# Properties

 Label 48.28.a.l.1.1 Level $48$ Weight $28$ Character 48.1 Self dual yes Analytic conductor $221.691$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,28,Mod(1,48)]

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([0, 0, 0]))

N = Newforms(chi, 28, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 48.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$221.690675922$$ 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: $$x^{4} - 2x^{3} - 360714331909x^{2} - 43287560841177118x + 8819337660421091919513$$ x^4 - 2*x^3 - 360714331909*x^2 - 43287560841177118*x + 8819337660421091919513 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{42}\cdot 3^{8}\cdot 5\cdot 7$$ Twist minimal: no (minimal twist has level 24) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-263517.$$ of defining polynomial Character $$\chi$$ $$=$$ 48.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.59432e6 q^{3} -2.94911e9 q^{5} +2.52860e11 q^{7} +2.54187e12 q^{9} +O(q^{10})$$ $$q-1.59432e6 q^{3} -2.94911e9 q^{5} +2.52860e11 q^{7} +2.54187e12 q^{9} +1.51354e14 q^{11} +9.28080e13 q^{13} +4.70183e15 q^{15} +1.39004e16 q^{17} +1.70678e17 q^{19} -4.03140e17 q^{21} +4.09502e18 q^{23} +1.24666e18 q^{25} -4.05256e18 q^{27} +4.32334e19 q^{29} +1.99593e20 q^{31} -2.41307e20 q^{33} -7.45710e20 q^{35} +1.37767e21 q^{37} -1.47966e20 q^{39} +6.82134e21 q^{41} -9.39297e21 q^{43} -7.49624e21 q^{45} +2.13749e21 q^{47} -1.77440e21 q^{49} -2.21618e22 q^{51} -2.93406e23 q^{53} -4.46358e23 q^{55} -2.72116e23 q^{57} -8.14136e23 q^{59} +5.59522e23 q^{61} +6.42735e23 q^{63} -2.73701e23 q^{65} +6.51018e24 q^{67} -6.52878e24 q^{69} -3.69127e24 q^{71} +1.27012e24 q^{73} -1.98757e24 q^{75} +3.82712e25 q^{77} +1.00060e25 q^{79} +6.46108e24 q^{81} -9.33551e25 q^{83} -4.09938e25 q^{85} -6.89281e25 q^{87} +3.02863e25 q^{89} +2.34674e25 q^{91} -3.18215e26 q^{93} -5.03347e26 q^{95} -6.11382e26 q^{97} +3.84721e26 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6377292 q^{3} + 4949050424 q^{5} + 88249731552 q^{7} + 10167463313316 q^{9}+O(q^{10})$$ 4 * q - 6377292 * q^3 + 4949050424 * q^5 + 88249731552 * q^7 + 10167463313316 * q^9 $$4 q - 6377292 q^{3} + 4949050424 q^{5} + 88249731552 q^{7} + 10167463313316 q^{9} + 111189633716848 q^{11} - 7339657642664 q^{13} - 78\!\cdots\!52 q^{15}+ \cdots + 28\!\cdots\!92 q^{99}+O(q^{100})$$ 4 * q - 6377292 * q^3 + 4949050424 * q^5 + 88249731552 * q^7 + 10167463313316 * q^9 + 111189633716848 * q^11 - 7339657642664 * q^13 - 7890384919142952 * q^15 + 32484737251067464 * q^17 + 32881723696992400 * q^19 - 140698576757179296 * q^21 + 321501199786814176 * q^23 + 17795887539272914076 * q^25 - 16210220612075905068 * q^27 + 143359861847958158808 * q^29 + 15263052153492355648 * q^31 - 177272190396346253904 * q^33 - 300802957612005610944 * q^35 + 1115572721950557665016 * q^37 + 11701784991824996472 * q^39 + 10700585339689986789480 * q^41 - 9887391055592056200208 * q^43 + 12579822155442748661496 * q^45 - 50192255945374906090176 * q^47 + 228398298477204573852836 * q^49 - 51791163748333632406872 * q^51 + 103627112374107145482488 * q^53 + 206030255590208876474912 * q^55 - 52424088369760014145200 * q^57 + 1996721345289804944085424 * q^59 - 1086322669153439941389800 * q^61 + 224318976991236366736608 * q^63 - 6329749346647807085694256 * q^65 + 176878326996309818449360 * q^67 - 512576757347712937522848 * q^69 - 10877481227380294006167136 * q^71 + 3243145341396450784281448 * q^73 - 28372392809276210188390548 * q^75 - 62677954987910524223224704 * q^77 - 120775774549766213536947712 * q^79 + 25844327556906693195728964 * q^81 - 66790155170591218991697520 * q^83 + 5164767199468201935151088 * q^85 - 228561925021022195625246984 * q^87 - 186629903639627419145275608 * q^89 - 532276318844313477179444928 * q^91 - 24334235098512392933786304 * q^93 - 750290459926244351858149408 * q^95 + 109476826727247656453691272 * q^97 + 282629130409273948562986992 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.59432e6 −0.577350
$$4$$ 0 0
$$5$$ −2.94911e9 −1.08043 −0.540214 0.841528i $$-0.681657\pi$$
−0.540214 + 0.841528i $$0.681657\pi$$
$$6$$ 0 0
$$7$$ 2.52860e11 0.986406 0.493203 0.869914i $$-0.335826\pi$$
0.493203 + 0.869914i $$0.335826\pi$$
$$8$$ 0 0
$$9$$ 2.54187e12 0.333333
$$10$$ 0 0
$$11$$ 1.51354e14 1.32188 0.660940 0.750439i $$-0.270158\pi$$
0.660940 + 0.750439i $$0.270158\pi$$
$$12$$ 0 0
$$13$$ 9.28080e13 0.0849865 0.0424933 0.999097i $$-0.486470\pi$$
0.0424933 + 0.999097i $$0.486470\pi$$
$$14$$ 0 0
$$15$$ 4.70183e15 0.623785
$$16$$ 0 0
$$17$$ 1.39004e16 0.340383 0.170191 0.985411i $$-0.445561\pi$$
0.170191 + 0.985411i $$0.445561\pi$$
