# Properties

 Label 8092.2.a.m.1.2 Level $8092$ Weight $2$ Character 8092.1 Self dual yes Analytic conductor $64.615$ 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] = [8092,2,Mod(1,8092)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8092, 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("8092.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8092 = 2^{2} \cdot 7 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8092.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: $$64.6149453156$$ 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 476) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 8092.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.61803 q^{3} -0.618034 q^{5} +1.00000 q^{7} -0.381966 q^{9} +O(q^{10})$$ $$q+1.61803 q^{3} -0.618034 q^{5} +1.00000 q^{7} -0.381966 q^{9} +0.763932 q^{11} +1.23607 q^{13} -1.00000 q^{15} -8.47214 q^{19} +1.61803 q^{21} +7.70820 q^{23} -4.61803 q^{25} -5.47214 q^{27} +5.70820 q^{29} -6.32624 q^{31} +1.23607 q^{33} -0.618034 q^{35} -0.472136 q^{37} +2.00000 q^{39} +0.0901699 q^{41} -12.0902 q^{43} +0.236068 q^{45} -8.47214 q^{47} +1.00000 q^{49} +10.7984 q^{53} -0.472136 q^{55} -13.7082 q^{57} +7.32624 q^{61} -0.381966 q^{63} -0.763932 q^{65} -13.0902 q^{67} +12.4721 q^{69} -10.9443 q^{71} -7.14590 q^{73} -7.47214 q^{75} +0.763932 q^{77} -2.94427 q^{79} -7.70820 q^{81} +15.4164 q^{83} +9.23607 q^{87} +2.00000 q^{89} +1.23607 q^{91} -10.2361 q^{93} +5.23607 q^{95} -15.0902 q^{97} -0.291796 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9 $$2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9} + 6 q^{11} - 2 q^{13} - 2 q^{15} - 8 q^{19} + q^{21} + 2 q^{23} - 7 q^{25} - 2 q^{27} - 2 q^{29} + 3 q^{31} - 2 q^{33} + q^{35} + 8 q^{37} + 4 q^{39} - 11 q^{41} - 13 q^{43} - 4 q^{45} - 8 q^{47} + 2 q^{49} - 3 q^{53} + 8 q^{55} - 14 q^{57} - q^{61} - 3 q^{63} - 6 q^{65} - 15 q^{67} + 16 q^{69} - 4 q^{71} - 21 q^{73} - 6 q^{75} + 6 q^{77} + 12 q^{79} - 2 q^{81} + 4 q^{83} + 14 q^{87} + 4 q^{89} - 2 q^{91} - 16 q^{93} + 6 q^{95} - 19 q^{97} - 14 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9 + 6 * q^11 - 2 * q^13 - 2 * q^15 - 8 * q^19 + q^21 + 2 * q^23 - 7 * q^25 - 2 * q^27 - 2 * q^29 + 3 * q^31 - 2 * q^33 + q^35 + 8 * q^37 + 4 * q^39 - 11 * q^41 - 13 * q^43 - 4 * q^45 - 8 * q^47 + 2 * q^49 - 3 * q^53 + 8 * q^55 - 14 * q^57 - q^61 - 3 * q^63 - 6 * q^65 - 15 * q^67 + 16 * q^69 - 4 * q^71 - 21 * q^73 - 6 * q^75 + 6 * q^77 + 12 * q^79 - 2 * q^81 + 4 * q^83 + 14 * q^87 + 4 * q^89 - 2 * q^91 - 16 * q^93 + 6 * q^95 - 19 * q^97 - 14 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.61803 0.934172 0.467086 0.884212i $$-0.345304\pi$$
0.467086 + 0.884212i $$0.345304\pi$$
$$4$$ 0 0
$$5$$ −0.618034 −0.276393 −0.138197 0.990405i $$-0.544131\pi$$
−0.138197 + 0.990405i $$0.544131\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −0.381966 −0.127322
$$10$$ 0 0
$$11$$ 0.763932 0.230334 0.115167 0.993346i $$-0.463260\pi$$
0.115167 + 0.993346i $$0.463260\pi$$
$$12$$ 0 0
$$13$$ 1.23607 0.342824 0.171412 0.985199i $$-0.445167\pi$$
0.171412 + 0.985199i $$0.445167\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 0 0
$$18$$ 0 0
$$19$$ −8.47214 −1.94364 −0.971821 0.235722i $$-0.924255\pi$$
−0.971821 + 0.235722i $$0.924255\pi$$
$$20$$ 0 0
$$21$$ 1.61803 0.353084
$$22$$ 0 0
$$23$$ 7.70820 1.60727 0.803636 0.595121i $$-0.202896\pi$$
0.803636 + 0.595121i $$0.202896\pi$$
$$24$$ 0 0
$$25$$ −4.61803 −0.923607
$$26$$ 0 0
$$27$$ −5.47214 −1.05311
$$28$$ 0 0
$$29$$ 5.70820 1.05999 0.529993 0.848002i $$-0.322194\pi$$
0.529993 + 0.848002i $$0.322194\pi$$
$$30$$ 0 0
$$31$$ −6.32624 −1.13623 −0.568113 0.822951i $$-0.692326\pi$$
−0.568113 + 0.822951i $$0.692326\pi$$
$$32$$ 0 0
$$33$$ 1.23607 0.215172
$$34$$ 0 0
$$35$$ −0.618034 −0.104467
$$36$$ 0 0
$$37$$ −0.472136 −0.0776187 −0.0388093 0.999247i $$-0.512356\pi$$
−0.0388093 + 0.999247i $$0.512356\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 0.0901699 0.0140822 0.00704109 0.999975i $$-0.497759\pi$$
0.00704109 + 0.999975i $$0.497759\pi$$
$$42$$ 0 0
$$43$$ −12.0902 −1.84373 −0.921867 0.387507i $$-0.873336\pi$$
−0.921867 + 0.387507i $$0.873336\pi$$
$$44$$ 0 0
$$45$$ 0.236068 0.0351909
$$46$$ 0 0
