# Properties

 Label 1815.4.a.o.1.2 Level $1815$ Weight $4$ Character 1815.1 Self dual yes Analytic conductor $107.088$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$107.088466660$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.70156 q^{2} +3.00000 q^{3} +5.70156 q^{4} +5.00000 q^{5} +11.1047 q^{6} -9.10469 q^{7} -8.50781 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+3.70156 q^{2} +3.00000 q^{3} +5.70156 q^{4} +5.00000 q^{5} +11.1047 q^{6} -9.10469 q^{7} -8.50781 q^{8} +9.00000 q^{9} +18.5078 q^{10} +17.1047 q^{12} +20.2984 q^{13} -33.7016 q^{14} +15.0000 q^{15} -77.1047 q^{16} -61.8062 q^{17} +33.3141 q^{18} -8.68594 q^{19} +28.5078 q^{20} -27.3141 q^{21} -127.748 q^{23} -25.5234 q^{24} +25.0000 q^{25} +75.1359 q^{26} +27.0000 q^{27} -51.9109 q^{28} -85.9109 q^{29} +55.5234 q^{30} -158.733 q^{31} -217.345 q^{32} -228.780 q^{34} -45.5234 q^{35} +51.3141 q^{36} +138.423 q^{37} -32.1515 q^{38} +60.8953 q^{39} -42.5391 q^{40} +28.2094 q^{41} -101.105 q^{42} +265.884 q^{43} +45.0000 q^{45} -472.869 q^{46} -515.602 q^{47} -231.314 q^{48} -260.105 q^{49} +92.5391 q^{50} -185.419 q^{51} +115.733 q^{52} -466.702 q^{53} +99.9422 q^{54} +77.4609 q^{56} -26.0578 q^{57} -318.005 q^{58} +373.475 q^{59} +85.5234 q^{60} +84.8172 q^{61} -587.559 q^{62} -81.9422 q^{63} -187.680 q^{64} +101.492 q^{65} -88.7657 q^{67} -352.392 q^{68} -383.245 q^{69} -168.508 q^{70} -536.695 q^{71} -76.5703 q^{72} +322.450 q^{73} +512.383 q^{74} +75.0000 q^{75} -49.5234 q^{76} +225.408 q^{78} +265.592 q^{79} -385.523 q^{80} +81.0000 q^{81} +104.419 q^{82} -520.758 q^{83} -155.733 q^{84} -309.031 q^{85} +984.187 q^{86} -257.733 q^{87} -464.713 q^{89} +166.570 q^{90} -184.811 q^{91} -728.366 q^{92} -476.198 q^{93} -1908.53 q^{94} -43.4297 q^{95} -652.036 q^{96} -204.481 q^{97} -962.794 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 6 q^{3} + 5 q^{4} + 10 q^{5} + 3 q^{6} + q^{7} + 15 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 + 6 * q^3 + 5 * q^4 + 10 * q^5 + 3 * q^6 + q^7 + 15 * q^8 + 18 * q^9 $$2 q + q^{2} + 6 q^{3} + 5 q^{4} + 10 q^{5} + 3 q^{6} + q^{7} + 15 q^{8} + 18 q^{9} + 5 q^{10} + 15 q^{12} + 47 q^{13} - 61 q^{14} + 30 q^{15} - 135 q^{16} - 98 q^{17} + 9 q^{18} - 75 q^{19} + 25 q^{20} + 3 q^{21} - 57 q^{23} + 45 q^{24} + 50 q^{25} + 3 q^{26} + 54 q^{27} - 59 q^{28} - 127 q^{29} + 15 q^{30} - 183 q^{31} - 249 q^{32} - 131 q^{34} + 5 q^{35} + 45 q^{36} - 229 q^{37} + 147 q^{38} + 141 q^{39} + 75 q^{40} + 18 q^{41} - 183 q^{42} + 186 q^{43} + 90 q^{45} - 664 q^{46} - 615 q^{47} - 405 q^{48} - 501 q^{49} + 25 q^{50} - 294 q^{51} + 97 q^{52} - 927 q^{53} + 27 q^{54} + 315 q^{56} - 225 q^{57} - 207 q^{58} - 380 q^{59} + 75 q^{60} + 509 q^{61} - 522 q^{62} + 9 q^{63} + 361 q^{64} + 235 q^{65} - 1138 q^{67} - 327 q^{68} - 171 q^{69} - 305 q^{70} - 273 q^{71} + 135 q^{72} + 440 q^{73} + 1505 q^{74} + 150 q^{75} - 3 q^{76} + 9 q^{78} + 973 q^{79} - 675 q^{80} + 162 q^{81} + 132 q^{82} + 15 q^{83} - 177 q^{84} - 490 q^{85} + 1200 q^{86} - 381 q^{87} - 1288 q^{89} + 45 q^{90} + 85 q^{91} - 778 q^{92} - 549 q^{93} - 1640 q^{94} - 375 q^{95} - 747 q^{96} - 76 q^{97} - 312 q^{98}+O(q^{100})$$ 2 * q + q^2 + 6 * q^3 + 5 * q^4 + 10 * q^5 + 3 * q^6 + q^7 + 15 * q^8 + 18 * q^9 + 5 * q^10 + 15 * q^12 + 47 * q^13 - 61 * q^14 + 30 * q^15 - 135 * q^16 - 98 * q^17 + 9 * q^18 - 75 * q^19 + 25 * q^20 + 3 * q^21 - 57 * q^23 + 45 * q^24 + 50 * q^25 + 3 * q^26 + 54 * q^27 - 59 * q^28 - 127 * q^29 + 15 * q^30 - 183 * q^31 - 249 * q^32 - 131 * q^34 + 5 * q^35 + 45 * q^36 - 229 * q^37 + 147 * q^38 + 141 * q^39 + 75 * q^40 + 18 * q^41 - 183 * q^42 + 186 * q^43 + 90 * q^45 - 664 * q^46 - 615 * q^47 - 405 * q^48 - 501 * q^49 + 25 * q^50 - 294 * q^51 + 97 * q^52 - 927 * q^53 + 27 * q^54 + 315 * q^56 - 225 * q^57 - 207 * q^58 - 380 * q^59 + 75 * q^60 + 509 * q^61 - 522 * q^62 + 9 * q^63 + 361 * q^64 + 235 * q^65 - 1138 * q^67 - 327 * q^68 - 171 * q^69 - 305 * q^70 - 273 * q^71 + 135 * q^72 + 440 * q^73 + 1505 * q^74 + 150 * q^75 - 3 * q^76 + 9 * q^78 + 973 * q^79 - 675 * q^80 + 162 * q^81 + 132 * q^82 + 15 * q^83 - 177 * q^84 - 490 * q^85 + 1200 * q^86 - 381 * q^87 - 1288 * q^89 + 45 * q^90 + 85 * q^91 - 778 * q^92 - 549 * q^93 - 1640 * q^94 - 375 * q^95 - 747 * q^96 - 76 * q^97 - 312 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.70156 1.30870 0.654350 0.756192i $$-0.272942\pi$$
0.654350 + 0.756192i $$0.272942\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 5.70156 0.712695
$$5$$ 5.00000 0.447214
$$6$$ 11.1047 0.755578
$$7$$ −9.10469 −0.491607 −0.245803 0.969320i $$-0.579052\pi$$
−0.245803 + 0.969320i $$0.579052\pi$$
$$8$$ −8.50781 −0.375996
$$9$$ 9.00000 0.333333
$$10$$ 18.5078 0.585268
$$11$$ 0 0
$$12$$ 17.1047 0.411475
$$13$$ 20.2984 0.433060 0.216530 0.976276i $$-0.430526\pi$$
0.216530 + 0.976276i $$0.430526\pi$$
$$14$$ −33.7016 −0.643366
$$15$$ 15.0000 0.258199
$$16$$ −77.1047 −1.20476
$$17$$ −61.8062 −0.881777 −0.440889 0.897562i $$-0.645337\pi$$
−0.440889 + 0.897562i $$0.645337\pi$$
$$18$$ 33.3141 0.436233
$$19$$ −8.68594 −0.104879 −0.0524393 0.998624i $$-0.516700\pi$$
−0.0524393 + 0.998624i $$0.516700\pi$$
$$20$$ 28.5078 0.318727
$$21$$ −27.3141 −0.283829
$$22$$ 0 0
