# Properties

 Label 9075.2.a.z.1.2 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 1.2 Root $$-0.618034$$ 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)

 $$f(q)$$ $$=$$ $$q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -0.763932 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -0.763932 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.61803 q^{12} -1.23607 q^{13} -0.472136 q^{14} +1.85410 q^{16} +2.00000 q^{17} +0.618034 q^{18} +5.00000 q^{19} +0.763932 q^{21} +2.38197 q^{23} +2.23607 q^{24} -0.763932 q^{26} -1.00000 q^{27} +1.23607 q^{28} -5.85410 q^{29} +7.00000 q^{31} +5.61803 q^{32} +1.23607 q^{34} -1.61803 q^{36} -7.47214 q^{37} +3.09017 q^{38} +1.23607 q^{39} +2.52786 q^{41} +0.472136 q^{42} -2.09017 q^{43} +1.47214 q^{46} +5.09017 q^{47} -1.85410 q^{48} -6.41641 q^{49} -2.00000 q^{51} +2.00000 q^{52} -7.61803 q^{53} -0.618034 q^{54} +1.70820 q^{56} -5.00000 q^{57} -3.61803 q^{58} -6.70820 q^{59} +7.00000 q^{61} +4.32624 q^{62} -0.763932 q^{63} -0.236068 q^{64} -9.38197 q^{67} -3.23607 q^{68} -2.38197 q^{69} -8.00000 q^{71} -2.23607 q^{72} -13.4721 q^{73} -4.61803 q^{74} -8.09017 q^{76} +0.763932 q^{78} -8.09017 q^{79} +1.00000 q^{81} +1.56231 q^{82} -5.70820 q^{83} -1.23607 q^{84} -1.29180 q^{86} +5.85410 q^{87} +10.8541 q^{89} +0.944272 q^{91} -3.85410 q^{92} -7.00000 q^{93} +3.14590 q^{94} -5.61803 q^{96} +11.4721 q^{97} -3.96556 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.618034 0.437016 0.218508 0.975835i $$-0.429881\pi$$
0.218508 + 0.975835i $$0.429881\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.61803 −0.809017
$$5$$ 0 0
$$6$$ −0.618034 −0.252311
$$7$$ −0.763932 −0.288739 −0.144370 0.989524i $$-0.546115\pi$$
−0.144370 + 0.989524i $$0.546115\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 1.61803 0.467086
$$13$$ −1.23607 −0.342824 −0.171412 0.985199i $$-0.554833\pi$$
−0.171412 + 0.985199i $$0.554833\pi$$
$$14$$ −0.472136 −0.126184
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0.618034 0.145672
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ 0.763932 0.166704
$$22$$ 0 0
$$23$$ 2.38197 0.496674 0.248337 0.968674i $$-0.420116\pi$$
0.248337 + 0.968674i $$0.420116\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ −0.763932 −0.149819
$$27$$ −1.00000 −0.192450
$$28$$ 1.23607 0.233595
$$29$$ −5.85410 −1.08708 −0.543540 0.839383i $$-0.682916\pi$$
−0.543540 + 0.839383i $$0.682916\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 5.61803 0.993137
$$33$$ 0 0
$$34$$ 1.23607 0.211984
$$35$$ 0 0
$$36$$ −1.61803 −0.269672
$$37$$ −7.47214 −1.22841 −0.614206 0.789146i $$-0.710524\pi$$
−0.614206 + 0.789146i $$0.710524\pi$$
$$38$$ 3.09017 0.501292
$$39$$ 1.23607 0.197929
$$40$$ 0 0
$$41$$ 2.52786 0.394786 0.197393 0.980324i $$-0.436752\pi$$
0.197393 + 0.980324i $$0.436752\pi$$
$$42$$ 0.472136 0.0728522
$$43$$ −2.09017 −0.318748 −0.159374 0.987218i $$-0.550948\pi$$
−0.159374 + 0.987218i $$0.550948\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 1.47214 0.217055
$$47$$ 5.09017 0.742478 0.371239 0.928537i $$-0.378933\pi$$
0.371239 + 0.928537i $$0.378933\pi$$
$$48$$ −1.85410 −0.267617
$$49$$ −6.41641 −0.916630
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 2.00000 0.277350
$$53$$ −7.61803 −1.04642 −0.523209 0.852205i $$-0.675265\pi$$
−0.523209 + 0.852205i $$0.675265\pi$$
$$54$$ −0.618034 −0.0841038
$$55$$ 0 0
$$56$$ 1.70820 0.228268
$$57$$ −5.00000 −0.662266
$$58$$ −3.61803 −0.475071
$$59$$ −6.70820 −0.873334 −0.436667 0.899623i $$-0.643841\pi$$
−0.436667 + 0.899623i $$0.643841\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 4.32624 0.549433
$$63$$ −0.763932 −0.0962464
$$64$$ −0.236068 −0.0295085
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.38197 −1.14619 −0.573095 0.819489i $$-0.694257\pi$$
−0.573095 + 0.819489i $$0.694257\pi$$
$$68$$ −3.23607 −0.392431
$$69$$ −2.38197 −0.286755
$$70$$ 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
$$73$$ −13.4721 −1.57679 −0.788397 0.615167i $$-0.789089\pi$$
−0.788397 + 0.615167i $$0.789089\pi$$
$$74$$ −4.61803 −0.536836
$$75$$ 0 0
$$76$$ −8.09017 −0.928006
$$77$$ 0 0
$$78$$ 0.763932 0.0864983
$$79$$ −8.09017 −0.910215 −0.455108 0.890436i $$-0.650399\pi$$
−0.455108 + 0.890436i $$0.650399\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 1.56231 0.172528
$$83$$ −5.70820 −0.626557 −0.313278 0.949661i $$-0.601427\pi$$
−0.313278 + 0.949661i $$0.601427\pi$$
$$84$$ −1.23607 −0.134866
$$85$$ 0 0
$$86$$ −1.29180 −0.139298
$$87$$ 5.85410 0.627626
$$88$$ 0 0
$$89$$ 10.8541 1.15053 0.575266 0.817966i $$-0.304898\pi$$
0.575266 + 0.817966i $$0.304898\pi$$
$$90$$ 0 0
$$91$$ 0.944272 0.0989866
