# Properties

 Label 39.4.a.c.1.3 Level $39$ Weight $4$ Character 39.1 Self dual yes Analytic conductor $2.301$ Analytic rank $0$ 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] = [39,4,Mod(1,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("39.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.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: $$2.30107449022$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.20905$$ of defining polynomial Character $$\chi$$ $$=$$ 39.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} -11.4322 q^{5} +12.6271 q^{6} -11.2543 q^{7} +7.22315 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} -11.4322 q^{5} +12.6271 q^{6} -11.2543 q^{7} +7.22315 q^{8} +9.00000 q^{9} -48.1187 q^{10} +25.8785 q^{11} +29.1483 q^{12} +13.0000 q^{13} -47.3699 q^{14} -34.2966 q^{15} -47.3262 q^{16} -20.3276 q^{17} +37.8814 q^{18} +154.712 q^{19} -111.076 q^{20} -33.7629 q^{21} +108.924 q^{22} -180.418 q^{23} +21.6695 q^{24} +5.69520 q^{25} +54.7176 q^{26} +27.0000 q^{27} -109.348 q^{28} -20.4522 q^{29} -144.356 q^{30} +266.424 q^{31} -256.984 q^{32} +77.6355 q^{33} -85.5599 q^{34} +128.661 q^{35} +87.4449 q^{36} +115.984 q^{37} +651.190 q^{38} +39.0000 q^{39} -82.5765 q^{40} +391.184 q^{41} -142.110 q^{42} +151.407 q^{43} +251.438 q^{44} -102.890 q^{45} -759.390 q^{46} -467.365 q^{47} -141.979 q^{48} -216.341 q^{49} +23.9714 q^{50} -60.9828 q^{51} +126.309 q^{52} +79.9842 q^{53} +113.644 q^{54} -295.848 q^{55} -81.2915 q^{56} +464.136 q^{57} -86.0843 q^{58} -873.710 q^{59} -333.229 q^{60} -187.068 q^{61} +1121.39 q^{62} -101.289 q^{63} -703.047 q^{64} -148.619 q^{65} +326.772 q^{66} -609.204 q^{67} -197.505 q^{68} -541.255 q^{69} +541.542 q^{70} +248.038 q^{71} +65.0084 q^{72} +852.765 q^{73} +488.181 q^{74} +17.0856 q^{75} +1503.20 q^{76} -291.244 q^{77} +164.153 q^{78} -331.221 q^{79} +541.043 q^{80} +81.0000 q^{81} +1646.51 q^{82} -435.432 q^{83} -328.044 q^{84} +232.389 q^{85} +637.281 q^{86} -61.3566 q^{87} +186.924 q^{88} +259.233 q^{89} -433.068 q^{90} -146.306 q^{91} -1752.96 q^{92} +799.273 q^{93} -1967.16 q^{94} -1768.70 q^{95} -770.951 q^{96} +1225.17 q^{97} -910.589 q^{98} +232.907 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9 $$3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9} - 4 q^{10} - 16 q^{11} + 30 q^{12} + 39 q^{13} - 176 q^{14} + 12 q^{15} - 110 q^{16} - 146 q^{17} + 18 q^{18} + 94 q^{19} - 244 q^{20} + 90 q^{21} - 56 q^{22} - 48 q^{23} - 18 q^{24} + 145 q^{25} + 26 q^{26} + 81 q^{27} + 80 q^{28} - 2 q^{29} - 12 q^{30} + 302 q^{31} + 154 q^{32} - 48 q^{33} + 164 q^{34} + 80 q^{35} + 90 q^{36} + 374 q^{37} + 312 q^{38} + 117 q^{39} - 516 q^{40} + 480 q^{41} - 528 q^{42} - 260 q^{43} + 712 q^{44} + 36 q^{45} - 1104 q^{46} - 24 q^{47} - 330 q^{48} + 447 q^{49} + 814 q^{50} - 438 q^{51} + 130 q^{52} - 678 q^{53} + 54 q^{54} - 1552 q^{55} + 96 q^{56} + 282 q^{57} - 628 q^{58} - 1788 q^{59} - 732 q^{60} + 230 q^{61} + 1952 q^{62} + 270 q^{63} - 750 q^{64} + 52 q^{65} - 168 q^{66} + 74 q^{67} - 460 q^{68} - 144 q^{69} + 1216 q^{70} - 948 q^{71} - 54 q^{72} - 222 q^{73} + 1724 q^{74} + 435 q^{75} + 2392 q^{76} + 112 q^{77} + 78 q^{78} - 24 q^{79} + 1100 q^{80} + 243 q^{81} + 564 q^{82} - 796 q^{83} + 240 q^{84} - 248 q^{85} + 1800 q^{86} - 6 q^{87} + 1608 q^{88} + 1436 q^{89} - 36 q^{90} + 390 q^{91} - 1296 q^{92} + 906 q^{93} - 1920 q^{94} - 4032 q^{95} + 462 q^{96} + 3242 q^{97} - 5070 q^{98} - 144 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9 - 4 * q^10 - 16 * q^11 + 30 * q^12 + 39 * q^13 - 176 * q^14 + 12 * q^15 - 110 * q^16 - 146 * q^17 + 18 * q^18 + 94 * q^19 - 244 * q^20 + 90 * q^21 - 56 * q^22 - 48 * q^23 - 18 * q^24 + 145 * q^25 + 26 * q^26 + 81 * q^27 + 80 * q^28 - 2 * q^29 - 12 * q^30 + 302 * q^31 + 154 * q^32 - 48 * q^33 + 164 * q^34 + 80 * q^35 + 90 * q^36 + 374 * q^37 + 312 * q^38 + 117 * q^39 - 516 * q^40 + 480 * q^41 - 528 * q^42 - 260 * q^43 + 712 * q^44 + 36 * q^45 - 1104 * q^46 - 24 * q^47 - 330 * q^48 + 447 * q^49 + 814 * q^50 - 438 * q^51 + 130 * q^52 - 678 * q^53 + 54 * q^54 - 1552 * q^55 + 96 * q^56 + 282 * q^57 - 628 * q^58 - 1788 * q^59 - 732 * q^60 + 230 * q^61 + 1952 * q^62 + 270 * q^63 - 750 * q^64 + 52 * q^65 - 168 * q^66 + 74 * q^67 - 460 * q^68 - 144 * q^69 + 1216 * q^70 - 948 * q^71 - 54 * q^72 - 222 * q^73 + 1724 * q^74 + 435 * q^75 + 2392 * q^76 + 112 * q^77 + 78 * q^78 - 24 * q^79 + 1100 * q^80 + 243 * q^81 + 564 * q^82 - 796 * q^83 + 240 * q^84 - 248 * q^85 + 1800 * q^86 - 6 * q^87 + 1608 * q^88 + 1436 * q^89 - 36 * q^90 + 390 * q^91 - 1296 * q^92 + 906 * q^93 - 1920 * q^94 - 4032 * q^95 + 462 * q^96 + 3242 * q^97 - 5070 * q^98 - 144 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.20905 1.48812 0.744062 0.668111i $$-0.232897\pi$$
0.744062 + 0.668111i $$0.232897\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 9.71610 1.21451
$$5$$ −11.4322 −1.02253 −0.511264 0.859424i $$-0.670822\pi$$
−0.511264 + 0.859424i $$0.670822\pi$$
$$6$$ 12.6271 0.859169
$$7$$ −11.2543 −0.607675 −0.303838 0.952724i $$-0.598268\pi$$
−0.303838 + 0.952724i $$0.598268\pi$$
$$8$$ 7.22315 0.319221
$$9$$ 9.00000 0.333333
$$10$$ −48.1187 −1.52165
$$11$$ 25.8785 0.709333 0.354666 0.934993i $$-0.384594\pi$$
0.354666 + 0.934993i $$0.384594\pi$$
$$12$$ 29.1483 0.701199
$$13$$ 13.0000 0.277350
$$14$$ −47.3699 −0.904296
$$15$$ −34.2966 −0.590356
$$16$$ −47.3262 −0.739472
$$17$$ −20.3276 −0.290010 −0.145005 0.989431i $$-0.546320\pi$$
