# Properties

 Label 48.22.a.l.1.2 Level $48$ Weight $22$ Character 48.1 Self dual yes Analytic conductor $134.149$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,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, 22, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ 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: $$134.149125258$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2295485x - 828958533$$ x^3 - x^2 - 2295485*x - 828958533 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{21}\cdot 3^{4}\cdot 5\cdot 7$$ Twist minimal: no (minimal twist has level 24) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1283.97$$ 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)

 $$f(q)$$ $$=$$ $$q+59049.0 q^{3} -8.90852e6 q^{5} -5.87822e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$$ $$q+59049.0 q^{3} -8.90852e6 q^{5} -5.87822e8 q^{7} +3.48678e9 q^{9} +1.42332e11 q^{11} +1.49035e11 q^{13} -5.26039e11 q^{15} +9.95199e12 q^{17} +5.02926e12 q^{19} -3.47103e13 q^{21} -3.25343e14 q^{23} -3.97475e14 q^{25} +2.05891e14 q^{27} +1.07038e15 q^{29} -3.98818e15 q^{31} +8.40458e15 q^{33} +5.23663e15 q^{35} -3.35659e16 q^{37} +8.80039e15 q^{39} +4.54703e16 q^{41} +8.38924e16 q^{43} -3.10621e16 q^{45} +5.34085e16 q^{47} -2.13011e17 q^{49} +5.87655e17 q^{51} +1.14205e18 q^{53} -1.26797e18 q^{55} +2.96973e17 q^{57} +5.97497e18 q^{59} +4.12289e17 q^{61} -2.04961e18 q^{63} -1.32769e18 q^{65} +2.42251e19 q^{67} -1.92112e19 q^{69} -7.25973e16 q^{71} -2.97771e19 q^{73} -2.34705e19 q^{75} -8.36661e19 q^{77} +5.78794e19 q^{79} +1.21577e19 q^{81} +2.48286e20 q^{83} -8.86575e19 q^{85} +6.32049e19 q^{87} -5.56792e19 q^{89} -8.76063e19 q^{91} -2.35498e20 q^{93} -4.48032e19 q^{95} -8.39559e20 q^{97} +4.96282e20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9}+O(q^{10})$$ 3 * q + 177147 * q^3 + 2080026 * q^5 + 1205282064 * q^7 + 10460353203 * q^9 $$3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9} - 13839247500 q^{11} + 718855551690 q^{13} + 122823455274 q^{15} + 2135189843046 q^{17} + 40122324686988 q^{19} + 71170700597136 q^{21} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots - 48\!\cdots\!00 q^{99}+O(q^{100})$$ 3 * q + 177147 * q^3 + 2080026 * q^5 + 1205282064 * q^7 + 10460353203 * q^9 - 13839247500 * q^11 + 718855551690 * q^13 + 122823455274 * q^15 + 2135189843046 * q^17 + 40122324686988 * q^19 + 71170700597136 * q^21 - 278424417682632 * q^23 + 1348043260553901 * q^25 + 617673396283947 * q^27 + 442708167991794 * q^29 - 8016070162990152 * q^31 - 817193725627500 * q^33 - 125384157242400 * q^35 + 27729341388737058 * q^37 + 42447701471742810 * q^39 - 125648125186340562 * q^41 + 229052541499074612 * q^43 + 7252602210474426 * q^45 + 448613782068047712 * q^47 + 365221903446092427 * q^49 + 126080825042023254 * q^51 + 1406206217208267066 * q^53 + 3437829264920292504 * q^55 + 2369183150441954412 * q^57 + 1844638981471622100 * q^59 - 3294066300350351382 * q^61 + 4202558699560283664 * q^63 - 19537666262756991444 * q^65 + 33491023693155020652 * q^67 - 16440683439741736968 * q^69 + 79431018431598881160 * q^71 - 46612822906958319618 * q^73 + 79600606492447300149 * q^75 - 255759806110876888128 * q^77 + 197919973704661098024 * q^79 + 36472996377170786403 * q^81 - 111848528551886940276 * q^83 - 531596971435705186956 * q^85 + 26141474611747443906 * q^87 - 731008175840243483058 * q^89 + 546304658547739915488 * q^91 - 473340927054405485448 * q^93 + 1952599110289053971304 * q^95 - 1594521920055126193722 * q^97 - 48254472304578247500 * 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
$$3$$ 59049.0 0.577350
$$4$$ 0 0
$$5$$ −8.90852e6 −0.407963 −0.203981 0.978975i $$-0.565388\pi$$
−0.203981 + 0.978975i $$0.565388\pi$$
$$6$$ 0 0
$$7$$ −5.87822e8 −0.786532 −0.393266 0.919425i $$-0.628655\pi$$
−0.393266 + 0.919425i $$0.628655\pi$$
$$8$$ 0 0
$$9$$ 3.48678e9 0.333333
$$10$$ 0 0
$$11$$ 1.42332e11 1.65455 0.827276 0.561796i $$-0.189889\pi$$
0.827276 + 0.561796i $$0.189889\pi$$
$$12$$ 0 0
$$13$$ 1.49035e11 0.299836 0.149918 0.988698i $$-0.452099\pi$$
0.149918 + 0.988698i $$0.452099\pi$$
$$14$$ 0 0
$$15$$ −5.26039e11 −0.235537
$$16$$ 0 0
$$17$$ 9.95199e12 1.19728 0.598641 0.801018i $$-0.295708\pi$$
0.598641 + 0.801018i $$0.295708\pi$$
$$18$$ 0 0
$$19$$ 5.02926e12 0.188188 0.0940938 0.995563i $$-0.470005\pi$$
0.0940938 + 0.995563i $$0.470005\pi$$
$$20$$ 0 0
$$21$$ −3.47103e13 −0.454105
$$22$$ 0 0
$$23$$ −3.25343e14 −1.63757 −0.818783 0.574103i $$-0.805351\pi$$
−0.818783 + 0.574103i $$0.805351\pi$$
$$24$$ 0 0
$$25$$ −3.97475e14 −0.833566
$$26$$ 0 0
$$27$$ 2.05891e14 0.192450
