# Properties

 Label 126.6.a.h.1.1 Level $126$ Weight $6$ Character 126.1 Self dual yes Analytic conductor $20.208$ 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] = [126,6,Mod(1,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("126.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 126.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: $$20.2083612964$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 126.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} +16.0000 q^{4} -44.0000 q^{5} -49.0000 q^{7} +64.0000 q^{8} +O(q^{10})$$ $$q+4.00000 q^{2} +16.0000 q^{4} -44.0000 q^{5} -49.0000 q^{7} +64.0000 q^{8} -176.000 q^{10} +470.000 q^{11} -1158.00 q^{13} -196.000 q^{14} +256.000 q^{16} -1204.00 q^{17} -2644.00 q^{19} -704.000 q^{20} +1880.00 q^{22} +1190.00 q^{23} -1189.00 q^{25} -4632.00 q^{26} -784.000 q^{28} -3614.00 q^{29} +5616.00 q^{31} +1024.00 q^{32} -4816.00 q^{34} +2156.00 q^{35} -6478.00 q^{37} -10576.0 q^{38} -2816.00 q^{40} -2856.00 q^{41} -13492.0 q^{43} +7520.00 q^{44} +4760.00 q^{46} +18372.0 q^{47} +2401.00 q^{49} -4756.00 q^{50} -18528.0 q^{52} +4374.00 q^{53} -20680.0 q^{55} -3136.00 q^{56} -14456.0 q^{58} -30248.0 q^{59} +19542.0 q^{61} +22464.0 q^{62} +4096.00 q^{64} +50952.0 q^{65} +54328.0 q^{67} -19264.0 q^{68} +8624.00 q^{70} +10730.0 q^{71} +35374.0 q^{73} -25912.0 q^{74} -42304.0 q^{76} -23030.0 q^{77} -49956.0 q^{79} -11264.0 q^{80} -11424.0 q^{82} +26948.0 q^{83} +52976.0 q^{85} -53968.0 q^{86} +30080.0 q^{88} -100776. q^{89} +56742.0 q^{91} +19040.0 q^{92} +73488.0 q^{94} +116336. q^{95} +77134.0 q^{97} +9604.00 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 0.707107
$$3$$ 0 0
$$4$$ 16.0000 0.500000
$$5$$ −44.0000 −0.787096 −0.393548 0.919304i $$-0.628752\pi$$
−0.393548 + 0.919304i $$0.628752\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 64.0000 0.353553
$$9$$ 0 0
$$10$$ −176.000 −0.556561
$$11$$ 470.000 1.17116 0.585580 0.810615i $$-0.300867\pi$$
0.585580 + 0.810615i $$0.300867\pi$$
$$12$$ 0 0
$$13$$ −1158.00 −1.90042 −0.950211 0.311606i $$-0.899133\pi$$
−0.950211 + 0.311606i $$0.899133\pi$$
$$14$$ −196.000 −0.267261
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ −1204.00 −1.01043 −0.505213 0.862995i $$-0.668586\pi$$
−0.505213 + 0.862995i $$0.668586\pi$$
$$18$$ 0 0
$$19$$ −2644.00 −1.68026 −0.840132 0.542382i $$-0.817522\pi$$
−0.840132 + 0.542382i $$0.817522\pi$$
$$20$$ −704.000 −0.393548
$$21$$ 0 0
$$22$$ 1880.00 0.828135
$$23$$ 1190.00 0.469059 0.234529 0.972109i $$-0.424645\pi$$
0.234529 + 0.972109i $$0.424645\pi$$
$$24$$ 0 0
$$25$$ −1189.00 −0.380480
$$26$$ −4632.00 −1.34380
$$27$$ 0 0
$$28$$ −784.000 −0.188982
$$29$$ −3614.00 −0.797982 −0.398991 0.916955i $$-0.630640\pi$$
−0.398991 + 0.916955i $$0.630640\pi$$
$$30$$ 0 0
$$31$$ 5616.00 1.04960 0.524799 0.851226i $$-0.324141\pi$$
0.524799 + 0.851226i $$0.324141\pi$$
$$32$$ 1024.00 0.176777
$$33$$ 0 0
$$34$$ −4816.00 −0.714479
$$35$$ 2156.00 0.297494
$$36$$ 0 0
$$37$$ −6478.00 −0.777923 −0.388962 0.921254i $$-0.627166\pi$$
−0.388962 + 0.921254i $$0.627166\pi$$
$$38$$ −10576.0 −1.18813
$$39$$ 0 0
$$40$$ −2816.00 −0.278280
$$41$$ −2856.00 −0.265337 −0.132669 0.991160i $$-0.542355\pi$$
−0.132669 + 0.991160i $$0.542355\pi$$
$$42$$ 0 0
$$43$$ −13492.0 −1.11277 −0.556385 0.830925i $$-0.687812\pi$$
−0.556385 + 0.830925i $$0.687812\pi$$
$$44$$ 7520.00 0.585580
$$45$$ 0 0
$$46$$ 4760.00 0.331675
$$47$$ 18372.0 1.21314 0.606571 0.795029i $$-0.292544\pi$$
0.606571 + 0.795029i $$0.292544\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ −4756.00 −0.269040
$$51$$ 0 0
$$52$$ −18528.0 −0.950211
$$53$$ 4374.00 0.213889 0.106945 0.994265i $$-0.465893\pi$$
0.106945 + 0.994265i $$0.465893\pi$$
$$54$$ 0 0
$$55$$ −20680.0 −0.921815
$$56$$ −3136.00 −0.133631
$$57$$ 0 0
$$58$$ −14456.0 −0.564259
$$59$$ −30248.0 −1.13127 −0.565635 0.824655i $$-0.691369\pi$$
−0.565635 + 0.824655i $$0.691369\pi$$
$$60$$ 0 0
$$61$$ 19542.0 0.672426 0.336213 0.941786i $$-0.390854\pi$$
0.336213 + 0.941786i $$0.390854\pi$$
$$62$$ 22464.0 0.742178
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ 50952.0 1.49581
$$66$$ 0 0
$$67$$ 54328.0 1.47855 0.739276 0.673402i $$-0.235168\pi$$
0.739276 + 0.673402i $$0.235168\pi$$
$$68$$ −19264.0 −0.505213
$$69$$ 0 0
$$70$$ 8624.00 0.210360
$$71$$ 10730.0 0.252612 0.126306 0.991991i $$-0.459688\pi$$
0.126306 + 0.991991i $$0.459688\pi$$
$$72$$ 0 0
$$73$$ 35374.0 0.776921 0.388461 0.921465i $$-0.373007\pi$$
0.388461 + 0.921465i $$0.373007\pi$$
$$74$$ −25912.0 −0.550075
$$75$$ 0 0
$$76$$ −42304.0 −0.840132
$$77$$ −23030.0 −0.442657
$$78$$ 0 0
$$79$$ −49956.0 −0.900575 −0.450288 0.892884i $$-0.648679\pi$$
