# Properties

 Label 725.2.a.c.1.1 Level $725$ Weight $2$ Character 725.1 Self dual yes Analytic conductor $5.789$ 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] = [725,2,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.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: $$5.78915414654$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 725.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +0.828427 q^{11} -3.65685 q^{12} +2.00000 q^{13} -2.00000 q^{14} +3.00000 q^{16} -2.82843 q^{17} -0.414214 q^{18} -4.82843 q^{19} +9.65685 q^{21} -0.343146 q^{22} +3.17157 q^{23} +3.17157 q^{24} -0.828427 q^{26} -4.00000 q^{27} -8.82843 q^{28} +1.00000 q^{29} +6.48528 q^{31} -4.41421 q^{32} +1.65685 q^{33} +1.17157 q^{34} -1.82843 q^{36} +8.48528 q^{37} +2.00000 q^{38} +4.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} +6.00000 q^{43} -1.51472 q^{44} -1.31371 q^{46} +11.6569 q^{47} +6.00000 q^{48} +16.3137 q^{49} -5.65685 q^{51} -3.65685 q^{52} +3.65685 q^{53} +1.65685 q^{54} +7.65685 q^{56} -9.65685 q^{57} -0.414214 q^{58} -3.65685 q^{61} -2.68629 q^{62} +4.82843 q^{63} -4.17157 q^{64} -0.686292 q^{66} -6.48528 q^{67} +5.17157 q^{68} +6.34315 q^{69} -15.3137 q^{71} +1.58579 q^{72} -8.48528 q^{73} -3.51472 q^{74} +8.82843 q^{76} +4.00000 q^{77} -1.65685 q^{78} -2.48528 q^{79} -11.0000 q^{81} +2.48528 q^{82} -7.17157 q^{83} -17.6569 q^{84} -2.48528 q^{86} +2.00000 q^{87} +1.31371 q^{88} -7.65685 q^{89} +9.65685 q^{91} -5.79899 q^{92} +12.9706 q^{93} -4.82843 q^{94} -8.82843 q^{96} +12.4853 q^{97} -6.75736 q^{98} +0.828427 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 4 * q^7 + 6 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9} - 4 q^{11} + 4 q^{12} + 4 q^{13} - 4 q^{14} + 6 q^{16} + 2 q^{18} - 4 q^{19} + 8 q^{21} - 12 q^{22} + 12 q^{23} + 12 q^{24} + 4 q^{26} - 8 q^{27} - 12 q^{28} + 2 q^{29} - 4 q^{31} - 6 q^{32} - 8 q^{33} + 8 q^{34} + 2 q^{36} + 4 q^{38} + 8 q^{39} - 12 q^{41} - 8 q^{42} + 12 q^{43} - 20 q^{44} + 20 q^{46} + 12 q^{47} + 12 q^{48} + 10 q^{49} + 4 q^{52} - 4 q^{53} - 8 q^{54} + 4 q^{56} - 8 q^{57} + 2 q^{58} + 4 q^{61} - 28 q^{62} + 4 q^{63} - 14 q^{64} - 24 q^{66} + 4 q^{67} + 16 q^{68} + 24 q^{69} - 8 q^{71} + 6 q^{72} - 24 q^{74} + 12 q^{76} + 8 q^{77} + 8 q^{78} + 12 q^{79} - 22 q^{81} - 12 q^{82} - 20 q^{83} - 24 q^{84} + 12 q^{86} + 4 q^{87} - 20 q^{88} - 4 q^{89} + 8 q^{91} + 28 q^{92} - 8 q^{93} - 4 q^{94} - 12 q^{96} + 8 q^{97} - 22 q^{98} - 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 4 * q^7 + 6 * q^8 + 2 * q^9 - 4 * q^11 + 4 * q^12 + 4 * q^13 - 4 * q^14 + 6 * q^16 + 2 * q^18 - 4 * q^19 + 8 * q^21 - 12 * q^22 + 12 * q^23 + 12 * q^24 + 4 * q^26 - 8 * q^27 - 12 * q^28 + 2 * q^29 - 4 * q^31 - 6 * q^32 - 8 * q^33 + 8 * q^34 + 2 * q^36 + 4 * q^38 + 8 * q^39 - 12 * q^41 - 8 * q^42 + 12 * q^43 - 20 * q^44 + 20 * q^46 + 12 * q^47 + 12 * q^48 + 10 * q^49 + 4 * q^52 - 4 * q^53 - 8 * q^54 + 4 * q^56 - 8 * q^57 + 2 * q^58 + 4 * q^61 - 28 * q^62 + 4 * q^63 - 14 * q^64 - 24 * q^66 + 4 * q^67 + 16 * q^68 + 24 * q^69 - 8 * q^71 + 6 * q^72 - 24 * q^74 + 12 * q^76 + 8 * q^77 + 8 * q^78 + 12 * q^79 - 22 * q^81 - 12 * q^82 - 20 * q^83 - 24 * q^84 + 12 * q^86 + 4 * q^87 - 20 * q^88 - 4 * q^89 + 8 * q^91 + 28 * q^92 - 8 * q^93 - 4 * q^94 - 12 * q^96 + 8 * q^97 - 22 * q^98 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.414214 −0.292893 −0.146447 0.989219i $$-0.546784\pi$$
−0.146447 + 0.989219i $$0.546784\pi$$
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ −1.82843 −0.914214
$$5$$ 0 0
$$6$$ −0.828427 −0.338204
$$7$$ 4.82843 1.82497 0.912487 0.409106i $$-0.134159\pi$$
0.912487 + 0.409106i $$0.134159\pi$$
$$8$$ 1.58579 0.560660
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0.828427 0.249780 0.124890 0.992171i $$-0.460142\pi$$
0.124890 + 0.992171i $$0.460142\pi$$
$$12$$ −3.65685 −1.05564
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ −2.82843 −0.685994 −0.342997 0.939336i $$-0.611442\pi$$
−0.342997 + 0.939336i $$0.611442\pi$$
$$18$$ −0.414214 −0.0976311
$$19$$ −4.82843 −1.10772 −0.553859 0.832611i $$-0.686845\pi$$
−0.553859 + 0.832611i $$0.686845\pi$$
$$20$$ 0 0
$$21$$ 9.65685 2.10730
$$22$$ −0.343146 −0.0731589
$$23$$ 3.17157 0.661319 0.330659 0.943750i $$-0.392729\pi$$
0.330659 + 0.943750i $$0.392729\pi$$
$$24$$ 3.17157 0.647395
$$25$$ 0 0
$$26$$ −0.828427 −0.162468
$$27$$ −4.00000 −0.769800
$$28$$ −8.82843 −1.66842
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 6.48528 1.16479 0.582395 0.812906i $$-0.302116\pi$$
0.582395 + 0.812906i $$0.302116\pi$$
$$32$$ −4.41421 −0.780330
$$33$$ 1.65685 0.288421
$$34$$ 1.17157 0.200923
$$35$$ 0 0
$$36$$ −1.82843 −0.304738
