# Properties

 Label 8.22.a.b.1.1 Level $8$ Weight $22$ Character 8.1 Self dual yes Analytic conductor $22.358$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,22,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 8.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: $$22.3581875430$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4963x + 96223$$ x^3 - x^2 - 4963*x + 96223 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{21}\cdot 3\cdot 5\cdot 7$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-78.2002$$ of defining polynomial Character $$\chi$$ $$=$$ 8.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-97085.2 q^{3} -3.39292e7 q^{5} -5.28731e8 q^{7} -1.03483e9 q^{9} +O(q^{10})$$ $$q-97085.2 q^{3} -3.39292e7 q^{5} -5.28731e8 q^{7} -1.03483e9 q^{9} -1.21955e11 q^{11} +4.33605e11 q^{13} +3.29402e12 q^{15} -1.31741e13 q^{17} +2.18808e13 q^{19} +5.13320e13 q^{21} +1.40779e14 q^{23} +6.74355e14 q^{25} +1.11601e15 q^{27} +1.17115e15 q^{29} +9.57156e14 q^{31} +1.18400e16 q^{33} +1.79394e16 q^{35} -3.53720e16 q^{37} -4.20966e16 q^{39} -1.71456e17 q^{41} +1.35679e17 q^{43} +3.51109e16 q^{45} -5.75514e17 q^{47} -2.78989e17 q^{49} +1.27901e18 q^{51} -9.62123e17 q^{53} +4.13782e18 q^{55} -2.12430e18 q^{57} +4.87671e18 q^{59} +4.59195e18 q^{61} +5.47145e17 q^{63} -1.47119e19 q^{65} -1.78611e19 q^{67} -1.36675e19 q^{69} +2.93629e19 q^{71} -8.32585e18 q^{73} -6.54699e19 q^{75} +6.44812e19 q^{77} -7.05718e19 q^{79} -9.75235e19 q^{81} +2.11929e20 q^{83} +4.46988e20 q^{85} -1.13702e20 q^{87} +2.62972e20 q^{89} -2.29260e20 q^{91} -9.29257e19 q^{93} -7.42397e20 q^{95} +3.84293e20 q^{97} +1.26202e20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 96764 q^{3} - 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9}+O(q^{10})$$ 3 * q + 96764 * q^3 - 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9 $$3 q + 96764 q^{3} - 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9} - 40335108684 q^{11} + 133734425946 q^{13} - 1223136458200 q^{15} + 7797732274422 q^{17} + 35788199781996 q^{19} + 198539224853088 q^{21} + 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 94\!\cdots\!60 q^{99}+O(q^{100})$$ 3 * q + 96764 * q^3 - 24111774 * q^5 + 295988280 * q^7 + 18844697239 * q^9 - 40335108684 * q^11 + 133734425946 * q^13 - 1223136458200 * q^15 + 7797732274422 * q^17 + 35788199781996 * q^19 + 198539224853088 * q^21 + 193770761479080 * q^23 + 1127564438439501 * q^25 + 5282002293508952 * q^27 + 5607343422466122 * q^29 + 11246757871503072 * q^31 + 10014149026970384 * q^33 + 5274251425350096 * q^35 - 24272499791100606 * q^37 - 52925308377862264 * q^39 - 298159108991869602 * q^41 - 33333932139754860 * q^43 - 927411477977893478 * q^45 - 120874283547603888 * q^47 - 850403331975639477 * q^49 + 3954388789815661240 * q^51 - 1138443393004854222 * q^53 + 6957552484263571704 * q^55 + 681697547133656944 * q^57 + 9225624498709937412 * q^59 - 6554902294063924182 * q^61 + 21785214559631141976 * q^63 - 20714581819561144452 * q^65 - 15793054074531629124 * q^67 - 72988310309689939168 * q^69 - 41139582493467997704 * q^71 - 19422167949903851970 * q^73 - 75231657393126995900 * q^75 + 68380237365358617888 * q^77 - 131735321299806049488 * q^79 + 500425796062282339147 * q^81 + 64013993832679681068 * q^83 + 390258202763001297252 * q^85 + 101898953973185066568 * q^87 + 429891446897537246766 * q^89 - 297181701588021496176 * q^91 + 640035369009914700160 * q^93 - 1036372406649019824696 * q^95 - 324059336514148638042 * q^97 - 949982905688477352860 * q^99

## 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
$$3$$ −97085.2 −0.949248 −0.474624 0.880189i $$-0.657416\pi$$
−0.474624 + 0.880189i $$0.657416\pi$$
$$4$$ 0 0
$$5$$ −3.39292e7 −1.55378 −0.776889 0.629638i $$-0.783203\pi$$
−0.776889 + 0.629638i $$0.783203\pi$$
$$6$$ 0 0
$$7$$ −5.28731e8 −0.707466 −0.353733 0.935346i $$-0.615088\pi$$
−0.353733 + 0.935346i $$0.615088\pi$$
$$8$$ 0 0
$$9$$ −1.03483e9 −0.0989284
$$10$$ 0 0
$$11$$ −1.21955e11 −1.41767 −0.708835 0.705375i $$-0.750779\pi$$
−0.708835 + 0.705375i $$0.750779\pi$$
$$12$$ 0 0
$$13$$ 4.33605e11 0.872346 0.436173 0.899863i $$-0.356334\pi$$
0.436173 + 0.899863i $$0.356334\pi$$
$$14$$ 0 0
$$15$$ 3.29402e12 1.47492
$$16$$ 0 0
$$17$$ −1.31741e13 −1.58492 −0.792461 0.609922i $$-0.791201\pi$$
−0.792461 + 0.609922i $$0.791201\pi$$
$$18$$ 0 0
$$19$$ 2.18808e13 0.818747 0.409373 0.912367i $$-0.365747\pi$$
0.409373 + 0.912367i $$0.365747\pi$$
$$20$$ 0 0
$$21$$ 5.13320e13 0.671561
$$22$$ 0 0
$$23$$ 1.40779e14 0.708589 0.354294 0.935134i $$-0.384721\pi$$
0.354294 + 0.935134i $$0.384721\pi$$
$$24$$ 0 0
