# Properties

 Label 289.4.a.g.1.5 Level $289$ Weight $4$ Character 289.1 Self dual yes Analytic conductor $17.052$ Analytic rank $1$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(1,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("289.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 289.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: $$17.0515519917$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + 38476 x^{4} - 32336 x^{3} - 57024 x^{2} - 15776 x + 1156$$ x^12 - 4*x^11 - 58*x^10 + 204*x^9 + 1191*x^8 - 3456*x^7 - 10364*x^6 + 21448*x^5 + 38476*x^4 - 32336*x^3 - 57024*x^2 - 15776*x + 1156 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$-1.47083$$ of defining polynomial Character $$\chi$$ $$=$$ 289.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.70547 q^{2} -4.44379 q^{3} -5.09138 q^{4} +6.80983 q^{5} +7.57875 q^{6} -13.8760 q^{7} +22.3269 q^{8} -7.25269 q^{9} +O(q^{10})$$ $$q-1.70547 q^{2} -4.44379 q^{3} -5.09138 q^{4} +6.80983 q^{5} +7.57875 q^{6} -13.8760 q^{7} +22.3269 q^{8} -7.25269 q^{9} -11.6140 q^{10} +30.8361 q^{11} +22.6250 q^{12} +66.0130 q^{13} +23.6651 q^{14} -30.2615 q^{15} +2.65318 q^{16} +12.3692 q^{18} -79.7150 q^{19} -34.6714 q^{20} +61.6622 q^{21} -52.5899 q^{22} +28.8986 q^{23} -99.2163 q^{24} -78.6262 q^{25} -112.583 q^{26} +152.212 q^{27} +70.6481 q^{28} +266.201 q^{29} +51.6100 q^{30} -35.7906 q^{31} -183.140 q^{32} -137.029 q^{33} -94.4935 q^{35} +36.9262 q^{36} -357.875 q^{37} +135.951 q^{38} -293.348 q^{39} +152.043 q^{40} +32.7452 q^{41} -105.163 q^{42} -516.157 q^{43} -156.998 q^{44} -49.3896 q^{45} -49.2857 q^{46} -210.602 q^{47} -11.7902 q^{48} -150.456 q^{49} +134.094 q^{50} -336.097 q^{52} +87.2324 q^{53} -259.593 q^{54} +209.989 q^{55} -309.809 q^{56} +354.237 q^{57} -453.997 q^{58} -310.728 q^{59} +154.073 q^{60} -365.247 q^{61} +61.0397 q^{62} +100.639 q^{63} +291.115 q^{64} +449.537 q^{65} +233.699 q^{66} -660.131 q^{67} -128.420 q^{69} +161.156 q^{70} +398.030 q^{71} -161.930 q^{72} +643.443 q^{73} +610.344 q^{74} +349.399 q^{75} +405.859 q^{76} -427.882 q^{77} +500.296 q^{78} -384.511 q^{79} +18.0677 q^{80} -480.576 q^{81} -55.8460 q^{82} -153.488 q^{83} -313.946 q^{84} +880.289 q^{86} -1182.94 q^{87} +688.475 q^{88} -599.053 q^{89} +84.2324 q^{90} -915.998 q^{91} -147.134 q^{92} +159.046 q^{93} +359.175 q^{94} -542.846 q^{95} +813.838 q^{96} +44.5693 q^{97} +256.598 q^{98} -223.645 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9}+O(q^{10})$$ 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 $$12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36} - 624 q^{38} - 1664 q^{42} - 1200 q^{43} - 1512 q^{47} - 1052 q^{49} - 2856 q^{50} + 792 q^{52} - 2504 q^{53} - 1424 q^{55} - 3408 q^{59} - 2808 q^{60} + 272 q^{64} + 272 q^{66} - 1080 q^{67} - 344 q^{69} + 2600 q^{70} + 248 q^{72} + 896 q^{76} + 848 q^{77} - 2404 q^{81} - 2960 q^{83} + 4768 q^{84} - 1200 q^{86} - 160 q^{87} - 2144 q^{89} + 3800 q^{93} + 5984 q^{94} + 3464 q^{98}+O(q^{100})$$ 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 - 8 * q^13 - 192 * q^15 - 184 * q^16 - 352 * q^19 - 256 * q^21 - 492 * q^25 - 784 * q^26 + 744 * q^30 + 24 * q^32 - 1400 * q^33 - 632 * q^35 - 856 * q^36 - 624 * q^38 - 1664 * q^42 - 1200 * q^43 - 1512 * q^47 - 1052 * q^49 - 2856 * q^50 + 792 * q^52 - 2504 * q^53 - 1424 * q^55 - 3408 * q^59 - 2808 * q^60 + 272 * q^64 + 272 * q^66 - 1080 * q^67 - 344 * q^69 + 2600 * q^70 + 248 * q^72 + 896 * q^76 + 848 * q^77 - 2404 * q^81 - 2960 * q^83 + 4768 * q^84 - 1200 * q^86 - 160 * q^87 - 2144 * q^89 + 3800 * q^93 + 5984 * q^94 + 3464 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.70547 −0.602974 −0.301487 0.953470i $$-0.597483\pi$$
−0.301487 + 0.953470i $$0.597483\pi$$
$$3$$ −4.44379 −0.855209 −0.427604 0.903966i $$-0.640642\pi$$
−0.427604 + 0.903966i $$0.640642\pi$$
$$4$$ −5.09138 −0.636422
$$5$$ 6.80983 0.609090 0.304545 0.952498i $$-0.401496\pi$$
0.304545 + 0.952498i $$0.401496\pi$$
$$6$$ 7.57875 0.515668
$$7$$ −13.8760 −0.749235 −0.374618 0.927179i $$-0.622226\pi$$
−0.374618 + 0.927179i $$0.622226\pi$$
$$8$$ 22.3269 0.986720
$$9$$ −7.25269 −0.268618
$$10$$ −11.6140 −0.367265
$$11$$ 30.8361 0.845221 0.422610 0.906311i $$-0.361114\pi$$
0.422610 + 0.906311i $$0.361114\pi$$
$$12$$ 22.6250 0.544274
$$13$$ 66.0130 1.40836 0.704181 0.710021i $$-0.251314\pi$$
0.704181 + 0.710021i $$0.251314\pi$$
$$14$$ 23.6651 0.451769
$$15$$ −30.2615 −0.520899
$$16$$ 2.65318 0.0414559
$$17$$ 0 0
$$18$$ 12.3692 0.161970
$$19$$ −79.7150 −0.962520 −0.481260 0.876578i $$-0.659821\pi$$
−0.481260 + 0.876578i $$0.659821\pi$$
$$20$$ −34.6714 −0.387639
$$21$$ 61.6622 0.640752
$$22$$ −52.5899 −0.509646
$$23$$ 28.8986 0.261990 0.130995 0.991383i $$-0.458183\pi$$
0.130995 + 0.991383i $$0.458183\pi$$
$$24$$ −99.2163 −0.843851
$$25$$ −78.6262 −0.629009
$$26$$ −112.583 −0.849205
$$27$$ 152.212 1.08493
$$28$$ 70.6481 0.476830
$$29$$ 266.201 1.70456 0.852281 0.523084i $$-0.175219\pi$$
0.852281 + 0.523084i $$0.175219\pi$$
$$30$$ 51.6100 0.314089
$$31$$ −35.7906 −0.207361 −0.103680 0.994611i $$-0.533062\pi$$
−0.103680 + 0.994611i $$0.533062\pi$$
$$32$$ −183.140 −1.01172
$$33$$ −137.029 −0.722840
$$34$$ 0 0
$$35$$ −94.4935 −0.456352
$$36$$ 36.9262 0.170955