$$47$$ −8.47214 −1.23579 −0.617894 0.786261i $$-0.712014\pi$$
−0.617894 + 0.786261i $$0.712014\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 10.7984 1.48327 0.741635 0.670803i $$-0.234050\pi$$
0.741635 + 0.670803i $$0.234050\pi$$
$$54$$ 0 0
$$55$$ −0.472136 −0.0636628
$$56$$ 0 0
$$57$$ −13.7082 −1.81570
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 7.32624 0.938029 0.469014 0.883191i $$-0.344609\pi$$
0.469014 + 0.883191i $$0.344609\pi$$
$$62$$ 0 0
$$63$$ −0.381966 −0.0481232
$$64$$ 0 0
$$65$$ −0.763932 −0.0947541
$$66$$ 0 0
$$67$$ −13.0902 −1.59922 −0.799609 0.600520i $$-0.794960\pi$$
−0.799609 + 0.600520i $$0.794960\pi$$
$$68$$ 0 0
$$69$$ 12.4721 1.50147
$$70$$ 0 0
$$71$$ −10.9443 −1.29885 −0.649423 0.760427i $$-0.724990\pi$$
−0.649423 + 0.760427i $$0.724990\pi$$
$$72$$ 0 0
$$73$$ −7.14590 −0.836364 −0.418182 0.908363i $$-0.637333\pi$$
−0.418182 + 0.908363i $$0.637333\pi$$
$$74$$ 0 0
$$75$$ −7.47214 −0.862808
$$76$$ 0 0
$$77$$ 0.763932 0.0870581
$$78$$ 0 0
$$79$$ −2.94427 −0.331256 −0.165628 0.986188i $$-0.552965\pi$$
−0.165628 + 0.986188i $$0.552965\pi$$
$$80$$ 0 0
$$81$$ −7.70820 −0.856467
$$82$$ 0 0
$$83$$ 15.4164 1.69217 0.846085 0.533048i $$-0.178953\pi$$
0.846085 + 0.533048i $$0.178953\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 9.23607 0.990210
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 1.23607 0.129575
$$92$$ 0 0
$$93$$ −10.2361 −1.06143
$$94$$ 0 0
$$95$$ 5.23607 0.537209
$$96$$ 0 0
$$97$$ −15.0902 −1.53217 −0.766087 0.642737i $$-0.777799\pi$$
−0.766087 + 0.642737i $$0.777799\pi$$
$$98$$ 0 0
$$99$$ −0.291796 −0.0293266
$$100$$ 0 0
$$101$$ −0.472136 −0.0469793 −0.0234896 0.999724i $$-0.507478\pi$$
−0.0234896 + 0.999724i $$0.507478\pi$$
$$102$$ 0 0
$$103$$ 15.4164 1.51902 0.759512 0.650493i $$-0.225438\pi$$
0.759512 + 0.650493i $$0.225438\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ −10.7639 −1.04059 −0.520294 0.853987i $$-0.674178\pi$$
−0.520294 + 0.853987i $$0.674178\pi$$
$$108$$ 0 0
$$109$$ 3.52786 0.337908 0.168954 0.985624i $$-0.445961\pi$$
0.168954 + 0.985624i $$0.445961\pi$$
$$110$$ 0 0
$$111$$ −0.763932 −0.0725092
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −4.76393 −0.444239
$$116$$ 0 0
$$117$$ −0.472136 −0.0436490
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.4164 −0.946946
$$122$$ 0 0
$$123$$ 0.145898 0.0131552
$$124$$ 0 0
$$125$$ 5.94427 0.531672
$$126$$ 0 0
$$127$$ −7.85410 −0.696939 −0.348469 0.937320i $$-0.613298\pi$$
−0.348469 + 0.937320i $$0.613298\pi$$
$$128$$ 0 0
$$129$$ −19.5623 −1.72236
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ −8.47214 −0.734627
$$134$$ 0 0
$$135$$ 3.38197 0.291073
$$136$$ 0 0
$$137$$ 2.38197 0.203505 0.101753 0.994810i $$-0.467555\pi$$
0.101753 + 0.994810i $$0.467555\pi$$
$$138$$ 0 0
$$139$$ 14.0344 1.19039 0.595193 0.803583i $$-0.297076\pi$$
0.595193 + 0.803583i $$0.297076\pi$$
$$140$$ 0 0
$$141$$ −13.7082 −1.15444
$$142$$ 0 0
$$143$$ 0.944272 0.0789640
$$144$$ 0 0
$$145$$ −3.52786 −0.292973
$$146$$ 0 0
$$147$$ 1.61803 0.133453
$$148$$ 0 0
$$149$$ −13.6180 −1.11563 −0.557816 0.829964i $$-0.688361\pi$$
−0.557816 + 0.829964i $$0.688361\pi$$
$$150$$ 0 0
$$151$$ −15.1459 −1.23256 −0.616278 0.787529i $$-0.711360\pi$$
−0.616278 + 0.787529i $$0.711360\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3.90983 0.314045
$$156$$ 0 0
$$157$$ −2.18034 −0.174010 −0.0870050 0.996208i $$-0.527730\pi$$
−0.0870050 + 0.996208i $$0.527730\pi$$
$$158$$ 0 0
$$159$$ 17.4721 1.38563
$$160$$ 0 0
$$161$$ 7.70820 0.607492
$$162$$ 0 0
$$163$$ −5.70820 −0.447101 −0.223551 0.974692i $$-0.571765\pi$$
−0.223551 + 0.974692i $$0.571765\pi$$
$$164$$ 0 0
$$165$$ −0.763932 −0.0594720
$$166$$ 0 0
$$167$$ −14.3820 −1.11291 −0.556455 0.830878i $$-0.687839\pi$$
−0.556455 + 0.830878i $$0.687839\pi$$
$$168$$ 0 0
$$169$$ −11.4721 −0.882472
$$170$$ 0 0
$$171$$ 3.23607 0.247468
$$172$$ 0 0
$$173$$ 11.9098 0.905488 0.452744 0.891641i $$-0.350445\pi$$
0.452744 + 0.891641i $$0.350445\pi$$
$$174$$ 0 0
$$175$$ −4.61803 −0.349091
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −18.3820 −1.37393 −0.686966 0.726689i $$-0.741058\pi$$
−0.686966 + 0.726689i $$0.741058\pi$$
$$180$$ 0 0
$$181$$ 12.4721 0.927047 0.463523 0.886085i $$-0.346585\pi$$
