# Properties

 Label 4600.2.a.v.1.1 Level $4600$ Weight $2$ Character 4600.1 Self dual yes Analytic conductor $36.731$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.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: $$36.7311849298$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.11491$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.47283 q^{3} -0.527166 q^{7} +3.11491 q^{9} +O(q^{10})$$ $$q-2.47283 q^{3} -0.527166 q^{7} +3.11491 q^{9} +3.11491 q^{11} -4.11491 q^{13} +4.39905 q^{17} -3.70265 q^{19} +1.30359 q^{21} +1.00000 q^{23} -0.284147 q^{27} -9.10170 q^{29} +4.83076 q^{31} -7.70265 q^{33} -9.74378 q^{37} +10.1755 q^{39} +6.93246 q^{41} +4.45963 q^{43} -0.642074 q^{47} -6.72210 q^{49} -10.8781 q^{51} +3.89134 q^{53} +9.15604 q^{57} +8.79811 q^{59} -3.45339 q^{61} -1.64207 q^{63} +8.60719 q^{67} -2.47283 q^{69} +12.3642 q^{71} +5.81756 q^{73} -1.64207 q^{77} -3.17548 q^{79} -8.64207 q^{81} -4.71585 q^{83} +22.5070 q^{87} -5.43171 q^{89} +2.16924 q^{91} -11.9457 q^{93} +4.06058 q^{97} +9.70265 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} - 7 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 - 7 * q^7 + 3 * q^9 $$3 q - 2 q^{3} - 7 q^{7} + 3 q^{9} + 3 q^{11} - 6 q^{13} + 5 q^{17} + 7 q^{19} - 6 q^{21} + 3 q^{23} + q^{27} - q^{29} + 10 q^{31} - 5 q^{33} - 2 q^{37} + 7 q^{39} - 10 q^{41} - 12 q^{43} - q^{47} + 6 q^{49} + 9 q^{51} - 10 q^{53} + 12 q^{57} + 10 q^{59} - 13 q^{61} - 4 q^{63} + 6 q^{67} - 2 q^{69} + 10 q^{71} - 7 q^{73} - 4 q^{77} + 14 q^{79} - 25 q^{81} - 16 q^{83} + 5 q^{87} - 20 q^{89} + 11 q^{91} - 25 q^{93} - 5 q^{97} + 11 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 - 7 * q^7 + 3 * q^9 + 3 * q^11 - 6 * q^13 + 5 * q^17 + 7 * q^19 - 6 * q^21 + 3 * q^23 + q^27 - q^29 + 10 * q^31 - 5 * q^33 - 2 * q^37 + 7 * q^39 - 10 * q^41 - 12 * q^43 - q^47 + 6 * q^49 + 9 * q^51 - 10 * q^53 + 12 * q^57 + 10 * q^59 - 13 * q^61 - 4 * q^63 + 6 * q^67 - 2 * q^69 + 10 * q^71 - 7 * q^73 - 4 * q^77 + 14 * q^79 - 25 * q^81 - 16 * q^83 + 5 * q^87 - 20 * q^89 + 11 * q^91 - 25 * q^93 - 5 * q^97 + 11 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.47283 −1.42769 −0.713846 0.700303i $$-0.753048\pi$$
−0.713846 + 0.700303i $$0.753048\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.527166 −0.199250 −0.0996250 0.995025i $$-0.531764\pi$$
−0.0996250 + 0.995025i $$0.531764\pi$$
$$8$$ 0 0
$$9$$ 3.11491 1.03830
$$10$$ 0 0
$$11$$ 3.11491 0.939180 0.469590 0.882885i $$-0.344402\pi$$
0.469590 + 0.882885i $$0.344402\pi$$
$$12$$ 0 0
$$13$$ −4.11491 −1.14127 −0.570635 0.821204i $$-0.693303\pi$$
−0.570635 + 0.821204i $$0.693303\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.39905 1.06693 0.533464 0.845823i $$-0.320890\pi$$
0.533464 + 0.845823i $$0.320890\pi$$
$$18$$ 0 0
$$19$$ −3.70265 −0.849446 −0.424723 0.905323i $$-0.639628\pi$$
−0.424723 + 0.905323i $$0.639628\pi$$
$$20$$ 0 0
$$21$$ 1.30359 0.284468
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −0.284147 −0.0546842
$$28$$ 0 0
$$29$$ −9.10170 −1.69014 −0.845072 0.534653i $$-0.820442\pi$$
−0.845072 + 0.534653i $$0.820442\pi$$
$$30$$ 0 0
$$31$$ 4.83076 0.867630 0.433815 0.901002i $$-0.357167\pi$$
0.433815 + 0.901002i $$0.357167\pi$$
$$32$$ 0 0
$$33$$ −7.70265 −1.34086
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.74378 −1.60187 −0.800934 0.598753i $$-0.795663\pi$$
−0.800934 + 0.598753i $$0.795663\pi$$
$$38$$ 0 0
$$39$$ 10.1755 1.62938
$$40$$ 0 0
$$41$$ 6.93246 1.08267 0.541334 0.840807i $$-0.317919\pi$$
0.541334 + 0.840807i $$0.317919\pi$$
$$42$$ 0 0
$$43$$ 4.45963 0.680087 0.340044 0.940410i $$-0.389558\pi$$
0.340044 + 0.940410i $$0.389558\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −0.642074 −0.0936561 −0.0468280 0.998903i $$-0.514911\pi$$
−0.0468280 + 0.998903i $$0.514911\pi$$
$$48$$ 0 0
$$49$$ −6.72210 −0.960299
$$50$$ 0 0
$$51$$ −10.8781 −1.52324
$$52$$ 0 0
$$53$$ 3.89134 0.534516 0.267258 0.963625i $$-0.413882\pi$$
0.267258 + 0.963625i $$0.413882\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 9.15604 1.21275
$$58$$ 0 0
$$59$$ 8.79811 1.14542 0.572708 0.819759i $$-0.305893\pi$$
0.572708 + 0.819759i $$0.305893\pi$$
$$60$$ 0 0
