# Properties

 Label 441.6.a.k.1.1 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ 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] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.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: $$70.7292645375$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+10.0000 q^{2} +68.0000 q^{4} -56.0000 q^{5} +360.000 q^{8} +O(q^{10})$$ $$q+10.0000 q^{2} +68.0000 q^{4} -56.0000 q^{5} +360.000 q^{8} -560.000 q^{10} -232.000 q^{11} +140.000 q^{13} +1424.00 q^{16} -1722.00 q^{17} +98.0000 q^{19} -3808.00 q^{20} -2320.00 q^{22} -1824.00 q^{23} +11.0000 q^{25} +1400.00 q^{26} -3418.00 q^{29} +7644.00 q^{31} +2720.00 q^{32} -17220.0 q^{34} -10398.0 q^{37} +980.000 q^{38} -20160.0 q^{40} -17962.0 q^{41} +10880.0 q^{43} -15776.0 q^{44} -18240.0 q^{46} +9324.00 q^{47} +110.000 q^{50} +9520.00 q^{52} -2262.00 q^{53} +12992.0 q^{55} -34180.0 q^{58} -2730.00 q^{59} -25648.0 q^{61} +76440.0 q^{62} -18368.0 q^{64} -7840.00 q^{65} -48404.0 q^{67} -117096. q^{68} +58560.0 q^{71} -68082.0 q^{73} -103980. q^{74} +6664.00 q^{76} +31784.0 q^{79} -79744.0 q^{80} -179620. q^{82} -20538.0 q^{83} +96432.0 q^{85} +108800. q^{86} -83520.0 q^{88} -50582.0 q^{89} -124032. q^{92} +93240.0 q^{94} -5488.00 q^{95} +58506.0 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 10.0000 1.76777 0.883883 0.467707i $$-0.154920\pi$$
0.883883 + 0.467707i $$0.154920\pi$$
$$3$$ 0 0
$$4$$ 68.0000 2.12500
$$5$$ −56.0000 −1.00176 −0.500879 0.865517i $$-0.666990\pi$$
−0.500879 + 0.865517i $$0.666990\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 360.000 1.98874
$$9$$ 0 0
$$10$$ −560.000 −1.77088
$$11$$ −232.000 −0.578104 −0.289052 0.957313i $$-0.593340\pi$$
−0.289052 + 0.957313i $$0.593340\pi$$
$$12$$ 0 0
$$13$$ 140.000 0.229757 0.114879 0.993380i $$-0.463352\pi$$
0.114879 + 0.993380i $$0.463352\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1424.00 1.39062
$$17$$ −1722.00 −1.44514 −0.722572 0.691296i $$-0.757040\pi$$
−0.722572 + 0.691296i $$0.757040\pi$$
$$18$$ 0 0
$$19$$ 98.0000 0.0622791 0.0311395 0.999515i $$-0.490086\pi$$
0.0311395 + 0.999515i $$0.490086\pi$$
$$20$$ −3808.00 −2.12874
$$21$$ 0 0
$$22$$ −2320.00 −1.02195
$$23$$ −1824.00 −0.718961 −0.359480 0.933153i $$-0.617046\pi$$
−0.359480 + 0.933153i $$0.617046\pi$$
$$24$$ 0 0
$$25$$ 11.0000 0.00352000
$$26$$ 1400.00 0.406158
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −3418.00 −0.754705 −0.377352 0.926070i $$-0.623165\pi$$
−0.377352 + 0.926070i $$0.623165\pi$$
$$30$$ 0 0
$$31$$ 7644.00 1.42862 0.714310 0.699830i $$-0.246741\pi$$
0.714310 + 0.699830i $$0.246741\pi$$
$$32$$ 2720.00 0.469563
$$33$$ 0 0
$$34$$ −17220.0 −2.55468
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10398.0 −1.24866 −0.624332 0.781159i $$-0.714629\pi$$
−0.624332 + 0.781159i $$0.714629\pi$$
$$38$$ 980.000 0.110095
$$39$$ 0 0
$$40$$ −20160.0 −1.99223
$$41$$ −17962.0 −1.66876 −0.834382 0.551186i $$-0.814175\pi$$
−0.834382 + 0.551186i $$0.814175\pi$$
$$42$$ 0 0
$$43$$ 10880.0 0.897342 0.448671 0.893697i $$-0.351898\pi$$
0.448671 + 0.893697i $$0.351898\pi$$
$$44$$ −15776.0 −1.22847
$$45$$ 0 0
$$46$$ −18240.0 −1.27096
$$47$$ 9324.00 0.615684 0.307842 0.951438i $$-0.400393\pi$$
0.307842 + 0.951438i $$0.400393\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 110.000 0.00622254
$$51$$ 0 0
$$52$$ 9520.00 0.488235
$$53$$ −2262.00 −0.110612 −0.0553061 0.998469i $$-0.517613\pi$$
−0.0553061 + 0.998469i $$0.517613\pi$$
$$54$$ 0 0
$$55$$ 12992.0 0.579121
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −34180.0 −1.33414
$$59$$ −2730.00 −0.102102 −0.0510508 0.998696i $$-0.516257\pi$$
−0.0510508 + 0.998696i $$0.516257\pi$$
$$60$$ 0 0
$$61$$ −25648.0 −0.882529 −0.441264 0.897377i $$-0.645470\pi$$
−0.441264 + 0.897377i $$0.645470\pi$$
$$62$$ 76440.0 2.52547
$$63$$ 0 0
$$64$$ −18368.0 −0.560547
$$65$$ −7840.00 −0.230161
$$66$$ 0 0
$$67$$ −48404.0 −1.31733 −0.658664 0.752437i $$-0.728878\pi$$
−0.658664 + 0.752437i $$0.728878\pi$$
$$68$$ −117096. −3.07093
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 58560.0 1.37865 0.689327 0.724450i $$-0.257906\pi$$
0.689327 + 0.724450i $$0.257906\pi$$
$$72$$ 0 0
$$73$$ −68082.0 −1.49529 −0.747645 0.664099i $$-0.768815\pi$$
−0.747645 + 0.664099i $$0.768815\pi$$
$$74$$ −103980. −2.20735
$$75$$ 0 0
$$76$$ 6664.00 0.132343
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 31784.0 0.572982 0.286491 0.958083i $$-0.407511\pi$$
0.286491 + 0.958083i $$0.407511\pi$$
$$80$$ −79744.0 −1.39307
$$81$$ 0 0
$$82$$ −179620. −2.94999
$$83$$ −20538.0 −0.327237 −0.163619 0.986524i $$-0.552317\pi$$
−0.163619 + 0.986524i $$0.552317\pi$$
$$84$$ 0 0
$$85$$ 96432.0 1.44768
