# Properties

 Label 48.9.g.c.31.1 Level $48$ Weight $9$ Character 48.31 Analytic conductor $19.554$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,9,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, 9, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ 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: $$19.5541732829$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{1801})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 451x^{2} + 450x + 202500$$ x^4 - x^3 + 451*x^2 + 450*x + 202500 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 $$10.8595 + 18.8093i$$ of defining polynomial Character $$\chi$$ $$=$$ 48.31 Dual form 48.9.g.c.31.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-46.7654i q^{3} -1084.52 q^{5} -426.979i q^{7} -2187.00 q^{9} +O(q^{10})$$ $$q-46.7654i q^{3} -1084.52 q^{5} -426.979i q^{7} -2187.00 q^{9} +14232.3i q^{11} +34213.5 q^{13} +50717.8i q^{15} +20078.0 q^{17} -196118. i q^{19} -19967.8 q^{21} +347985. i q^{23} +785551. q^{25} +102276. i q^{27} +1.00247e6 q^{29} +1.63986e6i q^{31} +665580. q^{33} +463066. i q^{35} -791050. q^{37} -1.60001e6i q^{39} +1.36054e6 q^{41} +1.50116e6i q^{43} +2.37184e6 q^{45} +1.49258e6i q^{47} +5.58249e6 q^{49} -938954. i q^{51} -8.94860e6 q^{53} -1.54352e7i q^{55} -9.17155e6 q^{57} +8.50216e6i q^{59} -1.78549e7 q^{61} +933804. i q^{63} -3.71051e7 q^{65} -3.49438e7i q^{67} +1.62736e7 q^{69} +3.84799e7i q^{71} +2.11205e7 q^{73} -3.67366e7i q^{75} +6.07690e6 q^{77} +3.67300e7i q^{79} +4.78297e6 q^{81} -2.94243e7i q^{83} -2.17749e7 q^{85} -4.68808e7i q^{87} -3.70040e7 q^{89} -1.46085e7i q^{91} +7.66885e7 q^{93} +2.12694e8i q^{95} +1.26423e8 q^{97} -3.11261e7i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 264 q^{5} - 8748 q^{9}+O(q^{10})$$ 4 * q - 264 * q^5 - 8748 * q^9 $$4 q - 264 q^{5} - 8748 q^{9} + 14632 q^{13} + 332904 q^{17} + 250128 q^{21} + 2604428 q^{25} + 2343576 q^{29} + 2002320 q^{33} + 4315784 q^{37} + 9035496 q^{41} + 577368 q^{45} + 3458884 q^{49} - 42186600 q^{53} + 2253744 q^{57} - 48148408 q^{61} - 125450832 q^{65} + 23514624 q^{69} - 21215480 q^{73} - 32354496 q^{77} + 19131876 q^{81} + 235297584 q^{85} - 12675576 q^{89} + 193564080 q^{93} + 263153800 q^{97}+O(q^{100})$$ 4 * q - 264 * q^5 - 8748 * q^9 + 14632 * q^13 + 332904 * q^17 + 250128 * q^21 + 2604428 * q^25 + 2343576 * q^29 + 2002320 * q^33 + 4315784 * q^37 + 9035496 * q^41 + 577368 * q^45 + 3458884 * q^49 - 42186600 * q^53 + 2253744 * q^57 - 48148408 * q^61 - 125450832 * q^65 + 23514624 * q^69 - 21215480 * q^73 - 32354496 * q^77 + 19131876 * q^81 + 235297584 * q^85 - 12675576 * q^89 + 193564080 * q^93 + 263153800 * 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
$$3$$ − 46.7654i − 0.577350i
$$4$$ 0 0
$$5$$ −1084.52 −1.73523 −0.867613 0.497240i $$-0.834347\pi$$
−0.867613 + 0.497240i $$0.834347\pi$$
$$6$$ 0 0
$$7$$ − 426.979i − 0.177834i −0.996039 0.0889170i $$-0.971659\pi$$
0.996039 0.0889170i $$-0.0283406\pi$$
$$8$$ 0 0
$$9$$ −2187.00 −0.333333
$$10$$ 0 0
$$11$$ 14232.3i 0.972087i 0.873935 + 0.486043i $$0.161560\pi$$
−0.873935 + 0.486043i $$0.838440\pi$$
$$12$$ 0 0
$$13$$ 34213.5 1.19791 0.598955 0.800783i $$-0.295583\pi$$
0.598955 + 0.800783i $$0.295583\pi$$
$$14$$ 0 0
$$15$$ 50717.8i 1.00183i
$$16$$ 0 0
$$17$$ 20078.0 0.240394 0.120197 0.992750i $$-0.461647\pi$$
0.120197 + 0.992750i $$0.461647\pi$$
$$18$$ 0 0
$$19$$ − 196118.i − 1.50489i −0.658657 0.752443i $$-0.728875\pi$$
0.658657 0.752443i $$-0.271125\pi$$
$$20$$ 0 0
$$21$$ −19967.8 −0.102672
$$22$$ 0 0
$$23$$ 347985.i 1.24351i 0.783212 + 0.621754i $$0.213580\pi$$
−0.783212 + 0.621754i $$0.786420\pi$$
$$24$$ 0 0
$$25$$ 785551. 2.01101
$$26$$ 0 0
$$27$$ 102276.i 0.192450i
$$28$$ 0 0
$$29$$ 1.00247e6 1.41735 0.708677 0.705533i $$-0.249292\pi$$
0.708677 + 0.705533i $$0.249292\pi$$
$$30$$ 0 0
$$31$$ 1.63986e6i 1.77566i 0.460175 + 0.887828i $$0.347787\pi$$
−0.460175 + 0.887828i $$0.652213\pi$$
$$32$$ 0 0
$$33$$ 665580. 0.561234
$$34$$ 0 0
$$35$$ 463066.i 0.308582i
$$36$$ 0 0
$$37$$ −791050. −0.422082 −0.211041 0.977477i $$-0.567685\pi$$
−0.211041 + 0.977477i $$0.567685\pi$$
$$38$$ 0 0
$$39$$ − 1.60001e6i − 0.691613i
$$40$$ 0 0
$$41$$ 1.36054e6 0.481478 0.240739 0.970590i $$-0.422610\pi$$
0.240739 + 0.970590i $$0.422610\pi$$
$$42$$ 0 0
$$43$$ 1.50116e6i 0.439091i 0.975602 + 0.219545i $$0.0704574\pi$$
−0.975602 + 0.219545i $$0.929543\pi$$
$$44$$ 0 0
$$45$$ 2.37184e6 0.578409
$$46$$ 0 0
$$47$$ 1.49258e6i 0.305878i 0.988236 + 0.152939i $$0.0488737\pi$$
−0.988236 + 0.152939i $$0.951126\pi$$
$$48$$ 0 0
$$49$$ 5.58249e6 0.968375
$$50$$ 0 0
