Properties

 Label 637.2.a.f.1.1 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ 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)

 $$f(q)$$ $$=$$ $$q-2.61803 q^{2} -2.23607 q^{3} +4.85410 q^{4} +2.23607 q^{5} +5.85410 q^{6} -7.47214 q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-2.61803 q^{2} -2.23607 q^{3} +4.85410 q^{4} +2.23607 q^{5} +5.85410 q^{6} -7.47214 q^{8} +2.00000 q^{9} -5.85410 q^{10} -3.00000 q^{11} -10.8541 q^{12} -1.00000 q^{13} -5.00000 q^{15} +9.85410 q^{16} +1.47214 q^{17} -5.23607 q^{18} +3.00000 q^{19} +10.8541 q^{20} +7.85410 q^{22} -8.23607 q^{23} +16.7082 q^{24} +2.61803 q^{26} +2.23607 q^{27} +4.47214 q^{29} +13.0902 q^{30} +5.00000 q^{31} -10.8541 q^{32} +6.70820 q^{33} -3.85410 q^{34} +9.70820 q^{36} +4.70820 q^{37} -7.85410 q^{38} +2.23607 q^{39} -16.7082 q^{40} -4.47214 q^{41} -8.00000 q^{43} -14.5623 q^{44} +4.47214 q^{45} +21.5623 q^{46} -7.47214 q^{47} -22.0344 q^{48} -3.29180 q^{51} -4.85410 q^{52} -7.47214 q^{53} -5.85410 q^{54} -6.70820 q^{55} -6.70820 q^{57} -11.7082 q^{58} -1.47214 q^{59} -24.2705 q^{60} +3.00000 q^{61} -13.0902 q^{62} +8.70820 q^{64} -2.23607 q^{65} -17.5623 q^{66} -3.00000 q^{67} +7.14590 q^{68} +18.4164 q^{69} -8.94427 q^{71} -14.9443 q^{72} -2.70820 q^{73} -12.3262 q^{74} +14.5623 q^{76} -5.85410 q^{78} -2.70820 q^{79} +22.0344 q^{80} -11.0000 q^{81} +11.7082 q^{82} +3.29180 q^{85} +20.9443 q^{86} -10.0000 q^{87} +22.4164 q^{88} +2.23607 q^{89} -11.7082 q^{90} -39.9787 q^{92} -11.1803 q^{93} +19.5623 q^{94} +6.70820 q^{95} +24.2705 q^{96} +9.41641 q^{97} -6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 3 q^{4} + 5 q^{6} - 6 q^{8} + 4 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 3 * q^4 + 5 * q^6 - 6 * q^8 + 4 * q^9 $$2 q - 3 q^{2} + 3 q^{4} + 5 q^{6} - 6 q^{8} + 4 q^{9} - 5 q^{10} - 6 q^{11} - 15 q^{12} - 2 q^{13} - 10 q^{15} + 13 q^{16} - 6 q^{17} - 6 q^{18} + 6 q^{19} + 15 q^{20} + 9 q^{22} - 12 q^{23} + 20 q^{24} + 3 q^{26} + 15 q^{30} + 10 q^{31} - 15 q^{32} - q^{34} + 6 q^{36} - 4 q^{37} - 9 q^{38} - 20 q^{40} - 16 q^{43} - 9 q^{44} + 23 q^{46} - 6 q^{47} - 15 q^{48} - 20 q^{51} - 3 q^{52} - 6 q^{53} - 5 q^{54} - 10 q^{58} + 6 q^{59} - 15 q^{60} + 6 q^{61} - 15 q^{62} + 4 q^{64} - 15 q^{66} - 6 q^{67} + 21 q^{68} + 10 q^{69} - 12 q^{72} + 8 q^{73} - 9 q^{74} + 9 q^{76} - 5 q^{78} + 8 q^{79} + 15 q^{80} - 22 q^{81} + 10 q^{82} + 20 q^{85} + 24 q^{86} - 20 q^{87} + 18 q^{88} - 10 q^{90} - 33 q^{92} + 19 q^{94} + 15 q^{96} - 8 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 3 * q^4 + 5 * q^6 - 6 * q^8 + 4 * q^9 - 5 * q^10 - 6 * q^11 - 15 * q^12 - 2 * q^13 - 10 * q^15 + 13 * q^16 - 6 * q^17 - 6 * q^18 + 6 * q^19 + 15 * q^20 + 9 * q^22 - 12 * q^23 + 20 * q^24 + 3 * q^26 + 15 * q^30 + 10 * q^31 - 15 * q^32 - q^34 + 6 * q^36 - 4 * q^37 - 9 * q^38 - 20 * q^40 - 16 * q^43 - 9 * q^44 + 23 * q^46 - 6 * q^47 - 15 * q^48 - 20 * q^51 - 3 * q^52 - 6 * q^53 - 5 * q^54 - 10 * q^58 + 6 * q^59 - 15 * q^60 + 6 * q^61 - 15 * q^62 + 4 * q^64 - 15 * q^66 - 6 * q^67 + 21 * q^68 + 10 * q^69 - 12 * q^72 + 8 * q^73 - 9 * q^74 + 9 * q^76 - 5 * q^78 + 8 * q^79 + 15 * q^80 - 22 * q^81 + 10 * q^82 + 20 * q^85 + 24 * q^86 - 20 * q^87 + 18 * q^88 - 10 * q^90 - 33 * q^92 + 19 * q^94 + 15 * q^96 - 8 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.61803 −1.85123 −0.925615 0.378467i $$-0.876451\pi$$
−0.925615 + 0.378467i $$0.876451\pi$$
$$3$$ −2.23607 −1.29099 −0.645497 0.763763i $$-0.723350\pi$$
−0.645497 + 0.763763i $$0.723350\pi$$
$$4$$ 4.85410 2.42705
$$5$$ 2.23607 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$6$$ 5.85410 2.38993
$$7$$ 0 0
$$8$$ −7.47214 −2.64180
$$9$$ 2.00000 0.666667
$$10$$ −5.85410 −1.85123
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ −10.8541 −3.13331
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ −5.00000 −1.29099
$$16$$ 9.85410 2.46353
$$17$$ 1.47214 0.357045 0.178523 0.983936i $$-0.442868\pi$$
0.178523 + 0.983936i $$0.442868\pi$$
$$18$$ −5.23607 −1.23415
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 10.8541 2.42705
$$21$$ 0 0
$$22$$ 7.85410 1.67450
$$23$$ −8.23607 −1.71734 −0.858669 0.512530i $$-0.828708\pi$$
−0.858669 + 0.512530i $$0.828708\pi$$
$$24$$ 16.7082 3.41055
$$25$$ 0 0
$$26$$ 2.61803 0.513439
$$27$$ 2.23607 0.430331
$$28$$ 0 0
$$29$$ 4.47214 0.830455 0.415227 0.909718i $$-0.363702\pi$$
0.415227 + 0.909718i $$0.363702\pi$$
$$30$$ 13.0902 2.38993
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ −10.8541 −1.91875
$$33$$ 6.70820 1.16775
$$34$$ −3.85410 −0.660973
$$35$$ 0 0
$$36$$ 9.70820 1.61803
$$37$$ 4.70820 0.774024 0.387012 0.922075i $$-0.373507\pi$$
0.387012 + 0.922075i $$0.373507\pi$$
$$38$$ −7.85410 −1.27410
$$39$$ 2.23607 0.358057
$$40$$ −16.7082 −2.64180
$$41$$ −4.47214 −0.698430 −0.349215 0.937043i $$-0.613552\pi$$
−0.349215 + 0.937043i $$0.613552\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −14.5623 −2.19535
