# Properties

 Label 784.6.a.d.1.1 Level $784$ Weight $6$ Character 784.1 Self dual yes Analytic conductor $125.741$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$125.740914733$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-12.0000 q^{3} -54.0000 q^{5} -99.0000 q^{9} +O(q^{10})$$ $$q-12.0000 q^{3} -54.0000 q^{5} -99.0000 q^{9} -540.000 q^{11} +418.000 q^{13} +648.000 q^{15} -594.000 q^{17} +836.000 q^{19} +4104.00 q^{23} -209.000 q^{25} +4104.00 q^{27} -594.000 q^{29} +4256.00 q^{31} +6480.00 q^{33} -298.000 q^{37} -5016.00 q^{39} -17226.0 q^{41} +12100.0 q^{43} +5346.00 q^{45} -1296.00 q^{47} +7128.00 q^{51} +19494.0 q^{53} +29160.0 q^{55} -10032.0 q^{57} -7668.00 q^{59} +34738.0 q^{61} -22572.0 q^{65} -21812.0 q^{67} -49248.0 q^{69} +46872.0 q^{71} -67562.0 q^{73} +2508.00 q^{75} +76912.0 q^{79} -25191.0 q^{81} +67716.0 q^{83} +32076.0 q^{85} +7128.00 q^{87} -29754.0 q^{89} -51072.0 q^{93} -45144.0 q^{95} +122398. q^{97} +53460.0 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −12.0000 −0.769800 −0.384900 0.922958i $$-0.625764\pi$$
−0.384900 + 0.922958i $$0.625764\pi$$
$$4$$ 0 0
$$5$$ −54.0000 −0.965981 −0.482991 0.875625i $$-0.660450\pi$$
−0.482991 + 0.875625i $$0.660450\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −99.0000 −0.407407
$$10$$ 0 0
$$11$$ −540.000 −1.34559 −0.672794 0.739830i $$-0.734906\pi$$
−0.672794 + 0.739830i $$0.734906\pi$$
$$12$$ 0 0
$$13$$ 418.000 0.685990 0.342995 0.939337i $$-0.388559\pi$$
0.342995 + 0.939337i $$0.388559\pi$$
$$14$$ 0 0
$$15$$ 648.000 0.743613
$$16$$ 0 0
$$17$$ −594.000 −0.498499 −0.249249 0.968439i $$-0.580184\pi$$
−0.249249 + 0.968439i $$0.580184\pi$$
$$18$$ 0 0
$$19$$ 836.000 0.531279 0.265639 0.964072i $$-0.414417\pi$$
0.265639 + 0.964072i $$0.414417\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4104.00 1.61766 0.808831 0.588041i $$-0.200101\pi$$
0.808831 + 0.588041i $$0.200101\pi$$
$$24$$ 0 0
$$25$$ −209.000 −0.0668800
$$26$$ 0 0
$$27$$ 4104.00 1.08342
$$28$$ 0 0
$$29$$ −594.000 −0.131157 −0.0655785 0.997847i $$-0.520889\pi$$
−0.0655785 + 0.997847i $$0.520889\pi$$
$$30$$ 0 0
$$31$$ 4256.00 0.795422 0.397711 0.917511i $$-0.369805\pi$$
0.397711 + 0.917511i $$0.369805\pi$$
$$32$$ 0 0
$$33$$ 6480.00 1.03583
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −298.000 −0.0357859 −0.0178930 0.999840i $$-0.505696\pi$$
−0.0178930 + 0.999840i $$0.505696\pi$$
$$38$$ 0 0
$$39$$ −5016.00 −0.528075
$$40$$ 0 0
$$41$$ −17226.0 −1.60039 −0.800193 0.599742i $$-0.795270\pi$$
−0.800193 + 0.599742i $$0.795270\pi$$
$$42$$ 0 0
$$43$$ 12100.0 0.997963 0.498981 0.866613i $$-0.333708\pi$$
0.498981 + 0.866613i $$0.333708\pi$$
$$44$$ 0 0
$$45$$ 5346.00 0.393548
$$46$$ 0 0
$$47$$ −1296.00 −0.0855777 −0.0427888 0.999084i $$-0.513624\pi$$
−0.0427888 + 0.999084i $$0.513624\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 7128.00 0.383745
$$52$$ 0 0
$$53$$ 19494.0 0.953260 0.476630 0.879104i $$-0.341858\pi$$
0.476630 + 0.879104i $$0.341858\pi$$
$$54$$ 0 0
$$55$$ 29160.0 1.29981
$$56$$ 0 0
$$57$$ −10032.0 −0.408978
$$58$$ 0 0
$$59$$ −7668.00 −0.286782 −0.143391 0.989666i $$-0.545801\pi$$
−0.143391 + 0.989666i $$0.545801\pi$$
$$60$$ 0 0
$$61$$ 34738.0 1.19531 0.597655 0.801754i $$-0.296099\pi$$
0.597655 + 0.801754i $$0.296099\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −22572.0 −0.662654
$$66$$ 0 0
$$67$$ −21812.0 −0.593620 −0.296810 0.954937i $$-0.595923\pi$$
−0.296810 + 0.954937i $$0.595923\pi$$
$$68$$ 0 0
$$69$$ −49248.0 −1.24528
$$70$$ 0 0
$$71$$ 46872.0 1.10349 0.551744 0.834014i $$-0.313963\pi$$
0.551744 + 0.834014i $$0.313963\pi$$
$$72$$ 0 0
$$73$$ −67562.0 −1.48387 −0.741934 0.670473i $$-0.766091\pi$$
−0.741934 + 0.670473i $$0.766091\pi$$
$$74$$ 0 0
$$75$$ 2508.00 0.0514842
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 76912.0 1.38652 0.693260 0.720687i $$-0.256174\pi$$
0.693260 + 0.720687i $$0.256174\pi$$
$$80$$ 0 0
$$81$$ −25191.0 −0.426612
$$82$$ 0 0
$$83$$ 67716.0 1.07894 0.539468 0.842006i $$-0.318625\pi$$
0.539468 + 0.842006i $$0.318625\pi$$
$$84$$ 0 0
$$85$$ 32076.0 0.481541
$$86$$ 0 0
$$87$$ 7128.00 0.100965
$$88$$ 0 0
$$89$$ −29754.0 −0.398172 −0.199086 0.979982i $$-0.563797\pi$$
−0.199086 + 0.979982i $$0.563797\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −51072.0 −0.612316
$$94$$ 0 0
$$95$$ −45144.0 −0.513205
$$96$$ 0 0
