# Properties

 Label 2320.2.a.k.1.2 Level $2320$ Weight $2$ Character 2320.1 Self dual yes Analytic conductor $18.525$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, 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("2320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2320 = 2^{4} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2320.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: $$18.5252932689$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2320.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} +1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} +1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} -0.828427 q^{11} -2.00000 q^{13} +2.00000 q^{15} +2.82843 q^{17} +4.82843 q^{19} +9.65685 q^{21} +3.17157 q^{23} +1.00000 q^{25} -4.00000 q^{27} +1.00000 q^{29} -6.48528 q^{31} -1.65685 q^{33} +4.82843 q^{35} -8.48528 q^{37} -4.00000 q^{39} -6.00000 q^{41} +6.00000 q^{43} +1.00000 q^{45} +11.6569 q^{47} +16.3137 q^{49} +5.65685 q^{51} -3.65685 q^{53} -0.828427 q^{55} +9.65685 q^{57} -3.65685 q^{61} +4.82843 q^{63} -2.00000 q^{65} -6.48528 q^{67} +6.34315 q^{69} +15.3137 q^{71} +8.48528 q^{73} +2.00000 q^{75} -4.00000 q^{77} +2.48528 q^{79} -11.0000 q^{81} -7.17157 q^{83} +2.82843 q^{85} +2.00000 q^{87} -7.65685 q^{89} -9.65685 q^{91} -12.9706 q^{93} +4.82843 q^{95} -12.4853 q^{97} -0.828427 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 $$2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{19} + 8 q^{21} + 12 q^{23} + 2 q^{25} - 8 q^{27} + 2 q^{29} + 4 q^{31} + 8 q^{33} + 4 q^{35} - 8 q^{39} - 12 q^{41} + 12 q^{43} + 2 q^{45} + 12 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{55} + 8 q^{57} + 4 q^{61} + 4 q^{63} - 4 q^{65} + 4 q^{67} + 24 q^{69} + 8 q^{71} + 4 q^{75} - 8 q^{77} - 12 q^{79} - 22 q^{81} - 20 q^{83} + 4 q^{87} - 4 q^{89} - 8 q^{91} + 8 q^{93} + 4 q^{95} - 8 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^19 + 8 * q^21 + 12 * q^23 + 2 * q^25 - 8 * q^27 + 2 * q^29 + 4 * q^31 + 8 * q^33 + 4 * q^35 - 8 * q^39 - 12 * q^41 + 12 * q^43 + 2 * q^45 + 12 * q^47 + 10 * q^49 + 4 * q^53 + 4 * q^55 + 8 * q^57 + 4 * q^61 + 4 * q^63 - 4 * q^65 + 4 * q^67 + 24 * q^69 + 8 * q^71 + 4 * q^75 - 8 * q^77 - 12 * q^79 - 22 * q^81 - 20 * q^83 + 4 * q^87 - 4 * q^89 - 8 * q^91 + 8 * q^93 + 4 * q^95 - 8 * q^97 + 4 * 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
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.82843 1.82497 0.912487 0.409106i $$-0.134159\pi$$
0.912487 + 0.409106i $$0.134159\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −0.828427 −0.249780 −0.124890 0.992171i $$-0.539858\pi$$
−0.124890 + 0.992171i $$0.539858\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ 2.82843 0.685994 0.342997 0.939336i $$-0.388558\pi$$
0.342997 + 0.939336i $$0.388558\pi$$
$$18$$ 0 0
$$19$$ 4.82843 1.10772 0.553859 0.832611i $$-0.313155\pi$$
0.553859 + 0.832611i $$0.313155\pi$$
$$20$$ 0 0
$$21$$ 9.65685 2.10730
$$22$$ 0 0
$$23$$ 3.17157 0.661319 0.330659 0.943750i $$-0.392729\pi$$
0.330659 + 0.943750i $$0.392729\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −6.48528 −1.16479 −0.582395 0.812906i $$-0.697884\pi$$
−0.582395 + 0.812906i $$0.697884\pi$$
$$32$$ 0 0
$$33$$ −1.65685 −0.288421
$$34$$ 0 0
$$35$$ 4.82843 0.816153
$$36$$ 0 0
$$37$$ −8.48528 −1.39497 −0.697486 0.716599i $$-0.745698\pi$$
−0.697486 + 0.716599i $$0.745698\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 11.6569 1.70033 0.850163 0.526519i $$-0.176503\pi$$
0.850163 + 0.526519i $$0.176503\pi$$
$$48$$ 0 0
$$49$$ 16.3137 2.33053
$$50$$ 0 0
$$51$$ 5.65685 0.792118
$$52$$ 0 0
$$53$$ −3.65685 −0.502308 −0.251154 0.967947i $$-0.580810\pi$$
−0.251154 + 0.967947i $$0.580810\pi$$
$$54$$ 0 0
$$55$$ −0.828427 −0.111705
$$56$$ 0 0
$$57$$ 9.65685 1.27908
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −3.65685 −0.468212 −0.234106 0.972211i $$-0.575216\pi$$
−0.234106 + 0.972211i $$0.575216\pi$$
$$62$$ 0 0
$$63$$ 4.82843 0.608325
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −6.48528 −0.792303 −0.396152 0.918185i $$-0.629655\pi$$
