# Properties

 Label 162.7.b.a.161.2 Level $162$ Weight $7$ Character 162.161 Analytic conductor $37.269$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,7,Mod(161,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("162.161");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 162.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$37.2687615464$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 161.2 Root $$-1.93185i$$ of defining polynomial Character $$\chi$$ $$=$$ 162.161 Dual form 162.7.b.a.161.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.65685i q^{2} -32.0000 q^{4} +208.528i q^{5} +4.19615 q^{7} +181.019i q^{8} +O(q^{10})$$ $$q-5.65685i q^{2} -32.0000 q^{4} +208.528i q^{5} +4.19615 q^{7} +181.019i q^{8} +1179.62 q^{10} -2261.95i q^{11} +2840.01 q^{13} -23.7370i q^{14} +1024.00 q^{16} +1965.86i q^{17} -281.295 q^{19} -6672.91i q^{20} -12795.5 q^{22} +16744.3i q^{23} -27859.1 q^{25} -16065.5i q^{26} -134.277 q^{28} +37114.1i q^{29} -24708.3 q^{31} -5792.62i q^{32} +11120.6 q^{34} +875.017i q^{35} -17016.7 q^{37} +1591.25i q^{38} -37747.7 q^{40} +116236. i q^{41} -30662.9 q^{43} +72382.5i q^{44} +94720.0 q^{46} -77666.6i q^{47} -117631. q^{49} +157595. i q^{50} -90880.2 q^{52} +138657. i q^{53} +471682. q^{55} +759.585i q^{56} +209949. q^{58} -152958. i q^{59} -16138.1 q^{61} +139771. i q^{62} -32768.0 q^{64} +592222. i q^{65} -474667. q^{67} -62907.4i q^{68} +4949.85 q^{70} -150338. i q^{71} +331690. q^{73} +96261.2i q^{74} +9001.45 q^{76} -9491.50i q^{77} -896112. q^{79} +213533. i q^{80} +657529. q^{82} +945090. i q^{83} -409937. q^{85} +173455. i q^{86} +409457. q^{88} +790302. i q^{89} +11917.1 q^{91} -535817. i q^{92} -439349. q^{94} -58658.1i q^{95} +1.39268e6 q^{97} +665424. i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 128 q^{4} - 4 q^{7}+O(q^{10})$$ 4 * q - 128 * q^4 - 4 * q^7 $$4 q - 128 q^{4} - 4 q^{7} + 2640 q^{10} + 3836 q^{13} + 4096 q^{16} + 10244 q^{19} - 27072 q^{22} - 25700 q^{25} + 128 q^{28} - 40096 q^{31} + 60528 q^{34} - 20200 q^{37} - 84480 q^{40} + 184940 q^{43} + 162720 q^{46} - 470484 q^{49} - 122752 q^{52} + 949860 q^{55} + 24624 q^{58} + 609056 q^{61} - 131072 q^{64} - 2008972 q^{67} + 8160 q^{70} + 525824 q^{73} - 327808 q^{76} - 848716 q^{79} + 705792 q^{82} - 987840 q^{85} + 866304 q^{88} + 35260 q^{91} - 1965408 q^{94} + 3621728 q^{97}+O(q^{100})$$ 4 * q - 128 * q^4 - 4 * q^7 + 2640 * q^10 + 3836 * q^13 + 4096 * q^16 + 10244 * q^19 - 27072 * q^22 - 25700 * q^25 + 128 * q^28 - 40096 * q^31 + 60528 * q^34 - 20200 * q^37 - 84480 * q^40 + 184940 * q^43 + 162720 * q^46 - 470484 * q^49 - 122752 * q^52 + 949860 * q^55 + 24624 * q^58 + 609056 * q^61 - 131072 * q^64 - 2008972 * q^67 + 8160 * q^70 + 525824 * q^73 - 327808 * q^76 - 848716 * q^79 + 705792 * q^82 - 987840 * q^85 + 866304 * q^88 + 35260 * q^91 - 1965408 * q^94 + 3621728 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 5.65685i − 0.707107i
$$3$$ 0 0
$$4$$ −32.0000 −0.500000
$$5$$ 208.528i 1.66823i 0.551592 + 0.834114i $$0.314020\pi$$
−0.551592 + 0.834114i $$0.685980\pi$$
$$6$$ 0 0
$$7$$ 4.19615 0.0122337 0.00611684 0.999981i $$-0.498053\pi$$
0.00611684 + 0.999981i $$0.498053\pi$$
$$8$$ 181.019i 0.353553i
$$9$$ 0 0
$$10$$ 1179.62 1.17962
$$11$$ − 2261.95i − 1.69944i −0.527235 0.849719i $$-0.676771\pi$$
0.527235 0.849719i $$-0.323229\pi$$
$$12$$ 0 0
$$13$$ 2840.01 1.29268 0.646338 0.763052i $$-0.276300\pi$$
0.646338 + 0.763052i $$0.276300\pi$$
$$14$$ − 23.7370i − 0.00865052i
$$15$$ 0 0
$$16$$ 1024.00 0.250000
$$17$$ 1965.86i 0.400134i 0.979782 + 0.200067i $$0.0641160\pi$$
−0.979782 + 0.200067i $$0.935884\pi$$
$$18$$ 0 0
$$19$$ −281.295 −0.0410111 −0.0205056 0.999790i $$-0.506528\pi$$
−0.0205056 + 0.999790i $$0.506528\pi$$
$$20$$ − 6672.91i − 0.834114i
$$21$$ 0 0
$$22$$ −12795.5 −1.20168
$$23$$ 16744.3i 1.37620i 0.725614 + 0.688102i $$0.241556\pi$$
−0.725614 + 0.688102i $$0.758444\pi$$
$$24$$ 0 0
$$25$$ −27859.1 −1.78298
$$26$$ − 16065.5i − 0.914059i
$$27$$ 0 0
$$28$$ −134.277 −0.00611684
$$29$$ 37114.1i 1.52176i 0.648895 + 0.760878i $$0.275231\pi$$
−0.648895 + 0.760878i $$0.724769\pi$$
$$30$$ 0 0
$$31$$ −24708.3 −0.829389 −0.414694 0.909961i $$-0.636112\pi$$
−0.414694 + 0.909961i $$0.636112\pi$$
$$32$$ − 5792.62i − 0.176777i
$$33$$ 0 0
$$34$$ 11120.6 0.282937
$$35$$ 875.017i 0.0204086i
$$36$$ 0 0
$$37$$ −17016.7 −0.335947 −0.167974 0.985791i $$-0.553722\pi$$
−0.167974 + 0.985791i $$0.553722\pi$$
$$38$$ 1591.25i 0.0289993i
$$39$$ 0 0
$$40$$ −37747.7 −0.589808
$$41$$ 116236.i 1.68651i 0.537516 + 0.843253i $$0.319363\pi$$
−0.537516 + 0.843253i $$0.680637\pi$$
$$42$$ 0 0
$$43$$ −30662.9 −0.385662 −0.192831 0.981232i $$-0.561767\pi$$
−0.192831 + 0.981232i $$0.561767\pi$$
$$44$$ 72382.5i 0.849719i
$$45$$ 0 0
$$46$$ 94720.0 0.973124
$$47$$ − 77666.6i − 0.748068i −0.927415 0.374034i $$-0.877974\pi$$
0.927415 0.374034i $$-0.122026\pi$$
$$48$$ 0 0
$$49$$ −117631. −0.999850
