Properties

 Label 637.2.a.n.1.2 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.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: $$5.08647060876$$ Analytic rank: $$0$$ 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$2.35100$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.17619 q^{2} +3.35100 q^{3} -0.616586 q^{4} +3.14862 q^{5} -3.94140 q^{6} +3.07759 q^{8} +8.22917 q^{9} +O(q^{10})$$ $$q-1.17619 q^{2} +3.35100 q^{3} -0.616586 q^{4} +3.14862 q^{5} -3.94140 q^{6} +3.07759 q^{8} +8.22917 q^{9} -3.70337 q^{10} -0.773390 q^{11} -2.06618 q^{12} +1.00000 q^{13} +10.5510 q^{15} -2.38665 q^{16} -5.75340 q^{17} -9.67904 q^{18} -1.22298 q^{19} -1.94140 q^{20} +0.909650 q^{22} -2.99182 q^{23} +10.3130 q^{24} +4.91383 q^{25} -1.17619 q^{26} +17.5229 q^{27} -2.46882 q^{29} -12.4100 q^{30} -6.13487 q^{31} -3.34804 q^{32} -2.59163 q^{33} +6.76707 q^{34} -5.07399 q^{36} +4.99933 q^{37} +1.43845 q^{38} +3.35100 q^{39} +9.69018 q^{40} -2.55981 q^{41} -2.73150 q^{43} +0.476861 q^{44} +25.9106 q^{45} +3.51894 q^{46} +5.37169 q^{47} -7.99766 q^{48} -5.77958 q^{50} -19.2796 q^{51} -0.616586 q^{52} -9.79015 q^{53} -20.6102 q^{54} -2.43511 q^{55} -4.09820 q^{57} +2.90379 q^{58} +2.50456 q^{59} -6.50561 q^{60} -10.9167 q^{61} +7.21575 q^{62} +8.71122 q^{64} +3.14862 q^{65} +3.04823 q^{66} +4.32518 q^{67} +3.54746 q^{68} -10.0256 q^{69} +10.6649 q^{71} +25.3260 q^{72} -5.17450 q^{73} -5.88014 q^{74} +16.4662 q^{75} +0.754072 q^{76} -3.94140 q^{78} -0.542038 q^{79} -7.51467 q^{80} +34.0318 q^{81} +3.01081 q^{82} +15.2259 q^{83} -18.1153 q^{85} +3.21275 q^{86} -8.27301 q^{87} -2.38018 q^{88} +9.23208 q^{89} -30.4757 q^{90} +1.84471 q^{92} -20.5579 q^{93} -6.31811 q^{94} -3.85070 q^{95} -11.2193 q^{96} -1.26291 q^{97} -6.36436 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 8 q^{3} + 4 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{9}+O(q^{10})$$ 6 * q + 8 * q^3 + 4 * q^4 + 6 * q^5 + 4 * q^6 + 6 * q^9 $$6 q + 8 q^{3} + 4 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{9} + 4 q^{10} + 4 q^{11} - 4 q^{12} + 6 q^{13} + 12 q^{15} + 16 q^{17} - 4 q^{18} + 2 q^{19} + 16 q^{20} - 12 q^{22} - 6 q^{23} + 12 q^{24} - 4 q^{25} + 20 q^{27} - 6 q^{29} + 6 q^{31} - 20 q^{32} + 4 q^{33} - 24 q^{36} + 8 q^{38} + 8 q^{39} + 4 q^{40} - 8 q^{41} + 2 q^{43} - 4 q^{44} + 14 q^{45} + 8 q^{46} + 30 q^{47} - 8 q^{48} + 8 q^{50} - 4 q^{51} + 4 q^{52} - 14 q^{53} - 48 q^{54} - 8 q^{55} + 4 q^{57} - 8 q^{58} + 24 q^{59} + 12 q^{60} + 28 q^{62} - 20 q^{64} + 6 q^{65} - 4 q^{66} + 16 q^{67} + 28 q^{68} - 20 q^{69} + 8 q^{71} + 28 q^{72} - 6 q^{73} - 12 q^{74} + 12 q^{75} - 16 q^{76} + 4 q^{78} - 22 q^{79} - 28 q^{80} + 46 q^{81} - 40 q^{82} + 50 q^{83} - 8 q^{85} - 16 q^{86} - 16 q^{87} - 44 q^{88} + 26 q^{89} - 40 q^{90} + 20 q^{92} + 16 q^{93} - 32 q^{94} - 6 q^{95} - 20 q^{96} - 14 q^{97} + 12 q^{99}+O(q^{100})$$ 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 + 6 * q^13 + 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 + 8 * q^39 + 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 + 4 * q^52 - 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 + 6 * q^65 - 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 + 4 * q^78 - 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 $$-0.636514\pi$$
−0.415845 + 0.909436i $$0.636514\pi$$