−0.450288 + 0.892884i $$0.648679\pi$$
$$80$$ −11264.0 −0.196774
$$81$$ 0 0
$$82$$ −11424.0 −0.187622
$$83$$ 26948.0 0.429370 0.214685 0.976683i $$-0.431128\pi$$
0.214685 + 0.976683i $$0.431128\pi$$
$$84$$ 0 0
$$85$$ 52976.0 0.795302
$$86$$ −53968.0 −0.786847
$$87$$ 0 0
$$88$$ 30080.0 0.414068
$$89$$ −100776. −1.34860 −0.674298 0.738459i $$-0.735554\pi$$
−0.674298 + 0.738459i $$0.735554\pi$$
$$90$$ 0 0
$$91$$ 56742.0 0.718292
$$92$$ 19040.0 0.234529
$$93$$ 0 0
$$94$$ 73488.0 0.857821
$$95$$ 116336. 1.32253
$$96$$ 0 0
$$97$$ 77134.0 0.832370 0.416185 0.909280i $$-0.363367\pi$$
0.416185 + 0.909280i $$0.363367\pi$$
$$98$$ 9604.00 0.101015
$$99$$ 0 0
$$100$$ −19024.0 −0.190240
$$101$$ −99464.0 −0.970203 −0.485101 0.874458i $$-0.661217\pi$$
−0.485101 + 0.874458i $$0.661217\pi$$
$$102$$ 0 0
$$103$$ 66944.0 0.621754 0.310877 0.950450i $$-0.399377\pi$$
0.310877 + 0.950450i $$0.399377\pi$$
$$104$$ −74112.0 −0.671901
$$105$$ 0 0
$$106$$ 17496.0 0.151243
$$107$$ 228198. 1.92687 0.963435 0.267942i $$-0.0863437\pi$$
0.963435 + 0.267942i $$0.0863437\pi$$
$$108$$ 0 0
$$109$$ −95186.0 −0.767374 −0.383687 0.923463i $$-0.625346\pi$$
−0.383687 + 0.923463i $$0.625346\pi$$
$$110$$ −82720.0 −0.651822
$$111$$ 0 0
$$112$$ −12544.0 −0.0944911
$$113$$ 261142. 1.92389 0.961946 0.273240i $$-0.0880954\pi$$
0.961946 + 0.273240i $$0.0880954\pi$$
$$114$$ 0 0
$$115$$ −52360.0 −0.369194
$$116$$ −57824.0 −0.398991
$$117$$ 0 0
$$118$$ −120992. −0.799929
$$119$$ 58996.0 0.381905
$$120$$ 0 0
$$121$$ 59849.0 0.371615
$$122$$ 78168.0 0.475477
$$123$$ 0 0
$$124$$ 89856.0 0.524799
$$125$$ 189816. 1.08657
$$126$$ 0 0
$$127$$ −167652. −0.922358 −0.461179 0.887307i $$-0.652573\pi$$
−0.461179 + 0.887307i $$0.652573\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 0 0
$$130$$ 203808. 1.05770
$$131$$ 83068.0 0.422917 0.211459 0.977387i $$-0.432179\pi$$
0.211459 + 0.977387i $$0.432179\pi$$
$$132$$ 0 0
$$133$$ 129556. 0.635080
$$134$$ 217312. 1.04549
$$135$$ 0 0
$$136$$ −77056.0 −0.357239
$$137$$ −162254. −0.738574 −0.369287 0.929315i $$-0.620398\pi$$
−0.369287 + 0.929315i $$0.620398\pi$$
$$138$$ 0 0
$$139$$ −58844.0 −0.258324 −0.129162 0.991623i $$-0.541229\pi$$
−0.129162 + 0.991623i $$0.541229\pi$$
$$140$$ 34496.0 0.148747
$$141$$ 0 0
$$142$$ 42920.0 0.178624
$$143$$ −544260. −2.22570
$$144$$ 0 0
$$145$$ 159016. 0.628088
$$146$$ 141496. 0.549366
$$147$$ 0 0
$$148$$ −103648. −0.388962
$$149$$ −430698. −1.58930 −0.794652 0.607065i $$-0.792347\pi$$
−0.794652 + 0.607065i $$0.792347\pi$$
$$150$$ 0 0
$$151$$ −500936. −1.78789 −0.893943 0.448180i $$-0.852072\pi$$
−0.893943 + 0.448180i $$0.852072\pi$$
$$152$$ −169216. −0.594063
$$153$$ 0 0
$$154$$ −92120.0 −0.313006
$$155$$ −247104. −0.826134
$$156$$ 0 0
$$157$$ −280258. −0.907421 −0.453711 0.891149i $$-0.649900\pi$$
−0.453711 + 0.891149i $$0.649900\pi$$
$$158$$ −199824. −0.636803
$$159$$ 0 0
$$160$$ −45056.0 −0.139140
$$161$$ −58310.0 −0.177288
$$162$$ 0 0
$$163$$ −120016. −0.353810 −0.176905 0.984228i $$-0.556609\pi$$
−0.176905 + 0.984228i $$0.556609\pi$$
$$164$$ −45696.0 −0.132669
$$165$$ 0 0
$$166$$ 107792. 0.303610
$$167$$ −546932. −1.51755 −0.758774 0.651355i $$-0.774201\pi$$
−0.758774 + 0.651355i $$0.774201\pi$$
$$168$$ 0 0
$$169$$ 969671. 2.61161
$$170$$ 211904. 0.562363
$$171$$ 0 0
$$172$$ −215872. −0.556385
$$173$$ 367096. 0.932533 0.466267 0.884644i $$-0.345599\pi$$
0.466267 + 0.884644i $$0.345599\pi$$
$$174$$ 0 0
$$175$$ 58261.0 0.143808
$$176$$ 120320. 0.292790
$$177$$ 0 0
$$178$$ −403104. −0.953602
$$179$$ 88890.0 0.207358 0.103679 0.994611i $$-0.466939\pi$$
0.103679 + 0.994611i $$0.466939\pi$$
$$180$$ 0 0
$$181$$ −782118. −1.77450 −0.887250 0.461290i $$-0.847387\pi$$
−0.887250 + 0.461290i $$0.847387\pi$$
$$182$$ 226968. 0.507909
$$183$$ 0 0
$$184$$ 76160.0 0.165837
$$185$$ 285032. 0.612300
$$186$$ 0 0
$$187$$ −565880. −1.18337
$$188$$ 293952. 0.606571
$$189$$ 0 0
$$190$$ 465344. 0.935169
$$191$$ −763350. −1.51405 −0.757025 0.653386i $$-0.773348\pi$$
−0.757025 + 0.653386i $$0.773348\pi$$
$$192$$ 0 0
$$193$$ −377002. −0.728535 −0.364267 0.931294i $$-0.618681\pi$$
−0.364267 + 0.931294i $$0.618681\pi$$
$$194$$ 308536. 0.588575
$$195$$ 0 0
$$196$$ 38416.0 0.0714286
$$197$$ 68678.0 0.126082 0.0630409 0.998011i $$-0.479920\pi$$
0.0630409 + 0.998011i $$0.479920\pi$$
$$198$$ 0 0
$$199$$ 182576. 0.326822 0.163411 0.986558i $$-0.447750\pi$$
0.163411 + 0.986558i $$0.447750\pi$$
$$200$$ −76096.0 −0.134520
$$201$$ 0 0
$$202$$ −397856. −0.686037
$$203$$ 177086. 0.301609
$$204$$ 0 0
$$205$$ 125664. 0.208846
$$206$$ 267776. 0.439646
$$207$$ 0 0
$$208$$ −296448. −0.475106
$$209$$ −1.24268e6 −1.96786
