# Properties

 Label 171.2.a.e.1.4 Level $171$ Weight $2$ Character 171.1 Self dual yes Analytic conductor $1.365$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,2,Mod(1,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, 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("171.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 171.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: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13068.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} - x + 1$$ x^4 - x^3 - 6*x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$0.328543$$ of defining polynomial Character $$\chi$$ $$=$$ 171.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.71519 q^{2} +5.37228 q^{4} -3.22060 q^{5} -2.37228 q^{7} +9.15640 q^{8} +O(q^{10})$$ $$q+2.71519 q^{2} +5.37228 q^{4} -3.22060 q^{5} -2.37228 q^{7} +9.15640 q^{8} -8.74456 q^{10} -2.20979 q^{11} +2.00000 q^{13} -6.44121 q^{14} +14.1168 q^{16} -3.22060 q^{17} +1.00000 q^{19} -17.3020 q^{20} -6.00000 q^{22} +1.01082 q^{23} +5.37228 q^{25} +5.43039 q^{26} -12.7446 q^{28} +1.01082 q^{29} +4.74456 q^{31} +20.0172 q^{32} -8.74456 q^{34} +7.64018 q^{35} +10.7446 q^{37} +2.71519 q^{38} -29.4891 q^{40} -5.43039 q^{41} -11.1168 q^{43} -11.8716 q^{44} +2.74456 q^{46} -4.23142 q^{47} -1.37228 q^{49} +14.5868 q^{50} +10.7446 q^{52} +9.84996 q^{53} +7.11684 q^{55} -21.7216 q^{56} +2.74456 q^{58} +10.8608 q^{59} -5.11684 q^{61} +12.8824 q^{62} +26.1168 q^{64} -6.44121 q^{65} -4.00000 q^{67} -17.3020 q^{68} +20.7446 q^{70} +2.02163 q^{71} -5.11684 q^{73} +29.1736 q^{74} +5.37228 q^{76} +5.24224 q^{77} -4.00000 q^{79} -45.4647 q^{80} -14.7446 q^{82} -11.8716 q^{83} +10.3723 q^{85} -30.1844 q^{86} -20.2337 q^{88} +9.84996 q^{89} -4.74456 q^{91} +5.43039 q^{92} -11.4891 q^{94} -3.22060 q^{95} +7.48913 q^{97} -3.72601 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{4} + 2 q^{7}+O(q^{10})$$ 4 * q + 10 * q^4 + 2 * q^7 $$4 q + 10 q^{4} + 2 q^{7} - 12 q^{10} + 8 q^{13} + 22 q^{16} + 4 q^{19} - 24 q^{22} + 10 q^{25} - 28 q^{28} - 4 q^{31} - 12 q^{34} + 20 q^{37} - 72 q^{40} - 10 q^{43} - 12 q^{46} + 6 q^{49} + 20 q^{52} - 6 q^{55} - 12 q^{58} + 14 q^{61} + 70 q^{64} - 16 q^{67} + 60 q^{70} + 14 q^{73} + 10 q^{76} - 16 q^{79} - 36 q^{82} + 30 q^{85} - 12 q^{88} + 4 q^{91} - 16 q^{97}+O(q^{100})$$ 4 * q + 10 * q^4 + 2 * q^7 - 12 * q^10 + 8 * q^13 + 22 * q^16 + 4 * q^19 - 24 * q^22 + 10 * q^25 - 28 * q^28 - 4 * q^31 - 12 * q^34 + 20 * q^37 - 72 * q^40 - 10 * q^43 - 12 * q^46 + 6 * q^49 + 20 * q^52 - 6 * q^55 - 12 * q^58 + 14 * q^61 + 70 * q^64 - 16 * q^67 + 60 * q^70 + 14 * q^73 + 10 * q^76 - 16 * q^79 - 36 * q^82 + 30 * q^85 - 12 * q^88 + 4 * q^91 - 16 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.71519 1.91993 0.959966 0.280116i $$-0.0903729\pi$$
0.959966 + 0.280116i $$0.0903729\pi$$
$$3$$ 0 0
$$4$$ 5.37228 2.68614
$$5$$ −3.22060 −1.44030 −0.720149 0.693820i $$-0.755926\pi$$
−0.720149 + 0.693820i $$0.755926\pi$$
$$6$$ 0 0
$$7$$ −2.37228 −0.896638 −0.448319 0.893874i $$-0.647977\pi$$
−0.448319 + 0.893874i $$0.647977\pi$$
$$8$$ 9.15640 3.23728
$$9$$ 0 0
$$10$$ −8.74456 −2.76527
$$11$$ −2.20979 −0.666276 −0.333138 0.942878i $$-0.608107\pi$$
−0.333138 + 0.942878i $$0.608107\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −6.44121 −1.72148
$$15$$ 0 0
$$16$$ 14.1168 3.52921
$$17$$ −3.22060 −0.781111 −0.390555 0.920579i $$-0.627717\pi$$
−0.390555 + 0.920579i $$0.627717\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ −17.3020 −3.86884
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ 1.01082 0.210770 0.105385 0.994432i $$-0.466393\pi$$
0.105385 + 0.994432i $$0.466393\pi$$
$$24$$ 0 0
$$25$$ 5.37228 1.07446
$$26$$ 5.43039 1.06499
$$27$$ 0 0
$$28$$ −12.7446 −2.40850
$$29$$ 1.01082 0.187704 0.0938519 0.995586i $$-0.470082\pi$$
0.0938519 + 0.995586i $$0.470082\pi$$
$$30$$ 0 0
$$31$$ 4.74456 0.852149 0.426074 0.904688i $$-0.359896\pi$$
0.426074 + 0.904688i $$0.359896\pi$$
$$32$$ 20.0172 3.53857
$$33$$ 0 0
$$34$$ −8.74456 −1.49968
$$35$$ 7.64018 1.29143
$$36$$ 0 0
$$37$$ 10.7446 1.76640 0.883198 0.469001i $$-0.155386\pi$$
0.883198 + 0.469001i $$0.155386\pi$$
$$38$$ 2.71519 0.440463
$$39$$ 0 0
$$40$$ −29.4891 −4.66264
$$41$$ −5.43039 −0.848084 −0.424042 0.905642i $$-0.639389\pi$$
−0.424042 + 0.905642i $$0.639389\pi$$
$$42$$ 0 0
$$43$$ −11.1168 −1.69530 −0.847651 0.530554i $$-0.821984\pi$$
−0.847651 + 0.530554i $$0.821984\pi$$
$$44$$ −11.8716 −1.78971
$$45$$ 0 0
$$46$$ 2.74456 0.404664
