# Properties

 Label 2738.2.a.i.1.2 Level $2738$ Weight $2$ Character 2738.1 Self dual yes Analytic conductor $21.863$ 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] = [2738,2,Mod(1,2738)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2738, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2738.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2738 = 2 \cdot 37^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2738.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: $$21.8630400734$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 2738.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} -1.73205 q^{5} +0.732051 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.46410 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} -1.73205 q^{5} +0.732051 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.46410 q^{9} -1.73205 q^{10} +1.26795 q^{11} +0.732051 q^{12} +3.46410 q^{13} -2.00000 q^{14} -1.26795 q^{15} +1.00000 q^{16} -4.26795 q^{17} -2.46410 q^{18} +4.73205 q^{19} -1.73205 q^{20} -1.46410 q^{21} +1.26795 q^{22} -1.26795 q^{23} +0.732051 q^{24} -2.00000 q^{25} +3.46410 q^{26} -4.00000 q^{27} -2.00000 q^{28} -8.66025 q^{29} -1.26795 q^{30} -4.73205 q^{31} +1.00000 q^{32} +0.928203 q^{33} -4.26795 q^{34} +3.46410 q^{35} -2.46410 q^{36} +4.73205 q^{38} +2.53590 q^{39} -1.73205 q^{40} +3.92820 q^{41} -1.46410 q^{42} -12.9282 q^{43} +1.26795 q^{44} +4.26795 q^{45} -1.26795 q^{46} +1.26795 q^{47} +0.732051 q^{48} -3.00000 q^{49} -2.00000 q^{50} -3.12436 q^{51} +3.46410 q^{52} +9.46410 q^{53} -4.00000 q^{54} -2.19615 q^{55} -2.00000 q^{56} +3.46410 q^{57} -8.66025 q^{58} -9.46410 q^{59} -1.26795 q^{60} +1.73205 q^{61} -4.73205 q^{62} +4.92820 q^{63} +1.00000 q^{64} -6.00000 q^{65} +0.928203 q^{66} -0.196152 q^{67} -4.26795 q^{68} -0.928203 q^{69} +3.46410 q^{70} -3.46410 q^{71} -2.46410 q^{72} -4.00000 q^{73} -1.46410 q^{75} +4.73205 q^{76} -2.53590 q^{77} +2.53590 q^{78} -16.7321 q^{79} -1.73205 q^{80} +4.46410 q^{81} +3.92820 q^{82} +11.6603 q^{83} -1.46410 q^{84} +7.39230 q^{85} -12.9282 q^{86} -6.33975 q^{87} +1.26795 q^{88} -17.1962 q^{89} +4.26795 q^{90} -6.92820 q^{91} -1.26795 q^{92} -3.46410 q^{93} +1.26795 q^{94} -8.19615 q^{95} +0.732051 q^{96} -4.26795 q^{97} -3.00000 q^{98} -3.12436 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} + 6 q^{11} - 2 q^{12} - 4 q^{14} - 6 q^{15} + 2 q^{16} - 12 q^{17} + 2 q^{18} + 6 q^{19} + 4 q^{21} + 6 q^{22} - 6 q^{23} - 2 q^{24} - 4 q^{25} - 8 q^{27} - 4 q^{28} - 6 q^{30} - 6 q^{31} + 2 q^{32} - 12 q^{33} - 12 q^{34} + 2 q^{36} + 6 q^{38} + 12 q^{39} - 6 q^{41} + 4 q^{42} - 12 q^{43} + 6 q^{44} + 12 q^{45} - 6 q^{46} + 6 q^{47} - 2 q^{48} - 6 q^{49} - 4 q^{50} + 18 q^{51} + 12 q^{53} - 8 q^{54} + 6 q^{55} - 4 q^{56} - 12 q^{59} - 6 q^{60} - 6 q^{62} - 4 q^{63} + 2 q^{64} - 12 q^{65} - 12 q^{66} + 10 q^{67} - 12 q^{68} + 12 q^{69} + 2 q^{72} - 8 q^{73} + 4 q^{75} + 6 q^{76} - 12 q^{77} + 12 q^{78} - 30 q^{79} + 2 q^{81} - 6 q^{82} + 6 q^{83} + 4 q^{84} - 6 q^{85} - 12 q^{86} - 30 q^{87} + 6 q^{88} - 24 q^{89} + 12 q^{90} - 6 q^{92} + 6 q^{94} - 6 q^{95} - 2 q^{96} - 12 q^{97} - 6 q^{98} + 18 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9 + 6 * q^11 - 2 * q^12 - 4 * q^14 - 6 * q^15 + 2 * q^16 - 12 * q^17 + 2 * q^18 + 6 * q^19 + 4 * q^21 + 6 * q^22 - 6 * q^23 - 2 * q^24 - 4 * q^25 - 8 * q^27 - 4 * q^28 - 6 * q^30 - 6 * q^31 + 2 * q^32 - 12 * q^33 - 12 * q^34 + 2 * q^36 + 6 * q^38 + 12 * q^39 - 6 * q^41 + 4 * q^42 - 12 * q^43 + 6 * q^44 + 12 * q^45 - 6 * q^46 + 6 * q^47 - 2 * q^48 - 6 * q^49 - 4 * q^50 + 18 * q^51 + 12 * q^53 - 8 * q^54 + 6 * q^55 - 4 * q^56 - 12 * q^59 - 6 * q^60 - 6 * q^62 - 4 * q^63 + 2 * q^64 - 12 * q^65 - 12 * q^66 + 10 * q^67 - 12 * q^68 + 12 * q^69 + 2 * q^72 - 8 * q^73 + 4 * q^75 + 6 * q^76 - 12 * q^77 + 12 * q^78 - 30 * q^79 + 2 * q^81 - 6 * q^82 + 6 * q^83 + 4 * q^84 - 6 * q^85 - 12 * q^86 - 30 * q^87 + 6 * q^88 - 24 * q^89 + 12 * q^90 - 6 * q^92 + 6 * q^94 - 6 * q^95 - 2 * q^96 - 12 * q^97 - 6 * q^98 + 18 * 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
$$3$$ 0.732051 0.422650 0.211325 0.977416i $$-0.432222\pi$$
0.211325 + 0.977416i $$0.432222\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.73205 −0.774597 −0.387298 0.921954i $$-0.626592\pi$$
−0.387298 + 0.921954i $$0.626592\pi$$
$$6$$ 0.732051 0.298858
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −2.46410 −0.821367
$$10$$ −1.73205 −0.547723
$$11$$ 1.26795 0.382301 0.191151 0.981561i $$-0.438778\pi$$
0.191151 + 0.981561i $$0.438778\pi$$
$$12$$ 0.732051 0.211325
$$13$$ 3.46410 0.960769 0.480384 0.877058i $$-0.340497\pi$$
0.480384 + 0.877058i $$0.340497\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ −1.26795 −0.327383
$$16$$ 1.00000 0.250000
$$17$$ −4.26795 −1.03513 −0.517565 0.855644i $$-0.673161\pi$$
−0.517565 + 0.855644i $$0.673161\pi$$
$$18$$ −2.46410 −0.580794
$$19$$ 4.73205 1.08561 0.542803 0.839860i $$-0.317363\pi$$
0.542803 + 0.839860i $$0.317363\pi$$
$$20$$ −1.73205 −0.387298
$$21$$ −1.46410 −0.319493
$$22$$ 1.26795 0.270328
$$23$$ −1.26795 −0.264386 −0.132193 0.991224i $$-0.542202\pi$$
−0.132193 + 0.991224i $$0.542202\pi$$
$$24$$ 0.732051 0.149429
$$25$$ −2.00000 −0.400000
$$26$$ 3.46410 0.679366
$$27$$ −4.00000 −0.769800
$$28$$ −2.00000 −0.377964
$$29$$ −8.66025 −1.60817 −0.804084 0.594515i $$-0.797344\pi$$
