Properties

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

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+16.0000 q^{2} +256.000 q^{4} +1310.00 q^{5} -5810.00 q^{7} +4096.00 q^{8} +O(q^{10})$$ $$q+16.0000 q^{2} +256.000 q^{4} +1310.00 q^{5} -5810.00 q^{7} +4096.00 q^{8} +20960.0 q^{10} +4498.00 q^{11} -28561.0 q^{13} -92960.0 q^{14} +65536.0 q^{16} +237498. q^{17} -913014. q^{19} +335360. q^{20} +71968.0 q^{22} -201544. q^{23} -237025. q^{25} -456976. q^{26} -1.48736e6 q^{28} -1.27683e6 q^{29} +4.16377e6 q^{31} +1.04858e6 q^{32} +3.79997e6 q^{34} -7.61110e6 q^{35} -1.84427e7 q^{37} -1.46082e7 q^{38} +5.36576e6 q^{40} +2.26017e7 q^{41} +1.17263e7 q^{43} +1.15149e6 q^{44} -3.22470e6 q^{46} -5.92915e7 q^{47} -6.59751e6 q^{49} -3.79240e6 q^{50} -7.31162e6 q^{52} -1.08159e8 q^{53} +5.89238e6 q^{55} -2.37978e7 q^{56} -2.04293e7 q^{58} +1.49202e7 q^{59} -5.70037e7 q^{61} +6.66203e7 q^{62} +1.67772e7 q^{64} -3.74149e7 q^{65} +2.20740e7 q^{67} +6.07995e7 q^{68} -1.21778e8 q^{70} -4.44162e7 q^{71} +2.65795e8 q^{73} -2.95083e8 q^{74} -2.33732e8 q^{76} -2.61334e7 q^{77} +4.76755e8 q^{79} +8.58522e7 q^{80} +3.61627e8 q^{82} +5.05316e8 q^{83} +3.11122e8 q^{85} +1.87621e8 q^{86} +1.84238e7 q^{88} -8.90841e8 q^{89} +1.65939e8 q^{91} -5.15953e7 q^{92} -9.48665e8 q^{94} -1.19605e9 q^{95} -8.02777e8 q^{97} -1.05560e8 q^{98} +O(q^{100})$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 16.0000 0.707107
$$3$$ 0 0
$$4$$ 256.000 0.500000
$$5$$ 1310.00 0.937360 0.468680 0.883368i $$-0.344730\pi$$
0.468680 + 0.883368i $$0.344730\pi$$
$$6$$ 0 0
$$7$$ −5810.00 −0.914608 −0.457304 0.889310i $$-0.651185\pi$$
−0.457304 + 0.889310i $$0.651185\pi$$
$$8$$ 4096.00 0.353553
$$9$$ 0 0
$$10$$ 20960.0 0.662813
$$11$$ 4498.00 0.0926302 0.0463151 0.998927i $$-0.485252\pi$$
0.0463151 + 0.998927i $$0.485252\pi$$
$$12$$ 0 0
$$13$$ −28561.0 −0.277350
$$14$$ −92960.0 −0.646725
$$15$$ 0 0
$$16$$ 65536.0 0.250000
$$17$$ 237498. 0.689668 0.344834 0.938664i $$-0.387935\pi$$
0.344834 + 0.938664i $$0.387935\pi$$
$$18$$ 0 0
$$19$$ −913014. −1.60726 −0.803630 0.595129i $$-0.797101\pi$$
−0.803630 + 0.595129i $$0.797101\pi$$
$$20$$ 335360. 0.468680
$$21$$ 0 0
$$22$$ 71968.0 0.0654994
$$23$$ −201544. −0.150174 −0.0750870 0.997177i $$-0.523923\pi$$
−0.0750870 + 0.997177i $$0.523923\pi$$
$$24$$ 0 0
$$25$$ −237025. −0.121357
$$26$$ −456976. −0.196116
$$27$$ 0 0
$$28$$ −1.48736e6 −0.457304
$$29$$ −1.27683e6 −0.335230 −0.167615 0.985852i $$-0.553607\pi$$
−0.167615 + 0.985852i $$0.553607\pi$$
$$30$$ 0 0
$$31$$ 4.16377e6 0.809765 0.404883 0.914369i $$-0.367312\pi$$
0.404883 + 0.914369i $$0.367312\pi$$
$$32$$ 1.04858e6 0.176777
$$33$$ 0 0
$$34$$ 3.79997e6 0.487669
$$35$$ −7.61110e6 −0.857317
$$36$$ 0 0
$$37$$ −1.84427e7 −1.61777 −0.808883 0.587969i $$-0.799928\pi$$
−0.808883 + 0.587969i $$0.799928\pi$$
$$38$$ −1.46082e7 −1.13650
$$39$$ 0 0
$$40$$ 5.36576e6 0.331407
$$41$$ 2.26017e7 1.24915 0.624573 0.780966i $$-0.285273\pi$$
0.624573 + 0.780966i $$0.285273\pi$$
$$42$$ 0 0
$$43$$ 1.17263e7 0.523062 0.261531 0.965195i $$-0.415773\pi$$
0.261531 + 0.965195i $$0.415773\pi$$
$$44$$ 1.15149e6 0.0463151
$$45$$ 0 0
$$46$$ −3.22470e6 −0.106189
$$47$$ −5.92915e7 −1.77236 −0.886181 0.463340i $$-0.846651\pi$$
−0.886181 + 0.463340i $$0.846651\pi$$
$$48$$ 0 0
$$49$$ −6.59751e6 −0.163492
$$50$$ −3.79240e6 −0.0858122
$$51$$ 0 0
$$52$$ −7.31162e6 −0.138675
$$53$$ −1.08159e8 −1.88287 −0.941434 0.337196i $$-0.890521\pi$$
−0.941434 + 0.337196i $$0.890521\pi$$
$$54$$ 0 0
$$55$$ 5.89238e6 0.0868278
$$56$$ −2.37978e7 −0.323363
$$57$$ 0 0
$$58$$ −2.04293e7 −0.237044
$$59$$ 1.49202e7 0.160302 0.0801511 0.996783i $$-0.474460\pi$$
0.0801511 + 0.996783i $$0.474460\pi$$
$$60$$ 0 0
$$61$$ −5.70037e7 −0.527132 −0.263566 0.964641i $$-0.584899\pi$$
−0.263566 + 0.964641i $$0.584899\pi$$
$$62$$ 6.66203e7 0.572590
$$63$$ 0 0
$$64$$ 1.67772e7 0.125000
$$65$$ −3.74149e7 −0.259977
$$66$$ 0 0
$$67$$ 2.20740e7 0.133827 0.0669136 0.997759i $$-0.478685\pi$$
0.0669136 + 0.997759i $$0.478685\pi$$
$$68$$ 6.07995e7 0.344834
$$69$$ 0 0
$$70$$ −1.21778e8 −0.606214
$$71$$ −4.44162e7 −0.207434 −0.103717 0.994607i $$-0.533074\pi$$
−0.103717 + 0.994607i $$0.533074\pi$$
$$72$$ 0 0
$$73$$ 2.65795e8 1.09545 0.547726 0.836658i $$-0.315494\pi$$
0.547726 + 0.836658i $$0.315494\pi$$
$$74$$ −2.95083e8 −1.14393
$$75$$ 0 0
$$76$$ −2.33732e8 −0.803630
$$77$$ −2.61334e7 −0.0847203
$$78$$ 0 0
$$79$$ 4.76755e8 1.37713 0.688563 0.725176i $$-0.258242\pi$$
0.688563 + 0.725176i $$0.258242\pi$$
$$80$$ 8.58522e7 0.234340
$$81$$ 0 0
$$82$$ 3.61627e8 0.883280
$$83$$ 5.05316e8 1.16872 0.584361 0.811494i $$-0.301345\pi$$
0.584361 + 0.811494i $$0.301345\pi$$
