# Properties

 Label 48.13.g.a.31.4 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 31.4 Root $$33.1599 - 57.4347i$$ of defining polynomial Character $$\chi$$ $$=$$ 48.31 Dual form 48.13.g.a.31.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+420.888i q^{3} +10306.8 q^{5} +25721.1i q^{7} -177147. q^{9} +O(q^{10})$$ $$q+420.888i q^{3} +10306.8 q^{5} +25721.1i q^{7} -177147. q^{9} -2.20777e6i q^{11} +5.65602e6 q^{13} +4.33799e6i q^{15} -2.37176e7 q^{17} -5.50828e7i q^{19} -1.08257e7 q^{21} +2.18200e7i q^{23} -1.37911e8 q^{25} -7.45591e7i q^{27} +3.88008e8 q^{29} -3.16995e8i q^{31} +9.29225e8 q^{33} +2.65101e8i q^{35} +1.17320e9 q^{37} +2.38055e9i q^{39} +3.32284e9 q^{41} -6.04589e9i q^{43} -1.82581e9 q^{45} -2.63294e9i q^{47} +1.31797e10 q^{49} -9.98246e9i q^{51} +2.86594e10 q^{53} -2.27550e10i q^{55} +2.31837e10 q^{57} -6.66570e10i q^{59} +4.54516e10 q^{61} -4.55641e9i q^{63} +5.82953e10 q^{65} +5.74377e10i q^{67} -9.18379e9 q^{69} -2.96341e9i q^{71} +2.94406e11 q^{73} -5.80453e10i q^{75} +5.67862e10 q^{77} -2.19409e11i q^{79} +3.13811e10 q^{81} -3.91018e10i q^{83} -2.44452e11 q^{85} +1.63308e11i q^{87} +1.53977e11 q^{89} +1.45479e11i q^{91} +1.33419e11 q^{93} -5.67725e11i q^{95} -1.21087e12 q^{97} +3.91100e11i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 21960 q^{5} - 708588 q^{9}+O(q^{10})$$ 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
$$5$$ 10306.8 0.659633 0.329816 0.944045i $$-0.393013\pi$$
0.329816 + 0.944045i $$0.393013\pi$$
$$6$$ 0 0
$$7$$ 25721.1i 0.218625i 0.994007 + 0.109313i $$0.0348650\pi$$
−0.994007 + 0.109313i $$0.965135\pi$$
$$8$$ 0 0
$$9$$ −177147. −0.333333
$$10$$ 0 0
$$11$$ − 2.20777e6i − 1.24623i −0.782130 0.623115i $$-0.785867\pi$$
0.782130 0.623115i $$-0.214133\pi$$
$$12$$ 0 0
$$13$$ 5.65602e6 1.17179 0.585897 0.810386i $$-0.300742\pi$$
0.585897 + 0.810386i $$0.300742\pi$$
$$14$$ 0 0
$$15$$ 4.33799e6i 0.380839i
$$16$$ 0 0
$$17$$ −2.37176e7 −0.982601 −0.491300 0.870990i $$-0.663478\pi$$
−0.491300 + 0.870990i $$0.663478\pi$$
$$18$$ 0 0
$$19$$ − 5.50828e7i − 1.17083i −0.810733 0.585415i $$-0.800931\pi$$
0.810733 0.585415i $$-0.199069\pi$$
$$20$$ 0 0
$$21$$ −1.08257e7 −0.126223
$$22$$ 0 0
$$23$$ 2.18200e7i 0.147397i 0.997281 + 0.0736984i $$0.0234802\pi$$
−0.997281 + 0.0736984i $$0.976520\pi$$
$$24$$ 0 0
$$25$$ −1.37911e8 −0.564885
$$26$$ 0 0
$$27$$ − 7.45591e7i − 0.192450i
$$28$$ 0 0
$$29$$ 3.88008e8 0.652309 0.326154 0.945317i $$-0.394247\pi$$
0.326154 + 0.945317i $$0.394247\pi$$
$$30$$ 0 0
$$31$$ − 3.16995e8i − 0.357176i −0.983924 0.178588i $$-0.942847\pi$$
0.983924 0.178588i $$-0.0571529\pi$$
$$32$$ 0 0
$$33$$ 9.29225e8 0.719511
$$34$$ 0 0
$$35$$ 2.65101e8i 0.144212i
$$36$$ 0 0
$$37$$ 1.17320e9 0.457259 0.228629 0.973514i $$-0.426576\pi$$
0.228629 + 0.973514i $$0.426576\pi$$
$$38$$ 0 0
$$39$$ 2.38055e9i 0.676535i
$$40$$ 0 0
$$41$$ 3.32284e9 0.699529 0.349765 0.936838i $$-0.386262\pi$$
0.349765 + 0.936838i $$0.386262\pi$$
$$42$$ 0 0
$$43$$ − 6.04589e9i − 0.956422i −0.878245 0.478211i $$-0.841285\pi$$
0.878245 0.478211i $$-0.158715\pi$$
$$44$$ 0 0
$$45$$ −1.82581e9 −0.219878
$$46$$ 0 0
$$47$$ − 2.63294e9i − 0.244261i −0.992514 0.122130i $$-0.961027\pi$$
0.992514 0.122130i $$-0.0389726\pi$$
$$48$$ 0 0
$$49$$ 1.31797e10 0.952203
$$50$$ 0 0
$$51$$ − 9.98246e9i − 0.567305i
$$52$$ 0 0
$$53$$ 2.86594e10 1.29304 0.646519 0.762898i $$-0.276224\pi$$
0.646519 + 0.762898i $$0.276224\pi$$
$$54$$ 0 0
$$55$$ − 2.27550e10i − 0.822054i
$$56$$ 0 0
$$57$$ 2.31837e10 0.675980
$$58$$ 0 0
$$59$$ − 6.66570e10i − 1.58028i −0.612927 0.790140i $$-0.710008\pi$$
0.612927 0.790140i $$-0.289992\pi$$
$$60$$ 0 0
$$61$$ 4.54516e10 0.882206 0.441103 0.897456i $$-0.354587\pi$$
0.441103 + 0.897456i $$0.354587\pi$$
$$62$$ 0 0
$$63$$ − 4.55641e9i − 0.0728751i
$$64$$ 0 0
$$65$$ 5.82953e10 0.772953
$$66$$ 0 0
$$67$$ 5.74377e10i 0.634962i 0.948265 + 0.317481i $$0.102837\pi$$
−0.948265 + 0.317481i $$0.897163\pi$$
$$68$$ 0 0
$$69$$ −9.18379e9 −0.0850996
$$70$$ 0 0
$$71$$ − 2.96341e9i − 0.0231335i −0.999933 0.0115667i $$-0.996318\pi$$
0.999933 0.0115667i $$-0.00368189\pi$$
$$72$$ 0 0
$$73$$ 2.94406e11 1.94540 0.972702 0.232058i $$-0.0745459\pi$$
0.972702 + 0.232058i $$0.0745459\pi$$
$$74$$ 0 0
$$75$$ − 5.80453e10i − 0.326136i
$$76$$ 0 0
$$77$$ 5.67862e10 0.272457
$$78$$ 0 0
$$79$$ − 2.19409e11i − 0.902594i −0.892374 0.451297i $$-0.850962\pi$$
0.892374 0.451297i $$-0.149038\pi$$
$$80$$ 0 0
$$81$$ 3.13811e10 0.111111
