# Properties

 Label 9075.2.a.z.1.1 Level $9075$ Weight $2$ Character 9075.1 Self dual yes Analytic conductor $72.464$ 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] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.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: $$72.4642398343$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 9075.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -5.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -5.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{12} +3.23607 q^{13} +8.47214 q^{14} -4.85410 q^{16} +2.00000 q^{17} -1.61803 q^{18} +5.00000 q^{19} +5.23607 q^{21} +4.61803 q^{23} -2.23607 q^{24} -5.23607 q^{26} -1.00000 q^{27} -3.23607 q^{28} +0.854102 q^{29} +7.00000 q^{31} +3.38197 q^{32} -3.23607 q^{34} +0.618034 q^{36} +1.47214 q^{37} -8.09017 q^{38} -3.23607 q^{39} +11.4721 q^{41} -8.47214 q^{42} +9.09017 q^{43} -7.47214 q^{46} -6.09017 q^{47} +4.85410 q^{48} +20.4164 q^{49} -2.00000 q^{51} +2.00000 q^{52} -5.38197 q^{53} +1.61803 q^{54} -11.7082 q^{56} -5.00000 q^{57} -1.38197 q^{58} +6.70820 q^{59} +7.00000 q^{61} -11.3262 q^{62} -5.23607 q^{63} +4.23607 q^{64} -11.6180 q^{67} +1.23607 q^{68} -4.61803 q^{69} -8.00000 q^{71} +2.23607 q^{72} -4.52786 q^{73} -2.38197 q^{74} +3.09017 q^{76} +5.23607 q^{78} +3.09017 q^{79} +1.00000 q^{81} -18.5623 q^{82} +7.70820 q^{83} +3.23607 q^{84} -14.7082 q^{86} -0.854102 q^{87} +4.14590 q^{89} -16.9443 q^{91} +2.85410 q^{92} -7.00000 q^{93} +9.85410 q^{94} -3.38197 q^{96} +2.52786 q^{97} -33.0344 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 6 * q^7 + 2 * q^9 $$2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 6 q^{7} + 2 q^{9} + q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} + 4 q^{17} - q^{18} + 10 q^{19} + 6 q^{21} + 7 q^{23} - 6 q^{26} - 2 q^{27} - 2 q^{28} - 5 q^{29} + 14 q^{31} + 9 q^{32} - 2 q^{34} - q^{36} - 6 q^{37} - 5 q^{38} - 2 q^{39} + 14 q^{41} - 8 q^{42} + 7 q^{43} - 6 q^{46} - q^{47} + 3 q^{48} + 14 q^{49} - 4 q^{51} + 4 q^{52} - 13 q^{53} + q^{54} - 10 q^{56} - 10 q^{57} - 5 q^{58} + 14 q^{61} - 7 q^{62} - 6 q^{63} + 4 q^{64} - 21 q^{67} - 2 q^{68} - 7 q^{69} - 16 q^{71} - 18 q^{73} - 7 q^{74} - 5 q^{76} + 6 q^{78} - 5 q^{79} + 2 q^{81} - 17 q^{82} + 2 q^{83} + 2 q^{84} - 16 q^{86} + 5 q^{87} + 15 q^{89} - 16 q^{91} - q^{92} - 14 q^{93} + 13 q^{94} - 9 q^{96} + 14 q^{97} - 37 q^{98}+O(q^{100})$$ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 6 * q^7 + 2 * q^9 + q^12 + 2 * q^13 + 8 * q^14 - 3 * q^16 + 4 * q^17 - q^18 + 10 * q^19 + 6 * q^21 + 7 * q^23 - 6 * q^26 - 2 * q^27 - 2 * q^28 - 5 * q^29 + 14 * q^31 + 9 * q^32 - 2 * q^34 - q^36 - 6 * q^37 - 5 * q^38 - 2 * q^39 + 14 * q^41 - 8 * q^42 + 7 * q^43 - 6 * q^46 - q^47 + 3 * q^48 + 14 * q^49 - 4 * q^51 + 4 * q^52 - 13 * q^53 + q^54 - 10 * q^56 - 10 * q^57 - 5 * q^58 + 14 * q^61 - 7 * q^62 - 6 * q^63 + 4 * q^64 - 21 * q^67 - 2 * q^68 - 7 * q^69 - 16 * q^71 - 18 * q^73 - 7 * q^74 - 5 * q^76 + 6 * q^78 - 5 * q^79 + 2 * q^81 - 17 * q^82 + 2 * q^83 + 2 * q^84 - 16 * q^86 + 5 * q^87 + 15 * q^89 - 16 * q^91 - q^92 - 14 * q^93 + 13 * q^94 - 9 * q^96 + 14 * q^97 - 37 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.61803 −1.14412 −0.572061 0.820211i $$-0.693856\pi$$
−0.572061 + 0.820211i $$0.693856\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ 1.61803 0.660560
$$7$$ −5.23607 −1.97905 −0.989524 0.144370i $$-0.953885\pi$$
−0.989524 + 0.144370i $$0.953885\pi$$
$$8$$ 2.23607 0.790569
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ −0.618034 −0.178411
$$13$$ 3.23607 0.897524 0.448762 0.893651i $$-0.351865\pi$$
0.448762 + 0.893651i $$0.351865\pi$$
$$14$$ 8.47214 2.26427
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −1.61803 −0.381374
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ 5.23607 1.14260
$$22$$ 0 0
$$23$$ 4.61803 0.962927 0.481463 0.876466i $$-0.340105\pi$$
0.481463 + 0.876466i $$0.340105\pi$$
$$24$$ −2.23607 −0.456435
$$25$$ 0 0
$$26$$ −5.23607 −1.02688
$$27$$ −1.00000 −0.192450
$$28$$ −3.23607 −0.611559
$$29$$ 0.854102 0.158603 0.0793014 0.996851i $$-0.474731\pi$$
0.0793014 + 0.996851i $$0.474731\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 3.38197 0.597853
$$33$$ 0 0
$$34$$ −3.23607 −0.554981
$$35$$ 0 0
$$36$$ 0.618034 0.103006
$$37$$ 1.47214 0.242018 0.121009 0.992651i $$-0.461387\pi$$
0.121009 + 0.992651i $$0.461387\pi$$
$$38$$ −8.09017 −1.31240
$$39$$ −3.23607 −0.518186
$$40$$ 0 0
$$41$$ 11.4721 1.79165 0.895823 0.444410i $$-0.146587\pi$$
0.895823 + 0.444410i $$0.146587\pi$$
$$42$$ −8.47214 −1.30728
$$43$$ 9.09017 1.38624 0.693119 0.720823i $$-0.256236\pi$$
0.693119 + 0.720823i $$0.256236\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −7.47214 −1.10171
$$47$$ −6.09017 −0.888343 −0.444171 0.895942i $$-0.646502\pi$$
−0.444171 + 0.895942i $$0.646502\pi$$
