# Properties

 Label 3822.2.a.bl.1.2 Level $3822$ Weight $2$ Character 3822.1 Self dual yes Analytic conductor $30.519$ 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] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.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: $$30.5188236525$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3822.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.41421 q^{11} +1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +0.414214 q^{17} -1.00000 q^{18} -3.82843 q^{19} -1.00000 q^{20} -2.41421 q^{22} -8.65685 q^{23} -1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +2.65685 q^{29} +1.00000 q^{30} +4.24264 q^{31} -1.00000 q^{32} +2.41421 q^{33} -0.414214 q^{34} +1.00000 q^{36} -3.24264 q^{37} +3.82843 q^{38} +1.00000 q^{39} +1.00000 q^{40} -6.82843 q^{41} -7.00000 q^{43} +2.41421 q^{44} -1.00000 q^{45} +8.65685 q^{46} -7.65685 q^{47} +1.00000 q^{48} +4.00000 q^{50} +0.414214 q^{51} +1.00000 q^{52} -13.6569 q^{53} -1.00000 q^{54} -2.41421 q^{55} -3.82843 q^{57} -2.65685 q^{58} +9.89949 q^{59} -1.00000 q^{60} +5.58579 q^{61} -4.24264 q^{62} +1.00000 q^{64} -1.00000 q^{65} -2.41421 q^{66} +1.41421 q^{67} +0.414214 q^{68} -8.65685 q^{69} +5.07107 q^{71} -1.00000 q^{72} +11.8284 q^{73} +3.24264 q^{74} -4.00000 q^{75} -3.82843 q^{76} -1.00000 q^{78} +10.2426 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.82843 q^{82} -11.6569 q^{83} -0.414214 q^{85} +7.00000 q^{86} +2.65685 q^{87} -2.41421 q^{88} -0.585786 q^{89} +1.00000 q^{90} -8.65685 q^{92} +4.24264 q^{93} +7.65685 q^{94} +3.82843 q^{95} -1.00000 q^{96} -7.31371 q^{97} +2.41421 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 6 q^{23} - 2 q^{24} - 8 q^{25} - 2 q^{26} + 2 q^{27} - 6 q^{29} + 2 q^{30} - 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 2 q^{37} + 2 q^{38} + 2 q^{39} + 2 q^{40} - 8 q^{41} - 14 q^{43} + 2 q^{44} - 2 q^{45} + 6 q^{46} - 4 q^{47} + 2 q^{48} + 8 q^{50} - 2 q^{51} + 2 q^{52} - 16 q^{53} - 2 q^{54} - 2 q^{55} - 2 q^{57} + 6 q^{58} - 2 q^{60} + 14 q^{61} + 2 q^{64} - 2 q^{65} - 2 q^{66} - 2 q^{68} - 6 q^{69} - 4 q^{71} - 2 q^{72} + 18 q^{73} - 2 q^{74} - 8 q^{75} - 2 q^{76} - 2 q^{78} + 12 q^{79} - 2 q^{80} + 2 q^{81} + 8 q^{82} - 12 q^{83} + 2 q^{85} + 14 q^{86} - 6 q^{87} - 2 q^{88} - 4 q^{89} + 2 q^{90} - 6 q^{92} + 4 q^{94} + 2 q^{95} - 2 q^{96} + 8 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^11 + 2 * q^12 + 2 * q^13 - 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 2 * q^19 - 2 * q^20 - 2 * q^22 - 6 * q^23 - 2 * q^24 - 8 * q^25 - 2 * q^26 + 2 * q^27 - 6 * q^29 + 2 * q^30 - 2 * q^32 + 2 * q^33 + 2 * q^34 + 2 * q^36 + 2 * q^37 + 2 * q^38 + 2 * q^39 + 2 * q^40 - 8 * q^41 - 14 * q^43 + 2 * q^44 - 2 * q^45 + 6 * q^46 - 4 * q^47 + 2 * q^48 + 8 * q^50 - 2 * q^51 + 2 * q^52 - 16 * q^53 - 2 * q^54 - 2 * q^55 - 2 * q^57 + 6 * q^58 - 2 * q^60 + 14 * q^61 + 2 * q^64 - 2 * q^65 - 2 * q^66 - 2 * q^68 - 6 * q^69 - 4 * q^71 - 2 * q^72 + 18 * q^73 - 2 * q^74 - 8 * q^75 - 2 * q^76 - 2 * q^78 + 12 * q^79 - 2 * q^80 + 2 * q^81 + 8 * q^82 - 12 * q^83 + 2 * q^85 + 14 * q^86 - 6 * q^87 - 2 * q^88 - 4 * q^89 + 2 * q^90 - 6 * q^92 + 4 * q^94 + 2 * q^95 - 2 * q^96 + 8 * q^97 + 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 2.41421 0.727913 0.363956 0.931416i $$-0.381426\pi$$
0.363956 + 0.931416i $$0.381426\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 0.414214 0.100462 0.0502308 0.998738i $$-0.484004\pi$$
0.0502308 + 0.998738i $$0.484004\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −3.82843 −0.878301 −0.439151 0.898413i $$-0.644721\pi$$
−0.439151 + 0.898413i $$0.644721\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −2.41421 −0.514712
$$23$$ −8.65685 −1.80508 −0.902539 0.430607i $$-0.858299\pi$$
−0.902539 + 0.430607i $$0.858299\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −4.00000 −0.800000
$$26$$ −1.00000 −0.196116
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 2.65685 0.493365 0.246683 0.969096i $$-0.420659\pi$$
0.246683 + 0.969096i $$0.420659\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 4.24264 0.762001 0.381000 0.924575i $$-0.375580\pi$$
0.381000 + 0.924575i $$0.375580\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 2.41421 0.420261
$$34$$ −0.414214 −0.0710370
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −3.24264 −0.533087 −0.266543 0.963823i $$-0.585882\pi$$
−0.266543 + 0.963823i $$0.585882\pi$$
$$38$$ 3.82843 0.621053
$$39$$ 1.00000 0.160128
$$40$$ 1.00000 0.158114
$$41$$ −6.82843 −1.06642 −0.533211 0.845983i $$-0.679015\pi$$
−0.533211 + 0.845983i $$0.679015\pi$$
$$42$$ 0 0
$$43$$ −7.00000 −1.06749 −0.533745 0.845645i $$-0.679216\pi$$
−0.533745 + 0.845645i $$0.679216\pi$$
$$44$$ 2.41421 0.363956
$$45$$ −1.00000 −0.149071
$$46$$ 8.65685 1.27638
$$47$$ −7.65685 −1.11687 −0.558433 0.829549i $$-0.688597\pi$$
−0.558433 + 0.829549i $$0.688597\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 4.00000 0.565685
$$51$$ 0.414214 0.0580015
