# Properties

 Label 24.22.a.c.1.3 Level $24$ Weight $22$ Character 24.1 Self dual yes Analytic conductor $67.075$ Analytic rank $1$ 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] = [24,22,Mod(1,24)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, 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("24.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 24.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: $$67.0745626289$$ Analytic rank: $$1$$ 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: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1671.27$$ of defining polynomial Character $$\chi$$ $$=$$ 24.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-59049.0 q^{3} +4.18179e7 q^{5} -6.87132e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$$ $$q-59049.0 q^{3} +4.18179e7 q^{5} -6.87132e8 q^{7} +3.48678e9 q^{9} +1.49871e9 q^{11} -8.84777e9 q^{13} -2.46931e12 q^{15} -9.41434e12 q^{17} -4.23870e13 q^{19} +4.05745e13 q^{21} +1.09316e14 q^{23} +1.27190e15 q^{25} -2.05891e14 q^{27} +1.18227e15 q^{29} -1.95307e15 q^{31} -8.84971e13 q^{33} -2.87345e16 q^{35} +4.20325e16 q^{37} +5.22452e14 q^{39} -6.92189e16 q^{41} +8.95366e16 q^{43} +1.45810e17 q^{45} -5.26493e17 q^{47} -8.63955e16 q^{49} +5.55907e17 q^{51} -2.24753e18 q^{53} +6.26728e16 q^{55} +2.50291e18 q^{57} -2.41265e17 q^{59} +3.17652e18 q^{61} -2.39588e18 q^{63} -3.69996e17 q^{65} -2.04213e19 q^{67} -6.45498e18 q^{69} -3.92177e19 q^{71} -4.75993e19 q^{73} -7.51046e19 q^{75} -1.02981e18 q^{77} +8.63125e18 q^{79} +1.21577e19 q^{81} +2.65834e20 q^{83} -3.93688e20 q^{85} -6.98121e19 q^{87} -4.57167e20 q^{89} +6.07959e18 q^{91} +1.15327e20 q^{93} -1.77254e21 q^{95} -9.21686e20 q^{97} +5.22566e18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 177147 q^{3} + 2080026 q^{5} - 1205282064 q^{7} + 10460353203 q^{9}+O(q^{10})$$ 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
$$5$$ 4.18179e7 1.91504 0.957520 0.288368i $$-0.0931127\pi$$
0.957520 + 0.288368i $$0.0931127\pi$$
$$6$$ 0 0
$$7$$ −6.87132e8 −0.919413 −0.459707 0.888071i $$-0.652045\pi$$
−0.459707 + 0.888071i $$0.652045\pi$$
$$8$$ 0 0
$$9$$ 3.48678e9 0.333333
$$10$$ 0 0
$$11$$ 1.49871e9 0.0174218 0.00871090 0.999962i $$-0.497227\pi$$
0.00871090 + 0.999962i $$0.497227\pi$$
$$12$$ 0 0
$$13$$ −8.84777e9 −0.0178004 −0.00890018 0.999960i $$-0.502833\pi$$
−0.00890018 + 0.999960i $$0.502833\pi$$
$$14$$ 0 0
$$15$$ −2.46931e12 −1.10565
$$16$$ 0 0
$$17$$ −9.41434e12 −1.13260 −0.566299 0.824200i $$-0.691625\pi$$
−0.566299 + 0.824200i $$0.691625\pi$$
$$18$$ 0 0
$$19$$ −4.23870e13 −1.58606 −0.793030 0.609183i $$-0.791498\pi$$
−0.793030 + 0.609183i $$0.791498\pi$$
$$20$$ 0 0
$$21$$ 4.05745e13 0.530823
$$22$$ 0 0
$$23$$ 1.09316e14 0.550225 0.275112 0.961412i $$-0.411285\pi$$
0.275112 + 0.961412i $$0.411285\pi$$
$$24$$ 0 0
$$25$$ 1.27190e15 2.66738
$$26$$ 0 0
$$27$$ −2.05891e14 −0.192450
$$28$$ 0 0
$$29$$ 1.18227e15 0.521842 0.260921 0.965360i $$-0.415974\pi$$
0.260921 + 0.965360i $$0.415974\pi$$
$$30$$ 0 0
$$31$$ −1.95307e15 −0.427977 −0.213988 0.976836i $$-0.568645\pi$$
−0.213988 + 0.976836i $$0.568645\pi$$
$$32$$ 0 0
$$33$$ −8.84971e13 −0.0100585
$$34$$ 0 0
$$35$$ −2.87345e16 −1.76071
$$36$$ 0 0
$$37$$ 4.20325e16 1.43704 0.718518 0.695509i $$-0.244821\pi$$
0.718518 + 0.695509i $$0.244821\pi$$
$$38$$ 0 0
$$39$$ 5.22452e14 0.0102770
$$40$$ 0 0
$$41$$ −6.92189e16 −0.805367 −0.402684 0.915339i $$-0.631923\pi$$
−0.402684 + 0.915339i $$0.631923\pi$$
$$42$$ 0 0
$$43$$ 8.95366e16 0.631803 0.315902 0.948792i $$-0.397693\pi$$
0.315902 + 0.948792i $$0.397693\pi$$
$$44$$ 0 0
$$45$$ 1.45810e17 0.638346
$$46$$ 0 0
$$47$$ −5.26493e17 −1.46004 −0.730021 0.683425i $$-0.760490\pi$$
−0.730021 + 0.683425i $$0.760490\pi$$
$$48$$ 0 0
$$49$$ −8.63955e16 −0.154679
$$50$$ 0 0
$$51$$ 5.55907e17 0.653906
$$52$$ 0 0
$$53$$ −2.24753e18 −1.76526 −0.882629 0.470070i $$-0.844229\pi$$
−0.882629 + 0.470070i $$0.844229\pi$$
$$54$$ 0 0
$$55$$ 6.26728e16 0.0333634
$$56$$ 0 0
$$57$$ 2.50291e18 0.915712
$$58$$ 0 0
$$59$$ −2.41265e17 −0.0614538 −0.0307269 0.999528i $$-0.509782\pi$$
−0.0307269 + 0.999528i $$0.509782\pi$$
$$60$$ 0 0
$$61$$ 3.17652e18 0.570148 0.285074 0.958505i $$-0.407982\pi$$
0.285074 + 0.958505i $$0.407982\pi$$
$$62$$ 0 0
$$63$$ −2.39588e18 −0.306471
$$64$$ 0 0
$$65$$ −3.69996e17 −0.0340884
$$66$$ 0 0
$$67$$ −2.04213e19 −1.36867 −0.684334 0.729169i $$-0.739907\pi$$
