# Properties

 Label 784.6.a.c.1.1 Level $784$ Weight $6$ Character 784.1 Self dual yes Analytic conductor $125.741$ 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] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.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: $$125.740914733$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-14.0000 q^{3} +56.0000 q^{5} -47.0000 q^{9} +O(q^{10})$$ $$q-14.0000 q^{3} +56.0000 q^{5} -47.0000 q^{9} -232.000 q^{11} +140.000 q^{13} -784.000 q^{15} +1722.00 q^{17} -98.0000 q^{19} -1824.00 q^{23} +11.0000 q^{25} +4060.00 q^{27} +3418.00 q^{29} -7644.00 q^{31} +3248.00 q^{33} -10398.0 q^{37} -1960.00 q^{39} +17962.0 q^{41} -10880.0 q^{43} -2632.00 q^{45} +9324.00 q^{47} -24108.0 q^{51} +2262.00 q^{53} -12992.0 q^{55} +1372.00 q^{57} -2730.00 q^{59} -25648.0 q^{61} +7840.00 q^{65} +48404.0 q^{67} +25536.0 q^{69} +58560.0 q^{71} -68082.0 q^{73} -154.000 q^{75} -31784.0 q^{79} -45419.0 q^{81} -20538.0 q^{83} +96432.0 q^{85} -47852.0 q^{87} +50582.0 q^{89} +107016. q^{93} -5488.00 q^{95} +58506.0 q^{97} +10904.0 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −14.0000 −0.898100 −0.449050 0.893507i $$-0.648238\pi$$
−0.449050 + 0.893507i $$0.648238\pi$$
$$4$$ 0 0
$$5$$ 56.0000 1.00176 0.500879 0.865517i $$-0.333010\pi$$
0.500879 + 0.865517i $$0.333010\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −47.0000 −0.193416
$$10$$ 0 0
$$11$$ −232.000 −0.578104 −0.289052 0.957313i $$-0.593340\pi$$
−0.289052 + 0.957313i $$0.593340\pi$$
$$12$$ 0 0
$$13$$ 140.000 0.229757 0.114879 0.993380i $$-0.463352\pi$$
0.114879 + 0.993380i $$0.463352\pi$$
$$14$$ 0 0
$$15$$ −784.000 −0.899680
$$16$$ 0 0
$$17$$ 1722.00 1.44514 0.722572 0.691296i $$-0.242960\pi$$
0.722572 + 0.691296i $$0.242960\pi$$
$$18$$ 0 0
$$19$$ −98.0000 −0.0622791 −0.0311395 0.999515i $$-0.509914\pi$$
−0.0311395 + 0.999515i $$0.509914\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1824.00 −0.718961 −0.359480 0.933153i $$-0.617046\pi$$
−0.359480 + 0.933153i $$0.617046\pi$$
$$24$$ 0 0
$$25$$ 11.0000 0.00352000
$$26$$ 0 0
$$27$$ 4060.00 1.07181
$$28$$ 0 0
$$29$$ 3418.00 0.754705 0.377352 0.926070i $$-0.376835\pi$$
0.377352 + 0.926070i $$0.376835\pi$$
$$30$$ 0 0
$$31$$ −7644.00 −1.42862 −0.714310 0.699830i $$-0.753259\pi$$
−0.714310 + 0.699830i $$0.753259\pi$$
$$32$$ 0 0
$$33$$ 3248.00 0.519196
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10398.0 −1.24866 −0.624332 0.781159i $$-0.714629\pi$$
−0.624332 + 0.781159i $$0.714629\pi$$
$$38$$ 0 0
$$39$$ −1960.00 −0.206345
$$40$$ 0 0
$$41$$ 17962.0 1.66876 0.834382 0.551186i $$-0.185825\pi$$
0.834382 + 0.551186i $$0.185825\pi$$
$$42$$ 0 0
$$43$$ −10880.0 −0.897342 −0.448671 0.893697i $$-0.648102\pi$$
−0.448671 + 0.893697i $$0.648102\pi$$
$$44$$ 0 0
$$45$$ −2632.00 −0.193756
$$46$$ 0 0
$$47$$ 9324.00 0.615684 0.307842 0.951438i $$-0.400393\pi$$
0.307842 + 0.951438i $$0.400393\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −24108.0 −1.29788
$$52$$ 0 0
$$53$$ 2262.00 0.110612 0.0553061 0.998469i $$-0.482387\pi$$
0.0553061 + 0.998469i $$0.482387\pi$$
$$54$$ 0 0
$$55$$ −12992.0 −0.579121
$$56$$ 0 0
$$57$$ 1372.00 0.0559329
$$58$$ 0 0
$$59$$ −2730.00 −0.102102 −0.0510508 0.998696i $$-0.516257\pi$$
−0.0510508 + 0.998696i $$0.516257\pi$$
$$60$$ 0 0
$$61$$ −25648.0 −0.882529 −0.441264 0.897377i $$-0.645470\pi$$
−0.441264 + 0.897377i $$0.645470\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 7840.00 0.230161
$$66$$ 0 0
$$67$$ 48404.0 1.31733 0.658664 0.752437i $$-0.271122\pi$$
0.658664 + 0.752437i $$0.271122\pi$$
$$68$$ 0 0
$$69$$ 25536.0 0.645699
$$70$$ 0 0
$$71$$ 58560.0 1.37865 0.689327 0.724450i $$-0.257906\pi$$
0.689327 + 0.724450i $$0.257906\pi$$
$$72$$ 0 0
$$73$$ −68082.0 −1.49529 −0.747645 0.664099i $$-0.768815\pi$$
−0.747645 + 0.664099i $$0.768815\pi$$
$$74$$ 0 0
$$75$$ −154.000 −0.00316131
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −31784.0 −0.572982 −0.286491 0.958083i $$-0.592489\pi$$
−0.286491 + 0.958083i $$0.592489\pi$$
$$80$$ 0 0
$$81$$ −45419.0 −0.769175
$$82$$ 0 0
$$83$$ −20538.0 −0.327237 −0.163619 0.986524i $$-0.552317\pi$$
−0.163619 + 0.986524i $$0.552317\pi$$
$$84$$ 0 0
$$85$$ 96432.0 1.44768
$$86$$ 0 0
