# Properties

 Label 126.6.a.e.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)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} +16.0000 q^{4} +54.0000 q^{5} +49.0000 q^{7} -64.0000 q^{8} +O(q^{10})$$ $$q-4.00000 q^{2} +16.0000 q^{4} +54.0000 q^{5} +49.0000 q^{7} -64.0000 q^{8} -216.000 q^{10} +594.000 q^{11} +26.0000 q^{13} -196.000 q^{14} +256.000 q^{16} -534.000 q^{17} -3004.00 q^{19} +864.000 q^{20} -2376.00 q^{22} +3510.00 q^{23} -209.000 q^{25} -104.000 q^{26} +784.000 q^{28} +4296.00 q^{29} +8036.00 q^{31} -1024.00 q^{32} +2136.00 q^{34} +2646.00 q^{35} -502.000 q^{37} +12016.0 q^{38} -3456.00 q^{40} +9870.00 q^{41} +9068.00 q^{43} +9504.00 q^{44} -14040.0 q^{46} +1140.00 q^{47} +2401.00 q^{49} +836.000 q^{50} +416.000 q^{52} +28356.0 q^{53} +32076.0 q^{55} -3136.00 q^{56} -17184.0 q^{58} -8196.00 q^{59} +29822.0 q^{61} -32144.0 q^{62} +4096.00 q^{64} +1404.00 q^{65} -62884.0 q^{67} -8544.00 q^{68} -10584.0 q^{70} -34398.0 q^{71} +56990.0 q^{73} +2008.00 q^{74} -48064.0 q^{76} +29106.0 q^{77} +49496.0 q^{79} +13824.0 q^{80} -39480.0 q^{82} -52512.0 q^{83} -28836.0 q^{85} -36272.0 q^{86} -38016.0 q^{88} -48282.0 q^{89} +1274.00 q^{91} +56160.0 q^{92} -4560.00 q^{94} -162216. q^{95} -83938.0 q^{97} -9604.00 q^{98} +O(q^{100})$$

## 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
$$5$$ 54.0000 0.965981 0.482991 0.875625i $$-0.339550\pi$$
0.482991 + 0.875625i $$0.339550\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ −64.0000 −0.353553
$$9$$ 0 0
$$10$$ −216.000 −0.683052
$$11$$ 594.000 1.48015 0.740073 0.672526i $$-0.234791\pi$$
0.740073 + 0.672526i $$0.234791\pi$$
$$12$$ 0 0
$$13$$ 26.0000 0.0426692 0.0213346 0.999772i $$-0.493208\pi$$
0.0213346 + 0.999772i $$0.493208\pi$$
$$14$$ −196.000 −0.267261
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ −534.000 −0.448145 −0.224073 0.974572i $$-0.571935\pi$$
−0.224073 + 0.974572i $$0.571935\pi$$
$$18$$ 0 0
$$19$$ −3004.00 −1.90904 −0.954522 0.298141i $$-0.903634\pi$$
−0.954522 + 0.298141i $$0.903634\pi$$
$$20$$ 864.000 0.482991
$$21$$ 0 0
$$22$$ −2376.00 −1.04662
$$23$$ 3510.00 1.38353 0.691763 0.722124i $$-0.256834\pi$$
0.691763 + 0.722124i $$0.256834\pi$$
$$24$$ 0 0
$$25$$ −209.000 −0.0668800
$$26$$ −104.000 −0.0301717
$$27$$ 0 0
$$28$$ 784.000 0.188982
$$29$$ 4296.00 0.948570 0.474285 0.880371i $$-0.342707\pi$$
0.474285 + 0.880371i $$0.342707\pi$$
$$30$$ 0 0
$$31$$ 8036.00 1.50188 0.750941 0.660370i $$-0.229600\pi$$
0.750941 + 0.660370i $$0.229600\pi$$
$$32$$ −1024.00 −0.176777
$$33$$ 0 0
$$34$$ 2136.00 0.316887
$$35$$ 2646.00 0.365107
$$36$$ 0 0
$$37$$ −502.000 −0.0602836 −0.0301418 0.999546i $$-0.509596\pi$$
−0.0301418 + 0.999546i $$0.509596\pi$$
$$38$$ 12016.0 1.34990
$$39$$ 0 0
$$40$$ −3456.00 −0.341526
$$41$$ 9870.00 0.916975 0.458488 0.888701i $$-0.348391\pi$$
0.458488 + 0.888701i $$0.348391\pi$$
$$42$$ 0 0
$$43$$ 9068.00 0.747895 0.373947 0.927450i $$-0.378004\pi$$
0.373947 + 0.927450i $$0.378004\pi$$
$$44$$ 9504.00 0.740073
$$45$$ 0 0
$$46$$ −14040.0 −0.978301
$$47$$ 1140.00 0.0752766 0.0376383 0.999291i $$-0.488017\pi$$
0.0376383 + 0.999291i $$0.488017\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 836.000 0.0472913
$$51$$ 0 0
$$52$$ 416.000 0.0213346
$$53$$ 28356.0 1.38661 0.693307 0.720643i $$-0.256153\pi$$
0.693307 + 0.720643i $$0.256153\pi$$
$$54$$ 0 0
$$55$$ 32076.0 1.42979
$$56$$ −3136.00 −0.133631
$$57$$ 0 0
$$58$$ −17184.0 −0.670740
$$59$$ −8196.00 −0.306529 −0.153265 0.988185i $$-0.548979\pi$$
−0.153265 + 0.988185i $$0.548979\pi$$
$$60$$ 0 0
$$61$$ 29822.0 1.02615 0.513077 0.858343i $$-0.328506\pi$$
0.513077 + 0.858343i $$0.328506\pi$$
$$62$$ −32144.0 −1.06199
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ 1404.00 0.0412177
$$66$$ 0 0
$$67$$ −62884.0 −1.71141 −0.855703 0.517467i $$-0.826875\pi$$
−0.855703 + 0.517467i $$0.826875\pi$$
$$68$$ −8544.00 −0.224073
$$69$$ 0 0
$$70$$ −10584.0 −0.258169
$$71$$ −34398.0 −0.809818 −0.404909 0.914357i $$-0.632697\pi$$
−0.404909 + 0.914357i $$0.632697\pi$$
$$72$$ 0 0
$$73$$ 56990.0 1.25167 0.625837 0.779954i $$-0.284757\pi$$
0.625837 + 0.779954i $$0.284757\pi$$
$$74$$ 2008.00 0.0426270
$$75$$ 0 0
$$76$$ −48064.0 −0.954522
$$77$$ 29106.0 0.559443
$$78$$ 0 0
$$79$$ 49496.0 0.892282 0.446141 0.894963i $$-0.352798\pi$$
0.446141 + 0.894963i $$0.352798\pi$$
$$80$$ 13824.0 0.241495
$$81$$ 0 0
$$82$$ −39480.0 −0.648399
$$83$$ −52512.0 −0.836688 −0.418344 0.908289i $$-0.637389\pi$$
−0.418344 + 0.908289i $$0.637389\pi$$
$$84$$ 0 0
$$85$$ −28836.0 −0.432900
$$86$$ −36272.0 −0.528841
