# Properties

 Label 48.22.a.l.1.1 Level $48$ Weight $22$ Character 48.1 Self dual yes Analytic conductor $134.149$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

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 1.1 Root $$-386.305$$ 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)

 $$f(q)$$ $$=$$ $$q+59049.0 q^{3} -3.08294e7 q^{5} +1.10597e9 q^{7} +3.48678e9 q^{9} +O(q^{10})$$ $$q+59049.0 q^{3} -3.08294e7 q^{5} +1.10597e9 q^{7} +3.48678e9 q^{9} -1.54673e11 q^{11} +5.78668e11 q^{13} -1.82045e12 q^{15} +1.59754e12 q^{17} -7.29390e12 q^{19} +6.53065e13 q^{21} +1.56234e14 q^{23} +4.73615e14 q^{25} +2.05891e14 q^{27} -1.80995e15 q^{29} -5.98096e15 q^{31} -9.13328e15 q^{33} -3.40965e16 q^{35} +1.92628e16 q^{37} +3.41698e16 q^{39} -1.01900e17 q^{41} +2.34697e17 q^{43} -1.07495e17 q^{45} -1.31288e17 q^{47} +6.64628e17 q^{49} +9.43330e16 q^{51} +2.51169e18 q^{53} +4.76847e18 q^{55} -4.30697e17 q^{57} -4.37160e18 q^{59} -6.88287e18 q^{61} +3.85629e18 q^{63} -1.78400e19 q^{65} -1.11554e19 q^{67} +9.22545e18 q^{69} +4.02859e19 q^{71} +3.07636e19 q^{73} +2.79665e19 q^{75} -1.71064e20 q^{77} +1.48672e20 q^{79} +1.21577e19 q^{81} -9.43003e19 q^{83} -4.92511e19 q^{85} -1.06876e20 q^{87} -2.18162e20 q^{89} +6.39991e20 q^{91} -3.53169e20 q^{93} +2.24866e20 q^{95} +1.66723e20 q^{97} -5.39311e20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9}+O(q^{10})$$ 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
$$5$$ −3.08294e7 −1.41182 −0.705911 0.708300i $$-0.749462\pi$$
−0.705911 + 0.708300i $$0.749462\pi$$
$$6$$ 0 0
$$7$$ 1.10597e9 1.47984 0.739920 0.672695i $$-0.234863\pi$$
0.739920 + 0.672695i $$0.234863\pi$$
$$8$$ 0 0
$$9$$ 3.48678e9 0.333333
$$10$$ 0 0
$$11$$ −1.54673e11 −1.79801 −0.899003 0.437943i $$-0.855707\pi$$
−0.899003 + 0.437943i $$0.855707\pi$$
$$12$$ 0 0
$$13$$ 5.78668e11 1.16419 0.582095 0.813120i $$-0.302233\pi$$
0.582095 + 0.813120i $$0.302233\pi$$
$$14$$ 0 0
$$15$$ −1.82045e12 −0.815116
$$16$$ 0 0
$$17$$ 1.59754e12 0.192193 0.0960964 0.995372i $$-0.469364\pi$$
0.0960964 + 0.995372i $$0.469364\pi$$
$$18$$ 0 0
$$19$$ −7.29390e12 −0.272927 −0.136464 0.990645i $$-0.543574\pi$$
−0.136464 + 0.990645i $$0.543574\pi$$
$$20$$ 0 0
$$21$$ 6.53065e13 0.854386
$$22$$ 0 0
$$23$$ 1.56234e14 0.786381 0.393190 0.919457i $$-0.371371\pi$$
0.393190 + 0.919457i $$0.371371\pi$$
$$24$$ 0 0
$$25$$ 4.73615e14 0.993242
$$26$$ 0 0
$$27$$ 2.05891e14 0.192450
$$28$$ 0 0
$$29$$ −1.80995e15 −0.798890 −0.399445 0.916757i $$-0.630797\pi$$
−0.399445 + 0.916757i $$0.630797\pi$$
$$30$$ 0 0
$$31$$ −5.98096e15 −1.31061 −0.655304 0.755365i $$-0.727459\pi$$
−0.655304 + 0.755365i $$0.727459\pi$$
$$32$$ 0 0
$$33$$ −9.13328e15 −1.03808
$$34$$ 0 0
$$35$$ −3.40965e16 −2.08927
$$36$$ 0 0
$$37$$ 1.92628e16 0.658568 0.329284 0.944231i $$-0.393193\pi$$
0.329284 + 0.944231i $$0.393193\pi$$
$$38$$ 0 0
$$39$$ 3.41698e16 0.672146
$$40$$ 0 0
$$41$$ −1.01900e17 −1.18561 −0.592805 0.805346i $$-0.701980\pi$$
−0.592805 + 0.805346i $$0.701980\pi$$
$$42$$ 0 0
$$43$$ 2.34697e17 1.65611 0.828053 0.560650i $$-0.189449\pi$$
0.828053 + 0.560650i $$0.189449\pi$$
$$44$$ 0 0
$$45$$ −1.07495e17 −0.470607
$$46$$ 0 0
$$47$$ −1.31288e17 −0.364080 −0.182040 0.983291i $$-0.558270\pi$$
−0.182040 + 0.983291i $$0.558270\pi$$
$$48$$ 0 0
$$49$$ 6.64628e17 1.18993
$$50$$ 0 0
$$51$$ 9.43330e16 0.110963
$$52$$ 0 0
$$53$$ 2.51169e18 1.97274 0.986368 0.164554i $$-0.0526184\pi$$
0.986368 + 0.164554i $$0.0526184\pi$$
$$54$$ 0 0
$$55$$ 4.76847e18 2.53846
$$56$$ 0 0
$$57$$ −4.30697e17 −0.157575
$$58$$ 0 0
$$59$$ −4.37160e18 −1.11351 −0.556755 0.830677i $$-0.687954\pi$$
−0.556755 + 0.830677i $$0.687954\pi$$
$$60$$ 0 0
$$61$$ −6.88287e18 −1.23540 −0.617698 0.786415i $$-0.711935\pi$$
−0.617698 + 0.786415i $$0.711935\pi$$
$$62$$ 0 0
$$63$$ 3.85629e18 0.493280
$$64$$ 0 0
$$65$$ −1.78400e19 −1.64363
$$66$$ 0 0
$$67$$ −1.11554e19 −0.747650 −0.373825 0.927499i $$-0.621954\pi$$
−0.373825 + 0.927499i $$0.621954\pi$$
$$68$$ 0 0
$$69$$ 9.22545e18 0.454017
$$70$$ 0 0
$$71$$ 4.02859e19 1.46872 0.734362 0.678758i $$-0.237481\pi$$
0.734362 + 0.678758i $$0.237481\pi$$
$$72$$ 0 0
$$73$$ 3.07636e19 0.837812 0.418906 0.908030i $$-0.362414\pi$$
0.418906 + 0.908030i $$0.362414\pi$$
$$74$$ 0 0
$$75$$ 2.79665e19 0.573449
$$76$$ 0 0
$$77$$ −1.71064e20 −2.66076
$$78$$ 0 0
