# Properties

 Label 6075.2.a.bq.1.3 Level $6075$ Weight $2$ Character 6075.1 Self dual yes Analytic conductor $48.509$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6075,2,Mod(1,6075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6075, 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("6075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6075 = 3^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6075.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: $$48.5091192279$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 243) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.87939$$ of defining polynomial Character $$\chi$$ $$=$$ 6075.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.879385 q^{2} -1.22668 q^{4} +2.18479 q^{7} -2.83750 q^{8} +O(q^{10})$$ $$q+0.879385 q^{2} -1.22668 q^{4} +2.18479 q^{7} -2.83750 q^{8} +0.162504 q^{11} -2.41147 q^{13} +1.92127 q^{14} -0.0418891 q^{16} -3.00000 q^{17} +3.59627 q^{19} +0.142903 q^{22} -2.83750 q^{23} -2.12061 q^{26} -2.68004 q^{28} +6.71688 q^{29} -5.18479 q^{31} +5.63816 q^{32} -2.63816 q^{34} +6.63816 q^{37} +3.16250 q^{38} -5.80066 q^{41} +6.22668 q^{43} -0.199340 q^{44} -2.49525 q^{46} -7.39693 q^{47} -2.22668 q^{49} +2.95811 q^{52} -1.40373 q^{53} -6.19934 q^{56} +5.90673 q^{58} -5.12061 q^{59} -3.78106 q^{61} -4.55943 q^{62} +5.04189 q^{64} +5.86484 q^{67} +3.68004 q^{68} -15.3182 q^{71} -8.68004 q^{73} +5.83750 q^{74} -4.41147 q^{76} +0.355037 q^{77} -1.26857 q^{79} -5.10101 q^{82} +8.47565 q^{83} +5.47565 q^{86} -0.461104 q^{88} +7.72193 q^{89} -5.26857 q^{91} +3.48070 q^{92} -6.50475 q^{94} +3.90673 q^{97} -1.95811 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} + 3 q^{7} - 6 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^7 - 6 * q^8 $$3 q - 3 q^{2} + 3 q^{4} + 3 q^{7} - 6 q^{8} + 3 q^{11} + 3 q^{13} - 3 q^{14} + 3 q^{16} - 9 q^{17} - 3 q^{19} - 6 q^{23} - 12 q^{26} + 12 q^{28} + 12 q^{29} - 12 q^{31} + 9 q^{34} + 3 q^{37} + 12 q^{38} - 3 q^{41} + 12 q^{43} - 15 q^{44} + 9 q^{46} + 6 q^{47} + 12 q^{52} - 18 q^{53} - 33 q^{56} - 9 q^{58} - 21 q^{59} + 6 q^{61} + 12 q^{62} + 12 q^{64} - 6 q^{67} - 9 q^{68} - 9 q^{71} - 6 q^{73} + 15 q^{74} - 3 q^{76} - 24 q^{77} + 6 q^{79} - 18 q^{82} + 6 q^{83} - 3 q^{86} + 36 q^{88} - 6 q^{91} - 24 q^{92} - 36 q^{94} - 15 q^{97} - 9 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^7 - 6 * q^8 + 3 * q^11 + 3 * q^13 - 3 * q^14 + 3 * q^16 - 9 * q^17 - 3 * q^19 - 6 * q^23 - 12 * q^26 + 12 * q^28 + 12 * q^29 - 12 * q^31 + 9 * q^34 + 3 * q^37 + 12 * q^38 - 3 * q^41 + 12 * q^43 - 15 * q^44 + 9 * q^46 + 6 * q^47 + 12 * q^52 - 18 * q^53 - 33 * q^56 - 9 * q^58 - 21 * q^59 + 6 * q^61 + 12 * q^62 + 12 * q^64 - 6 * q^67 - 9 * q^68 - 9 * q^71 - 6 * q^73 + 15 * q^74 - 3 * q^76 - 24 * q^77 + 6 * q^79 - 18 * q^82 + 6 * q^83 - 3 * q^86 + 36 * q^88 - 6 * q^91 - 24 * q^92 - 36 * q^94 - 15 * q^97 - 9 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.879385 0.621819 0.310910 0.950439i $$-0.399366\pi$$
0.310910 + 0.950439i $$0.399366\pi$$
$$3$$ 0 0
$$4$$ −1.22668 −0.613341
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.18479 0.825774 0.412887 0.910782i $$-0.364520\pi$$
0.412887 + 0.910782i $$0.364520\pi$$
$$8$$ −2.83750 −1.00321
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0.162504 0.0489967 0.0244984 0.999700i $$-0.492201\pi$$
0.0244984 + 0.999700i $$0.492201\pi$$
$$12$$ 0 0
$$13$$ −2.41147 −0.668823 −0.334411 0.942427i $$-0.608537\pi$$
−0.334411 + 0.942427i $$0.608537\pi$$
$$14$$ 1.92127 0.513482
$$15$$ 0 0
$$16$$ −0.0418891 −0.0104723
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 3.59627 0.825040 0.412520 0.910949i $$-0.364649\pi$$
0.412520 + 0.910949i $$0.364649\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0.142903 0.0304671
$$23$$ −2.83750 −0.591659 −0.295829 0.955241i $$-0.595596\pi$$
−0.295829 + 0.955241i $$0.595596\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.12061 −0.415887
$$27$$ 0 0
$$28$$ −2.68004 −0.506481
$$29$$ 6.71688 1.24729 0.623647 0.781706i $$-0.285650\pi$$
0.623647 + 0.781706i $$0.285650\pi$$
$$30$$ 0 0
$$31$$ −5.18479 −0.931216 −0.465608 0.884991i $$-0.654164\pi$$
−0.465608 + 0.884991i $$0.654164\pi$$
$$32$$ 5.63816 0.996695
$$33$$ 0 0
$$34$$ −2.63816 −0.452440
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.63816 1.09131 0.545653 0.838011i $$-0.316282\pi$$
0.545653 + 0.838011i $$0.316282\pi$$
$$38$$ 3.16250 0.513026
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.80066 −0.905911 −0.452955 0.891533i $$-0.649630\pi$$
−0.452955 + 0.891533i $$0.649630\pi$$
$$42$$ 0 0
$$43$$ 6.22668 0.949560 0.474780 0.880105i $$-0.342528\pi$$
0.474780 + 0.880105i $$0.342528\pi$$
$$44$$ −0.199340 −0.0300517
$$45$$ 0 0
$$46$$ −2.49525 −0.367905
$$47$$ −7.39693 −1.07895 −0.539476 0.842001i $$-0.681378\pi$$
−0.539476 + 0.842001i $$0.681378\pi$$
$$48$$ 0 0
$$49$$ −2.22668 −0.318097
