# Properties

 Label 39.4.a.b.1.2 Level $39$ Weight $4$ Character 39.1 Self dual yes Analytic conductor $2.301$ Analytic rank $0$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 14$$ x^2 - 14 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 $$3.74166$$ 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)

 $$f(q)$$ $$=$$ $$q+4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} +4.51669 q^{5} -14.2250 q^{6} -7.48331 q^{7} +30.7417 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} +4.51669 q^{5} -14.2250 q^{6} -7.48331 q^{7} +30.7417 q^{8} +9.00000 q^{9} +21.4166 q^{10} -66.8999 q^{11} -43.4499 q^{12} -13.0000 q^{13} -35.4833 q^{14} -13.5501 q^{15} +29.8999 q^{16} +96.9666 q^{17} +42.6749 q^{18} +31.4833 q^{19} +65.4166 q^{20} +22.4499 q^{21} -317.216 q^{22} +183.600 q^{23} -92.2250 q^{24} -104.600 q^{25} -61.6415 q^{26} -27.0000 q^{27} -108.383 q^{28} +112.200 q^{29} -64.2497 q^{30} -77.2831 q^{31} -104.158 q^{32} +200.700 q^{33} +459.783 q^{34} -33.7998 q^{35} +130.350 q^{36} +54.7664 q^{37} +149.283 q^{38} +39.0000 q^{39} +138.850 q^{40} +451.716 q^{41} +106.450 q^{42} -113.434 q^{43} -968.932 q^{44} +40.6502 q^{45} +870.566 q^{46} -42.2670 q^{47} -89.6997 q^{48} -287.000 q^{49} -495.975 q^{50} -290.900 q^{51} -188.283 q^{52} -530.999 q^{53} -128.025 q^{54} -302.166 q^{55} -230.049 q^{56} -94.4499 q^{57} +532.015 q^{58} +219.666 q^{59} -196.250 q^{60} +822.865 q^{61} -366.450 q^{62} -67.3498 q^{63} -733.082 q^{64} -58.7169 q^{65} +951.649 q^{66} -872.082 q^{67} +1404.40 q^{68} -550.799 q^{69} -160.267 q^{70} -100.299 q^{71} +276.675 q^{72} -165.634 q^{73} +259.684 q^{74} +313.799 q^{75} +455.983 q^{76} +500.633 q^{77} +184.925 q^{78} -545.266 q^{79} +135.048 q^{80} +81.0000 q^{81} +2141.88 q^{82} -454.534 q^{83} +325.150 q^{84} +437.968 q^{85} -537.864 q^{86} -336.601 q^{87} -2056.61 q^{88} -230.915 q^{89} +192.749 q^{90} +97.2831 q^{91} +2659.13 q^{92} +231.849 q^{93} -200.415 q^{94} +142.200 q^{95} +312.475 q^{96} -1089.16 q^{97} -1360.86 q^{98} -602.099 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9 $$2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9} - 32 q^{10} - 44 q^{11} - 42 q^{12} - 26 q^{13} - 56 q^{14} - 72 q^{15} - 30 q^{16} + 164 q^{17} + 18 q^{18} + 48 q^{19} + 56 q^{20} - 380 q^{22} + 8 q^{23} - 162 q^{24} + 150 q^{25} - 26 q^{26} - 54 q^{27} - 112 q^{28} + 404 q^{29} + 96 q^{30} + 40 q^{31} - 126 q^{32} + 132 q^{33} + 276 q^{34} + 112 q^{35} + 126 q^{36} - 100 q^{37} + 104 q^{38} + 78 q^{39} + 592 q^{40} + 200 q^{41} + 168 q^{42} - 616 q^{43} - 980 q^{44} + 216 q^{45} + 1352 q^{46} - 324 q^{47} + 90 q^{48} - 574 q^{49} - 1194 q^{50} - 492 q^{51} - 182 q^{52} - 164 q^{53} - 54 q^{54} + 144 q^{55} - 56 q^{56} - 144 q^{57} - 268 q^{58} + 140 q^{59} - 168 q^{60} + 628 q^{61} - 688 q^{62} - 194 q^{64} - 312 q^{65} + 1140 q^{66} - 472 q^{67} + 1372 q^{68} - 24 q^{69} - 560 q^{70} + 428 q^{71} + 486 q^{72} - 900 q^{73} + 684 q^{74} - 450 q^{75} + 448 q^{76} + 672 q^{77} + 78 q^{78} - 432 q^{79} - 1032 q^{80} + 162 q^{81} + 2832 q^{82} - 1388 q^{83} + 336 q^{84} + 1744 q^{85} + 840 q^{86} - 1212 q^{87} - 1524 q^{88} + 960 q^{89} - 288 q^{90} + 2744 q^{92} - 120 q^{93} + 572 q^{94} + 464 q^{95} + 378 q^{96} - 532 q^{97} - 574 q^{98} - 396 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9 - 32 * q^10 - 44 * q^11 - 42 * q^12 - 26 * q^13 - 56 * q^14 - 72 * q^15 - 30 * q^16 + 164 * q^17 + 18 * q^18 + 48 * q^19 + 56 * q^20 - 380 * q^22 + 8 * q^23 - 162 * q^24 + 150 * q^25 - 26 * q^26 - 54 * q^27 - 112 * q^28 + 404 * q^29 + 96 * q^30 + 40 * q^31 - 126 * q^32 + 132 * q^33 + 276 * q^34 + 112 * q^35 + 126 * q^36 - 100 * q^37 + 104 * q^38 + 78 * q^39 + 592 * q^40 + 200 * q^41 + 168 * q^42 - 616 * q^43 - 980 * q^44 + 216 * q^45 + 1352 * q^46 - 324 * q^47 + 90 * q^48 - 574 * q^49 - 1194 * q^50 - 492 * q^51 - 182 * q^52 - 164 * q^53 - 54 * q^54 + 144 * q^55 - 56 * q^56 - 144 * q^57 - 268 * q^58 + 140 * q^59 - 168 * q^60 + 628 * q^61 - 688 * q^62 - 194 * q^64 - 312 * q^65 + 1140 * q^66 - 472 * q^67 + 1372 * q^68 - 24 * q^69 - 560 * q^70 + 428 * q^71 + 486 * q^72 - 900 * q^73 + 684 * q^74 - 450 * q^75 + 448 * q^76 + 672 * q^77 + 78 * q^78 - 432 * q^79 - 1032 * q^80 + 162 * q^81 + 2832 * q^82 - 1388 * q^83 + 336 * q^84 + 1744 * q^85 + 840 * q^86 - 1212 * q^87 - 1524 * q^88 + 960 * q^89 - 288 * q^90 + 2744 * q^92 - 120 * q^93 + 572 * q^94 + 464 * q^95 + 378 * q^96 - 532 * q^97 - 574 * q^98 - 396 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.74166 1.67643 0.838215 0.545341i $$-0.183600\pi$$
0.838215 + 0.545341i $$0.183600\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 14.4833 1.81041
$$5$$ 4.51669 0.403985 0.201992 0.979387i $$-0.435258\pi$$
0.201992 + 0.979387i $$0.435258\pi$$
$$6$$ −14.2250 −0.967887
$$7$$ −7.48331 −0.404061 −0.202031 0.979379i $$-0.564754\pi$$
−0.202031 + 0.979379i $$0.564754\pi$$
$$8$$ 30.7417 1.35860
$$9$$ 9.00000 0.333333
$$10$$ 21.4166 0.677252
$$11$$ −66.8999 −1.83373 −0.916867 0.399193i $$-0.869290\pi$$
−0.916867 + 0.399193i $$0.869290\pi$$
$$12$$ −43.4499 −1.04524
$$13$$ −13.0000 −0.277350
$$14$$ −35.4833 −0.677380
$$15$$ −13.5501 −0.233241
$$16$$ 29.8999 0.467186
$$17$$ 96.9666 1.38340 0.691702 0.722183i $$-0.256861\pi$$
0.691702 + 0.722183i $$0.256861\pi$$
$$18$$ 42.6749 0.558810
$$19$$ 31.4833 0.380146 0.190073 0.981770i $$-0.439128\pi$$
0.190073 + 0.981770i $$0.439128\pi$$
$$20$$ 65.4166 0.731380
$$21$$ 22.4499 0.233285
$$22$$ −317.216 −3.07413
