# Properties

 Label 176.6.a.a.1.1 Level $176$ Weight $6$ Character 176.1 Self dual yes Analytic conductor $28.228$ 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] = [176,6,Mod(1,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, 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("176.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 176.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: $$28.2275522871$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 176.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-7.00000 q^{3} -79.0000 q^{5} +50.0000 q^{7} -194.000 q^{9} +O(q^{10})$$ $$q-7.00000 q^{3} -79.0000 q^{5} +50.0000 q^{7} -194.000 q^{9} -121.000 q^{11} -380.000 q^{13} +553.000 q^{15} -1154.00 q^{17} +1824.00 q^{19} -350.000 q^{21} -3591.00 q^{23} +3116.00 q^{25} +3059.00 q^{27} +8032.00 q^{29} +2945.00 q^{31} +847.000 q^{33} -3950.00 q^{35} +6979.00 q^{37} +2660.00 q^{39} -520.000 q^{41} +2486.00 q^{43} +15326.0 q^{45} +6920.00 q^{47} -14307.0 q^{49} +8078.00 q^{51} -13718.0 q^{53} +9559.00 q^{55} -12768.0 q^{57} +31779.0 q^{59} +34156.0 q^{61} -9700.00 q^{63} +30020.0 q^{65} +61503.0 q^{67} +25137.0 q^{69} +14971.0 q^{71} -36444.0 q^{73} -21812.0 q^{75} -6050.00 q^{77} +28538.0 q^{79} +25729.0 q^{81} -77482.0 q^{83} +91166.0 q^{85} -56224.0 q^{87} +36271.0 q^{89} -19000.0 q^{91} -20615.0 q^{93} -144096. q^{95} -49799.0 q^{97} +23474.0 q^{99} +O(q^{100})$$

## 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
$$3$$ −7.00000 −0.449050 −0.224525 0.974468i $$-0.572083\pi$$
−0.224525 + 0.974468i $$0.572083\pi$$
$$4$$ 0 0
$$5$$ −79.0000 −1.41319 −0.706597 0.707616i $$-0.749771\pi$$
−0.706597 + 0.707616i $$0.749771\pi$$
$$6$$ 0 0
$$7$$ 50.0000 0.385678 0.192839 0.981230i $$-0.438230\pi$$
0.192839 + 0.981230i $$0.438230\pi$$
$$8$$ 0 0
$$9$$ −194.000 −0.798354
$$10$$ 0 0
$$11$$ −121.000 −0.301511
$$12$$ 0 0
$$13$$ −380.000 −0.623627 −0.311814 0.950143i $$-0.600936\pi$$
−0.311814 + 0.950143i $$0.600936\pi$$
$$14$$ 0 0
$$15$$ 553.000 0.634595
$$16$$ 0 0
$$17$$ −1154.00 −0.968464 −0.484232 0.874940i $$-0.660901\pi$$
−0.484232 + 0.874940i $$0.660901\pi$$
$$18$$ 0 0
$$19$$ 1824.00 1.15915 0.579577 0.814918i $$-0.303218\pi$$
0.579577 + 0.814918i $$0.303218\pi$$
$$20$$ 0 0
$$21$$ −350.000 −0.173189
$$22$$ 0 0
$$23$$ −3591.00 −1.41545 −0.707727 0.706486i $$-0.750279\pi$$
−0.707727 + 0.706486i $$0.750279\pi$$
$$24$$ 0 0
$$25$$ 3116.00 0.997120
$$26$$ 0 0
$$27$$ 3059.00 0.807551
$$28$$ 0 0
$$29$$ 8032.00 1.77349 0.886745 0.462259i $$-0.152961\pi$$
0.886745 + 0.462259i $$0.152961\pi$$
$$30$$ 0 0
$$31$$ 2945.00 0.550403 0.275202 0.961387i $$-0.411255\pi$$
0.275202 + 0.961387i $$0.411255\pi$$
$$32$$ 0 0
$$33$$ 847.000 0.135394
$$34$$ 0 0
$$35$$ −3950.00 −0.545038
$$36$$ 0 0
$$37$$ 6979.00 0.838087 0.419043 0.907966i $$-0.362366\pi$$
0.419043 + 0.907966i $$0.362366\pi$$
$$38$$ 0 0
$$39$$ 2660.00 0.280040
$$40$$ 0 0
$$41$$ −520.000 −0.0483107 −0.0241554 0.999708i $$-0.507690\pi$$
−0.0241554 + 0.999708i $$0.507690\pi$$
$$42$$ 0 0
$$43$$ 2486.00 0.205036 0.102518 0.994731i $$-0.467310\pi$$
0.102518 + 0.994731i $$0.467310\pi$$
$$44$$ 0 0
$$45$$ 15326.0 1.12823
$$46$$ 0 0
$$47$$ 6920.00 0.456942 0.228471 0.973551i $$-0.426627\pi$$
0.228471 + 0.973551i $$0.426627\pi$$
$$48$$ 0 0
$$49$$ −14307.0 −0.851252
$$50$$ 0 0
$$51$$ 8078.00 0.434889
$$52$$ 0 0
$$53$$ −13718.0 −0.670812 −0.335406 0.942074i $$-0.608874\pi$$
−0.335406 + 0.942074i $$0.608874\pi$$
$$54$$ 0 0
$$55$$ 9559.00 0.426094
$$56$$ 0 0
$$57$$ −12768.0 −0.520518
$$58$$ 0 0
$$59$$ 31779.0 1.18853 0.594265 0.804269i $$-0.297443\pi$$
0.594265 + 0.804269i $$0.297443\pi$$
$$60$$ 0 0
$$61$$ 34156.0 1.17528 0.587641 0.809121i $$-0.300057\pi$$
0.587641 + 0.809121i $$0.300057\pi$$
$$62$$ 0 0
$$63$$ −9700.00 −0.307908
$$64$$ 0 0
$$65$$ 30020.0 0.881307
$$66$$ 0 0
$$67$$ 61503.0 1.67382 0.836911 0.547339i $$-0.184359\pi$$
0.836911 + 0.547339i $$0.184359\pi$$
$$68$$ 0 0
$$69$$ 25137.0 0.635610
$$70$$ 0 0
$$71$$ 14971.0 0.352456 0.176228 0.984349i $$-0.443610\pi$$
0.176228 + 0.984349i $$0.443610\pi$$
$$72$$ 0 0
$$73$$ −36444.0 −0.800422 −0.400211 0.916423i $$-0.631063\pi$$
−0.400211 + 0.916423i $$0.631063\pi$$
$$74$$ 0 0
$$75$$ −21812.0 −0.447757
$$76$$ 0 0
$$77$$ −6050.00 −0.116286
$$78$$ 0 0