$$86$$ 108800. 1.58629
$$87$$ 0 0
$$88$$ −83520.0 −1.14970
$$89$$ −50582.0 −0.676894 −0.338447 0.940985i $$-0.609902\pi$$
−0.338447 + 0.940985i $$0.609902\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −124032. −1.52779
$$93$$ 0 0
$$94$$ 93240.0 1.08839
$$95$$ −5488.00 −0.0623886
$$96$$ 0 0
$$97$$ 58506.0 0.631351 0.315676 0.948867i $$-0.397769\pi$$
0.315676 + 0.948867i $$0.397769\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 748.000 0.00748000
$$101$$ 38696.0 0.377453 0.188726 0.982030i $$-0.439564\pi$$
0.188726 + 0.982030i $$0.439564\pi$$
$$102$$ 0 0
$$103$$ −53060.0 −0.492804 −0.246402 0.969168i $$-0.579248\pi$$
−0.246402 + 0.969168i $$0.579248\pi$$
$$104$$ 50400.0 0.456927
$$105$$ 0 0
$$106$$ −22620.0 −0.195537
$$107$$ 146324. 1.23554 0.617769 0.786360i $$-0.288037\pi$$
0.617769 + 0.786360i $$0.288037\pi$$
$$108$$ 0 0
$$109$$ 92898.0 0.748928 0.374464 0.927241i $$-0.377827\pi$$
0.374464 + 0.927241i $$0.377827\pi$$
$$110$$ 129920. 1.02375
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 83354.0 0.614088 0.307044 0.951695i $$-0.400660\pi$$
0.307044 + 0.951695i $$0.400660\pi$$
$$114$$ 0 0
$$115$$ 102144. 0.720225
$$116$$ −232424. −1.60375
$$117$$ 0 0
$$118$$ −27300.0 −0.180492
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −107227. −0.665795
$$122$$ −256480. −1.56011
$$123$$ 0 0
$$124$$ 519792. 3.03582
$$125$$ 174384. 0.998232
$$126$$ 0 0
$$127$$ 60384.0 0.332210 0.166105 0.986108i $$-0.446881\pi$$
0.166105 + 0.986108i $$0.446881\pi$$
$$128$$ −270720. −1.46048
$$129$$ 0 0
$$130$$ −78400.0 −0.406872
$$131$$ −61586.0 −0.313548 −0.156774 0.987635i $$-0.550109\pi$$
−0.156774 + 0.987635i $$0.550109\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −484040. −2.32873
$$135$$ 0 0
$$136$$ −619920. −2.87401
$$137$$ 204462. 0.930703 0.465352 0.885126i $$-0.345928\pi$$
0.465352 + 0.885126i $$0.345928\pi$$
$$138$$ 0 0
$$139$$ 35406.0 0.155432 0.0777159 0.996976i $$-0.475237\pi$$
0.0777159 + 0.996976i $$0.475237\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 585600. 2.43714
$$143$$ −32480.0 −0.132824
$$144$$ 0 0
$$145$$ 191408. 0.756032
$$146$$ −680820. −2.64332
$$147$$ 0 0
$$148$$ −707064. −2.65341
$$149$$ 20226.0 0.0746353 0.0373177 0.999303i $$-0.488119\pi$$
0.0373177 + 0.999303i $$0.488119\pi$$
$$150$$ 0 0
$$151$$ 70904.0 0.253063 0.126531 0.991963i $$-0.459616\pi$$
0.126531 + 0.991963i $$0.459616\pi$$
$$152$$ 35280.0 0.123857
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −428064. −1.43113
$$156$$ 0 0
$$157$$ −293524. −0.950374 −0.475187 0.879885i $$-0.657620\pi$$
−0.475187 + 0.879885i $$0.657620\pi$$
$$158$$ 317840. 1.01290
$$159$$ 0 0
$$160$$ −152320. −0.470389
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 13192.0 0.0388903 0.0194452 0.999811i $$-0.493810\pi$$
0.0194452 + 0.999811i $$0.493810\pi$$
$$164$$ −1.22142e6 −3.54612
$$165$$ 0 0
$$166$$ −205380. −0.578479
$$167$$ 493612. 1.36960 0.684801 0.728730i $$-0.259889\pi$$
0.684801 + 0.728730i $$0.259889\pi$$
$$168$$ 0 0
$$169$$ −351693. −0.947212
$$170$$ 964320. 2.55917
$$171$$ 0 0
$$172$$ 739840. 1.90685
$$173$$ 240716. 0.611490 0.305745 0.952113i $$-0.401094\pi$$
0.305745 + 0.952113i $$0.401094\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −330368. −0.803926
$$177$$ 0 0
$$178$$ −505820. −1.19659
$$179$$ −294932. −0.688001 −0.344001 0.938969i $$-0.611782\pi$$
−0.344001 + 0.938969i $$0.611782\pi$$
$$180$$ 0 0
$$181$$ 336980. 0.764553 0.382277 0.924048i $$-0.375140\pi$$
0.382277 + 0.924048i $$0.375140\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −656640. −1.42982
$$185$$ 582288. 1.25086
$$186$$ 0 0
$$187$$ 399504. 0.835444
$$188$$ 634032. 1.30833
$$189$$ 0 0
$$190$$ −54880.0 −0.110288
$$191$$ −358264. −0.710591 −0.355296 0.934754i $$-0.615620\pi$$
−0.355296 + 0.934754i $$0.615620\pi$$
$$192$$ 0 0
$$193$$ −989554. −1.91226 −0.956128 0.292948i $$-0.905364\pi$$
−0.956128 + 0.292948i $$0.905364\pi$$
$$194$$ 585060. 1.11608
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 990050. 1.81757 0.908786 0.417263i $$-0.137011\pi$$
0.908786 + 0.417263i $$0.137011\pi$$
$$198$$ 0 0
$$199$$ 840756. 1.50500 0.752501 0.658591i $$-0.228847\pi$$
0.752501 + 0.658591i $$0.228847\pi$$
$$200$$ 3960.00 0.00700036
$$201$$ 0 0
$$202$$ 386960. 0.667249
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 1.00587e6 1.67170
$$206$$ −530600. −0.871163
$$207$$ 0 0
$$208$$ 199360. 0.319506
$$209$$ −22736.0 −0.0360038
$$210$$ 0 0
$$211$$ 1.15073e6 1.77938 0.889689 0.456568i $$-0.150921\pi$$
0.889689 + 0.456568i $$0.150921\pi$$
$$212$$ −153816. −0.235051
$$213$$ 0 0
$$214$$ 1.46324e6 2.18414
$$215$$ −609280. −0.898919
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 928980. 1.32393
