# Properties

 Label 729.2.a.e.1.4 Level $729$ Weight $2$ Character 729.1 Self dual yes Analytic conductor $5.821$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.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: $$5.82109430735$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.7459857.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8$$ x^6 - 3*x^5 - 6*x^4 + 13*x^3 + 12*x^2 - 12*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$3.45779$$ of defining polynomial Character $$\chi$$ $$=$$ 729.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.57840 q^{2} +0.491360 q^{4} -1.67851 q^{5} +2.77928 q^{7} -2.38124 q^{8} +O(q^{10})$$ $$q+1.57840 q^{2} +0.491360 q^{4} -1.67851 q^{5} +2.77928 q^{7} -2.38124 q^{8} -2.64936 q^{10} +4.15122 q^{11} +6.87605 q^{13} +4.38683 q^{14} -4.74129 q^{16} -0.976551 q^{17} +2.68529 q^{19} -0.824751 q^{20} +6.55231 q^{22} -1.61317 q^{23} -2.18261 q^{25} +10.8532 q^{26} +1.36563 q^{28} +8.22788 q^{29} +1.04407 q^{31} -2.72118 q^{32} -1.54139 q^{34} -4.66505 q^{35} -1.30834 q^{37} +4.23847 q^{38} +3.99694 q^{40} +4.84817 q^{41} +9.84023 q^{43} +2.03974 q^{44} -2.54623 q^{46} -12.4977 q^{47} +0.724408 q^{49} -3.44504 q^{50} +3.37861 q^{52} -7.34280 q^{53} -6.96786 q^{55} -6.61815 q^{56} +12.9869 q^{58} -9.05188 q^{59} -1.28574 q^{61} +1.64796 q^{62} +5.18745 q^{64} -11.5415 q^{65} -4.64630 q^{67} -0.479838 q^{68} -7.36333 q^{70} -5.62373 q^{71} -4.56144 q^{73} -2.06510 q^{74} +1.31944 q^{76} +11.5374 q^{77} +4.65680 q^{79} +7.95828 q^{80} +7.65237 q^{82} -5.76439 q^{83} +1.63915 q^{85} +15.5319 q^{86} -9.88507 q^{88} -4.54442 q^{89} +19.1105 q^{91} -0.792647 q^{92} -19.7264 q^{94} -4.50728 q^{95} +8.57544 q^{97} +1.14341 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10})$$ 6 * q + 3 * q^2 + 9 * q^4 - 3 * q^5 + 6 * q^7 + 6 * q^8 $$6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 24 q^{14} + 15 q^{16} - 9 q^{17} + 12 q^{19} - 21 q^{20} + 3 q^{22} - 12 q^{23} + 9 q^{25} + 24 q^{26} + 3 q^{28} + 21 q^{29} + 15 q^{31} + 30 q^{35} + 3 q^{37} + 15 q^{38} + 3 q^{40} - 12 q^{41} + 6 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 12 q^{49} - 24 q^{50} + 3 q^{52} - 9 q^{53} + 15 q^{55} + 12 q^{56} - 15 q^{58} + 6 q^{59} + 24 q^{61} - 30 q^{62} + 6 q^{64} - 15 q^{65} + 15 q^{67} + 36 q^{68} - 15 q^{70} + 12 q^{73} + 24 q^{74} + 9 q^{76} + 15 q^{77} + 24 q^{79} - 21 q^{80} - 21 q^{82} - 6 q^{83} - 18 q^{85} - 30 q^{86} - 21 q^{88} - 9 q^{89} + 18 q^{91} + 6 q^{92} - 6 q^{94} - 33 q^{95} - 21 q^{97} + 18 q^{98}+O(q^{100})$$ 6 * q + 3 * q^2 + 9 * q^4 - 3 * q^5 + 6 * q^7 + 6 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 + 24 * q^14 + 15 * q^16 - 9 * q^17 + 12 * q^19 - 21 * q^20 + 3 * q^22 - 12 * q^23 + 9 * q^25 + 24 * q^26 + 3 * q^28 + 21 * q^29 + 15 * q^31 + 30 * q^35 + 3 * q^37 + 15 * q^38 + 3 * q^40 - 12 * q^41 + 6 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 12 * q^49 - 24 * q^50 + 3 * q^52 - 9 * q^53 + 15 * q^55 + 12 * q^56 - 15 * q^58 + 6 * q^59 + 24 * q^61 - 30 * q^62 + 6 * q^64 - 15 * q^65 + 15 * q^67 + 36 * q^68 - 15 * q^70 + 12 * q^73 + 24 * q^74 + 9 * q^76 + 15 * q^77 + 24 * q^79 - 21 * q^80 - 21 * q^82 - 6 * q^83 - 18 * q^85 - 30 * q^86 - 21 * q^88 - 9 * q^89 + 18 * q^91 + 6 * q^92 - 6 * q^94 - 33 * q^95 - 21 * q^97 + 18 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.57840 1.11610 0.558050 0.829807i $$-0.311550\pi$$
0.558050 + 0.829807i $$0.311550\pi$$
$$3$$ 0 0
$$4$$ 0.491360 0.245680
$$5$$ −1.67851 −0.750651 −0.375326 0.926893i $$-0.622469\pi$$
−0.375326 + 0.926893i $$0.622469\pi$$
$$6$$ 0 0
$$7$$ 2.77928 1.05047 0.525235 0.850957i $$-0.323977\pi$$
0.525235 + 0.850957i $$0.323977\pi$$
$$8$$ −2.38124 −0.841897
$$9$$ 0 0
$$10$$ −2.64936 −0.837802
$$11$$ 4.15122 1.25164 0.625820 0.779967i $$-0.284764\pi$$
0.625820 + 0.779967i $$0.284764\pi$$
$$12$$ 0 0
$$13$$ 6.87605 1.90707 0.953536 0.301279i $$-0.0974137\pi$$
0.953536 + 0.301279i $$0.0974137\pi$$
$$14$$ 4.38683 1.17243
$$15$$ 0 0
$$16$$ −4.74129 −1.18532
$$17$$ −0.976551 −0.236848 −0.118424 0.992963i $$-0.537784\pi$$
−0.118424 + 0.992963i $$0.537784\pi$$
$$18$$ 0 0
$$19$$ 2.68529 0.616048 0.308024 0.951379i $$-0.400332\pi$$
0.308024 + 0.951379i $$0.400332\pi$$
$$20$$ −0.824751 −0.184420
$$21$$ 0 0
$$22$$ 6.55231 1.39696
$$23$$ −1.61317 −0.336369 −0.168185 0.985756i $$-0.553790\pi$$
−0.168185 + 0.985756i $$0.553790\pi$$
$$24$$ 0 0
$$25$$ −2.18261 −0.436522
$$26$$ 10.8532 2.12848
$$27$$ 0 0
$$28$$ 1.36563 0.258079
$$29$$ 8.22788 1.52788 0.763939 0.645288i $$-0.223263\pi$$
0.763939 + 0.645288i $$0.223263\pi$$
$$30$$ 0 0
$$31$$ 1.04407 0.187520 0.0937602 0.995595i $$-0.470111\pi$$
0.0937602 + 0.995595i $$0.470111\pi$$
$$32$$ −2.72118 −0.481041
$$33$$ 0 0
$$34$$ −1.54139 −0.264347
$$35$$ −4.66505 −0.788537
$$36$$ 0 0