−0.684334 + 0.729169i $$0.739907\pi$$
$$68$$ 0 0
$$69$$ −6.45498e18 −0.317672
$$70$$ 0 0
$$71$$ −3.92177e19 −1.42978 −0.714891 0.699236i $$-0.753524\pi$$
−0.714891 + 0.699236i $$0.753524\pi$$
$$72$$ 0 0
$$73$$ −4.75993e19 −1.29631 −0.648157 0.761507i $$-0.724460\pi$$
−0.648157 + 0.761507i $$0.724460\pi$$
$$74$$ 0 0
$$75$$ −7.51046e19 −1.54001
$$76$$ 0 0
$$77$$ −1.02981e18 −0.0160178
$$78$$ 0 0
$$79$$ 8.63125e18 0.102563 0.0512813 0.998684i $$-0.483669\pi$$
0.0512813 + 0.998684i $$0.483669\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ 0 0
$$83$$ 2.65834e20 1.88057 0.940287 0.340382i $$-0.110556\pi$$
0.940287 + 0.340382i $$0.110556\pi$$
$$84$$ 0 0
$$85$$ −3.93688e20 −2.16897
$$86$$ 0 0
$$87$$ −6.98121e19 −0.301286
$$88$$ 0 0
$$89$$ −4.57167e20 −1.55410 −0.777051 0.629438i $$-0.783285\pi$$
−0.777051 + 0.629438i $$0.783285\pi$$
$$90$$ 0 0
$$91$$ 6.07959e18 0.0163659
$$92$$ 0 0
$$93$$ 1.15327e20 0.247093
$$94$$ 0 0
$$95$$ −1.77254e21 −3.03737
$$96$$ 0 0
$$97$$ −9.21686e20 −1.26905 −0.634527 0.772901i $$-0.718805\pi$$
−0.634527 + 0.772901i $$0.718805\pi$$
$$98$$ 0 0
$$99$$ 5.22566e18 0.00580727
$$100$$ 0 0
$$101$$ 1.57744e21 1.42095 0.710475 0.703722i $$-0.248480\pi$$
0.710475 + 0.703722i $$0.248480\pi$$
$$102$$ 0 0
$$103$$ 1.20014e21 0.879919 0.439959 0.898018i $$-0.354993\pi$$
0.439959 + 0.898018i $$0.354993\pi$$
$$104$$ 0 0
$$105$$ 1.69674e21 1.01655
$$106$$ 0 0
$$107$$ −3.40780e21 −1.67473 −0.837365 0.546644i $$-0.815905\pi$$
−0.837365 + 0.546644i $$0.815905\pi$$
$$108$$ 0 0
$$109$$ 1.46243e21 0.591693 0.295846 0.955236i $$-0.404398\pi$$
0.295846 + 0.955236i $$0.404398\pi$$
$$110$$ 0 0
$$111$$ −2.48198e21 −0.829673
$$112$$ 0 0
$$113$$ −1.53398e21 −0.425105 −0.212552 0.977150i $$-0.568178\pi$$
−0.212552 + 0.977150i $$0.568178\pi$$
$$114$$ 0 0
$$115$$ 4.57136e21 1.05370
$$116$$ 0 0
$$117$$ −3.08503e19 −0.00593345
$$118$$ 0 0
$$119$$ 6.46889e21 1.04133
$$120$$ 0 0
$$121$$ −7.39800e21 −0.999696
$$122$$ 0 0
$$123$$ 4.08731e21 0.464979
$$124$$ 0 0
$$125$$ 3.32481e22 3.19309
$$126$$ 0 0
$$127$$ 1.28428e21 0.104405 0.0522024 0.998637i $$-0.483376\pi$$
0.0522024 + 0.998637i $$0.483376\pi$$
$$128$$ 0 0
$$129$$ −5.28705e21 −0.364772
$$130$$ 0 0
$$131$$ 1.22923e22 0.721577 0.360789 0.932648i $$-0.382508\pi$$
0.360789 + 0.932648i $$0.382508\pi$$
$$132$$ 0 0
$$133$$ 2.91254e22 1.45824
$$134$$ 0 0
$$135$$ −8.60995e21 −0.368549
$$136$$ 0 0
$$137$$ −1.02076e22 −0.374420 −0.187210 0.982320i $$-0.559945\pi$$
−0.187210 + 0.982320i $$0.559945\pi$$
$$138$$ 0 0
$$139$$ −5.48653e22 −1.72839 −0.864197 0.503154i $$-0.832173\pi$$
−0.864197 + 0.503154i $$0.832173\pi$$
$$140$$ 0 0
$$141$$ 3.10889e22 0.842955
$$142$$ 0 0
$$143$$ −1.32602e19 −0.000310114 0
$$144$$ 0 0
$$145$$ 4.94403e22 0.999348
$$146$$ 0 0
$$147$$ 5.10157e21 0.0893041
$$148$$ 0 0
$$149$$ −3.46082e22 −0.525682 −0.262841 0.964839i $$-0.584659\pi$$
−0.262841 + 0.964839i $$0.584659\pi$$
$$150$$ 0 0
$$151$$ −3.56145e21 −0.0470294 −0.0235147 0.999723i $$-0.507486\pi$$
−0.0235147 + 0.999723i $$0.507486\pi$$
$$152$$ 0 0
$$153$$ −3.28258e22 −0.377533
$$154$$ 0 0
$$155$$ −8.16734e22 −0.819592
$$156$$ 0 0
$$157$$ 2.21166e23 1.93987 0.969933 0.243374i $$-0.0782541\pi$$
0.969933 + 0.243374i $$0.0782541\pi$$
$$158$$ 0 0
$$159$$ 1.32714e23 1.01917
$$160$$ 0 0
$$161$$ −7.51143e22 −0.505884
$$162$$ 0 0
$$163$$ −3.21431e23 −1.90159 −0.950797 0.309814i $$-0.899733\pi$$
−0.950797 + 0.309814i $$0.899733\pi$$
$$164$$ 0 0
$$165$$ −3.70077e21 −0.0192624
$$166$$ 0 0
$$167$$ −1.92776e23 −0.884158 −0.442079 0.896976i $$-0.645759\pi$$
−0.442079 + 0.896976i $$0.645759\pi$$
$$168$$ 0 0
$$169$$ −2.46986e23 −0.999683
$$170$$ 0 0
$$171$$ −1.47794e23 −0.528687
$$172$$ 0 0
$$173$$ 1.48749e23 0.470946 0.235473 0.971881i $$-0.424336\pi$$
0.235473 + 0.971881i $$0.424336\pi$$
$$174$$ 0 0
$$175$$ −8.73966e23 −2.45242
$$176$$ 0 0
$$177$$ 1.42465e22 0.0354804
$$178$$ 0 0
$$179$$ −3.04507e23 −0.673970 −0.336985 0.941510i $$-0.609407\pi$$
−0.336985 + 0.941510i $$0.609407\pi$$
$$180$$ 0 0
$$181$$ −2.74401e23 −0.540457 −0.270228 0.962796i $$-0.587099\pi$$
−0.270228 + 0.962796i $$0.587099\pi$$
$$182$$ 0 0
$$183$$ −1.87570e23 −0.329175
$$184$$ 0 0
$$185$$ 1.75771e24 2.75198
$$186$$ 0 0
$$187$$ −1.41093e22 −0.0197319
$$188$$ 0 0
$$189$$ 1.41474e23 0.176941
$$190$$ 0 0
$$191$$ 3.88356e22 0.0434890 0.0217445 0.999764i $$-0.493078\pi$$
0.0217445 + 0.999764i $$0.493078\pi$$
$$192$$ 0 0