$$51$$ − 938954.i − 0.138792i
$$52$$ 0 0
$$53$$ −8.94860e6 −1.13410 −0.567050 0.823683i $$-0.691915\pi$$
−0.567050 + 0.823683i $$0.691915\pi$$
$$54$$ 0 0
$$55$$ − 1.54352e7i − 1.68679i
$$56$$ 0 0
$$57$$ −9.17155e6 −0.868847
$$58$$ 0 0
$$59$$ 8.50216e6i 0.701651i 0.936441 + 0.350826i $$0.114099\pi$$
−0.936441 + 0.350826i $$0.885901\pi$$
$$60$$ 0 0
$$61$$ −1.78549e7 −1.28955 −0.644774 0.764374i $$-0.723048\pi$$
−0.644774 + 0.764374i $$0.723048\pi$$
$$62$$ 0 0
$$63$$ 933804.i 0.0592780i
$$64$$ 0 0
$$65$$ −3.71051e7 −2.07864
$$66$$ 0 0
$$67$$ − 3.49438e7i − 1.73409i −0.498232 0.867044i $$-0.666017\pi$$
0.498232 0.867044i $$-0.333983\pi$$
$$68$$ 0 0
$$69$$ 1.62736e7 0.717940
$$70$$ 0 0
$$71$$ 3.84799e7i 1.51426i 0.653263 + 0.757131i $$0.273400\pi$$
−0.653263 + 0.757131i $$0.726600\pi$$
$$72$$ 0 0
$$73$$ 2.11205e7 0.743726 0.371863 0.928288i $$-0.378719\pi$$
0.371863 + 0.928288i $$0.378719\pi$$
$$74$$ 0 0
$$75$$ − 3.67366e7i − 1.16106i
$$76$$ 0 0
$$77$$ 6.07690e6 0.172870
$$78$$ 0 0
$$79$$ 3.67300e7i 0.943002i 0.881865 + 0.471501i $$0.156288\pi$$
−0.881865 + 0.471501i $$0.843712\pi$$
$$80$$ 0 0
$$81$$ 4.78297e6 0.111111
$$82$$ 0 0
$$83$$ − 2.94243e7i − 0.620003i −0.950736 0.310002i $$-0.899670\pi$$
0.950736 0.310002i $$-0.100330\pi$$
$$84$$ 0 0
$$85$$ −2.17749e7 −0.417139
$$86$$ 0 0
$$87$$ − 4.68808e7i − 0.818309i
$$88$$ 0 0
$$89$$ −3.70040e7 −0.589778 −0.294889 0.955531i $$-0.595283\pi$$
−0.294889 + 0.955531i $$0.595283\pi$$
$$90$$ 0 0
$$91$$ − 1.46085e7i − 0.213029i
$$92$$ 0 0
$$93$$ 7.66885e7 1.02518
$$94$$ 0 0
$$95$$ 2.12694e8i 2.61132i
$$96$$ 0 0
$$97$$ 1.26423e8 1.42803 0.714017 0.700129i $$-0.246874\pi$$
0.714017 + 0.700129i $$0.246874\pi$$
$$98$$ 0 0
$$99$$ − 3.11261e7i − 0.324029i
$$100$$ 0 0
$$101$$ 6.91121e7 0.664154 0.332077 0.943252i $$-0.392251\pi$$
0.332077 + 0.943252i $$0.392251\pi$$
$$102$$ 0 0
$$103$$ 1.20199e7i 0.106795i 0.998573 + 0.0533975i $$0.0170050\pi$$
−0.998573 + 0.0533975i $$0.982995\pi$$
$$104$$ 0 0
$$105$$ 2.16555e7 0.178160
$$106$$ 0 0
$$107$$ 2.48941e8i 1.89916i 0.313524 + 0.949580i $$0.398490\pi$$
−0.313524 + 0.949580i $$0.601510\pi$$
$$108$$ 0 0
$$109$$ 1.61675e8 1.14534 0.572672 0.819784i $$-0.305907\pi$$
0.572672 + 0.819784i $$0.305907\pi$$
$$110$$ 0 0
$$111$$ 3.69938e7i 0.243689i
$$112$$ 0 0
$$113$$ −7.05559e7 −0.432733 −0.216366 0.976312i $$-0.569421\pi$$
−0.216366 + 0.976312i $$0.569421\pi$$
$$114$$ 0 0
$$115$$ − 3.77395e8i − 2.15777i
$$116$$ 0 0
$$117$$ −7.48249e7 −0.399303
$$118$$ 0 0
$$119$$ − 8.57288e6i − 0.0427503i
$$120$$ 0 0
$$121$$ 1.18000e7 0.0550478
$$122$$ 0 0
$$123$$ − 6.36263e7i − 0.277982i
$$124$$ 0 0
$$125$$ −4.28304e8 −1.75433
$$126$$ 0 0
$$127$$ 9.17381e7i 0.352643i 0.984333 + 0.176321i $$0.0564198\pi$$
−0.984333 + 0.176321i $$0.943580\pi$$
$$128$$ 0 0
$$129$$ 7.02025e7 0.253509
$$130$$ 0 0
$$131$$ − 2.58852e8i − 0.878953i −0.898254 0.439477i $$-0.855164\pi$$
0.898254 0.439477i $$-0.144836\pi$$
$$132$$ 0 0
$$133$$ −8.37384e7 −0.267620
$$134$$ 0 0
$$135$$ − 1.10920e8i − 0.333944i
$$136$$ 0 0
$$137$$ 3.73710e8 1.06085 0.530423 0.847733i $$-0.322033\pi$$
0.530423 + 0.847733i $$0.322033\pi$$
$$138$$ 0 0
$$139$$ − 1.61105e7i − 0.0431569i −0.999767 0.0215785i $$-0.993131\pi$$
0.999767 0.0215785i $$-0.00686917\pi$$
$$140$$ 0 0
$$141$$ 6.98013e7 0.176598
$$142$$ 0 0
$$143$$ 4.86937e8i 1.16447i
$$144$$ 0 0
$$145$$ −1.08719e9 −2.45943
$$146$$ 0 0
$$147$$ − 2.61067e8i − 0.559092i
$$148$$ 0 0
$$149$$ −5.78564e8 −1.17383 −0.586917 0.809647i $$-0.699658\pi$$
−0.586917 + 0.809647i $$0.699658\pi$$
$$150$$ 0 0
$$151$$ 6.01470e8i 1.15693i 0.815708 + 0.578463i $$0.196347\pi$$
−0.815708 + 0.578463i $$0.803653\pi$$
$$152$$ 0 0
$$153$$ −4.39105e7 −0.0801314
$$154$$ 0 0
$$155$$ − 1.77845e9i − 3.08117i
$$156$$ 0 0
$$157$$ 1.28860e8 0.212090 0.106045 0.994361i $$-0.466181\pi$$
0.106045 + 0.994361i $$0.466181\pi$$
$$158$$ 0 0
$$159$$ 4.18484e8i 0.654773i
$$160$$ 0 0
$$161$$ 1.48582e8 0.221138
$$162$$ 0 0
$$163$$ − 2.84647e7i − 0.0403233i −0.999797 0.0201617i $$-0.993582\pi$$
0.999797 0.0201617i $$-0.00641809\pi$$
$$164$$ 0 0
$$165$$ −7.21832e8 −0.973869
$$166$$ 0 0
$$167$$ 178227.i 0 0.000229144i 1.00000 0.000114572i $$3.64693e-5\pi$$
−1.00000 0.000114572i $$0.999964\pi$$
$$168$$ 0 0
$$169$$ 3.54833e8 0.434987
$$170$$ 0 0
$$171$$ 4.28911e8i 0.501629i
$$172$$ 0 0
$$173$$ 1.60561e9 1.79248 0.896240 0.443569i $$-0.146288\pi$$
0.896240 + 0.443569i $$0.146288\pi$$
$$174$$ 0 0
$$175$$ − 3.35414e8i − 0.357626i
$$176$$ 0 0
$$177$$ 3.97607e8 0.405098
$$178$$ 0 0
$$179$$ 1.44579e9i 1.40829i 0.710054 + 0.704147i $$0.248671\pi$$