$$61$$ −3.45339 −0.442161 −0.221080 0.975256i $$-0.570958\pi$$
−0.221080 + 0.975256i $$0.570958\pi$$
$$62$$ 0 0
$$63$$ −1.64207 −0.206882
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.60719 1.05154 0.525768 0.850628i $$-0.323778\pi$$
0.525768 + 0.850628i $$0.323778\pi$$
$$68$$ 0 0
$$69$$ −2.47283 −0.297694
$$70$$ 0 0
$$71$$ 12.3642 1.46736 0.733678 0.679497i $$-0.237802\pi$$
0.733678 + 0.679497i $$0.237802\pi$$
$$72$$ 0 0
$$73$$ 5.81756 0.680893 0.340447 0.940264i $$-0.389422\pi$$
0.340447 + 0.940264i $$0.389422\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1.64207 −0.187132
$$78$$ 0 0
$$79$$ −3.17548 −0.357270 −0.178635 0.983915i $$-0.557168\pi$$
−0.178635 + 0.983915i $$0.557168\pi$$
$$80$$ 0 0
$$81$$ −8.64207 −0.960230
$$82$$ 0 0
$$83$$ −4.71585 −0.517632 −0.258816 0.965927i $$-0.583332\pi$$
−0.258816 + 0.965927i $$0.583332\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 22.5070 2.41300
$$88$$ 0 0
$$89$$ −5.43171 −0.575760 −0.287880 0.957667i $$-0.592950\pi$$
−0.287880 + 0.957667i $$0.592950\pi$$
$$90$$ 0 0
$$91$$ 2.16924 0.227398
$$92$$ 0 0
$$93$$ −11.9457 −1.23871
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.06058 0.412289 0.206144 0.978522i $$-0.433908\pi$$
0.206144 + 0.978522i $$0.433908\pi$$
$$98$$ 0 0
$$99$$ 9.70265 0.975153
$$100$$ 0 0
$$101$$ −15.5529 −1.54757 −0.773784 0.633450i $$-0.781638\pi$$
−0.773784 + 0.633450i $$0.781638\pi$$
$$102$$ 0 0
$$103$$ −17.9519 −1.76885 −0.884427 0.466678i $$-0.845451\pi$$
−0.884427 + 0.466678i $$0.845451\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.2298 1.18230 0.591150 0.806561i $$-0.298674\pi$$
0.591150 + 0.806561i $$0.298674\pi$$
$$108$$ 0 0
$$109$$ 9.10795 0.872383 0.436192 0.899854i $$-0.356327\pi$$
0.436192 + 0.899854i $$0.356327\pi$$
$$110$$ 0 0
$$111$$ 24.0947 2.28697
$$112$$ 0 0
$$113$$ 13.1755 1.23945 0.619723 0.784821i $$-0.287245\pi$$
0.619723 + 0.784821i $$0.287245\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −12.8176 −1.18498
$$118$$ 0 0
$$119$$ −2.31903 −0.212585
$$120$$ 0 0
$$121$$ −1.29735 −0.117941
$$122$$ 0 0
$$123$$ −17.1428 −1.54572
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −19.5613 −1.73579 −0.867894 0.496750i $$-0.834527\pi$$
−0.867894 + 0.496750i $$0.834527\pi$$
$$128$$ 0 0
$$129$$ −11.0279 −0.970955
$$130$$ 0 0
$$131$$ −4.15604 −0.363115 −0.181557 0.983380i $$-0.558114\pi$$
−0.181557 + 0.983380i $$0.558114\pi$$
$$132$$ 0 0
$$133$$ 1.95191 0.169252
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 11.1344 0.951272 0.475636 0.879642i $$-0.342218\pi$$
0.475636 + 0.879642i $$0.342218\pi$$
$$138$$ 0 0
$$139$$ −6.49452 −0.550858 −0.275429 0.961321i $$-0.588820\pi$$
−0.275429 + 0.961321i $$0.588820\pi$$
$$140$$ 0 0
$$141$$ 1.58774 0.133712
$$142$$ 0 0
$$143$$ −12.8176 −1.07186
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 16.6226 1.37101
$$148$$ 0 0
$$149$$ −5.47283 −0.448352 −0.224176 0.974549i $$-0.571969\pi$$
−0.224176 + 0.974549i $$0.571969\pi$$
$$150$$ 0 0
$$151$$ −0.0954606 −0.00776848 −0.00388424 0.999992i $$-0.501236\pi$$
−0.00388424 + 0.999992i $$0.501236\pi$$
$$152$$ 0 0
$$153$$ 13.7026 1.10779
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3.40530 0.271772 0.135886 0.990724i $$-0.456612\pi$$
0.135886 + 0.990724i $$0.456612\pi$$
$$158$$ 0 0
$$159$$ −9.62263 −0.763124
$$160$$ 0 0
$$161$$ −0.527166 −0.0415465
$$162$$ 0 0
$$163$$ −18.5745 −1.45487 −0.727435 0.686177i $$-0.759288\pi$$
−0.727435 + 0.686177i $$0.759288\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.4596 0.964155 0.482078 0.876129i $$-0.339882\pi$$
0.482078 + 0.876129i $$0.339882\pi$$
$$168$$ 0 0
$$169$$ 3.93246 0.302497
$$170$$ 0 0
$$171$$ −11.5334 −0.881982
$$172$$ 0 0
$$173$$ −3.49228 −0.265513 −0.132757 0.991149i $$-0.542383\pi$$
−0.132757 + 0.991149i $$0.542383\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −21.7563 −1.63530
$$178$$ 0 0
$$179$$ 6.38585 0.477301 0.238650 0.971106i $$-0.423295\pi$$
0.238650 + 0.971106i $$0.423295\pi$$
$$180$$ 0 0
$$181$$ −20.8781 −1.55186 −0.775930 0.630819i $$-0.782719\pi$$
−0.775930 + 0.630819i $$0.782719\pi$$
$$182$$ 0 0
$$183$$ 8.53965 0.631269
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 13.7026 1.00204
