# Properties

 Label 165.4.a.c.1.2 Level $165$ Weight $4$ Character 165.1 Self dual yes Analytic conductor $9.735$ 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] = [165,4,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("165.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 165.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: $$9.73531515095$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 165.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} -5.00000 q^{5} +4.68466 q^{6} -10.2462 q^{7} -21.1771 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} -5.00000 q^{5} +4.68466 q^{6} -10.2462 q^{7} -21.1771 q^{8} +9.00000 q^{9} -7.80776 q^{10} -11.0000 q^{11} -16.6847 q^{12} -40.8769 q^{13} -16.0000 q^{14} -15.0000 q^{15} +11.4233 q^{16} -98.7083 q^{17} +14.0540 q^{18} -39.6458 q^{19} +27.8078 q^{20} -30.7386 q^{21} -17.1771 q^{22} +61.6932 q^{23} -63.5312 q^{24} +25.0000 q^{25} -63.8314 q^{26} +27.0000 q^{27} +56.9848 q^{28} -149.093 q^{29} -23.4233 q^{30} +54.7386 q^{31} +187.255 q^{32} -33.0000 q^{33} -154.138 q^{34} +51.2311 q^{35} -50.0540 q^{36} +44.8939 q^{37} -61.9091 q^{38} -122.631 q^{39} +105.885 q^{40} +336.479 q^{41} -48.0000 q^{42} -2.36745 q^{43} +61.1771 q^{44} -45.0000 q^{45} +96.3371 q^{46} -333.295 q^{47} +34.2699 q^{48} -238.015 q^{49} +39.0388 q^{50} -296.125 q^{51} +227.339 q^{52} +640.064 q^{53} +42.1619 q^{54} +55.0000 q^{55} +216.985 q^{56} -118.938 q^{57} -232.816 q^{58} -370.773 q^{59} +83.4233 q^{60} -714.405 q^{61} +85.4773 q^{62} -92.2159 q^{63} +201.022 q^{64} +204.384 q^{65} -51.5312 q^{66} -404.985 q^{67} +548.972 q^{68} +185.080 q^{69} +80.0000 q^{70} +939.292 q^{71} -190.594 q^{72} -362.570 q^{73} +70.1042 q^{74} +75.0000 q^{75} +220.492 q^{76} +112.708 q^{77} -191.494 q^{78} +951.835 q^{79} -57.1165 q^{80} +81.0000 q^{81} +525.430 q^{82} +735.221 q^{83} +170.955 q^{84} +493.542 q^{85} -3.69690 q^{86} -447.278 q^{87} +232.948 q^{88} +385.879 q^{89} -70.2699 q^{90} +418.833 q^{91} -343.110 q^{92} +164.216 q^{93} -520.458 q^{94} +198.229 q^{95} +561.764 q^{96} -966.345 q^{97} -371.673 q^{98} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - q^2 + 6 * q^3 - 7 * q^4 - 10 * q^5 - 3 * q^6 - 4 * q^7 + 3 * q^8 + 18 * q^9 $$2 q - q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 18 q^{9} + 5 q^{10} - 22 q^{11} - 21 q^{12} - 90 q^{13} - 32 q^{14} - 30 q^{15} - 39 q^{16} - 16 q^{17} - 9 q^{18} - 170 q^{19} + 35 q^{20} - 12 q^{21} + 11 q^{22} - 124 q^{23} + 9 q^{24} + 50 q^{25} + 62 q^{26} + 54 q^{27} + 48 q^{28} - 158 q^{29} + 15 q^{30} + 60 q^{31} + 123 q^{32} - 66 q^{33} - 366 q^{34} + 20 q^{35} - 63 q^{36} - 372 q^{37} + 272 q^{38} - 270 q^{39} - 15 q^{40} + 38 q^{41} - 96 q^{42} - 516 q^{43} + 77 q^{44} - 90 q^{45} + 572 q^{46} + 224 q^{47} - 117 q^{48} - 542 q^{49} - 25 q^{50} - 48 q^{51} + 298 q^{52} + 472 q^{53} - 27 q^{54} + 110 q^{55} + 368 q^{56} - 510 q^{57} - 210 q^{58} + 248 q^{59} + 105 q^{60} + 72 q^{61} + 72 q^{62} - 36 q^{63} + 769 q^{64} + 450 q^{65} + 33 q^{66} - 744 q^{67} + 430 q^{68} - 372 q^{69} + 160 q^{70} + 2060 q^{71} + 27 q^{72} - 486 q^{73} + 1138 q^{74} + 150 q^{75} + 408 q^{76} + 44 q^{77} + 186 q^{78} + 642 q^{79} + 195 q^{80} + 162 q^{81} + 1290 q^{82} - 286 q^{83} + 144 q^{84} + 80 q^{85} + 1312 q^{86} - 474 q^{87} - 33 q^{88} + 244 q^{89} + 45 q^{90} + 112 q^{91} - 76 q^{92} + 180 q^{93} - 1948 q^{94} + 850 q^{95} + 369 q^{96} - 168 q^{97} + 407 q^{98} - 198 q^{99}+O(q^{100})$$ 2 * q - q^2 + 6 * q^3 - 7 * q^4 - 10 * q^5 - 3 * q^6 - 4 * q^7 + 3 * q^8 + 18 * q^9 + 5 * q^10 - 22 * q^11 - 21 * q^12 - 90 * q^13 - 32 * q^14 - 30 * q^15 - 39 * q^16 - 16 * q^17 - 9 * q^18 - 170 * q^19 + 35 * q^20 - 12 * q^21 + 11 * q^22 - 124 * q^23 + 9 * q^24 + 50 * q^25 + 62 * q^26 + 54 * q^27 + 48 * q^28 - 158 * q^29 + 15 * q^30 + 60 * q^31 + 123 * q^32 - 66 * q^33 - 366 * q^34 + 20 * q^35 - 63 * q^36 - 372 * q^37 + 272 * q^38 - 270 * q^39 - 15 * q^40 + 38 * q^41 - 96 * q^42 - 516 * q^43 + 77 * q^44 - 90 * q^45 + 572 * q^46 + 224 * q^47 - 117 * q^48 - 542 * q^49 - 25 * q^50 - 48 * q^51 + 298 * q^52 + 472 * q^53 - 27 * q^54 + 110 * q^55 + 368 * q^56 - 510 * q^57 - 210 * q^58 + 248 * q^59 + 105 * q^60 + 72 * q^61 + 72 * q^62 - 36 * q^63 + 769 * q^64 + 450 * q^65 + 33 * q^66 - 744 * q^67 + 430 * q^68 - 372 * q^69 + 160 * q^70 + 2060 * q^71 + 27 * q^72 - 486 * q^73 + 1138 * q^74 + 150 * q^75 + 408 * q^76 + 44 * q^77 + 186 * q^78 + 642 * q^79 + 195 * q^80 + 162 * q^81 + 1290 * q^82 - 286 * q^83 + 144 * q^84 + 80 * q^85 + 1312 * q^86 - 474 * q^87 - 33 * q^88 + 244 * q^89 + 45 * q^90 + 112 * q^91 - 76 * q^92 + 180 * q^93 - 1948 * q^94 + 850 * q^95 + 369 * q^96 - 168 * q^97 + 407 * q^98 - 198 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.56155 0.552092 0.276046 0.961144i $$-0.410976\pi$$
0.276046 + 0.961144i $$0.410976\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −5.56155 −0.695194
$$5$$ −5.00000 −0.447214
$$6$$ 4.68466 0.318751
$$7$$ −10.2462 −0.553243 −0.276622 0.960979i $$-0.589215\pi$$
−0.276622 + 0.960979i $$0.589215\pi$$
$$8$$ −21.1771 −0.935904
$$9$$ 9.00000 0.333333
$$10$$ −7.80776 −0.246903
$$11$$ −11.0000 −0.301511
$$12$$ −16.6847 −0.401371
$$13$$ −40.8769 −0.872093 −0.436047 0.899924i $$-0.643622\pi$$
−0.436047 + 0.899924i $$0.643622\pi$$
$$14$$ −16.0000 −0.305441
$$15$$ −15.0000 −0.258199
$$16$$ 11.4233 0.178489
$$17$$ −98.7083 −1.40825 −0.704126 0.710075i $$-0.748661\pi$$
−0.704126 + 0.710075i $$0.748661\pi$$
$$18$$ 14.0540 0.184031
