# Properties

 Label 8512.2.a.e.1.1 Level $8512$ Weight $2$ Character 8512.1 Self dual yes Analytic conductor $67.969$ Analytic rank $1$ 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] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8512 = 2^{6} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8512.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: $$67.9686622005$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1064) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 8512.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} +3.85410 q^{9} +O(q^{10})$$ $$q-2.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} +3.85410 q^{9} +5.85410 q^{11} -5.23607 q^{13} +9.47214 q^{15} -2.47214 q^{17} +1.00000 q^{19} -2.61803 q^{21} -5.70820 q^{23} +8.09017 q^{25} -2.23607 q^{27} +0.854102 q^{29} +4.47214 q^{31} -15.3262 q^{33} -3.61803 q^{35} +0.0901699 q^{37} +13.7082 q^{39} -6.09017 q^{41} -6.85410 q^{43} -13.9443 q^{45} -11.8541 q^{47} +1.00000 q^{49} +6.47214 q^{51} +3.38197 q^{53} -21.1803 q^{55} -2.61803 q^{57} -4.09017 q^{59} +7.85410 q^{61} +3.85410 q^{63} +18.9443 q^{65} +14.9443 q^{67} +14.9443 q^{69} +13.5623 q^{71} +8.76393 q^{73} -21.1803 q^{75} +5.85410 q^{77} -6.85410 q^{79} -5.70820 q^{81} -4.00000 q^{83} +8.94427 q^{85} -2.23607 q^{87} +1.09017 q^{89} -5.23607 q^{91} -11.7082 q^{93} -3.61803 q^{95} -3.56231 q^{97} +22.5623 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 5 q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 5 * q^5 + 2 * q^7 + q^9 $$2 q - 3 q^{3} - 5 q^{5} + 2 q^{7} + q^{9} + 5 q^{11} - 6 q^{13} + 10 q^{15} + 4 q^{17} + 2 q^{19} - 3 q^{21} + 2 q^{23} + 5 q^{25} - 5 q^{29} - 15 q^{33} - 5 q^{35} - 11 q^{37} + 14 q^{39} - q^{41} - 7 q^{43} - 10 q^{45} - 17 q^{47} + 2 q^{49} + 4 q^{51} + 9 q^{53} - 20 q^{55} - 3 q^{57} + 3 q^{59} + 9 q^{61} + q^{63} + 20 q^{65} + 12 q^{67} + 12 q^{69} + 7 q^{71} + 22 q^{73} - 20 q^{75} + 5 q^{77} - 7 q^{79} + 2 q^{81} - 8 q^{83} - 9 q^{89} - 6 q^{91} - 10 q^{93} - 5 q^{95} + 13 q^{97} + 25 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 - 5 * q^5 + 2 * q^7 + q^9 + 5 * q^11 - 6 * q^13 + 10 * q^15 + 4 * q^17 + 2 * q^19 - 3 * q^21 + 2 * q^23 + 5 * q^25 - 5 * q^29 - 15 * q^33 - 5 * q^35 - 11 * q^37 + 14 * q^39 - q^41 - 7 * q^43 - 10 * q^45 - 17 * q^47 + 2 * q^49 + 4 * q^51 + 9 * q^53 - 20 * q^55 - 3 * q^57 + 3 * q^59 + 9 * q^61 + q^63 + 20 * q^65 + 12 * q^67 + 12 * q^69 + 7 * q^71 + 22 * q^73 - 20 * q^75 + 5 * q^77 - 7 * q^79 + 2 * q^81 - 8 * q^83 - 9 * q^89 - 6 * q^91 - 10 * q^93 - 5 * q^95 + 13 * q^97 + 25 * 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
$$3$$ −2.61803 −1.51152 −0.755761 0.654847i $$-0.772733\pi$$
−0.755761 + 0.654847i $$0.772733\pi$$
$$4$$ 0 0
$$5$$ −3.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 3.85410 1.28470
$$10$$ 0 0
$$11$$ 5.85410 1.76508 0.882539 0.470239i $$-0.155832\pi$$
0.882539 + 0.470239i $$0.155832\pi$$
$$12$$ 0 0
$$13$$ −5.23607 −1.45222 −0.726112 0.687576i $$-0.758675\pi$$
−0.726112 + 0.687576i $$0.758675\pi$$
$$14$$ 0 0
$$15$$ 9.47214 2.44569
$$16$$ 0 0
$$17$$ −2.47214 −0.599581 −0.299791 0.954005i $$-0.596917\pi$$
−0.299791 + 0.954005i $$0.596917\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −2.61803 −0.571302
$$22$$ 0 0
$$23$$ −5.70820 −1.19024 −0.595121 0.803636i $$-0.702896\pi$$
−0.595121 + 0.803636i $$0.702896\pi$$
$$24$$ 0 0
$$25$$ 8.09017 1.61803
$$26$$ 0 0
$$27$$ −2.23607 −0.430331
$$28$$ 0 0
$$29$$ 0.854102 0.158603 0.0793014 0.996851i $$-0.474731\pi$$
0.0793014 + 0.996851i $$0.474731\pi$$
$$30$$ 0 0
$$31$$ 4.47214 0.803219 0.401610 0.915811i $$-0.368451\pi$$
0.401610 + 0.915811i $$0.368451\pi$$
$$32$$ 0 0
$$33$$ −15.3262 −2.66796
$$34$$ 0 0
$$35$$ −3.61803 −0.611559
$$36$$ 0 0
$$37$$ 0.0901699 0.0148238 0.00741192 0.999973i $$-0.497641\pi$$
0.00741192 + 0.999973i $$0.497641\pi$$
$$38$$ 0 0
$$39$$ 13.7082 2.19507
$$40$$ 0 0
$$41$$ −6.09017 −0.951125 −0.475562 0.879682i $$-0.657755\pi$$
−0.475562 + 0.879682i $$0.657755\pi$$
$$42$$ 0 0
$$43$$ −6.85410 −1.04524 −0.522620 0.852566i $$-0.675045\pi$$
−0.522620 + 0.852566i $$0.675045\pi$$
$$44$$ 0 0
$$45$$ −13.9443 −2.07869
$$46$$ 0 0
$$47$$ −11.8541 −1.72910 −0.864549 0.502548i $$-0.832396\pi$$
