# Properties

 Label 784.6.a.a.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 28) 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)

 $$f(q)$$ $$=$$ $$q-19.0000 q^{3} -19.0000 q^{5} +118.000 q^{9} +O(q^{10})$$ $$q-19.0000 q^{3} -19.0000 q^{5} +118.000 q^{9} +559.000 q^{11} -282.000 q^{13} +361.000 q^{15} -1259.00 q^{17} -1957.00 q^{19} +2977.00 q^{23} -2764.00 q^{25} +2375.00 q^{27} -62.0000 q^{29} +2037.00 q^{31} -10621.0 q^{33} +6023.00 q^{37} +5358.00 q^{39} +2178.00 q^{41} -23180.0 q^{43} -2242.00 q^{45} +26235.0 q^{47} +23921.0 q^{51} +30267.0 q^{53} -10621.0 q^{55} +37183.0 q^{57} +44965.0 q^{59} -27639.0 q^{61} +5358.00 q^{65} +58667.0 q^{67} -56563.0 q^{69} +9520.00 q^{71} +6785.00 q^{73} +52516.0 q^{75} +16929.0 q^{79} -73799.0 q^{81} -59572.0 q^{83} +23921.0 q^{85} +1178.00 q^{87} +51873.0 q^{89} -38703.0 q^{93} +37183.0 q^{95} -134110. q^{97} +65962.0 q^{99} +O(q^{100})$$

## 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
$$3$$ −19.0000 −1.21885 −0.609425 0.792844i $$-0.708600\pi$$
−0.609425 + 0.792844i $$0.708600\pi$$
$$4$$ 0 0
$$5$$ −19.0000 −0.339882 −0.169941 0.985454i $$-0.554358\pi$$
−0.169941 + 0.985454i $$0.554358\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 118.000 0.485597
$$10$$ 0 0
$$11$$ 559.000 1.39293 0.696466 0.717590i $$-0.254755\pi$$
0.696466 + 0.717590i $$0.254755\pi$$
$$12$$ 0 0
$$13$$ −282.000 −0.462797 −0.231399 0.972859i $$-0.574330\pi$$
−0.231399 + 0.972859i $$0.574330\pi$$
$$14$$ 0 0
$$15$$ 361.000 0.414266
$$16$$ 0 0
$$17$$ −1259.00 −1.05658 −0.528291 0.849063i $$-0.677167\pi$$
−0.528291 + 0.849063i $$0.677167\pi$$
$$18$$ 0 0
$$19$$ −1957.00 −1.24367 −0.621837 0.783146i $$-0.713614\pi$$
−0.621837 + 0.783146i $$0.713614\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2977.00 1.17344 0.586718 0.809791i $$-0.300420\pi$$
0.586718 + 0.809791i $$0.300420\pi$$
$$24$$ 0 0
$$25$$ −2764.00 −0.884480
$$26$$ 0 0
$$27$$ 2375.00 0.626981
$$28$$ 0 0
$$29$$ −62.0000 −0.0136898 −0.00684489 0.999977i $$-0.502179\pi$$
−0.00684489 + 0.999977i $$0.502179\pi$$
$$30$$ 0 0
$$31$$ 2037.00 0.380703 0.190352 0.981716i $$-0.439037\pi$$
0.190352 + 0.981716i $$0.439037\pi$$
$$32$$ 0 0
$$33$$ −10621.0 −1.69778
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6023.00 0.723283 0.361642 0.932317i $$-0.382216\pi$$
0.361642 + 0.932317i $$0.382216\pi$$
$$38$$ 0 0
$$39$$ 5358.00 0.564081
$$40$$ 0 0
$$41$$ 2178.00 0.202348 0.101174 0.994869i $$-0.467740\pi$$
0.101174 + 0.994869i $$0.467740\pi$$
$$42$$ 0 0
$$43$$ −23180.0 −1.91180 −0.955900 0.293694i $$-0.905115\pi$$
−0.955900 + 0.293694i $$0.905115\pi$$
$$44$$ 0 0
$$45$$ −2242.00 −0.165046
$$46$$ 0 0
$$47$$ 26235.0 1.73235 0.866177 0.499738i $$-0.166570\pi$$
0.866177 + 0.499738i $$0.166570\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 23921.0 1.28782
$$52$$ 0 0
$$53$$ 30267.0 1.48006 0.740031 0.672573i $$-0.234811\pi$$
0.740031 + 0.672573i $$0.234811\pi$$
$$54$$ 0 0
$$55$$ −10621.0 −0.473433
$$56$$ 0 0
$$57$$ 37183.0 1.51585
$$58$$ 0 0
$$59$$ 44965.0 1.68168 0.840842 0.541280i $$-0.182060\pi$$
0.840842 + 0.541280i $$0.182060\pi$$
$$60$$ 0 0
$$61$$ −27639.0 −0.951038 −0.475519 0.879706i $$-0.657740\pi$$
−0.475519 + 0.879706i $$0.657740\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 5358.00 0.157297
$$66$$ 0 0
$$67$$ 58667.0 1.59664 0.798320 0.602234i $$-0.205723\pi$$
0.798320 + 0.602234i $$0.205723\pi$$
$$68$$ 0 0
$$69$$ −56563.0 −1.43024
$$70$$ 0 0
$$71$$ 9520.00 0.224125 0.112063 0.993701i $$-0.464254\pi$$
0.112063 + 0.993701i $$0.464254\pi$$
$$72$$ 0 0
$$73$$ 6785.00 0.149019 0.0745097 0.997220i $$-0.476261\pi$$
0.0745097 + 0.997220i $$0.476261\pi$$
$$74$$ 0 0
$$75$$ 52516.0 1.07805
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16929.0 0.305185 0.152593 0.988289i $$-0.451238\pi$$
0.152593 + 0.988289i $$0.451238\pi$$
$$80$$ 0 0
$$81$$ −73799.0 −1.24979
$$82$$ 0 0
$$83$$ −59572.0 −0.949176 −0.474588 0.880208i $$-0.657403\pi$$
−0.474588 + 0.880208i $$0.657403\pi$$
$$84$$ 0 0
$$85$$ 23921.0 0.359114
$$86$$ 0 0
$$87$$ 1178.00 0.0166858
$$88$$ 0 0
$$89$$ 51873.0 0.694171 0.347085 0.937834i $$-0.387171\pi$$