$$188$$ 0 0
$$189$$ 0.149793 0.0108958
$$190$$ 0 0
$$191$$ −6.83700 −0.494708 −0.247354 0.968925i $$-0.579561\pi$$
−0.247354 + 0.968925i $$0.579561\pi$$
$$192$$ 0 0
$$193$$ −7.10170 −0.511192 −0.255596 0.966784i $$-0.582272\pi$$
−0.255596 + 0.966784i $$0.582272\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −0.850207 −0.0605748 −0.0302874 0.999541i $$-0.509642\pi$$
−0.0302874 + 0.999541i $$0.509642\pi$$
$$198$$ 0 0
$$199$$ 8.56829 0.607390 0.303695 0.952769i $$-0.401780\pi$$
0.303695 + 0.952769i $$0.401780\pi$$
$$200$$ 0 0
$$201$$ −21.2841 −1.50127
$$202$$ 0 0
$$203$$ 4.79811 0.336761
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 3.11491 0.216501
$$208$$ 0 0
$$209$$ −11.5334 −0.797783
$$210$$ 0 0
$$211$$ 9.63511 0.663309 0.331654 0.943401i $$-0.392393\pi$$
0.331654 + 0.943401i $$0.392393\pi$$
$$212$$ 0 0
$$213$$ −30.5745 −2.09493
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −2.54661 −0.172875
$$218$$ 0 0
$$219$$ −14.3859 −0.972106
$$220$$ 0 0
$$221$$ −18.1017 −1.21765
$$222$$ 0 0
$$223$$ −9.43171 −0.631594 −0.315797 0.948827i $$-0.602272\pi$$
−0.315797 + 0.948827i $$0.602272\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3.91774 0.260030 0.130015 0.991512i $$-0.458497\pi$$
0.130015 + 0.991512i $$0.458497\pi$$
$$228$$ 0 0
$$229$$ −22.0947 −1.46006 −0.730031 0.683414i $$-0.760494\pi$$
−0.730031 + 0.683414i $$0.760494\pi$$
$$230$$ 0 0
$$231$$ 4.06058 0.267166
$$232$$ 0 0
$$233$$ 12.5334 0.821091 0.410545 0.911840i $$-0.365338\pi$$
0.410545 + 0.911840i $$0.365338\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 7.85244 0.510071
$$238$$ 0 0
$$239$$ −12.4123 −0.802882 −0.401441 0.915885i $$-0.631491\pi$$
−0.401441 + 0.915885i $$0.631491\pi$$
$$240$$ 0 0
$$241$$ 5.74378 0.369989 0.184995 0.982740i $$-0.440773\pi$$
0.184995 + 0.982740i $$0.440773\pi$$
$$242$$ 0 0
$$243$$ 22.2229 1.42560
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 15.2361 0.969447
$$248$$ 0 0
$$249$$ 11.6615 0.739019
$$250$$ 0 0
$$251$$ −6.33624 −0.399940 −0.199970 0.979802i $$-0.564085\pi$$
−0.199970 + 0.979802i $$0.564085\pi$$
$$252$$ 0 0
$$253$$ 3.11491 0.195833
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −22.2772 −1.38961 −0.694806 0.719197i $$-0.744510\pi$$
−0.694806 + 0.719197i $$0.744510\pi$$
$$258$$ 0 0
$$259$$ 5.13659 0.319172
$$260$$ 0 0
$$261$$ −28.3510 −1.75488
$$262$$ 0 0
$$263$$ 4.66776 0.287827 0.143913 0.989590i $$-0.454031\pi$$
0.143913 + 0.989590i $$0.454031\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 13.4317 0.822007
$$268$$ 0 0
$$269$$ 9.70889 0.591962 0.295981 0.955194i $$-0.404354\pi$$
0.295981 + 0.955194i $$0.404354\pi$$
$$270$$ 0 0
$$271$$ −6.75698 −0.410457 −0.205229 0.978714i $$-0.565794\pi$$
−0.205229 + 0.978714i $$0.565794\pi$$
$$272$$ 0 0
$$273$$ −5.36417 −0.324654
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −19.0194 −1.14277 −0.571384 0.820683i $$-0.693593\pi$$
−0.571384 + 0.820683i $$0.693593\pi$$
$$278$$ 0 0
$$279$$ 15.0474 0.900863
$$280$$ 0 0
$$281$$ −32.8370 −1.95889 −0.979446 0.201708i $$-0.935351\pi$$
−0.979446 + 0.201708i $$0.935351\pi$$
$$282$$ 0 0
$$283$$ −13.7827 −0.819295 −0.409647 0.912244i $$-0.634348\pi$$
−0.409647 + 0.912244i $$0.634348\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −3.65456 −0.215722
$$288$$ 0 0
$$289$$ 2.35168 0.138334
$$290$$ 0 0
$$291$$ −10.0411 −0.588621
$$292$$ 0 0
$$293$$ −15.0668 −0.880213 −0.440106 0.897946i $$-0.645059\pi$$
−0.440106 + 0.897946i $$0.645059\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −0.885092 −0.0513583
$$298$$ 0 0
$$299$$ −4.11491 −0.237971
$$300$$ 0 0
$$301$$ −2.35097 −0.135507
$$302$$ 0 0
$$303$$ 38.4596 2.20945
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5.91302 0.337474 0.168737 0.985661i $$-0.446031\pi$$
0.168737 + 0.985661i $$0.446031\pi$$
$$308$$ 0 0
$$309$$ 44.3921 2.52538
$$310$$ 0 0
$$311$$ −25.7911 −1.46248 −0.731241 0.682119i $$-0.761058\pi$$
−0.731241 + 0.682119i $$0.761058\pi$$
$$312$$ 0 0
$$313$$ −16.7306 −0.945668 −0.472834 0.881152i $$-0.656769\pi$$
−0.472834 + 0.881152i $$0.656769\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −28.1623 −1.58175 −0.790876 0.611977i $$-0.790375\pi$$
