# Properties

 Label 126.6.a.j.1.1 Level $126$ Weight $6$ Character 126.1 Self dual yes Analytic conductor $20.208$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} +16.0000 q^{4} -26.0000 q^{5} -49.0000 q^{7} +64.0000 q^{8} +O(q^{10})$$ $$q+4.00000 q^{2} +16.0000 q^{4} -26.0000 q^{5} -49.0000 q^{7} +64.0000 q^{8} -104.000 q^{10} +470.000 q^{11} +642.000 q^{13} -196.000 q^{14} +256.000 q^{16} +1010.00 q^{17} +1532.00 q^{19} -416.000 q^{20} +1880.00 q^{22} -430.000 q^{23} -2449.00 q^{25} +2568.00 q^{26} -784.000 q^{28} +6736.00 q^{29} +2268.00 q^{31} +1024.00 q^{32} +4040.00 q^{34} +1274.00 q^{35} -9574.00 q^{37} +6128.00 q^{38} -1664.00 q^{40} +14406.0 q^{41} -9748.00 q^{43} +7520.00 q^{44} -1720.00 q^{46} +17004.0 q^{47} +2401.00 q^{49} -9796.00 q^{50} +10272.0 q^{52} +7596.00 q^{53} -12220.0 q^{55} -3136.00 q^{56} +26944.0 q^{58} -18908.0 q^{59} -36762.0 q^{61} +9072.00 q^{62} +4096.00 q^{64} -16692.0 q^{65} -36788.0 q^{67} +16160.0 q^{68} +5096.00 q^{70} +18326.0 q^{71} +36382.0 q^{73} -38296.0 q^{74} +24512.0 q^{76} -23030.0 q^{77} +29784.0 q^{79} -6656.00 q^{80} +57624.0 q^{82} -28240.0 q^{83} -26260.0 q^{85} -38992.0 q^{86} +30080.0 q^{88} -75954.0 q^{89} -31458.0 q^{91} -6880.00 q^{92} +68016.0 q^{94} -39832.0 q^{95} -80690.0 q^{97} +9604.00 q^{98} +O(q^{100})$$

## 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
$$5$$ −26.0000 −0.465102 −0.232551 0.972584i $$-0.574707\pi$$
−0.232551 + 0.972584i $$0.574707\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 64.0000 0.353553
$$9$$ 0 0
$$10$$ −104.000 −0.328877
$$11$$ 470.000 1.17116 0.585580 0.810615i $$-0.300867\pi$$
0.585580 + 0.810615i $$0.300867\pi$$
$$12$$ 0 0
$$13$$ 642.000 1.05360 0.526801 0.849989i $$-0.323391\pi$$
0.526801 + 0.849989i $$0.323391\pi$$
$$14$$ −196.000 −0.267261
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 1010.00 0.847616 0.423808 0.905752i $$-0.360693\pi$$
0.423808 + 0.905752i $$0.360693\pi$$
$$18$$ 0 0
$$19$$ 1532.00 0.973587 0.486793 0.873517i $$-0.338166\pi$$
0.486793 + 0.873517i $$0.338166\pi$$
$$20$$ −416.000 −0.232551
$$21$$ 0 0
$$22$$ 1880.00 0.828135
$$23$$ −430.000 −0.169492 −0.0847459 0.996403i $$-0.527008\pi$$
−0.0847459 + 0.996403i $$0.527008\pi$$
$$24$$ 0 0
$$25$$ −2449.00 −0.783680
$$26$$ 2568.00 0.745009
$$27$$ 0 0
$$28$$ −784.000 −0.188982
$$29$$ 6736.00 1.48733 0.743665 0.668553i $$-0.233086\pi$$
0.743665 + 0.668553i $$0.233086\pi$$
$$30$$ 0 0
$$31$$ 2268.00 0.423876 0.211938 0.977283i $$-0.432023\pi$$
0.211938 + 0.977283i $$0.432023\pi$$
$$32$$ 1024.00 0.176777
$$33$$ 0 0
$$34$$ 4040.00 0.599355
$$35$$ 1274.00 0.175792
$$36$$ 0 0
$$37$$ −9574.00 −1.14971 −0.574856 0.818255i $$-0.694942\pi$$
−0.574856 + 0.818255i $$0.694942\pi$$
$$38$$ 6128.00 0.688430
$$39$$ 0 0
$$40$$ −1664.00 −0.164438
$$41$$ 14406.0 1.33839 0.669197 0.743085i $$-0.266638\pi$$
0.669197 + 0.743085i $$0.266638\pi$$
$$42$$ 0 0
$$43$$ −9748.00 −0.803978 −0.401989 0.915644i $$-0.631681\pi$$
−0.401989 + 0.915644i $$0.631681\pi$$
$$44$$ 7520.00 0.585580
$$45$$ 0 0
$$46$$ −1720.00 −0.119849
$$47$$ 17004.0 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ −9796.00 −0.554145
$$51$$ 0 0
$$52$$ 10272.0 0.526801
$$53$$ 7596.00 0.371446 0.185723 0.982602i $$-0.440537\pi$$
0.185723 + 0.982602i $$0.440537\pi$$
$$54$$ 0 0
$$55$$ −12220.0 −0.544709
$$56$$ −3136.00 −0.133631
$$57$$ 0 0
$$58$$ 26944.0 1.05170
$$59$$ −18908.0 −0.707157 −0.353578 0.935405i $$-0.615035\pi$$
−0.353578 + 0.935405i $$0.615035\pi$$
$$60$$ 0 0
$$61$$ −36762.0 −1.26495 −0.632477 0.774579i $$-0.717962\pi$$
−0.632477 + 0.774579i $$0.717962\pi$$
$$62$$ 9072.00 0.299726
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ −16692.0 −0.490033
$$66$$ 0 0
$$67$$ −36788.0 −1.00120 −0.500598 0.865680i $$-0.666887\pi$$
−0.500598 + 0.865680i $$0.666887\pi$$
$$68$$ 16160.0 0.423808
$$69$$ 0 0
$$70$$ 5096.00 0.124304
$$71$$ 18326.0 0.431441 0.215721 0.976455i $$-0.430790\pi$$
0.215721 + 0.976455i $$0.430790\pi$$
$$72$$ 0 0
$$73$$ 36382.0 0.799060 0.399530 0.916720i $$-0.369173\pi$$
0.399530 + 0.916720i $$0.369173\pi$$
$$74$$ −38296.0 −0.812969
$$75$$ 0 0
$$76$$ 24512.0 0.486793
$$77$$ −23030.0 −0.442657
$$78$$ 0 0
$$79$$ 29784.0 0.536927 0.268464 0.963290i $$-0.413484\pi$$
0.268464 + 0.963290i $$0.413484\pi$$
$$80$$ −6656.00 −0.116276
$$81$$ 0 0
$$82$$ 57624.0 0.946387
$$83$$ −28240.0 −0.449955 −0.224978 0.974364i $$-0.572231\pi$$
−0.224978 + 0.974364i $$0.572231\pi$$
$$84$$ 0 0
$$85$$ −26260.0 −0.394228
$$86$$ −38992.0 −0.568499
$$87$$ 0 0
$$88$$ 30080.0 0.414068