$$23$$ −127.748 −1.15815 −0.579074 0.815275i $$-0.696586\pi$$
−0.579074 + 0.815275i $$0.696586\pi$$
$$24$$ −25.5234 −0.217081
$$25$$ 25.0000 0.200000
$$26$$ 75.1359 0.566745
$$27$$ 27.0000 0.192450
$$28$$ −51.9109 −0.350366
$$29$$ −85.9109 −0.550112 −0.275056 0.961428i $$-0.588696\pi$$
−0.275056 + 0.961428i $$0.588696\pi$$
$$30$$ 55.5234 0.337905
$$31$$ −158.733 −0.919653 −0.459827 0.888009i $$-0.652088\pi$$
−0.459827 + 0.888009i $$0.652088\pi$$
$$32$$ −217.345 −1.20067
$$33$$ 0 0
$$34$$ −228.780 −1.15398
$$35$$ −45.5234 −0.219853
$$36$$ 51.3141 0.237565
$$37$$ 138.423 0.615045 0.307523 0.951541i $$-0.400500\pi$$
0.307523 + 0.951541i $$0.400500\pi$$
$$38$$ −32.1515 −0.137254
$$39$$ 60.8953 0.250027
$$40$$ −42.5391 −0.168150
$$41$$ 28.2094 0.107453 0.0537264 0.998556i $$-0.482890\pi$$
0.0537264 + 0.998556i $$0.482890\pi$$
$$42$$ −101.105 −0.371447
$$43$$ 265.884 0.942953 0.471477 0.881879i $$-0.343721\pi$$
0.471477 + 0.881879i $$0.343721\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ −472.869 −1.51567
$$47$$ −515.602 −1.60017 −0.800087 0.599883i $$-0.795214\pi$$
−0.800087 + 0.599883i $$0.795214\pi$$
$$48$$ −231.314 −0.695569
$$49$$ −260.105 −0.758323
$$50$$ 92.5391 0.261740
$$51$$ −185.419 −0.509094
$$52$$ 115.733 0.308639
$$53$$ −466.702 −1.20955 −0.604777 0.796395i $$-0.706738\pi$$
−0.604777 + 0.796395i $$0.706738\pi$$
$$54$$ 99.9422 0.251859
$$55$$ 0 0
$$56$$ 77.4609 0.184842
$$57$$ −26.0578 −0.0605516
$$58$$ −318.005 −0.719932
$$59$$ 373.475 0.824107 0.412053 0.911160i $$-0.364812\pi$$
0.412053 + 0.911160i $$0.364812\pi$$
$$60$$ 85.5234 0.184017
$$61$$ 84.8172 0.178028 0.0890142 0.996030i $$-0.471628\pi$$
0.0890142 + 0.996030i $$0.471628\pi$$
$$62$$ −587.559 −1.20355
$$63$$ −81.9422 −0.163869
$$64$$ −187.680 −0.366562
$$65$$ 101.492 0.193670
$$66$$ 0 0
$$67$$ −88.7657 −0.161858 −0.0809288 0.996720i $$-0.525789\pi$$
−0.0809288 + 0.996720i $$0.525789\pi$$
$$68$$ −352.392 −0.628439
$$69$$ −383.245 −0.668657
$$70$$ −168.508 −0.287722
$$71$$ −536.695 −0.897099 −0.448549 0.893758i $$-0.648059\pi$$
−0.448549 + 0.893758i $$0.648059\pi$$
$$72$$ −76.5703 −0.125332
$$73$$ 322.450 0.516985 0.258493 0.966013i $$-0.416774\pi$$
0.258493 + 0.966013i $$0.416774\pi$$
$$74$$ 512.383 0.804909
$$75$$ 75.0000 0.115470
$$76$$ −49.5234 −0.0747464
$$77$$ 0 0
$$78$$ 225.408 0.327210
$$79$$ 265.592 0.378246 0.189123 0.981953i $$-0.439435\pi$$
0.189123 + 0.981953i $$0.439435\pi$$
$$80$$ −385.523 −0.538785
$$81$$ 81.0000 0.111111
$$82$$ 104.419 0.140623
$$83$$ −520.758 −0.688682 −0.344341 0.938845i $$-0.611898\pi$$
−0.344341 + 0.938845i $$0.611898\pi$$
$$84$$ −155.733 −0.202284
$$85$$ −309.031 −0.394343
$$86$$ 984.187 1.23404
$$87$$ −257.733 −0.317608
$$88$$ 0 0
$$89$$ −464.713 −0.553477 −0.276738 0.960945i $$-0.589254\pi$$
−0.276738 + 0.960945i $$0.589254\pi$$
$$90$$ 166.570 0.195089
$$91$$ −184.811 −0.212895
$$92$$ −728.366 −0.825406
$$93$$ −476.198 −0.530962
$$94$$ −1908.53 −2.09415
$$95$$ −43.4297 −0.0469031
$$96$$ −652.036 −0.693210
$$97$$ −204.481 −0.214040 −0.107020 0.994257i $$-0.534131\pi$$
−0.107020 + 0.994257i $$0.534131\pi$$
$$98$$ −962.794 −0.992417
$$99$$ 0 0
$$100$$ 142.539 0.142539
$$101$$ 27.2516 0.0268479 0.0134239 0.999910i $$-0.495727\pi$$
0.0134239 + 0.999910i $$0.495727\pi$$
$$102$$ −686.339 −0.666252
$$103$$ 1062.80 1.01671 0.508353 0.861149i $$-0.330255\pi$$
0.508353 + 0.861149i $$0.330255\pi$$
$$104$$ −172.695 −0.162828
$$105$$ −136.570 −0.126932
$$106$$ −1727.52 −1.58294
$$107$$ 1702.96 1.53861 0.769306 0.638881i $$-0.220602\pi$$
0.769306 + 0.638881i $$0.220602\pi$$
$$108$$ 153.942 0.137158
$$109$$ 556.334 0.488873 0.244437 0.969665i $$-0.421397\pi$$
0.244437 + 0.969665i $$0.421397\pi$$
$$110$$ 0 0
$$111$$ 415.270 0.355096
$$112$$ 702.014 0.592269
$$113$$ −2088.31 −1.73851 −0.869256 0.494363i $$-0.835401\pi$$
−0.869256 + 0.494363i $$0.835401\pi$$
$$114$$ −96.4546 −0.0792439
$$115$$ −638.742 −0.517939
$$116$$ −489.827 −0.392063
$$117$$ 182.686 0.144353
$$118$$ 1382.44 1.07851
$$119$$ 562.727 0.433488
$$120$$ −127.617 −0.0970817
$$121$$ 0 0
$$122$$ 313.956 0.232986
$$123$$ 84.6281 0.0620379
$$124$$ −905.025 −0.655433
$$125$$ 125.000 0.0894427
$$126$$ −303.314 −0.214455
$$127$$ −2179.16 −1.52259 −0.761296 0.648405i $$-0.775436\pi$$
−0.761296 + 0.648405i $$0.775436\pi$$
$$128$$ 1044.05 0.720955
$$129$$ 797.653 0.544414
$$130$$ 375.680 0.253456
$$131$$ −1951.11 −1.30129 −0.650645 0.759382i $$-0.725501\pi$$
−0.650645 + 0.759382i $$0.725501\pi$$
$$132$$ 0 0
$$133$$ 79.0828 0.0515590
$$134$$ −328.572 −0.211823
$$135$$ 135.000 0.0860663
$$136$$ 525.836 0.331545
$$137$$ −613.425 −0.382543 −0.191272 0.981537i $$-0.561261\pi$$
−0.191272 + 0.981537i $$0.561261\pi$$
$$138$$ −1418.61 −0.875071
$$139$$ −2130.63 −1.30013 −0.650064 0.759880i $$-0.725258\pi$$
−0.650064 + 0.759880i $$0.725258\pi$$
$$140$$ −259.555 −0.156688
$$141$$ −1546.80 −0.923861
$$142$$ −1986.61 −1.17403
$$143$$ 0 0
$$144$$ −693.942 −0.401587
$$145$$ −429.555 −0.246018
$$146$$ 1193.57 0.676578
$$147$$ −780.314 −0.437818
$$148$$ 789.230 0.438340
$$149$$ −570.900 −0.313892 −0.156946 0.987607i $$-0.550165\pi$$
−0.156946 + 0.987607i $$0.550165\pi$$
$$150$$ 277.617 0.151116
$$151$$ 2154.78 1.16128 0.580641 0.814159i $$-0.302802\pi$$