$$37$$ 8.48528 1.39497 0.697486 0.716599i $$-0.254302\pi$$
0.697486 + 0.716599i $$0.254302\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ −1.51472 −0.228352
$$45$$ 0 0
$$46$$ −1.31371 −0.193696
$$47$$ 11.6569 1.70033 0.850163 0.526519i $$-0.176503\pi$$
0.850163 + 0.526519i $$0.176503\pi$$
$$48$$ 6.00000 0.866025
$$49$$ 16.3137 2.33053
$$50$$ 0 0
$$51$$ −5.65685 −0.792118
$$52$$ −3.65685 −0.507114
$$53$$ 3.65685 0.502308 0.251154 0.967947i $$-0.419190\pi$$
0.251154 + 0.967947i $$0.419190\pi$$
$$54$$ 1.65685 0.225469
$$55$$ 0 0
$$56$$ 7.65685 1.02319
$$57$$ −9.65685 −1.27908
$$58$$ −0.414214 −0.0543889
$$59$$ 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 $$0.575216\pi$$
$$62$$ −2.68629 −0.341159
$$63$$ 4.82843 0.608325
$$64$$ −4.17157 −0.521447
$$65$$ 0 0
$$66$$ −0.686292 −0.0844766
$$67$$ −6.48528 −0.792303 −0.396152 0.918185i $$-0.629655\pi$$
−0.396152 + 0.918185i $$0.629655\pi$$
$$68$$ 5.17157 0.627145
$$69$$ 6.34315 0.763625
$$70$$ 0 0
$$71$$ −15.3137 −1.81740 −0.908701 0.417447i $$-0.862925\pi$$
−0.908701 + 0.417447i $$0.862925\pi$$
$$72$$ 1.58579 0.186887
$$73$$ −8.48528 −0.993127 −0.496564 0.868000i $$-0.665405\pi$$
−0.496564 + 0.868000i $$0.665405\pi$$
$$74$$ −3.51472 −0.408578
$$75$$ 0 0
$$76$$ 8.82843 1.01269
$$77$$ 4.00000 0.455842
$$78$$ −1.65685 −0.187602
$$79$$ −2.48528 −0.279616 −0.139808 0.990179i $$-0.544649\pi$$
−0.139808 + 0.990179i $$0.544649\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 2.48528 0.274453
$$83$$ −7.17157 −0.787182 −0.393591 0.919286i $$-0.628767\pi$$
−0.393591 + 0.919286i $$0.628767\pi$$
$$84$$ −17.6569 −1.92652
$$85$$ 0 0
$$86$$ −2.48528 −0.267995
$$87$$ 2.00000 0.214423
$$88$$ 1.31371 0.140042
$$89$$ −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
$$92$$ −5.79899 −0.604586
$$93$$ 12.9706 1.34498
$$94$$ −4.82843 −0.498014
$$95$$ 0 0
$$96$$ −8.82843 −0.901048
$$97$$ 12.4853 1.26769 0.633844 0.773461i $$-0.281476\pi$$
0.633844 + 0.773461i $$0.281476\pi$$
$$98$$ −6.75736 −0.682596
$$99$$ 0.828427 0.0832601
$$100$$ 0 0
$$101$$ 15.6569 1.55792 0.778958 0.627077i $$-0.215749\pi$$
0.778958 + 0.627077i $$0.215749\pi$$
$$102$$ 2.34315 0.232006
$$103$$ −16.1421 −1.59053 −0.795266 0.606261i $$-0.792669\pi$$
−0.795266 + 0.606261i $$0.792669\pi$$
$$104$$ 3.17157 0.310998
$$105$$ 0 0
$$106$$ −1.51472 −0.147122
$$107$$ −20.1421 −1.94721 −0.973607 0.228232i $$-0.926706\pi$$
−0.973607 + 0.228232i $$0.926706\pi$$
$$108$$ 7.31371 0.703762
$$109$$ 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
$$112$$ 14.4853 1.36873
$$113$$ 2.82843 0.266076 0.133038 0.991111i $$-0.457527\pi$$
0.133038 + 0.991111i $$0.457527\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −1.82843 −0.169765
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ −13.6569 −1.25192
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 1.51472 0.137136
$$123$$ −12.0000 −1.08200
$$124$$ −11.8579 −1.06487
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ −6.00000 −0.532414 −0.266207 0.963916i $$-0.585770\pi$$
−0.266207 + 0.963916i $$0.585770\pi$$
$$128$$ 10.5563 0.933058
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ −12.1421 −1.06086 −0.530432 0.847728i $$-0.677970\pi$$
−0.530432 + 0.847728i $$0.677970\pi$$
$$132$$ −3.02944 −0.263679
$$133$$ −23.3137 −2.02155
$$134$$ 2.68629 0.232060
$$135$$ 0 0
$$136$$ −4.48528 −0.384610
$$137$$ 5.17157 0.441837 0.220919 0.975292i $$-0.429094\pi$$
0.220919 + 0.975292i $$0.429094\pi$$
$$138$$ −2.62742 −0.223661
$$139$$ 21.6569 1.83691 0.918455 0.395525i $$-0.129437\pi$$
0.918455 + 0.395525i $$0.129437\pi$$
$$140$$ 0 0
$$141$$ 23.3137 1.96337
$$142$$ 6.34315 0.532305
$$143$$ 1.65685 0.138553
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ 3.51472 0.290880
$$147$$ 32.6274 2.69106
$$148$$ −15.5147 −1.27530
$$149$$ 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 $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ −7.65685 −0.621053
$$153$$ −2.82843 −0.228665
$$154$$ −1.65685 −0.133513
$$155$$ 0 0
$$156$$ −7.31371 −0.585565
$$157$$ −0.485281 −0.0387297 −0.0193648 0.999812i $$-0.506164\pi$$
−0.0193648 + 0.999812i $$0.506164\pi$$
$$158$$ 1.02944 0.0818976
$$159$$ 7.31371 0.580015
$$160$$ 0 0
$$161$$ 15.3137 1.20689
$$162$$ 4.55635 0.357981
$$163$$ 8.34315 0.653486 0.326743 0.945113i $$-0.394049\pi$$
0.326743 + 0.945113i $$0.394049\pi$$
$$164$$ 10.9706 0.856657
$$165$$ 0 0
$$166$$ 2.97056 0.230560
$$167$$ 2.48528 0.192317 0.0961584 0.995366i $$-0.469344\pi$$
0.0961584 + 0.995366i $$0.469344\pi$$
$$168$$ 15.3137 1.18148
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −4.82843 −0.369239
$$172$$ −10.9706 −0.836498
$$173$$ −17.3137 −1.31634 −0.658168 0.752871i $$-0.728669\pi$$
