# Properties

 Label 3724.2.a.e.1.2 Level $3724$ Weight $2$ Character 3724.1 Self dual yes Analytic conductor $29.736$ 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] = [3724,2,Mod(1,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, 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("3724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3724 = 2^{2} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3724.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: $$29.7362897127$$ Analytic rank: $$1$$ 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: no (minimal twist has level 532) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 3724.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.79129 q^{3} -3.00000 q^{5} +4.79129 q^{9} +O(q^{10})$$ $$q+2.79129 q^{3} -3.00000 q^{5} +4.79129 q^{9} -3.79129 q^{11} +1.00000 q^{13} -8.37386 q^{15} -3.79129 q^{17} -1.00000 q^{19} +4.58258 q^{23} +4.00000 q^{25} +5.00000 q^{27} +3.79129 q^{29} -7.37386 q^{31} -10.5826 q^{33} +5.00000 q^{37} +2.79129 q^{39} +3.79129 q^{41} +2.00000 q^{43} -14.3739 q^{45} -10.5826 q^{47} -10.5826 q^{51} -8.37386 q^{53} +11.3739 q^{55} -2.79129 q^{57} -12.1652 q^{59} +1.00000 q^{61} -3.00000 q^{65} -9.37386 q^{67} +12.7913 q^{69} -12.1652 q^{71} -16.3739 q^{73} +11.1652 q^{75} -10.0000 q^{79} -0.417424 q^{81} -14.3739 q^{83} +11.3739 q^{85} +10.5826 q^{87} -7.58258 q^{89} -20.5826 q^{93} +3.00000 q^{95} +7.00000 q^{97} -18.1652 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 6 q^{5} + 5 q^{9}+O(q^{10})$$ 2 * q + q^3 - 6 * q^5 + 5 * q^9 $$2 q + q^{3} - 6 q^{5} + 5 q^{9} - 3 q^{11} + 2 q^{13} - 3 q^{15} - 3 q^{17} - 2 q^{19} + 8 q^{25} + 10 q^{27} + 3 q^{29} - q^{31} - 12 q^{33} + 10 q^{37} + q^{39} + 3 q^{41} + 4 q^{43} - 15 q^{45} - 12 q^{47} - 12 q^{51} - 3 q^{53} + 9 q^{55} - q^{57} - 6 q^{59} + 2 q^{61} - 6 q^{65} - 5 q^{67} + 21 q^{69} - 6 q^{71} - 19 q^{73} + 4 q^{75} - 20 q^{79} - 10 q^{81} - 15 q^{83} + 9 q^{85} + 12 q^{87} - 6 q^{89} - 32 q^{93} + 6 q^{95} + 14 q^{97} - 18 q^{99}+O(q^{100})$$ 2 * q + q^3 - 6 * q^5 + 5 * q^9 - 3 * q^11 + 2 * q^13 - 3 * q^15 - 3 * q^17 - 2 * q^19 + 8 * q^25 + 10 * q^27 + 3 * q^29 - q^31 - 12 * q^33 + 10 * q^37 + q^39 + 3 * q^41 + 4 * q^43 - 15 * q^45 - 12 * q^47 - 12 * q^51 - 3 * q^53 + 9 * q^55 - q^57 - 6 * q^59 + 2 * q^61 - 6 * q^65 - 5 * q^67 + 21 * q^69 - 6 * q^71 - 19 * q^73 + 4 * q^75 - 20 * q^79 - 10 * q^81 - 15 * q^83 + 9 * q^85 + 12 * q^87 - 6 * q^89 - 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 $$-0.201749\pi$$
0.805775 + 0.592221i $$0.201749\pi$$
$$4$$ 0 0
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 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 $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ −8.37386 −2.16212
$$16$$ 0 0
$$17$$ −3.79129 −0.919522 −0.459761 0.888043i $$-0.652065\pi$$
−0.459761 + 0.888043i $$0.652065\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 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 $$-0.730374\pi$$
−0.662192 + 0.749334i $$0.730374\pi$$
$$32$$ 0 0
$$33$$ −10.5826 −1.84219
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 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 $$-0.404331\pi$$
0.296050 + 0.955172i $$0.404331\pi$$
$$42$$ 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 $$-0.780650\pi$$
−0.771814 + 0.635849i $$0.780650\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 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 $$-0.790900\pi$$
−0.791884 + 0.610672i $$0.790900\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 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 $$-0.907650\pi$$
−0.958208 + 0.286073i $$0.907650\pi$$
$$74$$ 0 0
$$75$$ 11.1652 1.28924
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 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 $$-0.789333\pi$$
−0.788868 + 0.614562i $$0.789333\pi$$
$$84$$ 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 $$-0.631642\pi$$
−0.401876 + 0.915694i $$0.631642\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 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 $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ −18.1652 −1.82567
$$100$$ 0 0
$$101$$ 7.74773 0.770928 0.385464 0.922723i $$-0.374041\pi$$
0.385464 + 0.922723i $$0.374041\pi$$
$$102$$ 0 0