−0.864549 + 0.502548i $$0.832396\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 6.47214 0.906280
$$52$$ 0 0
$$53$$ 3.38197 0.464549 0.232274 0.972650i $$-0.425383\pi$$
0.232274 + 0.972650i $$0.425383\pi$$
$$54$$ 0 0
$$55$$ −21.1803 −2.85596
$$56$$ 0 0
$$57$$ −2.61803 −0.346767
$$58$$ 0 0
$$59$$ −4.09017 −0.532495 −0.266247 0.963905i $$-0.585784\pi$$
−0.266247 + 0.963905i $$0.585784\pi$$
$$60$$ 0 0
$$61$$ 7.85410 1.00561 0.502807 0.864398i $$-0.332301\pi$$
0.502807 + 0.864398i $$0.332301\pi$$
$$62$$ 0 0
$$63$$ 3.85410 0.485571
$$64$$ 0 0
$$65$$ 18.9443 2.34975
$$66$$ 0 0
$$67$$ 14.9443 1.82573 0.912867 0.408258i $$-0.133864\pi$$
0.912867 + 0.408258i $$0.133864\pi$$
$$68$$ 0 0
$$69$$ 14.9443 1.79908
$$70$$ 0 0
$$71$$ 13.5623 1.60955 0.804775 0.593580i $$-0.202286\pi$$
0.804775 + 0.593580i $$0.202286\pi$$
$$72$$ 0 0
$$73$$ 8.76393 1.02574 0.512870 0.858466i $$-0.328582\pi$$
0.512870 + 0.858466i $$0.328582\pi$$
$$74$$ 0 0
$$75$$ −21.1803 −2.44569
$$76$$ 0 0
$$77$$ 5.85410 0.667137
$$78$$ 0 0
$$79$$ −6.85410 −0.771147 −0.385573 0.922677i $$-0.625996\pi$$
−0.385573 + 0.922677i $$0.625996\pi$$
$$80$$ 0 0
$$81$$ −5.70820 −0.634245
$$82$$ 0 0
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ 8.94427 0.970143
$$86$$ 0 0
$$87$$ −2.23607 −0.239732
$$88$$ 0 0
$$89$$ 1.09017 0.115558 0.0577789 0.998329i $$-0.481598\pi$$
0.0577789 + 0.998329i $$0.481598\pi$$
$$90$$ 0 0
$$91$$ −5.23607 −0.548889
$$92$$ 0 0
$$93$$ −11.7082 −1.21408
$$94$$ 0 0
$$95$$ −3.61803 −0.371202
$$96$$ 0 0
$$97$$ −3.56231 −0.361697 −0.180849 0.983511i $$-0.557884\pi$$
−0.180849 + 0.983511i $$0.557884\pi$$
$$98$$ 0 0
$$99$$ 22.5623 2.26760
$$100$$ 0 0
$$101$$ 10.9443 1.08900 0.544498 0.838762i $$-0.316720\pi$$
0.544498 + 0.838762i $$0.316720\pi$$
$$102$$ 0 0
$$103$$ 13.7082 1.35071 0.675355 0.737493i $$-0.263991\pi$$
0.675355 + 0.737493i $$0.263991\pi$$
$$104$$ 0 0
$$105$$ 9.47214 0.924386
$$106$$ 0 0
$$107$$ −12.1803 −1.17752 −0.588759 0.808309i $$-0.700383\pi$$
−0.588759 + 0.808309i $$0.700383\pi$$
$$108$$ 0 0
$$109$$ −9.56231 −0.915903 −0.457951 0.888977i $$-0.651417\pi$$
−0.457951 + 0.888977i $$0.651417\pi$$
$$110$$ 0 0
$$111$$ −0.236068 −0.0224066
$$112$$ 0 0
$$113$$ 16.9443 1.59398 0.796992 0.603991i $$-0.206424\pi$$
0.796992 + 0.603991i $$0.206424\pi$$
$$114$$ 0 0
$$115$$ 20.6525 1.92585
$$116$$ 0 0
$$117$$ −20.1803 −1.86567
$$118$$ 0 0
$$119$$ −2.47214 −0.226620
$$120$$ 0 0
$$121$$ 23.2705 2.11550
$$122$$ 0 0
$$123$$ 15.9443 1.43765
$$124$$ 0 0
$$125$$ −11.1803 −1.00000
$$126$$ 0 0
$$127$$ 2.14590 0.190418 0.0952088 0.995457i $$-0.469648\pi$$
0.0952088 + 0.995457i $$0.469648\pi$$
$$128$$ 0 0
$$129$$ 17.9443 1.57991
$$130$$ 0 0
$$131$$ 22.1803 1.93791 0.968953 0.247246i $$-0.0795257\pi$$
0.968953 + 0.247246i $$0.0795257\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ 8.09017 0.696291
$$136$$ 0 0
$$137$$ 18.5623 1.58588 0.792942 0.609297i $$-0.208548\pi$$
0.792942 + 0.609297i $$0.208548\pi$$
$$138$$ 0 0
$$139$$ 2.47214 0.209684 0.104842 0.994489i $$-0.466566\pi$$
0.104842 + 0.994489i $$0.466566\pi$$
$$140$$ 0 0
$$141$$ 31.0344 2.61357
$$142$$ 0 0
$$143$$ −30.6525 −2.56329
$$144$$ 0 0
$$145$$ −3.09017 −0.256625
$$146$$ 0 0
$$147$$ −2.61803 −0.215932
$$148$$ 0 0
$$149$$ 8.94427 0.732743 0.366372 0.930469i $$-0.380600\pi$$
0.366372 + 0.930469i $$0.380600\pi$$
$$150$$ 0 0
$$151$$ −20.0000 −1.62758 −0.813788 0.581161i $$-0.802599\pi$$
−0.813788 + 0.581161i $$0.802599\pi$$
$$152$$ 0 0
$$153$$ −9.52786 −0.770282
$$154$$ 0 0
$$155$$ −16.1803 −1.29964
$$156$$ 0 0
$$157$$ −18.6180 −1.48588 −0.742940 0.669358i $$-0.766569\pi$$
−0.742940 + 0.669358i $$0.766569\pi$$
$$158$$ 0 0
$$159$$ −8.85410 −0.702176
$$160$$ 0 0
$$161$$ −5.70820 −0.449869
$$162$$ 0 0
$$163$$ −1.56231 −0.122369 −0.0611846 0.998126i $$-0.519488\pi$$
−0.0611846 + 0.998126i $$0.519488\pi$$
$$164$$ 0 0
$$165$$ 55.4508 4.31684
$$166$$ 0 0
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ 14.4164 1.10895
$$170$$ 0 0
$$171$$ 3.85410 0.294731
$$172$$ 0 0
$$173$$ −0.180340 −0.0137110 −0.00685549 0.999977i $$-0.502182\pi$$
−0.00685549 + 0.999977i $$0.502182\pi$$
$$174$$ 0 0
$$175$$ 8.09017 0.611559
$$176$$ 0 0
$$177$$ 10.7082 0.804878
$$178$$ 0 0
$$179$$ −21.1246 −1.57893 −0.789464 0.613797i $$-0.789641\pi$$