$$87$$ 0 0
$$88$$ −38016.0 −0.523311
$$89$$ −48282.0 −0.646116 −0.323058 0.946379i $$-0.604711\pi$$
−0.323058 + 0.946379i $$0.604711\pi$$
$$90$$ 0 0
$$91$$ 1274.00 0.0161275
$$92$$ 56160.0 0.691763
$$93$$ 0 0
$$94$$ −4560.00 −0.0532286
$$95$$ −162216. −1.84410
$$96$$ 0 0
$$97$$ −83938.0 −0.905794 −0.452897 0.891563i $$-0.649609\pi$$
−0.452897 + 0.891563i $$0.649609\pi$$
$$98$$ −9604.00 −0.101015
$$99$$ 0 0
$$100$$ −3344.00 −0.0334400
$$101$$ −62694.0 −0.611537 −0.305768 0.952106i $$-0.598913\pi$$
−0.305768 + 0.952106i $$0.598913\pi$$
$$102$$ 0 0
$$103$$ −30988.0 −0.287806 −0.143903 0.989592i $$-0.545965\pi$$
−0.143903 + 0.989592i $$0.545965\pi$$
$$104$$ −1664.00 −0.0150859
$$105$$ 0 0
$$106$$ −113424. −0.980484
$$107$$ 118218. 0.998215 0.499108 0.866540i $$-0.333661\pi$$
0.499108 + 0.866540i $$0.333661\pi$$
$$108$$ 0 0
$$109$$ 207362. 1.67172 0.835859 0.548944i $$-0.184970\pi$$
0.835859 + 0.548944i $$0.184970\pi$$
$$110$$ −128304. −1.01102
$$111$$ 0 0
$$112$$ 12544.0 0.0944911
$$113$$ −136416. −1.00501 −0.502504 0.864575i $$-0.667588\pi$$
−0.502504 + 0.864575i $$0.667588\pi$$
$$114$$ 0 0
$$115$$ 189540. 1.33646
$$116$$ 68736.0 0.474285
$$117$$ 0 0
$$118$$ 32784.0 0.216749
$$119$$ −26166.0 −0.169383
$$120$$ 0 0
$$121$$ 191785. 1.19083
$$122$$ −119288. −0.725600
$$123$$ 0 0
$$124$$ 128576. 0.750941
$$125$$ −180036. −1.03059
$$126$$ 0 0
$$127$$ −128248. −0.705572 −0.352786 0.935704i $$-0.614766\pi$$
−0.352786 + 0.935704i $$0.614766\pi$$
$$128$$ −16384.0 −0.0883883
$$129$$ 0 0
$$130$$ −5616.00 −0.0291453
$$131$$ 370296. 1.88526 0.942629 0.333842i $$-0.108345\pi$$
0.942629 + 0.333842i $$0.108345\pi$$
$$132$$ 0 0
$$133$$ −147196. −0.721551
$$134$$ 251536. 1.21015
$$135$$ 0 0
$$136$$ 34176.0 0.158443
$$137$$ 12924.0 0.0588296 0.0294148 0.999567i $$-0.490636\pi$$
0.0294148 + 0.999567i $$0.490636\pi$$
$$138$$ 0 0
$$139$$ −177760. −0.780364 −0.390182 0.920738i $$-0.627588\pi$$
−0.390182 + 0.920738i $$0.627588\pi$$
$$140$$ 42336.0 0.182553
$$141$$ 0 0
$$142$$ 137592. 0.572628
$$143$$ 15444.0 0.0631567
$$144$$ 0 0
$$145$$ 231984. 0.916301
$$146$$ −227960. −0.885068
$$147$$ 0 0
$$148$$ −8032.00 −0.0301418
$$149$$ 309564. 1.14231 0.571156 0.820841i $$-0.306495\pi$$
0.571156 + 0.820841i $$0.306495\pi$$
$$150$$ 0 0
$$151$$ −300136. −1.07121 −0.535606 0.844468i $$-0.679917\pi$$
−0.535606 + 0.844468i $$0.679917\pi$$
$$152$$ 192256. 0.674949
$$153$$ 0 0
$$154$$ −116424. −0.395586
$$155$$ 433944. 1.45079
$$156$$ 0 0
$$157$$ 11726.0 0.0379665 0.0189833 0.999820i $$-0.493957\pi$$
0.0189833 + 0.999820i $$0.493957\pi$$
$$158$$ −197984. −0.630939
$$159$$ 0 0
$$160$$ −55296.0 −0.170763
$$161$$ 171990. 0.522924
$$162$$ 0 0
$$163$$ −269260. −0.793785 −0.396892 0.917865i $$-0.629911\pi$$
−0.396892 + 0.917865i $$0.629911\pi$$
$$164$$ 157920. 0.458488
$$165$$ 0 0
$$166$$ 210048. 0.591627
$$167$$ 41604.0 0.115437 0.0577184 0.998333i $$-0.481617\pi$$
0.0577184 + 0.998333i $$0.481617\pi$$
$$168$$ 0 0
$$169$$ −370617. −0.998179
$$170$$ 115344. 0.306107
$$171$$ 0 0
$$172$$ 145088. 0.373947
$$173$$ 286962. 0.728969 0.364485 0.931209i $$-0.381245\pi$$
0.364485 + 0.931209i $$0.381245\pi$$
$$174$$ 0 0
$$175$$ −10241.0 −0.0252783
$$176$$ 152064. 0.370037
$$177$$ 0 0
$$178$$ 193128. 0.456873
$$179$$ −420186. −0.980187 −0.490094 0.871670i $$-0.663037\pi$$
−0.490094 + 0.871670i $$0.663037\pi$$
$$180$$ 0 0
$$181$$ −16918.0 −0.0383842 −0.0191921 0.999816i $$-0.506109\pi$$
−0.0191921 + 0.999816i $$0.506109\pi$$
$$182$$ −5096.00 −0.0114038
$$183$$ 0 0
$$184$$ −224640. −0.489151
$$185$$ −27108.0 −0.0582329
$$186$$ 0 0
$$187$$ −317196. −0.663321
$$188$$ 18240.0 0.0376383
$$189$$ 0 0
$$190$$ 648864. 1.30398
$$191$$ −134742. −0.267251 −0.133626 0.991032i $$-0.542662\pi$$
−0.133626 + 0.991032i $$0.542662\pi$$
$$192$$ 0 0
$$193$$ −314650. −0.608043 −0.304022 0.952665i $$-0.598330\pi$$
−0.304022 + 0.952665i $$0.598330\pi$$
$$194$$ 335752. 0.640493
$$195$$ 0 0
$$196$$ 38416.0 0.0714286
$$197$$ 596628. 1.09531 0.547656 0.836703i $$-0.315520\pi$$
0.547656 + 0.836703i $$0.315520\pi$$
$$198$$ 0 0
$$199$$ −10096.0 −0.0180724 −0.00903622 0.999959i $$-0.502876\pi$$
−0.00903622 + 0.999959i $$0.502876\pi$$
$$200$$ 13376.0 0.0236457
$$201$$ 0 0
$$202$$ 250776. 0.432422
$$203$$ 210504. 0.358526
$$204$$ 0 0
$$205$$ 532980. 0.885781
$$206$$ 123952. 0.203510
$$207$$ 0 0
$$208$$ 6656.00 0.0106673
$$209$$ −1.78438e6 −2.82566
$$210$$ 0 0
$$211$$ −721324. −1.11538 −0.557692 0.830048i $$-0.688313\pi$$
−0.557692 + 0.830048i $$0.688313\pi$$
$$212$$ 453696. 0.693307
$$213$$ 0 0