$$45$$ 4.47214 0.666667
$$46$$ 21.5623 3.17919
$$47$$ −7.47214 −1.08992 −0.544962 0.838461i $$-0.683456\pi$$
−0.544962 + 0.838461i $$0.683456\pi$$
$$48$$ −22.0344 −3.18040
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −3.29180 −0.460944
$$52$$ −4.85410 −0.673143
$$53$$ −7.47214 −1.02638 −0.513188 0.858276i $$-0.671536\pi$$
−0.513188 + 0.858276i $$0.671536\pi$$
$$54$$ −5.85410 −0.796642
$$55$$ −6.70820 −0.904534
$$56$$ 0 0
$$57$$ −6.70820 −0.888523
$$58$$ −11.7082 −1.53736
$$59$$ −1.47214 −0.191656 −0.0958279 0.995398i $$-0.530550\pi$$
−0.0958279 + 0.995398i $$0.530550\pi$$
$$60$$ −24.2705 −3.13331
$$61$$ 3.00000 0.384111 0.192055 0.981384i $$-0.438485\pi$$
0.192055 + 0.981384i $$0.438485\pi$$
$$62$$ −13.0902 −1.66245
$$63$$ 0 0
$$64$$ 8.70820 1.08853
$$65$$ −2.23607 −0.277350
$$66$$ −17.5623 −2.16177
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ 7.14590 0.866567
$$69$$ 18.4164 2.21707
$$70$$ 0 0
$$71$$ −8.94427 −1.06149 −0.530745 0.847532i $$-0.678088\pi$$
−0.530745 + 0.847532i $$0.678088\pi$$
$$72$$ −14.9443 −1.76120
$$73$$ −2.70820 −0.316971 −0.158486 0.987361i $$-0.550661\pi$$
−0.158486 + 0.987361i $$0.550661\pi$$
$$74$$ −12.3262 −1.43290
$$75$$ 0 0
$$76$$ 14.5623 1.67041
$$77$$ 0 0
$$78$$ −5.85410 −0.662847
$$79$$ −2.70820 −0.304697 −0.152348 0.988327i $$-0.548684\pi$$
−0.152348 + 0.988327i $$0.548684\pi$$
$$80$$ 22.0344 2.46353
$$81$$ −11.0000 −1.22222
$$82$$ 11.7082 1.29295
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 3.29180 0.357045
$$86$$ 20.9443 2.25848
$$87$$ −10.0000 −1.07211
$$88$$ 22.4164 2.38960
$$89$$ 2.23607 0.237023 0.118511 0.992953i $$-0.462188\pi$$
0.118511 + 0.992953i $$0.462188\pi$$
$$90$$ −11.7082 −1.23415
$$91$$ 0 0
$$92$$ −39.9787 −4.16807
$$93$$ −11.1803 −1.15935
$$94$$ 19.5623 2.01770
$$95$$ 6.70820 0.688247
$$96$$ 24.2705 2.47710
$$97$$ 9.41641 0.956091 0.478046 0.878335i $$-0.341345\pi$$
0.478046 + 0.878335i $$0.341345\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 8.61803 0.853313
$$103$$ −2.70820 −0.266847 −0.133424 0.991059i $$-0.542597\pi$$
−0.133424 + 0.991059i $$0.542597\pi$$
$$104$$ 7.47214 0.732703
$$105$$ 0 0
$$106$$ 19.5623 1.90006
$$107$$ −9.76393 −0.943915 −0.471957 0.881621i $$-0.656452\pi$$
−0.471957 + 0.881621i $$0.656452\pi$$
$$108$$ 10.8541 1.04444
$$109$$ −2.70820 −0.259399 −0.129699 0.991553i $$-0.541401\pi$$
−0.129699 + 0.991553i $$0.541401\pi$$
$$110$$ 17.5623 1.67450
$$111$$ −10.5279 −0.999261
$$112$$ 0 0
$$113$$ 2.94427 0.276974 0.138487 0.990364i $$-0.455776\pi$$
0.138487 + 0.990364i $$0.455776\pi$$
$$114$$ 17.5623 1.64486
$$115$$ −18.4164 −1.71734
$$116$$ 21.7082 2.01556
$$117$$ −2.00000 −0.184900
$$118$$ 3.85410 0.354799
$$119$$ 0 0
$$120$$ 37.3607 3.41055
$$121$$ −2.00000 −0.181818
$$122$$ −7.85410 −0.711077
$$123$$ 10.0000 0.901670
$$124$$ 24.2705 2.17956
$$125$$ −11.1803 −1.00000
$$126$$ 0 0
$$127$$ −11.4164 −1.01304 −0.506521 0.862228i $$-0.669069\pi$$
−0.506521 + 0.862228i $$0.669069\pi$$
$$128$$ −1.09017 −0.0963583
$$129$$ 17.8885 1.57500
$$130$$ 5.85410 0.513439
$$131$$ −8.23607 −0.719589 −0.359794 0.933032i $$-0.617153\pi$$
−0.359794 + 0.933032i $$0.617153\pi$$
$$132$$ 32.5623 2.83418
$$133$$ 0 0
$$134$$ 7.85410 0.678491
$$135$$ 5.00000 0.430331
$$136$$ −11.0000 −0.943242
$$137$$ −8.23607 −0.703655 −0.351827 0.936065i $$-0.614440\pi$$
−0.351827 + 0.936065i $$0.614440\pi$$
$$138$$ −48.2148 −4.10431
$$139$$ −23.4164 −1.98615 −0.993077 0.117466i $$-0.962523\pi$$
−0.993077 + 0.117466i $$0.962523\pi$$
$$140$$ 0 0
$$141$$ 16.7082 1.40708
$$142$$ 23.4164 1.96506
$$143$$ 3.00000 0.250873
$$144$$ 19.7082 1.64235
$$145$$ 10.0000 0.830455
$$146$$ 7.09017 0.586787
$$147$$ 0 0
$$148$$ 22.8541 1.87860
$$149$$ 0.708204 0.0580183 0.0290092 0.999579i $$-0.490765\pi$$
0.0290092 + 0.999579i $$0.490765\pi$$
$$150$$ 0 0
$$151$$ 20.4164 1.66146 0.830732 0.556673i $$-0.187922\pi$$
0.830732 + 0.556673i $$0.187922\pi$$
$$152$$ −22.4164 −1.81821
$$153$$ 2.94427 0.238030
$$154$$ 0 0
$$155$$ 11.1803 0.898027
$$156$$ 10.8541 0.869024
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ 7.09017 0.564064
$$159$$ 16.7082 1.32505
$$160$$ −24.2705 −1.91875
$$161$$ 0 0
$$162$$ 28.7984 2.26261
$$163$$ −16.4164 −1.28583 −0.642916 0.765937i $$-0.722276\pi$$
−0.642916 + 0.765937i $$0.722276\pi$$
$$164$$ −21.7082 −1.69513
$$165$$ 15.0000 1.16775
$$166$$ 0 0
$$167$$ 22.4721 1.73895 0.869473 0.493980i $$-0.164459\pi$$
0.869473 + 0.493980i $$0.164459\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −8.61803 −0.660973
$$171$$ 6.00000 0.458831
$$172$$ −38.8328 −2.96097
$$173$$ −16.4164 −1.24812 −0.624058 0.781378i $$-0.714517\pi$$
−0.624058 + 0.781378i $$0.714517\pi$$
$$174$$ 26.1803 1.98473
$$175$$ 0 0
$$176$$ −29.5623 −2.22834
$$177$$ 3.29180 0.247427
$$178$$ −5.85410 −0.438783
$$179$$ −20.1246 −1.50418 −0.752092 0.659058i $$-0.770955\pi$$
−0.752092 + 0.659058i $$0.770955\pi$$
$$180$$ 21.7082 1.61803