$$79$$ 1.48672e20 1.76662 0.883312 0.468785i $$-0.155308\pi$$
0.883312 + 0.468785i $$0.155308\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ 0 0
$$83$$ −9.43003e19 −0.667104 −0.333552 0.942732i $$-0.608247\pi$$
−0.333552 + 0.942732i $$0.608247\pi$$
$$84$$ 0 0
$$85$$ −4.92511e19 −0.271342
$$86$$ 0 0
$$87$$ −1.06876e20 −0.461239
$$88$$ 0 0
$$89$$ −2.18162e20 −0.741626 −0.370813 0.928708i $$-0.620921\pi$$
−0.370813 + 0.928708i $$0.620921\pi$$
$$90$$ 0 0
$$91$$ 6.39991e20 1.72282
$$92$$ 0 0
$$93$$ −3.53169e20 −0.756680
$$94$$ 0 0
$$95$$ 2.24866e20 0.385325
$$96$$ 0 0
$$97$$ 1.66723e20 0.229557 0.114779 0.993391i $$-0.463384\pi$$
0.114779 + 0.993391i $$0.463384\pi$$
$$98$$ 0 0
$$99$$ −5.39311e20 −0.599335
$$100$$ 0 0
$$101$$ −1.30649e20 −0.117688 −0.0588440 0.998267i $$-0.518741\pi$$
−0.0588440 + 0.998267i $$0.518741\pi$$
$$102$$ 0 0
$$103$$ 1.79611e21 1.31687 0.658434 0.752639i $$-0.271219\pi$$
0.658434 + 0.752639i $$0.271219\pi$$
$$104$$ 0 0
$$105$$ −2.01336e21 −1.20624
$$106$$ 0 0
$$107$$ −1.52295e19 −0.00748437 −0.00374218 0.999993i $$-0.501191\pi$$
−0.00374218 + 0.999993i $$0.501191\pi$$
$$108$$ 0 0
$$109$$ −1.75681e20 −0.0710799 −0.0355400 0.999368i $$-0.511315\pi$$
−0.0355400 + 0.999368i $$0.511315\pi$$
$$110$$ 0 0
$$111$$ 1.13745e21 0.380224
$$112$$ 0 0
$$113$$ 5.54442e21 1.53650 0.768250 0.640150i $$-0.221128\pi$$
0.768250 + 0.640150i $$0.221128\pi$$
$$114$$ 0 0
$$115$$ −4.81660e21 −1.11023
$$116$$ 0 0
$$117$$ 2.01769e21 0.388064
$$118$$ 0 0
$$119$$ 1.76683e21 0.284415
$$120$$ 0 0
$$121$$ 1.65235e22 2.23282
$$122$$ 0 0
$$123$$ −6.01707e21 −0.684512
$$124$$ 0 0
$$125$$ 9.93400e19 0.00954045
$$126$$ 0 0
$$127$$ 7.59121e21 0.617123 0.308562 0.951204i $$-0.400152\pi$$
0.308562 + 0.951204i $$0.400152\pi$$
$$128$$ 0 0
$$129$$ 1.38586e22 0.956153
$$130$$ 0 0
$$131$$ 1.26195e22 0.740789 0.370395 0.928874i $$-0.379222\pi$$
0.370395 + 0.928874i $$0.379222\pi$$
$$132$$ 0 0
$$133$$ −8.06684e21 −0.403888
$$134$$ 0 0
$$135$$ −6.34750e21 −0.271705
$$136$$ 0 0
$$137$$ 2.04751e22 0.751036 0.375518 0.926815i $$-0.377465\pi$$
0.375518 + 0.926815i $$0.377465\pi$$
$$138$$ 0 0
$$139$$ 8.05555e21 0.253770 0.126885 0.991917i $$-0.459502\pi$$
0.126885 + 0.991917i $$0.459502\pi$$
$$140$$ 0 0
$$141$$ −7.75241e21 −0.210202
$$142$$ 0 0
$$143$$ −8.95042e22 −2.09322
$$144$$ 0 0
$$145$$ 5.57996e22 1.12789
$$146$$ 0 0
$$147$$ 3.92456e22 0.687004
$$148$$ 0 0
$$149$$ −1.91681e21 −0.0291155 −0.0145577 0.999894i $$-0.504634\pi$$
−0.0145577 + 0.999894i $$0.504634\pi$$
$$150$$ 0 0
$$151$$ −4.49230e22 −0.593213 −0.296606 0.955000i $$-0.595855\pi$$
−0.296606 + 0.955000i $$0.595855\pi$$
$$152$$ 0 0
$$153$$ 5.57027e21 0.0640643
$$154$$ 0 0
$$155$$ 1.84389e23 1.85035
$$156$$ 0 0
$$157$$ −1.35200e23 −1.18585 −0.592926 0.805257i $$-0.702027\pi$$
−0.592926 + 0.805257i $$0.702027\pi$$
$$158$$ 0 0
$$159$$ 1.48313e23 1.13896
$$160$$ 0 0
$$161$$ 1.72790e23 1.16372
$$162$$ 0 0
$$163$$ 2.95535e23 1.74839 0.874197 0.485572i $$-0.161389\pi$$
0.874197 + 0.485572i $$0.161389\pi$$
$$164$$ 0 0
$$165$$ 2.81574e23 1.46558
$$166$$ 0 0
$$167$$ 2.71414e23 1.24483 0.622413 0.782689i $$-0.286152\pi$$
0.622413 + 0.782689i $$0.286152\pi$$
$$168$$ 0 0
$$169$$ 8.77920e22 0.355340
$$170$$ 0 0
$$171$$ −2.54322e22 −0.0909757
$$172$$ 0 0
$$173$$ 3.84952e23 1.21877 0.609386 0.792873i $$-0.291416\pi$$
0.609386 + 0.792873i $$0.291416\pi$$
$$174$$ 0 0
$$175$$ 5.23805e23 1.46984
$$176$$ 0 0
$$177$$ −2.58138e23 −0.642885
$$178$$ 0 0
$$179$$ 5.60520e23 1.24061 0.620304 0.784362i $$-0.287009\pi$$
0.620304 + 0.784362i $$0.287009\pi$$
$$180$$ 0 0
$$181$$ −7.65962e23 −1.50863 −0.754314 0.656514i $$-0.772030\pi$$
−0.754314 + 0.656514i $$0.772030\pi$$
$$182$$ 0 0
$$183$$ −4.06427e23 −0.713256
$$184$$ 0 0
$$185$$ −5.93860e23 −0.929781
$$186$$ 0 0
$$187$$ −2.47096e23 −0.345564
$$188$$ 0 0
$$189$$ 2.27710e23 0.284795
$$190$$ 0 0
$$191$$ 4.28534e23 0.479883 0.239941 0.970787i $$-0.422872\pi$$
0.239941 + 0.970787i $$0.422872\pi$$
$$192$$ 0 0
$$193$$ 1.58535e24 1.59138 0.795689 0.605705i $$-0.207109\pi$$
0.795689 + 0.605705i $$0.207109\pi$$
$$194$$ 0 0
$$195$$ −1.05343e24 −0.948951
$$196$$ 0 0
$$197$$ 1.58432e24 1.28217 0.641086 0.767469i $$-0.278484\pi$$
0.641086 + 0.767469i $$0.278484\pi$$
$$198$$ 0 0
$$199$$ 1.61590e24 1.17614 0.588068 0.808811i $$-0.299889\pi$$
