# Properties

 Label 5415.2.a.p.1.2 Level $5415$ Weight $2$ Character 5415.1 Self dual yes Analytic conductor $43.239$ 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] = [5415,2,Mod(1,5415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5415, 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("5415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5415 = 3 \cdot 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5415.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: $$43.2389926945$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 5415.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} +0.585786 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} +0.585786 q^{7} -1.58579 q^{8} +1.00000 q^{9} -0.414214 q^{10} +1.41421 q^{11} -1.82843 q^{12} -5.41421 q^{13} +0.242641 q^{14} -1.00000 q^{15} +3.00000 q^{16} -1.17157 q^{17} +0.414214 q^{18} +1.82843 q^{20} +0.585786 q^{21} +0.585786 q^{22} +7.65685 q^{23} -1.58579 q^{24} +1.00000 q^{25} -2.24264 q^{26} +1.00000 q^{27} -1.07107 q^{28} +9.07107 q^{29} -0.414214 q^{30} -6.48528 q^{31} +4.41421 q^{32} +1.41421 q^{33} -0.485281 q^{34} -0.585786 q^{35} -1.82843 q^{36} -11.0711 q^{37} -5.41421 q^{39} +1.58579 q^{40} +7.41421 q^{41} +0.242641 q^{42} +0.585786 q^{43} -2.58579 q^{44} -1.00000 q^{45} +3.17157 q^{46} +0.343146 q^{47} +3.00000 q^{48} -6.65685 q^{49} +0.414214 q^{50} -1.17157 q^{51} +9.89949 q^{52} -4.00000 q^{53} +0.414214 q^{54} -1.41421 q^{55} -0.928932 q^{56} +3.75736 q^{58} -8.48528 q^{59} +1.82843 q^{60} +5.65685 q^{61} -2.68629 q^{62} +0.585786 q^{63} -4.17157 q^{64} +5.41421 q^{65} +0.585786 q^{66} -12.0000 q^{67} +2.14214 q^{68} +7.65685 q^{69} -0.242641 q^{70} +4.48528 q^{71} -1.58579 q^{72} -2.00000 q^{73} -4.58579 q^{74} +1.00000 q^{75} +0.828427 q^{77} -2.24264 q^{78} +11.3137 q^{79} -3.00000 q^{80} +1.00000 q^{81} +3.07107 q^{82} -10.4853 q^{83} -1.07107 q^{84} +1.17157 q^{85} +0.242641 q^{86} +9.07107 q^{87} -2.24264 q^{88} -10.7279 q^{89} -0.414214 q^{90} -3.17157 q^{91} -14.0000 q^{92} -6.48528 q^{93} +0.142136 q^{94} +4.41421 q^{96} +4.24264 q^{97} -2.75736 q^{98} +1.41421 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 4 * q^7 - 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 8 q^{13} - 8 q^{14} - 2 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{20} + 4 q^{21} + 4 q^{22} + 4 q^{23} - 6 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{27} + 12 q^{28} + 4 q^{29} + 2 q^{30} + 4 q^{31} + 6 q^{32} + 16 q^{34} - 4 q^{35} + 2 q^{36} - 8 q^{37} - 8 q^{39} + 6 q^{40} + 12 q^{41} - 8 q^{42} + 4 q^{43} - 8 q^{44} - 2 q^{45} + 12 q^{46} + 12 q^{47} + 6 q^{48} - 2 q^{49} - 2 q^{50} - 8 q^{51} - 8 q^{53} - 2 q^{54} - 16 q^{56} + 16 q^{58} - 2 q^{60} - 28 q^{62} + 4 q^{63} - 14 q^{64} + 8 q^{65} + 4 q^{66} - 24 q^{67} - 24 q^{68} + 4 q^{69} + 8 q^{70} - 8 q^{71} - 6 q^{72} - 4 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{77} + 4 q^{78} - 6 q^{80} + 2 q^{81} - 8 q^{82} - 4 q^{83} + 12 q^{84} + 8 q^{85} - 8 q^{86} + 4 q^{87} + 4 q^{88} + 4 q^{89} + 2 q^{90} - 12 q^{91} - 28 q^{92} + 4 q^{93} - 28 q^{94} + 6 q^{96} - 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 - 8 * q^13 - 8 * q^14 - 2 * q^15 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 2 * q^20 + 4 * q^21 + 4 * q^22 + 4 * q^23 - 6 * q^24 + 2 * q^25 + 4 * q^26 + 2 * q^27 + 12 * q^28 + 4 * q^29 + 2 * q^30 + 4 * q^31 + 6 * q^32 + 16 * q^34 - 4 * q^35 + 2 * q^36 - 8 * q^37 - 8 * q^39 + 6 * q^40 + 12 * q^41 - 8 * q^42 + 4 * q^43 - 8 * q^44 - 2 * q^45 + 12 * q^46 + 12 * q^47 + 6 * q^48 - 2 * q^49 - 2 * q^50 - 8 * q^51 - 8 * q^53 - 2 * q^54 - 16 * q^56 + 16 * q^58 - 2 * q^60 - 28 * q^62 + 4 * q^63 - 14 * q^64 + 8 * q^65 + 4 * q^66 - 24 * q^67 - 24 * q^68 + 4 * q^69 + 8 * q^70 - 8 * q^71 - 6 * q^72 - 4 * q^73 - 12 * q^74 + 2 * q^75 - 4 * q^77 + 4 * q^78 - 6 * q^80 + 2 * q^81 - 8 * q^82 - 4 * q^83 + 12 * q^84 + 8 * q^85 - 8 * q^86 + 4 * q^87 + 4 * q^88 + 4 * q^89 + 2 * q^90 - 12 * q^91 - 28 * q^92 + 4 * q^93 - 28 * q^94 + 6 * q^96 - 14 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.414214 0.292893 0.146447 0.989219i $$-0.453216\pi$$
0.146447 + 0.989219i $$0.453216\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.82843 −0.914214
$$5$$ −1.00000 −0.447214
$$6$$ 0.414214 0.169102
$$7$$ 0.585786 0.221406 0.110703 0.993854i $$-0.464690\pi$$
0.110703 + 0.993854i $$0.464690\pi$$
$$8$$ −1.58579 −0.560660
$$9$$ 1.00000 0.333333
$$10$$ −0.414214 −0.130986
$$11$$ 1.41421 0.426401 0.213201 0.977008i $$-0.431611\pi$$
0.213201 + 0.977008i $$0.431611\pi$$
$$12$$ −1.82843 −0.527821
$$13$$ −5.41421 −1.50163 −0.750816 0.660511i $$-0.770340\pi$$
−0.750816 + 0.660511i $$0.770340\pi$$
$$14$$ 0.242641 0.0648485
$$15$$ −1.00000 −0.258199
$$16$$ 3.00000 0.750000
$$17$$ −1.17157 −0.284148 −0.142074 0.989856i $$-0.545377\pi$$
−0.142074 + 0.989856i $$0.545377\pi$$
$$18$$ 0.414214 0.0976311
$$19$$ 0 0
$$20$$ 1.82843 0.408849
$$21$$ 0.585786 0.127829
$$22$$ 0.585786 0.124890
$$23$$ 7.65685 1.59656 0.798282 0.602284i $$-0.205742\pi$$
0.798282 + 0.602284i $$0.205742\pi$$
$$24$$ −1.58579 −0.323697
$$25$$ 1.00000 0.200000
$$26$$ −2.24264 −0.439818
$$27$$ 1.00000 0.192450
$$28$$ −1.07107 −0.202413
$$29$$ 9.07107 1.68446 0.842228 0.539122i $$-0.181244\pi$$
0.842228 + 0.539122i $$0.181244\pi$$