$$19$$ −39.6458 −0.478704 −0.239352 0.970933i $$-0.576935\pi$$
−0.239352 + 0.970933i $$0.576935\pi$$
$$20$$ 27.8078 0.310900
$$21$$ −30.7386 −0.319415
$$22$$ −17.1771 −0.166462
$$23$$ 61.6932 0.559301 0.279650 0.960102i $$-0.409781\pi$$
0.279650 + 0.960102i $$0.409781\pi$$
$$24$$ −63.5312 −0.540344
$$25$$ 25.0000 0.200000
$$26$$ −63.8314 −0.481476
$$27$$ 27.0000 0.192450
$$28$$ 56.9848 0.384612
$$29$$ −149.093 −0.954684 −0.477342 0.878718i $$-0.658400\pi$$
−0.477342 + 0.878718i $$0.658400\pi$$
$$30$$ −23.4233 −0.142550
$$31$$ 54.7386 0.317140 0.158570 0.987348i $$-0.449312\pi$$
0.158570 + 0.987348i $$0.449312\pi$$
$$32$$ 187.255 1.03445
$$33$$ −33.0000 −0.174078
$$34$$ −154.138 −0.777485
$$35$$ 51.2311 0.247418
$$36$$ −50.0540 −0.231731
$$37$$ 44.8939 0.199473 0.0997367 0.995014i $$-0.468200\pi$$
0.0997367 + 0.995014i $$0.468200\pi$$
$$38$$ −61.9091 −0.264289
$$39$$ −122.631 −0.503503
$$40$$ 105.885 0.418549
$$41$$ 336.479 1.28169 0.640844 0.767671i $$-0.278585\pi$$
0.640844 + 0.767671i $$0.278585\pi$$
$$42$$ −48.0000 −0.176347
$$43$$ −2.36745 −0.00839611 −0.00419806 0.999991i $$-0.501336\pi$$
−0.00419806 + 0.999991i $$0.501336\pi$$
$$44$$ 61.1771 0.209609
$$45$$ −45.0000 −0.149071
$$46$$ 96.3371 0.308786
$$47$$ −333.295 −1.03439 −0.517193 0.855869i $$-0.673023\pi$$
−0.517193 + 0.855869i $$0.673023\pi$$
$$48$$ 34.2699 0.103051
$$49$$ −238.015 −0.693922
$$50$$ 39.0388 0.110418
$$51$$ −296.125 −0.813055
$$52$$ 227.339 0.606274
$$53$$ 640.064 1.65886 0.829430 0.558610i $$-0.188665\pi$$
0.829430 + 0.558610i $$0.188665\pi$$
$$54$$ 42.1619 0.106250
$$55$$ 55.0000 0.134840
$$56$$ 216.985 0.517782
$$57$$ −118.938 −0.276380
$$58$$ −232.816 −0.527074
$$59$$ −370.773 −0.818144 −0.409072 0.912502i $$-0.634147\pi$$
−0.409072 + 0.912502i $$0.634147\pi$$
$$60$$ 83.4233 0.179498
$$61$$ −714.405 −1.49951 −0.749756 0.661715i $$-0.769829\pi$$
−0.749756 + 0.661715i $$0.769829\pi$$
$$62$$ 85.4773 0.175091
$$63$$ −92.2159 −0.184414
$$64$$ 201.022 0.392621
$$65$$ 204.384 0.390012
$$66$$ −51.5312 −0.0961069
$$67$$ −404.985 −0.738459 −0.369230 0.929338i $$-0.620378\pi$$
−0.369230 + 0.929338i $$0.620378\pi$$
$$68$$ 548.972 0.979009
$$69$$ 185.080 0.322912
$$70$$ 80.0000 0.136598
$$71$$ 939.292 1.57005 0.785024 0.619465i $$-0.212651\pi$$
0.785024 + 0.619465i $$0.212651\pi$$
$$72$$ −190.594 −0.311968
$$73$$ −362.570 −0.581310 −0.290655 0.956828i $$-0.593873\pi$$
−0.290655 + 0.956828i $$0.593873\pi$$
$$74$$ 70.1042 0.110128
$$75$$ 75.0000 0.115470
$$76$$ 220.492 0.332792
$$77$$ 112.708 0.166809
$$78$$ −191.494 −0.277980
$$79$$ 951.835 1.35557 0.677784 0.735261i $$-0.262941\pi$$
0.677784 + 0.735261i $$0.262941\pi$$
$$80$$ −57.1165 −0.0798227
$$81$$ 81.0000 0.111111
$$82$$ 525.430 0.707610
$$83$$ 735.221 0.972302 0.486151 0.873875i $$-0.338401\pi$$
0.486151 + 0.873875i $$0.338401\pi$$
$$84$$ 170.955 0.222056
$$85$$ 493.542 0.629789
$$86$$ −3.69690 −0.00463543
$$87$$ −447.278 −0.551187
$$88$$ 232.948 0.282186
$$89$$ 385.879 0.459585 0.229793 0.973240i $$-0.426195\pi$$
0.229793 + 0.973240i $$0.426195\pi$$
$$90$$ −70.2699 −0.0823011
$$91$$ 418.833 0.482480
$$92$$ −343.110 −0.388823
$$93$$ 164.216 0.183101
$$94$$ −520.458 −0.571076
$$95$$ 198.229 0.214083
$$96$$ 561.764 0.597238
$$97$$ −966.345 −1.01152 −0.505760 0.862674i $$-0.668788\pi$$
−0.505760 + 0.862674i $$0.668788\pi$$
$$98$$ −371.673 −0.383109
$$99$$ −99.0000 −0.100504
$$100$$ −139.039 −0.139039
$$101$$ 348.600 0.343436 0.171718 0.985146i $$-0.445068\pi$$
0.171718 + 0.985146i $$0.445068\pi$$
$$102$$ −462.415 −0.448881
$$103$$ −1536.38 −1.46975 −0.734873 0.678204i $$-0.762758\pi$$
−0.734873 + 0.678204i $$0.762758\pi$$
$$104$$ 865.653 0.816195
$$105$$ 153.693 0.142847
$$106$$ 999.494 0.915844
$$107$$ −779.180 −0.703983 −0.351991 0.936003i $$-0.614495\pi$$
−0.351991 + 0.936003i $$0.614495\pi$$
$$108$$ −150.162 −0.133790
$$109$$ −1501.79 −1.31968 −0.659842 0.751404i $$-0.729377\pi$$
−0.659842 + 0.751404i $$0.729377\pi$$
$$110$$ 85.8854 0.0744441
$$111$$ 134.682 0.115166
$$112$$ −117.045 −0.0987478
$$113$$ 170.000 0.141524 0.0707622 0.997493i $$-0.477457\pi$$
0.0707622 + 0.997493i $$0.477457\pi$$
$$114$$ −185.727 −0.152587
$$115$$ −308.466 −0.250127
$$116$$ 829.187 0.663691
$$117$$ −367.892 −0.290698
$$118$$ −578.981 −0.451691
$$119$$ 1011.39 0.779106
$$120$$ 317.656 0.241649
$$121$$ 121.000 0.0909091
$$122$$ −1115.58 −0.827869
$$123$$ 1009.44 0.739983
$$124$$ −304.432 −0.220474
$$125$$ −125.000 −0.0894427
$$126$$ −144.000 −0.101814
$$127$$ −1739.82 −1.21562 −0.607811 0.794082i $$-0.707952\pi$$
−0.607811 + 0.794082i $$0.707952\pi$$
$$128$$ −1184.13 −0.817683
$$129$$ −7.10235 −0.00484750
$$130$$ 319.157 0.215323
$$131$$ 312.837 0.208647 0.104323 0.994543i $$-0.466732\pi$$
0.104323 + 0.994543i $$0.466732\pi$$
$$132$$ 183.531 0.121018
$$133$$ 406.220 0.264840
$$134$$ −632.405 −0.407698
$$135$$ −135.000 −0.0860663
$$136$$ 2090.35 1.31799
$$137$$ −716.928 −0.447090 −0.223545 0.974694i $$-0.571763\pi$$
−0.223545 + 0.974694i $$0.571763\pi$$
$$138$$ 289.011 0.178277
$$139$$ −876.483 −0.534837 −0.267418 0.963581i $$-0.586171\pi$$
−0.267418 + 0.963581i $$0.586171\pi$$
$$140$$ −284.924 −0.172004
$$141$$ −999.886 −0.597203
$$142$$ 1466.75 0.866811
$$143$$ 449.646 0.262946
$$144$$ 102.810 0.0594963
$$145$$ 745.464 0.426948
$$146$$ −566.172 −0.320937
$$147$$ −714.045 −0.400636
$$148$$ −249.680 −0.138673
$$149$$ −2376.36 −1.30657 −0.653285 0.757112i $$-0.726610\pi$$
−0.653285 + 0.757112i $$0.726610\pi$$
