# Properties

 Label 8281.2.a.cc.1.5 Level $8281$ Weight $2$ Character 8281.1 Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.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: $$66.1241179138$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.4507648.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 637) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$2.35100$$ of defining polynomial Character $$\chi$$ $$=$$ 8281.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.17619 q^{2} -3.35100 q^{3} -0.616586 q^{4} +3.14862 q^{5} -3.94140 q^{6} -3.07759 q^{8} +8.22917 q^{9} +O(q^{10})$$ $$q+1.17619 q^{2} -3.35100 q^{3} -0.616586 q^{4} +3.14862 q^{5} -3.94140 q^{6} -3.07759 q^{8} +8.22917 q^{9} +3.70337 q^{10} +0.773390 q^{11} +2.06618 q^{12} -10.5510 q^{15} -2.38665 q^{16} +5.75340 q^{17} +9.67904 q^{18} -1.22298 q^{19} -1.94140 q^{20} +0.909650 q^{22} -2.99182 q^{23} +10.3130 q^{24} +4.91383 q^{25} -17.5229 q^{27} -2.46882 q^{29} -12.4100 q^{30} -6.13487 q^{31} +3.34804 q^{32} -2.59163 q^{33} +6.76707 q^{34} -5.07399 q^{36} -4.99933 q^{37} -1.43845 q^{38} -9.69018 q^{40} -2.55981 q^{41} -2.73150 q^{43} -0.476861 q^{44} +25.9106 q^{45} -3.51894 q^{46} +5.37169 q^{47} +7.99766 q^{48} +5.77958 q^{50} -19.2796 q^{51} -9.79015 q^{53} -20.6102 q^{54} +2.43511 q^{55} +4.09820 q^{57} -2.90379 q^{58} +2.50456 q^{59} +6.50561 q^{60} +10.9167 q^{61} -7.21575 q^{62} +8.71122 q^{64} -3.04823 q^{66} -4.32518 q^{67} -3.54746 q^{68} +10.0256 q^{69} -10.6649 q^{71} -25.3260 q^{72} -5.17450 q^{73} -5.88014 q^{74} -16.4662 q^{75} +0.754072 q^{76} -0.542038 q^{79} -7.51467 q^{80} +34.0318 q^{81} -3.01081 q^{82} +15.2259 q^{83} +18.1153 q^{85} -3.21275 q^{86} +8.27301 q^{87} -2.38018 q^{88} +9.23208 q^{89} +30.4757 q^{90} +1.84471 q^{92} +20.5579 q^{93} +6.31811 q^{94} -3.85070 q^{95} -11.2193 q^{96} -1.26291 q^{97} +6.36436 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{3} + 4 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{9}+O(q^{10})$$ 6 * q - 8 * q^3 + 4 * q^4 + 6 * q^5 + 4 * q^6 + 6 * q^9 $$6 q - 8 q^{3} + 4 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{9} - 4 q^{10} - 4 q^{11} + 4 q^{12} - 12 q^{15} - 16 q^{17} + 4 q^{18} + 2 q^{19} + 16 q^{20} - 12 q^{22} - 6 q^{23} + 12 q^{24} - 4 q^{25} - 20 q^{27} - 6 q^{29} + 6 q^{31} + 20 q^{32} + 4 q^{33} - 24 q^{36} - 8 q^{38} - 4 q^{40} - 8 q^{41} + 2 q^{43} + 4 q^{44} + 14 q^{45} - 8 q^{46} + 30 q^{47} + 8 q^{48} - 8 q^{50} - 4 q^{51} - 14 q^{53} - 48 q^{54} + 8 q^{55} - 4 q^{57} + 8 q^{58} + 24 q^{59} - 12 q^{60} - 28 q^{62} - 20 q^{64} + 4 q^{66} - 16 q^{67} - 28 q^{68} + 20 q^{69} - 8 q^{71} - 28 q^{72} - 6 q^{73} - 12 q^{74} - 12 q^{75} - 16 q^{76} - 22 q^{79} - 28 q^{80} + 46 q^{81} + 40 q^{82} + 50 q^{83} + 8 q^{85} + 16 q^{86} + 16 q^{87} - 44 q^{88} + 26 q^{89} + 40 q^{90} + 20 q^{92} - 16 q^{93} + 32 q^{94} - 6 q^{95} - 20 q^{96} - 14 q^{97} - 12 q^{99}+O(q^{100})$$ 6 * q - 8 * q^3 + 4 * q^4 + 6 * q^5 + 4 * q^6 + 6 * q^9 - 4 * q^10 - 4 * q^11 + 4 * q^12 - 12 * q^15 - 16 * q^17 + 4 * q^18 + 2 * q^19 + 16 * q^20 - 12 * q^22 - 6 * q^23 + 12 * q^24 - 4 * q^25 - 20 * q^27 - 6 * q^29 + 6 * q^31 + 20 * q^32 + 4 * q^33 - 24 * q^36 - 8 * q^38 - 4 * q^40 - 8 * q^41 + 2 * q^43 + 4 * q^44 + 14 * q^45 - 8 * q^46 + 30 * q^47 + 8 * q^48 - 8 * q^50 - 4 * q^51 - 14 * q^53 - 48 * q^54 + 8 * q^55 - 4 * q^57 + 8 * q^58 + 24 * q^59 - 12 * q^60 - 28 * q^62 - 20 * q^64 + 4 * q^66 - 16 * q^67 - 28 * q^68 + 20 * q^69 - 8 * q^71 - 28 * q^72 - 6 * q^73 - 12 * q^74 - 12 * q^75 - 16 * q^76 - 22 * q^79 - 28 * q^80 + 46 * q^81 + 40 * q^82 + 50 * q^83 + 8 * q^85 + 16 * q^86 + 16 * q^87 - 44 * q^88 + 26 * q^89 + 40 * q^90 + 20 * q^92 - 16 * q^93 + 32 * q^94 - 6 * q^95 - 20 * q^96 - 14 * q^97 - 12 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.17619 0.831689 0.415845 0.909436i $$-0.363486\pi$$
0.415845 + 0.909436i $$0.363486\pi$$
$$3$$ −3.35100 −1.93470 −0.967349 0.253447i $$-0.918436\pi$$
−0.967349 + 0.253447i $$0.918436\pi$$
$$4$$ −0.616586 −0.308293
$$5$$ 3.14862 1.40811 0.704054 0.710147i $$-0.251371\pi$$
0.704054 + 0.710147i $$0.251371\pi$$
$$6$$ −3.94140 −1.60907
$$7$$ 0 0
$$8$$ −3.07759 −1.08809
$$9$$ 8.22917 2.74306
$$10$$ 3.70337 1.17111
$$11$$ 0.773390 0.233186 0.116593 0.993180i $$-0.462803\pi$$
0.116593 + 0.993180i $$0.462803\pi$$
$$12$$ 2.06618 0.596454
$$13$$ 0 0
$$14$$ 0 0
$$15$$ −10.5510 −2.72426
$$16$$ −2.38665 −0.596663
$$17$$ 5.75340 1.39540 0.697702 0.716388i $$-0.254206\pi$$
0.697702 + 0.716388i $$0.254206\pi$$
$$18$$ 9.67904 2.28137
$$19$$ −1.22298 −0.280571 −0.140285 0.990111i $$-0.544802\pi$$
−0.140285 + 0.990111i $$0.544802\pi$$
$$20$$ −1.94140 −0.434109
$$21$$ 0 0
$$22$$ 0.909650 0.193938
$$23$$ −2.99182 −0.623838 −0.311919 0.950109i $$-0.600972\pi$$
−0.311919 + 0.950109i $$0.600972\pi$$
$$24$$ 10.3130 2.10513
$$25$$ 4.91383 0.982767
$$26$$ 0 0
$$27$$ −17.5229 −3.37229
$$28$$ 0 0
$$29$$ −2.46882 −0.458449 −0.229224 0.973374i $$-0.573619\pi$$
−0.229224 + 0.973374i $$0.573619\pi$$
$$30$$ −12.4100 −2.26574