−0.804084 + 0.594515i $$0.797344\pi$$
$$30$$ −1.26795 −0.231495
$$31$$ −4.73205 −0.849901 −0.424951 0.905216i $$-0.639709\pi$$
−0.424951 + 0.905216i $$0.639709\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0.928203 0.161579
$$34$$ −4.26795 −0.731947
$$35$$ 3.46410 0.585540
$$36$$ −2.46410 −0.410684
$$37$$ 0 0
$$38$$ 4.73205 0.767640
$$39$$ 2.53590 0.406069
$$40$$ −1.73205 −0.273861
$$41$$ 3.92820 0.613482 0.306741 0.951793i $$-0.400761\pi$$
0.306741 + 0.951793i $$0.400761\pi$$
$$42$$ −1.46410 −0.225916
$$43$$ −12.9282 −1.97153 −0.985766 0.168122i $$-0.946230\pi$$
−0.985766 + 0.168122i $$0.946230\pi$$
$$44$$ 1.26795 0.191151
$$45$$ 4.26795 0.636228
$$46$$ −1.26795 −0.186949
$$47$$ 1.26795 0.184949 0.0924747 0.995715i $$-0.470522\pi$$
0.0924747 + 0.995715i $$0.470522\pi$$
$$48$$ 0.732051 0.105662
$$49$$ −3.00000 −0.428571
$$50$$ −2.00000 −0.282843
$$51$$ −3.12436 −0.437497
$$52$$ 3.46410 0.480384
$$53$$ 9.46410 1.29999 0.649997 0.759937i $$-0.274770\pi$$
0.649997 + 0.759937i $$0.274770\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ −2.19615 −0.296129
$$56$$ −2.00000 −0.267261
$$57$$ 3.46410 0.458831
$$58$$ −8.66025 −1.13715
$$59$$ −9.46410 −1.23212 −0.616061 0.787699i $$-0.711272\pi$$
−0.616061 + 0.787699i $$0.711272\pi$$
$$60$$ −1.26795 −0.163692
$$61$$ 1.73205 0.221766 0.110883 0.993833i $$-0.464632\pi$$
0.110883 + 0.993833i $$0.464632\pi$$
$$62$$ −4.73205 −0.600971
$$63$$ 4.92820 0.620895
$$64$$ 1.00000 0.125000
$$65$$ −6.00000 −0.744208
$$66$$ 0.928203 0.114254
$$67$$ −0.196152 −0.0239638 −0.0119819 0.999928i $$-0.503814\pi$$
−0.0119819 + 0.999928i $$0.503814\pi$$
$$68$$ −4.26795 −0.517565
$$69$$ −0.928203 −0.111743
$$70$$ 3.46410 0.414039
$$71$$ −3.46410 −0.411113 −0.205557 0.978645i $$-0.565900\pi$$
−0.205557 + 0.978645i $$0.565900\pi$$
$$72$$ −2.46410 −0.290397
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 0 0
$$75$$ −1.46410 −0.169060
$$76$$ 4.73205 0.542803
$$77$$ −2.53590 −0.288992
$$78$$ 2.53590 0.287134
$$79$$ −16.7321 −1.88250 −0.941251 0.337707i $$-0.890349\pi$$
−0.941251 + 0.337707i $$0.890349\pi$$
$$80$$ −1.73205 −0.193649
$$81$$ 4.46410 0.496011
$$82$$ 3.92820 0.433797
$$83$$ 11.6603 1.27988 0.639940 0.768425i $$-0.278959\pi$$
0.639940 + 0.768425i $$0.278959\pi$$
$$84$$ −1.46410 −0.159747
$$85$$ 7.39230 0.801808
$$86$$ −12.9282 −1.39408
$$87$$ −6.33975 −0.679692
$$88$$ 1.26795 0.135164
$$89$$ −17.1962 −1.82279 −0.911394 0.411534i $$-0.864993\pi$$
−0.911394 + 0.411534i $$0.864993\pi$$
$$90$$ 4.26795 0.449881
$$91$$ −6.92820 −0.726273
$$92$$ −1.26795 −0.132193
$$93$$ −3.46410 −0.359211
$$94$$ 1.26795 0.130779
$$95$$ −8.19615 −0.840907
$$96$$ 0.732051 0.0747146
$$97$$ −4.26795 −0.433345 −0.216672 0.976244i $$-0.569520\pi$$
−0.216672 + 0.976244i $$0.569520\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ −3.12436 −0.314010
$$100$$ −2.00000 −0.200000
$$101$$ −5.53590 −0.550842 −0.275421 0.961324i $$-0.588817\pi$$
−0.275421 + 0.961324i $$0.588817\pi$$
$$102$$ −3.12436 −0.309357
$$103$$ −15.4641 −1.52372 −0.761862 0.647740i $$-0.775714\pi$$
−0.761862 + 0.647740i $$0.775714\pi$$
$$104$$ 3.46410 0.339683
$$105$$ 2.53590 0.247478
$$106$$ 9.46410 0.919235
$$107$$ 10.3923 1.00466 0.502331 0.864675i $$-0.332476\pi$$
0.502331 + 0.864675i $$0.332476\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ 14.6603 1.40420 0.702099 0.712080i $$-0.252246\pi$$
0.702099 + 0.712080i $$0.252246\pi$$
$$110$$ −2.19615 −0.209395
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ 3.46410 0.325875 0.162938 0.986636i $$-0.447903\pi$$
0.162938 + 0.986636i $$0.447903\pi$$
$$114$$ 3.46410 0.324443
$$115$$ 2.19615 0.204792
$$116$$ −8.66025 −0.804084
$$117$$ −8.53590 −0.789144
$$118$$ −9.46410 −0.871241
$$119$$ 8.53590 0.782485
$$120$$ −1.26795 −0.115747
$$121$$ −9.39230 −0.853846
$$122$$ 1.73205 0.156813
$$123$$ 2.87564 0.259288
$$124$$ −4.73205 −0.424951
$$125$$ 12.1244 1.08444
$$126$$ 4.92820 0.439039
$$127$$ −18.1962 −1.61465 −0.807324 0.590109i $$-0.799085\pi$$
−0.807324 + 0.590109i $$0.799085\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −9.46410 −0.833268
$$130$$ −6.00000 −0.526235
$$131$$ 20.1962 1.76455 0.882273 0.470738i $$-0.156012\pi$$
0.882273 + 0.470738i $$0.156012\pi$$
$$132$$ 0.928203 0.0807897
$$133$$ −9.46410 −0.820642
$$134$$ −0.196152 −0.0169450
$$135$$ 6.92820 0.596285
$$136$$ −4.26795 −0.365974
$$137$$ 19.3923 1.65680 0.828398 0.560140i $$-0.189253\pi$$
0.828398 + 0.560140i $$0.189253\pi$$
$$138$$ −0.928203 −0.0790139
$$139$$ −10.5885 −0.898101 −0.449051 0.893506i $$-0.648238\pi$$
−0.449051 + 0.893506i $$0.648238\pi$$
$$140$$ 3.46410 0.292770
$$141$$ 0.928203 0.0781688
$$142$$ −3.46410 −0.290701
$$143$$ 4.39230 0.367303
$$144$$ −2.46410 −0.205342
$$145$$ 15.0000 1.24568
$$146$$ −4.00000 −0.331042
$$147$$ −2.19615 −0.181136
$$148$$ 0 0
$$149$$ −2.07180 −0.169728 −0.0848641 0.996393i $$-0.527046\pi$$
−0.0848641 + 0.996393i $$0.527046\pi$$
$$150$$ −1.46410 −0.119543
$$151$$ 6.19615 0.504236 0.252118 0.967697i $$-0.418873\pi$$
0.252118 + 0.967697i $$0.418873\pi$$
$$152$$ 4.73205 0.383820
$$153$$ 10.5167 0.850222
$$154$$ −2.53590 −0.204349
$$155$$ 8.19615 0.658331
$$156$$ 2.53590 0.203034
$$157$$ 1.00000 0.0798087 0.0399043 0.999204i $$-0.487295\pi$$
0.0399043 + 0.999204i $$0.487295\pi$$
$$158$$ −16.7321 −1.33113
$$159$$ 6.92820 0.549442
$$160$$ −1.73205 −0.136931
