# Properties

 Label 16.22.a.f.1.1 Level $16$ Weight $22$ Character 16.1 Self dual yes Analytic conductor $44.716$ Analytic rank $1$ 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] = [16,22,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, 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("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$44.7163750859$$ Analytic rank: $$1$$ 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: no (minimal twist has level 8) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$21.2235$$ of defining polynomial Character $$\chi$$ $$=$$ 16.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-201833. q^{3} -2.11555e7 q^{5} -7.32981e8 q^{7} +3.02761e10 q^{9} +O(q^{10})$$ $$q-201833. q^{3} -2.11555e7 q^{5} -7.32981e8 q^{7} +3.02761e10 q^{9} +5.59634e9 q^{11} -6.30207e10 q^{13} +4.26988e12 q^{15} +1.35490e13 q^{17} -1.39028e13 q^{19} +1.47940e14 q^{21} +2.80711e14 q^{23} -2.92803e13 q^{25} -3.99948e15 q^{27} +1.19637e15 q^{29} -3.88487e15 q^{31} -1.12953e15 q^{33} +1.55066e16 q^{35} +2.72004e16 q^{37} +1.27197e16 q^{39} -6.89865e16 q^{41} -3.24557e16 q^{43} -6.40508e17 q^{45} +2.07417e17 q^{47} -2.12844e16 q^{49} -2.73464e18 q^{51} -6.38971e17 q^{53} -1.18394e17 q^{55} +2.80603e18 q^{57} +3.04610e18 q^{59} -5.64497e18 q^{61} -2.21918e19 q^{63} +1.33324e18 q^{65} +3.96718e18 q^{67} -5.66566e19 q^{69} +2.61878e19 q^{71} +1.37659e19 q^{73} +5.90972e18 q^{75} -4.10201e18 q^{77} +1.18258e20 q^{79} +4.90526e20 q^{81} +1.60950e19 q^{83} -2.86637e20 q^{85} -2.41466e20 q^{87} -2.30103e20 q^{89} +4.61930e19 q^{91} +7.84094e20 q^{93} +2.94120e20 q^{95} +2.72050e20 q^{97} +1.69436e20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 96764 q^{3} - 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9}+O(q^{10})$$ 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
$$3$$ −201833. −1.97342 −0.986708 0.162503i $$-0.948043\pi$$
−0.986708 + 0.162503i $$0.948043\pi$$
$$4$$ 0 0
$$5$$ −2.11555e7 −0.968811 −0.484406 0.874844i $$-0.660964\pi$$
−0.484406 + 0.874844i $$0.660964\pi$$
$$6$$ 0 0
$$7$$ −7.32981e8 −0.980761 −0.490381 0.871508i $$-0.663142\pi$$
−0.490381 + 0.871508i $$0.663142\pi$$
$$8$$ 0 0
$$9$$ 3.02761e10 2.89437
$$10$$ 0 0
$$11$$ 5.59634e9 0.0650551 0.0325275 0.999471i $$-0.489644\pi$$
0.0325275 + 0.999471i $$0.489644\pi$$
$$12$$ 0 0
$$13$$ −6.30207e10 −0.126788 −0.0633940 0.997989i $$-0.520192\pi$$
−0.0633940 + 0.997989i $$0.520192\pi$$
$$14$$ 0 0
$$15$$ 4.26988e12 1.91187
$$16$$ 0 0
$$17$$ 1.35490e13 1.63003 0.815013 0.579443i $$-0.196730\pi$$
0.815013 + 0.579443i $$0.196730\pi$$
$$18$$ 0 0
$$19$$ −1.39028e13 −0.520221 −0.260111 0.965579i $$-0.583759\pi$$
−0.260111 + 0.965579i $$0.583759\pi$$
$$20$$ 0 0
$$21$$ 1.47940e14 1.93545
$$22$$ 0 0
$$23$$ 2.80711e14 1.41292 0.706458 0.707754i $$-0.250292\pi$$
0.706458 + 0.707754i $$0.250292\pi$$
$$24$$ 0 0
$$25$$ −2.92803e13 −0.0614052
$$26$$ 0 0
$$27$$ −3.99948e15 −3.73838
$$28$$ 0 0
$$29$$ 1.19637e15 0.528063 0.264031 0.964514i $$-0.414948\pi$$
0.264031 + 0.964514i $$0.414948\pi$$
$$30$$ 0 0
$$31$$ −3.88487e15 −0.851292 −0.425646 0.904890i $$-0.639953\pi$$
−0.425646 + 0.904890i $$0.639953\pi$$
$$32$$ 0 0
$$33$$ −1.12953e15 −0.128381
$$34$$ 0 0
$$35$$ 1.55066e16 0.950173
$$36$$ 0 0
$$37$$ 2.72004e16 0.929944 0.464972 0.885325i $$-0.346064\pi$$
0.464972 + 0.885325i $$0.346064\pi$$
$$38$$ 0 0
$$39$$ 1.27197e16 0.250205
$$40$$ 0 0
$$41$$ −6.89865e16 −0.802663 −0.401332 0.915933i $$-0.631453\pi$$
−0.401332 + 0.915933i $$0.631453\pi$$
$$42$$ 0 0
$$43$$ −3.24557e16 −0.229020 −0.114510 0.993422i $$-0.536530\pi$$
−0.114510 + 0.993422i $$0.536530\pi$$
$$44$$ 0 0
$$45$$ −6.40508e17 −2.80410
$$46$$ 0 0
$$47$$ 2.07417e17 0.575196 0.287598 0.957751i $$-0.407143\pi$$
0.287598 + 0.957751i $$0.407143\pi$$
$$48$$ 0 0
$$49$$ −2.12844e16 −0.0381069
$$50$$ 0 0
$$51$$ −2.73464e18 −3.21672
$$52$$ 0 0
$$53$$ −6.38971e17 −0.501862 −0.250931 0.968005i $$-0.580737\pi$$
−0.250931 + 0.968005i $$0.580737\pi$$
$$54$$ 0 0
$$55$$ −1.18394e17 −0.0630261
$$56$$ 0 0
$$57$$ 2.80603e18 1.02661
$$58$$ 0 0
$$59$$ 3.04610e18 0.775887 0.387944 0.921683i $$-0.373186\pi$$
0.387944 + 0.921683i $$0.373186\pi$$
$$60$$ 0 0
$$61$$ −5.64497e18 −1.01321 −0.506604 0.862179i $$-0.669099\pi$$
−0.506604 + 0.862179i $$0.669099\pi$$
$$62$$ 0 0
$$63$$ −2.21918e19 −2.83869
$$64$$ 0 0
$$65$$ 1.33324e18 0.122834
$$66$$ 0 0