−0.396152 + 0.918185i $$0.629655\pi$$
$$68$$ 0 0
$$69$$ 6.34315 0.763625
$$70$$ 0 0
$$71$$ 15.3137 1.81740 0.908701 0.417447i $$-0.137075\pi$$
0.908701 + 0.417447i $$0.137075\pi$$
$$72$$ 0 0
$$73$$ 8.48528 0.993127 0.496564 0.868000i $$-0.334595\pi$$
0.496564 + 0.868000i $$0.334595\pi$$
$$74$$ 0 0
$$75$$ 2.00000 0.230940
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ 2.48528 0.279616 0.139808 0.990179i $$-0.455351\pi$$
0.139808 + 0.990179i $$0.455351\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ −7.17157 −0.787182 −0.393591 0.919286i $$-0.628767\pi$$
−0.393591 + 0.919286i $$0.628767\pi$$
$$84$$ 0 0
$$85$$ 2.82843 0.306786
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −7.65685 −0.811625 −0.405812 0.913956i $$-0.633011\pi$$
−0.405812 + 0.913956i $$0.633011\pi$$
$$90$$ 0 0
$$91$$ −9.65685 −1.01231
$$92$$ 0 0
$$93$$ −12.9706 −1.34498
$$94$$ 0 0
$$95$$ 4.82843 0.495386
$$96$$ 0 0
$$97$$ −12.4853 −1.26769 −0.633844 0.773461i $$-0.718524\pi$$
−0.633844 + 0.773461i $$0.718524\pi$$
$$98$$ 0 0
$$99$$ −0.828427 −0.0832601
$$100$$ 0 0
$$101$$ 15.6569 1.55792 0.778958 0.627077i $$-0.215749\pi$$
0.778958 + 0.627077i $$0.215749\pi$$
$$102$$ 0 0
$$103$$ −16.1421 −1.59053 −0.795266 0.606261i $$-0.792669\pi$$
−0.795266 + 0.606261i $$0.792669\pi$$
$$104$$ 0 0
$$105$$ 9.65685 0.942412
$$106$$ 0 0
$$107$$ −20.1421 −1.94721 −0.973607 0.228232i $$-0.926706\pi$$
−0.973607 + 0.228232i $$0.926706\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −16.9706 −1.61077
$$112$$ 0 0
$$113$$ −2.82843 −0.266076 −0.133038 0.991111i $$-0.542473\pi$$
−0.133038 + 0.991111i $$0.542473\pi$$
$$114$$ 0 0
$$115$$ 3.17157 0.295751
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 13.6569 1.25192
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 0 0
$$123$$ −12.0000 −1.08200
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −6.00000 −0.532414 −0.266207 0.963916i $$-0.585770\pi$$
−0.266207 + 0.963916i $$0.585770\pi$$
$$128$$ 0 0
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 12.1421 1.06086 0.530432 0.847728i $$-0.322030\pi$$
0.530432 + 0.847728i $$0.322030\pi$$
$$132$$ 0 0
$$133$$ 23.3137 2.02155
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ −5.17157 −0.441837 −0.220919 0.975292i $$-0.570906\pi$$
−0.220919 + 0.975292i $$0.570906\pi$$
$$138$$ 0 0
$$139$$ −21.6569 −1.83691 −0.918455 0.395525i $$-0.870563\pi$$
−0.918455 + 0.395525i $$0.870563\pi$$
$$140$$ 0 0
$$141$$ 23.3137 1.96337
$$142$$ 0 0
$$143$$ 1.65685 0.138553
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 0 0
$$147$$ 32.6274 2.69106
$$148$$ 0 0
$$149$$ 9.31371 0.763009 0.381504 0.924367i $$-0.375406\pi$$
0.381504 + 0.924367i $$0.375406\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 2.82843 0.228665
$$154$$ 0 0
$$155$$ −6.48528 −0.520910
$$156$$ 0 0
$$157$$ 0.485281 0.0387297 0.0193648 0.999812i $$-0.493836\pi$$
0.0193648 + 0.999812i $$0.493836\pi$$
$$158$$ 0 0
$$159$$ −7.31371 −0.580015
$$160$$ 0 0
$$161$$ 15.3137 1.20689
$$162$$ 0 0
$$163$$ 8.34315 0.653486 0.326743 0.945113i $$-0.394049\pi$$
0.326743 + 0.945113i $$0.394049\pi$$
$$164$$ 0 0
$$165$$ −1.65685 −0.128986
$$166$$ 0 0
$$167$$ 2.48528 0.192317 0.0961584 0.995366i $$-0.469344\pi$$
0.0961584 + 0.995366i $$0.469344\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 4.82843 0.369239
$$172$$ 0 0
$$173$$ 17.3137 1.31634 0.658168 0.752871i $$-0.271331\pi$$
0.658168 + 0.752871i $$0.271331\pi$$
$$174$$ 0 0
$$175$$ 4.82843 0.364995
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 23.3137 1.74255 0.871274 0.490797i $$-0.163294\pi$$
0.871274 + 0.490797i $$0.163294\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ −7.31371 −0.540645
$$184$$ 0 0
$$185$$ −8.48528 −0.623850
$$186$$ 0 0
$$187$$ −2.34315 −0.171348
$$188$$ 0 0
$$189$$ −19.3137 −1.40487
$$190$$ 0 0
$$191$$ 20.8284 1.50709 0.753546 0.657395i $$-0.228342\pi$$
0.753546 + 0.657395i $$0.228342\pi$$
$$192$$ 0 0
$$193$$ 4.48528 0.322858 0.161429 0.986884i $$-0.448390\pi$$
0.161429 + 0.986884i $$0.448390\pi$$
$$194$$ 0 0
$$195$$ −4.00000 −0.286446
$$196$$ 0 0
