Properties

 Label 1425.2.a.l.1.2 Level $1425$ Weight $2$ Character 1425.1 Self dual yes Analytic conductor $11.379$ 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] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, 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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.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: $$11.3786822880$$ 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 $$\chi$$ $$=$$ 1425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -0.585786 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -0.585786 q^{7} -1.58579 q^{8} +1.00000 q^{9} +1.41421 q^{11} -1.82843 q^{12} -5.41421 q^{13} -0.242641 q^{14} +3.00000 q^{16} +1.17157 q^{17} +0.414214 q^{18} +1.00000 q^{19} -0.585786 q^{21} +0.585786 q^{22} -7.65685 q^{23} -1.58579 q^{24} -2.24264 q^{26} +1.00000 q^{27} +1.07107 q^{28} -9.07107 q^{29} +6.48528 q^{31} +4.41421 q^{32} +1.41421 q^{33} +0.485281 q^{34} -1.82843 q^{36} -11.0711 q^{37} +0.414214 q^{38} -5.41421 q^{39} -7.41421 q^{41} -0.242641 q^{42} -0.585786 q^{43} -2.58579 q^{44} -3.17157 q^{46} -0.343146 q^{47} +3.00000 q^{48} -6.65685 q^{49} +1.17157 q^{51} +9.89949 q^{52} -4.00000 q^{53} +0.414214 q^{54} +0.928932 q^{56} +1.00000 q^{57} -3.75736 q^{58} +8.48528 q^{59} +5.65685 q^{61} +2.68629 q^{62} -0.585786 q^{63} -4.17157 q^{64} +0.585786 q^{66} -12.0000 q^{67} -2.14214 q^{68} -7.65685 q^{69} -4.48528 q^{71} -1.58579 q^{72} +2.00000 q^{73} -4.58579 q^{74} -1.82843 q^{76} -0.828427 q^{77} -2.24264 q^{78} -11.3137 q^{79} +1.00000 q^{81} -3.07107 q^{82} +10.4853 q^{83} +1.07107 q^{84} -0.242641 q^{86} -9.07107 q^{87} -2.24264 q^{88} +10.7279 q^{89} +3.17157 q^{91} +14.0000 q^{92} +6.48528 q^{93} -0.142136 q^{94} +4.41421 q^{96} +4.24264 q^{97} -2.75736 q^{98} +1.41421 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{12} - 8 q^{13} + 8 q^{14} + 6 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} - 4 q^{21} + 4 q^{22} - 4 q^{23} - 6 q^{24} + 4 q^{26} + 2 q^{27} - 12 q^{28} - 4 q^{29} - 4 q^{31} + 6 q^{32} - 16 q^{34} + 2 q^{36} - 8 q^{37} - 2 q^{38} - 8 q^{39} - 12 q^{41} + 8 q^{42} - 4 q^{43} - 8 q^{44} - 12 q^{46} - 12 q^{47} + 6 q^{48} - 2 q^{49} + 8 q^{51} - 8 q^{53} - 2 q^{54} + 16 q^{56} + 2 q^{57} - 16 q^{58} + 28 q^{62} - 4 q^{63} - 14 q^{64} + 4 q^{66} - 24 q^{67} + 24 q^{68} - 4 q^{69} + 8 q^{71} - 6 q^{72} + 4 q^{73} - 12 q^{74} + 2 q^{76} + 4 q^{77} + 4 q^{78} + 2 q^{81} + 8 q^{82} + 4 q^{83} - 12 q^{84} + 8 q^{86} - 4 q^{87} + 4 q^{88} - 4 q^{89} + 12 q^{91} + 28 q^{92} - 4 q^{93} + 28 q^{94} + 6 q^{96} - 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^12 - 8 * q^13 + 8 * q^14 + 6 * q^16 + 8 * q^17 - 2 * q^18 + 2 * q^19 - 4 * q^21 + 4 * q^22 - 4 * q^23 - 6 * q^24 + 4 * q^26 + 2 * q^27 - 12 * q^28 - 4 * q^29 - 4 * q^31 + 6 * q^32 - 16 * q^34 + 2 * q^36 - 8 * q^37 - 2 * q^38 - 8 * q^39 - 12 * q^41 + 8 * q^42 - 4 * q^43 - 8 * q^44 - 12 * q^46 - 12 * q^47 + 6 * q^48 - 2 * q^49 + 8 * q^51 - 8 * q^53 - 2 * q^54 + 16 * q^56 + 2 * q^57 - 16 * q^58 + 28 * q^62 - 4 * q^63 - 14 * q^64 + 4 * q^66 - 24 * q^67 + 24 * q^68 - 4 * q^69 + 8 * q^71 - 6 * q^72 + 4 * q^73 - 12 * q^74 + 2 * q^76 + 4 * q^77 + 4 * q^78 + 2 * q^81 + 8 * q^82 + 4 * q^83 - 12 * q^84 + 8 * q^86 - 4 * q^87 + 4 * q^88 - 4 * q^89 + 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
$$5$$ 0 0
$$6$$ 0.414214 0.169102
$$7$$ −0.585786 −0.221406 −0.110703 0.993854i $$-0.535310\pi$$
−0.110703 + 0.993854i $$0.535310\pi$$
$$8$$ −1.58579 −0.560660
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 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
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 1.17157 0.284148 0.142074 0.989856i $$-0.454623\pi$$
0.142074 + 0.989856i $$0.454623\pi$$
$$18$$ 0.414214 0.0976311
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −0.585786 −0.127829