−0.658168 + 0.752871i $$0.728669\pi$$
$$174$$ −0.828427 −0.0628029
$$175$$ 0 0
$$176$$ 2.48528 0.187335
$$177$$ 0 0
$$178$$ 3.17157 0.237719
$$179$$ −23.3137 −1.74255 −0.871274 0.490797i $$-0.836706\pi$$
−0.871274 + 0.490797i $$0.836706\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ −7.31371 −0.540645
$$184$$ 5.02944 0.370775
$$185$$ 0 0
$$186$$ −5.37258 −0.393937
$$187$$ −2.34315 −0.171348
$$188$$ −21.3137 −1.55446
$$189$$ −19.3137 −1.40487
$$190$$ 0 0
$$191$$ −20.8284 −1.50709 −0.753546 0.657395i $$-0.771658\pi$$
−0.753546 + 0.657395i $$0.771658\pi$$
$$192$$ −8.34315 −0.602115
$$193$$ −4.48528 −0.322858 −0.161429 0.986884i $$-0.551610\pi$$
−0.161429 + 0.986884i $$0.551610\pi$$
$$194$$ −5.17157 −0.371297
$$195$$ 0 0
$$196$$ −29.8284 −2.13060
$$197$$ 19.6569 1.40049 0.700246 0.713901i $$-0.253073\pi$$
0.700246 + 0.713901i $$0.253073\pi$$
$$198$$ −0.343146 −0.0243863
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ 0 0
$$201$$ −12.9706 −0.914873
$$202$$ −6.48528 −0.456303
$$203$$ 4.82843 0.338889
$$204$$ 10.3431 0.724165
$$205$$ 0 0
$$206$$ 6.68629 0.465856
$$207$$ 3.17157 0.220440
$$208$$ 6.00000 0.416025
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −0.828427 −0.0570313 −0.0285156 0.999593i $$-0.509078\pi$$
−0.0285156 + 0.999593i $$0.509078\pi$$
$$212$$ −6.68629 −0.459216
$$213$$ −30.6274 −2.09856
$$214$$ 8.34315 0.570326
$$215$$ 0 0
$$216$$ −6.34315 −0.431596
$$217$$ 31.3137 2.12571
$$218$$ −0.828427 −0.0561082
$$219$$ −16.9706 −1.14676
$$220$$ 0 0
$$221$$ −5.65685 −0.380521
$$222$$ −7.02944 −0.471785
$$223$$ 17.7990 1.19191 0.595954 0.803018i $$-0.296774\pi$$
0.595954 + 0.803018i $$0.296774\pi$$
$$224$$ −21.3137 −1.42408
$$225$$ 0 0
$$226$$ −1.17157 −0.0779319
$$227$$ −20.1421 −1.33688 −0.668440 0.743766i $$-0.733038\pi$$
−0.668440 + 0.743766i $$0.733038\pi$$
$$228$$ 17.6569 1.16935
$$229$$ −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
$$232$$ 1.58579 0.104112
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ −0.828427 −0.0541560
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −4.97056 −0.322873
$$238$$ 5.65685 0.366679
$$239$$ −0.686292 −0.0443925 −0.0221963 0.999754i $$-0.507066\pi$$
−0.0221963 + 0.999754i $$0.507066\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 4.27208 0.274620
$$243$$ −10.0000 −0.641500
$$244$$ 6.68629 0.428046
$$245$$ 0 0
$$246$$ 4.97056 0.316912
$$247$$ −9.65685 −0.614451
$$248$$ 10.2843 0.653052
$$249$$ −14.3431 −0.908960
$$250$$ 0 0
$$251$$ 8.82843 0.557245 0.278623 0.960401i $$-0.410122\pi$$
0.278623 + 0.960401i $$0.410122\pi$$
$$252$$ −8.82843 −0.556139
$$253$$ 2.62742 0.165184
$$254$$ 2.48528 0.155940
$$255$$ 0 0
$$256$$ 3.97056 0.248160
$$257$$ −6.68629 −0.417079 −0.208540 0.978014i $$-0.566871\pi$$
−0.208540 + 0.978014i $$0.566871\pi$$
$$258$$ −4.97056 −0.309454
$$259$$ 40.9706 2.54579
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 5.02944 0.310720
$$263$$ 19.6569 1.21209 0.606047 0.795429i $$-0.292754\pi$$
0.606047 + 0.795429i $$0.292754\pi$$
$$264$$ 2.62742 0.161706
$$265$$ 0 0
$$266$$ 9.65685 0.592100
$$267$$ −15.3137 −0.937184
$$268$$ 11.8579 0.724334
$$269$$ −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 $$-0.596194\pi$$
−0.297623 + 0.954683i $$0.596194\pi$$
$$272$$ −8.48528 −0.514496
$$273$$ 19.3137 1.16892
$$274$$ −2.14214 −0.129411
$$275$$ 0 0
$$276$$ −11.5980 −0.698116
$$277$$ 3.65685 0.219719 0.109860 0.993947i $$-0.464960\pi$$
0.109860 + 0.993947i $$0.464960\pi$$
$$278$$ −8.97056 −0.538019
$$279$$ 6.48528 0.388264
$$280$$ 0 0
$$281$$ −29.3137 −1.74871 −0.874355 0.485288i $$-0.838715\pi$$
−0.874355 + 0.485288i $$0.838715\pi$$
$$282$$ −9.65685 −0.575057
$$283$$ −4.82843 −0.287020 −0.143510 0.989649i $$-0.545839\pi$$
−0.143510 + 0.989649i $$0.545839\pi$$
$$284$$ 28.0000 1.66149
$$285$$ 0 0
$$286$$ −0.686292 −0.0405813
$$287$$ −28.9706 −1.71008
$$288$$ −4.41421 −0.260110
$$289$$ −9.00000 −0.529412
$$290$$ 0 0
$$291$$ 24.9706 1.46380
$$292$$ 15.5147 0.907930
$$293$$ −8.48528 −0.495715 −0.247858 0.968796i $$-0.579727\pi$$
−0.247858 + 0.968796i $$0.579727\pi$$
$$294$$ −13.5147 −0.788194
$$295$$ 0 0
$$296$$ 13.4558 0.782105
$$297$$ −3.31371 −0.192281
$$298$$ −3.85786 −0.223480
$$299$$ 6.34315 0.366834
$$300$$ 0 0
$$301$$ 28.9706 1.66984
$$302$$ 4.97056 0.286024
$$303$$ 31.3137 1.79893
$$304$$ −14.4853 −0.830788
$$305$$ 0 0
$$306$$ 1.17157 0.0669744
$$307$$ 22.9706 1.31100 0.655500 0.755195i $$-0.272458\pi$$
0.655500 + 0.755195i $$0.272458\pi$$
$$308$$ −7.31371 −0.416737
$$309$$ −32.2843 −1.83659
$$310$$ 0 0
$$311$$ 14.4853 0.821385 0.410692 0.911774i $$-0.365287\pi$$
0.410692 + 0.911774i $$0.365287\pi$$