$$18$$ 0 0
$$19$$ 1.70678e17 0.931115 0.465558 0.885018i $$-0.345854\pi$$
0.465558 + 0.885018i $$0.345854\pi$$
$$20$$ 0 0
$$21$$ −4.03140e17 −0.569502
$$22$$ 0 0
$$23$$ 4.09502e18 1.69407 0.847033 0.531541i $$-0.178387\pi$$
0.847033 + 0.531541i $$0.178387\pi$$
$$24$$ 0 0
$$25$$ 1.24666e18 0.167324
$$26$$ 0 0
$$27$$ −4.05256e18 −0.192450
$$28$$ 0 0
$$29$$ 4.32334e19 0.782433 0.391216 0.920299i $$-0.372054\pi$$
0.391216 + 0.920299i $$0.372054\pi$$
$$30$$ 0 0
$$31$$ 1.99593e20 1.46812 0.734060 0.679085i $$-0.237623\pi$$
0.734060 + 0.679085i $$0.237623\pi$$
$$32$$ 0 0
$$33$$ −2.41307e20 −0.763187
$$34$$ 0 0
$$35$$ −7.45710e20 −1.06574
$$36$$ 0 0
$$37$$ 1.37767e21 0.929867 0.464933 0.885346i $$-0.346078\pi$$
0.464933 + 0.885346i $$0.346078\pi$$
$$38$$ 0 0
$$39$$ −1.47966e20 −0.0490670
$$40$$ 0 0
$$41$$ 6.82134e21 1.15156 0.575781 0.817604i $$-0.304698\pi$$
0.575781 + 0.817604i $$0.304698\pi$$
$$42$$ 0 0
$$43$$ −9.39297e21 −0.833641 −0.416820 0.908989i $$-0.636856\pi$$
−0.416820 + 0.908989i $$0.636856\pi$$
$$44$$ 0 0
$$45$$ −7.49624e21 −0.360142
$$46$$ 0 0
$$47$$ 2.13749e21 0.0570929 0.0285464 0.999592i $$-0.490912\pi$$
0.0285464 + 0.999592i $$0.490912\pi$$
$$48$$ 0 0
$$49$$ −1.77440e21 −0.0270025
$$50$$ 0 0
$$51$$ −2.21618e22 −0.196520
$$52$$ 0 0
$$53$$ −2.93406e23 −1.54791 −0.773953 0.633244i $$-0.781723\pi$$
−0.773953 + 0.633244i $$0.781723\pi$$
$$54$$ 0 0
$$55$$ −4.46358e23 −1.42819
$$56$$ 0 0
$$57$$ −2.72116e23 −0.537580
$$58$$ 0 0
$$59$$ −8.14136e23 −1.00971 −0.504853 0.863205i $$-0.668453\pi$$
−0.504853 + 0.863205i $$0.668453\pi$$
$$60$$ 0 0
$$61$$ 5.59522e23 0.442450 0.221225 0.975223i $$-0.428994\pi$$
0.221225 + 0.975223i $$0.428994\pi$$
$$62$$ 0 0
$$63$$ 6.42735e23 0.328802
$$64$$ 0 0
$$65$$ −2.73701e23 −0.0918218
$$66$$ 0 0
$$67$$ 6.51018e24 1.45072 0.725359 0.688371i $$-0.241674\pi$$
0.725359 + 0.688371i $$0.241674\pi$$
$$68$$ 0 0
$$69$$ −6.52878e24 −0.978069
$$70$$ 0 0
$$71$$ −3.69127e24 −0.376000 −0.188000 0.982169i $$-0.560201\pi$$
−0.188000 + 0.982169i $$0.560201\pi$$
$$72$$ 0 0
$$73$$ 1.27012e24 0.0889176 0.0444588 0.999011i $$-0.485844\pi$$
0.0444588 + 0.999011i $$0.485844\pi$$
$$74$$ 0 0
$$75$$ −1.98757e24 −0.0966043
$$76$$ 0 0
$$77$$ 3.82712e25 1.30391
$$78$$ 0 0
$$79$$ 1.00060e25 0.241154 0.120577 0.992704i $$-0.461526\pi$$
0.120577 + 0.992704i $$0.461526\pi$$
$$80$$ 0 0
$$81$$ 6.46108e24 0.111111
$$82$$ 0 0
$$83$$ −9.33551e25 −1.15500 −0.577502 0.816389i $$-0.695973\pi$$
−0.577502 + 0.816389i $$0.695973\pi$$
$$84$$ 0 0
$$85$$ −4.09938e25 −0.367759
$$86$$ 0 0
$$87$$ −6.89281e25 −0.451738
$$88$$ 0 0
$$89$$ 3.02863e25 0.146043 0.0730215 0.997330i $$-0.476736\pi$$
0.0730215 + 0.997330i $$0.476736\pi$$
$$90$$ 0 0
$$91$$ 2.34674e25 0.0838312
$$92$$ 0 0
$$93$$ −3.18215e26 −0.847619
$$94$$ 0 0
$$95$$ −5.03347e26 −1.00600
$$96$$ 0 0
$$97$$ −6.11382e26 −0.922347 −0.461174 0.887310i $$-0.652571\pi$$
−0.461174 + 0.887310i $$0.652571\pi$$
$$98$$ 0 0
$$99$$ 3.84721e26 0.440626
$$100$$ 0 0
$$101$$ 3.53903e26 0.309418 0.154709 0.987960i $$-0.450556\pi$$
0.154709 + 0.987960i $$0.450556\pi$$
$$102$$ 0 0
$$103$$ −2.86663e27 −1.92339 −0.961697 0.274114i $$-0.911616\pi$$
−0.961697 + 0.274114i $$0.911616\pi$$
$$104$$ 0 0
$$105$$ 1.18890e27 0.615306
$$106$$ 0 0
$$107$$ 3.77279e27 1.51350 0.756748 0.653706i $$-0.226787\pi$$
0.756748 + 0.653706i $$0.226787\pi$$
$$108$$ 0 0
$$109$$ 1.19634e27 0.373764 0.186882 0.982382i $$-0.440162\pi$$
0.186882 + 0.982382i $$0.440162\pi$$
$$110$$ 0 0
$$111$$ −2.19644e27 −0.536859
$$112$$ 0 0
$$113$$ 6.03773e27 1.15962 0.579809 0.814753i $$-0.303127\pi$$
0.579809 + 0.814753i $$0.303127\pi$$
$$114$$ 0 0
$$115$$ −1.20767e28 −1.83031
$$116$$ 0 0
$$117$$ 2.35905e26 0.0283288
$$118$$ 0 0
$$119$$ 3.51485e27 0.335756
$$120$$ 0 0
$$121$$ 9.79795e27 0.747365
$$122$$ 0 0
$$123$$ −1.08754e28 −0.664855
$$124$$ 0 0
$$125$$ 1.82960e28 0.899647
$$126$$ 0 0
$$127$$ 2.19122e28 0.869631 0.434815 0.900520i $$-0.356814\pi$$
0.434815 + 0.900520i $$0.356814\pi$$
$$128$$ 0 0
$$129$$ 1.49754e28 0.481303
$$130$$ 0 0
$$131$$ −4.13755e28 −1.08039 −0.540196 0.841539i $$-0.681650\pi$$
−0.540196 + 0.841539i $$0.681650\pi$$
$$132$$ 0 0
$$133$$ 4.31575e28 0.918458
$$134$$ 0 0
$$135$$ 1.19514e28 0.207928
$$136$$ 0 0
$$137$$ 1.31599e29 1.87726 0.938632 0.344921i $$-0.112094\pi$$
0.938632 + 0.344921i $$0.112094\pi$$
$$138$$ 0 0
$$139$$ −1.31016e29 −1.53682 −0.768411 0.639956i $$-0.778953\pi$$
−0.768411 + 0.639956i $$0.778953\pi$$
$$140$$ 0 0
$$141$$ −3.40784e27 −0.0329626
$$142$$ 0 0
$$143$$ 1.40468e28 0.112342
$$144$$ 0 0
$$145$$ −1.27500e29 −0.845362
$$146$$ 0 0
$$147$$ 2.82896e27 0.0155899