$$48$$ 4.85410 0.700629
$$49$$ 20.4164 2.91663
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 2.00000 0.277350
$$53$$ −5.38197 −0.739270 −0.369635 0.929177i $$-0.620517\pi$$
−0.369635 + 0.929177i $$0.620517\pi$$
$$54$$ 1.61803 0.220187
$$55$$ 0 0
$$56$$ −11.7082 −1.56457
$$57$$ −5.00000 −0.662266
$$58$$ −1.38197 −0.181461
$$59$$ 6.70820 0.873334 0.436667 0.899623i $$-0.356159\pi$$
0.436667 + 0.899623i $$0.356159\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ −11.3262 −1.43843
$$63$$ −5.23607 −0.659683
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −11.6180 −1.41937 −0.709684 0.704520i $$-0.751162\pi$$
−0.709684 + 0.704520i $$0.751162\pi$$
$$68$$ 1.23607 0.149895
$$69$$ −4.61803 −0.555946
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 2.23607 0.263523
$$73$$ −4.52786 −0.529946 −0.264973 0.964256i $$-0.585363\pi$$
−0.264973 + 0.964256i $$0.585363\pi$$
$$74$$ −2.38197 −0.276898
$$75$$ 0 0
$$76$$ 3.09017 0.354467
$$77$$ 0 0
$$78$$ 5.23607 0.592868
$$79$$ 3.09017 0.347671 0.173836 0.984775i $$-0.444384\pi$$
0.173836 + 0.984775i $$0.444384\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −18.5623 −2.04986
$$83$$ 7.70820 0.846085 0.423043 0.906110i $$-0.360962\pi$$
0.423043 + 0.906110i $$0.360962\pi$$
$$84$$ 3.23607 0.353084
$$85$$ 0 0
$$86$$ −14.7082 −1.58603
$$87$$ −0.854102 −0.0915693
$$88$$ 0 0
$$89$$ 4.14590 0.439464 0.219732 0.975560i $$-0.429482\pi$$
0.219732 + 0.975560i $$0.429482\pi$$
$$90$$ 0 0
$$91$$ −16.9443 −1.77624
$$92$$ 2.85410 0.297561
$$93$$ −7.00000 −0.725866
$$94$$ 9.85410 1.01637
$$95$$ 0 0
$$96$$ −3.38197 −0.345170
$$97$$ 2.52786 0.256666 0.128333 0.991731i $$-0.459037\pi$$
0.128333 + 0.991731i $$0.459037\pi$$
$$98$$ −33.0344 −3.33698
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 19.5623 1.94652 0.973261 0.229702i $$-0.0737750\pi$$
0.973261 + 0.229702i $$0.0737750\pi$$
$$102$$ 3.23607 0.320418
$$103$$ 4.61803 0.455028 0.227514 0.973775i $$-0.426940\pi$$
0.227514 + 0.973775i $$0.426940\pi$$
$$104$$ 7.23607 0.709555
$$105$$ 0 0
$$106$$ 8.70820 0.845816
$$107$$ −3.85410 −0.372590 −0.186295 0.982494i $$-0.559648\pi$$
−0.186295 + 0.982494i $$0.559648\pi$$
$$108$$ −0.618034 −0.0594703
$$109$$ 10.8541 1.03963 0.519817 0.854278i $$-0.326000\pi$$
0.519817 + 0.854278i $$0.326000\pi$$
$$110$$ 0 0
$$111$$ −1.47214 −0.139729
$$112$$ 25.4164 2.40162
$$113$$ −3.47214 −0.326631 −0.163316 0.986574i $$-0.552219\pi$$
−0.163316 + 0.986574i $$0.552219\pi$$
$$114$$ 8.09017 0.757714
$$115$$ 0 0
$$116$$ 0.527864 0.0490109
$$117$$ 3.23607 0.299175
$$118$$ −10.8541 −0.999201
$$119$$ −10.4721 −0.959979
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −11.3262 −1.02543
$$123$$ −11.4721 −1.03441
$$124$$ 4.32624 0.388508
$$125$$ 0 0
$$126$$ 8.47214 0.754758
$$127$$ −1.94427 −0.172526 −0.0862631 0.996272i $$-0.527493\pi$$
−0.0862631 + 0.996272i $$0.527493\pi$$
$$128$$ −13.6180 −1.20368
$$129$$ −9.09017 −0.800345
$$130$$ 0 0
$$131$$ −9.18034 −0.802090 −0.401045 0.916058i $$-0.631353\pi$$
−0.401045 + 0.916058i $$0.631353\pi$$
$$132$$ 0 0
$$133$$ −26.1803 −2.27012
$$134$$ 18.7984 1.62393
$$135$$ 0 0
$$136$$ 4.47214 0.383482
$$137$$ 5.29180 0.452109 0.226054 0.974115i $$-0.427417\pi$$
0.226054 + 0.974115i $$0.427417\pi$$
$$138$$ 7.47214 0.636070
$$139$$ −1.18034 −0.100115 −0.0500576 0.998746i $$-0.515940\pi$$
−0.0500576 + 0.998746i $$0.515940\pi$$
$$140$$ 0 0
$$141$$ 6.09017 0.512885
$$142$$ 12.9443 1.08626
$$143$$ 0 0
$$144$$ −4.85410 −0.404508
$$145$$ 0 0
$$146$$ 7.32624 0.606324
$$147$$ −20.4164 −1.68392
$$148$$ 0.909830 0.0747876
$$149$$ −20.1246 −1.64867 −0.824336 0.566101i $$-0.808451\pi$$
−0.824336 + 0.566101i $$0.808451\pi$$
$$150$$ 0 0
$$151$$ −15.2361 −1.23989 −0.619947 0.784644i $$-0.712846\pi$$
−0.619947 + 0.784644i $$0.712846\pi$$
$$152$$ 11.1803 0.906845
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ 12.3262 0.983741 0.491870 0.870668i $$-0.336313\pi$$
0.491870 + 0.870668i $$0.336313\pi$$
$$158$$ −5.00000 −0.397779
$$159$$ 5.38197 0.426818
$$160$$ 0 0
$$161$$ −24.1803 −1.90568
$$162$$ −1.61803 −0.127125
$$163$$ −6.56231 −0.513999 −0.257000 0.966411i $$-0.582734\pi$$
−0.257000 + 0.966411i $$0.582734\pi$$
$$164$$ 7.09017 0.553649
$$165$$ 0 0
$$166$$ −12.4721 −0.968025
$$167$$ 9.03444 0.699106 0.349553 0.936917i $$-0.386333\pi$$
0.349553 + 0.936917i $$0.386333\pi$$
$$168$$ 11.7082 0.903308
$$169$$ −2.52786 −0.194451
$$170$$ 0 0
$$171$$ 5.00000 0.382360
$$172$$ 5.61803 0.428371
$$173$$ −16.5623 −1.25921 −0.629604 0.776916i $$-0.716783\pi$$
−0.629604 + 0.776916i $$0.716783\pi$$
$$174$$ 1.38197 0.104767
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −6.70820 −0.504219
$$178$$ −6.70820 −0.502801
$$179$$ −10.5279 −0.786890 −0.393445 0.919348i $$-0.628717\pi$$
−0.393445 + 0.919348i $$0.628717\pi$$
$$180$$ 0 0
$$181$$ −0.763932 −0.0567826 −0.0283913 0.999597i $$-0.509038\pi$$
