# Properties

 Label 1648.2.a.f.1.1 Level $1648$ Weight $2$ Character 1648.1 Self dual yes Analytic conductor $13.159$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1648,2,Mod(1,1648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1648 = 2^{4} \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1648.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: $$13.1593462531$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 103) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1648.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -2.61803 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -2.61803 q^{5} +1.00000 q^{7} -2.00000 q^{9} +0.381966 q^{11} +1.85410 q^{13} -2.61803 q^{15} -3.38197 q^{17} +0.854102 q^{19} +1.00000 q^{21} +4.47214 q^{23} +1.85410 q^{25} -5.00000 q^{27} -0.763932 q^{29} -6.70820 q^{31} +0.381966 q^{33} -2.61803 q^{35} -6.70820 q^{37} +1.85410 q^{39} -8.94427 q^{41} -4.70820 q^{43} +5.23607 q^{45} +7.09017 q^{47} -6.00000 q^{49} -3.38197 q^{51} -10.0902 q^{53} -1.00000 q^{55} +0.854102 q^{57} -8.61803 q^{59} +10.8541 q^{61} -2.00000 q^{63} -4.85410 q^{65} +12.4164 q^{67} +4.47214 q^{69} -7.09017 q^{71} -4.14590 q^{73} +1.85410 q^{75} +0.381966 q^{77} -13.5623 q^{79} +1.00000 q^{81} -9.32624 q^{83} +8.85410 q^{85} -0.763932 q^{87} -15.7082 q^{89} +1.85410 q^{91} -6.70820 q^{93} -2.23607 q^{95} +11.7082 q^{97} -0.763932 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 - 4 * q^9 $$2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} - 4 q^{9} + 3 q^{11} - 3 q^{13} - 3 q^{15} - 9 q^{17} - 5 q^{19} + 2 q^{21} - 3 q^{25} - 10 q^{27} - 6 q^{29} + 3 q^{33} - 3 q^{35} - 3 q^{39} + 4 q^{43} + 6 q^{45} + 3 q^{47} - 12 q^{49} - 9 q^{51} - 9 q^{53} - 2 q^{55} - 5 q^{57} - 15 q^{59} + 15 q^{61} - 4 q^{63} - 3 q^{65} - 2 q^{67} - 3 q^{71} - 15 q^{73} - 3 q^{75} + 3 q^{77} - 7 q^{79} + 2 q^{81} - 3 q^{83} + 11 q^{85} - 6 q^{87} - 18 q^{89} - 3 q^{91} + 10 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 - 4 * q^9 + 3 * q^11 - 3 * q^13 - 3 * q^15 - 9 * q^17 - 5 * q^19 + 2 * q^21 - 3 * q^25 - 10 * q^27 - 6 * q^29 + 3 * q^33 - 3 * q^35 - 3 * q^39 + 4 * q^43 + 6 * q^45 + 3 * q^47 - 12 * q^49 - 9 * q^51 - 9 * q^53 - 2 * q^55 - 5 * q^57 - 15 * q^59 + 15 * q^61 - 4 * q^63 - 3 * q^65 - 2 * q^67 - 3 * q^71 - 15 * q^73 - 3 * q^75 + 3 * q^77 - 7 * q^79 + 2 * q^81 - 3 * q^83 + 11 * q^85 - 6 * q^87 - 18 * q^89 - 3 * q^91 + 10 * q^97 - 6 * 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
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ −2.61803 −1.17082 −0.585410 0.810737i $$-0.699067\pi$$
−0.585410 + 0.810737i $$0.699067\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 0.381966 0.115167 0.0575835 0.998341i $$-0.481660\pi$$
0.0575835 + 0.998341i $$0.481660\pi$$
$$12$$ 0 0
$$13$$ 1.85410 0.514235 0.257118 0.966380i $$-0.417227\pi$$
0.257118 + 0.966380i $$0.417227\pi$$
$$14$$ 0 0
$$15$$ −2.61803 −0.675973
$$16$$ 0 0
$$17$$ −3.38197 −0.820247 −0.410124 0.912030i $$-0.634514\pi$$
−0.410124 + 0.912030i $$0.634514\pi$$
$$18$$ 0 0
$$19$$ 0.854102 0.195944 0.0979722 0.995189i $$-0.468764\pi$$
0.0979722 + 0.995189i $$0.468764\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 4.47214 0.932505 0.466252 0.884652i $$-0.345604\pi$$
0.466252 + 0.884652i $$0.345604\pi$$
$$24$$ 0 0
$$25$$ 1.85410 0.370820
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ −0.763932 −0.141859 −0.0709293 0.997481i $$-0.522596\pi$$
−0.0709293 + 0.997481i $$0.522596\pi$$
$$30$$ 0 0
$$31$$ −6.70820 −1.20483 −0.602414 0.798183i $$-0.705795\pi$$
−0.602414 + 0.798183i $$0.705795\pi$$
$$32$$ 0 0
$$33$$ 0.381966 0.0664917
$$34$$ 0 0
$$35$$ −2.61803 −0.442529
$$36$$ 0 0
$$37$$ −6.70820 −1.10282 −0.551411 0.834234i $$-0.685910\pi$$
−0.551411 + 0.834234i $$0.685910\pi$$
$$38$$ 0 0
$$39$$ 1.85410 0.296894
$$40$$ 0 0
$$41$$ −8.94427 −1.39686 −0.698430 0.715678i $$-0.746118\pi$$
−0.698430 + 0.715678i $$0.746118\pi$$
$$42$$ 0 0
$$43$$ −4.70820 −0.717994 −0.358997 0.933339i $$-0.616881\pi$$
−0.358997 + 0.933339i $$0.616881\pi$$
$$44$$ 0 0
$$45$$ 5.23607 0.780547
$$46$$ 0 0
$$47$$ 7.09017 1.03421 0.517104 0.855923i $$-0.327010\pi$$
0.517104 + 0.855923i $$0.327010\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −3.38197 −0.473570