$$214$$ −472872. −0.705845
$$215$$ 489672. 0.722452
$$216$$ 0 0
$$217$$ 393764. 0.567658
$$218$$ −829448. −1.18208
$$219$$ 0 0
$$220$$ 513216. 0.714897
$$221$$ −13884.0 −0.0191220
$$222$$ 0 0
$$223$$ −536584. −0.722563 −0.361281 0.932457i $$-0.617661\pi$$
−0.361281 + 0.932457i $$0.617661\pi$$
$$224$$ −50176.0 −0.0668153
$$225$$ 0 0
$$226$$ 545664. 0.710647
$$227$$ −1.48698e6 −1.91532 −0.957658 0.287908i $$-0.907040\pi$$
−0.957658 + 0.287908i $$0.907040\pi$$
$$228$$ 0 0
$$229$$ −1.10957e6 −1.39818 −0.699092 0.715032i $$-0.746412\pi$$
−0.699092 + 0.715032i $$0.746412\pi$$
$$230$$ −758160. −0.945021
$$231$$ 0 0
$$232$$ −274944. −0.335370
$$233$$ 1.38796e6 1.67489 0.837444 0.546523i $$-0.184049\pi$$
0.837444 + 0.546523i $$0.184049\pi$$
$$234$$ 0 0
$$235$$ 61560.0 0.0727158
$$236$$ −131136. −0.153265
$$237$$ 0 0
$$238$$ 104664. 0.119772
$$239$$ 1.56406e6 1.77117 0.885583 0.464481i $$-0.153759\pi$$
0.885583 + 0.464481i $$0.153759\pi$$
$$240$$ 0 0
$$241$$ 1.36319e6 1.51187 0.755934 0.654648i $$-0.227183\pi$$
0.755934 + 0.654648i $$0.227183\pi$$
$$242$$ −767140. −0.842047
$$243$$ 0 0
$$244$$ 477152. 0.513077
$$245$$ 129654. 0.137997
$$246$$ 0 0
$$247$$ −78104.0 −0.0814575
$$248$$ −514304. −0.530995
$$249$$ 0 0
$$250$$ 720144. 0.728734
$$251$$ −1.54847e6 −1.55138 −0.775690 0.631115i $$-0.782598\pi$$
−0.775690 + 0.631115i $$0.782598\pi$$
$$252$$ 0 0
$$253$$ 2.08494e6 2.04782
$$254$$ 512992. 0.498915
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ −1.54147e6 −1.45580 −0.727899 0.685684i $$-0.759503\pi$$
−0.727899 + 0.685684i $$0.759503\pi$$
$$258$$ 0 0
$$259$$ −24598.0 −0.0227851
$$260$$ 22464.0 0.0206088
$$261$$ 0 0
$$262$$ −1.48118e6 −1.33308
$$263$$ −2.13251e6 −1.90109 −0.950545 0.310588i $$-0.899474\pi$$
−0.950545 + 0.310588i $$0.899474\pi$$
$$264$$ 0 0
$$265$$ 1.53122e6 1.33944
$$266$$ 588784. 0.510213
$$267$$ 0 0
$$268$$ −1.00614e6 −0.855703
$$269$$ 386142. 0.325362 0.162681 0.986679i $$-0.447986\pi$$
0.162681 + 0.986679i $$0.447986\pi$$
$$270$$ 0 0
$$271$$ −1.17581e6 −0.972556 −0.486278 0.873804i $$-0.661646\pi$$
−0.486278 + 0.873804i $$0.661646\pi$$
$$272$$ −136704. −0.112036
$$273$$ 0 0
$$274$$ −51696.0 −0.0415988
$$275$$ −124146. −0.0989922
$$276$$ 0 0
$$277$$ 417038. 0.326570 0.163285 0.986579i $$-0.447791\pi$$
0.163285 + 0.986579i $$0.447791\pi$$
$$278$$ 711040. 0.551800
$$279$$ 0 0
$$280$$ −169344. −0.129085
$$281$$ −523932. −0.395830 −0.197915 0.980219i $$-0.563417\pi$$
−0.197915 + 0.980219i $$0.563417\pi$$
$$282$$ 0 0
$$283$$ −2.36724e6 −1.75702 −0.878510 0.477723i $$-0.841462\pi$$
−0.878510 + 0.477723i $$0.841462\pi$$
$$284$$ −550368. −0.404909
$$285$$ 0 0
$$286$$ −61776.0 −0.0446586
$$287$$ 483630. 0.346584
$$288$$ 0 0
$$289$$ −1.13470e6 −0.799166
$$290$$ −927936. −0.647922
$$291$$ 0 0
$$292$$ 911840. 0.625837
$$293$$ −2.44630e6 −1.66472 −0.832360 0.554236i $$-0.813011\pi$$
−0.832360 + 0.554236i $$0.813011\pi$$
$$294$$ 0 0
$$295$$ −442584. −0.296102
$$296$$ 32128.0 0.0213135
$$297$$ 0 0
$$298$$ −1.23826e6 −0.807737
$$299$$ 91260.0 0.0590340
$$300$$ 0 0
$$301$$ 444332. 0.282678
$$302$$ 1.20054e6 0.757462
$$303$$ 0 0
$$304$$ −769024. −0.477261
$$305$$ 1.61039e6 0.991245
$$306$$ 0 0
$$307$$ −969316. −0.586975 −0.293487 0.955963i $$-0.594816\pi$$
−0.293487 + 0.955963i $$0.594816\pi$$
$$308$$ 465696. 0.279721
$$309$$ 0 0
$$310$$ −1.73578e6 −1.02586
$$311$$ 2.44765e6 1.43499 0.717495 0.696564i $$-0.245289\pi$$
0.717495 + 0.696564i $$0.245289\pi$$
$$312$$ 0 0
$$313$$ 2.02541e6 1.16856 0.584281 0.811551i $$-0.301376\pi$$
0.584281 + 0.811551i $$0.301376\pi$$
$$314$$ −46904.0 −0.0268464
$$315$$ 0 0
$$316$$ 791936. 0.446141
$$317$$ 2.50379e6 1.39942 0.699712 0.714425i $$-0.253312\pi$$
0.699712 + 0.714425i $$0.253312\pi$$
$$318$$ 0 0
$$319$$ 2.55182e6 1.40402
$$320$$ 221184. 0.120748
$$321$$ 0 0
$$322$$ −687960. −0.369763
$$323$$ 1.60414e6 0.855529
$$324$$ 0 0
$$325$$ −5434.00 −0.00285372
$$326$$ 1.07704e6 0.561291
$$327$$ 0 0
$$328$$ −631680. −0.324200
$$329$$ 55860.0 0.0284519
$$330$$ 0 0
$$331$$ 1.28700e6 0.645665 0.322832 0.946456i $$-0.395365\pi$$
0.322832 + 0.946456i $$0.395365\pi$$
$$332$$ −840192. −0.418344
$$333$$ 0 0
$$334$$ −166416. −0.0816261
$$335$$ −3.39574e6 −1.65319
$$336$$ 0 0
$$337$$ −1.40639e6 −0.674574 −0.337287 0.941402i $$-0.609509\pi$$
−0.337287 + 0.941402i $$0.609509\pi$$
$$338$$ 1.48247e6 0.705819
$$339$$ 0 0
$$340$$ −461376. −0.216450
$$341$$ 4.77338e6 2.22300
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ −580352. −0.264421
$$345$$ 0 0
$$346$$ −1.14785e6 −0.515459