$$79$$ 28538.0 0.514465 0.257232 0.966350i $$-0.417189\pi$$
0.257232 + 0.966350i $$0.417189\pi$$
$$80$$ 0 0
$$81$$ 25729.0 0.435723
$$82$$ 0 0
$$83$$ −77482.0 −1.23454 −0.617271 0.786751i $$-0.711762\pi$$
−0.617271 + 0.786751i $$0.711762\pi$$
$$84$$ 0 0
$$85$$ 91166.0 1.36863
$$86$$ 0 0
$$87$$ −56224.0 −0.796386
$$88$$ 0 0
$$89$$ 36271.0 0.485383 0.242691 0.970104i $$-0.421970\pi$$
0.242691 + 0.970104i $$0.421970\pi$$
$$90$$ 0 0
$$91$$ −19000.0 −0.240519
$$92$$ 0 0
$$93$$ −20615.0 −0.247159
$$94$$ 0 0
$$95$$ −144096. −1.63811
$$96$$ 0 0
$$97$$ −49799.0 −0.537392 −0.268696 0.963225i $$-0.586593\pi$$
−0.268696 + 0.963225i $$0.586593\pi$$
$$98$$ 0 0
$$99$$ 23474.0 0.240713
$$100$$ 0 0
$$101$$ −153406. −1.49637 −0.748185 0.663490i $$-0.769074\pi$$
−0.748185 + 0.663490i $$0.769074\pi$$
$$102$$ 0 0
$$103$$ 134720. 1.25124 0.625618 0.780130i $$-0.284847\pi$$
0.625618 + 0.780130i $$0.284847\pi$$
$$104$$ 0 0
$$105$$ 27650.0 0.244750
$$106$$ 0 0
$$107$$ −169218. −1.42885 −0.714426 0.699711i $$-0.753312\pi$$
−0.714426 + 0.699711i $$0.753312\pi$$
$$108$$ 0 0
$$109$$ −233206. −1.88007 −0.940034 0.341081i $$-0.889207\pi$$
−0.940034 + 0.341081i $$0.889207\pi$$
$$110$$ 0 0
$$111$$ −48853.0 −0.376343
$$112$$ 0 0
$$113$$ 94329.0 0.694943 0.347471 0.937691i $$-0.387040\pi$$
0.347471 + 0.937691i $$0.387040\pi$$
$$114$$ 0 0
$$115$$ 283689. 2.00031
$$116$$ 0 0
$$117$$ 73720.0 0.497875
$$118$$ 0 0
$$119$$ −57700.0 −0.373515
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ 0 0
$$123$$ 3640.00 0.0216939
$$124$$ 0 0
$$125$$ 711.000 0.00407000
$$126$$ 0 0
$$127$$ 259480. 1.42756 0.713780 0.700370i $$-0.246981\pi$$
0.713780 + 0.700370i $$0.246981\pi$$
$$128$$ 0 0
$$129$$ −17402.0 −0.0920714
$$130$$ 0 0
$$131$$ 85410.0 0.434841 0.217420 0.976078i $$-0.430236\pi$$
0.217420 + 0.976078i $$0.430236\pi$$
$$132$$ 0 0
$$133$$ 91200.0 0.447060
$$134$$ 0 0
$$135$$ −241661. −1.14123
$$136$$ 0 0
$$137$$ −427703. −1.94689 −0.973444 0.228926i $$-0.926479\pi$$
−0.973444 + 0.228926i $$0.926479\pi$$
$$138$$ 0 0
$$139$$ −309690. −1.35953 −0.679767 0.733428i $$-0.737919\pi$$
−0.679767 + 0.733428i $$0.737919\pi$$
$$140$$ 0 0
$$141$$ −48440.0 −0.205190
$$142$$ 0 0
$$143$$ 45980.0 0.188031
$$144$$ 0 0
$$145$$ −634528. −2.50629
$$146$$ 0 0
$$147$$ 100149. 0.382255
$$148$$ 0 0
$$149$$ 449846. 1.65996 0.829981 0.557792i $$-0.188351\pi$$
0.829981 + 0.557792i $$0.188351\pi$$
$$150$$ 0 0
$$151$$ −405074. −1.44575 −0.722873 0.690981i $$-0.757179\pi$$
−0.722873 + 0.690981i $$0.757179\pi$$
$$152$$ 0 0
$$153$$ 223876. 0.773177
$$154$$ 0 0
$$155$$ −232655. −0.777827
$$156$$ 0 0
$$157$$ 339321. 1.09866 0.549328 0.835607i $$-0.314884\pi$$
0.549328 + 0.835607i $$0.314884\pi$$
$$158$$ 0 0
$$159$$ 96026.0 0.301228
$$160$$ 0 0
$$161$$ −179550. −0.545910
$$162$$ 0 0
$$163$$ −271396. −0.800082 −0.400041 0.916497i $$-0.631004\pi$$
−0.400041 + 0.916497i $$0.631004\pi$$
$$164$$ 0 0
$$165$$ −66913.0 −0.191338
$$166$$ 0 0
$$167$$ −72468.0 −0.201074 −0.100537 0.994933i $$-0.532056\pi$$
−0.100537 + 0.994933i $$0.532056\pi$$
$$168$$ 0 0
$$169$$ −226893. −0.611089
$$170$$ 0 0
$$171$$ −353856. −0.925414
$$172$$ 0 0
$$173$$ 479226. 1.21738 0.608689 0.793409i $$-0.291696\pi$$
0.608689 + 0.793409i $$0.291696\pi$$
$$174$$ 0 0
$$175$$ 155800. 0.384567
$$176$$ 0 0
$$177$$ −222453. −0.533710
$$178$$ 0 0
$$179$$ 40935.0 0.0954910 0.0477455 0.998860i $$-0.484796\pi$$
0.0477455 + 0.998860i $$0.484796\pi$$
$$180$$ 0 0
$$181$$ −90169.0 −0.204579 −0.102289 0.994755i $$-0.532617\pi$$
−0.102289 + 0.994755i $$0.532617\pi$$
$$182$$ 0 0
$$183$$ −239092. −0.527761
$$184$$ 0 0
$$185$$ −551341. −1.18438
$$186$$ 0 0
$$187$$ 139634. 0.292003
$$188$$ 0 0
$$189$$ 152950. 0.311455
$$190$$ 0 0
$$191$$ 260375. 0.516435 0.258218 0.966087i $$-0.416865\pi$$
0.258218 + 0.966087i $$0.416865\pi$$
$$192$$ 0 0
$$193$$ 524324. 1.01323 0.506613 0.862173i $$-0.330897\pi$$
0.506613 + 0.862173i $$0.330897\pi$$
$$194$$ 0 0
$$195$$ −210140. −0.395751
$$196$$ 0 0
$$197$$ 759582. 1.39447 0.697235 0.716843i $$-0.254413\pi$$
0.697235 + 0.716843i $$0.254413\pi$$
$$198$$ 0 0
$$199$$ 882736. 1.58015 0.790075 0.613011i $$-0.210042\pi$$
0.790075 + 0.613011i $$0.210042\pi$$
$$200$$ 0 0
$$201$$ −430521. −0.751630
$$202$$ 0 0
$$203$$ 401600. 0.683996
$$204$$ 0 0
$$205$$ 41080.0 0.0682725
