# Properties

 Label 48.13.g.a.31.1 Level $48$ Weight $13$ Character 48.31 Analytic conductor $43.872$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,13,Mod(31,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 13, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 13);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 48.g (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: $$43.8717032293$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 4333x^{2} + 4332x + 18766224$$ x^4 - x^3 + 4333*x^2 + 4332*x + 18766224 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 31.1 Root $$-32.6599 - 56.5686i$$ of defining polynomial Character $$\chi$$ $$=$$ 48.31 Dual form 48.13.g.a.31.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-420.888i q^{3} -21286.8 q^{5} +7111.90i q^{7} -177147. q^{9} +O(q^{10})$$ $$q-420.888i q^{3} -21286.8 q^{5} +7111.90i q^{7} -177147. q^{9} -1.88540e6i q^{11} -8.56106e6 q^{13} +8.95935e6i q^{15} +6.77647e6 q^{17} +645743. i q^{19} +2.99331e6 q^{21} +2.15694e8i q^{23} +2.08985e8 q^{25} +7.45591e7i q^{27} -3.66716e8 q^{29} -1.03858e9i q^{31} -7.93544e8 q^{33} -1.51389e8i q^{35} +3.52823e9 q^{37} +3.60325e9i q^{39} +5.66160e9 q^{41} -4.86430e9i q^{43} +3.77089e9 q^{45} +1.89030e10i q^{47} +1.37907e10 q^{49} -2.85214e9i q^{51} +1.61412e9 q^{53} +4.01341e10i q^{55} +2.71786e8 q^{57} -1.28375e10i q^{59} +3.86920e10 q^{61} -1.25985e9i q^{63} +1.82237e11 q^{65} +1.56544e11i q^{67} +9.07829e10 q^{69} +4.16779e10i q^{71} +1.00107e11 q^{73} -8.79596e10i q^{75} +1.34088e10 q^{77} -1.99076e11i q^{79} +3.13811e10 q^{81} +2.30690e11i q^{83} -1.44249e11 q^{85} +1.54347e11i q^{87} -4.73312e11 q^{89} -6.08854e10i q^{91} -4.37126e11 q^{93} -1.37458e10i q^{95} +9.29103e11 q^{97} +3.33993e11i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 21960 q^{5} - 708588 q^{9}+O(q^{10})$$ 4 * q - 21960 * q^5 - 708588 * q^9 $$4 q - 21960 q^{5} - 708588 q^{9} - 5810072 q^{13} - 33882264 q^{17} - 15664752 q^{21} + 142148300 q^{25} + 42583896 q^{29} + 271362960 q^{33} + 9402865736 q^{37} + 17968882536 q^{41} + 3890148120 q^{45} + 53940845764 q^{49} + 60546956760 q^{53} + 46910963376 q^{57} + 168287201672 q^{61} + 481064975280 q^{65} + 163198209024 q^{69} + 789026629000 q^{73} + 140389989696 q^{77} + 125524238436 q^{81} - 777401136720 q^{85} - 638670460536 q^{89} - 607412656080 q^{93} - 563542043000 q^{97}+O(q^{100})$$ 4 * q - 21960 * q^5 - 708588 * q^9 - 5810072 * q^13 - 33882264 * q^17 - 15664752 * q^21 + 142148300 * q^25 + 42583896 * q^29 + 271362960 * q^33 + 9402865736 * q^37 + 17968882536 * q^41 + 3890148120 * q^45 + 53940845764 * q^49 + 60546956760 * q^53 + 46910963376 * q^57 + 168287201672 * q^61 + 481064975280 * q^65 + 163198209024 * q^69 + 789026629000 * q^73 + 140389989696 * q^77 + 125524238436 * q^81 - 777401136720 * q^85 - 638670460536 * q^89 - 607412656080 * q^93 - 563542043000 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ − 420.888i − 0.577350i
$$4$$ 0 0
$$5$$ −21286.8 −1.36235 −0.681176 0.732119i $$-0.738531\pi$$
−0.681176 + 0.732119i $$0.738531\pi$$
$$6$$ 0 0
$$7$$ 7111.90i 0.0604501i 0.999543 + 0.0302251i $$0.00962240\pi$$
−0.999543 + 0.0302251i $$0.990378\pi$$
$$8$$ 0 0
$$9$$ −177147. −0.333333
$$10$$ 0 0
$$11$$ − 1.88540e6i − 1.06426i −0.846663 0.532130i $$-0.821392\pi$$
0.846663 0.532130i $$-0.178608\pi$$
$$12$$ 0 0
$$13$$ −8.56106e6 −1.77365 −0.886824 0.462107i $$-0.847093\pi$$
−0.886824 + 0.462107i $$0.847093\pi$$
$$14$$ 0 0
$$15$$ 8.95935e6i 0.786555i
$$16$$ 0 0
$$17$$ 6.77647e6 0.280744 0.140372 0.990099i $$-0.455170\pi$$
0.140372 + 0.990099i $$0.455170\pi$$
$$18$$ 0 0
$$19$$ 645743.i 0.0137258i 0.999976 + 0.00686291i $$0.00218455\pi$$
−0.999976 + 0.00686291i $$0.997815\pi$$
$$20$$ 0 0
$$21$$ 2.99331e6 0.0349009
$$22$$ 0 0
$$23$$ 2.15694e8i 1.45704i 0.685027 + 0.728518i $$0.259790\pi$$
−0.685027 + 0.728518i $$0.740210\pi$$
$$24$$ 0 0
$$25$$ 2.08985e8 0.856005
$$26$$ 0 0
$$27$$ 7.45591e7i 0.192450i
$$28$$ 0 0
$$29$$ −3.66716e8 −0.616513 −0.308257 0.951303i $$-0.599746\pi$$
−0.308257 + 0.951303i $$0.599746\pi$$
$$30$$ 0 0
$$31$$ − 1.03858e9i − 1.17022i −0.810952 0.585112i $$-0.801050\pi$$
0.810952 0.585112i $$-0.198950\pi$$
$$32$$ 0 0
$$33$$ −7.93544e8 −0.614451
$$34$$ 0 0
$$35$$ − 1.51389e8i − 0.0823544i
$$36$$ 0 0
$$37$$ 3.52823e9 1.37514 0.687570 0.726118i $$-0.258678\pi$$
0.687570 + 0.726118i $$0.258678\pi$$
$$38$$ 0 0
$$39$$ 3.60325e9i 1.02402i
$$40$$ 0 0
$$41$$ 5.66160e9 1.19189 0.595945 0.803025i $$-0.296778\pi$$
0.595945 + 0.803025i $$0.296778\pi$$
$$42$$ 0 0
$$43$$ − 4.86430e9i − 0.769502i −0.923020 0.384751i $$-0.874287\pi$$
0.923020 0.384751i $$-0.125713\pi$$
