# Properties

 Label 95.11.d.a.94.1 Level $95$ Weight $11$ Character 95.94 Analytic conductor $60.359$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,11,Mod(94,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 95.d (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: $$60.3589390040$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 5$$ x^2 - x + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$11$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 94.1 Root $$0.500000 - 2.17945i$$ of defining polynomial Character $$\chi$$ $$=$$ 95.94 Dual form 95.11.d.a.94.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1024.00 q^{4} +(1975.50 - 2421.37i) q^{5} -8486.78i q^{7} -59049.0 q^{9} +O(q^{10})$$ $$q-1024.00 q^{4} +(1975.50 - 2421.37i) q^{5} -8486.78i q^{7} -59049.0 q^{9} -203523. q^{11} +1.04858e6 q^{16} +1.85908e6i q^{17} -2.47610e6 q^{19} +(-2.02291e6 + 2.47948e6i) q^{20} -1.18025e7i q^{23} +(-1.96042e6 - 9.56683e6i) q^{25} +8.69046e6i q^{28} +(-2.05496e7 - 1.67656e7i) q^{35} +6.04662e7 q^{36} +2.03243e8i q^{43} +2.08408e8 q^{44} +(-1.16651e8 + 1.42979e8i) q^{45} -4.28715e7i q^{47} +2.10450e8 q^{49} +(-4.02060e8 + 4.92804e8i) q^{55} +1.60684e9 q^{61} +5.01136e8i q^{63} -1.07374e9 q^{64} -1.90370e9i q^{68} +2.70383e9i q^{73} +2.53553e9 q^{76} +1.72725e9i q^{77} +(2.07146e9 - 2.53899e9i) q^{80} +3.48678e9 q^{81} +7.57882e9i q^{83} +(4.50153e9 + 3.67262e9i) q^{85} +1.20857e10i q^{92} +(-4.89153e9 + 5.99555e9i) q^{95} +1.20178e10 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2048 q^{4} + 3951 q^{5} - 118098 q^{9}+O(q^{10})$$ 2 * q - 2048 * q^4 + 3951 * q^5 - 118098 * q^9 $$2 q - 2048 q^{4} + 3951 q^{5} - 118098 q^{9} - 407046 q^{11} + 2097152 q^{16} - 4952198 q^{19} - 4045824 q^{20} - 3920849 q^{25} - 41099223 q^{35} + 120932352 q^{36} + 416815104 q^{44} - 233302599 q^{45} + 420899756 q^{49} - 804119373 q^{55} + 3213673954 q^{61} - 2147483648 q^{64} + 5071050752 q^{76} + 4142923776 q^{80} + 6973568802 q^{81} + 9003051827 q^{85} - 9783067149 q^{95} + 24035659254 q^{99}+O(q^{100})$$ 2 * q - 2048 * q^4 + 3951 * q^5 - 118098 * q^9 - 407046 * q^11 + 2097152 * q^16 - 4952198 * q^19 - 4045824 * q^20 - 3920849 * q^25 - 41099223 * q^35 + 120932352 * q^36 + 416815104 * q^44 - 233302599 * q^45 + 420899756 * q^49 - 804119373 * q^55 + 3213673954 * q^61 - 2147483648 * q^64 + 5071050752 * q^76 + 4142923776 * q^80 + 6973568802 * q^81 + 9003051827 * q^85 - 9783067149 * q^95 + 24035659254 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −1024.00 −1.00000
$$5$$ 1975.50 2421.37i 0.632160 0.774838i
$$6$$ 0 0
$$7$$ 8486.78i 0.504955i −0.967603 0.252477i $$-0.918755\pi$$
0.967603 0.252477i $$-0.0812453\pi$$
$$8$$ 0 0
$$9$$ −59049.0 −1.00000
$$10$$ 0 0
$$11$$ −203523. −1.26372 −0.631859 0.775083i $$-0.717708\pi$$
−0.631859 + 0.775083i $$0.717708\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.04858e6 1.00000
$$17$$ 1.85908e6i 1.30935i 0.755912 + 0.654673i $$0.227194\pi$$
−0.755912 + 0.654673i $$0.772806\pi$$
$$18$$ 0 0
$$19$$ −2.47610e6 −1.00000
$$20$$ −2.02291e6 + 2.47948e6i −0.632160 + 0.774838i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.18025e7i 1.83372i −0.399206 0.916861i $$-0.630714\pi$$
0.399206 0.916861i $$-0.369286\pi$$
$$24$$ 0 0
$$25$$ −1.96042e6 9.56683e6i −0.200747 0.979643i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 8.69046e6i 0.504955i
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.05496e7 1.67656e7i −0.391258 0.319212i
$$36$$ 6.04662e7 1.00000