$$206$$ 0 0
$$207$$ 696654. 1.13003
$$208$$ 0 0
$$209$$ −220704. −0.349498
$$210$$ 0 0
$$211$$ 1.15285e6 1.78266 0.891328 0.453360i $$-0.149775\pi$$
0.891328 + 0.453360i $$0.149775\pi$$
$$212$$ 0 0
$$213$$ −104797. −0.158270
$$214$$ 0 0
$$215$$ −196394. −0.289756
$$216$$ 0 0
$$217$$ 147250. 0.212278
$$218$$ 0 0
$$219$$ 255108. 0.359430
$$220$$ 0 0
$$221$$ 438520. 0.603961
$$222$$ 0 0
$$223$$ 65893.0 0.0887314 0.0443657 0.999015i $$-0.485873\pi$$
0.0443657 + 0.999015i $$0.485873\pi$$
$$224$$ 0 0
$$225$$ −604504. −0.796055
$$226$$ 0 0
$$227$$ 314526. 0.405128 0.202564 0.979269i $$-0.435073\pi$$
0.202564 + 0.979269i $$0.435073\pi$$
$$228$$ 0 0
$$229$$ 1.03846e6 1.30859 0.654293 0.756241i $$-0.272966\pi$$
0.654293 + 0.756241i $$0.272966\pi$$
$$230$$ 0 0
$$231$$ 42350.0 0.0522184
$$232$$ 0 0
$$233$$ 509976. 0.615403 0.307702 0.951483i $$-0.400440\pi$$
0.307702 + 0.951483i $$0.400440\pi$$
$$234$$ 0 0
$$235$$ −546680. −0.645749
$$236$$ 0 0
$$237$$ −199766. −0.231021
$$238$$ 0 0
$$239$$ 444494. 0.503351 0.251676 0.967812i $$-0.419018\pi$$
0.251676 + 0.967812i $$0.419018\pi$$
$$240$$ 0 0
$$241$$ −283464. −0.314380 −0.157190 0.987568i $$-0.550244\pi$$
−0.157190 + 0.987568i $$0.550244\pi$$
$$242$$ 0 0
$$243$$ −923440. −1.00321
$$244$$ 0 0
$$245$$ 1.13025e6 1.20299
$$246$$ 0 0
$$247$$ −693120. −0.722880
$$248$$ 0 0
$$249$$ 542374. 0.554371
$$250$$ 0 0
$$251$$ 773807. 0.775262 0.387631 0.921815i $$-0.373294\pi$$
0.387631 + 0.921815i $$0.373294\pi$$
$$252$$ 0 0
$$253$$ 434511. 0.426775
$$254$$ 0 0
$$255$$ −638162. −0.614583
$$256$$ 0 0
$$257$$ −387714. −0.366167 −0.183083 0.983097i $$-0.558608\pi$$
−0.183083 + 0.983097i $$0.558608\pi$$
$$258$$ 0 0
$$259$$ 348950. 0.323232
$$260$$ 0 0
$$261$$ −1.55821e6 −1.41587
$$262$$ 0 0
$$263$$ 197602. 0.176158 0.0880789 0.996113i $$-0.471927\pi$$
0.0880789 + 0.996113i $$0.471927\pi$$
$$264$$ 0 0
$$265$$ 1.08372e6 0.947989
$$266$$ 0 0
$$267$$ −253897. −0.217961
$$268$$ 0 0
$$269$$ −262694. −0.221345 −0.110672 0.993857i $$-0.535300\pi$$
−0.110672 + 0.993857i $$0.535300\pi$$
$$270$$ 0 0
$$271$$ 159068. 0.131571 0.0657854 0.997834i $$-0.479045\pi$$
0.0657854 + 0.997834i $$0.479045\pi$$
$$272$$ 0 0
$$273$$ 133000. 0.108005
$$274$$ 0 0
$$275$$ −377036. −0.300643
$$276$$ 0 0
$$277$$ 1.29385e6 1.01318 0.506589 0.862188i $$-0.330906\pi$$
0.506589 + 0.862188i $$0.330906\pi$$
$$278$$ 0 0
$$279$$ −571330. −0.439417
$$280$$ 0 0
$$281$$ −1.78114e6 −1.34565 −0.672824 0.739802i $$-0.734919\pi$$
−0.672824 + 0.739802i $$0.734919\pi$$
$$282$$ 0 0
$$283$$ −1.98279e6 −1.47167 −0.735835 0.677161i $$-0.763210\pi$$
−0.735835 + 0.677161i $$0.763210\pi$$
$$284$$ 0 0
$$285$$ 1.00867e6 0.735593
$$286$$ 0 0
$$287$$ −26000.0 −0.0186324
$$288$$ 0 0
$$289$$ −88141.0 −0.0620774
$$290$$ 0 0
$$291$$ 348593. 0.241316
$$292$$ 0 0
$$293$$ 578360. 0.393577 0.196788 0.980446i $$-0.436949\pi$$
0.196788 + 0.980446i $$0.436949\pi$$
$$294$$ 0 0
$$295$$ −2.51054e6 −1.67962
$$296$$ 0 0
$$297$$ −370139. −0.243486
$$298$$ 0 0
$$299$$ 1.36458e6 0.882716
$$300$$ 0 0
$$301$$ 124300. 0.0790779
$$302$$ 0 0
$$303$$ 1.07384e6 0.671945
$$304$$ 0 0
$$305$$ −2.69832e6 −1.66090
$$306$$ 0 0
$$307$$ 3.07602e6 1.86270 0.931352 0.364120i $$-0.118630\pi$$
0.931352 + 0.364120i $$0.118630\pi$$
$$308$$ 0 0
$$309$$ −943040. −0.561868
$$310$$ 0 0
$$311$$ 3.13757e6 1.83947 0.919735 0.392540i $$-0.128403\pi$$
0.919735 + 0.392540i $$0.128403\pi$$
$$312$$ 0 0
$$313$$ 2.61784e6 1.51037 0.755183 0.655514i $$-0.227548\pi$$
0.755183 + 0.655514i $$0.227548\pi$$
$$314$$ 0 0
$$315$$ 766300. 0.435133
$$316$$ 0 0
$$317$$ −2.49220e6 −1.39294 −0.696472 0.717584i $$-0.745248\pi$$
−0.696472 + 0.717584i $$0.745248\pi$$
$$318$$ 0 0
$$319$$ −971872. −0.534727
$$320$$ 0 0
$$321$$ 1.18453e6 0.641626
$$322$$ 0 0
$$323$$ −2.10490e6 −1.12260
$$324$$ 0 0
$$325$$ −1.18408e6 −0.621831
$$326$$ 0 0
$$327$$ 1.63244e6 0.844245
$$328$$ 0 0
$$329$$ 346000. 0.176233
$$330$$ 0 0
$$331$$ 2.70125e6 1.35517 0.677586 0.735443i $$-0.263026\pi$$
0.677586 + 0.735443i $$0.263026\pi$$
$$332$$ 0 0
$$333$$ −1.35393e6 −0.669090
$$334$$ 0 0
$$335$$ −4.85874e6 −2.36544
$$336$$ 0 0
$$337$$ −1.42610e6 −0.684031 −0.342016 0.939694i $$-0.611110\pi$$
−0.342016 + 0.939694i $$0.611110\pi$$
$$338$$ 0 0
$$339$$ −660303. −0.312064
$$340$$ 0 0