$$23$$ 183.600 1.66448 0.832242 0.554412i $$-0.187057\pi$$
0.832242 + 0.554412i $$0.187057\pi$$
$$24$$ −92.2250 −0.784389
$$25$$ −104.600 −0.836796
$$26$$ −61.6415 −0.464958
$$27$$ −27.0000 −0.192450
$$28$$ −108.383 −0.731518
$$29$$ 112.200 0.718450 0.359225 0.933251i $$-0.383041\pi$$
0.359225 + 0.933251i $$0.383041\pi$$
$$30$$ −64.2497 −0.391011
$$31$$ −77.2831 −0.447757 −0.223878 0.974617i $$-0.571872\pi$$
−0.223878 + 0.974617i $$0.571872\pi$$
$$32$$ −104.158 −0.575398
$$33$$ 200.700 1.05871
$$34$$ 459.783 2.31918
$$35$$ −33.7998 −0.163234
$$36$$ 130.350 0.603471
$$37$$ 54.7664 0.243339 0.121669 0.992571i $$-0.461175\pi$$
0.121669 + 0.992571i $$0.461175\pi$$
$$38$$ 149.283 0.637287
$$39$$ 39.0000 0.160128
$$40$$ 138.850 0.548854
$$41$$ 451.716 1.72064 0.860319 0.509756i $$-0.170264\pi$$
0.860319 + 0.509756i $$0.170264\pi$$
$$42$$ 106.450 0.391085
$$43$$ −113.434 −0.402291 −0.201145 0.979561i $$-0.564466\pi$$
−0.201145 + 0.979561i $$0.564466\pi$$
$$44$$ −968.932 −3.31982
$$45$$ 40.6502 0.134662
$$46$$ 870.566 2.79039
$$47$$ −42.2670 −0.131176 −0.0655880 0.997847i $$-0.520892\pi$$
−0.0655880 + 0.997847i $$0.520892\pi$$
$$48$$ −89.6997 −0.269730
$$49$$ −287.000 −0.836735
$$50$$ −495.975 −1.40283
$$51$$ −290.900 −0.798708
$$52$$ −188.283 −0.502119
$$53$$ −530.999 −1.37619 −0.688097 0.725618i $$-0.741554\pi$$
−0.688097 + 0.725618i $$0.741554\pi$$
$$54$$ −128.025 −0.322629
$$55$$ −302.166 −0.740800
$$56$$ −230.049 −0.548958
$$57$$ −94.4499 −0.219477
$$58$$ 532.015 1.20443
$$59$$ 219.666 0.484714 0.242357 0.970187i $$-0.422080\pi$$
0.242357 + 0.970187i $$0.422080\pi$$
$$60$$ −196.250 −0.422262
$$61$$ 822.865 1.72717 0.863583 0.504207i $$-0.168215\pi$$
0.863583 + 0.504207i $$0.168215\pi$$
$$62$$ −366.450 −0.750632
$$63$$ −67.3498 −0.134687
$$64$$ −733.082 −1.43180
$$65$$ −58.7169 −0.112045
$$66$$ 951.649 1.77485
$$67$$ −872.082 −1.59018 −0.795088 0.606495i $$-0.792575\pi$$
−0.795088 + 0.606495i $$0.792575\pi$$
$$68$$ 1404.40 2.50453
$$69$$ −550.799 −0.960991
$$70$$ −160.267 −0.273651
$$71$$ −100.299 −0.167653 −0.0838263 0.996480i $$-0.526714\pi$$
−0.0838263 + 0.996480i $$0.526714\pi$$
$$72$$ 276.675 0.452867
$$73$$ −165.634 −0.265562 −0.132781 0.991145i $$-0.542391\pi$$
−0.132781 + 0.991145i $$0.542391\pi$$
$$74$$ 259.684 0.407941
$$75$$ 313.799 0.483125
$$76$$ 455.983 0.688221
$$77$$ 500.633 0.740940
$$78$$ 184.925 0.268443
$$79$$ −545.266 −0.776547 −0.388273 0.921544i $$-0.626928\pi$$
−0.388273 + 0.921544i $$0.626928\pi$$
$$80$$ 135.048 0.188736
$$81$$ 81.0000 0.111111
$$82$$ 2141.88 2.88453
$$83$$ −454.534 −0.601103 −0.300552 0.953766i $$-0.597171\pi$$
−0.300552 + 0.953766i $$0.597171\pi$$
$$84$$ 325.150 0.422342
$$85$$ 437.968 0.558874
$$86$$ −537.864 −0.674412
$$87$$ −336.601 −0.414797
$$88$$ −2056.61 −2.49132
$$89$$ −230.915 −0.275022 −0.137511 0.990500i $$-0.543910\pi$$
−0.137511 + 0.990500i $$0.543910\pi$$
$$90$$ 192.749 0.225751
$$91$$ 97.2831 0.112066
$$92$$ 2659.13 3.01341
$$93$$ 231.849 0.258512
$$94$$ −200.415 −0.219907
$$95$$ 142.200 0.153573
$$96$$ 312.475 0.332206
$$97$$ −1089.16 −1.14008 −0.570041 0.821616i $$-0.693073\pi$$
−0.570041 + 0.821616i $$0.693073\pi$$
$$98$$ −1360.86 −1.40273
$$99$$ −602.099 −0.611245
$$100$$ −1514.95 −1.51495
$$101$$ −77.2336 −0.0760894 −0.0380447 0.999276i $$-0.512113\pi$$
−0.0380447 + 0.999276i $$0.512113\pi$$
$$102$$ −1379.35 −1.33898
$$103$$ 1351.36 1.29275 0.646377 0.763018i $$-0.276283\pi$$
0.646377 + 0.763018i $$0.276283\pi$$
$$104$$ −399.642 −0.376808
$$105$$ 101.399 0.0942434
$$106$$ −2517.81 −2.30709
$$107$$ 1133.67 1.02426 0.512129 0.858908i $$-0.328857\pi$$
0.512129 + 0.858908i $$0.328857\pi$$
$$108$$ −391.049 −0.348414
$$109$$ 1017.66 0.894262 0.447131 0.894469i $$-0.352446\pi$$
0.447131 + 0.894469i $$0.352446\pi$$
$$110$$ −1432.77 −1.24190
$$111$$ −164.299 −0.140492
$$112$$ −223.750 −0.188772
$$113$$ 1570.06 1.30707 0.653536 0.756895i $$-0.273285\pi$$
0.653536 + 0.756895i $$0.273285\pi$$
$$114$$ −447.849 −0.367938
$$115$$ 829.261 0.672426
$$116$$ 1625.03 1.30069
$$117$$ −117.000 −0.0924500
$$118$$ 1041.58 0.812588
$$119$$ −725.632 −0.558979
$$120$$ −416.551 −0.316881
$$121$$ 3144.60 2.36258
$$122$$ 3901.75 2.89547
$$123$$ −1355.15 −0.993411
$$124$$ −1119.32 −0.810625
$$125$$ −1037.03 −0.742037
$$126$$ −319.350 −0.225793
$$127$$ −1248.16 −0.872099 −0.436050 0.899923i $$-0.643623\pi$$
−0.436050 + 0.899923i $$0.643623\pi$$
$$128$$ −2642.76 −1.82491
$$129$$ 340.301 0.232263
$$130$$ −278.415 −0.187836
$$131$$ −1274.80 −0.850227 −0.425113 0.905140i $$-0.639766\pi$$
−0.425113 + 0.905140i $$0.639766\pi$$
$$132$$ 2906.80 1.91670
$$133$$ −235.600 −0.153602
$$134$$ −4135.11 −2.66582
$$135$$ −121.951 −0.0777469
$$136$$ 2980.91 1.87950
$$137$$ 874.915 0.545613 0.272807 0.962069i $$-0.412048\pi$$
0.272807 + 0.962069i $$0.412048\pi$$
$$138$$ −2611.70 −1.61103
$$139$$ 310.334 0.189368 0.0946840 0.995507i $$-0.469816\pi$$
0.0946840 + 0.995507i $$0.469816\pi$$
$$140$$ −489.533 −0.295522
$$141$$ 126.801 0.0757345
$$142$$ −475.585 −0.281058
$$143$$ 869.699 0.508586
$$144$$ 269.099 0.155729
$$145$$ 506.773 0.290243
$$146$$ −785.380 −0.445195
$$147$$ 861.000 0.483089
$$148$$ 793.199 0.440544
$$149$$ 5.08064 0.00279344 0.00139672 0.999999i $$-0.499555\pi$$
0.00139672 + 0.999999i $$0.499555\pi$$
$$150$$ 1487.93 0.809924
$$151$$ 6.54894 0.00352944 0.00176472 0.999998i $$-0.499438\pi$$
0.00176472 + 0.999998i $$0.499438\pi$$
$$152$$ 967.849 0.516467
$$153$$ 872.700 0.461135
