# Properties

 Label 1815.4.a.bb.1.5 Level $1815$ Weight $4$ Character 1815.1 Self dual yes Analytic conductor $107.088$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.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: $$107.088466660$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 44x^{4} + 495x^{2} - 1200$$ x^6 - 44*x^4 + 495*x^2 - 1200 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 5$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$3.60164$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.60164 q^{2} +3.00000 q^{3} +4.97180 q^{4} +5.00000 q^{5} +10.8049 q^{6} +31.6129 q^{7} -10.9065 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+3.60164 q^{2} +3.00000 q^{3} +4.97180 q^{4} +5.00000 q^{5} +10.8049 q^{6} +31.6129 q^{7} -10.9065 q^{8} +9.00000 q^{9} +18.0082 q^{10} +14.9154 q^{12} +38.8162 q^{13} +113.858 q^{14} +15.0000 q^{15} -79.0556 q^{16} +138.261 q^{17} +32.4148 q^{18} +12.5997 q^{19} +24.8590 q^{20} +94.8387 q^{21} -179.716 q^{23} -32.7194 q^{24} +25.0000 q^{25} +139.802 q^{26} +27.0000 q^{27} +157.173 q^{28} +119.045 q^{29} +54.0246 q^{30} +75.6616 q^{31} -197.478 q^{32} +497.968 q^{34} +158.064 q^{35} +44.7462 q^{36} -133.716 q^{37} +45.3797 q^{38} +116.449 q^{39} -54.5324 q^{40} +47.0124 q^{41} +341.575 q^{42} -74.0172 q^{43} +45.0000 q^{45} -647.274 q^{46} -375.658 q^{47} -237.167 q^{48} +656.375 q^{49} +90.0410 q^{50} +414.784 q^{51} +192.986 q^{52} -76.7013 q^{53} +97.2443 q^{54} -344.785 q^{56} +37.7992 q^{57} +428.758 q^{58} +866.360 q^{59} +74.5770 q^{60} +442.580 q^{61} +272.506 q^{62} +284.516 q^{63} -78.7994 q^{64} +194.081 q^{65} -633.006 q^{67} +687.409 q^{68} -539.149 q^{69} +569.291 q^{70} -484.728 q^{71} -98.1583 q^{72} -436.143 q^{73} -481.598 q^{74} +75.0000 q^{75} +62.6434 q^{76} +419.405 q^{78} -472.439 q^{79} -395.278 q^{80} +81.0000 q^{81} +169.322 q^{82} +249.160 q^{83} +471.519 q^{84} +691.307 q^{85} -266.583 q^{86} +357.136 q^{87} +236.982 q^{89} +162.074 q^{90} +1227.09 q^{91} -893.515 q^{92} +226.985 q^{93} -1352.99 q^{94} +62.9986 q^{95} -592.434 q^{96} +1522.39 q^{97} +2364.03 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10})$$ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 $$6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100})$$ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 + 120 * q^12 + 116 * q^14 + 90 * q^15 + 164 * q^16 + 200 * q^20 + 56 * q^23 + 150 * q^25 + 292 * q^26 + 162 * q^27 + 576 * q^31 - 92 * q^34 + 360 * q^36 + 332 * q^37 + 496 * q^38 + 348 * q^42 + 270 * q^45 + 96 * q^47 + 492 * q^48 + 454 * q^49 + 308 * q^53 - 652 * q^56 + 1784 * q^58 + 2080 * q^59 + 600 * q^60 + 1928 * q^64 - 1168 * q^67 + 168 * q^69 + 580 * q^70 + 1064 * q^71 + 450 * q^75 + 876 * q^78 + 820 * q^80 + 486 * q^81 + 24 * q^82 + 5412 * q^86 - 684 * q^89 + 2744 * q^91 + 1368 * q^92 + 1728 * q^93 + 2812 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.60164 1.27337 0.636686 0.771123i $$-0.280305\pi$$
0.636686 + 0.771123i $$0.280305\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 4.97180 0.621475
$$5$$ 5.00000 0.447214
$$6$$ 10.8049 0.735181
$$7$$ 31.6129 1.70694 0.853468 0.521146i $$-0.174495\pi$$
0.853468 + 0.521146i $$0.174495\pi$$
$$8$$ −10.9065 −0.482003
$$9$$ 9.00000 0.333333
$$10$$ 18.0082 0.569469
$$11$$ 0 0
$$12$$ 14.9154 0.358809
$$13$$ 38.8162 0.828128 0.414064 0.910248i $$-0.364109\pi$$
0.414064 + 0.910248i $$0.364109\pi$$
$$14$$ 113.858 2.17356
$$15$$ 15.0000 0.258199
$$16$$ −79.0556 −1.23524
$$17$$ 138.261 1.97255 0.986274 0.165115i $$-0.0527995\pi$$
0.986274 + 0.165115i $$0.0527995\pi$$
$$18$$ 32.4148 0.424457
$$19$$ 12.5997 0.152136 0.0760678 0.997103i $$-0.475763\pi$$
0.0760678 + 0.997103i $$0.475763\pi$$
$$20$$ 24.8590 0.277932
$$21$$ 94.8387 0.985500
$$22$$ 0 0
$$23$$ −179.716 −1.62928 −0.814641 0.579966i $$-0.803066\pi$$
−0.814641 + 0.579966i $$0.803066\pi$$
$$24$$ −32.7194 −0.278284
$$25$$ 25.0000 0.200000
$$26$$ 139.802 1.05452
$$27$$ 27.0000 0.192450
$$28$$ 157.173 1.06082
$$29$$ 119.045 0.762280 0.381140 0.924517i $$-0.375532\pi$$
0.381140 + 0.924517i $$0.375532\pi$$
$$30$$ 54.0246 0.328783
$$31$$ 75.6616 0.438362 0.219181 0.975684i $$-0.429661\pi$$
0.219181 + 0.975684i $$0.429661\pi$$
$$32$$ −197.478 −1.09092
$$33$$ 0 0
$$34$$ 497.968 2.51179
$$35$$ 158.064 0.763365
$$36$$ 44.7462 0.207158
$$37$$ −133.716 −0.594131 −0.297066 0.954857i $$-0.596008\pi$$
−0.297066 + 0.954857i $$0.596008\pi$$
$$38$$ 45.3797 0.193725
$$39$$ 116.449 0.478120
$$40$$ −54.5324 −0.215558
$$41$$ 47.0124 0.179076 0.0895378 0.995983i $$-0.471461\pi$$
0.0895378 + 0.995983i $$0.471461\pi$$
$$42$$ 341.575 1.25491
$$43$$ −74.0172 −0.262501 −0.131250 0.991349i $$-0.541899\pi$$
−0.131250 + 0.991349i $$0.541899\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ −647.274 −2.07468
$$47$$ −375.658 −1.16586 −0.582930 0.812522i $$-0.698094\pi$$
−0.582930 + 0.812522i $$0.698094\pi$$
$$48$$ −237.167 −0.713168
$$49$$ 656.375 1.91363
$$50$$ 90.0410 0.254674
$$51$$ 414.784 1.13885
$$52$$ 192.986 0.514661