$$103$$ 13.0000 1.28093 0.640464 0.767988i $$-0.278742\pi$$
0.640464 + 0.767988i $$0.278742\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 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
$$119$$ 0 0
$$120$$ 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 $$-0.0827653\pi$$
0.966386 + 0.257095i $$0.0827653\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 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 $$-0.635657\pi$$
−0.413396 + 0.910551i $$0.635657\pi$$
$$140$$ 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
$$147$$ 0 0
$$148$$ 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 $$-0.203269\pi$$
0.802939 + 0.596061i $$0.203269\pi$$
$$158$$ 0 0
$$159$$ −23.3739 −1.85367
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 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 $$-0.673746\pi$$
−0.519136 + 0.854692i $$0.673746\pi$$
$$168$$ 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 $$-0.832516\pi$$
−0.864739 + 0.502222i $$0.832516\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 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 $$-0.386733\pi$$
0.348377 + 0.937355i $$0.386733\pi$$
$$182$$ 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
$$189$$ 0 0
$$190$$ 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 $$-0.181546\pi$$
0.841716 + 0.539921i $$0.181546\pi$$
$$200$$ 0 0
$$201$$ −26.1652 −1.84555
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 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
$$217$$ 0 0
$$218$$ 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 $$-0.207401\pi$$
0.795133 + 0.606435i $$0.207401\pi$$
$$224$$ 0 0
$$225$$ 19.1652 1.27768
$$226$$ 0 0
$$227$$ 3.79129 0.251637 0.125818 0.992053i $$-0.459844\pi$$
0.125818 + 0.992053i $$0.459844\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 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 $$-0.409087\pi$$
0.281745 + 0.959489i $$0.409087\pi$$
$$242$$ 0 0
$$243$$ −16.1652 −1.03699
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 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 $$-0.599998\pi$$
−0.309010 + 0.951059i $$0.599998\pi$$
$$252$$ 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 $$-0.613679\pi$$
−0.349590 + 0.936903i $$0.613679\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 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 $$-0.401836\pi$$
0.303527 + 0.952823i $$0.401836\pi$$
$$270$$ 0 0
$$271$$ 14.1216 0.857826 0.428913 0.903346i $$-0.358897\pi$$
0.428913 + 0.903346i $$0.358897\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 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 $$-0.439332\pi$$
0.189443 + 0.981892i $$0.439332\pi$$
$$284$$ 0 0
$$285$$ 8.37386 0.496025
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 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 $$-0.695713\pi$$
−0.576837 + 0.816859i $$0.695713\pi$$
$$294$$ 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
$$301$$ 0 0
$$302$$ 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 $$-0.214631\pi$$
0.781154 + 0.624338i $$0.214631\pi$$
$$308$$ 0 0
$$309$$ 36.2867 2.06428
$$310$$ 0 0
$$311$$ −26.2087 −1.48616 −0.743080 0.669203i $$-0.766636\pi$$
−0.743080 + 0.669203i $$0.766636\pi$$
$$312$$ 0 0
$$313$$ −12.7477 −0.720544 −0.360272 0.932847i $$-0.617316\pi$$
−0.360272 + 0.932847i $$0.617316\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 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
$$329$$ 0 0
$$330$$ 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
$$343$$ 0 0
$$344$$ 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 $$-0.486142\pi$$
0.0435225 + 0.999052i $$0.486142\pi$$
$$350$$ 0 0
$$351$$ 5.00000 0.266880
$$352$$ 0 0
$$353$$ 8.53901 0.454486 0.227243 0.973838i $$-0.427029\pi$$
0.227243 + 0.973838i $$0.427029\pi$$
$$354$$ 0 0
$$355$$ 36.4955 1.93698
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 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 $$-0.300259\pi$$
0.587127 + 0.809495i $$0.300259\pi$$
$$368$$ 0 0
$$369$$ 18.1652 0.945640
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 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 $$-0.680263\pi$$
−0.536525 + 0.843884i $$0.680263\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 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 $$-0.457924\pi$$