$$37$$ −1.30834 −0.215091 −0.107545 0.994200i $$-0.534299\pi$$
−0.107545 + 0.994200i $$0.534299\pi$$
$$38$$ 4.23847 0.687571
$$39$$ 0 0
$$40$$ 3.99694 0.631971
$$41$$ 4.84817 0.757157 0.378578 0.925569i $$-0.376413\pi$$
0.378578 + 0.925569i $$0.376413\pi$$
$$42$$ 0 0
$$43$$ 9.84023 1.50062 0.750310 0.661086i $$-0.229904\pi$$
0.750310 + 0.661086i $$0.229904\pi$$
$$44$$ 2.03974 0.307503
$$45$$ 0 0
$$46$$ −2.54623 −0.375422
$$47$$ −12.4977 −1.82298 −0.911488 0.411326i $$-0.865066\pi$$
−0.911488 + 0.411326i $$0.865066\pi$$
$$48$$ 0 0
$$49$$ 0.724408 0.103487
$$50$$ −3.44504 −0.487203
$$51$$ 0 0
$$52$$ 3.37861 0.468529
$$53$$ −7.34280 −1.00861 −0.504305 0.863525i $$-0.668251\pi$$
−0.504305 + 0.863525i $$0.668251\pi$$
$$54$$ 0 0
$$55$$ −6.96786 −0.939546
$$56$$ −6.61815 −0.884387
$$57$$ 0 0
$$58$$ 12.9869 1.70527
$$59$$ −9.05188 −1.17845 −0.589227 0.807968i $$-0.700568\pi$$
−0.589227 + 0.807968i $$0.700568\pi$$
$$60$$ 0 0
$$61$$ −1.28574 −0.164622 −0.0823112 0.996607i $$-0.526230\pi$$
−0.0823112 + 0.996607i $$0.526230\pi$$
$$62$$ 1.64796 0.209292
$$63$$ 0 0
$$64$$ 5.18745 0.648432
$$65$$ −11.5415 −1.43155
$$66$$ 0 0
$$67$$ −4.64630 −0.567636 −0.283818 0.958878i $$-0.591601\pi$$
−0.283818 + 0.958878i $$0.591601\pi$$
$$68$$ −0.479838 −0.0581889
$$69$$ 0 0
$$70$$ −7.36333 −0.880086
$$71$$ −5.62373 −0.667414 −0.333707 0.942677i $$-0.608300\pi$$
−0.333707 + 0.942677i $$0.608300\pi$$
$$72$$ 0 0
$$73$$ −4.56144 −0.533877 −0.266938 0.963714i $$-0.586012\pi$$
−0.266938 + 0.963714i $$0.586012\pi$$
$$74$$ −2.06510 −0.240063
$$75$$ 0 0
$$76$$ 1.31944 0.151351
$$77$$ 11.5374 1.31481
$$78$$ 0 0
$$79$$ 4.65680 0.523931 0.261966 0.965077i $$-0.415629\pi$$
0.261966 + 0.965077i $$0.415629\pi$$
$$80$$ 7.95828 0.889763
$$81$$ 0 0
$$82$$ 7.65237 0.845063
$$83$$ −5.76439 −0.632724 −0.316362 0.948639i $$-0.602461\pi$$
−0.316362 + 0.948639i $$0.602461\pi$$
$$84$$ 0 0
$$85$$ 1.63915 0.177791
$$86$$ 15.5319 1.67484
$$87$$ 0 0
$$88$$ −9.88507 −1.05375
$$89$$ −4.54442 −0.481707 −0.240854 0.970561i $$-0.577427\pi$$
−0.240854 + 0.970561i $$0.577427\pi$$
$$90$$ 0 0
$$91$$ 19.1105 2.00332
$$92$$ −0.792647 −0.0826391
$$93$$ 0 0
$$94$$ −19.7264 −2.03463
$$95$$ −4.50728 −0.462437
$$96$$ 0 0
$$97$$ 8.57544 0.870704 0.435352 0.900260i $$-0.356624\pi$$
0.435352 + 0.900260i $$0.356624\pi$$
$$98$$ 1.14341 0.115502
$$99$$ 0 0
$$100$$ −1.07245 −0.107245
$$101$$ 7.80570 0.776696 0.388348 0.921513i $$-0.373046\pi$$
0.388348 + 0.921513i $$0.373046\pi$$
$$102$$ 0 0
$$103$$ −2.16615 −0.213437 −0.106718 0.994289i $$-0.534034\pi$$
−0.106718 + 0.994289i $$0.534034\pi$$
$$104$$ −16.3735 −1.60556
$$105$$ 0 0
$$106$$ −11.5899 −1.12571
$$107$$ −12.5849 −1.21663 −0.608317 0.793695i $$-0.708155\pi$$
−0.608317 + 0.793695i $$0.708155\pi$$
$$108$$ 0 0
$$109$$ −12.2140 −1.16989 −0.584945 0.811073i $$-0.698884\pi$$
−0.584945 + 0.811073i $$0.698884\pi$$
$$110$$ −10.9981 −1.04863
$$111$$ 0 0
$$112$$ −13.1774 −1.24514
$$113$$ −0.450833 −0.0424108 −0.0212054 0.999775i $$-0.506750\pi$$
−0.0212054 + 0.999775i $$0.506750\pi$$
$$114$$ 0 0
$$115$$ 2.70772 0.252496
$$116$$ 4.04285 0.375369
$$117$$ 0 0
$$118$$ −14.2875 −1.31527
$$119$$ −2.71411 −0.248802
$$120$$ 0 0
$$121$$ 6.23265 0.566604
$$122$$ −2.02942 −0.183735
$$123$$ 0 0
$$124$$ 0.513014 0.0460700
$$125$$ 12.0561 1.07833
$$126$$ 0 0
$$127$$ −0.531069 −0.0471247 −0.0235624 0.999722i $$-0.507501\pi$$
−0.0235624 + 0.999722i $$0.507501\pi$$
$$128$$ 13.6303 1.20476
$$129$$ 0 0
$$130$$ −18.2171 −1.59775
$$131$$ 11.4160 0.997424 0.498712 0.866768i $$-0.333807\pi$$
0.498712 + 0.866768i $$0.333807\pi$$
$$132$$ 0 0
$$133$$ 7.46318 0.647139
$$134$$ −7.33374 −0.633539
$$135$$ 0 0
$$136$$ 2.32541 0.199402
$$137$$ 4.22924 0.361329 0.180664 0.983545i $$-0.442175\pi$$
0.180664 + 0.983545i $$0.442175\pi$$
$$138$$ 0 0
$$139$$ −11.1555 −0.946200 −0.473100 0.881009i $$-0.656865\pi$$
−0.473100 + 0.881009i $$0.656865\pi$$
$$140$$ −2.29222 −0.193728
$$141$$ 0 0
$$142$$ −8.87653 −0.744902
$$143$$ 28.5440 2.38697
$$144$$ 0 0
$$145$$ −13.8106 −1.14690
$$146$$ −7.19980 −0.595860
$$147$$ 0 0
$$148$$ −0.642868 −0.0528434
$$149$$ −19.4777 −1.59567 −0.797837 0.602873i $$-0.794023\pi$$
−0.797837 + 0.602873i $$0.794023\pi$$
$$150$$ 0 0
$$151$$ −1.24286 −0.101143 −0.0505713 0.998720i $$-0.516104\pi$$
−0.0505713 + 0.998720i $$0.516104\pi$$
$$152$$ −6.39433 −0.518649
$$153$$ 0 0
$$154$$ 18.2107 1.46746
$$155$$ −1.75248 −0.140762
$$156$$ 0 0
$$157$$ −3.53314 −0.281975 −0.140988 0.990011i $$-0.545028\pi$$
−0.140988 + 0.990011i $$0.545028\pi$$
$$158$$ 7.35031 0.584760
$$159$$ 0 0
$$160$$ 4.56752 0.361094
$$161$$ −4.48345 −0.353346
$$162$$ 0 0
$$163$$ 15.9509 1.24937 0.624685 0.780877i $$-0.285228\pi$$
0.624685 + 0.780877i $$0.285228\pi$$
$$164$$ 2.38220 0.186018
$$165$$ 0 0
$$166$$ −9.09854 −0.706184
$$167$$ 14.4846 1.12086 0.560428 0.828203i $$-0.310637\pi$$
