# Properties

 Label 6080.2.a.bx.1.3 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.34292$$ of defining polynomial Character $$\chi$$ $$=$$ 6080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.34292 q^{3} +1.00000 q^{5} +1.19656 q^{7} +2.48929 q^{9} +O(q^{10})$$ $$q+2.34292 q^{3} +1.00000 q^{5} +1.19656 q^{7} +2.48929 q^{9} -4.97858 q^{11} +6.63565 q^{13} +2.34292 q^{15} +1.48929 q^{17} -1.00000 q^{19} +2.80344 q^{21} -0.510711 q^{23} +1.00000 q^{25} -1.19656 q^{27} +7.88240 q^{29} -2.97858 q^{31} -11.6644 q^{33} +1.19656 q^{35} +7.14637 q^{37} +15.5468 q^{39} +1.66442 q^{41} +6.39312 q^{43} +2.48929 q^{45} -9.95715 q^{47} -5.56825 q^{49} +3.48929 q^{51} +11.4219 q^{53} -4.97858 q^{55} -2.34292 q^{57} +11.8396 q^{59} -3.66442 q^{61} +2.97858 q^{63} +6.63565 q^{65} -7.61423 q^{67} -1.19656 q^{69} +13.8396 q^{73} +2.34292 q^{75} -5.95715 q^{77} +12.6858 q^{79} -10.2713 q^{81} +8.68585 q^{83} +1.48929 q^{85} +18.4679 q^{87} -4.87819 q^{89} +7.93994 q^{91} -6.97858 q^{93} -1.00000 q^{95} -6.81079 q^{97} -12.3931 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 3 q^{5} - q^{7}+O(q^{10})$$ 3 * q + q^3 + 3 * q^5 - q^7 $$3 q + q^{3} + 3 q^{5} - q^{7} + 11 q^{13} + q^{15} - 3 q^{17} - 3 q^{19} + 13 q^{21} - 9 q^{23} + 3 q^{25} + q^{27} + 7 q^{29} + 6 q^{31} - 8 q^{33} - q^{35} + 20 q^{37} + 3 q^{39} - 22 q^{41} + 10 q^{43} + 12 q^{49} + 3 q^{51} + 7 q^{53} - q^{57} - 11 q^{59} + 16 q^{61} - 6 q^{63} + 11 q^{65} + q^{67} + q^{69} - 5 q^{73} + q^{75} + 12 q^{77} + 26 q^{79} - 13 q^{81} + 14 q^{83} - 3 q^{85} + 33 q^{87} - 6 q^{89} - 29 q^{91} - 6 q^{93} - 3 q^{95} + 8 q^{97} - 28 q^{99}+O(q^{100})$$ 3 * q + q^3 + 3 * q^5 - q^7 + 11 * q^13 + q^15 - 3 * q^17 - 3 * q^19 + 13 * q^21 - 9 * q^23 + 3 * q^25 + q^27 + 7 * q^29 + 6 * q^31 - 8 * q^33 - q^35 + 20 * q^37 + 3 * q^39 - 22 * q^41 + 10 * q^43 + 12 * q^49 + 3 * q^51 + 7 * q^53 - q^57 - 11 * q^59 + 16 * q^61 - 6 * q^63 + 11 * q^65 + q^67 + q^69 - 5 * q^73 + q^75 + 12 * q^77 + 26 * q^79 - 13 * q^81 + 14 * q^83 - 3 * q^85 + 33 * q^87 - 6 * q^89 - 29 * q^91 - 6 * q^93 - 3 * q^95 + 8 * q^97 - 28 * 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
$$3$$ 2.34292 1.35269 0.676344 0.736586i $$-0.263563\pi$$
0.676344 + 0.736586i $$0.263563\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 1.19656 0.452256 0.226128 0.974098i $$-0.427393\pi$$
0.226128 + 0.974098i $$0.427393\pi$$
$$8$$ 0 0
$$9$$ 2.48929 0.829763
$$10$$ 0 0
$$11$$ −4.97858 −1.50110 −0.750549 0.660815i $$-0.770211\pi$$
−0.750549 + 0.660815i $$0.770211\pi$$
$$12$$ 0 0
$$13$$ 6.63565 1.84040 0.920200 0.391449i $$-0.128026\pi$$
0.920200 + 0.391449i $$0.128026\pi$$
$$14$$ 0 0
$$15$$ 2.34292 0.604940
$$16$$ 0 0
$$17$$ 1.48929 0.361206 0.180603 0.983556i $$-0.442195\pi$$
0.180603 + 0.983556i $$0.442195\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.80344 0.611761
$$22$$ 0 0
$$23$$ −0.510711 −0.106491 −0.0532453 0.998581i $$-0.516957\pi$$
−0.0532453 + 0.998581i $$0.516957\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.19656 −0.230278
$$28$$ 0 0
$$29$$ 7.88240 1.46373 0.731863 0.681452i $$-0.238651\pi$$
0.731863 + 0.681452i $$0.238651\pi$$
$$30$$ 0 0
$$31$$ −2.97858 −0.534968 −0.267484 0.963562i $$-0.586192\pi$$
−0.267484 + 0.963562i $$0.586192\pi$$
$$32$$ 0 0
$$33$$ −11.6644 −2.03052
$$34$$ 0 0
$$35$$ 1.19656 0.202255
$$36$$ 0 0
$$37$$ 7.14637 1.17486 0.587428 0.809277i $$-0.300141\pi$$
0.587428 + 0.809277i $$0.300141\pi$$
$$38$$ 0 0
$$39$$ 15.5468 2.48948
$$40$$ 0 0
$$41$$ 1.66442 0.259939 0.129970 0.991518i $$-0.458512\pi$$
0.129970 + 0.991518i $$0.458512\pi$$
$$42$$ 0 0
$$43$$ 6.39312 0.974941 0.487470 0.873139i $$-0.337920\pi$$
0.487470 + 0.873139i $$0.337920\pi$$
$$44$$ 0 0
$$45$$ 2.48929 0.371081
$$46$$ 0 0
$$47$$ −9.95715 −1.45240 −0.726200 0.687483i $$-0.758715\pi$$
−0.726200 + 0.687483i $$0.758715\pi$$
$$48$$ 0 0
$$49$$ −5.56825 −0.795464
$$50$$ 0 0
$$51$$ 3.48929 0.488598
$$52$$ 0 0
$$53$$ 11.4219 1.56892 0.784458 0.620182i $$-0.212941\pi$$
0.784458 + 0.620182i $$0.212941\pi$$
$$54$$ 0 0
$$55$$ −4.97858 −0.671311
$$56$$ 0 0
$$57$$ −2.34292 −0.310328
$$58$$ 0 0
$$59$$ 11.8396 1.54138 0.770690 0.637211i $$-0.219912\pi$$
0.770690 + 0.637211i $$0.219912\pi$$
$$60$$ 0 0
