# Properties

 Label 6080.2.a.cf.1.4 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

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: $$4$$ Coefficient field: 4.4.17428.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ x^4 - x^3 - 6*x^2 + 4*x + 6 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 1.4 Root $$2.36865$$ 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)

 $$f(q)$$ $$=$$ $$q+2.36865 q^{3} -1.00000 q^{5} -3.61050 q^{7} +2.61050 q^{9} +O(q^{10})$$ $$q+2.36865 q^{3} -1.00000 q^{5} -3.61050 q^{7} +2.61050 q^{9} +2.20421 q^{11} -2.57286 q^{13} -2.36865 q^{15} +0.922589 q^{17} +1.00000 q^{19} -8.55201 q^{21} +7.23992 q^{23} +1.00000 q^{25} -0.922589 q^{27} -3.81471 q^{29} -4.48370 q^{31} +5.22100 q^{33} +3.61050 q^{35} +4.13397 q^{37} -6.09420 q^{39} -4.73730 q^{41} -5.42521 q^{43} -2.61050 q^{45} +0.279491 q^{47} +6.03571 q^{49} +2.18529 q^{51} -9.86914 q^{53} -2.20421 q^{55} +2.36865 q^{57} -1.33101 q^{59} +11.1299 q^{61} -9.42521 q^{63} +2.57286 q^{65} -13.7028 q^{67} +17.1488 q^{69} -11.1457 q^{71} -9.28931 q^{73} +2.36865 q^{75} -7.95830 q^{77} +4.85042 q^{79} -10.0168 q^{81} +9.42521 q^{83} -0.922589 q^{85} -9.03571 q^{87} -10.3667 q^{89} +9.28931 q^{91} -10.6203 q^{93} -1.00000 q^{95} +2.75815 q^{97} +5.75409 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} - 4 q^{5} - 5 q^{7} + q^{9}+O(q^{10})$$ 4 * q + q^3 - 4 * q^5 - 5 * q^7 + q^9 $$4 q + q^{3} - 4 q^{5} - 5 q^{7} + q^{9} + 6 q^{11} + q^{13} - q^{15} - q^{17} + 4 q^{19} - 5 q^{21} - 5 q^{23} + 4 q^{25} + q^{27} - 3 q^{29} - 16 q^{31} + 2 q^{33} + 5 q^{35} + 8 q^{37} - 13 q^{39} - 2 q^{41} - q^{45} + 2 q^{47} - 7 q^{49} + 21 q^{51} - 13 q^{53} - 6 q^{55} + q^{57} + 5 q^{59} + 2 q^{61} - 16 q^{63} - q^{65} - q^{67} + 11 q^{69} - 22 q^{71} + 9 q^{73} + q^{75} + 4 q^{77} - 24 q^{79} - 24 q^{81} + 16 q^{83} + q^{85} - 5 q^{87} - 9 q^{91} + 14 q^{93} - 4 q^{95} + 12 q^{97} - 10 q^{99}+O(q^{100})$$ 4 * q + q^3 - 4 * q^5 - 5 * q^7 + q^9 + 6 * q^11 + q^13 - q^15 - q^17 + 4 * q^19 - 5 * q^21 - 5 * q^23 + 4 * q^25 + q^27 - 3 * q^29 - 16 * q^31 + 2 * q^33 + 5 * q^35 + 8 * q^37 - 13 * q^39 - 2 * q^41 - q^45 + 2 * q^47 - 7 * q^49 + 21 * q^51 - 13 * q^53 - 6 * q^55 + q^57 + 5 * q^59 + 2 * q^61 - 16 * q^63 - q^65 - q^67 + 11 * q^69 - 22 * q^71 + 9 * q^73 + q^75 + 4 * q^77 - 24 * q^79 - 24 * q^81 + 16 * q^83 + q^85 - 5 * q^87 - 9 * q^91 + 14 * q^93 - 4 * q^95 + 12 * q^97 - 10 * 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
$$3$$ 2.36865 1.36754 0.683770 0.729697i $$-0.260339\pi$$
0.683770 + 0.729697i $$0.260339\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.61050 −1.36464 −0.682320 0.731053i $$-0.739029\pi$$
−0.682320 + 0.731053i $$0.739029\pi$$
$$8$$ 0 0
$$9$$ 2.61050 0.870167
$$10$$ 0 0
$$11$$ 2.20421 0.664594 0.332297 0.943175i $$-0.392176\pi$$
0.332297 + 0.943175i $$0.392176\pi$$
$$12$$ 0 0
$$13$$ −2.57286 −0.713583 −0.356791 0.934184i $$-0.616129\pi$$
−0.356791 + 0.934184i $$0.616129\pi$$
$$14$$ 0 0
$$15$$ −2.36865 −0.611583
$$16$$ 0 0
$$17$$ 0.922589 0.223761 0.111880 0.993722i $$-0.464313\pi$$
0.111880 + 0.993722i $$0.464313\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −8.55201 −1.86620
$$22$$ 0 0
$$23$$ 7.23992 1.50963 0.754814 0.655939i $$-0.227727\pi$$
0.754814 + 0.655939i $$0.227727\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −0.922589 −0.177552
$$28$$ 0 0
$$29$$ −3.81471 −0.708374 −0.354187 0.935175i $$-0.615242\pi$$
−0.354187 + 0.935175i $$0.615242\pi$$
$$30$$ 0 0
$$31$$ −4.48370 −0.805296 −0.402648 0.915355i $$-0.631910\pi$$
−0.402648 + 0.915355i $$0.631910\pi$$
$$32$$ 0 0
$$33$$ 5.22100 0.908859
$$34$$ 0 0
$$35$$ 3.61050 0.610286
$$36$$ 0 0
$$37$$ 4.13397 0.679621 0.339810 0.940494i $$-0.389637\pi$$
0.339810 + 0.940494i $$0.389637\pi$$
$$38$$ 0 0
$$39$$ −6.09420 −0.975853
$$40$$ 0 0
$$41$$ −4.73730 −0.739842 −0.369921 0.929063i $$-0.620615\pi$$
−0.369921 + 0.929063i $$0.620615\pi$$
$$42$$ 0 0
$$43$$ −5.42521 −0.827337 −0.413668 0.910428i $$-0.635753\pi$$
−0.413668 + 0.910428i $$0.635753\pi$$
$$44$$ 0 0
$$45$$ −2.61050 −0.389150
$$46$$ 0 0
$$47$$ 0.279491 0.0407680 0.0203840 0.999792i $$-0.493511\pi$$
0.0203840 + 0.999792i $$0.493511\pi$$
$$48$$ 0 0
$$49$$ 6.03571 0.862244
$$50$$ 0 0
$$51$$ 2.18529 0.306002
$$52$$ 0 0