$$148$$ 0 0
$$149$$ −9.91784e28 −0.455410 −0.227705 0.973730i $$-0.573122\pi$$
−0.227705 + 0.973730i $$0.573122\pi$$
$$150$$ 0 0
$$151$$ −3.24364e29 −1.24407 −0.622035 0.782990i $$-0.713694\pi$$
−0.622035 + 0.782990i $$0.713694\pi$$
$$152$$ 0 0
$$153$$ 3.53330e28 0.113461
$$154$$ 0 0
$$155$$ −5.88620e29 −1.58620
$$156$$ 0 0
$$157$$ 3.35983e29 0.761503 0.380752 0.924677i $$-0.375665\pi$$
0.380752 + 0.924677i $$0.375665\pi$$
$$158$$ 0 0
$$159$$ 4.67783e29 0.893683
$$160$$ 0 0
$$161$$ 1.03546e30 1.67104
$$162$$ 0 0
$$163$$ −2.71722e29 −0.371187 −0.185594 0.982627i $$-0.559421\pi$$
−0.185594 + 0.982627i $$0.559421\pi$$
$$164$$ 0 0
$$165$$ 7.11640e29 0.824569
$$166$$ 0 0
$$167$$ 8.17343e29 0.804882 0.402441 0.915446i $$-0.368162\pi$$
0.402441 + 0.915446i $$0.368162\pi$$
$$168$$ 0 0
$$169$$ −1.18392e30 −0.992777
$$170$$ 0 0
$$171$$ 4.33840e29 0.310372
$$172$$ 0 0
$$173$$ −2.06734e30 −1.26413 −0.632063 0.774917i $$-0.717791\pi$$
−0.632063 + 0.774917i $$0.717791\pi$$
$$174$$ 0 0
$$175$$ 3.15229e29 0.165049
$$176$$ 0 0
$$177$$ 1.29800e30 0.582954
$$178$$ 0 0
$$179$$ 2.84130e30 1.09648 0.548241 0.836320i $$-0.315297\pi$$
0.548241 + 0.836320i $$0.315297\pi$$
$$180$$ 0 0
$$181$$ −4.08892e30 −1.35815 −0.679074 0.734069i $$-0.737619\pi$$
−0.679074 + 0.734069i $$0.737619\pi$$
$$182$$ 0 0
$$183$$ −8.92058e29 −0.255449
$$184$$ 0 0
$$185$$ −4.06289e30 −1.00465
$$186$$ 0 0
$$187$$ 2.10388e30 0.449945
$$188$$ 0 0
$$189$$ −1.02473e30 −0.189834
$$190$$ 0 0
$$191$$ 2.18709e30 0.351493 0.175746 0.984435i $$-0.443766\pi$$
0.175746 + 0.984435i $$0.443766\pi$$
$$192$$ 0 0
$$193$$ 9.97718e30 1.39311 0.696553 0.717505i $$-0.254716\pi$$
0.696553 + 0.717505i $$0.254716\pi$$
$$194$$ 0 0
$$195$$ 4.36367e29 0.0530133
$$196$$ 0 0
$$197$$ 2.89658e30 0.306614 0.153307 0.988179i $$-0.451008\pi$$
0.153307 + 0.988179i $$0.451008\pi$$
$$198$$ 0 0
$$199$$ 6.81123e30 0.629085 0.314542 0.949243i $$-0.398149\pi$$
0.314542 + 0.949243i $$0.398149\pi$$
$$200$$ 0 0
$$201$$ −1.03793e31 −0.837572
$$202$$ 0 0
$$203$$ 1.09320e31 0.771796
$$204$$ 0 0
$$205$$ −2.01169e31 −1.24418
$$206$$ 0 0
$$207$$ 1.04090e31 0.564688
$$208$$ 0 0
$$209$$ 2.58327e31 1.23082
$$210$$ 0 0
$$211$$ 3.13701e31 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$212$$ 0 0
$$213$$ 5.88508e30 0.217084
$$214$$ 0 0
$$215$$ 2.77009e31 0.900688
$$216$$ 0 0
$$217$$ 5.04689e31 1.44816
$$218$$ 0 0
$$219$$ −2.02499e30 −0.0513366
$$220$$ 0 0
$$221$$ 1.29007e30 0.0289279
$$222$$ 0 0
$$223$$ 1.13345e31 0.225054 0.112527 0.993649i $$-0.464106\pi$$
0.112527 + 0.993649i $$0.464106\pi$$
$$224$$ 0 0
$$225$$ 3.16884e30 0.0557745
$$226$$ 0 0
$$227$$ −1.38884e31 −0.216922 −0.108461 0.994101i $$-0.534592\pi$$
−0.108461 + 0.994101i $$0.534592\pi$$
$$228$$ 0 0
$$229$$ −4.10593e31 −0.569682 −0.284841 0.958575i $$-0.591941\pi$$
−0.284841 + 0.958575i $$0.591941\pi$$
$$230$$ 0 0
$$231$$ −6.10167e31 −0.752813
$$232$$ 0 0
$$233$$ 8.01474e31 0.880207 0.440103 0.897947i $$-0.354942\pi$$
0.440103 + 0.897947i $$0.354942\pi$$
$$234$$ 0 0
$$235$$ −6.30367e30 −0.0616847
$$236$$ 0 0
$$237$$ −1.59528e31 −0.139230
$$238$$ 0 0
$$239$$ −1.22012e32 −0.950670 −0.475335 0.879805i $$-0.657673\pi$$
−0.475335 + 0.879805i $$0.657673\pi$$
$$240$$ 0 0
$$241$$ 1.79059e32 1.24671 0.623357 0.781937i $$-0.285768\pi$$
0.623357 + 0.781937i $$0.285768\pi$$
$$242$$ 0 0
$$243$$ −1.03011e31 −0.0641500
$$244$$ 0 0
$$245$$ 5.23289e30 0.0291742
$$246$$ 0 0
$$247$$ 1.58403e31 0.0791322
$$248$$ 0 0
$$249$$ 1.48838e32 0.666842
$$250$$ 0 0
$$251$$ 2.98031e31 0.119858 0.0599290 0.998203i $$-0.480913\pi$$
0.0599290 + 0.998203i $$0.480913\pi$$
$$252$$ 0 0
$$253$$ 6.19796e32 2.23935
$$254$$ 0 0
$$255$$ 6.53574e31 0.212326
$$256$$ 0 0
$$257$$ −4.62356e31 −0.135170 −0.0675849 0.997714i $$-0.521529\pi$$
−0.0675849 + 0.997714i $$0.521529\pi$$
$$258$$ 0 0
$$259$$ 3.48356e32 0.917226
$$260$$ 0 0
$$261$$ 1.09894e32 0.260811
$$262$$ 0 0
$$263$$ 3.44300e32 0.737113 0.368556 0.929605i $$-0.379852\pi$$
0.368556 + 0.929605i $$0.379852\pi$$
$$264$$ 0 0
$$265$$ 8.65285e32 1.67240
$$266$$ 0 0
$$267$$ −4.82861e31 −0.0843180
$$268$$ 0 0
$$269$$ 1.03375e33 1.63214 0.816072 0.577950i $$-0.196147\pi$$
0.816072 + 0.577950i $$0.196147\pi$$
$$270$$ 0 0
$$271$$ 9.90254e32 1.41469 0.707345 0.706869i $$-0.249893\pi$$
0.707345 + 0.706869i $$0.249893\pi$$
$$272$$ 0 0
$$273$$ −3.74146e31 −0.0484000
$$274$$ 0 0
$$275$$ 1.88686e32 0.221182
$$276$$ 0 0
$$277$$ −1.71524e33 −1.82326 −0.911631 0.411010i $$-0.865176\pi$$
−0.911631 + 0.411010i $$0.865176\pi$$
$$278$$ 0 0
$$279$$ 5.07337e32 0.489373
$$280$$ 0 0
$$281$$ 3.27334e32 0.286719 0.143359 0.989671i $$-0.454210\pi$$