−0.789464 + 0.613797i $$0.789641\pi$$
$$180$$ 0 0
$$181$$ 7.23607 0.537853 0.268926 0.963161i $$-0.413331\pi$$
0.268926 + 0.963161i $$0.413331\pi$$
$$182$$ 0 0
$$183$$ −20.5623 −1.52001
$$184$$ 0 0
$$185$$ −0.326238 −0.0239855
$$186$$ 0 0
$$187$$ −14.4721 −1.05831
$$188$$ 0 0
$$189$$ −2.23607 −0.162650
$$190$$ 0 0
$$191$$ 1.70820 0.123601 0.0618006 0.998089i $$-0.480316\pi$$
0.0618006 + 0.998089i $$0.480316\pi$$
$$192$$ 0 0
$$193$$ 7.23607 0.520864 0.260432 0.965492i $$-0.416135\pi$$
0.260432 + 0.965492i $$0.416135\pi$$
$$194$$ 0 0
$$195$$ −49.5967 −3.55170
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ 1.03444 0.0733296 0.0366648 0.999328i $$-0.488327\pi$$
0.0366648 + 0.999328i $$0.488327\pi$$
$$200$$ 0 0
$$201$$ −39.1246 −2.75964
$$202$$ 0 0
$$203$$ 0.854102 0.0599462
$$204$$ 0 0
$$205$$ 22.0344 1.53895
$$206$$ 0 0
$$207$$ −22.0000 −1.52911
$$208$$ 0 0
$$209$$ 5.85410 0.404937
$$210$$ 0 0
$$211$$ 13.8885 0.956127 0.478063 0.878325i $$-0.341339\pi$$
0.478063 + 0.878325i $$0.341339\pi$$
$$212$$ 0 0
$$213$$ −35.5066 −2.43287
$$214$$ 0 0
$$215$$ 24.7984 1.69124
$$216$$ 0 0
$$217$$ 4.47214 0.303588
$$218$$ 0 0
$$219$$ −22.9443 −1.55043
$$220$$ 0 0
$$221$$ 12.9443 0.870726
$$222$$ 0 0
$$223$$ 2.18034 0.146006 0.0730032 0.997332i $$-0.476742\pi$$
0.0730032 + 0.997332i $$0.476742\pi$$
$$224$$ 0 0
$$225$$ 31.1803 2.07869
$$226$$ 0 0
$$227$$ 12.9443 0.859142 0.429571 0.903033i $$-0.358665\pi$$
0.429571 + 0.903033i $$0.358665\pi$$
$$228$$ 0 0
$$229$$ 9.32624 0.616295 0.308148 0.951339i $$-0.400291\pi$$
0.308148 + 0.951339i $$0.400291\pi$$
$$230$$ 0 0
$$231$$ −15.3262 −1.00839
$$232$$ 0 0
$$233$$ −10.1459 −0.664680 −0.332340 0.943160i $$-0.607838\pi$$
−0.332340 + 0.943160i $$0.607838\pi$$
$$234$$ 0 0
$$235$$ 42.8885 2.79774
$$236$$ 0 0
$$237$$ 17.9443 1.16561
$$238$$ 0 0
$$239$$ −8.29180 −0.536352 −0.268176 0.963370i $$-0.586421\pi$$
−0.268176 + 0.963370i $$0.586421\pi$$
$$240$$ 0 0
$$241$$ 4.56231 0.293884 0.146942 0.989145i $$-0.453057\pi$$
0.146942 + 0.989145i $$0.453057\pi$$
$$242$$ 0 0
$$243$$ 21.6525 1.38901
$$244$$ 0 0
$$245$$ −3.61803 −0.231148
$$246$$ 0 0
$$247$$ −5.23607 −0.333163
$$248$$ 0 0
$$249$$ 10.4721 0.663645
$$250$$ 0 0
$$251$$ −29.2361 −1.84536 −0.922682 0.385562i $$-0.874008\pi$$
−0.922682 + 0.385562i $$0.874008\pi$$
$$252$$ 0 0
$$253$$ −33.4164 −2.10087
$$254$$ 0 0
$$255$$ −23.4164 −1.46639
$$256$$ 0 0
$$257$$ −2.67376 −0.166785 −0.0833923 0.996517i $$-0.526575\pi$$
−0.0833923 + 0.996517i $$0.526575\pi$$
$$258$$ 0 0
$$259$$ 0.0901699 0.00560289
$$260$$ 0 0
$$261$$ 3.29180 0.203757
$$262$$ 0 0
$$263$$ −25.2361 −1.55612 −0.778061 0.628188i $$-0.783797\pi$$
−0.778061 + 0.628188i $$0.783797\pi$$
$$264$$ 0 0
$$265$$ −12.2361 −0.751656
$$266$$ 0 0
$$267$$ −2.85410 −0.174668
$$268$$ 0 0
$$269$$ 29.4164 1.79355 0.896775 0.442487i $$-0.145904\pi$$
0.896775 + 0.442487i $$0.145904\pi$$
$$270$$ 0 0
$$271$$ 16.8541 1.02381 0.511907 0.859041i $$-0.328939\pi$$
0.511907 + 0.859041i $$0.328939\pi$$
$$272$$ 0 0
$$273$$ 13.7082 0.829658
$$274$$ 0 0
$$275$$ 47.3607 2.85596
$$276$$ 0 0
$$277$$ 18.2918 1.09905 0.549524 0.835478i $$-0.314809\pi$$
0.549524 + 0.835478i $$0.314809\pi$$
$$278$$ 0 0
$$279$$ 17.2361 1.03190
$$280$$ 0 0
$$281$$ −17.8885 −1.06714 −0.533571 0.845756i $$-0.679150\pi$$
−0.533571 + 0.845756i $$0.679150\pi$$
$$282$$ 0 0
$$283$$ −5.52786 −0.328597 −0.164299 0.986411i $$-0.552536\pi$$
−0.164299 + 0.986411i $$0.552536\pi$$
$$284$$ 0 0
$$285$$ 9.47214 0.561081
$$286$$ 0 0
$$287$$ −6.09017 −0.359491
$$288$$ 0 0
$$289$$ −10.8885 −0.640503
$$290$$ 0 0
$$291$$ 9.32624 0.546714
$$292$$ 0 0
$$293$$ −5.23607 −0.305894 −0.152947 0.988234i $$-0.548876\pi$$
−0.152947 + 0.988234i $$0.548876\pi$$
$$294$$ 0 0
$$295$$ 14.7984 0.861595
$$296$$ 0 0
$$297$$ −13.0902 −0.759569
$$298$$ 0 0
$$299$$ 29.8885 1.72850
$$300$$ 0 0
$$301$$ −6.85410 −0.395064
$$302$$ 0 0
$$303$$ −28.6525 −1.64604
$$304$$ 0 0
$$305$$ −28.4164 −1.62712
$$306$$ 0 0
$$307$$ −3.90983 −0.223146 −0.111573 0.993756i $$-0.535589\pi$$
−0.111573 + 0.993756i $$0.535589\pi$$
$$308$$ 0 0
$$309$$ −35.8885 −2.04163
$$310$$ 0 0
$$311$$ 25.4508 1.44319 0.721593 0.692318i $$-0.243410\pi$$
0.721593 + 0.692318i $$0.243410\pi$$
$$312$$ 0 0
$$313$$ −8.00000 −0.452187 −0.226093 0.974106i $$-0.572595\pi$$