$$92$$ −3.85410 −0.401818
$$93$$ −7.00000 −0.725866
$$94$$ 3.14590 0.324475
$$95$$ 0 0
$$96$$ −5.61803 −0.573388
$$97$$ 11.4721 1.16482 0.582409 0.812896i $$-0.302110\pi$$
0.582409 + 0.812896i $$0.302110\pi$$
$$98$$ −3.96556 −0.400582
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −0.562306 −0.0559515 −0.0279758 0.999609i $$-0.508906\pi$$
−0.0279758 + 0.999609i $$0.508906\pi$$
$$102$$ −1.23607 −0.122389
$$103$$ 2.38197 0.234702 0.117351 0.993090i $$-0.462560\pi$$
0.117351 + 0.993090i $$0.462560\pi$$
$$104$$ 2.76393 0.271026
$$105$$ 0 0
$$106$$ −4.70820 −0.457301
$$107$$ 2.85410 0.275916 0.137958 0.990438i $$-0.455946\pi$$
0.137958 + 0.990438i $$0.455946\pi$$
$$108$$ 1.61803 0.155695
$$109$$ 4.14590 0.397105 0.198553 0.980090i $$-0.436376\pi$$
0.198553 + 0.980090i $$0.436376\pi$$
$$110$$ 0 0
$$111$$ 7.47214 0.709224
$$112$$ −1.41641 −0.133838
$$113$$ 5.47214 0.514775 0.257388 0.966308i $$-0.417138\pi$$
0.257388 + 0.966308i $$0.417138\pi$$
$$114$$ −3.09017 −0.289421
$$115$$ 0 0
$$116$$ 9.47214 0.879466
$$117$$ −1.23607 −0.114275
$$118$$ −4.14590 −0.381661
$$119$$ −1.52786 −0.140059
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 4.32624 0.391679
$$123$$ −2.52786 −0.227930
$$124$$ −11.3262 −1.01713
$$125$$ 0 0
$$126$$ −0.472136 −0.0420612
$$127$$ 15.9443 1.41483 0.707413 0.706801i $$-0.249862\pi$$
0.707413 + 0.706801i $$0.249862\pi$$
$$128$$ −11.3820 −1.00603
$$129$$ 2.09017 0.184029
$$130$$ 0 0
$$131$$ 13.1803 1.15157 0.575786 0.817601i $$-0.304696\pi$$
0.575786 + 0.817601i $$0.304696\pi$$
$$132$$ 0 0
$$133$$ −3.81966 −0.331207
$$134$$ −5.79837 −0.500903
$$135$$ 0 0
$$136$$ −4.47214 −0.383482
$$137$$ 18.7082 1.59835 0.799175 0.601099i $$-0.205270\pi$$
0.799175 + 0.601099i $$0.205270\pi$$
$$138$$ −1.47214 −0.125317
$$139$$ 21.1803 1.79649 0.898246 0.439492i $$-0.144842\pi$$
0.898246 + 0.439492i $$0.144842\pi$$
$$140$$ 0 0
$$141$$ −5.09017 −0.428670
$$142$$ −4.94427 −0.414914
$$143$$ 0 0
$$144$$ 1.85410 0.154508
$$145$$ 0 0
$$146$$ −8.32624 −0.689084
$$147$$ 6.41641 0.529216
$$148$$ 12.0902 0.993806
$$149$$ 20.1246 1.64867 0.824336 0.566101i $$-0.191549\pi$$
0.824336 + 0.566101i $$0.191549\pi$$
$$150$$ 0 0
$$151$$ −10.7639 −0.875956 −0.437978 0.898986i $$-0.644305\pi$$
−0.437978 + 0.898986i $$0.644305\pi$$
$$152$$ −11.1803 −0.906845
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ −3.32624 −0.265463 −0.132731 0.991152i $$-0.542375\pi$$
−0.132731 + 0.991152i $$0.542375\pi$$
$$158$$ −5.00000 −0.397779
$$159$$ 7.61803 0.604149
$$160$$ 0 0
$$161$$ −1.81966 −0.143409
$$162$$ 0.618034 0.0485573
$$163$$ 13.5623 1.06228 0.531141 0.847284i $$-0.321763\pi$$
0.531141 + 0.847284i $$0.321763\pi$$
$$164$$ −4.09017 −0.319389
$$165$$ 0 0
$$166$$ −3.52786 −0.273815
$$167$$ −20.0344 −1.55031 −0.775156 0.631770i $$-0.782329\pi$$
−0.775156 + 0.631770i $$0.782329\pi$$
$$168$$ −1.70820 −0.131791
$$169$$ −11.4721 −0.882472
$$170$$ 0 0
$$171$$ 5.00000 0.382360
$$172$$ 3.38197 0.257872
$$173$$ 3.56231 0.270837 0.135419 0.990788i $$-0.456762\pi$$
0.135419 + 0.990788i $$0.456762\pi$$
$$174$$ 3.61803 0.274282
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.70820 0.504219
$$178$$ 6.70820 0.502801
$$179$$ −19.4721 −1.45542 −0.727708 0.685887i $$-0.759414\pi$$
−0.727708 + 0.685887i $$0.759414\pi$$
$$180$$ 0 0
$$181$$ −5.23607 −0.389194 −0.194597 0.980883i $$-0.562340\pi$$
−0.194597 + 0.980883i $$0.562340\pi$$
$$182$$ 0.583592 0.0432587
$$183$$ −7.00000 −0.517455
$$184$$ −5.32624 −0.392655
$$185$$ 0 0
$$186$$ −4.32624 −0.317215
$$187$$ 0 0
$$188$$ −8.23607 −0.600677
$$189$$ 0.763932 0.0555679
$$190$$ 0 0
$$191$$ 10.4164 0.753705 0.376852 0.926273i $$-0.377006\pi$$
0.376852 + 0.926273i $$0.377006\pi$$
$$192$$ 0.236068 0.0170367
$$193$$ −21.0344 −1.51409 −0.757046 0.653361i $$-0.773358\pi$$
−0.757046 + 0.653361i $$0.773358\pi$$
$$194$$ 7.09017 0.509045
$$195$$ 0 0
$$196$$ 10.3820 0.741569
$$197$$ 14.2361 1.01428 0.507139 0.861864i $$-0.330703\pi$$
0.507139 + 0.861864i $$0.330703\pi$$
$$198$$ 0 0
$$199$$ 19.7984 1.40347 0.701735 0.712438i $$-0.252409\pi$$
0.701735 + 0.712438i $$0.252409\pi$$
$$200$$ 0 0
$$201$$ 9.38197 0.661753
$$202$$ −0.347524 −0.0244517
$$203$$ 4.47214 0.313882
$$204$$ 3.23607 0.226570
$$205$$ 0 0
$$206$$ 1.47214 0.102569
$$207$$ 2.38197 0.165558
$$208$$ −2.29180 −0.158907
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −3.85410 −0.265327 −0.132664 0.991161i $$-0.542353\pi$$
−0.132664 + 0.991161i $$0.542353\pi$$
$$212$$ 12.3262 0.846569
$$213$$ 8.00000 0.548151
$$214$$ 1.76393 0.120580
$$215$$ 0 0
$$216$$ 2.23607 0.152145
$$217$$ −5.34752 −0.363014
$$218$$ 2.56231 0.173541
$$219$$ 13.4721 0.910363
$$220$$ 0 0