$$37$$ −357.875 −1.59011 −0.795057 0.606535i $$-0.792559\pi$$
−0.795057 + 0.606535i $$0.792559\pi$$
$$38$$ 135.951 0.580374
$$39$$ −293.348 −1.20444
$$40$$ 152.043 0.601001
$$41$$ 32.7452 0.124730 0.0623652 0.998053i $$-0.480136\pi$$
0.0623652 + 0.998053i $$0.480136\pi$$
$$42$$ −105.163 −0.386357
$$43$$ −516.157 −1.83054 −0.915270 0.402841i $$-0.868023\pi$$
−0.915270 + 0.402841i $$0.868023\pi$$
$$44$$ −156.998 −0.537917
$$45$$ −49.3896 −0.163613
$$46$$ −49.2857 −0.157973
$$47$$ −210.602 −0.653606 −0.326803 0.945093i $$-0.605971\pi$$
−0.326803 + 0.945093i $$0.605971\pi$$
$$48$$ −11.7902 −0.0354534
$$49$$ −150.456 −0.438647
$$50$$ 134.094 0.379276
$$51$$ 0 0
$$52$$ −336.097 −0.896313
$$53$$ 87.2324 0.226081 0.113040 0.993590i $$-0.463941\pi$$
0.113040 + 0.993590i $$0.463941\pi$$
$$54$$ −259.593 −0.654186
$$55$$ 209.989 0.514815
$$56$$ −309.809 −0.739285
$$57$$ 354.237 0.823155
$$58$$ −453.997 −1.02781
$$59$$ −310.728 −0.685651 −0.342825 0.939399i $$-0.611384\pi$$
−0.342825 + 0.939399i $$0.611384\pi$$
$$60$$ 154.073 0.331512
$$61$$ −365.247 −0.766641 −0.383321 0.923615i $$-0.625220\pi$$
−0.383321 + 0.923615i $$0.625220\pi$$
$$62$$ 61.0397 0.125033
$$63$$ 100.639 0.201258
$$64$$ 291.115 0.568583
$$65$$ 449.537 0.857819
$$66$$ 233.699 0.435854
$$67$$ −660.131 −1.20370 −0.601850 0.798609i $$-0.705569\pi$$
−0.601850 + 0.798609i $$0.705569\pi$$
$$68$$ 0 0
$$69$$ −128.420 −0.224056
$$70$$ 161.156 0.275168
$$71$$ 398.030 0.665316 0.332658 0.943047i $$-0.392054\pi$$
0.332658 + 0.943047i $$0.392054\pi$$
$$72$$ −161.930 −0.265051
$$73$$ 643.443 1.03163 0.515817 0.856699i $$-0.327489\pi$$
0.515817 + 0.856699i $$0.327489\pi$$
$$74$$ 610.344 0.958797
$$75$$ 349.399 0.537934
$$76$$ 405.859 0.612569
$$77$$ −427.882 −0.633269
$$78$$ 500.296 0.726248
$$79$$ −384.511 −0.547605 −0.273803 0.961786i $$-0.588282\pi$$
−0.273803 + 0.961786i $$0.588282\pi$$
$$80$$ 18.0677 0.0252504
$$81$$ −480.576 −0.659226
$$82$$ −55.8460 −0.0752092
$$83$$ −153.488 −0.202983 −0.101491 0.994836i $$-0.532361\pi$$
−0.101491 + 0.994836i $$0.532361\pi$$
$$84$$ −313.946 −0.407789
$$85$$ 0 0
$$86$$ 880.289 1.10377
$$87$$ −1182.94 −1.45776
$$88$$ 688.475 0.833996
$$89$$ −599.053 −0.713477 −0.356738 0.934204i $$-0.616111\pi$$
−0.356738 + 0.934204i $$0.616111\pi$$
$$90$$ 84.2324 0.0986542
$$91$$ −915.998 −1.05519
$$92$$ −147.134 −0.166737
$$93$$ 159.046 0.177336
$$94$$ 359.175 0.394107
$$95$$ −542.846 −0.586261
$$96$$ 813.838 0.865229
$$97$$ 44.5693 0.0466529 0.0233265 0.999728i $$-0.492574\pi$$
0.0233265 + 0.999728i $$0.492574\pi$$
$$98$$ 256.598 0.264492
$$99$$ −223.645 −0.227042
$$100$$ 400.316 0.400316
$$101$$ 1478.33 1.45643 0.728215 0.685348i $$-0.240350\pi$$
0.728215 + 0.685348i $$0.240350\pi$$
$$102$$ 0 0
$$103$$ 618.133 0.591325 0.295662 0.955293i $$-0.404460\pi$$
0.295662 + 0.955293i $$0.404460\pi$$
$$104$$ 1473.87 1.38966
$$105$$ 419.909 0.390276
$$106$$ −148.772 −0.136321
$$107$$ 138.979 0.125567 0.0627833 0.998027i $$-0.480002\pi$$
0.0627833 + 0.998027i $$0.480002\pi$$
$$108$$ −774.969 −0.690476
$$109$$ 823.466 0.723612 0.361806 0.932253i $$-0.382160\pi$$
0.361806 + 0.932253i $$0.382160\pi$$
$$110$$ −358.129 −0.310420
$$111$$ 1590.32 1.35988
$$112$$ −36.8156 −0.0310602
$$113$$ −602.474 −0.501558 −0.250779 0.968044i $$-0.580687\pi$$
−0.250779 + 0.968044i $$0.580687\pi$$
$$114$$ −604.140 −0.496341
$$115$$ 196.795 0.159576
$$116$$ −1355.33 −1.08482
$$117$$ −478.772 −0.378312
$$118$$ 529.937 0.413430
$$119$$ 0 0
$$120$$ −675.646 −0.513982
$$121$$ −380.136 −0.285602
$$122$$ 622.917 0.462265
$$123$$ −145.513 −0.106671
$$124$$ 182.223 0.131969
$$125$$ −1386.66 −0.992213
$$126$$ −171.636 −0.121354
$$127$$ 1506.98 1.05294 0.526468 0.850195i $$-0.323516\pi$$
0.526468 + 0.850195i $$0.323516\pi$$
$$128$$ 968.636 0.668876
$$129$$ 2293.70 1.56549
$$130$$ −766.671 −0.517243
$$131$$ −2783.58 −1.85651 −0.928254 0.371946i $$-0.878691\pi$$
−0.928254 + 0.371946i $$0.878691\pi$$
$$132$$ 697.667 0.460032
$$133$$ 1106.13 0.721154
$$134$$ 1125.83 0.725799
$$135$$ 1036.54 0.660822
$$136$$ 0 0
$$137$$ −629.783 −0.392744 −0.196372 0.980529i $$-0.562916\pi$$
−0.196372 + 0.980529i $$0.562916\pi$$
$$138$$ 219.015 0.135100
$$139$$ −2654.10 −1.61955 −0.809775 0.586740i $$-0.800411\pi$$
−0.809775 + 0.586740i $$0.800411\pi$$
$$140$$ 481.102 0.290432
$$141$$ 935.872 0.558969
$$142$$ −678.827 −0.401168
$$143$$ 2035.58 1.19038
$$144$$ −19.2427 −0.0111358
$$145$$ 1812.78 1.03823
$$146$$ −1097.37 −0.622048
$$147$$ 668.595 0.375134
$$148$$ 1822.08 1.01198
$$149$$ −2216.30 −1.21857 −0.609283 0.792953i $$-0.708543\pi$$
−0.609283 + 0.792953i $$0.708543\pi$$
$$150$$ −595.888 −0.324360
$$151$$ 2133.15 1.14963 0.574813 0.818285i $$-0.305075\pi$$
0.574813 + 0.818285i $$0.305075\pi$$
$$152$$ −1779.79 −0.949737
$$153$$ 0 0
$$154$$ 729.740 0.381845
$$155$$ −243.728 −0.126301
$$156$$ 1493.55 0.766534
$$157$$ −345.107 −0.175430 −0.0877152 0.996146i $$-0.527957\pi$$
−0.0877152 + 0.996146i $$0.527957\pi$$
$$158$$ 655.770 0.330192
$$159$$ −387.643 −0.193346
$$160$$ −1247.16 −0.616227
$$161$$ −400.998 −0.196292
$$162$$ 819.606 0.397496
$$163$$ −678.526 −0.326050 −0.163025 0.986622i $$-0.552125\pi$$
−0.163025 + 0.986622i $$0.552125\pi$$
$$164$$ −166.718 −0.0793812
$$165$$ −933.146 −0.440275
$$166$$ 261.770 0.122393