−0.0283913 + 0.999597i $$0.509038\pi$$
$$182$$ 27.4164 2.03224
$$183$$ −7.00000 −0.517455
$$184$$ 10.3262 0.761260
$$185$$ 0 0
$$186$$ 11.3262 0.830480
$$187$$ 0 0
$$188$$ −3.76393 −0.274513
$$189$$ 5.23607 0.380868
$$190$$ 0 0
$$191$$ −16.4164 −1.18785 −0.593925 0.804521i $$-0.702422\pi$$
−0.593925 + 0.804521i $$0.702422\pi$$
$$192$$ −4.23607 −0.305712
$$193$$ 8.03444 0.578332 0.289166 0.957279i $$-0.406622\pi$$
0.289166 + 0.957279i $$0.406622\pi$$
$$194$$ −4.09017 −0.293657
$$195$$ 0 0
$$196$$ 12.6180 0.901288
$$197$$ 9.76393 0.695651 0.347826 0.937559i $$-0.386920\pi$$
0.347826 + 0.937559i $$0.386920\pi$$
$$198$$ 0 0
$$199$$ −4.79837 −0.340148 −0.170074 0.985431i $$-0.554401\pi$$
−0.170074 + 0.985431i $$0.554401\pi$$
$$200$$ 0 0
$$201$$ 11.6180 0.819473
$$202$$ −31.6525 −2.22706
$$203$$ −4.47214 −0.313882
$$204$$ −1.23607 −0.0865421
$$205$$ 0 0
$$206$$ −7.47214 −0.520608
$$207$$ 4.61803 0.320976
$$208$$ −15.7082 −1.08917
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 2.85410 0.196484 0.0982422 0.995163i $$-0.468678\pi$$
0.0982422 + 0.995163i $$0.468678\pi$$
$$212$$ −3.32624 −0.228447
$$213$$ 8.00000 0.548151
$$214$$ 6.23607 0.426289
$$215$$ 0 0
$$216$$ −2.23607 −0.152145
$$217$$ −36.6525 −2.48813
$$218$$ −17.5623 −1.18947
$$219$$ 4.52786 0.305965
$$220$$ 0 0
$$221$$ 6.47214 0.435363
$$222$$ 2.38197 0.159867
$$223$$ −21.0344 −1.40857 −0.704285 0.709917i $$-0.748732\pi$$
−0.704285 + 0.709917i $$0.748732\pi$$
$$224$$ −17.7082 −1.18318
$$225$$ 0 0
$$226$$ 5.61803 0.373706
$$227$$ 8.05573 0.534677 0.267339 0.963603i $$-0.413856\pi$$
0.267339 + 0.963603i $$0.413856\pi$$
$$228$$ −3.09017 −0.204652
$$229$$ 6.90983 0.456614 0.228307 0.973589i $$-0.426681\pi$$
0.228307 + 0.973589i $$0.426681\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.90983 0.125386
$$233$$ 10.2705 0.672843 0.336422 0.941711i $$-0.390783\pi$$
0.336422 + 0.941711i $$0.390783\pi$$
$$234$$ −5.23607 −0.342292
$$235$$ 0 0
$$236$$ 4.14590 0.269875
$$237$$ −3.09017 −0.200728
$$238$$ 16.9443 1.09833
$$239$$ 22.0344 1.42529 0.712645 0.701525i $$-0.247497\pi$$
0.712645 + 0.701525i $$0.247497\pi$$
$$240$$ 0 0
$$241$$ 0.618034 0.0398111 0.0199055 0.999802i $$-0.493663\pi$$
0.0199055 + 0.999802i $$0.493663\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 4.32624 0.276959
$$245$$ 0 0
$$246$$ 18.5623 1.18349
$$247$$ 16.1803 1.02953
$$248$$ 15.6525 0.993933
$$249$$ −7.70820 −0.488488
$$250$$ 0 0
$$251$$ 16.4721 1.03971 0.519856 0.854254i $$-0.325986\pi$$
0.519856 + 0.854254i $$0.325986\pi$$
$$252$$ −3.23607 −0.203853
$$253$$ 0 0
$$254$$ 3.14590 0.197391
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ 9.56231 0.596480 0.298240 0.954491i $$-0.403600\pi$$
0.298240 + 0.954491i $$0.403600\pi$$
$$258$$ 14.7082 0.915693
$$259$$ −7.70820 −0.478964
$$260$$ 0 0
$$261$$ 0.854102 0.0528676
$$262$$ 14.8541 0.917689
$$263$$ 24.2148 1.49315 0.746574 0.665303i $$-0.231698\pi$$
0.746574 + 0.665303i $$0.231698\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 42.3607 2.59730
$$267$$ −4.14590 −0.253725
$$268$$ −7.18034 −0.438609
$$269$$ 0.854102 0.0520755 0.0260378 0.999661i $$-0.491711\pi$$
0.0260378 + 0.999661i $$0.491711\pi$$
$$270$$ 0 0
$$271$$ 18.3820 1.11662 0.558312 0.829631i $$-0.311449\pi$$
0.558312 + 0.829631i $$0.311449\pi$$
$$272$$ −9.70820 −0.588646
$$273$$ 16.9443 1.02551
$$274$$ −8.56231 −0.517268
$$275$$ 0 0
$$276$$ −2.85410 −0.171797
$$277$$ 22.6525 1.36106 0.680528 0.732722i $$-0.261751\pi$$
0.680528 + 0.732722i $$0.261751\pi$$
$$278$$ 1.90983 0.114544
$$279$$ 7.00000 0.419079
$$280$$ 0 0
$$281$$ −1.09017 −0.0650341 −0.0325170 0.999471i $$-0.510352\pi$$
−0.0325170 + 0.999471i $$0.510352\pi$$
$$282$$ −9.85410 −0.586803
$$283$$ −16.0344 −0.953149 −0.476574 0.879134i $$-0.658122\pi$$
−0.476574 + 0.879134i $$0.658122\pi$$
$$284$$ −4.94427 −0.293389
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −60.0689 −3.54575
$$288$$ 3.38197 0.199284
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −2.52786 −0.148186
$$292$$ −2.79837 −0.163762
$$293$$ 25.4721 1.48810 0.744049 0.668125i $$-0.232903\pi$$
0.744049 + 0.668125i $$0.232903\pi$$
$$294$$ 33.0344 1.92661
$$295$$ 0 0
$$296$$ 3.29180 0.191332
$$297$$ 0 0
$$298$$ 32.5623 1.88628
$$299$$ 14.9443 0.864250
$$300$$ 0 0
$$301$$ −47.5967 −2.74343
$$302$$ 24.6525 1.41859
$$303$$ −19.5623 −1.12383
$$304$$ −24.2705 −1.39201
$$305$$ 0 0
$$306$$ −3.23607 −0.184994
$$307$$ −28.1246 −1.60516 −0.802578 0.596547i $$-0.796539\pi$$
−0.802578 + 0.596547i $$0.796539\pi$$
$$308$$ 0 0
$$309$$ −4.61803 −0.262711
$$310$$ 0 0
$$311$$ 12.5279 0.710390 0.355195 0.934792i $$-0.384414\pi$$
0.355195 + 0.934792i $$0.384414\pi$$
$$312$$ −7.23607 −0.409662
$$313$$ −8.47214 −0.478873 −0.239437 0.970912i $$-0.576963\pi$$
−0.239437 + 0.970912i $$0.576963\pi$$
$$314$$ −19.9443 −1.12552
$$315$$ 0 0
$$316$$ 1.90983 0.107436