0.588068 + 0.808811i $$0.299889\pi$$
$$200$$ 0 0
$$201$$ −6.58713e23 −0.431656
$$202$$ 0 0
$$203$$ −2.00175e24 −1.18223
$$204$$ 0 0
$$205$$ 3.14150e24 1.67387
$$206$$ 0 0
$$207$$ 5.44754e23 0.262127
$$208$$ 0 0
$$209$$ 1.12817e24 0.490724
$$210$$ 0 0
$$211$$ 4.40467e24 1.73359 0.866797 0.498661i $$-0.166175\pi$$
0.866797 + 0.498661i $$0.166175\pi$$
$$212$$ 0 0
$$213$$ 2.37884e24 0.847968
$$214$$ 0 0
$$215$$ −7.23556e24 −2.33813
$$216$$ 0 0
$$217$$ −6.61477e24 −1.93949
$$218$$ 0 0
$$219$$ 1.81656e24 0.483711
$$220$$ 0 0
$$221$$ 9.24444e23 0.223749
$$222$$ 0 0
$$223$$ −2.38725e24 −0.525650 −0.262825 0.964843i $$-0.584654\pi$$
−0.262825 + 0.964843i $$0.584654\pi$$
$$224$$ 0 0
$$225$$ 1.65139e24 0.331081
$$226$$ 0 0
$$227$$ −1.41267e24 −0.258089 −0.129045 0.991639i $$-0.541191\pi$$
−0.129045 + 0.991639i $$0.541191\pi$$
$$228$$ 0 0
$$229$$ −6.13078e24 −1.02151 −0.510755 0.859726i $$-0.670634\pi$$
−0.510755 + 0.859726i $$0.670634\pi$$
$$230$$ 0 0
$$231$$ −1.01012e25 −1.53619
$$232$$ 0 0
$$233$$ 3.43514e24 0.477208 0.238604 0.971117i $$-0.423310\pi$$
0.238604 + 0.971117i $$0.423310\pi$$
$$234$$ 0 0
$$235$$ 4.04753e24 0.514016
$$236$$ 0 0
$$237$$ 8.77892e24 1.01996
$$238$$ 0 0
$$239$$ 2.65401e23 0.0282309 0.0141154 0.999900i $$-0.495507\pi$$
0.0141154 + 0.999900i $$0.495507\pi$$
$$240$$ 0 0
$$241$$ −4.64377e24 −0.452577 −0.226288 0.974060i $$-0.572659\pi$$
−0.226288 + 0.974060i $$0.572659\pi$$
$$242$$ 0 0
$$243$$ 7.17898e23 0.0641500
$$244$$ 0 0
$$245$$ −2.04901e25 −1.67996
$$246$$ 0 0
$$247$$ −4.22074e24 −0.317739
$$248$$ 0 0
$$249$$ −5.56834e24 −0.385152
$$250$$ 0 0
$$251$$ 9.13351e24 0.580850 0.290425 0.956898i $$-0.406203\pi$$
0.290425 + 0.956898i $$0.406203\pi$$
$$252$$ 0 0
$$253$$ −2.41651e25 −1.41392
$$254$$ 0 0
$$255$$ −2.90823e24 −0.156660
$$256$$ 0 0
$$257$$ 2.42762e25 1.20471 0.602355 0.798228i $$-0.294229\pi$$
0.602355 + 0.798228i $$0.294229\pi$$
$$258$$ 0 0
$$259$$ 2.13041e25 0.974575
$$260$$ 0 0
$$261$$ −6.31089e24 −0.266297
$$262$$ 0 0
$$263$$ 5.72262e24 0.222874 0.111437 0.993772i $$-0.464455\pi$$
0.111437 + 0.993772i $$0.464455\pi$$
$$264$$ 0 0
$$265$$ −7.74338e25 −2.78515
$$266$$ 0 0
$$267$$ −1.28823e25 −0.428178
$$268$$ 0 0
$$269$$ −2.61502e25 −0.803668 −0.401834 0.915712i $$-0.631627\pi$$
−0.401834 + 0.915712i $$0.631627\pi$$
$$270$$ 0 0
$$271$$ 1.46407e25 0.416279 0.208139 0.978099i $$-0.433259\pi$$
0.208139 + 0.978099i $$0.433259\pi$$
$$272$$ 0 0
$$273$$ 3.77908e25 0.994668
$$274$$ 0 0
$$275$$ −7.32554e25 −1.78586
$$276$$ 0 0
$$277$$ −4.49242e25 −1.01495 −0.507473 0.861668i $$-0.669420\pi$$
−0.507473 + 0.861668i $$0.669420\pi$$
$$278$$ 0 0
$$279$$ −2.08543e25 −0.436869
$$280$$ 0 0
$$281$$ 5.52906e25 1.07457 0.537286 0.843400i $$-0.319450\pi$$
0.537286 + 0.843400i $$0.319450\pi$$
$$282$$ 0 0
$$283$$ −4.21386e25 −0.760192 −0.380096 0.924947i $$-0.624109\pi$$
−0.380096 + 0.924947i $$0.624109\pi$$
$$284$$ 0 0
$$285$$ 1.32781e25 0.222467
$$286$$ 0 0
$$287$$ −1.12698e26 −1.75451
$$288$$ 0 0
$$289$$ −6.65398e25 −0.963062
$$290$$ 0 0
$$291$$ 9.84481e24 0.132535
$$292$$ 0 0
$$293$$ −6.37530e25 −0.798713 −0.399356 0.916796i $$-0.630766\pi$$
−0.399356 + 0.916796i $$0.630766\pi$$
$$294$$ 0 0
$$295$$ 1.34774e26 1.57208
$$296$$ 0 0
$$297$$ −3.18458e25 −0.346026
$$298$$ 0 0
$$299$$ 9.04075e25 0.915497
$$300$$ 0 0
$$301$$ 2.59568e26 2.45077
$$302$$ 0 0
$$303$$ −7.71470e24 −0.0679472
$$304$$ 0 0
$$305$$ 2.12195e26 1.74416
$$306$$ 0 0
$$307$$ −7.93138e25 −0.608689 −0.304345 0.952562i $$-0.598437\pi$$
−0.304345 + 0.952562i $$0.598437\pi$$
$$308$$ 0 0
$$309$$ 1.06058e26 0.760294
$$310$$ 0 0
$$311$$ −1.28045e26 −0.857783 −0.428892 0.903356i $$-0.641096\pi$$
−0.428892 + 0.903356i $$0.641096\pi$$
$$312$$ 0 0
$$313$$ 1.71687e26 1.07528 0.537640 0.843174i $$-0.319316\pi$$
0.537640 + 0.843174i $$0.319316\pi$$
$$314$$ 0 0
$$315$$ −1.18887e26 −0.696424
$$316$$ 0 0
$$317$$ −3.54215e26 −1.94153 −0.970766 0.240028i $$-0.922843\pi$$
−0.970766 + 0.240028i $$0.922843\pi$$
$$318$$ 0 0
$$319$$ 2.79950e26 1.43641
$$320$$ 0 0
$$321$$ −8.99285e23 −0.00432110
$$322$$ 0 0
$$323$$ −1.16523e25 −0.0524547
$$324$$ 0 0
$$325$$ 2.74066e26 1.15632
$$326$$ 0 0
$$327$$ −1.03738e25 −0.0410380
$$328$$ 0 0
$$329$$ −1.45201e26 −0.538780
$$330$$ 0 0
$$331$$ 2.50034e26 0.870573 0.435286 0.900292i $$-0.356647\pi$$
0.435286 + 0.900292i $$0.356647\pi$$