$$31$$ −6.13487 −1.10185 −0.550927 0.834553i $$-0.685726\pi$$
−0.550927 + 0.834553i $$0.685726\pi$$
$$32$$ 3.34804 0.591855
$$33$$ −2.59163 −0.451144
$$34$$ 6.76707 1.16054
$$35$$ 0 0
$$36$$ −5.07399 −0.845665
$$37$$ −4.99933 −0.821884 −0.410942 0.911661i $$-0.634800\pi$$
−0.410942 + 0.911661i $$0.634800\pi$$
$$38$$ −1.43845 −0.233348
$$39$$ 0 0
$$40$$ −9.69018 −1.53215
$$41$$ −2.55981 −0.399774 −0.199887 0.979819i $$-0.564058\pi$$
−0.199887 + 0.979819i $$0.564058\pi$$
$$42$$ 0 0
$$43$$ −2.73150 −0.416550 −0.208275 0.978070i $$-0.566785\pi$$
−0.208275 + 0.978070i $$0.566785\pi$$
$$44$$ −0.476861 −0.0718895
$$45$$ 25.9106 3.86252
$$46$$ −3.51894 −0.518839
$$47$$ 5.37169 0.783542 0.391771 0.920063i $$-0.371863\pi$$
0.391771 + 0.920063i $$0.371863\pi$$
$$48$$ 7.99766 1.15436
$$49$$ 0 0
$$50$$ 5.77958 0.817357
$$51$$ −19.2796 −2.69969
$$52$$ 0 0
$$53$$ −9.79015 −1.34478 −0.672390 0.740197i $$-0.734732\pi$$
−0.672390 + 0.740197i $$0.734732\pi$$
$$54$$ −20.6102 −2.80470
$$55$$ 2.43511 0.328351
$$56$$ 0 0
$$57$$ 4.09820 0.542820
$$58$$ −2.90379 −0.381287
$$59$$ 2.50456 0.326066 0.163033 0.986621i $$-0.447872\pi$$
0.163033 + 0.986621i $$0.447872\pi$$
$$60$$ 6.50561 0.839871
$$61$$ 10.9167 1.39774 0.698870 0.715248i $$-0.253686\pi$$
0.698870 + 0.715248i $$0.253686\pi$$
$$62$$ −7.21575 −0.916401
$$63$$ 0 0
$$64$$ 8.71122 1.08890
$$65$$ 0 0
$$66$$ −3.04823 −0.375212
$$67$$ −4.32518 −0.528404 −0.264202 0.964467i $$-0.585109\pi$$
−0.264202 + 0.964467i $$0.585109\pi$$
$$68$$ −3.54746 −0.430193
$$69$$ 10.0256 1.20694
$$70$$ 0 0
$$71$$ −10.6649 −1.26570 −0.632848 0.774276i $$-0.718114\pi$$
−0.632848 + 0.774276i $$0.718114\pi$$
$$72$$ −25.3260 −2.98470
$$73$$ −5.17450 −0.605630 −0.302815 0.953049i $$-0.597926\pi$$
−0.302815 + 0.953049i $$0.597926\pi$$
$$74$$ −5.88014 −0.683552
$$75$$ −16.4662 −1.90136
$$76$$ 0.754072 0.0864979
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −0.542038 −0.0609841 −0.0304920 0.999535i $$-0.509707\pi$$
−0.0304920 + 0.999535i $$0.509707\pi$$
$$80$$ −7.51467 −0.840165
$$81$$ 34.0318 3.78131
$$82$$ −3.01081 −0.332488
$$83$$ 15.2259 1.67125 0.835627 0.549297i $$-0.185104\pi$$
0.835627 + 0.549297i $$0.185104\pi$$
$$84$$ 0 0
$$85$$ 18.1153 1.96488
$$86$$ −3.21275 −0.346440
$$87$$ 8.27301 0.886960
$$88$$ −2.38018 −0.253728
$$89$$ 9.23208 0.978599 0.489299 0.872116i $$-0.337253\pi$$
0.489299 + 0.872116i $$0.337253\pi$$
$$90$$ 30.4757 3.21242
$$91$$ 0 0
$$92$$ 1.84471 0.192325
$$93$$ 20.5579 2.13176
$$94$$ 6.31811 0.651663
$$95$$ −3.85070 −0.395074
$$96$$ −11.2193 −1.14506
$$97$$ −1.26291 −0.128229 −0.0641145 0.997943i $$-0.520422\pi$$
−0.0641145 + 0.997943i $$0.520422\pi$$
$$98$$ 0 0
$$99$$ 6.36436 0.639642
$$100$$ −3.02980 −0.302980
$$101$$ −0.605447 −0.0602443 −0.0301221 0.999546i $$-0.509590\pi$$
−0.0301221 + 0.999546i $$0.509590\pi$$
$$102$$ −22.6764 −2.24530
$$103$$ −7.64804 −0.753584 −0.376792 0.926298i $$-0.622973\pi$$
−0.376792 + 0.926298i $$0.622973\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −11.5150 −1.11844
$$107$$ 4.82965 0.466900 0.233450 0.972369i $$-0.424999\pi$$
0.233450 + 0.972369i $$0.424999\pi$$
$$108$$ 10.8044 1.03965
$$109$$ −7.23092 −0.692596 −0.346298 0.938125i $$-0.612561\pi$$
−0.346298 + 0.938125i $$0.612561\pi$$
$$110$$ 2.86415 0.273086
$$111$$ 16.7527 1.59010
$$112$$ 0 0
$$113$$ −9.19375 −0.864875 −0.432438 0.901664i $$-0.642346\pi$$
−0.432438 + 0.901664i $$0.642346\pi$$
$$114$$ 4.82025 0.451458
$$115$$ −9.42012 −0.878430
$$116$$ 1.52224 0.141336
$$117$$ 0 0
$$118$$ 2.94583 0.271186
$$119$$ 0 0
$$120$$ 32.4718 2.96425
$$121$$ −10.4019 −0.945624
$$122$$ 12.8401 1.16249
$$123$$ 8.57790 0.773443
$$124$$ 3.78267 0.339694
$$125$$ −0.271305 −0.0242663
$$126$$ 0 0
$$127$$ −11.2118 −0.994888 −0.497444 0.867496i $$-0.665728\pi$$
−0.497444 + 0.867496i $$0.665728\pi$$
$$128$$ 3.54994 0.313773
$$129$$ 9.15325 0.805899
$$130$$ 0 0
$$131$$ −15.7380 −1.37503 −0.687517 0.726169i $$-0.741299\pi$$
−0.687517 + 0.726169i $$0.741299\pi$$
$$132$$ 1.59796 0.139084
$$133$$ 0 0
$$134$$ −5.08721 −0.439468
$$135$$ −55.1732 −4.74855
$$136$$ −17.7066 −1.51833
$$137$$ 18.6210 1.59090 0.795449 0.606020i $$-0.207235\pi$$
0.795449 + 0.606020i $$0.207235\pi$$
$$138$$ 11.7919 1.00380
$$139$$ 11.9137 1.01050 0.505252 0.862972i $$-0.331399\pi$$
0.505252 + 0.862972i $$0.331399\pi$$
$$140$$ 0 0
$$141$$ −18.0005 −1.51592
$$142$$ −12.5440 −1.05267
$$143$$ 0 0
$$144$$ −19.6402 −1.63668
$$145$$ −7.77339 −0.645545
$$146$$ −6.08618 −0.503696
$$147$$ 0 0
$$148$$ 3.08251 0.253381
$$149$$ −11.3753 −0.931904 −0.465952 0.884810i $$-0.654288\pi$$
−0.465952 + 0.884810i $$0.654288\pi$$
$$150$$ −19.3674 −1.58134
$$151$$ 3.69250 0.300492 0.150246 0.988649i $$-0.451993\pi$$
0.150246 + 0.988649i $$0.451993\pi$$
$$152$$ 3.76383 0.305287
$$153$$ 47.3457 3.82767
$$154$$ 0 0
$$155$$ −19.3164 −1.55153
$$156$$ 0 0
$$157$$ −1.12065 −0.0894374 −0.0447187 0.999000i $$-0.514239\pi$$
−0.0447187 + 0.999000i $$0.514239\pi$$
$$158$$ −0.637538 −0.0507198
$$159$$ 32.8068 2.60175
$$160$$ 10.5417 0.833396
$$161$$ 0 0
$$162$$ 40.0277 3.14488