0.580641 + 0.814159i $$0.302802\pi$$
$$152$$ 73.8983 0.0394339
$$153$$ −556.256 −0.293926
$$154$$ 0 0
$$155$$ −793.664 −0.411281
$$156$$ 347.198 0.178193
$$157$$ −157.105 −0.0798619 −0.0399310 0.999202i $$-0.512714\pi$$
−0.0399310 + 0.999202i $$0.512714\pi$$
$$158$$ 983.106 0.495011
$$159$$ −1400.10 −0.698337
$$160$$ −1086.73 −0.536958
$$161$$ 1163.11 0.569353
$$162$$ 299.827 0.145411
$$163$$ 1736.66 0.834512 0.417256 0.908789i $$-0.362992\pi$$
0.417256 + 0.908789i $$0.362992\pi$$
$$164$$ 160.837 0.0765811
$$165$$ 0 0
$$166$$ −1927.62 −0.901278
$$167$$ −1466.55 −0.679549 −0.339775 0.940507i $$-0.610351\pi$$
−0.339775 + 0.940507i $$0.610351\pi$$
$$168$$ 232.383 0.106719
$$169$$ −1784.97 −0.812459
$$170$$ −1143.90 −0.516076
$$171$$ −78.1735 −0.0349595
$$172$$ 1515.96 0.672038
$$173$$ −143.781 −0.0631878 −0.0315939 0.999501i $$-0.510058\pi$$
−0.0315939 + 0.999501i $$0.510058\pi$$
$$174$$ −954.014 −0.415653
$$175$$ −227.617 −0.0983214
$$176$$ 0 0
$$177$$ 1120.42 0.475798
$$178$$ −1720.16 −0.724335
$$179$$ −1412.93 −0.589984 −0.294992 0.955500i $$-0.595317\pi$$
−0.294992 + 0.955500i $$0.595317\pi$$
$$180$$ 256.570 0.106242
$$181$$ −2239.25 −0.919570 −0.459785 0.888030i $$-0.652073\pi$$
−0.459785 + 0.888030i $$0.652073\pi$$
$$182$$ −684.089 −0.278616
$$183$$ 254.452 0.102785
$$184$$ 1086.86 0.435458
$$185$$ 692.117 0.275057
$$186$$ −1762.68 −0.694870
$$187$$ 0 0
$$188$$ −2939.73 −1.14044
$$189$$ −245.827 −0.0946098
$$190$$ −160.758 −0.0613821
$$191$$ 1745.94 0.661425 0.330712 0.943732i $$-0.392711\pi$$
0.330712 + 0.943732i $$0.392711\pi$$
$$192$$ −563.039 −0.211635
$$193$$ −3985.04 −1.48627 −0.743134 0.669143i $$-0.766662\pi$$
−0.743134 + 0.669143i $$0.766662\pi$$
$$194$$ −756.900 −0.280115
$$195$$ 304.477 0.111815
$$196$$ −1483.00 −0.540453
$$197$$ 2262.00 0.818076 0.409038 0.912517i $$-0.365864\pi$$
0.409038 + 0.912517i $$0.365864\pi$$
$$198$$ 0 0
$$199$$ 2340.15 0.833613 0.416806 0.908995i $$-0.363149\pi$$
0.416806 + 0.908995i $$0.363149\pi$$
$$200$$ −212.695 −0.0751991
$$201$$ −266.297 −0.0934485
$$202$$ 100.873 0.0351358
$$203$$ 782.192 0.270439
$$204$$ −1057.18 −0.362829
$$205$$ 141.047 0.0480543
$$206$$ 3934.01 1.33056
$$207$$ −1149.74 −0.386049
$$208$$ −1565.10 −0.521733
$$209$$ 0 0
$$210$$ −505.523 −0.166116
$$211$$ 1524.44 0.497377 0.248689 0.968583i $$-0.420000\pi$$
0.248689 + 0.968583i $$0.420000\pi$$
$$212$$ −2660.93 −0.862044
$$213$$ −1610.09 −0.517940
$$214$$ 6303.62 2.01358
$$215$$ 1329.42 0.421701
$$216$$ −229.711 −0.0723604
$$217$$ 1445.21 0.452108
$$218$$ 2059.31 0.639788
$$219$$ 967.350 0.298482
$$220$$ 0 0
$$221$$ −1254.57 −0.381862
$$222$$ 1537.15 0.464715
$$223$$ 5083.42 1.52651 0.763253 0.646100i $$-0.223601\pi$$
0.763253 + 0.646100i $$0.223601\pi$$
$$224$$ 1978.86 0.590260
$$225$$ 225.000 0.0666667
$$226$$ −7730.01 −2.27519
$$227$$ 3910.66 1.14343 0.571717 0.820451i $$-0.306277\pi$$
0.571717 + 0.820451i $$0.306277\pi$$
$$228$$ −148.570 −0.0431549
$$229$$ 2733.31 0.788743 0.394371 0.918951i $$-0.370962\pi$$
0.394371 + 0.918951i $$0.370962\pi$$
$$230$$ −2364.34 −0.677827
$$231$$ 0 0
$$232$$ 730.914 0.206840
$$233$$ 4355.46 1.22462 0.612308 0.790619i $$-0.290241\pi$$
0.612308 + 0.790619i $$0.290241\pi$$
$$234$$ 676.223 0.188915
$$235$$ −2578.01 −0.715620
$$236$$ 2129.39 0.587337
$$237$$ 796.777 0.218381
$$238$$ 2082.97 0.567305
$$239$$ 4731.48 1.28056 0.640281 0.768141i $$-0.278818\pi$$
0.640281 + 0.768141i $$0.278818\pi$$
$$240$$ −1156.57 −0.311068
$$241$$ −1833.12 −0.489966 −0.244983 0.969527i $$-0.578782\pi$$
−0.244983 + 0.969527i $$0.578782\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 483.591 0.126880
$$245$$ −1300.52 −0.339132
$$246$$ 313.256 0.0811890
$$247$$ −176.311 −0.0454186
$$248$$ 1350.47 0.345786
$$249$$ −1562.27 −0.397611
$$250$$ 462.695 0.117054
$$251$$ −3168.31 −0.796741 −0.398370 0.917225i $$-0.630424\pi$$
−0.398370 + 0.917225i $$0.630424\pi$$
$$252$$ −467.198 −0.116789
$$253$$ 0 0
$$254$$ −8066.29 −1.99262
$$255$$ −927.094 −0.227674
$$256$$ 5366.07 1.31008
$$257$$ 5378.55 1.30547 0.652733 0.757588i $$-0.273622\pi$$
0.652733 + 0.757588i $$0.273622\pi$$
$$258$$ 2952.56 0.712475
$$259$$ −1260.30 −0.302360
$$260$$ 578.664 0.138028
$$261$$ −773.198 −0.183371
$$262$$ −7222.14 −1.70300
$$263$$ −4584.03 −1.07477 −0.537383 0.843338i $$-0.680587\pi$$
−0.537383 + 0.843338i $$0.680587\pi$$
$$264$$ 0 0
$$265$$ −2333.51 −0.540929
$$266$$ 292.730 0.0674752
$$267$$ −1394.14 −0.319550
$$268$$ −506.103 −0.115355
$$269$$ −1741.03 −0.394618 −0.197309 0.980341i $$-0.563220\pi$$
−0.197309 + 0.980341i $$0.563220\pi$$
$$270$$ 499.711 0.112635
$$271$$ −1340.58 −0.300495 −0.150248 0.988648i $$-0.548007\pi$$
−0.150248 + 0.988648i $$0.548007\pi$$
$$272$$ 4765.55 1.06233
$$273$$ −554.433 −0.122915
$$274$$ −2270.63 −0.500634
$$275$$ 0 0
$$276$$ −2185.10 −0.476548
$$277$$ 5220.77 1.13244 0.566220 0.824254i $$-0.308405\pi$$
0.566220 + 0.824254i $$0.308405\pi$$
$$278$$ −7886.66 −1.70148
$$279$$ −1428.60 −0.306551
$$280$$ 387.305 0.0826639
$$281$$ −2494.68 −0.529609 −0.264804 0.964302i $$-0.585307\pi$$
−0.264804 + 0.964302i $$0.585307\pi$$
$$282$$ −5725.59 −1.20906
$$283$$ −4329.43 −0.909391 −0.454696 0.890647i $$-0.650252\pi$$
−0.454696 + 0.890647i $$0.650252\pi$$