$$341$$ −356345. −0.165953
$$342$$ 0 0
$$343$$ −1.55570e6 −0.713987
$$344$$ 0 0
$$345$$ −1.98582e6 −0.898241
$$346$$ 0 0
$$347$$ −2.86374e6 −1.27676 −0.638381 0.769721i $$-0.720396\pi$$
−0.638381 + 0.769721i $$0.720396\pi$$
$$348$$ 0 0
$$349$$ 296350. 0.130239 0.0651195 0.997877i $$-0.479257\pi$$
0.0651195 + 0.997877i $$0.479257\pi$$
$$350$$ 0 0
$$351$$ −1.16242e6 −0.503611
$$352$$ 0 0
$$353$$ −2.12114e6 −0.906010 −0.453005 0.891508i $$-0.649648\pi$$
−0.453005 + 0.891508i $$0.649648\pi$$
$$354$$ 0 0
$$355$$ −1.18271e6 −0.498089
$$356$$ 0 0
$$357$$ 403900. 0.167727
$$358$$ 0 0
$$359$$ 3.47512e6 1.42310 0.711548 0.702638i $$-0.247994\pi$$
0.711548 + 0.702638i $$0.247994\pi$$
$$360$$ 0 0
$$361$$ 850877. 0.343636
$$362$$ 0 0
$$363$$ −102487. −0.0408227
$$364$$ 0 0
$$365$$ 2.87908e6 1.13115
$$366$$ 0 0
$$367$$ −1.56190e6 −0.605322 −0.302661 0.953098i $$-0.597875\pi$$
−0.302661 + 0.953098i $$0.597875\pi$$
$$368$$ 0 0
$$369$$ 100880. 0.0385691
$$370$$ 0 0
$$371$$ −685900. −0.258718
$$372$$ 0 0
$$373$$ 1.93773e6 0.721144 0.360572 0.932731i $$-0.382581\pi$$
0.360572 + 0.932731i $$0.382581\pi$$
$$374$$ 0 0
$$375$$ −4977.00 −0.00182764
$$376$$ 0 0
$$377$$ −3.05216e6 −1.10600
$$378$$ 0 0
$$379$$ −3.07495e6 −1.09961 −0.549806 0.835292i $$-0.685298\pi$$
−0.549806 + 0.835292i $$0.685298\pi$$
$$380$$ 0 0
$$381$$ −1.81636e6 −0.641046
$$382$$ 0 0
$$383$$ −4.31553e6 −1.50327 −0.751635 0.659579i $$-0.770734\pi$$
−0.751635 + 0.659579i $$0.770734\pi$$
$$384$$ 0 0
$$385$$ 477950. 0.164335
$$386$$ 0 0
$$387$$ −482284. −0.163691
$$388$$ 0 0
$$389$$ 2.36251e6 0.791590 0.395795 0.918339i $$-0.370469\pi$$
0.395795 + 0.918339i $$0.370469\pi$$
$$390$$ 0 0
$$391$$ 4.14401e6 1.37082
$$392$$ 0 0
$$393$$ −597870. −0.195265
$$394$$ 0 0
$$395$$ −2.25450e6 −0.727039
$$396$$ 0 0
$$397$$ 1.77598e6 0.565539 0.282769 0.959188i $$-0.408747\pi$$
0.282769 + 0.959188i $$0.408747\pi$$
$$398$$ 0 0
$$399$$ −638400. −0.200752
$$400$$ 0 0
$$401$$ 1.56967e6 0.487468 0.243734 0.969842i $$-0.421628\pi$$
0.243734 + 0.969842i $$0.421628\pi$$
$$402$$ 0 0
$$403$$ −1.11910e6 −0.343247
$$404$$ 0 0
$$405$$ −2.03259e6 −0.615761
$$406$$ 0 0
$$407$$ −844459. −0.252693
$$408$$ 0 0
$$409$$ 1.29485e6 0.382746 0.191373 0.981517i $$-0.438706\pi$$
0.191373 + 0.981517i $$0.438706\pi$$
$$410$$ 0 0
$$411$$ 2.99392e6 0.874250
$$412$$ 0 0
$$413$$ 1.58895e6 0.458390
$$414$$ 0 0
$$415$$ 6.12108e6 1.74465
$$416$$ 0 0
$$417$$ 2.16783e6 0.610499
$$418$$ 0 0
$$419$$ −272916. −0.0759441 −0.0379720 0.999279i $$-0.512090\pi$$
−0.0379720 + 0.999279i $$0.512090\pi$$
$$420$$ 0 0
$$421$$ −2.61801e6 −0.719890 −0.359945 0.932974i $$-0.617205\pi$$
−0.359945 + 0.932974i $$0.617205\pi$$
$$422$$ 0 0
$$423$$ −1.34248e6 −0.364802
$$424$$ 0 0
$$425$$ −3.59586e6 −0.965675
$$426$$ 0 0
$$427$$ 1.70780e6 0.453281
$$428$$ 0 0
$$429$$ −321860. −0.0844352
$$430$$ 0 0
$$431$$ −2.81037e6 −0.728735 −0.364368 0.931255i $$-0.618715\pi$$
−0.364368 + 0.931255i $$0.618715\pi$$
$$432$$ 0 0
$$433$$ 5.98509e6 1.53409 0.767046 0.641593i $$-0.221726\pi$$
0.767046 + 0.641593i $$0.221726\pi$$
$$434$$ 0 0
$$435$$ 4.44170e6 1.12545
$$436$$ 0 0
$$437$$ −6.54998e6 −1.64073
$$438$$ 0 0
$$439$$ 7.50486e6 1.85858 0.929290 0.369352i $$-0.120420\pi$$
0.929290 + 0.369352i $$0.120420\pi$$
$$440$$ 0 0
$$441$$ 2.77556e6 0.679601
$$442$$ 0 0
$$443$$ −1.56806e6 −0.379624 −0.189812 0.981820i $$-0.560788\pi$$
−0.189812 + 0.981820i $$0.560788\pi$$
$$444$$ 0 0
$$445$$ −2.86541e6 −0.685941
$$446$$ 0 0
$$447$$ −3.14892e6 −0.745406
$$448$$ 0 0
$$449$$ −4.04044e6 −0.945831 −0.472915 0.881108i $$-0.656798\pi$$
−0.472915 + 0.881108i $$0.656798\pi$$
$$450$$ 0 0
$$451$$ 62920.0 0.0145662
$$452$$ 0 0
$$453$$ 2.83552e6 0.649213
$$454$$ 0 0
$$455$$ 1.50100e6 0.339901
$$456$$ 0 0
$$457$$ 2.21132e6 0.495291 0.247645 0.968851i $$-0.420343\pi$$
0.247645 + 0.968851i $$0.420343\pi$$
$$458$$ 0 0
$$459$$ −3.53009e6 −0.782084
$$460$$ 0 0
$$461$$ −3.56735e6 −0.781795 −0.390898 0.920434i $$-0.627835\pi$$
−0.390898 + 0.920434i $$0.627835\pi$$
$$462$$ 0 0
$$463$$ −747757. −0.162109 −0.0810547 0.996710i $$-0.525829\pi$$
−0.0810547 + 0.996710i $$0.525829\pi$$
$$464$$ 0 0
$$465$$ 1.62858e6 0.349283
$$466$$ 0 0
$$467$$ 5.44511e6 1.15535 0.577676 0.816266i $$-0.303960\pi$$
0.577676 + 0.816266i $$0.303960\pi$$