0.560428 + 0.828203i $$0.310637\pi$$
$$168$$ 0 0
$$169$$ 34.2800 2.63692
$$170$$ 2.58724 0.198432
$$171$$ 0 0
$$172$$ 4.83509 0.368672
$$173$$ 12.6174 0.959283 0.479641 0.877465i $$-0.340767\pi$$
0.479641 + 0.877465i $$0.340767\pi$$
$$174$$ 0 0
$$175$$ −6.06609 −0.458554
$$176$$ −19.6821 −1.48360
$$177$$ 0 0
$$178$$ −7.17293 −0.537634
$$179$$ 0.295899 0.0221165 0.0110582 0.999939i $$-0.496480\pi$$
0.0110582 + 0.999939i $$0.496480\pi$$
$$180$$ 0 0
$$181$$ 1.42050 0.105585 0.0527925 0.998606i $$-0.483188\pi$$
0.0527925 + 0.998606i $$0.483188\pi$$
$$182$$ 30.1640 2.23591
$$183$$ 0 0
$$184$$ 3.84135 0.283188
$$185$$ 2.19607 0.161458
$$186$$ 0 0
$$187$$ −4.05388 −0.296449
$$188$$ −6.14087 −0.447869
$$189$$ 0 0
$$190$$ −7.11431 −0.516126
$$191$$ 20.6241 1.49231 0.746153 0.665774i $$-0.231899\pi$$
0.746153 + 0.665774i $$0.231899\pi$$
$$192$$ 0 0
$$193$$ −20.9559 −1.50844 −0.754221 0.656621i $$-0.771985\pi$$
−0.754221 + 0.656621i $$0.771985\pi$$
$$194$$ 13.5355 0.971793
$$195$$ 0 0
$$196$$ 0.355945 0.0254246
$$197$$ −9.59621 −0.683702 −0.341851 0.939754i $$-0.611054\pi$$
−0.341851 + 0.939754i $$0.611054\pi$$
$$198$$ 0 0
$$199$$ −10.6917 −0.757912 −0.378956 0.925415i $$-0.623717\pi$$
−0.378956 + 0.925415i $$0.623717\pi$$
$$200$$ 5.19733 0.367507
$$201$$ 0 0
$$202$$ 12.3206 0.866871
$$203$$ 22.8676 1.60499
$$204$$ 0 0
$$205$$ −8.13769 −0.568361
$$206$$ −3.41906 −0.238217
$$207$$ 0 0
$$208$$ −32.6013 −2.26049
$$209$$ 11.1472 0.771070
$$210$$ 0 0
$$211$$ −15.0502 −1.03610 −0.518051 0.855350i $$-0.673342\pi$$
−0.518051 + 0.855350i $$0.673342\pi$$
$$212$$ −3.60796 −0.247795
$$213$$ 0 0
$$214$$ −19.8641 −1.35788
$$215$$ −16.5169 −1.12644
$$216$$ 0 0
$$217$$ 2.90176 0.196984
$$218$$ −19.2786 −1.30571
$$219$$ 0 0
$$220$$ −3.42373 −0.230828
$$221$$ −6.71481 −0.451687
$$222$$ 0 0
$$223$$ −12.2881 −0.822873 −0.411436 0.911439i $$-0.634973\pi$$
−0.411436 + 0.911439i $$0.634973\pi$$
$$224$$ −7.56292 −0.505319
$$225$$ 0 0
$$226$$ −0.711597 −0.0473347
$$227$$ 3.75256 0.249066 0.124533 0.992215i $$-0.460257\pi$$
0.124533 + 0.992215i $$0.460257\pi$$
$$228$$ 0 0
$$229$$ −18.6024 −1.22928 −0.614641 0.788807i $$-0.710699\pi$$
−0.614641 + 0.788807i $$0.710699\pi$$
$$230$$ 4.27387 0.281811
$$231$$ 0 0
$$232$$ −19.5926 −1.28632
$$233$$ 0.545784 0.0357555 0.0178777 0.999840i $$-0.494309\pi$$
0.0178777 + 0.999840i $$0.494309\pi$$
$$234$$ 0 0
$$235$$ 20.9775 1.36842
$$236$$ −4.44773 −0.289522
$$237$$ 0 0
$$238$$ −4.28396 −0.277688
$$239$$ −20.0947 −1.29982 −0.649911 0.760011i $$-0.725194\pi$$
−0.649911 + 0.760011i $$0.725194\pi$$
$$240$$ 0 0
$$241$$ 15.2913 0.984999 0.492500 0.870313i $$-0.336083\pi$$
0.492500 + 0.870313i $$0.336083\pi$$
$$242$$ 9.83763 0.632387
$$243$$ 0 0
$$244$$ −0.631762 −0.0404444
$$245$$ −1.21592 −0.0776825
$$246$$ 0 0
$$247$$ 18.4642 1.17485
$$248$$ −2.48618 −0.157873
$$249$$ 0 0
$$250$$ 19.0294 1.20352
$$251$$ 12.7563 0.805171 0.402586 0.915382i $$-0.368112\pi$$
0.402586 + 0.915382i $$0.368112\pi$$
$$252$$ 0 0
$$253$$ −6.69663 −0.421013
$$254$$ −0.838241 −0.0525959
$$255$$ 0 0
$$256$$ 11.1391 0.696196
$$257$$ −13.1047 −0.817449 −0.408725 0.912658i $$-0.634026\pi$$
−0.408725 + 0.912658i $$0.634026\pi$$
$$258$$ 0 0
$$259$$ −3.63626 −0.225946
$$260$$ −5.67103 −0.351702
$$261$$ 0 0
$$262$$ 18.0191 1.11323
$$263$$ −9.51418 −0.586669 −0.293335 0.956010i $$-0.594765\pi$$
−0.293335 + 0.956010i $$0.594765\pi$$
$$264$$ 0 0
$$265$$ 12.3249 0.757115
$$266$$ 11.7799 0.722272
$$267$$ 0 0
$$268$$ −2.28301 −0.139457
$$269$$ 22.1408 1.34995 0.674973 0.737842i $$-0.264155\pi$$
0.674973 + 0.737842i $$0.264155\pi$$
$$270$$ 0 0
$$271$$ 27.9627 1.69861 0.849307 0.527899i $$-0.177020\pi$$
0.849307 + 0.527899i $$0.177020\pi$$
$$272$$ 4.63011 0.280742
$$273$$ 0 0
$$274$$ 6.67545 0.403279
$$275$$ −9.06051 −0.546369
$$276$$ 0 0
$$277$$ −18.8837 −1.13461 −0.567305 0.823508i $$-0.692014\pi$$
−0.567305 + 0.823508i $$0.692014\pi$$
$$278$$ −17.6079 −1.05605
$$279$$ 0 0
$$280$$ 11.1086 0.663867
$$281$$ −20.0017 −1.19320 −0.596602 0.802537i $$-0.703483\pi$$
−0.596602 + 0.802537i $$0.703483\pi$$
$$282$$ 0 0
$$283$$ 16.7450 0.995389 0.497695 0.867352i $$-0.334180\pi$$
0.497695 + 0.867352i $$0.334180\pi$$
$$284$$ −2.76328 −0.163970
$$285$$ 0 0
$$286$$ 45.0540 2.66410
$$287$$ 13.4744 0.795371
$$288$$ 0 0
$$289$$ −16.0463 −0.943903
$$290$$ −21.7986 −1.28006
$$291$$ 0 0
$$292$$ −2.24131 −0.131163
$$293$$ 19.5720 1.14341 0.571704 0.820460i $$-0.306282\pi$$
0.571704 + 0.820460i $$0.306282\pi$$
$$294$$ 0 0
$$295$$ 15.1936 0.884608
$$296$$ 3.11549 0.181084
$$297$$ 0 0
$$298$$ −30.7437 −1.78093
$$299$$ −11.0922 −0.641480
$$300$$ 0 0
$$301$$ 27.3488 1.57636
$$302$$ −1.96174 −0.112885
$$303$$ 0 0
$$304$$ −12.7317 −0.730214
$$305$$ 2.15813 0.123574
$$306$$ 0 0
$$307$$ 14.8995 0.850357 0.425179 0.905109i $$-0.360211\pi$$