$$3$$ 3.35100 1.93470 0.967349 0.253447i $$-0.0815644\pi$$
0.967349 + 0.253447i $$0.0815644\pi$$
$$4$$ −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 $$-0.537197\pi$$
−0.116593 + 0.993180i $$0.537197\pi$$
$$12$$ −2.06618 −0.596454
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 10.5510 2.72426
$$16$$ −2.38665 −0.596663
$$17$$ −5.75340 −1.39540 −0.697702 0.716388i $$-0.745794\pi$$
−0.697702 + 0.716388i $$0.745794\pi$$
$$18$$ −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
$$26$$ −1.17619 −0.230669
$$27$$ 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 $$-0.365200\pi$$
0.410942 + 0.911661i $$0.365200\pi$$
$$38$$ 1.43845 0.233348
$$39$$ 3.35100 0.536589
$$40$$ 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
$$52$$ −0.616586 −0.0855050
$$53$$ −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 $$-0.746314\pi$$
−0.698870 + 0.715248i $$0.746314\pi$$
$$62$$ 7.21575 0.916401
$$63$$ 0 0
$$64$$ 8.71122 1.08890
$$65$$ 3.14862 0.390539
$$66$$ 3.04823 0.375212
$$67$$ 4.32518 0.528404 0.264202 0.964467i $$-0.414891\pi$$
0.264202 + 0.964467i $$0.414891\pi$$
$$68$$ 3.54746 0.430193
$$69$$ −10.0256 −1.20694
$$70$$ 0 0
$$71$$ 10.6649 1.26570 0.632848 0.774276i $$-0.281886\pi$$
0.632848 + 0.774276i $$0.281886\pi$$
$$72$$ 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
$$78$$ −3.94140 −0.446275
$$79$$ −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 $$-0.490410\pi$$
0.0301221 + 0.999546i $$0.490410\pi$$
$$102$$ 22.6764 2.24530
$$103$$ 7.64804 0.753584 0.376792 0.926298i $$-0.377027\pi$$
0.376792 + 0.926298i $$0.377027\pi$$
$$104$$ 3.07759 0.301783
$$105$$ 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 $$-0.387439\pi$$
0.346298 + 0.938125i $$0.387439\pi$$
$$110$$ 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
$$117$$ 8.22917 0.760787
$$118$$ −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
$$130$$ −3.70337 −0.324807
$$131$$ 15.7380 1.37503 0.687517 0.726169i $$-0.258701\pi$$
0.687517 + 0.726169i $$0.258701\pi$$
$$132$$ 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 $$-0.792765\pi$$
−0.795449 + 0.606020i $$0.792765\pi$$
$$138$$ 11.7919 1.00380
$$139$$ −11.9137 −1.01050 −0.505252 0.862972i $$-0.668601\pi$$
−0.505252 + 0.862972i $$0.668601\pi$$
$$140$$ 0 0
$$141$$ 18.0005 1.51592
$$142$$ −12.5440 −1.05267
$$143$$ −0.773390 −0.0646741
$$144$$ −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 $$-0.345712\pi$$
0.465952 + 0.884810i $$0.345712\pi$$
$$150$$ −19.3674 −1.58134
$$151$$ −3.69250 −0.300492 −0.150246 0.988649i $$-0.548007\pi$$
−0.150246 + 0.988649i $$0.548007\pi$$
$$152$$ −3.76383 −0.305287
$$153$$ −47.3457 −3.82767
$$154$$ 0 0
$$155$$ −19.3164 −1.55153
$$156$$ −2.06618 −0.165426
$$157$$ 1.12065 0.0894374 0.0447187 0.999000i $$-0.485761\pi$$
0.0447187 + 0.999000i $$0.485761\pi$$
$$158$$ 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 $$-0.793429\pi$$
−0.796712 + 0.604359i $$0.793429\pi$$
$$164$$ 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
$$169$$ 1.00000 0.0769231
$$170$$ 21.3070 1.63417
$$171$$ −10.0641 −0.769622
$$172$$ 1.68420 0.128419
$$173$$ −25.9110 −1.96998 −0.984989 0.172616i $$-0.944778\pi$$
−0.984989 + 0.172616i $$0.944778\pi$$
$$174$$ 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 $$-0.534693\pi$$
−0.108775 + 0.994066i $$0.534693\pi$$