−0.226093 + 0.974106i $$0.572595\pi$$
$$314$$ 0 0
$$315$$ −13.9443 −0.785671
$$316$$ 0 0
$$317$$ 7.67376 0.431001 0.215501 0.976504i $$-0.430862\pi$$
0.215501 + 0.976504i $$0.430862\pi$$
$$318$$ 0 0
$$319$$ 5.00000 0.279946
$$320$$ 0 0
$$321$$ 31.8885 1.77984
$$322$$ 0 0
$$323$$ −2.47214 −0.137553
$$324$$ 0 0
$$325$$ −42.3607 −2.34975
$$326$$ 0 0
$$327$$ 25.0344 1.38441
$$328$$ 0 0
$$329$$ −11.8541 −0.653538
$$330$$ 0 0
$$331$$ 27.3050 1.50082 0.750408 0.660975i $$-0.229857\pi$$
0.750408 + 0.660975i $$0.229857\pi$$
$$332$$ 0 0
$$333$$ 0.347524 0.0190442
$$334$$ 0 0
$$335$$ −54.0689 −2.95410
$$336$$ 0 0
$$337$$ −9.23607 −0.503121 −0.251560 0.967842i $$-0.580944\pi$$
−0.251560 + 0.967842i $$0.580944\pi$$
$$338$$ 0 0
$$339$$ −44.3607 −2.40934
$$340$$ 0 0
$$341$$ 26.1803 1.41774
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −54.0689 −2.91097
$$346$$ 0 0
$$347$$ 2.47214 0.132711 0.0663556 0.997796i $$-0.478863\pi$$
0.0663556 + 0.997796i $$0.478863\pi$$
$$348$$ 0 0
$$349$$ 15.8885 0.850494 0.425247 0.905077i $$-0.360187\pi$$
0.425247 + 0.905077i $$0.360187\pi$$
$$350$$ 0 0
$$351$$ 11.7082 0.624938
$$352$$ 0 0
$$353$$ −24.3607 −1.29659 −0.648294 0.761390i $$-0.724517\pi$$
−0.648294 + 0.761390i $$0.724517\pi$$
$$354$$ 0 0
$$355$$ −49.0689 −2.60431
$$356$$ 0 0
$$357$$ 6.47214 0.342542
$$358$$ 0 0
$$359$$ −12.7639 −0.673655 −0.336827 0.941566i $$-0.609354\pi$$
−0.336827 + 0.941566i $$0.609354\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −60.9230 −3.19763
$$364$$ 0 0
$$365$$ −31.7082 −1.65968
$$366$$ 0 0
$$367$$ 5.67376 0.296168 0.148084 0.988975i $$-0.452689\pi$$
0.148084 + 0.988975i $$0.452689\pi$$
$$368$$ 0 0
$$369$$ −23.4721 −1.22191
$$370$$ 0 0
$$371$$ 3.38197 0.175583
$$372$$ 0 0
$$373$$ −9.03444 −0.467786 −0.233893 0.972262i $$-0.575146\pi$$
−0.233893 + 0.972262i $$0.575146\pi$$
$$374$$ 0 0
$$375$$ 29.2705 1.51152
$$376$$ 0 0
$$377$$ −4.47214 −0.230327
$$378$$ 0 0
$$379$$ −35.1246 −1.80423 −0.902115 0.431496i $$-0.857986\pi$$
−0.902115 + 0.431496i $$0.857986\pi$$
$$380$$ 0 0
$$381$$ −5.61803 −0.287821
$$382$$ 0 0
$$383$$ 26.0689 1.33206 0.666029 0.745926i $$-0.267993\pi$$
0.666029 + 0.745926i $$0.267993\pi$$
$$384$$ 0 0
$$385$$ −21.1803 −1.07945
$$386$$ 0 0
$$387$$ −26.4164 −1.34282
$$388$$ 0 0
$$389$$ −30.3607 −1.53935 −0.769674 0.638437i $$-0.779581\pi$$
−0.769674 + 0.638437i $$0.779581\pi$$
$$390$$ 0 0
$$391$$ 14.1115 0.713647
$$392$$ 0 0
$$393$$ −58.0689 −2.92919
$$394$$ 0 0
$$395$$ 24.7984 1.24774
$$396$$ 0 0
$$397$$ −36.5066 −1.83221 −0.916106 0.400935i $$-0.868685\pi$$
−0.916106 + 0.400935i $$0.868685\pi$$
$$398$$ 0 0
$$399$$ −2.61803 −0.131066
$$400$$ 0 0
$$401$$ 9.88854 0.493810 0.246905 0.969040i $$-0.420586\pi$$
0.246905 + 0.969040i $$0.420586\pi$$
$$402$$ 0 0
$$403$$ −23.4164 −1.16645
$$404$$ 0 0
$$405$$ 20.6525 1.02623
$$406$$ 0 0
$$407$$ 0.527864 0.0261652
$$408$$ 0 0
$$409$$ −9.09017 −0.449480 −0.224740 0.974419i $$-0.572153\pi$$
−0.224740 + 0.974419i $$0.572153\pi$$
$$410$$ 0 0
$$411$$ −48.5967 −2.39710
$$412$$ 0 0
$$413$$ −4.09017 −0.201264
$$414$$ 0 0
$$415$$ 14.4721 0.710409
$$416$$ 0 0
$$417$$ −6.47214 −0.316942
$$418$$ 0 0
$$419$$ −28.4721 −1.39095 −0.695477 0.718548i $$-0.744807\pi$$
−0.695477 + 0.718548i $$0.744807\pi$$
$$420$$ 0 0
$$421$$ 20.4721 0.997751 0.498875 0.866674i $$-0.333747\pi$$
0.498875 + 0.866674i $$0.333747\pi$$
$$422$$ 0 0
$$423$$ −45.6869 −2.22137
$$424$$ 0 0
$$425$$ −20.0000 −0.970143
$$426$$ 0 0
$$427$$ 7.85410 0.380087
$$428$$ 0 0
$$429$$ 80.2492 3.87447
$$430$$ 0 0
$$431$$ −23.0902 −1.11221 −0.556107 0.831111i $$-0.687706\pi$$
−0.556107 + 0.831111i $$0.687706\pi$$
$$432$$ 0 0
$$433$$ −18.2705 −0.878025 −0.439012 0.898481i $$-0.644672\pi$$
−0.439012 + 0.898481i $$0.644672\pi$$
$$434$$ 0 0
$$435$$ 8.09017 0.387894
$$436$$ 0 0
$$437$$ −5.70820 −0.273060
$$438$$ 0 0
$$439$$ −23.3050 −1.11228 −0.556142 0.831087i $$-0.687719\pi$$
−0.556142 + 0.831087i $$0.687719\pi$$
$$440$$ 0 0
$$441$$ 3.85410 0.183529
$$442$$ 0 0
$$443$$ 6.27051 0.297921 0.148960 0.988843i $$-0.452407\pi$$
0.148960 + 0.988843i $$0.452407\pi$$
$$444$$ 0 0
$$445$$ −3.94427 −0.186976
$$446$$ 0 0
$$447$$ −23.4164 −1.10756
$$448$$ 0 0
$$449$$ 26.1803 1.23553 0.617763 0.786364i $$-0.288039\pi$$