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 2.03243e8i 1.38252i 0.722605 + 0.691262i $$0.242945\pi$$
−0.722605 + 0.691262i $$0.757055\pi$$
$$44$$ 2.08408e8 1.26372
$$45$$ −1.16651e8 + 1.42979e8i −0.632160 + 0.774838i
$$46$$ 0 0
$$47$$ 4.28715e7i 0.186930i −0.995623 0.0934651i $$-0.970206\pi$$
0.995623 0.0934651i $$-0.0297943\pi$$
$$48$$ 0 0
$$49$$ 2.10450e8 0.745021
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −4.02060e8 + 4.92804e8i −0.798872 + 0.979176i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 1.60684e9 1.90249 0.951246 0.308435i $$-0.0998051\pi$$
0.951246 + 0.308435i $$0.0998051\pi$$
$$62$$ 0 0
$$63$$ 5.01136e8i 0.504955i
$$64$$ −1.07374e9 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 1.90370e9i 1.30935i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 2.70383e9i 1.30426i 0.758106 + 0.652131i $$0.226125\pi$$
−0.758106 + 0.652131i $$0.773875\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 2.53553e9 1.00000
$$77$$ 1.72725e9i 0.638120i
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 2.07146e9 2.53899e9i 0.632160 0.774838i
$$81$$ 3.48678e9 1.00000
$$82$$ 0 0
$$83$$ 7.57882e9i 1.92403i 0.273003 + 0.962013i $$0.411983\pi$$
−0.273003 + 0.962013i $$0.588017\pi$$
$$84$$ 0 0
$$85$$ 4.50153e9 + 3.67262e9i 1.01453 + 0.827716i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1.20857e10i 1.83372i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −4.89153e9 + 5.99555e9i −0.632160 + 0.774838i
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 1.20178e10 1.26372
$$100$$ 2.00747e9 + 9.79643e9i 0.200747 + 0.979643i
$$101$$ 9.98470e9 0.950010 0.475005 0.879983i $$-0.342446\pi$$
0.475005 + 0.879983i $$0.342446\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 8.89903e9i 0.504955i
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ −2.85781e10 2.33158e10i −1.42084 1.15921i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1.57776e10 0.661160
$$120$$ 0 0
$$121$$ 1.54842e10 0.596982
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −2.70376e10 1.41524e10i −0.885969 0.463744i
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 7.71148e10 1.99885 0.999427 0.0338381i $$-0.0107731\pi$$
0.999427 + 0.0338381i $$0.0107731\pi$$
$$132$$ 0 0
$$133$$ 2.10141e10i 0.504955i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 9.21504e10i 1.90939i 0.297587 + 0.954695i $$0.403818\pi$$
−0.297587 + 0.954695i $$0.596182\pi$$
$$138$$ 0 0
$$139$$ −7.41705e10 −1.42941 −0.714705 0.699426i $$-0.753439\pi$$
−0.714705 + 0.699426i $$0.753439\pi$$
$$140$$ 2.10428e10 + 1.71680e10i 0.391258 + 0.319212i
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −6.19174e10 −1.00000
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −5.61929e9 −0.0765156 −0.0382578 0.999268i $$-0.512181\pi$$
−0.0382578 + 0.999268i $$0.512181\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 1.09777e11i 1.30935i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1.88364e11i 1.97469i −0.158590 0.987344i $$-0.550695\pi$$
0.158590 0.987344i $$-0.449305\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1.00165e11 −0.925947
$$162$$ 0 0
$$163$$ 7.73148e10i 0.671931i −0.941874 0.335966i $$-0.890937\pi$$
0.941874 0.335966i $$-0.109063\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −1.37858e11 −1.00000
$$170$$ 0 0
$$171$$ 1.46211e11 1.00000
$$172$$ 2.08120e11i 1.38252i
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −8.11915e10 + 1.66377e10i −0.494676 + 0.101368i
$$176$$ −2.13409e11 −1.26372
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 1.19451e11 1.46411e11i 0.632160 0.774838i