$$87$$ −47852.0 −0.677801
$$88$$ 0 0
$$89$$ 50582.0 0.676894 0.338447 0.940985i $$-0.390098\pi$$
0.338447 + 0.940985i $$0.390098\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 107016. 1.28304
$$94$$ 0 0
$$95$$ −5488.00 −0.0623886
$$96$$ 0 0
$$97$$ 58506.0 0.631351 0.315676 0.948867i $$-0.397769\pi$$
0.315676 + 0.948867i $$0.397769\pi$$
$$98$$ 0 0
$$99$$ 10904.0 0.111814
$$100$$ 0 0
$$101$$ −38696.0 −0.377453 −0.188726 0.982030i $$-0.560436\pi$$
−0.188726 + 0.982030i $$0.560436\pi$$
$$102$$ 0 0
$$103$$ 53060.0 0.492804 0.246402 0.969168i $$-0.420752\pi$$
0.246402 + 0.969168i $$0.420752\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 146324. 1.23554 0.617769 0.786360i $$-0.288037\pi$$
0.617769 + 0.786360i $$0.288037\pi$$
$$108$$ 0 0
$$109$$ 92898.0 0.748928 0.374464 0.927241i $$-0.377827\pi$$
0.374464 + 0.927241i $$0.377827\pi$$
$$110$$ 0 0
$$111$$ 145572. 1.12143
$$112$$ 0 0
$$113$$ −83354.0 −0.614088 −0.307044 0.951695i $$-0.599340\pi$$
−0.307044 + 0.951695i $$0.599340\pi$$
$$114$$ 0 0
$$115$$ −102144. −0.720225
$$116$$ 0 0
$$117$$ −6580.00 −0.0444387
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −107227. −0.665795
$$122$$ 0 0
$$123$$ −251468. −1.49872
$$124$$ 0 0
$$125$$ −174384. −0.998232
$$126$$ 0 0
$$127$$ −60384.0 −0.332210 −0.166105 0.986108i $$-0.553119\pi$$
−0.166105 + 0.986108i $$0.553119\pi$$
$$128$$ 0 0
$$129$$ 152320. 0.805903
$$130$$ 0 0
$$131$$ −61586.0 −0.313548 −0.156774 0.987635i $$-0.550109\pi$$
−0.156774 + 0.987635i $$0.550109\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 227360. 1.07369
$$136$$ 0 0
$$137$$ −204462. −0.930703 −0.465352 0.885126i $$-0.654072\pi$$
−0.465352 + 0.885126i $$0.654072\pi$$
$$138$$ 0 0
$$139$$ −35406.0 −0.155432 −0.0777159 0.996976i $$-0.524763\pi$$
−0.0777159 + 0.996976i $$0.524763\pi$$
$$140$$ 0 0
$$141$$ −130536. −0.552946
$$142$$ 0 0
$$143$$ −32480.0 −0.132824
$$144$$ 0 0
$$145$$ 191408. 0.756032
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −20226.0 −0.0746353 −0.0373177 0.999303i $$-0.511881\pi$$
−0.0373177 + 0.999303i $$0.511881\pi$$
$$150$$ 0 0
$$151$$ −70904.0 −0.253063 −0.126531 0.991963i $$-0.540384\pi$$
−0.126531 + 0.991963i $$0.540384\pi$$
$$152$$ 0 0
$$153$$ −80934.0 −0.279513
$$154$$ 0 0
$$155$$ −428064. −1.43113
$$156$$ 0 0
$$157$$ −293524. −0.950374 −0.475187 0.879885i $$-0.657620\pi$$
−0.475187 + 0.879885i $$0.657620\pi$$
$$158$$ 0 0
$$159$$ −31668.0 −0.0993408
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −13192.0 −0.0388903 −0.0194452 0.999811i $$-0.506190\pi$$
−0.0194452 + 0.999811i $$0.506190\pi$$
$$164$$ 0 0
$$165$$ 181888. 0.520109
$$166$$ 0 0
$$167$$ 493612. 1.36960 0.684801 0.728730i $$-0.259889\pi$$
0.684801 + 0.728730i $$0.259889\pi$$
$$168$$ 0 0
$$169$$ −351693. −0.947212
$$170$$ 0 0
$$171$$ 4606.00 0.0120457
$$172$$ 0 0
$$173$$ −240716. −0.611490 −0.305745 0.952113i $$-0.598906\pi$$
−0.305745 + 0.952113i $$0.598906\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 38220.0 0.0916975
$$178$$ 0 0
$$179$$ −294932. −0.688001 −0.344001 0.938969i $$-0.611782\pi$$
−0.344001 + 0.938969i $$0.611782\pi$$
$$180$$ 0 0
$$181$$ 336980. 0.764553 0.382277 0.924048i $$-0.375140\pi$$
0.382277 + 0.924048i $$0.375140\pi$$
$$182$$ 0 0
$$183$$ 359072. 0.792600
$$184$$ 0 0
$$185$$ −582288. −1.25086
$$186$$ 0 0
$$187$$ −399504. −0.835444
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −358264. −0.710591 −0.355296 0.934754i $$-0.615620\pi$$
−0.355296 + 0.934754i $$0.615620\pi$$
$$192$$ 0 0
$$193$$ −989554. −1.91226 −0.956128 0.292948i $$-0.905364\pi$$
−0.956128 + 0.292948i $$0.905364\pi$$
$$194$$ 0 0
$$195$$ −109760. −0.206708
$$196$$ 0 0
$$197$$ −990050. −1.81757 −0.908786 0.417263i $$-0.862989\pi$$
−0.908786 + 0.417263i $$0.862989\pi$$
$$198$$ 0 0
$$199$$ −840756. −1.50500 −0.752501 0.658591i $$-0.771153\pi$$
−0.752501 + 0.658591i $$0.771153\pi$$
$$200$$ 0 0
$$201$$ −677656. −1.18309
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 1.00587e6 1.67170
$$206$$ 0 0
$$207$$ 85728.0 0.139058
$$208$$ 0 0
$$209$$ 22736.0 0.0360038
$$210$$ 0 0
$$211$$ −1.15073e6 −1.77938 −0.889689 0.456568i $$-0.849079\pi$$
−0.889689 + 0.456568i $$0.849079\pi$$
$$212$$ 0 0
$$213$$ −819840. −1.23817
$$214$$ 0 0
$$215$$ −609280. −0.898919