$$84$$ 0 0
$$85$$ 3.11122e8 0.646467
$$86$$ 1.87621e8 0.369861
$$87$$ 0 0
$$88$$ 1.84238e7 0.0327497
$$89$$ −8.90841e8 −1.50503 −0.752515 0.658575i $$-0.771159\pi$$
−0.752515 + 0.658575i $$0.771159\pi$$
$$90$$ 0 0
$$91$$ 1.65939e8 0.253667
$$92$$ −5.15953e7 −0.0750870
$$93$$ 0 0
$$94$$ −9.48665e8 −1.25325
$$95$$ −1.19605e9 −1.50658
$$96$$ 0 0
$$97$$ −8.02777e8 −0.920708 −0.460354 0.887735i $$-0.652278\pi$$
−0.460354 + 0.887735i $$0.652278\pi$$
$$98$$ −1.05560e8 −0.115607
$$99$$ 0 0
$$100$$ −6.06784e7 −0.0606784
$$101$$ −1.19998e9 −1.14743 −0.573717 0.819053i $$-0.694499\pi$$
−0.573717 + 0.819053i $$0.694499\pi$$
$$102$$ 0 0
$$103$$ −9.58027e8 −0.838707 −0.419353 0.907823i $$-0.637743\pi$$
−0.419353 + 0.907823i $$0.637743\pi$$
$$104$$ −1.16986e8 −0.0980581
$$105$$ 0 0
$$106$$ −1.73054e9 −1.33139
$$107$$ 2.39051e9 1.76304 0.881521 0.472145i $$-0.156520\pi$$
0.881521 + 0.472145i $$0.156520\pi$$
$$108$$ 0 0
$$109$$ −1.70171e9 −1.15469 −0.577346 0.816499i $$-0.695912\pi$$
−0.577346 + 0.816499i $$0.695912\pi$$
$$110$$ 9.42781e7 0.0613965
$$111$$ 0 0
$$112$$ −3.80764e8 −0.228652
$$113$$ 1.40793e9 0.812320 0.406160 0.913802i $$-0.366868\pi$$
0.406160 + 0.913802i $$0.366868\pi$$
$$114$$ 0 0
$$115$$ −2.64023e8 −0.140767
$$116$$ −3.26870e8 −0.167615
$$117$$ 0 0
$$118$$ 2.38722e8 0.113351
$$119$$ −1.37986e9 −0.630775
$$120$$ 0 0
$$121$$ −2.33772e9 −0.991420
$$122$$ −9.12060e8 −0.372738
$$123$$ 0 0
$$124$$ 1.06593e9 0.404883
$$125$$ −2.86910e9 −1.05111
$$126$$ 0 0
$$127$$ −3.31210e9 −1.12976 −0.564881 0.825172i $$-0.691078\pi$$
−0.564881 + 0.825172i $$0.691078\pi$$
$$128$$ 2.68435e8 0.0883883
$$129$$ 0 0
$$130$$ −5.98639e8 −0.183831
$$131$$ −3.06389e9 −0.908977 −0.454489 0.890753i $$-0.650178\pi$$
−0.454489 + 0.890753i $$0.650178\pi$$
$$132$$ 0 0
$$133$$ 5.30461e9 1.47001
$$134$$ 3.53184e8 0.0946302
$$135$$ 0 0
$$136$$ 9.72792e8 0.243834
$$137$$ −5.62781e8 −0.136489 −0.0682444 0.997669i $$-0.521740\pi$$
−0.0682444 + 0.997669i $$0.521740\pi$$
$$138$$ 0 0
$$139$$ −4.60597e8 −0.104654 −0.0523268 0.998630i $$-0.516664\pi$$
−0.0523268 + 0.998630i $$0.516664\pi$$
$$140$$ −1.94844e9 −0.428658
$$141$$ 0 0
$$142$$ −7.10660e8 −0.146678
$$143$$ −1.28467e8 −0.0256910
$$144$$ 0 0
$$145$$ −1.67265e9 −0.314232
$$146$$ 4.25271e9 0.774601
$$147$$ 0 0
$$148$$ −4.72132e9 −0.808883
$$149$$ 6.01717e8 0.100012 0.0500062 0.998749i $$-0.484076\pi$$
0.0500062 + 0.998749i $$0.484076\pi$$
$$150$$ 0 0
$$151$$ −1.26695e10 −1.98318 −0.991589 0.129428i $$-0.958686\pi$$
−0.991589 + 0.129428i $$0.958686\pi$$
$$152$$ −3.73971e9 −0.568252
$$153$$ 0 0
$$154$$ −4.18134e8 −0.0599063
$$155$$ 5.45454e9 0.759041
$$156$$ 0 0
$$157$$ −2.00733e8 −0.0263676 −0.0131838 0.999913i $$-0.504197\pi$$
−0.0131838 + 0.999913i $$0.504197\pi$$
$$158$$ 7.62809e9 0.973775
$$159$$ 0 0
$$160$$ 1.37363e9 0.165703
$$161$$ 1.17097e9 0.137350
$$162$$ 0 0
$$163$$ 6.32491e9 0.701795 0.350898 0.936414i $$-0.385877\pi$$
0.350898 + 0.936414i $$0.385877\pi$$
$$164$$ 5.78603e9 0.624573
$$165$$ 0 0
$$166$$ 8.08505e9 0.826412
$$167$$ −1.51400e10 −1.50627 −0.753134 0.657867i $$-0.771459\pi$$
−0.753134 + 0.657867i $$0.771459\pi$$
$$168$$ 0 0
$$169$$ 8.15731e8 0.0769231
$$170$$ 4.97796e9 0.457121
$$171$$ 0 0
$$172$$ 3.00193e9 0.261531
$$173$$ 1.63483e9 0.138760 0.0693802 0.997590i $$-0.477898\pi$$
0.0693802 + 0.997590i $$0.477898\pi$$
$$174$$ 0 0
$$175$$ 1.37712e9 0.110994
$$176$$ 2.94781e8 0.0231575
$$177$$ 0 0
$$178$$ −1.42535e10 −1.06422
$$179$$ 4.12980e9 0.300670 0.150335 0.988635i $$-0.451965\pi$$
0.150335 + 0.988635i $$0.451965\pi$$
$$180$$ 0 0
$$181$$ −2.13092e10 −1.47575 −0.737875 0.674937i $$-0.764171\pi$$
−0.737875 + 0.674937i $$0.764171\pi$$
$$182$$ 2.65503e9 0.179369
$$183$$ 0 0
$$184$$ −8.25524e8 −0.0530945
$$185$$ −2.41599e10 −1.51643
$$186$$ 0 0
$$187$$ 1.06827e9 0.0638840
$$188$$ −1.51786e10 −0.886181
$$189$$ 0 0
$$190$$ −1.91368e10 −1.06531
$$191$$ 3.08641e10 1.67804 0.839021 0.544099i $$-0.183128\pi$$
0.839021 + 0.544099i $$0.183128\pi$$
$$192$$ 0 0
$$193$$ −4.54917e9 −0.236007 −0.118003 0.993013i $$-0.537649\pi$$
−0.118003 + 0.993013i $$0.537649\pi$$
$$194$$ −1.28444e10 −0.651039
$$195$$ 0 0
$$196$$ −1.68896e9 −0.0817462
$$197$$ −2.26076e10 −1.06944 −0.534720 0.845030i $$-0.679583\pi$$
−0.534720 + 0.845030i $$0.679583\pi$$
$$198$$ 0 0
$$199$$ 1.05027e10 0.474749 0.237375 0.971418i $$-0.423713\pi$$
0.237375 + 0.971418i $$0.423713\pi$$
$$200$$ −9.70854e8 −0.0429061
$$201$$ 0 0
$$202$$ −1.91997e10 −0.811359
$$203$$ 7.41841e9 0.306604
$$204$$ 0 0
$$205$$ 2.96082e10 1.17090
$$206$$ −1.53284e10 −0.593055
$$207$$ 0 0
$$208$$ −1.87177e9 −0.0693375
$$209$$ −4.10674e9 −0.148881
$$210$$ 0 0