$$30$$ −0.414214 −0.0756247
$$31$$ −6.48528 −1.16479 −0.582395 0.812906i $$-0.697884\pi$$
−0.582395 + 0.812906i $$0.697884\pi$$
$$32$$ 4.41421 0.780330
$$33$$ 1.41421 0.246183
$$34$$ −0.485281 −0.0832251
$$35$$ −0.585786 −0.0990160
$$36$$ −1.82843 −0.304738
$$37$$ −11.0711 −1.82007 −0.910036 0.414529i $$-0.863946\pi$$
−0.910036 + 0.414529i $$0.863946\pi$$
$$38$$ 0 0
$$39$$ −5.41421 −0.866968
$$40$$ 1.58579 0.250735
$$41$$ 7.41421 1.15791 0.578953 0.815361i $$-0.303462\pi$$
0.578953 + 0.815361i $$0.303462\pi$$
$$42$$ 0.242641 0.0374403
$$43$$ 0.585786 0.0893316 0.0446658 0.999002i $$-0.485778\pi$$
0.0446658 + 0.999002i $$0.485778\pi$$
$$44$$ −2.58579 −0.389822
$$45$$ −1.00000 −0.149071
$$46$$ 3.17157 0.467623
$$47$$ 0.343146 0.0500530 0.0250265 0.999687i $$-0.492033\pi$$
0.0250265 + 0.999687i $$0.492033\pi$$
$$48$$ 3.00000 0.433013
$$49$$ −6.65685 −0.950979
$$50$$ 0.414214 0.0585786
$$51$$ −1.17157 −0.164053
$$52$$ 9.89949 1.37281
$$53$$ −4.00000 −0.549442 −0.274721 0.961524i $$-0.588586\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ 0.414214 0.0563673
$$55$$ −1.41421 −0.190693
$$56$$ −0.928932 −0.124134
$$57$$ 0 0
$$58$$ 3.75736 0.493365
$$59$$ −8.48528 −1.10469 −0.552345 0.833616i $$-0.686267\pi$$
−0.552345 + 0.833616i $$0.686267\pi$$
$$60$$ 1.82843 0.236049
$$61$$ 5.65685 0.724286 0.362143 0.932123i $$-0.382045\pi$$
0.362143 + 0.932123i $$0.382045\pi$$
$$62$$ −2.68629 −0.341159
$$63$$ 0.585786 0.0738022
$$64$$ −4.17157 −0.521447
$$65$$ 5.41421 0.671551
$$66$$ 0.585786 0.0721053
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 2.14214 0.259772
$$69$$ 7.65685 0.921777
$$70$$ −0.242641 −0.0290011
$$71$$ 4.48528 0.532305 0.266152 0.963931i $$-0.414248\pi$$
0.266152 + 0.963931i $$0.414248\pi$$
$$72$$ −1.58579 −0.186887
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ −4.58579 −0.533087
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0.828427 0.0944080
$$78$$ −2.24264 −0.253929
$$79$$ 11.3137 1.27289 0.636446 0.771321i $$-0.280404\pi$$
0.636446 + 0.771321i $$0.280404\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 1.00000 0.111111
$$82$$ 3.07107 0.339143
$$83$$ −10.4853 −1.15091 −0.575455 0.817834i $$-0.695175\pi$$
−0.575455 + 0.817834i $$0.695175\pi$$
$$84$$ −1.07107 −0.116863
$$85$$ 1.17157 0.127075
$$86$$ 0.242641 0.0261646
$$87$$ 9.07107 0.972521
$$88$$ −2.24264 −0.239066
$$89$$ −10.7279 −1.13716 −0.568579 0.822629i $$-0.692507\pi$$
−0.568579 + 0.822629i $$0.692507\pi$$
$$90$$ −0.414214 −0.0436619
$$91$$ −3.17157 −0.332471
$$92$$ −14.0000 −1.45960
$$93$$ −6.48528 −0.672492
$$94$$ 0.142136 0.0146602
$$95$$ 0 0
$$96$$ 4.41421 0.450524
$$97$$ 4.24264 0.430775 0.215387 0.976529i $$-0.430899\pi$$
0.215387 + 0.976529i $$0.430899\pi$$
$$98$$ −2.75736 −0.278535
$$99$$ 1.41421 0.142134
$$100$$ −1.82843 −0.182843
$$101$$ −4.82843 −0.480446 −0.240223 0.970718i $$-0.577221\pi$$
−0.240223 + 0.970718i $$0.577221\pi$$
$$102$$ −0.485281 −0.0480500
$$103$$ 1.65685 0.163255 0.0816274 0.996663i $$-0.473988\pi$$
0.0816274 + 0.996663i $$0.473988\pi$$
$$104$$ 8.58579 0.841906
$$105$$ −0.585786 −0.0571669
$$106$$ −1.65685 −0.160928
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −1.82843 −0.175940
$$109$$ −3.17157 −0.303782 −0.151891 0.988397i $$-0.548536\pi$$
−0.151891 + 0.988397i $$0.548536\pi$$
$$110$$ −0.585786 −0.0558525
$$111$$ −11.0711 −1.05082
$$112$$ 1.75736 0.166055
$$113$$ −12.4853 −1.17452 −0.587258 0.809400i $$-0.699793\pi$$
−0.587258 + 0.809400i $$0.699793\pi$$
$$114$$ 0 0
$$115$$ −7.65685 −0.714005
$$116$$ −16.5858 −1.53995
$$117$$ −5.41421 −0.500544
$$118$$ −3.51472 −0.323556
$$119$$ −0.686292 −0.0629122
$$120$$ 1.58579 0.144762
$$121$$ −9.00000 −0.818182
$$122$$ 2.34315 0.212138
$$123$$ 7.41421 0.668517
$$124$$ 11.8579 1.06487
$$125$$ −1.00000 −0.0894427
$$126$$ 0.242641 0.0216162
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −10.5563 −0.933058
$$129$$ 0.585786 0.0515756
$$130$$ 2.24264 0.196693
$$131$$ 16.7279 1.46153 0.730763 0.682632i $$-0.239165\pi$$
0.730763 + 0.682632i $$0.239165\pi$$
$$132$$ −2.58579 −0.225064
$$133$$ 0 0
$$134$$ −4.97056 −0.429391
$$135$$ −1.00000 −0.0860663
$$136$$ 1.85786 0.159311
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 3.17157 0.269982
$$139$$ 1.17157 0.0993715 0.0496858 0.998765i $$-0.484178\pi$$
0.0496858 + 0.998765i $$0.484178\pi$$
$$140$$ 1.07107 0.0905218
$$141$$ 0.343146 0.0288981
$$142$$ 1.85786 0.155909
$$143$$ −7.65685 −0.640298
$$144$$ 3.00000 0.250000
$$145$$ −9.07107 −0.753311
$$146$$ −0.828427 −0.0685611
$$147$$ −6.65685 −0.549048
$$148$$ 20.2426 1.66393
$$149$$ −3.65685 −0.299581 −0.149791 0.988718i $$-0.547860\pi$$
−0.149791 + 0.988718i $$0.547860\pi$$
$$150$$ 0.414214 0.0338204
$$151$$ −17.7990 −1.44846 −0.724231 0.689558i $$-0.757805\pi$$
−0.724231 + 0.689558i $$0.757805\pi$$
$$152$$ 0 0
$$153$$ −1.17157 −0.0947161
$$154$$ 0.343146 0.0276515
$$155$$ 6.48528 0.520910
$$156$$ 9.89949 0.792594
$$157$$ −21.7990 −1.73975 −0.869874 0.493273i $$-0.835800\pi$$
−0.869874 + 0.493273i $$0.835800\pi$$
$$158$$ 4.68629 0.372821
$$159$$ −4.00000 −0.317221
$$160$$ −4.41421 −0.348974
$$161$$ 4.48528 0.353490
$$162$$ 0.414214 0.0325437
$$163$$ 7.89949 0.618736 0.309368 0.950942i $$-0.399882\pi$$
0.309368 + 0.950942i $$0.399882\pi$$
$$164$$ −13.5563 −1.05857
