# Properties

 Label 26.8.a.a.1.1 Level $26$ Weight $8$ Character 26.1 Self dual yes Analytic conductor $8.122$ 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] = [26,8,Mod(1,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("26.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 26.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: $$8.12201066259$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 26.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-8.00000 q^{2} -39.0000 q^{3} +64.0000 q^{4} +385.000 q^{5} +312.000 q^{6} -293.000 q^{7} -512.000 q^{8} -666.000 q^{9} +O(q^{10})$$ $$q-8.00000 q^{2} -39.0000 q^{3} +64.0000 q^{4} +385.000 q^{5} +312.000 q^{6} -293.000 q^{7} -512.000 q^{8} -666.000 q^{9} -3080.00 q^{10} -5402.00 q^{11} -2496.00 q^{12} +2197.00 q^{13} +2344.00 q^{14} -15015.0 q^{15} +4096.00 q^{16} -21011.0 q^{17} +5328.00 q^{18} -27326.0 q^{19} +24640.0 q^{20} +11427.0 q^{21} +43216.0 q^{22} -63072.0 q^{23} +19968.0 q^{24} +70100.0 q^{25} -17576.0 q^{26} +111267. q^{27} -18752.0 q^{28} +122238. q^{29} +120120. q^{30} -208396. q^{31} -32768.0 q^{32} +210678. q^{33} +168088. q^{34} -112805. q^{35} -42624.0 q^{36} -442379. q^{37} +218608. q^{38} -85683.0 q^{39} -197120. q^{40} +58000.0 q^{41} -91416.0 q^{42} -202025. q^{43} -345728. q^{44} -256410. q^{45} +504576. q^{46} +588511. q^{47} -159744. q^{48} -737694. q^{49} -560800. q^{50} +819429. q^{51} +140608. q^{52} +1.68434e6 q^{53} -890136. q^{54} -2.07977e6 q^{55} +150016. q^{56} +1.06571e6 q^{57} -977904. q^{58} -442630. q^{59} -960960. q^{60} -1.08361e6 q^{61} +1.66717e6 q^{62} +195138. q^{63} +262144. q^{64} +845845. q^{65} -1.68542e6 q^{66} +3.44349e6 q^{67} -1.34470e6 q^{68} +2.45981e6 q^{69} +902440. q^{70} +2.08470e6 q^{71} +340992. q^{72} +5.93789e6 q^{73} +3.53903e6 q^{74} -2.73390e6 q^{75} -1.74886e6 q^{76} +1.58279e6 q^{77} +685464. q^{78} -6.60926e6 q^{79} +1.57696e6 q^{80} -2.88287e6 q^{81} -464000. q^{82} -142740. q^{83} +731328. q^{84} -8.08924e6 q^{85} +1.61620e6 q^{86} -4.76728e6 q^{87} +2.76582e6 q^{88} -6.98529e6 q^{89} +2.05128e6 q^{90} -643721. q^{91} -4.03661e6 q^{92} +8.12744e6 q^{93} -4.70809e6 q^{94} -1.05205e7 q^{95} +1.27795e6 q^{96} -200762. q^{97} +5.90155e6 q^{98} +3.59773e6 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −8.00000 −0.707107
$$3$$ −39.0000 −0.833950 −0.416975 0.908918i $$-0.636910\pi$$
−0.416975 + 0.908918i $$0.636910\pi$$
$$4$$ 64.0000 0.500000
$$5$$ 385.000 1.37742 0.688709 0.725038i $$-0.258178\pi$$
0.688709 + 0.725038i $$0.258178\pi$$
$$6$$ 312.000 0.589692
$$7$$ −293.000 −0.322868 −0.161434 0.986884i $$-0.551612\pi$$
−0.161434 + 0.986884i $$0.551612\pi$$
$$8$$ −512.000 −0.353553
$$9$$ −666.000 −0.304527
$$10$$ −3080.00 −0.973982
$$11$$ −5402.00 −1.22371 −0.611857 0.790968i $$-0.709577\pi$$
−0.611857 + 0.790968i $$0.709577\pi$$
$$12$$ −2496.00 −0.416975
$$13$$ 2197.00 0.277350
$$14$$ 2344.00 0.228302
$$15$$ −15015.0 −1.14870
$$16$$ 4096.00 0.250000
$$17$$ −21011.0 −1.03723 −0.518616 0.855008i $$-0.673552\pi$$
−0.518616 + 0.855008i $$0.673552\pi$$
$$18$$ 5328.00 0.215333
$$19$$ −27326.0 −0.913984 −0.456992 0.889471i $$-0.651073\pi$$
−0.456992 + 0.889471i $$0.651073\pi$$
$$20$$ 24640.0 0.688709
$$21$$ 11427.0 0.269256
$$22$$ 43216.0 0.865297
$$23$$ −63072.0 −1.08091 −0.540455 0.841373i $$-0.681748\pi$$
−0.540455 + 0.841373i $$0.681748\pi$$
$$24$$ 19968.0 0.294846
$$25$$ 70100.0 0.897280
$$26$$ −17576.0 −0.196116
$$27$$ 111267. 1.08791
$$28$$ −18752.0 −0.161434
$$29$$ 122238. 0.930708 0.465354 0.885125i $$-0.345927\pi$$
0.465354 + 0.885125i $$0.345927\pi$$
$$30$$ 120120. 0.812252
$$31$$ −208396. −1.25639 −0.628194 0.778057i $$-0.716205\pi$$
−0.628194 + 0.778057i $$0.716205\pi$$
$$32$$ −32768.0 −0.176777
$$33$$ 210678. 1.02052
$$34$$ 168088. 0.733433
$$35$$ −112805. −0.444724
$$36$$ −42624.0 −0.152263
$$37$$ −442379. −1.43578 −0.717891 0.696156i $$-0.754892\pi$$
−0.717891 + 0.696156i $$0.754892\pi$$
$$38$$ 218608. 0.646284
$$39$$ −85683.0 −0.231296
$$40$$ −197120. −0.486991
$$41$$ 58000.0 0.131427 0.0657135 0.997839i $$-0.479068\pi$$
0.0657135 + 0.997839i $$0.479068\pi$$
$$42$$ −91416.0 −0.190392
$$43$$ −202025. −0.387494 −0.193747 0.981051i $$-0.562064\pi$$
−0.193747 + 0.981051i $$0.562064\pi$$
$$44$$ −345728. −0.611857
$$45$$ −256410. −0.419461
$$46$$ 504576. 0.764318
$$47$$ 588511. 0.826822 0.413411 0.910545i $$-0.364337\pi$$
0.413411 + 0.910545i $$0.364337\pi$$
$$48$$ −159744. −0.208488
$$49$$ −737694. −0.895757
$$50$$ −560800. −0.634473
$$51$$ 819429. 0.864999
$$52$$ 140608. 0.138675
$$53$$ 1.68434e6 1.55404 0.777022 0.629474i $$-0.216729\pi$$
0.777022 + 0.629474i $$0.216729\pi$$
$$54$$ −890136. −0.769269
$$55$$ −2.07977e6 −1.68557
$$56$$ 150016. 0.114151
$$57$$ 1.06571e6 0.762217
$$58$$ −977904. −0.658110
$$59$$ −442630. −0.280581 −0.140291 0.990110i $$-0.544804\pi$$
−0.140291 + 0.990110i $$0.544804\pi$$
$$60$$ −960960. −0.574349
$$61$$ −1.08361e6 −0.611248 −0.305624 0.952152i $$-0.598865\pi$$
−0.305624 + 0.952152i $$0.598865\pi$$
$$62$$ 1.66717e6 0.888400
$$63$$ 195138. 0.0983218
$$64$$ 262144. 0.125000
$$65$$ 845845. 0.382027
$$66$$ −1.68542e6 −0.721615
$$67$$ 3.44349e6 1.39874 0.699369 0.714761i $$-0.253464\pi$$
0.699369 + 0.714761i $$0.253464\pi$$
$$68$$ −1.34470e6 −0.518616
$$69$$ 2.45981e6 0.901425
$$70$$ 902440. 0.314467
$$71$$ 2.08470e6 0.691258 0.345629 0.938371i $$-0.387666\pi$$
0.345629 + 0.938371i $$0.387666\pi$$
