# Properties

 Label 1815.4.a.q.1.1 Level $1815$ Weight $4$ Character 1815.1 Self dual yes Analytic conductor $107.088$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$3$$ Coefficient field: 3.3.1957.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 10$$ x^3 - x^2 - 9*x + 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.12946$$ 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)

 $$f(q)$$ $$=$$ $$q-4.59486 q^{2} -3.00000 q^{3} +13.1127 q^{4} +5.00000 q^{5} +13.7846 q^{6} -20.6383 q^{7} -23.4921 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.59486 q^{2} -3.00000 q^{3} +13.1127 q^{4} +5.00000 q^{5} +13.7846 q^{6} -20.6383 q^{7} -23.4921 q^{8} +9.00000 q^{9} -22.9743 q^{10} -39.3381 q^{12} +15.6584 q^{13} +94.8302 q^{14} -15.0000 q^{15} +3.04132 q^{16} -72.9507 q^{17} -41.3537 q^{18} -61.0513 q^{19} +65.5635 q^{20} +61.9150 q^{21} -13.6605 q^{23} +70.4764 q^{24} +25.0000 q^{25} -71.9483 q^{26} -27.0000 q^{27} -270.624 q^{28} +31.4663 q^{29} +68.9228 q^{30} -243.008 q^{31} +173.963 q^{32} +335.198 q^{34} -103.192 q^{35} +118.014 q^{36} -65.4018 q^{37} +280.522 q^{38} -46.9753 q^{39} -117.461 q^{40} +109.087 q^{41} -284.491 q^{42} +121.750 q^{43} +45.0000 q^{45} +62.7678 q^{46} -519.530 q^{47} -9.12396 q^{48} +82.9413 q^{49} -114.871 q^{50} +218.852 q^{51} +205.324 q^{52} -542.673 q^{53} +124.061 q^{54} +484.839 q^{56} +183.154 q^{57} -144.583 q^{58} +109.478 q^{59} -196.691 q^{60} +89.6156 q^{61} +1116.59 q^{62} -185.745 q^{63} -823.664 q^{64} +78.2922 q^{65} +488.446 q^{67} -956.581 q^{68} +40.9814 q^{69} +474.151 q^{70} +837.423 q^{71} -211.429 q^{72} -351.216 q^{73} +300.512 q^{74} -75.0000 q^{75} -800.547 q^{76} +215.845 q^{78} +831.205 q^{79} +15.2066 q^{80} +81.0000 q^{81} -501.238 q^{82} -1389.13 q^{83} +811.873 q^{84} -364.754 q^{85} -559.423 q^{86} -94.3988 q^{87} +1523.70 q^{89} -206.769 q^{90} -323.164 q^{91} -179.125 q^{92} +729.025 q^{93} +2387.17 q^{94} -305.256 q^{95} -521.888 q^{96} -426.612 q^{97} -381.103 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} + 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q - q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 + 3 * q^6 - 6 * q^7 + 3 * q^8 + 27 * q^9 $$3 q - q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} + 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9} - 5 q^{10} - 51 q^{12} + 20 q^{13} + 144 q^{14} - 45 q^{15} + 25 q^{16} - 32 q^{17} - 9 q^{18} - 116 q^{19} + 85 q^{20} + 18 q^{21} + 240 q^{23} - 9 q^{24} + 75 q^{25} + 302 q^{26} - 81 q^{27} - 160 q^{28} - 238 q^{29} + 15 q^{30} + 92 q^{31} - 197 q^{32} + 354 q^{34} - 30 q^{35} + 153 q^{36} - 90 q^{37} + 324 q^{38} - 60 q^{39} + 15 q^{40} + 46 q^{41} - 432 q^{42} + 134 q^{43} + 135 q^{45} + 240 q^{46} - 220 q^{47} - 75 q^{48} - 457 q^{49} - 25 q^{50} + 96 q^{51} + 1530 q^{52} - 798 q^{53} + 27 q^{54} + 688 q^{56} + 348 q^{57} - 978 q^{58} + 1236 q^{59} - 255 q^{60} - 342 q^{61} + 1792 q^{62} - 54 q^{63} - 1919 q^{64} + 100 q^{65} + 764 q^{67} - 1074 q^{68} - 720 q^{69} + 720 q^{70} + 1816 q^{71} + 27 q^{72} - 100 q^{73} + 1874 q^{74} - 225 q^{75} - 396 q^{76} - 906 q^{78} + 96 q^{79} + 125 q^{80} + 243 q^{81} - 910 q^{82} - 858 q^{83} + 480 q^{84} - 160 q^{85} + 188 q^{86} + 714 q^{87} + 838 q^{89} - 45 q^{90} + 332 q^{91} - 688 q^{92} - 276 q^{93} + 3112 q^{94} - 580 q^{95} + 591 q^{96} - 1322 q^{97} - 1017 q^{98}+O(q^{100})$$ 3 * q - q^2 - 9 * q^3 + 17 * q^4 + 15 * q^5 + 3 * q^6 - 6 * q^7 + 3 * q^8 + 27 * q^9 - 5 * q^10 - 51 * q^12 + 20 * q^13 + 144 * q^14 - 45 * q^15 + 25 * q^16 - 32 * q^17 - 9 * q^18 - 116 * q^19 + 85 * q^20 + 18 * q^21 + 240 * q^23 - 9 * q^24 + 75 * q^25 + 302 * q^26 - 81 * q^27 - 160 * q^28 - 238 * q^29 + 15 * q^30 + 92 * q^31 - 197 * q^32 + 354 * q^34 - 30 * q^35 + 153 * q^36 - 90 * q^37 + 324 * q^38 - 60 * q^39 + 15 * q^40 + 46 * q^41 - 432 * q^42 + 134 * q^43 + 135 * q^45 + 240 * q^46 - 220 * q^47 - 75 * q^48 - 457 * q^49 - 25 * q^50 + 96 * q^51 + 1530 * q^52 - 798 * q^53 + 27 * q^54 + 688 * q^56 + 348 * q^57 - 978 * q^58 + 1236 * q^59 - 255 * q^60 - 342 * q^61 + 1792 * q^62 - 54 * q^63 - 1919 * q^64 + 100 * q^65 + 764 * q^67 - 1074 * q^68 - 720 * q^69 + 720 * q^70 + 1816 * q^71 + 27 * q^72 - 100 * q^73 + 1874 * q^74 - 225 * q^75 - 396 * q^76 - 906 * q^78 + 96 * q^79 + 125 * q^80 + 243 * q^81 - 910 * q^82 - 858 * q^83 + 480 * q^84 - 160 * q^85 + 188 * q^86 + 714 * q^87 + 838 * q^89 - 45 * q^90 + 332 * q^91 - 688 * q^92 - 276 * q^93 + 3112 * q^94 - 580 * q^95 + 591 * q^96 - 1322 * q^97 - 1017 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.59486 −1.62453 −0.812263 0.583291i $$-0.801765\pi$$
−0.812263 + 0.583291i $$0.801765\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 13.1127 1.63909
$$5$$ 5.00000 0.447214
$$6$$ 13.7846 0.937921
$$7$$ −20.6383 −1.11437 −0.557183 0.830390i $$-0.688118\pi$$
−0.557183 + 0.830390i $$0.688118\pi$$
$$8$$ −23.4921 −1.03822
$$9$$ 9.00000 0.333333
$$10$$ −22.9743 −0.726511
$$11$$ 0 0
$$12$$ −39.3381 −0.946328
$$13$$ 15.6584 0.334067 0.167033 0.985951i $$-0.446581\pi$$
0.167033 + 0.985951i $$0.446581\pi$$
$$14$$ 94.8302 1.81032
$$15$$ −15.0000 −0.258199
$$16$$ 3.04132 0.0475206
$$17$$ −72.9507 −1.04077 −0.520387 0.853931i $$-0.674212\pi$$
−0.520387 + 0.853931i $$0.674212\pi$$
$$18$$ −41.3537 −0.541509