$$211$$ −5.66420e9 −0.196729 −0.0983643 0.995150i $$-0.531361\pi$$
−0.0983643 + 0.995150i $$0.531361\pi$$
$$212$$ −2.76886e10 −0.941434
$$213$$ 0 0
$$214$$ 3.82481e10 1.24666
$$215$$ 1.53615e10 0.490297
$$216$$ 0 0
$$217$$ −2.41915e10 −0.740618
$$218$$ −2.72274e10 −0.816491
$$219$$ 0 0
$$220$$ 1.50845e9 0.0434139
$$221$$ −6.78318e9 −0.191279
$$222$$ 0 0
$$223$$ −3.19607e10 −0.865454 −0.432727 0.901525i $$-0.642449\pi$$
−0.432727 + 0.901525i $$0.642449\pi$$
$$224$$ −6.09223e9 −0.161681
$$225$$ 0 0
$$226$$ 2.25268e10 0.574397
$$227$$ 5.07782e10 1.26929 0.634645 0.772804i $$-0.281146\pi$$
0.634645 + 0.772804i $$0.281146\pi$$
$$228$$ 0 0
$$229$$ 5.99836e10 1.44136 0.720681 0.693267i $$-0.243829\pi$$
0.720681 + 0.693267i $$0.243829\pi$$
$$230$$ −4.22436e9 −0.0995373
$$231$$ 0 0
$$232$$ −5.22991e9 −0.118522
$$233$$ −4.77228e10 −1.06078 −0.530389 0.847755i $$-0.677954\pi$$
−0.530389 + 0.847755i $$0.677954\pi$$
$$234$$ 0 0
$$235$$ −7.76719e10 −1.66134
$$236$$ 3.81956e9 0.0801511
$$237$$ 0 0
$$238$$ −2.20778e10 −0.446026
$$239$$ 8.71569e10 1.72787 0.863936 0.503602i $$-0.167992\pi$$
0.863936 + 0.503602i $$0.167992\pi$$
$$240$$ 0 0
$$241$$ −1.04205e11 −1.98981 −0.994903 0.100835i $$-0.967849\pi$$
−0.994903 + 0.100835i $$0.967849\pi$$
$$242$$ −3.74035e10 −0.701040
$$243$$ 0 0
$$244$$ −1.45930e10 −0.263566
$$245$$ −8.64273e9 −0.153251
$$246$$ 0 0
$$247$$ 2.60766e10 0.445774
$$248$$ 1.70548e10 0.286295
$$249$$ 0 0
$$250$$ −4.59055e10 −0.743250
$$251$$ −2.82027e9 −0.0448496 −0.0224248 0.999749i $$-0.507139\pi$$
−0.0224248 + 0.999749i $$0.507139\pi$$
$$252$$ 0 0
$$253$$ −9.06545e8 −0.0139106
$$254$$ −5.29937e10 −0.798863
$$255$$ 0 0
$$256$$ 4.29497e9 0.0625000
$$257$$ 1.41573e10 0.202433 0.101216 0.994864i $$-0.467727\pi$$
0.101216 + 0.994864i $$0.467727\pi$$
$$258$$ 0 0
$$259$$ 1.07152e11 1.47962
$$260$$ −9.57822e9 −0.129988
$$261$$ 0 0
$$262$$ −4.90223e10 −0.642744
$$263$$ 3.58497e10 0.462045 0.231023 0.972948i $$-0.425793\pi$$
0.231023 + 0.972948i $$0.425793\pi$$
$$264$$ 0 0
$$265$$ −1.41688e11 −1.76493
$$266$$ 8.48738e10 1.03946
$$267$$ 0 0
$$268$$ 5.65095e9 0.0669136
$$269$$ 7.14394e10 0.831864 0.415932 0.909396i $$-0.363455\pi$$
0.415932 + 0.909396i $$0.363455\pi$$
$$270$$ 0 0
$$271$$ −6.79344e9 −0.0765117 −0.0382558 0.999268i $$-0.512180\pi$$
−0.0382558 + 0.999268i $$0.512180\pi$$
$$272$$ 1.55647e10 0.172417
$$273$$ 0 0
$$274$$ −9.00450e9 −0.0965122
$$275$$ −1.06614e9 −0.0112413
$$276$$ 0 0
$$277$$ −6.93103e10 −0.707357 −0.353679 0.935367i $$-0.615069\pi$$
−0.353679 + 0.935367i $$0.615069\pi$$
$$278$$ −7.36955e9 −0.0740013
$$279$$ 0 0
$$280$$ −3.11751e10 −0.303107
$$281$$ −3.10369e10 −0.296961 −0.148480 0.988915i $$-0.547438\pi$$
−0.148480 + 0.988915i $$0.547438\pi$$
$$282$$ 0 0
$$283$$ 1.35312e11 1.25400 0.627001 0.779018i $$-0.284282\pi$$
0.627001 + 0.779018i $$0.284282\pi$$
$$284$$ −1.13706e10 −0.103717
$$285$$ 0 0
$$286$$ −2.05548e9 −0.0181663
$$287$$ −1.31316e11 −1.14248
$$288$$ 0 0
$$289$$ −6.21826e10 −0.524359
$$290$$ −2.67624e10 −0.222195
$$291$$ 0 0
$$292$$ 6.80434e10 0.547726
$$293$$ 7.55078e10 0.598532 0.299266 0.954170i $$-0.403258\pi$$
0.299266 + 0.954170i $$0.403258\pi$$
$$294$$ 0 0
$$295$$ 1.95454e10 0.150261
$$296$$ −7.55411e10 −0.571967
$$297$$ 0 0
$$298$$ 9.62747e9 0.0707195
$$299$$ 5.75630e9 0.0416508
$$300$$ 0 0
$$301$$ −6.81298e10 −0.478397
$$302$$ −2.02711e11 −1.40232
$$303$$ 0 0
$$304$$ −5.98353e10 −0.401815
$$305$$ −7.46749e10 −0.494112
$$306$$ 0 0
$$307$$ 1.42760e10 0.0917241 0.0458620 0.998948i $$-0.485397\pi$$
0.0458620 + 0.998948i $$0.485397\pi$$
$$308$$ −6.69015e9 −0.0423601
$$309$$ 0 0
$$310$$ 8.72726e10 0.536723
$$311$$ −3.58426e9 −0.0217259 −0.0108630 0.999941i $$-0.503458\pi$$
−0.0108630 + 0.999941i $$0.503458\pi$$
$$312$$ 0 0
$$313$$ 2.79830e11 1.64795 0.823977 0.566623i $$-0.191750\pi$$
0.823977 + 0.566623i $$0.191750\pi$$
$$314$$ −3.21173e9 −0.0186447
$$315$$ 0 0
$$316$$ 1.22049e11 0.688563
$$317$$ −2.40148e10 −0.133571 −0.0667855 0.997767i $$-0.521274\pi$$
−0.0667855 + 0.997767i $$0.521274\pi$$
$$318$$ 0 0
$$319$$ −5.74320e9 −0.0310524
$$320$$ 2.19782e10 0.117170
$$321$$ 0 0
$$322$$ 1.87355e10 0.0971213
$$323$$ −2.16839e11 −1.10848
$$324$$ 0 0
$$325$$ 6.76967e9 0.0336583
$$326$$ 1.01199e11 0.496244
$$327$$ 0 0
$$328$$ 9.25764e10 0.441640
$$329$$ 3.44484e11 1.62102
$$330$$ 0 0
$$331$$ 3.73009e11 1.70802 0.854010 0.520257i $$-0.174164\pi$$
0.854010 + 0.520257i $$0.174164\pi$$
$$332$$ 1.29361e11 0.584361
$$333$$ 0 0
$$334$$ −2.42240e11 −1.06509
$$335$$ 2.89170e10 0.125444
$$336$$ 0 0
$$337$$ 1.91157e11 0.807340 0.403670 0.914905i $$-0.367734\pi$$
0.403670 + 0.914905i $$0.367734\pi$$
