# Properties

 Label 6080.2.a.bp.1.2 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ 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] = [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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3040) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.571993$$ 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)

 $$f(q)$$ $$=$$ $$q-0.428007 q^{3} +1.00000 q^{5} -4.67282 q^{7} -2.81681 q^{9} +O(q^{10})$$ $$q-0.428007 q^{3} +1.00000 q^{5} -4.67282 q^{7} -2.81681 q^{9} +2.67282 q^{11} +2.24482 q^{13} -0.428007 q^{15} +2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} +0.672824 q^{23} +1.00000 q^{25} +2.48963 q^{27} -1.14399 q^{29} -1.14399 q^{31} -1.14399 q^{33} -4.67282 q^{35} -4.24482 q^{37} -0.960797 q^{39} +4.85601 q^{41} -5.52884 q^{43} -2.81681 q^{45} -4.67282 q^{47} +14.8353 q^{49} -0.856013 q^{51} +0.244817 q^{53} +2.67282 q^{55} -0.428007 q^{57} +0.287973 q^{59} +8.87448 q^{61} +13.1625 q^{63} +2.24482 q^{65} -9.20561 q^{67} -0.287973 q^{69} -9.14399 q^{71} +1.14399 q^{73} -0.428007 q^{75} -12.4896 q^{77} +2.00000 q^{79} +7.38485 q^{81} -0.183190 q^{83} +2.00000 q^{85} +0.489634 q^{87} +9.14399 q^{89} -10.4896 q^{91} +0.489634 q^{93} +1.00000 q^{95} +9.10083 q^{97} -7.52884 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 $$3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 4 q^{13} - 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} - 8 q^{23} + 3 q^{25} - 14 q^{27} - 2 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{35} - 2 q^{37} + 10 q^{39} + 16 q^{41} - 8 q^{43} + 3 q^{45} - 4 q^{47} + 3 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{55} - 2 q^{57} - 2 q^{59} - 2 q^{61} + 8 q^{63} - 4 q^{65} - 4 q^{67} + 2 q^{69} - 26 q^{71} + 2 q^{73} - 2 q^{75} - 16 q^{77} + 6 q^{79} + 15 q^{81} - 12 q^{83} + 6 q^{85} - 20 q^{87} + 26 q^{89} - 10 q^{91} - 20 q^{93} + 3 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 2 * q^11 - 4 * q^13 - 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 - 8 * q^23 + 3 * q^25 - 14 * q^27 - 2 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^35 - 2 * q^37 + 10 * q^39 + 16 * q^41 - 8 * q^43 + 3 * q^45 - 4 * q^47 + 3 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^55 - 2 * q^57 - 2 * q^59 - 2 * q^61 + 8 * q^63 - 4 * q^65 - 4 * q^67 + 2 * q^69 - 26 * q^71 + 2 * q^73 - 2 * q^75 - 16 * q^77 + 6 * q^79 + 15 * q^81 - 12 * q^83 + 6 * q^85 - 20 * q^87 + 26 * q^89 - 10 * q^91 - 20 * q^93 + 3 * q^95 + 18 * q^97 - 14 * 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
$$3$$ −0.428007 −0.247110 −0.123555 0.992338i $$-0.539430\pi$$
−0.123555 + 0.992338i $$0.539430\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.67282 −1.76616 −0.883081 0.469221i $$-0.844535\pi$$
−0.883081 + 0.469221i $$0.844535\pi$$
$$8$$ 0 0
$$9$$ −2.81681 −0.938937
$$10$$ 0 0
$$11$$ 2.67282 0.805887 0.402943 0.915225i $$-0.367987\pi$$
0.402943 + 0.915225i $$0.367987\pi$$
$$12$$ 0 0
$$13$$ 2.24482 0.622600 0.311300 0.950312i $$-0.399236\pi$$
0.311300 + 0.950312i $$0.399236\pi$$
$$14$$ 0 0
$$15$$ −0.428007 −0.110511
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 0.672824 0.140293 0.0701467 0.997537i $$-0.477653\pi$$
0.0701467 + 0.997537i $$0.477653\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 2.48963 0.479130
$$28$$ 0 0
$$29$$ −1.14399 −0.212433 −0.106216 0.994343i $$-0.533874\pi$$
−0.106216 + 0.994343i $$0.533874\pi$$
$$30$$ 0 0
$$31$$ −1.14399 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$32$$ 0 0
$$33$$ −1.14399 −0.199142
$$34$$ 0 0
$$35$$ −4.67282 −0.789851
$$36$$ 0 0
$$37$$ −4.24482 −0.697844 −0.348922 0.937152i $$-0.613452\pi$$
−0.348922 + 0.937152i $$0.613452\pi$$
$$38$$ 0 0
$$39$$ −0.960797 −0.153851
$$40$$ 0 0
$$41$$ 4.85601 0.758382 0.379191 0.925318i $$-0.376202\pi$$
0.379191 + 0.925318i $$0.376202\pi$$
$$42$$ 0 0
$$43$$ −5.52884 −0.843140 −0.421570 0.906796i $$-0.638521\pi$$
−0.421570 + 0.906796i $$0.638521\pi$$
$$44$$ 0 0
$$45$$ −2.81681 −0.419905
$$46$$ 0 0
$$47$$ −4.67282 −0.681601 −0.340801 0.940136i $$-0.610698\pi$$
−0.340801 + 0.940136i $$0.610698\pi$$
$$48$$ 0 0
$$49$$ 14.8353 2.11933
$$50$$ 0 0
$$51$$ −0.856013 −0.119866
$$52$$ 0 0
$$53$$ 0.244817 0.0336282 0.0168141 0.999859i $$-0.494648\pi$$