$$53$$ −9.86914 −1.35563 −0.677815 0.735232i $$-0.737073\pi$$
−0.677815 + 0.735232i $$0.737073\pi$$
$$54$$ 0 0
$$55$$ −2.20421 −0.297216
$$56$$ 0 0
$$57$$ 2.36865 0.313735
$$58$$ 0 0
$$59$$ −1.33101 −0.173283 −0.0866413 0.996240i $$-0.527613\pi$$
−0.0866413 + 0.996240i $$0.527613\pi$$
$$60$$ 0 0
$$61$$ 11.1299 1.42504 0.712519 0.701652i $$-0.247554\pi$$
0.712519 + 0.701652i $$0.247554\pi$$
$$62$$ 0 0
$$63$$ −9.42521 −1.18746
$$64$$ 0 0
$$65$$ 2.57286 0.319124
$$66$$ 0 0
$$67$$ −13.7028 −1.67406 −0.837030 0.547157i $$-0.815710\pi$$
−0.837030 + 0.547157i $$0.815710\pi$$
$$68$$ 0 0
$$69$$ 17.1488 2.06448
$$70$$ 0 0
$$71$$ −11.1457 −1.32275 −0.661377 0.750054i $$-0.730028\pi$$
−0.661377 + 0.750054i $$0.730028\pi$$
$$72$$ 0 0
$$73$$ −9.28931 −1.08723 −0.543616 0.839334i $$-0.682945\pi$$
−0.543616 + 0.839334i $$0.682945\pi$$
$$74$$ 0 0
$$75$$ 2.36865 0.273508
$$76$$ 0 0
$$77$$ −7.95830 −0.906932
$$78$$ 0 0
$$79$$ 4.85042 0.545715 0.272857 0.962054i $$-0.412031\pi$$
0.272857 + 0.962054i $$0.412031\pi$$
$$80$$ 0 0
$$81$$ −10.0168 −1.11298
$$82$$ 0 0
$$83$$ 9.42521 1.03455 0.517276 0.855819i $$-0.326946\pi$$
0.517276 + 0.855819i $$0.326946\pi$$
$$84$$ 0 0
$$85$$ −0.922589 −0.100069
$$86$$ 0 0
$$87$$ −9.03571 −0.968730
$$88$$ 0 0
$$89$$ −10.3667 −1.09887 −0.549435 0.835537i $$-0.685157\pi$$
−0.549435 + 0.835537i $$0.685157\pi$$
$$90$$ 0 0
$$91$$ 9.28931 0.973784
$$92$$ 0 0
$$93$$ −10.6203 −1.10128
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 2.75815 0.280048 0.140024 0.990148i $$-0.455282\pi$$
0.140024 + 0.990148i $$0.455282\pi$$
$$98$$ 0 0
$$99$$ 5.75409 0.578308
$$100$$ 0 0
$$101$$ −18.5292 −1.84373 −0.921863 0.387515i $$-0.873334\pi$$
−0.921863 + 0.387515i $$0.873334\pi$$
$$102$$ 0 0
$$103$$ −5.44606 −0.536616 −0.268308 0.963333i $$-0.586465\pi$$
−0.268308 + 0.963333i $$0.586465\pi$$
$$104$$ 0 0
$$105$$ 8.55201 0.834591
$$106$$ 0 0
$$107$$ 3.26077 0.315230 0.157615 0.987501i $$-0.449619\pi$$
0.157615 + 0.987501i $$0.449619\pi$$
$$108$$ 0 0
$$109$$ 15.6182 1.49595 0.747975 0.663726i $$-0.231026\pi$$
0.747975 + 0.663726i $$0.231026\pi$$
$$110$$ 0 0
$$111$$ 9.79193 0.929409
$$112$$ 0 0
$$113$$ −9.53329 −0.896816 −0.448408 0.893829i $$-0.648009\pi$$
−0.448408 + 0.893829i $$0.648009\pi$$
$$114$$ 0 0
$$115$$ −7.23992 −0.675126
$$116$$ 0 0
$$117$$ −6.71645 −0.620936
$$118$$ 0 0
$$119$$ −3.33101 −0.305353
$$120$$ 0 0
$$121$$ −6.14146 −0.558315
$$122$$ 0 0
$$123$$ −11.2210 −1.01176
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 1.92976 0.171239 0.0856193 0.996328i $$-0.472713\pi$$
0.0856193 + 0.996328i $$0.472713\pi$$
$$128$$ 0 0
$$129$$ −12.8504 −1.13142
$$130$$ 0 0
$$131$$ −14.8087 −1.29384 −0.646922 0.762556i $$-0.723944\pi$$
−0.646922 + 0.762556i $$0.723944\pi$$
$$132$$ 0 0
$$133$$ −3.61050 −0.313070
$$134$$ 0 0
$$135$$ 0.922589 0.0794038
$$136$$ 0 0
$$137$$ 17.5572 1.50002 0.750008 0.661428i $$-0.230049\pi$$
0.750008 + 0.661428i $$0.230049\pi$$
$$138$$ 0 0
$$139$$ 4.23779 0.359445 0.179722 0.983717i $$-0.442480\pi$$
0.179722 + 0.983717i $$0.442480\pi$$
$$140$$ 0 0
$$141$$ 0.662017 0.0557519
$$142$$ 0 0
$$143$$ −5.67112 −0.474243
$$144$$ 0 0
$$145$$ 3.81471 0.316794
$$146$$ 0 0
$$147$$ 14.2965 1.17915
$$148$$ 0 0
$$149$$ −6.06373 −0.496760 −0.248380 0.968663i $$-0.579898\pi$$
−0.248380 + 0.968663i $$0.579898\pi$$
$$150$$ 0 0
$$151$$ −6.85428 −0.557794 −0.278897 0.960321i $$-0.589969\pi$$
−0.278897 + 0.960321i $$0.589969\pi$$
$$152$$ 0 0
$$153$$ 2.40842 0.194709
$$154$$ 0 0
$$155$$ 4.48370 0.360140
$$156$$ 0 0
$$157$$ −8.85042 −0.706340 −0.353170 0.935559i $$-0.614896\pi$$
−0.353170 + 0.935559i $$0.614896\pi$$
$$158$$ 0 0
$$159$$ −23.3765 −1.85388
$$160$$ 0 0
$$161$$ −26.1397 −2.06010
$$162$$ 0 0
$$163$$ 16.6840 1.30680 0.653398 0.757015i $$-0.273343\pi$$
0.653398 + 0.757015i $$0.273343\pi$$
$$164$$ 0 0
$$165$$ −5.22100 −0.406454
$$166$$ 0 0
$$167$$ −10.5875 −0.819287 −0.409643 0.912246i $$-0.634347\pi$$
−0.409643 + 0.912246i $$0.634347\pi$$
$$168$$ 0 0
$$169$$ −6.38040 −0.490800
$$170$$ 0 0
$$171$$ 2.61050 0.199630
$$172$$ 0 0
$$173$$ 18.7879 1.42842 0.714208 0.699934i $$-0.246787\pi$$
0.714208 + 0.699934i $$0.246787\pi$$
$$174$$ 0 0
$$175$$ −3.61050 −0.272928
$$176$$ 0 0
$$177$$ −3.15269 −0.236971
$$178$$ 0 0
