# Properties

 Label 8550.2.a.co.1.3 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ Analytic rank $0$ 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] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.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: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 3$$ x^3 - x^2 - 4*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 950) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.713538$$ of defining polynomial Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +4.69527 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +4.69527 q^{7} +1.00000 q^{8} -6.40880 q^{11} -1.06379 q^{13} +4.69527 q^{14} +1.00000 q^{16} +1.91794 q^{17} -1.00000 q^{19} -6.40880 q^{22} +1.79560 q^{23} -1.06379 q^{26} +4.69527 q^{28} -2.93621 q^{29} -5.55465 q^{31} +1.00000 q^{32} +1.91794 q^{34} +11.4088 q^{37} -1.00000 q^{38} +1.14585 q^{41} +3.55465 q^{43} -6.40880 q^{44} +1.79560 q^{46} +10.8359 q^{47} +15.0455 q^{49} -1.06379 q^{52} -8.69527 q^{53} +4.69527 q^{56} -2.93621 q^{58} +5.63148 q^{59} -3.39053 q^{61} -5.55465 q^{62} +1.00000 q^{64} +8.82284 q^{67} +1.91794 q^{68} +1.42708 q^{71} +12.6132 q^{73} +11.4088 q^{74} -1.00000 q^{76} -30.0910 q^{77} -1.96345 q^{79} +1.14585 q^{82} +16.2447 q^{83} +3.55465 q^{86} -6.40880 q^{88} +10.0000 q^{89} -4.99477 q^{91} +1.79560 q^{92} +10.8359 q^{94} +14.9452 q^{97} +15.0455 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 $$3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 2 q^{13} - 2 q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 2 q^{22} + 14 q^{23} + 2 q^{26} - 2 q^{28} - 14 q^{29} - 4 q^{31} + 3 q^{32} - 4 q^{34} + 17 q^{37} - 3 q^{38} + 8 q^{41} - 2 q^{43} - 2 q^{44} + 14 q^{46} + 13 q^{47} + 25 q^{49} + 2 q^{52} - 10 q^{53} - 2 q^{56} - 14 q^{58} + 6 q^{59} + 22 q^{61} - 4 q^{62} + 3 q^{64} - 4 q^{68} + 2 q^{71} + 12 q^{73} + 17 q^{74} - 3 q^{76} - 50 q^{77} + 24 q^{79} + 8 q^{82} + 12 q^{83} - 2 q^{86} - 2 q^{88} + 30 q^{89} - 7 q^{91} + 14 q^{92} + 13 q^{94} + 25 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 2 * q^13 - 2 * q^14 + 3 * q^16 - 4 * q^17 - 3 * q^19 - 2 * q^22 + 14 * q^23 + 2 * q^26 - 2 * q^28 - 14 * q^29 - 4 * q^31 + 3 * q^32 - 4 * q^34 + 17 * q^37 - 3 * q^38 + 8 * q^41 - 2 * q^43 - 2 * q^44 + 14 * q^46 + 13 * q^47 + 25 * q^49 + 2 * q^52 - 10 * q^53 - 2 * q^56 - 14 * q^58 + 6 * q^59 + 22 * q^61 - 4 * q^62 + 3 * q^64 - 4 * q^68 + 2 * q^71 + 12 * q^73 + 17 * q^74 - 3 * q^76 - 50 * q^77 + 24 * q^79 + 8 * q^82 + 12 * q^83 - 2 * q^86 - 2 * q^88 + 30 * q^89 - 7 * q^91 + 14 * q^92 + 13 * q^94 + 25 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.69527 1.77464 0.887322 0.461151i $$-0.152563\pi$$
0.887322 + 0.461151i $$0.152563\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −6.40880 −1.93233 −0.966163 0.257931i $$-0.916959\pi$$
−0.966163 + 0.257931i $$0.916959\pi$$
$$12$$ 0 0
$$13$$ −1.06379 −0.295042 −0.147521 0.989059i $$-0.547129\pi$$
−0.147521 + 0.989059i $$0.547129\pi$$
$$14$$ 4.69527 1.25486
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.91794 0.465169 0.232584 0.972576i $$-0.425282\pi$$
0.232584 + 0.972576i $$0.425282\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −6.40880 −1.36636
$$23$$ 1.79560 0.374408 0.187204 0.982321i $$-0.440057\pi$$
0.187204 + 0.982321i $$0.440057\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.06379 −0.208626
$$27$$ 0 0
$$28$$ 4.69527 0.887322
$$29$$ −2.93621 −0.545241 −0.272620 0.962122i $$-0.587890\pi$$
−0.272620 + 0.962122i $$0.587890\pi$$
$$30$$ 0 0
$$31$$ −5.55465 −0.997645 −0.498822 0.866704i $$-0.666234\pi$$
−0.498822 + 0.866704i $$0.666234\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 1.91794 0.328924
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 11.4088 1.87560 0.937798 0.347182i $$-0.112861\pi$$
0.937798 + 0.347182i $$0.112861\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.14585 0.178951 0.0894757 0.995989i $$-0.471481\pi$$
0.0894757 + 0.995989i $$0.471481\pi$$
$$42$$ 0 0
$$43$$ 3.55465 0.542079 0.271040 0.962568i $$-0.412633\pi$$
0.271040 + 0.962568i $$0.412633\pi$$
$$44$$ −6.40880 −0.966163
$$45$$ 0 0
$$46$$ 1.79560 0.264747
$$47$$ 10.8359 1.58058 0.790288 0.612736i $$-0.209931\pi$$
0.790288 + 0.612736i $$0.209931\pi$$
$$48$$ 0 0
$$49$$ 15.0455 2.14936
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1.06379 −0.147521
$$53$$ −8.69527 −1.19439 −0.597193 0.802097i $$-0.703717\pi$$
−0.597193 + 0.802097i $$0.703717\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.69527 0.627431
$$57$$ 0 0
$$58$$ −2.93621 −0.385544
$$59$$ 5.63148 0.733156 0.366578 0.930387i $$-0.380529\pi$$