0.0168141 + 0.999859i $$0.494648\pi$$
$$54$$ 0 0
$$55$$ 2.67282 0.360403
$$56$$ 0 0
$$57$$ −0.428007 −0.0566909
$$58$$ 0 0
$$59$$ 0.287973 0.0374909 0.0187455 0.999824i $$-0.494033\pi$$
0.0187455 + 0.999824i $$0.494033\pi$$
$$60$$ 0 0
$$61$$ 8.87448 1.13626 0.568131 0.822938i $$-0.307667\pi$$
0.568131 + 0.822938i $$0.307667\pi$$
$$62$$ 0 0
$$63$$ 13.1625 1.65831
$$64$$ 0 0
$$65$$ 2.24482 0.278435
$$66$$ 0 0
$$67$$ −9.20561 −1.12464 −0.562322 0.826918i $$-0.690092\pi$$
−0.562322 + 0.826918i $$0.690092\pi$$
$$68$$ 0 0
$$69$$ −0.287973 −0.0346679
$$70$$ 0 0
$$71$$ −9.14399 −1.08519 −0.542596 0.839994i $$-0.682558\pi$$
−0.542596 + 0.839994i $$0.682558\pi$$
$$72$$ 0 0
$$73$$ 1.14399 0.133893 0.0669467 0.997757i $$-0.478674\pi$$
0.0669467 + 0.997757i $$0.478674\pi$$
$$74$$ 0 0
$$75$$ −0.428007 −0.0494220
$$76$$ 0 0
$$77$$ −12.4896 −1.42333
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 7.38485 0.820539
$$82$$ 0 0
$$83$$ −0.183190 −0.0201077 −0.0100538 0.999949i $$-0.503200\pi$$
−0.0100538 + 0.999949i $$0.503200\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ 0.489634 0.0524943
$$88$$ 0 0
$$89$$ 9.14399 0.969261 0.484630 0.874719i $$-0.338954\pi$$
0.484630 + 0.874719i $$0.338954\pi$$
$$90$$ 0 0
$$91$$ −10.4896 −1.09961
$$92$$ 0 0
$$93$$ 0.489634 0.0507727
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 9.10083 0.924049 0.462025 0.886867i $$-0.347123\pi$$
0.462025 + 0.886867i $$0.347123\pi$$
$$98$$ 0 0
$$99$$ −7.52884 −0.756677
$$100$$ 0 0
$$101$$ −2.67282 −0.265956 −0.132978 0.991119i $$-0.542454\pi$$
−0.132978 + 0.991119i $$0.542454\pi$$
$$102$$ 0 0
$$103$$ 6.91764 0.681615 0.340808 0.940133i $$-0.389299\pi$$
0.340808 + 0.940133i $$0.389299\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ 8.62967 0.834261 0.417131 0.908846i $$-0.363036\pi$$
0.417131 + 0.908846i $$0.363036\pi$$
$$108$$ 0 0
$$109$$ −11.6336 −1.11430 −0.557149 0.830412i $$-0.688105\pi$$
−0.557149 + 0.830412i $$0.688105\pi$$
$$110$$ 0 0
$$111$$ 1.81681 0.172444
$$112$$ 0 0
$$113$$ 6.61120 0.621929 0.310965 0.950422i $$-0.399348\pi$$
0.310965 + 0.950422i $$0.399348\pi$$
$$114$$ 0 0
$$115$$ 0.672824 0.0627411
$$116$$ 0 0
$$117$$ −6.32322 −0.584582
$$118$$ 0 0
$$119$$ −9.34565 −0.856714
$$120$$ 0 0
$$121$$ −3.85601 −0.350547
$$122$$ 0 0
$$123$$ −2.07841 −0.187404
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.91764 −0.791313 −0.395656 0.918399i $$-0.629483\pi$$
−0.395656 + 0.918399i $$0.629483\pi$$
$$128$$ 0 0
$$129$$ 2.36638 0.208348
$$130$$ 0 0
$$131$$ −6.77761 −0.592162 −0.296081 0.955163i $$-0.595680\pi$$
−0.296081 + 0.955163i $$0.595680\pi$$
$$132$$ 0 0
$$133$$ −4.67282 −0.405185
$$134$$ 0 0
$$135$$ 2.48963 0.214274
$$136$$ 0 0
$$137$$ −14.0369 −1.19926 −0.599628 0.800279i $$-0.704685\pi$$
−0.599628 + 0.800279i $$0.704685\pi$$
$$138$$ 0 0
$$139$$ −4.96080 −0.420769 −0.210385 0.977619i $$-0.567472\pi$$
−0.210385 + 0.977619i $$0.567472\pi$$
$$140$$ 0 0
$$141$$ 2.00000 0.168430
$$142$$ 0 0
$$143$$ 6.00000 0.501745
$$144$$ 0 0
$$145$$ −1.14399 −0.0950029
$$146$$ 0 0
$$147$$ −6.34960 −0.523706
$$148$$ 0 0
$$149$$ 0.874485 0.0716406 0.0358203 0.999358i $$-0.488596\pi$$
0.0358203 + 0.999358i $$0.488596\pi$$
$$150$$ 0 0
$$151$$ −21.4689 −1.74711 −0.873557 0.486721i $$-0.838193\pi$$
−0.873557 + 0.486721i $$0.838193\pi$$
$$152$$ 0 0
$$153$$ −5.63362 −0.455451
$$154$$ 0 0
$$155$$ −1.14399 −0.0918872
$$156$$ 0 0
$$157$$ −11.9216 −0.951447 −0.475723 0.879595i $$-0.657814\pi$$
−0.475723 + 0.879595i $$0.657814\pi$$
$$158$$ 0 0
$$159$$ −0.104783 −0.00830986
$$160$$ 0 0
$$161$$ −3.14399 −0.247781
$$162$$ 0 0
$$163$$ −2.47116 −0.193556 −0.0967782 0.995306i $$-0.530854\pi$$
−0.0967782 + 0.995306i $$0.530854\pi$$
$$164$$ 0 0
$$165$$ −1.14399 −0.0890592
$$166$$ 0 0
$$167$$ −19.4857 −1.50785 −0.753924 0.656962i $$-0.771841\pi$$
−0.753924 + 0.656962i $$0.771841\pi$$
$$168$$ 0 0
$$169$$ −7.96080 −0.612369
$$170$$ 0 0
$$171$$ −2.81681 −0.215407
$$172$$ 0 0
$$173$$ −21.3025 −1.61960 −0.809799 0.586707i $$-0.800424\pi$$
−0.809799 + 0.586707i $$0.800424\pi$$
$$174$$ 0 0
$$175$$ −4.67282 −0.353232
$$176$$ 0 0
$$177$$ −0.123254 −0.00926437
$$178$$ 0 0
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ 21.2672 1.58078 0.790391 0.612603i $$-0.209878\pi$$