$$219$$ 0 0
$$220$$ 883456. 1.23063
$$221$$ −241080. −0.332032
$$222$$ 0 0
$$223$$ 824264. 1.10995 0.554976 0.831866i $$-0.312727\pi$$
0.554976 + 0.831866i $$0.312727\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 833540. 1.08556
$$227$$ 74382.0 0.0958083 0.0479042 0.998852i $$-0.484746\pi$$
0.0479042 + 0.998852i $$0.484746\pi$$
$$228$$ 0 0
$$229$$ −1.13196e6 −1.42640 −0.713199 0.700961i $$-0.752755\pi$$
−0.713199 + 0.700961i $$0.752755\pi$$
$$230$$ 1.02144e6 1.27319
$$231$$ 0 0
$$232$$ −1.23048e6 −1.50091
$$233$$ 198726. 0.239809 0.119904 0.992785i $$-0.461741\pi$$
0.119904 + 0.992785i $$0.461741\pi$$
$$234$$ 0 0
$$235$$ −522144. −0.616766
$$236$$ −185640. −0.216966
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −482904. −0.546847 −0.273424 0.961894i $$-0.588156\pi$$
−0.273424 + 0.961894i $$0.588156\pi$$
$$240$$ 0 0
$$241$$ −805910. −0.893807 −0.446904 0.894582i $$-0.647473\pi$$
−0.446904 + 0.894582i $$0.647473\pi$$
$$242$$ −1.07227e6 −1.17697
$$243$$ 0 0
$$244$$ −1.74406e6 −1.87537
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 13720.0 0.0143091
$$248$$ 2.75184e6 2.84115
$$249$$ 0 0
$$250$$ 1.74384e6 1.76464
$$251$$ 430738. 0.431548 0.215774 0.976443i $$-0.430773\pi$$
0.215774 + 0.976443i $$0.430773\pi$$
$$252$$ 0 0
$$253$$ 423168. 0.415634
$$254$$ 603840. 0.587270
$$255$$ 0 0
$$256$$ −2.11942e6 −2.02124
$$257$$ −1.17691e6 −1.11150 −0.555751 0.831349i $$-0.687569\pi$$
−0.555751 + 0.831349i $$0.687569\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −533120. −0.489093
$$261$$ 0 0
$$262$$ −615860. −0.554279
$$263$$ −1.29098e6 −1.15088 −0.575438 0.817845i $$-0.695169\pi$$
−0.575438 + 0.817845i $$0.695169\pi$$
$$264$$ 0 0
$$265$$ 126672. 0.110807
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −3.29147e6 −2.79932
$$269$$ −1.27756e6 −1.07646 −0.538232 0.842797i $$-0.680907\pi$$
−0.538232 + 0.842797i $$0.680907\pi$$
$$270$$ 0 0
$$271$$ −1.65054e6 −1.36522 −0.682612 0.730781i $$-0.739156\pi$$
−0.682612 + 0.730781i $$0.739156\pi$$
$$272$$ −2.45213e6 −2.00965
$$273$$ 0 0
$$274$$ 2.04462e6 1.64527
$$275$$ −2552.00 −0.00203493
$$276$$ 0 0
$$277$$ −1.06409e6 −0.833257 −0.416628 0.909077i $$-0.636788\pi$$
−0.416628 + 0.909077i $$0.636788\pi$$
$$278$$ 354060. 0.274767
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 22342.0 0.0168794 0.00843969 0.999964i $$-0.497314\pi$$
0.00843969 + 0.999964i $$0.497314\pi$$
$$282$$ 0 0
$$283$$ 2.49574e6 1.85239 0.926196 0.377042i $$-0.123059\pi$$
0.926196 + 0.377042i $$0.123059\pi$$
$$284$$ 3.98208e6 2.92964
$$285$$ 0 0
$$286$$ −324800. −0.234802
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.54543e6 1.08844
$$290$$ 1.91408e6 1.33649
$$291$$ 0 0
$$292$$ −4.62958e6 −3.17749
$$293$$ −1.93178e6 −1.31458 −0.657291 0.753637i $$-0.728298\pi$$
−0.657291 + 0.753637i $$0.728298\pi$$
$$294$$ 0 0
$$295$$ 152880. 0.102281
$$296$$ −3.74328e6 −2.48326
$$297$$ 0 0
$$298$$ 202260. 0.131938
$$299$$ −255360. −0.165187
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 709040. 0.447356
$$303$$ 0 0
$$304$$ 139552. 0.0866068
$$305$$ 1.43629e6 0.884081
$$306$$ 0 0
$$307$$ 459074. 0.277995 0.138997 0.990293i $$-0.455612\pi$$
0.138997 + 0.990293i $$0.455612\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −4.28064e6 −2.52991
$$311$$ 667128. 0.391118 0.195559 0.980692i $$-0.437348\pi$$
0.195559 + 0.980692i $$0.437348\pi$$
$$312$$ 0 0
$$313$$ 111034. 0.0640612 0.0320306 0.999487i $$-0.489803\pi$$
0.0320306 + 0.999487i $$0.489803\pi$$
$$314$$ −2.93524e6 −1.68004
$$315$$ 0 0
$$316$$ 2.16131e6 1.21759
$$317$$ 68778.0 0.0384416 0.0192208 0.999815i $$-0.493881\pi$$
0.0192208 + 0.999815i $$0.493881\pi$$
$$318$$ 0 0
$$319$$ 792976. 0.436298
$$320$$ 1.02861e6 0.561533
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −168756. −0.0900022
$$324$$ 0 0
$$325$$ 1540.00 0.000808746 0
$$326$$ 131920. 0.0687490
$$327$$ 0 0
$$328$$ −6.46632e6 −3.31874
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −564448. −0.283174 −0.141587 0.989926i $$-0.545221\pi$$
−0.141587 + 0.989926i $$0.545221\pi$$
$$332$$ −1.39658e6 −0.695379
$$333$$ 0 0
$$334$$ 4.93612e6 2.42114
$$335$$ 2.71062e6 1.31965
$$336$$ 0 0
$$337$$ 2.07729e6 0.996376 0.498188 0.867069i $$-0.333999\pi$$
0.498188 + 0.867069i $$0.333999\pi$$
$$338$$ −3.51693e6 −1.67445
$$339$$ 0 0
$$340$$ 6.55738e6 3.07633
$$341$$ −1.77341e6 −0.825891
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 3.91680e6 1.78458
$$345$$ 0 0
$$346$$ 2.40716e6 1.08097
$$347$$ 53248.0 0.0237399 0.0118700 0.999930i $$-0.496222\pi$$
0.0118700 + 0.999930i $$0.496222\pi$$
$$348$$ 0 0
$$349$$ 2.27200e6 0.998494 0.499247 0.866460i $$-0.333610\pi$$