$$47$$ −4.23142 −0.617216 −0.308608 0.951189i $$-0.599863\pi$$
−0.308608 + 0.951189i $$0.599863\pi$$
$$48$$ 0 0
$$49$$ −1.37228 −0.196040
$$50$$ 14.5868 2.06288
$$51$$ 0 0
$$52$$ 10.7446 1.49000
$$53$$ 9.84996 1.35300 0.676498 0.736444i $$-0.263497\pi$$
0.676498 + 0.736444i $$0.263497\pi$$
$$54$$ 0 0
$$55$$ 7.11684 0.959635
$$56$$ −21.7216 −2.90267
$$57$$ 0 0
$$58$$ 2.74456 0.360379
$$59$$ 10.8608 1.41395 0.706976 0.707237i $$-0.250059\pi$$
0.706976 + 0.707237i $$0.250059\pi$$
$$60$$ 0 0
$$61$$ −5.11684 −0.655145 −0.327572 0.944826i $$-0.606231\pi$$
−0.327572 + 0.944826i $$0.606231\pi$$
$$62$$ 12.8824 1.63607
$$63$$ 0 0
$$64$$ 26.1168 3.26461
$$65$$ −6.44121 −0.798933
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −17.3020 −2.09817
$$69$$ 0 0
$$70$$ 20.7446 2.47945
$$71$$ 2.02163 0.239924 0.119962 0.992779i $$-0.461723\pi$$
0.119962 + 0.992779i $$0.461723\pi$$
$$72$$ 0 0
$$73$$ −5.11684 −0.598881 −0.299441 0.954115i $$-0.596800\pi$$
−0.299441 + 0.954115i $$0.596800\pi$$
$$74$$ 29.1736 3.39136
$$75$$ 0 0
$$76$$ 5.37228 0.616243
$$77$$ 5.24224 0.597408
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ −45.4647 −5.08311
$$81$$ 0 0
$$82$$ −14.7446 −1.62826
$$83$$ −11.8716 −1.30308 −0.651538 0.758616i $$-0.725876\pi$$
−0.651538 + 0.758616i $$0.725876\pi$$
$$84$$ 0 0
$$85$$ 10.3723 1.12503
$$86$$ −30.1844 −3.25487
$$87$$ 0 0
$$88$$ −20.2337 −2.15692
$$89$$ 9.84996 1.04409 0.522047 0.852917i $$-0.325169\pi$$
0.522047 + 0.852917i $$0.325169\pi$$
$$90$$ 0 0
$$91$$ −4.74456 −0.497365
$$92$$ 5.43039 0.566157
$$93$$ 0 0
$$94$$ −11.4891 −1.18501
$$95$$ −3.22060 −0.330427
$$96$$ 0 0
$$97$$ 7.48913 0.760405 0.380203 0.924903i $$-0.375854\pi$$
0.380203 + 0.924903i $$0.375854\pi$$
$$98$$ −3.72601 −0.376384
$$99$$ 0 0
$$100$$ 28.8614 2.88614
$$101$$ 12.8824 1.28185 0.640924 0.767604i $$-0.278551\pi$$
0.640924 + 0.767604i $$0.278551\pi$$
$$102$$ 0 0
$$103$$ −7.25544 −0.714899 −0.357450 0.933932i $$-0.616354\pi$$
−0.357450 + 0.933932i $$0.616354\pi$$
$$104$$ 18.3128 1.79572
$$105$$ 0 0
$$106$$ 26.7446 2.59766
$$107$$ 4.41957 0.427256 0.213628 0.976915i $$-0.431472\pi$$
0.213628 + 0.976915i $$0.431472\pi$$
$$108$$ 0 0
$$109$$ 5.25544 0.503380 0.251690 0.967808i $$-0.419014\pi$$
0.251690 + 0.967808i $$0.419014\pi$$
$$110$$ 19.3236 1.84243
$$111$$ 0 0
$$112$$ −33.4891 −3.16442
$$113$$ −11.8716 −1.11679 −0.558393 0.829577i $$-0.688582\pi$$
−0.558393 + 0.829577i $$0.688582\pi$$
$$114$$ 0 0
$$115$$ −3.25544 −0.303571
$$116$$ 5.43039 0.504199
$$117$$ 0 0
$$118$$ 29.4891 2.71469
$$119$$ 7.64018 0.700374
$$120$$ 0 0
$$121$$ −6.11684 −0.556077
$$122$$ −13.8932 −1.25783
$$123$$ 0 0
$$124$$ 25.4891 2.28899
$$125$$ −1.19897 −0.107239
$$126$$ 0 0
$$127$$ −12.7446 −1.13090 −0.565449 0.824784i $$-0.691297\pi$$
−0.565449 + 0.824784i $$0.691297\pi$$
$$128$$ 30.8780 2.72925
$$129$$ 0 0
$$130$$ −17.4891 −1.53390
$$131$$ −15.0922 −1.31861 −0.659306 0.751875i $$-0.729150\pi$$
−0.659306 + 0.751875i $$0.729150\pi$$
$$132$$ 0 0
$$133$$ −2.37228 −0.205703
$$134$$ −10.8608 −0.938228
$$135$$ 0 0
$$136$$ −29.4891 −2.52867
$$137$$ 12.0597 1.03033 0.515167 0.857090i $$-0.327730\pi$$
0.515167 + 0.857090i $$0.327730\pi$$
$$138$$ 0 0
$$139$$ 3.11684 0.264367 0.132184 0.991225i $$-0.457801\pi$$
0.132184 + 0.991225i $$0.457801\pi$$
$$140$$ 41.0452 3.46895
$$141$$ 0 0
$$142$$ 5.48913 0.460637
$$143$$ −4.41957 −0.369583
$$144$$ 0 0
$$145$$ −3.25544 −0.270349
$$146$$ −13.8932 −1.14981
$$147$$ 0 0
$$148$$ 57.7228 4.74479
$$149$$ 20.5226 1.68128 0.840638 0.541598i $$-0.182180\pi$$
0.840638 + 0.541598i $$0.182180\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 9.15640 0.742682
$$153$$ 0 0
$$154$$ 14.2337 1.14698
$$155$$ −15.2804 −1.22735
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ −10.8608 −0.864037
$$159$$ 0 0
$$160$$ −64.4674 −5.09659
$$161$$ −2.39794 −0.188984
$$162$$ 0 0
$$163$$ −21.4891 −1.68316 −0.841579 0.540134i $$-0.818374\pi$$
−0.841579 + 0.540134i $$0.818374\pi$$
$$164$$ −29.1736 −2.27807
$$165$$ 0 0
$$166$$ −32.2337 −2.50182
$$167$$ −6.44121 −0.498435 −0.249218 0.968447i $$-0.580173\pi$$
−0.249218 + 0.968447i $$0.580173\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 28.1628 2.15999
$$171$$ 0 0
$$172$$ −59.7228 −4.55382
$$173$$ −1.01082 −0.0768509 −0.0384255 0.999261i $$-0.512234\pi$$
−0.0384255 + 0.999261i $$0.512234\pi$$
$$174$$ 0 0
$$175$$ −12.7446 −0.963398
$$176$$ −31.1952 −2.35143
$$177$$ 0 0
$$178$$ 26.7446 2.00459
$$179$$ −4.41957 −0.330334 −0.165167 0.986266i $$-0.552816\pi$$