$$25$$ 6.74355e14 1.41423
$$26$$ 0 0
$$27$$ 1.11601e15 1.04316
$$28$$ 0 0
$$29$$ 1.17115e15 0.516934 0.258467 0.966020i $$-0.416783\pi$$
0.258467 + 0.966020i $$0.416783\pi$$
$$30$$ 0 0
$$31$$ 9.57156e14 0.209742 0.104871 0.994486i $$-0.466557\pi$$
0.104871 + 0.994486i $$0.466557\pi$$
$$32$$ 0 0
$$33$$ 1.18400e16 1.34572
$$34$$ 0 0
$$35$$ 1.79394e16 1.09925
$$36$$ 0 0
$$37$$ −3.53720e16 −1.20932 −0.604661 0.796483i $$-0.706691\pi$$
−0.604661 + 0.796483i $$0.706691\pi$$
$$38$$ 0 0
$$39$$ −4.20966e16 −0.828073
$$40$$ 0 0
$$41$$ −1.71456e17 −1.99490 −0.997451 0.0713504i $$-0.977269\pi$$
−0.997451 + 0.0713504i $$0.977269\pi$$
$$42$$ 0 0
$$43$$ 1.35679e17 0.957398 0.478699 0.877979i $$-0.341108\pi$$
0.478699 + 0.877979i $$0.341108\pi$$
$$44$$ 0 0
$$45$$ 3.51109e16 0.153713
$$46$$ 0 0
$$47$$ −5.75514e17 −1.59598 −0.797991 0.602669i $$-0.794104\pi$$
−0.797991 + 0.602669i $$0.794104\pi$$
$$48$$ 0 0
$$49$$ −2.78989e17 −0.499492
$$50$$ 0 0
$$51$$ 1.27901e18 1.50448
$$52$$ 0 0
$$53$$ −9.62123e17 −0.755673 −0.377837 0.925872i $$-0.623332\pi$$
−0.377837 + 0.925872i $$0.623332\pi$$
$$54$$ 0 0
$$55$$ 4.13782e18 2.20274
$$56$$ 0 0
$$57$$ −2.12430e18 −0.777194
$$58$$ 0 0
$$59$$ 4.87671e18 1.24217 0.621085 0.783743i $$-0.286692\pi$$
0.621085 + 0.783743i $$0.286692\pi$$
$$60$$ 0 0
$$61$$ 4.59195e18 0.824202 0.412101 0.911138i $$-0.364795\pi$$
0.412101 + 0.911138i $$0.364795\pi$$
$$62$$ 0 0
$$63$$ 5.47145e17 0.0699885
$$64$$ 0 0
$$65$$ −1.47119e19 −1.35543
$$66$$ 0 0
$$67$$ −1.78611e19 −1.19708 −0.598539 0.801094i $$-0.704252\pi$$
−0.598539 + 0.801094i $$0.704252\pi$$
$$68$$ 0 0
$$69$$ −1.36675e19 −0.672626
$$70$$ 0 0
$$71$$ 2.93629e19 1.07050 0.535250 0.844694i $$-0.320217\pi$$
0.535250 + 0.844694i $$0.320217\pi$$
$$72$$ 0 0
$$73$$ −8.32585e18 −0.226745 −0.113373 0.993553i $$-0.536165\pi$$
−0.113373 + 0.993553i $$0.536165\pi$$
$$74$$ 0 0
$$75$$ −6.54699e19 −1.34245
$$76$$ 0 0
$$77$$ 6.44812e19 1.00295
$$78$$ 0 0
$$79$$ −7.05718e19 −0.838584 −0.419292 0.907851i $$-0.637722\pi$$
−0.419292 + 0.907851i $$0.637722\pi$$
$$80$$ 0 0
$$81$$ −9.75235e19 −0.891285
$$82$$ 0 0
$$83$$ 2.11929e20 1.49924 0.749620 0.661869i $$-0.230236\pi$$
0.749620 + 0.661869i $$0.230236\pi$$
$$84$$ 0 0
$$85$$ 4.46988e20 2.46262
$$86$$ 0 0
$$87$$ −1.13702e20 −0.490698
$$88$$ 0 0
$$89$$ 2.62972e20 0.893952 0.446976 0.894546i $$-0.352501\pi$$
0.446976 + 0.894546i $$0.352501\pi$$
$$90$$ 0 0
$$91$$ −2.29260e20 −0.617155
$$92$$ 0 0
$$93$$ −9.29257e19 −0.199097
$$94$$ 0 0
$$95$$ −7.42397e20 −1.27215
$$96$$ 0 0
$$97$$ 3.84293e20 0.529126 0.264563 0.964368i $$-0.414772\pi$$
0.264563 + 0.964368i $$0.414772\pi$$
$$98$$ 0 0
$$99$$ 1.26202e20 0.140248
$$100$$ 0 0
$$101$$ −1.92132e21 −1.73071 −0.865356 0.501158i $$-0.832908\pi$$
−0.865356 + 0.501158i $$0.832908\pi$$
$$102$$ 0 0
$$103$$ 6.06205e20 0.444456 0.222228 0.974995i $$-0.428667\pi$$
0.222228 + 0.974995i $$0.428667\pi$$
$$104$$ 0 0
$$105$$ −1.74165e21 −1.04346
$$106$$ 0 0
$$107$$ 7.79821e20 0.383235 0.191618 0.981470i $$-0.438627\pi$$
0.191618 + 0.981470i $$0.438627\pi$$
$$108$$ 0 0
$$109$$ −2.58673e20 −0.104658 −0.0523291 0.998630i $$-0.516664\pi$$
−0.0523291 + 0.998630i $$0.516664\pi$$
$$110$$ 0 0
$$111$$ 3.43410e21 1.14795
$$112$$ 0 0
$$113$$ 1.56041e21 0.432428 0.216214 0.976346i $$-0.430629\pi$$
0.216214 + 0.976346i $$0.430629\pi$$
$$114$$ 0 0
$$115$$ −4.77651e21 −1.10099
$$116$$ 0 0
$$117$$ −4.48706e20 −0.0862998
$$118$$ 0 0
$$119$$ 6.96557e21 1.12128
$$120$$ 0 0
$$121$$ 7.47267e21 1.00979
$$122$$ 0 0
$$123$$ 1.66458e22 1.89366
$$124$$ 0 0
$$125$$ −6.70164e21 −0.643615
$$126$$ 0 0
$$127$$ −7.53317e21 −0.612405 −0.306202 0.951966i $$-0.599058\pi$$
−0.306202 + 0.951966i $$0.599058\pi$$
$$128$$ 0 0
$$129$$ −1.31724e22 −0.908808
$$130$$ 0 0
$$131$$ −1.41613e22 −0.831291 −0.415645 0.909527i $$-0.636444\pi$$
−0.415645 + 0.909527i $$0.636444\pi$$
$$132$$ 0 0
$$133$$ −1.15690e22 −0.579236
$$134$$ 0 0
$$135$$ −3.78654e22 −1.62083
$$136$$ 0 0
$$137$$ 2.76400e22 1.01385 0.506924 0.861991i $$-0.330782\pi$$
0.506924 + 0.861991i $$0.330782\pi$$
$$138$$ 0 0
$$139$$ 1.98007e21 0.0623770 0.0311885 0.999514i $$-0.490071\pi$$
0.0311885 + 0.999514i $$0.490071\pi$$
$$140$$ 0 0
$$141$$ 5.58738e22 1.51498
$$142$$ 0 0
$$143$$ −5.28801e22 −1.23670
$$144$$ 0 0
$$145$$ −3.97363e22 −0.803200
$$146$$ 0 0
$$147$$ 2.70857e22 0.474141
$$148$$ 0 0
$$149$$ 2.76024e22 0.419268 0.209634 0.977780i $$-0.432773\pi$$
0.209634 + 0.977780i $$0.432773\pi$$
$$150$$ 0 0
$$151$$ 4.72139e22 0.623465 0.311732 0.950170i $$-0.399091\pi$$
0.311732 + 0.950170i $$0.399091\pi$$
$$152$$ 0 0
$$153$$ 1.36329e22 0.156794
$$154$$ 0 0