−0.145005 + 0.989431i $$0.546320\pi$$
$$18$$ 37.8814 0.496041
$$19$$ 154.712 1.86807 0.934035 0.357181i $$-0.116262\pi$$
0.934035 + 0.357181i $$0.116262\pi$$
$$20$$ −111.076 −1.24187
$$21$$ −33.7629 −0.350841
$$22$$ 108.924 1.05558
$$23$$ −180.418 −1.63565 −0.817823 0.575471i $$-0.804819\pi$$
−0.817823 + 0.575471i $$0.804819\pi$$
$$24$$ 21.6695 0.184302
$$25$$ 5.69520 0.0455616
$$26$$ 54.7176 0.412731
$$27$$ 27.0000 0.192450
$$28$$ −109.348 −0.738029
$$29$$ −20.4522 −0.130961 −0.0654806 0.997854i $$-0.520858\pi$$
−0.0654806 + 0.997854i $$0.520858\pi$$
$$30$$ −144.356 −0.878523
$$31$$ 266.424 1.54359 0.771794 0.635873i $$-0.219360\pi$$
0.771794 + 0.635873i $$0.219360\pi$$
$$32$$ −256.984 −1.41965
$$33$$ 77.6355 0.409534
$$34$$ −85.5599 −0.431571
$$35$$ 128.661 0.621364
$$36$$ 87.4449 0.404837
$$37$$ 115.984 0.515340 0.257670 0.966233i $$-0.417045\pi$$
0.257670 + 0.966233i $$0.417045\pi$$
$$38$$ 651.190 2.77992
$$39$$ 39.0000 0.160128
$$40$$ −82.5765 −0.326412
$$41$$ 391.184 1.49006 0.745032 0.667029i $$-0.232434\pi$$
0.745032 + 0.667029i $$0.232434\pi$$
$$42$$ −142.110 −0.522095
$$43$$ 151.407 0.536963 0.268482 0.963285i $$-0.413478\pi$$
0.268482 + 0.963285i $$0.413478\pi$$
$$44$$ 251.438 0.861494
$$45$$ −102.890 −0.340842
$$46$$ −759.390 −2.43404
$$47$$ −467.365 −1.45047 −0.725236 0.688500i $$-0.758269\pi$$
−0.725236 + 0.688500i $$0.758269\pi$$
$$48$$ −141.979 −0.426934
$$49$$ −216.341 −0.630731
$$50$$ 23.9714 0.0678012
$$51$$ −60.9828 −0.167437
$$52$$ 126.309 0.336845
$$53$$ 79.9842 0.207296 0.103648 0.994614i $$-0.466949\pi$$
0.103648 + 0.994614i $$0.466949\pi$$
$$54$$ 113.644 0.286390
$$55$$ −295.848 −0.725312
$$56$$ −81.2915 −0.193983
$$57$$ 464.136 1.07853
$$58$$ −86.0843 −0.194887
$$59$$ −873.710 −1.92792 −0.963960 0.266045i $$-0.914283\pi$$
−0.963960 + 0.266045i $$0.914283\pi$$
$$60$$ −333.229 −0.716995
$$61$$ −187.068 −0.392649 −0.196325 0.980539i $$-0.562901\pi$$
−0.196325 + 0.980539i $$0.562901\pi$$
$$62$$ 1121.39 2.29705
$$63$$ −101.289 −0.202558
$$64$$ −703.047 −1.37314
$$65$$ −148.619 −0.283598
$$66$$ 326.772 0.609437
$$67$$ −609.204 −1.11084 −0.555418 0.831571i $$-0.687442\pi$$
−0.555418 + 0.831571i $$0.687442\pi$$
$$68$$ −197.505 −0.352221
$$69$$ −541.255 −0.944340
$$70$$ 541.542 0.924667
$$71$$ 248.038 0.414601 0.207301 0.978277i $$-0.433532\pi$$
0.207301 + 0.978277i $$0.433532\pi$$
$$72$$ 65.0084 0.106407
$$73$$ 852.765 1.36724 0.683621 0.729838i $$-0.260404\pi$$
0.683621 + 0.729838i $$0.260404\pi$$
$$74$$ 488.181 0.766890
$$75$$ 17.0856 0.0263050
$$76$$ 1503.20 2.26880
$$77$$ −291.244 −0.431044
$$78$$ 164.153 0.238291
$$79$$ −331.221 −0.471712 −0.235856 0.971788i $$-0.575789\pi$$
−0.235856 + 0.971788i $$0.575789\pi$$
$$80$$ 541.043 0.756130
$$81$$ 81.0000 0.111111
$$82$$ 1646.51 2.21740
$$83$$ −435.432 −0.575842 −0.287921 0.957654i $$-0.592964\pi$$
−0.287921 + 0.957654i $$0.592964\pi$$
$$84$$ −328.044 −0.426101
$$85$$ 232.389 0.296543
$$86$$ 637.281 0.799067
$$87$$ −61.3566 −0.0756105
$$88$$ 186.924 0.226434
$$89$$ 259.233 0.308749 0.154375 0.988012i $$-0.450664\pi$$
0.154375 + 0.988012i $$0.450664\pi$$
$$90$$ −433.068 −0.507216
$$91$$ −146.306 −0.168539
$$92$$ −1752.96 −1.98651
$$93$$ 799.273 0.891191
$$94$$ −1967.16 −2.15848
$$95$$ −1768.70 −1.91015
$$96$$ −770.951 −0.819634
$$97$$ 1225.17 1.28245 0.641223 0.767355i $$-0.278428\pi$$
0.641223 + 0.767355i $$0.278428\pi$$
$$98$$ −910.589 −0.938606
$$99$$ 232.907 0.236444
$$100$$ 55.3351 0.0553351
$$101$$ 645.416 0.635855 0.317927 0.948115i $$-0.397013\pi$$
0.317927 + 0.948115i $$0.397013\pi$$
$$102$$ −256.680 −0.249167
$$103$$ −511.137 −0.488969 −0.244484 0.969653i $$-0.578619\pi$$
−0.244484 + 0.969653i $$0.578619\pi$$
$$104$$ 93.9010 0.0885360
$$105$$ 385.984 0.358745
$$106$$ 336.657 0.308482
$$107$$ 608.195 0.549499 0.274750 0.961516i $$-0.411405\pi$$
0.274750 + 0.961516i $$0.411405\pi$$
$$108$$ 262.335 0.233733
$$109$$ −1300.04 −1.14239 −0.571197 0.820813i $$-0.693521\pi$$
−0.571197 + 0.820813i $$0.693521\pi$$
$$110$$ −1245.24 −1.07935
$$111$$ 347.951 0.297532
$$112$$ 532.623 0.449359
$$113$$ 42.1953 0.0351274 0.0175637 0.999846i $$-0.494409\pi$$
0.0175637 + 0.999846i $$0.494409\pi$$
$$114$$ 1953.57 1.60499
$$115$$ 2062.58 1.67249
$$116$$ −198.716 −0.159054
$$117$$ 117.000 0.0924500
$$118$$ −3677.49 −2.86899
$$119$$ 228.773 0.176232
$$120$$ −247.729 −0.188454
$$121$$ −661.303 −0.496847
$$122$$ −787.378 −0.584311
$$123$$ 1173.55 0.860289
$$124$$ 2588.61 1.87471
$$125$$ 1363.92 0.975939
$$126$$ −426.329 −0.301432
$$127$$ −311.018 −0.217310 −0.108655 0.994080i $$-0.534654\pi$$
−0.108655 + 0.994080i $$0.534654\pi$$
$$128$$ −903.291 −0.623753
$$129$$ 454.222 0.310016
$$130$$ −625.543 −0.422029
$$131$$ 2000.98 1.33456 0.667278 0.744809i $$-0.267459\pi$$
0.667278 + 0.744809i $$0.267459\pi$$
$$132$$ 754.314 0.497384
$$133$$ −1741.17 −1.13518
$$134$$ −2564.17 −1.65306
$$135$$ −308.669 −0.196785
$$136$$ −146.829 −0.0925773
$$137$$ 1038.53 0.647644 0.323822 0.946118i $$-0.395032\pi$$
0.323822 + 0.946118i $$0.395032\pi$$
$$138$$ −2278.17 −1.40529
$$139$$ −2858.46 −1.74426 −0.872128 0.489277i $$-0.837261\pi$$
−0.872128 + 0.489277i $$0.837261\pi$$
$$140$$ 1250.09 0.754655
$$141$$ −1402.09 −0.837430
$$142$$ 1044.00 0.616978
$$143$$ 336.421 0.196734
$$144$$ −425.936 −0.246491
$$145$$ 233.814 0.133911
$$146$$ 3589.33 2.03462
$$147$$ −649.022 −0.364153
$$148$$ 1126.91 0.625887
$$149$$ 743.479 0.408780 0.204390 0.978890i $$-0.434479\pi$$
0.204390 + 0.978890i $$0.434479\pi$$
$$150$$ 71.9141 0.0391451