$$89$$ −75954.0 −1.01643 −0.508213 0.861232i $$-0.669694\pi$$
−0.508213 + 0.861232i $$0.669694\pi$$
$$90$$ 0 0
$$91$$ −31458.0 −0.398224
$$92$$ −6880.00 −0.0847459
$$93$$ 0 0
$$94$$ 68016.0 0.793947
$$95$$ −39832.0 −0.452817
$$96$$ 0 0
$$97$$ −80690.0 −0.870744 −0.435372 0.900251i $$-0.643383\pi$$
−0.435372 + 0.900251i $$0.643383\pi$$
$$98$$ 9604.00 0.101015
$$99$$ 0 0
$$100$$ −39184.0 −0.391840
$$101$$ 31306.0 0.305368 0.152684 0.988275i $$-0.451208\pi$$
0.152684 + 0.988275i $$0.451208\pi$$
$$102$$ 0 0
$$103$$ 102908. 0.955776 0.477888 0.878421i $$-0.341402\pi$$
0.477888 + 0.878421i $$0.341402\pi$$
$$104$$ 41088.0 0.372505
$$105$$ 0 0
$$106$$ 30384.0 0.262652
$$107$$ −172482. −1.45641 −0.728206 0.685358i $$-0.759646\pi$$
−0.728206 + 0.685358i $$0.759646\pi$$
$$108$$ 0 0
$$109$$ −135470. −1.09214 −0.546068 0.837741i $$-0.683876\pi$$
−0.546068 + 0.837741i $$0.683876\pi$$
$$110$$ −48880.0 −0.385167
$$111$$ 0 0
$$112$$ −12544.0 −0.0944911
$$113$$ −135632. −0.999231 −0.499616 0.866247i $$-0.666525\pi$$
−0.499616 + 0.866247i $$0.666525\pi$$
$$114$$ 0 0
$$115$$ 11180.0 0.0788310
$$116$$ 107776. 0.743665
$$117$$ 0 0
$$118$$ −75632.0 −0.500035
$$119$$ −49490.0 −0.320369
$$120$$ 0 0
$$121$$ 59849.0 0.371615
$$122$$ −147048. −0.894457
$$123$$ 0 0
$$124$$ 36288.0 0.211938
$$125$$ 144924. 0.829593
$$126$$ 0 0
$$127$$ −275976. −1.51832 −0.759158 0.650907i $$-0.774389\pi$$
−0.759158 + 0.650907i $$0.774389\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 0 0
$$130$$ −66768.0 −0.346505
$$131$$ 50488.0 0.257045 0.128523 0.991707i $$-0.458976\pi$$
0.128523 + 0.991707i $$0.458976\pi$$
$$132$$ 0 0
$$133$$ −75068.0 −0.367981
$$134$$ −147152. −0.707953
$$135$$ 0 0
$$136$$ 64640.0 0.299677
$$137$$ 137212. 0.624584 0.312292 0.949986i $$-0.398903\pi$$
0.312292 + 0.949986i $$0.398903\pi$$
$$138$$ 0 0
$$139$$ 21040.0 0.0923653 0.0461826 0.998933i $$-0.485294\pi$$
0.0461826 + 0.998933i $$0.485294\pi$$
$$140$$ 20384.0 0.0878960
$$141$$ 0 0
$$142$$ 73304.0 0.305075
$$143$$ 301740. 1.23394
$$144$$ 0 0
$$145$$ −175136. −0.691760
$$146$$ 145528. 0.565021
$$147$$ 0 0
$$148$$ −153184. −0.574856
$$149$$ 240468. 0.887343 0.443672 0.896189i $$-0.353676\pi$$
0.443672 + 0.896189i $$0.353676\pi$$
$$150$$ 0 0
$$151$$ 325048. 1.16013 0.580063 0.814572i $$-0.303028\pi$$
0.580063 + 0.814572i $$0.303028\pi$$
$$152$$ 98048.0 0.344215
$$153$$ 0 0
$$154$$ −92120.0 −0.313006
$$155$$ −58968.0 −0.197146
$$156$$ 0 0
$$157$$ 537734. 1.74108 0.870539 0.492099i $$-0.163770\pi$$
0.870539 + 0.492099i $$0.163770\pi$$
$$158$$ 119136. 0.379665
$$159$$ 0 0
$$160$$ −26624.0 −0.0822192
$$161$$ 21070.0 0.0640619
$$162$$ 0 0
$$163$$ 403748. 1.19026 0.595129 0.803630i $$-0.297101\pi$$
0.595129 + 0.803630i $$0.297101\pi$$
$$164$$ 230496. 0.669197
$$165$$ 0 0
$$166$$ −112960. −0.318167
$$167$$ 702412. 1.94895 0.974475 0.224496i $$-0.0720734\pi$$
0.974475 + 0.224496i $$0.0720734\pi$$
$$168$$ 0 0
$$169$$ 40871.0 0.110077
$$170$$ −105040. −0.278761
$$171$$ 0 0
$$172$$ −155968. −0.401989
$$173$$ −726926. −1.84661 −0.923304 0.384069i $$-0.874523\pi$$
−0.923304 + 0.384069i $$0.874523\pi$$
$$174$$ 0 0
$$175$$ 120001. 0.296203
$$176$$ 120320. 0.292790
$$177$$ 0 0
$$178$$ −303816. −0.718721
$$179$$ 722418. 1.68522 0.842609 0.538526i $$-0.181019\pi$$
0.842609 + 0.538526i $$0.181019\pi$$
$$180$$ 0 0
$$181$$ −333486. −0.756626 −0.378313 0.925678i $$-0.623496\pi$$
−0.378313 + 0.925678i $$0.623496\pi$$
$$182$$ −125832. −0.281587
$$183$$ 0 0
$$184$$ −27520.0 −0.0599244
$$185$$ 248924. 0.534734
$$186$$ 0 0
$$187$$ 474700. 0.992694
$$188$$ 272064. 0.561405
$$189$$ 0 0
$$190$$ −159328. −0.320190
$$191$$ −669138. −1.32719 −0.663594 0.748093i $$-0.730970\pi$$
−0.663594 + 0.748093i $$0.730970\pi$$
$$192$$ 0 0
$$193$$ −115066. −0.222359 −0.111179 0.993800i $$-0.535463\pi$$
−0.111179 + 0.993800i $$0.535463\pi$$
$$194$$ −322760. −0.615709
$$195$$ 0 0
$$196$$ 38416.0 0.0714286
$$197$$ −213364. −0.391702 −0.195851 0.980634i $$-0.562747\pi$$
−0.195851 + 0.980634i $$0.562747\pi$$
$$198$$ 0 0
$$199$$ 795296. 1.42363 0.711813 0.702369i $$-0.247874\pi$$
0.711813 + 0.702369i $$0.247874\pi$$
$$200$$ −156736. −0.277073
$$201$$ 0 0
$$202$$ 125224. 0.215928
$$203$$ −330064. −0.562158
$$204$$ 0 0
$$205$$ −374556. −0.622490
$$206$$ 411632. 0.675836
$$207$$ 0 0
$$208$$ 164352. 0.263401
$$209$$ 720040. 1.14023
$$210$$ 0 0
$$211$$ 218468. 0.337817 0.168909 0.985632i $$-0.445976\pi$$
0.168909 + 0.985632i $$0.445976\pi$$
$$212$$ 121536. 0.185723
$$213$$ 0 0
$$214$$ −689928. −1.02984
$$215$$ 253448. 0.373932
