# Properties

 Label 441.6.a.g.1.1 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -28.0000 q^{4} -11.0000 q^{5} -120.000 q^{8} +O(q^{10})$$ $$q+2.00000 q^{2} -28.0000 q^{4} -11.0000 q^{5} -120.000 q^{8} -22.0000 q^{10} -269.000 q^{11} -308.000 q^{13} +656.000 q^{16} -1896.00 q^{17} -164.000 q^{19} +308.000 q^{20} -538.000 q^{22} +3264.00 q^{23} -3004.00 q^{25} -616.000 q^{26} -2417.00 q^{29} +2841.00 q^{31} +5152.00 q^{32} -3792.00 q^{34} -11328.0 q^{37} -328.000 q^{38} +1320.00 q^{40} +16856.0 q^{41} -7894.00 q^{43} +7532.00 q^{44} +6528.00 q^{46} -21102.0 q^{47} -6008.00 q^{50} +8624.00 q^{52} +29691.0 q^{53} +2959.00 q^{55} -4834.00 q^{58} +8163.00 q^{59} +15166.0 q^{61} +5682.00 q^{62} -10688.0 q^{64} +3388.00 q^{65} -32078.0 q^{67} +53088.0 q^{68} +38274.0 q^{71} +34866.0 q^{73} -22656.0 q^{74} +4592.00 q^{76} +13529.0 q^{79} -7216.00 q^{80} +33712.0 q^{82} +68103.0 q^{83} +20856.0 q^{85} -15788.0 q^{86} +32280.0 q^{88} +114922. q^{89} -91392.0 q^{92} -42204.0 q^{94} +1804.00 q^{95} +154959. q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.353553 0.176777 0.984251i $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ 0 0
$$4$$ −28.0000 −0.875000
$$5$$ −11.0000 −0.196774 −0.0983870 0.995148i $$-0.531368\pi$$
−0.0983870 + 0.995148i $$0.531368\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −120.000 −0.662913
$$9$$ 0 0
$$10$$ −22.0000 −0.0695701
$$11$$ −269.000 −0.670302 −0.335151 0.942164i $$-0.608787\pi$$
−0.335151 + 0.942164i $$0.608787\pi$$
$$12$$ 0 0
$$13$$ −308.000 −0.505466 −0.252733 0.967536i $$-0.581329\pi$$
−0.252733 + 0.967536i $$0.581329\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 656.000 0.640625
$$17$$ −1896.00 −1.59117 −0.795584 0.605843i $$-0.792836\pi$$
−0.795584 + 0.605843i $$0.792836\pi$$
$$18$$ 0 0
$$19$$ −164.000 −0.104222 −0.0521111 0.998641i $$-0.516595\pi$$
−0.0521111 + 0.998641i $$0.516595\pi$$
$$20$$ 308.000 0.172177
$$21$$ 0 0
$$22$$ −538.000 −0.236988
$$23$$ 3264.00 1.28656 0.643281 0.765630i $$-0.277573\pi$$
0.643281 + 0.765630i $$0.277573\pi$$
$$24$$ 0 0
$$25$$ −3004.00 −0.961280
$$26$$ −616.000 −0.178709
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2417.00 −0.533681 −0.266840 0.963741i $$-0.585980\pi$$
−0.266840 + 0.963741i $$0.585980\pi$$
$$30$$ 0 0
$$31$$ 2841.00 0.530966 0.265483 0.964115i $$-0.414469\pi$$
0.265483 + 0.964115i $$0.414469\pi$$
$$32$$ 5152.00 0.889408
$$33$$ 0 0
$$34$$ −3792.00 −0.562563
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −11328.0 −1.36034 −0.680172 0.733052i $$-0.738095\pi$$
−0.680172 + 0.733052i $$0.738095\pi$$
$$38$$ −328.000 −0.0368481
$$39$$ 0 0
$$40$$ 1320.00 0.130444
$$41$$ 16856.0 1.56601 0.783006 0.622015i $$-0.213686\pi$$
0.783006 + 0.622015i $$0.213686\pi$$
$$42$$ 0 0
$$43$$ −7894.00 −0.651067 −0.325534 0.945530i $$-0.605544\pi$$
−0.325534 + 0.945530i $$0.605544\pi$$
$$44$$ 7532.00 0.586514
$$45$$ 0 0
$$46$$ 6528.00 0.454868
$$47$$ −21102.0 −1.39341 −0.696705 0.717358i $$-0.745351\pi$$
−0.696705 + 0.717358i $$0.745351\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −6008.00 −0.339864
$$51$$ 0 0
$$52$$ 8624.00 0.442283
$$53$$ 29691.0 1.45189 0.725947 0.687750i $$-0.241402\pi$$
0.725947 + 0.687750i $$0.241402\pi$$
$$54$$ 0 0
$$55$$ 2959.00 0.131898
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −4834.00 −0.188685
$$59$$ 8163.00 0.305295 0.152648 0.988281i $$-0.451220\pi$$
0.152648 + 0.988281i $$0.451220\pi$$
$$60$$ 0 0
$$61$$ 15166.0 0.521851 0.260925 0.965359i $$-0.415972\pi$$
0.260925 + 0.965359i $$0.415972\pi$$
$$62$$ 5682.00 0.187725
$$63$$ 0 0
$$64$$ −10688.0 −0.326172
$$65$$ 3388.00 0.0994626
$$66$$ 0 0
$$67$$ −32078.0 −0.873012 −0.436506 0.899701i $$-0.643784\pi$$
−0.436506 + 0.899701i $$0.643784\pi$$
$$68$$ 53088.0 1.39227
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 38274.0 0.901069 0.450534 0.892759i $$-0.351233\pi$$
0.450534 + 0.892759i $$0.351233\pi$$
$$72$$ 0 0
$$73$$ 34866.0 0.765764 0.382882 0.923797i $$-0.374932\pi$$
0.382882 + 0.923797i $$0.374932\pi$$
$$74$$ −22656.0 −0.480954
$$75$$ 0 0
$$76$$ 4592.00 0.0911943
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 13529.0 0.243892 0.121946 0.992537i $$-0.461086\pi$$
0.121946 + 0.992537i $$0.461086\pi$$
$$80$$ −7216.00 −0.126058
$$81$$ 0 0
$$82$$ 33712.0 0.553669
$$83$$ 68103.0 1.08510 0.542552 0.840023i $$-0.317458\pi$$
0.542552 + 0.840023i $$0.317458\pi$$
$$84$$ 0 0
$$85$$ 20856.0 0.313100
$$86$$ −15788.0 −0.230187
$$87$$ 0 0
$$88$$ 32280.0 0.444352
$$89$$ 114922. 1.53790 0.768950 0.639309i $$-0.220779\pi$$
0.768950 + 0.639309i $$0.220779\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −91392.0 −1.12574