$$28$$ 0 0
$$29$$ 1.07038e15 0.472454 0.236227 0.971698i $$-0.424089\pi$$
0.236227 + 0.971698i $$0.424089\pi$$
$$30$$ 0 0
$$31$$ −3.98818e15 −0.873932 −0.436966 0.899478i $$-0.643947\pi$$
−0.436966 + 0.899478i $$0.643947\pi$$
$$32$$ 0 0
$$33$$ 8.40458e15 0.955256
$$34$$ 0 0
$$35$$ 5.23663e15 0.320876
$$36$$ 0 0
$$37$$ −3.35659e16 −1.14757 −0.573787 0.819004i $$-0.694526\pi$$
−0.573787 + 0.819004i $$0.694526\pi$$
$$38$$ 0 0
$$39$$ 8.80039e15 0.173111
$$40$$ 0 0
$$41$$ 4.54703e16 0.529051 0.264526 0.964379i $$-0.414785\pi$$
0.264526 + 0.964379i $$0.414785\pi$$
$$42$$ 0 0
$$43$$ 8.38924e16 0.591976 0.295988 0.955192i $$-0.404351\pi$$
0.295988 + 0.955192i $$0.404351\pi$$
$$44$$ 0 0
$$45$$ −3.10621e16 −0.135988
$$46$$ 0 0
$$47$$ 5.34085e16 0.148110 0.0740548 0.997254i $$-0.476406\pi$$
0.0740548 + 0.997254i $$0.476406\pi$$
$$48$$ 0 0
$$49$$ −2.13011e17 −0.381367
$$50$$ 0 0
$$51$$ 5.87655e17 0.691251
$$52$$ 0 0
$$53$$ 1.14205e18 0.896988 0.448494 0.893786i $$-0.351961\pi$$
0.448494 + 0.893786i $$0.351961\pi$$
$$54$$ 0 0
$$55$$ −1.26797e18 −0.674996
$$56$$ 0 0
$$57$$ 2.96973e17 0.108650
$$58$$ 0 0
$$59$$ 5.97497e18 1.52191 0.760956 0.648803i $$-0.224730\pi$$
0.760956 + 0.648803i $$0.224730\pi$$
$$60$$ 0 0
$$61$$ 4.12289e17 0.0740011 0.0370006 0.999315i $$-0.488220\pi$$
0.0370006 + 0.999315i $$0.488220\pi$$
$$62$$ 0 0
$$63$$ −2.04961e18 −0.262177
$$64$$ 0 0
$$65$$ −1.32769e18 −0.122322
$$66$$ 0 0
$$67$$ 2.42251e19 1.62360 0.811802 0.583933i $$-0.198487\pi$$
0.811802 + 0.583933i $$0.198487\pi$$
$$68$$ 0 0
$$69$$ −1.92112e19 −0.945449
$$70$$ 0 0
$$71$$ −7.25973e16 −0.00264672 −0.00132336 0.999999i $$-0.500421\pi$$
−0.00132336 + 0.999999i $$0.500421\pi$$
$$72$$ 0 0
$$73$$ −2.97771e19 −0.810947 −0.405473 0.914107i $$-0.632893\pi$$
−0.405473 + 0.914107i $$0.632893\pi$$
$$74$$ 0 0
$$75$$ −2.34705e19 −0.481260
$$76$$ 0 0
$$77$$ −8.36661e19 −1.30136
$$78$$ 0 0
$$79$$ 5.78794e19 0.687764 0.343882 0.939013i $$-0.388258\pi$$
0.343882 + 0.939013i $$0.388258\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ 0 0
$$83$$ 2.48286e20 1.75643 0.878217 0.478262i $$-0.158733\pi$$
0.878217 + 0.478262i $$0.158733\pi$$
$$84$$ 0 0
$$85$$ −8.86575e19 −0.488446
$$86$$ 0 0
$$87$$ 6.32049e19 0.272771
$$88$$ 0 0
$$89$$ −5.56792e19 −0.189277 −0.0946385 0.995512i $$-0.530170\pi$$
−0.0946385 + 0.995512i $$0.530170\pi$$
$$90$$ 0 0
$$91$$ −8.76063e19 −0.235831
$$92$$ 0 0
$$93$$ −2.35498e20 −0.504565
$$94$$ 0 0
$$95$$ −4.48032e19 −0.0767735
$$96$$ 0 0
$$97$$ −8.39559e20 −1.15597 −0.577987 0.816046i $$-0.696161\pi$$
−0.577987 + 0.816046i $$0.696161\pi$$
$$98$$ 0 0
$$99$$ 4.96282e20 0.551517
$$100$$ 0 0
$$101$$ −7.25262e20 −0.653312 −0.326656 0.945143i $$-0.605922\pi$$
−0.326656 + 0.945143i $$0.605922\pi$$
$$102$$ 0 0
$$103$$ 1.79494e21 1.31601 0.658004 0.753015i $$-0.271401\pi$$
0.658004 + 0.753015i $$0.271401\pi$$
$$104$$ 0 0
$$105$$ 3.09217e20 0.185258
$$106$$ 0 0
$$107$$ 1.59231e20 0.0782522 0.0391261 0.999234i $$-0.487543\pi$$
0.0391261 + 0.999234i $$0.487543\pi$$
$$108$$ 0 0
$$109$$ 2.97325e21 1.20297 0.601484 0.798885i $$-0.294577\pi$$
0.601484 + 0.798885i $$0.294577\pi$$
$$110$$ 0 0
$$111$$ −1.98204e21 −0.662552
$$112$$ 0 0
$$113$$ 1.38076e21 0.382644 0.191322 0.981527i $$-0.438723\pi$$
0.191322 + 0.981527i $$0.438723\pi$$
$$114$$ 0 0
$$115$$ 2.89832e21 0.668066
$$116$$ 0 0
$$117$$ 5.19654e20 0.0999455
$$118$$ 0 0
$$119$$ −5.85000e21 −0.941700
$$120$$ 0 0
$$121$$ 1.28582e22 1.73754
$$122$$ 0 0
$$123$$ 2.68498e21 0.305448
$$124$$ 0 0
$$125$$ 7.78883e21 0.748027
$$126$$ 0 0
$$127$$ −4.56871e21 −0.371411 −0.185706 0.982605i $$-0.559457\pi$$
−0.185706 + 0.982605i $$0.559457\pi$$
$$128$$ 0 0
$$129$$ 4.95376e21 0.341777
$$130$$ 0 0
$$131$$ 8.64013e21 0.507191 0.253595 0.967310i $$-0.418387\pi$$
0.253595 + 0.967310i $$0.418387\pi$$
$$132$$ 0 0
$$133$$ −2.95631e21 −0.148016
$$134$$ 0 0
$$135$$ −1.83419e21 −0.0785125
$$136$$ 0 0
$$137$$ 2.68277e22 0.984052 0.492026 0.870581i $$-0.336257\pi$$
0.492026 + 0.870581i $$0.336257\pi$$
$$138$$ 0 0
$$139$$ −3.90524e22 −1.23025 −0.615123 0.788431i $$-0.710894\pi$$
−0.615123 + 0.788431i $$0.710894\pi$$
$$140$$ 0 0
$$141$$ 3.15372e21 0.0855111
$$142$$ 0 0
$$143$$ 2.12126e22 0.496095
$$144$$ 0 0
$$145$$ −9.53551e21 −0.192744
$$146$$ 0 0
$$147$$ −1.25781e22 −0.220182
$$148$$ 0 0
$$149$$ 3.86793e22 0.587521 0.293760 0.955879i $$-0.405093\pi$$
0.293760 + 0.955879i $$0.405093\pi$$
$$150$$ 0 0
$$151$$ 1.07575e23 1.42054 0.710269 0.703930i $$-0.248573\pi$$
0.710269 + 0.703930i $$0.248573\pi$$
$$152$$ 0 0
$$153$$ 3.47004e22 0.399094
$$154$$ 0 0
$$155$$ 3.55288e22 0.356532
$$156$$ 0 0