$$317$$ 3.58359 0.201275 0.100637 0.994923i $$-0.467912\pi$$
0.100637 + 0.994923i $$0.467912\pi$$
$$318$$ −8.70820 −0.488332
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 3.85410 0.215115
$$322$$ 39.1246 2.18033
$$323$$ 10.0000 0.556415
$$324$$ 0.618034 0.0343352
$$325$$ 0 0
$$326$$ 10.6180 0.588079
$$327$$ −10.8541 −0.600233
$$328$$ 25.6525 1.41642
$$329$$ 31.8885 1.75807
$$330$$ 0 0
$$331$$ −27.5967 −1.51685 −0.758427 0.651758i $$-0.774032\pi$$
−0.758427 + 0.651758i $$0.774032\pi$$
$$332$$ 4.76393 0.261455
$$333$$ 1.47214 0.0806726
$$334$$ −14.6180 −0.799863
$$335$$ 0 0
$$336$$ −25.4164 −1.38658
$$337$$ −28.8541 −1.57178 −0.785892 0.618364i $$-0.787796\pi$$
−0.785892 + 0.618364i $$0.787796\pi$$
$$338$$ 4.09017 0.222476
$$339$$ 3.47214 0.188581
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −8.09017 −0.437466
$$343$$ −70.2492 −3.79310
$$344$$ 20.3262 1.09592
$$345$$ 0 0
$$346$$ 26.7984 1.44069
$$347$$ 28.1803 1.51280 0.756400 0.654109i $$-0.226956\pi$$
0.756400 + 0.654109i $$0.226956\pi$$
$$348$$ −0.527864 −0.0282965
$$349$$ 32.2361 1.72556 0.862779 0.505582i $$-0.168722\pi$$
0.862779 + 0.505582i $$0.168722\pi$$
$$350$$ 0 0
$$351$$ −3.23607 −0.172729
$$352$$ 0 0
$$353$$ 17.8328 0.949145 0.474573 0.880216i $$-0.342603\pi$$
0.474573 + 0.880216i $$0.342603\pi$$
$$354$$ 10.8541 0.576889
$$355$$ 0 0
$$356$$ 2.56231 0.135802
$$357$$ 10.4721 0.554244
$$358$$ 17.0344 0.900298
$$359$$ 3.29180 0.173734 0.0868672 0.996220i $$-0.472314\pi$$
0.0868672 + 0.996220i $$0.472314\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 1.23607 0.0649663
$$363$$ 0 0
$$364$$ −10.4721 −0.548889
$$365$$ 0 0
$$366$$ 11.3262 0.592032
$$367$$ 22.1246 1.15490 0.577448 0.816428i $$-0.304049\pi$$
0.577448 + 0.816428i $$0.304049\pi$$
$$368$$ −22.4164 −1.16854
$$369$$ 11.4721 0.597216
$$370$$ 0 0
$$371$$ 28.1803 1.46305
$$372$$ −4.32624 −0.224305
$$373$$ −32.7426 −1.69535 −0.847675 0.530516i $$-0.821998\pi$$
−0.847675 + 0.530516i $$0.821998\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −13.6180 −0.702296
$$377$$ 2.76393 0.142350
$$378$$ −8.47214 −0.435760
$$379$$ −5.00000 −0.256833 −0.128416 0.991720i $$-0.540989\pi$$
−0.128416 + 0.991720i $$0.540989\pi$$
$$380$$ 0 0
$$381$$ 1.94427 0.0996081
$$382$$ 26.5623 1.35905
$$383$$ 13.3607 0.682699 0.341349 0.939936i $$-0.389116\pi$$
0.341349 + 0.939936i $$0.389116\pi$$
$$384$$ 13.6180 0.694942
$$385$$ 0 0
$$386$$ −13.0000 −0.661683
$$387$$ 9.09017 0.462079
$$388$$ 1.56231 0.0793141
$$389$$ −21.0557 −1.06757 −0.533784 0.845621i $$-0.679230\pi$$
−0.533784 + 0.845621i $$0.679230\pi$$
$$390$$ 0 0
$$391$$ 9.23607 0.467088
$$392$$ 45.6525 2.30580
$$393$$ 9.18034 0.463087
$$394$$ −15.7984 −0.795911
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.72949 0.136989 0.0684946 0.997651i $$-0.478180\pi$$
0.0684946 + 0.997651i $$0.478180\pi$$
$$398$$ 7.76393 0.389171
$$399$$ 26.1803 1.31066
$$400$$ 0 0
$$401$$ −30.8885 −1.54250 −0.771250 0.636532i $$-0.780368\pi$$
−0.771250 + 0.636532i $$0.780368\pi$$
$$402$$ −18.7984 −0.937578
$$403$$ 22.6525 1.12840
$$404$$ 12.0902 0.601508
$$405$$ 0 0
$$406$$ 7.23607 0.359120
$$407$$ 0 0
$$408$$ −4.47214 −0.221404
$$409$$ −26.3050 −1.30070 −0.650348 0.759636i $$-0.725377\pi$$
−0.650348 + 0.759636i $$0.725377\pi$$
$$410$$ 0 0
$$411$$ −5.29180 −0.261025
$$412$$ 2.85410 0.140612
$$413$$ −35.1246 −1.72837
$$414$$ −7.47214 −0.367235
$$415$$ 0 0
$$416$$ 10.9443 0.536587
$$417$$ 1.18034 0.0578015
$$418$$ 0 0
$$419$$ −24.5967 −1.20163 −0.600815 0.799388i $$-0.705157\pi$$
−0.600815 + 0.799388i $$0.705157\pi$$
$$420$$ 0 0
$$421$$ −3.85410 −0.187837 −0.0939187 0.995580i $$-0.529939\pi$$
−0.0939187 + 0.995580i $$0.529939\pi$$
$$422$$ −4.61803 −0.224802
$$423$$ −6.09017 −0.296114
$$424$$ −12.0344 −0.584444
$$425$$ 0 0
$$426$$ −12.9443 −0.627152
$$427$$ −36.6525 −1.77374
$$428$$ −2.38197 −0.115137
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −15.8885 −0.765324 −0.382662 0.923888i $$-0.624993\pi$$
−0.382662 + 0.923888i $$0.624993\pi$$
$$432$$ 4.85410 0.233543
$$433$$ −1.03444 −0.0497121 −0.0248561 0.999691i $$-0.507913\pi$$
−0.0248561 + 0.999691i $$0.507913\pi$$
$$434$$ 59.3050 2.84673
$$435$$ 0 0
$$436$$ 6.70820 0.321265
$$437$$ 23.0902 1.10455
$$438$$ −7.32624 −0.350061
$$439$$ 25.3262 1.20876 0.604378 0.796698i $$-0.293422\pi$$
0.604378 + 0.796698i $$0.293422\pi$$
$$440$$ 0 0
$$441$$ 20.4164 0.972210
$$442$$ −10.4721 −0.498109
$$443$$ −8.27051 −0.392944 −0.196472 0.980509i $$-0.562948\pi$$
−0.196472 + 0.980509i $$0.562948\pi$$
$$444$$ −0.909830 −0.0431786
$$445$$ 0 0
$$446$$ 34.0344 1.61158
$$447$$ 20.1246 0.951861
$$448$$ −22.1803 −1.04792
$$449$$ −34.2705 −1.61733 −0.808663 0.588273i $$-0.799808\pi$$
−0.808663 + 0.588273i $$0.799808\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −2.14590 −0.100935
$$453$$ 15.2361 0.715853
$$454$$ −13.0344 −0.611737