$$182$$ 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 $$-0.500990\pi$$
−0.00310894 + 0.999995i $$0.500990\pi$$
$$194$$ 1.48542 0.106647
$$195$$ 10.5510 0.755575
$$196$$ 0 0
$$197$$ 15.0589 1.07290 0.536451 0.843932i $$-0.319765\pi$$
0.536451 + 0.843932i $$0.319765\pi$$
$$198$$ 7.48567 0.531983
$$199$$ 15.4466 1.09498 0.547492 0.836811i $$-0.315583\pi$$
0.547492 + 0.836811i $$0.315583\pi$$
$$200$$ 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
$$208$$ −2.38665 −0.165484
$$209$$ 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
$$221$$ −5.75340 −0.387015
$$222$$ −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 $$0.485446\pi$$
$$234$$ −9.67904 −0.632739
$$235$$ 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 $$-0.473731\pi$$
0.0824326 + 0.996597i $$0.473731\pi$$
$$240$$ −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
$$247$$ −1.22298 −0.0778163
$$248$$ −18.8806 −1.19892
$$249$$ 51.0218 3.23337
$$250$$ 0.319106 0.0201820
$$251$$ 18.0858 1.14157 0.570784 0.821100i $$-0.306639\pi$$
0.570784 + 0.821100i $$0.306639\pi$$
$$252$$ 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 $$-0.347944\pi$$
0.459737 + 0.888055i $$0.347944\pi$$
$$258$$ 10.7659 0.670257
$$259$$ 0 0
$$260$$ −1.94140 −0.120400
$$261$$ −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 $$-0.495705\pi$$
0.0134924 + 0.999909i $$0.495705\pi$$
$$270$$ −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 $$-0.586634\pi$$
−0.268821 + 0.963190i $$0.586634\pi$$
$$282$$ −21.1720 −1.26077
$$283$$ 27.7411 1.64904 0.824520 0.565833i $$-0.191445\pi$$
0.824520 + 0.565833i $$0.191445\pi$$
$$284$$ −6.57585 −0.390205
$$285$$ −12.9037 −0.764349
$$286$$ 0.909650 0.0537887
$$287$$ 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
$$299$$ −2.99182 −0.173021
$$300$$ −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 $$-0.301449\pi$$
0.584096 + 0.811685i $$0.301449\pi$$
$$312$$ 10.3130 0.583859
$$313$$ 16.5456 0.935213 0.467606 0.883937i $$-0.345117\pi$$
0.467606 + 0.883937i $$0.345117\pi$$
$$314$$ −1.31809 −0.0743841
$$315$$ 0 0
$$316$$ 0.334213 0.0188010
$$317$$ −21.2340 −1.19262 −0.596310 0.802754i $$-0.703367\pi$$
−0.596310 + 0.802754i $$0.703367\pi$$
$$318$$ 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
$$325$$ 4.91383 0.272570
$$326$$ 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 $$-0.352721\pi$$
0.446356 + 0.894855i $$0.352721\pi$$
$$332$$ −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 $$0.444040\pi$$
$$338$$ −1.17619 −0.0639761
$$339$$ −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
$$351$$ 17.5229 0.935306
$$352$$ 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 $$-0.425279\pi$$
0.232593 + 0.972574i $$0.425279\pi$$
$$360$$ 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 $$-0.671976\pi$$
−0.514374 + 0.857566i $$0.671976\pi$$
$$368$$ 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
$$377$$ −2.46882 −0.127151
$$378$$ 0 0
$$379$$ 7.39215 0.379709 0.189855 0.981812i $$-0.439198\pi$$
0.189855 + 0.981812i $$0.439198\pi$$
$$380$$ 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 $$0.520943\pi$$
$$390$$ −12.4100 −0.628403
$$391$$ 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 $$-0.0520561\pi$$
0.986657 + 0.162811i $$0.0520561\pi$$
$$402$$ −17.0472 −0.850239
$$403$$ −6.13487 −0.305599
$$404$$ −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