$$193$$ 4.72989e23 0.474787 0.237394 0.971414i $$-0.423707\pi$$
0.237394 + 0.971414i $$0.423707\pi$$
$$194$$ 0 0
$$195$$ 2.18479e22 0.0196809
$$196$$ 0 0
$$197$$ 9.09009e22 0.0735653 0.0367826 0.999323i $$-0.488289\pi$$
0.0367826 + 0.999323i $$0.488289\pi$$
$$198$$ 0 0
$$199$$ 9.81119e23 0.714109 0.357055 0.934084i $$-0.383781\pi$$
0.357055 + 0.934084i $$0.383781\pi$$
$$200$$ 0 0
$$201$$ 1.20586e24 0.790200
$$202$$ 0 0
$$203$$ −8.12378e23 −0.479789
$$204$$ 0 0
$$205$$ −2.89459e24 −1.54231
$$206$$ 0 0
$$207$$ 3.81160e23 0.183408
$$208$$ 0 0
$$209$$ −6.35256e22 −0.0276320
$$210$$ 0 0
$$211$$ 1.28388e24 0.505312 0.252656 0.967556i $$-0.418696\pi$$
0.252656 + 0.967556i $$0.418696\pi$$
$$212$$ 0 0
$$213$$ 2.31577e24 0.825485
$$214$$ 0 0
$$215$$ 3.74424e24 1.20993
$$216$$ 0 0
$$217$$ 1.34202e24 0.393488
$$218$$ 0 0
$$219$$ 2.81069e24 0.748427
$$220$$ 0 0
$$221$$ 8.32959e22 0.0201607
$$222$$ 0 0
$$223$$ 1.11219e24 0.244893 0.122446 0.992475i $$-0.460926\pi$$
0.122446 + 0.992475i $$0.460926\pi$$
$$224$$ 0 0
$$225$$ 4.43485e24 0.889125
$$226$$ 0 0
$$227$$ 1.79086e24 0.327182 0.163591 0.986528i $$-0.447692\pi$$
0.163591 + 0.986528i $$0.447692\pi$$
$$228$$ 0 0
$$229$$ −5.98363e24 −0.996993 −0.498496 0.866892i $$-0.666114\pi$$
−0.498496 + 0.866892i $$0.666114\pi$$
$$230$$ 0 0
$$231$$ 6.08092e22 0.00924790
$$232$$ 0 0
$$233$$ 8.20293e24 1.13955 0.569773 0.821802i $$-0.307031\pi$$
0.569773 + 0.821802i $$0.307031\pi$$
$$234$$ 0 0
$$235$$ −2.20169e25 −2.79604
$$236$$ 0 0
$$237$$ −5.09667e23 −0.0592146
$$238$$ 0 0
$$239$$ 7.01227e24 0.745900 0.372950 0.927851i $$-0.378346\pi$$
0.372950 + 0.927851i $$0.378346\pi$$
$$240$$ 0 0
$$241$$ −1.03693e25 −1.01058 −0.505288 0.862951i $$-0.668614\pi$$
−0.505288 + 0.862951i $$0.668614\pi$$
$$242$$ 0 0
$$243$$ −7.17898e23 −0.0641500
$$244$$ 0 0
$$245$$ −3.61288e24 −0.296217
$$246$$ 0 0
$$247$$ 3.75030e23 0.0282324
$$248$$ 0 0
$$249$$ −1.56972e25 −1.08575
$$250$$ 0 0
$$251$$ 2.28168e25 1.45104 0.725522 0.688199i $$-0.241598\pi$$
0.725522 + 0.688199i $$0.241598\pi$$
$$252$$ 0 0
$$253$$ 1.63832e23 0.00958591
$$254$$ 0 0
$$255$$ 2.32469e25 1.25226
$$256$$ 0 0
$$257$$ −1.57165e25 −0.779935 −0.389968 0.920829i $$-0.627514\pi$$
−0.389968 + 0.920829i $$0.627514\pi$$
$$258$$ 0 0
$$259$$ −2.88819e25 −1.32123
$$260$$ 0 0
$$261$$ 4.12233e24 0.173947
$$262$$ 0 0
$$263$$ 1.60373e24 0.0624593 0.0312296 0.999512i $$-0.490058\pi$$
0.0312296 + 0.999512i $$0.490058\pi$$
$$264$$ 0 0
$$265$$ −9.39870e25 −3.38054
$$266$$ 0 0
$$267$$ 2.69952e25 0.897261
$$268$$ 0 0
$$269$$ −4.92003e25 −1.51206 −0.756029 0.654538i $$-0.772863\pi$$
−0.756029 + 0.654538i $$0.772863\pi$$
$$270$$ 0 0
$$271$$ −7.98786e24 −0.227119 −0.113559 0.993531i $$-0.536225\pi$$
−0.113559 + 0.993531i $$0.536225\pi$$
$$272$$ 0 0
$$273$$ −3.58993e23 −0.00944885
$$274$$ 0 0
$$275$$ 1.90621e24 0.0464705
$$276$$ 0 0
$$277$$ 2.26388e25 0.511465 0.255733 0.966748i $$-0.417683\pi$$
0.255733 + 0.966748i $$0.417683\pi$$
$$278$$ 0 0
$$279$$ −6.80994e24 −0.142659
$$280$$ 0 0
$$281$$ 7.09532e24 0.137897 0.0689486 0.997620i $$-0.478036\pi$$
0.0689486 + 0.997620i $$0.478036\pi$$
$$282$$ 0 0
$$283$$ 1.98336e24 0.0357804 0.0178902 0.999840i $$-0.494305\pi$$
0.0178902 + 0.999840i $$0.494305\pi$$
$$284$$ 0 0
$$285$$ 1.04666e26 1.75362
$$286$$ 0 0
$$287$$ 4.75625e25 0.740465
$$288$$ 0 0
$$289$$ 1.95378e25 0.282780
$$290$$ 0 0
$$291$$ 5.44246e25 0.732688
$$292$$ 0 0
$$293$$ 1.17672e26 1.47422 0.737109 0.675774i $$-0.236190\pi$$
0.737109 + 0.675774i $$0.236190\pi$$
$$294$$ 0 0
$$295$$ −1.00892e25 −0.117687
$$296$$ 0 0
$$297$$ −3.08570e23 −0.00335283
$$298$$ 0 0
$$299$$ −9.67200e23 −0.00979419
$$300$$ 0 0
$$301$$ −6.15235e25 −0.580888
$$302$$ 0 0
$$303$$ −9.31464e25 −0.820386
$$304$$ 0 0
$$305$$ 1.32835e26 1.09186
$$306$$ 0 0
$$307$$ 7.71265e25 0.591903 0.295952 0.955203i $$-0.404363\pi$$
0.295952 + 0.955203i $$0.404363\pi$$
$$308$$ 0 0
$$309$$ −7.08673e25 −0.508021
$$310$$ 0 0
$$311$$ 7.96322e25 0.533463 0.266732 0.963771i $$-0.414056\pi$$
0.266732 + 0.963771i $$0.414056\pi$$
$$312$$ 0 0
$$313$$ −5.68844e25 −0.356269 −0.178134 0.984006i $$-0.557006\pi$$
−0.178134 + 0.984006i $$0.557006\pi$$
$$314$$ 0 0
$$315$$ −1.00191e26 −0.586904
$$316$$ 0 0
$$317$$ −1.51016e26 −0.827755 −0.413878 0.910333i $$-0.635826\pi$$
−0.413878 + 0.910333i $$0.635826\pi$$
$$318$$ 0 0
$$319$$ 1.77188e24 0.00909143
$$320$$ 0 0
$$321$$ 2.01227e26 0.966906