$$216$$ 0 0
$$217$$ −111132. −0.160210
$$218$$ −541880. −0.772257
$$219$$ 0 0
$$220$$ −195520. −0.272354
$$221$$ 648420. 0.893050
$$222$$ 0 0
$$223$$ 656888. 0.884564 0.442282 0.896876i $$-0.354169\pi$$
0.442282 + 0.896876i $$0.354169\pi$$
$$224$$ −50176.0 −0.0668153
$$225$$ 0 0
$$226$$ −542528. −0.706563
$$227$$ −971532. −1.25139 −0.625695 0.780068i $$-0.715184\pi$$
−0.625695 + 0.780068i $$0.715184\pi$$
$$228$$ 0 0
$$229$$ −459350. −0.578835 −0.289418 0.957203i $$-0.593462\pi$$
−0.289418 + 0.957203i $$0.593462\pi$$
$$230$$ 44720.0 0.0557420
$$231$$ 0 0
$$232$$ 431104. 0.525850
$$233$$ −1.23704e6 −1.49277 −0.746384 0.665515i $$-0.768212\pi$$
−0.746384 + 0.665515i $$0.768212\pi$$
$$234$$ 0 0
$$235$$ −442104. −0.522222
$$236$$ −302528. −0.353578
$$237$$ 0 0
$$238$$ −197960. −0.226535
$$239$$ −1.53433e6 −1.73749 −0.868746 0.495258i $$-0.835074\pi$$
−0.868746 + 0.495258i $$0.835074\pi$$
$$240$$ 0 0
$$241$$ −1.41990e6 −1.57476 −0.787380 0.616468i $$-0.788563\pi$$
−0.787380 + 0.616468i $$0.788563\pi$$
$$242$$ 239396. 0.262772
$$243$$ 0 0
$$244$$ −588192. −0.632477
$$245$$ −62426.0 −0.0664432
$$246$$ 0 0
$$247$$ 983544. 1.02577
$$248$$ 145152. 0.149863
$$249$$ 0 0
$$250$$ 579696. 0.586611
$$251$$ 1.61197e6 1.61500 0.807501 0.589866i $$-0.200819\pi$$
0.807501 + 0.589866i $$0.200819\pi$$
$$252$$ 0 0
$$253$$ −202100. −0.198502
$$254$$ −1.10390e6 −1.07361
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ −851442. −0.804123 −0.402061 0.915613i $$-0.631706\pi$$
−0.402061 + 0.915613i $$0.631706\pi$$
$$258$$ 0 0
$$259$$ 469126. 0.434550
$$260$$ −267072. −0.245016
$$261$$ 0 0
$$262$$ 201952. 0.181759
$$263$$ −1.06738e6 −0.951548 −0.475774 0.879568i $$-0.657832\pi$$
−0.475774 + 0.879568i $$0.657832\pi$$
$$264$$ 0 0
$$265$$ −197496. −0.172760
$$266$$ −300272. −0.260202
$$267$$ 0 0
$$268$$ −588608. −0.500598
$$269$$ 871870. 0.734634 0.367317 0.930096i $$-0.380276\pi$$
0.367317 + 0.930096i $$0.380276\pi$$
$$270$$ 0 0
$$271$$ −1.40737e6 −1.16409 −0.582044 0.813157i $$-0.697747\pi$$
−0.582044 + 0.813157i $$0.697747\pi$$
$$272$$ 258560. 0.211904
$$273$$ 0 0
$$274$$ 548848. 0.441647
$$275$$ −1.15103e6 −0.917814
$$276$$ 0 0
$$277$$ 888782. 0.695979 0.347989 0.937499i $$-0.386865\pi$$
0.347989 + 0.937499i $$0.386865\pi$$
$$278$$ 84160.0 0.0653121
$$279$$ 0 0
$$280$$ 81536.0 0.0621519
$$281$$ −2.00038e6 −1.51129 −0.755643 0.654984i $$-0.772676\pi$$
−0.755643 + 0.654984i $$0.772676\pi$$
$$282$$ 0 0
$$283$$ 150460. 0.111675 0.0558374 0.998440i $$-0.482217\pi$$
0.0558374 + 0.998440i $$0.482217\pi$$
$$284$$ 293216. 0.215721
$$285$$ 0 0
$$286$$ 1.20696e6 0.872525
$$287$$ −705894. −0.505865
$$288$$ 0 0
$$289$$ −399757. −0.281547
$$290$$ −700544. −0.489148
$$291$$ 0 0
$$292$$ 582112. 0.399530
$$293$$ 2.73669e6 1.86233 0.931165 0.364599i $$-0.118794\pi$$
0.931165 + 0.364599i $$0.118794\pi$$
$$294$$ 0 0
$$295$$ 491608. 0.328900
$$296$$ −612736. −0.406485
$$297$$ 0 0
$$298$$ 961872. 0.627446
$$299$$ −276060. −0.178577
$$300$$ 0 0
$$301$$ 477652. 0.303875
$$302$$ 1.30019e6 0.820333
$$303$$ 0 0
$$304$$ 392192. 0.243397
$$305$$ 955812. 0.588333
$$306$$ 0 0
$$307$$ 714436. 0.432631 0.216315 0.976324i $$-0.430596\pi$$
0.216315 + 0.976324i $$0.430596\pi$$
$$308$$ −368480. −0.221328
$$309$$ 0 0
$$310$$ −235872. −0.139403
$$311$$ 1.34019e6 0.785714 0.392857 0.919599i $$-0.371487\pi$$
0.392857 + 0.919599i $$0.371487\pi$$
$$312$$ 0 0
$$313$$ −2.59201e6 −1.49547 −0.747733 0.664000i $$-0.768858\pi$$
−0.747733 + 0.664000i $$0.768858\pi$$
$$314$$ 2.15094e6 1.23113
$$315$$ 0 0
$$316$$ 476544. 0.268464
$$317$$ −1.54753e6 −0.864951 −0.432475 0.901646i $$-0.642360\pi$$
−0.432475 + 0.901646i $$0.642360\pi$$
$$318$$ 0 0
$$319$$ 3.16592e6 1.74190
$$320$$ −106496. −0.0581378
$$321$$ 0 0
$$322$$ 84280.0 0.0452986
$$323$$ 1.54732e6 0.825228
$$324$$ 0 0
$$325$$ −1.57226e6 −0.825687
$$326$$ 1.61499e6 0.841640
$$327$$ 0 0
$$328$$ 921984. 0.473194
$$329$$ −833196. −0.424382
$$330$$ 0 0
$$331$$ −672892. −0.337579 −0.168789 0.985652i $$-0.553986\pi$$
−0.168789 + 0.985652i $$0.553986\pi$$
$$332$$ −451840. −0.224978
$$333$$ 0 0
$$334$$ 2.80965e6 1.37812
$$335$$ 956488. 0.465658
$$336$$ 0 0
$$337$$ −3.00127e6 −1.43956 −0.719780 0.694202i $$-0.755757\pi$$
−0.719780 + 0.694202i $$0.755757\pi$$
$$338$$ 163484. 0.0778365
$$339$$ 0 0
$$340$$ −420160. −0.197114
$$341$$ 1.06596e6 0.496427
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ −623872. −0.284249
$$345$$ 0 0
$$346$$ −2.90770e6 −1.30575
$$347$$ 3.29114e6 1.46731 0.733656 0.679521i $$-0.237812\pi$$