$$197$$ −19.6569 −1.40049 −0.700246 0.713901i $$-0.746927\pi$$
−0.700246 + 0.713901i $$0.746927\pi$$
$$198$$ 0 0
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 0 0
$$201$$ −12.9706 −0.914873
$$202$$ 0 0
$$203$$ 4.82843 0.338889
$$204$$ 0 0
$$205$$ −6.00000 −0.419058
$$206$$ 0 0
$$207$$ 3.17157 0.220440
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 0.828427 0.0570313 0.0285156 0.999593i $$-0.490922\pi$$
0.0285156 + 0.999593i $$0.490922\pi$$
$$212$$ 0 0
$$213$$ 30.6274 2.09856
$$214$$ 0 0
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ −31.3137 −2.12571
$$218$$ 0 0
$$219$$ 16.9706 1.14676
$$220$$ 0 0
$$221$$ −5.65685 −0.380521
$$222$$ 0 0
$$223$$ 17.7990 1.19191 0.595954 0.803018i $$-0.296774\pi$$
0.595954 + 0.803018i $$0.296774\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −20.1421 −1.33688 −0.668440 0.743766i $$-0.733038\pi$$
−0.668440 + 0.743766i $$0.733038\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 11.6569 0.760409
$$236$$ 0 0
$$237$$ 4.97056 0.322873
$$238$$ 0 0
$$239$$ 0.686292 0.0443925 0.0221963 0.999754i $$-0.492934\pi$$
0.0221963 + 0.999754i $$0.492934\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ 16.3137 1.04224
$$246$$ 0 0
$$247$$ −9.65685 −0.614451
$$248$$ 0 0
$$249$$ −14.3431 −0.908960
$$250$$ 0 0
$$251$$ −8.82843 −0.557245 −0.278623 0.960401i $$-0.589878\pi$$
−0.278623 + 0.960401i $$0.589878\pi$$
$$252$$ 0 0
$$253$$ −2.62742 −0.165184
$$254$$ 0 0
$$255$$ 5.65685 0.354246
$$256$$ 0 0
$$257$$ 6.68629 0.417079 0.208540 0.978014i $$-0.433129\pi$$
0.208540 + 0.978014i $$0.433129\pi$$
$$258$$ 0 0
$$259$$ −40.9706 −2.54579
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 0 0
$$263$$ 19.6569 1.21209 0.606047 0.795429i $$-0.292754\pi$$
0.606047 + 0.795429i $$0.292754\pi$$
$$264$$ 0 0
$$265$$ −3.65685 −0.224639
$$266$$ 0 0
$$267$$ −15.3137 −0.937184
$$268$$ 0 0
$$269$$ −21.3137 −1.29952 −0.649760 0.760140i $$-0.725131\pi$$
−0.649760 + 0.760140i $$0.725131\pi$$
$$270$$ 0 0
$$271$$ 9.79899 0.595246 0.297623 0.954683i $$-0.403806\pi$$
0.297623 + 0.954683i $$0.403806\pi$$
$$272$$ 0 0
$$273$$ −19.3137 −1.16892
$$274$$ 0 0
$$275$$ −0.828427 −0.0499560
$$276$$ 0 0
$$277$$ −3.65685 −0.219719 −0.109860 0.993947i $$-0.535040\pi$$
−0.109860 + 0.993947i $$0.535040\pi$$
$$278$$ 0 0
$$279$$ −6.48528 −0.388264
$$280$$ 0 0
$$281$$ −29.3137 −1.74871 −0.874355 0.485288i $$-0.838715\pi$$
−0.874355 + 0.485288i $$0.838715\pi$$
$$282$$ 0 0
$$283$$ −4.82843 −0.287020 −0.143510 0.989649i $$-0.545839\pi$$
−0.143510 + 0.989649i $$0.545839\pi$$
$$284$$ 0 0
$$285$$ 9.65685 0.572023
$$286$$ 0 0
$$287$$ −28.9706 −1.71008
$$288$$ 0 0
$$289$$ −9.00000 −0.529412
$$290$$ 0 0
$$291$$ −24.9706 −1.46380
$$292$$ 0 0
$$293$$ 8.48528 0.495715 0.247858 0.968796i $$-0.420273\pi$$
0.247858 + 0.968796i $$0.420273\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.31371 0.192281
$$298$$ 0 0
$$299$$ −6.34315 −0.366834
$$300$$ 0 0
$$301$$ 28.9706 1.66984
$$302$$ 0 0
$$303$$ 31.3137 1.79893
$$304$$ 0 0
$$305$$ −3.65685 −0.209391
$$306$$ 0 0
$$307$$ 22.9706 1.31100 0.655500 0.755195i $$-0.272458\pi$$
0.655500 + 0.755195i $$0.272458\pi$$
$$308$$ 0 0
$$309$$ −32.2843 −1.83659
$$310$$ 0 0
$$311$$ −14.4853 −0.821385 −0.410692 0.911774i $$-0.634713\pi$$
−0.410692 + 0.911774i $$0.634713\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 0 0
$$315$$ 4.82843 0.272051
$$316$$ 0 0
$$317$$ 2.82843 0.158860 0.0794301 0.996840i $$-0.474690\pi$$
0.0794301 + 0.996840i $$0.474690\pi$$
$$318$$ 0 0
$$319$$ −0.828427 −0.0463830
$$320$$ 0 0
$$321$$ −40.2843 −2.24845
$$322$$ 0 0
$$323$$ 13.6569 0.759888
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ 4.00000 0.221201
$$328$$ 0 0
$$329$$ 56.2843 3.10305
$$330$$ 0 0
$$331$$ −21.7990 −1.19818 −0.599090 0.800681i $$-0.704471\pi$$
−0.599090 + 0.800681i $$0.704471\pi$$
$$332$$ 0 0
$$333$$ −8.48528 −0.464991
$$334$$ 0 0
$$335$$ −6.48528 −0.354329
$$336$$ 0 0
$$337$$ −1.17157 −0.0638196 −0.0319098 0.999491i $$-0.510159\pi$$
−0.0319098 + 0.999491i $$0.510159\pi$$
$$338$$ 0 0
$$339$$ −5.65685 −0.307238
$$340$$ 0 0