$$154$$ 2373.83 1.24213
$$155$$ −349.063 −0.180887
$$156$$ 564.849 0.289898
$$157$$ −2297.60 −1.16795 −0.583975 0.811772i $$-0.698503\pi$$
−0.583975 + 0.811772i $$0.698503\pi$$
$$158$$ −2585.46 −1.30183
$$159$$ 1593.00 0.794546
$$160$$ −470.450 −0.232452
$$161$$ −1373.93 −0.672553
$$162$$ 384.074 0.186270
$$163$$ −1085.49 −0.521606 −0.260803 0.965392i $$-0.583987\pi$$
−0.260803 + 0.965392i $$0.583987\pi$$
$$164$$ 6542.34 3.11507
$$165$$ 906.497 0.427701
$$166$$ −2155.24 −1.00771
$$167$$ −109.066 −0.0505374 −0.0252687 0.999681i $$-0.508044\pi$$
−0.0252687 + 0.999681i $$0.508044\pi$$
$$168$$ 690.148 0.316941
$$169$$ 169.000 0.0769231
$$170$$ 2076.69 0.936912
$$171$$ 283.350 0.126715
$$172$$ −1642.90 −0.728313
$$173$$ −1889.17 −0.830236 −0.415118 0.909768i $$-0.636260\pi$$
−0.415118 + 0.909768i $$0.636260\pi$$
$$174$$ −1596.05 −0.695379
$$175$$ 782.751 0.338117
$$176$$ −2000.30 −0.856694
$$177$$ −658.999 −0.279850
$$178$$ −1094.92 −0.461054
$$179$$ 3427.86 1.43134 0.715672 0.698437i $$-0.246121\pi$$
0.715672 + 0.698437i $$0.246121\pi$$
$$180$$ 588.749 0.243793
$$181$$ 208.403 0.0855826 0.0427913 0.999084i $$-0.486375\pi$$
0.0427913 + 0.999084i $$0.486375\pi$$
$$182$$ 461.283 0.187871
$$183$$ −2468.60 −0.997180
$$184$$ 5644.15 2.26137
$$185$$ 247.363 0.0983052
$$186$$ 1099.35 0.433378
$$187$$ −6487.06 −2.53679
$$188$$ −612.166 −0.237483
$$189$$ 202.049 0.0777616
$$190$$ 674.265 0.257454
$$191$$ 957.735 0.362824 0.181412 0.983407i $$-0.441933\pi$$
0.181412 + 0.983407i $$0.441933\pi$$
$$192$$ 2199.25 0.826650
$$193$$ −512.730 −0.191228 −0.0956142 0.995418i $$-0.530482\pi$$
−0.0956142 + 0.995418i $$0.530482\pi$$
$$194$$ −5164.45 −1.91127
$$195$$ 176.151 0.0646893
$$196$$ −4156.71 −1.51484
$$197$$ −3870.35 −1.39975 −0.699876 0.714265i $$-0.746761\pi$$
−0.699876 + 0.714265i $$0.746761\pi$$
$$198$$ −2854.95 −1.02471
$$199$$ 2305.83 0.821388 0.410694 0.911773i $$-0.365286\pi$$
0.410694 + 0.911773i $$0.365286\pi$$
$$200$$ −3215.56 −1.13687
$$201$$ 2616.25 0.918088
$$202$$ −366.215 −0.127558
$$203$$ −839.630 −0.290298
$$204$$ −4213.19 −1.44599
$$205$$ 2040.26 0.695111
$$206$$ 6407.70 2.16721
$$207$$ 1652.40 0.554828
$$208$$ −388.699 −0.129574
$$209$$ −2106.23 −0.697086
$$210$$ 480.801 0.157992
$$211$$ −3672.40 −1.19819 −0.599096 0.800677i $$-0.704473\pi$$
−0.599096 + 0.800677i $$0.704473\pi$$
$$212$$ −7690.62 −2.49148
$$213$$ 300.898 0.0967942
$$214$$ 5375.46 1.71710
$$215$$ −512.345 −0.162519
$$216$$ −830.025 −0.261463
$$217$$ 578.334 0.180921
$$218$$ 4825.41 1.49917
$$219$$ 496.902 0.153322
$$220$$ −4376.36 −1.34116
$$221$$ −1260.57 −0.383687
$$222$$ −779.051 −0.235525
$$223$$ 5087.05 1.52760 0.763798 0.645455i $$-0.223332\pi$$
0.763798 + 0.645455i $$0.223332\pi$$
$$224$$ 779.449 0.232496
$$225$$ −941.396 −0.278932
$$226$$ 7444.71 2.19122
$$227$$ −2625.83 −0.767763 −0.383882 0.923382i $$-0.625413\pi$$
−0.383882 + 0.923382i $$0.625413\pi$$
$$228$$ −1367.95 −0.397345
$$229$$ 1678.73 0.484425 0.242213 0.970223i $$-0.422127\pi$$
0.242213 + 0.970223i $$0.422127\pi$$
$$230$$ 3932.07 1.12727
$$231$$ −1501.90 −0.427782
$$232$$ 3449.22 0.976088
$$233$$ 648.506 0.182339 0.0911696 0.995835i $$-0.470939\pi$$
0.0911696 + 0.995835i $$0.470939\pi$$
$$234$$ −554.774 −0.154986
$$235$$ −190.907 −0.0529931
$$236$$ 3181.50 0.877533
$$237$$ 1635.80 0.448340
$$238$$ −3440.70 −0.937089
$$239$$ 5219.69 1.41269 0.706347 0.707866i $$-0.250342\pi$$
0.706347 + 0.707866i $$0.250342\pi$$
$$240$$ −405.145 −0.108967
$$241$$ 6103.56 1.63139 0.815695 0.578483i $$-0.196355\pi$$
0.815695 + 0.578483i $$0.196355\pi$$
$$242$$ 14910.6 3.96070
$$243$$ −243.000 −0.0641500
$$244$$ 11917.8 3.12689
$$245$$ −1296.29 −0.338028
$$246$$ −6425.64 −1.66538
$$247$$ −409.283 −0.105433
$$248$$ −2375.81 −0.608323
$$249$$ 1363.60 0.347047
$$250$$ −4917.24 −1.24397
$$251$$ 6423.40 1.61530 0.807652 0.589660i $$-0.200738\pi$$
0.807652 + 0.589660i $$0.200738\pi$$
$$252$$ −975.449 −0.243839
$$253$$ −12282.8 −3.05222
$$254$$ −5918.36 −1.46201
$$255$$ −1313.90 −0.322666
$$256$$ −6666.39 −1.62754
$$257$$ −1230.23 −0.298597 −0.149299 0.988792i $$-0.547702\pi$$
−0.149299 + 0.988792i $$0.547702\pi$$
$$258$$ 1613.59 0.389372
$$259$$ −409.834 −0.0983238
$$260$$ −850.415 −0.202848
$$261$$ 1009.80 0.239483
$$262$$ −6044.66 −1.42534
$$263$$ −514.992 −0.120744 −0.0603722 0.998176i $$-0.519229\pi$$
−0.0603722 + 0.998176i $$0.519229\pi$$
$$264$$ 6169.84 1.43836
$$265$$ −2398.35 −0.555961
$$266$$ −1117.13 −0.257503
$$267$$ 692.745 0.158784
$$268$$ −12630.6 −2.87888
$$269$$ −5132.60 −1.16335 −0.581673 0.813423i $$-0.697602\pi$$
−0.581673 + 0.813423i $$0.697602\pi$$
$$270$$ −578.247 −0.130337
$$271$$ −4300.00 −0.963862 −0.481931 0.876209i $$-0.660064\pi$$
−0.481931 + 0.876209i $$0.660064\pi$$
$$272$$ 2899.29 0.646306
$$273$$ −291.849 −0.0647015
$$274$$ 4148.55 0.914682
$$275$$ 6997.70 1.53446
$$276$$ −7977.39 −1.73979
$$277$$ −1812.80 −0.393215 −0.196607 0.980482i $$-0.562992\pi$$
−0.196607 + 0.980482i $$0.562992\pi$$
$$278$$ 1471.50 0.317462
$$279$$ −695.548 −0.149252
$$280$$ −1039.06 −0.221771
$$281$$ −4073.08 −0.864696 −0.432348 0.901707i $$-0.642315\pi$$
−0.432348 + 0.901707i $$0.642315\pi$$
$$282$$ 601.246 0.126963
$$283$$ 6346.29 1.33303 0.666516 0.745491i $$-0.267785\pi$$
0.666516 + 0.745491i $$0.267785\pi$$
$$284$$ −1452.67 −0.303520
$$285$$ −426.601 −0.0886654
$$286$$ 4123.81 0.852609
$$287$$ −3380.33 −0.695243
$$288$$ −937.424 −0.191799
$$289$$ 4489.53 0.913806
$$290$$ 2402.94 0.486572