$$165$$ −1.41421 −0.110096
$$166$$ −4.34315 −0.337093
$$167$$ −10.0000 −0.773823 −0.386912 0.922117i $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ −0.928932 −0.0716687
$$169$$ 16.3137 1.25490
$$170$$ 0.485281 0.0372194
$$171$$ 0 0
$$172$$ −1.07107 −0.0816682
$$173$$ 6.14214 0.466978 0.233489 0.972359i $$-0.424986\pi$$
0.233489 + 0.972359i $$0.424986\pi$$
$$174$$ 3.75736 0.284845
$$175$$ 0.585786 0.0442813
$$176$$ 4.24264 0.319801
$$177$$ −8.48528 −0.637793
$$178$$ −4.44365 −0.333066
$$179$$ −17.1716 −1.28346 −0.641732 0.766929i $$-0.721784\pi$$
−0.641732 + 0.766929i $$0.721784\pi$$
$$180$$ 1.82843 0.136283
$$181$$ 19.1716 1.42501 0.712506 0.701666i $$-0.247560\pi$$
0.712506 + 0.701666i $$0.247560\pi$$
$$182$$ −1.31371 −0.0973786
$$183$$ 5.65685 0.418167
$$184$$ −12.1421 −0.895130
$$185$$ 11.0711 0.813961
$$186$$ −2.68629 −0.196968
$$187$$ −1.65685 −0.121161
$$188$$ −0.627417 −0.0457591
$$189$$ 0.585786 0.0426097
$$190$$ 0 0
$$191$$ 1.89949 0.137443 0.0687213 0.997636i $$-0.478108\pi$$
0.0687213 + 0.997636i $$0.478108\pi$$
$$192$$ −4.17157 −0.301057
$$193$$ 15.0711 1.08484 0.542420 0.840108i $$-0.317508\pi$$
0.542420 + 0.840108i $$0.317508\pi$$
$$194$$ 1.75736 0.126171
$$195$$ 5.41421 0.387720
$$196$$ 12.1716 0.869398
$$197$$ −14.8284 −1.05648 −0.528241 0.849095i $$-0.677148\pi$$
−0.528241 + 0.849095i $$0.677148\pi$$
$$198$$ 0.585786 0.0416300
$$199$$ −16.4853 −1.16861 −0.584305 0.811534i $$-0.698633\pi$$
−0.584305 + 0.811534i $$0.698633\pi$$
$$200$$ −1.58579 −0.112132
$$201$$ −12.0000 −0.846415
$$202$$ −2.00000 −0.140720
$$203$$ 5.31371 0.372949
$$204$$ 2.14214 0.149979
$$205$$ −7.41421 −0.517831
$$206$$ 0.686292 0.0478162
$$207$$ 7.65685 0.532188
$$208$$ −16.2426 −1.12622
$$209$$ 0 0
$$210$$ −0.242641 −0.0167438
$$211$$ 15.3137 1.05424 0.527120 0.849791i $$-0.323272\pi$$
0.527120 + 0.849791i $$0.323272\pi$$
$$212$$ 7.31371 0.502308
$$213$$ 4.48528 0.307326
$$214$$ −3.31371 −0.226520
$$215$$ −0.585786 −0.0399503
$$216$$ −1.58579 −0.107899
$$217$$ −3.79899 −0.257892
$$218$$ −1.31371 −0.0889756
$$219$$ −2.00000 −0.135147
$$220$$ 2.58579 0.174334
$$221$$ 6.34315 0.426686
$$222$$ −4.58579 −0.307778
$$223$$ −6.34315 −0.424768 −0.212384 0.977186i $$-0.568123\pi$$
−0.212384 + 0.977186i $$0.568123\pi$$
$$224$$ 2.58579 0.172770
$$225$$ 1.00000 0.0666667
$$226$$ −5.17157 −0.344008
$$227$$ 18.9706 1.25912 0.629560 0.776952i $$-0.283235\pi$$
0.629560 + 0.776952i $$0.283235\pi$$
$$228$$ 0 0
$$229$$ −1.65685 −0.109488 −0.0547440 0.998500i $$-0.517434\pi$$
−0.0547440 + 0.998500i $$0.517434\pi$$
$$230$$ −3.17157 −0.209127
$$231$$ 0.828427 0.0545065
$$232$$ −14.3848 −0.944407
$$233$$ −8.34315 −0.546578 −0.273289 0.961932i $$-0.588111\pi$$
−0.273289 + 0.961932i $$0.588111\pi$$
$$234$$ −2.24264 −0.146606
$$235$$ −0.343146 −0.0223844
$$236$$ 15.5147 1.00992
$$237$$ 11.3137 0.734904
$$238$$ −0.284271 −0.0184266
$$239$$ 2.58579 0.167261 0.0836303 0.996497i $$-0.473349\pi$$
0.0836303 + 0.996497i $$0.473349\pi$$
$$240$$ −3.00000 −0.193649
$$241$$ 14.9706 0.964339 0.482169 0.876078i $$-0.339849\pi$$
0.482169 + 0.876078i $$0.339849\pi$$
$$242$$ −3.72792 −0.239640
$$243$$ 1.00000 0.0641500
$$244$$ −10.3431 −0.662152
$$245$$ 6.65685 0.425291
$$246$$ 3.07107 0.195804
$$247$$ 0 0
$$248$$ 10.2843 0.653052
$$249$$ −10.4853 −0.664478
$$250$$ −0.414214 −0.0261972
$$251$$ 12.9289 0.816067 0.408033 0.912967i $$-0.366215\pi$$
0.408033 + 0.912967i $$0.366215\pi$$
$$252$$ −1.07107 −0.0674709
$$253$$ 10.8284 0.680777
$$254$$ −3.31371 −0.207921
$$255$$ 1.17157 0.0733667
$$256$$ 3.97056 0.248160
$$257$$ −1.17157 −0.0730807 −0.0365404 0.999332i $$-0.511634\pi$$
−0.0365404 + 0.999332i $$0.511634\pi$$
$$258$$ 0.242641 0.0151061
$$259$$ −6.48528 −0.402976
$$260$$ −9.89949 −0.613941
$$261$$ 9.07107 0.561485
$$262$$ 6.92893 0.428071
$$263$$ −32.1421 −1.98197 −0.990984 0.133977i $$-0.957225\pi$$
−0.990984 + 0.133977i $$0.957225\pi$$
$$264$$ −2.24264 −0.138025
$$265$$ 4.00000 0.245718
$$266$$ 0 0
$$267$$ −10.7279 −0.656538
$$268$$ 21.9411 1.34027
$$269$$ −8.38478 −0.511229 −0.255614 0.966779i $$-0.582278\pi$$
−0.255614 + 0.966779i $$0.582278\pi$$
$$270$$ −0.414214 −0.0252082
$$271$$ −30.8284 −1.87269 −0.936347 0.351076i $$-0.885816\pi$$
−0.936347 + 0.351076i $$0.885816\pi$$
$$272$$ −3.51472 −0.213111
$$273$$ −3.17157 −0.191952
$$274$$ −5.79899 −0.350330
$$275$$ 1.41421 0.0852803
$$276$$ −14.0000 −0.842701
$$277$$ 10.9706 0.659157 0.329579 0.944128i $$-0.393093\pi$$
0.329579 + 0.944128i $$0.393093\pi$$
$$278$$ 0.485281 0.0291052
$$279$$ −6.48528 −0.388264
$$280$$ 0.928932 0.0555143
$$281$$ 14.7279 0.878594 0.439297 0.898342i $$-0.355228\pi$$
0.439297 + 0.898342i $$0.355228\pi$$
$$282$$ 0.142136 0.00846405
$$283$$ 6.24264 0.371086 0.185543 0.982636i $$-0.440596\pi$$
0.185543 + 0.982636i $$0.440596\pi$$
$$284$$ −8.20101 −0.486640
$$285$$ 0 0
$$286$$ −3.17157 −0.187539
$$287$$ 4.34315 0.256368
$$288$$ 4.41421 0.260110
$$289$$ −15.6274 −0.919260
$$290$$ −3.75736 −0.220640
$$291$$ 4.24264 0.248708
$$292$$ 3.65685 0.214001
$$293$$ 31.7990 1.85772 0.928858 0.370435i $$-0.120791\pi$$
0.928858 + 0.370435i $$0.120791\pi$$
$$294$$ −2.75736 −0.160812
$$295$$ 8.48528 0.494032
$$296$$ 17.5563 1.02044
$$297$$ 1.41421 0.0820610
$$298$$ −1.51472 −0.0877453
$$299$$ −41.4558 −2.39745