$$221$$ −2.47214 −0.166294
$$222$$ 4.61803 0.309942
$$223$$ 8.03444 0.538026 0.269013 0.963137i $$-0.413303\pi$$
0.269013 + 0.963137i $$0.413303\pi$$
$$224$$ −4.29180 −0.286758
$$225$$ 0 0
$$226$$ 3.38197 0.224965
$$227$$ 25.9443 1.72198 0.860991 0.508620i $$-0.169844\pi$$
0.860991 + 0.508620i $$0.169844\pi$$
$$228$$ 8.09017 0.535785
$$229$$ 18.0902 1.19543 0.597716 0.801708i $$-0.296075\pi$$
0.597716 + 0.801708i $$0.296075\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 13.0902 0.859412
$$233$$ −23.2705 −1.52450 −0.762251 0.647282i $$-0.775906\pi$$
−0.762251 + 0.647282i $$0.775906\pi$$
$$234$$ −0.763932 −0.0499398
$$235$$ 0 0
$$236$$ 10.8541 0.706542
$$237$$ 8.09017 0.525513
$$238$$ −0.944272 −0.0612081
$$239$$ −7.03444 −0.455020 −0.227510 0.973776i $$-0.573058\pi$$
−0.227510 + 0.973776i $$0.573058\pi$$
$$240$$ 0 0
$$241$$ −1.61803 −0.104227 −0.0521134 0.998641i $$-0.516596\pi$$
−0.0521134 + 0.998641i $$0.516596\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ −11.3262 −0.725088
$$245$$ 0 0
$$246$$ −1.56231 −0.0996090
$$247$$ −6.18034 −0.393246
$$248$$ −15.6525 −0.993933
$$249$$ 5.70820 0.361743
$$250$$ 0 0
$$251$$ 7.52786 0.475155 0.237577 0.971369i $$-0.423647\pi$$
0.237577 + 0.971369i $$0.423647\pi$$
$$252$$ 1.23607 0.0778650
$$253$$ 0 0
$$254$$ 9.85410 0.618301
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ −10.5623 −0.658859 −0.329429 0.944180i $$-0.606856\pi$$
−0.329429 + 0.944180i $$0.606856\pi$$
$$258$$ 1.29180 0.0804237
$$259$$ 5.70820 0.354691
$$260$$ 0 0
$$261$$ −5.85410 −0.362360
$$262$$ 8.14590 0.503255
$$263$$ −27.2148 −1.67814 −0.839068 0.544027i $$-0.816899\pi$$
−0.839068 + 0.544027i $$0.816899\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.36068 −0.144743
$$267$$ −10.8541 −0.664260
$$268$$ 15.1803 0.927287
$$269$$ −5.85410 −0.356931 −0.178465 0.983946i $$-0.557113\pi$$
−0.178465 + 0.983946i $$0.557113\pi$$
$$270$$ 0 0
$$271$$ 20.6180 1.25246 0.626228 0.779640i $$-0.284598\pi$$
0.626228 + 0.779640i $$0.284598\pi$$
$$272$$ 3.70820 0.224843
$$273$$ −0.944272 −0.0571499
$$274$$ 11.5623 0.698504
$$275$$ 0 0
$$276$$ 3.85410 0.231990
$$277$$ −8.65248 −0.519877 −0.259938 0.965625i $$-0.583702\pi$$
−0.259938 + 0.965625i $$0.583702\pi$$
$$278$$ 13.0902 0.785096
$$279$$ 7.00000 0.419079
$$280$$ 0 0
$$281$$ 10.0902 0.601929 0.300965 0.953635i $$-0.402691\pi$$
0.300965 + 0.953635i $$0.402691\pi$$
$$282$$ −3.14590 −0.187336
$$283$$ 13.0344 0.774817 0.387409 0.921908i $$-0.373370\pi$$
0.387409 + 0.921908i $$0.373370\pi$$
$$284$$ 12.9443 0.768101
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1.93112 −0.113990
$$288$$ 5.61803 0.331046
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −11.4721 −0.672509
$$292$$ 21.7984 1.27565
$$293$$ 16.5279 0.965568 0.482784 0.875739i $$-0.339626\pi$$
0.482784 + 0.875739i $$0.339626\pi$$
$$294$$ 3.96556 0.231276
$$295$$ 0 0
$$296$$ 16.7082 0.971145
$$297$$ 0 0
$$298$$ 12.4377 0.720496
$$299$$ −2.94427 −0.170272
$$300$$ 0 0
$$301$$ 1.59675 0.0920350
$$302$$ −6.65248 −0.382807
$$303$$ 0.562306 0.0323036
$$304$$ 9.27051 0.531700
$$305$$ 0 0
$$306$$ 1.23607 0.0706613
$$307$$ 12.1246 0.691988 0.345994 0.938237i $$-0.387542\pi$$
0.345994 + 0.938237i $$0.387542\pi$$
$$308$$ 0 0
$$309$$ −2.38197 −0.135505
$$310$$ 0 0
$$311$$ 21.4721 1.21757 0.608787 0.793334i $$-0.291657\pi$$
0.608787 + 0.793334i $$0.291657\pi$$
$$312$$ −2.76393 −0.156477
$$313$$ 0.472136 0.0266867 0.0133434 0.999911i $$-0.495753\pi$$
0.0133434 + 0.999911i $$0.495753\pi$$
$$314$$ −2.05573 −0.116011
$$315$$ 0 0
$$316$$ 13.0902 0.736380
$$317$$ 30.4164 1.70836 0.854178 0.519981i $$-0.174061\pi$$
0.854178 + 0.519981i $$0.174061\pi$$
$$318$$ 4.70820 0.264023
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −2.85410 −0.159300
$$322$$ −1.12461 −0.0626722
$$323$$ 10.0000 0.556415
$$324$$ −1.61803 −0.0898908
$$325$$ 0 0
$$326$$ 8.38197 0.464234
$$327$$ −4.14590 −0.229269
$$328$$ −5.65248 −0.312106
$$329$$ −3.88854 −0.214382
$$330$$ 0 0
$$331$$ 21.5967 1.18706 0.593532 0.804810i $$-0.297733\pi$$
0.593532 + 0.804810i $$0.297733\pi$$
$$332$$ 9.23607 0.506895
$$333$$ −7.47214 −0.409471
$$334$$ −12.3820 −0.677511
$$335$$ 0 0
$$336$$ 1.41641 0.0772714
$$337$$ −22.1459 −1.20636 −0.603182 0.797604i $$-0.706101\pi$$
−0.603182 + 0.797604i $$0.706101\pi$$
$$338$$ −7.09017 −0.385654
$$339$$ −5.47214 −0.297206
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 3.09017 0.167097
$$343$$ 10.2492 0.553406
$$344$$ 4.67376 0.251992
$$345$$ 0 0
$$346$$ 2.20163 0.118360
$$347$$ 5.81966 0.312416 0.156208 0.987724i $$-0.450073\pi$$
0.156208 + 0.987724i $$0.450073\pi$$
$$348$$ −9.47214 −0.507760