$$338$$ 1.30517e10 0.0543928
$$339$$ 0 0
$$340$$ 7.96473e10 0.323233
$$341$$ 1.87286e10 0.0750087
$$342$$ 0 0
$$343$$ 2.72786e11 1.06414
$$344$$ 4.80310e10 0.184930
$$345$$ 0 0
$$346$$ 2.61573e10 0.0981184
$$347$$ −8.60398e10 −0.318579 −0.159289 0.987232i $$-0.550920\pi$$
−0.159289 + 0.987232i $$0.550920\pi$$
$$348$$ 0 0
$$349$$ −1.33612e11 −0.482094 −0.241047 0.970513i $$-0.577491\pi$$
−0.241047 + 0.970513i $$0.577491\pi$$
$$350$$ 2.20338e10 0.0784845
$$351$$ 0 0
$$352$$ 4.71649e9 0.0163749
$$353$$ 6.23799e10 0.213825 0.106912 0.994268i $$-0.465904\pi$$
0.106912 + 0.994268i $$0.465904\pi$$
$$354$$ 0 0
$$355$$ −5.81853e10 −0.194440
$$356$$ −2.28055e11 −0.752515
$$357$$ 0 0
$$358$$ 6.60767e10 0.212606
$$359$$ 3.82739e11 1.21612 0.608062 0.793890i $$-0.291947\pi$$
0.608062 + 0.793890i $$0.291947\pi$$
$$360$$ 0 0
$$361$$ 5.10907e11 1.58329
$$362$$ −3.40947e11 −1.04351
$$363$$ 0 0
$$364$$ 4.24805e10 0.126833
$$365$$ 3.48191e11 1.02683
$$366$$ 0 0
$$367$$ 2.59802e11 0.747560 0.373780 0.927517i $$-0.378062\pi$$
0.373780 + 0.927517i $$0.378062\pi$$
$$368$$ −1.32084e10 −0.0375435
$$369$$ 0 0
$$370$$ −3.86558e11 −1.07228
$$371$$ 6.28402e11 1.72209
$$372$$ 0 0
$$373$$ 4.70946e11 1.25974 0.629870 0.776700i $$-0.283108\pi$$
0.629870 + 0.776700i $$0.283108\pi$$
$$374$$ 1.70923e10 0.0451728
$$375$$ 0 0
$$376$$ −2.42858e11 −0.626624
$$377$$ 3.64677e10 0.0929762
$$378$$ 0 0
$$379$$ −3.60046e11 −0.896358 −0.448179 0.893944i $$-0.647927\pi$$
−0.448179 + 0.893944i $$0.647927\pi$$
$$380$$ −3.06188e11 −0.753291
$$381$$ 0 0
$$382$$ 4.93825e11 1.18655
$$383$$ 9.59380e10 0.227822 0.113911 0.993491i $$-0.463662\pi$$
0.113911 + 0.993491i $$0.463662\pi$$
$$384$$ 0 0
$$385$$ −3.42347e10 −0.0794134
$$386$$ −7.27868e10 −0.166882
$$387$$ 0 0
$$388$$ −2.05511e11 −0.460354
$$389$$ 4.60488e11 1.01964 0.509818 0.860282i $$-0.329713\pi$$
0.509818 + 0.860282i $$0.329713\pi$$
$$390$$ 0 0
$$391$$ −4.78663e10 −0.103570
$$392$$ −2.70234e10 −0.0578033
$$393$$ 0 0
$$394$$ −3.61721e11 −0.756208
$$395$$ 6.24550e11 1.29086
$$396$$ 0 0
$$397$$ 2.90299e11 0.586528 0.293264 0.956032i $$-0.405259\pi$$
0.293264 + 0.956032i $$0.405259\pi$$
$$398$$ 1.68044e11 0.335698
$$399$$ 0 0
$$400$$ −1.55337e10 −0.0303392
$$401$$ −6.85495e11 −1.32390 −0.661949 0.749549i $$-0.730270\pi$$
−0.661949 + 0.749549i $$0.730270\pi$$
$$402$$ 0 0
$$403$$ −1.18921e11 −0.224588
$$404$$ −3.07195e11 −0.573717
$$405$$ 0 0
$$406$$ 1.18694e11 0.216802
$$407$$ −8.29551e10 −0.149854
$$408$$ 0 0
$$409$$ 1.00030e12 1.76756 0.883779 0.467905i $$-0.154991\pi$$
0.883779 + 0.467905i $$0.154991\pi$$
$$410$$ 4.73731e11 0.827951
$$411$$ 0 0
$$412$$ −2.45255e11 −0.419353
$$413$$ −8.66861e10 −0.146614
$$414$$ 0 0
$$415$$ 6.61964e11 1.09551
$$416$$ −2.99484e10 −0.0490290
$$417$$ 0 0
$$418$$ −6.57078e10 −0.105275
$$419$$ 8.64798e11 1.37073 0.685364 0.728200i $$-0.259643\pi$$
0.685364 + 0.728200i $$0.259643\pi$$
$$420$$ 0 0
$$421$$ −9.57784e10 −0.148593 −0.0742965 0.997236i $$-0.523671\pi$$
−0.0742965 + 0.997236i $$0.523671\pi$$
$$422$$ −9.06272e10 −0.139108
$$423$$ 0 0
$$424$$ −4.43018e11 −0.665695
$$425$$ −5.62930e10 −0.0836959
$$426$$ 0 0
$$427$$ 3.31192e11 0.482119
$$428$$ 6.11969e11 0.881521
$$429$$ 0 0
$$430$$ 2.45783e11 0.346693
$$431$$ 1.27185e11 0.177536 0.0887682 0.996052i $$-0.471707\pi$$
0.0887682 + 0.996052i $$0.471707\pi$$
$$432$$ 0 0
$$433$$ −1.55264e11 −0.212264 −0.106132 0.994352i $$-0.533847\pi$$
−0.106132 + 0.994352i $$0.533847\pi$$
$$434$$ −3.87064e11 −0.523696
$$435$$ 0 0
$$436$$ −4.35638e11 −0.577346
$$437$$ 1.84012e11 0.241369
$$438$$ 0 0
$$439$$ −1.02610e12 −1.31855 −0.659277 0.751901i $$-0.729137\pi$$
−0.659277 + 0.751901i $$0.729137\pi$$
$$440$$ 2.41352e10 0.0306983
$$441$$ 0 0
$$442$$ −1.08531e11 −0.135255
$$443$$ −2.52039e11 −0.310922 −0.155461 0.987842i $$-0.549686\pi$$
−0.155461 + 0.987842i $$0.549686\pi$$
$$444$$ 0 0
$$445$$ −1.16700e12 −1.41075
$$446$$ −5.11371e11 −0.611969
$$447$$ 0 0
$$448$$ −9.74756e10 −0.114326
$$449$$ −7.66198e11 −0.889678 −0.444839 0.895611i $$-0.646739\pi$$
−0.444839 + 0.895611i $$0.646739\pi$$
$$450$$ 0 0
$$451$$ 1.01662e11 0.115709
$$452$$ 3.60429e11 0.406160
$$453$$ 0 0
$$454$$ 8.12451e11 0.897524
$$455$$ 2.17381e11 0.237777
$$456$$ 0 0
$$457$$ 1.75683e12 1.88411 0.942057 0.335454i $$-0.108890\pi$$
0.942057 + 0.335454i $$0.108890\pi$$
$$458$$ 9.59738e11 1.01920
$$459$$ 0 0
$$460$$ −6.75898e10 −0.0703835
$$461$$ 1.13127e12 1.16657 0.583287 0.812266i $$-0.301766\pi$$
0.583287 + 0.812266i $$0.301766\pi$$
$$462$$ 0 0
$$463$$ −2.71657e11 −0.274730 −0.137365 0.990521i $$-0.543863\pi$$
−0.137365 + 0.990521i $$0.543863\pi$$
$$464$$ −8.36786e10 −0.0838076
$$465$$ 0 0