$$50$$ 157595.i 1.26076i
$$51$$ 0 0
$$52$$ −90880.2 −0.646338
$$53$$ 138657.i 0.931354i 0.884955 + 0.465677i $$0.154189\pi$$
−0.884955 + 0.465677i $$0.845811\pi$$
$$54$$ 0 0
$$55$$ 471682. 2.83505
$$56$$ 759.585i 0.00432526i
$$57$$ 0 0
$$58$$ 209949. 1.07604
$$59$$ − 152958.i − 0.744758i −0.928081 0.372379i $$-0.878542\pi$$
0.928081 0.372379i $$-0.121458\pi$$
$$60$$ 0 0
$$61$$ −16138.1 −0.0710989 −0.0355495 0.999368i $$-0.511318\pi$$
−0.0355495 + 0.999368i $$0.511318\pi$$
$$62$$ 139771.i 0.586467i
$$63$$ 0 0
$$64$$ −32768.0 −0.125000
$$65$$ 592222.i 2.15648i
$$66$$ 0 0
$$67$$ −474667. −1.57821 −0.789105 0.614259i $$-0.789455\pi$$
−0.789105 + 0.614259i $$0.789455\pi$$
$$68$$ − 62907.4i − 0.200067i
$$69$$ 0 0
$$70$$ 4949.85 0.0144310
$$71$$ − 150338.i − 0.420043i −0.977697 0.210021i $$-0.932647\pi$$
0.977697 0.210021i $$-0.0673534\pi$$
$$72$$ 0 0
$$73$$ 331690. 0.852636 0.426318 0.904573i $$-0.359811\pi$$
0.426318 + 0.904573i $$0.359811\pi$$
$$74$$ 96261.2i 0.237551i
$$75$$ 0 0
$$76$$ 9001.45 0.0205056
$$77$$ − 9491.50i − 0.0207904i
$$78$$ 0 0
$$79$$ −896112. −1.81753 −0.908764 0.417310i $$-0.862973\pi$$
−0.908764 + 0.417310i $$0.862973\pi$$
$$80$$ 213533.i 0.417057i
$$81$$ 0 0
$$82$$ 657529. 1.19254
$$83$$ 945090.i 1.65287i 0.563032 + 0.826435i $$0.309635\pi$$
−0.563032 + 0.826435i $$0.690365\pi$$
$$84$$ 0 0
$$85$$ −409937. −0.667514
$$86$$ 173455.i 0.272704i
$$87$$ 0 0
$$88$$ 409457. 0.600842
$$89$$ 790302.i 1.12105i 0.828139 + 0.560523i $$0.189400\pi$$
−0.828139 + 0.560523i $$0.810600\pi$$
$$90$$ 0 0
$$91$$ 11917.1 0.0158142
$$92$$ − 535817.i − 0.688102i
$$93$$ 0 0
$$94$$ −439349. −0.528964
$$95$$ − 58658.1i − 0.0684159i
$$96$$ 0 0
$$97$$ 1.39268e6 1.52593 0.762965 0.646440i $$-0.223743\pi$$
0.762965 + 0.646440i $$0.223743\pi$$
$$98$$ 665424.i 0.707001i
$$99$$ 0 0
$$100$$ 891492. 0.891492
$$101$$ 131552.i 0.127683i 0.997960 + 0.0638414i $$0.0203352\pi$$
−0.997960 + 0.0638414i $$0.979665\pi$$
$$102$$ 0 0
$$103$$ 1.93461e6 1.77044 0.885221 0.465171i $$-0.154007\pi$$
0.885221 + 0.465171i $$0.154007\pi$$
$$104$$ 514096.i 0.457030i
$$105$$ 0 0
$$106$$ 784364. 0.658567
$$107$$ − 621916.i − 0.507669i −0.967248 0.253834i $$-0.918308\pi$$
0.967248 0.253834i $$-0.0816918\pi$$
$$108$$ 0 0
$$109$$ −1.34578e6 −1.03919 −0.519594 0.854413i $$-0.673917\pi$$
−0.519594 + 0.854413i $$0.673917\pi$$
$$110$$ − 2.66823e6i − 2.00468i
$$111$$ 0 0
$$112$$ 4296.86 0.00305842
$$113$$ 1.23932e6i 0.858914i 0.903087 + 0.429457i $$0.141295\pi$$
−0.903087 + 0.429457i $$0.858705\pi$$
$$114$$ 0 0
$$115$$ −3.49166e6 −2.29582
$$116$$ − 1.18765e6i − 0.760878i
$$117$$ 0 0
$$118$$ −865259. −0.526623
$$119$$ 8249.04i 0.00489511i
$$120$$ 0 0
$$121$$ −3.34487e6 −1.88809
$$122$$ 91290.9i 0.0502745i
$$123$$ 0 0
$$124$$ 790666. 0.414694
$$125$$ − 2.55116e6i − 1.30620i
$$126$$ 0 0
$$127$$ −1.65297e6 −0.806962 −0.403481 0.914988i $$-0.632200\pi$$
−0.403481 + 0.914988i $$0.632200\pi$$
$$128$$ 185364.i 0.0883883i
$$129$$ 0 0
$$130$$ 3.35012e6 1.52486
$$131$$ − 2.66126e6i − 1.18379i −0.806017 0.591893i $$-0.798381\pi$$
0.806017 0.591893i $$-0.201619\pi$$
$$132$$ 0 0
$$133$$ −1180.36 −0.000501717 0
$$134$$ 2.68512e6i 1.11596i
$$135$$ 0 0
$$136$$ −355858. −0.141469
$$137$$ 2.61015e6i 1.01509i 0.861626 + 0.507544i $$0.169447\pi$$
−0.861626 + 0.507544i $$0.830553\pi$$
$$138$$ 0 0
$$139$$ −701321. −0.261139 −0.130570 0.991439i $$-0.541681\pi$$
−0.130570 + 0.991439i $$0.541681\pi$$
$$140$$ − 28000.6i − 0.0102043i
$$141$$ 0 0
$$142$$ −850440. −0.297015
$$143$$ − 6.42396e6i − 2.19682i
$$144$$ 0 0
$$145$$ −7.73935e6 −2.53864
$$146$$ − 1.87632e6i − 0.602904i
$$147$$ 0 0
$$148$$ 544536. 0.167974
$$149$$ − 4.73344e6i − 1.43093i −0.698650 0.715464i $$-0.746215\pi$$
0.698650 0.715464i $$-0.253785\pi$$
$$150$$ 0 0
$$151$$ 3.73729e6 1.08549 0.542746 0.839897i $$-0.317385\pi$$
0.542746 + 0.839897i $$0.317385\pi$$
$$152$$ − 50919.9i − 0.0144996i
$$153$$ 0 0
$$154$$ −53692.0 −0.0147010
$$155$$ − 5.15239e6i − 1.38361i
$$156$$ 0 0
$$157$$ 3.20662e6 0.828607 0.414303 0.910139i $$-0.364025\pi$$
0.414303 + 0.910139i $$0.364025\pi$$
$$158$$ 5.06918e6i 1.28519i
$$159$$ 0 0
$$160$$ 1.20793e6 0.294904
$$161$$ 70261.6i 0.0168361i
$$162$$ 0 0
$$163$$ −964418. −0.222691 −0.111345 0.993782i $$-0.535516\pi$$
−0.111345 + 0.993782i $$0.535516\pi$$
$$164$$ − 3.71954e6i − 0.843253i
$$165$$ 0 0
$$166$$ 5.34624e6 1.16876
$$167$$ − 1.03499e6i − 0.222222i −0.993808 0.111111i $$-0.964559\pi$$
0.993808 0.111111i $$-0.0354410\pi$$
$$168$$ 0 0
$$169$$ 3.23883e6 0.671009
$$170$$ 2.31896e6i 0.472004i
$$171$$ 0 0
$$172$$ 981212. 0.192831
$$173$$ 648596.i 0.125267i 0.998037 + 0.0626334i $$0.0199499\pi$$
−0.998037 + 0.0626334i $$0.980050\pi$$
$$174$$ 0 0
$$175$$ −116901. −0.0218125
$$176$$ − 2.31624e6i − 0.424860i
$$177$$ 0 0
$$178$$ 4.47063e6 0.792699
$$179$$ 4.65189e6i 0.811092i 0.914075 + 0.405546i $$0.132919\pi$$
−0.914075 + 0.405546i $$0.867081\pi$$
$$180$$ 0 0