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.78366e11i 1.65464i
$$188$$ 4.39004e10i 0.186930i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4.79090e11 −1.88473 −0.942367 0.334582i $$-0.891405\pi$$
−0.942367 + 0.334582i $$0.891405\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −2.15501e11 −0.745021
$$197$$ 2.41155e11i 0.812765i 0.913703 + 0.406382i $$0.133210\pi$$
−0.913703 + 0.406382i $$0.866790\pi$$
$$198$$ 0 0
$$199$$ 6.66096e10 0.213438 0.106719 0.994289i $$-0.465966\pi$$
0.106719 + 0.994289i $$0.465966\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.96924e11i 1.83372i
$$208$$ 0 0
$$209$$ 5.03943e11 1.26372
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.92125e11 + 4.01506e11i 1.07123 + 0.873976i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 4.11709e11 5.04631e11i 0.798872 0.979176i
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 1.15761e11 + 5.64912e11i 0.200747 + 0.979643i
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 2.32468e11 0.369136 0.184568 0.982820i $$-0.440911\pi$$
0.184568 + 0.982820i $$0.440911\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 4.41959e11i 0.643580i −0.946811 0.321790i $$-0.895715\pi$$
0.946811 0.321790i $$-0.104285\pi$$
$$234$$ 0 0
$$235$$ −1.03808e11 8.46927e10i −0.144841 0.118170i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 8.33246e10 0.106852 0.0534261 0.998572i $$-0.482986\pi$$
0.0534261 + 0.998572i $$0.482986\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −1.64540e12 −1.90249
$$245$$ 4.15744e11 5.09577e11i 0.470972 0.577270i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 5.29646e11 0.531639 0.265820 0.964023i $$-0.414357\pi$$
0.265820 + 0.964023i $$0.414357\pi$$
$$252$$ 5.13163e11i 0.504955i
$$253$$ 2.40207e12i 2.31731i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.09951e12 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 7.87999e11i 0.626249i 0.949712 + 0.313124i $$0.101376\pi$$
−0.949712 + 0.313124i $$0.898624\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 2.83576e12 1.94010 0.970049 0.242910i $$-0.0781019\pi$$
0.970049 + 0.242910i $$0.0781019\pi$$
$$272$$ 1.94939e12i 1.30935i
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 3.98991e11 + 1.94707e12i 0.253688 + 1.23799i
$$276$$ 0 0
$$277$$ 3.15403e12i 1.93405i 0.254683 + 0.967024i $$0.418029\pi$$
−0.254683 + 0.967024i $$0.581971\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 2.72803e12i 1.50286i −0.659815 0.751428i $$-0.729365\pi$$
0.659815 0.751428i $$-0.270635\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.44020e12 −0.714386
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.76872e12i 1.30426i
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.72487e12 0.698112
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −2.59638e12 −1.00000
$$305$$ 3.17431e12 3.89074e12i 1.20268 1.47412i
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 1.76871e12i 0.638120i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −1.89531e12 −0.651446 −0.325723 0.945465i $$-0.605608\pi$$
−0.325723 + 0.945465i $$0.605608\pi$$
$$312$$ 0 0
$$313$$ 5.95513e12i 1.98230i 0.132744 + 0.991150i $$0.457621\pi$$
−0.132744 + 0.991150i $$0.542379\pi$$
$$314$$ 0 0
$$315$$ 1.21343e12 + 9.89993e11i 0.391258 + 0.319212i
$$316$$ 0 0
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −2.12118e12 + 2.59992e12i −0.632160 + 0.774838i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4.60327e12i 1.30935i
$$324$$ −3.57047e12 −1.00000