$$151$$ 2277.24 1.22728 0.613640 0.789586i $$-0.289705\pi$$
0.613640 + 0.789586i $$0.289705\pi$$
$$152$$ 1117.51 0.596328
$$153$$ −182.948 −0.0966700
$$154$$ −1225.86 −0.641447
$$155$$ −3045.82 −1.57836
$$156$$ 378.928 0.194478
$$157$$ 3173.51 1.61321 0.806605 0.591091i $$-0.201303\pi$$
0.806605 + 0.591091i $$0.201303\pi$$
$$158$$ −1394.12 −0.701966
$$159$$ 239.953 0.119682
$$160$$ 2937.89 1.45163
$$161$$ 2030.48 0.993941
$$162$$ 340.933 0.165347
$$163$$ −2314.65 −1.11225 −0.556126 0.831098i $$-0.687713\pi$$
−0.556126 + 0.831098i $$0.687713\pi$$
$$164$$ 3800.78 1.80970
$$165$$ −887.545 −0.418759
$$166$$ −1832.76 −0.856925
$$167$$ −2665.65 −1.23517 −0.617587 0.786502i $$-0.711890\pi$$
−0.617587 + 0.786502i $$0.711890\pi$$
$$168$$ −243.874 −0.111996
$$169$$ 169.000 0.0769231
$$170$$ 978.138 0.441293
$$171$$ 1392.41 0.622690
$$172$$ 1471.09 0.652148
$$173$$ −165.243 −0.0726198 −0.0363099 0.999341i $$-0.511560\pi$$
−0.0363099 + 0.999341i $$0.511560\pi$$
$$174$$ −258.253 −0.112518
$$175$$ −64.0954 −0.0276866
$$176$$ −1224.73 −0.524532
$$177$$ −2621.13 −1.11309
$$178$$ 1091.13 0.459457
$$179$$ 712.339 0.297446 0.148723 0.988879i $$-0.452484\pi$$
0.148723 + 0.988879i $$0.452484\pi$$
$$180$$ −999.688 −0.413957
$$181$$ 2206.53 0.906133 0.453066 0.891477i $$-0.350330\pi$$
0.453066 + 0.891477i $$0.350330\pi$$
$$182$$ −615.809 −0.250806
$$183$$ −561.204 −0.226696
$$184$$ −1303.19 −0.522132
$$185$$ −1325.95 −0.526949
$$186$$ 3364.18 1.32620
$$187$$ −526.048 −0.205714
$$188$$ −4540.96 −1.76162
$$189$$ −303.866 −0.116947
$$190$$ −7444.54 −2.84254
$$191$$ 1470.64 0.557129 0.278565 0.960417i $$-0.410141\pi$$
0.278565 + 0.960417i $$0.410141\pi$$
$$192$$ −2109.14 −0.792782
$$193$$ 369.560 0.137832 0.0689158 0.997622i $$-0.478046\pi$$
0.0689158 + 0.997622i $$0.478046\pi$$
$$194$$ 5156.80 1.90844
$$195$$ −445.856 −0.163735
$$196$$ −2101.99 −0.766031
$$197$$ −4273.41 −1.54552 −0.772761 0.634697i $$-0.781125\pi$$
−0.772761 + 0.634697i $$0.781125\pi$$
$$198$$ 980.315 0.351858
$$199$$ 4154.31 1.47985 0.739927 0.672687i $$-0.234860\pi$$
0.739927 + 0.672687i $$0.234860\pi$$
$$200$$ 41.1373 0.0145442
$$201$$ −1827.61 −0.641342
$$202$$ 2716.59 0.946230
$$203$$ 230.175 0.0795819
$$204$$ −592.515 −0.203355
$$205$$ −4472.09 −1.52363
$$206$$ −2151.40 −0.727646
$$207$$ −1623.77 −0.545215
$$208$$ −615.241 −0.205093
$$209$$ 4003.71 1.32508
$$210$$ 1624.63 0.533857
$$211$$ 1231.59 0.401830 0.200915 0.979609i $$-0.435608\pi$$
0.200915 + 0.979609i $$0.435608\pi$$
$$212$$ 777.134 0.251763
$$213$$ 744.114 0.239370
$$214$$ 2559.92 0.817723
$$215$$ −1730.92 −0.549059
$$216$$ 195.025 0.0614341
$$217$$ −2998.42 −0.938000
$$218$$ −5471.92 −1.70002
$$219$$ 2558.30 0.789377
$$220$$ −2874.49 −0.880901
$$221$$ −264.259 −0.0804343
$$222$$ 1464.54 0.442764
$$223$$ −2187.24 −0.656809 −0.328404 0.944537i $$-0.606511\pi$$
−0.328404 + 0.944537i $$0.606511\pi$$
$$224$$ 2892.17 0.862684
$$225$$ 51.2568 0.0151872
$$226$$ 177.602 0.0522739
$$227$$ 4138.67 1.21010 0.605051 0.796187i $$-0.293153\pi$$
0.605051 + 0.796187i $$0.293153\pi$$
$$228$$ 4509.59 1.30989
$$229$$ −835.354 −0.241056 −0.120528 0.992710i $$-0.538459\pi$$
−0.120528 + 0.992710i $$0.538459\pi$$
$$230$$ 8681.50 2.48887
$$231$$ −873.733 −0.248863
$$232$$ −147.729 −0.0418056
$$233$$ 3685.51 1.03625 0.518124 0.855305i $$-0.326630\pi$$
0.518124 + 0.855305i $$0.326630\pi$$
$$234$$ 492.459 0.137577
$$235$$ 5343.01 1.48315
$$236$$ −8489.05 −2.34148
$$237$$ −993.662 −0.272343
$$238$$ 962.917 0.262255
$$239$$ 3026.21 0.819034 0.409517 0.912303i $$-0.365697\pi$$
0.409517 + 0.912303i $$0.365697\pi$$
$$240$$ 1623.13 0.436552
$$241$$ 3265.58 0.872839 0.436420 0.899743i $$-0.356246\pi$$
0.436420 + 0.899743i $$0.356246\pi$$
$$242$$ −2783.46 −0.739370
$$243$$ 243.000 0.0641500
$$244$$ −1817.57 −0.476877
$$245$$ 2473.25 0.644940
$$246$$ 4939.53 1.28022
$$247$$ 2011.25 0.518110
$$248$$ 1924.42 0.492746
$$249$$ −1306.30 −0.332463
$$250$$ 5740.79 1.45232
$$251$$ −6363.16 −1.60016 −0.800078 0.599897i $$-0.795208\pi$$
−0.800078 + 0.599897i $$0.795208\pi$$
$$252$$ −984.131 −0.246010
$$253$$ −4668.96 −1.16022
$$254$$ −1309.09 −0.323385
$$255$$ 697.168 0.171209
$$256$$ 1822.38 0.444917
$$257$$ −6085.36 −1.47702 −0.738511 0.674242i $$-0.764471\pi$$
−0.738511 + 0.674242i $$0.764471\pi$$
$$258$$ 1911.84 0.461342
$$259$$ −1305.31 −0.313159
$$260$$ −1443.99 −0.344433
$$261$$ −184.070 −0.0436538
$$262$$ 8422.24 1.98598
$$263$$ 123.227 0.0288916 0.0144458 0.999896i $$-0.495402\pi$$
0.0144458 + 0.999896i $$0.495402\pi$$
$$264$$ 560.773 0.130732
$$265$$ −914.395 −0.211965
$$266$$ −7328.69 −1.68929
$$267$$ 777.700 0.178256
$$268$$ −5919.08 −1.34913
$$269$$ −1935.79 −0.438763 −0.219381 0.975639i $$-0.570404\pi$$
−0.219381 + 0.975639i $$0.570404\pi$$
$$270$$ −1299.20 −0.292841
$$271$$ −4612.69 −1.03395 −0.516976 0.856000i $$-0.672942\pi$$
−0.516976 + 0.856000i $$0.672942\pi$$
$$272$$ 962.028 0.214454
$$273$$ −438.918 −0.0973059
$$274$$ 4371.20 0.963774
$$275$$ 147.383 0.0323183
$$276$$ −5258.89 −1.14691
$$277$$ −5834.30 −1.26552 −0.632761 0.774347i $$-0.718078\pi$$
−0.632761 + 0.774347i $$0.718078\pi$$
$$278$$ −12031.4 −2.59567
$$279$$ 2397.82 0.514529
$$280$$ 929.341 0.198353
$$281$$ 4691.91 0.996071 0.498036 0.867157i $$-0.334055\pi$$
0.498036 + 0.867157i $$0.334055\pi$$
$$282$$ −5901.49 −1.24620
$$283$$ 3465.60 0.727945 0.363973 0.931410i $$-0.381420\pi$$
0.363973 + 0.931410i $$0.381420\pi$$
$$284$$ 2409.96 0.503539
$$285$$ −5306.09 −1.10283
