# Properties

 Label 338.8.a.d.1.1 Level $338$ Weight $8$ Character 338.1 Self dual yes Analytic conductor $105.586$ 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] = [338,8,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, 8, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$105.586138614$$ 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)

 $$f(q)$$ $$=$$ $$q+8.00000 q^{2} +12.0000 q^{3} +64.0000 q^{4} +210.000 q^{5} +96.0000 q^{6} -1016.00 q^{7} +512.000 q^{8} -2043.00 q^{9} +O(q^{10})$$ $$q+8.00000 q^{2} +12.0000 q^{3} +64.0000 q^{4} +210.000 q^{5} +96.0000 q^{6} -1016.00 q^{7} +512.000 q^{8} -2043.00 q^{9} +1680.00 q^{10} -1092.00 q^{11} +768.000 q^{12} -8128.00 q^{14} +2520.00 q^{15} +4096.00 q^{16} +14706.0 q^{17} -16344.0 q^{18} +39940.0 q^{19} +13440.0 q^{20} -12192.0 q^{21} -8736.00 q^{22} +68712.0 q^{23} +6144.00 q^{24} -34025.0 q^{25} -50760.0 q^{27} -65024.0 q^{28} -102570. q^{29} +20160.0 q^{30} -227552. q^{31} +32768.0 q^{32} -13104.0 q^{33} +117648. q^{34} -213360. q^{35} -130752. q^{36} -160526. q^{37} +319520. q^{38} +107520. q^{40} -10842.0 q^{41} -97536.0 q^{42} -630748. q^{43} -69888.0 q^{44} -429030. q^{45} +549696. q^{46} -472656. q^{47} +49152.0 q^{48} +208713. q^{49} -272200. q^{50} +176472. q^{51} -1.49402e6 q^{53} -406080. q^{54} -229320. q^{55} -520192. q^{56} +479280. q^{57} -820560. q^{58} -2.64066e6 q^{59} +161280. q^{60} +827702. q^{61} -1.82042e6 q^{62} +2.07569e6 q^{63} +262144. q^{64} -104832. q^{66} +126004. q^{67} +941184. q^{68} +824544. q^{69} -1.70688e6 q^{70} +1.41473e6 q^{71} -1.04602e6 q^{72} -980282. q^{73} -1.28421e6 q^{74} -408300. q^{75} +2.55616e6 q^{76} +1.10947e6 q^{77} -3.56680e6 q^{79} +860160. q^{80} +3.85892e6 q^{81} -86736.0 q^{82} -5.67289e6 q^{83} -780288. q^{84} +3.08826e6 q^{85} -5.04598e6 q^{86} -1.23084e6 q^{87} -559104. q^{88} +1.19512e7 q^{89} -3.43224e6 q^{90} +4.39757e6 q^{92} -2.73062e6 q^{93} -3.78125e6 q^{94} +8.38740e6 q^{95} +393216. q^{96} -8.68215e6 q^{97} +1.66970e6 q^{98} +2.23096e6 q^{99} +O(q^{100})$$

## 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
$$3$$ 12.0000 0.256600 0.128300 0.991735i $$-0.459048\pi$$
0.128300 + 0.991735i $$0.459048\pi$$
$$4$$ 64.0000 0.500000
$$5$$ 210.000 0.751319 0.375659 0.926758i $$-0.377416\pi$$
0.375659 + 0.926758i $$0.377416\pi$$
$$6$$ 96.0000 0.181444
$$7$$ −1016.00 −1.11957 −0.559784 0.828638i $$-0.689116\pi$$
−0.559784 + 0.828638i $$0.689116\pi$$
$$8$$ 512.000 0.353553
$$9$$ −2043.00 −0.934156
$$10$$ 1680.00 0.531263
$$11$$ −1092.00 −0.247371 −0.123685 0.992321i $$-0.539471\pi$$
−0.123685 + 0.992321i $$0.539471\pi$$
$$12$$ 768.000 0.128300
$$13$$ 0 0
$$14$$ −8128.00 −0.791654
$$15$$ 2520.00 0.192789
$$16$$ 4096.00 0.250000
$$17$$ 14706.0 0.725978 0.362989 0.931793i $$-0.381756\pi$$
0.362989 + 0.931793i $$0.381756\pi$$
$$18$$ −16344.0 −0.660548
$$19$$ 39940.0 1.33589 0.667945 0.744211i $$-0.267174\pi$$
0.667945 + 0.744211i $$0.267174\pi$$
$$20$$ 13440.0 0.375659
$$21$$ −12192.0 −0.287281
$$22$$ −8736.00 −0.174917
$$23$$ 68712.0 1.17757 0.588783 0.808291i $$-0.299607\pi$$
0.588783 + 0.808291i $$0.299607\pi$$
$$24$$ 6144.00 0.0907218
$$25$$ −34025.0 −0.435520
$$26$$ 0 0
$$27$$ −50760.0 −0.496305
$$28$$ −65024.0 −0.559784
$$29$$ −102570. −0.780957 −0.390479 0.920612i $$-0.627690\pi$$
−0.390479 + 0.920612i $$0.627690\pi$$
$$30$$ 20160.0 0.136322
$$31$$ −227552. −1.37188 −0.685938 0.727660i $$-0.740608\pi$$
−0.685938 + 0.727660i $$0.740608\pi$$
$$32$$ 32768.0 0.176777
$$33$$ −13104.0 −0.0634753
$$34$$ 117648. 0.513344
$$35$$ −213360. −0.841153
$$36$$ −130752. −0.467078
$$37$$ −160526. −0.521002 −0.260501 0.965474i $$-0.583888\pi$$
−0.260501 + 0.965474i $$0.583888\pi$$
$$38$$ 319520. 0.944616
$$39$$ 0 0
$$40$$ 107520. 0.265631
$$41$$ −10842.0 −0.0245678 −0.0122839 0.999925i $$-0.503910\pi$$
−0.0122839 + 0.999925i $$0.503910\pi$$
$$42$$ −97536.0 −0.203139
$$43$$ −630748. −1.20981 −0.604904 0.796299i $$-0.706788\pi$$
−0.604904 + 0.796299i $$0.706788\pi$$
$$44$$ −69888.0 −0.123685
$$45$$ −429030. −0.701849
$$46$$ 549696. 0.832665
$$47$$ −472656. −0.664053 −0.332026 0.943270i $$-0.607732\pi$$
−0.332026 + 0.943270i $$0.607732\pi$$
$$48$$ 49152.0 0.0641500
$$49$$ 208713. 0.253433
$$50$$ −272200. −0.307959
$$51$$ 176472. 0.186286
$$52$$ 0 0
$$53$$ −1.49402e6 −1.37845 −0.689224 0.724548i $$-0.742048\pi$$
−0.689224 + 0.724548i $$0.742048\pi$$
$$54$$ −406080. −0.350940
$$55$$ −229320. −0.185854
$$56$$ −520192. −0.395827
$$57$$ 479280. 0.342789
$$58$$ −820560. −0.552220
$$59$$ −2.64066e6 −1.67390 −0.836952 0.547277i $$-0.815665\pi$$
−0.836952 + 0.547277i $$0.815665\pi$$
$$60$$ 161280. 0.0963943
$$61$$ 827702. 0.466895 0.233448 0.972369i $$-0.424999\pi$$
0.233448 + 0.972369i $$0.424999\pi$$
$$62$$ −1.82042e6 −0.970063
$$63$$ 2.07569e6 1.04585
$$64$$ 262144. 0.125000
$$65$$ 0 0
$$66$$ −104832. −0.0448838
$$67$$ 126004. 0.0511826 0.0255913 0.999672i $$-0.491853\pi$$
0.0255913 + 0.999672i $$0.491853\pi$$
$$68$$ 941184. 0.362989
$$69$$ 824544. 0.302164
$$70$$ −1.70688e6 −0.594785
$$71$$ 1.41473e6 0.469104 0.234552 0.972104i $$-0.424638\pi$$
0.234552 + 0.972104i $$0.424638\pi$$
$$72$$ −1.04602e6 −0.330274
$$73$$ −980282. −0.294931 −0.147466 0.989067i $$-0.547112\pi$$
−0.147466 + 0.989067i $$0.547112\pi$$
$$74$$ −1.28421e6 −0.368404
$$75$$ −408300. −0.111754
$$76$$ 2.55616e6 0.667945
$$77$$ 1.10947e6 0.276948
$$78$$ 0 0
$$79$$ −3.56680e6 −0.813924 −0.406962 0.913445i $$-0.633412\pi$$