$$349$$ 27.7639 1.48617 0.743085 0.669197i $$-0.233362\pi$$
0.743085 + 0.669197i $$0.233362\pi$$
$$350$$ 0 0
$$351$$ 1.23607 0.0659764
$$352$$ 0 0
$$353$$ −35.8328 −1.90719 −0.953594 0.301095i $$-0.902648\pi$$
−0.953594 + 0.301095i $$0.902648\pi$$
$$354$$ 4.14590 0.220352
$$355$$ 0 0
$$356$$ −17.5623 −0.930800
$$357$$ 1.52786 0.0808631
$$358$$ −12.0344 −0.636040
$$359$$ 16.7082 0.881825 0.440913 0.897550i $$-0.354655\pi$$
0.440913 + 0.897550i $$0.354655\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ −3.23607 −0.170084
$$363$$ 0 0
$$364$$ −1.52786 −0.0800818
$$365$$ 0 0
$$366$$ −4.32624 −0.226136
$$367$$ −18.1246 −0.946097 −0.473049 0.881036i $$-0.656846\pi$$
−0.473049 + 0.881036i $$0.656846\pi$$
$$368$$ 4.41641 0.230221
$$369$$ 2.52786 0.131595
$$370$$ 0 0
$$371$$ 5.81966 0.302142
$$372$$ 11.3262 0.587238
$$373$$ 9.74265 0.504455 0.252228 0.967668i $$-0.418837\pi$$
0.252228 + 0.967668i $$0.418837\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −11.3820 −0.586980
$$377$$ 7.23607 0.372676
$$378$$ 0.472136 0.0242841
$$379$$ −5.00000 −0.256833 −0.128416 0.991720i $$-0.540989\pi$$
−0.128416 + 0.991720i $$0.540989\pi$$
$$380$$ 0 0
$$381$$ −15.9443 −0.816850
$$382$$ 6.43769 0.329381
$$383$$ −31.3607 −1.60246 −0.801228 0.598359i $$-0.795820\pi$$
−0.801228 + 0.598359i $$0.795820\pi$$
$$384$$ 11.3820 0.580834
$$385$$ 0 0
$$386$$ −13.0000 −0.661683
$$387$$ −2.09017 −0.106249
$$388$$ −18.5623 −0.942358
$$389$$ −38.9443 −1.97455 −0.987276 0.159013i $$-0.949169\pi$$
−0.987276 + 0.159013i $$0.949169\pi$$
$$390$$ 0 0
$$391$$ 4.76393 0.240922
$$392$$ 14.3475 0.724659
$$393$$ −13.1803 −0.664860
$$394$$ 8.79837 0.443256
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 36.2705 1.82036 0.910182 0.414208i $$-0.135941\pi$$
0.910182 + 0.414208i $$0.135941\pi$$
$$398$$ 12.2361 0.613339
$$399$$ 3.81966 0.191222
$$400$$ 0 0
$$401$$ 4.88854 0.244122 0.122061 0.992523i $$-0.461050\pi$$
0.122061 + 0.992523i $$0.461050\pi$$
$$402$$ 5.79837 0.289197
$$403$$ −8.65248 −0.431011
$$404$$ 0.909830 0.0452657
$$405$$ 0 0
$$406$$ 2.76393 0.137172
$$407$$ 0 0
$$408$$ 4.47214 0.221404
$$409$$ 36.3050 1.79516 0.897582 0.440847i $$-0.145322\pi$$
0.897582 + 0.440847i $$0.145322\pi$$
$$410$$ 0 0
$$411$$ −18.7082 −0.922808
$$412$$ −3.85410 −0.189878
$$413$$ 5.12461 0.252166
$$414$$ 1.47214 0.0723515
$$415$$ 0 0
$$416$$ −6.94427 −0.340471
$$417$$ −21.1803 −1.03721
$$418$$ 0 0
$$419$$ 24.5967 1.20163 0.600815 0.799388i $$-0.294843\pi$$
0.600815 + 0.799388i $$0.294843\pi$$
$$420$$ 0 0
$$421$$ 2.85410 0.139100 0.0695502 0.997578i $$-0.477844\pi$$
0.0695502 + 0.997578i $$0.477844\pi$$
$$422$$ −2.38197 −0.115952
$$423$$ 5.09017 0.247493
$$424$$ 17.0344 0.827266
$$425$$ 0 0
$$426$$ 4.94427 0.239551
$$427$$ −5.34752 −0.258785
$$428$$ −4.61803 −0.223221
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 19.8885 0.957997 0.478999 0.877816i $$-0.341000\pi$$
0.478999 + 0.877816i $$0.341000\pi$$
$$432$$ −1.85410 −0.0892055
$$433$$ 28.0344 1.34725 0.673625 0.739074i $$-0.264736\pi$$
0.673625 + 0.739074i $$0.264736\pi$$
$$434$$ −3.30495 −0.158643
$$435$$ 0 0
$$436$$ −6.70820 −0.321265
$$437$$ 11.9098 0.569724
$$438$$ 8.32624 0.397843
$$439$$ 9.67376 0.461703 0.230852 0.972989i $$-0.425849\pi$$
0.230852 + 0.972989i $$0.425849\pi$$
$$440$$ 0 0
$$441$$ −6.41641 −0.305543
$$442$$ −1.52786 −0.0726731
$$443$$ 25.2705 1.20064 0.600319 0.799761i $$-0.295040\pi$$
0.600319 + 0.799761i $$0.295040\pi$$
$$444$$ −12.0902 −0.573774
$$445$$ 0 0
$$446$$ 4.96556 0.235126
$$447$$ −20.1246 −0.951861
$$448$$ 0.180340 0.00852026
$$449$$ −0.729490 −0.0344268 −0.0172134 0.999852i $$-0.505479\pi$$
−0.0172134 + 0.999852i $$0.505479\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −8.85410 −0.416462
$$453$$ 10.7639 0.505734
$$454$$ 16.0344 0.752534
$$455$$ 0 0
$$456$$ 11.1803 0.523567
$$457$$ 1.47214 0.0688636 0.0344318 0.999407i $$-0.489038\pi$$
0.0344318 + 0.999407i $$0.489038\pi$$
$$458$$ 11.1803 0.522423
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 25.9443 1.20835 0.604173 0.796853i $$-0.293504\pi$$
0.604173 + 0.796853i $$0.293504\pi$$
$$462$$ 0 0
$$463$$ −33.2705 −1.54621 −0.773106 0.634277i $$-0.781298\pi$$
−0.773106 + 0.634277i $$0.781298\pi$$
$$464$$ −10.8541 −0.503889
$$465$$ 0 0
$$466$$ −14.3820 −0.666232
$$467$$ 4.88854 0.226215 0.113107 0.993583i $$-0.463920\pi$$
0.113107 + 0.993583i $$0.463920\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 7.16718 0.330950
$$470$$ 0 0
$$471$$ 3.32624 0.153265
$$472$$ 15.0000 0.690431
$$473$$ 0 0
$$474$$ 5.00000 0.229658
$$475$$ 0 0
$$476$$ 2.47214 0.113310
$$477$$ −7.61803 −0.348806
$$478$$ −4.34752 −0.198851