0.499247 + 0.866460i $$0.333610\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −631040. −0.271456
$$353$$ 4.00645e6 1.71129 0.855644 0.517565i $$-0.173162\pi$$
0.855644 + 0.517565i $$0.173162\pi$$
$$354$$ 0 0
$$355$$ −3.27936e6 −1.38108
$$356$$ −3.43958e6 −1.43840
$$357$$ 0 0
$$358$$ −2.94932e6 −1.21623
$$359$$ −73784.0 −0.0302152 −0.0151076 0.999886i $$-0.504809\pi$$
−0.0151076 + 0.999886i $$0.504809\pi$$
$$360$$ 0 0
$$361$$ −2.46650e6 −0.996121
$$362$$ 3.36980e6 1.35155
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3.81259e6 1.49792
$$366$$ 0 0
$$367$$ −1.40431e6 −0.544250 −0.272125 0.962262i $$-0.587726\pi$$
−0.272125 + 0.962262i $$0.587726\pi$$
$$368$$ −2.59738e6 −0.999805
$$369$$ 0 0
$$370$$ 5.82288e6 2.21123
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.60323e6 −0.596657 −0.298329 0.954463i $$-0.596429\pi$$
−0.298329 + 0.954463i $$0.596429\pi$$
$$374$$ 3.99504e6 1.47687
$$375$$ 0 0
$$376$$ 3.35664e6 1.22443
$$377$$ −478520. −0.173399
$$378$$ 0 0
$$379$$ −4.77012e6 −1.70581 −0.852906 0.522064i $$-0.825162\pi$$
−0.852906 + 0.522064i $$0.825162\pi$$
$$380$$ −373184. −0.132576
$$381$$ 0 0
$$382$$ −3.58264e6 −1.25616
$$383$$ −2.23079e6 −0.777072 −0.388536 0.921434i $$-0.627019\pi$$
−0.388536 + 0.921434i $$0.627019\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −9.89554e6 −3.38042
$$387$$ 0 0
$$388$$ 3.97841e6 1.34162
$$389$$ −4.84024e6 −1.62178 −0.810892 0.585196i $$-0.801018\pi$$
−0.810892 + 0.585196i $$0.801018\pi$$
$$390$$ 0 0
$$391$$ 3.14093e6 1.03900
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 9.90050e6 3.21304
$$395$$ −1.77990e6 −0.573989
$$396$$ 0 0
$$397$$ −995820. −0.317106 −0.158553 0.987350i $$-0.550683\pi$$
−0.158553 + 0.987350i $$0.550683\pi$$
$$398$$ 8.40756e6 2.66049
$$399$$ 0 0
$$400$$ 15664.0 0.00489500
$$401$$ 3.31605e6 1.02982 0.514909 0.857245i $$-0.327826\pi$$
0.514909 + 0.857245i $$0.327826\pi$$
$$402$$ 0 0
$$403$$ 1.07016e6 0.328236
$$404$$ 2.63133e6 0.802087
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2.41234e6 0.721858
$$408$$ 0 0
$$409$$ −3.07273e6 −0.908274 −0.454137 0.890932i $$-0.650052\pi$$
−0.454137 + 0.890932i $$0.650052\pi$$
$$410$$ 1.00587e7 2.95517
$$411$$ 0 0
$$412$$ −3.60808e6 −1.04721
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.15013e6 0.327813
$$416$$ 380800. 0.107886
$$417$$ 0 0
$$418$$ −227360. −0.0636463
$$419$$ 2.81438e6 0.783154 0.391577 0.920145i $$-0.371930\pi$$
0.391577 + 0.920145i $$0.371930\pi$$
$$420$$ 0 0
$$421$$ 3.05802e6 0.840883 0.420441 0.907320i $$-0.361875\pi$$
0.420441 + 0.907320i $$0.361875\pi$$
$$422$$ 1.15073e7 3.14552
$$423$$ 0 0
$$424$$ −814320. −0.219979
$$425$$ −18942.0 −0.00508690
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 9.95003e6 2.62552
$$429$$ 0 0
$$430$$ −6.09280e6 −1.58908
$$431$$ −1.93750e6 −0.502398 −0.251199 0.967936i $$-0.580825\pi$$
−0.251199 + 0.967936i $$0.580825\pi$$
$$432$$ 0 0
$$433$$ −3.94790e6 −1.01192 −0.505961 0.862557i $$-0.668862\pi$$
−0.505961 + 0.862557i $$0.668862\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6.31706e6 1.59147
$$437$$ −178752. −0.0447762
$$438$$ 0 0
$$439$$ 7.41770e6 1.83700 0.918498 0.395426i $$-0.129403\pi$$
0.918498 + 0.395426i $$0.129403\pi$$
$$440$$ 4.67712e6 1.15172
$$441$$ 0 0
$$442$$ −2.41080e6 −0.586956
$$443$$ −1.40269e6 −0.339589 −0.169794 0.985480i $$-0.554310\pi$$
−0.169794 + 0.985480i $$0.554310\pi$$
$$444$$ 0 0
$$445$$ 2.83259e6 0.678085
$$446$$ 8.24264e6 1.96214
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 590574. 0.138248 0.0691239 0.997608i $$-0.477980\pi$$
0.0691239 + 0.997608i $$0.477980\pi$$
$$450$$ 0 0
$$451$$ 4.16718e6 0.964720
$$452$$ 5.66807e6 1.30494
$$453$$ 0 0
$$454$$ 743820. 0.169367
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −2.90484e6 −0.650627 −0.325313 0.945606i $$-0.605470\pi$$
−0.325313 + 0.945606i $$0.605470\pi$$
$$458$$ −1.13196e7 −2.52154
$$459$$ 0 0
$$460$$ 6.94579e6 1.53048
$$461$$ −922684. −0.202209 −0.101105 0.994876i $$-0.532238\pi$$
−0.101105 + 0.994876i $$0.532238\pi$$
$$462$$ 0 0
$$463$$ 7.18235e6 1.55709 0.778546 0.627588i $$-0.215958\pi$$
0.778546 + 0.627588i $$0.215958\pi$$
$$464$$ −4.86723e6 −1.04951
$$465$$ 0 0
$$466$$ 1.98726e6 0.423926
$$467$$ −612570. −0.129976 −0.0649881 0.997886i $$-0.520701\pi$$
−0.0649881 + 0.997886i $$0.520701\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −5.22144e6 −1.09030
$$471$$ 0 0
$$472$$ −982800. −0.203053
$$473$$ −2.52416e6 −0.518757
$$474$$ 0 0
$$475$$ 1078.00 0.000219222 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −4.82904e6 −0.966699
$$479$$ 2.60330e6 0.518424 0.259212 0.965820i $$-0.416537\pi$$
0.259212 + 0.965820i $$0.416537\pi$$
$$480$$ 0 0