−0.406962 + 0.913445i $$0.633412\pi$$
$$80$$ 860160. 0.187830
$$81$$ 3.85892e6 0.806805
$$82$$ −86736.0 −0.0173720
$$83$$ −5.67289e6 −1.08901 −0.544504 0.838758i $$-0.683282\pi$$
−0.544504 + 0.838758i $$0.683282\pi$$
$$84$$ −780288. −0.143641
$$85$$ 3.08826e6 0.545441
$$86$$ −5.04598e6 −0.855463
$$87$$ −1.23084e6 −0.200394
$$88$$ −559104. −0.0874587
$$89$$ 1.19512e7 1.79699 0.898496 0.438982i $$-0.144661\pi$$
0.898496 + 0.438982i $$0.144661\pi$$
$$90$$ −3.43224e6 −0.496282
$$91$$ 0 0
$$92$$ 4.39757e6 0.588783
$$93$$ −2.73062e6 −0.352023
$$94$$ −3.78125e6 −0.469556
$$95$$ 8.38740e6 1.00368
$$96$$ 393216. 0.0453609
$$97$$ −8.68215e6 −0.965886 −0.482943 0.875652i $$-0.660432\pi$$
−0.482943 + 0.875652i $$0.660432\pi$$
$$98$$ 1.66970e6 0.179204
$$99$$ 2.23096e6 0.231083
$$100$$ −2.17760e6 −0.217760
$$101$$ −1.00795e7 −0.973455 −0.486727 0.873554i $$-0.661810\pi$$
−0.486727 + 0.873554i $$0.661810\pi$$
$$102$$ 1.41178e6 0.131724
$$103$$ 3.74799e6 0.337962 0.168981 0.985619i $$-0.445952\pi$$
0.168981 + 0.985619i $$0.445952\pi$$
$$104$$ 0 0
$$105$$ −2.56032e6 −0.215840
$$106$$ −1.19521e7 −0.974710
$$107$$ −1.79856e7 −1.41932 −0.709661 0.704543i $$-0.751152\pi$$
−0.709661 + 0.704543i $$0.751152\pi$$
$$108$$ −3.24864e6 −0.248152
$$109$$ −1.22570e7 −0.906552 −0.453276 0.891370i $$-0.649745\pi$$
−0.453276 + 0.891370i $$0.649745\pi$$
$$110$$ −1.83456e6 −0.131419
$$111$$ −1.92631e6 −0.133689
$$112$$ −4.16154e6 −0.279892
$$113$$ 1.65950e7 1.08194 0.540968 0.841043i $$-0.318058\pi$$
0.540968 + 0.841043i $$0.318058\pi$$
$$114$$ 3.83424e6 0.242389
$$115$$ 1.44295e7 0.884727
$$116$$ −6.56448e6 −0.390479
$$117$$ 0 0
$$118$$ −2.11253e7 −1.18363
$$119$$ −1.49413e7 −0.812782
$$120$$ 1.29024e6 0.0681610
$$121$$ −1.82947e7 −0.938808
$$122$$ 6.62162e6 0.330145
$$123$$ −130104. −0.00630410
$$124$$ −1.45633e7 −0.685938
$$125$$ −2.35515e7 −1.07853
$$126$$ 1.66055e7 0.739529
$$127$$ 1.16826e6 0.0506087 0.0253043 0.999680i $$-0.491945\pi$$
0.0253043 + 0.999680i $$0.491945\pi$$
$$128$$ 2.09715e6 0.0883883
$$129$$ −7.56898e6 −0.310437
$$130$$ 0 0
$$131$$ −7.92383e6 −0.307954 −0.153977 0.988074i $$-0.549208\pi$$
−0.153977 + 0.988074i $$0.549208\pi$$
$$132$$ −838656. −0.0317377
$$133$$ −4.05790e7 −1.49562
$$134$$ 1.00803e6 0.0361916
$$135$$ −1.06596e7 −0.372883
$$136$$ 7.52947e6 0.256672
$$137$$ 315654. 0.0104879 0.00524396 0.999986i $$-0.498331\pi$$
0.00524396 + 0.999986i $$0.498331\pi$$
$$138$$ 6.59635e6 0.213662
$$139$$ 3.92038e7 1.23816 0.619079 0.785329i $$-0.287506\pi$$
0.619079 + 0.785329i $$0.287506\pi$$
$$140$$ −1.36550e7 −0.420576
$$141$$ −5.67187e6 −0.170396
$$142$$ 1.13178e7 0.331706
$$143$$ 0 0
$$144$$ −8.36813e6 −0.233539
$$145$$ −2.15397e7 −0.586748
$$146$$ −7.84226e6 −0.208548
$$147$$ 2.50456e6 0.0650309
$$148$$ −1.02737e7 −0.260501
$$149$$ 2.18860e7 0.542020 0.271010 0.962577i $$-0.412642\pi$$
0.271010 + 0.962577i $$0.412642\pi$$
$$150$$ −3.26640e6 −0.0790224
$$151$$ 2.94154e7 0.695274 0.347637 0.937629i $$-0.386984\pi$$
0.347637 + 0.937629i $$0.386984\pi$$
$$152$$ 2.04493e7 0.472308
$$153$$ −3.00444e7 −0.678177
$$154$$ 8.87578e6 0.195832
$$155$$ −4.77859e7 −1.03072
$$156$$ 0 0
$$157$$ 6.05550e7 1.24882 0.624412 0.781095i $$-0.285339\pi$$
0.624412 + 0.781095i $$0.285339\pi$$
$$158$$ −2.85344e7 −0.575531
$$159$$ −1.79282e7 −0.353710
$$160$$ 6.88128e6 0.132816
$$161$$ −6.98114e7 −1.31837
$$162$$ 3.08714e7 0.570497
$$163$$ −5.70853e7 −1.03245 −0.516223 0.856454i $$-0.672663\pi$$
−0.516223 + 0.856454i $$0.672663\pi$$
$$164$$ −693888. −0.0122839
$$165$$ −2.75184e6 −0.0476902
$$166$$ −4.53831e7 −0.770045
$$167$$ 8.77265e7 1.45755 0.728775 0.684754i $$-0.240090\pi$$
0.728775 + 0.684754i $$0.240090\pi$$
$$168$$ −6.24230e6 −0.101569
$$169$$ 0 0
$$170$$ 2.47061e7 0.385685
$$171$$ −8.15974e7 −1.24793
$$172$$ −4.03679e7 −0.604904
$$173$$ 8.56954e6 0.125833 0.0629167 0.998019i $$-0.479960\pi$$
0.0629167 + 0.998019i $$0.479960\pi$$
$$174$$ −9.84672e6 −0.141700
$$175$$ 3.45694e7 0.487594
$$176$$ −4.47283e6 −0.0618427
$$177$$ −3.16879e7 −0.429524
$$178$$ 9.56095e7 1.27067
$$179$$ 1.88041e7 0.245056 0.122528 0.992465i $$-0.460900\pi$$
0.122528 + 0.992465i $$0.460900\pi$$
$$180$$ −2.74579e7 −0.350925
$$181$$ −5.99625e7 −0.751631 −0.375816 0.926694i $$-0.622637\pi$$
−0.375816 + 0.926694i $$0.622637\pi$$
$$182$$ 0 0
$$183$$ 9.93242e6 0.119805
$$184$$ 3.51805e7 0.416332
$$185$$ −3.37105e7 −0.391439
$$186$$ −2.18450e7 −0.248918
$$187$$ −1.60590e7 −0.179586
$$188$$ −3.02500e7 −0.332026
$$189$$ 5.15722e7 0.555647
$$190$$ 6.70992e7 0.709708
$$191$$ 9.39861e7 0.975993 0.487997 0.872845i $$-0.337728\pi$$
0.487997 + 0.872845i $$0.337728\pi$$
$$192$$ 3.14573e6 0.0320750
$$193$$ 3.51946e7 0.352391 0.176196 0.984355i $$-0.443621\pi$$
0.176196 + 0.984355i $$0.443621\pi$$
$$194$$ −6.94572e7 −0.682985
$$195$$ 0 0
$$196$$ 1.33576e7 0.126717
$$197$$ −1.02985e8 −0.959718 −0.479859 0.877346i $$-0.659312\pi$$
−0.479859 + 0.877346i $$0.659312\pi$$
$$198$$ 1.78476e7 0.163400
$$199$$ 8.36376e7 0.752342 0.376171 0.926550i $$-0.377240\pi$$
0.376171 + 0.926550i $$0.377240\pi$$
$$200$$ −1.74208e7 −0.153980
$$201$$ 1.51205e6 0.0131335
$$202$$ −8.06363e7 −0.688337
$$203$$ 1.04211e8 0.874335
$$204$$ 1.12942e7 0.0931430
$$205$$ −2.27682e6 −0.0184582
$$206$$ 2.99839e7 0.238975
$$207$$ −1.40379e8 −1.10003
$$208$$ 0 0
$$209$$ −4.36145e7 −0.330460
$$210$$ −2.04826e7 −0.152622
$$211$$ −9.74010e7 −0.713797 −0.356899 0.934143i $$-0.616166\pi$$
−0.356899 + 0.934143i $$0.616166\pi$$