0.366578 + 0.930387i $$0.380529\pi$$
$$60$$ 0 0
$$61$$ −3.39053 −0.434113 −0.217056 0.976159i $$-0.569646\pi$$
−0.217056 + 0.976159i $$0.569646\pi$$
$$62$$ −5.55465 −0.705441
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.82284 1.07788 0.538941 0.842344i $$-0.318825\pi$$
0.538941 + 0.842344i $$0.318825\pi$$
$$68$$ 1.91794 0.232584
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1.42708 0.169363 0.0846814 0.996408i $$-0.473013\pi$$
0.0846814 + 0.996408i $$0.473013\pi$$
$$72$$ 0 0
$$73$$ 12.6132 1.47626 0.738132 0.674656i $$-0.235708\pi$$
0.738132 + 0.674656i $$0.235708\pi$$
$$74$$ 11.4088 1.32625
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −30.0910 −3.42919
$$78$$ 0 0
$$79$$ −1.96345 −0.220906 −0.110453 0.993881i $$-0.535230\pi$$
−0.110453 + 0.993881i $$0.535230\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1.14585 0.126538
$$83$$ 16.2447 1.78309 0.891543 0.452937i $$-0.149624\pi$$
0.891543 + 0.452937i $$0.149624\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 3.55465 0.383308
$$87$$ 0 0
$$88$$ −6.40880 −0.683181
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −4.99477 −0.523594
$$92$$ 1.79560 0.187204
$$93$$ 0 0
$$94$$ 10.8359 1.11764
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.9452 1.51745 0.758727 0.651409i $$-0.225822\pi$$
0.758727 + 0.651409i $$0.225822\pi$$
$$98$$ 15.0455 1.51983
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.5364 −1.04841 −0.524204 0.851592i $$-0.675637\pi$$
−0.524204 + 0.851592i $$0.675637\pi$$
$$102$$ 0 0
$$103$$ 16.9817 1.67326 0.836630 0.547769i $$-0.184523\pi$$
0.836630 + 0.547769i $$0.184523\pi$$
$$104$$ −1.06379 −0.104313
$$105$$ 0 0
$$106$$ −8.69527 −0.844559
$$107$$ −1.79036 −0.173081 −0.0865405 0.996248i $$-0.527581\pi$$
−0.0865405 + 0.996248i $$0.527581\pi$$
$$108$$ 0 0
$$109$$ 2.41404 0.231223 0.115611 0.993295i $$-0.463117\pi$$
0.115611 + 0.993295i $$0.463117\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.69527 0.443661
$$113$$ 7.14585 0.672225 0.336112 0.941822i $$-0.390888\pi$$
0.336112 + 0.941822i $$0.390888\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.93621 −0.272620
$$117$$ 0 0
$$118$$ 5.63148 0.518420
$$119$$ 9.00523 0.825508
$$120$$ 0 0
$$121$$ 30.0728 2.73389
$$122$$ −3.39053 −0.306964
$$123$$ 0 0
$$124$$ −5.55465 −0.498822
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −9.26295 −0.821954 −0.410977 0.911646i $$-0.634812\pi$$
−0.410977 + 0.911646i $$0.634812\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −11.5181 −1.00634 −0.503171 0.864187i $$-0.667833\pi$$
−0.503171 + 0.864187i $$0.667833\pi$$
$$132$$ 0 0
$$133$$ −4.69527 −0.407131
$$134$$ 8.82284 0.762177
$$135$$ 0 0
$$136$$ 1.91794 0.164462
$$137$$ 15.1041 1.29043 0.645214 0.764002i $$-0.276768\pi$$
0.645214 + 0.764002i $$0.276768\pi$$
$$138$$ 0 0
$$139$$ −0.700500 −0.0594156 −0.0297078 0.999559i $$-0.509458\pi$$
−0.0297078 + 0.999559i $$0.509458\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1.42708 0.119758
$$143$$ 6.81761 0.570117
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 12.6132 1.04388
$$147$$ 0 0
$$148$$ 11.4088 0.937798
$$149$$ −12.9452 −1.06051 −0.530255 0.847838i $$-0.677904\pi$$
−0.530255 + 0.847838i $$0.677904\pi$$
$$150$$ 0 0
$$151$$ 5.70830 0.464535 0.232268 0.972652i $$-0.425385\pi$$
0.232268 + 0.972652i $$0.425385\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ −30.0910 −2.42480
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −9.26295 −0.739264 −0.369632 0.929178i $$-0.620516\pi$$
−0.369632 + 0.929178i $$0.620516\pi$$
$$158$$ −1.96345 −0.156204
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.43081 0.664441
$$162$$ 0 0
$$163$$ −4.28123 −0.335332 −0.167666 0.985844i $$-0.553623\pi$$
−0.167666 + 0.985844i $$0.553623\pi$$
$$164$$ 1.14585 0.0894757
$$165$$ 0 0
$$166$$ 16.2447 1.26083
$$167$$ −15.2264 −1.17825 −0.589127 0.808040i $$-0.700528\pi$$
−0.589127 + 0.808040i $$0.700528\pi$$
$$168$$ 0 0
$$169$$ −11.8684 −0.912950
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 3.55465 0.271040
$$173$$ −12.2630 −0.932335 −0.466168 0.884696i $$-0.654366\pi$$
−0.466168 + 0.884696i $$0.654366\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −6.40880 −0.483082
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ 1.57292 0.117566 0.0587829 0.998271i $$-0.481278\pi$$
0.0587829 + 0.998271i $$0.481278\pi$$
$$180$$ 0 0
$$181$$ 11.4088 0.848010 0.424005 0.905660i $$-0.360624\pi$$
0.424005 + 0.905660i $$0.360624\pi$$
$$182$$ −4.99477 −0.370237
$$183$$ 0 0
$$184$$ 1.79560 0.132373
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −12.2917 −0.898858
$$188$$ 10.8359 0.790288
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.35805 0.460053 0.230026 0.973184i $$-0.426119\pi$$