−0.710054 + 0.704147i $$0.751329\pi$$
$$180$$ 0 0
$$181$$ 1.78411e9 1.66229 0.831146 0.556054i $$-0.187685\pi$$
0.831146 + 0.556054i $$0.187685\pi$$
$$182$$ 0 0
$$183$$ 8.34990e8i 0.744521i
$$184$$ 0 0
$$185$$ 8.57907e8 0.732409
$$186$$ 0 0
$$187$$ 2.85756e8i 0.233684i
$$188$$ 0 0
$$189$$ 4.36697e7 0.0342242
$$190$$ 0 0
$$191$$ − 2.30763e9i − 1.73394i −0.498364 0.866968i $$-0.666066\pi$$
0.498364 0.866968i $$-0.333934\pi$$
$$192$$ 0 0
$$193$$ −5.47360e8 −0.394497 −0.197248 0.980354i $$-0.563201\pi$$
−0.197248 + 0.980354i $$0.563201\pi$$
$$194$$ 0 0
$$195$$ 1.73523e9i 1.20011i
$$196$$ 0 0
$$197$$ −3.43899e8 −0.228332 −0.114166 0.993462i $$-0.536420\pi$$
−0.114166 + 0.993462i $$0.536420\pi$$
$$198$$ 0 0
$$199$$ 1.12814e9i 0.719369i 0.933074 + 0.359684i $$0.117116\pi$$
−0.933074 + 0.359684i $$0.882884\pi$$
$$200$$ 0 0
$$201$$ −1.63416e9 −1.00118
$$202$$ 0 0
$$203$$ − 4.28033e8i − 0.252054i
$$204$$ 0 0
$$205$$ −1.47553e9 −0.835474
$$206$$ 0 0
$$207$$ − 7.61043e8i − 0.414503i
$$208$$ 0 0
$$209$$ 2.79122e9 1.46288
$$210$$ 0 0
$$211$$ 3.25742e9i 1.64340i 0.569919 + 0.821701i $$0.306975\pi$$
−0.569919 + 0.821701i $$0.693025\pi$$
$$212$$ 0 0
$$213$$ 1.79953e9 0.874259
$$214$$ 0 0
$$215$$ − 1.62804e9i − 0.761922i
$$216$$ 0 0
$$217$$ 7.00184e8 0.315772
$$218$$ 0 0
$$219$$ − 9.87709e8i − 0.429391i
$$220$$ 0 0
$$221$$ 6.86938e8 0.287971
$$222$$ 0 0
$$223$$ 7.96939e8i 0.322259i 0.986933 + 0.161130i $$0.0515137\pi$$
−0.986933 + 0.161130i $$0.948486\pi$$
$$224$$ 0 0
$$225$$ −1.71800e9 −0.670337
$$226$$ 0 0
$$227$$ − 2.32327e9i − 0.874977i −0.899224 0.437488i $$-0.855868\pi$$
0.899224 0.437488i $$-0.144132\pi$$
$$228$$ 0 0
$$229$$ −2.17161e9 −0.789659 −0.394830 0.918754i $$-0.629196\pi$$
−0.394830 + 0.918754i $$0.629196\pi$$
$$230$$ 0 0
$$231$$ − 2.84189e8i − 0.0998065i
$$232$$ 0 0
$$233$$ −1.99024e9 −0.675275 −0.337638 0.941276i $$-0.609628\pi$$
−0.337638 + 0.941276i $$0.609628\pi$$
$$234$$ 0 0
$$235$$ − 1.61873e9i − 0.530767i
$$236$$ 0 0
$$237$$ 1.71769e9 0.544443
$$238$$ 0 0
$$239$$ − 1.47132e9i − 0.450937i −0.974250 0.225469i $$-0.927609\pi$$
0.974250 0.225469i $$-0.0723914\pi$$
$$240$$ 0 0
$$241$$ −2.94516e9 −0.873052 −0.436526 0.899692i $$-0.643791\pi$$
−0.436526 + 0.899692i $$0.643791\pi$$
$$242$$ 0 0
$$243$$ − 2.23677e8i − 0.0641500i
$$244$$ 0 0
$$245$$ −6.05430e9 −1.68035
$$246$$ 0 0
$$247$$ − 6.70989e9i − 1.80272i
$$248$$ 0 0
$$249$$ −1.37604e9 −0.357959
$$250$$ 0 0
$$251$$ 2.41458e9i 0.608341i 0.952618 + 0.304170i $$0.0983792\pi$$
−0.952618 + 0.304170i $$0.901621\pi$$
$$252$$ 0 0
$$253$$ −4.95263e9 −1.20880
$$254$$ 0 0
$$255$$ 1.01831e9i 0.240835i
$$256$$ 0 0
$$257$$ 5.48578e8 0.125749 0.0628747 0.998021i $$-0.479973\pi$$
0.0628747 + 0.998021i $$0.479973\pi$$
$$258$$ 0 0
$$259$$ 3.37762e8i 0.0750606i
$$260$$ 0 0
$$261$$ −2.19240e9 −0.472451
$$262$$ 0 0
$$263$$ 2.76986e9i 0.578942i 0.957187 + 0.289471i $$0.0934794\pi$$
−0.957187 + 0.289471i $$0.906521\pi$$
$$264$$ 0 0
$$265$$ 9.70490e9 1.96792
$$266$$ 0 0
$$267$$ 1.73051e9i 0.340509i
$$268$$ 0 0
$$269$$ −2.47401e9 −0.472490 −0.236245 0.971694i $$-0.575917\pi$$
−0.236245 + 0.971694i $$0.575917\pi$$
$$270$$ 0 0
$$271$$ − 2.51223e9i − 0.465781i −0.972503 0.232890i $$-0.925182\pi$$
0.972503 0.232890i $$-0.0748183\pi$$
$$272$$ 0 0
$$273$$ −6.83170e8 −0.122992
$$274$$ 0 0
$$275$$ 1.11802e10i 1.95488i
$$276$$ 0 0
$$277$$ 1.06906e10 1.81585 0.907927 0.419127i $$-0.137664\pi$$
0.907927 + 0.419127i $$0.137664\pi$$
$$278$$ 0 0
$$279$$ − 3.58636e9i − 0.591885i
$$280$$ 0 0
$$281$$ −2.60700e9 −0.418134 −0.209067 0.977901i $$-0.567043\pi$$
−0.209067 + 0.977901i $$0.567043\pi$$
$$282$$ 0 0
$$283$$ 1.75548e8i 0.0273684i 0.999906 + 0.0136842i $$0.00435595\pi$$
−0.999906 + 0.0136842i $$0.995644\pi$$
$$284$$ 0 0
$$285$$ 9.94669e9 1.50765
$$286$$ 0 0
$$287$$ − 5.80923e8i − 0.0856232i
$$288$$ 0 0
$$289$$ −6.57263e9 −0.942211
$$290$$ 0 0
$$291$$ − 5.91221e9i − 0.824476i
$$292$$ 0 0
$$293$$ −1.83363e9 −0.248795 −0.124397 0.992232i $$-0.539700\pi$$
−0.124397 + 0.992232i $$0.539700\pi$$
$$294$$ 0 0
$$295$$ − 9.22073e9i − 1.21752i
$$296$$ 0 0
$$297$$ −1.45562e9 −0.187078
$$298$$ 0 0
$$299$$ 1.19058e10i 1.48961i
$$300$$ 0 0
$$301$$ 6.40966e8 0.0780852
$$302$$ 0 0
$$303$$ − 3.23205e9i − 0.383450i
$$304$$ 0 0
$$305$$ 1.93639e10 2.23766
$$306$$ 0 0
$$307$$ 1.39249e10i 1.56761i 0.621007 + 0.783805i $$0.286724\pi$$
−0.621007 + 0.783805i $$0.713276\pi$$
$$308$$ 0 0
$$309$$ 5.62113e8 0.0616581
$$310$$ 0 0
$$311$$ − 3.88787e9i − 0.415595i −0.978172 0.207798i $$-0.933370\pi$$
0.978172 0.207798i $$-0.0666295\pi$$
$$312$$ 0 0
$$313$$ −3.59882e9 −0.374958 −0.187479 0.982269i $$-0.560032\pi$$
