# Properties

 Label 532.2.a.c.1.2 Level $532$ Weight $2$ Character 532.1 Self dual yes Analytic conductor $4.248$ Analytic rank $0$ 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] = [532,2,Mod(1,532)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(532, 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("532.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$532 = 2^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 532.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: $$4.24804138753$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 532.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.79129 q^{3} +3.00000 q^{5} +1.00000 q^{7} +0.208712 q^{9} +O(q^{10})$$ $$q+1.79129 q^{3} +3.00000 q^{5} +1.00000 q^{7} +0.208712 q^{9} +0.791288 q^{11} -1.00000 q^{13} +5.37386 q^{15} -0.791288 q^{17} +1.00000 q^{19} +1.79129 q^{21} -4.58258 q^{23} +4.00000 q^{25} -5.00000 q^{27} -0.791288 q^{29} -6.37386 q^{31} +1.41742 q^{33} +3.00000 q^{35} +5.00000 q^{37} -1.79129 q^{39} +0.791288 q^{41} +2.00000 q^{43} +0.626136 q^{45} +1.41742 q^{47} +1.00000 q^{49} -1.41742 q^{51} +5.37386 q^{53} +2.37386 q^{55} +1.79129 q^{57} -6.16515 q^{59} -1.00000 q^{61} +0.208712 q^{63} -3.00000 q^{65} +4.37386 q^{67} -8.20871 q^{69} +6.16515 q^{71} +2.62614 q^{73} +7.16515 q^{75} +0.791288 q^{77} -10.0000 q^{79} -9.58258 q^{81} +0.626136 q^{83} -2.37386 q^{85} -1.41742 q^{87} -1.58258 q^{89} -1.00000 q^{91} -11.4174 q^{93} +3.00000 q^{95} -7.00000 q^{97} +0.165151 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 6 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q - q^3 + 6 * q^5 + 2 * q^7 + 5 * q^9 $$2 q - q^{3} + 6 q^{5} + 2 q^{7} + 5 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{15} + 3 q^{17} + 2 q^{19} - q^{21} + 8 q^{25} - 10 q^{27} + 3 q^{29} + q^{31} + 12 q^{33} + 6 q^{35} + 10 q^{37} + q^{39} - 3 q^{41} + 4 q^{43} + 15 q^{45} + 12 q^{47} + 2 q^{49} - 12 q^{51} - 3 q^{53} - 9 q^{55} - q^{57} + 6 q^{59} - 2 q^{61} + 5 q^{63} - 6 q^{65} - 5 q^{67} - 21 q^{69} - 6 q^{71} + 19 q^{73} - 4 q^{75} - 3 q^{77} - 20 q^{79} - 10 q^{81} + 15 q^{83} + 9 q^{85} - 12 q^{87} + 6 q^{89} - 2 q^{91} - 32 q^{93} + 6 q^{95} - 14 q^{97} - 18 q^{99}+O(q^{100})$$ 2 * q - q^3 + 6 * q^5 + 2 * q^7 + 5 * q^9 - 3 * q^11 - 2 * q^13 - 3 * q^15 + 3 * q^17 + 2 * q^19 - q^21 + 8 * q^25 - 10 * q^27 + 3 * q^29 + q^31 + 12 * q^33 + 6 * q^35 + 10 * q^37 + q^39 - 3 * q^41 + 4 * q^43 + 15 * q^45 + 12 * q^47 + 2 * q^49 - 12 * q^51 - 3 * q^53 - 9 * q^55 - q^57 + 6 * q^59 - 2 * q^61 + 5 * q^63 - 6 * q^65 - 5 * q^67 - 21 * q^69 - 6 * q^71 + 19 * q^73 - 4 * q^75 - 3 * q^77 - 20 * q^79 - 10 * q^81 + 15 * q^83 + 9 * q^85 - 12 * q^87 + 6 * q^89 - 2 * q^91 - 32 * q^93 + 6 * q^95 - 14 * q^97 - 18 * 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
$$3$$ 1.79129 1.03420 0.517100 0.855925i $$-0.327011\pi$$
0.517100 + 0.855925i $$0.327011\pi$$
$$4$$ 0 0
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 0.208712 0.0695707
$$10$$ 0 0
$$11$$ 0.791288 0.238582 0.119291 0.992859i $$-0.461938\pi$$
0.119291 + 0.992859i $$0.461938\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 5.37386 1.38753
$$16$$ 0 0
$$17$$ −0.791288 −0.191915 −0.0959577 0.995385i $$-0.530591\pi$$
−0.0959577 + 0.995385i $$0.530591\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 1.79129 0.390891
$$22$$ 0 0
$$23$$ −4.58258 −0.955533 −0.477767 0.878487i $$-0.658554\pi$$
−0.477767 + 0.878487i $$0.658554\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ −0.791288 −0.146938 −0.0734692 0.997297i $$-0.523407\pi$$
−0.0734692 + 0.997297i $$0.523407\pi$$
$$30$$ 0 0
$$31$$ −6.37386 −1.14478 −0.572390 0.819982i $$-0.693984\pi$$
−0.572390 + 0.819982i $$0.693984\pi$$
$$32$$ 0 0
$$33$$ 1.41742 0.246742
$$34$$ 0 0
$$35$$ 3.00000 0.507093
$$36$$ 0 0
$$37$$ 5.00000 0.821995 0.410997 0.911636i $$-0.365181\pi$$
0.410997 + 0.911636i $$0.365181\pi$$
$$38$$ 0 0
$$39$$ −1.79129 −0.286836
$$40$$ 0 0
$$41$$ 0.791288 0.123578 0.0617892 0.998089i $$-0.480319\pi$$
0.0617892 + 0.998089i $$0.480319\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 0 0
$$45$$ 0.626136 0.0933389
$$46$$ 0 0
$$47$$ 1.41742 0.206753 0.103376 0.994642i $$-0.467035\pi$$