0.230026 + 0.973184i $$0.426119\pi$$
$$192$$ 0 0
$$193$$ −14.4998 −1.04372 −0.521860 0.853031i $$-0.674762\pi$$
−0.521860 + 0.853031i $$0.674762\pi$$
$$194$$ 14.9452 1.07300
$$195$$ 0 0
$$196$$ 15.0455 1.07468
$$197$$ −5.14585 −0.366627 −0.183313 0.983055i $$-0.558682\pi$$
−0.183313 + 0.983055i $$0.558682\pi$$
$$198$$ 0 0
$$199$$ 3.87766 0.274880 0.137440 0.990510i $$-0.456113\pi$$
0.137440 + 0.990510i $$0.456113\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −10.5364 −0.741337
$$203$$ −13.7863 −0.967608
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 16.9817 1.18317
$$207$$ 0 0
$$208$$ −1.06379 −0.0737604
$$209$$ 6.40880 0.443306
$$210$$ 0 0
$$211$$ 17.1496 1.18063 0.590313 0.807174i $$-0.299004\pi$$
0.590313 + 0.807174i $$0.299004\pi$$
$$212$$ −8.69527 −0.597193
$$213$$ 0 0
$$214$$ −1.79036 −0.122387
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −26.0806 −1.77046
$$218$$ 2.41404 0.163499
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −2.04028 −0.137244
$$222$$ 0 0
$$223$$ −7.84635 −0.525430 −0.262715 0.964873i $$-0.584618\pi$$
−0.262715 + 0.964873i $$0.584618\pi$$
$$224$$ 4.69527 0.313716
$$225$$ 0 0
$$226$$ 7.14585 0.475335
$$227$$ 6.28646 0.417247 0.208624 0.977996i $$-0.433102\pi$$
0.208624 + 0.977996i $$0.433102\pi$$
$$228$$ 0 0
$$229$$ −3.14585 −0.207884 −0.103942 0.994583i $$-0.533146\pi$$
−0.103942 + 0.994583i $$0.533146\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −2.93621 −0.192772
$$233$$ 0.182394 0.0119490 0.00597451 0.999982i $$-0.498098\pi$$
0.00597451 + 0.999982i $$0.498098\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 5.63148 0.366578
$$237$$ 0 0
$$238$$ 9.00523 0.583723
$$239$$ 11.5039 0.744126 0.372063 0.928208i $$-0.378651\pi$$
0.372063 + 0.928208i $$0.378651\pi$$
$$240$$ 0 0
$$241$$ −0.445349 −0.0286874 −0.0143437 0.999897i $$-0.504566\pi$$
−0.0143437 + 0.999897i $$0.504566\pi$$
$$242$$ 30.0728 1.93315
$$243$$ 0 0
$$244$$ −3.39053 −0.217056
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.06379 0.0676872
$$248$$ −5.55465 −0.352721
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −13.2630 −0.837150 −0.418575 0.908182i $$-0.637470\pi$$
−0.418575 + 0.908182i $$0.637470\pi$$
$$252$$ 0 0
$$253$$ −11.5076 −0.723479
$$254$$ −9.26295 −0.581209
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −23.4816 −1.46474 −0.732370 0.680907i $$-0.761586\pi$$
−0.732370 + 0.680907i $$0.761586\pi$$
$$258$$ 0 0
$$259$$ 53.5674 3.32851
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −11.5181 −0.711591
$$263$$ 27.9269 1.72205 0.861023 0.508565i $$-0.169824\pi$$
0.861023 + 0.508565i $$0.169824\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.69527 −0.287885
$$267$$ 0 0
$$268$$ 8.82284 0.538941
$$269$$ −16.4816 −1.00490 −0.502449 0.864607i $$-0.667568\pi$$
−0.502449 + 0.864607i $$0.667568\pi$$
$$270$$ 0 0
$$271$$ −1.46736 −0.0891356 −0.0445678 0.999006i $$-0.514191\pi$$
−0.0445678 + 0.999006i $$0.514191\pi$$
$$272$$ 1.91794 0.116292
$$273$$ 0 0
$$274$$ 15.1041 0.912470
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −15.8359 −0.951486 −0.475743 0.879584i $$-0.657821\pi$$
−0.475743 + 0.879584i $$0.657821\pi$$
$$278$$ −0.700500 −0.0420132
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.25515 0.373151 0.186576 0.982441i $$-0.440261\pi$$
0.186576 + 0.982441i $$0.440261\pi$$
$$282$$ 0 0
$$283$$ 16.7005 0.992742 0.496371 0.868111i $$-0.334666\pi$$
0.496371 + 0.868111i $$0.334666\pi$$
$$284$$ 1.42708 0.0846814
$$285$$ 0 0
$$286$$ 6.81761 0.403134
$$287$$ 5.38006 0.317575
$$288$$ 0 0
$$289$$ −13.3215 −0.783618
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 12.6132 0.738132
$$293$$ 0.644516 0.0376530 0.0188265 0.999823i $$-0.494007\pi$$
0.0188265 + 0.999823i $$0.494007\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 11.4088 0.663123
$$297$$ 0 0
$$298$$ −12.9452 −0.749894
$$299$$ −1.91014 −0.110466
$$300$$ 0 0
$$301$$ 16.6900 0.961997
$$302$$ 5.70830 0.328476
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −26.6457 −1.52075 −0.760375 0.649485i $$-0.774985\pi$$
−0.760375 + 0.649485i $$0.774985\pi$$
$$308$$ −30.0910 −1.71460
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −11.5494 −0.654907 −0.327454 0.944867i $$-0.606191\pi$$
−0.327454 + 0.944867i $$0.606191\pi$$
$$312$$ 0 0
$$313$$ 23.0638 1.30364 0.651821 0.758373i $$-0.274005\pi$$
0.651821 + 0.758373i $$0.274005\pi$$
$$314$$ −9.26295 −0.522739
$$315$$ 0 0
$$316$$ −1.96345 −0.110453
$$317$$ −13.9232 −0.782003 −0.391002 0.920390i $$-0.627871\pi$$
−0.391002 + 0.920390i $$0.627871\pi$$
$$318$$ 0 0
$$319$$ 18.8176 1.05358