0.733656 + 0.679521i $$0.237812\pi$$
$$348$$ 0 0
$$349$$ −2.96059e6 −1.30111 −0.650557 0.759458i $$-0.725464\pi$$
−0.650557 + 0.759458i $$0.725464\pi$$
$$350$$ 480004. 0.209447
$$351$$ 0 0
$$352$$ 481280. 0.207034
$$353$$ 1.65899e6 0.708611 0.354306 0.935130i $$-0.384717\pi$$
0.354306 + 0.935130i $$0.384717\pi$$
$$354$$ 0 0
$$355$$ −476476. −0.200664
$$356$$ −1.21526e6 −0.508213
$$357$$ 0 0
$$358$$ 2.88967e6 1.19163
$$359$$ 2.33841e6 0.957603 0.478801 0.877923i $$-0.341071\pi$$
0.478801 + 0.877923i $$0.341071\pi$$
$$360$$ 0 0
$$361$$ −129075. −0.0521284
$$362$$ −1.33394e6 −0.535015
$$363$$ 0 0
$$364$$ −503328. −0.199112
$$365$$ −945932. −0.371645
$$366$$ 0 0
$$367$$ −1.77875e6 −0.689367 −0.344683 0.938719i $$-0.612014\pi$$
−0.344683 + 0.938719i $$0.612014\pi$$
$$368$$ −110080. −0.0423730
$$369$$ 0 0
$$370$$ 995696. 0.378114
$$371$$ −372204. −0.140393
$$372$$ 0 0
$$373$$ −1.38079e6 −0.513874 −0.256937 0.966428i $$-0.582713\pi$$
−0.256937 + 0.966428i $$0.582713\pi$$
$$374$$ 1.89880e6 0.701940
$$375$$ 0 0
$$376$$ 1.08826e6 0.396973
$$377$$ 4.32451e6 1.56705
$$378$$ 0 0
$$379$$ 4.18575e6 1.49684 0.748419 0.663226i $$-0.230813\pi$$
0.748419 + 0.663226i $$0.230813\pi$$
$$380$$ −637312. −0.226409
$$381$$ 0 0
$$382$$ −2.67655e6 −0.938463
$$383$$ −586216. −0.204202 −0.102101 0.994774i $$-0.532557\pi$$
−0.102101 + 0.994774i $$0.532557\pi$$
$$384$$ 0 0
$$385$$ 598780. 0.205881
$$386$$ −460264. −0.157231
$$387$$ 0 0
$$388$$ −1.29104e6 −0.435372
$$389$$ −1.50212e6 −0.503304 −0.251652 0.967818i $$-0.580974\pi$$
−0.251652 + 0.967818i $$0.580974\pi$$
$$390$$ 0 0
$$391$$ −434300. −0.143664
$$392$$ 153664. 0.0505076
$$393$$ 0 0
$$394$$ −853456. −0.276975
$$395$$ −774384. −0.249726
$$396$$ 0 0
$$397$$ −2.01988e6 −0.643205 −0.321603 0.946875i $$-0.604222\pi$$
−0.321603 + 0.946875i $$0.604222\pi$$
$$398$$ 3.18118e6 1.00666
$$399$$ 0 0
$$400$$ −626944. −0.195920
$$401$$ −2.67558e6 −0.830916 −0.415458 0.909612i $$-0.636379\pi$$
−0.415458 + 0.909612i $$0.636379\pi$$
$$402$$ 0 0
$$403$$ 1.45606e6 0.446597
$$404$$ 500896. 0.152684
$$405$$ 0 0
$$406$$ −1.32026e6 −0.397505
$$407$$ −4.49978e6 −1.34650
$$408$$ 0 0
$$409$$ −1.32378e6 −0.391297 −0.195649 0.980674i $$-0.562681\pi$$
−0.195649 + 0.980674i $$0.562681\pi$$
$$410$$ −1.49822e6 −0.440167
$$411$$ 0 0
$$412$$ 1.64653e6 0.477888
$$413$$ 926492. 0.267280
$$414$$ 0 0
$$415$$ 734240. 0.209275
$$416$$ 657408. 0.186252
$$417$$ 0 0
$$418$$ 2.88016e6 0.806261
$$419$$ −451212. −0.125558 −0.0627792 0.998027i $$-0.519996\pi$$
−0.0627792 + 0.998027i $$0.519996\pi$$
$$420$$ 0 0
$$421$$ −3.88005e6 −1.06692 −0.533460 0.845825i $$-0.679109\pi$$
−0.533460 + 0.845825i $$0.679109\pi$$
$$422$$ 873872. 0.238873
$$423$$ 0 0
$$424$$ 486144. 0.131326
$$425$$ −2.47349e6 −0.664260
$$426$$ 0 0
$$427$$ 1.80134e6 0.478107
$$428$$ −2.75971e6 −0.728206
$$429$$ 0 0
$$430$$ 1.01379e6 0.264410
$$431$$ 5.31469e6 1.37811 0.689056 0.724708i $$-0.258025\pi$$
0.689056 + 0.724708i $$0.258025\pi$$
$$432$$ 0 0
$$433$$ 2.68951e6 0.689373 0.344686 0.938718i $$-0.387985\pi$$
0.344686 + 0.938718i $$0.387985\pi$$
$$434$$ −444528. −0.113286
$$435$$ 0 0
$$436$$ −2.16752e6 −0.546068
$$437$$ −658760. −0.165015
$$438$$ 0 0
$$439$$ −643392. −0.159336 −0.0796681 0.996821i $$-0.525386\pi$$
−0.0796681 + 0.996821i $$0.525386\pi$$
$$440$$ −782080. −0.192584
$$441$$ 0 0
$$442$$ 2.59368e6 0.631482
$$443$$ −2.53601e6 −0.613961 −0.306981 0.951716i $$-0.599319\pi$$
−0.306981 + 0.951716i $$0.599319\pi$$
$$444$$ 0 0
$$445$$ 1.97480e6 0.472742
$$446$$ 2.62755e6 0.625481
$$447$$ 0 0
$$448$$ −200704. −0.0472456
$$449$$ 4.79178e6 1.12171 0.560856 0.827913i $$-0.310472\pi$$
0.560856 + 0.827913i $$0.310472\pi$$
$$450$$ 0 0
$$451$$ 6.77082e6 1.56747
$$452$$ −2.17011e6 −0.499616
$$453$$ 0 0
$$454$$ −3.88613e6 −0.884866
$$455$$ 817908. 0.185215
$$456$$ 0 0
$$457$$ 6.39833e6 1.43310 0.716549 0.697537i $$-0.245721\pi$$
0.716549 + 0.697537i $$0.245721\pi$$
$$458$$ −1.83740e6 −0.409298
$$459$$ 0 0
$$460$$ 178880. 0.0394155
$$461$$ −8.59151e6 −1.88286 −0.941428 0.337214i $$-0.890516\pi$$
−0.941428 + 0.337214i $$0.890516\pi$$
$$462$$ 0 0
$$463$$ −1.66558e6 −0.361089 −0.180544 0.983567i $$-0.557786\pi$$
−0.180544 + 0.983567i $$0.557786\pi$$
$$464$$ 1.72442e6 0.371832
$$465$$ 0 0
$$466$$ −4.94814e6 −1.05555
$$467$$ −7.88124e6 −1.67225 −0.836127 0.548536i $$-0.815185\pi$$
−0.836127 + 0.548536i $$0.815185\pi$$
$$468$$ 0 0
$$469$$ 1.80261e6 0.378417
$$470$$ −1.76842e6 −0.369266
$$471$$ 0 0
$$472$$ −1.21011e6 −0.250018
$$473$$ −4.58156e6 −0.941587
$$474$$ 0 0