0.790391 + 0.612603i $$0.209878\pi$$
$$182$$ 0 0
$$183$$ −3.79834 −0.280781
$$184$$ 0 0
$$185$$ −4.24482 −0.312085
$$186$$ 0 0
$$187$$ 5.34565 0.390912
$$188$$ 0 0
$$189$$ −11.6336 −0.846221
$$190$$ 0 0
$$191$$ −24.0369 −1.73925 −0.869626 0.493711i $$-0.835640\pi$$
−0.869626 + 0.493711i $$0.835640\pi$$
$$192$$ 0 0
$$193$$ 22.0801 1.58936 0.794680 0.607028i $$-0.207639\pi$$
0.794680 + 0.607028i $$0.207639\pi$$
$$194$$ 0 0
$$195$$ −0.960797 −0.0688041
$$196$$ 0 0
$$197$$ −25.5473 −1.82017 −0.910085 0.414421i $$-0.863984\pi$$
−0.910085 + 0.414421i $$0.863984\pi$$
$$198$$ 0 0
$$199$$ −16.9793 −1.20363 −0.601814 0.798636i $$-0.705555\pi$$
−0.601814 + 0.798636i $$0.705555\pi$$
$$200$$ 0 0
$$201$$ 3.94006 0.277911
$$202$$ 0 0
$$203$$ 5.34565 0.375191
$$204$$ 0 0
$$205$$ 4.85601 0.339159
$$206$$ 0 0
$$207$$ −1.89522 −0.131727
$$208$$ 0 0
$$209$$ 2.67282 0.184883
$$210$$ 0 0
$$211$$ 8.48963 0.584451 0.292225 0.956350i $$-0.405604\pi$$
0.292225 + 0.956350i $$0.405604\pi$$
$$212$$ 0 0
$$213$$ 3.91369 0.268161
$$214$$ 0 0
$$215$$ −5.52884 −0.377064
$$216$$ 0 0
$$217$$ 5.34565 0.362886
$$218$$ 0 0
$$219$$ −0.489634 −0.0330864
$$220$$ 0 0
$$221$$ 4.48963 0.302005
$$222$$ 0 0
$$223$$ 8.55126 0.572635 0.286317 0.958135i $$-0.407569\pi$$
0.286317 + 0.958135i $$0.407569\pi$$
$$224$$ 0 0
$$225$$ −2.81681 −0.187787
$$226$$ 0 0
$$227$$ −19.9832 −1.32633 −0.663166 0.748472i $$-0.730788\pi$$
−0.663166 + 0.748472i $$0.730788\pi$$
$$228$$ 0 0
$$229$$ −0.751230 −0.0496427 −0.0248213 0.999692i $$-0.507902\pi$$
−0.0248213 + 0.999692i $$0.507902\pi$$
$$230$$ 0 0
$$231$$ 5.34565 0.351718
$$232$$ 0 0
$$233$$ −15.2593 −0.999672 −0.499836 0.866120i $$-0.666606\pi$$
−0.499836 + 0.866120i $$0.666606\pi$$
$$234$$ 0 0
$$235$$ −4.67282 −0.304821
$$236$$ 0 0
$$237$$ −0.856013 −0.0556040
$$238$$ 0 0
$$239$$ 16.5266 1.06902 0.534508 0.845164i $$-0.320497\pi$$
0.534508 + 0.845164i $$0.320497\pi$$
$$240$$ 0 0
$$241$$ 4.48963 0.289203 0.144601 0.989490i $$-0.453810\pi$$
0.144601 + 0.989490i $$0.453810\pi$$
$$242$$ 0 0
$$243$$ −10.6297 −0.681893
$$244$$ 0 0
$$245$$ 14.8353 0.947791
$$246$$ 0 0
$$247$$ 2.24482 0.142834
$$248$$ 0 0
$$249$$ 0.0784065 0.00496881
$$250$$ 0 0
$$251$$ 20.0369 1.26472 0.632360 0.774674i $$-0.282086\pi$$
0.632360 + 0.774674i $$0.282086\pi$$
$$252$$ 0 0
$$253$$ 1.79834 0.113061
$$254$$ 0 0
$$255$$ −0.856013 −0.0536056
$$256$$ 0 0
$$257$$ −4.15850 −0.259400 −0.129700 0.991553i $$-0.541401\pi$$
−0.129700 + 0.991553i $$0.541401\pi$$
$$258$$ 0 0
$$259$$ 19.8353 1.23250
$$260$$ 0 0
$$261$$ 3.22239 0.199461
$$262$$ 0 0
$$263$$ −17.0761 −1.05296 −0.526480 0.850187i $$-0.676489\pi$$
−0.526480 + 0.850187i $$0.676489\pi$$
$$264$$ 0 0
$$265$$ 0.244817 0.0150390
$$266$$ 0 0
$$267$$ −3.91369 −0.239514
$$268$$ 0 0
$$269$$ −29.3826 −1.79149 −0.895744 0.444570i $$-0.853356\pi$$
−0.895744 + 0.444570i $$0.853356\pi$$
$$270$$ 0 0
$$271$$ 16.5944 1.00804 0.504020 0.863692i $$-0.331854\pi$$
0.504020 + 0.863692i $$0.331854\pi$$
$$272$$ 0 0
$$273$$ 4.48963 0.271725
$$274$$ 0 0
$$275$$ 2.67282 0.161177
$$276$$ 0 0
$$277$$ −6.36638 −0.382519 −0.191259 0.981540i $$-0.561257\pi$$
−0.191259 + 0.981540i $$0.561257\pi$$
$$278$$ 0 0
$$279$$ 3.22239 0.192920
$$280$$ 0 0
$$281$$ −0.280067 −0.0167074 −0.00835371 0.999965i $$-0.502659\pi$$
−0.00835371 + 0.999965i $$0.502659\pi$$
$$282$$ 0 0
$$283$$ 12.1832 0.724215 0.362108 0.932136i $$-0.382057\pi$$
0.362108 + 0.932136i $$0.382057\pi$$
$$284$$ 0 0
$$285$$ −0.428007 −0.0253529
$$286$$ 0 0
$$287$$ −22.6913 −1.33942
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −3.89522 −0.228342
$$292$$ 0 0
$$293$$ −25.5042 −1.48997 −0.744984 0.667082i $$-0.767543\pi$$
−0.744984 + 0.667082i $$0.767543\pi$$
$$294$$ 0 0
$$295$$ 0.287973 0.0167664
$$296$$ 0 0
$$297$$ 6.65435 0.386125
$$298$$ 0 0
$$299$$ 1.51037 0.0873467
$$300$$ 0 0
$$301$$ 25.8353 1.48912
$$302$$ 0 0
$$303$$ 1.14399 0.0657203
$$304$$ 0 0
$$305$$ 8.87448 0.508152
$$306$$ 0 0
$$307$$ −16.3496 −0.933121 −0.466560 0.884489i $$-0.654507\pi$$
−0.466560 + 0.884489i $$0.654507\pi$$
$$308$$ 0 0
$$309$$ −2.96080 −0.168434
$$310$$ 0 0
$$311$$ 10.3064 0.584425 0.292212 0.956353i $$-0.405609\pi$$
0.292212 + 0.956353i $$0.405609\pi$$
$$312$$ 0 0