$$286$$ 1416.01 0.292764
$$287$$ −4402.50 −0.905475
$$288$$ −2312.85 −0.473216
$$289$$ −4499.79 −0.915894
$$290$$ 984.133 0.199277
$$291$$ 3675.51 0.740420
$$292$$ 8285.55 1.66053
$$293$$ 2677.31 0.533822 0.266911 0.963721i $$-0.413997\pi$$
0.266911 + 0.963721i $$0.413997\pi$$
$$294$$ −2731.77 −0.541904
$$295$$ 9988.43 1.97135
$$296$$ 837.767 0.164507
$$297$$ 698.720 0.136511
$$298$$ 3129.34 0.608315
$$299$$ −2345.44 −0.453646
$$300$$ 166.005 0.0319477
$$301$$ −1703.98 −0.326299
$$302$$ 9585.02 1.82634
$$303$$ 1936.25 0.367111
$$304$$ −7321.93 −1.38139
$$305$$ 2138.60 0.401494
$$306$$ −770.039 −0.143857
$$307$$ 471.915 0.0877316 0.0438658 0.999037i $$-0.486033\pi$$
0.0438658 + 0.999037i $$0.486033\pi$$
$$308$$ −2829.76 −0.523508
$$309$$ −1533.41 −0.282306
$$310$$ −12820.0 −2.34880
$$311$$ −1518.52 −0.276872 −0.138436 0.990371i $$-0.544207\pi$$
−0.138436 + 0.990371i $$0.544207\pi$$
$$312$$ 281.703 0.0511163
$$313$$ 4049.86 0.731348 0.365674 0.930743i $$-0.380839\pi$$
0.365674 + 0.930743i $$0.380839\pi$$
$$314$$ 13357.5 2.40066
$$315$$ 1157.95 0.207121
$$316$$ −3218.17 −0.572900
$$317$$ 3253.96 0.576532 0.288266 0.957550i $$-0.406921\pi$$
0.288266 + 0.957550i $$0.406921\pi$$
$$318$$ 1009.97 0.178102
$$319$$ −529.272 −0.0928951
$$320$$ 8037.37 1.40407
$$321$$ 1824.59 0.317254
$$322$$ 8546.40 1.47911
$$323$$ −3144.92 −0.541759
$$324$$ 787.004 0.134946
$$325$$ 74.0375 0.0126365
$$326$$ −9742.46 −1.65517
$$327$$ −3900.11 −0.659561
$$328$$ 2825.58 0.475660
$$329$$ 5259.86 0.881415
$$330$$ −3735.72 −0.623165
$$331$$ −3422.45 −0.568322 −0.284161 0.958777i $$-0.591715\pi$$
−0.284161 + 0.958777i $$0.591715\pi$$
$$332$$ −4230.71 −0.699368
$$333$$ 1043.85 0.171780
$$334$$ −11219.8 −1.83809
$$335$$ 6964.54 1.13586
$$336$$ 1597.87 0.259437
$$337$$ −9301.67 −1.50354 −0.751772 0.659423i $$-0.770801\pi$$
−0.751772 + 0.659423i $$0.770801\pi$$
$$338$$ 711.329 0.114471
$$339$$ 126.586 0.0202808
$$340$$ 2257.92 0.360155
$$341$$ 6894.66 1.09492
$$342$$ 5860.71 0.926640
$$343$$ 6294.99 0.990955
$$344$$ 1093.64 0.171410
$$345$$ 6187.74 0.965613
$$346$$ −695.518 −0.108067
$$347$$ 216.898 0.0335554 0.0167777 0.999859i $$-0.494659\pi$$
0.0167777 + 0.999859i $$0.494659\pi$$
$$348$$ −596.147 −0.0918299
$$349$$ −4809.84 −0.737721 −0.368861 0.929485i $$-0.620252\pi$$
−0.368861 + 0.929485i $$0.620252\pi$$
$$350$$ −269.781 −0.0412011
$$351$$ 351.000 0.0533761
$$352$$ −6650.35 −1.00700
$$353$$ −2859.64 −0.431170 −0.215585 0.976485i $$-0.569166\pi$$
−0.215585 + 0.976485i $$0.569166\pi$$
$$354$$ −11032.5 −1.65641
$$355$$ −2835.62 −0.423941
$$356$$ 2518.74 0.374980
$$357$$ 686.319 0.101747
$$358$$ 2998.27 0.442636
$$359$$ 3686.04 0.541899 0.270949 0.962594i $$-0.412662\pi$$
0.270949 + 0.962594i $$0.412662\pi$$
$$360$$ −743.188 −0.108804
$$361$$ 17076.8 2.48969
$$362$$ 9287.39 1.34844
$$363$$ −1983.91 −0.286855
$$364$$ −1421.52 −0.204692
$$365$$ −9748.98 −1.39804
$$366$$ −2362.14 −0.337352
$$367$$ −3470.59 −0.493633 −0.246816 0.969062i $$-0.579384\pi$$
−0.246816 + 0.969062i $$0.579384\pi$$
$$368$$ 8538.52 1.20951
$$369$$ 3520.65 0.496688
$$370$$ −5580.98 −0.784166
$$371$$ −900.166 −0.125968
$$372$$ 7765.82 1.08236
$$373$$ −11963.4 −1.66070 −0.830352 0.557240i $$-0.811860\pi$$
−0.830352 + 0.557240i $$0.811860\pi$$
$$374$$ −2214.16 −0.306127
$$375$$ 4091.75 0.563459
$$376$$ −3375.85 −0.463021
$$377$$ −265.879 −0.0363221
$$378$$ −1278.99 −0.174032
$$379$$ 345.604 0.0468403 0.0234202 0.999726i $$-0.492544\pi$$
0.0234202 + 0.999726i $$0.492544\pi$$
$$380$$ −17184.8 −2.31990
$$381$$ −933.055 −0.125464
$$382$$ 6189.99 0.829078
$$383$$ −3386.40 −0.451793 −0.225897 0.974151i $$-0.572531\pi$$
−0.225897 + 0.974151i $$0.572531\pi$$
$$384$$ −2709.87 −0.360124
$$385$$ 3329.56 0.440754
$$386$$ 1555.49 0.205110
$$387$$ 1362.67 0.178988
$$388$$ 11903.9 1.55755
$$389$$ −1629.88 −0.212438 −0.106219 0.994343i $$-0.533874\pi$$
−0.106219 + 0.994343i $$0.533874\pi$$
$$390$$ −1876.63 −0.243659
$$391$$ 3667.47 0.474353
$$392$$ −1562.66 −0.201343
$$393$$ 6002.95 0.770506
$$394$$ −17987.0 −2.29993
$$395$$ 3786.58 0.482338
$$396$$ 2262.94 0.287165
$$397$$ 7938.94 1.00364 0.501819 0.864973i $$-0.332664\pi$$
0.501819 + 0.864973i $$0.332664\pi$$
$$398$$ 17485.7 2.20221
$$399$$ −5223.52 −0.655396
$$400$$ −269.532 −0.0336915
$$401$$ 214.402 0.0267001 0.0133500 0.999911i $$-0.495750\pi$$
0.0133500 + 0.999911i $$0.495750\pi$$
$$402$$ −7692.51 −0.954396
$$403$$ 3463.52 0.428114
$$404$$ 6270.93 0.772253
$$405$$ −926.008 −0.113614
$$406$$ 968.819 0.118428
$$407$$ 3001.48 0.365548
$$408$$ −440.488 −0.0534495
$$409$$ −4783.73 −0.578338 −0.289169 0.957278i $$-0.593379\pi$$
−0.289169 + 0.957278i $$0.593379\pi$$
$$410$$ −18823.2 −2.26735
$$411$$ 3115.58 0.373917
$$412$$ −4966.25 −0.593859
$$413$$ 9832.99 1.17155
$$414$$ −6834.51 −0.811347
$$415$$ 4977.95 0.588815
$$416$$ −3340.79 −0.393739
$$417$$ −8575.39 −1.00705
$$418$$ 16851.8 1.97189
$$419$$ −9903.67 −1.15472 −0.577358 0.816491i $$-0.695916\pi$$
−0.577358 + 0.816491i $$0.695916\pi$$
$$420$$ 3750.26 0.435700
$$421$$ −12120.6 −1.40314 −0.701572 0.712598i $$-0.747518\pi$$
−0.701572 + 0.712598i $$0.747518\pi$$
$$422$$ 5183.82 0.597973
$$423$$ −4206.28 −0.483491
$$424$$ 577.738 0.0661732
$$425$$ −115.770 −0.0132133
$$426$$ 3132.01 0.356213
$$427$$ 2105.32 0.238603
$$428$$ 5909.28 0.667374
$$429$$ 1009.26 0.113584
$$430$$ −7285.53 −0.817068
$$431$$ −13672.6 −1.52805 −0.764023 0.645189i $$-0.776779\pi$$
−0.764023 + 0.645189i $$0.776779\pi$$
$$432$$ −1277.81 −0.142311