0.103376 + 0.994642i $$0.467035\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −1.41742 −0.198479
$$52$$ 0 0
$$53$$ 5.37386 0.738157 0.369078 0.929398i $$-0.379673\pi$$
0.369078 + 0.929398i $$0.379673\pi$$
$$54$$ 0 0
$$55$$ 2.37386 0.320092
$$56$$ 0 0
$$57$$ 1.79129 0.237262
$$58$$ 0 0
$$59$$ −6.16515 −0.802634 −0.401317 0.915939i $$-0.631447\pi$$
−0.401317 + 0.915939i $$0.631447\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 0 0
$$63$$ 0.208712 0.0262953
$$64$$ 0 0
$$65$$ −3.00000 −0.372104
$$66$$ 0 0
$$67$$ 4.37386 0.534352 0.267176 0.963648i $$-0.413909\pi$$
0.267176 + 0.963648i $$0.413909\pi$$
$$68$$ 0 0
$$69$$ −8.20871 −0.988213
$$70$$ 0 0
$$71$$ 6.16515 0.731669 0.365834 0.930680i $$-0.380784\pi$$
0.365834 + 0.930680i $$0.380784\pi$$
$$72$$ 0 0
$$73$$ 2.62614 0.307366 0.153683 0.988120i $$-0.450887\pi$$
0.153683 + 0.988120i $$0.450887\pi$$
$$74$$ 0 0
$$75$$ 7.16515 0.827360
$$76$$ 0 0
$$77$$ 0.791288 0.0901756
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ −9.58258 −1.06473
$$82$$ 0 0
$$83$$ 0.626136 0.0687274 0.0343637 0.999409i $$-0.489060\pi$$
0.0343637 + 0.999409i $$0.489060\pi$$
$$84$$ 0 0
$$85$$ −2.37386 −0.257482
$$86$$ 0 0
$$87$$ −1.41742 −0.151964
$$88$$ 0 0
$$89$$ −1.58258 −0.167753 −0.0838763 0.996476i $$-0.526730\pi$$
−0.0838763 + 0.996476i $$0.526730\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ −11.4174 −1.18393
$$94$$ 0 0
$$95$$ 3.00000 0.307794
$$96$$ 0 0
$$97$$ −7.00000 −0.710742 −0.355371 0.934725i $$-0.615646\pi$$
−0.355371 + 0.934725i $$0.615646\pi$$
$$98$$ 0 0
$$99$$ 0.165151 0.0165983
$$100$$ 0 0
$$101$$ 19.7477 1.96497 0.982486 0.186336i $$-0.0596612\pi$$
0.982486 + 0.186336i $$0.0596612\pi$$
$$102$$ 0 0
$$103$$ −13.0000 −1.28093 −0.640464 0.767988i $$-0.721258\pi$$
−0.640464 + 0.767988i $$0.721258\pi$$
$$104$$ 0 0
$$105$$ 5.37386 0.524435
$$106$$ 0 0
$$107$$ −18.1652 −1.75609 −0.878046 0.478577i $$-0.841153\pi$$
−0.878046 + 0.478577i $$0.841153\pi$$
$$108$$ 0 0
$$109$$ 15.7477 1.50836 0.754179 0.656668i $$-0.228035\pi$$
0.754179 + 0.656668i $$0.228035\pi$$
$$110$$ 0 0
$$111$$ 8.95644 0.850108
$$112$$ 0 0
$$113$$ −5.37386 −0.505531 −0.252765 0.967528i $$-0.581340\pi$$
−0.252765 + 0.967528i $$0.581340\pi$$
$$114$$ 0 0
$$115$$ −13.7477 −1.28198
$$116$$ 0 0
$$117$$ −0.208712 −0.0192954
$$118$$ 0 0
$$119$$ −0.791288 −0.0725372
$$120$$ 0 0
$$121$$ −10.3739 −0.943079
$$122$$ 0 0
$$123$$ 1.41742 0.127805
$$124$$ 0 0
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ −20.7477 −1.84106 −0.920532 0.390668i $$-0.872244\pi$$
−0.920532 + 0.390668i $$0.872244\pi$$
$$128$$ 0 0
$$129$$ 3.58258 0.315428
$$130$$ 0 0
$$131$$ 19.1216 1.67066 0.835331 0.549748i $$-0.185276\pi$$
0.835331 + 0.549748i $$0.185276\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ −15.0000 −1.29099
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −17.7477 −1.50534 −0.752671 0.658396i $$-0.771235\pi$$
−0.752671 + 0.658396i $$0.771235\pi$$
$$140$$ 0 0
$$141$$ 2.53901 0.213824
$$142$$ 0 0
$$143$$ −0.791288 −0.0661708
$$144$$ 0 0
$$145$$ −2.37386 −0.197139
$$146$$ 0 0
$$147$$ 1.79129 0.147743
$$148$$ 0 0
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ 0 0
$$151$$ −0.373864 −0.0304246 −0.0152123 0.999884i $$-0.504842\pi$$
−0.0152123 + 0.999884i $$0.504842\pi$$
$$152$$ 0 0
$$153$$ −0.165151 −0.0133517
$$154$$ 0 0
$$155$$ −19.1216 −1.53588
$$156$$ 0 0
$$157$$ 21.1216 1.68569 0.842843 0.538159i $$-0.180880\pi$$
0.842843 + 0.538159i $$0.180880\pi$$
$$158$$ 0 0
$$159$$ 9.62614 0.763402
$$160$$ 0 0
$$161$$ −4.58258 −0.361158
$$162$$ 0 0
$$163$$ −9.37386 −0.734218 −0.367109 0.930178i $$-0.619652\pi$$
−0.367109 + 0.930178i $$0.619652\pi$$
$$164$$ 0 0
$$165$$ 4.25227 0.331039
$$166$$ 0 0
$$167$$ 22.5826 1.74749 0.873746 0.486382i $$-0.161684\pi$$
0.873746 + 0.486382i $$0.161684\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0.208712 0.0159606
$$172$$ 0 0
$$173$$ −4.74773 −0.360963 −0.180482 0.983578i $$-0.557766\pi$$
−0.180482 + 0.983578i $$0.557766\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ −11.0436 −0.830085
$$178$$ 0 0
$$179$$ −11.2087 −0.837778 −0.418889 0.908037i $$-0.637580\pi$$
−0.418889 + 0.908037i $$0.637580\pi$$
$$180$$ 0 0
