# Properties

 Label 1575.4.a.n.1.1 Level $1575$ Weight $4$ Character 1575.1 Self dual yes Analytic conductor $92.928$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.9280082590$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.23607 q^{2} +9.94427 q^{4} +7.00000 q^{7} -8.23607 q^{8} +O(q^{10})$$ $$q-4.23607 q^{2} +9.94427 q^{4} +7.00000 q^{7} -8.23607 q^{8} +41.5279 q^{11} -88.9706 q^{13} -29.6525 q^{14} -44.6656 q^{16} -120.387 q^{17} -112.138 q^{19} -175.915 q^{22} -115.279 q^{23} +376.885 q^{26} +69.6099 q^{28} +144.833 q^{29} -258.079 q^{31} +255.095 q^{32} +509.967 q^{34} -48.3344 q^{37} +475.023 q^{38} -200.885 q^{41} +218.217 q^{43} +412.964 q^{44} +488.328 q^{46} +575.659 q^{47} +49.0000 q^{49} -884.748 q^{52} -184.302 q^{53} -57.6525 q^{56} -613.522 q^{58} +151.502 q^{59} -529.830 q^{61} +1093.24 q^{62} -723.276 q^{64} -1.28485 q^{67} -1197.16 q^{68} +61.4226 q^{71} -484.800 q^{73} +204.748 q^{74} -1115.13 q^{76} +290.695 q^{77} +878.257 q^{79} +850.964 q^{82} +491.830 q^{83} -924.381 q^{86} -342.026 q^{88} +415.560 q^{89} -622.794 q^{91} -1146.36 q^{92} -2438.53 q^{94} +1031.70 q^{97} -207.567 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 2 q^{4} + 14 q^{7} - 12 q^{8} + O(q^{10})$$ $$2 q - 4 q^{2} + 2 q^{4} + 14 q^{7} - 12 q^{8} + 92 q^{11} - 8 q^{13} - 28 q^{14} + 18 q^{16} - 44 q^{17} - 108 q^{19} - 164 q^{22} - 320 q^{23} + 396 q^{26} + 14 q^{28} + 236 q^{29} - 60 q^{31} + 300 q^{32} + 528 q^{34} - 204 q^{37} + 476 q^{38} - 44 q^{41} - 136 q^{43} + 12 q^{44} + 440 q^{46} + 400 q^{47} + 98 q^{49} - 1528 q^{52} + 16 q^{53} - 84 q^{56} - 592 q^{58} + 464 q^{59} - 684 q^{61} + 1140 q^{62} - 1214 q^{64} - 736 q^{67} - 1804 q^{68} + 740 q^{71} - 424 q^{73} + 168 q^{74} - 1148 q^{76} + 644 q^{77} - 408 q^{79} + 888 q^{82} + 608 q^{83} - 1008 q^{86} - 532 q^{88} + 1332 q^{89} - 56 q^{91} + 480 q^{92} - 2480 q^{94} + 2448 q^{97} - 196 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.23607 −1.49768 −0.748838 0.662753i $$-0.769388\pi$$
−0.748838 + 0.662753i $$0.769388\pi$$
$$3$$ 0 0
$$4$$ 9.94427 1.24303
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ −8.23607 −0.363986
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 41.5279 1.13828 0.569142 0.822239i $$-0.307275\pi$$
0.569142 + 0.822239i $$0.307275\pi$$
$$12$$ 0 0
$$13$$ −88.9706 −1.89815 −0.949077 0.315044i $$-0.897981\pi$$
−0.949077 + 0.315044i $$0.897981\pi$$
$$14$$ −29.6525 −0.566068
$$15$$ 0 0
$$16$$ −44.6656 −0.697900
$$17$$ −120.387 −1.71754 −0.858769 0.512364i $$-0.828770\pi$$
−0.858769 + 0.512364i $$0.828770\pi$$
$$18$$ 0 0
$$19$$ −112.138 −1.35401 −0.677004 0.735979i $$-0.736722\pi$$
−0.677004 + 0.735979i $$0.736722\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −175.915 −1.70478
$$23$$ −115.279 −1.04510 −0.522549 0.852609i $$-0.675019\pi$$
−0.522549 + 0.852609i $$0.675019\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 376.885 2.84282