$$481$$ −1.45572e6 −0.286890
$$482$$ −8.05910e6 −1.58004
$$483$$ 0 0
$$484$$ −7.29144e6 −1.41482
$$485$$ −3.27634e6 −0.632461
$$486$$ 0 0
$$487$$ 5.46309e6 1.04380 0.521898 0.853008i $$-0.325224\pi$$
0.521898 + 0.853008i $$0.325224\pi$$
$$488$$ −9.23328e6 −1.75512
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1.64090e6 −0.307170 −0.153585 0.988135i $$-0.549082\pi$$
−0.153585 + 0.988135i $$0.549082\pi$$
$$492$$ 0 0
$$493$$ 5.88580e6 1.09066
$$494$$ 137200. 0.0252951
$$495$$ 0 0
$$496$$ 1.08851e7 1.98667
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 2.99796e6 0.538983 0.269491 0.963003i $$-0.413144\pi$$
0.269491 + 0.963003i $$0.413144\pi$$
$$500$$ 1.18581e7 2.12124
$$501$$ 0 0
$$502$$ 4.30738e6 0.762876
$$503$$ −6.89405e6 −1.21494 −0.607469 0.794343i $$-0.707815\pi$$
−0.607469 + 0.794343i $$0.707815\pi$$
$$504$$ 0 0
$$505$$ −2.16698e6 −0.378117
$$506$$ 4.23168e6 0.734745
$$507$$ 0 0
$$508$$ 4.10611e6 0.705946
$$509$$ 2.30476e6 0.394305 0.197152 0.980373i $$-0.436831\pi$$
0.197152 + 0.980373i $$0.436831\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.25312e7 −2.11260
$$513$$ 0 0
$$514$$ −1.17691e7 −1.96488
$$515$$ 2.97136e6 0.493671
$$516$$ 0 0
$$517$$ −2.16317e6 −0.355929
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −2.82240e6 −0.457731
$$521$$ −1.20960e7 −1.95231 −0.976155 0.217073i $$-0.930349\pi$$
−0.976155 + 0.217073i $$0.930349\pi$$
$$522$$ 0 0
$$523$$ −5.48443e6 −0.876753 −0.438377 0.898791i $$-0.644446\pi$$
−0.438377 + 0.898791i $$0.644446\pi$$
$$524$$ −4.18785e6 −0.666289
$$525$$ 0 0
$$526$$ −1.29098e7 −2.03448
$$527$$ −1.31630e7 −2.06456
$$528$$ 0 0
$$529$$ −3.10937e6 −0.483095
$$530$$ 1.26672e6 0.195880
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −2.51468e6 −0.383411
$$534$$ 0 0
$$535$$ −8.19414e6 −1.23771
$$536$$ −1.74254e7 −2.61982
$$537$$ 0 0
$$538$$ −1.27756e7 −1.90294
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −6.71799e6 −0.986839 −0.493420 0.869791i $$-0.664253\pi$$
−0.493420 + 0.869791i $$0.664253\pi$$
$$542$$ −1.65054e7 −2.41340
$$543$$ 0 0
$$544$$ −4.68384e6 −0.678586
$$545$$ −5.20229e6 −0.750245
$$546$$ 0 0
$$547$$ −5.00235e6 −0.714835 −0.357418 0.933945i $$-0.616343\pi$$
−0.357418 + 0.933945i $$0.616343\pi$$
$$548$$ 1.39034e7 1.97774
$$549$$ 0 0
$$550$$ −25520.0 −0.00359728
$$551$$ −334964. −0.0470023
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −1.06409e7 −1.47300
$$555$$ 0 0
$$556$$ 2.40761e6 0.330293
$$557$$ −9.01961e6 −1.23183 −0.615913 0.787814i $$-0.711213\pi$$
−0.615913 + 0.787814i $$0.711213\pi$$
$$558$$ 0 0
$$559$$ 1.52320e6 0.206171
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 223420. 0.0298388
$$563$$ 1.24051e7 1.64941 0.824707 0.565561i $$-0.191340\pi$$
0.824707 + 0.565561i $$0.191340\pi$$
$$564$$ 0 0
$$565$$ −4.66782e6 −0.615167
$$566$$ 2.49574e7 3.27460
$$567$$ 0 0
$$568$$ 2.10816e7 2.74178
$$569$$ −6.48804e6 −0.840103 −0.420052 0.907500i $$-0.637988\pi$$
−0.420052 + 0.907500i $$0.637988\pi$$
$$570$$ 0 0
$$571$$ −1.02285e7 −1.31287 −0.656435 0.754382i $$-0.727936\pi$$
−0.656435 + 0.754382i $$0.727936\pi$$
$$572$$ −2.20864e6 −0.282251
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −20064.0 −0.00253074
$$576$$ 0 0
$$577$$ −2.65338e6 −0.331787 −0.165894 0.986144i $$-0.553051\pi$$
−0.165894 + 0.986144i $$0.553051\pi$$
$$578$$ 1.54543e7 1.92411
$$579$$ 0 0
$$580$$ 1.30157e7 1.60657
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 524784. 0.0639454
$$584$$ −2.45095e7 −2.97374
$$585$$ 0 0
$$586$$ −1.93178e7 −2.32387
$$587$$ −1.43044e7 −1.71346 −0.856729 0.515766i $$-0.827507\pi$$
−0.856729 + 0.515766i $$0.827507\pi$$
$$588$$ 0 0
$$589$$ 749112. 0.0889731
$$590$$ 1.52880e6 0.180809
$$591$$ 0 0
$$592$$ −1.48068e7 −1.73642
$$593$$ −1.00265e7 −1.17088 −0.585442 0.810714i $$-0.699079\pi$$
−0.585442 + 0.810714i $$0.699079\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1.37537e6 0.158600
$$597$$ 0 0
$$598$$ −2.55360e6 −0.292011
$$599$$ 7.52292e6 0.856681 0.428341 0.903617i $$-0.359098\pi$$
0.428341 + 0.903617i $$0.359098\pi$$
$$600$$ 0 0
$$601$$ −3.38625e6 −0.382413 −0.191207 0.981550i $$-0.561240\pi$$
−0.191207 + 0.981550i $$0.561240\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 4.82147e6 0.537759
$$605$$ 6.00471e6 0.666966
$$606$$ 0 0
$$607$$ 6.90861e6 0.761060 0.380530 0.924769i $$-0.375742\pi$$
0.380530 + 0.924769i $$0.375742\pi$$
$$608$$ 266560. 0.0292439
$$609$$ 0 0
$$610$$ 1.43629e7 1.56285
$$611$$ 1.30536e6 0.141458
$$612$$ 0 0
$$613$$ −9.68896e6 −1.04142 −0.520710 0.853734i $$-0.674333\pi$$
−0.520710 + 0.853734i $$0.674333\pi$$
$$614$$ 4.59074e6 0.491430
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 7.84742e6 0.829877 0.414939 0.909849i $$-0.363803\pi$$