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.95811 0.410216
$$53$$ −1.40373 −0.192818 −0.0964088 0.995342i $$-0.530736\pi$$
−0.0964088 + 0.995342i $$0.530736\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −6.19934 −0.828422
$$57$$ 0 0
$$58$$ 5.90673 0.775591
$$59$$ −5.12061 −0.666647 −0.333324 0.942812i $$-0.608170\pi$$
−0.333324 + 0.942812i $$0.608170\pi$$
$$60$$ 0 0
$$61$$ −3.78106 −0.484115 −0.242058 0.970262i $$-0.577822\pi$$
−0.242058 + 0.970262i $$0.577822\pi$$
$$62$$ −4.55943 −0.579048
$$63$$ 0 0
$$64$$ 5.04189 0.630236
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.86484 0.716504 0.358252 0.933625i $$-0.383373\pi$$
0.358252 + 0.933625i $$0.383373\pi$$
$$68$$ 3.68004 0.446271
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −15.3182 −1.81794 −0.908968 0.416866i $$-0.863128\pi$$
−0.908968 + 0.416866i $$0.863128\pi$$
$$72$$ 0 0
$$73$$ −8.68004 −1.01592 −0.507961 0.861380i $$-0.669601\pi$$
−0.507961 + 0.861380i $$0.669601\pi$$
$$74$$ 5.83750 0.678595
$$75$$ 0 0
$$76$$ −4.41147 −0.506031
$$77$$ 0.355037 0.0404602
$$78$$ 0 0
$$79$$ −1.26857 −0.142725 −0.0713627 0.997450i $$-0.522735\pi$$
−0.0713627 + 0.997450i $$0.522735\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −5.10101 −0.563313
$$83$$ 8.47565 0.930324 0.465162 0.885226i $$-0.345996\pi$$
0.465162 + 0.885226i $$0.345996\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 5.47565 0.590455
$$87$$ 0 0
$$88$$ −0.461104 −0.0491538
$$89$$ 7.72193 0.818523 0.409262 0.912417i $$-0.365786\pi$$
0.409262 + 0.912417i $$0.365786\pi$$
$$90$$ 0 0
$$91$$ −5.26857 −0.552296
$$92$$ 3.48070 0.362889
$$93$$ 0 0
$$94$$ −6.50475 −0.670914
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 3.90673 0.396668 0.198334 0.980134i $$-0.436447\pi$$
0.198334 + 0.980134i $$0.436447\pi$$
$$98$$ −1.95811 −0.197799
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −8.11381 −0.807354 −0.403677 0.914902i $$-0.632268\pi$$
−0.403677 + 0.914902i $$0.632268\pi$$
$$102$$ 0 0
$$103$$ −18.6459 −1.83723 −0.918617 0.395148i $$-0.870693\pi$$
−0.918617 + 0.395148i $$0.870693\pi$$
$$104$$ 6.84255 0.670967
$$105$$ 0 0
$$106$$ −1.23442 −0.119898
$$107$$ −7.59627 −0.734359 −0.367179 0.930150i $$-0.619676\pi$$
−0.367179 + 0.930150i $$0.619676\pi$$
$$108$$ 0 0
$$109$$ −15.6382 −1.49786 −0.748932 0.662647i $$-0.769433\pi$$
−0.748932 + 0.662647i $$0.769433\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −0.0915189 −0.00864772
$$113$$ 2.31315 0.217603 0.108801 0.994064i $$-0.465299\pi$$
0.108801 + 0.994064i $$0.465299\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −8.23947 −0.765016
$$117$$ 0 0
$$118$$ −4.50299 −0.414534
$$119$$ −6.55438 −0.600839
$$120$$ 0 0
$$121$$ −10.9736 −0.997599
$$122$$ −3.32501 −0.301032
$$123$$ 0 0
$$124$$ 6.36009 0.571153
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −0.0418891 −0.00371705 −0.00185853 0.999998i $$-0.500592\pi$$
−0.00185853 + 0.999998i $$0.500592\pi$$
$$128$$ −6.84255 −0.604802
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 18.3482 1.60309 0.801546 0.597933i $$-0.204011\pi$$
0.801546 + 0.597933i $$0.204011\pi$$
$$132$$ 0 0
$$133$$ 7.85710 0.681297
$$134$$ 5.15745 0.445536
$$135$$ 0 0
$$136$$ 8.51249 0.729940
$$137$$ −14.3131 −1.22285 −0.611427 0.791301i $$-0.709404\pi$$
−0.611427 + 0.791301i $$0.709404\pi$$
$$138$$ 0 0
$$139$$ 10.4953 0.890196 0.445098 0.895482i $$-0.353169\pi$$
0.445098 + 0.895482i $$0.353169\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −13.4706 −1.13043
$$143$$ −0.391874 −0.0327701
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −7.63310 −0.631720
$$147$$ 0 0
$$148$$ −8.14290 −0.669343
$$149$$ −1.27126 −0.104146 −0.0520728 0.998643i $$-0.516583\pi$$
−0.0520728 + 0.998643i $$0.516583\pi$$
$$150$$ 0 0
$$151$$ 7.85710 0.639401 0.319701 0.947519i $$-0.396418\pi$$
0.319701 + 0.947519i $$0.396418\pi$$
$$152$$ −10.2044 −0.827686
$$153$$ 0 0
$$154$$ 0.312214 0.0251590
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 12.3523 0.985825 0.492912 0.870079i $$-0.335932\pi$$
0.492912 + 0.870079i $$0.335932\pi$$
$$158$$ −1.11556 −0.0887494
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −6.19934 −0.488576
$$162$$ 0 0
$$163$$ 13.7469 1.07674 0.538371 0.842708i $$-0.319040\pi$$
0.538371 + 0.842708i $$0.319040\pi$$
$$164$$ 7.11556 0.555632
$$165$$ 0 0
$$166$$ 7.45336 0.578493
$$167$$ 3.71688 0.287621 0.143810 0.989605i $$-0.454064\pi$$
0.143810 + 0.989605i $$0.454064\pi$$
$$168$$ 0 0
$$169$$ −7.18479 −0.552676
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −7.63816 −0.582404
$$173$$ 1.55943 0.118561 0.0592806 0.998241i $$-0.481119\pi$$
0.0592806 + 0.998241i $$0.481119\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −0.00680713 −0.000513107 0
$$177$$ 0 0
$$178$$ 6.79055 0.508974
$$179$$ −12.1925 −0.911313 −0.455656 0.890156i $$-0.650595\pi$$
−0.455656 + 0.890156i $$0.650595\pi$$
$$180$$ 0 0