0.617763 + 0.786364i $$0.288039\pi$$
$$450$$ 0 0
$$451$$ −35.6525 −1.67881
$$452$$ 0 0
$$453$$ 52.3607 2.46012
$$454$$ 0 0
$$455$$ 18.9443 0.888121
$$456$$ 0 0
$$457$$ 29.2705 1.36922 0.684608 0.728911i $$-0.259973\pi$$
0.684608 + 0.728911i $$0.259973\pi$$
$$458$$ 0 0
$$459$$ 5.52786 0.258019
$$460$$ 0 0
$$461$$ −26.0344 −1.21254 −0.606272 0.795257i $$-0.707336\pi$$
−0.606272 + 0.795257i $$0.707336\pi$$
$$462$$ 0 0
$$463$$ 5.23607 0.243341 0.121670 0.992571i $$-0.461175\pi$$
0.121670 + 0.992571i $$0.461175\pi$$
$$464$$ 0 0
$$465$$ 42.3607 1.96443
$$466$$ 0 0
$$467$$ 4.18034 0.193443 0.0967215 0.995311i $$-0.469164\pi$$
0.0967215 + 0.995311i $$0.469164\pi$$
$$468$$ 0 0
$$469$$ 14.9443 0.690062
$$470$$ 0 0
$$471$$ 48.7426 2.24594
$$472$$ 0 0
$$473$$ −40.1246 −1.84493
$$474$$ 0 0
$$475$$ 8.09017 0.371202
$$476$$ 0 0
$$477$$ 13.0344 0.596806
$$478$$ 0 0
$$479$$ −37.3951 −1.70863 −0.854313 0.519758i $$-0.826022\pi$$
−0.854313 + 0.519758i $$0.826022\pi$$
$$480$$ 0 0
$$481$$ −0.472136 −0.0215275
$$482$$ 0 0
$$483$$ 14.9443 0.679988
$$484$$ 0 0
$$485$$ 12.8885 0.585239
$$486$$ 0 0
$$487$$ 17.1459 0.776955 0.388477 0.921458i $$-0.373001\pi$$
0.388477 + 0.921458i $$0.373001\pi$$
$$488$$ 0 0
$$489$$ 4.09017 0.184964
$$490$$ 0 0
$$491$$ 5.88854 0.265746 0.132873 0.991133i $$-0.457580\pi$$
0.132873 + 0.991133i $$0.457580\pi$$
$$492$$ 0 0
$$493$$ −2.11146 −0.0950952
$$494$$ 0 0
$$495$$ −81.6312 −3.66905
$$496$$ 0 0
$$497$$ 13.5623 0.608353
$$498$$ 0 0
$$499$$ −31.0344 −1.38929 −0.694646 0.719352i $$-0.744439\pi$$
−0.694646 + 0.719352i $$0.744439\pi$$
$$500$$ 0 0
$$501$$ 5.23607 0.233930
$$502$$ 0 0
$$503$$ −7.14590 −0.318620 −0.159310 0.987229i $$-0.550927\pi$$
−0.159310 + 0.987229i $$0.550927\pi$$
$$504$$ 0 0
$$505$$ −39.5967 −1.76203
$$506$$ 0 0
$$507$$ −37.7426 −1.67621
$$508$$ 0 0
$$509$$ −4.76393 −0.211158 −0.105579 0.994411i $$-0.533670\pi$$
−0.105579 + 0.994411i $$0.533670\pi$$
$$510$$ 0 0
$$511$$ 8.76393 0.387694
$$512$$ 0 0
$$513$$ −2.23607 −0.0987248
$$514$$ 0 0
$$515$$ −49.5967 −2.18549
$$516$$ 0 0
$$517$$ −69.3951 −3.05199
$$518$$ 0 0
$$519$$ 0.472136 0.0207245
$$520$$ 0 0
$$521$$ 23.3050 1.02101 0.510504 0.859875i $$-0.329459\pi$$
0.510504 + 0.859875i $$0.329459\pi$$
$$522$$ 0 0
$$523$$ −33.8885 −1.48184 −0.740921 0.671592i $$-0.765611\pi$$
−0.740921 + 0.671592i $$0.765611\pi$$
$$524$$ 0 0
$$525$$ −21.1803 −0.924386
$$526$$ 0 0
$$527$$ −11.0557 −0.481595
$$528$$ 0 0
$$529$$ 9.58359 0.416678
$$530$$ 0 0
$$531$$ −15.7639 −0.684096
$$532$$ 0 0
$$533$$ 31.8885 1.38125
$$534$$ 0 0
$$535$$ 44.0689 1.90526
$$536$$ 0 0
$$537$$ 55.3050 2.38658
$$538$$ 0 0
$$539$$ 5.85410 0.252154
$$540$$ 0 0
$$541$$ −13.7082 −0.589362 −0.294681 0.955596i $$-0.595213\pi$$
−0.294681 + 0.955596i $$0.595213\pi$$
$$542$$ 0 0
$$543$$ −18.9443 −0.812977
$$544$$ 0 0
$$545$$ 34.5967 1.48196
$$546$$ 0 0
$$547$$ 25.7082 1.09920 0.549602 0.835427i $$-0.314779\pi$$
0.549602 + 0.835427i $$0.314779\pi$$
$$548$$ 0 0
$$549$$ 30.2705 1.29191
$$550$$ 0 0
$$551$$ 0.854102 0.0363860
$$552$$ 0 0
$$553$$ −6.85410 −0.291466
$$554$$ 0 0
$$555$$ 0.854102 0.0362546
$$556$$ 0 0
$$557$$ −1.81966 −0.0771015 −0.0385507 0.999257i $$-0.512274\pi$$
−0.0385507 + 0.999257i $$0.512274\pi$$
$$558$$ 0 0
$$559$$ 35.8885 1.51792
$$560$$ 0 0
$$561$$ 37.8885 1.59966
$$562$$ 0 0
$$563$$ −20.7984 −0.876547 −0.438273 0.898842i $$-0.644410\pi$$
−0.438273 + 0.898842i $$0.644410\pi$$
$$564$$ 0 0
$$565$$ −61.3050 −2.57912
$$566$$ 0 0
$$567$$ −5.70820 −0.239722
$$568$$ 0 0
$$569$$ 13.7082 0.574678 0.287339 0.957829i $$-0.407229\pi$$
0.287339 + 0.957829i $$0.407229\pi$$
$$570$$ 0 0
$$571$$ −34.2148 −1.43184 −0.715922 0.698180i $$-0.753993\pi$$
−0.715922 + 0.698180i $$0.753993\pi$$
$$572$$ 0 0
$$573$$ −4.47214 −0.186826
$$574$$ 0 0
$$575$$ −46.1803 −1.92585
$$576$$ 0 0
$$577$$ −18.9443 −0.788660 −0.394330 0.918969i $$-0.629023\pi$$
−0.394330 + 0.918969i $$0.629023\pi$$
$$578$$ 0 0
$$579$$ −18.9443 −0.787297
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ 19.7984 0.819965
$$584$$ 0 0
$$585$$ 73.0132 3.01872
$$586$$ 0 0
$$587$$ −21.1246 −0.871906 −0.435953 0.899969i $$-0.643589\pi$$
−0.435953 + 0.899969i $$0.643589\pi$$
$$588$$ 0 0
$$589$$ 4.47214 0.184271
$$590$$ 0 0
$$591$$ 47.1246 1.93845