$$150$$ 117.116 0.0637501
$$151$$ −92.8466 −0.0500381 −0.0250190 0.999687i $$-0.507965\pi$$
−0.0250190 + 0.999687i $$0.507965\pi$$
$$152$$ 839.583 0.448021
$$153$$ −888.375 −0.469417
$$154$$ 176.000 0.0920941
$$155$$ −273.693 −0.141829
$$156$$ 682.017 0.350032
$$157$$ −1881.24 −0.956301 −0.478150 0.878278i $$-0.658693\pi$$
−0.478150 + 0.878278i $$0.658693\pi$$
$$158$$ 1486.34 0.748398
$$159$$ 1920.19 0.957743
$$160$$ −936.274 −0.462618
$$161$$ −632.121 −0.309429
$$162$$ 126.486 0.0613436
$$163$$ −2465.49 −1.18474 −0.592369 0.805667i $$-0.701807\pi$$
−0.592369 + 0.805667i $$0.701807\pi$$
$$164$$ −1871.35 −0.891022
$$165$$ 165.000 0.0778499
$$166$$ 1148.09 0.536800
$$167$$ −1254.30 −0.581200 −0.290600 0.956845i $$-0.593855\pi$$
−0.290600 + 0.956845i $$0.593855\pi$$
$$168$$ 650.955 0.298942
$$169$$ −526.080 −0.239454
$$170$$ 770.691 0.347702
$$171$$ −356.813 −0.159568
$$172$$ 13.1667 0.00583693
$$173$$ −1206.71 −0.530314 −0.265157 0.964205i $$-0.585424\pi$$
−0.265157 + 0.964205i $$0.585424\pi$$
$$174$$ −698.449 −0.304306
$$175$$ −256.155 −0.110649
$$176$$ −125.656 −0.0538164
$$177$$ −1112.32 −0.472356
$$178$$ 602.570 0.253733
$$179$$ −1442.29 −0.602244 −0.301122 0.953586i $$-0.597361\pi$$
−0.301122 + 0.953586i $$0.597361\pi$$
$$180$$ 250.270 0.103633
$$181$$ 4261.81 1.75015 0.875076 0.483985i $$-0.160811\pi$$
0.875076 + 0.483985i $$0.160811\pi$$
$$182$$ 654.030 0.266373
$$183$$ −2143.22 −0.865744
$$184$$ −1306.48 −0.523451
$$185$$ −224.470 −0.0892072
$$186$$ 256.432 0.101089
$$187$$ 1085.79 0.424604
$$188$$ 1853.64 0.719099
$$189$$ −276.648 −0.106472
$$190$$ 309.545 0.118194
$$191$$ −852.223 −0.322852 −0.161426 0.986885i $$-0.551609\pi$$
−0.161426 + 0.986885i $$0.551609\pi$$
$$192$$ 603.065 0.226680
$$193$$ −2459.95 −0.917468 −0.458734 0.888574i $$-0.651697\pi$$
−0.458734 + 0.888574i $$0.651697\pi$$
$$194$$ −1509.00 −0.558452
$$195$$ 613.153 0.225173
$$196$$ 1323.73 0.482410
$$197$$ 3477.06 1.25751 0.628756 0.777602i $$-0.283564\pi$$
0.628756 + 0.777602i $$0.283564\pi$$
$$198$$ −154.594 −0.0554874
$$199$$ 3995.04 1.42312 0.711560 0.702626i $$-0.247989\pi$$
0.711560 + 0.702626i $$0.247989\pi$$
$$200$$ −529.427 −0.187181
$$201$$ −1214.95 −0.426350
$$202$$ 544.358 0.189608
$$203$$ 1527.64 0.528173
$$204$$ 1646.91 0.565231
$$205$$ −1682.40 −0.573188
$$206$$ −2399.14 −0.811436
$$207$$ 555.239 0.186434
$$208$$ −466.949 −0.155659
$$209$$ 436.104 0.144335
$$210$$ 240.000 0.0788646
$$211$$ 1046.13 0.341319 0.170660 0.985330i $$-0.445410\pi$$
0.170660 + 0.985330i $$0.445410\pi$$
$$212$$ −3559.75 −1.15323
$$213$$ 2817.88 0.906468
$$214$$ −1216.73 −0.388664
$$215$$ 11.8373 0.00375486
$$216$$ −571.781 −0.180115
$$217$$ −560.864 −0.175456
$$218$$ −2345.13 −0.728587
$$219$$ −1087.71 −0.335619
$$220$$ −305.885 −0.0937400
$$221$$ 4034.89 1.22813
$$222$$ 210.313 0.0635823
$$223$$ −506.265 −0.152027 −0.0760135 0.997107i $$-0.524219\pi$$
−0.0760135 + 0.997107i $$0.524219\pi$$
$$224$$ −1918.65 −0.572300
$$225$$ 225.000 0.0666667
$$226$$ 265.464 0.0781345
$$227$$ −4286.29 −1.25326 −0.626632 0.779315i $$-0.715567\pi$$
−0.626632 + 0.779315i $$0.715567\pi$$
$$228$$ 661.477 0.192138
$$229$$ 5709.37 1.64754 0.823769 0.566926i $$-0.191867\pi$$
0.823769 + 0.566926i $$0.191867\pi$$
$$230$$ −481.686 −0.138093
$$231$$ 338.125 0.0963073
$$232$$ 3157.35 0.893492
$$233$$ 2946.09 0.828348 0.414174 0.910198i $$-0.364071\pi$$
0.414174 + 0.910198i $$0.364071\pi$$
$$234$$ −574.483 −0.160492
$$235$$ 1666.48 0.462591
$$236$$ 2062.07 0.568769
$$237$$ 2855.51 0.782637
$$238$$ 1579.33 0.430139
$$239$$ −2078.89 −0.562646 −0.281323 0.959613i $$-0.590773\pi$$
−0.281323 + 0.959613i $$0.590773\pi$$
$$240$$ −171.349 −0.0460856
$$241$$ 1853.37 0.495378 0.247689 0.968840i $$-0.420329\pi$$
0.247689 + 0.968840i $$0.420329\pi$$
$$242$$ 188.948 0.0501902
$$243$$ 243.000 0.0641500
$$244$$ 3973.20 1.04245
$$245$$ 1190.08 0.310331
$$246$$ 1576.29 0.408539
$$247$$ 1620.60 0.417475
$$248$$ −1159.20 −0.296813
$$249$$ 2205.66 0.561359
$$250$$ −195.194 −0.0493806
$$251$$ −2358.39 −0.593068 −0.296534 0.955022i $$-0.595831\pi$$
−0.296534 + 0.955022i $$0.595831\pi$$
$$252$$ 512.864 0.128204
$$253$$ −678.625 −0.168635
$$254$$ −2716.82 −0.671135
$$255$$ 1480.62 0.363609
$$256$$ −3457.26 −0.844057
$$257$$ 5519.25 1.33962 0.669809 0.742534i $$-0.266376\pi$$
0.669809 + 0.742534i $$0.266376\pi$$
$$258$$ −11.0907 −0.00267627
$$259$$ −459.993 −0.110357
$$260$$ −1136.70 −0.271134
$$261$$ −1341.84 −0.318228
$$262$$ 488.512 0.115192
$$263$$ 2259.65 0.529795 0.264898 0.964277i $$-0.414662\pi$$
0.264898 + 0.964277i $$0.414662\pi$$
$$264$$ 698.844 0.162920
$$265$$ −3200.32 −0.741865
$$266$$ 634.333 0.146216
$$267$$ 1157.64 0.265342
$$268$$ 2252.34 0.513373
$$269$$ 7039.53 1.59557 0.797783 0.602944i $$-0.206006\pi$$
0.797783 + 0.602944i $$0.206006\pi$$
$$270$$ −210.810 −0.0475165
$$271$$ 5155.08 1.15553 0.577765 0.816203i $$-0.303925\pi$$
0.577765 + 0.816203i $$0.303925\pi$$
$$272$$ −1127.57 −0.251357
$$273$$ 1256.50 0.278560
$$274$$ −1119.52 −0.246835
$$275$$ −275.000 −0.0603023
$$276$$ −1029.33 −0.224487
$$277$$ 9074.52 1.96836 0.984179 0.177175i $$-0.0566960\pi$$
0.984179 + 0.177175i $$0.0566960\pi$$
$$278$$ −1368.67 −0.295279
$$279$$ 492.648 0.105713
$$280$$ −1084.92 −0.231559
$$281$$ −3407.79 −0.723459 −0.361729 0.932283i $$-0.617814\pi$$
−0.361729 + 0.932283i $$0.617814\pi$$
$$282$$ −1561.38 −0.329711
$$283$$ −8827.73 −1.85425 −0.927127 0.374746i $$-0.877730\pi$$
−0.927127 + 0.374746i $$0.877730\pi$$
