# Properties

 Label 532.2.a.c.1.1 Level $532$ Weight $2$ Character 532.1 Self dual yes Analytic conductor $4.248$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

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 1.1 Root $$2.79129$$ 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)

 $$f(q)$$ $$=$$ $$q-2.79129 q^{3} +3.00000 q^{5} +1.00000 q^{7} +4.79129 q^{9} +O(q^{10})$$ $$q-2.79129 q^{3} +3.00000 q^{5} +1.00000 q^{7} +4.79129 q^{9} -3.79129 q^{11} -1.00000 q^{13} -8.37386 q^{15} +3.79129 q^{17} +1.00000 q^{19} -2.79129 q^{21} +4.58258 q^{23} +4.00000 q^{25} -5.00000 q^{27} +3.79129 q^{29} +7.37386 q^{31} +10.5826 q^{33} +3.00000 q^{35} +5.00000 q^{37} +2.79129 q^{39} -3.79129 q^{41} +2.00000 q^{43} +14.3739 q^{45} +10.5826 q^{47} +1.00000 q^{49} -10.5826 q^{51} -8.37386 q^{53} -11.3739 q^{55} -2.79129 q^{57} +12.1652 q^{59} -1.00000 q^{61} +4.79129 q^{63} -3.00000 q^{65} -9.37386 q^{67} -12.7913 q^{69} -12.1652 q^{71} +16.3739 q^{73} -11.1652 q^{75} -3.79129 q^{77} -10.0000 q^{79} -0.417424 q^{81} +14.3739 q^{83} +11.3739 q^{85} -10.5826 q^{87} +7.58258 q^{89} -1.00000 q^{91} -20.5826 q^{93} +3.00000 q^{95} -7.00000 q^{97} -18.1652 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 6 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10})$$ 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
$$3$$ −2.79129 −1.61155 −0.805775 0.592221i $$-0.798251\pi$$
−0.805775 + 0.592221i $$0.798251\pi$$
$$4$$ 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
$$9$$ 4.79129 1.59710
$$10$$ 0 0
$$11$$ −3.79129 −1.14312 −0.571558 0.820562i $$-0.693661\pi$$
−0.571558 + 0.820562i $$0.693661\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ −8.37386 −2.16212
$$16$$ 0 0
$$17$$ 3.79129 0.919522 0.459761 0.888043i $$-0.347935\pi$$
0.459761 + 0.888043i $$0.347935\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −2.79129 −0.609109
$$22$$ 0 0
$$23$$ 4.58258 0.955533 0.477767 0.878487i $$-0.341446\pi$$
0.477767 + 0.878487i $$0.341446\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 3.79129 0.704024 0.352012 0.935995i $$-0.385498\pi$$
0.352012 + 0.935995i $$0.385498\pi$$
$$30$$ 0 0
$$31$$ 7.37386 1.32438 0.662192 0.749334i $$-0.269626\pi$$
0.662192 + 0.749334i $$0.269626\pi$$
$$32$$ 0 0
$$33$$ 10.5826 1.84219
$$34$$ 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
$$39$$ 2.79129 0.446964
$$40$$ 0 0
$$41$$ −3.79129 −0.592100 −0.296050 0.955172i $$-0.595669\pi$$
−0.296050 + 0.955172i $$0.595669\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 0 0
$$45$$ 14.3739 2.14273
$$46$$ 0 0
$$47$$ 10.5826 1.54363 0.771814 0.635849i $$-0.219350\pi$$
0.771814 + 0.635849i $$0.219350\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −10.5826 −1.48186
$$52$$ 0 0
$$53$$ −8.37386 −1.15024 −0.575119 0.818070i $$-0.695044\pi$$
−0.575119 + 0.818070i $$0.695044\pi$$
$$54$$ 0 0
$$55$$ −11.3739 −1.53365
$$56$$ 0 0
$$57$$ −2.79129 −0.369715
$$58$$ 0 0
$$59$$ 12.1652 1.58377 0.791884 0.610672i $$-0.209100\pi$$
0.791884 + 0.610672i $$0.209100\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 0 0
$$63$$ 4.79129 0.603646
$$64$$ 0 0
$$65$$ −3.00000 −0.372104
$$66$$ 0 0
$$67$$ −9.37386 −1.14520 −0.572600 0.819835i $$-0.694065\pi$$
−0.572600 + 0.819835i $$0.694065\pi$$
$$68$$ 0 0
$$69$$ −12.7913 −1.53989
$$70$$ 0 0
$$71$$ −12.1652 −1.44374 −0.721869 0.692030i $$-0.756717\pi$$
−0.721869 + 0.692030i $$0.756717\pi$$
$$72$$ 0 0
$$73$$ 16.3739 1.91642 0.958208 0.286073i $$-0.0923499\pi$$
0.958208 + 0.286073i $$0.0923499\pi$$
$$74$$ 0 0
$$75$$ −11.1652 −1.28924
$$76$$ 0 0
$$77$$ −3.79129 −0.432057
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ −0.417424 −0.0463805
$$82$$ 0 0
$$83$$ 14.3739 1.57774 0.788868 0.614562i $$-0.210667\pi$$
0.788868 + 0.614562i $$0.210667\pi$$
$$84$$ 0 0
$$85$$ 11.3739 1.23367
$$86$$ 0 0
$$87$$ −10.5826 −1.13457
$$88$$ 0 0
$$89$$ 7.58258 0.803751 0.401876 0.915694i $$-0.368358\pi$$