$$44$$ 0 0
$$45$$ 3.77089e9 0.454118
$$46$$ 0 0
$$47$$ 1.89030e10i 1.75365i 0.480811 + 0.876824i $$0.340342\pi$$
−0.480811 + 0.876824i $$0.659658\pi$$
$$48$$ 0 0
$$49$$ 1.37907e10 0.996346
$$50$$ 0 0
$$51$$ − 2.85214e9i − 0.162087i
$$52$$ 0 0
$$53$$ 1.61412e9 0.0728251 0.0364126 0.999337i $$-0.488407\pi$$
0.0364126 + 0.999337i $$0.488407\pi$$
$$54$$ 0 0
$$55$$ 4.01341e10i 1.44990i
$$56$$ 0 0
$$57$$ 2.71786e8 0.00792460
$$58$$ 0 0
$$59$$ − 1.28375e10i − 0.304346i −0.988354 0.152173i $$-0.951373\pi$$
0.988354 0.152173i $$-0.0486271\pi$$
$$60$$ 0 0
$$61$$ 3.86920e10 0.751004 0.375502 0.926822i $$-0.377470\pi$$
0.375502 + 0.926822i $$0.377470\pi$$
$$62$$ 0 0
$$63$$ − 1.25985e9i − 0.0201500i
$$64$$ 0 0
$$65$$ 1.82237e11 2.41633
$$66$$ 0 0
$$67$$ 1.56544e11i 1.73056i 0.501289 + 0.865280i $$0.332859\pi$$
−0.501289 + 0.865280i $$0.667141\pi$$
$$68$$ 0 0
$$69$$ 9.07829e10 0.841220
$$70$$ 0 0
$$71$$ 4.16779e10i 0.325354i 0.986679 + 0.162677i $$0.0520128\pi$$
−0.986679 + 0.162677i $$0.947987\pi$$
$$72$$ 0 0
$$73$$ 1.00107e11 0.661497 0.330748 0.943719i $$-0.392699\pi$$
0.330748 + 0.943719i $$0.392699\pi$$
$$74$$ 0 0
$$75$$ − 8.79596e10i − 0.494214i
$$76$$ 0 0
$$77$$ 1.34088e10 0.0643347
$$78$$ 0 0
$$79$$ − 1.99076e11i − 0.818947i −0.912322 0.409473i $$-0.865712\pi$$
0.912322 0.409473i $$-0.134288\pi$$
$$80$$ 0 0
$$81$$ 3.13811e10 0.111111
$$82$$ 0 0
$$83$$ 2.30690e11i 0.705604i 0.935698 + 0.352802i $$0.114771\pi$$
−0.935698 + 0.352802i $$0.885229\pi$$
$$84$$ 0 0
$$85$$ −1.44249e11 −0.382472
$$86$$ 0 0
$$87$$ 1.54347e11i 0.355944i
$$88$$ 0 0
$$89$$ −4.73312e11 −0.952375 −0.476187 0.879344i $$-0.657982\pi$$
−0.476187 + 0.879344i $$0.657982\pi$$
$$90$$ 0 0
$$91$$ − 6.08854e10i − 0.107217i
$$92$$ 0 0
$$93$$ −4.37126e11 −0.675630
$$94$$ 0 0
$$95$$ − 1.37458e10i − 0.0186994i
$$96$$ 0 0
$$97$$ 9.29103e11 1.11541 0.557704 0.830040i $$-0.311682\pi$$
0.557704 + 0.830040i $$0.311682\pi$$
$$98$$ 0 0
$$99$$ 3.33993e11i 0.354754i
$$100$$ 0 0
$$101$$ −4.30494e11 −0.405545 −0.202773 0.979226i $$-0.564995\pi$$
−0.202773 + 0.979226i $$0.564995\pi$$
$$102$$ 0 0
$$103$$ − 1.24450e12i − 1.04225i −0.853482 0.521123i $$-0.825513\pi$$
0.853482 0.521123i $$-0.174487\pi$$
$$104$$ 0 0
$$105$$ −6.37180e10 −0.0475473
$$106$$ 0 0
$$107$$ 3.40478e11i 0.226875i 0.993545 + 0.113437i $$0.0361861\pi$$
−0.993545 + 0.113437i $$0.963814\pi$$
$$108$$ 0 0
$$109$$ −3.31888e12 −1.97894 −0.989470 0.144738i $$-0.953766\pi$$
−0.989470 + 0.144738i $$0.953766\pi$$
$$110$$ 0 0
$$111$$ − 1.48499e12i − 0.793937i
$$112$$ 0 0
$$113$$ −3.87063e12 −1.85913 −0.929567 0.368654i $$-0.879819\pi$$
−0.929567 + 0.368654i $$0.879819\pi$$
$$114$$ 0 0
$$115$$ − 4.59142e12i − 1.98500i
$$116$$ 0 0
$$117$$ 1.51657e12 0.591216
$$118$$ 0 0
$$119$$ 4.81935e10i 0.0169710i
$$120$$ 0 0
$$121$$ −4.16314e11 −0.132650
$$122$$ 0 0
$$123$$ − 2.38290e12i − 0.688138i
$$124$$ 0 0
$$125$$ 7.48339e11 0.196173
$$126$$ 0 0
$$127$$ 8.23613e12i 1.96291i 0.191688 + 0.981456i $$0.438604\pi$$
−0.191688 + 0.981456i $$0.561396\pi$$
$$128$$ 0 0
$$129$$ −2.04733e12 −0.444272
$$130$$ 0 0
$$131$$ − 1.61593e12i − 0.319738i −0.987138 0.159869i $$-0.948893\pi$$
0.987138 0.159869i $$-0.0511072\pi$$
$$132$$ 0 0
$$133$$ −4.59246e9 −0.000829727 0
$$134$$ 0 0
$$135$$ − 1.58712e12i − 0.262185i
$$136$$ 0 0
$$137$$ −4.29703e11 −0.0649897 −0.0324948 0.999472i $$-0.510345\pi$$
−0.0324948 + 0.999472i $$0.510345\pi$$
$$138$$ 0 0
$$139$$ 1.70332e12i 0.236160i 0.993004 + 0.118080i $$0.0376739\pi$$
−0.993004 + 0.118080i $$0.962326\pi$$
$$140$$ 0 0
$$141$$ 7.95603e12 1.01247
$$142$$ 0 0
$$143$$ 1.61410e13i 1.88762i
$$144$$ 0 0
$$145$$ 7.80621e12 0.839908
$$146$$ 0 0
$$147$$ − 5.80435e12i − 0.575241i
$$148$$ 0 0
$$149$$ −5.75418e12 −0.525854 −0.262927 0.964816i $$-0.584688\pi$$
−0.262927 + 0.964816i $$0.584688\pi$$
$$150$$ 0 0
$$151$$ − 1.73435e13i − 1.46311i −0.681785 0.731553i $$-0.738796\pi$$
0.681785 0.731553i $$-0.261204\pi$$
$$152$$ 0 0
$$153$$ −1.20043e12 −0.0935812
$$154$$ 0 0
$$155$$ 2.21080e13i 1.59426i
$$156$$ 0 0
$$157$$ 1.18262e13 0.789672 0.394836 0.918752i $$-0.370801\pi$$
0.394836 + 0.918752i $$0.370801\pi$$
$$158$$ 0 0
$$159$$ − 6.79365e11i − 0.0420456i
$$160$$ 0 0
$$161$$ −1.53399e12 −0.0880780
$$162$$ 0 0
$$163$$ − 1.72346e13i − 0.918917i −0.888199 0.459458i $$-0.848044\pi$$
0.888199 0.459458i $$-0.151956\pi$$
$$164$$ 0 0
$$165$$ 1.68920e13 0.837099
$$166$$ 0 0
$$167$$ 3.25246e13i 1.49939i 0.661785 + 0.749694i $$0.269799\pi$$
−0.661785 + 0.749694i $$0.730201\pi$$
$$168$$ 0 0
$$169$$ 4.99937e13 2.14583
$$170$$ 0 0
$$171$$ − 1.14391e11i − 0.00457527i
$$172$$ 0 0
$$173$$ 2.94963e13 1.10025 0.550125 0.835082i $$-0.314580\pi$$