$$72$$ 340992. 0.107666
$$73$$ 5.93789e6 1.78650 0.893248 0.449564i $$-0.148421\pi$$
0.893248 + 0.449564i $$0.148421\pi$$
$$74$$ 3.53903e6 1.01525
$$75$$ −2.73390e6 −0.748287
$$76$$ −1.74886e6 −0.456992
$$77$$ 1.58279e6 0.395098
$$78$$ 685464. 0.163551
$$79$$ −6.60926e6 −1.50820 −0.754098 0.656762i $$-0.771926\pi$$
−0.754098 + 0.656762i $$0.771926\pi$$
$$80$$ 1.57696e6 0.344354
$$81$$ −2.88287e6 −0.602737
$$82$$ −464000. −0.0929329
$$83$$ −142740. −0.0274014 −0.0137007 0.999906i $$-0.504361\pi$$
−0.0137007 + 0.999906i $$0.504361\pi$$
$$84$$ 731328. 0.134628
$$85$$ −8.08924e6 −1.42870
$$86$$ 1.61620e6 0.274000
$$87$$ −4.76728e6 −0.776164
$$88$$ 2.76582e6 0.432648
$$89$$ −6.98529e6 −1.05031 −0.525157 0.851005i $$-0.675993\pi$$
−0.525157 + 0.851005i $$0.675993\pi$$
$$90$$ 2.05128e6 0.296603
$$91$$ −643721. −0.0895474
$$92$$ −4.03661e6 −0.540455
$$93$$ 8.12744e6 1.04776
$$94$$ −4.70809e6 −0.584652
$$95$$ −1.05205e7 −1.25894
$$96$$ 1.27795e6 0.147423
$$97$$ −200762. −0.0223347 −0.0111674 0.999938i $$-0.503555\pi$$
−0.0111674 + 0.999938i $$0.503555\pi$$
$$98$$ 5.90155e6 0.633395
$$99$$ 3.59773e6 0.372654
$$100$$ 4.48640e6 0.448640
$$101$$ −5.42144e6 −0.523588 −0.261794 0.965124i $$-0.584314\pi$$
−0.261794 + 0.965124i $$0.584314\pi$$
$$102$$ −6.55543e6 −0.611647
$$103$$ −1.71897e7 −1.55002 −0.775011 0.631948i $$-0.782255\pi$$
−0.775011 + 0.631948i $$0.782255\pi$$
$$104$$ −1.12486e6 −0.0980581
$$105$$ 4.39940e6 0.370877
$$106$$ −1.34747e7 −1.09887
$$107$$ 1.23582e7 0.975242 0.487621 0.873055i $$-0.337865\pi$$
0.487621 + 0.873055i $$0.337865\pi$$
$$108$$ 7.12109e6 0.543955
$$109$$ 1.70569e7 1.26156 0.630778 0.775964i $$-0.282736\pi$$
0.630778 + 0.775964i $$0.282736\pi$$
$$110$$ 1.66382e7 1.19188
$$111$$ 1.72528e7 1.19737
$$112$$ −1.20013e6 −0.0807169
$$113$$ 2.11250e7 1.37728 0.688639 0.725104i $$-0.258208\pi$$
0.688639 + 0.725104i $$0.258208\pi$$
$$114$$ −8.52571e6 −0.538969
$$115$$ −2.42827e7 −1.48886
$$116$$ 7.82323e6 0.465354
$$117$$ −1.46320e6 −0.0844605
$$118$$ 3.54104e6 0.198401
$$119$$ 6.15622e6 0.334888
$$120$$ 7.68768e6 0.406126
$$121$$ 9.69443e6 0.497478
$$122$$ 8.66886e6 0.432218
$$123$$ −2.26200e6 −0.109604
$$124$$ −1.33373e7 −0.628194
$$125$$ −3.08962e6 −0.141488
$$126$$ −1.56110e6 −0.0695240
$$127$$ −3.24008e7 −1.40360 −0.701800 0.712374i $$-0.747620\pi$$
−0.701800 + 0.712374i $$0.747620\pi$$
$$128$$ −2.09715e6 −0.0883883
$$129$$ 7.87898e6 0.323151
$$130$$ −6.76676e6 −0.270134
$$131$$ −2.64669e7 −1.02862 −0.514308 0.857605i $$-0.671951\pi$$
−0.514308 + 0.857605i $$0.671951\pi$$
$$132$$ 1.34834e7 0.510259
$$133$$ 8.00652e6 0.295096
$$134$$ −2.75479e7 −0.989057
$$135$$ 4.28378e7 1.49851
$$136$$ 1.07576e7 0.366717
$$137$$ 5.36201e7 1.78158 0.890791 0.454413i $$-0.150151\pi$$
0.890791 + 0.454413i $$0.150151\pi$$
$$138$$ −1.96785e7 −0.637403
$$139$$ 7.58784e6 0.239644 0.119822 0.992795i $$-0.461768\pi$$
0.119822 + 0.992795i $$0.461768\pi$$
$$140$$ −7.21952e6 −0.222362
$$141$$ −2.29519e7 −0.689529
$$142$$ −1.66776e7 −0.488793
$$143$$ −1.18682e7 −0.339397
$$144$$ −2.72794e6 −0.0761317
$$145$$ 4.70616e7 1.28197
$$146$$ −4.75031e7 −1.26324
$$147$$ 2.87701e7 0.747016
$$148$$ −2.83123e7 −0.717891
$$149$$ −5.70297e7 −1.41237 −0.706187 0.708026i $$-0.749586\pi$$
−0.706187 + 0.708026i $$0.749586\pi$$
$$150$$ 2.18712e7 0.529119
$$151$$ −2.00648e7 −0.474259 −0.237130 0.971478i $$-0.576207\pi$$
−0.237130 + 0.971478i $$0.576207\pi$$
$$152$$ 1.39909e7 0.323142
$$153$$ 1.39933e7 0.315865
$$154$$ −1.26623e7 −0.279376
$$155$$ −8.02325e7 −1.73057
$$156$$ −5.48371e6 −0.115648
$$157$$ −3.15314e7 −0.650272 −0.325136 0.945667i $$-0.605410\pi$$
−0.325136 + 0.945667i $$0.605410\pi$$
$$158$$ 5.28740e7 1.06646
$$159$$ −6.56891e7 −1.29600
$$160$$ −1.26157e7 −0.243495
$$161$$ 1.84801e7 0.348991
$$162$$ 2.30630e7 0.426199
$$163$$ −3.13938e7 −0.567789 −0.283895 0.958855i $$-0.591627\pi$$
−0.283895 + 0.958855i $$0.591627\pi$$
$$164$$ 3.71200e6 0.0657135
$$165$$ 8.11110e7 1.40568
$$166$$ 1.14192e6 0.0193757
$$167$$ 9.22170e7 1.53216 0.766079 0.642747i $$-0.222205\pi$$
0.766079 + 0.642747i $$0.222205\pi$$
$$168$$ −5.85062e6 −0.0951962
$$169$$ 4.82681e6 0.0769231
$$170$$ 6.47139e7 1.01024
$$171$$ 1.81991e7 0.278332
$$172$$ −1.29296e7 −0.193747
$$173$$ −6.57015e7 −0.964748 −0.482374 0.875965i $$-0.660225\pi$$
−0.482374 + 0.875965i $$0.660225\pi$$
$$174$$ 3.81383e7 0.548831
$$175$$ −2.05393e7 −0.289703
$$176$$ −2.21266e7 −0.305929
$$177$$ 1.72626e7 0.233991
$$178$$ 5.58823e7 0.742684
$$179$$ −3.20402e6 −0.0417551 −0.0208776 0.999782i $$-0.506646\pi$$
−0.0208776 + 0.999782i $$0.506646\pi$$
$$180$$ −1.64102e7 −0.209730
$$181$$ −4.45759e7 −0.558760 −0.279380 0.960181i $$-0.590129\pi$$
−0.279380 + 0.960181i $$0.590129\pi$$
$$182$$ 5.14977e6 0.0633195
$$183$$ 4.22607e7 0.509751
$$184$$ 3.22929e7 0.382159
$$185$$ −1.70316e8 −1.97767
$$186$$ −6.50196e7 −0.740881
$$187$$ 1.13501e8 1.26927
$$188$$ 3.76647e7 0.413411
$$189$$ −3.26012e7 −0.351251
$$190$$ 8.41641e7 0.890203
$$191$$ 1.86394e8 1.93559 0.967797 0.251733i $$-0.0810004\pi$$
0.967797 + 0.251733i $$0.0810004\pi$$
$$192$$ −1.02236e7 −0.104244
$$193$$ −1.52927e8 −1.53120 −0.765602 0.643314i $$-0.777559\pi$$
−0.765602 + 0.643314i $$0.777559\pi$$
$$194$$ 1.60610e6 0.0157930
$$195$$ −3.29880e7 −0.318592
$$196$$ −4.72124e7 −0.447878
$$197$$ 9.51837e7 0.887015 0.443507 0.896271i $$-0.353734\pi$$
0.443507 + 0.896271i $$0.353734\pi$$