0.401876 + 0.915694i $$0.368358\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ −20.5826 −2.13431
$$94$$ 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
$$99$$ −18.1652 −1.82567
$$100$$ 0 0
$$101$$ −7.74773 −0.770928 −0.385464 0.922723i $$-0.625959\pi$$
−0.385464 + 0.922723i $$0.625959\pi$$
$$102$$ 0 0
$$103$$ −13.0000 −1.28093 −0.640464 0.767988i $$-0.721258\pi$$
−0.640464 + 0.767988i $$0.721258\pi$$
$$104$$ 0 0
$$105$$ −8.37386 −0.817205
$$106$$ 0 0
$$107$$ 0.165151 0.0159658 0.00798289 0.999968i $$-0.497459\pi$$
0.00798289 + 0.999968i $$0.497459\pi$$
$$108$$ 0 0
$$109$$ −11.7477 −1.12523 −0.562614 0.826720i $$-0.690204\pi$$
−0.562614 + 0.826720i $$0.690204\pi$$
$$110$$ 0 0
$$111$$ −13.9564 −1.32469
$$112$$ 0 0
$$113$$ 8.37386 0.787747 0.393873 0.919165i $$-0.371135\pi$$
0.393873 + 0.919165i $$0.371135\pi$$
$$114$$ 0 0
$$115$$ 13.7477 1.28198
$$116$$ 0 0
$$117$$ −4.79129 −0.442955
$$118$$ 0 0
$$119$$ 3.79129 0.347547
$$120$$ 0 0
$$121$$ 3.37386 0.306715
$$122$$ 0 0
$$123$$ 10.5826 0.954199
$$124$$ 0 0
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ 6.74773 0.598764 0.299382 0.954133i $$-0.403220\pi$$
0.299382 + 0.954133i $$0.403220\pi$$
$$128$$ 0 0
$$129$$ −5.58258 −0.491518
$$130$$ 0 0
$$131$$ −22.1216 −1.93277 −0.966386 0.257095i $$-0.917235\pi$$
−0.966386 + 0.257095i $$0.917235\pi$$
$$132$$ 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
$$139$$ 9.74773 0.826791 0.413396 0.910551i $$-0.364343\pi$$
0.413396 + 0.910551i $$0.364343\pi$$
$$140$$ 0 0
$$141$$ −29.5390 −2.48763
$$142$$ 0 0
$$143$$ 3.79129 0.317043
$$144$$ 0 0
$$145$$ 11.3739 0.944548
$$146$$ 0 0
$$147$$ −2.79129 −0.230222
$$148$$ 0 0
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ 0 0
$$151$$ 13.3739 1.08835 0.544175 0.838972i $$-0.316843\pi$$
0.544175 + 0.838972i $$0.316843\pi$$
$$152$$ 0 0
$$153$$ 18.1652 1.46857
$$154$$ 0 0
$$155$$ 22.1216 1.77685
$$156$$ 0 0
$$157$$ −20.1216 −1.60588 −0.802939 0.596061i $$-0.796731\pi$$
−0.802939 + 0.596061i $$0.796731\pi$$
$$158$$ 0 0
$$159$$ 23.3739 1.85367
$$160$$ 0 0
$$161$$ 4.58258 0.361158
$$162$$ 0 0
$$163$$ 4.37386 0.342587 0.171294 0.985220i $$-0.445205\pi$$
0.171294 + 0.985220i $$0.445205\pi$$
$$164$$ 0 0
$$165$$ 31.7477 2.47156
$$166$$ 0 0
$$167$$ 13.4174 1.03827 0.519136 0.854692i $$-0.326254\pi$$
0.519136 + 0.854692i $$0.326254\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 4.79129 0.366399
$$172$$ 0 0
$$173$$ 22.7477 1.72948 0.864739 0.502222i $$-0.167484\pi$$
0.864739 + 0.502222i $$0.167484\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ −33.9564 −2.55232
$$178$$ 0 0
$$179$$ −15.7913 −1.18030 −0.590148 0.807295i $$-0.700931\pi$$
−0.590148 + 0.807295i $$0.700931\pi$$
$$180$$ 0 0
$$181$$ −9.37386 −0.696754 −0.348377 0.937355i $$-0.613267\pi$$
−0.348377 + 0.937355i $$0.613267\pi$$
$$182$$ 0 0
$$183$$ 2.79129 0.206338
$$184$$ 0 0
$$185$$ 15.0000 1.10282
$$186$$ 0 0
$$187$$ −14.3739 −1.05112
$$188$$ 0 0
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ 23.5390 1.70322 0.851612 0.524173i $$-0.175626\pi$$
0.851612 + 0.524173i $$0.175626\pi$$
$$192$$ 0 0
$$193$$ 7.37386 0.530782 0.265391 0.964141i $$-0.414499\pi$$
0.265391 + 0.964141i $$0.414499\pi$$
$$194$$ 0 0
$$195$$ 8.37386 0.599665
$$196$$ 0 0
$$197$$ −2.37386 −0.169131 −0.0845654 0.996418i $$-0.526950\pi$$
−0.0845654 + 0.996418i $$0.526950\pi$$
$$198$$ 0 0
$$199$$ −23.7477 −1.68343 −0.841716 0.539921i $$-0.818454\pi$$
−0.841716 + 0.539921i $$0.818454\pi$$
$$200$$ 0 0
$$201$$ 26.1652 1.84555
$$202$$ 0 0
$$203$$ 3.79129 0.266096
$$204$$ 0 0
$$205$$ −11.3739 −0.794385
$$206$$ 0 0
$$207$$ 21.9564 1.52608
$$208$$ 0 0
$$209$$ −3.79129 −0.262249
$$210$$ 0 0
$$211$$ −24.3739 −1.67797 −0.838983 0.544158i $$-0.816849\pi$$
−0.838983 + 0.544158i $$0.816849\pi$$
$$212$$ 0 0
$$213$$ 33.9564 2.32666
$$214$$ 0 0
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ 7.37386 0.500570
$$218$$ 0 0
$$219$$ −45.7042 −3.08840