−0.790876 + 0.611977i $$0.790375\pi$$
$$318$$ 0 0
$$319$$ −28.3510 −1.58735
$$320$$ 0 0
$$321$$ −30.2423 −1.68796
$$322$$ 0 0
$$323$$ −16.2882 −0.906297
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −22.5224 −1.24549
$$328$$ 0 0
$$329$$ 0.338479 0.0186610
$$330$$ 0 0
$$331$$ −25.8176 −1.41906 −0.709531 0.704675i $$-0.751093\pi$$
−0.709531 + 0.704675i $$0.751093\pi$$
$$332$$ 0 0
$$333$$ −30.3510 −1.66322
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −33.5676 −1.82854 −0.914271 0.405103i $$-0.867236\pi$$
−0.914271 + 0.405103i $$0.867236\pi$$
$$338$$ 0 0
$$339$$ −32.5808 −1.76955
$$340$$ 0 0
$$341$$ 15.0474 0.814861
$$342$$ 0 0
$$343$$ 7.23382 0.390590
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −7.63984 −0.410128 −0.205064 0.978749i $$-0.565740\pi$$
−0.205064 + 0.978749i $$0.565740\pi$$
$$348$$ 0 0
$$349$$ 3.01945 0.161627 0.0808136 0.996729i $$-0.474248\pi$$
0.0808136 + 0.996729i $$0.474248\pi$$
$$350$$ 0 0
$$351$$ 1.16924 0.0624094
$$352$$ 0 0
$$353$$ 25.1840 1.34041 0.670203 0.742177i $$-0.266207\pi$$
0.670203 + 0.742177i $$0.266207\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 5.73458 0.303506
$$358$$ 0 0
$$359$$ −5.32304 −0.280939 −0.140470 0.990085i $$-0.544861\pi$$
−0.140470 + 0.990085i $$0.544861\pi$$
$$360$$ 0 0
$$361$$ −5.29039 −0.278442
$$362$$ 0 0
$$363$$ 3.20813 0.168383
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.24230 −0.430245 −0.215122 0.976587i $$-0.569015\pi$$
−0.215122 + 0.976587i $$0.569015\pi$$
$$368$$ 0 0
$$369$$ 21.5940 1.12414
$$370$$ 0 0
$$371$$ −2.05138 −0.106502
$$372$$ 0 0
$$373$$ −1.91774 −0.0992970 −0.0496485 0.998767i $$-0.515810\pi$$
−0.0496485 + 0.998767i $$0.515810\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 37.4527 1.92891
$$378$$ 0 0
$$379$$ 30.1692 1.54969 0.774845 0.632151i $$-0.217828\pi$$
0.774845 + 0.632151i $$0.217828\pi$$
$$380$$ 0 0
$$381$$ 48.3719 2.47817
$$382$$ 0 0
$$383$$ 1.64903 0.0842617 0.0421309 0.999112i $$-0.486585\pi$$
0.0421309 + 0.999112i $$0.486585\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 13.8913 0.706136
$$388$$ 0 0
$$389$$ −16.6414 −0.843750 −0.421875 0.906654i $$-0.638628\pi$$
−0.421875 + 0.906654i $$0.638628\pi$$
$$390$$ 0 0
$$391$$ 4.39905 0.222470
$$392$$ 0 0
$$393$$ 10.2772 0.518415
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −36.0257 −1.80808 −0.904039 0.427450i $$-0.859412\pi$$
−0.904039 + 0.427450i $$0.859412\pi$$
$$398$$ 0 0
$$399$$ −4.82675 −0.241640
$$400$$ 0 0
$$401$$ −5.62263 −0.280781 −0.140390 0.990096i $$-0.544836\pi$$
−0.140390 + 0.990096i $$0.544836\pi$$
$$402$$ 0 0
$$403$$ −19.8781 −0.990200
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −30.3510 −1.50444
$$408$$ 0 0
$$409$$ −2.76546 −0.136743 −0.0683716 0.997660i $$-0.521780\pi$$
−0.0683716 + 0.997660i $$0.521780\pi$$
$$410$$ 0 0
$$411$$ −27.5334 −1.35812
$$412$$ 0 0
$$413$$ −4.63807 −0.228224
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 16.0599 0.786455
$$418$$ 0 0
$$419$$ −17.5962 −0.859632 −0.429816 0.902917i $$-0.641421\pi$$
−0.429816 + 0.902917i $$0.641421\pi$$
$$420$$ 0 0
$$421$$ 25.5676 1.24609 0.623044 0.782187i $$-0.285896\pi$$
0.623044 + 0.782187i $$0.285896\pi$$
$$422$$ 0 0
$$423$$ −2.00000 −0.0972433
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1.82051 0.0881006
$$428$$ 0 0
$$429$$ 31.6957 1.53028
$$430$$ 0 0
$$431$$ 17.9736 0.865757 0.432879 0.901452i $$-0.357498\pi$$
0.432879 + 0.901452i $$0.357498\pi$$
$$432$$ 0 0
$$433$$ −37.0382 −1.77994 −0.889971 0.456018i $$-0.849275\pi$$
−0.889971 + 0.456018i $$0.849275\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −3.70265 −0.177122
$$438$$ 0 0
$$439$$ 38.8168 1.85263 0.926313 0.376754i $$-0.122960\pi$$
0.926313 + 0.376754i $$0.122960\pi$$
$$440$$ 0 0
$$441$$ −20.9387 −0.997081
$$442$$ 0 0
$$443$$ −25.3726 −1.20549 −0.602745 0.797934i $$-0.705927\pi$$
−0.602745 + 0.797934i $$0.705927\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 13.5334 0.640108
$$448$$ 0 0
$$449$$ 33.4589 1.57902 0.789512 0.613735i $$-0.210334\pi$$
0.789512 + 0.613735i $$0.210334\pi$$
$$450$$ 0 0
$$451$$ 21.5940 1.01682
$$452$$ 0 0
$$453$$ 0.236058 0.0110910