$$97$$ 122398. 1.32082 0.660412 0.750903i $$-0.270382\pi$$
0.660412 + 0.750903i $$0.270382\pi$$
$$98$$ 0 0
$$99$$ 53460.0 0.548202
$$100$$ 0 0
$$101$$ −11286.0 −0.110087 −0.0550436 0.998484i $$-0.517530\pi$$
−0.0550436 + 0.998484i $$0.517530\pi$$
$$102$$ 0 0
$$103$$ −27256.0 −0.253145 −0.126572 0.991957i $$-0.540398\pi$$
−0.126572 + 0.991957i $$0.540398\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −122364. −1.03322 −0.516612 0.856220i $$-0.672807\pi$$
−0.516612 + 0.856220i $$0.672807\pi$$
$$108$$ 0 0
$$109$$ 99902.0 0.805393 0.402697 0.915334i $$-0.368073\pi$$
0.402697 + 0.915334i $$0.368073\pi$$
$$110$$ 0 0
$$111$$ 3576.00 0.0275480
$$112$$ 0 0
$$113$$ −29646.0 −0.218409 −0.109204 0.994019i $$-0.534830\pi$$
−0.109204 + 0.994019i $$0.534830\pi$$
$$114$$ 0 0
$$115$$ −221616. −1.56263
$$116$$ 0 0
$$117$$ −41382.0 −0.279477
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 130549. 0.810607
$$122$$ 0 0
$$123$$ 206712. 1.23198
$$124$$ 0 0
$$125$$ 180036. 1.03059
$$126$$ 0 0
$$127$$ −336512. −1.85136 −0.925681 0.378305i $$-0.876507\pi$$
−0.925681 + 0.378305i $$0.876507\pi$$
$$128$$ 0 0
$$129$$ −145200. −0.768232
$$130$$ 0 0
$$131$$ 100980. 0.514111 0.257056 0.966397i $$-0.417248\pi$$
0.257056 + 0.966397i $$0.417248\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −221616. −1.04657
$$136$$ 0 0
$$137$$ −317142. −1.44362 −0.721809 0.692092i $$-0.756689\pi$$
−0.721809 + 0.692092i $$0.756689\pi$$
$$138$$ 0 0
$$139$$ −148324. −0.651140 −0.325570 0.945518i $$-0.605556\pi$$
−0.325570 + 0.945518i $$0.605556\pi$$
$$140$$ 0 0
$$141$$ 15552.0 0.0658777
$$142$$ 0 0
$$143$$ −225720. −0.923060
$$144$$ 0 0
$$145$$ 32076.0 0.126695
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 196614. 0.725519 0.362759 0.931883i $$-0.381835\pi$$
0.362759 + 0.931883i $$0.381835\pi$$
$$150$$ 0 0
$$151$$ −74360.0 −0.265398 −0.132699 0.991156i $$-0.542364\pi$$
−0.132699 + 0.991156i $$0.542364\pi$$
$$152$$ 0 0
$$153$$ 58806.0 0.203092
$$154$$ 0 0
$$155$$ −229824. −0.768362
$$156$$ 0 0
$$157$$ −120878. −0.391380 −0.195690 0.980666i $$-0.562695\pi$$
−0.195690 + 0.980666i $$0.562695\pi$$
$$158$$ 0 0
$$159$$ −233928. −0.733820
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 111340. 0.328233 0.164116 0.986441i $$-0.447523\pi$$
0.164116 + 0.986441i $$0.447523\pi$$
$$164$$ 0 0
$$165$$ −349920. −1.00060
$$166$$ 0 0
$$167$$ −491832. −1.36466 −0.682332 0.731043i $$-0.739034\pi$$
−0.682332 + 0.731043i $$0.739034\pi$$
$$168$$ 0 0
$$169$$ −196569. −0.529417
$$170$$ 0 0
$$171$$ −82764.0 −0.216447
$$172$$ 0 0
$$173$$ −707454. −1.79714 −0.898572 0.438826i $$-0.855395\pi$$
−0.898572 + 0.438826i $$0.855395\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 92016.0 0.220765
$$178$$ 0 0
$$179$$ −493668. −1.15160 −0.575801 0.817590i $$-0.695310\pi$$
−0.575801 + 0.817590i $$0.695310\pi$$
$$180$$ 0 0
$$181$$ 559450. 1.26930 0.634651 0.772799i $$-0.281144\pi$$
0.634651 + 0.772799i $$0.281144\pi$$
$$182$$ 0 0
$$183$$ −416856. −0.920149
$$184$$ 0 0
$$185$$ 16092.0 0.0345685
$$186$$ 0 0
$$187$$ 320760. 0.670774
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 724032. 1.43607 0.718033 0.696009i $$-0.245043\pi$$
0.718033 + 0.696009i $$0.245043\pi$$
$$192$$ 0 0
$$193$$ 7106.00 0.0137319 0.00686597 0.999976i $$-0.497814\pi$$
0.00686597 + 0.999976i $$0.497814\pi$$
$$194$$ 0 0
$$195$$ 270864. 0.510111
$$196$$ 0 0
$$197$$ −530442. −0.973806 −0.486903 0.873456i $$-0.661873\pi$$
−0.486903 + 0.873456i $$0.661873\pi$$
$$198$$ 0 0
$$199$$ 56168.0 0.100544 0.0502720 0.998736i $$-0.483991\pi$$
0.0502720 + 0.998736i $$0.483991\pi$$
$$200$$ 0 0
$$201$$ 261744. 0.456969
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 930204. 1.54594
$$206$$ 0 0
$$207$$ −406296. −0.659047
$$208$$ 0 0
$$209$$ −451440. −0.714882
$$210$$ 0 0
$$211$$ 339196. 0.524499 0.262249 0.965000i $$-0.415536\pi$$
0.262249 + 0.965000i $$0.415536\pi$$
$$212$$ 0 0
$$213$$ −562464. −0.849465
$$214$$ 0 0
$$215$$ −653400. −0.964013
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 810744. 1.14228
$$220$$ 0 0
$$221$$ −248292. −0.341965
$$222$$ 0 0
$$223$$ 779360. 1.04948 0.524742 0.851261i $$-0.324162\pi$$
0.524742 + 0.851261i $$0.324162\pi$$
$$224$$ 0 0
$$225$$ 20691.0 0.0272474
$$226$$ 0 0
$$227$$ −744876. −0.959443 −0.479722 0.877421i $$-0.659262\pi$$
−0.479722 + 0.877421i $$0.659262\pi$$
$$228$$ 0 0