0.143359 + 0.989671i $$0.454210\pi$$
$$282$$ 0 0
$$283$$ 2.25086e32 0.179156 0.0895782 0.995980i $$-0.471448\pi$$
0.0895782 + 0.995980i $$0.471448\pi$$
$$284$$ 0 0
$$285$$ 8.02498e32 0.580816
$$286$$ 0 0
$$287$$ 1.72484e33 1.13591
$$288$$ 0 0
$$289$$ −1.47449e33 −0.884140
$$290$$ 0 0
$$291$$ 9.74741e32 0.532517
$$292$$ 0 0
$$293$$ 3.63890e33 1.81241 0.906207 0.422833i $$-0.138964\pi$$
0.906207 + 0.422833i $$0.138964\pi$$
$$294$$ 0 0
$$295$$ 2.40097e33 1.09091
$$296$$ 0 0
$$297$$ −6.13369e32 −0.254396
$$298$$ 0 0
$$299$$ 3.80050e32 0.143973
$$300$$ 0 0
$$301$$ −2.37510e33 −0.822308
$$302$$ 0 0
$$303$$ −5.64236e32 −0.178643
$$304$$ 0 0
$$305$$ −1.65009e33 −0.478035
$$306$$ 0 0
$$307$$ −3.46198e33 −0.918241 −0.459120 0.888374i $$-0.651835\pi$$
−0.459120 + 0.888374i $$0.651835\pi$$
$$308$$ 0 0
$$309$$ 4.57033e33 1.11047
$$310$$ 0 0
$$311$$ 7.31854e33 1.62989 0.814947 0.579536i $$-0.196766\pi$$
0.814947 + 0.579536i $$0.196766\pi$$
$$312$$ 0 0
$$313$$ 6.01872e32 0.122929 0.0614646 0.998109i $$-0.480423\pi$$
0.0614646 + 0.998109i $$0.480423\pi$$
$$314$$ 0 0
$$315$$ −1.89550e33 −0.355247
$$316$$ 0 0
$$317$$ −5.16567e33 −0.888847 −0.444423 0.895817i $$-0.646591\pi$$
−0.444423 + 0.895817i $$0.646591\pi$$
$$318$$ 0 0
$$319$$ 6.54354e33 1.03428
$$320$$ 0 0
$$321$$ −6.01504e33 −0.873818
$$322$$ 0 0
$$323$$ 2.37249e33 0.316936
$$324$$ 0 0
$$325$$ 1.15700e32 0.0142202
$$326$$ 0 0
$$327$$ −1.90736e33 −0.215793
$$328$$ 0 0
$$329$$ 5.40484e32 0.0563168
$$330$$ 0 0
$$331$$ 1.73903e34 1.66967 0.834834 0.550502i $$-0.185564\pi$$
0.834834 + 0.550502i $$0.185564\pi$$
$$332$$ 0 0
$$333$$ 3.50184e33 0.309956
$$334$$ 0 0
$$335$$ −1.91992e34 −1.56740
$$336$$ 0 0
$$337$$ 2.88416e33 0.217278 0.108639 0.994081i $$-0.465351\pi$$
0.108639 + 0.994081i $$0.465351\pi$$
$$338$$ 0 0
$$339$$ −9.62609e33 −0.669505
$$340$$ 0 0
$$341$$ 3.02091e34 1.94068
$$342$$ 0 0
$$343$$ −1.70647e34 −1.01304
$$344$$ 0 0
$$345$$ 1.92541e34 1.05673
$$346$$ 0 0
$$347$$ −2.65372e34 −1.34712 −0.673562 0.739131i $$-0.735237\pi$$
−0.673562 + 0.739131i $$0.735237\pi$$
$$348$$ 0 0
$$349$$ −1.77192e34 −0.832341 −0.416170 0.909287i $$-0.636628\pi$$
−0.416170 + 0.909287i $$0.636628\pi$$
$$350$$ 0 0
$$351$$ −3.76109e32 −0.0163557
$$352$$ 0 0
$$353$$ −1.17206e34 −0.472053 −0.236026 0.971747i $$-0.575845\pi$$
−0.236026 + 0.971747i $$0.575845\pi$$
$$354$$ 0 0
$$355$$ 1.08860e34 0.406241
$$356$$ 0 0
$$357$$ −5.60381e33 −0.193849
$$358$$ 0 0
$$359$$ −3.36159e34 −1.07838 −0.539188 0.842186i $$-0.681269\pi$$
−0.539188 + 0.842186i $$0.681269\pi$$
$$360$$ 0 0
$$361$$ −4.46969e33 −0.133024
$$362$$ 0 0
$$363$$ −1.56211e34 −0.431491
$$364$$ 0 0
$$365$$ −3.74573e33 −0.0960690
$$366$$ 0 0
$$367$$ 6.24921e34 1.48879 0.744395 0.667740i $$-0.232738\pi$$
0.744395 + 0.667740i $$0.232738\pi$$
$$368$$ 0 0
$$369$$ 1.73389e34 0.383854
$$370$$ 0 0
$$371$$ −7.41904e34 −1.52686
$$372$$ 0 0
$$373$$ −1.81566e34 −0.347509 −0.173754 0.984789i $$-0.555590\pi$$
−0.173754 + 0.984789i $$0.555590\pi$$
$$374$$ 0 0
$$375$$ −2.91698e34 −0.519411
$$376$$ 0 0
$$377$$ 4.01241e33 0.0664962
$$378$$ 0 0
$$379$$ −6.01249e34 −0.927738 −0.463869 0.885904i $$-0.653539\pi$$
−0.463869 + 0.885904i $$0.653539\pi$$
$$380$$ 0 0
$$381$$ −3.49350e34 −0.502081
$$382$$ 0 0
$$383$$ 1.27205e34 0.170342 0.0851709 0.996366i $$-0.472856\pi$$
0.0851709 + 0.996366i $$0.472856\pi$$
$$384$$ 0 0
$$385$$ −1.12866e35 −1.40878
$$386$$ 0 0
$$387$$ −2.38757e34 −0.277880
$$388$$ 0 0
$$389$$ −1.29479e34 −0.140566 −0.0702832 0.997527i $$-0.522390\pi$$
−0.0702832 + 0.997527i $$0.522390\pi$$
$$390$$ 0 0
$$391$$ 5.69225e34 0.576631
$$392$$ 0 0
$$393$$ 6.59660e34 0.623764
$$394$$ 0 0
$$395$$ −2.95087e34 −0.260549
$$396$$ 0 0
$$397$$ −2.21092e35 −1.82348 −0.911741 0.410765i $$-0.865262\pi$$
−0.911741 + 0.410765i $$0.865262\pi$$
$$398$$ 0 0
$$399$$ −6.88070e34 −0.530272
$$400$$ 0 0
$$401$$ −1.64003e34 −0.118142 −0.0590708 0.998254i $$-0.518814\pi$$
−0.0590708 + 0.998254i $$0.518814\pi$$
$$402$$ 0 0
$$403$$ 1.85238e34 0.124770
$$404$$ 0 0
$$405$$ −1.90544e34 −0.120047
$$406$$ 0 0
$$407$$ 2.08515e35 1.22917
$$408$$ 0 0
$$409$$ −1.30759e35 −0.721451 −0.360726 0.932672i $$-0.617471\pi$$
−0.360726 + 0.932672i $$0.617471\pi$$
$$410$$ 0 0
$$411$$ −2.09811e35 −1.08384
$$412$$ 0 0
$$413$$ −2.05862e35 −0.995980
$$414$$ 0 0
$$415$$ 2.75314e35 1.24790
$$416$$ 0 0
$$417$$ 2.08881e35 0.887285
$$418$$ 0 0
$$419$$ −3.17393e35 −1.26389 −0.631943 0.775015i $$-0.717742\pi$$
−0.631943 + 0.775015i $$0.717742\pi$$
$$420$$ 0 0
$$421$$ −4.30484e35 −1.60749 −0.803746 0.594973i $$-0.797163\pi$$
−0.803746 + 0.594973i $$0.797163\pi$$
$$422$$ 0 0
$$423$$ 5.43320e33 0.0190310
$$424$$ 0 0