$$291$$ 3267.49 0.658226
$$292$$ −2398.93 −0.480777
$$293$$ −8390.97 −1.67306 −0.836529 0.547923i $$-0.815419\pi$$
−0.836529 + 0.547923i $$0.815419\pi$$
$$294$$ 4082.57 0.809864
$$295$$ 992.164 0.195817
$$296$$ 1683.61 0.330601
$$297$$ 1806.30 0.352902
$$298$$ 24.0907 0.00468300
$$299$$ −2386.79 −0.461645
$$300$$ 4544.84 0.874656
$$301$$ 848.861 0.162550
$$302$$ 31.0528 0.00591685
$$303$$ 231.701 0.0439302
$$304$$ 941.348 0.177599
$$305$$ 3716.62 0.697748
$$306$$ 4138.04 0.773059
$$307$$ 4005.27 0.744603 0.372301 0.928112i $$-0.378569\pi$$
0.372301 + 0.928112i $$0.378569\pi$$
$$308$$ 7250.82 1.34141
$$309$$ −4054.09 −0.746372
$$310$$ −1655.14 −0.303244
$$311$$ 5836.53 1.06418 0.532088 0.846689i $$-0.321407\pi$$
0.532088 + 0.846689i $$0.321407\pi$$
$$312$$ 1198.92 0.217550
$$313$$ 1763.19 0.318407 0.159204 0.987246i $$-0.449107\pi$$
0.159204 + 0.987246i $$0.449107\pi$$
$$314$$ −10894.4 −1.95798
$$315$$ −304.198 −0.0544115
$$316$$ −7897.26 −1.40587
$$317$$ −6106.35 −1.08191 −0.540957 0.841050i $$-0.681938\pi$$
−0.540957 + 0.841050i $$0.681938\pi$$
$$318$$ 7553.44 1.33200
$$319$$ −7506.18 −1.31745
$$320$$ −3311.10 −0.578425
$$321$$ −3401.00 −0.591356
$$322$$ −6514.72 −1.12749
$$323$$ 3052.83 0.525895
$$324$$ 1173.15 0.201157
$$325$$ 1359.79 0.232086
$$326$$ −5147.00 −0.874436
$$327$$ −3052.99 −0.516302
$$328$$ 13886.5 2.33766
$$329$$ 316.297 0.0530031
$$330$$ 4298.30 0.717011
$$331$$ 7490.38 1.24383 0.621916 0.783084i $$-0.286354\pi$$
0.621916 + 0.783084i $$0.286354\pi$$
$$332$$ −6583.16 −1.08825
$$333$$ 492.898 0.0811130
$$334$$ −517.152 −0.0847224
$$335$$ −3938.92 −0.642406
$$336$$ 671.251 0.108987
$$337$$ 9462.46 1.52953 0.764767 0.644307i $$-0.222854\pi$$
0.764767 + 0.644307i $$0.222854\pi$$
$$338$$ 801.340 0.128956
$$339$$ −4710.19 −0.754639
$$340$$ 6343.22 1.01179
$$341$$ 5170.23 0.821066
$$342$$ 1343.55 0.212429
$$343$$ 4714.49 0.742153
$$344$$ −3487.14 −0.546553
$$345$$ −2487.78 −0.388226
$$346$$ −8957.79 −1.39183
$$347$$ 11460.3 1.77297 0.886487 0.462753i $$-0.153138\pi$$
0.886487 + 0.462753i $$0.153138\pi$$
$$348$$ −4875.09 −0.750955
$$349$$ −3673.96 −0.563503 −0.281751 0.959487i $$-0.590915\pi$$
−0.281751 + 0.959487i $$0.590915\pi$$
$$350$$ 3711.54 0.566829
$$351$$ 351.000 0.0533761
$$352$$ 6968.17 1.05513
$$353$$ 3388.65 0.510935 0.255467 0.966818i $$-0.417771\pi$$
0.255467 + 0.966818i $$0.417771\pi$$
$$354$$ −3124.75 −0.469148
$$355$$ −453.020 −0.0677290
$$356$$ −3344.41 −0.497903
$$357$$ 2176.90 0.322727
$$358$$ 16253.7 2.39955
$$359$$ −9673.98 −1.42221 −0.711105 0.703086i $$-0.751805\pi$$
−0.711105 + 0.703086i $$0.751805\pi$$
$$360$$ 1249.65 0.182951
$$361$$ −5867.80 −0.855489
$$362$$ 988.174 0.143473
$$363$$ −9433.79 −1.36404
$$364$$ 1408.98 0.202887
$$365$$ −748.117 −0.107283
$$366$$ −11705.2 −1.67170
$$367$$ −8715.98 −1.23970 −0.619851 0.784720i $$-0.712807\pi$$
−0.619851 + 0.784720i $$0.712807\pi$$
$$368$$ 5489.61 0.777624
$$369$$ 4065.44 0.573546
$$370$$ 1172.91 0.164802
$$371$$ 3973.63 0.556067
$$372$$ 3357.95 0.468014
$$373$$ 4667.99 0.647987 0.323994 0.946059i $$-0.394974\pi$$
0.323994 + 0.946059i $$0.394974\pi$$
$$374$$ −30759.4 −4.25276
$$375$$ 3111.09 0.428416
$$376$$ −1299.36 −0.178216
$$377$$ −1458.60 −0.199262
$$378$$ 958.049 0.130362
$$379$$ 10862.7 1.47225 0.736123 0.676848i $$-0.236655\pi$$
0.736123 + 0.676848i $$0.236655\pi$$
$$380$$ 2059.53 0.278031
$$381$$ 3744.49 0.503507
$$382$$ 4541.25 0.608248
$$383$$ 10054.2 1.34137 0.670686 0.741742i $$-0.266000\pi$$
0.670686 + 0.741742i $$0.266000\pi$$
$$384$$ 7928.27 1.05361
$$385$$ 2261.20 0.299329
$$386$$ −2431.19 −0.320581
$$387$$ −1020.90 −0.134097
$$388$$ −15774.7 −2.06402
$$389$$ 6418.50 0.836584 0.418292 0.908313i $$-0.362629\pi$$
0.418292 + 0.908313i $$0.362629\pi$$
$$390$$ 835.246 0.108447
$$391$$ 17803.0 2.30265
$$392$$ −8822.86 −1.13679
$$393$$ 3824.40 0.490879
$$394$$ −18351.9 −2.34658
$$395$$ −2462.79 −0.313713
$$396$$ −8720.39 −1.10661
$$397$$ −12019.9 −1.51955 −0.759775 0.650186i $$-0.774691\pi$$
−0.759775 + 0.650186i $$0.774691\pi$$
$$398$$ 10933.5 1.37700
$$399$$ 706.799 0.0886822
$$400$$ −3127.52 −0.390939
$$401$$ 3599.80 0.448293 0.224147 0.974555i $$-0.428041\pi$$
0.224147 + 0.974555i $$0.428041\pi$$
$$402$$ 12405.3 1.53911
$$403$$ 1004.68 0.124185
$$404$$ −1118.60 −0.137753
$$405$$ 365.852 0.0448872
$$406$$ −3981.24 −0.486664
$$407$$ −3663.87 −0.446219
$$408$$ −8942.74 −1.08513
$$409$$ −48.0968 −0.00581475 −0.00290737 0.999996i $$-0.500925\pi$$
−0.00290737 + 0.999996i $$0.500925\pi$$
$$410$$ 9674.20 1.16530
$$411$$ −2624.74 −0.315010
$$412$$ 19572.2 2.34042
$$413$$ −1643.83 −0.195854
$$414$$ 7835.10 0.930130
$$415$$ −2052.99 −0.242837
$$416$$ 1354.06 0.159587
$$417$$ −931.001 −0.109332
$$418$$ −9987.02 −1.16862
$$419$$ 723.462 0.0843518 0.0421759 0.999110i $$-0.486571\pi$$
0.0421759 + 0.999110i $$0.486571\pi$$
$$420$$ 1468.60 0.170620
$$421$$ −14845.5 −1.71859 −0.859295 0.511481i $$-0.829097\pi$$
−0.859295 + 0.511481i $$0.829097\pi$$
$$422$$ −17413.3 −2.00868
$$423$$ −380.403 −0.0437253
$$424$$ −16323.8 −1.86970
$$425$$ −10142.7 −1.15763
$$426$$ 1426.75 0.162269
$$427$$ −6157.76 −0.697880
$$428$$ 16419.2 1.85433
$$429$$ −2609.10 −0.293632
$$430$$ −2429.36 −0.272452
$$431$$ 1103.11 0.123283 0.0616417 0.998098i $$-0.480366\pi$$
0.0616417 + 0.998098i $$0.480366\pi$$
$$432$$ −807.297 −0.0899099
$$433$$ 8893.53 0.987057 0.493528 0.869730i $$-0.335707\pi$$
0.493528 + 0.869730i $$0.335707\pi$$
$$434$$ 2742.26 0.303301
$$435$$ −1520.32 −0.167572