$$341$$ 5.37258 0.290942
$$342$$ 0 0
$$343$$ 44.9706 2.42818
$$344$$ 0 0
$$345$$ 6.34315 0.341503
$$346$$ 0 0
$$347$$ 8.14214 0.437093 0.218546 0.975827i $$-0.429869\pi$$
0.218546 + 0.975827i $$0.429869\pi$$
$$348$$ 0 0
$$349$$ 20.6274 1.10416 0.552080 0.833791i $$-0.313834\pi$$
0.552080 + 0.833791i $$0.313834\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ 0 0
$$353$$ −4.34315 −0.231162 −0.115581 0.993298i $$-0.536873\pi$$
−0.115581 + 0.993298i $$0.536873\pi$$
$$354$$ 0 0
$$355$$ 15.3137 0.812767
$$356$$ 0 0
$$357$$ 27.3137 1.44559
$$358$$ 0 0
$$359$$ 3.85786 0.203610 0.101805 0.994804i $$-0.467538\pi$$
0.101805 + 0.994804i $$0.467538\pi$$
$$360$$ 0 0
$$361$$ 4.31371 0.227037
$$362$$ 0 0
$$363$$ −20.6274 −1.08266
$$364$$ 0 0
$$365$$ 8.48528 0.444140
$$366$$ 0 0
$$367$$ −18.0000 −0.939592 −0.469796 0.882775i $$-0.655673\pi$$
−0.469796 + 0.882775i $$0.655673\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −17.6569 −0.916698
$$372$$ 0 0
$$373$$ −6.97056 −0.360922 −0.180461 0.983582i $$-0.557759\pi$$
−0.180461 + 0.983582i $$0.557759\pi$$
$$374$$ 0 0
$$375$$ 2.00000 0.103280
$$376$$ 0 0
$$377$$ −2.00000 −0.103005
$$378$$ 0 0
$$379$$ −22.4853 −1.15499 −0.577496 0.816394i $$-0.695970\pi$$
−0.577496 + 0.816394i $$0.695970\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 0 0
$$383$$ −2.48528 −0.126992 −0.0634960 0.997982i $$-0.520225\pi$$
−0.0634960 + 0.997982i $$0.520225\pi$$
$$384$$ 0 0
$$385$$ −4.00000 −0.203859
$$386$$ 0 0
$$387$$ 6.00000 0.304997
$$388$$ 0 0
$$389$$ −29.3137 −1.48626 −0.743132 0.669145i $$-0.766661\pi$$
−0.743132 + 0.669145i $$0.766661\pi$$
$$390$$ 0 0
$$391$$ 8.97056 0.453661
$$392$$ 0 0
$$393$$ 24.2843 1.22498
$$394$$ 0 0
$$395$$ 2.48528 0.125048
$$396$$ 0 0
$$397$$ −19.6569 −0.986549 −0.493275 0.869874i $$-0.664200\pi$$
−0.493275 + 0.869874i $$0.664200\pi$$
$$398$$ 0 0
$$399$$ 46.6274 2.33429
$$400$$ 0 0
$$401$$ −6.68629 −0.333897 −0.166949 0.985966i $$-0.553391\pi$$
−0.166949 + 0.985966i $$0.553391\pi$$
$$402$$ 0 0
$$403$$ 12.9706 0.646110
$$404$$ 0 0
$$405$$ −11.0000 −0.546594
$$406$$ 0 0
$$407$$ 7.02944 0.348436
$$408$$ 0 0
$$409$$ −2.97056 −0.146885 −0.0734424 0.997299i $$-0.523399\pi$$
−0.0734424 + 0.997299i $$0.523399\pi$$
$$410$$ 0 0
$$411$$ −10.3431 −0.510190
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −7.17157 −0.352039
$$416$$ 0 0
$$417$$ −43.3137 −2.12108
$$418$$ 0 0
$$419$$ −28.9706 −1.41530 −0.707652 0.706561i $$-0.750246\pi$$
−0.707652 + 0.706561i $$0.750246\pi$$
$$420$$ 0 0
$$421$$ 18.9706 0.924569 0.462284 0.886732i $$-0.347030\pi$$
0.462284 + 0.886732i $$0.347030\pi$$
$$422$$ 0 0
$$423$$ 11.6569 0.566776
$$424$$ 0 0
$$425$$ 2.82843 0.137199
$$426$$ 0 0
$$427$$ −17.6569 −0.854475
$$428$$ 0 0
$$429$$ 3.31371 0.159987
$$430$$ 0 0
$$431$$ −3.31371 −0.159616 −0.0798079 0.996810i $$-0.525431\pi$$
−0.0798079 + 0.996810i $$0.525431\pi$$
$$432$$ 0 0
$$433$$ −29.1716 −1.40190 −0.700948 0.713212i $$-0.747240\pi$$
−0.700948 + 0.713212i $$0.747240\pi$$
$$434$$ 0 0
$$435$$ 2.00000 0.0958927
$$436$$ 0 0
$$437$$ 15.3137 0.732554
$$438$$ 0 0
$$439$$ 10.3431 0.493651 0.246826 0.969060i $$-0.420612\pi$$
0.246826 + 0.969060i $$0.420612\pi$$
$$440$$ 0 0
$$441$$ 16.3137 0.776843
$$442$$ 0 0
$$443$$ 7.65685 0.363788 0.181894 0.983318i $$-0.441777\pi$$
0.181894 + 0.983318i $$0.441777\pi$$
$$444$$ 0 0
$$445$$ −7.65685 −0.362970
$$446$$ 0 0
$$447$$ 18.6274 0.881047
$$448$$ 0 0
$$449$$ 11.6569 0.550121 0.275060 0.961427i $$-0.411302\pi$$
0.275060 + 0.961427i $$0.411302\pi$$
$$450$$ 0 0
$$451$$ 4.97056 0.234055
$$452$$ 0 0
$$453$$ 24.0000 1.12762
$$454$$ 0 0
$$455$$ −9.65685 −0.452720
$$456$$ 0 0
$$457$$ 19.6569 0.919509 0.459754 0.888046i $$-0.347937\pi$$
0.459754 + 0.888046i $$0.347937\pi$$
$$458$$ 0 0
$$459$$ −11.3137 −0.528079
$$460$$ 0 0
$$461$$ −35.6569 −1.66071 −0.830353 0.557238i $$-0.811861\pi$$
−0.830353 + 0.557238i $$0.811861\pi$$
$$462$$ 0 0
$$463$$ −21.7990 −1.01308 −0.506542 0.862215i $$-0.669077\pi$$
−0.506542 + 0.862215i $$0.669077\pi$$
$$464$$ 0 0
$$465$$ −12.9706 −0.601495
$$466$$ 0 0