$$163$$ 20.3435 1.59342 0.796712 0.604359i $$-0.206571\pi$$
0.796712 + 0.604359i $$0.206571\pi$$
$$164$$ 1.57834 0.123248
$$165$$ −8.16005 −0.635259
$$166$$ 17.9084 1.38996
$$167$$ 13.0063 1.00646 0.503228 0.864154i $$-0.332146\pi$$
0.503228 + 0.864154i $$0.332146\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 21.3070 1.63417
$$171$$ −10.0641 −0.769622
$$172$$ 1.68420 0.128419
$$173$$ 25.9110 1.96998 0.984989 0.172616i $$-0.0552221\pi$$
0.984989 + 0.172616i $$0.0552221\pi$$
$$174$$ 9.73060 0.737675
$$175$$ 0 0
$$176$$ −1.84581 −0.139133
$$177$$ −8.39278 −0.630840
$$178$$ 10.8587 0.813890
$$179$$ 9.64163 0.720649 0.360325 0.932827i $$-0.382666\pi$$
0.360325 + 0.932827i $$0.382666\pi$$
$$180$$ −15.9761 −1.19079
$$181$$ 2.92683 0.217550 0.108775 0.994066i $$-0.465307\pi$$
0.108775 + 0.994066i $$0.465307\pi$$
$$182$$ 0 0
$$183$$ −36.5818 −2.70421
$$184$$ 9.20760 0.678793
$$185$$ −15.7410 −1.15730
$$186$$ 24.1799 1.77296
$$187$$ 4.44962 0.325388
$$188$$ −3.31211 −0.241560
$$189$$ 0 0
$$190$$ −4.52914 −0.328579
$$191$$ −16.1364 −1.16759 −0.583796 0.811901i $$-0.698433\pi$$
−0.583796 + 0.811901i $$0.698433\pi$$
$$192$$ −29.1913 −2.10670
$$193$$ 0.0863817 0.00621789 0.00310894 0.999995i $$-0.499010\pi$$
0.00310894 + 0.999995i $$0.499010\pi$$
$$194$$ −1.48542 −0.106647
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −15.0589 −1.07290 −0.536451 0.843932i $$-0.680235\pi$$
−0.536451 + 0.843932i $$0.680235\pi$$
$$198$$ 7.48567 0.531983
$$199$$ −15.4466 −1.09498 −0.547492 0.836811i $$-0.684417\pi$$
−0.547492 + 0.836811i $$0.684417\pi$$
$$200$$ −15.1228 −1.06934
$$201$$ 14.4936 1.02230
$$202$$ −0.712119 −0.0501045
$$203$$ 0 0
$$204$$ 11.8875 0.832294
$$205$$ −8.05987 −0.562925
$$206$$ −8.99552 −0.626748
$$207$$ −24.6202 −1.71122
$$208$$ 0 0
$$209$$ −0.945840 −0.0654251
$$210$$ 0 0
$$211$$ −25.0561 −1.72493 −0.862466 0.506115i $$-0.831081\pi$$
−0.862466 + 0.506115i $$0.831081\pi$$
$$212$$ 6.03647 0.414586
$$213$$ 35.7382 2.44874
$$214$$ 5.68057 0.388316
$$215$$ −8.60047 −0.586547
$$216$$ 53.9285 3.66937
$$217$$ 0 0
$$218$$ −8.50491 −0.576025
$$219$$ 17.3397 1.17171
$$220$$ −1.50146 −0.101228
$$221$$ 0 0
$$222$$ 19.7043 1.32247
$$223$$ −20.6640 −1.38376 −0.691881 0.722012i $$-0.743218\pi$$
−0.691881 + 0.722012i $$0.743218\pi$$
$$224$$ 0 0
$$225$$ 40.4368 2.69579
$$226$$ −10.8136 −0.719307
$$227$$ 0.982378 0.0652027 0.0326013 0.999468i $$-0.489621\pi$$
0.0326013 + 0.999468i $$0.489621\pi$$
$$228$$ −2.52689 −0.167347
$$229$$ −14.2742 −0.943269 −0.471634 0.881794i $$-0.656336\pi$$
−0.471634 + 0.881794i $$0.656336\pi$$
$$230$$ −11.0798 −0.730581
$$231$$ 0 0
$$232$$ 7.59802 0.498835
$$233$$ 1.39534 0.0914119 0.0457060 0.998955i $$-0.485446\pi$$
0.0457060 + 0.998955i $$0.485446\pi$$
$$234$$ 0 0
$$235$$ 16.9134 1.10331
$$236$$ −1.54428 −0.100524
$$237$$ 1.81637 0.117986
$$238$$ 0 0
$$239$$ −2.54875 −0.164865 −0.0824326 0.996597i $$-0.526269\pi$$
−0.0824326 + 0.996597i $$0.526269\pi$$
$$240$$ 25.1816 1.62547
$$241$$ 29.6746 1.91151 0.955755 0.294164i $$-0.0950413\pi$$
0.955755 + 0.294164i $$0.0950413\pi$$
$$242$$ −12.2345 −0.786466
$$243$$ −61.4716 −3.94340
$$244$$ −6.73108 −0.430913
$$245$$ 0 0
$$246$$ 10.0892 0.643264
$$247$$ 0 0
$$248$$ 18.8806 1.19892
$$249$$ −51.0218 −3.23337
$$250$$ −0.319106 −0.0201820
$$251$$ −18.0858 −1.14157 −0.570784 0.821100i $$-0.693361\pi$$
−0.570784 + 0.821100i $$0.693361\pi$$
$$252$$ 0 0
$$253$$ −2.31384 −0.145470
$$254$$ −13.1872 −0.827437
$$255$$ −60.7043 −3.80145
$$256$$ −13.2470 −0.827940
$$257$$ −14.7403 −0.919473 −0.459737 0.888055i $$-0.652056\pi$$
−0.459737 + 0.888055i $$0.652056\pi$$
$$258$$ 10.7659 0.670257
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −20.3164 −1.25755
$$262$$ −18.5108 −1.14360
$$263$$ −8.15922 −0.503119 −0.251560 0.967842i $$-0.580943\pi$$
−0.251560 + 0.967842i $$0.580943\pi$$
$$264$$ 7.97597 0.490887
$$265$$ −30.8255 −1.89360
$$266$$ 0 0
$$267$$ −30.9367 −1.89329
$$268$$ 2.66684 0.162903
$$269$$ −0.442582 −0.0269847 −0.0134924 0.999909i $$-0.504295\pi$$
−0.0134924 + 0.999909i $$0.504295\pi$$
$$270$$ −64.8939 −3.94932
$$271$$ −21.0831 −1.28071 −0.640354 0.768080i $$-0.721212\pi$$
−0.640354 + 0.768080i $$0.721212\pi$$
$$272$$ −13.7314 −0.832586
$$273$$ 0 0
$$274$$ 21.9018 1.32313
$$275$$ 3.80031 0.229167
$$276$$ −6.18163 −0.372090
$$277$$ 3.59421 0.215955 0.107977 0.994153i $$-0.465563\pi$$
0.107977 + 0.994153i $$0.465563\pi$$
$$278$$ 14.0127 0.840426
$$279$$ −50.4849 −3.02245
$$280$$ 0 0
$$281$$ 9.01252 0.537642 0.268821 0.963190i $$-0.413366\pi$$
0.268821 + 0.963190i $$0.413366\pi$$
$$282$$ −21.1720 −1.26077
$$283$$ −27.7411 −1.64904 −0.824520 0.565833i $$-0.808555\pi$$
−0.824520 + 0.565833i $$0.808555\pi$$
$$284$$ 6.57585 0.390205
$$285$$ 12.9037 0.764349
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 27.5516 1.62349
$$289$$ 16.1016 0.947153
$$290$$ −9.14296 −0.536893
$$291$$ 4.23201 0.248085
$$292$$ 3.19052 0.186711
$$293$$ 12.7409 0.744333 0.372167 0.928166i $$-0.378615\pi$$
0.372167 + 0.928166i $$0.378615\pi$$
$$294$$ 0 0
$$295$$ 7.88593 0.459137