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 953148. 1.34292
$$220$$ 0 0
$$221$$ 241080. 0.332032
$$222$$ 0 0
$$223$$ −824264. −1.10995 −0.554976 0.831866i $$-0.687273\pi$$
−0.554976 + 0.831866i $$0.687273\pi$$
$$224$$ 0 0
$$225$$ −517.000 −0.000680823 0
$$226$$ 0 0
$$227$$ 74382.0 0.0958083 0.0479042 0.998852i $$-0.484746\pi$$
0.0479042 + 0.998852i $$0.484746\pi$$
$$228$$ 0 0
$$229$$ −1.13196e6 −1.42640 −0.713199 0.700961i $$-0.752755\pi$$
−0.713199 + 0.700961i $$0.752755\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −198726. −0.239809 −0.119904 0.992785i $$-0.538259\pi$$
−0.119904 + 0.992785i $$0.538259\pi$$
$$234$$ 0 0
$$235$$ 522144. 0.616766
$$236$$ 0 0
$$237$$ 444976. 0.514595
$$238$$ 0 0
$$239$$ −482904. −0.546847 −0.273424 0.961894i $$-0.588156\pi$$
−0.273424 + 0.961894i $$0.588156\pi$$
$$240$$ 0 0
$$241$$ −805910. −0.893807 −0.446904 0.894582i $$-0.647473\pi$$
−0.446904 + 0.894582i $$0.647473\pi$$
$$242$$ 0 0
$$243$$ −350714. −0.381011
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −13720.0 −0.0143091
$$248$$ 0 0
$$249$$ 287532. 0.293892
$$250$$ 0 0
$$251$$ 430738. 0.431548 0.215774 0.976443i $$-0.430773\pi$$
0.215774 + 0.976443i $$0.430773\pi$$
$$252$$ 0 0
$$253$$ 423168. 0.415634
$$254$$ 0 0
$$255$$ −1.35005e6 −1.30017
$$256$$ 0 0
$$257$$ 1.17691e6 1.11150 0.555751 0.831349i $$-0.312431\pi$$
0.555751 + 0.831349i $$0.312431\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −160646. −0.145972
$$262$$ 0 0
$$263$$ −1.29098e6 −1.15088 −0.575438 0.817845i $$-0.695169\pi$$
−0.575438 + 0.817845i $$0.695169\pi$$
$$264$$ 0 0
$$265$$ 126672. 0.110807
$$266$$ 0 0
$$267$$ −708148. −0.607919
$$268$$ 0 0
$$269$$ 1.27756e6 1.07646 0.538232 0.842797i $$-0.319093\pi$$
0.538232 + 0.842797i $$0.319093\pi$$
$$270$$ 0 0
$$271$$ 1.65054e6 1.36522 0.682612 0.730781i $$-0.260844\pi$$
0.682612 + 0.730781i $$0.260844\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2552.00 −0.00203493
$$276$$ 0 0
$$277$$ −1.06409e6 −0.833257 −0.416628 0.909077i $$-0.636788\pi$$
−0.416628 + 0.909077i $$0.636788\pi$$
$$278$$ 0 0
$$279$$ 359268. 0.276317
$$280$$ 0 0
$$281$$ −22342.0 −0.0168794 −0.00843969 0.999964i $$-0.502686\pi$$
−0.00843969 + 0.999964i $$0.502686\pi$$
$$282$$ 0 0
$$283$$ −2.49574e6 −1.85239 −0.926196 0.377042i $$-0.876941\pi$$
−0.926196 + 0.377042i $$0.876941\pi$$
$$284$$ 0 0
$$285$$ 76832.0 0.0560312
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.54543e6 1.08844
$$290$$ 0 0
$$291$$ −819084. −0.567017
$$292$$ 0 0
$$293$$ 1.93178e6 1.31458 0.657291 0.753637i $$-0.271702\pi$$
0.657291 + 0.753637i $$0.271702\pi$$
$$294$$ 0 0
$$295$$ −152880. −0.102281
$$296$$ 0 0
$$297$$ −941920. −0.619616
$$298$$ 0 0
$$299$$ −255360. −0.165187
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 541744. 0.338991
$$304$$ 0 0
$$305$$ −1.43629e6 −0.884081
$$306$$ 0 0
$$307$$ −459074. −0.277995 −0.138997 0.990293i $$-0.544388\pi$$
−0.138997 + 0.990293i $$0.544388\pi$$
$$308$$ 0 0
$$309$$ −742840. −0.442587
$$310$$ 0 0
$$311$$ 667128. 0.391118 0.195559 0.980692i $$-0.437348\pi$$
0.195559 + 0.980692i $$0.437348\pi$$
$$312$$ 0 0
$$313$$ 111034. 0.0640612 0.0320306 0.999487i $$-0.489803\pi$$
0.0320306 + 0.999487i $$0.489803\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −68778.0 −0.0384416 −0.0192208 0.999815i $$-0.506119\pi$$
−0.0192208 + 0.999815i $$0.506119\pi$$
$$318$$ 0 0
$$319$$ −792976. −0.436298
$$320$$ 0 0
$$321$$ −2.04854e6 −1.10964
$$322$$ 0 0
$$323$$ −168756. −0.0900022
$$324$$ 0 0
$$325$$ 1540.00 0.000808746 0
$$326$$ 0 0
$$327$$ −1.30057e6 −0.672613
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 564448. 0.283174 0.141587 0.989926i $$-0.454779\pi$$
0.141587 + 0.989926i $$0.454779\pi$$
$$332$$ 0 0
$$333$$ 488706. 0.241511
$$334$$ 0 0
$$335$$ 2.71062e6 1.31965
$$336$$ 0 0
$$337$$ 2.07729e6 0.996376 0.498188 0.867069i $$-0.333999\pi$$
0.498188 + 0.867069i $$0.333999\pi$$
$$338$$ 0 0
$$339$$ 1.16696e6 0.551512
$$340$$ 0 0
$$341$$ 1.77341e6 0.825891
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 1.43002e6 0.646834
$$346$$ 0 0
$$347$$ 53248.0 0.0237399 0.0118700 0.999930i $$-0.496222\pi$$
0.0118700 + 0.999930i $$0.496222\pi$$
$$348$$ 0 0
$$349$$ 2.27200e6 0.998494 0.499247 0.866460i $$-0.333610\pi$$