$$467$$ −10.9706 −0.507657 −0.253829 0.967249i $$-0.581690\pi$$
−0.253829 + 0.967249i $$0.581690\pi$$
$$468$$ 0 0
$$469$$ −31.3137 −1.44593
$$470$$ 0 0
$$471$$ 0.970563 0.0447212
$$472$$ 0 0
$$473$$ −4.97056 −0.228547
$$474$$ 0 0
$$475$$ 4.82843 0.221543
$$476$$ 0 0
$$477$$ −3.65685 −0.167436
$$478$$ 0 0
$$479$$ −7.17157 −0.327678 −0.163839 0.986487i $$-0.552388\pi$$
−0.163839 + 0.986487i $$0.552388\pi$$
$$480$$ 0 0
$$481$$ 16.9706 0.773791
$$482$$ 0 0
$$483$$ 30.6274 1.39360
$$484$$ 0 0
$$485$$ −12.4853 −0.566927
$$486$$ 0 0
$$487$$ −9.79899 −0.444035 −0.222017 0.975043i $$-0.571264\pi$$
−0.222017 + 0.975043i $$0.571264\pi$$
$$488$$ 0 0
$$489$$ 16.6863 0.754580
$$490$$ 0 0
$$491$$ 7.45584 0.336478 0.168239 0.985746i $$-0.446192\pi$$
0.168239 + 0.985746i $$0.446192\pi$$
$$492$$ 0 0
$$493$$ 2.82843 0.127386
$$494$$ 0 0
$$495$$ −0.828427 −0.0372350
$$496$$ 0 0
$$497$$ 73.9411 3.31671
$$498$$ 0 0
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 0 0
$$501$$ 4.97056 0.222068
$$502$$ 0 0
$$503$$ 30.0000 1.33763 0.668817 0.743427i $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$504$$ 0 0
$$505$$ 15.6569 0.696721
$$506$$ 0 0
$$507$$ −18.0000 −0.799408
$$508$$ 0 0
$$509$$ 0.627417 0.0278098 0.0139049 0.999903i $$-0.495574\pi$$
0.0139049 + 0.999903i $$0.495574\pi$$
$$510$$ 0 0
$$511$$ 40.9706 1.81243
$$512$$ 0 0
$$513$$ −19.3137 −0.852721
$$514$$ 0 0
$$515$$ −16.1421 −0.711307
$$516$$ 0 0
$$517$$ −9.65685 −0.424708
$$518$$ 0 0
$$519$$ 34.6274 1.51997
$$520$$ 0 0
$$521$$ −21.3137 −0.933771 −0.466885 0.884318i $$-0.654624\pi$$
−0.466885 + 0.884318i $$0.654624\pi$$
$$522$$ 0 0
$$523$$ −2.48528 −0.108674 −0.0543369 0.998523i $$-0.517304\pi$$
−0.0543369 + 0.998523i $$0.517304\pi$$
$$524$$ 0 0
$$525$$ 9.65685 0.421460
$$526$$ 0 0
$$527$$ −18.3431 −0.799040
$$528$$ 0 0
$$529$$ −12.9411 −0.562658
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ −20.1421 −0.870820
$$536$$ 0 0
$$537$$ 46.6274 2.01212
$$538$$ 0 0
$$539$$ −13.5147 −0.582120
$$540$$ 0 0
$$541$$ −5.02944 −0.216232 −0.108116 0.994138i $$-0.534482\pi$$
−0.108116 + 0.994138i $$0.534482\pi$$
$$542$$ 0 0
$$543$$ −12.0000 −0.514969
$$544$$ 0 0
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 2.48528 0.106263 0.0531315 0.998588i $$-0.483080\pi$$
0.0531315 + 0.998588i $$0.483080\pi$$
$$548$$ 0 0
$$549$$ −3.65685 −0.156071
$$550$$ 0 0
$$551$$ 4.82843 0.205698
$$552$$ 0 0
$$553$$ 12.0000 0.510292
$$554$$ 0 0
$$555$$ −16.9706 −0.720360
$$556$$ 0 0
$$557$$ −27.9411 −1.18390 −0.591952 0.805973i $$-0.701642\pi$$
−0.591952 + 0.805973i $$0.701642\pi$$
$$558$$ 0 0
$$559$$ −12.0000 −0.507546
$$560$$ 0 0
$$561$$ −4.68629 −0.197855
$$562$$ 0 0
$$563$$ −7.65685 −0.322698 −0.161349 0.986897i $$-0.551584\pi$$
−0.161349 + 0.986897i $$0.551584\pi$$
$$564$$ 0 0
$$565$$ −2.82843 −0.118993
$$566$$ 0 0
$$567$$ −53.1127 −2.23052
$$568$$ 0 0
$$569$$ 27.6569 1.15944 0.579718 0.814817i $$-0.303163\pi$$
0.579718 + 0.814817i $$0.303163\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 41.6569 1.74024
$$574$$ 0 0
$$575$$ 3.17157 0.132264
$$576$$ 0 0
$$577$$ 23.7990 0.990765 0.495382 0.868675i $$-0.335028\pi$$
0.495382 + 0.868675i $$0.335028\pi$$
$$578$$ 0 0
$$579$$ 8.97056 0.372804
$$580$$ 0 0
$$581$$ −34.6274 −1.43659
$$582$$ 0 0
$$583$$ 3.02944 0.125466
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ 29.7990 1.22994 0.614968 0.788552i $$-0.289169\pi$$
0.614968 + 0.788552i $$0.289169\pi$$
$$588$$ 0 0
$$589$$ −31.3137 −1.29026
$$590$$ 0 0
$$591$$ −39.3137 −1.61715
$$592$$ 0 0
$$593$$ −7.65685 −0.314429 −0.157215 0.987564i $$-0.550251\pi$$
−0.157215 + 0.987564i $$0.550251\pi$$
$$594$$ 0 0
$$595$$ 13.6569 0.559876
$$596$$ 0 0
$$597$$ −24.0000 −0.982255
$$598$$ 0 0
$$599$$ 37.7990 1.54442 0.772212 0.635364i $$-0.219150\pi$$
0.772212 + 0.635364i $$0.219150\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ −6.48528 −0.264101
$$604$$ 0 0
$$605$$ −10.3137 −0.419312
$$606$$ 0 0
$$607$$ −9.02944 −0.366494 −0.183247 0.983067i $$-0.558661\pi$$
−0.183247 + 0.983067i $$0.558661\pi$$
$$608$$ 0 0
$$609$$ 9.65685 0.391315
$$610$$ 0 0