$$167$$ −799.790 −0.370596 −0.185298 0.982682i $$-0.559325\pi$$
−0.185298 + 0.982682i $$0.559325\pi$$
$$168$$ 1376.73 0.632243
$$169$$ 2160.71 0.983483
$$170$$ 0 0
$$171$$ 578.148 0.258550
$$172$$ 2627.95 1.16500
$$173$$ 1885.48 0.828615 0.414307 0.910137i $$-0.364024\pi$$
0.414307 + 0.910137i $$0.364024\pi$$
$$174$$ 2017.47 0.878989
$$175$$ 1091.02 0.471276
$$176$$ 81.8136 0.0350394
$$177$$ 1380.81 0.586374
$$178$$ 1021.66 0.430208
$$179$$ −3588.86 −1.49857 −0.749286 0.662247i $$-0.769603\pi$$
−0.749286 + 0.662247i $$0.769603\pi$$
$$180$$ 251.461 0.104127
$$181$$ −461.347 −0.189457 −0.0947284 0.995503i $$-0.530198\pi$$
−0.0947284 + 0.995503i $$0.530198\pi$$
$$182$$ 1562.20 0.636255
$$183$$ 1623.08 0.655638
$$184$$ 645.218 0.258511
$$185$$ −2437.07 −0.968522
$$186$$ −271.248 −0.106929
$$187$$ 0 0
$$188$$ 1072.26 0.415969
$$189$$ −2112.10 −0.812870
$$190$$ 925.806 0.353500
$$191$$ −4519.91 −1.71230 −0.856149 0.516728i $$-0.827150\pi$$
−0.856149 + 0.516728i $$0.827150\pi$$
$$192$$ −1293.65 −0.486257
$$193$$ 2928.92 1.09238 0.546188 0.837663i $$-0.316078\pi$$
0.546188 + 0.837663i $$0.316078\pi$$
$$194$$ −76.0116 −0.0281305
$$195$$ −1997.65 −0.733614
$$196$$ 766.027 0.279165
$$197$$ −1926.13 −0.696606 −0.348303 0.937382i $$-0.613242\pi$$
−0.348303 + 0.937382i $$0.613242\pi$$
$$198$$ 381.419 0.136900
$$199$$ −3086.61 −1.09952 −0.549759 0.835323i $$-0.685280\pi$$
−0.549759 + 0.835323i $$0.685280\pi$$
$$200$$ −1755.48 −0.620656
$$201$$ 2933.49 1.02941
$$202$$ −2521.25 −0.878190
$$203$$ −3693.81 −1.27712
$$204$$ 0 0
$$205$$ 222.990 0.0759721
$$206$$ −1054.21 −0.356553
$$207$$ −209.593 −0.0703754
$$208$$ 175.144 0.0583849
$$209$$ −2458.10 −0.813541
$$210$$ −716.142 −0.235326
$$211$$ −3602.40 −1.17535 −0.587676 0.809097i $$-0.699957\pi$$
−0.587676 + 0.809097i $$0.699957\pi$$
$$212$$ −444.133 −0.143883
$$213$$ −1768.76 −0.568984
$$214$$ −237.024 −0.0757133
$$215$$ −3514.94 −1.11496
$$216$$ 3398.42 1.07053
$$217$$ 496.631 0.155362
$$218$$ −1404.39 −0.436319
$$219$$ −2859.33 −0.882262
$$220$$ −1069.13 −0.327640
$$221$$ 0 0
$$222$$ −2712.24 −0.819972
$$223$$ −1577.43 −0.473688 −0.236844 0.971548i $$-0.576113\pi$$
−0.236844 + 0.971548i $$0.576113\pi$$
$$224$$ 2541.26 0.758014
$$225$$ 570.252 0.168963
$$226$$ 1027.50 0.302426
$$227$$ 4292.90 1.25520 0.627599 0.778537i $$-0.284038\pi$$
0.627599 + 0.778537i $$0.284038\pi$$
$$228$$ −1803.55 −0.523874
$$229$$ 580.893 0.167627 0.0838133 0.996481i $$-0.473290\pi$$
0.0838133 + 0.996481i $$0.473290\pi$$
$$230$$ −335.627 −0.0962200
$$231$$ 1901.42 0.541577
$$232$$ 5943.45 1.68193
$$233$$ 671.778 0.188883 0.0944413 0.995530i $$-0.469894\pi$$
0.0944413 + 0.995530i $$0.469894\pi$$
$$234$$ 816.530 0.228112
$$235$$ −1434.16 −0.398105
$$236$$ 1582.04 0.436363
$$237$$ 1708.69 0.468317
$$238$$ 0 0
$$239$$ −273.635 −0.0740584 −0.0370292 0.999314i $$-0.511789\pi$$
−0.0370292 + 0.999314i $$0.511789\pi$$
$$240$$ −80.2891 −0.0215943
$$241$$ −6486.56 −1.73376 −0.866879 0.498518i $$-0.833878\pi$$
−0.866879 + 0.498518i $$0.833878\pi$$
$$242$$ 648.310 0.172211
$$243$$ −1974.14 −0.521158
$$244$$ 1859.61 0.487908
$$245$$ −1024.58 −0.267175
$$246$$ 248.168 0.0643195
$$247$$ −5262.22 −1.35558
$$248$$ −799.093 −0.204607
$$249$$ 682.071 0.173592
$$250$$ 2364.90 0.598279
$$251$$ −2527.58 −0.635615 −0.317808 0.948155i $$-0.602947\pi$$
−0.317808 + 0.948155i $$0.602947\pi$$
$$252$$ −512.389 −0.128085
$$253$$ 891.120 0.221440
$$254$$ −2570.11 −0.634893
$$255$$ 0 0
$$256$$ −3980.89 −0.971898
$$257$$ 1702.44 0.413211 0.206605 0.978424i $$-0.433758\pi$$
0.206605 + 0.978424i $$0.433758\pi$$
$$258$$ −3911.82 −0.943952
$$259$$ 4965.88 1.19137
$$260$$ −2288.76 −0.545935
$$261$$ −1930.67 −0.457877
$$262$$ 4747.31 1.11943
$$263$$ −4274.40 −1.00217 −0.501085 0.865398i $$-0.667066\pi$$
−0.501085 + 0.865398i $$0.667066\pi$$
$$264$$ −3059.44 −0.713241
$$265$$ 594.038 0.137704
$$266$$ −1886.46 −0.434837
$$267$$ 2662.07 0.610172
$$268$$ 3360.98 0.766061
$$269$$ 7996.34 1.81244 0.906218 0.422810i $$-0.138956\pi$$
0.906218 + 0.422810i $$0.138956\pi$$
$$270$$ −1767.78 −0.398458
$$271$$ 474.268 0.106309 0.0531545 0.998586i $$-0.483072\pi$$
0.0531545 + 0.998586i $$0.483072\pi$$
$$272$$ 0 0
$$273$$ 4070.51 0.902411
$$274$$ 1074.07 0.236815
$$275$$ −2424.52 −0.531652
$$276$$ 653.833 0.142595
$$277$$ −2826.25 −0.613043 −0.306521 0.951864i $$-0.599165\pi$$
−0.306521 + 0.951864i $$0.599165\pi$$
$$278$$ 4526.48 0.976547
$$279$$ 259.578 0.0557008
$$280$$ −2109.75 −0.450291
$$281$$ −2498.15 −0.530346 −0.265173 0.964201i $$-0.585429\pi$$
−0.265173 + 0.964201i $$0.585429\pi$$
$$282$$ −1596.10 −0.337044
$$283$$ 2417.07 0.507704 0.253852 0.967243i $$-0.418302\pi$$
0.253852 + 0.967243i $$0.418302\pi$$
$$284$$ −2026.52 −0.423422
$$285$$ 2412.29 0.501375
$$286$$ −3471.62 −0.717766
$$287$$ −454.374 −0.0934524
$$288$$ 1328.26 0.271766
$$289$$ 0 0
$$290$$ −3091.65 −0.626027
$$291$$ −198.057 −0.0398980
$$292$$ −3276.01 −0.656555
$$293$$ 6652.17 1.32636 0.663181 0.748459i $$-0.269206\pi$$
0.663181 + 0.748459i $$0.269206\pi$$
$$294$$ −1140.27 −0.226196
$$295$$ −2116.01 −0.417623
$$296$$ −7990.24 −1.56900
$$297$$ 4693.62 0.917008
$$298$$ 3779.83 0.734764
$$299$$ 1907.68 0.368977
$$300$$ −1778.92 −0.342353
$$301$$ 7162.21 1.37150
$$302$$ −3638.02 −0.693194