−0.187479 + 0.982269i $$0.560032\pi$$
$$314$$ 0 0
$$315$$ − 1.01273e9i − 0.102861i
$$316$$ 0 0
$$317$$ −8.64268e9 −0.855878 −0.427939 0.903808i $$-0.640760\pi$$
−0.427939 + 0.903808i $$0.640760\pi$$
$$318$$ 0 0
$$319$$ 1.42674e10i 1.37779i
$$320$$ 0 0
$$321$$ 1.16418e10 1.09648
$$322$$ 0 0
$$323$$ − 3.93766e9i − 0.361766i
$$324$$ 0 0
$$325$$ 2.68765e10 2.40901
$$326$$ 0 0
$$327$$ − 7.56078e9i − 0.661265i
$$328$$ 0 0
$$329$$ 6.37303e8 0.0543954
$$330$$ 0 0
$$331$$ − 9.59670e9i − 0.799484i −0.916628 0.399742i $$-0.869100\pi$$
0.916628 0.399742i $$-0.130900\pi$$
$$332$$ 0 0
$$333$$ 1.73003e9 0.140694
$$334$$ 0 0
$$335$$ 3.78971e10i 3.00904i
$$336$$ 0 0
$$337$$ −1.75447e10 −1.36027 −0.680137 0.733085i $$-0.738080\pi$$
−0.680137 + 0.733085i $$0.738080\pi$$
$$338$$ 0 0
$$339$$ 3.29957e9i 0.249838i
$$340$$ 0 0
$$341$$ −2.33390e10 −1.72609
$$342$$ 0 0
$$343$$ − 4.84506e9i − 0.350044i
$$344$$ 0 0
$$345$$ −1.76490e10 −1.24579
$$346$$ 0 0
$$347$$ − 1.71291e10i − 1.18146i −0.806871 0.590728i $$-0.798841\pi$$
0.806871 0.590728i $$-0.201159\pi$$
$$348$$ 0 0
$$349$$ 1.99032e9 0.134160 0.0670799 0.997748i $$-0.478632\pi$$
0.0670799 + 0.997748i $$0.478632\pi$$
$$350$$ 0 0
$$351$$ 3.49922e9i 0.230538i
$$352$$ 0 0
$$353$$ 2.53774e8 0.0163436 0.00817181 0.999967i $$-0.497399\pi$$
0.00817181 + 0.999967i $$0.497399\pi$$
$$354$$ 0 0
$$355$$ − 4.17321e10i − 2.62759i
$$356$$ 0 0
$$357$$ −4.00914e8 −0.0246819
$$358$$ 0 0
$$359$$ − 2.26677e10i − 1.36468i −0.731036 0.682339i $$-0.760963\pi$$
0.731036 0.682339i $$-0.239037\pi$$
$$360$$ 0 0
$$361$$ −2.14788e10 −1.26468
$$362$$ 0 0
$$363$$ − 5.51830e8i − 0.0317818i
$$364$$ 0 0
$$365$$ −2.29056e10 −1.29053
$$366$$ 0 0
$$367$$ 7.30278e9i 0.402554i 0.979534 + 0.201277i $$0.0645091\pi$$
−0.979534 + 0.201277i $$0.935491\pi$$
$$368$$ 0 0
$$369$$ −2.97551e9 −0.160493
$$370$$ 0 0
$$371$$ 3.82087e9i 0.201682i
$$372$$ 0 0
$$373$$ 1.11279e10 0.574879 0.287439 0.957799i $$-0.407196\pi$$
0.287439 + 0.957799i $$0.407196\pi$$
$$374$$ 0 0
$$375$$ 2.00298e10i 1.01286i
$$376$$ 0 0
$$377$$ 3.42979e10 1.69786
$$378$$ 0 0
$$379$$ − 5.68377e9i − 0.275473i −0.990469 0.137737i $$-0.956017\pi$$
0.990469 0.137737i $$-0.0439827\pi$$
$$380$$ 0 0
$$381$$ 4.29017e9 0.203598
$$382$$ 0 0
$$383$$ − 1.92155e9i − 0.0893009i −0.999003 0.0446504i $$-0.985783\pi$$
0.999003 0.0446504i $$-0.0142174\pi$$
$$384$$ 0 0
$$385$$ −6.59050e9 −0.299969
$$386$$ 0 0
$$387$$ − 3.28304e9i − 0.146364i
$$388$$ 0 0
$$389$$ −1.05878e10 −0.462389 −0.231194 0.972908i $$-0.574263\pi$$
−0.231194 + 0.972908i $$0.574263\pi$$
$$390$$ 0 0
$$391$$ 6.98683e9i 0.298932i
$$392$$ 0 0
$$393$$ −1.21053e10 −0.507464
$$394$$ 0 0
$$395$$ − 3.98343e10i − 1.63632i
$$396$$ 0 0
$$397$$ −2.38291e10 −0.959282 −0.479641 0.877465i $$-0.659233\pi$$
−0.479641 + 0.877465i $$0.659233\pi$$
$$398$$ 0 0
$$399$$ 3.91606e9i 0.154510i
$$400$$ 0 0
$$401$$ 1.59390e10 0.616428 0.308214 0.951317i $$-0.400269\pi$$
0.308214 + 0.951317i $$0.400269\pi$$
$$402$$ 0 0
$$403$$ 5.61052e10i 2.12708i
$$404$$ 0 0
$$405$$ −5.18721e9 −0.192803
$$406$$ 0 0
$$407$$ − 1.12585e10i − 0.410301i
$$408$$ 0 0
$$409$$ 8.96615e9 0.320415 0.160208 0.987083i $$-0.448784\pi$$
0.160208 + 0.987083i $$0.448784\pi$$
$$410$$ 0 0
$$411$$ − 1.74767e10i − 0.612480i
$$412$$ 0 0
$$413$$ 3.63025e9 0.124777
$$414$$ 0 0
$$415$$ 3.19112e10i 1.07585i
$$416$$ 0 0
$$417$$ −7.53415e8 −0.0249167
$$418$$ 0 0
$$419$$ − 1.06743e10i − 0.346326i −0.984893 0.173163i $$-0.944601\pi$$
0.984893 0.173163i $$-0.0553987\pi$$
$$420$$ 0 0
$$421$$ −2.60816e10 −0.830245 −0.415123 0.909765i $$-0.636261\pi$$
−0.415123 + 0.909765i $$0.636261\pi$$
$$422$$ 0 0
$$423$$ − 3.26428e9i − 0.101959i
$$424$$ 0 0
$$425$$ 1.57723e10 0.483436
$$426$$ 0 0
$$427$$ 7.62366e9i 0.229325i
$$428$$ 0 0
$$429$$ 2.27718e10 0.672308
$$430$$ 0 0
$$431$$ − 1.20312e9i − 0.0348658i −0.999848 0.0174329i $$-0.994451\pi$$
0.999848 0.0174329i $$-0.00554935\pi$$
$$432$$ 0 0
$$433$$ 1.13142e10 0.321865 0.160932 0.986965i $$-0.448550\pi$$
0.160932 + 0.986965i $$0.448550\pi$$
$$434$$ 0 0
$$435$$ 5.08430e10i 1.41995i
$$436$$ 0 0
$$437$$ 6.82462e10 1.87134
$$438$$ 0 0
$$439$$ 1.76972e10i 0.476482i 0.971206 + 0.238241i $$0.0765708\pi$$
−0.971206 + 0.238241i $$0.923429\pi$$
$$440$$ 0 0
$$441$$ −1.22089e10 −0.322792
$$442$$ 0 0
$$443$$ − 6.47192e10i − 1.68042i −0.542260 0.840211i $$-0.682431\pi$$
0.542260 0.840211i $$-0.317569\pi$$
$$444$$ 0 0
$$445$$ 4.01315e10 1.02340
$$446$$ 0 0
$$447$$ 2.70568e10i 0.677713i
$$448$$ 0 0
$$449$$ 6.07525e10 1.49478 0.747392 0.664383i $$-0.231306\pi$$