$$220$$ 0 0
$$221$$ −3.79129 −0.255030
$$222$$ 0 0
$$223$$ −23.7477 −1.59027 −0.795133 0.606435i $$-0.792599\pi$$
−0.795133 + 0.606435i $$0.792599\pi$$
$$224$$ 0 0
$$225$$ 19.1652 1.27768
$$226$$ 0 0
$$227$$ −3.79129 −0.251637 −0.125818 0.992053i $$-0.540156\pi$$
−0.125818 + 0.992053i $$0.540156\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 10.5826 0.696282
$$232$$ 0 0
$$233$$ −28.1216 −1.84231 −0.921153 0.389200i $$-0.872752\pi$$
−0.921153 + 0.389200i $$0.872752\pi$$
$$234$$ 0 0
$$235$$ 31.7477 2.07099
$$236$$ 0 0
$$237$$ 27.9129 1.81314
$$238$$ 0 0
$$239$$ −7.74773 −0.501159 −0.250579 0.968096i $$-0.580621\pi$$
−0.250579 + 0.968096i $$0.580621\pi$$
$$240$$ 0 0
$$241$$ −8.74773 −0.563491 −0.281745 0.959489i $$-0.590913\pi$$
−0.281745 + 0.959489i $$0.590913\pi$$
$$242$$ 0 0
$$243$$ 16.1652 1.03699
$$244$$ 0 0
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ −1.00000 −0.0636285
$$248$$ 0 0
$$249$$ −40.1216 −2.54260
$$250$$ 0 0
$$251$$ 9.79129 0.618021 0.309010 0.951059i $$-0.400002\pi$$
0.309010 + 0.951059i $$0.400002\pi$$
$$252$$ 0 0
$$253$$ −17.3739 −1.09229
$$254$$ 0 0
$$255$$ −31.7477 −1.98812
$$256$$ 0 0
$$257$$ 11.2087 0.699180 0.349590 0.936903i $$-0.386321\pi$$
0.349590 + 0.936903i $$0.386321\pi$$
$$258$$ 0 0
$$259$$ 5.00000 0.310685
$$260$$ 0 0
$$261$$ 18.1652 1.12439
$$262$$ 0 0
$$263$$ 6.79129 0.418769 0.209384 0.977833i $$-0.432854\pi$$
0.209384 + 0.977833i $$0.432854\pi$$
$$264$$ 0 0
$$265$$ −25.1216 −1.54321
$$266$$ 0 0
$$267$$ −21.1652 −1.29529
$$268$$ 0 0
$$269$$ −9.95644 −0.607055 −0.303527 0.952823i $$-0.598164\pi$$
−0.303527 + 0.952823i $$0.598164\pi$$
$$270$$ 0 0
$$271$$ −14.1216 −0.857826 −0.428913 0.903346i $$-0.641103\pi$$
−0.428913 + 0.903346i $$0.641103\pi$$
$$272$$ 0 0
$$273$$ 2.79129 0.168936
$$274$$ 0 0
$$275$$ −15.1652 −0.914493
$$276$$ 0 0
$$277$$ 3.25227 0.195410 0.0977051 0.995215i $$-0.468850\pi$$
0.0977051 + 0.995215i $$0.468850\pi$$
$$278$$ 0 0
$$279$$ 35.3303 2.11517
$$280$$ 0 0
$$281$$ 19.4174 1.15835 0.579173 0.815205i $$-0.303375\pi$$
0.579173 + 0.815205i $$0.303375\pi$$
$$282$$ 0 0
$$283$$ −6.37386 −0.378887 −0.189443 0.981892i $$-0.560668\pi$$
−0.189443 + 0.981892i $$0.560668\pi$$
$$284$$ 0 0
$$285$$ −8.37386 −0.496025
$$286$$ 0 0
$$287$$ −3.79129 −0.223793
$$288$$ 0 0
$$289$$ −2.62614 −0.154479
$$290$$ 0 0
$$291$$ 19.5390 1.14540
$$292$$ 0 0
$$293$$ 19.7477 1.15367 0.576837 0.816859i $$-0.304287\pi$$
0.576837 + 0.816859i $$0.304287\pi$$
$$294$$ 0 0
$$295$$ 36.4955 2.12485
$$296$$ 0 0
$$297$$ 18.9564 1.09996
$$298$$ 0 0
$$299$$ −4.58258 −0.265017
$$300$$ 0 0
$$301$$ 2.00000 0.115278
$$302$$ 0 0
$$303$$ 21.6261 1.24239
$$304$$ 0 0
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ −27.3739 −1.56231 −0.781154 0.624338i $$-0.785369\pi$$
−0.781154 + 0.624338i $$0.785369\pi$$
$$308$$ 0 0
$$309$$ 36.2867 2.06428
$$310$$ 0 0
$$311$$ 26.2087 1.48616 0.743080 0.669203i $$-0.233364\pi$$
0.743080 + 0.669203i $$0.233364\pi$$
$$312$$ 0 0
$$313$$ 12.7477 0.720544 0.360272 0.932847i $$-0.382684\pi$$
0.360272 + 0.932847i $$0.382684\pi$$
$$314$$ 0 0
$$315$$ 14.3739 0.809875
$$316$$ 0 0
$$317$$ 14.8348 0.833208 0.416604 0.909088i $$-0.363220\pi$$
0.416604 + 0.909088i $$0.363220\pi$$
$$318$$ 0 0
$$319$$ −14.3739 −0.804782
$$320$$ 0 0
$$321$$ −0.460985 −0.0257297
$$322$$ 0 0
$$323$$ 3.79129 0.210953
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 0 0
$$327$$ 32.7913 1.81336
$$328$$ 0 0
$$329$$ 10.5826 0.583436
$$330$$ 0 0
$$331$$ 28.3739 1.55957 0.779784 0.626048i $$-0.215329\pi$$
0.779784 + 0.626048i $$0.215329\pi$$
$$332$$ 0 0
$$333$$ 23.9564 1.31280
$$334$$ 0 0
$$335$$ −28.1216 −1.53645
$$336$$ 0 0
$$337$$ 1.37386 0.0748391 0.0374196 0.999300i $$-0.488086\pi$$
0.0374196 + 0.999300i $$0.488086\pi$$
$$338$$ 0 0
$$339$$ −23.3739 −1.26949
$$340$$ 0 0
$$341$$ −27.9564 −1.51393
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −38.3739 −2.06598
$$346$$ 0 0