0.347085 + 0.937834i $$0.387171\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −38703.0 −0.464021
$$94$$ 0 0
$$95$$ 37183.0 0.422703
$$96$$ 0 0
$$97$$ −134110. −1.44721 −0.723605 0.690214i $$-0.757516\pi$$
−0.723605 + 0.690214i $$0.757516\pi$$
$$98$$ 0 0
$$99$$ 65962.0 0.676403
$$100$$ 0 0
$$101$$ −122047. −1.19048 −0.595242 0.803546i $$-0.702944\pi$$
−0.595242 + 0.803546i $$0.702944\pi$$
$$102$$ 0 0
$$103$$ −80617.0 −0.748744 −0.374372 0.927279i $$-0.622142\pi$$
−0.374372 + 0.927279i $$0.622142\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1467.00 −0.0123871 −0.00619356 0.999981i $$-0.501971\pi$$
−0.00619356 + 0.999981i $$0.501971\pi$$
$$108$$ 0 0
$$109$$ 200527. 1.61662 0.808308 0.588761i $$-0.200384\pi$$
0.808308 + 0.588761i $$0.200384\pi$$
$$110$$ 0 0
$$111$$ −114437. −0.881574
$$112$$ 0 0
$$113$$ −722.000 −0.00531914 −0.00265957 0.999996i $$-0.500847\pi$$
−0.00265957 + 0.999996i $$0.500847\pi$$
$$114$$ 0 0
$$115$$ −56563.0 −0.398830
$$116$$ 0 0
$$117$$ −33276.0 −0.224733
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 151430. 0.940261
$$122$$ 0 0
$$123$$ −41382.0 −0.246632
$$124$$ 0 0
$$125$$ 111891. 0.640501
$$126$$ 0 0
$$127$$ −147288. −0.810323 −0.405161 0.914245i $$-0.632785\pi$$
−0.405161 + 0.914245i $$0.632785\pi$$
$$128$$ 0 0
$$129$$ 440420. 2.33020
$$130$$ 0 0
$$131$$ −53993.0 −0.274890 −0.137445 0.990509i $$-0.543889\pi$$
−0.137445 + 0.990509i $$0.543889\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −45125.0 −0.213100
$$136$$ 0 0
$$137$$ 122563. 0.557902 0.278951 0.960305i $$-0.410013\pi$$
0.278951 + 0.960305i $$0.410013\pi$$
$$138$$ 0 0
$$139$$ −128108. −0.562392 −0.281196 0.959650i $$-0.590731\pi$$
−0.281196 + 0.959650i $$0.590731\pi$$
$$140$$ 0 0
$$141$$ −498465. −2.11148
$$142$$ 0 0
$$143$$ −157638. −0.644645
$$144$$ 0 0
$$145$$ 1178.00 0.00465292
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 131955. 0.486923 0.243461 0.969911i $$-0.421717\pi$$
0.243461 + 0.969911i $$0.421717\pi$$
$$150$$ 0 0
$$151$$ −140125. −0.500119 −0.250059 0.968230i $$-0.580450\pi$$
−0.250059 + 0.968230i $$0.580450\pi$$
$$152$$ 0 0
$$153$$ −148562. −0.513073
$$154$$ 0 0
$$155$$ −38703.0 −0.129394
$$156$$ 0 0
$$157$$ −323339. −1.04691 −0.523455 0.852054i $$-0.675357\pi$$
−0.523455 + 0.852054i $$0.675357\pi$$
$$158$$ 0 0
$$159$$ −575073. −1.80397
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −122159. −0.360128 −0.180064 0.983655i $$-0.557630\pi$$
−0.180064 + 0.983655i $$0.557630\pi$$
$$164$$ 0 0
$$165$$ 201799. 0.577044
$$166$$ 0 0
$$167$$ 185404. 0.514432 0.257216 0.966354i $$-0.417195\pi$$
0.257216 + 0.966354i $$0.417195\pi$$
$$168$$ 0 0
$$169$$ −291769. −0.785819
$$170$$ 0 0
$$171$$ −230926. −0.603924
$$172$$ 0 0
$$173$$ 358625. 0.911015 0.455507 0.890232i $$-0.349458\pi$$
0.455507 + 0.890232i $$0.349458\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −854335. −2.04972
$$178$$ 0 0
$$179$$ 460551. 1.07435 0.537174 0.843471i $$-0.319492\pi$$
0.537174 + 0.843471i $$0.319492\pi$$
$$180$$ 0 0
$$181$$ 332538. 0.754475 0.377237 0.926117i $$-0.376874\pi$$
0.377237 + 0.926117i $$0.376874\pi$$
$$182$$ 0 0
$$183$$ 525141. 1.15917
$$184$$ 0 0
$$185$$ −114437. −0.245831
$$186$$ 0 0
$$187$$ −703781. −1.47175
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −987867. −1.95936 −0.979682 0.200558i $$-0.935725\pi$$
−0.979682 + 0.200558i $$0.935725\pi$$
$$192$$ 0 0
$$193$$ −344413. −0.665559 −0.332779 0.943005i $$-0.607986\pi$$
−0.332779 + 0.943005i $$0.607986\pi$$
$$194$$ 0 0
$$195$$ −101802. −0.191721
$$196$$ 0 0
$$197$$ 582362. 1.06912 0.534561 0.845130i $$-0.320477\pi$$
0.534561 + 0.845130i $$0.320477\pi$$
$$198$$ 0 0
$$199$$ −150955. −0.270218 −0.135109 0.990831i $$-0.543139\pi$$
−0.135109 + 0.990831i $$0.543139\pi$$
$$200$$ 0 0
$$201$$ −1.11467e6 −1.94606
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −41382.0 −0.0687744
$$206$$ 0 0
$$207$$ 351286. 0.569816
$$208$$ 0 0
$$209$$ −1.09396e6 −1.73236
$$210$$ 0 0
$$211$$ −272156. −0.420835 −0.210417 0.977612i $$-0.567482\pi$$
−0.210417 + 0.977612i $$0.567482\pi$$
$$212$$ 0 0
$$213$$ −180880. −0.273175
$$214$$ 0 0
$$215$$ 440420. 0.649787
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −128915. −0.181632