0.499247 + 0.866460i $$0.333610\pi$$
$$350$$ 0 0
$$351$$ 568400. 0.246256
$$352$$ 0 0
$$353$$ −4.00645e6 −1.71129 −0.855644 0.517565i $$-0.826838\pi$$
−0.855644 + 0.517565i $$0.826838\pi$$
$$354$$ 0 0
$$355$$ 3.27936e6 1.38108
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −73784.0 −0.0302152 −0.0151076 0.999886i $$-0.504809\pi$$
−0.0151076 + 0.999886i $$0.504809\pi$$
$$360$$ 0 0
$$361$$ −2.46650e6 −0.996121
$$362$$ 0 0
$$363$$ 1.50118e6 0.597951
$$364$$ 0 0
$$365$$ −3.81259e6 −1.49792
$$366$$ 0 0
$$367$$ 1.40431e6 0.544250 0.272125 0.962262i $$-0.412274\pi$$
0.272125 + 0.962262i $$0.412274\pi$$
$$368$$ 0 0
$$369$$ −844214. −0.322765
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.60323e6 −0.596657 −0.298329 0.954463i $$-0.596429\pi$$
−0.298329 + 0.954463i $$0.596429\pi$$
$$374$$ 0 0
$$375$$ 2.44138e6 0.896513
$$376$$ 0 0
$$377$$ 478520. 0.173399
$$378$$ 0 0
$$379$$ 4.77012e6 1.70581 0.852906 0.522064i $$-0.174838\pi$$
0.852906 + 0.522064i $$0.174838\pi$$
$$380$$ 0 0
$$381$$ 845376. 0.298358
$$382$$ 0 0
$$383$$ −2.23079e6 −0.777072 −0.388536 0.921434i $$-0.627019\pi$$
−0.388536 + 0.921434i $$0.627019\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 511360. 0.173560
$$388$$ 0 0
$$389$$ 4.84024e6 1.62178 0.810892 0.585196i $$-0.198982\pi$$
0.810892 + 0.585196i $$0.198982\pi$$
$$390$$ 0 0
$$391$$ −3.14093e6 −1.03900
$$392$$ 0 0
$$393$$ 862204. 0.281597
$$394$$ 0 0
$$395$$ −1.77990e6 −0.573989
$$396$$ 0 0
$$397$$ −995820. −0.317106 −0.158553 0.987350i $$-0.550683\pi$$
−0.158553 + 0.987350i $$0.550683\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3.31605e6 −1.02982 −0.514909 0.857245i $$-0.672174\pi$$
−0.514909 + 0.857245i $$0.672174\pi$$
$$402$$ 0 0
$$403$$ −1.07016e6 −0.328236
$$404$$ 0 0
$$405$$ −2.54346e6 −0.770527
$$406$$ 0 0
$$407$$ 2.41234e6 0.721858
$$408$$ 0 0
$$409$$ −3.07273e6 −0.908274 −0.454137 0.890932i $$-0.650052\pi$$
−0.454137 + 0.890932i $$0.650052\pi$$
$$410$$ 0 0
$$411$$ 2.86247e6 0.835865
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.15013e6 −0.327813
$$416$$ 0 0
$$417$$ 495684. 0.139593
$$418$$ 0 0
$$419$$ 2.81438e6 0.783154 0.391577 0.920145i $$-0.371930\pi$$
0.391577 + 0.920145i $$0.371930\pi$$
$$420$$ 0 0
$$421$$ 3.05802e6 0.840883 0.420441 0.907320i $$-0.361875\pi$$
0.420441 + 0.907320i $$0.361875\pi$$
$$422$$ 0 0
$$423$$ −438228. −0.119083
$$424$$ 0 0
$$425$$ 18942.0 0.00508690
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 454720. 0.119289
$$430$$ 0 0
$$431$$ −1.93750e6 −0.502398 −0.251199 0.967936i $$-0.580825\pi$$
−0.251199 + 0.967936i $$0.580825\pi$$
$$432$$ 0 0
$$433$$ −3.94790e6 −1.01192 −0.505961 0.862557i $$-0.668862\pi$$
−0.505961 + 0.862557i $$0.668862\pi$$
$$434$$ 0 0
$$435$$ −2.67971e6 −0.678993
$$436$$ 0 0
$$437$$ 178752. 0.0447762
$$438$$ 0 0
$$439$$ −7.41770e6 −1.83700 −0.918498 0.395426i $$-0.870597\pi$$
−0.918498 + 0.395426i $$0.870597\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −1.40269e6 −0.339589 −0.169794 0.985480i $$-0.554310\pi$$
−0.169794 + 0.985480i $$0.554310\pi$$
$$444$$ 0 0
$$445$$ 2.83259e6 0.678085
$$446$$ 0 0
$$447$$ 283164. 0.0670300
$$448$$ 0 0
$$449$$ −590574. −0.138248 −0.0691239 0.997608i $$-0.522020\pi$$
−0.0691239 + 0.997608i $$0.522020\pi$$
$$450$$ 0 0
$$451$$ −4.16718e6 −0.964720
$$452$$ 0 0
$$453$$ 992656. 0.227276
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −2.90484e6 −0.650627 −0.325313 0.945606i $$-0.605470\pi$$
−0.325313 + 0.945606i $$0.605470\pi$$
$$458$$ 0 0
$$459$$ 6.99132e6 1.54891
$$460$$ 0 0
$$461$$ 922684. 0.202209 0.101105 0.994876i $$-0.467762\pi$$
0.101105 + 0.994876i $$0.467762\pi$$
$$462$$ 0 0
$$463$$ −7.18235e6 −1.55709 −0.778546 0.627588i $$-0.784042\pi$$
−0.778546 + 0.627588i $$0.784042\pi$$
$$464$$ 0 0
$$465$$ 5.99290e6 1.28530
$$466$$ 0 0
$$467$$ −612570. −0.129976 −0.0649881 0.997886i $$-0.520701\pi$$
−0.0649881 + 0.997886i $$0.520701\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 4.10934e6 0.853531
$$472$$ 0 0
$$473$$ 2.52416e6 0.518757
$$474$$ 0 0
$$475$$ −1078.00 −0.000219222 0
$$476$$ 0 0
$$477$$ −106314. −0.0213941
$$478$$ 0 0
$$479$$ 2.60330e6 0.518424 0.259212 0.965820i $$-0.416537\pi$$
0.259212 + 0.965820i $$0.416537\pi$$