$$229$$ 272746. 0.343692 0.171846 0.985124i $$-0.445027\pi$$
0.171846 + 0.985124i $$0.445027\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −153846. −0.185651 −0.0928253 0.995682i $$-0.529590\pi$$
−0.0928253 + 0.995682i $$0.529590\pi$$
$$234$$ 0 0
$$235$$ 69984.0 0.0826664
$$236$$ 0 0
$$237$$ −922944. −1.06734
$$238$$ 0 0
$$239$$ −1.15474e6 −1.30764 −0.653820 0.756650i $$-0.726834\pi$$
−0.653820 + 0.756650i $$0.726834\pi$$
$$240$$ 0 0
$$241$$ −657074. −0.728738 −0.364369 0.931255i $$-0.618715\pi$$
−0.364369 + 0.931255i $$0.618715\pi$$
$$242$$ 0 0
$$243$$ −694980. −0.755017
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 349448. 0.364452
$$248$$ 0 0
$$249$$ −812592. −0.830566
$$250$$ 0 0
$$251$$ 1.34190e6 1.34442 0.672211 0.740359i $$-0.265345\pi$$
0.672211 + 0.740359i $$0.265345\pi$$
$$252$$ 0 0
$$253$$ −2.21616e6 −2.17671
$$254$$ 0 0
$$255$$ −384912. −0.370690
$$256$$ 0 0
$$257$$ −132354. −0.124998 −0.0624992 0.998045i $$-0.519907\pi$$
−0.0624992 + 0.998045i $$0.519907\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 58806.0 0.0534343
$$262$$ 0 0
$$263$$ −943272. −0.840906 −0.420453 0.907314i $$-0.638129\pi$$
−0.420453 + 0.907314i $$0.638129\pi$$
$$264$$ 0 0
$$265$$ −1.05268e6 −0.920831
$$266$$ 0 0
$$267$$ 357048. 0.306513
$$268$$ 0 0
$$269$$ −967518. −0.815227 −0.407613 0.913155i $$-0.633639\pi$$
−0.407613 + 0.913155i $$0.633639\pi$$
$$270$$ 0 0
$$271$$ −518320. −0.428721 −0.214360 0.976755i $$-0.568767\pi$$
−0.214360 + 0.976755i $$0.568767\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 112860. 0.0899929
$$276$$ 0 0
$$277$$ 2.22273e6 1.74055 0.870275 0.492566i $$-0.163941\pi$$
0.870275 + 0.492566i $$0.163941\pi$$
$$278$$ 0 0
$$279$$ −421344. −0.324061
$$280$$ 0 0
$$281$$ −196614. −0.148542 −0.0742709 0.997238i $$-0.523663\pi$$
−0.0742709 + 0.997238i $$0.523663\pi$$
$$282$$ 0 0
$$283$$ −1.55228e6 −1.15213 −0.576067 0.817403i $$-0.695413\pi$$
−0.576067 + 0.817403i $$0.695413\pi$$
$$284$$ 0 0
$$285$$ 541728. 0.395066
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.06702e6 −0.751499
$$290$$ 0 0
$$291$$ −1.46878e6 −1.01677
$$292$$ 0 0
$$293$$ 1.07217e6 0.729616 0.364808 0.931083i $$-0.381135\pi$$
0.364808 + 0.931083i $$0.381135\pi$$
$$294$$ 0 0
$$295$$ 414072. 0.277026
$$296$$ 0 0
$$297$$ −2.21616e6 −1.45784
$$298$$ 0 0
$$299$$ 1.71547e6 1.10970
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 135432. 0.0847451
$$304$$ 0 0
$$305$$ −1.87585e6 −1.15465
$$306$$ 0 0
$$307$$ 1.58589e6 0.960346 0.480173 0.877174i $$-0.340574\pi$$
0.480173 + 0.877174i $$0.340574\pi$$
$$308$$ 0 0
$$309$$ 327072. 0.194871
$$310$$ 0 0
$$311$$ −730728. −0.428405 −0.214203 0.976789i $$-0.568715\pi$$
−0.214203 + 0.976789i $$0.568715\pi$$
$$312$$ 0 0
$$313$$ −584858. −0.337435 −0.168717 0.985664i $$-0.553962\pi$$
−0.168717 + 0.985664i $$0.553962\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.48287e6 −1.38773 −0.693865 0.720105i $$-0.744094\pi$$
−0.693865 + 0.720105i $$0.744094\pi$$
$$318$$ 0 0
$$319$$ 320760. 0.176483
$$320$$ 0 0
$$321$$ 1.46837e6 0.795376
$$322$$ 0 0
$$323$$ −496584. −0.264842
$$324$$ 0 0
$$325$$ −87362.0 −0.0458790
$$326$$ 0 0
$$327$$ −1.19882e6 −0.619992
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −377948. −0.189610 −0.0948052 0.995496i $$-0.530223\pi$$
−0.0948052 + 0.995496i $$0.530223\pi$$
$$332$$ 0 0
$$333$$ 29502.0 0.0145794
$$334$$ 0 0
$$335$$ 1.17785e6 0.573426
$$336$$ 0 0
$$337$$ 639122. 0.306555 0.153278 0.988183i $$-0.451017\pi$$
0.153278 + 0.988183i $$0.451017\pi$$
$$338$$ 0 0
$$339$$ 355752. 0.168131
$$340$$ 0 0
$$341$$ −2.29824e6 −1.07031
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 2.65939e6 1.20291
$$346$$ 0 0
$$347$$ 2.90466e6 1.29501 0.647503 0.762063i $$-0.275813\pi$$
0.647503 + 0.762063i $$0.275813\pi$$
$$348$$ 0 0
$$349$$ 3.99157e6 1.75420 0.877102 0.480304i $$-0.159474\pi$$
0.877102 + 0.480304i $$0.159474\pi$$
$$350$$ 0 0
$$351$$ 1.71547e6 0.743217
$$352$$ 0 0
$$353$$ −1.42922e6 −0.610466 −0.305233 0.952278i $$-0.598734\pi$$
−0.305233 + 0.952278i $$0.598734\pi$$
$$354$$ 0 0
$$355$$ −2.53109e6 −1.06595
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1.16186e6 −0.475794 −0.237897 0.971290i $$-0.576458\pi$$
−0.237897 + 0.971290i $$0.576458\pi$$
$$360$$ 0 0
$$361$$ −1.77720e6 −0.717743
$$362$$ 0 0
$$363$$ −1.56659e6 −0.624005
$$364$$ 0 0
$$365$$ 3.64835e6 1.43339