$$347$$ 0.791288 0.0424786 0.0212393 0.999774i $$-0.493239\pi$$
0.0212393 + 0.999774i $$0.493239\pi$$
$$348$$ 0 0
$$349$$ −1.62614 −0.0870451 −0.0435225 0.999052i $$-0.513858\pi$$
−0.0435225 + 0.999052i $$0.513858\pi$$
$$350$$ 0 0
$$351$$ 5.00000 0.266880
$$352$$ 0 0
$$353$$ −8.53901 −0.454486 −0.227243 0.973838i $$-0.572971\pi$$
−0.227243 + 0.973838i $$0.572971\pi$$
$$354$$ 0 0
$$355$$ −36.4955 −1.93698
$$356$$ 0 0
$$357$$ −10.5826 −0.560089
$$358$$ 0 0
$$359$$ −0.626136 −0.0330462 −0.0165231 0.999863i $$-0.505260\pi$$
−0.0165231 + 0.999863i $$0.505260\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −9.41742 −0.494287
$$364$$ 0 0
$$365$$ 49.1216 2.57114
$$366$$ 0 0
$$367$$ −22.4955 −1.17425 −0.587127 0.809495i $$-0.699741\pi$$
−0.587127 + 0.809495i $$0.699741\pi$$
$$368$$ 0 0
$$369$$ −18.1652 −0.945640
$$370$$ 0 0
$$371$$ −8.37386 −0.434749
$$372$$ 0 0
$$373$$ 16.3739 0.847807 0.423903 0.905707i $$-0.360660\pi$$
0.423903 + 0.905707i $$0.360660\pi$$
$$374$$ 0 0
$$375$$ 8.37386 0.432424
$$376$$ 0 0
$$377$$ −3.79129 −0.195261
$$378$$ 0 0
$$379$$ 21.7477 1.11711 0.558553 0.829469i $$-0.311357\pi$$
0.558553 + 0.829469i $$0.311357\pi$$
$$380$$ 0 0
$$381$$ −18.8348 −0.964939
$$382$$ 0 0
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ 0 0
$$385$$ −11.3739 −0.579666
$$386$$ 0 0
$$387$$ 9.58258 0.487110
$$388$$ 0 0
$$389$$ −18.6261 −0.944383 −0.472191 0.881496i $$-0.656537\pi$$
−0.472191 + 0.881496i $$0.656537\pi$$
$$390$$ 0 0
$$391$$ 17.3739 0.878634
$$392$$ 0 0
$$393$$ 61.7477 3.11476
$$394$$ 0 0
$$395$$ −30.0000 −1.50946
$$396$$ 0 0
$$397$$ −5.25227 −0.263604 −0.131802 0.991276i $$-0.542076\pi$$
−0.131802 + 0.991276i $$0.542076\pi$$
$$398$$ 0 0
$$399$$ −2.79129 −0.139739
$$400$$ 0 0
$$401$$ −12.6261 −0.630519 −0.315260 0.949005i $$-0.602092\pi$$
−0.315260 + 0.949005i $$0.602092\pi$$
$$402$$ 0 0
$$403$$ −7.37386 −0.367318
$$404$$ 0 0
$$405$$ −1.25227 −0.0622259
$$406$$ 0 0
$$407$$ −18.9564 −0.939636
$$408$$ 0 0
$$409$$ 27.1216 1.34108 0.670538 0.741875i $$-0.266063\pi$$
0.670538 + 0.741875i $$0.266063\pi$$
$$410$$ 0 0
$$411$$ −16.7477 −0.826104
$$412$$ 0 0
$$413$$ 12.1652 0.598608
$$414$$ 0 0
$$415$$ 43.1216 2.11676
$$416$$ 0 0
$$417$$ −27.2087 −1.33242
$$418$$ 0 0
$$419$$ −25.7477 −1.25786 −0.628929 0.777462i $$-0.716507\pi$$
−0.628929 + 0.777462i $$0.716507\pi$$
$$420$$ 0 0
$$421$$ −19.0000 −0.926003 −0.463002 0.886357i $$-0.653228\pi$$
−0.463002 + 0.886357i $$0.653228\pi$$
$$422$$ 0 0
$$423$$ 50.7042 2.46532
$$424$$ 0 0
$$425$$ 15.1652 0.735618
$$426$$ 0 0
$$427$$ −1.00000 −0.0483934
$$428$$ 0 0
$$429$$ −10.5826 −0.510932
$$430$$ 0 0
$$431$$ −6.00000 −0.289010 −0.144505 0.989504i $$-0.546159\pi$$
−0.144505 + 0.989504i $$0.546159\pi$$
$$432$$ 0 0
$$433$$ 9.74773 0.468446 0.234223 0.972183i $$-0.424745\pi$$
0.234223 + 0.972183i $$0.424745\pi$$
$$434$$ 0 0
$$435$$ −31.7477 −1.52219
$$436$$ 0 0
$$437$$ 4.58258 0.219214
$$438$$ 0 0
$$439$$ 41.4955 1.98047 0.990235 0.139408i $$-0.0445200\pi$$
0.990235 + 0.139408i $$0.0445200\pi$$
$$440$$ 0 0
$$441$$ 4.79129 0.228157
$$442$$ 0 0
$$443$$ −14.0436 −0.667230 −0.333615 0.942709i $$-0.608268\pi$$
−0.333615 + 0.942709i $$0.608268\pi$$
$$444$$ 0 0
$$445$$ 22.7477 1.07835
$$446$$ 0 0
$$447$$ −8.37386 −0.396070
$$448$$ 0 0
$$449$$ 25.1216 1.18556 0.592781 0.805364i $$-0.298030\pi$$
0.592781 + 0.805364i $$0.298030\pi$$
$$450$$ 0 0
$$451$$ 14.3739 0.676839
$$452$$ 0 0
$$453$$ −37.3303 −1.75393
$$454$$ 0 0
$$455$$ −3.00000 −0.140642
$$456$$ 0 0
$$457$$ −39.8693 −1.86501 −0.932504 0.361160i $$-0.882381\pi$$
−0.932504 + 0.361160i $$0.882381\pi$$
$$458$$ 0 0
$$459$$ −18.9564 −0.884811
$$460$$ 0 0
$$461$$ −11.5390 −0.537426 −0.268713 0.963220i $$-0.586598\pi$$
−0.268713 + 0.963220i $$0.586598\pi$$
$$462$$ 0 0
$$463$$ −34.4955 −1.60314 −0.801570 0.597901i $$-0.796002\pi$$
−0.801570 + 0.597901i $$0.796002\pi$$
$$464$$ 0 0
$$465$$ −61.7477 −2.86348
$$466$$ 0 0