$$161$$ 2.53590 0.199857
$$162$$ 4.46410 0.350733
$$163$$ −0.928203 −0.0727025 −0.0363512 0.999339i $$-0.511574\pi$$
−0.0363512 + 0.999339i $$0.511574\pi$$
$$164$$ 3.92820 0.306741
$$165$$ −1.60770 −0.125159
$$166$$ 11.6603 0.905011
$$167$$ 13.8564 1.07224 0.536120 0.844141i $$-0.319889\pi$$
0.536120 + 0.844141i $$0.319889\pi$$
$$168$$ −1.46410 −0.112958
$$169$$ −1.00000 −0.0769231
$$170$$ 7.39230 0.566964
$$171$$ −11.6603 −0.891682
$$172$$ −12.9282 −0.985766
$$173$$ −12.4641 −0.947628 −0.473814 0.880625i $$-0.657123\pi$$
−0.473814 + 0.880625i $$0.657123\pi$$
$$174$$ −6.33975 −0.480615
$$175$$ 4.00000 0.302372
$$176$$ 1.26795 0.0955753
$$177$$ −6.92820 −0.520756
$$178$$ −17.1962 −1.28891
$$179$$ −2.53590 −0.189542 −0.0947710 0.995499i $$-0.530212\pi$$
−0.0947710 + 0.995499i $$0.530212\pi$$
$$180$$ 4.26795 0.318114
$$181$$ 11.3923 0.846783 0.423392 0.905947i $$-0.360839\pi$$
0.423392 + 0.905947i $$0.360839\pi$$
$$182$$ −6.92820 −0.513553
$$183$$ 1.26795 0.0937295
$$184$$ −1.26795 −0.0934745
$$185$$ 0 0
$$186$$ −3.46410 −0.254000
$$187$$ −5.41154 −0.395731
$$188$$ 1.26795 0.0924747
$$189$$ 8.00000 0.581914
$$190$$ −8.19615 −0.594611
$$191$$ 8.19615 0.593053 0.296526 0.955025i $$-0.404172\pi$$
0.296526 + 0.955025i $$0.404172\pi$$
$$192$$ 0.732051 0.0528312
$$193$$ −0.803848 −0.0578622 −0.0289311 0.999581i $$-0.509210\pi$$
−0.0289311 + 0.999581i $$0.509210\pi$$
$$194$$ −4.26795 −0.306421
$$195$$ −4.39230 −0.314539
$$196$$ −3.00000 −0.214286
$$197$$ −0.464102 −0.0330659 −0.0165329 0.999863i $$-0.505263\pi$$
−0.0165329 + 0.999863i $$0.505263\pi$$
$$198$$ −3.12436 −0.222038
$$199$$ 10.3923 0.736691 0.368345 0.929689i $$-0.379924\pi$$
0.368345 + 0.929689i $$0.379924\pi$$
$$200$$ −2.00000 −0.141421
$$201$$ −0.143594 −0.0101283
$$202$$ −5.53590 −0.389504
$$203$$ 17.3205 1.21566
$$204$$ −3.12436 −0.218749
$$205$$ −6.80385 −0.475201
$$206$$ −15.4641 −1.07744
$$207$$ 3.12436 0.217158
$$208$$ 3.46410 0.240192
$$209$$ 6.00000 0.415029
$$210$$ 2.53590 0.174994
$$211$$ 18.3923 1.26618 0.633089 0.774079i $$-0.281787\pi$$
0.633089 + 0.774079i $$0.281787\pi$$
$$212$$ 9.46410 0.649997
$$213$$ −2.53590 −0.173757
$$214$$ 10.3923 0.710403
$$215$$ 22.3923 1.52714
$$216$$ −4.00000 −0.272166
$$217$$ 9.46410 0.642465
$$218$$ 14.6603 0.992918
$$219$$ −2.92820 −0.197870
$$220$$ −2.19615 −0.148065
$$221$$ −14.7846 −0.994520
$$222$$ 0 0
$$223$$ 5.80385 0.388654 0.194327 0.980937i $$-0.437748\pi$$
0.194327 + 0.980937i $$0.437748\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 4.92820 0.328547
$$226$$ 3.46410 0.230429
$$227$$ 4.73205 0.314077 0.157039 0.987592i $$-0.449805\pi$$
0.157039 + 0.987592i $$0.449805\pi$$
$$228$$ 3.46410 0.229416
$$229$$ 6.60770 0.436649 0.218324 0.975876i $$-0.429941\pi$$
0.218324 + 0.975876i $$0.429941\pi$$
$$230$$ 2.19615 0.144810
$$231$$ −1.85641 −0.122143
$$232$$ −8.66025 −0.568574
$$233$$ −15.0000 −0.982683 −0.491341 0.870967i $$-0.663493\pi$$
−0.491341 + 0.870967i $$0.663493\pi$$
$$234$$ −8.53590 −0.558009
$$235$$ −2.19615 −0.143261
$$236$$ −9.46410 −0.616061
$$237$$ −12.2487 −0.795639
$$238$$ 8.53590 0.553300
$$239$$ −17.3205 −1.12037 −0.560185 0.828367i $$-0.689270\pi$$
−0.560185 + 0.828367i $$0.689270\pi$$
$$240$$ −1.26795 −0.0818458
$$241$$ 8.53590 0.549846 0.274923 0.961466i $$-0.411348\pi$$
0.274923 + 0.961466i $$0.411348\pi$$
$$242$$ −9.39230 −0.603760
$$243$$ 15.2679 0.979439
$$244$$ 1.73205 0.110883
$$245$$ 5.19615 0.331970
$$246$$ 2.87564 0.183344
$$247$$ 16.3923 1.04302
$$248$$ −4.73205 −0.300486
$$249$$ 8.53590 0.540941
$$250$$ 12.1244 0.766812
$$251$$ 17.3205 1.09326 0.546630 0.837374i $$-0.315910\pi$$
0.546630 + 0.837374i $$0.315910\pi$$
$$252$$ 4.92820 0.310448
$$253$$ −1.60770 −0.101075
$$254$$ −18.1962 −1.14173
$$255$$ 5.41154 0.338884
$$256$$ 1.00000 0.0625000
$$257$$ −17.1962 −1.07267 −0.536333 0.844006i $$-0.680191\pi$$
−0.536333 + 0.844006i $$0.680191\pi$$
$$258$$ −9.46410 −0.589209
$$259$$ 0 0
$$260$$ −6.00000 −0.372104
$$261$$ 21.3397 1.32090
$$262$$ 20.1962 1.24772
$$263$$ −14.5359 −0.896322 −0.448161 0.893953i $$-0.647921\pi$$
−0.448161 + 0.893953i $$0.647921\pi$$
$$264$$ 0.928203 0.0571270
$$265$$ −16.3923 −1.00697
$$266$$ −9.46410 −0.580281
$$267$$ −12.5885 −0.770401
$$268$$ −0.196152 −0.0119819
$$269$$ 16.3923 0.999456 0.499728 0.866182i $$-0.333433\pi$$
0.499728 + 0.866182i $$0.333433\pi$$
$$270$$ 6.92820 0.421637
$$271$$ 28.5885 1.73663 0.868313 0.496017i $$-0.165205\pi$$
0.868313 + 0.496017i $$0.165205\pi$$
$$272$$ −4.26795 −0.258782
$$273$$ −5.07180 −0.306959
$$274$$ 19.3923 1.17153
$$275$$ −2.53590 −0.152920
$$276$$ −0.928203 −0.0558713
$$277$$ −0.803848 −0.0482985 −0.0241493 0.999708i $$-0.507688\pi$$
−0.0241493 + 0.999708i $$0.507688\pi$$
$$278$$ −10.5885 −0.635053
$$279$$ 11.6603 0.698081
$$280$$ 3.46410 0.207020
$$281$$ −3.58846 −0.214069 −0.107035 0.994255i $$-0.534136\pi$$
−0.107035 + 0.994255i $$0.534136\pi$$
$$282$$ 0.928203 0.0552737
$$283$$ −19.8564 −1.18034 −0.590170 0.807279i $$-0.700939\pi$$
−0.590170 + 0.807279i $$0.700939\pi$$
$$284$$ −3.46410 −0.205557
$$285$$ −6.00000 −0.355409
$$286$$ 4.39230 0.259722
$$287$$ −7.85641 −0.463749
$$288$$ −2.46410 −0.145199
$$289$$ 1.21539 0.0714935
$$290$$ 15.0000 0.880830
$$291$$ −3.12436 −0.183153
$$292$$ −4.00000 −0.234082
$$293$$ 19.3923 1.13291 0.566455 0.824092i $$-0.308314\pi$$
0.566455 + 0.824092i $$0.308314\pi$$
