# Properties

 Label 1850.2.a.r.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ 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] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -3.44949 q^{3} +1.00000 q^{4} +3.44949 q^{6} +2.44949 q^{7} -1.00000 q^{8} +8.89898 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -3.44949 q^{3} +1.00000 q^{4} +3.44949 q^{6} +2.44949 q^{7} -1.00000 q^{8} +8.89898 q^{9} -1.44949 q^{11} -3.44949 q^{12} -4.44949 q^{13} -2.44949 q^{14} +1.00000 q^{16} +1.44949 q^{17} -8.89898 q^{18} -5.00000 q^{19} -8.44949 q^{21} +1.44949 q^{22} -2.00000 q^{23} +3.44949 q^{24} +4.44949 q^{26} -20.3485 q^{27} +2.44949 q^{28} +8.89898 q^{29} -0.449490 q^{31} -1.00000 q^{32} +5.00000 q^{33} -1.44949 q^{34} +8.89898 q^{36} +1.00000 q^{37} +5.00000 q^{38} +15.3485 q^{39} +1.00000 q^{41} +8.44949 q^{42} +10.8990 q^{43} -1.44949 q^{44} +2.00000 q^{46} +9.79796 q^{47} -3.44949 q^{48} -1.00000 q^{49} -5.00000 q^{51} -4.44949 q^{52} -6.00000 q^{53} +20.3485 q^{54} -2.44949 q^{56} +17.2474 q^{57} -8.89898 q^{58} -2.00000 q^{59} -1.55051 q^{61} +0.449490 q^{62} +21.7980 q^{63} +1.00000 q^{64} -5.00000 q^{66} -9.44949 q^{67} +1.44949 q^{68} +6.89898 q^{69} -12.4495 q^{71} -8.89898 q^{72} +6.79796 q^{73} -1.00000 q^{74} -5.00000 q^{76} -3.55051 q^{77} -15.3485 q^{78} -11.7980 q^{79} +43.4949 q^{81} -1.00000 q^{82} +1.44949 q^{83} -8.44949 q^{84} -10.8990 q^{86} -30.6969 q^{87} +1.44949 q^{88} -0.348469 q^{89} -10.8990 q^{91} -2.00000 q^{92} +1.55051 q^{93} -9.79796 q^{94} +3.44949 q^{96} -14.0000 q^{97} +1.00000 q^{98} -12.8990 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{17} - 8 q^{18} - 10 q^{19} - 12 q^{21} - 2 q^{22} - 4 q^{23} + 2 q^{24} + 4 q^{26} - 26 q^{27} + 8 q^{29} + 4 q^{31} - 2 q^{32} + 10 q^{33} + 2 q^{34} + 8 q^{36} + 2 q^{37} + 10 q^{38} + 16 q^{39} + 2 q^{41} + 12 q^{42} + 12 q^{43} + 2 q^{44} + 4 q^{46} - 2 q^{48} - 2 q^{49} - 10 q^{51} - 4 q^{52} - 12 q^{53} + 26 q^{54} + 10 q^{57} - 8 q^{58} - 4 q^{59} - 8 q^{61} - 4 q^{62} + 24 q^{63} + 2 q^{64} - 10 q^{66} - 14 q^{67} - 2 q^{68} + 4 q^{69} - 20 q^{71} - 8 q^{72} - 6 q^{73} - 2 q^{74} - 10 q^{76} - 12 q^{77} - 16 q^{78} - 4 q^{79} + 38 q^{81} - 2 q^{82} - 2 q^{83} - 12 q^{84} - 12 q^{86} - 32 q^{87} - 2 q^{88} + 14 q^{89} - 12 q^{91} - 4 q^{92} + 8 q^{93} + 2 q^{96} - 28 q^{97} + 2 q^{98} - 16 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^17 - 8 * q^18 - 10 * q^19 - 12 * q^21 - 2 * q^22 - 4 * q^23 + 2 * q^24 + 4 * q^26 - 26 * q^27 + 8 * q^29 + 4 * q^31 - 2 * q^32 + 10 * q^33 + 2 * q^34 + 8 * q^36 + 2 * q^37 + 10 * q^38 + 16 * q^39 + 2 * q^41 + 12 * q^42 + 12 * q^43 + 2 * q^44 + 4 * q^46 - 2 * q^48 - 2 * q^49 - 10 * q^51 - 4 * q^52 - 12 * q^53 + 26 * q^54 + 10 * q^57 - 8 * q^58 - 4 * q^59 - 8 * q^61 - 4 * q^62 + 24 * q^63 + 2 * q^64 - 10 * q^66 - 14 * q^67 - 2 * q^68 + 4 * q^69 - 20 * q^71 - 8 * q^72 - 6 * q^73 - 2 * q^74 - 10 * q^76 - 12 * q^77 - 16 * q^78 - 4 * q^79 + 38 * q^81 - 2 * q^82 - 2 * q^83 - 12 * q^84 - 12 * q^86 - 32 * q^87 - 2 * q^88 + 14 * q^89 - 12 * q^91 - 4 * q^92 + 8 * q^93 + 2 * q^96 - 28 * q^97 + 2 * q^98 - 16 * 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
$$3$$ −3.44949 −1.99156 −0.995782 0.0917517i $$-0.970753\pi$$
−0.995782 + 0.0917517i $$0.970753\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 3.44949 1.40825
$$7$$ 2.44949 0.925820 0.462910 0.886405i $$-0.346805\pi$$
0.462910 + 0.886405i $$0.346805\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 8.89898 2.96633
$$10$$ 0 0
$$11$$ −1.44949 −0.437038 −0.218519 0.975833i $$-0.570122\pi$$
−0.218519 + 0.975833i $$0.570122\pi$$
$$12$$ −3.44949 −0.995782
$$13$$ −4.44949 −1.23407 −0.617033 0.786937i $$-0.711666\pi$$
−0.617033 + 0.786937i $$0.711666\pi$$
$$14$$ −2.44949 −0.654654
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.44949 0.351553 0.175776 0.984430i $$-0.443756\pi$$
0.175776 + 0.984430i $$0.443756\pi$$
$$18$$ −8.89898 −2.09751
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ −8.44949 −1.84383
$$22$$ 1.44949 0.309032
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 3.44949 0.704124
$$25$$ 0 0
$$26$$ 4.44949 0.872617
$$27$$ −20.3485 −3.91606
$$28$$ 2.44949 0.462910
$$29$$ 8.89898 1.65250 0.826250 0.563304i $$-0.190470\pi$$
0.826250 + 0.563304i $$0.190470\pi$$
$$30$$ 0 0
$$31$$ −0.449490 −0.0807307 −0.0403654 0.999185i $$-0.512852\pi$$
−0.0403654 + 0.999185i $$0.512852\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 5.00000 0.870388
$$34$$ −1.44949 −0.248585
$$35$$ 0 0
$$36$$ 8.89898 1.48316
$$37$$ 1.00000 0.164399
$$38$$ 5.00000 0.811107
$$39$$ 15.3485 2.45772
$$40$$ 0 0
$$41$$ 1.00000 0.156174 0.0780869 0.996947i $$-0.475119\pi$$
0.0780869 + 0.996947i $$0.475119\pi$$
$$42$$ 8.44949 1.30378
$$43$$ 10.8990 1.66208 0.831039 0.556214i $$-0.187746\pi$$
0.831039 + 0.556214i $$0.187746\pi$$