$$181$$ −25.4164 −1.88919 −0.944593 0.328243i $$-0.893544\pi$$
−0.944593 + 0.328243i $$0.893544\pi$$
$$182$$ 0 0
$$183$$ −6.70820 −0.495885
$$184$$ 61.5410 4.53686
$$185$$ 10.5279 0.774024
$$186$$ 29.2705 2.14622
$$187$$ −4.41641 −0.322960
$$188$$ −36.2705 −2.64530
$$189$$ 0 0
$$190$$ −17.5623 −1.27410
$$191$$ 11.1803 0.808981 0.404491 0.914542i $$-0.367449\pi$$
0.404491 + 0.914542i $$0.367449\pi$$
$$192$$ −19.4721 −1.40528
$$193$$ 0.708204 0.0509776 0.0254888 0.999675i $$-0.491886\pi$$
0.0254888 + 0.999675i $$0.491886\pi$$
$$194$$ −24.6525 −1.76994
$$195$$ 5.00000 0.358057
$$196$$ 0 0
$$197$$ −9.05573 −0.645194 −0.322597 0.946536i $$-0.604556\pi$$
−0.322597 + 0.946536i $$0.604556\pi$$
$$198$$ 15.7082 1.11633
$$199$$ −20.7082 −1.46797 −0.733983 0.679168i $$-0.762341\pi$$
−0.733983 + 0.679168i $$0.762341\pi$$
$$200$$ 0 0
$$201$$ 6.70820 0.473160
$$202$$ 23.5623 1.65784
$$203$$ 0 0
$$204$$ −15.9787 −1.11873
$$205$$ −10.0000 −0.698430
$$206$$ 7.09017 0.493996
$$207$$ −16.4721 −1.14489
$$208$$ −9.85410 −0.683259
$$209$$ −9.00000 −0.622543
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −36.2705 −2.49107
$$213$$ 20.0000 1.37038
$$214$$ 25.5623 1.74740
$$215$$ −17.8885 −1.21999
$$216$$ −16.7082 −1.13685
$$217$$ 0 0
$$218$$ 7.09017 0.480207
$$219$$ 6.05573 0.409208
$$220$$ −32.5623 −2.19535
$$221$$ −1.47214 −0.0990266
$$222$$ 27.5623 1.84986
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −7.70820 −0.512742
$$227$$ 5.94427 0.394535 0.197268 0.980350i $$-0.436793\pi$$
0.197268 + 0.980350i $$0.436793\pi$$
$$228$$ −32.5623 −2.15649
$$229$$ 24.1246 1.59420 0.797100 0.603848i $$-0.206367\pi$$
0.797100 + 0.603848i $$0.206367\pi$$
$$230$$ 48.2148 3.17919
$$231$$ 0 0
$$232$$ −33.4164 −2.19389
$$233$$ 11.9443 0.782495 0.391248 0.920285i $$-0.372044\pi$$
0.391248 + 0.920285i $$0.372044\pi$$
$$234$$ 5.23607 0.342292
$$235$$ −16.7082 −1.08992
$$236$$ −7.14590 −0.465158
$$237$$ 6.05573 0.393362
$$238$$ 0 0
$$239$$ 19.4164 1.25594 0.627972 0.778236i $$-0.283885\pi$$
0.627972 + 0.778236i $$0.283885\pi$$
$$240$$ −49.2705 −3.18040
$$241$$ 4.70820 0.303282 0.151641 0.988436i $$-0.451544\pi$$
0.151641 + 0.988436i $$0.451544\pi$$
$$242$$ 5.23607 0.336587
$$243$$ 17.8885 1.14755
$$244$$ 14.5623 0.932256
$$245$$ 0 0
$$246$$ −26.1803 −1.66920
$$247$$ −3.00000 −0.190885
$$248$$ −37.3607 −2.37241
$$249$$ 0 0
$$250$$ 29.2705 1.85123
$$251$$ 1.52786 0.0964379 0.0482190 0.998837i $$-0.484645\pi$$
0.0482190 + 0.998837i $$0.484645\pi$$
$$252$$ 0 0
$$253$$ 24.7082 1.55339
$$254$$ 29.8885 1.87537
$$255$$ −7.36068 −0.460944
$$256$$ −14.5623 −0.910144
$$257$$ −0.0557281 −0.00347622 −0.00173811 0.999998i $$-0.500553\pi$$
−0.00173811 + 0.999998i $$0.500553\pi$$
$$258$$ −46.8328 −2.91568
$$259$$ 0 0
$$260$$ −10.8541 −0.673143
$$261$$ 8.94427 0.553637
$$262$$ 21.5623 1.33212
$$263$$ 26.1246 1.61091 0.805456 0.592655i $$-0.201920\pi$$
0.805456 + 0.592655i $$0.201920\pi$$
$$264$$ −50.1246 −3.08496
$$265$$ −16.7082 −1.02638
$$266$$ 0 0
$$267$$ −5.00000 −0.305995
$$268$$ −14.5623 −0.889534
$$269$$ 13.4721 0.821411 0.410705 0.911768i $$-0.365283\pi$$
0.410705 + 0.911768i $$0.365283\pi$$
$$270$$ −13.0902 −0.796642
$$271$$ −20.4164 −1.24021 −0.620104 0.784519i $$-0.712910\pi$$
−0.620104 + 0.784519i $$0.712910\pi$$
$$272$$ 14.5066 0.879590
$$273$$ 0 0
$$274$$ 21.5623 1.30263
$$275$$ 0 0
$$276$$ 89.3951 5.38095
$$277$$ −0.416408 −0.0250195 −0.0125098 0.999922i $$-0.503982\pi$$
−0.0125098 + 0.999922i $$0.503982\pi$$
$$278$$ 61.3050 3.67683
$$279$$ 10.0000 0.598684
$$280$$ 0 0
$$281$$ 26.9443 1.60736 0.803680 0.595061i $$-0.202872\pi$$
0.803680 + 0.595061i $$0.202872\pi$$
$$282$$ −43.7426 −2.60484
$$283$$ 26.1246 1.55295 0.776473 0.630150i $$-0.217007\pi$$
0.776473 + 0.630150i $$0.217007\pi$$
$$284$$ −43.4164 −2.57629
$$285$$ −15.0000 −0.888523
$$286$$ −7.85410 −0.464423
$$287$$ 0 0
$$288$$ −21.7082 −1.27917
$$289$$ −14.8328 −0.872519
$$290$$ −26.1803 −1.53736
$$291$$ −21.0557 −1.23431
$$292$$ −13.1459 −0.769305
$$293$$ −14.9443 −0.873054 −0.436527 0.899691i $$-0.643792\pi$$
−0.436527 + 0.899691i $$0.643792\pi$$
$$294$$ 0 0
$$295$$ −3.29180 −0.191656
$$296$$ −35.1803 −2.04482
$$297$$ −6.70820 −0.389249
$$298$$ −1.85410 −0.107405
$$299$$ 8.23607 0.476304
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −53.4508 −3.07575
$$303$$ 20.1246 1.15613
$$304$$ 29.5623 1.69551
$$305$$ 6.70820 0.384111
$$306$$ −7.70820 −0.440649
$$307$$ 19.4164 1.10815 0.554076 0.832466i $$-0.313072\pi$$
0.554076 + 0.832466i $$0.313072\pi$$
$$308$$ 0 0
$$309$$ 6.05573 0.344498
$$310$$ −29.2705 −1.66245
$$311$$ −27.7639 −1.57435 −0.787174 0.616731i $$-0.788457\pi$$
−0.787174 + 0.616731i $$0.788457\pi$$
$$312$$ −16.7082 −0.945916
$$313$$ −5.58359 −0.315603 −0.157802 0.987471i $$-0.550441\pi$$
−0.157802 + 0.987471i $$0.550441\pi$$
$$314$$ −18.3262 −1.03421
$$315$$ 0 0
$$316$$ −13.1459 −0.739515
$$317$$ −8.23607 −0.462584 −0.231292 0.972884i $$-0.574295\pi$$