$$366$$ 0 0
$$367$$ −1.08923e6 −0.422139 −0.211069 0.977471i $$-0.567695\pi$$
−0.211069 + 0.977471i $$0.567695\pi$$
$$368$$ 0 0
$$369$$ 1.70537e6 0.652009
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3.50577e6 1.30470 0.652350 0.757918i $$-0.273783\pi$$
0.652350 + 0.757918i $$0.273783\pi$$
$$374$$ 0 0
$$375$$ −2.16043e6 −0.793346
$$376$$ 0 0
$$377$$ −248292. −0.0899724
$$378$$ 0 0
$$379$$ −4.04385e6 −1.44610 −0.723048 0.690798i $$-0.757260\pi$$
−0.723048 + 0.690798i $$0.757260\pi$$
$$380$$ 0 0
$$381$$ 4.03814e6 1.42518
$$382$$ 0 0
$$383$$ 5.18746e6 1.80700 0.903499 0.428591i $$-0.140990\pi$$
0.903499 + 0.428591i $$0.140990\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.19790e6 −0.406577
$$388$$ 0 0
$$389$$ −950346. −0.318425 −0.159213 0.987244i $$-0.550896\pi$$
−0.159213 + 0.987244i $$0.550896\pi$$
$$390$$ 0 0
$$391$$ −2.43778e6 −0.806403
$$392$$ 0 0
$$393$$ −1.21176e6 −0.395763
$$394$$ 0 0
$$395$$ −4.15325e6 −1.33935
$$396$$ 0 0
$$397$$ 520738. 0.165822 0.0829112 0.996557i $$-0.473578\pi$$
0.0829112 + 0.996557i $$0.473578\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 764370. 0.237379 0.118690 0.992931i $$-0.462131\pi$$
0.118690 + 0.992931i $$0.462131\pi$$
$$402$$ 0 0
$$403$$ 1.77901e6 0.545651
$$404$$ 0 0
$$405$$ 1.36031e6 0.412099
$$406$$ 0 0
$$407$$ 160920. 0.0481531
$$408$$ 0 0
$$409$$ −2.64051e6 −0.780511 −0.390255 0.920707i $$-0.627613\pi$$
−0.390255 + 0.920707i $$0.627613\pi$$
$$410$$ 0 0
$$411$$ 3.80570e6 1.11130
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −3.65666e6 −1.04223
$$416$$ 0 0
$$417$$ 1.77989e6 0.501248
$$418$$ 0 0
$$419$$ −4.98020e6 −1.38584 −0.692918 0.721016i $$-0.743675\pi$$
−0.692918 + 0.721016i $$0.743675\pi$$
$$420$$ 0 0
$$421$$ −237994. −0.0654426 −0.0327213 0.999465i $$-0.510417\pi$$
−0.0327213 + 0.999465i $$0.510417\pi$$
$$422$$ 0 0
$$423$$ 128304. 0.0348650
$$424$$ 0 0
$$425$$ 124146. 0.0333396
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 2.70864e6 0.710572
$$430$$ 0 0
$$431$$ 3.88238e6 1.00671 0.503356 0.864079i $$-0.332098\pi$$
0.503356 + 0.864079i $$0.332098\pi$$
$$432$$ 0 0
$$433$$ 66958.0 0.0171626 0.00858129 0.999963i $$-0.497268\pi$$
0.00858129 + 0.999963i $$0.497268\pi$$
$$434$$ 0 0
$$435$$ −384912. −0.0975300
$$436$$ 0 0
$$437$$ 3.43094e6 0.859429
$$438$$ 0 0
$$439$$ −6.50135e6 −1.61006 −0.805031 0.593233i $$-0.797851\pi$$
−0.805031 + 0.593233i $$0.797851\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.60760e6 1.11549 0.557745 0.830012i $$-0.311667\pi$$
0.557745 + 0.830012i $$0.311667\pi$$
$$444$$ 0 0
$$445$$ 1.60672e6 0.384626
$$446$$ 0 0
$$447$$ −2.35937e6 −0.558505
$$448$$ 0 0
$$449$$ 3.77671e6 0.884092 0.442046 0.896992i $$-0.354253\pi$$
0.442046 + 0.896992i $$0.354253\pi$$
$$450$$ 0 0
$$451$$ 9.30204e6 2.15346
$$452$$ 0 0
$$453$$ 892320. 0.204303
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3.18069e6 −0.712412 −0.356206 0.934407i $$-0.615930\pi$$
−0.356206 + 0.934407i $$0.615930\pi$$
$$458$$ 0 0
$$459$$ −2.43778e6 −0.540085
$$460$$ 0 0
$$461$$ −6.68547e6 −1.46514 −0.732571 0.680691i $$-0.761680\pi$$
−0.732571 + 0.680691i $$0.761680\pi$$
$$462$$ 0 0
$$463$$ 4.35122e6 0.943318 0.471659 0.881781i $$-0.343655\pi$$
0.471659 + 0.881781i $$0.343655\pi$$
$$464$$ 0 0
$$465$$ 2.75789e6 0.591486
$$466$$ 0 0
$$467$$ 7.07994e6 1.50223 0.751117 0.660170i $$-0.229516\pi$$
0.751117 + 0.660170i $$0.229516\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 1.45054e6 0.301284
$$472$$ 0 0
$$473$$ −6.53400e6 −1.34285
$$474$$ 0 0
$$475$$ −174724. −0.0355319
$$476$$ 0 0
$$477$$ −1.92991e6 −0.388365
$$478$$ 0 0
$$479$$ 3.22186e6 0.641604 0.320802 0.947146i $$-0.396048\pi$$
0.320802 + 0.947146i $$0.396048\pi$$
$$480$$ 0 0
$$481$$ −124564. −0.0245488
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −6.60949e6 −1.27589
$$486$$ 0 0
$$487$$ −2.29710e6 −0.438891 −0.219446 0.975625i $$-0.570425\pi$$
−0.219446 + 0.975625i $$0.570425\pi$$
$$488$$ 0 0
$$489$$ −1.33608e6 −0.252674
$$490$$ 0 0
$$491$$ −2.82150e6 −0.528173 −0.264087 0.964499i $$-0.585070\pi$$
−0.264087 + 0.964499i $$0.585070\pi$$
$$492$$ 0 0
$$493$$ 352836. 0.0653816
$$494$$ 0 0
$$495$$ −2.88684e6 −0.529553
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.13628e6 0.743634 0.371817 0.928306i $$-0.378735\pi$$
0.371817 + 0.928306i $$0.378735\pi$$
$$500$$ 0 0
$$501$$ 5.90198e6 1.05052