$$332$$ 0 0
$$333$$ 6.71651e25 0.219523
$$334$$ 0 0
$$335$$ 3.43913e26 1.05555
$$336$$ 0 0
$$337$$ −1.04705e26 −0.301893 −0.150947 0.988542i $$-0.548232\pi$$
−0.150947 + 0.988542i $$0.548232\pi$$
$$338$$ 0 0
$$339$$ 3.27393e26 0.887099
$$340$$ 0 0
$$341$$ 9.25092e26 2.35648
$$342$$ 0 0
$$343$$ 1.17324e26 0.281061
$$344$$ 0 0
$$345$$ −2.84415e26 −0.640992
$$346$$ 0 0
$$347$$ −7.57106e25 −0.160582 −0.0802910 0.996771i $$-0.525585\pi$$
−0.0802910 + 0.996771i $$0.525585\pi$$
$$348$$ 0 0
$$349$$ −6.14974e26 −1.22798 −0.613988 0.789315i $$-0.710436\pi$$
−0.613988 + 0.789315i $$0.710436\pi$$
$$350$$ 0 0
$$351$$ 1.19143e26 0.224049
$$352$$ 0 0
$$353$$ −3.59357e25 −0.0636637 −0.0318319 0.999493i $$-0.510134\pi$$
−0.0318319 + 0.999493i $$0.510134\pi$$
$$354$$ 0 0
$$355$$ −1.24199e27 −2.07358
$$356$$ 0 0
$$357$$ 1.04330e26 0.164207
$$358$$ 0 0
$$359$$ 2.77822e26 0.412359 0.206180 0.978514i $$-0.433897\pi$$
0.206180 + 0.978514i $$0.433897\pi$$
$$360$$ 0 0
$$361$$ −6.61009e26 −0.925511
$$362$$ 0 0
$$363$$ 9.75693e26 1.28912
$$364$$ 0 0
$$365$$ −9.48422e26 −1.18284
$$366$$ 0 0
$$367$$ 2.81108e26 0.331039 0.165520 0.986206i $$-0.447070\pi$$
0.165520 + 0.986206i $$0.447070\pi$$
$$368$$ 0 0
$$369$$ −3.55302e26 −0.395203
$$370$$ 0 0
$$371$$ 2.77786e27 2.91933
$$372$$ 0 0
$$373$$ 1.46468e27 1.45479 0.727396 0.686218i $$-0.240730\pi$$
0.727396 + 0.686218i $$0.240730\pi$$
$$374$$ 0 0
$$375$$ 5.86593e24 0.00550818
$$376$$ 0 0
$$377$$ −1.04736e27 −0.930060
$$378$$ 0 0
$$379$$ −4.24665e26 −0.356726 −0.178363 0.983965i $$-0.557080\pi$$
−0.178363 + 0.983965i $$0.557080\pi$$
$$380$$ 0 0
$$381$$ 4.48253e26 0.356296
$$382$$ 0 0
$$383$$ −9.06730e26 −0.682167 −0.341083 0.940033i $$-0.610794\pi$$
−0.341083 + 0.940033i $$0.610794\pi$$
$$384$$ 0 0
$$385$$ 5.27380e27 3.75652
$$386$$ 0 0
$$387$$ 8.18337e26 0.552035
$$388$$ 0 0
$$389$$ 1.38481e26 0.0884951 0.0442476 0.999021i $$-0.485911\pi$$
0.0442476 + 0.999021i $$0.485911\pi$$
$$390$$ 0 0
$$391$$ 2.49589e26 0.151137
$$392$$ 0 0
$$393$$ 7.45171e26 0.427695
$$394$$ 0 0
$$395$$ −4.58346e27 −2.49416
$$396$$ 0 0
$$397$$ −3.07714e27 −1.58799 −0.793993 0.607927i $$-0.792001\pi$$
−0.793993 + 0.607927i $$0.792001\pi$$
$$398$$ 0 0
$$399$$ −4.76339e26 −0.233185
$$400$$ 0 0
$$401$$ −2.92618e27 −1.35920 −0.679602 0.733581i $$-0.737847\pi$$
−0.679602 + 0.733581i $$0.737847\pi$$
$$402$$ 0 0
$$403$$ −3.46099e27 −1.52580
$$404$$ 0 0
$$405$$ −3.74814e26 −0.156869
$$406$$ 0 0
$$407$$ −2.97943e27 −1.18411
$$408$$ 0 0
$$409$$ 3.30155e27 1.24630 0.623151 0.782102i $$-0.285852\pi$$
0.623151 + 0.782102i $$0.285852\pi$$
$$410$$ 0 0
$$411$$ 1.20904e27 0.433611
$$412$$ 0 0
$$413$$ −4.83487e27 −1.64782
$$414$$ 0 0
$$415$$ 2.90722e27 0.941832
$$416$$ 0 0
$$417$$ 4.75672e26 0.146514
$$418$$ 0 0
$$419$$ −3.60698e27 −1.05656 −0.528282 0.849069i $$-0.677164\pi$$
−0.528282 + 0.849069i $$0.677164\pi$$
$$420$$ 0 0
$$421$$ 1.13735e27 0.316908 0.158454 0.987366i $$-0.449349\pi$$
0.158454 + 0.987366i $$0.449349\pi$$
$$422$$ 0 0
$$423$$ −4.57772e26 −0.121360
$$424$$ 0 0
$$425$$ 7.56617e26 0.190894
$$426$$ 0 0
$$427$$ −7.61226e27 −1.82819
$$428$$ 0 0
$$429$$ −5.28514e27 −1.20852
$$430$$ 0 0
$$431$$ 1.65432e27 0.360254 0.180127 0.983643i $$-0.442349\pi$$
0.180127 + 0.983643i $$0.442349\pi$$
$$432$$ 0 0
$$433$$ −3.17892e27 −0.659412 −0.329706 0.944084i $$-0.606950\pi$$
−0.329706 + 0.944084i $$0.606950\pi$$
$$434$$ 0 0
$$435$$ 3.29491e27 0.651188
$$436$$ 0 0
$$437$$ −1.13955e27 −0.214625
$$438$$ 0 0
$$439$$ −2.89492e27 −0.519707 −0.259853 0.965648i $$-0.583674\pi$$
−0.259853 + 0.965648i $$0.583674\pi$$
$$440$$ 0 0
$$441$$ 2.31742e27 0.396642
$$442$$ 0 0
$$443$$ 3.00624e27 0.490665 0.245332 0.969439i $$-0.421103\pi$$
0.245332 + 0.969439i $$0.421103\pi$$
$$444$$ 0 0
$$445$$ 6.72582e27 1.04704
$$446$$ 0 0
$$447$$ −1.13186e26 −0.0168098
$$448$$ 0 0
$$449$$ −2.37879e27 −0.337109 −0.168554 0.985692i $$-0.553910\pi$$
−0.168554 + 0.985692i $$0.553910\pi$$
$$450$$ 0 0
$$451$$ 1.57611e28 2.13173
$$452$$ 0 0
$$453$$ −2.65266e27 −0.342491
$$454$$ 0 0
$$455$$ −1.97305e28 −2.43231
$$456$$ 0 0
$$457$$ −8.43469e27 −0.992999 −0.496500 0.868037i $$-0.665382\pi$$
−0.496500 + 0.868037i $$0.665382\pi$$
$$458$$ 0 0
$$459$$ 3.28919e26 0.0369875
$$460$$ 0 0
$$461$$ −1.82405e27 −0.195965 −0.0979823 0.995188i $$-0.531239\pi$$
−0.0979823 + 0.995188i $$0.531239\pi$$