−0.165167 + 0.986266i $$0.552816\pi$$
$$180$$ 0 0
$$181$$ 19.4891 1.44862 0.724308 0.689477i $$-0.242160\pi$$
0.724308 + 0.689477i $$0.242160\pi$$
$$182$$ −12.8824 −0.954908
$$183$$ 0 0
$$184$$ 9.25544 0.682320
$$185$$ −34.6040 −2.54413
$$186$$ 0 0
$$187$$ 7.11684 0.520435
$$188$$ −22.7324 −1.65793
$$189$$ 0 0
$$190$$ −8.74456 −0.634397
$$191$$ −21.5334 −1.55810 −0.779051 0.626960i $$-0.784299\pi$$
−0.779051 + 0.626960i $$0.784299\pi$$
$$192$$ 0 0
$$193$$ 16.2337 1.16853 0.584263 0.811564i $$-0.301384\pi$$
0.584263 + 0.811564i $$0.301384\pi$$
$$194$$ 20.3344 1.45993
$$195$$ 0 0
$$196$$ −7.37228 −0.526592
$$197$$ −23.7432 −1.69163 −0.845816 0.533475i $$-0.820886\pi$$
−0.845816 + 0.533475i $$0.820886\pi$$
$$198$$ 0 0
$$199$$ 0.883156 0.0626053 0.0313026 0.999510i $$-0.490034\pi$$
0.0313026 + 0.999510i $$0.490034\pi$$
$$200$$ 49.1908 3.47831
$$201$$ 0 0
$$202$$ 34.9783 2.46106
$$203$$ −2.39794 −0.168302
$$204$$ 0 0
$$205$$ 17.4891 1.22149
$$206$$ −19.6999 −1.37256
$$207$$ 0 0
$$208$$ 28.2337 1.95765
$$209$$ −2.20979 −0.152854
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 52.9168 3.63434
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 35.8029 2.44174
$$216$$ 0 0
$$217$$ −11.2554 −0.764069
$$218$$ 14.2695 0.966455
$$219$$ 0 0
$$220$$ 38.2337 2.57771
$$221$$ −6.44121 −0.433282
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −47.4864 −3.17282
$$225$$ 0 0
$$226$$ −32.2337 −2.14415
$$227$$ 19.3236 1.28255 0.641277 0.767310i $$-0.278405\pi$$
0.641277 + 0.767310i $$0.278405\pi$$
$$228$$ 0 0
$$229$$ 15.6277 1.03271 0.516354 0.856375i $$-0.327289\pi$$
0.516354 + 0.856375i $$0.327289\pi$$
$$230$$ −8.83915 −0.582836
$$231$$ 0 0
$$232$$ 9.25544 0.607649
$$233$$ −7.64018 −0.500525 −0.250262 0.968178i $$-0.580517\pi$$
−0.250262 + 0.968178i $$0.580517\pi$$
$$234$$ 0 0
$$235$$ 13.6277 0.888974
$$236$$ 58.3472 3.79808
$$237$$ 0 0
$$238$$ 20.7446 1.34467
$$239$$ −12.6943 −0.821123 −0.410562 0.911833i $$-0.634667\pi$$
−0.410562 + 0.911833i $$0.634667\pi$$
$$240$$ 0 0
$$241$$ −24.2337 −1.56103 −0.780515 0.625138i $$-0.785043\pi$$
−0.780515 + 0.625138i $$0.785043\pi$$
$$242$$ −16.6084 −1.06763
$$243$$ 0 0
$$244$$ −27.4891 −1.75981
$$245$$ 4.41957 0.282356
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 43.4431 2.75864
$$249$$ 0 0
$$250$$ −3.25544 −0.205892
$$251$$ 23.9313 1.51053 0.755266 0.655418i $$-0.227507\pi$$
0.755266 + 0.655418i $$0.227507\pi$$
$$252$$ 0 0
$$253$$ −2.23369 −0.140431
$$254$$ −34.6040 −2.17125
$$255$$ 0 0
$$256$$ 31.6060 1.97537
$$257$$ 5.43039 0.338738 0.169369 0.985553i $$-0.445827\pi$$
0.169369 + 0.985553i $$0.445827\pi$$
$$258$$ 0 0
$$259$$ −25.4891 −1.58382
$$260$$ −34.6040 −2.14605
$$261$$ 0 0
$$262$$ −40.9783 −2.53164
$$263$$ 13.0706 0.805966 0.402983 0.915208i $$-0.367973\pi$$
0.402983 + 0.915208i $$0.367973\pi$$
$$264$$ 0 0
$$265$$ −31.7228 −1.94872
$$266$$ −6.44121 −0.394936
$$267$$ 0 0
$$268$$ −21.4891 −1.31266
$$269$$ 7.82833 0.477302 0.238651 0.971105i $$-0.423295\pi$$
0.238651 + 0.971105i $$0.423295\pi$$
$$270$$ 0 0
$$271$$ 25.4891 1.54835 0.774177 0.632969i $$-0.218164\pi$$
0.774177 + 0.632969i $$0.218164\pi$$
$$272$$ −45.4647 −2.75671
$$273$$ 0 0
$$274$$ 32.7446 1.97817
$$275$$ −11.8716 −0.715884
$$276$$ 0 0
$$277$$ 9.11684 0.547778 0.273889 0.961761i $$-0.411690\pi$$
0.273889 + 0.961761i $$0.411690\pi$$
$$278$$ 8.46284 0.507567
$$279$$ 0 0
$$280$$ 69.9565 4.18070
$$281$$ 24.7540 1.47670 0.738350 0.674418i $$-0.235605\pi$$
0.738350 + 0.674418i $$0.235605\pi$$
$$282$$ 0 0
$$283$$ 6.37228 0.378793 0.189396 0.981901i $$-0.439347\pi$$
0.189396 + 0.981901i $$0.439347\pi$$
$$284$$ 10.8608 0.644469
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ 12.8824 0.760425
$$288$$ 0 0
$$289$$ −6.62772 −0.389866
$$290$$ −8.83915 −0.519053
$$291$$ 0 0
$$292$$ −27.4891 −1.60868
$$293$$ −25.1303 −1.46813 −0.734064 0.679080i $$-0.762379\pi$$
−0.734064 + 0.679080i $$0.762379\pi$$
$$294$$ 0 0
$$295$$ −34.9783 −2.03651
$$296$$ 98.3815 5.71831
$$297$$ 0 0
$$298$$ 55.7228 3.22794
$$299$$ 2.02163 0.116914
$$300$$ 0 0
$$301$$ 26.3723 1.52007
$$302$$ −10.8608 −0.624968
$$303$$ 0 0
$$304$$ 14.1168 0.809657
$$305$$ 16.4793 0.943603
$$306$$ 0 0
$$307$$ 25.4891 1.45474 0.727371 0.686245i $$-0.240742\pi$$
0.727371 + 0.686245i $$0.240742\pi$$
$$308$$ 28.1628 1.60472
$$309$$ 0 0
$$310$$ −41.4891 −2.35642
$$311$$ 12.6943 0.719825 0.359913 0.932986i $$-0.382806\pi$$
0.359913 + 0.932986i $$0.382806\pi$$
$$312$$ 0 0
$$313$$ 7.48913 0.423310 0.211655 0.977344i $$-0.432115\pi$$
0.211655 + 0.977344i $$0.432115\pi$$