$$53$$ −76.7013 −0.198788 −0.0993938 0.995048i $$-0.531690\pi$$
−0.0993938 + 0.995048i $$0.531690\pi$$
$$54$$ 97.2443 0.245060
$$55$$ 0 0
$$56$$ −344.785 −0.822747
$$57$$ 37.7992 0.0878355
$$58$$ 428.758 0.970666
$$59$$ 866.360 1.91170 0.955851 0.293851i $$-0.0949370\pi$$
0.955851 + 0.293851i $$0.0949370\pi$$
$$60$$ 74.5770 0.160464
$$61$$ 442.580 0.928961 0.464481 0.885583i $$-0.346241\pi$$
0.464481 + 0.885583i $$0.346241\pi$$
$$62$$ 272.506 0.558198
$$63$$ 284.516 0.568979
$$64$$ −78.7994 −0.153905
$$65$$ 194.081 0.370350
$$66$$ 0 0
$$67$$ −633.006 −1.15424 −0.577120 0.816660i $$-0.695823\pi$$
−0.577120 + 0.816660i $$0.695823\pi$$
$$68$$ 687.409 1.22589
$$69$$ −539.149 −0.940666
$$70$$ 569.291 0.972047
$$71$$ −484.728 −0.810235 −0.405117 0.914265i $$-0.632769\pi$$
−0.405117 + 0.914265i $$0.632769\pi$$
$$72$$ −98.1583 −0.160668
$$73$$ −436.143 −0.699270 −0.349635 0.936886i $$-0.613694\pi$$
−0.349635 + 0.936886i $$0.613694\pi$$
$$74$$ −481.598 −0.756550
$$75$$ 75.0000 0.115470
$$76$$ 62.6434 0.0945485
$$77$$ 0 0
$$78$$ 419.405 0.608825
$$79$$ −472.439 −0.672830 −0.336415 0.941714i $$-0.609214\pi$$
−0.336415 + 0.941714i $$0.609214\pi$$
$$80$$ −395.278 −0.552418
$$81$$ 81.0000 0.111111
$$82$$ 169.322 0.228030
$$83$$ 249.160 0.329504 0.164752 0.986335i $$-0.447318\pi$$
0.164752 + 0.986335i $$0.447318\pi$$
$$84$$ 471.519 0.612464
$$85$$ 691.307 0.882151
$$86$$ −266.583 −0.334261
$$87$$ 357.136 0.440103
$$88$$ 0 0
$$89$$ 236.982 0.282247 0.141124 0.989992i $$-0.454928\pi$$
0.141124 + 0.989992i $$0.454928\pi$$
$$90$$ 162.074 0.189823
$$91$$ 1227.09 1.41356
$$92$$ −893.515 −1.01256
$$93$$ 226.985 0.253089
$$94$$ −1352.99 −1.48457
$$95$$ 62.9986 0.0680371
$$96$$ −592.434 −0.629844
$$97$$ 1522.39 1.59356 0.796778 0.604272i $$-0.206536\pi$$
0.796778 + 0.604272i $$0.206536\pi$$
$$98$$ 2364.03 2.43676
$$99$$ 0 0
$$100$$ 124.295 0.124295
$$101$$ −1551.66 −1.52867 −0.764335 0.644819i $$-0.776933\pi$$
−0.764335 + 0.644819i $$0.776933\pi$$
$$102$$ 1493.90 1.45018
$$103$$ −1454.92 −1.39182 −0.695911 0.718128i $$-0.744999\pi$$
−0.695911 + 0.718128i $$0.744999\pi$$
$$104$$ −423.347 −0.399160
$$105$$ 474.193 0.440729
$$106$$ −276.251 −0.253130
$$107$$ 179.206 0.161912 0.0809558 0.996718i $$-0.474203\pi$$
0.0809558 + 0.996718i $$0.474203\pi$$
$$108$$ 134.239 0.119603
$$109$$ 2011.72 1.76777 0.883887 0.467700i $$-0.154917\pi$$
0.883887 + 0.467700i $$0.154917\pi$$
$$110$$ 0 0
$$111$$ −401.149 −0.343022
$$112$$ −2499.18 −2.10848
$$113$$ 2041.59 1.69962 0.849808 0.527092i $$-0.176718\pi$$
0.849808 + 0.527092i $$0.176718\pi$$
$$114$$ 136.139 0.111847
$$115$$ −898.582 −0.728637
$$116$$ 591.869 0.473739
$$117$$ 349.346 0.276043
$$118$$ 3120.32 2.43431
$$119$$ 4370.84 3.36701
$$120$$ −163.597 −0.124453
$$121$$ 0 0
$$122$$ 1594.02 1.18291
$$123$$ 141.037 0.103389
$$124$$ 376.175 0.272431
$$125$$ 125.000 0.0894427
$$126$$ 1024.72 0.724521
$$127$$ −804.825 −0.562336 −0.281168 0.959659i $$-0.590722\pi$$
−0.281168 + 0.959659i $$0.590722\pi$$
$$128$$ 1296.02 0.894943
$$129$$ −222.052 −0.151555
$$130$$ 699.009 0.471594
$$131$$ 1699.61 1.13356 0.566778 0.823871i $$-0.308190\pi$$
0.566778 + 0.823871i $$0.308190\pi$$
$$132$$ 0 0
$$133$$ 398.314 0.259686
$$134$$ −2279.86 −1.46978
$$135$$ 135.000 0.0860663
$$136$$ −1507.94 −0.950774
$$137$$ 1187.12 0.740311 0.370156 0.928970i $$-0.379304\pi$$
0.370156 + 0.928970i $$0.379304\pi$$
$$138$$ −1941.82 −1.19782
$$139$$ −2687.64 −1.64002 −0.820010 0.572350i $$-0.806032\pi$$
−0.820010 + 0.572350i $$0.806032\pi$$
$$140$$ 785.865 0.474412
$$141$$ −1126.98 −0.673110
$$142$$ −1745.82 −1.03173
$$143$$ 0 0
$$144$$ −711.500 −0.411748
$$145$$ 595.226 0.340902
$$146$$ −1570.83 −0.890430
$$147$$ 1969.12 1.10483
$$148$$ −664.812 −0.369238
$$149$$ 2893.54 1.59092 0.795462 0.606004i $$-0.207228\pi$$
0.795462 + 0.606004i $$0.207228\pi$$
$$150$$ 270.123 0.147036
$$151$$ 25.1251 0.0135408 0.00677038 0.999977i $$-0.497845\pi$$
0.00677038 + 0.999977i $$0.497845\pi$$
$$152$$ −137.419 −0.0733297
$$153$$ 1244.35 0.657516
$$154$$ 0 0
$$155$$ 378.308 0.196042
$$156$$ 578.959 0.297140
$$157$$ −1109.99 −0.564246 −0.282123 0.959378i $$-0.591039\pi$$
−0.282123 + 0.959378i $$0.591039\pi$$
$$158$$ −1701.55 −0.856762
$$159$$ −230.104 −0.114770
$$160$$ −987.390 −0.487875
$$161$$ −5681.36 −2.78108
$$162$$ 291.733 0.141486
$$163$$ −3513.43 −1.68830 −0.844151 0.536105i $$-0.819895\pi$$
−0.844151 + 0.536105i $$0.819895\pi$$
$$164$$ 233.736 0.111291
$$165$$ 0 0
$$166$$ 897.384 0.419581
$$167$$ −3213.84 −1.48919 −0.744594 0.667517i $$-0.767357\pi$$
−0.744594 + 0.667517i $$0.767357\pi$$
$$168$$ −1034.36 −0.475013
$$169$$ −690.305 −0.314203
$$170$$ 2489.84 1.12331
$$171$$ 113.398 0.0507118
$$172$$ −367.999 −0.163138
$$173$$ 2783.27 1.22317 0.611583 0.791180i $$-0.290533\pi$$
0.611583 + 0.791180i $$0.290533\pi$$
$$174$$ 1286.27 0.560414
$$175$$ 790.322 0.341387
$$176$$ 0 0
$$177$$ 2599.08 1.10372
$$178$$ 853.523 0.359406
$$179$$ 1532.11 0.639751 0.319876 0.947460i $$-0.396359\pi$$
0.319876 + 0.947460i $$0.396359\pi$$