$$157$$ 6.91463e22 0.606489 0.303244 0.952913i $$-0.401930\pi$$
0.303244 + 0.952913i $$0.401930\pi$$
$$158$$ 0 0
$$159$$ 6.74366e22 0.517876
$$160$$ 0 0
$$161$$ 1.91244e23 1.28800
$$162$$ 0 0
$$163$$ 2.70783e23 1.60196 0.800980 0.598691i $$-0.204312\pi$$
0.800980 + 0.598691i $$0.204312\pi$$
$$164$$ 0 0
$$165$$ −7.48724e22 −0.389709
$$166$$ 0 0
$$167$$ 2.30715e23 1.05816 0.529081 0.848571i $$-0.322537\pi$$
0.529081 + 0.848571i $$0.322537\pi$$
$$168$$ 0 0
$$169$$ −2.24853e23 −0.910098
$$170$$ 0 0
$$171$$ 1.75359e22 0.0627292
$$172$$ 0 0
$$173$$ −3.68393e23 −1.16635 −0.583173 0.812348i $$-0.698189\pi$$
−0.583173 + 0.812348i $$0.698189\pi$$
$$174$$ 0 0
$$175$$ 2.33645e23 0.655627
$$176$$ 0 0
$$177$$ 3.52816e23 0.878677
$$178$$ 0 0
$$179$$ 6.78508e23 1.50175 0.750876 0.660443i $$-0.229631\pi$$
0.750876 + 0.660443i $$0.229631\pi$$
$$180$$ 0 0
$$181$$ 9.41228e23 1.85383 0.926915 0.375272i $$-0.122451\pi$$
0.926915 + 0.375272i $$0.122451\pi$$
$$182$$ 0 0
$$183$$ 2.43452e22 0.0427246
$$184$$ 0 0
$$185$$ 2.99023e23 0.468168
$$186$$ 0 0
$$187$$ 1.41649e24 1.98096
$$188$$ 0 0
$$189$$ −1.21027e23 −0.151368
$$190$$ 0 0
$$191$$ −1.00355e24 −1.12380 −0.561899 0.827206i $$-0.689929\pi$$
−0.561899 + 0.827206i $$0.689929\pi$$
$$192$$ 0 0
$$193$$ −1.40841e24 −1.41377 −0.706885 0.707328i $$-0.749900\pi$$
−0.706885 + 0.707328i $$0.749900\pi$$
$$194$$ 0 0
$$195$$ −7.83985e22 −0.0706227
$$196$$ 0 0
$$197$$ 1.96986e24 1.59419 0.797096 0.603853i $$-0.206369\pi$$
0.797096 + 0.603853i $$0.206369\pi$$
$$198$$ 0 0
$$199$$ −1.33492e24 −0.971620 −0.485810 0.874064i $$-0.661475\pi$$
−0.485810 + 0.874064i $$0.661475\pi$$
$$200$$ 0 0
$$201$$ 1.43047e24 0.937388
$$202$$ 0 0
$$203$$ −6.29193e23 −0.371600
$$204$$ 0 0
$$205$$ −4.05073e23 −0.215833
$$206$$ 0 0
$$207$$ −1.13440e24 −0.545855
$$208$$ 0 0
$$209$$ 7.15826e23 0.311366
$$210$$ 0 0
$$211$$ 1.21610e24 0.478633 0.239316 0.970942i $$-0.423077\pi$$
0.239316 + 0.970942i $$0.423077\pi$$
$$212$$ 0 0
$$213$$ −4.28680e21 −0.00152808
$$214$$ 0 0
$$215$$ −7.47357e23 −0.241504
$$216$$ 0 0
$$217$$ 2.34434e24 0.687375
$$218$$ 0 0
$$219$$ −1.75831e24 −0.468200
$$220$$ 0 0
$$221$$ 1.48320e24 0.358989
$$222$$ 0 0
$$223$$ 5.72126e24 1.25977 0.629884 0.776689i $$-0.283102\pi$$
0.629884 + 0.776689i $$0.283102\pi$$
$$224$$ 0 0
$$225$$ −1.38591e24 −0.277855
$$226$$ 0 0
$$227$$ −5.60839e24 −1.02463 −0.512314 0.858798i $$-0.671212\pi$$
−0.512314 + 0.858798i $$0.671212\pi$$
$$228$$ 0 0
$$229$$ −2.51872e24 −0.419670 −0.209835 0.977737i $$-0.567293\pi$$
−0.209835 + 0.977737i $$0.567293\pi$$
$$230$$ 0 0
$$231$$ −4.94040e24 −0.751340
$$232$$ 0 0
$$233$$ 1.98939e24 0.276365 0.138182 0.990407i $$-0.455874\pi$$
0.138182 + 0.990407i $$0.455874\pi$$
$$234$$ 0 0
$$235$$ −4.75791e23 −0.0604232
$$236$$ 0 0
$$237$$ 3.41772e24 0.397081
$$238$$ 0 0
$$239$$ −7.32699e24 −0.779378 −0.389689 0.920947i $$-0.627417\pi$$
−0.389689 + 0.920947i $$0.627417\pi$$
$$240$$ 0 0
$$241$$ −1.88926e25 −1.84125 −0.920625 0.390448i $$-0.872320\pi$$
−0.920625 + 0.390448i $$0.872320\pi$$
$$242$$ 0 0
$$243$$ 7.17898e23 0.0641500
$$244$$ 0 0
$$245$$ 1.89761e24 0.155584
$$246$$ 0 0
$$247$$ 7.49537e23 0.0564255
$$248$$ 0 0
$$249$$ 1.46610e25 1.01408
$$250$$ 0 0
$$251$$ 1.65459e25 1.05224 0.526122 0.850409i $$-0.323645\pi$$
0.526122 + 0.850409i $$0.323645\pi$$
$$252$$ 0 0
$$253$$ −4.63068e25 −2.70944
$$254$$ 0 0
$$255$$ −5.23514e24 −0.282005
$$256$$ 0 0
$$257$$ −2.93804e25 −1.45801 −0.729004 0.684509i $$-0.760017\pi$$
−0.729004 + 0.684509i $$0.760017\pi$$
$$258$$ 0 0
$$259$$ 1.97308e25 0.902604
$$260$$ 0 0
$$261$$ 3.73219e24 0.157485
$$262$$ 0 0
$$263$$ −3.29044e25 −1.28150 −0.640749 0.767750i $$-0.721376\pi$$
−0.640749 + 0.767750i $$0.721376\pi$$
$$264$$ 0 0
$$265$$ −1.01739e25 −0.365938
$$266$$ 0 0
$$267$$ −3.28780e24 −0.109279
$$268$$ 0 0
$$269$$ −4.30191e24 −0.132209 −0.0661047 0.997813i $$-0.521057\pi$$
−0.0661047 + 0.997813i $$0.521057\pi$$
$$270$$ 0 0
$$271$$ 2.72610e25 0.775110 0.387555 0.921846i $$-0.373320\pi$$
0.387555 + 0.921846i $$0.373320\pi$$
$$272$$ 0 0
$$273$$ −5.17307e24 −0.136157
$$274$$ 0 0
$$275$$ −5.65736e25 −1.37918
$$276$$ 0 0
$$277$$ −4.61030e25 −1.04158 −0.520788 0.853686i $$-0.674362\pi$$
−0.520788 + 0.853686i $$0.674362\pi$$
$$278$$ 0 0
$$279$$ −1.39059e25 −0.291311
$$280$$ 0 0
$$281$$ −7.74125e24 −0.150451 −0.0752254 0.997167i $$-0.523968\pi$$
−0.0752254 + 0.997167i $$0.523968\pi$$
$$282$$ 0 0
$$283$$ 1.41856e25 0.255911 0.127955 0.991780i $$-0.459159\pi$$
0.127955 + 0.991780i $$0.459159\pi$$
$$284$$ 0 0
$$285$$ −2.64559e24 −0.0443252
$$286$$ 0 0
$$287$$ −2.67285e25 −0.416116