$$27$$ 0 0
$$28$$ 69.6099 0.469823
$$29$$ 144.833 0.927406 0.463703 0.885991i $$-0.346520\pi$$
0.463703 + 0.885991i $$0.346520\pi$$
$$30$$ 0 0
$$31$$ −258.079 −1.49524 −0.747618 0.664128i $$-0.768803\pi$$
−0.747618 + 0.664128i $$0.768803\pi$$
$$32$$ 255.095 1.40922
$$33$$ 0 0
$$34$$ 509.967 2.57231
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −48.3344 −0.214760 −0.107380 0.994218i $$-0.534246\pi$$
−0.107380 + 0.994218i $$0.534246\pi$$
$$38$$ 475.023 2.02787
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −200.885 −0.765196 −0.382598 0.923915i $$-0.624971\pi$$
−0.382598 + 0.923915i $$0.624971\pi$$
$$42$$ 0 0
$$43$$ 218.217 0.773901 0.386950 0.922101i $$-0.373528\pi$$
0.386950 + 0.922101i $$0.373528\pi$$
$$44$$ 412.964 1.41493
$$45$$ 0 0
$$46$$ 488.328 1.56522
$$47$$ 575.659 1.78657 0.893283 0.449496i $$-0.148396\pi$$
0.893283 + 0.449496i $$0.148396\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −884.748 −2.35947
$$53$$ −184.302 −0.477657 −0.238828 0.971062i $$-0.576763\pi$$
−0.238828 + 0.971062i $$0.576763\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −57.6525 −0.137574
$$57$$ 0 0
$$58$$ −613.522 −1.38895
$$59$$ 151.502 0.334302 0.167151 0.985931i $$-0.446543\pi$$
0.167151 + 0.985931i $$0.446543\pi$$
$$60$$ 0 0
$$61$$ −529.830 −1.11209 −0.556047 0.831151i $$-0.687683\pi$$
−0.556047 + 0.831151i $$0.687683\pi$$
$$62$$ 1093.24 2.23938
$$63$$ 0 0
$$64$$ −723.276 −1.41265
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −1.28485 −0.00234283 −0.00117142 0.999999i $$-0.500373\pi$$
−0.00117142 + 0.999999i $$0.500373\pi$$
$$68$$ −1197.16 −2.13496
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 61.4226 0.102669 0.0513347 0.998682i $$-0.483652\pi$$
0.0513347 + 0.998682i $$0.483652\pi$$
$$72$$ 0 0
$$73$$ −484.800 −0.777282 −0.388641 0.921389i $$-0.627055\pi$$
−0.388641 + 0.921389i $$0.627055\pi$$
$$74$$ 204.748 0.321641
$$75$$ 0 0
$$76$$ −1115.13 −1.68308
$$77$$ 290.695 0.430231
$$78$$ 0 0
$$79$$ 878.257 1.25078 0.625390 0.780312i $$-0.284940\pi$$
0.625390 + 0.780312i $$0.284940\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 850.964 1.14602
$$83$$ 491.830 0.650426 0.325213 0.945641i $$-0.394564\pi$$
0.325213 + 0.945641i $$0.394564\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −924.381 −1.15905
$$87$$ 0 0
$$88$$ −342.026 −0.414320
$$89$$ 415.560 0.494936 0.247468 0.968896i $$-0.420401\pi$$
0.247468 + 0.968896i $$0.420401\pi$$
$$90$$ 0 0
$$91$$ −622.794 −0.717435
$$92$$ −1146.36 −1.29909
$$93$$ 0 0
$$94$$ −2438.53 −2.67570
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1031.70 1.07993 0.539964 0.841688i $$-0.318438\pi$$
0.539964 + 0.841688i $$0.318438\pi$$
$$98$$ −207.567 −0.213954
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1447.19 −1.42576 −0.712878 0.701288i $$-0.752609\pi$$
−0.712878 + 0.701288i $$0.752609\pi$$