0.425179 + 0.905109i $$0.360211\pi$$
$$308$$ 5.66902 0.323023
$$309$$ 0 0
$$310$$ −2.76612 −0.157105
$$311$$ 4.59236 0.260409 0.130204 0.991487i $$-0.458437\pi$$
0.130204 + 0.991487i $$0.458437\pi$$
$$312$$ 0 0
$$313$$ 11.8687 0.670858 0.335429 0.942066i $$-0.391119\pi$$
0.335429 + 0.942066i $$0.391119\pi$$
$$314$$ −5.57672 −0.314713
$$315$$ 0 0
$$316$$ 2.28817 0.128719
$$317$$ −14.5173 −0.815375 −0.407688 0.913121i $$-0.633665\pi$$
−0.407688 + 0.913121i $$0.633665\pi$$
$$318$$ 0 0
$$319$$ 34.1557 1.91235
$$320$$ −8.70718 −0.486746
$$321$$ 0 0
$$322$$ −7.07670 −0.394369
$$323$$ −2.62232 −0.145910
$$324$$ 0 0
$$325$$ −15.0077 −0.832480
$$326$$ 25.1769 1.39442
$$327$$ 0 0
$$328$$ −11.5447 −0.637448
$$329$$ −34.7346 −1.91498
$$330$$ 0 0
$$331$$ −8.10570 −0.445530 −0.222765 0.974872i $$-0.571508\pi$$
−0.222765 + 0.974872i $$0.571508\pi$$
$$332$$ −2.83239 −0.155448
$$333$$ 0 0
$$334$$ 22.8626 1.25099
$$335$$ 7.79885 0.426097
$$336$$ 0 0
$$337$$ −18.8080 −1.02453 −0.512267 0.858826i $$-0.671194\pi$$
−0.512267 + 0.858826i $$0.671194\pi$$
$$338$$ 54.1077 2.94307
$$339$$ 0 0
$$340$$ 0.805412 0.0436796
$$341$$ 4.33416 0.234708
$$342$$ 0 0
$$343$$ −17.4416 −0.941760
$$344$$ −23.4320 −1.26337
$$345$$ 0 0
$$346$$ 19.9154 1.07066
$$347$$ −17.6176 −0.945764 −0.472882 0.881126i $$-0.656786\pi$$
−0.472882 + 0.881126i $$0.656786\pi$$
$$348$$ 0 0
$$349$$ −16.9893 −0.909416 −0.454708 0.890641i $$-0.650256\pi$$
−0.454708 + 0.890641i $$0.650256\pi$$
$$350$$ −9.57475 −0.511792
$$351$$ 0 0
$$352$$ −11.2962 −0.602090
$$353$$ −15.1556 −0.806651 −0.403326 0.915057i $$-0.632146\pi$$
−0.403326 + 0.915057i $$0.632146\pi$$
$$354$$ 0 0
$$355$$ 9.43948 0.500996
$$356$$ −2.23294 −0.118346
$$357$$ 0 0
$$358$$ 0.467048 0.0246842
$$359$$ 2.45096 0.129357 0.0646783 0.997906i $$-0.479398\pi$$
0.0646783 + 0.997906i $$0.479398\pi$$
$$360$$ 0 0
$$361$$ −11.7892 −0.620485
$$362$$ 2.24213 0.117844
$$363$$ 0 0
$$364$$ 9.39012 0.492176
$$365$$ 7.65642 0.400755
$$366$$ 0 0
$$367$$ −1.31353 −0.0685659 −0.0342829 0.999412i $$-0.510915\pi$$
−0.0342829 + 0.999412i $$0.510915\pi$$
$$368$$ 7.64850 0.398706
$$369$$ 0 0
$$370$$ 3.46628 0.180203
$$371$$ −20.4077 −1.05952
$$372$$ 0 0
$$373$$ −9.62617 −0.498424 −0.249212 0.968449i $$-0.580172\pi$$
−0.249212 + 0.968449i $$0.580172\pi$$
$$374$$ −6.39866 −0.330867
$$375$$ 0 0
$$376$$ 29.7601 1.53476
$$377$$ 56.5753 2.91377
$$378$$ 0 0
$$379$$ −8.56311 −0.439857 −0.219929 0.975516i $$-0.570582\pi$$
−0.219929 + 0.975516i $$0.570582\pi$$
$$380$$ −2.21470 −0.113611
$$381$$ 0 0
$$382$$ 32.5531 1.66556
$$383$$ 33.6346 1.71865 0.859324 0.511431i $$-0.170884\pi$$
0.859324 + 0.511431i $$0.170884\pi$$
$$384$$ 0 0
$$385$$ −19.3656 −0.986964
$$386$$ −33.0769 −1.68357
$$387$$ 0 0
$$388$$ 4.21363 0.213915
$$389$$ −14.9540 −0.758197 −0.379098 0.925356i $$-0.623766\pi$$
−0.379098 + 0.925356i $$0.623766\pi$$
$$390$$ 0 0
$$391$$ 1.57534 0.0796685
$$392$$ −1.72499 −0.0871252
$$393$$ 0 0
$$394$$ −15.1467 −0.763080
$$395$$ −7.81648 −0.393290
$$396$$ 0 0
$$397$$ −16.7788 −0.842102 −0.421051 0.907037i $$-0.638339\pi$$
−0.421051 + 0.907037i $$0.638339\pi$$
$$398$$ −16.8758 −0.845906
$$399$$ 0 0
$$400$$ 10.3484 0.517419
$$401$$ 13.1263 0.655496 0.327748 0.944765i $$-0.393710\pi$$
0.327748 + 0.944765i $$0.393710\pi$$
$$402$$ 0 0
$$403$$ 7.17907 0.357615
$$404$$ 3.83541 0.190819
$$405$$ 0 0
$$406$$ 36.0943 1.79133
$$407$$ −5.43123 −0.269216
$$408$$ 0 0
$$409$$ 25.4505 1.25845 0.629223 0.777225i $$-0.283373\pi$$
0.629223 + 0.777225i $$0.283373\pi$$
$$410$$ −12.8446 −0.634348
$$411$$ 0 0
$$412$$ −1.06436 −0.0524371
$$413$$ −25.1577 −1.23793
$$414$$ 0 0
$$415$$ 9.67558 0.474955
$$416$$ −18.7109 −0.917379
$$417$$ 0 0
$$418$$ 17.5948 0.860592
$$419$$ −12.6701 −0.618973 −0.309486 0.950904i $$-0.600157\pi$$
−0.309486 + 0.950904i $$0.600157\pi$$
$$420$$ 0 0
$$421$$ −17.6772 −0.861536 −0.430768 0.902463i $$-0.641757\pi$$
−0.430768 + 0.902463i $$0.641757\pi$$
$$422$$ −23.7554 −1.15639
$$423$$ 0 0
$$424$$ 17.4850 0.849146
$$425$$ 2.13143 0.103390
$$426$$ 0 0
$$427$$ −3.57344 −0.172931
$$428$$ −6.18374 −0.298902
$$429$$ 0 0
$$430$$ −26.0703 −1.25722
$$431$$ 15.6974 0.756117 0.378059 0.925782i $$-0.376592\pi$$
0.378059 + 0.925782i $$0.376592\pi$$
$$432$$ 0 0
$$433$$ −12.6258 −0.606759 −0.303380 0.952870i $$-0.598115\pi$$
−0.303380 + 0.952870i $$0.598115\pi$$
$$434$$ 4.58015 0.219854
$$435$$ 0 0
$$436$$ −6.00147 −0.287418
$$437$$ −4.33183 −0.207219
$$438$$ 0 0
$$439$$ 26.9369 1.28563 0.642814 0.766022i $$-0.277767\pi$$
0.642814 + 0.766022i $$0.277767\pi$$
$$440$$ 16.5922 0.791001
$$441$$ 0 0
$$442$$ −10.5987 −0.504128
$$443$$ 34.7279 1.64997 0.824986 0.565153i $$-0.191183\pi$$
0.824986 + 0.565153i $$0.191183\pi$$
$$444$$ 0 0
$$445$$ 7.62784 0.361594
$$446$$ −19.3956 −0.918408
$$447$$ 0 0