$$592$$ 0 0
$$593$$ −13.1246 −0.538963 −0.269482 0.963006i $$-0.586852\pi$$
−0.269482 + 0.963006i $$0.586852\pi$$
$$594$$ 0 0
$$595$$ 8.94427 0.366679
$$596$$ 0 0
$$597$$ −2.70820 −0.110839
$$598$$ 0 0
$$599$$ −42.7984 −1.74869 −0.874347 0.485301i $$-0.838710\pi$$
−0.874347 + 0.485301i $$0.838710\pi$$
$$600$$ 0 0
$$601$$ 7.88854 0.321780 0.160890 0.986972i $$-0.448563\pi$$
0.160890 + 0.986972i $$0.448563\pi$$
$$602$$ 0 0
$$603$$ 57.5967 2.34552
$$604$$ 0 0
$$605$$ −84.1935 −3.42295
$$606$$ 0 0
$$607$$ 9.23607 0.374880 0.187440 0.982276i $$-0.439981\pi$$
0.187440 + 0.982276i $$0.439981\pi$$
$$608$$ 0 0
$$609$$ −2.23607 −0.0906100
$$610$$ 0 0
$$611$$ 62.0689 2.51104
$$612$$ 0 0
$$613$$ −16.9443 −0.684373 −0.342186 0.939632i $$-0.611167\pi$$
−0.342186 + 0.939632i $$0.611167\pi$$
$$614$$ 0 0
$$615$$ −57.6869 −2.32616
$$616$$ 0 0
$$617$$ 31.9098 1.28464 0.642321 0.766436i $$-0.277972\pi$$
0.642321 + 0.766436i $$0.277972\pi$$
$$618$$ 0 0
$$619$$ −18.2918 −0.735209 −0.367605 0.929982i $$-0.619822\pi$$
−0.367605 + 0.929982i $$0.619822\pi$$
$$620$$ 0 0
$$621$$ 12.7639 0.512199
$$622$$ 0 0
$$623$$ 1.09017 0.0436767
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −15.3262 −0.612071
$$628$$ 0 0
$$629$$ −0.222912 −0.00888810
$$630$$ 0 0
$$631$$ 40.3607 1.60673 0.803367 0.595485i $$-0.203040\pi$$
0.803367 + 0.595485i $$0.203040\pi$$
$$632$$ 0 0
$$633$$ −36.3607 −1.44521
$$634$$ 0 0
$$635$$ −7.76393 −0.308102
$$636$$ 0 0
$$637$$ −5.23607 −0.207461
$$638$$ 0 0
$$639$$ 52.2705 2.06779
$$640$$ 0 0
$$641$$ −22.6525 −0.894719 −0.447360 0.894354i $$-0.647636\pi$$
−0.447360 + 0.894354i $$0.647636\pi$$
$$642$$ 0 0
$$643$$ −37.5279 −1.47995 −0.739977 0.672632i $$-0.765164\pi$$
−0.739977 + 0.672632i $$0.765164\pi$$
$$644$$ 0 0
$$645$$ −64.9230 −2.55634
$$646$$ 0 0
$$647$$ −20.2705 −0.796916 −0.398458 0.917187i $$-0.630455\pi$$
−0.398458 + 0.917187i $$0.630455\pi$$
$$648$$ 0 0
$$649$$ −23.9443 −0.939895
$$650$$ 0 0
$$651$$ −11.7082 −0.458881
$$652$$ 0 0
$$653$$ 32.6525 1.27779 0.638895 0.769294i $$-0.279392\pi$$
0.638895 + 0.769294i $$0.279392\pi$$
$$654$$ 0 0
$$655$$ −80.2492 −3.13560
$$656$$ 0 0
$$657$$ 33.7771 1.31777
$$658$$ 0 0
$$659$$ 8.94427 0.348419 0.174210 0.984709i $$-0.444263\pi$$
0.174210 + 0.984709i $$0.444263\pi$$
$$660$$ 0 0
$$661$$ 16.8328 0.654721 0.327360 0.944900i $$-0.393841\pi$$
0.327360 + 0.944900i $$0.393841\pi$$
$$662$$ 0 0
$$663$$ −33.8885 −1.31612
$$664$$ 0 0
$$665$$ −3.61803 −0.140301
$$666$$ 0 0
$$667$$ −4.87539 −0.188776
$$668$$ 0 0
$$669$$ −5.70820 −0.220692
$$670$$ 0 0
$$671$$ 45.9787 1.77499
$$672$$ 0 0
$$673$$ −25.2361 −0.972779 −0.486389 0.873742i $$-0.661686\pi$$
−0.486389 + 0.873742i $$0.661686\pi$$
$$674$$ 0 0
$$675$$ −18.0902 −0.696291
$$676$$ 0 0
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ −3.56231 −0.136709
$$680$$ 0 0
$$681$$ −33.8885 −1.29861
$$682$$ 0 0
$$683$$ −23.8885 −0.914070 −0.457035 0.889449i $$-0.651089\pi$$
−0.457035 + 0.889449i $$0.651089\pi$$
$$684$$ 0 0
$$685$$ −67.1591 −2.56602
$$686$$ 0 0
$$687$$ −24.4164 −0.931544
$$688$$ 0 0
$$689$$ −17.7082 −0.674629
$$690$$ 0 0
$$691$$ 38.6525 1.47041 0.735205 0.677845i $$-0.237086\pi$$
0.735205 + 0.677845i $$0.237086\pi$$
$$692$$ 0 0
$$693$$ 22.5623 0.857071
$$694$$ 0 0
$$695$$ −8.94427 −0.339276
$$696$$ 0 0
$$697$$ 15.0557 0.570276
$$698$$ 0 0
$$699$$ 26.5623 1.00468
$$700$$ 0 0
$$701$$ −39.2361 −1.48193 −0.740963 0.671546i $$-0.765631\pi$$
−0.740963 + 0.671546i $$0.765631\pi$$
$$702$$ 0 0
$$703$$ 0.0901699 0.00340082
$$704$$ 0 0
$$705$$ −112.284 −4.22885
$$706$$ 0 0
$$707$$ 10.9443 0.411602
$$708$$ 0 0
$$709$$ −18.1803 −0.682777 −0.341388 0.939922i $$-0.610897\pi$$
−0.341388 + 0.939922i $$0.610897\pi$$
$$710$$ 0 0
$$711$$ −26.4164 −0.990693
$$712$$ 0 0
$$713$$ −25.5279 −0.956026
$$714$$ 0 0
$$715$$ 110.902 4.14749
$$716$$ 0 0
$$717$$ 21.7082 0.810708
$$718$$ 0 0
$$719$$ 25.8885 0.965480 0.482740 0.875764i $$-0.339642\pi$$
0.482740 + 0.875764i $$0.339642\pi$$
$$720$$ 0 0
$$721$$ 13.7082 0.510520
$$722$$ 0 0
$$723$$ −11.9443 −0.444212
$$724$$ 0 0
$$725$$ 6.90983 0.256625
$$726$$ 0 0
$$727$$ 15.5623 0.577174 0.288587 0.957454i $$-0.406815\pi$$
0.288587 + 0.957454i $$0.406815\pi$$
$$728$$ 0 0
$$729$$ −39.5623 −1.46527
$$730$$ 0 0