$$466$$ −7.63565e11 −0.750083
$$467$$ 9.54617e11 0.928759 0.464380 0.885636i $$-0.346277\pi$$
0.464380 + 0.885636i $$0.346277\pi$$
$$468$$ 0 0
$$469$$ −1.28250e11 −0.122399
$$470$$ −1.24275e12 −1.17474
$$471$$ 0 0
$$472$$ 6.11130e10 0.0566754
$$473$$ 5.27449e10 0.0484513
$$474$$ 0 0
$$475$$ 2.16407e11 0.195052
$$476$$ −3.53245e11 −0.315388
$$477$$ 0 0
$$478$$ 1.39451e12 1.22179
$$479$$ −1.43680e12 −1.24705 −0.623527 0.781802i $$-0.714301\pi$$
−0.623527 + 0.781802i $$0.714301\pi$$
$$480$$ 0 0
$$481$$ 5.26741e11 0.448688
$$482$$ −1.66728e12 −1.40701
$$483$$ 0 0
$$484$$ −5.98455e11 −0.495710
$$485$$ −1.05164e12 −0.863035
$$486$$ 0 0
$$487$$ −1.22744e12 −0.988826 −0.494413 0.869227i $$-0.664617\pi$$
−0.494413 + 0.869227i $$0.664617\pi$$
$$488$$ −2.33487e11 −0.186369
$$489$$ 0 0
$$490$$ −1.38284e11 −0.108365
$$491$$ 1.00389e12 0.779506 0.389753 0.920919i $$-0.372560\pi$$
0.389753 + 0.920919i $$0.372560\pi$$
$$492$$ 0 0
$$493$$ −3.03246e11 −0.231198
$$494$$ 4.17225e11 0.315210
$$495$$ 0 0
$$496$$ 2.72877e11 0.202441
$$497$$ 2.58058e11 0.189721
$$498$$ 0 0
$$499$$ −7.58262e11 −0.547478 −0.273739 0.961804i $$-0.588260\pi$$
−0.273739 + 0.961804i $$0.588260\pi$$
$$500$$ −7.34489e11 −0.525557
$$501$$ 0 0
$$502$$ −4.51243e10 −0.0317134
$$503$$ −1.82032e12 −1.26792 −0.633959 0.773367i $$-0.718571\pi$$
−0.633959 + 0.773367i $$0.718571\pi$$
$$504$$ 0 0
$$505$$ −1.57197e12 −1.07556
$$506$$ −1.45047e10 −0.00983630
$$507$$ 0 0
$$508$$ −8.47899e11 −0.564881
$$509$$ −6.57012e11 −0.433854 −0.216927 0.976188i $$-0.569603\pi$$
−0.216927 + 0.976188i $$0.569603\pi$$
$$510$$ 0 0
$$511$$ −1.54427e12 −1.00191
$$512$$ 6.87195e10 0.0441942
$$513$$ 0 0
$$514$$ 2.26516e11 0.143141
$$515$$ −1.25502e12 −0.786170
$$516$$ 0 0
$$517$$ −2.66693e11 −0.164174
$$518$$ 1.71443e12 1.04625
$$519$$ 0 0
$$520$$ −1.53251e11 −0.0919157
$$521$$ −3.17678e11 −0.188894 −0.0944468 0.995530i $$-0.530108\pi$$
−0.0944468 + 0.995530i $$0.530108\pi$$
$$522$$ 0 0
$$523$$ −2.88365e12 −1.68533 −0.842666 0.538436i $$-0.819015\pi$$
−0.842666 + 0.538436i $$0.819015\pi$$
$$524$$ −7.84357e11 −0.454489
$$525$$ 0 0
$$526$$ 5.73595e11 0.326715
$$527$$ 9.88887e11 0.558469
$$528$$ 0 0
$$529$$ −1.76053e12 −0.977448
$$530$$ −2.26701e12 −1.24799
$$531$$ 0 0
$$532$$ 1.35798e12 0.735007
$$533$$ −6.45526e11 −0.346451
$$534$$ 0 0
$$535$$ 3.13156e12 1.65260
$$536$$ 9.04151e10 0.0473151
$$537$$ 0 0
$$538$$ 1.14303e12 0.588217
$$539$$ −2.96756e10 −0.0151443
$$540$$ 0 0
$$541$$ 2.16753e12 1.08787 0.543937 0.839126i $$-0.316933\pi$$
0.543937 + 0.839126i $$0.316933\pi$$
$$542$$ −1.08695e11 −0.0541019
$$543$$ 0 0
$$544$$ 2.49035e11 0.121917
$$545$$ −2.22924e12 −1.08236
$$546$$ 0 0
$$547$$ −9.14427e11 −0.436723 −0.218361 0.975868i $$-0.570071\pi$$
−0.218361 + 0.975868i $$0.570071\pi$$
$$548$$ −1.44072e11 −0.0682444
$$549$$ 0 0
$$550$$ −1.70582e10 −0.00794880
$$551$$ 1.16577e12 0.538803
$$552$$ 0 0
$$553$$ −2.76995e12 −1.25953
$$554$$ −1.10896e12 −0.500177
$$555$$ 0 0
$$556$$ −1.17913e11 −0.0523268
$$557$$ −1.75791e10 −0.00773834 −0.00386917 0.999993i $$-0.501232\pi$$
−0.00386917 + 0.999993i $$0.501232\pi$$
$$558$$ 0 0
$$559$$ −3.34915e11 −0.145071
$$560$$ −4.98801e11 −0.214329
$$561$$ 0 0
$$562$$ −4.96590e11 −0.209983
$$563$$ 3.70644e12 1.55478 0.777390 0.629019i $$-0.216543\pi$$
0.777390 + 0.629019i $$0.216543\pi$$
$$564$$ 0 0
$$565$$ 1.84438e12 0.761436
$$566$$ 2.16500e12 0.886713
$$567$$ 0 0
$$568$$ −1.81929e11 −0.0733389
$$569$$ 3.40051e12 1.36000 0.679999 0.733213i $$-0.261980\pi$$
0.679999 + 0.733213i $$0.261980\pi$$
$$570$$ 0 0
$$571$$ −4.46270e12 −1.75685 −0.878427 0.477877i $$-0.841406\pi$$
−0.878427 + 0.477877i $$0.841406\pi$$
$$572$$ −3.28876e10 −0.0128455
$$573$$ 0 0
$$574$$ −2.10105e12 −0.807854
$$575$$ 4.77710e10 0.0182246
$$576$$ 0 0
$$577$$ 3.47193e12 1.30401 0.652004 0.758215i $$-0.273929\pi$$
0.652004 + 0.758215i $$0.273929\pi$$
$$578$$ −9.94921e11 −0.370778
$$579$$ 0 0
$$580$$ −4.28199e11 −0.157116
$$581$$ −2.93588e12 −1.06892
$$582$$ 0 0
$$583$$ −4.86498e11 −0.174410
$$584$$ 1.08869e12 0.387301
$$585$$ 0 0
$$586$$ 1.20812e12 0.423226
$$587$$ −2.86266e12 −0.995171 −0.497586 0.867415i $$-0.665780\pi$$
−0.497586 + 0.867415i $$0.665780\pi$$
$$588$$ 0 0
$$589$$ −3.80158e12 −1.30150
$$590$$ 3.12726e11 0.106250
$$591$$ 0 0
$$592$$ −1.20866e12 −0.404442
$$593$$ 8.38217e11 0.278362 0.139181 0.990267i $$-0.455553\pi$$
0.139181 + 0.990267i $$0.455553\pi$$
$$594$$ 0 0
$$595$$ −1.80762e12 −0.591263
$$596$$ 1.54039e11 0.0500062
$$597$$ 0 0
$$598$$ 9.21008e10 0.0294515
$$599$$ −3.94489e12 −1.25203 −0.626013 0.779812i $$-0.715315\pi$$
−0.626013 + 0.779812i $$0.715315\pi$$
$$600$$ 0 0