$$296$$ 15.3859 0.894287
$$297$$ −13.5521 −0.786370
$$298$$ −13.3795 −0.775055
$$299$$ 0 0
$$300$$ 10.1528 0.586175
$$301$$ 0 0
$$302$$ 4.34307 0.249916
$$303$$ 2.02885 0.116555
$$304$$ 2.91883 0.167406
$$305$$ 34.3726 1.96817
$$306$$ 55.6874 3.18344
$$307$$ 10.1384 0.578626 0.289313 0.957235i $$-0.406573\pi$$
0.289313 + 0.957235i $$0.406573\pi$$
$$308$$ 0 0
$$309$$ 25.6286 1.45796
$$310$$ −22.7197 −1.29039
$$311$$ −20.6013 −1.16819 −0.584096 0.811685i $$-0.698551\pi$$
−0.584096 + 0.811685i $$0.698551\pi$$
$$312$$ 0 0
$$313$$ −16.5456 −0.935213 −0.467606 0.883937i $$-0.654883\pi$$
−0.467606 + 0.883937i $$0.654883\pi$$
$$314$$ −1.31809 −0.0743841
$$315$$ 0 0
$$316$$ 0.334213 0.0188010
$$317$$ 21.2340 1.19262 0.596310 0.802754i $$-0.296633\pi$$
0.596310 + 0.802754i $$0.296633\pi$$
$$318$$ 38.5869 2.16384
$$319$$ −1.90936 −0.106904
$$320$$ 27.4284 1.53329
$$321$$ −16.1841 −0.903310
$$322$$ 0 0
$$323$$ −7.03629 −0.391510
$$324$$ −20.9835 −1.16575
$$325$$ 0 0
$$326$$ 23.9277 1.32523
$$327$$ 24.2308 1.33996
$$328$$ 7.87804 0.434992
$$329$$ 0 0
$$330$$ −9.59774 −0.528338
$$331$$ −16.2415 −0.892712 −0.446356 0.894855i $$-0.647279\pi$$
−0.446356 + 0.894855i $$0.647279\pi$$
$$332$$ −9.38804 −0.515236
$$333$$ −41.1403 −2.25448
$$334$$ 15.2978 0.837058
$$335$$ −13.6184 −0.744050
$$336$$ 0 0
$$337$$ 6.42141 0.349797 0.174898 0.984586i $$-0.444040\pi$$
0.174898 + 0.984586i $$0.444040\pi$$
$$338$$ 0 0
$$339$$ 30.8082 1.67327
$$340$$ −11.1696 −0.605758
$$341$$ −4.74464 −0.256937
$$342$$ −11.8373 −0.640086
$$343$$ 0 0
$$344$$ 8.40645 0.453245
$$345$$ 31.5668 1.69950
$$346$$ 30.4762 1.63841
$$347$$ 14.4809 0.777376 0.388688 0.921369i $$-0.372928\pi$$
0.388688 + 0.921369i $$0.372928\pi$$
$$348$$ −5.10102 −0.273443
$$349$$ 1.74500 0.0934079 0.0467040 0.998909i $$-0.485128\pi$$
0.0467040 + 0.998909i $$0.485128\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.58934 0.138012
$$353$$ 15.9580 0.849358 0.424679 0.905344i $$-0.360387\pi$$
0.424679 + 0.905344i $$0.360387\pi$$
$$354$$ −9.87148 −0.524663
$$355$$ −33.5799 −1.78224
$$356$$ −5.69237 −0.301695
$$357$$ 0 0
$$358$$ 11.3404 0.599356
$$359$$ −8.81400 −0.465185 −0.232593 0.972574i $$-0.574721\pi$$
−0.232593 + 0.972574i $$0.574721\pi$$
$$360$$ −79.7422 −4.20278
$$361$$ −17.5043 −0.921280
$$362$$ 3.44250 0.180934
$$363$$ 34.8566 1.82950
$$364$$ 0 0
$$365$$ −16.2926 −0.852792
$$366$$ −43.0271 −2.24906
$$367$$ 19.7080 1.02875 0.514374 0.857566i $$-0.328024\pi$$
0.514374 + 0.857566i $$0.328024\pi$$
$$368$$ 7.14043 0.372221
$$369$$ −21.0651 −1.09660
$$370$$ −18.5144 −0.962515
$$371$$ 0 0
$$372$$ −12.6757 −0.657205
$$373$$ −0.365792 −0.0189400 −0.00947000 0.999955i $$-0.503014\pi$$
−0.00947000 + 0.999955i $$0.503014\pi$$
$$374$$ 5.23358 0.270622
$$375$$ 0.909143 0.0469479
$$376$$ −16.5319 −0.852566
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −7.39215 −0.379709 −0.189855 0.981812i $$-0.560802\pi$$
−0.189855 + 0.981812i $$0.560802\pi$$
$$380$$ 2.37429 0.121798
$$381$$ 37.5707 1.92481
$$382$$ −18.9795 −0.971073
$$383$$ −3.35364 −0.171363 −0.0856814 0.996323i $$-0.527307\pi$$
−0.0856814 + 0.996323i $$0.527307\pi$$
$$384$$ −11.8958 −0.607057
$$385$$ 0 0
$$386$$ 0.101601 0.00517135
$$387$$ −22.4780 −1.14262
$$388$$ 0.778692 0.0395321
$$389$$ −2.59344 −0.131492 −0.0657462 0.997836i $$-0.520943\pi$$
−0.0657462 + 0.997836i $$0.520943\pi$$
$$390$$ 0 0
$$391$$ −17.2131 −0.870506
$$392$$ 0 0
$$393$$ 52.7379 2.66027
$$394$$ −17.7121 −0.892321
$$395$$ −1.70668 −0.0858721
$$396$$ −3.92417 −0.197197
$$397$$ −33.4914 −1.68088 −0.840441 0.541902i $$-0.817704\pi$$
−0.840441 + 0.541902i $$0.817704\pi$$
$$398$$ −18.1681 −0.910686
$$399$$ 0 0
$$400$$ −11.7276 −0.586380
$$401$$ −39.5156 −1.97331 −0.986657 0.162811i $$-0.947944\pi$$
−0.986657 + 0.162811i $$0.947944\pi$$
$$402$$ 17.0472 0.850239
$$403$$ 0 0
$$404$$ 0.373310 0.0185729
$$405$$ 107.153 5.32449
$$406$$ 0 0
$$407$$ −3.86643 −0.191652
$$408$$ 59.3348 2.93751
$$409$$ −10.9799 −0.542922 −0.271461 0.962449i $$-0.587507\pi$$
−0.271461 + 0.962449i $$0.587507\pi$$
$$410$$ −9.47990 −0.468179
$$411$$ −62.3989 −3.07791
$$412$$ 4.71567 0.232324
$$413$$ 0 0
$$414$$ −28.9580 −1.42321
$$415$$ 47.9405 2.35331
$$416$$ 0 0
$$417$$ −39.9227 −1.95502
$$418$$ −1.11248 −0.0544134
$$419$$ −31.5621 −1.54191 −0.770954 0.636891i $$-0.780220\pi$$
−0.770954 + 0.636891i $$0.780220\pi$$
$$420$$ 0 0
$$421$$ 17.7055 0.862914 0.431457 0.902134i $$-0.358000\pi$$
0.431457 + 0.902134i $$0.358000\pi$$
$$422$$ −29.4706 −1.43461
$$423$$ 44.2046 2.14930
$$424$$ 30.1301 1.46325
$$425$$ 28.2712 1.37136
$$426$$ 42.0348 2.03659
$$427$$ 0 0
$$428$$ −2.97789 −0.143942
$$429$$ 0 0
$$430$$ −10.1158 −0.487825
$$431$$ 21.4816 1.03473 0.517367 0.855764i $$-0.326912\pi$$
0.517367 + 0.855764i $$0.326912\pi$$
$$432$$ 41.8212 2.01212
$$433$$ −34.7200 −1.66854 −0.834269 0.551358i $$-0.814110\pi$$
−0.834269 + 0.551358i $$0.814110\pi$$
$$434$$ 0 0
$$435$$ 26.0486 1.24893
$$436$$ 4.45848 0.213522
$$437$$ 3.65894 0.175031
$$438$$ 20.3948 0.974500
$$439$$ −0.728505 −0.0347697 −0.0173848 0.999849i $$-0.505534\pi$$