$$19$$ −61.0513 −0.737165 −0.368582 0.929595i $$-0.620157\pi$$
−0.368582 + 0.929595i $$0.620157\pi$$
$$20$$ 65.5635 0.733022
$$21$$ 61.9150 0.643379
$$22$$ 0 0
$$23$$ −13.6605 −0.123844 −0.0619218 0.998081i $$-0.519723\pi$$
−0.0619218 + 0.998081i $$0.519723\pi$$
$$24$$ 70.4764 0.599414
$$25$$ 25.0000 0.200000
$$26$$ −71.9483 −0.542701
$$27$$ −27.0000 −0.192450
$$28$$ −270.624 −1.82654
$$29$$ 31.4663 0.201487 0.100744 0.994912i $$-0.467878\pi$$
0.100744 + 0.994912i $$0.467878\pi$$
$$30$$ 68.9228 0.419451
$$31$$ −243.008 −1.40792 −0.703961 0.710239i $$-0.748587\pi$$
−0.703961 + 0.710239i $$0.748587\pi$$
$$32$$ 173.963 0.961016
$$33$$ 0 0
$$34$$ 335.198 1.69076
$$35$$ −103.192 −0.498360
$$36$$ 118.014 0.546363
$$37$$ −65.4018 −0.290594 −0.145297 0.989388i $$-0.546414\pi$$
−0.145297 + 0.989388i $$0.546414\pi$$
$$38$$ 280.522 1.19754
$$39$$ −46.9753 −0.192874
$$40$$ −117.461 −0.464304
$$41$$ 109.087 0.415524 0.207762 0.978179i $$-0.433382\pi$$
0.207762 + 0.978179i $$0.433382\pi$$
$$42$$ −284.491 −1.04519
$$43$$ 121.750 0.431783 0.215891 0.976417i $$-0.430734\pi$$
0.215891 + 0.976417i $$0.430734\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 62.7678 0.201187
$$47$$ −519.530 −1.61237 −0.806184 0.591665i $$-0.798471\pi$$
−0.806184 + 0.591665i $$0.798471\pi$$
$$48$$ −9.12396 −0.0274361
$$49$$ 82.9413 0.241811
$$50$$ −114.871 −0.324905
$$51$$ 218.852 0.600891
$$52$$ 205.324 0.547565
$$53$$ −542.673 −1.40645 −0.703226 0.710967i $$-0.748258\pi$$
−0.703226 + 0.710967i $$0.748258\pi$$
$$54$$ 124.061 0.312640
$$55$$ 0 0
$$56$$ 484.839 1.15695
$$57$$ 183.154 0.425602
$$58$$ −144.583 −0.327322
$$59$$ 109.478 0.241574 0.120787 0.992678i $$-0.461458\pi$$
0.120787 + 0.992678i $$0.461458\pi$$
$$60$$ −196.691 −0.423211
$$61$$ 89.6156 0.188100 0.0940501 0.995567i $$-0.470019\pi$$
0.0940501 + 0.995567i $$0.470019\pi$$
$$62$$ 1116.59 2.28721
$$63$$ −185.745 −0.371455
$$64$$ −823.664 −1.60872
$$65$$ 78.2922 0.149399
$$66$$ 0 0
$$67$$ 488.446 0.890644 0.445322 0.895371i $$-0.353089\pi$$
0.445322 + 0.895371i $$0.353089\pi$$
$$68$$ −956.581 −1.70592
$$69$$ 40.9814 0.0715011
$$70$$ 474.151 0.809599
$$71$$ 837.423 1.39977 0.699887 0.714254i $$-0.253234\pi$$
0.699887 + 0.714254i $$0.253234\pi$$
$$72$$ −211.429 −0.346072
$$73$$ −351.216 −0.563105 −0.281553 0.959546i $$-0.590849\pi$$
−0.281553 + 0.959546i $$0.590849\pi$$
$$74$$ 300.512 0.472078
$$75$$ −75.0000 −0.115470
$$76$$ −800.547 −1.20828
$$77$$ 0 0
$$78$$ 215.845 0.313328
$$79$$ 831.205 1.18377 0.591885 0.806022i $$-0.298384\pi$$
0.591885 + 0.806022i $$0.298384\pi$$
$$80$$ 15.2066 0.0212519
$$81$$ 81.0000 0.111111
$$82$$ −501.238 −0.675031
$$83$$ −1389.13 −1.83707 −0.918537 0.395335i $$-0.870629\pi$$
−0.918537 + 0.395335i $$0.870629\pi$$
$$84$$ 811.873 1.05456
$$85$$ −364.754 −0.465448
$$86$$ −559.423 −0.701443
$$87$$ −94.3988 −0.116329
$$88$$ 0 0
$$89$$ 1523.70 1.81474 0.907369 0.420335i $$-0.138088\pi$$
0.907369 + 0.420335i $$0.138088\pi$$
$$90$$ −206.769 −0.242170
$$91$$ −323.164 −0.372273
$$92$$ −179.125 −0.202990
$$93$$ 729.025 0.812864
$$94$$ 2387.17 2.61933
$$95$$ −305.256 −0.329670
$$96$$ −521.888 −0.554843
$$97$$ −426.612 −0.446555 −0.223278 0.974755i $$-0.571676\pi$$
−0.223278 + 0.974755i $$0.571676\pi$$
$$98$$ −381.103 −0.392829
$$99$$ 0 0
$$100$$ 327.818 0.327818
$$101$$ −74.1387 −0.0730403 −0.0365202 0.999333i $$-0.511627\pi$$
−0.0365202 + 0.999333i $$0.511627\pi$$
$$102$$ −1005.59 −0.976163
$$103$$ −69.3916 −0.0663821 −0.0331911 0.999449i $$-0.510567\pi$$
−0.0331911 + 0.999449i $$0.510567\pi$$
$$104$$ −367.850 −0.346833
$$105$$ 309.575 0.287728
$$106$$ 2493.51 2.28482
$$107$$ −1141.71 −1.03152 −0.515761 0.856733i $$-0.672491\pi$$
−0.515761 + 0.856733i $$0.672491\pi$$
$$108$$ −354.043 −0.315443
$$109$$ 2226.85 1.95682 0.978409 0.206680i $$-0.0662659\pi$$
0.978409 + 0.206680i $$0.0662659\pi$$
$$110$$ 0 0
$$111$$ 196.205 0.167775
$$112$$ −62.7678 −0.0529554
$$113$$ −1719.76 −1.43169 −0.715847 0.698257i $$-0.753959\pi$$
−0.715847 + 0.698257i $$0.753959\pi$$
$$114$$ −841.566 −0.691402
$$115$$ −68.3023 −0.0553845
$$116$$ 412.608 0.330256
$$117$$ 140.926 0.111356
$$118$$ −503.038 −0.392444
$$119$$ 1505.58 1.15980
$$120$$ 352.382 0.268066
$$121$$ 0 0
$$122$$ −411.771 −0.305574
$$123$$ −327.260 −0.239903
$$124$$ −3186.49 −2.30771
$$125$$ 125.000 0.0894427
$$126$$ 853.472 0.603439
$$127$$ 1601.63 1.11907 0.559534 0.828807i $$-0.310980\pi$$
0.559534 + 0.828807i $$0.310980\pi$$
$$128$$ 2392.91 1.65239
$$129$$ −365.249 −0.249290
$$130$$ −359.741 −0.242703
$$131$$ −2004.13 −1.33665 −0.668327 0.743868i $$-0.732989\pi$$
−0.668327 + 0.743868i $$0.732989\pi$$
$$132$$ 0 0
$$133$$ 1260.00 0.821471
$$134$$ −2244.34 −1.44687
$$135$$ −135.000 −0.0860663
$$136$$ 1713.77 1.08055
$$137$$ −1672.85 −1.04322 −0.521610 0.853184i $$-0.674669\pi$$
−0.521610 + 0.853184i $$0.674669\pi$$
$$138$$ −188.303 −0.116155
$$139$$ −2540.38 −1.55016 −0.775080 0.631863i $$-0.782291\pi$$
−0.775080 + 0.631863i $$0.782291\pi$$
$$140$$ −1353.12 −0.816855
$$141$$ 1558.59 0.930901
$$142$$ −3847.84 −2.27397
$$143$$ 0 0
$$144$$ 27.3719 0.0158402
$$145$$ 157.331 0.0901079
$$146$$ 1613.79 0.914780
$$147$$ −248.824 −0.139610
$$148$$ −857.594 −0.476310
$$149$$ −3090.68 −1.69932 −0.849658 0.527334i $$-0.823192\pi$$