0.463523 + 0.886085i $$0.346585\pi$$
$$182$$ 0 0
$$183$$ 11.8541 0.876280
$$184$$ 0 0
$$185$$ 0.291796 0.0214533
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −5.47214 −0.398039
$$190$$ 0 0
$$191$$ 20.7984 1.50492 0.752459 0.658639i $$-0.228868\pi$$
0.752459 + 0.658639i $$0.228868\pi$$
$$192$$ 0 0
$$193$$ 14.1803 1.02072 0.510362 0.859960i $$-0.329512\pi$$
0.510362 + 0.859960i $$0.329512\pi$$
$$194$$ 0 0
$$195$$ −1.23607 −0.0885167
$$196$$ 0 0
$$197$$ −9.70820 −0.691681 −0.345840 0.938293i $$-0.612406\pi$$
−0.345840 + 0.938293i $$0.612406\pi$$
$$198$$ 0 0
$$199$$ 12.0902 0.857049 0.428525 0.903530i $$-0.359033\pi$$
0.428525 + 0.903530i $$0.359033\pi$$
$$200$$ 0 0
$$201$$ −21.1803 −1.49395
$$202$$ 0 0
$$203$$ 5.70820 0.400637
$$204$$ 0 0
$$205$$ −0.0557281 −0.00389222
$$206$$ 0 0
$$207$$ −2.94427 −0.204641
$$208$$ 0 0
$$209$$ −6.47214 −0.447687
$$210$$ 0 0
$$211$$ 2.29180 0.157774 0.0788869 0.996884i $$-0.474863\pi$$
0.0788869 + 0.996884i $$0.474863\pi$$
$$212$$ 0 0
$$213$$ −17.7082 −1.21335
$$214$$ 0 0
$$215$$ 7.47214 0.509595
$$216$$ 0 0
$$217$$ −6.32624 −0.429453
$$218$$ 0 0
$$219$$ −11.5623 −0.781308
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 2.94427 0.197163 0.0985815 0.995129i $$-0.468569\pi$$
0.0985815 + 0.995129i $$0.468569\pi$$
$$224$$ 0 0
$$225$$ 1.76393 0.117595
$$226$$ 0 0
$$227$$ −25.7426 −1.70860 −0.854300 0.519781i $$-0.826014\pi$$
−0.854300 + 0.519781i $$0.826014\pi$$
$$228$$ 0 0
$$229$$ −19.2361 −1.27116 −0.635578 0.772037i $$-0.719238\pi$$
−0.635578 + 0.772037i $$0.719238\pi$$
$$230$$ 0 0
$$231$$ 1.23607 0.0813273
$$232$$ 0 0
$$233$$ −6.47214 −0.424004 −0.212002 0.977269i $$-0.567998\pi$$
−0.212002 + 0.977269i $$0.567998\pi$$
$$234$$ 0 0
$$235$$ 5.23607 0.341563
$$236$$ 0 0
$$237$$ −4.76393 −0.309451
$$238$$ 0 0
$$239$$ 11.9098 0.770383 0.385191 0.922837i $$-0.374135\pi$$
0.385191 + 0.922837i $$0.374135\pi$$
$$240$$ 0 0
$$241$$ −4.03444 −0.259881 −0.129941 0.991522i $$-0.541479\pi$$
−0.129941 + 0.991522i $$0.541479\pi$$
$$242$$ 0 0
$$243$$ 3.94427 0.253025
$$244$$ 0 0
$$245$$ −0.618034 −0.0394847
$$246$$ 0 0
$$247$$ −10.4721 −0.666326
$$248$$ 0 0
$$249$$ 24.9443 1.58078
$$250$$ 0 0
$$251$$ −17.4164 −1.09931 −0.549657 0.835390i $$-0.685242\pi$$
−0.549657 + 0.835390i $$0.685242\pi$$
$$252$$ 0 0
$$253$$ 5.88854 0.370210
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 25.7082 1.60363 0.801817 0.597570i $$-0.203867\pi$$
0.801817 + 0.597570i $$0.203867\pi$$
$$258$$ 0 0
$$259$$ −0.472136 −0.0293371
$$260$$ 0 0
$$261$$ −2.18034 −0.134960
$$262$$ 0 0
$$263$$ −16.9443 −1.04483 −0.522414 0.852692i $$-0.674969\pi$$
−0.522414 + 0.852692i $$0.674969\pi$$
$$264$$ 0 0
$$265$$ −6.67376 −0.409966
$$266$$ 0 0
$$267$$ 3.23607 0.198044
$$268$$ 0 0
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ 3.70820 0.225257 0.112629 0.993637i $$-0.464073\pi$$
0.112629 + 0.993637i $$0.464073\pi$$
$$272$$ 0 0
$$273$$ 2.00000 0.121046
$$274$$ 0 0
$$275$$ −3.52786 −0.212738
$$276$$ 0 0
$$277$$ −2.76393 −0.166069 −0.0830343 0.996547i $$-0.526461\pi$$
−0.0830343 + 0.996547i $$0.526461\pi$$
$$278$$ 0 0
$$279$$ 2.41641 0.144667
$$280$$ 0 0
$$281$$ −8.90983 −0.531516 −0.265758 0.964040i $$-0.585622\pi$$
−0.265758 + 0.964040i $$0.585622\pi$$
$$282$$ 0 0
$$283$$ 4.43769 0.263794 0.131897 0.991263i $$-0.457893\pi$$
0.131897 + 0.991263i $$0.457893\pi$$
$$284$$ 0 0
$$285$$ 8.47214 0.501846
$$286$$ 0 0
$$287$$ 0.0901699 0.00532256
$$288$$ 0 0
$$289$$ 0 0
$$290$$ 0 0
$$291$$ −24.4164 −1.43132
$$292$$ 0 0
$$293$$ −6.29180 −0.367571 −0.183785 0.982966i $$-0.558835\pi$$
−0.183785 + 0.982966i $$0.558835\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −4.18034 −0.242568
$$298$$ 0 0
$$299$$ 9.52786 0.551011
$$300$$ 0 0
$$301$$ −12.0902 −0.696866
$$302$$ 0 0
$$303$$ −0.763932 −0.0438867
$$304$$ 0 0
$$305$$ −4.52786 −0.259265
$$306$$ 0 0
$$307$$ 1.05573 0.0602536 0.0301268 0.999546i $$-0.490409\pi$$
0.0301268 + 0.999546i $$0.490409\pi$$
$$308$$ 0 0
$$309$$ 24.9443 1.41903
$$310$$ 0 0
$$311$$ −26.6180 −1.50937 −0.754685 0.656087i $$-0.772210\pi$$
−0.754685 + 0.656087i $$0.772210\pi$$
$$312$$ 0 0
$$313$$ 1.27051 0.0718135 0.0359067 0.999355i $$-0.488568\pi$$
0.0359067 + 0.999355i $$0.488568\pi$$
$$314$$ 0 0
$$315$$ 0.236068 0.0133009