−0.0173848 + 0.999849i $$0.505534\pi$$
$$440$$ −7.49428 −0.357276
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0.837291 0.0397809 0.0198904 0.999802i $$-0.493668\pi$$
0.0198904 + 0.999802i $$0.493668\pi$$
$$444$$ −10.3295 −0.490216
$$445$$ 29.0684 1.37797
$$446$$ −24.3047 −1.15086
$$447$$ 38.1187 1.80295
$$448$$ 0 0
$$449$$ −26.4312 −1.24737 −0.623683 0.781677i $$-0.714365\pi$$
−0.623683 + 0.781677i $$0.714365\pi$$
$$450$$ 47.5612 2.24206
$$451$$ −1.97973 −0.0932217
$$452$$ 5.66873 0.266635
$$453$$ −12.3736 −0.581361
$$454$$ 1.15546 0.0542284
$$455$$ 0 0
$$456$$ −12.6126 −0.590639
$$457$$ 2.40475 0.112489 0.0562447 0.998417i $$-0.482087\pi$$
0.0562447 + 0.998417i $$0.482087\pi$$
$$458$$ −16.7892 −0.784507
$$459$$ −100.816 −4.70571
$$460$$ 5.80831 0.270814
$$461$$ −22.2702 −1.03722 −0.518612 0.855010i $$-0.673551\pi$$
−0.518612 + 0.855010i $$0.673551\pi$$
$$462$$ 0 0
$$463$$ −32.3085 −1.50151 −0.750753 0.660583i $$-0.770309\pi$$
−0.750753 + 0.660583i $$0.770309\pi$$
$$464$$ 5.89221 0.273539
$$465$$ 64.7291 3.00174
$$466$$ 1.64118 0.0760263
$$467$$ 12.8744 0.595756 0.297878 0.954604i $$-0.403721\pi$$
0.297878 + 0.954604i $$0.403721\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 19.8934 0.917612
$$471$$ 3.75528 0.173034
$$472$$ −7.70802 −0.354791
$$473$$ −2.11251 −0.0971335
$$474$$ 2.13639 0.0981275
$$475$$ −6.00952 −0.275736
$$476$$ 0 0
$$477$$ −80.5649 −3.68881
$$478$$ −2.99781 −0.137117
$$479$$ 10.5419 0.481670 0.240835 0.970566i $$-0.422579\pi$$
0.240835 + 0.970566i $$0.422579\pi$$
$$480$$ −35.3252 −1.61237
$$481$$ 0 0
$$482$$ 34.9029 1.58978
$$483$$ 0 0
$$484$$ 6.41364 0.291529
$$485$$ −3.97643 −0.180560
$$486$$ −72.3020 −3.27969
$$487$$ −38.2416 −1.73289 −0.866446 0.499271i $$-0.833601\pi$$
−0.866446 + 0.499271i $$0.833601\pi$$
$$488$$ −33.5972 −1.52087
$$489$$ −68.1709 −3.08280
$$490$$ 0 0
$$491$$ −15.4291 −0.696306 −0.348153 0.937438i $$-0.613191\pi$$
−0.348153 + 0.937438i $$0.613191\pi$$
$$492$$ −5.28901 −0.238447
$$493$$ −14.2041 −0.639721
$$494$$ 0 0
$$495$$ 20.0390 0.900685
$$496$$ 14.6418 0.657436
$$497$$ 0 0
$$498$$ −60.0111 −2.68916
$$499$$ 38.8212 1.73788 0.868938 0.494921i $$-0.164803\pi$$
0.868938 + 0.494921i $$0.164803\pi$$
$$500$$ 0.167283 0.00748112
$$501$$ −43.5840 −1.94719
$$502$$ −21.2723 −0.949430
$$503$$ 27.0935 1.20804 0.604020 0.796969i $$-0.293565\pi$$
0.604020 + 0.796969i $$0.293565\pi$$
$$504$$ 0 0
$$505$$ −1.90633 −0.0848304
$$506$$ −2.72151 −0.120986
$$507$$ 0 0
$$508$$ 6.91304 0.306717
$$509$$ −15.7693 −0.698961 −0.349480 0.936944i $$-0.613642\pi$$
−0.349480 + 0.936944i $$0.613642\pi$$
$$510$$ −71.3995 −3.16162
$$511$$ 0 0
$$512$$ −22.6809 −1.00236
$$513$$ 21.4302 0.946167
$$514$$ −17.3373 −0.764716
$$515$$ −24.0808 −1.06113
$$516$$ −5.64376 −0.248453
$$517$$ 4.15441 0.182711
$$518$$ 0 0
$$519$$ −86.8277 −3.81131
$$520$$ 0 0
$$521$$ −7.30737 −0.320142 −0.160071 0.987106i $$-0.551172\pi$$
−0.160071 + 0.987106i $$0.551172\pi$$
$$522$$ −23.8958 −1.04589
$$523$$ −14.0826 −0.615788 −0.307894 0.951421i $$-0.599624\pi$$
−0.307894 + 0.951421i $$0.599624\pi$$
$$524$$ 9.70381 0.423913
$$525$$ 0 0
$$526$$ −9.59677 −0.418439
$$527$$ −35.2963 −1.53753
$$528$$ 6.18531 0.269181
$$529$$ −14.0490 −0.610827
$$530$$ −36.2565 −1.57488
$$531$$ 20.6105 0.894419
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −36.3873 −1.57463
$$535$$ 15.2067 0.657445
$$536$$ 13.3111 0.574953
$$537$$ −32.3091 −1.39424
$$538$$ −0.520559 −0.0224429
$$539$$ 0 0
$$540$$ 34.0190 1.46394
$$541$$ 39.1750 1.68426 0.842132 0.539271i $$-0.181300\pi$$
0.842132 + 0.539271i $$0.181300\pi$$
$$542$$ −24.7977 −1.06515
$$543$$ −9.80781 −0.420893
$$544$$ 19.2626 0.825877
$$545$$ −22.7674 −0.975250
$$546$$ 0 0
$$547$$ 38.7917 1.65862 0.829308 0.558792i $$-0.188735\pi$$
0.829308 + 0.558792i $$0.188735\pi$$
$$548$$ −11.4814 −0.490463
$$549$$ 89.8355 3.83408
$$550$$ 4.46987 0.190596
$$551$$ 3.01932 0.128627
$$552$$ −30.8546 −1.31326
$$553$$ 0 0
$$554$$ 4.22746 0.179607
$$555$$ 52.7480 2.23903
$$556$$ −7.34580 −0.311531
$$557$$ 27.4693 1.16391 0.581956 0.813220i $$-0.302287\pi$$
0.581956 + 0.813220i $$0.302287\pi$$
$$558$$ −59.3796 −2.51374
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −14.9107 −0.629528
$$562$$ 10.6004 0.447151
$$563$$ −14.1210 −0.595129 −0.297565 0.954702i $$-0.596174\pi$$
−0.297565 + 0.954702i $$0.596174\pi$$
$$564$$ 11.0989 0.467346
$$565$$ −28.9477 −1.21784
$$566$$ −32.6288 −1.37149
$$567$$ 0 0
$$568$$ 32.8223 1.37720
$$569$$ −8.28649 −0.347388 −0.173694 0.984800i $$-0.555570\pi$$
−0.173694 + 0.984800i $$0.555570\pi$$
$$570$$ 15.1771 0.635701
$$571$$ −23.5274 −0.984591 −0.492295 0.870428i $$-0.663842\pi$$
−0.492295 + 0.870428i $$0.663842\pi$$
$$572$$ 0 0
$$573$$ 54.0731 2.25894
$$574$$ 0 0
$$575$$ −14.7013 −0.613087
$$576$$ 71.6861 2.98692
$$577$$ −17.7732 −0.739906 −0.369953 0.929050i $$-0.620626\pi$$
−0.369953 + 0.929050i $$0.620626\pi$$
$$578$$ 18.9385 0.787737
$$579$$ −0.289465 −0.0120297
$$580$$ 4.79296 0.199017
$$581$$ 0 0
$$582$$ 4.97763 0.206329