−0.231292 + 0.972884i $$0.574295\pi$$
$$318$$ −43.7426 −2.45297
$$319$$ −13.4164 −0.751175
$$320$$ 19.4721 1.08853
$$321$$ 21.8328 1.21859
$$322$$ 0 0
$$323$$ 4.41641 0.245736
$$324$$ −53.3951 −2.96640
$$325$$ 0 0
$$326$$ 42.9787 2.38037
$$327$$ 6.05573 0.334883
$$328$$ 33.4164 1.84511
$$329$$ 0 0
$$330$$ −39.2705 −2.16177
$$331$$ −1.58359 −0.0870421 −0.0435210 0.999053i $$-0.513858\pi$$
−0.0435210 + 0.999053i $$0.513858\pi$$
$$332$$ 0 0
$$333$$ 9.41641 0.516016
$$334$$ −58.8328 −3.21919
$$335$$ −6.70820 −0.366508
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ −2.61803 −0.142402
$$339$$ −6.58359 −0.357572
$$340$$ 15.9787 0.866567
$$341$$ −15.0000 −0.812296
$$342$$ −15.7082 −0.849402
$$343$$ 0 0
$$344$$ 59.7771 3.22296
$$345$$ 41.1803 2.21707
$$346$$ 42.9787 2.31055
$$347$$ 23.0689 1.23840 0.619201 0.785232i $$-0.287456\pi$$
0.619201 + 0.785232i $$0.287456\pi$$
$$348$$ −48.5410 −2.60207
$$349$$ −29.4164 −1.57462 −0.787312 0.616555i $$-0.788528\pi$$
−0.787312 + 0.616555i $$0.788528\pi$$
$$350$$ 0 0
$$351$$ −2.23607 −0.119352
$$352$$ 32.5623 1.73558
$$353$$ 17.2918 0.920349 0.460175 0.887828i $$-0.347787\pi$$
0.460175 + 0.887828i $$0.347787\pi$$
$$354$$ −8.61803 −0.458043
$$355$$ −20.0000 −1.06149
$$356$$ 10.8541 0.575266
$$357$$ 0 0
$$358$$ 52.6869 2.78459
$$359$$ 11.9443 0.630395 0.315197 0.949026i $$-0.397929\pi$$
0.315197 + 0.949026i $$0.397929\pi$$
$$360$$ −33.4164 −1.76120
$$361$$ −10.0000 −0.526316
$$362$$ 66.5410 3.49732
$$363$$ 4.47214 0.234726
$$364$$ 0 0
$$365$$ −6.05573 −0.316971
$$366$$ 17.5623 0.917996
$$367$$ −12.7082 −0.663363 −0.331681 0.943391i $$-0.607616\pi$$
−0.331681 + 0.943391i $$0.607616\pi$$
$$368$$ −81.1591 −4.23071
$$369$$ −8.94427 −0.465620
$$370$$ −27.5623 −1.43290
$$371$$ 0 0
$$372$$ −54.2705 −2.81379
$$373$$ −1.58359 −0.0819953 −0.0409976 0.999159i $$-0.513054\pi$$
−0.0409976 + 0.999159i $$0.513054\pi$$
$$374$$ 11.5623 0.597873
$$375$$ 25.0000 1.29099
$$376$$ 55.8328 2.87936
$$377$$ −4.47214 −0.230327
$$378$$ 0 0
$$379$$ −15.4164 −0.791888 −0.395944 0.918275i $$-0.629583\pi$$
−0.395944 + 0.918275i $$0.629583\pi$$
$$380$$ 32.5623 1.67041
$$381$$ 25.5279 1.30783
$$382$$ −29.2705 −1.49761
$$383$$ 15.0000 0.766464 0.383232 0.923652i $$-0.374811\pi$$
0.383232 + 0.923652i $$0.374811\pi$$
$$384$$ 2.43769 0.124398
$$385$$ 0 0
$$386$$ −1.85410 −0.0943713
$$387$$ −16.0000 −0.813326
$$388$$ 45.7082 2.32048
$$389$$ 1.47214 0.0746403 0.0373201 0.999303i $$-0.488118\pi$$
0.0373201 + 0.999303i $$0.488118\pi$$
$$390$$ −13.0902 −0.662847
$$391$$ −12.1246 −0.613168
$$392$$ 0 0
$$393$$ 18.4164 0.928985
$$394$$ 23.7082 1.19440
$$395$$ −6.05573 −0.304697
$$396$$ −29.1246 −1.46357
$$397$$ −26.1246 −1.31116 −0.655578 0.755127i $$-0.727575\pi$$
−0.655578 + 0.755127i $$0.727575\pi$$
$$398$$ 54.2148 2.71754
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 14.2361 0.710915 0.355458 0.934692i $$-0.384325\pi$$
0.355458 + 0.934692i $$0.384325\pi$$
$$402$$ −17.5623 −0.875928
$$403$$ −5.00000 −0.249068
$$404$$ −43.6869 −2.17351
$$405$$ −24.5967 −1.22222
$$406$$ 0 0
$$407$$ −14.1246 −0.700131
$$408$$ 24.5967 1.21772
$$409$$ 8.70820 0.430593 0.215296 0.976549i $$-0.430928\pi$$
0.215296 + 0.976549i $$0.430928\pi$$
$$410$$ 26.1803 1.29295
$$411$$ 18.4164 0.908414
$$412$$ −13.1459 −0.647652
$$413$$ 0 0
$$414$$ 43.1246 2.11946
$$415$$ 0 0
$$416$$ 10.8541 0.532166
$$417$$ 52.3607 2.56411
$$418$$ 23.5623 1.15247
$$419$$ 32.9443 1.60943 0.804717 0.593659i $$-0.202317\pi$$
0.804717 + 0.593659i $$0.202317\pi$$
$$420$$ 0 0
$$421$$ 13.4164 0.653876 0.326938 0.945046i $$-0.393983\pi$$
0.326938 + 0.945046i $$0.393983\pi$$
$$422$$ −10.4721 −0.509776
$$423$$ −14.9443 −0.726615
$$424$$ 55.8328 2.71148
$$425$$ 0 0
$$426$$ −52.3607 −2.53688
$$427$$ 0 0
$$428$$ −47.3951 −2.29093
$$429$$ −6.70820 −0.323875
$$430$$ 46.8328 2.25848
$$431$$ −31.3607 −1.51059 −0.755295 0.655385i $$-0.772507\pi$$
−0.755295 + 0.655385i $$0.772507\pi$$
$$432$$ 22.0344 1.06013
$$433$$ 29.4164 1.41366 0.706831 0.707382i $$-0.250124\pi$$
0.706831 + 0.707382i $$0.250124\pi$$
$$434$$ 0 0
$$435$$ −22.3607 −1.07211
$$436$$ −13.1459 −0.629574
$$437$$ −24.7082 −1.18195
$$438$$ −15.8541 −0.757538
$$439$$ 24.1246 1.15140 0.575702 0.817659i $$-0.304729\pi$$
0.575702 + 0.817659i $$0.304729\pi$$
$$440$$ 50.1246 2.38960
$$441$$ 0 0
$$442$$ 3.85410 0.183321
$$443$$ 2.23607 0.106239 0.0531194 0.998588i $$-0.483084\pi$$
0.0531194 + 0.998588i $$0.483084\pi$$
$$444$$ −51.1033 −2.42526
$$445$$ 5.00000 0.237023
$$446$$ 10.4721 0.495870
$$447$$ −1.58359 −0.0749013
$$448$$ 0 0
$$449$$ −34.3607 −1.62158 −0.810790 0.585337i $$-0.800962\pi$$
−0.810790 + 0.585337i $$0.800962\pi$$
$$450$$ 0 0
$$451$$ 13.4164 0.631754
$$452$$ 14.2918 0.672230
$$453$$ −45.6525 −2.14494
$$454$$ −15.5623 −0.730375
$$455$$ 0 0
$$456$$ 50.1246 2.34730
$$457$$ −6.12461 −0.286497 −0.143249 0.989687i $$-0.545755\pi$$
−0.143249 + 0.989687i $$0.545755\pi$$