$$502$$ 0 0
$$503$$ 8.33263e6 1.46846 0.734230 0.678901i $$-0.237543\pi$$
0.734230 + 0.678901i $$0.237543\pi$$
$$504$$ 0 0
$$505$$ 609444. 0.106342
$$506$$ 0 0
$$507$$ 2.35883e6 0.407546
$$508$$ 0 0
$$509$$ −4.34101e6 −0.742670 −0.371335 0.928499i $$-0.621100\pi$$
−0.371335 + 0.928499i $$0.621100\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 3.43094e6 0.575599
$$514$$ 0 0
$$515$$ 1.47182e6 0.244533
$$516$$ 0 0
$$517$$ 699840. 0.115152
$$518$$ 0 0
$$519$$ 8.48945e6 1.38344
$$520$$ 0 0
$$521$$ 6.74185e6 1.08814 0.544070 0.839040i $$-0.316883\pi$$
0.544070 + 0.839040i $$0.316883\pi$$
$$522$$ 0 0
$$523$$ −7.72196e6 −1.23445 −0.617224 0.786787i $$-0.711743\pi$$
−0.617224 + 0.786787i $$0.711743\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.52806e6 −0.396517
$$528$$ 0 0
$$529$$ 1.04065e7 1.61683
$$530$$ 0 0
$$531$$ 759132. 0.116837
$$532$$ 0 0
$$533$$ −7.20047e6 −1.09785
$$534$$ 0 0
$$535$$ 6.60766e6 0.998075
$$536$$ 0 0
$$537$$ 5.92402e6 0.886504
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −682066. −0.100192 −0.0500960 0.998744i $$-0.515953\pi$$
−0.0500960 + 0.998744i $$0.515953\pi$$
$$542$$ 0 0
$$543$$ −6.71340e6 −0.977109
$$544$$ 0 0
$$545$$ −5.39471e6 −0.777995
$$546$$ 0 0
$$547$$ −2.15772e6 −0.308337 −0.154169 0.988045i $$-0.549270\pi$$
−0.154169 + 0.988045i $$0.549270\pi$$
$$548$$ 0 0
$$549$$ −3.43906e6 −0.486978
$$550$$ 0 0
$$551$$ −496584. −0.0696809
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −193104. −0.0266109
$$556$$ 0 0
$$557$$ −2.67597e6 −0.365463 −0.182731 0.983163i $$-0.558494\pi$$
−0.182731 + 0.983163i $$0.558494\pi$$
$$558$$ 0 0
$$559$$ 5.05780e6 0.684592
$$560$$ 0 0
$$561$$ −3.84912e6 −0.516362
$$562$$ 0 0
$$563$$ −3.55331e6 −0.472457 −0.236228 0.971698i $$-0.575911\pi$$
−0.236228 + 0.971698i $$0.575911\pi$$
$$564$$ 0 0
$$565$$ 1.60088e6 0.210979
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −1.29225e7 −1.67327 −0.836633 0.547764i $$-0.815479\pi$$
−0.836633 + 0.547764i $$0.815479\pi$$
$$570$$ 0 0
$$571$$ 6.08357e6 0.780851 0.390426 0.920634i $$-0.372328\pi$$
0.390426 + 0.920634i $$0.372328\pi$$
$$572$$ 0 0
$$573$$ −8.68838e6 −1.10548
$$574$$ 0 0
$$575$$ −857736. −0.108189
$$576$$ 0 0
$$577$$ 1.58241e7 1.97869 0.989347 0.145579i $$-0.0465047\pi$$
0.989347 + 0.145579i $$0.0465047\pi$$
$$578$$ 0 0
$$579$$ −85272.0 −0.0105709
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −1.05268e7 −1.28269
$$584$$ 0 0
$$585$$ 2.23463e6 0.269970
$$586$$ 0 0
$$587$$ 4.60220e6 0.551278 0.275639 0.961261i $$-0.411111\pi$$
0.275639 + 0.961261i $$0.411111\pi$$
$$588$$ 0 0
$$589$$ 3.55802e6 0.422590
$$590$$ 0 0
$$591$$ 6.36530e6 0.749636
$$592$$ 0 0
$$593$$ −8.61122e6 −1.00561 −0.502803 0.864401i $$-0.667698\pi$$
−0.502803 + 0.864401i $$0.667698\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −674016. −0.0773988
$$598$$ 0 0
$$599$$ 7.98228e6 0.908992 0.454496 0.890749i $$-0.349819\pi$$
0.454496 + 0.890749i $$0.349819\pi$$
$$600$$ 0 0
$$601$$ −1.01740e7 −1.14896 −0.574481 0.818518i $$-0.694796\pi$$
−0.574481 + 0.818518i $$0.694796\pi$$
$$602$$ 0 0
$$603$$ 2.15939e6 0.241845
$$604$$ 0 0
$$605$$ −7.04965e6 −0.783031
$$606$$ 0 0
$$607$$ −9.95843e6 −1.09703 −0.548516 0.836140i $$-0.684807\pi$$
−0.548516 + 0.836140i $$0.684807\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −541728. −0.0587054
$$612$$ 0 0
$$613$$ 4.19586e6 0.450993 0.225497 0.974244i $$-0.427600\pi$$
0.225497 + 0.974244i $$0.427600\pi$$
$$614$$ 0 0
$$615$$ −1.11624e7 −1.19007
$$616$$ 0 0
$$617$$ 9.12551e6 0.965038 0.482519 0.875885i $$-0.339722\pi$$
0.482519 + 0.875885i $$0.339722\pi$$
$$618$$ 0 0
$$619$$ 6.45734e6 0.677372 0.338686 0.940900i $$-0.390018\pi$$
0.338686 + 0.940900i $$0.390018\pi$$
$$620$$ 0 0
$$621$$ 1.68428e7 1.75261
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −9.06882e6 −0.928647
$$626$$ 0 0
$$627$$ 5.41728e6 0.550316
$$628$$ 0 0
$$629$$ 177012. 0.0178392
$$630$$ 0 0
$$631$$ 1.40514e7 1.40490 0.702450 0.711733i $$-0.252090\pi$$
0.702450 + 0.711733i $$0.252090\pi$$
$$632$$ 0 0
$$633$$ −4.07035e6 −0.403759
$$634$$ 0 0
$$635$$ 1.81716e7 1.78838
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −4.64033e6 −0.449569
$$640$$ 0 0
$$641$$ 8.47168e6 0.814375 0.407188 0.913345i $$-0.366510\pi$$
0.407188 + 0.913345i $$0.366510\pi$$
$$642$$ 0 0
$$643$$ 488564. 0.0466009 0.0233004 0.999729i $$-0.492583\pi$$