$$198$$ −2.87819e7 −0.263506
$$199$$ 1.78585e8 1.60642 0.803212 0.595693i $$-0.203122\pi$$
0.803212 + 0.595693i $$0.203122\pi$$
$$200$$ −3.58912e7 −0.317236
$$201$$ −1.34296e8 −1.16648
$$202$$ 4.33715e7 0.370232
$$203$$ −3.58157e7 −0.300495
$$204$$ 5.24435e7 0.432500
$$205$$ 2.23300e7 0.181030
$$206$$ 1.37517e8 1.09603
$$207$$ 4.20060e7 0.329166
$$208$$ 8.99891e6 0.0693375
$$209$$ 1.47615e8 1.11846
$$210$$ −3.51952e7 −0.262250
$$211$$ −1.33235e8 −0.976406 −0.488203 0.872730i $$-0.662348\pi$$
−0.488203 + 0.872730i $$0.662348\pi$$
$$212$$ 1.07798e8 0.777022
$$213$$ −8.13035e7 −0.576475
$$214$$ −9.88657e7 −0.689600
$$215$$ −7.77796e7 −0.533742
$$216$$ −5.69687e7 −0.384634
$$217$$ 6.10600e7 0.405647
$$218$$ −1.36455e8 −0.892054
$$219$$ −2.31578e8 −1.48985
$$220$$ −1.33105e8 −0.842783
$$221$$ −4.61612e7 −0.287676
$$222$$ −1.38022e8 −0.846669
$$223$$ 1.19394e8 0.720969 0.360484 0.932765i $$-0.382611\pi$$
0.360484 + 0.932765i $$0.382611\pi$$
$$224$$ 9.60102e6 0.0570755
$$225$$ −4.66866e7 −0.273246
$$226$$ −1.69000e8 −0.973883
$$227$$ 1.13656e7 0.0644911 0.0322456 0.999480i $$-0.489734\pi$$
0.0322456 + 0.999480i $$0.489734\pi$$
$$228$$ 6.82057e7 0.381109
$$229$$ −1.46559e7 −0.0806470 −0.0403235 0.999187i $$-0.512839\pi$$
−0.0403235 + 0.999187i $$0.512839\pi$$
$$230$$ 1.94262e8 1.05279
$$231$$ −6.17287e7 −0.329492
$$232$$ −6.25859e7 −0.329055
$$233$$ −2.46924e8 −1.27885 −0.639423 0.768855i $$-0.720827\pi$$
−0.639423 + 0.768855i $$0.720827\pi$$
$$234$$ 1.17056e7 0.0597226
$$235$$ 2.26577e8 1.13888
$$236$$ −2.83283e7 −0.140291
$$237$$ 2.57761e8 1.25776
$$238$$ −4.92498e7 −0.236802
$$239$$ −1.61239e7 −0.0763971 −0.0381985 0.999270i $$-0.512162\pi$$
−0.0381985 + 0.999270i $$0.512162\pi$$
$$240$$ −6.15014e7 −0.287175
$$241$$ 1.14256e8 0.525798 0.262899 0.964823i $$-0.415321\pi$$
0.262899 + 0.964823i $$0.415321\pi$$
$$242$$ −7.75555e7 −0.351770
$$243$$ −1.30909e8 −0.585258
$$244$$ −6.93509e7 −0.305624
$$245$$ −2.84012e8 −1.23383
$$246$$ 1.80960e7 0.0775014
$$247$$ −6.00352e7 −0.253493
$$248$$ 1.06699e8 0.444200
$$249$$ 5.56686e6 0.0228514
$$250$$ 2.47170e7 0.100047
$$251$$ 2.22704e8 0.888935 0.444467 0.895795i $$-0.353393\pi$$
0.444467 + 0.895795i $$0.353393\pi$$
$$252$$ 1.24888e7 0.0491609
$$253$$ 3.40715e8 1.32272
$$254$$ 2.59207e8 0.992494
$$255$$ 3.15480e8 1.19147
$$256$$ 1.67772e7 0.0625000
$$257$$ 2.82302e8 1.03741 0.518703 0.854955i $$-0.326415\pi$$
0.518703 + 0.854955i $$0.326415\pi$$
$$258$$ −6.30318e7 −0.228502
$$259$$ 1.29617e8 0.463567
$$260$$ 5.41341e7 0.191013
$$261$$ −8.14105e7 −0.283425
$$262$$ 2.11735e8 0.727342
$$263$$ −2.36490e8 −0.801619 −0.400809 0.916162i $$-0.631271\pi$$
−0.400809 + 0.916162i $$0.631271\pi$$
$$264$$ −1.07867e8 −0.360807
$$265$$ 6.48469e8 2.14057
$$266$$ −6.40521e7 −0.208664
$$267$$ 2.72426e8 0.875910
$$268$$ 2.20383e8 0.699369
$$269$$ −4.82172e8 −1.51032 −0.755160 0.655541i $$-0.772441\pi$$
−0.755160 + 0.655541i $$0.772441\pi$$
$$270$$ −3.42702e8 −1.05960
$$271$$ 4.66372e8 1.42344 0.711721 0.702462i $$-0.247916\pi$$
0.711721 + 0.702462i $$0.247916\pi$$
$$272$$ −8.60611e7 −0.259308
$$273$$ 2.51051e7 0.0746781
$$274$$ −4.28961e8 −1.25977
$$275$$ −3.78680e8 −1.09801
$$276$$ 1.57428e8 0.450712
$$277$$ −1.88709e8 −0.533475 −0.266738 0.963769i $$-0.585946\pi$$
−0.266738 + 0.963769i $$0.585946\pi$$
$$278$$ −6.07027e7 −0.169454
$$279$$ 1.38792e8 0.382603
$$280$$ 5.77562e7 0.157234
$$281$$ −7.15402e8 −1.92344 −0.961718 0.274040i $$-0.911640\pi$$
−0.961718 + 0.274040i $$0.911640\pi$$
$$282$$ 1.83615e8 0.487570
$$283$$ −4.04602e8 −1.06115 −0.530573 0.847639i $$-0.678023\pi$$
−0.530573 + 0.847639i $$0.678023\pi$$
$$284$$ 1.33421e8 0.345629
$$285$$ 4.10300e8 1.04989
$$286$$ 9.49456e7 0.239990
$$287$$ −1.69940e7 −0.0424335
$$288$$ 2.18235e7 0.0538332
$$289$$ 3.11234e7 0.0758482
$$290$$ −3.76493e8 −0.906492
$$291$$ 7.82972e6 0.0186260
$$292$$ 3.80025e8 0.893248
$$293$$ −8.11321e8 −1.88433 −0.942163 0.335156i $$-0.891211\pi$$
−0.942163 + 0.335156i $$0.891211\pi$$
$$294$$ −2.30161e8 −0.528220
$$295$$ −1.70413e8 −0.386478
$$296$$ 2.26498e8 0.507626
$$297$$ −6.01064e8 −1.33129
$$298$$ 4.56238e8 0.998699
$$299$$ −1.38569e8 −0.299790
$$300$$ −1.74970e8 −0.374144
$$301$$ 5.91933e7 0.125109
$$302$$ 1.60519e8 0.335352
$$303$$ 2.11436e8 0.436646
$$304$$ −1.11927e8 −0.228496
$$305$$ −4.17189e8 −0.841945
$$306$$ −1.11947e8 −0.223350
$$307$$ 4.60958e8 0.909237 0.454618 0.890686i $$-0.349776\pi$$
0.454618 + 0.890686i $$0.349776\pi$$
$$308$$ 1.01298e8 0.197549
$$309$$ 6.70398e8 1.29264
$$310$$ 6.41860e8 1.22370
$$311$$ −2.87718e8 −0.542383 −0.271192 0.962525i $$-0.587418\pi$$
−0.271192 + 0.962525i $$0.587418\pi$$
$$312$$ 4.38697e7 0.0817756
$$313$$ −9.56179e8 −1.76252 −0.881260 0.472632i $$-0.843304\pi$$
−0.881260 + 0.472632i $$0.843304\pi$$
$$314$$ 2.52252e8 0.459812
$$315$$ 7.51281e7 0.135430
$$316$$ −4.22992e8 −0.754098
$$317$$ 4.92761e8 0.868818 0.434409 0.900716i $$-0.356957\pi$$
0.434409 + 0.900716i $$0.356957\pi$$
$$318$$ 5.25513e8 0.916407
$$319$$ −6.60330e8 −1.13892
$$320$$ 1.00925e8 0.172177
$$321$$ −4.81970e8 −0.813303
$$322$$ −1.47841e8 −0.246774
$$323$$ 5.74147e8 0.948012
$$324$$ −1.84504e8 −0.301368
$$325$$ 1.54010e8 0.248861
$$326$$ 2.51150e8 0.401488
$$327$$ −6.65218e8 −1.05207
$$328$$ −2.96960e7 −0.0464665
$$329$$ −1.72434e8 −0.266954
$$330$$ −6.48888e8 −0.993965
$$331$$ −4.83358e8 −0.732607 −0.366304 0.930495i $$-0.619377\pi$$