$$455$$ 0 0
$$456$$ −11.1803 −0.523567
$$457$$ −7.47214 −0.349532 −0.174766 0.984610i $$-0.555917\pi$$
−0.174766 + 0.984610i $$0.555917\pi$$
$$458$$ −11.1803 −0.522423
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 8.05573 0.375193 0.187596 0.982246i $$-0.439930\pi$$
0.187596 + 0.982246i $$0.439930\pi$$
$$462$$ 0 0
$$463$$ 0.270510 0.0125717 0.00628583 0.999980i $$-0.497999\pi$$
0.00628583 + 0.999980i $$0.497999\pi$$
$$464$$ −4.14590 −0.192468
$$465$$ 0 0
$$466$$ −16.6180 −0.769816
$$467$$ −30.8885 −1.42935 −0.714676 0.699456i $$-0.753426\pi$$
−0.714676 + 0.699456i $$0.753426\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 60.8328 2.80900
$$470$$ 0 0
$$471$$ −12.3262 −0.567963
$$472$$ 15.0000 0.690431
$$473$$ 0 0
$$474$$ 5.00000 0.229658
$$475$$ 0 0
$$476$$ −6.47214 −0.296650
$$477$$ −5.38197 −0.246423
$$478$$ −35.6525 −1.63071
$$479$$ 6.58359 0.300812 0.150406 0.988624i $$-0.451942\pi$$
0.150406 + 0.988624i $$0.451942\pi$$
$$480$$ 0 0
$$481$$ 4.76393 0.217217
$$482$$ −1.00000 −0.0455488
$$483$$ 24.1803 1.10024
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 1.61803 0.0733955
$$487$$ 8.18034 0.370687 0.185343 0.982674i $$-0.440660\pi$$
0.185343 + 0.982674i $$0.440660\pi$$
$$488$$ 15.6525 0.708554
$$489$$ 6.56231 0.296758
$$490$$ 0 0
$$491$$ 7.20163 0.325005 0.162502 0.986708i $$-0.448043\pi$$
0.162502 + 0.986708i $$0.448043\pi$$
$$492$$ −7.09017 −0.319650
$$493$$ 1.70820 0.0769336
$$494$$ −26.1803 −1.17791
$$495$$ 0 0
$$496$$ −33.9787 −1.52569
$$497$$ 41.8885 1.87896
$$498$$ 12.4721 0.558890
$$499$$ 11.1803 0.500501 0.250250 0.968181i $$-0.419487\pi$$
0.250250 + 0.968181i $$0.419487\pi$$
$$500$$ 0 0
$$501$$ −9.03444 −0.403629
$$502$$ −26.6525 −1.18956
$$503$$ 13.8885 0.619260 0.309630 0.950857i $$-0.399795\pi$$
0.309630 + 0.950857i $$0.399795\pi$$
$$504$$ −11.7082 −0.521525
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 2.52786 0.112266
$$508$$ −1.20163 −0.0533135
$$509$$ 9.47214 0.419845 0.209923 0.977718i $$-0.432679\pi$$
0.209923 + 0.977718i $$0.432679\pi$$
$$510$$ 0 0
$$511$$ 23.7082 1.04879
$$512$$ 5.29180 0.233867
$$513$$ −5.00000 −0.220755
$$514$$ −15.4721 −0.682447
$$515$$ 0 0
$$516$$ −5.61803 −0.247320
$$517$$ 0 0
$$518$$ 12.4721 0.547994
$$519$$ 16.5623 0.727005
$$520$$ 0 0
$$521$$ 29.3607 1.28631 0.643157 0.765734i $$-0.277624\pi$$
0.643157 + 0.765734i $$0.277624\pi$$
$$522$$ −1.38197 −0.0604870
$$523$$ 3.76393 0.164585 0.0822926 0.996608i $$-0.473776\pi$$
0.0822926 + 0.996608i $$0.473776\pi$$
$$524$$ −5.67376 −0.247859
$$525$$ 0 0
$$526$$ −39.1803 −1.70834
$$527$$ 14.0000 0.609850
$$528$$ 0 0
$$529$$ −1.67376 −0.0727723
$$530$$ 0 0
$$531$$ 6.70820 0.291111
$$532$$ −16.1803 −0.701507
$$533$$ 37.1246 1.60805
$$534$$ 6.70820 0.290292
$$535$$ 0 0
$$536$$ −25.9787 −1.12211
$$537$$ 10.5279 0.454311
$$538$$ −1.38197 −0.0595808
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1.61803 −0.0695647 −0.0347824 0.999395i $$-0.511074\pi$$
−0.0347824 + 0.999395i $$0.511074\pi$$
$$542$$ −29.7426 −1.27756
$$543$$ 0.763932 0.0327835
$$544$$ 6.76393 0.290001
$$545$$ 0 0
$$546$$ −27.4164 −1.17331
$$547$$ 19.3607 0.827803 0.413901 0.910322i $$-0.364166\pi$$
0.413901 + 0.910322i $$0.364166\pi$$
$$548$$ 3.27051 0.139709
$$549$$ 7.00000 0.298753
$$550$$ 0 0
$$551$$ 4.27051 0.181930
$$552$$ −10.3262 −0.439514
$$553$$ −16.1803 −0.688058
$$554$$ −36.6525 −1.55721
$$555$$ 0 0
$$556$$ −0.729490 −0.0309373
$$557$$ −42.3951 −1.79634 −0.898169 0.439649i $$-0.855103\pi$$
−0.898169 + 0.439649i $$0.855103\pi$$
$$558$$ −11.3262 −0.479478
$$559$$ 29.4164 1.24418
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 1.76393 0.0744070
$$563$$ 20.6738 0.871295 0.435648 0.900117i $$-0.356519\pi$$
0.435648 + 0.900117i $$0.356519\pi$$
$$564$$ 3.76393 0.158490
$$565$$ 0 0
$$566$$ 25.9443 1.09052
$$567$$ −5.23607 −0.219894
$$568$$ −17.8885 −0.750587
$$569$$ −22.5623 −0.945861 −0.472931 0.881100i $$-0.656804\pi$$
−0.472931 + 0.881100i $$0.656804\pi$$
$$570$$ 0 0
$$571$$ −11.0902 −0.464109 −0.232055 0.972703i $$-0.574545\pi$$
−0.232055 + 0.972703i $$0.574545\pi$$
$$572$$ 0 0
$$573$$ 16.4164 0.685805
$$574$$ 97.1935 4.05678
$$575$$ 0 0
$$576$$ 4.23607 0.176503
$$577$$ 30.0132 1.24946 0.624732 0.780839i $$-0.285208\pi$$
0.624732 + 0.780839i $$0.285208\pi$$
$$578$$ 21.0344 0.874917
$$579$$ −8.03444 −0.333900
$$580$$ 0 0
$$581$$ −40.3607 −1.67444
$$582$$ 4.09017 0.169543
$$583$$ 0 0
$$584$$ −10.1246 −0.418959
$$585$$ 0 0
$$586$$ −41.2148 −1.70257
$$587$$ −21.6180 −0.892272 −0.446136 0.894965i $$-0.647200\pi$$
−0.446136 + 0.894965i $$0.647200\pi$$
$$588$$ −12.6180 −0.520359
$$589$$ 35.0000 1.44215
$$590$$ 0 0
$$591$$ −9.76393 −0.401634
$$592$$ −7.14590 −0.293695
$$593$$ 3.11146 0.127772 0.0638861 0.997957i $$-0.479651\pi$$
0.0638861 + 0.997957i $$0.479651\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −12.4377 −0.509468