$$67$$ 3.96718e18 0.265886 0.132943 0.991124i $$-0.457557\pi$$
0.132943 + 0.991124i $$0.457557\pi$$
$$68$$ 0 0
$$69$$ −5.66566e19 −2.78827
$$70$$ 0 0
$$71$$ 2.61878e19 0.954743 0.477371 0.878702i $$-0.341590\pi$$
0.477371 + 0.878702i $$0.341590\pi$$
$$72$$ 0 0
$$73$$ 1.37659e19 0.374899 0.187449 0.982274i $$-0.439978\pi$$
0.187449 + 0.982274i $$0.439978\pi$$
$$74$$ 0 0
$$75$$ 5.90972e18 0.121178
$$76$$ 0 0
$$77$$ −4.10201e18 −0.0638035
$$78$$ 0 0
$$79$$ 1.18258e20 1.40523 0.702615 0.711570i $$-0.252016\pi$$
0.702615 + 0.711570i $$0.252016\pi$$
$$80$$ 0 0
$$81$$ 4.90526e20 4.48301
$$82$$ 0 0
$$83$$ 1.60950e19 0.113860 0.0569301 0.998378i $$-0.481869\pi$$
0.0569301 + 0.998378i $$0.481869\pi$$
$$84$$ 0 0
$$85$$ −2.86637e20 −1.57919
$$86$$ 0 0
$$87$$ −2.41466e20 −1.04209
$$88$$ 0 0
$$89$$ −2.30103e20 −0.782217 −0.391109 0.920345i $$-0.627908\pi$$
−0.391109 + 0.920345i $$0.627908\pi$$
$$90$$ 0 0
$$91$$ 4.61930e19 0.124349
$$92$$ 0 0
$$93$$ 7.84094e20 1.67995
$$94$$ 0 0
$$95$$ 2.94120e20 0.503996
$$96$$ 0 0
$$97$$ 2.72050e20 0.374581 0.187290 0.982305i $$-0.440029\pi$$
0.187290 + 0.982305i $$0.440029\pi$$
$$98$$ 0 0
$$99$$ 1.69436e20 0.188293
$$100$$ 0 0
$$101$$ −1.18197e21 −1.06471 −0.532357 0.846520i $$-0.678693\pi$$
−0.532357 + 0.846520i $$0.678693\pi$$
$$102$$ 0 0
$$103$$ 2.39849e21 1.75852 0.879258 0.476345i $$-0.158039\pi$$
0.879258 + 0.476345i $$0.158039\pi$$
$$104$$ 0 0
$$105$$ −3.12974e21 −1.87509
$$106$$ 0 0
$$107$$ −4.01203e20 −0.197167 −0.0985835 0.995129i $$-0.531431\pi$$
−0.0985835 + 0.995129i $$0.531431\pi$$
$$108$$ 0 0
$$109$$ 4.25100e21 1.71994 0.859970 0.510345i $$-0.170482\pi$$
0.859970 + 0.510345i $$0.170482\pi$$
$$110$$ 0 0
$$111$$ −5.48993e21 −1.83517
$$112$$ 0 0
$$113$$ −3.36941e21 −0.933748 −0.466874 0.884324i $$-0.654620\pi$$
−0.466874 + 0.884324i $$0.654620\pi$$
$$114$$ 0 0
$$115$$ −5.93859e21 −1.36885
$$116$$ 0 0
$$117$$ −1.90802e21 −0.366971
$$118$$ 0 0
$$119$$ −9.93118e21 −1.59867
$$120$$ 0 0
$$121$$ −7.36893e21 −0.995768
$$122$$ 0 0
$$123$$ 1.39237e22 1.58399
$$124$$ 0 0
$$125$$ 1.07072e22 1.02830
$$126$$ 0 0
$$127$$ 1.15394e20 0.00938093 0.00469046 0.999989i $$-0.498507\pi$$
0.00469046 + 0.999989i $$0.498507\pi$$
$$128$$ 0 0
$$129$$ 6.55063e21 0.451951
$$130$$ 0 0
$$131$$ −1.13322e22 −0.665223 −0.332611 0.943064i $$-0.607930\pi$$
−0.332611 + 0.943064i $$0.607930\pi$$
$$132$$ 0 0
$$133$$ 1.01905e22 0.510213
$$134$$ 0 0
$$135$$ 8.46111e22 3.62178
$$136$$ 0 0
$$137$$ 7.18696e21 0.263621 0.131810 0.991275i $$-0.457921\pi$$
0.131810 + 0.991275i $$0.457921\pi$$
$$138$$ 0 0
$$139$$ −2.34620e22 −0.739109 −0.369555 0.929209i $$-0.620490\pi$$
−0.369555 + 0.929209i $$0.620490\pi$$
$$140$$ 0 0
$$141$$ −4.18635e22 −1.13510
$$142$$ 0 0
$$143$$ −3.52686e20 −0.00824820
$$144$$ 0 0
$$145$$ −2.53098e22 −0.511593
$$146$$ 0 0
$$147$$ 4.29590e21 0.0752007
$$148$$ 0 0
$$149$$ −2.57122e22 −0.390557 −0.195278 0.980748i $$-0.562561\pi$$
−0.195278 + 0.980748i $$0.562561\pi$$
$$150$$ 0 0
$$151$$ −1.38339e23 −1.82679 −0.913393 0.407078i $$-0.866548\pi$$
−0.913393 + 0.407078i $$0.866548\pi$$
$$152$$ 0 0
$$153$$ 4.10212e23 4.71790
$$154$$ 0 0
$$155$$ 8.21865e22 0.824741
$$156$$ 0 0
$$157$$ −1.14060e23 −1.00043 −0.500215 0.865901i $$-0.666746\pi$$
−0.500215 + 0.865901i $$0.666746\pi$$
$$158$$ 0 0
$$159$$ 1.28965e23 0.990382
$$160$$ 0 0
$$161$$ −2.05756e23 −1.38573
$$162$$ 0 0
$$163$$ −6.59877e22 −0.390385 −0.195192 0.980765i $$-0.562533\pi$$
−0.195192 + 0.980765i $$0.562533\pi$$
$$164$$ 0 0
$$165$$ 2.38957e22 0.124377
$$166$$ 0 0
$$167$$ −2.09416e22 −0.0960477 −0.0480239 0.998846i $$-0.515292\pi$$
−0.0480239 + 0.998846i $$0.515292\pi$$
$$168$$ 0 0
$$169$$ −2.43093e23 −0.983925
$$170$$ 0 0
$$171$$ −4.20922e23 −1.50571
$$172$$ 0 0
$$173$$ −1.69739e23 −0.537400 −0.268700 0.963224i $$-0.586594\pi$$
−0.268700 + 0.963224i $$0.586594\pi$$
$$174$$ 0 0
$$175$$ 2.14619e22 0.0602238
$$176$$ 0 0
$$177$$ −6.14804e23 −1.53115
$$178$$ 0 0
$$179$$ −5.58531e23 −1.23620 −0.618102 0.786098i $$-0.712098\pi$$
−0.618102 + 0.786098i $$0.712098\pi$$
$$180$$ 0 0
$$181$$ −2.50056e23 −0.492506 −0.246253 0.969206i $$-0.579199\pi$$
−0.246253 + 0.969206i $$0.579199\pi$$
$$182$$ 0 0
$$183$$ 1.13934e24 1.99948
$$184$$ 0 0
$$185$$ −5.75439e23 −0.900940
$$186$$ 0 0
$$187$$ 7.58250e22 0.106041
$$188$$ 0 0
$$189$$ 2.93154e24 3.66646
$$190$$ 0 0
$$191$$ 1.66807e23 0.186794 0.0933971 0.995629i $$-0.470227\pi$$
0.0933971 + 0.995629i $$0.470227\pi$$