$$155$$ −3.24756e22 −0.325892
$$156$$ 0 0
$$157$$ 1.89837e23 1.66508 0.832540 0.553965i $$-0.186886\pi$$
0.832540 + 0.553965i $$0.186886\pi$$
$$158$$ 0 0
$$159$$ 9.34079e22 0.717321
$$160$$ 0 0
$$161$$ −7.44340e22 −0.501302
$$162$$ 0 0
$$163$$ 1.59931e23 0.946158 0.473079 0.881020i $$-0.343143\pi$$
0.473079 + 0.881020i $$0.343143\pi$$
$$164$$ 0 0
$$165$$ −4.01721e23 −2.09095
$$166$$ 0 0
$$167$$ 3.26974e22 0.149965 0.0749825 0.997185i $$-0.476110\pi$$
0.0749825 + 0.997185i $$0.476110\pi$$
$$168$$ 0 0
$$169$$ −5.90514e22 −0.239012
$$170$$ 0 0
$$171$$ −2.26428e22 −0.0809973
$$172$$ 0 0
$$173$$ −5.17436e22 −0.163822 −0.0819111 0.996640i $$-0.526102\pi$$
−0.0819111 + 0.996640i $$0.526102\pi$$
$$174$$ 0 0
$$175$$ −3.56553e23 −1.00052
$$176$$ 0 0
$$177$$ −4.73457e23 −1.17913
$$178$$ 0 0
$$179$$ 3.75079e22 0.0830167 0.0415084 0.999138i $$-0.486784\pi$$
0.0415084 + 0.999138i $$0.486784\pi$$
$$180$$ 0 0
$$181$$ −3.65662e22 −0.0720203 −0.0360102 0.999351i $$-0.511465\pi$$
−0.0360102 + 0.999351i $$0.511465\pi$$
$$182$$ 0 0
$$183$$ −4.45810e23 −0.782372
$$184$$ 0 0
$$185$$ 1.20015e24 1.87902
$$186$$ 0 0
$$187$$ 1.60664e24 2.24690
$$188$$ 0 0
$$189$$ −5.90070e23 −0.737997
$$190$$ 0 0
$$191$$ −4.61401e23 −0.516688 −0.258344 0.966053i $$-0.583177\pi$$
−0.258344 + 0.966053i $$0.583177\pi$$
$$192$$ 0 0
$$193$$ 2.52365e23 0.253324 0.126662 0.991946i $$-0.459574\pi$$
0.126662 + 0.991946i $$0.459574\pi$$
$$194$$ 0 0
$$195$$ 1.42830e24 1.28664
$$196$$ 0 0
$$197$$ −7.91314e23 −0.640403 −0.320202 0.947349i $$-0.603751\pi$$
−0.320202 + 0.947349i $$0.603751\pi$$
$$198$$ 0 0
$$199$$ −2.46245e24 −1.79230 −0.896150 0.443752i $$-0.853647\pi$$
−0.896150 + 0.443752i $$0.853647\pi$$
$$200$$ 0 0
$$201$$ 1.73405e24 1.13632
$$202$$ 0 0
$$203$$ −6.19226e23 −0.365713
$$204$$ 0 0
$$205$$ 5.81736e24 3.09964
$$206$$ 0 0
$$207$$ −1.45681e23 −0.0700995
$$208$$ 0 0
$$209$$ −2.66846e24 −1.16071
$$210$$ 0 0
$$211$$ −1.79516e24 −0.706542 −0.353271 0.935521i $$-0.614931\pi$$
−0.353271 + 0.935521i $$0.614931\pi$$
$$212$$ 0 0
$$213$$ −2.85070e24 −1.01617
$$214$$ 0 0
$$215$$ −4.60347e24 −1.48758
$$216$$ 0 0
$$217$$ −5.06079e23 −0.148385
$$218$$ 0 0
$$219$$ 8.08316e23 0.215238
$$220$$ 0 0
$$221$$ −5.71236e24 −1.38260
$$222$$ 0 0
$$223$$ 3.76539e24 0.829103 0.414551 0.910026i $$-0.363939\pi$$
0.414551 + 0.910026i $$0.363939\pi$$
$$224$$ 0 0
$$225$$ −6.97841e23 −0.139907
$$226$$ 0 0
$$227$$ 8.37475e24 1.53003 0.765016 0.644011i $$-0.222731\pi$$
0.765016 + 0.644011i $$0.222731\pi$$
$$228$$ 0 0
$$229$$ 3.14314e24 0.523710 0.261855 0.965107i $$-0.415666\pi$$
0.261855 + 0.965107i $$0.415666\pi$$
$$230$$ 0 0
$$231$$ −6.26017e24 −0.952051
$$232$$ 0 0
$$233$$ −4.21087e24 −0.584971 −0.292486 0.956270i $$-0.594482\pi$$
−0.292486 + 0.956270i $$0.594482\pi$$
$$234$$ 0 0
$$235$$ 1.95267e25 2.47980
$$236$$ 0 0
$$237$$ 6.85147e24 0.796024
$$238$$ 0 0
$$239$$ −9.27921e24 −0.987036 −0.493518 0.869736i $$-0.664289\pi$$
−0.493518 + 0.869736i $$0.664289\pi$$
$$240$$ 0 0
$$241$$ 1.45105e24 0.141418 0.0707090 0.997497i $$-0.477474\pi$$
0.0707090 + 0.997497i $$0.477474\pi$$
$$242$$ 0 0
$$243$$ −2.20579e24 −0.197105
$$244$$ 0 0
$$245$$ 9.46588e24 0.776099
$$246$$ 0 0
$$247$$ 9.48760e24 0.714231
$$248$$ 0 0
$$249$$ −2.05752e25 −1.42315
$$250$$ 0 0
$$251$$ 2.04939e25 1.30332 0.651658 0.758513i $$-0.274074\pi$$
0.651658 + 0.758513i $$0.274074\pi$$
$$252$$ 0 0
$$253$$ −1.71686e25 −1.00454
$$254$$ 0 0
$$255$$ −4.33959e25 −2.33763
$$256$$ 0 0
$$257$$ 1.77825e25 0.882462 0.441231 0.897394i $$-0.354542\pi$$
0.441231 + 0.897394i $$0.354542\pi$$
$$258$$ 0 0
$$259$$ 1.87023e25 0.855554
$$260$$ 0 0
$$261$$ −1.21194e24 −0.0511394
$$262$$ 0 0
$$263$$ −3.76203e25 −1.46517 −0.732583 0.680678i $$-0.761685\pi$$
−0.732583 + 0.680678i $$0.761685\pi$$
$$264$$ 0 0
$$265$$ 3.26441e25 1.17415
$$266$$ 0 0
$$267$$ −2.55307e25 −0.848582
$$268$$ 0 0
$$269$$ −5.82547e25 −1.79033 −0.895163 0.445739i $$-0.852941\pi$$
−0.895163 + 0.445739i $$0.852941\pi$$
$$270$$ 0 0
$$271$$ 4.00531e25 1.13883 0.569414 0.822051i $$-0.307170\pi$$
0.569414 + 0.822051i $$0.307170\pi$$
$$272$$ 0 0
$$273$$ 2.22578e25 0.585834
$$274$$ 0 0
$$275$$ −8.22407e25 −2.00490
$$276$$ 0 0
$$277$$ 5.12820e25 1.15858 0.579292 0.815120i $$-0.303329\pi$$
0.579292 + 0.815120i $$0.303329\pi$$
$$278$$ 0 0
$$279$$ −9.90490e23 −0.0207494
$$280$$ 0 0
$$281$$ −1.22313e25 −0.237714 −0.118857 0.992911i $$-0.537923\pi$$
−0.118857 + 0.992911i $$0.537923\pi$$
$$282$$ 0 0
$$283$$ 8.45355e25 1.52504 0.762520 0.646964i $$-0.223962\pi$$
0.762520 + 0.646964i $$0.223962\pi$$
$$284$$ 0 0
$$285$$ 7.20758e25 1.20759
$$286$$ 0 0
