 + 0.924180i $$0.375250\pi$$
$$978$$ −2.06781e12 −0.0722745
$$979$$ 1.78604e12 0.0621397
$$980$$ 8.78570e12 0.304270
$$981$$ 6.25664e12 0.215690
$$982$$ −1.26981e13 −0.435749
$$983$$ −4.64998e13 −1.58840 −0.794201 0.607655i $$-0.792110\pi$$
−0.794201 + 0.607655i $$0.792110\pi$$
$$984$$ −9.66159e11 −0.0328527
$$985$$ 7.66548e12 0.259464
$$986$$ −1.52181e13 −0.512761
$$987$$ −1.57142e12 −0.0527065
$$988$$ 0 0
$$989$$ 6.88577e12 0.228860
$$990$$ 3.63669e12 0.120323
$$991$$ −1.55209e13 −0.511194 −0.255597 0.966783i $$-0.582272\pi$$
−0.255597 + 0.966783i $$0.582272\pi$$
$$992$$ −1.37301e12 −0.0450166
$$993$$ −1.66610e13 −0.543786
$$994$$ −5.89741e12 −0.191612
$$995$$ 2.16225e13 0.699361
$$996$$ −1.82675e13 −0.588184
$$997$$ −4.26334e13 −1.36654 −0.683268 0.730167i $$-0.739442\pi$$
−0.683268 + 0.730167i $$0.739442\pi$$
$$998$$ −3.15121e13 −1.00552
$$999$$ −1.00987e13 −0.320791
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 338.10.a.a.1.1 1
13.12 even 2 2.10.a.a.1.1 1
39.38 odd 2 18.10.a.a.1.1 1
52.51 odd 2 16.10.a.d.1.1 1
65.12 odd 4 50.10.b.a.49.2 2
65.38 odd 4 50.10.b.a.49.1 2
65.64 even 2 50.10.a.c.1.1 1
91.12 odd 6 98.10.c.b.67.1 2
91.25 even 6 98.10.c.c.79.1 2
91.38 odd 6 98.10.c.b.79.1 2
91.51 even 6 98.10.c.c.67.1 2
91.90 odd 2 98.10.a.c.1.1 1
104.51 odd 2 64.10.a.b.1.1 1
104.77 even 2 64.10.a.h.1.1 1
117.25 even 6 162.10.c.b.109.1 2
117.38 odd 6 162.10.c.i.109.1 2
117.77 odd 6 162.10.c.i.55.1 2
117.103 even 6 162.10.c.b.55.1 2
143.142 odd 2 242.10.a.a.1.1 1
156.155 even 2 144.10.a.d.1.1 1
208.51 odd 4 256.10.b.e.129.2 2
208.77 even 4 256.10.b.g.129.1 2
208.155 odd 4 256.10.b.e.129.1 2
208.181 even 4 256.10.b.g.129.2 2
260.103 even 4 400.10.c.d.49.2 2
260.207 even 4 400.10.c.d.49.1 2
260.259 odd 2 400.10.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2.10.a.a.1.1 1 13.12 even 2
16.10.a.d.1.1 1 52.51 odd 2
18.10.a.a.1.1 1 39.38 odd 2
50.10.a.c.1.1 1 65.64 even 2
50.10.b.a.49.1 2 65.38 odd 4
50.10.b.a.49.2 2 65.12 odd 4
64.10.a.b.1.1 1 104.51 odd 2
64.10.a.h.1.1 1 104.77 even 2
98.10.a.c.1.1 1 91.90 odd 2
98.10.c.b.67.1 2 91.12 odd 6
98.10.c.b.79.1 2 91.38 odd 6
98.10.c.c.67.1 2 91.51 even 6
98.10.c.c.79.1 2 91.25 even 6
144.10.a.d.1.1 1 156.155 even 2
162.10.c.b.55.1 2 117.103 even 6
162.10.c.b.109.1 2 117.25 even 6
162.10.c.i.55.1 2 117.77 odd 6
162.10.c.i.109.1 2 117.38 odd 6
242.10.a.a.1.1 1 143.142 odd 2
256.10.b.e.129.1 2 208.155 odd 4
256.10.b.e.129.2 2 208.51 odd 4
256.10.b.g.129.1 2 208.77 even 4
256.10.b.g.129.2 2 208.181 even 4
338.10.a.a.1.1 1 1.1 even 1 trivial
400.10.a.b.1.1 1 260.259 odd 2
400.10.c.d.49.1 2 260.207 even 4
400.10.c.d.49.2 2 260.103 even 4