$$314$$ 5.43039 0.306455
$$315$$ 0 0
$$316$$ −21.4891 −1.20886
$$317$$ −12.2479 −0.687911 −0.343955 0.938986i $$-0.611767\pi$$
−0.343955 + 0.938986i $$0.611767\pi$$
$$318$$ 0 0
$$319$$ −2.23369 −0.125063
$$320$$ −84.1120 −4.70200
$$321$$ 0 0
$$322$$ −6.51087 −0.362837
$$323$$ −3.22060 −0.179199
$$324$$ 0 0
$$325$$ 10.7446 0.596001
$$326$$ −58.3472 −3.23155
$$327$$ 0 0
$$328$$ −49.7228 −2.74548
$$329$$ 10.0381 0.553419
$$330$$ 0 0
$$331$$ 18.9783 1.04314 0.521569 0.853209i $$-0.325347\pi$$
0.521569 + 0.853209i $$0.325347\pi$$
$$332$$ −63.7775 −3.50025
$$333$$ 0 0
$$334$$ −17.4891 −0.956962
$$335$$ 12.8824 0.703841
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −24.4368 −1.32918
$$339$$ 0 0
$$340$$ 55.7228 3.02199
$$341$$ −10.4845 −0.567766
$$342$$ 0 0
$$343$$ 19.8614 1.07242
$$344$$ −101.790 −5.48816
$$345$$ 0 0
$$346$$ −2.74456 −0.147549
$$347$$ 32.3942 1.73901 0.869505 0.493924i $$-0.164438\pi$$
0.869505 + 0.493924i $$0.164438\pi$$
$$348$$ 0 0
$$349$$ 17.8614 0.956099 0.478050 0.878333i $$-0.341344\pi$$
0.478050 + 0.878333i $$0.341344\pi$$
$$350$$ −34.6040 −1.84966
$$351$$ 0 0
$$352$$ −44.2337 −2.35766
$$353$$ 14.9040 0.793262 0.396631 0.917978i $$-0.370179\pi$$
0.396631 + 0.917978i $$0.370179\pi$$
$$354$$ 0 0
$$355$$ −6.51087 −0.345561
$$356$$ 52.9168 2.80458
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ −4.60773 −0.243187 −0.121593 0.992580i $$-0.538800\pi$$
−0.121593 + 0.992580i $$0.538800\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 52.9168 2.78124
$$363$$ 0 0
$$364$$ −25.4891 −1.33599
$$365$$ 16.4793 0.862567
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 14.2695 0.743851
$$369$$ 0 0
$$370$$ −93.9565 −4.88457
$$371$$ −23.3669 −1.21315
$$372$$ 0 0
$$373$$ −24.2337 −1.25477 −0.627386 0.778708i $$-0.715875\pi$$
−0.627386 + 0.778708i $$0.715875\pi$$
$$374$$ 19.3236 0.999200
$$375$$ 0 0
$$376$$ −38.7446 −1.99810
$$377$$ 2.02163 0.104119
$$378$$ 0 0
$$379$$ 28.7446 1.47651 0.738255 0.674522i $$-0.235650\pi$$
0.738255 + 0.674522i $$0.235650\pi$$
$$380$$ −17.3020 −0.887573
$$381$$ 0 0
$$382$$ −58.4674 −2.99145
$$383$$ −17.3020 −0.884090 −0.442045 0.896993i $$-0.645747\pi$$
−0.442045 + 0.896993i $$0.645747\pi$$
$$384$$ 0 0
$$385$$ −16.8832 −0.860445
$$386$$ 44.0776 2.24349
$$387$$ 0 0
$$388$$ 40.2337 2.04256
$$389$$ 12.0597 0.611454 0.305727 0.952119i $$-0.401101\pi$$
0.305727 + 0.952119i $$0.401101\pi$$
$$390$$ 0 0
$$391$$ −3.25544 −0.164635
$$392$$ −12.5652 −0.634636
$$393$$ 0 0
$$394$$ −64.4674 −3.24782
$$395$$ 12.8824 0.648184
$$396$$ 0 0
$$397$$ −17.1168 −0.859070 −0.429535 0.903050i $$-0.641322\pi$$
−0.429535 + 0.903050i $$0.641322\pi$$
$$398$$ 2.39794 0.120198
$$399$$ 0 0
$$400$$ 75.8397 3.79198
$$401$$ 3.40876 0.170225 0.0851126 0.996371i $$-0.472875\pi$$
0.0851126 + 0.996371i $$0.472875\pi$$
$$402$$ 0 0
$$403$$ 9.48913 0.472687
$$404$$ 69.2079 3.44322
$$405$$ 0 0
$$406$$ −6.51087 −0.323129
$$407$$ −23.7432 −1.17691
$$408$$ 0 0
$$409$$ 7.48913 0.370313 0.185157 0.982709i $$-0.440721\pi$$
0.185157 + 0.982709i $$0.440721\pi$$
$$410$$ 47.4864 2.34519
$$411$$ 0 0
$$412$$ −38.9783 −1.92032
$$413$$ −25.7648 −1.26780
$$414$$ 0 0
$$415$$ 38.2337 1.87682
$$416$$ 40.0344 1.96285
$$417$$ 0 0
$$418$$ −6.00000 −0.293470
$$419$$ 11.8716 0.579965 0.289983 0.957032i $$-0.406350\pi$$
0.289983 + 0.957032i $$0.406350\pi$$
$$420$$ 0 0
$$421$$ −8.97825 −0.437573 −0.218787 0.975773i $$-0.570210\pi$$
−0.218787 + 0.975773i $$0.570210\pi$$
$$422$$ −10.8608 −0.528694
$$423$$ 0 0
$$424$$ 90.1902 4.38002
$$425$$ −17.3020 −0.839269
$$426$$ 0 0
$$427$$ 12.1386 0.587428
$$428$$ 23.7432 1.14767
$$429$$ 0 0
$$430$$ 97.2119 4.68798
$$431$$ −30.1844 −1.45393 −0.726966 0.686674i $$-0.759070\pi$$
−0.726966 + 0.686674i $$0.759070\pi$$
$$432$$ 0 0
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ −30.5607 −1.46696
$$435$$ 0 0
$$436$$ 28.2337 1.35215
$$437$$ 1.01082 0.0483539
$$438$$ 0 0
$$439$$ −18.2337 −0.870246 −0.435123 0.900371i $$-0.643295\pi$$
−0.435123 + 0.900371i $$0.643295\pi$$
$$440$$ 65.1647 3.10660
$$441$$ 0 0
$$442$$ −17.4891 −0.831873
$$443$$ −27.9746 −1.32911 −0.664557 0.747238i $$-0.731380\pi$$
−0.664557 + 0.747238i $$0.731380\pi$$
$$444$$ 0 0
$$445$$ −31.7228 −1.50381
$$446$$ −10.8608 −0.514273
$$447$$ 0 0
$$448$$ −61.9565 −2.92717
$$449$$ 22.3561 1.05505 0.527524 0.849540i $$-0.323120\pi$$
0.527524 + 0.849540i $$0.323120\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ −63.7775 −2.99984
$$453$$ 0 0
$$454$$ 52.4674 2.46242
$$455$$ 15.2804 0.716354