$$52$$ 1.00000 0.138675
$$53$$ −13.6569 −1.87591 −0.937957 0.346753i $$-0.887284\pi$$
−0.937957 + 0.346753i $$0.887284\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −2.41421 −0.325532
$$56$$ 0 0
$$57$$ −3.82843 −0.507088
$$58$$ −2.65685 −0.348862
$$59$$ 9.89949 1.28880 0.644402 0.764687i $$-0.277106\pi$$
0.644402 + 0.764687i $$0.277106\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 5.58579 0.715187 0.357593 0.933877i $$-0.383597\pi$$
0.357593 + 0.933877i $$0.383597\pi$$
$$62$$ −4.24264 −0.538816
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.00000 −0.124035
$$66$$ −2.41421 −0.297169
$$67$$ 1.41421 0.172774 0.0863868 0.996262i $$-0.472468\pi$$
0.0863868 + 0.996262i $$0.472468\pi$$
$$68$$ 0.414214 0.0502308
$$69$$ −8.65685 −1.04216
$$70$$ 0 0
$$71$$ 5.07107 0.601825 0.300913 0.953652i $$-0.402709\pi$$
0.300913 + 0.953652i $$0.402709\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 11.8284 1.38441 0.692206 0.721700i $$-0.256639\pi$$
0.692206 + 0.721700i $$0.256639\pi$$
$$74$$ 3.24264 0.376949
$$75$$ −4.00000 −0.461880
$$76$$ −3.82843 −0.439151
$$77$$ 0 0
$$78$$ −1.00000 −0.113228
$$79$$ 10.2426 1.15239 0.576194 0.817313i $$-0.304537\pi$$
0.576194 + 0.817313i $$0.304537\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 6.82843 0.754074
$$83$$ −11.6569 −1.27951 −0.639753 0.768581i $$-0.720963\pi$$
−0.639753 + 0.768581i $$0.720963\pi$$
$$84$$ 0 0
$$85$$ −0.414214 −0.0449278
$$86$$ 7.00000 0.754829
$$87$$ 2.65685 0.284845
$$88$$ −2.41421 −0.257356
$$89$$ −0.585786 −0.0620932 −0.0310466 0.999518i $$-0.509884\pi$$
−0.0310466 + 0.999518i $$0.509884\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −8.65685 −0.902539
$$93$$ 4.24264 0.439941
$$94$$ 7.65685 0.789744
$$95$$ 3.82843 0.392788
$$96$$ −1.00000 −0.102062
$$97$$ −7.31371 −0.742595 −0.371297 0.928514i $$-0.621087\pi$$
−0.371297 + 0.928514i $$0.621087\pi$$
$$98$$ 0 0
$$99$$ 2.41421 0.242638
$$100$$ −4.00000 −0.400000
$$101$$ 1.07107 0.106575 0.0532876 0.998579i $$-0.483030\pi$$
0.0532876 + 0.998579i $$0.483030\pi$$
$$102$$ −0.414214 −0.0410133
$$103$$ 8.07107 0.795266 0.397633 0.917545i $$-0.369832\pi$$
0.397633 + 0.917545i $$0.369832\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 13.6569 1.32647
$$107$$ 4.58579 0.443325 0.221662 0.975123i $$-0.428852\pi$$
0.221662 + 0.975123i $$0.428852\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 0.414214 0.0396745 0.0198372 0.999803i $$-0.493685\pi$$
0.0198372 + 0.999803i $$0.493685\pi$$
$$110$$ 2.41421 0.230186
$$111$$ −3.24264 −0.307778
$$112$$ 0 0
$$113$$ −18.2426 −1.71612 −0.858062 0.513547i $$-0.828331\pi$$
−0.858062 + 0.513547i $$0.828331\pi$$
$$114$$ 3.82843 0.358565
$$115$$ 8.65685 0.807256
$$116$$ 2.65685 0.246683
$$117$$ 1.00000 0.0924500
$$118$$ −9.89949 −0.911322
$$119$$ 0 0
$$120$$ 1.00000 0.0912871
$$121$$ −5.17157 −0.470143
$$122$$ −5.58579 −0.505713
$$123$$ −6.82843 −0.615699
$$124$$ 4.24264 0.381000
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ −2.92893 −0.259901 −0.129950 0.991521i $$-0.541482\pi$$
−0.129950 + 0.991521i $$0.541482\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −7.00000 −0.616316
$$130$$ 1.00000 0.0877058
$$131$$ −17.2426 −1.50650 −0.753248 0.657736i $$-0.771514\pi$$
−0.753248 + 0.657736i $$0.771514\pi$$
$$132$$ 2.41421 0.210130
$$133$$ 0 0
$$134$$ −1.41421 −0.122169
$$135$$ −1.00000 −0.0860663
$$136$$ −0.414214 −0.0355185
$$137$$ −7.24264 −0.618781 −0.309390 0.950935i $$-0.600125\pi$$
−0.309390 + 0.950935i $$0.600125\pi$$
$$138$$ 8.65685 0.736920
$$139$$ 6.82843 0.579180 0.289590 0.957151i $$-0.406481\pi$$
0.289590 + 0.957151i $$0.406481\pi$$
$$140$$ 0 0
$$141$$ −7.65685 −0.644823
$$142$$ −5.07107 −0.425555
$$143$$ 2.41421 0.201887
$$144$$ 1.00000 0.0833333
$$145$$ −2.65685 −0.220640
$$146$$ −11.8284 −0.978928
$$147$$ 0 0
$$148$$ −3.24264 −0.266543
$$149$$ −2.82843 −0.231714 −0.115857 0.993266i $$-0.536961\pi$$
−0.115857 + 0.993266i $$0.536961\pi$$
$$150$$ 4.00000 0.326599
$$151$$ −13.2426 −1.07767 −0.538835 0.842411i $$-0.681136\pi$$
−0.538835 + 0.842411i $$0.681136\pi$$
$$152$$ 3.82843 0.310526
$$153$$ 0.414214 0.0334872
$$154$$ 0 0
$$155$$ −4.24264 −0.340777
$$156$$ 1.00000 0.0800641
$$157$$ 2.89949 0.231405 0.115702 0.993284i $$-0.463088\pi$$
0.115702 + 0.993284i $$0.463088\pi$$
$$158$$ −10.2426 −0.814861
$$159$$ −13.6569 −1.08306
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −11.6569 −0.913035 −0.456518 0.889714i $$-0.650903\pi$$
−0.456518 + 0.889714i $$0.650903\pi$$
$$164$$ −6.82843 −0.533211
$$165$$ −2.41421 −0.187946
$$166$$ 11.6569 0.904747
$$167$$ −13.8284 −1.07008 −0.535038 0.844828i $$-0.679703\pi$$
−0.535038 + 0.844828i $$0.679703\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0.414214 0.0317687
$$171$$ −3.82843 −0.292767
$$172$$ −7.00000 −0.533745
$$173$$ −16.2426 −1.23491 −0.617453 0.786608i $$-0.711835\pi$$
−0.617453 + 0.786608i $$0.711835\pi$$
$$174$$ −2.65685 −0.201416
$$175$$ 0 0
$$176$$ 2.41421 0.181978
$$177$$ 9.89949 0.744092
$$178$$ 0.585786 0.0439065