$$82$$ 0 0
$$83$$ − 3.91018e10i − 0.119599i −0.998210 0.0597995i $$-0.980954\pi$$
0.998210 0.0597995i $$-0.0190461\pi$$
$$84$$ 0 0
$$85$$ −2.44452e11 −0.648156
$$86$$ 0 0
$$87$$ 1.63308e11i 0.376611i
$$88$$ 0 0
$$89$$ 1.53977e11 0.309825 0.154912 0.987928i $$-0.450490\pi$$
0.154912 + 0.987928i $$0.450490\pi$$
$$90$$ 0 0
$$91$$ 1.45479e11i 0.256184i
$$92$$ 0 0
$$93$$ 1.33419e11 0.206216
$$94$$ 0 0
$$95$$ − 5.67725e11i − 0.772318i
$$96$$ 0 0
$$97$$ −1.21087e12 −1.45368 −0.726840 0.686807i $$-0.759012\pi$$
−0.726840 + 0.686807i $$0.759012\pi$$
$$98$$ 0 0
$$99$$ 3.91100e11i 0.415410i
$$100$$ 0 0
$$101$$ −1.88925e12 −1.77976 −0.889878 0.456199i $$-0.849211\pi$$
−0.889878 + 0.456199i $$0.849211\pi$$
$$102$$ 0 0
$$103$$ 8.42992e11i 0.705993i 0.935625 + 0.352996i $$0.114837\pi$$
−0.935625 + 0.352996i $$0.885163\pi$$
$$104$$ 0 0
$$105$$ −1.11578e11 −0.0832611
$$106$$ 0 0
$$107$$ 1.79271e12i 1.19456i 0.802034 + 0.597279i $$0.203751\pi$$
−0.802034 + 0.597279i $$0.796249\pi$$
$$108$$ 0 0
$$109$$ 6.23866e11 0.371991 0.185996 0.982551i $$-0.440449\pi$$
0.185996 + 0.982551i $$0.440449\pi$$
$$110$$ 0 0
$$111$$ 4.93787e11i 0.263998i
$$112$$ 0 0
$$113$$ −6.52840e11 −0.313571 −0.156786 0.987633i $$-0.550113\pi$$
−0.156786 + 0.987633i $$0.550113\pi$$
$$114$$ 0 0
$$115$$ 2.24894e11i 0.0972277i
$$116$$ 0 0
$$117$$ −1.00195e12 −0.390598
$$118$$ 0 0
$$119$$ − 6.10042e11i − 0.214821i
$$120$$ 0 0
$$121$$ −1.73583e12 −0.553088
$$122$$ 0 0
$$123$$ 1.39854e12i 0.403873i
$$124$$ 0 0
$$125$$ −3.93772e12 −1.03225
$$126$$ 0 0
$$127$$ 6.71362e12i 1.60005i 0.599965 + 0.800026i $$0.295181\pi$$
−0.599965 + 0.800026i $$0.704819\pi$$
$$128$$ 0 0
$$129$$ 2.54464e12 0.552190
$$130$$ 0 0
$$131$$ − 7.69909e12i − 1.52339i −0.647935 0.761696i $$-0.724367\pi$$
0.647935 0.761696i $$-0.275633\pi$$
$$132$$ 0 0
$$133$$ 1.41679e12 0.255973
$$134$$ 0 0
$$135$$ − 7.68463e11i − 0.126946i
$$136$$ 0 0
$$137$$ −6.50551e12 −0.983916 −0.491958 0.870619i $$-0.663719\pi$$
−0.491958 + 0.870619i $$0.663719\pi$$
$$138$$ 0 0
$$139$$ 3.14897e12i 0.436596i 0.975882 + 0.218298i $$0.0700505\pi$$
−0.975882 + 0.218298i $$0.929949\pi$$
$$140$$ 0 0
$$141$$ 1.10817e12 0.141024
$$142$$ 0 0
$$143$$ − 1.24872e13i − 1.46032i
$$144$$ 0 0
$$145$$ 3.99911e12 0.430284
$$146$$ 0 0
$$147$$ 5.54719e12i 0.549755i
$$148$$ 0 0
$$149$$ 1.53701e13 1.40462 0.702309 0.711872i $$-0.252152\pi$$
0.702309 + 0.711872i $$0.252152\pi$$
$$150$$ 0 0
$$151$$ − 1.62889e13i − 1.37414i −0.726591 0.687071i $$-0.758896\pi$$
0.726591 0.687071i $$-0.241104\pi$$
$$152$$ 0 0
$$153$$ 4.20150e12 0.327534
$$154$$ 0 0
$$155$$ − 3.26719e12i − 0.235605i
$$156$$ 0 0
$$157$$ 2.68898e13 1.79552 0.897758 0.440489i $$-0.145195\pi$$
0.897758 + 0.440489i $$0.145195\pi$$
$$158$$ 0 0
$$159$$ 1.20624e13i 0.746536i
$$160$$ 0 0
$$161$$ −5.61234e11 −0.0322247
$$162$$ 0 0
$$163$$ 1.75586e13i 0.936193i 0.883677 + 0.468097i $$0.155060\pi$$
−0.883677 + 0.468097i $$0.844940\pi$$
$$164$$ 0 0
$$165$$ 9.57730e12 0.474613
$$166$$ 0 0
$$167$$ − 3.47614e13i − 1.60250i −0.598328 0.801251i $$-0.704168\pi$$
0.598328 0.801251i $$-0.295832\pi$$
$$168$$ 0 0
$$169$$ 8.69252e12 0.373100
$$170$$ 0 0
$$171$$ 9.75775e12i 0.390277i
$$172$$ 0 0
$$173$$ −2.88795e13 −1.07724 −0.538620 0.842549i $$-0.681054\pi$$
−0.538620 + 0.842549i $$0.681054\pi$$
$$174$$ 0 0
$$175$$ − 3.54722e12i − 0.123498i
$$176$$ 0 0
$$177$$ 2.80552e13 0.912375
$$178$$ 0 0
$$179$$ − 3.60079e13i − 1.09466i −0.836917 0.547330i $$-0.815644\pi$$
0.836917 0.547330i $$-0.184356\pi$$
$$180$$ 0 0
$$181$$ 3.45184e12 0.0981700 0.0490850 0.998795i $$-0.484369\pi$$
0.0490850 + 0.998795i $$0.484369\pi$$
$$182$$ 0 0
$$183$$ 1.91300e13i 0.509342i
$$184$$ 0 0
$$185$$ 1.20919e13 0.301623
$$186$$ 0 0
$$187$$ 5.23630e13i 1.22455i
$$188$$ 0 0
$$189$$ 1.91774e12 0.0420745
$$190$$ 0 0
$$191$$ − 7.34011e13i − 1.51183i −0.654671 0.755914i $$-0.727193\pi$$
0.654671 0.755914i $$-0.272807\pi$$
$$192$$ 0 0
$$193$$ −1.96054e13 −0.379343 −0.189672 0.981848i $$-0.560742\pi$$
−0.189672 + 0.981848i $$0.560742\pi$$
$$194$$ 0 0
$$195$$ 2.45358e13i 0.446265i
$$196$$ 0 0
$$197$$ −8.33525e13 −1.42601 −0.713003 0.701161i $$-0.752665\pi$$
−0.713003 + 0.701161i $$0.752665\pi$$
$$198$$ 0 0
$$199$$ − 1.92304e13i − 0.309650i −0.987942 0.154825i $$-0.950519\pi$$
0.987942 0.154825i $$-0.0494813\pi$$
$$200$$ 0 0
$$201$$ −2.41748e13 −0.366596
$$202$$ 0 0
$$203$$ 9.97998e12i 0.142611i
$$204$$ 0 0
$$205$$ 3.42477e13 0.461432
$$206$$ 0 0