$$731$$ 16.9443 0.626707
$$732$$ 0 0
$$733$$ 27.4377 1.01343 0.506717 0.862112i $$-0.330859\pi$$
0.506717 + 0.862112i $$0.330859\pi$$
$$734$$ 0 0
$$735$$ 9.47214 0.349385
$$736$$ 0 0
$$737$$ 87.4853 3.22256
$$738$$ 0 0
$$739$$ 32.5623 1.19782 0.598912 0.800815i $$-0.295600\pi$$
0.598912 + 0.800815i $$0.295600\pi$$
$$740$$ 0 0
$$741$$ 13.7082 0.503583
$$742$$ 0 0
$$743$$ 3.79837 0.139349 0.0696744 0.997570i $$-0.477804\pi$$
0.0696744 + 0.997570i $$0.477804\pi$$
$$744$$ 0 0
$$745$$ −32.3607 −1.18560
$$746$$ 0 0
$$747$$ −15.4164 −0.564057
$$748$$ 0 0
$$749$$ −12.1803 −0.445060
$$750$$ 0 0
$$751$$ 13.9098 0.507577 0.253788 0.967260i $$-0.418323\pi$$
0.253788 + 0.967260i $$0.418323\pi$$
$$752$$ 0 0
$$753$$ 76.5410 2.78931
$$754$$ 0 0
$$755$$ 72.3607 2.63347
$$756$$ 0 0
$$757$$ −21.5279 −0.782444 −0.391222 0.920296i $$-0.627947\pi$$
−0.391222 + 0.920296i $$0.627947\pi$$
$$758$$ 0 0
$$759$$ 87.4853 3.17551
$$760$$ 0 0
$$761$$ −10.9443 −0.396730 −0.198365 0.980128i $$-0.563563\pi$$
−0.198365 + 0.980128i $$0.563563\pi$$
$$762$$ 0 0
$$763$$ −9.56231 −0.346179
$$764$$ 0 0
$$765$$ 34.4721 1.24634
$$766$$ 0 0
$$767$$ 21.4164 0.773302
$$768$$ 0 0
$$769$$ 30.6525 1.10536 0.552678 0.833395i $$-0.313606\pi$$
0.552678 + 0.833395i $$0.313606\pi$$
$$770$$ 0 0
$$771$$ 7.00000 0.252099
$$772$$ 0 0
$$773$$ 37.7082 1.35627 0.678135 0.734937i $$-0.262789\pi$$
0.678135 + 0.734937i $$0.262789\pi$$
$$774$$ 0 0
$$775$$ 36.1803 1.29964
$$776$$ 0 0
$$777$$ −0.236068 −0.00846889
$$778$$ 0 0
$$779$$ −6.09017 −0.218203
$$780$$ 0 0
$$781$$ 79.3951 2.84098
$$782$$ 0 0
$$783$$ −1.90983 −0.0682518
$$784$$ 0 0
$$785$$ 67.3607 2.40421
$$786$$ 0 0
$$787$$ 28.6738 1.02211 0.511055 0.859548i $$-0.329255\pi$$
0.511055 + 0.859548i $$0.329255\pi$$
$$788$$ 0 0
$$789$$ 66.0689 2.35211
$$790$$ 0 0
$$791$$ 16.9443 0.602469
$$792$$ 0 0
$$793$$ −41.1246 −1.46038
$$794$$ 0 0
$$795$$ 32.0344 1.13614
$$796$$ 0 0
$$797$$ −49.7082 −1.76075 −0.880377 0.474274i $$-0.842711\pi$$
−0.880377 + 0.474274i $$0.842711\pi$$
$$798$$ 0 0
$$799$$ 29.3050 1.03673
$$800$$ 0 0
$$801$$ 4.20163 0.148457
$$802$$ 0 0
$$803$$ 51.3050 1.81051
$$804$$ 0 0
$$805$$ 20.6525 0.727904
$$806$$ 0 0
$$807$$ −77.0132 −2.71099
$$808$$ 0 0
$$809$$ 9.03444 0.317634 0.158817 0.987308i $$-0.449232\pi$$
0.158817 + 0.987308i $$0.449232\pi$$
$$810$$ 0 0
$$811$$ 5.45085 0.191405 0.0957026 0.995410i $$-0.469490\pi$$
0.0957026 + 0.995410i $$0.469490\pi$$
$$812$$ 0 0
$$813$$ −44.1246 −1.54752
$$814$$ 0 0
$$815$$ 5.65248 0.197998
$$816$$ 0 0
$$817$$ −6.85410 −0.239795
$$818$$ 0 0
$$819$$ −20.1803 −0.705158
$$820$$ 0 0
$$821$$ −33.4164 −1.16624 −0.583120 0.812386i $$-0.698168\pi$$
−0.583120 + 0.812386i $$0.698168\pi$$
$$822$$ 0 0
$$823$$ −29.4164 −1.02539 −0.512696 0.858570i $$-0.671353\pi$$
−0.512696 + 0.858570i $$0.671353\pi$$
$$824$$ 0 0
$$825$$ −123.992 −4.31684
$$826$$ 0 0
$$827$$ −18.0689 −0.628317 −0.314158 0.949371i $$-0.601722\pi$$
−0.314158 + 0.949371i $$0.601722\pi$$
$$828$$ 0 0
$$829$$ 31.4164 1.09114 0.545568 0.838066i $$-0.316314\pi$$
0.545568 + 0.838066i $$0.316314\pi$$
$$830$$ 0 0
$$831$$ −47.8885 −1.66124
$$832$$ 0 0
$$833$$ −2.47214 −0.0856544
$$834$$ 0 0
$$835$$ 7.23607 0.250414
$$836$$ 0 0
$$837$$ −10.0000 −0.345651
$$838$$ 0 0
$$839$$ −21.0557 −0.726924 −0.363462 0.931609i $$-0.618405\pi$$
−0.363462 + 0.931609i $$0.618405\pi$$
$$840$$ 0 0
$$841$$ −28.2705 −0.974845
$$842$$ 0 0
$$843$$ 46.8328 1.61301
$$844$$ 0 0
$$845$$ −52.1591 −1.79433
$$846$$ 0 0
$$847$$ 23.2705 0.799584
$$848$$ 0 0
$$849$$ 14.4721 0.496682
$$850$$ 0 0
$$851$$ −0.514708 −0.0176440
$$852$$ 0 0
$$853$$ −1.90983 −0.0653913 −0.0326957 0.999465i $$-0.510409\pi$$
−0.0326957 + 0.999465i $$0.510409\pi$$
$$854$$ 0 0
$$855$$ −13.9443 −0.476884
$$856$$ 0 0
$$857$$ 50.0000 1.70797 0.853984 0.520300i $$-0.174180\pi$$
0.853984 + 0.520300i $$0.174180\pi$$
$$858$$ 0 0
$$859$$ 7.59675 0.259198 0.129599 0.991567i $$-0.458631\pi$$
0.129599 + 0.991567i $$0.458631\pi$$
$$860$$ 0 0
$$861$$ 15.9443 0.543379
$$862$$ 0 0
$$863$$ 6.27051 0.213451 0.106725 0.994289i $$-0.465963\pi$$
0.106725 + 0.994289i $$0.465963\pi$$
$$864$$ 0 0
$$865$$ 0.652476 0.0221848
$$866$$ 0 0
$$867$$ 28.5066 0.968134
$$868$$ 0 0
$$869$$ −40.1246 −1.36113
$$870$$ 0 0