$$313$$ −16.9714 −0.959278 −0.479639 0.877466i $$-0.659232\pi$$
−0.479639 + 0.877466i $$0.659232\pi$$
$$314$$ 0 0
$$315$$ 13.1625 0.741620
$$316$$ 0 0
$$317$$ −31.3473 −1.76064 −0.880321 0.474378i $$-0.842673\pi$$
−0.880321 + 0.474378i $$0.842673\pi$$
$$318$$ 0 0
$$319$$ −3.05767 −0.171197
$$320$$ 0 0
$$321$$ −3.69356 −0.206154
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ 2.24482 0.124520
$$326$$ 0 0
$$327$$ 4.97927 0.275354
$$328$$ 0 0
$$329$$ 21.8353 1.20382
$$330$$ 0 0
$$331$$ −26.4112 −1.45169 −0.725846 0.687857i $$-0.758552\pi$$
−0.725846 + 0.687857i $$0.758552\pi$$
$$332$$ 0 0
$$333$$ 11.9568 0.655231
$$334$$ 0 0
$$335$$ −9.20561 −0.502956
$$336$$ 0 0
$$337$$ −23.7137 −1.29177 −0.645884 0.763435i $$-0.723511\pi$$
−0.645884 + 0.763435i $$0.723511\pi$$
$$338$$ 0 0
$$339$$ −2.82964 −0.153685
$$340$$ 0 0
$$341$$ −3.05767 −0.165582
$$342$$ 0 0
$$343$$ −36.6129 −1.97691
$$344$$ 0 0
$$345$$ −0.287973 −0.0155039
$$346$$ 0 0
$$347$$ −3.69356 −0.198280 −0.0991402 0.995073i $$-0.531609\pi$$
−0.0991402 + 0.995073i $$0.531609\pi$$
$$348$$ 0 0
$$349$$ 24.9714 1.33669 0.668343 0.743853i $$-0.267004\pi$$
0.668343 + 0.743853i $$0.267004\pi$$
$$350$$ 0 0
$$351$$ 5.58877 0.298307
$$352$$ 0 0
$$353$$ 12.8639 0.684677 0.342339 0.939577i $$-0.388781\pi$$
0.342339 + 0.939577i $$0.388781\pi$$
$$354$$ 0 0
$$355$$ −9.14399 −0.485312
$$356$$ 0 0
$$357$$ 4.00000 0.211702
$$358$$ 0 0
$$359$$ 32.0554 1.69182 0.845910 0.533326i $$-0.179058\pi$$
0.845910 + 0.533326i $$0.179058\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 1.65040 0.0866235
$$364$$ 0 0
$$365$$ 1.14399 0.0598790
$$366$$ 0 0
$$367$$ 20.4297 1.06642 0.533211 0.845982i $$-0.320985\pi$$
0.533211 + 0.845982i $$0.320985\pi$$
$$368$$ 0 0
$$369$$ −13.6785 −0.712073
$$370$$ 0 0
$$371$$ −1.14399 −0.0593928
$$372$$ 0 0
$$373$$ −5.50415 −0.284994 −0.142497 0.989795i $$-0.545513\pi$$
−0.142497 + 0.989795i $$0.545513\pi$$
$$374$$ 0 0
$$375$$ −0.428007 −0.0221022
$$376$$ 0 0
$$377$$ −2.56804 −0.132261
$$378$$ 0 0
$$379$$ −26.6913 −1.37104 −0.685520 0.728054i $$-0.740425\pi$$
−0.685520 + 0.728054i $$0.740425\pi$$
$$380$$ 0 0
$$381$$ 3.81681 0.195541
$$382$$ 0 0
$$383$$ 7.77365 0.397215 0.198608 0.980079i $$-0.436358\pi$$
0.198608 + 0.980079i $$0.436358\pi$$
$$384$$ 0 0
$$385$$ −12.4896 −0.636531
$$386$$ 0 0
$$387$$ 15.5737 0.791655
$$388$$ 0 0
$$389$$ 24.8145 1.25815 0.629074 0.777346i $$-0.283434\pi$$
0.629074 + 0.777346i $$0.283434\pi$$
$$390$$ 0 0
$$391$$ 1.34565 0.0680523
$$392$$ 0 0
$$393$$ 2.90086 0.146329
$$394$$ 0 0
$$395$$ 2.00000 0.100631
$$396$$ 0 0
$$397$$ 32.4482 1.62853 0.814263 0.580495i $$-0.197141\pi$$
0.814263 + 0.580495i $$0.197141\pi$$
$$398$$ 0 0
$$399$$ 2.00000 0.100125
$$400$$ 0 0
$$401$$ 0.481728 0.0240564 0.0120282 0.999928i $$-0.496171\pi$$
0.0120282 + 0.999928i $$0.496171\pi$$
$$402$$ 0 0
$$403$$ −2.56804 −0.127923
$$404$$ 0 0
$$405$$ 7.38485 0.366956
$$406$$ 0 0
$$407$$ −11.3456 −0.562383
$$408$$ 0 0
$$409$$ 16.9793 0.839571 0.419785 0.907623i $$-0.362105\pi$$
0.419785 + 0.907623i $$0.362105\pi$$
$$410$$ 0 0
$$411$$ 6.00791 0.296348
$$412$$ 0 0
$$413$$ −1.34565 −0.0662150
$$414$$ 0 0
$$415$$ −0.183190 −0.00899243
$$416$$ 0 0
$$417$$ 2.12325 0.103976
$$418$$ 0 0
$$419$$ 9.43196 0.460781 0.230391 0.973098i $$-0.426000\pi$$
0.230391 + 0.973098i $$0.426000\pi$$
$$420$$ 0 0
$$421$$ −10.6050 −0.516855 −0.258428 0.966031i $$-0.583204\pi$$
−0.258428 + 0.966031i $$0.583204\pi$$
$$422$$ 0 0
$$423$$ 13.1625 0.639981
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ −41.4689 −2.00682
$$428$$ 0 0
$$429$$ −2.56804 −0.123986
$$430$$ 0 0
$$431$$ 13.2593 0.638680 0.319340 0.947640i $$-0.396539\pi$$
0.319340 + 0.947640i $$0.396539\pi$$
$$432$$ 0 0
$$433$$ 20.5697 0.988518 0.494259 0.869315i $$-0.335439\pi$$
0.494259 + 0.869315i $$0.335439\pi$$
$$434$$ 0 0
$$435$$ 0.489634 0.0234762
$$436$$ 0 0
$$437$$ 0.672824 0.0321855
$$438$$ 0 0
$$439$$ −12.0784 −0.576471 −0.288235 0.957560i $$-0.593069\pi$$
−0.288235 + 0.957560i $$0.593069\pi$$
$$440$$ 0 0
$$441$$ −41.7882 −1.98991
$$442$$ 0 0
$$443$$ 2.01847 0.0959005 0.0479502 0.998850i $$-0.484731\pi$$
0.0479502 + 0.998850i $$0.484731\pi$$
$$444$$ 0 0
$$445$$ 9.14399 0.433467
$$446$$ 0 0
$$447$$ −0.374285 −0.0177031
$$448$$ 0 0