$$433$$ 7113.10 0.789455 0.394727 0.918798i $$-0.370839\pi$$
0.394727 + 0.918798i $$0.370839\pi$$
$$434$$ −12620.5 −1.39586
$$435$$ 701.441 0.0773138
$$436$$ −12631.3 −1.38745
$$437$$ −27912.9 −3.05550
$$438$$ 10768.0 1.17469
$$439$$ −6022.04 −0.654707 −0.327353 0.944902i $$-0.606157\pi$$
−0.327353 + 0.944902i $$0.606157\pi$$
$$440$$ −2136.96 −0.231535
$$441$$ −1947.07 −0.210244
$$442$$ −1112.28 −0.119696
$$443$$ −12994.4 −1.39364 −0.696821 0.717245i $$-0.745403\pi$$
−0.696821 + 0.717245i $$0.745403\pi$$
$$444$$ 3380.72 0.361356
$$445$$ −2963.61 −0.315704
$$446$$ −9206.20 −0.977413
$$447$$ 2230.44 0.236009
$$448$$ 7912.30 0.834422
$$449$$ 10984.3 1.15452 0.577260 0.816560i $$-0.304122\pi$$
0.577260 + 0.816560i $$0.304122\pi$$
$$450$$ 215.742 0.0226004
$$451$$ 10123.2 1.05695
$$452$$ 409.973 0.0426627
$$453$$ 6831.72 0.708570
$$454$$ 17419.9 1.80078
$$455$$ 1672.60 0.172335
$$456$$ 3352.52 0.344290
$$457$$ 9834.10 1.00661 0.503304 0.864109i $$-0.332118\pi$$
0.503304 + 0.864109i $$0.332118\pi$$
$$458$$ −3516.05 −0.358721
$$459$$ −548.845 −0.0558124
$$460$$ 20040.2 2.03126
$$461$$ 3401.42 0.343644 0.171822 0.985128i $$-0.445035\pi$$
0.171822 + 0.985128i $$0.445035\pi$$
$$462$$ −3677.59 −0.370339
$$463$$ 1739.42 0.174596 0.0872979 0.996182i $$-0.472177\pi$$
0.0872979 + 0.996182i $$0.472177\pi$$
$$464$$ 967.925 0.0968422
$$465$$ −9137.45 −0.911267
$$466$$ 15512.5 1.54207
$$467$$ −7958.82 −0.788630 −0.394315 0.918975i $$-0.629018\pi$$
−0.394315 + 0.918975i $$0.629018\pi$$
$$468$$ 1136.78 0.112282
$$469$$ 6856.16 0.675028
$$470$$ 22489.0 2.20711
$$471$$ 9520.54 0.931387
$$472$$ −6310.94 −0.615433
$$473$$ 3918.20 0.380886
$$474$$ −4182.37 −0.405280
$$475$$ 881.114 0.0851122
$$476$$ 2222.78 0.214036
$$477$$ 719.858 0.0690986
$$478$$ 12737.5 1.21882
$$479$$ 8431.98 0.804315 0.402158 0.915570i $$-0.368260\pi$$
0.402158 + 0.915570i $$0.368260\pi$$
$$480$$ 8813.66 0.838097
$$481$$ 1507.79 0.142930
$$482$$ 13745.0 1.29889
$$483$$ 6091.45 0.573852
$$484$$ −6425.29 −0.603427
$$485$$ −14006.4 −1.31133
$$486$$ 1022.80 0.0954632
$$487$$ −11684.7 −1.08723 −0.543617 0.839334i $$-0.682945\pi$$
−0.543617 + 0.839334i $$0.682945\pi$$
$$488$$ −1351.22 −0.125342
$$489$$ −6943.94 −0.642159
$$490$$ 10410.0 0.959750
$$491$$ −3954.70 −0.363489 −0.181745 0.983346i $$-0.558174\pi$$
−0.181745 + 0.983346i $$0.558174\pi$$
$$492$$ 11402.3 1.04483
$$493$$ 415.744 0.0379801
$$494$$ 8465.47 0.771011
$$495$$ −2662.63 −0.241771
$$496$$ −12608.8 −1.14144
$$497$$ −2791.49 −0.251943
$$498$$ −5498.27 −0.494746
$$499$$ −5690.37 −0.510493 −0.255246 0.966876i $$-0.582157\pi$$
−0.255246 + 0.966876i $$0.582157\pi$$
$$500$$ 13251.9 1.18529
$$501$$ −7996.95 −0.713128
$$502$$ −26782.8 −2.38123
$$503$$ 10859.1 0.962595 0.481298 0.876557i $$-0.340166\pi$$
0.481298 + 0.876557i $$0.340166\pi$$
$$504$$ −731.623 −0.0646609
$$505$$ −7378.53 −0.650178
$$506$$ −19651.9 −1.72655
$$507$$ 507.000 0.0444116
$$508$$ −3021.88 −0.263926
$$509$$ −18558.6 −1.61610 −0.808049 0.589115i $$-0.799476\pi$$
−0.808049 + 0.589115i $$0.799476\pi$$
$$510$$ 2934.41 0.254780
$$511$$ −9597.27 −0.830838
$$512$$ 14896.8 1.28584
$$513$$ 4177.22 0.359510
$$514$$ −25613.6 −2.19799
$$515$$ 5843.42 0.499984
$$516$$ 4413.27 0.376518
$$517$$ −12094.7 −1.02887
$$518$$ −5494.13 −0.466020
$$519$$ −495.730 −0.0419271
$$520$$ −1073.49 −0.0905305
$$521$$ 17297.5 1.45454 0.727271 0.686350i $$-0.240788\pi$$
0.727271 + 0.686350i $$0.240788\pi$$
$$522$$ −774.759 −0.0649622
$$523$$ −5016.11 −0.419386 −0.209693 0.977767i $$-0.567247\pi$$
−0.209693 + 0.977767i $$0.567247\pi$$
$$524$$ 19441.8 1.62084
$$525$$ −192.286 −0.0159849
$$526$$ 518.667 0.0429942
$$527$$ −5415.77 −0.447656
$$528$$ −3674.19 −0.302839
$$529$$ 20383.8 1.67533
$$530$$ −3848.73 −0.315431
$$531$$ −7863.39 −0.642640
$$532$$ −16917.4 −1.37869
$$533$$ 5085.39 0.413269
$$534$$ 3273.38 0.265268
$$535$$ −6953.01 −0.561878
$$536$$ −4400.37 −0.354603
$$537$$ 2137.02 0.171730
$$538$$ −8147.83 −0.652933
$$539$$ −5598.57 −0.447398
$$540$$ −2999.06 −0.238998
$$541$$ 17642.3 1.40204 0.701018 0.713144i $$-0.252729\pi$$
0.701018 + 0.713144i $$0.252729\pi$$
$$542$$ −19415.0 −1.53865
$$543$$ 6619.59 0.523156
$$544$$ 5223.86 0.411712
$$545$$ 14862.3 1.16813
$$546$$ −1847.43 −0.144803
$$547$$ −18414.9 −1.43943 −0.719713 0.694271i $$-0.755727\pi$$
−0.719713 + 0.694271i $$0.755727\pi$$
$$548$$ 10090.4 0.786571
$$549$$ −1683.61 −0.130883
$$550$$ 620.343 0.0480937
$$551$$ −3164.20 −0.244645
$$552$$ −3909.57 −0.301453
$$553$$ 3727.66 0.286648
$$554$$ −24556.9 −1.88325
$$555$$ −3977.84 −0.304234
$$556$$ −27773.1 −2.11842
$$557$$ 8179.15 0.622193 0.311096 0.950378i $$-0.399304\pi$$
0.311096 + 0.950378i $$0.399304\pi$$
$$558$$ 10092.5 0.765683
$$559$$ 1968.30 0.148927
$$560$$ −6089.05 −0.459481
$$561$$ −1578.14 −0.118769
$$562$$ 19748.5 1.48228
$$563$$ −1880.07 −0.140738 −0.0703690 0.997521i $$-0.522418\pi$$
−0.0703690 + 0.997521i $$0.522418\pi$$
$$564$$ −13622.9 −1.01707
$$565$$ −482.385 −0.0359187
$$566$$ 14586.9 1.08327
$$567$$ −911.598 −0.0675194
$$568$$ 1791.62 0.132350
$$569$$ 10118.3 0.745485 0.372743 0.927935i $$-0.378417\pi$$
0.372743 + 0.927935i $$0.378417\pi$$
$$570$$ −22333.6 −1.64114
$$571$$ 23428.9 1.71711 0.858555 0.512721i $$-0.171362\pi$$
0.858555 + 0.512721i $$0.171362\pi$$
$$572$$ 3268.70 0.238935
$$573$$ 4411.92 0.321659
$$574$$ −18530.3 −1.34746
$$575$$ −1027.52 −0.0745225
$$576$$ −6327.42 −0.457713
$$577$$ 20508.1 1.47966 0.739831 0.672793i $$-0.234906\pi$$