$$181$$ 3.43720e6 0.579654 0.289827 0.957079i $$-0.406402\pi$$
0.289827 + 0.957079i $$0.406402\pi$$
$$182$$ − 67413.3i − 0.0111823i
$$183$$ 0 0
$$184$$ −3.03104e6 −0.486562
$$185$$ − 3.54847e6i − 0.560437i
$$186$$ 0 0
$$187$$ 4.44668e6 0.680003
$$188$$ 2.48533e6i 0.374034i
$$189$$ 0 0
$$190$$ −331820. −0.0483774
$$191$$ 1.17509e7i 1.68644i 0.537565 + 0.843222i $$0.319344\pi$$
−0.537565 + 0.843222i $$0.680656\pi$$
$$192$$ 0 0
$$193$$ 6.95599e6 0.967581 0.483790 0.875184i $$-0.339260\pi$$
0.483790 + 0.875184i $$0.339260\pi$$
$$194$$ − 7.87816e6i − 1.07900i
$$195$$ 0 0
$$196$$ 3.76420e6 0.499925
$$197$$ 409645.i 0.0535807i 0.999641 + 0.0267904i $$0.00852866\pi$$
−0.999641 + 0.0267904i $$0.991471\pi$$
$$198$$ 0 0
$$199$$ −3.24847e6 −0.412212 −0.206106 0.978530i $$-0.566079\pi$$
−0.206106 + 0.978530i $$0.566079\pi$$
$$200$$ − 5.04304e6i − 0.630380i
$$201$$ 0 0
$$202$$ 744169. 0.0902854
$$203$$ 155736.i 0.0186167i
$$204$$ 0 0
$$205$$ −2.42385e7 −2.81348
$$206$$ − 1.09438e7i − 1.25189i
$$207$$ 0 0
$$208$$ 2.90817e6 0.323169
$$209$$ 636277.i 0.0696959i
$$210$$ 0 0
$$211$$ −4.99487e6 −0.531712 −0.265856 0.964013i $$-0.585655\pi$$
−0.265856 + 0.964013i $$0.585655\pi$$
$$212$$ − 4.43703e6i − 0.465677i
$$213$$ 0 0
$$214$$ −3.51809e6 −0.358976
$$215$$ − 6.39408e6i − 0.643373i
$$216$$ 0 0
$$217$$ −103680. −0.0101465
$$218$$ 7.61287e6i 0.734817i
$$219$$ 0 0
$$220$$ −1.50938e7 −1.41753
$$221$$ 5.58305e6i 0.517243i
$$222$$ 0 0
$$223$$ 5.68924e6 0.513027 0.256513 0.966541i $$-0.417426\pi$$
0.256513 + 0.966541i $$0.417426\pi$$
$$224$$ − 24306.7i − 0.00216263i
$$225$$ 0 0
$$226$$ 7.01068e6 0.607344
$$227$$ 1.16064e7i 0.992246i 0.868252 + 0.496123i $$0.165243\pi$$
−0.868252 + 0.496123i $$0.834757\pi$$
$$228$$ 0 0
$$229$$ −3.45690e6 −0.287859 −0.143930 0.989588i $$-0.545974\pi$$
−0.143930 + 0.989588i $$0.545974\pi$$
$$230$$ 1.97518e7i 1.62339i
$$231$$ 0 0
$$232$$ −6.71837e6 −0.538022
$$233$$ 1.27739e7i 1.00985i 0.863164 + 0.504924i $$0.168480\pi$$
−0.863164 + 0.504924i $$0.831520\pi$$
$$234$$ 0 0
$$235$$ 1.61957e7 1.24795
$$236$$ 4.89464e6i 0.372379i
$$237$$ 0 0
$$238$$ 46663.6 0.00346137
$$239$$ 1.29231e7i 0.946617i 0.880897 + 0.473309i $$0.156940\pi$$
−0.880897 + 0.473309i $$0.843060\pi$$
$$240$$ 0 0
$$241$$ 7.28032e6 0.520115 0.260057 0.965593i $$-0.416259\pi$$
0.260057 + 0.965593i $$0.416259\pi$$
$$242$$ 1.89214e7i 1.33508i
$$243$$ 0 0
$$244$$ 516419. 0.0355495
$$245$$ − 2.45295e7i − 1.66798i
$$246$$ 0 0
$$247$$ −798881. −0.0530141
$$248$$ − 4.47268e6i − 0.293233i
$$249$$ 0 0
$$250$$ −1.44316e7 −0.923620
$$251$$ 1.71927e7i 1.08723i 0.839334 + 0.543616i $$0.182945\pi$$
−0.839334 + 0.543616i $$0.817055\pi$$
$$252$$ 0 0
$$253$$ 3.78748e7 2.33878
$$254$$ 9.35059e6i 0.570608i
$$255$$ 0 0
$$256$$ 1.04858e6 0.0625000
$$257$$ 8.01111e6i 0.471947i 0.971759 + 0.235974i $$0.0758279\pi$$
−0.971759 + 0.235974i $$0.924172\pi$$
$$258$$ 0 0
$$259$$ −71404.8 −0.00410987
$$260$$ − 1.89511e7i − 1.07824i
$$261$$ 0 0
$$262$$ −1.50543e7 −0.837062
$$263$$ − 1.68209e7i − 0.924661i −0.886708 0.462331i $$-0.847013\pi$$
0.886708 0.462331i $$-0.152987\pi$$
$$264$$ 0 0
$$265$$ −2.89140e7 −1.55371
$$266$$ 6677.11i 0 0.000354768i
$$267$$ 0 0
$$268$$ 1.51893e7 0.789105
$$269$$ 3.33514e6i 0.171339i 0.996324 + 0.0856697i $$0.0273030\pi$$
−0.996324 + 0.0856697i $$0.972697\pi$$
$$270$$ 0 0
$$271$$ −1.47048e7 −0.738840 −0.369420 0.929263i $$-0.620444\pi$$
−0.369420 + 0.929263i $$0.620444\pi$$
$$272$$ 2.01304e6i 0.100033i
$$273$$ 0 0
$$274$$ 1.47652e7 0.717776
$$275$$ 6.30160e7i 3.03007i
$$276$$ 0 0
$$277$$ −1.09201e7 −0.513790 −0.256895 0.966439i $$-0.582699\pi$$
−0.256895 + 0.966439i $$0.582699\pi$$
$$278$$ 3.96727e6i 0.184653i
$$279$$ 0 0
$$280$$ −158395. −0.00721552
$$281$$ − 1.86747e7i − 0.841654i −0.907141 0.420827i $$-0.861740\pi$$
0.907141 0.420827i $$-0.138260\pi$$
$$282$$ 0 0
$$283$$ −1.46255e7 −0.645284 −0.322642 0.946521i $$-0.604571\pi$$
−0.322642 + 0.946521i $$0.604571\pi$$
$$284$$ 4.81081e6i 0.210021i
$$285$$ 0 0
$$286$$ −3.63394e7 −1.55339
$$287$$ 487743.i 0.0206322i
$$288$$ 0 0
$$289$$ 2.02730e7 0.839893
$$290$$ 4.37804e7i 1.79509i
$$291$$ 0 0
$$292$$ −1.06141e7 −0.426318
$$293$$ − 4.78429e7i − 1.90202i −0.309162 0.951010i $$-0.600048\pi$$
0.309162 0.951010i $$-0.399952\pi$$
$$294$$ 0 0
$$295$$ 3.18960e7 1.24243
$$296$$ − 3.08036e6i − 0.118775i
$$297$$ 0 0
$$298$$ −2.67764e7 −1.01182
$$299$$ 4.75539e7i 1.77899i
$$300$$ 0 0
$$301$$ −128666. −0.00471807
$$302$$ − 2.11413e7i − 0.767558i
$$303$$ 0 0
$$304$$ −288046. −0.0102528
$$305$$ − 3.36525e6i − 0.118609i
$$306$$ 0 0
$$307$$ 4.24782e7 1.46808 0.734042 0.679104i $$-0.237631\pi$$
0.734042 + 0.679104i $$0.237631\pi$$
$$308$$ 303728.i 0.0103952i
$$309$$ 0 0
$$310$$ −2.91463e7 −0.978360
$$311$$ − 3.09996e7i − 1.03056i −0.857021 0.515282i $$-0.827687\pi$$
0.857021 0.515282i $$-0.172313\pi$$
$$312$$ 0 0
$$313$$ −2.97828e7 −0.971254 −0.485627 0.874166i $$-0.661409\pi$$
−0.485627 + 0.874166i $$0.661409\pi$$