$$475$$ −3.75187e6 −0.762981
$$476$$ −791840. −0.160184
$$477$$ 0 0
$$478$$ −6.13730e6 −1.22859
$$479$$ −2.25396e6 −0.448857 −0.224429 0.974491i $$-0.572052\pi$$
−0.224429 + 0.974491i $$0.572052\pi$$
$$480$$ 0 0
$$481$$ −6.14651e6 −1.21134
$$482$$ −5.67959e6 −1.11352
$$483$$ 0 0
$$484$$ 957584. 0.185808
$$485$$ 2.09794e6 0.404985
$$486$$ 0 0
$$487$$ 6.29194e6 1.20216 0.601080 0.799189i $$-0.294737\pi$$
0.601080 + 0.799189i $$0.294737\pi$$
$$488$$ −2.35277e6 −0.447229
$$489$$ 0 0
$$490$$ −249704. −0.0469824
$$491$$ −460542. −0.0862116 −0.0431058 0.999071i $$-0.513725\pi$$
−0.0431058 + 0.999071i $$0.513725\pi$$
$$492$$ 0 0
$$493$$ 6.80336e6 1.26068
$$494$$ 3.93418e6 0.725331
$$495$$ 0 0
$$496$$ 580608. 0.105969
$$497$$ −897974. −0.163070
$$498$$ 0 0
$$499$$ 9.33924e6 1.67904 0.839519 0.543331i $$-0.182837\pi$$
0.839519 + 0.543331i $$0.182837\pi$$
$$500$$ 2.31878e6 0.414797
$$501$$ 0 0
$$502$$ 6.44789e6 1.14198
$$503$$ −2.89982e6 −0.511036 −0.255518 0.966804i $$-0.582246\pi$$
−0.255518 + 0.966804i $$0.582246\pi$$
$$504$$ 0 0
$$505$$ −813956. −0.142028
$$506$$ −808400. −0.140362
$$507$$ 0 0
$$508$$ −4.41562e6 −0.759158
$$509$$ 1.97477e6 0.337849 0.168925 0.985629i $$-0.445971\pi$$
0.168925 + 0.985629i $$0.445971\pi$$
$$510$$ 0 0
$$511$$ −1.78272e6 −0.302016
$$512$$ 262144. 0.0441942
$$513$$ 0 0
$$514$$ −3.40577e6 −0.568601
$$515$$ −2.67561e6 −0.444533
$$516$$ 0 0
$$517$$ 7.99188e6 1.31499
$$518$$ 1.87650e6 0.307273
$$519$$ 0 0
$$520$$ −1.06829e6 −0.173253
$$521$$ −4.31576e6 −0.696567 −0.348283 0.937389i $$-0.613235\pi$$
−0.348283 + 0.937389i $$0.613235\pi$$
$$522$$ 0 0
$$523$$ −1.01477e7 −1.62223 −0.811116 0.584885i $$-0.801140\pi$$
−0.811116 + 0.584885i $$0.801140\pi$$
$$524$$ 807808. 0.128523
$$525$$ 0 0
$$526$$ −4.26953e6 −0.672846
$$527$$ 2.29068e6 0.359284
$$528$$ 0 0
$$529$$ −6.25144e6 −0.971273
$$530$$ −789984. −0.122160
$$531$$ 0 0
$$532$$ −1.20109e6 −0.183991
$$533$$ 9.24865e6 1.41013
$$534$$ 0 0
$$535$$ 4.48453e6 0.677380
$$536$$ −2.35443e6 −0.353976
$$537$$ 0 0
$$538$$ 3.48748e6 0.519465
$$539$$ 1.12847e6 0.167309
$$540$$ 0 0
$$541$$ −1.57718e6 −0.231679 −0.115840 0.993268i $$-0.536956\pi$$
−0.115840 + 0.993268i $$0.536956\pi$$
$$542$$ −5.62949e6 −0.823134
$$543$$ 0 0
$$544$$ 1.03424e6 0.149839
$$545$$ 3.52222e6 0.507955
$$546$$ 0 0
$$547$$ −6.24229e6 −0.892022 −0.446011 0.895027i $$-0.647156\pi$$
−0.446011 + 0.895027i $$0.647156\pi$$
$$548$$ 2.19539e6 0.312292
$$549$$ 0 0
$$550$$ −4.60412e6 −0.648993
$$551$$ 1.03196e7 1.44804
$$552$$ 0 0
$$553$$ −1.45942e6 −0.202939
$$554$$ 3.55513e6 0.492131
$$555$$ 0 0
$$556$$ 336640. 0.0461826
$$557$$ −8.36277e6 −1.14212 −0.571061 0.820908i $$-0.693468\pi$$
−0.571061 + 0.820908i $$0.693468\pi$$
$$558$$ 0 0
$$559$$ −6.25822e6 −0.847073
$$560$$ 326144. 0.0439480
$$561$$ 0 0
$$562$$ −8.00152e6 −1.06864
$$563$$ 9.91051e6 1.31773 0.658863 0.752263i $$-0.271038\pi$$
0.658863 + 0.752263i $$0.271038\pi$$
$$564$$ 0 0
$$565$$ 3.52643e6 0.464745
$$566$$ 601840. 0.0789660
$$567$$ 0 0
$$568$$ 1.17286e6 0.152538
$$569$$ 7.44344e6 0.963814 0.481907 0.876222i $$-0.339944\pi$$
0.481907 + 0.876222i $$0.339944\pi$$
$$570$$ 0 0
$$571$$ −4.08068e6 −0.523773 −0.261886 0.965099i $$-0.584345\pi$$
−0.261886 + 0.965099i $$0.584345\pi$$
$$572$$ 4.82784e6 0.616968
$$573$$ 0 0
$$574$$ −2.82358e6 −0.357701
$$575$$ 1.05307e6 0.132827
$$576$$ 0 0
$$577$$ 5.65712e6 0.707385 0.353693 0.935362i $$-0.384926\pi$$
0.353693 + 0.935362i $$0.384926\pi$$
$$578$$ −1.59903e6 −0.199084
$$579$$ 0 0
$$580$$ −2.80218e6 −0.345880
$$581$$ 1.38376e6 0.170067
$$582$$ 0 0
$$583$$ 3.57012e6 0.435022
$$584$$ 2.32845e6 0.282510
$$585$$ 0 0
$$586$$ 1.09468e7 1.31687
$$587$$ −1.38451e7 −1.65845 −0.829223 0.558917i $$-0.811217\pi$$
−0.829223 + 0.558917i $$0.811217\pi$$
$$588$$ 0 0
$$589$$ 3.47458e6 0.412680
$$590$$ 1.96643e6 0.232567
$$591$$ 0 0
$$592$$ −2.45094e6 −0.287428
$$593$$ 1.22720e7 1.43311 0.716553 0.697533i $$-0.245719\pi$$
0.716553 + 0.697533i $$0.245719\pi$$
$$594$$ 0 0
$$595$$ 1.28674e6 0.149004
$$596$$ 3.84749e6 0.443672
$$597$$ 0 0
$$598$$ −1.10424e6 −0.126273
$$599$$ 920118. 0.104780 0.0523898 0.998627i $$-0.483316\pi$$
0.0523898 + 0.998627i $$0.483316\pi$$
$$600$$ 0 0
$$601$$ −1.30680e7 −1.47579 −0.737893 0.674917i $$-0.764179\pi$$
−0.737893 + 0.674917i $$0.764179\pi$$
$$602$$ 1.91061e6 0.214872
$$603$$ 0 0
$$604$$ 5.20077e6 0.580063
$$605$$ −1.55607e6 −0.172839
$$606$$ 0 0
$$607$$ 6.07692e6 0.669440 0.334720 0.942318i $$-0.391358\pi$$
0.334720 + 0.942318i $$0.391358\pi$$
$$608$$ 1.56877e6 0.172107
$$609$$ 0 0