$$179$$ −19.5563 −1.46171 −0.730855 0.682533i $$-0.760878\pi$$
−0.730855 + 0.682533i $$0.760878\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −18.1421 −1.34849 −0.674247 0.738506i $$-0.735532\pi$$
−0.674247 + 0.738506i $$0.735532\pi$$
$$182$$ 0 0
$$183$$ 5.58579 0.412913
$$184$$ 8.65685 0.638192
$$185$$ 3.24264 0.238404
$$186$$ −4.24264 −0.311086
$$187$$ 1.00000 0.0731272
$$188$$ −7.65685 −0.558433
$$189$$ 0 0
$$190$$ −3.82843 −0.277743
$$191$$ 5.48528 0.396901 0.198451 0.980111i $$-0.436409\pi$$
0.198451 + 0.980111i $$0.436409\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 12.3848 0.891476 0.445738 0.895164i $$-0.352941\pi$$
0.445738 + 0.895164i $$0.352941\pi$$
$$194$$ 7.31371 0.525094
$$195$$ −1.00000 −0.0716115
$$196$$ 0 0
$$197$$ 5.07107 0.361299 0.180649 0.983548i $$-0.442180\pi$$
0.180649 + 0.983548i $$0.442180\pi$$
$$198$$ −2.41421 −0.171571
$$199$$ −10.0711 −0.713919 −0.356960 0.934120i $$-0.616187\pi$$
−0.356960 + 0.934120i $$0.616187\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 1.41421 0.0997509
$$202$$ −1.07107 −0.0753601
$$203$$ 0 0
$$204$$ 0.414214 0.0290008
$$205$$ 6.82843 0.476918
$$206$$ −8.07107 −0.562338
$$207$$ −8.65685 −0.601693
$$208$$ 1.00000 0.0693375
$$209$$ −9.24264 −0.639327
$$210$$ 0 0
$$211$$ −16.3137 −1.12308 −0.561541 0.827449i $$-0.689791\pi$$
−0.561541 + 0.827449i $$0.689791\pi$$
$$212$$ −13.6569 −0.937957
$$213$$ 5.07107 0.347464
$$214$$ −4.58579 −0.313478
$$215$$ 7.00000 0.477396
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −0.414214 −0.0280541
$$219$$ 11.8284 0.799291
$$220$$ −2.41421 −0.162766
$$221$$ 0.414214 0.0278630
$$222$$ 3.24264 0.217632
$$223$$ 15.6569 1.04846 0.524230 0.851577i $$-0.324353\pi$$
0.524230 + 0.851577i $$0.324353\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ 18.2426 1.21348
$$227$$ 5.07107 0.336579 0.168289 0.985738i $$-0.446176\pi$$
0.168289 + 0.985738i $$0.446176\pi$$
$$228$$ −3.82843 −0.253544
$$229$$ 23.8995 1.57932 0.789662 0.613543i $$-0.210256\pi$$
0.789662 + 0.613543i $$0.210256\pi$$
$$230$$ −8.65685 −0.570816
$$231$$ 0 0
$$232$$ −2.65685 −0.174431
$$233$$ −10.8284 −0.709394 −0.354697 0.934981i $$-0.615416\pi$$
−0.354697 + 0.934981i $$0.615416\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 7.65685 0.499478
$$236$$ 9.89949 0.644402
$$237$$ 10.2426 0.665331
$$238$$ 0 0
$$239$$ 14.7279 0.952670 0.476335 0.879264i $$-0.341965\pi$$
0.476335 + 0.879264i $$0.341965\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 12.3431 0.795092 0.397546 0.917582i $$-0.369862\pi$$
0.397546 + 0.917582i $$0.369862\pi$$
$$242$$ 5.17157 0.332441
$$243$$ 1.00000 0.0641500
$$244$$ 5.58579 0.357593
$$245$$ 0 0
$$246$$ 6.82843 0.435365
$$247$$ −3.82843 −0.243597
$$248$$ −4.24264 −0.269408
$$249$$ −11.6569 −0.738723
$$250$$ −9.00000 −0.569210
$$251$$ −3.58579 −0.226333 −0.113166 0.993576i $$-0.536099\pi$$
−0.113166 + 0.993576i $$0.536099\pi$$
$$252$$ 0 0
$$253$$ −20.8995 −1.31394
$$254$$ 2.92893 0.183778
$$255$$ −0.414214 −0.0259391
$$256$$ 1.00000 0.0625000
$$257$$ −5.17157 −0.322594 −0.161297 0.986906i $$-0.551568\pi$$
−0.161297 + 0.986906i $$0.551568\pi$$
$$258$$ 7.00000 0.435801
$$259$$ 0 0
$$260$$ −1.00000 −0.0620174
$$261$$ 2.65685 0.164455
$$262$$ 17.2426 1.06525
$$263$$ 9.65685 0.595467 0.297734 0.954649i $$-0.403769\pi$$
0.297734 + 0.954649i $$0.403769\pi$$
$$264$$ −2.41421 −0.148585
$$265$$ 13.6569 0.838934
$$266$$ 0 0
$$267$$ −0.585786 −0.0358495
$$268$$ 1.41421 0.0863868
$$269$$ 16.7279 1.01992 0.509960 0.860198i $$-0.329660\pi$$
0.509960 + 0.860198i $$0.329660\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 14.2426 0.865179 0.432589 0.901591i $$-0.357600\pi$$
0.432589 + 0.901591i $$0.357600\pi$$
$$272$$ 0.414214 0.0251154
$$273$$ 0 0
$$274$$ 7.24264 0.437544
$$275$$ −9.65685 −0.582330
$$276$$ −8.65685 −0.521081
$$277$$ −16.5858 −0.996543 −0.498272 0.867021i $$-0.666032\pi$$
−0.498272 + 0.867021i $$0.666032\pi$$
$$278$$ −6.82843 −0.409542
$$279$$ 4.24264 0.254000
$$280$$ 0 0
$$281$$ 1.31371 0.0783693 0.0391846 0.999232i $$-0.487524\pi$$
0.0391846 + 0.999232i $$0.487524\pi$$
$$282$$ 7.65685 0.455959
$$283$$ 10.8284 0.643683 0.321842 0.946794i $$-0.395698\pi$$
0.321842 + 0.946794i $$0.395698\pi$$
$$284$$ 5.07107 0.300913
$$285$$ 3.82843 0.226776
$$286$$ −2.41421 −0.142755
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ −16.8284 −0.989907
$$290$$ 2.65685 0.156016
$$291$$ −7.31371 −0.428737
$$292$$ 11.8284 0.692206
$$293$$ −6.14214 −0.358827 −0.179414 0.983774i $$-0.557420\pi$$
−0.179414 + 0.983774i $$0.557420\pi$$
$$294$$ 0 0
$$295$$ −9.89949 −0.576371
$$296$$ 3.24264 0.188475
$$297$$ 2.41421 0.140087
$$298$$ 2.82843 0.163846
$$299$$ −8.65685 −0.500639
$$300$$ −4.00000 −0.230940
$$301$$ 0 0
$$302$$ 13.2426 0.762028
$$303$$ 1.07107 0.0615312
$$304$$ −3.82843 −0.219575
$$305$$ −5.58579 −0.319841
$$306$$ −0.414214 −0.0236790
$$307$$ 11.6569 0.665292 0.332646 0.943052i $$-0.392059\pi$$
0.332646 + 0.943052i $$0.392059\pi$$
$$308$$ 0 0