$$347$$ −1.25313e6 −0.558692 −0.279346 0.960191i $$-0.590118\pi$$
−0.279346 + 0.960191i $$0.590118\pi$$
$$348$$ 0 0
$$349$$ 1.66876e6 0.733381 0.366691 0.930343i $$-0.380491\pi$$
0.366691 + 0.930343i $$0.380491\pi$$
$$350$$ 40964.0 0.0178744
$$351$$ 0 0
$$352$$ −608256. −0.261655
$$353$$ −2.13687e6 −0.912728 −0.456364 0.889793i $$-0.650848\pi$$
−0.456364 + 0.889793i $$0.650848\pi$$
$$354$$ 0 0
$$355$$ −1.85749e6 −0.782269
$$356$$ −772512. −0.323058
$$357$$ 0 0
$$358$$ 1.68074e6 0.693097
$$359$$ 3.60907e6 1.47795 0.738973 0.673735i $$-0.235311\pi$$
0.738973 + 0.673735i $$0.235311\pi$$
$$360$$ 0 0
$$361$$ 6.54792e6 2.64445
$$362$$ 67672.0 0.0271417
$$363$$ 0 0
$$364$$ 20384.0 0.00806373
$$365$$ 3.07746e6 1.20909
$$366$$ 0 0
$$367$$ −1.88190e6 −0.729344 −0.364672 0.931136i $$-0.618819\pi$$
−0.364672 + 0.931136i $$0.618819\pi$$
$$368$$ 898560. 0.345882
$$369$$ 0 0
$$370$$ 108432. 0.0411769
$$371$$ 1.38944e6 0.524090
$$372$$ 0 0
$$373$$ 4.86186e6 1.80938 0.904692 0.426067i $$-0.140101\pi$$
0.904692 + 0.426067i $$0.140101\pi$$
$$374$$ 1.26878e6 0.469039
$$375$$ 0 0
$$376$$ −72960.0 −0.0266143
$$377$$ 111696. 0.0404748
$$378$$ 0 0
$$379$$ 251300. 0.0898658 0.0449329 0.998990i $$-0.485693\pi$$
0.0449329 + 0.998990i $$0.485693\pi$$
$$380$$ −2.59546e6 −0.922050
$$381$$ 0 0
$$382$$ 538968. 0.188975
$$383$$ −567720. −0.197759 −0.0988797 0.995099i $$-0.531526\pi$$
−0.0988797 + 0.995099i $$0.531526\pi$$
$$384$$ 0 0
$$385$$ 1.57172e6 0.540411
$$386$$ 1.25860e6 0.429951
$$387$$ 0 0
$$388$$ −1.34301e6 −0.452897
$$389$$ −2.34547e6 −0.785880 −0.392940 0.919564i $$-0.628542\pi$$
−0.392940 + 0.919564i $$0.628542\pi$$
$$390$$ 0 0
$$391$$ −1.87434e6 −0.620021
$$392$$ −153664. −0.0505076
$$393$$ 0 0
$$394$$ −2.38651e6 −0.774503
$$395$$ 2.67278e6 0.861928
$$396$$ 0 0
$$397$$ −5.25719e6 −1.67408 −0.837042 0.547139i $$-0.815717\pi$$
−0.837042 + 0.547139i $$0.815717\pi$$
$$398$$ 40384.0 0.0127791
$$399$$ 0 0
$$400$$ −53504.0 −0.0167200
$$401$$ −2.34140e6 −0.727136 −0.363568 0.931568i $$-0.618442\pi$$
−0.363568 + 0.931568i $$0.618442\pi$$
$$402$$ 0 0
$$403$$ 208936. 0.0640842
$$404$$ −1.00310e6 −0.305768
$$405$$ 0 0
$$406$$ −842016. −0.253516
$$407$$ −298188. −0.0892286
$$408$$ 0 0
$$409$$ 662318. 0.195775 0.0978877 0.995197i $$-0.468791\pi$$
0.0978877 + 0.995197i $$0.468791\pi$$
$$410$$ −2.13192e6 −0.626342
$$411$$ 0 0
$$412$$ −495808. −0.143903
$$413$$ −401604. −0.115857
$$414$$ 0 0
$$415$$ −2.83565e6 −0.808225
$$416$$ −26624.0 −0.00754293
$$417$$ 0 0
$$418$$ 7.13750e6 1.99805
$$419$$ 5.27752e6 1.46857 0.734285 0.678842i $$-0.237518\pi$$
0.734285 + 0.678842i $$0.237518\pi$$
$$420$$ 0 0
$$421$$ 1.74817e6 0.480706 0.240353 0.970686i $$-0.422737\pi$$
0.240353 + 0.970686i $$0.422737\pi$$
$$422$$ 2.88530e6 0.788695
$$423$$ 0 0
$$424$$ −1.81478e6 −0.490242
$$425$$ 111606. 0.0299720
$$426$$ 0 0
$$427$$ 1.46128e6 0.387849
$$428$$ 1.89149e6 0.499108
$$429$$ 0 0
$$430$$ −1.95869e6 −0.510851
$$431$$ −2.65575e6 −0.688643 −0.344321 0.938852i $$-0.611891\pi$$
−0.344321 + 0.938852i $$0.611891\pi$$
$$432$$ 0 0
$$433$$ −3.12026e6 −0.799781 −0.399891 0.916563i $$-0.630952\pi$$
−0.399891 + 0.916563i $$0.630952\pi$$
$$434$$ −1.57506e6 −0.401395
$$435$$ 0 0
$$436$$ 3.31779e6 0.835859
$$437$$ −1.05440e7 −2.64121
$$438$$ 0 0
$$439$$ −4.02131e6 −0.995879 −0.497939 0.867212i $$-0.665910\pi$$
−0.497939 + 0.867212i $$0.665910\pi$$
$$440$$ −2.05286e6 −0.505509
$$441$$ 0 0
$$442$$ 55536.0 0.0135213
$$443$$ −766146. −0.185482 −0.0927411 0.995690i $$-0.529563\pi$$
−0.0927411 + 0.995690i $$0.529563\pi$$
$$444$$ 0 0
$$445$$ −2.60723e6 −0.624136
$$446$$ 2.14634e6 0.510929
$$447$$ 0 0
$$448$$ 200704. 0.0472456
$$449$$ −3.01961e6 −0.706862 −0.353431 0.935461i $$-0.614985\pi$$
−0.353431 + 0.935461i $$0.614985\pi$$
$$450$$ 0 0
$$451$$ 5.86278e6 1.35726
$$452$$ −2.18266e6 −0.502504
$$453$$ 0 0
$$454$$ 5.94792e6 1.35433
$$455$$ 68796.0 0.0155788
$$456$$ 0 0
$$457$$ −223114. −0.0499731 −0.0249866 0.999688i $$-0.507954\pi$$
−0.0249866 + 0.999688i $$0.507954\pi$$
$$458$$ 4.43826e6 0.988666
$$459$$ 0 0
$$460$$ 3.03264e6 0.668230
$$461$$ −4.58050e6 −1.00383 −0.501916 0.864917i $$-0.667371\pi$$
−0.501916 + 0.864917i $$0.667371\pi$$
$$462$$ 0 0
$$463$$ −4.23654e6 −0.918458 −0.459229 0.888318i $$-0.651874\pi$$
−0.459229 + 0.888318i $$0.651874\pi$$
$$464$$ 1.09978e6 0.237142
$$465$$ 0 0
$$466$$ −5.55182e6 −1.18433
$$467$$ −2.74499e6 −0.582436 −0.291218 0.956657i $$-0.594061\pi$$
−0.291218 + 0.956657i $$0.594061\pi$$
$$468$$ 0 0
$$469$$ −3.08132e6 −0.646851
$$470$$ −246240. −0.0514179
$$471$$ 0 0