$$288$$ 0 0
$$289$$ 2.99502e25 0.433483
$$290$$ 0 0
$$291$$ −4.95751e25 −0.667401
$$292$$ 0 0
$$293$$ 8.61121e25 1.07883 0.539416 0.842039i $$-0.318645\pi$$
0.539416 + 0.842039i $$0.318645\pi$$
$$294$$ 0 0
$$295$$ −5.32282e25 −0.620884
$$296$$ 0 0
$$297$$ 2.93050e25 0.318419
$$298$$ 0 0
$$299$$ −4.84876e25 −0.491002
$$300$$ 0 0
$$301$$ −4.93138e25 −0.465608
$$302$$ 0 0
$$303$$ −4.28260e25 −0.377190
$$304$$ 0 0
$$305$$ −3.67288e24 −0.0301897
$$306$$ 0 0
$$307$$ 9.91709e25 0.761082 0.380541 0.924764i $$-0.375738\pi$$
0.380541 + 0.924764i $$0.375738\pi$$
$$308$$ 0 0
$$309$$ 1.05989e26 0.759797
$$310$$ 0 0
$$311$$ 1.15943e26 0.776709 0.388355 0.921510i $$-0.373044\pi$$
0.388355 + 0.921510i $$0.373044\pi$$
$$312$$ 0 0
$$313$$ 2.06648e26 1.29424 0.647122 0.762386i $$-0.275972\pi$$
0.647122 + 0.762386i $$0.275972\pi$$
$$314$$ 0 0
$$315$$ 1.82590e25 0.106959
$$316$$ 0 0
$$317$$ 2.90072e26 1.58995 0.794976 0.606641i $$-0.207483\pi$$
0.794976 + 0.606641i $$0.207483\pi$$
$$318$$ 0 0
$$319$$ 1.52350e26 0.781699
$$320$$ 0 0
$$321$$ 9.40241e24 0.0451790
$$322$$ 0 0
$$323$$ 5.00511e25 0.225314
$$324$$ 0 0
$$325$$ −5.92379e25 −0.249934
$$326$$ 0 0
$$327$$ 1.75568e26 0.694533
$$328$$ 0 0
$$329$$ −3.13947e25 −0.116493
$$330$$ 0 0
$$331$$ −2.56275e26 −0.892303 −0.446151 0.894958i $$-0.647206\pi$$
−0.446151 + 0.894958i $$0.647206\pi$$
$$332$$ 0 0
$$333$$ −1.17037e26 −0.382525
$$334$$ 0 0
$$335$$ −2.15810e26 −0.662370
$$336$$ 0 0
$$337$$ 1.42732e26 0.411535 0.205768 0.978601i $$-0.434031\pi$$
0.205768 + 0.978601i $$0.434031\pi$$
$$338$$ 0 0
$$339$$ 8.15325e25 0.220919
$$340$$ 0 0
$$341$$ −5.67648e26 −1.44597
$$342$$ 0 0
$$343$$ 4.53538e26 1.08649
$$344$$ 0 0
$$345$$ 1.71143e26 0.385708
$$346$$ 0 0
$$347$$ 4.74021e26 1.00540 0.502699 0.864462i $$-0.332341\pi$$
0.502699 + 0.864462i $$0.332341\pi$$
$$348$$ 0 0
$$349$$ 8.88859e26 1.77487 0.887434 0.460935i $$-0.152486\pi$$
0.887434 + 0.460935i $$0.152486\pi$$
$$350$$ 0 0
$$351$$ 3.06851e25 0.0577035
$$352$$ 0 0
$$353$$ −4.86433e26 −0.861764 −0.430882 0.902408i $$-0.641798\pi$$
−0.430882 + 0.902408i $$0.641798\pi$$
$$354$$ 0 0
$$355$$ 6.46734e23 0.00107976
$$356$$ 0 0
$$357$$ −3.45437e26 −0.543691
$$358$$ 0 0
$$359$$ 4.36541e26 0.647938 0.323969 0.946068i $$-0.394983\pi$$
0.323969 + 0.946068i $$0.394983\pi$$
$$360$$ 0 0
$$361$$ −6.88916e26 −0.964585
$$362$$ 0 0
$$363$$ 7.59267e26 1.00317
$$364$$ 0 0
$$365$$ 2.65270e26 0.330836
$$366$$ 0 0
$$367$$ 1.13289e27 1.33412 0.667060 0.745004i $$-0.267553\pi$$
0.667060 + 0.745004i $$0.267553\pi$$
$$368$$ 0 0
$$369$$ 1.58545e26 0.176350
$$370$$ 0 0
$$371$$ −6.71319e26 −0.705510
$$372$$ 0 0
$$373$$ 6.51576e26 0.647176 0.323588 0.946198i $$-0.395111\pi$$
0.323588 + 0.946198i $$0.395111\pi$$
$$374$$ 0 0
$$375$$ 4.59923e26 0.431874
$$376$$ 0 0
$$377$$ 1.59525e26 0.141659
$$378$$ 0 0
$$379$$ −1.71451e27 −1.44022 −0.720109 0.693861i $$-0.755908\pi$$
−0.720109 + 0.693861i $$0.755908\pi$$
$$380$$ 0 0
$$381$$ −2.69778e26 −0.214434
$$382$$ 0 0
$$383$$ 1.63419e27 1.22946 0.614731 0.788737i $$-0.289265\pi$$
0.614731 + 0.788737i $$0.289265\pi$$
$$384$$ 0 0
$$385$$ 7.45341e26 0.530906
$$386$$ 0 0
$$387$$ 2.92515e26 0.197325
$$388$$ 0 0
$$389$$ −2.64030e27 −1.68726 −0.843632 0.536922i $$-0.819587\pi$$
−0.843632 + 0.536922i $$0.819587\pi$$
$$390$$ 0 0
$$391$$ −3.23781e27 −1.96063
$$392$$ 0 0
$$393$$ 5.10191e26 0.292827
$$394$$ 0 0
$$395$$ −5.15620e26 −0.280582
$$396$$ 0 0
$$397$$ 2.94709e27 1.52087 0.760437 0.649411i $$-0.224985\pi$$
0.760437 + 0.649411i $$0.224985\pi$$
$$398$$ 0 0
$$399$$ −1.74567e26 −0.0854568
$$400$$ 0 0
$$401$$ 1.81255e27 0.841927 0.420963 0.907078i $$-0.361692\pi$$
0.420963 + 0.907078i $$0.361692\pi$$
$$402$$ 0 0
$$403$$ −5.94381e26 −0.262037
$$404$$ 0 0
$$405$$ −1.08307e26 −0.0453292
$$406$$ 0 0
$$407$$ −4.77752e27 −1.89872
$$408$$ 0 0
$$409$$ 2.59753e27 0.980542 0.490271 0.871570i $$-0.336898\pi$$
0.490271 + 0.871570i $$0.336898\pi$$
$$410$$ 0 0
$$411$$ 1.58415e27 0.568143
$$412$$ 0 0
$$413$$ −3.51222e27 −1.19703
$$414$$ 0 0
$$415$$ −2.21186e27 −0.716560
$$416$$ 0 0
$$417$$ −2.30600e27 −0.710283
$$418$$ 0 0
$$419$$ 3.40142e26 0.0996353 0.0498176 0.998758i $$-0.484136\pi$$
0.0498176 + 0.998758i $$0.484136\pi$$
$$420$$ 0 0
$$421$$ −4.22891e27 −1.17833 −0.589164 0.808014i $$-0.700543\pi$$
−0.589164 + 0.808014i $$0.700543\pi$$
$$422$$ 0 0
$$423$$ 1.86224e26 0.0493698
$$424$$ 0 0
$$425$$ −3.95567e27 −0.998014
$$426$$ 0 0
$$427$$ −2.42353e26 −0.0582043
$$428$$ 0 0
$$429$$ 1.25258e27 0.286421
$$430$$ 0 0
$$431$$ −7.72212e27 −1.68161 −0.840805 0.541339i $$-0.817918\pi$$
−0.840805 + 0.541339i $$0.817918\pi$$