$$449$$ −12.0369 −0.568058 −0.284029 0.958816i $$-0.591671\pi$$
−0.284029 + 0.958816i $$0.591671\pi$$
$$450$$ 0 0
$$451$$ 12.9793 0.611170
$$452$$ 0 0
$$453$$ 9.18883 0.431729
$$454$$ 0 0
$$455$$ −10.4896 −0.491762
$$456$$ 0 0
$$457$$ 38.6498 1.80796 0.903981 0.427572i $$-0.140631\pi$$
0.903981 + 0.427572i $$0.140631\pi$$
$$458$$ 0 0
$$459$$ 4.97927 0.232412
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ −12.8824 −0.598695 −0.299348 0.954144i $$-0.596769\pi$$
−0.299348 + 0.954144i $$0.596769\pi$$
$$464$$ 0 0
$$465$$ 0.489634 0.0227062
$$466$$ 0 0
$$467$$ −13.2488 −0.613080 −0.306540 0.951858i $$-0.599171\pi$$
−0.306540 + 0.951858i $$0.599171\pi$$
$$468$$ 0 0
$$469$$ 43.0162 1.98630
$$470$$ 0 0
$$471$$ 5.10252 0.235112
$$472$$ 0 0
$$473$$ −14.7776 −0.679475
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −0.689603 −0.0315747
$$478$$ 0 0
$$479$$ 24.9608 1.14049 0.570244 0.821475i $$-0.306849\pi$$
0.570244 + 0.821475i $$0.306849\pi$$
$$480$$ 0 0
$$481$$ −9.52884 −0.434478
$$482$$ 0 0
$$483$$ 1.34565 0.0612291
$$484$$ 0 0
$$485$$ 9.10083 0.413247
$$486$$ 0 0
$$487$$ −3.73671 −0.169327 −0.0846633 0.996410i $$-0.526981\pi$$
−0.0846633 + 0.996410i $$0.526981\pi$$
$$488$$ 0 0
$$489$$ 1.05767 0.0478297
$$490$$ 0 0
$$491$$ −41.9955 −1.89523 −0.947615 0.319416i $$-0.896513\pi$$
−0.947615 + 0.319416i $$0.896513\pi$$
$$492$$ 0 0
$$493$$ −2.28797 −0.103045
$$494$$ 0 0
$$495$$ −7.52884 −0.338396
$$496$$ 0 0
$$497$$ 42.7282 1.91662
$$498$$ 0 0
$$499$$ −12.0185 −0.538021 −0.269010 0.963137i $$-0.586697\pi$$
−0.269010 + 0.963137i $$0.586697\pi$$
$$500$$ 0 0
$$501$$ 8.34000 0.372604
$$502$$ 0 0
$$503$$ −14.9977 −0.668716 −0.334358 0.942446i $$-0.608519\pi$$
−0.334358 + 0.942446i $$0.608519\pi$$
$$504$$ 0 0
$$505$$ −2.67282 −0.118939
$$506$$ 0 0
$$507$$ 3.40727 0.151322
$$508$$ 0 0
$$509$$ −7.79834 −0.345655 −0.172828 0.984952i $$-0.555290\pi$$
−0.172828 + 0.984952i $$0.555290\pi$$
$$510$$ 0 0
$$511$$ −5.34565 −0.236478
$$512$$ 0 0
$$513$$ 2.48963 0.109920
$$514$$ 0 0
$$515$$ 6.91764 0.304828
$$516$$ 0 0
$$517$$ −12.4896 −0.549293
$$518$$ 0 0
$$519$$ 9.11761 0.400219
$$520$$ 0 0
$$521$$ −19.3456 −0.847548 −0.423774 0.905768i $$-0.639295\pi$$
−0.423774 + 0.905768i $$0.639295\pi$$
$$522$$ 0 0
$$523$$ 8.42801 0.368531 0.184266 0.982877i $$-0.441009\pi$$
0.184266 + 0.982877i $$0.441009\pi$$
$$524$$ 0 0
$$525$$ 2.00000 0.0872872
$$526$$ 0 0
$$527$$ −2.28797 −0.0996657
$$528$$ 0 0
$$529$$ −22.5473 −0.980318
$$530$$ 0 0
$$531$$ −0.811166 −0.0352016
$$532$$ 0 0
$$533$$ 10.9009 0.472169
$$534$$ 0 0
$$535$$ 8.62967 0.373093
$$536$$ 0 0
$$537$$ 2.56804 0.110819
$$538$$ 0 0
$$539$$ 39.6521 1.70794
$$540$$ 0 0
$$541$$ 40.6314 1.74688 0.873439 0.486933i $$-0.161884\pi$$
0.873439 + 0.486933i $$0.161884\pi$$
$$542$$ 0 0
$$543$$ −9.10252 −0.390627
$$544$$ 0 0
$$545$$ −11.6336 −0.498330
$$546$$ 0 0
$$547$$ −35.5226 −1.51884 −0.759419 0.650602i $$-0.774517\pi$$
−0.759419 + 0.650602i $$0.774517\pi$$
$$548$$ 0 0
$$549$$ −24.9977 −1.06688
$$550$$ 0 0
$$551$$ −1.14399 −0.0487355
$$552$$ 0 0
$$553$$ −9.34565 −0.397417
$$554$$ 0 0
$$555$$ 1.81681 0.0771193
$$556$$ 0 0
$$557$$ −3.67508 −0.155718 −0.0778592 0.996964i $$-0.524808\pi$$
−0.0778592 + 0.996964i $$0.524808\pi$$
$$558$$ 0 0
$$559$$ −12.4112 −0.524939
$$560$$ 0 0
$$561$$ −2.28797 −0.0965983
$$562$$ 0 0
$$563$$ 10.3865 0.437741 0.218870 0.975754i $$-0.429763\pi$$
0.218870 + 0.975754i $$0.429763\pi$$
$$564$$ 0 0
$$565$$ 6.61120 0.278135
$$566$$ 0 0
$$567$$ −34.5081 −1.44920
$$568$$ 0 0
$$569$$ 32.4033 1.35842 0.679209 0.733945i $$-0.262323\pi$$
0.679209 + 0.733945i $$0.262323\pi$$
$$570$$ 0 0
$$571$$ −6.70977 −0.280795 −0.140397 0.990095i $$-0.544838\pi$$
−0.140397 + 0.990095i $$0.544838\pi$$
$$572$$ 0 0
$$573$$ 10.2880 0.429786
$$574$$ 0 0
$$575$$ 0.672824 0.0280587
$$576$$ 0 0
$$577$$ 13.7199 0.571168 0.285584 0.958354i $$-0.407812\pi$$
0.285584 + 0.958354i $$0.407812\pi$$
$$578$$ 0 0
$$579$$ −9.45043 −0.392746
$$580$$ 0 0
$$581$$ 0.856013 0.0355134
$$582$$ 0 0
$$583$$ 0.654353 0.0271005
$$584$$ 0 0
$$585$$ −6.32322 −0.261433
$$586$$ 0 0
$$587$$ −6.10478 −0.251971 −0.125986 0.992032i $$-0.540209\pi$$
−0.125986 + 0.992032i $$0.540209\pi$$
$$588$$ 0 0
$$589$$ −1.14399 −0.0471371
$$590$$ 0 0