$$312$$ 6.34315 0.359110
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0.201010 0.0113437
$$315$$ 0 0
$$316$$ 4.54416 0.255629
$$317$$ −2.82843 −0.158860 −0.0794301 0.996840i $$-0.525310\pi$$
−0.0794301 + 0.996840i $$0.525310\pi$$
$$318$$ −3.02944 −0.169882
$$319$$ 0.828427 0.0463830
$$320$$ 0 0
$$321$$ −40.2843 −2.24845
$$322$$ −6.34315 −0.353490
$$323$$ 13.6569 0.759888
$$324$$ 20.1127 1.11737
$$325$$ 0 0
$$326$$ −3.45584 −0.191402
$$327$$ 4.00000 0.221201
$$328$$ −9.51472 −0.525362
$$329$$ 56.2843 3.10305
$$330$$ 0 0
$$331$$ 21.7990 1.19818 0.599090 0.800681i $$-0.295529\pi$$
0.599090 + 0.800681i $$0.295529\pi$$
$$332$$ 13.1127 0.719653
$$333$$ 8.48528 0.464991
$$334$$ −1.02944 −0.0563283
$$335$$ 0 0
$$336$$ 28.9706 1.58047
$$337$$ 1.17157 0.0638196 0.0319098 0.999491i $$-0.489841\pi$$
0.0319098 + 0.999491i $$0.489841\pi$$
$$338$$ 3.72792 0.202772
$$339$$ 5.65685 0.307238
$$340$$ 0 0
$$341$$ 5.37258 0.290942
$$342$$ 2.00000 0.108148
$$343$$ 44.9706 2.42818
$$344$$ 9.51472 0.512999
$$345$$ 0 0
$$346$$ 7.17157 0.385546
$$347$$ 8.14214 0.437093 0.218546 0.975827i $$-0.429869\pi$$
0.218546 + 0.975827i $$0.429869\pi$$
$$348$$ −3.65685 −0.196028
$$349$$ 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
$$352$$ −3.65685 −0.194911
$$353$$ 4.34315 0.231162 0.115581 0.993298i $$-0.463127\pi$$
0.115581 + 0.993298i $$0.463127\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ −27.3137 −1.44559
$$358$$ 9.65685 0.510381
$$359$$ −3.85786 −0.203610 −0.101805 0.994804i $$-0.532462\pi$$
−0.101805 + 0.994804i $$0.532462\pi$$
$$360$$ 0 0
$$361$$ 4.31371 0.227037
$$362$$ 2.48528 0.130623
$$363$$ −20.6274 −1.08266
$$364$$ −17.6569 −0.925471
$$365$$ 0 0
$$366$$ 3.02944 0.158351
$$367$$ −18.0000 −0.939592 −0.469796 0.882775i $$-0.655673\pi$$
−0.469796 + 0.882775i $$0.655673\pi$$
$$368$$ 9.51472 0.495989
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 17.6569 0.916698
$$372$$ −23.7157 −1.22960
$$373$$ 6.97056 0.360922 0.180461 0.983582i $$-0.442241\pi$$
0.180461 + 0.983582i $$0.442241\pi$$
$$374$$ 0.970563 0.0501866
$$375$$ 0 0
$$376$$ 18.4853 0.953306
$$377$$ 2.00000 0.103005
$$378$$ 8.00000 0.411476
$$379$$ 22.4853 1.15499 0.577496 0.816394i $$-0.304030\pi$$
0.577496 + 0.816394i $$0.304030\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 8.62742 0.441417
$$383$$ −2.48528 −0.126992 −0.0634960 0.997982i $$-0.520225\pi$$
−0.0634960 + 0.997982i $$0.520225\pi$$
$$384$$ 21.1127 1.07740
$$385$$ 0 0
$$386$$ 1.85786 0.0945628
$$387$$ 6.00000 0.304997
$$388$$ −22.8284 −1.15894
$$389$$ −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
$$392$$ 25.8701 1.30664
$$393$$ −24.2843 −1.22498
$$394$$ −8.14214 −0.410195
$$395$$ 0 0
$$396$$ −1.51472 −0.0761175
$$397$$ 19.6569 0.986549 0.493275 0.869874i $$-0.335800\pi$$
0.493275 + 0.869874i $$0.335800\pi$$
$$398$$ −4.97056 −0.249152
$$399$$ −46.6274 −2.33429
$$400$$ 0 0
$$401$$ −6.68629 −0.333897 −0.166949 0.985966i $$-0.553391\pi$$
−0.166949 + 0.985966i $$0.553391\pi$$
$$402$$ 5.37258 0.267960
$$403$$ 12.9706 0.646110
$$404$$ −28.6274 −1.42427
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ 7.02944 0.348436
$$408$$ −8.97056 −0.444109
$$409$$ −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
$$412$$ 29.5147 1.45409
$$413$$ 0 0
$$414$$ −1.31371 −0.0645653
$$415$$ 0 0
$$416$$ −8.82843 −0.432849
$$417$$ 43.3137 2.12108
$$418$$ 1.65685 0.0810394
$$419$$ 28.9706 1.41530 0.707652 0.706561i $$-0.249754\pi$$
0.707652 + 0.706561i $$0.249754\pi$$
$$420$$ 0 0
$$421$$ 18.9706 0.924569 0.462284 0.886732i $$-0.347030\pi$$
0.462284 + 0.886732i $$0.347030\pi$$
$$422$$ 0.343146 0.0167041
$$423$$ 11.6569 0.566776
$$424$$ 5.79899 0.281624
$$425$$ 0 0
$$426$$ 12.6863 0.614653
$$427$$ −17.6569 −0.854475
$$428$$ 36.8284 1.78017
$$429$$ 3.31371 0.159987
$$430$$ 0 0
$$431$$ 3.31371 0.159616 0.0798079 0.996810i $$-0.474569\pi$$
0.0798079 + 0.996810i $$0.474569\pi$$
$$432$$ −12.0000 −0.577350
$$433$$ 29.1716 1.40190 0.700948 0.713212i $$-0.252760\pi$$
0.700948 + 0.713212i $$0.252760\pi$$
$$434$$ −12.9706 −0.622607
$$435$$ 0 0
$$436$$ −3.65685 −0.175132
$$437$$ −15.3137 −0.732554
$$438$$ 7.02944 0.335880
$$439$$ −10.3431 −0.493651 −0.246826 0.969060i $$-0.579388\pi$$
−0.246826 + 0.969060i $$0.579388\pi$$
$$440$$ 0 0
$$441$$ 16.3137 0.776843
$$442$$ 2.34315 0.111452
$$443$$ 7.65685 0.363788 0.181894 0.983318i $$-0.441777\pi$$
0.181894 + 0.983318i $$0.441777\pi$$
$$444$$ −31.0294 −1.47259
$$445$$ 0 0
$$446$$ −7.37258 −0.349102
$$447$$ 18.6274 0.881047
$$448$$ −20.1421 −0.951626
$$449$$ 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
$$452$$ −5.17157 −0.243250
$$453$$ −24.0000 −1.12762
$$454$$ 8.34315 0.391563