0.739831 + 0.672793i $$0.234906\pi$$
$$578$$ −18939.8 −1.36296
$$579$$ 1108.68 0.0795771
$$580$$ 2271.76 0.162637
$$581$$ 4900.49 0.349925
$$582$$ 15470.4 1.10184
$$583$$ 2069.87 0.147042
$$584$$ 6159.65 0.436452
$$585$$ −1337.57 −0.0945327
$$586$$ 11268.9 0.794394
$$587$$ −5968.43 −0.419665 −0.209833 0.977737i $$-0.567292\pi$$
−0.209833 + 0.977737i $$0.567292\pi$$
$$588$$ −6305.97 −0.442268
$$589$$ 41219.0 2.88353
$$590$$ 42041.8 2.93361
$$591$$ −12820.2 −0.892308
$$592$$ −5489.06 −0.381080
$$593$$ −14659.5 −1.01517 −0.507584 0.861602i $$-0.669461\pi$$
−0.507584 + 0.861602i $$0.669461\pi$$
$$594$$ 2940.95 0.203146
$$595$$ −2615.38 −0.180202
$$596$$ 7223.72 0.496468
$$597$$ 12462.9 0.854394
$$598$$ −9872.07 −0.675082
$$599$$ 23635.9 1.61225 0.806125 0.591746i $$-0.201561\pi$$
0.806125 + 0.591746i $$0.201561\pi$$
$$600$$ 123.412 0.00839711
$$601$$ −11527.0 −0.782356 −0.391178 0.920315i $$-0.627932\pi$$
−0.391178 + 0.920315i $$0.627932\pi$$
$$602$$ −7172.15 −0.485573
$$603$$ −5482.83 −0.370279
$$604$$ 22125.9 1.49055
$$605$$ 7560.15 0.508039
$$606$$ 8149.77 0.546306
$$607$$ −5098.56 −0.340930 −0.170465 0.985364i $$-0.554527\pi$$
−0.170465 + 0.985364i $$0.554527\pi$$
$$608$$ −39758.4 −2.65200
$$609$$ 690.525 0.0459466
$$610$$ 9001.47 0.597473
$$611$$ −6075.74 −0.402288
$$612$$ −1777.55 −0.117407
$$613$$ 1516.39 0.0999128 0.0499564 0.998751i $$-0.484092\pi$$
0.0499564 + 0.998751i $$0.484092\pi$$
$$614$$ 1986.31 0.130556
$$615$$ −13416.3 −0.879668
$$616$$ −2103.70 −0.137598
$$617$$ 18539.3 1.20966 0.604832 0.796353i $$-0.293240\pi$$
0.604832 + 0.796353i $$0.293240\pi$$
$$618$$ −6454.20 −0.420107
$$619$$ 25684.9 1.66779 0.833897 0.551920i $$-0.186105\pi$$
0.833897 + 0.551920i $$0.186105\pi$$
$$620$$ −29593.5 −1.91694
$$621$$ −4871.30 −0.314780
$$622$$ −6391.51 −0.412020
$$623$$ −2917.49 −0.187619
$$624$$ −1845.72 −0.118410
$$625$$ −16304.5 −1.04349
$$626$$ 17046.1 1.08834
$$627$$ 12011.1 0.765038
$$628$$ 30834.2 1.95926
$$629$$ −2357.67 −0.149454
$$630$$ 4873.88 0.308222
$$631$$ −22410.9 −1.41389 −0.706945 0.707269i $$-0.749927\pi$$
−0.706945 + 0.707269i $$0.749927\pi$$
$$632$$ −2392.46 −0.150580
$$633$$ 3694.77 0.231997
$$634$$ 13696.1 0.857950
$$635$$ 3555.62 0.222206
$$636$$ 2331.40 0.145356
$$637$$ −2812.43 −0.174933
$$638$$ −2227.73 −0.138239
$$639$$ 2232.34 0.138200
$$640$$ 10326.6 0.637804
$$641$$ 6827.81 0.420721 0.210361 0.977624i $$-0.432536\pi$$
0.210361 + 0.977624i $$0.432536\pi$$
$$642$$ 7679.77 0.472113
$$643$$ −23264.3 −1.42684 −0.713418 0.700738i $$-0.752854\pi$$
−0.713418 + 0.700738i $$0.752854\pi$$
$$644$$ 19728.4 1.20715
$$645$$ −5192.76 −0.316999
$$646$$ −13237.1 −0.806204
$$647$$ 14745.9 0.896014 0.448007 0.894030i $$-0.352134\pi$$
0.448007 + 0.894030i $$0.352134\pi$$
$$648$$ 585.075 0.0354690
$$649$$ −22610.3 −1.36754
$$650$$ 311.628 0.0188047
$$651$$ −8995.26 −0.541554
$$652$$ −22489.3 −1.35084
$$653$$ 10909.0 0.653755 0.326878 0.945067i $$-0.394003\pi$$
0.326878 + 0.945067i $$0.394003\pi$$
$$654$$ −16415.8 −0.981509
$$655$$ −22875.7 −1.36462
$$656$$ −18513.2 −1.10186
$$657$$ 7674.89 0.455747
$$658$$ 22139.0 1.31166
$$659$$ −4182.99 −0.247263 −0.123631 0.992328i $$-0.539454\pi$$
−0.123631 + 0.992328i $$0.539454\pi$$
$$660$$ −8623.47 −0.508588
$$661$$ 2224.23 0.130881 0.0654406 0.997856i $$-0.479155\pi$$
0.0654406 + 0.997856i $$0.479155\pi$$
$$662$$ −14405.2 −0.845734
$$663$$ −792.776 −0.0464387
$$664$$ −3145.19 −0.183821
$$665$$ 19905.4 1.16075
$$666$$ 4393.63 0.255630
$$667$$ 3689.95 0.214206
$$668$$ −25899.7 −1.50013
$$669$$ −6561.72 −0.379209
$$670$$ 29314.1 1.69030
$$671$$ −4841.04 −0.278519
$$672$$ 8676.51 0.498071
$$673$$ −24152.5 −1.38337 −0.691687 0.722197i $$-0.743132\pi$$
−0.691687 + 0.722197i $$0.743132\pi$$
$$674$$ −39151.2 −2.23746
$$675$$ 153.770 0.00876833
$$676$$ 1642.02 0.0934240
$$677$$ −15310.7 −0.869187 −0.434593 0.900627i $$-0.643108\pi$$
−0.434593 + 0.900627i $$0.643108\pi$$
$$678$$ 532.806 0.0301804
$$679$$ −13788.4 −0.779310
$$680$$ 1678.58 0.0946628
$$681$$ 12416.0 0.698652
$$682$$ 29020.0 1.62937
$$683$$ 11399.6 0.638646 0.319323 0.947646i $$-0.396545\pi$$
0.319323 + 0.947646i $$0.396545\pi$$
$$684$$ 13528.8 0.756265
$$685$$ −11872.6 −0.662233
$$686$$ 26495.9 1.47466
$$687$$ −2506.06 −0.139174
$$688$$ −7165.54 −0.397069
$$689$$ 1039.79 0.0574935
$$690$$ 26044.5 1.43695
$$691$$ −3323.23 −0.182955 −0.0914773 0.995807i $$-0.529159\pi$$
−0.0914773 + 0.995807i $$0.529159\pi$$
$$692$$ −1605.52 −0.0881976
$$693$$ −2621.20 −0.143681
$$694$$ 912.936 0.0499345
$$695$$ 32678.5 1.78355
$$696$$ −443.188 −0.0241365
$$697$$ −7951.82 −0.432133
$$698$$ −20244.8 −1.09782
$$699$$ 11056.5 0.598279
$$700$$ −622.758 −0.0336257
$$701$$ −12670.4 −0.682673 −0.341336 0.939941i $$-0.610880\pi$$
−0.341336 + 0.939941i $$0.610880\pi$$
$$702$$ 1477.38 0.0794302
$$703$$ 17944.0 0.962692
$$704$$ −18193.8 −0.974012
$$705$$ 16029.0 0.856295
$$706$$ −12036.4 −0.641635
$$707$$ −7263.71 −0.386393
$$708$$ −25467.2 −1.35186
$$709$$ 13075.2 0.692594 0.346297 0.938125i $$-0.387439\pi$$
0.346297 + 0.938125i $$0.387439\pi$$
$$710$$ −11935.3 −0.630877
$$711$$ −2980.99 −0.157237
$$712$$ 1872.48 0.0985592
$$713$$ −48067.8 −2.52476
$$714$$ 2888.75 0.151413
$$715$$ −3846.03 −0.201165
$$716$$ 6921.16 0.361251
$$717$$ 9078.62 0.472869
$$718$$ 15514.7 0.806412
$$719$$ −2988.41 −0.155005 −0.0775026 0.996992i $$-0.524695\pi$$
−0.0775026 + 0.996992i $$0.524695\pi$$
$$720$$ 4869.38 0.252043
$$721$$ 5752.48 0.297134
$$722$$ 71877.0 3.70496