0.0233004 + 0.999729i $$0.492583\pi$$
$$644$$ 0 0
$$645$$ 7.84080e6 0.742098
$$646$$ 0 0
$$647$$ 2.48119e6 0.233023 0.116512 0.993189i $$-0.462829\pi$$
0.116512 + 0.993189i $$0.462829\pi$$
$$648$$ 0 0
$$649$$ 4.14072e6 0.385891
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −5.29130e6 −0.485601 −0.242800 0.970076i $$-0.578066\pi$$
−0.242800 + 0.970076i $$0.578066\pi$$
$$654$$ 0 0
$$655$$ −5.45292e6 −0.496622
$$656$$ 0 0
$$657$$ 6.68864e6 0.604539
$$658$$ 0 0
$$659$$ −4.72468e6 −0.423798 −0.211899 0.977292i $$-0.567965\pi$$
−0.211899 + 0.977292i $$0.567965\pi$$
$$660$$ 0 0
$$661$$ 6.17420e6 0.549639 0.274819 0.961496i $$-0.411382\pi$$
0.274819 + 0.961496i $$0.411382\pi$$
$$662$$ 0 0
$$663$$ 2.97950e6 0.263245
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2.43778e6 −0.212168
$$668$$ 0 0
$$669$$ −9.35232e6 −0.807893
$$670$$ 0 0
$$671$$ −1.87585e7 −1.60839
$$672$$ 0 0
$$673$$ −9.40925e6 −0.800787 −0.400394 0.916343i $$-0.631127\pi$$
−0.400394 + 0.916343i $$0.631127\pi$$
$$674$$ 0 0
$$675$$ −857736. −0.0724593
$$676$$ 0 0
$$677$$ −1.50086e7 −1.25854 −0.629272 0.777185i $$-0.716647\pi$$
−0.629272 + 0.777185i $$0.716647\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 8.93851e6 0.738580
$$682$$ 0 0
$$683$$ 1.29707e7 1.06393 0.531963 0.846768i $$-0.321455\pi$$
0.531963 + 0.846768i $$0.321455\pi$$
$$684$$ 0 0
$$685$$ 1.71257e7 1.39451
$$686$$ 0 0
$$687$$ −3.27295e6 −0.264574
$$688$$ 0 0
$$689$$ 8.14849e6 0.653927
$$690$$ 0 0
$$691$$ 2.26556e7 1.80501 0.902506 0.430677i $$-0.141725\pi$$
0.902506 + 0.430677i $$0.141725\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 8.00950e6 0.628989
$$696$$ 0 0
$$697$$ 1.02322e7 0.797791
$$698$$ 0 0
$$699$$ 1.84615e6 0.142914
$$700$$ 0 0
$$701$$ 1.90169e7 1.46166 0.730828 0.682562i $$-0.239134\pi$$
0.730828 + 0.682562i $$0.239134\pi$$
$$702$$ 0 0
$$703$$ −249128. −0.0190123
$$704$$ 0 0
$$705$$ −839808. −0.0636366
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.51311e7 1.13046 0.565231 0.824933i $$-0.308787\pi$$
0.565231 + 0.824933i $$0.308787\pi$$
$$710$$ 0 0
$$711$$ −7.61429e6 −0.564879
$$712$$ 0 0
$$713$$ 1.74666e7 1.28672
$$714$$ 0 0
$$715$$ 1.21889e7 0.891659
$$716$$ 0 0
$$717$$ 1.38568e7 1.00662
$$718$$ 0 0
$$719$$ −1.50323e7 −1.08443 −0.542217 0.840238i $$-0.682415\pi$$
−0.542217 + 0.840238i $$0.682415\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 7.88489e6 0.560983
$$724$$ 0 0
$$725$$ 124146. 0.00877178
$$726$$ 0 0
$$727$$ −7.41230e6 −0.520136 −0.260068 0.965590i $$-0.583745\pi$$
−0.260068 + 0.965590i $$0.583745\pi$$
$$728$$ 0 0
$$729$$ 1.44612e7 1.00782
$$730$$ 0 0
$$731$$ −7.18740e6 −0.497483
$$732$$ 0 0
$$733$$ 2.77928e6 0.191061 0.0955306 0.995426i $$-0.469545\pi$$
0.0955306 + 0.995426i $$0.469545\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.17785e7 0.798768
$$738$$ 0 0
$$739$$ 1.21046e7 0.815342 0.407671 0.913129i $$-0.366341\pi$$
0.407671 + 0.913129i $$0.366341\pi$$
$$740$$ 0 0
$$741$$ −4.19338e6 −0.280555
$$742$$ 0 0
$$743$$ −4.46926e6 −0.297005 −0.148502 0.988912i $$-0.547445\pi$$
−0.148502 + 0.988912i $$0.547445\pi$$
$$744$$ 0 0
$$745$$ −1.06172e7 −0.700838
$$746$$ 0 0
$$747$$ −6.70388e6 −0.439567
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −2.88463e7 −1.86634 −0.933168 0.359442i $$-0.882967\pi$$
−0.933168 + 0.359442i $$0.882967\pi$$
$$752$$ 0 0
$$753$$ −1.61028e7 −1.03494
$$754$$ 0 0
$$755$$ 4.01544e6 0.256369
$$756$$ 0 0
$$757$$ 9.60868e6 0.609430 0.304715 0.952444i $$-0.401439\pi$$
0.304715 + 0.952444i $$0.401439\pi$$
$$758$$ 0 0
$$759$$ 2.65939e7 1.67563
$$760$$ 0 0
$$761$$ −4.54588e6 −0.284549 −0.142274 0.989827i $$-0.545442\pi$$
−0.142274 + 0.989827i $$0.545442\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −3.17552e6 −0.196183
$$766$$ 0 0
$$767$$ −3.20522e6 −0.196730
$$768$$ 0 0
$$769$$ 2.15923e7 1.31669 0.658345 0.752716i $$-0.271257\pi$$
0.658345 + 0.752716i $$0.271257\pi$$
$$770$$ 0 0
$$771$$ 1.58825e6 0.0962238
$$772$$ 0 0
$$773$$ 1.48400e7 0.893276 0.446638 0.894715i $$-0.352621\pi$$
0.446638 + 0.894715i $$0.352621\pi$$
$$774$$ 0 0
$$775$$ −889504. −0.0531978
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.44009e7 −0.850251
$$780$$ 0 0
$$781$$ −2.53109e7 −1.48484
$$782$$ 0 0
$$783$$ −2.43778e6 −0.142098
$$784$$ 0 0
$$785$$ 6.52741e6 0.378065
$$786$$ 0 0
$$787$$ −2.48785e7 −1.43182 −0.715909 0.698194i $$-0.753987\pi$$