−0.366304 + 0.930495i $$0.619377\pi$$
$$332$$ −9.13536e6 −0.0137007
$$333$$ 2.94624e8 0.437234
$$334$$ −7.37736e8 −1.08340
$$335$$ 1.32574e9 1.92665
$$336$$ 4.68050e7 0.0673139
$$337$$ 1.30823e9 1.86200 0.930998 0.365025i $$-0.118940\pi$$
0.930998 + 0.365025i $$0.118940\pi$$
$$338$$ −3.86145e7 −0.0543928
$$339$$ −8.23874e8 −1.14858
$$340$$ −5.17711e8 −0.714350
$$341$$ 1.12576e9 1.53746
$$342$$ −1.45593e8 −0.196811
$$343$$ 4.57442e8 0.612078
$$344$$ 1.03437e8 0.137000
$$345$$ 9.47026e8 1.24164
$$346$$ 5.25612e8 0.682180
$$347$$ −8.94842e8 −1.14972 −0.574861 0.818251i $$-0.694944\pi$$
−0.574861 + 0.818251i $$0.694944\pi$$
$$348$$ −3.05106e8 −0.388082
$$349$$ −5.41626e8 −0.682041 −0.341020 0.940056i $$-0.610772\pi$$
−0.341020 + 0.940056i $$0.610772\pi$$
$$350$$ 1.64314e8 0.204851
$$351$$ 2.44454e8 0.301732
$$352$$ 1.77013e8 0.216324
$$353$$ 2.25334e8 0.272656 0.136328 0.990664i $$-0.456470\pi$$
0.136328 + 0.990664i $$0.456470\pi$$
$$354$$ −1.38101e8 −0.165457
$$355$$ 8.02611e8 0.952152
$$356$$ −4.47058e8 −0.525157
$$357$$ −2.40093e8 −0.279280
$$358$$ 2.56322e7 0.0295253
$$359$$ −4.38763e8 −0.500495 −0.250247 0.968182i $$-0.580512\pi$$
−0.250247 + 0.968182i $$0.580512\pi$$
$$360$$ 1.31282e8 0.148302
$$361$$ −1.47161e8 −0.164634
$$362$$ 3.56607e8 0.395103
$$363$$ −3.78083e8 −0.414872
$$364$$ −4.11981e7 −0.0447737
$$365$$ 2.28609e9 2.46075
$$366$$ −3.38086e8 −0.360448
$$367$$ 8.08568e8 0.853857 0.426929 0.904285i $$-0.359596\pi$$
0.426929 + 0.904285i $$0.359596\pi$$
$$368$$ −2.58343e8 −0.270227
$$369$$ −3.86280e7 −0.0400230
$$370$$ 1.36253e9 1.39843
$$371$$ −4.93510e8 −0.501750
$$372$$ 5.20156e8 0.523882
$$373$$ −1.17884e9 −1.17618 −0.588092 0.808794i $$-0.700121\pi$$
−0.588092 + 0.808794i $$0.700121\pi$$
$$374$$ −9.08011e8 −0.897513
$$375$$ 1.20495e8 0.117994
$$376$$ −3.01318e8 −0.292326
$$377$$ 2.68557e8 0.258132
$$378$$ 2.60810e8 0.248372
$$379$$ −1.79168e9 −1.69053 −0.845266 0.534345i $$-0.820558\pi$$
−0.845266 + 0.534345i $$0.820558\pi$$
$$380$$ −6.73313e8 −0.629469
$$381$$ 1.26363e9 1.17053
$$382$$ −1.49115e9 −1.36867
$$383$$ −1.19775e9 −1.08936 −0.544680 0.838644i $$-0.683349\pi$$
−0.544680 + 0.838644i $$0.683349\pi$$
$$384$$ 8.17889e7 0.0737115
$$385$$ 6.09373e8 0.544215
$$386$$ 1.22341e9 1.08273
$$387$$ 1.34549e8 0.118002
$$388$$ −1.28488e7 −0.0111674
$$389$$ −1.43672e8 −0.123751 −0.0618754 0.998084i $$-0.519708\pi$$
−0.0618754 + 0.998084i $$0.519708\pi$$
$$390$$ 2.63904e8 0.225278
$$391$$ 1.32521e9 1.12115
$$392$$ 3.77699e8 0.316698
$$393$$ 1.03221e9 0.857815
$$394$$ −7.61470e8 −0.627214
$$395$$ −2.54456e9 −2.07742
$$396$$ 2.30255e8 0.186327
$$397$$ −6.17334e8 −0.495169 −0.247584 0.968866i $$-0.579637\pi$$
−0.247584 + 0.968866i $$0.579637\pi$$
$$398$$ −1.42868e9 −1.13591
$$399$$ −3.12254e8 −0.246095
$$400$$ 2.87130e8 0.224320
$$401$$ 1.13305e9 0.877491 0.438746 0.898611i $$-0.355423\pi$$
0.438746 + 0.898611i $$0.355423\pi$$
$$402$$ 1.07437e9 0.824825
$$403$$ −4.57846e8 −0.348459
$$404$$ −3.46972e8 −0.261794
$$405$$ −1.10991e9 −0.830220
$$406$$ 2.86526e8 0.212482
$$407$$ 2.38973e9 1.75699
$$408$$ −4.19548e8 −0.305823
$$409$$ −1.04283e9 −0.753670 −0.376835 0.926280i $$-0.622988\pi$$
−0.376835 + 0.926280i $$0.622988\pi$$
$$410$$ −1.78640e8 −0.128007
$$411$$ −2.09119e9 −1.48575
$$412$$ −1.10014e9 −0.775011
$$413$$ 1.29691e8 0.0905906
$$414$$ −3.36048e8 −0.232755
$$415$$ −5.49549e7 −0.0377431
$$416$$ −7.19913e7 −0.0490290
$$417$$ −2.95926e8 −0.199851
$$418$$ −1.18092e9 −0.790867
$$419$$ 7.09302e8 0.471066 0.235533 0.971866i $$-0.424316\pi$$
0.235533 + 0.971866i $$0.424316\pi$$
$$420$$ 2.81561e8 0.185439
$$421$$ −1.19877e9 −0.782974 −0.391487 0.920184i $$-0.628039\pi$$
−0.391487 + 0.920184i $$0.628039\pi$$
$$422$$ 1.06588e9 0.690424
$$423$$ −3.91948e8 −0.251789
$$424$$ −8.62380e8 −0.549437
$$425$$ −1.47287e9 −0.930687
$$426$$ 6.50428e8 0.407629
$$427$$ 3.17497e8 0.197352
$$428$$ 7.90926e8 0.487621
$$429$$ 4.62860e8 0.283041
$$430$$ 6.22237e8 0.377412
$$431$$ 9.54153e8 0.574047 0.287024 0.957924i $$-0.407334\pi$$
0.287024 + 0.957924i $$0.407334\pi$$
$$432$$ 4.55750e8 0.271978
$$433$$ −3.81628e8 −0.225908 −0.112954 0.993600i $$-0.536031\pi$$
−0.112954 + 0.993600i $$0.536031\pi$$
$$434$$ −4.88480e8 −0.286836
$$435$$ −1.83540e9 −1.06910
$$436$$ 1.09164e9 0.630778
$$437$$ 1.72351e9 0.987933
$$438$$ 1.85262e9 1.05348
$$439$$ 1.11683e8 0.0630031 0.0315015 0.999504i $$-0.489971\pi$$
0.0315015 + 0.999504i $$0.489971\pi$$
$$440$$ 1.06484e9 0.595938
$$441$$ 4.91304e8 0.272782
$$442$$ 3.69289e8 0.203418
$$443$$ 1.45991e9 0.797837 0.398919 0.916986i $$-0.369386\pi$$
0.398919 + 0.916986i $$0.369386\pi$$
$$444$$ 1.10418e9 0.598685
$$445$$ −2.68934e9 −1.44672
$$446$$ −9.55154e8 −0.509802
$$447$$ 2.22416e9 1.17785
$$448$$ −7.68082e7 −0.0403585
$$449$$ −6.34009e8 −0.330547 −0.165273 0.986248i $$-0.552851\pi$$
−0.165273 + 0.986248i $$0.552851\pi$$
$$450$$ 3.73493e8 0.193214
$$451$$ −3.13316e8 −0.160829
$$452$$ 1.35200e9 0.688639
$$453$$ 7.82528e8 0.395509
$$454$$ −9.09244e7 −0.0456021
$$455$$ −2.47833e8 −0.123344
$$456$$ −5.45646e8 −0.269484
$$457$$ 6.04376e8 0.296211 0.148105 0.988972i $$-0.452683\pi$$
0.148105 + 0.988972i $$0.452683\pi$$
$$458$$ 1.17247e8 0.0570261
$$459$$ −2.33783e9 −1.12841
$$460$$ −1.55409e9 −0.744432
$$461$$ 2.20565e9 1.04853 0.524267 0.851554i $$-0.324339\pi$$
0.524267 + 0.851554i $$0.324339\pi$$
$$462$$ 4.93829e8 0.232986
$$463$$ 1.04925e9 0.491299 0.245650 0.969359i $$-0.420999\pi$$