$$61$$ −3.66442 −0.469181 −0.234591 0.972094i $$-0.575375\pi$$
−0.234591 + 0.972094i $$0.575375\pi$$
$$62$$ 0 0
$$63$$ 2.97858 0.375265
$$64$$ 0 0
$$65$$ 6.63565 0.823052
$$66$$ 0 0
$$67$$ −7.61423 −0.930226 −0.465113 0.885251i $$-0.653986\pi$$
−0.465113 + 0.885251i $$0.653986\pi$$
$$68$$ 0 0
$$69$$ −1.19656 −0.144049
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 13.8396 1.61980 0.809899 0.586570i $$-0.199522\pi$$
0.809899 + 0.586570i $$0.199522\pi$$
$$74$$ 0 0
$$75$$ 2.34292 0.270537
$$76$$ 0 0
$$77$$ −5.95715 −0.678881
$$78$$ 0 0
$$79$$ 12.6858 1.42727 0.713635 0.700518i $$-0.247048\pi$$
0.713635 + 0.700518i $$0.247048\pi$$
$$80$$ 0 0
$$81$$ −10.2713 −1.14126
$$82$$ 0 0
$$83$$ 8.68585 0.953395 0.476698 0.879067i $$-0.341834\pi$$
0.476698 + 0.879067i $$0.341834\pi$$
$$84$$ 0 0
$$85$$ 1.48929 0.161536
$$86$$ 0 0
$$87$$ 18.4679 1.97996
$$88$$ 0 0
$$89$$ −4.87819 −0.517087 −0.258544 0.966000i $$-0.583243\pi$$
−0.258544 + 0.966000i $$0.583243\pi$$
$$90$$ 0 0
$$91$$ 7.93994 0.832332
$$92$$ 0 0
$$93$$ −6.97858 −0.723645
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −6.81079 −0.691531 −0.345765 0.938321i $$-0.612381\pi$$
−0.345765 + 0.938321i $$0.612381\pi$$
$$98$$ 0 0
$$99$$ −12.3931 −1.24555
$$100$$ 0 0
$$101$$ 2.29273 0.228135 0.114068 0.993473i $$-0.463612\pi$$
0.114068 + 0.993473i $$0.463612\pi$$
$$102$$ 0 0
$$103$$ −6.51806 −0.642243 −0.321122 0.947038i $$-0.604060\pi$$
−0.321122 + 0.947038i $$0.604060\pi$$
$$104$$ 0 0
$$105$$ 2.80344 0.273588
$$106$$ 0 0
$$107$$ −7.71462 −0.745800 −0.372900 0.927872i $$-0.621637\pi$$
−0.372900 + 0.927872i $$0.621637\pi$$
$$108$$ 0 0
$$109$$ −15.5468 −1.48912 −0.744558 0.667558i $$-0.767340\pi$$
−0.744558 + 0.667558i $$0.767340\pi$$
$$110$$ 0 0
$$111$$ 16.7434 1.58921
$$112$$ 0 0
$$113$$ −0.753250 −0.0708598 −0.0354299 0.999372i $$-0.511280\pi$$
−0.0354299 + 0.999372i $$0.511280\pi$$
$$114$$ 0 0
$$115$$ −0.510711 −0.0476241
$$116$$ 0 0
$$117$$ 16.5181 1.52709
$$118$$ 0 0
$$119$$ 1.78202 0.163357
$$120$$ 0 0
$$121$$ 13.7862 1.25329
$$122$$ 0 0
$$123$$ 3.89962 0.351617
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 10.4177 0.924419 0.462210 0.886771i $$-0.347057\pi$$
0.462210 + 0.886771i $$0.347057\pi$$
$$128$$ 0 0
$$129$$ 14.9786 1.31879
$$130$$ 0 0
$$131$$ 17.9572 1.56892 0.784462 0.620177i $$-0.212939\pi$$
0.784462 + 0.620177i $$0.212939\pi$$
$$132$$ 0 0
$$133$$ −1.19656 −0.103755
$$134$$ 0 0
$$135$$ −1.19656 −0.102983
$$136$$ 0 0
$$137$$ −9.83956 −0.840650 −0.420325 0.907374i $$-0.638084\pi$$
−0.420325 + 0.907374i $$0.638084\pi$$
$$138$$ 0 0
$$139$$ 0.978577 0.0830018 0.0415009 0.999138i $$-0.486786\pi$$
0.0415009 + 0.999138i $$0.486786\pi$$
$$140$$ 0 0
$$141$$ −23.3288 −1.96464
$$142$$ 0 0
$$143$$ −33.0361 −2.76262
$$144$$ 0 0
$$145$$ 7.88240 0.654598
$$146$$ 0 0
$$147$$ −13.0460 −1.07601
$$148$$ 0 0
$$149$$ 8.24989 0.675857 0.337928 0.941172i $$-0.390274\pi$$
0.337928 + 0.941172i $$0.390274\pi$$
$$150$$ 0 0
$$151$$ 13.8568 1.12765 0.563824 0.825895i $$-0.309330\pi$$
0.563824 + 0.825895i $$0.309330\pi$$
$$152$$ 0 0
$$153$$ 3.70727 0.299715
$$154$$ 0 0
$$155$$ −2.97858 −0.239245
$$156$$ 0 0
$$157$$ 15.7073 1.25358 0.626788 0.779190i $$-0.284369\pi$$
0.626788 + 0.779190i $$0.284369\pi$$
$$158$$ 0 0
$$159$$ 26.7606 2.12225
$$160$$ 0 0
$$161$$ −0.611096 −0.0481611
$$162$$ 0 0
$$163$$ −5.80765 −0.454891 −0.227445 0.973791i $$-0.573037\pi$$
−0.227445 + 0.973791i $$0.573037\pi$$
$$164$$ 0 0
$$165$$ −11.6644 −0.908074
$$166$$ 0 0
$$167$$ −13.8322 −1.07037 −0.535184 0.844735i $$-0.679758\pi$$
−0.535184 + 0.844735i $$0.679758\pi$$
$$168$$ 0 0
$$169$$ 31.0319 2.38707
$$170$$ 0 0
$$171$$ −2.48929 −0.190361
$$172$$ 0 0
$$173$$ −14.8108 −1.12604 −0.563022 0.826442i $$-0.690361\pi$$
−0.563022 + 0.826442i $$0.690361\pi$$
$$174$$ 0 0
$$175$$ 1.19656 0.0904513
$$176$$ 0 0
$$177$$ 27.7392 2.08500
$$178$$ 0 0
$$179$$ −15.6644 −1.17081 −0.585407 0.810740i $$-0.699065\pi$$
−0.585407 + 0.810740i $$0.699065\pi$$
$$180$$ 0 0
$$181$$ 14.7862 1.09905 0.549526 0.835477i $$-0.314808\pi$$
0.549526 + 0.835477i $$0.314808\pi$$
$$182$$ 0 0
$$183$$ −8.58546 −0.634656
$$184$$ 0 0
$$185$$ 7.14637 0.525411
$$186$$ 0 0
$$187$$ −7.41454 −0.542205
$$188$$ 0 0
$$189$$ −1.43175 −0.104144
$$190$$ 0 0