$$597$$ 4.79837 0.196384
$$598$$ −24.1803 −0.988808
$$599$$ −17.7639 −0.725815 −0.362907 0.931825i $$-0.618216\pi$$
−0.362907 + 0.931825i $$0.618216\pi$$
$$600$$ 0 0
$$601$$ −10.3607 −0.422621 −0.211310 0.977419i $$-0.567773\pi$$
−0.211310 + 0.977419i $$0.567773\pi$$
$$602$$ 77.0132 3.13882
$$603$$ −11.6180 −0.473123
$$604$$ −9.41641 −0.383148
$$605$$ 0 0
$$606$$ 31.6525 1.28579
$$607$$ −33.0000 −1.33943 −0.669714 0.742619i $$-0.733583\pi$$
−0.669714 + 0.742619i $$0.733583\pi$$
$$608$$ 16.9098 0.685784
$$609$$ 4.47214 0.181220
$$610$$ 0 0
$$611$$ −19.7082 −0.797309
$$612$$ 1.23607 0.0499651
$$613$$ 3.76393 0.152024 0.0760119 0.997107i $$-0.475781\pi$$
0.0760119 + 0.997107i $$0.475781\pi$$
$$614$$ 45.5066 1.83650
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.2492 1.70089 0.850445 0.526064i $$-0.176333\pi$$
0.850445 + 0.526064i $$0.176333\pi$$
$$618$$ 7.47214 0.300573
$$619$$ 42.8885 1.72384 0.861918 0.507048i $$-0.169263\pi$$
0.861918 + 0.507048i $$0.169263\pi$$
$$620$$ 0 0
$$621$$ −4.61803 −0.185315
$$622$$ −20.2705 −0.812773
$$623$$ −21.7082 −0.869721
$$624$$ 15.7082 0.628831
$$625$$ 0 0
$$626$$ 13.7082 0.547890
$$627$$ 0 0
$$628$$ 7.61803 0.303993
$$629$$ 2.94427 0.117396
$$630$$ 0 0
$$631$$ −39.6312 −1.57769 −0.788846 0.614590i $$-0.789321\pi$$
−0.788846 + 0.614590i $$0.789321\pi$$
$$632$$ 6.90983 0.274858
$$633$$ −2.85410 −0.113440
$$634$$ −5.79837 −0.230283
$$635$$ 0 0
$$636$$ 3.32624 0.131894
$$637$$ 66.0689 2.61774
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 4.23607 0.167315 0.0836573 0.996495i $$-0.473340\pi$$
0.0836573 + 0.996495i $$0.473340\pi$$
$$642$$ −6.23607 −0.246118
$$643$$ −13.6738 −0.539241 −0.269620 0.962967i $$-0.586898\pi$$
−0.269620 + 0.962967i $$0.586898\pi$$
$$644$$ −14.9443 −0.588887
$$645$$ 0 0
$$646$$ −16.1803 −0.636607
$$647$$ 31.5967 1.24220 0.621098 0.783733i $$-0.286687\pi$$
0.621098 + 0.783733i $$0.286687\pi$$
$$648$$ 2.23607 0.0878410
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 36.6525 1.43652
$$652$$ −4.05573 −0.158835
$$653$$ 43.5623 1.70472 0.852362 0.522952i $$-0.175169\pi$$
0.852362 + 0.522952i $$0.175169\pi$$
$$654$$ 17.5623 0.686741
$$655$$ 0 0
$$656$$ −55.6869 −2.17421
$$657$$ −4.52786 −0.176649
$$658$$ −51.5967 −2.01145
$$659$$ 42.0344 1.63743 0.818715 0.574201i $$-0.194687\pi$$
0.818715 + 0.574201i $$0.194687\pi$$
$$660$$ 0 0
$$661$$ 15.0902 0.586940 0.293470 0.955968i $$-0.405190\pi$$
0.293470 + 0.955968i $$0.405190\pi$$
$$662$$ 44.6525 1.73547
$$663$$ −6.47214 −0.251357
$$664$$ 17.2361 0.668889
$$665$$ 0 0
$$666$$ −2.38197 −0.0922993
$$667$$ 3.94427 0.152723
$$668$$ 5.58359 0.216036
$$669$$ 21.0344 0.813239
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 17.7082 0.683109
$$673$$ −35.3050 −1.36091 −0.680453 0.732792i $$-0.738217\pi$$
−0.680453 + 0.732792i $$0.738217\pi$$
$$674$$ 46.6869 1.79831
$$675$$ 0 0
$$676$$ −1.56231 −0.0600887
$$677$$ 5.94427 0.228457 0.114228 0.993455i $$-0.463560\pi$$
0.114228 + 0.993455i $$0.463560\pi$$
$$678$$ −5.61803 −0.215759
$$679$$ −13.2361 −0.507954
$$680$$ 0 0
$$681$$ −8.05573 −0.308696
$$682$$ 0 0
$$683$$ −15.7082 −0.601058 −0.300529 0.953773i $$-0.597163\pi$$
−0.300529 + 0.953773i $$0.597163\pi$$
$$684$$ 3.09017 0.118156
$$685$$ 0 0
$$686$$ 113.666 4.33977
$$687$$ −6.90983 −0.263626
$$688$$ −44.1246 −1.68224
$$689$$ −17.4164 −0.663512
$$690$$ 0 0
$$691$$ −29.3050 −1.11481 −0.557406 0.830240i $$-0.688203\pi$$
−0.557406 + 0.830240i $$0.688203\pi$$
$$692$$ −10.2361 −0.389117
$$693$$ 0 0
$$694$$ −45.5967 −1.73083
$$695$$ 0 0
$$696$$ −1.90983 −0.0723919
$$697$$ 22.9443 0.869076
$$698$$ −52.1591 −1.97425
$$699$$ −10.2705 −0.388466
$$700$$ 0 0
$$701$$ 40.4164 1.52651 0.763253 0.646099i $$-0.223601\pi$$
0.763253 + 0.646099i $$0.223601\pi$$
$$702$$ 5.23607 0.197623
$$703$$ 7.36068 0.277613
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −28.8541 −1.08594
$$707$$ −102.430 −3.85226
$$708$$ −4.14590 −0.155812
$$709$$ 18.8197 0.706787 0.353394 0.935475i $$-0.385028\pi$$
0.353394 + 0.935475i $$0.385028\pi$$
$$710$$ 0 0
$$711$$ 3.09017 0.115890
$$712$$ 9.27051 0.347427
$$713$$ 32.3262 1.21063
$$714$$ −16.9443 −0.634123
$$715$$ 0 0
$$716$$ −6.50658 −0.243162
$$717$$ −22.0344 −0.822891
$$718$$ −5.32624 −0.198773
$$719$$ 26.9098 1.00357 0.501784 0.864993i $$-0.332677\pi$$
0.501784 + 0.864993i $$0.332677\pi$$
$$720$$ 0 0
$$721$$ −24.1803 −0.900523
$$722$$ −9.70820 −0.361302
$$723$$ −0.618034 −0.0229849
$$724$$ −0.472136 −0.0175468
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 21.6738 0.803835 0.401918 0.915676i $$-0.368344\pi$$
0.401918 + 0.915676i $$0.368344\pi$$
$$728$$ −37.8885 −1.40424
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 18.1803 0.672424
$$732$$ −4.32624 −0.159902
$$733$$ −4.85410 −0.179290 −0.0896452 0.995974i $$-0.528573\pi$$
−0.0896452 + 0.995974i $$0.528573\pi$$