$$181$$ −16.8726 −1.25413 −0.627064 0.778967i $$-0.715744\pi$$
−0.627064 + 0.778967i $$0.715744\pi$$
$$182$$ −4.63310 −0.343428
$$183$$ 0 0
$$184$$ 8.05138 0.593556
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −0.487511 −0.0356504
$$188$$ 9.07367 0.661766
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.4757 −1.26449 −0.632247 0.774767i $$-0.717867\pi$$
−0.632247 + 0.774767i $$0.717867\pi$$
$$192$$ 0 0
$$193$$ 1.99226 0.143406 0.0717030 0.997426i $$-0.477157\pi$$
0.0717030 + 0.997426i $$0.477157\pi$$
$$194$$ 3.43552 0.246656
$$195$$ 0 0
$$196$$ 2.73143 0.195102
$$197$$ −21.1925 −1.50991 −0.754953 0.655779i $$-0.772340\pi$$
−0.754953 + 0.655779i $$0.772340\pi$$
$$198$$ 0 0
$$199$$ −3.08378 −0.218603 −0.109302 0.994009i $$-0.534861\pi$$
−0.109302 + 0.994009i $$0.534861\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −7.13516 −0.502028
$$203$$ 14.6750 1.02998
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −16.3969 −1.14243
$$207$$ 0 0
$$208$$ 0.101014 0.00700409
$$209$$ 0.584407 0.0404243
$$210$$ 0 0
$$211$$ 1.00774 0.0693757 0.0346879 0.999398i $$-0.488956\pi$$
0.0346879 + 0.999398i $$0.488956\pi$$
$$212$$ 1.72193 0.118263
$$213$$ 0 0
$$214$$ −6.68004 −0.456638
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −11.3277 −0.768974
$$218$$ −13.7520 −0.931400
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 7.23442 0.486640
$$222$$ 0 0
$$223$$ −18.2841 −1.22439 −0.612195 0.790707i $$-0.709713\pi$$
−0.612195 + 0.790707i $$0.709713\pi$$
$$224$$ 12.3182 0.823044
$$225$$ 0 0
$$226$$ 2.03415 0.135310
$$227$$ 2.64496 0.175552 0.0877762 0.996140i $$-0.472024\pi$$
0.0877762 + 0.996140i $$0.472024\pi$$
$$228$$ 0 0
$$229$$ −3.46286 −0.228832 −0.114416 0.993433i $$-0.536500\pi$$
−0.114416 + 0.993433i $$0.536500\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −19.0591 −1.25129
$$233$$ −6.12567 −0.401306 −0.200653 0.979662i $$-0.564306\pi$$
−0.200653 + 0.979662i $$0.564306\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 6.28136 0.408882
$$237$$ 0 0
$$238$$ −5.76382 −0.373613
$$239$$ −28.9513 −1.87270 −0.936352 0.351062i $$-0.885821\pi$$
−0.936352 + 0.351062i $$0.885821\pi$$
$$240$$ 0 0
$$241$$ 22.3259 1.43814 0.719070 0.694937i $$-0.244568\pi$$
0.719070 + 0.694937i $$0.244568\pi$$
$$242$$ −9.65002 −0.620326
$$243$$ 0 0
$$244$$ 4.63816 0.296927
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −8.67230 −0.551805
$$248$$ 14.7118 0.934202
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −22.7219 −1.43420 −0.717098 0.696972i $$-0.754530\pi$$
−0.717098 + 0.696972i $$0.754530\pi$$
$$252$$ 0 0
$$253$$ −0.461104 −0.0289894
$$254$$ −0.0368366 −0.00231134
$$255$$ 0 0
$$256$$ −16.1010 −1.00631
$$257$$ 19.5895 1.22196 0.610978 0.791647i $$-0.290776\pi$$
0.610978 + 0.791647i $$0.290776\pi$$
$$258$$ 0 0
$$259$$ 14.5030 0.901172
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 16.1352 0.996834
$$263$$ −17.8007 −1.09764 −0.548818 0.835942i $$-0.684922\pi$$
−0.548818 + 0.835942i $$0.684922\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 6.90941 0.423643
$$267$$ 0 0
$$268$$ −7.19429 −0.439461
$$269$$ 22.7888 1.38946 0.694729 0.719272i $$-0.255524\pi$$
0.694729 + 0.719272i $$0.255524\pi$$
$$270$$ 0 0
$$271$$ −3.44562 −0.209307 −0.104653 0.994509i $$-0.533373\pi$$
−0.104653 + 0.994509i $$0.533373\pi$$
$$272$$ 0.125667 0.00761969
$$273$$ 0 0
$$274$$ −12.5868 −0.760395
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.61350 0.157030 0.0785151 0.996913i $$-0.474982\pi$$
0.0785151 + 0.996913i $$0.474982\pi$$
$$278$$ 9.22937 0.553541
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −13.6851 −0.816384 −0.408192 0.912896i $$-0.633841\pi$$
−0.408192 + 0.912896i $$0.633841\pi$$
$$282$$ 0 0
$$283$$ −22.8803 −1.36009 −0.680047 0.733169i $$-0.738041\pi$$
−0.680047 + 0.733169i $$0.738041\pi$$
$$284$$ 18.7906 1.11501
$$285$$ 0 0
$$286$$ −0.344608 −0.0203771
$$287$$ −12.6732 −0.748078
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 10.6477 0.623107
$$293$$ −24.2814 −1.41853 −0.709266 0.704941i $$-0.750974\pi$$
−0.709266 + 0.704941i $$0.750974\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −18.8357 −1.09481
$$297$$ 0 0
$$298$$ −1.11793 −0.0647597
$$299$$ 6.84255 0.395715
$$300$$ 0 0
$$301$$ 13.6040 0.784122
$$302$$ 6.90941 0.397592
$$303$$ 0 0
$$304$$ −0.150644 −0.00864004
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 16.1489 0.921666 0.460833 0.887487i $$-0.347551\pi$$
0.460833 + 0.887487i $$0.347551\pi$$
$$308$$ −0.435518 −0.0248159
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.7101 1.06095 0.530475 0.847700i $$-0.322013\pi$$
0.530475 + 0.847700i $$0.322013\pi$$
$$312$$ 0 0
$$313$$ −2.77332 −0.156757 −0.0783786 0.996924i $$-0.524974\pi$$
−0.0783786 + 0.996924i $$0.524974\pi$$
$$314$$ 10.8625 0.613005
$$315$$ 0 0
$$316$$ 1.55613 0.0875393
$$317$$ −17.4825 −0.981913 −0.490956 0.871184i $$-0.663353\pi$$