$$191$$ −6.36748 −0.460735 −0.230367 0.973104i $$-0.573993\pi$$
−0.230367 + 0.973104i $$0.573993\pi$$
$$192$$ 0 0
$$193$$ −19.5970 −1.41062 −0.705312 0.708897i $$-0.749193\pi$$
−0.705312 + 0.708897i $$0.749193\pi$$
$$194$$ 0 0
$$195$$ 15.5468 1.11333
$$196$$ 0 0
$$197$$ −10.7862 −0.768487 −0.384244 0.923232i $$-0.625538\pi$$
−0.384244 + 0.923232i $$0.625538\pi$$
$$198$$ 0 0
$$199$$ 2.80344 0.198731 0.0993654 0.995051i $$-0.468319\pi$$
0.0993654 + 0.995051i $$0.468319\pi$$
$$200$$ 0 0
$$201$$ −17.8396 −1.25831
$$202$$ 0 0
$$203$$ 9.43175 0.661979
$$204$$ 0 0
$$205$$ 1.66442 0.116248
$$206$$ 0 0
$$207$$ −1.27131 −0.0883620
$$208$$ 0 0
$$209$$ 4.97858 0.344375
$$210$$ 0 0
$$211$$ 11.2541 0.774764 0.387382 0.921919i $$-0.373379\pi$$
0.387382 + 0.921919i $$0.373379\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 6.39312 0.436007
$$216$$ 0 0
$$217$$ −3.56404 −0.241943
$$218$$ 0 0
$$219$$ 32.4250 2.19108
$$220$$ 0 0
$$221$$ 9.88240 0.664762
$$222$$ 0 0
$$223$$ 17.4966 1.17166 0.585831 0.810433i $$-0.300768\pi$$
0.585831 + 0.810433i $$0.300768\pi$$
$$224$$ 0 0
$$225$$ 2.48929 0.165953
$$226$$ 0 0
$$227$$ 3.84942 0.255495 0.127748 0.991807i $$-0.459225\pi$$
0.127748 + 0.991807i $$0.459225\pi$$
$$228$$ 0 0
$$229$$ 14.6430 0.967637 0.483818 0.875168i $$-0.339250\pi$$
0.483818 + 0.875168i $$0.339250\pi$$
$$230$$ 0 0
$$231$$ −13.9572 −0.918313
$$232$$ 0 0
$$233$$ 11.9572 0.783339 0.391670 0.920106i $$-0.371898\pi$$
0.391670 + 0.920106i $$0.371898\pi$$
$$234$$ 0 0
$$235$$ −9.95715 −0.649533
$$236$$ 0 0
$$237$$ 29.7220 1.93065
$$238$$ 0 0
$$239$$ 15.5897 1.00841 0.504206 0.863583i $$-0.331785\pi$$
0.504206 + 0.863583i $$0.331785\pi$$
$$240$$ 0 0
$$241$$ −16.0575 −1.03436 −0.517178 0.855878i $$-0.673018\pi$$
−0.517178 + 0.855878i $$0.673018\pi$$
$$242$$ 0 0
$$243$$ −20.4752 −1.31349
$$244$$ 0 0
$$245$$ −5.56825 −0.355742
$$246$$ 0 0
$$247$$ −6.63565 −0.422217
$$248$$ 0 0
$$249$$ 20.3503 1.28965
$$250$$ 0 0
$$251$$ −23.9143 −1.50946 −0.754729 0.656037i $$-0.772232\pi$$
−0.754729 + 0.656037i $$0.772232\pi$$
$$252$$ 0 0
$$253$$ 2.54262 0.159853
$$254$$ 0 0
$$255$$ 3.48929 0.218508
$$256$$ 0 0
$$257$$ −24.4324 −1.52405 −0.762025 0.647548i $$-0.775794\pi$$
−0.762025 + 0.647548i $$0.775794\pi$$
$$258$$ 0 0
$$259$$ 8.55104 0.531336
$$260$$ 0 0
$$261$$ 19.6216 1.21455
$$262$$ 0 0
$$263$$ −14.4935 −0.893707 −0.446854 0.894607i $$-0.647456\pi$$
−0.446854 + 0.894607i $$0.647456\pi$$
$$264$$ 0 0
$$265$$ 11.4219 0.701641
$$266$$ 0 0
$$267$$ −11.4292 −0.699458
$$268$$ 0 0
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ −6.76060 −0.410677 −0.205339 0.978691i $$-0.565830\pi$$
−0.205339 + 0.978691i $$0.565830\pi$$
$$272$$ 0 0
$$273$$ 18.6027 1.12589
$$274$$ 0 0
$$275$$ −4.97858 −0.300219
$$276$$ 0 0
$$277$$ 25.4292 1.52789 0.763947 0.645279i $$-0.223259\pi$$
0.763947 + 0.645279i $$0.223259\pi$$
$$278$$ 0 0
$$279$$ −7.41454 −0.443897
$$280$$ 0 0
$$281$$ 9.07896 0.541605 0.270803 0.962635i $$-0.412711\pi$$
0.270803 + 0.962635i $$0.412711\pi$$
$$282$$ 0 0
$$283$$ −12.0147 −0.714199 −0.357100 0.934066i $$-0.616234\pi$$
−0.357100 + 0.934066i $$0.616234\pi$$
$$284$$ 0 0
$$285$$ −2.34292 −0.138783
$$286$$ 0 0
$$287$$ 1.99158 0.117559
$$288$$ 0 0
$$289$$ −14.7820 −0.869531
$$290$$ 0 0
$$291$$ −15.9572 −0.935425
$$292$$ 0 0
$$293$$ −23.6718 −1.38292 −0.691460 0.722415i $$-0.743032\pi$$
−0.691460 + 0.722415i $$0.743032\pi$$
$$294$$ 0 0
$$295$$ 11.8396 0.689326
$$296$$ 0 0
$$297$$ 5.95715 0.345669
$$298$$ 0 0
$$299$$ −3.38890 −0.195985
$$300$$ 0 0
$$301$$ 7.64973 0.440923
$$302$$ 0 0
$$303$$ 5.37169 0.308596
$$304$$ 0 0
$$305$$ −3.66442 −0.209824
$$306$$ 0 0
$$307$$ 16.0246 0.914570 0.457285 0.889320i $$-0.348822\pi$$
0.457285 + 0.889320i $$0.348822\pi$$
$$308$$ 0 0
$$309$$ −15.2713 −0.868754
$$310$$ 0 0
$$311$$ −27.5468 −1.56204 −0.781019 0.624508i $$-0.785300\pi$$
−0.781019 + 0.624508i $$0.785300\pi$$
$$312$$ 0 0
$$313$$ 10.1751 0.575133 0.287566 0.957761i $$-0.407154\pi$$
0.287566 + 0.957761i $$0.407154\pi$$
$$314$$ 0 0
$$315$$ 2.97858 0.167824
$$316$$ 0 0
$$317$$ −29.6289 −1.66413 −0.832063 0.554681i $$-0.812840\pi$$
−0.832063 + 0.554681i $$0.812840\pi$$
$$318$$ 0 0
$$319$$ −39.2432 −2.19719
$$320$$ 0 0
$$321$$ −18.0748 −1.00883
$$322$$ 0 0
$$323$$ −1.48929 −0.0828662