0.131802 + 0.991276i $$0.457924\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 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 $$-0.733937\pi$$
−0.670538 + 0.741875i $$0.733937\pi$$
$$410$$ 0 0
$$411$$ 16.7477 0.826104
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 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 $$-0.283493\pi$$
0.628929 + 0.777462i $$0.283493\pi$$
$$420$$ 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
$$427$$ 0 0
$$428$$ 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 $$-0.575255\pi$$
−0.234223 + 0.972183i $$0.575255\pi$$
$$434$$ 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 $$-0.955480\pi$$
−0.990235 + 0.139408i $$0.955480\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 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
$$455$$ 0 0
$$456$$ 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 $$-0.413402\pi$$
0.268713 + 0.963220i $$0.413402\pi$$
$$462$$ 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 $$-0.791639\pi$$
−0.793301 + 0.608830i $$0.791639\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 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 $$-0.593140\pi$$
−0.288451 + 0.957495i $$0.593140\pi$$
$$480$$ 0 0
$$481$$ 5.00000 0.227980
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 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
$$497$$ 0 0
$$498$$ 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 $$-0.454927\pi$$
0.141127 + 0.989991i $$0.454927\pi$$
$$504$$ 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 $$-0.502330\pi$$
−0.00732021 + 0.999973i $$0.502330\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 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 $$-0.405042\pi$$
0.293914 + 0.955832i $$0.405042\pi$$
$$522$$ 0 0
$$523$$ −0.252273 −0.0110311 −0.00551556 0.999985i $$-0.501756\pi$$
−0.00551556 + 0.999985i $$0.501756\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 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
$$539$$ 0 0
$$540$$ 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
$$553$$ 0 0
$$554$$ 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 $$-0.417480\pi$$
0.256350 + 0.966584i $$0.417480\pi$$
$$564$$ 0 0
$$565$$ −25.1216 −1.05687
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 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 $$-0.340171\pi$$
0.481282 + 0.876566i $$0.340171\pi$$
$$578$$ 0 0
$$579$$ 20.5826 0.855383
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 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 $$-0.439429\pi$$
0.189143 + 0.981950i $$0.439429\pi$$
$$588$$ 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 $$-0.297684\pi$$
0.593655 + 0.804720i $$0.297684\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 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 $$-0.576203\pi$$
−0.237120 + 0.971480i $$0.576203\pi$$
$$602$$ 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 $$-0.680710\pi$$
−0.537709 + 0.843131i $$0.680710\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 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 $$-0.528016\pi$$
−0.0879002 + 0.996129i $$0.528016\pi$$
$$620$$ 0 0
$$621$$ 22.9129 0.919462
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 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
$$637$$ 0 0
$$638$$ 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 $$-0.649830\pi$$
−0.453516 + 0.891248i $$0.649830\pi$$
$$644$$ 0 0
$$645$$ −16.7477 −0.659441
$$646$$ 0 0
$$647$$ 13.4174 0.527493 0.263747 0.964592i $$-0.415042\pi$$
0.263747 + 0.964592i $$0.415042\pi$$
$$648$$ 0 0
$$649$$ 46.1216 1.81043
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 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 $$-0.270038\pi$$
0.661223 + 0.750189i $$0.270038\pi$$
$$662$$ 0 0
$$663$$ −10.5826 −0.410993
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 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 $$-0.361354\pi$$
0.421927 + 0.906630i $$0.361354\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 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 $$-0.877813\pi$$
−0.927225 + 0.374504i $$0.877813\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 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
$$707$$ 0 0