0.245650 + 0.969359i $$0.420999\pi$$
$$464$$ 5.00687e8 0.232677
$$465$$ 3.12907e9 1.44321
$$466$$ 1.97539e9 0.904281
$$467$$ −2.01461e9 −0.915337 −0.457668 0.889123i $$-0.651315\pi$$
−0.457668 + 0.889123i $$0.651315\pi$$
$$468$$ −9.36449e7 −0.0422303
$$469$$ −1.00894e9 −0.451607
$$470$$ −1.81261e9 −0.805309
$$471$$ 1.22973e9 0.542295
$$472$$ 2.26627e8 0.0992005
$$473$$ 1.09134e9 0.474183
$$474$$ −2.06209e9 −0.889371
$$475$$ −1.91555e9 −0.820099
$$476$$ 3.93998e8 0.167444
$$477$$ −1.12177e9 −0.473248
$$478$$ 1.28991e8 0.0540209
$$479$$ 3.67842e9 1.52928 0.764639 0.644458i $$-0.222917\pi$$
0.764639 + 0.644458i $$0.222917\pi$$
$$480$$ 4.92012e8 0.203063
$$481$$ −9.71907e8 −0.398214
$$482$$ −9.14048e8 −0.371796
$$483$$ −7.20724e8 −0.291041
$$484$$ 6.20444e8 0.248739
$$485$$ −7.72934e7 −0.0307642
$$486$$ 1.04727e9 0.413840
$$487$$ −1.91497e8 −0.0751294 −0.0375647 0.999294i $$-0.511960\pi$$
−0.0375647 + 0.999294i $$0.511960\pi$$
$$488$$ 5.54807e8 0.216109
$$489$$ 1.22436e9 0.473508
$$490$$ 2.27210e9 0.872450
$$491$$ −3.22321e8 −0.122886 −0.0614431 0.998111i $$-0.519570\pi$$
−0.0614431 + 0.998111i $$0.519570\pi$$
$$492$$ −1.44768e8 −0.0548018
$$493$$ −2.56834e9 −0.965359
$$494$$ 4.80282e8 0.179247
$$495$$ 1.38513e9 0.513300
$$496$$ −8.53590e8 −0.314097
$$497$$ −6.10819e8 −0.223185
$$498$$ −4.45349e7 −0.0161584
$$499$$ 3.86695e9 1.39321 0.696604 0.717455i $$-0.254693\pi$$
0.696604 + 0.717455i $$0.254693\pi$$
$$500$$ −1.97736e8 −0.0707442
$$501$$ −3.59646e9 −1.27774
$$502$$ −1.78163e9 −0.628572
$$503$$ −3.43814e8 −0.120458 −0.0602290 0.998185i $$-0.519183\pi$$
−0.0602290 + 0.998185i $$0.519183\pi$$
$$504$$ −9.99107e7 −0.0347620
$$505$$ −2.08725e9 −0.721199
$$506$$ −2.72572e9 −0.935307
$$507$$ −1.88246e8 −0.0641500
$$508$$ −2.07365e9 −0.701800
$$509$$ 2.11533e9 0.710993 0.355497 0.934678i $$-0.384312\pi$$
0.355497 + 0.934678i $$0.384312\pi$$
$$510$$ −2.52384e9 −0.842493
$$511$$ −1.73980e9 −0.576802
$$512$$ −1.34218e8 −0.0441942
$$513$$ −3.04048e9 −0.994333
$$514$$ −2.25842e9 −0.733556
$$515$$ −6.61803e9 −2.13503
$$516$$ 5.04254e8 0.161576
$$517$$ −3.17914e9 −1.01179
$$518$$ −1.03694e9 −0.327792
$$519$$ 2.56236e9 0.804552
$$520$$ −4.33073e8 −0.135067
$$521$$ −1.40622e9 −0.435634 −0.217817 0.975990i $$-0.569894\pi$$
−0.217817 + 0.975990i $$0.569894\pi$$
$$522$$ 6.51284e8 0.200412
$$523$$ 2.18120e9 0.666712 0.333356 0.942801i $$-0.391819\pi$$
0.333356 + 0.942801i $$0.391819\pi$$
$$524$$ −1.69388e9 −0.514308
$$525$$ 8.01033e8 0.241598
$$526$$ 1.89192e9 0.566830
$$527$$ 4.37861e9 1.30316
$$528$$ 8.62937e8 0.255129
$$529$$ 5.73252e8 0.168365
$$530$$ −5.18775e9 −1.51361
$$531$$ 2.94792e8 0.0854445
$$532$$ 5.12417e8 0.147548
$$533$$ 1.27426e8 0.0364513
$$534$$ −2.17941e9 −0.619362
$$535$$ 4.75791e9 1.34332
$$536$$ −1.76306e9 −0.494529
$$537$$ 1.24957e8 0.0348217
$$538$$ 3.85737e9 1.06796
$$539$$ 3.98502e9 1.09615
$$540$$ 2.74162e9 0.749254
$$541$$ 2.54634e8 0.0691395 0.0345698 0.999402i $$-0.488994\pi$$
0.0345698 + 0.999402i $$0.488994\pi$$
$$542$$ −3.73097e9 −1.00653
$$543$$ 1.73846e9 0.465978
$$544$$ 6.88488e8 0.183358
$$545$$ 6.56689e9 1.73769
$$546$$ −2.00841e8 −0.0528054
$$547$$ 2.15158e9 0.562085 0.281043 0.959695i $$-0.409320\pi$$
0.281043 + 0.959695i $$0.409320\pi$$
$$548$$ 3.43169e9 0.890791
$$549$$ 7.21683e8 0.186142
$$550$$ 3.02944e9 0.776414
$$551$$ −3.34028e9 −0.850652
$$552$$ −1.25942e9 −0.318702
$$553$$ 1.93651e9 0.486948
$$554$$ 1.50967e9 0.377224
$$555$$ 6.64232e9 1.64928
$$556$$ 4.85622e8 0.119822
$$557$$ −7.71518e9 −1.89170 −0.945852 0.324599i $$-0.894771\pi$$
−0.945852 + 0.324599i $$0.894771\pi$$
$$558$$ −1.11033e9 −0.270542
$$559$$ −4.43849e8 −0.107472
$$560$$ −4.62049e8 −0.111181
$$561$$ −4.42656e9 −1.05851
$$562$$ 5.72321e9 1.36008
$$563$$ −8.12996e7 −0.0192003 −0.00960017 0.999954i $$-0.503056\pi$$
−0.00960017 + 0.999954i $$0.503056\pi$$
$$564$$ −1.46892e9 −0.344764
$$565$$ 8.13312e9 1.89709
$$566$$ 3.23681e9 0.750343
$$567$$ 8.44681e8 0.194604
$$568$$ −1.06737e9 −0.244397
$$569$$ −5.08814e9 −1.15789 −0.578944 0.815367i $$-0.696535\pi$$
−0.578944 + 0.815367i $$0.696535\pi$$
$$570$$ −3.28240e9 −0.742385
$$571$$ −5.61762e9 −1.26277 −0.631387 0.775468i $$-0.717514\pi$$
−0.631387 + 0.775468i $$0.717514\pi$$
$$572$$ −7.59564e8 −0.169699
$$573$$ −7.26935e9 −1.61419
$$574$$ 1.35952e8 0.0300050
$$575$$ −4.42135e9 −0.969878
$$576$$ −1.74588e8 −0.0380658
$$577$$ 4.12728e9 0.894435 0.447218 0.894425i $$-0.352415\pi$$
0.447218 + 0.894425i $$0.352415\pi$$
$$578$$ −2.48988e8 −0.0536328
$$579$$ 5.96415e9 1.27695
$$580$$ 3.01194e9 0.640987
$$581$$ 4.18228e7 0.00884702
$$582$$ −6.26377e7 −0.0131706
$$583$$ −9.09878e9 −1.90171
$$584$$ −3.04020e9 −0.631622
$$585$$ −5.63333e8 −0.116337
$$586$$ 6.49057e9 1.33242
$$587$$ 1.86734e9 0.381056 0.190528 0.981682i $$-0.438980\pi$$
0.190528 + 0.981682i $$0.438980\pi$$
$$588$$ 1.84128e9 0.373508
$$589$$ 5.69463e9 1.14832
$$590$$ 1.36330e9 0.273281
$$591$$ −3.71216e9 −0.739726
$$592$$ −1.81198e9 −0.358945
$$593$$ 3.31544e9 0.652905 0.326453 0.945214i $$-0.394147\pi$$
0.326453 + 0.945214i $$0.394147\pi$$
$$594$$ 4.80851e9 0.941366
$$595$$ 2.37015e9 0.461281
$$596$$ −3.64990e9 −0.706187
$$597$$ −6.96483e9 −1.33968
$$598$$ 1.10855e9 0.211984
$$599$$ 1.93367e9 0.367610 0.183805 0.982963i $$-0.441158\pi$$
0.183805 + 0.982963i $$0.441158\pi$$
$$600$$ 1.39976e9 0.264559
$$601$$ −5.88820e9 −1.10643 −0.553213 0.833040i $$-0.686598\pi$$