$$432$$ 0 0
$$433$$ 3.14232e27 0.651819 0.325910 0.945401i $$-0.394330\pi$$
0.325910 + 0.945401i $$0.394330\pi$$
$$434$$ 0 0
$$435$$ −5.63062e26 −0.111281
$$436$$ 0 0
$$437$$ −1.63623e27 −0.308170
$$438$$ 0 0
$$439$$ 3.34873e27 0.601177 0.300588 0.953754i $$-0.402817\pi$$
0.300588 + 0.953754i $$0.402817\pi$$
$$440$$ 0 0
$$441$$ −7.42724e26 −0.127122
$$442$$ 0 0
$$443$$ −5.82708e27 −0.951068 −0.475534 0.879697i $$-0.657745\pi$$
−0.475534 + 0.879697i $$0.657745\pi$$
$$444$$ 0 0
$$445$$ 4.96019e26 0.0772179
$$446$$ 0 0
$$447$$ 2.28398e27 0.339205
$$448$$ 0 0
$$449$$ −1.14216e28 −1.61860 −0.809300 0.587396i $$-0.800153\pi$$
−0.809300 + 0.587396i $$0.800153\pi$$
$$450$$ 0 0
$$451$$ 6.47190e27 0.875343
$$452$$ 0 0
$$453$$ 6.35219e27 0.820148
$$454$$ 0 0
$$455$$ 7.80443e26 0.0962103
$$456$$ 0 0
$$457$$ 1.70411e27 0.200622 0.100311 0.994956i $$-0.468016\pi$$
0.100311 + 0.994956i $$0.468016\pi$$
$$458$$ 0 0
$$459$$ 2.04903e27 0.230417
$$460$$ 0 0
$$461$$ −1.57190e28 −1.68875 −0.844374 0.535753i $$-0.820028\pi$$
−0.844374 + 0.535753i $$0.820028\pi$$
$$462$$ 0 0
$$463$$ 2.45032e27 0.251549 0.125774 0.992059i $$-0.459858\pi$$
0.125774 + 0.992059i $$0.459858\pi$$
$$464$$ 0 0
$$465$$ 2.09794e27 0.205844
$$466$$ 0 0
$$467$$ −4.79077e26 −0.0449344 −0.0224672 0.999748i $$-0.507152\pi$$
−0.0224672 + 0.999748i $$0.507152\pi$$
$$468$$ 0 0
$$469$$ −1.42400e28 −1.27702
$$470$$ 0 0
$$471$$ 4.08302e27 0.350156
$$472$$ 0 0
$$473$$ 1.19406e28 0.979455
$$474$$ 0 0
$$475$$ −1.99901e27 −0.156867
$$476$$ 0 0
$$477$$ 3.98207e27 0.298996
$$478$$ 0 0
$$479$$ 5.29156e27 0.380243 0.190121 0.981761i $$-0.439112\pi$$
0.190121 + 0.981761i $$0.439112\pi$$
$$480$$ 0 0
$$481$$ −5.00252e27 −0.344085
$$482$$ 0 0
$$483$$ 1.12927e28 0.743626
$$484$$ 0 0
$$485$$ 7.47922e27 0.471594
$$486$$ 0 0
$$487$$ 2.99405e28 1.80803 0.904013 0.427506i $$-0.140608\pi$$
0.904013 + 0.427506i $$0.140608\pi$$
$$488$$ 0 0
$$489$$ 1.59895e28 0.924893
$$490$$ 0 0
$$491$$ −1.72464e27 −0.0955749 −0.0477875 0.998858i $$-0.515217\pi$$
−0.0477875 + 0.998858i $$0.515217\pi$$
$$492$$ 0 0
$$493$$ 1.06524e28 0.565660
$$494$$ 0 0
$$495$$ −4.42114e27 −0.224999
$$496$$ 0 0
$$497$$ 4.26743e25 0.00208173
$$498$$ 0 0
$$499$$ 1.82237e28 0.852276 0.426138 0.904658i $$-0.359874\pi$$
0.426138 + 0.904658i $$0.359874\pi$$
$$500$$ 0 0
$$501$$ 1.36235e28 0.610930
$$502$$ 0 0
$$503$$ −3.98628e28 −1.71437 −0.857183 0.515011i $$-0.827788\pi$$
−0.857183 + 0.515011i $$0.827788\pi$$
$$504$$ 0 0
$$505$$ 6.46101e27 0.266527
$$506$$ 0 0
$$507$$ −1.32773e28 −0.525445
$$508$$ 0 0
$$509$$ −1.48470e28 −0.563771 −0.281886 0.959448i $$-0.590960\pi$$
−0.281886 + 0.959448i $$0.590960\pi$$
$$510$$ 0 0
$$511$$ 1.75036e28 0.637836
$$512$$ 0 0
$$513$$ 1.03548e27 0.0362167
$$514$$ 0 0
$$515$$ −1.59902e28 −0.536882
$$516$$ 0 0
$$517$$ 7.60176e27 0.245055
$$518$$ 0 0
$$519$$ −2.17532e28 −0.673390
$$520$$ 0 0
$$521$$ 4.40264e28 1.30893 0.654465 0.756092i $$-0.272894\pi$$
0.654465 + 0.756092i $$0.272894\pi$$
$$522$$ 0 0
$$523$$ 3.71320e28 1.06043 0.530214 0.847864i $$-0.322112\pi$$
0.530214 + 0.847864i $$0.322112\pi$$
$$524$$ 0 0
$$525$$ 1.37965e28 0.378526
$$526$$ 0 0
$$527$$ −3.96904e28 −1.04634
$$528$$ 0 0
$$529$$ 6.63763e28 1.68162
$$530$$ 0 0
$$531$$ 2.08334e28 0.507304
$$532$$ 0 0
$$533$$ 6.77669e27 0.158629
$$534$$ 0 0
$$535$$ −1.41851e27 −0.0319240
$$536$$ 0 0
$$537$$ 4.00652e28 0.867037
$$538$$ 0 0
$$539$$ −3.03184e28 −0.630992
$$540$$ 0 0
$$541$$ −4.12076e28 −0.824909 −0.412454 0.910978i $$-0.635328\pi$$
−0.412454 + 0.910978i $$0.635328\pi$$
$$542$$ 0 0
$$543$$ 5.55786e28 1.07031
$$544$$ 0 0
$$545$$ −2.64873e28 −0.490766
$$546$$ 0 0
$$547$$ 2.13981e28 0.381512 0.190756 0.981637i $$-0.438906\pi$$
0.190756 + 0.981637i $$0.438906\pi$$
$$548$$ 0 0
$$549$$ 1.43756e27 0.0246670
$$550$$ 0 0
$$551$$ 5.38322e27 0.0889099
$$552$$ 0 0
$$553$$ −3.40228e28 −0.540948
$$554$$ 0 0
$$555$$ 1.76570e28 0.270297
$$556$$ 0 0
$$557$$ −9.11122e28 −1.34307 −0.671533 0.740975i $$-0.734364\pi$$
−0.671533 + 0.740975i $$0.734364\pi$$
$$558$$ 0 0
$$559$$ 1.25029e28 0.177496
$$560$$ 0 0
$$561$$ 8.36423e28 1.14371
$$562$$ 0 0
$$563$$ −7.72184e28 −1.01714 −0.508572 0.861019i $$-0.669826\pi$$
−0.508572 + 0.861019i $$0.669826\pi$$
$$564$$ 0 0
$$565$$ −1.23005e28 −0.156104
$$566$$ 0 0
$$567$$ −7.14654e27 −0.0873925
$$568$$ 0 0
$$569$$ −8.18518e28 −0.964605 −0.482303 0.876005i $$-0.660199\pi$$
−0.482303 + 0.876005i $$0.660199\pi$$
$$570$$ 0 0
$$571$$ −3.83570e27 −0.0435678 −0.0217839 0.999763i $$-0.506935\pi$$
−0.0217839 + 0.999763i $$0.506935\pi$$
$$572$$ 0 0
$$573$$ −5.92586e28 −0.648825
$$574$$ 0 0