$$102$$ 0 0
$$103$$ 163.567 0.156473 0.0782364 0.996935i $$-0.475071\pi$$
0.0782364 + 0.996935i $$0.475071\pi$$
$$104$$ 732.768 0.690902
$$105$$ 0 0
$$106$$ 780.715 0.715375
$$107$$ −129.653 −0.117141 −0.0585703 0.998283i $$-0.518654\pi$$
−0.0585703 + 0.998283i $$0.518654\pi$$
$$108$$ 0 0
$$109$$ 566.681 0.497965 0.248983 0.968508i $$-0.419904\pi$$
0.248983 + 0.968508i $$0.419904\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −312.659 −0.263782
$$113$$ 809.890 0.674230 0.337115 0.941463i $$-0.390549\pi$$
0.337115 + 0.941463i $$0.390549\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1440.26 1.15280
$$117$$ 0 0
$$118$$ −641.771 −0.500676
$$119$$ −842.709 −0.649168
$$120$$ 0 0
$$121$$ 393.563 0.295690
$$122$$ 2244.39 1.66556
$$123$$ 0 0
$$124$$ −2566.41 −1.85863
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2584.25 1.80563 0.902816 0.430028i $$-0.141496\pi$$
0.902816 + 0.430028i $$0.141496\pi$$
$$128$$ 1023.08 0.706473
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1421.10 0.947804 0.473902 0.880578i $$-0.342845\pi$$
0.473902 + 0.880578i $$0.342845\pi$$
$$132$$ 0 0
$$133$$ −784.964 −0.511767
$$134$$ 5.44272 0.00350880
$$135$$ 0 0
$$136$$ 991.515 0.625160
$$137$$ 104.878 0.0654037 0.0327019 0.999465i $$-0.489589\pi$$
0.0327019 + 0.999465i $$0.489589\pi$$
$$138$$ 0 0
$$139$$ −913.160 −0.557217 −0.278609 0.960405i $$-0.589873\pi$$
−0.278609 + 0.960405i $$0.589873\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −260.190 −0.153765
$$143$$ −3694.76 −2.16064
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2053.65 1.16412
$$147$$ 0 0
$$148$$ −480.650 −0.266954
$$149$$ −1781.45 −0.979476 −0.489738 0.871870i $$-0.662908\pi$$
−0.489738 + 0.871870i $$0.662908\pi$$
$$150$$ 0 0
$$151$$ 1407.53 0.758564 0.379282 0.925281i $$-0.376171\pi$$
0.379282 + 0.925281i $$0.376171\pi$$
$$152$$ 923.574 0.492841
$$153$$ 0 0
$$154$$ −1231.40 −0.644346
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1598.94 0.812798 0.406399 0.913696i $$-0.366784\pi$$
0.406399 + 0.913696i $$0.366784\pi$$
$$158$$ −3720.36 −1.87326
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −806.950 −0.395010
$$162$$ 0 0
$$163$$ 204.892 0.0984562 0.0492281 0.998788i $$-0.484324\pi$$
0.0492281 + 0.998788i $$0.484324\pi$$
$$164$$ −1997.66 −0.951165
$$165$$ 0 0
$$166$$ −2083.42 −0.974127
$$167$$ −1165.94 −0.540259 −0.270129 0.962824i $$-0.587067\pi$$
−0.270129 + 0.962824i $$0.587067\pi$$
$$168$$ 0 0
$$169$$ 5718.76 2.60299
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2170.01 0.961985
$$173$$ −2538.00 −1.11538 −0.557690 0.830049i $$-0.688312\pi$$
−0.557690 + 0.830049i $$0.688312\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1854.87 −0.794409
$$177$$ 0 0
$$178$$ −1760.34 −0.741254
$$179$$ −392.255 −0.163791 −0.0818954 0.996641i $$-0.526097\pi$$
−0.0818954 + 0.996641i $$0.526097\pi$$
$$180$$ 0 0