$$284$$ −5223.92 −1.09149
$$285$$ 594.688 0.123601
$$286$$ 702.146 0.145170
$$287$$ −3447.64 −0.709085
$$288$$ 1685.29 0.344815
$$289$$ 4830.33 0.983174
$$290$$ 1164.08 0.235715
$$291$$ −2899.03 −0.584001
$$292$$ 2016.45 0.404123
$$293$$ −4528.29 −0.902886 −0.451443 0.892300i $$-0.649091\pi$$
−0.451443 + 0.892300i $$0.649091\pi$$
$$294$$ −1115.02 −0.221188
$$295$$ 1853.86 0.365885
$$296$$ −950.722 −0.186688
$$297$$ −297.000 −0.0580259
$$298$$ −3710.81 −0.721347
$$299$$ −2521.83 −0.487762
$$300$$ −417.116 −0.0802741
$$301$$ 24.2574 0.00464509
$$302$$ −144.985 −0.0276256
$$303$$ 1045.80 0.198283
$$304$$ −452.886 −0.0854434
$$305$$ 3572.03 0.670602
$$306$$ −1387.24 −0.259162
$$307$$ 568.106 0.105614 0.0528071 0.998605i $$-0.483183\pi$$
0.0528071 + 0.998605i $$0.483183\pi$$
$$308$$ −626.833 −0.115965
$$309$$ −4609.14 −0.848559
$$310$$ −427.386 −0.0783029
$$311$$ −6853.59 −1.24962 −0.624809 0.780778i $$-0.714823\pi$$
−0.624809 + 0.780778i $$0.714823\pi$$
$$312$$ 2596.96 0.471230
$$313$$ −1138.92 −0.205673 −0.102837 0.994698i $$-0.532792\pi$$
−0.102837 + 0.994698i $$0.532792\pi$$
$$314$$ −2937.65 −0.527966
$$315$$ 461.080 0.0824727
$$316$$ −5293.68 −0.942382
$$317$$ 3207.48 0.568297 0.284148 0.958780i $$-0.408289\pi$$
0.284148 + 0.958780i $$0.408289\pi$$
$$318$$ 2998.48 0.528763
$$319$$ 1640.02 0.287848
$$320$$ −1005.11 −0.175585
$$321$$ −2337.54 −0.406445
$$322$$ −987.091 −0.170834
$$323$$ 3913.37 0.674136
$$324$$ −450.486 −0.0772438
$$325$$ −1021.92 −0.174419
$$326$$ −3850.00 −0.654085
$$327$$ −4505.37 −0.761920
$$328$$ −7125.65 −1.19954
$$329$$ 3415.02 0.572267
$$330$$ 257.656 0.0429803
$$331$$ −9135.12 −1.51695 −0.758477 0.651700i $$-0.774056\pi$$
−0.758477 + 0.651700i $$0.774056\pi$$
$$332$$ −4088.97 −0.675938
$$333$$ 404.045 0.0664911
$$334$$ −1958.65 −0.320876
$$335$$ 2024.92 0.330249
$$336$$ −351.136 −0.0570121
$$337$$ −3470.05 −0.560907 −0.280453 0.959868i $$-0.590485\pi$$
−0.280453 + 0.959868i $$0.590485\pi$$
$$338$$ −821.501 −0.132200
$$339$$ 510.000 0.0817091
$$340$$ −2744.86 −0.437826
$$341$$ −602.125 −0.0956214
$$342$$ −557.182 −0.0880963
$$343$$ 5953.20 0.937151
$$344$$ 50.1357 0.00785795
$$345$$ −925.398 −0.144411
$$346$$ −1884.34 −0.292782
$$347$$ −89.3315 −0.0138201 −0.00691004 0.999976i $$-0.502200\pi$$
−0.00691004 + 0.999976i $$0.502200\pi$$
$$348$$ 2487.56 0.383182
$$349$$ −149.375 −0.0229107 −0.0114554 0.999934i $$-0.503646\pi$$
−0.0114554 + 0.999934i $$0.503646\pi$$
$$350$$ −400.000 −0.0610883
$$351$$ −1103.68 −0.167834
$$352$$ −2059.80 −0.311897
$$353$$ 7867.64 1.18627 0.593133 0.805104i $$-0.297891\pi$$
0.593133 + 0.805104i $$0.297891\pi$$
$$354$$ −1736.94 −0.260784
$$355$$ −4696.46 −0.702147
$$356$$ −2146.09 −0.319501
$$357$$ 3034.16 0.449817
$$358$$ −2252.21 −0.332494
$$359$$ 4974.22 0.731279 0.365639 0.930757i $$-0.380850\pi$$
0.365639 + 0.930757i $$0.380850\pi$$
$$360$$ 952.969 0.139516
$$361$$ −5287.21 −0.770842
$$362$$ 6655.04 0.966246
$$363$$ 363.000 0.0524864
$$364$$ −2329.36 −0.335417
$$365$$ 1812.85 0.259970
$$366$$ −3346.74 −0.477970
$$367$$ −13266.7 −1.88696 −0.943479 0.331433i $$-0.892468\pi$$
−0.943479 + 0.331433i $$0.892468\pi$$
$$368$$ 704.739 0.0998290
$$369$$ 3028.31 0.427229
$$370$$ −350.521 −0.0492506
$$371$$ −6558.23 −0.917754
$$372$$ −913.295 −0.127291
$$373$$ −4632.77 −0.643099 −0.321549 0.946893i $$-0.604204\pi$$
−0.321549 + 0.946893i $$0.604204\pi$$
$$374$$ 1695.52 0.234421
$$375$$ −375.000 −0.0516398
$$376$$ 7058.22 0.968085
$$377$$ 6094.45 0.832573
$$378$$ −432.000 −0.0587822
$$379$$ 6503.31 0.881406 0.440703 0.897653i $$-0.354729\pi$$
0.440703 + 0.897653i $$0.354729\pi$$
$$380$$ −1102.46 −0.148829
$$381$$ −5219.45 −0.701839
$$382$$ −1330.79 −0.178244
$$383$$ 12734.5 1.69897 0.849484 0.527614i $$-0.176913\pi$$
0.849484 + 0.527614i $$0.176913\pi$$
$$384$$ −3552.39 −0.472090
$$385$$ −563.542 −0.0745993
$$386$$ −3841.35 −0.506527
$$387$$ −21.3071 −0.00279870
$$388$$ 5374.38 0.703203
$$389$$ 12024.6 1.56728 0.783639 0.621216i $$-0.213361\pi$$
0.783639 + 0.621216i $$0.213361\pi$$
$$390$$ 957.471 0.124317
$$391$$ −6089.63 −0.787636
$$392$$ 5040.47 0.649444
$$393$$ 938.511 0.120462
$$394$$ 5429.61 0.694263
$$395$$ −4759.18 −0.606228
$$396$$ 550.594 0.0698696
$$397$$ −5223.65 −0.660371 −0.330186 0.943916i $$-0.607111\pi$$
−0.330186 + 0.943916i $$0.607111\pi$$
$$398$$ 6238.46 0.785693
$$399$$ 1218.66 0.152905
$$400$$ 285.582 0.0356978
$$401$$ 9648.18 1.20151 0.600757 0.799432i $$-0.294866\pi$$
0.600757 + 0.799432i $$0.294866\pi$$
$$402$$ −1897.22 −0.235384
$$403$$ −2237.55 −0.276576
$$404$$ −1938.76 −0.238755
$$405$$ −405.000 −0.0496904
$$406$$ 2385.48 0.291600
$$407$$ −493.833 −0.0601435
$$408$$ 6271.06 0.760941
$$409$$ −2010.47 −0.243060 −0.121530 0.992588i $$-0.538780\pi$$
−0.121530 + 0.992588i $$0.538780\pi$$
$$410$$ −2627.15 −0.316453
$$411$$ −2150.78 −0.258127
$$412$$ 8544.65 1.02176
$$413$$ 3799.02 0.452633
$$414$$ 867.034 0.102929
$$415$$ −3676.11 −0.434827
$$416$$ −7654.39 −0.902133
$$417$$ −2629.45 −0.308788
$$418$$ 681.000 0.0796861
$$419$$ 4435.27 0.517129 0.258565 0.965994i $$-0.416750\pi$$
0.258565 + 0.965994i $$0.416750\pi$$
$$420$$ −854.773 −0.0993063
$$421$$ 15217.9 1.76170 0.880852 0.473392i $$-0.156971\pi$$
0.880852 + 0.473392i $$0.156971\pi$$
$$422$$ 1633.58 0.188440
$$423$$ −2999.66 −0.344795
$$424$$ −13554.7 −1.55253
$$425$$ −2467.71 −0.281650
$$426$$ 4400.26 0.500454
$$427$$ 7319.95 0.829595
$$428$$ 4333.45 0.489405
$$429$$ 1348.94 0.151812
$$430$$ 18.4845 0.00207303