$$575$$ 1.29316e29 1.36502
$$576$$ 0 0
$$577$$ 1.15551e29 1.17606 0.588028 0.808841i $$-0.299905\pi$$
0.588028 + 0.808841i $$0.299905\pi$$
$$578$$ 0 0
$$579$$ −8.31655e28 −0.816241
$$580$$ 0 0
$$581$$ −1.45948e29 −1.38149
$$582$$ 0 0
$$583$$ 1.62550e29 1.48411
$$584$$ 0 0
$$585$$ −4.62935e27 −0.0407740
$$586$$ 0 0
$$587$$ −1.04413e29 −0.887271 −0.443636 0.896207i $$-0.646312\pi$$
−0.443636 + 0.896207i $$0.646312\pi$$
$$588$$ 0 0
$$589$$ −2.00576e28 −0.164463
$$590$$ 0 0
$$591$$ 1.16318e29 0.920407
$$592$$ 0 0
$$593$$ 1.30270e29 0.994879 0.497439 0.867499i $$-0.334274\pi$$
0.497439 + 0.867499i $$0.334274\pi$$
$$594$$ 0 0
$$595$$ 5.21148e28 0.384179
$$596$$ 0 0
$$597$$ −7.88255e28 −0.560965
$$598$$ 0 0
$$599$$ −1.35818e29 −0.933205 −0.466603 0.884467i $$-0.654522\pi$$
−0.466603 + 0.884467i $$0.654522\pi$$
$$600$$ 0 0
$$601$$ −5.67794e28 −0.376712 −0.188356 0.982101i $$-0.560316\pi$$
−0.188356 + 0.982101i $$0.560316\pi$$
$$602$$ 0 0
$$603$$ 8.44677e28 0.541201
$$604$$ 0 0
$$605$$ −1.14548e29 −0.708853
$$606$$ 0 0
$$607$$ −2.93166e29 −1.75239 −0.876197 0.481953i $$-0.839927\pi$$
−0.876197 + 0.481953i $$0.839927\pi$$
$$608$$ 0 0
$$609$$ −3.71532e28 −0.214543
$$610$$ 0 0
$$611$$ 7.95976e27 0.0444086
$$612$$ 0 0
$$613$$ −1.03374e29 −0.557282 −0.278641 0.960395i $$-0.589884\pi$$
−0.278641 + 0.960395i $$0.589884\pi$$
$$614$$ 0 0
$$615$$ −2.39192e28 −0.124611
$$616$$ 0 0
$$617$$ −2.61665e29 −1.31750 −0.658750 0.752362i $$-0.728915\pi$$
−0.658750 + 0.752362i $$0.728915\pi$$
$$618$$ 0 0
$$619$$ −2.52208e29 −1.22746 −0.613729 0.789517i $$-0.710331\pi$$
−0.613729 + 0.789517i $$0.710331\pi$$
$$620$$ 0 0
$$621$$ −6.69852e28 −0.315150
$$622$$ 0 0
$$623$$ 3.27294e28 0.148872
$$624$$ 0 0
$$625$$ 1.20144e29 0.528399
$$626$$ 0 0
$$627$$ 4.22688e28 0.179767
$$628$$ 0 0
$$629$$ −3.34048e29 −1.37397
$$630$$ 0 0
$$631$$ −4.18556e29 −1.66512 −0.832559 0.553936i $$-0.813125\pi$$
−0.832559 + 0.553936i $$0.813125\pi$$
$$632$$ 0 0
$$633$$ 7.18093e28 0.276339
$$634$$ 0 0
$$635$$ 4.07005e28 0.151522
$$636$$ 0 0
$$637$$ −3.17462e28 −0.114348
$$638$$ 0 0
$$639$$ −2.53131e26 −0.000882239 0
$$640$$ 0 0
$$641$$ 4.16106e29 1.40344 0.701721 0.712451i $$-0.252415\pi$$
0.701721 + 0.712451i $$0.252415\pi$$
$$642$$ 0 0
$$643$$ −1.16723e29 −0.381015 −0.190508 0.981686i $$-0.561013\pi$$
−0.190508 + 0.981686i $$0.561013\pi$$
$$644$$ 0 0
$$645$$ −4.41307e28 −0.139432
$$646$$ 0 0
$$647$$ −2.12199e29 −0.649007 −0.324503 0.945885i $$-0.605197\pi$$
−0.324503 + 0.945885i $$0.605197\pi$$
$$648$$ 0 0
$$649$$ 8.50432e29 2.51808
$$650$$ 0 0
$$651$$ 1.38431e29 0.396856
$$652$$ 0 0
$$653$$ 2.19143e29 0.608329 0.304165 0.952619i $$-0.401623\pi$$
0.304165 + 0.952619i $$0.401623\pi$$
$$654$$ 0 0
$$655$$ −7.69707e28 −0.206915
$$656$$ 0 0
$$657$$ −1.03826e29 −0.270316
$$658$$ 0 0
$$659$$ 7.81698e28 0.197125 0.0985627 0.995131i $$-0.468576\pi$$
0.0985627 + 0.995131i $$0.468576\pi$$
$$660$$ 0 0
$$661$$ −4.98165e29 −1.21691 −0.608454 0.793589i $$-0.708210\pi$$
−0.608454 + 0.793589i $$0.708210\pi$$
$$662$$ 0 0
$$663$$ 8.75814e28 0.207262
$$664$$ 0 0
$$665$$ 2.63363e28 0.0603849
$$666$$ 0 0
$$667$$ −3.48241e29 −0.773674
$$668$$ 0 0
$$669$$ 3.37835e29 0.727328
$$670$$ 0 0
$$671$$ 5.86820e28 0.122439
$$672$$ 0 0
$$673$$ 7.45598e29 1.50781 0.753904 0.656984i $$-0.228168\pi$$
0.753904 + 0.656984i $$0.228168\pi$$
$$674$$ 0 0
$$675$$ −8.18367e28 −0.160420
$$676$$ 0 0
$$677$$ 1.07199e29 0.203708 0.101854 0.994799i $$-0.467523\pi$$
0.101854 + 0.994799i $$0.467523\pi$$
$$678$$ 0 0
$$679$$ 4.93511e29 0.909210
$$680$$ 0 0
$$681$$ −3.31170e29 −0.591569
$$682$$ 0 0
$$683$$ 6.20112e29 1.07412 0.537060 0.843544i $$-0.319535\pi$$
0.537060 + 0.843544i $$0.319535\pi$$
$$684$$ 0 0
$$685$$ −2.38995e29 −0.401457
$$686$$ 0 0
$$687$$ −1.48728e29 −0.242297
$$688$$ 0 0
$$689$$ 1.70205e29 0.268950
$$690$$ 0 0
$$691$$ 1.11755e30 1.71296 0.856482 0.516177i $$-0.172645\pi$$
0.856482 + 0.516177i $$0.172645\pi$$
$$692$$ 0 0
$$693$$ −2.91726e29 −0.433786
$$694$$ 0 0
$$695$$ 3.47899e29 0.501895
$$696$$ 0 0
$$697$$ 4.52520e29 0.633423
$$698$$ 0 0
$$699$$ 1.17471e29 0.159559
$$700$$ 0 0
$$701$$ −9.02930e29 −1.19019 −0.595093 0.803657i $$-0.702885\pi$$
−0.595093 + 0.803657i $$0.702885\pi$$
$$702$$ 0 0
$$703$$ −1.68812e29 −0.215959
$$704$$ 0 0
$$705$$ −2.80950e28 −0.0348853
$$706$$ 0 0
$$707$$ 4.26325e29 0.513851
$$708$$ 0 0
$$709$$ 1.25054e29 0.146322 0.0731612 0.997320i $$-0.476691\pi$$
0.0731612 + 0.997320i $$0.476691\pi$$
$$710$$ 0 0
$$711$$ 2.01813e29 0.229255
$$712$$ 0 0
$$713$$ 1.29753e30 1.43112
$$714$$ 0 0
$$715$$ −1.88973e29 −0.202388
$$716$$ 0 0
$$717$$ −4.32652e29 −0.449974