$$287$$ 9.06541e25 1.41133
$$288$$ 0 0
$$289$$ 1.04466e26 1.51198
$$290$$ 0 0
$$291$$ −3.73092e25 −0.502272
$$292$$ 0 0
$$293$$ −2.19428e25 −0.274905 −0.137453 0.990508i $$-0.543891\pi$$
−0.137453 + 0.990508i $$0.543891\pi$$
$$294$$ 0 0
$$295$$ −1.65463e26 −1.93006
$$296$$ 0 0
$$297$$ −1.36103e26 −1.47885
$$298$$ 0 0
$$299$$ 6.10422e25 0.618135
$$300$$ 0 0
$$301$$ −7.17375e25 −0.677327
$$302$$ 0 0
$$303$$ 1.86531e26 1.64287
$$304$$ 0 0
$$305$$ −1.55801e26 −1.28063
$$306$$ 0 0
$$307$$ 3.79642e25 0.291354 0.145677 0.989332i $$-0.453464\pi$$
0.145677 + 0.989332i $$0.453464\pi$$
$$308$$ 0 0
$$309$$ −5.88535e25 −0.421899
$$310$$ 0 0
$$311$$ 7.02783e25 0.470801 0.235401 0.971898i $$-0.424360\pi$$
0.235401 + 0.971898i $$0.424360\pi$$
$$312$$ 0 0
$$313$$ −9.09673e25 −0.569731 −0.284865 0.958568i $$-0.591949\pi$$
−0.284865 + 0.958568i $$0.591949\pi$$
$$314$$ 0 0
$$315$$ −1.85642e25 −0.108747
$$316$$ 0 0
$$317$$ −2.99033e25 −0.163907 −0.0819533 0.996636i $$-0.526116\pi$$
−0.0819533 + 0.996636i $$0.526116\pi$$
$$318$$ 0 0
$$319$$ −1.42828e26 −0.732841
$$320$$ 0 0
$$321$$ −7.57091e25 −0.363785
$$322$$ 0 0
$$323$$ −2.88260e26 −1.29765
$$324$$ 0 0
$$325$$ 2.92404e26 1.23369
$$326$$ 0 0
$$327$$ 2.51133e25 0.0993465
$$328$$ 0 0
$$329$$ 3.04292e26 1.12910
$$330$$ 0 0
$$331$$ −6.61150e25 −0.230200 −0.115100 0.993354i $$-0.536719\pi$$
−0.115100 + 0.993354i $$0.536719\pi$$
$$332$$ 0 0
$$333$$ 3.66039e25 0.119636
$$334$$ 0 0
$$335$$ 6.06013e26 1.85999
$$336$$ 0 0
$$337$$ −1.11607e26 −0.321793 −0.160896 0.986971i $$-0.551438\pi$$
−0.160896 + 0.986971i $$0.551438\pi$$
$$338$$ 0 0
$$339$$ −1.51492e26 −0.410482
$$340$$ 0 0
$$341$$ −1.16730e26 −0.297345
$$342$$ 0 0
$$343$$ 4.42831e26 1.06084
$$344$$ 0 0
$$345$$ 4.63728e26 1.04511
$$346$$ 0 0
$$347$$ −5.94900e26 −1.26178 −0.630891 0.775872i $$-0.717310\pi$$
−0.630891 + 0.775872i $$0.717310\pi$$
$$348$$ 0 0
$$349$$ 4.45113e26 0.888799 0.444400 0.895829i $$-0.353417\pi$$
0.444400 + 0.895829i $$0.353417\pi$$
$$350$$ 0 0
$$351$$ 4.83908e26 0.909993
$$352$$ 0 0
$$353$$ 2.37403e26 0.420584 0.210292 0.977639i $$-0.432559\pi$$
0.210292 + 0.977639i $$0.432559\pi$$
$$354$$ 0 0
$$355$$ −9.96262e26 −1.66332
$$356$$ 0 0
$$357$$ −6.76254e26 −1.06437
$$358$$ 0 0
$$359$$ −3.45652e26 −0.513036 −0.256518 0.966539i $$-0.582575\pi$$
−0.256518 + 0.966539i $$0.582575\pi$$
$$360$$ 0 0
$$361$$ −2.35442e26 −0.329654
$$362$$ 0 0
$$363$$ −7.25485e26 −0.958537
$$364$$ 0 0
$$365$$ 2.82490e26 0.352312
$$366$$ 0 0
$$367$$ −8.76689e26 −1.03241 −0.516205 0.856465i $$-0.672656\pi$$
−0.516205 + 0.856465i $$0.672656\pi$$
$$368$$ 0 0
$$369$$ 1.77427e26 0.197353
$$370$$ 0 0
$$371$$ 5.08705e26 0.534613
$$372$$ 0 0
$$373$$ −1.54618e27 −1.53573 −0.767867 0.640609i $$-0.778682\pi$$
−0.767867 + 0.640609i $$0.778682\pi$$
$$374$$ 0 0
$$375$$ 6.50630e26 0.610950
$$376$$ 0 0
$$377$$ 5.07818e26 0.450945
$$378$$ 0 0
$$379$$ 1.48367e27 1.24631 0.623156 0.782097i $$-0.285850\pi$$
0.623156 + 0.782097i $$0.285850\pi$$
$$380$$ 0 0
$$381$$ 7.31359e26 0.581324
$$382$$ 0 0
$$383$$ 1.86137e27 1.40038 0.700190 0.713956i $$-0.253099\pi$$
0.700190 + 0.713956i $$0.253099\pi$$
$$384$$ 0 0
$$385$$ −2.18780e27 −1.55837
$$386$$ 0 0
$$387$$ −1.40404e26 −0.0947139
$$388$$ 0 0
$$389$$ −4.43791e26 −0.283601 −0.141801 0.989895i $$-0.545289\pi$$
−0.141801 + 0.989895i $$0.545289\pi$$
$$390$$ 0 0
$$391$$ −1.85463e27 −1.12306
$$392$$ 0 0
$$393$$ 1.37485e27 0.789101
$$394$$ 0 0
$$395$$ 2.39445e27 1.30297
$$396$$ 0 0
$$397$$ −3.85978e26 −0.199188 −0.0995939 0.995028i $$-0.531754\pi$$
−0.0995939 + 0.995028i $$0.531754\pi$$
$$398$$ 0 0
$$399$$ 1.12318e27 0.549838
$$400$$ 0 0
$$401$$ −8.65938e26 −0.402227 −0.201113 0.979568i $$-0.564456\pi$$
−0.201113 + 0.979568i $$0.564456\pi$$
$$402$$ 0 0
$$403$$ 4.15028e26 0.182967
$$404$$ 0 0
$$405$$ 3.30890e27 1.38486
$$406$$ 0 0
$$407$$ 4.31378e27 1.71442
$$408$$ 0 0
$$409$$ 6.19682e26 0.233924 0.116962 0.993136i $$-0.462684\pi$$
0.116962 + 0.993136i $$0.462684\pi$$
$$410$$ 0 0
$$411$$ −2.68344e27 −0.962393
$$412$$ 0 0
$$413$$ −2.57847e27 −0.878793
$$414$$ 0 0
$$415$$ −7.19060e27 −2.32948
$$416$$ 0 0
$$417$$ −1.92235e26 −0.0592112
$$418$$ 0 0
$$419$$ −2.87299e27 −0.841562 −0.420781 0.907162i $$-0.638244\pi$$
−0.420781 + 0.907162i $$0.638244\pi$$
$$420$$ 0 0
$$421$$ 2.25957e27 0.629598 0.314799 0.949158i $$-0.398063\pi$$
0.314799 + 0.949158i $$0.398063\pi$$
$$422$$ 0 0
$$423$$ 5.95557e26 0.157888
$$424$$ 0 0
$$425$$ −8.88404e27 −2.24144
$$426$$ 0 0
$$427$$ −2.42791e27 −0.583095
$$428$$ 0 0
$$429$$ 5.13387e27 1.17393
$$430$$ 0 0
$$431$$ −3.01840e27 −0.657303 −0.328652 0.944451i $$-0.606594\pi$$