$$52$$ 0 0
$$53$$ −10.0902 −1.38599 −0.692996 0.720942i $$-0.743710\pi$$
−0.692996 + 0.720942i $$0.743710\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 0.854102 0.113129
$$58$$ 0 0
$$59$$ −8.61803 −1.12197 −0.560986 0.827825i $$-0.689578\pi$$
−0.560986 + 0.827825i $$0.689578\pi$$
$$60$$ 0 0
$$61$$ 10.8541 1.38973 0.694863 0.719142i $$-0.255465\pi$$
0.694863 + 0.719142i $$0.255465\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 0 0
$$65$$ −4.85410 −0.602077
$$66$$ 0 0
$$67$$ 12.4164 1.51691 0.758453 0.651728i $$-0.225956\pi$$
0.758453 + 0.651728i $$0.225956\pi$$
$$68$$ 0 0
$$69$$ 4.47214 0.538382
$$70$$ 0 0
$$71$$ −7.09017 −0.841448 −0.420724 0.907189i $$-0.638224\pi$$
−0.420724 + 0.907189i $$0.638224\pi$$
$$72$$ 0 0
$$73$$ −4.14590 −0.485241 −0.242620 0.970121i $$-0.578007\pi$$
−0.242620 + 0.970121i $$0.578007\pi$$
$$74$$ 0 0
$$75$$ 1.85410 0.214093
$$76$$ 0 0
$$77$$ 0.381966 0.0435291
$$78$$ 0 0
$$79$$ −13.5623 −1.52588 −0.762939 0.646470i $$-0.776245\pi$$
−0.762939 + 0.646470i $$0.776245\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −9.32624 −1.02369 −0.511844 0.859079i $$-0.671037\pi$$
−0.511844 + 0.859079i $$0.671037\pi$$
$$84$$ 0 0
$$85$$ 8.85410 0.960362
$$86$$ 0 0
$$87$$ −0.763932 −0.0819021
$$88$$ 0 0
$$89$$ −15.7082 −1.66507 −0.832533 0.553975i $$-0.813110\pi$$
−0.832533 + 0.553975i $$0.813110\pi$$
$$90$$ 0 0
$$91$$ 1.85410 0.194363
$$92$$ 0 0
$$93$$ −6.70820 −0.695608
$$94$$ 0 0
$$95$$ −2.23607 −0.229416
$$96$$ 0 0
$$97$$ 11.7082 1.18879 0.594394 0.804174i $$-0.297392\pi$$
0.594394 + 0.804174i $$0.297392\pi$$
$$98$$ 0 0
$$99$$ −0.763932 −0.0767781
$$100$$ 0 0
$$101$$ 13.0902 1.30252 0.651260 0.758854i $$-0.274241\pi$$
0.651260 + 0.758854i $$0.274241\pi$$
$$102$$ 0 0
$$103$$ 1.00000 0.0985329
$$104$$ 0 0
$$105$$ −2.61803 −0.255494
$$106$$ 0 0
$$107$$ −4.09017 −0.395412 −0.197706 0.980261i $$-0.563349\pi$$
−0.197706 + 0.980261i $$0.563349\pi$$
$$108$$ 0 0
$$109$$ 10.5623 1.01169 0.505843 0.862626i $$-0.331182\pi$$
0.505843 + 0.862626i $$0.331182\pi$$
$$110$$ 0 0
$$111$$ −6.70820 −0.636715
$$112$$ 0 0
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 0 0
$$115$$ −11.7082 −1.09180
$$116$$ 0 0
$$117$$ −3.70820 −0.342824
$$118$$ 0 0
$$119$$ −3.38197 −0.310024
$$120$$ 0 0
$$121$$ −10.8541 −0.986737
$$122$$ 0 0
$$123$$ −8.94427 −0.806478
$$124$$ 0 0
$$125$$ 8.23607 0.736656
$$126$$ 0 0
$$127$$ 18.2705 1.62125 0.810623 0.585569i $$-0.199129\pi$$
0.810623 + 0.585569i $$0.199129\pi$$
$$128$$ 0 0
$$129$$ −4.70820 −0.414534
$$130$$ 0 0
$$131$$ −2.23607 −0.195366 −0.0976831 0.995218i $$-0.531143\pi$$
−0.0976831 + 0.995218i $$0.531143\pi$$
$$132$$ 0 0
$$133$$ 0.854102 0.0740600
$$134$$ 0 0
$$135$$ 13.0902 1.12662
$$136$$ 0 0
$$137$$ 0.708204 0.0605059 0.0302530 0.999542i $$-0.490369\pi$$
0.0302530 + 0.999542i $$0.490369\pi$$
$$138$$ 0 0
$$139$$ 17.8541 1.51437 0.757183 0.653203i $$-0.226575\pi$$
0.757183 + 0.653203i $$0.226575\pi$$
$$140$$ 0 0
$$141$$ 7.09017 0.597100
$$142$$ 0 0
$$143$$ 0.708204 0.0592230
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ −6.00000 −0.494872
$$148$$ 0 0
$$149$$ 1.47214 0.120602 0.0603010 0.998180i $$-0.480794\pi$$
0.0603010 + 0.998180i $$0.480794\pi$$
$$150$$ 0 0
$$151$$ 19.0000 1.54620 0.773099 0.634285i $$-0.218706\pi$$
0.773099 + 0.634285i $$0.218706\pi$$
$$152$$ 0 0
$$153$$ 6.76393 0.546831
$$154$$ 0 0
$$155$$ 17.5623 1.41064
$$156$$ 0 0
$$157$$ 3.29180 0.262714 0.131357 0.991335i $$-0.458067\pi$$
0.131357 + 0.991335i $$0.458067\pi$$
$$158$$ 0 0
$$159$$ −10.0902 −0.800203
$$160$$ 0 0
$$161$$ 4.47214 0.352454
$$162$$ 0 0
$$163$$ 10.7082 0.838731 0.419366 0.907817i $$-0.362253\pi$$
0.419366 + 0.907817i $$0.362253\pi$$
$$164$$ 0 0
$$165$$ −1.00000 −0.0778499
$$166$$ 0 0
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 0 0
$$169$$ −9.56231 −0.735562
$$170$$ 0 0
$$171$$ −1.70820 −0.130630
$$172$$ 0 0
$$173$$ −13.0344 −0.990990 −0.495495 0.868611i $$-0.665013\pi$$
−0.495495 + 0.868611i $$0.665013\pi$$
$$174$$ 0 0
$$175$$ 1.85410 0.140157
$$176$$ 0 0
$$177$$ −8.61803 −0.647771
$$178$$ 0 0
$$179$$ 1.14590 0.0856484 0.0428242 0.999083i $$-0.486364\pi$$
0.0428242 + 0.999083i $$0.486364\pi$$
$$180$$ 0 0
$$181$$ −2.85410 −0.212144 −0.106072 0.994358i $$-0.533827\pi$$