$$723$$ 9796.73 0.503934
$$724$$ 21438.9 1.10051
$$725$$ −116.479 −0.00596680
$$726$$ −8350.37 −0.426875
$$727$$ −5507.46 −0.280963 −0.140482 0.990083i $$-0.544865\pi$$
−0.140482 + 0.990083i $$0.544865\pi$$
$$728$$ −1056.79 −0.0538011
$$729$$ 729.000 0.0370370
$$730$$ −41033.9 −2.08046
$$731$$ −3077.75 −0.155725
$$732$$ −5452.71 −0.275325
$$733$$ −36585.2 −1.84353 −0.921764 0.387751i $$-0.873252\pi$$
−0.921764 + 0.387751i $$0.873252\pi$$
$$734$$ −14607.9 −0.734587
$$735$$ 7419.75 0.372356
$$736$$ 46364.6 2.32204
$$737$$ −15765.3 −0.787953
$$738$$ 14818.6 0.739133
$$739$$ 6425.89 0.319865 0.159933 0.987128i $$-0.448872\pi$$
0.159933 + 0.987128i $$0.448872\pi$$
$$740$$ −12883.0 −0.639986
$$741$$ 6033.76 0.299131
$$742$$ −3788.84 −0.187457
$$743$$ 20411.0 1.00782 0.503908 0.863757i $$-0.331895\pi$$
0.503908 + 0.863757i $$0.331895\pi$$
$$744$$ 5773.27 0.284487
$$745$$ −8499.60 −0.417988
$$746$$ −50354.6 −2.47133
$$747$$ −3918.89 −0.191947
$$748$$ −5111.13 −0.249842
$$749$$ −6844.81 −0.333917
$$750$$ 17222.4 0.838496
$$751$$ −24259.5 −1.17875 −0.589375 0.807860i $$-0.700626\pi$$
−0.589375 + 0.807860i $$0.700626\pi$$
$$752$$ 22118.6 1.07258
$$753$$ −19089.5 −0.923850
$$754$$ −1119.10 −0.0540518
$$755$$ −26033.9 −1.25493
$$756$$ −2952.39 −0.142034
$$757$$ 9295.39 0.446297 0.223148 0.974785i $$-0.428367\pi$$
0.223148 + 0.974785i $$0.428367\pi$$
$$758$$ 1454.66 0.0697042
$$759$$ −14006.9 −0.669851
$$760$$ −12775.6 −0.609761
$$761$$ −21974.7 −1.04676 −0.523378 0.852101i $$-0.675328\pi$$
−0.523378 + 0.852101i $$0.675328\pi$$
$$762$$ −3927.27 −0.186706
$$763$$ 14631.0 0.694204
$$764$$ 14288.9 0.676641
$$765$$ 2091.50 0.0988476
$$766$$ −14253.5 −0.672324
$$767$$ −11358.2 −0.534709
$$768$$ 5467.13 0.256873
$$769$$ 22987.4 1.07795 0.538977 0.842320i $$-0.318811\pi$$
0.538977 + 0.842320i $$0.318811\pi$$
$$770$$ 14014.3 0.655897
$$771$$ −18256.1 −0.852759
$$772$$ 3590.68 0.167398
$$773$$ 31970.9 1.48760 0.743799 0.668404i $$-0.233022\pi$$
0.743799 + 0.668404i $$0.233022\pi$$
$$774$$ 5735.53 0.266356
$$775$$ 1517.34 0.0703283
$$776$$ 8849.59 0.409384
$$777$$ −3915.94 −0.180803
$$778$$ −6860.25 −0.316133
$$779$$ 60520.8 2.78354
$$780$$ −4331.98 −0.198859
$$781$$ 6418.85 0.294090
$$782$$ 15436.6 0.705896
$$783$$ −552.209 −0.0252035
$$784$$ 10238.6 0.466408
$$785$$ −36280.2 −1.64955
$$786$$ 25266.7 1.14661
$$787$$ 6087.26 0.275715 0.137857 0.990452i $$-0.455978\pi$$
0.137857 + 0.990452i $$0.455978\pi$$
$$788$$ −41520.9 −1.87706
$$789$$ 369.680 0.0166805
$$790$$ 15937.9 0.717779
$$791$$ −474.878 −0.0213460
$$792$$ 1682.32 0.0754780
$$793$$ −2431.88 −0.108901
$$794$$ 33415.4 1.49354
$$795$$ −2743.19 −0.122378
$$796$$ 40363.6 1.79730
$$797$$ 23080.0 1.02577 0.512883 0.858458i $$-0.328577\pi$$
0.512883 + 0.858458i $$0.328577\pi$$
$$798$$ −21986.1 −0.975311
$$799$$ 9500.41 0.420651
$$800$$ −1463.57 −0.0646813
$$801$$ 2333.10 0.102916
$$802$$ 902.429 0.0397330
$$803$$ 22068.3 0.969829
$$804$$ −17757.3 −0.778918
$$805$$ −23212.9 −1.01633
$$806$$ 14578.1 0.637087
$$807$$ −5807.37 −0.253320
$$808$$ 4661.94 0.202978
$$809$$ −32377.8 −1.40710 −0.703550 0.710646i $$-0.748403\pi$$
−0.703550 + 0.710646i $$0.748403\pi$$
$$810$$ −3897.61 −0.169072
$$811$$ 26352.8 1.14103 0.570513 0.821288i $$-0.306744\pi$$
0.570513 + 0.821288i $$0.306744\pi$$
$$812$$ 2236.40 0.0966532
$$813$$ −13838.1 −0.596952
$$814$$ 12633.4 0.543980
$$815$$ 26461.5 1.13731
$$816$$ 2886.08 0.123815
$$817$$ 23424.5 1.00308
$$818$$ −20135.0 −0.860638
$$819$$ −1316.75 −0.0561796
$$820$$ −43451.3 −1.85047
$$821$$ 35355.3 1.50294 0.751468 0.659770i $$-0.229346\pi$$
0.751468 + 0.659770i $$0.229346\pi$$
$$822$$ 13113.6 0.556435
$$823$$ −12663.3 −0.536347 −0.268173 0.963371i $$-0.586420\pi$$
−0.268173 + 0.963371i $$0.586420\pi$$
$$824$$ −3692.02 −0.156089
$$825$$ 442.149 0.0186590
$$826$$ 41387.6 1.74341
$$827$$ −16295.2 −0.685176 −0.342588 0.939486i $$-0.611303\pi$$
−0.342588 + 0.939486i $$0.611303\pi$$
$$828$$ −15776.7 −0.662170
$$829$$ 13638.9 0.571411 0.285705 0.958318i $$-0.407772\pi$$
0.285705 + 0.958318i $$0.407772\pi$$
$$830$$ 20952.4 0.876229
$$831$$ −17502.9 −0.730649
$$832$$ −9139.61 −0.380840
$$833$$ 4397.69 0.182918
$$834$$ −36094.2 −1.49861
$$835$$ 30474.2 1.26300
$$836$$ 38900.5 1.60933
$$837$$ 7193.46 0.297064
$$838$$ −41685.0 −1.71836
$$839$$ −1890.31 −0.0777838 −0.0388919 0.999243i $$-0.512383\pi$$
−0.0388919 + 0.999243i $$0.512383\pi$$
$$840$$ 2788.02 0.114519
$$841$$ −23970.7 −0.982849
$$842$$ −51016.4 −2.08805
$$843$$ 14075.7 0.575082
$$844$$ 11966.3 0.488028
$$845$$ −1932.04 −0.0786559
$$846$$ −17704.5 −0.719494
$$847$$ 7442.50 0.301921
$$848$$ −3785.35 −0.153289
$$849$$ 10396.8 0.420279
$$850$$ −487.280 −0.0196630
$$851$$ −20925.6 −0.842914
$$852$$ 7229.89 0.290718
$$853$$ 1620.21 0.0650351 0.0325175 0.999471i $$-0.489648\pi$$
0.0325175 + 0.999471i $$0.489648\pi$$
$$854$$ 8861.39 0.355071
$$855$$ −15918.3 −0.636718
$$856$$ 4393.08 0.175412
$$857$$ −14508.4 −0.578292 −0.289146 0.957285i $$-0.593371\pi$$
−0.289146 + 0.957285i $$0.593371\pi$$
$$858$$ 4248.03 0.169027
$$859$$ 29639.8 1.17730 0.588648 0.808389i $$-0.299660\pi$$
0.588648 + 0.808389i $$0.299660\pi$$
$$860$$ −16817.8 −0.666839
$$861$$ −13207.5 −0.522776
$$862$$ −57548.8 −2.27392
$$863$$ 21528.8 0.849186 0.424593 0.905384i $$-0.360417\pi$$
0.424593 + 0.905384i $$0.360417\pi$$
$$864$$ −6938.56 −0.273211
$$865$$ 1889.10 0.0742557
$$866$$ 29939.4 1.17481
$$867$$ −13499.4 −0.528792
$$868$$ −29132.9 −1.13921
$$869$$ −8571.50 −0.334601