$$93$$ 0 0
$$94$$ −42204.0 −0.492645
$$95$$ 1804.00 0.0205082
$$96$$ 0 0
$$97$$ 154959. 1.67220 0.836099 0.548579i $$-0.184831\pi$$
0.836099 + 0.548579i $$0.184831\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 84112.0 0.841120
$$101$$ 107570. 1.04927 0.524636 0.851327i $$-0.324202\pi$$
0.524636 + 0.851327i $$0.324202\pi$$
$$102$$ 0 0
$$103$$ 8936.00 0.0829947 0.0414973 0.999139i $$-0.486787\pi$$
0.0414973 + 0.999139i $$0.486787\pi$$
$$104$$ 36960.0 0.335080
$$105$$ 0 0
$$106$$ 59382.0 0.513322
$$107$$ −193667. −1.63530 −0.817648 0.575719i $$-0.804722\pi$$
−0.817648 + 0.575719i $$0.804722\pi$$
$$108$$ 0 0
$$109$$ 205110. 1.65356 0.826781 0.562524i $$-0.190169\pi$$
0.826781 + 0.562524i $$0.190169\pi$$
$$110$$ 5918.00 0.0466330
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −46664.0 −0.343784 −0.171892 0.985116i $$-0.554988\pi$$
−0.171892 + 0.985116i $$0.554988\pi$$
$$114$$ 0 0
$$115$$ −35904.0 −0.253162
$$116$$ 67676.0 0.466971
$$117$$ 0 0
$$118$$ 16326.0 0.107938
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −88690.0 −0.550695
$$122$$ 30332.0 0.184502
$$123$$ 0 0
$$124$$ −79548.0 −0.464596
$$125$$ 67419.0 0.385929
$$126$$ 0 0
$$127$$ −304365. −1.67450 −0.837250 0.546820i $$-0.815838\pi$$
−0.837250 + 0.546820i $$0.815838\pi$$
$$128$$ −186240. −1.00473
$$129$$ 0 0
$$130$$ 6776.00 0.0351654
$$131$$ 13303.0 0.0677285 0.0338642 0.999426i $$-0.489219\pi$$
0.0338642 + 0.999426i $$0.489219\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −64156.0 −0.308656
$$135$$ 0 0
$$136$$ 227520. 1.05481
$$137$$ 398262. 1.81287 0.906437 0.422342i $$-0.138792\pi$$
0.906437 + 0.422342i $$0.138792\pi$$
$$138$$ 0 0
$$139$$ 230286. 1.01095 0.505476 0.862841i $$-0.331317\pi$$
0.505476 + 0.862841i $$0.331317\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 76548.0 0.318576
$$143$$ 82852.0 0.338815
$$144$$ 0 0
$$145$$ 26587.0 0.105015
$$146$$ 69732.0 0.270738
$$147$$ 0 0
$$148$$ 317184. 1.19030
$$149$$ 97134.0 0.358431 0.179216 0.983810i $$-0.442644\pi$$
0.179216 + 0.983810i $$0.442644\pi$$
$$150$$ 0 0
$$151$$ −29047.0 −0.103671 −0.0518357 0.998656i $$-0.516507\pi$$
−0.0518357 + 0.998656i $$0.516507\pi$$
$$152$$ 19680.0 0.0690901
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −31251.0 −0.104480
$$156$$ 0 0
$$157$$ −576500. −1.86660 −0.933298 0.359104i $$-0.883082\pi$$
−0.933298 + 0.359104i $$0.883082\pi$$
$$158$$ 27058.0 0.0862289
$$159$$ 0 0
$$160$$ −56672.0 −0.175012
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −265232. −0.781910 −0.390955 0.920410i $$-0.627855\pi$$
−0.390955 + 0.920410i $$0.627855\pi$$
$$164$$ −471968. −1.37026
$$165$$ 0 0
$$166$$ 136206. 0.383642
$$167$$ 363790. 1.00939 0.504696 0.863297i $$-0.331605\pi$$
0.504696 + 0.863297i $$0.331605\pi$$
$$168$$ 0 0
$$169$$ −276429. −0.744504
$$170$$ 41712.0 0.110698
$$171$$ 0 0
$$172$$ 221032. 0.569684
$$173$$ 164846. 0.418758 0.209379 0.977835i $$-0.432856\pi$$
0.209379 + 0.977835i $$0.432856\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −176464. −0.429412
$$177$$ 0 0
$$178$$ 229844. 0.543730
$$179$$ −30628.0 −0.0714473 −0.0357237 0.999362i $$-0.511374\pi$$
−0.0357237 + 0.999362i $$0.511374\pi$$
$$180$$ 0 0
$$181$$ −651392. −1.47790 −0.738952 0.673759i $$-0.764679\pi$$
−0.738952 + 0.673759i $$0.764679\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −391680. −0.852878
$$185$$ 124608. 0.267680
$$186$$ 0 0
$$187$$ 510024. 1.06656
$$188$$ 590856. 1.21923
$$189$$ 0 0
$$190$$ 3608.00 0.00725074
$$191$$ 757360. 1.50217 0.751085 0.660206i $$-0.229531\pi$$
0.751085 + 0.660206i $$0.229531\pi$$
$$192$$ 0 0
$$193$$ −160339. −0.309846 −0.154923 0.987927i $$-0.549513\pi$$
−0.154923 + 0.987927i $$0.549513\pi$$
$$194$$ 309918. 0.591211
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 61738.0 0.113341 0.0566705 0.998393i $$-0.481952\pi$$
0.0566705 + 0.998393i $$0.481952\pi$$
$$198$$ 0 0
$$199$$ 370908. 0.663947 0.331974 0.943289i $$-0.392286\pi$$
0.331974 + 0.943289i $$0.392286\pi$$
$$200$$ 360480. 0.637245
$$201$$ 0 0
$$202$$ 215140. 0.370973
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −185416. −0.308150
$$206$$ 17872.0 0.0293430
$$207$$ 0 0
$$208$$ −202048. −0.323814
$$209$$ 44116.0 0.0698603
$$210$$ 0 0
$$211$$ 217450. 0.336243 0.168122 0.985766i $$-0.446230\pi$$
0.168122 + 0.985766i $$0.446230\pi$$
$$212$$ −831348. −1.27041
$$213$$ 0 0
$$214$$ −387334. −0.578164
$$215$$ 86834.0 0.128113
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 410220. 0.584623
$$219$$ 0 0
$$220$$ −82852.0 −0.115411
$$221$$ 583968. 0.804282
$$222$$ 0 0
$$223$$ 589771. 0.794184 0.397092 0.917779i $$-0.370019\pi$$