$$22$$ 0.585786 0.124890
$$23$$ −7.65685 −1.59656 −0.798282 0.602284i $$-0.794258\pi$$
−0.798282 + 0.602284i $$0.794258\pi$$
$$24$$ −1.58579 −0.323697
$$25$$ 0 0
$$26$$ −2.24264 −0.439818
$$27$$ 1.00000 0.192450
$$28$$ 1.07107 0.202413
$$29$$ −9.07107 −1.68446 −0.842228 0.539122i $$-0.818756\pi$$
−0.842228 + 0.539122i $$0.818756\pi$$
$$30$$ 0 0
$$31$$ 6.48528 1.16479 0.582395 0.812906i $$-0.302116\pi$$
0.582395 + 0.812906i $$0.302116\pi$$
$$32$$ 4.41421 0.780330
$$33$$ 1.41421 0.246183
$$34$$ 0.485281 0.0832251
$$35$$ 0 0
$$36$$ −1.82843 −0.304738
$$37$$ −11.0711 −1.82007 −0.910036 0.414529i $$-0.863946\pi$$
−0.910036 + 0.414529i $$0.863946\pi$$
$$38$$ 0.414214 0.0671943
$$39$$ −5.41421 −0.866968
$$40$$ 0 0
$$41$$ −7.41421 −1.15791 −0.578953 0.815361i $$-0.696538\pi$$
−0.578953 + 0.815361i $$0.696538\pi$$
$$42$$ −0.242641 −0.0374403
$$43$$ −0.585786 −0.0893316 −0.0446658 0.999002i $$-0.514222\pi$$
−0.0446658 + 0.999002i $$0.514222\pi$$
$$44$$ −2.58579 −0.389822
$$45$$ 0 0
$$46$$ −3.17157 −0.467623
$$47$$ −0.343146 −0.0500530 −0.0250265 0.999687i $$-0.507967\pi$$
−0.0250265 + 0.999687i $$0.507967\pi$$
$$48$$ 3.00000 0.433013
$$49$$ −6.65685 −0.950979
$$50$$ 0 0
$$51$$ 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
$$55$$ 0 0
$$56$$ 0.928932 0.124134
$$57$$ 1.00000 0.132453
$$58$$ −3.75736 −0.493365
$$59$$ 8.48528 1.10469 0.552345 0.833616i $$-0.313733\pi$$
0.552345 + 0.833616i $$0.313733\pi$$
$$60$$ 0 0
$$61$$ 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
$$65$$ 0 0
$$66$$ 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 0
$$71$$ −4.48528 −0.532305 −0.266152 0.963931i $$-0.585752\pi$$
−0.266152 + 0.963931i $$0.585752\pi$$
$$72$$ −1.58579 −0.186887
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −4.58579 −0.533087
$$75$$ 0 0
$$76$$ −1.82843 −0.209735
$$77$$ −0.828427 −0.0944080
$$78$$ −2.24264 −0.253929
$$79$$ −11.3137 −1.27289 −0.636446 0.771321i $$-0.719596\pi$$
−0.636446 + 0.771321i $$0.719596\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −3.07107 −0.339143
$$83$$ 10.4853 1.15091 0.575455 0.817834i $$-0.304825\pi$$
0.575455 + 0.817834i $$0.304825\pi$$
$$84$$ 1.07107 0.116863
$$85$$ 0 0
$$86$$ −0.242641 −0.0261646
$$87$$ −9.07107 −0.972521
$$88$$ −2.24264 −0.239066
$$89$$ 10.7279 1.13716 0.568579 0.822629i $$-0.307493\pi$$
0.568579 + 0.822629i $$0.307493\pi$$
$$90$$ 0 0
$$91$$ 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
$$100$$ 0 0
$$101$$ −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 0
$$106$$ −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 $$-0.451464\pi$$
0.151891 + 0.988397i $$0.451464\pi$$
$$110$$ 0 0
$$111$$ −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 $$0.699793\pi$$
$$114$$ 0.414214 0.0387947
$$115$$ 0 0
$$116$$ 16.5858 1.53995
$$117$$ −5.41421 −0.500544
$$118$$ 3.51472 0.323556
$$119$$ −0.686292 −0.0629122
$$120$$ 0 0
$$121$$ −9.00000 −0.818182
$$122$$ 2.34315 0.212138
$$123$$ −7.41421 −0.668517
$$124$$ −11.8579 −1.06487
$$125$$ 0 0
$$126$$ −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
$$130$$ 0 0
$$131$$ 16.7279 1.46153 0.730763 0.682632i $$-0.239165\pi$$
0.730763 + 0.682632i $$0.239165\pi$$
$$132$$ −2.58579 −0.225064
$$133$$ −0.585786 −0.0507941
$$134$$ −4.97056 −0.429391
$$135$$ 0 0
$$136$$ −1.85786 −0.159311
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ −3.17157 −0.269982
$$139$$ 1.17157 0.0993715 0.0496858 0.998765i $$-0.484178\pi$$
0.0496858 + 0.998765i $$0.484178\pi$$
$$140$$ 0 0
$$141$$ −0.343146 −0.0288981
$$142$$ −1.85786 −0.155909
$$143$$ −7.65685 −0.640298
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ 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 0
$$151$$ 17.7990 1.44846 0.724231 0.689558i $$-0.242195\pi$$
0.724231 + 0.689558i $$0.242195\pi$$
$$152$$ −1.58579 −0.128624
$$153$$ 1.17157 0.0947161
$$154$$ −0.343146 −0.0276515
$$155$$ 0 0