$$479$$ 33.4164 1.52683 0.763417 0.645906i $$-0.223520\pi$$
0.763417 + 0.645906i $$0.223520\pi$$
$$480$$ 0 0
$$481$$ 9.23607 0.421128
$$482$$ −1.00000 −0.0455488
$$483$$ 1.81966 0.0827974
$$484$$ 0 0
$$485$$ 0 0
$$486$$ −0.618034 −0.0280346
$$487$$ −14.1803 −0.642573 −0.321286 0.946982i $$-0.604115\pi$$
−0.321286 + 0.946982i $$0.604115\pi$$
$$488$$ −15.6525 −0.708554
$$489$$ −13.5623 −0.613309
$$490$$ 0 0
$$491$$ 31.7984 1.43504 0.717520 0.696538i $$-0.245277\pi$$
0.717520 + 0.696538i $$0.245277\pi$$
$$492$$ 4.09017 0.184399
$$493$$ −11.7082 −0.527311
$$494$$ −3.81966 −0.171855
$$495$$ 0 0
$$496$$ 12.9787 0.582761
$$497$$ 6.11146 0.274136
$$498$$ 3.52786 0.158087
$$499$$ −11.1803 −0.500501 −0.250250 0.968181i $$-0.580513\pi$$
−0.250250 + 0.968181i $$0.580513\pi$$
$$500$$ 0 0
$$501$$ 20.0344 0.895073
$$502$$ 4.65248 0.207650
$$503$$ −21.8885 −0.975962 −0.487981 0.872854i $$-0.662266\pi$$
−0.487981 + 0.872854i $$0.662266\pi$$
$$504$$ 1.70820 0.0760895
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 11.4721 0.509495
$$508$$ −25.7984 −1.14462
$$509$$ 0.527864 0.0233972 0.0116986 0.999932i $$-0.496276\pi$$
0.0116986 + 0.999932i $$0.496276\pi$$
$$510$$ 0 0
$$511$$ 10.2918 0.455282
$$512$$ 18.7082 0.826794
$$513$$ −5.00000 −0.220755
$$514$$ −6.52786 −0.287932
$$515$$ 0 0
$$516$$ −3.38197 −0.148883
$$517$$ 0 0
$$518$$ 3.52786 0.155005
$$519$$ −3.56231 −0.156368
$$520$$ 0 0
$$521$$ −15.3607 −0.672964 −0.336482 0.941690i $$-0.609237\pi$$
−0.336482 + 0.941690i $$0.609237\pi$$
$$522$$ −3.61803 −0.158357
$$523$$ 8.23607 0.360138 0.180069 0.983654i $$-0.442368\pi$$
0.180069 + 0.983654i $$0.442368\pi$$
$$524$$ −21.3262 −0.931641
$$525$$ 0 0
$$526$$ −16.8197 −0.733372
$$527$$ 14.0000 0.609850
$$528$$ 0 0
$$529$$ −17.3262 −0.753315
$$530$$ 0 0
$$531$$ −6.70820 −0.291111
$$532$$ 6.18034 0.267952
$$533$$ −3.12461 −0.135342
$$534$$ −6.70820 −0.290292
$$535$$ 0 0
$$536$$ 20.9787 0.906142
$$537$$ 19.4721 0.840285
$$538$$ −3.61803 −0.155985
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0.618034 0.0265714 0.0132857 0.999912i $$-0.495771\pi$$
0.0132857 + 0.999912i $$0.495771\pi$$
$$542$$ 12.7426 0.547344
$$543$$ 5.23607 0.224701
$$544$$ 11.2361 0.481742
$$545$$ 0 0
$$546$$ −0.583592 −0.0249754
$$547$$ −25.3607 −1.08434 −0.542172 0.840267i $$-0.682398\pi$$
−0.542172 + 0.840267i $$0.682398\pi$$
$$548$$ −30.2705 −1.29309
$$549$$ 7.00000 0.298753
$$550$$ 0 0
$$551$$ −29.2705 −1.24697
$$552$$ 5.32624 0.226700
$$553$$ 6.18034 0.262815
$$554$$ −5.34752 −0.227195
$$555$$ 0 0
$$556$$ −34.2705 −1.45339
$$557$$ 31.3951 1.33025 0.665127 0.746730i $$-0.268377\pi$$
0.665127 + 0.746730i $$0.268377\pi$$
$$558$$ 4.32624 0.183144
$$559$$ 2.58359 0.109274
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.23607 0.263053
$$563$$ 36.3262 1.53097 0.765484 0.643455i $$-0.222500\pi$$
0.765484 + 0.643455i $$0.222500\pi$$
$$564$$ 8.23607 0.346801
$$565$$ 0 0
$$566$$ 8.05573 0.338608
$$567$$ −0.763932 −0.0320821
$$568$$ 17.8885 0.750587
$$569$$ −2.43769 −0.102193 −0.0510967 0.998694i $$-0.516272\pi$$
−0.0510967 + 0.998694i $$0.516272\pi$$
$$570$$ 0 0
$$571$$ 0.0901699 0.00377349 0.00188675 0.999998i $$-0.499399\pi$$
0.00188675 + 0.999998i $$0.499399\pi$$
$$572$$ 0 0
$$573$$ −10.4164 −0.435152
$$574$$ −1.19350 −0.0498155
$$575$$ 0 0
$$576$$ −0.236068 −0.00983617
$$577$$ −46.0132 −1.91555 −0.957776 0.287514i $$-0.907171\pi$$
−0.957776 + 0.287514i $$0.907171\pi$$
$$578$$ −8.03444 −0.334189
$$579$$ 21.0344 0.874162
$$580$$ 0 0
$$581$$ 4.36068 0.180911
$$582$$ −7.09017 −0.293897
$$583$$ 0 0
$$584$$ 30.1246 1.24657
$$585$$ 0 0
$$586$$ 10.2148 0.421969
$$587$$ −19.3820 −0.799979 −0.399990 0.916520i $$-0.630986\pi$$
−0.399990 + 0.916520i $$0.630986\pi$$
$$588$$ −10.3820 −0.428145
$$589$$ 35.0000 1.44215
$$590$$ 0 0
$$591$$ −14.2361 −0.585594
$$592$$ −13.8541 −0.569400
$$593$$ 38.8885 1.59696 0.798481 0.602021i $$-0.205638\pi$$
0.798481 + 0.602021i $$0.205638\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −32.5623 −1.33380
$$597$$ −19.7984 −0.810294
$$598$$ −1.81966 −0.0744114
$$599$$ −22.2361 −0.908541 −0.454271 0.890864i $$-0.650100\pi$$
−0.454271 + 0.890864i $$0.650100\pi$$
$$600$$ 0 0
$$601$$ 34.3607 1.40160 0.700801 0.713357i $$-0.252826\pi$$
0.700801 + 0.713357i $$0.252826\pi$$
$$602$$ 0.986844 0.0402208
$$603$$ −9.38197 −0.382063
$$604$$ 17.4164 0.708664
$$605$$ 0 0
$$606$$ 0.347524 0.0141172
$$607$$ −33.0000 −1.33943 −0.669714 0.742619i $$-0.733583\pi$$
−0.669714 + 0.742619i $$0.733583\pi$$
$$608$$ 28.0902 1.13921
$$609$$ −4.47214 −0.181220
$$610$$ 0 0
$$611$$ −6.29180 −0.254539
$$612$$ −3.23607 −0.130810
$$613$$ 8.23607 0.332652 0.166326 0.986071i $$-0.446810\pi$$