$$314$$ − 1.81394e7i − 0.585913i
$$315$$ 0 0
$$316$$ 2.86756e7 0.908764
$$317$$ 2.17911e6i 0.0684072i 0.999415 + 0.0342036i $$0.0108895\pi$$
−0.999415 + 0.0342036i $$0.989111\pi$$
$$318$$ 0 0
$$319$$ 8.39504e7 2.58613
$$320$$ − 6.83306e6i − 0.208528i
$$321$$ 0 0
$$322$$ 397459. 0.0119049
$$323$$ − 552987.i − 0.0164099i
$$324$$ 0 0
$$325$$ −7.91201e7 −2.30482
$$326$$ 5.45557e6i 0.157466i
$$327$$ 0 0
$$328$$ −2.10409e7 −0.596270
$$329$$ − 325901.i − 0.00915162i
$$330$$ 0 0
$$331$$ 1.33671e7 0.368598 0.184299 0.982870i $$-0.440999\pi$$
0.184299 + 0.982870i $$0.440999\pi$$
$$332$$ − 3.02429e7i − 0.826435i
$$333$$ 0 0
$$334$$ −5.85480e6 −0.157135
$$335$$ − 9.89816e7i − 2.63281i
$$336$$ 0 0
$$337$$ 4.45488e7 1.16398 0.581991 0.813195i $$-0.302274\pi$$
0.581991 + 0.813195i $$0.302274\pi$$
$$338$$ − 1.83216e7i − 0.474475i
$$339$$ 0 0
$$340$$ 1.31180e7 0.333757
$$341$$ 5.58891e7i 1.40950i
$$342$$ 0 0
$$343$$ −987272. −0.0244655
$$344$$ − 5.55057e6i − 0.136352i
$$345$$ 0 0
$$346$$ 3.66902e6 0.0885771
$$347$$ − 2.08986e7i − 0.500182i −0.968222 0.250091i $$-0.919540\pi$$
0.968222 0.250091i $$-0.0804605\pi$$
$$348$$ 0 0
$$349$$ 6.45154e7 1.51770 0.758852 0.651263i $$-0.225761\pi$$
0.758852 + 0.651263i $$0.225761\pi$$
$$350$$ 661293.i 0.0154237i
$$351$$ 0 0
$$352$$ −1.31026e7 −0.300421
$$353$$ − 4.20921e6i − 0.0956923i −0.998855 0.0478461i $$-0.984764\pi$$
0.998855 0.0478461i $$-0.0152357\pi$$
$$354$$ 0 0
$$355$$ 3.13497e7 0.700727
$$356$$ − 2.52897e7i − 0.560523i
$$357$$ 0 0
$$358$$ 2.63151e7 0.573529
$$359$$ − 5.61320e7i − 1.21319i −0.795012 0.606593i $$-0.792536\pi$$
0.795012 0.606593i $$-0.207464\pi$$
$$360$$ 0 0
$$361$$ −4.69668e7 −0.998318
$$362$$ − 1.94437e7i − 0.409877i
$$363$$ 0 0
$$364$$ −381347. −0.00790709
$$365$$ 6.91668e7i 1.42239i
$$366$$ 0 0
$$367$$ 3.35103e7 0.677922 0.338961 0.940800i $$-0.389925\pi$$
0.338961 + 0.940800i $$0.389925\pi$$
$$368$$ 1.71461e7i 0.344051i
$$369$$ 0 0
$$370$$ −2.00732e7 −0.396289
$$371$$ 581827.i 0.0113939i
$$372$$ 0 0
$$373$$ 2.22777e7 0.429283 0.214642 0.976693i $$-0.431142\pi$$
0.214642 + 0.976693i $$0.431142\pi$$
$$374$$ − 2.51542e7i − 0.480835i
$$375$$ 0 0
$$376$$ 1.40592e7 0.264482
$$377$$ 1.05404e8i 1.96714i
$$378$$ 0 0
$$379$$ 7.72205e7 1.41845 0.709226 0.704981i $$-0.249044\pi$$
0.709226 + 0.704981i $$0.249044\pi$$
$$380$$ 1.87706e6i 0.0342080i
$$381$$ 0 0
$$382$$ 6.64733e7 1.19250
$$383$$ − 3.32594e7i − 0.591996i −0.955189 0.295998i $$-0.904348\pi$$
0.955189 0.295998i $$-0.0956521\pi$$
$$384$$ 0 0
$$385$$ 1.97925e6 0.0346831
$$386$$ − 3.93490e7i − 0.684183i
$$387$$ 0 0
$$388$$ −4.45656e7 −0.762965
$$389$$ − 1.50959e7i − 0.256454i −0.991745 0.128227i $$-0.959071\pi$$
0.991745 0.128227i $$-0.0409286\pi$$
$$390$$ 0 0
$$391$$ −3.29169e7 −0.550666
$$392$$ − 2.12936e7i − 0.353500i
$$393$$ 0 0
$$394$$ 2.31730e6 0.0378873
$$395$$ − 1.86865e8i − 3.03205i
$$396$$ 0 0
$$397$$ −8.91350e7 −1.42455 −0.712273 0.701902i $$-0.752334\pi$$
−0.712273 + 0.701902i $$0.752334\pi$$
$$398$$ 1.83761e7i 0.291478i
$$399$$ 0 0
$$400$$ −2.85277e7 −0.445746
$$401$$ − 1.12043e8i − 1.73761i −0.495158 0.868803i $$-0.664890\pi$$
0.495158 0.868803i $$-0.335110\pi$$
$$402$$ 0 0
$$403$$ −7.01718e7 −1.07213
$$404$$ − 4.20966e6i − 0.0638414i
$$405$$ 0 0
$$406$$ 880978. 0.0131640
$$407$$ 3.84911e7i 0.570922i
$$408$$ 0 0
$$409$$ −6.67941e7 −0.976266 −0.488133 0.872769i $$-0.662322\pi$$
−0.488133 + 0.872769i $$0.662322\pi$$
$$410$$ 1.37113e8i 1.98943i
$$411$$ 0 0
$$412$$ −6.19075e7 −0.885221
$$413$$ − 641833.i − 0.00911113i
$$414$$ 0 0
$$415$$ −1.97078e8 −2.75736
$$416$$ − 1.64511e7i − 0.228515i
$$417$$ 0 0
$$418$$ 3.59933e6 0.0492824
$$419$$ 4.73571e7i 0.643789i 0.946776 + 0.321894i $$0.104320\pi$$
−0.946776 + 0.321894i $$0.895680\pi$$
$$420$$ 0 0
$$421$$ 6.61569e7 0.886602 0.443301 0.896373i $$-0.353807\pi$$
0.443301 + 0.896373i $$0.353807\pi$$
$$422$$ 2.82552e7i 0.375977i
$$423$$ 0 0
$$424$$ −2.50996e7 −0.329283
$$425$$ − 5.47671e7i − 0.713432i
$$426$$ 0 0
$$427$$ −67717.9 −0.000869801 0
$$428$$ 1.99013e7i 0.253834i
$$429$$ 0 0
$$430$$ −3.61704e7 −0.454933
$$431$$ − 9.06666e7i − 1.13244i −0.824254 0.566221i $$-0.808405\pi$$
0.824254 0.566221i $$-0.191595\pi$$
$$432$$ 0 0
$$433$$ −699414. −0.00861530 −0.00430765 0.999991i $$-0.501371\pi$$
−0.00430765 + 0.999991i $$0.501371\pi$$
$$434$$ 586502.i 0.00717464i
$$435$$ 0 0
$$436$$ 4.30649e7 0.519594
$$437$$ − 4.71009e6i − 0.0564397i
$$438$$ 0 0
$$439$$ −2.57296e7 −0.304116 −0.152058 0.988372i $$-0.548590\pi$$
−0.152058 + 0.988372i $$0.548590\pi$$
$$440$$ 8.53835e7i 1.00234i
$$441$$ 0 0
$$442$$ 3.15825e7 0.365746
$$443$$ 165578.i 0.00190455i 1.00000 0.000952273i $$0.000303118\pi$$
−1.00000 0.000952273i $$0.999697\pi$$
$$444$$ 0 0
$$445$$ −1.64801e8 −1.87016
$$446$$ − 3.21832e7i − 0.362765i
$$447$$ 0 0
$$448$$ −137500. −0.00152921
$$449$$ − 4.30456e7i − 0.475542i −0.971321 0.237771i $$-0.923583\pi$$
0.971321 0.237771i $$-0.0764169\pi$$