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 27.5264 1.28763 0.643816 0.765180i $$-0.277350\pi$$
0.643816 + 0.765180i $$0.277350\pi$$
$$458$$ 0 0
$$459$$ −1.24998 −0.0583440
$$460$$ 0 0
$$461$$ 36.3161 1.69141 0.845704 0.533652i $$-0.179181\pi$$
0.845704 + 0.533652i $$0.179181\pi$$
$$462$$ 0 0
$$463$$ 22.5544 1.04819 0.524095 0.851660i $$-0.324404\pi$$
0.524095 + 0.851660i $$0.324404\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −22.8370 −1.05677 −0.528385 0.849005i $$-0.677202\pi$$
−0.528385 + 0.849005i $$0.677202\pi$$
$$468$$ 0 0
$$469$$ −4.53742 −0.209518
$$470$$ 0 0
$$471$$ −8.42074 −0.388007
$$472$$ 0 0
$$473$$ 13.8913 0.638724
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.1212 0.554989
$$478$$ 0 0
$$479$$ −35.5264 −1.62324 −0.811622 0.584182i $$-0.801415\pi$$
−0.811622 + 0.584182i $$0.801415\pi$$
$$480$$ 0 0
$$481$$ 40.0947 1.82816
$$482$$ 0 0
$$483$$ 1.30359 0.0593156
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 10.4945 0.475552 0.237776 0.971320i $$-0.423582\pi$$
0.237776 + 0.971320i $$0.423582\pi$$
$$488$$ 0 0
$$489$$ 45.9317 2.07711
$$490$$ 0 0
$$491$$ 42.1685 1.90304 0.951519 0.307589i $$-0.0995221\pi$$
0.951519 + 0.307589i $$0.0995221\pi$$
$$492$$ 0 0
$$493$$ −40.0389 −1.80326
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −6.51797 −0.292371
$$498$$ 0 0
$$499$$ 30.1560 1.34997 0.674985 0.737832i $$-0.264150\pi$$
0.674985 + 0.737832i $$0.264150\pi$$
$$500$$ 0 0
$$501$$ −30.8106 −1.37652
$$502$$ 0 0
$$503$$ −22.9729 −1.02431 −0.512155 0.858893i $$-0.671153\pi$$
−0.512155 + 0.858893i $$0.671153\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9.72433 −0.431873
$$508$$ 0 0
$$509$$ 11.8176 0.523804 0.261902 0.965094i $$-0.415650\pi$$
0.261902 + 0.965094i $$0.415650\pi$$
$$510$$ 0 0
$$511$$ −3.06682 −0.135668
$$512$$ 0 0
$$513$$ 1.05210 0.0464512
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −2.00000 −0.0879599
$$518$$ 0 0
$$519$$ 8.63583 0.379071
$$520$$ 0 0
$$521$$ −35.7438 −1.56596 −0.782982 0.622045i $$-0.786302\pi$$
−0.782982 + 0.622045i $$0.786302\pi$$
$$522$$ 0 0
$$523$$ −14.3899 −0.629225 −0.314612 0.949220i $$-0.601875\pi$$
−0.314612 + 0.949220i $$0.601875\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 21.2508 0.925698
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 27.4053 1.18929
$$532$$ 0 0
$$533$$ −28.5264 −1.23562
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −15.7911 −0.681438
$$538$$ 0 0
$$539$$ −20.9387 −0.901894
$$540$$ 0 0
$$541$$ 14.5209 0.624303 0.312152 0.950032i $$-0.398950\pi$$
0.312152 + 0.950032i $$0.398950\pi$$
$$542$$ 0 0
$$543$$ 51.6282 2.21558
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −19.6289 −0.839270 −0.419635 0.907693i $$-0.637842\pi$$
−0.419635 + 0.907693i $$0.637842\pi$$
$$548$$ 0 0
$$549$$ −10.7570 −0.459097
$$550$$ 0 0
$$551$$ 33.7004 1.43569
$$552$$ 0 0
$$553$$ 1.67401 0.0711860
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1.13659 −0.0481588 −0.0240794 0.999710i $$-0.507665\pi$$
−0.0240794 + 0.999710i $$0.507665\pi$$
$$558$$ 0 0
$$559$$ −18.3510 −0.776163
$$560$$ 0 0
$$561$$ −33.8844 −1.43060
$$562$$ 0 0
$$563$$ −25.8913 −1.09119 −0.545595 0.838049i $$-0.683696\pi$$
−0.545595 + 0.838049i $$0.683696\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 4.55581 0.191326
$$568$$ 0 0
$$569$$ −44.0558 −1.84692 −0.923459 0.383698i $$-0.874650\pi$$
−0.923459 + 0.383698i $$0.874650\pi$$
$$570$$ 0 0
$$571$$ −41.8991 −1.75342 −0.876711 0.481017i $$-0.840268\pi$$
−0.876711 + 0.481017i $$0.840268\pi$$
$$572$$ 0 0
$$573$$ 16.9068 0.706291
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −1.34544 −0.0560114 −0.0280057 0.999608i $$-0.508916\pi$$
−0.0280057 + 0.999608i $$0.508916\pi$$
$$578$$ 0 0
$$579$$ 17.5613 0.729824
$$580$$ 0 0
$$581$$ 2.48604 0.103138
$$582$$ 0 0
$$583$$ 12.1212 0.502007
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −25.0411 −1.03356 −0.516779 0.856119i $$-0.672869\pi$$
−0.516779 + 0.856119i $$0.672869\pi$$
$$588$$ 0 0
$$589$$ −17.8866 −0.737005
$$590$$ 0 0
$$591$$ 2.10242 0.0864821
$$592$$ 0 0
$$593$$ 22.5947 0.927853 0.463927 0.885874i $$-0.346440\pi$$
0.463927 + 0.885874i $$0.346440\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −21.1880 −0.867166