$$294$$ −2.19615 −0.128082
$$295$$ 16.3923 0.954397
$$296$$ 0 0
$$297$$ −5.07180 −0.294295
$$298$$ −2.07180 −0.120016
$$299$$ −4.39230 −0.254014
$$300$$ −1.46410 −0.0845299
$$301$$ 25.8564 1.49034
$$302$$ 6.19615 0.356549
$$303$$ −4.05256 −0.232813
$$304$$ 4.73205 0.271402
$$305$$ −3.00000 −0.171780
$$306$$ 10.5167 0.601197
$$307$$ 22.0000 1.25561 0.627803 0.778372i $$-0.283954\pi$$
0.627803 + 0.778372i $$0.283954\pi$$
$$308$$ −2.53590 −0.144496
$$309$$ −11.3205 −0.644001
$$310$$ 8.19615 0.465510
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 2.53590 0.143567
$$313$$ 0.124356 0.00702900 0.00351450 0.999994i $$-0.498881\pi$$
0.00351450 + 0.999994i $$0.498881\pi$$
$$314$$ 1.00000 0.0564333
$$315$$ −8.53590 −0.480943
$$316$$ −16.7321 −0.941251
$$317$$ 13.3923 0.752187 0.376093 0.926582i $$-0.377267\pi$$
0.376093 + 0.926582i $$0.377267\pi$$
$$318$$ 6.92820 0.388514
$$319$$ −10.9808 −0.614805
$$320$$ −1.73205 −0.0968246
$$321$$ 7.60770 0.424620
$$322$$ 2.53590 0.141320
$$323$$ −20.1962 −1.12374
$$324$$ 4.46410 0.248006
$$325$$ −6.92820 −0.384308
$$326$$ −0.928203 −0.0514084
$$327$$ 10.7321 0.593484
$$328$$ 3.92820 0.216899
$$329$$ −2.53590 −0.139809
$$330$$ −1.60770 −0.0885007
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 11.6603 0.639940
$$333$$ 0 0
$$334$$ 13.8564 0.758189
$$335$$ 0.339746 0.0185623
$$336$$ −1.46410 −0.0798733
$$337$$ −30.1769 −1.64384 −0.821921 0.569602i $$-0.807097\pi$$
−0.821921 + 0.569602i $$0.807097\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 2.53590 0.137731
$$340$$ 7.39230 0.400904
$$341$$ −6.00000 −0.324918
$$342$$ −11.6603 −0.630514
$$343$$ 20.0000 1.07990
$$344$$ −12.9282 −0.697042
$$345$$ 1.60770 0.0865554
$$346$$ −12.4641 −0.670074
$$347$$ −15.1244 −0.811918 −0.405959 0.913891i $$-0.633062\pi$$
−0.405959 + 0.913891i $$0.633062\pi$$
$$348$$ −6.33975 −0.339846
$$349$$ −9.39230 −0.502759 −0.251379 0.967889i $$-0.580884\pi$$
−0.251379 + 0.967889i $$0.580884\pi$$
$$350$$ 4.00000 0.213809
$$351$$ −13.8564 −0.739600
$$352$$ 1.26795 0.0675819
$$353$$ −16.2679 −0.865856 −0.432928 0.901429i $$-0.642519\pi$$
−0.432928 + 0.901429i $$0.642519\pi$$
$$354$$ −6.92820 −0.368230
$$355$$ 6.00000 0.318447
$$356$$ −17.1962 −0.911394
$$357$$ 6.24871 0.330717
$$358$$ −2.53590 −0.134026
$$359$$ 0.679492 0.0358622 0.0179311 0.999839i $$-0.494292\pi$$
0.0179311 + 0.999839i $$0.494292\pi$$
$$360$$ 4.26795 0.224941
$$361$$ 3.39230 0.178542
$$362$$ 11.3923 0.598766
$$363$$ −6.87564 −0.360878
$$364$$ −6.92820 −0.363137
$$365$$ 6.92820 0.362639
$$366$$ 1.26795 0.0662768
$$367$$ −8.39230 −0.438075 −0.219037 0.975716i $$-0.570292\pi$$
−0.219037 + 0.975716i $$0.570292\pi$$
$$368$$ −1.26795 −0.0660964
$$369$$ −9.67949 −0.503894
$$370$$ 0 0
$$371$$ −18.9282 −0.982703
$$372$$ −3.46410 −0.179605
$$373$$ −7.00000 −0.362446 −0.181223 0.983442i $$-0.558006\pi$$
−0.181223 + 0.983442i $$0.558006\pi$$
$$374$$ −5.41154 −0.279824
$$375$$ 8.87564 0.458336
$$376$$ 1.26795 0.0653895
$$377$$ −30.0000 −1.54508
$$378$$ 8.00000 0.411476
$$379$$ 6.78461 0.348502 0.174251 0.984701i $$-0.444250\pi$$
0.174251 + 0.984701i $$0.444250\pi$$
$$380$$ −8.19615 −0.420454
$$381$$ −13.3205 −0.682430
$$382$$ 8.19615 0.419352
$$383$$ 3.46410 0.177007 0.0885037 0.996076i $$-0.471792\pi$$
0.0885037 + 0.996076i $$0.471792\pi$$
$$384$$ 0.732051 0.0373573
$$385$$ 4.39230 0.223853
$$386$$ −0.803848 −0.0409148
$$387$$ 31.8564 1.61935
$$388$$ −4.26795 −0.216672
$$389$$ 25.9808 1.31728 0.658638 0.752460i $$-0.271133\pi$$
0.658638 + 0.752460i $$0.271133\pi$$
$$390$$ −4.39230 −0.222413
$$391$$ 5.41154 0.273673
$$392$$ −3.00000 −0.151523
$$393$$ 14.7846 0.745785
$$394$$ −0.464102 −0.0233811
$$395$$ 28.9808 1.45818
$$396$$ −3.12436 −0.157005
$$397$$ 23.0000 1.15434 0.577168 0.816625i $$-0.304158\pi$$
0.577168 + 0.816625i $$0.304158\pi$$
$$398$$ 10.3923 0.520919
$$399$$ −6.92820 −0.346844
$$400$$ −2.00000 −0.100000
$$401$$ 10.3923 0.518967 0.259483 0.965748i $$-0.416448\pi$$
0.259483 + 0.965748i $$0.416448\pi$$
$$402$$ −0.143594 −0.00716179
$$403$$ −16.3923 −0.816559
$$404$$ −5.53590 −0.275421
$$405$$ −7.73205 −0.384209
$$406$$ 17.3205 0.859602
$$407$$ 0 0
$$408$$ −3.12436 −0.154679
$$409$$ −34.5167 −1.70674 −0.853370 0.521307i $$-0.825445\pi$$
−0.853370 + 0.521307i $$0.825445\pi$$
$$410$$ −6.80385 −0.336018
$$411$$ 14.1962 0.700245
$$412$$ −15.4641 −0.761862
$$413$$ 18.9282 0.931396
$$414$$ 3.12436 0.153554
$$415$$ −20.1962 −0.991390
$$416$$ 3.46410 0.169842
$$417$$ −7.75129 −0.379582
$$418$$ 6.00000 0.293470
$$419$$ −31.8564 −1.55629 −0.778144 0.628086i $$-0.783838\pi$$
−0.778144 + 0.628086i $$0.783838\pi$$
$$420$$ 2.53590 0.123739
$$421$$ 12.1244 0.590905 0.295452 0.955357i $$-0.404530\pi$$
0.295452 + 0.955357i $$0.404530\pi$$
$$422$$ 18.3923 0.895323
$$423$$ −3.12436 −0.151911
$$424$$ 9.46410 0.459617
$$425$$ 8.53590 0.414052
$$426$$ −2.53590 −0.122865
$$427$$ −3.46410 −0.167640
$$428$$ 10.3923 0.502331
$$429$$ 3.21539 0.155241
$$430$$ 22.3923 1.07985
$$431$$ 0.339746 0.0163650 0.00818249 0.999967i $$-0.497395\pi$$
0.00818249 + 0.999967i $$0.497395\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −25.7846 −1.23913 −0.619565 0.784946i $$-0.712691\pi$$
−0.619565 + 0.784946i $$0.712691\pi$$
$$434$$ 9.46410 0.454291
$$435$$ 10.9808 0.526487
$$436$$ 14.6603 0.702099
$$437$$ −6.00000 −0.287019
$$438$$ −2.92820 −0.139915