$$583$$ −7.57160 −0.313584
$$584$$ 15.9250 0.658982
$$585$$ 0 0
$$586$$ 14.9857 0.619054
$$587$$ −6.64096 −0.274102 −0.137051 0.990564i $$-0.543762\pi$$
−0.137051 + 0.990564i $$0.543762\pi$$
$$588$$ 0 0
$$589$$ 7.50282 0.309148
$$590$$ 9.27532 0.381859
$$591$$ 50.4623 2.07574
$$592$$ 11.9317 0.490388
$$593$$ 34.0504 1.39828 0.699141 0.714984i $$-0.253566\pi$$
0.699141 + 0.714984i $$0.253566\pi$$
$$594$$ −15.9398 −0.654016
$$595$$ 0 0
$$596$$ 7.01387 0.287299
$$597$$ 51.7616 2.11846
$$598$$ 0 0
$$599$$ 9.38441 0.383437 0.191718 0.981450i $$-0.438594\pi$$
0.191718 + 0.981450i $$0.438594\pi$$
$$600$$ 50.6764 2.06885
$$601$$ −8.80294 −0.359079 −0.179540 0.983751i $$-0.557461\pi$$
−0.179540 + 0.983751i $$0.557461\pi$$
$$602$$ 0 0
$$603$$ −35.5926 −1.44944
$$604$$ −2.27675 −0.0926394
$$605$$ −32.7516 −1.33154
$$606$$ 2.38631 0.0969371
$$607$$ −17.7720 −0.721342 −0.360671 0.932693i $$-0.617452\pi$$
−0.360671 + 0.932693i $$0.617452\pi$$
$$608$$ −4.09458 −0.166057
$$609$$ 0 0
$$610$$ 40.4286 1.63691
$$611$$ 0 0
$$612$$ −29.1927 −1.18004
$$613$$ −10.6081 −0.428457 −0.214228 0.976784i $$-0.568724\pi$$
−0.214228 + 0.976784i $$0.568724\pi$$
$$614$$ 11.9246 0.481237
$$615$$ 27.0086 1.08909
$$616$$ 0 0
$$617$$ −49.3483 −1.98669 −0.993344 0.115188i $$-0.963253\pi$$
−0.993344 + 0.115188i $$0.963253\pi$$
$$618$$ 30.1440 1.21257
$$619$$ 7.42203 0.298317 0.149158 0.988813i $$-0.452344\pi$$
0.149158 + 0.988813i $$0.452344\pi$$
$$620$$ 11.9102 0.478325
$$621$$ 52.4255 2.10376
$$622$$ −24.2309 −0.971573
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −25.4234 −1.01694
$$626$$ −19.4607 −0.777806
$$627$$ 3.16951 0.126578
$$628$$ 0.690975 0.0275729
$$629$$ −28.7631 −1.14686
$$630$$ 0 0
$$631$$ −35.5184 −1.41396 −0.706982 0.707231i $$-0.749944\pi$$
−0.706982 + 0.707231i $$0.749944\pi$$
$$632$$ 1.66817 0.0663564
$$633$$ 83.9628 3.33722
$$634$$ 24.9752 0.991890
$$635$$ −35.3018 −1.40091
$$636$$ −20.2282 −0.802099
$$637$$ 0 0
$$638$$ −2.24576 −0.0889106
$$639$$ −87.7637 −3.47188
$$640$$ 11.1774 0.441827
$$641$$ 37.1554 1.46755 0.733775 0.679393i $$-0.237757\pi$$
0.733775 + 0.679393i $$0.237757\pi$$
$$642$$ −19.0356 −0.751274
$$643$$ 39.7694 1.56835 0.784176 0.620538i $$-0.213086\pi$$
0.784176 + 0.620538i $$0.213086\pi$$
$$644$$ 0 0
$$645$$ 28.8201 1.13479
$$646$$ −8.27599 −0.325614
$$647$$ −22.9754 −0.903256 −0.451628 0.892206i $$-0.649157\pi$$
−0.451628 + 0.892206i $$0.649157\pi$$
$$648$$ −104.736 −4.11442
$$649$$ 1.93700 0.0760340
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −12.5435 −0.491241
$$653$$ 18.1757 0.711268 0.355634 0.934625i $$-0.384265\pi$$
0.355634 + 0.934625i $$0.384265\pi$$
$$654$$ 28.4999 1.11443
$$655$$ −49.5530 −1.93619
$$656$$ 6.10936 0.238531
$$657$$ −42.5819 −1.66128
$$658$$ 0 0
$$659$$ 14.1044 0.549431 0.274716 0.961526i $$-0.411416\pi$$
0.274716 + 0.961526i $$0.411416\pi$$
$$660$$ 5.03137 0.195846
$$661$$ −12.8557 −0.500027 −0.250013 0.968242i $$-0.580435\pi$$
−0.250013 + 0.968242i $$0.580435\pi$$
$$662$$ −19.1030 −0.742459
$$663$$ 0 0
$$664$$ −46.8590 −1.81848
$$665$$ 0 0
$$666$$ −48.3887 −1.87502
$$667$$ 7.38627 0.285997
$$668$$ −8.01948 −0.310283
$$669$$ 69.2449 2.67716
$$670$$ −16.0177 −0.618819
$$671$$ 8.44287 0.325933
$$672$$ 0 0
$$673$$ −45.6138 −1.75828 −0.879141 0.476561i $$-0.841883\pi$$
−0.879141 + 0.476561i $$0.841883\pi$$
$$674$$ 7.55278 0.290922
$$675$$ −86.1048 −3.31418
$$676$$ 0 0
$$677$$ −10.2469 −0.393821 −0.196910 0.980421i $$-0.563091\pi$$
−0.196910 + 0.980421i $$0.563091\pi$$
$$678$$ 36.2362 1.39164
$$679$$ 0 0
$$680$$ −55.7515 −2.13797
$$681$$ −3.29194 −0.126148
$$682$$ −5.58058 −0.213692
$$683$$ −2.90040 −0.110981 −0.0554904 0.998459i $$-0.517672\pi$$
−0.0554904 + 0.998459i $$0.517672\pi$$
$$684$$ 6.20539 0.237269
$$685$$ 58.6305 2.24016
$$686$$ 0 0
$$687$$ 47.8329 1.82494
$$688$$ 6.51914 0.248540
$$689$$ 0 0
$$690$$ 37.1284 1.41345
$$691$$ 6.91470 0.263048 0.131524 0.991313i $$-0.458013\pi$$
0.131524 + 0.991313i $$0.458013\pi$$
$$692$$ −15.9764 −0.607330
$$693$$ 0 0
$$694$$ 17.0323 0.646536
$$695$$ 37.5117 1.42290
$$696$$ −25.4610 −0.965095
$$697$$ −14.7276 −0.557847
$$698$$ 2.05245 0.0776864
$$699$$ −4.67579 −0.176855
$$700$$ 0 0
$$701$$ −26.0973 −0.985682 −0.492841 0.870119i $$-0.664042\pi$$
−0.492841 + 0.870119i $$0.664042\pi$$
$$702$$ 0 0
$$703$$ 6.11408 0.230597
$$704$$ 6.73717 0.253916
$$705$$ −56.6769 −2.13457
$$706$$ 18.7696 0.706402
$$707$$ 0 0
$$708$$ 5.17487 0.194483
$$709$$ 3.38472 0.127116 0.0635580 0.997978i $$-0.479755\pi$$
0.0635580 + 0.997978i $$0.479755\pi$$
$$710$$ −39.4962 −1.48227
$$711$$ −4.46053 −0.167283
$$712$$ −28.4126 −1.06481
$$713$$ 18.3544 0.687378
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −5.94489 −0.222171
$$717$$ 8.54087 0.318964
$$718$$ −10.3669 −0.386890
$$719$$ 2.41574 0.0900918 0.0450459 0.998985i $$-0.485657\pi$$
0.0450459 + 0.998985i $$0.485657\pi$$
$$720$$ −61.8395 −2.30462
$$721$$ 0 0
$$722$$ −20.5883 −0.766219
$$723$$ −99.4395 −3.69819
$$724$$ −1.80464 −0.0670691
$$725$$ −12.1314 −0.450548
$$726$$ 40.9979 1.52157