$$456$$ 0 0
$$457$$ −8.37228 −0.391639 −0.195819 0.980640i $$-0.562737\pi$$
−0.195819 + 0.980640i $$0.562737\pi$$
$$458$$ 42.4323 1.98273
$$459$$ 0 0
$$460$$ −17.4891 −0.815435
$$461$$ −26.9638 −1.25583 −0.627914 0.778282i $$-0.716091\pi$$
−0.627914 + 0.778282i $$0.716091\pi$$
$$462$$ 0 0
$$463$$ −2.37228 −0.110249 −0.0551246 0.998479i $$-0.517556\pi$$
−0.0551246 + 0.998479i $$0.517556\pi$$
$$464$$ 14.2695 0.662447
$$465$$ 0 0
$$466$$ −20.7446 −0.960973
$$467$$ −36.4374 −1.68612 −0.843062 0.537816i $$-0.819249\pi$$
−0.843062 + 0.537816i $$0.819249\pi$$
$$468$$ 0 0
$$469$$ 9.48913 0.438167
$$470$$ 37.0019 1.70677
$$471$$ 0 0
$$472$$ 99.4456 4.57736
$$473$$ 24.5659 1.12954
$$474$$ 0 0
$$475$$ 5.37228 0.246497
$$476$$ 41.0452 1.88130
$$477$$ 0 0
$$478$$ −34.4674 −1.57650
$$479$$ 33.5932 1.53491 0.767455 0.641103i $$-0.221523\pi$$
0.767455 + 0.641103i $$0.221523\pi$$
$$480$$ 0 0
$$481$$ 21.4891 0.979820
$$482$$ −65.7992 −2.99707
$$483$$ 0 0
$$484$$ −32.8614 −1.49370
$$485$$ −24.1195 −1.09521
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −46.8519 −2.12088
$$489$$ 0 0
$$490$$ 12.0000 0.542105
$$491$$ −1.01082 −0.0456175 −0.0228087 0.999740i $$-0.507261\pi$$
−0.0228087 + 0.999740i $$0.507261\pi$$
$$492$$ 0 0
$$493$$ −3.25544 −0.146618
$$494$$ 5.43039 0.244325
$$495$$ 0 0
$$496$$ 66.9783 3.00741
$$497$$ −4.79588 −0.215125
$$498$$ 0 0
$$499$$ −11.1168 −0.497658 −0.248829 0.968547i $$-0.580046\pi$$
−0.248829 + 0.968547i $$0.580046\pi$$
$$500$$ −6.44121 −0.288059
$$501$$ 0 0
$$502$$ 64.9783 2.90012
$$503$$ −31.5715 −1.40770 −0.703852 0.710346i $$-0.748538\pi$$
−0.703852 + 0.710346i $$0.748538\pi$$
$$504$$ 0 0
$$505$$ −41.4891 −1.84624
$$506$$ −6.06490 −0.269618
$$507$$ 0 0
$$508$$ −68.4674 −3.03775
$$509$$ 24.7540 1.09720 0.548601 0.836084i $$-0.315161\pi$$
0.548601 + 0.836084i $$0.315161\pi$$
$$510$$ 0 0
$$511$$ 12.1386 0.536980
$$512$$ 24.0604 1.06333
$$513$$ 0 0
$$514$$ 14.7446 0.650355
$$515$$ 23.3669 1.02967
$$516$$ 0 0
$$517$$ 9.35053 0.411236
$$518$$ −69.2079 −3.04082
$$519$$ 0 0
$$520$$ −58.9783 −2.58637
$$521$$ 16.2912 0.713729 0.356864 0.934156i $$-0.383846\pi$$
0.356864 + 0.934156i $$0.383846\pi$$
$$522$$ 0 0
$$523$$ −18.2337 −0.797304 −0.398652 0.917102i $$-0.630522\pi$$
−0.398652 + 0.917102i $$0.630522\pi$$
$$524$$ −81.0795 −3.54198
$$525$$ 0 0
$$526$$ 35.4891 1.54740
$$527$$ −15.2804 −0.665623
$$528$$ 0 0
$$529$$ −21.9783 −0.955576
$$530$$ −86.1336 −3.74140
$$531$$ 0 0
$$532$$ −12.7446 −0.552547
$$533$$ −10.8608 −0.470433
$$534$$ 0 0
$$535$$ −14.2337 −0.615376
$$536$$ −36.6256 −1.58198
$$537$$ 0 0
$$538$$ 21.2554 0.916387
$$539$$ 3.03245 0.130617
$$540$$ 0 0
$$541$$ 21.1168 0.907884 0.453942 0.891031i $$-0.350017\pi$$
0.453942 + 0.891031i $$0.350017\pi$$
$$542$$ 69.2079 2.97274
$$543$$ 0 0
$$544$$ −64.4674 −2.76402
$$545$$ −16.9257 −0.725016
$$546$$ 0 0
$$547$$ 22.2337 0.950644 0.475322 0.879812i $$-0.342332\pi$$
0.475322 + 0.879812i $$0.342332\pi$$
$$548$$ 64.7884 2.76762
$$549$$ 0 0
$$550$$ −32.2337 −1.37445
$$551$$ 1.01082 0.0430622
$$552$$ 0 0
$$553$$ 9.48913 0.403519
$$554$$ 24.7540 1.05170
$$555$$ 0 0
$$556$$ 16.7446 0.710128
$$557$$ −20.5226 −0.869570 −0.434785 0.900534i $$-0.643176\pi$$
−0.434785 + 0.900534i $$0.643176\pi$$
$$558$$ 0 0
$$559$$ −22.2337 −0.940385
$$560$$ 107.855 4.55771
$$561$$ 0 0
$$562$$ 67.2119 2.83516
$$563$$ −38.6472 −1.62879 −0.814393 0.580313i $$-0.802930\pi$$
−0.814393 + 0.580313i $$0.802930\pi$$
$$564$$ 0 0
$$565$$ 38.2337 1.60850
$$566$$ 17.3020 0.727257
$$567$$ 0 0
$$568$$ 18.5109 0.776699
$$569$$ 3.03245 0.127127 0.0635634 0.997978i $$-0.479753\pi$$
0.0635634 + 0.997978i $$0.479753\pi$$
$$570$$ 0 0
$$571$$ 2.51087 0.105077 0.0525384 0.998619i $$-0.483269\pi$$
0.0525384 + 0.998619i $$0.483269\pi$$
$$572$$ −23.7432 −0.992753
$$573$$ 0 0
$$574$$ 34.9783 1.45996
$$575$$ 5.43039 0.226463
$$576$$ 0 0
$$577$$ −41.1168 −1.71172 −0.855858 0.517210i $$-0.826970\pi$$
−0.855858 + 0.517210i $$0.826970\pi$$
$$578$$ −17.9955 −0.748516
$$579$$ 0 0
$$580$$ −17.4891 −0.726196
$$581$$ 28.1628 1.16839
$$582$$ 0 0
$$583$$ −21.7663 −0.901469
$$584$$ −46.8519 −1.93874
$$585$$ 0 0
$$586$$ −68.2337 −2.81871
$$587$$ −1.83348 −0.0756758 −0.0378379 0.999284i $$-0.512047\pi$$
−0.0378379 + 0.999284i $$0.512047\pi$$
$$588$$ 0 0
$$589$$ 4.74456 0.195496
$$590$$ −94.9728 −3.90997
$$591$$ 0 0
$$592$$ 151.679 6.23398
$$593$$ 4.04326 0.166037 0.0830185 0.996548i $$-0.473544\pi$$
0.0830185 + 0.996548i $$0.473544\pi$$
$$594$$ 0 0
$$595$$ −24.6060 −1.00875