$$425$$ 1.73291e34 0.0569540
$$426$$ 0 0
$$427$$ 1.41480e35 0.436435
$$428$$ 0 0
$$429$$ −2.23952e34 −0.0648606
$$430$$ 0 0
$$431$$ −5.03774e35 −1.37023 −0.685114 0.728436i $$-0.740248\pi$$
−0.685114 + 0.728436i $$0.740248\pi$$
$$432$$ 0 0
$$433$$ −7.57241e35 −1.93485 −0.967426 0.253154i $$-0.918532\pi$$
−0.967426 + 0.253154i $$0.918532\pi$$
$$434$$ 0 0
$$435$$ 2.03276e35 0.488070
$$436$$ 0 0
$$437$$ 6.98929e35 1.57737
$$438$$ 0 0
$$439$$ −4.23974e35 −0.899638 −0.449819 0.893120i $$-0.648511\pi$$
−0.449819 + 0.893120i $$0.648511\pi$$
$$440$$ 0 0
$$441$$ −4.51028e33 −0.00900083
$$442$$ 0 0
$$443$$ 8.43326e35 1.58323 0.791617 0.611017i $$-0.209239\pi$$
0.791617 + 0.611017i $$0.209239\pi$$
$$444$$ 0 0
$$445$$ −8.93175e34 −0.157789
$$446$$ 0 0
$$447$$ 1.58122e35 0.262931
$$448$$ 0 0
$$449$$ 5.18891e35 0.812364 0.406182 0.913792i $$-0.366860\pi$$
0.406182 + 0.913792i $$0.366860\pi$$
$$450$$ 0 0
$$451$$ 1.03243e36 1.52223
$$452$$ 0 0
$$453$$ 5.17142e35 0.718264
$$454$$ 0 0
$$455$$ −6.92078e34 −0.0905736
$$456$$ 0 0
$$457$$ 9.52583e35 1.17499 0.587495 0.809228i $$-0.300114\pi$$
0.587495 + 0.809228i $$0.300114\pi$$
$$458$$ 0 0
$$459$$ −5.63322e34 −0.0655067
$$460$$ 0 0
$$461$$ 1.04963e36 1.15099 0.575497 0.817804i $$-0.304809\pi$$
0.575497 + 0.817804i $$0.304809\pi$$
$$462$$ 0 0
$$463$$ 5.25044e35 0.543066 0.271533 0.962429i $$-0.412469\pi$$
0.271533 + 0.962429i $$0.412469\pi$$
$$464$$ 0 0
$$465$$ 9.38450e35 0.915791
$$466$$ 0 0
$$467$$ −1.53832e36 −1.41667 −0.708336 0.705876i $$-0.750554\pi$$
−0.708336 + 0.705876i $$0.750554\pi$$
$$468$$ 0 0
$$469$$ 1.64616e36 1.43100
$$470$$ 0 0
$$471$$ −5.35665e35 −0.439654
$$472$$ 0 0
$$473$$ −1.42166e36 −1.10197
$$474$$ 0 0
$$475$$ 2.12777e35 0.155797
$$476$$ 0 0
$$477$$ −7.45798e35 −0.515968
$$478$$ 0 0
$$479$$ −1.71162e35 −0.111912 −0.0559561 0.998433i $$-0.517821\pi$$
−0.0559561 + 0.998433i $$0.517821\pi$$
$$480$$ 0 0
$$481$$ 1.27858e35 0.0790261
$$482$$ 0 0
$$483$$ −1.65087e36 −0.964774
$$484$$ 0 0
$$485$$ 1.80303e36 0.996529
$$486$$ 0 0
$$487$$ −1.41363e35 −0.0739084 −0.0369542 0.999317i $$-0.511766\pi$$
−0.0369542 + 0.999317i $$0.511766\pi$$
$$488$$ 0 0
$$489$$ 4.33213e35 0.214305
$$490$$ 0 0
$$491$$ −1.38190e35 −0.0646959 −0.0323480 0.999477i $$-0.510298\pi$$
−0.0323480 + 0.999477i $$0.510298\pi$$
$$492$$ 0 0
$$493$$ 6.00963e35 0.266327
$$494$$ 0 0
$$495$$ −1.13458e36 −0.476065
$$496$$ 0 0
$$497$$ −9.33374e35 −0.370889
$$498$$ 0 0
$$499$$ 2.66489e36 1.00305 0.501524 0.865144i $$-0.332773\pi$$
0.501524 + 0.865144i $$0.332773\pi$$
$$500$$ 0 0
$$501$$ −1.30311e36 −0.464699
$$502$$ 0 0
$$503$$ 1.79792e36 0.607578 0.303789 0.952739i $$-0.401748\pi$$
0.303789 + 0.952739i $$0.401748\pi$$
$$504$$ 0 0
$$505$$ −1.04370e36 −0.334304
$$506$$ 0 0
$$507$$ 1.88755e36 0.573180
$$508$$ 0 0
$$509$$ 2.19200e36 0.631176 0.315588 0.948896i $$-0.397798\pi$$
0.315588 + 0.948896i $$0.397798\pi$$
$$510$$ 0 0
$$511$$ 3.21163e35 0.0877089
$$512$$ 0 0
$$513$$ −6.91681e35 −0.179193
$$514$$ 0 0
$$515$$ 8.45399e36 2.07809
$$516$$ 0 0
$$517$$ 3.23516e35 0.0754699
$$518$$ 0 0
$$519$$ 3.29601e36 0.729843
$$520$$ 0 0
$$521$$ 1.51669e36 0.318852 0.159426 0.987210i $$-0.449036\pi$$
0.159426 + 0.987210i $$0.449036\pi$$
$$522$$ 0 0
$$523$$ 3.10531e36 0.619915 0.309958 0.950750i $$-0.399685\pi$$
0.309958 + 0.950750i $$0.399685\pi$$
$$524$$ 0 0
$$525$$ −5.02577e35 −0.0952911
$$526$$ 0 0
$$527$$ 2.77442e36 0.499723
$$528$$ 0 0
$$529$$ 1.09260e37 1.86986
$$530$$ 0 0
$$531$$ −2.06942e36 −0.336569
$$532$$ 0 0
$$533$$ 6.33074e35 0.0978673
$$534$$ 0 0
$$535$$ −1.11264e37 −1.63522
$$536$$ 0 0
$$537$$ −4.52996e36 −0.633055
$$538$$ 0 0
$$539$$ −2.68562e35 −0.0356940
$$540$$ 0 0
$$541$$ −1.30208e37 −1.64617 −0.823087 0.567916i $$-0.807750\pi$$
−0.823087 + 0.567916i $$0.807750\pi$$
$$542$$ 0 0
$$543$$ 6.51905e36 0.784128
$$544$$ 0 0
$$545$$ −3.52814e36 −0.403825
$$546$$ 0 0
$$547$$ 4.39612e36 0.478895 0.239448 0.970909i $$-0.423034\pi$$
0.239448 + 0.970909i $$0.423034\pi$$
$$548$$ 0 0
$$549$$ 1.42223e36 0.147483
$$550$$ 0 0
$$551$$ 7.37899e36 0.728535
$$552$$ 0 0
$$553$$ 2.53011e36 0.237875
$$554$$ 0 0
$$555$$ 6.47755e36 0.580037
$$556$$ 0 0
$$557$$ 3.10948e36 0.265242 0.132621 0.991167i $$-0.457661\pi$$
0.132621 + 0.991167i $$0.457661\pi$$
$$558$$ 0 0
$$559$$ −8.71743e35 −0.0708482
$$560$$ 0 0
$$561$$ −3.35426e36 −0.259776
$$562$$ 0 0
$$563$$ −9.78955e36 −0.722603 −0.361301 0.932449i $$-0.617667\pi$$
−0.361301 + 0.932449i $$0.617667\pi$$
$$564$$ 0 0
$$565$$ −1.78059e37 −1.25288
$$566$$ 0 0
$$567$$ 1.63375e36 0.109601
$$568$$ 0 0
$$569$$ −2.38541e37 −1.52597 −0.762986 0.646415i $$-0.776267\pi$$
−0.762986 + 0.646415i $$0.776267\pi$$