$$192$$ 0 0
$$193$$ 1.33244e24 1.33750 0.668751 0.743487i $$-0.266829\pi$$
0.668751 + 0.743487i $$0.266829\pi$$
$$194$$ 0 0
$$195$$ −2.69091e23 −0.242402
$$196$$ 0 0
$$197$$ −6.38870e23 −0.517032 −0.258516 0.966007i $$-0.583233\pi$$
−0.258516 + 0.966007i $$0.583233\pi$$
$$198$$ 0 0
$$199$$ −1.84623e24 −1.34378 −0.671889 0.740652i $$-0.734517\pi$$
−0.671889 + 0.740652i $$0.734517\pi$$
$$200$$ 0 0
$$201$$ −8.00707e23 −0.524705
$$202$$ 0 0
$$203$$ −8.76915e23 −0.517904
$$204$$ 0 0
$$205$$ 1.45945e24 0.777629
$$206$$ 0 0
$$207$$ 8.49883e24 4.08951
$$208$$ 0 0
$$209$$ −7.78046e22 −0.0338430
$$210$$ 0 0
$$211$$ −3.35146e24 −1.31907 −0.659536 0.751673i $$-0.729247\pi$$
−0.659536 + 0.751673i $$0.729247\pi$$
$$212$$ 0 0
$$213$$ −5.28556e24 −1.88410
$$214$$ 0 0
$$215$$ 6.86619e23 0.221877
$$216$$ 0 0
$$217$$ 2.84754e24 0.834915
$$218$$ 0 0
$$219$$ −2.77841e24 −0.739831
$$220$$ 0 0
$$221$$ −8.53869e23 −0.206668
$$222$$ 0 0
$$223$$ −4.56926e22 −0.0100611 −0.00503054 0.999987i $$-0.501601\pi$$
−0.00503054 + 0.999987i $$0.501601\pi$$
$$224$$ 0 0
$$225$$ −8.86493e23 −0.177729
$$226$$ 0 0
$$227$$ 6.05295e24 1.10585 0.552924 0.833232i $$-0.313512\pi$$
0.552924 + 0.833232i $$0.313512\pi$$
$$228$$ 0 0
$$229$$ 5.16931e24 0.861311 0.430655 0.902516i $$-0.358282\pi$$
0.430655 + 0.902516i $$0.358282\pi$$
$$230$$ 0 0
$$231$$ 8.27921e23 0.125911
$$232$$ 0 0
$$233$$ 5.21547e24 0.724530 0.362265 0.932075i $$-0.382003\pi$$
0.362265 + 0.932075i $$0.382003\pi$$
$$234$$ 0 0
$$235$$ −4.38801e24 −0.557257
$$236$$ 0 0
$$237$$ −2.38684e25 −2.77310
$$238$$ 0 0
$$239$$ −1.57703e24 −0.167749 −0.0838747 0.996476i $$-0.526730\pi$$
−0.0838747 + 0.996476i $$0.526730\pi$$
$$240$$ 0 0
$$241$$ 9.93088e24 0.967851 0.483926 0.875109i $$-0.339211\pi$$
0.483926 + 0.875109i $$0.339211\pi$$
$$242$$ 0 0
$$243$$ −5.71684e25 −5.10846
$$244$$ 0 0
$$245$$ 4.50284e23 0.0369184
$$246$$ 0 0
$$247$$ 8.76162e23 0.0659578
$$248$$ 0 0
$$249$$ −3.24850e24 −0.224693
$$250$$ 0 0
$$251$$ 2.38505e25 1.51679 0.758393 0.651798i $$-0.225985\pi$$
0.758393 + 0.651798i $$0.225985\pi$$
$$252$$ 0 0
$$253$$ 1.57095e24 0.0919174
$$254$$ 0 0
$$255$$ 5.78528e25 3.11639
$$256$$ 0 0
$$257$$ −3.76190e25 −1.86685 −0.933425 0.358772i $$-0.883195\pi$$
−0.933425 + 0.358772i $$0.883195\pi$$
$$258$$ 0 0
$$259$$ −1.99374e25 −0.912054
$$260$$ 0 0
$$261$$ 3.62214e25 1.52841
$$262$$ 0 0
$$263$$ 2.52809e25 0.984592 0.492296 0.870428i $$-0.336158\pi$$
0.492296 + 0.870428i $$0.336158\pi$$
$$264$$ 0 0
$$265$$ 1.35178e25 0.486209
$$266$$ 0 0
$$267$$ 4.64424e25 1.54364
$$268$$ 0 0
$$269$$ 4.38239e25 1.34683 0.673414 0.739266i $$-0.264827\pi$$
0.673414 + 0.739266i $$0.264827\pi$$
$$270$$ 0 0
$$271$$ −2.43576e24 −0.0692560 −0.0346280 0.999400i $$-0.511025\pi$$
−0.0346280 + 0.999400i $$0.511025\pi$$
$$272$$ 0 0
$$273$$ −9.32326e24 −0.245392
$$274$$ 0 0
$$275$$ −1.63862e23 −0.00399472
$$276$$ 0 0
$$277$$ 4.43475e25 1.00192 0.500958 0.865472i $$-0.332981\pi$$
0.500958 + 0.865472i $$0.332981\pi$$
$$278$$ 0 0
$$279$$ −1.17619e26 −2.46396
$$280$$ 0 0
$$281$$ −8.89794e25 −1.72931 −0.864655 0.502365i $$-0.832463\pi$$
−0.864655 + 0.502365i $$0.832463\pi$$
$$282$$ 0 0
$$283$$ 2.34320e25 0.422719 0.211359 0.977408i $$-0.432211\pi$$
0.211359 + 0.977408i $$0.432211\pi$$
$$284$$ 0 0
$$285$$ −5.93631e25 −0.994594
$$286$$ 0 0
$$287$$ 5.05658e25 0.787221
$$288$$ 0 0
$$289$$ 1.14484e26 1.65698
$$290$$ 0 0
$$291$$ −5.49086e25 −0.739204
$$292$$ 0 0
$$293$$ 5.82209e25 0.729405 0.364703 0.931124i $$-0.381171\pi$$
0.364703 + 0.931124i $$0.381171\pi$$
$$294$$ 0 0
$$295$$ −6.44420e25 −0.751688
$$296$$ 0 0
$$297$$ −2.23824e25 −0.243201
$$298$$ 0 0
$$299$$ −1.76906e25 −0.179141
$$300$$ 0 0
$$301$$ 2.37894e25 0.224614
$$302$$ 0 0
$$303$$ 2.38561e26 2.10112
$$304$$ 0 0
$$305$$ 1.19422e26 0.981607
$$306$$ 0 0
$$307$$ −1.92535e26 −1.47760 −0.738799 0.673926i $$-0.764607\pi$$
−0.738799 + 0.673926i $$0.764607\pi$$
$$308$$ 0 0
$$309$$ −4.84093e26 −3.47028
$$310$$ 0 0
$$311$$ −1.71812e26 −1.15098 −0.575491 0.817808i $$-0.695189\pi$$
−0.575491 + 0.817808i $$0.695189\pi$$
$$312$$ 0 0
$$313$$ −3.10727e25 −0.194609 −0.0973045 0.995255i $$-0.531022\pi$$
−0.0973045 + 0.995255i $$0.531022\pi$$
$$314$$ 0 0
$$315$$ 4.69480e26 2.75015
$$316$$ 0 0
$$317$$ −2.10055e26 −1.15136 −0.575680 0.817675i $$-0.695262\pi$$
−0.575680 + 0.817675i $$0.695262\pi$$
$$318$$ 0 0
$$319$$ 6.69528e24 0.0343532
$$320$$ 0 0