$$303$$ −6569.40 −1.24555
$$304$$ −211.498 −0.0399021
$$305$$ −2487.27 −0.466954
$$306$$ 0 0
$$307$$ −2511.18 −0.466842 −0.233421 0.972376i $$-0.574992\pi$$
−0.233421 + 0.972376i $$0.574992\pi$$
$$308$$ 2178.51 0.403027
$$309$$ −2746.85 −0.505706
$$310$$ 415.670 0.0761563
$$311$$ 3585.10 0.653674 0.326837 0.945081i $$-0.394017\pi$$
0.326837 + 0.945081i $$0.394017\pi$$
$$312$$ −6549.56 −1.18845
$$313$$ −2612.32 −0.471747 −0.235874 0.971784i $$-0.575795\pi$$
−0.235874 + 0.971784i $$0.575795\pi$$
$$314$$ 588.570 0.105780
$$315$$ 685.332 0.122584
$$316$$ 1957.69 0.348508
$$317$$ −3903.80 −0.691669 −0.345834 0.938296i $$-0.612404\pi$$
−0.345834 + 0.938296i $$0.612404\pi$$
$$318$$ 661.112 0.116583
$$319$$ 8208.60 1.44073
$$320$$ 1982.44 0.346318
$$321$$ −617.594 −0.107386
$$322$$ 683.890 0.118359
$$323$$ 0 0
$$324$$ 2446.79 0.419546
$$325$$ −5190.35 −0.885873
$$326$$ 1157.20 0.196600
$$327$$ −3659.31 −0.618839
$$328$$ 731.101 0.123074
$$329$$ 2922.32 0.489704
$$330$$ 1591.45 0.265474
$$331$$ −6236.03 −1.03554 −0.517769 0.855520i $$-0.673237\pi$$
−0.517769 + 0.855520i $$0.673237\pi$$
$$332$$ 781.468 0.129183
$$333$$ 2595.55 0.427134
$$334$$ 1364.02 0.223460
$$335$$ −4495.38 −0.733161
$$336$$ 163.601 0.0265630
$$337$$ 3842.08 0.621043 0.310521 0.950566i $$-0.399496\pi$$
0.310521 + 0.950566i $$0.399496\pi$$
$$338$$ −3685.02 −0.593014
$$339$$ 2677.27 0.428936
$$340$$ 0 0
$$341$$ −1103.64 −0.175265
$$342$$ −986.014 −0.155899
$$343$$ 6847.21 1.07788
$$344$$ −11524.2 −1.80623
$$345$$ −874.516 −0.136471
$$346$$ −3215.62 −0.499633
$$347$$ 8301.86 1.28434 0.642172 0.766561i $$-0.278034\pi$$
0.642172 + 0.766561i $$0.278034\pi$$
$$348$$ 6022.81 0.927748
$$349$$ 4929.17 0.756024 0.378012 0.925801i $$-0.376608\pi$$
0.378012 + 0.925801i $$0.376608\pi$$
$$350$$ −1860.70 −0.284167
$$351$$ 10048.0 1.52798
$$352$$ −5647.33 −0.855124
$$353$$ −9607.54 −1.44861 −0.724303 0.689482i $$-0.757838\pi$$
−0.724303 + 0.689482i $$0.757838\pi$$
$$354$$ −2354.93 −0.353568
$$355$$ 2710.52 0.405237
$$356$$ 3050.00 0.454073
$$357$$ 0 0
$$358$$ 6120.69 0.903600
$$359$$ −1962.76 −0.288553 −0.144277 0.989537i $$-0.546085\pi$$
−0.144277 + 0.989537i $$0.546085\pi$$
$$360$$ −1102.72 −0.161440
$$361$$ −504.522 −0.0735561
$$362$$ 786.813 0.114238
$$363$$ 1689.25 0.244249
$$364$$ 4663.69 0.671549
$$365$$ 4381.74 0.628358
$$366$$ −2768.12 −0.395333
$$367$$ 3458.29 0.491883 0.245941 0.969285i $$-0.420903\pi$$
0.245941 + 0.969285i $$0.420903\pi$$
$$368$$ 76.6732 0.0108610
$$369$$ −237.491 −0.0335049
$$370$$ 4156.34 0.583994
$$371$$ −1210.44 −0.169388
$$372$$ −809.763 −0.112861
$$373$$ 2661.22 0.369418 0.184709 0.982793i $$-0.440866\pi$$
0.184709 + 0.982793i $$0.440866\pi$$
$$374$$ 0 0
$$375$$ 6162.03 0.848549
$$376$$ −4702.10 −0.644926
$$377$$ 17572.7 2.40064
$$378$$ 3602.11 0.490140
$$379$$ 5583.03 0.756677 0.378339 0.925667i $$-0.376495\pi$$
0.378339 + 0.925667i $$0.376495\pi$$
$$380$$ 2763.83 0.373110
$$381$$ −6696.71 −0.900480
$$382$$ 7708.56 1.03247
$$383$$ 7832.13 1.04492 0.522459 0.852665i $$-0.325015\pi$$
0.522459 + 0.852665i $$0.325015\pi$$
$$384$$ −4304.42 −0.572029
$$385$$ −2913.81 −0.385718
$$386$$ −4995.18 −0.658674
$$387$$ 3743.53 0.491717
$$388$$ −226.919 −0.0296910
$$389$$ 7432.95 0.968807 0.484403 0.874845i $$-0.339037\pi$$
0.484403 + 0.874845i $$0.339037\pi$$
$$390$$ 3406.93 0.442350
$$391$$ 0 0
$$392$$ −3359.22 −0.432821
$$393$$ 12369.7 1.58770
$$394$$ 3284.96 0.420035
$$395$$ −2618.45 −0.333541
$$396$$ 1138.66 0.144494
$$397$$ −9540.26 −1.20608 −0.603038 0.797713i $$-0.706043\pi$$
−0.603038 + 0.797713i $$0.706043\pi$$
$$398$$ 5264.12 0.662981
$$399$$ −4915.40 −0.616737
$$400$$ −208.609 −0.0260761
$$401$$ −8663.30 −1.07886 −0.539432 0.842029i $$-0.681361\pi$$
−0.539432 + 0.842029i $$0.681361\pi$$
$$402$$ −5002.97 −0.620710
$$403$$ −2362.64 −0.292039
$$404$$ −7526.75 −0.926905
$$405$$ −3272.64 −0.401528
$$406$$ 6299.68 0.770069
$$407$$ −11035.4 −1.34400
$$408$$ 0 0
$$409$$ 5279.17 0.638235 0.319117 0.947715i $$-0.396614\pi$$
0.319117 + 0.947715i $$0.396614\pi$$
$$410$$ −380.302 −0.0458092
$$411$$ 2798.62 0.335878
$$412$$ −3147.15 −0.376332
$$413$$ 4311.68 0.513714
$$414$$ 357.454 0.0424346
$$415$$ −1045.23 −0.123635
$$416$$ −12089.6 −1.42486
$$417$$ 11794.3 1.38505
$$418$$ 4192.21 0.490544
$$419$$ −9408.24 −1.09695 −0.548476 0.836166i $$-0.684792\pi$$
−0.548476 + 0.836166i $$0.684792\pi$$
$$420$$ −2137.92 −0.248380
$$421$$ −8191.97 −0.948343 −0.474171 0.880433i $$-0.657252\pi$$
−0.474171 + 0.880433i $$0.657252\pi$$
$$422$$ 6143.77 0.708706
$$423$$ 1527.43 0.175570
$$424$$ 1947.63 0.223079
$$425$$ 0 0
$$426$$ 3016.57 0.343083
$$427$$ 5068.18 0.574395
$$428$$ −707.595 −0.0799133
$$429$$ −9045.70 −1.01802
$$430$$ 5994.62 0.672294
$$431$$ −1950.29 −0.217964 −0.108982 0.994044i $$-0.534759\pi$$
−0.108982 + 0.994044i $$0.534759\pi$$
$$432$$ 403.845 0.0449769
$$433$$ 11152.3 1.23775 0.618876 0.785489i $$-0.287588\pi$$
0.618876 + 0.785489i $$0.287588\pi$$
$$434$$ −846.988 −0.0936791
$$435$$ −8055.64 −0.887905
$$436$$ −4192.58 −0.460523
$$437$$ −2303.65 −0.252171
$$438$$ 4876.49 0.531981
$$439$$ −470.158 −0.0511149 −0.0255574 0.999673i $$-0.508136\pi$$
−0.0255574 + 0.999673i $$0.508136\pi$$
$$440$$ 4688.40 0.507979
$$441$$ 1091.21 0.117829
$$442$$ 0 0