$$601$$ 4.99865e12 1.56285 0.781426 0.623998i $$-0.214492\pi$$
0.781426 + 0.623998i $$0.214492\pi$$
$$602$$ −1.09008e12 −0.338278
$$603$$ 0 0
$$604$$ −3.24338e12 −0.991589
$$605$$ −3.06241e12 −0.929317
$$606$$ 0 0
$$607$$ −3.95582e11 −0.118274 −0.0591368 0.998250i $$-0.518835\pi$$
−0.0591368 + 0.998250i $$0.518835\pi$$
$$608$$ −9.57365e11 −0.284126
$$609$$ 0 0
$$610$$ −1.19480e12 −0.349390
$$611$$ 1.69343e12 0.491565
$$612$$ 0 0
$$613$$ 5.03617e12 1.44055 0.720275 0.693688i $$-0.244016\pi$$
0.720275 + 0.693688i $$0.244016\pi$$
$$614$$ 2.28416e11 0.0648587
$$615$$ 0 0
$$616$$ −1.07042e11 −0.0299531
$$617$$ −4.06829e12 −1.13013 −0.565065 0.825046i $$-0.691149\pi$$
−0.565065 + 0.825046i $$0.691149\pi$$
$$618$$ 0 0
$$619$$ −4.73136e12 −1.29532 −0.647662 0.761928i $$-0.724253\pi$$
−0.647662 + 0.761928i $$0.724253\pi$$
$$620$$ 1.39636e12 0.379521
$$621$$ 0 0
$$622$$ −5.73482e10 −0.0153626
$$623$$ 5.17578e12 1.37651
$$624$$ 0 0
$$625$$ −3.29558e12 −0.863916
$$626$$ 4.47728e12 1.16528
$$627$$ 0 0
$$628$$ −5.13877e10 −0.0131838
$$629$$ −4.38010e12 −1.11572
$$630$$ 0 0
$$631$$ 3.45019e12 0.866384 0.433192 0.901302i $$-0.357387\pi$$
0.433192 + 0.901302i $$0.357387\pi$$
$$632$$ 1.95279e12 0.486888
$$633$$ 0 0
$$634$$ −3.84237e11 −0.0944490
$$635$$ −4.33886e12 −1.05899
$$636$$ 0 0
$$637$$ 1.88431e11 0.0453446
$$638$$ −9.18912e10 −0.0219574
$$639$$ 0 0
$$640$$ 3.51650e11 0.0828517
$$641$$ 3.87461e12 0.906498 0.453249 0.891384i $$-0.350265\pi$$
0.453249 + 0.891384i $$0.350265\pi$$
$$642$$ 0 0
$$643$$ 2.77752e12 0.640778 0.320389 0.947286i $$-0.396186\pi$$
0.320389 + 0.947286i $$0.396186\pi$$
$$644$$ 2.99768e11 0.0686751
$$645$$ 0 0
$$646$$ −3.46942e12 −0.783810
$$647$$ −6.12025e12 −1.37309 −0.686546 0.727086i $$-0.740874\pi$$
−0.686546 + 0.727086i $$0.740874\pi$$
$$648$$ 0 0
$$649$$ 6.71109e10 0.0148488
$$650$$ 1.08315e11 0.0238000
$$651$$ 0 0
$$652$$ 1.61918e12 0.350898
$$653$$ −2.50039e12 −0.538143 −0.269072 0.963120i $$-0.586717\pi$$
−0.269072 + 0.963120i $$0.586717\pi$$
$$654$$ 0 0
$$655$$ −4.01370e12 −0.852038
$$656$$ 1.48122e12 0.312286
$$657$$ 0 0
$$658$$ 5.51174e12 1.14623
$$659$$ 6.13676e12 1.26752 0.633760 0.773529i $$-0.281511\pi$$
0.633760 + 0.773529i $$0.281511\pi$$
$$660$$ 0 0
$$661$$ −6.28369e12 −1.28029 −0.640145 0.768254i $$-0.721126\pi$$
−0.640145 + 0.768254i $$0.721126\pi$$
$$662$$ 5.96814e12 1.20775
$$663$$ 0 0
$$664$$ 2.06977e12 0.413206
$$665$$ 6.94904e12 1.37793
$$666$$ 0 0
$$667$$ 2.57338e11 0.0503429
$$668$$ −3.87585e12 −0.753134
$$669$$ 0 0
$$670$$ 4.62671e11 0.0887025
$$671$$ −2.56403e11 −0.0488283
$$672$$ 0 0
$$673$$ −7.98616e12 −1.50062 −0.750309 0.661087i $$-0.770095\pi$$
−0.750309 + 0.661087i $$0.770095\pi$$
$$674$$ 3.05852e12 0.570876
$$675$$ 0 0
$$676$$ 2.08827e11 0.0384615
$$677$$ 3.64428e12 0.666749 0.333374 0.942794i $$-0.391813\pi$$
0.333374 + 0.942794i $$0.391813\pi$$
$$678$$ 0 0
$$679$$ 4.66413e12 0.842087
$$680$$ 1.27436e12 0.228560
$$681$$ 0 0
$$682$$ 2.99658e11 0.0530391
$$683$$ 1.11273e12 0.195658 0.0978291 0.995203i $$-0.468810\pi$$
0.0978291 + 0.995203i $$0.468810\pi$$
$$684$$ 0 0
$$685$$ −7.37244e11 −0.127939
$$686$$ 4.36458e12 0.752460
$$687$$ 0 0
$$688$$ 7.68495e11 0.130766
$$689$$ 3.08912e12 0.522214
$$690$$ 0 0
$$691$$ −2.75811e12 −0.460214 −0.230107 0.973165i $$-0.573908\pi$$
−0.230107 + 0.973165i $$0.573908\pi$$
$$692$$ 4.18517e11 0.0693802
$$693$$ 0 0
$$694$$ −1.37664e12 −0.225269
$$695$$ −6.03382e11 −0.0980982
$$696$$ 0 0
$$697$$ 5.36785e12 0.861495
$$698$$ −2.13779e12 −0.340892
$$699$$ 0 0
$$700$$ 3.52542e11 0.0554969
$$701$$ −8.08880e12 −1.26518 −0.632591 0.774486i $$-0.718009\pi$$
−0.632591 + 0.774486i $$0.718009\pi$$
$$702$$ 0 0
$$703$$ 1.68384e13 2.60017
$$704$$ 7.54639e10 0.0115788
$$705$$ 0 0
$$706$$ 9.98078e11 0.151197
$$707$$ 6.97189e12 1.04945
$$708$$ 0 0
$$709$$ 2.19552e11 0.0326310 0.0163155 0.999867i $$-0.494806\pi$$
0.0163155 + 0.999867i $$0.494806\pi$$
$$710$$ −9.30965e11 −0.137490
$$711$$ 0 0
$$712$$ −3.64888e12 −0.532108
$$713$$ −8.39183e11 −0.121606
$$714$$ 0 0
$$715$$ −1.68292e11 −0.0240817
$$716$$ 1.05723e12 0.150335
$$717$$ 0 0
$$718$$ 6.12382e12 0.859929
$$719$$ 8.58532e12 1.19805 0.599027 0.800729i $$-0.295554\pi$$
0.599027 + 0.800729i $$0.295554\pi$$
$$720$$ 0 0
$$721$$ 5.56614e12 0.767088
$$722$$ 8.17451e12 1.11955
$$723$$ 0 0
$$724$$ −5.45515e12 −0.737875
$$725$$ 3.02642e11 0.0406825
$$726$$ 0 0
$$727$$ 7.59563e11 0.100846 0.0504230 0.998728i $$-0.483943\pi$$
0.0504230 + 0.998728i $$0.483943\pi$$
$$728$$ 6.79688e11 0.0896847
$$729$$ 0 0
$$730$$ 5.57106e12 0.726080
$$731$$ 2.78497e12 0.360739
$$732$$ 0 0
$$733$$ −7.83005e12 −1.00184 −0.500918 0.865495i $$-0.667004\pi$$