$$324$$ 0 0
$$325$$ 6.63565 0.368080
$$326$$ 0 0
$$327$$ −36.4250 −2.01431
$$328$$ 0 0
$$329$$ −11.9143 −0.656857
$$330$$ 0 0
$$331$$ −30.4679 −1.67467 −0.837333 0.546694i $$-0.815886\pi$$
−0.837333 + 0.546694i $$0.815886\pi$$
$$332$$ 0 0
$$333$$ 17.7894 0.974851
$$334$$ 0 0
$$335$$ −7.61423 −0.416010
$$336$$ 0 0
$$337$$ 26.1396 1.42392 0.711958 0.702222i $$-0.247808\pi$$
0.711958 + 0.702222i $$0.247808\pi$$
$$338$$ 0 0
$$339$$ −1.76481 −0.0958512
$$340$$ 0 0
$$341$$ 14.8291 0.803039
$$342$$ 0 0
$$343$$ −15.0386 −0.812010
$$344$$ 0 0
$$345$$ −1.19656 −0.0644205
$$346$$ 0 0
$$347$$ 7.31415 0.392644 0.196322 0.980539i $$-0.437100\pi$$
0.196322 + 0.980539i $$0.437100\pi$$
$$348$$ 0 0
$$349$$ −0.628308 −0.0336325 −0.0168163 0.999859i $$-0.505353\pi$$
−0.0168163 + 0.999859i $$0.505353\pi$$
$$350$$ 0 0
$$351$$ −7.93994 −0.423803
$$352$$ 0 0
$$353$$ −8.23267 −0.438181 −0.219090 0.975705i $$-0.570309\pi$$
−0.219090 + 0.975705i $$0.570309\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4.17513 0.220972
$$358$$ 0 0
$$359$$ 12.1323 0.640318 0.320159 0.947364i $$-0.396264\pi$$
0.320159 + 0.947364i $$0.396264\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 32.3001 1.69531
$$364$$ 0 0
$$365$$ 13.8396 0.724396
$$366$$ 0 0
$$367$$ 17.9572 0.937356 0.468678 0.883369i $$-0.344730\pi$$
0.468678 + 0.883369i $$0.344730\pi$$
$$368$$ 0 0
$$369$$ 4.14323 0.215688
$$370$$ 0 0
$$371$$ 13.6669 0.709552
$$372$$ 0 0
$$373$$ −29.6289 −1.53413 −0.767064 0.641571i $$-0.778283\pi$$
−0.767064 + 0.641571i $$0.778283\pi$$
$$374$$ 0 0
$$375$$ 2.34292 0.120988
$$376$$ 0 0
$$377$$ 52.3049 2.69384
$$378$$ 0 0
$$379$$ 6.80344 0.349469 0.174735 0.984616i $$-0.444093\pi$$
0.174735 + 0.984616i $$0.444093\pi$$
$$380$$ 0 0
$$381$$ 24.4078 1.25045
$$382$$ 0 0
$$383$$ −20.2253 −1.03347 −0.516733 0.856147i $$-0.672852\pi$$
−0.516733 + 0.856147i $$0.672852\pi$$
$$384$$ 0 0
$$385$$ −5.95715 −0.303605
$$386$$ 0 0
$$387$$ 15.9143 0.808970
$$388$$ 0 0
$$389$$ −0.393115 −0.0199317 −0.00996587 0.999950i $$-0.503172\pi$$
−0.00996587 + 0.999950i $$0.503172\pi$$
$$390$$ 0 0
$$391$$ −0.760597 −0.0384650
$$392$$ 0 0
$$393$$ 42.0722 2.12226
$$394$$ 0 0
$$395$$ 12.6858 0.638294
$$396$$ 0 0
$$397$$ −17.4292 −0.874748 −0.437374 0.899280i $$-0.644091\pi$$
−0.437374 + 0.899280i $$0.644091\pi$$
$$398$$ 0 0
$$399$$ −2.80344 −0.140348
$$400$$ 0 0
$$401$$ 13.1281 0.655585 0.327792 0.944750i $$-0.393695\pi$$
0.327792 + 0.944750i $$0.393695\pi$$
$$402$$ 0 0
$$403$$ −19.7648 −0.984555
$$404$$ 0 0
$$405$$ −10.2713 −0.510385
$$406$$ 0 0
$$407$$ −35.5787 −1.76357
$$408$$ 0 0
$$409$$ −18.7862 −0.928919 −0.464460 0.885594i $$-0.653751\pi$$
−0.464460 + 0.885594i $$0.653751\pi$$
$$410$$ 0 0
$$411$$ −23.0533 −1.13714
$$412$$ 0 0
$$413$$ 14.1667 0.697098
$$414$$ 0 0
$$415$$ 8.68585 0.426371
$$416$$ 0 0
$$417$$ 2.29273 0.112276
$$418$$ 0 0
$$419$$ 5.22219 0.255121 0.127560 0.991831i $$-0.459285\pi$$
0.127560 + 0.991831i $$0.459285\pi$$
$$420$$ 0 0
$$421$$ 7.68164 0.374380 0.187190 0.982324i $$-0.440062\pi$$
0.187190 + 0.982324i $$0.440062\pi$$
$$422$$ 0 0
$$423$$ −24.7862 −1.20515
$$424$$ 0 0
$$425$$ 1.48929 0.0722411
$$426$$ 0 0
$$427$$ −4.38469 −0.212190
$$428$$ 0 0
$$429$$ −77.4011 −3.73696
$$430$$ 0 0
$$431$$ −17.7073 −0.852929 −0.426465 0.904504i $$-0.640241\pi$$
−0.426465 + 0.904504i $$0.640241\pi$$
$$432$$ 0 0
$$433$$ 17.9901 0.864551 0.432275 0.901742i $$-0.357711\pi$$
0.432275 + 0.901742i $$0.357711\pi$$
$$434$$ 0 0
$$435$$ 18.4679 0.885466
$$436$$ 0 0
$$437$$ 0.510711 0.0244306
$$438$$ 0 0
$$439$$ 14.8438 0.708454 0.354227 0.935159i $$-0.384744\pi$$
0.354227 + 0.935159i $$0.384744\pi$$
$$440$$ 0 0
$$441$$ −13.8610 −0.660047
$$442$$ 0 0
$$443$$ −22.4078 −1.06463 −0.532314 0.846547i $$-0.678677\pi$$
−0.532314 + 0.846547i $$0.678677\pi$$
$$444$$ 0 0
$$445$$ −4.87819 −0.231249
$$446$$ 0 0
$$447$$ 19.3288 0.914223
$$448$$ 0 0
$$449$$ −40.7434 −1.92280 −0.961400 0.275156i $$-0.911271\pi$$
−0.961400 + 0.275156i $$0.911271\pi$$
$$450$$ 0 0
$$451$$ −8.28646 −0.390194
$$452$$ 0 0
$$453$$ 32.4653 1.52536
$$454$$ 0 0
$$455$$ 7.93994 0.372230
$$456$$ 0 0
$$457$$ 8.01721 0.375029 0.187515 0.982262i $$-0.439957\pi$$
0.187515 + 0.982262i $$0.439957\pi$$
$$458$$ 0 0
$$459$$ −1.78202 −0.0831775