$$480$$ 0 0
$$481$$ −1.45572e6 −0.286890
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3.27634e6 0.632461
$$486$$ 0 0
$$487$$ −5.46309e6 −1.04380 −0.521898 0.853008i $$-0.674776\pi$$
−0.521898 + 0.853008i $$0.674776\pi$$
$$488$$ 0 0
$$489$$ 184688. 0.0349274
$$490$$ 0 0
$$491$$ −1.64090e6 −0.307170 −0.153585 0.988135i $$-0.549082\pi$$
−0.153585 + 0.988135i $$0.549082\pi$$
$$492$$ 0 0
$$493$$ 5.88580e6 1.09066
$$494$$ 0 0
$$495$$ 610624. 0.112011
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −2.99796e6 −0.538983 −0.269491 0.963003i $$-0.586856\pi$$
−0.269491 + 0.963003i $$0.586856\pi$$
$$500$$ 0 0
$$501$$ −6.91057e6 −1.23004
$$502$$ 0 0
$$503$$ −6.89405e6 −1.21494 −0.607469 0.794343i $$-0.707815\pi$$
−0.607469 + 0.794343i $$0.707815\pi$$
$$504$$ 0 0
$$505$$ −2.16698e6 −0.378117
$$506$$ 0 0
$$507$$ 4.92370e6 0.850691
$$508$$ 0 0
$$509$$ −2.30476e6 −0.394305 −0.197152 0.980373i $$-0.563169\pi$$
−0.197152 + 0.980373i $$0.563169\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −397880. −0.0667511
$$514$$ 0 0
$$515$$ 2.97136e6 0.493671
$$516$$ 0 0
$$517$$ −2.16317e6 −0.355929
$$518$$ 0 0
$$519$$ 3.37002e6 0.549180
$$520$$ 0 0
$$521$$ 1.20960e7 1.95231 0.976155 0.217073i $$-0.0696509\pi$$
0.976155 + 0.217073i $$0.0696509\pi$$
$$522$$ 0 0
$$523$$ 5.48443e6 0.876753 0.438377 0.898791i $$-0.355554\pi$$
0.438377 + 0.898791i $$0.355554\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1.31630e7 −2.06456
$$528$$ 0 0
$$529$$ −3.10937e6 −0.483095
$$530$$ 0 0
$$531$$ 128310. 0.0197480
$$532$$ 0 0
$$533$$ 2.51468e6 0.383411
$$534$$ 0 0
$$535$$ 8.19414e6 1.23771
$$536$$ 0 0
$$537$$ 4.12905e6 0.617894
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −6.71799e6 −0.986839 −0.493420 0.869791i $$-0.664253\pi$$
−0.493420 + 0.869791i $$0.664253\pi$$
$$542$$ 0 0
$$543$$ −4.71772e6 −0.686646
$$544$$ 0 0
$$545$$ 5.20229e6 0.750245
$$546$$ 0 0
$$547$$ 5.00235e6 0.714835 0.357418 0.933945i $$-0.383657\pi$$
0.357418 + 0.933945i $$0.383657\pi$$
$$548$$ 0 0
$$549$$ 1.20546e6 0.170695
$$550$$ 0 0
$$551$$ −334964. −0.0470023
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 8.15203e6 1.12340
$$556$$ 0 0
$$557$$ 9.01961e6 1.23183 0.615913 0.787814i $$-0.288787\pi$$
0.615913 + 0.787814i $$0.288787\pi$$
$$558$$ 0 0
$$559$$ −1.52320e6 −0.206171
$$560$$ 0 0
$$561$$ 5.59306e6 0.750312
$$562$$ 0 0
$$563$$ 1.24051e7 1.64941 0.824707 0.565561i $$-0.191340\pi$$
0.824707 + 0.565561i $$0.191340\pi$$
$$564$$ 0 0
$$565$$ −4.66782e6 −0.615167
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.48804e6 0.840103 0.420052 0.907500i $$-0.362012\pi$$
0.420052 + 0.907500i $$0.362012\pi$$
$$570$$ 0 0
$$571$$ 1.02285e7 1.31287 0.656435 0.754382i $$-0.272064\pi$$
0.656435 + 0.754382i $$0.272064\pi$$
$$572$$ 0 0
$$573$$ 5.01570e6 0.638182
$$574$$ 0 0
$$575$$ −20064.0 −0.00253074
$$576$$ 0 0
$$577$$ −2.65338e6 −0.331787 −0.165894 0.986144i $$-0.553051\pi$$
−0.165894 + 0.986144i $$0.553051\pi$$
$$578$$ 0 0
$$579$$ 1.38538e7 1.71740
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −524784. −0.0639454
$$584$$ 0 0
$$585$$ −368480. −0.0445168
$$586$$ 0 0
$$587$$ −1.43044e7 −1.71346 −0.856729 0.515766i $$-0.827507\pi$$
−0.856729 + 0.515766i $$0.827507\pi$$
$$588$$ 0 0
$$589$$ 749112. 0.0889731
$$590$$ 0 0
$$591$$ 1.38607e7 1.63236
$$592$$ 0 0
$$593$$ 1.00265e7 1.17088 0.585442 0.810714i $$-0.300921\pi$$
0.585442 + 0.810714i $$0.300921\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1.17706e7 1.35164
$$598$$ 0 0
$$599$$ 7.52292e6 0.856681 0.428341 0.903617i $$-0.359098\pi$$
0.428341 + 0.903617i $$0.359098\pi$$
$$600$$ 0 0
$$601$$ −3.38625e6 −0.382413 −0.191207 0.981550i $$-0.561240\pi$$
−0.191207 + 0.981550i $$0.561240\pi$$
$$602$$ 0 0
$$603$$ −2.27499e6 −0.254792
$$604$$ 0 0
$$605$$ −6.00471e6 −0.666966
$$606$$ 0 0
$$607$$ −6.90861e6 −0.761060 −0.380530 0.924769i $$-0.624258\pi$$
−0.380530 + 0.924769i $$0.624258\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.30536e6 0.141458
$$612$$ 0 0
$$613$$ −9.68896e6 −1.04142 −0.520710 0.853734i $$-0.674333\pi$$
−0.520710 + 0.853734i $$0.674333\pi$$
$$614$$ 0 0
$$615$$ −1.40822e7 −1.50135
$$616$$ 0 0
$$617$$ −7.84742e6 −0.829877 −0.414939 0.909849i $$-0.636197\pi$$