$$284$$ −3060.00 −0.639358
$$285$$ −130.289 −0.0270795
$$286$$ 0 0
$$287$$ −256.837 −0.0528245
$$288$$ −1956.11 −0.400225
$$289$$ −1092.99 −0.222468
$$290$$ −1590.02 −0.321963
$$291$$ −613.444 −0.123576
$$292$$ 1838.47 0.368453
$$293$$ 4330.93 0.863534 0.431767 0.901985i $$-0.357890\pi$$
0.431767 + 0.901985i $$0.357890\pi$$
$$294$$ −2888.38 −0.572972
$$295$$ 1867.37 0.368552
$$296$$ −1177.68 −0.231254
$$297$$ 0 0
$$298$$ −2113.22 −0.410791
$$299$$ −2593.09 −0.501547
$$300$$ 427.617 0.0822950
$$301$$ −2420.79 −0.463562
$$302$$ 7976.06 1.51977
$$303$$ 81.7547 0.0155006
$$304$$ 669.727 0.126353
$$305$$ 424.086 0.0796167
$$306$$ −2059.02 −0.384661
$$307$$ 2052.56 0.381582 0.190791 0.981631i $$-0.438895\pi$$
0.190791 + 0.981631i $$0.438895\pi$$
$$308$$ 0 0
$$309$$ 3188.39 0.586995
$$310$$ −2937.80 −0.538244
$$311$$ 2627.76 0.479121 0.239561 0.970881i $$-0.422997\pi$$
0.239561 + 0.970881i $$0.422997\pi$$
$$312$$ −518.086 −0.0940091
$$313$$ 6878.40 1.24214 0.621070 0.783755i $$-0.286698\pi$$
0.621070 + 0.783755i $$0.286698\pi$$
$$314$$ −581.533 −0.104515
$$315$$ −409.711 −0.0732844
$$316$$ 1514.29 0.269574
$$317$$ −3026.05 −0.536151 −0.268076 0.963398i $$-0.586388\pi$$
−0.268076 + 0.963398i $$0.586388\pi$$
$$318$$ −5182.57 −0.913913
$$319$$ 0 0
$$320$$ −938.398 −0.163931
$$321$$ 5108.88 0.888318
$$322$$ 4305.32 0.745112
$$323$$ 536.845 0.0924795
$$324$$ 461.827 0.0791884
$$325$$ 507.461 0.0866119
$$326$$ 6428.34 1.09213
$$327$$ 1669.00 0.282251
$$328$$ −240.000 −0.0404018
$$329$$ 4694.39 0.786657
$$330$$ 0 0
$$331$$ −4326.50 −0.718447 −0.359223 0.933252i $$-0.616958\pi$$
−0.359223 + 0.933252i $$0.616958\pi$$
$$332$$ −2969.13 −0.490820
$$333$$ 1245.81 0.205015
$$334$$ −5428.51 −0.889326
$$335$$ −443.828 −0.0723849
$$336$$ 2106.04 0.341946
$$337$$ 1759.36 0.284387 0.142194 0.989839i $$-0.454584\pi$$
0.142194 + 0.989839i $$0.454584\pi$$
$$338$$ −6607.19 −1.06327
$$339$$ −6264.93 −1.00373
$$340$$ −1761.96 −0.281046
$$341$$ 0 0
$$342$$ −289.364 −0.0457515
$$343$$ 5491.08 0.864403
$$344$$ −2262.09 −0.354546
$$345$$ −1916.23 −0.299032
$$346$$ −532.215 −0.0826939
$$347$$ 11730.7 1.81481 0.907404 0.420260i $$-0.138061\pi$$
0.907404 + 0.420260i $$0.138061\pi$$
$$348$$ −1469.48 −0.226357
$$349$$ 3152.38 0.483504 0.241752 0.970338i $$-0.422278\pi$$
0.241752 + 0.970338i $$0.422278\pi$$
$$350$$ −842.539 −0.128673
$$351$$ 548.058 0.0833423
$$352$$ 0 0
$$353$$ −6881.03 −1.03751 −0.518754 0.854923i $$-0.673604\pi$$
−0.518754 + 0.854923i $$0.673604\pi$$
$$354$$ 4147.32 0.622677
$$355$$ −2683.48 −0.401195
$$356$$ −2649.59 −0.394460
$$357$$ 1688.18 0.250274
$$358$$ −5230.04 −0.772112
$$359$$ 2858.96 0.420307 0.210154 0.977668i $$-0.432604\pi$$
0.210154 + 0.977668i $$0.432604\pi$$
$$360$$ −382.851 −0.0560501
$$361$$ −6783.55 −0.989001
$$362$$ −8288.72 −1.20344
$$363$$ 0 0
$$364$$ −1053.71 −0.151729
$$365$$ 1612.25 0.231203
$$366$$ 941.868 0.134514
$$367$$ −10973.4 −1.56079 −0.780393 0.625289i $$-0.784981\pi$$
−0.780393 + 0.625289i $$0.784981\pi$$
$$368$$ 9850.00 1.39529
$$369$$ 253.884 0.0358176
$$370$$ 2561.91 0.359966
$$371$$ 4249.17 0.594625
$$372$$ −2715.07 −0.378414
$$373$$ −10443.4 −1.44970 −0.724852 0.688905i $$-0.758092\pi$$
−0.724852 + 0.688905i $$0.758092\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 4386.64 0.601659
$$377$$ −1743.86 −0.238231
$$378$$ −909.942 −0.123816
$$379$$ −7405.86 −1.00373 −0.501865 0.864946i $$-0.667352\pi$$
−0.501865 + 0.864946i $$0.667352\pi$$
$$380$$ −247.617 −0.0334276
$$381$$ −6537.48 −0.879069
$$382$$ 6462.72 0.865607
$$383$$ 14155.0 1.88848 0.944238 0.329264i $$-0.106801\pi$$
0.944238 + 0.329264i $$0.106801\pi$$
$$384$$ 3132.16 0.416244
$$385$$ 0 0
$$386$$ −14750.9 −1.94508
$$387$$ 2392.96 0.314318
$$388$$ −1165.86 −0.152546
$$389$$ −13031.6 −1.69853 −0.849263 0.527970i $$-0.822953\pi$$
−0.849263 + 0.527970i $$0.822953\pi$$
$$390$$ 1127.04 0.146333
$$391$$ 7895.65 1.02123
$$392$$ 2212.92 0.285126
$$393$$ −5853.32 −0.751300
$$394$$ 8372.94 1.07062
$$395$$ 1327.96 0.169157
$$396$$ 0 0
$$397$$ 2281.02 0.288366 0.144183 0.989551i $$-0.453945\pi$$
0.144183 + 0.989551i $$0.453945\pi$$
$$398$$ 8662.22 1.09095
$$399$$ 237.248 0.0297676
$$400$$ −1927.62 −0.240952
$$401$$ −13017.7 −1.62114 −0.810568 0.585645i $$-0.800841\pi$$
−0.810568 + 0.585645i $$0.800841\pi$$
$$402$$ −985.715 −0.122296
$$403$$ −3222.03 −0.398265
$$404$$ 155.377 0.0191343
$$405$$ 405.000 0.0496904
$$406$$ 2895.33 0.353924
$$407$$ 0 0
$$408$$ 1577.51 0.191417
$$409$$ 3548.46 0.428998 0.214499 0.976724i $$-0.431188\pi$$
0.214499 + 0.976724i $$0.431188\pi$$
$$410$$ 522.094 0.0628887
$$411$$ −1840.27 −0.220861
$$412$$ 6059.61 0.724601
$$413$$ −3400.37 −0.405136
$$414$$ −4255.82 −0.505222
$$415$$ −2603.79 −0.307988
$$416$$ −4411.77 −0.519964
$$417$$ −6391.89 −0.750629
$$418$$ 0 0
$$419$$ −14801.3 −1.72575 −0.862876 0.505416i $$-0.831339\pi$$
−0.862876 + 0.505416i $$0.831339\pi$$
$$420$$ −778.664 −0.0904641
$$421$$ 11542.8 1.33625 0.668123 0.744051i $$-0.267098\pi$$
0.668123 + 0.744051i $$0.267098\pi$$
$$422$$ 5642.80 0.650918
$$423$$ −4640.41 −0.533392
$$424$$ 3970.61 0.454787
$$425$$ −1545.16 −0.176355
$$426$$ −5959.83 −0.677828
$$427$$ −772.234 −0.0875200
$$428$$ 9709.54 1.09656
$$429$$ 0 0
$$430$$ 4920.94 0.551881