$$460$$ 0 0
$$461$$ 25.1281 1.17033 0.585166 0.810914i $$-0.301029\pi$$
0.585166 + 0.810914i $$0.301029\pi$$
$$462$$ 0 0
$$463$$ −31.7795 −1.47692 −0.738459 0.674298i $$-0.764446\pi$$
−0.738459 + 0.674298i $$0.764446\pi$$
$$464$$ 0 0
$$465$$ −6.97858 −0.323624
$$466$$ 0 0
$$467$$ −20.9210 −0.968110 −0.484055 0.875038i $$-0.660837\pi$$
−0.484055 + 0.875038i $$0.660837\pi$$
$$468$$ 0 0
$$469$$ −9.11087 −0.420701
$$470$$ 0 0
$$471$$ 36.8009 1.69570
$$472$$ 0 0
$$473$$ −31.8286 −1.46348
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 28.4324 1.30183
$$478$$ 0 0
$$479$$ 27.7220 1.26665 0.633324 0.773886i $$-0.281690\pi$$
0.633324 + 0.773886i $$0.281690\pi$$
$$480$$ 0 0
$$481$$ 47.4208 2.16220
$$482$$ 0 0
$$483$$ −1.43175 −0.0651469
$$484$$ 0 0
$$485$$ −6.81079 −0.309262
$$486$$ 0 0
$$487$$ 3.20390 0.145183 0.0725914 0.997362i $$-0.476873\pi$$
0.0725914 + 0.997362i $$0.476873\pi$$
$$488$$ 0 0
$$489$$ −13.6069 −0.615325
$$490$$ 0 0
$$491$$ 21.5212 0.971238 0.485619 0.874171i $$-0.338594\pi$$
0.485619 + 0.874171i $$0.338594\pi$$
$$492$$ 0 0
$$493$$ 11.7392 0.528706
$$494$$ 0 0
$$495$$ −12.3931 −0.557029
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 23.1709 1.03727 0.518637 0.854995i $$-0.326440\pi$$
0.518637 + 0.854995i $$0.326440\pi$$
$$500$$ 0 0
$$501$$ −32.4078 −1.44787
$$502$$ 0 0
$$503$$ −1.88240 −0.0839322 −0.0419661 0.999119i $$-0.513362\pi$$
−0.0419661 + 0.999119i $$0.513362\pi$$
$$504$$ 0 0
$$505$$ 2.29273 0.102025
$$506$$ 0 0
$$507$$ 72.7054 3.22896
$$508$$ 0 0
$$509$$ −5.46365 −0.242172 −0.121086 0.992642i $$-0.538638\pi$$
−0.121086 + 0.992642i $$0.538638\pi$$
$$510$$ 0 0
$$511$$ 16.5598 0.732564
$$512$$ 0 0
$$513$$ 1.19656 0.0528293
$$514$$ 0 0
$$515$$ −6.51806 −0.287220
$$516$$ 0 0
$$517$$ 49.5725 2.18019
$$518$$ 0 0
$$519$$ −34.7005 −1.52318
$$520$$ 0 0
$$521$$ 11.2222 0.491653 0.245827 0.969314i $$-0.420941\pi$$
0.245827 + 0.969314i $$0.420941\pi$$
$$522$$ 0 0
$$523$$ 29.1867 1.27624 0.638122 0.769935i $$-0.279711\pi$$
0.638122 + 0.769935i $$0.279711\pi$$
$$524$$ 0 0
$$525$$ 2.80344 0.122352
$$526$$ 0 0
$$527$$ −4.43596 −0.193233
$$528$$ 0 0
$$529$$ −22.7392 −0.988660
$$530$$ 0 0
$$531$$ 29.4721 1.27898
$$532$$ 0 0
$$533$$ 11.0445 0.478392
$$534$$ 0 0
$$535$$ −7.71462 −0.333532
$$536$$ 0 0
$$537$$ −36.7005 −1.58375
$$538$$ 0 0
$$539$$ 27.7220 1.19407
$$540$$ 0 0
$$541$$ −20.2499 −0.870611 −0.435305 0.900283i $$-0.643360\pi$$
−0.435305 + 0.900283i $$0.643360\pi$$
$$542$$ 0 0
$$543$$ 34.6430 1.48667
$$544$$ 0 0
$$545$$ −15.5468 −0.665953
$$546$$ 0 0
$$547$$ −42.8830 −1.83355 −0.916773 0.399409i $$-0.869215\pi$$
−0.916773 + 0.399409i $$0.869215\pi$$
$$548$$ 0 0
$$549$$ −9.12181 −0.389309
$$550$$ 0 0
$$551$$ −7.88240 −0.335802
$$552$$ 0 0
$$553$$ 15.1793 0.645491
$$554$$ 0 0
$$555$$ 16.7434 0.710717
$$556$$ 0 0
$$557$$ −28.2583 −1.19734 −0.598671 0.800995i $$-0.704304\pi$$
−0.598671 + 0.800995i $$0.704304\pi$$
$$558$$ 0 0
$$559$$ 42.4225 1.79428
$$560$$ 0 0
$$561$$ −17.3717 −0.733433
$$562$$ 0 0
$$563$$ 20.2253 0.852396 0.426198 0.904630i $$-0.359853\pi$$
0.426198 + 0.904630i $$0.359853\pi$$
$$564$$ 0 0
$$565$$ −0.753250 −0.0316895
$$566$$ 0 0
$$567$$ −12.2902 −0.516140
$$568$$ 0 0
$$569$$ −31.1365 −1.30531 −0.652655 0.757655i $$-0.726345\pi$$
−0.652655 + 0.757655i $$0.726345\pi$$
$$570$$ 0 0
$$571$$ −27.4868 −1.15029 −0.575143 0.818053i $$-0.695053\pi$$
−0.575143 + 0.818053i $$0.695053\pi$$
$$572$$ 0 0
$$573$$ −14.9185 −0.623230
$$574$$ 0 0
$$575$$ −0.510711 −0.0212981
$$576$$ 0 0
$$577$$ 23.7820 0.990058 0.495029 0.868876i $$-0.335157\pi$$
0.495029 + 0.868876i $$0.335157\pi$$
$$578$$ 0 0
$$579$$ −45.9143 −1.90813
$$580$$ 0 0
$$581$$ 10.3931 0.431179
$$582$$ 0 0
$$583$$ −56.8647 −2.35510
$$584$$ 0 0
$$585$$ 16.5181 0.682938
$$586$$ 0 0
$$587$$ −24.0147 −0.991192 −0.495596 0.868553i $$-0.665050\pi$$
−0.495596 + 0.868553i $$0.665050\pi$$
$$588$$ 0 0
$$589$$ 2.97858 0.122730
$$590$$ 0 0
$$591$$ −25.2713 −1.03952
$$592$$ 0 0
$$593$$ 46.6148 1.91424 0.957121 0.289688i $$-0.0935515\pi$$
0.957121 + 0.289688i $$0.0935515\pi$$
$$594$$ 0 0
$$595$$ 1.78202 0.0730557
$$596$$ 0 0
$$597$$ 6.56825 0.268821
$$598$$ 0 0
$$599$$ −44.6002 −1.82231 −0.911156 0.412061i $$-0.864809\pi$$
−0.911156 + 0.412061i $$0.864809\pi$$