$$179$$ −8.91136 −0.666066 −0.333033 0.942915i $$-0.608072\pi$$
−0.333033 + 0.942915i $$0.608072\pi$$
$$180$$ 0 0
$$181$$ −6.96740 −0.517883 −0.258941 0.965893i $$-0.583374\pi$$
−0.258941 + 0.965893i $$0.583374\pi$$
$$182$$ 0 0
$$183$$ 26.3629 1.94880
$$184$$ 0 0
$$185$$ −4.13397 −0.303936
$$186$$ 0 0
$$187$$ 2.03358 0.148710
$$188$$ 0 0
$$189$$ 3.33101 0.242295
$$190$$ 0 0
$$191$$ −23.9278 −1.73136 −0.865678 0.500600i $$-0.833113\pi$$
−0.865678 + 0.500600i $$0.833113\pi$$
$$192$$ 0 0
$$193$$ −9.89961 −0.712589 −0.356295 0.934374i $$-0.615960\pi$$
−0.356295 + 0.934374i $$0.615960\pi$$
$$194$$ 0 0
$$195$$ 6.09420 0.436415
$$196$$ 0 0
$$197$$ 12.3015 0.876447 0.438224 0.898866i $$-0.355608\pi$$
0.438224 + 0.898866i $$0.355608\pi$$
$$198$$ 0 0
$$199$$ −9.07741 −0.643481 −0.321740 0.946828i $$-0.604268\pi$$
−0.321740 + 0.946828i $$0.604268\pi$$
$$200$$ 0 0
$$201$$ −32.4571 −2.28934
$$202$$ 0 0
$$203$$ 13.7730 0.966676
$$204$$ 0 0
$$205$$ 4.73730 0.330867
$$206$$ 0 0
$$207$$ 18.8998 1.31363
$$208$$ 0 0
$$209$$ 2.20421 0.152468
$$210$$ 0 0
$$211$$ 5.40629 0.372184 0.186092 0.982532i $$-0.440418\pi$$
0.186092 + 0.982532i $$0.440418\pi$$
$$212$$ 0 0
$$213$$ −26.4003 −1.80892
$$214$$ 0 0
$$215$$ 5.42521 0.369996
$$216$$ 0 0
$$217$$ 16.1884 1.09894
$$218$$ 0 0
$$219$$ −22.0031 −1.48683
$$220$$ 0 0
$$221$$ −2.37369 −0.159672
$$222$$ 0 0
$$223$$ 2.85546 0.191216 0.0956079 0.995419i $$-0.469520\pi$$
0.0956079 + 0.995419i $$0.469520\pi$$
$$224$$ 0 0
$$225$$ 2.61050 0.174033
$$226$$ 0 0
$$227$$ 3.73013 0.247577 0.123789 0.992309i $$-0.460496\pi$$
0.123789 + 0.992309i $$0.460496\pi$$
$$228$$ 0 0
$$229$$ 24.3744 1.61071 0.805353 0.592796i $$-0.201976\pi$$
0.805353 + 0.592796i $$0.201976\pi$$
$$230$$ 0 0
$$231$$ −18.8504 −1.24027
$$232$$ 0 0
$$233$$ −20.4319 −1.33854 −0.669270 0.743020i $$-0.733393\pi$$
−0.669270 + 0.743020i $$0.733393\pi$$
$$234$$ 0 0
$$235$$ −0.279491 −0.0182320
$$236$$ 0 0
$$237$$ 11.4889 0.746287
$$238$$ 0 0
$$239$$ −16.2189 −1.04911 −0.524556 0.851376i $$-0.675769\pi$$
−0.524556 + 0.851376i $$0.675769\pi$$
$$240$$ 0 0
$$241$$ 9.43290 0.607626 0.303813 0.952732i $$-0.401740\pi$$
0.303813 + 0.952732i $$0.401740\pi$$
$$242$$ 0 0
$$243$$ −20.9585 −1.34449
$$244$$ 0 0
$$245$$ −6.03571 −0.385607
$$246$$ 0 0
$$247$$ −2.57286 −0.163707
$$248$$ 0 0
$$249$$ 22.3250 1.41479
$$250$$ 0 0
$$251$$ −1.37582 −0.0868411 −0.0434205 0.999057i $$-0.513826\pi$$
−0.0434205 + 0.999057i $$0.513826\pi$$
$$252$$ 0 0
$$253$$ 15.9583 1.00329
$$254$$ 0 0
$$255$$ −2.18529 −0.136848
$$256$$ 0 0
$$257$$ 10.3080 0.642997 0.321499 0.946910i $$-0.395813\pi$$
0.321499 + 0.946910i $$0.395813\pi$$
$$258$$ 0 0
$$259$$ −14.9257 −0.927438
$$260$$ 0 0
$$261$$ −9.95830 −0.616403
$$262$$ 0 0
$$263$$ −15.6371 −0.964225 −0.482113 0.876109i $$-0.660130\pi$$
−0.482113 + 0.876109i $$0.660130\pi$$
$$264$$ 0 0
$$265$$ 9.86914 0.606256
$$266$$ 0 0
$$267$$ −24.5551 −1.50275
$$268$$ 0 0
$$269$$ 16.7944 1.02397 0.511986 0.858994i $$-0.328910\pi$$
0.511986 + 0.858994i $$0.328910\pi$$
$$270$$ 0 0
$$271$$ −20.9068 −1.27000 −0.634998 0.772514i $$-0.718999\pi$$
−0.634998 + 0.772514i $$0.718999\pi$$
$$272$$ 0 0
$$273$$ 22.0031 1.33169
$$274$$ 0 0
$$275$$ 2.20421 0.132919
$$276$$ 0 0
$$277$$ −20.4798 −1.23051 −0.615257 0.788327i $$-0.710948\pi$$
−0.615257 + 0.788327i $$0.710948\pi$$
$$278$$ 0 0
$$279$$ −11.7047 −0.700742
$$280$$ 0 0
$$281$$ 16.1467 0.963231 0.481616 0.876383i $$-0.340050\pi$$
0.481616 + 0.876383i $$0.340050\pi$$
$$282$$ 0 0
$$283$$ −5.19897 −0.309047 −0.154523 0.987989i $$-0.549384\pi$$
−0.154523 + 0.987989i $$0.549384\pi$$
$$284$$ 0 0
$$285$$ −2.36865 −0.140307
$$286$$ 0 0
$$287$$ 17.1040 1.00962
$$288$$ 0 0
$$289$$ −16.1488 −0.949931
$$290$$ 0 0
$$291$$ 6.53309 0.382976
$$292$$ 0 0
$$293$$ 7.79386 0.455322 0.227661 0.973740i $$-0.426892\pi$$
0.227661 + 0.973740i $$0.426892\pi$$
$$294$$ 0 0
$$295$$ 1.33101 0.0774943
$$296$$ 0 0
$$297$$ −2.03358 −0.118000
$$298$$ 0 0
$$299$$ −18.6273 −1.07724
$$300$$ 0 0
$$301$$ 19.5877 1.12902
$$302$$ 0 0
$$303$$ −43.8892 −2.52137
$$304$$ 0 0
$$305$$ −11.1299 −0.637297
$$306$$ 0 0
$$307$$ 6.27298 0.358018 0.179009 0.983847i $$-0.442711\pi$$
0.179009 + 0.983847i $$0.442711\pi$$
$$308$$ 0 0
$$309$$ −12.8998 −0.733844
$$310$$ 0 0
$$311$$ −29.2634 −1.65938 −0.829688 0.558227i $$-0.811482\pi$$