−0.849658 + 0.527334i $$0.823192\pi$$
$$150$$ 344.614 0.187584
$$151$$ −1358.74 −0.732267 −0.366134 0.930562i $$-0.619319\pi$$
−0.366134 + 0.930562i $$0.619319\pi$$
$$152$$ 1434.22 0.765335
$$153$$ −656.557 −0.346925
$$154$$ 0 0
$$155$$ −1215.04 −0.629642
$$156$$ −615.973 −0.316137
$$157$$ −1011.95 −0.514411 −0.257205 0.966357i $$-0.582802\pi$$
−0.257205 + 0.966357i $$0.582802\pi$$
$$158$$ −3819.27 −1.92307
$$159$$ 1628.02 0.812015
$$160$$ 869.813 0.429780
$$161$$ 281.929 0.138007
$$162$$ −372.183 −0.180503
$$163$$ −2816.37 −1.35334 −0.676672 0.736285i $$-0.736578\pi$$
−0.676672 + 0.736285i $$0.736578\pi$$
$$164$$ 1430.42 0.681081
$$165$$ 0 0
$$166$$ 6382.87 2.98438
$$167$$ −3448.89 −1.59810 −0.799052 0.601262i $$-0.794665\pi$$
−0.799052 + 0.601262i $$0.794665\pi$$
$$168$$ −1454.52 −0.667966
$$169$$ −1951.81 −0.888399
$$170$$ 1675.99 0.756133
$$171$$ −549.462 −0.245722
$$172$$ 1596.47 0.707730
$$173$$ 2287.85 1.00545 0.502723 0.864448i $$-0.332332\pi$$
0.502723 + 0.864448i $$0.332332\pi$$
$$174$$ 433.749 0.188979
$$175$$ −515.959 −0.222873
$$176$$ 0 0
$$177$$ −328.435 −0.139473
$$178$$ −7001.17 −2.94809
$$179$$ 3249.06 1.35668 0.678340 0.734748i $$-0.262700\pi$$
0.678340 + 0.734748i $$0.262700\pi$$
$$180$$ 590.072 0.244341
$$181$$ 1170.45 0.480655 0.240328 0.970692i $$-0.422745\pi$$
0.240328 + 0.970692i $$0.422745\pi$$
$$182$$ 1484.89 0.604767
$$183$$ −268.847 −0.108600
$$184$$ 320.913 0.128576
$$185$$ −327.009 −0.129958
$$186$$ −3349.76 −1.32052
$$187$$ 0 0
$$188$$ −6812.44 −2.64281
$$189$$ 557.235 0.214460
$$190$$ 1402.61 0.535558
$$191$$ −2760.35 −1.04572 −0.522859 0.852419i $$-0.675134\pi$$
−0.522859 + 0.852419i $$0.675134\pi$$
$$192$$ 2470.99 0.928794
$$193$$ 1250.61 0.466430 0.233215 0.972425i $$-0.425075\pi$$
0.233215 + 0.972425i $$0.425075\pi$$
$$194$$ 1960.22 0.725441
$$195$$ −234.877 −0.0862557
$$196$$ 1087.58 0.396350
$$197$$ 143.991 0.0520756 0.0260378 0.999661i $$-0.491711\pi$$
0.0260378 + 0.999661i $$0.491711\pi$$
$$198$$ 0 0
$$199$$ 761.249 0.271174 0.135587 0.990765i $$-0.456708\pi$$
0.135587 + 0.990765i $$0.456708\pi$$
$$200$$ −587.303 −0.207643
$$201$$ −1465.34 −0.514213
$$202$$ 340.657 0.118656
$$203$$ −649.411 −0.224531
$$204$$ 2869.74 0.984913
$$205$$ 545.434 0.185828
$$206$$ 318.844 0.107840
$$207$$ −122.944 −0.0412812
$$208$$ 47.6223 0.0158751
$$209$$ 0 0
$$210$$ −1422.45 −0.467422
$$211$$ −3976.58 −1.29743 −0.648717 0.761029i $$-0.724694\pi$$
−0.648717 + 0.761029i $$0.724694\pi$$
$$212$$ −7115.91 −2.30530
$$213$$ −2512.27 −0.808159
$$214$$ 5245.97 1.67573
$$215$$ 608.749 0.193099
$$216$$ 634.287 0.199805
$$217$$ 5015.29 1.56894
$$218$$ −10232.0 −3.17890
$$219$$ 1053.65 0.325109
$$220$$ 0 0
$$221$$ −1142.29 −0.347688
$$222$$ −901.535 −0.272555
$$223$$ 908.084 0.272690 0.136345 0.990661i $$-0.456464\pi$$
0.136345 + 0.990661i $$0.456464\pi$$
$$224$$ −3590.30 −1.07092
$$225$$ 225.000 0.0666667
$$226$$ 7902.05 2.32583
$$227$$ 2062.15 0.602951 0.301475 0.953474i $$-0.402521\pi$$
0.301475 + 0.953474i $$0.402521\pi$$
$$228$$ 2401.64 0.697599
$$229$$ 4077.47 1.17662 0.588312 0.808634i $$-0.299793\pi$$
0.588312 + 0.808634i $$0.299793\pi$$
$$230$$ 313.839 0.0899737
$$231$$ 0 0
$$232$$ −739.209 −0.209187
$$233$$ −1682.76 −0.473138 −0.236569 0.971615i $$-0.576023\pi$$
−0.236569 + 0.971615i $$0.576023\pi$$
$$234$$ −647.534 −0.180900
$$235$$ −2597.65 −0.721073
$$236$$ 1435.56 0.395961
$$237$$ −2493.62 −0.683450
$$238$$ −6917.93 −1.88413
$$239$$ −4024.96 −1.08934 −0.544672 0.838649i $$-0.683346\pi$$
−0.544672 + 0.838649i $$0.683346\pi$$
$$240$$ −45.6198 −0.0122698
$$241$$ 2784.27 0.744194 0.372097 0.928194i $$-0.378639\pi$$
0.372097 + 0.928194i $$0.378639\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 1175.10 0.308313
$$245$$ 414.707 0.108141
$$246$$ 1503.71 0.389729
$$247$$ −955.968 −0.246262
$$248$$ 5708.78 1.46173
$$249$$ 4167.40 1.06064
$$250$$ −574.357 −0.145302
$$251$$ −1827.60 −0.459591 −0.229796 0.973239i $$-0.573806\pi$$
−0.229796 + 0.973239i $$0.573806\pi$$
$$252$$ −2435.62 −0.608848
$$253$$ 0 0
$$254$$ −7359.26 −1.81796
$$255$$ 1094.26 0.268727
$$256$$ −4405.79 −1.07563
$$257$$ 585.171 0.142031 0.0710155 0.997475i $$-0.477376\pi$$
0.0710155 + 0.997475i $$0.477376\pi$$
$$258$$ 1678.27 0.404978
$$259$$ 1349.78 0.323828
$$260$$ 1026.62 0.244878
$$261$$ 283.196 0.0671625
$$262$$ 9208.69 2.17143
$$263$$ 238.098 0.0558241 0.0279120 0.999610i $$-0.491114\pi$$
0.0279120 + 0.999610i $$0.491114\pi$$
$$264$$ 0 0
$$265$$ −2713.37 −0.628984
$$266$$ −5789.51 −1.33450
$$267$$ −4571.09 −1.04774
$$268$$ 6404.84 1.45984
$$269$$ 4618.46 1.04681 0.523406 0.852083i $$-0.324661\pi$$
0.523406 + 0.852083i $$0.324661\pi$$
$$270$$ 620.306 0.139817
$$271$$ 143.439 0.0321525 0.0160762 0.999871i $$-0.494883\pi$$
0.0160762 + 0.999871i $$0.494883\pi$$
$$272$$ −221.867 −0.0494582
$$273$$ 969.493 0.214932
$$274$$ 7686.51 1.69474
$$275$$ 0 0
$$276$$ 537.376 0.117197
$$277$$ 8602.51 1.86597 0.932987 0.359911i $$-0.117193\pi$$
0.932987 + 0.359911i $$0.117193\pi$$
$$278$$ 11672.7 2.51828
$$279$$ −2187.07 −0.469307
$$280$$ 2424.19 0.517404
$$281$$ 2992.81 0.635360 0.317680 0.948198i $$-0.397096\pi$$
0.317680 + 0.948198i $$0.397096\pi$$
$$282$$ −7161.50 −1.51227
$$283$$ 6858.89 1.44070 0.720351 0.693610i $$-0.243981\pi$$