$$431$$ −5622.11 −0.628324 −0.314162 0.949369i $$-0.601723\pi$$
−0.314162 + 0.949369i $$0.601723\pi$$
$$432$$ 308.429 0.0343502
$$433$$ −14306.3 −1.58780 −0.793898 0.608051i $$-0.791951\pi$$
−0.793898 + 0.608051i $$0.791951\pi$$
$$434$$ −875.818 −0.0968678
$$435$$ 2236.39 0.246498
$$436$$ 8352.29 0.917436
$$437$$ −2445.88 −0.267740
$$438$$ −1698.52 −0.185293
$$439$$ 4384.20 0.476643 0.238322 0.971186i $$-0.423403\pi$$
0.238322 + 0.971186i $$0.423403\pi$$
$$440$$ −1164.74 −0.126197
$$441$$ −2142.14 −0.231307
$$442$$ 6300.69 0.678039
$$443$$ −10090.0 −1.08214 −0.541071 0.840977i $$-0.681981\pi$$
−0.541071 + 0.840977i $$0.681981\pi$$
$$444$$ −749.040 −0.0800627
$$445$$ −1929.39 −0.205533
$$446$$ −790.560 −0.0839330
$$447$$ −7129.07 −0.754348
$$448$$ −2059.71 −0.217215
$$449$$ 9582.52 1.00719 0.503594 0.863941i $$-0.332011\pi$$
0.503594 + 0.863941i $$0.332011\pi$$
$$450$$ 351.349 0.0368062
$$451$$ −3701.27 −0.386444
$$452$$ −945.464 −0.0983869
$$453$$ −278.540 −0.0288895
$$454$$ −6693.27 −0.691918
$$455$$ −2094.17 −0.215772
$$456$$ 2518.75 0.258665
$$457$$ 9999.34 1.02352 0.511761 0.859128i $$-0.328993\pi$$
0.511761 + 0.859128i $$0.328993\pi$$
$$458$$ 8915.49 0.909593
$$459$$ −2665.12 −0.271018
$$460$$ 1715.55 0.173887
$$461$$ −11115.8 −1.12302 −0.561512 0.827468i $$-0.689780\pi$$
−0.561512 + 0.827468i $$0.689780\pi$$
$$462$$ 528.000 0.0531705
$$463$$ −1567.16 −0.157305 −0.0786524 0.996902i $$-0.525062\pi$$
−0.0786524 + 0.996902i $$0.525062\pi$$
$$464$$ −1703.13 −0.170401
$$465$$ −821.080 −0.0818853
$$466$$ 4600.48 0.457325
$$467$$ 12648.8 1.25335 0.626675 0.779281i $$-0.284415\pi$$
0.626675 + 0.779281i $$0.284415\pi$$
$$468$$ 2046.05 0.202091
$$469$$ 4149.56 0.408548
$$470$$ 2602.29 0.255393
$$471$$ −5643.72 −0.552120
$$472$$ 7851.88 0.765704
$$473$$ 26.0420 0.00253152
$$474$$ 4459.02 0.432088
$$475$$ −991.146 −0.0957408
$$476$$ −5624.88 −0.541630
$$477$$ 5760.58 0.552953
$$478$$ −3246.30 −0.310633
$$479$$ −10719.2 −1.02249 −0.511247 0.859434i $$-0.670816\pi$$
−0.511247 + 0.859434i $$0.670816\pi$$
$$480$$ −2808.82 −0.267093
$$481$$ −1835.12 −0.173959
$$482$$ 2894.14 0.273494
$$483$$ −1896.36 −0.178649
$$484$$ −672.948 −0.0631995
$$485$$ 4831.72 0.452365
$$486$$ 379.457 0.0354167
$$487$$ 7161.20 0.666335 0.333167 0.942868i $$-0.391883\pi$$
0.333167 + 0.942868i $$0.391883\pi$$
$$488$$ 15129.0 1.40340
$$489$$ −7396.48 −0.684009
$$490$$ 1858.37 0.171331
$$491$$ 14567.3 1.33893 0.669463 0.742845i $$-0.266524\pi$$
0.669463 + 0.742845i $$0.266524\pi$$
$$492$$ −5614.04 −0.514432
$$493$$ 14716.7 1.34444
$$494$$ 2530.65 0.230485
$$495$$ 495.000 0.0449467
$$496$$ 625.295 0.0566060
$$497$$ −9624.18 −0.868619
$$498$$ 3444.26 0.309922
$$499$$ −4638.99 −0.416172 −0.208086 0.978111i $$-0.566723\pi$$
−0.208086 + 0.978111i $$0.566723\pi$$
$$500$$ 695.194 0.0621801
$$501$$ −3762.89 −0.335556
$$502$$ −3682.75 −0.327428
$$503$$ −12206.3 −1.08201 −0.541006 0.841019i $$-0.681956\pi$$
−0.541006 + 0.841019i $$0.681956\pi$$
$$504$$ 1952.86 0.172594
$$505$$ −1743.00 −0.153589
$$506$$ −1059.71 −0.0931024
$$507$$ −1578.24 −0.138249
$$508$$ 9676.09 0.845093
$$509$$ 10018.6 0.872427 0.436214 0.899843i $$-0.356319\pi$$
0.436214 + 0.899843i $$0.356319\pi$$
$$510$$ 2312.07 0.200746
$$511$$ 3714.97 0.321606
$$512$$ 4074.36 0.351686
$$513$$ −1070.44 −0.0921267
$$514$$ 8618.61 0.739592
$$515$$ 7681.89 0.657291
$$516$$ 39.5001 0.00336995
$$517$$ 3666.25 0.311879
$$518$$ −718.303 −0.0609274
$$519$$ −3620.12 −0.306177
$$520$$ −4328.27 −0.365014
$$521$$ 1054.72 0.0886916 0.0443458 0.999016i $$-0.485880\pi$$
0.0443458 + 0.999016i $$0.485880\pi$$
$$522$$ −2095.35 −0.175691
$$523$$ −16234.2 −1.35730 −0.678652 0.734460i $$-0.737436\pi$$
−0.678652 + 0.734460i $$0.737436\pi$$
$$524$$ −1739.86 −0.145050
$$525$$ −768.466 −0.0638830
$$526$$ 3528.57 0.292496
$$527$$ −5403.16 −0.446613
$$528$$ −376.969 −0.0310709
$$529$$ −8360.95 −0.687183
$$530$$ −4997.47 −0.409578
$$531$$ −3336.95 −0.272715
$$532$$ −2259.21 −0.184115
$$533$$ −13754.2 −1.11775
$$534$$ 1807.71 0.146493
$$535$$ 3895.90 0.314831
$$536$$ 8576.40 0.691127
$$537$$ −4326.86 −0.347706
$$538$$ 10992.6 0.880900
$$539$$ 2618.17 0.209225
$$540$$ 750.810 0.0598328
$$541$$ 675.936 0.0537167 0.0268584 0.999639i $$-0.491450\pi$$
0.0268584 + 0.999639i $$0.491450\pi$$
$$542$$ 8049.93 0.637960
$$543$$ 12785.4 1.01045
$$544$$ −18483.6 −1.45676
$$545$$ 7508.96 0.590181
$$546$$ 1962.09 0.153791
$$547$$ 13058.2 1.02071 0.510355 0.859964i $$-0.329514\pi$$
0.510355 + 0.859964i $$0.329514\pi$$
$$548$$ 3987.23 0.310814
$$549$$ −6429.65 −0.499837
$$550$$ −429.427 −0.0332924
$$551$$ 5910.91 0.457011
$$552$$ −3919.44 −0.302215
$$553$$ −9752.70 −0.749959
$$554$$ 14170.3 1.08672
$$555$$ −673.409 −0.0515038
$$556$$ 4874.61 0.371815
$$557$$ −6710.48 −0.510471 −0.255236 0.966879i $$-0.582153\pi$$
−0.255236 + 0.966879i $$0.582153\pi$$
$$558$$ 769.295 0.0583636
$$559$$ 96.7741 0.00732219
$$560$$ 585.227 0.0441614
$$561$$ 3257.37 0.245145
$$562$$ −5321.45 −0.399416
$$563$$ −20820.5 −1.55858 −0.779288 0.626666i $$-0.784419\pi$$
−0.779288 + 0.626666i $$0.784419\pi$$
$$564$$ 5560.92 0.415172
$$565$$ −850.000 −0.0632916
$$566$$ −13785.0 −1.02372
$$567$$ −829.943 −0.0614715
$$568$$ −19891.5 −1.46941
$$569$$ 3251.08 0.239530 0.119765 0.992802i $$-0.461786\pi$$
0.119765 + 0.992802i $$0.461786\pi$$
$$570$$ 928.636 0.0682391
$$571$$ −4637.50 −0.339883 −0.169941 0.985454i $$-0.554358\pi$$
−0.169941 + 0.985454i $$0.554358\pi$$
$$572$$ −2500.73 −0.182798