$$181$$ 4.37386 0.325107 0.162553 0.986700i $$-0.448027\pi$$
0.162553 + 0.986700i $$0.448027\pi$$
$$182$$ 0 0
$$183$$ −1.79129 −0.132416
$$184$$ 0 0
$$185$$ 15.0000 1.10282
$$186$$ 0 0
$$187$$ −0.626136 −0.0457876
$$188$$ 0 0
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ −8.53901 −0.617861 −0.308931 0.951085i $$-0.599971\pi$$
−0.308931 + 0.951085i $$0.599971\pi$$
$$192$$ 0 0
$$193$$ −6.37386 −0.458801 −0.229400 0.973332i $$-0.573677\pi$$
−0.229400 + 0.973332i $$0.573677\pi$$
$$194$$ 0 0
$$195$$ −5.37386 −0.384830
$$196$$ 0 0
$$197$$ 11.3739 0.810354 0.405177 0.914238i $$-0.367210\pi$$
0.405177 + 0.914238i $$0.367210\pi$$
$$198$$ 0 0
$$199$$ 3.74773 0.265669 0.132835 0.991138i $$-0.457592\pi$$
0.132835 + 0.991138i $$0.457592\pi$$
$$200$$ 0 0
$$201$$ 7.83485 0.552628
$$202$$ 0 0
$$203$$ −0.791288 −0.0555375
$$204$$ 0 0
$$205$$ 2.37386 0.165798
$$206$$ 0 0
$$207$$ −0.956439 −0.0664771
$$208$$ 0 0
$$209$$ 0.791288 0.0547345
$$210$$ 0 0
$$211$$ −10.6261 −0.731533 −0.365767 0.930707i $$-0.619193\pi$$
−0.365767 + 0.930707i $$0.619193\pi$$
$$212$$ 0 0
$$213$$ 11.0436 0.756692
$$214$$ 0 0
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ −6.37386 −0.432686
$$218$$ 0 0
$$219$$ 4.70417 0.317878
$$220$$ 0 0
$$221$$ 0.791288 0.0532278
$$222$$ 0 0
$$223$$ 3.74773 0.250966 0.125483 0.992096i $$-0.459952\pi$$
0.125483 + 0.992096i $$0.459952\pi$$
$$224$$ 0 0
$$225$$ 0.834849 0.0556566
$$226$$ 0 0
$$227$$ 0.791288 0.0525196 0.0262598 0.999655i $$-0.491640\pi$$
0.0262598 + 0.999655i $$0.491640\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 1.41742 0.0932597
$$232$$ 0 0
$$233$$ 13.1216 0.859624 0.429812 0.902918i $$-0.358580\pi$$
0.429812 + 0.902918i $$0.358580\pi$$
$$234$$ 0 0
$$235$$ 4.25227 0.277388
$$236$$ 0 0
$$237$$ −17.9129 −1.16357
$$238$$ 0 0
$$239$$ 19.7477 1.27737 0.638687 0.769467i $$-0.279478\pi$$
0.638687 + 0.769467i $$0.279478\pi$$
$$240$$ 0 0
$$241$$ 18.7477 1.20765 0.603824 0.797118i $$-0.293643\pi$$
0.603824 + 0.797118i $$0.293643\pi$$
$$242$$ 0 0
$$243$$ −2.16515 −0.138895
$$244$$ 0 0
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ −1.00000 −0.0636285
$$248$$ 0 0
$$249$$ 1.12159 0.0710779
$$250$$ 0 0
$$251$$ 5.20871 0.328771 0.164385 0.986396i $$-0.447436\pi$$
0.164385 + 0.986396i $$0.447436\pi$$
$$252$$ 0 0
$$253$$ −3.62614 −0.227973
$$254$$ 0 0
$$255$$ −4.25227 −0.266288
$$256$$ 0 0
$$257$$ 15.7913 0.985033 0.492517 0.870303i $$-0.336077\pi$$
0.492517 + 0.870303i $$0.336077\pi$$
$$258$$ 0 0
$$259$$ 5.00000 0.310685
$$260$$ 0 0
$$261$$ −0.165151 −0.0102226
$$262$$ 0 0
$$263$$ 2.20871 0.136195 0.0680975 0.997679i $$-0.478307\pi$$
0.0680975 + 0.997679i $$0.478307\pi$$
$$264$$ 0 0
$$265$$ 16.1216 0.990341
$$266$$ 0 0
$$267$$ −2.83485 −0.173490
$$268$$ 0 0
$$269$$ 12.9564 0.789968 0.394984 0.918688i $$-0.370750\pi$$
0.394984 + 0.918688i $$0.370750\pi$$
$$270$$ 0 0
$$271$$ 27.1216 1.64752 0.823760 0.566939i $$-0.191873\pi$$
0.823760 + 0.566939i $$0.191873\pi$$
$$272$$ 0 0
$$273$$ −1.79129 −0.108414
$$274$$ 0 0
$$275$$ 3.16515 0.190866
$$276$$ 0 0
$$277$$ 30.7477 1.84745 0.923726 0.383054i $$-0.125128\pi$$
0.923726 + 0.383054i $$0.125128\pi$$
$$278$$ 0 0
$$279$$ −1.33030 −0.0796431
$$280$$ 0 0
$$281$$ 28.5826 1.70509 0.852547 0.522651i $$-0.175057\pi$$
0.852547 + 0.522651i $$0.175057\pi$$
$$282$$ 0 0
$$283$$ 7.37386 0.438331 0.219165 0.975688i $$-0.429667\pi$$
0.219165 + 0.975688i $$0.429667\pi$$
$$284$$ 0 0
$$285$$ 5.37386 0.318320
$$286$$ 0 0
$$287$$ 0.791288 0.0467082
$$288$$ 0 0
$$289$$ −16.3739 −0.963168
$$290$$ 0 0
$$291$$ −12.5390 −0.735050
$$292$$ 0 0
$$293$$ −7.74773 −0.452627 −0.226314 0.974055i $$-0.572667\pi$$
−0.226314 + 0.974055i $$0.572667\pi$$
$$294$$ 0 0
$$295$$ −18.4955 −1.07685
$$296$$ 0 0
$$297$$ −3.95644 −0.229576
$$298$$ 0 0
$$299$$ 4.58258 0.265017
$$300$$ 0 0
$$301$$ 2.00000 0.115278
$$302$$ 0 0
$$303$$ 35.3739 2.03218
$$304$$ 0 0
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ −13.6261 −0.777685 −0.388842 0.921304i $$-0.627125\pi$$
−0.388842 + 0.921304i $$0.627125\pi$$
$$308$$ 0 0
$$309$$ −23.2867 −1.32474
$$310$$ 0 0
$$311$$ 30.7913 1.74601 0.873007 0.487708i $$-0.162167\pi$$
0.873007 + 0.487708i $$0.162167\pi$$
$$312$$ 0 0