$$210$$ 0 0
$$211$$ 232652. 0.359750 0.179875 0.983689i $$-0.442431\pi$$
0.179875 + 0.983689i $$0.442431\pi$$
$$212$$ 69984.0 0.106945
$$213$$ 0 0
$$214$$ 912792. 1.36250
$$215$$ 593648. 0.875856
$$216$$ 0 0
$$217$$ −275184. −0.396711
$$218$$ −380744. −0.542615
$$219$$ 0 0
$$220$$ −330880. −0.460908
$$221$$ 1.39423e6 1.92023
$$222$$ 0 0
$$223$$ 167144. 0.225076 0.112538 0.993647i $$-0.464102\pi$$
0.112538 + 0.993647i $$0.464102\pi$$
$$224$$ −50176.0 −0.0668153
$$225$$ 0 0
$$226$$ 1.04457e6 1.36040
$$227$$ −415728. −0.535482 −0.267741 0.963491i $$-0.586277\pi$$
−0.267741 + 0.963491i $$0.586277\pi$$
$$228$$ 0 0
$$229$$ 473482. 0.596643 0.298322 0.954465i $$-0.403573\pi$$
0.298322 + 0.954465i $$0.403573\pi$$
$$230$$ −209440. −0.261060
$$231$$ 0 0
$$232$$ −231296. −0.282129
$$233$$ 1.55655e6 1.87833 0.939166 0.343465i $$-0.111601\pi$$
0.939166 + 0.343465i $$0.111601\pi$$
$$234$$ 0 0
$$235$$ −808368. −0.954859
$$236$$ −483968. −0.565635
$$237$$ 0 0
$$238$$ 235984. 0.270048
$$239$$ −655890. −0.742739 −0.371370 0.928485i $$-0.621112\pi$$
−0.371370 + 0.928485i $$0.621112\pi$$
$$240$$ 0 0
$$241$$ −889474. −0.986485 −0.493243 0.869892i $$-0.664189\pi$$
−0.493243 + 0.869892i $$0.664189\pi$$
$$242$$ 239396. 0.262772
$$243$$ 0 0
$$244$$ 312672. 0.336213
$$245$$ −105644. −0.112442
$$246$$ 0 0
$$247$$ 3.06175e6 3.19321
$$248$$ 359424. 0.371089
$$249$$ 0 0
$$250$$ 759264. 0.768321
$$251$$ −131832. −0.132080 −0.0660399 0.997817i $$-0.521036\pi$$
−0.0660399 + 0.997817i $$0.521036\pi$$
$$252$$ 0 0
$$253$$ 559300. 0.549343
$$254$$ −670608. −0.652205
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 1.46482e6 1.38341 0.691704 0.722181i $$-0.256860\pi$$
0.691704 + 0.722181i $$0.256860\pi$$
$$258$$ 0 0
$$259$$ 317422. 0.294027
$$260$$ 815232. 0.747907
$$261$$ 0 0
$$262$$ 332272. 0.299048
$$263$$ −1.47969e6 −1.31911 −0.659556 0.751656i $$-0.729255\pi$$
−0.659556 + 0.751656i $$0.729255\pi$$
$$264$$ 0 0
$$265$$ −192456. −0.168351
$$266$$ 518224. 0.449069
$$267$$ 0 0
$$268$$ 869248. 0.739276
$$269$$ 187852. 0.158283 0.0791417 0.996863i $$-0.474782\pi$$
0.0791417 + 0.996863i $$0.474782\pi$$
$$270$$ 0 0
$$271$$ 193800. 0.160299 0.0801495 0.996783i $$-0.474460\pi$$
0.0801495 + 0.996783i $$0.474460\pi$$
$$272$$ −308224. −0.252606
$$273$$ 0 0
$$274$$ −649016. −0.522251
$$275$$ −558830. −0.445603
$$276$$ 0 0
$$277$$ −617062. −0.483203 −0.241601 0.970376i $$-0.577673\pi$$
−0.241601 + 0.970376i $$0.577673\pi$$
$$278$$ −235376. −0.182663
$$279$$ 0 0
$$280$$ 137984. 0.105180
$$281$$ 1.73129e6 1.30799 0.653994 0.756499i $$-0.273092\pi$$
0.653994 + 0.756499i $$0.273092\pi$$
$$282$$ 0 0
$$283$$ 356020. 0.264246 0.132123 0.991233i $$-0.457821\pi$$
0.132123 + 0.991233i $$0.457821\pi$$
$$284$$ 171680. 0.126306
$$285$$ 0 0
$$286$$ −2.17704e6 −1.57381
$$287$$ 139944. 0.100288
$$288$$ 0 0
$$289$$ 29759.0 0.0209592
$$290$$ 636064. 0.444126
$$291$$ 0 0
$$292$$ 565984. 0.388461
$$293$$ −536664. −0.365202 −0.182601 0.983187i $$-0.558452\pi$$
−0.182601 + 0.983187i $$0.558452\pi$$
$$294$$ 0 0
$$295$$ 1.33091e6 0.890419
$$296$$ −414592. −0.275037
$$297$$ 0 0
$$298$$ −1.72279e6 −1.12381
$$299$$ −1.37802e6 −0.891410
$$300$$ 0 0
$$301$$ 661108. 0.420587
$$302$$ −2.00374e6 −1.26423
$$303$$ 0 0
$$304$$ −676864. −0.420066
$$305$$ −859848. −0.529264
$$306$$ 0 0
$$307$$ −2.88398e6 −1.74641 −0.873205 0.487353i $$-0.837963\pi$$
−0.873205 + 0.487353i $$0.837963\pi$$
$$308$$ −368480. −0.221328
$$309$$ 0 0
$$310$$ −988416. −0.584165
$$311$$ 1.02958e6 0.603614 0.301807 0.953369i $$-0.402410\pi$$
0.301807 + 0.953369i $$0.402410\pi$$
$$312$$ 0 0
$$313$$ 1.39297e6 0.803676 0.401838 0.915711i $$-0.368372\pi$$
0.401838 + 0.915711i $$0.368372\pi$$
$$314$$ −1.12103e6 −0.641644
$$315$$ 0 0
$$316$$ −799296. −0.450288
$$317$$ 780030. 0.435977 0.217988 0.975951i $$-0.430051\pi$$
0.217988 + 0.975951i $$0.430051\pi$$
$$318$$ 0 0
$$319$$ −1.69858e6 −0.934565
$$320$$ −180224. −0.0983870
$$321$$ 0 0
$$322$$ −233240. −0.125361
$$323$$ 3.18338e6 1.69778
$$324$$ 0 0
$$325$$ 1.37686e6 0.723073
$$326$$ −480064. −0.250181
$$327$$ 0 0
$$328$$ −182784. −0.0938110
$$329$$ −900228. −0.458525
$$330$$ 0 0
$$331$$ −1.41204e6 −0.708399 −0.354200 0.935170i $$-0.615247\pi$$
−0.354200 + 0.935170i $$0.615247\pi$$
$$332$$ 431168. 0.214685
$$333$$ 0 0
$$334$$ −2.18773e6 −1.07307
$$335$$ −2.39043e6 −1.16376
$$336$$ 0 0
$$337$$ −634662. −0.304416 −0.152208 0.988348i $$-0.548638\pi$$
−0.152208 + 0.988348i $$0.548638\pi$$
$$338$$ 3.87868e6 1.84668
$$339$$ 0 0
$$340$$ 847616. 0.397651
$$341$$ 2.63952e6 1.22925
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ −863488. −0.393423
$$345$$ 0 0