$$448$$ 14.4174 0.681158
$$449$$ −20.7461 −0.979070 −0.489535 0.871984i $$-0.662834\pi$$
−0.489535 + 0.871984i $$0.662834\pi$$
$$450$$ 0 0
$$451$$ 20.1258 0.947688
$$452$$ −0.221521 −0.0104195
$$453$$ 0 0
$$454$$ 5.92306 0.277983
$$455$$ −32.0771 −1.50380
$$456$$ 0 0
$$457$$ −7.20773 −0.337164 −0.168582 0.985688i $$-0.553919\pi$$
−0.168582 + 0.985688i $$0.553919\pi$$
$$458$$ −29.3621 −1.37200
$$459$$ 0 0
$$460$$ 1.33046 0.0620332
$$461$$ 23.1268 1.07712 0.538562 0.842586i $$-0.318968\pi$$
0.538562 + 0.842586i $$0.318968\pi$$
$$462$$ 0 0
$$463$$ 4.97584 0.231247 0.115623 0.993293i $$-0.463113\pi$$
0.115623 + 0.993293i $$0.463113\pi$$
$$464$$ −39.0107 −1.81103
$$465$$ 0 0
$$466$$ 0.861467 0.0399067
$$467$$ −12.4814 −0.577569 −0.288784 0.957394i $$-0.593251\pi$$
−0.288784 + 0.957394i $$0.593251\pi$$
$$468$$ 0 0
$$469$$ −12.9134 −0.596284
$$470$$ 33.1110 1.52729
$$471$$ 0 0
$$472$$ 21.5547 0.992137
$$473$$ 40.8490 1.87824
$$474$$ 0 0
$$475$$ −5.86094 −0.268919
$$476$$ −1.33361 −0.0611257
$$477$$ 0 0
$$478$$ −31.7176 −1.45073
$$479$$ 28.4713 1.30089 0.650443 0.759555i $$-0.274583\pi$$
0.650443 + 0.759555i $$0.274583\pi$$
$$480$$ 0 0
$$481$$ −8.99624 −0.410193
$$482$$ 24.1359 1.09936
$$483$$ 0 0
$$484$$ 3.06247 0.139203
$$485$$ −14.3939 −0.653595
$$486$$ 0 0
$$487$$ 29.6841 1.34511 0.672557 0.740045i $$-0.265196\pi$$
0.672557 + 0.740045i $$0.265196\pi$$
$$488$$ 3.06166 0.138595
$$489$$ 0 0
$$490$$ −1.91922 −0.0867015
$$491$$ 12.4050 0.559828 0.279914 0.960025i $$-0.409694\pi$$
0.279914 + 0.960025i $$0.409694\pi$$
$$492$$ 0 0
$$493$$ −8.03494 −0.361876
$$494$$ 29.1439 1.31125
$$495$$ 0 0
$$496$$ −4.95023 −0.222272
$$497$$ −15.6299 −0.701099
$$498$$ 0 0
$$499$$ −2.27982 −0.102059 −0.0510294 0.998697i $$-0.516250\pi$$
−0.0510294 + 0.998697i $$0.516250\pi$$
$$500$$ 5.92387 0.264923
$$501$$ 0 0
$$502$$ 20.1346 0.898652
$$503$$ 41.2812 1.84064 0.920320 0.391167i $$-0.127929\pi$$
0.920320 + 0.391167i $$0.127929\pi$$
$$504$$ 0 0
$$505$$ −13.1019 −0.583028
$$506$$ −10.5700 −0.469893
$$507$$ 0 0
$$508$$ −0.260946 −0.0115776
$$509$$ 16.4242 0.727988 0.363994 0.931401i $$-0.381413\pi$$
0.363994 + 0.931401i $$0.381413\pi$$
$$510$$ 0 0
$$511$$ −12.6775 −0.560821
$$512$$ −9.67844 −0.427731
$$513$$ 0 0
$$514$$ −20.6845 −0.912355
$$515$$ 3.63589 0.160217
$$516$$ 0 0
$$517$$ −51.8807 −2.28171
$$518$$ −5.73949 −0.252179
$$519$$ 0 0
$$520$$ 27.4831 1.20521
$$521$$ −9.29672 −0.407297 −0.203648 0.979044i $$-0.565280\pi$$
−0.203648 + 0.979044i $$0.565280\pi$$
$$522$$ 0 0
$$523$$ −22.7471 −0.994662 −0.497331 0.867561i $$-0.665686\pi$$
−0.497331 + 0.867561i $$0.665686\pi$$
$$524$$ 5.60938 0.245047
$$525$$ 0 0
$$526$$ −15.0172 −0.654782
$$527$$ −1.01959 −0.0444139
$$528$$ 0 0
$$529$$ −20.3977 −0.886856
$$530$$ 19.4537 0.845016
$$531$$ 0 0
$$532$$ 3.66710 0.158989
$$533$$ 33.3362 1.44395
$$534$$ 0 0
$$535$$ 21.1239 0.913267
$$536$$ 11.0640 0.477891
$$537$$ 0 0
$$538$$ 34.9471 1.50668
$$539$$ 3.00718 0.129528
$$540$$ 0 0
$$541$$ 2.38959 0.102737 0.0513683 0.998680i $$-0.483642\pi$$
0.0513683 + 0.998680i $$0.483642\pi$$
$$542$$ 44.1365 1.89582
$$543$$ 0 0
$$544$$ 2.65737 0.113934
$$545$$ 20.5013 0.878179
$$546$$ 0 0
$$547$$ 29.6668 1.26846 0.634230 0.773144i $$-0.281317\pi$$
0.634230 + 0.773144i $$0.281317\pi$$
$$548$$ 2.07808 0.0887712
$$549$$ 0 0
$$550$$ −14.3011 −0.609803
$$551$$ 22.0942 0.941246
$$552$$ 0 0
$$553$$ 12.9426 0.550374
$$554$$ −29.8061 −1.26634
$$555$$ 0 0
$$556$$ −5.48138 −0.232462
$$557$$ 8.41413 0.356518 0.178259 0.983984i $$-0.442954\pi$$
0.178259 + 0.983984i $$0.442954\pi$$
$$558$$ 0 0
$$559$$ 67.6619 2.86179
$$560$$ 22.1183 0.934669
$$561$$ 0 0
$$562$$ −31.5708 −1.33174
$$563$$ −27.2265 −1.14746 −0.573730 0.819044i $$-0.694504\pi$$
−0.573730 + 0.819044i $$0.694504\pi$$
$$564$$ 0 0
$$565$$ 0.756727 0.0318357
$$566$$ 26.4304 1.11095
$$567$$ 0 0
$$568$$ 13.3915 0.561894
$$569$$ −21.9493 −0.920163 −0.460082 0.887877i $$-0.652180\pi$$
−0.460082 + 0.887877i $$0.652180\pi$$
$$570$$ 0 0
$$571$$ −44.8535 −1.87706 −0.938530 0.345199i $$-0.887811\pi$$
−0.938530 + 0.345199i $$0.887811\pi$$
$$572$$ 14.0254 0.586430
$$573$$ 0 0
$$574$$ 21.2681 0.887713
$$575$$ 3.52092 0.146833
$$576$$ 0 0
$$577$$ 12.0191 0.500361 0.250181 0.968199i $$-0.419510\pi$$
0.250181 + 0.968199i $$0.419510\pi$$
$$578$$ −25.3276 −1.05349
$$579$$ 0 0
$$580$$ −6.78595 −0.281771
$$581$$ −16.0209 −0.664658
$$582$$ 0 0
$$583$$ −30.4816 −1.26242
$$584$$ 10.8619 0.449469
$$585$$ 0 0
$$586$$ 30.8925 1.27616
$$587$$ −17.0299 −0.702898 −0.351449 0.936207i $$-0.614311\pi$$
−0.351449 + 0.936207i $$0.614311\pi$$
$$588$$ 0 0
$$589$$ 2.80363 0.115521
$$590$$ 23.9817 0.987311
$$591$$ 0 0
$$592$$ 6.20324 0.254951
$$593$$ −14.9284 −0.613037 −0.306519 0.951865i $$-0.599164\pi$$
−0.306519 + 0.951865i $$0.599164\pi$$
$$594$$ 0 0