$$598$$ 0 0
$$599$$ 14.7111 0.601080 0.300540 0.953769i $$-0.402833\pi$$
0.300540 + 0.953769i $$0.402833\pi$$
$$600$$ 0 0
$$601$$ 19.2904 0.786871 0.393436 0.919352i $$-0.371286\pi$$
0.393436 + 0.919352i $$0.371286\pi$$
$$602$$ 0 0
$$603$$ 26.8106 1.09181
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 20.7547 0.842409 0.421205 0.906966i $$-0.361607\pi$$
0.421205 + 0.906966i $$0.361607\pi$$
$$608$$ 0 0
$$609$$ −11.8649 −0.480791
$$610$$ 0 0
$$611$$ 2.64207 0.106887
$$612$$ 0 0
$$613$$ 43.3091 1.74924 0.874619 0.484810i $$-0.161111\pi$$
0.874619 + 0.484810i $$0.161111\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −13.5621 −0.545988 −0.272994 0.962016i $$-0.588014\pi$$
−0.272994 + 0.962016i $$0.588014\pi$$
$$618$$ 0 0
$$619$$ −12.9045 −0.518677 −0.259339 0.965786i $$-0.583505\pi$$
−0.259339 + 0.965786i $$0.583505\pi$$
$$620$$ 0 0
$$621$$ −0.284147 −0.0114024
$$622$$ 0 0
$$623$$ 2.86341 0.114720
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 28.5202 1.13899
$$628$$ 0 0
$$629$$ −42.8634 −1.70908
$$630$$ 0 0
$$631$$ 45.0404 1.79303 0.896515 0.443013i $$-0.146090\pi$$
0.896515 + 0.443013i $$0.146090\pi$$
$$632$$ 0 0
$$633$$ −23.8260 −0.947000
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 27.6608 1.09596
$$638$$ 0 0
$$639$$ 38.5132 1.52356
$$640$$ 0 0
$$641$$ −3.08074 −0.121682 −0.0608410 0.998147i $$-0.519378\pi$$
−0.0608410 + 0.998147i $$0.519378\pi$$
$$642$$ 0 0
$$643$$ −4.52493 −0.178446 −0.0892229 0.996012i $$-0.528438\pi$$
−0.0892229 + 0.996012i $$0.528438\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −34.8719 −1.37096 −0.685478 0.728094i $$-0.740407\pi$$
−0.685478 + 0.728094i $$0.740407\pi$$
$$648$$ 0 0
$$649$$ 27.4053 1.07575
$$650$$ 0 0
$$651$$ 6.29735 0.246813
$$652$$ 0 0
$$653$$ 35.3726 1.38424 0.692119 0.721783i $$-0.256677\pi$$
0.692119 + 0.721783i $$0.256677\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 18.1212 0.706973
$$658$$ 0 0
$$659$$ −2.10866 −0.0821419 −0.0410710 0.999156i $$-0.513077\pi$$
−0.0410710 + 0.999156i $$0.513077\pi$$
$$660$$ 0 0
$$661$$ 20.0342 0.779239 0.389619 0.920976i $$-0.372607\pi$$
0.389619 + 0.920976i $$0.372607\pi$$
$$662$$ 0 0
$$663$$ 44.7625 1.73843
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9.10170 −0.352419
$$668$$ 0 0
$$669$$ 23.3230 0.901721
$$670$$ 0 0
$$671$$ −10.7570 −0.415269
$$672$$ 0 0
$$673$$ 5.07530 0.195638 0.0978191 0.995204i $$-0.468813\pi$$
0.0978191 + 0.995204i $$0.468813\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 13.4876 0.518369 0.259184 0.965828i $$-0.416546\pi$$
0.259184 + 0.965828i $$0.416546\pi$$
$$678$$ 0 0
$$679$$ −2.14060 −0.0821486
$$680$$ 0 0
$$681$$ −9.68793 −0.371242
$$682$$ 0 0
$$683$$ −25.4379 −0.973356 −0.486678 0.873581i $$-0.661792\pi$$
−0.486678 + 0.873581i $$0.661792\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 54.6366 2.08452
$$688$$ 0 0
$$689$$ −16.0125 −0.610027
$$690$$ 0 0
$$691$$ 22.6506 0.861668 0.430834 0.902431i $$-0.358220\pi$$
0.430834 + 0.902431i $$0.358220\pi$$
$$692$$ 0 0
$$693$$ −5.11491 −0.194299
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 30.4963 1.15513
$$698$$ 0 0
$$699$$ −30.9930 −1.17226
$$700$$ 0 0
$$701$$ −41.8099 −1.57914 −0.789569 0.613662i $$-0.789696\pi$$
−0.789569 + 0.613662i $$0.789696\pi$$
$$702$$ 0 0
$$703$$ 36.0778 1.36070
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.19894 0.308353
$$708$$ 0 0
$$709$$ 2.87117 0.107829 0.0539146 0.998546i $$-0.482830\pi$$
0.0539146 + 0.998546i $$0.482830\pi$$
$$710$$ 0 0
$$711$$ −9.89134 −0.370954
$$712$$ 0 0
$$713$$ 4.83076 0.180913
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 30.6935 1.14627
$$718$$ 0 0
$$719$$ −50.1623 −1.87074 −0.935369 0.353674i $$-0.884932\pi$$
−0.935369 + 0.353674i $$0.884932\pi$$
$$720$$ 0 0
$$721$$ 9.46364 0.352444
$$722$$ 0 0
$$723$$ −14.2034 −0.528230
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 26.8198 0.994691 0.497345 0.867553i $$-0.334308\pi$$
0.497345 + 0.867553i $$0.334308\pi$$
$$728$$ 0 0
$$729$$ −29.0272 −1.07508
$$730$$ 0 0
$$731$$ 19.6182 0.725604
$$732$$ 0 0
$$733$$ −35.9861 −1.32918 −0.664588 0.747210i $$-0.731393\pi$$