$$467$$ 34.2867 1.58660 0.793301 0.608830i $$-0.208361\pi$$
0.793301 + 0.608830i $$0.208361\pi$$
$$468$$ 0 0
$$469$$ −9.37386 −0.432845
$$470$$ 0 0
$$471$$ 56.1652 2.58795
$$472$$ 0 0
$$473$$ −7.58258 −0.348647
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −40.1216 −1.83704
$$478$$ 0 0
$$479$$ 12.6261 0.576903 0.288451 0.957495i $$-0.406860\pi$$
0.288451 + 0.957495i $$0.406860\pi$$
$$480$$ 0 0
$$481$$ −5.00000 −0.227980
$$482$$ 0 0
$$483$$ −12.7913 −0.582024
$$484$$ 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
$$489$$ −12.2087 −0.552097
$$490$$ 0 0
$$491$$ −15.1652 −0.684394 −0.342197 0.939628i $$-0.611171\pi$$
−0.342197 + 0.939628i $$0.611171\pi$$
$$492$$ 0 0
$$493$$ 14.3739 0.647366
$$494$$ 0 0
$$495$$ −54.4955 −2.44939
$$496$$ 0 0
$$497$$ −12.1652 −0.545682
$$498$$ 0 0
$$499$$ 17.6261 0.789054 0.394527 0.918884i $$-0.370908\pi$$
0.394527 + 0.918884i $$0.370908\pi$$
$$500$$ 0 0
$$501$$ −37.4519 −1.67323
$$502$$ 0 0
$$503$$ −6.33030 −0.282254 −0.141127 0.989991i $$-0.545073\pi$$
−0.141127 + 0.989991i $$0.545073\pi$$
$$504$$ 0 0
$$505$$ −23.2432 −1.03431
$$506$$ 0 0
$$507$$ 33.4955 1.48759
$$508$$ 0 0
$$509$$ 0.330303 0.0146404 0.00732021 0.999973i $$-0.497670\pi$$
0.00732021 + 0.999973i $$0.497670\pi$$
$$510$$ 0 0
$$511$$ 16.3739 0.724337
$$512$$ 0 0
$$513$$ −5.00000 −0.220755
$$514$$ 0 0
$$515$$ −39.0000 −1.71855
$$516$$ 0 0
$$517$$ −40.1216 −1.76455
$$518$$ 0 0
$$519$$ −63.4955 −2.78714
$$520$$ 0 0
$$521$$ −13.4174 −0.587828 −0.293914 0.955832i $$-0.594958\pi$$
−0.293914 + 0.955832i $$0.594958\pi$$
$$522$$ 0 0
$$523$$ 0.252273 0.0110311 0.00551556 0.999985i $$-0.498244\pi$$
0.00551556 + 0.999985i $$0.498244\pi$$
$$524$$ 0 0
$$525$$ −11.1652 −0.487287
$$526$$ 0 0
$$527$$ 27.9564 1.21780
$$528$$ 0 0
$$529$$ −2.00000 −0.0869565
$$530$$ 0 0
$$531$$ 58.2867 2.52943
$$532$$ 0 0
$$533$$ 3.79129 0.164219
$$534$$ 0 0
$$535$$ 0.495454 0.0214204
$$536$$ 0 0
$$537$$ 44.0780 1.90211
$$538$$ 0 0
$$539$$ −3.79129 −0.163302
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 0 0
$$543$$ 26.1652 1.12285
$$544$$ 0 0
$$545$$ −35.2432 −1.50965
$$546$$ 0 0
$$547$$ −12.3739 −0.529068 −0.264534 0.964376i $$-0.585218\pi$$
−0.264534 + 0.964376i $$0.585218\pi$$
$$548$$ 0 0
$$549$$ −4.79129 −0.204487
$$550$$ 0 0
$$551$$ 3.79129 0.161514
$$552$$ 0 0
$$553$$ −10.0000 −0.425243
$$554$$ 0 0
$$555$$ −41.8693 −1.77725
$$556$$ 0 0
$$557$$ 0.626136 0.0265303 0.0132651 0.999912i $$-0.495777\pi$$
0.0132651 + 0.999912i $$0.495777\pi$$
$$558$$ 0 0
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ 40.1216 1.69393
$$562$$ 0 0
$$563$$ −12.1652 −0.512700 −0.256350 0.966584i $$-0.582520\pi$$
−0.256350 + 0.966584i $$0.582520\pi$$
$$564$$ 0 0
$$565$$ 25.1216 1.05687
$$566$$ 0 0
$$567$$ −0.417424 −0.0175302
$$568$$ 0 0
$$569$$ 30.1652 1.26459 0.632294 0.774728i $$-0.282113\pi$$
0.632294 + 0.774728i $$0.282113\pi$$
$$570$$ 0 0
$$571$$ 32.4955 1.35989 0.679946 0.733262i $$-0.262003\pi$$
0.679946 + 0.733262i $$0.262003\pi$$
$$572$$ 0 0
$$573$$ −65.7042 −2.74483
$$574$$ 0 0
$$575$$ 18.3303 0.764426
$$576$$ 0 0
$$577$$ −23.1216 −0.962564 −0.481282 0.876566i $$-0.659829\pi$$
−0.481282 + 0.876566i $$0.659829\pi$$
$$578$$ 0 0
$$579$$ −20.5826 −0.855383
$$580$$ 0 0
$$581$$ 14.3739 0.596328
$$582$$ 0 0
$$583$$ 31.7477 1.31486
$$584$$ 0 0
$$585$$ −14.3739 −0.594286
$$586$$ 0 0
$$587$$ −9.16515 −0.378286 −0.189143 0.981950i $$-0.560571\pi$$
−0.189143 + 0.981950i $$0.560571\pi$$
$$588$$ 0 0
$$589$$ 7.37386 0.303835
$$590$$ 0 0
$$591$$ 6.62614 0.272563
$$592$$ 0 0
$$593$$ −28.9129 −1.18731 −0.593655 0.804720i $$-0.702316\pi$$
−0.593655 + 0.804720i $$0.702316\pi$$
$$594$$ 0 0
$$595$$ 11.3739 0.466283
$$596$$ 0 0
$$597$$ 66.2867 2.71294
$$598$$ 0 0
$$599$$ −20.3739 −0.832453 −0.416227 0.909261i $$-0.636648\pi$$
−0.416227 + 0.909261i $$0.636648\pi$$
$$600$$ 0 0
$$601$$ 11.6261 0.474240 0.237120 0.971480i $$-0.423797\pi$$
0.237120 + 0.971480i $$0.423797\pi$$