$$870$$ 2952.40 0.115053
$$871$$ −7919.65 −0.308091
$$872$$ −9390.36 −0.364676
$$873$$ 11026.5 0.427482
$$874$$ −117487. −4.54696
$$875$$ −15349.9 −0.593054
$$876$$ 24856.7 0.958708
$$877$$ 14865.3 0.572366 0.286183 0.958175i $$-0.407613\pi$$
0.286183 + 0.958175i $$0.407613\pi$$
$$878$$ −25347.1 −0.974285
$$879$$ 8031.92 0.308202
$$880$$ 14001.4 0.536348
$$881$$ −21336.0 −0.815921 −0.407961 0.913000i $$-0.633760\pi$$
−0.407961 + 0.913000i $$0.633760\pi$$
$$882$$ −8195.30 −0.312869
$$883$$ 37538.2 1.43065 0.715323 0.698794i $$-0.246280\pi$$
0.715323 + 0.698794i $$0.246280\pi$$
$$884$$ −2567.57 −0.0976884
$$885$$ 29965.3 1.13816
$$886$$ −54694.1 −2.07391
$$887$$ 34575.0 1.30881 0.654406 0.756144i $$-0.272919\pi$$
0.654406 + 0.756144i $$0.272919\pi$$
$$888$$ 2513.30 0.0949784
$$889$$ 3500.29 0.132054
$$890$$ −12474.0 −0.469807
$$891$$ 2096.16 0.0788148
$$892$$ −21251.4 −0.797703
$$893$$ −72306.9 −2.70958
$$894$$ 9388.02 0.351211
$$895$$ −8143.61 −0.304146
$$896$$ 10165.9 0.379039
$$897$$ −7036.32 −0.261913
$$898$$ 46233.3 1.71807
$$899$$ −5448.96 −0.202150
$$900$$ 498.016 0.0184450
$$901$$ −1625.89 −0.0601178
$$902$$ 42609.2 1.57287
$$903$$ −5111.95 −0.188389
$$904$$ 304.783 0.0112134
$$905$$ −25225.5 −0.926545
$$906$$ 28755.0 1.05444
$$907$$ −10424.8 −0.381641 −0.190820 0.981625i $$-0.561115\pi$$
−0.190820 + 0.981625i $$0.561115\pi$$
$$908$$ 40211.7 1.46968
$$909$$ 5808.75 0.211952
$$910$$ 7040.05 0.256456
$$911$$ 10961.8 0.398661 0.199331 0.979932i $$-0.436123\pi$$
0.199331 + 0.979932i $$0.436123\pi$$
$$912$$ −21965.8 −0.797543
$$913$$ −11268.3 −0.408464
$$914$$ 41392.2 1.49796
$$915$$ 6415.80 0.231803
$$916$$ −8116.38 −0.292765
$$917$$ −22519.7 −0.810976
$$918$$ −2310.12 −0.0830558
$$919$$ −10779.2 −0.386914 −0.193457 0.981109i $$-0.561970\pi$$
−0.193457 + 0.981109i $$0.561970\pi$$
$$920$$ 14898.3 0.533895
$$921$$ 1415.74 0.0506519
$$922$$ 14316.7 0.511384
$$923$$ 3224.49 0.114990
$$924$$ −8489.28 −0.302248
$$925$$ 660.549 0.0234797
$$926$$ 7321.32 0.259820
$$927$$ −4600.23 −0.162990
$$928$$ 5255.88 0.185919
$$929$$ 5429.07 0.191735 0.0958675 0.995394i $$-0.469437\pi$$
0.0958675 + 0.995394i $$0.469437\pi$$
$$930$$ −38460.0 −1.35608
$$931$$ −33470.5 −1.17825
$$932$$ 35808.8 1.25854
$$933$$ −4555.55 −0.159852
$$934$$ −33499.1 −1.17358
$$935$$ 6013.88 0.210348
$$936$$ 845.109 0.0295120
$$937$$ −21300.1 −0.742631 −0.371315 0.928507i $$-0.621093\pi$$
−0.371315 + 0.928507i $$0.621093\pi$$
$$938$$ 28857.9 1.00453
$$939$$ 12149.6 0.422244
$$940$$ 51913.2 1.80130
$$941$$ 26851.2 0.930207 0.465103 0.885256i $$-0.346017\pi$$
0.465103 + 0.885256i $$0.346017\pi$$
$$942$$ 40072.4 1.38602
$$943$$ −70576.7 −2.43722
$$944$$ 41349.4 1.42564
$$945$$ 3473.86 0.119582
$$946$$ 16491.9 0.566805
$$947$$ −8021.68 −0.275258 −0.137629 0.990484i $$-0.543948\pi$$
−0.137629 + 0.990484i $$0.543948\pi$$
$$948$$ −9654.52 −0.330764
$$949$$ 11085.9 0.379204
$$950$$ 3708.65 0.126658
$$951$$ 9761.88 0.332861
$$952$$ 1652.46 0.0562569
$$953$$ 35715.0 1.21398 0.606990 0.794709i $$-0.292377\pi$$
0.606990 + 0.794709i $$0.292377\pi$$
$$954$$ 3029.92 0.102827
$$955$$ −16812.6 −0.569680
$$956$$ 29402.9 0.994726
$$957$$ −1587.82 −0.0536330
$$958$$ 35490.6 1.19692
$$959$$ −11687.9 −0.393557
$$960$$ 24112.1 0.810641
$$961$$ 41190.9 1.38266
$$962$$ 6346.35 0.212697
$$963$$ 5473.76 0.183166
$$964$$ 31728.7 1.06007
$$965$$ −4224.88 −0.140936
$$966$$ 25639.2 0.853963
$$967$$ 53338.8 1.77380 0.886898 0.461965i $$-0.152855\pi$$
0.886898 + 0.461965i $$0.152855\pi$$
$$968$$ −4776.69 −0.158604
$$969$$ −9434.77 −0.312785
$$970$$ −58953.6 −1.95143
$$971$$ 23112.9 0.763882 0.381941 0.924187i $$-0.375256\pi$$
0.381941 + 0.924187i $$0.375256\pi$$
$$972$$ 2361.01 0.0779110
$$973$$ 32170.0 1.05994
$$974$$ −49181.3 −1.61794
$$975$$ 222.113 0.00729569
$$976$$ 8853.22 0.290353
$$977$$ −52874.6 −1.73143 −0.865715 0.500538i $$-0.833136\pi$$
−0.865715 + 0.500538i $$0.833136\pi$$
$$978$$ −29227.4 −0.955612
$$979$$ 6708.57 0.219006
$$980$$ 24030.4 0.783287
$$981$$ −11700.3 −0.380798
$$982$$ −16645.5 −0.540917
$$983$$ 45173.1 1.46572 0.732858 0.680381i $$-0.238186\pi$$
0.732858 + 0.680381i $$0.238186\pi$$
$$984$$ 8476.73 0.274622
$$985$$ 48854.5 1.58034
$$986$$ 1749.89 0.0565190
$$987$$ 15779.6 0.508885
$$988$$ 19541.6 0.629251
$$989$$ −27316.7 −0.878281
$$990$$ −11207.2 −0.359785
$$991$$ 60485.6 1.93884 0.969418 0.245414i $$-0.0789239\pi$$
0.969418 + 0.245414i $$0.0789239\pi$$
$$992$$ −68466.7 −2.19135
$$993$$ −10267.3 −0.328121
$$994$$ −11749.5 −0.374922
$$995$$ −47492.8 −1.51319
$$996$$ −12692.1 −0.403780
$$997$$ −18108.1 −0.575214 −0.287607 0.957749i $$-0.592860\pi$$
−0.287607 + 0.957749i $$0.592860\pi$$
$$998$$ −23951.1 −0.759677
$$999$$ 3131.56 0.0991773
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 39.4.a.c.1.3 3
3.2 odd 2 117.4.a.f.1.1 3
4.3 odd 2 624.4.a.t.1.1 3
5.4 even 2 975.4.a.l.1.1 3
7.6 odd 2 1911.4.a.k.1.3 3
8.3 odd 2 2496.4.a.bp.1.3 3
8.5 even 2 2496.4.a.bl.1.3 3
12.11 even 2 1872.4.a.bk.1.3 3
13.5 odd 4 507.4.b.g.337.1 6
13.8 odd 4 507.4.b.g.337.6 6
13.12 even 2 507.4.a.h.1.1 3
39.38 odd 2 1521.4.a.u.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 1.1 even 1 trivial
117.4.a.f.1.1 3 3.2 odd 2
507.4.a.h.1.1 3 13.12 even 2
507.4.b.g.337.1 6 13.5 odd 4
507.4.b.g.337.6 6 13.8 odd 4
624.4.a.t.1.1 3 4.3 odd 2
975.4.a.l.1.1 3 5.4 even 2
1521.4.a.u.1.3 3 39.38 odd 2
1872.4.a.bk.1.3 3 12.11 even 2
1911.4.a.k.1.3 3 7.6 odd 2
2496.4.a.bl.1.3 3 8.5 even 2
2496.4.a.bp.1.3 3 8.3 odd 2