$$600$$ 0 0
$$601$$ −17.5787 −0.717051 −0.358526 0.933520i $$-0.616720\pi$$
−0.358526 + 0.933520i $$0.616720\pi$$
$$602$$ 0 0
$$603$$ −18.9540 −0.771867
$$604$$ 0 0
$$605$$ 13.7862 0.560490
$$606$$ 0 0
$$607$$ 37.2467 1.51180 0.755899 0.654688i $$-0.227200\pi$$
0.755899 + 0.654688i $$0.227200\pi$$
$$608$$ 0 0
$$609$$ 22.0979 0.895451
$$610$$ 0 0
$$611$$ −66.0722 −2.67300
$$612$$ 0 0
$$613$$ 14.8866 0.601265 0.300632 0.953740i $$-0.402802\pi$$
0.300632 + 0.953740i $$0.402802\pi$$
$$614$$ 0 0
$$615$$ 3.89962 0.157248
$$616$$ 0 0
$$617$$ 3.70727 0.149249 0.0746245 0.997212i $$-0.476224\pi$$
0.0746245 + 0.997212i $$0.476224\pi$$
$$618$$ 0 0
$$619$$ −17.7648 −0.714028 −0.357014 0.934099i $$-0.616205\pi$$
−0.357014 + 0.934099i $$0.616205\pi$$
$$620$$ 0 0
$$621$$ 0.611096 0.0245224
$$622$$ 0 0
$$623$$ −5.83704 −0.233856
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 11.6644 0.465832
$$628$$ 0 0
$$629$$ 10.6430 0.424364
$$630$$ 0 0
$$631$$ 40.3074 1.60461 0.802307 0.596912i $$-0.203606\pi$$
0.802307 + 0.596912i $$0.203606\pi$$
$$632$$ 0 0
$$633$$ 26.3675 1.04801
$$634$$ 0 0
$$635$$ 10.4177 0.413413
$$636$$ 0 0
$$637$$ −36.9490 −1.46397
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −27.6728 −1.09301 −0.546506 0.837455i $$-0.684042\pi$$
−0.546506 + 0.837455i $$0.684042\pi$$
$$642$$ 0 0
$$643$$ −19.0277 −0.750379 −0.375190 0.926948i $$-0.622422\pi$$
−0.375190 + 0.926948i $$0.622422\pi$$
$$644$$ 0 0
$$645$$ 14.9786 0.589781
$$646$$ 0 0
$$647$$ −32.0403 −1.25964 −0.629818 0.776743i $$-0.716870\pi$$
−0.629818 + 0.776743i $$0.716870\pi$$
$$648$$ 0 0
$$649$$ −58.9442 −2.31376
$$650$$ 0 0
$$651$$ −8.35027 −0.327273
$$652$$ 0 0
$$653$$ −25.3142 −0.990619 −0.495310 0.868716i $$-0.664945\pi$$
−0.495310 + 0.868716i $$0.664945\pi$$
$$654$$ 0 0
$$655$$ 17.9572 0.701644
$$656$$ 0 0
$$657$$ 34.4507 1.34405
$$658$$ 0 0
$$659$$ 0.860981 0.0335391 0.0167695 0.999859i $$-0.494662\pi$$
0.0167695 + 0.999859i $$0.494662\pi$$
$$660$$ 0 0
$$661$$ −4.55356 −0.177113 −0.0885564 0.996071i $$-0.528225\pi$$
−0.0885564 + 0.996071i $$0.528225\pi$$
$$662$$ 0 0
$$663$$ 23.1537 0.899216
$$664$$ 0 0
$$665$$ −1.19656 −0.0464005
$$666$$ 0 0
$$667$$ −4.02563 −0.155873
$$668$$ 0 0
$$669$$ 40.9933 1.58489
$$670$$ 0 0
$$671$$ 18.2436 0.704287
$$672$$ 0 0
$$673$$ −11.7465 −0.452795 −0.226398 0.974035i $$-0.572695\pi$$
−0.226398 + 0.974035i $$0.572695\pi$$
$$674$$ 0 0
$$675$$ −1.19656 −0.0460555
$$676$$ 0 0
$$677$$ 22.2854 0.856497 0.428248 0.903661i $$-0.359131\pi$$
0.428248 + 0.903661i $$0.359131\pi$$
$$678$$ 0 0
$$679$$ −8.14950 −0.312749
$$680$$ 0 0
$$681$$ 9.01890 0.345605
$$682$$ 0 0
$$683$$ 7.23833 0.276967 0.138483 0.990365i $$-0.455777\pi$$
0.138483 + 0.990365i $$0.455777\pi$$
$$684$$ 0 0
$$685$$ −9.83956 −0.375950
$$686$$ 0 0
$$687$$ 34.3074 1.30891
$$688$$ 0 0
$$689$$ 75.7917 2.88743
$$690$$ 0 0
$$691$$ −32.7434 −1.24562 −0.622809 0.782374i $$-0.714008\pi$$
−0.622809 + 0.782374i $$0.714008\pi$$
$$692$$ 0 0
$$693$$ −14.8291 −0.563310
$$694$$ 0 0
$$695$$ 0.978577 0.0371195
$$696$$ 0 0
$$697$$ 2.47881 0.0938915
$$698$$ 0 0
$$699$$ 28.0147 1.05961
$$700$$ 0 0
$$701$$ 29.4208 1.11121 0.555604 0.831447i $$-0.312487\pi$$
0.555604 + 0.831447i $$0.312487\pi$$
$$702$$ 0 0
$$703$$ −7.14637 −0.269530
$$704$$ 0 0
$$705$$ −23.3288 −0.878615
$$706$$ 0 0
$$707$$ 2.74338 0.103176
$$708$$ 0 0
$$709$$ 5.32885 0.200129 0.100065 0.994981i $$-0.468095\pi$$
0.100065 + 0.994981i $$0.468095\pi$$
$$710$$ 0 0
$$711$$ 31.5787 1.18429
$$712$$ 0 0
$$713$$ 1.52119 0.0569691
$$714$$ 0 0
$$715$$ −33.0361 −1.23548
$$716$$ 0 0
$$717$$ 36.5254 1.36407
$$718$$ 0 0
$$719$$ 8.45317 0.315250 0.157625 0.987499i $$-0.449616\pi$$
0.157625 + 0.987499i $$0.449616\pi$$
$$720$$ 0 0
$$721$$ −7.79923 −0.290459
$$722$$ 0 0
$$723$$ −37.6216 −1.39916
$$724$$ 0 0
$$725$$ 7.88240 0.292745
$$726$$ 0 0
$$727$$ 2.56825 0.0952511 0.0476256 0.998865i $$-0.484835\pi$$
0.0476256 + 0.998865i $$0.484835\pi$$
$$728$$ 0 0
$$729$$ −17.1579 −0.635479
$$730$$ 0 0
$$731$$ 9.52119 0.352154
$$732$$ 0 0
$$733$$ −35.5212 −1.31201 −0.656003 0.754759i $$-0.727754\pi$$
−0.656003 + 0.754759i $$0.727754\pi$$
$$734$$ 0 0
$$735$$ −13.0460 −0.481208
$$736$$ 0 0
$$737$$ 37.9080 1.39636
$$738$$ 0 0