$$212$$ −9.56172e7 −0.689224
$$213$$ 1.69767e7 0.120372
$$214$$ −1.43885e8 −1.00361
$$215$$ −1.32457e8 −0.908951
$$216$$ −2.59891e7 −0.175470
$$217$$ 2.31193e8 1.53591
$$218$$ −9.80562e7 −0.641029
$$219$$ −1.17634e7 −0.0756794
$$220$$ −1.46765e7 −0.0929271
$$221$$ 0 0
$$222$$ −1.54105e7 −0.0945325
$$223$$ 1.46457e7 0.0884390 0.0442195 0.999022i $$-0.485920\pi$$
0.0442195 + 0.999022i $$0.485920\pi$$
$$224$$ −3.32923e7 −0.197914
$$225$$ 6.95131e7 0.406844
$$226$$ 1.32760e8 0.765045
$$227$$ 1.84541e8 1.04713 0.523567 0.851985i $$-0.324601\pi$$
0.523567 + 0.851985i $$0.324601\pi$$
$$228$$ 3.06739e7 0.171395
$$229$$ 8.75461e6 0.0481740 0.0240870 0.999710i $$-0.492332\pi$$
0.0240870 + 0.999710i $$0.492332\pi$$
$$230$$ 1.15436e8 0.625597
$$231$$ 1.33137e7 0.0710650
$$232$$ −5.25158e7 −0.276110
$$233$$ −1.19556e8 −0.619193 −0.309597 0.950868i $$-0.600194\pi$$
−0.309597 + 0.950868i $$0.600194\pi$$
$$234$$ 0 0
$$235$$ −9.92578e7 −0.498915
$$236$$ −1.69002e8 −0.836952
$$237$$ −4.28016e7 −0.208853
$$238$$ −1.19530e8 −0.574723
$$239$$ −3.96209e8 −1.87729 −0.938646 0.344883i $$-0.887919\pi$$
−0.938646 + 0.344883i $$0.887919\pi$$
$$240$$ 1.03219e7 0.0481971
$$241$$ 2.56606e8 1.18089 0.590443 0.807080i $$-0.298953\pi$$
0.590443 + 0.807080i $$0.298953\pi$$
$$242$$ −1.46358e8 −0.663837
$$243$$ 1.57319e8 0.703331
$$244$$ 5.29729e7 0.233448
$$245$$ 4.38297e7 0.190409
$$246$$ −1.04083e6 −0.00445767
$$247$$ 0 0
$$248$$ −1.16507e8 −0.485031
$$249$$ −6.80747e7 −0.279440
$$250$$ −1.88412e8 −0.762638
$$251$$ −7.34775e7 −0.293290 −0.146645 0.989189i $$-0.546847\pi$$
−0.146645 + 0.989189i $$0.546847\pi$$
$$252$$ 1.32844e8 0.522926
$$253$$ −7.50335e7 −0.291295
$$254$$ 9.34605e6 0.0357857
$$255$$ 3.70591e7 0.139960
$$256$$ 1.67772e7 0.0625000
$$257$$ −2.02701e8 −0.744886 −0.372443 0.928055i $$-0.621480\pi$$
−0.372443 + 0.928055i $$0.621480\pi$$
$$258$$ −6.05518e7 −0.219512
$$259$$ 1.63094e8 0.583297
$$260$$ 0 0
$$261$$ 2.09551e8 0.729536
$$262$$ −6.33906e7 −0.217756
$$263$$ 1.54254e8 0.522867 0.261434 0.965221i $$-0.415805\pi$$
0.261434 + 0.965221i $$0.415805\pi$$
$$264$$ −6.70925e6 −0.0224419
$$265$$ −3.13744e8 −1.03565
$$266$$ −3.24632e8 −1.05756
$$267$$ 1.43414e8 0.461108
$$268$$ 8.06426e6 0.0255913
$$269$$ −6.24018e8 −1.95463 −0.977315 0.211793i $$-0.932070\pi$$
−0.977315 + 0.211793i $$0.932070\pi$$
$$270$$ −8.52768e7 −0.263668
$$271$$ 3.87983e8 1.18419 0.592094 0.805869i $$-0.298302\pi$$
0.592094 + 0.805869i $$0.298302\pi$$
$$272$$ 6.02358e7 0.181494
$$273$$ 0 0
$$274$$ 2.52523e6 0.00741608
$$275$$ 3.71553e7 0.107735
$$276$$ 5.27708e7 0.151082
$$277$$ 4.53952e8 1.28331 0.641654 0.766994i $$-0.278248\pi$$
0.641654 + 0.766994i $$0.278248\pi$$
$$278$$ 3.13630e8 0.875510
$$279$$ 4.64889e8 1.28155
$$280$$ −1.09240e8 −0.297392
$$281$$ −3.33770e8 −0.897377 −0.448689 0.893688i $$-0.648109\pi$$
−0.448689 + 0.893688i $$0.648109\pi$$
$$282$$ −4.53750e7 −0.120488
$$283$$ 5.37695e8 1.41021 0.705104 0.709104i $$-0.250900\pi$$
0.705104 + 0.709104i $$0.250900\pi$$
$$284$$ 9.05426e7 0.234552
$$285$$ 1.00649e8 0.257544
$$286$$ 0 0
$$287$$ 1.10155e7 0.0275053
$$288$$ −6.69450e7 −0.165137
$$289$$ −1.94072e8 −0.472956
$$290$$ −1.72318e8 −0.414894
$$291$$ −1.04186e8 −0.247847
$$292$$ −6.27380e7 −0.147466
$$293$$ −3.35600e8 −0.779445 −0.389722 0.920932i $$-0.627429\pi$$
−0.389722 + 0.920932i $$0.627429\pi$$
$$294$$ 2.00364e7 0.0459838
$$295$$ −5.54539e8 −1.25764
$$296$$ −8.21893e7 −0.184202
$$297$$ 5.54299e7 0.122771
$$298$$ 1.75088e8 0.383266
$$299$$ 0 0
$$300$$ −2.61312e7 −0.0558772
$$301$$ 6.40840e8 1.35446
$$302$$ 2.35324e8 0.491633
$$303$$ −1.20954e8 −0.249789
$$304$$ 1.63594e8 0.333972
$$305$$ 1.73817e8 0.350787
$$306$$ −2.40355e8 −0.479543
$$307$$ −2.15029e8 −0.424143 −0.212072 0.977254i $$-0.568021\pi$$
−0.212072 + 0.977254i $$0.568021\pi$$
$$308$$ 7.10062e7 0.138474
$$309$$ 4.49759e7 0.0867212
$$310$$ −3.82287e8 −0.728826
$$311$$ 7.92062e8 1.49313 0.746565 0.665313i $$-0.231702\pi$$
0.746565 + 0.665313i $$0.231702\pi$$
$$312$$ 0 0
$$313$$ −1.18457e8 −0.218352 −0.109176 0.994022i $$-0.534821\pi$$
−0.109176 + 0.994022i $$0.534821\pi$$
$$314$$ 4.84440e8 0.883051
$$315$$ 4.35894e8 0.785768
$$316$$ −2.28275e8 −0.406962
$$317$$ 5.07310e7 0.0894470 0.0447235 0.998999i $$-0.485759\pi$$
0.0447235 + 0.998999i $$0.485759\pi$$
$$318$$ −1.43426e8 −0.250111
$$319$$ 1.12006e8 0.193186
$$320$$ 5.50502e7 0.0939149
$$321$$ −2.15827e8 −0.364198
$$322$$ −5.58491e8 −0.932225
$$323$$ 5.87358e8 0.969826
$$324$$ 2.46971e8 0.403402
$$325$$ 0 0
$$326$$ −4.56682e8 −0.730050
$$327$$ −1.47084e8 −0.232621
$$328$$ −5.55110e6 −0.00868602
$$329$$ 4.80218e8 0.743453
$$330$$ −2.20147e7 −0.0337221
$$331$$ −2.73757e8 −0.414923 −0.207461 0.978243i $$-0.566520\pi$$
−0.207461 + 0.978243i $$0.566520\pi$$
$$332$$ −3.63065e8 −0.544504
$$333$$ 3.27955e8 0.486697
$$334$$ 7.01812e8 1.03064
$$335$$ 2.64608e7 0.0384545
$$336$$ −4.99384e7 −0.0718203
$$337$$ −9.18512e7 −0.130732 −0.0653658 0.997861i $$-0.520821\pi$$
−0.0653658 + 0.997861i $$0.520821\pi$$
$$338$$ 0 0
$$339$$ 1.99140e8 0.277625
$$340$$ 1.97649e8 0.272720
$$341$$ 2.48487e8 0.339362
$$342$$ −6.52779e8 −0.882419
$$343$$ 6.24667e8 0.835833
$$344$$ −3.22943e8 −0.427732
$$345$$ 1.73154e8 0.227021
$$346$$ 6.85563e7 0.0889777
$$347$$ −1.36700e9 −1.75637 −0.878187 0.478318i $$-0.841247\pi$$
−0.878187 + 0.478318i $$0.841247\pi$$
$$348$$ −7.87738e7 −0.100197
$$349$$ −1.13143e9 −1.42475 −0.712377 0.701797i $$-0.752381\pi$$