$$346$$ 1.46838e6 0.659401
$$347$$ −3.07942e6 −1.37292 −0.686460 0.727167i $$-0.740837\pi$$
−0.686460 + 0.727167i $$0.740837\pi$$
$$348$$ 0 0
$$349$$ −2.60671e6 −1.14559 −0.572796 0.819698i $$-0.694141\pi$$
−0.572796 + 0.819698i $$0.694141\pi$$
$$350$$ 233044. 0.101688
$$351$$ 0 0
$$352$$ 481280. 0.207034
$$353$$ 63132.0 0.0269658 0.0134829 0.999909i $$-0.495708\pi$$
0.0134829 + 0.999909i $$0.495708\pi$$
$$354$$ 0 0
$$355$$ −472120. −0.198830
$$356$$ −1.61242e6 −0.674298
$$357$$ 0 0
$$358$$ 355560. 0.146624
$$359$$ −479270. −0.196266 −0.0981328 0.995173i $$-0.531287\pi$$
−0.0981328 + 0.995173i $$0.531287\pi$$
$$360$$ 0 0
$$361$$ 4.51464e6 1.82329
$$362$$ −3.12847e6 −1.25476
$$363$$ 0 0
$$364$$ 907872. 0.359146
$$365$$ −1.55646e6 −0.611512
$$366$$ 0 0
$$367$$ −1.33451e6 −0.517199 −0.258599 0.965985i $$-0.583261\pi$$
−0.258599 + 0.965985i $$0.583261\pi$$
$$368$$ 304640. 0.117265
$$369$$ 0 0
$$370$$ 1.14013e6 0.432962
$$371$$ −214326. −0.0808426
$$372$$ 0 0
$$373$$ −1.69759e6 −0.631774 −0.315887 0.948797i $$-0.602302\pi$$
−0.315887 + 0.948797i $$0.602302\pi$$
$$374$$ −2.26352e6 −0.836769
$$375$$ 0 0
$$376$$ 1.17581e6 0.428911
$$377$$ 4.18501e6 1.51650
$$378$$ 0 0
$$379$$ 2.51074e6 0.897850 0.448925 0.893569i $$-0.351807\pi$$
0.448925 + 0.893569i $$0.351807\pi$$
$$380$$ 1.86138e6 0.661264
$$381$$ 0 0
$$382$$ −3.05340e6 −1.07060
$$383$$ −559144. −0.194772 −0.0973860 0.995247i $$-0.531048\pi$$
−0.0973860 + 0.995247i $$0.531048\pi$$
$$384$$ 0 0
$$385$$ 1.01332e6 0.348413
$$386$$ −1.50801e6 −0.515152
$$387$$ 0 0
$$388$$ 1.23414e6 0.416185
$$389$$ −4.51055e6 −1.51132 −0.755658 0.654966i $$-0.772683\pi$$
−0.755658 + 0.654966i $$0.772683\pi$$
$$390$$ 0 0
$$391$$ −1.43276e6 −0.473949
$$392$$ 153664. 0.0505076
$$393$$ 0 0
$$394$$ 274712. 0.0891532
$$395$$ 2.19806e6 0.708839
$$396$$ 0 0
$$397$$ 5.19862e6 1.65543 0.827717 0.561145i $$-0.189639\pi$$
0.827717 + 0.561145i $$0.189639\pi$$
$$398$$ 730304. 0.231098
$$399$$ 0 0
$$400$$ −304384. −0.0951200
$$401$$ 6.34816e6 1.97146 0.985728 0.168346i $$-0.0538426\pi$$
0.985728 + 0.168346i $$0.0538426\pi$$
$$402$$ 0 0
$$403$$ −6.50333e6 −1.99468
$$404$$ −1.59142e6 −0.485101
$$405$$ 0 0
$$406$$ 708344. 0.213270
$$407$$ −3.04466e6 −0.911072
$$408$$ 0 0
$$409$$ −181642. −0.0536918 −0.0268459 0.999640i $$-0.508546\pi$$
−0.0268459 + 0.999640i $$0.508546\pi$$
$$410$$ 502656. 0.147676
$$411$$ 0 0
$$412$$ 1.07110e6 0.310877
$$413$$ 1.48215e6 0.427580
$$414$$ 0 0
$$415$$ −1.18571e6 −0.337955
$$416$$ −1.18579e6 −0.335950
$$417$$ 0 0
$$418$$ −4.97072e6 −1.39149
$$419$$ 5.62699e6 1.56582 0.782909 0.622136i $$-0.213735\pi$$
0.782909 + 0.622136i $$0.213735\pi$$
$$420$$ 0 0
$$421$$ −4.42671e6 −1.21724 −0.608619 0.793462i $$-0.708276\pi$$
−0.608619 + 0.793462i $$0.708276\pi$$
$$422$$ 930608. 0.254382
$$423$$ 0 0
$$424$$ 279936. 0.0756213
$$425$$ 1.43156e6 0.384447
$$426$$ 0 0
$$427$$ −957558. −0.254153
$$428$$ 3.65117e6 0.963435
$$429$$ 0 0
$$430$$ 2.37459e6 0.619324
$$431$$ 1.44163e6 0.373817 0.186909 0.982377i $$-0.440153\pi$$
0.186909 + 0.982377i $$0.440153\pi$$
$$432$$ 0 0
$$433$$ −3.89661e6 −0.998775 −0.499387 0.866379i $$-0.666442\pi$$
−0.499387 + 0.866379i $$0.666442\pi$$
$$434$$ −1.10074e6 −0.280517
$$435$$ 0 0
$$436$$ −1.52298e6 −0.383687
$$437$$ −3.14636e6 −0.788143
$$438$$ 0 0
$$439$$ 5.11207e6 1.26601 0.633003 0.774149i $$-0.281822\pi$$
0.633003 + 0.774149i $$0.281822\pi$$
$$440$$ −1.32352e6 −0.325911
$$441$$ 0 0
$$442$$ 5.57693e6 1.35781
$$443$$ −5.44070e6 −1.31718 −0.658591 0.752501i $$-0.728847\pi$$
−0.658591 + 0.752501i $$0.728847\pi$$
$$444$$ 0 0
$$445$$ 4.43414e6 1.06147
$$446$$ 668576. 0.159153
$$447$$ 0 0
$$448$$ −200704. −0.0472456
$$449$$ 1.31525e6 0.307887 0.153943 0.988080i $$-0.450803\pi$$
0.153943 + 0.988080i $$0.450803\pi$$
$$450$$ 0 0
$$451$$ −1.34232e6 −0.310753
$$452$$ 4.17827e6 0.961946
$$453$$ 0 0
$$454$$ −1.66291e6 −0.378643
$$455$$ −2.49665e6 −0.565365
$$456$$ 0 0
$$457$$ 2.77604e6 0.621778 0.310889 0.950446i $$-0.399373\pi$$
0.310889 + 0.950446i $$0.399373\pi$$
$$458$$ 1.89393e6 0.421891
$$459$$ 0 0
$$460$$ −837760. −0.184597
$$461$$ −138080. −0.0302607 −0.0151303 0.999886i $$-0.504816\pi$$
−0.0151303 + 0.999886i $$0.504816\pi$$
$$462$$ 0 0
$$463$$ −364076. −0.0789295 −0.0394648 0.999221i $$-0.512565\pi$$
−0.0394648 + 0.999221i $$0.512565\pi$$
$$464$$ −925184. −0.199496
$$465$$ 0 0
$$466$$ 6.22618e6 1.32818
$$467$$ 5.73897e6 1.21770 0.608852 0.793284i $$-0.291630\pi$$
0.608852 + 0.793284i $$0.291630\pi$$
$$468$$ 0 0
$$469$$ −2.66207e6 −0.558840
$$470$$ −3.23347e6 −0.675188
$$471$$ 0 0
$$472$$ −1.93587e6 −0.399965
$$473$$ −6.34124e6 −1.30323
$$474$$ 0 0