$$734$$ −35.7984 −1.32134
$$735$$ 0 0
$$736$$ 15.6180 0.575688
$$737$$ 0 0
$$738$$ −18.5623 −0.683288
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ −16.1803 −0.594400
$$742$$ −45.5967 −1.67391
$$743$$ 31.6525 1.16122 0.580608 0.814183i $$-0.302815\pi$$
0.580608 + 0.814183i $$0.302815\pi$$
$$744$$ −15.6525 −0.573848
$$745$$ 0 0
$$746$$ 52.9787 1.93969
$$747$$ 7.70820 0.282028
$$748$$ 0 0
$$749$$ 20.1803 0.737374
$$750$$ 0 0
$$751$$ −10.1115 −0.368972 −0.184486 0.982835i $$-0.559062\pi$$
−0.184486 + 0.982835i $$0.559062\pi$$
$$752$$ 29.5623 1.07803
$$753$$ −16.4721 −0.600278
$$754$$ −4.47214 −0.162866
$$755$$ 0 0
$$756$$ 3.23607 0.117695
$$757$$ −1.81966 −0.0661367 −0.0330683 0.999453i $$-0.510528\pi$$
−0.0330683 + 0.999453i $$0.510528\pi$$
$$758$$ 8.09017 0.293848
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 9.96556 0.361251 0.180626 0.983552i $$-0.442188\pi$$
0.180626 + 0.983552i $$0.442188\pi$$
$$762$$ −3.14590 −0.113964
$$763$$ −56.8328 −2.05749
$$764$$ −10.1459 −0.367066
$$765$$ 0 0
$$766$$ −21.6180 −0.781091
$$767$$ 21.7082 0.783838
$$768$$ −13.5623 −0.489388
$$769$$ 17.2361 0.621549 0.310774 0.950484i $$-0.399412\pi$$
0.310774 + 0.950484i $$0.399412\pi$$
$$770$$ 0 0
$$771$$ −9.56231 −0.344378
$$772$$ 4.96556 0.178714
$$773$$ 27.5066 0.989343 0.494671 0.869080i $$-0.335288\pi$$
0.494671 + 0.869080i $$0.335288\pi$$
$$774$$ −14.7082 −0.528675
$$775$$ 0 0
$$776$$ 5.65248 0.202912
$$777$$ 7.70820 0.276530
$$778$$ 34.0689 1.22143
$$779$$ 57.3607 2.05516
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −14.9443 −0.534406
$$783$$ −0.854102 −0.0305231
$$784$$ −99.1033 −3.53940
$$785$$ 0 0
$$786$$ −14.8541 −0.529828
$$787$$ −32.7984 −1.16914 −0.584568 0.811345i $$-0.698736\pi$$
−0.584568 + 0.811345i $$0.698736\pi$$
$$788$$ 6.03444 0.214968
$$789$$ −24.2148 −0.862069
$$790$$ 0 0
$$791$$ 18.1803 0.646418
$$792$$ 0 0
$$793$$ 22.6525 0.804413
$$794$$ −4.41641 −0.156732
$$795$$ 0 0
$$796$$ −2.96556 −0.105111
$$797$$ 15.2918 0.541663 0.270832 0.962627i $$-0.412701\pi$$
0.270832 + 0.962627i $$0.412701\pi$$
$$798$$ −42.3607 −1.49955
$$799$$ −12.1803 −0.430909
$$800$$ 0 0
$$801$$ 4.14590 0.146488
$$802$$ 49.9787 1.76481
$$803$$ 0 0
$$804$$ 7.18034 0.253231
$$805$$ 0 0
$$806$$ −36.6525 −1.29103
$$807$$ −0.854102 −0.0300658
$$808$$ 43.7426 1.53886
$$809$$ 29.6738 1.04327 0.521637 0.853168i $$-0.325322\pi$$
0.521637 + 0.853168i $$0.325322\pi$$
$$810$$ 0 0
$$811$$ 43.6312 1.53210 0.766049 0.642782i $$-0.222220\pi$$
0.766049 + 0.642782i $$0.222220\pi$$
$$812$$ −2.76393 −0.0969950
$$813$$ −18.3820 −0.644684
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 9.70820 0.339855
$$817$$ 45.4508 1.59012
$$818$$ 42.5623 1.48816
$$819$$ −16.9443 −0.592081
$$820$$ 0 0
$$821$$ 20.8197 0.726611 0.363306 0.931670i $$-0.381648\pi$$
0.363306 + 0.931670i $$0.381648\pi$$
$$822$$ 8.56231 0.298645
$$823$$ 16.5279 0.576125 0.288063 0.957612i $$-0.406989\pi$$
0.288063 + 0.957612i $$0.406989\pi$$
$$824$$ 10.3262 0.359732
$$825$$ 0 0
$$826$$ 56.8328 1.97747
$$827$$ −51.4164 −1.78792 −0.893962 0.448143i $$-0.852086\pi$$
−0.893962 + 0.448143i $$0.852086\pi$$
$$828$$ 2.85410 0.0991869
$$829$$ −32.8885 −1.14227 −0.571133 0.820857i $$-0.693496\pi$$
−0.571133 + 0.820857i $$0.693496\pi$$
$$830$$ 0 0
$$831$$ −22.6525 −0.785806
$$832$$ 13.7082 0.475246
$$833$$ 40.8328 1.41477
$$834$$ −1.90983 −0.0661320
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −7.00000 −0.241955
$$838$$ 39.7984 1.37481
$$839$$ 36.3050 1.25339 0.626693 0.779266i $$-0.284408\pi$$
0.626693 + 0.779266i $$0.284408\pi$$
$$840$$ 0 0
$$841$$ −28.2705 −0.974845
$$842$$ 6.23607 0.214909
$$843$$ 1.09017 0.0375474
$$844$$ 1.76393 0.0607170
$$845$$ 0 0
$$846$$ 9.85410 0.338791
$$847$$ 0 0
$$848$$ 26.1246 0.897123
$$849$$ 16.0344 0.550301
$$850$$ 0 0
$$851$$ 6.79837 0.233045
$$852$$ 4.94427 0.169388
$$853$$ 23.3607 0.799854 0.399927 0.916547i $$-0.369035\pi$$
0.399927 + 0.916547i $$0.369035\pi$$
$$854$$ 59.3050 2.02937
$$855$$ 0 0
$$856$$ −8.61803 −0.294558
$$857$$ −41.0132 −1.40098 −0.700491 0.713661i $$-0.747036\pi$$
−0.700491 + 0.713661i $$0.747036\pi$$
$$858$$ 0 0
$$859$$ −39.2705 −1.33989 −0.669946 0.742410i $$-0.733683\pi$$
−0.669946 + 0.742410i $$0.733683\pi$$
$$860$$ 0 0
$$861$$ 60.0689 2.04714
$$862$$ 25.7082 0.875625
$$863$$ 23.0344 0.784102 0.392051 0.919944i $$-0.371766\pi$$
0.392051 + 0.919944i $$0.371766\pi$$
$$864$$ −3.38197 −0.115057
$$865$$ 0 0
$$866$$ 1.67376 0.0568768
$$867$$ 13.0000 0.441503
$$868$$ −22.6525 −0.768875
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −37.5967 −1.27392
$$872$$ 24.2705 0.821903
$$873$$ 2.52786 0.0855552
$$874$$ −37.3607 −1.26374
$$875$$ 0 0
$$876$$ 2.79837 0.0945483
$$877$$ 7.52786 0.254198 0.127099 0.991890i $$-0.459433\pi$$
0.127099 + 0.991890i $$0.459433\pi$$
$$878$$ −40.9787 −1.38296
$$879$$ −25.4721 −0.859154
$$880$$ 0 0