−0.664588 + 0.747210i $$0.731393\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 26.8106 0.987581
$$738$$ 0 0
$$739$$ 5.84396 0.214974 0.107487 0.994207i $$-0.465720\pi$$
0.107487 + 0.994207i $$0.465720\pi$$
$$740$$ 0 0
$$741$$ −37.6762 −1.38407
$$742$$ 0 0
$$743$$ −14.2012 −0.520991 −0.260495 0.965475i $$-0.583886\pi$$
−0.260495 + 0.965475i $$0.583886\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −14.6894 −0.537459
$$748$$ 0 0
$$749$$ −6.44714 −0.235574
$$750$$ 0 0
$$751$$ 33.0404 1.20566 0.602831 0.797869i $$-0.294039\pi$$
0.602831 + 0.797869i $$0.294039\pi$$
$$752$$ 0 0
$$753$$ 15.6685 0.570991
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −18.2034 −0.661614 −0.330807 0.943698i $$-0.607321\pi$$
−0.330807 + 0.943698i $$0.607321\pi$$
$$758$$ 0 0
$$759$$ −7.70265 −0.279588
$$760$$ 0 0
$$761$$ 34.1010 1.23616 0.618080 0.786115i $$-0.287911\pi$$
0.618080 + 0.786115i $$0.287911\pi$$
$$762$$ 0 0
$$763$$ −4.80140 −0.173822
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −36.2034 −1.30723
$$768$$ 0 0
$$769$$ 35.1366 1.26706 0.633529 0.773719i $$-0.281606\pi$$
0.633529 + 0.773719i $$0.281606\pi$$
$$770$$ 0 0
$$771$$ 55.0878 1.98394
$$772$$ 0 0
$$773$$ −1.13659 −0.0408803 −0.0204401 0.999791i $$-0.506507\pi$$
−0.0204401 + 0.999791i $$0.506507\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −12.7019 −0.455679
$$778$$ 0 0
$$779$$ −25.6685 −0.919669
$$780$$ 0 0
$$781$$ 38.5132 1.37811
$$782$$ 0 0
$$783$$ 2.58622 0.0924241
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 8.33848 0.297235 0.148617 0.988895i $$-0.452518\pi$$
0.148617 + 0.988895i $$0.452518\pi$$
$$788$$ 0 0
$$789$$ −11.5426 −0.410928
$$790$$ 0 0
$$791$$ −6.94567 −0.246960
$$792$$ 0 0
$$793$$ 14.2104 0.504625
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0.190921 0.00676278 0.00338139 0.999994i $$-0.498924\pi$$
0.00338139 + 0.999994i $$0.498924\pi$$
$$798$$ 0 0
$$799$$ −2.82452 −0.0999242
$$800$$ 0 0
$$801$$ −16.9193 −0.597813
$$802$$ 0 0
$$803$$ 18.1212 0.639482
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −24.0085 −0.845138
$$808$$ 0 0
$$809$$ −14.5202 −0.510503 −0.255252 0.966875i $$-0.582158\pi$$
−0.255252 + 0.966875i $$0.582158\pi$$
$$810$$ 0 0
$$811$$ −51.5474 −1.81007 −0.905037 0.425332i $$-0.860157\pi$$
−0.905037 + 0.425332i $$0.860157\pi$$
$$812$$ 0 0
$$813$$ 16.7089 0.586006
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −16.5124 −0.577697
$$818$$ 0 0
$$819$$ 6.75698 0.236108
$$820$$ 0 0
$$821$$ 13.7827 0.481019 0.240509 0.970647i $$-0.422686\pi$$
0.240509 + 0.970647i $$0.422686\pi$$
$$822$$ 0 0
$$823$$ 37.1531 1.29508 0.647538 0.762034i $$-0.275799\pi$$
0.647538 + 0.762034i $$0.275799\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 39.8510 1.38576 0.692878 0.721055i $$-0.256343\pi$$
0.692878 + 0.721055i $$0.256343\pi$$
$$828$$ 0 0
$$829$$ 20.5933 0.715234 0.357617 0.933868i $$-0.383589\pi$$
0.357617 + 0.933868i $$0.383589\pi$$
$$830$$ 0 0
$$831$$ 47.0319 1.63152
$$832$$ 0 0
$$833$$ −29.5709 −1.02457
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1.37265 −0.0474456
$$838$$ 0 0
$$839$$ −36.5669 −1.26243 −0.631214 0.775609i $$-0.717443\pi$$
−0.631214 + 0.775609i $$0.717443\pi$$
$$840$$ 0 0
$$841$$ 53.8410 1.85659
$$842$$ 0 0
$$843$$ 81.2005 2.79669
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.683919 0.0234998
$$848$$ 0 0
$$849$$ 34.0823 1.16970
$$850$$ 0 0
$$851$$ −9.74378 −0.334012
$$852$$ 0 0
$$853$$ 19.3208 0.661532 0.330766 0.943713i $$-0.392693\pi$$
0.330766 + 0.943713i $$0.392693\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −54.6017 −1.86516 −0.932580 0.360963i $$-0.882448\pi$$
−0.932580 + 0.360963i $$0.882448\pi$$
$$858$$ 0 0
$$859$$ −32.9541 −1.12438 −0.562190 0.827008i $$-0.690041\pi$$
−0.562190 + 0.827008i $$0.690041\pi$$
$$860$$ 0 0
$$861$$ 9.03712 0.307984
$$862$$ 0 0
$$863$$ 54.6840 1.86147 0.930733 0.365701i $$-0.119171\pi$$
0.930733 + 0.365701i $$0.119171\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −5.81532 −0.197499
$$868$$ 0 0
$$869$$ −9.89134 −0.335541
$$870$$ 0 0
$$871$$ −35.4178 −1.20009
$$872$$ 0 0
$$873$$ 12.6483 0.428081
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −16.7500 −0.565608 −0.282804 0.959178i $$-0.591265\pi$$