$$596$$ 110.253 4.51614
$$597$$ 0 0
$$598$$ 5.48913 0.224467
$$599$$ 12.8824 0.526361 0.263181 0.964747i $$-0.415229\pi$$
0.263181 + 0.964747i $$0.415229\pi$$
$$600$$ 0 0
$$601$$ −25.2554 −1.03019 −0.515095 0.857133i $$-0.672244\pi$$
−0.515095 + 0.857133i $$0.672244\pi$$
$$602$$ 71.6059 2.91844
$$603$$ 0 0
$$604$$ −21.4891 −0.874380
$$605$$ 19.6999 0.800916
$$606$$ 0 0
$$607$$ 28.7446 1.16671 0.583353 0.812219i $$-0.301740\pi$$
0.583353 + 0.812219i $$0.301740\pi$$
$$608$$ 20.0172 0.811804
$$609$$ 0 0
$$610$$ 44.7446 1.81165
$$611$$ −8.46284 −0.342370
$$612$$ 0 0
$$613$$ 2.60597 0.105254 0.0526271 0.998614i $$-0.483241\pi$$
0.0526271 + 0.998614i $$0.483241\pi$$
$$614$$ 69.2079 2.79300
$$615$$ 0 0
$$616$$ 48.0000 1.93398
$$617$$ 3.22060 0.129657 0.0648283 0.997896i $$-0.479350\pi$$
0.0648283 + 0.997896i $$0.479350\pi$$
$$618$$ 0 0
$$619$$ 6.97825 0.280480 0.140240 0.990118i $$-0.455213\pi$$
0.140240 + 0.990118i $$0.455213\pi$$
$$620$$ −82.0903 −3.29683
$$621$$ 0 0
$$622$$ 34.4674 1.38202
$$623$$ −23.3669 −0.936174
$$624$$ 0 0
$$625$$ −23.0000 −0.920000
$$626$$ 20.3344 0.812727
$$627$$ 0 0
$$628$$ 10.7446 0.428755
$$629$$ −34.6040 −1.37975
$$630$$ 0 0
$$631$$ 15.1168 0.601792 0.300896 0.953657i $$-0.402714\pi$$
0.300896 + 0.953657i $$0.402714\pi$$
$$632$$ −36.6256 −1.45689
$$633$$ 0 0
$$634$$ −33.2554 −1.32074
$$635$$ 41.0452 1.62883
$$636$$ 0 0
$$637$$ −2.74456 −0.108744
$$638$$ −6.06490 −0.240112
$$639$$ 0 0
$$640$$ −99.4456 −3.93093
$$641$$ −24.7540 −0.977724 −0.488862 0.872361i $$-0.662588\pi$$
−0.488862 + 0.872361i $$0.662588\pi$$
$$642$$ 0 0
$$643$$ −35.1168 −1.38487 −0.692437 0.721479i $$-0.743463\pi$$
−0.692437 + 0.721479i $$0.743463\pi$$
$$644$$ −12.8824 −0.507638
$$645$$ 0 0
$$646$$ −8.74456 −0.344050
$$647$$ −17.1138 −0.672814 −0.336407 0.941717i $$-0.609212\pi$$
−0.336407 + 0.941717i $$0.609212\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 29.1736 1.14428
$$651$$ 0 0
$$652$$ −115.446 −4.52120
$$653$$ 14.0814 0.551047 0.275524 0.961294i $$-0.411149\pi$$
0.275524 + 0.961294i $$0.411149\pi$$
$$654$$ 0 0
$$655$$ 48.6060 1.89919
$$656$$ −76.6600 −2.99307
$$657$$ 0 0
$$658$$ 27.2554 1.06253
$$659$$ 11.2371 0.437735 0.218867 0.975755i $$-0.429764\pi$$
0.218867 + 0.975755i $$0.429764\pi$$
$$660$$ 0 0
$$661$$ −6.74456 −0.262333 −0.131167 0.991360i $$-0.541872\pi$$
−0.131167 + 0.991360i $$0.541872\pi$$
$$662$$ 51.5296 2.00276
$$663$$ 0 0
$$664$$ −108.701 −4.21842
$$665$$ 7.64018 0.296273
$$666$$ 0 0
$$667$$ 1.02175 0.0395623
$$668$$ −34.6040 −1.33887
$$669$$ 0 0
$$670$$ 34.9783 1.35133
$$671$$ 11.3071 0.436507
$$672$$ 0 0
$$673$$ −25.2554 −0.973526 −0.486763 0.873534i $$-0.661822\pi$$
−0.486763 + 0.873534i $$0.661822\pi$$
$$674$$ 38.0127 1.46420
$$675$$ 0 0
$$676$$ −48.3505 −1.85964
$$677$$ 44.4539 1.70850 0.854252 0.519860i $$-0.174016\pi$$
0.854252 + 0.519860i $$0.174016\pi$$
$$678$$ 0 0
$$679$$ −17.7663 −0.681808
$$680$$ 94.9728 3.64204
$$681$$ 0 0
$$682$$ −28.4674 −1.09007
$$683$$ 34.2277 1.30968 0.654842 0.755765i $$-0.272735\pi$$
0.654842 + 0.755765i $$0.272735\pi$$
$$684$$ 0 0
$$685$$ −38.8397 −1.48399
$$686$$ 53.9276 2.05896
$$687$$ 0 0
$$688$$ −156.935 −5.98308
$$689$$ 19.6999 0.750507
$$690$$ 0 0
$$691$$ 27.1168 1.03157 0.515787 0.856717i $$-0.327500\pi$$
0.515787 + 0.856717i $$0.327500\pi$$
$$692$$ −5.43039 −0.206432
$$693$$ 0 0
$$694$$ 87.9565 3.33878
$$695$$ −10.0381 −0.380767
$$696$$ 0 0
$$697$$ 17.4891 0.662448
$$698$$ 48.4972 1.83565
$$699$$ 0 0
$$700$$ −68.4674 −2.58782
$$701$$ −30.5607 −1.15426 −0.577131 0.816652i $$-0.695828\pi$$
−0.577131 + 0.816652i $$0.695828\pi$$
$$702$$ 0 0
$$703$$ 10.7446 0.405239
$$704$$ −57.7126 −2.17513
$$705$$ 0 0
$$706$$ 40.4674 1.52301
$$707$$ −30.5607 −1.14935
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ −17.6783 −0.663454
$$711$$ 0 0
$$712$$ 90.1902 3.38002
$$713$$ 4.79588 0.179607
$$714$$ 0 0
$$715$$ 14.2337 0.532310
$$716$$ −23.7432 −0.887325
$$717$$ 0 0
$$718$$ −12.5109 −0.466902
$$719$$ −38.8354 −1.44832 −0.724158 0.689634i $$-0.757771\pi$$
−0.724158 + 0.689634i $$0.757771\pi$$
$$720$$ 0 0
$$721$$ 17.2119 0.641006
$$722$$ 2.71519 0.101049
$$723$$ 0 0
$$724$$ 104.701 3.89118
$$725$$ 5.43039 0.201680
$$726$$ 0 0
$$727$$ −37.3505 −1.38525 −0.692627 0.721296i $$-0.743547\pi$$
−0.692627 + 0.721296i $$0.743547\pi$$
$$728$$ −43.4431 −1.61011
$$729$$ 0 0
$$730$$ 44.7446 1.65607
$$731$$ 35.8029 1.32422
$$732$$ 0 0
$$733$$ 18.4674 0.682108 0.341054 0.940044i $$-0.389216\pi$$
0.341054 + 0.940044i $$0.389216\pi$$
$$734$$ 21.7216 0.801757