$$322$$ 0 0
$$323$$ 3.99045e26 1.79637
$$324$$ 0 0
$$325$$ −1.12535e25 −0.0474802
$$326$$ 0 0
$$327$$ −8.63549e25 −0.341614
$$328$$ 0 0
$$329$$ 3.61770e26 1.34238
$$330$$ 0 0
$$331$$ −3.29882e26 −1.14859 −0.574295 0.818649i $$-0.694724\pi$$
−0.574295 + 0.818649i $$0.694724\pi$$
$$332$$ 0 0
$$333$$ 1.46558e26 0.479012
$$334$$ 0 0
$$335$$ −8.53977e26 −2.62105
$$336$$ 0 0
$$337$$ 4.68433e26 1.35062 0.675310 0.737534i $$-0.264010\pi$$
0.675310 + 0.737534i $$0.264010\pi$$
$$338$$ 0 0
$$339$$ 9.05800e25 0.245434
$$340$$ 0 0
$$341$$ −2.92708e24 −0.00745613
$$342$$ 0 0
$$343$$ 4.43160e26 1.06163
$$344$$ 0 0
$$345$$ −2.69934e26 −0.608355
$$346$$ 0 0
$$347$$ −6.05861e26 −1.28503 −0.642515 0.766273i $$-0.722109\pi$$
−0.642515 + 0.766273i $$0.722109\pi$$
$$348$$ 0 0
$$349$$ 8.48298e25 0.169388 0.0846938 0.996407i $$-0.473009\pi$$
0.0846938 + 0.996407i $$0.473009\pi$$
$$350$$ 0 0
$$351$$ 1.82168e24 0.00342568
$$352$$ 0 0
$$353$$ −6.10413e26 −1.08141 −0.540704 0.841213i $$-0.681842\pi$$
−0.540704 + 0.841213i $$0.681842\pi$$
$$354$$ 0 0
$$355$$ −1.64000e27 −2.73809
$$356$$ 0 0
$$357$$ −3.81982e26 −0.601210
$$358$$ 0 0
$$359$$ −6.20073e26 −0.920346 −0.460173 0.887829i $$-0.652213\pi$$
−0.460173 + 0.887829i $$0.652213\pi$$
$$360$$ 0 0
$$361$$ 1.08245e27 1.51558
$$362$$ 0 0
$$363$$ 4.36845e26 0.577175
$$364$$ 0 0
$$365$$ −1.99050e27 −2.48249
$$366$$ 0 0
$$367$$ 1.05311e27 1.24017 0.620084 0.784536i $$-0.287099\pi$$
0.620084 + 0.784536i $$0.287099\pi$$
$$368$$ 0 0
$$369$$ −2.41351e26 −0.268456
$$370$$ 0 0
$$371$$ 1.54435e27 1.62300
$$372$$ 0 0
$$373$$ 9.54206e26 0.947762 0.473881 0.880589i $$-0.342853\pi$$
0.473881 + 0.880589i $$0.342853\pi$$
$$374$$ 0 0
$$375$$ −1.96326e27 −1.84353
$$376$$ 0 0
$$377$$ −1.04605e25 −0.00928897
$$378$$ 0 0
$$379$$ 7.26138e26 0.609968 0.304984 0.952357i $$-0.401349\pi$$
0.304984 + 0.952357i $$0.401349\pi$$
$$380$$ 0 0
$$381$$ −7.58355e25 −0.0602782
$$382$$ 0 0
$$383$$ 1.16288e27 0.874879 0.437439 0.899248i $$-0.355885\pi$$
0.437439 + 0.899248i $$0.355885\pi$$
$$384$$ 0 0
$$385$$ −4.30645e25 −0.0306748
$$386$$ 0 0
$$387$$ 3.12195e26 0.210601
$$388$$ 0 0
$$389$$ 1.26901e27 0.810951 0.405476 0.914106i $$-0.367106\pi$$
0.405476 + 0.914106i $$0.367106\pi$$
$$390$$ 0 0
$$391$$ −1.02913e27 −0.623184
$$392$$ 0 0
$$393$$ −7.25845e26 −0.416603
$$394$$ 0 0
$$395$$ 3.60941e26 0.196412
$$396$$ 0 0
$$397$$ 3.15060e27 1.62590 0.812948 0.582337i $$-0.197862\pi$$
0.812948 + 0.582337i $$0.197862\pi$$
$$398$$ 0 0
$$399$$ −1.71983e27 −0.841918
$$400$$ 0 0
$$401$$ −2.77564e26 −0.128928 −0.0644640 0.997920i $$-0.520534\pi$$
−0.0644640 + 0.997920i $$0.520534\pi$$
$$402$$ 0 0
$$403$$ 1.72803e25 0.00761814
$$404$$ 0 0
$$405$$ 5.08409e26 0.212782
$$406$$ 0 0
$$407$$ 6.29944e25 0.0250357
$$408$$ 0 0
$$409$$ −1.19865e27 −0.452478 −0.226239 0.974072i $$-0.572643\pi$$
−0.226239 + 0.974072i $$0.572643\pi$$
$$410$$ 0 0
$$411$$ 6.02751e26 0.216172
$$412$$ 0 0
$$413$$ 1.65781e26 0.0565015
$$414$$ 0 0
$$415$$ 1.11166e28 3.60137
$$416$$ 0 0
$$417$$ 3.23974e27 0.997888
$$418$$ 0 0
$$419$$ 5.80111e27 1.69928 0.849638 0.527366i $$-0.176820\pi$$
0.849638 + 0.527366i $$0.176820\pi$$
$$420$$ 0 0
$$421$$ 5.69555e26 0.158699 0.0793494 0.996847i $$-0.474716\pi$$
0.0793494 + 0.996847i $$0.474716\pi$$
$$422$$ 0 0
$$423$$ −1.83577e27 −0.486680
$$424$$ 0 0
$$425$$ −1.19741e28 −3.02107
$$426$$ 0 0
$$427$$ −2.18269e27 −0.524202
$$428$$ 0 0
$$429$$ 7.83002e23 0.000179045 0
$$430$$ 0 0
$$431$$ −1.82453e27 −0.397320 −0.198660 0.980069i $$-0.563659\pi$$
−0.198660 + 0.980069i $$0.563659\pi$$
$$432$$ 0 0
$$433$$ 4.12366e27 0.855381 0.427690 0.903925i $$-0.359327\pi$$
0.427690 + 0.903925i $$0.359327\pi$$
$$434$$ 0 0
$$435$$ −2.91940e27 −0.576974
$$436$$ 0 0
$$437$$ −4.63356e27 −0.872689
$$438$$ 0 0
$$439$$ 6.93325e27 1.24468 0.622342 0.782745i $$-0.286181\pi$$
0.622342 + 0.782745i $$0.286181\pi$$
$$440$$ 0 0
$$441$$ −3.01242e26 −0.0515598
$$442$$ 0 0
$$443$$ −1.53594e27 −0.250689 −0.125345 0.992113i $$-0.540004\pi$$
−0.125345 + 0.992113i $$0.540004\pi$$
$$444$$ 0 0
$$445$$ −1.91178e28 −2.97617
$$446$$ 0 0
$$447$$ 2.04358e27 0.303502
$$448$$ 0 0
$$449$$ −1.32087e27 −0.187186 −0.0935929 0.995611i $$-0.529835\pi$$
−0.0935929 + 0.995611i $$0.529835\pi$$
$$450$$ 0 0
$$451$$ −1.03739e26 −0.0140310
$$452$$ 0 0
$$453$$ 2.10300e26 0.0271524
$$454$$ 0 0
$$455$$ 2.54236e26 0.0313413
$$456$$ 0 0
$$457$$ 2.28794e27 0.269354 0.134677 0.990890i $$-0.457000\pi$$