$$44$$ −1.44949 −0.218519
$$45$$ 0 0
$$46$$ 2.00000 0.294884
$$47$$ 9.79796 1.42918 0.714590 0.699544i $$-0.246613\pi$$
0.714590 + 0.699544i $$0.246613\pi$$
$$48$$ −3.44949 −0.497891
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −5.00000 −0.700140
$$52$$ −4.44949 −0.617033
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 20.3485 2.76908
$$55$$ 0 0
$$56$$ −2.44949 −0.327327
$$57$$ 17.2474 2.28448
$$58$$ −8.89898 −1.16849
$$59$$ −2.00000 −0.260378 −0.130189 0.991489i $$-0.541558\pi$$
−0.130189 + 0.991489i $$0.541558\pi$$
$$60$$ 0 0
$$61$$ −1.55051 −0.198522 −0.0992612 0.995061i $$-0.531648\pi$$
−0.0992612 + 0.995061i $$0.531648\pi$$
$$62$$ 0.449490 0.0570853
$$63$$ 21.7980 2.74628
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −5.00000 −0.615457
$$67$$ −9.44949 −1.15444 −0.577219 0.816589i $$-0.695862\pi$$
−0.577219 + 0.816589i $$0.695862\pi$$
$$68$$ 1.44949 0.175776
$$69$$ 6.89898 0.830540
$$70$$ 0 0
$$71$$ −12.4495 −1.47748 −0.738741 0.673989i $$-0.764579\pi$$
−0.738741 + 0.673989i $$0.764579\pi$$
$$72$$ −8.89898 −1.04875
$$73$$ 6.79796 0.795641 0.397820 0.917463i $$-0.369767\pi$$
0.397820 + 0.917463i $$0.369767\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −5.00000 −0.573539
$$77$$ −3.55051 −0.404618
$$78$$ −15.3485 −1.73787
$$79$$ −11.7980 −1.32737 −0.663687 0.748010i $$-0.731009\pi$$
−0.663687 + 0.748010i $$0.731009\pi$$
$$80$$ 0 0
$$81$$ 43.4949 4.83277
$$82$$ −1.00000 −0.110432
$$83$$ 1.44949 0.159102 0.0795511 0.996831i $$-0.474651\pi$$
0.0795511 + 0.996831i $$0.474651\pi$$
$$84$$ −8.44949 −0.921915
$$85$$ 0 0
$$86$$ −10.8990 −1.17527
$$87$$ −30.6969 −3.29106
$$88$$ 1.44949 0.154516
$$89$$ −0.348469 −0.0369377 −0.0184688 0.999829i $$-0.505879\pi$$
−0.0184688 + 0.999829i $$0.505879\pi$$
$$90$$ 0 0
$$91$$ −10.8990 −1.14252
$$92$$ −2.00000 −0.208514
$$93$$ 1.55051 0.160780
$$94$$ −9.79796 −1.01058
$$95$$ 0 0
$$96$$ 3.44949 0.352062
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −12.8990 −1.29640
$$100$$ 0 0
$$101$$ 10.8990 1.08449 0.542244 0.840221i $$-0.317575\pi$$
0.542244 + 0.840221i $$0.317575\pi$$
$$102$$ 5.00000 0.495074
$$103$$ −1.55051 −0.152776 −0.0763882 0.997078i $$-0.524339\pi$$
−0.0763882 + 0.997078i $$0.524339\pi$$
$$104$$ 4.44949 0.436308
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 13.4495 1.30021 0.650106 0.759844i $$-0.274725\pi$$
0.650106 + 0.759844i $$0.274725\pi$$
$$108$$ −20.3485 −1.95803
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −3.44949 −0.327411
$$112$$ 2.44949 0.231455
$$113$$ 9.44949 0.888933 0.444467 0.895795i $$-0.353393\pi$$
0.444467 + 0.895795i $$0.353393\pi$$
$$114$$ −17.2474 −1.61537
$$115$$ 0 0
$$116$$ 8.89898 0.826250
$$117$$ −39.5959 −3.66064
$$118$$ 2.00000 0.184115
$$119$$ 3.55051 0.325475
$$120$$ 0 0
$$121$$ −8.89898 −0.808998
$$122$$ 1.55051 0.140377
$$123$$ −3.44949 −0.311030
$$124$$ −0.449490 −0.0403654
$$125$$ 0 0
$$126$$ −21.7980 −1.94192
$$127$$ −8.24745 −0.731843 −0.365921 0.930646i $$-0.619246\pi$$
−0.365921 + 0.930646i $$0.619246\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −37.5959 −3.31014
$$130$$ 0 0
$$131$$ −20.6969 −1.80830 −0.904150 0.427215i $$-0.859495\pi$$
−0.904150 + 0.427215i $$0.859495\pi$$
$$132$$ 5.00000 0.435194
$$133$$ −12.2474 −1.06199
$$134$$ 9.44949 0.816312
$$135$$ 0 0
$$136$$ −1.44949 −0.124293
$$137$$ 19.6969 1.68282 0.841412 0.540395i $$-0.181725\pi$$
0.841412 + 0.540395i $$0.181725\pi$$
$$138$$ −6.89898 −0.587280
$$139$$ −17.4495 −1.48005 −0.740023 0.672581i $$-0.765186\pi$$
−0.740023 + 0.672581i $$0.765186\pi$$
$$140$$ 0 0
$$141$$ −33.7980 −2.84630
$$142$$ 12.4495 1.04474
$$143$$ 6.44949 0.539333
$$144$$ 8.89898 0.741582
$$145$$ 0 0
$$146$$ −6.79796 −0.562603
$$147$$ 3.44949 0.284509
$$148$$ 1.00000 0.0821995
$$149$$ −19.3485 −1.58509 −0.792544 0.609814i $$-0.791244\pi$$
−0.792544 + 0.609814i $$0.791244\pi$$
$$150$$ 0 0
$$151$$ −14.0000 −1.13930 −0.569652 0.821886i $$-0.692922\pi$$
−0.569652 + 0.821886i $$0.692922\pi$$
$$152$$ 5.00000 0.405554
$$153$$ 12.8990 1.04282
$$154$$ 3.55051 0.286108
$$155$$ 0 0
$$156$$ 15.3485 1.22886
$$157$$ −11.5505 −0.921831 −0.460916 0.887444i $$-0.652479\pi$$
−0.460916 + 0.887444i $$0.652479\pi$$
$$158$$ 11.7980 0.938595
$$159$$ 20.6969 1.64137
$$160$$ 0 0
$$161$$ −4.89898 −0.386094
$$162$$ −43.4949 −3.41728
$$163$$ −19.8990 −1.55861 −0.779304 0.626646i $$-0.784427\pi$$
−0.779304 + 0.626646i $$0.784427\pi$$
$$164$$ 1.00000 0.0780869
$$165$$ 0 0
$$166$$ −1.44949 −0.112502
$$167$$ 5.55051 0.429511 0.214756 0.976668i $$-0.431104\pi$$
0.214756 + 0.976668i $$0.431104\pi$$
$$168$$ 8.44949 0.651892
$$169$$ 6.79796 0.522920
$$170$$ 0 0
$$171$$ −44.4949 −3.40261
$$172$$ 10.8990 0.831039
$$173$$ −22.6969 −1.72562 −0.862808 0.505532i $$-0.831296\pi$$
−0.862808 + 0.505532i $$0.831296\pi$$
$$174$$ 30.6969 2.32713
$$175$$ 0 0
$$176$$ −1.44949 −0.109259