−0.712377 + 0.701797i $$0.752381\pi$$
$$350$$ 2.76555e8 0.344781
$$351$$ 0 0
$$352$$ −3.57827e7 −0.0437294
$$353$$ 4.48395e7 0.0542562 0.0271281 0.999632i $$-0.491364\pi$$
0.0271281 + 0.999632i $$0.491364\pi$$
$$354$$ −2.53503e8 −0.303719
$$355$$ 2.97093e8 0.352446
$$356$$ 7.64876e8 0.898496
$$357$$ −1.79296e8 −0.208560
$$358$$ 1.50432e8 0.173281
$$359$$ −3.98281e8 −0.454317 −0.227158 0.973858i $$-0.572943\pi$$
−0.227158 + 0.973858i $$0.572943\pi$$
$$360$$ −2.19663e8 −0.248141
$$361$$ 7.01332e8 0.784600
$$362$$ −4.79700e8 −0.531483
$$363$$ −2.19536e8 −0.240898
$$364$$ 0 0
$$365$$ −2.05859e8 −0.221588
$$366$$ 7.94594e7 0.0847152
$$367$$ 1.63472e9 1.72628 0.863140 0.504964i $$-0.168494\pi$$
0.863140 + 0.504964i $$0.168494\pi$$
$$368$$ 2.81444e8 0.294391
$$369$$ 2.21502e7 0.0229501
$$370$$ −2.69684e8 −0.276789
$$371$$ 1.51792e9 1.54327
$$372$$ −1.74760e8 −0.176012
$$373$$ −1.54633e9 −1.54284 −0.771421 0.636325i $$-0.780454\pi$$
−0.771421 + 0.636325i $$0.780454\pi$$
$$374$$ −1.28472e8 −0.126986
$$375$$ −2.82618e8 −0.276752
$$376$$ −2.42000e8 −0.234778
$$377$$ 0 0
$$378$$ 4.12577e8 0.392902
$$379$$ 1.05688e9 0.997216 0.498608 0.866828i $$-0.333845\pi$$
0.498608 + 0.866828i $$0.333845\pi$$
$$380$$ 5.36794e8 0.501839
$$381$$ 1.40191e7 0.0129862
$$382$$ 7.51889e8 0.690132
$$383$$ −2.24910e8 −0.204556 −0.102278 0.994756i $$-0.532613\pi$$
−0.102278 + 0.994756i $$0.532613\pi$$
$$384$$ 2.51658e7 0.0226805
$$385$$ 2.32989e8 0.208077
$$386$$ 2.81556e8 0.249178
$$387$$ 1.28862e9 1.13015
$$388$$ −5.55657e8 −0.482943
$$389$$ 1.01788e9 0.876746 0.438373 0.898793i $$-0.355555\pi$$
0.438373 + 0.898793i $$0.355555\pi$$
$$390$$ 0 0
$$391$$ 1.01048e9 0.854887
$$392$$ 1.06861e8 0.0896021
$$393$$ −9.50859e7 −0.0790210
$$394$$ −8.23883e8 −0.678623
$$395$$ −7.49028e8 −0.611517
$$396$$ 1.42781e8 0.115541
$$397$$ 1.47565e9 1.18363 0.591817 0.806072i $$-0.298411\pi$$
0.591817 + 0.806072i $$0.298411\pi$$
$$398$$ 6.69100e8 0.531986
$$399$$ −4.86948e8 −0.383776
$$400$$ −1.39366e8 −0.108880
$$401$$ −2.74912e8 −0.212906 −0.106453 0.994318i $$-0.533949\pi$$
−0.106453 + 0.994318i $$0.533949\pi$$
$$402$$ 1.20964e7 0.00928676
$$403$$ 0 0
$$404$$ −6.45090e8 −0.486727
$$405$$ 8.10373e8 0.606167
$$406$$ 8.33689e8 0.618248
$$407$$ 1.75294e8 0.128881
$$408$$ 9.03537e7 0.0658620
$$409$$ 1.63427e9 1.18112 0.590558 0.806995i $$-0.298908\pi$$
0.590558 + 0.806995i $$0.298908\pi$$
$$410$$ −1.82146e7 −0.0130519
$$411$$ 3.78785e6 0.00269120
$$412$$ 2.39871e8 0.168981
$$413$$ 2.68291e9 1.87405
$$414$$ −1.12303e9 −0.777839
$$415$$ −1.19131e9 −0.818192
$$416$$ 0 0
$$417$$ 4.70445e8 0.317712
$$418$$ −3.48916e8 −0.233670
$$419$$ −1.11280e9 −0.739039 −0.369519 0.929223i $$-0.620478\pi$$
−0.369519 + 0.929223i $$0.620478\pi$$
$$420$$ −1.63860e8 −0.107920
$$421$$ −9.22528e8 −0.602549 −0.301274 0.953537i $$-0.597412\pi$$
−0.301274 + 0.953537i $$0.597412\pi$$
$$422$$ −7.79208e8 −0.504731
$$423$$ 9.65636e8 0.620329
$$424$$ −7.64937e8 −0.487355
$$425$$ −5.00372e8 −0.316178
$$426$$ 1.35814e8 0.0851159
$$427$$ −8.40945e8 −0.522721
$$428$$ −1.15108e9 −0.709661
$$429$$ 0 0
$$430$$ −1.05966e9 −0.642726
$$431$$ 9.81508e8 0.590505 0.295252 0.955419i $$-0.404596\pi$$
0.295252 + 0.955419i $$0.404596\pi$$
$$432$$ −2.07913e8 −0.124076
$$433$$ 2.84998e9 1.68707 0.843537 0.537071i $$-0.180469\pi$$
0.843537 + 0.537071i $$0.180469\pi$$
$$434$$ 1.84954e9 1.08605
$$435$$ −2.58476e8 −0.150560
$$436$$ −7.84450e8 −0.453276
$$437$$ 2.74436e9 1.57310
$$438$$ −9.41071e7 −0.0535134
$$439$$ −1.05622e9 −0.595838 −0.297919 0.954591i $$-0.596292\pi$$
−0.297919 + 0.954591i $$0.596292\pi$$
$$440$$ −1.17412e8 −0.0657094
$$441$$ −4.26401e8 −0.236746
$$442$$ 0 0
$$443$$ 1.82325e9 0.996401 0.498201 0.867062i $$-0.333994\pi$$
0.498201 + 0.867062i $$0.333994\pi$$
$$444$$ −1.23284e8 −0.0668446
$$445$$ 2.50975e9 1.35011
$$446$$ 1.17166e8 0.0625358
$$447$$ 2.62633e8 0.139082
$$448$$ −2.66338e8 −0.139946
$$449$$ −1.84846e9 −0.963713 −0.481856 0.876250i $$-0.660037\pi$$
−0.481856 + 0.876250i $$0.660037\pi$$
$$450$$ 5.56105e8 0.287682
$$451$$ 1.18395e7 0.00607735
$$452$$ 1.06208e9 0.540968
$$453$$ 3.52985e8 0.178407
$$454$$ 1.47633e9 0.740435
$$455$$ 0 0
$$456$$ 2.45391e8 0.121194
$$457$$ 2.98066e9 1.46085 0.730425 0.682993i $$-0.239322\pi$$
0.730425 + 0.682993i $$0.239322\pi$$
$$458$$ 7.00369e7 0.0340642
$$459$$ −7.46477e8 −0.360306
$$460$$ 9.23489e8 0.442364
$$461$$ 2.52781e9 1.20169 0.600843 0.799367i $$-0.294832\pi$$
0.600843 + 0.799367i $$0.294832\pi$$
$$462$$ 1.06509e8 0.0502505
$$463$$ 8.90291e8 0.416868 0.208434 0.978036i $$-0.433163\pi$$
0.208434 + 0.978036i $$0.433163\pi$$
$$464$$ −4.20127e8 −0.195239
$$465$$ −5.73431e8 −0.264482
$$466$$ −9.56450e8 −0.437836
$$467$$ 2.65667e9 1.20706 0.603529 0.797341i $$-0.293761\pi$$
0.603529 + 0.797341i $$0.293761\pi$$
$$468$$ 0 0
$$469$$ −1.28020e8 −0.0573024
$$470$$ −7.94062e8 −0.352786
$$471$$ 7.26660e8 0.320448
$$472$$ −1.35202e9 −0.591814
$$473$$ 6.88777e8 0.299271
$$474$$ −3.42413e8 −0.147681
$$475$$ −1.35896e9 −0.581806
$$476$$ −9.56243e8 −0.406391
$$477$$ 3.05228e9 1.28769
$$478$$ −3.16967e9 −1.32745
$$479$$ −1.30093e9 −0.540855 −0.270428 0.962740i $$-0.587165\pi$$
−0.270428 + 0.962740i $$0.587165\pi$$
$$480$$ 8.25754e7 0.0340805
$$481$$ 0 0
$$482$$ 2.05285e9 0.835012
$$483$$ −8.37737e8 −0.338293
$$484$$ −1.17086e9 −0.469404
$$485$$ −1.82325e9 −0.725689
$$486$$ 1.25855e9 0.497330
$$487$$ 1.07447e9 0.421542 0.210771 0.977535i $$-0.432402\pi$$