$$436$$ 14739.1 1.61898
$$437$$ 5780.32 0.632747
$$438$$ 2356.14 0.257034
$$439$$ −10901.7 −1.18521 −0.592607 0.805492i $$-0.701901\pi$$
−0.592607 + 0.805492i $$0.701901\pi$$
$$440$$ −9289.08 −1.00645
$$441$$ −2583.00 −0.278912
$$442$$ −5977.17 −0.643224
$$443$$ −3781.37 −0.405550 −0.202775 0.979225i $$-0.564996\pi$$
−0.202775 + 0.979225i $$0.564996\pi$$
$$444$$ −2379.60 −0.254348
$$445$$ −1042.97 −0.111105
$$446$$ 24121.0 2.56091
$$447$$ −15.2419 −0.00161279
$$448$$ 5485.88 0.578535
$$449$$ 106.834 0.0112289 0.00561447 0.999984i $$-0.498213\pi$$
0.00561447 + 0.999984i $$0.498213\pi$$
$$450$$ −4463.78 −0.467610
$$451$$ −30219.7 −3.15519
$$452$$ 22739.7 2.36634
$$453$$ −19.6468 −0.00203772
$$454$$ −12450.8 −1.28710
$$455$$ 439.397 0.0452731
$$456$$ −2903.55 −0.298182
$$457$$ −1237.64 −0.126684 −0.0633419 0.997992i $$-0.520176\pi$$
−0.0633419 + 0.997992i $$0.520176\pi$$
$$458$$ 7959.95 0.812105
$$459$$ −2618.10 −0.266236
$$460$$ 12010.5 1.21737
$$461$$ 8790.90 0.888141 0.444071 0.895992i $$-0.353534\pi$$
0.444071 + 0.895992i $$0.353534\pi$$
$$462$$ −7121.49 −0.717146
$$463$$ −3861.55 −0.387606 −0.193803 0.981040i $$-0.562082\pi$$
−0.193803 + 0.981040i $$0.562082\pi$$
$$464$$ 3354.77 0.335650
$$465$$ 1047.19 0.104435
$$466$$ 3074.99 0.305679
$$467$$ −8991.38 −0.890945 −0.445473 0.895296i $$-0.646964\pi$$
−0.445473 + 0.895296i $$0.646964\pi$$
$$468$$ −1694.55 −0.167373
$$469$$ 6526.06 0.642528
$$470$$ −905.214 −0.0888391
$$471$$ 6892.79 0.674316
$$472$$ 6752.91 0.658533
$$473$$ 7588.71 0.737694
$$474$$ 7756.39 0.751609
$$475$$ −3293.14 −0.318105
$$476$$ −10509.6 −1.01198
$$477$$ −4778.99 −0.458731
$$478$$ 24750.0 2.36828
$$479$$ 4179.82 0.398707 0.199354 0.979928i $$-0.436116\pi$$
0.199354 + 0.979928i $$0.436116\pi$$
$$480$$ 1411.35 0.134206
$$481$$ −711.963 −0.0674901
$$482$$ 28941.0 2.73491
$$483$$ 4121.80 0.388299
$$484$$ 45544.2 4.27725
$$485$$ −4919.41 −0.460575
$$486$$ −1152.22 −0.107543
$$487$$ −18443.8 −1.71616 −0.858078 0.513519i $$-0.828342\pi$$
−0.858078 + 0.513519i $$0.828342\pi$$
$$488$$ 25296.2 2.34653
$$489$$ 3256.46 0.301149
$$490$$ −6146.56 −0.566680
$$491$$ 8093.26 0.743877 0.371939 0.928257i $$-0.378693\pi$$
0.371939 + 0.928257i $$0.378693\pi$$
$$492$$ −19627.0 −1.79849
$$493$$ 10879.7 0.993907
$$494$$ −1940.68 −0.176752
$$495$$ −2719.49 −0.246933
$$496$$ −2310.76 −0.209185
$$497$$ 750.571 0.0677418
$$498$$ 6465.73 0.581800
$$499$$ −10941.6 −0.981591 −0.490796 0.871275i $$-0.663294\pi$$
−0.490796 + 0.871275i $$0.663294\pi$$
$$500$$ −15019.6 −1.34340
$$501$$ 327.197 0.0291778
$$502$$ 30457.5 2.70794
$$503$$ −9260.11 −0.820851 −0.410425 0.911894i $$-0.634620\pi$$
−0.410425 + 0.911894i $$0.634620\pi$$
$$504$$ −2070.45 −0.182986
$$505$$ −348.840 −0.0307389
$$506$$ −58240.8 −5.11684
$$507$$ −507.000 −0.0444116
$$508$$ −18077.5 −1.57886
$$509$$ −9996.40 −0.870497 −0.435248 0.900310i $$-0.643339\pi$$
−0.435248 + 0.900310i $$0.643339\pi$$
$$510$$ −6230.08 −0.540927
$$511$$ 1239.49 0.107303
$$512$$ −10467.7 −0.903538
$$513$$ −850.049 −0.0731591
$$514$$ −5833.32 −0.500577
$$515$$ 6103.68 0.522253
$$516$$ 4928.69 0.420491
$$517$$ 2827.66 0.240542
$$518$$ −1943.29 −0.164833
$$519$$ 5667.51 0.479337
$$520$$ −1805.06 −0.152225
$$521$$ 11427.9 0.960973 0.480486 0.877002i $$-0.340460\pi$$
0.480486 + 0.877002i $$0.340460\pi$$
$$522$$ 4788.14 0.401477
$$523$$ −4810.47 −0.402193 −0.201097 0.979571i $$-0.564451\pi$$
−0.201097 + 0.979571i $$0.564451\pi$$
$$524$$ −18463.3 −1.53926
$$525$$ −2348.25 −0.195212
$$526$$ −2441.92 −0.202419
$$527$$ −7493.88 −0.619428
$$528$$ 6000.90 0.494613
$$529$$ 21541.8 1.77051
$$530$$ −11372.2 −0.932030
$$531$$ 1977.00 0.161571
$$532$$ −3412.26 −0.278083
$$533$$ −5872.31 −0.477219
$$534$$ 3284.76 0.266190
$$535$$ 5120.41 0.413785
$$536$$ −26809.2 −2.16042
$$537$$ −10283.6 −0.826386
$$538$$ −24337.0 −1.95027
$$539$$ 19200.3 1.53435
$$540$$ −1766.25 −0.140754
$$541$$ 2411.99 0.191681 0.0958406 0.995397i $$-0.469446\pi$$
0.0958406 + 0.995397i $$0.469446\pi$$
$$542$$ −20389.1 −1.61585
$$543$$ −625.208 −0.0494111
$$544$$ −10099.9 −0.796008
$$545$$ 4596.47 0.361268
$$546$$ −1383.85 −0.108468
$$547$$ −4396.34 −0.343646 −0.171823 0.985128i $$-0.554966\pi$$
−0.171823 + 0.985128i $$0.554966\pi$$
$$548$$ 12671.7 0.987786
$$549$$ 7405.79 0.575722
$$550$$ 33180.7 2.57242
$$551$$ 3532.43 0.273116
$$552$$ −16932.5 −1.30560
$$553$$ 4080.40 0.313772
$$554$$ −8595.67 −0.659197
$$555$$ −742.088 −0.0567565
$$556$$ 4494.66 0.342835
$$557$$ −17488.0 −1.33032 −0.665160 0.746701i $$-0.731637\pi$$
−0.665160 + 0.746701i $$0.731637\pi$$
$$558$$ −3298.05 −0.250211
$$559$$ 1474.64 0.111575
$$560$$ −1010.61 −0.0762608
$$561$$ 19461.2 1.46462
$$562$$ −19313.2 −1.44960
$$563$$ −6881.77 −0.515154 −0.257577 0.966258i $$-0.582924\pi$$
−0.257577 + 0.966258i $$0.582924\pi$$
$$564$$ 1836.50 0.137111
$$565$$ 7091.49 0.528037
$$566$$ 30092.0 2.23473
$$567$$ −606.148 −0.0448957
$$568$$ −3083.36 −0.227773
$$569$$ 14733.5 1.08552 0.542758 0.839889i $$-0.317380\pi$$
0.542758 + 0.839889i $$0.317380\pi$$
$$570$$ −2022.79 −0.148641
$$571$$ 4488.51 0.328964 0.164482 0.986380i $$-0.447405\pi$$
0.164482 + 0.986380i $$0.447405\pi$$
$$572$$ 12596.1 0.920752
$$573$$ −2873.21 −0.209476
$$574$$ −16028.4 −1.16553
$$575$$ −19204.4 −1.39284
$$576$$ −6597.74 −0.477267
$$577$$ −10552.2 −0.761338 −0.380669 0.924711i $$-0.624306\pi$$
−0.380669 + 0.924711i $$0.624306\pi$$
$$578$$ 21287.8 1.53193
$$579$$ 1538.19 0.110406
$$580$$ 7339.75 0.525460