$$468$$ 0 0
$$469$$ 3.07515e6 0.645556
$$470$$ 0 0
$$471$$ −2.37525e6 −0.493352
$$472$$ 0 0
$$473$$ −300806. −0.0618207
$$474$$ 0 0
$$475$$ 5.68358e6 1.15581
$$476$$ 0 0
$$477$$ 2.66129e6 0.535546
$$478$$ 0 0
$$479$$ 6.22046e6 1.23875 0.619375 0.785095i $$-0.287386\pi$$
0.619375 + 0.785095i $$0.287386\pi$$
$$480$$ 0 0
$$481$$ −2.65202e6 −0.522654
$$482$$ 0 0
$$483$$ 1.25685e6 0.245141
$$484$$ 0 0
$$485$$ 3.93412e6 0.759440
$$486$$ 0 0
$$487$$ 3.34398e6 0.638913 0.319457 0.947601i $$-0.396500\pi$$
0.319457 + 0.947601i $$0.396500\pi$$
$$488$$ 0 0
$$489$$ 1.89977e6 0.359277
$$490$$ 0 0
$$491$$ −5.58646e6 −1.04576 −0.522881 0.852406i $$-0.675143\pi$$
−0.522881 + 0.852406i $$0.675143\pi$$
$$492$$ 0 0
$$493$$ −9.26893e6 −1.71756
$$494$$ 0 0
$$495$$ −1.85445e6 −0.340174
$$496$$ 0 0
$$497$$ 748550. 0.135935
$$498$$ 0 0
$$499$$ 8.29348e6 1.49103 0.745514 0.666490i $$-0.232204\pi$$
0.745514 + 0.666490i $$0.232204\pi$$
$$500$$ 0 0
$$501$$ 507276. 0.0902922
$$502$$ 0 0
$$503$$ −5.29951e6 −0.933933 −0.466967 0.884275i $$-0.654653\pi$$
−0.466967 + 0.884275i $$0.654653\pi$$
$$504$$ 0 0
$$505$$ 1.21191e7 2.11466
$$506$$ 0 0
$$507$$ 1.58825e6 0.274410
$$508$$ 0 0
$$509$$ 24415.0 0.00417698 0.00208849 0.999998i $$-0.499335\pi$$
0.00208849 + 0.999998i $$0.499335\pi$$
$$510$$ 0 0
$$511$$ −1.82220e6 −0.308705
$$512$$ 0 0
$$513$$ 5.57962e6 0.936076
$$514$$ 0 0
$$515$$ −1.06429e7 −1.76824
$$516$$ 0 0
$$517$$ −837320. −0.137773
$$518$$ 0 0
$$519$$ −3.35458e6 −0.546663
$$520$$ 0 0
$$521$$ 4.76275e6 0.768712 0.384356 0.923185i $$-0.374424\pi$$
0.384356 + 0.923185i $$0.374424\pi$$
$$522$$ 0 0
$$523$$ −735248. −0.117538 −0.0587692 0.998272i $$-0.518718\pi$$
−0.0587692 + 0.998272i $$0.518718\pi$$
$$524$$ 0 0
$$525$$ −1.09060e6 −0.172690
$$526$$ 0 0
$$527$$ −3.39853e6 −0.533046
$$528$$ 0 0
$$529$$ 6.45894e6 1.00351
$$530$$ 0 0
$$531$$ −6.16513e6 −0.948868
$$532$$ 0 0
$$533$$ 197600. 0.0301279
$$534$$ 0 0
$$535$$ 1.33682e7 2.01925
$$536$$ 0 0
$$537$$ −286545. −0.0428802
$$538$$ 0 0
$$539$$ 1.73115e6 0.256662
$$540$$ 0 0
$$541$$ −3.19649e6 −0.469548 −0.234774 0.972050i $$-0.575435\pi$$
−0.234774 + 0.972050i $$0.575435\pi$$
$$542$$ 0 0
$$543$$ 631183. 0.0918662
$$544$$ 0 0
$$545$$ 1.84233e7 2.65690
$$546$$ 0 0
$$547$$ −8.85902e6 −1.26595 −0.632976 0.774171i $$-0.718167\pi$$
−0.632976 + 0.774171i $$0.718167\pi$$
$$548$$ 0 0
$$549$$ −6.62626e6 −0.938292
$$550$$ 0 0
$$551$$ 1.46504e7 2.05575
$$552$$ 0 0
$$553$$ 1.42690e6 0.198418
$$554$$ 0 0
$$555$$ 3.85939e6 0.531846
$$556$$ 0 0
$$557$$ −1.74512e6 −0.238335 −0.119167 0.992874i $$-0.538023\pi$$
−0.119167 + 0.992874i $$0.538023\pi$$
$$558$$ 0 0
$$559$$ −944680. −0.127866
$$560$$ 0 0
$$561$$ −977438. −0.131124
$$562$$ 0 0
$$563$$ 1.32333e7 1.75953 0.879764 0.475410i $$-0.157700\pi$$
0.879764 + 0.475410i $$0.157700\pi$$
$$564$$ 0 0
$$565$$ −7.45199e6 −0.982090
$$566$$ 0 0
$$567$$ 1.28645e6 0.168049
$$568$$ 0 0
$$569$$ 1.04156e7 1.34867 0.674335 0.738426i $$-0.264430\pi$$
0.674335 + 0.738426i $$0.264430\pi$$
$$570$$ 0 0
$$571$$ 2.48163e6 0.318527 0.159264 0.987236i $$-0.449088\pi$$
0.159264 + 0.987236i $$0.449088\pi$$
$$572$$ 0 0
$$573$$ −1.82262e6 −0.231905
$$574$$ 0 0
$$575$$ −1.11896e7 −1.41138
$$576$$ 0 0
$$577$$ −1.31244e7 −1.64112 −0.820562 0.571557i $$-0.806339\pi$$
−0.820562 + 0.571557i $$0.806339\pi$$
$$578$$ 0 0
$$579$$ −3.67027e6 −0.454989
$$580$$ 0 0
$$581$$ −3.87410e6 −0.476135
$$582$$ 0 0
$$583$$ 1.65988e6 0.202258
$$584$$ 0 0
$$585$$ −5.82388e6 −0.703595
$$586$$ 0 0
$$587$$ 4.86010e6 0.582170 0.291085 0.956697i $$-0.405984\pi$$
0.291085 + 0.956697i $$0.405984\pi$$
$$588$$ 0 0
$$589$$ 5.37168e6 0.638002
$$590$$ 0 0
$$591$$ −5.31707e6 −0.626187
$$592$$ 0 0
$$593$$ −1.58559e6 −0.185163 −0.0925814 0.995705i $$-0.529512\pi$$
−0.0925814 + 0.995705i $$0.529512\pi$$
$$594$$ 0 0
$$595$$ 4.55830e6 0.527850
$$596$$ 0 0
$$597$$ −6.17915e6 −0.709566
$$598$$ 0 0
$$599$$ −9.04294e6 −1.02978 −0.514888 0.857258i $$-0.672166\pi$$
−0.514888 + 0.857258i $$0.672166\pi$$
$$600$$ 0 0
$$601$$ 729186. 0.0823478 0.0411739 0.999152i $$-0.486890\pi$$
0.0411739 + 0.999152i $$0.486890\pi$$
$$602$$ 0 0
$$603$$ −1.19316e7 −1.33630
$$604$$ 0 0
$$605$$ −1.15664e6 −0.128472
$$606$$ 0 0
$$607$$ 3.91130e6 0.430873 0.215437 0.976518i $$-0.430883\pi$$