0.414939 + 0.909849i $$0.363803\pi$$
$$618$$ 0 0
$$619$$ 1.01972e7 1.06968 0.534840 0.844953i $$-0.320372\pi$$
0.534840 + 0.844953i $$0.320372\pi$$
$$620$$ −2.91084e7 −3.04115
$$621$$ 0 0
$$622$$ 6.67128e6 0.691406
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −9.79988e6 −1.00351
$$626$$ 1.11034e6 0.113245
$$627$$ 0 0
$$628$$ −1.99596e7 −2.01954
$$629$$ 1.79054e7 1.80450
$$630$$ 0 0
$$631$$ −8.36258e6 −0.836116 −0.418058 0.908420i $$-0.637289\pi$$
−0.418058 + 0.908420i $$0.637289\pi$$
$$632$$ 1.14422e7 1.13951
$$633$$ 0 0
$$634$$ 687780. 0.0679558
$$635$$ −3.38150e6 −0.332794
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 7.92976e6 0.771273
$$639$$ 0 0
$$640$$ 1.51603e7 1.46305
$$641$$ −1.10283e6 −0.106014 −0.0530070 0.998594i $$-0.516881\pi$$
−0.0530070 + 0.998594i $$0.516881\pi$$
$$642$$ 0 0
$$643$$ −1.71354e7 −1.63443 −0.817217 0.576330i $$-0.804484\pi$$
−0.817217 + 0.576330i $$0.804484\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −1.68756e6 −0.159103
$$647$$ −54964.0 −0.00516200 −0.00258100 0.999997i $$-0.500822\pi$$
−0.00258100 + 0.999997i $$0.500822\pi$$
$$648$$ 0 0
$$649$$ 633360. 0.0590254
$$650$$ 15400.0 0.00142968
$$651$$ 0 0
$$652$$ 897056. 0.0826420
$$653$$ 485166. 0.0445254 0.0222627 0.999752i $$-0.492913\pi$$
0.0222627 + 0.999752i $$0.492913\pi$$
$$654$$ 0 0
$$655$$ 3.44882e6 0.314099
$$656$$ −2.55779e7 −2.32063
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 2.72136e6 0.244103 0.122051 0.992524i $$-0.461053\pi$$
0.122051 + 0.992524i $$0.461053\pi$$
$$660$$ 0 0
$$661$$ 2.14525e6 0.190974 0.0954869 0.995431i $$-0.469559\pi$$
0.0954869 + 0.995431i $$0.469559\pi$$
$$662$$ −5.64448e6 −0.500586
$$663$$ 0 0
$$664$$ −7.39368e6 −0.650789
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.23443e6 0.542603
$$668$$ 3.35656e7 2.91041
$$669$$ 0 0
$$670$$ 2.71062e7 2.33283
$$671$$ 5.95034e6 0.510194
$$672$$ 0 0
$$673$$ 2.92796e6 0.249188 0.124594 0.992208i $$-0.460237\pi$$
0.124594 + 0.992208i $$0.460237\pi$$
$$674$$ 2.07729e7 1.76136
$$675$$ 0 0
$$676$$ −2.39151e7 −2.01282
$$677$$ −1.34992e7 −1.13198 −0.565988 0.824414i $$-0.691505\pi$$
−0.565988 + 0.824414i $$0.691505\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 3.47155e7 2.87906
$$681$$ 0 0
$$682$$ −1.77341e7 −1.45998
$$683$$ 5.42972e6 0.445375 0.222688 0.974890i $$-0.428517\pi$$
0.222688 + 0.974890i $$0.428517\pi$$
$$684$$ 0 0
$$685$$ −1.14499e7 −0.932340
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 1.54931e7 1.24787
$$689$$ −316680. −0.0254140
$$690$$ 0 0
$$691$$ −2.08280e7 −1.65940 −0.829702 0.558207i $$-0.811490\pi$$
−0.829702 + 0.558207i $$0.811490\pi$$
$$692$$ 1.63687e7 1.29942
$$693$$ 0 0
$$694$$ 532480. 0.0419667
$$695$$ −1.98274e6 −0.155705
$$696$$ 0 0
$$697$$ 3.09306e7 2.41160
$$698$$ 2.27200e7 1.76510
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −2.35141e7 −1.80731 −0.903655 0.428261i $$-0.859126\pi$$
−0.903655 + 0.428261i $$0.859126\pi$$
$$702$$ 0 0
$$703$$ −1.01900e6 −0.0777656
$$704$$ 4.26138e6 0.324055
$$705$$ 0 0
$$706$$ 4.00645e7 3.02516
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.95747e7 −1.46244 −0.731221 0.682140i $$-0.761049\pi$$
−0.731221 + 0.682140i $$0.761049\pi$$
$$710$$ −3.27936e7 −2.44142
$$711$$ 0 0
$$712$$ −1.82095e7 −1.34617
$$713$$ −1.39427e7 −1.02712
$$714$$ 0 0
$$715$$ 1.81888e6 0.133057
$$716$$ −2.00554e7 −1.46200
$$717$$ 0 0
$$718$$ −737840. −0.0534135
$$719$$ −2.61152e7 −1.88396 −0.941978 0.335674i $$-0.891036\pi$$
−0.941978 + 0.335674i $$0.891036\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −2.46650e7 −1.76091
$$723$$ 0 0
$$724$$ 2.29146e7 1.62468
$$725$$ −37598.0 −0.00265656
$$726$$ 0 0
$$727$$ −1.54126e7 −1.08154 −0.540768 0.841172i $$-0.681866\pi$$
−0.540768 + 0.841172i $$0.681866\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 3.81259e7 2.64797
$$731$$ −1.87354e7 −1.29679
$$732$$ 0 0
$$733$$ 1.69868e7 1.16776 0.583878 0.811841i $$-0.301535\pi$$
0.583878 + 0.811841i $$0.301535\pi$$
$$734$$ −1.40431e7 −0.962107
$$735$$ 0 0
$$736$$ −4.96128e6 −0.337597
$$737$$ 1.12297e7 0.761554
$$738$$ 0 0
$$739$$ 2.01511e6 0.135734 0.0678669 0.997694i $$-0.478381\pi$$
0.0678669 + 0.997694i $$0.478381\pi$$
$$740$$ 3.95956e7 2.65808
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.51381e7 1.00600 0.503001 0.864286i $$-0.332229\pi$$
0.503001 + 0.864286i $$0.332229\pi$$
$$744$$ 0 0
$$745$$ −1.13266e6 −0.0747666
$$746$$ −1.60323e7 −1.05475
$$747$$ 0 0
$$748$$ 2.71663e7 1.77532
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 7.21401e6 0.466742 0.233371 0.972388i $$-0.425024\pi$$
0.233371 + 0.972388i $$0.425024\pi$$
$$752$$ 1.32774e7 0.856185