$$458$$ −63.1591 −2.95123
$$459$$ 3.29180 0.153648
$$460$$ −89.3951 −4.16807
$$461$$ 34.3607 1.60034 0.800168 0.599776i $$-0.204744\pi$$
0.800168 + 0.599776i $$0.204744\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ 44.0689 2.04585
$$465$$ −25.0000 −1.15935
$$466$$ −31.2705 −1.44858
$$467$$ 9.65248 0.446663 0.223332 0.974743i $$-0.428307\pi$$
0.223332 + 0.974743i $$0.428307\pi$$
$$468$$ −9.70820 −0.448762
$$469$$ 0 0
$$470$$ 43.7426 2.01770
$$471$$ −15.6525 −0.721228
$$472$$ 11.0000 0.506316
$$473$$ 24.0000 1.10352
$$474$$ −15.8541 −0.728203
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −14.9443 −0.684251
$$478$$ −50.8328 −2.32504
$$479$$ 23.8328 1.08895 0.544475 0.838777i $$-0.316729\pi$$
0.544475 + 0.838777i $$0.316729\pi$$
$$480$$ 54.2705 2.47710
$$481$$ −4.70820 −0.214676
$$482$$ −12.3262 −0.561445
$$483$$ 0 0
$$484$$ −9.70820 −0.441282
$$485$$ 21.0557 0.956091
$$486$$ −46.8328 −2.12438
$$487$$ −21.8328 −0.989339 −0.494670 0.869081i $$-0.664711\pi$$
−0.494670 + 0.869081i $$0.664711\pi$$
$$488$$ −22.4164 −1.01474
$$489$$ 36.7082 1.66000
$$490$$ 0 0
$$491$$ −25.5279 −1.15206 −0.576028 0.817430i $$-0.695398\pi$$
−0.576028 + 0.817430i $$0.695398\pi$$
$$492$$ 48.5410 2.18840
$$493$$ 6.58359 0.296510
$$494$$ 7.85410 0.353373
$$495$$ −13.4164 −0.603023
$$496$$ 49.2705 2.21231
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 26.4164 1.18256 0.591280 0.806466i $$-0.298623\pi$$
0.591280 + 0.806466i $$0.298623\pi$$
$$500$$ −54.2705 −2.42705
$$501$$ −50.2492 −2.24497
$$502$$ −4.00000 −0.178529
$$503$$ 20.9443 0.933859 0.466929 0.884295i $$-0.345360\pi$$
0.466929 + 0.884295i $$0.345360\pi$$
$$504$$ 0 0
$$505$$ −20.1246 −0.895533
$$506$$ −64.6869 −2.87568
$$507$$ −2.23607 −0.0993073
$$508$$ −55.4164 −2.45871
$$509$$ −20.2361 −0.896948 −0.448474 0.893796i $$-0.648032\pi$$
−0.448474 + 0.893796i $$0.648032\pi$$
$$510$$ 19.2705 0.853313
$$511$$ 0 0
$$512$$ 40.3050 1.78124
$$513$$ 6.70820 0.296174
$$514$$ 0.145898 0.00643529
$$515$$ −6.05573 −0.266847
$$516$$ 86.8328 3.82260
$$517$$ 22.4164 0.985872
$$518$$ 0 0
$$519$$ 36.7082 1.61131
$$520$$ 16.7082 0.732703
$$521$$ −17.9443 −0.786153 −0.393076 0.919506i $$-0.628589\pi$$
−0.393076 + 0.919506i $$0.628589\pi$$
$$522$$ −23.4164 −1.02491
$$523$$ 32.7082 1.43023 0.715115 0.699007i $$-0.246374\pi$$
0.715115 + 0.699007i $$0.246374\pi$$
$$524$$ −39.9787 −1.74648
$$525$$ 0 0
$$526$$ −68.3951 −2.98217
$$527$$ 7.36068 0.320636
$$528$$ 66.1033 2.87678
$$529$$ 44.8328 1.94925
$$530$$ 43.7426 1.90006
$$531$$ −2.94427 −0.127771
$$532$$ 0 0
$$533$$ 4.47214 0.193710
$$534$$ 13.0902 0.566467
$$535$$ −21.8328 −0.943915
$$536$$ 22.4164 0.968241
$$537$$ 45.0000 1.94189
$$538$$ −35.2705 −1.52062
$$539$$ 0 0
$$540$$ 24.2705 1.04444
$$541$$ 1.29180 0.0555387 0.0277693 0.999614i $$-0.491160\pi$$
0.0277693 + 0.999614i $$0.491160\pi$$
$$542$$ 53.4508 2.29591
$$543$$ 56.8328 2.43893
$$544$$ −15.9787 −0.685082
$$545$$ −6.05573 −0.259399
$$546$$ 0 0
$$547$$ −4.58359 −0.195980 −0.0979901 0.995187i $$-0.531241\pi$$
−0.0979901 + 0.995187i $$0.531241\pi$$
$$548$$ −39.9787 −1.70781
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 13.4164 0.571558
$$552$$ −137.610 −5.85707
$$553$$ 0 0
$$554$$ 1.09017 0.0463169
$$555$$ −23.5410 −0.999261
$$556$$ −113.666 −4.82050
$$557$$ −18.7082 −0.792692 −0.396346 0.918101i $$-0.629722\pi$$
−0.396346 + 0.918101i $$0.629722\pi$$
$$558$$ −26.1803 −1.10830
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 9.87539 0.416939
$$562$$ −70.5410 −2.97559
$$563$$ 12.5967 0.530890 0.265445 0.964126i $$-0.414481\pi$$
0.265445 + 0.964126i $$0.414481\pi$$
$$564$$ 81.1033 3.41507
$$565$$ 6.58359 0.276974
$$566$$ −68.3951 −2.87486
$$567$$ 0 0
$$568$$ 66.8328 2.80424
$$569$$ 25.4721 1.06785 0.533924 0.845533i $$-0.320717\pi$$
0.533924 + 0.845533i $$0.320717\pi$$
$$570$$ 39.2705 1.64486
$$571$$ −36.1246 −1.51177 −0.755884 0.654706i $$-0.772793\pi$$
−0.755884 + 0.654706i $$0.772793\pi$$
$$572$$ 14.5623 0.608881
$$573$$ −25.0000 −1.04439
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 17.4164 0.725684
$$577$$ −19.2918 −0.803128 −0.401564 0.915831i $$-0.631533\pi$$
−0.401564 + 0.915831i $$0.631533\pi$$
$$578$$ 38.8328 1.61523
$$579$$ −1.58359 −0.0658118
$$580$$ 48.5410 2.01556
$$581$$ 0 0
$$582$$ 55.1246 2.28499
$$583$$ 22.4164 0.928393
$$584$$ 20.2361 0.837374
$$585$$ −4.47214 −0.184900
$$586$$ 39.1246 1.61622
$$587$$ −6.11146 −0.252247 −0.126123 0.992015i $$-0.540254\pi$$
−0.126123 + 0.992015i $$0.540254\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ 8.61803 0.354799
$$591$$ 20.2492 0.832942
$$592$$ 46.3951 1.90683
$$593$$ 27.7639 1.14013 0.570064 0.821600i $$-0.306918\pi$$
0.570064 + 0.821600i $$0.306918\pi$$
$$594$$ 17.5623 0.720590
$$595$$ 0 0
$$596$$ 3.43769 0.140813
$$597$$ 46.3050 1.89514
$$598$$ −21.5623 −0.881748
$$599$$ −17.0689 −0.697416 −0.348708 0.937231i $$-0.613379\pi$$