$$881$$ −17.1459 −0.577660 −0.288830 0.957380i $$-0.593266\pi$$
−0.288830 + 0.957380i $$0.593266\pi$$
$$882$$ −33.0344 −1.11233
$$883$$ 23.2361 0.781956 0.390978 0.920400i $$-0.372137\pi$$
0.390978 + 0.920400i $$0.372137\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 13.3820 0.449576
$$887$$ 0.167184 0.00561350 0.00280675 0.999996i $$-0.499107\pi$$
0.00280675 + 0.999996i $$0.499107\pi$$
$$888$$ −3.29180 −0.110465
$$889$$ 10.1803 0.341438
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −13.0000 −0.435272
$$893$$ −30.4508 −1.01900
$$894$$ −32.5623 −1.08905
$$895$$ 0 0
$$896$$ 71.3050 2.38213
$$897$$ −14.9443 −0.498975
$$898$$ 55.4508 1.85042
$$899$$ 5.97871 0.199401
$$900$$ 0 0
$$901$$ −10.7639 −0.358599
$$902$$ 0 0
$$903$$ 47.5967 1.58392
$$904$$ −7.76393 −0.258225
$$905$$ 0 0
$$906$$ −24.6525 −0.819024
$$907$$ 44.8885 1.49050 0.745250 0.666785i $$-0.232330\pi$$
0.745250 + 0.666785i $$0.232330\pi$$
$$908$$ 4.97871 0.165224
$$909$$ 19.5623 0.648841
$$910$$ 0 0
$$911$$ 52.9017 1.75271 0.876356 0.481664i $$-0.159968\pi$$
0.876356 + 0.481664i $$0.159968\pi$$
$$912$$ 24.2705 0.803677
$$913$$ 0 0
$$914$$ 12.0902 0.399907
$$915$$ 0 0
$$916$$ 4.27051 0.141102
$$917$$ 48.0689 1.58737
$$918$$ 3.23607 0.106806
$$919$$ −22.2361 −0.733500 −0.366750 0.930319i $$-0.619530\pi$$
−0.366750 + 0.930319i $$0.619530\pi$$
$$920$$ 0 0
$$921$$ 28.1246 0.926737
$$922$$ −13.0344 −0.429266
$$923$$ −25.8885 −0.852132
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −0.437694 −0.0143835
$$927$$ 4.61803 0.151676
$$928$$ 2.88854 0.0948211
$$929$$ −10.8541 −0.356112 −0.178056 0.984020i $$-0.556981\pi$$
−0.178056 + 0.984020i $$0.556981\pi$$
$$930$$ 0 0
$$931$$ 102.082 3.34560
$$932$$ 6.34752 0.207920
$$933$$ −12.5279 −0.410144
$$934$$ 49.9787 1.63535
$$935$$ 0 0
$$936$$ 7.23607 0.236518
$$937$$ 17.2016 0.561953 0.280976 0.959715i $$-0.409342\pi$$
0.280976 + 0.959715i $$0.409342\pi$$
$$938$$ −98.4296 −3.21384
$$939$$ 8.47214 0.276478
$$940$$ 0 0
$$941$$ −9.70820 −0.316478 −0.158239 0.987401i $$-0.550582\pi$$
−0.158239 + 0.987401i $$0.550582\pi$$
$$942$$ 19.9443 0.649819
$$943$$ 52.9787 1.72522
$$944$$ −32.5623 −1.05981
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.56231 0.148255 0.0741275 0.997249i $$-0.476383\pi$$
0.0741275 + 0.997249i $$0.476383\pi$$
$$948$$ −1.90983 −0.0620284
$$949$$ −14.6525 −0.475639
$$950$$ 0 0
$$951$$ −3.58359 −0.116206
$$952$$ −23.4164 −0.758930
$$953$$ 49.0902 1.59019 0.795093 0.606487i $$-0.207422\pi$$
0.795093 + 0.606487i $$0.207422\pi$$
$$954$$ 8.70820 0.281939
$$955$$ 0 0
$$956$$ 13.6180 0.440439
$$957$$ 0 0
$$958$$ −10.6525 −0.344166
$$959$$ −27.7082 −0.894745
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ −7.70820 −0.248522
$$963$$ −3.85410 −0.124197
$$964$$ 0.381966 0.0123023
$$965$$ 0 0
$$966$$ −39.1246 −1.25881
$$967$$ −32.6738 −1.05072 −0.525359 0.850881i $$-0.676069\pi$$
−0.525359 + 0.850881i $$0.676069\pi$$
$$968$$ 0 0
$$969$$ −10.0000 −0.321246
$$970$$ 0 0
$$971$$ −31.5410 −1.01220 −0.506100 0.862475i $$-0.668913\pi$$
−0.506100 + 0.862475i $$0.668913\pi$$
$$972$$ −0.618034 −0.0198234
$$973$$ 6.18034 0.198133
$$974$$ −13.2361 −0.424111
$$975$$ 0 0
$$976$$ −33.9787 −1.08763
$$977$$ −46.0902 −1.47456 −0.737278 0.675590i $$-0.763889\pi$$
−0.737278 + 0.675590i $$0.763889\pi$$
$$978$$ −10.6180 −0.339527
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.8541 0.346545
$$982$$ −11.6525 −0.371845
$$983$$ −19.0000 −0.606006 −0.303003 0.952990i $$-0.597989\pi$$
−0.303003 + 0.952990i $$0.597989\pi$$
$$984$$ −25.6525 −0.817771
$$985$$ 0 0
$$986$$ −2.76393 −0.0880215
$$987$$ −31.8885 −1.01502
$$988$$ 10.0000 0.318142
$$989$$ 41.9787 1.33485
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 23.6738 0.751643
$$993$$ 27.5967 0.875756
$$994$$ −67.7771 −2.14976
$$995$$ 0 0
$$996$$ −4.76393 −0.150951
$$997$$ −11.2918 −0.357615 −0.178807 0.983884i $$-0.557224\pi$$
−0.178807 + 0.983884i $$0.557224\pi$$
$$998$$ −18.0902 −0.572634
$$999$$ −1.47214 −0.0465763
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 9075.2.a.z.1.1 2
5.4 even 2 9075.2.a.by.1.2 2
11.3 even 5 825.2.n.b.526.1 4
11.4 even 5 825.2.n.b.676.1 yes 4
11.10 odd 2 9075.2.a.bt.1.2 2
55.3 odd 20 825.2.bx.c.724.2 8
55.4 even 10 825.2.n.d.676.1 yes 4
55.14 even 10 825.2.n.d.526.1 yes 4
55.37 odd 20 825.2.bx.c.49.2 8
55.47 odd 20 825.2.bx.c.724.1 8
55.48 odd 20 825.2.bx.c.49.1 8
55.54 odd 2 9075.2.a.bc.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.n.b.526.1 4 11.3 even 5
825.2.n.b.676.1 yes 4 11.4 even 5
825.2.n.d.526.1 yes 4 55.14 even 10
825.2.n.d.676.1 yes 4 55.4 even 10
825.2.bx.c.49.1 8 55.48 odd 20
825.2.bx.c.49.2 8 55.37 odd 20
825.2.bx.c.724.1 8 55.47 odd 20
825.2.bx.c.724.2 8 55.3 odd 20
9075.2.a.z.1.1 2 1.1 even 1 trivial
9075.2.a.bc.1.1 2 55.54 odd 2
9075.2.a.bt.1.2 2 11.10 odd 2
9075.2.a.by.1.2 2 5.4 even 2