$$431$$ 12712.6 1.42076 0.710378 0.703820i $$-0.248524\pi$$
0.710378 + 0.703820i $$0.248524\pi$$
$$432$$ −2081.83 −0.231856
$$433$$ −16385.4 −1.81855 −0.909273 0.416200i $$-0.863361\pi$$
−0.909273 + 0.416200i $$0.863361\pi$$
$$434$$ 5349.54 0.591674
$$435$$ −1288.66 −0.142038
$$436$$ 3171.97 0.348418
$$437$$ 1109.62 0.121465
$$438$$ 3580.71 0.390623
$$439$$ 6603.62 0.717935 0.358968 0.933350i $$-0.383129\pi$$
0.358968 + 0.933350i $$0.383129\pi$$
$$440$$ 0 0
$$441$$ −2340.94 −0.252774
$$442$$ −4643.87 −0.499743
$$443$$ −11641.0 −1.24849 −0.624244 0.781230i $$-0.714593\pi$$
−0.624244 + 0.781230i $$0.714593\pi$$
$$444$$ 2367.69 0.253076
$$445$$ −2323.56 −0.247522
$$446$$ 18816.6 1.99774
$$447$$ −1712.70 −0.181226
$$448$$ 1708.76 0.180204
$$449$$ 13886.8 1.45960 0.729799 0.683662i $$-0.239614\pi$$
0.729799 + 0.683662i $$0.239614\pi$$
$$450$$ 832.851 0.0872467
$$451$$ 0 0
$$452$$ −11906.6 −1.23903
$$453$$ 6464.35 0.670467
$$454$$ 14475.6 1.49641
$$455$$ −924.055 −0.0952096
$$456$$ 221.695 0.0227672
$$457$$ 3625.85 0.371138 0.185569 0.982631i $$-0.440587\pi$$
0.185569 + 0.982631i $$0.440587\pi$$
$$458$$ 10117.5 1.03223
$$459$$ −1668.77 −0.169698
$$460$$ −3641.83 −0.369133
$$461$$ 18329.1 1.85178 0.925890 0.377794i $$-0.123317\pi$$
0.925890 + 0.377794i $$0.123317\pi$$
$$462$$ 0 0
$$463$$ 10760.7 1.08011 0.540056 0.841629i $$-0.318403\pi$$
0.540056 + 0.841629i $$0.318403\pi$$
$$464$$ 6624.14 0.662754
$$465$$ −2380.99 −0.237453
$$466$$ 16122.0 1.60265
$$467$$ −7964.29 −0.789172 −0.394586 0.918859i $$-0.629112\pi$$
−0.394586 + 0.918859i $$0.629112\pi$$
$$468$$ 1041.60 0.102880
$$469$$ 808.184 0.0795703
$$470$$ −9542.66 −0.936532
$$471$$ −471.314 −0.0461083
$$472$$ −3177.45 −0.309861
$$473$$ 0 0
$$474$$ 2949.32 0.285795
$$475$$ −217.149 −0.0209757
$$476$$ 3208.42 0.308945
$$477$$ −4200.31 −0.403185
$$478$$ 17513.9 1.67587
$$479$$ 822.463 0.0784536 0.0392268 0.999230i $$-0.487511\pi$$
0.0392268 + 0.999230i $$0.487511\pi$$
$$480$$ −3260.18 −0.310013
$$481$$ 2809.78 0.266351
$$482$$ −6785.41 −0.641218
$$483$$ 3489.33 0.328716
$$484$$ 0 0
$$485$$ −1022.41 −0.0957218
$$486$$ 899.480 0.0839531
$$487$$ 6709.58 0.624312 0.312156 0.950031i $$-0.398949\pi$$
0.312156 + 0.950031i $$0.398949\pi$$
$$488$$ −721.609 −0.0669379
$$489$$ 5209.97 0.481806
$$490$$ −4813.97 −0.443822
$$491$$ 12689.3 1.16631 0.583156 0.812360i $$-0.301818\pi$$
0.583156 + 0.812360i $$0.301818\pi$$
$$492$$ 482.512 0.0442141
$$493$$ 5309.83 0.485077
$$494$$ −652.626 −0.0594394
$$495$$ 0 0
$$496$$ 12239.0 1.10796
$$497$$ 4886.44 0.441020
$$498$$ −5782.85 −0.520353
$$499$$ −9810.79 −0.880143 −0.440072 0.897963i $$-0.645047\pi$$
−0.440072 + 0.897963i $$0.645047\pi$$
$$500$$ 712.695 0.0637454
$$501$$ −4399.64 −0.392338
$$502$$ −11727.7 −1.04269
$$503$$ −12865.7 −1.14047 −0.570233 0.821483i $$-0.693147\pi$$
−0.570233 + 0.821483i $$0.693147\pi$$
$$504$$ 697.149 0.0616140
$$505$$ 136.258 0.0120067
$$506$$ 0 0
$$507$$ −5354.92 −0.469074
$$508$$ −12424.6 −1.08514
$$509$$ 2551.78 0.222211 0.111106 0.993809i $$-0.464561\pi$$
0.111106 + 0.993809i $$0.464561\pi$$
$$510$$ −3431.70 −0.297957
$$511$$ −2935.81 −0.254153
$$512$$ 11510.4 0.993541
$$513$$ −234.520 −0.0201839
$$514$$ 19909.0 1.70846
$$515$$ 5313.99 0.454684
$$516$$ 4547.87 0.388001
$$517$$ 0 0
$$518$$ −4665.09 −0.395699
$$519$$ −431.344 −0.0364815
$$520$$ −863.476 −0.0728191
$$521$$ 9761.13 0.820812 0.410406 0.911903i $$-0.365387\pi$$
0.410406 + 0.911903i $$0.365387\pi$$
$$522$$ −2862.04 −0.239977
$$523$$ 4548.79 0.380315 0.190157 0.981754i $$-0.439100\pi$$
0.190157 + 0.981754i $$0.439100\pi$$
$$524$$ −11124.4 −0.927423
$$525$$ −682.851 −0.0567659
$$526$$ −16968.1 −1.40655
$$527$$ 9810.68 0.810930
$$528$$ 0 0
$$529$$ 4152.66 0.341305
$$530$$ −8637.62 −0.707914
$$531$$ 3361.27 0.274702
$$532$$ 450.895 0.0367458
$$533$$ 572.606 0.0465334
$$534$$ −5160.49 −0.418195
$$535$$ 8514.80 0.688088
$$536$$ 755.202 0.0608577
$$537$$ −4238.78 −0.340628
$$538$$ −6444.52 −0.516437
$$539$$ 0 0
$$540$$ 769.711 0.0613390
$$541$$ −9184.79 −0.729917 −0.364958 0.931024i $$-0.618917\pi$$
−0.364958 + 0.931024i $$0.618917\pi$$
$$542$$ −4962.23 −0.393258
$$543$$ −6717.75 −0.530914
$$544$$ 13433.3 1.05873
$$545$$ 2781.67 0.218631
$$546$$ −2052.27 −0.160859
$$547$$ 20966.6 1.63888 0.819439 0.573167i $$-0.194285\pi$$
0.819439 + 0.573167i $$0.194285\pi$$
$$548$$ −3497.48 −0.272637
$$549$$ 763.355 0.0593428
$$550$$ 0 0
$$551$$ 746.217 0.0576950
$$552$$ 3260.58 0.251412
$$553$$ −2418.13 −0.185948
$$554$$ 19325.0 1.48202
$$555$$ 2076.35 0.158804
$$556$$ −12147.9 −0.926595
$$557$$ −4422.49 −0.336422 −0.168211 0.985751i $$-0.553799\pi$$
−0.168211 + 0.985751i $$0.553799\pi$$
$$558$$ −5288.03 −0.401183
$$559$$ 5397.04 0.408355
$$560$$ 3510.07 0.264871
$$561$$ 0 0
$$562$$ −9234.21 −0.693099
$$563$$ −6066.46 −0.454122 −0.227061 0.973881i $$-0.572912\pi$$
−0.227061 + 0.973881i $$0.572912\pi$$
$$564$$ −8819.20 −0.658432
$$565$$ −10441.6 −0.777486
$$566$$ −16025.6 −1.19012
$$567$$ −737.480 −0.0546230
$$568$$ 4566.10 0.337305
$$569$$ 17341.9 1.27770 0.638850 0.769331i $$-0.279410\pi$$
0.638850 + 0.769331i $$0.279410\pi$$
$$570$$ −482.273 −0.0354390
$$571$$ 621.014 0.0455142 0.0227571 0.999741i $$-0.492756\pi$$