−0.106072 + 0.994358i $$0.533827\pi$$
$$182$$ 0 0
$$183$$ 10.8541 0.802358
$$184$$ 0 0
$$185$$ 17.5623 1.29121
$$186$$ 0 0
$$187$$ −1.29180 −0.0944655
$$188$$ 0 0
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ −3.38197 −0.244710 −0.122355 0.992486i $$-0.539045\pi$$
−0.122355 + 0.992486i $$0.539045\pi$$
$$192$$ 0 0
$$193$$ −20.1246 −1.44860 −0.724301 0.689484i $$-0.757837\pi$$
−0.724301 + 0.689484i $$0.757837\pi$$
$$194$$ 0 0
$$195$$ −4.85410 −0.347609
$$196$$ 0 0
$$197$$ 10.4164 0.742138 0.371069 0.928605i $$-0.378991\pi$$
0.371069 + 0.928605i $$0.378991\pi$$
$$198$$ 0 0
$$199$$ −3.41641 −0.242183 −0.121091 0.992641i $$-0.538639\pi$$
−0.121091 + 0.992641i $$0.538639\pi$$
$$200$$ 0 0
$$201$$ 12.4164 0.875786
$$202$$ 0 0
$$203$$ −0.763932 −0.0536175
$$204$$ 0 0
$$205$$ 23.4164 1.63547
$$206$$ 0 0
$$207$$ −8.94427 −0.621670
$$208$$ 0 0
$$209$$ 0.326238 0.0225663
$$210$$ 0 0
$$211$$ −8.14590 −0.560787 −0.280393 0.959885i $$-0.590465\pi$$
−0.280393 + 0.959885i $$0.590465\pi$$
$$212$$ 0 0
$$213$$ −7.09017 −0.485810
$$214$$ 0 0
$$215$$ 12.3262 0.840642
$$216$$ 0 0
$$217$$ −6.70820 −0.455383
$$218$$ 0 0
$$219$$ −4.14590 −0.280154
$$220$$ 0 0
$$221$$ −6.27051 −0.421800
$$222$$ 0 0
$$223$$ 5.70820 0.382250 0.191125 0.981566i $$-0.438786\pi$$
0.191125 + 0.981566i $$0.438786\pi$$
$$224$$ 0 0
$$225$$ −3.70820 −0.247214
$$226$$ 0 0
$$227$$ 14.9443 0.991886 0.495943 0.868355i $$-0.334822\pi$$
0.495943 + 0.868355i $$0.334822\pi$$
$$228$$ 0 0
$$229$$ −6.70820 −0.443291 −0.221645 0.975127i $$-0.571143\pi$$
−0.221645 + 0.975127i $$0.571143\pi$$
$$230$$ 0 0
$$231$$ 0.381966 0.0251315
$$232$$ 0 0
$$233$$ 8.88854 0.582308 0.291154 0.956676i $$-0.405961\pi$$
0.291154 + 0.956676i $$0.405961\pi$$
$$234$$ 0 0
$$235$$ −18.5623 −1.21087
$$236$$ 0 0
$$237$$ −13.5623 −0.880966
$$238$$ 0 0
$$239$$ −24.3262 −1.57353 −0.786767 0.617250i $$-0.788247\pi$$
−0.786767 + 0.617250i $$0.788247\pi$$
$$240$$ 0 0
$$241$$ 25.2705 1.62782 0.813908 0.580993i $$-0.197336\pi$$
0.813908 + 0.580993i $$0.197336\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 15.7082 1.00356
$$246$$ 0 0
$$247$$ 1.58359 0.100762
$$248$$ 0 0
$$249$$ −9.32624 −0.591026
$$250$$ 0 0
$$251$$ −6.76393 −0.426936 −0.213468 0.976950i $$-0.568476\pi$$
−0.213468 + 0.976950i $$0.568476\pi$$
$$252$$ 0 0
$$253$$ 1.70820 0.107394
$$254$$ 0 0
$$255$$ 8.85410 0.554465
$$256$$ 0 0
$$257$$ −4.52786 −0.282440 −0.141220 0.989978i $$-0.545103\pi$$
−0.141220 + 0.989978i $$0.545103\pi$$
$$258$$ 0 0
$$259$$ −6.70820 −0.416828
$$260$$ 0 0
$$261$$ 1.52786 0.0945724
$$262$$ 0 0
$$263$$ 18.3820 1.13348 0.566740 0.823896i $$-0.308204\pi$$
0.566740 + 0.823896i $$0.308204\pi$$
$$264$$ 0 0
$$265$$ 26.4164 1.62275
$$266$$ 0 0
$$267$$ −15.7082 −0.961326
$$268$$ 0 0
$$269$$ −3.32624 −0.202804 −0.101402 0.994846i $$-0.532333\pi$$
−0.101402 + 0.994846i $$0.532333\pi$$
$$270$$ 0 0
$$271$$ −1.00000 −0.0607457 −0.0303728 0.999539i $$-0.509669\pi$$
−0.0303728 + 0.999539i $$0.509669\pi$$
$$272$$ 0 0
$$273$$ 1.85410 0.112215
$$274$$ 0 0
$$275$$ 0.708204 0.0427063
$$276$$ 0 0
$$277$$ −8.70820 −0.523225 −0.261613 0.965173i $$-0.584254\pi$$
−0.261613 + 0.965173i $$0.584254\pi$$
$$278$$ 0 0
$$279$$ 13.4164 0.803219
$$280$$ 0 0
$$281$$ −22.5279 −1.34390 −0.671950 0.740597i $$-0.734543\pi$$
−0.671950 + 0.740597i $$0.734543\pi$$
$$282$$ 0 0
$$283$$ 6.29180 0.374008 0.187004 0.982359i $$-0.440122\pi$$
0.187004 + 0.982359i $$0.440122\pi$$
$$284$$ 0 0
$$285$$ −2.23607 −0.132453
$$286$$ 0 0
$$287$$ −8.94427 −0.527964
$$288$$ 0 0
$$289$$ −5.56231 −0.327194
$$290$$ 0 0
$$291$$ 11.7082 0.686347
$$292$$ 0 0
$$293$$ 9.65248 0.563904 0.281952 0.959429i $$-0.409018\pi$$
0.281952 + 0.959429i $$0.409018\pi$$
$$294$$ 0 0
$$295$$ 22.5623 1.31363
$$296$$ 0 0
$$297$$ −1.90983 −0.110820
$$298$$ 0 0
$$299$$ 8.29180 0.479527
$$300$$ 0 0
$$301$$ −4.70820 −0.271376
$$302$$ 0 0
$$303$$ 13.0902 0.752011
$$304$$ 0 0
$$305$$ −28.4164 −1.62712
$$306$$ 0 0
$$307$$ −3.85410 −0.219965 −0.109983 0.993934i $$-0.535080\pi$$
−0.109983 + 0.993934i $$0.535080\pi$$
$$308$$ 0 0
$$309$$ 1.00000 0.0568880
$$310$$ 0 0
$$311$$ 32.8885 1.86494 0.932469 0.361250i $$-0.117650\pi$$
0.932469 + 0.361250i $$0.117650\pi$$
$$312$$ 0 0
$$313$$ 29.7082 1.67921 0.839603 0.543200i $$-0.182787\pi$$