$$207$$ − 3.86535e12i − 0.0491323i
$$208$$ 0 0
$$209$$ −1.21610e14 −1.45912
$$210$$ 0 0
$$211$$ 1.67562e14i 1.89880i 0.314062 + 0.949402i $$0.398310\pi$$
−0.314062 + 0.949402i $$0.601690\pi$$
$$212$$ 0 0
$$213$$ 1.24726e12 0.0133561
$$214$$ 0 0
$$215$$ − 6.23135e13i − 0.630887i
$$216$$ 0 0
$$217$$ 8.15344e12 0.0780877
$$218$$ 0 0
$$219$$ 1.23912e14i 1.12318i
$$220$$ 0 0
$$221$$ −1.34147e14 −1.15141
$$222$$ 0 0
$$223$$ 1.42965e14i 1.16252i 0.813718 + 0.581261i $$0.197440\pi$$
−0.813718 + 0.581261i $$0.802560\pi$$
$$224$$ 0 0
$$225$$ 2.44306e13 0.188295
$$226$$ 0 0
$$227$$ − 3.24397e13i − 0.237095i −0.992948 0.118547i $$-0.962176\pi$$
0.992948 0.118547i $$-0.0378238\pi$$
$$228$$ 0 0
$$229$$ −2.11741e14 −1.46822 −0.734110 0.679030i $$-0.762400\pi$$
−0.734110 + 0.679030i $$0.762400\pi$$
$$230$$ 0 0
$$231$$ 2.39007e13i 0.157303i
$$232$$ 0 0
$$233$$ 4.89837e13 0.306137 0.153069 0.988216i $$-0.451084\pi$$
0.153069 + 0.988216i $$0.451084\pi$$
$$234$$ 0 0
$$235$$ − 2.71371e13i − 0.161122i
$$236$$ 0 0
$$237$$ 9.23468e13 0.521113
$$238$$ 0 0
$$239$$ − 1.66870e14i − 0.895347i −0.894197 0.447674i $$-0.852253\pi$$
0.894197 0.447674i $$-0.147747\pi$$
$$240$$ 0 0
$$241$$ 9.85623e13 0.503047 0.251524 0.967851i $$-0.419068\pi$$
0.251524 + 0.967851i $$0.419068\pi$$
$$242$$ 0 0
$$243$$ 1.32079e13i 0.0641500i
$$244$$ 0 0
$$245$$ 1.35840e14 0.628104
$$246$$ 0 0
$$247$$ − 3.11549e14i − 1.37197i
$$248$$ 0 0
$$249$$ 1.64575e13 0.0690505
$$250$$ 0 0
$$251$$ 3.73220e14i 1.49253i 0.665649 + 0.746265i $$0.268155\pi$$
−0.665649 + 0.746265i $$0.731845\pi$$
$$252$$ 0 0
$$253$$ 4.81736e13 0.183690
$$254$$ 0 0
$$255$$ − 1.02887e14i − 0.374213i
$$256$$ 0 0
$$257$$ −2.93272e14 −1.01782 −0.508912 0.860819i $$-0.669952\pi$$
−0.508912 + 0.860819i $$0.669952\pi$$
$$258$$ 0 0
$$259$$ 3.01760e13i 0.0999683i
$$260$$ 0 0
$$261$$ −6.87345e13 −0.217436
$$262$$ 0 0
$$263$$ − 6.00708e14i − 1.81522i −0.419817 0.907609i $$-0.637906\pi$$
0.419817 0.907609i $$-0.362094\pi$$
$$264$$ 0 0
$$265$$ 2.95385e14 0.852930
$$266$$ 0 0
$$267$$ 6.48072e13i 0.178878i
$$268$$ 0 0
$$269$$ 5.21692e14 1.37689 0.688447 0.725287i $$-0.258293\pi$$
0.688447 + 0.725287i $$0.258293\pi$$
$$270$$ 0 0
$$271$$ − 3.63124e14i − 0.916726i −0.888765 0.458363i $$-0.848436\pi$$
0.888765 0.458363i $$-0.151564\pi$$
$$272$$ 0 0
$$273$$ −6.12304e13 −0.147908
$$274$$ 0 0
$$275$$ 3.04477e14i 0.703976i
$$276$$ 0 0
$$277$$ 2.94181e14 0.651233 0.325616 0.945502i $$-0.394428\pi$$
0.325616 + 0.945502i $$0.394428\pi$$
$$278$$ 0 0
$$279$$ 5.61547e13i 0.119059i
$$280$$ 0 0
$$281$$ 5.70665e14 1.15916 0.579580 0.814916i $$-0.303217\pi$$
0.579580 + 0.814916i $$0.303217\pi$$
$$282$$ 0 0
$$283$$ 4.19893e14i 0.817373i 0.912675 + 0.408687i $$0.134013\pi$$
−0.912675 + 0.408687i $$0.865987\pi$$
$$284$$ 0 0
$$285$$ 2.38949e14 0.445898
$$286$$ 0 0
$$287$$ 8.54669e13i 0.152935i
$$288$$ 0 0
$$289$$ −2.00978e13 −0.0344954
$$290$$ 0 0
$$291$$ − 5.09643e14i − 0.839282i
$$292$$ 0 0
$$293$$ 3.50267e14 0.553596 0.276798 0.960928i $$-0.410727\pi$$
0.276798 + 0.960928i $$0.410727\pi$$
$$294$$ 0 0
$$295$$ − 6.87018e14i − 1.04240i
$$296$$ 0 0
$$297$$ −1.64609e14 −0.239837
$$298$$ 0 0
$$299$$ 1.23415e14i 0.172719i
$$300$$ 0 0
$$301$$ 1.55507e14 0.209098
$$302$$ 0 0
$$303$$ − 7.95162e14i − 1.02754i
$$304$$ 0 0
$$305$$ 4.68459e14 0.581932
$$306$$ 0 0
$$307$$ 7.25089e14i 0.866086i 0.901373 + 0.433043i $$0.142560\pi$$
−0.901373 + 0.433043i $$0.857440\pi$$
$$308$$ 0 0
$$309$$ −3.54806e14 −0.407605
$$310$$ 0 0
$$311$$ 1.30645e15i 1.44388i 0.691955 + 0.721941i $$0.256750\pi$$
−0.691955 + 0.721941i $$0.743250\pi$$
$$312$$ 0 0
$$313$$ −9.51812e14 −1.01224 −0.506122 0.862462i $$-0.668921\pi$$
−0.506122 + 0.862462i $$0.668921\pi$$
$$314$$ 0 0
$$315$$ − 4.69618e13i − 0.0480708i
$$316$$ 0 0
$$317$$ −8.04849e14 −0.793157 −0.396578 0.918001i $$-0.629802\pi$$
−0.396578 + 0.918001i $$0.629802\pi$$
$$318$$ 0 0
$$319$$ − 8.56634e14i − 0.812926i
$$320$$ 0 0
$$321$$ −7.54530e14 −0.689678
$$322$$ 0 0
$$323$$ 1.30643e15i 1.15046i
$$324$$ 0 0
$$325$$ −7.80030e14 −0.661928
$$326$$ 0 0
$$327$$ 2.62578e14i 0.214769i
$$328$$ 0 0
$$329$$ 6.77220e13 0.0534016
$$330$$ 0 0
$$331$$ − 1.68149e15i − 1.27858i −0.768967 0.639288i $$-0.779229\pi$$
0.768967 0.639288i $$-0.220771\pi$$
$$332$$ 0 0
$$333$$ −2.07829e14 −0.152420
$$334$$ 0 0
$$335$$ 5.91996e14i 0.418842i
$$336$$ 0 0
$$337$$ −8.00353e14 −0.546389 −0.273195 0.961959i $$-0.588080\pi$$
−0.273195 + 0.961959i $$0.588080\pi$$