0.397092 + 0.917779i $$0.370019\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −93328.0 −0.121546
$$227$$ 387045. 0.498536 0.249268 0.968434i $$-0.419810\pi$$
0.249268 + 0.968434i $$0.419810\pi$$
$$228$$ 0 0
$$229$$ −232732. −0.293270 −0.146635 0.989191i $$-0.546844\pi$$
−0.146635 + 0.989191i $$0.546844\pi$$
$$230$$ −71808.0 −0.0895062
$$231$$ 0 0
$$232$$ 290040. 0.353784
$$233$$ −42096.0 −0.0507985 −0.0253993 0.999677i $$-0.508086\pi$$
−0.0253993 + 0.999677i $$0.508086\pi$$
$$234$$ 0 0
$$235$$ 232122. 0.274187
$$236$$ −228564. −0.267133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 313416. 0.354917 0.177458 0.984128i $$-0.443212\pi$$
0.177458 + 0.984128i $$0.443212\pi$$
$$240$$ 0 0
$$241$$ 857807. 0.951365 0.475682 0.879617i $$-0.342201\pi$$
0.475682 + 0.879617i $$0.342201\pi$$
$$242$$ −177380. −0.194700
$$243$$ 0 0
$$244$$ −424648. −0.456620
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 50512.0 0.0526808
$$248$$ −340920. −0.351984
$$249$$ 0 0
$$250$$ 134838. 0.136446
$$251$$ −454517. −0.455371 −0.227686 0.973735i $$-0.573116\pi$$
−0.227686 + 0.973735i $$0.573116\pi$$
$$252$$ 0 0
$$253$$ −878016. −0.862385
$$254$$ −608730. −0.592026
$$255$$ 0 0
$$256$$ −30464.0 −0.0290527
$$257$$ −878182. −0.829376 −0.414688 0.909964i $$-0.636109\pi$$
−0.414688 + 0.909964i $$0.636109\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −94864.0 −0.0870298
$$261$$ 0 0
$$262$$ 26606.0 0.0239456
$$263$$ −1.96093e6 −1.74813 −0.874065 0.485809i $$-0.838525\pi$$
−0.874065 + 0.485809i $$0.838525\pi$$
$$264$$ 0 0
$$265$$ −326601. −0.285695
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 898184. 0.763886
$$269$$ 1.05380e6 0.887923 0.443962 0.896046i $$-0.353573\pi$$
0.443962 + 0.896046i $$0.353573\pi$$
$$270$$ 0 0
$$271$$ −105059. −0.0868981 −0.0434490 0.999056i $$-0.513835\pi$$
−0.0434490 + 0.999056i $$0.513835\pi$$
$$272$$ −1.24378e6 −1.01934
$$273$$ 0 0
$$274$$ 796524. 0.640948
$$275$$ 808076. 0.644348
$$276$$ 0 0
$$277$$ −427592. −0.334834 −0.167417 0.985886i $$-0.553543\pi$$
−0.167417 + 0.985886i $$0.553543\pi$$
$$278$$ 460572. 0.357426
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −638878. −0.482672 −0.241336 0.970442i $$-0.577586\pi$$
−0.241336 + 0.970442i $$0.577586\pi$$
$$282$$ 0 0
$$283$$ −2.45142e6 −1.81950 −0.909750 0.415157i $$-0.863727\pi$$
−0.909750 + 0.415157i $$0.863727\pi$$
$$284$$ −1.07167e6 −0.788435
$$285$$ 0 0
$$286$$ 165704. 0.119789
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2.17496e6 1.53182
$$290$$ 53174.0 0.0371282
$$291$$ 0 0
$$292$$ −976248. −0.670044
$$293$$ −1.71617e6 −1.16786 −0.583930 0.811804i $$-0.698486\pi$$
−0.583930 + 0.811804i $$0.698486\pi$$
$$294$$ 0 0
$$295$$ −89793.0 −0.0600741
$$296$$ 1.35936e6 0.901790
$$297$$ 0 0
$$298$$ 194268. 0.126725
$$299$$ −1.00531e6 −0.650314
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −58094.0 −0.0366534
$$303$$ 0 0
$$304$$ −107584. −0.0667673
$$305$$ −166826. −0.102687
$$306$$ 0 0
$$307$$ 1.80897e6 1.09543 0.547715 0.836665i $$-0.315498\pi$$
0.547715 + 0.836665i $$0.315498\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −62502.0 −0.0369394
$$311$$ −1.52146e6 −0.891987 −0.445993 0.895036i $$-0.647150\pi$$
−0.445993 + 0.895036i $$0.647150\pi$$
$$312$$ 0 0
$$313$$ −1.34840e6 −0.777961 −0.388980 0.921246i $$-0.627173\pi$$
−0.388980 + 0.921246i $$0.627173\pi$$
$$314$$ −1.15300e6 −0.659941
$$315$$ 0 0
$$316$$ −378812. −0.213406
$$317$$ 49695.0 0.0277757 0.0138878 0.999904i $$-0.495579\pi$$
0.0138878 + 0.999904i $$0.495579\pi$$
$$318$$ 0 0
$$319$$ 650173. 0.357727
$$320$$ 117568. 0.0641821
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 310944. 0.165835
$$324$$ 0 0
$$325$$ 925232. 0.485895
$$326$$ −530464. −0.276447
$$327$$ 0 0
$$328$$ −2.02272e6 −1.03813
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1.58784e6 0.796591 0.398296 0.917257i $$-0.369602\pi$$
0.398296 + 0.917257i $$0.369602\pi$$
$$332$$ −1.90688e6 −0.949465
$$333$$ 0 0
$$334$$ 727580. 0.356874
$$335$$ 352858. 0.171786
$$336$$ 0 0
$$337$$ 214825. 0.103041 0.0515205 0.998672i $$-0.483593\pi$$
0.0515205 + 0.998672i $$0.483593\pi$$
$$338$$ −552858. −0.263222
$$339$$ 0 0
$$340$$ −583968. −0.273963
$$341$$ −764229. −0.355908
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 947280. 0.431601
$$345$$ 0 0
$$346$$ 329692. 0.148053
$$347$$ −2.58860e6 −1.15409 −0.577046 0.816711i $$-0.695795\pi$$
−0.577046 + 0.816711i $$0.695795\pi$$
$$348$$ 0 0
$$349$$ −24878.0 −0.0109333 −0.00546666 0.999985i $$-0.501740\pi$$
−0.00546666 + 0.999985i $$0.501740\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.38589e6 −0.596172