$$177$$ 6.89898 0.518559
$$178$$ 0.348469 0.0261189
$$179$$ 18.7980 1.40503 0.702513 0.711671i $$-0.252061\pi$$
0.702513 + 0.711671i $$0.252061\pi$$
$$180$$ 0 0
$$181$$ −2.89898 −0.215479 −0.107740 0.994179i $$-0.534361\pi$$
−0.107740 + 0.994179i $$0.534361\pi$$
$$182$$ 10.8990 0.807886
$$183$$ 5.34847 0.395370
$$184$$ 2.00000 0.147442
$$185$$ 0 0
$$186$$ −1.55051 −0.113689
$$187$$ −2.10102 −0.153642
$$188$$ 9.79796 0.714590
$$189$$ −49.8434 −3.62557
$$190$$ 0 0
$$191$$ 11.7980 0.853670 0.426835 0.904329i $$-0.359628\pi$$
0.426835 + 0.904329i $$0.359628\pi$$
$$192$$ −3.44949 −0.248945
$$193$$ 7.44949 0.536226 0.268113 0.963387i $$-0.413600\pi$$
0.268113 + 0.963387i $$0.413600\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −1.00000 −0.0714286
$$197$$ 2.65153 0.188914 0.0944569 0.995529i $$-0.469889\pi$$
0.0944569 + 0.995529i $$0.469889\pi$$
$$198$$ 12.8990 0.916691
$$199$$ 23.5959 1.67267 0.836335 0.548219i $$-0.184694\pi$$
0.836335 + 0.548219i $$0.184694\pi$$
$$200$$ 0 0
$$201$$ 32.5959 2.29914
$$202$$ −10.8990 −0.766850
$$203$$ 21.7980 1.52992
$$204$$ −5.00000 −0.350070
$$205$$ 0 0
$$206$$ 1.55051 0.108029
$$207$$ −17.7980 −1.23704
$$208$$ −4.44949 −0.308517
$$209$$ 7.24745 0.501317
$$210$$ 0 0
$$211$$ −9.44949 −0.650530 −0.325265 0.945623i $$-0.605453\pi$$
−0.325265 + 0.945623i $$0.605453\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 42.9444 2.94250
$$214$$ −13.4495 −0.919388
$$215$$ 0 0
$$216$$ 20.3485 1.38454
$$217$$ −1.10102 −0.0747421
$$218$$ −14.0000 −0.948200
$$219$$ −23.4495 −1.58457
$$220$$ 0 0
$$221$$ −6.44949 −0.433840
$$222$$ 3.44949 0.231515
$$223$$ −2.20204 −0.147460 −0.0737298 0.997278i $$-0.523490\pi$$
−0.0737298 + 0.997278i $$0.523490\pi$$
$$224$$ −2.44949 −0.163663
$$225$$ 0 0
$$226$$ −9.44949 −0.628571
$$227$$ −12.6969 −0.842725 −0.421363 0.906892i $$-0.638448\pi$$
−0.421363 + 0.906892i $$0.638448\pi$$
$$228$$ 17.2474 1.14224
$$229$$ −13.7980 −0.911795 −0.455897 0.890032i $$-0.650682\pi$$
−0.455897 + 0.890032i $$0.650682\pi$$
$$230$$ 0 0
$$231$$ 12.2474 0.805823
$$232$$ −8.89898 −0.584247
$$233$$ −20.4949 −1.34267 −0.671333 0.741156i $$-0.734278\pi$$
−0.671333 + 0.741156i $$0.734278\pi$$
$$234$$ 39.5959 2.58847
$$235$$ 0 0
$$236$$ −2.00000 −0.130189
$$237$$ 40.6969 2.64355
$$238$$ −3.55051 −0.230145
$$239$$ −8.89898 −0.575627 −0.287814 0.957686i $$-0.592928\pi$$
−0.287814 + 0.957686i $$0.592928\pi$$
$$240$$ 0 0
$$241$$ −7.44949 −0.479864 −0.239932 0.970790i $$-0.577125\pi$$
−0.239932 + 0.970790i $$0.577125\pi$$
$$242$$ 8.89898 0.572048
$$243$$ −88.9898 −5.70870
$$244$$ −1.55051 −0.0992612
$$245$$ 0 0
$$246$$ 3.44949 0.219931
$$247$$ 22.2474 1.41557
$$248$$ 0.449490 0.0285426
$$249$$ −5.00000 −0.316862
$$250$$ 0 0
$$251$$ −11.2020 −0.707067 −0.353533 0.935422i $$-0.615020\pi$$
−0.353533 + 0.935422i $$0.615020\pi$$
$$252$$ 21.7980 1.37314
$$253$$ 2.89898 0.182257
$$254$$ 8.24745 0.517491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.89898 −0.180833 −0.0904167 0.995904i $$-0.528820\pi$$
−0.0904167 + 0.995904i $$0.528820\pi$$
$$258$$ 37.5959 2.34062
$$259$$ 2.44949 0.152204
$$260$$ 0 0
$$261$$ 79.1918 4.90185
$$262$$ 20.6969 1.27866
$$263$$ −19.7980 −1.22079 −0.610397 0.792095i $$-0.708990\pi$$
−0.610397 + 0.792095i $$0.708990\pi$$
$$264$$ −5.00000 −0.307729
$$265$$ 0 0
$$266$$ 12.2474 0.750939
$$267$$ 1.20204 0.0735637
$$268$$ −9.44949 −0.577219
$$269$$ 4.00000 0.243884 0.121942 0.992537i $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ 0 0
$$271$$ −14.0454 −0.853198 −0.426599 0.904441i $$-0.640288\pi$$
−0.426599 + 0.904441i $$0.640288\pi$$
$$272$$ 1.44949 0.0878882
$$273$$ 37.5959 2.27541
$$274$$ −19.6969 −1.18994
$$275$$ 0 0
$$276$$ 6.89898 0.415270
$$277$$ 13.7980 0.829039 0.414520 0.910040i $$-0.363950\pi$$
0.414520 + 0.910040i $$0.363950\pi$$
$$278$$ 17.4495 1.04655
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −0.898979 −0.0536286 −0.0268143 0.999640i $$-0.508536\pi$$
−0.0268143 + 0.999640i $$0.508536\pi$$
$$282$$ 33.7980 2.01264
$$283$$ −17.6969 −1.05197 −0.525987 0.850493i $$-0.676304\pi$$
−0.525987 + 0.850493i $$0.676304\pi$$
$$284$$ −12.4495 −0.738741
$$285$$ 0 0
$$286$$ −6.44949 −0.381366
$$287$$ 2.44949 0.144589
$$288$$ −8.89898 −0.524377
$$289$$ −14.8990 −0.876411
$$290$$ 0 0
$$291$$ 48.2929 2.83098
$$292$$ 6.79796 0.397820
$$293$$ −16.0454 −0.937383 −0.468691 0.883362i $$-0.655274\pi$$
−0.468691 + 0.883362i $$0.655274\pi$$
$$294$$ −3.44949 −0.201178
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 29.4949 1.71147
$$298$$ 19.3485 1.12083
$$299$$ 8.89898 0.514641
$$300$$ 0 0
$$301$$ 26.6969 1.53879
$$302$$ 14.0000 0.805609
$$303$$ −37.5959 −2.15983
$$304$$ −5.00000 −0.286770
$$305$$ 0 0
$$306$$ −12.8990 −0.737386
$$307$$ −22.3485 −1.27549 −0.637747 0.770246i $$-0.720134\pi$$
−0.637747 + 0.770246i $$0.720134\pi$$
$$308$$ −3.55051 −0.202309