−0.328652 + 0.944451i $$0.606594\pi$$
$$432$$ 0 0
$$433$$ 6.44418e27 1.33673 0.668367 0.743832i $$-0.266994\pi$$
0.668367 + 0.743832i $$0.266994\pi$$
$$434$$ 0 0
$$435$$ 3.85781e27 0.762436
$$436$$ 0 0
$$437$$ 3.08034e27 0.580155
$$438$$ 0 0
$$439$$ −8.50049e27 −1.52604 −0.763021 0.646374i $$-0.776285\pi$$
−0.763021 + 0.646374i $$0.776285\pi$$
$$440$$ 0 0
$$441$$ 2.88705e26 0.0494139
$$442$$ 0 0
$$443$$ 6.00246e27 0.979694 0.489847 0.871809i $$-0.337053\pi$$
0.489847 + 0.871809i $$0.337053\pi$$
$$444$$ 0 0
$$445$$ −8.92243e27 −1.38900
$$446$$ 0 0
$$447$$ −2.67979e27 −0.397989
$$448$$ 0 0
$$449$$ 8.58205e27 1.21620 0.608100 0.793861i $$-0.291932\pi$$
0.608100 + 0.793861i $$0.291932\pi$$
$$450$$ 0 0
$$451$$ 2.09098e28 2.82811
$$452$$ 0 0
$$453$$ −4.58377e27 −0.591823
$$454$$ 0 0
$$455$$ 7.77863e27 0.958922
$$456$$ 0 0
$$457$$ 1.09251e28 1.28619 0.643093 0.765788i $$-0.277651\pi$$
0.643093 + 0.765788i $$0.277651\pi$$
$$458$$ 0 0
$$459$$ −1.47025e28 −1.65332
$$460$$ 0 0
$$461$$ −7.29500e27 −0.783728 −0.391864 0.920023i $$-0.628170\pi$$
−0.391864 + 0.920023i $$0.628170\pi$$
$$462$$ 0 0
$$463$$ −1.13930e28 −1.16960 −0.584800 0.811178i $$-0.698827\pi$$
−0.584800 + 0.811178i $$0.698827\pi$$
$$464$$ 0 0
$$465$$ 3.15290e27 0.309353
$$466$$ 0 0
$$467$$ 1.51114e28 1.41735 0.708677 0.705533i $$-0.249292\pi$$
0.708677 + 0.705533i $$0.249292\pi$$
$$468$$ 0 0
$$469$$ 9.44371e27 0.846892
$$470$$ 0 0
$$471$$ −1.84304e28 −1.58057
$$472$$ 0 0
$$473$$ −1.65466e28 −1.35727
$$474$$ 0 0
$$475$$ 1.47554e28 1.15789
$$476$$ 0 0
$$477$$ 9.95630e26 0.0747576
$$478$$ 0 0
$$479$$ 3.89042e27 0.279559 0.139779 0.990183i $$-0.455361\pi$$
0.139779 + 0.990183i $$0.455361\pi$$
$$480$$ 0 0
$$481$$ −1.53375e28 −1.05495
$$482$$ 0 0
$$483$$ 7.22644e27 0.475860
$$484$$ 0 0
$$485$$ −1.30388e28 −0.822145
$$486$$ 0 0
$$487$$ 1.89704e28 1.14557 0.572787 0.819704i $$-0.305862\pi$$
0.572787 + 0.819704i $$0.305862\pi$$
$$488$$ 0 0
$$489$$ −1.55270e28 −0.898138
$$490$$ 0 0
$$491$$ −1.33589e28 −0.740313 −0.370157 0.928969i $$-0.620696\pi$$
−0.370157 + 0.928969i $$0.620696\pi$$
$$492$$ 0 0
$$493$$ −1.54289e28 −0.819300
$$494$$ 0 0
$$495$$ −4.28193e27 −0.217914
$$496$$ 0 0
$$497$$ −1.55251e28 −0.757343
$$498$$ 0 0
$$499$$ −2.86135e28 −1.33818 −0.669090 0.743181i $$-0.733316\pi$$
−0.669090 + 0.743181i $$0.733316\pi$$
$$500$$ 0 0
$$501$$ −3.17443e27 −0.142354
$$502$$ 0 0
$$503$$ 9.44655e27 0.406265 0.203133 0.979151i $$-0.434888\pi$$
0.203133 + 0.979151i $$0.434888\pi$$
$$504$$ 0 0
$$505$$ 6.51888e28 2.68914
$$506$$ 0 0
$$507$$ 5.73302e27 0.226882
$$508$$ 0 0
$$509$$ 2.06060e28 0.782452 0.391226 0.920295i $$-0.372051\pi$$
0.391226 + 0.920295i $$0.372051\pi$$
$$510$$ 0 0
$$511$$ 4.40214e27 0.160415
$$512$$ 0 0
$$513$$ 2.44192e28 0.854080
$$514$$ 0 0
$$515$$ −2.05681e28 −0.690586
$$516$$ 0 0
$$517$$ 7.01865e28 2.26258
$$518$$ 0 0
$$519$$ 5.02354e27 0.155508
$$520$$ 0 0
$$521$$ −4.30969e28 −1.28130 −0.640649 0.767834i $$-0.721335\pi$$
−0.640649 + 0.767834i $$0.721335\pi$$
$$522$$ 0 0
$$523$$ 1.08864e28 0.310897 0.155448 0.987844i $$-0.450318\pi$$
0.155448 + 0.987844i $$0.450318\pi$$
$$524$$ 0 0
$$525$$ 3.46160e28 0.949739
$$526$$ 0 0
$$527$$ −1.26097e28 −0.332425
$$528$$ 0 0
$$529$$ −1.96530e28 −0.497902
$$530$$ 0 0
$$531$$ −5.04655e27 −0.122886
$$532$$ 0 0
$$533$$ −7.43441e28 −1.74025
$$534$$ 0 0
$$535$$ −2.64587e28 −0.595462
$$536$$ 0 0
$$537$$ −3.64146e27 −0.0788035
$$538$$ 0 0
$$539$$ 3.40240e28 0.708114
$$540$$ 0 0
$$541$$ 3.99516e28 0.799765 0.399883 0.916566i $$-0.369051\pi$$
0.399883 + 0.916566i $$0.369051\pi$$
$$542$$ 0 0
$$543$$ 3.55004e27 0.0683651
$$544$$ 0 0
$$545$$ 8.77658e27 0.162616
$$546$$ 0 0
$$547$$ −5.58498e28 −0.995760 −0.497880 0.867246i $$-0.665888\pi$$
−0.497880 + 0.867246i $$0.665888\pi$$
$$548$$ 0 0
$$549$$ −4.75187e27 −0.0815370
$$550$$ 0 0
$$551$$ 2.56257e28 0.423238
$$552$$ 0 0
$$553$$ 3.73135e28 0.593270
$$554$$ 0 0
$$555$$ −1.16516e29 −1.78365
$$556$$ 0 0
$$557$$ −8.87710e28 −1.30855 −0.654277 0.756255i $$-0.727027\pi$$
−0.654277 + 0.756255i $$0.727027\pi$$
$$558$$ 0 0
$$559$$ 5.88309e28 0.835183
$$560$$ 0 0
$$561$$ −1.55981e29 −2.13286
$$562$$ 0 0
$$563$$ 7.63115e28 1.00520 0.502599 0.864520i $$-0.332377\pi$$
0.502599 + 0.864520i $$0.332377\pi$$
$$564$$ 0 0
$$565$$ −5.29434e28 −0.671898
$$566$$ 0 0
$$567$$ 5.15637e28 0.630554
$$568$$ 0 0
$$569$$ 1.48073e29 1.74500 0.872502 0.488610i $$-0.162496\pi$$
0.872502 + 0.488610i $$0.162496\pi$$
$$570$$ 0 0
$$571$$ −7.51419e28 −0.853499 −0.426749 0.904370i $$-0.640341\pi$$
−0.426749 + 0.904370i $$0.640341\pi$$
$$572$$ 0 0
$$573$$ 4.47952e28 0.490465
$$574$$ 0 0