0.210771 + 0.977535i $$0.432402\pi$$
$$488$$ 4.23783e8 0.165072
$$489$$ −6.85024e8 −0.264926
$$490$$ 3.50638e8 0.134640
$$491$$ −7.83344e8 −0.298653 −0.149327 0.988788i $$-0.547711\pi$$
−0.149327 + 0.988788i $$0.547711\pi$$
$$492$$ −8.32666e6 −0.00315205
$$493$$ −1.50839e9 −0.566958
$$494$$ 0 0
$$495$$ 4.68501e8 0.173617
$$496$$ −9.32053e8 −0.342969
$$497$$ −1.43736e9 −0.525193
$$498$$ −5.44598e8 −0.197594
$$499$$ 6.23188e8 0.224526 0.112263 0.993679i $$-0.464190\pi$$
0.112263 + 0.993679i $$0.464190\pi$$
$$500$$ −1.50730e9 −0.539267
$$501$$ 1.05272e9 0.374007
$$502$$ −5.87820e8 −0.207387
$$503$$ −2.70927e9 −0.949215 −0.474607 0.880198i $$-0.657410\pi$$
−0.474607 + 0.880198i $$0.657410\pi$$
$$504$$ 1.06275e9 0.369764
$$505$$ −2.11670e9 −0.731375
$$506$$ −6.00268e8 −0.205977
$$507$$ 0 0
$$508$$ 7.47684e7 0.0253043
$$509$$ −3.49943e9 −1.17621 −0.588106 0.808784i $$-0.700126\pi$$
−0.588106 + 0.808784i $$0.700126\pi$$
$$510$$ 2.96473e8 0.0989668
$$511$$ 9.95967e8 0.330196
$$512$$ 1.34218e8 0.0441942
$$513$$ −2.02735e9 −0.663008
$$514$$ −1.62161e9 −0.526714
$$515$$ 7.87078e8 0.253918
$$516$$ −4.84414e8 −0.155218
$$517$$ 5.16140e8 0.164267
$$518$$ 1.30476e9 0.412453
$$519$$ 1.02835e8 0.0322889
$$520$$ 0 0
$$521$$ −1.37683e9 −0.426530 −0.213265 0.976994i $$-0.568410\pi$$
−0.213265 + 0.976994i $$0.568410\pi$$
$$522$$ 1.67640e9 0.515860
$$523$$ −2.86154e9 −0.874669 −0.437334 0.899299i $$-0.644077\pi$$
−0.437334 + 0.899299i $$0.644077\pi$$
$$524$$ −5.07125e8 −0.153977
$$525$$ 4.14833e8 0.125117
$$526$$ 1.23403e9 0.369723
$$527$$ −3.34638e9 −0.995951
$$528$$ −5.36740e7 −0.0158688
$$529$$ 1.31651e9 0.386661
$$530$$ −2.50995e9 −0.732318
$$531$$ 5.39487e9 1.56369
$$532$$ −2.59706e9 −0.747810
$$533$$ 0 0
$$534$$ 1.14731e9 0.326053
$$535$$ −3.77697e9 −1.06636
$$536$$ 6.45140e7 0.0180958
$$537$$ 2.25649e8 0.0628815
$$538$$ −4.99215e9 −1.38213
$$539$$ −2.27915e8 −0.0626919
$$540$$ −6.82214e8 −0.186442
$$541$$ −5.34467e9 −1.45121 −0.725605 0.688111i $$-0.758440\pi$$
−0.725605 + 0.688111i $$0.758440\pi$$
$$542$$ 3.10387e9 0.837347
$$543$$ −7.19550e8 −0.192869
$$544$$ 4.81886e8 0.128336
$$545$$ −2.57398e9 −0.681109
$$546$$ 0 0
$$547$$ −3.37135e9 −0.880740 −0.440370 0.897816i $$-0.645153\pi$$
−0.440370 + 0.897816i $$0.645153\pi$$
$$548$$ 2.02019e7 0.00524396
$$549$$ −1.69100e9 −0.436153
$$550$$ 2.97242e8 0.0761801
$$551$$ −4.09665e9 −1.04327
$$552$$ 4.22167e8 0.106831
$$553$$ 3.62387e9 0.911244
$$554$$ 3.63162e9 0.907436
$$555$$ −4.04526e8 −0.100443
$$556$$ 2.50904e9 0.619079
$$557$$ 5.61106e9 1.37579 0.687894 0.725811i $$-0.258535\pi$$
0.687894 + 0.725811i $$0.258535\pi$$
$$558$$ 3.71911e9 0.906190
$$559$$ 0 0
$$560$$ −8.73923e8 −0.210288
$$561$$ −1.92707e8 −0.0460817
$$562$$ −2.67016e9 −0.634542
$$563$$ 6.69690e9 1.58159 0.790795 0.612081i $$-0.209667\pi$$
0.790795 + 0.612081i $$0.209667\pi$$
$$564$$ −3.63000e8 −0.0851980
$$565$$ 3.48494e9 0.812879
$$566$$ 4.30156e9 0.997168
$$567$$ −3.92066e9 −0.903273
$$568$$ 7.24341e8 0.165853
$$569$$ 1.96850e9 0.447964 0.223982 0.974593i $$-0.428094\pi$$
0.223982 + 0.974593i $$0.428094\pi$$
$$570$$ 8.05190e8 0.182111
$$571$$ 1.02926e9 0.231365 0.115682 0.993286i $$-0.463094\pi$$
0.115682 + 0.993286i $$0.463094\pi$$
$$572$$ 0 0
$$573$$ 1.12783e9 0.250440
$$574$$ 8.81238e7 0.0194492
$$575$$ −2.33793e9 −0.512853
$$576$$ −5.35560e8 −0.116770
$$577$$ −3.31179e9 −0.717708 −0.358854 0.933394i $$-0.616832\pi$$
−0.358854 + 0.933394i $$0.616832\pi$$
$$578$$ −1.55258e9 −0.334431
$$579$$ 4.22335e8 0.0904236
$$580$$ −1.37854e9 −0.293374
$$581$$ 5.76366e9 1.21922
$$582$$ −8.33486e8 −0.175254
$$583$$ 1.63147e9 0.340988
$$584$$ −5.01904e8 −0.104274
$$585$$ 0 0
$$586$$ −2.68480e9 −0.551151
$$587$$ 5.59411e8 0.114156 0.0570778 0.998370i $$-0.481822\pi$$
0.0570778 + 0.998370i $$0.481822\pi$$
$$588$$ 1.60292e8 0.0325155
$$589$$ −9.08843e9 −1.83267
$$590$$ −4.43631e9 −0.889282
$$591$$ −1.23582e9 −0.246264
$$592$$ −6.57514e8 −0.130250
$$593$$ 3.02459e9 0.595628 0.297814 0.954624i $$-0.403742\pi$$
0.297814 + 0.954624i $$0.403742\pi$$
$$594$$ 4.43439e8 0.0868124
$$595$$ −3.13767e9 −0.610658
$$596$$ 1.40071e9 0.271010
$$597$$ 1.00365e9 0.193051
$$598$$ 0 0
$$599$$ −5.63246e9 −1.07079 −0.535395 0.844602i $$-0.679837\pi$$
−0.535395 + 0.844602i $$0.679837\pi$$
$$600$$ −2.09050e8 −0.0395112
$$601$$ 3.40792e8 0.0640366 0.0320183 0.999487i $$-0.489807\pi$$
0.0320183 + 0.999487i $$0.489807\pi$$
$$602$$ 5.12672e9 0.957749
$$603$$ −2.57426e8 −0.0478126
$$604$$ 1.88259e9 0.347637
$$605$$ −3.84189e9 −0.705344
$$606$$ −9.67636e8 −0.176627
$$607$$ 3.85420e9 0.699477 0.349739 0.936847i $$-0.386270\pi$$
0.349739 + 0.936847i $$0.386270\pi$$
$$608$$ 1.30875e9 0.236154
$$609$$ 1.25053e9 0.224355
$$610$$ 1.39054e9 0.248044
$$611$$ 0 0
$$612$$ −1.92284e9 −0.339088
$$613$$ −9.22245e9 −1.61709 −0.808545 0.588434i $$-0.799745\pi$$
−0.808545 + 0.588434i $$0.799745\pi$$
$$614$$ −1.72023e9 −0.299915
$$615$$ −2.73218e7 −0.00473639
$$616$$ 5.68050e8 0.0979160
$$617$$ −6.53611e9 −1.12027 −0.560133 0.828402i $$-0.689250\pi$$
−0.560133 + 0.828402i $$0.689250\pi$$
$$618$$ 3.59807e8 0.0613211
$$619$$ −1.36559e9 −0.231420 −0.115710 0.993283i $$-0.536914\pi$$
−0.115710 + 0.993283i $$0.536914\pi$$
$$620$$ −3.05830e9 −0.515358
$$621$$ −3.48782e9 −0.584431
$$622$$ 6.33649e9 1.05580
$$623$$ −1.21424e10 −2.01186
$$624$$ 0 0
$$625$$ −2.28761e9 −0.374802
$$626$$ −9.47659e8 −0.154398
$$627$$ −5.23374e8 −0.0847960
$$628$$ 3.87552e9 0.624412
$$629$$ −2.36070e9 −0.378236