0.215437 + 0.976518i $$0.430883\pi$$
$$608$$ 0 0
$$609$$ −2.81120e6 −0.307149
$$610$$ 0 0
$$611$$ −2.62960e6 −0.284962
$$612$$ 0 0
$$613$$ −5.52184e6 −0.593516 −0.296758 0.954953i $$-0.595906\pi$$
−0.296758 + 0.954953i $$0.595906\pi$$
$$614$$ 0 0
$$615$$ −287560. −0.0306578
$$616$$ 0 0
$$617$$ −4.88539e6 −0.516638 −0.258319 0.966060i $$-0.583169\pi$$
−0.258319 + 0.966060i $$0.583169\pi$$
$$618$$ 0 0
$$619$$ 4.11150e6 0.431295 0.215647 0.976471i $$-0.430814\pi$$
0.215647 + 0.976471i $$0.430814\pi$$
$$620$$ 0 0
$$621$$ −1.09849e7 −1.14305
$$622$$ 0 0
$$623$$ 1.81355e6 0.187202
$$624$$ 0 0
$$625$$ −9.79367e6 −1.00287
$$626$$ 0 0
$$627$$ 1.54493e6 0.156942
$$628$$ 0 0
$$629$$ −8.05377e6 −0.811657
$$630$$ 0 0
$$631$$ −8.24910e6 −0.824771 −0.412385 0.911009i $$-0.635304\pi$$
−0.412385 + 0.911009i $$0.635304\pi$$
$$632$$ 0 0
$$633$$ −8.06996e6 −0.800502
$$634$$ 0 0
$$635$$ −2.04989e7 −2.01742
$$636$$ 0 0
$$637$$ 5.43666e6 0.530864
$$638$$ 0 0
$$639$$ −2.90437e6 −0.281385
$$640$$ 0 0
$$641$$ −4.29330e6 −0.412711 −0.206355 0.978477i $$-0.566160\pi$$
−0.206355 + 0.978477i $$0.566160\pi$$
$$642$$ 0 0
$$643$$ 1.63045e7 1.55518 0.777588 0.628774i $$-0.216443\pi$$
0.777588 + 0.628774i $$0.216443\pi$$
$$644$$ 0 0
$$645$$ 1.37476e6 0.130115
$$646$$ 0 0
$$647$$ −4.42624e6 −0.415695 −0.207847 0.978161i $$-0.566646\pi$$
−0.207847 + 0.978161i $$0.566646\pi$$
$$648$$ 0 0
$$649$$ −3.84526e6 −0.358355
$$650$$ 0 0
$$651$$ −1.03075e6 −0.0953237
$$652$$ 0 0
$$653$$ −6.27529e6 −0.575905 −0.287952 0.957645i $$-0.592975\pi$$
−0.287952 + 0.957645i $$0.592975\pi$$
$$654$$ 0 0
$$655$$ −6.74739e6 −0.614515
$$656$$ 0 0
$$657$$ 7.07014e6 0.639020
$$658$$ 0 0
$$659$$ 1.09748e7 0.984422 0.492211 0.870476i $$-0.336189\pi$$
0.492211 + 0.870476i $$0.336189\pi$$
$$660$$ 0 0
$$661$$ 2.02025e7 1.79846 0.899229 0.437478i $$-0.144128\pi$$
0.899229 + 0.437478i $$0.144128\pi$$
$$662$$ 0 0
$$663$$ −3.06964e6 −0.271209
$$664$$ 0 0
$$665$$ −7.20480e6 −0.631783
$$666$$ 0 0
$$667$$ −2.88429e7 −2.51029
$$668$$ 0 0
$$669$$ −461251. −0.0398448
$$670$$ 0 0
$$671$$ −4.13288e6 −0.354361
$$672$$ 0 0
$$673$$ 1.14233e7 0.972200 0.486100 0.873903i $$-0.338419\pi$$
0.486100 + 0.873903i $$0.338419\pi$$
$$674$$ 0 0
$$675$$ 9.53184e6 0.805225
$$676$$ 0 0
$$677$$ −2.43918e6 −0.204537 −0.102268 0.994757i $$-0.532610\pi$$
−0.102268 + 0.994757i $$0.532610\pi$$
$$678$$ 0 0
$$679$$ −2.48995e6 −0.207260
$$680$$ 0 0
$$681$$ −2.20168e6 −0.181923
$$682$$ 0 0
$$683$$ −1.01384e6 −0.0831606 −0.0415803 0.999135i $$-0.513239\pi$$
−0.0415803 + 0.999135i $$0.513239\pi$$
$$684$$ 0 0
$$685$$ 3.37885e7 2.75133
$$686$$ 0 0
$$687$$ −7.26924e6 −0.587621
$$688$$ 0 0
$$689$$ 5.21284e6 0.418337
$$690$$ 0 0
$$691$$ 8.03186e6 0.639913 0.319957 0.947432i $$-0.396332\pi$$
0.319957 + 0.947432i $$0.396332\pi$$
$$692$$ 0 0
$$693$$ 1.17370e6 0.0928376
$$694$$ 0 0
$$695$$ 2.44655e7 1.92129
$$696$$ 0 0
$$697$$ 600080. 0.0467872
$$698$$ 0 0
$$699$$ −3.56983e6 −0.276347
$$700$$ 0 0
$$701$$ −259806. −0.0199689 −0.00998445 0.999950i $$-0.503178\pi$$
−0.00998445 + 0.999950i $$0.503178\pi$$
$$702$$ 0 0
$$703$$ 1.27297e7 0.971471
$$704$$ 0 0
$$705$$ 3.82676e6 0.289974
$$706$$ 0 0
$$707$$ −7.67030e6 −0.577117
$$708$$ 0 0
$$709$$ −1.92848e7 −1.44079 −0.720393 0.693566i $$-0.756039\pi$$
−0.720393 + 0.693566i $$0.756039\pi$$
$$710$$ 0 0
$$711$$ −5.53637e6 −0.410725
$$712$$ 0 0
$$713$$ −1.05755e7 −0.779071
$$714$$ 0 0
$$715$$ −3.63242e6 −0.265724
$$716$$ 0 0
$$717$$ −3.11146e6 −0.226030
$$718$$ 0 0
$$719$$ −926119. −0.0668105 −0.0334052 0.999442i $$-0.510635\pi$$
−0.0334052 + 0.999442i $$0.510635\pi$$
$$720$$ 0 0
$$721$$ 6.73600e6 0.482574
$$722$$ 0 0
$$723$$ 1.98425e6 0.141173
$$724$$ 0 0
$$725$$ 2.50277e7 1.76838
$$726$$ 0 0
$$727$$ −2.02599e7 −1.42168 −0.710840 0.703354i $$-0.751685\pi$$
−0.710840 + 0.703354i $$0.751685\pi$$
$$728$$ 0 0
$$729$$ 211933. 0.0147700
$$730$$ 0 0
$$731$$ −2.86884e6 −0.198570
$$732$$ 0 0
$$733$$ 1.10982e7 0.762944 0.381472 0.924380i $$-0.375417\pi$$
0.381472 + 0.924380i $$0.375417\pi$$
$$734$$ 0 0
$$735$$ −7.91177e6 −0.540201
$$736$$ 0 0
$$737$$ −7.44186e6 −0.504676
$$738$$ 0 0
$$739$$ −624962. −0.0420962 −0.0210481 0.999778i $$-0.506700\pi$$
−0.0210481 + 0.999778i $$0.506700\pi$$
$$740$$ 0 0
$$741$$ 4.85184e6 0.324609
$$742$$ 0 0