$$610$$ 3.82325e6 0.416014
$$611$$ 1.09166e7 1.18300
$$612$$ 0 0
$$613$$ 1.40826e7 1.51367 0.756835 0.653606i $$-0.226745\pi$$
0.756835 + 0.653606i $$0.226745\pi$$
$$614$$ 2.85774e6 0.305916
$$615$$ 0 0
$$616$$ −1.47392e6 −0.156503
$$617$$ −3.45617e6 −0.365496 −0.182748 0.983160i $$-0.558499\pi$$
−0.182748 + 0.983160i $$0.558499\pi$$
$$618$$ 0 0
$$619$$ 1.29450e6 0.135793 0.0678964 0.997692i $$-0.478371\pi$$
0.0678964 + 0.997692i $$0.478371\pi$$
$$620$$ −943488. −0.0985728
$$621$$ 0 0
$$622$$ 5.36075e6 0.555584
$$623$$ 3.72175e6 0.384173
$$624$$ 0 0
$$625$$ 3.88510e6 0.397834
$$626$$ −1.03681e7 −1.05745
$$627$$ 0 0
$$628$$ 8.60374e6 0.870539
$$629$$ −9.66974e6 −0.974514
$$630$$ 0 0
$$631$$ −1.51131e7 −1.51106 −0.755529 0.655116i $$-0.772620\pi$$
−0.755529 + 0.655116i $$0.772620\pi$$
$$632$$ 1.90618e6 0.189832
$$633$$ 0 0
$$634$$ −6.19013e6 −0.611613
$$635$$ 7.17538e6 0.706172
$$636$$ 0 0
$$637$$ 1.54144e6 0.150515
$$638$$ 1.26637e7 1.23171
$$639$$ 0 0
$$640$$ −425984. −0.0411096
$$641$$ 2.22343e6 0.213736 0.106868 0.994273i $$-0.465918\pi$$
0.106868 + 0.994273i $$0.465918\pi$$
$$642$$ 0 0
$$643$$ 2.02821e6 0.193458 0.0967288 0.995311i $$-0.469162\pi$$
0.0967288 + 0.995311i $$0.469162\pi$$
$$644$$ 337120. 0.0320310
$$645$$ 0 0
$$646$$ 6.18928e6 0.583524
$$647$$ −1.31471e7 −1.23472 −0.617360 0.786681i $$-0.711798\pi$$
−0.617360 + 0.786681i $$0.711798\pi$$
$$648$$ 0 0
$$649$$ −8.88676e6 −0.828193
$$650$$ −6.28903e6 −0.583849
$$651$$ 0 0
$$652$$ 6.45997e6 0.595129
$$653$$ 1.07280e7 0.984550 0.492275 0.870440i $$-0.336165\pi$$
0.492275 + 0.870440i $$0.336165\pi$$
$$654$$ 0 0
$$655$$ −1.31269e6 −0.119552
$$656$$ 3.68794e6 0.334598
$$657$$ 0 0
$$658$$ −3.33278e6 −0.300084
$$659$$ 1.71881e7 1.54176 0.770878 0.636983i $$-0.219818\pi$$
0.770878 + 0.636983i $$0.219818\pi$$
$$660$$ 0 0
$$661$$ 1.48793e7 1.32459 0.662293 0.749245i $$-0.269583\pi$$
0.662293 + 0.749245i $$0.269583\pi$$
$$662$$ −2.69157e6 −0.238704
$$663$$ 0 0
$$664$$ −1.80736e6 −0.159083
$$665$$ 1.95177e6 0.171149
$$666$$ 0 0
$$667$$ −2.89648e6 −0.252090
$$668$$ 1.12386e7 0.974475
$$669$$ 0 0
$$670$$ 3.82595e6 0.329270
$$671$$ −1.72781e7 −1.48146
$$672$$ 0 0
$$673$$ −1.02649e7 −0.873611 −0.436806 0.899556i $$-0.643890\pi$$
−0.436806 + 0.899556i $$0.643890\pi$$
$$674$$ −1.20051e7 −1.01792
$$675$$ 0 0
$$676$$ 653936. 0.0550387
$$677$$ 5.21750e6 0.437513 0.218756 0.975780i $$-0.429800\pi$$
0.218756 + 0.975780i $$0.429800\pi$$
$$678$$ 0 0
$$679$$ 3.95381e6 0.329110
$$680$$ −1.68064e6 −0.139381
$$681$$ 0 0
$$682$$ 4.26384e6 0.351027
$$683$$ −1.36733e7 −1.12156 −0.560780 0.827965i $$-0.689499\pi$$
−0.560780 + 0.827965i $$0.689499\pi$$
$$684$$ 0 0
$$685$$ −3.56751e6 −0.290495
$$686$$ −470596. −0.0381802
$$687$$ 0 0
$$688$$ −2.49549e6 −0.200995
$$689$$ 4.87663e6 0.391356
$$690$$ 0 0
$$691$$ −2.09599e7 −1.66991 −0.834956 0.550317i $$-0.814507\pi$$
−0.834956 + 0.550317i $$0.814507\pi$$
$$692$$ −1.16308e7 −0.923304
$$693$$ 0 0
$$694$$ 1.31646e7 1.03755
$$695$$ −547040. −0.0429593
$$696$$ 0 0
$$697$$ 1.45501e7 1.13444
$$698$$ −1.18424e7 −0.920026
$$699$$ 0 0
$$700$$ 1.92002e6 0.148102
$$701$$ 6.76672e6 0.520096 0.260048 0.965596i $$-0.416262\pi$$
0.260048 + 0.965596i $$0.416262\pi$$
$$702$$ 0 0
$$703$$ −1.46674e7 −1.11934
$$704$$ 1.92512e6 0.146395
$$705$$ 0 0
$$706$$ 6.63598e6 0.501064
$$707$$ −1.53399e6 −0.115418
$$708$$ 0 0
$$709$$ 1.52983e7 1.14295 0.571477 0.820618i $$-0.306371\pi$$
0.571477 + 0.820618i $$0.306371\pi$$
$$710$$ −1.90590e6 −0.141891
$$711$$ 0 0
$$712$$ −4.86106e6 −0.359361
$$713$$ −975240. −0.0718435
$$714$$ 0 0
$$715$$ −7.84524e6 −0.573906
$$716$$ 1.15587e7 0.842609
$$717$$ 0 0
$$718$$ 9.35366e6 0.677127
$$719$$ 932736. 0.0672878 0.0336439 0.999434i $$-0.489289\pi$$
0.0336439 + 0.999434i $$0.489289\pi$$
$$720$$ 0 0
$$721$$ −5.04249e6 −0.361249
$$722$$ −516300. −0.0368603
$$723$$ 0 0
$$724$$ −5.33578e6 −0.378313
$$725$$ −1.64965e7 −1.16559
$$726$$ 0 0
$$727$$ −2.19675e7 −1.54150 −0.770751 0.637137i $$-0.780119\pi$$
−0.770751 + 0.637137i $$0.780119\pi$$
$$728$$ −2.01331e6 −0.140794
$$729$$ 0 0
$$730$$ −3.78373e6 −0.262792
$$731$$ −9.84548e6 −0.681465
$$732$$ 0 0
$$733$$ −3.41626e6 −0.234850 −0.117425 0.993082i $$-0.537464\pi$$
−0.117425 + 0.993082i $$0.537464\pi$$
$$734$$ −7.11501e6 −0.487456
$$735$$ 0 0
$$736$$ −440320. −0.0299622
$$737$$ −1.72904e7 −1.17256
$$738$$ 0 0
$$739$$ −2.00714e7 −1.35197 −0.675983 0.736917i $$-0.736281\pi$$
−0.675983 + 0.736917i $$0.736281\pi$$
$$740$$ 3.98278e6 0.267367
$$741$$ 0 0
$$742$$ −1.48882e6 −0.0992730