−0.490956 + 0.871184i $$0.663353\pi$$
$$318$$ 0 0
$$319$$ 1.09152 0.0611133
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −5.45161 −0.303806
$$323$$ −10.7888 −0.600305
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12.0888 0.669538
$$327$$ 0 0
$$328$$ 16.4593 0.908816
$$329$$ −16.1607 −0.890971
$$330$$ 0 0
$$331$$ −32.4593 −1.78413 −0.892064 0.451910i $$-0.850743\pi$$
−0.892064 + 0.451910i $$0.850743\pi$$
$$332$$ −10.3969 −0.570605
$$333$$ 0 0
$$334$$ 3.26857 0.178848
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −8.28581 −0.451357 −0.225678 0.974202i $$-0.572460\pi$$
−0.225678 + 0.974202i $$0.572460\pi$$
$$338$$ −6.31820 −0.343665
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −0.842549 −0.0456266
$$342$$ 0 0
$$343$$ −20.1584 −1.08845
$$344$$ −17.6682 −0.952605
$$345$$ 0 0
$$346$$ 1.37134 0.0737237
$$347$$ −14.9632 −0.803265 −0.401632 0.915801i $$-0.631557\pi$$
−0.401632 + 0.915801i $$0.631557\pi$$
$$348$$ 0 0
$$349$$ 33.6459 1.80102 0.900512 0.434832i $$-0.143192\pi$$
0.900512 + 0.434832i $$0.143192\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0.916222 0.0488348
$$353$$ 15.7537 0.838486 0.419243 0.907874i $$-0.362296\pi$$
0.419243 + 0.907874i $$0.362296\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −9.47235 −0.502034
$$357$$ 0 0
$$358$$ −10.7219 −0.566672
$$359$$ 18.1257 0.956636 0.478318 0.878187i $$-0.341247\pi$$
0.478318 + 0.878187i $$0.341247\pi$$
$$360$$ 0 0
$$361$$ −6.06687 −0.319309
$$362$$ −14.8375 −0.779841
$$363$$ 0 0
$$364$$ 6.46286 0.338746
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 19.1429 0.999251 0.499626 0.866241i $$-0.333471\pi$$
0.499626 + 0.866241i $$0.333471\pi$$
$$368$$ 0.118860 0.00619601
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −3.06687 −0.159224
$$372$$ 0 0
$$373$$ 15.2499 0.789610 0.394805 0.918765i $$-0.370812\pi$$
0.394805 + 0.918765i $$0.370812\pi$$
$$374$$ −0.428710 −0.0221681
$$375$$ 0 0
$$376$$ 20.9887 1.08241
$$377$$ −16.1976 −0.834218
$$378$$ 0 0
$$379$$ 9.84760 0.505837 0.252919 0.967488i $$-0.418610\pi$$
0.252919 + 0.967488i $$0.418610\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −15.3678 −0.786287
$$383$$ −28.3901 −1.45067 −0.725334 0.688397i $$-0.758315\pi$$
−0.725334 + 0.688397i $$0.758315\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 1.75196 0.0891726
$$387$$ 0 0
$$388$$ −4.79231 −0.243293
$$389$$ 10.8844 0.551863 0.275931 0.961177i $$-0.411014\pi$$
0.275931 + 0.961177i $$0.411014\pi$$
$$390$$ 0 0
$$391$$ 8.51249 0.430495
$$392$$ 6.31820 0.319117
$$393$$ 0 0
$$394$$ −18.6364 −0.938888
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 18.1070 0.908764 0.454382 0.890807i $$-0.349860\pi$$
0.454382 + 0.890807i $$0.349860\pi$$
$$398$$ −2.71183 −0.135932
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1.43376 0.0715987 0.0357993 0.999359i $$-0.488602\pi$$
0.0357993 + 0.999359i $$0.488602\pi$$
$$402$$ 0 0
$$403$$ 12.5030 0.622818
$$404$$ 9.95306 0.495183
$$405$$ 0 0
$$406$$ 12.9050 0.640463
$$407$$ 1.07873 0.0534704
$$408$$ 0 0
$$409$$ 8.60401 0.425441 0.212720 0.977113i $$-0.431768\pi$$
0.212720 + 0.977113i $$0.431768\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 22.8726 1.12685
$$413$$ −11.1875 −0.550500
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −13.5963 −0.666612
$$417$$ 0 0
$$418$$ 0.513919 0.0251366
$$419$$ −12.3114 −0.601451 −0.300725 0.953711i $$-0.597229\pi$$
−0.300725 + 0.953711i $$0.597229\pi$$
$$420$$ 0 0
$$421$$ 11.1165 0.541785 0.270892 0.962610i $$-0.412681\pi$$
0.270892 + 0.962610i $$0.412681\pi$$
$$422$$ 0.886192 0.0431392
$$423$$ 0 0
$$424$$ 3.98309 0.193436
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −8.26083 −0.399770
$$428$$ 9.31820 0.450412
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −36.8958 −1.77721 −0.888604 0.458675i $$-0.848324\pi$$
−0.888604 + 0.458675i $$0.848324\pi$$
$$432$$ 0 0
$$433$$ 37.9982 1.82608 0.913040 0.407871i $$-0.133729\pi$$
0.913040 + 0.407871i $$0.133729\pi$$
$$434$$ −9.96141 −0.478163
$$435$$ 0 0
$$436$$ 19.1830 0.918701
$$437$$ −10.2044 −0.488142
$$438$$ 0 0
$$439$$ 0.202029 0.00964231 0.00482115 0.999988i $$-0.498465\pi$$
0.00482115 + 0.999988i $$0.498465\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6.36184 0.302602
$$443$$ −21.2294 −1.00864 −0.504319 0.863517i $$-0.668256\pi$$
−0.504319 + 0.863517i $$0.668256\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −16.0787 −0.761350
$$447$$ 0 0
$$448$$ 11.0155 0.520433
$$449$$ −33.2594 −1.56961 −0.784804 0.619744i $$-0.787236\pi$$
−0.784804 + 0.619744i $$0.787236\pi$$
$$450$$ 0 0
$$451$$ −0.942629 −0.0443867
$$452$$ −2.83750 −0.133465
$$453$$ 0 0
$$454$$ 2.32594 0.109162
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0.0341483 0.00159739 0.000798695 1.00000i $$-0.499746\pi$$
0.000798695 1.00000i $$0.499746\pi$$
$$458$$ −3.04519 −0.142292