$$739$$ 37.0508 1.36294 0.681468 0.731848i $$-0.261342\pi$$
0.681468 + 0.731848i $$0.261342\pi$$
$$740$$ 0 0
$$741$$ −15.5468 −0.571127
$$742$$ 0 0
$$743$$ 23.2039 0.851269 0.425634 0.904895i $$-0.360051\pi$$
0.425634 + 0.904895i $$0.360051\pi$$
$$744$$ 0 0
$$745$$ 8.24989 0.302252
$$746$$ 0 0
$$747$$ 21.6216 0.791092
$$748$$ 0 0
$$749$$ −9.23098 −0.337293
$$750$$ 0 0
$$751$$ 51.2285 1.86935 0.934677 0.355499i $$-0.115689\pi$$
0.934677 + 0.355499i $$0.115689\pi$$
$$752$$ 0 0
$$753$$ −56.0294 −2.04182
$$754$$ 0 0
$$755$$ 13.8568 0.504299
$$756$$ 0 0
$$757$$ 43.3717 1.57637 0.788185 0.615438i $$-0.211021\pi$$
0.788185 + 0.615438i $$0.211021\pi$$
$$758$$ 0 0
$$759$$ 5.95715 0.216231
$$760$$ 0 0
$$761$$ 24.7753 0.898104 0.449052 0.893506i $$-0.351762\pi$$
0.449052 + 0.893506i $$0.351762\pi$$
$$762$$ 0 0
$$763$$ −18.6027 −0.673462
$$764$$ 0 0
$$765$$ 3.70727 0.134037
$$766$$ 0 0
$$767$$ 78.5632 2.83675
$$768$$ 0 0
$$769$$ −11.4893 −0.414314 −0.207157 0.978308i $$-0.566421\pi$$
−0.207157 + 0.978308i $$0.566421\pi$$
$$770$$ 0 0
$$771$$ −57.2432 −2.06156
$$772$$ 0 0
$$773$$ −27.0863 −0.974227 −0.487113 0.873339i $$-0.661950\pi$$
−0.487113 + 0.873339i $$0.661950\pi$$
$$774$$ 0 0
$$775$$ −2.97858 −0.106994
$$776$$ 0 0
$$777$$ 20.0344 0.718731
$$778$$ 0 0
$$779$$ −1.66442 −0.0596342
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −9.43175 −0.337063
$$784$$ 0 0
$$785$$ 15.7073 0.560616
$$786$$ 0 0
$$787$$ −13.4563 −0.479666 −0.239833 0.970814i $$-0.577093\pi$$
−0.239833 + 0.970814i $$0.577093\pi$$
$$788$$ 0 0
$$789$$ −33.9572 −1.20891
$$790$$ 0 0
$$791$$ −0.901307 −0.0320468
$$792$$ 0 0
$$793$$ −24.3158 −0.863481
$$794$$ 0 0
$$795$$ 26.7606 0.949101
$$796$$ 0 0
$$797$$ −30.4496 −1.07858 −0.539290 0.842120i $$-0.681307\pi$$
−0.539290 + 0.842120i $$0.681307\pi$$
$$798$$ 0 0
$$799$$ −14.8291 −0.524615
$$800$$ 0 0
$$801$$ −12.1432 −0.429060
$$802$$ 0 0
$$803$$ −68.9013 −2.43147
$$804$$ 0 0
$$805$$ −0.611096 −0.0215383
$$806$$ 0 0
$$807$$ 32.8009 1.15465
$$808$$ 0 0
$$809$$ −29.7047 −1.04436 −0.522182 0.852834i $$-0.674882\pi$$
−0.522182 + 0.852834i $$0.674882\pi$$
$$810$$ 0 0
$$811$$ −32.2902 −1.13386 −0.566931 0.823765i $$-0.691870\pi$$
−0.566931 + 0.823765i $$0.691870\pi$$
$$812$$ 0 0
$$813$$ −15.8396 −0.555518
$$814$$ 0 0
$$815$$ −5.80765 −0.203433
$$816$$ 0 0
$$817$$ −6.39312 −0.223667
$$818$$ 0 0
$$819$$ 19.7648 0.690638
$$820$$ 0 0
$$821$$ −9.06427 −0.316345 −0.158173 0.987411i $$-0.550560\pi$$
−0.158173 + 0.987411i $$0.550560\pi$$
$$822$$ 0 0
$$823$$ −7.35448 −0.256361 −0.128181 0.991751i $$-0.540914\pi$$
−0.128181 + 0.991751i $$0.540914\pi$$
$$824$$ 0 0
$$825$$ −11.6644 −0.406103
$$826$$ 0 0
$$827$$ −2.02204 −0.0703132 −0.0351566 0.999382i $$-0.511193\pi$$
−0.0351566 + 0.999382i $$0.511193\pi$$
$$828$$ 0 0
$$829$$ 36.3675 1.26309 0.631547 0.775337i $$-0.282420\pi$$
0.631547 + 0.775337i $$0.282420\pi$$
$$830$$ 0 0
$$831$$ 59.5787 2.06676
$$832$$ 0 0
$$833$$ −8.29273 −0.287326
$$834$$ 0 0
$$835$$ −13.8322 −0.478683
$$836$$ 0 0
$$837$$ 3.56404 0.123191
$$838$$ 0 0
$$839$$ 25.8223 0.891486 0.445743 0.895161i $$-0.352939\pi$$
0.445743 + 0.895161i $$0.352939\pi$$
$$840$$ 0 0
$$841$$ 33.1323 1.14249
$$842$$ 0 0
$$843$$ 21.2713 0.732623
$$844$$ 0 0
$$845$$ 31.0319 1.06753
$$846$$ 0 0
$$847$$ 16.4960 0.566810
$$848$$ 0 0
$$849$$ −28.1495 −0.966088
$$850$$ 0 0
$$851$$ −3.64973 −0.125111
$$852$$ 0 0
$$853$$ −8.54262 −0.292494 −0.146247 0.989248i $$-0.546719\pi$$
−0.146247 + 0.989248i $$0.546719\pi$$
$$854$$ 0 0
$$855$$ −2.48929 −0.0851319
$$856$$ 0 0
$$857$$ 27.8322 0.950730 0.475365 0.879789i $$-0.342316\pi$$
0.475365 + 0.879789i $$0.342316\pi$$
$$858$$ 0 0
$$859$$ −33.4868 −1.14255 −0.571277 0.820757i $$-0.693552\pi$$
−0.571277 + 0.820757i $$0.693552\pi$$
$$860$$ 0 0
$$861$$ 4.66611 0.159021
$$862$$ 0 0
$$863$$ −29.2614 −0.996071 −0.498036 0.867157i $$-0.665945\pi$$
−0.498036 + 0.867157i $$0.665945\pi$$
$$864$$ 0 0
$$865$$ −14.8108 −0.503582
$$866$$ 0 0
$$867$$ −34.6331 −1.17620
$$868$$ 0 0
$$869$$ −63.1575 −2.14247
$$870$$ 0 0
$$871$$ −50.5254 −1.71199
$$872$$ 0 0
$$873$$ −16.9540 −0.573807
$$874$$ 0 0
$$875$$ 1.19656 0.0404510
$$876$$ 0 0
$$877$$ 46.7852 1.57982 0.789911 0.613221i $$-0.210127\pi$$
0.789911 + 0.613221i $$0.210127\pi$$
$$878$$ 0 0