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −3.63841e11 −0.0943913
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 7.76071e12i 1.92403i
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −4.60956e12 3.76076e12i −1.01453 0.827716i
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 4.18335e12i 0.881157i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 9.27491e12i 1.84358i −0.387687 0.921791i $$-0.626726\pi$$
0.387687 0.921791i $$-0.373274\pi$$
$$348$$ 0 0
$$349$$ −9.39511e12 −1.81457 −0.907287 0.420512i $$-0.861851\pi$$
−0.907287 + 0.420512i $$0.861851\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6.86062e12i 1.25167i 0.779955 + 0.625835i $$0.215242\pi$$
−0.779955 + 0.625835i $$0.784758\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1.17823e13 −1.97587 −0.987935 0.154871i $$-0.950504\pi$$
−0.987935 + 0.154871i $$0.950504\pi$$
$$360$$ 0 0
$$361$$ 6.13107e12 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 6.54696e12 + 5.34141e12i 1.01059 + 0.824502i
$$366$$ 0 0
$$367$$ 1.25495e13i 1.88493i 0.334301 + 0.942466i $$0.391500\pi$$
−0.334301 + 0.942466i $$0.608500\pi$$
$$368$$ 1.23758e13i 1.83372i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 5.00893e12 6.13944e12i 0.632160 0.774838i
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 4.18232e12 + 3.41219e12i 0.494440 + 0.403394i
$$386$$ 0 0
$$387$$ 1.20013e13i 1.38252i
$$388$$ 0 0
$$389$$ −1.48910e13 −1.67177 −0.835884 0.548906i $$-0.815044\pi$$
−0.835884 + 0.548906i $$0.815044\pi$$
$$390$$ 0 0
$$391$$ 2.19418e13 2.40098
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −1.23063e13 −1.26372
$$397$$ 1.93498e13i 1.96211i 0.193732 + 0.981054i $$0.437941\pi$$
−0.193732 + 0.981054i $$0.562059\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −2.05565e12 1.00315e13i −0.200747 0.979643i
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −1.02243e13 −0.950010
$$405$$ 6.88814e12 8.44279e12i 0.632160 0.774838i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.83511e13 + 1.49720e13i 1.49081 + 1.21629i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −1.40486e13 −1.08783 −0.543916 0.839140i $$-0.683059\pi$$
−0.543916 + 0.839140i $$0.683059\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 2.53152e12i 0.186930i
$$424$$ 0 0
$$425$$ 1.77855e13 3.64459e12i 1.28269 0.262848i
$$426$$ 0 0
$$427$$ 1.36369e13i 0.960672i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.92241e13i 1.83372i
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ −1.24269e13 −0.745021
$$442$$ 0 0
$$443$$ 3.30278e13i 1.93580i 0.251335 + 0.967900i $$0.419130\pi$$
−0.251335 + 0.967900i $$0.580870\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 9.11261e12i 0.504955i
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.23713e13i 1.62397i −0.583675 0.811987i $$-0.698386\pi$$
0.583675 0.811987i $$-0.301614\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 2.92640e13 + 2.38754e13i 1.42084 + 1.15921i
$$461$$ 2.38839e13 1.14710 0.573548 0.819172i $$-0.305566\pi$$
0.573548 + 0.819172i $$0.305566\pi$$
$$462$$ 0 0
$$463$$ 2.30301e12i 0.108241i 0.998534 + 0.0541204i $$0.0172355\pi$$
−0.998534 + 0.0541204i $$0.982765\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 3.76252e13i 1.69393i 0.531652 + 0.846963i $$0.321571\pi$$
−0.531652 + 0.846963i $$0.678429\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 4.13645e13i 1.74712i
$$474$$ 0 0
$$475$$ 4.85421e12 + 2.36884e13i 0.200747 + 0.979643i
$$476$$ −1.61563e13 −0.661160