$$602$$ 0 0
$$603$$ −44.9129 −1.82899
$$604$$ 0 0
$$605$$ 10.1216 0.411501
$$606$$ 0 0
$$607$$ 26.4955 1.07542 0.537709 0.843131i $$-0.319290\pi$$
0.537709 + 0.843131i $$0.319290\pi$$
$$608$$ 0 0
$$609$$ −10.5826 −0.428828
$$610$$ 0 0
$$611$$ −10.5826 −0.428125
$$612$$ 0 0
$$613$$ 34.3739 1.38835 0.694174 0.719808i $$-0.255770\pi$$
0.694174 + 0.719808i $$0.255770\pi$$
$$614$$ 0 0
$$615$$ 31.7477 1.28019
$$616$$ 0 0
$$617$$ −6.95644 −0.280056 −0.140028 0.990148i $$-0.544719\pi$$
−0.140028 + 0.990148i $$0.544719\pi$$
$$618$$ 0 0
$$619$$ 4.37386 0.175800 0.0879002 0.996129i $$-0.471984\pi$$
0.0879002 + 0.996129i $$0.471984\pi$$
$$620$$ 0 0
$$621$$ −22.9129 −0.919462
$$622$$ 0 0
$$623$$ 7.58258 0.303789
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 10.5826 0.422627
$$628$$ 0 0
$$629$$ 18.9564 0.755843
$$630$$ 0 0
$$631$$ −25.0000 −0.995234 −0.497617 0.867397i $$-0.665792\pi$$
−0.497617 + 0.867397i $$0.665792\pi$$
$$632$$ 0 0
$$633$$ 68.0345 2.70413
$$634$$ 0 0
$$635$$ 20.2432 0.803326
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ −58.2867 −2.30579
$$640$$ 0 0
$$641$$ −20.2087 −0.798196 −0.399098 0.916908i $$-0.630677\pi$$
−0.399098 + 0.916908i $$0.630677\pi$$
$$642$$ 0 0
$$643$$ 23.0000 0.907031 0.453516 0.891248i $$-0.350170\pi$$
0.453516 + 0.891248i $$0.350170\pi$$
$$644$$ 0 0
$$645$$ −16.7477 −0.659441
$$646$$ 0 0
$$647$$ −13.4174 −0.527493 −0.263747 0.964592i $$-0.584958\pi$$
−0.263747 + 0.964592i $$0.584958\pi$$
$$648$$ 0 0
$$649$$ −46.1216 −1.81043
$$650$$ 0 0
$$651$$ −20.5826 −0.806695
$$652$$ 0 0
$$653$$ 19.4174 0.759863 0.379931 0.925015i $$-0.375948\pi$$
0.379931 + 0.925015i $$0.375948\pi$$
$$654$$ 0 0
$$655$$ −66.3648 −2.59309
$$656$$ 0 0
$$657$$ 78.4519 3.06070
$$658$$ 0 0
$$659$$ −36.9564 −1.43962 −0.719809 0.694172i $$-0.755771\pi$$
−0.719809 + 0.694172i $$0.755771\pi$$
$$660$$ 0 0
$$661$$ −34.0000 −1.32245 −0.661223 0.750189i $$-0.729962\pi$$
−0.661223 + 0.750189i $$0.729962\pi$$
$$662$$ 0 0
$$663$$ 10.5826 0.410993
$$664$$ 0 0
$$665$$ 3.00000 0.116335
$$666$$ 0 0
$$667$$ 17.3739 0.672719
$$668$$ 0 0
$$669$$ 66.2867 2.56279
$$670$$ 0 0
$$671$$ 3.79129 0.146361
$$672$$ 0 0
$$673$$ −17.1216 −0.659989 −0.329994 0.943983i $$-0.607047\pi$$
−0.329994 + 0.943983i $$0.607047\pi$$
$$674$$ 0 0
$$675$$ −20.0000 −0.769800
$$676$$ 0 0
$$677$$ −21.9564 −0.843855 −0.421927 0.906630i $$-0.638646\pi$$
−0.421927 + 0.906630i $$0.638646\pi$$
$$678$$ 0 0
$$679$$ −7.00000 −0.268635
$$680$$ 0 0
$$681$$ 10.5826 0.405525
$$682$$ 0 0
$$683$$ −36.1652 −1.38382 −0.691911 0.721983i $$-0.743231\pi$$
−0.691911 + 0.721983i $$0.743231\pi$$
$$684$$ 0 0
$$685$$ 18.0000 0.687745
$$686$$ 0 0
$$687$$ 61.4083 2.34287
$$688$$ 0 0
$$689$$ 8.37386 0.319019
$$690$$ 0 0
$$691$$ 48.7477 1.85445 0.927225 0.374504i $$-0.122187\pi$$
0.927225 + 0.374504i $$0.122187\pi$$
$$692$$ 0 0
$$693$$ −18.1652 −0.690037
$$694$$ 0 0
$$695$$ 29.2432 1.10926
$$696$$ 0 0
$$697$$ −14.3739 −0.544449
$$698$$ 0 0
$$699$$ 78.4955 2.96897
$$700$$ 0 0
$$701$$ 42.3303 1.59879 0.799397 0.600804i $$-0.205153\pi$$
0.799397 + 0.600804i $$0.205153\pi$$
$$702$$ 0 0
$$703$$ 5.00000 0.188579
$$704$$ 0 0
$$705$$ −88.6170 −3.33751
$$706$$ 0 0
$$707$$ −7.74773 −0.291383
$$708$$ 0 0
$$709$$ 31.2432 1.17336 0.586681 0.809818i $$-0.300434\pi$$
0.586681 + 0.809818i $$0.300434\pi$$
$$710$$ 0 0
$$711$$ −47.9129 −1.79687
$$712$$ 0 0
$$713$$ 33.7913 1.26549
$$714$$ 0 0
$$715$$ 11.3739 0.425358
$$716$$ 0 0
$$717$$ 21.6261 0.807643
$$718$$ 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
$$723$$ 24.4174 0.908094
$$724$$ 0 0
$$725$$ 15.1652 0.563220
$$726$$ 0 0
$$727$$ 11.0000 0.407967 0.203984 0.978974i $$-0.434611\pi$$
0.203984 + 0.978974i $$0.434611\pi$$
$$728$$ 0 0
$$729$$ −43.8693 −1.62479
$$730$$ 0 0
$$731$$ 7.58258 0.280452
$$732$$ 0 0
$$733$$ 41.4955 1.53267 0.766335 0.642441i $$-0.222078\pi$$
0.766335 + 0.642441i $$0.222078\pi$$
$$734$$ 0 0
$$735$$ −8.37386 −0.308875