−0.829688 + 0.558227i $$0.811482\pi$$
$$312$$ 0 0
$$313$$ −20.4725 −1.15717 −0.578586 0.815621i $$-0.696395\pi$$
−0.578586 + 0.815621i $$0.696395\pi$$
$$314$$ 0 0
$$315$$ 9.42521 0.531050
$$316$$ 0 0
$$317$$ 4.64814 0.261066 0.130533 0.991444i $$-0.458331\pi$$
0.130533 + 0.991444i $$0.458331\pi$$
$$318$$ 0 0
$$319$$ −8.40842 −0.470781
$$320$$ 0 0
$$321$$ 7.72362 0.431090
$$322$$ 0 0
$$323$$ 0.922589 0.0513342
$$324$$ 0 0
$$325$$ −2.57286 −0.142717
$$326$$ 0 0
$$327$$ 36.9940 2.04577
$$328$$ 0 0
$$329$$ −1.00910 −0.0556337
$$330$$ 0 0
$$331$$ −26.3891 −1.45047 −0.725237 0.688499i $$-0.758270\pi$$
−0.725237 + 0.688499i $$0.758270\pi$$
$$332$$ 0 0
$$333$$ 10.7917 0.591383
$$334$$ 0 0
$$335$$ 13.7028 0.748662
$$336$$ 0 0
$$337$$ 17.1441 0.933899 0.466949 0.884284i $$-0.345353\pi$$
0.466949 + 0.884284i $$0.345353\pi$$
$$338$$ 0 0
$$339$$ −22.5810 −1.22643
$$340$$ 0 0
$$341$$ −9.88302 −0.535195
$$342$$ 0 0
$$343$$ 3.48157 0.187987
$$344$$ 0 0
$$345$$ −17.1488 −0.923262
$$346$$ 0 0
$$347$$ −2.84273 −0.152606 −0.0763029 0.997085i $$-0.524312\pi$$
−0.0763029 + 0.997085i $$0.524312\pi$$
$$348$$ 0 0
$$349$$ 17.6487 0.944711 0.472355 0.881408i $$-0.343404\pi$$
0.472355 + 0.881408i $$0.343404\pi$$
$$350$$ 0 0
$$351$$ 2.37369 0.126698
$$352$$ 0 0
$$353$$ −1.88999 −0.100594 −0.0502970 0.998734i $$-0.516017\pi$$
−0.0502970 + 0.998734i $$0.516017\pi$$
$$354$$ 0 0
$$355$$ 11.1457 0.591553
$$356$$ 0 0
$$357$$ −7.88999 −0.417583
$$358$$ 0 0
$$359$$ 28.8234 1.52124 0.760620 0.649198i $$-0.224895\pi$$
0.760620 + 0.649198i $$0.224895\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −14.5470 −0.763518
$$364$$ 0 0
$$365$$ 9.28931 0.486225
$$366$$ 0 0
$$367$$ 8.92716 0.465994 0.232997 0.972477i $$-0.425147\pi$$
0.232997 + 0.972477i $$0.425147\pi$$
$$368$$ 0 0
$$369$$ −12.3667 −0.643786
$$370$$ 0 0
$$371$$ 35.6325 1.84995
$$372$$ 0 0
$$373$$ 11.7939 0.610663 0.305331 0.952246i $$-0.401233\pi$$
0.305331 + 0.952246i $$0.401233\pi$$
$$374$$ 0 0
$$375$$ −2.36865 −0.122317
$$376$$ 0 0
$$377$$ 9.81471 0.505483
$$378$$ 0 0
$$379$$ −10.6354 −0.546304 −0.273152 0.961971i $$-0.588066\pi$$
−0.273152 + 0.961971i $$0.588066\pi$$
$$380$$ 0 0
$$381$$ 4.57093 0.234176
$$382$$ 0 0
$$383$$ −2.86358 −0.146322 −0.0731611 0.997320i $$-0.523309\pi$$
−0.0731611 + 0.997320i $$0.523309\pi$$
$$384$$ 0 0
$$385$$ 7.95830 0.405592
$$386$$ 0 0
$$387$$ −14.1625 −0.719921
$$388$$ 0 0
$$389$$ −3.45536 −0.175194 −0.0875969 0.996156i $$-0.527919\pi$$
−0.0875969 + 0.996156i $$0.527919\pi$$
$$390$$ 0 0
$$391$$ 6.67947 0.337795
$$392$$ 0 0
$$393$$ −35.0767 −1.76938
$$394$$ 0 0
$$395$$ −4.85042 −0.244051
$$396$$ 0 0
$$397$$ 0.854282 0.0428752 0.0214376 0.999770i $$-0.493176\pi$$
0.0214376 + 0.999770i $$0.493176\pi$$
$$398$$ 0 0
$$399$$ −8.55201 −0.428136
$$400$$ 0 0
$$401$$ 16.0336 0.800679 0.400339 0.916367i $$-0.368892\pi$$
0.400339 + 0.916367i $$0.368892\pi$$
$$402$$ 0 0
$$403$$ 11.5359 0.574646
$$404$$ 0 0
$$405$$ 10.0168 0.497738
$$406$$ 0 0
$$407$$ 9.11214 0.451672
$$408$$ 0 0
$$409$$ 13.8452 0.684600 0.342300 0.939591i $$-0.388794\pi$$
0.342300 + 0.939591i $$0.388794\pi$$
$$410$$ 0 0
$$411$$ 41.5870 2.05133
$$412$$ 0 0
$$413$$ 4.80561 0.236468
$$414$$ 0 0
$$415$$ −9.42521 −0.462665
$$416$$ 0 0
$$417$$ 10.0378 0.491555
$$418$$ 0 0
$$419$$ −16.1783 −0.790362 −0.395181 0.918603i $$-0.629318\pi$$
−0.395181 + 0.918603i $$0.629318\pi$$
$$420$$ 0 0
$$421$$ 1.71593 0.0836295 0.0418147 0.999125i $$-0.486686\pi$$
0.0418147 + 0.999125i $$0.486686\pi$$
$$422$$ 0 0
$$423$$ 0.729612 0.0354750
$$424$$ 0 0
$$425$$ 0.922589 0.0447522
$$426$$ 0 0
$$427$$ −40.1845 −1.94467
$$428$$ 0 0
$$429$$ −13.4329 −0.648546
$$430$$ 0 0
$$431$$ −30.8087 −1.48400 −0.742002 0.670398i $$-0.766123\pi$$
−0.742002 + 0.670398i $$0.766123\pi$$
$$432$$ 0 0
$$433$$ 30.3084 1.45653 0.728265 0.685296i $$-0.240327\pi$$
0.728265 + 0.685296i $$0.240327\pi$$
$$434$$ 0 0
$$435$$ 9.03571 0.433229
$$436$$ 0 0
$$437$$ 7.23992 0.346332
$$438$$ 0 0
$$439$$ 0.722955 0.0345048 0.0172524 0.999851i $$-0.494508\pi$$
0.0172524 + 0.999851i $$0.494508\pi$$
$$440$$ 0 0
$$441$$ 15.7562 0.750296
$$442$$ 0 0
$$443$$ −2.32159 −0.110302 −0.0551510 0.998478i $$-0.517564\pi$$
−0.0551510 + 0.998478i $$0.517564\pi$$
$$444$$ 0 0
$$445$$ 10.3667 0.491430