$$180$$ 223.731 0.0926441
$$181$$ −23.1715 −0.00951561 −0.00475780 0.999989i $$-0.501514\pi$$
−0.00475780 + 0.999989i $$0.501514\pi$$
$$182$$ 4419.54 1.79999
$$183$$ 1327.74 0.536336
$$184$$ 1960.07 0.785318
$$185$$ −668.582 −0.265703
$$186$$ 817.518 0.322276
$$187$$ 0 0
$$188$$ −1867.70 −0.724553
$$189$$ 853.548 0.328500
$$190$$ 226.898 0.0866365
$$191$$ −4258.34 −1.61321 −0.806604 0.591092i $$-0.798697\pi$$
−0.806604 + 0.591092i $$0.798697\pi$$
$$192$$ −236.398 −0.0888572
$$193$$ −122.310 −0.0456169 −0.0228085 0.999740i $$-0.507261\pi$$
−0.0228085 + 0.999740i $$0.507261\pi$$
$$194$$ 5483.09 2.02919
$$195$$ 582.243 0.213822
$$196$$ 3263.37 1.18927
$$197$$ 1512.84 0.547134 0.273567 0.961853i $$-0.411797\pi$$
0.273567 + 0.961853i $$0.411797\pi$$
$$198$$ 0 0
$$199$$ 1398.64 0.498227 0.249114 0.968474i $$-0.419861\pi$$
0.249114 + 0.968474i $$0.419861\pi$$
$$200$$ −272.662 −0.0964005
$$201$$ −1899.02 −0.666400
$$202$$ −5588.51 −1.94657
$$203$$ 3763.36 1.30116
$$204$$ 2062.23 0.707768
$$205$$ 235.062 0.0800851
$$206$$ −5240.10 −1.77231
$$207$$ −1617.45 −0.543094
$$208$$ −3068.64 −1.02294
$$209$$ 0 0
$$210$$ 1707.87 0.561212
$$211$$ 609.235 0.198775 0.0993873 0.995049i $$-0.468312\pi$$
0.0993873 + 0.995049i $$0.468312\pi$$
$$212$$ −381.344 −0.123542
$$213$$ −1454.19 −0.467789
$$214$$ 645.437 0.206174
$$215$$ −370.086 −0.117394
$$216$$ −294.475 −0.0927614
$$217$$ 2391.88 0.748256
$$218$$ 7245.47 2.25103
$$219$$ −1308.43 −0.403723
$$220$$ 0 0
$$221$$ 5366.78 1.63352
$$222$$ −1444.79 −0.436794
$$223$$ −2983.97 −0.896060 −0.448030 0.894019i $$-0.647874\pi$$
−0.448030 + 0.894019i $$0.647874\pi$$
$$224$$ −6242.85 −1.86213
$$225$$ 225.000 0.0666667
$$226$$ 7353.07 2.16424
$$227$$ −762.600 −0.222976 −0.111488 0.993766i $$-0.535562\pi$$
−0.111488 + 0.993766i $$0.535562\pi$$
$$228$$ 187.930 0.0545876
$$229$$ −4537.22 −1.30929 −0.654646 0.755935i $$-0.727182\pi$$
−0.654646 + 0.755935i $$0.727182\pi$$
$$230$$ −3236.37 −0.927825
$$231$$ 0 0
$$232$$ −1298.36 −0.367421
$$233$$ 399.725 0.112390 0.0561950 0.998420i $$-0.482103\pi$$
0.0561950 + 0.998420i $$0.482103\pi$$
$$234$$ 1258.22 0.351505
$$235$$ −1878.29 −0.521388
$$236$$ 4307.37 1.18808
$$237$$ −1417.32 −0.388458
$$238$$ 15742.2 4.28746
$$239$$ −3303.30 −0.894029 −0.447015 0.894527i $$-0.647513\pi$$
−0.447015 + 0.894527i $$0.647513\pi$$
$$240$$ −1185.83 −0.318939
$$241$$ 2398.59 0.641107 0.320553 0.947230i $$-0.396131\pi$$
0.320553 + 0.947230i $$0.396131\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 2200.42 0.577326
$$245$$ 3281.87 0.855801
$$246$$ 507.965 0.131653
$$247$$ 489.073 0.125988
$$248$$ −825.202 −0.211292
$$249$$ 747.480 0.190239
$$250$$ 450.205 0.113894
$$251$$ 2613.23 0.657153 0.328577 0.944477i $$-0.393431\pi$$
0.328577 + 0.944477i $$0.393431\pi$$
$$252$$ 1414.56 0.353606
$$253$$ 0 0
$$254$$ −2898.69 −0.716063
$$255$$ 2073.92 0.509310
$$256$$ 5298.18 1.29350
$$257$$ 5774.18 1.40149 0.700746 0.713411i $$-0.252851\pi$$
0.700746 + 0.713411i $$0.252851\pi$$
$$258$$ −799.750 −0.192986
$$259$$ −4227.16 −1.01414
$$260$$ 964.932 0.230164
$$261$$ 1071.41 0.254093
$$262$$ 6121.39 1.44344
$$263$$ −6772.15 −1.58779 −0.793895 0.608055i $$-0.791950\pi$$
−0.793895 + 0.608055i $$0.791950\pi$$
$$264$$ 0 0
$$265$$ −383.507 −0.0889005
$$266$$ 1434.58 0.330676
$$267$$ 710.945 0.162956
$$268$$ −3147.18 −0.717331
$$269$$ −3059.42 −0.693442 −0.346721 0.937968i $$-0.612705\pi$$
−0.346721 + 0.937968i $$0.612705\pi$$
$$270$$ 486.221 0.109594
$$271$$ −1420.57 −0.318427 −0.159213 0.987244i $$-0.550896\pi$$
−0.159213 + 0.987244i $$0.550896\pi$$
$$272$$ −10930.3 −2.43658
$$273$$ 3681.27 0.816120
$$274$$ 4275.58 0.942691
$$275$$ 0 0
$$276$$ −2680.54 −0.584601
$$277$$ 614.227 0.133232 0.0666161 0.997779i $$-0.478780\pi$$
0.0666161 + 0.997779i $$0.478780\pi$$
$$278$$ −9679.91 −2.08835
$$279$$ 680.955 0.146121
$$280$$ −1723.93 −0.367944
$$281$$ 5797.52 1.23079 0.615393 0.788220i $$-0.288997\pi$$
0.615393 + 0.788220i $$0.288997\pi$$
$$282$$ −4058.96 −0.857119
$$283$$ 4077.91 0.856560 0.428280 0.903646i $$-0.359120\pi$$
0.428280 + 0.903646i $$0.359120\pi$$
$$284$$ −2409.97 −0.503541
$$285$$ 188.996 0.0392812
$$286$$ 0 0
$$287$$ 1486.20 0.305671
$$288$$ −1777.30 −0.363641
$$289$$ 14203.2 2.89095
$$290$$ 2143.79 0.434095
$$291$$ 4567.16 0.920040
$$292$$ −2168.42 −0.434579
$$293$$ −5029.67 −1.00286 −0.501428 0.865200i $$-0.667192\pi$$
−0.501428 + 0.865200i $$0.667192\pi$$
$$294$$ 7092.08 1.40686
$$295$$ 4331.80 0.854939
$$296$$ 1458.37 0.286373
$$297$$ 0 0
$$298$$ 10421.5 2.02584
$$299$$ −6975.90 −1.34925
$$300$$ 372.885 0.0717618
$$301$$ −2339.90 −0.448072
$$302$$ 90.4917 0.0172424
$$303$$ −4654.97 −0.882579
$$304$$ −996.079 −0.187924
$$305$$ 2212.90 0.415444
$$306$$ 4481.71 0.837263
$$307$$ 3667.16 0.681746 0.340873 0.940109i $$-0.389277\pi$$
0.340873 + 0.940109i $$0.389277\pi$$
$$308$$ 0 0
$$309$$ −4364.76 −0.803568
$$310$$ 1362.53 0.249634
$$311$$ −2328.46 −0.424549 −0.212274 0.977210i $$-0.568087\pi$$
−0.212274 + 0.977210i $$0.568087\pi$$
$$312$$ −1270.04 −0.230455