$$462$$ 0 0
$$463$$ 1.00432e28 1.03103 0.515516 0.856880i $$-0.327600\pi$$
0.515516 + 0.856880i $$0.327600\pi$$
$$464$$ 0 0
$$465$$ 1.08880e28 1.06830
$$466$$ 0 0
$$467$$ 1.94652e28 1.82571 0.912856 0.408282i $$-0.133872\pi$$
0.912856 + 0.408282i $$0.133872\pi$$
$$468$$ 0 0
$$469$$ −1.23375e28 −1.10640
$$470$$ 0 0
$$471$$ −7.98343e27 −0.684652
$$472$$ 0 0
$$473$$ −3.63012e28 −2.97769
$$474$$ 0 0
$$475$$ −3.45450e27 −0.271083
$$476$$ 0 0
$$477$$ 8.75771e27 0.657579
$$478$$ 0 0
$$479$$ −1.75628e28 −1.26204 −0.631018 0.775768i $$-0.717362\pi$$
−0.631018 + 0.775768i $$0.717362\pi$$
$$480$$ 0 0
$$481$$ 1.11467e28 0.766699
$$482$$ 0 0
$$483$$ 1.02031e28 0.671873
$$484$$ 0 0
$$485$$ −5.13996e27 −0.324094
$$486$$ 0 0
$$487$$ 1.86000e28 1.12320 0.561602 0.827407i $$-0.310185\pi$$
0.561602 + 0.827407i $$0.310185\pi$$
$$488$$ 0 0
$$489$$ 1.74510e28 1.00944
$$490$$ 0 0
$$491$$ −3.81388e27 −0.211354 −0.105677 0.994400i $$-0.533701\pi$$
−0.105677 + 0.994400i $$0.533701\pi$$
$$492$$ 0 0
$$493$$ −2.89146e27 −0.153541
$$494$$ 0 0
$$495$$ 1.66266e28 0.846155
$$496$$ 0 0
$$497$$ 4.45551e28 2.17348
$$498$$ 0 0
$$499$$ 9.64495e27 0.451070 0.225535 0.974235i $$-0.427587\pi$$
0.225535 + 0.974235i $$0.427587\pi$$
$$500$$ 0 0
$$501$$ 1.60267e28 0.718701
$$502$$ 0 0
$$503$$ −1.64738e27 −0.0708485 −0.0354243 0.999372i $$-0.511278\pi$$
−0.0354243 + 0.999372i $$0.511278\pi$$
$$504$$ 0 0
$$505$$ 4.02784e27 0.166155
$$506$$ 0 0
$$507$$ 5.18403e27 0.205156
$$508$$ 0 0
$$509$$ 2.39315e28 0.908725 0.454363 0.890817i $$-0.349867\pi$$
0.454363 + 0.890817i $$0.349867\pi$$
$$510$$ 0 0
$$511$$ 3.40236e28 1.23983
$$512$$ 0 0
$$513$$ −1.50175e27 −0.0525249
$$514$$ 0 0
$$515$$ −5.53730e28 −1.85918
$$516$$ 0 0
$$517$$ 2.03067e28 0.654618
$$518$$ 0 0
$$519$$ 2.27310e28 0.703659
$$520$$ 0 0
$$521$$ 1.46945e28 0.436876 0.218438 0.975851i $$-0.429904\pi$$
0.218438 + 0.975851i $$0.429904\pi$$
$$522$$ 0 0
$$523$$ −2.10247e27 −0.0600430 −0.0300215 0.999549i $$-0.509558\pi$$
−0.0300215 + 0.999549i $$0.509558\pi$$
$$524$$ 0 0
$$525$$ 3.09302e28 0.848612
$$526$$ 0 0
$$527$$ −9.55480e27 −0.251890
$$528$$ 0 0
$$529$$ −1.50626e28 −0.381605
$$530$$ 0 0
$$531$$ −1.52428e28 −0.371170
$$532$$ 0 0
$$533$$ −5.89660e28 −1.38028
$$534$$ 0 0
$$535$$ 4.69515e26 0.0105666
$$536$$ 0 0
$$537$$ 3.30982e28 0.716265
$$538$$ 0 0
$$539$$ −1.02800e29 −2.13949
$$540$$ 0 0
$$541$$ −7.08289e27 −0.141788 −0.0708939 0.997484i $$-0.522585\pi$$
−0.0708939 + 0.997484i $$0.522585\pi$$
$$542$$ 0 0
$$543$$ −4.52293e28 −0.871007
$$544$$ 0 0
$$545$$ 5.41614e27 0.100352
$$546$$ 0 0
$$547$$ 6.08725e28 1.08531 0.542655 0.839955i $$-0.317419\pi$$
0.542655 + 0.839955i $$0.317419\pi$$
$$548$$ 0 0
$$549$$ −2.39991e28 −0.411799
$$550$$ 0 0
$$551$$ 1.32016e28 0.218039
$$552$$ 0 0
$$553$$ 1.64427e29 2.61432
$$554$$ 0 0
$$555$$ −3.50668e28 −0.536809
$$556$$ 0 0
$$557$$ −1.08630e29 −1.60129 −0.800643 0.599142i $$-0.795508\pi$$
−0.800643 + 0.599142i $$0.795508\pi$$
$$558$$ 0 0
$$559$$ 1.35811e29 1.92802
$$560$$ 0 0
$$561$$ −1.45908e28 −0.199511
$$562$$ 0 0
$$563$$ 7.40541e28 0.975463 0.487732 0.872994i $$-0.337824\pi$$
0.487732 + 0.872994i $$0.337824\pi$$
$$564$$ 0 0
$$565$$ −1.70931e29 −2.16927
$$566$$ 0 0
$$567$$ 1.34460e28 0.164427
$$568$$ 0 0
$$569$$ 2.96450e28 0.349359 0.174680 0.984625i $$-0.444111\pi$$
0.174680 + 0.984625i $$0.444111\pi$$
$$570$$ 0 0
$$571$$ 1.55764e29 1.76924 0.884622 0.466309i $$-0.154416\pi$$
0.884622 + 0.466309i $$0.154416\pi$$
$$572$$ 0 0
$$573$$ 2.53045e28 0.277060
$$574$$ 0 0
$$575$$ 7.39947e28 0.781067
$$576$$ 0 0
$$577$$ 1.06756e29 1.08654 0.543270 0.839558i $$-0.317186\pi$$
0.543270 + 0.839558i $$0.317186\pi$$
$$578$$ 0 0
$$579$$ 9.36133e28 0.918783
$$580$$ 0 0
$$581$$ −1.04294e29 −0.987207
$$582$$ 0 0
$$583$$ −3.88490e29 −3.54699
$$584$$ 0 0
$$585$$ −6.22042e28 −0.547877
$$586$$ 0 0
$$587$$ −1.34711e29 −1.14473 −0.572365 0.819999i $$-0.693974\pi$$
−0.572365 + 0.819999i $$0.693974\pi$$
$$588$$ 0 0
$$589$$ 4.36245e28 0.357700
$$590$$ 0 0
$$591$$ 9.35523e28 0.740263
$$592$$ 0 0
$$593$$ 1.79961e28 0.137437 0.0687185 0.997636i $$-0.478109\pi$$
0.0687185 + 0.997636i $$0.478109\pi$$
$$594$$ 0 0
$$595$$ −5.44704e28 −0.401543
$$596$$ 0 0
$$597$$ 9.54174e28 0.679043
$$598$$ 0 0
$$599$$ −4.20344e28 −0.288817 −0.144409 0.989518i $$-0.546128\pi$$