$$300$$ −1.82843 −0.105564
$$301$$ 0.343146 0.0197786
$$302$$ −7.37258 −0.424244
$$303$$ −4.82843 −0.277386
$$304$$ 0 0
$$305$$ −5.65685 −0.323911
$$306$$ −0.485281 −0.0277417
$$307$$ −7.79899 −0.445112 −0.222556 0.974920i $$-0.571440\pi$$
−0.222556 + 0.974920i $$0.571440\pi$$
$$308$$ −1.51472 −0.0863091
$$309$$ 1.65685 0.0942551
$$310$$ 2.68629 0.152571
$$311$$ −32.2426 −1.82831 −0.914156 0.405362i $$-0.867145\pi$$
−0.914156 + 0.405362i $$0.867145\pi$$
$$312$$ 8.58579 0.486074
$$313$$ −9.51472 −0.537804 −0.268902 0.963168i $$-0.586661\pi$$
−0.268902 + 0.963168i $$0.586661\pi$$
$$314$$ −9.02944 −0.509561
$$315$$ −0.585786 −0.0330053
$$316$$ −20.6863 −1.16369
$$317$$ −11.3137 −0.635441 −0.317721 0.948184i $$-0.602917\pi$$
−0.317721 + 0.948184i $$0.602917\pi$$
$$318$$ −1.65685 −0.0929118
$$319$$ 12.8284 0.718254
$$320$$ 4.17157 0.233198
$$321$$ −8.00000 −0.446516
$$322$$ 1.85786 0.103535
$$323$$ 0 0
$$324$$ −1.82843 −0.101579
$$325$$ −5.41421 −0.300327
$$326$$ 3.27208 0.181224
$$327$$ −3.17157 −0.175388
$$328$$ −11.7574 −0.649192
$$329$$ 0.201010 0.0110820
$$330$$ −0.585786 −0.0322465
$$331$$ −7.17157 −0.394185 −0.197093 0.980385i $$-0.563150\pi$$
−0.197093 + 0.980385i $$0.563150\pi$$
$$332$$ 19.1716 1.05218
$$333$$ −11.0711 −0.606691
$$334$$ −4.14214 −0.226648
$$335$$ 12.0000 0.655630
$$336$$ 1.75736 0.0958718
$$337$$ −28.2426 −1.53847 −0.769237 0.638963i $$-0.779364\pi$$
−0.769237 + 0.638963i $$0.779364\pi$$
$$338$$ 6.75736 0.367552
$$339$$ −12.4853 −0.678107
$$340$$ −2.14214 −0.116174
$$341$$ −9.17157 −0.496669
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ −0.928932 −0.0500847
$$345$$ −7.65685 −0.412231
$$346$$ 2.54416 0.136775
$$347$$ 10.4853 0.562879 0.281440 0.959579i $$-0.409188\pi$$
0.281440 + 0.959579i $$0.409188\pi$$
$$348$$ −16.5858 −0.889091
$$349$$ 29.3137 1.56913 0.784563 0.620049i $$-0.212887\pi$$
0.784563 + 0.620049i $$0.212887\pi$$
$$350$$ 0.242641 0.0129697
$$351$$ −5.41421 −0.288989
$$352$$ 6.24264 0.332734
$$353$$ 3.65685 0.194635 0.0973174 0.995253i $$-0.468974\pi$$
0.0973174 + 0.995253i $$0.468974\pi$$
$$354$$ −3.51472 −0.186805
$$355$$ −4.48528 −0.238054
$$356$$ 19.6152 1.03960
$$357$$ −0.686292 −0.0363224
$$358$$ −7.11270 −0.375918
$$359$$ 9.89949 0.522475 0.261238 0.965275i $$-0.415869\pi$$
0.261238 + 0.965275i $$0.415869\pi$$
$$360$$ 1.58579 0.0835783
$$361$$ 0 0
$$362$$ 7.94113 0.417376
$$363$$ −9.00000 −0.472377
$$364$$ 5.79899 0.303950
$$365$$ 2.00000 0.104685
$$366$$ 2.34315 0.122478
$$367$$ −19.4142 −1.01341 −0.506707 0.862118i $$-0.669137\pi$$
−0.506707 + 0.862118i $$0.669137\pi$$
$$368$$ 22.9706 1.19742
$$369$$ 7.41421 0.385969
$$370$$ 4.58579 0.238404
$$371$$ −2.34315 −0.121650
$$372$$ 11.8579 0.614802
$$373$$ 9.89949 0.512576 0.256288 0.966600i $$-0.417500\pi$$
0.256288 + 0.966600i $$0.417500\pi$$
$$374$$ −0.686292 −0.0354873
$$375$$ −1.00000 −0.0516398
$$376$$ −0.544156 −0.0280627
$$377$$ −49.1127 −2.52943
$$378$$ 0.242641 0.0124801
$$379$$ −24.1421 −1.24010 −0.620049 0.784563i $$-0.712887\pi$$
−0.620049 + 0.784563i $$0.712887\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0.786797 0.0402560
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ −10.5563 −0.538701
$$385$$ −0.828427 −0.0422206
$$386$$ 6.24264 0.317742
$$387$$ 0.585786 0.0297772
$$388$$ −7.75736 −0.393820
$$389$$ 14.9706 0.759038 0.379519 0.925184i $$-0.376090\pi$$
0.379519 + 0.925184i $$0.376090\pi$$
$$390$$ 2.24264 0.113561
$$391$$ −8.97056 −0.453661
$$392$$ 10.5563 0.533176
$$393$$ 16.7279 0.843812
$$394$$ −6.14214 −0.309436
$$395$$ −11.3137 −0.569254
$$396$$ −2.58579 −0.129941
$$397$$ 35.6569 1.78957 0.894783 0.446501i $$-0.147330\pi$$
0.894783 + 0.446501i $$0.147330\pi$$
$$398$$ −6.82843 −0.342278
$$399$$ 0 0
$$400$$ 3.00000 0.150000
$$401$$ −30.0416 −1.50021 −0.750104 0.661320i $$-0.769996\pi$$
−0.750104 + 0.661320i $$0.769996\pi$$
$$402$$ −4.97056 −0.247909
$$403$$ 35.1127 1.74909
$$404$$ 8.82843 0.439231
$$405$$ −1.00000 −0.0496904
$$406$$ 2.20101 0.109234
$$407$$ −15.6569 −0.776081
$$408$$ 1.85786 0.0919780
$$409$$ −9.51472 −0.470473 −0.235236 0.971938i $$-0.575586\pi$$
−0.235236 + 0.971938i $$0.575586\pi$$
$$410$$ −3.07107 −0.151669
$$411$$ −14.0000 −0.690569
$$412$$ −3.02944 −0.149250
$$413$$ −4.97056 −0.244585
$$414$$ 3.17157 0.155874
$$415$$ 10.4853 0.514702
$$416$$ −23.8995 −1.17177
$$417$$ 1.17157 0.0573722
$$418$$ 0 0
$$419$$ −34.8701 −1.70351 −0.851757 0.523937i $$-0.824463\pi$$
−0.851757 + 0.523937i $$0.824463\pi$$
$$420$$ 1.07107 0.0522628
$$421$$ 14.6863 0.715766 0.357883 0.933766i $$-0.383499\pi$$
0.357883 + 0.933766i $$0.383499\pi$$
$$422$$ 6.34315 0.308780
$$423$$ 0.343146 0.0166843
$$424$$ 6.34315 0.308050
$$425$$ −1.17157 −0.0568296
$$426$$ 1.85786 0.0900138
$$427$$ 3.31371 0.160362
$$428$$ 14.6274 0.707043
$$429$$ −7.65685 −0.369676
$$430$$ −0.242641 −0.0117012
$$431$$ −3.51472 −0.169298 −0.0846490 0.996411i $$-0.526977\pi$$
−0.0846490 + 0.996411i $$0.526977\pi$$
$$432$$ 3.00000 0.144338
$$433$$ −0.928932 −0.0446416 −0.0223208 0.999751i $$-0.507106\pi$$
−0.0223208 + 0.999751i $$0.507106\pi$$
$$434$$ −1.57359 −0.0755349
$$435$$ −9.07107 −0.434924
$$436$$ 5.79899 0.277721
$$437$$ 0 0
$$438$$ −0.828427 −0.0395838
$$439$$ −0.970563 −0.0463224 −0.0231612 0.999732i $$-0.507373\pi$$