$$455$$ 0 0
$$456$$ −15.3137 −0.717130
$$457$$ −19.6569 −0.919509 −0.459754 0.888046i $$-0.652063\pi$$
−0.459754 + 0.888046i $$0.652063\pi$$
$$458$$ 0.828427 0.0387099
$$459$$ 11.3137 0.528079
$$460$$ 0 0
$$461$$ −35.6569 −1.66071 −0.830353 0.557238i $$-0.811861\pi$$
−0.830353 + 0.557238i $$0.811861\pi$$
$$462$$ −3.31371 −0.154168
$$463$$ −21.7990 −1.01308 −0.506542 0.862215i $$-0.669077\pi$$
−0.506542 + 0.862215i $$0.669077\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ 7.45584 0.345385
$$467$$ −10.9706 −0.507657 −0.253829 0.967249i $$-0.581690\pi$$
−0.253829 + 0.967249i $$0.581690\pi$$
$$468$$ −3.65685 −0.169038
$$469$$ −31.3137 −1.44593
$$470$$ 0 0
$$471$$ −0.970563 −0.0447212
$$472$$ 0 0
$$473$$ 4.97056 0.228547
$$474$$ 2.05887 0.0945672
$$475$$ 0 0
$$476$$ 24.9706 1.14452
$$477$$ 3.65685 0.167436
$$478$$ 0.284271 0.0130023
$$479$$ 7.17157 0.327678 0.163839 0.986487i $$-0.447612\pi$$
0.163839 + 0.986487i $$0.447612\pi$$
$$480$$ 0 0
$$481$$ 16.9706 0.773791
$$482$$ −4.14214 −0.188669
$$483$$ 30.6274 1.39360
$$484$$ 18.8579 0.857176
$$485$$ 0 0
$$486$$ 4.14214 0.187891
$$487$$ −9.79899 −0.444035 −0.222017 0.975043i $$-0.571264\pi$$
−0.222017 + 0.975043i $$0.571264\pi$$
$$488$$ −5.79899 −0.262508
$$489$$ 16.6863 0.754580
$$490$$ 0 0
$$491$$ −7.45584 −0.336478 −0.168239 0.985746i $$-0.553808\pi$$
−0.168239 + 0.985746i $$0.553808\pi$$
$$492$$ 21.9411 0.989182
$$493$$ −2.82843 −0.127386
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ 19.4558 0.873593
$$497$$ −73.9411 −3.31671
$$498$$ 5.94113 0.266228
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 4.97056 0.222068
$$502$$ −3.65685 −0.163213
$$503$$ 30.0000 1.33763 0.668817 0.743427i $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$504$$ 7.65685 0.341063
$$505$$ 0 0
$$506$$ −1.08831 −0.0483814
$$507$$ −18.0000 −0.799408
$$508$$ 10.9706 0.486740
$$509$$ 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
$$512$$ −22.7574 −1.00574
$$513$$ 19.3137 0.852721
$$514$$ 2.76955 0.122160
$$515$$ 0 0
$$516$$ −21.9411 −0.965904
$$517$$ 9.65685 0.424708
$$518$$ −16.9706 −0.745644
$$519$$ −34.6274 −1.51997
$$520$$ 0 0
$$521$$ −21.3137 −0.933771 −0.466885 0.884318i $$-0.654624\pi$$
−0.466885 + 0.884318i $$0.654624\pi$$
$$522$$ −0.414214 −0.0181296
$$523$$ −2.48528 −0.108674 −0.0543369 0.998523i $$-0.517304\pi$$
−0.0543369 + 0.998523i $$0.517304\pi$$
$$524$$ 22.2010 0.969856
$$525$$ 0 0
$$526$$ −8.14214 −0.355014
$$527$$ −18.3431 −0.799040
$$528$$ 4.97056 0.216316
$$529$$ −12.9411 −0.562658
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 42.6274 1.84813
$$533$$ −12.0000 −0.519778
$$534$$ 6.34315 0.274495
$$535$$ 0 0
$$536$$ −10.2843 −0.444213
$$537$$ −46.6274 −2.01212
$$538$$ 8.82843 0.380621
$$539$$ 13.5147 0.582120
$$540$$ 0 0
$$541$$ −5.02944 −0.216232 −0.108116 0.994138i $$-0.534482\pi$$
−0.108116 + 0.994138i $$0.534482\pi$$
$$542$$ 4.05887 0.174344
$$543$$ −12.0000 −0.514969
$$544$$ 12.4853 0.535302
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 2.48528 0.106263 0.0531315 0.998588i $$-0.483080\pi$$
0.0531315 + 0.998588i $$0.483080\pi$$
$$548$$ −9.45584 −0.403934
$$549$$ −3.65685 −0.156071
$$550$$ 0 0
$$551$$ −4.82843 −0.205698
$$552$$ 10.0589 0.428134
$$553$$ −12.0000 −0.510292
$$554$$ −1.51472 −0.0643542
$$555$$ 0 0
$$556$$ −39.5980 −1.67933
$$557$$ 27.9411 1.18390 0.591952 0.805973i $$-0.298358\pi$$
0.591952 + 0.805973i $$0.298358\pi$$
$$558$$ −2.68629 −0.113720
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ −4.68629 −0.197855
$$562$$ 12.1421 0.512185
$$563$$ −7.65685 −0.322698 −0.161349 0.986897i $$-0.551584\pi$$
−0.161349 + 0.986897i $$0.551584\pi$$
$$564$$ −42.6274 −1.79494
$$565$$ 0 0
$$566$$ 2.00000 0.0840663
$$567$$ −53.1127 −2.23052
$$568$$ −24.2843 −1.01895
$$569$$ 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 $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ −3.02944 −0.126667
$$573$$ −41.6569 −1.74024
$$574$$ 12.0000 0.500870
$$575$$ 0 0
$$576$$ −4.17157 −0.173816
$$577$$ −23.7990 −0.990765 −0.495382 0.868675i $$-0.664972\pi$$
−0.495382 + 0.868675i $$0.664972\pi$$
$$578$$ 3.72792 0.155061
$$579$$ −8.97056 −0.372804
$$580$$ 0 0
$$581$$ −34.6274 −1.43659
$$582$$ −10.3431 −0.428737
$$583$$ 3.02944 0.125466
$$584$$ −13.4558 −0.556807
$$585$$ 0 0
$$586$$ 3.51472 0.145192
$$587$$ 29.7990 1.22994 0.614968 0.788552i $$-0.289169\pi$$
0.614968 + 0.788552i $$0.289169\pi$$
$$588$$ −59.6569 −2.46021
$$589$$ −31.3137 −1.29026
$$590$$ 0 0
$$591$$ 39.3137 1.61715
$$592$$ 25.4558 1.04623
$$593$$ 7.65685 0.314429 0.157215 0.987564i $$-0.449749\pi$$
0.157215 + 0.987564i $$0.449749\pi$$
$$594$$ 1.37258 0.0563178
$$595$$ 0 0