$$591$$ 10.9344 0.449782
$$592$$ 0 0
$$593$$ −26.4033 −1.08425 −0.542127 0.840296i $$-0.682381\pi$$
−0.542127 + 0.840296i $$0.682381\pi$$
$$594$$ 0 0
$$595$$ −9.34565 −0.383134
$$596$$ 0 0
$$597$$ 7.26724 0.297428
$$598$$ 0 0
$$599$$ 7.34565 0.300135 0.150068 0.988676i $$-0.452051\pi$$
0.150068 + 0.988676i $$0.452051\pi$$
$$600$$ 0 0
$$601$$ −18.7776 −0.765955 −0.382977 0.923758i $$-0.625101\pi$$
−0.382977 + 0.923758i $$0.625101\pi$$
$$602$$ 0 0
$$603$$ 25.9305 1.05597
$$604$$ 0 0
$$605$$ −3.85601 −0.156769
$$606$$ 0 0
$$607$$ 5.20561 0.211289 0.105645 0.994404i $$-0.466309\pi$$
0.105645 + 0.994404i $$0.466309\pi$$
$$608$$ 0 0
$$609$$ −2.28797 −0.0927133
$$610$$ 0 0
$$611$$ −10.4896 −0.424365
$$612$$ 0 0
$$613$$ −28.8145 −1.16381 −0.581904 0.813257i $$-0.697692\pi$$
−0.581904 + 0.813257i $$0.697692\pi$$
$$614$$ 0 0
$$615$$ −2.07841 −0.0838094
$$616$$ 0 0
$$617$$ 38.5266 1.55102 0.775511 0.631334i $$-0.217492\pi$$
0.775511 + 0.631334i $$0.217492\pi$$
$$618$$ 0 0
$$619$$ 36.4218 1.46392 0.731958 0.681350i $$-0.238607\pi$$
0.731958 + 0.681350i $$0.238607\pi$$
$$620$$ 0 0
$$621$$ 1.67508 0.0672188
$$622$$ 0 0
$$623$$ −42.7282 −1.71187
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −1.14399 −0.0456864
$$628$$ 0 0
$$629$$ −8.48963 −0.338504
$$630$$ 0 0
$$631$$ 10.7098 0.426349 0.213175 0.977014i $$-0.431620\pi$$
0.213175 + 0.977014i $$0.431620\pi$$
$$632$$ 0 0
$$633$$ −3.63362 −0.144423
$$634$$ 0 0
$$635$$ −8.91764 −0.353886
$$636$$ 0 0
$$637$$ 33.3025 1.31949
$$638$$ 0 0
$$639$$ 25.7569 1.01893
$$640$$ 0 0
$$641$$ −30.6050 −1.20882 −0.604412 0.796672i $$-0.706592\pi$$
−0.604412 + 0.796672i $$0.706592\pi$$
$$642$$ 0 0
$$643$$ −13.4425 −0.530121 −0.265061 0.964232i $$-0.585392\pi$$
−0.265061 + 0.964232i $$0.585392\pi$$
$$644$$ 0 0
$$645$$ 2.36638 0.0931761
$$646$$ 0 0
$$647$$ −33.5288 −1.31815 −0.659077 0.752075i $$-0.729053\pi$$
−0.659077 + 0.752075i $$0.729053\pi$$
$$648$$ 0 0
$$649$$ 0.769701 0.0302134
$$650$$ 0 0
$$651$$ −2.28797 −0.0896727
$$652$$ 0 0
$$653$$ 23.2593 0.910208 0.455104 0.890438i $$-0.349602\pi$$
0.455104 + 0.890438i $$0.349602\pi$$
$$654$$ 0 0
$$655$$ −6.77761 −0.264823
$$656$$ 0 0
$$657$$ −3.22239 −0.125718
$$658$$ 0 0
$$659$$ −30.9793 −1.20678 −0.603390 0.797446i $$-0.706184\pi$$
−0.603390 + 0.797446i $$0.706184\pi$$
$$660$$ 0 0
$$661$$ 10.3743 0.403513 0.201756 0.979436i $$-0.435335\pi$$
0.201756 + 0.979436i $$0.435335\pi$$
$$662$$ 0 0
$$663$$ −1.92159 −0.0746285
$$664$$ 0 0
$$665$$ −4.67282 −0.181204
$$666$$ 0 0
$$667$$ −0.769701 −0.0298030
$$668$$ 0 0
$$669$$ −3.66000 −0.141504
$$670$$ 0 0
$$671$$ 23.7199 0.915698
$$672$$ 0 0
$$673$$ −26.0880 −1.00562 −0.502809 0.864397i $$-0.667700\pi$$
−0.502809 + 0.864397i $$0.667700\pi$$
$$674$$ 0 0
$$675$$ 2.48963 0.0958261
$$676$$ 0 0
$$677$$ −12.5249 −0.481370 −0.240685 0.970603i $$-0.577372\pi$$
−0.240685 + 0.970603i $$0.577372\pi$$
$$678$$ 0 0
$$679$$ −42.5266 −1.63202
$$680$$ 0 0
$$681$$ 8.55295 0.327750
$$682$$ 0 0
$$683$$ −25.9339 −0.992331 −0.496166 0.868228i $$-0.665259\pi$$
−0.496166 + 0.868228i $$0.665259\pi$$
$$684$$ 0 0
$$685$$ −14.0369 −0.536324
$$686$$ 0 0
$$687$$ 0.321532 0.0122672
$$688$$ 0 0
$$689$$ 0.549569 0.0209369
$$690$$ 0 0
$$691$$ 5.52093 0.210026 0.105013 0.994471i $$-0.466512\pi$$
0.105013 + 0.994471i $$0.466512\pi$$
$$692$$ 0 0
$$693$$ 35.1809 1.33641
$$694$$ 0 0
$$695$$ −4.96080 −0.188174
$$696$$ 0 0
$$697$$ 9.71203 0.367869
$$698$$ 0 0
$$699$$ 6.53110 0.247029
$$700$$ 0 0
$$701$$ 22.9687 0.867516 0.433758 0.901029i $$-0.357187\pi$$
0.433758 + 0.901029i $$0.357187\pi$$
$$702$$ 0 0
$$703$$ −4.24482 −0.160096
$$704$$ 0 0
$$705$$ 2.00000 0.0753244
$$706$$ 0 0
$$707$$ 12.4896 0.469721
$$708$$ 0 0
$$709$$ −16.2386 −0.609854 −0.304927 0.952376i $$-0.598632\pi$$
−0.304927 + 0.952376i $$0.598632\pi$$
$$710$$ 0 0
$$711$$ −5.63362 −0.211277
$$712$$ 0 0
$$713$$ −0.769701 −0.0288255
$$714$$ 0 0
$$715$$ 6.00000 0.224387
$$716$$ 0 0
$$717$$ −7.07349 −0.264164
$$718$$ 0 0
$$719$$ 13.1704 0.491172 0.245586 0.969375i $$-0.421020\pi$$
0.245586 + 0.969375i $$0.421020\pi$$
$$720$$ 0 0
$$721$$ −32.3249 −1.20384
$$722$$ 0 0
$$723$$ −1.92159 −0.0714648
$$724$$ 0 0
$$725$$ −1.14399 −0.0424866
$$726$$ 0 0
$$727$$ −40.7961 −1.51304 −0.756521 0.653969i $$-0.773103\pi$$