$$439$$ −30.2487 −1.44369 −0.721846 0.692054i $$-0.756706\pi$$
−0.721846 + 0.692054i $$0.756706\pi$$
$$440$$ −2.19615 −0.104697
$$441$$ 7.39230 0.352015
$$442$$ −14.7846 −0.703232
$$443$$ 21.4641 1.01979 0.509895 0.860237i $$-0.329684\pi$$
0.509895 + 0.860237i $$0.329684\pi$$
$$444$$ 0 0
$$445$$ 29.7846 1.41193
$$446$$ 5.80385 0.274820
$$447$$ −1.51666 −0.0717356
$$448$$ −2.00000 −0.0944911
$$449$$ 27.4641 1.29611 0.648056 0.761593i $$-0.275582\pi$$
0.648056 + 0.761593i $$0.275582\pi$$
$$450$$ 4.92820 0.232318
$$451$$ 4.98076 0.234535
$$452$$ 3.46410 0.162938
$$453$$ 4.53590 0.213115
$$454$$ 4.73205 0.222086
$$455$$ 12.0000 0.562569
$$456$$ 3.46410 0.162221
$$457$$ 13.9808 0.653992 0.326996 0.945026i $$-0.393964\pi$$
0.326996 + 0.945026i $$0.393964\pi$$
$$458$$ 6.60770 0.308757
$$459$$ 17.0718 0.796843
$$460$$ 2.19615 0.102396
$$461$$ 32.5359 1.51535 0.757674 0.652633i $$-0.226336\pi$$
0.757674 + 0.652633i $$0.226336\pi$$
$$462$$ −1.85641 −0.0863678
$$463$$ −30.0000 −1.39422 −0.697109 0.716965i $$-0.745531\pi$$
−0.697109 + 0.716965i $$0.745531\pi$$
$$464$$ −8.66025 −0.402042
$$465$$ 6.00000 0.278243
$$466$$ −15.0000 −0.694862
$$467$$ 9.46410 0.437946 0.218973 0.975731i $$-0.429729\pi$$
0.218973 + 0.975731i $$0.429729\pi$$
$$468$$ −8.53590 −0.394572
$$469$$ 0.392305 0.0181150
$$470$$ −2.19615 −0.101301
$$471$$ 0.732051 0.0337311
$$472$$ −9.46410 −0.435621
$$473$$ −16.3923 −0.753719
$$474$$ −12.2487 −0.562602
$$475$$ −9.46410 −0.434243
$$476$$ 8.53590 0.391242
$$477$$ −23.3205 −1.06777
$$478$$ −17.3205 −0.792222
$$479$$ −2.53590 −0.115868 −0.0579341 0.998320i $$-0.518451\pi$$
−0.0579341 + 0.998320i $$0.518451\pi$$
$$480$$ −1.26795 −0.0578737
$$481$$ 0 0
$$482$$ 8.53590 0.388800
$$483$$ 1.85641 0.0844694
$$484$$ −9.39230 −0.426923
$$485$$ 7.39230 0.335667
$$486$$ 15.2679 0.692568
$$487$$ 22.0526 0.999297 0.499648 0.866228i $$-0.333463\pi$$
0.499648 + 0.866228i $$0.333463\pi$$
$$488$$ 1.73205 0.0784063
$$489$$ −0.679492 −0.0307277
$$490$$ 5.19615 0.234738
$$491$$ 15.8038 0.713218 0.356609 0.934254i $$-0.383933\pi$$
0.356609 + 0.934254i $$0.383933\pi$$
$$492$$ 2.87564 0.129644
$$493$$ 36.9615 1.66466
$$494$$ 16.3923 0.737525
$$495$$ 5.41154 0.243231
$$496$$ −4.73205 −0.212475
$$497$$ 6.92820 0.310772
$$498$$ 8.53590 0.382503
$$499$$ 9.80385 0.438880 0.219440 0.975626i $$-0.429577\pi$$
0.219440 + 0.975626i $$0.429577\pi$$
$$500$$ 12.1244 0.542218
$$501$$ 10.1436 0.453182
$$502$$ 17.3205 0.773052
$$503$$ 9.80385 0.437132 0.218566 0.975822i $$-0.429862\pi$$
0.218566 + 0.975822i $$0.429862\pi$$
$$504$$ 4.92820 0.219520
$$505$$ 9.58846 0.426681
$$506$$ −1.60770 −0.0714708
$$507$$ −0.732051 −0.0325115
$$508$$ −18.1962 −0.807324
$$509$$ −0.464102 −0.0205709 −0.0102855 0.999947i $$-0.503274\pi$$
−0.0102855 + 0.999947i $$0.503274\pi$$
$$510$$ 5.41154 0.239627
$$511$$ 8.00000 0.353899
$$512$$ 1.00000 0.0441942
$$513$$ −18.9282 −0.835701
$$514$$ −17.1962 −0.758490
$$515$$ 26.7846 1.18027
$$516$$ −9.46410 −0.416634
$$517$$ 1.60770 0.0707064
$$518$$ 0 0
$$519$$ −9.12436 −0.400515
$$520$$ −6.00000 −0.263117
$$521$$ −18.9282 −0.829260 −0.414630 0.909990i $$-0.636089\pi$$
−0.414630 + 0.909990i $$0.636089\pi$$
$$522$$ 21.3397 0.934015
$$523$$ −38.4449 −1.68108 −0.840538 0.541752i $$-0.817761\pi$$
−0.840538 + 0.541752i $$0.817761\pi$$
$$524$$ 20.1962 0.882273
$$525$$ 2.92820 0.127797
$$526$$ −14.5359 −0.633795
$$527$$ 20.1962 0.879758
$$528$$ 0.928203 0.0403949
$$529$$ −21.3923 −0.930100
$$530$$ −16.3923 −0.712036
$$531$$ 23.3205 1.01202
$$532$$ −9.46410 −0.410321
$$533$$ 13.6077 0.589415
$$534$$ −12.5885 −0.544756
$$535$$ −18.0000 −0.778208
$$536$$ −0.196152 −0.00847249
$$537$$ −1.85641 −0.0801099
$$538$$ 16.3923 0.706722
$$539$$ −3.80385 −0.163843
$$540$$ 6.92820 0.298142
$$541$$ 19.0526 0.819133 0.409567 0.912280i $$-0.365680\pi$$
0.409567 + 0.912280i $$0.365680\pi$$
$$542$$ 28.5885 1.22798
$$543$$ 8.33975 0.357893
$$544$$ −4.26795 −0.182987
$$545$$ −25.3923 −1.08769
$$546$$ −5.07180 −0.217053
$$547$$ 24.3397 1.04069 0.520346 0.853955i $$-0.325803\pi$$
0.520346 + 0.853955i $$0.325803\pi$$
$$548$$ 19.3923 0.828398
$$549$$ −4.26795 −0.182152
$$550$$ −2.53590 −0.108131
$$551$$ −40.9808 −1.74584
$$552$$ −0.928203 −0.0395070
$$553$$ 33.4641 1.42304
$$554$$ −0.803848 −0.0341522
$$555$$ 0 0
$$556$$ −10.5885 −0.449051
$$557$$ −35.4449 −1.50185 −0.750924 0.660389i $$-0.770391\pi$$
−0.750924 + 0.660389i $$0.770391\pi$$
$$558$$ 11.6603 0.493618
$$559$$ −44.7846 −1.89419
$$560$$ 3.46410 0.146385
$$561$$ −3.96152 −0.167256
$$562$$ −3.58846 −0.151370
$$563$$ −12.5885 −0.530540 −0.265270 0.964174i $$-0.585461\pi$$
−0.265270 + 0.964174i $$0.585461\pi$$
$$564$$ 0.928203 0.0390844
$$565$$ −6.00000 −0.252422
$$566$$ −19.8564 −0.834627
$$567$$ −8.92820 −0.374949
$$568$$ −3.46410 −0.145350
$$569$$ −30.1244 −1.26288 −0.631439 0.775425i $$-0.717536\pi$$
−0.631439 + 0.775425i $$0.717536\pi$$
$$570$$ −6.00000 −0.251312
$$571$$ −35.3731 −1.48032 −0.740158 0.672433i $$-0.765249\pi$$
−0.740158 + 0.672433i $$0.765249\pi$$
$$572$$ 4.39230 0.183651
$$573$$ 6.00000 0.250654
$$574$$ −7.85641 −0.327920
$$575$$ 2.53590 0.105754
$$576$$ −2.46410 −0.102671
$$577$$ −12.9282 −0.538208 −0.269104 0.963111i $$-0.586728\pi$$
−0.269104 + 0.963111i $$0.586728\pi$$
$$578$$ 1.21539 0.0505536
$$579$$ −0.588457 −0.0244554
$$580$$ 15.0000 0.622841
$$581$$ −23.3205 −0.967498