$$596$$ −17.0294 −0.697553
$$597$$ 24.0000 0.982255
$$598$$ −2.62742 −0.107443
$$599$$ −37.7990 −1.54442 −0.772212 0.635364i $$-0.780850\pi$$
−0.772212 + 0.635364i $$0.780850\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ −12.0000 −0.489083
$$603$$ −6.48528 −0.264101
$$604$$ 21.9411 0.892772
$$605$$ 0 0
$$606$$ −12.9706 −0.526893
$$607$$ −9.02944 −0.366494 −0.183247 0.983067i $$-0.558661\pi$$
−0.183247 + 0.983067i $$0.558661\pi$$
$$608$$ 21.3137 0.864385
$$609$$ 9.65685 0.391315
$$610$$ 0 0
$$611$$ 23.3137 0.943172
$$612$$ 5.17157 0.209048
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ −9.51472 −0.383983
$$615$$ 0 0
$$616$$ 6.34315 0.255573
$$617$$ 9.17157 0.369234 0.184617 0.982811i $$-0.440896\pi$$
0.184617 + 0.982811i $$0.440896\pi$$
$$618$$ 13.3726 0.537924
$$619$$ −9.79899 −0.393855 −0.196927 0.980418i $$-0.563096\pi$$
−0.196927 + 0.980418i $$0.563096\pi$$
$$620$$ 0 0
$$621$$ −12.6863 −0.509083
$$622$$ −6.00000 −0.240578
$$623$$ −36.9706 −1.48119
$$624$$ 12.0000 0.480384
$$625$$ 0 0
$$626$$ −2.48528 −0.0993318
$$627$$ −8.00000 −0.319489
$$628$$ 0.887302 0.0354072
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ 36.9706 1.47177 0.735887 0.677104i $$-0.236765\pi$$
0.735887 + 0.677104i $$0.236765\pi$$
$$632$$ −3.94113 −0.156770
$$633$$ −1.65685 −0.0658540
$$634$$ 1.17157 0.0465291
$$635$$ 0 0
$$636$$ −13.3726 −0.530257
$$637$$ 32.6274 1.29275
$$638$$ −0.343146 −0.0135853
$$639$$ −15.3137 −0.605801
$$640$$ 0 0
$$641$$ 0.627417 0.0247815 0.0123907 0.999923i $$-0.496056\pi$$
0.0123907 + 0.999923i $$0.496056\pi$$
$$642$$ 16.6863 0.658555
$$643$$ −19.4558 −0.767264 −0.383632 0.923486i $$-0.625327\pi$$
−0.383632 + 0.923486i $$0.625327\pi$$
$$644$$ −28.0000 −1.10335
$$645$$ 0 0
$$646$$ −5.65685 −0.222566
$$647$$ 41.1127 1.61631 0.808153 0.588972i $$-0.200467\pi$$
0.808153 + 0.588972i $$0.200467\pi$$
$$648$$ −17.4437 −0.685251
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 62.6274 2.45456
$$652$$ −15.2548 −0.597425
$$653$$ 17.1716 0.671976 0.335988 0.941866i $$-0.390930\pi$$
0.335988 + 0.941866i $$0.390930\pi$$
$$654$$ −1.65685 −0.0647881
$$655$$ 0 0
$$656$$ −18.0000 −0.702782
$$657$$ −8.48528 −0.331042
$$658$$ −23.3137 −0.908863
$$659$$ 1.79899 0.0700787 0.0350393 0.999386i $$-0.488844\pi$$
0.0350393 + 0.999386i $$0.488844\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ −9.02944 −0.350939
$$663$$ −11.3137 −0.439388
$$664$$ −11.3726 −0.441342
$$665$$ 0 0
$$666$$ −3.51472 −0.136193
$$667$$ 3.17157 0.122804
$$668$$ −4.54416 −0.175819
$$669$$ 35.5980 1.37630
$$670$$ 0 0
$$671$$ −3.02944 −0.116950
$$672$$ −42.6274 −1.64439
$$673$$ −22.9706 −0.885450 −0.442725 0.896657i $$-0.645988\pi$$
−0.442725 + 0.896657i $$0.645988\pi$$
$$674$$ −0.485281 −0.0186923
$$675$$ 0 0
$$676$$ 16.4558 0.632917
$$677$$ 36.7696 1.41317 0.706584 0.707629i $$-0.250235\pi$$
0.706584 + 0.707629i $$0.250235\pi$$
$$678$$ −2.34315 −0.0899880
$$679$$ 60.2843 2.31350
$$680$$ 0 0
$$681$$ −40.2843 −1.54370
$$682$$ −2.22540 −0.0852148
$$683$$ −11.8579 −0.453729 −0.226864 0.973926i $$-0.572847\pi$$
−0.226864 + 0.973926i $$0.572847\pi$$
$$684$$ 8.82843 0.337563
$$685$$ 0 0
$$686$$ −18.6274 −0.711198
$$687$$ −4.00000 −0.152610
$$688$$ 18.0000 0.686244
$$689$$ 7.31371 0.278630
$$690$$ 0 0
$$691$$ −44.9706 −1.71076 −0.855380 0.518000i $$-0.826677\pi$$
−0.855380 + 0.518000i $$0.826677\pi$$
$$692$$ 31.6569 1.20341
$$693$$ 4.00000 0.151947
$$694$$ −3.37258 −0.128022
$$695$$ 0 0
$$696$$ 3.17157 0.120218
$$697$$ 16.9706 0.642806
$$698$$ −8.54416 −0.323401
$$699$$ −36.0000 −1.36165
$$700$$ 0 0
$$701$$ 6.68629 0.252538 0.126269 0.991996i $$-0.459700\pi$$
0.126269 + 0.991996i $$0.459700\pi$$
$$702$$ 3.31371 0.125068
$$703$$ −40.9706 −1.54523
$$704$$ −3.45584 −0.130247
$$705$$ 0 0
$$706$$ −1.79899 −0.0677059
$$707$$ 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
$$712$$ −12.1421 −0.455046
$$713$$ 20.5685 0.770298
$$714$$ 11.3137 0.423405
$$715$$ 0 0
$$716$$ 42.6274 1.59306
$$717$$ −1.37258 −0.0512601
$$718$$ 1.59798 0.0596361
$$719$$ −34.6274 −1.29138 −0.645692 0.763598i $$-0.723431\pi$$
−0.645692 + 0.763598i $$0.723431\pi$$
$$720$$ 0 0
$$721$$ −77.9411 −2.90268
$$722$$ −1.78680 −0.0664977
$$723$$ 20.0000 0.743808
$$724$$ 10.9706 0.407718
$$725$$ 0 0
$$726$$ 8.54416 0.317103
$$727$$ 23.9411 0.887927 0.443964 0.896045i $$-0.353572\pi$$
0.443964 + 0.896045i $$0.353572\pi$$
$$728$$ 15.3137 0.567564
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −16.9706 −0.627679
$$732$$ 13.3726 0.494265
$$733$$ 22.8284 0.843187 0.421594 0.906785i $$-0.361471\pi$$
0.421594 + 0.906785i $$0.361471\pi$$
$$734$$ 7.45584 0.275200
$$735$$ 0 0
$$736$$ −14.0000 −0.516047