$$443$$ −13132.9 −1.40849 −0.704246 0.709956i $$-0.748715\pi$$
−0.704246 + 0.709956i $$0.748715\pi$$
$$444$$ −8096.93 −0.865457
$$445$$ −4079.45 −0.434572
$$446$$ 2690.25 0.285621
$$447$$ 9848.78 1.04213
$$448$$ −4039.51 −0.426002
$$449$$ 10433.0 1.09658 0.548291 0.836288i $$-0.315279\pi$$
0.548291 + 0.836288i $$0.315279\pi$$
$$450$$ −972.546 −0.101881
$$451$$ 1009.73 0.105425
$$452$$ 3067.43 0.319203
$$453$$ −9479.28 −0.983169
$$454$$ −7321.40 −0.756851
$$455$$ −6237.79 −0.642708
$$456$$ 7909.02 0.812224
$$457$$ −253.385 −0.0259362 −0.0129681 0.999916i $$-0.504128\pi$$
−0.0129681 + 0.999916i $$0.504128\pi$$
$$458$$ −990.694 −0.101074
$$459$$ 0 0
$$460$$ −1001.96 −0.101558
$$461$$ 15560.2 1.57204 0.786019 0.618202i $$-0.212139\pi$$
0.786019 + 0.618202i $$0.212139\pi$$
$$462$$ −3242.81 −0.326557
$$463$$ 10727.9 1.07682 0.538409 0.842684i $$-0.319026\pi$$
0.538409 + 0.842684i $$0.319026\pi$$
$$464$$ 706.279 0.0706641
$$465$$ 1083.08 0.108014
$$466$$ −1145.70 −0.113891
$$467$$ −1960.48 −0.194262 −0.0971308 0.995272i $$-0.530967\pi$$
−0.0971308 + 0.995272i $$0.530967\pi$$
$$468$$ 2437.61 0.240766
$$469$$ 9160.00 0.901854
$$470$$ 2445.92 0.240047
$$471$$ 1533.59 0.150030
$$472$$ −6937.61 −0.676545
$$473$$ −15916.3 −1.54721
$$474$$ −2914.11 −0.282383
$$475$$ 6267.68 0.605434
$$476$$ 0 0
$$477$$ −632.670 −0.0607295
$$478$$ 466.675 0.0446553
$$479$$ 4294.19 0.409617 0.204808 0.978802i $$-0.434343\pi$$
0.204808 + 0.978802i $$0.434343\pi$$
$$480$$ 5542.10 0.527002
$$481$$ −23624.4 −2.23946
$$482$$ 11062.6 1.04541
$$483$$ 1781.95 0.167871
$$484$$ 1935.42 0.181764
$$485$$ 303.510 0.0284158
$$486$$ 3366.84 0.314244
$$487$$ 18013.5 1.67612 0.838060 0.545578i $$-0.183690\pi$$
0.838060 + 0.545578i $$0.183690\pi$$
$$488$$ −8154.85 −0.756460
$$489$$ 3015.23 0.278841
$$490$$ 1747.39 0.161100
$$491$$ 13733.2 1.26226 0.631129 0.775678i $$-0.282592\pi$$
0.631129 + 0.775678i $$0.282592\pi$$
$$492$$ 740.862 0.0678875
$$493$$ 0 0
$$494$$ 8974.55 0.817377
$$495$$ −1522.98 −0.138289
$$496$$ −94.9587 −0.00859632
$$497$$ −5523.07 −0.498478
$$498$$ −1163.25 −0.104672
$$499$$ −7747.71 −0.695060 −0.347530 0.937669i $$-0.612980\pi$$
−0.347530 + 0.937669i $$0.612980\pi$$
$$500$$ 7060.01 0.631467
$$501$$ 3554.10 0.316937
$$502$$ 4310.71 0.383259
$$503$$ −19203.7 −1.70229 −0.851143 0.524934i $$-0.824090\pi$$
−0.851143 + 0.524934i $$0.824090\pi$$
$$504$$ 2246.95 0.198586
$$505$$ 10067.2 0.887098
$$506$$ −1519.78 −0.133522
$$507$$ −9601.76 −0.841083
$$508$$ −7672.61 −0.670112
$$509$$ 151.325 0.0131775 0.00658876 0.999978i $$-0.497903\pi$$
0.00658876 + 0.999978i $$0.497903\pi$$
$$510$$ 0 0
$$511$$ −8928.43 −0.772936
$$512$$ −959.802 −0.0828470
$$513$$ −12133.6 −1.04427
$$514$$ −2903.45 −0.249155
$$515$$ 4209.38 0.360170
$$516$$ −11678.1 −0.996315
$$517$$ −6494.14 −0.552441
$$518$$ −8469.15 −0.718365
$$519$$ −8378.68 −0.708638
$$520$$ 10036.8 0.846427
$$521$$ 8548.62 0.718852 0.359426 0.933174i $$-0.382973\pi$$
0.359426 + 0.933174i $$0.382973\pi$$
$$522$$ 3292.70 0.276088
$$523$$ 17874.0 1.49441 0.747205 0.664593i $$-0.231395\pi$$
0.747205 + 0.664593i $$0.231395\pi$$
$$524$$ 14172.3 1.18152
$$525$$ −4848.26 −0.403039
$$526$$ 7289.85 0.604283
$$527$$ 0 0
$$528$$ −363.563 −0.0299660
$$529$$ −11331.9 −0.931361
$$530$$ −1013.11 −0.0830317
$$531$$ 2253.62 0.184178
$$532$$ −5631.71 −0.458958
$$533$$ 2161.61 0.175666
$$534$$ −4540.07 −0.367918
$$535$$ 946.425 0.0764813
$$536$$ −14738.7 −1.18771
$$537$$ 15948.2 1.28159
$$538$$ −13637.5 −1.09285
$$539$$ −4639.47 −0.370753
$$540$$ −5277.41 −0.420562
$$541$$ −2122.24 −0.168655 −0.0843273 0.996438i $$-0.526874\pi$$
−0.0843273 + 0.996438i $$0.526874\pi$$
$$542$$ −808.849 −0.0641015
$$543$$ 2050.13 0.162025
$$544$$ 0 0
$$545$$ 5607.67 0.440745
$$546$$ −6942.12 −0.544130
$$547$$ −284.066 −0.0222044 −0.0111022 0.999938i $$-0.503534\pi$$
−0.0111022 + 0.999938i $$0.503534\pi$$
$$548$$ 3206.46 0.249951
$$549$$ 2649.03 0.205934
$$550$$ 4134.95 0.320572
$$551$$ −21220.2 −1.64067
$$552$$ −2867.21 −0.221081
$$553$$ 5335.48 0.410285
$$554$$ 4820.08 0.369649
$$555$$ 10829.8 0.828289
$$556$$ 13513.0 1.03072
$$557$$ 9915.36 0.754268 0.377134 0.926159i $$-0.376910\pi$$
0.377134 + 0.926159i $$0.376910\pi$$
$$558$$ −442.702 −0.0335862
$$559$$ −34073.1 −2.57806
$$560$$ −250.708 −0.0189185
$$561$$ 0 0
$$562$$ 4260.52 0.319785
$$563$$ −5048.52 −0.377922 −0.188961 0.981985i $$-0.560512\pi$$
−0.188961 + 0.981985i $$0.560512\pi$$
$$564$$ −4764.88 −0.355741
$$565$$ −4102.75 −0.305494
$$566$$ −4122.24 −0.306132
$$567$$ 6668.48 0.493915
$$568$$ 8886.78 0.656481
$$569$$ 10964.5 0.807828 0.403914 0.914797i $$-0.367650\pi$$
0.403914 + 0.914797i $$0.367650\pi$$
$$570$$ −4114.09 −0.302316
$$571$$ −6757.57 −0.495263 −0.247632 0.968854i $$-0.579652\pi$$
−0.247632 + 0.968854i $$0.579652\pi$$
$$572$$ −10363.9 −0.757582
$$573$$ 20085.5 1.46437
$$574$$ 774.920 0.0563494
$$575$$ −2272.19 −0.164794
$$576$$ −2111.36 −0.152732
$$577$$ −18050.9 −1.30237 −0.651187 0.758917i $$-0.725729\pi$$
−0.651187 + 0.758917i $$0.725729\pi$$
$$578$$ 0 0
$$579$$ −13015.5 −0.934209
$$580$$ −9229.57 −0.660754
$$581$$ 2129.81 0.152082
$$582$$ 337.780 0.0240574
$$583$$ 2689.90 0.191088
$$584$$ 14366.1 1.01793
$$585$$ −3260.36 −0.230426
$$586$$ −11345.1 −0.799762