$$611$$ −23.3137 −0.943172
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 0 0
$$615$$ −12.0000 −0.483887
$$616$$ 0 0
$$617$$ −9.17157 −0.369234 −0.184617 0.982811i $$-0.559104\pi$$
−0.184617 + 0.982811i $$0.559104\pi$$
$$618$$ 0 0
$$619$$ 9.79899 0.393855 0.196927 0.980418i $$-0.436904\pi$$
0.196927 + 0.980418i $$0.436904\pi$$
$$620$$ 0 0
$$621$$ −12.6863 −0.509083
$$622$$ 0 0
$$623$$ −36.9706 −1.48119
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −8.00000 −0.319489
$$628$$ 0 0
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ −36.9706 −1.47177 −0.735887 0.677104i $$-0.763235\pi$$
−0.735887 + 0.677104i $$0.763235\pi$$
$$632$$ 0 0
$$633$$ 1.65685 0.0658540
$$634$$ 0 0
$$635$$ −6.00000 −0.238103
$$636$$ 0 0
$$637$$ −32.6274 −1.29275
$$638$$ 0 0
$$639$$ 15.3137 0.605801
$$640$$ 0 0
$$641$$ 0.627417 0.0247815 0.0123907 0.999923i $$-0.496056\pi$$
0.0123907 + 0.999923i $$0.496056\pi$$
$$642$$ 0 0
$$643$$ −19.4558 −0.767264 −0.383632 0.923486i $$-0.625327\pi$$
−0.383632 + 0.923486i $$0.625327\pi$$
$$644$$ 0 0
$$645$$ 12.0000 0.472500
$$646$$ 0 0
$$647$$ 41.1127 1.61631 0.808153 0.588972i $$-0.200467\pi$$
0.808153 + 0.588972i $$0.200467\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −62.6274 −2.45456
$$652$$ 0 0
$$653$$ −17.1716 −0.671976 −0.335988 0.941866i $$-0.609070\pi$$
−0.335988 + 0.941866i $$0.609070\pi$$
$$654$$ 0 0
$$655$$ 12.1421 0.474432
$$656$$ 0 0
$$657$$ 8.48528 0.331042
$$658$$ 0 0
$$659$$ −1.79899 −0.0700787 −0.0350393 0.999386i $$-0.511156\pi$$
−0.0350393 + 0.999386i $$0.511156\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ 0 0
$$663$$ −11.3137 −0.439388
$$664$$ 0 0
$$665$$ 23.3137 0.904067
$$666$$ 0 0
$$667$$ 3.17157 0.122804
$$668$$ 0 0
$$669$$ 35.5980 1.37630
$$670$$ 0 0
$$671$$ 3.02944 0.116950
$$672$$ 0 0
$$673$$ 22.9706 0.885450 0.442725 0.896657i $$-0.354012\pi$$
0.442725 + 0.896657i $$0.354012\pi$$
$$674$$ 0 0
$$675$$ −4.00000 −0.153960
$$676$$ 0 0
$$677$$ −36.7696 −1.41317 −0.706584 0.707629i $$-0.749765\pi$$
−0.706584 + 0.707629i $$0.749765\pi$$
$$678$$ 0 0
$$679$$ −60.2843 −2.31350
$$680$$ 0 0
$$681$$ −40.2843 −1.54370
$$682$$ 0 0
$$683$$ −11.8579 −0.453729 −0.226864 0.973926i $$-0.572847\pi$$
−0.226864 + 0.973926i $$0.572847\pi$$
$$684$$ 0 0
$$685$$ −5.17157 −0.197596
$$686$$ 0 0
$$687$$ −4.00000 −0.152610
$$688$$ 0 0
$$689$$ 7.31371 0.278630
$$690$$ 0 0
$$691$$ 44.9706 1.71076 0.855380 0.518000i $$-0.173323\pi$$
0.855380 + 0.518000i $$0.173323\pi$$
$$692$$ 0 0
$$693$$ −4.00000 −0.151947
$$694$$ 0 0
$$695$$ −21.6569 −0.821491
$$696$$ 0 0
$$697$$ −16.9706 −0.642806
$$698$$ 0 0
$$699$$ 36.0000 1.36165
$$700$$ 0 0
$$701$$ 6.68629 0.252538 0.126269 0.991996i $$-0.459700\pi$$
0.126269 + 0.991996i $$0.459700\pi$$
$$702$$ 0 0
$$703$$ −40.9706 −1.54523
$$704$$ 0 0
$$705$$ 23.3137 0.878045
$$706$$ 0 0
$$707$$ 75.5980 2.84315
$$708$$ 0 0
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ 2.48528 0.0932053
$$712$$ 0 0
$$713$$ −20.5685 −0.770298
$$714$$ 0 0
$$715$$ 1.65685 0.0619628
$$716$$ 0 0
$$717$$ 1.37258 0.0512601
$$718$$ 0 0
$$719$$ 34.6274 1.29138 0.645692 0.763598i $$-0.276569\pi$$
0.645692 + 0.763598i $$0.276569\pi$$
$$720$$ 0 0
$$721$$ −77.9411 −2.90268
$$722$$ 0 0
$$723$$ 20.0000 0.743808
$$724$$ 0 0
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ 23.9411 0.887927 0.443964 0.896045i $$-0.353572\pi$$
0.443964 + 0.896045i $$0.353572\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 16.9706 0.627679
$$732$$ 0 0
$$733$$ −22.8284 −0.843187 −0.421594 0.906785i $$-0.638529\pi$$
−0.421594 + 0.906785i $$0.638529\pi$$
$$734$$ 0 0
$$735$$ 32.6274 1.20348
$$736$$ 0 0
$$737$$ 5.37258 0.197902
$$738$$ 0 0
$$739$$ −14.4853 −0.532850 −0.266425 0.963856i $$-0.585842\pi$$
−0.266425 + 0.963856i $$0.585842\pi$$
$$740$$ 0 0
$$741$$ −19.3137 −0.709507
$$742$$ 0 0
$$743$$ 52.6274 1.93071 0.965356 0.260935i $$-0.0840309\pi$$
0.965356 + 0.260935i $$0.0840309\pi$$
$$744$$ 0 0
$$745$$ 9.31371 0.341228
$$746$$ 0 0
$$747$$ −7.17157 −0.262394
$$748$$ 0 0
$$749$$ −97.2548 −3.55361
$$750$$ 0 0