$$718$$ 0 0
$$719$$ 7.25984e29 0.733287 0.366643 0.930362i $$-0.380507\pi$$
0.366643 + 0.930362i $$0.380507\pi$$
$$720$$ 0 0
$$721$$ −1.05510e30 −1.03508
$$722$$ 0 0
$$723$$ −1.11559e30 −1.06305
$$724$$ 0 0
$$725$$ −4.25450e29 −0.393822
$$726$$ 0 0
$$727$$ 3.73992e29 0.336318 0.168159 0.985760i $$-0.446218\pi$$
0.168159 + 0.985760i $$0.446218\pi$$
$$728$$ 0 0
$$729$$ 4.23912e28 0.0370370
$$730$$ 0 0
$$731$$ 8.34897e29 0.708762
$$732$$ 0 0
$$733$$ −1.77085e30 −1.46080 −0.730401 0.683019i $$-0.760667\pi$$
−0.730401 + 0.683019i $$0.760667\pi$$
$$734$$ 0 0
$$735$$ 1.12052e29 0.0898263
$$736$$ 0 0
$$737$$ 3.44802e30 2.68634
$$738$$ 0 0
$$739$$ 1.88567e30 1.42791 0.713953 0.700194i $$-0.246903\pi$$
0.713953 + 0.700194i $$0.246903\pi$$
$$740$$ 0 0
$$741$$ 4.42594e28 0.0325773
$$742$$ 0 0
$$743$$ 8.39029e29 0.600336 0.300168 0.953886i $$-0.402957\pi$$
0.300168 + 0.953886i $$0.402957\pi$$
$$744$$ 0 0
$$745$$ −3.44576e29 −0.239687
$$746$$ 0 0
$$747$$ 8.65719e29 0.585478
$$748$$ 0 0
$$749$$ −9.35992e28 −0.0615479
$$750$$ 0 0
$$751$$ −1.32480e30 −0.847095 −0.423547 0.905874i $$-0.639215\pi$$
−0.423547 + 0.905874i $$0.639215\pi$$
$$752$$ 0 0
$$753$$ 9.77020e29 0.607514
$$754$$ 0 0
$$755$$ −9.58333e29 −0.579527
$$756$$ 0 0
$$757$$ −8.91190e29 −0.524159 −0.262080 0.965046i $$-0.584408\pi$$
−0.262080 + 0.965046i $$0.584408\pi$$
$$758$$ 0 0
$$759$$ −2.73437e30 −1.56429
$$760$$ 0 0
$$761$$ 2.22977e30 1.24086 0.620429 0.784263i $$-0.286959\pi$$
0.620429 + 0.784263i $$0.286959\pi$$
$$762$$ 0 0
$$763$$ −1.74774e30 −0.946172
$$764$$ 0 0
$$765$$ −3.09130e29 −0.162815
$$766$$ 0 0
$$767$$ 8.90482e29 0.456325
$$768$$ 0 0
$$769$$ −1.65220e30 −0.823827 −0.411913 0.911223i $$-0.635139\pi$$
−0.411913 + 0.911223i $$0.635139\pi$$
$$770$$ 0 0
$$771$$ −1.73488e30 −0.841782
$$772$$ 0 0
$$773$$ −7.95379e29 −0.375569 −0.187784 0.982210i $$-0.560131\pi$$
−0.187784 + 0.982210i $$0.560131\pi$$
$$774$$ 0 0
$$775$$ 1.58521e30 0.728480
$$776$$ 0 0
$$777$$ 1.16508e30 0.521119
$$778$$ 0 0
$$779$$ 2.28682e29 0.0995609
$$780$$ 0 0
$$781$$ −1.03329e28 −0.00437913
$$782$$ 0 0
$$783$$ 2.20382e29 0.0909238
$$784$$ 0 0
$$785$$ −6.15992e29 −0.247425
$$786$$ 0 0
$$787$$ −2.00720e30 −0.784977 −0.392489 0.919757i $$-0.628386\pi$$
−0.392489 + 0.919757i $$0.628386\pi$$
$$788$$ 0 0
$$789$$ −1.94297e30 −0.739873
$$790$$ 0 0
$$791$$ −8.11641e29 −0.300962
$$792$$ 0 0
$$793$$ 6.14457e28 0.0221882
$$794$$ 0 0
$$795$$ −6.00761e29 −0.211274
$$796$$ 0 0
$$797$$ 3.69070e30 1.26414 0.632071 0.774910i $$-0.282205\pi$$
0.632071 + 0.774910i $$0.282205\pi$$
$$798$$ 0 0
$$799$$ 5.31521e29 0.177329
$$800$$ 0 0
$$801$$ −1.94141e29 −0.0630923
$$802$$ 0 0
$$803$$ −4.23824e30 −1.34175
$$804$$ 0 0
$$805$$ −1.70370e30 −0.525455
$$806$$ 0 0
$$807$$ −2.54024e29 −0.0763311
$$808$$ 0 0
$$809$$ 4.72614e30 1.38372 0.691859 0.722033i $$-0.256792\pi$$
0.691859 + 0.722033i $$0.256792\pi$$
$$810$$ 0 0
$$811$$ −2.65583e30 −0.757673 −0.378837 0.925464i $$-0.623676\pi$$
−0.378837 + 0.925464i $$0.623676\pi$$
$$812$$ 0 0
$$813$$ 1.60973e30 0.447510
$$814$$ 0 0
$$815$$ −2.41228e30 −0.653540
$$816$$ 0 0
$$817$$ 4.21917e29 0.111403
$$818$$ 0 0
$$819$$ −3.05464e29 −0.0786103
$$820$$ 0 0
$$821$$ −5.10610e30 −1.28082 −0.640408 0.768035i $$-0.721235\pi$$
−0.640408 + 0.768035i $$0.721235\pi$$
$$822$$ 0 0
$$823$$ 4.35473e30 1.06479 0.532394 0.846497i $$-0.321293\pi$$
0.532394 + 0.846497i $$0.321293\pi$$
$$824$$ 0 0
$$825$$ −3.34061e30 −0.796269
$$826$$ 0 0
$$827$$ 4.07345e30 0.946573 0.473287 0.880909i $$-0.343068\pi$$
0.473287 + 0.880909i $$0.343068\pi$$
$$828$$ 0 0
$$829$$ −2.77876e30 −0.629548 −0.314774 0.949167i $$-0.601929\pi$$
−0.314774 + 0.949167i $$0.601929\pi$$
$$830$$ 0 0
$$831$$ −2.72233e30 −0.601354
$$832$$ 0 0
$$833$$ −2.11988e30 −0.456604
$$834$$ 0 0
$$835$$ −2.05533e30 −0.431691
$$836$$ 0 0
$$837$$ −8.21132e29 −0.168188
$$838$$ 0 0
$$839$$ −3.15950e30 −0.631128 −0.315564 0.948904i $$-0.602194\pi$$
−0.315564 + 0.948904i $$0.602194\pi$$
$$840$$ 0 0
$$841$$ −3.98713e30 −0.776787
$$842$$ 0 0
$$843$$ −4.57113e29 −0.0868628
$$844$$ 0 0
$$845$$ 2.00311e30 0.371286
$$846$$ 0 0
$$847$$ −7.55836e30 −1.36663
$$848$$ 0 0
$$849$$ 8.37643e29 0.147750
$$850$$ 0 0
$$851$$ 1.09204e31 1.87923
$$852$$ 0 0
$$853$$ −1.12969e31 −1.89668 −0.948338 0.317260i $$-0.897237\pi$$
−0.948338 + 0.317260i $$0.897237\pi$$
$$854$$ 0 0
$$855$$ −1.56219e29 −0.0255912
$$856$$ 0 0
$$857$$ −1.09526e31 −1.75073 −0.875366 0.483462i $$-0.839379\pi$$
−0.875366 + 0.483462i $$0.839379\pi$$
$$858$$ 0 0
$$859$$ 8.09790e29 0.126312 0.0631559 0.998004i $$-0.479883\pi$$
0.0631559 + 0.998004i $$0.479883\pi$$