$$735$$ 0 0
$$736$$ 20.2337 0.745824
$$737$$ 8.83915 0.325594
$$738$$ 0 0
$$739$$ 27.1168 0.997509 0.498755 0.866743i $$-0.333791\pi$$
0.498755 + 0.866743i $$0.333791\pi$$
$$740$$ −185.902 −6.83390
$$741$$ 0 0
$$742$$ −63.4456 −2.32916
$$743$$ 28.5391 1.04700 0.523498 0.852027i $$-0.324627\pi$$
0.523498 + 0.852027i $$0.324627\pi$$
$$744$$ 0 0
$$745$$ −66.0951 −2.42154
$$746$$ −65.7992 −2.40908
$$747$$ 0 0
$$748$$ 38.2337 1.39796
$$749$$ −10.4845 −0.383094
$$750$$ 0 0
$$751$$ 24.4674 0.892827 0.446414 0.894827i $$-0.352701\pi$$
0.446414 + 0.894827i $$0.352701\pi$$
$$752$$ −59.7343 −2.17829
$$753$$ 0 0
$$754$$ 5.48913 0.199902
$$755$$ 12.8824 0.468839
$$756$$ 0 0
$$757$$ −5.11684 −0.185975 −0.0929874 0.995667i $$-0.529642\pi$$
−0.0929874 + 0.995667i $$0.529642\pi$$
$$758$$ 78.0471 2.83480
$$759$$ 0 0
$$760$$ −29.4891 −1.06968
$$761$$ 0.822662 0.0298215 0.0149107 0.999889i $$-0.495254\pi$$
0.0149107 + 0.999889i $$0.495254\pi$$
$$762$$ 0 0
$$763$$ −12.4674 −0.451349
$$764$$ −115.683 −4.18528
$$765$$ 0 0
$$766$$ −46.9783 −1.69739
$$767$$ 21.7216 0.784320
$$768$$ 0 0
$$769$$ 6.88316 0.248213 0.124106 0.992269i $$-0.460394\pi$$
0.124106 + 0.992269i $$0.460394\pi$$
$$770$$ −45.8411 −1.65200
$$771$$ 0 0
$$772$$ 87.2119 3.13883
$$773$$ 5.43039 0.195318 0.0976588 0.995220i $$-0.468865\pi$$
0.0976588 + 0.995220i $$0.468865\pi$$
$$774$$ 0 0
$$775$$ 25.4891 0.915596
$$776$$ 68.5734 2.46164
$$777$$ 0 0
$$778$$ 32.7446 1.17395
$$779$$ −5.43039 −0.194564
$$780$$ 0 0
$$781$$ −4.46738 −0.159855
$$782$$ −8.83915 −0.316087
$$783$$ 0 0
$$784$$ −19.3723 −0.691867
$$785$$ −6.44121 −0.229896
$$786$$ 0 0
$$787$$ −49.9565 −1.78076 −0.890378 0.455221i $$-0.849560\pi$$
−0.890378 + 0.455221i $$0.849560\pi$$
$$788$$ −127.555 −4.54396
$$789$$ 0 0
$$790$$ 34.9783 1.24447
$$791$$ 28.1628 1.00135
$$792$$ 0 0
$$793$$ −10.2337 −0.363409
$$794$$ −46.4756 −1.64936
$$795$$ 0 0
$$796$$ 4.74456 0.168167
$$797$$ −7.45202 −0.263964 −0.131982 0.991252i $$-0.542134\pi$$
−0.131982 + 0.991252i $$0.542134\pi$$
$$798$$ 0 0
$$799$$ 13.6277 0.482114
$$800$$ 107.538 3.80204
$$801$$ 0 0
$$802$$ 9.25544 0.326821
$$803$$ 11.3071 0.399020
$$804$$ 0 0
$$805$$ 7.72281 0.272193
$$806$$ 25.7648 0.907527
$$807$$ 0 0
$$808$$ 117.957 4.14970
$$809$$ −3.59691 −0.126461 −0.0632303 0.997999i $$-0.520140\pi$$
−0.0632303 + 0.997999i $$0.520140\pi$$
$$810$$ 0 0
$$811$$ −36.7446 −1.29028 −0.645138 0.764066i $$-0.723200\pi$$
−0.645138 + 0.764066i $$0.723200\pi$$
$$812$$ −12.8824 −0.452084
$$813$$ 0 0
$$814$$ −64.4674 −2.25958
$$815$$ 69.2079 2.42425
$$816$$ 0 0
$$817$$ −11.1168 −0.388929
$$818$$ 20.3344 0.710977
$$819$$ 0 0
$$820$$ 93.9565 3.28110
$$821$$ −29.3617 −1.02473 −0.512366 0.858767i $$-0.671231\pi$$
−0.512366 + 0.858767i $$0.671231\pi$$
$$822$$ 0 0
$$823$$ 8.60597 0.299985 0.149993 0.988687i $$-0.452075\pi$$
0.149993 + 0.988687i $$0.452075\pi$$
$$824$$ −66.4337 −2.31433
$$825$$ 0 0
$$826$$ −69.9565 −2.43410
$$827$$ −28.5391 −0.992401 −0.496200 0.868208i $$-0.665272\pi$$
−0.496200 + 0.868208i $$0.665272\pi$$
$$828$$ 0 0
$$829$$ −1.25544 −0.0436031 −0.0218016 0.999762i $$-0.506940\pi$$
−0.0218016 + 0.999762i $$0.506940\pi$$
$$830$$ 103.812 3.60336
$$831$$ 0 0
$$832$$ 52.2337 1.81088
$$833$$ 4.41957 0.153129
$$834$$ 0 0
$$835$$ 20.7446 0.717895
$$836$$ −11.8716 −0.410588
$$837$$ 0 0
$$838$$ 32.2337 1.11349
$$839$$ 30.1844 1.04208 0.521041 0.853532i $$-0.325544\pi$$
0.521041 + 0.853532i $$0.325544\pi$$
$$840$$ 0 0
$$841$$ −27.9783 −0.964767
$$842$$ −24.3777 −0.840111
$$843$$ 0 0
$$844$$ −21.4891 −0.739686
$$845$$ 28.9854 0.997129
$$846$$ 0 0
$$847$$ 14.5109 0.498600
$$848$$ 139.050 4.77501
$$849$$ 0 0
$$850$$ −46.9783 −1.61134
$$851$$ 10.8608 0.372303
$$852$$ 0 0
$$853$$ −38.4674 −1.31710 −0.658549 0.752538i $$-0.728829\pi$$
−0.658549 + 0.752538i $$0.728829\pi$$
$$854$$ 32.9586 1.12782
$$855$$ 0 0
$$856$$ 40.4674 1.38315
$$857$$ −35.2385 −1.20372 −0.601862 0.798600i $$-0.705574\pi$$
−0.601862 + 0.798600i $$0.705574\pi$$
$$858$$ 0 0
$$859$$ 3.11684 0.106345 0.0531727 0.998585i $$-0.483067\pi$$
0.0531727 + 0.998585i $$0.483067\pi$$
$$860$$ 192.343 6.55886
$$861$$ 0 0
$$862$$ −81.9565 −2.79145
$$863$$ −14.9040 −0.507340 −0.253670 0.967291i $$-0.581638\pi$$
−0.253670 + 0.967291i $$0.581638\pi$$
$$864$$ 0 0
$$865$$ 3.25544 0.110688
$$866$$ −59.7343 −2.02985
$$867$$ 0 0
$$868$$ −60.4674 −2.05240
$$869$$ 8.83915 0.299847
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 48.1209 1.62958
$$873$$ 0 0
$$874$$ 2.74456 0.0928362