$$475$$ 3.14372e6 0.639307
$$476$$ 943936. 0.190952
$$477$$ 0 0
$$478$$ −2.62356e6 −0.525196
$$479$$ −5.51996e6 −1.09925 −0.549625 0.835411i $$-0.685230\pi$$
−0.549625 + 0.835411i $$0.685230\pi$$
$$480$$ 0 0
$$481$$ 7.50152e6 1.47838
$$482$$ −3.55790e6 −0.697550
$$483$$ 0 0
$$484$$ 957584. 0.185808
$$485$$ −3.39390e6 −0.655155
$$486$$ 0 0
$$487$$ 4.23022e6 0.808241 0.404121 0.914706i $$-0.367578\pi$$
0.404121 + 0.914706i $$0.367578\pi$$
$$488$$ 1.25069e6 0.237738
$$489$$ 0 0
$$490$$ −422576. −0.0795087
$$491$$ 7.21423e6 1.35047 0.675237 0.737601i $$-0.264041\pi$$
0.675237 + 0.737601i $$0.264041\pi$$
$$492$$ 0 0
$$493$$ 4.35126e6 0.806301
$$494$$ 1.22470e7 2.25794
$$495$$ 0 0
$$496$$ 1.43770e6 0.262399
$$497$$ −525770. −0.0954783
$$498$$ 0 0
$$499$$ −224804. −0.0404159 −0.0202080 0.999796i $$-0.506433\pi$$
−0.0202080 + 0.999796i $$0.506433\pi$$
$$500$$ 3.03706e6 0.543285
$$501$$ 0 0
$$502$$ −527328. −0.0933946
$$503$$ −5.06983e6 −0.893457 −0.446728 0.894670i $$-0.647411\pi$$
−0.446728 + 0.894670i $$0.647411\pi$$
$$504$$ 0 0
$$505$$ 4.37642e6 0.763643
$$506$$ 2.23720e6 0.388444
$$507$$ 0 0
$$508$$ −2.68243e6 −0.461179
$$509$$ −5.48135e6 −0.937763 −0.468881 0.883261i $$-0.655343\pi$$
−0.468881 + 0.883261i $$0.655343\pi$$
$$510$$ 0 0
$$511$$ −1.73333e6 −0.293649
$$512$$ 262144. 0.0441942
$$513$$ 0 0
$$514$$ 5.85926e6 0.978217
$$515$$ −2.94554e6 −0.489380
$$516$$ 0 0
$$517$$ 8.63484e6 1.42078
$$518$$ 1.26969e6 0.207909
$$519$$ 0 0
$$520$$ 3.26093e6 0.528850
$$521$$ −7.88932e6 −1.27334 −0.636671 0.771136i $$-0.719689\pi$$
−0.636671 + 0.771136i $$0.719689\pi$$
$$522$$ 0 0
$$523$$ −9.74950e6 −1.55858 −0.779288 0.626666i $$-0.784419\pi$$
−0.779288 + 0.626666i $$0.784419\pi$$
$$524$$ 1.32909e6 0.211459
$$525$$ 0 0
$$526$$ −5.91876e6 −0.932752
$$527$$ −6.76166e6 −1.06054
$$528$$ 0 0
$$529$$ −5.02024e6 −0.779984
$$530$$ −769824. −0.119042
$$531$$ 0 0
$$532$$ 2.07290e6 0.317540
$$533$$ 3.30725e6 0.504253
$$534$$ 0 0
$$535$$ −1.00407e7 −1.51663
$$536$$ 3.47699e6 0.522747
$$537$$ 0 0
$$538$$ 751408. 0.111923
$$539$$ 1.12847e6 0.167309
$$540$$ 0 0
$$541$$ −4.17454e6 −0.613219 −0.306610 0.951835i $$-0.599195\pi$$
−0.306610 + 0.951835i $$0.599195\pi$$
$$542$$ 775200. 0.113348
$$543$$ 0 0
$$544$$ −1.23290e6 −0.178620
$$545$$ 4.18818e6 0.603997
$$546$$ 0 0
$$547$$ −2.72887e6 −0.389955 −0.194978 0.980808i $$-0.562463\pi$$
−0.194978 + 0.980808i $$0.562463\pi$$
$$548$$ −2.59606e6 −0.369287
$$549$$ 0 0
$$550$$ −2.23532e6 −0.315089
$$551$$ 9.55542e6 1.34082
$$552$$ 0 0
$$553$$ 2.44784e6 0.340385
$$554$$ −2.46825e6 −0.341676
$$555$$ 0 0
$$556$$ −941504. −0.129162
$$557$$ 7.35103e6 1.00395 0.501973 0.864883i $$-0.332608\pi$$
0.501973 + 0.864883i $$0.332608\pi$$
$$558$$ 0 0
$$559$$ 1.56237e7 2.11473
$$560$$ 551936. 0.0743736
$$561$$ 0 0
$$562$$ 6.92516e6 0.924888
$$563$$ 1.37821e7 1.83250 0.916248 0.400612i $$-0.131202\pi$$
0.916248 + 0.400612i $$0.131202\pi$$
$$564$$ 0 0
$$565$$ −1.14902e7 −1.51429
$$566$$ 1.42408e6 0.186850
$$567$$ 0 0
$$568$$ 686720. 0.0893118
$$569$$ −2.71217e6 −0.351186 −0.175593 0.984463i $$-0.556184\pi$$
−0.175593 + 0.984463i $$0.556184\pi$$
$$570$$ 0 0
$$571$$ 4.94398e6 0.634580 0.317290 0.948329i $$-0.397227\pi$$
0.317290 + 0.948329i $$0.397227\pi$$
$$572$$ −8.70816e6 −1.11285
$$573$$ 0 0
$$574$$ 559776. 0.0709144
$$575$$ −1.41491e6 −0.178468
$$576$$ 0 0
$$577$$ −1.32683e7 −1.65911 −0.829556 0.558424i $$-0.811406\pi$$
−0.829556 + 0.558424i $$0.811406\pi$$
$$578$$ 119036. 0.0148204
$$579$$ 0 0
$$580$$ 2.54426e6 0.314044
$$581$$ −1.32045e6 −0.162286
$$582$$ 0 0
$$583$$ 2.05578e6 0.250499
$$584$$ 2.26394e6 0.274683
$$585$$ 0 0
$$586$$ −2.14666e6 −0.258237
$$587$$ −1.67205e6 −0.200287 −0.100144 0.994973i $$-0.531930\pi$$
−0.100144 + 0.994973i $$0.531930\pi$$
$$588$$ 0 0
$$589$$ −1.48487e7 −1.76360
$$590$$ 5.32365e6 0.629621
$$591$$ 0 0
$$592$$ −1.65837e6 −0.194481
$$593$$ −5.11693e6 −0.597548 −0.298774 0.954324i $$-0.596578\pi$$
−0.298774 + 0.954324i $$0.596578\pi$$
$$594$$ 0 0
$$595$$ −2.59582e6 −0.300596
$$596$$ −6.89117e6 −0.794652
$$597$$ 0 0
$$598$$ −5.51208e6 −0.630322
$$599$$ 7.15931e6 0.815275 0.407638 0.913144i $$-0.366353\pi$$
0.407638 + 0.913144i $$0.366353\pi$$
$$600$$ 0 0
$$601$$ −9.63384e6 −1.08796 −0.543980 0.839098i $$-0.683083\pi$$
−0.543980 + 0.839098i $$0.683083\pi$$
$$602$$ 2.64443e6 0.297400
$$603$$ 0 0
$$604$$ −8.01498e6 −0.893943
$$605$$ −2.63336e6 −0.292497
$$606$$ 0 0
$$607$$ 5.52115e6 0.608216 0.304108 0.952638i $$-0.401642\pi$$
0.304108 + 0.952638i $$0.401642\pi$$
$$608$$ −2.70746e6 −0.297031
$$609$$ 0 0
$$610$$ −3.43939e6 −0.374246
$$611$$ −2.12748e7 −2.30548
$$612$$ 0 0