$$313$$ −14.7477 −0.833591 −0.416795 0.909000i $$-0.636847\pi$$
−0.416795 + 0.909000i $$0.636847\pi$$
$$314$$ 0 0
$$315$$ 0.626136 0.0352788
$$316$$ 0 0
$$317$$ 33.1652 1.86274 0.931370 0.364073i $$-0.118614\pi$$
0.931370 + 0.364073i $$0.118614\pi$$
$$318$$ 0 0
$$319$$ −0.626136 −0.0350569
$$320$$ 0 0
$$321$$ −32.5390 −1.81615
$$322$$ 0 0
$$323$$ −0.791288 −0.0440284
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 0 0
$$327$$ 28.2087 1.55995
$$328$$ 0 0
$$329$$ 1.41742 0.0781451
$$330$$ 0 0
$$331$$ 14.6261 0.803925 0.401963 0.915656i $$-0.368328\pi$$
0.401963 + 0.915656i $$0.368328\pi$$
$$332$$ 0 0
$$333$$ 1.04356 0.0571868
$$334$$ 0 0
$$335$$ 13.1216 0.716909
$$336$$ 0 0
$$337$$ −12.3739 −0.674047 −0.337024 0.941496i $$-0.609420\pi$$
−0.337024 + 0.941496i $$0.609420\pi$$
$$338$$ 0 0
$$339$$ −9.62614 −0.522820
$$340$$ 0 0
$$341$$ −5.04356 −0.273124
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −24.6261 −1.32583
$$346$$ 0 0
$$347$$ −3.79129 −0.203527 −0.101763 0.994809i $$-0.532448\pi$$
−0.101763 + 0.994809i $$0.532448\pi$$
$$348$$ 0 0
$$349$$ −15.3739 −0.822944 −0.411472 0.911422i $$-0.634985\pi$$
−0.411472 + 0.911422i $$0.634985\pi$$
$$350$$ 0 0
$$351$$ 5.00000 0.266880
$$352$$ 0 0
$$353$$ 23.5390 1.25286 0.626428 0.779480i $$-0.284516\pi$$
0.626428 + 0.779480i $$0.284516\pi$$
$$354$$ 0 0
$$355$$ 18.4955 0.981637
$$356$$ 0 0
$$357$$ −1.41742 −0.0750180
$$358$$ 0 0
$$359$$ −14.3739 −0.758624 −0.379312 0.925269i $$-0.623839\pi$$
−0.379312 + 0.925269i $$0.623839\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −18.5826 −0.975332
$$364$$ 0 0
$$365$$ 7.87841 0.412375
$$366$$ 0 0
$$367$$ 32.4955 1.69625 0.848124 0.529797i $$-0.177732\pi$$
0.848124 + 0.529797i $$0.177732\pi$$
$$368$$ 0 0
$$369$$ 0.165151 0.00859744
$$370$$ 0 0
$$371$$ 5.37386 0.278997
$$372$$ 0 0
$$373$$ 2.62614 0.135976 0.0679881 0.997686i $$-0.478342\pi$$
0.0679881 + 0.997686i $$0.478342\pi$$
$$374$$ 0 0
$$375$$ −5.37386 −0.277505
$$376$$ 0 0
$$377$$ 0.791288 0.0407534
$$378$$ 0 0
$$379$$ −5.74773 −0.295241 −0.147620 0.989044i $$-0.547161\pi$$
−0.147620 + 0.989044i $$0.547161\pi$$
$$380$$ 0 0
$$381$$ −37.1652 −1.90403
$$382$$ 0 0
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ 0 0
$$385$$ 2.37386 0.120983
$$386$$ 0 0
$$387$$ 0.417424 0.0212189
$$388$$ 0 0
$$389$$ −32.3739 −1.64142 −0.820710 0.571345i $$-0.806422\pi$$
−0.820710 + 0.571345i $$0.806422\pi$$
$$390$$ 0 0
$$391$$ 3.62614 0.183382
$$392$$ 0 0
$$393$$ 34.2523 1.72780
$$394$$ 0 0
$$395$$ −30.0000 −1.50946
$$396$$ 0 0
$$397$$ −32.7477 −1.64356 −0.821781 0.569804i $$-0.807019\pi$$
−0.821781 + 0.569804i $$0.807019\pi$$
$$398$$ 0 0
$$399$$ 1.79129 0.0896766
$$400$$ 0 0
$$401$$ −26.3739 −1.31705 −0.658524 0.752560i $$-0.728819\pi$$
−0.658524 + 0.752560i $$0.728819\pi$$
$$402$$ 0 0
$$403$$ 6.37386 0.317505
$$404$$ 0 0
$$405$$ −28.7477 −1.42849
$$406$$ 0 0
$$407$$ 3.95644 0.196113
$$408$$ 0 0
$$409$$ −14.1216 −0.698268 −0.349134 0.937073i $$-0.613524\pi$$
−0.349134 + 0.937073i $$0.613524\pi$$
$$410$$ 0 0
$$411$$ 10.7477 0.530146
$$412$$ 0 0
$$413$$ −6.16515 −0.303367
$$414$$ 0 0
$$415$$ 1.87841 0.0922075
$$416$$ 0 0
$$417$$ −31.7913 −1.55683
$$418$$ 0 0
$$419$$ 1.74773 0.0853821 0.0426910 0.999088i $$-0.486407\pi$$
0.0426910 + 0.999088i $$0.486407\pi$$
$$420$$ 0 0
$$421$$ −19.0000 −0.926003 −0.463002 0.886357i $$-0.653228\pi$$
−0.463002 + 0.886357i $$0.653228\pi$$
$$422$$ 0 0
$$423$$ 0.295834 0.0143839
$$424$$ 0 0
$$425$$ −3.16515 −0.153532
$$426$$ 0 0
$$427$$ −1.00000 −0.0483934
$$428$$ 0 0
$$429$$ −1.41742 −0.0684339
$$430$$ 0 0
$$431$$ −6.00000 −0.289010 −0.144505 0.989504i $$-0.546159\pi$$
−0.144505 + 0.989504i $$0.546159\pi$$
$$432$$ 0 0
$$433$$ −17.7477 −0.852901 −0.426451 0.904511i $$-0.640236\pi$$
−0.426451 + 0.904511i $$0.640236\pi$$
$$434$$ 0 0
$$435$$ −4.25227 −0.203881
$$436$$ 0 0
$$437$$ −4.58258 −0.219214
$$438$$ 0 0
$$439$$ −13.4955 −0.644103 −0.322051 0.946722i $$-0.604372\pi$$
−0.322051 + 0.946722i $$0.604372\pi$$
$$440$$ 0 0
$$441$$ 0.208712 0.00993867
$$442$$ 0 0
$$443$$ −36.9564 −1.75585 −0.877927 0.478795i $$-0.841074\pi$$
−0.877927 + 0.478795i $$0.841074\pi$$
$$444$$ 0 0
$$445$$ −4.74773 −0.225064
$$446$$ 0 0
$$447$$ 5.37386 0.254175
$$448$$ 0 0