$$743$$ 3.87764e6 0.257689 0.128844 0.991665i $$-0.458873\pi$$
0.128844 + 0.991665i $$0.458873\pi$$
$$744$$ 0 0
$$745$$ −6.25217e6 −0.412705
$$746$$ −5.52318e6 −0.363364
$$747$$ 0 0
$$748$$ 7.59520e6 0.496347
$$749$$ 8.45162e6 0.550472
$$750$$ 0 0
$$751$$ 965112. 0.0624422 0.0312211 0.999513i $$-0.490060\pi$$
0.0312211 + 0.999513i $$0.490060\pi$$
$$752$$ 4.35302e6 0.280703
$$753$$ 0 0
$$754$$ 1.72980e7 1.10807
$$755$$ −8.45125e6 −0.539577
$$756$$ 0 0
$$757$$ 1.51809e6 0.0962848 0.0481424 0.998840i $$-0.484670\pi$$
0.0481424 + 0.998840i $$0.484670\pi$$
$$758$$ 1.67430e7 1.05842
$$759$$ 0 0
$$760$$ −2.54925e6 −0.160095
$$761$$ −2.78210e7 −1.74145 −0.870724 0.491772i $$-0.836349\pi$$
−0.870724 + 0.491772i $$0.836349\pi$$
$$762$$ 0 0
$$763$$ 6.63803e6 0.412789
$$764$$ −1.07062e7 −0.663594
$$765$$ 0 0
$$766$$ −2.34486e6 −0.144393
$$767$$ −1.21389e7 −0.745062
$$768$$ 0 0
$$769$$ 1.29011e7 0.786704 0.393352 0.919388i $$-0.371315\pi$$
0.393352 + 0.919388i $$0.371315\pi$$
$$770$$ 2.39512e6 0.145580
$$771$$ 0 0
$$772$$ −1.84106e6 −0.111179
$$773$$ −2.03741e7 −1.22640 −0.613198 0.789929i $$-0.710117\pi$$
−0.613198 + 0.789929i $$0.710117\pi$$
$$774$$ 0 0
$$775$$ −5.55433e6 −0.332183
$$776$$ −5.16416e6 −0.307854
$$777$$ 0 0
$$778$$ −6.00848e6 −0.355890
$$779$$ 2.20700e7 1.30304
$$780$$ 0 0
$$781$$ 8.61322e6 0.505287
$$782$$ −1.73720e6 −0.101586
$$783$$ 0 0
$$784$$ 614656. 0.0357143
$$785$$ −1.39811e7 −0.809779
$$786$$ 0 0
$$787$$ −8.69990e6 −0.500700 −0.250350 0.968155i $$-0.580546\pi$$
−0.250350 + 0.968155i $$0.580546\pi$$
$$788$$ −3.41382e6 −0.195851
$$789$$ 0 0
$$790$$ −3.09754e6 −0.176583
$$791$$ 6.64597e6 0.377674
$$792$$ 0 0
$$793$$ −2.36012e7 −1.33276
$$794$$ −8.07953e6 −0.454815
$$795$$ 0 0
$$796$$ 1.27247e7 0.711813
$$797$$ −3.56281e7 −1.98677 −0.993383 0.114845i $$-0.963363\pi$$
−0.993383 + 0.114845i $$0.963363\pi$$
$$798$$ 0 0
$$799$$ 1.71740e7 0.951712
$$800$$ −2.50778e6 −0.138536
$$801$$ 0 0
$$802$$ −1.07023e7 −0.587546
$$803$$ 1.70995e7 0.935827
$$804$$ 0 0
$$805$$ −547820. −0.0297953
$$806$$ 5.82422e6 0.315792
$$807$$ 0 0
$$808$$ 2.00358e6 0.107964
$$809$$ 2.12912e7 1.14375 0.571873 0.820342i $$-0.306217\pi$$
0.571873 + 0.820342i $$0.306217\pi$$
$$810$$ 0 0
$$811$$ 1.83458e7 0.979455 0.489728 0.871875i $$-0.337096\pi$$
0.489728 + 0.871875i $$0.337096\pi$$
$$812$$ −5.28102e6 −0.281079
$$813$$ 0 0
$$814$$ −1.79991e7 −0.952117
$$815$$ −1.04974e7 −0.553592
$$816$$ 0 0
$$817$$ −1.49339e7 −0.782743
$$818$$ −5.29511e6 −0.276689
$$819$$ 0 0
$$820$$ −5.99290e6 −0.311245
$$821$$ 2.50359e7 1.29630 0.648149 0.761514i $$-0.275544\pi$$
0.648149 + 0.761514i $$0.275544\pi$$
$$822$$ 0 0
$$823$$ 3.30923e6 0.170305 0.0851525 0.996368i $$-0.472862\pi$$
0.0851525 + 0.996368i $$0.472862\pi$$
$$824$$ 6.58611e6 0.337918
$$825$$ 0 0
$$826$$ 3.70597e6 0.188996
$$827$$ −1.25220e7 −0.636664 −0.318332 0.947979i $$-0.603123\pi$$
−0.318332 + 0.947979i $$0.603123\pi$$
$$828$$ 0 0
$$829$$ 8.72677e6 0.441029 0.220515 0.975384i $$-0.429226\pi$$
0.220515 + 0.975384i $$0.429226\pi$$
$$830$$ 2.93696e6 0.147980
$$831$$ 0 0
$$832$$ 2.62963e6 0.131700
$$833$$ 2.42501e6 0.121088
$$834$$ 0 0
$$835$$ −1.82627e7 −0.906461
$$836$$ 1.15206e7 0.570113
$$837$$ 0 0
$$838$$ −1.80485e6 −0.0887832
$$839$$ 3.17349e7 1.55644 0.778221 0.627991i $$-0.216123\pi$$
0.778221 + 0.627991i $$0.216123\pi$$
$$840$$ 0 0
$$841$$ 2.48625e7 1.21215
$$842$$ −1.55202e7 −0.754427
$$843$$ 0 0
$$844$$ 3.49549e6 0.168909
$$845$$ −1.06265e6 −0.0511973
$$846$$ 0 0
$$847$$ −2.93260e6 −0.140457
$$848$$ 1.94458e6 0.0928614
$$849$$ 0 0
$$850$$ −9.89396e6 −0.469702
$$851$$ 4.11682e6 0.194867
$$852$$ 0 0
$$853$$ 3.40388e7 1.60178 0.800888 0.598814i $$-0.204361\pi$$
0.800888 + 0.598814i $$0.204361\pi$$
$$854$$ 7.20535e6 0.338073
$$855$$ 0 0
$$856$$ −1.10388e7 −0.514920
$$857$$ 2.84100e7 1.32135 0.660676 0.750671i $$-0.270270\pi$$
0.660676 + 0.750671i $$0.270270\pi$$
$$858$$ 0 0
$$859$$ 1.44582e7 0.668545 0.334272 0.942477i $$-0.391509\pi$$
0.334272 + 0.942477i $$0.391509\pi$$
$$860$$ 4.05517e6 0.186966
$$861$$ 0 0
$$862$$ 2.12587e7 0.974472
$$863$$ −1.46943e7 −0.671619 −0.335809 0.941930i $$-0.609010\pi$$
−0.335809 + 0.941930i $$0.609010\pi$$
$$864$$ 0 0
$$865$$ 1.89001e7 0.858862
$$866$$ 1.07581e7 0.487460
$$867$$ 0 0
$$868$$ −1.77811e6 −0.0801050
$$869$$ 1.39985e7 0.628827
$$870$$ 0 0
$$871$$ −2.36179e7 −1.05486
$$872$$ −8.67008e6 −0.386129
$$873$$ 0 0
$$874$$ −2.63504e6 −0.116683
$$875$$ −7.10128e6 −0.313557
$$876$$ 0 0
$$877$$ 2.47228e7 1.08542 0.542711 0.839920i $$-0.317398\pi$$