$$316$$ 0 0
$$317$$ 28.1803 1.58277 0.791383 0.611321i $$-0.209362\pi$$
0.791383 + 0.611321i $$0.209362\pi$$
$$318$$ 0 0
$$319$$ 4.36068 0.244151
$$320$$ 0 0
$$321$$ −17.4164 −0.972089
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −5.70820 −0.316634
$$326$$ 0 0
$$327$$ 5.70820 0.315664
$$328$$ 0 0
$$329$$ −8.47214 −0.467084
$$330$$ 0 0
$$331$$ 18.0344 0.991263 0.495631 0.868533i $$-0.334937\pi$$
0.495631 + 0.868533i $$0.334937\pi$$
$$332$$ 0 0
$$333$$ 0.180340 0.00988256
$$334$$ 0 0
$$335$$ 8.09017 0.442013
$$336$$ 0 0
$$337$$ −9.52786 −0.519016 −0.259508 0.965741i $$-0.583560\pi$$
−0.259508 + 0.965741i $$0.583560\pi$$
$$338$$ 0 0
$$339$$ −9.70820 −0.527277
$$340$$ 0 0
$$341$$ −4.83282 −0.261712
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −7.70820 −0.414996
$$346$$ 0 0
$$347$$ −30.9443 −1.66118 −0.830588 0.556888i $$-0.811995\pi$$
−0.830588 + 0.556888i $$0.811995\pi$$
$$348$$ 0 0
$$349$$ −15.5279 −0.831188 −0.415594 0.909550i $$-0.636426\pi$$
−0.415594 + 0.909550i $$0.636426\pi$$
$$350$$ 0 0
$$351$$ −6.76393 −0.361032
$$352$$ 0 0
$$353$$ −12.7639 −0.679356 −0.339678 0.940542i $$-0.610318\pi$$
−0.339678 + 0.940542i $$0.610318\pi$$
$$354$$ 0 0
$$355$$ 6.76393 0.358992
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 19.5623 1.03246 0.516230 0.856450i $$-0.327335\pi$$
0.516230 + 0.856450i $$0.327335\pi$$
$$360$$ 0 0
$$361$$ 52.7771 2.77774
$$362$$ 0 0
$$363$$ −16.8541 −0.884611
$$364$$ 0 0
$$365$$ 4.41641 0.231165
$$366$$ 0 0
$$367$$ 19.5066 1.01824 0.509118 0.860697i $$-0.329972\pi$$
0.509118 + 0.860697i $$0.329972\pi$$
$$368$$ 0 0
$$369$$ −0.0344419 −0.00179297
$$370$$ 0 0
$$371$$ 10.7984 0.560624
$$372$$ 0 0
$$373$$ −16.8541 −0.872672 −0.436336 0.899784i $$-0.643724\pi$$
−0.436336 + 0.899784i $$0.643724\pi$$
$$374$$ 0 0
$$375$$ 9.61803 0.496673
$$376$$ 0 0
$$377$$ 7.05573 0.363388
$$378$$ 0 0
$$379$$ −10.6525 −0.547181 −0.273590 0.961846i $$-0.588211\pi$$
−0.273590 + 0.961846i $$0.588211\pi$$
$$380$$ 0 0
$$381$$ −12.7082 −0.651061
$$382$$ 0 0
$$383$$ −30.6525 −1.56627 −0.783134 0.621853i $$-0.786380\pi$$
−0.783134 + 0.621853i $$0.786380\pi$$
$$384$$ 0 0
$$385$$ −0.472136 −0.0240623
$$386$$ 0 0
$$387$$ 4.61803 0.234748
$$388$$ 0 0
$$389$$ −21.5623 −1.09325 −0.546626 0.837377i $$-0.684088\pi$$
−0.546626 + 0.837377i $$0.684088\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 12.9443 0.652952
$$394$$ 0 0
$$395$$ 1.81966 0.0915570
$$396$$ 0 0
$$397$$ 6.79837 0.341201 0.170600 0.985340i $$-0.445429\pi$$
0.170600 + 0.985340i $$0.445429\pi$$
$$398$$ 0 0
$$399$$ −13.7082 −0.686269
$$400$$ 0 0
$$401$$ 10.4721 0.522954 0.261477 0.965210i $$-0.415791\pi$$
0.261477 + 0.965210i $$0.415791\pi$$
$$402$$ 0 0
$$403$$ −7.81966 −0.389525
$$404$$ 0 0
$$405$$ 4.76393 0.236722
$$406$$ 0 0
$$407$$ −0.360680 −0.0178782
$$408$$ 0 0
$$409$$ −8.18034 −0.404492 −0.202246 0.979335i $$-0.564824\pi$$
−0.202246 + 0.979335i $$0.564824\pi$$
$$410$$ 0 0
$$411$$ 3.85410 0.190109
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −9.52786 −0.467704
$$416$$ 0 0
$$417$$ 22.7082 1.11203
$$418$$ 0 0
$$419$$ −37.4508 −1.82959 −0.914797 0.403914i $$-0.867649\pi$$
−0.914797 + 0.403914i $$0.867649\pi$$
$$420$$ 0 0
$$421$$ −29.2148 −1.42384 −0.711921 0.702260i $$-0.752174\pi$$
−0.711921 + 0.702260i $$0.752174\pi$$
$$422$$ 0 0
$$423$$ 3.23607 0.157343
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 7.32624 0.354542
$$428$$ 0 0
$$429$$ 1.52786 0.0737660
$$430$$ 0 0
$$431$$ −5.05573 −0.243526 −0.121763 0.992559i $$-0.538855\pi$$
−0.121763 + 0.992559i $$0.538855\pi$$
$$432$$ 0 0
$$433$$ 16.9443 0.814290 0.407145 0.913364i $$-0.366524\pi$$
0.407145 + 0.913364i $$0.366524\pi$$
$$434$$ 0 0
$$435$$ −5.70820 −0.273687
$$436$$ 0 0
$$437$$ −65.3050 −3.12396
$$438$$ 0 0
$$439$$ 3.32624 0.158753 0.0793763 0.996845i $$-0.474707\pi$$
0.0793763 + 0.996845i $$0.474707\pi$$
$$440$$ 0 0
$$441$$ −0.381966 −0.0181889
$$442$$ 0 0
$$443$$ 25.8885 1.23000 0.615001 0.788526i $$-0.289156\pi$$
0.615001 + 0.788526i $$0.289156\pi$$
$$444$$ 0 0
$$445$$ −1.23607 −0.0585952
$$446$$ 0 0
$$447$$ −22.0344 −1.04219
$$448$$ 0 0
$$449$$ −17.7082 −0.835702 −0.417851 0.908516i $$-0.637217\pi$$
−0.417851 + 0.908516i $$0.637217\pi$$
$$450$$ 0 0
$$451$$ 0.0688837 0.00324361