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 3.07491e13 1.21942 0.609711 0.792624i $$-0.291285\pi$$
0.609711 + 0.792624i $$0.291285\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −1.58558e13 −0.596982
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1.38065e13 0.483811 0.241905 0.970300i $$-0.422228\pi$$
0.241905 + 0.970300i $$0.422228\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 2.37412e13 2.90996e13i 0.798872 0.979176i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −2.31588e13 −0.748536 −0.374268 0.927321i $$-0.622106\pi$$
−0.374268 + 0.927321i $$0.622106\pi$$
$$500$$ 2.76865e13 + 1.44920e13i 0.885969 + 0.463744i
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 5.96791e13i 1.85345i 0.375734 + 0.926727i $$0.377391\pi$$
−0.375734 + 0.926727i $$0.622609\pi$$
$$504$$ 0 0
$$505$$ 1.97248e13 2.41766e13i 0.600558 0.736104i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 2.29468e13 0.658593
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 8.72534e12i 0.236227i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ −7.89655e13 −1.99885
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −9.78718e13 −2.36254
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.15184e13i 0.504955i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −4.28314e13 −0.941496
$$540$$ 0 0
$$541$$ 7.59982e13 1.63990 0.819949 0.572436i $$-0.194001\pi$$
0.819949 + 0.572436i $$0.194001\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 9.43620e13i 1.90939i
$$549$$ −9.48821e13 −1.90249
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 7.59505e13 1.42941
$$557$$ 8.56199e13i 1.59698i −0.602010 0.798488i $$-0.705633\pi$$
0.602010 0.798488i $$-0.294367\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −2.15478e13 1.75800e13i −0.391258 0.319212i
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.95916e13i 0.504955i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ −1.06969e14 −1.76229 −0.881143 0.472850i $$-0.843225\pi$$
−0.881143 + 0.472850i $$0.843225\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.12912e14 + 2.31378e13i −1.79639 + 0.368115i
$$576$$ 6.34034e13 1.00000
$$577$$ 7.48265e13i 1.16998i 0.811042 + 0.584988i $$0.198901\pi$$
−0.811042 + 0.584988i $$0.801099\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 6.43197e13 0.971546
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1.38365e14i 1.98535i −0.120823 0.992674i $$-0.538554\pi$$
0.120823 0.992674i $$-0.461446\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 1.45486e14i 1.98402i −0.126152 0.992011i $$-0.540263\pi$$
0.126152 0.992011i $$-0.459737\pi$$
$$594$$ 0 0
$$595$$ 3.11687e13 3.82034e13i 0.417959 0.512292i
$$596$$ 5.75415e12 0.0765156
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 3.05890e13 3.74929e13i 0.377388 0.462565i
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 1.12412e14i 1.30935i
$$613$$ 6.02984e13i 0.696632i 0.937377 + 0.348316i $$0.113246\pi$$
−0.937377 + 0.348316i $$0.886754\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8.47903e13i 0.948244i 0.880459 + 0.474122i $$0.157234\pi$$
−0.880459 + 0.474122i $$0.842766\pi$$
$$618$$ 0 0
$$619$$ 5.72825e13 0.630332 0.315166 0.949037i $$-0.397940\pi$$
0.315166 + 0.949037i $$0.397940\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −8.76809e13 + 3.75101e13i −0.919401 + 0.393322i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 1.92884e14i 1.97469i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −1.98866e14 −1.98799 −0.993994 0.109430i $$-0.965097\pi$$