0.0227571 + 0.999741i $$0.492756\pi$$
$$572$$ 0 0
$$573$$ 5237.83 0.381874
$$574$$ −950.700 −0.0691314
$$575$$ −3193.71 −0.231629
$$576$$ −1689.12 −0.122187
$$577$$ 4458.77 0.321700 0.160850 0.986979i $$-0.448576\pi$$
0.160850 + 0.986979i $$0.448576\pi$$
$$578$$ −4045.76 −0.291144
$$579$$ −11955.1 −0.858097
$$580$$ −2449.13 −0.175336
$$581$$ 4741.34 0.338561
$$582$$ −2270.70 −0.161724
$$583$$ 0 0
$$584$$ −2743.34 −0.194384
$$585$$ 913.430 0.0645567
$$586$$ 16031.2 1.13011
$$587$$ 1326.96 0.0933038 0.0466519 0.998911i $$-0.485145\pi$$
0.0466519 + 0.998911i $$0.485145\pi$$
$$588$$ −4449.01 −0.312031
$$589$$ 1378.74 0.0964519
$$590$$ 6912.20 0.482324
$$591$$ 6786.01 0.472317
$$592$$ −10673.1 −0.740982
$$593$$ −13296.2 −0.920756 −0.460378 0.887723i $$-0.652286\pi$$
−0.460378 + 0.887723i $$0.652286\pi$$
$$594$$ 0 0
$$595$$ 2813.63 0.193862
$$596$$ −3255.02 −0.223710
$$597$$ 7020.45 0.481287
$$598$$ −9598.50 −0.656374
$$599$$ −7420.72 −0.506181 −0.253091 0.967443i $$-0.581447\pi$$
−0.253091 + 0.967443i $$0.581447\pi$$
$$600$$ −638.086 −0.0434162
$$601$$ 5189.31 0.352207 0.176104 0.984372i $$-0.443651\pi$$
0.176104 + 0.984372i $$0.443651\pi$$
$$602$$ −8960.72 −0.606664
$$603$$ −798.891 −0.0539525
$$604$$ 12285.6 0.827641
$$605$$ 0 0
$$606$$ 302.620 0.0202857
$$607$$ −26545.6 −1.77505 −0.887523 0.460763i $$-0.847576\pi$$
−0.887523 + 0.460763i $$0.847576\pi$$
$$608$$ 1887.85 0.125925
$$609$$ 2346.58 0.156138
$$610$$ 1569.78 0.104194
$$611$$ −10465.9 −0.692971
$$612$$ −3171.53 −0.209480
$$613$$ −26097.9 −1.71955 −0.859776 0.510671i $$-0.829397\pi$$
−0.859776 + 0.510671i $$0.829397\pi$$
$$614$$ 7597.67 0.499377
$$615$$ 423.141 0.0277442
$$616$$ 0 0
$$617$$ 12275.1 0.800933 0.400466 0.916311i $$-0.368848\pi$$
0.400466 + 0.916311i $$0.368848\pi$$
$$618$$ 11802.0 0.768200
$$619$$ 6067.21 0.393961 0.196980 0.980407i $$-0.436886\pi$$
0.196980 + 0.980407i $$0.436886\pi$$
$$620$$ −4525.12 −0.293118
$$621$$ −3449.21 −0.222886
$$622$$ 9726.82 0.627026
$$623$$ 4231.06 0.272093
$$624$$ −4695.31 −0.301223
$$625$$ 625.000 0.0400000
$$626$$ 25460.8 1.62559
$$627$$ 0 0
$$628$$ −895.742 −0.0569172
$$629$$ −8555.43 −0.542333
$$630$$ −1516.57 −0.0959073
$$631$$ 3074.16 0.193947 0.0969733 0.995287i $$-0.469084\pi$$
0.0969733 + 0.995287i $$0.469084\pi$$
$$632$$ −2259.61 −0.142219
$$633$$ 4573.31 0.287161
$$634$$ −11201.1 −0.701661
$$635$$ −10895.8 −0.680924
$$636$$ −7982.78 −0.497701
$$637$$ −5279.72 −0.328399
$$638$$ 0 0
$$639$$ −4830.26 −0.299033
$$640$$ 5220.27 0.322421
$$641$$ −10825.9 −0.667080 −0.333540 0.942736i $$-0.608243\pi$$
−0.333540 + 0.942736i $$0.608243\pi$$
$$642$$ 18910.8 1.16254
$$643$$ −117.142 −0.00718450 −0.00359225 0.999994i $$-0.501143\pi$$
−0.00359225 + 0.999994i $$0.501143\pi$$
$$644$$ 6631.54 0.405775
$$645$$ 3988.27 0.243469
$$646$$ 1987.17 0.121028
$$647$$ 23811.6 1.44688 0.723439 0.690388i $$-0.242560\pi$$
0.723439 + 0.690388i $$0.242560\pi$$
$$648$$ −689.133 −0.0417773
$$649$$ 0 0
$$650$$ 1878.40 0.113349
$$651$$ 4335.64 0.261025
$$652$$ 9901.65 0.594753
$$653$$ −17773.3 −1.06512 −0.532560 0.846392i $$-0.678770\pi$$
−0.532560 + 0.846392i $$0.678770\pi$$
$$654$$ 6177.92 0.369382
$$655$$ −9755.53 −0.581954
$$656$$ −2175.07 −0.129455
$$657$$ 2902.05 0.172328
$$658$$ 17376.6 1.02950
$$659$$ −13015.6 −0.769370 −0.384685 0.923048i $$-0.625690\pi$$
−0.384685 + 0.923048i $$0.625690\pi$$
$$660$$ 0 0
$$661$$ 15769.6 0.927935 0.463968 0.885852i $$-0.346425\pi$$
0.463968 + 0.885852i $$0.346425\pi$$
$$662$$ −16014.8 −0.940231
$$663$$ −3763.71 −0.220468
$$664$$ 4430.51 0.258941
$$665$$ 395.414 0.0230579
$$666$$ 4611.45 0.268303
$$667$$ 10975.0 0.637111
$$668$$ −8361.60 −0.484311
$$669$$ 15250.3 0.881329
$$670$$ −1642.86 −0.0947301
$$671$$ 0 0
$$672$$ 5936.58 0.340787
$$673$$ −31519.1 −1.80531 −0.902653 0.430370i $$-0.858383\pi$$
−0.902653 + 0.430370i $$0.858383\pi$$
$$674$$ 6512.38 0.372177
$$675$$ 675.000 0.0384900
$$676$$ −10177.1 −0.579036
$$677$$ 12418.6 0.705000 0.352500 0.935812i $$-0.385332\pi$$
0.352500 + 0.935812i $$0.385332\pi$$
$$678$$ −23190.0 −1.31358
$$679$$ 1861.74 0.105224
$$680$$ 2629.18 0.148271
$$681$$ 11732.0 0.660162
$$682$$ 0 0
$$683$$ 23543.0 1.31896 0.659478 0.751724i $$-0.270777\pi$$
0.659478 + 0.751724i $$0.270777\pi$$
$$684$$ −445.711 −0.0249155
$$685$$ −3067.12 −0.171079
$$686$$ 20325.6 1.13124
$$687$$ 8199.92 0.455381
$$688$$ −20500.9 −1.13603
$$689$$ −9473.31 −0.523809
$$690$$ −7093.03 −0.391344
$$691$$ −9642.84 −0.530870 −0.265435 0.964129i $$-0.585516\pi$$
−0.265435 + 0.964129i $$0.585516\pi$$
$$692$$ −819.778 −0.0450336
$$693$$ 0 0
$$694$$ 43422.0 2.37504
$$695$$ −10653.2 −0.581435
$$696$$ 2192.74 0.119419
$$697$$ −1743.52 −0.0947494
$$698$$ 11668.7 0.632762
$$699$$ 13066.4 0.707032
$$700$$ −1297.77 −0.0700732
$$701$$ 30593.3 1.64835 0.824176 0.566334i $$-0.191639\pi$$
0.824176 + 0.566334i $$0.191639\pi$$
$$702$$ 2028.67 0.109070
$$703$$ −1202.34 −0.0645050
$$704$$ 0 0
$$705$$ −7734.02 −0.413163
$$706$$ −25470.6 −1.35779
$$707$$ −248.117 −0.0131986
$$708$$ 6388.17 0.339099
$$709$$ −30248.4 −1.60226 −0.801130 0.598491i $$-0.795767\pi$$
−0.801130 + 0.598491i $$0.795767\pi$$
$$710$$ −9933.05 −0.525044
$$711$$ 2390.33 0.126082
$$712$$ 3953.69 0.208105
$$713$$ 20277.9 1.06509