$$338$$ 0 0
$$339$$ − 2.74773e14i − 0.181040i
$$340$$ 0 0
$$341$$ −6.99852e14 −0.445123
$$342$$ 0 0
$$343$$ 6.95009e14i 0.426801i
$$344$$ 0 0
$$345$$ −9.46551e13 −0.0561344
$$346$$ 0 0
$$347$$ 2.02247e15i 1.15853i 0.815141 + 0.579263i $$0.196660\pi$$
−0.815141 + 0.579263i $$0.803340\pi$$
$$348$$ 0 0
$$349$$ −1.47943e15 −0.818732 −0.409366 0.912370i $$-0.634250\pi$$
−0.409366 + 0.912370i $$0.634250\pi$$
$$350$$ 0 0
$$351$$ − 4.21708e14i − 0.225512i
$$352$$ 0 0
$$353$$ 3.77287e13 0.0194995 0.00974976 0.999952i $$-0.496897\pi$$
0.00974976 + 0.999952i $$0.496897\pi$$
$$354$$ 0 0
$$355$$ − 3.05431e13i − 0.0152596i
$$356$$ 0 0
$$357$$ 2.56759e14 0.124027
$$358$$ 0 0
$$359$$ 2.92222e15i 1.36504i 0.730865 + 0.682522i $$0.239117\pi$$
−0.730865 + 0.682522i $$0.760883\pi$$
$$360$$ 0 0
$$361$$ −8.20797e14 −0.370845
$$362$$ 0 0
$$363$$ − 7.30590e14i − 0.319326i
$$364$$ 0 0
$$365$$ 3.03437e15 1.28325
$$366$$ 0 0
$$367$$ 1.63345e15i 0.668514i 0.942482 + 0.334257i $$0.108485\pi$$
−0.942482 + 0.334257i $$0.891515\pi$$
$$368$$ 0 0
$$369$$ −5.88631e14 −0.233176
$$370$$ 0 0
$$371$$ 7.37149e14i 0.282691i
$$372$$ 0 0
$$373$$ −1.73734e15 −0.645108 −0.322554 0.946551i $$-0.604541\pi$$
−0.322554 + 0.946551i $$0.604541\pi$$
$$374$$ 0 0
$$375$$ − 1.65734e15i − 0.595969i
$$376$$ 0 0
$$377$$ 2.19458e15 0.764371
$$378$$ 0 0
$$379$$ − 2.47588e15i − 0.835401i −0.908585 0.417700i $$-0.862836\pi$$
0.908585 0.417700i $$-0.137164\pi$$
$$380$$ 0 0
$$381$$ −2.82568e15 −0.923791
$$382$$ 0 0
$$383$$ − 1.13485e15i − 0.359538i −0.983709 0.179769i $$-0.942465\pi$$
0.983709 0.179769i $$-0.0575350\pi$$
$$384$$ 0 0
$$385$$ 5.85282e14 0.179722
$$386$$ 0 0
$$387$$ 1.07101e15i 0.318807i
$$388$$ 0 0
$$389$$ −4.47360e15 −1.29110 −0.645549 0.763719i $$-0.723371\pi$$
−0.645549 + 0.763719i $$0.723371\pi$$
$$390$$ 0 0
$$391$$ − 5.17518e14i − 0.144832i
$$392$$ 0 0
$$393$$ 3.24046e15 0.879530
$$394$$ 0 0
$$395$$ − 2.26140e15i − 0.595380i
$$396$$ 0 0
$$397$$ 7.57834e15 1.93567 0.967834 0.251591i $$-0.0809536\pi$$
0.967834 + 0.251591i $$0.0809536\pi$$
$$398$$ 0 0
$$399$$ 5.96309e14i 0.147786i
$$400$$ 0 0
$$401$$ 2.42681e15 0.583673 0.291836 0.956468i $$-0.405734\pi$$
0.291836 + 0.956468i $$0.405734\pi$$
$$402$$ 0 0
$$403$$ − 1.79293e15i − 0.418536i
$$404$$ 0 0
$$405$$ 3.23437e14 0.0732925
$$406$$ 0 0
$$407$$ − 2.59016e15i − 0.569849i
$$408$$ 0 0
$$409$$ 3.65591e15 0.781009 0.390505 0.920601i $$-0.372301\pi$$
0.390505 + 0.920601i $$0.372301\pi$$
$$410$$ 0 0
$$411$$ − 2.73809e15i − 0.568064i
$$412$$ 0 0
$$413$$ 1.71449e15 0.345489
$$414$$ 0 0
$$415$$ − 4.03012e14i − 0.0788914i
$$416$$ 0 0
$$417$$ −1.32537e15 −0.252069
$$418$$ 0 0
$$419$$ − 9.72429e15i − 1.79711i −0.438865 0.898553i $$-0.644620\pi$$
0.438865 0.898553i $$-0.355380\pi$$
$$420$$ 0 0
$$421$$ 4.09655e15 0.735743 0.367871 0.929877i $$-0.380087\pi$$
0.367871 + 0.929877i $$0.380087\pi$$
$$422$$ 0 0
$$423$$ 4.66418e14i 0.0814203i
$$424$$ 0 0
$$425$$ 3.27093e15 0.555056
$$426$$ 0 0
$$427$$ 1.16906e15i 0.192873i
$$428$$ 0 0
$$429$$ 5.25572e15 0.843118
$$430$$ 0 0
$$431$$ 7.54740e15i 1.17743i 0.808342 + 0.588713i $$0.200365\pi$$
−0.808342 + 0.588713i $$0.799635\pi$$
$$432$$ 0 0
$$433$$ −8.06264e15 −1.22335 −0.611674 0.791110i $$-0.709503\pi$$
−0.611674 + 0.791110i $$0.709503\pi$$
$$434$$ 0 0
$$435$$ 1.68318e15i 0.248425i
$$436$$ 0 0
$$437$$ 1.20191e15 0.172577
$$438$$ 0 0
$$439$$ 8.34545e15i 1.16590i 0.812506 + 0.582952i $$0.198102\pi$$
−0.812506 + 0.582952i $$0.801898\pi$$
$$440$$ 0 0
$$441$$ −2.33475e15 −0.317401
$$442$$ 0 0
$$443$$ − 5.63523e15i − 0.745572i −0.927917 0.372786i $$-0.878403\pi$$
0.927917 0.372786i $$-0.121597\pi$$
$$444$$ 0 0
$$445$$ 1.58701e15 0.204371
$$446$$ 0 0
$$447$$ 6.46909e15i 0.810957i
$$448$$ 0 0
$$449$$ −1.21818e15 −0.148674 −0.0743368 0.997233i $$-0.523684\pi$$
−0.0743368 + 0.997233i $$0.523684\pi$$
$$450$$ 0 0
$$451$$ − 7.33607e15i − 0.871774i
$$452$$ 0 0
$$453$$ 6.85583e15 0.793361
$$454$$ 0 0
$$455$$ 1.49942e15i 0.168987i
$$456$$ 0 0
$$457$$ 1.04213e15 0.114400 0.0571998 0.998363i $$-0.481783\pi$$
0.0571998 + 0.998363i $$0.481783\pi$$
$$458$$ 0 0
$$459$$ 1.76836e15i 0.189102i
$$460$$ 0 0
$$461$$ −1.16311e16 −1.21175 −0.605877 0.795558i $$-0.707178\pi$$
−0.605877 + 0.795558i $$0.707178\pi$$
$$462$$ 0 0
$$463$$ 1.67988e15i 0.170527i 0.996358 + 0.0852636i $$0.0271732\pi$$
−0.996358 + 0.0852636i $$0.972827\pi$$
$$464$$ 0 0
$$465$$ 1.37512e15 0.136027
$$466$$ 0 0