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.9864 −0.697986 −0.348993 0.937125i $$-0.613476\pi$$
−0.348993 + 0.937125i $$0.613476\pi$$
$$462$$ 0 0
$$463$$ 30.4424 1.41478 0.707390 0.706823i $$-0.249872\pi$$
0.707390 + 0.706823i $$0.249872\pi$$
$$464$$ −0.281364 −0.0130620
$$465$$ 0 0
$$466$$ −5.38682 −0.249540
$$467$$ 0.510734 0.0236339 0.0118170 0.999930i $$-0.496238\pi$$
0.0118170 + 0.999930i $$0.496238\pi$$
$$468$$ 0 0
$$469$$ 12.8135 0.591670
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 14.5297 0.668785
$$473$$ 1.01186 0.0465254
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 8.04013 0.368519
$$477$$ 0 0
$$478$$ −25.4593 −1.16448
$$479$$ −15.4507 −0.705959 −0.352980 0.935631i $$-0.614832\pi$$
−0.352980 + 0.935631i $$0.614832\pi$$
$$480$$ 0 0
$$481$$ −16.0077 −0.729890
$$482$$ 19.6331 0.894263
$$483$$ 0 0
$$484$$ 13.4611 0.611868
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −29.5107 −1.33726 −0.668629 0.743596i $$-0.733119\pi$$
−0.668629 + 0.743596i $$0.733119\pi$$
$$488$$ 10.7287 0.485667
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2.15745 0.0973644 0.0486822 0.998814i $$-0.484498\pi$$
0.0486822 + 0.998814i $$0.484498\pi$$
$$492$$ 0 0
$$493$$ −20.1506 −0.907539
$$494$$ −7.62630 −0.343123
$$495$$ 0 0
$$496$$ 0.217186 0.00975194
$$497$$ −33.4671 −1.50120
$$498$$ 0 0
$$499$$ 7.49525 0.335534 0.167767 0.985827i $$-0.446344\pi$$
0.167767 + 0.985827i $$0.446344\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −19.9813 −0.891811
$$503$$ −28.5963 −1.27504 −0.637522 0.770432i $$-0.720041\pi$$
−0.637522 + 0.770432i $$0.720041\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −0.405488 −0.0180261
$$507$$ 0 0
$$508$$ 0.0513845 0.00227982
$$509$$ −1.69190 −0.0749923 −0.0374962 0.999297i $$-0.511938\pi$$
−0.0374962 + 0.999297i $$0.511938\pi$$
$$510$$ 0 0
$$511$$ −18.9641 −0.838922
$$512$$ −0.473897 −0.0209435
$$513$$ 0 0
$$514$$ 17.2267 0.759836
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1.20203 −0.0528652
$$518$$ 12.7537 0.560366
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.4037 −0.981525 −0.490763 0.871293i $$-0.663282\pi$$
−0.490763 + 0.871293i $$0.663282\pi$$
$$522$$ 0 0
$$523$$ −2.42871 −0.106200 −0.0531000 0.998589i $$-0.516910\pi$$
−0.0531000 + 0.998589i $$0.516910\pi$$
$$524$$ −22.5074 −0.983242
$$525$$ 0 0
$$526$$ −15.6536 −0.682531
$$527$$ 15.5544 0.677559
$$528$$ 0 0
$$529$$ −14.9486 −0.649940
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −9.63816 −0.417867
$$533$$ 13.9881 0.605894
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −16.6415 −0.718801
$$537$$ 0 0
$$538$$ 20.0401 0.863992
$$539$$ −0.361844 −0.0155857
$$540$$ 0 0
$$541$$ 38.9394 1.67414 0.837069 0.547098i $$-0.184267\pi$$
0.837069 + 0.547098i $$0.184267\pi$$
$$542$$ −3.03003 −0.130151
$$543$$ 0 0
$$544$$ −16.9145 −0.725202
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 14.6723 0.627342 0.313671 0.949532i $$-0.398441\pi$$
0.313671 + 0.949532i $$0.398441\pi$$
$$548$$ 17.5577 0.750027
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 24.1557 1.02907
$$552$$ 0 0
$$553$$ −2.77156 −0.117859
$$554$$ 2.29828 0.0976444
$$555$$ 0 0
$$556$$ −12.8743 −0.545993
$$557$$ 11.1070 0.470619 0.235309 0.971921i $$-0.424390\pi$$
0.235309 + 0.971921i $$0.424390\pi$$
$$558$$ 0 0
$$559$$ −15.0155 −0.635087
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −12.0345 −0.507644
$$563$$ −16.3081 −0.687304 −0.343652 0.939097i $$-0.611664\pi$$
−0.343652 + 0.939097i $$0.611664\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −20.1206 −0.845733
$$567$$ 0 0
$$568$$ 43.4653 1.82376
$$569$$ 36.0164 1.50989 0.754943 0.655790i $$-0.227664\pi$$
0.754943 + 0.655790i $$0.227664\pi$$
$$570$$ 0 0
$$571$$ 39.1584 1.63873 0.819364 0.573274i $$-0.194327\pi$$
0.819364 + 0.573274i $$0.194327\pi$$
$$572$$ 0.480704 0.0200993
$$573$$ 0 0
$$574$$ −11.1447 −0.465169
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −11.8057 −0.491478 −0.245739 0.969336i $$-0.579031\pi$$
−0.245739 + 0.969336i $$0.579031\pi$$
$$578$$ −7.03508 −0.292621
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 18.5175 0.768237
$$582$$ 0 0
$$583$$ −0.228112 −0.00944744
$$584$$ 24.6296 1.01918
$$585$$ 0 0
$$586$$ −21.3527 −0.882071
$$587$$ 39.9614 1.64938 0.824692 0.565582i $$-0.191349\pi$$
0.824692 + 0.565582i $$0.191349\pi$$
$$588$$ 0 0
$$589$$ −18.6459 −0.768291
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −0.278066 −0.0114284
$$593$$ 29.2995 1.20319 0.601594 0.798802i $$-0.294533\pi$$
0.601594 + 0.798802i $$0.294533\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1.55943 0.0638767
$$597$$ 0 0
$$598$$ 6.01724 0.246063
$$599$$ 10.0719 0.411527 0.205764 0.978602i $$-0.434032\pi$$
0.205764 + 0.978602i $$0.434032\pi$$
$$600$$ 0 0
$$601$$ −30.4192 −1.24083 −0.620413 0.784275i $$-0.713035\pi$$
−0.620413 + 0.784275i $$0.713035\pi$$
$$602$$ 11.9632 0.487582
$$603$$ 0 0
$$604$$ −9.63816 −0.392171