−0.500918 + 0.865495i $$0.667004\pi$$
$$734$$ 4.15684e12 0.528605
$$735$$ 0 0
$$736$$ −2.11334e11 −0.0265473
$$737$$ 9.92889e10 0.0123964
$$738$$ 0 0
$$739$$ 5.41643e12 0.668056 0.334028 0.942563i $$-0.391592\pi$$
0.334028 + 0.942563i $$0.391592\pi$$
$$740$$ −6.18493e12 −0.758215
$$741$$ 0 0
$$742$$ 1.00544e13 1.21770
$$743$$ −2.66408e12 −0.320699 −0.160349 0.987060i $$-0.551262\pi$$
−0.160349 + 0.987060i $$0.551262\pi$$
$$744$$ 0 0
$$745$$ 7.88249e11 0.0937476
$$746$$ 7.53513e12 0.890771
$$747$$ 0 0
$$748$$ 2.73476e11 0.0319420
$$749$$ −1.38888e13 −1.61249
$$750$$ 0 0
$$751$$ 5.82882e12 0.668653 0.334326 0.942457i $$-0.391491\pi$$
0.334326 + 0.942457i $$0.391491\pi$$
$$752$$ −3.88573e12 −0.443090
$$753$$ 0 0
$$754$$ 5.83482e11 0.0657441
$$755$$ −1.65970e13 −1.85895
$$756$$ 0 0
$$757$$ 5.67869e12 0.628517 0.314258 0.949338i $$-0.398244\pi$$
0.314258 + 0.949338i $$0.398244\pi$$
$$758$$ −5.76074e12 −0.633821
$$759$$ 0 0
$$760$$ −4.89901e12 −0.532657
$$761$$ 1.12490e13 1.21586 0.607931 0.793990i $$-0.292000\pi$$
0.607931 + 0.793990i $$0.292000\pi$$
$$762$$ 0 0
$$763$$ 9.88694e12 1.05609
$$764$$ 7.90120e12 0.839021
$$765$$ 0 0
$$766$$ 1.53501e12 0.161095
$$767$$ −4.26135e11 −0.0444598
$$768$$ 0 0
$$769$$ 5.02943e12 0.518621 0.259311 0.965794i $$-0.416505\pi$$
0.259311 + 0.965794i $$0.416505\pi$$
$$770$$ −5.47756e11 −0.0561537
$$771$$ 0 0
$$772$$ −1.16459e12 −0.118003
$$773$$ −1.22620e13 −1.23524 −0.617621 0.786476i $$-0.711903\pi$$
−0.617621 + 0.786476i $$0.711903\pi$$
$$774$$ 0 0
$$775$$ −9.86918e11 −0.0982705
$$776$$ −3.28817e12 −0.325520
$$777$$ 0 0
$$778$$ 7.36781e12 0.720992
$$779$$ −2.06356e13 −2.00770
$$780$$ 0 0
$$781$$ −1.99784e11 −0.0192146
$$782$$ −7.65861e11 −0.0732351
$$783$$ 0 0
$$784$$ −4.32374e11 −0.0408731
$$785$$ −2.62960e11 −0.0247159
$$786$$ 0 0
$$787$$ −1.13978e13 −1.05909 −0.529547 0.848281i $$-0.677638\pi$$
−0.529547 + 0.848281i $$0.677638\pi$$
$$788$$ −5.78754e12 −0.534720
$$789$$ 0 0
$$790$$ 9.99279e12 0.912778
$$791$$ −8.18005e12 −0.742954
$$792$$ 0 0
$$793$$ 1.62808e12 0.146200
$$794$$ 4.64479e12 0.414738
$$795$$ 0 0
$$796$$ 2.68870e12 0.237375
$$797$$ −9.66670e12 −0.848625 −0.424313 0.905516i $$-0.639484\pi$$
−0.424313 + 0.905516i $$0.639484\pi$$
$$798$$ 0 0
$$799$$ −1.40816e13 −1.22234
$$800$$ −2.48539e11 −0.0214531
$$801$$ 0 0
$$802$$ −1.09679e13 −0.936137
$$803$$ 1.19554e12 0.101472
$$804$$ 0 0
$$805$$ 1.53397e12 0.128747
$$806$$ −1.90274e12 −0.158808
$$807$$ 0 0
$$808$$ −4.91512e12 −0.405679
$$809$$ −4.89988e12 −0.402177 −0.201088 0.979573i $$-0.564448\pi$$
−0.201088 + 0.979573i $$0.564448\pi$$
$$810$$ 0 0
$$811$$ −9.97393e12 −0.809604 −0.404802 0.914404i $$-0.632660\pi$$
−0.404802 + 0.914404i $$0.632660\pi$$
$$812$$ 1.89911e12 0.153302
$$813$$ 0 0
$$814$$ −1.32728e12 −0.105963
$$815$$ 8.28564e12 0.657835
$$816$$ 0 0
$$817$$ −1.07063e13 −0.840697
$$818$$ 1.60047e13 1.24985
$$819$$ 0 0
$$820$$ 7.57970e12 0.585450
$$821$$ 6.20376e12 0.476552 0.238276 0.971197i $$-0.423418\pi$$
0.238276 + 0.971197i $$0.423418\pi$$
$$822$$ 0 0
$$823$$ 2.05255e13 1.55953 0.779765 0.626072i $$-0.215339\pi$$
0.779765 + 0.626072i $$0.215339\pi$$
$$824$$ −3.92408e12 −0.296528
$$825$$ 0 0
$$826$$ −1.38698e12 −0.103671
$$827$$ 1.52447e13 1.13330 0.566649 0.823959i $$-0.308239\pi$$
0.566649 + 0.823959i $$0.308239\pi$$
$$828$$ 0 0
$$829$$ 1.59195e12 0.117067 0.0585334 0.998285i $$-0.481358\pi$$
0.0585334 + 0.998285i $$0.481358\pi$$
$$830$$ 1.05914e13 0.774645
$$831$$ 0 0
$$832$$ −4.79174e11 −0.0346688
$$833$$ −1.56689e12 −0.112755
$$834$$ 0 0
$$835$$ −1.98334e13 −1.41192
$$836$$ −1.05132e12 −0.0744404
$$837$$ 0 0
$$838$$ 1.38368e13 0.969251
$$839$$ 5.56668e11 0.0387853 0.0193927 0.999812i $$-0.493827\pi$$
0.0193927 + 0.999812i $$0.493827\pi$$
$$840$$ 0 0
$$841$$ −1.28768e13 −0.887621
$$842$$ −1.53245e12 −0.105071
$$843$$ 0 0
$$844$$ −1.45004e12 −0.0983643
$$845$$ 1.06861e12 0.0721046
$$846$$ 0 0
$$847$$ 1.35821e13 0.906760
$$848$$ −7.08829e12 −0.470717
$$849$$ 0 0
$$850$$ −9.00687e11 −0.0591819
$$851$$ 3.71701e12 0.242946
$$852$$ 0 0
$$853$$ −1.76959e13 −1.14446 −0.572231 0.820093i $$-0.693922\pi$$
−0.572231 + 0.820093i $$0.693922\pi$$
$$854$$ 5.29907e12 0.340909
$$855$$ 0 0
$$856$$ 9.79151e12 0.623330
$$857$$ −1.34064e13 −0.848983 −0.424492 0.905432i $$-0.639547\pi$$
−0.424492 + 0.905432i $$0.639547\pi$$
$$858$$ 0 0
$$859$$ 2.16215e13 1.35493 0.677466 0.735554i $$-0.263078\pi$$
0.677466 + 0.735554i $$0.263078\pi$$
$$860$$ 3.93253e12 0.245149
$$861$$ 0 0
$$862$$ 2.03496e12 0.125537
$$863$$ 1.09637e13 0.672838 0.336419 0.941712i $$-0.390784\pi$$
0.336419 + 0.941712i $$0.390784\pi$$
$$864$$ 0 0
$$865$$ 2.14163e12 0.130068
$$866$$ −2.48423e12 −0.150093