$$309$$ 8.07107 0.459147
$$310$$ 4.24264 0.240966
$$311$$ −21.7990 −1.23611 −0.618054 0.786136i $$-0.712079\pi$$
−0.618054 + 0.786136i $$0.712079\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 21.0711 1.19101 0.595504 0.803353i $$-0.296953\pi$$
0.595504 + 0.803353i $$0.296953\pi$$
$$314$$ −2.89949 −0.163628
$$315$$ 0 0
$$316$$ 10.2426 0.576194
$$317$$ 18.0416 1.01332 0.506659 0.862146i $$-0.330880\pi$$
0.506659 + 0.862146i $$0.330880\pi$$
$$318$$ 13.6569 0.765838
$$319$$ 6.41421 0.359127
$$320$$ −1.00000 −0.0559017
$$321$$ 4.58579 0.255954
$$322$$ 0 0
$$323$$ −1.58579 −0.0882355
$$324$$ 1.00000 0.0555556
$$325$$ −4.00000 −0.221880
$$326$$ 11.6569 0.645613
$$327$$ 0.414214 0.0229061
$$328$$ 6.82843 0.377037
$$329$$ 0 0
$$330$$ 2.41421 0.132898
$$331$$ −29.8995 −1.64342 −0.821712 0.569902i $$-0.806981\pi$$
−0.821712 + 0.569902i $$0.806981\pi$$
$$332$$ −11.6569 −0.639753
$$333$$ −3.24264 −0.177696
$$334$$ 13.8284 0.756658
$$335$$ −1.41421 −0.0772667
$$336$$ 0 0
$$337$$ −17.4853 −0.952484 −0.476242 0.879314i $$-0.658001\pi$$
−0.476242 + 0.879314i $$0.658001\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ −18.2426 −0.990804
$$340$$ −0.414214 −0.0224639
$$341$$ 10.2426 0.554670
$$342$$ 3.82843 0.207018
$$343$$ 0 0
$$344$$ 7.00000 0.377415
$$345$$ 8.65685 0.466069
$$346$$ 16.2426 0.873210
$$347$$ −22.5858 −1.21247 −0.606234 0.795286i $$-0.707321\pi$$
−0.606234 + 0.795286i $$0.707321\pi$$
$$348$$ 2.65685 0.142422
$$349$$ −14.7279 −0.788368 −0.394184 0.919032i $$-0.628973\pi$$
−0.394184 + 0.919032i $$0.628973\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ −2.41421 −0.128678
$$353$$ −19.4558 −1.03553 −0.517765 0.855523i $$-0.673236\pi$$
−0.517765 + 0.855523i $$0.673236\pi$$
$$354$$ −9.89949 −0.526152
$$355$$ −5.07107 −0.269144
$$356$$ −0.585786 −0.0310466
$$357$$ 0 0
$$358$$ 19.5563 1.03359
$$359$$ −29.4558 −1.55462 −0.777310 0.629118i $$-0.783416\pi$$
−0.777310 + 0.629118i $$0.783416\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −4.34315 −0.228587
$$362$$ 18.1421 0.953529
$$363$$ −5.17157 −0.271437
$$364$$ 0 0
$$365$$ −11.8284 −0.619128
$$366$$ −5.58579 −0.291974
$$367$$ −4.00000 −0.208798 −0.104399 0.994535i $$-0.533292\pi$$
−0.104399 + 0.994535i $$0.533292\pi$$
$$368$$ −8.65685 −0.451270
$$369$$ −6.82843 −0.355474
$$370$$ −3.24264 −0.168577
$$371$$ 0 0
$$372$$ 4.24264 0.219971
$$373$$ 4.24264 0.219676 0.109838 0.993950i $$-0.464967\pi$$
0.109838 + 0.993950i $$0.464967\pi$$
$$374$$ −1.00000 −0.0517088
$$375$$ 9.00000 0.464758
$$376$$ 7.65685 0.394872
$$377$$ 2.65685 0.136835
$$378$$ 0 0
$$379$$ −24.3848 −1.25256 −0.626281 0.779597i $$-0.715424\pi$$
−0.626281 + 0.779597i $$0.715424\pi$$
$$380$$ 3.82843 0.196394
$$381$$ −2.92893 −0.150054
$$382$$ −5.48528 −0.280651
$$383$$ 11.3431 0.579608 0.289804 0.957086i $$-0.406410\pi$$
0.289804 + 0.957086i $$0.406410\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −12.3848 −0.630369
$$387$$ −7.00000 −0.355830
$$388$$ −7.31371 −0.371297
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 1.00000 0.0506370
$$391$$ −3.58579 −0.181341
$$392$$ 0 0
$$393$$ −17.2426 −0.869776
$$394$$ −5.07107 −0.255477
$$395$$ −10.2426 −0.515363
$$396$$ 2.41421 0.121319
$$397$$ 6.68629 0.335575 0.167788 0.985823i $$-0.446338\pi$$
0.167788 + 0.985823i $$0.446338\pi$$
$$398$$ 10.0711 0.504817
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −21.4558 −1.07145 −0.535727 0.844391i $$-0.679962\pi$$
−0.535727 + 0.844391i $$0.679962\pi$$
$$402$$ −1.41421 −0.0705346
$$403$$ 4.24264 0.211341
$$404$$ 1.07107 0.0532876
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −7.82843 −0.388041
$$408$$ −0.414214 −0.0205066
$$409$$ 33.9706 1.67974 0.839868 0.542791i $$-0.182632\pi$$
0.839868 + 0.542791i $$0.182632\pi$$
$$410$$ −6.82843 −0.337232
$$411$$ −7.24264 −0.357253
$$412$$ 8.07107 0.397633
$$413$$ 0 0
$$414$$ 8.65685 0.425461
$$415$$ 11.6569 0.572212
$$416$$ −1.00000 −0.0490290
$$417$$ 6.82843 0.334390
$$418$$ 9.24264 0.452072
$$419$$ 10.5563 0.515711 0.257856 0.966183i $$-0.416984\pi$$
0.257856 + 0.966183i $$0.416984\pi$$
$$420$$ 0 0
$$421$$ −33.1127 −1.61381 −0.806907 0.590678i $$-0.798860\pi$$
−0.806907 + 0.590678i $$0.798860\pi$$
$$422$$ 16.3137 0.794139
$$423$$ −7.65685 −0.372289
$$424$$ 13.6569 0.663235
$$425$$ −1.65685 −0.0803692
$$426$$ −5.07107 −0.245694
$$427$$ 0 0
$$428$$ 4.58579 0.221662
$$429$$ 2.41421 0.116559
$$430$$ −7.00000 −0.337570
$$431$$ 15.1716 0.730789 0.365394 0.930853i $$-0.380934\pi$$
0.365394 + 0.930853i $$0.380934\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 26.6274 1.27963 0.639816 0.768528i $$-0.279011\pi$$
0.639816 + 0.768528i $$0.279011\pi$$
$$434$$ 0 0
$$435$$ −2.65685 −0.127386
$$436$$ 0.414214 0.0198372
$$437$$ 33.1421 1.58540
$$438$$ −11.8284 −0.565184
$$439$$ −38.2132 −1.82382 −0.911908 0.410394i $$-0.865391\pi$$
−0.911908 + 0.410394i $$0.865391\pi$$
$$440$$ 2.41421 0.115093
$$441$$ 0 0
$$442$$ −0.414214 −0.0197021
$$443$$ 0.828427 0.0393598 0.0196799 0.999806i $$-0.493735\pi$$