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 23.0743 0.936556 0.468278 0.883581i $$-0.344875\pi$$
0.468278 + 0.883581i $$0.344875\pi$$
$$608$$ 20.2763 0.822313
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 17.8375 0.721628
$$612$$ 0 0
$$613$$ −0.765578 −0.0309214 −0.0154607 0.999880i $$-0.504921\pi$$
−0.0154607 + 0.999880i $$0.504921\pi$$
$$614$$ 14.2011 0.573110
$$615$$ 0 0
$$616$$ −1.00742 −0.0405900
$$617$$ 9.28817 0.373928 0.186964 0.982367i $$-0.440135\pi$$
0.186964 + 0.982367i $$0.440135\pi$$
$$618$$ 0 0
$$619$$ −35.0823 −1.41008 −0.705039 0.709168i $$-0.749071\pi$$
−0.705039 + 0.709168i $$0.749071\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 16.4534 0.659720
$$623$$ 16.8708 0.675915
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −2.43882 −0.0974747
$$627$$ 0 0
$$628$$ −15.1524 −0.604647
$$629$$ −19.9145 −0.794042
$$630$$ 0 0
$$631$$ 35.7621 1.42367 0.711833 0.702349i $$-0.247865\pi$$
0.711833 + 0.702349i $$0.247865\pi$$
$$632$$ 3.59956 0.143183
$$633$$ 0 0
$$634$$ −15.3738 −0.610572
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5.36959 0.212751
$$638$$ 0.959866 0.0380014
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.92633 0.115583 0.0577915 0.998329i $$-0.481594\pi$$
0.0577915 + 0.998329i $$0.481594\pi$$
$$642$$ 0 0
$$643$$ 20.2517 0.798647 0.399324 0.916810i $$-0.369245\pi$$
0.399324 + 0.916810i $$0.369245\pi$$
$$644$$ 7.60462 0.299664
$$645$$ 0 0
$$646$$ −9.48751 −0.373281
$$647$$ −10.7219 −0.421523 −0.210761 0.977538i $$-0.567594\pi$$
−0.210761 + 0.977538i $$0.567594\pi$$
$$648$$ 0 0
$$649$$ −0.832119 −0.0326635
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −16.8631 −0.660409
$$653$$ −35.7270 −1.39811 −0.699053 0.715070i $$-0.746395\pi$$
−0.699053 + 0.715070i $$0.746395\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0.242984 0.00948694
$$657$$ 0 0
$$658$$ −14.2115 −0.554023
$$659$$ 30.8658 1.20236 0.601180 0.799114i $$-0.294698\pi$$
0.601180 + 0.799114i $$0.294698\pi$$
$$660$$ 0 0
$$661$$ −9.84793 −0.383040 −0.191520 0.981489i $$-0.561342\pi$$
−0.191520 + 0.981489i $$0.561342\pi$$
$$662$$ −28.5443 −1.10940
$$663$$ 0 0
$$664$$ −24.0496 −0.933307
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −19.0591 −0.737972
$$668$$ −4.55943 −0.176410
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −0.614437 −0.0237201
$$672$$ 0 0
$$673$$ 19.6973 0.759274 0.379637 0.925135i $$-0.376049\pi$$
0.379637 + 0.925135i $$0.376049\pi$$
$$674$$ −7.28642 −0.280662
$$675$$ 0 0
$$676$$ 8.81345 0.338979
$$677$$ 28.4570 1.09369 0.546845 0.837234i $$-0.315829\pi$$
0.546845 + 0.837234i $$0.315829\pi$$
$$678$$ 0 0
$$679$$ 8.53539 0.327558
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −0.740925 −0.0283715
$$683$$ −12.5107 −0.478710 −0.239355 0.970932i $$-0.576936\pi$$
−0.239355 + 0.970932i $$0.576936\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −17.7270 −0.676819
$$687$$ 0 0
$$688$$ −0.260830 −0.00994405
$$689$$ 3.38507 0.128961
$$690$$ 0 0
$$691$$ 42.6255 1.62155 0.810775 0.585358i $$-0.199046\pi$$
0.810775 + 0.585358i $$0.199046\pi$$
$$692$$ −1.91292 −0.0727185
$$693$$ 0 0
$$694$$ −13.1584 −0.499485
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 17.4020 0.659147
$$698$$ 29.5877 1.11991
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 51.7701 1.95533 0.977665 0.210167i $$-0.0674008\pi$$
0.977665 + 0.210167i $$0.0674008\pi$$
$$702$$ 0 0
$$703$$ 23.8726 0.900371
$$704$$ 0.819326 0.0308795
$$705$$ 0 0
$$706$$ 13.8536 0.521387
$$707$$ −17.7270 −0.666692
$$708$$ 0 0
$$709$$ 15.1584 0.569285 0.284643 0.958634i $$-0.408125\pi$$
0.284643 + 0.958634i $$0.408125\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −21.9110 −0.821148
$$713$$ 14.7118 0.550962
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 14.9564 0.558945
$$717$$ 0 0
$$718$$ 15.9394 0.594855
$$719$$ −2.61493 −0.0975206 −0.0487603 0.998811i $$-0.515527\pi$$
−0.0487603 + 0.998811i $$0.515527\pi$$
$$720$$ 0 0
$$721$$ −40.7374 −1.51714
$$722$$ −5.33511 −0.198552
$$723$$ 0 0
$$724$$ 20.6973 0.769208
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 4.09926 0.152033 0.0760166 0.997107i $$-0.475780\pi$$
0.0760166 + 0.997107i $$0.475780\pi$$
$$728$$ 14.9495 0.554067
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −18.6800 −0.690906
$$732$$ 0 0
$$733$$ 38.2080 1.41125 0.705623 0.708588i $$-0.250667\pi$$
0.705623 + 0.708588i $$0.250667\pi$$
$$734$$ 16.8340 0.621354
$$735$$ 0 0
$$736$$ −15.9982 −0.589703
$$737$$ 0.953058 0.0351064
$$738$$ 0 0
$$739$$ −24.2094 −0.890559 −0.445279 0.895392i $$-0.646896\pi$$
−0.445279 + 0.895392i $$0.646896\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −2.69696 −0.0990084
$$743$$ 3.31139 0.121483 0.0607416 0.998154i $$-0.480653\pi$$
0.0607416 + 0.998154i $$0.480653\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 13.4105 0.490995
$$747$$ 0 0
$$748$$ 0.598021 0.0218658