$$353$$ 1.73601e6 0.741506 0.370753 0.928731i $$-0.379100\pi$$
0.370753 + 0.928731i $$0.379100\pi$$
$$354$$ 0 0
$$355$$ −421014. −0.177307
$$356$$ −3.21782e6 −1.34566
$$357$$ 0 0
$$358$$ −61256.0 −0.0252604
$$359$$ −862426. −0.353172 −0.176586 0.984285i $$-0.556505\pi$$
−0.176586 + 0.984285i $$0.556505\pi$$
$$360$$ 0 0
$$361$$ −2.44920e6 −0.989138
$$362$$ −1.30278e6 −0.522518
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −383526. −0.150682
$$366$$ 0 0
$$367$$ 3.11542e6 1.20740 0.603700 0.797211i $$-0.293692\pi$$
0.603700 + 0.797211i $$0.293692\pi$$
$$368$$ 2.14118e6 0.824203
$$369$$ 0 0
$$370$$ 249216. 0.0946393
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.79694e6 −0.668748 −0.334374 0.942440i $$-0.608525\pi$$
−0.334374 + 0.942440i $$0.608525\pi$$
$$374$$ 1.02005e6 0.377087
$$375$$ 0 0
$$376$$ 2.53224e6 0.923709
$$377$$ 744436. 0.269758
$$378$$ 0 0
$$379$$ 3.45466e6 1.23540 0.617699 0.786415i $$-0.288065\pi$$
0.617699 + 0.786415i $$0.288065\pi$$
$$380$$ −50512.0 −0.0179447
$$381$$ 0 0
$$382$$ 1.51472e6 0.531097
$$383$$ −2.15504e6 −0.750685 −0.375343 0.926886i $$-0.622475\pi$$
−0.375343 + 0.926886i $$0.622475\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −320678. −0.109547
$$387$$ 0 0
$$388$$ −4.33885e6 −1.46317
$$389$$ 462774. 0.155058 0.0775291 0.996990i $$-0.475297\pi$$
0.0775291 + 0.996990i $$0.475297\pi$$
$$390$$ 0 0
$$391$$ −6.18854e6 −2.04714
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 123476. 0.0400721
$$395$$ −148819. −0.0479916
$$396$$ 0 0
$$397$$ −4.06621e6 −1.29483 −0.647416 0.762136i $$-0.724151\pi$$
−0.647416 + 0.762136i $$0.724151\pi$$
$$398$$ 741816. 0.234741
$$399$$ 0 0
$$400$$ −1.97062e6 −0.615820
$$401$$ 5.06863e6 1.57409 0.787045 0.616895i $$-0.211610\pi$$
0.787045 + 0.616895i $$0.211610\pi$$
$$402$$ 0 0
$$403$$ −875028. −0.268386
$$404$$ −3.01196e6 −0.918112
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3.04723e6 0.911842
$$408$$ 0 0
$$409$$ 2.87734e6 0.850515 0.425258 0.905072i $$-0.360183\pi$$
0.425258 + 0.905072i $$0.360183\pi$$
$$410$$ −370832. −0.108948
$$411$$ 0 0
$$412$$ −250208. −0.0726203
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −749133. −0.213520
$$416$$ −1.58682e6 −0.449566
$$417$$ 0 0
$$418$$ 88232.0 0.0246993
$$419$$ 3.41342e6 0.949850 0.474925 0.880026i $$-0.342475\pi$$
0.474925 + 0.880026i $$0.342475\pi$$
$$420$$ 0 0
$$421$$ −1.30737e6 −0.359496 −0.179748 0.983713i $$-0.557528\pi$$
−0.179748 + 0.983713i $$0.557528\pi$$
$$422$$ 434900. 0.118880
$$423$$ 0 0
$$424$$ −3.56292e6 −0.962479
$$425$$ 5.69558e6 1.52956
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 5.42268e6 1.43088
$$429$$ 0 0
$$430$$ 173668. 0.0452948
$$431$$ −1.93547e6 −0.501872 −0.250936 0.968004i $$-0.580738\pi$$
−0.250936 + 0.968004i $$0.580738\pi$$
$$432$$ 0 0
$$433$$ 516670. 0.132432 0.0662161 0.997805i $$-0.478907\pi$$
0.0662161 + 0.997805i $$0.478907\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −5.74308e6 −1.44687
$$437$$ −535296. −0.134088
$$438$$ 0 0
$$439$$ 2.91530e6 0.721975 0.360987 0.932571i $$-0.382440\pi$$
0.360987 + 0.932571i $$0.382440\pi$$
$$440$$ −355080. −0.0874369
$$441$$ 0 0
$$442$$ 1.16794e6 0.284357
$$443$$ 1.78379e6 0.431852 0.215926 0.976410i $$-0.430723\pi$$
0.215926 + 0.976410i $$0.430723\pi$$
$$444$$ 0 0
$$445$$ −1.26414e6 −0.302619
$$446$$ 1.17954e6 0.280787
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −4.00158e6 −0.936733 −0.468366 0.883534i $$-0.655157\pi$$
−0.468366 + 0.883534i $$0.655157\pi$$
$$450$$ 0 0
$$451$$ −4.53426e6 −1.04970
$$452$$ 1.30659e6 0.300811
$$453$$ 0 0
$$454$$ 774090. 0.176259
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.16766e6 −0.261534 −0.130767 0.991413i $$-0.541744\pi$$
−0.130767 + 0.991413i $$0.541744\pi$$
$$458$$ −465464. −0.103687
$$459$$ 0 0
$$460$$ 1.00531e6 0.221517
$$461$$ −3.61358e6 −0.791928 −0.395964 0.918266i $$-0.629589\pi$$
−0.395964 + 0.918266i $$0.629589\pi$$
$$462$$ 0 0
$$463$$ −1.80111e6 −0.390471 −0.195235 0.980756i $$-0.562547\pi$$
−0.195235 + 0.980756i $$0.562547\pi$$
$$464$$ −1.58555e6 −0.341889
$$465$$ 0 0
$$466$$ −84192.0 −0.0179600
$$467$$ 2.36975e6 0.502817 0.251409 0.967881i $$-0.419106\pi$$
0.251409 + 0.967881i $$0.419106\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 464244. 0.0969397
$$471$$ 0 0
$$472$$ −979560. −0.202384
$$473$$ 2.12349e6 0.436412
$$474$$ 0 0
$$475$$ 492656. 0.100187
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 626832. 0.125482
$$479$$ −518146. −0.103184 −0.0515921 0.998668i $$-0.516430\pi$$
−0.0515921 + 0.998668i $$0.516430\pi$$
$$480$$ 0 0
$$481$$ 3.48902e6 0.687609