0.550125 + 0.835082i $$0.314580\pi$$
$$174$$ 0 0
$$175$$ 1.48628e12i 0.0517456i
$$176$$ 0 0
$$177$$ −5.40314e12 −0.175714
$$178$$ 0 0
$$179$$ 2.17325e13i 0.660682i 0.943862 + 0.330341i $$0.107164\pi$$
−0.943862 + 0.330341i $$0.892836\pi$$
$$180$$ 0 0
$$181$$ −1.24786e12 −0.0354891 −0.0177446 0.999843i $$-0.505649\pi$$
−0.0177446 + 0.999843i $$0.505649\pi$$
$$182$$ 0 0
$$183$$ − 1.62850e13i − 0.433592i
$$184$$ 0 0
$$185$$ −7.51046e13 −1.87343
$$186$$ 0 0
$$187$$ − 1.27764e13i − 0.298784i
$$188$$ 0 0
$$189$$ −5.30257e11 −0.0116336
$$190$$ 0 0
$$191$$ 6.52191e13i 1.34330i 0.740866 + 0.671652i $$0.234415\pi$$
−0.740866 + 0.671652i $$0.765585\pi$$
$$192$$ 0 0
$$193$$ −1.62613e13 −0.314638 −0.157319 0.987548i $$-0.550285\pi$$
−0.157319 + 0.987548i $$0.550285\pi$$
$$194$$ 0 0
$$195$$ − 7.67015e13i − 1.39507i
$$196$$ 0 0
$$197$$ −8.69576e13 −1.48768 −0.743841 0.668356i $$-0.766998\pi$$
−0.743841 + 0.668356i $$0.766998\pi$$
$$198$$ 0 0
$$199$$ − 1.03084e13i − 0.165986i −0.996550 0.0829932i $$-0.973552\pi$$
0.996550 0.0829932i $$-0.0264480\pi$$
$$200$$ 0 0
$$201$$ 6.58874e13 0.999139
$$202$$ 0 0
$$203$$ − 2.60805e12i − 0.0372683i
$$204$$ 0 0
$$205$$ −1.20517e14 −1.62378
$$206$$ 0 0
$$207$$ − 3.82095e13i − 0.485678i
$$208$$ 0 0
$$209$$ 1.21749e12 0.0146078
$$210$$ 0 0
$$211$$ − 1.21337e14i − 1.37499i −0.726189 0.687495i $$-0.758710\pi$$
0.726189 0.687495i $$-0.241290\pi$$
$$212$$ 0 0
$$213$$ 1.75417e13 0.187843
$$214$$ 0 0
$$215$$ 1.03545e14i 1.04833i
$$216$$ 0 0
$$217$$ 7.38627e12 0.0707402
$$218$$ 0 0
$$219$$ − 4.21339e13i − 0.381915i
$$220$$ 0 0
$$221$$ −5.80137e13 −0.497940
$$222$$ 0 0
$$223$$ 1.52486e14i 1.23994i 0.784624 + 0.619972i $$0.212856\pi$$
−0.784624 + 0.619972i $$0.787144\pi$$
$$224$$ 0 0
$$225$$ −3.70212e13 −0.285335
$$226$$ 0 0
$$227$$ − 2.60931e14i − 1.90709i −0.301249 0.953545i $$-0.597404\pi$$
0.301249 0.953545i $$-0.402596\pi$$
$$228$$ 0 0
$$229$$ 1.11017e14 0.769798 0.384899 0.922959i $$-0.374236\pi$$
0.384899 + 0.922959i $$0.374236\pi$$
$$230$$ 0 0
$$231$$ − 5.64360e12i − 0.0371436i
$$232$$ 0 0
$$233$$ −1.48696e14 −0.929318 −0.464659 0.885490i $$-0.653823\pi$$
−0.464659 + 0.885490i $$0.653823\pi$$
$$234$$ 0 0
$$235$$ − 4.02383e14i − 2.38909i
$$236$$ 0 0
$$237$$ −8.37887e13 −0.472819
$$238$$ 0 0
$$239$$ 3.25593e14i 1.74698i 0.486841 + 0.873491i $$0.338149\pi$$
−0.486841 + 0.873491i $$0.661851\pi$$
$$240$$ 0 0
$$241$$ −1.46403e14 −0.747220 −0.373610 0.927586i $$-0.621880\pi$$
−0.373610 + 0.927586i $$0.621880\pi$$
$$242$$ 0 0
$$243$$ − 1.32079e13i − 0.0641500i
$$244$$ 0 0
$$245$$ −2.93559e14 −1.35737
$$246$$ 0 0
$$247$$ − 5.52825e12i − 0.0243448i
$$248$$ 0 0
$$249$$ 9.70949e13 0.407381
$$250$$ 0 0
$$251$$ 4.56243e14i 1.82454i 0.409589 + 0.912270i $$0.365672\pi$$
−0.409589 + 0.912270i $$0.634328\pi$$
$$252$$ 0 0
$$253$$ 4.06669e14 1.55067
$$254$$ 0 0
$$255$$ 6.07127e13i 0.220820i
$$256$$ 0 0
$$257$$ 4.70340e14 1.63235 0.816175 0.577804i $$-0.196090\pi$$
0.816175 + 0.577804i $$0.196090\pi$$
$$258$$ 0 0
$$259$$ 2.50924e13i 0.0831274i
$$260$$ 0 0
$$261$$ 6.49627e13 0.205504
$$262$$ 0 0
$$263$$ − 1.42061e14i − 0.429280i −0.976693 0.214640i $$-0.931142\pi$$
0.976693 0.214640i $$-0.0688578\pi$$
$$264$$ 0 0
$$265$$ −3.43594e13 −0.0992135
$$266$$ 0 0
$$267$$ 1.99212e14i 0.549854i
$$268$$ 0 0
$$269$$ 2.94994e14 0.778573 0.389286 0.921117i $$-0.372722\pi$$
0.389286 + 0.921117i $$0.372722\pi$$
$$270$$ 0 0
$$271$$ 4.70773e14i 1.18849i 0.804284 + 0.594246i $$0.202549\pi$$
−0.804284 + 0.594246i $$0.797451\pi$$
$$272$$ 0 0
$$273$$ −2.56259e13 −0.0619019
$$274$$ 0 0
$$275$$ − 3.94022e14i − 0.911012i
$$276$$ 0 0
$$277$$ 4.15318e14 0.919395 0.459698 0.888075i $$-0.347958\pi$$
0.459698 + 0.888075i $$0.347958\pi$$
$$278$$ 0 0
$$279$$ 1.83981e14i 0.390075i
$$280$$ 0 0
$$281$$ 6.73345e14 1.36773 0.683864 0.729609i $$-0.260298\pi$$
0.683864 + 0.729609i $$0.260298\pi$$
$$282$$ 0 0
$$283$$ 3.70518e14i 0.721259i 0.932709 + 0.360629i $$0.117438\pi$$
−0.932709 + 0.360629i $$0.882562\pi$$
$$284$$ 0 0
$$285$$ −5.78544e12 −0.0107961
$$286$$ 0 0
$$287$$ 4.02647e13i 0.0720499i
$$288$$ 0 0
$$289$$ −5.36702e14 −0.921183
$$290$$ 0 0
$$291$$ − 3.91049e14i − 0.643981i
$$292$$ 0 0
$$293$$ 2.56469e14 0.405350 0.202675 0.979246i $$-0.435037\pi$$
0.202675 + 0.979246i $$0.435037\pi$$
$$294$$ 0 0
$$295$$ 2.73268e14i 0.414626i
$$296$$ 0 0
$$297$$ 1.40574e14 0.204817
$$298$$ 0 0
$$299$$ − 1.84657e15i − 2.58427i
$$300$$ 0 0
$$301$$ 3.45944e13 0.0465165
$$302$$ 0 0
$$303$$ 1.81190e14i 0.234142i
$$304$$ 0 0
$$305$$ −8.23627e14 −1.02313
$$306$$ 0 0
$$307$$ 8.24545e14i 0.984882i 0.870346 + 0.492441i $$0.163895\pi$$