$$449$$ −16.1216 −0.760825 −0.380412 0.924817i $$-0.624218\pi$$
−0.380412 + 0.924817i $$0.624218\pi$$
$$450$$ 0 0
$$451$$ 0.626136 0.0294836
$$452$$ 0 0
$$453$$ −0.669697 −0.0314651
$$454$$ 0 0
$$455$$ −3.00000 −0.140642
$$456$$ 0 0
$$457$$ 28.8693 1.35045 0.675225 0.737612i $$-0.264047\pi$$
0.675225 + 0.737612i $$0.264047\pi$$
$$458$$ 0 0
$$459$$ 3.95644 0.184671
$$460$$ 0 0
$$461$$ 20.5390 0.956597 0.478299 0.878197i $$-0.341254\pi$$
0.478299 + 0.878197i $$0.341254\pi$$
$$462$$ 0 0
$$463$$ 20.4955 0.952505 0.476252 0.879309i $$-0.341995\pi$$
0.476252 + 0.879309i $$0.341995\pi$$
$$464$$ 0 0
$$465$$ −34.2523 −1.58841
$$466$$ 0 0
$$467$$ −25.2867 −1.17013 −0.585065 0.810986i $$-0.698931\pi$$
−0.585065 + 0.810986i $$0.698931\pi$$
$$468$$ 0 0
$$469$$ 4.37386 0.201966
$$470$$ 0 0
$$471$$ 37.8348 1.74334
$$472$$ 0 0
$$473$$ 1.58258 0.0727669
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 1.12159 0.0513541
$$478$$ 0 0
$$479$$ 26.3739 1.20505 0.602526 0.798099i $$-0.294161\pi$$
0.602526 + 0.798099i $$0.294161\pi$$
$$480$$ 0 0
$$481$$ −5.00000 −0.227980
$$482$$ 0 0
$$483$$ −8.20871 −0.373509
$$484$$ 0 0
$$485$$ −21.0000 −0.953561
$$486$$ 0 0
$$487$$ 11.0000 0.498458 0.249229 0.968445i $$-0.419823\pi$$
0.249229 + 0.968445i $$0.419823\pi$$
$$488$$ 0 0
$$489$$ −16.7913 −0.759328
$$490$$ 0 0
$$491$$ 3.16515 0.142841 0.0714206 0.997446i $$-0.477247\pi$$
0.0714206 + 0.997446i $$0.477247\pi$$
$$492$$ 0 0
$$493$$ 0.626136 0.0281998
$$494$$ 0 0
$$495$$ 0.495454 0.0222690
$$496$$ 0 0
$$497$$ 6.16515 0.276545
$$498$$ 0 0
$$499$$ 31.3739 1.40449 0.702244 0.711937i $$-0.252182\pi$$
0.702244 + 0.711937i $$0.252182\pi$$
$$500$$ 0 0
$$501$$ 40.4519 1.80726
$$502$$ 0 0
$$503$$ 30.3303 1.35236 0.676181 0.736736i $$-0.263634\pi$$
0.676181 + 0.736736i $$0.263634\pi$$
$$504$$ 0 0
$$505$$ 59.2432 2.63629
$$506$$ 0 0
$$507$$ −21.4955 −0.954647
$$508$$ 0 0
$$509$$ −36.3303 −1.61031 −0.805156 0.593063i $$-0.797919\pi$$
−0.805156 + 0.593063i $$0.797919\pi$$
$$510$$ 0 0
$$511$$ 2.62614 0.116173
$$512$$ 0 0
$$513$$ −5.00000 −0.220755
$$514$$ 0 0
$$515$$ −39.0000 −1.71855
$$516$$ 0 0
$$517$$ 1.12159 0.0493275
$$518$$ 0 0
$$519$$ −8.50455 −0.373308
$$520$$ 0 0
$$521$$ −22.5826 −0.989361 −0.494680 0.869075i $$-0.664715\pi$$
−0.494680 + 0.869075i $$0.664715\pi$$
$$522$$ 0 0
$$523$$ 27.7477 1.21332 0.606662 0.794960i $$-0.292508\pi$$
0.606662 + 0.794960i $$0.292508\pi$$
$$524$$ 0 0
$$525$$ 7.16515 0.312713
$$526$$ 0 0
$$527$$ 5.04356 0.219701
$$528$$ 0 0
$$529$$ −2.00000 −0.0869565
$$530$$ 0 0
$$531$$ −1.28674 −0.0558398
$$532$$ 0 0
$$533$$ −0.791288 −0.0342745
$$534$$ 0 0
$$535$$ −54.4955 −2.35604
$$536$$ 0 0
$$537$$ −20.0780 −0.866431
$$538$$ 0 0
$$539$$ 0.791288 0.0340832
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 0 0
$$543$$ 7.83485 0.336226
$$544$$ 0 0
$$545$$ 47.2432 2.02368
$$546$$ 0 0
$$547$$ 1.37386 0.0587422 0.0293711 0.999569i $$-0.490650\pi$$
0.0293711 + 0.999569i $$0.490650\pi$$
$$548$$ 0 0
$$549$$ −0.208712 −0.00890762
$$550$$ 0 0
$$551$$ −0.791288 −0.0337100
$$552$$ 0 0
$$553$$ −10.0000 −0.425243
$$554$$ 0 0
$$555$$ 26.8693 1.14054
$$556$$ 0 0
$$557$$ 14.3739 0.609040 0.304520 0.952506i $$-0.401504\pi$$
0.304520 + 0.952506i $$0.401504\pi$$
$$558$$ 0 0
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ −1.12159 −0.0473536
$$562$$ 0 0
$$563$$ 6.16515 0.259830 0.129915 0.991525i $$-0.458530\pi$$
0.129915 + 0.991525i $$0.458530\pi$$
$$564$$ 0 0
$$565$$ −16.1216 −0.678240
$$566$$ 0 0
$$567$$ −9.58258 −0.402430
$$568$$ 0 0
$$569$$ 11.8348 0.496143 0.248071 0.968742i $$-0.420203\pi$$
0.248071 + 0.968742i $$0.420203\pi$$
$$570$$ 0 0
$$571$$ −22.4955 −0.941405 −0.470703 0.882292i $$-0.656000\pi$$
−0.470703 + 0.882292i $$0.656000\pi$$
$$572$$ 0 0
$$573$$ −15.2958 −0.638993
$$574$$ 0 0
$$575$$ −18.3303 −0.764426
$$576$$ 0 0
$$577$$ 18.1216 0.754412 0.377206 0.926129i $$-0.376885\pi$$
0.377206 + 0.926129i $$0.376885\pi$$
$$578$$ 0 0
$$579$$ −11.4174 −0.474492
$$580$$ 0 0
$$581$$ 0.626136 0.0259765
$$582$$ 0 0
$$583$$ 4.25227 0.176111
$$584$$ 0 0
$$585$$ −0.626136 −0.0258876
$$586$$ 0 0
$$587$$ 9.16515 0.378286 0.189143 0.981950i $$-0.439429\pi$$
0.189143 + 0.981950i $$0.439429\pi$$
$$588$$ 0 0