$$482$$ 1.71561e6 0.336358
$$483$$ 0 0
$$484$$ 2.48332e6 0.481858
$$485$$ −1.70455e6 −0.329045
$$486$$ 0 0
$$487$$ 2.82613e6 0.539970 0.269985 0.962865i $$-0.412981\pi$$
0.269985 + 0.962865i $$0.412981\pi$$
$$488$$ −1.81992e6 −0.345942
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −9.34747e6 −1.74981 −0.874904 0.484296i $$-0.839076\pi$$
−0.874904 + 0.484296i $$0.839076\pi$$
$$492$$ 0 0
$$493$$ 4.58263e6 0.849176
$$494$$ 101024. 0.0186255
$$495$$ 0 0
$$496$$ 1.86370e6 0.340150
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 8.17185e6 1.46916 0.734580 0.678522i $$-0.237379\pi$$
0.734580 + 0.678522i $$0.237379\pi$$
$$500$$ −1.88773e6 −0.337688
$$501$$ 0 0
$$502$$ −909034. −0.160998
$$503$$ 7.37713e6 1.30007 0.650036 0.759903i $$-0.274754\pi$$
0.650036 + 0.759903i $$0.274754\pi$$
$$504$$ 0 0
$$505$$ −1.18327e6 −0.206469
$$506$$ −1.75603e6 −0.304899
$$507$$ 0 0
$$508$$ 8.52222e6 1.46519
$$509$$ 326315. 0.0558268 0.0279134 0.999610i $$-0.491114\pi$$
0.0279134 + 0.999610i $$0.491114\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 5.89875e6 0.994455
$$513$$ 0 0
$$514$$ −1.75636e6 −0.293229
$$515$$ −98296.0 −0.0163312
$$516$$ 0 0
$$517$$ 5.67644e6 0.934006
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −406560. −0.0659350
$$521$$ 2.16703e6 0.349760 0.174880 0.984590i $$-0.444046\pi$$
0.174880 + 0.984590i $$0.444046\pi$$
$$522$$ 0 0
$$523$$ −723404. −0.115645 −0.0578225 0.998327i $$-0.518416\pi$$
−0.0578225 + 0.998327i $$0.518416\pi$$
$$524$$ −372484. −0.0592624
$$525$$ 0 0
$$526$$ −3.92187e6 −0.618057
$$527$$ −5.38654e6 −0.844857
$$528$$ 0 0
$$529$$ 4.21735e6 0.655241
$$530$$ −653202. −0.101008
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −5.19165e6 −0.791566
$$534$$ 0 0
$$535$$ 2.13034e6 0.321784
$$536$$ 3.84936e6 0.578731
$$537$$ 0 0
$$538$$ 2.10759e6 0.313928
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 5.99964e6 0.881317 0.440659 0.897675i $$-0.354745\pi$$
0.440659 + 0.897675i $$0.354745\pi$$
$$542$$ −210118. −0.0307231
$$543$$ 0 0
$$544$$ −9.76819e6 −1.41520
$$545$$ −2.25621e6 −0.325378
$$546$$ 0 0
$$547$$ 7.01570e6 1.00254 0.501271 0.865290i $$-0.332866\pi$$
0.501271 + 0.865290i $$0.332866\pi$$
$$548$$ −1.11513e7 −1.58626
$$549$$ 0 0
$$550$$ 1.61615e6 0.227811
$$551$$ 396388. 0.0556213
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −855184. −0.118382
$$555$$ 0 0
$$556$$ −6.44801e6 −0.884583
$$557$$ 8.91872e6 1.21805 0.609025 0.793151i $$-0.291561\pi$$
0.609025 + 0.793151i $$0.291561\pi$$
$$558$$ 0 0
$$559$$ 2.43135e6 0.329093
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −1.27776e6 −0.170650
$$563$$ 1.33482e7 1.77481 0.887407 0.460987i $$-0.152505\pi$$
0.887407 + 0.460987i $$0.152505\pi$$
$$564$$ 0 0
$$565$$ 513304. 0.0676478
$$566$$ −4.90284e6 −0.643290
$$567$$ 0 0
$$568$$ −4.59288e6 −0.597330
$$569$$ −1.10215e6 −0.142712 −0.0713558 0.997451i $$-0.522733\pi$$
−0.0713558 + 0.997451i $$0.522733\pi$$
$$570$$ 0 0
$$571$$ 1.89348e6 0.243036 0.121518 0.992589i $$-0.461224\pi$$
0.121518 + 0.992589i $$0.461224\pi$$
$$572$$ −2.31986e6 −0.296463
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −9.80506e6 −1.23675
$$576$$ 0 0
$$577$$ 2.82951e6 0.353811 0.176906 0.984228i $$-0.443391\pi$$
0.176906 + 0.984228i $$0.443391\pi$$
$$578$$ 4.34992e6 0.541579
$$579$$ 0 0
$$580$$ −744436. −0.0918877
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −7.98688e6 −0.973208
$$584$$ −4.18392e6 −0.507635
$$585$$ 0 0
$$586$$ −3.43234e6 −0.412901
$$587$$ 1.06799e7 1.27930 0.639649 0.768667i $$-0.279080\pi$$
0.639649 + 0.768667i $$0.279080\pi$$
$$588$$ 0 0
$$589$$ −465924. −0.0553384
$$590$$ −179586. −0.0212394
$$591$$ 0 0
$$592$$ −7.43117e6 −0.871471
$$593$$ 1.46997e7 1.71661 0.858304 0.513141i $$-0.171518\pi$$
0.858304 + 0.513141i $$0.171518\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −2.71975e6 −0.313627
$$597$$ 0 0
$$598$$ −2.01062e6 −0.229921
$$599$$ −8.49163e6 −0.966994 −0.483497 0.875346i $$-0.660634\pi$$
−0.483497 + 0.875346i $$0.660634\pi$$
$$600$$ 0 0
$$601$$ 8.62947e6 0.974536 0.487268 0.873253i $$-0.337994\pi$$
0.487268 + 0.873253i $$0.337994\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 813316. 0.0907125
$$605$$ 975590. 0.108362
$$606$$ 0 0
$$607$$ 1.05807e7 1.16559 0.582793 0.812621i $$-0.301960\pi$$
0.582793 + 0.812621i $$0.301960\pi$$
$$608$$ −844928. −0.0926959
$$609$$ 0 0
$$610$$ −333652. −0.0363052
$$611$$ 6.49942e6 0.704322
$$612$$ 0 0
$$613$$ 3.84784e6 0.413586 0.206793 0.978385i $$-0.433697\pi$$
0.206793 + 0.978385i $$0.433697\pi$$
$$614$$ 3.61794e6 0.387293