−0.870346 + 0.492441i $$0.836105\pi$$
$$308$$ 0 0
$$309$$ −5.23793e14 −0.601740
$$310$$ 0 0
$$311$$ 7.97009e14i 0.880848i 0.897790 + 0.440424i $$0.145172\pi$$
−0.897790 + 0.440424i $$0.854828\pi$$
$$312$$ 0 0
$$313$$ 1.79978e14 0.191406 0.0957028 0.995410i $$-0.469490\pi$$
0.0957028 + 0.995410i $$0.469490\pi$$
$$314$$ 0 0
$$315$$ 2.68181e13i 0.0274515i
$$316$$ 0 0
$$317$$ 1.63990e15 1.61608 0.808039 0.589130i $$-0.200529\pi$$
0.808039 + 0.589130i $$0.200529\pi$$
$$318$$ 0 0
$$319$$ 6.91408e14i 0.656131i
$$320$$ 0 0
$$321$$ 1.43303e14 0.130986
$$322$$ 0 0
$$323$$ 4.37586e12i 0.00385343i
$$324$$ 0 0
$$325$$ −1.78914e15 −1.51825
$$326$$ 0 0
$$327$$ 1.39688e15i 1.14254i
$$328$$ 0 0
$$329$$ −1.34436e14 −0.106008
$$330$$ 0 0
$$331$$ − 2.09770e15i − 1.59505i −0.603283 0.797527i $$-0.706141\pi$$
0.603283 0.797527i $$-0.293859\pi$$
$$332$$ 0 0
$$333$$ −6.25016e14 −0.458380
$$334$$ 0 0
$$335$$ − 3.33231e15i − 2.35763i
$$336$$ 0 0
$$337$$ −1.36274e15 −0.930321 −0.465160 0.885226i $$-0.654003\pi$$
−0.465160 + 0.885226i $$0.654003\pi$$
$$338$$ 0 0
$$339$$ 1.62910e15i 1.07337i
$$340$$ 0 0
$$341$$ −1.95814e15 −1.24542
$$342$$ 0 0
$$343$$ 1.96516e14i 0.120679i
$$344$$ 0 0
$$345$$ −1.93247e15 −1.14604
$$346$$ 0 0
$$347$$ − 2.21109e15i − 1.26657i −0.773917 0.633287i $$-0.781705\pi$$
0.773917 0.633287i $$-0.218295\pi$$
$$348$$ 0 0
$$349$$ −8.25554e14 −0.456870 −0.228435 0.973559i $$-0.573361\pi$$
−0.228435 + 0.973559i $$0.573361\pi$$
$$350$$ 0 0
$$351$$ − 6.38305e14i − 0.341339i
$$352$$ 0 0
$$353$$ 9.06854e14 0.468694 0.234347 0.972153i $$-0.424705\pi$$
0.234347 + 0.972153i $$0.424705\pi$$
$$354$$ 0 0
$$355$$ − 8.87187e14i − 0.443246i
$$356$$ 0 0
$$357$$ 2.02841e13 0.00979820
$$358$$ 0 0
$$359$$ 5.13417e14i 0.239830i 0.992784 + 0.119915i $$0.0382622\pi$$
−0.992784 + 0.119915i $$0.961738\pi$$
$$360$$ 0 0
$$361$$ 2.21290e15 0.999812
$$362$$ 0 0
$$363$$ 1.75222e14i 0.0765858i
$$364$$ 0 0
$$365$$ −2.13096e15 −0.901192
$$366$$ 0 0
$$367$$ − 6.68311e14i − 0.273516i −0.990605 0.136758i $$-0.956332\pi$$
0.990605 0.136758i $$-0.0436682\pi$$
$$368$$ 0 0
$$369$$ −1.00294e15 −0.397297
$$370$$ 0 0
$$371$$ 1.14795e13i 0.00440229i
$$372$$ 0 0
$$373$$ 9.11568e13 0.0338483 0.0169241 0.999857i $$-0.494613\pi$$
0.0169241 + 0.999857i $$0.494613\pi$$
$$374$$ 0 0
$$375$$ − 3.14967e14i − 0.113260i
$$376$$ 0 0
$$377$$ 3.13948e15 1.09348
$$378$$ 0 0
$$379$$ 3.53114e15i 1.19146i 0.803184 + 0.595731i $$0.203137\pi$$
−0.803184 + 0.595731i $$0.796863\pi$$
$$380$$ 0 0
$$381$$ 3.46649e15 1.13329
$$382$$ 0 0
$$383$$ − 2.41943e15i − 0.766515i −0.923642 0.383257i $$-0.874802\pi$$
0.923642 0.383257i $$-0.125198\pi$$
$$384$$ 0 0
$$385$$ −2.85430e14 −0.0876465
$$386$$ 0 0
$$387$$ 8.61697e14i 0.256501i
$$388$$ 0 0
$$389$$ 4.88362e15 1.40943 0.704717 0.709489i $$-0.251074\pi$$
0.704717 + 0.709489i $$0.251074\pi$$
$$390$$ 0 0
$$391$$ 1.46164e15i 0.409053i
$$392$$ 0 0
$$393$$ −6.80125e14 −0.184601
$$394$$ 0 0
$$395$$ 4.23768e15i 1.11569i
$$396$$ 0 0
$$397$$ −3.63632e15 −0.928792 −0.464396 0.885628i $$-0.653729\pi$$
−0.464396 + 0.885628i $$0.653729\pi$$
$$398$$ 0 0
$$399$$ 1.93291e12i 0 0.000479043i
$$400$$ 0 0
$$401$$ 2.72304e14 0.0654919 0.0327459 0.999464i $$-0.489575\pi$$
0.0327459 + 0.999464i $$0.489575\pi$$
$$402$$ 0 0
$$403$$ 8.89134e15i 2.07557i
$$404$$ 0 0
$$405$$ −6.68001e14 −0.151373
$$406$$ 0 0
$$407$$ − 6.65214e15i − 1.46351i
$$408$$ 0 0
$$409$$ 1.47132e15 0.314317 0.157158 0.987573i $$-0.449767\pi$$
0.157158 + 0.987573i $$0.449767\pi$$
$$410$$ 0 0
$$411$$ 1.80857e14i 0.0375218i
$$412$$ 0 0
$$413$$ 9.12988e13 0.0183977
$$414$$ 0 0
$$415$$ − 4.91065e15i − 0.961281i
$$416$$ 0 0
$$417$$ 7.16906e14 0.136347
$$418$$ 0 0
$$419$$ 6.30146e15i 1.16455i 0.812993 + 0.582273i $$0.197837\pi$$
−0.812993 + 0.582273i $$0.802163\pi$$
$$420$$ 0 0
$$421$$ 3.25314e15 0.584266 0.292133 0.956378i $$-0.405635\pi$$
0.292133 + 0.956378i $$0.405635\pi$$
$$422$$ 0 0
$$423$$ − 3.34860e15i − 0.584550i
$$424$$ 0 0
$$425$$ 1.41618e15 0.240318
$$426$$ 0 0
$$427$$ 2.75174e14i 0.0453983i
$$428$$ 0 0
$$429$$ 6.79358e15 1.08982
$$430$$ 0 0
$$431$$ 1.03071e16i 1.60795i 0.594660 + 0.803977i $$0.297287\pi$$
−0.594660 + 0.803977i $$0.702713\pi$$
$$432$$ 0 0
$$433$$ 9.64406e15 1.46330 0.731648 0.681682i $$-0.238751\pi$$
0.731648 + 0.681682i $$0.238751\pi$$
$$434$$ 0 0
$$435$$ − 3.28554e15i − 0.484921i
$$436$$ 0 0
$$437$$ −1.39283e14 −0.0199990
$$438$$ 0 0
$$439$$ − 3.83012e15i − 0.535087i −0.963546 0.267544i $$-0.913788\pi$$
0.963546 0.267544i $$-0.0862120\pi$$
$$440$$ 0 0
$$441$$ −2.44298e15 −0.332115
$$442$$ 0 0
$$443$$ 6.98519e15i 0.924179i 0.886833 + 0.462089i $$0.152900\pi$$
−0.886833 + 0.462089i $$0.847100\pi$$