$$313$$ 5698.19 1.02901 0.514506 0.857487i $$-0.327975\pi$$
0.514506 + 0.857487i $$0.327975\pi$$
$$314$$ −3997.77 −0.718495
$$315$$ 1422.58 0.254455
$$316$$ −2348.87 −0.418147
$$317$$ 3389.61 0.600567 0.300283 0.953850i $$-0.402919\pi$$
0.300283 + 0.953850i $$0.402919\pi$$
$$318$$ −828.752 −0.146145
$$319$$ 0 0
$$320$$ −393.997 −0.0688285
$$321$$ 537.619 0.0934797
$$322$$ −20462.2 −3.54135
$$323$$ 1742.06 0.300095
$$324$$ 402.716 0.0690528
$$325$$ 970.404 0.165626
$$326$$ −12654.1 −2.14984
$$327$$ 6035.15 1.02063
$$328$$ −512.739 −0.0863149
$$329$$ −11875.6 −1.99005
$$330$$ 0 0
$$331$$ 10574.4 1.75595 0.877977 0.478702i $$-0.158893\pi$$
0.877977 + 0.478702i $$0.158893\pi$$
$$332$$ 1238.77 0.204779
$$333$$ −1203.45 −0.198044
$$334$$ −11575.1 −1.89629
$$335$$ −3165.03 −0.516191
$$336$$ −7497.53 −1.21733
$$337$$ −1929.14 −0.311831 −0.155916 0.987770i $$-0.549833\pi$$
−0.155916 + 0.987770i $$0.549833\pi$$
$$338$$ −2486.23 −0.400098
$$339$$ 6124.77 0.981274
$$340$$ 3437.04 0.548235
$$341$$ 0 0
$$342$$ 408.417 0.0645750
$$343$$ 9906.69 1.55951
$$344$$ 807.267 0.126526
$$345$$ −2695.75 −0.420679
$$346$$ 10024.3 1.55755
$$347$$ −10407.4 −1.61008 −0.805039 0.593221i $$-0.797856\pi$$
−0.805039 + 0.593221i $$0.797856\pi$$
$$348$$ 1775.61 0.273513
$$349$$ −11338.1 −1.73901 −0.869503 0.493928i $$-0.835561\pi$$
−0.869503 + 0.493928i $$0.835561\pi$$
$$350$$ 2846.46 0.434713
$$351$$ 1048.04 0.159373
$$352$$ 0 0
$$353$$ −3084.18 −0.465027 −0.232513 0.972593i $$-0.574695\pi$$
−0.232513 + 0.972593i $$0.574695\pi$$
$$354$$ 9360.95 1.40545
$$355$$ −2423.64 −0.362348
$$356$$ 1178.23 0.175410
$$357$$ 13112.5 1.94395
$$358$$ 5518.12 0.814641
$$359$$ 2292.87 0.337083 0.168542 0.985695i $$-0.446094\pi$$
0.168542 + 0.985695i $$0.446094\pi$$
$$360$$ −490.791 −0.0718527
$$361$$ −6700.25 −0.976855
$$362$$ −83.4554 −0.0121169
$$363$$ 0 0
$$364$$ 6100.86 0.878494
$$365$$ −2180.71 −0.312723
$$366$$ 4782.05 0.682955
$$367$$ −1121.90 −0.159572 −0.0797858 0.996812i $$-0.525424\pi$$
−0.0797858 + 0.996812i $$0.525424\pi$$
$$368$$ 14207.6 2.01256
$$369$$ 423.112 0.0596919
$$370$$ −2407.99 −0.338339
$$371$$ −2424.75 −0.339318
$$372$$ 1128.52 0.157288
$$373$$ −10327.8 −1.43365 −0.716826 0.697253i $$-0.754406\pi$$
−0.716826 + 0.697253i $$0.754406\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 4097.11 0.561947
$$377$$ 4620.88 0.631266
$$378$$ 3074.17 0.418302
$$379$$ −3836.25 −0.519934 −0.259967 0.965617i $$-0.583712\pi$$
−0.259967 + 0.965617i $$0.583712\pi$$
$$380$$ 313.217 0.0422834
$$381$$ −2414.48 −0.324665
$$382$$ −15337.0 −2.05421
$$383$$ −9206.79 −1.22832 −0.614158 0.789183i $$-0.710504\pi$$
−0.614158 + 0.789183i $$0.710504\pi$$
$$384$$ 3888.05 0.516696
$$385$$ 0 0
$$386$$ −440.517 −0.0580873
$$387$$ −666.155 −0.0875002
$$388$$ 7569.01 0.990356
$$389$$ −3057.27 −0.398483 −0.199241 0.979950i $$-0.563848\pi$$
−0.199241 + 0.979950i $$0.563848\pi$$
$$390$$ 2097.03 0.272275
$$391$$ −24847.9 −3.21384
$$392$$ −7158.73 −0.922374
$$393$$ 5098.84 0.654459
$$394$$ 5448.70 0.696704
$$395$$ −2362.19 −0.300899
$$396$$ 0 0
$$397$$ −9682.72 −1.22409 −0.612043 0.790825i $$-0.709652\pi$$
−0.612043 + 0.790825i $$0.709652\pi$$
$$398$$ 5037.41 0.634428
$$399$$ 1194.94 0.149930
$$400$$ −1976.39 −0.247049
$$401$$ −924.364 −0.115114 −0.0575568 0.998342i $$-0.518331\pi$$
−0.0575568 + 0.998342i $$0.518331\pi$$
$$402$$ −6839.58 −0.848575
$$403$$ 2936.89 0.363020
$$404$$ −7714.54 −0.950031
$$405$$ 405.000 0.0496904
$$406$$ 13554.3 1.65687
$$407$$ 0 0
$$408$$ −4523.83 −0.548929
$$409$$ 4949.78 0.598413 0.299207 0.954188i $$-0.403278\pi$$
0.299207 + 0.954188i $$0.403278\pi$$
$$410$$ 846.608 0.101978
$$411$$ 3561.36 0.427419
$$412$$ −7233.58 −0.864983
$$413$$ 27388.1 3.26315
$$414$$ −5825.46 −0.691560
$$415$$ 1245.80 0.147359
$$416$$ −7665.34 −0.903423
$$417$$ −8062.92 −0.946865
$$418$$ 0 0
$$419$$ −8647.87 −1.00830 −0.504148 0.863617i $$-0.668193\pi$$
−0.504148 + 0.863617i $$0.668193\pi$$
$$420$$ 2357.60 0.273902
$$421$$ −9322.71 −1.07924 −0.539621 0.841908i $$-0.681432\pi$$
−0.539621 + 0.841908i $$0.681432\pi$$
$$422$$ 2194.24 0.253114
$$423$$ −3380.93 −0.388620
$$424$$ 836.541 0.0958161
$$425$$ 3456.54 0.394510
$$426$$ −5237.45 −0.595670
$$427$$ 13991.2 1.58568
$$428$$ 890.979 0.100624
$$429$$ 0 0
$$430$$ −1332.92 −0.149486
$$431$$ −8160.16 −0.911975 −0.455987 0.889986i $$-0.650714\pi$$
−0.455987 + 0.889986i $$0.650714\pi$$
$$432$$ −2134.50 −0.237723
$$433$$ −7199.03 −0.798991 −0.399496 0.916735i $$-0.630815\pi$$
−0.399496 + 0.916735i $$0.630815\pi$$
$$434$$ 8614.70 0.952808
$$435$$ 1785.68 0.196820
$$436$$ 10001.9 1.09863
$$437$$ −2264.38 −0.247872
$$438$$ −4712.49 −0.514090
$$439$$ −10297.5 −1.11953 −0.559766 0.828651i $$-0.689109\pi$$
−0.559766 + 0.828651i $$0.689109\pi$$
$$440$$ 0 0
$$441$$ 5907.37 0.637876
$$442$$ 19329.2 2.08008
$$443$$ 12847.2 1.37786 0.688928 0.724830i $$-0.258082\pi$$
0.688928 + 0.724830i $$0.258082\pi$$
$$444$$ −1994.44 −0.213180
$$445$$ 1184.91 0.126225
$$446$$ −10747.2 −1.14102
$$447$$ 8680.61 0.918520
$$448$$ −2491.08 −0.262706