$$581$$ 3401.42 0.242882
$$582$$ 15493.3 1.10347
$$583$$ 35523.8 2.52357
$$584$$ −5091.86 −0.360793
$$585$$ −528.452 −0.0373484
$$586$$ −39787.1 −2.80476
$$587$$ −1637.20 −0.115118 −0.0575591 0.998342i $$-0.518332\pi$$
−0.0575591 + 0.998342i $$0.518332\pi$$
$$588$$ 12470.1 0.874591
$$589$$ −2433.13 −0.170213
$$590$$ 4704.50 0.328273
$$591$$ 11611.1 0.808147
$$592$$ 1637.51 0.113685
$$593$$ 14024.0 0.971155 0.485577 0.874194i $$-0.338609\pi$$
0.485577 + 0.874194i $$0.338609\pi$$
$$594$$ 8564.84 0.591616
$$595$$ −3277.45 −0.225819
$$596$$ 73.5845 0.00505728
$$597$$ −6917.50 −0.474229
$$598$$ −11317.4 −0.773915
$$599$$ 365.860 0.0249560 0.0124780 0.999922i $$-0.496028\pi$$
0.0124780 + 0.999922i $$0.496028\pi$$
$$600$$ 9646.69 0.656374
$$601$$ −10128.9 −0.687465 −0.343733 0.939068i $$-0.611691\pi$$
−0.343733 + 0.939068i $$0.611691\pi$$
$$602$$ 4025.01 0.272503
$$603$$ −7848.74 −0.530058
$$604$$ 94.8504 0.00638975
$$605$$ 14203.1 0.954446
$$606$$ 1098.65 0.0736459
$$607$$ 2324.97 0.155465 0.0777327 0.996974i $$-0.475232\pi$$
0.0777327 + 0.996974i $$0.475232\pi$$
$$608$$ −3279.25 −0.218735
$$609$$ 2518.89 0.167603
$$610$$ 17623.0 1.16973
$$611$$ 549.471 0.0363817
$$612$$ 12639.6 0.834845
$$613$$ 633.133 0.0417162 0.0208581 0.999782i $$-0.493360\pi$$
0.0208581 + 0.999782i $$0.493360\pi$$
$$614$$ 18991.6 1.24827
$$615$$ −6120.77 −0.401323
$$616$$ 15390.3 1.00664
$$617$$ −2981.85 −0.194562 −0.0972810 0.995257i $$-0.531015\pi$$
−0.0972810 + 0.995257i $$0.531015\pi$$
$$618$$ −19223.1 −1.25124
$$619$$ 15158.6 0.984292 0.492146 0.870513i $$-0.336213\pi$$
0.492146 + 0.870513i $$0.336213\pi$$
$$620$$ −5055.60 −0.327480
$$621$$ −4957.19 −0.320330
$$622$$ 27674.8 1.78402
$$623$$ 1728.01 0.111126
$$624$$ 1166.10 0.0748096
$$625$$ 8391.01 0.537025
$$626$$ 8360.45 0.533787
$$627$$ 6318.69 0.402463
$$628$$ −33276.8 −2.11447
$$629$$ 5310.51 0.336636
$$630$$ −1442.40 −0.0912170
$$631$$ 5562.30 0.350922 0.175461 0.984486i $$-0.443858\pi$$
0.175461 + 0.984486i $$0.443858\pi$$
$$632$$ −16762.4 −1.05502
$$633$$ 11017.2 0.691776
$$634$$ −28954.2 −1.81375
$$635$$ −5637.56 −0.352315
$$636$$ 23071.9 1.43846
$$637$$ 3731.00 0.232068
$$638$$ −35591.7 −2.20861
$$639$$ −902.693 −0.0558842
$$640$$ −11936.5 −0.737237
$$641$$ −24140.7 −1.48752 −0.743761 0.668446i $$-0.766960\pi$$
−0.743761 + 0.668446i $$0.766960\pi$$
$$642$$ −16126.4 −0.991366
$$643$$ −1749.69 −0.107311 −0.0536555 0.998560i $$-0.517087\pi$$
−0.0536555 + 0.998560i $$0.517087\pi$$
$$644$$ −19899.1 −1.21760
$$645$$ 1537.03 0.0938305
$$646$$ 14475.5 0.881626
$$647$$ 1489.51 0.0905083 0.0452542 0.998976i $$-0.485590\pi$$
0.0452542 + 0.998976i $$0.485590\pi$$
$$648$$ 2490.07 0.150956
$$649$$ −14695.7 −0.888836
$$650$$ 6447.68 0.389075
$$651$$ −1735.00 −0.104455
$$652$$ −15721.4 −0.944323
$$653$$ 10668.2 0.639327 0.319663 0.947531i $$-0.396430\pi$$
0.319663 + 0.947531i $$0.396430\pi$$
$$654$$ −14476.2 −0.865544
$$655$$ −5757.86 −0.343478
$$656$$ 13506.3 0.803858
$$657$$ −1490.71 −0.0885205
$$658$$ 1499.77 0.0888559
$$659$$ −13933.1 −0.823608 −0.411804 0.911272i $$-0.635101\pi$$
−0.411804 + 0.911272i $$0.635101\pi$$
$$660$$ 13129.1 0.774317
$$661$$ −2349.69 −0.138264 −0.0691320 0.997608i $$-0.522023\pi$$
−0.0691320 + 0.997608i $$0.522023\pi$$
$$662$$ 35516.8 2.08520
$$663$$ 3781.70 0.221522
$$664$$ −13973.1 −0.816660
$$665$$ −1064.13 −0.0620529
$$666$$ 2337.15 0.135980
$$667$$ 20599.9 1.19585
$$668$$ −1579.63 −0.0914937
$$669$$ −15261.1 −0.881958
$$670$$ −18677.0 −1.07695
$$671$$ −55049.6 −3.16716
$$672$$ −2338.35 −0.134232
$$673$$ −32421.0 −1.85697 −0.928483 0.371374i $$-0.878887\pi$$
−0.928483 + 0.371374i $$0.878887\pi$$
$$674$$ 44867.7 2.56415
$$675$$ 2824.19 0.161042
$$676$$ 2447.68 0.139263
$$677$$ −1071.74 −0.0608421 −0.0304211 0.999537i $$-0.509685\pi$$
−0.0304211 + 0.999537i $$0.509685\pi$$
$$678$$ −22334.1 −1.26510
$$679$$ 8150.56 0.460663
$$680$$ 13463.9 0.759287
$$681$$ 7877.48 0.443268
$$682$$ 24515.5 1.37646
$$683$$ −305.487 −0.0171144 −0.00855721 0.999963i $$-0.502724\pi$$
−0.00855721 + 0.999963i $$0.502724\pi$$
$$684$$ 4103.84 0.229407
$$685$$ 3951.72 0.220419
$$686$$ 22354.5 1.24417
$$687$$ −5036.18 −0.279683
$$688$$ −3391.66 −0.187944
$$689$$ 6902.99 0.381688
$$690$$ −11796.2 −0.650833
$$691$$ 2180.81 0.120061 0.0600303 0.998197i $$-0.480880\pi$$
0.0600303 + 0.998197i $$0.480880\pi$$
$$692$$ −27361.4 −1.50307
$$693$$ 4505.70 0.246980
$$694$$ 54340.9 2.97227
$$695$$ 1401.68 0.0765018
$$696$$ −10347.7 −0.563545
$$697$$ 43801.4 2.38034
$$698$$ −17420.6 −0.944672
$$699$$ −1945.52 −0.105274
$$700$$ 11336.8 0.612132
$$701$$ −15168.3 −0.817259 −0.408629 0.912700i $$-0.633993\pi$$
−0.408629 + 0.912700i $$0.633993\pi$$
$$702$$ 1664.32 0.0894812
$$703$$ 1724.23 0.0925043
$$704$$ 49043.1 2.62554
$$705$$ 572.720 0.0305956
$$706$$ 16067.8 0.856545
$$707$$ 577.963 0.0307448
$$708$$ −9544.49 −0.506644
$$709$$ −8988.17 −0.476104 −0.238052 0.971252i $$-0.576509\pi$$
−0.238052 + 0.971252i $$0.576509\pi$$
$$710$$ −2148.07 −0.113543
$$711$$ −4907.39 −0.258849
$$712$$ −7098.71 −0.373645
$$713$$ −14189.1 −0.745284
$$714$$ 10322.1 0.541029
$$715$$ 3928.15 0.205461
$$716$$ 49646.8 2.59132
$$717$$ −15659.1 −0.815619
$$718$$ −45870.7 −2.38423
$$719$$ 8448.18 0.438198 0.219099 0.975703i $$-0.429688\pi$$
0.219099 + 0.975703i $$0.429688\pi$$
$$720$$ 1215.44 0.0629120
$$721$$ −10112.7 −0.522352
$$722$$ −27823.1 −1.43417
$$723$$ −18310.7 −0.941883
$$724$$ 3018.36 0.154940
$$725$$ −11736.1 −0.601197