0.839603 + 0.543200i $$0.182787\pi$$
$$314$$ 0 0
$$315$$ 5.23607 0.295019
$$316$$ 0 0
$$317$$ 1.58359 0.0889434 0.0444717 0.999011i $$-0.485840\pi$$
0.0444717 + 0.999011i $$0.485840\pi$$
$$318$$ 0 0
$$319$$ −0.291796 −0.0163374
$$320$$ 0 0
$$321$$ −4.09017 −0.228291
$$322$$ 0 0
$$323$$ −2.88854 −0.160723
$$324$$ 0 0
$$325$$ 3.43769 0.190689
$$326$$ 0 0
$$327$$ 10.5623 0.584097
$$328$$ 0 0
$$329$$ 7.09017 0.390894
$$330$$ 0 0
$$331$$ 22.8541 1.25618 0.628088 0.778143i $$-0.283838\pi$$
0.628088 + 0.778143i $$0.283838\pi$$
$$332$$ 0 0
$$333$$ 13.4164 0.735215
$$334$$ 0 0
$$335$$ −32.5066 −1.77602
$$336$$ 0 0
$$337$$ −22.5623 −1.22905 −0.614524 0.788898i $$-0.710652\pi$$
−0.614524 + 0.788898i $$0.710652\pi$$
$$338$$ 0 0
$$339$$ −15.0000 −0.814688
$$340$$ 0 0
$$341$$ −2.56231 −0.138757
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 0 0
$$345$$ −11.7082 −0.630349
$$346$$ 0 0
$$347$$ −7.47214 −0.401125 −0.200563 0.979681i $$-0.564277\pi$$
−0.200563 + 0.979681i $$0.564277\pi$$
$$348$$ 0 0
$$349$$ −11.4164 −0.611106 −0.305553 0.952175i $$-0.598841\pi$$
−0.305553 + 0.952175i $$0.598841\pi$$
$$350$$ 0 0
$$351$$ −9.27051 −0.494823
$$352$$ 0 0
$$353$$ 4.03444 0.214732 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$354$$ 0 0
$$355$$ 18.5623 0.985185
$$356$$ 0 0
$$357$$ −3.38197 −0.178993
$$358$$ 0 0
$$359$$ −30.3262 −1.60056 −0.800279 0.599628i $$-0.795315\pi$$
−0.800279 + 0.599628i $$0.795315\pi$$
$$360$$ 0 0
$$361$$ −18.2705 −0.961606
$$362$$ 0 0
$$363$$ −10.8541 −0.569693
$$364$$ 0 0
$$365$$ 10.8541 0.568130
$$366$$ 0 0
$$367$$ −36.5623 −1.90854 −0.954268 0.298951i $$-0.903363\pi$$
−0.954268 + 0.298951i $$0.903363\pi$$
$$368$$ 0 0
$$369$$ 17.8885 0.931240
$$370$$ 0 0
$$371$$ −10.0902 −0.523856
$$372$$ 0 0
$$373$$ 37.6869 1.95135 0.975677 0.219212i $$-0.0703485\pi$$
0.975677 + 0.219212i $$0.0703485\pi$$
$$374$$ 0 0
$$375$$ 8.23607 0.425309
$$376$$ 0 0
$$377$$ −1.41641 −0.0729487
$$378$$ 0 0
$$379$$ −5.00000 −0.256833 −0.128416 0.991720i $$-0.540989\pi$$
−0.128416 + 0.991720i $$0.540989\pi$$
$$380$$ 0 0
$$381$$ 18.2705 0.936027
$$382$$ 0 0
$$383$$ −23.1803 −1.18446 −0.592230 0.805769i $$-0.701752\pi$$
−0.592230 + 0.805769i $$0.701752\pi$$
$$384$$ 0 0
$$385$$ −1.00000 −0.0509647
$$386$$ 0 0
$$387$$ 9.41641 0.478663
$$388$$ 0 0
$$389$$ −19.4164 −0.984451 −0.492225 0.870468i $$-0.663816\pi$$
−0.492225 + 0.870468i $$0.663816\pi$$
$$390$$ 0 0
$$391$$ −15.1246 −0.764884
$$392$$ 0 0
$$393$$ −2.23607 −0.112795
$$394$$ 0 0
$$395$$ 35.5066 1.78653
$$396$$ 0 0
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ 0 0
$$399$$ 0.854102 0.0427586
$$400$$ 0 0
$$401$$ 11.8885 0.593686 0.296843 0.954926i $$-0.404066\pi$$
0.296843 + 0.954926i $$0.404066\pi$$
$$402$$ 0 0
$$403$$ −12.4377 −0.619566
$$404$$ 0 0
$$405$$ −2.61803 −0.130091
$$406$$ 0 0
$$407$$ −2.56231 −0.127009
$$408$$ 0 0
$$409$$ 23.2918 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$410$$ 0 0
$$411$$ 0.708204 0.0349331
$$412$$ 0 0
$$413$$ −8.61803 −0.424066
$$414$$ 0 0
$$415$$ 24.4164 1.19855
$$416$$ 0 0
$$417$$ 17.8541 0.874319
$$418$$ 0 0
$$419$$ −7.09017 −0.346377 −0.173189 0.984889i $$-0.555407\pi$$
−0.173189 + 0.984889i $$0.555407\pi$$
$$420$$ 0 0
$$421$$ 3.00000 0.146211 0.0731055 0.997324i $$-0.476709\pi$$
0.0731055 + 0.997324i $$0.476709\pi$$
$$422$$ 0 0
$$423$$ −14.1803 −0.689472
$$424$$ 0 0
$$425$$ −6.27051 −0.304164
$$426$$ 0 0
$$427$$ 10.8541 0.525267
$$428$$ 0 0
$$429$$ 0.708204 0.0341924
$$430$$ 0 0
$$431$$ 10.3607 0.499056 0.249528 0.968368i $$-0.419724\pi$$
0.249528 + 0.968368i $$0.419724\pi$$
$$432$$ 0 0
$$433$$ −12.4164 −0.596694 −0.298347 0.954457i $$-0.596435\pi$$
−0.298347 + 0.954457i $$0.596435\pi$$
$$434$$ 0 0
$$435$$ 2.00000 0.0958927
$$436$$ 0 0
$$437$$ 3.81966 0.182719
$$438$$ 0 0
$$439$$ −9.43769 −0.450437 −0.225218 0.974308i $$-0.572310\pi$$
−0.225218 + 0.974308i $$0.572310\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ 0 0
$$443$$ 35.5623 1.68962 0.844808 0.535069i $$-0.179715\pi$$
0.844808 + 0.535069i $$0.179715\pi$$
$$444$$ 0 0
$$445$$ 41.1246 1.94949
$$446$$ 0 0
$$447$$ 1.47214 0.0696296
$$448$$ 0 0
$$449$$ 31.3607 1.48000 0.740001 0.672606i $$-0.234825\pi$$
0.740001 + 0.672606i $$0.234825\pi$$
$$450$$ 0 0
$$451$$ −3.41641 −0.160872