$$450$$ 0 0
$$451$$ 2.62920e8 2.86611
$$452$$ − 3.96584e7i − 0.429457i
$$453$$ 0 0
$$454$$ 6.56556e7 0.701624
$$455$$ 2.48506e6i 0.0263816i
$$456$$ 0 0
$$457$$ −1.35491e8 −1.41959 −0.709793 0.704411i $$-0.751211\pi$$
−0.709793 + 0.704411i $$0.751211\pi$$
$$458$$ 1.95552e7i 0.203547i
$$459$$ 0 0
$$460$$ 1.11733e8 1.14791
$$461$$ 3.20349e7i 0.326980i 0.986545 + 0.163490i $$0.0522751\pi$$
−0.986545 + 0.163490i $$0.947725\pi$$
$$462$$ 0 0
$$463$$ 1.29333e8 1.30306 0.651532 0.758621i $$-0.274127\pi$$
0.651532 + 0.758621i $$0.274127\pi$$
$$464$$ 3.80048e7i 0.380439i
$$465$$ 0 0
$$466$$ 7.22602e7 0.714071
$$467$$ 8.49294e7i 0.833887i 0.908932 + 0.416944i $$0.136899\pi$$
−0.908932 + 0.416944i $$0.863101\pi$$
$$468$$ 0 0
$$469$$ −1.99178e6 −0.0193073
$$470$$ − 9.16168e7i − 0.882432i
$$471$$ 0 0
$$472$$ 2.76883e7 0.263312
$$473$$ 6.93579e7i 0.655410i
$$474$$ 0 0
$$475$$ 7.83664e6 0.0731222
$$476$$ − 263969.i − 0.00244755i
$$477$$ 0 0
$$478$$ 7.31043e7 0.669360
$$479$$ 1.15719e8i 1.05292i 0.850199 + 0.526461i $$0.176482\pi$$
−0.850199 + 0.526461i $$0.823518\pi$$
$$480$$ 0 0
$$481$$ −4.83277e7 −0.434271
$$482$$ − 4.11837e7i − 0.367777i
$$483$$ 0 0
$$484$$ 1.07036e8 0.944046
$$485$$ 2.90412e8i 2.54560i
$$486$$ 0 0
$$487$$ 4.52839e7 0.392064 0.196032 0.980597i $$-0.437194\pi$$
0.196032 + 0.980597i $$0.437194\pi$$
$$488$$ − 2.92131e6i − 0.0251373i
$$489$$ 0 0
$$490$$ −1.38760e8 −1.17944
$$491$$ 2.04312e8i 1.72603i 0.505175 + 0.863017i $$0.331428\pi$$
−0.505175 + 0.863017i $$0.668572\pi$$
$$492$$ 0 0
$$493$$ −7.29611e7 −0.608906
$$494$$ 4.51915e6i 0.0374866i
$$495$$ 0 0
$$496$$ −2.53013e7 −0.207347
$$497$$ − 630841.i − 0.00513867i
$$498$$ 0 0
$$499$$ −2.92990e7 −0.235804 −0.117902 0.993025i $$-0.537617\pi$$
−0.117902 + 0.993025i $$0.537617\pi$$
$$500$$ 8.16373e7i 0.653098i
$$501$$ 0 0
$$502$$ 9.72564e7 0.768789
$$503$$ − 1.64834e8i − 1.29522i −0.761973 0.647609i $$-0.775769\pi$$
0.761973 0.647609i $$-0.224231\pi$$
$$504$$ 0 0
$$505$$ −2.74323e7 −0.213004
$$506$$ − 2.14252e8i − 1.65376i
$$507$$ 0 0
$$508$$ 5.28949e7 0.403481
$$509$$ − 1.67339e8i − 1.26895i −0.772944 0.634474i $$-0.781217\pi$$
0.772944 0.634474i $$-0.218783\pi$$
$$510$$ 0 0
$$511$$ 1.39182e6 0.0104309
$$512$$ − 5.93164e6i − 0.0441942i
$$513$$ 0 0
$$514$$ 4.53177e7 0.333717
$$515$$ 4.03421e8i 2.95350i
$$516$$ 0 0
$$517$$ −1.75678e8 −1.27130
$$518$$ 403927.i 0.00290612i
$$519$$ 0 0
$$520$$ −1.07204e8 −0.762430
$$521$$ − 6.01419e7i − 0.425270i −0.977132 0.212635i $$-0.931796\pi$$
0.977132 0.212635i $$-0.0682045\pi$$
$$522$$ 0 0
$$523$$ 1.09251e8 0.763698 0.381849 0.924225i $$-0.375287\pi$$
0.381849 + 0.924225i $$0.375287\pi$$
$$524$$ 8.51602e7i 0.591893i
$$525$$ 0 0
$$526$$ −9.51535e7 −0.653834
$$527$$ − 4.85731e7i − 0.331867i
$$528$$ 0 0
$$529$$ −1.32335e8 −0.893940
$$530$$ 1.63562e8i 1.09864i
$$531$$ 0 0
$$532$$ 37771.5 0.000250859 0
$$533$$ 3.30110e8i 2.18011i
$$534$$ 0 0
$$535$$ 1.29687e8 0.846907
$$536$$ − 8.59239e7i − 0.557981i
$$537$$ 0 0
$$538$$ 1.88664e7 0.121155
$$539$$ 2.66077e8i 1.69918i
$$540$$ 0 0
$$541$$ 4.52159e7 0.285561 0.142781 0.989754i $$-0.454396\pi$$
0.142781 + 0.989754i $$0.454396\pi$$
$$542$$ 8.31827e7i 0.522439i
$$543$$ 0 0
$$544$$ 1.13875e7 0.0707343
$$545$$ − 2.80633e8i − 1.73360i
$$546$$ 0 0
$$547$$ 5.11451e7 0.312494 0.156247 0.987718i $$-0.450060\pi$$
0.156247 + 0.987718i $$0.450060\pi$$
$$548$$ − 8.35248e7i − 0.507544i
$$549$$ 0 0
$$550$$ 3.56473e8 2.14258
$$551$$ − 1.04400e7i − 0.0624089i
$$552$$ 0 0
$$553$$ −3.76022e6 −0.0222351
$$554$$ 6.17732e7i 0.363304i
$$555$$ 0 0
$$556$$ 2.24423e7 0.130570
$$557$$ − 2.43306e7i − 0.140795i −0.997519 0.0703974i $$-0.977573\pi$$
0.997519 0.0703974i $$-0.0224267\pi$$
$$558$$ 0 0
$$559$$ −8.70827e7 −0.498536
$$560$$ 896018.i 0.00510214i
$$561$$ 0 0
$$562$$ −1.05640e8 −0.595139
$$563$$ 1.44398e8i 0.809166i 0.914501 + 0.404583i $$0.132583\pi$$
−0.914501 + 0.404583i $$0.867417\pi$$
$$564$$ 0 0
$$565$$ −2.58434e8 −1.43286
$$566$$ 8.27342e7i 0.456284i
$$567$$ 0 0
$$568$$ 2.72141e7 0.148508
$$569$$ − 8.31553e7i − 0.451391i −0.974198 0.225696i $$-0.927534\pi$$
0.974198 0.225696i $$-0.0724655\pi$$
$$570$$ 0 0
$$571$$ −2.26080e8 −1.21438 −0.607188 0.794558i $$-0.707703\pi$$
−0.607188 + 0.794558i $$0.707703\pi$$
$$572$$ 2.05567e8i 1.09841i
$$573$$ 0 0
$$574$$ 2.75909e6 0.0145892
$$575$$ − 4.66481e8i − 2.45375i
$$576$$ 0 0
$$577$$ 4.11006e7 0.213954 0.106977 0.994261i $$-0.465883\pi$$
0.106977 + 0.994261i $$0.465883\pi$$
$$578$$ − 1.14681e8i − 0.593894i
$$579$$ 0 0
$$580$$ 2.47659e8 1.26932
$$581$$ 3.96574e6i 0.0202207i
$$582$$ 0 0
$$583$$ 3.13636e8 1.58278
$$584$$ 6.00423e7i 0.301452i
$$585$$ 0 0
$$586$$ −2.70640e8 −1.34493
$$587$$ − 1.14693e8i − 0.567052i −0.958965 0.283526i $$-0.908496\pi$$
0.958965 0.283526i $$-0.0915042\pi$$
$$588$$ 0 0
$$589$$ 6.95034e6 0.0340142
$$590$$ − 1.80431e8i − 0.878528i
$$591$$ 0 0
$$592$$ −1.74251e7 −0.0839868
$$593$$ 2.90487e8i 1.39304i 0.717539 + 0.696518i $$0.245269\pi$$