−0.715909 + 0.698194i $$0.753987\pi$$
$$788$$ 0 0
$$789$$ 1.13193e7 0.647330
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.45205e7 0.819970
$$794$$ 0 0
$$795$$ 1.26321e7 0.708856
$$796$$ 0 0
$$797$$ −3.16080e7 −1.76259 −0.881294 0.472568i $$-0.843327\pi$$
−0.881294 + 0.472568i $$0.843327\pi$$
$$798$$ 0 0
$$799$$ 769824. 0.0426604
$$800$$ 0 0
$$801$$ 2.94565e6 0.162218
$$802$$ 0 0
$$803$$ 3.64835e7 1.99668
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.16102e7 0.627562
$$808$$ 0 0
$$809$$ −3.10009e6 −0.166534 −0.0832669 0.996527i $$-0.526535\pi$$
−0.0832669 + 0.996527i $$0.526535\pi$$
$$810$$ 0 0
$$811$$ 1.87180e6 0.0999328 0.0499664 0.998751i $$-0.484089\pi$$
0.0499664 + 0.998751i $$0.484089\pi$$
$$812$$ 0 0
$$813$$ 6.21984e6 0.330030
$$814$$ 0 0
$$815$$ −6.01236e6 −0.317067
$$816$$ 0 0
$$817$$ 1.01156e7 0.530196
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.00184e7 −1.03650 −0.518252 0.855228i $$-0.673417\pi$$
−0.518252 + 0.855228i $$0.673417\pi$$
$$822$$ 0 0
$$823$$ −1.53118e7 −0.787999 −0.394000 0.919111i $$-0.628909\pi$$
−0.394000 + 0.919111i $$0.628909\pi$$
$$824$$ 0 0
$$825$$ −1.35432e6 −0.0692766
$$826$$ 0 0
$$827$$ −9.59310e6 −0.487748 −0.243874 0.969807i $$-0.578418\pi$$
−0.243874 + 0.969807i $$0.578418\pi$$
$$828$$ 0 0
$$829$$ −2.52209e7 −1.27460 −0.637302 0.770615i $$-0.719949\pi$$
−0.637302 + 0.770615i $$0.719949\pi$$
$$830$$ 0 0
$$831$$ −2.66727e7 −1.33988
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 2.65589e7 1.31824
$$836$$ 0 0
$$837$$ 1.74666e7 0.861778
$$838$$ 0 0
$$839$$ −1.77623e7 −0.871154 −0.435577 0.900151i $$-0.643456\pi$$
−0.435577 + 0.900151i $$0.643456\pi$$
$$840$$ 0 0
$$841$$ −2.01583e7 −0.982798
$$842$$ 0 0
$$843$$ 2.35937e6 0.114348
$$844$$ 0 0
$$845$$ 1.06147e7 0.511407
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 1.86273e7 0.886913
$$850$$ 0 0
$$851$$ −1.22299e6 −0.0578895
$$852$$ 0 0
$$853$$ 486970. 0.0229155 0.0114578 0.999934i $$-0.496353\pi$$
0.0114578 + 0.999934i $$0.496353\pi$$
$$854$$ 0 0
$$855$$ 4.46926e6 0.209084
$$856$$ 0 0
$$857$$ 1.92634e6 0.0895945 0.0447972 0.998996i $$-0.485736\pi$$
0.0447972 + 0.998996i $$0.485736\pi$$
$$858$$ 0 0
$$859$$ 2.23538e7 1.03364 0.516820 0.856094i $$-0.327116\pi$$
0.516820 + 0.856094i $$0.327116\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −1.85838e7 −0.849390 −0.424695 0.905337i $$-0.639619\pi$$
−0.424695 + 0.905337i $$0.639619\pi$$
$$864$$ 0 0
$$865$$ 3.82025e7 1.73601
$$866$$ 0 0
$$867$$ 1.28043e7 0.578504
$$868$$ 0 0
$$869$$ −4.15325e7 −1.86569
$$870$$ 0 0
$$871$$ −9.11742e6 −0.407217
$$872$$ 0 0
$$873$$ −1.21174e7 −0.538114
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −2.91048e7 −1.27781 −0.638905 0.769286i $$-0.720612\pi$$
−0.638905 + 0.769286i $$0.720612\pi$$
$$878$$ 0 0
$$879$$ −1.28660e7 −0.561659
$$880$$ 0 0
$$881$$ 3.14696e6 0.136600 0.0683001 0.997665i $$-0.478242\pi$$
0.0683001 + 0.997665i $$0.478242\pi$$
$$882$$ 0 0
$$883$$ −1.59995e7 −0.690566 −0.345283 0.938499i $$-0.612217\pi$$
−0.345283 + 0.938499i $$0.612217\pi$$
$$884$$ 0 0
$$885$$ −4.96886e6 −0.213255
$$886$$ 0 0
$$887$$ −3.45874e7 −1.47608 −0.738039 0.674758i $$-0.764248\pi$$
−0.738039 + 0.674758i $$0.764248\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1.36031e7 0.574044
$$892$$ 0 0
$$893$$ −1.08346e6 −0.0454656
$$894$$ 0 0
$$895$$ 2.66581e7 1.11243
$$896$$ 0 0
$$897$$ −2.05857e7 −0.854248
$$898$$ 0 0
$$899$$ −2.52806e6 −0.104325
$$900$$ 0 0
$$901$$ −1.15794e7 −0.475199
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −3.02103e7 −1.22612
$$906$$ 0 0
$$907$$ −1.74396e7 −0.703914 −0.351957 0.936016i $$-0.614484\pi$$
−0.351957 + 0.936016i $$0.614484\pi$$
$$908$$ 0 0
$$909$$ 1.11731e6 0.0448503
$$910$$ 0 0
$$911$$ 2.59589e6 0.103631 0.0518155 0.998657i $$-0.483499\pi$$
0.0518155 + 0.998657i $$0.483499\pi$$
$$912$$ 0 0
$$913$$ −3.65666e7 −1.45180
$$914$$ 0 0
$$915$$ 2.25102e7 0.888847
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.76411e7 0.689028 0.344514 0.938781i $$-0.388044\pi$$
0.344514 + 0.938781i $$0.388044\pi$$
$$920$$ 0 0
$$921$$ −1.90307e7 −0.739275
$$922$$ 0 0
$$923$$ 1.95925e7 0.756982
$$924$$ 0 0
$$925$$ 62282.0 0.00239336
$$926$$ 0 0
$$927$$ 2.69834e6 0.103133
$$928$$ 0 0
$$929$$ −3.96785e7 −1.50840 −0.754199 0.656646i $$-0.771975\pi$$