$$416$$ −3.34804 −0.164151
$$417$$ −39.9227 −1.95502
$$418$$ −1.11248 −0.0544134
$$419$$ 31.5621 1.54191 0.770954 0.636891i $$-0.219780\pi$$
0.770954 + 0.636891i $$0.219780\pi$$
$$420$$ 0 0
$$421$$ −17.7055 −0.862914 −0.431457 0.902134i $$-0.642000\pi$$
−0.431457 + 0.902134i $$0.642000\pi$$
$$422$$ 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
$$429$$ −2.59163 −0.125125
$$430$$ 10.1158 0.487825
$$431$$ −21.4816 −1.03473 −0.517367 0.855764i $$-0.673088\pi$$
−0.517367 + 0.855764i $$0.673088\pi$$
$$432$$ −41.8212 −2.01212
$$433$$ 34.7200 1.66854 0.834269 0.551358i $$-0.185890\pi$$
0.834269 + 0.551358i $$0.185890\pi$$
$$434$$ 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 $$-0.494466\pi$$
0.0173848 + 0.999849i $$0.494466\pi$$
$$440$$ −7.49428 −0.357276
$$441$$ 0 0
$$442$$ 6.76707 0.321877
$$443$$ 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 $$-0.285635\pi$$
0.623683 + 0.781677i $$0.285635\pi$$
$$450$$ −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 $$-0.517913\pi$$
−0.0562447 + 0.998417i $$0.517913\pi$$
$$458$$ 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 $$-0.229691\pi$$
0.750753 + 0.660583i $$0.229691\pi$$
$$464$$ 5.89221 0.273539
$$465$$ −64.7291 −3.00174
$$466$$ −1.64118 −0.0760263
$$467$$ −12.8744 −0.595756 −0.297878 0.954604i $$-0.596279\pi$$
−0.297878 + 0.954604i $$0.596279\pi$$
$$468$$ −5.07399 −0.234545
$$469$$ 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
$$481$$ 4.99933 0.227950
$$482$$ −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 $$-0.166399\pi$$
0.866446 + 0.499271i $$0.166399\pi$$
$$488$$ −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
$$494$$ 1.43845 0.0647190
$$495$$ −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 $$-0.835197\pi$$
−0.868938 + 0.494921i $$0.835197\pi$$
$$500$$ 0.167283 0.00748112
$$501$$ 43.5840 1.94719
$$502$$ −21.2723 −0.949430
$$503$$ −27.0935 −1.20804 −0.604020 0.796969i $$-0.706435\pi$$
−0.604020 + 0.796969i $$0.706435\pi$$
$$504$$ 0 0
$$505$$ 1.90633 0.0848304
$$506$$ −2.72151 −0.120986
$$507$$ 3.35100 0.148823
$$508$$ 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
$$520$$ 9.69018 0.424943
$$521$$ 7.30737 0.320142 0.160071 0.987106i $$-0.448828\pi$$
0.160071 + 0.987106i $$0.448828\pi$$
$$522$$ 23.8958 1.04589
$$523$$ 14.0826 0.615788 0.307894 0.951421i $$-0.400376\pi$$
0.307894 + 0.951421i $$0.400376\pi$$
$$524$$ −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
$$533$$ −2.55981 −0.110877
$$534$$ −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 $$-0.818700\pi$$
−0.842132 + 0.539271i $$0.818700\pi$$
$$542$$ 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 $$-0.697713\pi$$
−0.581956 + 0.813220i $$0.697713\pi$$
$$558$$ 59.3796 2.51374
$$559$$ −2.73150 −0.115530
$$560$$ 0 0
$$561$$ 14.9107 0.629528
$$562$$ 10.6004 0.447151
$$563$$ 14.1210 0.595129 0.297565 0.954702i $$-0.403826\pi$$
0.297565 + 0.954702i $$0.403826\pi$$
$$564$$ −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 $$0.663842\pi$$
$$572$$ 0.476861 0.0199386
$$573$$ −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
$$585$$ 25.9106 1.07127
$$586$$ −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
$$598$$ 3.51894 0.143900
$$599$$ 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 $$-0.442539\pi$$
0.179540 + 0.983751i $$0.442539\pi$$