$$321$$ 8.09759e25 0.389093
$$322$$ 0 0
$$323$$ −1.88369e26 −0.847974
$$324$$ 0 0
$$325$$ 1.84526e24 0.00778544
$$326$$ 0 0
$$327$$ −8.57992e26 −3.39416
$$328$$ 0 0
$$329$$ −1.52033e26 −0.564130
$$330$$ 0 0
$$331$$ 2.36914e26 0.824891 0.412446 0.910982i $$-0.364675\pi$$
0.412446 + 0.910982i $$0.364675\pi$$
$$332$$ 0 0
$$333$$ 8.23523e26 2.69160
$$334$$ 0 0
$$335$$ −8.39278e25 −0.257594
$$336$$ 0 0
$$337$$ −6.34353e25 −0.182902 −0.0914508 0.995810i $$-0.529150\pi$$
−0.0914508 + 0.995810i $$0.529150\pi$$
$$338$$ 0 0
$$339$$ 6.80057e26 1.84267
$$340$$ 0 0
$$341$$ −2.17411e25 −0.0553809
$$342$$ 0 0
$$343$$ 4.25005e26 1.01814
$$344$$ 0 0
$$345$$ 1.19860e27 2.70131
$$346$$ 0 0
$$347$$ 2.30760e26 0.489442 0.244721 0.969593i $$-0.421304\pi$$
0.244721 + 0.969593i $$0.421304\pi$$
$$348$$ 0 0
$$349$$ −6.64125e26 −1.32612 −0.663060 0.748566i $$-0.730742\pi$$
−0.663060 + 0.748566i $$0.730742\pi$$
$$350$$ 0 0
$$351$$ 2.52050e26 0.473982
$$352$$ 0 0
$$353$$ −7.72081e26 −1.36782 −0.683909 0.729567i $$-0.739722\pi$$
−0.683909 + 0.729567i $$0.739722\pi$$
$$354$$ 0 0
$$355$$ −5.54017e26 −0.924965
$$356$$ 0 0
$$357$$ 2.00444e27 3.15483
$$358$$ 0 0
$$359$$ −8.35001e26 −1.23935 −0.619677 0.784857i $$-0.712736\pi$$
−0.619677 + 0.784857i $$0.712736\pi$$
$$360$$ 0 0
$$361$$ −5.20923e26 −0.729370
$$362$$ 0 0
$$363$$ 1.48729e27 1.96506
$$364$$ 0 0
$$365$$ −2.91225e26 −0.363206
$$366$$ 0 0
$$367$$ 4.88336e26 0.575075 0.287538 0.957769i $$-0.407163\pi$$
0.287538 + 0.957769i $$0.407163\pi$$
$$368$$ 0 0
$$369$$ −2.08864e27 −2.32321
$$370$$ 0 0
$$371$$ 4.68353e26 0.492207
$$372$$ 0 0
$$373$$ −1.33813e27 −1.32909 −0.664547 0.747247i $$-0.731375\pi$$
−0.664547 + 0.747247i $$0.731375\pi$$
$$374$$ 0 0
$$375$$ −2.16106e27 −2.02927
$$376$$ 0 0
$$377$$ −7.53960e25 −0.0669520
$$378$$ 0 0
$$379$$ −7.26568e25 −0.0610329 −0.0305165 0.999534i $$-0.509715\pi$$
−0.0305165 + 0.999534i $$0.509715\pi$$
$$380$$ 0 0
$$381$$ −2.32904e25 −0.0185125
$$382$$ 0 0
$$383$$ −1.24472e27 −0.936449 −0.468225 0.883610i $$-0.655106\pi$$
−0.468225 + 0.883610i $$0.655106\pi$$
$$384$$ 0 0
$$385$$ 8.67803e25 0.0618135
$$386$$ 0 0
$$387$$ −9.82635e26 −0.662868
$$388$$ 0 0
$$389$$ 2.14241e27 1.36909 0.684544 0.728972i $$-0.260002\pi$$
0.684544 + 0.728972i $$0.260002\pi$$
$$390$$ 0 0
$$391$$ 3.80336e27 2.30309
$$392$$ 0 0
$$393$$ 2.28722e27 1.31276
$$394$$ 0 0
$$395$$ −2.50182e27 −1.36140
$$396$$ 0 0
$$397$$ 2.44762e27 1.26312 0.631560 0.775327i $$-0.282415\pi$$
0.631560 + 0.775327i $$0.282415\pi$$
$$398$$ 0 0
$$399$$ −2.05677e27 −1.00686
$$400$$ 0 0
$$401$$ 2.80970e27 1.30510 0.652550 0.757746i $$-0.273699\pi$$
0.652550 + 0.757746i $$0.273699\pi$$
$$402$$ 0 0
$$403$$ 2.44827e26 0.107934
$$404$$ 0 0
$$405$$ −1.03774e28 −4.34319
$$406$$ 0 0
$$407$$ 1.52223e26 0.0604976
$$408$$ 0 0
$$409$$ −3.81600e27 −1.44050 −0.720250 0.693714i $$-0.755973\pi$$
−0.720250 + 0.693714i $$0.755973\pi$$
$$410$$ 0 0
$$411$$ −1.45057e27 −0.520233
$$412$$ 0 0
$$413$$ −2.23274e27 −0.760960
$$414$$ 0 0
$$415$$ −3.40499e26 −0.110309
$$416$$ 0 0
$$417$$ 4.73539e27 1.45857
$$418$$ 0 0
$$419$$ 4.13782e26 0.121206 0.0606031 0.998162i $$-0.480698\pi$$
0.0606031 + 0.998162i $$0.480698\pi$$
$$420$$ 0 0
$$421$$ 4.16773e27 1.16128 0.580641 0.814160i $$-0.302802\pi$$
0.580641 + 0.814160i $$0.302802\pi$$
$$422$$ 0 0
$$423$$ 6.27978e27 1.66483
$$424$$ 0 0
$$425$$ −3.96719e26 −0.100092
$$426$$ 0 0
$$427$$ 4.13766e27 0.993715
$$428$$ 0 0
$$429$$ 7.11835e25 0.0162771
$$430$$ 0 0
$$431$$ −3.31148e27 −0.721125 −0.360563 0.932735i $$-0.617415\pi$$
−0.360563 + 0.932735i $$0.617415\pi$$
$$432$$ 0 0
$$433$$ 7.71199e27 1.59972 0.799859 0.600188i $$-0.204908\pi$$
0.799859 + 0.600188i $$0.204908\pi$$
$$434$$ 0 0
$$435$$ 5.10835e27 1.00959
$$436$$ 0 0
$$437$$ −3.90265e27 −0.735029
$$438$$ 0 0
$$439$$ 7.40474e27 1.32933 0.664665 0.747142i $$-0.268574\pi$$
0.664665 + 0.747142i $$0.268574\pi$$
$$440$$ 0 0
$$441$$ −6.44411e26 −0.110295
$$442$$ 0 0
$$443$$ −2.59541e27 −0.423611 −0.211805 0.977312i $$-0.567934\pi$$
−0.211805 + 0.977312i $$0.567934\pi$$
$$444$$ 0 0
$$445$$ 4.86795e27 0.757821
$$446$$ 0 0
$$447$$ 5.18958e27 0.770731
$$448$$ 0 0
$$449$$ −8.63721e27 −1.22402 −0.612008 0.790852i $$-0.709638\pi$$
−0.612008 + 0.790852i $$0.709638\pi$$
$$450$$ 0 0
$$451$$ −3.86072e26 −0.0522173
$$452$$ 0 0
$$453$$ 2.79214e28 3.60501
$$454$$ 0 0
$$455$$ −9.77238e26 −0.120470
$$456$$ 0 0
$$457$$ −7.22330e27 −0.850385 −0.425193 0.905103i $$-0.639794\pi$$