$$736$$ 0 0
$$737$$ 35.5390 1.30910
$$738$$ 0 0
$$739$$ −35.7477 −1.31500 −0.657501 0.753454i $$-0.728386\pi$$
−0.657501 + 0.753454i $$0.728386\pi$$
$$740$$ 0 0
$$741$$ 2.79129 0.102541
$$742$$ 0 0
$$743$$ 21.3303 0.782533 0.391266 0.920277i $$-0.372037\pi$$
0.391266 + 0.920277i $$0.372037\pi$$
$$744$$ 0 0
$$745$$ 9.00000 0.329734
$$746$$ 0 0
$$747$$ 68.8693 2.51980
$$748$$ 0 0
$$749$$ 0.165151 0.00603450
$$750$$ 0 0
$$751$$ 11.6261 0.424244 0.212122 0.977243i $$-0.431963\pi$$
0.212122 + 0.977243i $$0.431963\pi$$
$$752$$ 0 0
$$753$$ −27.3303 −0.995972
$$754$$ 0 0
$$755$$ 40.1216 1.46017
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 48.4955 1.76027
$$760$$ 0 0
$$761$$ −39.3303 −1.42572 −0.712861 0.701305i $$-0.752601\pi$$
−0.712861 + 0.701305i $$0.752601\pi$$
$$762$$ 0 0
$$763$$ −11.7477 −0.425296
$$764$$ 0 0
$$765$$ 54.4955 1.97029
$$766$$ 0 0
$$767$$ −12.1652 −0.439258
$$768$$ 0 0
$$769$$ −13.4955 −0.486659 −0.243329 0.969944i $$-0.578240\pi$$
−0.243329 + 0.969944i $$0.578240\pi$$
$$770$$ 0 0
$$771$$ −31.2867 −1.12676
$$772$$ 0 0
$$773$$ 16.7477 0.602374 0.301187 0.953565i $$-0.402617\pi$$
0.301187 + 0.953565i $$0.402617\pi$$
$$774$$ 0 0
$$775$$ 29.4955 1.05951
$$776$$ 0 0
$$777$$ −13.9564 −0.500684
$$778$$ 0 0
$$779$$ −3.79129 −0.135837
$$780$$ 0 0
$$781$$ 46.1216 1.65036
$$782$$ 0 0
$$783$$ −18.9564 −0.677448
$$784$$ 0 0
$$785$$ −60.3648 −2.15451
$$786$$ 0 0
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ 0 0
$$789$$ −18.9564 −0.674867
$$790$$ 0 0
$$791$$ 8.37386 0.297740
$$792$$ 0 0
$$793$$ 1.00000 0.0355110
$$794$$ 0 0
$$795$$ 70.1216 2.48696
$$796$$ 0 0
$$797$$ −32.8693 −1.16429 −0.582145 0.813085i $$-0.697787\pi$$
−0.582145 + 0.813085i $$0.697787\pi$$
$$798$$ 0 0
$$799$$ 40.1216 1.41940
$$800$$ 0 0
$$801$$ 36.3303 1.28367
$$802$$ 0 0
$$803$$ −62.0780 −2.19069
$$804$$ 0 0
$$805$$ 13.7477 0.484544
$$806$$ 0 0
$$807$$ 27.7913 0.978300
$$808$$ 0 0
$$809$$ −22.9129 −0.805574 −0.402787 0.915294i $$-0.631958\pi$$
−0.402787 + 0.915294i $$0.631958\pi$$
$$810$$ 0 0
$$811$$ −31.4955 −1.10595 −0.552977 0.833196i $$-0.686508\pi$$
−0.552977 + 0.833196i $$0.686508\pi$$
$$812$$ 0 0
$$813$$ 39.4174 1.38243
$$814$$ 0 0
$$815$$ 13.1216 0.459629
$$816$$ 0 0
$$817$$ 2.00000 0.0699711
$$818$$ 0 0
$$819$$ −4.79129 −0.167421
$$820$$ 0 0
$$821$$ 42.3303 1.47734 0.738669 0.674068i $$-0.235455\pi$$
0.738669 + 0.674068i $$0.235455\pi$$
$$822$$ 0 0
$$823$$ −54.2432 −1.89080 −0.945399 0.325915i $$-0.894328\pi$$
−0.945399 + 0.325915i $$0.894328\pi$$
$$824$$ 0 0
$$825$$ 42.3303 1.47375
$$826$$ 0 0
$$827$$ 39.6606 1.37913 0.689567 0.724222i $$-0.257801\pi$$
0.689567 + 0.724222i $$0.257801\pi$$
$$828$$ 0 0
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ −9.07803 −0.314913
$$832$$ 0 0
$$833$$ 3.79129 0.131360
$$834$$ 0 0
$$835$$ 40.2523 1.39299
$$836$$ 0 0
$$837$$ −36.8693 −1.27439
$$838$$ 0 0
$$839$$ 8.83485 0.305013 0.152506 0.988302i $$-0.451266\pi$$
0.152506 + 0.988302i $$0.451266\pi$$
$$840$$ 0 0
$$841$$ −14.6261 −0.504350
$$842$$ 0 0
$$843$$ −54.1996 −1.86673
$$844$$ 0 0
$$845$$ −36.0000 −1.23844
$$846$$ 0 0
$$847$$ 3.37386 0.115927
$$848$$ 0 0
$$849$$ 17.7913 0.610595
$$850$$ 0 0
$$851$$ 22.9129 0.785443
$$852$$ 0 0
$$853$$ 9.12159 0.312317 0.156159 0.987732i $$-0.450089\pi$$
0.156159 + 0.987732i $$0.450089\pi$$
$$854$$ 0 0
$$855$$ 14.3739 0.491576
$$856$$ 0 0
$$857$$ −11.7042 −0.399807 −0.199903 0.979816i $$-0.564063\pi$$
−0.199903 + 0.979816i $$0.564063\pi$$
$$858$$ 0 0
$$859$$ 7.37386 0.251593 0.125796 0.992056i $$-0.459851\pi$$
0.125796 + 0.992056i $$0.459851\pi$$
$$860$$ 0 0
$$861$$ 10.5826 0.360653
$$862$$ 0 0
$$863$$ −38.7042 −1.31751 −0.658753 0.752360i $$-0.728916\pi$$
−0.658753 + 0.752360i $$0.728916\pi$$
$$864$$ 0 0
$$865$$ 68.2432 2.32034
$$866$$ 0 0
$$867$$ 7.33030 0.248950
$$868$$ 0 0
$$869$$ 37.9129 1.28611
$$870$$ 0 0
$$871$$ 9.37386 0.317621
$$872$$ 0 0
$$873$$ −33.5390 −1.13512