−0.0231612 + 0.999732i $$0.507373\pi$$
$$440$$ 2.24264 0.106914
$$441$$ −6.65685 −0.316993
$$442$$ 2.62742 0.124973
$$443$$ −1.31371 −0.0624162 −0.0312081 0.999513i $$-0.509935\pi$$
−0.0312081 + 0.999513i $$0.509935\pi$$
$$444$$ 20.2426 0.960673
$$445$$ 10.7279 0.508552
$$446$$ −2.62742 −0.124412
$$447$$ −3.65685 −0.172963
$$448$$ −2.44365 −0.115452
$$449$$ −7.89949 −0.372800 −0.186400 0.982474i $$-0.559682\pi$$
−0.186400 + 0.982474i $$0.559682\pi$$
$$450$$ 0.414214 0.0195262
$$451$$ 10.4853 0.493733
$$452$$ 22.8284 1.07376
$$453$$ −17.7990 −0.836269
$$454$$ 7.85786 0.368788
$$455$$ 3.17157 0.148686
$$456$$ 0 0
$$457$$ 28.8284 1.34854 0.674268 0.738486i $$-0.264459\pi$$
0.674268 + 0.738486i $$0.264459\pi$$
$$458$$ −0.686292 −0.0320683
$$459$$ −1.17157 −0.0546843
$$460$$ 14.0000 0.652753
$$461$$ 10.6863 0.497710 0.248855 0.968541i $$-0.419946\pi$$
0.248855 + 0.968541i $$0.419946\pi$$
$$462$$ 0.343146 0.0159646
$$463$$ −8.10051 −0.376462 −0.188231 0.982125i $$-0.560275\pi$$
−0.188231 + 0.982125i $$0.560275\pi$$
$$464$$ 27.2132 1.26334
$$465$$ 6.48528 0.300748
$$466$$ −3.45584 −0.160089
$$467$$ 16.3431 0.756271 0.378135 0.925750i $$-0.376565\pi$$
0.378135 + 0.925750i $$0.376565\pi$$
$$468$$ 9.89949 0.457604
$$469$$ −7.02944 −0.324589
$$470$$ −0.142136 −0.00655623
$$471$$ −21.7990 −1.00444
$$472$$ 13.4558 0.619355
$$473$$ 0.828427 0.0380911
$$474$$ 4.68629 0.215248
$$475$$ 0 0
$$476$$ 1.25483 0.0575152
$$477$$ −4.00000 −0.183147
$$478$$ 1.07107 0.0489895
$$479$$ −38.1838 −1.74466 −0.872330 0.488917i $$-0.837392\pi$$
−0.872330 + 0.488917i $$0.837392\pi$$
$$480$$ −4.41421 −0.201480
$$481$$ 59.9411 2.73308
$$482$$ 6.20101 0.282448
$$483$$ 4.48528 0.204087
$$484$$ 16.4558 0.747993
$$485$$ −4.24264 −0.192648
$$486$$ 0.414214 0.0187891
$$487$$ 2.82843 0.128168 0.0640841 0.997944i $$-0.479587\pi$$
0.0640841 + 0.997944i $$0.479587\pi$$
$$488$$ −8.97056 −0.406078
$$489$$ 7.89949 0.357228
$$490$$ 2.75736 0.124565
$$491$$ 21.8995 0.988310 0.494155 0.869374i $$-0.335477\pi$$
0.494155 + 0.869374i $$0.335477\pi$$
$$492$$ −13.5563 −0.611167
$$493$$ −10.6274 −0.478635
$$494$$ 0 0
$$495$$ −1.41421 −0.0635642
$$496$$ −19.4558 −0.873593
$$497$$ 2.62742 0.117856
$$498$$ −4.34315 −0.194621
$$499$$ −23.7990 −1.06539 −0.532695 0.846308i $$-0.678821\pi$$
−0.532695 + 0.846308i $$0.678821\pi$$
$$500$$ 1.82843 0.0817697
$$501$$ −10.0000 −0.446767
$$502$$ 5.35534 0.239020
$$503$$ 0.828427 0.0369377 0.0184689 0.999829i $$-0.494121\pi$$
0.0184689 + 0.999829i $$0.494121\pi$$
$$504$$ −0.928932 −0.0413779
$$505$$ 4.82843 0.214862
$$506$$ 4.48528 0.199395
$$507$$ 16.3137 0.724517
$$508$$ 14.6274 0.648987
$$509$$ −32.3848 −1.43543 −0.717715 0.696337i $$-0.754812\pi$$
−0.717715 + 0.696337i $$0.754812\pi$$
$$510$$ 0.485281 0.0214886
$$511$$ −1.17157 −0.0518273
$$512$$ 22.7574 1.00574
$$513$$ 0 0
$$514$$ −0.485281 −0.0214048
$$515$$ −1.65685 −0.0730097
$$516$$ −1.07107 −0.0471511
$$517$$ 0.485281 0.0213427
$$518$$ −2.68629 −0.118029
$$519$$ 6.14214 0.269610
$$520$$ −8.58579 −0.376512
$$521$$ 24.3848 1.06832 0.534158 0.845385i $$-0.320629\pi$$
0.534158 + 0.845385i $$0.320629\pi$$
$$522$$ 3.75736 0.164455
$$523$$ 15.7990 0.690842 0.345421 0.938448i $$-0.387736\pi$$
0.345421 + 0.938448i $$0.387736\pi$$
$$524$$ −30.5858 −1.33615
$$525$$ 0.585786 0.0255658
$$526$$ −13.3137 −0.580505
$$527$$ 7.59798 0.330973
$$528$$ 4.24264 0.184637
$$529$$ 35.6274 1.54902
$$530$$ 1.65685 0.0719691
$$531$$ −8.48528 −0.368230
$$532$$ 0 0
$$533$$ −40.1421 −1.73875
$$534$$ −4.44365 −0.192296
$$535$$ 8.00000 0.345870
$$536$$ 19.0294 0.821946
$$537$$ −17.1716 −0.741008
$$538$$ −3.47309 −0.149735
$$539$$ −9.41421 −0.405499
$$540$$ 1.82843 0.0786830
$$541$$ −32.6274 −1.40276 −0.701381 0.712786i $$-0.747433\pi$$
−0.701381 + 0.712786i $$0.747433\pi$$
$$542$$ −12.7696 −0.548499
$$543$$ 19.1716 0.822731
$$544$$ −5.17157 −0.221729
$$545$$ 3.17157 0.135855
$$546$$ −1.31371 −0.0562215
$$547$$ 34.1421 1.45981 0.729906 0.683547i $$-0.239564\pi$$
0.729906 + 0.683547i $$0.239564\pi$$
$$548$$ 25.5980 1.09349
$$549$$ 5.65685 0.241429
$$550$$ 0.585786 0.0249780
$$551$$ 0 0
$$552$$ −12.1421 −0.516804
$$553$$ 6.62742 0.281826
$$554$$ 4.54416 0.193063
$$555$$ 11.0711 0.469941
$$556$$ −2.14214 −0.0908468
$$557$$ 10.0000 0.423714 0.211857 0.977301i $$-0.432049\pi$$
0.211857 + 0.977301i $$0.432049\pi$$
$$558$$ −2.68629 −0.113720
$$559$$ −3.17157 −0.134143
$$560$$ −1.75736 −0.0742620
$$561$$ −1.65685 −0.0699524
$$562$$ 6.10051 0.257334
$$563$$ −42.2843 −1.78207 −0.891035 0.453935i $$-0.850020\pi$$
−0.891035 + 0.453935i $$0.850020\pi$$
$$564$$ −0.627417 −0.0264190
$$565$$ 12.4853 0.525260
$$566$$ 2.58579 0.108689
$$567$$ 0.585786 0.0246007
$$568$$ −7.11270 −0.298442
$$569$$ 6.72792 0.282049 0.141025 0.990006i $$-0.454960\pi$$
0.141025 + 0.990006i $$0.454960\pi$$
$$570$$ 0 0
$$571$$ 19.7990 0.828562 0.414281 0.910149i $$-0.364033\pi$$
0.414281 + 0.910149i $$0.364033\pi$$
$$572$$ 14.0000 0.585369
$$573$$ 1.89949 0.0793525
$$574$$ 1.79899 0.0750884
$$575$$ 7.65685 0.319313
$$576$$ −4.17157 −0.173816
$$577$$ −37.7990 −1.57359 −0.786796 0.617213i $$-0.788262\pi$$
−0.786796 + 0.617213i $$0.788262\pi$$
$$578$$ −6.47309 −0.269245
$$579$$ 15.0711 0.626332
$$580$$ 16.5858 0.688687
$$581$$ −6.14214 −0.254819
$$582$$ 1.75736 0.0728449
$$583$$ −5.65685 −0.234283