$$595$$ 4.55566 0.186764
$$596$$ −9.57056 −0.392025
$$597$$ 0 0
$$598$$ −17.5080 −0.715956
$$599$$ 13.5348 0.553017 0.276508 0.961011i $$-0.410823\pi$$
0.276508 + 0.961011i $$0.410823\pi$$
$$600$$ 0 0
$$601$$ −45.3173 −1.84853 −0.924265 0.381752i $$-0.875321\pi$$
−0.924265 + 0.381752i $$0.875321\pi$$
$$602$$ 43.1674 1.75937
$$603$$ 0 0
$$604$$ −0.610691 −0.0248487
$$605$$ −10.4615 −0.425322
$$606$$ 0 0
$$607$$ 24.4368 0.991859 0.495929 0.868363i $$-0.334827\pi$$
0.495929 + 0.868363i $$0.334827\pi$$
$$608$$ −7.30715 −0.296344
$$609$$ 0 0
$$610$$ 3.40640 0.137921
$$611$$ −85.9348 −3.47655
$$612$$ 0 0
$$613$$ −25.1996 −1.01780 −0.508901 0.860825i $$-0.669948\pi$$
−0.508901 + 0.860825i $$0.669948\pi$$
$$614$$ 23.5174 0.949084
$$615$$ 0 0
$$616$$ −27.4734 −1.10693
$$617$$ 4.04185 0.162719 0.0813594 0.996685i $$-0.474074\pi$$
0.0813594 + 0.996685i $$0.474074\pi$$
$$618$$ 0 0
$$619$$ −44.7700 −1.79946 −0.899729 0.436449i $$-0.856236\pi$$
−0.899729 + 0.436449i $$0.856236\pi$$
$$620$$ −0.861097 −0.0345825
$$621$$ 0 0
$$622$$ 7.24860 0.290642
$$623$$ −12.6302 −0.506019
$$624$$ 0 0
$$625$$ −9.32314 −0.372926
$$626$$ 18.7336 0.748745
$$627$$ 0 0
$$628$$ −1.73604 −0.0692757
$$629$$ 1.27767 0.0509439
$$630$$ 0 0
$$631$$ 31.4116 1.25048 0.625238 0.780434i $$-0.285002\pi$$
0.625238 + 0.780434i $$0.285002\pi$$
$$632$$ −11.0890 −0.441096
$$633$$ 0 0
$$634$$ −22.9142 −0.910041
$$635$$ 0.891403 0.0353742
$$636$$ 0 0
$$637$$ 4.98106 0.197357
$$638$$ 53.9116 2.13438
$$639$$ 0 0
$$640$$ −22.8785 −0.904351
$$641$$ 48.6406 1.92119 0.960594 0.277954i $$-0.0896564\pi$$
0.960594 + 0.277954i $$0.0896564\pi$$
$$642$$ 0 0
$$643$$ −28.0324 −1.10549 −0.552744 0.833351i $$-0.686419\pi$$
−0.552744 + 0.833351i $$0.686419\pi$$
$$644$$ −2.20299 −0.0868099
$$645$$ 0 0
$$646$$ −4.13908 −0.162850
$$647$$ −37.5519 −1.47632 −0.738159 0.674627i $$-0.764304\pi$$
−0.738159 + 0.674627i $$0.764304\pi$$
$$648$$ 0 0
$$649$$ −37.5763 −1.47500
$$650$$ −23.6883 −0.929131
$$651$$ 0 0
$$652$$ 7.83762 0.306945
$$653$$ −5.11222 −0.200057 −0.100028 0.994985i $$-0.531893\pi$$
−0.100028 + 0.994985i $$0.531893\pi$$
$$654$$ 0 0
$$655$$ −19.1619 −0.748718
$$656$$ −22.9866 −0.897474
$$657$$ 0 0
$$658$$ −54.8253 −2.13731
$$659$$ 35.0059 1.36364 0.681818 0.731522i $$-0.261189\pi$$
0.681818 + 0.731522i $$0.261189\pi$$
$$660$$ 0 0
$$661$$ 1.33346 0.0518656 0.0259328 0.999664i $$-0.491744\pi$$
0.0259328 + 0.999664i $$0.491744\pi$$
$$662$$ −12.7941 −0.497256
$$663$$ 0 0
$$664$$ 13.7264 0.532689
$$665$$ −12.5270 −0.485776
$$666$$ 0 0
$$667$$ −13.2730 −0.513931
$$668$$ 7.11717 0.275372
$$669$$ 0 0
$$670$$ 12.3097 0.475567
$$671$$ −5.33740 −0.206048
$$672$$ 0 0
$$673$$ −3.57953 −0.137981 −0.0689904 0.997617i $$-0.521978\pi$$
−0.0689904 + 0.997617i $$0.521978\pi$$
$$674$$ −29.6866 −1.14348
$$675$$ 0 0
$$676$$ 16.8438 0.647839
$$677$$ −36.9389 −1.41968 −0.709839 0.704364i $$-0.751232\pi$$
−0.709839 + 0.704364i $$0.751232\pi$$
$$678$$ 0 0
$$679$$ 23.8336 0.914648
$$680$$ −3.90321 −0.149681
$$681$$ 0 0
$$682$$ 6.84106 0.261958
$$683$$ 38.0166 1.45466 0.727332 0.686286i $$-0.240760\pi$$
0.727332 + 0.686286i $$0.240760\pi$$
$$684$$ 0 0
$$685$$ −7.09881 −0.271232
$$686$$ −27.5300 −1.05110
$$687$$ 0 0
$$688$$ −46.6553 −1.77872
$$689$$ −50.4894 −1.92349
$$690$$ 0 0
$$691$$ 0.550464 0.0209406 0.0104703 0.999945i $$-0.496667\pi$$
0.0104703 + 0.999945i $$0.496667\pi$$
$$692$$ 6.19968 0.235677
$$693$$ 0 0
$$694$$ −27.8077 −1.05557
$$695$$ 18.7246 0.710266
$$696$$ 0 0
$$697$$ −4.73449 −0.179331
$$698$$ −26.8160 −1.01500
$$699$$ 0 0
$$700$$ −2.98064 −0.112657
$$701$$ 19.0242 0.718534 0.359267 0.933235i $$-0.383027\pi$$
0.359267 + 0.933235i $$0.383027\pi$$
$$702$$ 0 0
$$703$$ −3.51328 −0.132506
$$704$$ 21.5343 0.811603
$$705$$ 0 0
$$706$$ −23.9217 −0.900304
$$707$$ 21.6942 0.815896
$$708$$ 0 0
$$709$$ 12.1568 0.456558 0.228279 0.973596i $$-0.426690\pi$$
0.228279 + 0.973596i $$0.426690\pi$$
$$710$$ 14.8993 0.559161
$$711$$ 0 0
$$712$$ 10.8214 0.405548
$$713$$ −1.68426 −0.0630761
$$714$$ 0 0
$$715$$ −47.9113 −1.79178
$$716$$ 0.145393 0.00543358
$$717$$ 0 0
$$718$$ 3.86860 0.144375
$$719$$ 9.77667 0.364608 0.182304 0.983242i $$-0.441644\pi$$
0.182304 + 0.983242i $$0.441644\pi$$
$$720$$ 0 0
$$721$$ −6.02033 −0.224209
$$722$$ −18.6082 −0.692524
$$723$$ 0 0
$$724$$ 0.697978 0.0259401
$$725$$ −17.9583 −0.666953
$$726$$ 0 0
$$727$$ 4.33493 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$728$$ −45.5067 −1.68659
$$729$$ 0 0
$$730$$ 12.0849 0.447283
$$731$$ −9.60949 −0.355420
$$732$$ 0 0
$$733$$ 29.1937 1.07829 0.539146 0.842212i $$-0.318747\pi$$
0.539146 + 0.842212i $$0.318747\pi$$
$$734$$ −2.07329 −0.0765264
$$735$$ 0 0
$$736$$ 4.38972 0.161807
$$737$$ −19.2878 −0.710476
$$738$$ 0 0
$$739$$ 41.5553 1.52864 0.764319 0.644838i $$-0.223075\pi$$