$$874$$ 0 0
$$875$$ −3.00000 −0.101419
$$876$$ 0 0
$$877$$ −38.7477 −1.30842 −0.654209 0.756314i $$-0.726998\pi$$
−0.654209 + 0.756314i $$0.726998\pi$$
$$878$$ 0 0
$$879$$ −55.1216 −1.85921
$$880$$ 0 0
$$881$$ 2.37386 0.0799775 0.0399887 0.999200i $$-0.487268\pi$$
0.0399887 + 0.999200i $$0.487268\pi$$
$$882$$ 0 0
$$883$$ 24.2523 0.816154 0.408077 0.912948i $$-0.366199\pi$$
0.408077 + 0.912948i $$0.366199\pi$$
$$884$$ 0 0
$$885$$ −101.869 −3.42430
$$886$$ 0 0
$$887$$ 4.58258 0.153868 0.0769339 0.997036i $$-0.475487\pi$$
0.0769339 + 0.997036i $$0.475487\pi$$
$$888$$ 0 0
$$889$$ 6.74773 0.226312
$$890$$ 0 0
$$891$$ 1.58258 0.0530183
$$892$$ 0 0
$$893$$ 10.5826 0.354132
$$894$$ 0 0
$$895$$ −47.3739 −1.58353
$$896$$ 0 0
$$897$$ 12.7913 0.427089
$$898$$ 0 0
$$899$$ 27.9564 0.932399
$$900$$ 0 0
$$901$$ −31.7477 −1.05767
$$902$$ 0 0
$$903$$ −5.58258 −0.185776
$$904$$ 0 0
$$905$$ −28.1216 −0.934793
$$906$$ 0 0
$$907$$ 31.2432 1.03741 0.518706 0.854952i $$-0.326414\pi$$
0.518706 + 0.854952i $$0.326414\pi$$
$$908$$ 0 0
$$909$$ −37.1216 −1.23125
$$910$$ 0 0
$$911$$ 3.49545 0.115810 0.0579048 0.998322i $$-0.481558\pi$$
0.0579048 + 0.998322i $$0.481558\pi$$
$$912$$ 0 0
$$913$$ −54.4955 −1.80354
$$914$$ 0 0
$$915$$ 8.37386 0.276831
$$916$$ 0 0
$$917$$ −22.1216 −0.730519
$$918$$ 0 0
$$919$$ −34.4955 −1.13790 −0.568950 0.822372i $$-0.692650\pi$$
−0.568950 + 0.822372i $$0.692650\pi$$
$$920$$ 0 0
$$921$$ 76.4083 2.51774
$$922$$ 0 0
$$923$$ 12.1652 0.400421
$$924$$ 0 0
$$925$$ 20.0000 0.657596
$$926$$ 0 0
$$927$$ −62.2867 −2.04577
$$928$$ 0 0
$$929$$ 47.8693 1.57054 0.785271 0.619153i $$-0.212524\pi$$
0.785271 + 0.619153i $$0.212524\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ −73.1561 −2.39502
$$934$$ 0 0
$$935$$ −43.1216 −1.41023
$$936$$ 0 0
$$937$$ −15.3739 −0.502242 −0.251121 0.967956i $$-0.580799\pi$$
−0.251121 + 0.967956i $$0.580799\pi$$
$$938$$ 0 0
$$939$$ −35.5826 −1.16119
$$940$$ 0 0
$$941$$ 27.4955 0.896326 0.448163 0.893952i $$-0.352078\pi$$
0.448163 + 0.893952i $$0.352078\pi$$
$$942$$ 0 0
$$943$$ −17.3739 −0.565771
$$944$$ 0 0
$$945$$ −15.0000 −0.487950
$$946$$ 0 0
$$947$$ −52.6170 −1.70982 −0.854912 0.518773i $$-0.826389\pi$$
−0.854912 + 0.518773i $$0.826389\pi$$
$$948$$ 0 0
$$949$$ −16.3739 −0.531518
$$950$$ 0 0
$$951$$ −41.4083 −1.34276
$$952$$ 0 0
$$953$$ 21.4610 0.695189 0.347595 0.937645i $$-0.386999\pi$$
0.347595 + 0.937645i $$0.386999\pi$$
$$954$$ 0 0
$$955$$ 70.6170 2.28511
$$956$$ 0 0
$$957$$ 40.1216 1.29695
$$958$$ 0 0
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ 23.3739 0.753996
$$962$$ 0 0
$$963$$ 0.791288 0.0254989
$$964$$ 0 0
$$965$$ 22.1216 0.712119
$$966$$ 0 0
$$967$$ 12.1216 0.389804 0.194902 0.980823i $$-0.437561\pi$$
0.194902 + 0.980823i $$0.437561\pi$$
$$968$$ 0 0
$$969$$ −10.5826 −0.339961
$$970$$ 0 0
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 0 0
$$973$$ 9.74773 0.312498
$$974$$ 0 0
$$975$$ 11.1652 0.357571
$$976$$ 0 0
$$977$$ −34.4174 −1.10111 −0.550555 0.834799i $$-0.685584\pi$$
−0.550555 + 0.834799i $$0.685584\pi$$
$$978$$ 0 0
$$979$$ −28.7477 −0.918781
$$980$$ 0 0
$$981$$ −56.2867 −1.79710
$$982$$ 0 0
$$983$$ 8.66970 0.276520 0.138260 0.990396i $$-0.455849\pi$$
0.138260 + 0.990396i $$0.455849\pi$$
$$984$$ 0 0
$$985$$ −7.12159 −0.226913
$$986$$ 0 0
$$987$$ −29.5390 −0.940237
$$988$$ 0 0
$$989$$ 9.16515 0.291435
$$990$$ 0 0
$$991$$ −8.74773 −0.277881 −0.138940 0.990301i $$-0.544370\pi$$
−0.138940 + 0.990301i $$0.544370\pi$$
$$992$$ 0 0
$$993$$ −79.1996 −2.51332
$$994$$ 0 0
$$995$$ −71.2432 −2.25856
$$996$$ 0 0
$$997$$ −2.87841 −0.0911601 −0.0455801 0.998961i $$-0.514514\pi$$
−0.0455801 + 0.998961i $$0.514514\pi$$
$$998$$ 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.1.1 2
3.2 odd 2 4788.2.a.g.1.2 2
4.3 odd 2 2128.2.a.k.1.2 2
7.6 odd 2 3724.2.a.e.1.2 2
8.3 odd 2 8512.2.a.m.1.1 2
8.5 even 2 8512.2.a.t.1.2 2

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