−0.425193 + 0.905103i $$0.639794\pi$$
$$458$$ 0 0
$$459$$ −5.41890e28 −6.09366
$$460$$ 0 0
$$461$$ −6.67275e27 −0.716878 −0.358439 0.933553i $$-0.616691\pi$$
−0.358439 + 0.933553i $$0.616691\pi$$
$$462$$ 0 0
$$463$$ 7.22059e27 0.741263 0.370632 0.928780i $$-0.379141\pi$$
0.370632 + 0.928780i $$0.379141\pi$$
$$464$$ 0 0
$$465$$ −1.65879e28 −1.62756
$$466$$ 0 0
$$467$$ 3.14336e27 0.294827 0.147413 0.989075i $$-0.452905\pi$$
0.147413 + 0.989075i $$0.452905\pi$$
$$468$$ 0 0
$$469$$ −2.90787e27 −0.260771
$$470$$ 0 0
$$471$$ 2.30210e28 1.97427
$$472$$ 0 0
$$473$$ −1.81634e26 −0.0148989
$$474$$ 0 0
$$475$$ 4.07076e26 0.0319443
$$476$$ 0 0
$$477$$ −1.93456e28 −1.45257
$$478$$ 0 0
$$479$$ −1.98918e28 −1.42939 −0.714696 0.699435i $$-0.753435\pi$$
−0.714696 + 0.699435i $$0.753435\pi$$
$$480$$ 0 0
$$481$$ −1.71419e27 −0.117906
$$482$$ 0 0
$$483$$ 4.15282e28 2.73463
$$484$$ 0 0
$$485$$ −5.75536e27 −0.362898
$$486$$ 0 0
$$487$$ −1.49710e28 −0.904058 −0.452029 0.892003i $$-0.649300\pi$$
−0.452029 + 0.892003i $$0.649300\pi$$
$$488$$ 0 0
$$489$$ 1.33185e28 0.770392
$$490$$ 0 0
$$491$$ 5.87261e27 0.325444 0.162722 0.986672i $$-0.447973\pi$$
0.162722 + 0.986672i $$0.447973\pi$$
$$492$$ 0 0
$$493$$ 1.62096e28 0.860756
$$494$$ 0 0
$$495$$ −3.58450e27 −0.182421
$$496$$ 0 0
$$497$$ −1.91952e28 −0.936375
$$498$$ 0 0
$$499$$ −3.56403e28 −1.66681 −0.833404 0.552665i $$-0.813611\pi$$
−0.833404 + 0.552665i $$0.813611\pi$$
$$500$$ 0 0
$$501$$ 4.22670e27 0.189542
$$502$$ 0 0
$$503$$ 1.66833e28 0.717495 0.358748 0.933435i $$-0.383204\pi$$
0.358748 + 0.933435i $$0.383204\pi$$
$$504$$ 0 0
$$505$$ 2.50053e28 1.03151
$$506$$ 0 0
$$507$$ 4.90641e28 1.94169
$$508$$ 0 0
$$509$$ −3.66847e28 −1.39299 −0.696495 0.717562i $$-0.745258\pi$$
−0.696495 + 0.717562i $$0.745258\pi$$
$$510$$ 0 0
$$511$$ −1.00901e28 −0.367686
$$512$$ 0 0
$$513$$ 5.56037e28 1.94478
$$514$$ 0 0
$$515$$ −5.07413e28 −1.70367
$$516$$ 0 0
$$517$$ 1.16078e27 0.0374194
$$518$$ 0 0
$$519$$ 3.42589e28 1.06051
$$520$$ 0 0
$$521$$ −3.16512e28 −0.941009 −0.470505 0.882398i $$-0.655928\pi$$
−0.470505 + 0.882398i $$0.655928\pi$$
$$522$$ 0 0
$$523$$ −4.41639e28 −1.26125 −0.630623 0.776090i $$-0.717200\pi$$
−0.630623 + 0.776090i $$0.717200\pi$$
$$524$$ 0 0
$$525$$ −4.33171e27 −0.118847
$$526$$ 0 0
$$527$$ −5.26362e28 −1.38763
$$528$$ 0 0
$$529$$ 3.93269e28 0.996334
$$530$$ 0 0
$$531$$ 9.22243e28 2.24571
$$532$$ 0 0
$$533$$ 4.34758e27 0.101768
$$534$$ 0 0
$$535$$ 8.48766e27 0.191018
$$536$$ 0 0
$$537$$ 1.12730e29 2.43954
$$538$$ 0 0
$$539$$ −1.19115e26 −0.00247905
$$540$$ 0 0
$$541$$ 8.65965e28 1.73352 0.866760 0.498725i $$-0.166198\pi$$
0.866760 + 0.498725i $$0.166198\pi$$
$$542$$ 0 0
$$543$$ 5.04694e28 0.971919
$$544$$ 0 0
$$545$$ −8.99322e28 −1.66630
$$546$$ 0 0
$$547$$ 6.71643e28 1.19749 0.598745 0.800940i $$-0.295666\pi$$
0.598745 + 0.800940i $$0.295666\pi$$
$$548$$ 0 0
$$549$$ −1.70908e29 −2.93260
$$550$$ 0 0
$$551$$ −1.66328e28 −0.274710
$$552$$ 0 0
$$553$$ −8.66811e28 −1.37820
$$554$$ 0 0
$$555$$ 1.16142e29 1.77793
$$556$$ 0 0
$$557$$ 6.28272e28 0.926122 0.463061 0.886326i $$-0.346751\pi$$
0.463061 + 0.886326i $$0.346751\pi$$
$$558$$ 0 0
$$559$$ 2.04538e27 0.0290369
$$560$$ 0 0
$$561$$ −1.53040e28 −0.209264
$$562$$ 0 0
$$563$$ −9.00019e28 −1.18553 −0.592766 0.805375i $$-0.701964\pi$$
−0.592766 + 0.805375i $$0.701964\pi$$
$$564$$ 0 0
$$565$$ 7.12816e28 0.904625
$$566$$ 0 0
$$567$$ −3.59547e29 −4.39676
$$568$$ 0 0
$$569$$ −1.19277e29 −1.40565 −0.702824 0.711364i $$-0.748078\pi$$
−0.702824 + 0.711364i $$0.748078\pi$$
$$570$$ 0 0
$$571$$ 1.14674e29 1.30253 0.651265 0.758851i $$-0.274239\pi$$
0.651265 + 0.758851i $$0.274239\pi$$
$$572$$ 0 0
$$573$$ −3.36671e28 −0.368623
$$574$$ 0 0
$$575$$ −8.21928e27 −0.0867604
$$576$$ 0 0
$$577$$ −2.51949e28 −0.256429 −0.128214 0.991746i $$-0.540925\pi$$
−0.128214 + 0.991746i $$0.540925\pi$$
$$578$$ 0 0
$$579$$ −2.68929e29 −2.63945
$$580$$ 0 0
$$581$$ −1.17974e28 −0.111670
$$582$$ 0 0
$$583$$ −3.57590e27 −0.0326487
$$584$$ 0 0
$$585$$ 4.03653e28 0.355526
$$586$$ 0 0
$$587$$ 4.22050e28 0.358644 0.179322 0.983790i $$-0.442610\pi$$
0.179322 + 0.983790i $$0.442610\pi$$
$$588$$ 0 0
$$589$$ 5.40104e28 0.442860
$$590$$ 0 0
$$591$$ 1.28945e29 1.02032
$$592$$ 0 0
$$593$$ −1.61710e29 −1.23499 −0.617495 0.786575i $$-0.711852\pi$$
−0.617495 + 0.786575i $$0.711852\pi$$
$$594$$ 0 0
$$595$$ 2.10100e29 1.54881