$$743$$ −46436.0 −0.00308591 −0.00154295 0.999999i $$-0.500491\pi$$
−0.00154295 + 0.999999i $$0.500491\pi$$
$$744$$ 0 0
$$745$$ −3.55378e7 −2.34585
$$746$$ 0 0
$$747$$ 1.50315e7 0.985601
$$748$$ 0 0
$$749$$ −8.46090e6 −0.551077
$$750$$ 0 0
$$751$$ 6.12144e6 0.396053 0.198027 0.980197i $$-0.436547\pi$$
0.198027 + 0.980197i $$0.436547\pi$$
$$752$$ 0 0
$$753$$ −5.41665e6 −0.348131
$$754$$ 0 0
$$755$$ 3.20008e7 2.04312
$$756$$ 0 0
$$757$$ −3.26458e6 −0.207056 −0.103528 0.994627i $$-0.533013\pi$$
−0.103528 + 0.994627i $$0.533013\pi$$
$$758$$ 0 0
$$759$$ −3.04158e6 −0.191644
$$760$$ 0 0
$$761$$ 1.60311e7 1.00346 0.501732 0.865023i $$-0.332696\pi$$
0.501732 + 0.865023i $$0.332696\pi$$
$$762$$ 0 0
$$763$$ −1.16603e7 −0.725101
$$764$$ 0 0
$$765$$ −1.76862e7 −1.09265
$$766$$ 0 0
$$767$$ −1.20760e7 −0.741200
$$768$$ 0 0
$$769$$ 2.64617e7 1.61362 0.806811 0.590810i $$-0.201192\pi$$
0.806811 + 0.590810i $$0.201192\pi$$
$$770$$ 0 0
$$771$$ 2.71400e6 0.164427
$$772$$ 0 0
$$773$$ −2.63836e7 −1.58813 −0.794063 0.607836i $$-0.792038\pi$$
−0.794063 + 0.607836i $$0.792038\pi$$
$$774$$ 0 0
$$775$$ 9.17662e6 0.548818
$$776$$ 0 0
$$777$$ −2.44265e6 −0.145147
$$778$$ 0 0
$$779$$ −948480. −0.0559995
$$780$$ 0 0
$$781$$ −1.81149e6 −0.106269
$$782$$ 0 0
$$783$$ 2.45699e7 1.43218
$$784$$ 0 0
$$785$$ −2.68064e7 −1.55261
$$786$$ 0 0
$$787$$ 5.68115e6 0.326964 0.163482 0.986546i $$-0.447727\pi$$
0.163482 + 0.986546i $$0.447727\pi$$
$$788$$ 0 0
$$789$$ −1.38321e6 −0.0791037
$$790$$ 0 0
$$791$$ 4.71645e6 0.268024
$$792$$ 0 0
$$793$$ −1.29793e7 −0.732939
$$794$$ 0 0
$$795$$ −7.58605e6 −0.425695
$$796$$ 0 0
$$797$$ 9.99383e6 0.557296 0.278648 0.960393i $$-0.410114\pi$$
0.278648 + 0.960393i $$0.410114\pi$$
$$798$$ 0 0
$$799$$ −7.98568e6 −0.442532
$$800$$ 0 0
$$801$$ −7.03657e6 −0.387507
$$802$$ 0 0
$$803$$ 4.40972e6 0.241336
$$804$$ 0 0
$$805$$ 1.41844e7 0.771477
$$806$$ 0 0
$$807$$ 1.83886e6 0.0993950
$$808$$ 0 0
$$809$$ 2.32455e7 1.24873 0.624364 0.781134i $$-0.285358\pi$$
0.624364 + 0.781134i $$0.285358\pi$$
$$810$$ 0 0
$$811$$ −1.27367e7 −0.679991 −0.339995 0.940427i $$-0.610426\pi$$
−0.339995 + 0.940427i $$0.610426\pi$$
$$812$$ 0 0
$$813$$ −1.11348e6 −0.0590819
$$814$$ 0 0
$$815$$ 2.14403e7 1.13067
$$816$$ 0 0
$$817$$ 4.53446e6 0.237668
$$818$$ 0 0
$$819$$ 3.68600e6 0.192020
$$820$$ 0 0
$$821$$ −7.85748e6 −0.406842 −0.203421 0.979091i $$-0.565206\pi$$
−0.203421 + 0.979091i $$0.565206\pi$$
$$822$$ 0 0
$$823$$ 1.09499e7 0.563524 0.281762 0.959484i $$-0.409081\pi$$
0.281762 + 0.959484i $$0.409081\pi$$
$$824$$ 0 0
$$825$$ 2.63925e6 0.135004
$$826$$ 0 0
$$827$$ −2.20638e7 −1.12180 −0.560901 0.827883i $$-0.689545\pi$$
−0.560901 + 0.827883i $$0.689545\pi$$
$$828$$ 0 0
$$829$$ 7.05255e6 0.356418 0.178209 0.983993i $$-0.442970\pi$$
0.178209 + 0.983993i $$0.442970\pi$$
$$830$$ 0 0
$$831$$ −9.05698e6 −0.454968
$$832$$ 0 0
$$833$$ 1.65103e7 0.824407
$$834$$ 0 0
$$835$$ 5.72497e6 0.284156
$$836$$ 0 0
$$837$$ 9.00876e6 0.444479
$$838$$ 0 0
$$839$$ 2.26195e7 1.10937 0.554686 0.832060i $$-0.312838\pi$$
0.554686 + 0.832060i $$0.312838\pi$$
$$840$$ 0 0
$$841$$ 4.40019e7 2.14527
$$842$$ 0 0
$$843$$ 1.24680e7 0.604264
$$844$$ 0 0
$$845$$ 1.79245e7 0.863588
$$846$$ 0 0
$$847$$ 732050. 0.0350616
$$848$$ 0 0
$$849$$ 1.38795e7 0.660853
$$850$$ 0 0
$$851$$ −2.50616e7 −1.18627
$$852$$ 0 0
$$853$$ 9.46645e6 0.445466 0.222733 0.974880i $$-0.428502\pi$$
0.222733 + 0.974880i $$0.428502\pi$$
$$854$$ 0 0
$$855$$ 2.79546e7 1.30779
$$856$$ 0 0
$$857$$ 941480. 0.0437884 0.0218942 0.999760i $$-0.493030\pi$$
0.0218942 + 0.999760i $$0.493030\pi$$
$$858$$ 0 0
$$859$$ 806423. 0.0372889 0.0186445 0.999826i $$-0.494065\pi$$
0.0186445 + 0.999826i $$0.494065\pi$$
$$860$$ 0 0
$$861$$ 182000. 0.00836688
$$862$$ 0 0
$$863$$ −1.19485e7 −0.546119 −0.273059 0.961997i $$-0.588036\pi$$
−0.273059 + 0.961997i $$0.588036\pi$$
$$864$$ 0 0
$$865$$ −3.78589e7 −1.72039
$$866$$ 0 0
$$867$$ 616987. 0.0278759
$$868$$ 0 0
$$869$$ −3.45310e6 −0.155117
$$870$$ 0 0
$$871$$ −2.33711e7 −1.04384
$$872$$ 0 0
$$873$$ 9.66101e6 0.429029
$$874$$ 0 0
$$875$$ 35550.0 0.00156971
$$876$$ 0 0
$$877$$ 7.84853e6 0.344580 0.172290 0.985046i $$-0.444883\pi$$
0.172290 + 0.985046i $$0.444883\pi$$
$$878$$ 0 0
$$879$$ −4.04852e6 −0.176736
$$880$$ 0 0