$$630$$ 3.48716e9 0.555622
$$631$$ −1.54079e9 −0.244141 −0.122070 0.992521i $$-0.538953\pi$$
−0.122070 + 0.992521i $$0.538953\pi$$
$$632$$ −1.82620e9 −0.287766
$$633$$ −1.16881e9 −0.183160
$$634$$ 4.05848e8 0.0632486
$$635$$ 2.45334e8 0.0380233
$$636$$ −1.14741e9 −0.176855
$$637$$ 0 0
$$638$$ 8.96052e8 0.136603
$$639$$ −2.89029e9 −0.438216
$$640$$ 4.40402e8 0.0664078
$$641$$ −4.54018e9 −0.680879 −0.340440 0.940266i $$-0.610576\pi$$
−0.340440 + 0.940266i $$0.610576\pi$$
$$642$$ −1.72661e9 −0.257527
$$643$$ −1.14054e10 −1.69189 −0.845944 0.533272i $$-0.820962\pi$$
−0.845944 + 0.533272i $$0.820962\pi$$
$$644$$ −4.46793e9 −0.659183
$$645$$ −1.58948e9 −0.233237
$$646$$ 4.69886e9 0.685770
$$647$$ −1.26393e10 −1.83468 −0.917338 0.398109i $$-0.869666\pi$$
−0.917338 + 0.398109i $$0.869666\pi$$
$$648$$ 1.97577e9 0.285248
$$649$$ 2.88360e9 0.414075
$$650$$ 0 0
$$651$$ 2.77431e9 0.394114
$$652$$ −3.65346e9 −0.516223
$$653$$ −1.05004e10 −1.47575 −0.737873 0.674940i $$-0.764170\pi$$
−0.737873 + 0.674940i $$0.764170\pi$$
$$654$$ −1.17667e9 −0.164488
$$655$$ −1.66400e9 −0.231371
$$656$$ −4.44088e7 −0.00614194
$$657$$ 2.00272e9 0.275512
$$658$$ 3.84175e9 0.525700
$$659$$ 9.64818e9 1.31325 0.656624 0.754219i $$-0.271984\pi$$
0.656624 + 0.754219i $$0.271984\pi$$
$$660$$ −1.76118e8 −0.0238451
$$661$$ 6.58299e9 0.886580 0.443290 0.896378i $$-0.353811\pi$$
0.443290 + 0.896378i $$0.353811\pi$$
$$662$$ −2.19006e9 −0.293395
$$663$$ 0 0
$$664$$ −2.90452e9 −0.385023
$$665$$ −8.52160e9 −1.12369
$$666$$ 2.62364e9 0.344147
$$667$$ −7.04779e9 −0.919629
$$668$$ 5.61450e9 0.728775
$$669$$ 1.75749e8 0.0226935
$$670$$ 2.11687e8 0.0271914
$$671$$ −9.03851e8 −0.115496
$$672$$ −3.99507e8 −0.0507846
$$673$$ −8.54649e9 −1.08077 −0.540387 0.841416i $$-0.681722\pi$$
−0.540387 + 0.841416i $$0.681722\pi$$
$$674$$ −7.34810e8 −0.0924411
$$675$$ 1.72711e9 0.216151
$$676$$ 0 0
$$677$$ 8.71305e9 1.07922 0.539610 0.841915i $$-0.318572\pi$$
0.539610 + 0.841915i $$0.318572\pi$$
$$678$$ 1.59312e9 0.196311
$$679$$ 8.82106e9 1.08138
$$680$$ 1.58119e9 0.192842
$$681$$ 2.21449e9 0.268695
$$682$$ 1.98789e9 0.239965
$$683$$ −1.46109e10 −1.75470 −0.877351 0.479849i $$-0.840692\pi$$
−0.877351 + 0.479849i $$0.840692\pi$$
$$684$$ −5.22223e9 −0.623965
$$685$$ 6.62873e7 0.00787977
$$686$$ 4.99734e9 0.591023
$$687$$ 1.05055e8 0.0123615
$$688$$ −2.58354e9 −0.302452
$$689$$ 0 0
$$690$$ 1.38523e9 0.160528
$$691$$ 1.47348e10 1.69891 0.849454 0.527662i $$-0.176931\pi$$
0.849454 + 0.527662i $$0.176931\pi$$
$$692$$ 5.48451e8 0.0629167
$$693$$ −2.26665e9 −0.258713
$$694$$ −1.09360e10 −1.24194
$$695$$ 8.23279e9 0.930252
$$696$$ −6.30190e8 −0.0708499
$$697$$ −1.59442e8 −0.0178357
$$698$$ −9.05146e9 −1.00745
$$699$$ −1.43467e9 −0.158885
$$700$$ 2.21244e9 0.243797
$$701$$ 1.31502e9 0.144185 0.0720923 0.997398i $$-0.477032\pi$$
0.0720923 + 0.997398i $$0.477032\pi$$
$$702$$ 0 0
$$703$$ −6.41141e9 −0.696001
$$704$$ −2.86261e8 −0.0309213
$$705$$ −1.19109e9 −0.128022
$$706$$ 3.58716e8 0.0383649
$$707$$ 1.02408e10 1.08985
$$708$$ −2.02803e9 −0.214762
$$709$$ −6.64028e8 −0.0699721 −0.0349860 0.999388i $$-0.511139\pi$$
−0.0349860 + 0.999388i $$0.511139\pi$$
$$710$$ 2.37674e9 0.249217
$$711$$ 7.28697e9 0.760332
$$712$$ 6.11901e9 0.635333
$$713$$ −1.56356e10 −1.61547
$$714$$ −1.43436e9 −0.147474
$$715$$ 0 0
$$716$$ 1.20346e9 0.122528
$$717$$ −4.75451e9 −0.481713
$$718$$ −3.18624e9 −0.321250
$$719$$ 4.95034e9 0.496689 0.248344 0.968672i $$-0.420114\pi$$
0.248344 + 0.968672i $$0.420114\pi$$
$$720$$ −1.75731e9 −0.175462
$$721$$ −3.80796e9 −0.378372
$$722$$ 5.61065e9 0.554796
$$723$$ 3.07928e9 0.303015
$$724$$ −3.83760e9 −0.375816
$$725$$ 3.48994e9 0.340123
$$726$$ −1.75629e9 −0.170341
$$727$$ 8.81101e9 0.850463 0.425231 0.905085i $$-0.360193\pi$$
0.425231 + 0.905085i $$0.360193\pi$$
$$728$$ 0 0
$$729$$ −6.55163e9 −0.626330
$$730$$ −1.64687e9 −0.156686
$$731$$ −9.27578e9 −0.878293
$$732$$ 6.35675e8 0.0599027
$$733$$ 1.49414e8 0.0140129 0.00700643 0.999975i $$-0.497770\pi$$
0.00700643 + 0.999975i $$0.497770\pi$$
$$734$$ 1.30777e10 1.22066
$$735$$ 5.25957e8 0.0488590
$$736$$ 2.25155e9 0.208166
$$737$$ −1.37596e8 −0.0126611
$$738$$ 1.77202e8 0.0162282
$$739$$ 4.70806e9 0.429127 0.214564 0.976710i $$-0.431167\pi$$
0.214564 + 0.976710i $$0.431167\pi$$
$$740$$ −2.15747e9 −0.195719
$$741$$ 0 0
$$742$$ 1.21434e10 1.09125
$$743$$ −1.69676e9 −0.151761 −0.0758805 0.997117i $$-0.524177\pi$$
−0.0758805 + 0.997117i $$0.524177\pi$$
$$744$$ −1.39808e9 −0.124459
$$745$$ 4.59607e9 0.407230
$$746$$ −1.23707e10 −1.09095
$$747$$ 1.15897e10 1.01730
$$748$$ −1.02777e9 −0.0897928
$$749$$ 1.82733e10 1.58903
$$750$$ −2.26094e9 −0.195693
$$751$$ 1.06650e10 0.918800 0.459400 0.888229i $$-0.348064\pi$$
0.459400 + 0.888229i $$0.348064\pi$$
$$752$$ −1.93600e9 −0.166013
$$753$$ −8.81731e8 −0.0752581
$$754$$ 0 0
$$755$$ 6.17724e9 0.522373
$$756$$ 3.30062e9 0.277824
$$757$$ 6.22876e9 0.521874 0.260937 0.965356i $$-0.415968\pi$$
0.260937 + 0.965356i $$0.415968\pi$$
$$758$$ 8.45506e9 0.705138
$$759$$ −9.00402e8 −0.0747464
$$760$$ 4.29435e9 0.354854
$$761$$ 8.38334e9 0.689558 0.344779 0.938684i $$-0.387954\pi$$
0.344779 + 0.938684i $$0.387954\pi$$
$$762$$ 1.12153e8 0.00918263
$$763$$ 1.24531e10 1.01495
$$764$$ 6.01511e9 0.487997
$$765$$ −6.30932e9 −0.509527
$$766$$ −1.79928e9 −0.144643
$$767$$ 0 0
$$768$$ 2.01327e8 0.0160375
$$769$$ 1.18649e10 0.940852 0.470426 0.882439i $$-0.344100\pi$$
0.470426 + 0.882439i $$0.344100\pi$$