0.720351 + 0.693610i $$0.243981\pi$$
$$284$$ 10980.9 2.29435
$$285$$ 915.769 0.190335
$$286$$ 0 0
$$287$$ −2251.37 −0.463046
$$288$$ 1565.66 0.320339
$$289$$ 408.809 0.0832096
$$290$$ −722.915 −0.146383
$$291$$ 1279.84 0.257819
$$292$$ −4605.39 −0.922979
$$293$$ −4049.70 −0.807461 −0.403731 0.914878i $$-0.632287\pi$$
−0.403731 + 0.914878i $$0.632287\pi$$
$$294$$ 1143.31 0.226800
$$295$$ 547.392 0.108035
$$296$$ 1536.43 0.301699
$$297$$ 0 0
$$298$$ 14201.2 2.76059
$$299$$ −213.901 −0.0413720
$$300$$ −983.453 −0.189266
$$301$$ −2512.71 −0.481164
$$302$$ 6243.20 1.18959
$$303$$ 222.416 0.0421699
$$304$$ −185.677 −0.0350305
$$305$$ 448.078 0.0841209
$$306$$ 3016.78 0.563588
$$307$$ 9572.69 1.77962 0.889808 0.456335i $$-0.150838\pi$$
0.889808 + 0.456335i $$0.150838\pi$$
$$308$$ 0 0
$$309$$ 208.175 0.0383257
$$310$$ 5582.94 1.02287
$$311$$ 5396.42 0.983932 0.491966 0.870614i $$-0.336278\pi$$
0.491966 + 0.870614i $$0.336278\pi$$
$$312$$ 1103.55 0.200244
$$313$$ 9755.04 1.76162 0.880811 0.473469i $$-0.156998\pi$$
0.880811 + 0.473469i $$0.156998\pi$$
$$314$$ 4649.77 0.835674
$$315$$ −928.726 −0.166120
$$316$$ 10899.3 1.94030
$$317$$ −4353.75 −0.771391 −0.385695 0.922626i $$-0.626038\pi$$
−0.385695 + 0.922626i $$0.626038\pi$$
$$318$$ −7480.52 −1.31914
$$319$$ 0 0
$$320$$ −4118.32 −0.719440
$$321$$ 3425.12 0.595549
$$322$$ −1295.42 −0.224196
$$323$$ 4453.74 0.767221
$$324$$ 1062.13 0.182121
$$325$$ 391.461 0.0668134
$$326$$ 12940.8 2.19854
$$327$$ −6680.54 −1.12977
$$328$$ −2562.68 −0.431404
$$329$$ 10722.2 1.79677
$$330$$ 0 0
$$331$$ 5387.64 0.894656 0.447328 0.894370i $$-0.352376\pi$$
0.447328 + 0.894370i $$0.352376\pi$$
$$332$$ −18215.3 −3.01113
$$333$$ −588.616 −0.0968648
$$334$$ 15847.2 2.59616
$$335$$ 2442.23 0.398308
$$336$$ 188.303 0.0305738
$$337$$ 4500.27 0.727434 0.363717 0.931509i $$-0.381508\pi$$
0.363717 + 0.931509i $$0.381508\pi$$
$$338$$ 8968.30 1.44323
$$339$$ 5159.28 0.826589
$$340$$ −4782.91 −0.762910
$$341$$ 0 0
$$342$$ 2524.70 0.399181
$$343$$ 5367.18 0.844899
$$344$$ −2860.16 −0.448283
$$345$$ 204.907 0.0319763
$$346$$ −10512.4 −1.63337
$$347$$ −5906.32 −0.913740 −0.456870 0.889533i $$-0.651030\pi$$
−0.456870 + 0.889533i $$0.651030\pi$$
$$348$$ −1237.82 −0.190673
$$349$$ −3636.26 −0.557721 −0.278860 0.960332i $$-0.589957\pi$$
−0.278860 + 0.960332i $$0.589957\pi$$
$$350$$ 2370.76 0.362063
$$351$$ −422.778 −0.0642912
$$352$$ 0 0
$$353$$ 210.408 0.0317248 0.0158624 0.999874i $$-0.494951\pi$$
0.0158624 + 0.999874i $$0.494951\pi$$
$$354$$ 1509.11 0.226577
$$355$$ 4187.12 0.625998
$$356$$ 19979.8 2.97451
$$357$$ −4516.75 −0.669612
$$358$$ −14928.9 −2.20396
$$359$$ −2499.68 −0.367488 −0.183744 0.982974i $$-0.558822\pi$$
−0.183744 + 0.982974i $$0.558822\pi$$
$$360$$ −1057.15 −0.154768
$$361$$ −3131.74 −0.456588
$$362$$ −5378.03 −0.780837
$$363$$ 0 0
$$364$$ −4237.56 −0.610188
$$365$$ −1756.08 −0.251828
$$366$$ 1235.31 0.176423
$$367$$ 5748.70 0.817656 0.408828 0.912612i $$-0.365938\pi$$
0.408828 + 0.912612i $$0.365938\pi$$
$$368$$ −41.5458 −0.00588512
$$369$$ 981.781 0.138508
$$370$$ 1502.56 0.211120
$$371$$ 11199.9 1.56730
$$372$$ 9559.48 1.33236
$$373$$ −4467.78 −0.620196 −0.310098 0.950705i $$-0.600362\pi$$
−0.310098 + 0.950705i $$0.600362\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 12204.9 1.67398
$$377$$ 492.712 0.0673103
$$378$$ −2560.42 −0.348396
$$379$$ 7804.08 1.05770 0.528851 0.848715i $$-0.322623\pi$$
0.528851 + 0.848715i $$0.322623\pi$$
$$380$$ −4002.74 −0.540358
$$381$$ −4804.89 −0.646094
$$382$$ 12683.4 1.69880
$$383$$ −11161.1 −1.48904 −0.744522 0.667597i $$-0.767323\pi$$
−0.744522 + 0.667597i $$0.767323\pi$$
$$384$$ −7178.74 −0.954007
$$385$$ 0 0
$$386$$ −5746.38 −0.757728
$$387$$ 1095.75 0.143928
$$388$$ −5594.04 −0.731944
$$389$$ 8490.24 1.10661 0.553306 0.832978i $$-0.313366\pi$$
0.553306 + 0.832978i $$0.313366\pi$$
$$390$$ 1079.22 0.140125
$$391$$ 996.540 0.128893
$$392$$ −1948.47 −0.251052
$$393$$ 6012.39 0.771717
$$394$$ −661.616 −0.0845983
$$395$$ 4156.03 0.529398
$$396$$ 0 0
$$397$$ 6019.74 0.761013 0.380507 0.924778i $$-0.375750\pi$$
0.380507 + 0.924778i $$0.375750\pi$$
$$398$$ −3497.83 −0.440529
$$399$$ −3779.99 −0.474277
$$400$$ 76.0330 0.00950413
$$401$$ −10398.8 −1.29499 −0.647495 0.762069i $$-0.724184\pi$$
−0.647495 + 0.762069i $$0.724184\pi$$
$$402$$ 6733.01 0.835354
$$403$$ −3805.13 −0.470340
$$404$$ −972.158 −0.119720
$$405$$ 405.000 0.0496904
$$406$$ 2983.95 0.364756
$$407$$ 0 0
$$408$$ −5141.30 −0.623854
$$409$$ 4733.68 0.572287 0.286144 0.958187i $$-0.407627\pi$$
0.286144 + 0.958187i $$0.407627\pi$$
$$410$$ −2506.19 −0.301883
$$411$$ 5018.55 0.602304
$$412$$ −909.911 −0.108806
$$413$$ −2259.45 −0.269202
$$414$$ 564.910 0.0670624
$$415$$ −6945.67 −0.821565
$$416$$ 2723.98 0.321044
$$417$$ 7621.14 0.894985
$$418$$ 0 0
$$419$$ −8117.57 −0.946466 −0.473233 0.880937i $$-0.656913\pi$$
−0.473233 + 0.880937i $$0.656913\pi$$
$$420$$ 4059.37 0.471611
$$421$$ −9484.27 −1.09795 −0.548973 0.835840i $$-0.684981\pi$$
−0.548973 + 0.835840i $$0.684981\pi$$
$$422$$ 18271.8 2.10772
$$423$$ −4675.77 −0.537456
$$424$$ 12748.5 1.46020
$$425$$ −1823.77 −0.208155
$$426$$ 11543.5 1.31288
$$427$$ −1849.52 −0.209612
$$428$$ −14970.8 −1.69075
$$429$$ 0 0
$$430$$ −2797.11 −0.313695