$$751$$ −16.1421 −0.589035 −0.294517 0.955646i $$-0.595159\pi$$
−0.294517 + 0.955646i $$0.595159\pi$$
$$752$$ 0 0
$$753$$ −17.6569 −0.643452
$$754$$ 0 0
$$755$$ 12.0000 0.436725
$$756$$ 0 0
$$757$$ 19.5147 0.709275 0.354637 0.935004i $$-0.384604\pi$$
0.354637 + 0.935004i $$0.384604\pi$$
$$758$$ 0 0
$$759$$ −5.25483 −0.190738
$$760$$ 0 0
$$761$$ 8.62742 0.312744 0.156372 0.987698i $$-0.450020\pi$$
0.156372 + 0.987698i $$0.450020\pi$$
$$762$$ 0 0
$$763$$ 9.65685 0.349602
$$764$$ 0 0
$$765$$ 2.82843 0.102262
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −15.6569 −0.564601 −0.282300 0.959326i $$-0.591097\pi$$
−0.282300 + 0.959326i $$0.591097\pi$$
$$770$$ 0 0
$$771$$ 13.3726 0.481602
$$772$$ 0 0
$$773$$ −8.48528 −0.305194 −0.152597 0.988288i $$-0.548764\pi$$
−0.152597 + 0.988288i $$0.548764\pi$$
$$774$$ 0 0
$$775$$ −6.48528 −0.232958
$$776$$ 0 0
$$777$$ −81.9411 −2.93962
$$778$$ 0 0
$$779$$ −28.9706 −1.03798
$$780$$ 0 0
$$781$$ −12.6863 −0.453951
$$782$$ 0 0
$$783$$ −4.00000 −0.142948
$$784$$ 0 0
$$785$$ 0.485281 0.0173204
$$786$$ 0 0
$$787$$ −17.7990 −0.634465 −0.317233 0.948348i $$-0.602754\pi$$
−0.317233 + 0.948348i $$0.602754\pi$$
$$788$$ 0 0
$$789$$ 39.3137 1.39961
$$790$$ 0 0
$$791$$ −13.6569 −0.485582
$$792$$ 0 0
$$793$$ 7.31371 0.259717
$$794$$ 0 0
$$795$$ −7.31371 −0.259391
$$796$$ 0 0
$$797$$ −5.85786 −0.207496 −0.103748 0.994604i $$-0.533084\pi$$
−0.103748 + 0.994604i $$0.533084\pi$$
$$798$$ 0 0
$$799$$ 32.9706 1.16641
$$800$$ 0 0
$$801$$ −7.65685 −0.270542
$$802$$ 0 0
$$803$$ −7.02944 −0.248063
$$804$$ 0 0
$$805$$ 15.3137 0.539737
$$806$$ 0 0
$$807$$ −42.6274 −1.50056
$$808$$ 0 0
$$809$$ 42.2843 1.48664 0.743318 0.668938i $$-0.233251\pi$$
0.743318 + 0.668938i $$0.233251\pi$$
$$810$$ 0 0
$$811$$ −37.6569 −1.32231 −0.661155 0.750249i $$-0.729934\pi$$
−0.661155 + 0.750249i $$0.729934\pi$$
$$812$$ 0 0
$$813$$ 19.5980 0.687331
$$814$$ 0 0
$$815$$ 8.34315 0.292248
$$816$$ 0 0
$$817$$ 28.9706 1.01355
$$818$$ 0 0
$$819$$ −9.65685 −0.337438
$$820$$ 0 0
$$821$$ −22.6863 −0.791757 −0.395879 0.918303i $$-0.629560\pi$$
−0.395879 + 0.918303i $$0.629560\pi$$
$$822$$ 0 0
$$823$$ 30.9706 1.07957 0.539783 0.841804i $$-0.318506\pi$$
0.539783 + 0.841804i $$0.318506\pi$$
$$824$$ 0 0
$$825$$ −1.65685 −0.0576843
$$826$$ 0 0
$$827$$ −17.3137 −0.602057 −0.301028 0.953615i $$-0.597330\pi$$
−0.301028 + 0.953615i $$0.597330\pi$$
$$828$$ 0 0
$$829$$ 20.6274 0.716420 0.358210 0.933641i $$-0.383387\pi$$
0.358210 + 0.933641i $$0.383387\pi$$
$$830$$ 0 0
$$831$$ −7.31371 −0.253710
$$832$$ 0 0
$$833$$ 46.1421 1.59873
$$834$$ 0 0
$$835$$ 2.48528 0.0860067
$$836$$ 0 0
$$837$$ 25.9411 0.896656
$$838$$ 0 0
$$839$$ −2.48528 −0.0858014 −0.0429007 0.999079i $$-0.513660\pi$$
−0.0429007 + 0.999079i $$0.513660\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ −58.6274 −2.01924
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ −49.7990 −1.71111
$$848$$ 0 0
$$849$$ −9.65685 −0.331422
$$850$$ 0 0
$$851$$ −26.9117 −0.922521
$$852$$ 0 0
$$853$$ 51.1127 1.75007 0.875033 0.484064i $$-0.160840\pi$$
0.875033 + 0.484064i $$0.160840\pi$$
$$854$$ 0 0
$$855$$ 4.82843 0.165129
$$856$$ 0 0
$$857$$ 3.37258 0.115205 0.0576026 0.998340i $$-0.481654\pi$$
0.0576026 + 0.998340i $$0.481654\pi$$
$$858$$ 0 0
$$859$$ 56.4264 1.92524 0.962622 0.270848i $$-0.0873041\pi$$
0.962622 + 0.270848i $$0.0873041\pi$$
$$860$$ 0 0
$$861$$ −57.9411 −1.97463
$$862$$ 0 0
$$863$$ 36.1421 1.23029 0.615146 0.788413i $$-0.289097\pi$$
0.615146 + 0.788413i $$0.289097\pi$$
$$864$$ 0 0
$$865$$ 17.3137 0.588684
$$866$$ 0 0
$$867$$ −18.0000 −0.611312
$$868$$ 0 0
$$869$$ −2.05887 −0.0698425
$$870$$ 0 0
$$871$$ 12.9706 0.439491
$$872$$ 0 0
$$873$$ −12.4853 −0.422563
$$874$$ 0 0
$$875$$ 4.82843 0.163231
$$876$$ 0 0
$$877$$ 38.2843 1.29277 0.646384 0.763012i $$-0.276280\pi$$
0.646384 + 0.763012i $$0.276280\pi$$
$$878$$ 0 0
$$879$$ 16.9706 0.572403
$$880$$ 0 0
$$881$$ 29.3137 0.987604 0.493802 0.869574i $$-0.335607\pi$$
0.493802 + 0.869574i $$0.335607\pi$$
$$882$$ 0 0
$$883$$ 14.4853 0.487469 0.243734 0.969842i $$-0.421628\pi$$