$$596$$ 0 0
$$597$$ 3.72629e29 2.65183
$$598$$ 0 0
$$599$$ 8.18099e28 0.562113 0.281057 0.959691i $$-0.409315\pi$$
0.281057 + 0.959691i $$0.409315\pi$$
$$600$$ 0 0
$$601$$ 2.09704e28 0.139131 0.0695657 0.997577i $$-0.477839\pi$$
0.0695657 + 0.997577i $$0.477839\pi$$
$$602$$ 0 0
$$603$$ 1.20111e29 0.769574
$$604$$ 0 0
$$605$$ 1.55894e29 0.964711
$$606$$ 0 0
$$607$$ −1.39211e29 −0.832131 −0.416066 0.909335i $$-0.636591\pi$$
−0.416066 + 0.909335i $$0.636591\pi$$
$$608$$ 0 0
$$609$$ 1.76990e29 1.02204
$$610$$ 0 0
$$611$$ −1.30715e28 −0.0729280
$$612$$ 0 0
$$613$$ −1.57634e29 −0.849795 −0.424898 0.905241i $$-0.639690\pi$$
−0.424898 + 0.905241i $$0.639690\pi$$
$$614$$ 0 0
$$615$$ −2.94564e29 −1.53459
$$616$$ 0 0
$$617$$ 1.96390e29 0.988836 0.494418 0.869224i $$-0.335381\pi$$
0.494418 + 0.869224i $$0.335381\pi$$
$$618$$ 0 0
$$619$$ 2.32870e29 1.13334 0.566671 0.823944i $$-0.308231\pi$$
0.566671 + 0.823944i $$0.308231\pi$$
$$620$$ 0 0
$$621$$ −1.12270e30 −5.28202
$$622$$ 0 0
$$623$$ 1.68661e29 0.767168
$$624$$ 0 0
$$625$$ −2.12554e29 −0.934824
$$626$$ 0 0
$$627$$ 1.57035e28 0.0667864
$$628$$ 0 0
$$629$$ 3.68539e29 1.51583
$$630$$ 0 0
$$631$$ 2.44143e29 0.971261 0.485631 0.874164i $$-0.338590\pi$$
0.485631 + 0.874164i $$0.338590\pi$$
$$632$$ 0 0
$$633$$ 6.76435e29 2.60308
$$634$$ 0 0
$$635$$ −2.44123e27 −0.00908835
$$636$$ 0 0
$$637$$ 1.34136e27 0.00483150
$$638$$ 0 0
$$639$$ 7.92865e29 2.76338
$$640$$ 0 0
$$641$$ −3.44014e28 −0.116029 −0.0580145 0.998316i $$-0.518477\pi$$
−0.0580145 + 0.998316i $$0.518477\pi$$
$$642$$ 0 0
$$643$$ −4.55690e29 −1.48749 −0.743745 0.668463i $$-0.766952\pi$$
−0.743745 + 0.668463i $$0.766952\pi$$
$$644$$ 0 0
$$645$$ −1.38582e29 −0.437855
$$646$$ 0 0
$$647$$ −4.99700e29 −1.52832 −0.764161 0.645026i $$-0.776847\pi$$
−0.764161 + 0.645026i $$0.776847\pi$$
$$648$$ 0 0
$$649$$ 1.70470e28 0.0504754
$$650$$ 0 0
$$651$$ −5.74726e29 −1.64763
$$652$$ 0 0
$$653$$ 2.87688e29 0.798607 0.399303 0.916819i $$-0.369252\pi$$
0.399303 + 0.916819i $$0.369252\pi$$
$$654$$ 0 0
$$655$$ 2.39740e29 0.644475
$$656$$ 0 0
$$657$$ 4.16778e29 1.08510
$$658$$ 0 0
$$659$$ −4.60817e29 −1.16207 −0.581034 0.813879i $$-0.697352\pi$$
−0.581034 + 0.813879i $$0.697352\pi$$
$$660$$ 0 0
$$661$$ 2.94237e29 0.718756 0.359378 0.933192i $$-0.382989\pi$$
0.359378 + 0.933192i $$0.382989\pi$$
$$662$$ 0 0
$$663$$ 1.72339e29 0.407841
$$664$$ 0 0
$$665$$ −2.15585e29 −0.494300
$$666$$ 0 0
$$667$$ 3.35833e29 0.746109
$$668$$ 0 0
$$669$$ 9.22226e27 0.0198547
$$670$$ 0 0
$$671$$ −3.15912e28 −0.0659143
$$672$$ 0 0
$$673$$ −4.64032e29 −0.938403 −0.469202 0.883091i $$-0.655458\pi$$
−0.469202 + 0.883091i $$0.655458\pi$$
$$674$$ 0 0
$$675$$ 1.17106e29 0.229556
$$676$$ 0 0
$$677$$ −3.90684e29 −0.742411 −0.371206 0.928551i $$-0.621056\pi$$
−0.371206 + 0.928551i $$0.621056\pi$$
$$678$$ 0 0
$$679$$ −1.99408e29 −0.367374
$$680$$ 0 0
$$681$$ −1.22168e30 −2.18230
$$682$$ 0 0
$$683$$ −1.80255e29 −0.312227 −0.156113 0.987739i $$-0.549897\pi$$
−0.156113 + 0.987739i $$0.549897\pi$$
$$684$$ 0 0
$$685$$ −1.52044e29 −0.255399
$$686$$ 0 0
$$687$$ −1.04334e30 −1.69972
$$688$$ 0 0
$$689$$ 4.02684e28 0.0636301
$$690$$ 0 0
$$691$$ 9.69163e28 0.148552 0.0742758 0.997238i $$-0.476335\pi$$
0.0742758 + 0.997238i $$0.476335\pi$$
$$692$$ 0 0
$$693$$ −1.24193e29 −0.184671
$$694$$ 0 0
$$695$$ 4.96350e29 0.716057
$$696$$ 0 0
$$697$$ −9.34700e29 −1.30836
$$698$$ 0 0
$$699$$ −1.05265e30 −1.42980
$$700$$ 0 0
$$701$$ 5.44452e29 0.717663 0.358831 0.933402i $$-0.383175\pi$$
0.358831 + 0.933402i $$0.383175\pi$$
$$702$$ 0 0
$$703$$ −3.78160e29 −0.483777
$$704$$ 0 0
$$705$$ 8.85645e29 1.09970
$$706$$ 0 0
$$707$$ 8.66363e29 1.04423
$$708$$ 0 0
$$709$$ 1.34223e30 1.57052 0.785258 0.619168i $$-0.212530\pi$$
0.785258 + 0.619168i $$0.212530\pi$$
$$710$$ 0 0
$$711$$ 3.58041e30 4.06725
$$712$$ 0 0
$$713$$ −1.09052e30 −1.20281
$$714$$ 0 0
$$715$$ 7.46126e27 0.00799095
$$716$$ 0 0
$$717$$ 3.18296e29 0.331039
$$718$$ 0 0
$$719$$ 9.27979e29 0.937313 0.468657 0.883380i $$-0.344738\pi$$
0.468657 + 0.883380i $$0.344738\pi$$
$$720$$ 0 0
$$721$$ −1.75805e30 −1.72469
$$722$$ 0 0
$$723$$ −2.00438e30 −1.90997
$$724$$ 0 0
$$725$$ −3.50300e28 −0.0324258
$$726$$ 0 0
$$727$$ −1.97796e29 −0.177871 −0.0889357 0.996037i $$-0.528347\pi$$
−0.0889357 + 0.996037i $$0.528347\pi$$
$$728$$ 0 0
$$729$$ 6.40738e30 5.59811
$$730$$ 0 0
$$731$$ −4.39744e29 −0.373308
$$732$$ 0 0