$$156$$ 9.89949 0.792594
$$157$$ 21.7990 1.73975 0.869874 0.493273i $$-0.164200\pi$$
0.869874 + 0.493273i $$0.164200\pi$$
$$158$$ −4.68629 −0.372821
$$159$$ −4.00000 −0.317221
$$160$$ 0 0
$$161$$ 4.48528 0.353490
$$162$$ 0.414214 0.0325437
$$163$$ −7.89949 −0.618736 −0.309368 0.950942i $$-0.600118\pi$$
−0.309368 + 0.950942i $$0.600118\pi$$
$$164$$ 13.5563 1.05857
$$165$$ 0 0
$$166$$ 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 0
$$171$$ 1.00000 0.0764719
$$172$$ 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 0
$$176$$ 4.24264 0.319801
$$177$$ 8.48528 0.637793
$$178$$ 4.44365 0.333066
$$179$$ 17.1716 1.28346 0.641732 0.766929i $$-0.278216\pi$$
0.641732 + 0.766929i $$0.278216\pi$$
$$180$$ 0 0
$$181$$ −19.1716 −1.42501 −0.712506 0.701666i $$-0.752440\pi$$
−0.712506 + 0.701666i $$0.752440\pi$$
$$182$$ 1.31371 0.0973786
$$183$$ 5.65685 0.418167
$$184$$ 12.1421 0.895130
$$185$$ 0 0
$$186$$ 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
$$195$$ 0 0
$$196$$ 12.1716 0.869398
$$197$$ 14.8284 1.05648 0.528241 0.849095i $$-0.322852\pi$$
0.528241 + 0.849095i $$0.322852\pi$$
$$198$$ 0.585786 0.0416300
$$199$$ −16.4853 −1.16861 −0.584305 0.811534i $$-0.698633\pi$$
−0.584305 + 0.811534i $$0.698633\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ −2.00000 −0.140720
$$203$$ 5.31371 0.372949
$$204$$ −2.14214 −0.149979
$$205$$ 0 0
$$206$$ 0.686292 0.0478162
$$207$$ −7.65685 −0.532188
$$208$$ −16.2426 −1.12622
$$209$$ 1.41421 0.0978232
$$210$$ 0 0
$$211$$ −15.3137 −1.05424 −0.527120 0.849791i $$-0.676728\pi$$
−0.527120 + 0.849791i $$0.676728\pi$$
$$212$$ 7.31371 0.502308
$$213$$ −4.48528 −0.307326
$$214$$ −3.31371 −0.226520
$$215$$ 0 0
$$216$$ −1.58579 −0.107899
$$217$$ −3.79899 −0.257892
$$218$$ 1.31371 0.0889756
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ −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
$$225$$ 0 0
$$226$$ −5.17157 −0.344008
$$227$$ 18.9706 1.25912 0.629560 0.776952i $$-0.283235\pi$$
0.629560 + 0.776952i $$0.283235\pi$$
$$228$$ −1.82843 −0.121091
$$229$$ −1.65685 −0.109488 −0.0547440 0.998500i $$-0.517434\pi$$
−0.0547440 + 0.998500i $$0.517434\pi$$
$$230$$ 0 0
$$231$$ −0.828427 −0.0545065
$$232$$ 14.3848 0.944407
$$233$$ 8.34315 0.546578 0.273289 0.961932i $$-0.411889\pi$$
0.273289 + 0.961932i $$0.411889\pi$$
$$234$$ −2.24264 −0.146606
$$235$$ 0 0
$$236$$ −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 $$0.473349\pi$$
$$240$$ 0 0
$$241$$ −14.9706 −0.964339 −0.482169 0.876078i $$-0.660151\pi$$
−0.482169 + 0.876078i $$0.660151\pi$$
$$242$$ −3.72792 −0.239640
$$243$$ 1.00000 0.0641500
$$244$$ −10.3431 −0.662152
$$245$$ 0 0
$$246$$ −3.07107 −0.195804
$$247$$ −5.41421 −0.344498
$$248$$ −10.2843 −0.653052
$$249$$ 10.4853 0.664478
$$250$$ 0 0
$$251$$ 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
$$255$$ 0 0
$$256$$ 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
$$260$$ 0 0
$$261$$ −9.07107 −0.561485
$$262$$ 6.92893 0.428071
$$263$$ 32.1421 1.98197 0.990984 0.133977i $$-0.0427747\pi$$
0.990984 + 0.133977i $$0.0427747\pi$$
$$264$$ −2.24264 −0.138025
$$265$$ 0 0
$$266$$ −0.242641 −0.0148773
$$267$$ 10.7279 0.656538
$$268$$ 21.9411 1.34027
$$269$$ 8.38478 0.511229 0.255614 0.966779i $$-0.417722\pi$$
0.255614 + 0.966779i $$0.417722\pi$$
$$270$$ 0 0
$$271$$ −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
$$275$$ 0 0
$$276$$ 14.0000 0.842701
$$277$$ −10.9706 −0.659157 −0.329579 0.944128i $$-0.606907\pi$$
−0.329579 + 0.944128i $$0.606907\pi$$
$$278$$ 0.485281 0.0291052
$$279$$ 6.48528 0.388264
$$280$$ 0 0
$$281$$ −14.7279 −0.878594 −0.439297 0.898342i $$-0.644772\pi$$
−0.439297 + 0.898342i $$0.644772\pi$$
$$282$$ −0.142136 −0.00846405
$$283$$ −6.24264 −0.371086 −0.185543 0.982636i $$-0.559404\pi$$
−0.185543 + 0.982636i $$0.559404\pi$$
$$284$$ 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
$$290$$ 0 0