$$431$$ −9335.16 −1.04329 −0.521646 0.853162i $$-0.674682\pi$$
−0.521646 + 0.853162i $$0.674682\pi$$
$$432$$ −82.1157 −0.00914535
$$433$$ −2983.02 −0.331074 −0.165537 0.986204i $$-0.552936\pi$$
−0.165537 + 0.986204i $$0.552936\pi$$
$$434$$ −23044.5 −2.54878
$$435$$ −471.994 −0.0520238
$$436$$ 29200.0 3.20739
$$437$$ 833.988 0.0912931
$$438$$ −4841.36 −0.528148
$$439$$ 5232.32 0.568850 0.284425 0.958698i $$-0.408197\pi$$
0.284425 + 0.958698i $$0.408197\pi$$
$$440$$ 0 0
$$441$$ 746.472 0.0806038
$$442$$ 5248.68 0.564828
$$443$$ 7517.71 0.806269 0.403135 0.915141i $$-0.367921\pi$$
0.403135 + 0.915141i $$0.367921\pi$$
$$444$$ 2572.78 0.274997
$$445$$ 7618.49 0.811575
$$446$$ −4172.52 −0.442992
$$447$$ 9272.03 0.981101
$$448$$ 16999.1 1.79270
$$449$$ 16070.9 1.68916 0.844581 0.535428i $$-0.179850\pi$$
0.844581 + 0.535428i $$0.179850\pi$$
$$450$$ −1033.84 −0.108302
$$451$$ 0 0
$$452$$ −22550.7 −2.34667
$$453$$ 4076.21 0.422775
$$454$$ −9475.29 −0.979510
$$455$$ −1615.82 −0.166485
$$456$$ −4302.67 −0.441867
$$457$$ −9718.51 −0.994776 −0.497388 0.867528i $$-0.665707\pi$$
−0.497388 + 0.867528i $$0.665707\pi$$
$$458$$ −18735.4 −1.91146
$$459$$ 1969.67 0.200297
$$460$$ −895.627 −0.0907801
$$461$$ 14538.0 1.46877 0.734385 0.678733i $$-0.237471\pi$$
0.734385 + 0.678733i $$0.237471\pi$$
$$462$$ 0 0
$$463$$ −9978.17 −1.00157 −0.500783 0.865573i $$-0.666955\pi$$
−0.500783 + 0.865573i $$0.666955\pi$$
$$464$$ 95.6990 0.00957481
$$465$$ 3645.12 0.363524
$$466$$ 7732.03 0.768625
$$467$$ 15188.2 1.50498 0.752489 0.658605i $$-0.228853\pi$$
0.752489 + 0.658605i $$0.228853\pi$$
$$468$$ 1847.92 0.182522
$$469$$ −10080.7 −0.992503
$$470$$ 11935.8 1.17140
$$471$$ 3035.85 0.296995
$$472$$ −2571.88 −0.250806
$$473$$ 0 0
$$474$$ 11457.8 1.11028
$$475$$ −1526.28 −0.147433
$$476$$ 19742.3 1.90102
$$477$$ −4884.06 −0.468817
$$478$$ 18494.1 1.76967
$$479$$ −11330.8 −1.08083 −0.540415 0.841399i $$-0.681733\pi$$
−0.540415 + 0.841399i $$0.681733\pi$$
$$480$$ −2609.44 −0.248133
$$481$$ −1024.09 −0.0970779
$$482$$ −12793.3 −1.20896
$$483$$ −845.788 −0.0796784
$$484$$ 0 0
$$485$$ −2133.06 −0.199706
$$486$$ 1116.55 0.104213
$$487$$ 19086.9 1.77599 0.887997 0.459850i $$-0.152097\pi$$
0.887997 + 0.459850i $$0.152097\pi$$
$$488$$ −2105.26 −0.195288
$$489$$ 8449.11 0.781353
$$490$$ −1905.52 −0.175679
$$491$$ 8112.85 0.745677 0.372839 0.927896i $$-0.378384\pi$$
0.372839 + 0.927896i $$0.378384\pi$$
$$492$$ −4291.27 −0.393222
$$493$$ −2295.49 −0.209703
$$494$$ 4392.53 0.400060
$$495$$ 0 0
$$496$$ −739.066 −0.0669053
$$497$$ −17283.0 −1.55986
$$498$$ −19148.6 −1.72303
$$499$$ 18329.1 1.64433 0.822167 0.569246i $$-0.192765\pi$$
0.822167 + 0.569246i $$0.192765\pi$$
$$500$$ 1639.09 0.146604
$$501$$ 10346.7 0.922666
$$502$$ 8397.58 0.746618
$$503$$ −7739.57 −0.686064 −0.343032 0.939324i $$-0.611454\pi$$
−0.343032 + 0.939324i $$0.611454\pi$$
$$504$$ 4363.55 0.385651
$$505$$ −370.693 −0.0326646
$$506$$ 0 0
$$507$$ 5855.44 0.512918
$$508$$ 21001.7 1.83425
$$509$$ 15914.9 1.38589 0.692943 0.720993i $$-0.256314\pi$$
0.692943 + 0.720993i $$0.256314\pi$$
$$510$$ −5027.97 −0.436554
$$511$$ 7248.51 0.627505
$$512$$ 1100.65 0.0950048
$$513$$ 1648.38 0.141867
$$514$$ −2688.78 −0.230733
$$515$$ −346.958 −0.0296870
$$516$$ −4789.40 −0.408608
$$517$$ 0 0
$$518$$ −6202.07 −0.526068
$$519$$ −6863.56 −0.580495
$$520$$ −1839.25 −0.155109
$$521$$ 2274.50 0.191262 0.0956312 0.995417i $$-0.469513\pi$$
0.0956312 + 0.995417i $$0.469513\pi$$
$$522$$ −1301.25 −0.109107
$$523$$ −10971.1 −0.917274 −0.458637 0.888624i $$-0.651662\pi$$
−0.458637 + 0.888624i $$0.651662\pi$$
$$524$$ −26279.6 −2.19089
$$525$$ 1547.88 0.128676
$$526$$ −1094.02 −0.0906877
$$527$$ 17727.6 1.46533
$$528$$ 0 0
$$529$$ −11980.4 −0.984663
$$530$$ 12467.5 1.02180
$$531$$ 985.306 0.0805247
$$532$$ 16522.0 1.34646
$$533$$ 1708.13 0.138813
$$534$$ 21003.5 1.70208
$$535$$ −5708.53 −0.461310
$$536$$ −11474.6 −0.924680
$$537$$ −9747.17 −0.783280
$$538$$ −21221.2 −1.70058
$$539$$ 0 0
$$540$$ −1770.21 −0.141070
$$541$$ −5313.05 −0.422229 −0.211115 0.977461i $$-0.567709\pi$$
−0.211115 + 0.977461i $$0.567709\pi$$
$$542$$ −659.084 −0.0522326
$$543$$ −3511.34 −0.277506
$$544$$ −12690.7 −1.00020
$$545$$ 11134.2 0.875115
$$546$$ −4454.68 −0.349162
$$547$$ −20685.1 −1.61688 −0.808439 0.588581i $$-0.799687\pi$$
−0.808439 + 0.588581i $$0.799687\pi$$
$$548$$ −21935.6 −1.70993
$$549$$ 806.541 0.0627000
$$550$$ 0 0
$$551$$ −1921.06 −0.148529
$$552$$ −962.739 −0.0742335
$$553$$ −17154.7 −1.31915
$$554$$ −39527.3 −3.03132
$$555$$ 981.027 0.0750311
$$556$$ −33311.2 −2.54085
$$557$$ 10853.8 0.825659 0.412830 0.910808i $$-0.364540\pi$$
0.412830 + 0.910808i $$0.364540\pi$$
$$558$$ 10049.3 0.762402
$$559$$ 1906.41 0.144244
$$560$$ −313.839 −0.0236824
$$561$$ 0 0
$$562$$ −13751.5 −1.03216
$$563$$ 15381.2 1.15141 0.575704 0.817658i $$-0.304728\pi$$
0.575704 + 0.817658i $$0.304728\pi$$
$$564$$ 20437.3 1.52583
$$565$$ −8598.80 −0.640273
$$566$$ −31515.6 −2.34046
$$567$$ −1671.71 −0.123818
$$568$$ −19672.9 −1.45327
$$569$$ 1348.88 0.0993814 0.0496907 0.998765i $$-0.484176\pi$$
0.0496907 + 0.998765i $$0.484176\pi$$
$$570$$ −4207.83 −0.309204
$$571$$ −3463.51 −0.253841 −0.126920 0.991913i $$-0.540509\pi$$
−0.126920 + 0.991913i $$0.540509\pi$$
$$572$$ 0 0
$$573$$ 8281.05 0.603745