$$749$$ −16.5963 −0.606414
$$750$$ 0 0
$$751$$ 13.7110 0.500322 0.250161 0.968204i $$-0.419516\pi$$
0.250161 + 0.968204i $$0.419516\pi$$
$$752$$ 0.309850 0.0112991
$$753$$ 0 0
$$754$$ −14.2439 −0.518733
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −12.3833 −0.450079 −0.225040 0.974350i $$-0.572251\pi$$
−0.225040 + 0.974350i $$0.572251\pi$$
$$758$$ 8.65984 0.314539
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 7.58265 0.274871 0.137435 0.990511i $$-0.456114\pi$$
0.137435 + 0.990511i $$0.456114\pi$$
$$762$$ 0 0
$$763$$ −34.1661 −1.23690
$$764$$ 21.4371 0.775566
$$765$$ 0 0
$$766$$ −24.9659 −0.902053
$$767$$ 12.3482 0.445869
$$768$$ 0 0
$$769$$ 3.21719 0.116015 0.0580073 0.998316i $$-0.481525\pi$$
0.0580073 + 0.998316i $$0.481525\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −2.44387 −0.0879567
$$773$$ 0.184468 0.00663486 0.00331743 0.999994i $$-0.498944\pi$$
0.00331743 + 0.999994i $$0.498944\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −11.0853 −0.397940
$$777$$ 0 0
$$778$$ 9.57161 0.343159
$$779$$ −20.8607 −0.747413
$$780$$ 0 0
$$781$$ −2.48927 −0.0890729
$$782$$ 7.48576 0.267690
$$783$$ 0 0
$$784$$ 0.0932736 0.00333120
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0.478016 0.0170394 0.00851971 0.999964i $$-0.497288\pi$$
0.00851971 + 0.999964i $$0.497288\pi$$
$$788$$ 25.9965 0.926087
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5.05375 0.179691
$$792$$ 0 0
$$793$$ 9.11793 0.323787
$$794$$ 15.9230 0.565087
$$795$$ 0 0
$$796$$ 3.78281 0.134078
$$797$$ −14.4989 −0.513576 −0.256788 0.966468i $$-0.582664\pi$$
−0.256788 + 0.966468i $$0.582664\pi$$
$$798$$ 0 0
$$799$$ 22.1908 0.785053
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 1.26083 0.0445215
$$803$$ −1.41054 −0.0497769
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 10.9949 0.387281
$$807$$ 0 0
$$808$$ 23.0229 0.809943
$$809$$ −14.8743 −0.522954 −0.261477 0.965210i $$-0.584210\pi$$
−0.261477 + 0.965210i $$0.584210\pi$$
$$810$$ 0 0
$$811$$ 21.5963 0.758347 0.379174 0.925325i $$-0.376208\pi$$
0.379174 + 0.925325i $$0.376208\pi$$
$$812$$ −18.0015 −0.631730
$$813$$ 0 0
$$814$$ 0.948615 0.0332490
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 22.3928 0.783425
$$818$$ 7.56624 0.264547
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −3.28136 −0.114520 −0.0572602 0.998359i $$-0.518236\pi$$
−0.0572602 + 0.998359i $$0.518236\pi$$
$$822$$ 0 0
$$823$$ −13.7314 −0.478648 −0.239324 0.970940i $$-0.576926\pi$$
−0.239324 + 0.970940i $$0.576926\pi$$
$$824$$ 52.9077 1.84313
$$825$$ 0 0
$$826$$ −9.83811 −0.342311
$$827$$ −20.2327 −0.703559 −0.351779 0.936083i $$-0.614423\pi$$
−0.351779 + 0.936083i $$0.614423\pi$$
$$828$$ 0 0
$$829$$ 25.5276 0.886612 0.443306 0.896370i $$-0.353806\pi$$
0.443306 + 0.896370i $$0.353806\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −12.1584 −0.421516
$$833$$ 6.68004 0.231450
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −0.716881 −0.0247939
$$837$$ 0 0
$$838$$ −10.8265 −0.373994
$$839$$ 17.3800 0.600025 0.300012 0.953935i $$-0.403009\pi$$
0.300012 + 0.953935i $$0.403009\pi$$
$$840$$ 0 0
$$841$$ 16.1165 0.555741
$$842$$ 9.77568 0.336892
$$843$$ 0 0
$$844$$ −1.23618 −0.0425510
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −23.9750 −0.823792
$$848$$ 0.0588011 0.00201924
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −18.8357 −0.645681
$$852$$ 0 0
$$853$$ −13.3027 −0.455476 −0.227738 0.973722i $$-0.573133\pi$$
−0.227738 + 0.973722i $$0.573133\pi$$
$$854$$ −7.26445 −0.248584
$$855$$ 0 0
$$856$$ 21.5544 0.736713
$$857$$ 19.9213 0.680498 0.340249 0.940335i $$-0.389489\pi$$
0.340249 + 0.940335i $$0.389489\pi$$
$$858$$ 0 0
$$859$$ 26.3446 0.898866 0.449433 0.893314i $$-0.351626\pi$$
0.449433 + 0.893314i $$0.351626\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −32.4456 −1.10510
$$863$$ −38.2995 −1.30373 −0.651866 0.758334i $$-0.726013\pi$$
−0.651866 + 0.758334i $$0.726013\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 33.4151 1.13549
$$867$$ 0 0
$$868$$ 13.8955 0.471643
$$869$$ −0.206148 −0.00699308
$$870$$ 0 0
$$871$$ −14.1429 −0.479214
$$872$$ 44.3732 1.50267
$$873$$ 0 0
$$874$$ −8.97359 −0.303536
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 27.0574 0.913662 0.456831 0.889553i $$-0.348984\pi$$
0.456831 + 0.889553i $$0.348984\pi$$
$$878$$ 0.177661 0.00599577
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.5776 1.03019 0.515093 0.857134i $$-0.327757\pi$$
0.515093 + 0.857134i $$0.327757\pi$$
$$882$$ 0 0
$$883$$ −44.1052 −1.48426 −0.742130 0.670256i $$-0.766184\pi$$
−0.742130 + 0.670256i $$0.766184\pi$$
$$884$$ −8.87433 −0.298476
$$885$$ 0 0
$$886$$ −18.6688 −0.627190
$$887$$ 7.88444 0.264734 0.132367 0.991201i $$-0.457742\pi$$
0.132367 + 0.991201i $$0.457742\pi$$
$$888$$ 0 0
$$889$$ −0.0915189 −0.00306945
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 22.4287 0.750969