$$291$$ 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
$$295$$ 0 0
$$296$$ 17.5563 1.02044
$$297$$ 1.41421 0.0820610
$$298$$ −1.51472 −0.0877453
$$299$$ 41.4558 2.39745
$$300$$ 0 0
$$301$$ 0.343146 0.0197786
$$302$$ 7.37258 0.424244
$$303$$ −4.82843 −0.277386
$$304$$ 3.00000 0.172062
$$305$$ 0 0
$$306$$ 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
$$310$$ 0 0
$$311$$ −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 $$-0.413339\pi$$
0.268902 + 0.963168i $$0.413339\pi$$
$$314$$ 9.02944 0.509561
$$315$$ 0 0
$$316$$ 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
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 1.85786 0.103535
$$323$$ 1.17157 0.0651881
$$324$$ −1.82843 −0.101579
$$325$$ 0 0
$$326$$ −3.27208 −0.181224
$$327$$ 3.17157 0.175388
$$328$$ 11.7574 0.649192
$$329$$ 0.201010 0.0110820
$$330$$ 0 0
$$331$$ 7.17157 0.394185 0.197093 0.980385i $$-0.436850\pi$$
0.197093 + 0.980385i $$0.436850\pi$$
$$332$$ −19.1716 −1.05218
$$333$$ −11.0711 −0.606691
$$334$$ −4.14214 −0.226648
$$335$$ 0 0
$$336$$ −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
$$340$$ 0 0
$$341$$ 9.17157 0.496669
$$342$$ 0.414214 0.0223981
$$343$$ 8.00000 0.431959
$$344$$ 0.928932 0.0500847
$$345$$ 0 0
$$346$$ 2.54416 0.136775
$$347$$ −10.4853 −0.562879 −0.281440 0.959579i $$-0.590812\pi$$
−0.281440 + 0.959579i $$0.590812\pi$$
$$348$$ 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 0
$$351$$ −5.41421 −0.288989
$$352$$ 6.24264 0.332734
$$353$$ −3.65685 −0.194635 −0.0973174 0.995253i $$-0.531026\pi$$
−0.0973174 + 0.995253i $$0.531026\pi$$
$$354$$ 3.51472 0.186805
$$355$$ 0 0
$$356$$ −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 $$0.415869\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −7.94113 −0.417376
$$363$$ −9.00000 −0.472377
$$364$$ −5.79899 −0.303950
$$365$$ 0 0
$$366$$ 2.34315 0.122478
$$367$$ 19.4142 1.01341 0.506707 0.862118i $$-0.330863\pi$$
0.506707 + 0.862118i $$0.330863\pi$$
$$368$$ −22.9706 −1.19742
$$369$$ −7.41421 −0.385969
$$370$$ 0 0
$$371$$ 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
$$375$$ 0 0
$$376$$ 0.544156 0.0280627
$$377$$ 49.1127 2.52943
$$378$$ −0.242641 −0.0124801
$$379$$ 24.1421 1.24010 0.620049 0.784563i $$-0.287113\pi$$
0.620049 + 0.784563i $$0.287113\pi$$
$$380$$ 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 0
$$386$$ 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 $$0.376090\pi$$
$$390$$ 0 0
$$391$$ −8.97056 −0.453661
$$392$$ 10.5563 0.533176
$$393$$ 16.7279 0.843812
$$394$$ 6.14214 0.309436
$$395$$ 0 0
$$396$$ −2.58579 −0.129941
$$397$$ −35.6569 −1.78957 −0.894783 0.446501i $$-0.852670\pi$$
−0.894783 + 0.446501i $$0.852670\pi$$
$$398$$ −6.82843 −0.342278
$$399$$ −0.585786 −0.0293260
$$400$$ 0 0
$$401$$ 30.0416 1.50021 0.750104 0.661320i $$-0.230004\pi$$
0.750104 + 0.661320i $$0.230004\pi$$
$$402$$ −4.97056 −0.247909
$$403$$ −35.1127 −1.74909
$$404$$ 8.82843 0.439231
$$405$$ 0 0
$$406$$ 2.20101 0.109234
$$407$$ −15.6569 −0.776081
$$408$$ −1.85786 −0.0919780
$$409$$ 9.51472 0.470473 0.235236 0.971938i $$-0.424414\pi$$
0.235236 + 0.971938i $$0.424414\pi$$
$$410$$ 0 0
$$411$$ 14.0000 0.690569
$$412$$ −3.02944 −0.149250
$$413$$ −4.97056 −0.244585
$$414$$ −3.17157 −0.155874
$$415$$ 0 0
$$416$$ −23.8995 −1.17177
$$417$$ 1.17157 0.0573722
$$418$$ 0.585786 0.0286518
$$419$$ −34.8701 −1.70351 −0.851757 0.523937i $$-0.824463\pi$$
−0.851757 + 0.523937i $$0.824463\pi$$
$$420$$ 0 0
$$421$$ −14.6863 −0.715766 −0.357883 0.933766i $$-0.616501\pi$$
−0.357883 + 0.933766i $$0.616501\pi$$
$$422$$ −6.34315 −0.308780
$$423$$ −0.343146 −0.0166843
$$424$$ 6.34315 0.308050
$$425$$ 0 0
$$426$$ −1.85786 −0.0900138
$$427$$ −3.31371 −0.160362
$$428$$ 14.6274 0.707043
$$429$$ −7.65685 −0.369676
$$430$$ 0 0
$$431$$ 3.51472 0.169298 0.0846490 0.996411i $$-0.473023\pi$$
0.0846490 + 0.996411i $$0.473023\pi$$