$$727$$ −17.0150 −0.631050 −0.315525 0.948917i $$-0.602181\pi$$
−0.315525 + 0.948917i $$0.602181\pi$$
$$728$$ 0 0
$$729$$ 103.896 3.84799
$$730$$ −19.1631 −0.709258
$$731$$ −15.7154 −0.581256
$$732$$ 22.5558 0.833687
$$733$$ 4.18453 0.154559 0.0772795 0.997009i $$-0.475377\pi$$
0.0772795 + 0.997009i $$0.475377\pi$$
$$734$$ 23.1803 0.855599
$$735$$ 0 0
$$736$$ −10.0167 −0.369221
$$737$$ −3.34505 −0.123216
$$738$$ −24.7765 −0.912034
$$739$$ 28.0794 1.03292 0.516458 0.856312i $$-0.327250\pi$$
0.516458 + 0.856312i $$0.327250\pi$$
$$740$$ 9.70567 0.356788
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 26.5210 0.972961 0.486481 0.873691i $$-0.338280\pi$$
0.486481 + 0.873691i $$0.338280\pi$$
$$744$$ −63.2689 −2.31955
$$745$$ −35.8167 −1.31222
$$746$$ −0.430240 −0.0157522
$$747$$ 125.296 4.58435
$$748$$ −2.74357 −0.100315
$$749$$ 0 0
$$750$$ 1.06932 0.0390461
$$751$$ −23.7162 −0.865418 −0.432709 0.901534i $$-0.642442\pi$$
−0.432709 + 0.901534i $$0.642442\pi$$
$$752$$ −12.8204 −0.467510
$$753$$ 60.6056 2.20859
$$754$$ 0 0
$$755$$ 11.6263 0.423125
$$756$$ 0 0
$$757$$ −52.3661 −1.90328 −0.951639 0.307219i $$-0.900602\pi$$
−0.951639 + 0.307219i $$0.900602\pi$$
$$758$$ −8.69454 −0.315800
$$759$$ 7.75368 0.281441
$$760$$ 11.8509 0.429877
$$761$$ 23.8695 0.865268 0.432634 0.901570i $$-0.357584\pi$$
0.432634 + 0.901570i $$0.357584\pi$$
$$762$$ 44.1902 1.60084
$$763$$ 0 0
$$764$$ 9.94949 0.359960
$$765$$ 149.074 5.38978
$$766$$ −3.94450 −0.142521
$$767$$ 0 0
$$768$$ 44.3908 1.60181
$$769$$ 41.7599 1.50590 0.752950 0.658077i $$-0.228630\pi$$
0.752950 + 0.658077i $$0.228630\pi$$
$$770$$ 0 0
$$771$$ 49.3946 1.77890
$$772$$ −0.0532617 −0.00191693
$$773$$ 2.97178 0.106888 0.0534438 0.998571i $$-0.482980\pi$$
0.0534438 + 0.998571i $$0.482980\pi$$
$$774$$ −26.4383 −0.950306
$$775$$ −30.1457 −1.08287
$$776$$ 3.88672 0.139525
$$777$$ 0 0
$$778$$ −3.05037 −0.109361
$$779$$ 3.13059 0.112165
$$780$$ 0 0
$$781$$ −8.24815 −0.295142
$$782$$ −20.2459 −0.723990
$$783$$ 43.2610 1.54602
$$784$$ 0 0
$$785$$ −3.52850 −0.125937
$$786$$ 62.0296 2.21252
$$787$$ −48.6142 −1.73291 −0.866454 0.499256i $$-0.833607\pi$$
−0.866454 + 0.499256i $$0.833607\pi$$
$$788$$ 9.28509 0.330768
$$789$$ 27.3415 0.973384
$$790$$ −2.00737 −0.0714189
$$791$$ 0 0
$$792$$ −19.5869 −0.695990
$$793$$ 0 0
$$794$$ −39.3921 −1.39797
$$795$$ 103.296 3.66354
$$796$$ 9.52418 0.337576
$$797$$ −25.6470 −0.908463 −0.454232 0.890884i $$-0.650086\pi$$
−0.454232 + 0.890884i $$0.650086\pi$$
$$798$$ 0 0
$$799$$ 30.9055 1.09336
$$800$$ 16.4517 0.581655
$$801$$ 75.9724 2.68435
$$802$$ −46.4777 −1.64118
$$803$$ −4.00191 −0.141224
$$804$$ −8.93657 −0.315169
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.48309 0.0522073
$$808$$ 1.86332 0.0655514
$$809$$ −11.9205 −0.419101 −0.209550 0.977798i $$-0.567200\pi$$
−0.209550 + 0.977798i $$0.567200\pi$$
$$810$$ 126.032 4.42832
$$811$$ −10.5564 −0.370685 −0.185343 0.982674i $$-0.559339\pi$$
−0.185343 + 0.982674i $$0.559339\pi$$
$$812$$ 0 0
$$813$$ 70.6494 2.47778
$$814$$ −4.54764 −0.159395
$$815$$ 64.0540 2.24371
$$816$$ 46.0137 1.61080
$$817$$ 3.34057 0.116872
$$818$$ −12.9144 −0.451543
$$819$$ 0 0
$$820$$ 4.96960 0.173546
$$821$$ 46.2139 1.61288 0.806438 0.591319i $$-0.201393\pi$$
0.806438 + 0.591319i $$0.201393\pi$$
$$822$$ −73.3927 −2.55986
$$823$$ 41.2171 1.43674 0.718368 0.695663i $$-0.244889\pi$$
0.718368 + 0.695663i $$0.244889\pi$$
$$824$$ 23.5375 0.819969
$$825$$ −12.7348 −0.443369
$$826$$ 0 0
$$827$$ −5.94430 −0.206704 −0.103352 0.994645i $$-0.532957\pi$$
−0.103352 + 0.994645i $$0.532957\pi$$
$$828$$ 15.1805 0.527558
$$829$$ −34.8106 −1.20902 −0.604511 0.796596i $$-0.706632\pi$$
−0.604511 + 0.796596i $$0.706632\pi$$
$$830$$ 56.3869 1.95722
$$831$$ −12.0442 −0.417808
$$832$$ 0 0
$$833$$ 0 0
$$834$$ −46.9565 −1.62597
$$835$$ 40.9519 1.41720
$$836$$ 0.583191 0.0201701
$$837$$ 107.501 3.71578
$$838$$ −37.1229 −1.28239
$$839$$ 10.7896 0.372500 0.186250 0.982502i $$-0.440367\pi$$
0.186250 + 0.982502i $$0.440367\pi$$
$$840$$ 0 0
$$841$$ −22.9049 −0.789825
$$842$$ 20.8250 0.717676
$$843$$ −30.2009 −1.04018
$$844$$ 15.4492 0.531784
$$845$$ 0 0
$$846$$ 51.9928 1.78755
$$847$$ 0 0
$$848$$ 23.3657 0.802381
$$849$$ 92.9605 3.19039
$$850$$ 33.2523 1.14054
$$851$$ 14.9571 0.512722
$$852$$ −22.0356 −0.754929
$$853$$ 39.1053 1.33894 0.669470 0.742839i $$-0.266521\pi$$
0.669470 + 0.742839i $$0.266521\pi$$
$$854$$ 0 0
$$855$$ −31.6881 −1.08371
$$856$$ −14.8637 −0.508030
$$857$$ 33.3654 1.13974 0.569870 0.821735i $$-0.306994\pi$$
0.569870 + 0.821735i $$0.306994\pi$$
$$858$$ 0 0
$$859$$ 29.4914 1.00623 0.503116 0.864219i $$-0.332187\pi$$
0.503116 + 0.864219i $$0.332187\pi$$
$$860$$ 5.30293 0.180828
$$861$$ 0 0
$$862$$ 25.2664 0.860578
$$863$$ 24.6224 0.838157 0.419079 0.907950i $$-0.362353\pi$$
0.419079 + 0.907950i $$0.362353\pi$$
$$864$$ −58.6675 −1.99591
$$865$$ 81.5841 2.77394
$$866$$ −40.8372 −1.38771
$$867$$ −53.9564 −1.83246
$$868$$ 0 0
$$869$$ −0.419207 −0.0142206
$$870$$ 30.6380 1.03873
$$871$$ 0 0
$$872$$ 22.2538 0.753609
$$873$$ −10.3927 −0.351740