−0.414939 + 0.909849i $$0.636197\pi$$
$$618$$ 0 0
$$619$$ −1.01972e7 −1.06968 −0.534840 0.844953i $$-0.679628\pi$$
−0.534840 + 0.844953i $$0.679628\pi$$
$$620$$ 0 0
$$621$$ −7.40544e6 −0.770587
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −9.79988e6 −1.00351
$$626$$ 0 0
$$627$$ −318304. −0.0323350
$$628$$ 0 0
$$629$$ −1.79054e7 −1.80450
$$630$$ 0 0
$$631$$ 8.36258e6 0.836116 0.418058 0.908420i $$-0.362711\pi$$
0.418058 + 0.908420i $$0.362711\pi$$
$$632$$ 0 0
$$633$$ 1.61102e7 1.59806
$$634$$ 0 0
$$635$$ −3.38150e6 −0.332794
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −2.75232e6 −0.266653
$$640$$ 0 0
$$641$$ 1.10283e6 0.106014 0.0530070 0.998594i $$-0.483119\pi$$
0.0530070 + 0.998594i $$0.483119\pi$$
$$642$$ 0 0
$$643$$ 1.71354e7 1.63443 0.817217 0.576330i $$-0.195516\pi$$
0.817217 + 0.576330i $$0.195516\pi$$
$$644$$ 0 0
$$645$$ 8.52992e6 0.807320
$$646$$ 0 0
$$647$$ −54964.0 −0.00516200 −0.00258100 0.999997i $$-0.500822\pi$$
−0.00258100 + 0.999997i $$0.500822\pi$$
$$648$$ 0 0
$$649$$ 633360. 0.0590254
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −485166. −0.0445254 −0.0222627 0.999752i $$-0.507087\pi$$
−0.0222627 + 0.999752i $$0.507087\pi$$
$$654$$ 0 0
$$655$$ −3.44882e6 −0.314099
$$656$$ 0 0
$$657$$ 3.19985e6 0.289212
$$658$$ 0 0
$$659$$ 2.72136e6 0.244103 0.122051 0.992524i $$-0.461053\pi$$
0.122051 + 0.992524i $$0.461053\pi$$
$$660$$ 0 0
$$661$$ 2.14525e6 0.190974 0.0954869 0.995431i $$-0.469559\pi$$
0.0954869 + 0.995431i $$0.469559\pi$$
$$662$$ 0 0
$$663$$ −3.37512e6 −0.298198
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.23443e6 −0.542603
$$668$$ 0 0
$$669$$ 1.15397e7 0.996848
$$670$$ 0 0
$$671$$ 5.95034e6 0.510194
$$672$$ 0 0
$$673$$ 2.92796e6 0.249188 0.124594 0.992208i $$-0.460237\pi$$
0.124594 + 0.992208i $$0.460237\pi$$
$$674$$ 0 0
$$675$$ 44660.0 0.00377276
$$676$$ 0 0
$$677$$ 1.34992e7 1.13198 0.565988 0.824414i $$-0.308495\pi$$
0.565988 + 0.824414i $$0.308495\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −1.04135e6 −0.0860455
$$682$$ 0 0
$$683$$ 5.42972e6 0.445375 0.222688 0.974890i $$-0.428517\pi$$
0.222688 + 0.974890i $$0.428517\pi$$
$$684$$ 0 0
$$685$$ −1.14499e7 −0.932340
$$686$$ 0 0
$$687$$ 1.58474e7 1.28105
$$688$$ 0 0
$$689$$ 316680. 0.0254140
$$690$$ 0 0
$$691$$ 2.08280e7 1.65940 0.829702 0.558207i $$-0.188510\pi$$
0.829702 + 0.558207i $$0.188510\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1.98274e6 −0.155705
$$696$$ 0 0
$$697$$ 3.09306e7 2.41160
$$698$$ 0 0
$$699$$ 2.78216e6 0.215372
$$700$$ 0 0
$$701$$ 2.35141e7 1.80731 0.903655 0.428261i $$-0.140874\pi$$
0.903655 + 0.428261i $$0.140874\pi$$
$$702$$ 0 0
$$703$$ 1.01900e6 0.0777656
$$704$$ 0 0
$$705$$ −7.31002e6 −0.553918
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.95747e7 −1.46244 −0.731221 0.682140i $$-0.761049\pi$$
−0.731221 + 0.682140i $$0.761049\pi$$
$$710$$ 0 0
$$711$$ 1.49385e6 0.110824
$$712$$ 0 0
$$713$$ 1.39427e7 1.02712
$$714$$ 0 0
$$715$$ −1.81888e6 −0.133057
$$716$$ 0 0
$$717$$ 6.76066e6 0.491124
$$718$$ 0 0
$$719$$ −2.61152e7 −1.88396 −0.941978 0.335674i $$-0.891036\pi$$
−0.941978 + 0.335674i $$0.891036\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 1.12827e7 0.802729
$$724$$ 0 0
$$725$$ 37598.0 0.00265656
$$726$$ 0 0
$$727$$ 1.54126e7 1.08154 0.540768 0.841172i $$-0.318134\pi$$
0.540768 + 0.841172i $$0.318134\pi$$
$$728$$ 0 0
$$729$$ 1.59468e7 1.11136
$$730$$ 0 0
$$731$$ −1.87354e7 −1.29679
$$732$$ 0 0
$$733$$ 1.69868e7 1.16776 0.583878 0.811841i $$-0.301535\pi$$
0.583878 + 0.811841i $$0.301535\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.12297e7 −0.761554
$$738$$ 0 0
$$739$$ −2.01511e6 −0.135734 −0.0678669 0.997694i $$-0.521619\pi$$
−0.0678669 + 0.997694i $$0.521619\pi$$
$$740$$ 0 0
$$741$$ 192080. 0.0128510
$$742$$ 0 0
$$743$$ 1.51381e7 1.00600 0.503001 0.864286i $$-0.332229\pi$$
0.503001 + 0.864286i $$0.332229\pi$$
$$744$$ 0 0
$$745$$ −1.13266e6 −0.0747666
$$746$$ 0 0
$$747$$ 965286. 0.0632928
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −7.21401e6 −0.466742 −0.233371 0.972388i $$-0.574976\pi$$
−0.233371 + 0.972388i $$0.574976\pi$$
$$752$$ 0 0
$$753$$ −6.03033e6 −0.387573
$$754$$ 0 0
$$755$$ −3.97062e6 −0.253508
$$756$$ 0 0