$$432$$ 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
$$435$$ 0 0
$$436$$ −5.79899 −0.277721
$$437$$ −7.65685 −0.366277
$$438$$ 0.828427 0.0395838
$$439$$ 0.970563 0.0463224 0.0231612 0.999732i $$-0.492627\pi$$
0.0231612 + 0.999732i $$0.492627\pi$$
$$440$$ 0 0
$$441$$ −6.65685 −0.316993
$$442$$ −2.62742 −0.124973
$$443$$ 1.31371 0.0624162 0.0312081 0.999513i $$-0.490065\pi$$
0.0312081 + 0.999513i $$0.490065\pi$$
$$444$$ 20.2426 0.960673
$$445$$ 0 0
$$446$$ −2.62742 −0.124412
$$447$$ −3.65685 −0.172963
$$448$$ 2.44365 0.115452
$$449$$ 7.89949 0.372800 0.186400 0.982474i $$-0.440318\pi$$
0.186400 + 0.982474i $$0.440318\pi$$
$$450$$ 0 0
$$451$$ −10.4853 −0.493733
$$452$$ 22.8284 1.07376
$$453$$ 17.7990 0.836269
$$454$$ 7.85786 0.368788
$$455$$ 0 0
$$456$$ −1.58579 −0.0742613
$$457$$ −28.8284 −1.34854 −0.674268 0.738486i $$-0.735541\pi$$
−0.674268 + 0.738486i $$0.735541\pi$$
$$458$$ −0.686292 −0.0320683
$$459$$ 1.17157 0.0546843
$$460$$ 0 0
$$461$$ 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 $$-0.439725\pi$$
0.188231 + 0.982125i $$0.439725\pi$$
$$464$$ −27.2132 −1.26334
$$465$$ 0 0
$$466$$ 3.45584 0.160089
$$467$$ −16.3431 −0.756271 −0.378135 0.925750i $$-0.623435\pi$$
−0.378135 + 0.925750i $$0.623435\pi$$
$$468$$ 9.89949 0.457604
$$469$$ 7.02944 0.324589
$$470$$ 0 0
$$471$$ 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 $$0.837392\pi$$
$$480$$ 0 0
$$481$$ 59.9411 2.73308
$$482$$ −6.20101 −0.282448
$$483$$ 4.48528 0.204087
$$484$$ 16.4558 0.747993
$$485$$ 0 0
$$486$$ 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
$$490$$ 0 0
$$491$$ 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
$$494$$ −2.24264 −0.100901
$$495$$ 0 0
$$496$$ 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 $$0.678821\pi$$
$$500$$ 0 0
$$501$$ −10.0000 −0.446767
$$502$$ 5.35534 0.239020
$$503$$ −0.828427 −0.0369377 −0.0184689 0.999829i $$-0.505879\pi$$
−0.0184689 + 0.999829i $$0.505879\pi$$
$$504$$ 0.928932 0.0413779
$$505$$ 0 0
$$506$$ −4.48528 −0.199395
$$507$$ 16.3137 0.724517
$$508$$ 14.6274 0.648987
$$509$$ 32.3848 1.43543 0.717715 0.696337i $$-0.245188\pi$$
0.717715 + 0.696337i $$0.245188\pi$$
$$510$$ 0 0
$$511$$ −1.17157 −0.0518273
$$512$$ 22.7574 1.00574
$$513$$ 1.00000 0.0441511
$$514$$ −0.485281 −0.0214048
$$515$$ 0 0
$$516$$ 1.07107 0.0471511
$$517$$ −0.485281 −0.0213427
$$518$$ 2.68629 0.118029
$$519$$ 6.14214 0.269610
$$520$$ 0 0
$$521$$ −24.3848 −1.06832 −0.534158 0.845385i $$-0.679371\pi$$
−0.534158 + 0.845385i $$0.679371\pi$$
$$522$$ −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 0
$$526$$ 13.3137 0.580505
$$527$$ 7.59798 0.330973
$$528$$ 4.24264 0.184637
$$529$$ 35.6274 1.54902
$$530$$ 0 0
$$531$$ 8.48528 0.368230
$$532$$ 1.07107 0.0464367
$$533$$ 40.1421 1.73875
$$534$$ 4.44365 0.192296
$$535$$ 0 0
$$536$$ 19.0294 0.821946
$$537$$ 17.1716 0.741008
$$538$$ 3.47309 0.149735
$$539$$ −9.41421 −0.405499
$$540$$ 0 0
$$541$$ −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
$$545$$ 0 0
$$546$$ 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 0
$$551$$ −9.07107 −0.386440
$$552$$ 12.1421 0.516804
$$553$$ 6.62742 0.281826
$$554$$ −4.54416 −0.193063
$$555$$ 0 0
$$556$$ −2.14214 −0.0908468
$$557$$ −10.0000 −0.423714 −0.211857 0.977301i $$-0.567951\pi$$
−0.211857 + 0.977301i $$0.567951\pi$$
$$558$$ 2.68629 0.113720
$$559$$ 3.17157 0.134143
$$560$$ 0 0
$$561$$ 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
$$565$$ 0 0
$$566$$ −2.58579 −0.108689
$$567$$ −0.585786 −0.0246007
$$568$$ 7.11270 0.298442
$$569$$ −6.72792 −0.282049 −0.141025 0.990006i $$-0.545040\pi$$
−0.141025 + 0.990006i $$0.545040\pi$$
$$570$$ 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
$$575$$ 0 0
$$576$$ −4.17157 −0.173816
$$577$$ 37.7990 1.57359 0.786796 0.617213i $$-0.211738\pi$$
0.786796 + 0.617213i $$0.211738\pi$$
$$578$$ −6.47309 −0.269245
$$579$$ 15.0711 0.626332
$$580$$ 0 0