$$449$$ 12493.1 1.31311 0.656555 0.754278i $$-0.272013\pi$$
0.656555 + 0.754278i $$0.272013\pi$$
$$450$$ 810.369 0.0848914
$$451$$ 0 0
$$452$$ 10150.4 1.05627
$$453$$ 75.3754 0.00781776
$$454$$ −2746.61 −0.283931
$$455$$ 6135.46 0.632164
$$456$$ −412.256 −0.0423369
$$457$$ −14968.1 −1.53212 −0.766061 0.642767i $$-0.777786\pi$$
−0.766061 + 0.642767i $$0.777786\pi$$
$$458$$ −16341.4 −1.66722
$$459$$ 3733.06 0.379617
$$460$$ −4467.57 −0.452830
$$461$$ −13649.5 −1.37900 −0.689501 0.724285i $$-0.742170\pi$$
−0.689501 + 0.724285i $$0.742170\pi$$
$$462$$ 0 0
$$463$$ −4655.26 −0.467275 −0.233638 0.972324i $$-0.575063\pi$$
−0.233638 + 0.972324i $$0.575063\pi$$
$$464$$ −9411.19 −0.941602
$$465$$ 1134.92 0.113185
$$466$$ 1439.67 0.143114
$$467$$ −1053.94 −0.104434 −0.0522170 0.998636i $$-0.516629\pi$$
−0.0522170 + 0.998636i $$0.516629\pi$$
$$468$$ 1736.88 0.171554
$$469$$ −20011.2 −1.97021
$$470$$ −6764.93 −0.663921
$$471$$ −3329.96 −0.325768
$$472$$ −9448.93 −0.921445
$$473$$ 0 0
$$474$$ −5104.66 −0.494652
$$475$$ 314.993 0.0304271
$$476$$ 21731.0 2.09252
$$477$$ −690.312 −0.0662625
$$478$$ −11897.3 −1.13843
$$479$$ −2045.84 −0.195150 −0.0975749 0.995228i $$-0.531109\pi$$
−0.0975749 + 0.995228i $$0.531109\pi$$
$$480$$ −2962.17 −0.281675
$$481$$ −5190.36 −0.492017
$$482$$ 8638.85 0.816367
$$483$$ −17044.1 −1.60566
$$484$$ 0 0
$$485$$ 7611.93 0.712660
$$486$$ 875.198 0.0816868
$$487$$ 11771.6 1.09532 0.547661 0.836700i $$-0.315518\pi$$
0.547661 + 0.836700i $$0.315518\pi$$
$$488$$ −4826.99 −0.447762
$$489$$ −10540.3 −0.974742
$$490$$ 11820.1 1.08975
$$491$$ −7886.33 −0.724858 −0.362429 0.932011i $$-0.618052\pi$$
−0.362429 + 0.932011i $$0.618052\pi$$
$$492$$ 701.209 0.0642540
$$493$$ 16459.4 1.50364
$$494$$ 1761.46 0.160429
$$495$$ 0 0
$$496$$ −5981.48 −0.541484
$$497$$ −15323.7 −1.38302
$$498$$ 2692.15 0.242245
$$499$$ 13390.6 1.20129 0.600645 0.799516i $$-0.294910\pi$$
0.600645 + 0.799516i $$0.294910\pi$$
$$500$$ 621.475 0.0555864
$$501$$ −9641.52 −0.859784
$$502$$ 9411.90 0.836800
$$503$$ −28.9098 −0.00256267 −0.00128134 0.999999i $$-0.500408\pi$$
−0.00128134 + 0.999999i $$0.500408\pi$$
$$504$$ −3103.07 −0.274249
$$505$$ −7758.29 −0.683642
$$506$$ 0 0
$$507$$ −2070.92 −0.181405
$$508$$ −4001.43 −0.349478
$$509$$ 8883.68 0.773599 0.386800 0.922164i $$-0.373580\pi$$
0.386800 + 0.922164i $$0.373580\pi$$
$$510$$ 7469.52 0.648541
$$511$$ −13787.7 −1.19361
$$512$$ 8714.00 0.752164
$$513$$ 340.193 0.0292785
$$514$$ 20796.5 1.78462
$$515$$ −7274.60 −0.622441
$$516$$ −1104.00 −0.0941876
$$517$$ 0 0
$$518$$ −15224.7 −1.29138
$$519$$ 8349.80 0.706196
$$520$$ −2116.74 −0.178510
$$521$$ −1371.72 −0.115348 −0.0576738 0.998335i $$-0.518368\pi$$
−0.0576738 + 0.998335i $$0.518368\pi$$
$$522$$ 3858.82 0.323555
$$523$$ 6234.89 0.521286 0.260643 0.965435i $$-0.416065\pi$$
0.260643 + 0.965435i $$0.416065\pi$$
$$524$$ 8450.14 0.704477
$$525$$ 2370.97 0.197100
$$526$$ −24390.8 −2.02185
$$527$$ 10461.1 0.864691
$$528$$ 0 0
$$529$$ 20131.0 1.65456
$$530$$ −1381.25 −0.113203
$$531$$ 7797.24 0.637234
$$532$$ 1980.34 0.161388
$$533$$ 1824.84 0.148298
$$534$$ 2560.57 0.207503
$$535$$ 896.032 0.0724090
$$536$$ 6903.87 0.556346
$$537$$ 4596.34 0.369361
$$538$$ −11018.9 −0.883009
$$539$$ 0 0
$$540$$ 671.193 0.0534881
$$541$$ −11629.7 −0.924212 −0.462106 0.886825i $$-0.652906\pi$$
−0.462106 + 0.886825i $$0.652906\pi$$
$$542$$ −5116.39 −0.405475
$$543$$ −69.5145 −0.00549384
$$544$$ −27303.6 −2.15190
$$545$$ 10058.6 0.790573
$$546$$ 13258.6 1.03922
$$547$$ 22397.1 1.75070 0.875349 0.483492i $$-0.160632\pi$$
0.875349 + 0.483492i $$0.160632\pi$$
$$548$$ 5902.13 0.460085
$$549$$ 3983.22 0.309654
$$550$$ 0 0
$$551$$ 1499.94 0.115970
$$552$$ 5880.22 0.453403
$$553$$ −14935.2 −1.14848
$$554$$ 2212.22 0.169654
$$555$$ −2005.75 −0.153404
$$556$$ −13362.4 −1.01923
$$557$$ 3202.74 0.243635 0.121817 0.992553i $$-0.461128\pi$$
0.121817 + 0.992553i $$0.461128\pi$$
$$558$$ 2452.55 0.186066
$$559$$ −2873.07 −0.217384
$$560$$ −12495.9 −0.942942
$$561$$ 0 0
$$562$$ 20880.6 1.56725
$$563$$ 19681.1 1.47329 0.736644 0.676281i $$-0.236410\pi$$
0.736644 + 0.676281i $$0.236410\pi$$
$$564$$ −5603.10 −0.418321
$$565$$ 10208.0 0.760092
$$566$$ 14687.2 1.09072
$$567$$ 2560.64 0.189660
$$568$$ 5286.68 0.390535
$$569$$ −21352.3 −1.57317 −0.786585 0.617482i $$-0.788153\pi$$
−0.786585 + 0.617482i $$0.788153\pi$$
$$570$$ 680.695 0.0500196
$$571$$ 8197.73 0.600813 0.300407 0.953811i $$-0.402878\pi$$
0.300407 + 0.953811i $$0.402878\pi$$
$$572$$ 0 0
$$573$$ −12775.0 −0.931387
$$574$$ 5352.75 0.389232
$$575$$ −4492.91 −0.325856
$$576$$ −709.195 −0.0513017
$$577$$ −17324.5 −1.24996 −0.624982 0.780640i $$-0.714894\pi$$
−0.624982 + 0.780640i $$0.714894\pi$$
$$578$$ 51154.9 3.68125
$$579$$ −366.930 −0.0263370
$$580$$ 2959.35 0.211862
$$581$$ 7876.67 0.562443
$$582$$ 16449.3 1.17155
$$583$$ 0 0
$$584$$ 4756.78 0.337050
$$585$$ 1746.73 0.123450
$$586$$ −18115.1 −1.27701
$$587$$ −829.014 −0.0582914 −0.0291457 0.999575i $$-0.509279\pi$$
−0.0291457 + 0.999575i $$0.509279\pi$$