$$220$$ 0 0
$$221$$ 355038. 0.488983
$$222$$ 0 0
$$223$$ −939112. −1.26461 −0.632303 0.774721i $$-0.717890\pi$$
−0.632303 + 0.774721i $$0.717890\pi$$
$$224$$ 0 0
$$225$$ −326152. −0.429501
$$226$$ 0 0
$$227$$ 481713. 0.620474 0.310237 0.950659i $$-0.399592\pi$$
0.310237 + 0.950659i $$0.399592\pi$$
$$228$$ 0 0
$$229$$ −523147. −0.659227 −0.329614 0.944116i $$-0.606918\pi$$
−0.329614 + 0.944116i $$0.606918\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1.22603e6 −1.47949 −0.739743 0.672889i $$-0.765053\pi$$
−0.739743 + 0.672889i $$0.765053\pi$$
$$234$$ 0 0
$$235$$ −498465. −0.588796
$$236$$ 0 0
$$237$$ −321651. −0.371975
$$238$$ 0 0
$$239$$ −1.32568e6 −1.50121 −0.750607 0.660749i $$-0.770239\pi$$
−0.750607 + 0.660749i $$0.770239\pi$$
$$240$$ 0 0
$$241$$ −271871. −0.301523 −0.150761 0.988570i $$-0.548173\pi$$
−0.150761 + 0.988570i $$0.548173\pi$$
$$242$$ 0 0
$$243$$ 825056. 0.896330
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 551874. 0.575569
$$248$$ 0 0
$$249$$ 1.13187e6 1.15690
$$250$$ 0 0
$$251$$ −781368. −0.782837 −0.391418 0.920213i $$-0.628015\pi$$
−0.391418 + 0.920213i $$0.628015\pi$$
$$252$$ 0 0
$$253$$ 1.66414e6 1.63452
$$254$$ 0 0
$$255$$ −454499. −0.437706
$$256$$ 0 0
$$257$$ −1.00337e6 −0.947608 −0.473804 0.880630i $$-0.657119\pi$$
−0.473804 + 0.880630i $$0.657119\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −7316.00 −0.00664772
$$262$$ 0 0
$$263$$ 1.10958e6 0.989169 0.494584 0.869130i $$-0.335320\pi$$
0.494584 + 0.869130i $$0.335320\pi$$
$$264$$ 0 0
$$265$$ −575073. −0.503047
$$266$$ 0 0
$$267$$ −985587. −0.846090
$$268$$ 0 0
$$269$$ −1.72416e6 −1.45277 −0.726383 0.687290i $$-0.758800\pi$$
−0.726383 + 0.687290i $$0.758800\pi$$
$$270$$ 0 0
$$271$$ 831399. 0.687680 0.343840 0.939028i $$-0.388272\pi$$
0.343840 + 0.939028i $$0.388272\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.54508e6 −1.23202
$$276$$ 0 0
$$277$$ −696757. −0.545609 −0.272805 0.962069i $$-0.587951\pi$$
−0.272805 + 0.962069i $$0.587951\pi$$
$$278$$ 0 0
$$279$$ 240366. 0.184868
$$280$$ 0 0
$$281$$ −2.26355e6 −1.71011 −0.855054 0.518539i $$-0.826476\pi$$
−0.855054 + 0.518539i $$0.826476\pi$$
$$282$$ 0 0
$$283$$ 423985. 0.314691 0.157346 0.987544i $$-0.449706\pi$$
0.157346 + 0.987544i $$0.449706\pi$$
$$284$$ 0 0
$$285$$ −706477. −0.515212
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 165224. 0.116367
$$290$$ 0 0
$$291$$ 2.54809e6 1.76393
$$292$$ 0 0
$$293$$ −1.03724e6 −0.705845 −0.352923 0.935653i $$-0.614812\pi$$
−0.352923 + 0.935653i $$0.614812\pi$$
$$294$$ 0 0
$$295$$ −854335. −0.571575
$$296$$ 0 0
$$297$$ 1.32762e6 0.873342
$$298$$ 0 0
$$299$$ −839514. −0.543063
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 2.31889e6 1.45102
$$304$$ 0 0
$$305$$ 525141. 0.323241
$$306$$ 0 0
$$307$$ −152684. −0.0924587 −0.0462293 0.998931i $$-0.514720\pi$$
−0.0462293 + 0.998931i $$0.514720\pi$$
$$308$$ 0 0
$$309$$ 1.53172e6 0.912608
$$310$$ 0 0
$$311$$ 1.66634e6 0.976931 0.488466 0.872583i $$-0.337557\pi$$
0.488466 + 0.872583i $$0.337557\pi$$
$$312$$ 0 0
$$313$$ −64471.0 −0.0371966 −0.0185983 0.999827i $$-0.505920\pi$$
−0.0185983 + 0.999827i $$0.505920\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.28934e6 0.720643 0.360322 0.932828i $$-0.382667\pi$$
0.360322 + 0.932828i $$0.382667\pi$$
$$318$$ 0 0
$$319$$ −34658.0 −0.0190690
$$320$$ 0 0
$$321$$ 27873.0 0.0150981
$$322$$ 0 0
$$323$$ 2.46386e6 1.31405
$$324$$ 0 0
$$325$$ 779448. 0.409335
$$326$$ 0 0
$$327$$ −3.81001e6 −1.97041
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2.64238e6 1.32564 0.662819 0.748780i $$-0.269360\pi$$
0.662819 + 0.748780i $$0.269360\pi$$
$$332$$ 0 0
$$333$$ 710714. 0.351224
$$334$$ 0 0
$$335$$ −1.11467e6 −0.542670
$$336$$ 0 0
$$337$$ −3.00561e6 −1.44164 −0.720822 0.693120i $$-0.756235\pi$$
−0.720822 + 0.693120i $$0.756235\pi$$
$$338$$ 0 0
$$339$$ 13718.0 0.00648323
$$340$$ 0 0
$$341$$ 1.13868e6 0.530294
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 1.07470e6 0.486114
$$346$$ 0 0
$$347$$ 185307. 0.0826168 0.0413084 0.999146i $$-0.486847\pi$$
0.0413084 + 0.999146i $$0.486847\pi$$
$$348$$ 0 0
$$349$$ −2.82147e6 −1.23997 −0.619987 0.784612i $$-0.712862\pi$$