$$446$$ 0 0
$$447$$ −14.3629 −0.679340
$$448$$ 0 0
$$449$$ 39.8039 1.87846 0.939230 0.343287i $$-0.111540\pi$$
0.939230 + 0.343287i $$0.111540\pi$$
$$450$$ 0 0
$$451$$ −10.4420 −0.491695
$$452$$ 0 0
$$453$$ −16.2354 −0.762805
$$454$$ 0 0
$$455$$ −9.28931 −0.435489
$$456$$ 0 0
$$457$$ −17.9657 −0.840398 −0.420199 0.907432i $$-0.638040\pi$$
−0.420199 + 0.907432i $$0.638040\pi$$
$$458$$ 0 0
$$459$$ −0.851171 −0.0397293
$$460$$ 0 0
$$461$$ 34.8840 1.62471 0.812355 0.583163i $$-0.198185\pi$$
0.812355 + 0.583163i $$0.198185\pi$$
$$462$$ 0 0
$$463$$ −18.4118 −0.855671 −0.427836 0.903857i $$-0.640724\pi$$
−0.427836 + 0.903857i $$0.640724\pi$$
$$464$$ 0 0
$$465$$ 10.6203 0.492505
$$466$$ 0 0
$$467$$ 17.7541 0.821561 0.410781 0.911734i $$-0.365256\pi$$
0.410781 + 0.911734i $$0.365256\pi$$
$$468$$ 0 0
$$469$$ 49.4738 2.28449
$$470$$ 0 0
$$471$$ −20.9635 −0.965949
$$472$$ 0 0
$$473$$ −11.9583 −0.549843
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −25.7634 −1.17962
$$478$$ 0 0
$$479$$ 23.7545 1.08537 0.542685 0.839936i $$-0.317408\pi$$
0.542685 + 0.839936i $$0.317408\pi$$
$$480$$ 0 0
$$481$$ −10.6361 −0.484965
$$482$$ 0 0
$$483$$ −61.9158 −2.81727
$$484$$ 0 0
$$485$$ −2.75815 −0.125241
$$486$$ 0 0
$$487$$ 28.9581 1.31222 0.656108 0.754667i $$-0.272201\pi$$
0.656108 + 0.754667i $$0.272201\pi$$
$$488$$ 0 0
$$489$$ 39.5187 1.78710
$$490$$ 0 0
$$491$$ 2.40416 0.108498 0.0542491 0.998527i $$-0.482723\pi$$
0.0542491 + 0.998527i $$0.482723\pi$$
$$492$$ 0 0
$$493$$ −3.51941 −0.158506
$$494$$ 0 0
$$495$$ −5.75409 −0.258627
$$496$$ 0 0
$$497$$ 40.2416 1.80508
$$498$$ 0 0
$$499$$ 12.2756 0.549533 0.274766 0.961511i $$-0.411400\pi$$
0.274766 + 0.961511i $$0.411400\pi$$
$$500$$ 0 0
$$501$$ −25.0781 −1.12041
$$502$$ 0 0
$$503$$ 0.488277 0.0217712 0.0108856 0.999941i $$-0.496535\pi$$
0.0108856 + 0.999941i $$0.496535\pi$$
$$504$$ 0 0
$$505$$ 18.5292 0.824540
$$506$$ 0 0
$$507$$ −15.1129 −0.671188
$$508$$ 0 0
$$509$$ 14.6246 0.648223 0.324111 0.946019i $$-0.394935\pi$$
0.324111 + 0.946019i $$0.394935\pi$$
$$510$$ 0 0
$$511$$ 33.5390 1.48368
$$512$$ 0 0
$$513$$ −0.922589 −0.0407333
$$514$$ 0 0
$$515$$ 5.44606 0.239982
$$516$$ 0 0
$$517$$ 0.616058 0.0270942
$$518$$ 0 0
$$519$$ 44.5019 1.95342
$$520$$ 0 0
$$521$$ 14.5450 0.637229 0.318615 0.947884i $$-0.396782\pi$$
0.318615 + 0.947884i $$0.396782\pi$$
$$522$$ 0 0
$$523$$ −22.0590 −0.964573 −0.482287 0.876014i $$-0.660194\pi$$
−0.482287 + 0.876014i $$0.660194\pi$$
$$524$$ 0 0
$$525$$ −8.55201 −0.373240
$$526$$ 0 0
$$527$$ −4.13661 −0.180194
$$528$$ 0 0
$$529$$ 29.4164 1.27897
$$530$$ 0 0
$$531$$ −3.47460 −0.150785
$$532$$ 0 0
$$533$$ 12.1884 0.527938
$$534$$ 0 0
$$535$$ −3.26077 −0.140975
$$536$$ 0 0
$$537$$ −21.1079 −0.910872
$$538$$ 0 0
$$539$$ 13.3040 0.573042
$$540$$ 0 0
$$541$$ −19.2388 −0.827139 −0.413570 0.910472i $$-0.635718\pi$$
−0.413570 + 0.910472i $$0.635718\pi$$
$$542$$ 0 0
$$543$$ −16.5033 −0.708226
$$544$$ 0 0
$$545$$ −15.6182 −0.669010
$$546$$ 0 0
$$547$$ 44.3833 1.89769 0.948846 0.315740i $$-0.102253\pi$$
0.948846 + 0.315740i $$0.102253\pi$$
$$548$$ 0 0
$$549$$ 29.0546 1.24002
$$550$$ 0 0
$$551$$ −3.81471 −0.162512
$$552$$ 0 0
$$553$$ −17.5124 −0.744705
$$554$$ 0 0
$$555$$ −9.79193 −0.415644
$$556$$ 0 0
$$557$$ 16.1030 0.682307 0.341154 0.940008i $$-0.389182\pi$$
0.341154 + 0.940008i $$0.389182\pi$$
$$558$$ 0 0
$$559$$ 13.9583 0.590373
$$560$$ 0 0
$$561$$ 4.81684 0.203367
$$562$$ 0 0
$$563$$ 0.107683 0.00453828 0.00226914 0.999997i $$-0.499278\pi$$
0.00226914 + 0.999997i $$0.499278\pi$$
$$564$$ 0 0
$$565$$ 9.53329 0.401068
$$566$$ 0 0
$$567$$ 36.1656 1.51881
$$568$$ 0 0
$$569$$ 17.7842 0.745554 0.372777 0.927921i $$-0.378406\pi$$
0.372777 + 0.927921i $$0.378406\pi$$
$$570$$ 0 0
$$571$$ −26.5902 −1.11276 −0.556382 0.830927i $$-0.687811\pi$$
−0.556382 + 0.830927i $$0.687811\pi$$
$$572$$ 0 0
$$573$$ −56.6766 −2.36770
$$574$$ 0 0
$$575$$ 7.23992 0.301925
$$576$$ 0 0
$$577$$ 33.3565 1.38865 0.694324 0.719663i $$-0.255704\pi$$
0.694324 + 0.719663i $$0.255704\pi$$
$$578$$ 0 0
$$579$$ −23.4487 −0.974495
$$580$$ 0 0
$$581$$ −34.0297 −1.41179
$$582$$ 0 0
$$583$$ −21.7537 −0.900944
$$584$$ 0 0
$$585$$ 6.71645 0.277691
$$586$$ 0 0
$$587$$ −29.0322 −1.19829 −0.599143 0.800642i $$-0.704492\pi$$
−0.599143 + 0.800642i $$0.704492\pi$$