−0.348708 + 0.937231i $$0.613379\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ −6.00000 −0.244339
$$604$$ 99.1033 4.03246
$$605$$ −4.47214 −0.181818
$$606$$ −52.6869 −2.14026
$$607$$ 24.1246 0.979188 0.489594 0.871951i $$-0.337145\pi$$
0.489594 + 0.871951i $$0.337145\pi$$
$$608$$ −32.5623 −1.32058
$$609$$ 0 0
$$610$$ −17.5623 −0.711077
$$611$$ 7.47214 0.302290
$$612$$ 14.2918 0.577712
$$613$$ −18.1246 −0.732046 −0.366023 0.930606i $$-0.619281\pi$$
−0.366023 + 0.930606i $$0.619281\pi$$
$$614$$ −50.8328 −2.05145
$$615$$ 22.3607 0.901670
$$616$$ 0 0
$$617$$ −4.47214 −0.180041 −0.0900207 0.995940i $$-0.528693\pi$$
−0.0900207 + 0.995940i $$0.528693\pi$$
$$618$$ −15.8541 −0.637746
$$619$$ 17.0000 0.683288 0.341644 0.939829i $$-0.389016\pi$$
0.341644 + 0.939829i $$0.389016\pi$$
$$620$$ 54.2705 2.17956
$$621$$ −18.4164 −0.739025
$$622$$ 72.6869 2.91448
$$623$$ 0 0
$$624$$ 22.0344 0.882084
$$625$$ −25.0000 −1.00000
$$626$$ 14.6180 0.584254
$$627$$ 20.1246 0.803700
$$628$$ 33.9787 1.35590
$$629$$ 6.93112 0.276362
$$630$$ 0 0
$$631$$ 22.8328 0.908960 0.454480 0.890757i $$-0.349825\pi$$
0.454480 + 0.890757i $$0.349825\pi$$
$$632$$ 20.2361 0.804948
$$633$$ −8.94427 −0.355503
$$634$$ 21.5623 0.856349
$$635$$ −25.5279 −1.01304
$$636$$ 81.1033 3.21596
$$637$$ 0 0
$$638$$ 35.1246 1.39060
$$639$$ −17.8885 −0.707660
$$640$$ −2.43769 −0.0963583
$$641$$ 11.9443 0.471770 0.235885 0.971781i $$-0.424201\pi$$
0.235885 + 0.971781i $$0.424201\pi$$
$$642$$ −57.1591 −2.25589
$$643$$ 34.8328 1.37367 0.686836 0.726812i $$-0.258999\pi$$
0.686836 + 0.726812i $$0.258999\pi$$
$$644$$ 0 0
$$645$$ 40.0000 1.57500
$$646$$ −11.5623 −0.454913
$$647$$ 20.2361 0.795562 0.397781 0.917480i $$-0.369780\pi$$
0.397781 + 0.917480i $$0.369780\pi$$
$$648$$ 82.1935 3.22887
$$649$$ 4.41641 0.173359
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −79.6869 −3.12078
$$653$$ 4.52786 0.177189 0.0885945 0.996068i $$-0.471762\pi$$
0.0885945 + 0.996068i $$0.471762\pi$$
$$654$$ −15.8541 −0.619944
$$655$$ −18.4164 −0.719589
$$656$$ −44.0689 −1.72060
$$657$$ −5.41641 −0.211314
$$658$$ 0 0
$$659$$ −8.94427 −0.348419 −0.174210 0.984709i $$-0.555737\pi$$
−0.174210 + 0.984709i $$0.555737\pi$$
$$660$$ 72.8115 2.83418
$$661$$ −6.70820 −0.260919 −0.130459 0.991454i $$-0.541645\pi$$
−0.130459 + 0.991454i $$0.541645\pi$$
$$662$$ 4.14590 0.161135
$$663$$ 3.29180 0.127843
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −24.6525 −0.955264
$$667$$ −36.8328 −1.42617
$$668$$ 109.082 4.22051
$$669$$ 8.94427 0.345806
$$670$$ 17.5623 0.678491
$$671$$ −9.00000 −0.347441
$$672$$ 0 0
$$673$$ −9.41641 −0.362976 −0.181488 0.983393i $$-0.558091\pi$$
−0.181488 + 0.983393i $$0.558091\pi$$
$$674$$ −47.1246 −1.81517
$$675$$ 0 0
$$676$$ 4.85410 0.186696
$$677$$ −2.88854 −0.111016 −0.0555079 0.998458i $$-0.517678\pi$$
−0.0555079 + 0.998458i $$0.517678\pi$$
$$678$$ 17.2361 0.661947
$$679$$ 0 0
$$680$$ −24.5967 −0.943242
$$681$$ −13.2918 −0.509343
$$682$$ 39.2705 1.50375
$$683$$ −13.4721 −0.515497 −0.257748 0.966212i $$-0.582981\pi$$
−0.257748 + 0.966212i $$0.582981\pi$$
$$684$$ 29.1246 1.11361
$$685$$ −18.4164 −0.703655
$$686$$ 0 0
$$687$$ −53.9443 −2.05810
$$688$$ −78.8328 −3.00547
$$689$$ 7.47214 0.284666
$$690$$ −107.812 −4.10431
$$691$$ −51.8328 −1.97181 −0.985907 0.167297i $$-0.946496\pi$$
−0.985907 + 0.167297i $$0.946496\pi$$
$$692$$ −79.6869 −3.02924
$$693$$ 0 0
$$694$$ −60.3951 −2.29257
$$695$$ −52.3607 −1.98615
$$696$$ 74.7214 2.83231
$$697$$ −6.58359 −0.249371
$$698$$ 77.0132 2.91499
$$699$$ −26.7082 −1.01020
$$700$$ 0 0
$$701$$ −22.3607 −0.844551 −0.422276 0.906467i $$-0.638769\pi$$
−0.422276 + 0.906467i $$0.638769\pi$$
$$702$$ 5.85410 0.220949
$$703$$ 14.1246 0.532720
$$704$$ −26.1246 −0.984608
$$705$$ 37.3607 1.40708
$$706$$ −45.2705 −1.70378
$$707$$ 0 0
$$708$$ 15.9787 0.600517
$$709$$ −50.1246 −1.88247 −0.941235 0.337753i $$-0.890333\pi$$
−0.941235 + 0.337753i $$0.890333\pi$$
$$710$$ 52.3607 1.96506
$$711$$ −5.41641 −0.203131
$$712$$ −16.7082 −0.626166
$$713$$ −41.1803 −1.54222
$$714$$ 0 0
$$715$$ 6.70820 0.250873
$$716$$ −97.6869 −3.65073
$$717$$ −43.4164 −1.62142
$$718$$ −31.2705 −1.16701
$$719$$ 24.7082 0.921461 0.460730 0.887540i $$-0.347588\pi$$
0.460730 + 0.887540i $$0.347588\pi$$
$$720$$ 44.0689 1.64235
$$721$$ 0 0
$$722$$ 26.1803 0.974331
$$723$$ −10.5279 −0.391535
$$724$$ −123.374 −4.58515
$$725$$ 0 0
$$726$$ −11.7082 −0.434532
$$727$$ −38.8328 −1.44023 −0.720115 0.693855i $$-0.755911\pi$$
−0.720115 + 0.693855i $$0.755911\pi$$
$$728$$ 0 0
$$729$$ −7.00000 −0.259259
$$730$$ 15.8541 0.586787
$$731$$ −11.7771 −0.435591
$$732$$ −32.5623 −1.20354
$$733$$ 28.7082 1.06036 0.530181 0.847885i $$-0.322124\pi$$
0.530181 + 0.847885i $$0.322124\pi$$
$$734$$ 33.2705 1.22804
$$735$$ 0 0
$$736$$ 89.3951 3.29515
$$737$$ 9.00000 0.331519
$$738$$ 23.4164 0.861970
$$739$$ −17.8328 −0.655991 −0.327995 0.944679i $$-0.606373\pi$$
−0.327995 + 0.944679i $$0.606373\pi$$