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 8.43081 0.469831
$$323$$ −1.91794 −0.106717
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −4.28123 −0.237115
$$327$$ 0 0
$$328$$ 1.14585 0.0632689
$$329$$ 50.8773 2.80496
$$330$$ 0 0
$$331$$ −0.735546 −0.0404292 −0.0202146 0.999796i $$-0.506435\pi$$
−0.0202146 + 0.999796i $$0.506435\pi$$
$$332$$ 16.2447 0.891543
$$333$$ 0 0
$$334$$ −15.2264 −0.833152
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.28123 0.124266 0.0621332 0.998068i $$-0.480210\pi$$
0.0621332 + 0.998068i $$0.480210\pi$$
$$338$$ −11.8684 −0.645553
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 35.5987 1.92778
$$342$$ 0 0
$$343$$ 37.7758 2.03970
$$344$$ 3.55465 0.191654
$$345$$ 0 0
$$346$$ −12.2630 −0.659261
$$347$$ −2.56246 −0.137560 −0.0687799 0.997632i $$-0.521911\pi$$
−0.0687799 + 0.997632i $$0.521911\pi$$
$$348$$ 0 0
$$349$$ −5.67176 −0.303602 −0.151801 0.988411i $$-0.548507\pi$$
−0.151801 + 0.988411i $$0.548507\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −6.40880 −0.341590
$$353$$ −23.6665 −1.25964 −0.629821 0.776740i $$-0.716872\pi$$
−0.629821 + 0.776740i $$0.716872\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ 1.57292 0.0831316
$$359$$ −31.9724 −1.68744 −0.843720 0.536784i $$-0.819639\pi$$
−0.843720 + 0.536784i $$0.819639\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 11.4088 0.599633
$$363$$ 0 0
$$364$$ −4.99477 −0.261797
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −1.14585 −0.0598128 −0.0299064 0.999553i $$-0.509521\pi$$
−0.0299064 + 0.999553i $$0.509521\pi$$
$$368$$ 1.79560 0.0936020
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −40.8266 −2.11961
$$372$$ 0 0
$$373$$ −32.8579 −1.70132 −0.850658 0.525719i $$-0.823796\pi$$
−0.850658 + 0.525719i $$0.823796\pi$$
$$374$$ −12.2917 −0.635588
$$375$$ 0 0
$$376$$ 10.8359 0.558818
$$377$$ 3.12351 0.160869
$$378$$ 0 0
$$379$$ −18.2865 −0.939312 −0.469656 0.882849i $$-0.655622\pi$$
−0.469656 + 0.882849i $$0.655622\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 6.35805 0.325306
$$383$$ 20.8542 1.06560 0.532799 0.846242i $$-0.321140\pi$$
0.532799 + 0.846242i $$0.321140\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −14.4998 −0.738022
$$387$$ 0 0
$$388$$ 14.9452 0.758727
$$389$$ 24.9086 1.26292 0.631459 0.775409i $$-0.282456\pi$$
0.631459 + 0.775409i $$0.282456\pi$$
$$390$$ 0 0
$$391$$ 3.44385 0.174163
$$392$$ 15.0455 0.759913
$$393$$ 0 0
$$394$$ −5.14585 −0.259244
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 24.7445 1.24189 0.620946 0.783853i $$-0.286749\pi$$
0.620946 + 0.783853i $$0.286749\pi$$
$$398$$ 3.87766 0.194369
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −15.6457 −0.781308 −0.390654 0.920538i $$-0.627751\pi$$
−0.390654 + 0.920538i $$0.627751\pi$$
$$402$$ 0 0
$$403$$ 5.90897 0.294347
$$404$$ −10.5364 −0.524204
$$405$$ 0 0
$$406$$ −13.7863 −0.684202
$$407$$ −73.1168 −3.62426
$$408$$ 0 0
$$409$$ 30.5804 1.51210 0.756052 0.654512i $$-0.227126\pi$$
0.756052 + 0.654512i $$0.227126\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 16.9817 0.836630
$$413$$ 26.4413 1.30109
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.06379 −0.0521565
$$417$$ 0 0
$$418$$ 6.40880 0.313465
$$419$$ −1.96345 −0.0959210 −0.0479605 0.998849i $$-0.515272\pi$$
−0.0479605 + 0.998849i $$0.515272\pi$$
$$420$$ 0 0
$$421$$ −25.5949 −1.24742 −0.623710 0.781656i $$-0.714376\pi$$
−0.623710 + 0.781656i $$0.714376\pi$$
$$422$$ 17.1496 0.834829
$$423$$ 0 0
$$424$$ −8.69527 −0.422279
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −15.9194 −0.770396
$$428$$ −1.79036 −0.0865405
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.3540 0.739575 0.369788 0.929116i $$-0.379430\pi$$
0.369788 + 0.929116i $$0.379430\pi$$
$$432$$ 0 0
$$433$$ −16.6640 −0.800819 −0.400409 0.916336i $$-0.631132\pi$$
−0.400409 + 0.916336i $$0.631132\pi$$
$$434$$ −26.0806 −1.25191
$$435$$ 0 0
$$436$$ 2.41404 0.115611
$$437$$ −1.79560 −0.0858951
$$438$$ 0 0
$$439$$ −19.5987 −0.935393 −0.467697 0.883889i $$-0.654916\pi$$
−0.467697 + 0.883889i $$0.654916\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −2.04028 −0.0970462
$$443$$ 13.0183 0.618517 0.309258 0.950978i $$-0.399919\pi$$
0.309258 + 0.950978i $$0.399919\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −7.84635 −0.371535
$$447$$ 0 0
$$448$$ 4.69527 0.221830
$$449$$ −6.77359 −0.319666 −0.159833 0.987144i $$-0.551095\pi$$
−0.159833 + 0.987144i $$0.551095\pi$$
$$450$$ 0 0
$$451$$ −7.34352 −0.345793
$$452$$ 7.14585 0.336112
$$453$$ 0 0
$$454$$ 6.28646 0.295038
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 27.3189 1.27793 0.638963 0.769237i $$-0.279364\pi$$
0.638963 + 0.769237i $$0.279364\pi$$