0.0196799 + 0.999806i $$0.493735\pi$$
$$444$$ −3.24264 −0.153889
$$445$$ 0.585786 0.0277689
$$446$$ −15.6569 −0.741374
$$447$$ −2.82843 −0.133780
$$448$$ 0 0
$$449$$ 18.5563 0.875728 0.437864 0.899041i $$-0.355735\pi$$
0.437864 + 0.899041i $$0.355735\pi$$
$$450$$ 4.00000 0.188562
$$451$$ −16.4853 −0.776262
$$452$$ −18.2426 −0.858062
$$453$$ −13.2426 −0.622194
$$454$$ −5.07107 −0.237997
$$455$$ 0 0
$$456$$ 3.82843 0.179283
$$457$$ −22.2426 −1.04047 −0.520233 0.854024i $$-0.674155\pi$$
−0.520233 + 0.854024i $$0.674155\pi$$
$$458$$ −23.8995 −1.11675
$$459$$ 0.414214 0.0193338
$$460$$ 8.65685 0.403628
$$461$$ 23.4853 1.09382 0.546909 0.837192i $$-0.315804\pi$$
0.546909 + 0.837192i $$0.315804\pi$$
$$462$$ 0 0
$$463$$ 31.8701 1.48113 0.740564 0.671986i $$-0.234559\pi$$
0.740564 + 0.671986i $$0.234559\pi$$
$$464$$ 2.65685 0.123341
$$465$$ −4.24264 −0.196748
$$466$$ 10.8284 0.501617
$$467$$ −9.10051 −0.421121 −0.210561 0.977581i $$-0.567529\pi$$
−0.210561 + 0.977581i $$0.567529\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 0 0
$$470$$ −7.65685 −0.353184
$$471$$ 2.89949 0.133602
$$472$$ −9.89949 −0.455661
$$473$$ −16.8995 −0.777040
$$474$$ −10.2426 −0.470460
$$475$$ 15.3137 0.702641
$$476$$ 0 0
$$477$$ −13.6569 −0.625304
$$478$$ −14.7279 −0.673639
$$479$$ 9.48528 0.433394 0.216697 0.976239i $$-0.430472\pi$$
0.216697 + 0.976239i $$0.430472\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ −3.24264 −0.147852
$$482$$ −12.3431 −0.562215
$$483$$ 0 0
$$484$$ −5.17157 −0.235071
$$485$$ 7.31371 0.332098
$$486$$ −1.00000 −0.0453609
$$487$$ 37.5980 1.70373 0.851864 0.523764i $$-0.175473\pi$$
0.851864 + 0.523764i $$0.175473\pi$$
$$488$$ −5.58579 −0.252857
$$489$$ −11.6569 −0.527141
$$490$$ 0 0
$$491$$ 34.7696 1.56913 0.784564 0.620048i $$-0.212887\pi$$
0.784564 + 0.620048i $$0.212887\pi$$
$$492$$ −6.82843 −0.307849
$$493$$ 1.10051 0.0495643
$$494$$ 3.82843 0.172249
$$495$$ −2.41421 −0.108511
$$496$$ 4.24264 0.190500
$$497$$ 0 0
$$498$$ 11.6569 0.522356
$$499$$ 22.7279 1.01744 0.508721 0.860932i $$-0.330119\pi$$
0.508721 + 0.860932i $$0.330119\pi$$
$$500$$ 9.00000 0.402492
$$501$$ −13.8284 −0.617809
$$502$$ 3.58579 0.160041
$$503$$ 11.0711 0.493635 0.246817 0.969062i $$-0.420615\pi$$
0.246817 + 0.969062i $$0.420615\pi$$
$$504$$ 0 0
$$505$$ −1.07107 −0.0476619
$$506$$ 20.8995 0.929096
$$507$$ 1.00000 0.0444116
$$508$$ −2.92893 −0.129950
$$509$$ 5.14214 0.227921 0.113961 0.993485i $$-0.463646\pi$$
0.113961 + 0.993485i $$0.463646\pi$$
$$510$$ 0.414214 0.0183417
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ −3.82843 −0.169029
$$514$$ 5.17157 0.228108
$$515$$ −8.07107 −0.355654
$$516$$ −7.00000 −0.308158
$$517$$ −18.4853 −0.812982
$$518$$ 0 0
$$519$$ −16.2426 −0.712973
$$520$$ 1.00000 0.0438529
$$521$$ 10.5563 0.462482 0.231241 0.972896i $$-0.425721\pi$$
0.231241 + 0.972896i $$0.425721\pi$$
$$522$$ −2.65685 −0.116287
$$523$$ 5.27208 0.230532 0.115266 0.993335i $$-0.463228\pi$$
0.115266 + 0.993335i $$0.463228\pi$$
$$524$$ −17.2426 −0.753248
$$525$$ 0 0
$$526$$ −9.65685 −0.421059
$$527$$ 1.75736 0.0765518
$$528$$ 2.41421 0.105065
$$529$$ 51.9411 2.25831
$$530$$ −13.6569 −0.593216
$$531$$ 9.89949 0.429601
$$532$$ 0 0
$$533$$ −6.82843 −0.295772
$$534$$ 0.585786 0.0253495
$$535$$ −4.58579 −0.198261
$$536$$ −1.41421 −0.0610847
$$537$$ −19.5563 −0.843919
$$538$$ −16.7279 −0.721192
$$539$$ 0 0
$$540$$ −1.00000 −0.0430331
$$541$$ −25.3848 −1.09138 −0.545688 0.837988i $$-0.683732\pi$$
−0.545688 + 0.837988i $$0.683732\pi$$
$$542$$ −14.2426 −0.611774
$$543$$ −18.1421 −0.778554
$$544$$ −0.414214 −0.0177593
$$545$$ −0.414214 −0.0177430
$$546$$ 0 0
$$547$$ −25.7990 −1.10309 −0.551543 0.834147i $$-0.685961\pi$$
−0.551543 + 0.834147i $$0.685961\pi$$
$$548$$ −7.24264 −0.309390
$$549$$ 5.58579 0.238396
$$550$$ 9.65685 0.411770
$$551$$ −10.1716 −0.433324
$$552$$ 8.65685 0.368460
$$553$$ 0 0
$$554$$ 16.5858 0.704663
$$555$$ 3.24264 0.137642
$$556$$ 6.82843 0.289590
$$557$$ −8.48528 −0.359533 −0.179766 0.983709i $$-0.557534\pi$$
−0.179766 + 0.983709i $$0.557534\pi$$
$$558$$ −4.24264 −0.179605
$$559$$ −7.00000 −0.296068
$$560$$ 0 0
$$561$$ 1.00000 0.0422200
$$562$$ −1.31371 −0.0554154
$$563$$ −8.41421 −0.354617 −0.177308 0.984155i $$-0.556739\pi$$
−0.177308 + 0.984155i $$0.556739\pi$$
$$564$$ −7.65685 −0.322412
$$565$$ 18.2426 0.767474
$$566$$ −10.8284 −0.455153
$$567$$ 0 0
$$568$$ −5.07107 −0.212777
$$569$$ −18.4853 −0.774943 −0.387472 0.921882i $$-0.626651\pi$$
−0.387472 + 0.921882i $$0.626651\pi$$
$$570$$ −3.82843 −0.160355
$$571$$ 30.2843 1.26736 0.633679 0.773596i $$-0.281544\pi$$
0.633679 + 0.773596i $$0.281544\pi$$
$$572$$ 2.41421 0.100943
$$573$$ 5.48528 0.229151
$$574$$ 0 0
$$575$$ 34.6274 1.44406
$$576$$ 1.00000 0.0416667
$$577$$ −21.1716 −0.881384 −0.440692 0.897658i $$-0.645267\pi$$
−0.440692 + 0.897658i $$0.645267\pi$$
$$578$$ 16.8284 0.699970
$$579$$ 12.3848 0.514694
$$580$$ −2.65685 −0.110320
$$581$$ 0 0
$$582$$ 7.31371 0.303163
$$583$$ −32.9706 −1.36550
$$584$$ −11.8284 −0.489464