$$574$$ 10344.7 0.752231
$$575$$ −341.511 −0.0247687
$$576$$ −7412.97 −0.536239
$$577$$ 12052.6 0.869598 0.434799 0.900528i $$-0.356819\pi$$
0.434799 + 0.900528i $$0.356819\pi$$
$$578$$ −1878.42 −0.135176
$$579$$ −3751.83 −0.269293
$$580$$ 2063.04 0.147695
$$581$$ 28669.4 2.04717
$$582$$ −5880.66 −0.418834
$$583$$ 0 0
$$584$$ 8250.80 0.584624
$$585$$ 704.630 0.0497997
$$586$$ 18607.8 1.31174
$$587$$ −11133.1 −0.782813 −0.391407 0.920218i $$-0.628011\pi$$
−0.391407 + 0.920218i $$0.628011\pi$$
$$588$$ −3262.75 −0.228833
$$589$$ 14836.0 1.03787
$$590$$ −2515.19 −0.175506
$$591$$ −431.972 −0.0300659
$$592$$ −198.908 −0.0138092
$$593$$ 7939.69 0.549821 0.274911 0.961470i $$-0.411352\pi$$
0.274911 + 0.961470i $$0.411352\pi$$
$$594$$ 0 0
$$595$$ 7527.91 0.518680
$$596$$ −40527.1 −2.78533
$$597$$ −2283.75 −0.156562
$$598$$ 982.846 0.0672100
$$599$$ −19474.7 −1.32840 −0.664202 0.747553i $$-0.731229\pi$$
−0.664202 + 0.747553i $$0.731229\pi$$
$$600$$ 1761.91 0.119883
$$601$$ 19946.1 1.35377 0.676887 0.736087i $$-0.263329\pi$$
0.676887 + 0.736087i $$0.263329\pi$$
$$602$$ 11545.6 0.781664
$$603$$ 4396.01 0.296881
$$604$$ −17816.7 −1.20025
$$605$$ 0 0
$$606$$ −1021.97 −0.0685061
$$607$$ −1427.44 −0.0954496 −0.0477248 0.998861i $$-0.515197\pi$$
−0.0477248 + 0.998861i $$0.515197\pi$$
$$608$$ −10620.6 −0.708427
$$609$$ 1948.23 0.129633
$$610$$ −2058.85 −0.136657
$$611$$ −8135.03 −0.538638
$$612$$ −8609.23 −0.568640
$$613$$ 8029.40 0.529045 0.264522 0.964380i $$-0.414786\pi$$
0.264522 + 0.964380i $$0.414786\pi$$
$$614$$ −43985.1 −2.89103
$$615$$ −1636.30 −0.107288
$$616$$ 0 0
$$617$$ 20795.5 1.35688 0.678440 0.734655i $$-0.262656\pi$$
0.678440 + 0.734655i $$0.262656\pi$$
$$618$$ −956.533 −0.0622612
$$619$$ 1677.43 0.108920 0.0544602 0.998516i $$-0.482656\pi$$
0.0544602 + 0.998516i $$0.482656\pi$$
$$620$$ −15932.5 −1.03204
$$621$$ 368.832 0.0238337
$$622$$ −24795.8 −1.59842
$$623$$ −31446.6 −2.02228
$$624$$ −142.867 −0.00916547
$$625$$ 625.000 0.0400000
$$626$$ −44823.0 −2.86180
$$627$$ 0 0
$$628$$ −13269.4 −0.843165
$$629$$ 4771.11 0.302443
$$630$$ 4267.36 0.269866
$$631$$ −25225.2 −1.59144 −0.795719 0.605666i $$-0.792907\pi$$
−0.795719 + 0.605666i $$0.792907\pi$$
$$632$$ −19526.8 −1.22901
$$633$$ 11929.7 0.749074
$$634$$ 20004.8 1.25315
$$635$$ 8008.15 0.500462
$$636$$ 21347.7 1.33096
$$637$$ 1298.73 0.0807812
$$638$$ 0 0
$$639$$ 7536.81 0.466591
$$640$$ 11964.6 0.738971
$$641$$ 15165.3 0.934468 0.467234 0.884134i $$-0.345251\pi$$
0.467234 + 0.884134i $$0.345251\pi$$
$$642$$ −15737.9 −0.967486
$$643$$ 27156.1 1.66553 0.832763 0.553630i $$-0.186758\pi$$
0.832763 + 0.553630i $$0.186758\pi$$
$$644$$ 3696.85 0.226206
$$645$$ −1826.25 −0.111486
$$646$$ −20464.3 −1.24637
$$647$$ 29154.9 1.77156 0.885778 0.464110i $$-0.153626\pi$$
0.885778 + 0.464110i $$0.153626\pi$$
$$648$$ −1902.86 −0.115357
$$649$$ 0 0
$$650$$ −1798.71 −0.108540
$$651$$ −15045.9 −0.905828
$$652$$ −36930.2 −2.21825
$$653$$ −19141.7 −1.14713 −0.573564 0.819161i $$-0.694440\pi$$
−0.573564 + 0.819161i $$0.694440\pi$$
$$654$$ 30696.1 1.83534
$$655$$ −10020.7 −0.597770
$$656$$ 331.768 0.0197460
$$657$$ −3160.94 −0.187702
$$658$$ −49267.2 −2.91890
$$659$$ 24939.6 1.47422 0.737110 0.675773i $$-0.236190\pi$$
0.737110 + 0.675773i $$0.236190\pi$$
$$660$$ 0 0
$$661$$ 22617.7 1.33090 0.665452 0.746440i $$-0.268239\pi$$
0.665452 + 0.746440i $$0.268239\pi$$
$$662$$ −24755.4 −1.45339
$$663$$ 3426.88 0.200738
$$664$$ 32633.7 1.90728
$$665$$ 6299.99 0.367373
$$666$$ 2704.61 0.157359
$$667$$ −429.843 −0.0249529
$$668$$ −45224.3 −2.61943
$$669$$ −2724.25 −0.157438
$$670$$ −11221.7 −0.647062
$$671$$ 0 0
$$672$$ 10770.9 0.618298
$$673$$ 13855.8 0.793615 0.396807 0.917902i $$-0.370118\pi$$
0.396807 + 0.917902i $$0.370118\pi$$
$$674$$ −20678.1 −1.18174
$$675$$ −675.000 −0.0384900
$$676$$ −25593.5 −1.45616
$$677$$ 24992.8 1.41884 0.709419 0.704787i $$-0.248957\pi$$
0.709419 + 0.704787i $$0.248957\pi$$
$$678$$ −23706.1 −1.34282
$$679$$ 8804.57 0.497626
$$680$$ 8568.84 0.483235
$$681$$ −6186.46 −0.348114
$$682$$ 0 0
$$683$$ −14420.5 −0.807887 −0.403943 0.914784i $$-0.632361\pi$$
−0.403943 + 0.914784i $$0.632361\pi$$
$$684$$ −7204.93 −0.402759
$$685$$ −8364.25 −0.466543
$$686$$ −24661.4 −1.37256
$$687$$ −12232.4 −0.679324
$$688$$ 370.280 0.0205186
$$689$$ −8497.42 −0.469849
$$690$$ −941.517 −0.0519463
$$691$$ 30552.4 1.68201 0.841005 0.541027i $$-0.181964\pi$$
0.841005 + 0.541027i $$0.181964\pi$$
$$692$$ 29999.9 1.64801
$$693$$ 0 0
$$694$$ 27138.7 1.48440
$$695$$ −12701.9 −0.693253
$$696$$ 2217.63 0.120774
$$697$$ −7957.96 −0.432467
$$698$$ 16708.1 0.906032
$$699$$ 5048.27 0.273166
$$700$$ −6765.61 −0.365309
$$701$$ 9151.47 0.493076 0.246538 0.969133i $$-0.420707\pi$$
0.246538 + 0.969133i $$0.420707\pi$$
$$702$$ 1942.60 0.104443
$$703$$ 3992.86 0.214216
$$704$$ 0 0
$$705$$ 7792.95 0.416311
$$706$$ −966.793 −0.0515378
$$707$$ 1530.10 0.0813937
$$708$$ −4306.67 −0.228608
$$709$$ −6261.96 −0.331697 −0.165848 0.986151i $$-0.553036\pi$$
−0.165848 + 0.986151i $$0.553036\pi$$
$$710$$ −19239.2 −1.01695
$$711$$ 7480.85 0.394590
$$712$$ −35794.9 −1.88409
$$713$$ 3319.60 0.174362
$$714$$ 20753.8 1.08780
$$715$$ 0 0
$$716$$ 42603.9 2.22372
$$717$$ 12074.9 0.628933
$$718$$ 11485.7 0.596995
$$719$$ −18228.7 −0.945500 −0.472750 0.881197i $$-0.656739\pi$$