$$708$$ 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 $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 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 $$-0.565389\pi$$
−0.203984 + 0.978974i $$0.565389\pi$$
$$728$$ 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 $$-0.777922\pi$$
−0.766335 + 0.642441i $$0.777922\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 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 0
$$750$$ 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 $$-0.247399\pi$$
0.712861 + 0.701305i $$0.247399\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 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 $$-0.421760\pi$$
0.243329 + 0.969944i $$0.421760\pi$$
$$770$$ 0 0
$$771$$ −31.2867 −1.12676
$$772$$ 0 0
$$773$$ −16.7477 −0.602374 −0.301187 0.953565i $$-0.597383\pi$$
−0.301187 + 0.953565i $$0.597383\pi$$
$$774$$ 0 0
$$775$$ −29.4955 −1.05951
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 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 $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 0 0
$$789$$ 18.9564 0.674867
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 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 $$-0.302213\pi$$
0.582145 + 0.813085i $$0.302213\pi$$
$$798$$ 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
$$805$$ 0 0
$$806$$ 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 $$-0.313492\pi$$
0.552977 + 0.833196i $$0.313492\pi$$
$$812$$ 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
$$819$$ 0 0
$$820$$ 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 $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ 9.07803 0.314913
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 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 $$-0.548734\pi$$
−0.152506 + 0.988302i $$0.548734\pi$$
$$840$$ 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
$$847$$ 0 0
$$848$$ 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 $$-0.549911\pi$$
−0.156159 + 0.987732i $$0.549911\pi$$
$$854$$ 0 0
$$855$$ 14.3739 0.491576
$$856$$ 0 0
$$857$$ 11.7042 0.399807 0.199903 0.979816i $$-0.435937\pi$$
0.199903 + 0.979816i $$0.435937\pi$$
$$858$$ 0 0
$$859$$ −7.37386 −0.251593 −0.125796 0.992056i $$-0.540149\pi$$
−0.125796 + 0.992056i $$0.540149\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 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
$$875$$ 0 0
$$876$$ 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 $$-0.512732\pi$$
−0.0399887 + 0.999200i $$0.512732\pi$$
$$882$$ 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 $$-0.524513\pi$$
−0.0769339 + 0.997036i $$0.524513\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 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
$$903$$ 0 0
$$904$$ 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
$$917$$ 0 0
$$918$$ 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 $$-0.787476\pi$$
−0.785271 + 0.619153i $$0.787476\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 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 $$-0.419201\pi$$
0.251121 + 0.967956i $$0.419201\pi$$
$$938$$ 0 0
$$939$$ −35.5826 −1.16119
$$940$$ 0 0
$$941$$ −27.4955 −0.896326 −0.448163 0.893952i $$-0.647922\pi$$
−0.448163 + 0.893952i $$0.647922\pi$$
$$942$$ 0 0
$$943$$ 17.3739 0.565771
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 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
$$959$$ 0 0
$$960$$ 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 $$-0.577373\pi$$
−0.240686 + 0.970603i $$0.577373\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 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 $$-0.544151\pi$$
−0.138260 + 0.990396i $$0.544151\pi$$
$$984$$ 0 0
$$985$$ 7.12159 0.226913
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 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 $$-0.485486\pi$$
0.0455801 + 0.998961i $$0.485486\pi$$
$$998$$ 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 3724.2.a.e.1.2 2
7.6 odd 2 532.2.a.c.1.1 2
21.20 even 2 4788.2.a.g.1.2 2
28.27 even 2 2128.2.a.k.1.2 2
56.13 odd 2 8512.2.a.t.1.2 2
56.27 even 2 8512.2.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
532.2.a.c.1.1 2 7.6 odd 2
2128.2.a.k.1.2 2 28.27 even 2
3724.2.a.e.1.2 2 1.1 even 1 trivial
4788.2.a.g.1.2 2 21.20 even 2
8512.2.a.m.1.1 2 56.27 even 2
8512.2.a.t.1.2 2 56.13 odd 2