$$881$$ −1.73933e7 −0.754991 −0.377496 0.926011i $$-0.623215\pi$$
−0.377496 + 0.926011i $$0.623215\pi$$
$$882$$ 0 0
$$883$$ 4.31619e7 1.86294 0.931470 0.363818i $$-0.118527\pi$$
0.931470 + 0.363818i $$0.118527\pi$$
$$884$$ 0 0
$$885$$ 1.75738e7 0.754236
$$886$$ 0 0
$$887$$ 9.25652e6 0.395038 0.197519 0.980299i $$-0.436712\pi$$
0.197519 + 0.980299i $$0.436712\pi$$
$$888$$ 0 0
$$889$$ 1.29740e7 0.550579
$$890$$ 0 0
$$891$$ −3.11321e6 −0.131375
$$892$$ 0 0
$$893$$ 1.26221e7 0.529666
$$894$$ 0 0
$$895$$ −3.23386e6 −0.134947
$$896$$ 0 0
$$897$$ −9.55206e6 −0.396384
$$898$$ 0 0
$$899$$ 2.36542e7 0.976135
$$900$$ 0 0
$$901$$ 1.58306e7 0.649658
$$902$$ 0 0
$$903$$ −870100. −0.0355099
$$904$$ 0 0
$$905$$ 7.12335e6 0.289110
$$906$$ 0 0
$$907$$ −4.47293e7 −1.80540 −0.902700 0.430270i $$-0.858418\pi$$
−0.902700 + 0.430270i $$0.858418\pi$$
$$908$$ 0 0
$$909$$ 2.97608e7 1.19463
$$910$$ 0 0
$$911$$ −2.57577e7 −1.02828 −0.514139 0.857707i $$-0.671888\pi$$
−0.514139 + 0.857707i $$0.671888\pi$$
$$912$$ 0 0
$$913$$ 9.37532e6 0.372228
$$914$$ 0 0
$$915$$ 1.88883e7 0.745829
$$916$$ 0 0
$$917$$ 4.27050e6 0.167709
$$918$$ 0 0
$$919$$ 3.63488e7 1.41972 0.709858 0.704344i $$-0.248759\pi$$
0.709858 + 0.704344i $$0.248759\pi$$
$$920$$ 0 0
$$921$$ −2.15322e7 −0.836447
$$922$$ 0 0
$$923$$ −5.68898e6 −0.219801
$$924$$ 0 0
$$925$$ 2.17466e7 0.835673
$$926$$ 0 0
$$927$$ −2.61357e7 −0.998929
$$928$$ 0 0
$$929$$ −3.96617e6 −0.150776 −0.0753880 0.997154i $$-0.524020\pi$$
−0.0753880 + 0.997154i $$0.524020\pi$$
$$930$$ 0 0
$$931$$ −2.60960e7 −0.986732
$$932$$ 0 0
$$933$$ −2.19630e7 −0.826014
$$934$$ 0 0
$$935$$ −1.10311e7 −0.412657
$$936$$ 0 0
$$937$$ −3.50528e7 −1.30429 −0.652145 0.758094i $$-0.726131\pi$$
−0.652145 + 0.758094i $$0.726131\pi$$
$$938$$ 0 0
$$939$$ −1.83249e7 −0.678230
$$940$$ 0 0
$$941$$ −1.40738e7 −0.518130 −0.259065 0.965860i $$-0.583414\pi$$
−0.259065 + 0.965860i $$0.583414\pi$$
$$942$$ 0 0
$$943$$ 1.86732e6 0.0683816
$$944$$ 0 0
$$945$$ −1.20830e7 −0.440146
$$946$$ 0 0
$$947$$ 3.73759e7 1.35431 0.677153 0.735842i $$-0.263214\pi$$
0.677153 + 0.735842i $$0.263214\pi$$
$$948$$ 0 0
$$949$$ 1.38487e7 0.499165
$$950$$ 0 0
$$951$$ 1.74454e7 0.625502
$$952$$ 0 0
$$953$$ −3.18424e7 −1.13572 −0.567862 0.823124i $$-0.692229\pi$$
−0.567862 + 0.823124i $$0.692229\pi$$
$$954$$ 0 0
$$955$$ −2.05696e7 −0.729824
$$956$$ 0 0
$$957$$ 6.80310e6 0.240119
$$958$$ 0 0
$$959$$ −2.13852e7 −0.750872
$$960$$ 0 0
$$961$$ −1.99561e7 −0.697056
$$962$$ 0 0
$$963$$ 3.28283e7 1.14073
$$964$$ 0 0
$$965$$ −4.14216e7 −1.43189
$$966$$ 0 0
$$967$$ 3.16276e7 1.08768 0.543838 0.839190i $$-0.316971\pi$$
0.543838 + 0.839190i $$0.316971\pi$$
$$968$$ 0 0
$$969$$ 1.47343e7 0.504103
$$970$$ 0 0
$$971$$ 2.73412e7 0.930614 0.465307 0.885149i $$-0.345944\pi$$
0.465307 + 0.885149i $$0.345944\pi$$
$$972$$ 0 0
$$973$$ −1.54845e7 −0.524343
$$974$$ 0 0
$$975$$ 8.28856e6 0.279234
$$976$$ 0 0
$$977$$ −5.81630e6 −0.194944 −0.0974721 0.995238i $$-0.531076\pi$$
−0.0974721 + 0.995238i $$0.531076\pi$$
$$978$$ 0 0
$$979$$ −4.38879e6 −0.146348
$$980$$ 0 0
$$981$$ 4.52420e7 1.50096
$$982$$ 0 0
$$983$$ 3.81817e7 1.26029 0.630146 0.776476i $$-0.282995\pi$$
0.630146 + 0.776476i $$0.282995\pi$$
$$984$$ 0 0
$$985$$ −6.00070e7 −1.97066
$$986$$ 0 0
$$987$$ −2.42200e6 −0.0791373
$$988$$ 0 0
$$989$$ −8.92723e6 −0.290219
$$990$$ 0 0
$$991$$ −5.44564e6 −0.176143 −0.0880714 0.996114i $$-0.528070\pi$$
−0.0880714 + 0.996114i $$0.528070\pi$$
$$992$$ 0 0
$$993$$ −1.89087e7 −0.608541
$$994$$ 0 0
$$995$$ −6.97361e7 −2.23306
$$996$$ 0 0
$$997$$ −3.77967e6 −0.120425 −0.0602125 0.998186i $$-0.519178\pi$$
−0.0602125 + 0.998186i $$0.519178\pi$$
$$998$$ 0 0
$$999$$ 2.13488e7 0.676798
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 176.6.a.a.1.1 1
4.3 odd 2 44.6.a.a.1.1 1
8.3 odd 2 704.6.a.d.1.1 1
8.5 even 2 704.6.a.g.1.1 1
12.11 even 2 396.6.a.e.1.1 1
20.3 even 4 1100.6.b.a.749.2 2
20.7 even 4 1100.6.b.a.749.1 2
20.19 odd 2 1100.6.a.a.1.1 1
44.43 even 2 484.6.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
44.6.a.a.1.1 1 4.3 odd 2
176.6.a.a.1.1 1 1.1 even 1 trivial
396.6.a.e.1.1 1 12.11 even 2
484.6.a.b.1.1 1 44.43 even 2
704.6.a.d.1.1 1 8.3 odd 2
704.6.a.g.1.1 1 8.5 even 2
1100.6.a.a.1.1 1 20.19 odd 2
1100.6.b.a.749.1 2 20.7 even 4
1100.6.b.a.749.2 2 20.3 even 4