$$587$$ −7447.06 −0.523634 −0.261817 0.965118i $$-0.584322\pi$$
−0.261817 + 0.965118i $$0.584322\pi$$
$$588$$ −3404.07 −0.238744
$$589$$ 2853.04 0.199589
$$590$$ 3608.78 0.251816
$$591$$ 8559.34 0.595743
$$592$$ −949.505 −0.0659196
$$593$$ −1229.32 −0.0851298 −0.0425649 0.999094i $$-0.513553\pi$$
−0.0425649 + 0.999094i $$0.513553\pi$$
$$594$$ −8004.82 −0.552932
$$595$$ 0 0
$$596$$ 11284.0 0.775523
$$597$$ 13716.3 0.940317
$$598$$ −3253.49 −0.222484
$$599$$ 10655.6 0.726840 0.363420 0.931625i $$-0.381609\pi$$
0.363420 + 0.931625i $$0.381609\pi$$
$$600$$ 7800.99 0.530790
$$601$$ −2134.49 −0.144871 −0.0724356 0.997373i $$-0.523077\pi$$
−0.0724356 + 0.997373i $$0.523077\pi$$
$$602$$ −12214.9 −0.826982
$$603$$ 4787.73 0.323336
$$604$$ −10860.7 −0.731647
$$605$$ −2588.67 −0.173957
$$606$$ 11203.9 0.751036
$$607$$ 12364.9 0.826812 0.413406 0.910547i $$-0.364339\pi$$
0.413406 + 0.910547i $$0.364339\pi$$
$$608$$ 14599.0 0.973797
$$609$$ 16414.5 1.09220
$$610$$ 4241.96 0.281561
$$611$$ −13902.5 −0.920513
$$612$$ 0 0
$$613$$ 4888.87 0.322120 0.161060 0.986945i $$-0.448509\pi$$
0.161060 + 0.986945i $$0.448509\pi$$
$$614$$ 4282.73 0.281493
$$615$$ −990.920 −0.0649720
$$616$$ −9553.30 −0.624859
$$617$$ −24750.0 −1.61491 −0.807453 0.589932i $$-0.799155\pi$$
−0.807453 + 0.589932i $$0.799155\pi$$
$$618$$ 4684.67 0.304927
$$619$$ −10794.8 −0.700937 −0.350469 0.936574i $$-0.613978\pi$$
−0.350469 + 0.936574i $$0.613978\pi$$
$$620$$ 1240.91 0.0803809
$$621$$ 4398.72 0.284242
$$622$$ −6114.28 −0.394148
$$623$$ 8312.47 0.534562
$$624$$ −778.304 −0.0499313
$$625$$ 385.346 0.0246621
$$626$$ 4455.22 0.284451
$$627$$ 10923.3 0.695748
$$628$$ 1757.07 0.111648
$$629$$ 0 0
$$630$$ −1168.81 −0.0739152
$$631$$ −7579.90 −0.478211 −0.239106 0.970994i $$-0.576854\pi$$
−0.239106 + 0.970994i $$0.576854\pi$$
$$632$$ −8584.94 −0.540333
$$633$$ 16008.3 1.00517
$$634$$ 6657.80 0.417058
$$635$$ 10262.3 0.641333
$$636$$ 1973.64 0.123050
$$637$$ −9932.03 −0.617773
$$638$$ −13999.5 −0.868723
$$639$$ −2886.79 −0.178716
$$640$$ 6596.25 0.407406
$$641$$ 3181.53 0.196042 0.0980208 0.995184i $$-0.468749\pi$$
0.0980208 + 0.995184i $$0.468749\pi$$
$$642$$ 1053.29 0.0647507
$$643$$ −10552.8 −0.647219 −0.323609 0.946191i $$-0.604896\pi$$
−0.323609 + 0.946191i $$0.604896\pi$$
$$644$$ 2041.63 0.124925
$$645$$ 15619.7 0.953526
$$646$$ 0 0
$$647$$ −10952.0 −0.665483 −0.332742 0.943018i $$-0.607974\pi$$
−0.332742 + 0.943018i $$0.607974\pi$$
$$648$$ −10729.8 −0.650471
$$649$$ −9581.64 −0.579526
$$650$$ 8851.97 0.534158
$$651$$ −2206.93 −0.132867
$$652$$ 3454.63 0.207506
$$653$$ 6526.76 0.391136 0.195568 0.980690i $$-0.437345\pi$$
0.195568 + 0.980690i $$0.437345\pi$$
$$654$$ 6240.84 0.373144
$$655$$ −18955.7 −1.13078
$$656$$ 86.8789 0.00517081
$$657$$ −4666.69 −0.277116
$$658$$ −4983.92 −0.295279
$$659$$ −25717.9 −1.52022 −0.760111 0.649793i $$-0.774855\pi$$
−0.760111 + 0.649793i $$0.774855\pi$$
$$660$$ 4751.00 0.280201
$$661$$ 11808.9 0.694878 0.347439 0.937703i $$-0.387051\pi$$
0.347439 + 0.937703i $$0.387051\pi$$
$$662$$ 10635.3 0.624403
$$663$$ 0 0
$$664$$ −3426.93 −0.200287
$$665$$ 7532.54 0.439247
$$666$$ −4426.64 −0.257550
$$667$$ 7692.85 0.446579
$$668$$ 4072.03 0.235856
$$669$$ 7009.76 0.405102
$$670$$ 7666.73 0.442077
$$671$$ −11262.8 −0.647981
$$672$$ −11292.8 −0.648260
$$673$$ 6282.85 0.359860 0.179930 0.983679i $$-0.442413\pi$$
0.179930 + 0.983679i $$0.442413\pi$$
$$674$$ −6552.54 −0.374472
$$675$$ −11967.8 −0.682433
$$676$$ −11001.0 −0.625910
$$677$$ 18502.8 1.05040 0.525200 0.850979i $$-0.323991\pi$$
0.525200 + 0.850979i $$0.323991\pi$$
$$678$$ −4566.00 −0.258638
$$679$$ −618.446 −0.0349540
$$680$$ 0 0
$$681$$ −19076.8 −1.07346
$$682$$ 1882.22 0.105680
$$683$$ −137.518 −0.00770424 −0.00385212 0.999993i $$-0.501226\pi$$
−0.00385212 + 0.999993i $$0.501226\pi$$
$$684$$ −2943.57 −0.164547
$$685$$ −4288.71 −0.239217
$$686$$ −11677.7 −0.649936
$$687$$ −2581.37 −0.143356
$$688$$ −1369.46 −0.0758867
$$689$$ 5758.47 0.318404
$$690$$ 1491.46 0.0822882
$$691$$ −303.235 −0.0166940 −0.00834702 0.999965i $$-0.502657\pi$$
−0.00834702 + 0.999965i $$0.502657\pi$$
$$692$$ −9599.69 −0.527349
$$693$$ 3103.30 0.170108
$$694$$ −14158.6 −0.774426
$$695$$ −18074.0 −0.986452
$$696$$ −26411.5 −1.43840
$$697$$ 0 0
$$698$$ −8406.54 −0.455863
$$699$$ −2985.24 −0.161534
$$700$$ −5554.79 −0.299931
$$701$$ −23434.3 −1.26263 −0.631313 0.775528i $$-0.717484\pi$$
−0.631313 + 0.775528i $$0.717484\pi$$
$$702$$ −17136.5 −0.921331
$$703$$ 28528.0 1.53052
$$704$$ 8976.83 0.480578
$$705$$ 6373.13 0.340463
$$706$$ 16385.4 0.873472
$$707$$ −20513.4 −1.09121
$$708$$ −7030.24 −0.373182
$$709$$ −52.1860 −0.00276430 −0.00138215 0.999999i $$-0.500440\pi$$
−0.00138215 + 0.999999i $$0.500440\pi$$
$$710$$ −4622.70 −0.244348
$$711$$ 2788.74 0.147097
$$712$$ −13375.0 −0.704002
$$713$$ −1034.30 −0.0543265
$$714$$ 0 0
$$715$$ 13862.0 0.725046
$$716$$ 18272.3 0.953725
$$717$$ 1215.98 0.0633354
$$718$$ 3347.43 0.173990
$$719$$ −2244.80 −0.116435 −0.0582175 0.998304i $$-0.518542\pi$$
−0.0582175 + 0.998304i $$0.518542\pi$$
$$720$$ −131.039 −0.00678271
$$721$$ −8577.23 −0.443041
$$722$$ 860.445 0.0443524
$$723$$ 28824.9 1.48272
$$724$$ 2348.89 0.120575
$$725$$ −20930.4 −1.07219
$$726$$ −2880.96 −0.147276