0.747392 + 0.664383i $$0.231306\pi$$
$$450$$ 0 0
$$451$$ 1.93637e10i 0.468038i
$$452$$ 0 0
$$453$$ 2.81279e10 0.667952
$$454$$ 0 0
$$455$$ 1.58431e10i 0.369653i
$$456$$ 0 0
$$457$$ 4.77740e10 1.09528 0.547642 0.836713i $$-0.315526\pi$$
0.547642 + 0.836713i $$0.315526\pi$$
$$458$$ 0 0
$$459$$ 2.05349e9i 0.0462639i
$$460$$ 0 0
$$461$$ −7.17546e9 −0.158872 −0.0794358 0.996840i $$-0.525312\pi$$
−0.0794358 + 0.996840i $$0.525312\pi$$
$$462$$ 0 0
$$463$$ − 5.86354e9i − 0.127596i −0.997963 0.0637978i $$-0.979679\pi$$
0.997963 0.0637978i $$-0.0203213\pi$$
$$464$$ 0 0
$$465$$ −8.31699e10 −1.77891
$$466$$ 0 0
$$467$$ 7.53384e10i 1.58398i 0.610537 + 0.791988i $$0.290954\pi$$
−0.610537 + 0.791988i $$0.709046\pi$$
$$468$$ 0 0
$$469$$ −1.49203e10 −0.308380
$$470$$ 0 0
$$471$$ − 6.02620e9i − 0.122450i
$$472$$ 0 0
$$473$$ −2.13650e10 −0.426834
$$474$$ 0 0
$$475$$ − 1.54061e11i − 3.02634i
$$476$$ 0 0
$$477$$ 1.95706e10 0.378033
$$478$$ 0 0
$$479$$ − 1.97647e10i − 0.375446i −0.982222 0.187723i $$-0.939889\pi$$
0.982222 0.187723i $$-0.0601107\pi$$
$$480$$ 0 0
$$481$$ −2.70646e10 −0.505617
$$482$$ 0 0
$$483$$ − 6.94850e9i − 0.127674i
$$484$$ 0 0
$$485$$ −1.37108e11 −2.47796
$$486$$ 0 0
$$487$$ 5.47310e10i 0.973011i 0.873677 + 0.486506i $$0.161729\pi$$
−0.873677 + 0.486506i $$0.838271\pi$$
$$488$$ 0 0
$$489$$ −1.33116e9 −0.0232807
$$490$$ 0 0
$$491$$ − 2.44118e10i − 0.420024i −0.977699 0.210012i $$-0.932650\pi$$
0.977699 0.210012i $$-0.0673502\pi$$
$$492$$ 0 0
$$493$$ 2.01275e10 0.340724
$$494$$ 0 0
$$495$$ 3.37568e10i 0.562263i
$$496$$ 0 0
$$497$$ 1.64301e10 0.269287
$$498$$ 0 0
$$499$$ 3.29856e10i 0.532014i 0.963971 + 0.266007i $$0.0857044\pi$$
−0.963971 + 0.266007i $$0.914296\pi$$
$$500$$ 0 0
$$501$$ 8.33485e6 0.000132296 0
$$502$$ 0 0
$$503$$ − 7.46160e9i − 0.116563i −0.998300 0.0582814i $$-0.981438\pi$$
0.998300 0.0582814i $$-0.0185621\pi$$
$$504$$ 0 0
$$505$$ −7.49533e10 −1.15246
$$506$$ 0 0
$$507$$ − 1.65939e10i − 0.251140i
$$508$$ 0 0
$$509$$ 2.46374e10 0.367049 0.183524 0.983015i $$-0.441249\pi$$
0.183524 + 0.983015i $$0.441249\pi$$
$$510$$ 0 0
$$511$$ − 9.01802e9i − 0.132260i
$$512$$ 0 0
$$513$$ 2.00582e10 0.289616
$$514$$ 0 0
$$515$$ − 1.30357e10i − 0.185313i
$$516$$ 0 0
$$517$$ −2.12429e10 −0.297339
$$518$$ 0 0
$$519$$ − 7.50867e10i − 1.03489i
$$520$$ 0 0
$$521$$ −2.99951e10 −0.407099 −0.203549 0.979065i $$-0.565248\pi$$
−0.203549 + 0.979065i $$0.565248\pi$$
$$522$$ 0 0
$$523$$ 1.04737e11i 1.39988i 0.714199 + 0.699942i $$0.246791\pi$$
−0.714199 + 0.699942i $$0.753209\pi$$
$$524$$ 0 0
$$525$$ −1.56858e10 −0.206475
$$526$$ 0 0
$$527$$ 3.29250e10i 0.426858i
$$528$$ 0 0
$$529$$ −4.27824e10 −0.546314
$$530$$ 0 0
$$531$$ − 1.85942e10i − 0.233884i
$$532$$ 0 0
$$533$$ 4.65489e10 0.576767
$$534$$ 0 0
$$535$$ − 2.69981e11i − 3.29547i
$$536$$ 0 0
$$537$$ 6.76130e10 0.813079
$$538$$ 0 0
$$539$$ 7.94518e10i 0.941344i
$$540$$ 0 0
$$541$$ −5.69348e10 −0.664644 −0.332322 0.943166i $$-0.607832\pi$$
−0.332322 + 0.943166i $$0.607832\pi$$
$$542$$ 0 0
$$543$$ − 8.34346e10i − 0.959725i
$$544$$ 0 0
$$545$$ −1.75339e11 −1.98743
$$546$$ 0 0
$$547$$ 1.13814e11i 1.27130i 0.771979 + 0.635648i $$0.219267\pi$$
−0.771979 + 0.635648i $$0.780733\pi$$
$$548$$ 0 0
$$549$$ 3.90486e10 0.429849
$$550$$ 0 0
$$551$$ − 1.96602e11i − 2.13296i
$$552$$ 0 0
$$553$$ 1.56830e10 0.167698
$$554$$ 0 0
$$555$$ − 4.01204e10i − 0.422856i
$$556$$ 0 0
$$557$$ −1.32489e11 −1.37645 −0.688225 0.725497i $$-0.741610\pi$$
−0.688225 + 0.725497i $$0.741610\pi$$
$$558$$ 0 0
$$559$$ 5.13600e10i 0.525991i
$$560$$ 0 0
$$561$$ 1.33635e10 0.134918
$$562$$ 0 0
$$563$$ 1.68664e11i 1.67876i 0.543545 + 0.839380i $$0.317082\pi$$
−0.543545 + 0.839380i $$0.682918\pi$$
$$564$$ 0 0
$$565$$ 7.65191e10 0.750889
$$566$$ 0 0
$$567$$ − 2.04223e9i − 0.0197593i
$$568$$ 0 0
$$569$$ 1.36419e11 1.30145 0.650725 0.759314i $$-0.274465\pi$$
0.650725 + 0.759314i $$0.274465\pi$$
$$570$$ 0 0
$$571$$ − 5.62677e10i − 0.529315i −0.964342 0.264658i $$-0.914741\pi$$
0.964342 0.264658i $$-0.0852590\pi$$
$$572$$ 0 0
$$573$$ −1.07917e11 −1.00109
$$574$$ 0 0
$$575$$ 2.73360e11i 2.50071i
$$576$$ 0 0
$$577$$ 1.35733e11 1.22457 0.612285 0.790637i $$-0.290250\pi$$
0.612285 + 0.790637i $$0.290250\pi$$
$$578$$ 0 0
$$579$$ 2.55975e10i 0.227763i
$$580$$ 0 0
$$581$$ −1.25636e10 −0.110258
$$582$$ 0 0
$$583$$ − 1.27359e11i − 1.10244i
$$584$$ 0 0
$$585$$ 8.11489e10 0.692882
$$586$$ 0 0
$$587$$ 1.54607e11i 1.30220i 0.758993 + 0.651098i $$0.225691\pi$$
−0.758993 + 0.651098i $$0.774309\pi$$
$$588$$ 0 0
$$589$$ 3.21606e11 2.67216
$$590$$ 0 0
$$591$$ 1.60826e10i 0.131827i
$$592$$ 0 0