−0.619987 + 0.784612i $$0.712862\pi$$
$$350$$ 0 0
$$351$$ −669750. −0.290165
$$352$$ 0 0
$$353$$ −1.30752e6 −0.558486 −0.279243 0.960220i $$-0.590084\pi$$
−0.279243 + 0.960220i $$0.590084\pi$$
$$354$$ 0 0
$$355$$ −180880. −0.0761763
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −2.89605e6 −1.18596 −0.592980 0.805217i $$-0.702049\pi$$
−0.592980 + 0.805217i $$0.702049\pi$$
$$360$$ 0 0
$$361$$ 1.35375e6 0.546727
$$362$$ 0 0
$$363$$ −2.87717e6 −1.14604
$$364$$ 0 0
$$365$$ −128915. −0.0506490
$$366$$ 0 0
$$367$$ −4.71893e6 −1.82885 −0.914427 0.404752i $$-0.867358\pi$$
−0.914427 + 0.404752i $$0.867358\pi$$
$$368$$ 0 0
$$369$$ 257004. 0.0982594
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 438079. 0.163035 0.0815174 0.996672i $$-0.474023\pi$$
0.0815174 + 0.996672i $$0.474023\pi$$
$$374$$ 0 0
$$375$$ −2.12593e6 −0.780676
$$376$$ 0 0
$$377$$ 17484.0 0.00633560
$$378$$ 0 0
$$379$$ 549632. 0.196550 0.0982752 0.995159i $$-0.468667\pi$$
0.0982752 + 0.995159i $$0.468667\pi$$
$$380$$ 0 0
$$381$$ 2.79847e6 0.987662
$$382$$ 0 0
$$383$$ 3.53000e6 1.22964 0.614820 0.788667i $$-0.289229\pi$$
0.614820 + 0.788667i $$0.289229\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2.73524e6 −0.928363
$$388$$ 0 0
$$389$$ 3.13395e6 1.05007 0.525035 0.851081i $$-0.324052\pi$$
0.525035 + 0.851081i $$0.324052\pi$$
$$390$$ 0 0
$$391$$ −3.74804e6 −1.23983
$$392$$ 0 0
$$393$$ 1.02587e6 0.335050
$$394$$ 0 0
$$395$$ −321651. −0.103727
$$396$$ 0 0
$$397$$ 2.55391e6 0.813260 0.406630 0.913593i $$-0.366704\pi$$
0.406630 + 0.913593i $$0.366704\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −702837. −0.218270 −0.109135 0.994027i $$-0.534808\pi$$
−0.109135 + 0.994027i $$0.534808\pi$$
$$402$$ 0 0
$$403$$ −574434. −0.176188
$$404$$ 0 0
$$405$$ 1.40218e6 0.424782
$$406$$ 0 0
$$407$$ 3.36686e6 1.00749
$$408$$ 0 0
$$409$$ 5.47715e6 1.61900 0.809499 0.587121i $$-0.199739\pi$$
0.809499 + 0.587121i $$0.199739\pi$$
$$410$$ 0 0
$$411$$ −2.32870e6 −0.679999
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.13187e6 0.322608
$$416$$ 0 0
$$417$$ 2.43405e6 0.685472
$$418$$ 0 0
$$419$$ 5.80976e6 1.61668 0.808339 0.588718i $$-0.200367\pi$$
0.808339 + 0.588718i $$0.200367\pi$$
$$420$$ 0 0
$$421$$ 1.69370e6 0.465726 0.232863 0.972510i $$-0.425191\pi$$
0.232863 + 0.972510i $$0.425191\pi$$
$$422$$ 0 0
$$423$$ 3.09573e6 0.841225
$$424$$ 0 0
$$425$$ 3.47988e6 0.934526
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 2.99512e6 0.785726
$$430$$ 0 0
$$431$$ 1.46468e6 0.379794 0.189897 0.981804i $$-0.439185\pi$$
0.189897 + 0.981804i $$0.439185\pi$$
$$432$$ 0 0
$$433$$ 3.23418e6 0.828980 0.414490 0.910054i $$-0.363960\pi$$
0.414490 + 0.910054i $$0.363960\pi$$
$$434$$ 0 0
$$435$$ −22382.0 −0.00567121
$$436$$ 0 0
$$437$$ −5.82599e6 −1.45937
$$438$$ 0 0
$$439$$ −2.54217e6 −0.629570 −0.314785 0.949163i $$-0.601932\pi$$
−0.314785 + 0.949163i $$0.601932\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −65939.0 −0.0159637 −0.00798184 0.999968i $$-0.502541\pi$$
−0.00798184 + 0.999968i $$0.502541\pi$$
$$444$$ 0 0
$$445$$ −985587. −0.235936
$$446$$ 0 0
$$447$$ −2.50714e6 −0.593486
$$448$$ 0 0
$$449$$ −5.32399e6 −1.24630 −0.623149 0.782103i $$-0.714147\pi$$
−0.623149 + 0.782103i $$0.714147\pi$$
$$450$$ 0 0
$$451$$ 1.21750e6 0.281857
$$452$$ 0 0
$$453$$ 2.66238e6 0.609570
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −4.05825e6 −0.908967 −0.454484 0.890755i $$-0.650176\pi$$
−0.454484 + 0.890755i $$0.650176\pi$$
$$458$$ 0 0
$$459$$ −2.99012e6 −0.662457
$$460$$ 0 0
$$461$$ −3.73021e6 −0.817487 −0.408744 0.912649i $$-0.634033\pi$$
−0.408744 + 0.912649i $$0.634033\pi$$
$$462$$ 0 0
$$463$$ 3.45186e6 0.748342 0.374171 0.927360i $$-0.377927\pi$$
0.374171 + 0.927360i $$0.377927\pi$$
$$464$$ 0 0
$$465$$ 735357. 0.157712
$$466$$ 0 0
$$467$$ 3.92062e6 0.831884 0.415942 0.909391i $$-0.363452\pi$$
0.415942 + 0.909391i $$0.363452\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 6.14344e6 1.27603
$$472$$ 0 0
$$473$$ −1.29576e7 −2.66301
$$474$$ 0 0
$$475$$ 5.40915e6 1.10001
$$476$$ 0 0
$$477$$ 3.57151e6 0.718713
$$478$$ 0 0
$$479$$ −5.98674e6 −1.19221 −0.596103 0.802908i $$-0.703285\pi$$