$$588$$ 9790.10 0.686627
$$589$$ 953.316 0.0666905
$$590$$ 15601.6 1.08866
$$591$$ 4538.52 0.315888
$$592$$ 10571.0 0.733897
$$593$$ −20619.8 −1.42791 −0.713957 0.700189i $$-0.753099\pi$$
−0.713957 + 0.700189i $$0.753099\pi$$
$$594$$ 0 0
$$595$$ 21854.2 1.50577
$$596$$ 14386.1 0.988720
$$597$$ 4195.93 0.287652
$$598$$ −25124.7 −1.71810
$$599$$ 6047.56 0.412515 0.206258 0.978498i $$-0.433871\pi$$
0.206258 + 0.978498i $$0.433871\pi$$
$$600$$ −817.985 −0.0556569
$$601$$ 3645.81 0.247447 0.123724 0.992317i $$-0.460516\pi$$
0.123724 + 0.992317i $$0.460516\pi$$
$$602$$ −8427.47 −0.570562
$$603$$ −5697.06 −0.384746
$$604$$ 124.917 0.00841525
$$605$$ 0 0
$$606$$ −16765.5 −1.12385
$$607$$ −7474.16 −0.499780 −0.249890 0.968274i $$-0.580395\pi$$
−0.249890 + 0.968274i $$0.580395\pi$$
$$608$$ −2488.17 −0.165968
$$609$$ 11290.1 0.751227
$$610$$ 7970.08 0.529015
$$611$$ −14581.6 −0.965482
$$612$$ 6186.68 0.408630
$$613$$ −10709.8 −0.705652 −0.352826 0.935689i $$-0.614779\pi$$
−0.352826 + 0.935689i $$0.614779\pi$$
$$614$$ 13207.8 0.868116
$$615$$ 705.186 0.0462371
$$616$$ 0 0
$$617$$ −26312.2 −1.71684 −0.858418 0.512950i $$-0.828552\pi$$
−0.858418 + 0.512950i $$0.828552\pi$$
$$618$$ −15720.3 −1.02324
$$619$$ −20408.5 −1.32518 −0.662590 0.748983i $$-0.730543\pi$$
−0.662590 + 0.748983i $$0.730543\pi$$
$$620$$ 1880.87 0.121835
$$621$$ −4852.34 −0.313555
$$622$$ −8386.26 −0.540608
$$623$$ 7491.68 0.481778
$$624$$ −9205.91 −0.590595
$$625$$ 625.000 0.0400000
$$626$$ 20522.8 1.31032
$$627$$ 0 0
$$628$$ −5518.64 −0.350665
$$629$$ −18487.8 −1.17195
$$630$$ 5123.62 0.324016
$$631$$ −13017.6 −0.821273 −0.410636 0.911799i $$-0.634693\pi$$
−0.410636 + 0.911799i $$0.634693\pi$$
$$632$$ 5152.64 0.324306
$$633$$ 1827.70 0.114763
$$634$$ 12208.2 0.764745
$$635$$ −4024.13 −0.251484
$$636$$ −1144.03 −0.0713267
$$637$$ 25478.0 1.58473
$$638$$ 0 0
$$639$$ −4362.56 −0.270078
$$640$$ 6480.08 0.400231
$$641$$ 22044.5 1.35836 0.679178 0.733974i $$-0.262337\pi$$
0.679178 + 0.733974i $$0.262337\pi$$
$$642$$ 1936.31 0.119034
$$643$$ −22105.6 −1.35577 −0.677884 0.735169i $$-0.737103\pi$$
−0.677884 + 0.735169i $$0.737103\pi$$
$$644$$ −28246.6 −1.72837
$$645$$ −1110.26 −0.0677774
$$646$$ 6274.26 0.382132
$$647$$ −25733.7 −1.56367 −0.781836 0.623484i $$-0.785717\pi$$
−0.781836 + 0.623484i $$0.785717\pi$$
$$648$$ −883.424 −0.0535558
$$649$$ 0 0
$$650$$ 3495.05 0.210903
$$651$$ 7175.65 0.432006
$$652$$ −17468.1 −1.04924
$$653$$ −13778.1 −0.825697 −0.412849 0.910800i $$-0.635466\pi$$
−0.412849 + 0.910800i $$0.635466\pi$$
$$654$$ 21736.4 1.29964
$$655$$ 8498.06 0.506942
$$656$$ −3716.59 −0.221202
$$657$$ −3925.29 −0.233090
$$658$$ −42771.8 −2.53407
$$659$$ 9441.03 0.558074 0.279037 0.960280i $$-0.409985\pi$$
0.279037 + 0.960280i $$0.409985\pi$$
$$660$$ 0 0
$$661$$ 12160.3 0.715553 0.357776 0.933807i $$-0.383535\pi$$
0.357776 + 0.933807i $$0.383535\pi$$
$$662$$ 38085.1 2.23598
$$663$$ 16100.3 0.943115
$$664$$ −2717.46 −0.158822
$$665$$ 1991.57 0.116135
$$666$$ −4334.38 −0.252183
$$667$$ −21394.4 −1.24197
$$668$$ −15978.6 −0.925494
$$669$$ −8951.91 −0.517341
$$670$$ −11399.3 −0.657304
$$671$$ 0 0
$$672$$ −18728.5 −1.07510
$$673$$ −26325.7 −1.50784 −0.753922 0.656964i $$-0.771841\pi$$
−0.753922 + 0.656964i $$0.771841\pi$$
$$674$$ −6948.08 −0.397077
$$675$$ 675.000 0.0384900
$$676$$ −3432.06 −0.195270
$$677$$ 21595.7 1.22598 0.612991 0.790090i $$-0.289966\pi$$
0.612991 + 0.790090i $$0.289966\pi$$
$$678$$ 22059.2 1.24953
$$679$$ 48127.0 2.72010
$$680$$ −7539.72 −0.425199
$$681$$ −2287.80 −0.128735
$$682$$ 0 0
$$683$$ 15507.2 0.868766 0.434383 0.900728i $$-0.356966\pi$$
0.434383 + 0.900728i $$0.356966\pi$$
$$684$$ 563.790 0.0315162
$$685$$ 5935.61 0.331077
$$686$$ 35680.3 1.98583
$$687$$ −13611.7 −0.755921
$$688$$ 5851.48 0.324252
$$689$$ −2977.25 −0.164622
$$690$$ −9709.11 −0.535680
$$691$$ −16616.7 −0.914804 −0.457402 0.889260i $$-0.651220\pi$$
−0.457402 + 0.889260i $$0.651220\pi$$
$$692$$ 13837.9 0.760168
$$693$$ 0 0
$$694$$ −37483.6 −2.05023
$$695$$ −13438.2 −0.733439
$$696$$ −3895.09 −0.212131
$$697$$ 6500.00 0.353235
$$698$$ −40835.6 −2.21440
$$699$$ 1199.18 0.0648884
$$700$$ 3929.33 0.212164
$$701$$ −15303.7 −0.824552 −0.412276 0.911059i $$-0.635266\pi$$
−0.412276 + 0.911059i $$0.635266\pi$$
$$702$$ 3774.65 0.202942
$$703$$ −1684.79 −0.0903884
$$704$$ 0 0
$$705$$ −5634.88 −0.301024
$$706$$ −11108.1 −0.592152
$$707$$ −49052.4 −2.60934
$$708$$ 12922.1 0.685936
$$709$$ −15341.9 −0.812664 −0.406332 0.913726i $$-0.633192\pi$$
−0.406332 + 0.913726i $$0.633192\pi$$
$$710$$ −8729.08 −0.461404
$$711$$ −4251.95 −0.224277
$$712$$ −2584.63 −0.136044
$$713$$ −13597.6 −0.714215
$$714$$ 47226.6 2.47537
$$715$$ 0 0
$$716$$ 7617.36 0.397590
$$717$$ −9909.91 −0.516168
$$718$$ 8258.08 0.429232
$$719$$ −4365.48 −0.226432 −0.113216 0.993570i $$-0.536115\pi$$
−0.113216 + 0.993570i $$0.536115\pi$$
$$720$$ −3557.50 −0.184139
$$721$$ −45994.2 −2.37575
$$722$$ −24131.9 −1.24390
$$723$$ 7195.77 0.370143
$$724$$ −115.204 −0.00591372
$$725$$ 2976.13 0.152456