$$581$$ −6.14214 −0.254819
$$582$$ 1.75736 0.0728449
$$583$$ −5.65685 −0.234283
$$584$$ −3.17157 −0.131241
$$585$$ 0 0
$$586$$ 13.1716 0.544113
$$587$$ −11.6569 −0.481130 −0.240565 0.970633i $$-0.577333\pi$$
−0.240565 + 0.970633i $$0.577333\pi$$
$$588$$ 12.1716 0.501947
$$589$$ 6.48528 0.267221
$$590$$ 0 0
$$591$$ 14.8284 0.609960
$$592$$ −33.2132 −1.36505
$$593$$ 29.3137 1.20377 0.601885 0.798583i $$-0.294417\pi$$
0.601885 + 0.798583i $$0.294417\pi$$
$$594$$ 0.585786 0.0240351
$$595$$ 0 0
$$596$$ 6.68629 0.273881
$$597$$ −16.4853 −0.674698
$$598$$ 17.1716 0.702198
$$599$$ −21.9411 −0.896490 −0.448245 0.893911i $$-0.647951\pi$$
−0.448245 + 0.893911i $$0.647951\pi$$
$$600$$ 0 0
$$601$$ −28.8284 −1.17594 −0.587968 0.808884i $$-0.700072\pi$$
−0.587968 + 0.808884i $$0.700072\pi$$
$$602$$ 0.142136 0.00579302
$$603$$ −12.0000 −0.488678
$$604$$ −32.5442 −1.32420
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ 2.14214 0.0869466 0.0434733 0.999055i $$-0.486158\pi$$
0.0434733 + 0.999055i $$0.486158\pi$$
$$608$$ 4.41421 0.179020
$$609$$ 5.31371 0.215322
$$610$$ 0 0
$$611$$ 1.85786 0.0751611
$$612$$ −2.14214 −0.0865907
$$613$$ −25.1127 −1.01429 −0.507146 0.861860i $$-0.669300\pi$$
−0.507146 + 0.861860i $$0.669300\pi$$
$$614$$ −3.23045 −0.130370
$$615$$ 0 0
$$616$$ 1.31371 0.0529308
$$617$$ −10.1421 −0.408307 −0.204154 0.978939i $$-0.565444\pi$$
−0.204154 + 0.978939i $$0.565444\pi$$
$$618$$ 0.686292 0.0276067
$$619$$ −43.7990 −1.76043 −0.880215 0.474575i $$-0.842602\pi$$
−0.880215 + 0.474575i $$0.842602\pi$$
$$620$$ 0 0
$$621$$ −7.65685 −0.307259
$$622$$ −13.3553 −0.535500
$$623$$ −6.28427 −0.251774
$$624$$ −16.2426 −0.650226
$$625$$ 0 0
$$626$$ 3.94113 0.157519
$$627$$ 1.41421 0.0564782
$$628$$ −39.8579 −1.59050
$$629$$ −12.9706 −0.517170
$$630$$ 0 0
$$631$$ 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
$$635$$ 0 0
$$636$$ 7.31371 0.290007
$$637$$ 36.0416 1.42802
$$638$$ −5.31371 −0.210372
$$639$$ −4.48528 −0.177435
$$640$$ 0 0
$$641$$ −8.58579 −0.339118 −0.169559 0.985520i $$-0.554234\pi$$
−0.169559 + 0.985520i $$0.554234\pi$$
$$642$$ −3.31371 −0.130782
$$643$$ −6.04163 −0.238259 −0.119129 0.992879i $$-0.538010\pi$$
−0.119129 + 0.992879i $$0.538010\pi$$
$$644$$ −8.20101 −0.323165
$$645$$ 0 0
$$646$$ 0.485281 0.0190931
$$647$$ −45.1127 −1.77356 −0.886782 0.462189i $$-0.847064\pi$$
−0.886782 + 0.462189i $$0.847064\pi$$
$$648$$ −1.58579 −0.0622956
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ −3.79899 −0.148894
$$652$$ 14.4437 0.565657
$$653$$ −42.4264 −1.66027 −0.830137 0.557560i $$-0.811738\pi$$
−0.830137 + 0.557560i $$0.811738\pi$$
$$654$$ 1.31371 0.0513701
$$655$$ 0 0
$$656$$ −22.2426 −0.868429
$$657$$ 2.00000 0.0780274
$$658$$ 0.0832611 0.00324586
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ 18.4853 0.718994 0.359497 0.933146i $$-0.382948\pi$$
0.359497 + 0.933146i $$0.382948\pi$$
$$662$$ 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
$$670$$ 0 0
$$671$$ 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
$$675$$ 0 0
$$676$$ −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
$$680$$ 0 0
$$681$$ 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 $$0.534517\pi$$
$$684$$ −1.82843 −0.0699117
$$685$$ 0 0
$$686$$ 3.31371 0.126518
$$687$$ −1.65685 −0.0632129
$$688$$ −1.75736 −0.0669987
$$689$$ 21.6569 0.825060
$$690$$ 0 0
$$691$$ −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
$$695$$ 0 0
$$696$$ 14.3848 0.545254
$$697$$ −8.68629 −0.329017
$$698$$ 12.1421 0.459587
$$699$$ 8.34315 0.315567
$$700$$ 0 0
$$701$$ −0.343146 −0.0129604 −0.00648022 0.999979i $$-0.502063\pi$$
−0.00648022 + 0.999979i $$0.502063\pi$$
$$702$$ −2.24264 −0.0846430
$$703$$ −11.0711 −0.417553
$$704$$ −5.89949 −0.222346
$$705$$ 0 0
$$706$$ −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 $$0.730767\pi$$
$$710$$ 0 0
$$711$$ −11.3137 −0.424297
$$712$$ −17.0122 −0.637559