$$452$$ 0 0
$$453$$ −24.5066 −1.15142
$$454$$ 0 0
$$455$$ −0.763932 −0.0358137
$$456$$ 0 0
$$457$$ 0.618034 0.0289104 0.0144552 0.999896i $$-0.495399\pi$$
0.0144552 + 0.999896i $$0.495399\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −12.9443 −0.602875 −0.301437 0.953486i $$-0.597466\pi$$
−0.301437 + 0.953486i $$0.597466\pi$$
$$462$$ 0 0
$$463$$ −13.0344 −0.605762 −0.302881 0.953028i $$-0.597948\pi$$
−0.302881 + 0.953028i $$0.597948\pi$$
$$464$$ 0 0
$$465$$ 6.32624 0.293372
$$466$$ 0 0
$$467$$ 10.4721 0.484593 0.242296 0.970202i $$-0.422099\pi$$
0.242296 + 0.970202i $$0.422099\pi$$
$$468$$ 0 0
$$469$$ −13.0902 −0.604448
$$470$$ 0 0
$$471$$ −3.52786 −0.162555
$$472$$ 0 0
$$473$$ −9.23607 −0.424675
$$474$$ 0 0
$$475$$ 39.1246 1.79516
$$476$$ 0 0
$$477$$ −4.12461 −0.188853
$$478$$ 0 0
$$479$$ 24.2705 1.10895 0.554474 0.832201i $$-0.312920\pi$$
0.554474 + 0.832201i $$0.312920\pi$$
$$480$$ 0 0
$$481$$ −0.583592 −0.0266095
$$482$$ 0 0
$$483$$ 12.4721 0.567502
$$484$$ 0 0
$$485$$ 9.32624 0.423483
$$486$$ 0 0
$$487$$ 22.3607 1.01326 0.506630 0.862164i $$-0.330891\pi$$
0.506630 + 0.862164i $$0.330891\pi$$
$$488$$ 0 0
$$489$$ −9.23607 −0.417669
$$490$$ 0 0
$$491$$ −1.09017 −0.0491987 −0.0245993 0.999697i $$-0.507831\pi$$
−0.0245993 + 0.999697i $$0.507831\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0.180340 0.00810568
$$496$$ 0 0
$$497$$ −10.9443 −0.490918
$$498$$ 0 0
$$499$$ −17.0557 −0.763519 −0.381760 0.924262i $$-0.624682\pi$$
−0.381760 + 0.924262i $$0.624682\pi$$
$$500$$ 0 0
$$501$$ −23.2705 −1.03965
$$502$$ 0 0
$$503$$ 10.6180 0.473435 0.236717 0.971579i $$-0.423928\pi$$
0.236717 + 0.971579i $$0.423928\pi$$
$$504$$ 0 0
$$505$$ 0.291796 0.0129848
$$506$$ 0 0
$$507$$ −18.5623 −0.824381
$$508$$ 0 0
$$509$$ −0.763932 −0.0338607 −0.0169303 0.999857i $$-0.505389\pi$$
−0.0169303 + 0.999857i $$0.505389\pi$$
$$510$$ 0 0
$$511$$ −7.14590 −0.316116
$$512$$ 0 0
$$513$$ 46.3607 2.04687
$$514$$ 0 0
$$515$$ −9.52786 −0.419848
$$516$$ 0 0
$$517$$ −6.47214 −0.284644
$$518$$ 0 0
$$519$$ 19.2705 0.845881
$$520$$ 0 0
$$521$$ 26.7426 1.17162 0.585808 0.810450i $$-0.300777\pi$$
0.585808 + 0.810450i $$0.300777\pi$$
$$522$$ 0 0
$$523$$ 34.1803 1.49460 0.747301 0.664486i $$-0.231349\pi$$
0.747301 + 0.664486i $$0.231349\pi$$
$$524$$ 0 0
$$525$$ −7.47214 −0.326111
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 36.4164 1.58332
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0.111456 0.00482770
$$534$$ 0 0
$$535$$ 6.65248 0.287612
$$536$$ 0 0
$$537$$ −29.7426 −1.28349
$$538$$ 0 0
$$539$$ 0.763932 0.0329049
$$540$$ 0 0
$$541$$ 5.23607 0.225116 0.112558 0.993645i $$-0.464096\pi$$
0.112558 + 0.993645i $$0.464096\pi$$
$$542$$ 0 0
$$543$$ 20.1803 0.866021
$$544$$ 0 0
$$545$$ −2.18034 −0.0933955
$$546$$ 0 0
$$547$$ 40.6525 1.73817 0.869087 0.494659i $$-0.164707\pi$$
0.869087 + 0.494659i $$0.164707\pi$$
$$548$$ 0 0
$$549$$ −2.79837 −0.119432
$$550$$ 0 0
$$551$$ −48.3607 −2.06023
$$552$$ 0 0
$$553$$ −2.94427 −0.125203
$$554$$ 0 0
$$555$$ 0.472136 0.0200411
$$556$$ 0 0
$$557$$ 34.3607 1.45591 0.727954 0.685626i $$-0.240471\pi$$
0.727954 + 0.685626i $$0.240471\pi$$
$$558$$ 0 0
$$559$$ −14.9443 −0.632075
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −29.1246 −1.22746 −0.613728 0.789518i $$-0.710331\pi$$
−0.613728 + 0.789518i $$0.710331\pi$$
$$564$$ 0 0
$$565$$ 3.70820 0.156005
$$566$$ 0 0
$$567$$ −7.70820 −0.323714
$$568$$ 0 0
$$569$$ −43.5066 −1.82389 −0.911945 0.410312i $$-0.865420\pi$$
−0.911945 + 0.410312i $$0.865420\pi$$
$$570$$ 0 0
$$571$$ −8.47214 −0.354548 −0.177274 0.984162i $$-0.556728\pi$$
−0.177274 + 0.984162i $$0.556728\pi$$
$$572$$ 0 0
$$573$$ 33.6525 1.40585
$$574$$ 0 0
$$575$$ −35.5967 −1.48449
$$576$$ 0 0
$$577$$ 17.1246 0.712907 0.356453 0.934313i $$-0.383986\pi$$
0.356453 + 0.934313i $$0.383986\pi$$
$$578$$ 0 0
$$579$$ 22.9443 0.953531
$$580$$ 0 0
$$581$$ 15.4164 0.639580
$$582$$ 0 0
$$583$$ 8.24922 0.341648
$$584$$ 0 0
$$585$$ 0.291796 0.0120643
$$586$$ 0 0
$$587$$ −42.8328 −1.76790 −0.883950 0.467582i $$-0.845125\pi$$
−0.883950 + 0.467582i $$0.845125\pi$$
$$588$$ 0 0
$$589$$ 53.5967 2.20842
$$590$$ 0 0
$$591$$ −15.7082 −0.646149
$$592$$ 0 0
$$593$$ −22.3607 −0.918243 −0.459122 0.888373i $$-0.651836\pi$$