$$582$$ −3.12436 −0.129509
$$583$$ 12.0000 0.496989
$$584$$ −4.00000 −0.165521
$$585$$ 14.7846 0.611268
$$586$$ 19.3923 0.801089
$$587$$ −8.87564 −0.366337 −0.183169 0.983082i $$-0.558635\pi$$
−0.183169 + 0.983082i $$0.558635\pi$$
$$588$$ −2.19615 −0.0905678
$$589$$ −22.3923 −0.922659
$$590$$ 16.3923 0.674861
$$591$$ −0.339746 −0.0139753
$$592$$ 0 0
$$593$$ −25.6410 −1.05295 −0.526475 0.850191i $$-0.676487\pi$$
−0.526475 + 0.850191i $$0.676487\pi$$
$$594$$ −5.07180 −0.208098
$$595$$ −14.7846 −0.606110
$$596$$ −2.07180 −0.0848641
$$597$$ 7.60770 0.311362
$$598$$ −4.39230 −0.179615
$$599$$ 30.3397 1.23965 0.619824 0.784741i $$-0.287204\pi$$
0.619824 + 0.784741i $$0.287204\pi$$
$$600$$ −1.46410 −0.0597717
$$601$$ −33.3923 −1.36210 −0.681050 0.732237i $$-0.738476\pi$$
−0.681050 + 0.732237i $$0.738476\pi$$
$$602$$ 25.8564 1.05383
$$603$$ 0.483340 0.0196831
$$604$$ 6.19615 0.252118
$$605$$ 16.2679 0.661386
$$606$$ −4.05256 −0.164624
$$607$$ 41.9090 1.70103 0.850516 0.525949i $$-0.176290\pi$$
0.850516 + 0.525949i $$0.176290\pi$$
$$608$$ 4.73205 0.191910
$$609$$ 12.6795 0.513799
$$610$$ −3.00000 −0.121466
$$611$$ 4.39230 0.177694
$$612$$ 10.5167 0.425111
$$613$$ 33.7846 1.36455 0.682274 0.731097i $$-0.260991\pi$$
0.682274 + 0.731097i $$0.260991\pi$$
$$614$$ 22.0000 0.887848
$$615$$ −4.98076 −0.200844
$$616$$ −2.53590 −0.102174
$$617$$ −47.3205 −1.90505 −0.952526 0.304457i $$-0.901525\pi$$
−0.952526 + 0.304457i $$0.901525\pi$$
$$618$$ −11.3205 −0.455378
$$619$$ −9.60770 −0.386166 −0.193083 0.981182i $$-0.561849\pi$$
−0.193083 + 0.981182i $$0.561849\pi$$
$$620$$ 8.19615 0.329165
$$621$$ 5.07180 0.203524
$$622$$ −18.0000 −0.721734
$$623$$ 34.3923 1.37790
$$624$$ 2.53590 0.101517
$$625$$ −11.0000 −0.440000
$$626$$ 0.124356 0.00497025
$$627$$ 4.39230 0.175412
$$628$$ 1.00000 0.0399043
$$629$$ 0 0
$$630$$ −8.53590 −0.340078
$$631$$ 16.9808 0.675993 0.337997 0.941147i $$-0.390251\pi$$
0.337997 + 0.941147i $$0.390251\pi$$
$$632$$ −16.7321 −0.665565
$$633$$ 13.4641 0.535150
$$634$$ 13.3923 0.531876
$$635$$ 31.5167 1.25070
$$636$$ 6.92820 0.274721
$$637$$ −10.3923 −0.411758
$$638$$ −10.9808 −0.434733
$$639$$ 8.53590 0.337675
$$640$$ −1.73205 −0.0684653
$$641$$ 28.8564 1.13976 0.569880 0.821728i $$-0.306990\pi$$
0.569880 + 0.821728i $$0.306990\pi$$
$$642$$ 7.60770 0.300252
$$643$$ −16.7321 −0.659848 −0.329924 0.944008i $$-0.607023\pi$$
−0.329924 + 0.944008i $$0.607023\pi$$
$$644$$ 2.53590 0.0999284
$$645$$ 16.3923 0.645446
$$646$$ −20.1962 −0.794607
$$647$$ −36.9282 −1.45180 −0.725899 0.687802i $$-0.758576\pi$$
−0.725899 + 0.687802i $$0.758576\pi$$
$$648$$ 4.46410 0.175366
$$649$$ −12.0000 −0.471041
$$650$$ −6.92820 −0.271746
$$651$$ 6.92820 0.271538
$$652$$ −0.928203 −0.0363512
$$653$$ −7.98076 −0.312311 −0.156156 0.987732i $$-0.549910\pi$$
−0.156156 + 0.987732i $$0.549910\pi$$
$$654$$ 10.7321 0.419656
$$655$$ −34.9808 −1.36681
$$656$$ 3.92820 0.153371
$$657$$ 9.85641 0.384535
$$658$$ −2.53590 −0.0988596
$$659$$ 2.53590 0.0987846 0.0493923 0.998779i $$-0.484272\pi$$
0.0493923 + 0.998779i $$0.484272\pi$$
$$660$$ −1.60770 −0.0625794
$$661$$ 1.73205 0.0673690 0.0336845 0.999433i $$-0.489276\pi$$
0.0336845 + 0.999433i $$0.489276\pi$$
$$662$$ 0 0
$$663$$ −10.8231 −0.420334
$$664$$ 11.6603 0.452506
$$665$$ 16.3923 0.635666
$$666$$ 0 0
$$667$$ 10.9808 0.425177
$$668$$ 13.8564 0.536120
$$669$$ 4.24871 0.164265
$$670$$ 0.339746 0.0131255
$$671$$ 2.19615 0.0847815
$$672$$ −1.46410 −0.0564789
$$673$$ 16.0000 0.616755 0.308377 0.951264i $$-0.400214\pi$$
0.308377 + 0.951264i $$0.400214\pi$$
$$674$$ −30.1769 −1.16237
$$675$$ 8.00000 0.307920
$$676$$ −1.00000 −0.0384615
$$677$$ −20.0718 −0.771422 −0.385711 0.922620i $$-0.626044\pi$$
−0.385711 + 0.922620i $$0.626044\pi$$
$$678$$ 2.53590 0.0973906
$$679$$ 8.53590 0.327578
$$680$$ 7.39230 0.283482
$$681$$ 3.46410 0.132745
$$682$$ −6.00000 −0.229752
$$683$$ 27.4641 1.05088 0.525442 0.850829i $$-0.323900\pi$$
0.525442 + 0.850829i $$0.323900\pi$$
$$684$$ −11.6603 −0.445841
$$685$$ −33.5885 −1.28335
$$686$$ 20.0000 0.763604
$$687$$ 4.83717 0.184549
$$688$$ −12.9282 −0.492883
$$689$$ 32.7846 1.24899
$$690$$ 1.60770 0.0612039
$$691$$ 24.9808 0.950313 0.475156 0.879901i $$-0.342391\pi$$
0.475156 + 0.879901i $$0.342391\pi$$
$$692$$ −12.4641 −0.473814
$$693$$ 6.24871 0.237369
$$694$$ −15.1244 −0.574113
$$695$$ 18.3397 0.695666
$$696$$ −6.33975 −0.240307
$$697$$ −16.7654 −0.635034
$$698$$ −9.39230 −0.355504
$$699$$ −10.9808 −0.415331
$$700$$ 4.00000 0.151186
$$701$$ −5.07180 −0.191559 −0.0957796 0.995403i $$-0.530534\pi$$
−0.0957796 + 0.995403i $$0.530534\pi$$
$$702$$ −13.8564 −0.522976
$$703$$ 0 0
$$704$$ 1.26795 0.0477876
$$705$$ −1.60770 −0.0605493
$$706$$ −16.2679 −0.612252
$$707$$ 11.0718 0.416398
$$708$$ −6.92820 −0.260378
$$709$$ −45.7128 −1.71678 −0.858390 0.512997i $$-0.828535\pi$$
−0.858390 + 0.512997i $$0.828535\pi$$
$$710$$ 6.00000 0.225176
$$711$$ 41.2295 1.54623
$$712$$ −17.1962 −0.644453
$$713$$ 6.00000 0.224702
$$714$$ 6.24871 0.233852
$$715$$ −7.60770 −0.284512
$$716$$ −2.53590 −0.0947710
$$717$$ −12.6795 −0.473524
$$718$$ 0.679492 0.0253584
$$719$$ 40.7321 1.51905 0.759525 0.650479i $$-0.225432\pi$$
0.759525 + 0.650479i $$0.225432\pi$$
$$720$$ 4.26795 0.159057
$$721$$ 30.9282 1.15183
$$722$$ 3.39230 0.126249
$$723$$ 6.24871 0.232392
$$724$$ 11.3923 0.423392
$$725$$ 17.3205 0.643268