$$753$$ 0 0
$$754$$ −4.78520e6 −0.306529
$$755$$ −3.97062e6 −0.253508
$$756$$ 0 0
$$757$$ −1.09697e7 −0.695755 −0.347877 0.937540i $$-0.613097\pi$$
−0.347877 + 0.937540i $$0.613097\pi$$
$$758$$ −4.77012e7 −3.01548
$$759$$ 0 0
$$760$$ −1.97568e6 −0.124075
$$761$$ 1.92442e7 1.20459 0.602293 0.798275i $$-0.294254\pi$$
0.602293 + 0.798275i $$0.294254\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2.43620e7 −1.51001
$$765$$ 0 0
$$766$$ −2.23079e7 −1.37368
$$767$$ −382200. −0.0234586
$$768$$ 0 0
$$769$$ −8.21185e6 −0.500755 −0.250378 0.968148i $$-0.580555\pi$$
−0.250378 + 0.968148i $$0.580555\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −6.72897e7 −4.06355
$$773$$ 1.86187e7 1.12073 0.560363 0.828247i $$-0.310662\pi$$
0.560363 + 0.828247i $$0.310662\pi$$
$$774$$ 0 0
$$775$$ 84084.0 0.00502874
$$776$$ 2.10622e7 1.25559
$$777$$ 0 0
$$778$$ −4.84024e7 −2.86694
$$779$$ −1.76028e6 −0.103929
$$780$$ 0 0
$$781$$ −1.35859e7 −0.797006
$$782$$ 3.14093e7 1.83671
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.64373e7 0.952045
$$786$$ 0 0
$$787$$ −2.62501e7 −1.51075 −0.755377 0.655291i $$-0.772546\pi$$
−0.755377 + 0.655291i $$0.772546\pi$$
$$788$$ 6.73234e7 3.86234
$$789$$ 0 0
$$790$$ −1.77990e7 −1.01468
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −3.59072e6 −0.202768
$$794$$ −9.95820e6 −0.560570
$$795$$ 0 0
$$796$$ 5.71714e7 3.19813
$$797$$ −1.00373e7 −0.559720 −0.279860 0.960041i $$-0.590288\pi$$
−0.279860 + 0.960041i $$0.590288\pi$$
$$798$$ 0 0
$$799$$ −1.60559e7 −0.889751
$$800$$ 29920.0 0.00165286
$$801$$ 0 0
$$802$$ 3.31605e7 1.82048
$$803$$ 1.57950e7 0.864433
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 1.07016e7 0.580245
$$807$$ 0 0
$$808$$ 1.39306e7 0.750655
$$809$$ −1.40884e7 −0.756816 −0.378408 0.925639i $$-0.623528\pi$$
−0.378408 + 0.925639i $$0.623528\pi$$
$$810$$ 0 0
$$811$$ −1.81433e7 −0.968646 −0.484323 0.874889i $$-0.660934\pi$$
−0.484323 + 0.874889i $$0.660934\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 2.41234e7 1.27608
$$815$$ −738752. −0.0389587
$$816$$ 0 0
$$817$$ 1.06624e6 0.0558856
$$818$$ −3.07273e7 −1.60562
$$819$$ 0 0
$$820$$ 6.83993e7 3.55236
$$821$$ 2.13669e7 1.10633 0.553164 0.833072i $$-0.313420\pi$$
0.553164 + 0.833072i $$0.313420\pi$$
$$822$$ 0 0
$$823$$ 1.78017e7 0.916142 0.458071 0.888916i $$-0.348541\pi$$
0.458071 + 0.888916i $$0.348541\pi$$
$$824$$ −1.91016e7 −0.980058
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1.62921e7 −0.828350 −0.414175 0.910197i $$-0.635930\pi$$
−0.414175 + 0.910197i $$0.635930\pi$$
$$828$$ 0 0
$$829$$ 2.08499e6 0.105370 0.0526851 0.998611i $$-0.483222\pi$$
0.0526851 + 0.998611i $$0.483222\pi$$
$$830$$ 1.15013e7 0.579497
$$831$$ 0 0
$$832$$ −2.57152e6 −0.128790
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −2.76423e7 −1.37201
$$836$$ −1.54605e6 −0.0765081
$$837$$ 0 0
$$838$$ 2.81438e7 1.38443
$$839$$ −2.27850e7 −1.11749 −0.558745 0.829340i $$-0.688717\pi$$
−0.558745 + 0.829340i $$0.688717\pi$$
$$840$$ 0 0
$$841$$ −8.82842e6 −0.430421
$$842$$ 3.05802e7 1.48648
$$843$$ 0 0
$$844$$ 7.82498e7 3.78118
$$845$$ 1.96948e7 0.948877
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −3.22109e6 −0.153820
$$849$$ 0 0
$$850$$ −189420. −0.00899246
$$851$$ 1.89660e7 0.897740
$$852$$ 0 0
$$853$$ 2.26975e7 1.06808 0.534042 0.845458i $$-0.320672\pi$$
0.534042 + 0.845458i $$0.320672\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 5.26766e7 2.45716
$$857$$ 2.52900e7 1.17624 0.588120 0.808774i $$-0.299868\pi$$
0.588120 + 0.808774i $$0.299868\pi$$
$$858$$ 0 0
$$859$$ 1.03947e7 0.480652 0.240326 0.970692i $$-0.422746\pi$$
0.240326 + 0.970692i $$0.422746\pi$$
$$860$$ −4.14310e7 −1.91020
$$861$$ 0 0
$$862$$ −1.93750e7 −0.888122
$$863$$ −4.33399e7 −1.98089 −0.990447 0.137892i $$-0.955967\pi$$
−0.990447 + 0.137892i $$0.955967\pi$$
$$864$$ 0 0
$$865$$ −1.34801e7 −0.612566
$$866$$ −3.94790e7 −1.78884
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −7.37389e6 −0.331243
$$870$$ 0 0
$$871$$ −6.77656e6 −0.302666
$$872$$ 3.34433e7 1.48942
$$873$$ 0 0
$$874$$ −1.78752e6 −0.0791539
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 3.71659e7 1.63172 0.815861 0.578248i $$-0.196264\pi$$
0.815861 + 0.578248i $$0.196264\pi$$
$$878$$ 7.41770e7 3.24738
$$879$$ 0 0
$$880$$ 1.85006e7 0.805340
$$881$$ 9.04785e6 0.392740 0.196370 0.980530i $$-0.437085\pi$$
0.196370 + 0.980530i $$0.437085\pi$$
$$882$$ 0 0
$$883$$ 3.29679e7 1.42295 0.711474 0.702712i $$-0.248028\pi$$
0.711474 + 0.702712i $$0.248028\pi$$
$$884$$ −1.63934e7 −0.705569
$$885$$ 0 0
$$886$$ −1.40269e7 −0.600313
$$887$$ −1.61099e7 −0.687517 −0.343758 0.939058i $$-0.611700\pi$$