−0.717539 + 0.696518i $$0.754731\pi$$
$$594$$ 0 0
$$595$$ −1.72016e6 −0.00816616
$$596$$ 1.51470e8i 0.715464i
$$597$$ 0 0
$$598$$ 2.69005e8 1.25793
$$599$$ 3.57152e8i 1.66178i 0.556439 + 0.830888i $$0.312167\pi$$
−0.556439 + 0.830888i $$0.687833\pi$$
$$600$$ 0 0
$$601$$ −8.40094e7 −0.386994 −0.193497 0.981101i $$-0.561983\pi$$
−0.193497 + 0.981101i $$0.561983\pi$$
$$602$$ 727845.i 0.00333618i
$$603$$ 0 0
$$604$$ −1.19593e8 −0.542746
$$605$$ − 6.97500e8i − 3.14977i
$$606$$ 0 0
$$607$$ 5.16287e7 0.230847 0.115424 0.993316i $$-0.463177\pi$$
0.115424 + 0.993316i $$0.463177\pi$$
$$608$$ 1.62944e6i 0.00724981i
$$609$$ 0 0
$$610$$ −1.90368e7 −0.0838694
$$611$$ − 2.20574e8i − 0.967009i
$$612$$ 0 0
$$613$$ 4.44326e7 0.192895 0.0964473 0.995338i $$-0.469252\pi$$
0.0964473 + 0.995338i $$0.469252\pi$$
$$614$$ − 2.40293e8i − 1.03809i
$$615$$ 0 0
$$616$$ 1.71814e6 0.00735051
$$617$$ − 1.65408e8i − 0.704208i −0.935961 0.352104i $$-0.885466\pi$$
0.935961 0.352104i $$-0.114534\pi$$
$$618$$ 0 0
$$619$$ 2.68162e8 1.13064 0.565320 0.824871i $$-0.308753\pi$$
0.565320 + 0.824871i $$0.308753\pi$$
$$620$$ 1.64876e8i 0.691805i
$$621$$ 0 0
$$622$$ −1.75360e8 −0.728718
$$623$$ 3.31623e6i 0.0137145i
$$624$$ 0 0
$$625$$ 9.66915e7 0.396049
$$626$$ 1.68477e8i 0.686780i
$$627$$ 0 0
$$628$$ −1.02612e8 −0.414303
$$629$$ − 3.34525e7i − 0.134424i
$$630$$ 0 0
$$631$$ 1.13998e8 0.453741 0.226871 0.973925i $$-0.427151\pi$$
0.226871 + 0.973925i $$0.427151\pi$$
$$632$$ − 1.62214e8i − 0.642593i
$$633$$ 0 0
$$634$$ 1.23269e7 0.0483712
$$635$$ − 3.44691e8i − 1.34620i
$$636$$ 0 0
$$637$$ −3.34074e8 −1.29248
$$638$$ − 4.74895e8i − 1.82867i
$$639$$ 0 0
$$640$$ −3.86536e7 −0.147452
$$641$$ 3.65639e7i 0.138828i 0.997588 + 0.0694142i $$0.0221130\pi$$
−0.997588 + 0.0694142i $$0.977887\pi$$
$$642$$ 0 0
$$643$$ −2.82268e8 −1.06177 −0.530883 0.847445i $$-0.678140\pi$$
−0.530883 + 0.847445i $$0.678140\pi$$
$$644$$ − 2.24837e6i − 0.00841803i
$$645$$ 0 0
$$646$$ −3.12817e6 −0.0116036
$$647$$ 1.45187e8i 0.536062i 0.963410 + 0.268031i $$0.0863729\pi$$
−0.963410 + 0.268031i $$0.913627\pi$$
$$648$$ 0 0
$$649$$ −3.45983e8 −1.26567
$$650$$ 4.47571e8i 1.62975i
$$651$$ 0 0
$$652$$ 3.08614e7 0.111345
$$653$$ 9.91905e7i 0.356230i 0.984010 + 0.178115i $$0.0569999\pi$$
−0.984010 + 0.178115i $$0.943000\pi$$
$$654$$ 0 0
$$655$$ 5.54948e8 1.97482
$$656$$ 1.19025e8i 0.421627i
$$657$$ 0 0
$$658$$ −1.84357e6 −0.00647117
$$659$$ 1.69578e8i 0.592535i 0.955105 + 0.296268i $$0.0957420\pi$$
−0.955105 + 0.296268i $$0.904258\pi$$
$$660$$ 0 0
$$661$$ 5.38050e8 1.86302 0.931511 0.363713i $$-0.118491\pi$$
0.931511 + 0.363713i $$0.118491\pi$$
$$662$$ − 7.56157e7i − 0.260638i
$$663$$ 0 0
$$664$$ −1.71080e8 −0.584378
$$665$$ − 246138.i 0 0.000836978i
$$666$$ 0 0
$$667$$ −6.21449e8 −2.09425
$$668$$ 3.31198e7i 0.111111i
$$669$$ 0 0
$$670$$ −5.59924e8 −1.86168
$$671$$ 3.65036e7i 0.120828i
$$672$$ 0 0
$$673$$ −3.85365e7 −0.126423 −0.0632116 0.998000i $$-0.520134\pi$$
−0.0632116 + 0.998000i $$0.520134\pi$$
$$674$$ − 2.52006e8i − 0.823060i
$$675$$ 0 0
$$676$$ −1.03643e8 −0.335504
$$677$$ − 1.90802e8i − 0.614916i −0.951562 0.307458i $$-0.900522\pi$$
0.951562 0.307458i $$-0.0994784\pi$$
$$678$$ 0 0
$$679$$ 5.84388e6 0.0186677
$$680$$ − 7.42066e7i − 0.236002i
$$681$$ 0 0
$$682$$ 3.16156e8 0.996664
$$683$$ − 5.45896e8i − 1.71336i −0.515850 0.856679i $$-0.672524\pi$$
0.515850 0.856679i $$-0.327476\pi$$
$$684$$ 0 0
$$685$$ −5.44291e8 −1.69340
$$686$$ 5.58486e6i 0.0172997i
$$687$$ 0 0
$$688$$ −3.13988e7 −0.0964156
$$689$$ 3.93788e8i 1.20394i
$$690$$ 0 0
$$691$$ −1.70447e8 −0.516601 −0.258301 0.966065i $$-0.583162\pi$$
−0.258301 + 0.966065i $$0.583162\pi$$
$$692$$ − 2.07551e7i − 0.0626334i
$$693$$ 0 0
$$694$$ −1.18220e8 −0.353682
$$695$$ − 1.46245e8i − 0.435640i
$$696$$ 0 0
$$697$$ −2.28503e8 −0.674828
$$698$$ − 3.64954e8i − 1.07318i
$$699$$ 0 0
$$700$$ 3.74084e6 0.0109062
$$701$$ − 8.02305e7i − 0.232909i −0.993196 0.116454i $$-0.962847\pi$$
0.993196 0.116454i $$-0.0371529\pi$$
$$702$$ 0 0
$$703$$ 4.78673e6 0.0137776
$$704$$ 7.41197e7i 0.212430i
$$705$$ 0 0
$$706$$ −2.38109e7 −0.0676647
$$707$$ 552011.i 0.00156203i
$$708$$ 0 0
$$709$$ 4.01132e8 1.12551 0.562754 0.826625i $$-0.309742\pi$$
0.562754 + 0.826625i $$0.309742\pi$$
$$710$$ − 1.77341e8i − 0.495489i
$$711$$ 0 0
$$712$$ −1.43060e8 −0.396349
$$713$$ − 4.13723e8i − 1.14141i
$$714$$ 0 0
$$715$$ 1.33958e9 3.66480
$$716$$ − 1.48860e8i − 0.405546i
$$717$$ 0 0
$$718$$ −3.17531e8 −0.857852
$$719$$ − 1.02302e8i − 0.275232i −0.990486 0.137616i $$-0.956056\pi$$
0.990486 0.137616i $$-0.0439439\pi$$
$$720$$ 0 0
$$721$$ 8.11792e6 0.0216590
$$722$$ 2.65684e8i 0.705917i
$$723$$ 0 0
$$724$$ −1.09990e8 −0.289827
$$725$$ − 1.03397e9i − 2.71327i
$$726$$ 0 0
$$727$$ −5.87675e8 −1.52944 −0.764722 0.644360i $$-0.777124\pi$$
−0.764722 + 0.644360i $$0.777124\pi$$
$$728$$ 2.15723e6i 0.00559115i
$$729$$ 0 0
$$730$$ 3.91266e8 1.00578