$$444$$ 0 0
$$445$$ 1.00753e16 1.29747
$$446$$ 0 0
$$447$$ 2.42187e15i 0.303602i
$$448$$ 0 0
$$449$$ −6.54024e15 −0.798207 −0.399103 0.916906i $$-0.630679\pi$$
−0.399103 + 0.916906i $$0.630679\pi$$
$$450$$ 0 0
$$451$$ − 1.06744e16i − 1.26848i
$$452$$ 0 0
$$453$$ −7.29969e15 −0.844725
$$454$$ 0 0
$$455$$ 1.29605e15i 0.146068i
$$456$$ 0 0
$$457$$ −3.93711e15 −0.432196 −0.216098 0.976372i $$-0.569333\pi$$
−0.216098 + 0.976372i $$0.569333\pi$$
$$458$$ 0 0
$$459$$ 5.05247e14i 0.0540291i
$$460$$ 0 0
$$461$$ −6.78296e15 −0.706665 −0.353333 0.935498i $$-0.614952\pi$$
−0.353333 + 0.935498i $$0.614952\pi$$
$$462$$ 0 0
$$463$$ − 1.21932e16i − 1.23774i −0.785492 0.618872i $$-0.787590\pi$$
0.785492 0.618872i $$-0.212410\pi$$
$$464$$ 0 0
$$465$$ 9.30499e15 0.920446
$$466$$ 0 0
$$467$$ − 9.69994e14i − 0.0935121i −0.998906 0.0467560i $$-0.985112\pi$$
0.998906 0.0467560i $$-0.0148883\pi$$
$$468$$ 0 0
$$469$$ −1.11332e15 −0.104613
$$470$$ 0 0
$$471$$ − 4.97750e15i − 0.455917i
$$472$$ 0 0
$$473$$ −9.17117e15 −0.818951
$$474$$ 0 0
$$475$$ 1.34951e14i 0.0117494i
$$476$$ 0 0
$$477$$ −2.85937e14 −0.0242750
$$478$$ 0 0
$$479$$ 6.76625e15i 0.560190i 0.959972 + 0.280095i $$0.0903659\pi$$
−0.959972 + 0.280095i $$0.909634\pi$$
$$480$$ 0 0
$$481$$ −3.02054e16 −2.43901
$$482$$ 0 0
$$483$$ 6.45639e14i 0.0508518i
$$484$$ 0 0
$$485$$ −1.97776e16 −1.51958
$$486$$ 0 0
$$487$$ − 1.15311e16i − 0.864367i −0.901786 0.432184i $$-0.857743\pi$$
0.901786 0.432184i $$-0.142257\pi$$
$$488$$ 0 0
$$489$$ −7.25385e15 −0.530537
$$490$$ 0 0
$$491$$ 1.32213e16i 0.943596i 0.881707 + 0.471798i $$0.156395\pi$$
−0.881707 + 0.471798i $$0.843605\pi$$
$$492$$ 0 0
$$493$$ −2.48504e15 −0.173082
$$494$$ 0 0
$$495$$ − 7.10964e15i − 0.483299i
$$496$$ 0 0
$$497$$ −2.96409e14 −0.0196677
$$498$$ 0 0
$$499$$ 4.97689e15i 0.322370i 0.986924 + 0.161185i $$0.0515316\pi$$
−0.986924 + 0.161185i $$0.948468\pi$$
$$500$$ 0 0
$$501$$ 1.36892e16 0.865672
$$502$$ 0 0
$$503$$ − 1.01640e16i − 0.627562i −0.949495 0.313781i $$-0.898404\pi$$
0.949495 0.313781i $$-0.101596\pi$$
$$504$$ 0 0
$$505$$ 9.16383e15 0.552496
$$506$$ 0 0
$$507$$ − 2.10417e16i − 1.23889i
$$508$$ 0 0
$$509$$ −1.39231e16 −0.800626 −0.400313 0.916379i $$-0.631099\pi$$
−0.400313 + 0.916379i $$0.631099\pi$$
$$510$$ 0 0
$$511$$ 7.11951e14i 0.0399876i
$$512$$ 0 0
$$513$$ −4.81460e13 −0.00264153
$$514$$ 0 0
$$515$$ 2.64913e16i 1.41991i
$$516$$ 0 0
$$517$$ 3.56397e16 1.86634
$$518$$ 0 0
$$519$$ − 1.24147e16i − 0.635230i
$$520$$ 0 0
$$521$$ 1.75513e16 0.877572 0.438786 0.898592i $$-0.355409\pi$$
0.438786 + 0.898592i $$0.355409\pi$$
$$522$$ 0 0
$$523$$ 1.65340e16i 0.807919i 0.914777 + 0.403959i $$0.132366\pi$$
−0.914777 + 0.403959i $$0.867634\pi$$
$$524$$ 0 0
$$525$$ 6.25559e14 0.0298753
$$526$$ 0 0
$$527$$ − 7.03790e15i − 0.328533i
$$528$$ 0 0
$$529$$ −2.46091e16 −1.12295
$$530$$ 0 0
$$531$$ 2.27412e15i 0.101449i
$$532$$ 0 0
$$533$$ −4.84693e16 −2.11399
$$534$$ 0 0
$$535$$ − 7.24767e15i − 0.309083i
$$536$$ 0 0
$$537$$ 9.14697e15 0.381445
$$538$$ 0 0
$$539$$ − 2.60010e16i − 1.06037i
$$540$$ 0 0
$$541$$ −3.40764e16 −1.35916 −0.679580 0.733602i $$-0.737838\pi$$
−0.679580 + 0.733602i $$0.737838\pi$$
$$542$$ 0 0
$$543$$ 5.25211e14i 0.0204897i
$$544$$ 0 0
$$545$$ 7.06482e16 2.69601
$$546$$ 0 0
$$547$$ − 3.02110e16i − 1.12782i −0.825835 0.563912i $$-0.809296\pi$$
0.825835 0.563912i $$-0.190704\pi$$
$$548$$ 0 0
$$549$$ −6.85417e15 −0.250335
$$550$$ 0 0
$$551$$ − 2.36805e14i − 0.00846215i
$$552$$ 0 0
$$553$$ 1.41581e15 0.0495054
$$554$$ 0 0
$$555$$ 3.16107e16i 1.08162i
$$556$$ 0 0
$$557$$ 1.86821e16 0.625596 0.312798 0.949820i $$-0.398734\pi$$
0.312798 + 0.949820i $$0.398734\pi$$
$$558$$ 0 0
$$559$$ 4.16436e16i 1.36483i
$$560$$ 0 0
$$561$$ −5.37742e15 −0.172503
$$562$$ 0 0
$$563$$ 1.77453e16i 0.557228i 0.960403 + 0.278614i $$0.0898750\pi$$
−0.960403 + 0.278614i $$0.910125\pi$$
$$564$$ 0 0
$$565$$ 8.23931e16 2.53280
$$566$$ 0 0
$$567$$ 2.23179e14i 0.00671668i
$$568$$ 0 0
$$569$$ −2.04279e16 −0.601935 −0.300968 0.953634i $$-0.597310\pi$$
−0.300968 + 0.953634i $$0.597310\pi$$
$$570$$ 0 0
$$571$$ − 5.67402e15i − 0.163709i −0.996644 0.0818547i $$-0.973916\pi$$
0.996644 0.0818547i $$-0.0260844\pi$$
$$572$$ 0 0
$$573$$ 2.74500e16 0.775557
$$574$$ 0 0
$$575$$ 4.50768e16i 1.24723i
$$576$$ 0 0
$$577$$ 1.70028e16 0.460750 0.230375 0.973102i $$-0.426005\pi$$
0.230375 + 0.973102i $$0.426005\pi$$
$$578$$ 0 0
$$579$$ 6.84418e15i 0.181656i
$$580$$ 0 0
$$581$$ −1.64065e15 −0.0426538
$$582$$ 0 0
$$583$$ − 3.04327e15i − 0.0775049i
$$584$$ 0 0
$$585$$ −3.22828e16 −0.805445
$$586$$ 0 0
$$587$$ 3.27992e16i 0.801742i 0.916134 + 0.400871i $$0.131292\pi$$