$$713$$ −49.6569 −1.85966
$$714$$ −0.284271 −0.0106386
$$715$$ 0 0
$$716$$ −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 $$0.599199\pi$$
$$720$$ 0 0
$$721$$ −0.970563 −0.0361456
$$722$$ 0.414214 0.0154154
$$723$$ −14.9706 −0.556761
$$724$$ 35.0538 1.30277
$$725$$ 0 0
$$726$$ −3.72792 −0.138356
$$727$$ 4.58579 0.170077 0.0850387 0.996378i $$-0.472899\pi$$
0.0850387 + 0.996378i $$0.472899\pi$$
$$728$$ −5.02944 −0.186403
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −0.686292 −0.0253834
$$732$$ −10.3431 −0.382294
$$733$$ 1.31371 0.0485229 0.0242615 0.999706i $$-0.492277\pi$$
0.0242615 + 0.999706i $$0.492277\pi$$
$$734$$ 8.04163 0.296822
$$735$$ 0 0
$$736$$ −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 $$0.556839\pi$$
$$740$$ 0 0
$$741$$ −5.41421 −0.198896
$$742$$ 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
$$745$$ 0 0
$$746$$ 4.10051 0.150130
$$747$$ 10.4853 0.383636
$$748$$ −3.02944 −0.110767
$$749$$ 4.68629 0.171233
$$750$$ 0 0
$$751$$ −37.1127 −1.35426 −0.677131 0.735863i $$-0.736777\pi$$
−0.677131 + 0.735863i $$0.736777\pi$$
$$752$$ −1.02944 −0.0375397
$$753$$ 12.9289 0.471156
$$754$$ 20.3431 0.740854
$$755$$ 0 0
$$756$$ 1.07107 0.0389544
$$757$$ −16.8284 −0.611640 −0.305820 0.952089i $$-0.598931\pi$$
−0.305820 + 0.952089i $$0.598931\pi$$
$$758$$ 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
$$765$$ 0 0
$$766$$ −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 0
$$771$$ −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
$$775$$ 0 0
$$776$$ −6.72792 −0.241518
$$777$$ 6.48528 0.232658
$$778$$ 6.20101 0.222317
$$779$$ −7.41421 −0.265642
$$780$$ 0 0
$$781$$ −6.34315 −0.226976
$$782$$ −3.71573 −0.132874
$$783$$ −9.07107 −0.324174
$$784$$ −19.9706 −0.713234
$$785$$ 0 0
$$786$$ 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
$$790$$ 0 0
$$791$$ 7.31371 0.260046
$$792$$ −2.24264 −0.0796888
$$793$$ −30.6274 −1.08761
$$794$$ −14.7696 −0.524152
$$795$$ 0 0
$$796$$ 30.1421 1.06836
$$797$$ −10.8284 −0.383563 −0.191781 0.981438i $$-0.561426\pi$$
−0.191781 + 0.981438i $$0.561426\pi$$
$$798$$ −0.242641 −0.00858939
$$799$$ −0.402020 −0.0142225
$$800$$ 0 0
$$801$$ 10.7279 0.379052
$$802$$ 12.4437 0.439401
$$803$$ 2.82843 0.0998130
$$804$$ 21.9411 0.773804
$$805$$ 0 0
$$806$$ −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 0
$$811$$ −15.3137 −0.537737 −0.268869 0.963177i $$-0.586650\pi$$
−0.268869 + 0.963177i $$0.586650\pi$$
$$812$$ −9.71573 −0.340955
$$813$$ −30.8284 −1.08120
$$814$$ −6.48528 −0.227309
$$815$$ 0 0
$$816$$ 3.51472 0.123040
$$817$$ −0.585786 −0.0204941
$$818$$ 3.94113 0.137798
$$819$$ 3.17157 0.110824
$$820$$ 0 0
$$821$$ 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 $$-0.484860\pi$$
0.0475447 + 0.998869i $$0.484860\pi$$
$$824$$ −2.62742 −0.0915304
$$825$$ 0 0
$$826$$ −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 $$-0.732988\pi$$
−0.668325 + 0.743870i $$0.732988\pi$$
$$830$$ 0 0
$$831$$ −10.9706 −0.380565
$$832$$ 22.5858 0.783021
$$833$$ −7.79899 −0.270219
$$834$$ 0.485281 0.0168039
$$835$$ 0 0
$$836$$ −2.58579 −0.0894313
$$837$$ 6.48528 0.224164
$$838$$ −14.4437 −0.498948
$$839$$ 27.1127 0.936034 0.468017 0.883719i $$-0.344969\pi$$
0.468017 + 0.883719i $$0.344969\pi$$
$$840$$ 0 0
$$841$$ 53.2843 1.83739
$$842$$ −6.08326 −0.209643
$$843$$ −14.7279 −0.507257
$$844$$ 28.0000 0.963800
$$845$$ 0 0
$$846$$ −0.142136 −0.00488672
$$847$$ 5.27208 0.181151
$$848$$ −12.0000 −0.412082
$$849$$ −6.24264 −0.214247
$$850$$ 0 0
$$851$$ 84.7696 2.90586
$$852$$ 8.20101 0.280962
$$853$$ 9.51472 0.325778 0.162889 0.986644i $$-0.447919\pi$$
0.162889 + 0.986644i $$0.447919\pi$$
$$854$$ −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 $$0.354074\pi$$
$$860$$ 0 0
$$861$$ 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
$$865$$ 0 0
$$866$$ −0.384776 −0.0130752
$$867$$ −15.6274 −0.530735