$$727$$ −2971.90 −0.151612 −0.0758059 0.997123i $$-0.524153\pi$$
−0.0758059 + 0.997123i $$0.524153\pi$$
$$728$$ −20451.4 −1.04118
$$729$$ 21748.2 1.10492
$$730$$ −7472.91 −0.378883
$$731$$ 0 0
$$732$$ −8263.73 −0.417263
$$733$$ 31704.8 1.59760 0.798802 0.601594i $$-0.205468\pi$$
0.798802 + 0.601594i $$0.205468\pi$$
$$734$$ −5898.00 −0.296593
$$735$$ 4553.02 0.228491
$$736$$ −5292.50 −0.265060
$$737$$ −20355.9 −1.01739
$$738$$ 405.034 0.0202026
$$739$$ −36985.6 −1.84105 −0.920526 0.390681i $$-0.872240\pi$$
−0.920526 + 0.390681i $$0.872240\pi$$
$$740$$ 12408.0 0.616389
$$741$$ 23384.2 1.15930
$$742$$ 2064.36 0.102136
$$743$$ 10601.9 0.523481 0.261741 0.965138i $$-0.415704\pi$$
0.261741 + 0.965138i $$0.415704\pi$$
$$744$$ 3551.01 0.174981
$$745$$ −15092.6 −0.742217
$$746$$ −4538.63 −0.222750
$$747$$ 1113.21 0.0545248
$$748$$ 0 0
$$749$$ −1928.48 −0.0940789
$$750$$ −10509.1 −0.511653
$$751$$ 29161.5 1.41694 0.708468 0.705743i $$-0.249387\pi$$
0.708468 + 0.705743i $$0.249387\pi$$
$$752$$ −558.765 −0.0270958
$$753$$ 11232.0 0.543583
$$754$$ −29969.7 −1.44752
$$755$$ 14526.4 0.700225
$$756$$ 10753.5 0.517329
$$757$$ −34447.9 −1.65394 −0.826969 0.562248i $$-0.809937\pi$$
−0.826969 + 0.562248i $$0.809937\pi$$
$$758$$ −9521.67 −0.456257
$$759$$ −3959.96 −0.189377
$$760$$ −12120.1 −0.578476
$$761$$ −14414.5 −0.686632 −0.343316 0.939220i $$-0.611550\pi$$
−0.343316 + 0.939220i $$0.611550\pi$$
$$762$$ 11421.0 0.542966
$$763$$ −11426.4 −0.542156
$$764$$ 23012.6 1.08975
$$765$$ 0 0
$$766$$ −13357.5 −0.630058
$$767$$ −20512.1 −0.965644
$$768$$ 17690.3 0.831175
$$769$$ −7049.33 −0.330566 −0.165283 0.986246i $$-0.552854\pi$$
−0.165283 + 0.986246i $$0.552854\pi$$
$$770$$ 4969.40 0.232578
$$771$$ −7565.28 −0.353381
$$772$$ −14912.3 −0.695212
$$773$$ 26768.3 1.24552 0.622760 0.782413i $$-0.286011\pi$$
0.622760 + 0.782413i $$0.286011\pi$$
$$774$$ −6384.47 −0.296492
$$775$$ 2814.08 0.130432
$$776$$ 995.097 0.0460334
$$777$$ −22067.3 −1.01887
$$778$$ −12676.7 −0.584165
$$779$$ −2610.29 −0.120055
$$780$$ 10170.8 0.466889
$$781$$ 12273.7 0.562339
$$782$$ 0 0
$$783$$ 40519.0 1.84934
$$784$$ −399.186 −0.0181845
$$785$$ −2350.12 −0.106853
$$786$$ −21096.1 −0.957343
$$787$$ 26085.5 1.18151 0.590755 0.806851i $$-0.298830\pi$$
0.590755 + 0.806851i $$0.298830\pi$$
$$788$$ 9806.68 0.443336
$$789$$ 18994.6 0.857065
$$790$$ 4465.69 0.201116
$$791$$ 8359.95 0.375785
$$792$$ −4993.30 −0.224027
$$793$$ −24111.1 −1.07971
$$794$$ 16270.6 0.727232
$$795$$ −2639.78 −0.117765
$$796$$ 15715.1 0.699758
$$797$$ 18108.6 0.804818 0.402409 0.915460i $$-0.368173\pi$$
0.402409 + 0.915460i $$0.368173\pi$$
$$798$$ 8383.06 0.371876
$$799$$ 0 0
$$800$$ 14399.6 0.636379
$$801$$ 4344.75 0.191653
$$802$$ 14775.0 0.650527
$$803$$ 19841.2 0.871958
$$804$$ −14935.5 −0.655142
$$805$$ −2730.73 −0.119560
$$806$$ 4029.41 0.176092
$$807$$ −35534.1 −1.55001
$$808$$ 33006.6 1.43709
$$809$$ −7304.62 −0.317450 −0.158725 0.987323i $$-0.550738\pi$$
−0.158725 + 0.987323i $$0.550738\pi$$
$$810$$ 5581.38 0.242111
$$811$$ −19511.7 −0.844817 −0.422409 0.906405i $$-0.638815\pi$$
−0.422409 + 0.906405i $$0.638815\pi$$
$$812$$ 18806.6 0.812786
$$813$$ −2107.55 −0.0909163
$$814$$ 18820.6 0.810395
$$815$$ −4620.65 −0.198594
$$816$$ 0 0
$$817$$ 41145.5 1.76193
$$818$$ −9003.45 −0.384839
$$819$$ 6643.45 0.283444
$$820$$ −1135.32 −0.0483503
$$821$$ 29364.3 1.24826 0.624131 0.781320i $$-0.285453\pi$$
0.624131 + 0.781320i $$0.285453\pi$$
$$822$$ −4772.96 −0.202526
$$823$$ 28112.4 1.19069 0.595345 0.803470i $$-0.297015\pi$$
0.595345 + 0.803470i $$0.297015\pi$$
$$824$$ 13801.0 0.583472
$$825$$ 10774.1 0.454673
$$826$$ −7353.43 −0.309756
$$827$$ 39360.5 1.65502 0.827509 0.561453i $$-0.189757\pi$$
0.827509 + 0.561453i $$0.189757\pi$$
$$828$$ 1067.12 0.0447885
$$829$$ 15525.3 0.650443 0.325221 0.945638i $$-0.394561\pi$$
0.325221 + 0.945638i $$0.394561\pi$$
$$830$$ 1782.61 0.0745485
$$831$$ 12559.3 0.524280
$$832$$ 19217.3 0.800771
$$833$$ 0 0
$$834$$ −20114.7 −0.835151
$$835$$ −5446.44 −0.225727
$$836$$ 12515.1 0.517756
$$837$$ −5447.75 −0.224972
$$838$$ 16045.5 0.661433
$$839$$ −30007.9 −1.23479 −0.617394 0.786654i $$-0.711811\pi$$
−0.617394 + 0.786654i $$0.711811\pi$$
$$840$$ 9375.29 0.385093
$$841$$ 46474.0 1.90553
$$842$$ 13971.1 0.571826
$$843$$ 11101.3 0.453556
$$844$$ 18341.2 0.748020
$$845$$ 14714.1 0.599029
$$846$$ −2604.99 −0.105864
$$847$$ 5274.78 0.213983
$$848$$ 231.443 0.00937239
$$849$$ −10741.0 −0.434193
$$850$$ 0 0
$$851$$ −10342.1 −0.416595
$$852$$ 9005.44 0.362114
$$853$$ 4351.43 0.174666 0.0873331 0.996179i $$-0.472166\pi$$
0.0873331 + 0.996179i $$0.472166\pi$$
$$854$$ −8643.62 −0.346345
$$855$$ 3937.09 0.157480
$$856$$ 3102.98 0.123899
$$857$$ −25861.5 −1.03082 −0.515410 0.856944i $$-0.672360\pi$$
−0.515410 + 0.856944i $$0.672360\pi$$
$$858$$ 15427.2 0.613840
$$859$$ 19084.0 0.758017 0.379009 0.925393i $$-0.376265\pi$$
0.379009 + 0.925393i $$0.376265\pi$$
$$860$$ 17895.9 0.709588
$$861$$ 2019.14 0.0799213
$$862$$ 3326.16 0.131426
$$863$$ 24790.2 0.977831 0.488916 0.872331i $$-0.337393\pi$$
0.488916 + 0.872331i $$0.337393\pi$$
$$864$$ −27876.1 −1.09765
$$865$$ 12839.8 0.504701
$$866$$ −19019.9 −0.746332
$$867$$ 0 0
$$868$$ −2528.54 −0.0988757
$$869$$ −11856.8 −0.462847
$$870$$ 13738.6 0.535383