$$867$$ 0 0
$$868$$ −6.19302e12 −0.370309
$$869$$ 2.14445e12 0.127563
$$870$$ 0 0
$$871$$ −6.30456e11 −0.0371170
$$872$$ −6.97021e12 −0.408245
$$873$$ 0 0
$$874$$ 2.94420e12 0.170673
$$875$$ 1.66695e13 0.961358
$$876$$ 0 0
$$877$$ 1.20955e13 0.690440 0.345220 0.938522i $$-0.387804\pi$$
0.345220 + 0.938522i $$0.387804\pi$$
$$878$$ −1.64175e13 −0.932358
$$879$$ 0 0
$$880$$ 3.86163e11 0.0217069
$$881$$ 3.33493e13 1.86507 0.932534 0.361083i $$-0.117593\pi$$
0.932534 + 0.361083i $$0.117593\pi$$
$$882$$ 0 0
$$883$$ −1.01455e13 −0.561628 −0.280814 0.959762i $$-0.590604\pi$$
−0.280814 + 0.959762i $$0.590604\pi$$
$$884$$ −1.73649e12 −0.0956397
$$885$$ 0 0
$$886$$ −4.03263e12 −0.219855
$$887$$ −3.04816e13 −1.65341 −0.826707 0.562633i $$-0.809789\pi$$
−0.826707 + 0.562633i $$0.809789\pi$$
$$888$$ 0 0
$$889$$ 1.92433e13 1.03329
$$890$$ −1.86720e13 −0.997554
$$891$$ 0 0
$$892$$ −8.18194e12 −0.432727
$$893$$ 5.41340e13 2.84865
$$894$$ 0 0
$$895$$ 5.41003e12 0.281836
$$896$$ −1.55961e12 −0.0808407
$$897$$ 0 0
$$898$$ −1.22592e13 −0.629097
$$899$$ −5.31644e12 −0.271458
$$900$$ 0 0
$$901$$ −2.56875e13 −1.29855
$$902$$ 1.62660e12 0.0818183
$$903$$ 0 0
$$904$$ 5.76687e12 0.287198
$$905$$ −2.79150e13 −1.38331
$$906$$ 0 0
$$907$$ −2.08823e12 −0.102458 −0.0512289 0.998687i $$-0.516314\pi$$
−0.0512289 + 0.998687i $$0.516314\pi$$
$$908$$ 1.29992e13 0.634645
$$909$$ 0 0
$$910$$ 3.47809e12 0.168134
$$911$$ 1.70747e13 0.821336 0.410668 0.911785i $$-0.365296\pi$$
0.410668 + 0.911785i $$0.365296\pi$$
$$912$$ 0 0
$$913$$ 2.27291e12 0.108259
$$914$$ 2.81093e13 1.33227
$$915$$ 0 0
$$916$$ 1.53558e13 0.720681
$$917$$ 1.78012e13 0.831358
$$918$$ 0 0
$$919$$ −3.86177e12 −0.178594 −0.0892970 0.996005i $$-0.528462\pi$$
−0.0892970 + 0.996005i $$0.528462\pi$$
$$920$$ −1.08144e12 −0.0497687
$$921$$ 0 0
$$922$$ 1.81003e13 0.824893
$$923$$ 1.26857e12 0.0575318
$$924$$ 0 0
$$925$$ 4.37137e12 0.196327
$$926$$ −4.34651e12 −0.194263
$$927$$ 0 0
$$928$$ −1.33886e12 −0.0592609
$$929$$ −2.72392e13 −1.19984 −0.599921 0.800059i $$-0.704801\pi$$
−0.599921 + 0.800059i $$0.704801\pi$$
$$930$$ 0 0
$$931$$ 6.02362e12 0.262775
$$932$$ −1.22170e13 −0.530389
$$933$$ 0 0
$$934$$ 1.52739e13 0.656732
$$935$$ 1.39943e12 0.0598823
$$936$$ 0 0
$$937$$ −1.33830e13 −0.567184 −0.283592 0.958945i $$-0.591526\pi$$
−0.283592 + 0.958945i $$0.591526\pi$$
$$938$$ −2.05200e12 −0.0865495
$$939$$ 0 0
$$940$$ −1.98840e13 −0.830670
$$941$$ −5.78614e12 −0.240567 −0.120283 0.992740i $$-0.538380\pi$$
−0.120283 + 0.992740i $$0.538380\pi$$
$$942$$ 0 0
$$943$$ −4.55523e12 −0.187589
$$944$$ 9.77807e11 0.0400755
$$945$$ 0 0
$$946$$ 8.43919e11 0.0342603
$$947$$ 4.03191e13 1.62905 0.814527 0.580125i $$-0.196996\pi$$
0.814527 + 0.580125i $$0.196996\pi$$
$$948$$ 0 0
$$949$$ −7.59136e12 −0.303824
$$950$$ 3.46251e12 0.137923
$$951$$ 0 0
$$952$$ −5.65192e12 −0.223013
$$953$$ −1.46625e13 −0.575824 −0.287912 0.957657i $$-0.592961\pi$$
−0.287912 + 0.957657i $$0.592961\pi$$
$$954$$ 0 0
$$955$$ 4.04319e13 1.57293
$$956$$ 2.23122e13 0.863936
$$957$$ 0 0
$$958$$ −2.29887e13 −0.881801
$$959$$ 3.26976e12 0.124834
$$960$$ 0 0
$$961$$ −9.10264e12 −0.344280
$$962$$ 8.42785e12 0.317270
$$963$$ 0 0
$$964$$ −2.66764e13 −0.994903
$$965$$ −5.95942e12 −0.221223
$$966$$ 0 0
$$967$$ 1.87662e13 0.690172 0.345086 0.938571i $$-0.387850\pi$$
0.345086 + 0.938571i $$0.387850\pi$$
$$968$$ −9.57528e12 −0.350520
$$969$$ 0 0
$$970$$ −1.68262e13 −0.610258
$$971$$ −2.66964e13 −0.963755 −0.481877 0.876239i $$-0.660045\pi$$
−0.481877 + 0.876239i $$0.660045\pi$$
$$972$$ 0 0
$$973$$ 2.67607e12 0.0957171
$$974$$ −1.96390e13 −0.699206
$$975$$ 0 0
$$976$$ −3.73580e12 −0.131783
$$977$$ −1.44408e13 −0.507068 −0.253534 0.967327i $$-0.581593\pi$$
−0.253534 + 0.967327i $$0.581593\pi$$
$$978$$ 0 0
$$979$$ −4.00700e12 −0.139411
$$980$$ −2.21254e12 −0.0766256
$$981$$ 0 0
$$982$$ 1.60622e13 0.551194
$$983$$ −3.96507e13 −1.35444 −0.677220 0.735781i $$-0.736815\pi$$
−0.677220 + 0.735781i $$0.736815\pi$$
$$984$$ 0 0
$$985$$ −2.96159e13 −1.00245
$$986$$ −4.85193e12 −0.163481
$$987$$ 0 0
$$988$$ 6.67561e12 0.222887
$$989$$ −2.36337e12 −0.0785503
$$990$$ 0 0
$$991$$ −4.04833e13 −1.33335 −0.666675 0.745348i $$-0.732283\pi$$
−0.666675 + 0.745348i $$0.732283\pi$$
$$992$$ 4.36603e12 0.143148
$$993$$ 0 0
$$994$$ 4.12893e12 0.134153
$$995$$ 1.37586e13 0.445011
$$996$$ 0 0
$$997$$ −2.52148e13 −0.808217 −0.404109 0.914711i $$-0.632418\pi$$
−0.404109 + 0.914711i $$0.632418\pi$$
$$998$$ −1.21322e13 −0.387125
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

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

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