$$868$$ 6.94618 0.235769
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 64.9706 2.20144
$$872$$ −5.02944 −0.170318
$$873$$ 4.24264 0.143592
$$874$$ −3.17157 −0.107280
$$875$$ 0 0
$$876$$ −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
$$880$$ 0 0
$$881$$ −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 $$-0.786249\pi$$
−0.782878 + 0.622176i $$0.786249\pi$$
$$884$$ 11.5980 0.390082
$$885$$ 0 0
$$886$$ 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
$$890$$ 0 0
$$891$$ 1.41421 0.0473779
$$892$$ 11.5980 0.388329
$$893$$ −0.343146 −0.0114829
$$894$$ −1.51472 −0.0506598
$$895$$ 0 0
$$896$$ 6.18377 0.206585
$$897$$ 41.4558 1.38417
$$898$$ 3.27208 0.109191
$$899$$ −58.8284 −1.96204
$$900$$ 0 0
$$901$$ −4.68629 −0.156123
$$902$$ −4.34315 −0.144611
$$903$$ 0.343146 0.0114192
$$904$$ 19.7990 0.658505
$$905$$ 0 0
$$906$$ 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
$$910$$ 0 0
$$911$$ −23.3137 −0.772418 −0.386209 0.922411i $$-0.626216\pi$$
−0.386209 + 0.922411i $$0.626216\pi$$
$$912$$ 3.00000 0.0993399
$$913$$ 14.8284 0.490749
$$914$$ −11.9411 −0.394977
$$915$$ 0 0
$$916$$ 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 $$0.563419\pi$$
$$920$$ 0 0
$$921$$ −7.79899 −0.256985
$$922$$ 4.42641 0.145776
$$923$$ 24.2843 0.799327
$$924$$ 1.51472 0.0498306
$$925$$ 0 0
$$926$$ 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 $$0.819868\pi$$
$$930$$ 0 0
$$931$$ −6.65685 −0.218170
$$932$$ −15.2548 −0.499689
$$933$$ −32.2426 −1.05558
$$934$$ −6.76955 −0.221507
$$935$$ 0 0
$$936$$ 8.58579 0.280635
$$937$$ 18.7696 0.613175 0.306587 0.951843i $$-0.400813\pi$$
0.306587 + 0.951843i $$0.400813\pi$$
$$938$$ 2.91169 0.0950700
$$939$$ 9.51472 0.310501
$$940$$ 0 0
$$941$$ −17.5563 −0.572321 −0.286160 0.958182i $$-0.592379\pi$$
−0.286160 + 0.958182i $$0.592379\pi$$
$$942$$ 9.02944 0.294195
$$943$$ 56.7696 1.84867
$$944$$ 25.4558 0.828517
$$945$$ 0 0
$$946$$ −0.343146 −0.0111566
$$947$$ −12.8284 −0.416868 −0.208434 0.978036i $$-0.566837\pi$$
−0.208434 + 0.978036i $$0.566837\pi$$
$$948$$ 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
$$955$$ 0 0
$$956$$ −4.72792 −0.152912
$$957$$ −12.8284 −0.414684
$$958$$ −15.8162 −0.510999
$$959$$ −8.20101 −0.264824
$$960$$ 0 0
$$961$$ 11.0589 0.356738
$$962$$ 24.8284 0.800501
$$963$$ −8.00000 −0.257796
$$964$$ 27.3726 0.881612
$$965$$ 0 0
$$966$$ 1.85786 0.0597758
$$967$$ 27.8995 0.897187 0.448594 0.893736i $$-0.351925\pi$$
0.448594 + 0.893736i $$0.351925\pi$$
$$968$$ 14.2721 0.458722
$$969$$ 1.17157 0.0376363
$$970$$ 0 0
$$971$$ 17.6569 0.566635 0.283318 0.959026i $$-0.408565\pi$$
0.283318 + 0.959026i $$0.408565\pi$$
$$972$$ −1.82843 −0.0586468
$$973$$ −0.686292 −0.0220015
$$974$$ 1.17157 0.0375396
$$975$$ 0 0
$$976$$ 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
$$980$$ 0 0
$$981$$ 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
$$985$$ 0 0
$$986$$ −4.40202 −0.140189
$$987$$ 0.201010 0.00639822
$$988$$ 9.89949 0.314945
$$989$$ 4.48528 0.142624
$$990$$ 0 0
$$991$$ −61.6569 −1.95859 −0.979297 0.202427i $$-0.935117\pi$$
−0.979297 + 0.202427i $$0.935117\pi$$
$$992$$ 28.6274 0.908921
$$993$$ 7.17157 0.227583
$$994$$ 1.08831 0.0345192
$$995$$ 0 0
$$996$$ −19.1716 −0.607475
$$997$$ 50.4853 1.59888 0.799442 0.600743i $$-0.205128\pi$$
0.799442 + 0.600743i $$0.205128\pi$$
$$998$$ −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 1425.2.a.l.1.2 2
3.2 odd 2 4275.2.a.x.1.1 2
5.2 odd 4 1425.2.c.j.799.3 4
5.3 odd 4 1425.2.c.j.799.2 4
5.4 even 2 285.2.a.f.1.1 2
15.14 odd 2 855.2.a.e.1.2 2
20.19 odd 2 4560.2.a.bj.1.2 2
95.94 odd 2 5415.2.a.p.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.1 2 5.4 even 2
855.2.a.e.1.2 2 15.14 odd 2
1425.2.a.l.1.2 2 1.1 even 1 trivial
1425.2.c.j.799.2 4 5.3 odd 4
1425.2.c.j.799.3 4 5.2 odd 4
4275.2.a.x.1.1 2 3.2 odd 2
4560.2.a.bj.1.2 2 20.19 odd 2
5415.2.a.p.1.2 2 95.94 odd 2