$$452$$ 0 0
$$453$$ 19.0000 0.892698
$$454$$ 0 0
$$455$$ −4.85410 −0.227564
$$456$$ 0 0
$$457$$ −3.14590 −0.147159 −0.0735795 0.997289i $$-0.523442\pi$$
−0.0735795 + 0.997289i $$0.523442\pi$$
$$458$$ 0 0
$$459$$ 16.9098 0.789283
$$460$$ 0 0
$$461$$ −39.2148 −1.82641 −0.913207 0.407495i $$-0.866402\pi$$
−0.913207 + 0.407495i $$0.866402\pi$$
$$462$$ 0 0
$$463$$ −5.41641 −0.251722 −0.125861 0.992048i $$-0.540169\pi$$
−0.125861 + 0.992048i $$0.540169\pi$$
$$464$$ 0 0
$$465$$ 17.5623 0.814432
$$466$$ 0 0
$$467$$ −27.6525 −1.27960 −0.639802 0.768540i $$-0.720984\pi$$
−0.639802 + 0.768540i $$0.720984\pi$$
$$468$$ 0 0
$$469$$ 12.4164 0.573336
$$470$$ 0 0
$$471$$ 3.29180 0.151678
$$472$$ 0 0
$$473$$ −1.79837 −0.0826893
$$474$$ 0 0
$$475$$ 1.58359 0.0726602
$$476$$ 0 0
$$477$$ 20.1803 0.923994
$$478$$ 0 0
$$479$$ 14.1803 0.647916 0.323958 0.946071i $$-0.394986\pi$$
0.323958 + 0.946071i $$0.394986\pi$$
$$480$$ 0 0
$$481$$ −12.4377 −0.567110
$$482$$ 0 0
$$483$$ 4.47214 0.203489
$$484$$ 0 0
$$485$$ −30.6525 −1.39186
$$486$$ 0 0
$$487$$ 23.0000 1.04223 0.521115 0.853487i $$-0.325516\pi$$
0.521115 + 0.853487i $$0.325516\pi$$
$$488$$ 0 0
$$489$$ 10.7082 0.484242
$$490$$ 0 0
$$491$$ 35.7771 1.61460 0.807299 0.590143i $$-0.200929\pi$$
0.807299 + 0.590143i $$0.200929\pi$$
$$492$$ 0 0
$$493$$ 2.58359 0.116359
$$494$$ 0 0
$$495$$ 2.00000 0.0898933
$$496$$ 0 0
$$497$$ −7.09017 −0.318038
$$498$$ 0 0
$$499$$ 20.2705 0.907433 0.453716 0.891146i $$-0.350098\pi$$
0.453716 + 0.891146i $$0.350098\pi$$
$$500$$ 0 0
$$501$$ −9.00000 −0.402090
$$502$$ 0 0
$$503$$ 31.3607 1.39830 0.699152 0.714973i $$-0.253561\pi$$
0.699152 + 0.714973i $$0.253561\pi$$
$$504$$ 0 0
$$505$$ −34.2705 −1.52502
$$506$$ 0 0
$$507$$ −9.56231 −0.424677
$$508$$ 0 0
$$509$$ 24.3820 1.08071 0.540356 0.841437i $$-0.318290\pi$$
0.540356 + 0.841437i $$0.318290\pi$$
$$510$$ 0 0
$$511$$ −4.14590 −0.183404
$$512$$ 0 0
$$513$$ −4.27051 −0.188548
$$514$$ 0 0
$$515$$ −2.61803 −0.115364
$$516$$ 0 0
$$517$$ 2.70820 0.119107
$$518$$ 0 0
$$519$$ −13.0344 −0.572148
$$520$$ 0 0
$$521$$ −5.18034 −0.226955 −0.113477 0.993541i $$-0.536199\pi$$
−0.113477 + 0.993541i $$0.536199\pi$$
$$522$$ 0 0
$$523$$ 37.4164 1.63611 0.818053 0.575143i $$-0.195054\pi$$
0.818053 + 0.575143i $$0.195054\pi$$
$$524$$ 0 0
$$525$$ 1.85410 0.0809196
$$526$$ 0 0
$$527$$ 22.6869 0.988258
$$528$$ 0 0
$$529$$ −3.00000 −0.130435
$$530$$ 0 0
$$531$$ 17.2361 0.747982
$$532$$ 0 0
$$533$$ −16.5836 −0.718315
$$534$$ 0 0
$$535$$ 10.7082 0.462956
$$536$$ 0 0
$$537$$ 1.14590 0.0494492
$$538$$ 0 0
$$539$$ −2.29180 −0.0987146
$$540$$ 0 0
$$541$$ 15.8541 0.681621 0.340811 0.940132i $$-0.389299\pi$$
0.340811 + 0.940132i $$0.389299\pi$$
$$542$$ 0 0
$$543$$ −2.85410 −0.122481
$$544$$ 0 0
$$545$$ −27.6525 −1.18450
$$546$$ 0 0
$$547$$ −31.2705 −1.33703 −0.668515 0.743698i $$-0.733070\pi$$
−0.668515 + 0.743698i $$0.733070\pi$$
$$548$$ 0 0
$$549$$ −21.7082 −0.926484
$$550$$ 0 0
$$551$$ −0.652476 −0.0277964
$$552$$ 0 0
$$553$$ −13.5623 −0.576728
$$554$$ 0 0
$$555$$ 17.5623 0.745478
$$556$$ 0 0
$$557$$ 28.6869 1.21550 0.607752 0.794127i $$-0.292072\pi$$
0.607752 + 0.794127i $$0.292072\pi$$
$$558$$ 0 0
$$559$$ −8.72949 −0.369218
$$560$$ 0 0
$$561$$ −1.29180 −0.0545397
$$562$$ 0 0
$$563$$ 25.7984 1.08727 0.543636 0.839321i $$-0.317047\pi$$
0.543636 + 0.839321i $$0.317047\pi$$
$$564$$ 0 0
$$565$$ 39.2705 1.65212
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 42.1591 1.76740 0.883700 0.468054i $$-0.155045\pi$$
0.883700 + 0.468054i $$0.155045\pi$$
$$570$$ 0 0
$$571$$ 2.43769 0.102014 0.0510072 0.998698i $$-0.483757\pi$$
0.0510072 + 0.998698i $$0.483757\pi$$
$$572$$ 0 0
$$573$$ −3.38197 −0.141284
$$574$$ 0 0
$$575$$ 8.29180 0.345792
$$576$$ 0 0
$$577$$ 16.8328 0.700759 0.350380 0.936608i $$-0.386053\pi$$
0.350380 + 0.936608i $$0.386053\pi$$
$$578$$ 0 0
$$579$$ −20.1246 −0.836350
$$580$$ 0 0
$$581$$ −9.32624 −0.386918
$$582$$ 0 0
$$583$$ −3.85410 −0.159621
$$584$$ 0 0
$$585$$ 9.70820 0.401385
$$586$$ 0 0
$$587$$ −11.0689 −0.456862 −0.228431 0.973560i $$-0.573359\pi$$
−0.228431 + 0.973560i $$0.573359\pi$$
$$588$$ 0 0
$$589$$ −5.72949 −0.236080
$$590$$ 0 0
$$591$$ 10.4164 0.428474
$$592$$ 0 0
$$593$$ −20.1803 −0.828707 −0.414354 0.910116i $$-0.635992\pi$$