$$458$$ −3.14585 −0.146996
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 27.7993 1.29474 0.647372 0.762174i $$-0.275868\pi$$
0.647372 + 0.762174i $$0.275868\pi$$
$$462$$ 0 0
$$463$$ −38.2369 −1.77702 −0.888509 0.458859i $$-0.848258\pi$$
−0.888509 + 0.458859i $$0.848258\pi$$
$$464$$ −2.93621 −0.136310
$$465$$ 0 0
$$466$$ 0.182394 0.00844923
$$467$$ 23.4711 1.08611 0.543056 0.839696i $$-0.317267\pi$$
0.543056 + 0.839696i $$0.317267\pi$$
$$468$$ 0 0
$$469$$ 41.4256 1.91286
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 5.63148 0.259210
$$473$$ −22.7811 −1.04747
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 9.00523 0.412754
$$477$$ 0 0
$$478$$ 11.5039 0.526176
$$479$$ 19.6900 0.899660 0.449830 0.893114i $$-0.351484\pi$$
0.449830 + 0.893114i $$0.351484\pi$$
$$480$$ 0 0
$$481$$ −12.1365 −0.553379
$$482$$ −0.445349 −0.0202851
$$483$$ 0 0
$$484$$ 30.0728 1.36694
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −16.3357 −0.740242 −0.370121 0.928984i $$-0.620684\pi$$
−0.370121 + 0.928984i $$0.620684\pi$$
$$488$$ −3.39053 −0.153482
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −27.1093 −1.22343 −0.611713 0.791080i $$-0.709519\pi$$
−0.611713 + 0.791080i $$0.709519\pi$$
$$492$$ 0 0
$$493$$ −5.63148 −0.253629
$$494$$ 1.06379 0.0478621
$$495$$ 0 0
$$496$$ −5.55465 −0.249411
$$497$$ 6.70050 0.300558
$$498$$ 0 0
$$499$$ −4.69003 −0.209955 −0.104977 0.994475i $$-0.533477\pi$$
−0.104977 + 0.994475i $$0.533477\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −13.2630 −0.591955
$$503$$ 5.19136 0.231471 0.115736 0.993280i $$-0.463077\pi$$
0.115736 + 0.993280i $$0.463077\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −11.5076 −0.511577
$$507$$ 0 0
$$508$$ −9.26295 −0.410977
$$509$$ 4.51811 0.200262 0.100131 0.994974i $$-0.468074\pi$$
0.100131 + 0.994974i $$0.468074\pi$$
$$510$$ 0 0
$$511$$ 59.2223 2.61984
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −23.4816 −1.03573
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −69.4450 −3.05419
$$518$$ 53.5674 2.35361
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −14.4453 −0.632862 −0.316431 0.948615i $$-0.602485\pi$$
−0.316431 + 0.948615i $$0.602485\pi$$
$$522$$ 0 0
$$523$$ 18.2134 0.796415 0.398208 0.917295i $$-0.369632\pi$$
0.398208 + 0.917295i $$0.369632\pi$$
$$524$$ −11.5181 −0.503171
$$525$$ 0 0
$$526$$ 27.9269 1.21767
$$527$$ −10.6535 −0.464073
$$528$$ 0 0
$$529$$ −19.7758 −0.859819
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −4.69527 −0.203566
$$533$$ −1.21894 −0.0527981
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 8.82284 0.381089
$$537$$ 0 0
$$538$$ −16.4816 −0.710571
$$539$$ −96.4237 −4.15326
$$540$$ 0 0
$$541$$ 37.3905 1.60754 0.803772 0.594937i $$-0.202823\pi$$
0.803772 + 0.594937i $$0.202823\pi$$
$$542$$ −1.46736 −0.0630284
$$543$$ 0 0
$$544$$ 1.91794 0.0822310
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −29.5621 −1.26399 −0.631993 0.774974i $$-0.717763\pi$$
−0.631993 + 0.774974i $$0.717763\pi$$
$$548$$ 15.1041 0.645214
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2.93621 0.125087
$$552$$ 0 0
$$553$$ −9.21894 −0.392029
$$554$$ −15.8359 −0.672802
$$555$$ 0 0
$$556$$ −0.700500 −0.0297078
$$557$$ 14.3723 0.608972 0.304486 0.952517i $$-0.401515\pi$$
0.304486 + 0.952517i $$0.401515\pi$$
$$558$$ 0 0
$$559$$ −3.78139 −0.159936
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.25515 0.263858
$$563$$ −6.39053 −0.269329 −0.134664 0.990891i $$-0.542996\pi$$
−0.134664 + 0.990891i $$0.542996\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 16.7005 0.701974
$$567$$ 0 0
$$568$$ 1.42708 0.0598788
$$569$$ 44.9086 1.88267 0.941334 0.337476i $$-0.109573\pi$$
0.941334 + 0.337476i $$0.109573\pi$$
$$570$$ 0 0
$$571$$ 12.4193 0.519730 0.259865 0.965645i $$-0.416322\pi$$
0.259865 + 0.965645i $$0.416322\pi$$
$$572$$ 6.81761 0.285058
$$573$$ 0 0
$$574$$ 5.38006 0.224559
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 4.20440 0.175032 0.0875158 0.996163i $$-0.472107\pi$$
0.0875158 + 0.996163i $$0.472107\pi$$
$$578$$ −13.3215 −0.554102
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 76.2731 3.16434
$$582$$ 0 0
$$583$$ 55.7262 2.30795
$$584$$ 12.6132 0.521938
$$585$$ 0 0
$$586$$ 0.644516 0.0266247
$$587$$ 24.7915 1.02326 0.511628 0.859207i $$-0.329043\pi$$
0.511628 + 0.859207i $$0.329043\pi$$
$$588$$ 0 0
$$589$$ 5.55465 0.228875
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 11.4088 0.468899
$$593$$ 4.67176 0.191846 0.0959231 0.995389i $$-0.469420\pi$$
0.0959231 + 0.995389i $$0.469420\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −12.9452 −0.530255
$$597$$ 0 0
$$598$$ −1.91014 −0.0781113