$$588$$ 0 0
$$589$$ −4.48370 −0.184748
$$590$$ 0 0
$$591$$ 29.1380 1.19858
$$592$$ 0 0
$$593$$ 0.722955 0.0296882 0.0148441 0.999890i $$-0.495275\pi$$
0.0148441 + 0.999890i $$0.495275\pi$$
$$594$$ 0 0
$$595$$ 3.33101 0.136558
$$596$$ 0 0
$$597$$ −21.5012 −0.879986
$$598$$ 0 0
$$599$$ 20.7516 0.847889 0.423945 0.905688i $$-0.360645\pi$$
0.423945 + 0.905688i $$0.360645\pi$$
$$600$$ 0 0
$$601$$ 0.919478 0.0375063 0.0187532 0.999824i $$-0.494030\pi$$
0.0187532 + 0.999824i $$0.494030\pi$$
$$602$$ 0 0
$$603$$ −35.7711 −1.45671
$$604$$ 0 0
$$605$$ 6.14146 0.249686
$$606$$ 0 0
$$607$$ 3.23416 0.131271 0.0656353 0.997844i $$-0.479093\pi$$
0.0656353 + 0.997844i $$0.479093\pi$$
$$608$$ 0 0
$$609$$ 32.6234 1.32197
$$610$$ 0 0
$$611$$ −0.719092 −0.0290913
$$612$$ 0 0
$$613$$ 47.0392 1.89990 0.949948 0.312408i $$-0.101136\pi$$
0.949948 + 0.312408i $$0.101136\pi$$
$$614$$ 0 0
$$615$$ 11.2210 0.452474
$$616$$ 0 0
$$617$$ −32.1510 −1.29435 −0.647174 0.762342i $$-0.724049\pi$$
−0.647174 + 0.762342i $$0.724049\pi$$
$$618$$ 0 0
$$619$$ −25.7727 −1.03589 −0.517946 0.855413i $$-0.673303\pi$$
−0.517946 + 0.855413i $$0.673303\pi$$
$$620$$ 0 0
$$621$$ −6.67947 −0.268038
$$622$$ 0 0
$$623$$ 37.4290 1.49956
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 5.22100 0.208507
$$628$$ 0 0
$$629$$ 3.81396 0.152072
$$630$$ 0 0
$$631$$ −35.0133 −1.39386 −0.696929 0.717140i $$-0.745451\pi$$
−0.696929 + 0.717140i $$0.745451\pi$$
$$632$$ 0 0
$$633$$ 12.8056 0.508977
$$634$$ 0 0
$$635$$ −1.92976 −0.0765802
$$636$$ 0 0
$$637$$ −15.5290 −0.615283
$$638$$ 0 0
$$639$$ −29.0959 −1.15102
$$640$$ 0 0
$$641$$ 28.0340 1.10728 0.553638 0.832758i $$-0.313239\pi$$
0.553638 + 0.832758i $$0.313239\pi$$
$$642$$ 0 0
$$643$$ 19.1831 0.756508 0.378254 0.925702i $$-0.376525\pi$$
0.378254 + 0.925702i $$0.376525\pi$$
$$644$$ 0 0
$$645$$ 12.8504 0.505985
$$646$$ 0 0
$$647$$ 18.0046 0.707833 0.353916 0.935277i $$-0.384850\pi$$
0.353916 + 0.935277i $$0.384850\pi$$
$$648$$ 0 0
$$649$$ −2.93382 −0.115163
$$650$$ 0 0
$$651$$ 38.3446 1.50285
$$652$$ 0 0
$$653$$ 22.8130 0.892741 0.446370 0.894848i $$-0.352716\pi$$
0.446370 + 0.894848i $$0.352716\pi$$
$$654$$ 0 0
$$655$$ 14.8087 0.578624
$$656$$ 0 0
$$657$$ −24.2497 −0.946072
$$658$$ 0 0
$$659$$ 46.1401 1.79736 0.898682 0.438601i $$-0.144526\pi$$
0.898682 + 0.438601i $$0.144526\pi$$
$$660$$ 0 0
$$661$$ 8.44511 0.328477 0.164238 0.986421i $$-0.447483\pi$$
0.164238 + 0.986421i $$0.447483\pi$$
$$662$$ 0 0
$$663$$ −5.62244 −0.218358
$$664$$ 0 0
$$665$$ 3.61050 0.140009
$$666$$ 0 0
$$667$$ −27.6182 −1.06938
$$668$$ 0 0
$$669$$ 6.76359 0.261495
$$670$$ 0 0
$$671$$ 24.5327 0.947073
$$672$$ 0 0
$$673$$ 10.8334 0.417598 0.208799 0.977959i $$-0.433045\pi$$
0.208799 + 0.977959i $$0.433045\pi$$
$$674$$ 0 0
$$675$$ −0.922589 −0.0355105
$$676$$ 0 0
$$677$$ −4.74202 −0.182251 −0.0911254 0.995839i $$-0.529046\pi$$
−0.0911254 + 0.995839i $$0.529046\pi$$
$$678$$ 0 0
$$679$$ −9.95830 −0.382164
$$680$$ 0 0
$$681$$ 8.83536 0.338572
$$682$$ 0 0
$$683$$ 8.84963 0.338622 0.169311 0.985563i $$-0.445846\pi$$
0.169311 + 0.985563i $$0.445846\pi$$
$$684$$ 0 0
$$685$$ −17.5572 −0.670828
$$686$$ 0 0
$$687$$ 57.7344 2.20271
$$688$$ 0 0
$$689$$ 25.3919 0.967355
$$690$$ 0 0
$$691$$ 2.41571 0.0918980 0.0459490 0.998944i $$-0.485369\pi$$
0.0459490 + 0.998944i $$0.485369\pi$$
$$692$$ 0 0
$$693$$ −20.7751 −0.789182
$$694$$ 0 0
$$695$$ −4.23779 −0.160749
$$696$$ 0 0
$$697$$ −4.37058 −0.165548
$$698$$ 0 0
$$699$$ −48.3960 −1.83051
$$700$$ 0 0
$$701$$ 3.62173 0.136791 0.0683955 0.997658i $$-0.478212\pi$$
0.0683955 + 0.997658i $$0.478212\pi$$
$$702$$ 0 0
$$703$$ 4.13397 0.155916
$$704$$ 0 0
$$705$$ −0.662017 −0.0249330
$$706$$ 0 0
$$707$$ 66.8998 2.51602
$$708$$ 0 0
$$709$$ 36.9909 1.38922 0.694611 0.719385i $$-0.255576\pi$$
0.694611 + 0.719385i $$0.255576\pi$$
$$710$$ 0 0
$$711$$ 12.6620 0.474863
$$712$$ 0 0
$$713$$ −32.4616 −1.21570
$$714$$ 0 0
$$715$$ 5.67112 0.212088
$$716$$ 0 0
$$717$$ −38.4168 −1.43470
$$718$$ 0 0
$$719$$ −27.8175 −1.03742 −0.518709 0.854951i $$-0.673587\pi$$
−0.518709 + 0.854951i $$0.673587\pi$$
$$720$$ 0 0
$$721$$ 19.6630 0.732288
$$722$$ 0 0
$$723$$ 22.3432 0.830953
$$724$$ 0 0
$$725$$ −3.81471 −0.141675
$$726$$ 0 0
$$727$$ −37.5947 −1.39431 −0.697154 0.716921i $$-0.745551\pi$$