$$467$$ − 1.24813e16i − 1.20325i −0.798777 0.601627i $$-0.794519\pi$$
0.798777 0.601627i $$-0.205481\pi$$
$$468$$ 0 0
$$469$$ −1.47736e15 −0.138819
$$470$$ 0 0
$$471$$ 1.13176e16i 1.03664i
$$472$$ 0 0
$$473$$ −1.33479e16 −1.19192
$$474$$ 0 0
$$475$$ 7.59654e15i 0.661385i
$$476$$ 0 0
$$477$$ −5.07692e15 −0.431013
$$478$$ 0 0
$$479$$ − 9.23267e15i − 0.764389i −0.924082 0.382195i $$-0.875168\pi$$
0.924082 0.382195i $$-0.124832\pi$$
$$480$$ 0 0
$$481$$ 6.63565e15 0.535813
$$482$$ 0 0
$$483$$ − 2.36217e14i − 0.0186049i
$$484$$ 0 0
$$485$$ −1.24802e16 −0.958894
$$486$$ 0 0
$$487$$ 1.80780e16i 1.35511i 0.735470 + 0.677557i $$0.236961\pi$$
−0.735470 + 0.677557i $$0.763039\pi$$
$$488$$ 0 0
$$489$$ −7.39023e15 −0.540511
$$490$$ 0 0
$$491$$ − 3.01221e15i − 0.214979i −0.994206 0.107490i $$-0.965719\pi$$
0.994206 0.107490i $$-0.0342813\pi$$
$$492$$ 0 0
$$493$$ −9.20263e15 −0.640959
$$494$$ 0 0
$$495$$ 4.03097e15i 0.274018i
$$496$$ 0 0
$$497$$ 7.62219e13 0.00505757
$$498$$ 0 0
$$499$$ − 3.33196e15i − 0.215822i −0.994161 0.107911i $$-0.965584\pi$$
0.994161 0.107911i $$-0.0344162\pi$$
$$500$$ 0 0
$$501$$ 1.46307e16 0.925205
$$502$$ 0 0
$$503$$ − 9.62787e15i − 0.594459i −0.954806 0.297230i $$-0.903937\pi$$
0.954806 0.297230i $$-0.0960627\pi$$
$$504$$ 0 0
$$505$$ −1.94720e16 −1.17398
$$506$$ 0 0
$$507$$ 3.65858e15i 0.215409i
$$508$$ 0 0
$$509$$ −2.53124e16 −1.45555 −0.727774 0.685817i $$-0.759445\pi$$
−0.727774 + 0.685817i $$0.759445\pi$$
$$510$$ 0 0
$$511$$ 7.57244e15i 0.425315i
$$512$$ 0 0
$$513$$ −4.10692e15 −0.225327
$$514$$ 0 0
$$515$$ 8.68852e15i 0.465696i
$$516$$ 0 0
$$517$$ −5.81293e15 −0.304405
$$518$$ 0 0
$$519$$ − 1.21550e16i − 0.621945i
$$520$$ 0 0
$$521$$ −2.07262e15 −0.103632 −0.0518158 0.998657i $$-0.516501\pi$$
−0.0518158 + 0.998657i $$0.516501\pi$$
$$522$$ 0 0
$$523$$ 1.96180e16i 0.958617i 0.877646 + 0.479309i $$0.159113\pi$$
−0.877646 + 0.479309i $$0.840887\pi$$
$$524$$ 0 0
$$525$$ 1.49299e15 0.0713017
$$526$$ 0 0
$$527$$ 7.51836e15i 0.350961i
$$528$$ 0 0
$$529$$ 2.14385e16 0.978274
$$530$$ 0 0
$$531$$ 1.18081e16i 0.526760i
$$532$$ 0 0
$$533$$ 1.87940e16 0.819704
$$534$$ 0 0
$$535$$ 1.84770e16i 0.787969i
$$536$$ 0 0
$$537$$ 1.51553e16 0.632002
$$538$$ 0 0
$$539$$ − 2.90978e16i − 1.18666i
$$540$$ 0 0
$$541$$ 3.69504e16 1.47379 0.736895 0.676007i $$-0.236291\pi$$
0.736895 + 0.676007i $$0.236291\pi$$
$$542$$ 0 0
$$543$$ 1.45284e15i 0.0566785i
$$544$$ 0 0
$$545$$ 6.43004e15 0.245377
$$546$$ 0 0
$$547$$ − 1.90846e16i − 0.712458i −0.934399 0.356229i $$-0.884062\pi$$
0.934399 0.356229i $$-0.115938\pi$$
$$548$$ 0 0
$$549$$ −8.05161e15 −0.294069
$$550$$ 0 0
$$551$$ − 2.13726e16i − 0.763743i
$$552$$ 0 0
$$553$$ 5.64343e15 0.197330
$$554$$ 0 0
$$555$$ 5.08934e15i 0.174142i
$$556$$ 0 0
$$557$$ −1.48841e16 −0.498415 −0.249208 0.968450i $$-0.580170\pi$$
−0.249208 + 0.968450i $$0.580170\pi$$
$$558$$ 0 0
$$559$$ − 3.41957e16i − 1.12073i
$$560$$ 0 0
$$561$$ −2.20390e16 −0.706992
$$562$$ 0 0
$$563$$ 1.17234e15i 0.0368133i 0.999831 + 0.0184067i $$0.00585935\pi$$
−0.999831 + 0.0184067i $$0.994141\pi$$
$$564$$ 0 0
$$565$$ −6.72866e15 −0.206842
$$566$$ 0 0
$$567$$ 8.07154e14i 0.0242917i
$$568$$ 0 0
$$569$$ 6.41843e16 1.89128 0.945639 0.325219i $$-0.105438\pi$$
0.945639 + 0.325219i $$0.105438\pi$$
$$570$$ 0 0
$$571$$ − 9.82359e15i − 0.283435i −0.989907 0.141718i $$-0.954738\pi$$
0.989907 0.141718i $$-0.0452625\pi$$
$$572$$ 0 0
$$573$$ 3.08937e16 0.872855
$$574$$ 0 0
$$575$$ − 3.00923e15i − 0.0832622i
$$576$$ 0 0
$$577$$ 2.13311e16 0.578040 0.289020 0.957323i $$-0.406671\pi$$
0.289020 + 0.957323i $$0.406671\pi$$
$$578$$ 0 0
$$579$$ − 8.25169e15i − 0.219014i
$$580$$ 0 0
$$581$$ 1.00574e15 0.0261474
$$582$$ 0 0
$$583$$ − 6.32733e16i − 1.61142i
$$584$$ 0 0
$$585$$ −1.03268e16 −0.257651
$$586$$ 0 0
$$587$$ 7.72848e16i 1.88915i 0.328302 + 0.944573i $$0.393524\pi$$
−0.328302 + 0.944573i $$0.606476\pi$$
$$588$$ 0 0
$$589$$ −1.74610e16 −0.418193
$$590$$ 0 0
$$591$$ − 3.50821e16i − 0.823305i
$$592$$ 0 0
$$593$$ 4.15519e16 0.955570 0.477785 0.878477i $$-0.341440\pi$$
0.477785 + 0.878477i $$0.341440\pi$$
$$594$$ 0 0
$$595$$ − 6.28755e15i − 0.141703i
$$596$$ 0 0
$$597$$ 8.09386e15 0.178776
$$598$$ 0 0
$$599$$ 5.90470e15i 0.127831i 0.997955 + 0.0639156i $$0.0203588\pi$$
−0.997955 + 0.0639156i $$0.979641\pi$$
$$600$$ 0 0
$$601$$ −6.32387e16 −1.34195 −0.670975 0.741480i $$-0.734124\pi$$
−0.670975 + 0.741480i $$0.734124\pi$$
$$602$$ 0 0
$$603$$ − 1.01749e16i − 0.211654i
$$604$$ 0 0