$$770$$ 1.86391e9 0.147132
$$771$$ −2.43241e9 −0.191138
$$772$$ 2.25245e9 0.176196
$$773$$ −5.56680e9 −0.433488 −0.216744 0.976228i $$-0.569544\pi$$
−0.216744 + 0.976228i $$0.569544\pi$$
$$774$$ 1.03089e10 0.799136
$$775$$ 7.74246e9 0.597479
$$776$$ −4.44526e9 −0.341492
$$777$$ 1.95713e9 0.149674
$$778$$ 8.14306e9 0.619953
$$779$$ −4.33029e8 −0.0328198
$$780$$ 0 0
$$781$$ −1.54488e9 −0.116042
$$782$$ 8.08383e9 0.604496
$$783$$ 5.20645e9 0.387593
$$784$$ 8.54888e8 0.0633583
$$785$$ 1.27165e10 0.938264
$$786$$ −7.60687e8 −0.0558763
$$787$$ −1.34611e8 −0.00984395 −0.00492198 0.999988i $$-0.501567\pi$$
−0.00492198 + 0.999988i $$0.501567\pi$$
$$788$$ −6.59106e9 −0.479859
$$789$$ 1.85105e9 0.134168
$$790$$ −5.99222e9 −0.432408
$$791$$ −1.68605e10 −1.21130
$$792$$ 1.14225e9 0.0817001
$$793$$ 0 0
$$794$$ 1.18052e10 0.836955
$$795$$ −3.76493e9 −0.265749
$$796$$ 5.35280e9 0.376171
$$797$$ −7.41548e9 −0.518842 −0.259421 0.965764i $$-0.583532\pi$$
−0.259421 + 0.965764i $$0.583532\pi$$
$$798$$ −3.89559e9 −0.271371
$$799$$ −6.95088e9 −0.482088
$$800$$ −1.11493e9 −0.0769898
$$801$$ −2.44163e10 −1.67867
$$802$$ −2.19930e9 −0.150548
$$803$$ 1.07047e9 0.0729574
$$804$$ 9.67711e7 0.00656673
$$805$$ −1.46604e10 −0.990513
$$806$$ 0 0
$$807$$ −7.48822e9 −0.501558
$$808$$ −5.16072e9 −0.344168
$$809$$ −1.41542e10 −0.939863 −0.469932 0.882703i $$-0.655721\pi$$
−0.469932 + 0.882703i $$0.655721\pi$$
$$810$$ 6.48299e9 0.428625
$$811$$ 2.63708e10 1.73600 0.868001 0.496563i $$-0.165405\pi$$
0.868001 + 0.496563i $$0.165405\pi$$
$$812$$ 6.66951e9 0.437168
$$813$$ 4.65580e9 0.303863
$$814$$ 1.40236e9 0.0911324
$$815$$ −1.19879e10 −0.775697
$$816$$ 7.22829e8 0.0465715
$$817$$ −2.51921e10 −1.61617
$$818$$ 1.30742e10 0.835176
$$819$$ 0 0
$$820$$ −1.45716e8 −0.00922912
$$821$$ −8.06264e9 −0.508483 −0.254241 0.967141i $$-0.581826\pi$$
−0.254241 + 0.967141i $$0.581826\pi$$
$$822$$ 3.03028e7 0.00190297
$$823$$ −2.34202e10 −1.46451 −0.732253 0.681033i $$-0.761531\pi$$
−0.732253 + 0.681033i $$0.761531\pi$$
$$824$$ 1.91897e9 0.119488
$$825$$ 4.45864e8 0.0276448
$$826$$ 2.14633e10 1.32515
$$827$$ −5.55722e9 −0.341655 −0.170828 0.985301i $$-0.554644\pi$$
−0.170828 + 0.985301i $$0.554644\pi$$
$$828$$ −8.98423e9 −0.550015
$$829$$ 2.84256e10 1.73288 0.866440 0.499281i $$-0.166403\pi$$
0.866440 + 0.499281i $$0.166403\pi$$
$$830$$ −9.53046e9 −0.578549
$$831$$ 5.44743e9 0.329297
$$832$$ 0 0
$$833$$ 3.06933e9 0.183987
$$834$$ 3.76356e9 0.224656
$$835$$ 1.84226e10 1.09508
$$836$$ −2.79133e9 −0.165230
$$837$$ 1.15505e10 0.680868
$$838$$ −8.90238e9 −0.522579
$$839$$ −1.04036e10 −0.608156 −0.304078 0.952647i $$-0.598348\pi$$
−0.304078 + 0.952647i $$0.598348\pi$$
$$840$$ −1.31088e9 −0.0763109
$$841$$ −6.72927e9 −0.390105
$$842$$ −7.38023e9 −0.426066
$$843$$ −4.00524e9 −0.230267
$$844$$ −6.23367e9 −0.356899
$$845$$ 0 0
$$846$$ 7.72509e9 0.438639
$$847$$ 1.85874e10 1.05106
$$848$$ −6.11950e9 −0.344612
$$849$$ 6.45234e9 0.361860
$$850$$ −4.00297e9 −0.223572
$$851$$ −1.10301e10 −0.613514
$$852$$ 1.08651e9 0.0601860
$$853$$ 1.80580e10 0.996205 0.498102 0.867118i $$-0.334030\pi$$
0.498102 + 0.867118i $$0.334030\pi$$
$$854$$ −6.72756e9 −0.369620
$$855$$ −1.71355e10 −0.937593
$$856$$ −9.20861e9 −0.501806
$$857$$ −6.34034e9 −0.344096 −0.172048 0.985089i $$-0.555038\pi$$
−0.172048 + 0.985089i $$0.555038\pi$$
$$858$$ 0 0
$$859$$ 1.21489e10 0.653973 0.326987 0.945029i $$-0.393967\pi$$
0.326987 + 0.945029i $$0.393967\pi$$
$$860$$ −8.47725e9 −0.454476
$$861$$ 1.32186e8 0.00705786
$$862$$ 7.85206e9 0.417550
$$863$$ 2.87111e10 1.52059 0.760295 0.649578i $$-0.225054\pi$$
0.760295 + 0.649578i $$0.225054\pi$$
$$864$$ −1.66330e9 −0.0877351
$$865$$ 1.79960e9 0.0945411
$$866$$ 2.27998e10 1.19294
$$867$$ −2.32887e9 −0.121361
$$868$$ 1.47963e10 0.767954
$$869$$ 3.89495e9 0.201341
$$870$$ −2.06781e9 −0.106462
$$871$$ 0 0
$$872$$ −6.27560e9 −0.320514
$$873$$ 1.77376e10 0.902289
$$874$$ 2.19549e10 1.11235
$$875$$ 2.39283e10 1.20749
$$876$$ −7.52857e8 −0.0378397
$$877$$ −2.46021e10 −1.23161 −0.615806 0.787898i $$-0.711169\pi$$
−0.615806 + 0.787898i $$0.711169\pi$$
$$878$$ −8.44975e9 −0.421321
$$879$$ −4.02720e9 −0.200006
$$880$$ −9.39295e8 −0.0464636
$$881$$ −1.25378e10 −0.617738 −0.308869 0.951105i $$-0.599951\pi$$
−0.308869 + 0.951105i $$0.599951\pi$$
$$882$$ −3.41121e9 −0.167405
$$883$$ 1.93097e10 0.943873 0.471937 0.881633i $$-0.343555\pi$$
0.471937 + 0.881633i $$0.343555\pi$$
$$884$$ 0 0
$$885$$ −6.65446e9 −0.322709
$$886$$ 1.45860e10 0.704562
$$887$$ 3.20268e10 1.54092 0.770462 0.637486i $$-0.220026\pi$$
0.770462 + 0.637486i $$0.220026\pi$$
$$888$$ −9.86272e8 −0.0472663
$$889$$ −1.18695e9 −0.0566599
$$890$$ 2.00780e10 0.954675
$$891$$ −4.21394e9 −0.199580
$$892$$ 9.37327e8 0.0442195
$$893$$ −1.88779e10 −0.887101
$$894$$ 2.10106e9 0.0983461
$$895$$ 3.94885e9 0.184115
$$896$$ −2.13071e9 −0.0989568
$$897$$ 0 0
$$898$$ −1.47877e10 −0.681448
$$899$$ 2.33400e10 1.07138
$$900$$ 4.44884e9 0.203422
$$901$$ −2.19710e10 −1.00072
$$902$$ 9.47157e7 0.00429733
$$903$$ 7.69008e9 0.347555
$$904$$ 8.49662e9 0.382522
$$905$$ −1.25921e10 −0.564715
$$906$$ 2.82388e9 0.126153
$$907$$ 2.33703e9 0.104002 0.0520008 0.998647i $$-0.483440\pi$$
0.0520008 + 0.998647i $$0.483440\pi$$
$$908$$ 1.18106e10 0.523567
$$909$$ 2.05925e10 0.909359
$$910$$ 0 0
$$911$$ 2.20343e10 0.965573 0.482786 0.875738i $$-0.339625\pi$$
0.482786 + 0.875738i $$0.339625\pi$$
$$912$$ 1.96313e9 0.0856973
$$913$$ 6.19480e9 0.269389