−0.472750 + 0.881197i $$0.656739\pi$$
$$720$$ 136.859 0.00708396
$$721$$ 1432.13 0.0739740
$$722$$ 14389.9 0.741740
$$723$$ −8352.81 −0.429660
$$724$$ 15347.7 0.787836
$$725$$ 786.656 0.0402975
$$726$$ 0 0
$$727$$ −7233.66 −0.369026 −0.184513 0.982830i $$-0.559071\pi$$
−0.184513 + 0.982830i $$0.559071\pi$$
$$728$$ 7591.81 0.386499
$$729$$ 729.000 0.0370370
$$730$$ 8068.93 0.409102
$$731$$ −8881.73 −0.449388
$$732$$ −3525.31 −0.178004
$$733$$ 13444.8 0.677485 0.338743 0.940879i $$-0.389998\pi$$
0.338743 + 0.940879i $$0.389998\pi$$
$$734$$ −26414.4 −1.32830
$$735$$ −1244.12 −0.0624355
$$736$$ −2376.41 −0.119016
$$737$$ 0 0
$$738$$ −4511.14 −0.225010
$$739$$ −18490.9 −0.920432 −0.460216 0.887807i $$-0.652228\pi$$
−0.460216 + 0.887807i $$0.652228\pi$$
$$740$$ −4287.97 −0.213012
$$741$$ 2867.90 0.142180
$$742$$ −51461.8 −2.54612
$$743$$ 25160.9 1.24235 0.621173 0.783674i $$-0.286657\pi$$
0.621173 + 0.783674i $$0.286657\pi$$
$$744$$ −17126.3 −0.843927
$$745$$ −15453.4 −0.759957
$$746$$ 20528.8 1.00752
$$747$$ −12502.2 −0.612358
$$748$$ 0 0
$$749$$ 23562.9 1.14949
$$750$$ 1723.07 0.0838902
$$751$$ −13419.5 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$752$$ −1580.06 −0.0766207
$$753$$ 5482.81 0.265345
$$754$$ −2263.94 −0.109347
$$755$$ −6793.68 −0.327480
$$756$$ 7306.86 0.351518
$$757$$ −7014.90 −0.336804 −0.168402 0.985718i $$-0.553861\pi$$
−0.168402 + 0.985718i $$0.553861\pi$$
$$758$$ −35858.6 −1.71826
$$759$$ 0 0
$$760$$ 7171.12 0.342268
$$761$$ 30156.9 1.43651 0.718256 0.695779i $$-0.244941\pi$$
0.718256 + 0.695779i $$0.244941\pi$$
$$762$$ 22077.8 1.04960
$$763$$ −45958.4 −2.18061
$$764$$ −36195.7 −1.71402
$$765$$ −3282.78 −0.155149
$$766$$ 51283.5 2.41899
$$767$$ 1714.26 0.0807019
$$768$$ 13217.4 0.621017
$$769$$ −11292.2 −0.529530 −0.264765 0.964313i $$-0.585294\pi$$
−0.264765 + 0.964313i $$0.585294\pi$$
$$770$$ 0 0
$$771$$ −1755.51 −0.0820016
$$772$$ 16398.9 0.764519
$$773$$ 8524.10 0.396624 0.198312 0.980139i $$-0.436454\pi$$
0.198312 + 0.980139i $$0.436454\pi$$
$$774$$ −5034.80 −0.233814
$$775$$ −6075.21 −0.281584
$$776$$ 10022.0 0.463621
$$777$$ −4049.35 −0.186962
$$778$$ −39011.4 −1.79772
$$779$$ −6659.89 −0.306310
$$780$$ −3079.87 −0.141381
$$781$$ 0 0
$$782$$ −4578.96 −0.209390
$$783$$ −849.589 −0.0387763
$$784$$ 252.251 0.0114910
$$785$$ −5059.76 −0.230052
$$786$$ −27626.1 −1.25368
$$787$$ 14983.9 0.678676 0.339338 0.940665i $$-0.389797\pi$$
0.339338 + 0.940665i $$0.389797\pi$$
$$788$$ 1888.10 0.0853565
$$789$$ −714.293 −0.0322300
$$790$$ −19096.3 −0.860022
$$791$$ 35493.0 1.59543
$$792$$ 0 0
$$793$$ 1403.24 0.0628380
$$794$$ −27659.9 −1.23629
$$795$$ 8140.10 0.363144
$$796$$ 9982.04 0.444477
$$797$$ 37172.3 1.65208 0.826041 0.563610i $$-0.190588\pi$$
0.826041 + 0.563610i $$0.190588\pi$$
$$798$$ 17368.5 0.770475
$$799$$ 37900.1 1.67811
$$800$$ 4349.06 0.192203
$$801$$ 13713.3 0.604913
$$802$$ 47781.0 2.10375
$$803$$ 0 0
$$804$$ −19214.5 −0.842841
$$805$$ 1409.65 0.0617186
$$806$$ 17484.0 0.764080
$$807$$ −13855.4 −0.604378
$$808$$ 1741.68 0.0758316
$$809$$ −23797.1 −1.03419 −0.517096 0.855928i $$-0.672987\pi$$
−0.517096 + 0.855928i $$0.672987\pi$$
$$810$$ −1860.92 −0.0807234
$$811$$ −8988.35 −0.389178 −0.194589 0.980885i $$-0.562337\pi$$
−0.194589 + 0.980885i $$0.562337\pi$$
$$812$$ −8515.54 −0.368026
$$813$$ −430.318 −0.0185633
$$814$$ 0 0
$$815$$ −14081.8 −0.605234
$$816$$ 665.600 0.0285547
$$817$$ −7432.98 −0.318295
$$818$$ −21750.6 −0.929696
$$819$$ −2908.48 −0.124091
$$820$$ 7152.11 0.304589
$$821$$ 25156.8 1.06940 0.534702 0.845041i $$-0.320424\pi$$
0.534702 + 0.845041i $$0.320424\pi$$
$$822$$ −23059.5 −0.978459
$$823$$ 1318.51 0.0558447 0.0279224 0.999610i $$-0.491111\pi$$
0.0279224 + 0.999610i $$0.491111\pi$$
$$824$$ 1630.16 0.0689189
$$825$$ 0 0
$$826$$ 10381.9 0.437326
$$827$$ −124.982 −0.00525519 −0.00262760 0.999997i $$-0.500836\pi$$
−0.00262760 + 0.999997i $$0.500836\pi$$
$$828$$ −1612.13 −0.0676635
$$829$$ −8886.80 −0.372318 −0.186159 0.982520i $$-0.559604\pi$$
−0.186159 + 0.982520i $$0.559604\pi$$
$$830$$ 31914.3 1.33465
$$831$$ −25807.5 −1.07732
$$832$$ −12897.3 −0.537419
$$833$$ −6050.63 −0.251671
$$834$$ −35018.0 −1.45393
$$835$$ −17244.5 −0.714694
$$836$$ 0 0
$$837$$ 6561.22 0.270955
$$838$$ 37299.1 1.53756
$$839$$ 2995.21 0.123249 0.0616247 0.998099i $$-0.480372\pi$$
0.0616247 + 0.998099i $$0.480372\pi$$
$$840$$ −7272.58 −0.298724
$$841$$ −23398.9 −0.959403
$$842$$ 43578.8 1.78364
$$843$$ −8978.43 −0.366825
$$844$$ −52143.6 −2.12661
$$845$$ −9759.07 −0.397304
$$846$$ 21484.5 0.873111
$$847$$ 0 0
$$848$$ −1650.44 −0.0668355
$$849$$ −20576.7 −0.831789
$$850$$ 8379.95 0.338153
$$851$$ 893.418 0.0359882
$$852$$ −32942.7 −1.32464
$$853$$ −18130.5 −0.727757 −0.363878 0.931446i $$-0.618548\pi$$
−0.363878 + 0.931446i $$0.618548\pi$$
$$854$$ 8498.27 0.340521
$$855$$ −2747.31 −0.109890
$$856$$ 26821.1 1.07094
$$857$$ −26394.1 −1.05205 −0.526024 0.850470i $$-0.676318\pi$$
−0.526024 + 0.850470i $$0.676318\pi$$
$$858$$ 0 0
$$859$$ −29456.2 −1.17000 −0.585002 0.811032i $$-0.698906\pi$$
−0.585002 + 0.811032i $$0.698906\pi$$
$$860$$ 7982.34 0.316506
$$861$$ 6754.12 0.267340
$$862$$ 42893.7 1.69486
$$863$$ −762.616 −0.0300808 −0.0150404 0.999887i $$-0.504788\pi$$