−0.343758 + 0.939058i $$0.611700\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 2.83259e7 1.19870
$$891$$ 0 0
$$892$$ 5.60500e7 2.35865
$$893$$ 913752. 0.0383442
$$894$$ 0 0
$$895$$ 1.65162e7 0.689211
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 5.90574e6 0.244390
$$899$$ −2.61272e7 −1.07819
$$900$$ 0 0
$$901$$ 3.89516e6 0.159850
$$902$$ 4.16718e7 1.70540
$$903$$ 0 0
$$904$$ 3.00074e7 1.22126
$$905$$ −1.88709e7 −0.765898
$$906$$ 0 0
$$907$$ −4.47286e7 −1.80537 −0.902686 0.430300i $$-0.858408\pi$$
−0.902686 + 0.430300i $$0.858408\pi$$
$$908$$ 5.05798e6 0.203593
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 6.60518e6 0.263687 0.131844 0.991271i $$-0.457910\pi$$
0.131844 + 0.991271i $$0.457910\pi$$
$$912$$ 0 0
$$913$$ 4.76482e6 0.189177
$$914$$ −2.90484e7 −1.15016
$$915$$ 0 0
$$916$$ −7.69730e7 −3.03110
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −3.08930e7 −1.20662 −0.603311 0.797506i $$-0.706152\pi$$
−0.603311 + 0.797506i $$0.706152\pi$$
$$920$$ 3.67718e7 1.43234
$$921$$ 0 0
$$922$$ −9.22684e6 −0.357459
$$923$$ 8.19840e6 0.316756
$$924$$ 0 0
$$925$$ −114378. −0.00439530
$$926$$ 7.18235e7 2.75258
$$927$$ 0 0
$$928$$ −9.29696e6 −0.354381
$$929$$ −4.87215e6 −0.185217 −0.0926087 0.995703i $$-0.529521\pi$$
−0.0926087 + 0.995703i $$0.529521\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 1.35134e7 0.509593
$$933$$ 0 0
$$934$$ −6.12570e6 −0.229767
$$935$$ −2.23722e7 −0.836913
$$936$$ 0 0
$$937$$ 3.25004e7 1.20932 0.604658 0.796485i $$-0.293310\pi$$
0.604658 + 0.796485i $$0.293310\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −3.55058e7 −1.31063
$$941$$ −2.64040e6 −0.0972066 −0.0486033 0.998818i $$-0.515477\pi$$
−0.0486033 + 0.998818i $$0.515477\pi$$
$$942$$ 0 0
$$943$$ 3.27627e7 1.19978
$$944$$ −3.88752e6 −0.141985
$$945$$ 0 0
$$946$$ −2.52416e7 −0.917042
$$947$$ 4.08179e7 1.47903 0.739513 0.673142i $$-0.235056\pi$$
0.739513 + 0.673142i $$0.235056\pi$$
$$948$$ 0 0
$$949$$ −9.53148e6 −0.343554
$$950$$ 10780.0 0.000387534 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 6.71983e6 0.239677 0.119838 0.992793i $$-0.461762\pi$$
0.119838 + 0.992793i $$0.461762\pi$$
$$954$$ 0 0
$$955$$ 2.00628e7 0.711841
$$956$$ −3.28375e7 −1.16205
$$957$$ 0 0
$$958$$ 2.60330e7 0.916454
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 2.98016e7 1.04095
$$962$$ −1.45572e7 −0.507154
$$963$$ 0 0
$$964$$ −5.48019e7 −1.89934
$$965$$ 5.54150e7 1.91562
$$966$$ 0 0
$$967$$ −2.78979e6 −0.0959413 −0.0479707 0.998849i $$-0.515275\pi$$
−0.0479707 + 0.998849i $$0.515275\pi$$
$$968$$ −3.86017e7 −1.32409
$$969$$ 0 0
$$970$$ −3.27634e7 −1.11804
$$971$$ 3.33594e7 1.13545 0.567727 0.823217i $$-0.307823\pi$$
0.567727 + 0.823217i $$0.307823\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 5.46309e7 1.84519
$$975$$ 0 0
$$976$$ −3.65228e7 −1.22727
$$977$$ 7.60033e6 0.254739 0.127370 0.991855i $$-0.459347\pi$$
0.127370 + 0.991855i $$0.459347\pi$$
$$978$$ 0 0
$$979$$ 1.17350e7 0.391316
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −1.64090e7 −0.543004
$$983$$ −5.79760e6 −0.191366 −0.0956829 0.995412i $$-0.530503\pi$$
−0.0956829 + 0.995412i $$0.530503\pi$$
$$984$$ 0 0
$$985$$ −5.54428e7 −1.82077
$$986$$ 5.88580e7 1.92803
$$987$$ 0 0
$$988$$ 932960. 0.0304068
$$989$$ −1.98451e7 −0.645153
$$990$$ 0 0
$$991$$ 1.26825e7 0.410224 0.205112 0.978739i $$-0.434244\pi$$
0.205112 + 0.978739i $$0.434244\pi$$
$$992$$ 2.07917e7 0.670827
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.70823e7 −1.50765
$$996$$ 0 0
$$997$$ −1.44400e7 −0.460077 −0.230039 0.973182i $$-0.573885\pi$$
−0.230039 + 0.973182i $$0.573885\pi$$
$$998$$ 2.99796e7 0.952796
$$999$$ 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 441.6.a.k.1.1 1
3.2 odd 2 49.6.a.a.1.1 1
7.6 odd 2 63.6.a.e.1.1 1
12.11 even 2 784.6.a.c.1.1 1
21.2 odd 6 49.6.c.b.18.1 2
21.5 even 6 49.6.c.c.18.1 2
21.11 odd 6 49.6.c.b.30.1 2
21.17 even 6 49.6.c.c.30.1 2
21.20 even 2 7.6.a.a.1.1 1
28.27 even 2 1008.6.a.y.1.1 1
84.83 odd 2 112.6.a.g.1.1 1
105.62 odd 4 175.6.b.a.99.1 2
105.83 odd 4 175.6.b.a.99.2 2
105.104 even 2 175.6.a.b.1.1 1
168.83 odd 2 448.6.a.c.1.1 1
168.125 even 2 448.6.a.m.1.1 1
231.230 odd 2 847.6.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 21.20 even 2
49.6.a.a.1.1 1 3.2 odd 2
49.6.c.b.18.1 2 21.2 odd 6
49.6.c.b.30.1 2 21.11 odd 6
49.6.c.c.18.1 2 21.5 even 6
49.6.c.c.30.1 2 21.17 even 6
63.6.a.e.1.1 1 7.6 odd 2
112.6.a.g.1.1 1 84.83 odd 2
175.6.a.b.1.1 1 105.104 even 2
175.6.b.a.99.1 2 105.62 odd 4
175.6.b.a.99.2 2 105.83 odd 4
441.6.a.k.1.1 1 1.1 even 1 trivial
448.6.a.c.1.1 1 168.83 odd 2
448.6.a.m.1.1 1 168.125 even 2
784.6.a.c.1.1 1 12.11 even 2
847.6.a.b.1.1 1 231.230 odd 2
1008.6.a.y.1.1 1 28.27 even 2