$$860$$ 0 0
$$861$$ −1.57829e30 −0.240245
$$862$$ 0 0
$$863$$ 9.41039e30 1.39796 0.698979 0.715142i $$-0.253638\pi$$
0.698979 + 0.715142i $$0.253638\pi$$
$$864$$ 0 0
$$865$$ 3.28184e30 0.475825
$$866$$ 0 0
$$867$$ 1.76853e30 0.250272
$$868$$ 0 0
$$869$$ 8.23811e30 1.13794
$$870$$ 0 0
$$871$$ 3.61040e30 0.486816
$$872$$ 0 0
$$873$$ −2.92736e30 −0.385324
$$874$$ 0 0
$$875$$ −4.57845e30 −0.588347
$$876$$ 0 0
$$877$$ 9.07878e30 1.13902 0.569510 0.821984i $$-0.307133\pi$$
0.569510 + 0.821984i $$0.307133\pi$$
$$878$$ 0 0
$$879$$ 5.08483e30 0.622864
$$880$$ 0 0
$$881$$ 3.03572e30 0.363090 0.181545 0.983383i $$-0.441890\pi$$
0.181545 + 0.983383i $$0.441890\pi$$
$$882$$ 0 0
$$883$$ −6.61886e28 −0.00773030 −0.00386515 0.999993i $$-0.501230\pi$$
−0.00386515 + 0.999993i $$0.501230\pi$$
$$884$$ 0 0
$$885$$ −3.14307e30 −0.358467
$$886$$ 0 0
$$887$$ 3.50101e30 0.389937 0.194969 0.980809i $$-0.437539\pi$$
0.194969 + 0.980809i $$0.437539\pi$$
$$888$$ 0 0
$$889$$ 2.68559e30 0.292127
$$890$$ 0 0
$$891$$ 1.73043e30 0.183839
$$892$$ 0 0
$$893$$ 2.68605e29 0.0278724
$$894$$ 0 0
$$895$$ −6.04451e30 −0.612659
$$896$$ 0 0
$$897$$ −2.86314e30 −0.283480
$$898$$ 0 0
$$899$$ −4.26888e30 −0.412892
$$900$$ 0 0
$$901$$ 1.13656e31 1.07395
$$902$$ 0 0
$$903$$ −2.91193e30 −0.268819
$$904$$ 0 0
$$905$$ −8.38495e30 −0.756293
$$906$$ 0 0
$$907$$ −5.24604e29 −0.0462333 −0.0231167 0.999733i $$-0.507359\pi$$
−0.0231167 + 0.999733i $$0.507359\pi$$
$$908$$ 0 0
$$909$$ −2.52883e30 −0.217771
$$910$$ 0 0
$$911$$ 1.16565e31 0.980902 0.490451 0.871469i $$-0.336832\pi$$
0.490451 + 0.871469i $$0.336832\pi$$
$$912$$ 0 0
$$913$$ 3.53391e31 2.90611
$$914$$ 0 0
$$915$$ −2.16880e29 −0.0174300
$$916$$ 0 0
$$917$$ −5.07886e30 −0.398922
$$918$$ 0 0
$$919$$ 7.15232e30 0.549078 0.274539 0.961576i $$-0.411475\pi$$
0.274539 + 0.961576i $$0.411475\pi$$
$$920$$ 0 0
$$921$$ 5.85594e30 0.439411
$$922$$ 0 0
$$923$$ −1.08196e28 −0.000793582 0
$$924$$ 0 0
$$925$$ 1.33416e31 0.956579
$$926$$ 0 0
$$927$$ 6.25856e30 0.438669
$$928$$ 0 0
$$929$$ 9.72648e30 0.666486 0.333243 0.942841i $$-0.391857\pi$$
0.333243 + 0.942841i $$0.391857\pi$$
$$930$$ 0 0
$$931$$ −1.07129e30 −0.0717686
$$932$$ 0 0
$$933$$ 6.84629e30 0.448433
$$934$$ 0 0
$$935$$ −1.26188e31 −0.808160
$$936$$ 0 0
$$937$$ −1.38653e31 −0.868288 −0.434144 0.900844i $$-0.642949\pi$$
−0.434144 + 0.900844i $$0.642949\pi$$
$$938$$ 0 0
$$939$$ 1.22024e31 0.747233
$$940$$ 0 0
$$941$$ 2.03835e31 1.22064 0.610320 0.792155i $$-0.291041\pi$$
0.610320 + 0.792155i $$0.291041\pi$$
$$942$$ 0 0
$$943$$ −1.47934e31 −0.866356
$$944$$ 0 0
$$945$$ 1.07817e30 0.0617526
$$946$$ 0 0
$$947$$ 3.05609e30 0.171195 0.0855975 0.996330i $$-0.472720\pi$$
0.0855975 + 0.996330i $$0.472720\pi$$
$$948$$ 0 0
$$949$$ −4.43784e30 −0.243151
$$950$$ 0 0
$$951$$ 1.71285e31 0.917960
$$952$$ 0 0
$$953$$ −1.54521e31 −0.810053 −0.405026 0.914305i $$-0.632738\pi$$
−0.405026 + 0.914305i $$0.632738\pi$$
$$954$$ 0 0
$$955$$ 8.94014e30 0.458468
$$956$$ 0 0
$$957$$ 8.99610e30 0.451314
$$958$$ 0 0
$$959$$ −1.57699e31 −0.773988
$$960$$ 0 0
$$961$$ −4.91989e30 −0.236243
$$962$$ 0 0
$$963$$ 5.55203e29 0.0260841
$$964$$ 0 0
$$965$$ 1.25469e31 0.576766
$$966$$ 0 0
$$967$$ −2.30504e31 −1.03681 −0.518406 0.855135i $$-0.673474\pi$$
−0.518406 + 0.855135i $$0.673474\pi$$
$$968$$ 0 0
$$969$$ 2.95547e30 0.130085
$$970$$ 0 0
$$971$$ −5.03493e29 −0.0216866 −0.0108433 0.999941i $$-0.503452\pi$$
−0.0108433 + 0.999941i $$0.503452\pi$$
$$972$$ 0 0
$$973$$ 2.29558e31 0.967628
$$974$$ 0 0
$$975$$ −3.49794e30 −0.144299
$$976$$ 0 0
$$977$$ −3.24415e31 −1.30981 −0.654905 0.755711i $$-0.727291\pi$$
−0.654905 + 0.755711i $$0.727291\pi$$
$$978$$ 0 0
$$979$$ −7.92495e30 −0.313169
$$980$$ 0 0
$$981$$ 1.03671e31 0.400989
$$982$$ 0 0
$$983$$ −1.25577e31 −0.475444 −0.237722 0.971333i $$-0.576401\pi$$
−0.237722 + 0.971333i $$0.576401\pi$$
$$984$$ 0 0
$$985$$ −1.75486e31 −0.650371
$$986$$ 0 0
$$987$$ −1.85383e30 −0.0672572
$$988$$ 0 0
$$989$$ −2.72938e31 −0.969399
$$990$$ 0 0
$$991$$ 1.56643e31 0.544676 0.272338 0.962202i $$-0.412203\pi$$
0.272338 + 0.962202i $$0.412203\pi$$
$$992$$ 0 0
$$993$$ −1.51328e31 −0.515171
$$994$$ 0 0
$$995$$ 1.18921e31 0.396385
$$996$$ 0 0
$$997$$ 4.16202e31 1.35833 0.679165 0.733986i $$-0.262342\pi$$
0.679165 + 0.733986i $$0.262342\pi$$
$$998$$ 0 0
$$999$$ −6.91093e30 −0.220851
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 48.22.a.l.1.2 3
4.3 odd 2 24.22.a.c.1.2 3
12.11 even 2 72.22.a.d.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
24.22.a.c.1.2 3 4.3 odd 2
48.22.a.l.1.2 3 1.1 even 1 trivial
72.22.a.d.1.2 3 12.11 even 2