−0.916134 + 0.400871i $$0.868708\pi$$
$$588$$ 0 0
$$589$$ 6.70655e14 0.0160623
$$590$$ 0 0
$$591$$ 3.65995e16i 0.858914i
$$592$$ 0 0
$$593$$ 6.54880e16 1.50603 0.753015 0.658003i $$-0.228599\pi$$
0.753015 + 0.658003i $$0.228599\pi$$
$$594$$ 0 0
$$595$$ − 1.02588e15i − 0.0231205i
$$596$$ 0 0
$$597$$ −4.33868e15 −0.0958322
$$598$$ 0 0
$$599$$ 1.46157e16i 0.316417i 0.987406 + 0.158209i $$0.0505718\pi$$
−0.987406 + 0.158209i $$0.949428\pi$$
$$600$$ 0 0
$$601$$ 3.63481e16 0.771320 0.385660 0.922641i $$-0.373974\pi$$
0.385660 + 0.922641i $$0.373974\pi$$
$$602$$ 0 0
$$603$$ − 2.77312e16i − 0.576853i
$$604$$ 0 0
$$605$$ 8.86197e15 0.180717
$$606$$ 0 0
$$607$$ 4.19339e16i 0.838364i 0.907902 + 0.419182i $$0.137683\pi$$
−0.907902 + 0.419182i $$0.862317\pi$$
$$608$$ 0 0
$$609$$ −1.09770e15 −0.0215169
$$610$$ 0 0
$$611$$ − 1.61829e17i − 3.11036i
$$612$$ 0 0
$$613$$ −2.03819e16 −0.384134 −0.192067 0.981382i $$-0.561519\pi$$
−0.192067 + 0.981382i $$0.561519\pi$$
$$614$$ 0 0
$$615$$ 5.07243e16i 0.937487i
$$616$$ 0 0
$$617$$ −3.74912e16 −0.679545 −0.339772 0.940508i $$-0.610350\pi$$
−0.339772 + 0.940508i $$0.610350\pi$$
$$618$$ 0 0
$$619$$ − 2.18233e16i − 0.387951i −0.981006 0.193975i $$-0.937862\pi$$
0.981006 0.193975i $$-0.0621382\pi$$
$$620$$ 0 0
$$621$$ −1.60819e16 −0.280407
$$622$$ 0 0
$$623$$ − 3.36615e15i − 0.0575712i
$$624$$ 0 0
$$625$$ −6.69516e16 −1.12326
$$626$$ 0 0
$$627$$ − 5.12426e14i − 0.00843384i
$$628$$ 0 0
$$629$$ 2.39089e16 0.386062
$$630$$ 0 0
$$631$$ 4.54812e16i 0.720537i 0.932849 + 0.360268i $$0.117315\pi$$
−0.932849 + 0.360268i $$0.882685\pi$$
$$632$$ 0 0
$$633$$ −5.10695e16 −0.793851
$$634$$ 0 0
$$635$$ − 1.75320e17i − 2.67418i
$$636$$ 0 0
$$637$$ −1.18063e17 −1.76717
$$638$$ 0 0
$$639$$ − 7.38311e15i − 0.108451i
$$640$$ 0 0
$$641$$ 7.57002e16 1.09131 0.545656 0.838009i $$-0.316281\pi$$
0.545656 + 0.838009i $$0.316281\pi$$
$$642$$ 0 0
$$643$$ 5.74737e16i 0.813212i 0.913604 + 0.406606i $$0.133288\pi$$
−0.913604 + 0.406606i $$0.866712\pi$$
$$644$$ 0 0
$$645$$ 4.35810e16 0.605255
$$646$$ 0 0
$$647$$ − 5.03250e16i − 0.686053i −0.939326 0.343027i $$-0.888548\pi$$
0.939326 0.343027i $$-0.111452\pi$$
$$648$$ 0 0
$$649$$ −2.42038e16 −0.323903
$$650$$ 0 0
$$651$$ − 3.10879e15i − 0.0408419i
$$652$$ 0 0
$$653$$ 1.84738e15 0.0238275 0.0119137 0.999929i $$-0.496208\pi$$
0.0119137 + 0.999929i $$0.496208\pi$$
$$654$$ 0 0
$$655$$ 3.43979e16i 0.435596i
$$656$$ 0 0
$$657$$ −1.77337e16 −0.220499
$$658$$ 0 0
$$659$$ 8.93078e16i 1.09038i 0.838313 + 0.545189i $$0.183542\pi$$
−0.838313 + 0.545189i $$0.816458\pi$$
$$660$$ 0 0
$$661$$ 1.13618e16 0.136219 0.0681095 0.997678i $$-0.478303\pi$$
0.0681095 + 0.997678i $$0.478303\pi$$
$$662$$ 0 0
$$663$$ 2.44173e16i 0.287486i
$$664$$ 0 0
$$665$$ 9.77586e13 0.00113038
$$666$$ 0 0
$$667$$ − 7.90984e16i − 0.898282i
$$668$$ 0 0
$$669$$ 6.41797e16 0.715881
$$670$$ 0 0
$$671$$ − 7.29500e16i − 0.799264i
$$672$$ 0 0
$$673$$ −1.40873e17 −1.51614 −0.758069 0.652174i $$-0.773857\pi$$
−0.758069 + 0.652174i $$0.773857\pi$$
$$674$$ 0 0
$$675$$ 1.55818e16i 0.164738i
$$676$$ 0 0
$$677$$ −1.06344e17 −1.10454 −0.552268 0.833667i $$-0.686238\pi$$
−0.552268 + 0.833667i $$0.686238\pi$$
$$678$$ 0 0
$$679$$ 6.60769e15i 0.0674265i
$$680$$ 0 0
$$681$$ −1.09823e17 −1.10106
$$682$$ 0 0
$$683$$ − 7.82367e16i − 0.770702i −0.922770 0.385351i $$-0.874080\pi$$
0.922770 0.385351i $$-0.125920\pi$$
$$684$$ 0 0
$$685$$ 9.14697e15 0.0885389
$$686$$ 0 0
$$687$$ − 4.67258e16i − 0.444443i
$$688$$ 0 0
$$689$$ −1.38186e16 −0.129166
$$690$$ 0 0
$$691$$ 1.63249e17i 1.49963i 0.661650 + 0.749813i $$0.269857\pi$$
−0.661650 + 0.749813i $$0.730143\pi$$
$$692$$ 0 0
$$693$$ −2.37533e15 −0.0214449
$$694$$ 0 0
$$695$$ − 3.62581e16i − 0.321733i
$$696$$ 0 0
$$697$$ 3.83657e16 0.334616
$$698$$ 0 0
$$699$$ 6.25845e16i 0.536542i
$$700$$ 0 0
$$701$$ −1.46995e17 −1.23878 −0.619391 0.785083i $$-0.712620\pi$$
−0.619391 + 0.785083i $$0.712620\pi$$
$$702$$ 0 0
$$703$$ 2.27833e15i 0.0188749i
$$704$$ 0 0
$$705$$ −1.69358e17 −1.37934
$$706$$ 0 0
$$707$$ − 3.06163e15i − 0.0245153i
$$708$$ 0 0
$$709$$ 5.55937e16 0.437672 0.218836 0.975762i $$-0.429774\pi$$
0.218836 + 0.975762i $$0.429774\pi$$
$$710$$ 0 0
$$711$$ 3.52657e16i 0.272982i
$$712$$ 0 0
$$713$$ 2.24015e17 1.70506
$$714$$ 0 0
$$715$$ − 3.43590e17i − 2.57161i
$$716$$ 0 0
$$717$$ 1.37038e17 1.00862
$$718$$ 0 0
$$719$$ 1.11386e16i 0.0806229i 0.999187 + 0.0403115i $$0.0128350\pi$$
−0.999187 + 0.0403115i $$0.987165\pi$$
$$720$$ 0 0
$$721$$ 8.85072e15 0.0630038
$$722$$ 0 0
$$723$$ 6.16194e16i 0.431408i
$$724$$ 0 0
$$725$$ −7.66384e16 −0.527738
$$726$$ 0 0
$$727$$ 4.35822e16i 0.295191i 0.989048 + 0.147595i $$0.0471533\pi$$