−0.596103 + 0.802908i $$0.703285\pi$$
$$480$$ 0 0
$$481$$ −1.69849e6 −0.334734
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2.54809e6 0.491881
$$486$$ 0 0
$$487$$ −71873.0 −0.0137323 −0.00686615 0.999976i $$-0.502186\pi$$
−0.00686615 + 0.999976i $$0.502186\pi$$
$$488$$ 0 0
$$489$$ 2.32102e6 0.438942
$$490$$ 0 0
$$491$$ −1.01122e6 −0.189295 −0.0946477 0.995511i $$-0.530172\pi$$
−0.0946477 + 0.995511i $$0.530172\pi$$
$$492$$ 0 0
$$493$$ 78058.0 0.0144644
$$494$$ 0 0
$$495$$ −1.25328e6 −0.229898
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −7.80469e6 −1.40315 −0.701575 0.712596i $$-0.747519\pi$$
−0.701575 + 0.712596i $$0.747519\pi$$
$$500$$ 0 0
$$501$$ −3.52268e6 −0.627016
$$502$$ 0 0
$$503$$ 7.89298e6 1.39098 0.695490 0.718536i $$-0.255187\pi$$
0.695490 + 0.718536i $$0.255187\pi$$
$$504$$ 0 0
$$505$$ 2.31889e6 0.404625
$$506$$ 0 0
$$507$$ 5.54361e6 0.957796
$$508$$ 0 0
$$509$$ −6.62650e6 −1.13368 −0.566839 0.823829i $$-0.691834\pi$$
−0.566839 + 0.823829i $$0.691834\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −4.64788e6 −0.779760
$$514$$ 0 0
$$515$$ 1.53172e6 0.254485
$$516$$ 0 0
$$517$$ 1.46654e7 2.41305
$$518$$ 0 0
$$519$$ −6.81387e6 −1.11039
$$520$$ 0 0
$$521$$ −7.27862e6 −1.17477 −0.587387 0.809306i $$-0.699843\pi$$
−0.587387 + 0.809306i $$0.699843\pi$$
$$522$$ 0 0
$$523$$ 4.75278e6 0.759790 0.379895 0.925030i $$-0.375960\pi$$
0.379895 + 0.925030i $$0.375960\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.56458e6 −0.402245
$$528$$ 0 0
$$529$$ 2.42619e6 0.376951
$$530$$ 0 0
$$531$$ 5.30587e6 0.816621
$$532$$ 0 0
$$533$$ −614196. −0.0936459
$$534$$ 0 0
$$535$$ 27873.0 0.00421017
$$536$$ 0 0
$$537$$ −8.75047e6 −1.30947
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −9.25161e6 −1.35902 −0.679508 0.733668i $$-0.737807\pi$$
−0.679508 + 0.733668i $$0.737807\pi$$
$$542$$ 0 0
$$543$$ −6.31822e6 −0.919592
$$544$$ 0 0
$$545$$ −3.81001e6 −0.549459
$$546$$ 0 0
$$547$$ −4.66834e6 −0.667104 −0.333552 0.942732i $$-0.608247\pi$$
−0.333552 + 0.942732i $$0.608247\pi$$
$$548$$ 0 0
$$549$$ −3.26140e6 −0.461821
$$550$$ 0 0
$$551$$ 121334. 0.0170256
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 2.17430e6 0.299632
$$556$$ 0 0
$$557$$ −6.50094e6 −0.887848 −0.443924 0.896065i $$-0.646414\pi$$
−0.443924 + 0.896065i $$0.646414\pi$$
$$558$$ 0 0
$$559$$ 6.53676e6 0.884775
$$560$$ 0 0
$$561$$ 1.33718e7 1.79384
$$562$$ 0 0
$$563$$ 1.06422e7 1.41501 0.707505 0.706708i $$-0.249821\pi$$
0.707505 + 0.706708i $$0.249821\pi$$
$$564$$ 0 0
$$565$$ 13718.0 0.00180788
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 9.71001e6 1.25730 0.628650 0.777688i $$-0.283608\pi$$
0.628650 + 0.777688i $$0.283608\pi$$
$$570$$ 0 0
$$571$$ −1.15693e7 −1.48497 −0.742485 0.669862i $$-0.766353\pi$$
−0.742485 + 0.669862i $$0.766353\pi$$
$$572$$ 0 0
$$573$$ 1.87695e7 2.38817
$$574$$ 0 0
$$575$$ −8.22843e6 −1.03788
$$576$$ 0 0
$$577$$ 5.72550e6 0.715936 0.357968 0.933734i $$-0.383470\pi$$
0.357968 + 0.933734i $$0.383470\pi$$
$$578$$ 0 0
$$579$$ 6.54385e6 0.811216
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 1.69193e7 2.06163
$$584$$ 0 0
$$585$$ 632244. 0.0763827
$$586$$ 0 0
$$587$$ 2.35929e6 0.282609 0.141305 0.989966i $$-0.454870\pi$$
0.141305 + 0.989966i $$0.454870\pi$$
$$588$$ 0 0
$$589$$ −3.98641e6 −0.473471
$$590$$ 0 0
$$591$$ −1.10649e7 −1.30310
$$592$$ 0 0
$$593$$ −2.91024e6 −0.339854 −0.169927 0.985457i $$-0.554353\pi$$
−0.169927 + 0.985457i $$0.554353\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 2.86814e6 0.329356
$$598$$ 0 0
$$599$$ −9.99466e6 −1.13815 −0.569077 0.822284i $$-0.692699\pi$$
−0.569077 + 0.822284i $$0.692699\pi$$
$$600$$ 0 0
$$601$$ −6.12412e6 −0.691604 −0.345802 0.938308i $$-0.612393\pi$$
−0.345802 + 0.938308i $$0.612393\pi$$
$$602$$ 0 0
$$603$$ 6.92271e6 0.775323
$$604$$ 0 0
$$605$$ −2.87717e6 −0.319578
$$606$$ 0 0
$$607$$ 1.77047e7 1.95037 0.975185 0.221391i $$-0.0710599\pi$$
0.975185 + 0.221391i $$0.0710599\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −7.39827e6 −0.801728
$$612$$ 0 0
$$613$$ −8.83215e6 −0.949326 −0.474663 0.880168i $$-0.657430\pi$$
−0.474663 + 0.880168i $$0.657430\pi$$