$$570$$ 0 0
$$571$$ −2.74250e37 −1.67324 −0.836622 0.547781i $$-0.815473\pi$$
−0.836622 + 0.547781i $$0.815473\pi$$
$$572$$ 0 0
$$573$$ −3.48693e36 −0.202934
$$574$$ 0 0
$$575$$ 5.10509e36 0.283457
$$576$$ 0 0
$$577$$ 3.11547e37 1.65063 0.825316 0.564672i $$-0.190997\pi$$
0.825316 + 0.564672i $$0.190997\pi$$
$$578$$ 0 0
$$579$$ −1.59069e37 −0.804310
$$580$$ 0 0
$$581$$ −2.36057e37 −1.13930
$$582$$ 0 0
$$583$$ −4.44080e37 −2.04614
$$584$$ 0 0
$$585$$ −6.95710e35 −0.0306073
$$586$$ 0 0
$$587$$ 4.69405e36 0.197212 0.0986059 0.995127i $$-0.468562\pi$$
0.0986059 + 0.995127i $$0.468562\pi$$
$$588$$ 0 0
$$589$$ 3.40660e37 1.36699
$$590$$ 0 0
$$591$$ −4.61809e36 −0.177024
$$592$$ 0 0
$$593$$ 3.58866e37 1.31430 0.657149 0.753761i $$-0.271762\pi$$
0.657149 + 0.753761i $$0.271762\pi$$
$$594$$ 0 0
$$595$$ −1.03657e37 −0.362760
$$596$$ 0 0
$$597$$ −1.08593e37 −0.363202
$$598$$ 0 0
$$599$$ −1.35541e37 −0.433321 −0.216660 0.976247i $$-0.569516\pi$$
−0.216660 + 0.976247i $$0.569516\pi$$
$$600$$ 0 0
$$601$$ −2.72933e37 −0.834163 −0.417081 0.908869i $$-0.636947\pi$$
−0.417081 + 0.908869i $$0.636947\pi$$
$$602$$ 0 0
$$603$$ 1.65480e37 0.483573
$$604$$ 0 0
$$605$$ −2.88952e37 −0.807474
$$606$$ 0 0
$$607$$ −6.35953e37 −1.69972 −0.849861 0.527007i $$-0.823314\pi$$
−0.849861 + 0.527007i $$0.823314\pi$$
$$608$$ 0 0
$$609$$ −1.74291e37 −0.445597
$$610$$ 0 0
$$611$$ 1.98376e35 0.00485212
$$612$$ 0 0
$$613$$ 4.82418e37 1.12903 0.564517 0.825421i $$-0.309062\pi$$
0.564517 + 0.825421i $$0.309062\pi$$
$$614$$ 0 0
$$615$$ 3.20728e37 0.718327
$$616$$ 0 0
$$617$$ 2.26067e37 0.484604 0.242302 0.970201i $$-0.422097\pi$$
0.242302 + 0.970201i $$0.422097\pi$$
$$618$$ 0 0
$$619$$ −2.23444e37 −0.458506 −0.229253 0.973367i $$-0.573628\pi$$
−0.229253 + 0.973367i $$0.573628\pi$$
$$620$$ 0 0
$$621$$ −1.65953e37 −0.326023
$$622$$ 0 0
$$623$$ 7.65817e36 0.144058
$$624$$ 0 0
$$625$$ −6.32453e37 −1.13933
$$626$$ 0 0
$$627$$ −4.11857e37 −0.710616
$$628$$ 0 0
$$629$$ 1.91501e37 0.316511
$$630$$ 0 0
$$631$$ −5.33027e37 −0.844021 −0.422010 0.906591i $$-0.638675\pi$$
−0.422010 + 0.906591i $$0.638675\pi$$
$$632$$ 0 0
$$633$$ −5.00140e37 −0.758825
$$634$$ 0 0
$$635$$ −6.46213e37 −0.939573
$$636$$ 0 0
$$637$$ −1.64678e35 −0.00229485
$$638$$ 0 0
$$639$$ −9.38272e36 −0.125333
$$640$$ 0 0
$$641$$ 1.19868e38 1.53504 0.767522 0.641023i $$-0.221490\pi$$
0.767522 + 0.641023i $$0.221490\pi$$
$$642$$ 0 0
$$643$$ 8.41805e37 1.03363 0.516813 0.856098i $$-0.327118\pi$$
0.516813 + 0.856098i $$0.327118\pi$$
$$644$$ 0 0
$$645$$ −4.41642e37 −0.520013
$$646$$ 0 0
$$647$$ 1.54778e38 1.74784 0.873919 0.486072i $$-0.161571\pi$$
0.873919 + 0.486072i $$0.161571\pi$$
$$648$$ 0 0
$$649$$ −1.23222e38 −1.33471
$$650$$ 0 0
$$651$$ −8.04637e37 −0.836097
$$652$$ 0 0
$$653$$ −1.36035e38 −1.35620 −0.678099 0.734970i $$-0.737196\pi$$
−0.678099 + 0.734970i $$0.737196\pi$$
$$654$$ 0 0
$$655$$ 1.22021e38 1.16728
$$656$$ 0 0
$$657$$ 3.22849e36 0.0296392
$$658$$ 0 0
$$659$$ 1.52293e38 1.34192 0.670959 0.741494i $$-0.265883\pi$$
0.670959 + 0.741494i $$0.265883\pi$$
$$660$$ 0 0
$$661$$ 1.83977e38 1.55612 0.778062 0.628188i $$-0.216203\pi$$
0.778062 + 0.628188i $$0.216203\pi$$
$$662$$ 0 0
$$663$$ −2.05679e36 −0.0167016
$$664$$ 0 0
$$665$$ −1.27276e38 −0.992327
$$666$$ 0 0
$$667$$ 1.77042e38 1.32549
$$668$$ 0 0
$$669$$ −1.80708e37 −0.129935
$$670$$ 0 0
$$671$$ 8.46857e37 0.584866
$$672$$ 0 0
$$673$$ 9.36686e37 0.621428 0.310714 0.950503i $$-0.399432\pi$$
0.310714 + 0.950503i $$0.399432\pi$$
$$674$$ 0 0
$$675$$ −5.05215e36 −0.0322014
$$676$$ 0 0
$$677$$ 2.69907e38 1.65298 0.826488 0.562954i $$-0.190335\pi$$
0.826488 + 0.562954i $$0.190335\pi$$
$$678$$ 0 0
$$679$$ −1.54594e38 −0.909809
$$680$$ 0 0
$$681$$ 2.21427e37 0.125240
$$682$$ 0 0
$$683$$ −1.91566e38 −1.04145 −0.520726 0.853724i $$-0.674339\pi$$
−0.520726 + 0.853724i $$0.674339\pi$$
$$684$$ 0 0
$$685$$ −3.88100e38 −2.02825
$$686$$ 0 0
$$687$$ 6.54618e37 0.328906
$$688$$ 0 0
$$689$$ −2.72304e37 −0.131551
$$690$$ 0 0
$$691$$ −6.26889e37 −0.291230 −0.145615 0.989341i $$-0.546516\pi$$
−0.145615 + 0.989341i $$0.546516\pi$$
$$692$$ 0 0
$$693$$ 9.72803e37 0.434637
$$694$$ 0 0
$$695$$ 3.86380e38 1.66043
$$696$$ 0 0
$$697$$ 9.48195e37 0.391972
$$698$$ 0 0
$$699$$ −1.27781e38 −0.508188
$$700$$ 0 0
$$701$$ 1.06445e38 0.407317 0.203659 0.979042i $$-0.434717\pi$$
0.203659 + 0.979042i $$0.434717\pi$$
$$702$$ 0 0
$$703$$ 2.35137e38 0.865813
$$704$$ 0 0
$$705$$ 1.00501e37 0.0356137
$$706$$ 0 0
$$707$$ 8.94878e37 0.305212
$$708$$ 0 0
$$709$$ 7.15469e37 0.234891 0.117446 0.993079i $$-0.462529\pi$$
0.117446 + 0.993079i $$0.462529\pi$$
$$710$$ 0 0