−0.414354 + 0.910116i $$0.635992\pi$$
$$594$$ 0 0
$$595$$ 8.85410 0.362983
$$596$$ 0 0
$$597$$ −3.41641 −0.139824
$$598$$ 0 0
$$599$$ −20.4508 −0.835599 −0.417800 0.908539i $$-0.637199\pi$$
−0.417800 + 0.908539i $$0.637199\pi$$
$$600$$ 0 0
$$601$$ −16.5623 −0.675591 −0.337795 0.941220i $$-0.609681\pi$$
−0.337795 + 0.941220i $$0.609681\pi$$
$$602$$ 0 0
$$603$$ −24.8328 −1.01127
$$604$$ 0 0
$$605$$ 28.4164 1.15529
$$606$$ 0 0
$$607$$ 7.70820 0.312866 0.156433 0.987689i $$-0.450000\pi$$
0.156433 + 0.987689i $$0.450000\pi$$
$$608$$ 0 0
$$609$$ −0.763932 −0.0309561
$$610$$ 0 0
$$611$$ 13.1459 0.531826
$$612$$ 0 0
$$613$$ −2.41641 −0.0975978 −0.0487989 0.998809i $$-0.515539\pi$$
−0.0487989 + 0.998809i $$0.515539\pi$$
$$614$$ 0 0
$$615$$ 23.4164 0.944241
$$616$$ 0 0
$$617$$ −30.2705 −1.21864 −0.609322 0.792923i $$-0.708558\pi$$
−0.609322 + 0.792923i $$0.708558\pi$$
$$618$$ 0 0
$$619$$ −31.6869 −1.27360 −0.636802 0.771027i $$-0.719743\pi$$
−0.636802 + 0.771027i $$0.719743\pi$$
$$620$$ 0 0
$$621$$ −22.3607 −0.897303
$$622$$ 0 0
$$623$$ −15.7082 −0.629336
$$624$$ 0 0
$$625$$ −30.8328 −1.23331
$$626$$ 0 0
$$627$$ 0.326238 0.0130287
$$628$$ 0 0
$$629$$ 22.6869 0.904587
$$630$$ 0 0
$$631$$ −8.72949 −0.347516 −0.173758 0.984788i $$-0.555591\pi$$
−0.173758 + 0.984788i $$0.555591\pi$$
$$632$$ 0 0
$$633$$ −8.14590 −0.323770
$$634$$ 0 0
$$635$$ −47.8328 −1.89819
$$636$$ 0 0
$$637$$ −11.1246 −0.440773
$$638$$ 0 0
$$639$$ 14.1803 0.560966
$$640$$ 0 0
$$641$$ −15.0000 −0.592464 −0.296232 0.955116i $$-0.595730\pi$$
−0.296232 + 0.955116i $$0.595730\pi$$
$$642$$ 0 0
$$643$$ −5.00000 −0.197181 −0.0985904 0.995128i $$-0.531433\pi$$
−0.0985904 + 0.995128i $$0.531433\pi$$
$$644$$ 0 0
$$645$$ 12.3262 0.485345
$$646$$ 0 0
$$647$$ −43.7426 −1.71970 −0.859850 0.510546i $$-0.829443\pi$$
−0.859850 + 0.510546i $$0.829443\pi$$
$$648$$ 0 0
$$649$$ −3.29180 −0.129214
$$650$$ 0 0
$$651$$ −6.70820 −0.262915
$$652$$ 0 0
$$653$$ −0.763932 −0.0298950 −0.0149475 0.999888i $$-0.504758\pi$$
−0.0149475 + 0.999888i $$0.504758\pi$$
$$654$$ 0 0
$$655$$ 5.85410 0.228739
$$656$$ 0 0
$$657$$ 8.29180 0.323494
$$658$$ 0 0
$$659$$ 12.5967 0.490700 0.245350 0.969435i $$-0.421097\pi$$
0.245350 + 0.969435i $$0.421097\pi$$
$$660$$ 0 0
$$661$$ −11.4377 −0.444875 −0.222437 0.974947i $$-0.571401\pi$$
−0.222437 + 0.974947i $$0.571401\pi$$
$$662$$ 0 0
$$663$$ −6.27051 −0.243526
$$664$$ 0 0
$$665$$ −2.23607 −0.0867110
$$666$$ 0 0
$$667$$ −3.41641 −0.132284
$$668$$ 0 0
$$669$$ 5.70820 0.220692
$$670$$ 0 0
$$671$$ 4.14590 0.160051
$$672$$ 0 0
$$673$$ −37.7082 −1.45354 −0.726772 0.686879i $$-0.758980\pi$$
−0.726772 + 0.686879i $$0.758980\pi$$
$$674$$ 0 0
$$675$$ −9.27051 −0.356822
$$676$$ 0 0
$$677$$ 10.9656 0.421441 0.210720 0.977546i $$-0.432419\pi$$
0.210720 + 0.977546i $$0.432419\pi$$
$$678$$ 0 0
$$679$$ 11.7082 0.449320
$$680$$ 0 0
$$681$$ 14.9443 0.572666
$$682$$ 0 0
$$683$$ −42.7639 −1.63632 −0.818158 0.574993i $$-0.805005\pi$$
−0.818158 + 0.574993i $$0.805005\pi$$
$$684$$ 0 0
$$685$$ −1.85410 −0.0708416
$$686$$ 0 0
$$687$$ −6.70820 −0.255934
$$688$$ 0 0
$$689$$ −18.7082 −0.712726
$$690$$ 0 0
$$691$$ −1.14590 −0.0435920 −0.0217960 0.999762i $$-0.506938\pi$$
−0.0217960 + 0.999762i $$0.506938\pi$$
$$692$$ 0 0
$$693$$ −0.763932 −0.0290194
$$694$$ 0 0
$$695$$ −46.7426 −1.77305
$$696$$ 0 0
$$697$$ 30.2492 1.14577
$$698$$ 0 0
$$699$$ 8.88854 0.336196
$$700$$ 0 0
$$701$$ −1.20163 −0.0453848 −0.0226924 0.999742i $$-0.507224\pi$$
−0.0226924 + 0.999742i $$0.507224\pi$$
$$702$$ 0 0
$$703$$ −5.72949 −0.216092
$$704$$ 0 0
$$705$$ −18.5623 −0.699097
$$706$$ 0 0
$$707$$ 13.0902 0.492307
$$708$$ 0 0
$$709$$ −47.9787 −1.80188 −0.900939 0.433945i $$-0.857121\pi$$
−0.900939 + 0.433945i $$0.857121\pi$$
$$710$$ 0 0
$$711$$ 27.1246 1.01725
$$712$$ 0 0
$$713$$ −30.0000 −1.12351
$$714$$ 0 0
$$715$$ −1.85410 −0.0693395
$$716$$ 0 0
$$717$$ −24.3262 −0.908480
$$718$$ 0 0
$$719$$ −23.6738 −0.882882 −0.441441 0.897290i $$-0.645533\pi$$
−0.441441 + 0.897290i $$0.645533\pi$$
$$720$$ 0 0
$$721$$ 1.00000 0.0372419
$$722$$ 0 0
$$723$$ 25.2705 0.939820
$$724$$ 0 0
$$725$$ −1.41641 −0.0526041
$$726$$ 0 0
$$727$$ 38.2705 1.41937 0.709687 0.704517i $$-0.248836\pi$$
0.709687 + 0.704517i $$0.248836\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 15.9230 0.588933