$$613$$ 4.79890e6 0.515811 0.257906 0.966170i $$-0.416968\pi$$
0.257906 + 0.966170i $$0.416968\pi$$
$$614$$ −1.15359e7 −1.23490
$$615$$ 0 0
$$616$$ −1.47392e6 −0.156503
$$617$$ −2.71826e6 −0.287461 −0.143730 0.989617i $$-0.545910\pi$$
−0.143730 + 0.989617i $$0.545910\pi$$
$$618$$ 0 0
$$619$$ −4.34335e6 −0.455615 −0.227807 0.973706i $$-0.573156\pi$$
−0.227807 + 0.973706i $$0.573156\pi$$
$$620$$ −3.95366e6 −0.413067
$$621$$ 0 0
$$622$$ 4.11832e6 0.426819
$$623$$ 4.93802e6 0.509722
$$624$$ 0 0
$$625$$ −4.63628e6 −0.474755
$$626$$ 5.57188e6 0.568285
$$627$$ 0 0
$$628$$ −4.48413e6 −0.453711
$$629$$ 7.79951e6 0.786033
$$630$$ 0 0
$$631$$ 1.11952e6 0.111933 0.0559667 0.998433i $$-0.482176\pi$$
0.0559667 + 0.998433i $$0.482176\pi$$
$$632$$ −3.19718e6 −0.318401
$$633$$ 0 0
$$634$$ 3.12012e6 0.308282
$$635$$ 7.37669e6 0.725984
$$636$$ 0 0
$$637$$ −2.78036e6 −0.271489
$$638$$ −6.79432e6 −0.660837
$$639$$ 0 0
$$640$$ −720896. −0.0695701
$$641$$ 3.96588e6 0.381237 0.190618 0.981664i $$-0.438951\pi$$
0.190618 + 0.981664i $$0.438951\pi$$
$$642$$ 0 0
$$643$$ 1.92086e7 1.83218 0.916092 0.400968i $$-0.131326\pi$$
0.916092 + 0.400968i $$0.131326\pi$$
$$644$$ −932960. −0.0886438
$$645$$ 0 0
$$646$$ 1.27335e7 1.20051
$$647$$ −4.72739e6 −0.443977 −0.221989 0.975049i $$-0.571255\pi$$
−0.221989 + 0.975049i $$0.571255\pi$$
$$648$$ 0 0
$$649$$ −1.42166e7 −1.32490
$$650$$ 5.50745e6 0.511290
$$651$$ 0 0
$$652$$ −1.92026e6 −0.176905
$$653$$ −1.21159e7 −1.11192 −0.555958 0.831210i $$-0.687648\pi$$
−0.555958 + 0.831210i $$0.687648\pi$$
$$654$$ 0 0
$$655$$ −3.65499e6 −0.332877
$$656$$ −731136. −0.0663344
$$657$$ 0 0
$$658$$ −3.60091e6 −0.324226
$$659$$ −5.91457e6 −0.530530 −0.265265 0.964176i $$-0.585459\pi$$
−0.265265 + 0.964176i $$0.585459\pi$$
$$660$$ 0 0
$$661$$ 1.41779e7 1.26214 0.631072 0.775724i $$-0.282615\pi$$
0.631072 + 0.775724i $$0.282615\pi$$
$$662$$ −5.64818e6 −0.500914
$$663$$ 0 0
$$664$$ 1.72467e6 0.151805
$$665$$ −5.70046e6 −0.499869
$$666$$ 0 0
$$667$$ −4.30066e6 −0.374301
$$668$$ −8.75091e6 −0.758774
$$669$$ 0 0
$$670$$ −9.56173e6 −0.822904
$$671$$ 9.18474e6 0.787518
$$672$$ 0 0
$$673$$ 1.34245e7 1.14251 0.571256 0.820772i $$-0.306456\pi$$
0.571256 + 0.820772i $$0.306456\pi$$
$$674$$ −2.53865e6 −0.215255
$$675$$ 0 0
$$676$$ 1.55147e7 1.30580
$$677$$ 8.67312e6 0.727283 0.363642 0.931539i $$-0.381533\pi$$
0.363642 + 0.931539i $$0.381533\pi$$
$$678$$ 0 0
$$679$$ −3.77957e6 −0.314606
$$680$$ 3.39046e6 0.281182
$$681$$ 0 0
$$682$$ 1.05581e7 0.869209
$$683$$ −1.49777e7 −1.22855 −0.614275 0.789092i $$-0.710552\pi$$
−0.614275 + 0.789092i $$0.710552\pi$$
$$684$$ 0 0
$$685$$ 7.13918e6 0.581329
$$686$$ −470596. −0.0381802
$$687$$ 0 0
$$688$$ −3.45395e6 −0.278192
$$689$$ −5.06509e6 −0.406480
$$690$$ 0 0
$$691$$ −8.76742e6 −0.698517 −0.349258 0.937026i $$-0.613566\pi$$
−0.349258 + 0.937026i $$0.613566\pi$$
$$692$$ 5.87354e6 0.466267
$$693$$ 0 0
$$694$$ −1.23177e7 −0.970802
$$695$$ 2.58914e6 0.203326
$$696$$ 0 0
$$697$$ 3.43862e6 0.268104
$$698$$ −1.04269e7 −0.810056
$$699$$ 0 0
$$700$$ 932176. 0.0719040
$$701$$ 1.99459e7 1.53306 0.766529 0.642209i $$-0.221982\pi$$
0.766529 + 0.642209i $$0.221982\pi$$
$$702$$ 0 0
$$703$$ 1.71278e7 1.30712
$$704$$ 1.92512e6 0.146395
$$705$$ 0 0
$$706$$ 252528. 0.0190677
$$707$$ 4.87374e6 0.366702
$$708$$ 0 0
$$709$$ −2.66387e7 −1.99020 −0.995100 0.0988699i $$-0.968477\pi$$
−0.995100 + 0.0988699i $$0.968477\pi$$
$$710$$ −1.88848e6 −0.140594
$$711$$ 0 0
$$712$$ −6.44966e6 −0.476801
$$713$$ 6.68304e6 0.492323
$$714$$ 0 0
$$715$$ 2.39474e7 1.75184
$$716$$ 1.42224e6 0.103679
$$717$$ 0 0
$$718$$ −1.91708e6 −0.138781
$$719$$ 1.17408e6 0.0846985 0.0423492 0.999103i $$-0.486516\pi$$
0.0423492 + 0.999103i $$0.486516\pi$$
$$720$$ 0 0
$$721$$ −3.28026e6 −0.235001
$$722$$ 1.80585e7 1.28926
$$723$$ 0 0
$$724$$ −1.25139e7 −0.887250
$$725$$ 4.29705e6 0.303616
$$726$$ 0 0
$$727$$ 1.05734e7 0.741957 0.370979 0.928641i $$-0.379022\pi$$
0.370979 + 0.928641i $$0.379022\pi$$
$$728$$ 3.63149e6 0.253955
$$729$$ 0 0
$$730$$ −6.22582e6 −0.432404
$$731$$ 1.62444e7 1.12437
$$732$$ 0 0
$$733$$ 2.31670e7 1.59261 0.796306 0.604894i $$-0.206785\pi$$
0.796306 + 0.604894i $$0.206785\pi$$
$$734$$ −5.33805e6 −0.365715
$$735$$ 0 0
$$736$$ 1.21856e6 0.0829187
$$737$$ 2.55342e7 1.73162
$$738$$ 0 0
$$739$$ −1.55334e7 −1.04630 −0.523148 0.852242i $$-0.675243\pi$$
−0.523148 + 0.852242i $$0.675243\pi$$
$$740$$ 4.56051e6 0.306150
$$741$$ 0 0
$$742$$ −857304. −0.0571643
$$743$$ −2.54630e7 −1.69215 −0.846074 0.533066i $$-0.821040\pi$$
−0.846074 + 0.533066i $$0.821040\pi$$
$$744$$ 0 0
$$745$$ 1.89507e7 1.25094
$$746$$ −6.79038e6 −0.446732
$$747$$ 0 0