$$602$$ 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 $$-0.382548\pi$$
0.360671 + 0.932693i $$0.382548\pi$$
$$608$$ 4.09458 0.166057
$$609$$ 0 0
$$610$$ 40.4286 1.63691
$$611$$ 5.37169 0.217315
$$612$$ 29.1927 1.18004
$$613$$ 10.6081 0.428457 0.214228 0.976784i $$-0.431276\pi$$
0.214228 + 0.976784i $$0.431276\pi$$
$$614$$ −11.9246 −0.481237
$$615$$ −27.0086 −1.08909
$$616$$ 0 0
$$617$$ 49.3483 1.98669 0.993344 0.115188i $$-0.0367469\pi$$
0.993344 + 0.115188i $$0.0367469\pi$$
$$618$$ −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
$$624$$ −7.99766 −0.320163
$$625$$ −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 $$-0.250056\pi$$
0.706982 + 0.707231i $$0.250056\pi$$
$$632$$ −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 $$-0.350843\pi$$
0.451628 + 0.892206i $$0.350843\pi$$
$$648$$ 104.736 4.11442
$$649$$ −1.93700 −0.0760340
$$650$$ −5.77958 −0.226694
$$651$$ 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
$$663$$ −19.2796 −0.748758
$$664$$ 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
$$676$$ −0.616586 −0.0237148
$$677$$ 10.2469 0.393821 0.196910 0.980421i $$-0.436909\pi$$
0.196910 + 0.980421i $$0.436909\pi$$
$$678$$ 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 $$-0.482328\pi$$
0.0554904 + 0.998459i $$0.482328\pi$$
$$684$$ 6.20539 0.237269
$$685$$ −58.6305 −2.24016
$$686$$ 0 0
$$687$$ −47.8329 −1.82494
$$688$$ 6.51914 0.248540
$$689$$ −9.79015 −0.372975
$$690$$ 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 $$0.664042\pi$$
$$702$$ −20.6102 −0.777884
$$703$$ −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 $$-0.520245\pi$$
−0.0635580 + 0.997978i $$0.520245\pi$$
$$710$$ −39.4962 −1.48227
$$711$$ −4.46053 −0.167283
$$712$$ 28.4126 1.06481
$$713$$ 18.3544 0.687378
$$714$$ 0 0
$$715$$ −2.43511 −0.0910681
$$716$$ −5.94489 −0.222171
$$717$$ 8.54087 0.318964
$$718$$ −10.3669 −0.386890
$$719$$ −2.41574 −0.0900918 −0.0450459 0.998985i $$-0.514343\pi$$
−0.0450459 + 0.998985i $$0.514343\pi$$
$$720$$ −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 $$-0.397819\pi$$
0.315525 + 0.948917i $$0.397819\pi$$
$$728$$ 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 $$-0.672750\pi$$
−0.516458 + 0.856312i $$0.672750\pi$$
$$740$$ −9.70567 −0.356788
$$741$$ −4.09820 −0.150551
$$742$$ 0 0
$$743$$ −26.5210 −0.972961 −0.486481 0.873691i $$-0.661720\pi$$
−0.486481 + 0.873691i $$0.661720\pi$$
$$744$$ −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
$$754$$ 2.90379 0.105750
$$755$$ −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
$$767$$ 2.50456 0.0904345
$$768$$ −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
$$780$$ −6.50561 −0.232938
$$781$$ −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
$$793$$ −10.9167 −0.387664
$$794$$ 39.3921 1.39797
$$795$$ −103.296 −3.66354
$$796$$ −9.52418 −0.337576
$$797$$ 25.6470 0.908463 0.454232 0.890884i $$-0.349914\pi$$
0.454232 + 0.890884i $$0.349914\pi$$
$$798$$ 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
$$806$$ 7.21575 0.254164
$$807$$ 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 $$-0.798607\pi$$
−0.806438 + 0.591319i $$0.798607\pi$$
$$822$$ 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 $$-0.467043\pi$$
0.103352 + 0.994645i $$0.467043\pi$$
$$828$$ 15.1805 0.527558
$$829$$ 34.8106 1.20902 0.604511 0.796596i $$-0.293368\pi$$