$$309$$ 5.34847 0.304264
$$310$$ 0 0
$$311$$ −3.55051 −0.201331 −0.100665 0.994920i $$-0.532097\pi$$
−0.100665 + 0.994920i $$0.532097\pi$$
$$312$$ −15.3485 −0.868936
$$313$$ 0.898979 0.0508133 0.0254067 0.999677i $$-0.491912\pi$$
0.0254067 + 0.999677i $$0.491912\pi$$
$$314$$ 11.5505 0.651833
$$315$$ 0 0
$$316$$ −11.7980 −0.663687
$$317$$ 14.4495 0.811564 0.405782 0.913970i $$-0.366999\pi$$
0.405782 + 0.913970i $$0.366999\pi$$
$$318$$ −20.6969 −1.16063
$$319$$ −12.8990 −0.722204
$$320$$ 0 0
$$321$$ −46.3939 −2.58945
$$322$$ 4.89898 0.273009
$$323$$ −7.24745 −0.403259
$$324$$ 43.4949 2.41638
$$325$$ 0 0
$$326$$ 19.8990 1.10210
$$327$$ −48.2929 −2.67060
$$328$$ −1.00000 −0.0552158
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ 10.1010 0.555202 0.277601 0.960696i $$-0.410461\pi$$
0.277601 + 0.960696i $$0.410461\pi$$
$$332$$ 1.44949 0.0795511
$$333$$ 8.89898 0.487661
$$334$$ −5.55051 −0.303710
$$335$$ 0 0
$$336$$ −8.44949 −0.460957
$$337$$ 13.6969 0.746120 0.373060 0.927807i $$-0.378309\pi$$
0.373060 + 0.927807i $$0.378309\pi$$
$$338$$ −6.79796 −0.369760
$$339$$ −32.5959 −1.77037
$$340$$ 0 0
$$341$$ 0.651531 0.0352824
$$342$$ 44.4949 2.40601
$$343$$ −19.5959 −1.05808
$$344$$ −10.8990 −0.587634
$$345$$ 0 0
$$346$$ 22.6969 1.22019
$$347$$ −0.797959 −0.0428367 −0.0214183 0.999771i $$-0.506818\pi$$
−0.0214183 + 0.999771i $$0.506818\pi$$
$$348$$ −30.6969 −1.64553
$$349$$ −7.55051 −0.404170 −0.202085 0.979368i $$-0.564772\pi$$
−0.202085 + 0.979368i $$0.564772\pi$$
$$350$$ 0 0
$$351$$ 90.5403 4.83268
$$352$$ 1.44949 0.0772581
$$353$$ −5.10102 −0.271500 −0.135750 0.990743i $$-0.543344\pi$$
−0.135750 + 0.990743i $$0.543344\pi$$
$$354$$ −6.89898 −0.366677
$$355$$ 0 0
$$356$$ −0.348469 −0.0184688
$$357$$ −12.2474 −0.648204
$$358$$ −18.7980 −0.993503
$$359$$ 27.3939 1.44579 0.722897 0.690956i $$-0.242810\pi$$
0.722897 + 0.690956i $$0.242810\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 2.89898 0.152367
$$363$$ 30.6969 1.61117
$$364$$ −10.8990 −0.571262
$$365$$ 0 0
$$366$$ −5.34847 −0.279569
$$367$$ −10.2474 −0.534912 −0.267456 0.963570i $$-0.586183\pi$$
−0.267456 + 0.963570i $$0.586183\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ 8.89898 0.463262
$$370$$ 0 0
$$371$$ −14.6969 −0.763027
$$372$$ 1.55051 0.0803902
$$373$$ −24.0454 −1.24502 −0.622512 0.782610i $$-0.713888\pi$$
−0.622512 + 0.782610i $$0.713888\pi$$
$$374$$ 2.10102 0.108641
$$375$$ 0 0
$$376$$ −9.79796 −0.505291
$$377$$ −39.5959 −2.03929
$$378$$ 49.8434 2.56367
$$379$$ 23.0454 1.18376 0.591882 0.806025i $$-0.298385\pi$$
0.591882 + 0.806025i $$0.298385\pi$$
$$380$$ 0 0
$$381$$ 28.4495 1.45751
$$382$$ −11.7980 −0.603636
$$383$$ −9.34847 −0.477684 −0.238842 0.971058i $$-0.576768\pi$$
−0.238842 + 0.971058i $$0.576768\pi$$
$$384$$ 3.44949 0.176031
$$385$$ 0 0
$$386$$ −7.44949 −0.379169
$$387$$ 96.9898 4.93027
$$388$$ −14.0000 −0.710742
$$389$$ −21.1464 −1.07217 −0.536083 0.844165i $$-0.680097\pi$$
−0.536083 + 0.844165i $$0.680097\pi$$
$$390$$ 0 0
$$391$$ −2.89898 −0.146608
$$392$$ 1.00000 0.0505076
$$393$$ 71.3939 3.60134
$$394$$ −2.65153 −0.133582
$$395$$ 0 0
$$396$$ −12.8990 −0.648198
$$397$$ 0.651531 0.0326994 0.0163497 0.999866i $$-0.494795\pi$$
0.0163497 + 0.999866i $$0.494795\pi$$
$$398$$ −23.5959 −1.18276
$$399$$ 42.2474 2.11502
$$400$$ 0 0
$$401$$ −37.9444 −1.89485 −0.947426 0.319975i $$-0.896326\pi$$
−0.947426 + 0.319975i $$0.896326\pi$$
$$402$$ −32.5959 −1.62574
$$403$$ 2.00000 0.0996271
$$404$$ 10.8990 0.542244
$$405$$ 0 0
$$406$$ −21.7980 −1.08181
$$407$$ −1.44949 −0.0718485
$$408$$ 5.00000 0.247537
$$409$$ 0.146428 0.00724041 0.00362020 0.999993i $$-0.498848\pi$$
0.00362020 + 0.999993i $$0.498848\pi$$
$$410$$ 0 0
$$411$$ −67.9444 −3.35145
$$412$$ −1.55051 −0.0763882
$$413$$ −4.89898 −0.241063
$$414$$ 17.7980 0.874722
$$415$$ 0 0
$$416$$ 4.44949 0.218154
$$417$$ 60.1918 2.94761
$$418$$ −7.24745 −0.354484
$$419$$ −16.5505 −0.808545 −0.404273 0.914639i $$-0.632475\pi$$
−0.404273 + 0.914639i $$0.632475\pi$$
$$420$$ 0 0
$$421$$ 23.1010 1.12587 0.562937 0.826500i $$-0.309671\pi$$
0.562937 + 0.826500i $$0.309671\pi$$
$$422$$ 9.44949 0.459994
$$423$$ 87.1918 4.23941
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −42.9444 −2.08066
$$427$$ −3.79796 −0.183796
$$428$$ 13.4495 0.650106
$$429$$ −22.2474 −1.07412
$$430$$ 0 0
$$431$$ 8.69694 0.418917 0.209458 0.977818i $$-0.432830\pi$$
0.209458 + 0.977818i $$0.432830\pi$$
$$432$$ −20.3485 −0.979016
$$433$$ −19.0000 −0.913082 −0.456541 0.889702i $$-0.650912\pi$$
−0.456541 + 0.889702i $$0.650912\pi$$
$$434$$ 1.10102 0.0528507
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 10.0000 0.478365
$$438$$ 23.4495 1.12046
$$439$$ 19.5959 0.935262 0.467631 0.883924i $$-0.345108\pi$$
0.467631 + 0.883924i $$0.345108\pi$$
$$440$$ 0 0
$$441$$ −8.89898 −0.423761
$$442$$ 6.44949 0.306771