$$726$$ −44731.8 −2.28671
$$727$$ −8624.18 −0.439963 −0.219982 0.975504i $$-0.570600\pi$$
−0.219982 + 0.975504i $$0.570600\pi$$
$$728$$ 2990.64 0.152254
$$729$$ 729.000 0.0370370
$$730$$ −3547.31 −0.179852
$$731$$ −10999.3 −0.556530
$$732$$ −35753.5 −1.80531
$$733$$ 31124.2 1.56835 0.784174 0.620541i $$-0.213087\pi$$
0.784174 + 0.620541i $$0.213087\pi$$
$$734$$ −41328.2 −2.07827
$$735$$ 3888.87 0.195161
$$736$$ −19123.4 −0.957742
$$737$$ 58342.2 2.91596
$$738$$ 19276.9 0.961509
$$739$$ −17671.1 −0.879626 −0.439813 0.898089i $$-0.644955\pi$$
−0.439813 + 0.898089i $$0.644955\pi$$
$$740$$ 3582.63 0.177973
$$741$$ 1227.85 0.0608720
$$742$$ 18841.6 0.932206
$$743$$ −21331.1 −1.05325 −0.526623 0.850099i $$-0.676542\pi$$
−0.526623 + 0.850099i $$0.676542\pi$$
$$744$$ 7127.43 0.351215
$$745$$ 22.9477 0.00112851
$$746$$ 22134.0 1.08630
$$747$$ −4090.81 −0.200368
$$748$$ −93954.1 −4.59265
$$749$$ −8483.58 −0.413863
$$750$$ 14751.7 0.718208
$$751$$ 11712.9 0.569122 0.284561 0.958658i $$-0.408152\pi$$
0.284561 + 0.958658i $$0.408152\pi$$
$$752$$ −1263.78 −0.0612835
$$753$$ −19270.2 −0.932596
$$754$$ −6916.20 −0.334049
$$755$$ 29.5795 0.00142584
$$756$$ 2926.35 0.140781
$$757$$ −16610.9 −0.797537 −0.398768 0.917052i $$-0.630562\pi$$
−0.398768 + 0.917052i $$0.630562\pi$$
$$758$$ 51507.4 2.46812
$$759$$ 36848.4 1.76220
$$760$$ 4371.47 0.208645
$$761$$ −29365.5 −1.39882 −0.699408 0.714723i $$-0.746553\pi$$
−0.699408 + 0.714723i $$0.746553\pi$$
$$762$$ 17755.1 0.844093
$$763$$ −7615.50 −0.361336
$$764$$ 13871.2 0.656861
$$765$$ 3941.71 0.186291
$$766$$ 47673.5 2.24871
$$767$$ −2855.66 −0.134435
$$768$$ 19999.2 0.939659
$$769$$ −28599.9 −1.34114 −0.670572 0.741844i $$-0.733951\pi$$
−0.670572 + 0.741844i $$0.733951\pi$$
$$770$$ 10721.8 0.501803
$$771$$ 3690.68 0.172395
$$772$$ −7426.03 −0.346203
$$773$$ 13491.8 0.627772 0.313886 0.949461i $$-0.398369\pi$$
0.313886 + 0.949461i $$0.398369\pi$$
$$774$$ −4840.78 −0.224804
$$775$$ 8083.78 0.374681
$$776$$ −33482.7 −1.54892
$$777$$ 1229.50 0.0567673
$$778$$ 30434.3 1.40247
$$779$$ 14221.5 0.654093
$$780$$ 2551.25 0.117114
$$781$$ 6710.01 0.307430
$$782$$ 84415.9 3.86024
$$783$$ −3029.41 −0.138266
$$784$$ −8581.27 −0.390911
$$785$$ −10377.5 −0.471834
$$786$$ 18134.0 0.822923
$$787$$ 25876.7 1.17205 0.586025 0.810293i $$-0.300692\pi$$
0.586025 + 0.810293i $$0.300692\pi$$
$$788$$ −56055.5 −2.53413
$$789$$ 1544.98 0.0697118
$$790$$ −11677.7 −0.525918
$$791$$ −11749.3 −0.528137
$$792$$ −18509.5 −0.830438
$$793$$ −10697.3 −0.479030
$$794$$ −56994.2 −2.54742
$$795$$ 7195.06 0.320984
$$796$$ 33396.1 1.48705
$$797$$ 21936.4 0.974938 0.487469 0.873140i $$-0.337920\pi$$
0.487469 + 0.873140i $$0.337920\pi$$
$$798$$ 3351.40 0.148669
$$799$$ −4098.49 −0.181469
$$800$$ 10894.9 0.481491
$$801$$ −2078.23 −0.0916739
$$802$$ 17069.0 0.751531
$$803$$ 11080.9 0.486969
$$804$$ 37891.9 1.66212
$$805$$ −6205.62 −0.271701
$$806$$ 4763.85 0.208188
$$807$$ 15397.8 0.671658
$$808$$ −2374.29 −0.103375
$$809$$ 5583.23 0.242640 0.121320 0.992613i $$-0.461287\pi$$
0.121320 + 0.992613i $$0.461287\pi$$
$$810$$ 1734.74 0.0752502
$$811$$ 12925.4 0.559647 0.279823 0.960052i $$-0.409724\pi$$
0.279823 + 0.960052i $$0.409724\pi$$
$$812$$ −12160.6 −0.525559
$$813$$ 12900.0 0.556486
$$814$$ −17372.8 −0.748054
$$815$$ −4902.80 −0.210721
$$816$$ −8697.87 −0.373145
$$817$$ −3571.27 −0.152929
$$818$$ −228.058 −0.00974801
$$819$$ 875.548 0.0373555
$$820$$ 29549.7 1.25844
$$821$$ −10153.2 −0.431608 −0.215804 0.976437i $$-0.569237\pi$$
−0.215804 + 0.976437i $$0.569237\pi$$
$$822$$ −12445.6 −0.528092
$$823$$ 3282.93 0.139047 0.0695235 0.997580i $$-0.477852\pi$$
0.0695235 + 0.997580i $$0.477852\pi$$
$$824$$ 41543.1 1.75634
$$825$$ −20993.1 −0.885922
$$826$$ −7794.49 −0.328335
$$827$$ −17689.3 −0.743795 −0.371897 0.928274i $$-0.621293\pi$$
−0.371897 + 0.928274i $$0.621293\pi$$
$$828$$ 23932.2 1.00447
$$829$$ 38181.5 1.59964 0.799818 0.600243i $$-0.204930\pi$$
0.799818 + 0.600243i $$0.204930\pi$$
$$830$$ −9734.56 −0.407098
$$831$$ 5438.40 0.227023
$$832$$ 9530.06 0.397110
$$833$$ −27829.4 −1.15754
$$834$$ −4414.49 −0.183287
$$835$$ −492.615 −0.0204163
$$836$$ −30505.2 −1.26201
$$837$$ 2086.64 0.0861708
$$838$$ 3430.41 0.141410
$$839$$ 43895.2 1.80623 0.903117 0.429395i $$-0.141273\pi$$
0.903117 + 0.429395i $$0.141273\pi$$
$$840$$ 3117.18 0.128039
$$841$$ −11800.1 −0.483829
$$842$$ −70392.4 −2.88109
$$843$$ 12219.2 0.499233
$$844$$ −53188.5 −2.16922
$$845$$ 763.320 0.0310757
$$846$$ −1803.74 −0.0733024
$$847$$ −23532.0 −0.954627
$$848$$ −15876.8 −0.642938
$$849$$ −19038.9 −0.769626
$$850$$ −48093.0 −1.94068
$$851$$ 10055.1 0.405034
$$852$$ 4358.00 0.175238
$$853$$ −19955.2 −0.800998 −0.400499 0.916297i $$-0.631163\pi$$
−0.400499 + 0.916297i $$0.631163\pi$$
$$854$$ −29198.0 −1.16995
$$855$$ 1279.80 0.0511910
$$856$$ 34850.8 1.39156
$$857$$ 26030.4 1.03755 0.518776 0.854910i $$-0.326388\pi$$
0.518776 + 0.854910i $$0.326388\pi$$
$$858$$ −12371.4 −0.492254
$$859$$ −45617.4 −1.81193 −0.905964 0.423354i $$-0.860853\pi$$
−0.905964 + 0.423354i $$0.860853\pi$$
$$860$$ −7420.45 −0.294227
$$861$$ 10141.0 0.401399
$$862$$ 5230.59 0.206676
$$863$$ −2010.93 −0.0793195 −0.0396597 0.999213i $$-0.512627\pi$$
−0.0396597 + 0.999213i $$0.512627\pi$$
$$864$$ 2812.27 0.110735
$$865$$ −8532.78 −0.335403
$$866$$ 42170.1 1.65473
$$867$$ −13468.6 −0.527586
$$868$$ 8376.19 0.327542
$$869$$ 36478.2 1.42398
$$870$$ −7208.83 −0.280922
$$871$$ 11337.1 0.441035