$$871$$ −43577.2 −1.69524
$$872$$ 18385.5 0.714003
$$873$$ −323.248 −0.0125318
$$874$$ 3928.81 0.152053
$$875$$ 19241.3 0.743401
$$876$$ 14557.9 0.561491
$$877$$ 14394.0 0.554219 0.277109 0.960838i $$-0.410624\pi$$
0.277109 + 0.960838i $$0.410624\pi$$
$$878$$ 801.839 0.0308209
$$879$$ −29560.9 −1.13432
$$880$$ 557.137 0.0213421
$$881$$ 15328.6 0.586190 0.293095 0.956083i $$-0.405315\pi$$
0.293095 + 0.956083i $$0.405315\pi$$
$$882$$ −1861.02 −0.0710475
$$883$$ 8385.90 0.319602 0.159801 0.987149i $$-0.448915\pi$$
0.159801 + 0.987149i $$0.448915\pi$$
$$884$$ 0 0
$$885$$ 9403.11 0.357155
$$886$$ 22397.7 0.849283
$$887$$ −9215.90 −0.348861 −0.174430 0.984669i $$-0.555808\pi$$
−0.174430 + 0.984669i $$0.555808\pi$$
$$888$$ 35507.0 1.34182
$$889$$ −20910.9 −0.788897
$$890$$ 6957.37 0.262035
$$891$$ −14819.1 −0.557191
$$892$$ 8031.28 0.301465
$$893$$ 16788.1 0.629108
$$894$$ −16796.8 −0.628376
$$895$$ −24439.6 −0.912765
$$896$$ −13440.8 −0.501146
$$897$$ −8477.36 −0.315553
$$898$$ −17793.2 −0.661211
$$899$$ −9527.49 −0.353459
$$900$$ −2903.37 −0.107532
$$901$$ 0 0
$$902$$ −1722.07 −0.0635684
$$903$$ −31827.4 −1.17292
$$904$$ −13451.4 −0.494897
$$905$$ −3141.70 −0.115396
$$906$$ 16166.6 0.592825
$$907$$ 7468.98 0.273433 0.136716 0.990610i $$-0.456345\pi$$
0.136716 + 0.990610i $$0.456345\pi$$
$$908$$ −21856.8 −0.798836
$$909$$ −10721.9 −0.391224
$$910$$ 10638.4 0.387536
$$911$$ 30698.3 1.11644 0.558222 0.829692i $$-0.311484\pi$$
0.558222 + 0.829692i $$0.311484\pi$$
$$912$$ 939.854 0.0341246
$$913$$ −4732.98 −0.171565
$$914$$ 432.140 0.0156389
$$915$$ 11052.9 0.399343
$$916$$ −2957.55 −0.106681
$$917$$ 38625.1 1.39096
$$918$$ 0 0
$$919$$ −24550.8 −0.881237 −0.440619 0.897694i $$-0.645241\pi$$
−0.440619 + 0.897694i $$0.645241\pi$$
$$920$$ 4393.82 0.157457
$$921$$ 11159.2 0.399247
$$922$$ −26537.4 −0.947898
$$923$$ 26275.1 0.937006
$$924$$ −9680.86 −0.344672
$$925$$ 28138.3 1.00020
$$926$$ −18296.0 −0.649293
$$927$$ −4483.13 −0.158841
$$928$$ −48752.1 −1.72453
$$929$$ −41655.0 −1.47110 −0.735552 0.677468i $$-0.763077\pi$$
−0.735552 + 0.677468i $$0.763077\pi$$
$$930$$ −1847.15 −0.0651296
$$931$$ 11993.6 0.422206
$$932$$ −3420.28 −0.120209
$$933$$ −15931.5 −0.559027
$$934$$ 3343.54 0.117135
$$935$$ 0 0
$$936$$ −10689.5 −0.373288
$$937$$ 83.7368 0.00291949 0.00145975 0.999999i $$-0.499535\pi$$
0.00145975 + 0.999999i $$0.499535\pi$$
$$938$$ −15622.1 −0.543795
$$939$$ 11608.6 0.403442
$$940$$ 7301.88 0.253363
$$941$$ −49294.1 −1.70770 −0.853848 0.520523i $$-0.825737\pi$$
−0.853848 + 0.520523i $$0.825737\pi$$
$$942$$ −2615.48 −0.0904639
$$943$$ 946.293 0.0326782
$$944$$ −824.418 −0.0284243
$$945$$ −14383.0 −0.495111
$$946$$ 27144.7 0.932927
$$947$$ −9445.27 −0.324108 −0.162054 0.986782i $$-0.551812\pi$$
−0.162054 + 0.986782i $$0.551812\pi$$
$$948$$ −8699.57 −0.298047
$$949$$ 42475.5 1.45291
$$950$$ −10689.3 −0.365061
$$951$$ 17347.7 0.591521
$$952$$ 0 0
$$953$$ 41693.4 1.41719 0.708595 0.705616i $$-0.249329\pi$$
0.708595 + 0.705616i $$0.249329\pi$$
$$954$$ 1079.00 0.0366183
$$955$$ −30779.8 −1.04294
$$956$$ 1393.18 0.0471324
$$957$$ −36477.3 −1.23213
$$958$$ −7323.60 −0.246988
$$959$$ 8738.88 0.294258
$$960$$ −8809.56 −0.296174
$$961$$ −28510.0 −0.957002
$$962$$ 40290.6 1.35033
$$963$$ −1007.97 −0.0337295
$$964$$ 33025.5 1.10340
$$965$$ 19945.5 0.665355
$$966$$ −3039.06 −0.101222
$$967$$ 32380.0 1.07681 0.538403 0.842688i $$-0.319028\pi$$
0.538403 + 0.842688i $$0.319028\pi$$
$$968$$ −8487.28 −0.281809
$$969$$ 0 0
$$970$$ −517.626 −0.0171340
$$971$$ 5401.25 0.178511 0.0892556 0.996009i $$-0.471551\pi$$
0.0892556 + 0.996009i $$0.471551\pi$$
$$972$$ 10051.1 0.331676
$$973$$ 36828.3 1.21342
$$974$$ −30721.5 −1.01066
$$975$$ 23064.8 0.757606
$$976$$ −969.066 −0.0317818
$$977$$ −25086.2 −0.821473 −0.410737 0.911754i $$-0.634728\pi$$
−0.410737 + 0.911754i $$0.634728\pi$$
$$978$$ −5142.37 −0.168134
$$979$$ −18472.4 −0.603045
$$980$$ 5216.52 0.170036
$$981$$ −5972.35 −0.194375
$$982$$ −23421.4 −0.761108
$$983$$ −59631.6 −1.93485 −0.967423 0.253166i $$-0.918528\pi$$
−0.967423 + 0.253166i $$0.918528\pi$$
$$984$$ −3248.86 −0.105254
$$985$$ −13116.7 −0.424296
$$986$$ 0 0
$$987$$ −12986.2 −0.418799
$$988$$ 26792.0 0.862719
$$989$$ −14916.2 −0.479584
$$990$$ 2597.40 0.0833846
$$991$$ −40063.8 −1.28423 −0.642114 0.766609i $$-0.721942\pi$$
−0.642114 + 0.766609i $$0.721942\pi$$
$$992$$ 6554.70 0.209790
$$993$$ 27711.6 0.885601
$$994$$ 9419.43 0.300569
$$995$$ −21019.3 −0.669706
$$996$$ −3472.68 −0.110478
$$997$$ −10818.4 −0.343652 −0.171826 0.985127i $$-0.554967\pi$$
−0.171826 + 0.985127i $$0.554967\pi$$
$$998$$ 13213.5 0.419103
$$999$$ −54472.8 −1.72517
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 289.4.a.g.1.5 12
17.4 even 4 289.4.b.e.288.8 12
17.11 odd 16 17.4.d.a.2.2 12
17.13 even 4 289.4.b.e.288.7 12
17.14 odd 16 17.4.d.a.9.2 yes 12
17.16 even 2 inner 289.4.a.g.1.6 12
51.11 even 16 153.4.l.a.19.2 12
51.14 even 16 153.4.l.a.145.2 12

By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.d.a.2.2 12 17.11 odd 16
17.4.d.a.9.2 yes 12 17.14 odd 16
153.4.l.a.19.2 12 51.11 even 16
153.4.l.a.145.2 12 51.14 even 16
289.4.a.g.1.5 12 1.1 even 1 trivial
289.4.a.g.1.6 12 17.16 even 2 inner
289.4.b.e.288.7 12 17.13 even 4
289.4.b.e.288.8 12 17.4 even 4