$$914$$ 2.38453e10 1.03298
$$915$$ 2.08581e9 0.0900121
$$916$$ 5.60295e8 0.0240870
$$917$$ 8.05061e9 0.344775
$$918$$ −5.97181e9 −0.254775
$$919$$ −1.43277e10 −0.608938 −0.304469 0.952522i $$-0.598479\pi$$
−0.304469 + 0.952522i $$0.598479\pi$$
$$920$$ 7.38791e9 0.312798
$$921$$ −2.58035e9 −0.108835
$$922$$ 2.02225e10 0.849720
$$923$$ 0 0
$$924$$ 8.52074e8 0.0355325
$$925$$ 5.46190e9 0.226907
$$926$$ 7.12233e9 0.294770
$$927$$ −7.65715e9 −0.315710
$$928$$ −3.36101e9 −0.138055
$$929$$ −1.31280e10 −0.537208 −0.268604 0.963251i $$-0.586562\pi$$
−0.268604 + 0.963251i $$0.586562\pi$$
$$930$$ −4.58745e9 −0.187017
$$931$$ 8.33600e9 0.338558
$$932$$ −7.65160e9 −0.309597
$$933$$ 9.50474e9 0.383137
$$934$$ 2.12533e10 0.853519
$$935$$ −3.37238e9 −0.134926
$$936$$ 0 0
$$937$$ −3.87626e10 −1.53930 −0.769652 0.638463i $$-0.779571\pi$$
−0.769652 + 0.638463i $$0.779571\pi$$
$$938$$ −1.02416e9 −0.0405189
$$939$$ −1.42149e9 −0.0560291
$$940$$ −6.35250e9 −0.249458
$$941$$ −2.06279e10 −0.807035 −0.403517 0.914972i $$-0.632212\pi$$
−0.403517 + 0.914972i $$0.632212\pi$$
$$942$$ 5.81328e9 0.226591
$$943$$ −7.44976e8 −0.0289302
$$944$$ −1.08161e10 −0.418476
$$945$$ 1.08302e10 0.417468
$$946$$ 5.51021e9 0.211617
$$947$$ 2.11705e10 0.810040 0.405020 0.914308i $$-0.367264\pi$$
0.405020 + 0.914308i $$0.367264\pi$$
$$948$$ −2.73930e9 −0.104427
$$949$$ 0 0
$$950$$ −1.08717e10 −0.411399
$$951$$ 6.08771e8 0.0229521
$$952$$ −7.64994e9 −0.287362
$$953$$ 2.14876e10 0.804196 0.402098 0.915597i $$-0.368281\pi$$
0.402098 + 0.915597i $$0.368281\pi$$
$$954$$ 2.44182e10 0.910531
$$955$$ 1.97371e10 0.733282
$$956$$ −2.53574e10 −0.938646
$$957$$ 1.34408e9 0.0495715
$$958$$ −1.04075e10 −0.382442
$$959$$ −3.20704e8 −0.0117419
$$960$$ 6.60603e8 0.0240986
$$961$$ 2.42673e10 0.882043
$$962$$ 0 0
$$963$$ 3.67445e10 1.32587
$$964$$ 1.64228e10 0.590443
$$965$$ 7.39086e9 0.264758
$$966$$ −6.70189e9 −0.239209
$$967$$ −3.92625e10 −1.39632 −0.698161 0.715941i $$-0.745998\pi$$
−0.698161 + 0.715941i $$0.745998\pi$$
$$968$$ −9.36689e9 −0.331919
$$969$$ 7.04829e9 0.248857
$$970$$ −1.45860e10 −0.513139
$$971$$ −5.62647e10 −1.97228 −0.986140 0.165917i $$-0.946941\pi$$
−0.986140 + 0.165917i $$0.946941\pi$$
$$972$$ 1.00684e10 0.351665
$$973$$ −3.98310e10 −1.38620
$$974$$ 8.59573e9 0.298076
$$975$$ 0 0
$$976$$ 3.39027e9 0.116724
$$977$$ 8.43437e9 0.289349 0.144674 0.989479i $$-0.453787\pi$$
0.144674 + 0.989479i $$0.453787\pi$$
$$978$$ −5.48019e9 −0.187331
$$979$$ −1.30507e10 −0.444523
$$980$$ 2.80510e9 0.0952045
$$981$$ 2.50411e10 0.846861
$$982$$ −6.26675e9 −0.211180
$$983$$ 2.24230e10 0.752932 0.376466 0.926430i $$-0.377139\pi$$
0.376466 + 0.926430i $$0.377139\pi$$
$$984$$ −6.66132e7 −0.00222883
$$985$$ −2.16269e10 −0.721054
$$986$$ −1.20672e10 −0.400900
$$987$$ 5.76262e9 0.190770
$$988$$ 0 0
$$989$$ −4.33400e10 −1.42463
$$990$$ 3.74801e9 0.122766
$$991$$ 3.46728e10 1.13170 0.565849 0.824509i $$-0.308548\pi$$
0.565849 + 0.824509i $$0.308548\pi$$
$$992$$ −7.45642e9 −0.242516
$$993$$ −3.28508e9 −0.106469
$$994$$ −1.14989e10 −0.371368
$$995$$ 1.75639e10 0.565249
$$996$$ −4.35678e9 −0.139720
$$997$$ −2.96474e10 −0.947444 −0.473722 0.880674i $$-0.657090\pi$$
−0.473722 + 0.880674i $$0.657090\pi$$
$$998$$ 4.98550e9 0.158764
$$999$$ 8.14830e9 0.258576
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 338.8.a.d.1.1 1
13.5 odd 4 338.8.b.d.337.1 2
13.8 odd 4 338.8.b.d.337.2 2
13.12 even 2 2.8.a.a.1.1 1
39.38 odd 2 18.8.a.b.1.1 1
52.51 odd 2 16.8.a.b.1.1 1
65.12 odd 4 50.8.b.c.49.1 2
65.38 odd 4 50.8.b.c.49.2 2
65.64 even 2 50.8.a.g.1.1 1
91.12 odd 6 98.8.c.e.67.1 2
91.25 even 6 98.8.c.d.79.1 2
91.38 odd 6 98.8.c.e.79.1 2
91.51 even 6 98.8.c.d.67.1 2
91.90 odd 2 98.8.a.a.1.1 1
104.51 odd 2 64.8.a.e.1.1 1
104.77 even 2 64.8.a.c.1.1 1
117.25 even 6 162.8.c.l.109.1 2
117.38 odd 6 162.8.c.a.109.1 2
117.77 odd 6 162.8.c.a.55.1 2
117.103 even 6 162.8.c.l.55.1 2
143.142 odd 2 242.8.a.e.1.1 1
156.155 even 2 144.8.a.i.1.1 1
195.38 even 4 450.8.c.g.199.1 2
195.77 even 4 450.8.c.g.199.2 2
195.194 odd 2 450.8.a.c.1.1 1
208.51 odd 4 256.8.b.f.129.1 2
208.77 even 4 256.8.b.b.129.2 2
208.155 odd 4 256.8.b.f.129.2 2
208.181 even 4 256.8.b.b.129.1 2
221.220 even 2 578.8.a.b.1.1 1
260.103 even 4 400.8.c.j.49.1 2
260.207 even 4 400.8.c.j.49.2 2
260.259 odd 2 400.8.a.l.1.1 1
312.77 odd 2 576.8.a.g.1.1 1
312.155 even 2 576.8.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 1 13.12 even 2
16.8.a.b.1.1 1 52.51 odd 2
18.8.a.b.1.1 1 39.38 odd 2
50.8.a.g.1.1 1 65.64 even 2
50.8.b.c.49.1 2 65.12 odd 4
50.8.b.c.49.2 2 65.38 odd 4
64.8.a.c.1.1 1 104.77 even 2
64.8.a.e.1.1 1 104.51 odd 2
98.8.a.a.1.1 1 91.90 odd 2
98.8.c.d.67.1 2 91.51 even 6
98.8.c.d.79.1 2 91.25 even 6
98.8.c.e.67.1 2 91.12 odd 6
98.8.c.e.79.1 2 91.38 odd 6
144.8.a.i.1.1 1 156.155 even 2
162.8.c.a.55.1 2 117.77 odd 6
162.8.c.a.109.1 2 117.38 odd 6
162.8.c.l.55.1 2 117.103 even 6
162.8.c.l.109.1 2 117.25 even 6
242.8.a.e.1.1 1 143.142 odd 2
256.8.b.b.129.1 2 208.181 even 4
256.8.b.b.129.2 2 208.77 even 4
256.8.b.f.129.1 2 208.51 odd 4
256.8.b.f.129.2 2 208.155 odd 4
338.8.a.d.1.1 1 1.1 even 1 trivial
338.8.b.d.337.1 2 13.5 odd 4
338.8.b.d.337.2 2 13.8 odd 4
400.8.a.l.1.1 1 260.259 odd 2
400.8.c.j.49.1 2 260.103 even 4
400.8.c.j.49.2 2 260.207 even 4
450.8.a.c.1.1 1 195.194 odd 2
450.8.c.g.199.1 2 195.38 even 4
450.8.c.g.199.2 2 195.77 even 4
576.8.a.f.1.1 1 312.155 even 2
576.8.a.g.1.1 1 312.77 odd 2
578.8.a.b.1.1 1 221.220 even 2