0.243734 + 0.969842i $$0.421628\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 6.68629 0.224504 0.112252 0.993680i $$-0.464194\pi$$
0.112252 + 0.993680i $$0.464194\pi$$
$$888$$ 0 0
$$889$$ −28.9706 −0.971641
$$890$$ 0 0
$$891$$ 9.11270 0.305287
$$892$$ 0 0
$$893$$ 56.2843 1.88348
$$894$$ 0 0
$$895$$ 23.3137 0.779291
$$896$$ 0 0
$$897$$ −12.6863 −0.423583
$$898$$ 0 0
$$899$$ −6.48528 −0.216296
$$900$$ 0 0
$$901$$ −10.3431 −0.344580
$$902$$ 0 0
$$903$$ 57.9411 1.92816
$$904$$ 0 0
$$905$$ −6.00000 −0.199447
$$906$$ 0 0
$$907$$ 10.0000 0.332045 0.166022 0.986122i $$-0.446908\pi$$
0.166022 + 0.986122i $$0.446908\pi$$
$$908$$ 0 0
$$909$$ 15.6569 0.519305
$$910$$ 0 0
$$911$$ −32.1421 −1.06492 −0.532458 0.846456i $$-0.678732\pi$$
−0.532458 + 0.846456i $$0.678732\pi$$
$$912$$ 0 0
$$913$$ 5.94113 0.196623
$$914$$ 0 0
$$915$$ −7.31371 −0.241784
$$916$$ 0 0
$$917$$ 58.6274 1.93605
$$918$$ 0 0
$$919$$ 36.0000 1.18753 0.593765 0.804638i $$-0.297641\pi$$
0.593765 + 0.804638i $$0.297641\pi$$
$$920$$ 0 0
$$921$$ 45.9411 1.51381
$$922$$ 0 0
$$923$$ −30.6274 −1.00811
$$924$$ 0 0
$$925$$ −8.48528 −0.278994
$$926$$ 0 0
$$927$$ −16.1421 −0.530177
$$928$$ 0 0
$$929$$ −4.62742 −0.151821 −0.0759103 0.997115i $$-0.524186\pi$$
−0.0759103 + 0.997115i $$0.524186\pi$$
$$930$$ 0 0
$$931$$ 78.7696 2.58157
$$932$$ 0 0
$$933$$ −28.9706 −0.948454
$$934$$ 0 0
$$935$$ −2.34315 −0.0766291
$$936$$ 0 0
$$937$$ 19.6569 0.642161 0.321081 0.947052i $$-0.395954\pi$$
0.321081 + 0.947052i $$0.395954\pi$$
$$938$$ 0 0
$$939$$ −12.0000 −0.391605
$$940$$ 0 0
$$941$$ −27.9411 −0.910855 −0.455427 0.890273i $$-0.650514\pi$$
−0.455427 + 0.890273i $$0.650514\pi$$
$$942$$ 0 0
$$943$$ −19.0294 −0.619684
$$944$$ 0 0
$$945$$ −19.3137 −0.628275
$$946$$ 0 0
$$947$$ −44.9117 −1.45943 −0.729717 0.683749i $$-0.760348\pi$$
−0.729717 + 0.683749i $$0.760348\pi$$
$$948$$ 0 0
$$949$$ −16.9706 −0.550888
$$950$$ 0 0
$$951$$ 5.65685 0.183436
$$952$$ 0 0
$$953$$ 29.3137 0.949564 0.474782 0.880103i $$-0.342527\pi$$
0.474782 + 0.880103i $$0.342527\pi$$
$$954$$ 0 0
$$955$$ 20.8284 0.673992
$$956$$ 0 0
$$957$$ −1.65685 −0.0535585
$$958$$ 0 0
$$959$$ −24.9706 −0.806342
$$960$$ 0 0
$$961$$ 11.0589 0.356738
$$962$$ 0 0
$$963$$ −20.1421 −0.649071
$$964$$ 0 0
$$965$$ 4.48528 0.144386
$$966$$ 0 0
$$967$$ −14.9706 −0.481421 −0.240710 0.970597i $$-0.577380\pi$$
−0.240710 + 0.970597i $$0.577380\pi$$
$$968$$ 0 0
$$969$$ 27.3137 0.877443
$$970$$ 0 0
$$971$$ −28.1421 −0.903124 −0.451562 0.892240i $$-0.649133\pi$$
−0.451562 + 0.892240i $$0.649133\pi$$
$$972$$ 0 0
$$973$$ −104.569 −3.35231
$$974$$ 0 0
$$975$$ −4.00000 −0.128103
$$976$$ 0 0
$$977$$ −2.68629 −0.0859421 −0.0429710 0.999076i $$-0.513682\pi$$
−0.0429710 + 0.999076i $$0.513682\pi$$
$$978$$ 0 0
$$979$$ 6.34315 0.202728
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ 9.31371 0.297061 0.148531 0.988908i $$-0.452546\pi$$
0.148531 + 0.988908i $$0.452546\pi$$
$$984$$ 0 0
$$985$$ −19.6569 −0.626319
$$986$$ 0 0
$$987$$ 112.569 3.58310
$$988$$ 0 0
$$989$$ 19.0294 0.605101
$$990$$ 0 0
$$991$$ −52.0000 −1.65183 −0.825917 0.563791i $$-0.809342\pi$$
−0.825917 + 0.563791i $$0.809342\pi$$
$$992$$ 0 0
$$993$$ −43.5980 −1.38354
$$994$$ 0 0
$$995$$ −12.0000 −0.380426
$$996$$ 0 0
$$997$$ 6.82843 0.216258 0.108129 0.994137i $$-0.465514\pi$$
0.108129 + 0.994137i $$0.465514\pi$$
$$998$$ 0 0
$$999$$ 33.9411 1.07385
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 2320.2.a.k.1.2 2
4.3 odd 2 145.2.a.b.1.2 2
8.3 odd 2 9280.2.a.be.1.1 2
8.5 even 2 9280.2.a.w.1.2 2
12.11 even 2 1305.2.a.n.1.1 2
20.3 even 4 725.2.b.c.349.2 4
20.7 even 4 725.2.b.c.349.3 4
20.19 odd 2 725.2.a.c.1.1 2
28.27 even 2 7105.2.a.e.1.2 2
60.59 even 2 6525.2.a.p.1.2 2
116.115 odd 2 4205.2.a.d.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 2 4.3 odd 2
725.2.a.c.1.1 2 20.19 odd 2
725.2.b.c.349.2 4 20.3 even 4
725.2.b.c.349.3 4 20.7 even 4
1305.2.a.n.1.1 2 12.11 even 2
2320.2.a.k.1.2 2 1.1 even 1 trivial
4205.2.a.d.1.1 2 116.115 odd 2
6525.2.a.p.1.2 2 60.59 even 2
7105.2.a.e.1.2 2 28.27 even 2
9280.2.a.w.1.2 2 8.5 even 2
9280.2.a.be.1.1 2 8.3 odd 2