$$599$$ −39.2369 −1.60318 −0.801588 0.597877i $$-0.796011\pi$$
−0.801588 + 0.597877i $$0.796011\pi$$
$$600$$ 0 0
$$601$$ −42.4267 −1.73062 −0.865311 0.501235i $$-0.832879\pi$$
−0.865311 + 0.501235i $$0.832879\pi$$
$$602$$ 16.6900 0.680235
$$603$$ 0 0
$$604$$ 5.70830 0.232268
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 2.98173 0.121025 0.0605123 0.998167i $$-0.480727\pi$$
0.0605123 + 0.998167i $$0.480727\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −11.5271 −0.466336
$$612$$ 0 0
$$613$$ 28.9452 1.16908 0.584542 0.811363i $$-0.301274\pi$$
0.584542 + 0.811363i $$0.301274\pi$$
$$614$$ −26.6457 −1.07533
$$615$$ 0 0
$$616$$ −30.0910 −1.21240
$$617$$ −15.0365 −0.605349 −0.302674 0.953094i $$-0.597879\pi$$
−0.302674 + 0.953094i $$0.597879\pi$$
$$618$$ 0 0
$$619$$ 8.25515 0.331803 0.165901 0.986142i $$-0.446947\pi$$
0.165901 + 0.986142i $$0.446947\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −11.5494 −0.463089
$$623$$ 46.9527 1.88112
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 23.0638 0.921814
$$627$$ 0 0
$$628$$ −9.26295 −0.369632
$$629$$ 21.8814 0.872468
$$630$$ 0 0
$$631$$ −9.09103 −0.361908 −0.180954 0.983492i $$-0.557919\pi$$
−0.180954 + 0.983492i $$0.557919\pi$$
$$632$$ −1.96345 −0.0781020
$$633$$ 0 0
$$634$$ −13.9232 −0.552960
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −16.0052 −0.634150
$$638$$ 18.8176 0.744996
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −30.1171 −1.18955 −0.594777 0.803891i $$-0.702760\pi$$
−0.594777 + 0.803891i $$0.702760\pi$$
$$642$$ 0 0
$$643$$ 35.3174 1.39278 0.696392 0.717662i $$-0.254788\pi$$
0.696392 + 0.717662i $$0.254788\pi$$
$$644$$ 8.43081 0.332220
$$645$$ 0 0
$$646$$ −1.91794 −0.0754603
$$647$$ 4.12234 0.162066 0.0810330 0.996711i $$-0.474178\pi$$
0.0810330 + 0.996711i $$0.474178\pi$$
$$648$$ 0 0
$$649$$ −36.0910 −1.41670
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.28123 −0.167666
$$653$$ 8.62741 0.337617 0.168808 0.985649i $$-0.446008\pi$$
0.168808 + 0.985649i $$0.446008\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 1.14585 0.0447379
$$657$$ 0 0
$$658$$ 50.8773 1.98340
$$659$$ 37.3853 1.45632 0.728162 0.685405i $$-0.240375\pi$$
0.728162 + 0.685405i $$0.240375\pi$$
$$660$$ 0 0
$$661$$ −12.8997 −0.501739 −0.250869 0.968021i $$-0.580716\pi$$
−0.250869 + 0.968021i $$0.580716\pi$$
$$662$$ −0.735546 −0.0285878
$$663$$ 0 0
$$664$$ 16.2447 0.630416
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −5.27226 −0.204143
$$668$$ −15.2264 −0.589127
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 21.7292 0.838848
$$672$$ 0 0
$$673$$ −10.4349 −0.402235 −0.201118 0.979567i $$-0.564457\pi$$
−0.201118 + 0.979567i $$0.564457\pi$$
$$674$$ 2.28123 0.0878696
$$675$$ 0 0
$$676$$ −11.8684 −0.456475
$$677$$ 12.4401 0.478112 0.239056 0.971006i $$-0.423162\pi$$
0.239056 + 0.971006i $$0.423162\pi$$
$$678$$ 0 0
$$679$$ 70.1716 2.69294
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 35.5987 1.36314
$$683$$ 17.5364 0.671011 0.335505 0.942038i $$-0.391093\pi$$
0.335505 + 0.942038i $$0.391093\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 37.7758 1.44229
$$687$$ 0 0
$$688$$ 3.55465 0.135520
$$689$$ 9.24992 0.352394
$$690$$ 0 0
$$691$$ −25.2734 −0.961446 −0.480723 0.876872i $$-0.659626\pi$$
−0.480723 + 0.876872i $$0.659626\pi$$
$$692$$ −12.2630 −0.466168
$$693$$ 0 0
$$694$$ −2.56246 −0.0972695
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.19767 0.0832426
$$698$$ −5.67176 −0.214679
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 16.0545 0.606370 0.303185 0.952932i $$-0.401950\pi$$
0.303185 + 0.952932i $$0.401950\pi$$
$$702$$ 0 0
$$703$$ −11.4088 −0.430291
$$704$$ −6.40880 −0.241541
$$705$$ 0 0
$$706$$ −23.6665 −0.890701
$$707$$ −49.4711 −1.86055
$$708$$ 0 0
$$709$$ 13.5076 0.507290 0.253645 0.967297i $$-0.418371\pi$$
0.253645 + 0.967297i $$0.418371\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ −9.97392 −0.373526
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 1.57292 0.0587829
$$717$$ 0 0
$$718$$ −31.9724 −1.19320
$$719$$ −0.803402 −0.0299619 −0.0149809 0.999888i $$-0.504769\pi$$
−0.0149809 + 0.999888i $$0.504769\pi$$
$$720$$ 0 0
$$721$$ 79.7337 2.96944
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ 11.4088 0.424005
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −51.6860 −1.91693 −0.958463 0.285217i $$-0.907934\pi$$
−0.958463 + 0.285217i $$0.907934\pi$$
$$728$$ −4.99477 −0.185118
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 6.81761 0.252158
$$732$$ 0 0
$$733$$ 22.3723 0.826338 0.413169 0.910654i $$-0.364422\pi$$
0.413169 + 0.910654i $$0.364422\pi$$
$$734$$ −1.14585 −0.0422940