$$732$$ 0 0
$$733$$ 1.29180 0.0477136 0.0238568 0.999715i $$-0.492405\pi$$
0.0238568 + 0.999715i $$0.492405\pi$$
$$734$$ 0 0
$$735$$ 15.7082 0.579406
$$736$$ 0 0
$$737$$ 4.74265 0.174698
$$738$$ 0 0
$$739$$ −36.8328 −1.35492 −0.677459 0.735561i $$-0.736919\pi$$
−0.677459 + 0.735561i $$0.736919\pi$$
$$740$$ 0 0
$$741$$ 1.58359 0.0581747
$$742$$ 0 0
$$743$$ −17.2918 −0.634374 −0.317187 0.948363i $$-0.602738\pi$$
−0.317187 + 0.948363i $$0.602738\pi$$
$$744$$ 0 0
$$745$$ −3.85410 −0.141203
$$746$$ 0 0
$$747$$ 18.6525 0.682458
$$748$$ 0 0
$$749$$ −4.09017 −0.149452
$$750$$ 0 0
$$751$$ 3.87539 0.141415 0.0707075 0.997497i $$-0.477474\pi$$
0.0707075 + 0.997497i $$0.477474\pi$$
$$752$$ 0 0
$$753$$ −6.76393 −0.246491
$$754$$ 0 0
$$755$$ −49.7426 −1.81032
$$756$$ 0 0
$$757$$ −34.7082 −1.26149 −0.630746 0.775990i $$-0.717251\pi$$
−0.630746 + 0.775990i $$0.717251\pi$$
$$758$$ 0 0
$$759$$ 1.70820 0.0620039
$$760$$ 0 0
$$761$$ 1.47214 0.0533649 0.0266824 0.999644i $$-0.491506\pi$$
0.0266824 + 0.999644i $$0.491506\pi$$
$$762$$ 0 0
$$763$$ 10.5623 0.382381
$$764$$ 0 0
$$765$$ −17.7082 −0.640241
$$766$$ 0 0
$$767$$ −15.9787 −0.576958
$$768$$ 0 0
$$769$$ −42.3951 −1.52881 −0.764404 0.644738i $$-0.776966\pi$$
−0.764404 + 0.644738i $$0.776966\pi$$
$$770$$ 0 0
$$771$$ −4.52786 −0.163067
$$772$$ 0 0
$$773$$ −25.4721 −0.916169 −0.458085 0.888909i $$-0.651464\pi$$
−0.458085 + 0.888909i $$0.651464\pi$$
$$774$$ 0 0
$$775$$ −12.4377 −0.446775
$$776$$ 0 0
$$777$$ −6.70820 −0.240655
$$778$$ 0 0
$$779$$ −7.63932 −0.273707
$$780$$ 0 0
$$781$$ −2.70820 −0.0969072
$$782$$ 0 0
$$783$$ 3.81966 0.136504
$$784$$ 0 0
$$785$$ −8.61803 −0.307591
$$786$$ 0 0
$$787$$ −16.5836 −0.591141 −0.295571 0.955321i $$-0.595510\pi$$
−0.295571 + 0.955321i $$0.595510\pi$$
$$788$$ 0 0
$$789$$ 18.3820 0.654415
$$790$$ 0 0
$$791$$ −15.0000 −0.533339
$$792$$ 0 0
$$793$$ 20.1246 0.714646
$$794$$ 0 0
$$795$$ 26.4164 0.936893
$$796$$ 0 0
$$797$$ −9.87539 −0.349804 −0.174902 0.984586i $$-0.555961\pi$$
−0.174902 + 0.984586i $$0.555961\pi$$
$$798$$ 0 0
$$799$$ −23.9787 −0.848306
$$800$$ 0 0
$$801$$ 31.4164 1.11004
$$802$$ 0 0
$$803$$ −1.58359 −0.0558838
$$804$$ 0 0
$$805$$ −11.7082 −0.412660
$$806$$ 0 0
$$807$$ −3.32624 −0.117089
$$808$$ 0 0
$$809$$ −14.9443 −0.525413 −0.262706 0.964876i $$-0.584615\pi$$
−0.262706 + 0.964876i $$0.584615\pi$$
$$810$$ 0 0
$$811$$ 45.5410 1.59916 0.799581 0.600559i $$-0.205055\pi$$
0.799581 + 0.600559i $$0.205055\pi$$
$$812$$ 0 0
$$813$$ −1.00000 −0.0350715
$$814$$ 0 0
$$815$$ −28.0344 −0.982004
$$816$$ 0 0
$$817$$ −4.02129 −0.140687
$$818$$ 0 0
$$819$$ −3.70820 −0.129575
$$820$$ 0 0
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ 0 0
$$823$$ −23.5623 −0.821330 −0.410665 0.911786i $$-0.634703\pi$$
−0.410665 + 0.911786i $$0.634703\pi$$
$$824$$ 0 0
$$825$$ 0.708204 0.0246565
$$826$$ 0 0
$$827$$ 6.70820 0.233267 0.116634 0.993175i $$-0.462790\pi$$
0.116634 + 0.993175i $$0.462790\pi$$
$$828$$ 0 0
$$829$$ −19.2705 −0.669292 −0.334646 0.942344i $$-0.608617\pi$$
−0.334646 + 0.942344i $$0.608617\pi$$
$$830$$ 0 0
$$831$$ −8.70820 −0.302084
$$832$$ 0 0
$$833$$ 20.2918 0.703069
$$834$$ 0 0
$$835$$ 23.5623 0.815407
$$836$$ 0 0
$$837$$ 33.5410 1.15935
$$838$$ 0 0
$$839$$ −21.3820 −0.738187 −0.369094 0.929392i $$-0.620332\pi$$
−0.369094 + 0.929392i $$0.620332\pi$$
$$840$$ 0 0
$$841$$ −28.4164 −0.979876
$$842$$ 0 0
$$843$$ −22.5279 −0.775901
$$844$$ 0 0
$$845$$ 25.0344 0.861211
$$846$$ 0 0
$$847$$ −10.8541 −0.372951
$$848$$ 0 0
$$849$$ 6.29180 0.215934
$$850$$ 0 0
$$851$$ −30.0000 −1.02839
$$852$$ 0 0
$$853$$ −10.7295 −0.367371 −0.183685 0.982985i $$-0.558803\pi$$
−0.183685 + 0.982985i $$0.558803\pi$$
$$854$$ 0 0
$$855$$ 4.47214 0.152944
$$856$$ 0 0
$$857$$ 3.76393 0.128573 0.0642867 0.997931i $$-0.479523\pi$$
0.0642867 + 0.997931i $$0.479523\pi$$
$$858$$ 0 0
$$859$$ 9.56231 0.326262 0.163131 0.986604i $$-0.447841\pi$$
0.163131 + 0.986604i $$0.447841\pi$$
$$860$$ 0 0
$$861$$ −8.94427 −0.304820
$$862$$ 0 0
$$863$$ 8.29180 0.282256 0.141128 0.989991i $$-0.454927\pi$$
0.141128 + 0.989991i $$0.454927\pi$$
$$864$$ 0 0
$$865$$ 34.1246 1.16027
$$866$$ 0 0
$$867$$ −5.56231 −0.188906
$$868$$ 0 0
$$869$$ −5.18034 −0.175731
$$870$$ 0 0
$$871$$ 23.0213 0.780047