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.51001e7 1.59686 0.798428 0.602090i $$-0.205665\pi$$
0.798428 + 0.602090i $$0.205665\pi$$
$$618$$ 0 0
$$619$$ 9.93102e6 1.04176 0.520879 0.853630i $$-0.325604\pi$$
0.520879 + 0.853630i $$0.325604\pi$$
$$620$$ 875028. 0.0914203
$$621$$ 0 0
$$622$$ −3.04291e6 −0.315365
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 8.64589e6 0.885339
$$626$$ −2.69680e6 −0.275051
$$627$$ 0 0
$$628$$ 1.61420e7 1.63327
$$629$$ 2.14779e7 2.16454
$$630$$ 0 0
$$631$$ −9.25224e6 −0.925068 −0.462534 0.886602i $$-0.653060\pi$$
−0.462534 + 0.886602i $$0.653060\pi$$
$$632$$ −1.62348e6 −0.161679
$$633$$ 0 0
$$634$$ 99390.0 0.00982018
$$635$$ 3.34802e6 0.329498
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 1.30035e6 0.126476
$$639$$ 0 0
$$640$$ 2.04864e6 0.197704
$$641$$ −5.00428e6 −0.481057 −0.240529 0.970642i $$-0.577321\pi$$
−0.240529 + 0.970642i $$0.577321\pi$$
$$642$$ 0 0
$$643$$ 1.26137e7 1.20314 0.601569 0.798821i $$-0.294543\pi$$
0.601569 + 0.798821i $$0.294543\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 621888. 0.0586315
$$647$$ 1.25383e7 1.17755 0.588774 0.808298i $$-0.299611\pi$$
0.588774 + 0.808298i $$0.299611\pi$$
$$648$$ 0 0
$$649$$ −2.19585e6 −0.204640
$$650$$ 1.85046e6 0.171790
$$651$$ 0 0
$$652$$ 7.42650e6 0.684171
$$653$$ 8.66066e6 0.794819 0.397409 0.917641i $$-0.369909\pi$$
0.397409 + 0.917641i $$0.369909\pi$$
$$654$$ 0 0
$$655$$ −146333. −0.0133272
$$656$$ 1.10575e7 1.00323
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −7.94177e6 −0.712367 −0.356183 0.934416i $$-0.615922\pi$$
−0.356183 + 0.934416i $$0.615922\pi$$
$$660$$ 0 0
$$661$$ −2.11416e6 −0.188206 −0.0941032 0.995562i $$-0.529998\pi$$
−0.0941032 + 0.995562i $$0.529998\pi$$
$$662$$ 3.17567e6 0.281638
$$663$$ 0 0
$$664$$ −8.17236e6 −0.719329
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −7.88909e6 −0.686613
$$668$$ −1.01861e7 −0.883217
$$669$$ 0 0
$$670$$ 705716. 0.0607355
$$671$$ −4.07965e6 −0.349798
$$672$$ 0 0
$$673$$ −442307. −0.0376432 −0.0188216 0.999823i $$-0.505991\pi$$
−0.0188216 + 0.999823i $$0.505991\pi$$
$$674$$ 429650. 0.0364305
$$675$$ 0 0
$$676$$ 7.74001e6 0.651441
$$677$$ −1.07561e7 −0.901949 −0.450975 0.892537i $$-0.648923\pi$$
−0.450975 + 0.892537i $$0.648923\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −2.50272e6 −0.207558
$$681$$ 0 0
$$682$$ −1.52846e6 −0.125832
$$683$$ 1.14886e7 0.942356 0.471178 0.882038i $$-0.343829\pi$$
0.471178 + 0.882038i $$0.343829\pi$$
$$684$$ 0 0
$$685$$ −4.38088e6 −0.356726
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −5.17846e6 −0.417090
$$689$$ −9.14483e6 −0.733884
$$690$$ 0 0
$$691$$ 1.01388e7 0.807779 0.403890 0.914808i $$-0.367658\pi$$
0.403890 + 0.914808i $$0.367658\pi$$
$$692$$ −4.61569e6 −0.366413
$$693$$ 0 0
$$694$$ −5.17719e6 −0.408033
$$695$$ −2.53315e6 −0.198929
$$696$$ 0 0
$$697$$ −3.19590e7 −2.49179
$$698$$ −49756.0 −0.00386551
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.96839e7 1.51292 0.756459 0.654041i $$-0.226927\pi$$
0.756459 + 0.654041i $$0.226927\pi$$
$$702$$ 0 0
$$703$$ 1.85779e6 0.141778
$$704$$ 2.87507e6 0.218634
$$705$$ 0 0
$$706$$ 3.47202e6 0.262162
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −2.01717e7 −1.50705 −0.753524 0.657420i $$-0.771648\pi$$
−0.753524 + 0.657420i $$0.771648\pi$$
$$710$$ −842028. −0.0626875
$$711$$ 0 0
$$712$$ −1.37906e7 −1.01949
$$713$$ 9.27302e6 0.683121
$$714$$ 0 0
$$715$$ −911372. −0.0666700
$$716$$ 857584. 0.0625164
$$717$$ 0 0
$$718$$ −1.72485e6 −0.124865
$$719$$ −4.15735e6 −0.299912 −0.149956 0.988693i $$-0.547913\pi$$
−0.149956 + 0.988693i $$0.547913\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −4.89841e6 −0.349713
$$723$$ 0 0
$$724$$ 1.82390e7 1.29317
$$725$$ 7.26067e6 0.513017
$$726$$ 0 0
$$727$$ −1.54433e7 −1.08369 −0.541845 0.840479i $$-0.682274\pi$$
−0.541845 + 0.840479i $$0.682274\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −767052. −0.0532743
$$731$$ 1.49670e7 1.03596
$$732$$ 0 0
$$733$$ −6.20414e6 −0.426502 −0.213251 0.976997i $$-0.568405\pi$$
−0.213251 + 0.976997i $$0.568405\pi$$
$$734$$ 6.23084e6 0.426880
$$735$$ 0 0
$$736$$ 1.68161e7 1.14428
$$737$$ 8.62898e6 0.585182
$$738$$ 0 0
$$739$$ 2.18984e7 1.47503 0.737517 0.675328i $$-0.235998\pi$$
0.737517 + 0.675328i $$0.235998\pi$$
$$740$$ −3.48902e6 −0.234220
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 2.75483e6 0.183073 0.0915363 0.995802i $$-0.470822\pi$$
0.0915363 + 0.995802i $$0.470822\pi$$
$$744$$ 0 0
$$745$$ −1.06847e6 −0.0705299
$$746$$ −3.59389e6 −0.236438
$$747$$ 0 0
$$748$$ −1.42807e7 −0.933243
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 1.29121e7 0.835404 0.417702 0.908584i $$-0.362836\pi$$