$$573$$ −2556.67 −0.186399
$$574$$ −5383.67 −0.391481
$$575$$ 1542.33 0.111860
$$576$$ 1809.20 0.130874
$$577$$ 14462.4 1.04346 0.521730 0.853111i $$-0.325287\pi$$
0.521730 + 0.853111i $$0.325287\pi$$
$$578$$ 7542.82 0.542803
$$579$$ −7379.86 −0.529700
$$580$$ −4145.94 −0.296812
$$581$$ −7533.23 −0.537920
$$582$$ −4526.99 −0.322423
$$583$$ −7040.71 −0.500165
$$584$$ 7678.18 0.544050
$$585$$ 1839.46 0.130004
$$586$$ −7071.17 −0.498476
$$587$$ −22759.7 −1.60033 −0.800166 0.599779i $$-0.795255\pi$$
−0.800166 + 0.599779i $$0.795255\pi$$
$$588$$ 3971.20 0.278520
$$589$$ −2170.16 −0.151816
$$590$$ 2894.91 0.202002
$$591$$ 10431.2 0.726025
$$592$$ 512.836 0.0356038
$$593$$ −14956.4 −1.03573 −0.517864 0.855463i $$-0.673273\pi$$
−0.517864 + 0.855463i $$0.673273\pi$$
$$594$$ −463.781 −0.0320356
$$595$$ −5056.93 −0.348427
$$596$$ 13216.2 0.908319
$$597$$ 11985.1 0.821638
$$598$$ −3937.96 −0.269290
$$599$$ 2150.77 0.146708 0.0733539 0.997306i $$-0.476630\pi$$
0.0733539 + 0.997306i $$0.476630\pi$$
$$600$$ −1588.28 −0.108069
$$601$$ 27759.8 1.88410 0.942050 0.335472i $$-0.108896\pi$$
0.942050 + 0.335472i $$0.108896\pi$$
$$602$$ 37.8792 0.00256452
$$603$$ −3644.86 −0.246153
$$604$$ 516.371 0.0347862
$$605$$ −605.000 −0.0406558
$$606$$ 1633.07 0.109470
$$607$$ −10991.5 −0.734974 −0.367487 0.930029i $$-0.619782\pi$$
−0.367487 + 0.930029i $$0.619782\pi$$
$$608$$ −7423.87 −0.495194
$$609$$ 4582.91 0.304941
$$610$$ 5577.91 0.370234
$$611$$ 13624.1 0.902081
$$612$$ 4940.74 0.326336
$$613$$ 10646.1 0.701457 0.350728 0.936477i $$-0.385934\pi$$
0.350728 + 0.936477i $$0.385934\pi$$
$$614$$ 887.128 0.0583088
$$615$$ −5047.19 −0.330930
$$616$$ −2386.83 −0.156117
$$617$$ 7199.92 0.469786 0.234893 0.972021i $$-0.424526\pi$$
0.234893 + 0.972021i $$0.424526\pi$$
$$618$$ −7197.41 −0.468483
$$619$$ 12186.9 0.791332 0.395666 0.918395i $$-0.370514\pi$$
0.395666 + 0.918395i $$0.370514\pi$$
$$620$$ 1522.16 0.0985990
$$621$$ 1665.72 0.107637
$$622$$ −10702.2 −0.689905
$$623$$ −3953.80 −0.254262
$$624$$ −1400.85 −0.0898698
$$625$$ 625.000 0.0400000
$$626$$ −1778.49 −0.113551
$$627$$ 1308.31 0.0833317
$$628$$ 10462.6 0.664814
$$629$$ −4431.40 −0.280909
$$630$$ 720.000 0.0455325
$$631$$ 7370.64 0.465009 0.232505 0.972595i $$-0.425308\pi$$
0.232505 + 0.972595i $$0.425308\pi$$
$$632$$ −20157.1 −1.26868
$$633$$ 3138.38 0.197061
$$634$$ 5008.65 0.313752
$$635$$ 8699.09 0.543642
$$636$$ −10679.3 −0.665818
$$637$$ 9729.32 0.605164
$$638$$ 2560.98 0.158919
$$639$$ 8453.63 0.523349
$$640$$ 5920.66 0.365679
$$641$$ −25014.9 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$642$$ −3650.19 −0.224395
$$643$$ −21668.2 −1.32894 −0.664472 0.747313i $$-0.731343\pi$$
−0.664472 + 0.747313i $$0.731343\pi$$
$$644$$ 3515.58 0.215113
$$645$$ 35.5118 0.00216787
$$646$$ 6110.94 0.372185
$$647$$ −27625.3 −1.67861 −0.839305 0.543661i $$-0.817038\pi$$
−0.839305 + 0.543661i $$0.817038\pi$$
$$648$$ −1715.34 −0.103989
$$649$$ 4078.50 0.246680
$$650$$ −1595.79 −0.0962952
$$651$$ −1682.59 −0.101299
$$652$$ 13712.0 0.823623
$$653$$ −14314.0 −0.857810 −0.428905 0.903350i $$-0.641101\pi$$
−0.428905 + 0.903350i $$0.641101\pi$$
$$654$$ −7035.38 −0.420650
$$655$$ −1564.19 −0.0933096
$$656$$ 3843.70 0.228767
$$657$$ −3263.13 −0.193770
$$658$$ 5332.73 0.315944
$$659$$ −28327.8 −1.67450 −0.837249 0.546822i $$-0.815837\pi$$
−0.837249 + 0.546822i $$0.815837\pi$$
$$660$$ −917.656 −0.0541208
$$661$$ −32190.9 −1.89422 −0.947112 0.320905i $$-0.896013\pi$$
−0.947112 + 0.320905i $$0.896013\pi$$
$$662$$ −14265.0 −0.837498
$$663$$ 12104.7 0.709059
$$664$$ −15569.8 −0.909981
$$665$$ −2031.10 −0.118440
$$666$$ 630.938 0.0367092
$$667$$ −9198.01 −0.533955
$$668$$ 6975.84 0.404047
$$669$$ −1518.80 −0.0877729
$$670$$ 3162.03 0.182328
$$671$$ 7858.46 0.452120
$$672$$ −5755.95 −0.330418
$$673$$ −6207.38 −0.355538 −0.177769 0.984072i $$-0.556888\pi$$
−0.177769 + 0.984072i $$0.556888\pi$$
$$674$$ −5418.66 −0.309672
$$675$$ 675.000 0.0384900
$$676$$ 2925.82 0.166467
$$677$$ 28831.1 1.63674 0.818368 0.574695i $$-0.194879\pi$$
0.818368 + 0.574695i $$0.194879\pi$$
$$678$$ 796.392 0.0451110
$$679$$ 9901.37 0.559617
$$680$$ −10451.8 −0.589422
$$681$$ −12858.9 −0.723573
$$682$$ −940.250 −0.0527918
$$683$$ 3193.10 0.178888 0.0894441 0.995992i $$-0.471491\pi$$
0.0894441 + 0.995992i $$0.471491\pi$$
$$684$$ 1984.43 0.110931
$$685$$ 3584.64 0.199945
$$686$$ 9296.24 0.517394
$$687$$ 17128.1 0.951206
$$688$$ −27.0441 −0.00149861
$$689$$ −26163.8 −1.44668
$$690$$ −1445.06 −0.0797281
$$691$$ 7682.49 0.422946 0.211473 0.977384i $$-0.432174\pi$$
0.211473 + 0.977384i $$0.432174\pi$$
$$692$$ 6711.17 0.368671
$$693$$ 1014.37 0.0556031
$$694$$ −139.496 −0.00762996
$$695$$ 4382.41 0.239186
$$696$$ 9472.05 0.515858
$$697$$ −33213.3 −1.80494
$$698$$ −233.256 −0.0126488
$$699$$ 8838.28 0.478247
$$700$$ 1424.62 0.0769223
$$701$$ 26551.6 1.43058 0.715292 0.698825i $$-0.246293\pi$$
0.715292 + 0.698825i $$0.246293\pi$$
$$702$$ −1723.45 −0.0926601
$$703$$ −1779.86 −0.0954887
$$704$$ −2211.24 −0.118380
$$705$$ 4999.43 0.267077
$$706$$ 12285.7 0.654928
$$707$$ −3571.83 −0.190004
$$708$$ 6186.22 0.328379
$$709$$ −16304.6 −0.863655 −0.431828 0.901956i $$-0.642131\pi$$
−0.431828 + 0.901956i $$0.642131\pi$$
$$710$$ −7333.77 −0.387650
$$711$$ 8566.52 0.451856
$$712$$ −8171.79 −0.430127
$$713$$ 3377.00 0.177377
$$714$$ 4738.00 0.248341
$$715$$ −2248.23 −0.117593
$$716$$ 8021.36 0.418676
$$717$$ −6236.68 −0.324844
$$718$$ 7767.50 0.403733