−0.282804 + 0.959178i $$0.591265\pi$$
$$878$$ 0 0
$$879$$ 37.2577 1.25667
$$880$$ 0 0
$$881$$ −31.7049 −1.06816 −0.534082 0.845432i $$-0.679343\pi$$
−0.534082 + 0.845432i $$0.679343\pi$$
$$882$$ 0 0
$$883$$ −51.9185 −1.74720 −0.873599 0.486646i $$-0.838220\pi$$
−0.873599 + 0.486646i $$0.838220\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −36.2897 −1.21849 −0.609244 0.792983i $$-0.708527\pi$$
−0.609244 + 0.792983i $$0.708527\pi$$
$$888$$ 0 0
$$889$$ 10.3121 0.345856
$$890$$ 0 0
$$891$$ −26.9193 −0.901829
$$892$$ 0 0
$$893$$ 2.37737 0.0795558
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 10.1755 0.339749
$$898$$ 0 0
$$899$$ −43.9681 −1.46642
$$900$$ 0 0
$$901$$ 17.1182 0.570290
$$902$$ 0 0
$$903$$ 5.81355 0.193463
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 25.4442 0.844861 0.422430 0.906395i $$-0.361177\pi$$
0.422430 + 0.906395i $$0.361177\pi$$
$$908$$ 0 0
$$909$$ −48.4457 −1.60684
$$910$$ 0 0
$$911$$ 47.9597 1.58897 0.794487 0.607281i $$-0.207740\pi$$
0.794487 + 0.607281i $$0.207740\pi$$
$$912$$ 0 0
$$913$$ −14.6894 −0.486150
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 2.19092 0.0723506
$$918$$ 0 0
$$919$$ 2.78562 0.0918892 0.0459446 0.998944i $$-0.485370\pi$$
0.0459446 + 0.998944i $$0.485370\pi$$
$$920$$ 0 0
$$921$$ −14.6219 −0.481808
$$922$$ 0 0
$$923$$ −50.8774 −1.67465
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −55.9185 −1.83661
$$928$$ 0 0
$$929$$ 40.2508 1.32059 0.660293 0.751008i $$-0.270432\pi$$
0.660293 + 0.751008i $$0.270432\pi$$
$$930$$ 0 0
$$931$$ 24.8896 0.815722
$$932$$ 0 0
$$933$$ 63.7772 2.08797
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 38.9868 1.27364 0.636822 0.771011i $$-0.280249\pi$$
0.636822 + 0.771011i $$0.280249\pi$$
$$938$$ 0 0
$$939$$ 41.3719 1.35012
$$940$$ 0 0
$$941$$ 38.3682 1.25077 0.625383 0.780318i $$-0.284943\pi$$
0.625383 + 0.780318i $$0.284943\pi$$
$$942$$ 0 0
$$943$$ 6.93246 0.225752
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −15.0217 −0.488139 −0.244070 0.969758i $$-0.578483\pi$$
−0.244070 + 0.969758i $$0.578483\pi$$
$$948$$ 0 0
$$949$$ −23.9387 −0.777083
$$950$$ 0 0
$$951$$ 69.6406 2.25825
$$952$$ 0 0
$$953$$ 12.1303 0.392940 0.196470 0.980510i $$-0.437052\pi$$
0.196470 + 0.980510i $$0.437052\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 70.1072 2.26624
$$958$$ 0 0
$$959$$ −5.86965 −0.189541
$$960$$ 0 0
$$961$$ −7.66376 −0.247218
$$962$$ 0 0
$$963$$ 38.0947 1.22759
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −21.2882 −0.684581 −0.342290 0.939594i $$-0.611203\pi$$
−0.342290 + 0.939594i $$0.611203\pi$$
$$968$$ 0 0
$$969$$ 40.2779 1.29391
$$970$$ 0 0
$$971$$ 11.4006 0.365862 0.182931 0.983126i $$-0.441442\pi$$
0.182931 + 0.983126i $$0.441442\pi$$
$$972$$ 0 0
$$973$$ 3.42369 0.109758
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −41.2849 −1.32082 −0.660411 0.750904i $$-0.729618\pi$$
−0.660411 + 0.750904i $$0.729618\pi$$
$$978$$ 0 0
$$979$$ −16.9193 −0.540742
$$980$$ 0 0
$$981$$ 28.3704 0.905798
$$982$$ 0 0
$$983$$ 42.8448 1.36654 0.683268 0.730168i $$-0.260558\pi$$
0.683268 + 0.730168i $$0.260558\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −0.837003 −0.0266421
$$988$$ 0 0
$$989$$ 4.45963 0.141808
$$990$$ 0 0
$$991$$ 8.73307 0.277415 0.138707 0.990333i $$-0.455705\pi$$
0.138707 + 0.990333i $$0.455705\pi$$
$$992$$ 0 0
$$993$$ 63.8425 2.02598
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 44.0808 1.39605 0.698027 0.716072i $$-0.254062\pi$$
0.698027 + 0.716072i $$0.254062\pi$$
$$998$$ 0 0
$$999$$ 2.76867 0.0875968
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 4600.2.a.v.1.1 3
4.3 odd 2 9200.2.a.ci.1.3 3
5.2 odd 4 4600.2.e.q.4049.6 6
5.3 odd 4 4600.2.e.q.4049.1 6
5.4 even 2 920.2.a.i.1.3 3
15.14 odd 2 8280.2.a.bl.1.1 3
20.19 odd 2 1840.2.a.q.1.1 3
40.19 odd 2 7360.2.a.cf.1.3 3
40.29 even 2 7360.2.a.bw.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.3 3 5.4 even 2
1840.2.a.q.1.1 3 20.19 odd 2
4600.2.a.v.1.1 3 1.1 even 1 trivial
4600.2.e.q.4049.1 6 5.3 odd 4
4600.2.e.q.4049.6 6 5.2 odd 4
7360.2.a.bw.1.1 3 40.29 even 2
7360.2.a.cf.1.3 3 40.19 odd 2
8280.2.a.bl.1.1 3 15.14 odd 2
9200.2.a.ci.1.3 3 4.3 odd 2