$$748$$ −9.05408e6 −0.591685
$$749$$ −1.11817e7 −0.728288
$$750$$ 0 0
$$751$$ 9.88512e6 0.639561 0.319781 0.947492i $$-0.396391\pi$$
0.319781 + 0.947492i $$0.396391\pi$$
$$752$$ 4.70323e6 0.303286
$$753$$ 0 0
$$754$$ 1.67400e7 1.07233
$$755$$ 2.20412e7 1.40724
$$756$$ 0 0
$$757$$ −6.41980e6 −0.407176 −0.203588 0.979057i $$-0.565260\pi$$
−0.203588 + 0.979057i $$0.565260\pi$$
$$758$$ 1.00430e7 0.634876
$$759$$ 0 0
$$760$$ 7.44550e6 0.467585
$$761$$ 5.05052e6 0.316136 0.158068 0.987428i $$-0.449473\pi$$
0.158068 + 0.987428i $$0.449473\pi$$
$$762$$ 0 0
$$763$$ 4.66411e6 0.290040
$$764$$ −1.22136e7 −0.757025
$$765$$ 0 0
$$766$$ −2.23658e6 −0.137725
$$767$$ 3.50272e7 2.14989
$$768$$ 0 0
$$769$$ 2.28169e6 0.139136 0.0695682 0.997577i $$-0.477838\pi$$
0.0695682 + 0.997577i $$0.477838\pi$$
$$770$$ 4.05328e6 0.246365
$$771$$ 0 0
$$772$$ −6.03203e6 −0.364267
$$773$$ −2.73777e7 −1.64797 −0.823984 0.566613i $$-0.808254\pi$$
−0.823984 + 0.566613i $$0.808254\pi$$
$$774$$ 0 0
$$775$$ −6.67742e6 −0.399351
$$776$$ 4.93658e6 0.294287
$$777$$ 0 0
$$778$$ −1.80422e7 −1.06866
$$779$$ 7.55126e6 0.445837
$$780$$ 0 0
$$781$$ 5.04310e6 0.295849
$$782$$ −5.73104e6 −0.335133
$$783$$ 0 0
$$784$$ 614656. 0.0357143
$$785$$ 1.23314e7 0.714227
$$786$$ 0 0
$$787$$ 2.04263e7 1.17558 0.587791 0.809013i $$-0.299998\pi$$
0.587791 + 0.809013i $$0.299998\pi$$
$$788$$ 1.09885e6 0.0630409
$$789$$ 0 0
$$790$$ 8.79226e6 0.501225
$$791$$ −1.27960e7 −0.727163
$$792$$ 0 0
$$793$$ −2.26296e7 −1.27789
$$794$$ 2.07945e7 1.17057
$$795$$ 0 0
$$796$$ 2.92122e6 0.163411
$$797$$ −2.31557e7 −1.29126 −0.645628 0.763652i $$-0.723404\pi$$
−0.645628 + 0.763652i $$0.723404\pi$$
$$798$$ 0 0
$$799$$ −2.21199e7 −1.22579
$$800$$ −1.21754e6 −0.0672600
$$801$$ 0 0
$$802$$ 2.53926e7 1.39403
$$803$$ 1.66258e7 0.909899
$$804$$ 0 0
$$805$$ 2.56564e6 0.139542
$$806$$ −2.60133e7 −1.41045
$$807$$ 0 0
$$808$$ −6.36570e6 −0.343018
$$809$$ −1.14894e7 −0.617203 −0.308601 0.951191i $$-0.599861\pi$$
−0.308601 + 0.951191i $$0.599861\pi$$
$$810$$ 0 0
$$811$$ 1.72443e7 0.920648 0.460324 0.887751i $$-0.347733\pi$$
0.460324 + 0.887751i $$0.347733\pi$$
$$812$$ 2.83338e6 0.150804
$$813$$ 0 0
$$814$$ −1.21786e7 −0.644225
$$815$$ 5.28070e6 0.278482
$$816$$ 0 0
$$817$$ 3.56728e7 1.86975
$$818$$ −726568. −0.0379658
$$819$$ 0 0
$$820$$ 2.01062e6 0.104423
$$821$$ −1.37077e6 −0.0709752 −0.0354876 0.999370i $$-0.511298\pi$$
−0.0354876 + 0.999370i $$0.511298\pi$$
$$822$$ 0 0
$$823$$ 8.56851e6 0.440967 0.220483 0.975391i $$-0.429237\pi$$
0.220483 + 0.975391i $$0.429237\pi$$
$$824$$ 4.28442e6 0.219823
$$825$$ 0 0
$$826$$ 5.92861e6 0.302345
$$827$$ 1.33258e6 0.0677533 0.0338766 0.999426i $$-0.489215\pi$$
0.0338766 + 0.999426i $$0.489215\pi$$
$$828$$ 0 0
$$829$$ 6.25659e6 0.316193 0.158096 0.987424i $$-0.449464\pi$$
0.158096 + 0.987424i $$0.449464\pi$$
$$830$$ −4.74285e6 −0.238970
$$831$$ 0 0
$$832$$ −4.74317e6 −0.237553
$$833$$ −2.89080e6 −0.144346
$$834$$ 0 0
$$835$$ 2.40650e7 1.19446
$$836$$ −1.98829e7 −0.983929
$$837$$ 0 0
$$838$$ 2.25080e7 1.10720
$$839$$ 1.61258e6 0.0790891 0.0395445 0.999218i $$-0.487409\pi$$
0.0395445 + 0.999218i $$0.487409\pi$$
$$840$$ 0 0
$$841$$ −7.45015e6 −0.363225
$$842$$ −1.77068e7 −0.860718
$$843$$ 0 0
$$844$$ 3.72243e6 0.179875
$$845$$ −4.26655e7 −2.05558
$$846$$ 0 0
$$847$$ −2.93260e6 −0.140457
$$848$$ 1.11974e6 0.0534723
$$849$$ 0 0
$$850$$ 5.72622e6 0.271845
$$851$$ −7.70882e6 −0.364892
$$852$$ 0 0
$$853$$ −3.44919e7 −1.62310 −0.811548 0.584286i $$-0.801375\pi$$
−0.811548 + 0.584286i $$0.801375\pi$$
$$854$$ −3.83023e6 −0.179713
$$855$$ 0 0
$$856$$ 1.46047e7 0.681251
$$857$$ 1.85487e7 0.862704 0.431352 0.902184i $$-0.358037\pi$$
0.431352 + 0.902184i $$0.358037\pi$$
$$858$$ 0 0
$$859$$ −2.56435e7 −1.18575 −0.592877 0.805293i $$-0.702008\pi$$
−0.592877 + 0.805293i $$0.702008\pi$$
$$860$$ 9.49837e6 0.437928
$$861$$ 0 0
$$862$$ 5.76650e6 0.264329
$$863$$ 2.49899e7 1.14219 0.571093 0.820885i $$-0.306519\pi$$
0.571093 + 0.820885i $$0.306519\pi$$
$$864$$ 0 0
$$865$$ −1.61522e7 −0.733993
$$866$$ −1.55865e7 −0.706241
$$867$$ 0 0
$$868$$ −4.40294e6 −0.198355
$$869$$ −2.34793e7 −1.05472
$$870$$ 0 0
$$871$$ −6.29118e7 −2.80987
$$872$$ −6.09190e6 −0.271308
$$873$$ 0 0
$$874$$ −1.25854e7 −0.557301
$$875$$ −9.30098e6 −0.410685
$$876$$ 0 0
$$877$$ −3.46337e7 −1.52055 −0.760274 0.649602i $$-0.774935\pi$$
−0.760274 + 0.649602i $$0.774935\pi$$
$$878$$ 2.04483e7 0.895201
$$879$$ 0 0
$$880$$ −5.29408e6 −0.230454
$$881$$ −2.92434e7 −1.26937 −0.634685 0.772771i $$-0.718870\pi$$
−0.634685 + 0.772771i $$0.718870\pi$$
$$882$$ 0 0
$$883$$ −3.76532e7 −1.62518 −0.812588 0.582839i $$-0.801942\pi$$