$$719$$ −3973.62 −0.206107 −0.103053 0.994676i $$-0.532861\pi$$
−0.103053 + 0.994676i $$0.532861\pi$$
$$720$$ −514.048 −0.0266076
$$721$$ 15742.1 0.813128
$$722$$ −8256.25 −0.425576
$$723$$ 5560.11 0.286007
$$724$$ −23702.3 −1.21670
$$725$$ −3727.32 −0.190937
$$726$$ 566.844 0.0289773
$$727$$ −10780.4 −0.549961 −0.274980 0.961450i $$-0.588671\pi$$
−0.274980 + 0.961450i $$0.588671\pi$$
$$728$$ −8869.67 −0.451555
$$729$$ 729.000 0.0370370
$$730$$ 2830.86 0.143527
$$731$$ 233.687 0.0118238
$$732$$ 11919.6 0.601860
$$733$$ −9211.46 −0.464165 −0.232083 0.972696i $$-0.574554\pi$$
−0.232083 + 0.972696i $$0.574554\pi$$
$$734$$ −20716.6 −1.04177
$$735$$ 3570.23 0.179170
$$736$$ 11552.3 0.578566
$$737$$ 4454.83 0.222654
$$738$$ 4728.87 0.235870
$$739$$ 11084.7 0.551768 0.275884 0.961191i $$-0.411029\pi$$
0.275884 + 0.961191i $$0.411029\pi$$
$$740$$ 1248.40 0.0620163
$$741$$ 4861.80 0.241029
$$742$$ −10241.0 −0.506685
$$743$$ −27420.4 −1.35391 −0.676955 0.736024i $$-0.736701\pi$$
−0.676955 + 0.736024i $$0.736701\pi$$
$$744$$ −3477.61 −0.171365
$$745$$ 11881.8 0.584316
$$746$$ −7234.32 −0.355050
$$747$$ 6616.99 0.324101
$$748$$ −6038.69 −0.295182
$$749$$ 7983.64 0.389474
$$750$$ −585.582 −0.0285099
$$751$$ −11290.8 −0.548614 −0.274307 0.961642i $$-0.588448\pi$$
−0.274307 + 0.961642i $$0.588448\pi$$
$$752$$ −3807.33 −0.184626
$$753$$ −7075.16 −0.342408
$$754$$ 9516.81 0.459657
$$755$$ 464.233 0.0223777
$$756$$ 1538.59 0.0740185
$$757$$ 3739.19 0.179528 0.0897642 0.995963i $$-0.471389\pi$$
0.0897642 + 0.995963i $$0.471389\pi$$
$$758$$ 10155.3 0.486617
$$759$$ −2035.87 −0.0973617
$$760$$ −4197.92 −0.200361
$$761$$ −15621.1 −0.744107 −0.372053 0.928211i $$-0.621346\pi$$
−0.372053 + 0.928211i $$0.621346\pi$$
$$762$$ −8150.45 −0.387480
$$763$$ 15387.7 0.730106
$$764$$ 4739.69 0.224445
$$765$$ 4441.87 0.209930
$$766$$ 19885.7 0.937987
$$767$$ 15156.0 0.713498
$$768$$ −10371.8 −0.487317
$$769$$ 40241.7 1.88706 0.943531 0.331284i $$-0.107482\pi$$
0.943531 + 0.331284i $$0.107482\pi$$
$$770$$ −880.000 −0.0411857
$$771$$ 16557.8 0.773428
$$772$$ 13681.2 0.637818
$$773$$ 22821.4 1.06187 0.530936 0.847412i $$-0.321840\pi$$
0.530936 + 0.847412i $$0.321840\pi$$
$$774$$ −33.2721 −0.00154514
$$775$$ 1368.47 0.0634281
$$776$$ 20464.4 0.946685
$$777$$ −1379.98 −0.0637148
$$778$$ 18777.0 0.865282
$$779$$ −13340.0 −0.613549
$$780$$ −3410.09 −0.156539
$$781$$ −10332.2 −0.473387
$$782$$ −9509.28 −0.434848
$$783$$ −4025.51 −0.183729
$$784$$ −2718.92 −0.123857
$$785$$ 9406.19 0.427671
$$786$$ 1465.53 0.0665062
$$787$$ −29454.3 −1.33410 −0.667048 0.745015i $$-0.732442\pi$$
−0.667048 + 0.745015i $$0.732442\pi$$
$$788$$ −19337.8 −0.874216
$$789$$ 6778.96 0.305878
$$790$$ −7431.70 −0.334694
$$791$$ −1741.86 −0.0782974
$$792$$ 2096.53 0.0940619
$$793$$ 29202.7 1.30771
$$794$$ −8157.00 −0.364586
$$795$$ −9600.97 −0.428316
$$796$$ −22218.6 −0.989344
$$797$$ −27440.3 −1.21955 −0.609777 0.792573i $$-0.708741\pi$$
−0.609777 + 0.792573i $$0.708741\pi$$
$$798$$ 1903.00 0.0844179
$$799$$ 32899.0 1.45668
$$800$$ 4681.37 0.206889
$$801$$ 3472.91 0.153195
$$802$$ 15066.1 0.663346
$$803$$ 3988.27 0.175272
$$804$$ 6757.03 0.296396
$$805$$ 3160.61 0.138381
$$806$$ −3494.05 −0.152695
$$807$$ 21118.6 0.921201
$$808$$ −7382.34 −0.321423
$$809$$ −5060.18 −0.219909 −0.109954 0.993937i $$-0.535070\pi$$
−0.109954 + 0.993937i $$0.535070\pi$$
$$810$$ −632.429 −0.0274337
$$811$$ 30480.1 1.31973 0.659865 0.751384i $$-0.270613\pi$$
0.659865 + 0.751384i $$0.270613\pi$$
$$812$$ −8496.03 −0.367183
$$813$$ 15465.2 0.667146
$$814$$ −771.146 −0.0332048
$$815$$ 12327.5 0.529831
$$816$$ −3382.72 −0.145121
$$817$$ 93.8596 0.00401925
$$818$$ −3139.46 −0.134192
$$819$$ 3769.50 0.160827
$$820$$ 9356.73 0.398477
$$821$$ 37909.0 1.61149 0.805745 0.592263i $$-0.201765\pi$$
0.805745 + 0.592263i $$0.201765\pi$$
$$822$$ −3358.56 −0.142510
$$823$$ 23636.0 1.00109 0.500546 0.865710i $$-0.333133\pi$$
0.500546 + 0.865710i $$0.333133\pi$$
$$824$$ 32536.0 1.37554
$$825$$ −825.000 −0.0348155
$$826$$ 5932.36 0.249895
$$827$$ −42634.3 −1.79267 −0.896336 0.443376i $$-0.853781\pi$$
−0.896336 + 0.443376i $$0.853781\pi$$
$$828$$ −3087.99 −0.129608
$$829$$ −45152.5 −1.89169 −0.945845 0.324619i $$-0.894764\pi$$
−0.945845 + 0.324619i $$0.894764\pi$$
$$830$$ −5740.44 −0.240064
$$831$$ 27223.6 1.13643
$$832$$ −8217.15 −0.342402
$$833$$ 23494.1 0.977217
$$834$$ −4106.02 −0.170480
$$835$$ 6271.49 0.259921
$$836$$ −2425.42 −0.100341
$$837$$ 1477.94 0.0610337
$$838$$ 6925.91 0.285503
$$839$$ 30431.5 1.25222 0.626110 0.779734i $$-0.284646\pi$$
0.626110 + 0.779734i $$0.284646\pi$$
$$840$$ −3254.77 −0.133691
$$841$$ −2160.34 −0.0885784
$$842$$ 23763.6 0.972623
$$843$$ −10223.4 −0.417689
$$844$$ −5818.09 −0.237283
$$845$$ 2630.40 0.107087
$$846$$ −4684.13 −0.190359
$$847$$ −1239.79 −0.0502949
$$848$$ 7311.64 0.296088
$$849$$ −26483.2 −1.07055
$$850$$ −3853.46 −0.155497
$$851$$ 2769.65 0.111566
$$852$$ −15671.8 −0.630171
$$853$$ 10367.2 0.416139 0.208070 0.978114i $$-0.433282\pi$$
0.208070 + 0.978114i $$0.433282\pi$$
$$854$$ 11430.5 0.458013
$$855$$ 1784.06 0.0713610
$$856$$ 16500.8 0.658860
$$857$$ 12947.1 0.516063 0.258032 0.966136i $$-0.416926\pi$$
0.258032 + 0.966136i $$0.416926\pi$$
$$858$$ 2106.44 0.0838142
$$859$$ −20383.5 −0.809636 −0.404818 0.914397i $$-0.632665\pi$$
−0.404818 + 0.914397i $$0.632665\pi$$
$$860$$ −65.8335 −0.00261035
$$861$$ −10342.9 −0.409391
$$862$$ −8779.22 −0.346893
$$863$$ 9056.42 0.357224 0.178612 0.983920i $$-0.442839\pi$$