$$443$$ 12.5505 0.596293 0.298146 0.954520i $$-0.403632\pi$$
0.298146 + 0.954520i $$0.403632\pi$$
$$444$$ −3.44949 −0.163706
$$445$$ 0 0
$$446$$ 2.20204 0.104270
$$447$$ 66.7423 3.15680
$$448$$ 2.44949 0.115728
$$449$$ 8.75255 0.413058 0.206529 0.978440i $$-0.433783\pi$$
0.206529 + 0.978440i $$0.433783\pi$$
$$450$$ 0 0
$$451$$ −1.44949 −0.0682538
$$452$$ 9.44949 0.444467
$$453$$ 48.2929 2.26900
$$454$$ 12.6969 0.595897
$$455$$ 0 0
$$456$$ −17.2474 −0.807686
$$457$$ 9.24745 0.432577 0.216289 0.976329i $$-0.430605\pi$$
0.216289 + 0.976329i $$0.430605\pi$$
$$458$$ 13.7980 0.644736
$$459$$ −29.4949 −1.37670
$$460$$ 0 0
$$461$$ 38.6969 1.80230 0.901148 0.433511i $$-0.142726\pi$$
0.901148 + 0.433511i $$0.142726\pi$$
$$462$$ −12.2474 −0.569803
$$463$$ −14.4495 −0.671525 −0.335762 0.941947i $$-0.608994\pi$$
−0.335762 + 0.941947i $$0.608994\pi$$
$$464$$ 8.89898 0.413125
$$465$$ 0 0
$$466$$ 20.4949 0.949408
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ −39.5959 −1.83032
$$469$$ −23.1464 −1.06880
$$470$$ 0 0
$$471$$ 39.8434 1.83589
$$472$$ 2.00000 0.0920575
$$473$$ −15.7980 −0.726391
$$474$$ −40.6969 −1.86927
$$475$$ 0 0
$$476$$ 3.55051 0.162737
$$477$$ −53.3939 −2.44474
$$478$$ 8.89898 0.407030
$$479$$ −18.2474 −0.833747 −0.416874 0.908964i $$-0.636874\pi$$
−0.416874 + 0.908964i $$0.636874\pi$$
$$480$$ 0 0
$$481$$ −4.44949 −0.202879
$$482$$ 7.44949 0.339315
$$483$$ 16.8990 0.768930
$$484$$ −8.89898 −0.404499
$$485$$ 0 0
$$486$$ 88.9898 4.03666
$$487$$ −42.7423 −1.93684 −0.968420 0.249323i $$-0.919792\pi$$
−0.968420 + 0.249323i $$0.919792\pi$$
$$488$$ 1.55051 0.0701883
$$489$$ 68.6413 3.10407
$$490$$ 0 0
$$491$$ −22.2020 −1.00196 −0.500982 0.865458i $$-0.667028\pi$$
−0.500982 + 0.865458i $$0.667028\pi$$
$$492$$ −3.44949 −0.155515
$$493$$ 12.8990 0.580941
$$494$$ −22.2474 −1.00096
$$495$$ 0 0
$$496$$ −0.449490 −0.0201827
$$497$$ −30.4949 −1.36788
$$498$$ 5.00000 0.224055
$$499$$ 22.0000 0.984855 0.492428 0.870353i $$-0.336110\pi$$
0.492428 + 0.870353i $$0.336110\pi$$
$$500$$ 0 0
$$501$$ −19.1464 −0.855399
$$502$$ 11.2020 0.499972
$$503$$ 9.79796 0.436869 0.218435 0.975852i $$-0.429905\pi$$
0.218435 + 0.975852i $$0.429905\pi$$
$$504$$ −21.7980 −0.970958
$$505$$ 0 0
$$506$$ −2.89898 −0.128875
$$507$$ −23.4495 −1.04143
$$508$$ −8.24745 −0.365921
$$509$$ 14.6515 0.649418 0.324709 0.945814i $$-0.394734\pi$$
0.324709 + 0.945814i $$0.394734\pi$$
$$510$$ 0 0
$$511$$ 16.6515 0.736620
$$512$$ −1.00000 −0.0441942
$$513$$ 101.742 4.49203
$$514$$ 2.89898 0.127869
$$515$$ 0 0
$$516$$ −37.5959 −1.65507
$$517$$ −14.2020 −0.624605
$$518$$ −2.44949 −0.107624
$$519$$ 78.2929 3.43667
$$520$$ 0 0
$$521$$ 19.8990 0.871790 0.435895 0.899997i $$-0.356432\pi$$
0.435895 + 0.899997i $$0.356432\pi$$
$$522$$ −79.1918 −3.46613
$$523$$ −40.3939 −1.76630 −0.883150 0.469090i $$-0.844582\pi$$
−0.883150 + 0.469090i $$0.844582\pi$$
$$524$$ −20.6969 −0.904150
$$525$$ 0 0
$$526$$ 19.7980 0.863232
$$527$$ −0.651531 −0.0283811
$$528$$ 5.00000 0.217597
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ −17.7980 −0.772366
$$532$$ −12.2474 −0.530994
$$533$$ −4.44949 −0.192729
$$534$$ −1.20204 −0.0520174
$$535$$ 0 0
$$536$$ 9.44949 0.408156
$$537$$ −64.8434 −2.79820
$$538$$ −4.00000 −0.172452
$$539$$ 1.44949 0.0624339
$$540$$ 0 0
$$541$$ 10.6515 0.457945 0.228973 0.973433i $$-0.426463\pi$$
0.228973 + 0.973433i $$0.426463\pi$$
$$542$$ 14.0454 0.603302
$$543$$ 10.0000 0.429141
$$544$$ −1.44949 −0.0621464
$$545$$ 0 0
$$546$$ −37.5959 −1.60896
$$547$$ 25.6969 1.09872 0.549361 0.835585i $$-0.314871\pi$$
0.549361 + 0.835585i $$0.314871\pi$$
$$548$$ 19.6969 0.841412
$$549$$ −13.7980 −0.588883
$$550$$ 0 0
$$551$$ −44.4949 −1.89555
$$552$$ −6.89898 −0.293640
$$553$$ −28.8990 −1.22891
$$554$$ −13.7980 −0.586219
$$555$$ 0 0
$$556$$ −17.4495 −0.740023
$$557$$ 2.20204 0.0933035 0.0466517 0.998911i $$-0.485145\pi$$
0.0466517 + 0.998911i $$0.485145\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −48.4949 −2.05112
$$560$$ 0 0
$$561$$ 7.24745 0.305988
$$562$$ 0.898979 0.0379212
$$563$$ 1.59592 0.0672599 0.0336300 0.999434i $$-0.489293\pi$$
0.0336300 + 0.999434i $$0.489293\pi$$
$$564$$ −33.7980 −1.42315
$$565$$ 0 0
$$566$$ 17.6969 0.743858
$$567$$ 106.540 4.47427
$$568$$ 12.4495 0.522369
$$569$$ −17.0454 −0.714581 −0.357290 0.933993i $$-0.616299\pi$$
−0.357290 + 0.933993i $$0.616299\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 6.44949 0.269667
$$573$$ −40.6969 −1.70014
$$574$$ −2.44949 −0.102240
$$575$$ 0 0
$$576$$ 8.89898 0.370791
$$577$$ 23.0454 0.959393 0.479696 0.877435i $$-0.340747\pi$$
0.479696 + 0.877435i $$0.340747\pi$$
$$578$$ 14.8990 0.619716
$$579$$ −25.6969 −1.06793
$$580$$ 0 0
$$581$$ 3.55051 0.147300
$$582$$ −48.2929 −2.00180
$$583$$ 8.69694 0.360190
$$584$$ −6.79796 −0.281302
$$585$$ 0 0
$$586$$ 16.0454 0.662830
$$587$$ 35.6969 1.47337 0.736685 0.676236i $$-0.236390\pi$$