$$879$$ −55.4611 −1.87066
$$880$$ 0 0
$$881$$ −30.7581 −1.03627 −0.518133 0.855300i $$-0.673373\pi$$
−0.518133 + 0.855300i $$0.673373\pi$$
$$882$$ 0 0
$$883$$ −0.435961 −0.0146713 −0.00733563 0.999973i $$-0.502335\pi$$
−0.00733563 + 0.999973i $$0.502335\pi$$
$$884$$ 0 0
$$885$$ 27.7392 0.932442
$$886$$ 0 0
$$887$$ −2.15310 −0.0722939 −0.0361469 0.999346i $$-0.511508\pi$$
−0.0361469 + 0.999346i $$0.511508\pi$$
$$888$$ 0 0
$$889$$ 12.4653 0.418074
$$890$$ 0 0
$$891$$ 51.1365 1.71314
$$892$$ 0 0
$$893$$ 9.95715 0.333203
$$894$$ 0 0
$$895$$ −15.6644 −0.523604
$$896$$ 0 0
$$897$$ −7.93994 −0.265107
$$898$$ 0 0
$$899$$ −23.4783 −0.783047
$$900$$ 0 0
$$901$$ 17.0105 0.566701
$$902$$ 0 0
$$903$$ 17.9227 0.596431
$$904$$ 0 0
$$905$$ 14.7862 0.491511
$$906$$ 0 0
$$907$$ 36.9002 1.22525 0.612626 0.790373i $$-0.290113\pi$$
0.612626 + 0.790373i $$0.290113\pi$$
$$908$$ 0 0
$$909$$ 5.70727 0.189298
$$910$$ 0 0
$$911$$ −36.9504 −1.22422 −0.612111 0.790772i $$-0.709679\pi$$
−0.612111 + 0.790772i $$0.709679\pi$$
$$912$$ 0 0
$$913$$ −43.2432 −1.43114
$$914$$ 0 0
$$915$$ −8.58546 −0.283827
$$916$$ 0 0
$$917$$ 21.4868 0.709556
$$918$$ 0 0
$$919$$ 19.1537 0.631823 0.315911 0.948789i $$-0.397690\pi$$
0.315911 + 0.948789i $$0.397690\pi$$
$$920$$ 0 0
$$921$$ 37.5443 1.23713
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 7.14637 0.234971
$$926$$ 0 0
$$927$$ −16.2253 −0.532910
$$928$$ 0 0
$$929$$ −56.9185 −1.86744 −0.933718 0.358009i $$-0.883456\pi$$
−0.933718 + 0.358009i $$0.883456\pi$$
$$930$$ 0 0
$$931$$ 5.56825 0.182492
$$932$$ 0 0
$$933$$ −64.5401 −2.11295
$$934$$ 0 0
$$935$$ −7.41454 −0.242481
$$936$$ 0 0
$$937$$ −24.1176 −0.787888 −0.393944 0.919135i $$-0.628890\pi$$
−0.393944 + 0.919135i $$0.628890\pi$$
$$938$$ 0 0
$$939$$ 23.8396 0.777975
$$940$$ 0 0
$$941$$ −3.19656 −0.104205 −0.0521024 0.998642i $$-0.516592\pi$$
−0.0521024 + 0.998642i $$0.516592\pi$$
$$942$$ 0 0
$$943$$ −0.850040 −0.0276811
$$944$$ 0 0
$$945$$ −1.43175 −0.0465748
$$946$$ 0 0
$$947$$ −3.63504 −0.118123 −0.0590614 0.998254i $$-0.518811\pi$$
−0.0590614 + 0.998254i $$0.518811\pi$$
$$948$$ 0 0
$$949$$ 91.8345 2.98107
$$950$$ 0 0
$$951$$ −69.4183 −2.25104
$$952$$ 0 0
$$953$$ −48.0821 −1.55753 −0.778766 0.627315i $$-0.784154\pi$$
−0.778766 + 0.627315i $$0.784154\pi$$
$$954$$ 0 0
$$955$$ −6.36748 −0.206047
$$956$$ 0 0
$$957$$ −91.9437 −2.97212
$$958$$ 0 0
$$959$$ −11.7736 −0.380189
$$960$$ 0 0
$$961$$ −22.1281 −0.713809
$$962$$ 0 0
$$963$$ −19.2039 −0.618837
$$964$$ 0 0
$$965$$ −19.5970 −0.630850
$$966$$ 0 0
$$967$$ −23.4637 −0.754540 −0.377270 0.926103i $$-0.623137\pi$$
−0.377270 + 0.926103i $$0.623137\pi$$
$$968$$ 0 0
$$969$$ −3.48929 −0.112092
$$970$$ 0 0
$$971$$ −17.4868 −0.561177 −0.280589 0.959828i $$-0.590530\pi$$
−0.280589 + 0.959828i $$0.590530\pi$$
$$972$$ 0 0
$$973$$ 1.17092 0.0375381
$$974$$ 0 0
$$975$$ 15.5468 0.497897
$$976$$ 0 0
$$977$$ 20.2316 0.647266 0.323633 0.946183i $$-0.395096\pi$$
0.323633 + 0.946183i $$0.395096\pi$$
$$978$$ 0 0
$$979$$ 24.2865 0.776199
$$980$$ 0 0
$$981$$ −38.7005 −1.23561
$$982$$ 0 0
$$983$$ 28.1249 0.897046 0.448523 0.893771i $$-0.351950\pi$$
0.448523 + 0.893771i $$0.351950\pi$$
$$984$$ 0 0
$$985$$ −10.7862 −0.343678
$$986$$ 0 0
$$987$$ −27.9143 −0.888522
$$988$$ 0 0
$$989$$ −3.26504 −0.103822
$$990$$ 0 0
$$991$$ −33.9718 −1.07915 −0.539576 0.841937i $$-0.681415\pi$$
−0.539576 + 0.841937i $$0.681415\pi$$
$$992$$ 0 0
$$993$$ −71.3839 −2.26530
$$994$$ 0 0
$$995$$ 2.80344 0.0888751
$$996$$ 0 0
$$997$$ 23.8223 0.754461 0.377231 0.926119i $$-0.376876\pi$$
0.377231 + 0.926119i $$0.376876\pi$$
$$998$$ 0 0
$$999$$ −8.55104 −0.270543
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 6080.2.a.bx.1.3 3
4.3 odd 2 6080.2.a.br.1.1 3
8.3 odd 2 1520.2.a.q.1.3 3
8.5 even 2 760.2.a.i.1.1 3
24.5 odd 2 6840.2.a.bm.1.2 3
40.13 odd 4 3800.2.d.n.3649.1 6
40.19 odd 2 7600.2.a.bp.1.1 3
40.29 even 2 3800.2.a.w.1.3 3
40.37 odd 4 3800.2.d.n.3649.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.1 3 8.5 even 2
1520.2.a.q.1.3 3 8.3 odd 2
3800.2.a.w.1.3 3 40.29 even 2
3800.2.d.n.3649.1 6 40.13 odd 4
3800.2.d.n.3649.6 6 40.37 odd 4
6080.2.a.br.1.1 3 4.3 odd 2
6080.2.a.bx.1.3 3 1.1 even 1 trivial
6840.2.a.bm.1.2 3 24.5 odd 2
7600.2.a.bp.1.1 3 40.19 odd 2