$$737$$ −5.37258 −0.197902
$$738$$ 2.48528 0.0914845
$$739$$ 14.4853 0.532850 0.266425 0.963856i $$-0.414158\pi$$
0.266425 + 0.963856i $$0.414158\pi$$
$$740$$ 0 0
$$741$$ −19.3137 −0.709507
$$742$$ −7.31371 −0.268495
$$743$$ 52.6274 1.93071 0.965356 0.260935i $$-0.0840309\pi$$
0.965356 + 0.260935i $$0.0840309\pi$$
$$744$$ 20.5685 0.754079
$$745$$ 0 0
$$746$$ −2.88730 −0.105712
$$747$$ −7.17157 −0.262394
$$748$$ 4.28427 0.156648
$$749$$ −97.2548 −3.55361
$$750$$ 0 0
$$751$$ 16.1421 0.589035 0.294517 0.955646i $$-0.404841\pi$$
0.294517 + 0.955646i $$0.404841\pi$$
$$752$$ 34.9706 1.27525
$$753$$ 17.6569 0.643452
$$754$$ −0.828427 −0.0301695
$$755$$ 0 0
$$756$$ 35.3137 1.28435
$$757$$ −19.5147 −0.709275 −0.354637 0.935004i $$-0.615396\pi$$
−0.354637 + 0.935004i $$0.615396\pi$$
$$758$$ −9.31371 −0.338289
$$759$$ 5.25483 0.190738
$$760$$ 0 0
$$761$$ 8.62742 0.312744 0.156372 0.987698i $$-0.450020\pi$$
0.156372 + 0.987698i $$0.450020\pi$$
$$762$$ 4.97056 0.180064
$$763$$ 9.65685 0.349602
$$764$$ 38.0833 1.37780
$$765$$ 0 0
$$766$$ 1.02944 0.0371951
$$767$$ 0 0
$$768$$ 7.94113 0.286551
$$769$$ −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
$$772$$ 8.20101 0.295161
$$773$$ 8.48528 0.305194 0.152597 0.988288i $$-0.451236\pi$$
0.152597 + 0.988288i $$0.451236\pi$$
$$774$$ −2.48528 −0.0893316
$$775$$ 0 0
$$776$$ 19.7990 0.710742
$$777$$ 81.9411 2.93962
$$778$$ 12.1421 0.435317
$$779$$ 28.9706 1.03798
$$780$$ 0 0
$$781$$ −12.6863 −0.453951
$$782$$ 3.71573 0.132874
$$783$$ −4.00000 −0.142948
$$784$$ 48.9411 1.74790
$$785$$ 0 0
$$786$$ 10.0589 0.358788
$$787$$ −17.7990 −0.634465 −0.317233 0.948348i $$-0.602754\pi$$
−0.317233 + 0.948348i $$0.602754\pi$$
$$788$$ −35.9411 −1.28035
$$789$$ 39.3137 1.39961
$$790$$ 0 0
$$791$$ 13.6569 0.485582
$$792$$ 1.31371 0.0466806
$$793$$ −7.31371 −0.259717
$$794$$ −8.14214 −0.288954
$$795$$ 0 0
$$796$$ −21.9411 −0.777683
$$797$$ 5.85786 0.207496 0.103748 0.994604i $$-0.466916\pi$$
0.103748 + 0.994604i $$0.466916\pi$$
$$798$$ 19.3137 0.683698
$$799$$ −32.9706 −1.16641
$$800$$ 0 0
$$801$$ −7.65685 −0.270542
$$802$$ 2.76955 0.0977963
$$803$$ −7.02944 −0.248063
$$804$$ 23.7157 0.836389
$$805$$ 0 0
$$806$$ −5.37258 −0.189241
$$807$$ −42.6274 −1.50056
$$808$$ 24.8284 0.873461
$$809$$ 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 $$-0.270066\pi$$
0.661155 + 0.750249i $$0.270066\pi$$
$$812$$ −8.82843 −0.309817
$$813$$ −19.5980 −0.687331
$$814$$ −2.91169 −0.102055
$$815$$ 0 0
$$816$$ −16.9706 −0.594089
$$817$$ −28.9706 −1.01355
$$818$$ 1.23045 0.0430216
$$819$$ 9.65685 0.337438
$$820$$ 0 0
$$821$$ −22.6863 −0.791757 −0.395879 0.918303i $$-0.629560\pi$$
−0.395879 + 0.918303i $$0.629560\pi$$
$$822$$ −4.28427 −0.149431
$$823$$ 30.9706 1.07957 0.539783 0.841804i $$-0.318506\pi$$
0.539783 + 0.841804i $$0.318506\pi$$
$$824$$ −25.5980 −0.891748
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −17.3137 −0.602057 −0.301028 0.953615i $$-0.597330\pi$$
−0.301028 + 0.953615i $$0.597330\pi$$
$$828$$ −5.79899 −0.201529
$$829$$ 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
$$832$$ −8.34315 −0.289247
$$833$$ −46.1421 −1.59873
$$834$$ −17.9411 −0.621250
$$835$$ 0 0
$$836$$ 7.31371 0.252950
$$837$$ −25.9411 −0.896656
$$838$$ −12.0000 −0.414533
$$839$$ 2.48528 0.0858014 0.0429007 0.999079i $$-0.486340\pi$$
0.0429007 + 0.999079i $$0.486340\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −7.85786 −0.270800
$$843$$ −58.6274 −2.01924
$$844$$ 1.51472 0.0521388
$$845$$ 0 0
$$846$$ −4.82843 −0.166005
$$847$$ −49.7990 −1.71111
$$848$$ 10.9706 0.376731
$$849$$ −9.65685 −0.331422
$$850$$ 0 0
$$851$$ 26.9117 0.922521
$$852$$ 56.0000 1.91853
$$853$$ −51.1127 −1.75007 −0.875033 0.484064i $$-0.839160\pi$$
−0.875033 + 0.484064i $$0.839160\pi$$
$$854$$ 7.31371 0.250270
$$855$$ 0 0
$$856$$ −31.9411 −1.09173
$$857$$ −3.37258 −0.115205 −0.0576026 0.998340i $$-0.518346\pi$$
−0.0576026 + 0.998340i $$0.518346\pi$$
$$858$$ −1.37258 −0.0468592
$$859$$ −56.4264 −1.92524 −0.962622 0.270848i $$-0.912696\pi$$
−0.962622 + 0.270848i $$0.912696\pi$$
$$860$$ 0 0
$$861$$ −57.9411 −1.97463
$$862$$ −1.37258 −0.0467504
$$863$$ 36.1421 1.23029 0.615146 0.788413i $$-0.289097\pi$$
0.615146 + 0.788413i $$0.289097\pi$$
$$864$$ 17.6569 0.600698
$$865$$ 0 0
$$866$$ −12.0833 −0.410606
$$867$$ −18.0000 −0.611312
$$868$$ −57.2548 −1.94336
$$869$$ −2.05887 −0.0698425
$$870$$ 0 0
$$871$$ −12.9706 −0.439491
$$872$$ 3.17157 0.107403
$$873$$ 12.4853 0.422563
$$874$$ 6.34315 0.214560
$$875$$ 0 0
$$876$$ 31.0294 1.04839
$$877$$ −38.2843 −1.29277 −0.646384 0.763012i $$-0.723720\pi$$
−0.646384 + 0.763012i $$0.723720\pi$$
$$878$$ 4.28427 0.144587
$$879$$ −16.9706 −0.572403