$$605$$ −1.78908e16 −0.364835
$$606$$ 0 0
$$607$$ − 9.38322e15i − 0.187594i −0.995591 0.0937971i $$-0.970099\pi$$
0.995591 0.0937971i $$-0.0299005\pi$$
$$608$$ 0 0
$$609$$ −4.20046e15 −0.0823366
$$610$$ 0 0
$$611$$ − 1.48920e16i − 0.286223i
$$612$$ 0 0
$$613$$ 4.29006e16 0.808538 0.404269 0.914640i $$-0.367526\pi$$
0.404269 + 0.914640i $$0.367526\pi$$
$$614$$ 0 0
$$615$$ 1.44145e16i 0.266408i
$$616$$ 0 0
$$617$$ 8.82180e16 1.59899 0.799496 0.600672i $$-0.205100\pi$$
0.799496 + 0.600672i $$0.205100\pi$$
$$618$$ 0 0
$$619$$ 4.76892e16i 0.847766i 0.905717 + 0.423883i $$0.139333\pi$$
−0.905717 + 0.423883i $$0.860667\pi$$
$$620$$ 0 0
$$621$$ 1.62688e15 0.0283665
$$622$$ 0 0
$$623$$ 3.96046e15i 0.0677356i
$$624$$ 0 0
$$625$$ −6.91535e15 −0.116020
$$626$$ 0 0
$$627$$ − 5.11843e16i − 0.842426i
$$628$$ 0 0
$$629$$ −2.78255e16 −0.449303
$$630$$ 0 0
$$631$$ − 6.79965e15i − 0.107723i −0.998548 0.0538617i $$-0.982847\pi$$
0.998548 0.0538617i $$-0.0171530\pi$$
$$632$$ 0 0
$$633$$ −7.05248e16 −1.09628
$$634$$ 0 0
$$635$$ 6.91956e16i 1.05545i
$$636$$ 0 0
$$637$$ 7.45448e16 1.11579
$$638$$ 0 0
$$639$$ 5.24959e14i 0.00771116i
$$640$$ 0 0
$$641$$ −3.57968e16 −0.516055 −0.258027 0.966138i $$-0.583072\pi$$
−0.258027 + 0.966138i $$0.583072\pi$$
$$642$$ 0 0
$$643$$ 4.41166e16i 0.624218i 0.950046 + 0.312109i $$0.101035\pi$$
−0.950046 + 0.312109i $$0.898965\pi$$
$$644$$ 0 0
$$645$$ 2.62270e16 0.364243
$$646$$ 0 0
$$647$$ 1.27368e17i 1.73634i 0.496267 + 0.868170i $$0.334704\pi$$
−0.496267 + 0.868170i $$0.665296\pi$$
$$648$$ 0 0
$$649$$ −1.47164e17 −1.96939
$$650$$ 0 0
$$651$$ 3.43169e15i 0.0450840i
$$652$$ 0 0
$$653$$ −9.19791e16 −1.18634 −0.593171 0.805076i $$-0.702124\pi$$
−0.593171 + 0.805076i $$0.702124\pi$$
$$654$$ 0 0
$$655$$ − 7.93526e16i − 1.00488i
$$656$$ 0 0
$$657$$ −5.21532e16 −0.648468
$$658$$ 0 0
$$659$$ 1.36757e17i 1.66970i 0.550477 + 0.834850i $$0.314446\pi$$
−0.550477 + 0.834850i $$0.685554\pi$$
$$660$$ 0 0
$$661$$ −2.59246e16 −0.310816 −0.155408 0.987850i $$-0.549669\pi$$
−0.155408 + 0.987850i $$0.549669\pi$$
$$662$$ 0 0
$$663$$ − 5.64610e16i − 0.664764i
$$664$$ 0 0
$$665$$ 1.46025e16 0.168848
$$666$$ 0 0
$$667$$ 8.46635e15i 0.0961482i
$$668$$ 0 0
$$669$$ −6.01723e16 −0.671182
$$670$$ 0 0
$$671$$ − 1.00347e17i − 1.09943i
$$672$$ 0 0
$$673$$ −6.31003e16 −0.679111 −0.339555 0.940586i $$-0.610277\pi$$
−0.339555 + 0.940586i $$0.610277\pi$$
$$674$$ 0 0
$$675$$ 1.02825e16i 0.108712i
$$676$$ 0 0
$$677$$ −1.78154e16 −0.185039 −0.0925196 0.995711i $$-0.529492\pi$$
−0.0925196 + 0.995711i $$0.529492\pi$$
$$678$$ 0 0
$$679$$ − 3.11450e16i − 0.317811i
$$680$$ 0 0
$$681$$ 1.36535e16 0.136887
$$682$$ 0 0
$$683$$ 8.17507e16i 0.805318i 0.915350 + 0.402659i $$0.131914\pi$$
−0.915350 + 0.402659i $$0.868086\pi$$
$$684$$ 0 0
$$685$$ −6.70507e16 −0.649023
$$686$$ 0 0
$$687$$ − 8.91191e16i − 0.847677i
$$688$$ 0 0
$$689$$ 1.62098e17 1.51517
$$690$$ 0 0
$$691$$ 1.18406e17i 1.08769i 0.839184 + 0.543847i $$0.183033\pi$$
−0.839184 + 0.543847i $$0.816967\pi$$
$$692$$ 0 0
$$693$$ −1.00595e16 −0.0908191
$$694$$ 0 0
$$695$$ 3.24557e16i 0.287993i
$$696$$ 0 0
$$697$$ −7.88097e16 −0.687358
$$698$$ 0 0
$$699$$ 2.06167e16i 0.176748i
$$700$$ 0 0
$$701$$ −3.50131e16 −0.295068 −0.147534 0.989057i $$-0.547134\pi$$
−0.147534 + 0.989057i $$0.547134\pi$$
$$702$$ 0 0
$$703$$ − 6.46231e16i − 0.535373i
$$704$$ 0 0
$$705$$ 1.14217e16 0.0930241
$$706$$ 0 0
$$707$$ − 4.85934e16i − 0.389100i
$$708$$ 0 0
$$709$$ 6.04581e16 0.475967 0.237984 0.971269i $$-0.423514\pi$$
0.237984 + 0.971269i $$0.423514\pi$$
$$710$$ 0 0
$$711$$ 3.88677e16i 0.300865i
$$712$$ 0 0
$$713$$ 6.91683e15 0.0526466
$$714$$ 0 0
$$715$$ − 1.28703e17i − 0.963277i
$$716$$ 0 0
$$717$$ 7.02337e16 0.516929
$$718$$ 0 0
$$719$$ − 1.15269e17i − 0.834330i −0.908831 0.417165i $$-0.863024\pi$$
0.908831 0.417165i $$-0.136976\pi$$
$$720$$ 0 0
$$721$$ −2.16826e16 −0.154348
$$722$$ 0 0
$$723$$ 4.14837e16i 0.290434i
$$724$$ 0 0
$$725$$ −5.35108e16 −0.368479
$$726$$ 0 0
$$727$$ 7.77948e16i 0.526919i 0.964670 + 0.263460i $$0.0848636\pi$$
−0.964670 + 0.263460i $$0.915136\pi$$
$$728$$ 0 0
$$729$$ −5.55906e15 −0.0370370
$$730$$ 0 0
$$731$$ 1.43394e17i 0.939781i
$$732$$ 0 0
$$733$$ 2.20976e17 1.42469 0.712345 0.701829i $$-0.247633\pi$$
0.712345 + 0.701829i $$0.247633\pi$$
$$734$$ 0 0
$$735$$ 5.71735e16i 0.362636i
$$736$$ 0 0
$$737$$ 1.26809e17 0.791309
$$738$$ 0 0
$$739$$ − 2.10708e17i − 1.29365i −0.762640 0.646823i $$-0.776097\pi$$