$$589$$ −6.37386 −0.262630
$$590$$ 0 0
$$591$$ 20.3739 0.838069
$$592$$ 0 0
$$593$$ 16.9129 0.694529 0.347264 0.937767i $$-0.387111\pi$$
0.347264 + 0.937767i $$0.387111\pi$$
$$594$$ 0 0
$$595$$ −2.37386 −0.0973189
$$596$$ 0 0
$$597$$ 6.71326 0.274755
$$598$$ 0 0
$$599$$ −6.62614 −0.270737 −0.135368 0.990795i $$-0.543222\pi$$
−0.135368 + 0.990795i $$0.543222\pi$$
$$600$$ 0 0
$$601$$ 25.3739 1.03502 0.517511 0.855677i $$-0.326859\pi$$
0.517511 + 0.855677i $$0.326859\pi$$
$$602$$ 0 0
$$603$$ 0.912878 0.0371753
$$604$$ 0 0
$$605$$ −31.1216 −1.26527
$$606$$ 0 0
$$607$$ −28.4955 −1.15659 −0.578297 0.815826i $$-0.696283\pi$$
−0.578297 + 0.815826i $$0.696283\pi$$
$$608$$ 0 0
$$609$$ −1.41742 −0.0574369
$$610$$ 0 0
$$611$$ −1.41742 −0.0573428
$$612$$ 0 0
$$613$$ 20.6261 0.833082 0.416541 0.909117i $$-0.363242\pi$$
0.416541 + 0.909117i $$0.363242\pi$$
$$614$$ 0 0
$$615$$ 4.25227 0.171468
$$616$$ 0 0
$$617$$ 15.9564 0.642382 0.321191 0.947014i $$-0.395917\pi$$
0.321191 + 0.947014i $$0.395917\pi$$
$$618$$ 0 0
$$619$$ −9.37386 −0.376767 −0.188384 0.982096i $$-0.560325\pi$$
−0.188384 + 0.982096i $$0.560325\pi$$
$$620$$ 0 0
$$621$$ 22.9129 0.919462
$$622$$ 0 0
$$623$$ −1.58258 −0.0634046
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 1.41742 0.0566065
$$628$$ 0 0
$$629$$ −3.95644 −0.157754
$$630$$ 0 0
$$631$$ −25.0000 −0.995234 −0.497617 0.867397i $$-0.665792\pi$$
−0.497617 + 0.867397i $$0.665792\pi$$
$$632$$ 0 0
$$633$$ −19.0345 −0.756552
$$634$$ 0 0
$$635$$ −62.2432 −2.47005
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ 1.28674 0.0509027
$$640$$ 0 0
$$641$$ −24.7913 −0.979197 −0.489598 0.871948i $$-0.662857\pi$$
−0.489598 + 0.871948i $$0.662857\pi$$
$$642$$ 0 0
$$643$$ 23.0000 0.907031 0.453516 0.891248i $$-0.350170\pi$$
0.453516 + 0.891248i $$0.350170\pi$$
$$644$$ 0 0
$$645$$ 10.7477 0.423191
$$646$$ 0 0
$$647$$ −22.5826 −0.887813 −0.443906 0.896073i $$-0.646408\pi$$
−0.443906 + 0.896073i $$0.646408\pi$$
$$648$$ 0 0
$$649$$ −4.87841 −0.191494
$$650$$ 0 0
$$651$$ −11.4174 −0.447484
$$652$$ 0 0
$$653$$ 28.5826 1.11852 0.559261 0.828991i $$-0.311085\pi$$
0.559261 + 0.828991i $$0.311085\pi$$
$$654$$ 0 0
$$655$$ 57.3648 2.24143
$$656$$ 0 0
$$657$$ 0.548107 0.0213837
$$658$$ 0 0
$$659$$ −14.0436 −0.547059 −0.273530 0.961864i $$-0.588191\pi$$
−0.273530 + 0.961864i $$0.588191\pi$$
$$660$$ 0 0
$$661$$ −34.0000 −1.32245 −0.661223 0.750189i $$-0.729962\pi$$
−0.661223 + 0.750189i $$0.729962\pi$$
$$662$$ 0 0
$$663$$ 1.41742 0.0550482
$$664$$ 0 0
$$665$$ 3.00000 0.116335
$$666$$ 0 0
$$667$$ 3.62614 0.140405
$$668$$ 0 0
$$669$$ 6.71326 0.259550
$$670$$ 0 0
$$671$$ −0.791288 −0.0305473
$$672$$ 0 0
$$673$$ 24.1216 0.929819 0.464909 0.885358i $$-0.346087\pi$$
0.464909 + 0.885358i $$0.346087\pi$$
$$674$$ 0 0
$$675$$ −20.0000 −0.769800
$$676$$ 0 0
$$677$$ 0.956439 0.0367589 0.0183795 0.999831i $$-0.494149\pi$$
0.0183795 + 0.999831i $$0.494149\pi$$
$$678$$ 0 0
$$679$$ −7.00000 −0.268635
$$680$$ 0 0
$$681$$ 1.41742 0.0543158
$$682$$ 0 0
$$683$$ −17.8348 −0.682432 −0.341216 0.939985i $$-0.610839\pi$$
−0.341216 + 0.939985i $$0.610839\pi$$
$$684$$ 0 0
$$685$$ 18.0000 0.687745
$$686$$ 0 0
$$687$$ −39.4083 −1.50352
$$688$$ 0 0
$$689$$ −5.37386 −0.204728
$$690$$ 0 0
$$691$$ 21.2523 0.808475 0.404237 0.914654i $$-0.367537\pi$$
0.404237 + 0.914654i $$0.367537\pi$$
$$692$$ 0 0
$$693$$ 0.165151 0.00627358
$$694$$ 0 0
$$695$$ −53.2432 −2.01963
$$696$$ 0 0
$$697$$ −0.626136 −0.0237166
$$698$$ 0 0
$$699$$ 23.5045 0.889024
$$700$$ 0 0
$$701$$ 5.66970 0.214142 0.107071 0.994251i $$-0.465853\pi$$
0.107071 + 0.994251i $$0.465853\pi$$
$$702$$ 0 0
$$703$$ 5.00000 0.188579
$$704$$ 0 0
$$705$$ 7.61704 0.286875
$$706$$ 0 0
$$707$$ 19.7477 0.742690
$$708$$ 0 0
$$709$$ −51.2432 −1.92448 −0.962239 0.272206i $$-0.912247\pi$$
−0.962239 + 0.272206i $$0.912247\pi$$
$$710$$ 0 0
$$711$$ −2.08712 −0.0782732
$$712$$ 0 0
$$713$$ 29.2087 1.09387
$$714$$ 0 0
$$715$$ −2.37386 −0.0887775
$$716$$ 0 0
$$717$$ 35.3739 1.32106
$$718$$ 0 0
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 0 0
$$721$$ −13.0000 −0.484145
$$722$$ 0 0
$$723$$ 33.5826 1.24895
$$724$$ 0 0
$$725$$ −3.16515 −0.117551
$$726$$ 0 0
$$727$$ 11.0000 0.407967 0.203984 0.978974i $$-0.434611\pi$$