$$875$$ 2.84429 0.0961547
$$876$$ 0 0
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ −49.5080 −1.67081
$$879$$ 0 0
$$880$$ 100.467 3.38675
$$881$$ −0.822662 −0.0277162 −0.0138581 0.999904i $$-0.504411\pi$$
−0.0138581 + 0.999904i $$0.504411\pi$$
$$882$$ 0 0
$$883$$ 3.11684 0.104890 0.0524451 0.998624i $$-0.483299\pi$$
0.0524451 + 0.998624i $$0.483299\pi$$
$$884$$ −34.6040 −1.16386
$$885$$ 0 0
$$886$$ −75.9565 −2.55181
$$887$$ 21.7216 0.729338 0.364669 0.931137i $$-0.381182\pi$$
0.364669 + 0.931137i $$0.381182\pi$$
$$888$$ 0 0
$$889$$ 30.2337 1.01401
$$890$$ −86.1336 −2.88721
$$891$$ 0 0
$$892$$ −21.4891 −0.719509
$$893$$ −4.23142 −0.141599
$$894$$ 0 0
$$895$$ 14.2337 0.475780
$$896$$ −73.2512 −2.44715
$$897$$ 0 0
$$898$$ 60.7011 2.02562
$$899$$ 4.79588 0.159952
$$900$$ 0 0
$$901$$ −31.7228 −1.05684
$$902$$ 32.5823 1.08487
$$903$$ 0 0
$$904$$ −108.701 −3.61534
$$905$$ −62.7667 −2.08644
$$906$$ 0 0
$$907$$ 23.2554 0.772184 0.386092 0.922460i $$-0.373825\pi$$
0.386092 + 0.922460i $$0.373825\pi$$
$$908$$ 103.812 3.44512
$$909$$ 0 0
$$910$$ 41.4891 1.37535
$$911$$ −37.0019 −1.22593 −0.612964 0.790111i $$-0.710023\pi$$
−0.612964 + 0.790111i $$0.710023\pi$$
$$912$$ 0 0
$$913$$ 26.2337 0.868208
$$914$$ −22.7324 −0.751920
$$915$$ 0 0
$$916$$ 83.9565 2.77400
$$917$$ 35.8029 1.18232
$$918$$ 0 0
$$919$$ 18.9783 0.626035 0.313017 0.949747i $$-0.398660\pi$$
0.313017 + 0.949747i $$0.398660\pi$$
$$920$$ −29.8081 −0.982743
$$921$$ 0 0
$$922$$ −73.2119 −2.41111
$$923$$ 4.04326 0.133086
$$924$$ 0 0
$$925$$ 57.7228 1.89791
$$926$$ −6.44121 −0.211671
$$927$$ 0 0
$$928$$ 20.2337 0.664203
$$929$$ 4.79588 0.157348 0.0786739 0.996900i $$-0.474931\pi$$
0.0786739 + 0.996900i $$0.474931\pi$$
$$930$$ 0 0
$$931$$ −1.37228 −0.0449747
$$932$$ −41.0452 −1.34448
$$933$$ 0 0
$$934$$ −98.9348 −3.23724
$$935$$ −22.9205 −0.749581
$$936$$ 0 0
$$937$$ −5.11684 −0.167160 −0.0835800 0.996501i $$-0.526635\pi$$
−0.0835800 + 0.996501i $$0.526635\pi$$
$$938$$ 25.7648 0.841251
$$939$$ 0 0
$$940$$ 73.2119 2.38791
$$941$$ 24.7540 0.806957 0.403479 0.914989i $$-0.367801\pi$$
0.403479 + 0.914989i $$0.367801\pi$$
$$942$$ 0 0
$$943$$ −5.48913 −0.178751
$$944$$ 153.320 4.99014
$$945$$ 0 0
$$946$$ 66.7011 2.16864
$$947$$ −9.84996 −0.320081 −0.160040 0.987110i $$-0.551162\pi$$
−0.160040 + 0.987110i $$0.551162\pi$$
$$948$$ 0 0
$$949$$ −10.2337 −0.332200
$$950$$ 14.5868 0.473258
$$951$$ 0 0
$$952$$ 69.9565 2.26730
$$953$$ 9.47365 0.306882 0.153441 0.988158i $$-0.450965\pi$$
0.153441 + 0.988158i $$0.450965\pi$$
$$954$$ 0 0
$$955$$ 69.3505 2.24413
$$956$$ −68.1971 −2.20565
$$957$$ 0 0
$$958$$ 91.2119 2.94692
$$959$$ −28.6091 −0.923837
$$960$$ 0 0
$$961$$ −8.48913 −0.273843
$$962$$ 58.3472 1.88119
$$963$$ 0 0
$$964$$ −130.190 −4.19314
$$965$$ −52.2823 −1.68303
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ −56.0083 −1.80017
$$969$$ 0 0
$$970$$ −65.4891 −2.10273
$$971$$ −51.5296 −1.65366 −0.826832 0.562448i $$-0.809860\pi$$
−0.826832 + 0.562448i $$0.809860\pi$$
$$972$$ 0 0
$$973$$ −7.39403 −0.237042
$$974$$ 21.7216 0.696004
$$975$$ 0 0
$$976$$ −72.2337 −2.31214
$$977$$ −20.7107 −0.662595 −0.331298 0.943526i $$-0.607486\pi$$
−0.331298 + 0.943526i $$0.607486\pi$$
$$978$$ 0 0
$$979$$ −21.7663 −0.695654
$$980$$ 23.7432 0.758448
$$981$$ 0 0
$$982$$ −2.74456 −0.0875825
$$983$$ 52.2823 1.66755 0.833773 0.552108i $$-0.186176\pi$$
0.833773 + 0.552108i $$0.186176\pi$$
$$984$$ 0 0
$$985$$ 76.4674 2.43645
$$986$$ −8.83915 −0.281496
$$987$$ 0 0
$$988$$ 10.7446 0.341830
$$989$$ −11.2371 −0.357319
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 94.9728 3.01539
$$993$$ 0 0
$$994$$ −13.0217 −0.413025
$$995$$ −2.84429 −0.0901702
$$996$$ 0 0
$$997$$ −19.3505 −0.612837 −0.306419 0.951897i $$-0.599131\pi$$
−0.306419 + 0.951897i $$0.599131\pi$$
$$998$$ −30.1844 −0.955470
$$999$$ 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 171.2.a.e.1.4 yes 4
3.2 odd 2 inner 171.2.a.e.1.1 4
4.3 odd 2 2736.2.a.bf.1.1 4
5.4 even 2 4275.2.a.bp.1.1 4
7.6 odd 2 8379.2.a.bw.1.4 4
12.11 even 2 2736.2.a.bf.1.4 4
15.14 odd 2 4275.2.a.bp.1.4 4
19.18 odd 2 3249.2.a.bf.1.1 4
21.20 even 2 8379.2.a.bw.1.1 4
57.56 even 2 3249.2.a.bf.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.1 4 3.2 odd 2 inner
171.2.a.e.1.4 yes 4 1.1 even 1 trivial
2736.2.a.bf.1.1 4 4.3 odd 2
2736.2.a.bf.1.4 4 12.11 even 2
3249.2.a.bf.1.1 4 19.18 odd 2
3249.2.a.bf.1.4 4 57.56 even 2
4275.2.a.bp.1.1 4 5.4 even 2
4275.2.a.bp.1.4 4 15.14 odd 2
8379.2.a.bw.1.1 4 21.20 even 2
8379.2.a.bw.1.4 4 7.6 odd 2