$$575$$ 9.49348e28 1.00210
$$576$$ 0 0
$$577$$ 1.56608e28 0.159392 0.0796961 0.996819i $$-0.474605\pi$$
0.0796961 + 0.996819i $$0.474605\pi$$
$$578$$ 0 0
$$579$$ −2.45009e28 −0.240468
$$580$$ 0 0
$$581$$ −1.12054e29 −1.06066
$$582$$ 0 0
$$583$$ 1.17335e29 1.07129
$$584$$ 0 0
$$585$$ 1.52242e28 0.134091
$$586$$ 0 0
$$587$$ 1.19163e29 1.01260 0.506302 0.862356i $$-0.331012\pi$$
0.506302 + 0.862356i $$0.331012\pi$$
$$588$$ 0 0
$$589$$ 2.09433e28 0.171725
$$590$$ 0 0
$$591$$ 7.68248e28 0.607901
$$592$$ 0 0
$$593$$ 3.34770e28 0.255666 0.127833 0.991796i $$-0.459198\pi$$
0.127833 + 0.991796i $$0.459198\pi$$
$$594$$ 0 0
$$595$$ −2.36337e29 −1.74222
$$596$$ 0 0
$$597$$ 2.39068e29 1.70134
$$598$$ 0 0
$$599$$ 1.39785e29 0.960461 0.480231 0.877142i $$-0.340553\pi$$
0.480231 + 0.877142i $$0.340553\pi$$
$$600$$ 0 0
$$601$$ −1.02954e27 −0.00683061 −0.00341531 0.999994i $$-0.501087\pi$$
−0.00341531 + 0.999994i $$0.501087\pi$$
$$602$$ 0 0
$$603$$ 1.84831e28 0.118425
$$604$$ 0 0
$$605$$ −2.53542e29 −1.56898
$$606$$ 0 0
$$607$$ 1.61045e29 0.962647 0.481323 0.876543i $$-0.340156\pi$$
0.481323 + 0.876543i $$0.340156\pi$$
$$608$$ 0 0
$$609$$ 6.01176e28 0.347152
$$610$$ 0 0
$$611$$ −2.49546e29 −1.39225
$$612$$ 0 0
$$613$$ −1.08394e29 −0.584345 −0.292173 0.956366i $$-0.594378\pi$$
−0.292173 + 0.956366i $$0.594378\pi$$
$$614$$ 0 0
$$615$$ −5.64780e29 −2.94232
$$616$$ 0 0
$$617$$ −1.69220e29 −0.852033 −0.426017 0.904715i $$-0.640084\pi$$
−0.426017 + 0.904715i $$0.640084\pi$$
$$618$$ 0 0
$$619$$ 4.72000e28 0.229715 0.114858 0.993382i $$-0.463359\pi$$
0.114858 + 0.993382i $$0.463359\pi$$
$$620$$ 0 0
$$621$$ 1.57110e29 0.739168
$$622$$ 0 0
$$623$$ −1.39041e29 −0.632441
$$624$$ 0 0
$$625$$ −9.41762e28 −0.414191
$$626$$ 0 0
$$627$$ 2.59068e29 1.10180
$$628$$ 0 0
$$629$$ 4.65995e29 1.91668
$$630$$ 0 0
$$631$$ −4.15217e29 −1.65184 −0.825918 0.563790i $$-0.809343\pi$$
−0.825918 + 0.563790i $$0.809343\pi$$
$$632$$ 0 0
$$633$$ 1.74284e29 0.670684
$$634$$ 0 0
$$635$$ 2.55595e29 0.951541
$$636$$ 0 0
$$637$$ −1.20971e29 −0.435730
$$638$$ 0 0
$$639$$ −3.03855e28 −0.105903
$$640$$ 0 0
$$641$$ 9.51731e28 0.321000 0.160500 0.987036i $$-0.448689\pi$$
0.160500 + 0.987036i $$0.448689\pi$$
$$642$$ 0 0
$$643$$ −3.25707e29 −1.06319 −0.531596 0.846998i $$-0.678408\pi$$
−0.531596 + 0.846998i $$0.678408\pi$$
$$644$$ 0 0
$$645$$ 4.46929e29 1.41209
$$646$$ 0 0
$$647$$ −3.06325e28 −0.0936888 −0.0468444 0.998902i $$-0.514916\pi$$
−0.0468444 + 0.998902i $$0.514916\pi$$
$$648$$ 0 0
$$649$$ −5.94737e29 −1.76099
$$650$$ 0 0
$$651$$ 4.91327e28 0.140854
$$652$$ 0 0
$$653$$ −8.66491e28 −0.240534 −0.120267 0.992742i $$-0.538375\pi$$
−0.120267 + 0.992742i $$0.538375\pi$$
$$654$$ 0 0
$$655$$ 4.80480e29 1.29164
$$656$$ 0 0
$$657$$ 8.61580e27 0.0224316
$$658$$ 0 0
$$659$$ 7.32944e29 1.84831 0.924154 0.382021i $$-0.124772\pi$$
0.924154 + 0.382021i $$0.124772\pi$$
$$660$$ 0 0
$$661$$ −5.06204e29 −1.23655 −0.618273 0.785964i $$-0.712167\pi$$
−0.618273 + 0.785964i $$0.712167\pi$$
$$662$$ 0 0
$$663$$ 5.54586e29 1.31243
$$664$$ 0 0
$$665$$ 3.92529e29 0.900004
$$666$$ 0 0
$$667$$ 1.64873e29 0.366293
$$668$$ 0 0
$$669$$ −3.65563e29 −0.787024
$$670$$ 0 0
$$671$$ −5.60009e29 −1.16845
$$672$$ 0 0
$$673$$ 4.74688e29 0.959953 0.479977 0.877281i $$-0.340645\pi$$
0.479977 + 0.877281i $$0.340645\pi$$
$$674$$ 0 0
$$675$$ 7.52588e29 1.47526
$$676$$ 0 0
$$677$$ 6.45182e29 1.22603 0.613015 0.790071i $$-0.289956\pi$$
0.613015 + 0.790071i $$0.289956\pi$$
$$678$$ 0 0
$$679$$ −2.03188e29 −0.374339
$$680$$ 0 0
$$681$$ −8.13064e29 −1.45238
$$682$$ 0 0
$$683$$ −5.14621e29 −0.891394 −0.445697 0.895184i $$-0.647044\pi$$
−0.445697 + 0.895184i $$0.647044\pi$$
$$684$$ 0 0
$$685$$ −9.37805e29 −1.57529
$$686$$ 0 0
$$687$$ −3.05152e29 −0.497131
$$688$$ 0 0
$$689$$ −4.17181e29 −0.659209
$$690$$ 0 0
$$691$$ −1.06801e30 −1.63702 −0.818512 0.574489i $$-0.805201\pi$$
−0.818512 + 0.574489i $$0.805201\pi$$
$$692$$ 0 0
$$693$$ −6.67268e28 −0.0992205
$$694$$ 0 0
$$695$$ −6.71821e28 −0.0969199
$$696$$ 0 0
$$697$$ 2.25878e30 3.16177
$$698$$ 0 0
$$699$$ 4.08813e29 0.555283
$$700$$ 0 0
$$701$$ −1.04529e30 −1.37784 −0.688919 0.724839i $$-0.741914\pi$$
−0.688919 + 0.724839i $$0.741914\pi$$
$$702$$ 0 0
$$703$$ −7.73967e29 −0.990128
$$704$$ 0 0
$$705$$ −1.89576e30 −2.35395
$$706$$ 0 0
$$707$$ 1.01586e30 1.22442
$$708$$ 0 0
$$709$$ 8.68030e28 0.101566 0.0507831 0.998710i $$-0.483828\pi$$
0.0507831 + 0.998710i $$0.483828\pi$$
$$710$$ 0 0
$$711$$ 7.30295e28 0.0829598
$$712$$ 0 0
$$713$$ 1.34747e29 0.148621
$$714$$ 0 0
$$715$$ 1.79418e30 1.92155
$$716$$ 0 0
$$717$$ 9.00873e29 0.936942
$$718$$ 0 0