0.203984 + 0.978974i $$0.434611\pi$$
$$728$$ 0 0
$$729$$ 24.8693 0.921086
$$730$$ 0 0
$$731$$ −1.58258 −0.0585337
$$732$$ 0 0
$$733$$ −13.4955 −0.498466 −0.249233 0.968444i $$-0.580178\pi$$
−0.249233 + 0.968444i $$0.580178\pi$$
$$734$$ 0 0
$$735$$ 5.37386 0.198218
$$736$$ 0 0
$$737$$ 3.46099 0.127487
$$738$$ 0 0
$$739$$ −8.25227 −0.303565 −0.151782 0.988414i $$-0.548501\pi$$
−0.151782 + 0.988414i $$0.548501\pi$$
$$740$$ 0 0
$$741$$ −1.79129 −0.0658046
$$742$$ 0 0
$$743$$ −15.3303 −0.562414 −0.281207 0.959647i $$-0.590735\pi$$
−0.281207 + 0.959647i $$0.590735\pi$$
$$744$$ 0 0
$$745$$ 9.00000 0.329734
$$746$$ 0 0
$$747$$ 0.130682 0.00478141
$$748$$ 0 0
$$749$$ −18.1652 −0.663740
$$750$$ 0 0
$$751$$ 25.3739 0.925905 0.462953 0.886383i $$-0.346790\pi$$
0.462953 + 0.886383i $$0.346790\pi$$
$$752$$ 0 0
$$753$$ 9.33030 0.340015
$$754$$ 0 0
$$755$$ −1.12159 −0.0408189
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ −6.49545 −0.235770
$$760$$ 0 0
$$761$$ −2.66970 −0.0967764 −0.0483882 0.998829i $$-0.515408\pi$$
−0.0483882 + 0.998829i $$0.515408\pi$$
$$762$$ 0 0
$$763$$ 15.7477 0.570106
$$764$$ 0 0
$$765$$ −0.495454 −0.0179132
$$766$$ 0 0
$$767$$ 6.16515 0.222611
$$768$$ 0 0
$$769$$ 41.4955 1.49636 0.748182 0.663493i $$-0.230927\pi$$
0.748182 + 0.663493i $$0.230927\pi$$
$$770$$ 0 0
$$771$$ 28.2867 1.01872
$$772$$ 0 0
$$773$$ −10.7477 −0.386569 −0.193284 0.981143i $$-0.561914\pi$$
−0.193284 + 0.981143i $$0.561914\pi$$
$$774$$ 0 0
$$775$$ −25.4955 −0.915824
$$776$$ 0 0
$$777$$ 8.95644 0.321310
$$778$$ 0 0
$$779$$ 0.791288 0.0283508
$$780$$ 0 0
$$781$$ 4.87841 0.174563
$$782$$ 0 0
$$783$$ 3.95644 0.141392
$$784$$ 0 0
$$785$$ 63.3648 2.26159
$$786$$ 0 0
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ 0 0
$$789$$ 3.95644 0.140853
$$790$$ 0 0
$$791$$ −5.37386 −0.191073
$$792$$ 0 0
$$793$$ 1.00000 0.0355110
$$794$$ 0 0
$$795$$ 28.8784 1.02421
$$796$$ 0 0
$$797$$ 35.8693 1.27056 0.635278 0.772283i $$-0.280885\pi$$
0.635278 + 0.772283i $$0.280885\pi$$
$$798$$ 0 0
$$799$$ −1.12159 −0.0396790
$$800$$ 0 0
$$801$$ −0.330303 −0.0116707
$$802$$ 0 0
$$803$$ 2.07803 0.0733321
$$804$$ 0 0
$$805$$ −13.7477 −0.484544
$$806$$ 0 0
$$807$$ 23.2087 0.816985
$$808$$ 0 0
$$809$$ 22.9129 0.805574 0.402787 0.915294i $$-0.368042\pi$$
0.402787 + 0.915294i $$0.368042\pi$$
$$810$$ 0 0
$$811$$ 23.4955 0.825037 0.412518 0.910949i $$-0.364649\pi$$
0.412518 + 0.910949i $$0.364649\pi$$
$$812$$ 0 0
$$813$$ 48.5826 1.70387
$$814$$ 0 0
$$815$$ −28.1216 −0.985056
$$816$$ 0 0
$$817$$ 2.00000 0.0699711
$$818$$ 0 0
$$819$$ −0.208712 −0.00729299
$$820$$ 0 0
$$821$$ 5.66970 0.197874 0.0989369 0.995094i $$-0.468456\pi$$
0.0989369 + 0.995094i $$0.468456\pi$$
$$822$$ 0 0
$$823$$ 28.2432 0.984495 0.492248 0.870455i $$-0.336175\pi$$
0.492248 + 0.870455i $$0.336175\pi$$
$$824$$ 0 0
$$825$$ 5.66970 0.197394
$$826$$ 0 0
$$827$$ −33.6606 −1.17049 −0.585247 0.810855i $$-0.699002\pi$$
−0.585247 + 0.810855i $$0.699002\pi$$
$$828$$ 0 0
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ 55.0780 1.91064
$$832$$ 0 0
$$833$$ −0.791288 −0.0274165
$$834$$ 0 0
$$835$$ 67.7477 2.34451
$$836$$ 0 0
$$837$$ 31.8693 1.10156
$$838$$ 0 0
$$839$$ 27.1652 0.937845 0.468923 0.883239i $$-0.344642\pi$$
0.468923 + 0.883239i $$0.344642\pi$$
$$840$$ 0 0
$$841$$ −28.3739 −0.978409
$$842$$ 0 0
$$843$$ 51.1996 1.76341
$$844$$ 0 0
$$845$$ −36.0000 −1.23844
$$846$$ 0 0
$$847$$ −10.3739 −0.356450
$$848$$ 0 0
$$849$$ 13.2087 0.453322
$$850$$ 0 0
$$851$$ −22.9129 −0.785443
$$852$$ 0 0
$$853$$ −32.1216 −1.09982 −0.549911 0.835223i $$-0.685338\pi$$
−0.549911 + 0.835223i $$0.685338\pi$$
$$854$$ 0 0
$$855$$ 0.626136 0.0214134
$$856$$ 0 0
$$857$$ 38.7042 1.32211 0.661055 0.750338i $$-0.270109\pi$$
0.661055 + 0.750338i $$0.270109\pi$$
$$858$$ 0 0
$$859$$ −6.37386 −0.217473 −0.108737 0.994071i $$-0.534681\pi$$
−0.108737 + 0.994071i $$0.534681\pi$$
$$860$$ 0 0
$$861$$ 1.41742 0.0483057
$$862$$ 0 0
$$863$$ 11.7042 0.398414 0.199207 0.979957i $$-0.436163\pi$$
0.199207 + 0.979957i $$0.436163\pi$$
$$864$$ 0 0
$$865$$ −14.2432 −0.484283
$$866$$ 0 0
$$867$$ −29.3303 −0.996109
$$868$$ 0 0
$$869$$ −7.91288 −0.268426
$$870$$ 0 0
$$871$$ −4.37386 −0.148203