−0.697154 + 0.716921i $$0.745551\pi$$
$$728$$ 0 0
$$729$$ −19.5930 −0.725665
$$730$$ 0 0
$$731$$ −5.00524 −0.185125
$$732$$ 0 0
$$733$$ 19.7201 0.728378 0.364189 0.931325i $$-0.381346\pi$$
0.364189 + 0.931325i $$0.381346\pi$$
$$734$$ 0 0
$$735$$ −14.2965 −0.527334
$$736$$ 0 0
$$737$$ −30.2038 −1.11257
$$738$$ 0 0
$$739$$ −48.3629 −1.77906 −0.889528 0.456880i $$-0.848967\pi$$
−0.889528 + 0.456880i $$0.848967\pi$$
$$740$$ 0 0
$$741$$ −6.09420 −0.223876
$$742$$ 0 0
$$743$$ 41.0765 1.50695 0.753475 0.657476i $$-0.228376\pi$$
0.753475 + 0.657476i $$0.228376\pi$$
$$744$$ 0 0
$$745$$ 6.06373 0.222158
$$746$$ 0 0
$$747$$ 24.6045 0.900232
$$748$$ 0 0
$$749$$ −11.7730 −0.430176
$$750$$ 0 0
$$751$$ −0.0508527 −0.00185564 −0.000927821 1.00000i $$-0.500295\pi$$
−0.000927821 1.00000i $$0.500295\pi$$
$$752$$ 0 0
$$753$$ −3.25884 −0.118759
$$754$$ 0 0
$$755$$ 6.85428 0.249453
$$756$$ 0 0
$$757$$ −27.7109 −1.00717 −0.503585 0.863946i $$-0.667986\pi$$
−0.503585 + 0.863946i $$0.667986\pi$$
$$758$$ 0 0
$$759$$ 37.7996 1.37204
$$760$$ 0 0
$$761$$ −13.4140 −0.486256 −0.243128 0.969994i $$-0.578174\pi$$
−0.243128 + 0.969994i $$0.578174\pi$$
$$762$$ 0 0
$$763$$ −56.3895 −2.04144
$$764$$ 0 0
$$765$$ −2.40842 −0.0870766
$$766$$ 0 0
$$767$$ 3.42450 0.123651
$$768$$ 0 0
$$769$$ −36.5362 −1.31753 −0.658765 0.752349i $$-0.728921\pi$$
−0.658765 + 0.752349i $$0.728921\pi$$
$$770$$ 0 0
$$771$$ 24.4161 0.879325
$$772$$ 0 0
$$773$$ −46.0124 −1.65495 −0.827475 0.561503i $$-0.810223\pi$$
−0.827475 + 0.561503i $$0.810223\pi$$
$$774$$ 0 0
$$775$$ −4.48370 −0.161059
$$776$$ 0 0
$$777$$ −35.3538 −1.26831
$$778$$ 0 0
$$779$$ −4.73730 −0.169731
$$780$$ 0 0
$$781$$ −24.5675 −0.879094
$$782$$ 0 0
$$783$$ 3.51941 0.125773
$$784$$ 0 0
$$785$$ 8.85042 0.315885
$$786$$ 0 0
$$787$$ −28.4971 −1.01581 −0.507907 0.861412i $$-0.669581\pi$$
−0.507907 + 0.861412i $$0.669581\pi$$
$$788$$ 0 0
$$789$$ −37.0388 −1.31862
$$790$$ 0 0
$$791$$ 34.4199 1.22383
$$792$$ 0 0
$$793$$ −28.6357 −1.01688
$$794$$ 0 0
$$795$$ 23.3765 0.829080
$$796$$ 0 0
$$797$$ 37.2978 1.32116 0.660578 0.750758i $$-0.270311\pi$$
0.660578 + 0.750758i $$0.270311\pi$$
$$798$$ 0 0
$$799$$ 0.257856 0.00912228
$$800$$ 0 0
$$801$$ −27.0623 −0.956200
$$802$$ 0 0
$$803$$ −20.4756 −0.722568
$$804$$ 0 0
$$805$$ 26.1397 0.921304
$$806$$ 0 0
$$807$$ 39.7800 1.40032
$$808$$ 0 0
$$809$$ −20.6609 −0.726398 −0.363199 0.931712i $$-0.618315\pi$$
−0.363199 + 0.931712i $$0.618315\pi$$
$$810$$ 0 0
$$811$$ −31.4210 −1.10334 −0.551671 0.834062i $$-0.686010\pi$$
−0.551671 + 0.834062i $$0.686010\pi$$
$$812$$ 0 0
$$813$$ −49.5208 −1.73677
$$814$$ 0 0
$$815$$ −16.6840 −0.584417
$$816$$ 0 0
$$817$$ −5.42521 −0.189804
$$818$$ 0 0
$$819$$ 24.2497 0.847354
$$820$$ 0 0
$$821$$ −40.0297 −1.39705 −0.698523 0.715587i $$-0.746159\pi$$
−0.698523 + 0.715587i $$0.746159\pi$$
$$822$$ 0 0
$$823$$ 55.9610 1.95068 0.975338 0.220714i $$-0.0708389\pi$$
0.975338 + 0.220714i $$0.0708389\pi$$
$$824$$ 0 0
$$825$$ 5.22100 0.181772
$$826$$ 0 0
$$827$$ 36.5868 1.27225 0.636123 0.771587i $$-0.280537\pi$$
0.636123 + 0.771587i $$0.280537\pi$$
$$828$$ 0 0
$$829$$ −42.3474 −1.47078 −0.735392 0.677642i $$-0.763002\pi$$
−0.735392 + 0.677642i $$0.763002\pi$$
$$830$$ 0 0
$$831$$ −48.5096 −1.68278
$$832$$ 0 0
$$833$$ 5.56848 0.192936
$$834$$ 0 0
$$835$$ 10.5875 0.366396
$$836$$ 0 0
$$837$$ 4.13661 0.142982
$$838$$ 0 0
$$839$$ −17.6966 −0.610954 −0.305477 0.952199i $$-0.598816\pi$$
−0.305477 + 0.952199i $$0.598816\pi$$
$$840$$ 0 0
$$841$$ −14.4480 −0.498207
$$842$$ 0 0
$$843$$ 38.2459 1.31726
$$844$$ 0 0
$$845$$ 6.38040 0.219492
$$846$$ 0 0
$$847$$ 22.1737 0.761899
$$848$$ 0 0
$$849$$ −12.3145 −0.422634
$$850$$ 0 0
$$851$$ 29.9296 1.02597
$$852$$ 0 0
$$853$$ −17.2272 −0.589849 −0.294924 0.955521i $$-0.595294\pi$$
−0.294924 + 0.955521i $$0.595294\pi$$
$$854$$ 0 0
$$855$$ −2.61050 −0.0892772
$$856$$ 0 0
$$857$$ −36.3377 −1.24127 −0.620637 0.784098i $$-0.713126\pi$$
−0.620637 + 0.784098i $$0.713126\pi$$
$$858$$ 0 0
$$859$$ 35.7915 1.22119 0.610595 0.791943i $$-0.290930\pi$$
0.610595 + 0.791943i $$0.290930\pi$$
$$860$$ 0 0
$$861$$ 40.5134 1.38069
$$862$$ 0 0
$$863$$ −11.9672 −0.407368 −0.203684 0.979037i $$-0.565292\pi$$
−0.203684 + 0.979037i $$0.565292\pi$$
$$864$$ 0 0