0.166326 + 0.986071i $$0.446810\pi$$
$$614$$ 7.49342 0.302410
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −38.2492 −1.53986 −0.769928 0.638131i $$-0.779708\pi$$
−0.769928 + 0.638131i $$0.779708\pi$$
$$618$$ −1.47214 −0.0592180
$$619$$ 7.11146 0.285834 0.142917 0.989735i $$-0.454352\pi$$
0.142917 + 0.989735i $$0.454352\pi$$
$$620$$ 0 0
$$621$$ −2.38197 −0.0955850
$$622$$ 13.2705 0.532099
$$623$$ −8.29180 −0.332204
$$624$$ 2.29180 0.0917453
$$625$$ 0 0
$$626$$ 0.291796 0.0116625
$$627$$ 0 0
$$628$$ 5.38197 0.214764
$$629$$ −14.9443 −0.595867
$$630$$ 0 0
$$631$$ 38.6312 1.53788 0.768942 0.639319i $$-0.220784\pi$$
0.768942 + 0.639319i $$0.220784\pi$$
$$632$$ 18.0902 0.719588
$$633$$ 3.85410 0.153187
$$634$$ 18.7984 0.746579
$$635$$ 0 0
$$636$$ −12.3262 −0.488767
$$637$$ 7.93112 0.314242
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −0.236068 −0.00932412 −0.00466206 0.999989i $$-0.501484\pi$$
−0.00466206 + 0.999989i $$0.501484\pi$$
$$642$$ −1.76393 −0.0696168
$$643$$ −29.3262 −1.15651 −0.578257 0.815855i $$-0.696267\pi$$
−0.578257 + 0.815855i $$0.696267\pi$$
$$644$$ 2.94427 0.116021
$$645$$ 0 0
$$646$$ 6.18034 0.243162
$$647$$ −17.5967 −0.691800 −0.345900 0.938271i $$-0.612426\pi$$
−0.345900 + 0.938271i $$0.612426\pi$$
$$648$$ −2.23607 −0.0878410
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 5.34752 0.209586
$$652$$ −21.9443 −0.859404
$$653$$ 23.4377 0.917188 0.458594 0.888646i $$-0.348353\pi$$
0.458594 + 0.888646i $$0.348353\pi$$
$$654$$ −2.56231 −0.100194
$$655$$ 0 0
$$656$$ 4.68692 0.182993
$$657$$ −13.4721 −0.525598
$$658$$ −2.40325 −0.0936885
$$659$$ 12.9656 0.505066 0.252533 0.967588i $$-0.418736\pi$$
0.252533 + 0.967588i $$0.418736\pi$$
$$660$$ 0 0
$$661$$ 3.90983 0.152075 0.0760374 0.997105i $$-0.475773\pi$$
0.0760374 + 0.997105i $$0.475773\pi$$
$$662$$ 13.3475 0.518766
$$663$$ 2.47214 0.0960098
$$664$$ 12.7639 0.495337
$$665$$ 0 0
$$666$$ −4.61803 −0.178945
$$667$$ −13.9443 −0.539924
$$668$$ 32.4164 1.25423
$$669$$ −8.03444 −0.310629
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 4.29180 0.165560
$$673$$ 27.3050 1.05253 0.526264 0.850321i $$-0.323592\pi$$
0.526264 + 0.850321i $$0.323592\pi$$
$$674$$ −13.6869 −0.527200
$$675$$ 0 0
$$676$$ 18.5623 0.713935
$$677$$ −11.9443 −0.459056 −0.229528 0.973302i $$-0.573718\pi$$
−0.229528 + 0.973302i $$0.573718\pi$$
$$678$$ −3.38197 −0.129884
$$679$$ −8.76393 −0.336329
$$680$$ 0 0
$$681$$ −25.9443 −0.994187
$$682$$ 0 0
$$683$$ −2.29180 −0.0876931 −0.0438466 0.999038i $$-0.513961\pi$$
−0.0438466 + 0.999038i $$0.513961\pi$$
$$684$$ −8.09017 −0.309335
$$685$$ 0 0
$$686$$ 6.33437 0.241847
$$687$$ −18.0902 −0.690183
$$688$$ −3.87539 −0.147748
$$689$$ 9.41641 0.358737
$$690$$ 0 0
$$691$$ 33.3050 1.26698 0.633490 0.773751i $$-0.281622\pi$$
0.633490 + 0.773751i $$0.281622\pi$$
$$692$$ −5.76393 −0.219112
$$693$$ 0 0
$$694$$ 3.59675 0.136531
$$695$$ 0 0
$$696$$ −13.0902 −0.496182
$$697$$ 5.05573 0.191499
$$698$$ 17.1591 0.649480
$$699$$ 23.2705 0.880172
$$700$$ 0 0
$$701$$ 13.5836 0.513045 0.256523 0.966538i $$-0.417423\pi$$
0.256523 + 0.966538i $$0.417423\pi$$
$$702$$ 0.763932 0.0288328
$$703$$ −37.3607 −1.40908
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −22.1459 −0.833472
$$707$$ 0.429563 0.0161554
$$708$$ −10.8541 −0.407922
$$709$$ 41.1803 1.54656 0.773280 0.634065i $$-0.218615\pi$$
0.773280 + 0.634065i $$0.218615\pi$$
$$710$$ 0 0
$$711$$ −8.09017 −0.303405
$$712$$ −24.2705 −0.909576
$$713$$ 16.6738 0.624437
$$714$$ 0.944272 0.0353385
$$715$$ 0 0
$$716$$ 31.5066 1.17746
$$717$$ 7.03444 0.262706
$$718$$ 10.3262 0.385372
$$719$$ 38.0902 1.42052 0.710262 0.703938i $$-0.248577\pi$$
0.710262 + 0.703938i $$0.248577\pi$$
$$720$$ 0 0
$$721$$ −1.81966 −0.0677677
$$722$$ 3.70820 0.138005
$$723$$ 1.61803 0.0601753
$$724$$ 8.47214 0.314864
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 37.3262 1.38435 0.692177 0.721728i $$-0.256652\pi$$
0.692177 + 0.721728i $$0.256652\pi$$
$$728$$ −2.11146 −0.0782558
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −4.18034 −0.154615
$$732$$ 11.3262 0.418630
$$733$$ 1.85410 0.0684828 0.0342414 0.999414i $$-0.489098\pi$$
0.0342414 + 0.999414i $$0.489098\pi$$
$$734$$ −11.2016 −0.413460
$$735$$ 0 0
$$736$$ 13.3820 0.493266
$$737$$ 0 0
$$738$$ 1.56231 0.0575093
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 6.18034 0.227040
$$742$$ 3.59675 0.132041
$$743$$ 0.347524 0.0127494 0.00637471 0.999980i $$-0.497971\pi$$
0.00637471 + 0.999980i $$0.497971\pi$$
$$744$$ 15.6525 0.573848
$$745$$ 0 0
$$746$$ 6.02129 0.220455
$$747$$ −5.70820 −0.208852
$$748$$ 0 0
$$749$$ −2.18034 −0.0796679
$$750$$ 0 0