0.736685 + 0.676236i $$0.236390\pi$$
$$588$$ 3.44949 0.142255
$$589$$ 2.24745 0.0926045
$$590$$ 0 0
$$591$$ −9.14643 −0.376234
$$592$$ 1.00000 0.0410997
$$593$$ 12.3939 0.508956 0.254478 0.967079i $$-0.418096\pi$$
0.254478 + 0.967079i $$0.418096\pi$$
$$594$$ −29.4949 −1.21019
$$595$$ 0 0
$$596$$ −19.3485 −0.792544
$$597$$ −81.3939 −3.33123
$$598$$ −8.89898 −0.363906
$$599$$ −14.9444 −0.610611 −0.305306 0.952254i $$-0.598759\pi$$
−0.305306 + 0.952254i $$0.598759\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ −26.6969 −1.08809
$$603$$ −84.0908 −3.42444
$$604$$ −14.0000 −0.569652
$$605$$ 0 0
$$606$$ 37.5959 1.52723
$$607$$ −44.7423 −1.81604 −0.908018 0.418931i $$-0.862405\pi$$
−0.908018 + 0.418931i $$0.862405\pi$$
$$608$$ 5.00000 0.202777
$$609$$ −75.1918 −3.04693
$$610$$ 0 0
$$611$$ −43.5959 −1.76370
$$612$$ 12.8990 0.521410
$$613$$ −14.4949 −0.585443 −0.292722 0.956198i $$-0.594561\pi$$
−0.292722 + 0.956198i $$0.594561\pi$$
$$614$$ 22.3485 0.901911
$$615$$ 0 0
$$616$$ 3.55051 0.143054
$$617$$ −4.89898 −0.197225 −0.0986127 0.995126i $$-0.531441\pi$$
−0.0986127 + 0.995126i $$0.531441\pi$$
$$618$$ −5.34847 −0.215147
$$619$$ −23.1010 −0.928508 −0.464254 0.885702i $$-0.653678\pi$$
−0.464254 + 0.885702i $$0.653678\pi$$
$$620$$ 0 0
$$621$$ 40.6969 1.63311
$$622$$ 3.55051 0.142362
$$623$$ −0.853572 −0.0341976
$$624$$ 15.3485 0.614431
$$625$$ 0 0
$$626$$ −0.898979 −0.0359304
$$627$$ −25.0000 −0.998404
$$628$$ −11.5505 −0.460916
$$629$$ 1.44949 0.0577949
$$630$$ 0 0
$$631$$ 2.00000 0.0796187 0.0398094 0.999207i $$-0.487325\pi$$
0.0398094 + 0.999207i $$0.487325\pi$$
$$632$$ 11.7980 0.469298
$$633$$ 32.5959 1.29557
$$634$$ −14.4495 −0.573863
$$635$$ 0 0
$$636$$ 20.6969 0.820687
$$637$$ 4.44949 0.176295
$$638$$ 12.8990 0.510675
$$639$$ −110.788 −4.38270
$$640$$ 0 0
$$641$$ −17.5959 −0.694997 −0.347498 0.937681i $$-0.612969\pi$$
−0.347498 + 0.937681i $$0.612969\pi$$
$$642$$ 46.3939 1.83102
$$643$$ 36.2929 1.43125 0.715625 0.698484i $$-0.246142\pi$$
0.715625 + 0.698484i $$0.246142\pi$$
$$644$$ −4.89898 −0.193047
$$645$$ 0 0
$$646$$ 7.24745 0.285147
$$647$$ 15.7526 0.619297 0.309648 0.950851i $$-0.399789\pi$$
0.309648 + 0.950851i $$0.399789\pi$$
$$648$$ −43.4949 −1.70864
$$649$$ 2.89898 0.113795
$$650$$ 0 0
$$651$$ 3.79796 0.148854
$$652$$ −19.8990 −0.779304
$$653$$ −34.0454 −1.33230 −0.666150 0.745818i $$-0.732059\pi$$
−0.666150 + 0.745818i $$0.732059\pi$$
$$654$$ 48.2929 1.88840
$$655$$ 0 0
$$656$$ 1.00000 0.0390434
$$657$$ 60.4949 2.36013
$$658$$ −24.0000 −0.935617
$$659$$ 11.4495 0.446009 0.223004 0.974817i $$-0.428414\pi$$
0.223004 + 0.974817i $$0.428414\pi$$
$$660$$ 0 0
$$661$$ 6.20204 0.241231 0.120616 0.992699i $$-0.461513\pi$$
0.120616 + 0.992699i $$0.461513\pi$$
$$662$$ −10.1010 −0.392587
$$663$$ 22.2474 0.864019
$$664$$ −1.44949 −0.0562511
$$665$$ 0 0
$$666$$ −8.89898 −0.344828
$$667$$ −17.7980 −0.689140
$$668$$ 5.55051 0.214756
$$669$$ 7.59592 0.293675
$$670$$ 0 0
$$671$$ 2.24745 0.0867618
$$672$$ 8.44949 0.325946
$$673$$ 13.7980 0.531872 0.265936 0.963991i $$-0.414319\pi$$
0.265936 + 0.963991i $$0.414319\pi$$
$$674$$ −13.6969 −0.527586
$$675$$ 0 0
$$676$$ 6.79796 0.261460
$$677$$ 4.40408 0.169263 0.0846313 0.996412i $$-0.473029\pi$$
0.0846313 + 0.996412i $$0.473029\pi$$
$$678$$ 32.5959 1.25184
$$679$$ −34.2929 −1.31604
$$680$$ 0 0
$$681$$ 43.7980 1.67834
$$682$$ −0.651531 −0.0249484
$$683$$ 7.00000 0.267848 0.133924 0.990992i $$-0.457242\pi$$
0.133924 + 0.990992i $$0.457242\pi$$
$$684$$ −44.4949 −1.70130
$$685$$ 0 0
$$686$$ 19.5959 0.748176
$$687$$ 47.5959 1.81590
$$688$$ 10.8990 0.415520
$$689$$ 26.6969 1.01707
$$690$$ 0 0
$$691$$ 44.3485 1.68710 0.843548 0.537054i $$-0.180463\pi$$
0.843548 + 0.537054i $$0.180463\pi$$
$$692$$ −22.6969 −0.862808
$$693$$ −31.5959 −1.20023
$$694$$ 0.797959 0.0302901
$$695$$ 0 0
$$696$$ 30.6969 1.16356
$$697$$ 1.44949 0.0549033
$$698$$ 7.55051 0.285791
$$699$$ 70.6969 2.67400
$$700$$ 0 0
$$701$$ −9.75255 −0.368349 −0.184174 0.982894i $$-0.558961\pi$$
−0.184174 + 0.982894i $$0.558961\pi$$
$$702$$ −90.5403 −3.41722
$$703$$ −5.00000 −0.188579
$$704$$ −1.44949 −0.0546297
$$705$$ 0 0
$$706$$ 5.10102 0.191979
$$707$$ 26.6969 1.00404
$$708$$ 6.89898 0.259280
$$709$$ 7.10102 0.266684 0.133342 0.991070i $$-0.457429\pi$$
0.133342 + 0.991070i $$0.457429\pi$$
$$710$$ 0 0
$$711$$ −104.990 −3.93742
$$712$$ 0.348469 0.0130594
$$713$$ 0.898979 0.0336670
$$714$$ 12.2474 0.458349
$$715$$ 0 0
$$716$$ 18.7980 0.702513
$$717$$ 30.6969 1.14640
$$718$$ −27.3939 −1.02233
$$719$$ −15.3485 −0.572401 −0.286201 0.958170i $$-0.592392\pi$$
−0.286201 + 0.958170i $$0.592392\pi$$
$$720$$ 0 0
$$721$$ −3.79796 −0.141443
$$722$$ −6.00000 −0.223297
$$723$$ 25.6969 0.955679
$$724$$ −2.89898 −0.107740
$$725$$ 0 0
$$726$$ −30.6969 −1.13927
$$727$$ −36.0000 −1.33517 −0.667583 0.744535i $$-0.732671\pi$$