$$865$$ −18.7879 −0.638807
$$866$$ 0 0
$$867$$ −38.2509 −1.29907
$$868$$ 0 0
$$869$$ 10.6913 0.362679
$$870$$ 0 0
$$871$$ 35.2553 1.19458
$$872$$ 0 0
$$873$$ 7.20015 0.243688
$$874$$ 0 0
$$875$$ 3.61050 0.122057
$$876$$ 0 0
$$877$$ 53.5704 1.80894 0.904471 0.426534i $$-0.140266\pi$$
0.904471 + 0.426534i $$0.140266\pi$$
$$878$$ 0 0
$$879$$ 18.4609 0.622671
$$880$$ 0 0
$$881$$ −52.7852 −1.77838 −0.889189 0.457540i $$-0.848731\pi$$
−0.889189 + 0.457540i $$0.848731\pi$$
$$882$$ 0 0
$$883$$ −47.2665 −1.59065 −0.795323 0.606186i $$-0.792699\pi$$
−0.795323 + 0.606186i $$0.792699\pi$$
$$884$$ 0 0
$$885$$ 3.15269 0.105977
$$886$$ 0 0
$$887$$ 30.3904 1.02041 0.510204 0.860053i $$-0.329570\pi$$
0.510204 + 0.860053i $$0.329570\pi$$
$$888$$ 0 0
$$889$$ −6.96740 −0.233679
$$890$$ 0 0
$$891$$ −22.0791 −0.739678
$$892$$ 0 0
$$893$$ 0.279491 0.00935282
$$894$$ 0 0
$$895$$ 8.91136 0.297874
$$896$$ 0 0
$$897$$ −44.1215 −1.47317
$$898$$ 0 0
$$899$$ 17.1040 0.570451
$$900$$ 0 0
$$901$$ −9.10516 −0.303337
$$902$$ 0 0
$$903$$ 46.3964 1.54398
$$904$$ 0 0
$$905$$ 6.96740 0.231604
$$906$$ 0 0
$$907$$ −22.1389 −0.735111 −0.367556 0.930001i $$-0.619805\pi$$
−0.367556 + 0.930001i $$0.619805\pi$$
$$908$$ 0 0
$$909$$ −48.3705 −1.60435
$$910$$ 0 0
$$911$$ −43.3159 −1.43512 −0.717560 0.696497i $$-0.754741\pi$$
−0.717560 + 0.696497i $$0.754741\pi$$
$$912$$ 0 0
$$913$$ 20.7751 0.687557
$$914$$ 0 0
$$915$$ −26.3629 −0.871529
$$916$$ 0 0
$$917$$ 53.4669 1.76563
$$918$$ 0 0
$$919$$ −1.68725 −0.0556571 −0.0278286 0.999613i $$-0.508859\pi$$
−0.0278286 + 0.999613i $$0.508859\pi$$
$$920$$ 0 0
$$921$$ 14.8585 0.489604
$$922$$ 0 0
$$923$$ 28.6764 0.943894
$$924$$ 0 0
$$925$$ 4.13397 0.135924
$$926$$ 0 0
$$927$$ −14.2169 −0.466946
$$928$$ 0 0
$$929$$ −29.4360 −0.965764 −0.482882 0.875685i $$-0.660410\pi$$
−0.482882 + 0.875685i $$0.660410\pi$$
$$930$$ 0 0
$$931$$ 6.03571 0.197812
$$932$$ 0 0
$$933$$ −69.3148 −2.26926
$$934$$ 0 0
$$935$$ −2.03358 −0.0665052
$$936$$ 0 0
$$937$$ −49.5540 −1.61886 −0.809430 0.587217i $$-0.800224\pi$$
−0.809430 + 0.587217i $$0.800224\pi$$
$$938$$ 0 0
$$939$$ −48.4921 −1.58248
$$940$$ 0 0
$$941$$ 57.9201 1.88814 0.944071 0.329743i $$-0.106962\pi$$
0.944071 + 0.329743i $$0.106962\pi$$
$$942$$ 0 0
$$943$$ −34.2977 −1.11689
$$944$$ 0 0
$$945$$ −3.33101 −0.108358
$$946$$ 0 0
$$947$$ −26.6304 −0.865370 −0.432685 0.901545i $$-0.642434\pi$$
−0.432685 + 0.901545i $$0.642434\pi$$
$$948$$ 0 0
$$949$$ 23.9001 0.775829
$$950$$ 0 0
$$951$$ 11.0098 0.357018
$$952$$ 0 0
$$953$$ −0.0289722 −0.000938500 0 −0.000469250 1.00000i $$-0.500149\pi$$
−0.000469250 1.00000i $$0.500149\pi$$
$$954$$ 0 0
$$955$$ 23.9278 0.774286
$$956$$ 0 0
$$957$$ −19.9166 −0.643812
$$958$$ 0 0
$$959$$ −63.3904 −2.04698
$$960$$ 0 0
$$961$$ −10.8964 −0.351498
$$962$$ 0 0
$$963$$ 8.51224 0.274303
$$964$$ 0 0
$$965$$ 9.89961 0.318680
$$966$$ 0 0
$$967$$ 23.9367 0.769751 0.384876 0.922968i $$-0.374244\pi$$
0.384876 + 0.922968i $$0.374244\pi$$
$$968$$ 0 0
$$969$$ 2.18529 0.0702016
$$970$$ 0 0
$$971$$ 18.5867 0.596477 0.298238 0.954491i $$-0.403601\pi$$
0.298238 + 0.954491i $$0.403601\pi$$
$$972$$ 0 0
$$973$$ −15.3005 −0.490513
$$974$$ 0 0
$$975$$ −6.09420 −0.195171
$$976$$ 0 0
$$977$$ −23.6982 −0.758172 −0.379086 0.925361i $$-0.623762\pi$$
−0.379086 + 0.925361i $$0.623762\pi$$
$$978$$ 0 0
$$979$$ −22.8504 −0.730303
$$980$$ 0 0
$$981$$ 40.7713 1.30173
$$982$$ 0 0
$$983$$ −31.0827 −0.991385 −0.495693 0.868498i $$-0.665086\pi$$
−0.495693 + 0.868498i $$0.665086\pi$$
$$984$$ 0 0
$$985$$ −12.3015 −0.391959
$$986$$ 0 0
$$987$$ −2.39021 −0.0760813
$$988$$ 0 0
$$989$$ −39.2781 −1.24897
$$990$$ 0 0
$$991$$ −3.58772 −0.113968 −0.0569838 0.998375i $$-0.518148\pi$$
−0.0569838 + 0.998375i $$0.518148\pi$$
$$992$$ 0 0
$$993$$ −62.5064 −1.98358
$$994$$ 0 0
$$995$$ 9.07741 0.287773
$$996$$ 0 0
$$997$$ 13.8409 0.438346 0.219173 0.975686i $$-0.429664\pi$$
0.219173 + 0.975686i $$0.429664\pi$$
$$998$$ 0 0
$$999$$ −3.81396 −0.120668
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.cf.1.4 4
4.3 odd 2 6080.2.a.cd.1.1 4
8.3 odd 2 3040.2.a.t.1.4 yes 4
8.5 even 2 3040.2.a.r.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.r.1.1 4 8.5 even 2
3040.2.a.t.1.4 yes 4 8.3 odd 2
6080.2.a.cd.1.1 4 4.3 odd 2
6080.2.a.cf.1.4 4 1.1 even 1 trivial