$$711$$ 2.54338e37 0.0803845
$$712$$ 0 0
$$713$$ 8.17335e38 2.48709
$$714$$ 0 0
$$715$$ −4.14256e37 −0.121377
$$716$$ 0 0
$$717$$ 1.94526e38 0.548870
$$718$$ 0 0
$$719$$ 2.71010e38 0.736453 0.368227 0.929736i $$-0.379965\pi$$
0.368227 + 0.929736i $$0.379965\pi$$
$$720$$ 0 0
$$721$$ −7.24854e38 −1.89725
$$722$$ 0 0
$$723$$ −2.85478e38 −0.719791
$$724$$ 0 0
$$725$$ 5.38973e37 0.130919
$$726$$ 0 0
$$727$$ −2.02421e38 −0.473741 −0.236870 0.971541i $$-0.576122\pi$$
−0.236870 + 0.971541i $$0.576122\pi$$
$$728$$ 0 0
$$729$$ 1.64232e37 0.0370370
$$730$$ 0 0
$$731$$ −1.30566e38 −0.283757
$$732$$ 0 0
$$733$$ −3.86048e38 −0.808607 −0.404304 0.914625i $$-0.632486\pi$$
−0.404304 + 0.914625i $$0.632486\pi$$
$$734$$ 0 0
$$735$$ −8.34292e36 −0.0168438
$$736$$ 0 0
$$737$$ 9.85340e38 1.91767
$$738$$ 0 0
$$739$$ 6.27102e38 1.17663 0.588313 0.808633i $$-0.299792\pi$$
0.588313 + 0.808633i $$0.299792\pi$$
$$740$$ 0 0
$$741$$ −2.52545e37 −0.0456870
$$742$$ 0 0
$$743$$ −7.62105e38 −1.32943 −0.664716 0.747096i $$-0.731447\pi$$
−0.664716 + 0.747096i $$0.731447\pi$$
$$744$$ 0 0
$$745$$ 2.92488e38 0.492037
$$746$$ 0 0
$$747$$ −2.37296e38 −0.385002
$$748$$ 0 0
$$749$$ 9.53986e38 1.49292
$$750$$ 0 0
$$751$$ −6.28777e38 −0.949200 −0.474600 0.880202i $$-0.657407\pi$$
−0.474600 + 0.880202i $$0.657407\pi$$
$$752$$ 0 0
$$753$$ −4.75158e37 −0.0692000
$$754$$ 0 0
$$755$$ 9.56586e38 1.34413
$$756$$ 0 0
$$757$$ −8.24507e38 −1.11789 −0.558946 0.829204i $$-0.688794\pi$$
−0.558946 + 0.829204i $$0.688794\pi$$
$$758$$ 0 0
$$759$$ −9.88156e38 −1.29289
$$760$$ 0 0
$$761$$ 1.41435e39 1.78593 0.892963 0.450130i $$-0.148622\pi$$
0.892963 + 0.450130i $$0.148622\pi$$
$$762$$ 0 0
$$763$$ 3.02507e38 0.368684
$$764$$ 0 0
$$765$$ −1.04201e38 −0.122586
$$766$$ 0 0
$$767$$ −7.55583e37 −0.0858113
$$768$$ 0 0
$$769$$ 7.77007e38 0.851960 0.425980 0.904732i $$-0.359929\pi$$
0.425980 + 0.904732i $$0.359929\pi$$
$$770$$ 0 0
$$771$$ 7.37145e37 0.0780403
$$772$$ 0 0
$$773$$ −4.61034e38 −0.471314 −0.235657 0.971836i $$-0.575724\pi$$
−0.235657 + 0.971836i $$0.575724\pi$$
$$774$$ 0 0
$$775$$ 2.48823e38 0.245651
$$776$$ 0 0
$$777$$ −5.55392e38 −0.529561
$$778$$ 0 0
$$779$$ 1.16425e39 1.07224
$$780$$ 0 0
$$781$$ −5.58688e38 −0.497027
$$782$$ 0 0
$$783$$ −1.75206e38 −0.150579
$$784$$ 0 0
$$785$$ −9.90850e38 −0.822749
$$786$$ 0 0
$$787$$ 1.74475e38 0.139983 0.0699914 0.997548i $$-0.477703\pi$$
0.0699914 + 0.997548i $$0.477703\pi$$
$$788$$ 0 0
$$789$$ −5.48925e38 −0.425572
$$790$$ 0 0
$$791$$ 1.52670e39 1.14385
$$792$$ 0 0
$$793$$ 5.19281e37 0.0376023
$$794$$ 0 0
$$795$$ −1.37954e39 −0.965560
$$796$$ 0 0
$$797$$ −6.47929e38 −0.438369 −0.219185 0.975683i $$-0.570340\pi$$
−0.219185 + 0.975683i $$0.570340\pi$$
$$798$$ 0 0
$$799$$ 2.97119e37 0.0194334
$$800$$ 0 0
$$801$$ 7.69836e37 0.0486810
$$802$$ 0 0
$$803$$ 1.92238e38 0.117538
$$804$$ 0 0
$$805$$ −3.05370e39 −1.80543
$$806$$ 0 0
$$807$$ −1.64813e39 −0.942319
$$808$$ 0 0
$$809$$ −2.06930e39 −1.14424 −0.572122 0.820168i $$-0.693880\pi$$
−0.572122 + 0.820168i $$0.693880\pi$$
$$810$$ 0 0
$$811$$ 4.70862e38 0.251833 0.125916 0.992041i $$-0.459813\pi$$
0.125916 + 0.992041i $$0.459813\pi$$
$$812$$ 0 0
$$813$$ −1.57878e39 −0.816771
$$814$$ 0 0
$$815$$ 8.01339e38 0.401041
$$816$$ 0 0
$$817$$ −1.60317e39 −0.776216
$$818$$ 0 0
$$819$$ 5.96509e37 0.0279437
$$820$$ 0 0
$$821$$ 1.51803e39 0.688093 0.344046 0.938953i $$-0.388202\pi$$
0.344046 + 0.938953i $$0.388202\pi$$
$$822$$ 0 0
$$823$$ −3.42558e39 −1.50257 −0.751287 0.659975i $$-0.770567\pi$$
−0.751287 + 0.659975i $$0.770567\pi$$
$$824$$ 0 0
$$825$$ −3.00827e38 −0.127699
$$826$$ 0 0
$$827$$ −3.46077e39 −1.42183 −0.710916 0.703277i $$-0.751719\pi$$
−0.710916 + 0.703277i $$0.751719\pi$$
$$828$$ 0 0
$$829$$ −2.65696e39 −1.05657 −0.528286 0.849067i $$-0.677165\pi$$
−0.528286 + 0.849067i $$0.677165\pi$$
$$830$$ 0 0
$$831$$ 2.73465e39 1.05266
$$832$$ 0 0
$$833$$ −2.46649e37 −0.00919118
$$834$$ 0 0
$$835$$ −2.41043e39 −0.869616
$$836$$ 0 0
$$837$$ −8.08860e38 −0.282540
$$838$$ 0 0
$$839$$ −2.26620e39 −0.766499 −0.383249 0.923645i $$-0.625195\pi$$
−0.383249 + 0.923645i $$0.625195\pi$$
$$840$$ 0 0
$$841$$ −1.18400e39 −0.387799
$$842$$ 0 0
$$843$$ −5.21877e38 −0.165537
$$844$$ 0 0
$$845$$ 3.49151e39 1.07262
$$846$$ 0 0
$$847$$ 2.47751e39 0.737206
$$848$$ 0 0
$$849$$ −3.58860e38 −0.103436
$$850$$ 0 0
$$851$$ 5.64157e39 1.57526
$$852$$ 0 0
$$853$$ −7.54148e38 −0.204007 −0.102003 0.994784i $$-0.532525\pi$$
−0.102003 + 0.994784i $$0.532525\pi$$
$$854$$ 0 0
$$855$$ −1.27944e39 −0.335334
$$856$$ 0 0
$$857$$ −1.01461e39 −0.257666 −0.128833 0.991666i $$-0.541123\pi$$
−0.128833 + 0.991666i $$0.541123\pi$$