0.178612 + 0.983920i $$0.442839\pi$$
$$864$$ 5055.88 0.199079
$$865$$ 6033.54 0.237164
$$866$$ −22340.0 −0.876610
$$867$$ 14491.0 0.567636
$$868$$ 3119.27 0.121976
$$869$$ −10470.2 −0.408719
$$870$$ 3492.24 0.136090
$$871$$ 16554.5 0.644005
$$872$$ 31803.6 1.23510
$$873$$ −8697.10 −0.337173
$$874$$ −3819.37 −0.147817
$$875$$ 1280.78 0.0494836
$$876$$ 6049.36 0.233321
$$877$$ −2867.88 −0.110424 −0.0552118 0.998475i $$-0.517583\pi$$
−0.0552118 + 0.998475i $$0.517583\pi$$
$$878$$ 6846.16 0.263151
$$879$$ −13584.9 −0.521281
$$880$$ 628.281 0.0240674
$$881$$ −11862.5 −0.453640 −0.226820 0.973937i $$-0.572833\pi$$
−0.226820 + 0.973937i $$0.572833\pi$$
$$882$$ −3345.06 −0.127703
$$883$$ 33463.8 1.27537 0.637683 0.770299i $$-0.279893\pi$$
0.637683 + 0.770299i $$0.279893\pi$$
$$884$$ −22440.3 −0.853787
$$885$$ 5561.59 0.211244
$$886$$ −15756.0 −0.597442
$$887$$ 2420.75 0.0916357 0.0458178 0.998950i $$-0.485411\pi$$
0.0458178 + 0.998950i $$0.485411\pi$$
$$888$$ −2852.17 −0.107784
$$889$$ 17826.5 0.672534
$$890$$ −3012.85 −0.113473
$$891$$ −891.000 −0.0335013
$$892$$ 2815.62 0.105688
$$893$$ 13213.8 0.495165
$$894$$ −11132.4 −0.416470
$$895$$ 7211.44 0.269332
$$896$$ 12132.9 0.452378
$$897$$ −7565.48 −0.281610
$$898$$ 14963.6 0.556060
$$899$$ −8161.14 −0.302769
$$900$$ −1251.35 −0.0463463
$$901$$ −63179.7 −2.33609
$$902$$ −5779.73 −0.213352
$$903$$ 72.7722 0.00268185
$$904$$ −3600.10 −0.132453
$$905$$ −21309.0 −0.782692
$$906$$ −434.955 −0.0159497
$$907$$ −38154.1 −1.39679 −0.698393 0.715714i $$-0.746101\pi$$
−0.698393 + 0.715714i $$0.746101\pi$$
$$908$$ 23838.4 0.871262
$$909$$ 3137.40 0.114479
$$910$$ −3270.15 −0.119126
$$911$$ −35758.0 −1.30045 −0.650227 0.759740i $$-0.725326\pi$$
−0.650227 + 0.759740i $$0.725326\pi$$
$$912$$ −1358.66 −0.0493308
$$913$$ −8087.44 −0.293160
$$914$$ 15614.5 0.565079
$$915$$ 10716.1 0.387172
$$916$$ −31753.0 −1.14536
$$917$$ −3205.39 −0.115432
$$918$$ −4161.73 −0.149627
$$919$$ 17387.5 0.624115 0.312058 0.950063i $$-0.398982\pi$$
0.312058 + 0.950063i $$0.398982\pi$$
$$920$$ 6532.41 0.234095
$$921$$ 1704.32 0.0609764
$$922$$ −17357.9 −0.620013
$$923$$ −38395.3 −1.36923
$$924$$ −1880.50 −0.0669523
$$925$$ 1122.35 0.0398947
$$926$$ −2447.20 −0.0868467
$$927$$ −13827.4 −0.489915
$$928$$ −27918.3 −0.987569
$$929$$ 6955.93 0.245658 0.122829 0.992428i $$-0.460803\pi$$
0.122829 + 0.992428i $$0.460803\pi$$
$$930$$ −1282.16 −0.0452082
$$931$$ 9436.31 0.332183
$$932$$ −16384.9 −0.575863
$$933$$ −20560.8 −0.721467
$$934$$ 19751.7 0.691965
$$935$$ −5428.96 −0.189889
$$936$$ 7790.88 0.272065
$$937$$ −16074.5 −0.560438 −0.280219 0.959936i $$-0.590407\pi$$
−0.280219 + 0.959936i $$0.590407\pi$$
$$938$$ 6479.76 0.225556
$$939$$ −3416.77 −0.118746
$$940$$ −9268.20 −0.321591
$$941$$ −687.126 −0.0238041 −0.0119021 0.999929i $$-0.503789\pi$$
−0.0119021 + 0.999929i $$0.503789\pi$$
$$942$$ −8812.96 −0.304821
$$943$$ 20758.5 0.716849
$$944$$ −4235.44 −0.146030
$$945$$ 1383.24 0.0476156
$$946$$ 40.6659 0.00139763
$$947$$ 35352.0 1.21308 0.606540 0.795053i $$-0.292557\pi$$
0.606540 + 0.795053i $$0.292557\pi$$
$$948$$ −15881.0 −0.544085
$$949$$ 14820.7 0.506956
$$950$$ −1547.73 −0.0528578
$$951$$ 9622.44 0.328106
$$952$$ −21418.2 −0.729168
$$953$$ −19390.7 −0.659103 −0.329552 0.944138i $$-0.606898\pi$$
−0.329552 + 0.944138i $$0.606898\pi$$
$$954$$ 8995.45 0.305281
$$955$$ 4261.12 0.144384
$$956$$ 11561.9 0.391148
$$957$$ 4920.06 0.166189
$$958$$ −16738.7 −0.564511
$$959$$ 7345.80 0.247349
$$960$$ −3015.33 −0.101374
$$961$$ −26794.7 −0.899422
$$962$$ −2865.64 −0.0960416
$$963$$ −7012.62 −0.234661
$$964$$ −10307.6 −0.344384
$$965$$ 12299.8 0.410304
$$966$$ −2961.27 −0.0986308
$$967$$ 28643.6 0.952551 0.476275 0.879296i $$-0.341987\pi$$
0.476275 + 0.879296i $$0.341987\pi$$
$$968$$ −2562.43 −0.0850821
$$969$$ 11740.1 0.389213
$$970$$ 7544.99 0.249747
$$971$$ −19574.8 −0.646946 −0.323473 0.946237i $$-0.604851\pi$$
−0.323473 + 0.946237i $$0.604851\pi$$
$$972$$ −1351.46 −0.0445967
$$973$$ 8980.63 0.295895
$$974$$ 11182.6 0.367878
$$975$$ −3065.77 −0.100701
$$976$$ −8160.86 −0.267646
$$977$$ 50095.5 1.64043 0.820213 0.572058i $$-0.193855\pi$$
0.820213 + 0.572058i $$0.193855\pi$$
$$978$$ −11550.0 −0.377636
$$979$$ −4244.67 −0.138570
$$980$$ −6618.67 −0.215740
$$981$$ −13516.1 −0.439895
$$982$$ 22747.6 0.739211
$$983$$ 14445.4 0.468706 0.234353 0.972152i $$-0.424703\pi$$
0.234353 + 0.972152i $$0.424703\pi$$
$$984$$ −21376.9 −0.692553
$$985$$ −17385.3 −0.562377
$$986$$ 22980.9 0.742253
$$987$$ 10245.0 0.330399
$$988$$ −9013.05 −0.290226
$$989$$ −146.056 −0.00469595
$$990$$ 772.969 0.0248147
$$991$$ 29120.1 0.933430 0.466715 0.884408i $$-0.345437\pi$$
0.466715 + 0.884408i $$0.345437\pi$$
$$992$$ 10250.1 0.328064
$$993$$ −27405.4 −0.875814
$$994$$ −15028.7 −0.479558
$$995$$ −19975.2 −0.636438
$$996$$ −12266.9 −0.390253
$$997$$ −9137.45 −0.290257 −0.145128 0.989413i $$-0.546360\pi$$
−0.145128 + 0.989413i $$0.546360\pi$$
$$998$$ −7244.03 −0.229765
$$999$$ 1212.14 0.0383887
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 165.4.a.c.1.2 2
3.2 odd 2 495.4.a.d.1.1 2
5.2 odd 4 825.4.c.j.199.3 4
5.3 odd 4 825.4.c.j.199.2 4
5.4 even 2 825.4.a.m.1.1 2
11.10 odd 2 1815.4.a.n.1.1 2
15.14 odd 2 2475.4.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.2 2 1.1 even 1 trivial
495.4.a.d.1.1 2 3.2 odd 2
825.4.a.m.1.1 2 5.4 even 2
825.4.c.j.199.2 4 5.3 odd 4
825.4.c.j.199.3 4 5.2 odd 4
1815.4.a.n.1.1 2 11.10 odd 2
2475.4.a.n.1.2 2 15.14 odd 2