$$740$$ 51.1033 1.87860
$$741$$ 6.70820 0.246432
$$742$$ 0 0
$$743$$ 32.9443 1.20861 0.604304 0.796754i $$-0.293451\pi$$
0.604304 + 0.796754i $$0.293451\pi$$
$$744$$ 83.5410 3.06276
$$745$$ 1.58359 0.0580183
$$746$$ 4.14590 0.151792
$$747$$ 0 0
$$748$$ −21.4377 −0.783840
$$749$$ 0 0
$$750$$ −65.4508 −2.38993
$$751$$ −10.1246 −0.369452 −0.184726 0.982790i $$-0.559140\pi$$
−0.184726 + 0.982790i $$0.559140\pi$$
$$752$$ −73.6312 −2.68505
$$753$$ −3.41641 −0.124501
$$754$$ 11.7082 0.426388
$$755$$ 45.6525 1.66146
$$756$$ 0 0
$$757$$ −52.8328 −1.92024 −0.960121 0.279586i $$-0.909803\pi$$
−0.960121 + 0.279586i $$0.909803\pi$$
$$758$$ 40.3607 1.46597
$$759$$ −55.2492 −2.00542
$$760$$ −50.1246 −1.81821
$$761$$ −33.5410 −1.21586 −0.607931 0.793990i $$-0.708000\pi$$
−0.607931 + 0.793990i $$0.708000\pi$$
$$762$$ −66.8328 −2.42110
$$763$$ 0 0
$$764$$ 54.2705 1.96344
$$765$$ 6.58359 0.238030
$$766$$ −39.2705 −1.41890
$$767$$ 1.47214 0.0531557
$$768$$ 32.5623 1.17499
$$769$$ 46.0000 1.65880 0.829401 0.558653i $$-0.188682\pi$$
0.829401 + 0.558653i $$0.188682\pi$$
$$770$$ 0 0
$$771$$ 0.124612 0.00448778
$$772$$ 3.43769 0.123725
$$773$$ −47.0689 −1.69295 −0.846475 0.532428i $$-0.821280\pi$$
−0.846475 + 0.532428i $$0.821280\pi$$
$$774$$ 41.8885 1.50565
$$775$$ 0 0
$$776$$ −70.3607 −2.52580
$$777$$ 0 0
$$778$$ −3.85410 −0.138176
$$779$$ −13.4164 −0.480693
$$780$$ 24.2705 0.869024
$$781$$ 26.8328 0.960154
$$782$$ 31.7426 1.13511
$$783$$ 10.0000 0.357371
$$784$$ 0 0
$$785$$ 15.6525 0.558661
$$786$$ −48.2148 −1.71976
$$787$$ 20.4164 0.727766 0.363883 0.931445i $$-0.381451\pi$$
0.363883 + 0.931445i $$0.381451\pi$$
$$788$$ −43.9574 −1.56592
$$789$$ −58.4164 −2.07968
$$790$$ 15.8541 0.564064
$$791$$ 0 0
$$792$$ 44.8328 1.59306
$$793$$ −3.00000 −0.106533
$$794$$ 68.3951 2.42725
$$795$$ 37.3607 1.32505
$$796$$ −100.520 −3.56283
$$797$$ 9.05573 0.320770 0.160385 0.987055i $$-0.448726\pi$$
0.160385 + 0.987055i $$0.448726\pi$$
$$798$$ 0 0
$$799$$ −11.0000 −0.389152
$$800$$ 0 0
$$801$$ 4.47214 0.158015
$$802$$ −37.2705 −1.31607
$$803$$ 8.12461 0.286711
$$804$$ 32.5623 1.14838
$$805$$ 0 0
$$806$$ 13.0902 0.461082
$$807$$ −30.1246 −1.06044
$$808$$ 67.2492 2.36582
$$809$$ 22.4164 0.788119 0.394059 0.919085i $$-0.371070\pi$$
0.394059 + 0.919085i $$0.371070\pi$$
$$810$$ 64.3951 2.26261
$$811$$ −14.8328 −0.520851 −0.260425 0.965494i $$-0.583863\pi$$
−0.260425 + 0.965494i $$0.583863\pi$$
$$812$$ 0 0
$$813$$ 45.6525 1.60110
$$814$$ 36.9787 1.29610
$$815$$ −36.7082 −1.28583
$$816$$ −32.4377 −1.13555
$$817$$ −24.0000 −0.839654
$$818$$ −22.7984 −0.797126
$$819$$ 0 0
$$820$$ −48.5410 −1.69513
$$821$$ 38.2361 1.33445 0.667224 0.744857i $$-0.267482\pi$$
0.667224 + 0.744857i $$0.267482\pi$$
$$822$$ −48.2148 −1.68168
$$823$$ 34.1246 1.18951 0.594755 0.803907i $$-0.297249\pi$$
0.594755 + 0.803907i $$0.297249\pi$$
$$824$$ 20.2361 0.704957
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 26.8328 0.933068 0.466534 0.884503i $$-0.345502\pi$$
0.466534 + 0.884503i $$0.345502\pi$$
$$828$$ −79.9574 −2.77871
$$829$$ −25.0000 −0.868286 −0.434143 0.900844i $$-0.642949\pi$$
−0.434143 + 0.900844i $$0.642949\pi$$
$$830$$ 0 0
$$831$$ 0.931116 0.0323001
$$832$$ −8.70820 −0.301903
$$833$$ 0 0
$$834$$ −137.082 −4.74676
$$835$$ 50.2492 1.73895
$$836$$ −43.6869 −1.51094
$$837$$ 11.1803 0.386449
$$838$$ −86.2492 −2.97943
$$839$$ −5.88854 −0.203295 −0.101648 0.994820i $$-0.532411\pi$$
−0.101648 + 0.994820i $$0.532411\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ −35.1246 −1.21047
$$843$$ −60.2492 −2.07509
$$844$$ 19.4164 0.668340
$$845$$ 2.23607 0.0769231
$$846$$ 39.1246 1.34513
$$847$$ 0 0
$$848$$ −73.6312 −2.52851
$$849$$ −58.4164 −2.00485
$$850$$ 0 0
$$851$$ −38.7771 −1.32926
$$852$$ 97.0820 3.32598
$$853$$ −52.2492 −1.78898 −0.894490 0.447089i $$-0.852461\pi$$
−0.894490 + 0.447089i $$0.852461\pi$$
$$854$$ 0 0
$$855$$ 13.4164 0.458831
$$856$$ 72.9574 2.49363
$$857$$ −1.36068 −0.0464799 −0.0232400 0.999730i $$-0.507398\pi$$
−0.0232400 + 0.999730i $$0.507398\pi$$
$$858$$ 17.5623 0.599567
$$859$$ 20.7082 0.706555 0.353277 0.935519i $$-0.385067\pi$$
0.353277 + 0.935519i $$0.385067\pi$$
$$860$$ −86.8328 −2.96097
$$861$$ 0 0
$$862$$ 82.1033 2.79645
$$863$$ 23.9443 0.815072 0.407536 0.913189i $$-0.366388\pi$$
0.407536 + 0.913189i $$0.366388\pi$$
$$864$$ −24.2705 −0.825700
$$865$$ −36.7082 −1.24812
$$866$$ −77.0132 −2.61701
$$867$$ 33.1672 1.12642
$$868$$ 0 0
$$869$$ 8.12461 0.275609
$$870$$ 58.5410 1.98473
$$871$$ 3.00000 0.101651
$$872$$ 20.2361 0.685280
$$873$$ 18.8328 0.637394
$$874$$ 64.6869 2.18807
$$875$$ 0 0
$$876$$ 29.3951 0.993169
$$877$$ 32.1246 1.08477 0.542386 0.840130i $$-0.317521\pi$$
0.542386 + 0.840130i $$0.317521\pi$$
$$878$$ −63.1591 −2.13151
$$879$$ 33.4164 1.12711
$$880$$ −66.1033 −2.22834
$$881$$ 43.3050 1.45898 0.729490 0.683991i $$-0.239757\pi$$