$$751$$ −45.8885 −1.67450 −0.837248 0.546823i $$-0.815837\pi$$
−0.837248 + 0.546823i $$0.815837\pi$$
$$752$$ 9.43769 0.344157
$$753$$ −7.52786 −0.274331
$$754$$ 4.47214 0.162866
$$755$$ 0 0
$$756$$ −1.23607 −0.0449554
$$757$$ −24.1803 −0.878849 −0.439425 0.898279i $$-0.644818\pi$$
−0.439425 + 0.898279i $$0.644818\pi$$
$$758$$ −3.09017 −0.112240
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 39.0344 1.41500 0.707499 0.706715i $$-0.249824\pi$$
0.707499 + 0.706715i $$0.249824\pi$$
$$762$$ −9.85410 −0.356976
$$763$$ −3.16718 −0.114660
$$764$$ −16.8541 −0.609760
$$765$$ 0 0
$$766$$ −19.3820 −0.700299
$$767$$ 8.29180 0.299399
$$768$$ 6.56231 0.236797
$$769$$ 12.7639 0.460279 0.230140 0.973158i $$-0.426082\pi$$
0.230140 + 0.973158i $$0.426082\pi$$
$$770$$ 0 0
$$771$$ 10.5623 0.380392
$$772$$ 34.0344 1.22493
$$773$$ −10.5066 −0.377895 −0.188948 0.981987i $$-0.560508\pi$$
−0.188948 + 0.981987i $$0.560508\pi$$
$$774$$ −1.29180 −0.0464327
$$775$$ 0 0
$$776$$ −25.6525 −0.920870
$$777$$ −5.70820 −0.204781
$$778$$ −24.0689 −0.862911
$$779$$ 12.6393 0.452851
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 2.94427 0.105287
$$783$$ 5.85410 0.209209
$$784$$ −11.8967 −0.424881
$$785$$ 0 0
$$786$$ −8.14590 −0.290555
$$787$$ −8.20163 −0.292356 −0.146178 0.989258i $$-0.546697\pi$$
−0.146178 + 0.989258i $$0.546697\pi$$
$$788$$ −23.0344 −0.820568
$$789$$ 27.2148 0.968872
$$790$$ 0 0
$$791$$ −4.18034 −0.148636
$$792$$ 0 0
$$793$$ −8.65248 −0.307258
$$794$$ 22.4164 0.795529
$$795$$ 0 0
$$796$$ −32.0344 −1.13543
$$797$$ 28.7082 1.01690 0.508448 0.861092i $$-0.330219\pi$$
0.508448 + 0.861092i $$0.330219\pi$$
$$798$$ 2.36068 0.0835672
$$799$$ 10.1803 0.360155
$$800$$ 0 0
$$801$$ 10.8541 0.383511
$$802$$ 3.02129 0.106685
$$803$$ 0 0
$$804$$ −15.1803 −0.535369
$$805$$ 0 0
$$806$$ −5.34752 −0.188359
$$807$$ 5.85410 0.206074
$$808$$ 1.25735 0.0442336
$$809$$ 45.3262 1.59359 0.796793 0.604253i $$-0.206528\pi$$
0.796793 + 0.604253i $$0.206528\pi$$
$$810$$ 0 0
$$811$$ −34.6312 −1.21607 −0.608033 0.793912i $$-0.708041\pi$$
−0.608033 + 0.793912i $$0.708041\pi$$
$$812$$ −7.23607 −0.253936
$$813$$ −20.6180 −0.723106
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −3.70820 −0.129813
$$817$$ −10.4508 −0.365629
$$818$$ 22.4377 0.784516
$$819$$ 0.944272 0.0329955
$$820$$ 0 0
$$821$$ 43.1803 1.50700 0.753502 0.657445i $$-0.228363\pi$$
0.753502 + 0.657445i $$0.228363\pi$$
$$822$$ −11.5623 −0.403282
$$823$$ 25.4721 0.887903 0.443951 0.896051i $$-0.353576\pi$$
0.443951 + 0.896051i $$0.353576\pi$$
$$824$$ −5.32624 −0.185548
$$825$$ 0 0
$$826$$ 3.16718 0.110200
$$827$$ −24.5836 −0.854855 −0.427428 0.904050i $$-0.640580\pi$$
−0.427428 + 0.904050i $$0.640580\pi$$
$$828$$ −3.85410 −0.133939
$$829$$ 2.88854 0.100323 0.0501616 0.998741i $$-0.484026\pi$$
0.0501616 + 0.998741i $$0.484026\pi$$
$$830$$ 0 0
$$831$$ 8.65248 0.300151
$$832$$ 0.291796 0.0101162
$$833$$ −12.8328 −0.444631
$$834$$ −13.0902 −0.453276
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −7.00000 −0.241955
$$838$$ 15.2016 0.525131
$$839$$ −26.3050 −0.908148 −0.454074 0.890964i $$-0.650030\pi$$
−0.454074 + 0.890964i $$0.650030\pi$$
$$840$$ 0 0
$$841$$ 5.27051 0.181742
$$842$$ 1.76393 0.0607891
$$843$$ −10.0902 −0.347524
$$844$$ 6.23607 0.214654
$$845$$ 0 0
$$846$$ 3.14590 0.108158
$$847$$ 0 0
$$848$$ −14.1246 −0.485041
$$849$$ −13.0344 −0.447341
$$850$$ 0 0
$$851$$ −17.7984 −0.610120
$$852$$ −12.9443 −0.443463
$$853$$ −21.3607 −0.731376 −0.365688 0.930738i $$-0.619166\pi$$
−0.365688 + 0.930738i $$0.619166\pi$$
$$854$$ −3.30495 −0.113093
$$855$$ 0 0
$$856$$ −6.38197 −0.218131
$$857$$ 35.0132 1.19603 0.598013 0.801486i $$-0.295957\pi$$
0.598013 + 0.801486i $$0.295957\pi$$
$$858$$ 0 0
$$859$$ −5.72949 −0.195488 −0.0977438 0.995212i $$-0.531163\pi$$
−0.0977438 + 0.995212i $$0.531163\pi$$
$$860$$ 0 0
$$861$$ 1.93112 0.0658123
$$862$$ 12.2918 0.418660
$$863$$ −6.03444 −0.205415 −0.102707 0.994712i $$-0.532751\pi$$
−0.102707 + 0.994712i $$0.532751\pi$$
$$864$$ −5.61803 −0.191129
$$865$$ 0 0
$$866$$ 17.3262 0.588770
$$867$$ 13.0000 0.441503
$$868$$ 8.65248 0.293684
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 11.5967 0.392941
$$872$$ −9.27051 −0.313939
$$873$$ 11.4721 0.388273
$$874$$ 7.36068 0.248979
$$875$$ 0 0
$$876$$ −21.7984 −0.736499
$$877$$ 16.4721 0.556225 0.278112 0.960549i $$-0.410291\pi$$
0.278112 + 0.960549i $$0.410291\pi$$
$$878$$ 5.97871 0.201772
$$879$$ −16.5279 −0.557471
$$880$$ 0 0
$$881$$ −23.8541 −0.803665 −0.401833 0.915713i $$-0.631627\pi$$
−0.401833 + 0.915713i $$0.631627\pi$$
$$882$$ −3.96556 −0.133527
$$883$$ 18.7639 0.631457 0.315728 0.948850i $$-0.397751\pi$$