−0.553213 + 0.833040i $$0.686598\pi$$
$$602$$ −4.73547e8 −0.0884657
$$603$$ −2.29336e9 −0.425953
$$604$$ −1.28415e9 −0.237130
$$605$$ 3.73236e9 0.685235
$$606$$ −1.69149e9 −0.308756
$$607$$ 7.94197e9 1.44135 0.720673 0.693276i $$-0.243833\pi$$
0.720673 + 0.693276i $$0.243833\pi$$
$$608$$ 8.95418e8 0.161571
$$609$$ 1.39681e9 0.250598
$$610$$ 3.33751e9 0.595345
$$611$$ 1.29296e9 0.229319
$$612$$ 8.95573e8 0.157932
$$613$$ −2.36146e8 −0.0414065 −0.0207033 0.999786i $$-0.506591\pi$$
−0.0207033 + 0.999786i $$0.506591\pi$$
$$614$$ −3.68766e9 −0.642927
$$615$$ −8.70870e8 −0.150970
$$616$$ −8.10386e8 −0.139688
$$617$$ 1.27029e9 0.217723 0.108862 0.994057i $$-0.465279\pi$$
0.108862 + 0.994057i $$0.465279\pi$$
$$618$$ −5.36318e9 −0.914035
$$619$$ 1.63555e9 0.277170 0.138585 0.990351i $$-0.455745\pi$$
0.138585 + 0.990351i $$0.455745\pi$$
$$620$$ −5.13488e9 −0.865285
$$621$$ −7.01783e9 −1.17593
$$622$$ 2.30175e9 0.383523
$$623$$ 2.04669e9 0.339112
$$624$$ −3.50958e8 −0.0578241
$$625$$ −6.66607e9 −1.09217
$$626$$ 7.64943e9 1.24629
$$627$$ −5.75699e9 −0.932736
$$628$$ −2.01801e9 −0.325136
$$629$$ 9.29483e9 1.48924
$$630$$ −6.01025e8 −0.0957636
$$631$$ 1.68242e9 0.266582 0.133291 0.991077i $$-0.457445\pi$$
0.133291 + 0.991077i $$0.457445\pi$$
$$632$$ 3.38394e9 0.533228
$$633$$ 5.19618e9 0.814275
$$634$$ −3.94209e9 −0.614347
$$635$$ −1.24743e10 −1.93334
$$636$$ −4.20410e9 −0.647998
$$637$$ −1.62071e9 −0.248438
$$638$$ 5.28264e9 0.805338
$$639$$ −1.38841e9 −0.210507
$$640$$ −8.07404e8 −0.121748
$$641$$ −1.70575e9 −0.255807 −0.127903 0.991787i $$-0.540825\pi$$
−0.127903 + 0.991787i $$0.540825\pi$$
$$642$$ 3.85576e9 0.575092
$$643$$ −1.45635e9 −0.216036 −0.108018 0.994149i $$-0.534450\pi$$
−0.108018 + 0.994149i $$0.534450\pi$$
$$644$$ 1.18273e9 0.174495
$$645$$ 3.03341e9 0.445114
$$646$$ −4.59317e9 −0.670346
$$647$$ −3.56464e9 −0.517430 −0.258715 0.965954i $$-0.583299\pi$$
−0.258715 + 0.965954i $$0.583299\pi$$
$$648$$ 1.47603e9 0.213100
$$649$$ 2.39109e9 0.343352
$$650$$ −1.23208e9 −0.175971
$$651$$ −2.38134e9 −0.338289
$$652$$ −2.00920e9 −0.283895
$$653$$ −5.86806e9 −0.824705 −0.412352 0.911024i $$-0.635293\pi$$
−0.412352 + 0.911024i $$0.635293\pi$$
$$654$$ 5.32174e9 0.743929
$$655$$ −1.01898e10 −1.41683
$$656$$ 2.37568e8 0.0328567
$$657$$ −3.95463e9 −0.544036
$$658$$ 1.37947e9 0.188765
$$659$$ −2.73239e9 −0.371915 −0.185958 0.982558i $$-0.559539\pi$$
−0.185958 + 0.982558i $$0.559539\pi$$
$$660$$ 5.19111e9 0.702839
$$661$$ 8.50066e9 1.14485 0.572424 0.819958i $$-0.306003\pi$$
0.572424 + 0.819958i $$0.306003\pi$$
$$662$$ 3.86687e9 0.518032
$$663$$ 1.80029e9 0.239908
$$664$$ 7.30829e7 0.00968785
$$665$$ 3.08251e9 0.406470
$$666$$ −2.35700e9 −0.309171
$$667$$ −7.70980e9 −1.00601
$$668$$ 5.90189e9 0.766079
$$669$$ −4.65638e9 −0.601252
$$670$$ −1.06059e10 −1.36235
$$671$$ 5.85365e9 0.747994
$$672$$ −3.74440e8 −0.0475981
$$673$$ −3.85727e7 −0.00487784 −0.00243892 0.999997i $$-0.500776\pi$$
−0.00243892 + 0.999997i $$0.500776\pi$$
$$674$$ −1.04658e10 −1.31663
$$675$$ 7.79982e9 0.976160
$$676$$ 3.08916e8 0.0384615
$$677$$ −7.34428e9 −0.909681 −0.454840 0.890573i $$-0.650304\pi$$
−0.454840 + 0.890573i $$0.650304\pi$$
$$678$$ 6.59100e9 0.812170
$$679$$ 5.88233e7 0.00721116
$$680$$ 4.14169e9 0.505122
$$681$$ −4.43257e8 −0.0537824
$$682$$ −9.00604e9 −1.08715
$$683$$ −7.49577e9 −0.900210 −0.450105 0.892976i $$-0.648613\pi$$
−0.450105 + 0.892976i $$0.648613\pi$$
$$684$$ 1.16474e9 0.139166
$$685$$ 2.06438e10 2.45398
$$686$$ −3.65954e9 −0.432805
$$687$$ 5.71580e8 0.0672556
$$688$$ −8.27494e8 −0.0968736
$$689$$ 3.70049e9 0.431014
$$690$$ −7.57621e9 −0.877971
$$691$$ 1.66382e10 1.91838 0.959188 0.282769i $$-0.0912530\pi$$
0.959188 + 0.282769i $$0.0912530\pi$$
$$692$$ −4.20490e9 −0.482374
$$693$$ −1.05414e9 −0.120318
$$694$$ 7.15873e9 0.812976
$$695$$ 2.92132e9 0.330090
$$696$$ 2.44085e9 0.274415
$$697$$ −1.21864e9 −0.136320
$$698$$ 4.33300e9 0.482275
$$699$$ 9.63005e9 1.06649
$$700$$ −1.31452e9 −0.144851
$$701$$ 3.26804e9 0.358323 0.179161 0.983820i $$-0.442662\pi$$
0.179161 + 0.983820i $$0.442662\pi$$
$$702$$ −1.95563e9 −0.213357
$$703$$ 1.20884e10 1.31228
$$704$$ −1.41610e9 −0.152964
$$705$$ −8.83649e9 −0.949769
$$706$$ −1.80267e9 −0.192797
$$707$$ 1.58848e9 0.169050
$$708$$ 1.10480e9 0.116995
$$709$$ 4.48613e9 0.472727 0.236363 0.971665i $$-0.424044\pi$$
0.236363 + 0.971665i $$0.424044\pi$$
$$710$$ −6.42089e9 −0.673273
$$711$$ 4.40176e9 0.459286
$$712$$ 3.57647e9 0.371342
$$713$$ 1.31440e10 1.35804
$$714$$ 1.92074e9 0.197481
$$715$$ −4.56925e9 −0.467492
$$716$$ −2.05057e8 −0.0208776
$$717$$ 6.28831e8 0.0637114
$$718$$ 3.51011e9 0.353903
$$719$$ 5.42385e9 0.544198 0.272099 0.962269i $$-0.412282\pi$$
0.272099 + 0.962269i $$0.412282\pi$$
$$720$$ −1.05026e9 −0.104865
$$721$$ 5.03658e9 0.500452
$$722$$ 1.17729e9 0.116414
$$723$$ −4.45598e9 −0.438490
$$724$$ −2.85286e9 −0.279380
$$725$$ 8.56888e9 0.835105
$$726$$ 3.02466e9 0.293359
$$727$$ −1.50827e10 −1.45582 −0.727911 0.685672i $$-0.759508\pi$$
−0.727911 + 0.685672i $$0.759508\pi$$
$$728$$ 3.29585e8 0.0316598
$$729$$ 1.14103e10 1.09081
$$730$$ −1.82887e10 −1.74001
$$731$$ 4.24475e9 0.401921
$$732$$ 2.70469e9 0.254875
$$733$$ −6.75596e9 −0.633612 −0.316806 0.948490i $$-0.602610\pi$$
−0.316806 + 0.948490i $$0.602610\pi$$
$$734$$ −6.46854e9 −0.603768
$$735$$ 1.10765e10 1.02895
$$736$$ 2.06674e9 0.191080