$$593$$ 1.44789e11 1.17089 0.585447 0.810711i $$-0.300919\pi$$
0.585447 + 0.810711i $$0.300919\pi$$
$$594$$ 0 0
$$595$$ 9.29743e9i 0.0741814i
$$596$$ 0 0
$$597$$ 5.27580e10 0.415328
$$598$$ 0 0
$$599$$ 2.20988e10i 0.171657i 0.996310 + 0.0858284i $$0.0273537\pi$$
−0.996310 + 0.0858284i $$0.972646\pi$$
$$600$$ 0 0
$$601$$ −1.29762e11 −0.994602 −0.497301 0.867578i $$-0.665676\pi$$
−0.497301 + 0.867578i $$0.665676\pi$$
$$602$$ 0 0
$$603$$ 7.64221e10i 0.578029i
$$604$$ 0 0
$$605$$ −1.27973e10 −0.0955204
$$606$$ 0 0
$$607$$ 1.15676e10i 0.0852099i 0.999092 + 0.0426050i $$0.0135657\pi$$
−0.999092 + 0.0426050i $$0.986434\pi$$
$$608$$ 0 0
$$609$$ −2.00171e10 −0.145523
$$610$$ 0 0
$$611$$ 5.10665e10i 0.366414i
$$612$$ 0 0
$$613$$ −1.57261e11 −1.11373 −0.556865 0.830603i $$-0.687996\pi$$
−0.556865 + 0.830603i $$0.687996\pi$$
$$614$$ 0 0
$$615$$ 6.90037e10i 0.482361i
$$616$$ 0 0
$$617$$ −3.94877e10 −0.272471 −0.136236 0.990676i $$-0.543500\pi$$
−0.136236 + 0.990676i $$0.543500\pi$$
$$618$$ 0 0
$$619$$ − 1.02727e11i − 0.699718i −0.936802 0.349859i $$-0.886230\pi$$
0.936802 0.349859i $$-0.113770\pi$$
$$620$$ 0 0
$$621$$ −3.55904e10 −0.239313
$$622$$ 0 0
$$623$$ 1.57999e10i 0.104883i
$$624$$ 0 0
$$625$$ 1.57647e11 1.03315
$$626$$ 0 0
$$627$$ − 1.30532e11i − 0.844594i
$$628$$ 0 0
$$629$$ −1.58827e10 −0.101466
$$630$$ 0 0
$$631$$ 2.04152e10i 0.128777i 0.997925 + 0.0643883i $$0.0205096\pi$$
−0.997925 + 0.0643883i $$0.979490\pi$$
$$632$$ 0 0
$$633$$ 1.52334e11 0.948819
$$634$$ 0 0
$$635$$ − 9.94915e10i − 0.611915i
$$636$$ 0 0
$$637$$ 1.90996e11 1.16003
$$638$$ 0 0
$$639$$ − 8.41556e10i − 0.504754i
$$640$$ 0 0
$$641$$ −2.17056e11 −1.28570 −0.642851 0.765992i $$-0.722248\pi$$
−0.642851 + 0.765992i $$0.722248\pi$$
$$642$$ 0 0
$$643$$ − 1.02152e11i − 0.597590i −0.954317 0.298795i $$-0.903415\pi$$
0.954317 0.298795i $$-0.0965848\pi$$
$$644$$ 0 0
$$645$$ −7.61357e10 −0.439896
$$646$$ 0 0
$$647$$ − 9.09290e10i − 0.518902i −0.965756 0.259451i $$-0.916458\pi$$
0.965756 0.259451i $$-0.0835416\pi$$
$$648$$ 0 0
$$649$$ −1.21005e11 −0.682065
$$650$$ 0 0
$$651$$ − 3.27444e10i − 0.182311i
$$652$$ 0 0
$$653$$ 1.71409e11 0.942717 0.471359 0.881942i $$-0.343764\pi$$
0.471359 + 0.881942i $$0.343764\pi$$
$$654$$ 0 0
$$655$$ 2.80729e11i 1.52518i
$$656$$ 0 0
$$657$$ −4.61906e10 −0.247909
$$658$$ 0 0
$$659$$ 2.27365e9i 0.0120554i 0.999982 + 0.00602769i $$0.00191869\pi$$
−0.999982 + 0.00602769i $$0.998081\pi$$
$$660$$ 0 0
$$661$$ −2.92200e11 −1.53065 −0.765323 0.643647i $$-0.777421\pi$$
−0.765323 + 0.643647i $$0.777421\pi$$
$$662$$ 0 0
$$663$$ − 3.21249e10i − 0.166260i
$$664$$ 0 0
$$665$$ 9.08157e10 0.464381
$$666$$ 0 0
$$667$$ 3.48843e11i 1.76249i
$$668$$ 0 0
$$669$$ 3.72691e10 0.186056
$$670$$ 0 0
$$671$$ − 2.54116e11i − 1.25355i
$$672$$ 0 0
$$673$$ −1.51294e10 −0.0737501 −0.0368751 0.999320i $$-0.511740\pi$$
−0.0368751 + 0.999320i $$0.511740\pi$$
$$674$$ 0 0
$$675$$ 8.03429e10i 0.387019i
$$676$$ 0 0
$$677$$ 3.80775e11 1.81265 0.906324 0.422584i $$-0.138877\pi$$
0.906324 + 0.422584i $$0.138877\pi$$
$$678$$ 0 0
$$679$$ − 5.39799e10i − 0.253953i
$$680$$ 0 0
$$681$$ −1.08649e11 −0.505168
$$682$$ 0 0
$$683$$ − 4.18285e11i − 1.92216i −0.276269 0.961080i $$-0.589098\pi$$
0.276269 0.961080i $$-0.410902\pi$$
$$684$$ 0 0
$$685$$ −4.05295e11 −1.84081
$$686$$ 0 0
$$687$$ 1.01556e11i 0.455910i
$$688$$ 0 0
$$689$$ −3.06163e11 −1.35855
$$690$$ 0 0
$$691$$ − 8.10077e10i − 0.355315i −0.984092 0.177658i $$-0.943148\pi$$
0.984092 0.177658i $$-0.0568520\pi$$
$$692$$ 0 0
$$693$$ −1.32902e10 −0.0576233
$$694$$ 0 0
$$695$$ 1.74721e10i 0.0748870i
$$696$$ 0 0
$$697$$ 2.73169e10 0.115745
$$698$$ 0 0
$$699$$ 9.30741e10i 0.389870i
$$700$$ 0 0
$$701$$ −3.22597e11 −1.33594 −0.667971 0.744187i $$-0.732837\pi$$
−0.667971 + 0.744187i $$0.732837\pi$$
$$702$$ 0 0
$$703$$ 1.55139e11i 0.635186i
$$704$$ 0 0
$$705$$ −7.57006e10 −0.306438
$$706$$ 0 0
$$707$$ − 2.95094e10i − 0.118109i
$$708$$ 0 0
$$709$$ −2.99903e11 −1.18685 −0.593424 0.804890i $$-0.702224\pi$$
−0.593424 + 0.804890i $$0.702224\pi$$
$$710$$ 0 0
$$711$$ − 8.03286e10i − 0.314334i
$$712$$ 0 0
$$713$$ −5.70645e11 −2.20804
$$714$$ 0 0
$$715$$ − 5.28092e11i − 2.02062i
$$716$$ 0 0
$$717$$ −6.88070e10 −0.260349
$$718$$ 0 0
$$719$$ − 1.62338e11i − 0.607442i −0.952761 0.303721i $$-0.901771\pi$$
0.952761 0.303721i $$-0.0982290\pi$$
$$720$$ 0 0
$$721$$ 5.13223e9 0.0189918
$$722$$ 0 0
$$723$$ 1.37731e11i 0.504057i
$$724$$ 0 0
$$725$$ 7.87489e11 2.85031
$$726$$ 0 0
$$727$$ − 2.20900e11i − 0.790786i −0.918512 0.395393i $$-0.870608\pi$$
0.918512 0.395393i $$-0.129392\pi$$
$$728$$ 0 0