$$872$$ 31284.7 1.21495
$$873$$ −9802.48 −0.380027
$$874$$ 27408.3 1.06076
$$875$$ 7760.41 0.299828
$$876$$ 7196.79 0.277576
$$877$$ −36767.3 −1.41567 −0.707836 0.706376i $$-0.750329\pi$$
−0.707836 + 0.706376i $$0.750329\pi$$
$$878$$ −51692.0 −1.98693
$$879$$ 25172.9 0.965940
$$880$$ −9034.72 −0.346091
$$881$$ 35401.2 1.35380 0.676899 0.736076i $$-0.263323\pi$$
0.676899 + 0.736076i $$0.263323\pi$$
$$882$$ −12247.7 −0.467575
$$883$$ −11928.0 −0.454596 −0.227298 0.973825i $$-0.572989\pi$$
−0.227298 + 0.973825i $$0.572989\pi$$
$$884$$ −18257.2 −0.694633
$$885$$ −2976.49 −0.113055
$$886$$ −17930.0 −0.679875
$$887$$ −32939.3 −1.24689 −0.623447 0.781866i $$-0.714268\pi$$
−0.623447 + 0.781866i $$0.714268\pi$$
$$888$$ −5050.83 −0.190872
$$889$$ 9340.40 0.352381
$$890$$ −4945.41 −0.186259
$$891$$ −5418.89 −0.203748
$$892$$ 73677.3 2.76558
$$893$$ −1330.70 −0.0498660
$$894$$ −72.2720 −0.00270373
$$895$$ 15482.6 0.578241
$$896$$ 19776.6 0.737376
$$897$$ 7160.38 0.266531
$$898$$ 506.569 0.0188245
$$899$$ −8671.18 −0.321691
$$900$$ −13634.5 −0.504983
$$901$$ −51489.2 −1.90383
$$902$$ −143292. −5.28946
$$903$$ −2546.58 −0.0938483
$$904$$ 48266.4 1.77579
$$905$$ 941.289 0.0345740
$$906$$ −93.1585 −0.00341610
$$907$$ 1500.89 0.0549461 0.0274731 0.999623i $$-0.491254\pi$$
0.0274731 + 0.999623i $$0.491254\pi$$
$$908$$ −38030.7 −1.38997
$$909$$ −695.102 −0.0253631
$$910$$ 2083.47 0.0758971
$$911$$ 19728.2 0.717481 0.358740 0.933437i $$-0.383206\pi$$
0.358740 + 0.933437i $$0.383206\pi$$
$$912$$ −2824.04 −0.102537
$$913$$ 30408.3 1.10226
$$914$$ −5868.48 −0.212376
$$915$$ −11149.9 −0.402845
$$916$$ 24313.5 0.877011
$$917$$ 9539.72 0.343543
$$918$$ −12414.1 −0.446326
$$919$$ 19992.0 0.717602 0.358801 0.933414i $$-0.383186\pi$$
0.358801 + 0.933414i $$0.383186\pi$$
$$920$$ 25492.9 0.913560
$$921$$ −12015.8 −0.429897
$$922$$ 41683.4 1.48891
$$923$$ 1303.89 0.0464984
$$924$$ −21752.5 −0.774463
$$925$$ −5728.54 −0.203625
$$926$$ −18310.2 −0.649794
$$927$$ 12162.3 0.430918
$$928$$ −11686.6 −0.413395
$$929$$ −1923.62 −0.0679354 −0.0339677 0.999423i $$-0.510814\pi$$
−0.0339677 + 0.999423i $$0.510814\pi$$
$$930$$ 4965.42 0.175078
$$931$$ −9035.71 −0.318081
$$932$$ 9392.52 0.330110
$$933$$ −17509.6 −0.614403
$$934$$ −42634.0 −1.49361
$$935$$ −29300.0 −1.02483
$$936$$ −3596.77 −0.125603
$$937$$ −10252.9 −0.357468 −0.178734 0.983897i $$-0.557200\pi$$
−0.178734 + 0.983897i $$0.557200\pi$$
$$938$$ 30944.4 1.07715
$$939$$ −5289.58 −0.183833
$$940$$ −2764.96 −0.0959394
$$941$$ 11043.0 0.382563 0.191282 0.981535i $$-0.438736\pi$$
0.191282 + 0.981535i $$0.438736\pi$$
$$942$$ 32683.2 1.13044
$$943$$ 82934.8 2.86398
$$944$$ 6568.00 0.226451
$$945$$ 912.594 0.0314145
$$946$$ 35983.1 1.23669
$$947$$ 32105.2 1.10167 0.550833 0.834615i $$-0.314310\pi$$
0.550833 + 0.834615i $$0.314310\pi$$
$$948$$ 23691.8 0.811680
$$949$$ 2153.24 0.0736535
$$950$$ −15614.9 −0.533280
$$951$$ 18319.0 0.624643
$$952$$ −22307.1 −0.759431
$$953$$ −9473.37 −0.322007 −0.161003 0.986954i $$-0.551473\pi$$
−0.161003 + 0.986954i $$0.551473\pi$$
$$954$$ −22660.3 −0.769031
$$955$$ 4325.79 0.146575
$$956$$ 75598.4 2.55756
$$957$$ 22518.5 0.760628
$$958$$ 19819.3 0.668404
$$959$$ −6547.26 −0.220461
$$960$$ 9933.30 0.333954
$$961$$ −23818.3 −0.799514
$$962$$ −3375.89 −0.113142
$$963$$ 10203.0 0.341419
$$964$$ 88399.8 2.95349
$$965$$ −2315.84 −0.0772534
$$966$$ 19544.2 0.650956
$$967$$ −5310.75 −0.176610 −0.0883052 0.996093i $$-0.528145\pi$$
−0.0883052 + 0.996093i $$0.528145\pi$$
$$968$$ 96670.1 3.20981
$$969$$ −9158.49 −0.303626
$$970$$ −23326.2 −0.772122
$$971$$ 24271.2 0.802164 0.401082 0.916042i $$-0.368634\pi$$
0.401082 + 0.916042i $$0.368634\pi$$
$$972$$ −3519.45 −0.116138
$$973$$ −2322.32 −0.0765163
$$974$$ −87454.2 −2.87702
$$975$$ −4079.38 −0.133995
$$976$$ 24603.6 0.806907
$$977$$ −49602.5 −1.62428 −0.812142 0.583460i $$-0.801699\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$978$$ 15441.0 0.504856
$$979$$ 15448.2 0.504317
$$980$$ −18774.6 −0.611971
$$981$$ 9158.98 0.298087
$$982$$ 38375.5 1.24706
$$983$$ 47385.7 1.53751 0.768753 0.639545i $$-0.220877\pi$$
0.768753 + 0.639545i $$0.220877\pi$$
$$984$$ −41659.5 −1.34965
$$985$$ −17481.2 −0.565478
$$986$$ 51587.7 1.66621
$$987$$ −948.891 −0.0306014
$$988$$ −5927.78 −0.190878
$$989$$ −20826.4 −0.669607
$$990$$ −12894.9 −0.413966
$$991$$ −8947.33 −0.286802 −0.143401 0.989665i $$-0.545804\pi$$
−0.143401 + 0.989665i $$0.545804\pi$$
$$992$$ 8049.67 0.257638
$$993$$ −22471.1 −0.718127
$$994$$ 3558.95 0.113564
$$995$$ 10414.7 0.331828
$$996$$ 19749.5 0.628299
$$997$$ −14908.6 −0.473582 −0.236791 0.971561i $$-0.576096\pi$$
−0.236791 + 0.971561i $$0.576096\pi$$
$$998$$ −51881.4 −1.64557
$$999$$ −1478.69 −0.0468306
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.b.1.2 2
3.2 odd 2 117.4.a.c.1.1 2
4.3 odd 2 624.4.a.r.1.1 2
5.4 even 2 975.4.a.j.1.1 2
7.6 odd 2 1911.4.a.h.1.2 2
8.3 odd 2 2496.4.a.s.1.2 2
8.5 even 2 2496.4.a.bc.1.2 2
12.11 even 2 1872.4.a.t.1.2 2
13.5 odd 4 507.4.b.f.337.1 4
13.8 odd 4 507.4.b.f.337.4 4
13.12 even 2 507.4.a.f.1.1 2
39.38 odd 2 1521.4.a.s.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.2 2 1.1 even 1 trivial
117.4.a.c.1.1 2 3.2 odd 2
507.4.a.f.1.1 2 13.12 even 2
507.4.b.f.337.1 4 13.5 odd 4
507.4.b.f.337.4 4 13.8 odd 4
624.4.a.r.1.1 2 4.3 odd 2
975.4.a.j.1.1 2 5.4 even 2
1521.4.a.s.1.2 2 39.38 odd 2
1872.4.a.t.1.2 2 12.11 even 2
1911.4.a.h.1.2 2 7.6 odd 2
2496.4.a.s.1.2 2 8.3 odd 2
2496.4.a.bc.1.2 2 8.5 even 2