0.134677 + 0.990890i $$0.457000\pi$$
$$458$$ 0 0
$$459$$ 1.93833e27 0.217969
$$460$$ 0 0
$$461$$ 8.47625e27 0.910635 0.455317 0.890329i $$-0.349526\pi$$
0.455317 + 0.890329i $$0.349526\pi$$
$$462$$ 0 0
$$463$$ −9.11529e27 −0.935772 −0.467886 0.883789i $$-0.654984\pi$$
−0.467886 + 0.883789i $$0.654984\pi$$
$$464$$ 0 0
$$465$$ 4.82273e27 0.473192
$$466$$ 0 0
$$467$$ 1.41181e28 1.32419 0.662093 0.749421i $$-0.269668\pi$$
0.662093 + 0.749421i $$0.269668\pi$$
$$468$$ 0 0
$$469$$ 1.40321e28 1.25837
$$470$$ 0 0
$$471$$ −1.30596e28 −1.11998
$$472$$ 0 0
$$473$$ 1.34189e26 0.0110072
$$474$$ 0 0
$$475$$ −5.39121e28 −4.23062
$$476$$ 0 0
$$477$$ −7.83664e27 −0.588419
$$478$$ 0 0
$$479$$ 2.20208e28 1.58238 0.791189 0.611572i $$-0.209463\pi$$
0.791189 + 0.611572i $$0.209463\pi$$
$$480$$ 0 0
$$481$$ −3.71894e26 −0.0255797
$$482$$ 0 0
$$483$$ 4.43542e27 0.292072
$$484$$ 0 0
$$485$$ −3.85430e28 −2.43029
$$486$$ 0 0
$$487$$ −1.76771e28 −1.06748 −0.533738 0.845650i $$-0.679213\pi$$
−0.533738 + 0.845650i $$0.679213\pi$$
$$488$$ 0 0
$$489$$ 1.89802e28 1.09789
$$490$$ 0 0
$$491$$ −5.25882e27 −0.291429 −0.145715 0.989327i $$-0.546548\pi$$
−0.145715 + 0.989327i $$0.546548\pi$$
$$492$$ 0 0
$$493$$ −1.11303e28 −0.591038
$$494$$ 0 0
$$495$$ 2.18527e26 0.0111211
$$496$$ 0 0
$$497$$ 2.69478e28 1.31456
$$498$$ 0 0
$$499$$ −1.70312e28 −0.796508 −0.398254 0.917275i $$-0.630384\pi$$
−0.398254 + 0.917275i $$0.630384\pi$$
$$500$$ 0 0
$$501$$ 1.13832e28 0.510469
$$502$$ 0 0
$$503$$ 7.74942e27 0.333277 0.166639 0.986018i $$-0.446709\pi$$
0.166639 + 0.986018i $$0.446709\pi$$
$$504$$ 0 0
$$505$$ 6.59654e28 2.72118
$$506$$ 0 0
$$507$$ 1.45843e28 0.577167
$$508$$ 0 0
$$509$$ −2.30034e28 −0.873483 −0.436742 0.899587i $$-0.643868\pi$$
−0.436742 + 0.899587i $$0.643868\pi$$
$$510$$ 0 0
$$511$$ 3.27070e28 1.19185
$$512$$ 0 0
$$513$$ 8.72710e27 0.305237
$$514$$ 0 0
$$515$$ 5.01876e28 1.68508
$$516$$ 0 0
$$517$$ −7.89058e26 −0.0254366
$$518$$ 0 0
$$519$$ −8.78350e27 −0.271901
$$520$$ 0 0
$$521$$ −6.09418e27 −0.181184 −0.0905918 0.995888i $$-0.528876\pi$$
−0.0905918 + 0.995888i $$0.528876\pi$$
$$522$$ 0 0
$$523$$ 2.18205e28 0.623155 0.311578 0.950221i $$-0.399143\pi$$
0.311578 + 0.950221i $$0.399143\pi$$
$$524$$ 0 0
$$525$$ 5.16068e28 1.41591
$$526$$ 0 0
$$527$$ 1.83869e28 0.484726
$$528$$ 0 0
$$529$$ −2.75217e28 −0.697253
$$530$$ 0 0
$$531$$ −8.41241e26 −0.0204846
$$532$$ 0 0
$$533$$ 6.12433e26 0.0143358
$$534$$ 0 0
$$535$$ −1.42507e29 −3.20718
$$536$$ 0 0
$$537$$ 1.79808e28 0.389117
$$538$$ 0 0
$$539$$ −1.29481e26 −0.00269479
$$540$$ 0 0
$$541$$ −6.59608e27 −0.132043 −0.0660214 0.997818i $$-0.521031\pi$$
−0.0660214 + 0.997818i $$0.521031\pi$$
$$542$$ 0 0
$$543$$ 1.62031e28 0.312033
$$544$$ 0 0
$$545$$ 6.11557e28 1.13311
$$546$$ 0 0
$$547$$ 4.42458e28 0.788870 0.394435 0.918924i $$-0.370940\pi$$
0.394435 + 0.918924i $$0.370940\pi$$
$$548$$ 0 0
$$549$$ 1.10758e28 0.190049
$$550$$ 0 0
$$551$$ −5.01130e28 −0.827673
$$552$$ 0 0
$$553$$ −5.93081e27 −0.0942975
$$554$$ 0 0
$$555$$ −1.03791e29 −1.58886
$$556$$ 0 0
$$557$$ 4.40727e28 0.649666 0.324833 0.945771i $$-0.394692\pi$$
0.324833 + 0.945771i $$0.394692\pi$$
$$558$$ 0 0
$$559$$ −7.92199e26 −0.0112463
$$560$$ 0 0
$$561$$ 8.33141e26 0.0113922
$$562$$ 0 0
$$563$$ 6.93890e28 0.914013 0.457006 0.889463i $$-0.348922\pi$$
0.457006 + 0.889463i $$0.348922\pi$$
$$564$$ 0 0
$$565$$ −6.41479e28 −0.814093
$$566$$ 0 0
$$567$$ −8.35392e27 −0.102157
$$568$$ 0 0
$$569$$ −1.87409e28 −0.220858 −0.110429 0.993884i $$-0.535222\pi$$
−0.110429 + 0.993884i $$0.535222\pi$$
$$570$$ 0 0
$$571$$ −3.73641e28 −0.424400 −0.212200 0.977226i $$-0.568063\pi$$
−0.212200 + 0.977226i $$0.568063\pi$$
$$572$$ 0 0
$$573$$ −2.29320e27 −0.0251084
$$574$$ 0 0
$$575$$ 1.39039e29 1.46766
$$576$$ 0 0
$$577$$ −1.05657e29 −1.07536 −0.537680 0.843149i $$-0.680699\pi$$
−0.537680 + 0.843149i $$0.680699\pi$$
$$578$$ 0 0
$$579$$ −2.79295e28 −0.274118
$$580$$ 0 0
$$581$$ −1.82663e29 −1.72903
$$582$$ 0 0
$$583$$ −3.36838e27 −0.0307540
$$584$$ 0 0
$$585$$ −1.29009e27 −0.0113628
$$586$$ 0 0
$$587$$ −1.31403e29 −1.11662 −0.558310 0.829632i $$-0.688550\pi$$
−0.558310 + 0.829632i $$0.688550\pi$$
$$588$$ 0 0
$$589$$ 8.27847e28 0.678797
$$590$$ 0 0
$$591$$ −5.36761e27 −0.0424729
$$592$$ 0 0
$$593$$ −2.29428e29 −1.75215 −0.876075 0.482174i $$-0.839847\pi$$
−0.876075 + 0.482174i $$0.839847\pi$$
$$594$$ 0 0