$$726$$ −6.87564 −0.255179
$$727$$ 15.7128 0.582756 0.291378 0.956608i $$-0.405886\pi$$
0.291378 + 0.956608i $$0.405886\pi$$
$$728$$ −6.92820 −0.256776
$$729$$ −2.21539 −0.0820515
$$730$$ 6.92820 0.256424
$$731$$ 55.1769 2.04079
$$732$$ 1.26795 0.0468648
$$733$$ −49.5692 −1.83088 −0.915440 0.402453i $$-0.868158\pi$$
−0.915440 + 0.402453i $$0.868158\pi$$
$$734$$ −8.39230 −0.309766
$$735$$ 3.80385 0.140307
$$736$$ −1.26795 −0.0467372
$$737$$ −0.248711 −0.00916140
$$738$$ −9.67949 −0.356307
$$739$$ −1.41154 −0.0519244 −0.0259622 0.999663i $$-0.508265\pi$$
−0.0259622 + 0.999663i $$0.508265\pi$$
$$740$$ 0 0
$$741$$ 12.0000 0.440831
$$742$$ −18.9282 −0.694876
$$743$$ 43.5167 1.59647 0.798236 0.602345i $$-0.205767\pi$$
0.798236 + 0.602345i $$0.205767\pi$$
$$744$$ −3.46410 −0.127000
$$745$$ 3.58846 0.131471
$$746$$ −7.00000 −0.256288
$$747$$ −28.7321 −1.05125
$$748$$ −5.41154 −0.197866
$$749$$ −20.7846 −0.759453
$$750$$ 8.87564 0.324093
$$751$$ −50.3923 −1.83884 −0.919421 0.393276i $$-0.871342\pi$$
−0.919421 + 0.393276i $$0.871342\pi$$
$$752$$ 1.26795 0.0462373
$$753$$ 12.6795 0.462066
$$754$$ −30.0000 −1.09254
$$755$$ −10.7321 −0.390579
$$756$$ 8.00000 0.290957
$$757$$ −6.80385 −0.247290 −0.123645 0.992327i $$-0.539458\pi$$
−0.123645 + 0.992327i $$0.539458\pi$$
$$758$$ 6.78461 0.246428
$$759$$ −1.17691 −0.0427193
$$760$$ −8.19615 −0.297306
$$761$$ −32.3205 −1.17162 −0.585809 0.810449i $$-0.699223\pi$$
−0.585809 + 0.810449i $$0.699223\pi$$
$$762$$ −13.3205 −0.482551
$$763$$ −29.3205 −1.06147
$$764$$ 8.19615 0.296526
$$765$$ −18.2154 −0.658579
$$766$$ 3.46410 0.125163
$$767$$ −32.7846 −1.18378
$$768$$ 0.732051 0.0264156
$$769$$ 34.3923 1.24022 0.620109 0.784516i $$-0.287088\pi$$
0.620109 + 0.784516i $$0.287088\pi$$
$$770$$ 4.39230 0.158288
$$771$$ −12.5885 −0.453362
$$772$$ −0.803848 −0.0289311
$$773$$ 20.0718 0.721932 0.360966 0.932579i $$-0.382447\pi$$
0.360966 + 0.932579i $$0.382447\pi$$
$$774$$ 31.8564 1.14505
$$775$$ 9.46410 0.339961
$$776$$ −4.26795 −0.153210
$$777$$ 0 0
$$778$$ 25.9808 0.931455
$$779$$ 18.5885 0.666001
$$780$$ −4.39230 −0.157270
$$781$$ −4.39230 −0.157169
$$782$$ 5.41154 0.193516
$$783$$ 34.6410 1.23797
$$784$$ −3.00000 −0.107143
$$785$$ −1.73205 −0.0618195
$$786$$ 14.7846 0.527350
$$787$$ 21.1769 0.754875 0.377438 0.926035i $$-0.376805\pi$$
0.377438 + 0.926035i $$0.376805\pi$$
$$788$$ −0.464102 −0.0165329
$$789$$ −10.6410 −0.378830
$$790$$ 28.9808 1.03109
$$791$$ −6.92820 −0.246339
$$792$$ −3.12436 −0.111019
$$793$$ 6.00000 0.213066
$$794$$ 23.0000 0.816239
$$795$$ −12.0000 −0.425596
$$796$$ 10.3923 0.368345
$$797$$ −32.7846 −1.16129 −0.580645 0.814157i $$-0.697200\pi$$
−0.580645 + 0.814157i $$0.697200\pi$$
$$798$$ −6.92820 −0.245256
$$799$$ −5.41154 −0.191447
$$800$$ −2.00000 −0.0707107
$$801$$ 42.3731 1.49718
$$802$$ 10.3923 0.366965
$$803$$ −5.07180 −0.178980
$$804$$ −0.143594 −0.00506415
$$805$$ −4.39230 −0.154808
$$806$$ −16.3923 −0.577394
$$807$$ 12.0000 0.422420
$$808$$ −5.53590 −0.194752
$$809$$ 39.7128 1.39623 0.698114 0.715987i $$-0.254023\pi$$
0.698114 + 0.715987i $$0.254023\pi$$
$$810$$ −7.73205 −0.271677
$$811$$ −0.392305 −0.0137757 −0.00688784 0.999976i $$-0.502192\pi$$
−0.00688784 + 0.999976i $$0.502192\pi$$
$$812$$ 17.3205 0.607831
$$813$$ 20.9282 0.733984
$$814$$ 0 0
$$815$$ 1.60770 0.0563151
$$816$$ −3.12436 −0.109374
$$817$$ −61.1769 −2.14031
$$818$$ −34.5167 −1.20685
$$819$$ 17.0718 0.596537
$$820$$ −6.80385 −0.237601
$$821$$ 9.46410 0.330299 0.165150 0.986269i $$-0.447189\pi$$
0.165150 + 0.986269i $$0.447189\pi$$
$$822$$ 14.1962 0.495148
$$823$$ 34.1962 1.19200 0.596001 0.802983i $$-0.296755\pi$$
0.596001 + 0.802983i $$0.296755\pi$$
$$824$$ −15.4641 −0.538718
$$825$$ −1.85641 −0.0646318
$$826$$ 18.9282 0.658596
$$827$$ 25.8564 0.899115 0.449558 0.893251i $$-0.351582\pi$$
0.449558 + 0.893251i $$0.351582\pi$$
$$828$$ 3.12436 0.108579
$$829$$ −10.3923 −0.360940 −0.180470 0.983581i $$-0.557762\pi$$
−0.180470 + 0.983581i $$0.557762\pi$$
$$830$$ −20.1962 −0.701019
$$831$$ −0.588457 −0.0204134
$$832$$ 3.46410 0.120096
$$833$$ 12.8038 0.443627
$$834$$ −7.75129 −0.268405
$$835$$ −24.0000 −0.830554
$$836$$ 6.00000 0.207514
$$837$$ 18.9282 0.654254
$$838$$ −31.8564 −1.10046
$$839$$ 1.01924 0.0351880 0.0175940 0.999845i $$-0.494399\pi$$
0.0175940 + 0.999845i $$0.494399\pi$$
$$840$$ 2.53590 0.0874968
$$841$$ 46.0000 1.58621
$$842$$ 12.1244 0.417833
$$843$$ −2.62693 −0.0904764
$$844$$ 18.3923 0.633089
$$845$$ 1.73205 0.0595844
$$846$$ −3.12436 −0.107418
$$847$$ 18.7846 0.645447
$$848$$ 9.46410 0.324999
$$849$$ −14.5359 −0.498871
$$850$$ 8.53590 0.292779
$$851$$ 0 0
$$852$$ −2.53590 −0.0868784
$$853$$ −13.7321 −0.470176 −0.235088 0.971974i $$-0.575538\pi$$
−0.235088 + 0.971974i $$0.575538\pi$$
$$854$$ −3.46410 −0.118539
$$855$$ 20.1962 0.690694
$$856$$ 10.3923 0.355202
$$857$$ 14.6603 0.500785 0.250392 0.968144i $$-0.419440\pi$$
0.250392 + 0.968144i $$0.419440\pi$$
$$858$$ 3.21539 0.109772
$$859$$ −30.0000 −1.02359 −0.511793 0.859109i $$-0.671019\pi$$
−0.511793 + 0.859109i $$0.671019\pi$$
$$860$$ 22.3923 0.763571
$$861$$ −5.75129 −0.196003
$$862$$ 0.339746 0.0115718
$$863$$ −16.3923 −0.558001 −0.279000 0.960291i $$-0.590003\pi$$
−0.279000 + 0.960291i $$0.590003\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 21.5885 0.734030
$$866$$ −25.7846 −0.876197