$$472$$ 524544. 0.108374
$$473$$ 5.38639e6 1.10699
$$474$$ 0 0
$$475$$ 627836. 0.127677
$$476$$ −418656. −0.0846915
$$477$$ 0 0
$$478$$ −6.25625e6 −1.25240
$$479$$ −2.67628e6 −0.532957 −0.266478 0.963841i $$-0.585860\pi$$
−0.266478 + 0.963841i $$0.585860\pi$$
$$480$$ 0 0
$$481$$ −13052.0 −0.00257226
$$482$$ −5.45276e6 −1.06905
$$483$$ 0 0
$$484$$ 3.06856e6 0.595417
$$485$$ −4.53265e6 −0.874980
$$486$$ 0 0
$$487$$ −7.92959e6 −1.51506 −0.757528 0.652803i $$-0.773593\pi$$
−0.757528 + 0.652803i $$0.773593\pi$$
$$488$$ −1.90861e6 −0.362800
$$489$$ 0 0
$$490$$ −518616. −0.0975789
$$491$$ −7.10567e6 −1.33015 −0.665076 0.746775i $$-0.731601\pi$$
−0.665076 + 0.746775i $$0.731601\pi$$
$$492$$ 0 0
$$493$$ −2.29406e6 −0.425097
$$494$$ 312416. 0.0575991
$$495$$ 0 0
$$496$$ 2.05722e6 0.375470
$$497$$ −1.68550e6 −0.306082
$$498$$ 0 0
$$499$$ −1.31352e6 −0.236149 −0.118075 0.993005i $$-0.537672\pi$$
−0.118075 + 0.993005i $$0.537672\pi$$
$$500$$ −2.88058e6 −0.515293
$$501$$ 0 0
$$502$$ 6.19387e6 1.09699
$$503$$ 3.64608e6 0.642549 0.321274 0.946986i $$-0.395889\pi$$
0.321274 + 0.946986i $$0.395889\pi$$
$$504$$ 0 0
$$505$$ −3.38548e6 −0.590733
$$506$$ −8.33976e6 −1.44803
$$507$$ 0 0
$$508$$ −2.05197e6 −0.352786
$$509$$ 9.65410e6 1.65165 0.825824 0.563928i $$-0.190711\pi$$
0.825824 + 0.563928i $$0.190711\pi$$
$$510$$ 0 0
$$511$$ 2.79251e6 0.473089
$$512$$ −262144. −0.0441942
$$513$$ 0 0
$$514$$ 6.16586e6 1.02940
$$515$$ −1.67335e6 −0.278016
$$516$$ 0 0
$$517$$ 677160. 0.111420
$$518$$ 98392.0 0.0161115
$$519$$ 0 0
$$520$$ −89856.0 −0.0145727
$$521$$ 7.67870e6 1.23935 0.619674 0.784859i $$-0.287265\pi$$
0.619674 + 0.784859i $$0.287265\pi$$
$$522$$ 0 0
$$523$$ −9.06510e6 −1.44917 −0.724584 0.689187i $$-0.757968\pi$$
−0.724584 + 0.689187i $$0.757968\pi$$
$$524$$ 5.92474e6 0.942629
$$525$$ 0 0
$$526$$ 8.53006e6 1.34427
$$527$$ −4.29122e6 −0.673061
$$528$$ 0 0
$$529$$ 5.88376e6 0.914146
$$530$$ −6.12490e6 −0.947129
$$531$$ 0 0
$$532$$ −2.35514e6 −0.360775
$$533$$ 256620. 0.0391266
$$534$$ 0 0
$$535$$ 6.38377e6 0.964257
$$536$$ 4.02458e6 0.605074
$$537$$ 0 0
$$538$$ −1.54457e6 −0.230065
$$539$$ 1.42619e6 0.211450
$$540$$ 0 0
$$541$$ 7.33108e6 1.07690 0.538449 0.842658i $$-0.319010\pi$$
0.538449 + 0.842658i $$0.319010\pi$$
$$542$$ 4.70325e6 0.687701
$$543$$ 0 0
$$544$$ 546816. 0.0792217
$$545$$ 1.11975e7 1.61485
$$546$$ 0 0
$$547$$ −3.16498e6 −0.452275 −0.226138 0.974095i $$-0.572610\pi$$
−0.226138 + 0.974095i $$0.572610\pi$$
$$548$$ 206784. 0.0294148
$$549$$ 0 0
$$550$$ 496584. 0.0699981
$$551$$ −1.29052e7 −1.81086
$$552$$ 0 0
$$553$$ 2.42530e6 0.337251
$$554$$ −1.66815e6 −0.230920
$$555$$ 0 0
$$556$$ −2.84416e6 −0.390182
$$557$$ −118092. −0.0161281 −0.00806404 0.999967i $$-0.502567\pi$$
−0.00806404 + 0.999967i $$0.502567\pi$$
$$558$$ 0 0
$$559$$ 235768. 0.0319121
$$560$$ 677376. 0.0912767
$$561$$ 0 0
$$562$$ 2.09573e6 0.279894
$$563$$ 6.43544e6 0.855672 0.427836 0.903856i $$-0.359276\pi$$
0.427836 + 0.903856i $$0.359276\pi$$
$$564$$ 0 0
$$565$$ −7.36646e6 −0.970818
$$566$$ 9.46898e6 1.24240
$$567$$ 0 0
$$568$$ 2.20147e6 0.286314
$$569$$ 3.22976e6 0.418206 0.209103 0.977894i $$-0.432946\pi$$
0.209103 + 0.977894i $$0.432946\pi$$
$$570$$ 0 0
$$571$$ −228556. −0.0293361 −0.0146680 0.999892i $$-0.504669\pi$$
−0.0146680 + 0.999892i $$0.504669\pi$$
$$572$$ 247104. 0.0315784
$$573$$ 0 0
$$574$$ −1.93452e6 −0.245072
$$575$$ −733590. −0.0925303
$$576$$ 0 0
$$577$$ 1.50817e7 1.88587 0.942933 0.332983i $$-0.108055\pi$$
0.942933 + 0.332983i $$0.108055\pi$$
$$578$$ 4.53880e6 0.565095
$$579$$ 0 0
$$580$$ 3.71174e6 0.458150
$$581$$ −2.57309e6 −0.316238
$$582$$ 0 0
$$583$$ 1.68435e7 2.05239
$$584$$ −3.64736e6 −0.442534
$$585$$ 0 0
$$586$$ 9.78521e6 1.17713
$$587$$ 8.90044e6 1.06614 0.533072 0.846070i $$-0.321037\pi$$
0.533072 + 0.846070i $$0.321037\pi$$
$$588$$ 0 0
$$589$$ −2.41401e7 −2.86716
$$590$$ 1.77034e6 0.209375
$$591$$ 0 0
$$592$$ −128512. −0.0150709
$$593$$ 1.34870e6 0.157499 0.0787495 0.996894i $$-0.474907\pi$$
0.0787495 + 0.996894i $$0.474907\pi$$
$$594$$ 0 0
$$595$$ −1.41296e6 −0.163621
$$596$$ 4.95302e6 0.571156
$$597$$ 0 0
$$598$$ −365040. −0.0417434
$$599$$ 1.18444e7 1.34879 0.674395 0.738371i $$-0.264405\pi$$
0.674395 + 0.738371i $$0.264405\pi$$
$$600$$ 0 0
$$601$$ −9.62671e6 −1.08716 −0.543578 0.839359i $$-0.682931\pi$$
−0.543578 + 0.839359i $$0.682931\pi$$
$$602$$ −1.77733e6 −0.199883
$$603$$ 0 0
$$604$$ −4.80218e6 −0.535606
$$605$$ 1.03564e7 1.15032
$$606$$ 0 0
$$607$$ −641512. −0.0706697 −0.0353348 0.999376i $$-0.511250\pi$$
−0.0353348 + 0.999376i $$0.511250\pi$$