$$585$$ −1.00000 −0.0413449
$$586$$ 6.14214 0.253729
$$587$$ 37.1127 1.53180 0.765902 0.642957i $$-0.222293\pi$$
0.765902 + 0.642957i $$0.222293\pi$$
$$588$$ 0 0
$$589$$ −16.2426 −0.669266
$$590$$ 9.89949 0.407556
$$591$$ 5.07107 0.208596
$$592$$ −3.24264 −0.133272
$$593$$ −7.41421 −0.304465 −0.152233 0.988345i $$-0.548646\pi$$
−0.152233 + 0.988345i $$0.548646\pi$$
$$594$$ −2.41421 −0.0990564
$$595$$ 0 0
$$596$$ −2.82843 −0.115857
$$597$$ −10.0711 −0.412181
$$598$$ 8.65685 0.354005
$$599$$ 20.5147 0.838209 0.419104 0.907938i $$-0.362344\pi$$
0.419104 + 0.907938i $$0.362344\pi$$
$$600$$ 4.00000 0.163299
$$601$$ 29.4558 1.20153 0.600764 0.799426i $$-0.294863\pi$$
0.600764 + 0.799426i $$0.294863\pi$$
$$602$$ 0 0
$$603$$ 1.41421 0.0575912
$$604$$ −13.2426 −0.538835
$$605$$ 5.17157 0.210254
$$606$$ −1.07107 −0.0435092
$$607$$ −35.1838 −1.42807 −0.714033 0.700113i $$-0.753133\pi$$
−0.714033 + 0.700113i $$0.753133\pi$$
$$608$$ 3.82843 0.155263
$$609$$ 0 0
$$610$$ 5.58579 0.226162
$$611$$ −7.65685 −0.309763
$$612$$ 0.414214 0.0167436
$$613$$ 6.07107 0.245208 0.122604 0.992456i $$-0.460875\pi$$
0.122604 + 0.992456i $$0.460875\pi$$
$$614$$ −11.6569 −0.470432
$$615$$ 6.82843 0.275349
$$616$$ 0 0
$$617$$ −25.5858 −1.03004 −0.515022 0.857177i $$-0.672216\pi$$
−0.515022 + 0.857177i $$0.672216\pi$$
$$618$$ −8.07107 −0.324666
$$619$$ −21.2843 −0.855487 −0.427744 0.903900i $$-0.640691\pi$$
−0.427744 + 0.903900i $$0.640691\pi$$
$$620$$ −4.24264 −0.170389
$$621$$ −8.65685 −0.347388
$$622$$ 21.7990 0.874060
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 11.0000 0.440000
$$626$$ −21.0711 −0.842169
$$627$$ −9.24264 −0.369116
$$628$$ 2.89949 0.115702
$$629$$ −1.34315 −0.0535547
$$630$$ 0 0
$$631$$ 0.615224 0.0244917 0.0122458 0.999925i $$-0.496102\pi$$
0.0122458 + 0.999925i $$0.496102\pi$$
$$632$$ −10.2426 −0.407430
$$633$$ −16.3137 −0.648412
$$634$$ −18.0416 −0.716525
$$635$$ 2.92893 0.116231
$$636$$ −13.6569 −0.541529
$$637$$ 0 0
$$638$$ −6.41421 −0.253941
$$639$$ 5.07107 0.200608
$$640$$ 1.00000 0.0395285
$$641$$ −42.5269 −1.67971 −0.839856 0.542809i $$-0.817361\pi$$
−0.839856 + 0.542809i $$0.817361\pi$$
$$642$$ −4.58579 −0.180987
$$643$$ −15.6863 −0.618607 −0.309303 0.950963i $$-0.600096\pi$$
−0.309303 + 0.950963i $$0.600096\pi$$
$$644$$ 0 0
$$645$$ 7.00000 0.275625
$$646$$ 1.58579 0.0623919
$$647$$ 11.0294 0.433612 0.216806 0.976215i $$-0.430436\pi$$
0.216806 + 0.976215i $$0.430436\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 23.8995 0.938137
$$650$$ 4.00000 0.156893
$$651$$ 0 0
$$652$$ −11.6569 −0.456518
$$653$$ −21.4853 −0.840784 −0.420392 0.907343i $$-0.638107\pi$$
−0.420392 + 0.907343i $$0.638107\pi$$
$$654$$ −0.414214 −0.0161970
$$655$$ 17.2426 0.673726
$$656$$ −6.82843 −0.266605
$$657$$ 11.8284 0.461471
$$658$$ 0 0
$$659$$ −11.4558 −0.446256 −0.223128 0.974789i $$-0.571627\pi$$
−0.223128 + 0.974789i $$0.571627\pi$$
$$660$$ −2.41421 −0.0939731
$$661$$ −34.9706 −1.36020 −0.680099 0.733121i $$-0.738063\pi$$
−0.680099 + 0.733121i $$0.738063\pi$$
$$662$$ 29.8995 1.16208
$$663$$ 0.414214 0.0160867
$$664$$ 11.6569 0.452374
$$665$$ 0 0
$$666$$ 3.24264 0.125650
$$667$$ −23.0000 −0.890564
$$668$$ −13.8284 −0.535038
$$669$$ 15.6569 0.605329
$$670$$ 1.41421 0.0546358
$$671$$ 13.4853 0.520594
$$672$$ 0 0
$$673$$ 38.1127 1.46914 0.734568 0.678535i $$-0.237385\pi$$
0.734568 + 0.678535i $$0.237385\pi$$
$$674$$ 17.4853 0.673508
$$675$$ −4.00000 −0.153960
$$676$$ 1.00000 0.0384615
$$677$$ −38.7696 −1.49003 −0.745017 0.667045i $$-0.767559\pi$$
−0.745017 + 0.667045i $$0.767559\pi$$
$$678$$ 18.2426 0.700604
$$679$$ 0 0
$$680$$ 0.414214 0.0158844
$$681$$ 5.07107 0.194324
$$682$$ −10.2426 −0.392211
$$683$$ 23.5858 0.902485 0.451243 0.892401i $$-0.350981\pi$$
0.451243 + 0.892401i $$0.350981\pi$$
$$684$$ −3.82843 −0.146384
$$685$$ 7.24264 0.276727
$$686$$ 0 0
$$687$$ 23.8995 0.911823
$$688$$ −7.00000 −0.266872
$$689$$ −13.6569 −0.520285
$$690$$ −8.65685 −0.329561
$$691$$ 23.1716 0.881488 0.440744 0.897633i $$-0.354715\pi$$
0.440744 + 0.897633i $$0.354715\pi$$
$$692$$ −16.2426 −0.617453
$$693$$ 0 0
$$694$$ 22.5858 0.857345
$$695$$ −6.82843 −0.259017
$$696$$ −2.65685 −0.100708
$$697$$ −2.82843 −0.107134
$$698$$ 14.7279 0.557460
$$699$$ −10.8284 −0.409569
$$700$$ 0 0
$$701$$ 34.8284 1.31545 0.657726 0.753257i $$-0.271519\pi$$
0.657726 + 0.753257i $$0.271519\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ 12.4142 0.468211
$$704$$ 2.41421 0.0909891
$$705$$ 7.65685 0.288374
$$706$$ 19.4558 0.732230
$$707$$ 0 0
$$708$$ 9.89949 0.372046
$$709$$ 40.8284 1.53334 0.766672 0.642039i $$-0.221911\pi$$
0.766672 + 0.642039i $$0.221911\pi$$
$$710$$ 5.07107 0.190314
$$711$$ 10.2426 0.384129
$$712$$ 0.585786 0.0219533
$$713$$ −36.7279 −1.37547
$$714$$ 0 0
$$715$$ −2.41421 −0.0902865
$$716$$ −19.5563 −0.730855
$$717$$ 14.7279 0.550024
$$718$$ 29.4558 1.09928
$$719$$ −10.5858 −0.394783 −0.197392 0.980325i $$-0.563247\pi$$
−0.197392 + 0.980325i $$0.563247\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ 4.34315 0.161635