$$726$$ 0 0
$$727$$ 33165.7 1.69195 0.845975 0.533222i $$-0.179019\pi$$
0.845975 + 0.533222i $$0.179019\pi$$
$$728$$ −13383.2 −0.681340
$$729$$ 729.000 0.0370370
$$730$$ −7854.15 −0.398212
$$731$$ −10233.7 −0.517795
$$732$$ 6601.27 0.333320
$$733$$ 14858.5 0.748717 0.374359 0.927284i $$-0.377863\pi$$
0.374359 + 0.927284i $$0.377863\pi$$
$$734$$ −4040.68 −0.203194
$$735$$ 9845.62 0.494097
$$736$$ 35490.0 1.77742
$$737$$ 0 0
$$738$$ 1523.90 0.0760100
$$739$$ 19969.8 0.994048 0.497024 0.867737i $$-0.334426\pi$$
0.497024 + 0.867737i $$0.334426\pi$$
$$740$$ −3324.06 −0.165128
$$741$$ 1467.22 0.0727391
$$742$$ −8733.08 −0.432077
$$743$$ 11701.0 0.577750 0.288875 0.957367i $$-0.406719\pi$$
0.288875 + 0.957367i $$0.406719\pi$$
$$744$$ −2475.60 −0.121989
$$745$$ 14467.7 0.711483
$$746$$ −37196.9 −1.82557
$$747$$ 2242.44 0.109835
$$748$$ 0 0
$$749$$ 5665.23 0.276373
$$750$$ 1350.61 0.0657566
$$751$$ 29554.0 1.43600 0.718002 0.696041i $$-0.245057\pi$$
0.718002 + 0.696041i $$0.245057\pi$$
$$752$$ 29697.9 1.44012
$$753$$ 7839.68 0.379408
$$754$$ 16642.7 0.803836
$$755$$ 125.626 0.00605561
$$756$$ 4243.67 0.204155
$$757$$ −10649.1 −0.511292 −0.255646 0.966770i $$-0.582288\pi$$
−0.255646 + 0.966770i $$0.582288\pi$$
$$758$$ −13816.8 −0.662069
$$759$$ 0 0
$$760$$ −687.093 −0.0327940
$$761$$ 9031.14 0.430195 0.215098 0.976593i $$-0.430993\pi$$
0.215098 + 0.976593i $$0.430993\pi$$
$$762$$ −8696.07 −0.413419
$$763$$ 63596.2 3.01748
$$764$$ −21171.6 −1.00257
$$765$$ 6221.77 0.294050
$$766$$ −33159.5 −1.56410
$$767$$ 33628.8 1.58313
$$768$$ 15894.5 0.746803
$$769$$ −15844.9 −0.743017 −0.371509 0.928429i $$-0.621159\pi$$
−0.371509 + 0.928429i $$0.621159\pi$$
$$770$$ 0 0
$$771$$ 17322.5 0.809151
$$772$$ −608.101 −0.0283498
$$773$$ 18651.1 0.867832 0.433916 0.900953i $$-0.357131\pi$$
0.433916 + 0.900953i $$0.357131\pi$$
$$774$$ −2399.25 −0.111420
$$775$$ 1891.54 0.0876725
$$776$$ −16603.9 −0.768098
$$777$$ −12681.5 −0.585516
$$778$$ −11011.2 −0.507417
$$779$$ 592.343 0.0272438
$$780$$ 2894.80 0.132885
$$781$$ 0 0
$$782$$ −89493.0 −4.09241
$$783$$ 3214.22 0.146701
$$784$$ −51890.1 −2.36380
$$785$$ −5549.94 −0.252339
$$786$$ 18364.2 0.833369
$$787$$ 25474.1 1.15382 0.576908 0.816809i $$-0.304259\pi$$
0.576908 + 0.816809i $$0.304259\pi$$
$$788$$ 7521.54 0.340030
$$789$$ −20316.5 −0.916711
$$790$$ −8507.77 −0.383156
$$791$$ 64540.6 2.90114
$$792$$ 0 0
$$793$$ 17179.3 0.769299
$$794$$ −34873.7 −1.55872
$$795$$ −1150.52 −0.0513267
$$796$$ 6953.78 0.309636
$$797$$ −22591.3 −1.00405 −0.502024 0.864854i $$-0.667411\pi$$
−0.502024 + 0.864854i $$0.667411\pi$$
$$798$$ 4303.75 0.190916
$$799$$ −51939.1 −2.29972
$$800$$ −4936.95 −0.218184
$$801$$ 2132.84 0.0940824
$$802$$ −3329.22 −0.146582
$$803$$ 0 0
$$804$$ −9441.55 −0.414151
$$805$$ −28406.8 −1.24374
$$806$$ 10577.6 0.462260
$$807$$ −9178.25 −0.400359
$$808$$ 16923.1 0.736823
$$809$$ −10434.9 −0.453489 −0.226745 0.973954i $$-0.572808\pi$$
−0.226745 + 0.973954i $$0.572808\pi$$
$$810$$ 1458.66 0.0632743
$$811$$ 12216.2 0.528937 0.264469 0.964394i $$-0.414803\pi$$
0.264469 + 0.964394i $$0.414803\pi$$
$$812$$ 18710.7 0.808641
$$813$$ −4261.72 −0.183844
$$814$$ 0 0
$$815$$ −17567.2 −0.755032
$$816$$ −32791.0 −1.40676
$$817$$ −932.597 −0.0399357
$$818$$ 17827.3 0.762002
$$819$$ 11043.8 0.471187
$$820$$ 1168.68 0.0497709
$$821$$ −21314.2 −0.906054 −0.453027 0.891497i $$-0.649656\pi$$
−0.453027 + 0.891497i $$0.649656\pi$$
$$822$$ 12826.7 0.544263
$$823$$ 30062.2 1.27327 0.636635 0.771165i $$-0.280326\pi$$
0.636635 + 0.771165i $$0.280326\pi$$
$$824$$ 15868.1 0.670861
$$825$$ 0 0
$$826$$ 98642.2 4.15521
$$827$$ −30802.9 −1.29519 −0.647595 0.761985i $$-0.724225\pi$$
−0.647595 + 0.761985i $$0.724225\pi$$
$$828$$ −8041.63 −0.337519
$$829$$ −15852.0 −0.664130 −0.332065 0.943257i $$-0.607745\pi$$
−0.332065 + 0.943257i $$0.607745\pi$$
$$830$$ 4486.92 0.187642
$$831$$ 1842.68 0.0769217
$$832$$ −3058.69 −0.127453
$$833$$ 90751.3 3.77473
$$834$$ −29039.7 −1.20571
$$835$$ −16069.2 −0.665985
$$836$$ 0 0
$$837$$ 2042.86 0.0843629
$$838$$ −31146.5 −1.28394
$$839$$ 28627.5 1.17799 0.588994 0.808137i $$-0.299524\pi$$
0.588994 + 0.808137i $$0.299524\pi$$
$$840$$ −5171.78 −0.212432
$$841$$ −10217.2 −0.418928
$$842$$ −33577.0 −1.37428
$$843$$ 17392.6 0.710595
$$844$$ 3029.00 0.123534
$$845$$ −3451.53 −0.140516
$$846$$ −12176.9 −0.494858
$$847$$ 0 0
$$848$$ 6063.67 0.245551
$$849$$ 12233.7 0.494535
$$850$$ 12449.2 0.502358
$$851$$ 24031.0 0.968006
$$852$$ −7229.92 −0.290720
$$853$$ 22333.6 0.896468 0.448234 0.893916i $$-0.352053\pi$$
0.448234 + 0.893916i $$0.352053\pi$$
$$854$$ 50391.4 2.01916
$$855$$ 566.988 0.0226790
$$856$$ −1954.51 −0.0780418
$$857$$ 20331.4 0.810392 0.405196 0.914230i $$-0.367203\pi$$
0.405196 + 0.914230i $$0.367203\pi$$
$$858$$ 0 0
$$859$$ −41004.7 −1.62871 −0.814355 0.580367i $$-0.802909\pi$$
−0.814355 + 0.580367i $$0.802909\pi$$
$$860$$ −1840.00 −0.0729574
$$861$$ 4458.59 0.176479
$$862$$ −29390.0 −1.16128
$$863$$ 27576.4 1.08773 0.543865 0.839173i $$-0.316960\pi$$