$$608$$ 3.07610e6 0.337474
$$609$$ 0 0
$$610$$ −6.44155e6 −0.700916
$$611$$ 29640.0 0.00321200
$$612$$ 0 0
$$613$$ 3.72964e6 0.400881 0.200441 0.979706i $$-0.435763\pi$$
0.200441 + 0.979706i $$0.435763\pi$$
$$614$$ 3.87726e6 0.415054
$$615$$ 0 0
$$616$$ −1.86278e6 −0.197793
$$617$$ −1.18580e7 −1.25400 −0.627000 0.779019i $$-0.715718\pi$$
−0.627000 + 0.779019i $$0.715718\pi$$
$$618$$ 0 0
$$619$$ 2.60636e6 0.273406 0.136703 0.990612i $$-0.456349\pi$$
0.136703 + 0.990612i $$0.456349\pi$$
$$620$$ 6.94310e6 0.725395
$$621$$ 0 0
$$622$$ −9.79061e6 −1.01469
$$623$$ −2.36582e6 −0.244209
$$624$$ 0 0
$$625$$ −9.06882e6 −0.928647
$$626$$ −8.10164e6 −0.826299
$$627$$ 0 0
$$628$$ 187616. 0.0189833
$$629$$ 268068. 0.0270158
$$630$$ 0 0
$$631$$ 5.15540e6 0.515453 0.257726 0.966218i $$-0.417027\pi$$
0.257726 + 0.966218i $$0.417027\pi$$
$$632$$ −3.16774e6 −0.315470
$$633$$ 0 0
$$634$$ −1.00152e7 −0.989542
$$635$$ −6.92539e6 −0.681569
$$636$$ 0 0
$$637$$ 62426.0 0.00609561
$$638$$ −1.02073e7 −0.992794
$$639$$ 0 0
$$640$$ −884736. −0.0853815
$$641$$ 1.42517e7 1.37000 0.685002 0.728541i $$-0.259801\pi$$
0.685002 + 0.728541i $$0.259801\pi$$
$$642$$ 0 0
$$643$$ −1.24310e7 −1.18571 −0.592857 0.805308i $$-0.702000\pi$$
−0.592857 + 0.805308i $$0.702000\pi$$
$$644$$ 2.75184e6 0.261462
$$645$$ 0 0
$$646$$ −6.41654e6 −0.604951
$$647$$ 5.71643e6 0.536864 0.268432 0.963299i $$-0.413495\pi$$
0.268432 + 0.963299i $$0.413495\pi$$
$$648$$ 0 0
$$649$$ −4.86842e6 −0.453708
$$650$$ 21736.0 0.00201788
$$651$$ 0 0
$$652$$ −4.30816e6 −0.396892
$$653$$ −2.48479e6 −0.228038 −0.114019 0.993479i $$-0.536372\pi$$
−0.114019 + 0.993479i $$0.536372\pi$$
$$654$$ 0 0
$$655$$ 1.99960e7 1.82112
$$656$$ 2.52672e6 0.229244
$$657$$ 0 0
$$658$$ −223440. −0.0201185
$$659$$ 2.87481e6 0.257867 0.128933 0.991653i $$-0.458845\pi$$
0.128933 + 0.991653i $$0.458845\pi$$
$$660$$ 0 0
$$661$$ −8.18274e6 −0.728442 −0.364221 0.931313i $$-0.618665\pi$$
−0.364221 + 0.931313i $$0.618665\pi$$
$$662$$ −5.14798e6 −0.456554
$$663$$ 0 0
$$664$$ 3.36077e6 0.295814
$$665$$ −7.94858e6 −0.697005
$$666$$ 0 0
$$667$$ 1.50790e7 1.31237
$$668$$ 665664. 0.0577184
$$669$$ 0 0
$$670$$ 1.35829e7 1.16898
$$671$$ 1.77143e7 1.51886
$$672$$ 0 0
$$673$$ 1.52187e7 1.29521 0.647603 0.761978i $$-0.275772\pi$$
0.647603 + 0.761978i $$0.275772\pi$$
$$674$$ 5.62554e6 0.476996
$$675$$ 0 0
$$676$$ −5.92987e6 −0.499090
$$677$$ −1.85942e7 −1.55922 −0.779609 0.626266i $$-0.784582\pi$$
−0.779609 + 0.626266i $$0.784582\pi$$
$$678$$ 0 0
$$679$$ −4.11296e6 −0.342358
$$680$$ 1.84550e6 0.153053
$$681$$ 0 0
$$682$$ −1.90935e7 −1.57190
$$683$$ −1362.00 −0.000111719 0 −5.58593e−5 1.00000i $$-0.500018\pi$$
−5.58593e−5 1.00000i $$0.500018\pi$$
$$684$$ 0 0
$$685$$ 697896. 0.0568282
$$686$$ −470596. −0.0381802
$$687$$ 0 0
$$688$$ 2.32141e6 0.186974
$$689$$ 737256. 0.0591657
$$690$$ 0 0
$$691$$ 1.83515e7 1.46210 0.731048 0.682327i $$-0.239032\pi$$
0.731048 + 0.682327i $$0.239032\pi$$
$$692$$ 4.59139e6 0.364485
$$693$$ 0 0
$$694$$ 5.01252e6 0.395055
$$695$$ −9.59904e6 −0.753817
$$696$$ 0 0
$$697$$ −5.27058e6 −0.410938
$$698$$ −6.67503e6 −0.518579
$$699$$ 0 0
$$700$$ −163856. −0.0126391
$$701$$ −1.00448e7 −0.772051 −0.386025 0.922488i $$-0.626152\pi$$
−0.386025 + 0.922488i $$0.626152\pi$$
$$702$$ 0 0
$$703$$ 1.50801e6 0.115084
$$704$$ 2.43302e6 0.185018
$$705$$ 0 0
$$706$$ 8.54748e6 0.645396
$$707$$ −3.07201e6 −0.231139
$$708$$ 0 0
$$709$$ −2.11149e6 −0.157752 −0.0788759 0.996884i $$-0.525133\pi$$
−0.0788759 + 0.996884i $$0.525133\pi$$
$$710$$ 7.42997e6 0.553148
$$711$$ 0 0
$$712$$ 3.09005e6 0.228436
$$713$$ 2.82064e7 2.07789
$$714$$ 0 0
$$715$$ 833976. 0.0610082
$$716$$ −6.72298e6 −0.490094
$$717$$ 0 0
$$718$$ −1.44363e7 −1.04507
$$719$$ −296016. −0.0213547 −0.0106773 0.999943i $$-0.503399\pi$$
−0.0106773 + 0.999943i $$0.503399\pi$$
$$720$$ 0 0
$$721$$ −1.51841e6 −0.108781
$$722$$ −2.61917e7 −1.86991
$$723$$ 0 0
$$724$$ −270688. −0.0191921
$$725$$ −897864. −0.0634403
$$726$$ 0 0
$$727$$ −90220.0 −0.00633092 −0.00316546 0.999995i $$-0.501008\pi$$
−0.00316546 + 0.999995i $$0.501008\pi$$
$$728$$ −81536.0 −0.00570192
$$729$$ 0 0
$$730$$ −1.23098e7 −0.854959
$$731$$ −4.84231e6 −0.335166
$$732$$ 0 0
$$733$$ −1.40664e7 −0.966992 −0.483496 0.875347i $$-0.660633\pi$$
−0.483496 + 0.875347i $$0.660633\pi$$
$$734$$ 7.52762e6 0.515724
$$735$$ 0 0
$$736$$ −3.59424e6 −0.244575
$$737$$ −3.73531e7 −2.53313
$$738$$ 0 0
$$739$$ 2.20018e7 1.48199 0.740997 0.671508i $$-0.234353\pi$$
0.740997 + 0.671508i $$0.234353\pi$$
$$740$$ −433728. −0.0291164