0.417702 + 0.908584i $$0.362836\pi$$
$$752$$ −1.38429e7 −0.892653
$$753$$ 0 0
$$754$$ 1.48887e6 0.0953738
$$755$$ 319517. 0.0203998
$$756$$ 0 0
$$757$$ −2.64315e7 −1.67642 −0.838209 0.545349i $$-0.816397\pi$$
−0.838209 + 0.545349i $$0.816397\pi$$
$$758$$ 6.90931e6 0.436779
$$759$$ 0 0
$$760$$ −216480. −0.0135951
$$761$$ −1.22214e7 −0.764996 −0.382498 0.923956i $$-0.624936\pi$$
−0.382498 + 0.923956i $$0.624936\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2.12061e7 −1.31440
$$765$$ 0 0
$$766$$ −4.31008e6 −0.265407
$$767$$ −2.51420e6 −0.154316
$$768$$ 0 0
$$769$$ 6.10654e6 0.372374 0.186187 0.982514i $$-0.440387\pi$$
0.186187 + 0.982514i $$0.440387\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 4.48949e6 0.271115
$$773$$ 3.02220e6 0.181918 0.0909588 0.995855i $$-0.471007\pi$$
0.0909588 + 0.995855i $$0.471007\pi$$
$$774$$ 0 0
$$775$$ −8.53436e6 −0.510407
$$776$$ −1.85951e7 −1.10852
$$777$$ 0 0
$$778$$ 925548. 0.0548214
$$779$$ −2.76438e6 −0.163213
$$780$$ 0 0
$$781$$ −1.02957e7 −0.603988
$$782$$ −1.23771e7 −0.723772
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 6.34150e6 0.367297
$$786$$ 0 0
$$787$$ −2.08285e7 −1.19873 −0.599365 0.800476i $$-0.704580\pi$$
−0.599365 + 0.800476i $$0.704580\pi$$
$$788$$ −1.72866e6 −0.0991734
$$789$$ 0 0
$$790$$ −297638. −0.0169676
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −4.67113e6 −0.263778
$$794$$ −8.13242e6 −0.457793
$$795$$ 0 0
$$796$$ −1.03854e7 −0.580954
$$797$$ −2.32328e7 −1.29556 −0.647778 0.761829i $$-0.724302\pi$$
−0.647778 + 0.761829i $$0.724302\pi$$
$$798$$ 0 0
$$799$$ 4.00094e7 2.21715
$$800$$ −1.54766e7 −0.854970
$$801$$ 0 0
$$802$$ 1.01373e7 0.556525
$$803$$ −9.37895e6 −0.513293
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −1.75006e6 −0.0948887
$$807$$ 0 0
$$808$$ −1.29084e7 −0.695575
$$809$$ −1.08668e7 −0.583753 −0.291876 0.956456i $$-0.594280\pi$$
−0.291876 + 0.956456i $$0.594280\pi$$
$$810$$ 0 0
$$811$$ −2.22632e7 −1.18860 −0.594299 0.804244i $$-0.702570\pi$$
−0.594299 + 0.804244i $$0.702570\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 6.09446e6 0.322385
$$815$$ 2.91755e6 0.153860
$$816$$ 0 0
$$817$$ 1.29462e6 0.0678556
$$818$$ 5.75467e6 0.300703
$$819$$ 0 0
$$820$$ 5.19165e6 0.269631
$$821$$ 1.23881e7 0.641426 0.320713 0.947176i $$-0.396078\pi$$
0.320713 + 0.947176i $$0.396078\pi$$
$$822$$ 0 0
$$823$$ 1.69481e6 0.0872210 0.0436105 0.999049i $$-0.486114\pi$$
0.0436105 + 0.999049i $$0.486114\pi$$
$$824$$ −1.07232e6 −0.0550182
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 378495. 0.0192440 0.00962202 0.999954i $$-0.496937\pi$$
0.00962202 + 0.999954i $$0.496937\pi$$
$$828$$ 0 0
$$829$$ −1.04287e7 −0.527043 −0.263521 0.964654i $$-0.584884\pi$$
−0.263521 + 0.964654i $$0.584884\pi$$
$$830$$ −1.49827e6 −0.0754907
$$831$$ 0 0
$$832$$ 3.29190e6 0.164869
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −4.00169e6 −0.198622
$$836$$ −1.23525e6 −0.0611278
$$837$$ 0 0
$$838$$ 6.82685e6 0.335823
$$839$$ −3.04082e7 −1.49137 −0.745686 0.666297i $$-0.767878\pi$$
−0.745686 + 0.666297i $$0.767878\pi$$
$$840$$ 0 0
$$841$$ −1.46693e7 −0.715185
$$842$$ −2.61475e6 −0.127101
$$843$$ 0 0
$$844$$ −6.08860e6 −0.294213
$$845$$ 3.04072e6 0.146499
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 1.94773e7 0.930120
$$849$$ 0 0
$$850$$ 1.13912e7 0.540780
$$851$$ −3.69746e7 −1.75017
$$852$$ 0 0
$$853$$ 2.80315e7 1.31909 0.659544 0.751666i $$-0.270750\pi$$
0.659544 + 0.751666i $$0.270750\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 2.32400e7 1.08406
$$857$$ −1.88030e7 −0.874529 −0.437264 0.899333i $$-0.644053\pi$$
−0.437264 + 0.899333i $$0.644053\pi$$
$$858$$ 0 0
$$859$$ 7.86323e6 0.363595 0.181798 0.983336i $$-0.441808\pi$$
0.181798 + 0.983336i $$0.441808\pi$$
$$860$$ −2.43135e6 −0.112099
$$861$$ 0 0
$$862$$ −3.87094e6 −0.177438
$$863$$ 1.12858e7 0.515827 0.257913 0.966168i $$-0.416965\pi$$
0.257913 + 0.966168i $$0.416965\pi$$
$$864$$ 0 0
$$865$$ −1.81331e6 −0.0824007
$$866$$ 1.03334e6 0.0468218
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3.63930e6 −0.163481
$$870$$ 0 0
$$871$$ 9.88002e6 0.441278
$$872$$ −2.46132e7 −1.09617
$$873$$ 0 0
$$874$$ −1.07059e6 −0.0474073
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.34150e7 0.588968 0.294484 0.955656i $$-0.404852\pi$$
0.294484 + 0.955656i $$0.404852\pi$$
$$878$$ 5.83060e6 0.255257
$$879$$ 0 0
$$880$$ 1.94110e6 0.0844972
$$881$$ −3.18547e7 −1.38272 −0.691359 0.722511i $$-0.742988\pi$$
−0.691359 + 0.722511i $$0.742988\pi$$
$$882$$ 0 0
$$883$$ −3.05922e7 −1.32041 −0.660205 0.751086i $$-0.729530\pi$$