0.315728 + 0.948850i $$0.397751\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 15.6180 0.524698
$$887$$ 53.8328 1.80753 0.903765 0.428030i $$-0.140792\pi$$
0.903765 + 0.428030i $$0.140792\pi$$
$$888$$ −16.7082 −0.560691
$$889$$ −12.1803 −0.408515
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −13.0000 −0.435272
$$893$$ 25.4508 0.851680
$$894$$ −12.4377 −0.415979
$$895$$ 0 0
$$896$$ 8.69505 0.290481
$$897$$ 2.94427 0.0983064
$$898$$ −0.450850 −0.0150450
$$899$$ −40.9787 −1.36672
$$900$$ 0 0
$$901$$ −15.2361 −0.507587
$$902$$ 0 0
$$903$$ −1.59675 −0.0531364
$$904$$ −12.2361 −0.406966
$$905$$ 0 0
$$906$$ 6.65248 0.221014
$$907$$ 9.11146 0.302541 0.151270 0.988492i $$-0.451664\pi$$
0.151270 + 0.988492i $$0.451664\pi$$
$$908$$ −41.9787 −1.39311
$$909$$ −0.562306 −0.0186505
$$910$$ 0 0
$$911$$ −58.9017 −1.95150 −0.975750 0.218887i $$-0.929757\pi$$
−0.975750 + 0.218887i $$0.929757\pi$$
$$912$$ −9.27051 −0.306977
$$913$$ 0 0
$$914$$ 0.909830 0.0300945
$$915$$ 0 0
$$916$$ −29.2705 −0.967125
$$917$$ −10.0689 −0.332504
$$918$$ −1.23607 −0.0407963
$$919$$ −17.7639 −0.585978 −0.292989 0.956116i $$-0.594650\pi$$
−0.292989 + 0.956116i $$0.594650\pi$$
$$920$$ 0 0
$$921$$ −12.1246 −0.399520
$$922$$ 16.0344 0.528066
$$923$$ 9.88854 0.325485
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −20.5623 −0.675719
$$927$$ 2.38197 0.0782340
$$928$$ −32.8885 −1.07962
$$929$$ −4.14590 −0.136023 −0.0680113 0.997685i $$-0.521665\pi$$
−0.0680113 + 0.997685i $$0.521665\pi$$
$$930$$ 0 0
$$931$$ −32.0820 −1.05145
$$932$$ 37.6525 1.23335
$$933$$ −21.4721 −0.702966
$$934$$ 3.02129 0.0988595
$$935$$ 0 0
$$936$$ 2.76393 0.0903419
$$937$$ 41.7984 1.36549 0.682747 0.730655i $$-0.260785\pi$$
0.682747 + 0.730655i $$0.260785\pi$$
$$938$$ 4.42956 0.144630
$$939$$ −0.472136 −0.0154076
$$940$$ 0 0
$$941$$ 3.70820 0.120884 0.0604420 0.998172i $$-0.480749\pi$$
0.0604420 + 0.998172i $$0.480749\pi$$
$$942$$ 2.05573 0.0669792
$$943$$ 6.02129 0.196080
$$944$$ −12.4377 −0.404812
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −15.5623 −0.505707 −0.252853 0.967505i $$-0.581369\pi$$
−0.252853 + 0.967505i $$0.581369\pi$$
$$948$$ −13.0902 −0.425149
$$949$$ 16.6525 0.540562
$$950$$ 0 0
$$951$$ −30.4164 −0.986320
$$952$$ 3.41641 0.110726
$$953$$ 37.9098 1.22802 0.614010 0.789298i $$-0.289555\pi$$
0.614010 + 0.789298i $$0.289555\pi$$
$$954$$ −4.70820 −0.152434
$$955$$ 0 0
$$956$$ 11.3820 0.368119
$$957$$ 0 0
$$958$$ 20.6525 0.667251
$$959$$ −14.2918 −0.461506
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 5.70820 0.184040
$$963$$ 2.85410 0.0919722
$$964$$ 2.61803 0.0843212
$$965$$ 0 0
$$966$$ 1.12461 0.0361838
$$967$$ −48.3262 −1.55407 −0.777034 0.629459i $$-0.783276\pi$$
−0.777034 + 0.629459i $$0.783276\pi$$
$$968$$ 0 0
$$969$$ −10.0000 −0.321246
$$970$$ 0 0
$$971$$ 35.5410 1.14057 0.570283 0.821448i $$-0.306834\pi$$
0.570283 + 0.821448i $$0.306834\pi$$
$$972$$ 1.61803 0.0518985
$$973$$ −16.1803 −0.518718
$$974$$ −8.76393 −0.280814
$$975$$ 0 0
$$976$$ 12.9787 0.415439
$$977$$ −34.9098 −1.11686 −0.558432 0.829550i $$-0.688597\pi$$
−0.558432 + 0.829550i $$0.688597\pi$$
$$978$$ −8.38197 −0.268026
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 4.14590 0.132368
$$982$$ 19.6525 0.627136
$$983$$ −19.0000 −0.606006 −0.303003 0.952990i $$-0.597989\pi$$
−0.303003 + 0.952990i $$0.597989\pi$$
$$984$$ 5.65248 0.180194
$$985$$ 0 0
$$986$$ −7.23607 −0.230443
$$987$$ 3.88854 0.123774
$$988$$ 10.0000 0.318142
$$989$$ −4.97871 −0.158314
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 39.3262 1.24861
$$993$$ −21.5967 −0.685352
$$994$$ 3.77709 0.119802
$$995$$ 0 0
$$996$$ −9.23607 −0.292656
$$997$$ −24.7082 −0.782517 −0.391258 0.920281i $$-0.627960\pi$$
−0.391258 + 0.920281i $$0.627960\pi$$
$$998$$ −6.90983 −0.218727
$$999$$ 7.47214 0.236408
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.1.2 2
5.4 even 2 9075.2.a.by.1.1 2
11.5 even 5 825.2.n.b.751.1 yes 4
11.9 even 5 825.2.n.b.301.1 4
11.10 odd 2 9075.2.a.bt.1.1 2
55.9 even 10 825.2.n.d.301.1 yes 4
55.27 odd 20 825.2.bx.c.124.1 8
55.38 odd 20 825.2.bx.c.124.2 8
55.42 odd 20 825.2.bx.c.499.2 8
55.49 even 10 825.2.n.d.751.1 yes 4
55.53 odd 20 825.2.bx.c.499.1 8
55.54 odd 2 9075.2.a.bc.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.n.b.301.1 4 11.9 even 5
825.2.n.b.751.1 yes 4 11.5 even 5
825.2.n.d.301.1 yes 4 55.9 even 10
825.2.n.d.751.1 yes 4 55.49 even 10
825.2.bx.c.124.1 8 55.27 odd 20
825.2.bx.c.124.2 8 55.38 odd 20
825.2.bx.c.499.1 8 55.53 odd 20
825.2.bx.c.499.2 8 55.42 odd 20
9075.2.a.z.1.2 2 1.1 even 1 trivial
9075.2.a.bc.1.2 2 55.54 odd 2
9075.2.a.bt.1.1 2 11.10 odd 2
9075.2.a.by.1.1 2 5.4 even 2