0.729490 + 0.683991i $$0.239757\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −7.14590 −0.240343
$$885$$ 7.36068 0.247427
$$886$$ −5.85410 −0.196672
$$887$$ −20.2361 −0.679461 −0.339730 0.940523i $$-0.610336\pi$$
−0.339730 + 0.940523i $$0.610336\pi$$
$$888$$ 78.6656 2.63985
$$889$$ 0 0
$$890$$ −13.0902 −0.438783
$$891$$ 33.0000 1.10554
$$892$$ −19.4164 −0.650109
$$893$$ −22.4164 −0.750136
$$894$$ 4.14590 0.138660
$$895$$ −45.0000 −1.50418
$$896$$ 0 0
$$897$$ −18.4164 −0.614906
$$898$$ 89.9574 3.00192
$$899$$ 22.3607 0.745770
$$900$$ 0 0
$$901$$ −11.0000 −0.366463
$$902$$ −35.1246 −1.16952
$$903$$ 0 0
$$904$$ −22.0000 −0.731709
$$905$$ −56.8328 −1.88919
$$906$$ 119.520 3.97078
$$907$$ 18.7082 0.621196 0.310598 0.950541i $$-0.399471\pi$$
0.310598 + 0.950541i $$0.399471\pi$$
$$908$$ 28.8541 0.957557
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ −34.2492 −1.13473 −0.567364 0.823467i $$-0.692037\pi$$
−0.567364 + 0.823467i $$0.692037\pi$$
$$912$$ −66.1033 −2.18890
$$913$$ 0 0
$$914$$ 16.0344 0.530372
$$915$$ −15.0000 −0.495885
$$916$$ 117.103 3.86920
$$917$$ 0 0
$$918$$ −8.61803 −0.284438
$$919$$ 20.1246 0.663850 0.331925 0.943306i $$-0.392302\pi$$
0.331925 + 0.943306i $$0.392302\pi$$
$$920$$ 137.610 4.53686
$$921$$ −43.4164 −1.43062
$$922$$ −89.9574 −2.96259
$$923$$ 8.94427 0.294404
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 62.8328 2.06481
$$927$$ −5.41641 −0.177898
$$928$$ −48.5410 −1.59344
$$929$$ −24.8197 −0.814307 −0.407153 0.913360i $$-0.633479\pi$$
−0.407153 + 0.913360i $$0.633479\pi$$
$$930$$ 65.4508 2.14622
$$931$$ 0 0
$$932$$ 57.9787 1.89916
$$933$$ 62.0820 2.03247
$$934$$ −25.2705 −0.826876
$$935$$ −9.87539 −0.322960
$$936$$ 14.9443 0.488469
$$937$$ 25.4164 0.830318 0.415159 0.909749i $$-0.363726\pi$$
0.415159 + 0.909749i $$0.363726\pi$$
$$938$$ 0 0
$$939$$ 12.4853 0.407442
$$940$$ −81.1033 −2.64530
$$941$$ 50.2361 1.63765 0.818825 0.574044i $$-0.194626\pi$$
0.818825 + 0.574044i $$0.194626\pi$$
$$942$$ 40.9787 1.33516
$$943$$ 36.8328 1.19944
$$944$$ −14.5066 −0.472149
$$945$$ 0 0
$$946$$ −62.8328 −2.04287
$$947$$ −22.5279 −0.732057 −0.366029 0.930604i $$-0.619283\pi$$
−0.366029 + 0.930604i $$0.619283\pi$$
$$948$$ 29.3951 0.954709
$$949$$ 2.70820 0.0879120
$$950$$ 0 0
$$951$$ 18.4164 0.597193
$$952$$ 0 0
$$953$$ −41.7771 −1.35329 −0.676646 0.736308i $$-0.736567\pi$$
−0.676646 + 0.736308i $$0.736567\pi$$
$$954$$ 39.1246 1.26671
$$955$$ 25.0000 0.808981
$$956$$ 94.2492 3.04824
$$957$$ 30.0000 0.969762
$$958$$ −62.3951 −2.01589
$$959$$ 0 0
$$960$$ −43.5410 −1.40528
$$961$$ −6.00000 −0.193548
$$962$$ 12.3262 0.397414
$$963$$ −19.5279 −0.629277
$$964$$ 22.8541 0.736081
$$965$$ 1.58359 0.0509776
$$966$$ 0 0
$$967$$ 16.5836 0.533292 0.266646 0.963794i $$-0.414084\pi$$
0.266646 + 0.963794i $$0.414084\pi$$
$$968$$ 14.9443 0.480327
$$969$$ −9.87539 −0.317243
$$970$$ −55.1246 −1.76994
$$971$$ −11.2918 −0.362371 −0.181185 0.983449i $$-0.557993\pi$$
−0.181185 + 0.983449i $$0.557993\pi$$
$$972$$ 86.8328 2.78516
$$973$$ 0 0
$$974$$ 57.1591 1.83149
$$975$$ 0 0
$$976$$ 29.5623 0.946266
$$977$$ −2.34752 −0.0751040 −0.0375520 0.999295i $$-0.511956\pi$$
−0.0375520 + 0.999295i $$0.511956\pi$$
$$978$$ −96.1033 −3.07305
$$979$$ −6.70820 −0.214395
$$980$$ 0 0
$$981$$ −5.41641 −0.172933
$$982$$ 66.8328 2.13272
$$983$$ 7.47214 0.238324 0.119162 0.992875i $$-0.461979\pi$$
0.119162 + 0.992875i $$0.461979\pi$$
$$984$$ −74.7214 −2.38203
$$985$$ −20.2492 −0.645194
$$986$$ −17.2361 −0.548908
$$987$$ 0 0
$$988$$ −14.5623 −0.463289
$$989$$ 65.8885 2.09513
$$990$$ 35.1246 1.11633
$$991$$ 30.7082 0.975478 0.487739 0.872989i $$-0.337822\pi$$
0.487739 + 0.872989i $$0.337822\pi$$
$$992$$ −54.2705 −1.72309
$$993$$ 3.54102 0.112371
$$994$$ 0 0
$$995$$ −46.3050 −1.46797
$$996$$ 0 0
$$997$$ 26.4164 0.836616 0.418308 0.908305i $$-0.362623\pi$$
0.418308 + 0.908305i $$0.362623\pi$$
$$998$$ −69.1591 −2.18919
$$999$$ 10.5279 0.333087
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.f.1.1 2
3.2 odd 2 5733.2.a.v.1.2 2
7.2 even 3 91.2.e.b.53.2 4
7.3 odd 6 637.2.e.h.79.2 4
7.4 even 3 91.2.e.b.79.2 yes 4
7.5 odd 6 637.2.e.h.508.2 4
7.6 odd 2 637.2.a.e.1.1 2
13.12 even 2 8281.2.a.z.1.2 2
21.2 odd 6 819.2.j.c.235.1 4
21.11 odd 6 819.2.j.c.352.1 4
21.20 even 2 5733.2.a.w.1.2 2
28.11 odd 6 1456.2.r.j.625.1 4
28.23 odd 6 1456.2.r.j.417.1 4
91.25 even 6 1183.2.e.d.170.1 4
91.51 even 6 1183.2.e.d.508.1 4
91.90 odd 2 8281.2.a.ba.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.2 4 7.2 even 3
91.2.e.b.79.2 yes 4 7.4 even 3
637.2.a.e.1.1 2 7.6 odd 2
637.2.a.f.1.1 2 1.1 even 1 trivial
637.2.e.h.79.2 4 7.3 odd 6
637.2.e.h.508.2 4 7.5 odd 6
819.2.j.c.235.1 4 21.2 odd 6
819.2.j.c.352.1 4 21.11 odd 6
1183.2.e.d.170.1 4 91.25 even 6
1183.2.e.d.508.1 4 91.51 even 6
1456.2.r.j.417.1 4 28.23 odd 6
1456.2.r.j.625.1 4 28.11 odd 6
5733.2.a.v.1.2 2 3.2 odd 2
5733.2.a.w.1.2 2 21.20 even 2
8281.2.a.z.1.2 2 13.12 even 2
8281.2.a.ba.1.2 2 91.90 odd 2