0.762640 0.646823i $$-0.223903\pi$$
$$740$$ 0 0
$$741$$ 1.31128e17 0.792108
$$742$$ 0 0
$$743$$ 1.48321e17i 0.881594i 0.897607 + 0.440797i $$0.145304\pi$$
−0.897607 + 0.440797i $$0.854696\pi$$
$$744$$ 0 0
$$745$$ 1.58416e17 0.926532
$$746$$ 0 0
$$747$$ 6.92676e15i 0.0398663i
$$748$$ 0 0
$$749$$ −4.61103e16 −0.261160
$$750$$ 0 0
$$751$$ 8.57383e16i 0.477898i 0.971032 + 0.238949i $$0.0768029\pi$$
−0.971032 + 0.238949i $$0.923197\pi$$
$$752$$ 0 0
$$753$$ −1.57084e17 −0.861712
$$754$$ 0 0
$$755$$ − 1.67886e17i − 0.906428i
$$756$$ 0 0
$$757$$ 1.75448e16 0.0932335 0.0466168 0.998913i $$-0.485156\pi$$
0.0466168 + 0.998913i $$0.485156\pi$$
$$758$$ 0 0
$$759$$ 2.02757e16i 0.106054i
$$760$$ 0 0
$$761$$ 2.12682e17 1.09502 0.547511 0.836798i $$-0.315575\pi$$
0.547511 + 0.836798i $$0.315575\pi$$
$$762$$ 0 0
$$763$$ 1.60465e16i 0.0813267i
$$764$$ 0 0
$$765$$ 4.33039e16 0.216052
$$766$$ 0 0
$$767$$ − 3.77014e17i − 1.85176i
$$768$$ 0 0
$$769$$ 7.21877e16 0.349064 0.174532 0.984651i $$-0.444159\pi$$
0.174532 + 0.984651i $$0.444159\pi$$
$$770$$ 0 0
$$771$$ − 1.23435e17i − 0.587641i
$$772$$ 0 0
$$773$$ −1.73938e17 −0.815299 −0.407649 0.913139i $$-0.633651\pi$$
−0.407649 + 0.913139i $$0.633651\pi$$
$$774$$ 0 0
$$775$$ 4.37172e16i 0.201763i
$$776$$ 0 0
$$777$$ −1.27007e16 −0.0577167
$$778$$ 0 0
$$779$$ − 1.83031e17i − 0.819031i
$$780$$ 0 0
$$781$$ −6.54253e15 −0.0288296
$$782$$ 0 0
$$783$$ − 2.89296e16i − 0.125537i
$$784$$ 0 0
$$785$$ 2.77146e17 1.18438
$$786$$ 0 0
$$787$$ 1.47492e16i 0.0620754i 0.999518 + 0.0310377i $$0.00988119\pi$$
−0.999518 + 0.0310377i $$0.990119\pi$$
$$788$$ 0 0
$$789$$ 2.52831e17 1.04802
$$790$$ 0 0
$$791$$ − 1.67917e16i − 0.0685546i
$$792$$ 0 0
$$793$$ 2.57075e17 1.03376
$$794$$ 0 0
$$795$$ 1.24324e17i 0.492439i
$$796$$ 0 0
$$797$$ −3.41145e17 −1.33103 −0.665517 0.746382i $$-0.731789\pi$$
−0.665517 + 0.746382i $$0.731789\pi$$
$$798$$ 0 0
$$799$$ 6.24470e16i 0.240011i
$$800$$ 0 0
$$801$$ −2.72766e16 −0.103275
$$802$$ 0 0
$$803$$ − 6.49982e17i − 2.42442i
$$804$$ 0 0
$$805$$ −5.78450e15 −0.0212564
$$806$$ 0 0
$$807$$ 2.19574e17i 0.794950i
$$808$$ 0 0
$$809$$ −1.84056e16 −0.0656537 −0.0328269 0.999461i $$-0.510451\pi$$
−0.0328269 + 0.999461i $$0.510451\pi$$
$$810$$ 0 0
$$811$$ 3.79008e17i 1.33206i 0.745926 + 0.666029i $$0.232007\pi$$
−0.745926 + 0.666029i $$0.767993\pi$$
$$812$$ 0 0
$$813$$ 1.52835e17 0.529272
$$814$$ 0 0
$$815$$ 1.80973e17i 0.617544i
$$816$$ 0 0
$$817$$ −3.33024e17 −1.11981
$$818$$ 0 0
$$819$$ − 2.57711e16i − 0.0853946i
$$820$$ 0 0
$$821$$ −4.57140e17 −1.49276 −0.746381 0.665519i $$-0.768210\pi$$
−0.746381 + 0.665519i $$0.768210\pi$$
$$822$$ 0 0
$$823$$ 3.70495e17i 1.19229i 0.802876 + 0.596147i $$0.203302\pi$$
−0.802876 + 0.596147i $$0.796698\pi$$
$$824$$ 0 0
$$825$$ −1.28151e17 −0.406441
$$826$$ 0 0
$$827$$ − 1.82136e17i − 0.569328i −0.958627 0.284664i $$-0.908118\pi$$
0.958627 0.284664i $$-0.0918821\pi$$
$$828$$ 0 0
$$829$$ 2.11760e17 0.652404 0.326202 0.945300i $$-0.394231\pi$$
0.326202 + 0.945300i $$0.394231\pi$$
$$830$$ 0 0
$$831$$ 1.23817e17i 0.375989i
$$832$$ 0 0
$$833$$ −3.12591e17 −0.935636
$$834$$ 0 0
$$835$$ − 3.58277e17i − 1.05706i
$$836$$ 0 0
$$837$$ −2.36349e16 −0.0687385
$$838$$ 0 0
$$839$$ 1.42876e17i 0.409626i 0.978801 + 0.204813i $$0.0656587\pi$$
−0.978801 + 0.204813i $$0.934341\pi$$
$$840$$ 0 0
$$841$$ −2.03264e17 −0.574493
$$842$$ 0 0
$$843$$ 2.40186e17i 0.669241i
$$844$$ 0 0
$$845$$ 8.95917e16 0.246109
$$846$$ 0 0
$$847$$ − 4.46473e16i − 0.120919i
$$848$$ 0 0
$$849$$ −1.76728e17 −0.471911
$$850$$ 0 0
$$851$$ 2.55993e16i 0.0673985i
$$852$$ 0 0
$$853$$ 7.50904e16 0.194935 0.0974675 0.995239i $$-0.468926\pi$$
0.0974675 + 0.995239i $$0.468926\pi$$
$$854$$ 0 0
$$855$$ 1.00571e17i 0.257439i
$$856$$ 0 0
$$857$$ −1.57808e16 −0.0398330 −0.0199165 0.999802i $$-0.506340\pi$$
−0.0199165 + 0.999802i $$0.506340\pi$$
$$858$$ 0 0
$$859$$ 6.52158e17i 1.62328i 0.584157 + 0.811641i $$0.301425\pi$$
−0.584157 + 0.811641i $$0.698575\pi$$
$$860$$ 0 0
$$861$$ −3.59720e16 −0.0882970
$$862$$ 0 0
$$863$$ 3.77726e17i 0.914350i 0.889377 + 0.457175i $$0.151139\pi$$
−0.889377 + 0.457175i $$0.848861\pi$$
$$864$$ 0 0
$$865$$ −2.97654e17 −0.710583
$$866$$ 0 0
$$867$$ − 8.45892e15i − 0.0199159i
$$868$$ 0 0
$$869$$ −4.84405e17 −1.12484
$$870$$ 0 0
$$871$$ 3.24869e17i 0.744045i
$$872$$ 0 0
$$873$$ 2.14503e17 0.484560
$$874$$ 0 0
$$875$$ − 1.01282e17i − 0.225676i
$$876$$ 0 0
$$877$$ 2.37602e17 0.522220 0.261110 0.965309i $$-0.415912\pi$$