$$584$$ 3.17157 0.131241
$$585$$ 5.41421 0.223850
$$586$$ 13.1716 0.544113
$$587$$ 11.6569 0.481130 0.240565 0.970633i $$-0.422667\pi$$
0.240565 + 0.970633i $$0.422667\pi$$
$$588$$ 12.1716 0.501947
$$589$$ 0 0
$$590$$ 3.51472 0.144699
$$591$$ −14.8284 −0.609960
$$592$$ −33.2132 −1.36505
$$593$$ −29.3137 −1.20377 −0.601885 0.798583i $$-0.705583\pi$$
−0.601885 + 0.798583i $$0.705583\pi$$
$$594$$ 0.585786 0.0240351
$$595$$ 0.686292 0.0281352
$$596$$ 6.68629 0.273881
$$597$$ −16.4853 −0.674698
$$598$$ −17.1716 −0.702198
$$599$$ 21.9411 0.896490 0.448245 0.893911i $$-0.352049\pi$$
0.448245 + 0.893911i $$0.352049\pi$$
$$600$$ −1.58579 −0.0647395
$$601$$ 28.8284 1.17594 0.587968 0.808884i $$-0.299928\pi$$
0.587968 + 0.808884i $$0.299928\pi$$
$$602$$ 0.142136 0.00579302
$$603$$ −12.0000 −0.488678
$$604$$ 32.5442 1.32420
$$605$$ 9.00000 0.365902
$$606$$ −2.00000 −0.0812444
$$607$$ 2.14214 0.0869466 0.0434733 0.999055i $$-0.486158\pi$$
0.0434733 + 0.999055i $$0.486158\pi$$
$$608$$ 0 0
$$609$$ 5.31371 0.215322
$$610$$ −2.34315 −0.0948712
$$611$$ −1.85786 −0.0751611
$$612$$ 2.14214 0.0865907
$$613$$ 25.1127 1.01429 0.507146 0.861860i $$-0.330700\pi$$
0.507146 + 0.861860i $$0.330700\pi$$
$$614$$ −3.23045 −0.130370
$$615$$ −7.41421 −0.298970
$$616$$ −1.31371 −0.0529308
$$617$$ 10.1421 0.408307 0.204154 0.978939i $$-0.434556\pi$$
0.204154 + 0.978939i $$0.434556\pi$$
$$618$$ 0.686292 0.0276067
$$619$$ −43.7990 −1.76043 −0.880215 0.474575i $$-0.842602\pi$$
−0.880215 + 0.474575i $$0.842602\pi$$
$$620$$ −11.8579 −0.476223
$$621$$ 7.65685 0.307259
$$622$$ −13.3553 −0.535500
$$623$$ −6.28427 −0.251774
$$624$$ −16.2426 −0.650226
$$625$$ 1.00000 0.0400000
$$626$$ −3.94113 −0.157519
$$627$$ 0 0
$$628$$ 39.8579 1.59050
$$629$$ 12.9706 0.517170
$$630$$ −0.242641 −0.00966704
$$631$$ 22.6274 0.900783 0.450392 0.892831i $$-0.351284\pi$$
0.450392 + 0.892831i $$0.351284\pi$$
$$632$$ −17.9411 −0.713660
$$633$$ 15.3137 0.608665
$$634$$ −4.68629 −0.186116
$$635$$ 8.00000 0.317470
$$636$$ 7.31371 0.290007
$$637$$ 36.0416 1.42802
$$638$$ 5.31371 0.210372
$$639$$ 4.48528 0.177435
$$640$$ 10.5563 0.417276
$$641$$ 8.58579 0.339118 0.169559 0.985520i $$-0.445766\pi$$
0.169559 + 0.985520i $$0.445766\pi$$
$$642$$ −3.31371 −0.130782
$$643$$ 6.04163 0.238259 0.119129 0.992879i $$-0.461990\pi$$
0.119129 + 0.992879i $$0.461990\pi$$
$$644$$ −8.20101 −0.323165
$$645$$ −0.585786 −0.0230653
$$646$$ 0 0
$$647$$ 45.1127 1.77356 0.886782 0.462189i $$-0.152936\pi$$
0.886782 + 0.462189i $$0.152936\pi$$
$$648$$ −1.58579 −0.0622956
$$649$$ −12.0000 −0.471041
$$650$$ −2.24264 −0.0879636
$$651$$ −3.79899 −0.148894
$$652$$ −14.4437 −0.565657
$$653$$ 42.4264 1.66027 0.830137 0.557560i $$-0.188262\pi$$
0.830137 + 0.557560i $$0.188262\pi$$
$$654$$ −1.31371 −0.0513701
$$655$$ −16.7279 −0.653614
$$656$$ 22.2426 0.868429
$$657$$ −2.00000 −0.0780274
$$658$$ 0.0832611 0.00324586
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 2.58579 0.100652
$$661$$ −18.4853 −0.718994 −0.359497 0.933146i $$-0.617052\pi$$
−0.359497 + 0.933146i $$0.617052\pi$$
$$662$$ −2.97056 −0.115454
$$663$$ 6.34315 0.246347
$$664$$ 16.6274 0.645269
$$665$$ 0 0
$$666$$ −4.58579 −0.177696
$$667$$ 69.4558 2.68934
$$668$$ 18.2843 0.707440
$$669$$ −6.34315 −0.245240
$$670$$ 4.97056 0.192030
$$671$$ 8.00000 0.308837
$$672$$ 2.58579 0.0997489
$$673$$ −21.8995 −0.844163 −0.422082 0.906558i $$-0.638700\pi$$
−0.422082 + 0.906558i $$0.638700\pi$$
$$674$$ −11.6985 −0.450609
$$675$$ 1.00000 0.0384900
$$676$$ −29.8284 −1.14725
$$677$$ 44.9706 1.72836 0.864180 0.503184i $$-0.167838\pi$$
0.864180 + 0.503184i $$0.167838\pi$$
$$678$$ −5.17157 −0.198613
$$679$$ 2.48528 0.0953763
$$680$$ −1.85786 −0.0712458
$$681$$ 18.9706 0.726954
$$682$$ −3.79899 −0.145471
$$683$$ −5.65685 −0.216454 −0.108227 0.994126i $$-0.534517\pi$$
−0.108227 + 0.994126i $$0.534517\pi$$
$$684$$ 0 0
$$685$$ 14.0000 0.534913
$$686$$ −3.31371 −0.126518
$$687$$ −1.65685 −0.0632129
$$688$$ 1.75736 0.0669987
$$689$$ 21.6569 0.825060
$$690$$ −3.17157 −0.120740
$$691$$ −23.1127 −0.879248 −0.439624 0.898182i $$-0.644888\pi$$
−0.439624 + 0.898182i $$0.644888\pi$$
$$692$$ −11.2304 −0.426918
$$693$$ 0.828427 0.0314693
$$694$$ 4.34315 0.164864
$$695$$ −1.17157 −0.0444403
$$696$$ −14.3848 −0.545254
$$697$$ −8.68629 −0.329017
$$698$$ 12.1421 0.459587
$$699$$ −8.34315 −0.315567
$$700$$ −1.07107 −0.0404826
$$701$$ −0.343146 −0.0129604 −0.00648022 0.999979i $$-0.502063\pi$$
−0.00648022 + 0.999979i $$0.502063\pi$$
$$702$$ −2.24264 −0.0846430
$$703$$ 0 0
$$704$$ −5.89949 −0.222346
$$705$$ −0.343146 −0.0129236
$$706$$ 1.51472 0.0570072
$$707$$ −2.82843 −0.106374
$$708$$ 15.5147 0.583079
$$709$$ −35.3137 −1.32623 −0.663117 0.748516i $$-0.730767\pi$$
−0.663117 + 0.748516i $$0.730767\pi$$
$$710$$ −1.85786 −0.0697244
$$711$$ 11.3137 0.424297
$$712$$ 17.0122 0.637559
$$713$$ −49.6569 −1.85966
$$714$$ −0.284271 −0.0106386
$$715$$ 7.65685 0.286350
$$716$$ 31.3970 1.17336
$$717$$ 2.58579 0.0965680
$$718$$ 4.10051 0.153029
$$719$$ −16.4437 −0.613245 −0.306622 0.951831i $$-0.599199\pi$$
−0.306622 + 0.951831i $$0.599199\pi$$
$$720$$ −3.00000 −0.111803
$$721$$ 0.970563 0.0361456
$$722$$ 0 0
$$723$$ 14.9706 0.556761
$$724$$ −35.0538 −1.30277
$$725$$ 9.07107 0.336891
$$726$$ −3.72792 −0.138356