$$874$$ 4.30359 0.145571
$$875$$ 0 0
$$876$$ −10.6914 −0.361230
$$877$$ 4.62556 0.156194 0.0780971 0.996946i $$-0.475116\pi$$
0.0780971 + 0.996946i $$0.475116\pi$$
$$878$$ −0.856858 −0.0289176
$$879$$ −42.6948 −1.44006
$$880$$ −5.81177 −0.195915
$$881$$ 42.1720 1.42081 0.710405 0.703793i $$-0.248512\pi$$
0.710405 + 0.703793i $$0.248512\pi$$
$$882$$ 0 0
$$883$$ 20.7992 0.699950 0.349975 0.936759i $$-0.386190\pi$$
0.349975 + 0.936759i $$0.386190\pi$$
$$884$$ 0 0
$$885$$ −26.4257 −0.888291
$$886$$ 0.984810 0.0330853
$$887$$ −34.4889 −1.15802 −0.579012 0.815319i $$-0.696562\pi$$
−0.579012 + 0.815319i $$0.696562\pi$$
$$888$$ −51.5581 −1.73018
$$889$$ 0 0
$$890$$ 34.1898 1.14605
$$891$$ 26.3198 0.881748
$$892$$ 12.7411 0.426604
$$893$$ −6.56947 −0.219839
$$894$$ 44.8347 1.49950
$$895$$ 30.3579 1.01475
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −31.0880 −1.03742
$$899$$ 15.1459 0.505144
$$900$$ −24.9327 −0.831091
$$901$$ −56.3267 −1.87651
$$902$$ −2.32853 −0.0775315
$$903$$ 0 0
$$904$$ 28.2946 0.941065
$$905$$ 9.21550 0.306334
$$906$$ −14.5536 −0.483512
$$907$$ 13.3619 0.443675 0.221838 0.975084i $$-0.428795\pi$$
0.221838 + 0.975084i $$0.428795\pi$$
$$908$$ −0.605720 −0.0201015
$$909$$ −4.98233 −0.165254
$$910$$ 0 0
$$911$$ 5.29058 0.175285 0.0876424 0.996152i $$-0.472067\pi$$
0.0876424 + 0.996152i $$0.472067\pi$$
$$912$$ −9.78097 −0.323880
$$913$$ 11.7755 0.389713
$$914$$ 2.82843 0.0935562
$$915$$ −115.182 −3.80781
$$916$$ 8.80129 0.290803
$$917$$ 0 0
$$918$$ −118.579 −3.91369
$$919$$ −18.8306 −0.621164 −0.310582 0.950547i $$-0.600524\pi$$
−0.310582 + 0.950547i $$0.600524\pi$$
$$920$$ 28.9913 0.955814
$$921$$ −33.9736 −1.11947
$$922$$ −26.1939 −0.862649
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −24.5659 −0.807721
$$926$$ −38.0009 −1.24879
$$927$$ −62.9371 −2.06712
$$928$$ −8.26571 −0.271335
$$929$$ −44.9537 −1.47488 −0.737442 0.675411i $$-0.763966\pi$$
−0.737442 + 0.675411i $$0.763966\pi$$
$$930$$ 76.1335 2.49652
$$931$$ 0 0
$$932$$ −0.860348 −0.0281816
$$933$$ 69.0348 2.26010
$$934$$ 15.1427 0.495484
$$935$$ 14.0102 0.458182
$$936$$ 0 0
$$937$$ −37.9272 −1.23903 −0.619514 0.784986i $$-0.712670\pi$$
−0.619514 + 0.784986i $$0.712670\pi$$
$$938$$ 0 0
$$939$$ 55.4442 1.80935
$$940$$ −10.4286 −0.340143
$$941$$ 35.7869 1.16662 0.583310 0.812250i $$-0.301757\pi$$
0.583310 + 0.812250i $$0.301757\pi$$
$$942$$ 4.41691 0.143911
$$943$$ 7.65848 0.249394
$$944$$ −5.97752 −0.194552
$$945$$ 0 0
$$946$$ −2.48471 −0.0807849
$$947$$ 14.7443 0.479125 0.239563 0.970881i $$-0.422996\pi$$
0.239563 + 0.970881i $$0.422996\pi$$
$$948$$ −1.11995 −0.0363742
$$949$$ 0 0
$$950$$ −7.06831 −0.229326
$$951$$ −71.1551 −2.30736
$$952$$ 0 0
$$953$$ 0.649669 0.0210448 0.0105224 0.999945i $$-0.496651\pi$$
0.0105224 + 0.999945i $$0.496651\pi$$
$$954$$ −94.7593 −3.06795
$$955$$ −50.8076 −1.64409
$$956$$ 1.57153 0.0508268
$$957$$ 6.39826 0.206826
$$958$$ 12.3992 0.400600
$$959$$ 0 0
$$960$$ −91.9123 −2.96646
$$961$$ 6.63659 0.214083
$$962$$ 0 0
$$963$$ 39.7440 1.28073
$$964$$ −18.2969 −0.589305
$$965$$ 0.271983 0.00875545
$$966$$ 0 0
$$967$$ 13.0802 0.420632 0.210316 0.977633i $$-0.432551\pi$$
0.210316 + 0.977633i $$0.432551\pi$$
$$968$$ 32.0127 1.02893
$$969$$ 23.5786 0.757453
$$970$$ −4.67702 −0.150170
$$971$$ 29.1203 0.934515 0.467258 0.884121i $$-0.345242\pi$$
0.467258 + 0.884121i $$0.345242\pi$$
$$972$$ 37.9025 1.21572
$$973$$ 0 0
$$974$$ −44.9792 −1.44123
$$975$$ 0 0
$$976$$ −26.0544 −0.833980
$$977$$ 18.5000 0.591868 0.295934 0.955208i $$-0.404369\pi$$
0.295934 + 0.955208i $$0.404369\pi$$
$$978$$ −80.1817 −2.56393
$$979$$ 7.14000 0.228195
$$980$$ 0 0
$$981$$ −59.5045 −1.89983
$$982$$ −18.1475 −0.579111
$$983$$ −11.0462 −0.352318 −0.176159 0.984362i $$-0.556367\pi$$
−0.176159 + 0.984362i $$0.556367\pi$$
$$984$$ −26.3993 −0.841578
$$985$$ −47.4148 −1.51076
$$986$$ −16.7067 −0.532049
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.17216 0.259860
$$990$$ 23.5696 0.749090
$$991$$ −40.4757 −1.28575 −0.642877 0.765969i $$-0.722259\pi$$
−0.642877 + 0.765969i $$0.722259\pi$$
$$992$$ −20.5398 −0.652138
$$993$$ 54.4251 1.72713
$$994$$ 0 0
$$995$$ −48.6357 −1.54185
$$996$$ 31.4593 0.996826
$$997$$ 12.0623 0.382018 0.191009 0.981588i $$-0.438824\pi$$
0.191009 + 0.981588i $$0.438824\pi$$
$$998$$ 45.6610 1.44537
$$999$$ 87.6029 2.77163
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 8281.2.a.cc.1.5 6
7.6 odd 2 8281.2.a.cd.1.5 6
13.12 even 2 637.2.a.m.1.2 6
39.38 odd 2 5733.2.a.bu.1.5 6
91.12 odd 6 637.2.e.n.508.5 12
91.25 even 6 637.2.e.o.79.5 12
91.38 odd 6 637.2.e.n.79.5 12
91.51 even 6 637.2.e.o.508.5 12
91.90 odd 2 637.2.a.n.1.2 yes 6
273.272 even 2 5733.2.a.br.1.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.2 6 13.12 even 2
637.2.a.n.1.2 yes 6 91.90 odd 2
637.2.e.n.79.5 12 91.38 odd 6
637.2.e.n.508.5 12 91.12 odd 6
637.2.e.o.79.5 12 91.25 even 6
637.2.e.o.508.5 12 91.51 even 6
5733.2.a.br.1.5 6 273.272 even 2
5733.2.a.bu.1.5 6 39.38 odd 2
8281.2.a.cc.1.5 6 1.1 even 1 trivial
8281.2.a.cd.1.5 6 7.6 odd 2