−0.754199 + 0.656646i $$0.771975\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 8.76874e6 0.329787
$$934$$ 0 0
$$935$$ −1.73210e7 −0.647955
$$936$$ 0 0
$$937$$ −3.93413e7 −1.46386 −0.731930 0.681380i $$-0.761380\pi$$
−0.731930 + 0.681380i $$0.761380\pi$$
$$938$$ 0 0
$$939$$ 7.01830e6 0.259757
$$940$$ 0 0
$$941$$ −4.62506e7 −1.70272 −0.851361 0.524581i $$-0.824222\pi$$
−0.851361 + 0.524581i $$0.824222\pi$$
$$942$$ 0 0
$$943$$ −7.06955e7 −2.58888
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 3.79025e7 1.37339 0.686693 0.726947i $$-0.259062\pi$$
0.686693 + 0.726947i $$0.259062\pi$$
$$948$$ 0 0
$$949$$ −2.82409e7 −1.01792
$$950$$ 0 0
$$951$$ 2.97944e7 1.06828
$$952$$ 0 0
$$953$$ −2.66462e7 −0.950394 −0.475197 0.879879i $$-0.657623\pi$$
−0.475197 + 0.879879i $$0.657623\pi$$
$$954$$ 0 0
$$955$$ −3.90977e7 −1.38721
$$956$$ 0 0
$$957$$ −3.84912e6 −0.135857
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −1.05156e7 −0.367304
$$962$$ 0 0
$$963$$ 1.21140e7 0.420943
$$964$$ 0 0
$$965$$ −383724. −0.0132648
$$966$$ 0 0
$$967$$ −4.09790e7 −1.40927 −0.704637 0.709568i $$-0.748890\pi$$
−0.704637 + 0.709568i $$0.748890\pi$$
$$968$$ 0 0
$$969$$ 5.95901e6 0.203875
$$970$$ 0 0
$$971$$ −2.72034e7 −0.925922 −0.462961 0.886379i $$-0.653213\pi$$
−0.462961 + 0.886379i $$0.653213\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 1.04834e6 0.0353177
$$976$$ 0 0
$$977$$ 2.53555e7 0.849839 0.424919 0.905231i $$-0.360302\pi$$
0.424919 + 0.905231i $$0.360302\pi$$
$$978$$ 0 0
$$979$$ 1.60672e7 0.535775
$$980$$ 0 0
$$981$$ −9.89030e6 −0.328123
$$982$$ 0 0
$$983$$ 1.19139e7 0.393252 0.196626 0.980479i $$-0.437002\pi$$
0.196626 + 0.980479i $$0.437002\pi$$
$$984$$ 0 0
$$985$$ 2.86439e7 0.940678
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 4.96584e7 1.61437
$$990$$ 0 0
$$991$$ −2.91931e7 −0.944268 −0.472134 0.881527i $$-0.656516\pi$$
−0.472134 + 0.881527i $$0.656516\pi$$
$$992$$ 0 0
$$993$$ 4.53538e6 0.145962
$$994$$ 0 0
$$995$$ −3.03307e6 −0.0971237
$$996$$ 0 0
$$997$$ 1.73001e7 0.551201 0.275601 0.961272i $$-0.411123\pi$$
0.275601 + 0.961272i $$0.411123\pi$$
$$998$$ 0 0
$$999$$ −1.22299e6 −0.0387713
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 784.6.a.d.1.1 1
4.3 odd 2 196.6.a.e.1.1 1
7.6 odd 2 16.6.a.b.1.1 1
21.20 even 2 144.6.a.c.1.1 1
28.3 even 6 196.6.e.g.177.1 2
28.11 odd 6 196.6.e.d.177.1 2
28.19 even 6 196.6.e.g.165.1 2
28.23 odd 6 196.6.e.d.165.1 2
28.27 even 2 4.6.a.a.1.1 1
35.13 even 4 400.6.c.f.49.2 2
35.27 even 4 400.6.c.f.49.1 2
35.34 odd 2 400.6.a.d.1.1 1
56.13 odd 2 64.6.a.b.1.1 1
56.27 even 2 64.6.a.f.1.1 1
84.83 odd 2 36.6.a.a.1.1 1
112.13 odd 4 256.6.b.c.129.2 2
112.27 even 4 256.6.b.g.129.2 2
112.69 odd 4 256.6.b.c.129.1 2
112.83 even 4 256.6.b.g.129.1 2
140.27 odd 4 100.6.c.b.49.2 2
140.83 odd 4 100.6.c.b.49.1 2
140.139 even 2 100.6.a.b.1.1 1
168.83 odd 2 576.6.a.bc.1.1 1
168.125 even 2 576.6.a.bd.1.1 1
252.83 odd 6 324.6.e.d.109.1 2
252.139 even 6 324.6.e.a.217.1 2
252.167 odd 6 324.6.e.d.217.1 2
252.223 even 6 324.6.e.a.109.1 2
308.307 odd 2 484.6.a.a.1.1 1
364.83 odd 4 676.6.d.a.337.1 2
364.307 odd 4 676.6.d.a.337.2 2
364.363 even 2 676.6.a.a.1.1 1
420.83 even 4 900.6.d.a.649.2 2
420.167 even 4 900.6.d.a.649.1 2
420.419 odd 2 900.6.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4.6.a.a.1.1 1 28.27 even 2
16.6.a.b.1.1 1 7.6 odd 2
36.6.a.a.1.1 1 84.83 odd 2
64.6.a.b.1.1 1 56.13 odd 2
64.6.a.f.1.1 1 56.27 even 2
100.6.a.b.1.1 1 140.139 even 2
100.6.c.b.49.1 2 140.83 odd 4
100.6.c.b.49.2 2 140.27 odd 4
144.6.a.c.1.1 1 21.20 even 2
196.6.a.e.1.1 1 4.3 odd 2
196.6.e.d.165.1 2 28.23 odd 6
196.6.e.d.177.1 2 28.11 odd 6
196.6.e.g.165.1 2 28.19 even 6
196.6.e.g.177.1 2 28.3 even 6
256.6.b.c.129.1 2 112.69 odd 4
256.6.b.c.129.2 2 112.13 odd 4
256.6.b.g.129.1 2 112.83 even 4
256.6.b.g.129.2 2 112.27 even 4
324.6.e.a.109.1 2 252.223 even 6
324.6.e.a.217.1 2 252.139 even 6
324.6.e.d.109.1 2 252.83 odd 6
324.6.e.d.217.1 2 252.167 odd 6
400.6.a.d.1.1 1 35.34 odd 2
400.6.c.f.49.1 2 35.27 even 4
400.6.c.f.49.2 2 35.13 even 4
484.6.a.a.1.1 1 308.307 odd 2
576.6.a.bc.1.1 1 168.83 odd 2
576.6.a.bd.1.1 1 168.125 even 2
676.6.a.a.1.1 1 364.363 even 2
676.6.d.a.337.1 2 364.83 odd 4
676.6.d.a.337.2 2 364.307 odd 4
784.6.a.d.1.1 1 1.1 even 1 trivial
900.6.a.h.1.1 1 420.419 odd 2
900.6.d.a.649.1 2 420.167 even 4
900.6.d.a.649.2 2 420.83 even 4