$$727$$ −4.58579 −0.170077 −0.0850387 0.996378i $$-0.527101\pi$$
−0.0850387 + 0.996378i $$0.527101\pi$$
$$728$$ 5.02944 0.186403
$$729$$ 1.00000 0.0370370
$$730$$ 0.828427 0.0306615
$$731$$ −0.686292 −0.0253834
$$732$$ −10.3431 −0.382294
$$733$$ −1.31371 −0.0485229 −0.0242615 0.999706i $$-0.507723\pi$$
−0.0242615 + 0.999706i $$0.507723\pi$$
$$734$$ −8.04163 −0.296822
$$735$$ 6.65685 0.245542
$$736$$ 33.7990 1.24585
$$737$$ −16.9706 −0.625119
$$738$$ 3.07107 0.113048
$$739$$ −9.65685 −0.355233 −0.177617 0.984100i $$-0.556839\pi$$
−0.177617 + 0.984100i $$0.556839\pi$$
$$740$$ −20.2426 −0.744134
$$741$$ 0 0
$$742$$ −0.970563 −0.0356305
$$743$$ 15.3137 0.561805 0.280903 0.959736i $$-0.409366\pi$$
0.280903 + 0.959736i $$0.409366\pi$$
$$744$$ 10.2843 0.377040
$$745$$ 3.65685 0.133977
$$746$$ 4.10051 0.150130
$$747$$ −10.4853 −0.383636
$$748$$ 3.02944 0.110767
$$749$$ −4.68629 −0.171233
$$750$$ −0.414214 −0.0151249
$$751$$ 37.1127 1.35426 0.677131 0.735863i $$-0.263223\pi$$
0.677131 + 0.735863i $$0.263223\pi$$
$$752$$ 1.02944 0.0375397
$$753$$ 12.9289 0.471156
$$754$$ −20.3431 −0.740854
$$755$$ 17.7990 0.647772
$$756$$ −1.07107 −0.0389544
$$757$$ 16.8284 0.611640 0.305820 0.952089i $$-0.401069\pi$$
0.305820 + 0.952089i $$0.401069\pi$$
$$758$$ −10.0000 −0.363216
$$759$$ 10.8284 0.393047
$$760$$ 0 0
$$761$$ −10.2843 −0.372805 −0.186402 0.982474i $$-0.559683\pi$$
−0.186402 + 0.982474i $$0.559683\pi$$
$$762$$ −3.31371 −0.120043
$$763$$ −1.85786 −0.0672592
$$764$$ −3.47309 −0.125652
$$765$$ 1.17157 0.0423583
$$766$$ −4.97056 −0.179594
$$767$$ 45.9411 1.65884
$$768$$ 3.97056 0.143275
$$769$$ −35.6569 −1.28582 −0.642910 0.765942i $$-0.722273\pi$$
−0.642910 + 0.765942i $$0.722273\pi$$
$$770$$ −0.343146 −0.0123661
$$771$$ −1.17157 −0.0421932
$$772$$ −27.5563 −0.991775
$$773$$ 32.9706 1.18587 0.592934 0.805251i $$-0.297969\pi$$
0.592934 + 0.805251i $$0.297969\pi$$
$$774$$ 0.242641 0.00872154
$$775$$ −6.48528 −0.232958
$$776$$ −6.72792 −0.241518
$$777$$ −6.48528 −0.232658
$$778$$ 6.20101 0.222317
$$779$$ 0 0
$$780$$ −9.89949 −0.354459
$$781$$ 6.34315 0.226976
$$782$$ −3.71573 −0.132874
$$783$$ 9.07107 0.324174
$$784$$ −19.9706 −0.713234
$$785$$ 21.7990 0.778039
$$786$$ 6.92893 0.247147
$$787$$ −13.4558 −0.479649 −0.239825 0.970816i $$-0.577090\pi$$
−0.239825 + 0.970816i $$0.577090\pi$$
$$788$$ 27.1127 0.965850
$$789$$ −32.1421 −1.14429
$$790$$ −4.68629 −0.166731
$$791$$ −7.31371 −0.260046
$$792$$ −2.24264 −0.0796888
$$793$$ −30.6274 −1.08761
$$794$$ 14.7696 0.524152
$$795$$ 4.00000 0.141865
$$796$$ 30.1421 1.06836
$$797$$ −10.8284 −0.383563 −0.191781 0.981438i $$-0.561426\pi$$
−0.191781 + 0.981438i $$0.561426\pi$$
$$798$$ 0 0
$$799$$ −0.402020 −0.0142225
$$800$$ 4.41421 0.156066
$$801$$ −10.7279 −0.379052
$$802$$ −12.4437 −0.439401
$$803$$ −2.82843 −0.0998130
$$804$$ 21.9411 0.773804
$$805$$ −4.48528 −0.158085
$$806$$ 14.5442 0.512296
$$807$$ −8.38478 −0.295158
$$808$$ 7.65685 0.269367
$$809$$ 49.3137 1.73378 0.866889 0.498502i $$-0.166116\pi$$
0.866889 + 0.498502i $$0.166116\pi$$
$$810$$ −0.414214 −0.0145540
$$811$$ 15.3137 0.537737 0.268869 0.963177i $$-0.413350\pi$$
0.268869 + 0.963177i $$0.413350\pi$$
$$812$$ −9.71573 −0.340955
$$813$$ −30.8284 −1.08120
$$814$$ −6.48528 −0.227309
$$815$$ −7.89949 −0.276707
$$816$$ −3.51472 −0.123040
$$817$$ 0 0
$$818$$ −3.94113 −0.137798
$$819$$ −3.17157 −0.110824
$$820$$ 13.5563 0.473408
$$821$$ 51.4558 1.79582 0.897911 0.440178i $$-0.145085\pi$$
0.897911 + 0.440178i $$0.145085\pi$$
$$822$$ −5.79899 −0.202263
$$823$$ −2.72792 −0.0950894 −0.0475447 0.998869i $$-0.515140\pi$$
−0.0475447 + 0.998869i $$0.515140\pi$$
$$824$$ −2.62742 −0.0915304
$$825$$ 1.41421 0.0492366
$$826$$ −2.05887 −0.0716374
$$827$$ −48.6274 −1.69094 −0.845470 0.534022i $$-0.820680\pi$$
−0.845470 + 0.534022i $$0.820680\pi$$
$$828$$ −14.0000 −0.486534
$$829$$ 38.4853 1.33665 0.668325 0.743870i $$-0.267012\pi$$
0.668325 + 0.743870i $$0.267012\pi$$
$$830$$ 4.34315 0.150753
$$831$$ 10.9706 0.380565
$$832$$ 22.5858 0.783021
$$833$$ 7.79899 0.270219
$$834$$ 0.485281 0.0168039
$$835$$ 10.0000 0.346064
$$836$$ 0 0
$$837$$ −6.48528 −0.224164
$$838$$ −14.4437 −0.498948
$$839$$ −27.1127 −0.936034 −0.468017 0.883719i $$-0.655031\pi$$
−0.468017 + 0.883719i $$0.655031\pi$$
$$840$$ 0.928932 0.0320512
$$841$$ 53.2843 1.83739
$$842$$ 6.08326 0.209643
$$843$$ 14.7279 0.507257
$$844$$ −28.0000 −0.963800
$$845$$ −16.3137 −0.561209
$$846$$ 0.142136 0.00488672
$$847$$ −5.27208 −0.181151
$$848$$ −12.0000 −0.412082
$$849$$ 6.24264 0.214247
$$850$$ −0.485281 −0.0166450
$$851$$ −84.7696 −2.90586
$$852$$ −8.20101 −0.280962
$$853$$ −9.51472 −0.325778 −0.162889 0.986644i $$-0.552081\pi$$
−0.162889 + 0.986644i $$0.552081\pi$$
$$854$$ 1.37258 0.0469688
$$855$$ 0 0
$$856$$ 12.6863 0.433609
$$857$$ −37.9411 −1.29604 −0.648022 0.761622i $$-0.724404\pi$$
−0.648022 + 0.761622i $$0.724404\pi$$
$$858$$ −3.17157 −0.108276
$$859$$ 25.9411 0.885100 0.442550 0.896744i $$-0.354074\pi$$
0.442550 + 0.896744i $$0.354074\pi$$
$$860$$ 1.07107 0.0365231
$$861$$ 4.34315 0.148014
$$862$$ −1.45584 −0.0495862
$$863$$ 31.3137 1.06593 0.532966 0.846137i $$-0.321078\pi$$
0.532966 + 0.846137i $$0.321078\pi$$
$$864$$ 4.41421 0.150175
$$865$$ −6.14214 −0.208839
$$866$$ −0.384776 −0.0130752
$$867$$ −15.6274 −0.530735