−0.756521 + 0.653969i $$0.773103\pi$$
$$728$$ 0 0
$$729$$ −17.6050 −0.652036
$$730$$ 0 0
$$731$$ −11.0577 −0.408983
$$732$$ 0 0
$$733$$ 27.8722 1.02948 0.514742 0.857345i $$-0.327888\pi$$
0.514742 + 0.857345i $$0.327888\pi$$
$$734$$ 0 0
$$735$$ −6.34960 −0.234209
$$736$$ 0 0
$$737$$ −24.6050 −0.906336
$$738$$ 0 0
$$739$$ 36.0739 1.32700 0.663500 0.748177i $$-0.269070\pi$$
0.663500 + 0.748177i $$0.269070\pi$$
$$740$$ 0 0
$$741$$ −0.960797 −0.0352958
$$742$$ 0 0
$$743$$ 19.8890 0.729657 0.364828 0.931075i $$-0.381128\pi$$
0.364828 + 0.931075i $$0.381128\pi$$
$$744$$ 0 0
$$745$$ 0.874485 0.0320386
$$746$$ 0 0
$$747$$ 0.516011 0.0188798
$$748$$ 0 0
$$749$$ −40.3249 −1.47344
$$750$$ 0 0
$$751$$ 17.1025 0.624080 0.312040 0.950069i $$-0.398988\pi$$
0.312040 + 0.950069i $$0.398988\pi$$
$$752$$ 0 0
$$753$$ −8.57595 −0.312525
$$754$$ 0 0
$$755$$ −21.4689 −0.781333
$$756$$ 0 0
$$757$$ −10.6992 −0.388869 −0.194435 0.980915i $$-0.562287\pi$$
−0.194435 + 0.980915i $$0.562287\pi$$
$$758$$ 0 0
$$759$$ −0.769701 −0.0279384
$$760$$ 0 0
$$761$$ −48.4218 −1.75529 −0.877644 0.479312i $$-0.840886\pi$$
−0.877644 + 0.479312i $$0.840886\pi$$
$$762$$ 0 0
$$763$$ 54.3619 1.96803
$$764$$ 0 0
$$765$$ −5.63362 −0.203684
$$766$$ 0 0
$$767$$ 0.646447 0.0233418
$$768$$ 0 0
$$769$$ −49.4011 −1.78145 −0.890724 0.454545i $$-0.849802\pi$$
−0.890724 + 0.454545i $$0.849802\pi$$
$$770$$ 0 0
$$771$$ 1.77987 0.0641004
$$772$$ 0 0
$$773$$ −32.8129 −1.18020 −0.590098 0.807331i $$-0.700911\pi$$
−0.590098 + 0.807331i $$0.700911\pi$$
$$774$$ 0 0
$$775$$ −1.14399 −0.0410932
$$776$$ 0 0
$$777$$ −8.48963 −0.304564
$$778$$ 0 0
$$779$$ 4.85601 0.173985
$$780$$ 0 0
$$781$$ −24.4403 −0.874541
$$782$$ 0 0
$$783$$ −2.84811 −0.101783
$$784$$ 0 0
$$785$$ −11.9216 −0.425500
$$786$$ 0 0
$$787$$ −30.1400 −1.07438 −0.537188 0.843462i $$-0.680513\pi$$
−0.537188 + 0.843462i $$0.680513\pi$$
$$788$$ 0 0
$$789$$ 7.30871 0.260197
$$790$$ 0 0
$$791$$ −30.8930 −1.09843
$$792$$ 0 0
$$793$$ 19.9216 0.707437
$$794$$ 0 0
$$795$$ −0.104783 −0.00371628
$$796$$ 0 0
$$797$$ −1.31040 −0.0464166 −0.0232083 0.999731i $$-0.507388\pi$$
−0.0232083 + 0.999731i $$0.507388\pi$$
$$798$$ 0 0
$$799$$ −9.34565 −0.330625
$$800$$ 0 0
$$801$$ −25.7569 −0.910074
$$802$$ 0 0
$$803$$ 3.05767 0.107903
$$804$$ 0 0
$$805$$ −3.14399 −0.110811
$$806$$ 0 0
$$807$$ 12.5759 0.442694
$$808$$ 0 0
$$809$$ 35.5058 1.24832 0.624159 0.781297i $$-0.285442\pi$$
0.624159 + 0.781297i $$0.285442\pi$$
$$810$$ 0 0
$$811$$ 26.9299 0.945637 0.472818 0.881160i $$-0.343237\pi$$
0.472818 + 0.881160i $$0.343237\pi$$
$$812$$ 0 0
$$813$$ −7.10252 −0.249096
$$814$$ 0 0
$$815$$ −2.47116 −0.0865611
$$816$$ 0 0
$$817$$ −5.52884 −0.193430
$$818$$ 0 0
$$819$$ 29.5473 1.03247
$$820$$ 0 0
$$821$$ −13.1440 −0.458728 −0.229364 0.973341i $$-0.573665\pi$$
−0.229364 + 0.973341i $$0.573665\pi$$
$$822$$ 0 0
$$823$$ 40.3928 1.40800 0.704001 0.710198i $$-0.251395\pi$$
0.704001 + 0.710198i $$0.251395\pi$$
$$824$$ 0 0
$$825$$ −1.14399 −0.0398285
$$826$$ 0 0
$$827$$ 27.3658 0.951602 0.475801 0.879553i $$-0.342158\pi$$
0.475801 + 0.879553i $$0.342158\pi$$
$$828$$ 0 0
$$829$$ 29.3456 1.01922 0.509608 0.860407i $$-0.329790\pi$$
0.509608 + 0.860407i $$0.329790\pi$$
$$830$$ 0 0
$$831$$ 2.72485 0.0945241
$$832$$ 0 0
$$833$$ 29.6706 1.02802
$$834$$ 0 0
$$835$$ −19.4857 −0.674330
$$836$$ 0 0
$$837$$ −2.84811 −0.0984450
$$838$$ 0 0
$$839$$ 49.6336 1.71354 0.856771 0.515696i $$-0.172467\pi$$
0.856771 + 0.515696i $$0.172467\pi$$
$$840$$ 0 0
$$841$$ −27.6913 −0.954872
$$842$$ 0 0
$$843$$ 0.119871 0.00412857
$$844$$ 0 0
$$845$$ −7.96080 −0.273860
$$846$$ 0 0
$$847$$ 18.0185 0.619122
$$848$$ 0 0
$$849$$ −5.21449 −0.178961
$$850$$ 0 0
$$851$$ −2.85601 −0.0979029
$$852$$ 0 0
$$853$$ −31.1730 −1.06734 −0.533672 0.845692i $$-0.679188\pi$$
−0.533672 + 0.845692i $$0.679188\pi$$
$$854$$ 0 0
$$855$$ −2.81681 −0.0963329
$$856$$ 0 0
$$857$$ 48.4755 1.65589 0.827946 0.560808i $$-0.189509\pi$$
0.827946 + 0.560808i $$0.189509\pi$$
$$858$$ 0 0
$$859$$ 25.8353 0.881488 0.440744 0.897633i $$-0.354715\pi$$
0.440744 + 0.897633i $$0.354715\pi$$
$$860$$ 0 0
$$861$$ 9.71203 0.330985
$$862$$ 0 0
$$863$$ 38.3865 1.30669 0.653347 0.757059i $$-0.273364\pi$$