$$867$$ 0.889727 0.0302167
$$868$$ 9.46410 0.321233
$$869$$ −21.2154 −0.719683
$$870$$ 10.9808 0.372283
$$871$$ −0.679492 −0.0230237
$$872$$ 14.6603 0.496459
$$873$$ 10.5167 0.355935
$$874$$ −6.00000 −0.202953
$$875$$ −24.2487 −0.819756
$$876$$ −2.92820 −0.0989348
$$877$$ −24.1769 −0.816396 −0.408198 0.912893i $$-0.633843\pi$$
−0.408198 + 0.912893i $$0.633843\pi$$
$$878$$ −30.2487 −1.02084
$$879$$ 14.1962 0.478824
$$880$$ −2.19615 −0.0740323
$$881$$ −49.3923 −1.66407 −0.832035 0.554724i $$-0.812824\pi$$
−0.832035 + 0.554724i $$0.812824\pi$$
$$882$$ 7.39230 0.248912
$$883$$ −22.0526 −0.742128 −0.371064 0.928607i $$-0.621007\pi$$
−0.371064 + 0.928607i $$0.621007\pi$$
$$884$$ −14.7846 −0.497260
$$885$$ 12.0000 0.403376
$$886$$ 21.4641 0.721101
$$887$$ 7.94744 0.266849 0.133424 0.991059i $$-0.457403\pi$$
0.133424 + 0.991059i $$0.457403\pi$$
$$888$$ 0 0
$$889$$ 36.3923 1.22056
$$890$$ 29.7846 0.998382
$$891$$ 5.66025 0.189626
$$892$$ 5.80385 0.194327
$$893$$ 6.00000 0.200782
$$894$$ −1.51666 −0.0507247
$$895$$ 4.39230 0.146819
$$896$$ −2.00000 −0.0668153
$$897$$ −3.21539 −0.107359
$$898$$ 27.4641 0.916489
$$899$$ 40.9808 1.36678
$$900$$ 4.92820 0.164273
$$901$$ −40.3923 −1.34566
$$902$$ 4.98076 0.165841
$$903$$ 18.9282 0.629891
$$904$$ 3.46410 0.115214
$$905$$ −19.7321 −0.655916
$$906$$ 4.53590 0.150695
$$907$$ 42.5885 1.41413 0.707063 0.707150i $$-0.250020\pi$$
0.707063 + 0.707150i $$0.250020\pi$$
$$908$$ 4.73205 0.157039
$$909$$ 13.6410 0.452444
$$910$$ 12.0000 0.397796
$$911$$ −18.3397 −0.607623 −0.303811 0.952732i $$-0.598259\pi$$
−0.303811 + 0.952732i $$0.598259\pi$$
$$912$$ 3.46410 0.114708
$$913$$ 14.7846 0.489299
$$914$$ 13.9808 0.462443
$$915$$ −2.19615 −0.0726026
$$916$$ 6.60770 0.218324
$$917$$ −40.3923 −1.33387
$$918$$ 17.0718 0.563453
$$919$$ 3.12436 0.103063 0.0515315 0.998671i $$-0.483590\pi$$
0.0515315 + 0.998671i $$0.483590\pi$$
$$920$$ 2.19615 0.0724050
$$921$$ 16.1051 0.530682
$$922$$ 32.5359 1.07151
$$923$$ −12.0000 −0.394985
$$924$$ −1.85641 −0.0610713
$$925$$ 0 0
$$926$$ −30.0000 −0.985861
$$927$$ 38.1051 1.25154
$$928$$ −8.66025 −0.284287
$$929$$ 4.60770 0.151174 0.0755868 0.997139i $$-0.475917\pi$$
0.0755868 + 0.997139i $$0.475917\pi$$
$$930$$ 6.00000 0.196748
$$931$$ −14.1962 −0.465260
$$932$$ −15.0000 −0.491341
$$933$$ −13.1769 −0.431393
$$934$$ 9.46410 0.309675
$$935$$ 9.37307 0.306532
$$936$$ −8.53590 −0.279005
$$937$$ 6.60770 0.215864 0.107932 0.994158i $$-0.465577\pi$$
0.107932 + 0.994158i $$0.465577\pi$$
$$938$$ 0.392305 0.0128092
$$939$$ 0.0910347 0.00297080
$$940$$ −2.19615 −0.0716306
$$941$$ −11.7846 −0.384167 −0.192084 0.981379i $$-0.561524\pi$$
−0.192084 + 0.981379i $$0.561524\pi$$
$$942$$ 0.732051 0.0238515
$$943$$ −4.98076 −0.162196
$$944$$ −9.46410 −0.308030
$$945$$ −13.8564 −0.450749
$$946$$ −16.3923 −0.532960
$$947$$ 25.1769 0.818140 0.409070 0.912503i $$-0.365853\pi$$
0.409070 + 0.912503i $$0.365853\pi$$
$$948$$ −12.2487 −0.397820
$$949$$ −13.8564 −0.449798
$$950$$ −9.46410 −0.307056
$$951$$ 9.80385 0.317912
$$952$$ 8.53590 0.276650
$$953$$ 7.85641 0.254494 0.127247 0.991871i $$-0.459386\pi$$
0.127247 + 0.991871i $$0.459386\pi$$
$$954$$ −23.3205 −0.755029
$$955$$ −14.1962 −0.459377
$$956$$ −17.3205 −0.560185
$$957$$ −8.03848 −0.259847
$$958$$ −2.53590 −0.0819312
$$959$$ −38.7846 −1.25242
$$960$$ −1.26795 −0.0409229
$$961$$ −8.60770 −0.277668
$$962$$ 0 0
$$963$$ −25.6077 −0.825196
$$964$$ 8.53590 0.274923
$$965$$ 1.39230 0.0448199
$$966$$ 1.85641 0.0597289
$$967$$ −48.0000 −1.54358 −0.771788 0.635880i $$-0.780637\pi$$
−0.771788 + 0.635880i $$0.780637\pi$$
$$968$$ −9.39230 −0.301880
$$969$$ −14.7846 −0.474950
$$970$$ 7.39230 0.237353
$$971$$ −58.9808 −1.89278 −0.946391 0.323022i $$-0.895301\pi$$
−0.946391 + 0.323022i $$0.895301\pi$$
$$972$$ 15.2679 0.489720
$$973$$ 21.1769 0.678901
$$974$$ 22.0526 0.706610
$$975$$ −5.07180 −0.162427
$$976$$ 1.73205 0.0554416
$$977$$ 3.46410 0.110826 0.0554132 0.998464i $$-0.482352\pi$$
0.0554132 + 0.998464i $$0.482352\pi$$
$$978$$ −0.679492 −0.0217278
$$979$$ −21.8038 −0.696854
$$980$$ 5.19615 0.165985
$$981$$ −36.1244 −1.15336
$$982$$ 15.8038 0.504321
$$983$$ −20.8756 −0.665830 −0.332915 0.942957i $$-0.608032\pi$$
−0.332915 + 0.942957i $$0.608032\pi$$
$$984$$ 2.87564 0.0916722
$$985$$ 0.803848 0.0256127
$$986$$ 36.9615 1.17709
$$987$$ −1.85641 −0.0590901
$$988$$ 16.3923 0.521509
$$989$$ 16.3923 0.521245
$$990$$ 5.41154 0.171990
$$991$$ 52.6410 1.67220 0.836098 0.548579i $$-0.184831\pi$$
0.836098 + 0.548579i $$0.184831\pi$$
$$992$$ −4.73205 −0.150243
$$993$$ 0 0
$$994$$ 6.92820 0.219749
$$995$$ −18.0000 −0.570638
$$996$$ 8.53590 0.270470
$$997$$ −10.3923 −0.329128 −0.164564 0.986366i $$-0.552622\pi$$
−0.164564 + 0.986366i $$0.552622\pi$$
$$998$$ 9.80385 0.310335
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2738.2.a.i.1.2 2
37.23 odd 12 74.2.e.b.11.1 4
37.29 odd 12 74.2.e.b.27.1 yes 4
37.36 even 2 2738.2.a.e.1.2 2
111.23 even 12 666.2.s.a.307.2 4
111.29 even 12 666.2.s.a.397.2 4
148.23 even 12 592.2.w.e.529.1 4
148.103 even 12 592.2.w.e.545.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.1 4 37.23 odd 12
74.2.e.b.27.1 yes 4 37.29 odd 12
592.2.w.e.529.1 4 148.23 even 12
592.2.w.e.545.1 4 148.103 even 12
666.2.s.a.307.2 4 111.23 even 12
666.2.s.a.397.2 4 111.29 even 12
2738.2.a.e.1.2 2 37.36 even 2
2738.2.a.i.1.2 2 1.1 even 1 trivial