$$595$$ 2.70516e29 1.99418
$$596$$ 0 0
$$597$$ −5.79341e28 −0.412291
$$598$$ 0 0
$$599$$ 1.80553e27 0.0124058 0.00620288 0.999981i $$-0.498026\pi$$
0.00620288 + 0.999981i $$0.498026\pi$$
$$600$$ 0 0
$$601$$ 9.79897e28 0.650128 0.325064 0.945692i $$-0.394614\pi$$
0.325064 + 0.945692i $$0.394614\pi$$
$$602$$ 0 0
$$603$$ −7.12047e28 −0.456222
$$604$$ 0 0
$$605$$ −3.09369e29 −1.91446
$$606$$ 0 0
$$607$$ −9.51352e28 −0.568670 −0.284335 0.958725i $$-0.591773\pi$$
−0.284335 + 0.958725i $$0.591773\pi$$
$$608$$ 0 0
$$609$$ 4.79701e28 0.277006
$$610$$ 0 0
$$611$$ 4.65829e27 0.0259893
$$612$$ 0 0
$$613$$ −1.06132e29 −0.572151 −0.286075 0.958207i $$-0.592351\pi$$
−0.286075 + 0.958207i $$0.592351\pi$$
$$614$$ 0 0
$$615$$ 1.70923e29 0.890453
$$616$$ 0 0
$$617$$ 1.24152e29 0.625113 0.312557 0.949899i $$-0.398815\pi$$
0.312557 + 0.949899i $$0.398815\pi$$
$$618$$ 0 0
$$619$$ 3.70280e28 0.180210 0.0901049 0.995932i $$-0.471280\pi$$
0.0901049 + 0.995932i $$0.471280\pi$$
$$620$$ 0 0
$$621$$ −2.25071e28 −0.105891
$$622$$ 0 0
$$623$$ 3.14134e29 1.42886
$$624$$ 0 0
$$625$$ 7.83875e29 3.44752
$$626$$ 0 0
$$627$$ 3.75112e27 0.0159534
$$628$$ 0 0
$$629$$ −3.95708e29 −1.62758
$$630$$ 0 0
$$631$$ −3.98518e29 −1.58540 −0.792702 0.609610i $$-0.791326\pi$$
−0.792702 + 0.609610i $$0.791326\pi$$
$$632$$ 0 0
$$633$$ −7.58120e28 −0.291742
$$634$$ 0 0
$$635$$ 5.37060e28 0.199939
$$636$$ 0 0
$$637$$ 7.64408e26 0.00275335
$$638$$ 0 0
$$639$$ −1.36744e29 −0.476594
$$640$$ 0 0
$$641$$ 3.11810e29 1.05167 0.525836 0.850586i $$-0.323753\pi$$
0.525836 + 0.850586i $$0.323753\pi$$
$$642$$ 0 0
$$643$$ 5.91629e29 1.93123 0.965616 0.259973i $$-0.0837138\pi$$
0.965616 + 0.259973i $$0.0837138\pi$$
$$644$$ 0 0
$$645$$ −2.21093e29 −0.698552
$$646$$ 0 0
$$647$$ 2.57623e29 0.787934 0.393967 0.919125i $$-0.371102\pi$$
0.393967 + 0.919125i $$0.371102\pi$$
$$648$$ 0 0
$$649$$ −3.61586e26 −0.00107064
$$650$$ 0 0
$$651$$ −7.92448e28 −0.227180
$$652$$ 0 0
$$653$$ −4.06486e29 −1.12838 −0.564192 0.825644i $$-0.690812\pi$$
−0.564192 + 0.825644i $$0.690812\pi$$
$$654$$ 0 0
$$655$$ 5.14037e29 1.38185
$$656$$ 0 0
$$657$$ −1.65968e29 −0.432105
$$658$$ 0 0
$$659$$ −1.76184e29 −0.444294 −0.222147 0.975013i $$-0.571306\pi$$
−0.222147 + 0.975013i $$0.571306\pi$$
$$660$$ 0 0
$$661$$ −2.99970e29 −0.732760 −0.366380 0.930465i $$-0.619403\pi$$
−0.366380 + 0.930465i $$0.619403\pi$$
$$662$$ 0 0
$$663$$ −4.91854e27 −0.0116398
$$664$$ 0 0
$$665$$ 1.21797e30 2.79259
$$666$$ 0 0
$$667$$ 1.29241e29 0.287130
$$668$$ 0 0
$$669$$ −6.56734e28 −0.141389
$$670$$ 0 0
$$671$$ 4.76066e27 0.00993301
$$672$$ 0 0
$$673$$ 5.25225e29 1.06215 0.531077 0.847324i $$-0.321788\pi$$
0.531077 + 0.847324i $$0.321788\pi$$
$$674$$ 0 0
$$675$$ −2.61874e29 −0.513337
$$676$$ 0 0
$$677$$ 3.62662e29 0.689162 0.344581 0.938757i $$-0.388021\pi$$
0.344581 + 0.938757i $$0.388021\pi$$
$$678$$ 0 0
$$679$$ 6.33320e29 1.16678
$$680$$ 0 0
$$681$$ −1.05748e29 −0.188898
$$682$$ 0 0
$$683$$ −2.66967e29 −0.462424 −0.231212 0.972903i $$-0.574269\pi$$
−0.231212 + 0.972903i $$0.574269\pi$$
$$684$$ 0 0
$$685$$ −4.26863e29 −0.717030
$$686$$ 0 0
$$687$$ 3.53327e29 0.575614
$$688$$ 0 0
$$689$$ 1.98856e28 0.0314222
$$690$$ 0 0
$$691$$ −5.15096e29 −0.789531 −0.394765 0.918782i $$-0.629174\pi$$
−0.394765 + 0.918782i $$0.629174\pi$$
$$692$$ 0 0
$$693$$ −3.59072e27 −0.00533928
$$694$$ 0 0
$$695$$ −2.29436e30 −3.30994
$$696$$ 0 0
$$697$$ 6.51650e29 0.912158
$$698$$ 0 0
$$699$$ −4.84375e29 −0.657917
$$700$$ 0 0
$$701$$ −4.60995e29 −0.607655 −0.303827 0.952727i $$-0.598265\pi$$
−0.303827 + 0.952727i $$0.598265\pi$$
$$702$$ 0 0
$$703$$ −1.78163e30 −2.27922
$$704$$ 0 0
$$705$$ 1.30007e30 1.61429
$$706$$ 0 0
$$707$$ −1.08391e30 −1.30644
$$708$$ 0 0
$$709$$ −1.05247e30 −1.23147 −0.615733 0.787955i $$-0.711140\pi$$
−0.615733 + 0.787955i $$0.711140\pi$$
$$710$$ 0 0
$$711$$ 3.00953e28 0.0341875
$$712$$ 0 0
$$713$$ −2.13501e29 −0.235483
$$714$$ 0 0
$$715$$ −5.54514e26 −0.000593881 0
$$716$$ 0 0
$$717$$ −4.14068e29 −0.430646
$$718$$ 0 0
$$719$$ 2.88703e29 0.291607 0.145804 0.989314i $$-0.453423\pi$$
0.145804 + 0.989314i $$0.453423\pi$$
$$720$$ 0 0
$$721$$ −8.24658e29 −0.809009
$$722$$ 0 0
$$723$$ 6.12295e29 0.583456
$$724$$ 0 0
$$725$$ 1.50374e30 1.39195
$$726$$ 0 0
$$727$$ 1.96219e30 1.76454 0.882268 0.470748i $$-0.156016\pi$$
0.882268 + 0.470748i $$0.156016\pi$$
$$728$$ 0 0
$$729$$ 4.23912e28 0.0370370
$$730$$ 0 0