0.653347 + 0.757059i $$0.273364\pi$$
$$864$$ 0 0
$$865$$ −21.3025 −0.724306
$$866$$ 0 0
$$867$$ 5.56409 0.188966
$$868$$ 0 0
$$869$$ 5.34565 0.181339
$$870$$ 0 0
$$871$$ −20.6649 −0.700204
$$872$$ 0 0
$$873$$ −25.6353 −0.867624
$$874$$ 0 0
$$875$$ −4.67282 −0.157970
$$876$$ 0 0
$$877$$ 17.2610 0.582863 0.291432 0.956592i $$-0.405868\pi$$
0.291432 + 0.956592i $$0.405868\pi$$
$$878$$ 0 0
$$879$$ 10.9159 0.368186
$$880$$ 0 0
$$881$$ −13.8168 −0.465500 −0.232750 0.972537i $$-0.574772\pi$$
−0.232750 + 0.972537i $$0.574772\pi$$
$$882$$ 0 0
$$883$$ −51.2073 −1.72326 −0.861632 0.507534i $$-0.830557\pi$$
−0.861632 + 0.507534i $$0.830557\pi$$
$$884$$ 0 0
$$885$$ −0.123254 −0.00414315
$$886$$ 0 0
$$887$$ 39.6538 1.33144 0.665722 0.746200i $$-0.268124\pi$$
0.665722 + 0.746200i $$0.268124\pi$$
$$888$$ 0 0
$$889$$ 41.6706 1.39759
$$890$$ 0 0
$$891$$ 19.7384 0.661261
$$892$$ 0 0
$$893$$ −4.67282 −0.156370
$$894$$ 0 0
$$895$$ −6.00000 −0.200558
$$896$$ 0 0
$$897$$ −0.646447 −0.0215842
$$898$$ 0 0
$$899$$ 1.30871 0.0436478
$$900$$ 0 0
$$901$$ 0.489634 0.0163121
$$902$$ 0 0
$$903$$ −11.0577 −0.367976
$$904$$ 0 0
$$905$$ 21.2672 0.706947
$$906$$ 0 0
$$907$$ 4.62967 0.153726 0.0768628 0.997042i $$-0.475510\pi$$
0.0768628 + 0.997042i $$0.475510\pi$$
$$908$$ 0 0
$$909$$ 7.52884 0.249716
$$910$$ 0 0
$$911$$ −15.9506 −0.528468 −0.264234 0.964459i $$-0.585119\pi$$
−0.264234 + 0.964459i $$0.585119\pi$$
$$912$$ 0 0
$$913$$ −0.489634 −0.0162045
$$914$$ 0 0
$$915$$ −3.79834 −0.125569
$$916$$ 0 0
$$917$$ 31.6706 1.04585
$$918$$ 0 0
$$919$$ −32.1233 −1.05965 −0.529824 0.848107i $$-0.677742\pi$$
−0.529824 + 0.848107i $$0.677742\pi$$
$$920$$ 0 0
$$921$$ 6.99774 0.230583
$$922$$ 0 0
$$923$$ −20.5266 −0.675640
$$924$$ 0 0
$$925$$ −4.24482 −0.139569
$$926$$ 0 0
$$927$$ −19.4857 −0.639994
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 14.8353 0.486207
$$932$$ 0 0
$$933$$ −4.41123 −0.144417
$$934$$ 0 0
$$935$$ 5.34565 0.174821
$$936$$ 0 0
$$937$$ −20.9872 −0.685621 −0.342811 0.939405i $$-0.611379\pi$$
−0.342811 + 0.939405i $$0.611379\pi$$
$$938$$ 0 0
$$939$$ 7.26386 0.237047
$$940$$ 0 0
$$941$$ 3.51827 0.114692 0.0573462 0.998354i $$-0.481736\pi$$
0.0573462 + 0.998354i $$0.481736\pi$$
$$942$$ 0 0
$$943$$ 3.26724 0.106396
$$944$$ 0 0
$$945$$ −11.6336 −0.378442
$$946$$ 0 0
$$947$$ 0.112689 0.00366190 0.00183095 0.999998i $$-0.499417\pi$$
0.00183095 + 0.999998i $$0.499417\pi$$
$$948$$ 0 0
$$949$$ 2.56804 0.0833621
$$950$$ 0 0
$$951$$ 13.4169 0.435072
$$952$$ 0 0
$$953$$ −31.6353 −1.02477 −0.512384 0.858756i $$-0.671238\pi$$
−0.512384 + 0.858756i $$0.671238\pi$$
$$954$$ 0 0
$$955$$ −24.0369 −0.777817
$$956$$ 0 0
$$957$$ 1.30871 0.0423044
$$958$$ 0 0
$$959$$ 65.5922 2.11808
$$960$$ 0 0
$$961$$ −29.6913 −0.957784
$$962$$ 0 0
$$963$$ −24.3081 −0.783319
$$964$$ 0 0
$$965$$ 22.0801 0.710784
$$966$$ 0 0
$$967$$ 31.6442 1.01761 0.508804 0.860882i $$-0.330088\pi$$
0.508804 + 0.860882i $$0.330088\pi$$
$$968$$ 0 0
$$969$$ −0.856013 −0.0274991
$$970$$ 0 0
$$971$$ 55.4689 1.78008 0.890041 0.455881i $$-0.150676\pi$$
0.890041 + 0.455881i $$0.150676\pi$$
$$972$$ 0 0
$$973$$ 23.1809 0.743146
$$974$$ 0 0
$$975$$ −0.960797 −0.0307701
$$976$$ 0 0
$$977$$ −2.82076 −0.0902442 −0.0451221 0.998981i $$-0.514368\pi$$
−0.0451221 + 0.998981i $$0.514368\pi$$
$$978$$ 0 0
$$979$$ 24.4403 0.781114
$$980$$ 0 0
$$981$$ 32.7697 1.04626
$$982$$ 0 0
$$983$$ 24.9467 0.795675 0.397838 0.917456i $$-0.369761\pi$$
0.397838 + 0.917456i $$0.369761\pi$$
$$984$$ 0 0
$$985$$ −25.5473 −0.814005
$$986$$ 0 0
$$987$$ −9.34565 −0.297475
$$988$$ 0 0
$$989$$ −3.71993 −0.118287
$$990$$ 0 0
$$991$$ 31.0241 0.985514 0.492757 0.870167i $$-0.335989\pi$$
0.492757 + 0.870167i $$0.335989\pi$$
$$992$$ 0 0
$$993$$ 11.3042 0.358727
$$994$$ 0 0
$$995$$ −16.9793 −0.538279
$$996$$ 0 0
$$997$$ −55.7859 −1.76676 −0.883379 0.468660i $$-0.844737\pi$$
−0.883379 + 0.468660i $$0.844737\pi$$
$$998$$ 0 0
$$999$$ −10.5680 −0.334358
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.bp.1.2 3
4.3 odd 2 6080.2.a.bz.1.2 3
8.3 odd 2 3040.2.a.k.1.2 3
8.5 even 2 3040.2.a.n.1.2 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.k.1.2 3 8.3 odd 2
3040.2.a.n.1.2 yes 3 8.5 even 2
6080.2.a.bp.1.2 3 1.1 even 1 trivial
6080.2.a.bz.1.2 3 4.3 odd 2