$$723$$ 12.3431 0.459047
$$724$$ −18.1421 −0.674247
$$725$$ −10.6274 −0.394692
$$726$$ 5.17157 0.191935
$$727$$ −7.78680 −0.288796 −0.144398 0.989520i $$-0.546125\pi$$
−0.144398 + 0.989520i $$0.546125\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 11.8284 0.437790
$$731$$ −2.89949 −0.107242
$$732$$ 5.58579 0.206457
$$733$$ −15.2721 −0.564087 −0.282044 0.959402i $$-0.591012\pi$$
−0.282044 + 0.959402i $$0.591012\pi$$
$$734$$ 4.00000 0.147643
$$735$$ 0 0
$$736$$ 8.65685 0.319096
$$737$$ 3.41421 0.125764
$$738$$ 6.82843 0.251358
$$739$$ 34.2843 1.26117 0.630584 0.776121i $$-0.282816\pi$$
0.630584 + 0.776121i $$0.282816\pi$$
$$740$$ 3.24264 0.119202
$$741$$ −3.82843 −0.140641
$$742$$ 0 0
$$743$$ 52.6274 1.93071 0.965356 0.260935i $$-0.0840309\pi$$
0.965356 + 0.260935i $$0.0840309\pi$$
$$744$$ −4.24264 −0.155543
$$745$$ 2.82843 0.103626
$$746$$ −4.24264 −0.155334
$$747$$ −11.6569 −0.426502
$$748$$ 1.00000 0.0365636
$$749$$ 0 0
$$750$$ −9.00000 −0.328634
$$751$$ 16.4437 0.600037 0.300019 0.953933i $$-0.403007\pi$$
0.300019 + 0.953933i $$0.403007\pi$$
$$752$$ −7.65685 −0.279217
$$753$$ −3.58579 −0.130673
$$754$$ −2.65685 −0.0967569
$$755$$ 13.2426 0.481949
$$756$$ 0 0
$$757$$ −5.07107 −0.184311 −0.0921555 0.995745i $$-0.529376\pi$$
−0.0921555 + 0.995745i $$0.529376\pi$$
$$758$$ 24.3848 0.885695
$$759$$ −20.8995 −0.758604
$$760$$ −3.82843 −0.138872
$$761$$ −6.82843 −0.247530 −0.123765 0.992312i $$-0.539497\pi$$
−0.123765 + 0.992312i $$0.539497\pi$$
$$762$$ 2.92893 0.106104
$$763$$ 0 0
$$764$$ 5.48528 0.198451
$$765$$ −0.414214 −0.0149759
$$766$$ −11.3431 −0.409845
$$767$$ 9.89949 0.357450
$$768$$ 1.00000 0.0360844
$$769$$ 0.857864 0.0309354 0.0154677 0.999880i $$-0.495076\pi$$
0.0154677 + 0.999880i $$0.495076\pi$$
$$770$$ 0 0
$$771$$ −5.17157 −0.186250
$$772$$ 12.3848 0.445738
$$773$$ 8.02944 0.288799 0.144399 0.989519i $$-0.453875\pi$$
0.144399 + 0.989519i $$0.453875\pi$$
$$774$$ 7.00000 0.251610
$$775$$ −16.9706 −0.609601
$$776$$ 7.31371 0.262547
$$777$$ 0 0
$$778$$ 8.00000 0.286814
$$779$$ 26.1421 0.936639
$$780$$ −1.00000 −0.0358057
$$781$$ 12.2426 0.438076
$$782$$ 3.58579 0.128227
$$783$$ 2.65685 0.0949482
$$784$$ 0 0
$$785$$ −2.89949 −0.103487
$$786$$ 17.2426 0.615025
$$787$$ 10.1716 0.362577 0.181289 0.983430i $$-0.441973\pi$$
0.181289 + 0.983430i $$0.441973\pi$$
$$788$$ 5.07107 0.180649
$$789$$ 9.65685 0.343793
$$790$$ 10.2426 0.364417
$$791$$ 0 0
$$792$$ −2.41421 −0.0857853
$$793$$ 5.58579 0.198357
$$794$$ −6.68629 −0.237288
$$795$$ 13.6569 0.484359
$$796$$ −10.0711 −0.356960
$$797$$ −6.72792 −0.238315 −0.119158 0.992875i $$-0.538019\pi$$
−0.119158 + 0.992875i $$0.538019\pi$$
$$798$$ 0 0
$$799$$ −3.17157 −0.112202
$$800$$ 4.00000 0.141421
$$801$$ −0.585786 −0.0206977
$$802$$ 21.4558 0.757632
$$803$$ 28.5563 1.00773
$$804$$ 1.41421 0.0498755
$$805$$ 0 0
$$806$$ −4.24264 −0.149441
$$807$$ 16.7279 0.588851
$$808$$ −1.07107 −0.0376800
$$809$$ −13.5563 −0.476616 −0.238308 0.971190i $$-0.576593\pi$$
−0.238308 + 0.971190i $$0.576593\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −21.3431 −0.749459 −0.374730 0.927134i $$-0.622264\pi$$
−0.374730 + 0.927134i $$0.622264\pi$$
$$812$$ 0 0
$$813$$ 14.2426 0.499511
$$814$$ 7.82843 0.274386
$$815$$ 11.6569 0.408322
$$816$$ 0.414214 0.0145004
$$817$$ 26.7990 0.937578
$$818$$ −33.9706 −1.18775
$$819$$ 0 0
$$820$$ 6.82843 0.238459
$$821$$ 28.4853 0.994143 0.497072 0.867710i $$-0.334409\pi$$
0.497072 + 0.867710i $$0.334409\pi$$
$$822$$ 7.24264 0.252616
$$823$$ −20.5269 −0.715523 −0.357762 0.933813i $$-0.616460\pi$$
−0.357762 + 0.933813i $$0.616460\pi$$
$$824$$ −8.07107 −0.281169
$$825$$ −9.65685 −0.336209
$$826$$ 0 0
$$827$$ 27.5269 0.957205 0.478602 0.878032i $$-0.341144\pi$$
0.478602 + 0.878032i $$0.341144\pi$$
$$828$$ −8.65685 −0.300846
$$829$$ 51.5269 1.78960 0.894802 0.446464i $$-0.147317\pi$$
0.894802 + 0.446464i $$0.147317\pi$$
$$830$$ −11.6569 −0.404615
$$831$$ −16.5858 −0.575355
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ −6.82843 −0.236449
$$835$$ 13.8284 0.478552
$$836$$ −9.24264 −0.319663
$$837$$ 4.24264 0.146647
$$838$$ −10.5563 −0.364663
$$839$$ 56.6274 1.95500 0.977498 0.210946i $$-0.0676543\pi$$
0.977498 + 0.210946i $$0.0676543\pi$$
$$840$$ 0 0
$$841$$ −21.9411 −0.756591
$$842$$ 33.1127 1.14114
$$843$$ 1.31371 0.0452465
$$844$$ −16.3137 −0.561541
$$845$$ −1.00000 −0.0344010
$$846$$ 7.65685 0.263248
$$847$$ 0 0
$$848$$ −13.6569 −0.468978
$$849$$ 10.8284 0.371631
$$850$$ 1.65685 0.0568296
$$851$$ 28.0711 0.962264
$$852$$ 5.07107 0.173732
$$853$$ 33.6569 1.15239 0.576194 0.817313i $$-0.304537\pi$$
0.576194 + 0.817313i $$0.304537\pi$$
$$854$$ 0 0
$$855$$ 3.82843 0.130929
$$856$$ −4.58579 −0.156739
$$857$$ −44.8284 −1.53131 −0.765655 0.643252i $$-0.777585\pi$$
−0.765655 + 0.643252i $$0.777585\pi$$
$$858$$ −2.41421 −0.0824199
$$859$$ −48.9706 −1.67085 −0.835427 0.549601i $$-0.814780\pi$$
−0.835427 + 0.549601i $$0.814780\pi$$
$$860$$ 7.00000 0.238698