$$737$$ −1.86017e10 −1.71166
$$738$$ 3.09024e8 0.0283006
$$739$$ 1.08154e10 0.985797 0.492899 0.870087i $$-0.335937\pi$$
0.492899 + 0.870087i $$0.335937\pi$$
$$740$$ −1.09002e10 −0.988836
$$741$$ 2.34137e9 0.211401
$$742$$ 3.94808e9 0.354791
$$743$$ −3.71897e9 −0.332630 −0.166315 0.986073i $$-0.553187\pi$$
−0.166315 + 0.986073i $$0.553187\pi$$
$$744$$ −4.16125e9 −0.370441
$$745$$ −2.19565e10 −1.94543
$$746$$ 9.43075e9 0.831687
$$747$$ 9.50648e7 0.00834445
$$748$$ 7.26409e9 0.634637
$$749$$ −3.62096e9 −0.314874
$$750$$ −9.63963e8 −0.0834346
$$751$$ −2.15786e10 −1.85902 −0.929510 0.368797i $$-0.879770\pi$$
−0.929510 + 0.368797i $$0.879770\pi$$
$$752$$ 2.41054e9 0.206706
$$753$$ −8.68546e9 −0.741328
$$754$$ −2.14846e9 −0.182527
$$755$$ −7.72496e9 −0.653253
$$756$$ −2.08648e9 −0.175626
$$757$$ 7.42446e9 0.622056 0.311028 0.950401i $$-0.399327\pi$$
0.311028 + 0.950401i $$0.399327\pi$$
$$758$$ 1.43335e10 1.19539
$$759$$ −1.32879e10 −1.10309
$$760$$ 5.38650e9 0.445102
$$761$$ 8.57002e9 0.704913 0.352457 0.935828i $$-0.385346\pi$$
0.352457 + 0.935828i $$0.385346\pi$$
$$762$$ −1.01091e10 −0.827691
$$763$$ −4.99766e9 −0.407315
$$764$$ 1.19292e10 0.967797
$$765$$ 5.38743e9 0.435078
$$766$$ 9.58202e9 0.770294
$$767$$ −9.72458e8 −0.0778193
$$768$$ −6.54311e8 −0.0521219
$$769$$ 7.81741e9 0.619899 0.309949 0.950753i $$-0.399688\pi$$
0.309949 + 0.950753i $$0.399688\pi$$
$$770$$ −4.87498e9 −0.384818
$$771$$ −1.10098e10 −0.865144
$$772$$ −9.78732e9 −0.765602
$$773$$ 1.30864e10 1.01904 0.509522 0.860457i $$-0.329822\pi$$
0.509522 + 0.860457i $$0.329822\pi$$
$$774$$ −1.07639e9 −0.0834403
$$775$$ −1.46086e10 −1.12733
$$776$$ 1.02790e8 0.00789651
$$777$$ −5.05506e9 −0.386592
$$778$$ 1.14937e9 0.0875051
$$779$$ −1.58491e9 −0.120122
$$780$$ −2.11123e9 −0.159296
$$781$$ −1.12616e10 −0.845903
$$782$$ −1.06016e10 −0.792775
$$783$$ 1.36011e10 1.01253
$$784$$ −3.02159e9 −0.223939
$$785$$ −1.21396e10 −0.895696
$$786$$ −8.25767e9 −0.606567
$$787$$ 5.35561e9 0.391649 0.195825 0.980639i $$-0.437262\pi$$
0.195825 + 0.980639i $$0.437262\pi$$
$$788$$ 6.09176e9 0.443507
$$789$$ 9.22311e9 0.668510
$$790$$ 2.03565e10 1.46895
$$791$$ −6.18962e9 −0.444679
$$792$$ −1.84204e9 −0.131753
$$793$$ −2.38069e9 −0.169530
$$794$$ 4.93867e9 0.350137
$$795$$ −2.52903e10 −1.78513
$$796$$ 1.14295e10 0.803212
$$797$$ 1.21863e10 0.852641 0.426320 0.904572i $$-0.359810\pi$$
0.426320 + 0.904572i $$0.359810\pi$$
$$798$$ 2.49803e9 0.174016
$$799$$ −1.23652e10 −0.857606
$$800$$ −2.29704e9 −0.158618
$$801$$ 4.65220e9 0.319849
$$802$$ −9.06438e9 −0.620480
$$803$$ −3.20765e10 −2.18616
$$804$$ −8.59494e9 −0.583239
$$805$$ 7.11484e9 0.480706
$$806$$ 3.66277e9 0.246398
$$807$$ 1.88047e10 1.25953
$$808$$ 2.77578e9 0.185116
$$809$$ 1.32472e10 0.879636 0.439818 0.898087i $$-0.355043\pi$$
0.439818 + 0.898087i $$0.355043\pi$$
$$810$$ 8.87924e9 0.587054
$$811$$ 1.45473e10 0.957658 0.478829 0.877908i $$-0.341061\pi$$
0.478829 + 0.877908i $$0.341061\pi$$
$$812$$ −2.29221e9 −0.150248
$$813$$ −1.81885e10 −1.18708
$$814$$ −1.91179e10 −1.24238
$$815$$ −1.20866e10 −0.782083
$$816$$ 3.35638e9 0.216250
$$817$$ 5.52054e9 0.354164
$$818$$ 8.34262e9 0.532925
$$819$$ 4.28718e8 0.0272696
$$820$$ 1.42912e9 0.0905149
$$821$$ 6.51876e9 0.411115 0.205558 0.978645i $$-0.434099\pi$$
0.205558 + 0.978645i $$0.434099\pi$$
$$822$$ 1.67295e10 1.05059
$$823$$ −6.77944e9 −0.423930 −0.211965 0.977277i $$-0.567986\pi$$
−0.211965 + 0.977277i $$0.567986\pi$$
$$824$$ 8.80112e9 0.548015
$$825$$ 1.47685e10 0.915690
$$826$$ −1.03752e9 −0.0640572
$$827$$ −7.96808e9 −0.489874 −0.244937 0.969539i $$-0.578767\pi$$
−0.244937 + 0.969539i $$0.578767\pi$$
$$828$$ 2.68838e9 0.164583
$$829$$ 3.74439e9 0.228265 0.114133 0.993466i $$-0.463591\pi$$
0.114133 + 0.993466i $$0.463591\pi$$
$$830$$ 4.39639e8 0.0266884
$$831$$ 7.35966e9 0.444892
$$832$$ 5.75930e8 0.0346688
$$833$$ 1.54997e10 0.929106
$$834$$ 2.36741e9 0.141316
$$835$$ 3.55035e10 2.11042
$$836$$ 9.44736e9 0.559228
$$837$$ −2.31876e10 −1.36684
$$838$$ −5.67442e9 −0.333094
$$839$$ −8.06205e9 −0.471280 −0.235640 0.971840i $$-0.575719\pi$$
−0.235640 + 0.971840i $$0.575719\pi$$
$$840$$ −2.25249e9 −0.131125
$$841$$ −2.30775e9 −0.133783
$$842$$ 9.59013e9 0.553646
$$843$$ 2.79007e10 1.60405
$$844$$ −8.52706e9 −0.488203
$$845$$ 1.85832e9 0.105955
$$846$$ 3.13559e9 0.178042
$$847$$ −2.84047e9 −0.160619
$$848$$ 6.89904e9 0.388511
$$849$$ 1.57795e10 0.884943
$$850$$ 1.17830e10 0.658095
$$851$$ 2.79017e10 1.55195
$$852$$ −5.20342e9 −0.288238
$$853$$ −3.31854e9 −0.183074 −0.0915368 0.995802i $$-0.529178\pi$$
−0.0915368 + 0.995802i $$0.529178\pi$$
$$854$$ −2.53998e9 −0.139549
$$855$$ 7.00666e9 0.383380
$$856$$ −6.32740e9 −0.344800
$$857$$ −1.81939e10 −0.987398 −0.493699 0.869633i $$-0.664356\pi$$
−0.493699 + 0.869633i $$0.664356\pi$$
$$858$$ −3.70288e9 −0.200140
$$859$$ −1.91859e10 −1.03278 −0.516388 0.856355i $$-0.672724\pi$$
−0.516388 + 0.856355i $$0.672724\pi$$
$$860$$ −4.97790e9 −0.266871
$$861$$ 6.62766e8 0.0353874
$$862$$ −7.63322e9 −0.405913
$$863$$ −2.77943e10 −1.47203 −0.736017 0.676963i $$-0.763296\pi$$
−0.736017 + 0.676963i $$0.763296\pi$$
$$864$$ −3.64600e9 −0.192317
$$865$$ −2.52951e10 −1.32886
$$866$$ 3.05302e9 0.159741
$$867$$ −1.21381e9 −0.0632536
$$868$$ 3.90784e9 0.202823
$$869$$ 3.57032e10 1.84560
$$870$$ 1.46832e10 0.755969
$$871$$ 7.56534e9 0.387940
$$872$$ −8.73311e9 −0.446027
$$873$$ 1.33707e8 0.00680152