−0.667583 + 0.744535i $$0.732671\pi$$
$$728$$ 10.8990 0.403943
$$729$$ 176.485 6.53647
$$730$$ 0 0
$$731$$ 15.7980 0.584309
$$732$$ 5.34847 0.197685
$$733$$ −23.5959 −0.871535 −0.435768 0.900059i $$-0.643523\pi$$
−0.435768 + 0.900059i $$0.643523\pi$$
$$734$$ 10.2474 0.378240
$$735$$ 0 0
$$736$$ 2.00000 0.0737210
$$737$$ 13.6969 0.504533
$$738$$ −8.89898 −0.327576
$$739$$ 7.59592 0.279420 0.139710 0.990192i $$-0.455383\pi$$
0.139710 + 0.990192i $$0.455383\pi$$
$$740$$ 0 0
$$741$$ −76.7423 −2.81920
$$742$$ 14.6969 0.539542
$$743$$ 45.5959 1.67275 0.836376 0.548156i $$-0.184670\pi$$
0.836376 + 0.548156i $$0.184670\pi$$
$$744$$ −1.55051 −0.0568445
$$745$$ 0 0
$$746$$ 24.0454 0.880365
$$747$$ 12.8990 0.471949
$$748$$ −2.10102 −0.0768209
$$749$$ 32.9444 1.20376
$$750$$ 0 0
$$751$$ −1.30306 −0.0475494 −0.0237747 0.999717i $$-0.507568\pi$$
−0.0237747 + 0.999717i $$0.507568\pi$$
$$752$$ 9.79796 0.357295
$$753$$ 38.6413 1.40817
$$754$$ 39.5959 1.44200
$$755$$ 0 0
$$756$$ −49.8434 −1.81279
$$757$$ −5.79796 −0.210730 −0.105365 0.994434i $$-0.533601\pi$$
−0.105365 + 0.994434i $$0.533601\pi$$
$$758$$ −23.0454 −0.837047
$$759$$ −10.0000 −0.362977
$$760$$ 0 0
$$761$$ −43.6969 −1.58401 −0.792006 0.610513i $$-0.790963\pi$$
−0.792006 + 0.610513i $$0.790963\pi$$
$$762$$ −28.4495 −1.03062
$$763$$ 34.2929 1.24148
$$764$$ 11.7980 0.426835
$$765$$ 0 0
$$766$$ 9.34847 0.337774
$$767$$ 8.89898 0.321324
$$768$$ −3.44949 −0.124473
$$769$$ 28.7526 1.03684 0.518422 0.855125i $$-0.326520\pi$$
0.518422 + 0.855125i $$0.326520\pi$$
$$770$$ 0 0
$$771$$ 10.0000 0.360141
$$772$$ 7.44949 0.268113
$$773$$ 22.0454 0.792918 0.396459 0.918052i $$-0.370239\pi$$
0.396459 + 0.918052i $$0.370239\pi$$
$$774$$ −96.9898 −3.48623
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ −8.44949 −0.303124
$$778$$ 21.1464 0.758136
$$779$$ −5.00000 −0.179144
$$780$$ 0 0
$$781$$ 18.0454 0.645715
$$782$$ 2.89898 0.103667
$$783$$ −181.081 −6.47129
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ −71.3939 −2.54654
$$787$$ −1.30306 −0.0464491 −0.0232246 0.999730i $$-0.507393\pi$$
−0.0232246 + 0.999730i $$0.507393\pi$$
$$788$$ 2.65153 0.0944569
$$789$$ 68.2929 2.43129
$$790$$ 0 0
$$791$$ 23.1464 0.822992
$$792$$ 12.8990 0.458345
$$793$$ 6.89898 0.244990
$$794$$ −0.651531 −0.0231220
$$795$$ 0 0
$$796$$ 23.5959 0.836335
$$797$$ 20.2474 0.717201 0.358601 0.933491i $$-0.383254\pi$$
0.358601 + 0.933491i $$0.383254\pi$$
$$798$$ −42.2474 −1.49554
$$799$$ 14.2020 0.502432
$$800$$ 0 0
$$801$$ −3.10102 −0.109569
$$802$$ 37.9444 1.33986
$$803$$ −9.85357 −0.347725
$$804$$ 32.5959 1.14957
$$805$$ 0 0
$$806$$ −2.00000 −0.0704470
$$807$$ −13.7980 −0.485711
$$808$$ −10.8990 −0.383425
$$809$$ 19.3939 0.681852 0.340926 0.940090i $$-0.389259\pi$$
0.340926 + 0.940090i $$0.389259\pi$$
$$810$$ 0 0
$$811$$ −13.7980 −0.484512 −0.242256 0.970212i $$-0.577887\pi$$
−0.242256 + 0.970212i $$0.577887\pi$$
$$812$$ 21.7980 0.764958
$$813$$ 48.4495 1.69920
$$814$$ 1.44949 0.0508046
$$815$$ 0 0
$$816$$ −5.00000 −0.175035
$$817$$ −54.4949 −1.90654
$$818$$ −0.146428 −0.00511974
$$819$$ −96.9898 −3.38910
$$820$$ 0 0
$$821$$ −40.0454 −1.39759 −0.698797 0.715320i $$-0.746281\pi$$
−0.698797 + 0.715320i $$0.746281\pi$$
$$822$$ 67.9444 2.36983
$$823$$ 29.8434 1.04027 0.520137 0.854083i $$-0.325881\pi$$
0.520137 + 0.854083i $$0.325881\pi$$
$$824$$ 1.55051 0.0540146
$$825$$ 0 0
$$826$$ 4.89898 0.170457
$$827$$ −27.0000 −0.938882 −0.469441 0.882964i $$-0.655545\pi$$
−0.469441 + 0.882964i $$0.655545\pi$$
$$828$$ −17.7980 −0.618522
$$829$$ 6.65153 0.231017 0.115509 0.993306i $$-0.463150\pi$$
0.115509 + 0.993306i $$0.463150\pi$$
$$830$$ 0 0
$$831$$ −47.5959 −1.65108
$$832$$ −4.44949 −0.154258
$$833$$ −1.44949 −0.0502218
$$834$$ −60.1918 −2.08427
$$835$$ 0 0
$$836$$ 7.24745 0.250658
$$837$$ 9.14643 0.316147
$$838$$ 16.5505 0.571728
$$839$$ 13.3485 0.460840 0.230420 0.973091i $$-0.425990\pi$$
0.230420 + 0.973091i $$0.425990\pi$$
$$840$$ 0 0
$$841$$ 50.1918 1.73075
$$842$$ −23.1010 −0.796114
$$843$$ 3.10102 0.106805
$$844$$ −9.44949 −0.325265
$$845$$ 0 0
$$846$$ −87.1918 −2.99772
$$847$$ −21.7980 −0.748987
$$848$$ −6.00000 −0.206041
$$849$$ 61.0454 2.09507
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 42.9444 1.47125
$$853$$ 4.65153 0.159265 0.0796327 0.996824i $$-0.474625\pi$$
0.0796327 + 0.996824i $$0.474625\pi$$
$$854$$ 3.79796 0.129963
$$855$$ 0 0
$$856$$ −13.4495 −0.459694
$$857$$ −2.14643 −0.0733206 −0.0366603 0.999328i $$-0.511672\pi$$
−0.0366603 + 0.999328i $$0.511672\pi$$
$$858$$ 22.2474 0.759515
$$859$$ 39.4949 1.34755 0.673774 0.738937i $$-0.264672\pi$$
0.673774 + 0.738937i $$0.264672\pi$$
$$860$$ 0 0
$$861$$ −8.44949 −0.287958
$$862$$ −8.69694 −0.296219
$$863$$ 6.24745 0.212666 0.106333 0.994331i $$-0.466089\pi$$
0.106333 + 0.994331i $$0.466089\pi$$
$$864$$ 20.3485 0.692269