$$729$$ −1.04604e10 −0.0370370
$$730$$ 0 0
$$731$$ 3.01403e10i 0.105555i
$$732$$ 0 0
$$733$$ 1.15228e11 0.399154 0.199577 0.979882i $$-0.436043\pi$$
0.199577 + 0.979882i $$0.436043\pi$$
$$734$$ 0 0
$$735$$ 2.83132e11i 0.970151i
$$736$$ 0 0
$$737$$ 4.97332e11 1.68568
$$738$$ 0 0
$$739$$ 5.64364e11i 1.89226i 0.323781 + 0.946132i $$0.395046\pi$$
−0.323781 + 0.946132i $$0.604954\pi$$
$$740$$ 0 0
$$741$$ −3.13791e11 −1.04080
$$742$$ 0 0
$$743$$ 1.89167e11i 0.620713i 0.950620 + 0.310357i $$0.100448\pi$$
−0.950620 + 0.310357i $$0.899552\pi$$
$$744$$ 0 0
$$745$$ 6.27462e11 2.03687
$$746$$ 0 0
$$747$$ 6.43510e10i 0.206668i
$$748$$ 0 0
$$749$$ 1.06293e11 0.337735
$$750$$ 0 0
$$751$$ − 3.66355e11i − 1.15171i −0.817553 0.575853i $$-0.804670\pi$$
0.817553 0.575853i $$-0.195330\pi$$
$$752$$ 0 0
$$753$$ 1.12919e11 0.351226
$$754$$ 0 0
$$755$$ − 6.52304e11i − 2.00753i
$$756$$ 0 0
$$757$$ 1.30666e11 0.397904 0.198952 0.980009i $$-0.436246\pi$$
0.198952 + 0.980009i $$0.436246\pi$$
$$758$$ 0 0
$$759$$ 2.31612e11i 0.697900i
$$760$$ 0 0
$$761$$ −3.92845e11 −1.17134 −0.585670 0.810550i $$-0.699168\pi$$
−0.585670 + 0.810550i $$0.699168\pi$$
$$762$$ 0 0
$$763$$ − 6.90318e10i − 0.203681i
$$764$$ 0 0
$$765$$ 4.76217e10 0.139046
$$766$$ 0 0
$$767$$ 2.90889e11i 0.840515i
$$768$$ 0 0
$$769$$ 5.36177e11 1.53322 0.766608 0.642116i $$-0.221943\pi$$
0.766608 + 0.642116i $$0.221943\pi$$
$$770$$ 0 0
$$771$$ − 2.56545e10i − 0.0726015i
$$772$$ 0 0
$$773$$ 3.88295e11 1.08754 0.543769 0.839235i $$-0.316997\pi$$
0.543769 + 0.839235i $$0.316997\pi$$
$$774$$ 0 0
$$775$$ 1.28819e12i 3.57086i
$$776$$ 0 0
$$777$$ 1.57956e10 0.0433362
$$778$$ 0 0
$$779$$ − 2.66827e11i − 0.724570i
$$780$$ 0 0
$$781$$ −5.47659e11 −1.47199
$$782$$ 0 0
$$783$$ 1.02528e11i 0.272770i
$$784$$ 0 0
$$785$$ −1.39751e11 −0.368025
$$786$$ 0 0
$$787$$ − 3.56234e11i − 0.928616i −0.885674 0.464308i $$-0.846303\pi$$
0.885674 0.464308i $$-0.153697\pi$$
$$788$$ 0 0
$$789$$ 1.29534e11 0.334253
$$790$$ 0 0
$$791$$ 3.01259e10i 0.0769546i
$$792$$ 0 0
$$793$$ −6.10877e11 −1.54476
$$794$$ 0 0
$$795$$ − 4.53853e11i − 1.13618i
$$796$$ 0 0
$$797$$ 5.81694e11 1.44165 0.720827 0.693115i $$-0.243762\pi$$
0.720827 + 0.693115i $$0.243762\pi$$
$$798$$ 0 0
$$799$$ 2.99681e10i 0.0735312i
$$800$$ 0 0
$$801$$ 8.09278e10 0.196593
$$802$$ 0 0
$$803$$ 3.00594e11i 0.722966i
$$804$$ 0 0
$$805$$ −1.61140e11 −0.383725
$$806$$ 0 0
$$807$$ 1.15698e11i 0.272792i
$$808$$ 0 0
$$809$$ 5.29788e11 1.23682 0.618412 0.785854i $$-0.287776\pi$$
0.618412 + 0.785854i $$0.287776\pi$$
$$810$$ 0 0
$$811$$ 1.27857e11i 0.295556i 0.989021 + 0.147778i $$0.0472121\pi$$
−0.989021 + 0.147778i $$0.952788\pi$$
$$812$$ 0 0
$$813$$ −1.17485e11 −0.268919
$$814$$ 0 0
$$815$$ 3.08705e10i 0.0699701i
$$816$$ 0 0
$$817$$ 2.94406e11 0.660781
$$818$$ 0 0
$$819$$ 3.19487e10i 0.0710097i
$$820$$ 0 0
$$821$$ −5.23718e11 −1.15272 −0.576361 0.817195i $$-0.695528\pi$$
−0.576361 + 0.817195i $$0.695528\pi$$
$$822$$ 0 0
$$823$$ 4.16147e11i 0.907083i 0.891235 + 0.453541i $$0.149840\pi$$
−0.891235 + 0.453541i $$0.850160\pi$$
$$824$$ 0 0
$$825$$ 5.22847e11 1.12865
$$826$$ 0 0
$$827$$ 1.32442e11i 0.283141i 0.989928 + 0.141570i $$0.0452152\pi$$
−0.989928 + 0.141570i $$0.954785\pi$$
$$828$$ 0 0
$$829$$ 1.50571e11 0.318804 0.159402 0.987214i $$-0.449043\pi$$
0.159402 + 0.987214i $$0.449043\pi$$
$$830$$ 0 0
$$831$$ − 4.99948e11i − 1.04838i
$$832$$ 0 0
$$833$$ 1.12085e11 0.232792
$$834$$ 0 0
$$835$$ − 1.93290e8i 0 0.000397616i
$$836$$ 0 0
$$837$$ −1.67718e11 −0.341725
$$838$$ 0 0
$$839$$ 5.30040e11i 1.06970i 0.844948 + 0.534848i $$0.179631\pi$$
−0.844948 + 0.534848i $$0.820369\pi$$
$$840$$ 0 0
$$841$$ 5.04694e11 1.00889
$$842$$ 0 0
$$843$$ 1.21917e11i 0.241410i
$$844$$ 0 0
$$845$$ −3.84822e11 −0.754802
$$846$$ 0 0
$$847$$ − 5.03835e9i − 0.00978936i
$$848$$ 0 0
$$849$$ 8.20955e9 0.0158011
$$850$$ 0 0
$$851$$ − 2.75273e11i − 0.524863i
$$852$$ 0 0
$$853$$ 7.37919e11 1.39384 0.696919 0.717150i $$-0.254554\pi$$
0.696919 + 0.717150i $$0.254554\pi$$
$$854$$ 0 0
$$855$$ − 4.65161e11i − 0.870440i
$$856$$ 0 0
$$857$$ −2.91969e11 −0.541270 −0.270635 0.962682i $$-0.587234\pi$$
−0.270635 + 0.962682i $$0.587234\pi$$
$$858$$ 0 0
$$859$$ − 3.42628e11i − 0.629289i −0.949210 0.314645i $$-0.898115\pi$$
0.949210 0.314645i $$-0.101885\pi$$
$$860$$ 0 0
$$861$$ −2.71671e10 −0.0494346
$$862$$ 0 0
$$863$$ − 6.00307e11i − 1.08226i −0.840940 0.541128i $$-0.817997\pi$$
0.840940 0.541128i $$-0.182003\pi$$
$$864$$ 0 0
$$865$$ −1.74131e12 −3.11036
$$866$$ 0 0
$$867$$ 3.07372e11i 0.543986i
$$868$$ 0 0
$$869$$ −5.22753e11 −0.916680
$$870$$ 0 0