0.604511 + 0.796596i $$0.293368\pi$$
$$830$$ −56.3869 −1.95722
$$831$$ 12.0442 0.417808
$$832$$ 8.71122 0.302007
$$833$$ 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
$$845$$ 3.14862 0.108316
$$846$$ −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 $$-0.693006\pi$$
−0.569870 + 0.821735i $$0.693006\pi$$
$$858$$ 3.04823 0.104065
$$859$$ −29.4914 −1.00623 −0.503116 0.864219i $$-0.667813\pi$$
−0.503116 + 0.864219i $$0.667813\pi$$
$$860$$ 5.30293 0.180828
$$861$$ 0 0
$$862$$ 25.2664 0.860578
$$863$$ −24.6224 −0.838157 −0.419079 0.907950i $$-0.637647\pi$$
−0.419079 + 0.907950i $$0.637647\pi$$
$$864$$ −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
$$871$$ 4.32518 0.146553
$$872$$ 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 $$-0.524884\pi$$
−0.0780971 + 0.996946i $$0.524884\pi$$
$$878$$ −0.856858 −0.0289176
$$879$$ 42.6948 1.44006
$$880$$ 5.81177 0.195915
$$881$$ −42.1720 −1.42081 −0.710405 0.703793i $$-0.751488\pi$$
−0.710405 + 0.703793i $$0.751488\pi$$
$$882$$ 0 0
$$883$$ 20.7992 0.699950 0.349975 0.936759i $$-0.386190\pi$$
0.349975 + 0.936759i $$0.386190\pi$$
$$884$$ 3.54746 0.119314
$$885$$ 26.4257 0.888291
$$886$$ −0.984810 −0.0330853
$$887$$ 34.4889 1.15802 0.579012 0.815319i $$-0.303438\pi$$
0.579012 + 0.815319i $$0.303438\pi$$
$$888$$ 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
$$897$$ −10.0256 −0.334744
$$898$$ −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
$$923$$ 10.6649 0.351041
$$924$$ 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
$$936$$ 25.3260 0.827808
$$937$$ 37.9272 1.23903 0.619514 0.784986i $$-0.287330\pi$$
0.619514 + 0.784986i $$0.287330\pi$$
$$938$$ 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 $$-0.577004\pi$$
−0.239563 + 0.970881i $$0.577004\pi$$
$$948$$ 1.11995 0.0363742
$$949$$ −5.17450 −0.167971
$$950$$ 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
$$962$$ −5.88014 −0.189583
$$963$$ 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 $$-0.567449\pi$$
−0.210316 + 0.977633i $$0.567449\pi$$
$$968$$ −32.0127 −1.02893
$$969$$ 23.5786 0.757453
$$970$$ 4.67702 0.150170
$$971$$ −29.1203 −0.934515 −0.467258 0.884121i $$-0.654758\pi$$
−0.467258 + 0.884121i $$0.654758\pi$$
$$972$$ −37.9025 −1.21572
$$973$$ 0 0
$$974$$ −44.9792 −1.44123
$$975$$ 16.4662 0.527342
$$976$$ 26.0544 0.833980
$$977$$ −18.5000 −0.591868 −0.295934 0.955208i $$-0.595631\pi$$
−0.295934 + 0.955208i $$0.595631\pi$$
$$978$$ 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
$$988$$ 0.754072 0.0239902
$$989$$ 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 $$-0.561176\pi$$
−0.191009 + 0.981588i $$0.561176\pi$$
$$998$$ 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 637.2.a.n.1.2 yes 6
3.2 odd 2 5733.2.a.br.1.5 6
7.2 even 3 637.2.e.n.508.5 12
7.3 odd 6 637.2.e.o.79.5 12
7.4 even 3 637.2.e.n.79.5 12
7.5 odd 6 637.2.e.o.508.5 12
7.6 odd 2 637.2.a.m.1.2 6
13.12 even 2 8281.2.a.cd.1.5 6
21.20 even 2 5733.2.a.bu.1.5 6
91.90 odd 2 8281.2.a.cc.1.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.2 6 7.6 odd 2
637.2.a.n.1.2 yes 6 1.1 even 1 trivial
637.2.e.n.79.5 12 7.4 even 3
637.2.e.n.508.5 12 7.2 even 3
637.2.e.o.79.5 12 7.3 odd 6
637.2.e.o.508.5 12 7.5 odd 6
5733.2.a.br.1.5 6 3.2 odd 2
5733.2.a.bu.1.5 6 21.20 even 2
8281.2.a.cc.1.5 6 91.90 odd 2
8281.2.a.cd.1.5 6 13.12 even 2