−0.144409 + 0.989518i $$0.546128\pi$$
$$600$$ 0 0
$$601$$ −2.69124e28 −0.178554 −0.0892772 0.996007i $$-0.528456\pi$$
−0.0892772 + 0.996007i $$0.528456\pi$$
$$602$$ 0 0
$$603$$ −3.88964e28 −0.249217
$$604$$ 0 0
$$605$$ −5.09408e29 −3.15235
$$606$$ 0 0
$$607$$ 1.40422e29 0.839373 0.419687 0.907669i $$-0.362140\pi$$
0.419687 + 0.907669i $$0.362140\pi$$
$$608$$ 0 0
$$609$$ −1.18201e29 −0.682560
$$610$$ 0 0
$$611$$ −7.59720e28 −0.423859
$$612$$ 0 0
$$613$$ −2.71738e29 −1.46493 −0.732464 0.680806i $$-0.761630\pi$$
−0.732464 + 0.680806i $$0.761630\pi$$
$$614$$ 0 0
$$615$$ 1.85503e29 0.966410
$$616$$ 0 0
$$617$$ 2.15284e29 1.08397 0.541984 0.840389i $$-0.317673\pi$$
0.541984 + 0.840389i $$0.317673\pi$$
$$618$$ 0 0
$$619$$ −1.12642e29 −0.548212 −0.274106 0.961699i $$-0.588382\pi$$
−0.274106 + 0.961699i $$0.588382\pi$$
$$620$$ 0 0
$$621$$ 3.21672e28 0.151339
$$622$$ 0 0
$$623$$ −2.41282e29 −1.09749
$$624$$ 0 0
$$625$$ −2.28900e29 −1.00671
$$626$$ 0 0
$$627$$ 6.66172e28 0.283320
$$628$$ 0 0
$$629$$ 3.07730e28 0.126572
$$630$$ 0 0
$$631$$ −4.46959e29 −1.77811 −0.889056 0.457798i $$-0.848638\pi$$
−0.889056 + 0.457798i $$0.848638\pi$$
$$632$$ 0 0
$$633$$ 2.60091e29 1.00089
$$634$$ 0 0
$$635$$ −2.34033e29 −0.871269
$$636$$ 0 0
$$637$$ 3.84599e29 1.38530
$$638$$ 0 0
$$639$$ 1.40468e29 0.489575
$$640$$ 0 0
$$641$$ −2.05893e29 −0.694435 −0.347218 0.937785i $$-0.612873\pi$$
−0.347218 + 0.937785i $$0.612873\pi$$
$$642$$ 0 0
$$643$$ −1.28958e29 −0.420951 −0.210476 0.977599i $$-0.567501\pi$$
−0.210476 + 0.977599i $$0.567501\pi$$
$$644$$ 0 0
$$645$$ −4.27253e29 −1.34992
$$646$$ 0 0
$$647$$ −2.06845e28 −0.0632629 −0.0316315 0.999500i $$-0.510070\pi$$
−0.0316315 + 0.999500i $$0.510070\pi$$
$$648$$ 0 0
$$649$$ 6.76168e29 2.00210
$$650$$ 0 0
$$651$$ −3.90596e29 −1.11977
$$652$$ 0 0
$$653$$ −1.59343e28 −0.0442328 −0.0221164 0.999755i $$-0.507040\pi$$
−0.0221164 + 0.999755i $$0.507040\pi$$
$$654$$ 0 0
$$655$$ −3.89053e29 −1.04586
$$656$$ 0 0
$$657$$ 1.07266e29 0.279271
$$658$$ 0 0
$$659$$ 5.85081e29 1.47543 0.737716 0.675111i $$-0.235904\pi$$
0.737716 + 0.675111i $$0.235904\pi$$
$$660$$ 0 0
$$661$$ 6.08260e29 1.48585 0.742923 0.669377i $$-0.233439\pi$$
0.742923 + 0.669377i $$0.233439\pi$$
$$662$$ 0 0
$$663$$ 5.45875e28 0.129182
$$664$$ 0 0
$$665$$ 2.48696e29 0.570219
$$666$$ 0 0
$$667$$ −2.82775e29 −0.628231
$$668$$ 0 0
$$669$$ −1.40965e29 −0.303484
$$670$$ 0 0
$$671$$ 1.06459e30 2.22125
$$672$$ 0 0
$$673$$ 1.17199e29 0.237009 0.118505 0.992953i $$-0.462190\pi$$
0.118505 + 0.992953i $$0.462190\pi$$
$$674$$ 0 0
$$675$$ 9.75131e28 0.191150
$$676$$ 0 0
$$677$$ −1.66070e29 −0.315581 −0.157790 0.987473i $$-0.550437\pi$$
−0.157790 + 0.987473i $$0.550437\pi$$
$$678$$ 0 0
$$679$$ 1.84391e29 0.339708
$$680$$ 0 0
$$681$$ −8.34169e28 −0.149008
$$682$$ 0 0
$$683$$ 3.97118e29 0.687863 0.343931 0.938995i $$-0.388241\pi$$
0.343931 + 0.938995i $$0.388241\pi$$
$$684$$ 0 0
$$685$$ −6.31236e29 −1.06033
$$686$$ 0 0
$$687$$ −3.62016e29 −0.589770
$$688$$ 0 0
$$689$$ 1.45343e30 2.29664
$$690$$ 0 0
$$691$$ 5.43975e29 0.833795 0.416897 0.908954i $$-0.363117\pi$$
0.416897 + 0.908954i $$0.363117\pi$$
$$692$$ 0 0
$$693$$ −5.96463e29 −0.886920
$$694$$ 0 0
$$695$$ −2.48348e29 −0.358278
$$696$$ 0 0
$$697$$ −1.62788e29 −0.227866
$$698$$ 0 0
$$699$$ 2.02842e29 0.275516
$$700$$ 0 0
$$701$$ 5.32602e29 0.702043 0.351022 0.936367i $$-0.385834\pi$$
0.351022 + 0.936367i $$0.385834\pi$$
$$702$$ 0 0
$$703$$ −1.40501e29 −0.179741
$$704$$ 0 0
$$705$$ 2.39002e29 0.296768
$$706$$ 0 0
$$707$$ −1.44494e29 −0.174159
$$708$$ 0 0
$$709$$ −1.45514e30 −1.70262 −0.851312 0.524660i $$-0.824193\pi$$
−0.851312 + 0.524660i $$0.824193\pi$$
$$710$$ 0 0
$$711$$ 5.18387e29 0.588875
$$712$$ 0 0
$$713$$ −9.34428e29 −1.03064
$$714$$ 0 0
$$715$$ 2.75936e30 2.95526
$$716$$ 0 0
$$717$$ 1.56717e28 0.0162991
$$718$$ 0 0
$$719$$ −5.61730e29 −0.567381 −0.283690 0.958916i $$-0.591559\pi$$
−0.283690 + 0.958916i $$0.591559\pi$$
$$720$$ 0 0
$$721$$ 1.98645e30 1.94875
$$722$$ 0 0
$$723$$ −2.74210e29 −0.261295
$$724$$ 0 0
$$725$$ −8.57218e29 −0.793491
$$726$$ 0 0
$$727$$ 9.24645e29 0.831503 0.415751 0.909478i $$-0.363519\pi$$
0.415751 + 0.909478i $$0.363519\pi$$
$$728$$ 0 0
$$729$$ 4.23912e28 0.0370370
$$730$$ 0 0
$$731$$ 3.74937e29 0.318292
$$732$$ 0 0
$$733$$ −1.10815e30 −0.914124 −0.457062 0.889435i $$-0.651098\pi$$