$$874$$ −1.37880e10 −0.698574
$$875$$ 9.05260e8 0.0456820
$$876$$ −1.48210e10 −0.744925
$$877$$ −3.40401e10 −1.70409 −0.852043 0.523471i $$-0.824637\pi$$
−0.852043 + 0.523471i $$0.824637\pi$$
$$878$$ −8.93465e8 −0.0445499
$$879$$ 3.16415e10 1.57143
$$880$$ −8.51874e9 −0.421392
$$881$$ 3.24476e10 1.59870 0.799350 0.600866i $$-0.205177\pi$$
0.799350 + 0.600866i $$0.205177\pi$$
$$882$$ −3.93043e9 −0.192886
$$883$$ 1.15866e10 0.566362 0.283181 0.959066i $$-0.408610\pi$$
0.283181 + 0.959066i $$0.408610\pi$$
$$884$$ −2.95431e9 −0.143838
$$885$$ 6.64609e9 0.322303
$$886$$ −1.16793e10 −0.564156
$$887$$ −2.86160e10 −1.37682 −0.688408 0.725323i $$-0.741690\pi$$
−0.688408 + 0.725323i $$0.741690\pi$$
$$888$$ −8.83342e9 −0.423335
$$889$$ 9.49344e9 0.453177
$$890$$ 2.15147e10 1.02299
$$891$$ 1.55733e10 0.737578
$$892$$ 7.64123e9 0.360484
$$893$$ −1.60817e10 −0.755702
$$894$$ −1.77933e10 −0.832865
$$895$$ −1.23355e9 −0.0575142
$$896$$ 6.14466e8 0.0285377
$$897$$ 5.40420e9 0.250010
$$898$$ 5.07207e9 0.233732
$$899$$ −2.54739e10 −1.16933
$$900$$ −2.98794e9 −0.136623
$$901$$ −3.53896e10 −1.61190
$$902$$ 2.50653e9 0.113723
$$903$$ −2.30854e9 −0.104335
$$904$$ −1.08160e10 −0.486942
$$905$$ −1.71617e10 −0.769646
$$906$$ −6.26022e9 −0.279667
$$907$$ 2.28278e10 1.01587 0.507936 0.861395i $$-0.330409\pi$$
0.507936 + 0.861395i $$0.330409\pi$$
$$908$$ 7.27395e8 0.0322456
$$909$$ 3.61068e9 0.159446
$$910$$ 1.98266e9 0.0872175
$$911$$ −1.76175e10 −0.772024 −0.386012 0.922494i $$-0.626148\pi$$
−0.386012 + 0.922494i $$0.626148\pi$$
$$912$$ 4.36516e9 0.190554
$$913$$ 7.71081e8 0.0335315
$$914$$ −4.83501e9 −0.209453
$$915$$ 1.62704e10 0.702140
$$916$$ −9.37978e8 −0.0403235
$$917$$ 7.75480e9 0.332107
$$918$$ 1.87026e10 0.797910
$$919$$ −4.93202e9 −0.209614 −0.104807 0.994493i $$-0.533423\pi$$
−0.104807 + 0.994493i $$0.533423\pi$$
$$920$$ 1.24328e10 0.526393
$$921$$ −1.79774e10 −0.758258
$$922$$ −1.76452e10 −0.741425
$$923$$ 4.58010e9 0.191721
$$924$$ −3.95063e9 −0.164746
$$925$$ −3.10108e10 −1.28830
$$926$$ −8.39401e9 −0.347401
$$927$$ 1.14483e10 0.472023
$$928$$ −4.00549e9 −0.164527
$$929$$ −2.68352e10 −1.09812 −0.549061 0.835783i $$-0.685014\pi$$
−0.549061 + 0.835783i $$0.685014\pi$$
$$930$$ −2.50325e10 −1.02050
$$931$$ 2.01582e10 0.818707
$$932$$ −1.58032e10 −0.639423
$$933$$ 1.12210e10 0.452321
$$934$$ 1.61168e10 0.647241
$$935$$ 4.36980e10 1.74832
$$936$$ 7.49159e8 0.0298613
$$937$$ −2.08650e10 −0.828570 −0.414285 0.910147i $$-0.635968\pi$$
−0.414285 + 0.910147i $$0.635968\pi$$
$$938$$ 8.07153e9 0.319335
$$939$$ 3.72910e10 1.46985
$$940$$ 1.45009e10 0.569440
$$941$$ −3.07099e10 −1.20147 −0.600737 0.799447i $$-0.705126\pi$$
−0.600737 + 0.799447i $$0.705126\pi$$
$$942$$ −9.83781e9 −0.383460
$$943$$ −3.65818e9 −0.142061
$$944$$ −1.81301e9 −0.0701453
$$945$$ −1.25515e10 −0.483820
$$946$$ −8.73071e9 −0.335298
$$947$$ −1.03377e9 −0.0395548 −0.0197774 0.999804i $$-0.506296\pi$$
−0.0197774 + 0.999804i $$0.506296\pi$$
$$948$$ 1.64967e10 0.628880
$$949$$ 1.30455e10 0.495485
$$950$$ 1.53244e10 0.579898
$$951$$ −1.92177e10 −0.724552
$$952$$ −3.15199e9 −0.118401
$$953$$ −1.78629e9 −0.0668540 −0.0334270 0.999441i $$-0.510642\pi$$
−0.0334270 + 0.999441i $$0.510642\pi$$
$$954$$ 8.97414e9 0.334637
$$955$$ 7.17615e10 2.66612
$$956$$ −1.03193e9 −0.0381985
$$957$$ 2.57529e10 0.949803
$$958$$ −2.94273e10 −1.08136
$$959$$ −1.57107e10 −0.575215
$$960$$ −3.93609e9 −0.143587
$$961$$ 1.59163e10 0.578508
$$962$$ 7.77525e9 0.281580
$$963$$ −8.23057e9 −0.296987
$$964$$ 7.31238e9 0.262899
$$965$$ −5.88768e10 −2.10911
$$966$$ 5.76579e9 0.205797
$$967$$ −1.53418e10 −0.545613 −0.272806 0.962069i $$-0.587952\pi$$
−0.272806 + 0.962069i $$0.587952\pi$$
$$968$$ −4.96355e9 −0.175885
$$969$$ −2.23917e10 −0.790595
$$970$$ 6.18347e8 0.0217536
$$971$$ −5.22182e8 −0.0183043 −0.00915217 0.999958i $$-0.502913\pi$$
−0.00915217 + 0.999958i $$0.502913\pi$$
$$972$$ −8.37817e9 −0.292629
$$973$$ −2.22324e9 −0.0773733
$$974$$ 1.53197e9 0.0531245
$$975$$ −6.00638e9 −0.207537
$$976$$ −4.43846e9 −0.152812
$$977$$ 5.68857e10 1.95152 0.975758 0.218851i $$-0.0702309\pi$$
0.975758 + 0.218851i $$0.0702309\pi$$
$$978$$ −9.79486e9 −0.334821
$$979$$ 3.77345e10 1.28528
$$980$$ −1.81768e10 −0.616916
$$981$$ −1.13599e10 −0.384177
$$982$$ 2.57857e9 0.0868936
$$983$$ −1.15736e10 −0.388624 −0.194312 0.980940i $$-0.562247\pi$$
−0.194312 + 0.980940i $$0.562247\pi$$
$$984$$ 1.15814e9 0.0387507
$$985$$ 3.66457e10 1.22179
$$986$$ 2.05467e10 0.682612
$$987$$ 6.72492e9 0.222626
$$988$$ −3.84225e9 −0.126747
$$989$$ 1.27421e10 0.418846
$$990$$ −1.10810e10 −0.362958
$$991$$ 1.38509e10 0.452086 0.226043 0.974117i $$-0.427421\pi$$
0.226043 + 0.974117i $$0.427421\pi$$
$$992$$ 6.82872e9 0.222100
$$993$$ 1.88510e10 0.610958
$$994$$ 4.88655e9 0.157816
$$995$$ 6.87554e10 2.21272
$$996$$ 3.56279e8 0.0114257
$$997$$ 5.45316e9 0.174267 0.0871334 0.996197i $$-0.472229\pi$$
0.0871334 + 0.996197i $$0.472229\pi$$
$$998$$ −3.09356e10 −0.985147
$$999$$ −4.92222e10 −1.56200
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 26.8.a.a.1.1 1
3.2 odd 2 234.8.a.d.1.1 1
4.3 odd 2 208.8.a.c.1.1 1
13.5 odd 4 338.8.b.b.337.2 2
13.8 odd 4 338.8.b.b.337.1 2
13.12 even 2 338.8.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.a.1.1 1 1.1 even 1 trivial
208.8.a.c.1.1 1 4.3 odd 2
234.8.a.d.1.1 1 3.2 odd 2
338.8.a.c.1.1 1 13.12 even 2
338.8.b.b.337.1 2 13.8 odd 4
338.8.b.b.337.2 2 13.5 odd 4