−0.459122 + 0.888373i $$0.651836\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 19.5623 0.800632
$$598$$ 0 0
$$599$$ 36.0902 1.47460 0.737302 0.675563i $$-0.236099\pi$$
0.737302 + 0.675563i $$0.236099\pi$$
$$600$$ 0 0
$$601$$ −48.2492 −1.96813 −0.984063 0.177818i $$-0.943096\pi$$
−0.984063 + 0.177818i $$0.943096\pi$$
$$602$$ 0 0
$$603$$ 5.00000 0.203616
$$604$$ 0 0
$$605$$ 6.43769 0.261729
$$606$$ 0 0
$$607$$ 39.8541 1.61763 0.808814 0.588064i $$-0.200110\pi$$
0.808814 + 0.588064i $$0.200110\pi$$
$$608$$ 0 0
$$609$$ 9.23607 0.374264
$$610$$ 0 0
$$611$$ −10.4721 −0.423657
$$612$$ 0 0
$$613$$ −17.5623 −0.709335 −0.354667 0.934993i $$-0.615406\pi$$
−0.354667 + 0.934993i $$0.615406\pi$$
$$614$$ 0 0
$$615$$ −0.0901699 −0.00363600
$$616$$ 0 0
$$617$$ 35.0132 1.40958 0.704788 0.709418i $$-0.251042\pi$$
0.704788 + 0.709418i $$0.251042\pi$$
$$618$$ 0 0
$$619$$ 32.0000 1.28619 0.643094 0.765787i $$-0.277650\pi$$
0.643094 + 0.765787i $$0.277650\pi$$
$$620$$ 0 0
$$621$$ −42.1803 −1.69264
$$622$$ 0 0
$$623$$ 2.00000 0.0801283
$$624$$ 0 0
$$625$$ 19.4164 0.776656
$$626$$ 0 0
$$627$$ −10.4721 −0.418217
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −37.0902 −1.47654 −0.738268 0.674507i $$-0.764356\pi$$
−0.738268 + 0.674507i $$0.764356\pi$$
$$632$$ 0 0
$$633$$ 3.70820 0.147388
$$634$$ 0 0
$$635$$ 4.85410 0.192629
$$636$$ 0 0
$$637$$ 1.23607 0.0489748
$$638$$ 0 0
$$639$$ 4.18034 0.165372
$$640$$ 0 0
$$641$$ −21.0557 −0.831651 −0.415826 0.909444i $$-0.636507\pi$$
−0.415826 + 0.909444i $$0.636507\pi$$
$$642$$ 0 0
$$643$$ −3.14590 −0.124062 −0.0620311 0.998074i $$-0.519758\pi$$
−0.0620311 + 0.998074i $$0.519758\pi$$
$$644$$ 0 0
$$645$$ 12.0902 0.476050
$$646$$ 0 0
$$647$$ −20.7639 −0.816314 −0.408157 0.912912i $$-0.633828\pi$$
−0.408157 + 0.912912i $$0.633828\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −10.2361 −0.401183
$$652$$ 0 0
$$653$$ 32.4721 1.27073 0.635366 0.772211i $$-0.280849\pi$$
0.635366 + 0.772211i $$0.280849\pi$$
$$654$$ 0 0
$$655$$ −4.94427 −0.193189
$$656$$ 0 0
$$657$$ 2.72949 0.106488
$$658$$ 0 0
$$659$$ 1.74265 0.0678838 0.0339419 0.999424i $$-0.489194\pi$$
0.0339419 + 0.999424i $$0.489194\pi$$
$$660$$ 0 0
$$661$$ 30.8328 1.19926 0.599629 0.800278i $$-0.295315\pi$$
0.599629 + 0.800278i $$0.295315\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 5.23607 0.203046
$$666$$ 0 0
$$667$$ 44.0000 1.70369
$$668$$ 0 0
$$669$$ 4.76393 0.184184
$$670$$ 0 0
$$671$$ 5.59675 0.216060
$$672$$ 0 0
$$673$$ 14.4721 0.557860 0.278930 0.960311i $$-0.410020\pi$$
0.278930 + 0.960311i $$0.410020\pi$$
$$674$$ 0 0
$$675$$ 25.2705 0.972662
$$676$$ 0 0
$$677$$ 33.7771 1.29816 0.649079 0.760721i $$-0.275154\pi$$
0.649079 + 0.760721i $$0.275154\pi$$
$$678$$ 0 0
$$679$$ −15.0902 −0.579108
$$680$$ 0 0
$$681$$ −41.6525 −1.59613
$$682$$ 0 0
$$683$$ −28.1803 −1.07829 −0.539145 0.842213i $$-0.681253\pi$$
−0.539145 + 0.842213i $$0.681253\pi$$
$$684$$ 0 0
$$685$$ −1.47214 −0.0562474
$$686$$ 0 0
$$687$$ −31.1246 −1.18748
$$688$$ 0 0
$$689$$ 13.3475 0.508500
$$690$$ 0 0
$$691$$ −8.49342 −0.323105 −0.161553 0.986864i $$-0.551650\pi$$
−0.161553 + 0.986864i $$0.551650\pi$$
$$692$$ 0 0
$$693$$ −0.291796 −0.0110844
$$694$$ 0 0
$$695$$ −8.67376 −0.329015
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −10.4721 −0.396093
$$700$$ 0 0
$$701$$ −45.4164 −1.71535 −0.857677 0.514189i $$-0.828093\pi$$
−0.857677 + 0.514189i $$0.828093\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 8.47214 0.319079
$$706$$ 0 0
$$707$$ −0.472136 −0.0177565
$$708$$ 0 0
$$709$$ −40.6525 −1.52674 −0.763368 0.645964i $$-0.776456\pi$$
−0.763368 + 0.645964i $$0.776456\pi$$
$$710$$ 0 0
$$711$$ 1.12461 0.0421762
$$712$$ 0 0
$$713$$ −48.7639 −1.82622
$$714$$ 0 0
$$715$$ −0.583592 −0.0218251
$$716$$ 0 0
$$717$$ 19.2705 0.719670
$$718$$ 0 0
$$719$$ 39.1459 1.45990 0.729948 0.683503i $$-0.239544\pi$$
0.729948 + 0.683503i $$0.239544\pi$$
$$720$$ 0 0
$$721$$ 15.4164 0.574137
$$722$$ 0 0
$$723$$ −6.52786 −0.242774
$$724$$ 0 0
$$725$$ −26.3607 −0.979011
$$726$$ 0 0
$$727$$ 16.4721 0.610918 0.305459 0.952205i $$-0.401190\pi$$
0.305459 + 0.952205i $$0.401190\pi$$
$$728$$ 0 0
$$729$$ 29.5066 1.09284
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 36.0000 1.32969 0.664845 0.746981i $$-0.268498\pi$$