$$893$$ −26.6013 −0.890179
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −14.9495 −0.499429
$$897$$ 0 0
$$898$$ −29.2478 −0.976013
$$899$$ −34.8256 −1.16150
$$900$$ 0 0
$$901$$ 4.21120 0.140295
$$902$$ −0.828934 −0.0276005
$$903$$ 0 0
$$904$$ −6.56355 −0.218300
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12.9067 0.428561 0.214280 0.976772i $$-0.431259\pi$$
0.214280 + 0.976772i $$0.431259\pi$$
$$908$$ −3.24453 −0.107673
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 21.1857 0.701914 0.350957 0.936391i $$-0.385856\pi$$
0.350957 + 0.936391i $$0.385856\pi$$
$$912$$ 0 0
$$913$$ 1.37733 0.0455828
$$914$$ 0.0300295 0.000993287 0
$$915$$ 0 0
$$916$$ 4.24783 0.140352
$$917$$ 40.0871 1.32379
$$918$$ 0 0
$$919$$ 31.4688 1.03806 0.519031 0.854756i $$-0.326293\pi$$
0.519031 + 0.854756i $$0.326293\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −13.1788 −0.434021
$$923$$ 36.9394 1.21588
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 26.7706 0.879737
$$927$$ 0 0
$$928$$ 37.8708 1.24317
$$929$$ −2.32676 −0.0763386 −0.0381693 0.999271i $$-0.512153\pi$$
−0.0381693 + 0.999271i $$0.512153\pi$$
$$930$$ 0 0
$$931$$ −8.00774 −0.262443
$$932$$ 7.51424 0.246137
$$933$$ 0 0
$$934$$ 0.449132 0.0146960
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 10.9982 0.359297 0.179649 0.983731i $$-0.442504\pi$$
0.179649 + 0.983731i $$0.442504\pi$$
$$938$$ 11.2680 0.367912
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −24.1037 −0.785758 −0.392879 0.919590i $$-0.628521\pi$$
−0.392879 + 0.919590i $$0.628521\pi$$
$$942$$ 0 0
$$943$$ 16.4593 0.535990
$$944$$ 0.214498 0.00698131
$$945$$ 0 0
$$946$$ 0.889814 0.0289304
$$947$$ 11.9195 0.387332 0.193666 0.981067i $$-0.437962\pi$$
0.193666 + 0.981067i $$0.437962\pi$$
$$948$$ 0 0
$$949$$ 20.9317 0.679472
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 18.5980 0.602765
$$953$$ 36.8289 1.19301 0.596503 0.802611i $$-0.296556\pi$$
0.596503 + 0.802611i $$0.296556\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 35.5140 1.14861
$$957$$ 0 0
$$958$$ −13.5871 −0.438979
$$959$$ −31.2713 −1.00980
$$960$$ 0 0
$$961$$ −4.11793 −0.132836
$$962$$ −14.0770 −0.453860
$$963$$ 0 0
$$964$$ −27.3868 −0.882070
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −53.5604 −1.72239 −0.861193 0.508279i $$-0.830282\pi$$
−0.861193 + 0.508279i $$0.830282\pi$$
$$968$$ 31.1375 1.00080
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 53.2327 1.70832 0.854159 0.520012i $$-0.174073\pi$$
0.854159 + 0.520012i $$0.174073\pi$$
$$972$$ 0 0
$$973$$ 22.9299 0.735100
$$974$$ −25.9513 −0.831533
$$975$$ 0 0
$$976$$ 0.158385 0.00506978
$$977$$ −13.4570 −0.430527 −0.215264 0.976556i $$-0.569061\pi$$
−0.215264 + 0.976556i $$0.569061\pi$$
$$978$$ 0 0
$$979$$ 1.25484 0.0401050
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 1.89723 0.0605431
$$983$$ 10.2412 0.326644 0.163322 0.986573i $$-0.447779\pi$$
0.163322 + 0.986573i $$0.447779\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −17.7202 −0.564325
$$987$$ 0 0
$$988$$ 10.6382 0.338445
$$989$$ −17.6682 −0.561816
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ −29.2327 −0.928138
$$993$$ 0 0
$$994$$ −29.4305 −0.933478
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −38.5016 −1.21936 −0.609678 0.792649i $$-0.708701\pi$$
−0.609678 + 0.792649i $$0.708701\pi$$
$$998$$ 6.59121 0.208641
$$999$$ 0 0
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 6075.2.a.bq.1.3 3
3.2 odd 2 6075.2.a.bv.1.1 3
5.4 even 2 243.2.a.f.1.1 yes 3
15.14 odd 2 243.2.a.e.1.3 3
20.19 odd 2 3888.2.a.bk.1.3 3
45.4 even 6 243.2.c.e.163.3 6
45.14 odd 6 243.2.c.f.163.1 6
45.29 odd 6 243.2.c.f.82.1 6
45.34 even 6 243.2.c.e.82.3 6
60.59 even 2 3888.2.a.bd.1.1 3
135.4 even 18 729.2.e.b.406.1 6
135.14 odd 18 729.2.e.a.163.1 6
135.29 odd 18 729.2.e.a.568.1 6
135.34 even 18 729.2.e.b.325.1 6
135.49 even 18 729.2.e.c.649.1 6
135.59 odd 18 729.2.e.h.649.1 6
135.74 odd 18 729.2.e.g.325.1 6
135.79 even 18 729.2.e.i.568.1 6
135.94 even 18 729.2.e.i.163.1 6
135.104 odd 18 729.2.e.g.406.1 6
135.119 odd 18 729.2.e.h.82.1 6
135.124 even 18 729.2.e.c.82.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
243.2.a.e.1.3 3 15.14 odd 2
243.2.a.f.1.1 yes 3 5.4 even 2
243.2.c.e.82.3 6 45.34 even 6
243.2.c.e.163.3 6 45.4 even 6
243.2.c.f.82.1 6 45.29 odd 6
243.2.c.f.163.1 6 45.14 odd 6
729.2.e.a.163.1 6 135.14 odd 18
729.2.e.a.568.1 6 135.29 odd 18
729.2.e.b.325.1 6 135.34 even 18
729.2.e.b.406.1 6 135.4 even 18
729.2.e.c.82.1 6 135.124 even 18
729.2.e.c.649.1 6 135.49 even 18
729.2.e.g.325.1 6 135.74 odd 18
729.2.e.g.406.1 6 135.104 odd 18
729.2.e.h.82.1 6 135.119 odd 18
729.2.e.h.649.1 6 135.59 odd 18
729.2.e.i.163.1 6 135.94 even 18
729.2.e.i.568.1 6 135.79 even 18
3888.2.a.bd.1.1 3 60.59 even 2
3888.2.a.bk.1.3 3 20.19 odd 2
6075.2.a.bq.1.3 3 1.1 even 1 trivial
6075.2.a.bv.1.1 3 3.2 odd 2