$$733$$ −8.95031e28 −0.0738323 −0.0369161 0.999318i $$-0.511753\pi$$
−0.0369161 + 0.999318i $$0.511753\pi$$
$$734$$ 0 0
$$735$$ −9.08821e28 −0.0728553
$$736$$ 0 0
$$737$$ 2.22017e28 0.0172973
$$738$$ 0 0
$$739$$ −1.59180e29 −0.120537 −0.0602687 0.998182i $$-0.519196\pi$$
−0.0602687 + 0.998182i $$0.519196\pi$$
$$740$$ 0 0
$$741$$ −1.76838e29 −0.130162
$$742$$ 0 0
$$743$$ −9.50012e28 −0.0679746 −0.0339873 0.999422i $$-0.510821\pi$$
−0.0339873 + 0.999422i $$0.510821\pi$$
$$744$$ 0 0
$$745$$ 5.43956e29 0.378376
$$746$$ 0 0
$$747$$ 4.87295e29 0.329553
$$748$$ 0 0
$$749$$ 2.94074e29 0.193374
$$750$$ 0 0
$$751$$ −2.97935e30 −1.90503 −0.952516 0.304488i $$-0.901514\pi$$
−0.952516 + 0.304488i $$0.901514\pi$$
$$752$$ 0 0
$$753$$ −4.81382e30 −2.99325
$$754$$ 0 0
$$755$$ 2.92664e30 1.76981
$$756$$ 0 0
$$757$$ 8.10128e29 0.476482 0.238241 0.971206i $$-0.423429\pi$$
0.238241 + 0.971206i $$0.423429\pi$$
$$758$$ 0 0
$$759$$ −3.17070e29 −0.181391
$$760$$ 0 0
$$761$$ 8.84872e29 0.492426 0.246213 0.969216i $$-0.420814\pi$$
0.246213 + 0.969216i $$0.420814\pi$$
$$762$$ 0 0
$$763$$ −3.11590e30 −1.68685
$$764$$ 0 0
$$765$$ −8.67826e30 −4.57075
$$766$$ 0 0
$$767$$ −1.91968e29 −0.0983732
$$768$$ 0 0
$$769$$ 2.59763e29 0.129524 0.0647620 0.997901i $$-0.479371\pi$$
0.0647620 + 0.997901i $$0.479371\pi$$
$$770$$ 0 0
$$771$$ 7.59275e30 3.68407
$$772$$ 0 0
$$773$$ 3.97566e30 1.87726 0.938630 0.344925i $$-0.112096\pi$$
0.938630 + 0.344925i $$0.112096\pi$$
$$774$$ 0 0
$$775$$ 1.13750e29 0.0522738
$$776$$ 0 0
$$777$$ 4.02402e30 1.79986
$$778$$ 0 0
$$779$$ 9.59102e29 0.417562
$$780$$ 0 0
$$781$$ 1.46556e29 0.0621109
$$782$$ 0 0
$$783$$ −4.78484e30 −1.97410
$$784$$ 0 0
$$785$$ 2.41300e30 0.969228
$$786$$ 0 0
$$787$$ −1.54080e30 −0.602577 −0.301288 0.953533i $$-0.597417\pi$$
−0.301288 + 0.953533i $$0.597417\pi$$
$$788$$ 0 0
$$789$$ −5.10251e30 −1.94301
$$790$$ 0 0
$$791$$ 2.46971e30 0.915784
$$792$$ 0 0
$$793$$ 3.55750e29 0.128463
$$794$$ 0 0
$$795$$ −2.72833e30 −0.959493
$$796$$ 0 0
$$797$$ 1.29587e29 0.0443862 0.0221931 0.999754i $$-0.492935\pi$$
0.0221931 + 0.999754i $$0.492935\pi$$
$$798$$ 0 0
$$799$$ 2.81029e30 0.937585
$$800$$ 0 0
$$801$$ −6.96663e30 −2.26403
$$802$$ 0 0
$$803$$ 7.70386e28 0.0243891
$$804$$ 0 0
$$805$$ 4.35287e30 1.34251
$$806$$ 0 0
$$807$$ −8.84510e30 −2.65785
$$808$$ 0 0
$$809$$ −2.25920e30 −0.661449 −0.330724 0.943727i $$-0.607293\pi$$
−0.330724 + 0.943727i $$0.607293\pi$$
$$810$$ 0 0
$$811$$ −4.17856e30 −1.19209 −0.596043 0.802953i $$-0.703261\pi$$
−0.596043 + 0.802953i $$0.703261\pi$$
$$812$$ 0 0
$$813$$ 4.91617e29 0.136671
$$814$$ 0 0
$$815$$ 1.39600e30 0.378209
$$816$$ 0 0
$$817$$ 4.51224e29 0.119141
$$818$$ 0 0
$$819$$ 1.39855e30 0.359911
$$820$$ 0 0
$$821$$ −1.32971e29 −0.0333543 −0.0166772 0.999861i $$-0.505309\pi$$
−0.0166772 + 0.999861i $$0.505309\pi$$
$$822$$ 0 0
$$823$$ 1.12434e30 0.274915 0.137457 0.990508i $$-0.456107\pi$$
0.137457 + 0.990508i $$0.456107\pi$$
$$824$$ 0 0
$$825$$ 3.30728e28 0.00788324
$$826$$ 0 0
$$827$$ 3.53339e30 0.821077 0.410539 0.911843i $$-0.365341\pi$$
0.410539 + 0.911843i $$0.365341\pi$$
$$828$$ 0 0
$$829$$ 3.22988e30 0.731751 0.365875 0.930664i $$-0.380770\pi$$
0.365875 + 0.930664i $$0.380770\pi$$
$$830$$ 0 0
$$831$$ −8.95077e30 −1.97720
$$832$$ 0 0
$$833$$ −2.88384e29 −0.0621152
$$834$$ 0 0
$$835$$ 4.43031e29 0.0930521
$$836$$ 0 0
$$837$$ 1.55374e31 3.18245
$$838$$ 0 0
$$839$$ −1.24571e30 −0.248838 −0.124419 0.992230i $$-0.539707\pi$$
−0.124419 + 0.992230i $$0.539707\pi$$
$$840$$ 0 0
$$841$$ −3.70155e30 −0.721150
$$842$$ 0 0
$$843$$ 1.79590e31 3.41265
$$844$$ 0 0
$$845$$ 5.14276e30 0.953237
$$846$$ 0 0
$$847$$ 5.40129e30 0.976611
$$848$$ 0 0
$$849$$ −4.72934e30 −0.834200
$$850$$ 0 0
$$851$$ 7.63544e30 1.31393
$$852$$ 0 0
$$853$$ −6.54885e30 −1.09951 −0.549757 0.835325i $$-0.685280\pi$$
−0.549757 + 0.835325i $$0.685280\pi$$
$$854$$ 0 0
$$855$$ 8.90483e30 1.45875
$$856$$ 0 0
$$857$$ 1.91673e30 0.306381 0.153191 0.988197i $$-0.451045\pi$$
0.153191 + 0.988197i $$0.451045\pi$$
$$858$$ 0 0
$$859$$ 2.28266e30 0.356051 0.178026 0.984026i $$-0.443029\pi$$
0.178026 + 0.984026i $$0.443029\pi$$
$$860$$ 0 0
$$861$$ −1.02058e31 −1.55352
$$862$$ 0 0
$$863$$ 7.87662e30 1.17011 0.585055 0.810994i $$-0.301073\pi$$
0.585055 + 0.810994i $$0.301073\pi$$
$$864$$ 0 0
$$865$$ 3.59092e30 0.520639
$$866$$ 0 0
$$867$$ −2.31067e31 −3.26992
$$868$$ 0 0
$$869$$ 6.61814e29 0.0914173
$$870$$ 0 0