0.664845 + 0.746981i $$0.268498\pi$$
$$734$$ 0 0
$$735$$ −1.00000 −0.0368856
$$736$$ 0 0
$$737$$ −10.0000 −0.368355
$$738$$ 0 0
$$739$$ −1.43769 −0.0528864 −0.0264432 0.999650i $$-0.508418\pi$$
−0.0264432 + 0.999650i $$0.508418\pi$$
$$740$$ 0 0
$$741$$ −16.9443 −0.622463
$$742$$ 0 0
$$743$$ −4.65248 −0.170683 −0.0853414 0.996352i $$-0.527198\pi$$
−0.0853414 + 0.996352i $$0.527198\pi$$
$$744$$ 0 0
$$745$$ 8.41641 0.308353
$$746$$ 0 0
$$747$$ −5.88854 −0.215451
$$748$$ 0 0
$$749$$ −10.7639 −0.393306
$$750$$ 0 0
$$751$$ −10.9443 −0.399362 −0.199681 0.979861i $$-0.563991\pi$$
−0.199681 + 0.979861i $$0.563991\pi$$
$$752$$ 0 0
$$753$$ −28.1803 −1.02695
$$754$$ 0 0
$$755$$ 9.36068 0.340670
$$756$$ 0 0
$$757$$ −9.43769 −0.343019 −0.171509 0.985182i $$-0.554864\pi$$
−0.171509 + 0.985182i $$0.554864\pi$$
$$758$$ 0 0
$$759$$ 9.52786 0.345840
$$760$$ 0 0
$$761$$ 30.1803 1.09404 0.547018 0.837121i $$-0.315763\pi$$
0.547018 + 0.837121i $$0.315763\pi$$
$$762$$ 0 0
$$763$$ 3.52786 0.127717
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −7.12461 −0.256920 −0.128460 0.991715i $$-0.541003\pi$$
−0.128460 + 0.991715i $$0.541003\pi$$
$$770$$ 0 0
$$771$$ 41.5967 1.49807
$$772$$ 0 0
$$773$$ 5.41641 0.194815 0.0974073 0.995245i $$-0.468945\pi$$
0.0974073 + 0.995245i $$0.468945\pi$$
$$774$$ 0 0
$$775$$ 29.2148 1.04943
$$776$$ 0 0
$$777$$ −0.763932 −0.0274059
$$778$$ 0 0
$$779$$ −0.763932 −0.0273707
$$780$$ 0 0
$$781$$ −8.36068 −0.299169
$$782$$ 0 0
$$783$$ −31.2361 −1.11629
$$784$$ 0 0
$$785$$ 1.34752 0.0480952
$$786$$ 0 0
$$787$$ 12.0000 0.427754 0.213877 0.976861i $$-0.431391\pi$$
0.213877 + 0.976861i $$0.431391\pi$$
$$788$$ 0 0
$$789$$ −27.4164 −0.976050
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ 9.05573 0.321578
$$794$$ 0 0
$$795$$ −10.7984 −0.382979
$$796$$ 0 0
$$797$$ 15.4164 0.546077 0.273039 0.962003i $$-0.411971\pi$$
0.273039 + 0.962003i $$0.411971\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −0.763932 −0.0269922
$$802$$ 0 0
$$803$$ −5.45898 −0.192643
$$804$$ 0 0
$$805$$ −4.76393 −0.167907
$$806$$ 0 0
$$807$$ 16.1803 0.569575
$$808$$ 0 0
$$809$$ 53.4853 1.88044 0.940221 0.340564i $$-0.110618\pi$$
0.940221 + 0.340564i $$0.110618\pi$$
$$810$$ 0 0
$$811$$ 2.96556 0.104135 0.0520674 0.998644i $$-0.483419\pi$$
0.0520674 + 0.998644i $$0.483419\pi$$
$$812$$ 0 0
$$813$$ 6.00000 0.210429
$$814$$ 0 0
$$815$$ 3.52786 0.123576
$$816$$ 0 0
$$817$$ 102.430 3.58356
$$818$$ 0 0
$$819$$ −0.472136 −0.0164978
$$820$$ 0 0
$$821$$ −14.3607 −0.501191 −0.250596 0.968092i $$-0.580626\pi$$
−0.250596 + 0.968092i $$0.580626\pi$$
$$822$$ 0 0
$$823$$ −47.4853 −1.65523 −0.827617 0.561294i $$-0.810304\pi$$
−0.827617 + 0.561294i $$0.810304\pi$$
$$824$$ 0 0
$$825$$ −5.70820 −0.198734
$$826$$ 0 0
$$827$$ −27.5279 −0.957238 −0.478619 0.878023i $$-0.658862\pi$$
−0.478619 + 0.878023i $$0.658862\pi$$
$$828$$ 0 0
$$829$$ −6.18034 −0.214652 −0.107326 0.994224i $$-0.534229\pi$$
−0.107326 + 0.994224i $$0.534229\pi$$
$$830$$ 0 0
$$831$$ −4.47214 −0.155137
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 8.88854 0.307601
$$836$$ 0 0
$$837$$ 34.6180 1.19657
$$838$$ 0 0
$$839$$ 13.5279 0.467034 0.233517 0.972353i $$-0.424977\pi$$
0.233517 + 0.972353i $$0.424977\pi$$
$$840$$ 0 0
$$841$$ 3.58359 0.123572
$$842$$ 0 0
$$843$$ −14.4164 −0.496527
$$844$$ 0 0
$$845$$ 7.09017 0.243909
$$846$$ 0 0
$$847$$ −10.4164 −0.357912
$$848$$ 0 0
$$849$$ 7.18034 0.246429
$$850$$ 0 0
$$851$$ −3.63932 −0.124754
$$852$$ 0 0
$$853$$ 38.3607 1.31344 0.656722 0.754132i $$-0.271942\pi$$
0.656722 + 0.754132i $$0.271942\pi$$
$$854$$ 0 0
$$855$$ −2.00000 −0.0683986
$$856$$ 0 0
$$857$$ 15.0902 0.515470 0.257735 0.966216i $$-0.417024\pi$$
0.257735 + 0.966216i $$0.417024\pi$$
$$858$$ 0 0
$$859$$ 14.9443 0.509892 0.254946 0.966955i $$-0.417942\pi$$
0.254946 + 0.966955i $$0.417942\pi$$
$$860$$ 0 0
$$861$$ 0.145898 0.00497219
$$862$$ 0 0
$$863$$ −46.3262 −1.57696 −0.788482 0.615058i $$-0.789133\pi$$
−0.788482 + 0.615058i $$0.789133\pi$$
$$864$$ 0 0
$$865$$ −7.36068 −0.250271
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −2.24922 −0.0762997
$$870$$ 0 0
$$871$$ −16.1803 −0.548250
$$872$$ 0 0
$$873$$ 5.76393 0.195080
$$874$$ 0 0