$$872$$ 0 0
$$873$$ −23.4164 −0.792525
$$874$$ 0 0
$$875$$ 8.23607 0.278430
$$876$$ 0 0
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ 0 0
$$879$$ 9.65248 0.325570
$$880$$ 0 0
$$881$$ −5.88854 −0.198390 −0.0991950 0.995068i $$-0.531627\pi$$
−0.0991950 + 0.995068i $$0.531627\pi$$
$$882$$ 0 0
$$883$$ −46.1246 −1.55222 −0.776108 0.630600i $$-0.782809\pi$$
−0.776108 + 0.630600i $$0.782809\pi$$
$$884$$ 0 0
$$885$$ 22.5623 0.758424
$$886$$ 0 0
$$887$$ 58.1935 1.95395 0.976973 0.213362i $$-0.0684414\pi$$
0.976973 + 0.213362i $$0.0684414\pi$$
$$888$$ 0 0
$$889$$ 18.2705 0.612773
$$890$$ 0 0
$$891$$ 0.381966 0.0127963
$$892$$ 0 0
$$893$$ 6.05573 0.202647
$$894$$ 0 0
$$895$$ −3.00000 −0.100279
$$896$$ 0 0
$$897$$ 8.29180 0.276855
$$898$$ 0 0
$$899$$ 5.12461 0.170915
$$900$$ 0 0
$$901$$ 34.1246 1.13686
$$902$$ 0 0
$$903$$ −4.70820 −0.156679
$$904$$ 0 0
$$905$$ 7.47214 0.248382
$$906$$ 0 0
$$907$$ −33.1246 −1.09988 −0.549942 0.835203i $$-0.685350\pi$$
−0.549942 + 0.835203i $$0.685350\pi$$
$$908$$ 0 0
$$909$$ −26.1803 −0.868347
$$910$$ 0 0
$$911$$ −19.0344 −0.630639 −0.315320 0.948986i $$-0.602112\pi$$
−0.315320 + 0.948986i $$0.602112\pi$$
$$912$$ 0 0
$$913$$ −3.56231 −0.117895
$$914$$ 0 0
$$915$$ −28.4164 −0.939417
$$916$$ 0 0
$$917$$ −2.23607 −0.0738415
$$918$$ 0 0
$$919$$ −18.9787 −0.626050 −0.313025 0.949745i $$-0.601342\pi$$
−0.313025 + 0.949745i $$0.601342\pi$$
$$920$$ 0 0
$$921$$ −3.85410 −0.126997
$$922$$ 0 0
$$923$$ −13.1459 −0.432703
$$924$$ 0 0
$$925$$ −12.4377 −0.408949
$$926$$ 0 0
$$927$$ −2.00000 −0.0656886
$$928$$ 0 0
$$929$$ −2.94427 −0.0965984 −0.0482992 0.998833i $$-0.515380\pi$$
−0.0482992 + 0.998833i $$0.515380\pi$$
$$930$$ 0 0
$$931$$ −5.12461 −0.167952
$$932$$ 0 0
$$933$$ 32.8885 1.07672
$$934$$ 0 0
$$935$$ 3.38197 0.110602
$$936$$ 0 0
$$937$$ −11.0000 −0.359354 −0.179677 0.983726i $$-0.557505\pi$$
−0.179677 + 0.983726i $$0.557505\pi$$
$$938$$ 0 0
$$939$$ 29.7082 0.969491
$$940$$ 0 0
$$941$$ 50.3951 1.64283 0.821417 0.570328i $$-0.193184\pi$$
0.821417 + 0.570328i $$0.193184\pi$$
$$942$$ 0 0
$$943$$ −40.0000 −1.30258
$$944$$ 0 0
$$945$$ 13.0902 0.425823
$$946$$ 0 0
$$947$$ −35.0132 −1.13777 −0.568887 0.822415i $$-0.692626\pi$$
−0.568887 + 0.822415i $$0.692626\pi$$
$$948$$ 0 0
$$949$$ −7.68692 −0.249528
$$950$$ 0 0
$$951$$ 1.58359 0.0513515
$$952$$ 0 0
$$953$$ −31.3607 −1.01587 −0.507936 0.861395i $$-0.669591\pi$$
−0.507936 + 0.861395i $$0.669591\pi$$
$$954$$ 0 0
$$955$$ 8.85410 0.286512
$$956$$ 0 0
$$957$$ −0.291796 −0.00943243
$$958$$ 0 0
$$959$$ 0.708204 0.0228691
$$960$$ 0 0
$$961$$ 14.0000 0.451613
$$962$$ 0 0
$$963$$ 8.18034 0.263608
$$964$$ 0 0
$$965$$ 52.6869 1.69605
$$966$$ 0 0
$$967$$ −12.4164 −0.399285 −0.199642 0.979869i $$-0.563978\pi$$
−0.199642 + 0.979869i $$0.563978\pi$$
$$968$$ 0 0
$$969$$ −2.88854 −0.0927934
$$970$$ 0 0
$$971$$ −23.0132 −0.738527 −0.369264 0.929325i $$-0.620390\pi$$
−0.369264 + 0.929325i $$0.620390\pi$$
$$972$$ 0 0
$$973$$ 17.8541 0.572376
$$974$$ 0 0
$$975$$ 3.43769 0.110094
$$976$$ 0 0
$$977$$ 4.25735 0.136205 0.0681024 0.997678i $$-0.478306\pi$$
0.0681024 + 0.997678i $$0.478306\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ −21.1246 −0.674457
$$982$$ 0 0
$$983$$ 12.6525 0.403551 0.201776 0.979432i $$-0.435329\pi$$
0.201776 + 0.979432i $$0.435329\pi$$
$$984$$ 0 0
$$985$$ −27.2705 −0.868911
$$986$$ 0 0
$$987$$ 7.09017 0.225683
$$988$$ 0 0
$$989$$ −21.0557 −0.669533
$$990$$ 0 0
$$991$$ 9.27051 0.294487 0.147244 0.989100i $$-0.452960\pi$$
0.147244 + 0.989100i $$0.452960\pi$$
$$992$$ 0 0
$$993$$ 22.8541 0.725253
$$994$$ 0 0
$$995$$ 8.94427 0.283552
$$996$$ 0 0
$$997$$ −52.7082 −1.66929 −0.834643 0.550792i $$-0.814326\pi$$
−0.834643 + 0.550792i $$0.814326\pi$$
$$998$$ 0 0
$$999$$ 33.5410 1.06119
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 1648.2.a.f.1.1 2
4.3 odd 2 103.2.a.a.1.2 2
8.3 odd 2 6592.2.a.t.1.2 2
8.5 even 2 6592.2.a.h.1.2 2
12.11 even 2 927.2.a.b.1.1 2
20.19 odd 2 2575.2.a.g.1.1 2
28.27 even 2 5047.2.a.a.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.2 2 4.3 odd 2
927.2.a.b.1.1 2 12.11 even 2
1648.2.a.f.1.1 2 1.1 even 1 trivial
2575.2.a.g.1.1 2 20.19 odd 2
5047.2.a.a.1.2 2 28.27 even 2
6592.2.a.h.1.2 2 8.5 even 2
6592.2.a.t.1.2 2 8.3 odd 2