$$735$$ 0 0
$$736$$ 1.79560 0.0661866
$$737$$ −56.5438 −2.08282
$$738$$ 0 0
$$739$$ 33.4271 1.22963 0.614817 0.788669i $$-0.289230\pi$$
0.614817 + 0.788669i $$0.289230\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −40.8266 −1.49879
$$743$$ 14.4998 0.531947 0.265974 0.963980i $$-0.414307\pi$$
0.265974 + 0.963980i $$0.414307\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −32.8579 −1.20301
$$747$$ 0 0
$$748$$ −12.2917 −0.449429
$$749$$ −8.40623 −0.307157
$$750$$ 0 0
$$751$$ −26.6169 −0.971266 −0.485633 0.874163i $$-0.661411\pi$$
−0.485633 + 0.874163i $$0.661411\pi$$
$$752$$ 10.8359 0.395144
$$753$$ 0 0
$$754$$ 3.12351 0.113751
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 31.0362 1.12803 0.564015 0.825764i $$-0.309256\pi$$
0.564015 + 0.825764i $$0.309256\pi$$
$$758$$ −18.2865 −0.664194
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 27.8083 1.00805 0.504025 0.863689i $$-0.331852\pi$$
0.504025 + 0.863689i $$0.331852\pi$$
$$762$$ 0 0
$$763$$ 11.3345 0.410338
$$764$$ 6.35805 0.230026
$$765$$ 0 0
$$766$$ 20.8542 0.753491
$$767$$ −5.99070 −0.216312
$$768$$ 0 0
$$769$$ −19.2279 −0.693376 −0.346688 0.937980i $$-0.612694\pi$$
−0.346688 + 0.937980i $$0.612694\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14.4998 −0.521860
$$773$$ 6.00373 0.215939 0.107970 0.994154i $$-0.465565\pi$$
0.107970 + 0.994154i $$0.465565\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 14.9452 0.536501
$$777$$ 0 0
$$778$$ 24.9086 0.893018
$$779$$ −1.14585 −0.0410543
$$780$$ 0 0
$$781$$ −9.14585 −0.327264
$$782$$ 3.44385 0.123152
$$783$$ 0 0
$$784$$ 15.0455 0.537340
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −22.2797 −0.794187 −0.397093 0.917778i $$-0.629981\pi$$
−0.397093 + 0.917778i $$0.629981\pi$$
$$788$$ −5.14585 −0.183313
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 33.5517 1.19296
$$792$$ 0 0
$$793$$ 3.60680 0.128081
$$794$$ 24.7445 0.878150
$$795$$ 0 0
$$796$$ 3.87766 0.137440
$$797$$ 26.6259 0.943138 0.471569 0.881829i $$-0.343688\pi$$
0.471569 + 0.881829i $$0.343688\pi$$
$$798$$ 0 0
$$799$$ 20.7826 0.735234
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −15.6457 −0.552468
$$803$$ −80.8355 −2.85262
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 5.90897 0.208135
$$807$$ 0 0
$$808$$ −10.5364 −0.370669
$$809$$ −0.651250 −0.0228967 −0.0114484 0.999934i $$-0.503644\pi$$
−0.0114484 + 0.999934i $$0.503644\pi$$
$$810$$ 0 0
$$811$$ −13.0780 −0.459230 −0.229615 0.973281i $$-0.573747\pi$$
−0.229615 + 0.973281i $$0.573747\pi$$
$$812$$ −13.7863 −0.483804
$$813$$ 0 0
$$814$$ −73.1168 −2.56274
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −3.55465 −0.124362
$$818$$ 30.5804 1.06922
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −14.6640 −0.511776 −0.255888 0.966706i $$-0.582368\pi$$
−0.255888 + 0.966706i $$0.582368\pi$$
$$822$$ 0 0
$$823$$ 18.2850 0.637374 0.318687 0.947860i $$-0.396758\pi$$
0.318687 + 0.947860i $$0.396758\pi$$
$$824$$ 16.9817 0.591587
$$825$$ 0 0
$$826$$ 26.4413 0.920010
$$827$$ 40.1910 1.39758 0.698790 0.715327i $$-0.253722\pi$$
0.698790 + 0.715327i $$0.253722\pi$$
$$828$$ 0 0
$$829$$ 28.3760 0.985539 0.492769 0.870160i $$-0.335985\pi$$
0.492769 + 0.870160i $$0.335985\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.06379 −0.0368802
$$833$$ 28.8564 0.999815
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 6.40880 0.221653
$$837$$ 0 0
$$838$$ −1.96345 −0.0678264
$$839$$ −6.93471 −0.239413 −0.119706 0.992809i $$-0.538195\pi$$
−0.119706 + 0.992809i $$0.538195\pi$$
$$840$$ 0 0
$$841$$ −20.3787 −0.702712
$$842$$ −25.5949 −0.882060
$$843$$ 0 0
$$844$$ 17.1496 0.590313
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 141.200 4.85167
$$848$$ −8.69527 −0.298597
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 20.4856 0.702238
$$852$$ 0 0
$$853$$ −21.9739 −0.752373 −0.376186 0.926544i $$-0.622765\pi$$
−0.376186 + 0.926544i $$0.622765\pi$$
$$854$$ −15.9194 −0.544752
$$855$$ 0 0
$$856$$ −1.79036 −0.0611934
$$857$$ 13.6091 0.464879 0.232440 0.972611i $$-0.425329\pi$$
0.232440 + 0.972611i $$0.425329\pi$$
$$858$$ 0 0
$$859$$ 5.17192 0.176464 0.0882319 0.996100i $$-0.471878\pi$$
0.0882319 + 0.996100i $$0.471878\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 15.3540 0.522959
$$863$$ 15.9635 0.543402 0.271701 0.962382i $$-0.412414\pi$$
0.271701 + 0.962382i $$0.412414\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −16.6640 −0.566264
$$867$$ 0 0
$$868$$ −26.0806 −0.885232
$$869$$ 12.5834 0.426862
$$870$$ 0 0
$$871$$ −9.38563 −0.318020
$$872$$ 2.41404 0.0817496
$$873$$ 0 0
$$874$$ −1.79560 −0.0607370
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 15.8866 0.536453 0.268227 0.963356i $$-0.413562\pi$$