−0.457062 + 0.889435i $$0.651098\pi$$
$$734$$ 0 0
$$735$$ −1.20992e30 −0.969928
$$736$$ 0 0
$$737$$ 1.72543e30 1.34428
$$738$$ 0 0
$$739$$ −2.39445e30 −1.81317 −0.906585 0.422023i $$-0.861320\pi$$
−0.906585 + 0.422023i $$0.861320\pi$$
$$740$$ 0 0
$$741$$ −2.49231e29 −0.183447
$$742$$ 0 0
$$743$$ 1.95615e30 1.39965 0.699825 0.714314i $$-0.253261\pi$$
0.699825 + 0.714314i $$0.253261\pi$$
$$744$$ 0 0
$$745$$ 5.90942e28 0.0411059
$$746$$ 0 0
$$747$$ −3.28805e29 −0.222368
$$748$$ 0 0
$$749$$ −1.68434e28 −0.0110757
$$750$$ 0 0
$$751$$ 1.69999e30 1.08699 0.543497 0.839411i $$-0.317100\pi$$
0.543497 + 0.839411i $$0.317100\pi$$
$$752$$ 0 0
$$753$$ 5.39325e29 0.335354
$$754$$ 0 0
$$755$$ 1.38495e30 0.837511
$$756$$ 0 0
$$757$$ −1.95617e30 −1.15054 −0.575269 0.817964i $$-0.695103\pi$$
−0.575269 + 0.817964i $$0.695103\pi$$
$$758$$ 0 0
$$759$$ −1.42693e30 −0.816325
$$760$$ 0 0
$$761$$ 6.47879e29 0.360541 0.180271 0.983617i $$-0.442303\pi$$
0.180271 + 0.983617i $$0.442303\pi$$
$$762$$ 0 0
$$763$$ −1.94298e29 −0.105187
$$764$$ 0 0
$$765$$ −1.71728e29 −0.0904474
$$766$$ 0 0
$$767$$ −2.52970e30 −1.29634
$$768$$ 0 0
$$769$$ 3.06878e30 1.53017 0.765083 0.643931i $$-0.222698\pi$$
0.765083 + 0.643931i $$0.222698\pi$$
$$770$$ 0 0
$$771$$ 1.43348e30 0.695540
$$772$$ 0 0
$$773$$ 3.32643e30 1.57070 0.785351 0.619050i $$-0.212482\pi$$
0.785351 + 0.619050i $$0.212482\pi$$
$$774$$ 0 0
$$775$$ −2.83267e30 −1.30175
$$776$$ 0 0
$$777$$ 1.25798e30 0.562671
$$778$$ 0 0
$$779$$ 7.43245e29 0.323585
$$780$$ 0 0
$$781$$ −6.23113e30 −2.64077
$$782$$ 0 0
$$783$$ −3.72652e29 −0.153746
$$784$$ 0 0
$$785$$ 4.16814e30 1.67421
$$786$$ 0 0
$$787$$ −3.66843e30 −1.43465 −0.717324 0.696740i $$-0.754633\pi$$
−0.717324 + 0.696740i $$0.754633\pi$$
$$788$$ 0 0
$$789$$ 3.37915e29 0.128676
$$790$$ 0 0
$$791$$ 6.13198e30 2.27378
$$792$$ 0 0
$$793$$ −3.98290e30 −1.43824
$$794$$ 0 0
$$795$$ −4.57239e30 −1.60801
$$796$$ 0 0
$$797$$ 9.14635e29 0.313282 0.156641 0.987656i $$-0.449933\pi$$
0.156641 + 0.987656i $$0.449933\pi$$
$$798$$ 0 0
$$799$$ −2.09737e29 −0.0699736
$$800$$ 0 0
$$801$$ −7.60685e29 −0.247209
$$802$$ 0 0
$$803$$ −4.75829e30 −1.50639
$$804$$ 0 0
$$805$$ −5.32702e30 −1.64296
$$806$$ 0 0
$$807$$ −1.54415e30 −0.463998
$$808$$ 0 0
$$809$$ −1.04306e30 −0.305385 −0.152693 0.988274i $$-0.548794\pi$$
−0.152693 + 0.988274i $$0.548794\pi$$
$$810$$ 0 0
$$811$$ −4.59742e28 −0.0131158 −0.00655791 0.999978i $$-0.502087\pi$$
−0.00655791 + 0.999978i $$0.502087\pi$$
$$812$$ 0 0
$$813$$ 8.64519e29 0.240339
$$814$$ 0 0
$$815$$ −9.11117e30 −2.46842
$$816$$ 0 0
$$817$$ −1.71185e30 −0.451996
$$818$$ 0 0
$$819$$ 2.23151e30 0.574272
$$820$$ 0 0
$$821$$ 2.75888e30 0.692037 0.346018 0.938228i $$-0.387533\pi$$
0.346018 + 0.938228i $$0.387533\pi$$
$$822$$ 0 0
$$823$$ 2.37654e30 0.581094 0.290547 0.956861i $$-0.406163\pi$$
0.290547 + 0.956861i $$0.406163\pi$$
$$824$$ 0 0
$$825$$ −4.32566e30 −1.03106
$$826$$ 0 0
$$827$$ 3.99308e30 0.927897 0.463949 0.885862i $$-0.346432\pi$$
0.463949 + 0.885862i $$0.346432\pi$$
$$828$$ 0 0
$$829$$ 7.15523e30 1.62107 0.810534 0.585691i $$-0.199177\pi$$
0.810534 + 0.585691i $$0.199177\pi$$
$$830$$ 0 0
$$831$$ −2.65273e30 −0.585979
$$832$$ 0 0
$$833$$ 1.06177e30 0.228695
$$834$$ 0 0
$$835$$ −8.36752e30 −1.75747
$$836$$ 0 0
$$837$$ −1.23143e30 −0.252227
$$838$$ 0 0
$$839$$ 5.19244e30 1.03722 0.518610 0.855011i $$-0.326450\pi$$
0.518610 + 0.855011i $$0.326450\pi$$
$$840$$ 0 0
$$841$$ −1.85694e30 −0.361775
$$842$$ 0 0
$$843$$ 3.26486e30 0.620404
$$844$$ 0 0
$$845$$ −2.70657e30 −0.501677
$$846$$ 0 0
$$847$$ 1.82745e31 3.30422
$$848$$ 0 0
$$849$$ −2.48825e30 −0.438897
$$850$$ 0 0
$$851$$ 3.00950e30 0.517885
$$852$$ 0 0
$$853$$ 9.61096e30 1.61362 0.806811 0.590809i $$-0.201191\pi$$
0.806811 + 0.590809i $$0.201191\pi$$
$$854$$ 0 0
$$855$$ 7.84061e29 0.128442
$$856$$ 0 0
$$857$$ −4.96150e30 −0.793075 −0.396537 0.918019i $$-0.629788\pi$$
−0.396537 + 0.918019i $$0.629788\pi$$
$$858$$ 0 0
$$859$$ −4.43332e30 −0.691513 −0.345757 0.938324i $$-0.612378\pi$$
−0.345757 + 0.938324i $$0.612378\pi$$
$$860$$ 0 0
$$861$$ −6.65471e30 −1.01297
$$862$$ 0 0
$$863$$ 3.32513e30 0.493965 0.246982 0.969020i $$-0.420561\pi$$
0.246982 + 0.969020i $$0.420561\pi$$
$$864$$ 0 0
$$865$$ −1.18678e31 −1.72069
$$866$$ 0 0
$$867$$ −3.92911e30 −0.556024