0.261110 + 0.965309i $$0.415912\pi$$
$$878$$ 0 0
$$879$$ 1.47423e17i 0.319619i
$$880$$ 0 0
$$881$$ −1.87143e17 −0.400238 −0.200119 0.979772i $$-0.564133\pi$$
−0.200119 + 0.979772i $$0.564133\pi$$
$$882$$ 0 0
$$883$$ − 5.84655e17i − 1.23349i −0.787163 0.616745i $$-0.788451\pi$$
0.787163 0.616745i $$-0.211549\pi$$
$$884$$ 0 0
$$885$$ 2.89158e17 0.601832
$$886$$ 0 0
$$887$$ − 2.95742e17i − 0.607254i −0.952791 0.303627i $$-0.901802\pi$$
0.952791 0.303627i $$-0.0981977\pi$$
$$888$$ 0 0
$$889$$ −1.72681e17 −0.349812
$$890$$ 0 0
$$891$$ − 6.92822e16i − 0.138470i
$$892$$ 0 0
$$893$$ −1.45030e17 −0.285988
$$894$$ 0 0
$$895$$ − 3.71124e17i − 0.722074i
$$896$$ 0 0
$$897$$ −5.19437e16 −0.0997191
$$898$$ 0 0
$$899$$ − 1.22997e17i − 0.232989i
$$900$$ 0 0
$$901$$ −6.79731e17 −1.27054
$$902$$ 0 0
$$903$$ 6.54509e16i 0.120723i
$$904$$ 0 0
$$905$$ 3.55772e16 0.0647561
$$906$$ 0 0
$$907$$ − 8.53775e14i − 0.00153356i −1.00000 0.000766778i $$-0.999756\pi$$
1.00000 0.000766778i $$-0.000244073\pi$$
$$908$$ 0 0
$$909$$ 3.34674e17 0.593252
$$910$$ 0 0
$$911$$ − 3.24702e17i − 0.568035i −0.958819 0.284018i $$-0.908333\pi$$
0.958819 0.284018i $$-0.0916674\pi$$
$$912$$ 0 0
$$913$$ −8.63277e16 −0.149048
$$914$$ 0 0
$$915$$ 1.97169e17i 0.335979i
$$916$$ 0 0
$$917$$ 1.98029e17 0.333052
$$918$$ 0 0
$$919$$ 1.05696e18i 1.75454i 0.479996 + 0.877271i $$0.340638\pi$$
−0.479996 + 0.877271i $$0.659362\pi$$
$$920$$ 0 0
$$921$$ −3.05182e17 −0.500035
$$922$$ 0 0
$$923$$ − 1.67611e16i − 0.0271077i
$$924$$ 0 0
$$925$$ −1.61798e17 −0.258299
$$926$$ 0 0
$$927$$ − 1.49334e17i − 0.235331i
$$928$$ 0 0
$$929$$ 2.48690e17 0.386869 0.193435 0.981113i $$-0.438037\pi$$
0.193435 + 0.981113i $$0.438037\pi$$
$$930$$ 0 0
$$931$$ − 7.25975e17i − 1.11487i
$$932$$ 0 0
$$933$$ −5.49871e17 −0.833625
$$934$$ 0 0
$$935$$ 5.39693e17i 0.807751i
$$936$$ 0 0
$$937$$ 4.48490e17 0.662697 0.331348 0.943508i $$-0.392496\pi$$
0.331348 + 0.943508i $$0.392496\pi$$
$$938$$ 0 0
$$939$$ − 4.00607e17i − 0.584419i
$$940$$ 0 0
$$941$$ 2.19998e17 0.316869 0.158435 0.987369i $$-0.449355\pi$$
0.158435 + 0.987369i $$0.449355\pi$$
$$942$$ 0 0
$$943$$ 7.25043e16i 0.103108i
$$944$$ 0 0
$$945$$ 1.97657e16 0.0277537
$$946$$ 0 0
$$947$$ − 5.50630e17i − 0.763414i −0.924283 0.381707i $$-0.875336\pi$$
0.924283 0.381707i $$-0.124664\pi$$
$$948$$ 0 0
$$949$$ 1.66517e18 2.27961
$$950$$ 0 0
$$951$$ − 3.38752e17i − 0.457929i
$$952$$ 0 0
$$953$$ 9.59450e17 1.28075 0.640376 0.768062i $$-0.278779\pi$$
0.640376 + 0.768062i $$0.278779\pi$$
$$954$$ 0 0
$$955$$ − 7.56528e17i − 0.997251i
$$956$$ 0 0
$$957$$ 3.60547e17 0.469343
$$958$$ 0 0
$$959$$ − 1.67329e17i − 0.215109i
$$960$$ 0 0
$$961$$ 6.87177e17 0.872425
$$962$$ 0 0
$$963$$ − 3.17573e17i − 0.398186i
$$964$$ 0 0
$$965$$ −2.02068e17 −0.250227
$$966$$ 0 0
$$967$$ 1.36871e17i 0.167399i 0.996491 + 0.0836993i $$0.0266735\pi$$
−0.996491 + 0.0836993i $$0.973326\pi$$
$$968$$ 0 0
$$969$$ −5.49862e17 −0.664218
$$970$$ 0 0
$$971$$ 1.45694e18i 1.73831i 0.494538 + 0.869156i $$0.335337\pi$$
−0.494538 + 0.869156i $$0.664663\pi$$
$$972$$ 0 0
$$973$$ −8.09949e16 −0.0954510
$$974$$ 0 0
$$975$$ − 3.28305e17i − 0.382165i
$$976$$ 0 0
$$977$$ 1.57226e18 1.80783 0.903915 0.427713i $$-0.140681\pi$$
0.903915 + 0.427713i $$0.140681\pi$$
$$978$$ 0 0
$$979$$ − 3.39947e17i − 0.386113i
$$980$$ 0 0
$$981$$ −1.10516e17 −0.123997
$$982$$ 0 0
$$983$$ 1.13899e18i 1.26240i 0.775618 + 0.631202i $$0.217438\pi$$
−0.775618 + 0.631202i $$0.782562\pi$$
$$984$$ 0 0
$$985$$ −8.59094e17 −0.940640
$$986$$ 0 0
$$987$$ 2.85034e16i 0.0308314i
$$988$$ 0 0
$$989$$ 1.31921e17 0.140973
$$990$$ 0 0
$$991$$ − 1.72529e18i − 1.82146i −0.413001 0.910731i $$-0.635519\pi$$
0.413001 0.910731i $$-0.364481\pi$$
$$992$$ 0 0
$$993$$ 7.07720e17 0.738187
$$994$$ 0 0
$$995$$ − 1.98203e17i − 0.204255i
$$996$$ 0 0
$$997$$ 9.17404e17 0.934092 0.467046 0.884233i $$-0.345318\pi$$
0.467046 + 0.884233i $$0.345318\pi$$
$$998$$ 0 0
$$999$$ − 8.74728e16i − 0.0879995i
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.31.4 yes 4
3.2 odd 2 144.13.g.h.127.2 4
4.3 odd 2 inner 48.13.g.a.31.2 4
8.3 odd 2 192.13.g.c.127.3 4
8.5 even 2 192.13.g.c.127.1 4
12.11 even 2 144.13.g.h.127.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
48.13.g.a.31.2 4 4.3 odd 2 inner
48.13.g.a.31.4 yes 4 1.1 even 1 trivial
144.13.g.h.127.1 4 12.11 even 2
144.13.g.h.127.2 4 3.2 odd 2
192.13.g.c.127.1 4 8.5 even 2
192.13.g.c.127.3 4 8.3 odd 2