$$865$$ 0 0
$$866$$ 19.0000 0.645646
$$867$$ 51.3939 1.74543
$$868$$ −1.10102 −0.0373711
$$869$$ 17.1010 0.580112
$$870$$ 0 0
$$871$$ 42.0454 1.42465
$$872$$ −14.0000 −0.474100
$$873$$ −124.586 −4.21659
$$874$$ −10.0000 −0.338255
$$875$$ 0 0
$$876$$ −23.4495 −0.792285
$$877$$ −19.3485 −0.653351 −0.326676 0.945136i $$-0.605928\pi$$
−0.326676 + 0.945136i $$0.605928\pi$$
$$878$$ −19.5959 −0.661330
$$879$$ 55.3485 1.86686
$$880$$ 0 0
$$881$$ −34.2929 −1.15536 −0.577678 0.816265i $$-0.696041\pi$$
−0.577678 + 0.816265i $$0.696041\pi$$
$$882$$ 8.89898 0.299644
$$883$$ −35.8990 −1.20810 −0.604048 0.796948i $$-0.706447\pi$$
−0.604048 + 0.796948i $$0.706447\pi$$
$$884$$ −6.44949 −0.216920
$$885$$ 0 0
$$886$$ −12.5505 −0.421643
$$887$$ −21.3031 −0.715287 −0.357643 0.933858i $$-0.616420\pi$$
−0.357643 + 0.933858i $$0.616420\pi$$
$$888$$ 3.44949 0.115757
$$889$$ −20.2020 −0.677555
$$890$$ 0 0
$$891$$ −63.0454 −2.11210
$$892$$ −2.20204 −0.0737298
$$893$$ −48.9898 −1.63938
$$894$$ −66.7423 −2.23220
$$895$$ 0 0
$$896$$ −2.44949 −0.0818317
$$897$$ −30.6969 −1.02494
$$898$$ −8.75255 −0.292076
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ −8.69694 −0.289737
$$902$$ 1.44949 0.0482627
$$903$$ −92.0908 −3.06459
$$904$$ −9.44949 −0.314285
$$905$$ 0 0
$$906$$ −48.2929 −1.60442
$$907$$ 25.1010 0.833466 0.416733 0.909029i $$-0.363175\pi$$
0.416733 + 0.909029i $$0.363175\pi$$
$$908$$ −12.6969 −0.421363
$$909$$ 96.9898 3.21695
$$910$$ 0 0
$$911$$ 8.24745 0.273250 0.136625 0.990623i $$-0.456374\pi$$
0.136625 + 0.990623i $$0.456374\pi$$
$$912$$ 17.2474 0.571120
$$913$$ −2.10102 −0.0695336
$$914$$ −9.24745 −0.305878
$$915$$ 0 0
$$916$$ −13.7980 −0.455897
$$917$$ −50.6969 −1.67416
$$918$$ 29.4949 0.973477
$$919$$ −32.2020 −1.06225 −0.531124 0.847294i $$-0.678230\pi$$
−0.531124 + 0.847294i $$0.678230\pi$$
$$920$$ 0 0
$$921$$ 77.0908 2.54023
$$922$$ −38.6969 −1.27442
$$923$$ 55.3939 1.82331
$$924$$ 12.2474 0.402911
$$925$$ 0 0
$$926$$ 14.4495 0.474840
$$927$$ −13.7980 −0.453184
$$928$$ −8.89898 −0.292123
$$929$$ 13.7980 0.452696 0.226348 0.974046i $$-0.427321\pi$$
0.226348 + 0.974046i $$0.427321\pi$$
$$930$$ 0 0
$$931$$ 5.00000 0.163868
$$932$$ −20.4949 −0.671333
$$933$$ 12.2474 0.400963
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 39.5959 1.29423
$$937$$ 24.5959 0.803514 0.401757 0.915746i $$-0.368400\pi$$
0.401757 + 0.915746i $$0.368400\pi$$
$$938$$ 23.1464 0.755758
$$939$$ −3.10102 −0.101198
$$940$$ 0 0
$$941$$ 17.3939 0.567024 0.283512 0.958969i $$-0.408500\pi$$
0.283512 + 0.958969i $$0.408500\pi$$
$$942$$ −39.8434 −1.29817
$$943$$ −2.00000 −0.0651290
$$944$$ −2.00000 −0.0650945
$$945$$ 0 0
$$946$$ 15.7980 0.513636
$$947$$ −32.0000 −1.03986 −0.519930 0.854209i $$-0.674042\pi$$
−0.519930 + 0.854209i $$0.674042\pi$$
$$948$$ 40.6969 1.32178
$$949$$ −30.2474 −0.981874
$$950$$ 0 0
$$951$$ −49.8434 −1.61628
$$952$$ −3.55051 −0.115073
$$953$$ 23.4949 0.761074 0.380537 0.924766i $$-0.375739\pi$$
0.380537 + 0.924766i $$0.375739\pi$$
$$954$$ 53.3939 1.72869
$$955$$ 0 0
$$956$$ −8.89898 −0.287814
$$957$$ 44.4949 1.43832
$$958$$ 18.2474 0.589548
$$959$$ 48.2474 1.55799
$$960$$ 0 0
$$961$$ −30.7980 −0.993483
$$962$$ 4.44949 0.143457
$$963$$ 119.687 3.85685
$$964$$ −7.44949 −0.239932
$$965$$ 0 0
$$966$$ −16.8990 −0.543716
$$967$$ 44.0000 1.41494 0.707472 0.706741i $$-0.249835\pi$$
0.707472 + 0.706741i $$0.249835\pi$$
$$968$$ 8.89898 0.286024
$$969$$ 25.0000 0.803116
$$970$$ 0 0
$$971$$ 39.5403 1.26891 0.634454 0.772960i $$-0.281225\pi$$
0.634454 + 0.772960i $$0.281225\pi$$
$$972$$ −88.9898 −2.85435
$$973$$ −42.7423 −1.37026
$$974$$ 42.7423 1.36955
$$975$$ 0 0
$$976$$ −1.55051 −0.0496306
$$977$$ −49.7423 −1.59140 −0.795699 0.605692i $$-0.792896\pi$$
−0.795699 + 0.605692i $$0.792896\pi$$
$$978$$ −68.6413 −2.19491
$$979$$ 0.505103 0.0161431
$$980$$ 0 0
$$981$$ 124.586 3.97772
$$982$$ 22.2020 0.708496
$$983$$ 30.4949 0.972636 0.486318 0.873782i $$-0.338340\pi$$
0.486318 + 0.873782i $$0.338340\pi$$
$$984$$ 3.44949 0.109966
$$985$$ 0 0
$$986$$ −12.8990 −0.410787
$$987$$ −82.7878 −2.63516
$$988$$ 22.2474 0.707786
$$989$$ −21.7980 −0.693135
$$990$$ 0 0
$$991$$ −11.5505 −0.366914 −0.183457 0.983028i $$-0.558729\pi$$
−0.183457 + 0.983028i $$0.558729\pi$$
$$992$$ 0.449490 0.0142713
$$993$$ −34.8434 −1.10572
$$994$$ 30.4949 0.967239
$$995$$ 0 0
$$996$$ −5.00000 −0.158431
$$997$$ −44.4495 −1.40773 −0.703865 0.710334i $$-0.748544\pi$$
−0.703865 + 0.710334i $$0.748544\pi$$
$$998$$ −22.0000 −0.696398
$$999$$ −20.3485 −0.643797
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 1850.2.a.r.1.1 2
5.2 odd 4 1850.2.b.k.149.2 4
5.3 odd 4 1850.2.b.k.149.3 4
5.4 even 2 1850.2.a.w.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.r.1.1 2 1.1 even 1 trivial
1850.2.a.w.1.2 yes 2 5.4 even 2
1850.2.b.k.149.2 4 5.2 odd 4
1850.2.b.k.149.3 4 5.3 odd 4