$$861$$ 0 0
$$862$$ −15.1716 −0.516746
$$863$$ −10.9706 −0.373442 −0.186721 0.982413i $$-0.559786\pi$$
−0.186721 + 0.982413i $$0.559786\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 16.2426 0.552266
$$866$$ −26.6274 −0.904836
$$867$$ −16.8284 −0.571523
$$868$$ 0 0
$$869$$ 24.7279 0.838837
$$870$$ 2.65685 0.0900758
$$871$$ 1.41421 0.0479188
$$872$$ −0.414214 −0.0140270
$$873$$ −7.31371 −0.247532
$$874$$ −33.1421 −1.12105
$$875$$ 0 0
$$876$$ 11.8284 0.399646
$$877$$ −8.48528 −0.286528 −0.143264 0.989685i $$-0.545760\pi$$
−0.143264 + 0.989685i $$0.545760\pi$$
$$878$$ 38.2132 1.28963
$$879$$ −6.14214 −0.207169
$$880$$ −2.41421 −0.0813831
$$881$$ 41.3848 1.39429 0.697144 0.716931i $$-0.254454\pi$$
0.697144 + 0.716931i $$0.254454\pi$$
$$882$$ 0 0
$$883$$ 22.0294 0.741350 0.370675 0.928763i $$-0.379126\pi$$
0.370675 + 0.928763i $$0.379126\pi$$
$$884$$ 0.414214 0.0139315
$$885$$ −9.89949 −0.332768
$$886$$ −0.828427 −0.0278316
$$887$$ −59.1127 −1.98481 −0.992405 0.123013i $$-0.960744\pi$$
−0.992405 + 0.123013i $$0.960744\pi$$
$$888$$ 3.24264 0.108816
$$889$$ 0 0
$$890$$ −0.585786 −0.0196356
$$891$$ 2.41421 0.0808792
$$892$$ 15.6569 0.524230
$$893$$ 29.3137 0.980946
$$894$$ 2.82843 0.0945968
$$895$$ 19.5563 0.653697
$$896$$ 0 0
$$897$$ −8.65685 −0.289044
$$898$$ −18.5563 −0.619233
$$899$$ 11.2721 0.375945
$$900$$ −4.00000 −0.133333
$$901$$ −5.65685 −0.188457
$$902$$ 16.4853 0.548900
$$903$$ 0 0
$$904$$ 18.2426 0.606741
$$905$$ 18.1421 0.603065
$$906$$ 13.2426 0.439957
$$907$$ 51.1716 1.69912 0.849562 0.527489i $$-0.176866\pi$$
0.849562 + 0.527489i $$0.176866\pi$$
$$908$$ 5.07107 0.168289
$$909$$ 1.07107 0.0355251
$$910$$ 0 0
$$911$$ −9.97056 −0.330339 −0.165170 0.986265i $$-0.552817\pi$$
−0.165170 + 0.986265i $$0.552817\pi$$
$$912$$ −3.82843 −0.126772
$$913$$ −28.1421 −0.931369
$$914$$ 22.2426 0.735721
$$915$$ −5.58579 −0.184660
$$916$$ 23.8995 0.789662
$$917$$ 0 0
$$918$$ −0.414214 −0.0136711
$$919$$ −12.1421 −0.400532 −0.200266 0.979742i $$-0.564181\pi$$
−0.200266 + 0.979742i $$0.564181\pi$$
$$920$$ −8.65685 −0.285408
$$921$$ 11.6569 0.384106
$$922$$ −23.4853 −0.773447
$$923$$ 5.07107 0.166916
$$924$$ 0 0
$$925$$ 12.9706 0.426469
$$926$$ −31.8701 −1.04732
$$927$$ 8.07107 0.265089
$$928$$ −2.65685 −0.0872155
$$929$$ 1.85786 0.0609546 0.0304773 0.999535i $$-0.490297\pi$$
0.0304773 + 0.999535i $$0.490297\pi$$
$$930$$ 4.24264 0.139122
$$931$$ 0 0
$$932$$ −10.8284 −0.354697
$$933$$ −21.7990 −0.713667
$$934$$ 9.10051 0.297778
$$935$$ −1.00000 −0.0327035
$$936$$ −1.00000 −0.0326860
$$937$$ −41.6985 −1.36223 −0.681115 0.732176i $$-0.738505\pi$$
−0.681115 + 0.732176i $$0.738505\pi$$
$$938$$ 0 0
$$939$$ 21.0711 0.687628
$$940$$ 7.65685 0.249739
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ −2.89949 −0.0944706
$$943$$ 59.1127 1.92497
$$944$$ 9.89949 0.322201
$$945$$ 0 0
$$946$$ 16.8995 0.549450
$$947$$ 21.2426 0.690293 0.345147 0.938549i $$-0.387829\pi$$
0.345147 + 0.938549i $$0.387829\pi$$
$$948$$ 10.2426 0.332666
$$949$$ 11.8284 0.383967
$$950$$ −15.3137 −0.496842
$$951$$ 18.0416 0.585040
$$952$$ 0 0
$$953$$ 36.2843 1.17536 0.587681 0.809092i $$-0.300041\pi$$
0.587681 + 0.809092i $$0.300041\pi$$
$$954$$ 13.6569 0.442157
$$955$$ −5.48528 −0.177500
$$956$$ 14.7279 0.476335
$$957$$ 6.41421 0.207342
$$958$$ −9.48528 −0.306456
$$959$$ 0 0
$$960$$ −1.00000 −0.0322749
$$961$$ −13.0000 −0.419355
$$962$$ 3.24264 0.104547
$$963$$ 4.58579 0.147775
$$964$$ 12.3431 0.397546
$$965$$ −12.3848 −0.398680
$$966$$ 0 0
$$967$$ −1.58579 −0.0509955 −0.0254977 0.999675i $$-0.508117\pi$$
−0.0254977 + 0.999675i $$0.508117\pi$$
$$968$$ 5.17157 0.166221
$$969$$ −1.58579 −0.0509428
$$970$$ −7.31371 −0.234829
$$971$$ 31.5147 1.01136 0.505678 0.862722i $$-0.331242\pi$$
0.505678 + 0.862722i $$0.331242\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −37.5980 −1.20472
$$975$$ −4.00000 −0.128103
$$976$$ 5.58579 0.178797
$$977$$ −47.8701 −1.53150 −0.765749 0.643139i $$-0.777632\pi$$
−0.765749 + 0.643139i $$0.777632\pi$$
$$978$$ 11.6569 0.372745
$$979$$ −1.41421 −0.0451985
$$980$$ 0 0
$$981$$ 0.414214 0.0132248
$$982$$ −34.7696 −1.10954
$$983$$ 13.6863 0.436525 0.218262 0.975890i $$-0.429961\pi$$
0.218262 + 0.975890i $$0.429961\pi$$
$$984$$ 6.82843 0.217682
$$985$$ −5.07107 −0.161578
$$986$$ −1.10051 −0.0350472
$$987$$ 0 0
$$988$$ −3.82843 −0.121798
$$989$$ 60.5980 1.92690
$$990$$ 2.41421 0.0767287
$$991$$ 37.3137 1.18531 0.592655 0.805457i $$-0.298080\pi$$
0.592655 + 0.805457i $$0.298080\pi$$
$$992$$ −4.24264 −0.134704
$$993$$ −29.8995 −0.948832
$$994$$ 0 0
$$995$$ 10.0711 0.319274
$$996$$ −11.6569 −0.369362
$$997$$ 58.2843 1.84588 0.922941 0.384942i $$-0.125779\pi$$
0.922941 + 0.384942i $$0.125779\pi$$
$$998$$ −22.7279 −0.719440
$$999$$ −3.24264 −0.102593
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 3822.2.a.bl.1.2 yes 2
7.6 odd 2 3822.2.a.bj.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3822.2.a.bj.1.2 2 7.6 odd 2
3822.2.a.bl.1.2 yes 2 1.1 even 1 trivial