$$871$$ − 1.19555e12i − 2.07728i
$$872$$ 0 0
$$873$$ −2.76487e11 −0.476011
$$874$$ 0 0
$$875$$ 1.82877e11i 0.311980i
$$876$$ 0 0
$$877$$ 3.40566e11 0.575709 0.287854 0.957674i $$-0.407058\pi$$
0.287854 + 0.957674i $$0.407058\pi$$
$$878$$ 0 0
$$879$$ 8.57504e10i 0.143642i
$$880$$ 0 0
$$881$$ −8.36651e11 −1.38880 −0.694402 0.719587i $$-0.744331\pi$$
−0.694402 + 0.719587i $$0.744331\pi$$
$$882$$ 0 0
$$883$$ − 6.58705e11i − 1.08355i −0.840524 0.541774i $$-0.817753\pi$$
0.840524 0.541774i $$-0.182247\pi$$
$$884$$ 0 0
$$885$$ −4.31211e11 −0.702937
$$886$$ 0 0
$$887$$ 1.84018e11i 0.297281i 0.988891 + 0.148640i $$0.0474897\pi$$
−0.988891 + 0.148640i $$0.952510\pi$$
$$888$$ 0 0
$$889$$ 3.91703e10 0.0627119
$$890$$ 0 0
$$891$$ 6.80727e10i 0.108010i
$$892$$ 0 0
$$893$$ 2.92723e11 0.460311
$$894$$ 0 0
$$895$$ − 1.56799e12i − 2.44371i
$$896$$ 0 0
$$897$$ 5.56778e11 0.860027
$$898$$ 0 0
$$899$$ 1.64390e12i 2.51673i
$$900$$ 0 0
$$901$$ −1.79670e11 −0.272631
$$902$$ 0 0
$$903$$ − 2.99750e10i − 0.0450825i
$$904$$ 0 0
$$905$$ −1.93490e12 −2.88445
$$906$$ 0 0
$$907$$ 7.74263e11i 1.14409i 0.820223 + 0.572044i $$0.193849\pi$$
−0.820223 + 0.572044i $$0.806151\pi$$
$$908$$ 0 0
$$909$$ −1.51148e11 −0.221385
$$910$$ 0 0
$$911$$ 2.57326e11i 0.373602i 0.982398 + 0.186801i $$0.0598120\pi$$
−0.982398 + 0.186801i $$0.940188\pi$$
$$912$$ 0 0
$$913$$ 4.18776e11 0.602697
$$914$$ 0 0
$$915$$ − 9.05560e11i − 1.29191i
$$916$$ 0 0
$$917$$ −1.10524e11 −0.156308
$$918$$ 0 0
$$919$$ 3.23480e11i 0.453508i 0.973952 + 0.226754i $$0.0728114\pi$$
−0.973952 + 0.226754i $$0.927189\pi$$
$$920$$ 0 0
$$921$$ 6.51202e11 0.905060
$$922$$ 0 0
$$923$$ 1.31653e12i 1.81395i
$$924$$ 0 0
$$925$$ −6.21411e11 −0.848812
$$926$$ 0 0
$$927$$ − 2.62874e10i − 0.0355983i
$$928$$ 0 0
$$929$$ 1.13798e12 1.52782 0.763910 0.645322i $$-0.223277\pi$$
0.763910 + 0.645322i $$0.223277\pi$$
$$930$$ 0 0
$$931$$ − 1.09483e12i − 1.45729i
$$932$$ 0 0
$$933$$ −1.81818e11 −0.239944
$$934$$ 0 0
$$935$$ − 3.09907e11i − 0.405495i
$$936$$ 0 0
$$937$$ 1.34507e12 1.74497 0.872483 0.488645i $$-0.162509\pi$$
0.872483 + 0.488645i $$0.162509\pi$$
$$938$$ 0 0
$$939$$ 1.68300e11i 0.216482i
$$940$$ 0 0
$$941$$ −1.54350e10 −0.0196856 −0.00984279 0.999952i $$-0.503133\pi$$
−0.00984279 + 0.999952i $$0.503133\pi$$
$$942$$ 0 0
$$943$$ 4.73448e11i 0.598722i
$$944$$ 0 0
$$945$$ −4.73605e10 −0.0593867
$$946$$ 0 0
$$947$$ − 1.10486e12i − 1.37374i −0.726779 0.686872i $$-0.758983\pi$$
0.726779 0.686872i $$-0.241017\pi$$
$$948$$ 0 0
$$949$$ 7.22607e11 0.890917
$$950$$ 0 0
$$951$$ 4.04178e11i 0.494141i
$$952$$ 0 0
$$953$$ −3.94710e11 −0.478527 −0.239264 0.970955i $$-0.576906\pi$$
−0.239264 + 0.970955i $$0.576906\pi$$
$$954$$ 0 0
$$955$$ 2.50267e12i 3.00877i
$$956$$ 0 0
$$957$$ 6.67222e11 0.795468
$$958$$ 0 0
$$959$$ − 1.59566e11i − 0.188654i
$$960$$ 0 0
$$961$$ −1.83624e12 −2.15295
$$962$$ 0 0
$$963$$ − 5.44434e11i − 0.633053i
$$964$$ 0 0
$$965$$ 5.93621e11 0.684541
$$966$$ 0 0
$$967$$ − 6.43683e11i − 0.736150i −0.929796 0.368075i $$-0.880017\pi$$
0.929796 0.368075i $$-0.119983\pi$$
$$968$$ 0 0
$$969$$ −1.84146e11 −0.208866
$$970$$ 0 0
$$971$$ 9.15239e11i 1.02957i 0.857318 + 0.514787i $$0.172129\pi$$
−0.857318 + 0.514787i $$0.827871\pi$$
$$972$$ 0 0
$$973$$ −6.87886e9 −0.00767477
$$974$$ 0 0
$$975$$ − 1.25689e12i − 1.39084i
$$976$$ 0 0
$$977$$ 1.39678e12 1.53303 0.766515 0.642226i $$-0.221989\pi$$
0.766515 + 0.642226i $$0.221989\pi$$
$$978$$ 0 0
$$979$$ − 5.26653e11i − 0.573316i
$$980$$ 0 0
$$981$$ −3.53583e11 −0.381782
$$982$$ 0 0
$$983$$ − 4.88577e11i − 0.523261i −0.965168 0.261631i $$-0.915740\pi$$
0.965168 0.261631i $$-0.0842602\pi$$
$$984$$ 0 0
$$985$$ 3.72964e11 0.396207
$$986$$ 0 0
$$987$$ − 2.98037e10i − 0.0314052i
$$988$$ 0 0
$$989$$ −5.22382e11 −0.546013
$$990$$ 0 0
$$991$$ 7.71826e11i 0.800248i 0.916461 + 0.400124i $$0.131033\pi$$
−0.916461 + 0.400124i $$0.868967\pi$$
$$992$$ 0 0
$$993$$ −4.48793e11 −0.461582
$$994$$ 0 0
$$995$$ − 1.22349e12i − 1.24827i
$$996$$ 0 0
$$997$$ 1.24478e12 1.25983 0.629916 0.776663i $$-0.283089\pi$$
0.629916 + 0.776663i $$0.283089\pi$$
$$998$$ 0 0
$$999$$ − 8.09054e10i − 0.0812298i
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 48.9.g.c.31.1 4
3.2 odd 2 144.9.g.i.127.3 4
4.3 odd 2 inner 48.9.g.c.31.3 yes 4
8.3 odd 2 192.9.g.c.127.2 4
8.5 even 2 192.9.g.c.127.4 4
12.11 even 2 144.9.g.i.127.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.c.31.1 4 1.1 even 1 trivial
48.9.g.c.31.3 yes 4 4.3 odd 2 inner
144.9.g.i.127.3 4 3.2 odd 2
144.9.g.i.127.4 4 12.11 even 2
192.9.g.c.127.2 4 8.3 odd 2
192.9.g.c.127.4 4 8.5 even 2