$$871$$ −78.2492 −2.65137
$$872$$ 0 0
$$873$$ −13.7295 −0.464673
$$874$$ 0 0
$$875$$ −11.1803 −0.377964
$$876$$ 0 0
$$877$$ 34.4377 1.16288 0.581439 0.813590i $$-0.302490\pi$$
0.581439 + 0.813590i $$0.302490\pi$$
$$878$$ 0 0
$$879$$ 13.7082 0.462366
$$880$$ 0 0
$$881$$ 17.5967 0.592849 0.296425 0.955056i $$-0.404206\pi$$
0.296425 + 0.955056i $$0.404206\pi$$
$$882$$ 0 0
$$883$$ 19.9656 0.671895 0.335947 0.941881i $$-0.390944\pi$$
0.335947 + 0.941881i $$0.390944\pi$$
$$884$$ 0 0
$$885$$ −38.7426 −1.30232
$$886$$ 0 0
$$887$$ −17.4164 −0.584786 −0.292393 0.956298i $$-0.594451\pi$$
−0.292393 + 0.956298i $$0.594451\pi$$
$$888$$ 0 0
$$889$$ 2.14590 0.0719711
$$890$$ 0 0
$$891$$ −33.4164 −1.11949
$$892$$ 0 0
$$893$$ −11.8541 −0.396682
$$894$$ 0 0
$$895$$ 76.4296 2.55476
$$896$$ 0 0
$$897$$ −78.2492 −2.61267
$$898$$ 0 0
$$899$$ 3.81966 0.127393
$$900$$ 0 0
$$901$$ −8.36068 −0.278535
$$902$$ 0 0
$$903$$ 17.9443 0.597148
$$904$$ 0 0
$$905$$ −26.1803 −0.870264
$$906$$ 0 0
$$907$$ −29.0132 −0.963366 −0.481683 0.876346i $$-0.659974\pi$$
−0.481683 + 0.876346i $$0.659974\pi$$
$$908$$ 0 0
$$909$$ 42.1803 1.39903
$$910$$ 0 0
$$911$$ −43.9787 −1.45708 −0.728540 0.685003i $$-0.759801\pi$$
−0.728540 + 0.685003i $$0.759801\pi$$
$$912$$ 0 0
$$913$$ −23.4164 −0.774970
$$914$$ 0 0
$$915$$ 74.3951 2.45943
$$916$$ 0 0
$$917$$ 22.1803 0.732459
$$918$$ 0 0
$$919$$ 39.5967 1.30618 0.653088 0.757282i $$-0.273473\pi$$
0.653088 + 0.757282i $$0.273473\pi$$
$$920$$ 0 0
$$921$$ 10.2361 0.337290
$$922$$ 0 0
$$923$$ −71.0132 −2.33743
$$924$$ 0 0
$$925$$ 0.729490 0.0239855
$$926$$ 0 0
$$927$$ 52.8328 1.73526
$$928$$ 0 0
$$929$$ −2.76393 −0.0906817 −0.0453408 0.998972i $$-0.514437\pi$$
−0.0453408 + 0.998972i $$0.514437\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ −66.6312 −2.18141
$$934$$ 0 0
$$935$$ 52.3607 1.71238
$$936$$ 0 0
$$937$$ −25.8885 −0.845742 −0.422871 0.906190i $$-0.638978\pi$$
−0.422871 + 0.906190i $$0.638978\pi$$
$$938$$ 0 0
$$939$$ 20.9443 0.683490
$$940$$ 0 0
$$941$$ 58.5410 1.90838 0.954191 0.299197i $$-0.0967188\pi$$
0.954191 + 0.299197i $$0.0967188\pi$$
$$942$$ 0 0
$$943$$ 34.7639 1.13207
$$944$$ 0 0
$$945$$ 8.09017 0.263173
$$946$$ 0 0
$$947$$ −12.6869 −0.412269 −0.206135 0.978524i $$-0.566089\pi$$
−0.206135 + 0.978524i $$0.566089\pi$$
$$948$$ 0 0
$$949$$ −45.8885 −1.48961
$$950$$ 0 0
$$951$$ −20.0902 −0.651468
$$952$$ 0 0
$$953$$ −36.4721 −1.18145 −0.590724 0.806874i $$-0.701158\pi$$
−0.590724 + 0.806874i $$0.701158\pi$$
$$954$$ 0 0
$$955$$ −6.18034 −0.199991
$$956$$ 0 0
$$957$$ −13.0902 −0.423145
$$958$$ 0 0
$$959$$ 18.5623 0.599408
$$960$$ 0 0
$$961$$ −11.0000 −0.354839
$$962$$ 0 0
$$963$$ −46.9443 −1.51276
$$964$$ 0 0
$$965$$ −26.1803 −0.842775
$$966$$ 0 0
$$967$$ −15.4164 −0.495758 −0.247879 0.968791i $$-0.579734\pi$$
−0.247879 + 0.968791i $$0.579734\pi$$
$$968$$ 0 0
$$969$$ 6.47214 0.207915
$$970$$ 0 0
$$971$$ 4.43769 0.142412 0.0712062 0.997462i $$-0.477315\pi$$
0.0712062 + 0.997462i $$0.477315\pi$$
$$972$$ 0 0
$$973$$ 2.47214 0.0792530
$$974$$ 0 0
$$975$$ 110.902 3.55170
$$976$$ 0 0
$$977$$ −2.06888 −0.0661895 −0.0330947 0.999452i $$-0.510536\pi$$
−0.0330947 + 0.999452i $$0.510536\pi$$
$$978$$ 0 0
$$979$$ 6.38197 0.203969
$$980$$ 0 0
$$981$$ −36.8541 −1.17666
$$982$$ 0 0
$$983$$ −31.0557 −0.990524 −0.495262 0.868744i $$-0.664928\pi$$
−0.495262 + 0.868744i $$0.664928\pi$$
$$984$$ 0 0
$$985$$ 65.1246 2.07504
$$986$$ 0 0
$$987$$ 31.0344 0.987837
$$988$$ 0 0
$$989$$ 39.1246 1.24409
$$990$$ 0 0
$$991$$ 23.6738 0.752022 0.376011 0.926615i $$-0.377296\pi$$
0.376011 + 0.926615i $$0.377296\pi$$
$$992$$ 0 0
$$993$$ −71.4853 −2.26852
$$994$$ 0 0
$$995$$ −3.74265 −0.118650
$$996$$ 0 0
$$997$$ 14.4508 0.457663 0.228832 0.973466i $$-0.426510\pi$$
0.228832 + 0.973466i $$0.426510\pi$$
$$998$$ 0 0
$$999$$ −0.201626 −0.00637917
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 8512.2.a.e.1.1 2
4.3 odd 2 8512.2.a.ba.1.2 2
8.3 odd 2 2128.2.a.d.1.1 2
8.5 even 2 1064.2.a.e.1.2 2
24.5 odd 2 9576.2.a.bc.1.1 2
56.13 odd 2 7448.2.a.y.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.a.e.1.2 2 8.5 even 2
2128.2.a.d.1.1 2 8.3 odd 2
7448.2.a.y.1.1 2 56.13 odd 2
8512.2.a.e.1.1 2 1.1 even 1 trivial
8512.2.a.ba.1.2 2 4.3 odd 2
9576.2.a.bc.1.1 2 24.5 odd 2