$$757$$ −1.09697e7 −0.695755 −0.347877 0.937540i $$-0.613097\pi$$
−0.347877 + 0.937540i $$0.613097\pi$$
$$758$$ 0 0
$$759$$ −5.92435e6 −0.373281
$$760$$ 0 0
$$761$$ −1.92442e7 −1.20459 −0.602293 0.798275i $$-0.705746\pi$$
−0.602293 + 0.798275i $$0.705746\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −4.53230e6 −0.280005
$$766$$ 0 0
$$767$$ −382200. −0.0234586
$$768$$ 0 0
$$769$$ −8.21185e6 −0.500755 −0.250378 0.968148i $$-0.580555\pi$$
−0.250378 + 0.968148i $$0.580555\pi$$
$$770$$ 0 0
$$771$$ −1.64767e7 −0.998241
$$772$$ 0 0
$$773$$ −1.86187e7 −1.12073 −0.560363 0.828247i $$-0.689338\pi$$
−0.560363 + 0.828247i $$0.689338\pi$$
$$774$$ 0 0
$$775$$ −84084.0 −0.00502874
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.76028e6 −0.103929
$$780$$ 0 0
$$781$$ −1.35859e7 −0.797006
$$782$$ 0 0
$$783$$ 1.38771e7 0.808898
$$784$$ 0 0
$$785$$ −1.64373e7 −0.952045
$$786$$ 0 0
$$787$$ 2.62501e7 1.51075 0.755377 0.655291i $$-0.227454\pi$$
0.755377 + 0.655291i $$0.227454\pi$$
$$788$$ 0 0
$$789$$ 1.80737e7 1.03360
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −3.59072e6 −0.202768
$$794$$ 0 0
$$795$$ −1.77341e6 −0.0995155
$$796$$ 0 0
$$797$$ 1.00373e7 0.559720 0.279860 0.960041i $$-0.409712\pi$$
0.279860 + 0.960041i $$0.409712\pi$$
$$798$$ 0 0
$$799$$ 1.60559e7 0.889751
$$800$$ 0 0
$$801$$ −2.37735e6 −0.130922
$$802$$ 0 0
$$803$$ 1.57950e7 0.864433
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −1.78858e7 −0.966772
$$808$$ 0 0
$$809$$ 1.40884e7 0.756816 0.378408 0.925639i $$-0.376472\pi$$
0.378408 + 0.925639i $$0.376472\pi$$
$$810$$ 0 0
$$811$$ 1.81433e7 0.968646 0.484323 0.874889i $$-0.339066\pi$$
0.484323 + 0.874889i $$0.339066\pi$$
$$812$$ 0 0
$$813$$ −2.31076e7 −1.22611
$$814$$ 0 0
$$815$$ −738752. −0.0389587
$$816$$ 0 0
$$817$$ 1.06624e6 0.0558856
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.13669e7 −1.10633 −0.553164 0.833072i $$-0.686580\pi$$
−0.553164 + 0.833072i $$0.686580\pi$$
$$822$$ 0 0
$$823$$ −1.78017e7 −0.916142 −0.458071 0.888916i $$-0.651459\pi$$
−0.458071 + 0.888916i $$0.651459\pi$$
$$824$$ 0 0
$$825$$ 35728.0 0.00182757
$$826$$ 0 0
$$827$$ −1.62921e7 −0.828350 −0.414175 0.910197i $$-0.635930\pi$$
−0.414175 + 0.910197i $$0.635930\pi$$
$$828$$ 0 0
$$829$$ 2.08499e6 0.105370 0.0526851 0.998611i $$-0.483222\pi$$
0.0526851 + 0.998611i $$0.483222\pi$$
$$830$$ 0 0
$$831$$ 1.48973e7 0.748348
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 2.76423e7 1.37201
$$836$$ 0 0
$$837$$ −3.10346e7 −1.53120
$$838$$ 0 0
$$839$$ −2.27850e7 −1.11749 −0.558745 0.829340i $$-0.688717\pi$$
−0.558745 + 0.829340i $$0.688717\pi$$
$$840$$ 0 0
$$841$$ −8.82842e6 −0.430421
$$842$$ 0 0
$$843$$ 312788. 0.0151594
$$844$$ 0 0
$$845$$ −1.96948e7 −0.948877
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 3.49403e7 1.66363
$$850$$ 0 0
$$851$$ 1.89660e7 0.897740
$$852$$ 0 0
$$853$$ 2.26975e7 1.06808 0.534042 0.845458i $$-0.320672\pi$$
0.534042 + 0.845458i $$0.320672\pi$$
$$854$$ 0 0
$$855$$ 257936. 0.0120669
$$856$$ 0 0
$$857$$ −2.52900e7 −1.17624 −0.588120 0.808774i $$-0.700132\pi$$
−0.588120 + 0.808774i $$0.700132\pi$$
$$858$$ 0 0
$$859$$ −1.03947e7 −0.480652 −0.240326 0.970692i $$-0.577254\pi$$
−0.240326 + 0.970692i $$0.577254\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −4.33399e7 −1.98089 −0.990447 0.137892i $$-0.955967\pi$$
−0.990447 + 0.137892i $$0.955967\pi$$
$$864$$ 0 0
$$865$$ −1.34801e7 −0.612566
$$866$$ 0 0
$$867$$ −2.16360e7 −0.977527
$$868$$ 0 0
$$869$$ 7.37389e6 0.331243
$$870$$ 0 0
$$871$$ 6.77656e6 0.302666
$$872$$ 0 0
$$873$$ −2.74978e6 −0.122113
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 3.71659e7 1.63172 0.815861 0.578248i $$-0.196264\pi$$
0.815861 + 0.578248i $$0.196264\pi$$
$$878$$ 0 0
$$879$$ −2.70449e7 −1.18063
$$880$$ 0 0
$$881$$ −9.04785e6 −0.392740 −0.196370 0.980530i $$-0.562915\pi$$
−0.196370 + 0.980530i $$0.562915\pi$$
$$882$$ 0 0
$$883$$ −3.29679e7 −1.42295 −0.711474 0.702712i $$-0.751972\pi$$
−0.711474 + 0.702712i $$0.751972\pi$$
$$884$$ 0 0
$$885$$ 2.14032e6 0.0918588
$$886$$ 0 0
$$887$$ −1.61099e7 −0.687517 −0.343758 0.939058i $$-0.611700\pi$$
−0.343758 + 0.939058i $$0.611700\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1.05372e7 0.444663
$$892$$ 0 0