$$872$$ 0 0
$$873$$ −1.46099 −0.0494469
$$874$$ 0 0
$$875$$ −3.00000 −0.101419
$$876$$ 0 0
$$877$$ −11.2523 −0.379962 −0.189981 0.981788i $$-0.560843\pi$$
−0.189981 + 0.981788i $$0.560843\pi$$
$$878$$ 0 0
$$879$$ −13.8784 −0.468107
$$880$$ 0 0
$$881$$ −11.3739 −0.383195 −0.191598 0.981474i $$-0.561367\pi$$
−0.191598 + 0.981474i $$0.561367\pi$$
$$882$$ 0 0
$$883$$ 51.7477 1.74145 0.870725 0.491771i $$-0.163650\pi$$
0.870725 + 0.491771i $$0.163650\pi$$
$$884$$ 0 0
$$885$$ −33.1307 −1.11368
$$886$$ 0 0
$$887$$ −4.58258 −0.153868 −0.0769339 0.997036i $$-0.524513\pi$$
−0.0769339 + 0.997036i $$0.524513\pi$$
$$888$$ 0 0
$$889$$ −20.7477 −0.695856
$$890$$ 0 0
$$891$$ −7.58258 −0.254026
$$892$$ 0 0
$$893$$ 1.41742 0.0474323
$$894$$ 0 0
$$895$$ −33.6261 −1.12400
$$896$$ 0 0
$$897$$ 8.20871 0.274081
$$898$$ 0 0
$$899$$ 5.04356 0.168212
$$900$$ 0 0
$$901$$ −4.25227 −0.141664
$$902$$ 0 0
$$903$$ 3.58258 0.119221
$$904$$ 0 0
$$905$$ 13.1216 0.436176
$$906$$ 0 0
$$907$$ −51.2432 −1.70150 −0.850751 0.525569i $$-0.823852\pi$$
−0.850751 + 0.525569i $$0.823852\pi$$
$$908$$ 0 0
$$909$$ 4.12159 0.136705
$$910$$ 0 0
$$911$$ −51.4955 −1.70612 −0.853060 0.521812i $$-0.825256\pi$$
−0.853060 + 0.521812i $$0.825256\pi$$
$$912$$ 0 0
$$913$$ 0.495454 0.0163971
$$914$$ 0 0
$$915$$ −5.37386 −0.177654
$$916$$ 0 0
$$917$$ 19.1216 0.631451
$$918$$ 0 0
$$919$$ 20.4955 0.676083 0.338041 0.941131i $$-0.390236\pi$$
0.338041 + 0.941131i $$0.390236\pi$$
$$920$$ 0 0
$$921$$ −24.4083 −0.804282
$$922$$ 0 0
$$923$$ −6.16515 −0.202928
$$924$$ 0 0
$$925$$ 20.0000 0.657596
$$926$$ 0 0
$$927$$ −2.71326 −0.0891151
$$928$$ 0 0
$$929$$ −20.8693 −0.684700 −0.342350 0.939572i $$-0.611223\pi$$
−0.342350 + 0.939572i $$0.611223\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ 55.1561 1.80573
$$934$$ 0 0
$$935$$ −1.87841 −0.0614306
$$936$$ 0 0
$$937$$ −1.62614 −0.0531236 −0.0265618 0.999647i $$-0.508456\pi$$
−0.0265618 + 0.999647i $$0.508456\pi$$
$$938$$ 0 0
$$939$$ −26.4174 −0.862100
$$940$$ 0 0
$$941$$ −27.4955 −0.896326 −0.448163 0.893952i $$-0.647922\pi$$
−0.448163 + 0.893952i $$0.647922\pi$$
$$942$$ 0 0
$$943$$ −3.62614 −0.118083
$$944$$ 0 0
$$945$$ −15.0000 −0.487950
$$946$$ 0 0
$$947$$ 43.6170 1.41736 0.708682 0.705528i $$-0.249290\pi$$
0.708682 + 0.705528i $$0.249290\pi$$
$$948$$ 0 0
$$949$$ −2.62614 −0.0852480
$$950$$ 0 0
$$951$$ 59.4083 1.92645
$$952$$ 0 0
$$953$$ 53.5390 1.73430 0.867149 0.498048i $$-0.165950\pi$$
0.867149 + 0.498048i $$0.165950\pi$$
$$954$$ 0 0
$$955$$ −25.6170 −0.828948
$$956$$ 0 0
$$957$$ −1.12159 −0.0362559
$$958$$ 0 0
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ 9.62614 0.310521
$$962$$ 0 0
$$963$$ −3.79129 −0.122173
$$964$$ 0 0
$$965$$ −19.1216 −0.615546
$$966$$ 0 0
$$967$$ −29.1216 −0.936487 −0.468244 0.883599i $$-0.655113\pi$$
−0.468244 + 0.883599i $$0.655113\pi$$
$$968$$ 0 0
$$969$$ −1.41742 −0.0455342
$$970$$ 0 0
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 0 0
$$973$$ −17.7477 −0.568966
$$974$$ 0 0
$$975$$ −7.16515 −0.229468
$$976$$ 0 0
$$977$$ −43.5826 −1.39433 −0.697165 0.716911i $$-0.745556\pi$$
−0.697165 + 0.716911i $$0.745556\pi$$
$$978$$ 0 0
$$979$$ −1.25227 −0.0400228
$$980$$ 0 0
$$981$$ 3.28674 0.104938
$$982$$ 0 0
$$983$$ 45.3303 1.44581 0.722906 0.690946i $$-0.242806\pi$$
0.722906 + 0.690946i $$0.242806\pi$$
$$984$$ 0 0
$$985$$ 34.1216 1.08720
$$986$$ 0 0
$$987$$ 2.53901 0.0808177
$$988$$ 0 0
$$989$$ −9.16515 −0.291435
$$990$$ 0 0
$$991$$ 18.7477 0.595541 0.297771 0.954637i $$-0.403757\pi$$
0.297771 + 0.954637i $$0.403757\pi$$
$$992$$ 0 0
$$993$$ 26.1996 0.831420
$$994$$ 0 0
$$995$$ 11.2432 0.356433
$$996$$ 0 0
$$997$$ −44.1216 −1.39734 −0.698672 0.715442i $$-0.746225\pi$$
−0.698672 + 0.715442i $$0.746225\pi$$
$$998$$ 0 0
$$999$$ −25.0000 −0.790965
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 532.2.a.c.1.2 2
3.2 odd 2 4788.2.a.g.1.1 2
4.3 odd 2 2128.2.a.k.1.1 2
7.6 odd 2 3724.2.a.e.1.1 2
8.3 odd 2 8512.2.a.m.1.2 2
8.5 even 2 8512.2.a.t.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
532.2.a.c.1.2 2 1.1 even 1 trivial
2128.2.a.k.1.1 2 4.3 odd 2
3724.2.a.e.1.1 2 7.6 odd 2
4788.2.a.g.1.1 2 3.2 odd 2
8512.2.a.m.1.2 2 8.3 odd 2
8512.2.a.t.1.1 2 8.5 even 2