$$714$$ 6248.90 0.327534
$$715$$ 0 0
$$716$$ −8055.90 −0.420479
$$717$$ 14194.4 0.739332
$$718$$ 10582.6 0.550056
$$719$$ 30471.8 1.58054 0.790269 0.612761i $$-0.209941\pi$$
0.790269 + 0.612761i $$0.209941\pi$$
$$720$$ −3469.71 −0.179595
$$721$$ −9676.45 −0.499819
$$722$$ −25109.7 −1.29430
$$723$$ −5499.37 −0.282882
$$724$$ −12767.2 −0.655373
$$725$$ −2147.77 −0.110022
$$726$$ 0 0
$$727$$ 25149.9 1.28302 0.641511 0.767114i $$-0.278308\pi$$
0.641511 + 0.767114i $$0.278308\pi$$
$$728$$ 1572.34 0.0800476
$$729$$ 729.000 0.0370370
$$730$$ 5967.84 0.302575
$$731$$ −16433.3 −0.831475
$$732$$ 1450.77 0.0732542
$$733$$ −9974.73 −0.502626 −0.251313 0.967906i $$-0.580862\pi$$
−0.251313 + 0.967906i $$0.580862\pi$$
$$734$$ −40618.8 −2.04260
$$735$$ −3901.57 −0.195798
$$736$$ 27765.5 1.39056
$$737$$ 0 0
$$738$$ 939.769 0.0468745
$$739$$ −11579.9 −0.576418 −0.288209 0.957568i $$-0.593060\pi$$
−0.288209 + 0.957568i $$0.593060\pi$$
$$740$$ 3946.15 0.196031
$$741$$ −528.933 −0.0262225
$$742$$ 15728.6 0.778186
$$743$$ −20323.7 −1.00351 −0.501753 0.865011i $$-0.667311\pi$$
−0.501753 + 0.865011i $$0.667311\pi$$
$$744$$ 4051.41 0.199639
$$745$$ −2854.50 −0.140377
$$746$$ −38656.9 −1.89723
$$747$$ −4686.82 −0.229561
$$748$$ 0 0
$$749$$ −15504.9 −0.756392
$$750$$ 1388.09 0.0675810
$$751$$ 8023.94 0.389877 0.194939 0.980815i $$-0.437549\pi$$
0.194939 + 0.980815i $$0.437549\pi$$
$$752$$ 39755.3 1.92783
$$753$$ −9504.93 −0.459998
$$754$$ −6455.00 −0.311773
$$755$$ 10773.9 0.519342
$$756$$ −1401.60 −0.0674279
$$757$$ 542.713 0.0260571 0.0130285 0.999915i $$-0.495853\pi$$
0.0130285 + 0.999915i $$0.495853\pi$$
$$758$$ −27413.2 −1.31358
$$759$$ 0 0
$$760$$ 369.492 0.0176354
$$761$$ 16292.2 0.776073 0.388037 0.921644i $$-0.373153\pi$$
0.388037 + 0.921644i $$0.373153\pi$$
$$762$$ −24198.9 −1.15044
$$763$$ −5065.25 −0.240333
$$764$$ 9954.61 0.471394
$$765$$ −2781.28 −0.131448
$$766$$ 52395.6 2.47145
$$767$$ 7580.96 0.356887
$$768$$ 16098.2 0.756373
$$769$$ −7107.01 −0.333271 −0.166636 0.986019i $$-0.553290\pi$$
−0.166636 + 0.986019i $$0.553290\pi$$
$$770$$ 0 0
$$771$$ 16135.7 0.753711
$$772$$ −22721.0 −1.05926
$$773$$ −19776.7 −0.920204 −0.460102 0.887866i $$-0.652187\pi$$
−0.460102 + 0.887866i $$0.652187\pi$$
$$774$$ 8857.69 0.411348
$$775$$ −3968.32 −0.183931
$$776$$ 1739.69 0.0804783
$$777$$ −3780.91 −0.174568
$$778$$ −48237.1 −2.22286
$$779$$ −245.025 −0.0112695
$$780$$ 1735.99 0.0796904
$$781$$ 0 0
$$782$$ 29226.2 1.33648
$$783$$ −2319.60 −0.105869
$$784$$ 20055.3 0.913597
$$785$$ −785.523 −0.0357153
$$786$$ −21666.4 −0.983226
$$787$$ 19943.8 0.903327 0.451664 0.892188i $$-0.350831\pi$$
0.451664 + 0.892188i $$0.350831\pi$$
$$788$$ 12896.9 0.583039
$$789$$ −13752.1 −0.620517
$$790$$ 4915.53 0.221376
$$791$$ 19013.4 0.854664
$$792$$ 0 0
$$793$$ 1721.66 0.0770969
$$794$$ 8443.34 0.377384
$$795$$ −7000.52 −0.312306
$$796$$ 13342.5 0.594112
$$797$$ −24333.8 −1.08149 −0.540744 0.841187i $$-0.681857\pi$$
−0.540744 + 0.841187i $$0.681857\pi$$
$$798$$ 878.189 0.0389568
$$799$$ 31867.4 1.41100
$$800$$ −5433.63 −0.240135
$$801$$ −4182.41 −0.184492
$$802$$ −48186.0 −2.12158
$$803$$ 0 0
$$804$$ −1518.31 −0.0666003
$$805$$ 5815.55 0.254622
$$806$$ −11926.5 −0.521209
$$807$$ −5223.08 −0.227833
$$808$$ −231.851 −0.0100947
$$809$$ −15308.7 −0.665298 −0.332649 0.943051i $$-0.607942\pi$$
−0.332649 + 0.943051i $$0.607942\pi$$
$$810$$ 1499.13 0.0650298
$$811$$ 22639.1 0.980229 0.490115 0.871658i $$-0.336955\pi$$
0.490115 + 0.871658i $$0.336955\pi$$
$$812$$ 4459.72 0.192741
$$813$$ −4021.73 −0.173491
$$814$$ 0 0
$$815$$ 8683.28 0.373205
$$816$$ 14296.7 0.613337
$$817$$ −2309.46 −0.0988955
$$818$$ 13134.8 0.561429
$$819$$ −1663.30 −0.0709650
$$820$$ 804.187 0.0342481
$$821$$ −35341.0 −1.50232 −0.751162 0.660118i $$-0.770506\pi$$
−0.751162 + 0.660118i $$0.770506\pi$$
$$822$$ −6811.89 −0.289041
$$823$$ 28121.5 1.19107 0.595536 0.803329i $$-0.296940\pi$$
0.595536 + 0.803329i $$0.296940\pi$$
$$824$$ −9042.09 −0.382277
$$825$$ 0 0
$$826$$ −12586.7 −0.530202
$$827$$ −5952.92 −0.250306 −0.125153 0.992137i $$-0.539942\pi$$
−0.125153 + 0.992137i $$0.539942\pi$$
$$828$$ −6555.29 −0.275135
$$829$$ 18696.9 0.783316 0.391658 0.920111i $$-0.371902\pi$$
0.391658 + 0.920111i $$0.371902\pi$$
$$830$$ −9638.09 −0.403064
$$831$$ 15662.3 0.653814
$$832$$ −3809.60 −0.158743
$$833$$ 16076.1 0.668672
$$834$$ −23660.0 −0.982348
$$835$$ −7332.73 −0.303904
$$836$$ 0 0
$$837$$ −4285.79 −0.176987
$$838$$ −54787.9 −2.25849
$$839$$ −25774.1 −1.06057 −0.530287 0.847818i $$-0.677916\pi$$
−0.530287 + 0.847818i $$0.677916\pi$$
$$840$$ 1161.91 0.0477260
$$841$$ −17008.3 −0.697376
$$842$$ 42726.2 1.74874
$$843$$ −7484.04 −0.305770
$$844$$ 8691.68 0.354478
$$845$$ −8924.87 −0.363343
$$846$$ −17176.8 −0.698049
$$847$$ 0 0
$$848$$ 35984.9 1.45722
$$849$$ −12988.3 −0.525037
$$850$$ −5719.49 −0.230796
$$851$$ −17683.4 −0.712313
$$852$$ −9180.00 −0.369134
$$853$$ −18664.1 −0.749174 −0.374587 0.927192i $$-0.622215\pi$$
−0.374587 + 0.927192i $$0.622215\pi$$
$$854$$ −2858.47 −0.114537
$$855$$ −390.867 −0.0156344
$$856$$ −14488.5 −0.578511
$$857$$ 37192.5 1.48246 0.741232 0.671249i $$-0.234242\pi$$
0.741232 + 0.671249i $$0.234242\pi$$
$$858$$ 0 0