$$741$$ 0 0
$$742$$ −5.55778e6 −0.370588
$$743$$ 9.42981e6 0.626658 0.313329 0.949645i $$-0.398556\pi$$
0.313329 + 0.949645i $$0.398556\pi$$
$$744$$ 0 0
$$745$$ 1.67165e7 1.10345
$$746$$ −1.94474e7 −1.27943
$$747$$ 0 0
$$748$$ −5.07514e6 −0.331660
$$749$$ 5.79268e6 0.377290
$$750$$ 0 0
$$751$$ 5.06420e6 0.327651 0.163825 0.986489i $$-0.447617\pi$$
0.163825 + 0.986489i $$0.447617\pi$$
$$752$$ 291840. 0.0188192
$$753$$ 0 0
$$754$$ −446784. −0.0286200
$$755$$ −1.62073e7 −1.03477
$$756$$ 0 0
$$757$$ −9.41479e6 −0.597133 −0.298566 0.954389i $$-0.596508\pi$$
−0.298566 + 0.954389i $$0.596508\pi$$
$$758$$ −1.00520e6 −0.0635447
$$759$$ 0 0
$$760$$ 1.03818e7 0.651988
$$761$$ −1.81025e7 −1.13313 −0.566563 0.824019i $$-0.691727\pi$$
−0.566563 + 0.824019i $$0.691727\pi$$
$$762$$ 0 0
$$763$$ 1.01607e7 0.631850
$$764$$ −2.15587e6 −0.133626
$$765$$ 0 0
$$766$$ 2.27088e6 0.139837
$$767$$ −213096. −0.0130794
$$768$$ 0 0
$$769$$ 2.37970e7 1.45113 0.725566 0.688152i $$-0.241578\pi$$
0.725566 + 0.688152i $$0.241578\pi$$
$$770$$ −6.28690e6 −0.382129
$$771$$ 0 0
$$772$$ −5.03440e6 −0.304022
$$773$$ −9.76453e6 −0.587763 −0.293882 0.955842i $$-0.594947\pi$$
−0.293882 + 0.955842i $$0.594947\pi$$
$$774$$ 0 0
$$775$$ −1.67952e6 −0.100446
$$776$$ 5.37203e6 0.320246
$$777$$ 0 0
$$778$$ 9.38189e6 0.555701
$$779$$ −2.96495e7 −1.75055
$$780$$ 0 0
$$781$$ −2.04324e7 −1.19865
$$782$$ 7.49736e6 0.438421
$$783$$ 0 0
$$784$$ 614656. 0.0357143
$$785$$ 633204. 0.0366749
$$786$$ 0 0
$$787$$ −2.69301e6 −0.154989 −0.0774945 0.996993i $$-0.524692\pi$$
−0.0774945 + 0.996993i $$0.524692\pi$$
$$788$$ 9.54605e6 0.547656
$$789$$ 0 0
$$790$$ −1.06911e7 −0.609475
$$791$$ −6.68438e6 −0.379857
$$792$$ 0 0
$$793$$ 775372. 0.0437852
$$794$$ 2.10287e7 1.18376
$$795$$ 0 0
$$796$$ −161536. −0.00903622
$$797$$ −2.69834e7 −1.50470 −0.752352 0.658762i $$-0.771081\pi$$
−0.752352 + 0.658762i $$0.771081\pi$$
$$798$$ 0 0
$$799$$ −608760. −0.0337349
$$800$$ 214016. 0.0118228
$$801$$ 0 0
$$802$$ 9.36562e6 0.514163
$$803$$ 3.38521e7 1.85266
$$804$$ 0 0
$$805$$ 9.28746e6 0.505135
$$806$$ −835744. −0.0453143
$$807$$ 0 0
$$808$$ 4.01242e6 0.216211
$$809$$ 1.30813e7 0.702718 0.351359 0.936241i $$-0.385720\pi$$
0.351359 + 0.936241i $$0.385720\pi$$
$$810$$ 0 0
$$811$$ 2.12063e7 1.13217 0.566086 0.824346i $$-0.308457\pi$$
0.566086 + 0.824346i $$0.308457\pi$$
$$812$$ 3.36806e6 0.179263
$$813$$ 0 0
$$814$$ 1.19275e6 0.0630942
$$815$$ −1.45400e7 −0.766781
$$816$$ 0 0
$$817$$ −2.72403e7 −1.42776
$$818$$ −2.64927e6 −0.138434
$$819$$ 0 0
$$820$$ 8.52768e6 0.442890
$$821$$ 3.76237e7 1.94806 0.974032 0.226411i $$-0.0726993\pi$$
0.974032 + 0.226411i $$0.0726993\pi$$
$$822$$ 0 0
$$823$$ 4.75582e6 0.244752 0.122376 0.992484i $$-0.460949\pi$$
0.122376 + 0.992484i $$0.460949\pi$$
$$824$$ 1.98323e6 0.101755
$$825$$ 0 0
$$826$$ 1.60642e6 0.0819234
$$827$$ −2.26167e7 −1.14991 −0.574957 0.818184i $$-0.694981\pi$$
−0.574957 + 0.818184i $$0.694981\pi$$
$$828$$ 0 0
$$829$$ −2.44896e7 −1.23764 −0.618821 0.785532i $$-0.712389\pi$$
−0.618821 + 0.785532i $$0.712389\pi$$
$$830$$ 1.13426e7 0.571501
$$831$$ 0 0
$$832$$ 106496. 0.00533366
$$833$$ −1.28213e6 −0.0640208
$$834$$ 0 0
$$835$$ 2.24662e6 0.111510
$$836$$ −2.85500e7 −1.41283
$$837$$ 0 0
$$838$$ −2.11101e7 −1.03844
$$839$$ 3.13107e7 1.53563 0.767816 0.640670i $$-0.221343\pi$$
0.767816 + 0.640670i $$0.221343\pi$$
$$840$$ 0 0
$$841$$ −2.05553e6 −0.100215
$$842$$ −6.99270e6 −0.339910
$$843$$ 0 0
$$844$$ −1.15412e7 −0.557692
$$845$$ −2.00133e7 −0.964223
$$846$$ 0 0
$$847$$ 9.39746e6 0.450093
$$848$$ 7.25914e6 0.346653
$$849$$ 0 0
$$850$$ −446424. −0.0211934
$$851$$ −1.76202e6 −0.0834040
$$852$$ 0 0
$$853$$ −1.59565e7 −0.750870 −0.375435 0.926849i $$-0.622507\pi$$
−0.375435 + 0.926849i $$0.622507\pi$$
$$854$$ −5.84511e6 −0.274251
$$855$$ 0 0
$$856$$ −7.56595e6 −0.352922
$$857$$ 6.51800e6 0.303153 0.151577 0.988446i $$-0.451565\pi$$
0.151577 + 0.988446i $$0.451565\pi$$
$$858$$ 0 0
$$859$$ −1.77405e7 −0.820321 −0.410161 0.912013i $$-0.634527\pi$$
−0.410161 + 0.912013i $$0.634527\pi$$
$$860$$ 7.83475e6 0.361226
$$861$$ 0 0
$$862$$ 1.06230e7 0.486944
$$863$$ 8.57437e6 0.391900 0.195950 0.980614i $$-0.437221\pi$$
0.195950 + 0.980614i $$0.437221\pi$$
$$864$$ 0 0
$$865$$ 1.54959e7 0.704171
$$866$$ 1.24810e7 0.565531
$$867$$ 0 0
$$868$$ 6.30022e6 0.283829
$$869$$ 2.94006e7 1.32071
$$870$$ 0 0
$$871$$ −1.63498e6 −0.0730244
$$872$$ −1.32712e7 −0.591041
$$873$$ 0 0
$$874$$ 4.21762e7 1.86762
$$875$$ −8.82176e6 −0.389525
$$876$$ 0 0