$$868$$ 6.94618 0.235769
$$869$$ 16.0000 0.542763
$$870$$ −3.75736 −0.127386
$$871$$ 64.9706 2.20144
$$872$$ 5.02944 0.170318
$$873$$ 4.24264 0.143592
$$874$$ 0 0
$$875$$ −0.585786 −0.0198032
$$876$$ 3.65685 0.123554
$$877$$ −17.8995 −0.604423 −0.302211 0.953241i $$-0.597725\pi$$
−0.302211 + 0.953241i $$0.597725\pi$$
$$878$$ −0.402020 −0.0135675
$$879$$ 31.7990 1.07255
$$880$$ −4.24264 −0.143019
$$881$$ −47.4558 −1.59883 −0.799414 0.600781i $$-0.794857\pi$$
−0.799414 + 0.600781i $$0.794857\pi$$
$$882$$ −2.75736 −0.0928451
$$883$$ 46.5269 1.56576 0.782878 0.622176i $$-0.213751\pi$$
0.782878 + 0.622176i $$0.213751\pi$$
$$884$$ −11.5980 −0.390082
$$885$$ 8.48528 0.285230
$$886$$ −0.544156 −0.0182813
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 17.5563 0.589153
$$889$$ −4.68629 −0.157173
$$890$$ 4.44365 0.148952
$$891$$ 1.41421 0.0473779
$$892$$ 11.5980 0.388329
$$893$$ 0 0
$$894$$ −1.51472 −0.0506598
$$895$$ 17.1716 0.573982
$$896$$ −6.18377 −0.206585
$$897$$ −41.4558 −1.38417
$$898$$ −3.27208 −0.109191
$$899$$ −58.8284 −1.96204
$$900$$ −1.82843 −0.0609476
$$901$$ 4.68629 0.156123
$$902$$ 4.34315 0.144611
$$903$$ 0.343146 0.0114192
$$904$$ 19.7990 0.658505
$$905$$ −19.1716 −0.637285
$$906$$ −7.37258 −0.244938
$$907$$ −38.1421 −1.26649 −0.633244 0.773952i $$-0.718277\pi$$
−0.633244 + 0.773952i $$0.718277\pi$$
$$908$$ −34.6863 −1.15111
$$909$$ −4.82843 −0.160149
$$910$$ 1.31371 0.0435490
$$911$$ 23.3137 0.772418 0.386209 0.922411i $$-0.373784\pi$$
0.386209 + 0.922411i $$0.373784\pi$$
$$912$$ 0 0
$$913$$ −14.8284 −0.490749
$$914$$ 11.9411 0.394977
$$915$$ −5.65685 −0.187010
$$916$$ 3.02944 0.100095
$$917$$ 9.79899 0.323591
$$918$$ −0.485281 −0.0160167
$$919$$ −12.0000 −0.395843 −0.197922 0.980218i $$-0.563419\pi$$
−0.197922 + 0.980218i $$0.563419\pi$$
$$920$$ 12.1421 0.400314
$$921$$ −7.79899 −0.256985
$$922$$ 4.42641 0.145776
$$923$$ −24.2843 −0.799327
$$924$$ −1.51472 −0.0498306
$$925$$ −11.0711 −0.364014
$$926$$ −3.35534 −0.110263
$$927$$ 1.65685 0.0544182
$$928$$ 40.0416 1.31443
$$929$$ −51.4558 −1.68821 −0.844106 0.536177i $$-0.819868\pi$$
−0.844106 + 0.536177i $$0.819868\pi$$
$$930$$ 2.68629 0.0880870
$$931$$ 0 0
$$932$$ 15.2548 0.499689
$$933$$ −32.2426 −1.05558
$$934$$ 6.76955 0.221507
$$935$$ 1.65685 0.0541849
$$936$$ 8.58579 0.280635
$$937$$ −18.7696 −0.613175 −0.306587 0.951843i $$-0.599187\pi$$
−0.306587 + 0.951843i $$0.599187\pi$$
$$938$$ −2.91169 −0.0950700
$$939$$ −9.51472 −0.310501
$$940$$ 0.627417 0.0204641
$$941$$ 17.5563 0.572321 0.286160 0.958182i $$-0.407621\pi$$
0.286160 + 0.958182i $$0.407621\pi$$
$$942$$ −9.02944 −0.294195
$$943$$ 56.7696 1.84867
$$944$$ −25.4558 −0.828517
$$945$$ −0.585786 −0.0190556
$$946$$ 0.343146 0.0111566
$$947$$ 12.8284 0.416868 0.208434 0.978036i $$-0.433163\pi$$
0.208434 + 0.978036i $$0.433163\pi$$
$$948$$ −20.6863 −0.671860
$$949$$ 10.8284 0.351506
$$950$$ 0 0
$$951$$ −11.3137 −0.366872
$$952$$ 1.08831 0.0352724
$$953$$ 5.85786 0.189755 0.0948774 0.995489i $$-0.469754\pi$$
0.0948774 + 0.995489i $$0.469754\pi$$
$$954$$ −1.65685 −0.0536426
$$955$$ −1.89949 −0.0614662
$$956$$ −4.72792 −0.152912
$$957$$ 12.8284 0.414684
$$958$$ −15.8162 −0.510999
$$959$$ −8.20101 −0.264824
$$960$$ 4.17157 0.134637
$$961$$ 11.0589 0.356738
$$962$$ 24.8284 0.800501
$$963$$ −8.00000 −0.257796
$$964$$ −27.3726 −0.881612
$$965$$ −15.0711 −0.485155
$$966$$ 1.85786 0.0597758
$$967$$ −27.8995 −0.897187 −0.448594 0.893736i $$-0.648075\pi$$
−0.448594 + 0.893736i $$0.648075\pi$$
$$968$$ 14.2721 0.458722
$$969$$ 0 0
$$970$$ −1.75736 −0.0564254
$$971$$ −17.6569 −0.566635 −0.283318 0.959026i $$-0.591435\pi$$
−0.283318 + 0.959026i $$0.591435\pi$$
$$972$$ −1.82843 −0.0586468
$$973$$ 0.686292 0.0220015
$$974$$ 1.17157 0.0375396
$$975$$ −5.41421 −0.173394
$$976$$ 16.9706 0.543214
$$977$$ −39.5980 −1.26685 −0.633426 0.773803i $$-0.718352\pi$$
−0.633426 + 0.773803i $$0.718352\pi$$
$$978$$ 3.27208 0.104630
$$979$$ −15.1716 −0.484886
$$980$$ −12.1716 −0.388807
$$981$$ −3.17157 −0.101261
$$982$$ 9.07107 0.289469
$$983$$ −31.9411 −1.01876 −0.509382 0.860541i $$-0.670126\pi$$
−0.509382 + 0.860541i $$0.670126\pi$$
$$984$$ −11.7574 −0.374811
$$985$$ 14.8284 0.472473
$$986$$ −4.40202 −0.140189
$$987$$ 0.201010 0.00639822
$$988$$ 0 0
$$989$$ 4.48528 0.142624
$$990$$ −0.585786 −0.0186175
$$991$$ 61.6569 1.95859 0.979297 0.202427i $$-0.0648830\pi$$
0.979297 + 0.202427i $$0.0648830\pi$$
$$992$$ −28.6274 −0.908921
$$993$$ −7.17157 −0.227583
$$994$$ 1.08831 0.0345192
$$995$$ 16.4853 0.522619
$$996$$ 19.1716 0.607475
$$997$$ −50.4853 −1.59888 −0.799442 0.600743i $$-0.794872\pi$$
−0.799442 + 0.600743i $$0.794872\pi$$
$$998$$ −9.85786 −0.312045
$$999$$ −11.0711 −0.350273
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 5415.2.a.p.1.2 2
19.18 odd 2 285.2.a.f.1.1 2
57.56 even 2 855.2.a.e.1.2 2
76.75 even 2 4560.2.a.bj.1.2 2
95.18 even 4 1425.2.c.j.799.3 4
95.37 even 4 1425.2.c.j.799.2 4
95.94 odd 2 1425.2.a.l.1.2 2
285.284 even 2 4275.2.a.x.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.1 2 19.18 odd 2
855.2.a.e.1.2 2 57.56 even 2
1425.2.a.l.1.2 2 95.94 odd 2
1425.2.c.j.799.2 4 95.37 even 4
1425.2.c.j.799.3 4 95.18 even 4
4275.2.a.x.1.1 2 285.284 even 2
4560.2.a.bj.1.2 2 76.75 even 2
5415.2.a.p.1.2 2 1.1 even 1 trivial