0.268227 + 0.963356i $$0.413562\pi$$
$$878$$ −19.5987 −0.661423
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 16.1458 0.543967 0.271984 0.962302i $$-0.412320\pi$$
0.271984 + 0.962302i $$0.412320\pi$$
$$882$$ 0 0
$$883$$ −38.2887 −1.28852 −0.644259 0.764808i $$-0.722834\pi$$
−0.644259 + 0.764808i $$0.722834\pi$$
$$884$$ −2.04028 −0.0686221
$$885$$ 0 0
$$886$$ 13.0183 0.437357
$$887$$ 19.2809 0.647389 0.323695 0.946162i $$-0.395075\pi$$
0.323695 + 0.946162i $$0.395075\pi$$
$$888$$ 0 0
$$889$$ −43.4920 −1.45868
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −7.84635 −0.262715
$$893$$ −10.8359 −0.362609
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 4.69527 0.156858
$$897$$ 0 0
$$898$$ −6.77359 −0.226038
$$899$$ 16.3096 0.543957
$$900$$ 0 0
$$901$$ −16.6770 −0.555591
$$902$$ −7.34352 −0.244512
$$903$$ 0 0
$$904$$ 7.14585 0.237667
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −43.0205 −1.42847 −0.714236 0.699905i $$-0.753226\pi$$
−0.714236 + 0.699905i $$0.753226\pi$$
$$908$$ 6.28646 0.208624
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 25.7733 0.853906 0.426953 0.904274i $$-0.359587\pi$$
0.426953 + 0.904274i $$0.359587\pi$$
$$912$$ 0 0
$$913$$ −104.109 −3.44550
$$914$$ 27.3189 0.903630
$$915$$ 0 0
$$916$$ −3.14585 −0.103942
$$917$$ −54.0806 −1.78590
$$918$$ 0 0
$$919$$ −29.5897 −0.976074 −0.488037 0.872823i $$-0.662287\pi$$
−0.488037 + 0.872823i $$0.662287\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 27.7993 0.915522
$$923$$ −1.51811 −0.0499691
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −38.2369 −1.25654
$$927$$ 0 0
$$928$$ −2.93621 −0.0963859
$$929$$ 1.42334 0.0466983 0.0233491 0.999727i $$-0.492567\pi$$
0.0233491 + 0.999727i $$0.492567\pi$$
$$930$$ 0 0
$$931$$ −15.0455 −0.493097
$$932$$ 0.182394 0.00597451
$$933$$ 0 0
$$934$$ 23.4711 0.767998
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −28.2276 −0.922155 −0.461077 0.887360i $$-0.652537\pi$$
−0.461077 + 0.887360i $$0.652537\pi$$
$$938$$ 41.4256 1.35259
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 15.8672 0.517256 0.258628 0.965977i $$-0.416730\pi$$
0.258628 + 0.965977i $$0.416730\pi$$
$$942$$ 0 0
$$943$$ 2.05748 0.0670009
$$944$$ 5.63148 0.183289
$$945$$ 0 0
$$946$$ −22.7811 −0.740676
$$947$$ 43.6718 1.41914 0.709571 0.704634i $$-0.248889\pi$$
0.709571 + 0.704634i $$0.248889\pi$$
$$948$$ 0 0
$$949$$ −13.4178 −0.435559
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 9.00523 0.291861
$$953$$ 16.0261 0.519136 0.259568 0.965725i $$-0.416420\pi$$
0.259568 + 0.965725i $$0.416420\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 11.5039 0.372063
$$957$$ 0 0
$$958$$ 19.6900 0.636156
$$959$$ 70.9176 2.29005
$$960$$ 0 0
$$961$$ −0.145848 −0.00470478
$$962$$ −12.1365 −0.391298
$$963$$ 0 0
$$964$$ −0.445349 −0.0143437
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 11.5987 0.372988 0.186494 0.982456i $$-0.440288\pi$$
0.186494 + 0.982456i $$0.440288\pi$$
$$968$$ 30.0728 0.966575
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 27.0362 0.867633 0.433817 0.901001i $$-0.357167\pi$$
0.433817 + 0.901001i $$0.357167\pi$$
$$972$$ 0 0
$$973$$ −3.28903 −0.105442
$$974$$ −16.3357 −0.523430
$$975$$ 0 0
$$976$$ −3.39053 −0.108528
$$977$$ 14.1537 0.452815 0.226408 0.974033i $$-0.427302\pi$$
0.226408 + 0.974033i $$0.427302\pi$$
$$978$$ 0 0
$$979$$ −64.0880 −2.04826
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −27.1093 −0.865093
$$983$$ −32.8542 −1.04788 −0.523942 0.851754i $$-0.675539\pi$$
−0.523942 + 0.851754i $$0.675539\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −5.63148 −0.179343
$$987$$ 0 0
$$988$$ 1.06379 0.0338436
$$989$$ 6.38273 0.202959
$$990$$ 0 0
$$991$$ 47.9709 1.52385 0.761923 0.647667i $$-0.224255\pi$$
0.761923 + 0.647667i $$0.224255\pi$$
$$992$$ −5.55465 −0.176360
$$993$$ 0 0
$$994$$ 6.70050 0.212527
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −44.1716 −1.39893 −0.699464 0.714668i $$-0.746578\pi$$
−0.699464 + 0.714668i $$0.746578\pi$$
$$998$$ −4.69003 −0.148460
$$999$$ 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 8550.2.a.co.1.3 3
3.2 odd 2 950.2.a.k.1.1 3
5.4 even 2 8550.2.a.cj.1.1 3
12.11 even 2 7600.2.a.cb.1.3 3
15.2 even 4 950.2.b.g.799.3 6
15.8 even 4 950.2.b.g.799.4 6
15.14 odd 2 950.2.a.m.1.3 yes 3
60.59 even 2 7600.2.a.bm.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.1 3 3.2 odd 2
950.2.a.m.1.3 yes 3 15.14 odd 2
950.2.b.g.799.3 6 15.2 even 4
950.2.b.g.799.4 6 15.8 even 4
7600.2.a.bm.1.1 3 60.59 even 2
7600.2.a.cb.1.3 3 12.11 even 2
8550.2.a.cj.1.1 3 5.4 even 2
8550.2.a.co.1.3 3 1.1 even 1 trivial