$$859$$ 46306.0 1.83928 0.919640 0.392763i $$-0.128481\pi$$
0.919640 + 0.392763i $$0.128481\pi$$
$$860$$ 7579.78 0.300545
$$861$$ −770.512 −0.0304983
$$862$$ 47056.6 1.85934
$$863$$ −19587.1 −0.772599 −0.386299 0.922373i $$-0.626247\pi$$
−0.386299 + 0.922373i $$0.626247\pi$$
$$864$$ −5868.32 −0.231070
$$865$$ −718.907 −0.0282584
$$866$$ −60651.4 −2.37993
$$867$$ −3278.96 −0.128442
$$868$$ 8239.97 0.322215
$$869$$ 0 0
$$870$$ −4770.07 −0.185886
$$871$$ −1801.80 −0.0700939
$$872$$ −4733.19 −0.183814
$$873$$ −1840.33 −0.0713468
$$874$$ 4107.31 0.158961
$$875$$ −1138.09 −0.0439707
$$876$$ 5515.41 0.212726
$$877$$ 26580.8 1.02345 0.511727 0.859148i $$-0.329006\pi$$
0.511727 + 0.859148i $$0.329006\pi$$
$$878$$ 24443.7 0.939562
$$879$$ 12992.8 0.498561
$$880$$ 0 0
$$881$$ −731.459 −0.0279722 −0.0139861 0.999902i $$-0.504452\pi$$
−0.0139861 + 0.999902i $$0.504452\pi$$
$$882$$ −8665.14 −0.330806
$$883$$ −45291.1 −1.72612 −0.863062 0.505099i $$-0.831456\pi$$
−0.863062 + 0.505099i $$0.831456\pi$$
$$884$$ −7153.01 −0.272151
$$885$$ 5602.12 0.212783
$$886$$ −43089.8 −1.63390
$$887$$ −37730.7 −1.42827 −0.714134 0.700009i $$-0.753179\pi$$
−0.714134 + 0.700009i $$0.753179\pi$$
$$888$$ −3533.04 −0.133515
$$889$$ 19840.6 0.748516
$$890$$ −8600.81 −0.323932
$$891$$ 0 0
$$892$$ 28983.4 1.08793
$$893$$ 4478.48 0.167824
$$894$$ −6339.67 −0.237170
$$895$$ −7064.64 −0.263849
$$896$$ −9505.79 −0.354426
$$897$$ −7779.28 −0.289568
$$898$$ 51402.9 1.91017
$$899$$ 13636.9 0.505913
$$900$$ 1282.85 0.0475130
$$901$$ 28845.1 1.06656
$$902$$ 0 0
$$903$$ −7262.38 −0.267638
$$904$$ 17767.0 0.653673
$$905$$ −11196.2 −0.411244
$$906$$ 23928.2 0.877440
$$907$$ 25846.2 0.946208 0.473104 0.881007i $$-0.343134\pi$$
0.473104 + 0.881007i $$0.343134\pi$$
$$908$$ 22296.9 0.814920
$$909$$ 245.264 0.00894928
$$910$$ −3420.45 −0.124601
$$911$$ −30349.8 −1.10377 −0.551885 0.833920i $$-0.686091\pi$$
−0.551885 + 0.833920i $$0.686091\pi$$
$$912$$ 2009.18 0.0729502
$$913$$ 0 0
$$914$$ 13421.3 0.485708
$$915$$ 1272.26 0.0459667
$$916$$ 15584.1 0.562133
$$917$$ 17764.2 0.639723
$$918$$ −6177.05 −0.222084
$$919$$ 41031.3 1.47279 0.736397 0.676550i $$-0.236526\pi$$
0.736397 + 0.676550i $$0.236526\pi$$
$$920$$ 5434.30 0.194743
$$921$$ 6157.68 0.220307
$$922$$ 67846.2 2.42342
$$923$$ −10894.1 −0.388497
$$924$$ 0 0
$$925$$ 3460.59 0.123009
$$926$$ 39831.4 1.41354
$$927$$ 9565.18 0.338902
$$928$$ 18672.3 0.660506
$$929$$ −3859.82 −0.136315 −0.0681575 0.997675i $$-0.521712\pi$$
−0.0681575 + 0.997675i $$0.521712\pi$$
$$930$$ −8813.39 −0.310755
$$931$$ 2259.25 0.0795317
$$932$$ 24832.9 0.872778
$$933$$ 7883.29 0.276621
$$934$$ −29480.3 −1.03279
$$935$$ 0 0
$$936$$ −1554.26 −0.0542762
$$937$$ 22026.5 0.767955 0.383977 0.923343i $$-0.374554\pi$$
0.383977 + 0.923343i $$0.374554\pi$$
$$938$$ 2991.54 0.104134
$$939$$ 20635.2 0.717150
$$940$$ −14698.7 −0.510019
$$941$$ −24778.0 −0.858386 −0.429193 0.903213i $$-0.641202\pi$$
−0.429193 + 0.903213i $$0.641202\pi$$
$$942$$ −1744.60 −0.0603419
$$943$$ −3603.70 −0.124446
$$944$$ −28796.7 −0.992851
$$945$$ −1229.13 −0.0423108
$$946$$ 0 0
$$947$$ 34055.7 1.16860 0.584299 0.811539i $$-0.301370\pi$$
0.584299 + 0.811539i $$0.301370\pi$$
$$948$$ 4542.87 0.155639
$$949$$ 6545.23 0.223885
$$950$$ −803.789 −0.0274509
$$951$$ −9078.15 −0.309547
$$952$$ −4787.57 −0.162990
$$953$$ −30211.9 −1.02693 −0.513463 0.858112i $$-0.671637\pi$$
−0.513463 + 0.858112i $$0.671637\pi$$
$$954$$ −15547.7 −0.527648
$$955$$ 8729.72 0.295798
$$956$$ 26976.8 0.912650
$$957$$ 0 0
$$958$$ 3044.40 0.102672
$$959$$ 5585.04 0.188061
$$960$$ −2815.19 −0.0946459
$$961$$ −4594.90 −0.154238
$$962$$ 10400.6 0.348574
$$963$$ 15326.6 0.512871
$$964$$ −10451.7 −0.349196
$$965$$ −19925.2 −0.664679
$$966$$ 12916.0 0.430191
$$967$$ 34883.2 1.16005 0.580024 0.814599i $$-0.303043\pi$$
0.580024 + 0.814599i $$0.303043\pi$$
$$968$$ 0 0
$$969$$ 1610.54 0.0533931
$$970$$ −3784.50 −0.125271
$$971$$ 19753.0 0.652835 0.326418 0.945226i $$-0.394158\pi$$
0.326418 + 0.945226i $$0.394158\pi$$
$$972$$ 1385.48 0.0457194
$$973$$ 19398.7 0.639152
$$974$$ 24835.9 0.817038
$$975$$ 1522.38 0.0500054
$$976$$ −6539.80 −0.214482
$$977$$ −2208.90 −0.0723326 −0.0361663 0.999346i $$-0.511515\pi$$
−0.0361663 + 0.999346i $$0.511515\pi$$
$$978$$ 19285.0 0.630539
$$979$$ 0 0
$$980$$ −7415.02 −0.241698
$$981$$ 5007.01 0.162958
$$982$$ 46970.2 1.52635
$$983$$ −40306.0 −1.30779 −0.653897 0.756583i $$-0.726867\pi$$
−0.653897 + 0.756583i $$0.726867\pi$$
$$984$$ −720.000 −0.0233260
$$985$$ 11310.0 0.365855
$$986$$ 19654.7 0.634820
$$987$$ 14083.2 0.454177
$$988$$ −1005.25 −0.0323696
$$989$$ −33966.3 −1.09208
$$990$$ 0 0
$$991$$ −15199.4 −0.487211 −0.243605 0.969874i $$-0.578330\pi$$
−0.243605 + 0.969874i $$0.578330\pi$$
$$992$$ 34499.8 1.10420
$$993$$ −12979.5 −0.414796
$$994$$ 18087.5 0.577163
$$995$$ 11700.8 0.372803
$$996$$ −8907.40 −0.283375
$$997$$ −48073.6 −1.52709 −0.763543 0.645757i $$-0.776542\pi$$
−0.763543 + 0.645757i $$0.776542\pi$$
$$998$$ −36315.3 −1.15184
$$999$$ 3737.43 0.118365
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.o.1.2 yes 2
11.10 odd 2 1815.4.a.i.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1815.4.a.i.1.1 2 11.10 odd 2
1815.4.a.o.1.2 yes 2 1.1 even 1 trivial