$$868$$ 0 0
$$869$$ −2.29955e31 −3.17640
$$870$$ 0 0
$$871$$ −6.45525e30 −0.870408
$$872$$ 0 0
$$873$$ 5.81326e29 0.0765192
$$874$$ 0 0
$$875$$ 1.09867e29 0.0141183
$$876$$ 0 0
$$877$$ 4.83563e30 0.606677 0.303339 0.952883i $$-0.401899\pi$$
0.303339 + 0.952883i $$0.401899\pi$$
$$878$$ 0 0
$$879$$ −3.76455e30 −0.461137
$$880$$ 0 0
$$881$$ 1.40504e30 0.168051 0.0840257 0.996464i $$-0.473222\pi$$
0.0840257 + 0.996464i $$0.473222\pi$$
$$882$$ 0 0
$$883$$ 3.49210e30 0.407849 0.203924 0.978987i $$-0.434630\pi$$
0.203924 + 0.978987i $$0.434630\pi$$
$$884$$ 0 0
$$885$$ 7.95825e30 0.907640
$$886$$ 0 0
$$887$$ −6.00845e30 −0.669213 −0.334607 0.942358i $$-0.608603\pi$$
−0.334607 + 0.942358i $$0.608603\pi$$
$$888$$ 0 0
$$889$$ 8.39567e30 0.913244
$$890$$ 0 0
$$891$$ −1.88046e30 −0.199778
$$892$$ 0 0
$$893$$ 9.57600e29 0.0993673
$$894$$ 0 0
$$895$$ −1.72805e31 −1.75152
$$896$$ 0 0
$$897$$ 5.33847e30 0.528563
$$898$$ 0 0
$$899$$ 1.08252e31 1.04703
$$900$$ 0 0
$$901$$ 4.01251e30 0.379146
$$902$$ 0 0
$$903$$ 1.53272e31 1.41495
$$904$$ 0 0
$$905$$ 2.36141e31 2.12991
$$906$$ 0 0
$$907$$ 1.46441e31 1.29058 0.645290 0.763938i $$-0.276737\pi$$
0.645290 + 0.763938i $$0.276737\pi$$
$$908$$ 0 0
$$909$$ −4.55545e29 −0.0392293
$$910$$ 0 0
$$911$$ −8.68894e30 −0.731179 −0.365590 0.930776i $$-0.619133\pi$$
−0.365590 + 0.930776i $$0.619133\pi$$
$$912$$ 0 0
$$913$$ 1.45857e31 1.19946
$$914$$ 0 0
$$915$$ 1.25299e31 1.00699
$$916$$ 0 0
$$917$$ 1.39569e31 1.09625
$$918$$ 0 0
$$919$$ −1.84101e30 −0.141333 −0.0706664 0.997500i $$-0.522513\pi$$
−0.0706664 + 0.997500i $$0.522513\pi$$
$$920$$ 0 0
$$921$$ −4.68340e30 −0.351427
$$922$$ 0 0
$$923$$ 2.33122e31 1.70988
$$924$$ 0 0
$$925$$ 9.12313e30 0.654118
$$926$$ 0 0
$$927$$ 6.26265e30 0.438956
$$928$$ 0 0
$$929$$ −1.19812e31 −0.820983 −0.410491 0.911864i $$-0.634643\pi$$
−0.410491 + 0.911864i $$0.634643\pi$$
$$930$$ 0 0
$$931$$ −4.84773e30 −0.324763
$$932$$ 0 0
$$933$$ −7.56091e30 −0.495242
$$934$$ 0 0
$$935$$ 7.61781e30 0.487875
$$936$$ 0 0
$$937$$ −3.39787e30 −0.212785 −0.106392 0.994324i $$-0.533930\pi$$
−0.106392 + 0.994324i $$0.533930\pi$$
$$938$$ 0 0
$$939$$ 1.01379e31 0.620813
$$940$$ 0 0
$$941$$ 2.36473e31 1.41609 0.708045 0.706167i $$-0.249577\pi$$
0.708045 + 0.706167i $$0.249577\pi$$
$$942$$ 0 0
$$943$$ −1.59202e31 −0.932341
$$944$$ 0 0
$$945$$ −7.02016e30 −0.402080
$$946$$ 0 0
$$947$$ −3.07342e31 −1.72166 −0.860829 0.508894i $$-0.830055\pi$$
−0.860829 + 0.508894i $$0.830055\pi$$
$$948$$ 0 0
$$949$$ 1.78019e31 0.975373
$$950$$ 0 0
$$951$$ −2.09160e31 −1.12094
$$952$$ 0 0
$$953$$ 2.23214e31 1.17016 0.585082 0.810974i $$-0.301062\pi$$
0.585082 + 0.810974i $$0.301062\pi$$
$$954$$ 0 0
$$955$$ −1.32115e31 −0.677509
$$956$$ 0 0
$$957$$ 1.65307e31 0.829310
$$958$$ 0 0
$$959$$ 2.26449e31 1.11141
$$960$$ 0 0
$$961$$ 1.49463e31 0.717693
$$962$$ 0 0
$$963$$ −5.31019e28 −0.00249479
$$964$$ 0 0
$$965$$ −4.88754e31 −2.24674
$$966$$ 0 0
$$967$$ −1.97944e31 −0.890356 −0.445178 0.895442i $$-0.646860\pi$$
−0.445178 + 0.895442i $$0.646860\pi$$
$$968$$ 0 0
$$969$$ −6.88055e29 −0.0302847
$$970$$ 0 0
$$971$$ −2.03370e31 −0.875961 −0.437980 0.898985i $$-0.644306\pi$$
−0.437980 + 0.898985i $$0.644306\pi$$
$$972$$ 0 0
$$973$$ 8.90921e30 0.375538
$$974$$ 0 0
$$975$$ 1.61833e31 0.667604
$$976$$ 0 0
$$977$$ −4.80480e31 −1.93991 −0.969956 0.243280i $$-0.921777\pi$$
−0.969956 + 0.243280i $$0.921777\pi$$
$$978$$ 0 0
$$979$$ 3.37438e31 1.33345
$$980$$ 0 0
$$981$$ −6.12562e29 −0.0236933
$$982$$ 0 0
$$983$$ 1.07092e31 0.405458 0.202729 0.979235i $$-0.435019\pi$$
0.202729 + 0.979235i $$0.435019\pi$$
$$984$$ 0 0
$$985$$ −4.88435e31 −1.81020
$$986$$ 0 0
$$987$$ −8.57395e30 −0.311065
$$988$$ 0 0
$$989$$ 3.66676e31 1.30233
$$990$$ 0 0
$$991$$ 2.08415e31 0.724694 0.362347 0.932043i $$-0.381976\pi$$
0.362347 + 0.932043i $$0.381976\pi$$
$$992$$ 0 0
$$993$$ 1.47643e31 0.502625
$$994$$ 0 0
$$995$$ −4.98173e31 −1.66050
$$996$$ 0 0
$$997$$ −1.63103e31 −0.532306 −0.266153 0.963931i $$-0.585753\pi$$
−0.266153 + 0.963931i $$0.585753\pi$$
$$998$$ 0 0
$$999$$ 3.96603e30 0.126741
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.1.1 3
4.3 odd 2 24.22.a.c.1.1 3
12.11 even 2 72.22.a.d.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
24.22.a.c.1.1 3 4.3 odd 2
48.22.a.l.1.1 3 1.1 even 1 trivial
72.22.a.d.1.3 3 12.11 even 2