−0.989048 + 0.147595i $$0.952847\pi$$
$$728$$ 0 0
$$729$$ −5.55906e15 −0.0370370
$$730$$ 0 0
$$731$$ − 3.29628e16i − 0.216033i
$$732$$ 0 0
$$733$$ 1.02849e17 0.663095 0.331547 0.943439i $$-0.392429\pi$$
0.331547 + 0.943439i $$0.392429\pi$$
$$734$$ 0 0
$$735$$ 1.23556e17i 0.783680i
$$736$$ 0 0
$$737$$ 2.95148e17 1.84177
$$738$$ 0 0
$$739$$ 2.89187e16i 0.177546i 0.996052 + 0.0887732i $$0.0282946\pi$$
−0.996052 + 0.0887732i $$0.971705\pi$$
$$740$$ 0 0
$$741$$ −2.32677e15 −0.0140555
$$742$$ 0 0
$$743$$ 1.84464e17i 1.09642i 0.836339 + 0.548212i $$0.184691\pi$$
−0.836339 + 0.548212i $$0.815309\pi$$
$$744$$ 0 0
$$745$$ 1.22488e17 0.716399
$$746$$ 0 0
$$747$$ − 4.08661e16i − 0.235201i
$$748$$ 0 0
$$749$$ −2.42144e15 −0.0137146
$$750$$ 0 0
$$751$$ 1.07917e17i 0.601522i 0.953700 + 0.300761i $$0.0972406\pi$$
−0.953700 + 0.300761i $$0.902759\pi$$
$$752$$ 0 0
$$753$$ 1.92027e17 1.05340
$$754$$ 0 0
$$755$$ 3.69188e17i 1.99327i
$$756$$ 0 0
$$757$$ 2.20280e17 1.17058 0.585288 0.810825i $$-0.300981\pi$$
0.585288 + 0.810825i $$0.300981\pi$$
$$758$$ 0 0
$$759$$ − 1.71162e17i − 0.895277i
$$760$$ 0 0
$$761$$ 1.15648e17 0.595430 0.297715 0.954655i $$-0.403775\pi$$
0.297715 + 0.954655i $$0.403775\pi$$
$$762$$ 0 0
$$763$$ − 2.36035e16i − 0.119627i
$$764$$ 0 0
$$765$$ 2.55533e16 0.127491
$$766$$ 0 0
$$767$$ 1.09902e17i 0.539803i
$$768$$ 0 0
$$769$$ −9.40988e14 −0.00455016 −0.00227508 0.999997i $$-0.500724\pi$$
−0.00227508 + 0.999997i $$0.500724\pi$$
$$770$$ 0 0
$$771$$ − 1.97961e17i − 0.942438i
$$772$$ 0 0
$$773$$ −2.12581e17 −0.996434 −0.498217 0.867052i $$-0.666012\pi$$
−0.498217 + 0.867052i $$0.666012\pi$$
$$774$$ 0 0
$$775$$ − 2.17048e17i − 1.00172i
$$776$$ 0 0
$$777$$ 1.05611e16 0.0479936
$$778$$ 0 0
$$779$$ 3.65594e15i 0.0163597i
$$780$$ 0 0
$$781$$ 7.85796e16 0.346261
$$782$$ 0 0
$$783$$ − 2.73421e16i − 0.118648i
$$784$$ 0 0
$$785$$ −2.51741e17 −1.07581
$$786$$ 0 0
$$787$$ 3.88757e17i 1.63617i 0.575095 + 0.818087i $$0.304965\pi$$
−0.575095 + 0.818087i $$0.695035\pi$$
$$788$$ 0 0
$$789$$ −5.97919e16 −0.247845
$$790$$ 0 0
$$791$$ − 2.75275e16i − 0.112385i
$$792$$ 0 0
$$793$$ −3.31245e17 −1.33202
$$794$$ 0 0
$$795$$ 1.44615e16i 0.0572809i
$$796$$ 0 0
$$797$$ −2.11775e16 −0.0826276 −0.0413138 0.999146i $$-0.513154\pi$$
−0.0413138 + 0.999146i $$0.513154\pi$$
$$798$$ 0 0
$$799$$ 1.28095e17i 0.492326i
$$800$$ 0 0
$$801$$ 8.38459e16 0.317458
$$802$$ 0 0
$$803$$ − 1.88742e17i − 0.704005i
$$804$$ 0 0
$$805$$ 3.26537e16 0.119993
$$806$$ 0 0
$$807$$ − 1.24159e17i − 0.449509i
$$808$$ 0 0
$$809$$ 2.74837e17 0.980357 0.490179 0.871622i $$-0.336932\pi$$
0.490179 + 0.871622i $$0.336932\pi$$
$$810$$ 0 0
$$811$$ 3.56743e17i 1.25381i 0.779098 + 0.626903i $$0.215678\pi$$
−0.779098 + 0.626903i $$0.784322\pi$$
$$812$$ 0 0
$$813$$ 1.98143e17 0.686176
$$814$$ 0 0
$$815$$ 3.66869e17i 1.25189i
$$816$$ 0 0
$$817$$ 3.14109e15 0.0105620
$$818$$ 0 0
$$819$$ 1.07857e16i 0.0357391i
$$820$$ 0 0
$$821$$ −1.70191e17 −0.555748 −0.277874 0.960617i $$-0.589630\pi$$
−0.277874 + 0.960617i $$0.589630\pi$$
$$822$$ 0 0
$$823$$ − 4.53033e17i − 1.45791i −0.684562 0.728955i $$-0.740006\pi$$
0.684562 0.728955i $$-0.259994\pi$$
$$824$$ 0 0
$$825$$ −1.65839e17 −0.525973
$$826$$ 0 0
$$827$$ 4.07186e17i 1.27280i 0.771360 + 0.636399i $$0.219577\pi$$
−0.771360 + 0.636399i $$0.780423\pi$$
$$828$$ 0 0
$$829$$ 2.99377e17 0.922342 0.461171 0.887311i $$-0.347429\pi$$
0.461171 + 0.887311i $$0.347429\pi$$
$$830$$ 0 0
$$831$$ − 1.74803e17i − 0.530813i
$$832$$ 0 0
$$833$$ 9.34523e16 0.279718
$$834$$ 0 0
$$835$$ − 6.92344e17i − 2.04269i
$$836$$ 0 0
$$837$$ 7.74355e16 0.225210
$$838$$ 0 0
$$839$$ − 6.54448e17i − 1.87631i −0.346222 0.938153i $$-0.612536\pi$$
0.346222 0.938153i $$-0.387464\pi$$
$$840$$ 0 0
$$841$$ −2.19334e17 −0.619911
$$842$$ 0 0
$$843$$ − 2.83403e17i − 0.789659i
$$844$$ 0 0
$$845$$ −1.06420e18 −2.92337
$$846$$ 0 0
$$847$$ − 2.96078e15i − 0.00801873i
$$848$$ 0 0
$$849$$ 1.55947e17 0.416419
$$850$$ 0 0
$$851$$ 7.61017e17i 2.00363i
$$852$$ 0 0
$$853$$ 9.94589e16 0.258196 0.129098 0.991632i $$-0.458792\pi$$
0.129098 + 0.991632i $$0.458792\pi$$
$$854$$ 0 0
$$855$$ 2.43502e15i 0.00623313i
$$856$$ 0 0
$$857$$ 1.81107e17 0.457142 0.228571 0.973527i $$-0.426595\pi$$
0.228571 + 0.973527i $$0.426595\pi$$
$$858$$ 0 0
$$859$$ − 2.11062e17i − 0.525353i −0.964884 0.262677i $$-0.915395\pi$$
0.964884 0.262677i $$-0.0846052\pi$$
$$860$$ 0 0
$$861$$ 1.69470e16 0.0415980
$$862$$ 0 0
$$863$$ 3.64870e17i 0.883229i 0.897205 + 0.441614i $$0.145594\pi$$
−0.897205 + 0.441614i $$0.854406\pi$$
$$864$$ 0 0
$$865$$ −6.27881e17 −1.49893
$$866$$ 0 0
$$867$$ 2.25892e17i 0.531845i