$$719$$ −1.16156e30 −1.17325 −0.586624 0.809859i $$-0.699543\pi$$
−0.586624 + 0.809859i $$0.699543\pi$$
$$720$$ 0 0
$$721$$ −3.20520e29 −0.314438
$$722$$ 0 0
$$723$$ −1.40876e29 −0.134241
$$724$$ 0 0
$$725$$ 7.89774e29 0.731061
$$726$$ 0 0
$$727$$ 2.77822e29 0.249836 0.124918 0.992167i $$-0.460133\pi$$
0.124918 + 0.992167i $$0.460133\pi$$
$$728$$ 0 0
$$729$$ 1.23428e30 1.07839
$$730$$ 0 0
$$731$$ −1.78745e30 −1.51740
$$732$$ 0 0
$$733$$ 2.16010e30 1.78189 0.890947 0.454107i $$-0.150042\pi$$
0.890947 + 0.454107i $$0.150042\pi$$
$$734$$ 0 0
$$735$$ −9.18996e29 −0.736710
$$736$$ 0 0
$$737$$ 2.17824e30 1.69706
$$738$$ 0 0
$$739$$ 1.74302e30 1.31988 0.659942 0.751317i $$-0.270581\pi$$
0.659942 + 0.751317i $$0.270581\pi$$
$$740$$ 0 0
$$741$$ −9.21105e29 −0.677982
$$742$$ 0 0
$$743$$ −5.89639e29 −0.421895 −0.210947 0.977497i $$-0.567655\pi$$
−0.210947 + 0.977497i $$0.567655\pi$$
$$744$$ 0 0
$$745$$ −9.36530e29 −0.651449
$$746$$ 0 0
$$747$$ −2.19310e29 −0.148317
$$748$$ 0 0
$$749$$ −4.12316e29 −0.271126
$$750$$ 0 0
$$751$$ −8.31754e28 −0.0531833 −0.0265917 0.999646i $$-0.508465\pi$$
−0.0265917 + 0.999646i $$0.508465\pi$$
$$752$$ 0 0
$$753$$ −1.98965e30 −1.23717
$$754$$ 0 0
$$755$$ −1.60193e30 −0.968726
$$756$$ 0 0
$$757$$ −2.67582e29 −0.157380 −0.0786902 0.996899i $$-0.525074\pi$$
−0.0786902 + 0.996899i $$0.525074\pi$$
$$758$$ 0 0
$$759$$ 1.66681e30 0.953561
$$760$$ 0 0
$$761$$ 2.95107e30 1.64225 0.821127 0.570745i $$-0.193346\pi$$
0.821127 + 0.570745i $$0.193346\pi$$
$$762$$ 0 0
$$763$$ 1.36769e29 0.0740421
$$764$$ 0 0
$$765$$ −4.62555e29 −0.243623
$$766$$ 0 0
$$767$$ 2.11457e30 1.08360
$$768$$ 0 0
$$769$$ −2.62362e30 −1.30820 −0.654100 0.756408i $$-0.726953\pi$$
−0.654100 + 0.756408i $$0.726953\pi$$
$$770$$ 0 0
$$771$$ −1.72642e30 −0.837675
$$772$$ 0 0
$$773$$ 9.36482e29 0.442196 0.221098 0.975252i $$-0.429036\pi$$
0.221098 + 0.975252i $$0.429036\pi$$
$$774$$ 0 0
$$775$$ 6.45464e29 0.296622
$$776$$ 0 0
$$777$$ −1.81572e30 −0.812133
$$778$$ 0 0
$$779$$ −3.75158e30 −1.63332
$$780$$ 0 0
$$781$$ −3.58094e30 −1.51762
$$782$$ 0 0
$$783$$ 1.30702e30 0.539242
$$784$$ 0 0
$$785$$ −6.44103e30 −2.58716
$$786$$ 0 0
$$787$$ 2.42850e30 0.949736 0.474868 0.880057i $$-0.342496\pi$$
0.474868 + 0.880057i $$0.342496\pi$$
$$788$$ 0 0
$$789$$ 3.65237e30 1.39081
$$790$$ 0 0
$$791$$ −8.25036e29 −0.305929
$$792$$ 0 0
$$793$$ 1.99109e30 0.718990
$$794$$ 0 0
$$795$$ −3.16926e30 −1.11456
$$796$$ 0 0
$$797$$ −1.99411e29 −0.0683025 −0.0341513 0.999417i $$-0.510873\pi$$
−0.0341513 + 0.999417i $$0.510873\pi$$
$$798$$ 0 0
$$799$$ 7.58189e30 2.52951
$$800$$ 0 0
$$801$$ −2.72130e29 −0.0884373
$$802$$ 0 0
$$803$$ 1.01537e30 0.321450
$$804$$ 0 0
$$805$$ 2.52549e30 0.778913
$$806$$ 0 0
$$807$$ 5.65567e30 1.69946
$$808$$ 0 0
$$809$$ 3.77765e30 1.10602 0.553010 0.833175i $$-0.313479\pi$$
0.553010 + 0.833175i $$0.313479\pi$$
$$810$$ 0 0
$$811$$ 2.72778e30 0.778199 0.389099 0.921196i $$-0.372786\pi$$
0.389099 + 0.921196i $$0.372786\pi$$
$$812$$ 0 0
$$813$$ −3.88856e30 −1.08103
$$814$$ 0 0
$$815$$ −5.42635e30 −1.47012
$$816$$ 0 0
$$817$$ 2.96875e30 0.783867
$$818$$ 0 0
$$819$$ 2.37245e29 0.0610542
$$820$$ 0 0
$$821$$ −1.90653e30 −0.478234 −0.239117 0.970991i $$-0.576858\pi$$
−0.239117 + 0.970991i $$0.576858\pi$$
$$822$$ 0 0
$$823$$ −3.39146e30 −0.829255 −0.414628 0.909991i $$-0.636088\pi$$
−0.414628 + 0.909991i $$0.636088\pi$$
$$824$$ 0 0
$$825$$ 7.98435e30 1.90315
$$826$$ 0 0
$$827$$ −3.99616e30 −0.928613 −0.464307 0.885675i $$-0.653696\pi$$
−0.464307 + 0.885675i $$0.653696\pi$$
$$828$$ 0 0
$$829$$ −2.80835e30 −0.636251 −0.318126 0.948049i $$-0.603053\pi$$
−0.318126 + 0.948049i $$0.603053\pi$$
$$830$$ 0 0
$$831$$ −4.97872e30 −1.09978
$$832$$ 0 0
$$833$$ 3.67544e30 0.791656
$$834$$ 0 0
$$835$$ −1.10940e30 −0.233012
$$836$$ 0 0
$$837$$ 1.06820e30 0.218793
$$838$$ 0 0
$$839$$ 5.63065e30 1.12475 0.562377 0.826881i $$-0.309887\pi$$
0.562377 + 0.826881i $$0.309887\pi$$
$$840$$ 0 0
$$841$$ −3.76124e30 −0.732779
$$842$$ 0 0
$$843$$ 1.18748e30 0.225650
$$844$$ 0 0
$$845$$ 2.00357e30 0.371372
$$846$$ 0 0
$$847$$ −3.95103e30 −0.714389
$$848$$ 0 0
$$849$$ −8.20714e30 −1.44764
$$850$$ 0 0
$$851$$ −4.97962e30 −0.856911
$$852$$ 0 0
$$853$$ 4.25429e30 0.714269 0.357135 0.934053i $$-0.383754\pi$$
0.357135 + 0.934053i $$0.383754\pi$$
$$854$$ 0 0
$$855$$ 7.68252e29 0.125852
$$856$$ 0 0
$$857$$ −5.42388e29 −0.0866984 −0.0433492 0.999060i $$-0.513803\pi$$
−0.0433492 + 0.999060i $$0.513803\pi$$
$$858$$ 0 0
$$859$$ −3.77718e30 −0.589169 −0.294585 0.955625i $$-0.595181\pi$$
−0.294585 + 0.955625i $$0.595181\pi$$
$$860$$ 0 0
$$861$$ −8.80116e30 −1.33970