0.764319 + 0.644838i $$0.223075\pi$$
$$740$$ 1.07906 0.0396670
$$741$$ 0 0
$$742$$ −32.2116 −1.18253
$$743$$ 21.3774 0.784259 0.392130 0.919910i $$-0.371738\pi$$
0.392130 + 0.919910i $$0.371738\pi$$
$$744$$ 0 0
$$745$$ 32.6935 1.19780
$$746$$ −15.1940 −0.556291
$$747$$ 0 0
$$748$$ −1.99191 −0.0728316
$$749$$ −34.9771 −1.27804
$$750$$ 0 0
$$751$$ −39.9676 −1.45844 −0.729220 0.684280i $$-0.760117\pi$$
−0.729220 + 0.684280i $$0.760117\pi$$
$$752$$ 59.2552 2.16081
$$753$$ 0 0
$$754$$ 89.2986 3.25206
$$755$$ 2.08615 0.0759228
$$756$$ 0 0
$$757$$ 6.68348 0.242915 0.121458 0.992597i $$-0.461243\pi$$
0.121458 + 0.992597i $$0.461243\pi$$
$$758$$ −13.5161 −0.490925
$$759$$ 0 0
$$760$$ 10.7329 0.389324
$$761$$ 42.0250 1.52341 0.761703 0.647927i $$-0.224364\pi$$
0.761703 + 0.647927i $$0.224364\pi$$
$$762$$ 0 0
$$763$$ −33.9461 −1.22893
$$764$$ 10.1338 0.366630
$$765$$ 0 0
$$766$$ 53.0890 1.91818
$$767$$ −62.2411 −2.24740
$$768$$ 0 0
$$769$$ 2.41323 0.0870233 0.0435117 0.999053i $$-0.486145\pi$$
0.0435117 + 0.999053i $$0.486145\pi$$
$$770$$ −30.5668 −1.10155
$$771$$ 0 0
$$772$$ −10.2969 −0.370594
$$773$$ 1.39780 0.0502754 0.0251377 0.999684i $$-0.491998\pi$$
0.0251377 + 0.999684i $$0.491998\pi$$
$$774$$ 0 0
$$775$$ −2.27880 −0.0818569
$$776$$ −20.4202 −0.733043
$$777$$ 0 0
$$778$$ −23.6034 −0.846224
$$779$$ 13.0187 0.466445
$$780$$ 0 0
$$781$$ −23.3454 −0.835363
$$782$$ 2.48653 0.0889181
$$783$$ 0 0
$$784$$ −3.43462 −0.122665
$$785$$ 5.93040 0.211665
$$786$$ 0 0
$$787$$ −39.6515 −1.41342 −0.706712 0.707501i $$-0.749822\pi$$
−0.706712 + 0.707501i $$0.749822\pi$$
$$788$$ −4.71519 −0.167972
$$789$$ 0 0
$$790$$ −12.3376 −0.438951
$$791$$ −1.25299 −0.0445513
$$792$$ 0 0
$$793$$ −8.84082 −0.313947
$$794$$ −26.4837 −0.939871
$$795$$ 0 0
$$796$$ −5.25345 −0.186204
$$797$$ −3.29387 −0.116675 −0.0583374 0.998297i $$-0.518580\pi$$
−0.0583374 + 0.998297i $$0.518580\pi$$
$$798$$ 0 0
$$799$$ 12.2046 0.431769
$$800$$ 5.93927 0.209985
$$801$$ 0 0
$$802$$ 20.7186 0.731599
$$803$$ −18.9356 −0.668222
$$804$$ 0 0
$$805$$ 7.52551 0.265239
$$806$$ 11.3315 0.399134
$$807$$ 0 0
$$808$$ −18.5873 −0.653898
$$809$$ −6.54436 −0.230087 −0.115044 0.993360i $$-0.536701\pi$$
−0.115044 + 0.993360i $$0.536701\pi$$
$$810$$ 0 0
$$811$$ −44.7516 −1.57144 −0.785721 0.618581i $$-0.787708\pi$$
−0.785721 + 0.618581i $$0.787708\pi$$
$$812$$ 11.2362 0.394314
$$813$$ 0 0
$$814$$ −8.57268 −0.300472
$$815$$ −26.7737 −0.937841
$$816$$ 0 0
$$817$$ 26.4239 0.924454
$$818$$ 40.1712 1.40455
$$819$$ 0 0
$$820$$ −3.99853 −0.139635
$$821$$ −49.6840 −1.73398 −0.866991 0.498324i $$-0.833949\pi$$
−0.866991 + 0.498324i $$0.833949\pi$$
$$822$$ 0 0
$$823$$ 10.6206 0.370211 0.185106 0.982719i $$-0.440737\pi$$
0.185106 + 0.982719i $$0.440737\pi$$
$$824$$ 5.15812 0.179692
$$825$$ 0 0
$$826$$ −39.7090 −1.38165
$$827$$ −16.4008 −0.570311 −0.285156 0.958481i $$-0.592045\pi$$
−0.285156 + 0.958481i $$0.592045\pi$$
$$828$$ 0 0
$$829$$ −2.95645 −0.102682 −0.0513409 0.998681i $$-0.516350\pi$$
−0.0513409 + 0.998681i $$0.516350\pi$$
$$830$$ 15.2720 0.530098
$$831$$ 0 0
$$832$$ 35.6692 1.23661
$$833$$ −0.707421 −0.0245107
$$834$$ 0 0
$$835$$ −24.3126 −0.841372
$$836$$ 5.47730 0.189436
$$837$$ 0 0
$$838$$ −19.9985 −0.690835
$$839$$ −32.4464 −1.12018 −0.560088 0.828433i $$-0.689233\pi$$
−0.560088 + 0.828433i $$0.689233\pi$$
$$840$$ 0 0
$$841$$ 38.6980 1.33441
$$842$$ −27.9018 −0.961561
$$843$$ 0 0
$$844$$ −7.39508 −0.254549
$$845$$ −57.5392 −1.97941
$$846$$ 0 0
$$847$$ 17.3223 0.595201
$$848$$ 34.8143 1.19553
$$849$$ 0 0
$$850$$ 3.36426 0.115393
$$851$$ 2.11058 0.0723498
$$852$$ 0 0
$$853$$ 28.3331 0.970107 0.485053 0.874485i $$-0.338800\pi$$
0.485053 + 0.874485i $$0.338800\pi$$
$$854$$ −5.64033 −0.193008
$$855$$ 0 0
$$856$$ 29.9678 1.02428
$$857$$ 14.0035 0.478349 0.239175 0.970977i $$-0.423123\pi$$
0.239175 + 0.970977i $$0.423123\pi$$
$$858$$ 0 0
$$859$$ −2.15977 −0.0736905 −0.0368452 0.999321i $$-0.511731\pi$$
−0.0368452 + 0.999321i $$0.511731\pi$$
$$860$$ −8.11574 −0.276744
$$861$$ 0 0
$$862$$ 24.7768 0.843903
$$863$$ −14.9487 −0.508859 −0.254430 0.967091i $$-0.581888\pi$$
−0.254430 + 0.967091i $$0.581888\pi$$
$$864$$ 0 0
$$865$$ −21.1784 −0.720087
$$866$$ −19.9287 −0.677204
$$867$$ 0 0
$$868$$ 1.42581 0.0483951
$$869$$ 19.3314 0.655773
$$870$$ 0 0
$$871$$ −31.9482 −1.08252
$$872$$ 29.0845 0.984926
$$873$$ 0 0
$$874$$ −6.83737 −0.231278
$$875$$ 33.5072 1.13275
$$876$$ 0 0
$$877$$ −1.88708 −0.0637221 −0.0318611 0.999492i $$-0.510143\pi$$
−0.0318611 + 0.999492i $$0.510143\pi$$
$$878$$ 42.5173 1.43489
$$879$$ 0 0
$$880$$ 33.0366 1.11366
$$881$$ 46.8258 1.57760 0.788800 0.614649i $$-0.210703\pi$$
0.788800 + 0.614649i $$0.210703\pi$$
$$882$$ 0 0
$$883$$ −30.0635 −1.01172 −0.505858 0.862617i $$-0.668824\pi$$
−0.505858 + 0.862617i $$0.668824\pi$$
$$884$$ −3.29939 −0.110970