$$731$$ −8.42928e29 −0.715580
$$732$$ 0 0
$$733$$ −7.69033e28 −0.0634386 −0.0317193 0.999497i $$-0.510098\pi$$
−0.0317193 + 0.999497i $$0.510098\pi$$
$$734$$ 0 0
$$735$$ 2.13337e29 0.171021
$$736$$ 0 0
$$737$$ −3.06055e28 −0.0238447
$$738$$ 0 0
$$739$$ −4.37247e29 −0.331101 −0.165550 0.986201i $$-0.552940\pi$$
−0.165550 + 0.986201i $$0.552940\pi$$
$$740$$ 0 0
$$741$$ −2.21452e28 −0.0163000
$$742$$ 0 0
$$743$$ 1.93922e30 1.38754 0.693770 0.720197i $$-0.255948\pi$$
0.693770 + 0.720197i $$0.255948\pi$$
$$744$$ 0 0
$$745$$ −1.44724e30 −1.00670
$$746$$ 0 0
$$747$$ 9.26906e29 0.626858
$$748$$ 0 0
$$749$$ 2.34161e30 1.53977
$$750$$ 0 0
$$751$$ −9.91312e29 −0.633857 −0.316928 0.948449i $$-0.602651\pi$$
−0.316928 + 0.948449i $$0.602651\pi$$
$$752$$ 0 0
$$753$$ −1.34731e30 −0.837761
$$754$$ 0 0
$$755$$ −1.48933e29 −0.0900631
$$756$$ 0 0
$$757$$ −2.12609e30 −1.25048 −0.625238 0.780434i $$-0.714998\pi$$
−0.625238 + 0.780434i $$0.714998\pi$$
$$758$$ 0 0
$$759$$ −9.67411e27 −0.00553442
$$760$$ 0 0
$$761$$ −8.13487e29 −0.452701 −0.226351 0.974046i $$-0.572680\pi$$
−0.226351 + 0.974046i $$0.572680\pi$$
$$762$$ 0 0
$$763$$ −1.00488e30 −0.544010
$$764$$ 0 0
$$765$$ −1.37271e30 −0.722990
$$766$$ 0 0
$$767$$ 2.13466e27 0.00109390
$$768$$ 0 0
$$769$$ 3.51201e30 1.75117 0.875587 0.483061i $$-0.160475\pi$$
0.875587 + 0.483061i $$0.160475\pi$$
$$770$$ 0 0
$$771$$ 9.28045e29 0.450296
$$772$$ 0 0
$$773$$ −1.85796e30 −0.877308 −0.438654 0.898656i $$-0.644545\pi$$
−0.438654 + 0.898656i $$0.644545\pi$$
$$774$$ 0 0
$$775$$ −2.48412e30 −1.14157
$$776$$ 0 0
$$777$$ 1.70545e30 0.762812
$$778$$ 0 0
$$779$$ 2.93398e30 1.27736
$$780$$ 0 0
$$781$$ −5.87758e28 −0.0249094
$$782$$ 0 0
$$783$$ −2.43420e29 −0.100429
$$784$$ 0 0
$$785$$ 9.24870e30 3.71492
$$786$$ 0 0
$$787$$ −3.95934e30 −1.54842 −0.774209 0.632930i $$-0.781852\pi$$
−0.774209 + 0.632930i $$0.781852\pi$$
$$788$$ 0 0
$$789$$ −9.46989e28 −0.0360609
$$790$$ 0 0
$$791$$ 1.05405e30 0.390847
$$792$$ 0 0
$$793$$ −2.81051e28 −0.0101488
$$794$$ 0 0
$$795$$ 5.54984e30 1.95175
$$796$$ 0 0
$$797$$ 1.59540e30 0.546460 0.273230 0.961949i $$-0.411908\pi$$
0.273230 + 0.961949i $$0.411908\pi$$
$$798$$ 0 0
$$799$$ 4.95658e30 1.65364
$$800$$ 0 0
$$801$$ −1.59404e30 −0.518034
$$802$$ 0 0
$$803$$ −7.13373e28 −0.0225841
$$804$$ 0 0
$$805$$ −3.14112e30 −0.968787
$$806$$ 0 0
$$807$$ 2.90523e30 0.872987
$$808$$ 0 0
$$809$$ −1.96445e30 −0.575152 −0.287576 0.957758i $$-0.592849\pi$$
−0.287576 + 0.957758i $$0.592849\pi$$
$$810$$ 0 0
$$811$$ 5.85777e30 1.67114 0.835571 0.549382i $$-0.185137\pi$$
0.835571 + 0.549382i $$0.185137\pi$$
$$812$$ 0 0
$$813$$ 4.71675e29 0.131127
$$814$$ 0 0
$$815$$ −1.34416e31 −3.64163
$$816$$ 0 0
$$817$$ −3.79518e30 −1.00208
$$818$$ 0 0
$$819$$ 2.11982e28 0.00545529
$$820$$ 0 0
$$821$$ −4.07728e30 −1.02274 −0.511372 0.859359i $$-0.670863\pi$$
−0.511372 + 0.859359i $$0.670863\pi$$
$$822$$ 0 0
$$823$$ 7.57530e30 1.85226 0.926129 0.377206i $$-0.123115\pi$$
0.926129 + 0.377206i $$0.123115\pi$$
$$824$$ 0 0
$$825$$ −1.12560e29 −0.0268298
$$826$$ 0 0
$$827$$ 8.09644e30 1.88142 0.940711 0.339209i $$-0.110159\pi$$
0.940711 + 0.339209i $$0.110159\pi$$
$$828$$ 0 0
$$829$$ −4.67371e30 −1.05886 −0.529431 0.848353i $$-0.677594\pi$$
−0.529431 + 0.848353i $$0.677594\pi$$
$$830$$ 0 0
$$831$$ −1.33680e30 −0.295295
$$832$$ 0 0
$$833$$ 8.13356e29 0.175190
$$834$$ 0 0
$$835$$ −8.06149e30 −1.69320
$$836$$ 0 0
$$837$$ 4.02120e29 0.0823642
$$838$$ 0 0
$$839$$ −1.78537e30 −0.356637 −0.178319 0.983973i $$-0.557066\pi$$
−0.178319 + 0.983973i $$0.557066\pi$$
$$840$$ 0 0
$$841$$ −3.73507e30 −0.727681
$$842$$ 0 0
$$843$$ −4.18972e29 −0.0796150
$$844$$ 0 0
$$845$$ −1.03285e31 −1.91443
$$846$$ 0 0
$$847$$ 5.08341e30 0.919134
$$848$$ 0 0
$$849$$ −1.17116e29 −0.0206578
$$850$$ 0 0
$$851$$ 4.59481e30 0.790692
$$852$$ 0 0
$$853$$ −3.78820e29 −0.0636017 −0.0318008 0.999494i $$-0.510124\pi$$
−0.0318008 + 0.999494i $$0.510124\pi$$
$$854$$ 0 0
$$855$$ −6.18045e30 −1.01246
$$856$$ 0 0
$$857$$ 6.68323e30 1.06829 0.534144 0.845394i $$-0.320634\pi$$
0.534144 + 0.845394i $$0.320634\pi$$
$$858$$ 0 0
$$859$$ 1.08420e30 0.169114 0.0845569 0.996419i $$-0.473053\pi$$
0.0845569 + 0.996419i $$0.473053\pi$$
$$860$$ 0 0
$$861$$ −2.80852e30 −0.427508
$$862$$ 0 0
$$863$$ 2.95381e29 0.0438803 0.0219401 0.999759i $$-0.493016\pi$$
0.0219401 + 0.999759i $$0.493016\pi$$
$$864$$ 0 0
$$865$$ 6.22039e30 0.901880