$$871$$ −2.50014e29 −0.0337112
$$872$$ 0 0
$$873$$ 8.23662e30 1.08418
$$874$$ 0 0
$$875$$ −7.84817e30 −1.00852
$$876$$ 0 0
$$877$$ −1.25742e31 −1.57756 −0.788778 0.614678i $$-0.789286\pi$$
−0.788778 + 0.614678i $$0.789286\pi$$
$$878$$ 0 0
$$879$$ −1.17509e31 −1.43942
$$880$$ 0 0
$$881$$ −5.18000e29 −0.0619560 −0.0309780 0.999520i $$-0.509862\pi$$
−0.0309780 + 0.999520i $$0.509862\pi$$
$$882$$ 0 0
$$883$$ 4.26502e30 0.498120 0.249060 0.968488i $$-0.419878\pi$$
0.249060 + 0.968488i $$0.419878\pi$$
$$884$$ 0 0
$$885$$ 1.30065e31 1.48339
$$886$$ 0 0
$$887$$ −1.39563e31 −1.55444 −0.777218 0.629232i $$-0.783370\pi$$
−0.777218 + 0.629232i $$0.783370\pi$$
$$888$$ 0 0
$$889$$ −8.45819e28 −0.00920045
$$890$$ 0 0
$$891$$ 2.74515e30 0.291643
$$892$$ 0 0
$$893$$ −2.88366e30 −0.299229
$$894$$ 0 0
$$895$$ 1.18160e31 1.19765
$$896$$ 0 0
$$897$$ 3.57054e30 0.353520
$$898$$ 0 0
$$899$$ −4.64773e30 −0.449536
$$900$$ 0 0
$$901$$ −8.65743e30 −0.818048
$$902$$ 0 0
$$903$$ −4.80149e30 −0.443256
$$904$$ 0 0
$$905$$ 5.29006e30 0.477145
$$906$$ 0 0
$$907$$ 7.99799e30 0.704862 0.352431 0.935838i $$-0.385355\pi$$
0.352431 + 0.935838i $$0.385355\pi$$
$$908$$ 0 0
$$909$$ −3.57855e31 −3.08168
$$910$$ 0 0
$$911$$ −6.04758e30 −0.508907 −0.254454 0.967085i $$-0.581896\pi$$
−0.254454 + 0.967085i $$0.581896\pi$$
$$912$$ 0 0
$$913$$ 9.00733e28 0.00740718
$$914$$ 0 0
$$915$$ −2.41034e31 −1.93712
$$916$$ 0 0
$$917$$ 8.30632e30 0.652425
$$918$$ 0 0
$$919$$ −2.36741e31 −1.81744 −0.908719 0.417408i $$-0.862939\pi$$
−0.908719 + 0.417408i $$0.862939\pi$$
$$920$$ 0 0
$$921$$ 3.88598e31 2.91591
$$922$$ 0 0
$$923$$ −1.65037e30 −0.121050
$$924$$ 0 0
$$925$$ −7.96435e29 −0.0571034
$$926$$ 0 0
$$927$$ 7.26169e31 5.08980
$$928$$ 0 0
$$929$$ −2.41968e31 −1.65803 −0.829016 0.559226i $$-0.811099\pi$$
−0.829016 + 0.559226i $$0.811099\pi$$
$$930$$ 0 0
$$931$$ 2.95912e29 0.0198240
$$932$$ 0 0
$$933$$ 3.46773e31 2.27137
$$934$$ 0 0
$$935$$ −1.60412e30 −0.102734
$$936$$ 0 0
$$937$$ −8.58736e30 −0.537767 −0.268883 0.963173i $$-0.586655\pi$$
−0.268883 + 0.963173i $$0.586655\pi$$
$$938$$ 0 0
$$939$$ 6.27148e30 0.384044
$$940$$ 0 0
$$941$$ −5.47302e30 −0.327745 −0.163872 0.986482i $$-0.552399\pi$$
−0.163872 + 0.986482i $$0.552399\pi$$
$$942$$ 0 0
$$943$$ −1.93652e31 −1.13410
$$944$$ 0 0
$$945$$ −6.20183e31 −3.55211
$$946$$ 0 0
$$947$$ −2.44243e31 −1.36819 −0.684096 0.729392i $$-0.739803\pi$$
−0.684096 + 0.729392i $$0.739803\pi$$
$$948$$ 0 0
$$949$$ −8.67536e29 −0.0475327
$$950$$ 0 0
$$951$$ 4.23960e31 2.27211
$$952$$ 0 0
$$953$$ 1.04469e31 0.547661 0.273831 0.961778i $$-0.411709\pi$$
0.273831 + 0.961778i $$0.411709\pi$$
$$954$$ 0 0
$$955$$ −3.52889e30 −0.180968
$$956$$ 0 0
$$957$$ −1.35133e30 −0.0677931
$$958$$ 0 0
$$959$$ −5.26791e30 −0.258549
$$960$$ 0 0
$$961$$ −5.73329e30 −0.275301
$$962$$ 0 0
$$963$$ −1.21469e31 −0.570675
$$964$$ 0 0
$$965$$ −2.81884e31 −1.29579
$$966$$ 0 0
$$967$$ 3.41008e31 1.53386 0.766931 0.641729i $$-0.221783\pi$$
0.766931 + 0.641729i $$0.221783\pi$$
$$968$$ 0 0
$$969$$ 3.80190e31 1.67340
$$970$$ 0 0
$$971$$ −3.17245e31 −1.36645 −0.683224 0.730209i $$-0.739423\pi$$
−0.683224 + 0.730209i $$0.739423\pi$$
$$972$$ 0 0
$$973$$ 1.71972e31 0.724890
$$974$$ 0 0
$$975$$ −3.72435e29 −0.0153639
$$976$$ 0 0
$$977$$ 2.33987e31 0.944710 0.472355 0.881408i $$-0.343404\pi$$
0.472355 + 0.881408i $$0.343404\pi$$
$$978$$ 0 0
$$979$$ −1.28774e30 −0.0508872
$$980$$ 0 0
$$981$$ 1.28704e32 4.97814
$$982$$ 0 0
$$983$$ −1.15166e28 −0.000436024 0 −0.000218012 1.00000i $$-0.500069\pi$$
−0.000218012 1.00000i $$0.500069\pi$$
$$984$$ 0 0
$$985$$ 1.35156e31 0.500906
$$986$$ 0 0
$$987$$ 3.06852e31 1.11326
$$988$$ 0 0
$$989$$ −9.11067e30 −0.323586
$$990$$ 0 0
$$991$$ 1.52057e31 0.528730 0.264365 0.964423i $$-0.414838\pi$$
0.264365 + 0.964423i $$0.414838\pi$$
$$992$$ 0 0
$$993$$ −4.78170e31 −1.62785
$$994$$ 0 0
$$995$$ 3.90579e31 1.30187
$$996$$ 0 0
$$997$$ −4.15395e31 −1.35569 −0.677847 0.735203i $$-0.737087\pi$$
−0.677847 + 0.735203i $$0.737087\pi$$
$$998$$ 0 0
$$999$$ −1.08787e32 −3.47649
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 16.22.a.f.1.1 3
4.3 odd 2 8.22.a.b.1.3 3
8.3 odd 2 64.22.a.l.1.1 3
8.5 even 2 64.22.a.m.1.3 3
12.11 even 2 72.22.a.f.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.3 3 4.3 odd 2
16.22.a.f.1.1 3 1.1 even 1 trivial
64.22.a.l.1.1 3 8.3 odd 2
64.22.a.m.1.3 3 8.5 even 2
72.22.a.f.1.2 3 12.11 even 2