$$862$$ 0 0
$$863$$ 9.65328e30 1.43404 0.717021 0.697052i $$-0.245505\pi$$
0.717021 + 0.697052i $$0.245505\pi$$
$$864$$ 0 0
$$865$$ 1.75562e30 0.254543
$$866$$ 0 0
$$867$$ −1.01421e31 −1.43524
$$868$$ 0 0
$$869$$ 8.60655e30 1.18883
$$870$$ 0 0
$$871$$ −7.74465e30 −1.04427
$$872$$ 0 0
$$873$$ −3.97677e29 −0.0523456
$$874$$ 0 0
$$875$$ 3.54337e30 0.455336
$$876$$ 0 0
$$877$$ −8.26757e30 −1.03725 −0.518623 0.855003i $$-0.673555\pi$$
−0.518623 + 0.855003i $$0.673555\pi$$
$$878$$ 0 0
$$879$$ 2.13032e30 0.260953
$$880$$ 0 0
$$881$$ 3.71573e30 0.444423 0.222212 0.974998i $$-0.428672\pi$$
0.222212 + 0.974998i $$0.428672\pi$$
$$882$$ 0 0
$$883$$ 3.53555e29 0.0412923 0.0206462 0.999787i $$-0.493428\pi$$
0.0206462 + 0.999787i $$0.493428\pi$$
$$884$$ 0 0
$$885$$ 1.60640e31 1.83210
$$886$$ 0 0
$$887$$ 7.78376e28 0.00866944 0.00433472 0.999991i $$-0.498620\pi$$
0.00433472 + 0.999991i $$0.498620\pi$$
$$888$$ 0 0
$$889$$ 3.98302e30 0.433256
$$890$$ 0 0
$$891$$ 1.18934e31 1.26355
$$892$$ 0 0
$$893$$ −1.25927e31 −1.30671
$$894$$ 0 0
$$895$$ −1.27261e30 −0.128990
$$896$$ 0 0
$$897$$ −5.92630e30 −0.586763
$$898$$ 0 0
$$899$$ 1.12098e30 0.108423
$$900$$ 0 0
$$901$$ 1.26751e31 1.19768
$$902$$ 0 0
$$903$$ 6.96465e30 0.642951
$$904$$ 0 0
$$905$$ 1.24066e30 0.111904
$$906$$ 0 0
$$907$$ 7.49286e30 0.660345 0.330173 0.943921i $$-0.392893\pi$$
0.330173 + 0.943921i $$0.392893\pi$$
$$908$$ 0 0
$$909$$ 1.98823e30 0.171217
$$910$$ 0 0
$$911$$ 1.85880e31 1.56419 0.782096 0.623159i $$-0.214151\pi$$
0.782096 + 0.623159i $$0.214151\pi$$
$$912$$ 0 0
$$913$$ −2.58457e31 −2.12543
$$914$$ 0 0
$$915$$ 1.51260e31 1.21563
$$916$$ 0 0
$$917$$ 7.48750e30 0.588110
$$918$$ 0 0
$$919$$ −1.21998e31 −0.936573 −0.468286 0.883577i $$-0.655128\pi$$
−0.468286 + 0.883577i $$0.655128\pi$$
$$920$$ 0 0
$$921$$ −3.68576e30 −0.276567
$$922$$ 0 0
$$923$$ 1.27319e31 0.933847
$$924$$ 0 0
$$925$$ −2.38533e31 −1.71025
$$926$$ 0 0
$$927$$ −6.27317e29 −0.0439693
$$928$$ 0 0
$$929$$ −1.53725e31 −1.05337 −0.526684 0.850061i $$-0.676565\pi$$
−0.526684 + 0.850061i $$0.676565\pi$$
$$930$$ 0 0
$$931$$ −6.10449e30 −0.408957
$$932$$ 0 0
$$933$$ −6.82298e30 −0.446907
$$934$$ 0 0
$$935$$ −5.45122e31 −3.49118
$$936$$ 0 0
$$937$$ 1.67237e31 1.04729 0.523645 0.851937i $$-0.324572\pi$$
0.523645 + 0.851937i $$0.324572\pi$$
$$938$$ 0 0
$$939$$ 8.83157e30 0.540816
$$940$$ 0 0
$$941$$ 2.77704e30 0.166299 0.0831497 0.996537i $$-0.473502\pi$$
0.0831497 + 0.996537i $$0.473502\pi$$
$$942$$ 0 0
$$943$$ −2.41373e31 −1.41357
$$944$$ 0 0
$$945$$ 2.00206e31 1.14668
$$946$$ 0 0
$$947$$ 1.06419e30 0.0596132 0.0298066 0.999556i $$-0.490511\pi$$
0.0298066 + 0.999556i $$0.490511\pi$$
$$948$$ 0 0
$$949$$ −3.61013e30 −0.197800
$$950$$ 0 0
$$951$$ 2.90316e30 0.155588
$$952$$ 0 0
$$953$$ 2.76399e31 1.44897 0.724487 0.689289i $$-0.242077\pi$$
0.724487 + 0.689289i $$0.242077\pi$$
$$954$$ 0 0
$$955$$ 1.56550e31 0.802818
$$956$$ 0 0
$$957$$ 1.38664e31 0.695648
$$958$$ 0 0
$$959$$ −1.46142e31 −0.717263
$$960$$ 0 0
$$961$$ −1.99094e31 −0.956008
$$962$$ 0 0
$$963$$ −8.06980e29 −0.0379129
$$964$$ 0 0
$$965$$ −8.56254e30 −0.393610
$$966$$ 0 0
$$967$$ 1.28234e31 0.576800 0.288400 0.957510i $$-0.406877\pi$$
0.288400 + 0.957510i $$0.406877\pi$$
$$968$$ 0 0
$$969$$ 2.79858e31 1.23179
$$970$$ 0 0
$$971$$ 7.42766e30 0.319926 0.159963 0.987123i $$-0.448862\pi$$
0.159963 + 0.987123i $$0.448862\pi$$
$$972$$ 0 0
$$973$$ −1.04692e30 −0.0441296
$$974$$ 0 0
$$975$$ −2.83881e31 −1.17108
$$976$$ 0 0
$$977$$ −2.93176e31 −1.18368 −0.591842 0.806054i $$-0.701599\pi$$
−0.591842 + 0.806054i $$0.701599\pi$$
$$978$$ 0 0
$$979$$ −3.20706e31 −1.26733
$$980$$ 0 0
$$981$$ 2.67682e29 0.0103537
$$982$$ 0 0
$$983$$ 4.75461e31 1.80013 0.900064 0.435758i $$-0.143520\pi$$
0.900064 + 0.435758i $$0.143520\pi$$
$$984$$ 0 0
$$985$$ 2.68487e31 0.995044
$$986$$ 0 0
$$987$$ −2.95423e31 −1.07180
$$988$$ 0 0
$$989$$ 1.91006e31 0.678401
$$990$$ 0 0
$$991$$ 4.88614e31 1.69900 0.849498 0.527592i $$-0.176905\pi$$
0.849498 + 0.527592i $$0.176905\pi$$
$$992$$ 0 0
$$993$$ 6.41878e30 0.218517
$$994$$ 0 0
$$995$$ 8.35491e31 2.78484
$$996$$ 0 0
$$997$$ 5.22919e31 1.70661 0.853306 0.521410i $$-0.174594\pi$$
0.853306 + 0.521410i $$0.174594\pi$$
$$998$$ 0 0
$$999$$ −3.94756e31 −1.26151
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 8.22.a.b.1.1 3
3.2 odd 2 72.22.a.f.1.3 3
4.3 odd 2 16.22.a.f.1.3 3
8.3 odd 2 64.22.a.m.1.1 3
8.5 even 2 64.22.a.l.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.1 3 1.1 even 1 trivial
16.22.a.f.1.3 3 4.3 odd 2
64.22.a.l.1.3 3 8.5 even 2
64.22.a.m.1.1 3 8.3 odd 2
72.22.a.f.1.3 3 3.2 odd 2