$$614$$ 0 0
$$615$$ 786258. 0.0838257
$$616$$ 0 0
$$617$$ 1.07392e6 0.113569 0.0567843 0.998386i $$-0.481915\pi$$
0.0567843 + 0.998386i $$0.481915\pi$$
$$618$$ 0 0
$$619$$ −352211. −0.0369468 −0.0184734 0.999829i $$-0.505881\pi$$
−0.0184734 + 0.999829i $$0.505881\pi$$
$$620$$ 0 0
$$621$$ 7.07038e6 0.735722
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 6.51157e6 0.666785
$$626$$ 0 0
$$627$$ 2.07853e7 2.11148
$$628$$ 0 0
$$629$$ −7.58296e6 −0.764209
$$630$$ 0 0
$$631$$ −775808. −0.0775677 −0.0387838 0.999248i $$-0.512348\pi$$
−0.0387838 + 0.999248i $$0.512348\pi$$
$$632$$ 0 0
$$633$$ 5.17096e6 0.512935
$$634$$ 0 0
$$635$$ 2.79847e6 0.275414
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 1.12336e6 0.108835
$$640$$ 0 0
$$641$$ −1.18421e7 −1.13837 −0.569187 0.822208i $$-0.692742\pi$$
−0.569187 + 0.822208i $$0.692742\pi$$
$$642$$ 0 0
$$643$$ 307460. 0.0293266 0.0146633 0.999892i $$-0.495332\pi$$
0.0146633 + 0.999892i $$0.495332\pi$$
$$644$$ 0 0
$$645$$ −8.36798e6 −0.791993
$$646$$ 0 0
$$647$$ 1.30708e7 1.22756 0.613780 0.789477i $$-0.289648\pi$$
0.613780 + 0.789477i $$0.289648\pi$$
$$648$$ 0 0
$$649$$ 2.51354e7 2.34247
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 927663. 0.0851348 0.0425674 0.999094i $$-0.486446\pi$$
0.0425674 + 0.999094i $$0.486446\pi$$
$$654$$ 0 0
$$655$$ 1.02587e6 0.0934303
$$656$$ 0 0
$$657$$ 800630. 0.0723633
$$658$$ 0 0
$$659$$ 1.90355e7 1.70746 0.853731 0.520715i $$-0.174334\pi$$
0.853731 + 0.520715i $$0.174334\pi$$
$$660$$ 0 0
$$661$$ 2.17236e6 0.193388 0.0966939 0.995314i $$-0.469173\pi$$
0.0966939 + 0.995314i $$0.469173\pi$$
$$662$$ 0 0
$$663$$ −6.74572e6 −0.595998
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −184574. −0.0160641
$$668$$ 0 0
$$669$$ 1.78431e7 1.54137
$$670$$ 0 0
$$671$$ −1.54502e7 −1.32473
$$672$$ 0 0
$$673$$ 1.32268e7 1.12569 0.562844 0.826563i $$-0.309707\pi$$
0.562844 + 0.826563i $$0.309707\pi$$
$$674$$ 0 0
$$675$$ −6.56450e6 −0.554552
$$676$$ 0 0
$$677$$ 1.59662e7 1.33884 0.669422 0.742882i $$-0.266542\pi$$
0.669422 + 0.742882i $$0.266542\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −9.15255e6 −0.756265
$$682$$ 0 0
$$683$$ −1.40029e7 −1.14859 −0.574297 0.818647i $$-0.694724\pi$$
−0.574297 + 0.818647i $$0.694724\pi$$
$$684$$ 0 0
$$685$$ −2.32870e6 −0.189621
$$686$$ 0 0
$$687$$ 9.93979e6 0.803499
$$688$$ 0 0
$$689$$ −8.53529e6 −0.684968
$$690$$ 0 0
$$691$$ −1.05246e7 −0.838514 −0.419257 0.907868i $$-0.637709\pi$$
−0.419257 + 0.907868i $$0.637709\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 2.43405e6 0.191147
$$696$$ 0 0
$$697$$ −2.74210e6 −0.213797
$$698$$ 0 0
$$699$$ 2.32946e7 1.80327
$$700$$ 0 0
$$701$$ −1.30811e7 −1.00542 −0.502710 0.864455i $$-0.667664\pi$$
−0.502710 + 0.864455i $$0.667664\pi$$
$$702$$ 0 0
$$703$$ −1.17870e7 −0.899529
$$704$$ 0 0
$$705$$ 9.47084e6 0.717655
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 488519. 0.0364977 0.0182489 0.999833i $$-0.494191\pi$$
0.0182489 + 0.999833i $$0.494191\pi$$
$$710$$ 0 0
$$711$$ 1.99762e6 0.148197
$$712$$ 0 0
$$713$$ 6.06415e6 0.446731
$$714$$ 0 0
$$715$$ 2.99512e6 0.219104
$$716$$ 0 0
$$717$$ 2.51878e7 1.82976
$$718$$ 0 0
$$719$$ −8.40850e6 −0.606592 −0.303296 0.952896i $$-0.598087\pi$$
−0.303296 + 0.952896i $$0.598087\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 5.16555e6 0.367511
$$724$$ 0 0
$$725$$ 171368. 0.0121083
$$726$$ 0 0
$$727$$ −2.44145e7 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$728$$ 0 0
$$729$$ 2.25709e6 0.157301
$$730$$ 0 0
$$731$$ 2.91836e7 2.01997
$$732$$ 0 0
$$733$$ −1.13916e7 −0.783111 −0.391556 0.920154i $$-0.628063\pi$$
−0.391556 + 0.920154i $$0.628063\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 3.27949e7 2.22401
$$738$$ 0 0
$$739$$ 2.18963e6 0.147489 0.0737445 0.997277i $$-0.476505\pi$$
0.0737445 + 0.997277i $$0.476505\pi$$
$$740$$ 0 0
$$741$$ −1.04856e7 −0.701533
$$742$$ 0 0
$$743$$ 7.06982e6 0.469825 0.234913 0.972016i $$-0.424520\pi$$
0.234913 + 0.972016i $$0.424520\pi$$
$$744$$ 0 0
$$745$$ −2.50714e6 −0.165496
$$746$$ 0 0
$$747$$ −7.02950e6 −0.460917
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −6.55880e6 −0.424350 −0.212175 0.977232i $$-0.568055\pi$$