$$875$$ 5.94427 0.200953
$$876$$ 0 0
$$877$$ −18.2918 −0.617670 −0.308835 0.951116i $$-0.599939\pi$$
−0.308835 + 0.951116i $$0.599939\pi$$
$$878$$ 0 0
$$879$$ −10.1803 −0.343374
$$880$$ 0 0
$$881$$ 19.6738 0.662826 0.331413 0.943486i $$-0.392475\pi$$
0.331413 + 0.943486i $$0.392475\pi$$
$$882$$ 0 0
$$883$$ 14.9787 0.504074 0.252037 0.967718i $$-0.418900\pi$$
0.252037 + 0.967718i $$0.418900\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 16.5066 0.554237 0.277118 0.960836i $$-0.410621\pi$$
0.277118 + 0.960836i $$0.410621\pi$$
$$888$$ 0 0
$$889$$ −7.85410 −0.263418
$$890$$ 0 0
$$891$$ −5.88854 −0.197274
$$892$$ 0 0
$$893$$ 71.7771 2.40193
$$894$$ 0 0
$$895$$ 11.3607 0.379746
$$896$$ 0 0
$$897$$ 15.4164 0.514739
$$898$$ 0 0
$$899$$ −36.1115 −1.20438
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ −19.5623 −0.650993
$$904$$ 0 0
$$905$$ −7.70820 −0.256229
$$906$$ 0 0
$$907$$ 31.3050 1.03946 0.519732 0.854329i $$-0.326032\pi$$
0.519732 + 0.854329i $$0.326032\pi$$
$$908$$ 0 0
$$909$$ 0.180340 0.00598150
$$910$$ 0 0
$$911$$ 14.4721 0.479483 0.239742 0.970837i $$-0.422937\pi$$
0.239742 + 0.970837i $$0.422937\pi$$
$$912$$ 0 0
$$913$$ 11.7771 0.389765
$$914$$ 0 0
$$915$$ −7.32624 −0.242198
$$916$$ 0 0
$$917$$ 8.00000 0.264183
$$918$$ 0 0
$$919$$ −30.1591 −0.994855 −0.497428 0.867505i $$-0.665722\pi$$
−0.497428 + 0.867505i $$0.665722\pi$$
$$920$$ 0 0
$$921$$ 1.70820 0.0562872
$$922$$ 0 0
$$923$$ −13.5279 −0.445275
$$924$$ 0 0
$$925$$ 2.18034 0.0716891
$$926$$ 0 0
$$927$$ −5.88854 −0.193405
$$928$$ 0 0
$$929$$ −21.2705 −0.697863 −0.348931 0.937148i $$-0.613455\pi$$
−0.348931 + 0.937148i $$0.613455\pi$$
$$930$$ 0 0
$$931$$ −8.47214 −0.277663
$$932$$ 0 0
$$933$$ −43.0689 −1.41001
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −5.30495 −0.173305 −0.0866526 0.996239i $$-0.527617\pi$$
−0.0866526 + 0.996239i $$0.527617\pi$$
$$938$$ 0 0
$$939$$ 2.05573 0.0670862
$$940$$ 0 0
$$941$$ −45.9787 −1.49886 −0.749432 0.662082i $$-0.769673\pi$$
−0.749432 + 0.662082i $$0.769673\pi$$
$$942$$ 0 0
$$943$$ 0.695048 0.0226339
$$944$$ 0 0
$$945$$ 3.38197 0.110015
$$946$$ 0 0
$$947$$ −6.87539 −0.223420 −0.111710 0.993741i $$-0.535633\pi$$
−0.111710 + 0.993741i $$0.535633\pi$$
$$948$$ 0 0
$$949$$ −8.83282 −0.286725
$$950$$ 0 0
$$951$$ 45.5967 1.47858
$$952$$ 0 0
$$953$$ 32.3262 1.04715 0.523575 0.851980i $$-0.324598\pi$$
0.523575 + 0.851980i $$0.324598\pi$$
$$954$$ 0 0
$$955$$ −12.8541 −0.415949
$$956$$ 0 0
$$957$$ 7.05573 0.228079
$$958$$ 0 0
$$959$$ 2.38197 0.0769177
$$960$$ 0 0
$$961$$ 9.02129 0.291009
$$962$$ 0 0
$$963$$ 4.11146 0.132490
$$964$$ 0 0
$$965$$ −8.76393 −0.282121
$$966$$ 0 0
$$967$$ 37.7984 1.21551 0.607757 0.794123i $$-0.292070\pi$$
0.607757 + 0.794123i $$0.292070\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 21.1246 0.677921 0.338961 0.940801i $$-0.389925\pi$$
0.338961 + 0.940801i $$0.389925\pi$$
$$972$$ 0 0
$$973$$ 14.0344 0.449924
$$974$$ 0 0
$$975$$ −9.23607 −0.295791
$$976$$ 0 0
$$977$$ −2.21478 −0.0708571 −0.0354286 0.999372i $$-0.511280\pi$$
−0.0354286 + 0.999372i $$0.511280\pi$$
$$978$$ 0 0
$$979$$ 1.52786 0.0488307
$$980$$ 0 0
$$981$$ −1.34752 −0.0430231
$$982$$ 0 0
$$983$$ −6.90983 −0.220389 −0.110195 0.993910i $$-0.535147\pi$$
−0.110195 + 0.993910i $$0.535147\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ −13.7082 −0.436337
$$988$$ 0 0
$$989$$ −93.1935 −2.96338
$$990$$ 0 0
$$991$$ 28.2492 0.897366 0.448683 0.893691i $$-0.351893\pi$$
0.448683 + 0.893691i $$0.351893\pi$$
$$992$$ 0 0
$$993$$ 29.1803 0.926010
$$994$$ 0 0
$$995$$ −7.47214 −0.236883
$$996$$ 0 0
$$997$$ −29.0344 −0.919530 −0.459765 0.888041i $$-0.652066\pi$$
−0.459765 + 0.888041i $$0.652066\pi$$
$$998$$ 0 0
$$999$$ 2.58359 0.0817412
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 8092.2.a.m.1.2 2
17.16 even 2 476.2.a.b.1.1 2
51.50 odd 2 4284.2.a.m.1.1 2
68.67 odd 2 1904.2.a.j.1.2 2
119.118 odd 2 3332.2.a.l.1.2 2
136.67 odd 2 7616.2.a.p.1.1 2
136.101 even 2 7616.2.a.u.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.b.1.1 2 17.16 even 2
1904.2.a.j.1.2 2 68.67 odd 2
3332.2.a.l.1.2 2 119.118 odd 2
4284.2.a.m.1.1 2 51.50 odd 2
7616.2.a.p.1.1 2 136.67 odd 2
7616.2.a.u.1.2 2 136.101 even 2
8092.2.a.m.1.2 2 1.1 even 1 trivial