$$731$$ − 6.02788e7i − 0.154317i
$$732$$ 0 0
$$733$$ −1.12015e8 −0.284424 −0.142212 0.989836i $$-0.545421\pi$$
−0.142212 + 0.989836i $$0.545421\pi$$
$$734$$ − 1.89563e8i − 0.479363i
$$735$$ 0 0
$$736$$ 9.69933e7 0.243281
$$737$$ 1.07367e9i 2.68207i
$$738$$ 0 0
$$739$$ 6.86968e8 1.70217 0.851086 0.525027i $$-0.175945\pi$$
0.851086 + 0.525027i $$0.175945\pi$$
$$740$$ 1.13551e8i 0.280218i
$$741$$ 0 0
$$742$$ 3.29131e6 0.00805670
$$743$$ 5.57615e8i 1.35946i 0.733460 + 0.679732i $$0.237904\pi$$
−0.733460 + 0.679732i $$0.762096\pi$$
$$744$$ 0 0
$$745$$ 9.87057e8 2.38711
$$746$$ − 1.26022e8i − 0.303549i
$$747$$ 0 0
$$748$$ −1.42294e8 −0.340001
$$749$$ − 2.60966e6i − 0.00621066i
$$750$$ 0 0
$$751$$ 4.04996e8 0.956160 0.478080 0.878316i $$-0.341333\pi$$
0.478080 + 0.878316i $$0.341333\pi$$
$$752$$ − 7.95306e7i − 0.187017i
$$753$$ 0 0
$$754$$ 5.96257e8 1.39098
$$755$$ 7.79332e8i 1.81085i
$$756$$ 0 0
$$757$$ 6.51769e8 1.50247 0.751235 0.660035i $$-0.229458\pi$$
0.751235 + 0.660035i $$0.229458\pi$$
$$758$$ − 4.36825e8i − 1.00300i
$$759$$ 0 0
$$760$$ 1.06182e7 0.0241887
$$761$$ 3.11208e8i 0.706149i 0.935595 + 0.353074i $$0.114864\pi$$
−0.935595 + 0.353074i $$0.885136\pi$$
$$762$$ 0 0
$$763$$ −5.64709e6 −0.0127131
$$764$$ − 3.76030e8i − 0.843222i
$$765$$ 0 0
$$766$$ −1.88144e8 −0.418604
$$767$$ − 4.34401e8i − 0.962730i
$$768$$ 0 0
$$769$$ −4.37211e8 −0.961417 −0.480708 0.876880i $$-0.659620\pi$$
−0.480708 + 0.876880i $$0.659620\pi$$
$$770$$ − 1.11963e7i − 0.0245247i
$$771$$ 0 0
$$772$$ −2.22592e8 −0.483790
$$773$$ − 5.41906e7i − 0.117324i −0.998278 0.0586619i $$-0.981317\pi$$
0.998278 0.0586619i $$-0.0186834\pi$$
$$774$$ 0 0
$$775$$ 6.88352e8 1.47879
$$776$$ 2.52101e8i 0.539498i
$$777$$ 0 0
$$778$$ −8.53951e7 −0.181340
$$779$$ − 3.26966e7i − 0.0691656i
$$780$$ 0 0
$$781$$ −3.40057e8 −0.713837
$$782$$ 1.86206e8i 0.389380i
$$783$$ 0 0
$$784$$ −1.20455e8 −0.249963
$$785$$ 6.68671e8i 1.38230i
$$786$$ 0 0
$$787$$ −4.02320e7 −0.0825367 −0.0412684 0.999148i $$-0.513140\pi$$
−0.0412684 + 0.999148i $$0.513140\pi$$
$$788$$ − 1.31086e7i − 0.0267904i
$$789$$ 0 0
$$790$$ −1.05707e9 −2.14398
$$791$$ 5.20039e6i 0.0105077i
$$792$$ 0 0
$$793$$ −4.58323e7 −0.0919078
$$794$$ 5.04223e8i 1.00731i
$$795$$ 0 0
$$796$$ 1.03951e8 0.206106
$$797$$ 8.94029e7i 0.176594i 0.996094 + 0.0882972i $$0.0281425\pi$$
−0.996094 + 0.0882972i $$0.971857\pi$$
$$798$$ 0 0
$$799$$ 1.52682e8 0.299327
$$800$$ 1.61377e8i 0.315190i
$$801$$ 0 0
$$802$$ −6.33810e8 −1.22867
$$803$$ − 7.50266e8i − 1.44900i
$$804$$ 0 0
$$805$$ −1.46515e7 −0.0280864
$$806$$ 3.96952e8i 0.758111i
$$807$$ 0 0
$$808$$ −2.38134e7 −0.0451427
$$809$$ − 7.82985e8i − 1.47879i −0.673270 0.739397i $$-0.735111\pi$$
0.673270 0.739397i $$-0.264889\pi$$
$$810$$ 0 0
$$811$$ −6.76238e6 −0.0126776 −0.00633880 0.999980i $$-0.502018\pi$$
−0.00633880 + 0.999980i $$0.502018\pi$$
$$812$$ − 4.98357e6i − 0.00930834i
$$813$$ 0 0
$$814$$ 2.17738e8 0.403703
$$815$$ − 2.01109e8i − 0.371499i
$$816$$ 0 0
$$817$$ 8.62532e6 0.0158165
$$818$$ 3.77844e8i 0.690324i
$$819$$ 0 0
$$820$$ 7.75631e8 1.40674
$$821$$ 2.93295e7i 0.0530000i 0.999649 + 0.0265000i $$0.00843619\pi$$
−0.999649 + 0.0265000i $$0.991564\pi$$
$$822$$ 0 0
$$823$$ −8.90952e8 −1.59829 −0.799144 0.601140i $$-0.794713\pi$$
−0.799144 + 0.601140i $$0.794713\pi$$
$$824$$ 3.50202e8i 0.625946i
$$825$$ 0 0
$$826$$ −3.63076e6 −0.00644254
$$827$$ − 4.36886e8i − 0.772417i −0.922411 0.386209i $$-0.873784\pi$$
0.922411 0.386209i $$-0.126216\pi$$
$$828$$ 0 0
$$829$$ 2.08489e8 0.365948 0.182974 0.983118i $$-0.441428\pi$$
0.182974 + 0.983118i $$0.441428\pi$$
$$830$$ 1.11484e9i 1.94975i
$$831$$ 0 0
$$832$$ −9.30614e7 −0.161584
$$833$$ − 2.31247e8i − 0.400074i
$$834$$ 0 0
$$835$$ 2.15825e8 0.370718
$$836$$ − 2.03609e7i − 0.0348480i
$$837$$ 0 0
$$838$$ 2.67892e8 0.455227
$$839$$ 1.11428e9i 1.88672i 0.331767 + 0.943361i $$0.392355\pi$$
−0.331767 + 0.943361i $$0.607645\pi$$
$$840$$ 0 0
$$841$$ −7.82634e8 −1.31574
$$842$$ − 3.74240e8i − 0.626922i
$$843$$ 0 0
$$844$$ 1.59836e8 0.265856
$$845$$ 6.75389e8i 1.11940i
$$846$$ 0 0
$$847$$ −1.40356e7 −0.0230983
$$848$$ 1.41985e8i 0.232839i
$$849$$ 0 0
$$850$$ −3.09809e8 −0.504473
$$851$$ − 2.84933e8i − 0.462332i
$$852$$ 0 0
$$853$$ −7.03808e7 −0.113399 −0.0566993 0.998391i $$-0.518058\pi$$
−0.0566993 + 0.998391i $$0.518058\pi$$
$$854$$ 383071.i 0 0.000615043i
$$855$$ 0 0
$$856$$ 1.12579e8 0.179488
$$857$$ 1.58916e8i 0.252479i 0.992000 + 0.126240i $$0.0402908\pi$$
−0.992000 + 0.126240i $$0.959709\pi$$
$$858$$ 0 0
$$859$$ 3.14241e8 0.495774 0.247887 0.968789i $$-0.420264\pi$$
0.247887 + 0.968789i $$0.420264\pi$$
$$860$$ 2.04611e8i 0.321686i
$$861$$ 0 0
$$862$$ −5.12888e8 −0.800757
$$863$$ 3.90351e8i 0.607328i 0.952779 + 0.303664i $$0.0982101\pi$$
−0.952779 + 0.303664i $$0.901790\pi$$
$$864$$ 0 0
$$865$$ −1.35251e8 −0.208974
$$866$$ 3.95648e6i 0.00609194i
$$867$$ 0 0
$$868$$ 3.31776e6 0.00507324
$$869$$ 2.02696e9i 3.08878i