−0.212175 + 0.977232i $$0.568055\pi$$
$$752$$ 0 0
$$753$$ 1.48460e7 0.954161
$$754$$ 0 0
$$755$$ 2.66238e6 0.169982
$$756$$ 0 0
$$757$$ 2.17588e6 0.138005 0.0690026 0.997616i $$-0.478018\pi$$
0.0690026 + 0.997616i $$0.478018\pi$$
$$758$$ 0 0
$$759$$ −3.16187e7 −1.99223
$$760$$ 0 0
$$761$$ −1.58632e7 −0.992954 −0.496477 0.868050i $$-0.665373\pi$$
−0.496477 + 0.868050i $$0.665373\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 2.82268e6 0.174384
$$766$$ 0 0
$$767$$ −1.26801e7 −0.778279
$$768$$ 0 0
$$769$$ 1.86634e7 1.13809 0.569044 0.822307i $$-0.307313\pi$$
0.569044 + 0.822307i $$0.307313\pi$$
$$770$$ 0 0
$$771$$ 1.90640e7 1.15499
$$772$$ 0 0
$$773$$ 2.26397e7 1.36277 0.681384 0.731926i $$-0.261378\pi$$
0.681384 + 0.731926i $$0.261378\pi$$
$$774$$ 0 0
$$775$$ −5.63027e6 −0.336725
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −4.26235e6 −0.251655
$$780$$ 0 0
$$781$$ 5.32168e6 0.312192
$$782$$ 0 0
$$783$$ −147250. −0.00858323
$$784$$ 0 0
$$785$$ 6.14344e6 0.355826
$$786$$ 0 0
$$787$$ −9.66053e6 −0.555986 −0.277993 0.960583i $$-0.589669\pi$$
−0.277993 + 0.960583i $$0.589669\pi$$
$$788$$ 0 0
$$789$$ −2.10821e7 −1.20565
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 7.79420e6 0.440138
$$794$$ 0 0
$$795$$ 1.09264e7 0.613139
$$796$$ 0 0
$$797$$ −1.15546e7 −0.644330 −0.322165 0.946684i $$-0.604411\pi$$
−0.322165 + 0.946684i $$0.604411\pi$$
$$798$$ 0 0
$$799$$ −3.30299e7 −1.83037
$$800$$ 0 0
$$801$$ 6.12101e6 0.337087
$$802$$ 0 0
$$803$$ 3.79282e6 0.207574
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 3.27589e7 1.77070
$$808$$ 0 0
$$809$$ 1.85307e7 0.995453 0.497727 0.867334i $$-0.334168\pi$$
0.497727 + 0.867334i $$0.334168\pi$$
$$810$$ 0 0
$$811$$ −1.31357e7 −0.701298 −0.350649 0.936507i $$-0.614039\pi$$
−0.350649 + 0.936507i $$0.614039\pi$$
$$812$$ 0 0
$$813$$ −1.57966e7 −0.838179
$$814$$ 0 0
$$815$$ 2.32102e6 0.122401
$$816$$ 0 0
$$817$$ 4.53633e7 2.37766
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −4.20868e6 −0.217915 −0.108958 0.994046i $$-0.534751\pi$$
−0.108958 + 0.994046i $$0.534751\pi$$
$$822$$ 0 0
$$823$$ −1.10437e7 −0.568350 −0.284175 0.958772i $$-0.591720\pi$$
−0.284175 + 0.958772i $$0.591720\pi$$
$$824$$ 0 0
$$825$$ 2.93564e7 1.50165
$$826$$ 0 0
$$827$$ −1.74824e7 −0.888867 −0.444433 0.895812i $$-0.646595\pi$$
−0.444433 + 0.895812i $$0.646595\pi$$
$$828$$ 0 0
$$829$$ −2.35418e7 −1.18974 −0.594871 0.803821i $$-0.702797\pi$$
−0.594871 + 0.803821i $$0.702797\pi$$
$$830$$ 0 0
$$831$$ 1.32384e7 0.665016
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −3.52268e6 −0.174846
$$836$$ 0 0
$$837$$ 4.83788e6 0.238694
$$838$$ 0 0
$$839$$ 2.64326e7 1.29639 0.648195 0.761474i $$-0.275524\pi$$
0.648195 + 0.761474i $$0.275524\pi$$
$$840$$ 0 0
$$841$$ −2.05073e7 −0.999813
$$842$$ 0 0
$$843$$ 4.30074e7 2.08437
$$844$$ 0 0
$$845$$ 5.54361e6 0.267086
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −8.05572e6 −0.383561
$$850$$ 0 0
$$851$$ 1.79305e7 0.848727
$$852$$ 0 0
$$853$$ −3.86472e7 −1.81863 −0.909317 0.416103i $$-0.863395\pi$$
−0.909317 + 0.416103i $$0.863395\pi$$
$$854$$ 0 0
$$855$$ 4.38759e6 0.205263
$$856$$ 0 0
$$857$$ −2.46708e7 −1.14744 −0.573721 0.819051i $$-0.694501\pi$$
−0.573721 + 0.819051i $$0.694501\pi$$
$$858$$ 0 0
$$859$$ 2.74324e7 1.26847 0.634235 0.773140i $$-0.281315\pi$$
0.634235 + 0.773140i $$0.281315\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 6.10970e6 0.279250 0.139625 0.990204i $$-0.455410\pi$$
0.139625 + 0.990204i $$0.455410\pi$$
$$864$$ 0 0
$$865$$ −6.81388e6 −0.309638
$$866$$ 0 0
$$867$$ −3.13926e6 −0.141834
$$868$$ 0 0
$$869$$ 9.46331e6 0.425103
$$870$$ 0 0
$$871$$ −1.65441e7 −0.738920
$$872$$ 0 0
$$873$$ −1.58250e7 −0.702761
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7.62612e6 −0.334815 −0.167408 0.985888i $$-0.553540\pi$$
−0.167408 + 0.985888i $$0.553540\pi$$
$$878$$ 0 0
$$879$$ 1.97075e7 0.860320
$$880$$ 0 0
$$881$$ 3.22357e6 0.139925 0.0699627 0.997550i $$-0.477712\pi$$
0.0699627 + 0.997550i $$0.477712\pi$$
$$882$$ 0 0
$$883$$ −7.31409e6 −0.315688 −0.157844 0.987464i $$-0.550454\pi$$
−0.157844 + 0.987464i $$0.550454\pi$$
$$884$$ 0 0