$$893$$ −913752. −0.0383442
$$894$$ 0 0
$$895$$ −1.65162e7 −0.689211
$$896$$ 0 0
$$897$$ 3.57504e6 0.148354
$$898$$ 0 0
$$899$$ −2.61272e7 −1.07819
$$900$$ 0 0
$$901$$ 3.89516e6 0.159850
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 1.88709e7 0.765898
$$906$$ 0 0
$$907$$ 4.47286e7 1.80537 0.902686 0.430300i $$-0.141592\pi$$
0.902686 + 0.430300i $$0.141592\pi$$
$$908$$ 0 0
$$909$$ 1.81871e6 0.0730053
$$910$$ 0 0
$$911$$ 6.60518e6 0.263687 0.131844 0.991271i $$-0.457910\pi$$
0.131844 + 0.991271i $$0.457910\pi$$
$$912$$ 0 0
$$913$$ 4.76482e6 0.189177
$$914$$ 0 0
$$915$$ 2.01080e7 0.793993
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 3.08930e7 1.20662 0.603311 0.797506i $$-0.293848\pi$$
0.603311 + 0.797506i $$0.293848\pi$$
$$920$$ 0 0
$$921$$ 6.42704e6 0.249667
$$922$$ 0 0
$$923$$ 8.19840e6 0.316756
$$924$$ 0 0
$$925$$ −114378. −0.00439530
$$926$$ 0 0
$$927$$ −2.49382e6 −0.0953160
$$928$$ 0 0
$$929$$ 4.87215e6 0.185217 0.0926087 0.995703i $$-0.470479\pi$$
0.0926087 + 0.995703i $$0.470479\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −9.33979e6 −0.351264
$$934$$ 0 0
$$935$$ −2.23722e7 −0.836913
$$936$$ 0 0
$$937$$ 3.25004e7 1.20932 0.604658 0.796485i $$-0.293310\pi$$
0.604658 + 0.796485i $$0.293310\pi$$
$$938$$ 0 0
$$939$$ −1.55448e6 −0.0575334
$$940$$ 0 0
$$941$$ 2.64040e6 0.0972066 0.0486033 0.998818i $$-0.484523\pi$$
0.0486033 + 0.998818i $$0.484523\pi$$
$$942$$ 0 0
$$943$$ −3.27627e7 −1.19978
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.08179e7 1.47903 0.739513 0.673142i $$-0.235056\pi$$
0.739513 + 0.673142i $$0.235056\pi$$
$$948$$ 0 0
$$949$$ −9.53148e6 −0.343554
$$950$$ 0 0
$$951$$ 962892. 0.0345244
$$952$$ 0 0
$$953$$ −6.71983e6 −0.239677 −0.119838 0.992793i $$-0.538238\pi$$
−0.119838 + 0.992793i $$0.538238\pi$$
$$954$$ 0 0
$$955$$ −2.00628e7 −0.711841
$$956$$ 0 0
$$957$$ 1.11017e7 0.391840
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 2.98016e7 1.04095
$$962$$ 0 0
$$963$$ −6.87723e6 −0.238972
$$964$$ 0 0
$$965$$ −5.54150e7 −1.91562
$$966$$ 0 0
$$967$$ 2.78979e6 0.0959413 0.0479707 0.998849i $$-0.484725\pi$$
0.0479707 + 0.998849i $$0.484725\pi$$
$$968$$ 0 0
$$969$$ 2.36258e6 0.0808310
$$970$$ 0 0
$$971$$ 3.33594e7 1.13545 0.567727 0.823217i $$-0.307823\pi$$
0.567727 + 0.823217i $$0.307823\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −21560.0 −0.000726335 0
$$976$$ 0 0
$$977$$ −7.60033e6 −0.254739 −0.127370 0.991855i $$-0.540653\pi$$
−0.127370 + 0.991855i $$0.540653\pi$$
$$978$$ 0 0
$$979$$ −1.17350e7 −0.391316
$$980$$ 0 0
$$981$$ −4.36621e6 −0.144854
$$982$$ 0 0
$$983$$ −5.79760e6 −0.191366 −0.0956829 0.995412i $$-0.530503\pi$$
−0.0956829 + 0.995412i $$0.530503\pi$$
$$984$$ 0 0
$$985$$ −5.54428e7 −1.82077
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.98451e7 0.645153
$$990$$ 0 0
$$991$$ −1.26825e7 −0.410224 −0.205112 0.978739i $$-0.565756\pi$$
−0.205112 + 0.978739i $$0.565756\pi$$
$$992$$ 0 0
$$993$$ −7.90227e6 −0.254319
$$994$$ 0 0
$$995$$ −4.70823e7 −1.50765
$$996$$ 0 0
$$997$$ −1.44400e7 −0.460077 −0.230039 0.973182i $$-0.573885\pi$$
−0.230039 + 0.973182i $$0.573885\pi$$
$$998$$ 0 0
$$999$$ −4.22159e7 −1.33833
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 784.6.a.c.1.1 1
4.3 odd 2 49.6.a.a.1.1 1
7.6 odd 2 112.6.a.g.1.1 1
12.11 even 2 441.6.a.k.1.1 1
21.20 even 2 1008.6.a.y.1.1 1
28.3 even 6 49.6.c.c.30.1 2
28.11 odd 6 49.6.c.b.30.1 2
28.19 even 6 49.6.c.c.18.1 2
28.23 odd 6 49.6.c.b.18.1 2
28.27 even 2 7.6.a.a.1.1 1
56.13 odd 2 448.6.a.c.1.1 1
56.27 even 2 448.6.a.m.1.1 1
84.83 odd 2 63.6.a.e.1.1 1
140.27 odd 4 175.6.b.a.99.1 2
140.83 odd 4 175.6.b.a.99.2 2
140.139 even 2 175.6.a.b.1.1 1
308.307 odd 2 847.6.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 28.27 even 2
49.6.a.a.1.1 1 4.3 odd 2
49.6.c.b.18.1 2 28.23 odd 6
49.6.c.b.30.1 2 28.11 odd 6
49.6.c.c.18.1 2 28.19 even 6
49.6.c.c.30.1 2 28.3 even 6
63.6.a.e.1.1 1 84.83 odd 2
112.6.a.g.1.1 1 7.6 odd 2
175.6.a.b.1.1 1 140.139 even 2
175.6.b.a.99.1 2 140.27 odd 4
175.6.b.a.99.2 2 140.83 odd 4
441.6.a.k.1.1 1 12.11 even 2
448.6.a.c.1.1 1 56.13 odd 2
448.6.a.m.1.1 1 56.27 even 2
784.6.a.c.1.1 1 1.1 even 1 trivial
847.6.a.b.1.1 1 308.307 odd 2
1008.6.a.y.1.1 1 21.20 even 2