0.543865 + 0.839173i $$0.316960\pi$$
$$864$$ −5331.90 −0.209948
$$865$$ 13916.3 0.547017
$$866$$ −25928.3 −1.01741
$$867$$ 42609.7 1.66909
$$868$$ 11892.0 0.465023
$$869$$ 0 0
$$870$$ 6431.37 0.250625
$$871$$ −24570.9 −0.955858
$$872$$ −21940.7 −0.852072
$$873$$ 13701.5 0.531185
$$874$$ −8155.47 −0.315633
$$875$$ 3951.61 0.152673
$$876$$ −6505.25 −0.250904
$$877$$ −35904.7 −1.38246 −0.691229 0.722636i $$-0.742931\pi$$
−0.691229 + 0.722636i $$0.742931\pi$$
$$878$$ −37088.0 −1.42558
$$879$$ −15089.0 −0.578999
$$880$$ 0 0
$$881$$ −8200.35 −0.313595 −0.156797 0.987631i $$-0.550117\pi$$
−0.156797 + 0.987631i $$0.550117\pi$$
$$882$$ 21276.2 0.812254
$$883$$ −37817.9 −1.44131 −0.720654 0.693295i $$-0.756158\pi$$
−0.720654 + 0.693295i $$0.756158\pi$$
$$884$$ 26682.6 1.01519
$$885$$ 12995.4 0.493599
$$886$$ 46271.1 1.75452
$$887$$ 7045.46 0.266701 0.133350 0.991069i $$-0.457426\pi$$
0.133350 + 0.991069i $$0.457426\pi$$
$$888$$ 4375.12 0.165337
$$889$$ −25442.8 −0.959871
$$890$$ 4267.61 0.160731
$$891$$ 0 0
$$892$$ −14835.7 −0.556879
$$893$$ −4733.19 −0.177369
$$894$$ 31264.4 1.16962
$$895$$ 7660.56 0.286105
$$896$$ 40970.8 1.52761
$$897$$ −20927.7 −0.778992
$$898$$ 44995.7 1.67208
$$899$$ 9007.15 0.334155
$$900$$ 1118.66 0.0414317
$$901$$ −10604.8 −0.392118
$$902$$ 0 0
$$903$$ −7019.70 −0.258694
$$904$$ −22266.6 −0.819220
$$905$$ −115.858 −0.00425551
$$906$$ 271.475 0.00995492
$$907$$ −31139.3 −1.13998 −0.569992 0.821651i $$-0.693054\pi$$
−0.569992 + 0.821651i $$0.693054\pi$$
$$908$$ −3791.50 −0.138574
$$909$$ −13964.9 −0.509557
$$910$$ 22097.7 0.804980
$$911$$ 5053.79 0.183798 0.0918988 0.995768i $$-0.470706\pi$$
0.0918988 + 0.995768i $$0.470706\pi$$
$$912$$ −2988.24 −0.108498
$$913$$ 0 0
$$914$$ −53909.8 −1.95096
$$915$$ 6638.71 0.239857
$$916$$ −22558.2 −0.813693
$$917$$ 53729.7 1.93491
$$918$$ 13445.1 0.483394
$$919$$ 32201.2 1.15584 0.577922 0.816092i $$-0.303864\pi$$
0.577922 + 0.816092i $$0.303864\pi$$
$$920$$ 9800.36 0.351205
$$921$$ 11001.5 0.393606
$$922$$ −49160.5 −1.75598
$$923$$ −18815.3 −0.670978
$$924$$ 0 0
$$925$$ −3342.91 −0.118826
$$926$$ −16766.6 −0.595015
$$927$$ −13094.3 −0.463940
$$928$$ −23508.8 −0.831588
$$929$$ 18704.0 0.660557 0.330279 0.943883i $$-0.392857\pi$$
0.330279 + 0.943883i $$0.392857\pi$$
$$930$$ 4087.59 0.144126
$$931$$ 8270.14 0.291131
$$932$$ 1987.36 0.0698477
$$933$$ −6985.37 −0.245113
$$934$$ −3795.92 −0.132983
$$935$$ 0 0
$$936$$ −3810.13 −0.133053
$$937$$ −27848.6 −0.970944 −0.485472 0.874252i $$-0.661352\pi$$
−0.485472 + 0.874252i $$0.661352\pi$$
$$938$$ −72073.0 −2.50881
$$939$$ 17094.6 0.594101
$$940$$ −9338.50 −0.324030
$$941$$ 24235.0 0.839573 0.419786 0.907623i $$-0.362105\pi$$
0.419786 + 0.907623i $$0.362105\pi$$
$$942$$ −11993.3 −0.414823
$$943$$ −8448.90 −0.291765
$$944$$ −68490.6 −2.36142
$$945$$ 4267.74 0.146910
$$946$$ 0 0
$$947$$ −35474.1 −1.21727 −0.608635 0.793451i $$-0.708282\pi$$
−0.608635 + 0.793451i $$0.708282\pi$$
$$948$$ −7046.62 −0.241417
$$949$$ −16929.4 −0.579085
$$950$$ 1134.49 0.0387450
$$951$$ 10168.8 0.346737
$$952$$ −47670.5 −1.62291
$$953$$ 6194.48 0.210555 0.105277 0.994443i $$-0.466427\pi$$
0.105277 + 0.994443i $$0.466427\pi$$
$$954$$ −2486.25 −0.0843768
$$955$$ −21291.7 −0.721449
$$956$$ −16423.4 −0.555617
$$957$$ 0 0
$$958$$ −7368.37 −0.248498
$$959$$ 37528.3 1.26366
$$960$$ −1181.99 −0.0397381
$$961$$ −24066.3 −0.807839
$$962$$ −18693.8 −0.626520
$$963$$ 1612.86 0.0539705
$$964$$ 11925.3 0.398432
$$965$$ −611.550 −0.0204005
$$966$$ −61386.6 −2.04460
$$967$$ 14858.3 0.494115 0.247058 0.969001i $$-0.420536\pi$$
0.247058 + 0.969001i $$0.420536\pi$$
$$968$$ 0 0
$$969$$ 5226.17 0.173260
$$970$$ 27415.4 0.907481
$$971$$ 3584.21 0.118458 0.0592290 0.998244i $$-0.481136\pi$$
0.0592290 + 0.998244i $$0.481136\pi$$
$$972$$ 1208.15 0.0398677
$$973$$ −84964.1 −2.79941
$$974$$ 42397.0 1.39475
$$975$$ 2911.21 0.0956240
$$976$$ −34988.5 −1.14749
$$977$$ 39576.7 1.29598 0.647989 0.761650i $$-0.275610\pi$$
0.647989 + 0.761650i $$0.275610\pi$$
$$978$$ −37962.4 −1.24121
$$979$$ 0 0
$$980$$ 16316.8 0.531859
$$981$$ 18105.4 0.589258
$$982$$ −28403.7 −0.923014
$$983$$ −74.1406 −0.00240561 −0.00120281 0.999999i $$-0.500383\pi$$
−0.00120281 + 0.999999i $$0.500383\pi$$
$$984$$ −1538.22 −0.0498340
$$985$$ 7564.20 0.244686
$$986$$ 59280.7 1.91469
$$987$$ −35626.9 −1.14895
$$988$$ 2431.57 0.0782983
$$989$$ 13302.1 0.427687
$$990$$ 0 0
$$991$$ −44698.7 −1.43280 −0.716399 0.697691i $$-0.754211\pi$$
−0.716399 + 0.697691i $$0.754211\pi$$
$$992$$ −14941.5 −0.478219
$$993$$ 31723.2 1.01380
$$994$$ −55190.3 −1.76110
$$995$$ 6993.22 0.222814
$$996$$ 3716.32 0.118229
$$997$$ −7086.74 −0.225115 −0.112557 0.993645i $$-0.535904\pi$$
−0.112557 + 0.993645i $$0.535904\pi$$
$$998$$ 48228.0 1.52969
$$999$$ −3610.34 −0.114341
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 1815.4.a.bb.1.5 yes 6
11.10 odd 2 inner 1815.4.a.bb.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1815.4.a.bb.1.2 6 11.10 odd 2 inner
1815.4.a.bb.1.5 yes 6 1.1 even 1 trivial