$$880$$ 0 0
$$881$$ 29.3137 0.987604 0.493802 0.869574i $$-0.335607\pi$$
0.493802 + 0.869574i $$0.335607\pi$$
$$882$$ −6.75736 −0.227532
$$883$$ 14.4853 0.487469 0.243734 0.969842i $$-0.421628\pi$$
0.243734 + 0.969842i $$0.421628\pi$$
$$884$$ 10.3431 0.347878
$$885$$ 0 0
$$886$$ −3.17157 −0.106551
$$887$$ 6.68629 0.224504 0.112252 0.993680i $$-0.464194\pi$$
0.112252 + 0.993680i $$0.464194\pi$$
$$888$$ 26.9117 0.903097
$$889$$ −28.9706 −0.971641
$$890$$ 0 0
$$891$$ −9.11270 −0.305287
$$892$$ −32.5442 −1.08966
$$893$$ −56.2843 −1.88348
$$894$$ −7.71573 −0.258053
$$895$$ 0 0
$$896$$ 50.9706 1.70281
$$897$$ 12.6863 0.423583
$$898$$ −4.82843 −0.161127
$$899$$ 6.48528 0.216296
$$900$$ 0 0
$$901$$ −10.3431 −0.344580
$$902$$ 2.05887 0.0685530
$$903$$ 57.9411 1.92816
$$904$$ 4.48528 0.149178
$$905$$ 0 0
$$906$$ 9.94113 0.330272
$$907$$ 10.0000 0.332045 0.166022 0.986122i $$-0.446908\pi$$
0.166022 + 0.986122i $$0.446908\pi$$
$$908$$ 36.8284 1.22219
$$909$$ 15.6569 0.519305
$$910$$ 0 0
$$911$$ 32.1421 1.06492 0.532458 0.846456i $$-0.321268\pi$$
0.532458 + 0.846456i $$0.321268\pi$$
$$912$$ −28.9706 −0.959311
$$913$$ −5.94113 −0.196623
$$914$$ 8.14214 0.269318
$$915$$ 0 0
$$916$$ 3.65685 0.120826
$$917$$ −58.6274 −1.93605
$$918$$ −4.68629 −0.154671
$$919$$ −36.0000 −1.18753 −0.593765 0.804638i $$-0.702359\pi$$
−0.593765 + 0.804638i $$0.702359\pi$$
$$920$$ 0 0
$$921$$ 45.9411 1.51381
$$922$$ 14.7696 0.486409
$$923$$ −30.6274 −1.00811
$$924$$ −14.6274 −0.481207
$$925$$ 0 0
$$926$$ 9.02944 0.296726
$$927$$ −16.1421 −0.530177
$$928$$ −4.41421 −0.144904
$$929$$ −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
$$932$$ 32.9117 1.07806
$$933$$ 28.9706 0.948454
$$934$$ 4.54416 0.148689
$$935$$ 0 0
$$936$$ 3.17157 0.103666
$$937$$ −19.6569 −0.642161 −0.321081 0.947052i $$-0.604046\pi$$
−0.321081 + 0.947052i $$0.604046\pi$$
$$938$$ 12.9706 0.423504
$$939$$ 12.0000 0.391605
$$940$$ 0 0
$$941$$ −27.9411 −0.910855 −0.455427 0.890273i $$-0.650514\pi$$
−0.455427 + 0.890273i $$0.650514\pi$$
$$942$$ 0.402020 0.0130985
$$943$$ −19.0294 −0.619684
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −2.05887 −0.0669398
$$947$$ −44.9117 −1.45943 −0.729717 0.683749i $$-0.760348\pi$$
−0.729717 + 0.683749i $$0.760348\pi$$
$$948$$ 9.08831 0.295175
$$949$$ −16.9706 −0.550888
$$950$$ 0 0
$$951$$ −5.65685 −0.183436
$$952$$ −21.6569 −0.701903
$$953$$ −29.3137 −0.949564 −0.474782 0.880103i $$-0.657473\pi$$
−0.474782 + 0.880103i $$0.657473\pi$$
$$954$$ −1.51472 −0.0490408
$$955$$ 0 0
$$956$$ 1.25483 0.0405842
$$957$$ 1.65685 0.0535585
$$958$$ −2.97056 −0.0959745
$$959$$ 24.9706 0.806342
$$960$$ 0 0
$$961$$ 11.0589 0.356738
$$962$$ −7.02944 −0.226638
$$963$$ −20.1421 −0.649071
$$964$$ −18.2843 −0.588897
$$965$$ 0 0
$$966$$ −12.6863 −0.408175
$$967$$ −14.9706 −0.481421 −0.240710 0.970597i $$-0.577380\pi$$
−0.240710 + 0.970597i $$0.577380\pi$$
$$968$$ −16.3553 −0.525681
$$969$$ 27.3137 0.877443
$$970$$ 0 0
$$971$$ 28.1421 0.903124 0.451562 0.892240i $$-0.350867\pi$$
0.451562 + 0.892240i $$0.350867\pi$$
$$972$$ 18.2843 0.586468
$$973$$ 104.569 3.35231
$$974$$ 4.05887 0.130055
$$975$$ 0 0
$$976$$ −10.9706 −0.351159
$$977$$ 2.68629 0.0859421 0.0429710 0.999076i $$-0.486318\pi$$
0.0429710 + 0.999076i $$0.486318\pi$$
$$978$$ −6.91169 −0.221011
$$979$$ −6.34315 −0.202728
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 3.08831 0.0985520
$$983$$ 9.31371 0.297061 0.148531 0.988908i $$-0.452546\pi$$
0.148531 + 0.988908i $$0.452546\pi$$
$$984$$ −19.0294 −0.606636
$$985$$ 0 0
$$986$$ 1.17157 0.0373105
$$987$$ 112.569 3.58310
$$988$$ 17.6569 0.561739
$$989$$ 19.0294 0.605101
$$990$$ 0 0
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ −28.6274 −0.908921
$$993$$ 43.5980 1.38354
$$994$$ 30.6274 0.971443
$$995$$ 0 0
$$996$$ 26.2254 0.830983
$$997$$ −6.82843 −0.216258 −0.108129 0.994137i $$-0.534486\pi$$
−0.108129 + 0.994137i $$0.534486\pi$$
$$998$$ −14.9117 −0.472021
$$999$$ −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 725.2.a.c.1.1 2
3.2 odd 2 6525.2.a.p.1.2 2
5.2 odd 4 725.2.b.c.349.2 4
5.3 odd 4 725.2.b.c.349.3 4
5.4 even 2 145.2.a.b.1.2 2
15.14 odd 2 1305.2.a.n.1.1 2
20.19 odd 2 2320.2.a.k.1.2 2
35.34 odd 2 7105.2.a.e.1.2 2
40.19 odd 2 9280.2.a.w.1.2 2
40.29 even 2 9280.2.a.be.1.1 2
145.144 even 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 5.4 even 2
725.2.a.c.1.1 2 1.1 even 1 trivial
725.2.b.c.349.2 4 5.2 odd 4
725.2.b.c.349.3 4 5.3 odd 4
1305.2.a.n.1.1 2 15.14 odd 2
2320.2.a.k.1.2 2 20.19 odd 2
4205.2.a.d.1.1 2 145.144 even 2
6525.2.a.p.1.2 2 3.2 odd 2
7105.2.a.e.1.2 2 35.34 odd 2
9280.2.a.w.1.2 2 40.19 odd 2
9280.2.a.be.1.1 2 40.29 even 2