$$866$$ 0 0
$$867$$ −1.15369e30 −0.163263
$$868$$ 0 0
$$869$$ 1.29357e28 0.00178683
$$870$$ 0 0
$$871$$ 1.80683e29 0.0243628
$$872$$ 0 0
$$873$$ −3.21372e30 −0.423018
$$874$$ 0 0
$$875$$ −2.28458e31 −2.93577
$$876$$ 0 0
$$877$$ −1.13789e31 −1.42759 −0.713797 0.700353i $$-0.753026\pi$$
−0.713797 + 0.700353i $$0.753026\pi$$
$$878$$ 0 0
$$879$$ −6.94839e30 −0.851140
$$880$$ 0 0
$$881$$ 1.24451e30 0.148851 0.0744257 0.997227i $$-0.476288\pi$$
0.0744257 + 0.997227i $$0.476288\pi$$
$$882$$ 0 0
$$883$$ 3.51456e30 0.410472 0.205236 0.978712i $$-0.434204\pi$$
0.205236 + 0.978712i $$0.434204\pi$$
$$884$$ 0 0
$$885$$ 5.95759e29 0.0679463
$$886$$ 0 0
$$887$$ 9.29130e30 1.03485 0.517426 0.855728i $$-0.326890\pi$$
0.517426 + 0.855728i $$0.326890\pi$$
$$888$$ 0 0
$$889$$ −8.82470e29 −0.0959912
$$890$$ 0 0
$$891$$ 1.82208e28 0.00193576
$$892$$ 0 0
$$893$$ 2.23164e31 2.31571
$$894$$ 0 0
$$895$$ −1.27339e31 −1.29068
$$896$$ 0 0
$$897$$ 5.71122e28 0.00565468
$$898$$ 0 0
$$899$$ −2.30906e30 −0.223336
$$900$$ 0 0
$$901$$ 2.11590e31 1.99933
$$902$$ 0 0
$$903$$ 3.63290e30 0.335376
$$904$$ 0 0
$$905$$ −1.14749e31 −1.03500
$$906$$ 0 0
$$907$$ −1.98499e29 −0.0174937 −0.00874685 0.999962i $$-0.502784\pi$$
−0.00874685 + 0.999962i $$0.502784\pi$$
$$908$$ 0 0
$$909$$ 5.50020e30 0.473650
$$910$$ 0 0
$$911$$ −7.61422e29 −0.0640741 −0.0320371 0.999487i $$-0.510199\pi$$
−0.0320371 + 0.999487i $$0.510199\pi$$
$$912$$ 0 0
$$913$$ 3.98407e29 0.0327630
$$914$$ 0 0
$$915$$ −7.84380e30 −0.630383
$$916$$ 0 0
$$917$$ −8.44640e30 −0.663428
$$918$$ 0 0
$$919$$ 2.14472e31 1.64648 0.823242 0.567691i $$-0.192163\pi$$
0.823242 + 0.567691i $$0.192163\pi$$
$$920$$ 0 0
$$921$$ −4.55424e30 −0.341736
$$922$$ 0 0
$$923$$ 3.46989e29 0.0254506
$$924$$ 0 0
$$925$$ 5.34613e31 3.83311
$$926$$ 0 0
$$927$$ 4.18464e30 0.293306
$$928$$ 0 0
$$929$$ −1.20678e31 −0.826916 −0.413458 0.910523i $$-0.635679\pi$$
−0.413458 + 0.910523i $$0.635679\pi$$
$$930$$ 0 0
$$931$$ 3.66204e30 0.245331
$$932$$ 0 0
$$933$$ −4.70220e30 −0.307995
$$934$$ 0 0
$$935$$ −5.90023e29 −0.0377874
$$936$$ 0 0
$$937$$ 1.48955e31 0.932803 0.466401 0.884573i $$-0.345550\pi$$
0.466401 + 0.884573i $$0.345550\pi$$
$$938$$ 0 0
$$939$$ 3.35897e30 0.205692
$$940$$ 0 0
$$941$$ −8.63339e29 −0.0517000 −0.0258500 0.999666i $$-0.508229\pi$$
−0.0258500 + 0.999666i $$0.508229\pi$$
$$942$$ 0 0
$$943$$ −7.56671e30 −0.443133
$$944$$ 0 0
$$945$$ 5.91617e30 0.338849
$$946$$ 0 0
$$947$$ −1.29303e31 −0.724328 −0.362164 0.932114i $$-0.617962\pi$$
−0.362164 + 0.932114i $$0.617962\pi$$
$$948$$ 0 0
$$949$$ 4.21147e29 0.0230749
$$950$$ 0 0
$$951$$ 8.91737e30 0.477905
$$952$$ 0 0
$$953$$ −1.85893e31 −0.974513 −0.487257 0.873259i $$-0.662002\pi$$
−0.487257 + 0.873259i $$0.662002\pi$$
$$954$$ 0 0
$$955$$ 1.62402e30 0.0832831
$$956$$ 0 0
$$957$$ −1.04628e29 −0.00524894
$$958$$ 0 0
$$959$$ 7.01400e30 0.344247
$$960$$ 0 0
$$961$$ −1.70110e31 −0.816836
$$962$$ 0 0
$$963$$ −1.18823e31 −0.558244
$$964$$ 0 0
$$965$$ 1.97794e31 0.909236
$$966$$ 0 0
$$967$$ −2.45390e31 −1.10377 −0.551886 0.833919i $$-0.686092\pi$$
−0.551886 + 0.833919i $$0.686092\pi$$
$$968$$ 0 0
$$969$$ −2.35632e31 −1.03713
$$970$$ 0 0
$$971$$ 3.38258e31 1.45696 0.728478 0.685069i $$-0.240228\pi$$
0.728478 + 0.685069i $$0.240228\pi$$
$$972$$ 0 0
$$973$$ 3.76997e31 1.58911
$$974$$ 0 0
$$975$$ 6.64509e29 0.0274127
$$976$$ 0 0
$$977$$ −6.12303e30 −0.247214 −0.123607 0.992331i $$-0.539446\pi$$
−0.123607 + 0.992331i $$0.539446\pi$$
$$978$$ 0 0
$$979$$ −6.85158e29 −0.0270753
$$980$$ 0 0
$$981$$ 5.09917e30 0.197231
$$982$$ 0 0
$$983$$ −2.33411e31 −0.883710 −0.441855 0.897087i $$-0.645679\pi$$
−0.441855 + 0.897087i $$0.645679\pi$$
$$984$$ 0 0
$$985$$ 3.80129e30 0.140880
$$986$$ 0 0
$$987$$ −2.13622e31 −0.775024
$$988$$ 0 0
$$989$$ 9.78775e30 0.347634
$$990$$ 0 0
$$991$$ −1.93811e31 −0.673913 −0.336957 0.941520i $$-0.609398\pi$$
−0.336957 + 0.941520i $$0.609398\pi$$
$$992$$ 0 0
$$993$$ 1.94792e31 0.663138
$$994$$ 0 0
$$995$$ 4.10284e31 1.36755
$$996$$ 0 0
$$997$$ −4.93033e31 −1.60908 −0.804538 0.593901i $$-0.797587\pi$$
−0.804538 + 0.593901i $$0.797587\pi$$
$$998$$ 0 0
$$999$$ −8.65412e30 −0.276558
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 24.22.a.c.1.3 3
3.2 odd 2 72.22.a.d.1.1 3
4.3 odd 2 48.22.a.l.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
24.22.a.c.1.3 3 1.1 even 1 trivial
48.22.a.l.1.3 3 4.3 odd 2
72.22.a.d.1.1 3 3.2 odd 2