−0.812588 + 0.582839i $$0.801942\pi$$
$$884$$ 2.23077e7 0.960117
$$885$$ 0 0
$$886$$ −2.17628e7 −0.931388
$$887$$ 2.08201e6 0.0888534 0.0444267 0.999013i $$-0.485854\pi$$
0.0444267 + 0.999013i $$0.485854\pi$$
$$888$$ 0 0
$$889$$ 8.21495e6 0.348618
$$890$$ 1.77366e7 0.750576
$$891$$ 0 0
$$892$$ 2.67430e6 0.112538
$$893$$ −4.85756e7 −2.03840
$$894$$ 0 0
$$895$$ −3.91116e6 −0.163210
$$896$$ −802816. −0.0334077
$$897$$ 0 0
$$898$$ 5.26098e6 0.217709
$$899$$ −2.02962e7 −0.837560
$$900$$ 0 0
$$901$$ −5.26630e6 −0.216119
$$902$$ −5.36928e6 −0.219735
$$903$$ 0 0
$$904$$ 1.67131e7 0.680198
$$905$$ 3.44132e7 1.39670
$$906$$ 0 0
$$907$$ 1.62350e7 0.655291 0.327645 0.944801i $$-0.393745\pi$$
0.327645 + 0.944801i $$0.393745\pi$$
$$908$$ −6.65165e6 −0.267741
$$909$$ 0 0
$$910$$ −9.98659e6 −0.399773
$$911$$ 2.58656e7 1.03259 0.516294 0.856412i $$-0.327311\pi$$
0.516294 + 0.856412i $$0.327311\pi$$
$$912$$ 0 0
$$913$$ 1.26656e7 0.502860
$$914$$ 1.11042e7 0.439664
$$915$$ 0 0
$$916$$ 7.57571e6 0.298322
$$917$$ −4.07033e6 −0.159848
$$918$$ 0 0
$$919$$ −1.23266e7 −0.481453 −0.240726 0.970593i $$-0.577386\pi$$
−0.240726 + 0.970593i $$0.577386\pi$$
$$920$$ −3.35104e6 −0.130530
$$921$$ 0 0
$$922$$ −552320. −0.0213975
$$923$$ −1.24253e7 −0.480069
$$924$$ 0 0
$$925$$ 7.70234e6 0.295984
$$926$$ −1.45630e6 −0.0558116
$$927$$ 0 0
$$928$$ −3.70074e6 −0.141065
$$929$$ 4.15128e7 1.57813 0.789064 0.614310i $$-0.210566\pi$$
0.789064 + 0.614310i $$0.210566\pi$$
$$930$$ 0 0
$$931$$ −6.34824e6 −0.240038
$$932$$ 2.49047e7 0.939166
$$933$$ 0 0
$$934$$ 2.29559e7 0.861046
$$935$$ 2.48987e7 0.931425
$$936$$ 0 0
$$937$$ 2.26895e7 0.844260 0.422130 0.906535i $$-0.361283\pi$$
0.422130 + 0.906535i $$0.361283\pi$$
$$938$$ −1.06483e7 −0.395160
$$939$$ 0 0
$$940$$ −1.29339e7 −0.477430
$$941$$ −1.58213e7 −0.582464 −0.291232 0.956652i $$-0.594065\pi$$
−0.291232 + 0.956652i $$0.594065\pi$$
$$942$$ 0 0
$$943$$ −3.39864e6 −0.124459
$$944$$ −7.74349e6 −0.282818
$$945$$ 0 0
$$946$$ −2.53650e7 −0.921523
$$947$$ −5.17579e7 −1.87543 −0.937716 0.347402i $$-0.887064\pi$$
−0.937716 + 0.347402i $$0.887064\pi$$
$$948$$ 0 0
$$949$$ −4.09631e7 −1.47648
$$950$$ 1.25749e7 0.452058
$$951$$ 0 0
$$952$$ 3.77574e6 0.135024
$$953$$ −2.29818e7 −0.819695 −0.409848 0.912154i $$-0.634418\pi$$
−0.409848 + 0.912154i $$0.634418\pi$$
$$954$$ 0 0
$$955$$ 3.35874e7 1.19170
$$956$$ −1.04942e7 −0.371370
$$957$$ 0 0
$$958$$ −2.20798e7 −0.777288
$$959$$ 7.95045e6 0.279155
$$960$$ 0 0
$$961$$ 2.91030e6 0.101655
$$962$$ 3.00061e7 1.04537
$$963$$ 0 0
$$964$$ −1.42316e7 −0.493243
$$965$$ 1.65881e7 0.573427
$$966$$ 0 0
$$967$$ 3.20783e7 1.10318 0.551588 0.834117i $$-0.314022\pi$$
0.551588 + 0.834117i $$0.314022\pi$$
$$968$$ 3.83034e6 0.131386
$$969$$ 0 0
$$970$$ −1.35756e7 −0.463265
$$971$$ −7.31101e6 −0.248845 −0.124423 0.992229i $$-0.539708\pi$$
−0.124423 + 0.992229i $$0.539708\pi$$
$$972$$ 0 0
$$973$$ 2.88336e6 0.0976374
$$974$$ 1.69209e7 0.571513
$$975$$ 0 0
$$976$$ 5.00275e6 0.168106
$$977$$ −7.58600e6 −0.254259 −0.127129 0.991886i $$-0.540576\pi$$
−0.127129 + 0.991886i $$0.540576\pi$$
$$978$$ 0 0
$$979$$ −4.73647e7 −1.57942
$$980$$ −1.69030e6 −0.0562211
$$981$$ 0 0
$$982$$ 2.88569e7 0.954929
$$983$$ 2.47823e7 0.818007 0.409004 0.912533i $$-0.365876\pi$$
0.409004 + 0.912533i $$0.365876\pi$$
$$984$$ 0 0
$$985$$ −3.02183e6 −0.0992384
$$986$$ 1.74050e7 0.570141
$$987$$ 0 0
$$988$$ 4.89880e7 1.59661
$$989$$ −1.60555e7 −0.521954
$$990$$ 0 0
$$991$$ −7.63530e6 −0.246969 −0.123484 0.992347i $$-0.539407\pi$$
−0.123484 + 0.992347i $$0.539407\pi$$
$$992$$ 5.75078e6 0.185544
$$993$$ 0 0
$$994$$ −2.10308e6 −0.0675134
$$995$$ −8.03334e6 −0.257240
$$996$$ 0 0
$$997$$ −2.89785e7 −0.923289 −0.461644 0.887065i $$-0.652740\pi$$
−0.461644 + 0.887065i $$0.652740\pi$$
$$998$$ −899216. −0.0285784
$$999$$ 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 126.6.a.h.1.1 1
3.2 odd 2 42.6.a.b.1.1 1
4.3 odd 2 1008.6.a.g.1.1 1
7.6 odd 2 882.6.a.v.1.1 1
12.11 even 2 336.6.a.o.1.1 1
15.2 even 4 1050.6.g.l.799.1 2
15.8 even 4 1050.6.g.l.799.2 2
15.14 odd 2 1050.6.a.o.1.1 1
21.2 odd 6 294.6.e.o.67.1 2
21.5 even 6 294.6.e.k.67.1 2
21.11 odd 6 294.6.e.o.79.1 2
21.17 even 6 294.6.e.k.79.1 2
21.20 even 2 294.6.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.a.b.1.1 1 3.2 odd 2
126.6.a.h.1.1 1 1.1 even 1 trivial
294.6.a.f.1.1 1 21.20 even 2
294.6.e.k.67.1 2 21.5 even 6
294.6.e.k.79.1 2 21.17 even 6
294.6.e.o.67.1 2 21.2 odd 6
294.6.e.o.79.1 2 21.11 odd 6
336.6.a.o.1.1 1 12.11 even 2
882.6.a.v.1.1 1 7.6 odd 2
1008.6.a.g.1.1 1 4.3 odd 2
1050.6.a.o.1.1 1 15.14 odd 2
1050.6.g.l.799.1 2 15.2 even 4
1050.6.g.l.799.2 2 15.8 even 4