−0.993994 + 0.109430i $$0.965097\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 1.11824e14i 1.01737i −0.860952 0.508686i $$-0.830131\pi$$
0.860952 0.508686i $$-0.169869\pi$$
$$644$$ 1.02569e14 0.925947
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 5.95300e13i 0.525066i 0.964923 + 0.262533i $$0.0845579\pi$$
−0.964923 + 0.262533i $$0.915442\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 7.91704e13i 0.671931i
$$653$$ 2.71577e13i 0.228732i −0.993439 0.114366i $$-0.963516\pi$$
0.993439 0.114366i $$-0.0364837\pi$$
$$654$$ 0 0
$$655$$ 1.52340e14 1.86723e14i 1.26360 1.54879i
$$656$$ 0 0
$$657$$ 1.59658e14i 1.30426i
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 5.08829e13 + 4.15134e13i 0.391258 + 0.319212i
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −3.27028e14 −2.40421
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 1.41167e14 1.00000
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ −1.49720e14 −1.00000
$$685$$ 2.23130e14 + 1.82043e14i 1.47947 + 1.20704i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2.13115e14i 1.38252i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1.69827e14 1.07799 0.538997 0.842307i $$-0.318803\pi$$
0.538997 + 0.842307i $$0.318803\pi$$
$$692$$ 0 0
$$693$$ 1.01993e14i 0.638120i
$$694$$ 0 0
$$695$$ −1.46524e14 + 1.79594e14i −0.903616 + 1.10756i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 8.31401e13 1.70370e13i 0.494676 0.101368i
$$701$$ 3.30849e14 1.95452 0.977258 0.212054i $$-0.0680151\pi$$
0.977258 + 0.212054i $$0.0680151\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 2.18531e14 1.26372
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.47379e13i 0.479712i
$$708$$ 0 0
$$709$$ 1.12103e14 0.625729 0.312864 0.949798i $$-0.398711\pi$$
0.312864 + 0.949798i $$0.398711\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 1.93381e14 1.00640 0.503199 0.864170i $$-0.332156\pi$$
0.503199 + 0.864170i $$0.332156\pi$$
$$720$$ −1.22318e14 + 1.49925e14i −0.632160 + 0.774838i
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 3.88917e14i 1.91507i 0.288312 + 0.957536i $$0.406906\pi$$
−0.288312 + 0.957536i $$0.593094\pi$$
$$728$$ 0 0
$$729$$ −2.05891e14 −1.00000
$$730$$ 0 0
$$731$$ −3.77845e14 −1.81020
$$732$$ 0 0
$$733$$ 2.12619e14i 1.00480i 0.864634 + 0.502402i $$0.167550\pi$$
−0.864634 + 0.502402i $$0.832450\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 4.17769e14 1.89546 0.947728 0.319080i $$-0.103374\pi$$
0.947728 + 0.319080i $$0.103374\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −1.11009e13 + 1.36064e13i −0.0483701 + 0.0592872i
$$746$$ 0 0
$$747$$ 4.47522e14i 1.92403i
$$748$$ 3.87447e14i 1.65464i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 4.49540e13i 0.186930i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 4.54743e14i 1.82931i 0.404241 + 0.914653i $$0.367536\pi$$
−0.404241 + 0.914653i $$0.632464\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −3.58961e14 −1.40645 −0.703224 0.710968i $$-0.748257\pi$$
−0.703224 + 0.710968i $$0.748257\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 4.90588e14 1.88473
$$765$$ −2.65811e14 2.16865e14i −1.01453 0.827716i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 3.35620e14 1.24800 0.624002 0.781423i $$-0.285506\pi$$
0.624002 + 0.781423i $$0.285506\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 2.20673e14 0.745021
$$785$$ −4.56098e14 3.72112e14i −1.53006 1.24832i
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 2.46942e14i 0.812765i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$