$$877$$ −3.83551e7 −1.68393 −0.841964 0.539533i $$-0.818601\pi$$
−0.841964 + 0.539533i $$0.818601\pi$$
$$878$$ 1.60852e7 0.704193
$$879$$ 0 0
$$880$$ 8.21146e6 0.357449
$$881$$ 3.91651e7 1.70004 0.850020 0.526751i $$-0.176590\pi$$
0.850020 + 0.526751i $$0.176590\pi$$
$$882$$ 0 0
$$883$$ 1.72766e7 0.745688 0.372844 0.927894i $$-0.378383\pi$$
0.372844 + 0.927894i $$0.378383\pi$$
$$884$$ −222144. −0.00956101
$$885$$ 0 0
$$886$$ 3.06458e6 0.131156
$$887$$ 2.33351e7 0.995865 0.497932 0.867216i $$-0.334093\pi$$
0.497932 + 0.867216i $$0.334093\pi$$
$$888$$ 0 0
$$889$$ −6.28415e6 −0.266681
$$890$$ 1.04289e7 0.441331
$$891$$ 0 0
$$892$$ −8.58534e6 −0.361281
$$893$$ −3.42456e6 −0.143706
$$894$$ 0 0
$$895$$ −2.26900e7 −0.946843
$$896$$ −802816. −0.0334077
$$897$$ 0 0
$$898$$ 1.20784e7 0.499827
$$899$$ 3.45227e7 1.42464
$$900$$ 0 0
$$901$$ −1.51421e7 −0.621404
$$902$$ −2.34511e7 −0.959726
$$903$$ 0 0
$$904$$ 8.73062e6 0.355324
$$905$$ −913572. −0.0370784
$$906$$ 0 0
$$907$$ −3.16449e7 −1.27728 −0.638640 0.769506i $$-0.720503\pi$$
−0.638640 + 0.769506i $$0.720503\pi$$
$$908$$ −2.37917e7 −0.957658
$$909$$ 0 0
$$910$$ −275184. −0.0110159
$$911$$ −2.29551e6 −0.0916396 −0.0458198 0.998950i $$-0.514590\pi$$
−0.0458198 + 0.998950i $$0.514590\pi$$
$$912$$ 0 0
$$913$$ −3.11921e7 −1.23842
$$914$$ 892456. 0.0353363
$$915$$ 0 0
$$916$$ −1.77531e7 −0.699092
$$917$$ 1.81445e7 0.712561
$$918$$ 0 0
$$919$$ −2.80612e7 −1.09602 −0.548008 0.836473i $$-0.684614\pi$$
−0.548008 + 0.836473i $$0.684614\pi$$
$$920$$ −1.21306e7 −0.472510
$$921$$ 0 0
$$922$$ 1.83220e7 0.709816
$$923$$ −894348. −0.0345543
$$924$$ 0 0
$$925$$ 104918. 0.00403177
$$926$$ 1.69462e7 0.649448
$$927$$ 0 0
$$928$$ −4.39910e6 −0.167685
$$929$$ 6.79025e6 0.258135 0.129067 0.991636i $$-0.458802\pi$$
0.129067 + 0.991636i $$0.458802\pi$$
$$930$$ 0 0
$$931$$ −7.21260e6 −0.272721
$$932$$ 2.22073e7 0.837444
$$933$$ 0 0
$$934$$ 1.09800e7 0.411844
$$935$$ −1.71286e7 −0.640756
$$936$$ 0 0
$$937$$ 3.83161e7 1.42571 0.712857 0.701310i $$-0.247401\pi$$
0.712857 + 0.701310i $$0.247401\pi$$
$$938$$ 1.23253e7 0.457393
$$939$$ 0 0
$$940$$ 984960. 0.0363579
$$941$$ 1.16868e7 0.430250 0.215125 0.976587i $$-0.430984\pi$$
0.215125 + 0.976587i $$0.430984\pi$$
$$942$$ 0 0
$$943$$ 3.46437e7 1.26866
$$944$$ −2.09818e6 −0.0766323
$$945$$ 0 0
$$946$$ −2.15456e7 −0.782763
$$947$$ 2.08201e7 0.754410 0.377205 0.926130i $$-0.376885\pi$$
0.377205 + 0.926130i $$0.376885\pi$$
$$948$$ 0 0
$$949$$ 1.48174e6 0.0534080
$$950$$ −2.51134e6 −0.0902812
$$951$$ 0 0
$$952$$ 1.67462e6 0.0598859
$$953$$ −1.41556e7 −0.504888 −0.252444 0.967612i $$-0.581234\pi$$
−0.252444 + 0.967612i $$0.581234\pi$$
$$954$$ 0 0
$$955$$ −7.27607e6 −0.258160
$$956$$ 2.50250e7 0.885583
$$957$$ 0 0
$$958$$ 1.07051e7 0.376857
$$959$$ 633276. 0.0222355
$$960$$ 0 0
$$961$$ 3.59481e7 1.25565
$$962$$ 52208.0 0.00181886
$$963$$ 0 0
$$964$$ 2.18110e7 0.755934
$$965$$ −1.69911e7 −0.587358
$$966$$ 0 0
$$967$$ 5.29558e7 1.82116 0.910578 0.413337i $$-0.135637\pi$$
0.910578 + 0.413337i $$0.135637\pi$$
$$968$$ −1.22742e7 −0.421023
$$969$$ 0 0
$$970$$ 1.81306e7 0.618704
$$971$$ 1.07845e7 0.367072 0.183536 0.983013i $$-0.441246\pi$$
0.183536 + 0.983013i $$0.441246\pi$$
$$972$$ 0 0
$$973$$ −8.71024e6 −0.294950
$$974$$ 3.17184e7 1.07131
$$975$$ 0 0
$$976$$ 7.63443e6 0.256538
$$977$$ −4.78226e7 −1.60286 −0.801432 0.598086i $$-0.795928\pi$$
−0.801432 + 0.598086i $$0.795928\pi$$
$$978$$ 0 0
$$979$$ −2.86795e7 −0.956346
$$980$$ 2.07446e6 0.0689987
$$981$$ 0 0
$$982$$ 2.84227e7 0.940560
$$983$$ −2.96662e7 −0.979216 −0.489608 0.871943i $$-0.662860\pi$$
−0.489608 + 0.871943i $$0.662860\pi$$
$$984$$ 0 0
$$985$$ 3.22179e7 1.05805
$$986$$ 9.17626e6 0.300589
$$987$$ 0 0
$$988$$ −1.24966e6 −0.0407287
$$989$$ 3.18287e7 1.03473
$$990$$ 0 0
$$991$$ −1.39263e7 −0.450456 −0.225228 0.974306i $$-0.572313\pi$$
−0.225228 + 0.974306i $$0.572313\pi$$
$$992$$ −8.22886e6 −0.265498
$$993$$ 0 0
$$994$$ 6.74201e6 0.216433
$$995$$ −545184. −0.0174576
$$996$$ 0 0
$$997$$ −3.59999e6 −0.114700 −0.0573499 0.998354i $$-0.518265\pi$$
−0.0573499 + 0.998354i $$0.518265\pi$$
$$998$$ 5.25410e6 0.166983
$$999$$ 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.e.1.1 1
3.2 odd 2 126.6.a.g.1.1 yes 1
4.3 odd 2 1008.6.a.w.1.1 1
7.6 odd 2 882.6.a.b.1.1 1
12.11 even 2 1008.6.a.f.1.1 1
21.20 even 2 882.6.a.w.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
126.6.a.e.1.1 1 1.1 even 1 trivial
126.6.a.g.1.1 yes 1 3.2 odd 2
882.6.a.b.1.1 1 7.6 odd 2
882.6.a.w.1.1 1 21.20 even 2
1008.6.a.f.1.1 1 12.11 even 2
1008.6.a.w.1.1 1 4.3 odd 2