$$868$$ 0 0
$$869$$ −3.75338e17 −0.871573
$$870$$ 0 0
$$871$$ − 1.34018e18i − 3.06940i
$$872$$ 0 0
$$873$$ −1.64588e17 −0.371803
$$874$$ 0 0
$$875$$ 5.32211e15i 0.0118587i
$$876$$ 0 0
$$877$$ 2.20060e17 0.483663 0.241832 0.970318i $$-0.422252\pi$$
0.241832 + 0.970318i $$0.422252\pi$$
$$878$$ 0 0
$$879$$ − 1.07945e17i − 0.234029i
$$880$$ 0 0
$$881$$ −5.23854e17 −1.12035 −0.560176 0.828373i $$-0.689267\pi$$
−0.560176 + 0.828373i $$0.689267\pi$$
$$882$$ 0 0
$$883$$ 5.19490e17i 1.09601i 0.836476 + 0.548003i $$0.184612\pi$$
−0.836476 + 0.548003i $$0.815388\pi$$
$$884$$ 0 0
$$885$$ 1.15015e17 0.239385
$$886$$ 0 0
$$887$$ − 6.43189e17i − 1.32068i −0.750968 0.660339i $$-0.770413\pi$$
0.750968 0.660339i $$-0.229587\pi$$
$$888$$ 0 0
$$889$$ −5.85745e16 −0.118658
$$890$$ 0 0
$$891$$ − 5.91659e16i − 0.118251i
$$892$$ 0 0
$$893$$ −1.22065e16 −0.0240703
$$894$$ 0 0
$$895$$ − 4.62615e17i − 0.900081i
$$896$$ 0 0
$$897$$ −7.77198e17 −1.49203
$$898$$ 0 0
$$899$$ 3.80864e17i 0.721459i
$$900$$ 0 0
$$901$$ 1.09380e16 0.0204452
$$902$$ 0 0
$$903$$ − 1.45604e16i − 0.0268563i
$$904$$ 0 0
$$905$$ 2.65630e16 0.0483487
$$906$$ 0 0
$$907$$ 8.20905e17i 1.47452i 0.675612 + 0.737258i $$0.263880\pi$$
−0.675612 + 0.737258i $$0.736120\pi$$
$$908$$ 0 0
$$909$$ 7.62608e16 0.135182
$$910$$ 0 0
$$911$$ 7.92986e16i 0.138725i 0.997592 + 0.0693625i $$0.0220965\pi$$
−0.997592 + 0.0693625i $$0.977903\pi$$
$$912$$ 0 0
$$913$$ 4.34944e17 0.750946
$$914$$ 0 0
$$915$$ 3.46655e17i 0.590706i
$$916$$ 0 0
$$917$$ 1.14923e16 0.0193282
$$918$$ 0 0
$$919$$ 7.38065e17i 1.22518i 0.790399 + 0.612592i $$0.209873\pi$$
−0.790399 + 0.612592i $$0.790127\pi$$
$$920$$ 0 0
$$921$$ 3.47041e17 0.568622
$$922$$ 0 0
$$923$$ − 3.56807e17i − 0.577063i
$$924$$ 0 0
$$925$$ 7.37349e17 1.17713
$$926$$ 0 0
$$927$$ 2.20459e17i 0.347415i
$$928$$ 0 0
$$929$$ 1.11243e18 1.73052 0.865261 0.501322i $$-0.167153\pi$$
0.865261 + 0.501322i $$0.167153\pi$$
$$930$$ 0 0
$$931$$ 8.90526e15i 0.0136757i
$$932$$ 0 0
$$933$$ 3.35452e17 0.508558
$$934$$ 0 0
$$935$$ 2.71967e17i 0.407049i
$$936$$ 0 0
$$937$$ −3.87919e17 −0.573197 −0.286598 0.958051i $$-0.592525\pi$$
−0.286598 + 0.958051i $$0.592525\pi$$
$$938$$ 0 0
$$939$$ − 7.57508e16i − 0.110508i
$$940$$ 0 0
$$941$$ 3.15087e17 0.453829 0.226914 0.973915i $$-0.427136\pi$$
0.226914 + 0.973915i $$0.427136\pi$$
$$942$$ 0 0
$$943$$ 1.22117e18i 1.73663i
$$944$$ 0 0
$$945$$ 1.12874e16 0.0158491
$$946$$ 0 0
$$947$$ − 3.79195e16i − 0.0525730i −0.999654 0.0262865i $$-0.991632\pi$$
0.999654 0.0262865i $$-0.00836821\pi$$
$$948$$ 0 0
$$949$$ −8.57023e17 −1.17326
$$950$$ 0 0
$$951$$ − 6.90215e17i − 0.933043i
$$952$$ 0 0
$$953$$ 6.56690e17 0.876603 0.438302 0.898828i $$-0.355580\pi$$
0.438302 + 0.898828i $$0.355580\pi$$
$$954$$ 0 0
$$955$$ − 1.38830e18i − 1.83005i
$$956$$ 0 0
$$957$$ 2.91006e17 0.378817
$$958$$ 0 0
$$959$$ − 3.05600e15i − 0.00392864i
$$960$$ 0 0
$$961$$ −2.90984e17 −0.369427
$$962$$ 0 0
$$963$$ − 6.03146e16i − 0.0756249i
$$964$$ 0 0
$$965$$ 3.46150e17 0.428647
$$966$$ 0 0
$$967$$ 6.24136e17i 0.763345i 0.924298 + 0.381672i $$0.124652\pi$$
−0.924298 + 0.381672i $$0.875348\pi$$
$$968$$ 0 0
$$969$$ 1.84175e15 0.00222478
$$970$$ 0 0
$$971$$ − 9.04996e17i − 1.07977i −0.841739 0.539885i $$-0.818468\pi$$
0.841739 0.539885i $$-0.181532\pi$$
$$972$$ 0 0
$$973$$ −1.21138e16 −0.0142759
$$974$$ 0 0
$$975$$ 7.53027e17i 0.876562i
$$976$$ 0 0
$$977$$ −1.37695e17 −0.158326 −0.0791628 0.996862i $$-0.525225\pi$$
−0.0791628 + 0.996862i $$0.525225\pi$$
$$978$$ 0 0
$$979$$ 8.92384e17i 1.01357i
$$980$$ 0 0
$$981$$ 5.87930e17 0.659647
$$982$$ 0 0
$$983$$ − 1.06312e18i − 1.17831i −0.808018 0.589157i $$-0.799460\pi$$
0.808018 0.589157i $$-0.200540\pi$$
$$984$$ 0 0
$$985$$ 1.85105e18 2.02675
$$986$$ 0 0
$$987$$ 5.65825e16i 0.0612039i
$$988$$ 0 0
$$989$$ 1.04920e18 1.12119
$$990$$ 0 0
$$991$$ 9.28788e17i 0.980561i 0.871565 + 0.490281i $$0.163106\pi$$
−0.871565 + 0.490281i $$0.836894\pi$$
$$992$$ 0 0
$$993$$ −8.82898e17 −0.920905
$$994$$ 0 0
$$995$$ 2.19432e17i 0.226132i
$$996$$ 0 0
$$997$$ −9.52810e17 −0.970142 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$998$$ 0 0
$$999$$ 2.63062e17i 0.264646i
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 48.13.g.a.31.1 4
3.2 odd 2 144.13.g.h.127.4 4
4.3 odd 2 inner 48.13.g.a.31.3 yes 4
8.3 odd 2 192.13.g.c.127.2 4
8.5 even 2 192.13.g.c.127.4 4
12.11 even 2 144.13.g.h.127.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
48.13.g.a.31.1 4 1.1 even 1 trivial
48.13.g.a.31.3 yes 4 4.3 odd 2 inner
144.13.g.h.127.3 4 12.11 even 2
144.13.g.h.127.4 4 3.2 odd 2
192.13.g.c.127.2 4 8.3 odd 2
192.13.g.c.127.4 4 8.5 even 2