−0.0150404 + 0.999887i $$0.504788\pi$$
$$864$$ −4696.99 −0.184948
$$865$$ 11439.3 0.449649
$$866$$ 13706.6 0.537838
$$867$$ −1226.43 −0.0480411
$$868$$ 65764.0 2.57163
$$869$$ 0 0
$$870$$ 2168.74 0.0845141
$$871$$ 7648.29 0.297535
$$872$$ −52313.3 −2.03160
$$873$$ −3839.51 −0.148852
$$874$$ −3832.06 −0.148308
$$875$$ −2579.79 −0.0996719
$$876$$ 13816.2 0.532882
$$877$$ −44767.2 −1.72369 −0.861847 0.507168i $$-0.830692\pi$$
−0.861847 + 0.507168i $$0.830692\pi$$
$$878$$ −24041.8 −0.924112
$$879$$ 12149.1 0.466188
$$880$$ 0 0
$$881$$ −32057.9 −1.22595 −0.612973 0.790104i $$-0.710027\pi$$
−0.612973 + 0.790104i $$0.710027\pi$$
$$882$$ −3429.93 −0.130943
$$883$$ 7078.95 0.269791 0.134896 0.990860i $$-0.456930\pi$$
0.134896 + 0.990860i $$0.456930\pi$$
$$884$$ −14978.6 −0.569891
$$885$$ −1642.18 −0.0623742
$$886$$ −34542.8 −1.30981
$$887$$ 25148.1 0.951964 0.475982 0.879455i $$-0.342093\pi$$
0.475982 + 0.879455i $$0.342093\pi$$
$$888$$ −4609.28 −0.174186
$$889$$ −33055.0 −1.24705
$$890$$ −35005.9 −1.31843
$$891$$ 0 0
$$892$$ 11907.4 0.446963
$$893$$ 31718.0 1.18858
$$894$$ −42603.6 −1.59382
$$895$$ 16245.3 0.606726
$$896$$ −49385.8 −1.84137
$$897$$ 641.704 0.0238862
$$898$$ −73843.6 −2.74409
$$899$$ −7646.56 −0.283679
$$900$$ 2950.36 0.109273
$$901$$ 39588.4 1.46380
$$902$$ 0 0
$$903$$ 7538.14 0.277800
$$904$$ 40400.8 1.48641
$$905$$ 5852.23 0.214955
$$906$$ −18729.6 −0.686809
$$907$$ 1269.76 0.0464848 0.0232424 0.999730i $$-0.492601\pi$$
0.0232424 + 0.999730i $$0.492601\pi$$
$$908$$ 27040.4 0.988289
$$909$$ −667.248 −0.0243468
$$910$$ 7424.47 0.270460
$$911$$ −33783.1 −1.22863 −0.614316 0.789060i $$-0.710568\pi$$
−0.614316 + 0.789060i $$0.710568\pi$$
$$912$$ 557.030 0.0202249
$$913$$ 0 0
$$914$$ 44655.1 1.61604
$$915$$ −1344.23 −0.0485672
$$916$$ 53466.6 1.92859
$$917$$ 41361.9 1.48952
$$918$$ −9050.35 −0.325388
$$919$$ −39262.5 −1.40930 −0.704652 0.709553i $$-0.748897\pi$$
−0.704652 + 0.709553i $$0.748897\pi$$
$$920$$ 1604.57 0.0575010
$$921$$ −28718.1 −1.02746
$$922$$ −66800.1 −2.38606
$$923$$ 13112.7 0.467618
$$924$$ 0 0
$$925$$ −1635.04 −0.0581189
$$926$$ 45848.3 1.62707
$$927$$ −624.525 −0.0221274
$$928$$ 5473.95 0.193633
$$929$$ −21175.0 −0.747825 −0.373913 0.927464i $$-0.621984\pi$$
−0.373913 + 0.927464i $$0.621984\pi$$
$$930$$ −16748.8 −0.590554
$$931$$ −5063.68 −0.178255
$$932$$ −22065.5 −0.775515
$$933$$ −16189.3 −0.568073
$$934$$ −69787.5 −2.44488
$$935$$ 0 0
$$936$$ −3310.65 −0.115611
$$937$$ 5135.11 0.179036 0.0895180 0.995985i $$-0.471467\pi$$
0.0895180 + 0.995985i $$0.471467\pi$$
$$938$$ 46319.4 1.61235
$$939$$ −29265.1 −1.01707
$$940$$ −34062.2 −1.18190
$$941$$ 9702.77 0.336133 0.168067 0.985776i $$-0.446248\pi$$
0.168067 + 0.985776i $$0.446248\pi$$
$$942$$ −13949.3 −0.482477
$$943$$ −1490.18 −0.0514600
$$944$$ 332.959 0.0114798
$$945$$ 2786.18 0.0959094
$$946$$ 0 0
$$947$$ −699.579 −0.0240055 −0.0120028 0.999928i $$-0.503821\pi$$
−0.0120028 + 0.999928i $$0.503821\pi$$
$$948$$ −32698.0 −1.12024
$$949$$ −5499.49 −0.188115
$$950$$ 7013.05 0.239509
$$951$$ 13061.2 0.445363
$$952$$ −35369.3 −1.20412
$$953$$ −42039.3 −1.42895 −0.714473 0.699663i $$-0.753333\pi$$
−0.714473 + 0.699663i $$0.753333\pi$$
$$954$$ 22441.6 0.761606
$$955$$ −13801.8 −0.467659
$$956$$ −52778.1 −1.78553
$$957$$ 0 0
$$958$$ 52063.4 1.75584
$$959$$ 34524.9 1.16253
$$960$$ 12355.0 0.415369
$$961$$ 29262.0 0.982243
$$962$$ 4705.55 0.157706
$$963$$ −10275.3 −0.343841
$$964$$ 36509.3 1.21980
$$965$$ 6253.05 0.208594
$$966$$ 3886.27 0.129440
$$967$$ 32794.8 1.09060 0.545299 0.838242i $$-0.316416\pi$$
0.545299 + 0.838242i $$0.316416\pi$$
$$968$$ 0 0
$$969$$ −13361.2 −0.442955
$$970$$ 9801.10 0.324427
$$971$$ −3322.53 −0.109810 −0.0549048 0.998492i $$-0.517486\pi$$
−0.0549048 + 0.998492i $$0.517486\pi$$
$$972$$ −3186.39 −0.105148
$$973$$ 52429.3 1.72745
$$974$$ −87701.4 −2.88515
$$975$$ −1174.38 −0.0385747
$$976$$ 272.550 0.00893864
$$977$$ 22192.5 0.726716 0.363358 0.931650i $$-0.381630\pi$$
0.363358 + 0.931650i $$0.381630\pi$$
$$978$$ −38822.4 −1.26933
$$979$$ 0 0
$$980$$ 5437.92 0.177253
$$981$$ 20041.6 0.652272
$$982$$ −37277.4 −1.21137
$$983$$ 7383.09 0.239556 0.119778 0.992801i $$-0.461782\pi$$
0.119778 + 0.992801i $$0.461782\pi$$
$$984$$ 7688.04 0.249071
$$985$$ 719.953 0.0232889
$$986$$ 10547.4 0.340668
$$987$$ −32166.7 −1.03736
$$988$$ −12535.3 −0.403645
$$989$$ −1663.16 −0.0534735
$$990$$ 0 0
$$991$$ 46260.8 1.48287 0.741434 0.671026i $$-0.234146\pi$$
0.741434 + 0.671026i $$0.234146\pi$$
$$992$$ −42274.3 −1.35304
$$993$$ −16162.9 −0.516530
$$994$$ 79413.1 2.53403
$$995$$ 3806.25 0.121273
$$996$$ 54645.9 1.73847
$$997$$ −41196.8 −1.30864 −0.654320 0.756217i $$-0.727045\pi$$
−0.654320 + 0.756217i $$0.727045\pi$$
$$998$$ −84219.5 −2.67127
$$999$$ 1765.85 0.0559249
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.q.1.1 3
11.10 odd 2 165.4.a.g.1.3 3
33.32 even 2 495.4.a.i.1.1 3
55.32 even 4 825.4.c.m.199.6 6
55.43 even 4 825.4.c.m.199.1 6
55.54 odd 2 825.4.a.p.1.1 3
165.164 even 2 2475.4.a.z.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.g.1.3 3 11.10 odd 2
495.4.a.i.1.1 3 33.32 even 2
825.4.a.p.1.1 3 55.54 odd 2
825.4.c.m.199.1 6 55.43 even 4
825.4.c.m.199.6 6 55.32 even 4
1815.4.a.q.1.1 3 1.1 even 1 trivial
2475.4.a.z.1.3 3 165.164 even 2