−0.660205 + 0.751086i $$0.729530\pi$$
$$884$$ −1.63511e7 −0.703747
$$885$$ 0 0
$$886$$ 3.56758e6 0.152683
$$887$$ 4.63772e6 0.197923 0.0989613 0.995091i $$-0.468448\pi$$
0.0989613 + 0.995091i $$0.468448\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −2.52828e6 −0.106992
$$891$$ 0 0
$$892$$ −1.65136e7 −0.694911
$$893$$ 3.46073e6 0.145224
$$894$$ 0 0
$$895$$ 336908. 0.0140590
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −8.00316e6 −0.331185
$$899$$ −6.86670e6 −0.283367
$$900$$ 0 0
$$901$$ −5.62941e7 −2.31021
$$902$$ −9.06853e6 −0.371125
$$903$$ 0 0
$$904$$ 5.59968e6 0.227899
$$905$$ 7.16531e6 0.290813
$$906$$ 0 0
$$907$$ 604376. 0.0243943 0.0121972 0.999926i $$-0.496117\pi$$
0.0121972 + 0.999926i $$0.496117\pi$$
$$908$$ −1.08373e7 −0.436219
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 2.44059e7 0.974315 0.487157 0.873314i $$-0.338034\pi$$
0.487157 + 0.873314i $$0.338034\pi$$
$$912$$ 0 0
$$913$$ −1.83197e7 −0.727347
$$914$$ −2.33533e6 −0.0924661
$$915$$ 0 0
$$916$$ 6.51650e6 0.256611
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 3.67095e7 1.43380 0.716902 0.697174i $$-0.245560\pi$$
0.716902 + 0.697174i $$0.245560\pi$$
$$920$$ 4.30848e6 0.167824
$$921$$ 0 0
$$922$$ −7.22716e6 −0.279989
$$923$$ −1.17884e7 −0.455460
$$924$$ 0 0
$$925$$ 3.40293e7 1.30767
$$926$$ −3.60222e6 −0.138052
$$927$$ 0 0
$$928$$ −1.24524e7 −0.474660
$$929$$ 2.29089e7 0.870892 0.435446 0.900215i $$-0.356591\pi$$
0.435446 + 0.900215i $$0.356591\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 1.17869e6 0.0444487
$$933$$ 0 0
$$934$$ 4.73950e6 0.177773
$$935$$ −5.61026e6 −0.209872
$$936$$ 0 0
$$937$$ −5.99611e6 −0.223111 −0.111555 0.993758i $$-0.535583\pi$$
−0.111555 + 0.993758i $$0.535583\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −6.49942e6 −0.239914
$$941$$ 1.16516e7 0.428954 0.214477 0.976729i $$-0.431195\pi$$
0.214477 + 0.976729i $$0.431195\pi$$
$$942$$ 0 0
$$943$$ 5.50180e7 2.01477
$$944$$ 5.35493e6 0.195580
$$945$$ 0 0
$$946$$ 4.24697e6 0.154295
$$947$$ 1.10926e6 0.0401939 0.0200969 0.999798i $$-0.493603\pi$$
0.0200969 + 0.999798i $$0.493603\pi$$
$$948$$ 0 0
$$949$$ −1.07387e7 −0.387068
$$950$$ 985312. 0.0354213
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −1.05743e7 −0.377155 −0.188578 0.982058i $$-0.560388\pi$$
−0.188578 + 0.982058i $$0.560388\pi$$
$$954$$ 0 0
$$955$$ −8.33096e6 −0.295588
$$956$$ −8.77565e6 −0.310552
$$957$$ 0 0
$$958$$ −1.03629e6 −0.0364811
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.05579e7 −0.718075
$$962$$ 6.97805e6 0.243106
$$963$$ 0 0
$$964$$ −2.40186e7 −0.832444
$$965$$ 1.76373e6 0.0609696
$$966$$ 0 0
$$967$$ 6.32666e6 0.217575 0.108787 0.994065i $$-0.465303\pi$$
0.108787 + 0.994065i $$0.465303\pi$$
$$968$$ 1.06428e7 0.365063
$$969$$ 0 0
$$970$$ −3.40910e6 −0.116335
$$971$$ −3.92395e7 −1.33560 −0.667798 0.744343i $$-0.732763\pi$$
−0.667798 + 0.744343i $$0.732763\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 5.65225e6 0.190908
$$975$$ 0 0
$$976$$ 9.94890e6 0.334311
$$977$$ 1.55074e6 0.0519760 0.0259880 0.999662i $$-0.491727\pi$$
0.0259880 + 0.999662i $$0.491727\pi$$
$$978$$ 0 0
$$979$$ −3.09140e7 −1.03086
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −1.86949e7 −0.618651
$$983$$ 4.87484e7 1.60908 0.804538 0.593901i $$-0.202413\pi$$
0.804538 + 0.593901i $$0.202413\pi$$
$$984$$ 0 0
$$985$$ −679118. −0.0223026
$$986$$ 9.16526e6 0.300229
$$987$$ 0 0
$$988$$ −1.41434e6 −0.0460957
$$989$$ −2.57660e7 −0.837638
$$990$$ 0 0
$$991$$ −1.92552e6 −0.0622820 −0.0311410 0.999515i $$-0.509914\pi$$
−0.0311410 + 0.999515i $$0.509914\pi$$
$$992$$ 1.46368e7 0.472246
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.07999e6 −0.130648
$$996$$ 0 0
$$997$$ 5.42564e7 1.72867 0.864337 0.502913i $$-0.167739\pi$$
0.864337 + 0.502913i $$0.167739\pi$$
$$998$$ 1.63437e7 0.519427
$$999$$ 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.g.1.1 1
3.2 odd 2 147.6.a.c.1.1 1
7.2 even 3 63.6.e.a.46.1 2
7.4 even 3 63.6.e.a.37.1 2
7.6 odd 2 441.6.a.h.1.1 1
21.2 odd 6 21.6.e.a.4.1 2
21.5 even 6 147.6.e.g.67.1 2
21.11 odd 6 21.6.e.a.16.1 yes 2
21.17 even 6 147.6.e.g.79.1 2
21.20 even 2 147.6.a.d.1.1 1
84.11 even 6 336.6.q.b.289.1 2
84.23 even 6 336.6.q.b.193.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.a.4.1 2 21.2 odd 6
21.6.e.a.16.1 yes 2 21.11 odd 6
63.6.e.a.37.1 2 7.4 even 3
63.6.e.a.46.1 2 7.2 even 3
147.6.a.c.1.1 1 3.2 odd 2
147.6.a.d.1.1 1 21.20 even 2
147.6.e.g.67.1 2 21.5 even 6
147.6.e.g.79.1 2 21.17 even 6
336.6.q.b.193.1 2 84.23 even 6
336.6.q.b.289.1 2 84.11 even 6
441.6.a.g.1.1 1 1.1 even 1 trivial
441.6.a.h.1.1 1 7.6 odd 2