0.542711 + 0.839920i $$0.317398\pi$$
$$878$$ −2.57357e6 −0.112668
$$879$$ 0 0
$$880$$ −3.12832e6 −0.136177
$$881$$ −3.50211e6 −0.152016 −0.0760081 0.997107i $$-0.524217\pi$$
−0.0760081 + 0.997107i $$0.524217\pi$$
$$882$$ 0 0
$$883$$ −767908. −0.0331442 −0.0165721 0.999863i $$-0.505275\pi$$
−0.0165721 + 0.999863i $$0.505275\pi$$
$$884$$ 1.03747e7 0.446525
$$885$$ 0 0
$$886$$ −1.01440e7 −0.434136
$$887$$ −1.14209e7 −0.487405 −0.243703 0.969850i $$-0.578362\pi$$
−0.243703 + 0.969850i $$0.578362\pi$$
$$888$$ 0 0
$$889$$ 1.35228e7 0.573869
$$890$$ 7.89922e6 0.334279
$$891$$ 0 0
$$892$$ 1.05102e7 0.442282
$$893$$ 2.60501e7 1.09315
$$894$$ 0 0
$$895$$ −1.87829e7 −0.783798
$$896$$ −802816. −0.0334077
$$897$$ 0 0
$$898$$ 1.91671e7 0.793170
$$899$$ 1.52772e7 0.630443
$$900$$ 0 0
$$901$$ 7.67196e6 0.314843
$$902$$ 2.70833e7 1.10837
$$903$$ 0 0
$$904$$ −8.68045e6 −0.353282
$$905$$ 8.67064e6 0.351908
$$906$$ 0 0
$$907$$ −1.31900e7 −0.532387 −0.266194 0.963920i $$-0.585766\pi$$
−0.266194 + 0.963920i $$0.585766\pi$$
$$908$$ −1.55445e7 −0.625695
$$909$$ 0 0
$$910$$ 3.27163e6 0.130967
$$911$$ 2.99072e7 1.19393 0.596965 0.802267i $$-0.296373\pi$$
0.596965 + 0.802267i $$0.296373\pi$$
$$912$$ 0 0
$$913$$ −1.32728e7 −0.526970
$$914$$ 2.55933e7 1.01335
$$915$$ 0 0
$$916$$ −7.34960e6 −0.289418
$$917$$ −2.47391e6 −0.0971540
$$918$$ 0 0
$$919$$ 2.53866e7 0.991552 0.495776 0.868450i $$-0.334884\pi$$
0.495776 + 0.868450i $$0.334884\pi$$
$$920$$ 715520. 0.0278710
$$921$$ 0 0
$$922$$ −3.43660e7 −1.33138
$$923$$ 1.17653e7 0.454568
$$924$$ 0 0
$$925$$ 2.34467e7 0.901006
$$926$$ −6.66234e6 −0.255328
$$927$$ 0 0
$$928$$ 6.89766e6 0.262925
$$929$$ 2.29559e7 0.872680 0.436340 0.899782i $$-0.356274\pi$$
0.436340 + 0.899782i $$0.356274\pi$$
$$930$$ 0 0
$$931$$ 3.67833e6 0.139084
$$932$$ −1.97926e7 −0.746384
$$933$$ 0 0
$$934$$ −3.15249e7 −1.18246
$$935$$ −1.23422e7 −0.461704
$$936$$ 0 0
$$937$$ 1.16444e7 0.433280 0.216640 0.976252i $$-0.430490\pi$$
0.216640 + 0.976252i $$0.430490\pi$$
$$938$$ 7.21045e6 0.267581
$$939$$ 0 0
$$940$$ −7.07366e6 −0.261111
$$941$$ 2.04943e7 0.754500 0.377250 0.926111i $$-0.376870\pi$$
0.377250 + 0.926111i $$0.376870\pi$$
$$942$$ 0 0
$$943$$ −6.19458e6 −0.226847
$$944$$ −4.84045e6 −0.176789
$$945$$ 0 0
$$946$$ −1.83262e7 −0.665803
$$947$$ −2.91160e7 −1.05501 −0.527505 0.849552i $$-0.676872\pi$$
−0.527505 + 0.849552i $$0.676872\pi$$
$$948$$ 0 0
$$949$$ 2.33572e7 0.841891
$$950$$ −1.50075e7 −0.539509
$$951$$ 0 0
$$952$$ −3.16736e6 −0.113267
$$953$$ −5.20790e6 −0.185751 −0.0928753 0.995678i $$-0.529606\pi$$
−0.0928753 + 0.995678i $$0.529606\pi$$
$$954$$ 0 0
$$955$$ 1.73976e7 0.617278
$$956$$ −2.45492e7 −0.868746
$$957$$ 0 0
$$958$$ −9.01586e6 −0.317390
$$959$$ −6.72339e6 −0.236070
$$960$$ 0 0
$$961$$ −2.34853e7 −0.820329
$$962$$ −2.45860e7 −0.856546
$$963$$ 0 0
$$964$$ −2.27184e7 −0.787380
$$965$$ 2.99172e6 0.103419
$$966$$ 0 0
$$967$$ −4.28147e6 −0.147240 −0.0736202 0.997286i $$-0.523455\pi$$
−0.0736202 + 0.997286i $$0.523455\pi$$
$$968$$ 3.83034e6 0.131386
$$969$$ 0 0
$$970$$ 8.39176e6 0.286367
$$971$$ −1.20741e7 −0.410966 −0.205483 0.978661i $$-0.565876\pi$$
−0.205483 + 0.978661i $$0.565876\pi$$
$$972$$ 0 0
$$973$$ −1.03096e6 −0.0349108
$$974$$ 2.51678e7 0.850056
$$975$$ 0 0
$$976$$ −9.41107e6 −0.316238
$$977$$ 1.06568e7 0.357183 0.178591 0.983923i $$-0.442846\pi$$
0.178591 + 0.983923i $$0.442846\pi$$
$$978$$ 0 0
$$979$$ −3.56984e7 −1.19040
$$980$$ −998816. −0.0332216
$$981$$ 0 0
$$982$$ −1.84217e6 −0.0609608
$$983$$ 3.55409e7 1.17312 0.586562 0.809904i $$-0.300481\pi$$
0.586562 + 0.809904i $$0.300481\pi$$
$$984$$ 0 0
$$985$$ 5.54746e6 0.182181
$$986$$ 2.72134e7 0.891438
$$987$$ 0 0
$$988$$ 1.57367e7 0.512887
$$989$$ 4.19164e6 0.136268
$$990$$ 0 0
$$991$$ 2.83700e7 0.917647 0.458823 0.888527i $$-0.348271\pi$$
0.458823 + 0.888527i $$0.348271\pi$$
$$992$$ 2.32243e6 0.0749314
$$993$$ 0 0
$$994$$ −3.59190e6 −0.115308
$$995$$ −2.06777e7 −0.662132
$$996$$ 0 0
$$997$$ −6.15275e6 −0.196034 −0.0980169 0.995185i $$-0.531250\pi$$
−0.0980169 + 0.995185i $$0.531250\pi$$
$$998$$ 3.73570e7 1.18726
$$999$$ 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.j.1.1 yes 1
3.2 odd 2 126.6.a.d.1.1 1
4.3 odd 2 1008.6.a.i.1.1 1
7.6 odd 2 882.6.a.u.1.1 1
12.11 even 2 1008.6.a.r.1.1 1
21.20 even 2 882.6.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
126.6.a.d.1.1 1 3.2 odd 2
126.6.a.j.1.1 yes 1 1.1 even 1 trivial
882.6.a.c.1.1 1 21.20 even 2
882.6.a.u.1.1 1 7.6 odd 2
1008.6.a.i.1.1 1 4.3 odd 2
1008.6.a.r.1.1 1 12.11 even 2