$$885$$ 0 0
$$886$$ 54.8147 1.84153
$$887$$ −29.9139 −1.00441 −0.502205 0.864749i $$-0.667478\pi$$
−0.502205 + 0.864749i $$0.667478\pi$$
$$888$$ 0 0
$$889$$ −1.47599 −0.0495031
$$890$$ 12.0398 0.403576
$$891$$ 0 0
$$892$$ −6.03788 −0.202163
$$893$$ −33.5599 −1.12304
$$894$$ 0 0
$$895$$ −0.496668 −0.0166018
$$896$$ 37.8823 1.26556
$$897$$ 0 0
$$898$$ −32.7458 −1.09274
$$899$$ 8.59047 0.286508
$$900$$ 0 0
$$901$$ 7.17062 0.238888
$$902$$ 31.7667 1.05772
$$903$$ 0 0
$$904$$ 1.07354 0.0357055
$$905$$ −2.38432 −0.0792576
$$906$$ 0 0
$$907$$ 26.7652 0.888725 0.444363 0.895847i $$-0.353430\pi$$
0.444363 + 0.895847i $$0.353430\pi$$
$$908$$ 1.84386 0.0611905
$$909$$ 0 0
$$910$$ −50.6306 −1.67839
$$911$$ 0.441137 0.0146155 0.00730776 0.999973i $$-0.497674\pi$$
0.00730776 + 0.999973i $$0.497674\pi$$
$$912$$ 0 0
$$913$$ −23.9293 −0.791943
$$914$$ −11.3767 −0.376308
$$915$$ 0 0
$$916$$ −9.14048 −0.302010
$$917$$ 31.7284 1.04776
$$918$$ 0 0
$$919$$ 49.0749 1.61883 0.809416 0.587236i $$-0.199784\pi$$
0.809416 + 0.587236i $$0.199784\pi$$
$$920$$ −6.44774 −0.212576
$$921$$ 0 0
$$922$$ 36.5035 1.20218
$$923$$ −38.6691 −1.27281
$$924$$ 0 0
$$925$$ 2.85561 0.0938919
$$926$$ 7.85389 0.258095
$$927$$ 0 0
$$928$$ −22.3895 −0.734972
$$929$$ 32.4312 1.06403 0.532017 0.846734i $$-0.321434\pi$$
0.532017 + 0.846734i $$0.321434\pi$$
$$930$$ 0 0
$$931$$ 1.94524 0.0637528
$$932$$ 0.268176 0.00878441
$$933$$ 0 0
$$934$$ −19.7006 −0.644625
$$935$$ 6.80447 0.222530
$$936$$ 0 0
$$937$$ 48.6157 1.58821 0.794103 0.607783i $$-0.207941\pi$$
0.794103 + 0.607783i $$0.207941\pi$$
$$938$$ −20.3825 −0.665513
$$939$$ 0 0
$$940$$ 10.3075 0.336193
$$941$$ −0.710434 −0.0231595 −0.0115797 0.999933i $$-0.503686\pi$$
−0.0115797 + 0.999933i $$0.503686\pi$$
$$942$$ 0 0
$$943$$ −7.82092 −0.254684
$$944$$ 42.9175 1.39685
$$945$$ 0 0
$$946$$ 64.4762 2.09630
$$947$$ 26.1077 0.848387 0.424194 0.905571i $$-0.360558\pi$$
0.424194 + 0.905571i $$0.360558\pi$$
$$948$$ 0 0
$$949$$ −31.3647 −1.01814
$$950$$ −9.25094 −0.300140
$$951$$ 0 0
$$952$$ 6.46296 0.209466
$$953$$ −51.0054 −1.65223 −0.826114 0.563503i $$-0.809453\pi$$
−0.826114 + 0.563503i $$0.809453\pi$$
$$954$$ 0 0
$$955$$ −34.6177 −1.12020
$$956$$ −9.87375 −0.319340
$$957$$ 0 0
$$958$$ 44.9392 1.45192
$$959$$ 11.7543 0.379565
$$960$$ 0 0
$$961$$ −29.9099 −0.964836
$$962$$ −14.1997 −0.457817
$$963$$ 0 0
$$964$$ 7.51353 0.241995
$$965$$ 35.1747 1.13231
$$966$$ 0 0
$$967$$ 10.0048 0.321733 0.160867 0.986976i $$-0.448571\pi$$
0.160867 + 0.986976i $$0.448571\pi$$
$$968$$ −14.8414 −0.477022
$$969$$ 0 0
$$970$$ −22.7195 −0.729478
$$971$$ 44.6269 1.43215 0.716073 0.698025i $$-0.245938\pi$$
0.716073 + 0.698025i $$0.245938\pi$$
$$972$$ 0 0
$$973$$ −31.0044 −0.993954
$$974$$ 46.8535 1.50128
$$975$$ 0 0
$$976$$ 6.09607 0.195130
$$977$$ −39.0856 −1.25046 −0.625230 0.780440i $$-0.714995\pi$$
−0.625230 + 0.780440i $$0.714995\pi$$
$$978$$ 0 0
$$979$$ −18.8649 −0.602925
$$980$$ −0.597456 −0.0190850
$$981$$ 0 0
$$982$$ 19.5800 0.624824
$$983$$ 6.79459 0.216714 0.108357 0.994112i $$-0.465441\pi$$
0.108357 + 0.994112i $$0.465441\pi$$
$$984$$ 0 0
$$985$$ 16.1073 0.513222
$$986$$ −12.6824 −0.403890
$$987$$ 0 0
$$988$$ 9.07255 0.288636
$$989$$ −15.8740 −0.504763
$$990$$ 0 0
$$991$$ 17.2046 0.546522 0.273261 0.961940i $$-0.411898\pi$$
0.273261 + 0.961940i $$0.411898\pi$$
$$992$$ −2.84110 −0.0902049
$$993$$ 0 0
$$994$$ −24.6704 −0.782497
$$995$$ 17.9460 0.568928
$$996$$ 0 0
$$997$$ 7.33368 0.232260 0.116130 0.993234i $$-0.462951\pi$$
0.116130 + 0.993234i $$0.462951\pi$$
$$998$$ −3.59848 −0.113908
$$999$$ 0 0
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 729.2.a.e.1.4 yes 6
3.2 odd 2 729.2.a.b.1.3 6
9.2 odd 6 729.2.c.d.244.4 12
9.4 even 3 729.2.c.a.487.3 12
9.5 odd 6 729.2.c.d.487.4 12
9.7 even 3 729.2.c.a.244.3 12
27.2 odd 18 729.2.e.j.568.2 12
27.4 even 9 729.2.e.l.406.2 12
27.5 odd 18 729.2.e.t.649.1 12
27.7 even 9 729.2.e.l.325.2 12
27.11 odd 18 729.2.e.t.82.1 12
27.13 even 9 729.2.e.u.163.1 12
27.14 odd 18 729.2.e.j.163.2 12
27.16 even 9 729.2.e.k.82.2 12
27.20 odd 18 729.2.e.s.325.1 12
27.22 even 9 729.2.e.k.649.2 12
27.23 odd 18 729.2.e.s.406.1 12
27.25 even 9 729.2.e.u.568.1 12

By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.3 6 3.2 odd 2
729.2.a.e.1.4 yes 6 1.1 even 1 trivial
729.2.c.a.244.3 12 9.7 even 3
729.2.c.a.487.3 12 9.4 even 3
729.2.c.d.244.4 12 9.2 odd 6
729.2.c.d.487.4 12 9.5 odd 6
729.2.e.j.163.2 12 27.14 odd 18
729.2.e.j.568.2 12 27.2 odd 18
729.2.e.k.82.2 12 27.16 even 9
729.2.e.k.649.2 12 27.22 even 9
729.2.e.l.325.2 12 27.7 even 9
729.2.e.l.406.2 12 27.4 even 9
729.2.e.s.325.1 12 27.20 odd 18
729.2.e.s.406.1 12 27.23 odd 18
729.2.e.t.82.1 12 27.11 odd 18
729.2.e.t.649.1 12 27.5 odd 18
729.2.e.u.163.1 12 27.13 even 9
729.2.e.u.568.1 12 27.25 even 9