$$858$$ 0 0
$$859$$ −7.17513e39 −1.76572 −0.882861 0.469635i $$-0.844386\pi$$
−0.882861 + 0.469635i $$0.844386\pi$$
$$860$$ 0 0
$$861$$ −2.74995e39 −0.655817
$$862$$ 0 0
$$863$$ 4.96688e39 1.14799 0.573994 0.818859i $$-0.305393\pi$$
0.573994 + 0.818859i $$0.305393\pi$$
$$864$$ 0 0
$$865$$ 6.09682e39 1.36580
$$866$$ 0 0
$$867$$ 2.35081e39 0.510458
$$868$$ 0 0
$$869$$ 1.51444e39 0.318776
$$870$$ 0 0
$$871$$ 6.04196e38 0.123291
$$872$$ 0 0
$$873$$ −1.55405e39 −0.307449
$$874$$ 0 0
$$875$$ 4.62633e39 0.887417
$$876$$ 0 0
$$877$$ 8.89158e39 1.65381 0.826903 0.562344i $$-0.190100\pi$$
0.826903 + 0.562344i $$0.190100\pi$$
$$878$$ 0 0
$$879$$ −5.80159e39 −1.04640
$$880$$ 0 0
$$881$$ 8.48498e39 1.48414 0.742072 0.670320i $$-0.233843\pi$$
0.742072 + 0.670320i $$0.233843\pi$$
$$882$$ 0 0
$$883$$ −8.65208e39 −1.46775 −0.733873 0.679287i $$-0.762289\pi$$
−0.733873 + 0.679287i $$0.762289\pi$$
$$884$$ 0 0
$$885$$ −3.82793e39 −0.629839
$$886$$ 0 0
$$887$$ −8.11389e38 −0.129497 −0.0647486 0.997902i $$-0.520625\pi$$
−0.0647486 + 0.997902i $$0.520625\pi$$
$$888$$ 0 0
$$889$$ 5.54070e39 0.857809
$$890$$ 0 0
$$891$$ 9.77909e38 0.146875
$$892$$ 0 0
$$893$$ 3.64821e38 0.0531601
$$894$$ 0 0
$$895$$ −8.37931e39 −1.18467
$$896$$ 0 0
$$897$$ −6.05923e38 −0.0831227
$$898$$ 0 0
$$899$$ 8.62907e39 1.14870
$$900$$ 0 0
$$901$$ −4.07846e39 −0.526880
$$902$$ 0 0
$$903$$ 3.78668e39 0.474760
$$904$$ 0 0
$$905$$ 1.20587e40 1.46738
$$906$$ 0 0
$$907$$ −4.26511e39 −0.503769 −0.251884 0.967757i $$-0.581050\pi$$
−0.251884 + 0.967757i $$0.581050\pi$$
$$908$$ 0 0
$$909$$ 8.99574e38 0.103139
$$910$$ 0 0
$$911$$ 1.06535e40 1.18576 0.592878 0.805292i $$-0.297991\pi$$
0.592878 + 0.805292i $$0.297991\pi$$
$$912$$ 0 0
$$913$$ −1.41296e40 −1.52678
$$914$$ 0 0
$$915$$ 2.63078e39 0.275994
$$916$$ 0 0
$$917$$ −1.04622e40 −1.06570
$$918$$ 0 0
$$919$$ −1.28657e40 −1.27254 −0.636271 0.771465i $$-0.719524\pi$$
−0.636271 + 0.771465i $$0.719524\pi$$
$$920$$ 0 0
$$921$$ 5.51952e39 0.530147
$$922$$ 0 0
$$923$$ −3.42579e38 −0.0319550
$$924$$ 0 0
$$925$$ 1.71748e39 0.155589
$$926$$ 0 0
$$927$$ −7.28658e39 −0.641132
$$928$$ 0 0
$$929$$ 3.77009e39 0.322211 0.161105 0.986937i $$-0.448494\pi$$
0.161105 + 0.986937i $$0.448494\pi$$
$$930$$ 0 0
$$931$$ −3.02850e38 −0.0251424
$$932$$ 0 0
$$933$$ −1.16681e40 −0.941020
$$934$$ 0 0
$$935$$ −6.20457e39 −0.486133
$$936$$ 0 0
$$937$$ 2.50990e40 1.91061 0.955307 0.295617i $$-0.0955251\pi$$
0.955307 + 0.295617i $$0.0955251\pi$$
$$938$$ 0 0
$$939$$ −9.59578e38 −0.0709733
$$940$$ 0 0
$$941$$ 1.48642e39 0.106827 0.0534136 0.998572i $$-0.482990\pi$$
0.0534136 + 0.998572i $$0.482990\pi$$
$$942$$ 0 0
$$943$$ 2.79335e40 1.95082
$$944$$ 0 0
$$945$$ 3.02203e39 0.205102
$$946$$ 0 0
$$947$$ 1.95808e40 1.29153 0.645766 0.763536i $$-0.276538\pi$$
0.645766 + 0.763536i $$0.276538\pi$$
$$948$$ 0 0
$$949$$ 1.17878e38 0.00755679
$$950$$ 0 0
$$951$$ 8.23575e39 0.513176
$$952$$ 0 0
$$953$$ 2.88838e40 1.74945 0.874723 0.484623i $$-0.161043\pi$$
0.874723 + 0.484623i $$0.161043\pi$$
$$954$$ 0 0
$$955$$ −6.44996e39 −0.379762
$$956$$ 0 0
$$957$$ −1.04325e40 −0.597143
$$958$$ 0 0
$$959$$ 3.32761e40 1.85174
$$960$$ 0 0
$$961$$ 2.13545e40 1.15537
$$962$$ 0 0
$$963$$ 9.58992e39 0.504499
$$964$$ 0 0
$$965$$ −2.94238e40 −1.50515
$$966$$ 0 0
$$967$$ 2.69517e40 1.34069 0.670346 0.742049i $$-0.266146\pi$$
0.670346 + 0.742049i $$0.266146\pi$$
$$968$$ 0 0
$$969$$ −3.78252e39 −0.182983
$$970$$ 0 0
$$971$$ −5.85142e39 −0.275297 −0.137649 0.990481i $$-0.543954\pi$$
−0.137649 + 0.990481i $$0.543954\pi$$
$$972$$ 0 0
$$973$$ −3.31286e40 −1.51593
$$974$$ 0 0
$$975$$ −1.84463e38 −0.00821006
$$976$$ 0 0
$$977$$ 6.94583e39 0.300710 0.150355 0.988632i $$-0.451958\pi$$
0.150355 + 0.988632i $$0.451958\pi$$
$$978$$ 0 0
$$979$$ 4.58394e39 0.193051
$$980$$ 0 0
$$981$$ 3.04094e39 0.124588
$$982$$ 0 0
$$983$$ −1.80358e40 −0.718892 −0.359446 0.933166i $$-0.617034\pi$$
−0.359446 + 0.933166i $$0.617034\pi$$
$$984$$ 0 0
$$985$$ −8.54233e39 −0.331274
$$986$$ 0 0
$$987$$ −8.61705e38 −0.0325145
$$988$$ 0 0
$$989$$ −3.84644e40 −1.41224
$$990$$ 0 0
$$991$$ 4.92955e40 1.76122 0.880608 0.473846i $$-0.157135\pi$$
0.880608 + 0.473846i $$0.157135\pi$$
$$992$$ 0 0
$$993$$ −2.77258e40 −0.963983
$$994$$ 0 0
$$995$$ −2.00871e40 −0.679681
$$996$$ 0 0
$$997$$ 8.75195e39 0.288217 0.144109 0.989562i $$-0.453968\pi$$
0.144109 + 0.989562i $$0.453968\pi$$
$$998$$ 0 0
$$999$$ −5.58307e39 −0.178953
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 48.28.a.l.1.1 4
4.3 odd 2 24.28.a.d.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
24.28.a.d.1.1 4 4.3 odd 2
48.28.a.l.1.1 4 1.1 even 1 trivial