$$885$$ 1.62324e7 0.696664
$$886$$ 0 0
$$887$$ −4.32134e7 −1.84421 −0.922103 0.386944i $$-0.873531\pi$$
−0.922103 + 0.386944i $$0.873531\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.12536e7 −1.74088
$$892$$ 0 0
$$893$$ −5.13419e7 −2.15448
$$894$$ 0 0
$$895$$ −8.75047e6 −0.365152
$$896$$ 0 0
$$897$$ 1.59508e7 0.661912
$$898$$ 0 0
$$899$$ −126294. −0.00521175
$$900$$ 0 0
$$901$$ −3.81062e7 −1.56381
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −6.31822e6 −0.256433
$$906$$ 0 0
$$907$$ 1.48978e7 0.601317 0.300659 0.953732i $$-0.402794\pi$$
0.300659 + 0.953732i $$0.402794\pi$$
$$908$$ 0 0
$$909$$ −1.44015e7 −0.578095
$$910$$ 0 0
$$911$$ −1.65823e7 −0.661987 −0.330993 0.943633i $$-0.607384\pi$$
−0.330993 + 0.943633i $$0.607384\pi$$
$$912$$ 0 0
$$913$$ −3.33007e7 −1.32214
$$914$$ 0 0
$$915$$ −9.97768e6 −0.393982
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.94331e7 0.759021 0.379510 0.925187i $$-0.376092\pi$$
0.379510 + 0.925187i $$0.376092\pi$$
$$920$$ 0 0
$$921$$ 2.90100e6 0.112693
$$922$$ 0 0
$$923$$ −2.68464e6 −0.103725
$$924$$ 0 0
$$925$$ −1.66476e7 −0.639730
$$926$$ 0 0
$$927$$ −9.51281e6 −0.363588
$$928$$ 0 0
$$929$$ −4.61166e7 −1.75315 −0.876573 0.481269i $$-0.840176\pi$$
−0.876573 + 0.481269i $$0.840176\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −3.16606e7 −1.19073
$$934$$ 0 0
$$935$$ 1.33718e7 0.500221
$$936$$ 0 0
$$937$$ −1.59157e7 −0.592210 −0.296105 0.955155i $$-0.595688\pi$$
−0.296105 + 0.955155i $$0.595688\pi$$
$$938$$ 0 0
$$939$$ 1.22495e6 0.0453371
$$940$$ 0 0
$$941$$ 1.07301e7 0.395031 0.197516 0.980300i $$-0.436713\pi$$
0.197516 + 0.980300i $$0.436713\pi$$
$$942$$ 0 0
$$943$$ 6.48391e6 0.237442
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −2.27632e7 −0.824820 −0.412410 0.910998i $$-0.635313\pi$$
−0.412410 + 0.910998i $$0.635313\pi$$
$$948$$ 0 0
$$949$$ −1.91337e6 −0.0689657
$$950$$ 0 0
$$951$$ −2.44975e7 −0.878356
$$952$$ 0 0
$$953$$ 2.69718e7 0.962005 0.481002 0.876719i $$-0.340273\pi$$
0.481002 + 0.876719i $$0.340273\pi$$
$$954$$ 0 0
$$955$$ 1.87695e7 0.665953
$$956$$ 0 0
$$957$$ 658502. 0.0232422
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.44798e7 −0.855065
$$962$$ 0 0
$$963$$ −173106. −0.00601515
$$964$$ 0 0
$$965$$ 6.54385e6 0.226212
$$966$$ 0 0
$$967$$ −1.02880e7 −0.353805 −0.176902 0.984228i $$-0.556608\pi$$
−0.176902 + 0.984228i $$0.556608\pi$$
$$968$$ 0 0
$$969$$ −4.68134e7 −1.60162
$$970$$ 0 0
$$971$$ 2.85803e7 0.972791 0.486395 0.873739i $$-0.338312\pi$$
0.486395 + 0.873739i $$0.338312\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −1.48095e7 −0.498918
$$976$$ 0 0
$$977$$ −2.84299e6 −0.0952881 −0.0476441 0.998864i $$-0.515171\pi$$
−0.0476441 + 0.998864i $$0.515171\pi$$
$$978$$ 0 0
$$979$$ 2.89970e7 0.966933
$$980$$ 0 0
$$981$$ 2.36622e7 0.785023
$$982$$ 0 0
$$983$$ −5.31666e7 −1.75491 −0.877455 0.479659i $$-0.840760\pi$$
−0.877455 + 0.479659i $$0.840760\pi$$
$$984$$ 0 0
$$985$$ −1.10649e7 −0.363376
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −6.90069e7 −2.24337
$$990$$ 0 0
$$991$$ −5.41874e7 −1.75273 −0.876363 0.481652i $$-0.840037\pi$$
−0.876363 + 0.481652i $$0.840037\pi$$
$$992$$ 0 0
$$993$$ −5.02052e7 −1.61575
$$994$$ 0 0
$$995$$ 2.86814e6 0.0918424
$$996$$ 0 0
$$997$$ −6.55175e6 −0.208747 −0.104373 0.994538i $$-0.533284\pi$$
−0.104373 + 0.994538i $$0.533284\pi$$
$$998$$ 0 0
$$999$$ 1.43046e7 0.453485
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.a.1.1 1
4.3 odd 2 196.6.a.g.1.1 1
7.3 odd 6 112.6.i.a.65.1 2
7.5 odd 6 112.6.i.a.81.1 2
7.6 odd 2 784.6.a.k.1.1 1
28.3 even 6 28.6.e.a.9.1 2
28.11 odd 6 196.6.e.b.177.1 2
28.19 even 6 28.6.e.a.25.1 yes 2
28.23 odd 6 196.6.e.b.165.1 2
28.27 even 2 196.6.a.b.1.1 1
84.47 odd 6 252.6.k.c.109.1 2
84.59 odd 6 252.6.k.c.37.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
28.6.e.a.9.1 2 28.3 even 6
28.6.e.a.25.1 yes 2 28.19 even 6
112.6.i.a.65.1 2 7.3 odd 6
112.6.i.a.81.1 2 7.5 odd 6
196.6.a.b.1.1 1 28.27 even 2
196.6.a.g.1.1 1 4.3 odd 2
196.6.e.b.165.1 2 28.23 odd 6
196.6.e.b.177.1 2 28.11 odd 6
252.6.k.c.37.1 2 84.59 odd 6
252.6.k.c.109.1 2 84.47 odd 6
784.6.a.a.1.1 1 1.1 even 1 trivial
784.6.a.k.1.1 1 7.6 odd 2