$$870$$ 0 0
$$871$$ −1.34806e9 −2.04011
$$872$$ − 2.43612e8i − 0.367408i
$$873$$ 0 0
$$874$$ −2.66443e7 −0.0399089
$$875$$ − 1.07051e7i − 0.0159796i
$$876$$ 0 0
$$877$$ −1.15579e9 −1.71349 −0.856745 0.515741i $$-0.827517\pi$$
−0.856745 + 0.515741i $$0.827517\pi$$
$$878$$ 1.45549e8i 0.215043i
$$879$$ 0 0
$$880$$ 4.83002e8 0.708763
$$881$$ 7.82859e8i 1.14487i 0.819951 + 0.572434i $$0.194001\pi$$
−0.819951 + 0.572434i $$0.805999\pi$$
$$882$$ 0 0
$$883$$ 1.01047e9 1.46772 0.733860 0.679300i $$-0.237717\pi$$
0.733860 + 0.679300i $$0.237717\pi$$
$$884$$ − 1.78658e8i − 0.258622i
$$885$$ 0 0
$$886$$ 936650. 0.00134672
$$887$$ 3.28343e8i 0.470498i 0.971935 + 0.235249i $$0.0755905\pi$$
−0.971935 + 0.235249i $$0.924410\pi$$
$$888$$ 0 0
$$889$$ −6.93610e6 −0.00987211
$$890$$ 9.32253e8i 1.32240i
$$891$$ 0 0
$$892$$ −1.82056e8 −0.256513
$$893$$ 2.18473e7i 0.0306791i
$$894$$ 0 0
$$895$$ −9.70051e8 −1.35309
$$896$$ 777815.i 0.00108131i
$$897$$ 0 0
$$898$$ −2.43502e8 −0.336259
$$899$$ − 9.17028e8i − 1.26213i
$$900$$ 0 0
$$901$$ −2.72580e8 −0.372666
$$902$$ − 1.48730e9i − 2.02665i
$$903$$ 0 0
$$904$$ −2.24342e8 −0.303672
$$905$$ 7.16753e8i 0.966994i
$$906$$ 0 0
$$907$$ 2.15438e8 0.288736 0.144368 0.989524i $$-0.453885\pi$$
0.144368 + 0.989524i $$0.453885\pi$$
$$908$$ − 3.71404e8i − 0.496123i
$$909$$ 0 0
$$910$$ 1.40576e7 0.0186546
$$911$$ − 7.52234e8i − 0.994942i −0.867481 0.497471i $$-0.834262\pi$$
0.867481 0.497471i $$-0.165738\pi$$
$$912$$ 0 0
$$913$$ 2.13775e9 2.80895
$$914$$ 7.66452e8i 1.00380i
$$915$$ 0 0
$$916$$ 1.10621e8 0.143930
$$917$$ − 1.11670e7i − 0.0144820i
$$918$$ 0 0
$$919$$ 6.81968e8 0.878653 0.439327 0.898327i $$-0.355217\pi$$
0.439327 + 0.898327i $$0.355217\pi$$
$$920$$ − 6.32058e8i − 0.811696i
$$921$$ 0 0
$$922$$ 1.81217e8 0.231210
$$923$$ − 4.26961e8i − 0.542979i
$$924$$ 0 0
$$925$$ 4.74072e8 0.598989
$$926$$ − 7.31617e8i − 0.921406i
$$927$$ 0 0
$$928$$ 2.14988e8 0.269011
$$929$$ − 1.11639e8i − 0.139242i −0.997574 0.0696208i $$-0.977821\pi$$
0.997574 0.0696208i $$-0.0221789\pi$$
$$930$$ 0 0
$$931$$ 3.30892e7 0.0410050
$$932$$ − 4.08765e8i − 0.504924i
$$933$$ 0 0
$$934$$ 4.80433e8 0.589647
$$935$$ 9.27259e8i 1.13440i
$$936$$ 0 0
$$937$$ 7.66443e8 0.931668 0.465834 0.884872i $$-0.345754\pi$$
0.465834 + 0.884872i $$0.345754\pi$$
$$938$$ 1.12672e7i 0.0136523i
$$939$$ 0 0
$$940$$ −5.18263e8 −0.623974
$$941$$ − 5.32996e8i − 0.639669i −0.947473 0.319834i $$-0.896373\pi$$
0.947473 0.319834i $$-0.103627\pi$$
$$942$$ 0 0
$$943$$ −1.94628e9 −2.32098
$$944$$ − 1.56629e8i − 0.186189i
$$945$$ 0 0
$$946$$ 3.92348e8 0.463445
$$947$$ − 3.13628e7i − 0.0369288i −0.999830 0.0184644i $$-0.994122\pi$$
0.999830 0.0184644i $$-0.00587773\pi$$
$$948$$ 0 0
$$949$$ 9.42001e8 1.10218
$$950$$ − 4.43308e7i − 0.0517052i
$$951$$ 0 0
$$952$$ −1.49324e6 −0.00173068
$$953$$ − 6.21244e8i − 0.717767i −0.933382 0.358883i $$-0.883158\pi$$
0.933382 0.358883i $$-0.116842\pi$$
$$954$$ 0 0
$$955$$ −2.45040e9 −2.81337
$$956$$ − 4.13541e8i − 0.473309i
$$957$$ 0 0
$$958$$ 6.54603e8 0.744529
$$959$$ 1.09526e7i 0.0124183i
$$960$$ 0 0
$$961$$ −2.77002e8 −0.312114
$$962$$ 2.73383e8i 0.307076i
$$963$$ 0 0
$$964$$ −2.32970e8 −0.260057
$$965$$ 1.45052e9i 1.61415i
$$966$$ 0 0
$$967$$ −6.29984e8 −0.696707 −0.348354 0.937363i $$-0.613259\pi$$
−0.348354 + 0.937363i $$0.613259\pi$$
$$968$$ − 6.05486e8i − 0.667541i
$$969$$ 0 0
$$970$$ 1.64282e9 1.80001
$$971$$ − 1.38333e8i − 0.151101i −0.997142 0.0755507i $$-0.975929\pi$$
0.997142 0.0755507i $$-0.0240715\pi$$
$$972$$ 0 0
$$973$$ −2.94285e6 −0.00319470
$$974$$ − 2.56165e8i − 0.277231i
$$975$$ 0 0
$$976$$ −1.65254e7 −0.0177747
$$977$$ − 8.23975e8i − 0.883549i −0.897126 0.441774i $$-0.854349\pi$$
0.897126 0.441774i $$-0.145651\pi$$
$$978$$ 0 0
$$979$$ 1.78763e9 1.90515
$$980$$ 7.84944e8i 0.833989i
$$981$$ 0 0
$$982$$ 1.15576e9 1.22049
$$983$$ 1.28601e9i 1.35389i 0.736032 + 0.676947i $$0.236697\pi$$
−0.736032 + 0.676947i $$0.763303\pi$$
$$984$$ 0 0
$$985$$ −8.54226e7 −0.0893849
$$986$$ 4.12730e8i 0.430562i
$$987$$ 0 0
$$988$$ 2.55642e7 0.0265070
$$989$$ − 5.13428e8i − 0.530751i
$$990$$ 0 0
$$991$$ −1.70244e8 −0.174924 −0.0874621 0.996168i $$-0.527876\pi$$
−0.0874621 + 0.996168i $$0.527876\pi$$
$$992$$ 1.43126e8i 0.146617i
$$993$$ 0 0
$$994$$ −3.56858e6 −0.00363359
$$995$$ − 6.77399e8i − 0.687663i
$$996$$ 0 0
$$997$$ −7.01078e8 −0.707426 −0.353713 0.935354i $$-0.615081\pi$$
−0.353713 + 0.935354i $$0.615081\pi$$
$$998$$ 1.65740e8i 0.166738i
$$999$$ 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 162.7.b.a.161.2 4
3.2 odd 2 inner 162.7.b.a.161.3 yes 4
9.2 odd 6 162.7.d.f.53.4 8
9.4 even 3 162.7.d.f.107.4 8
9.5 odd 6 162.7.d.f.107.1 8
9.7 even 3 162.7.d.f.53.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
162.7.b.a.161.2 4 1.1 even 1 trivial
162.7.b.a.161.3 yes 4 3.2 odd 2 inner
162.7.d.f.53.1 8 9.7 even 3
162.7.d.f.53.4 8 9.2 odd 6
162.7.d.f.107.1 8 9.5 odd 6
162.7.d.f.107.4 8 9.4 even 3