$$181$$ −2978.08 −1.22298 −0.611489 0.791253i $$-0.709429\pi$$
−0.611489 + 0.791253i $$0.709429\pi$$
$$182$$ 2638.20 1.07448
$$183$$ 0 0
$$184$$ 949.443 0.380401
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4999.41 −1.95504
$$188$$ 5724.51 2.22076
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1097.37 0.415722 0.207861 0.978158i $$-0.433350\pi$$
0.207861 + 0.978158i $$0.433350\pi$$
$$192$$ 0 0
$$193$$ −3500.31 −1.30548 −0.652740 0.757582i $$-0.726381\pi$$
−0.652740 + 0.757582i $$0.726381\pi$$
$$194$$ −4370.34 −1.61738
$$195$$ 0 0
$$196$$ 487.269 0.177576
$$197$$ 1573.96 0.569237 0.284618 0.958641i $$-0.408133\pi$$
0.284618 + 0.958641i $$0.408133\pi$$
$$198$$ 0 0
$$199$$ −3396.62 −1.20995 −0.604976 0.796244i $$-0.706817\pi$$
−0.604976 + 0.796244i $$0.706817\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6130.42 2.13532
$$203$$ 1013.83 0.350527
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −692.879 −0.234346
$$207$$ 0 0
$$208$$ 3973.93 1.32472
$$209$$ −4656.84 −1.54125
$$210$$ 0 0
$$211$$ 3337.81 1.08903 0.544513 0.838753i $$-0.316715\pi$$
0.544513 + 0.838753i $$0.316715\pi$$
$$212$$ −1832.75 −0.593744
$$213$$ 0 0
$$214$$ 549.220 0.175439
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1806.55 −0.565146
$$218$$ −2400.50 −0.745791
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 10710.9 3.26015
$$222$$ 0 0
$$223$$ −127.328 −0.0382356 −0.0191178 0.999817i $$-0.506086\pi$$
−0.0191178 + 0.999817i $$0.506086\pi$$
$$224$$ 1785.67 0.532633
$$225$$ 0 0
$$226$$ −3430.75 −1.00978
$$227$$ 3844.12 1.12398 0.561990 0.827144i $$-0.310036\pi$$
0.561990 + 0.827144i $$0.310036\pi$$
$$228$$ 0 0
$$229$$ 2536.95 0.732080 0.366040 0.930599i $$-0.380713\pi$$
0.366040 + 0.930599i $$0.380713\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1192.85 −0.337563
$$233$$ 3987.44 1.12114 0.560570 0.828107i $$-0.310582\pi$$
0.560570 + 0.828107i $$0.310582\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 1506.57 0.415549
$$237$$ 0 0
$$238$$ 3569.77 0.972244
$$239$$ 3367.18 0.911317 0.455659 0.890155i $$-0.349404\pi$$
0.455659 + 0.890155i $$0.349404\pi$$
$$240$$ 0 0
$$241$$ −939.551 −0.251128 −0.125564 0.992086i $$-0.540074\pi$$
−0.125564 + 0.992086i $$0.540074\pi$$
$$242$$ −1667.16 −0.442848
$$243$$ 0 0
$$244$$ −5268.77 −1.38237
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 9976.96 2.57012
$$248$$ 2125.56 0.544246
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1403.96 0.353056 0.176528 0.984296i $$-0.443513\pi$$
0.176528 + 0.984296i $$0.443513\pi$$
$$252$$ 0 0
$$253$$ −4787.28 −1.18962
$$254$$ −10947.1 −2.70425
$$255$$ 0 0
$$256$$ 1452.36 0.354579
$$257$$ −1964.86 −0.476905 −0.238453 0.971154i $$-0.576640\pi$$
−0.238453 + 0.971154i $$0.576640\pi$$
$$258$$ 0 0
$$259$$ −338.341 −0.0811717
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −6019.89 −1.41950
$$263$$ −393.821 −0.0923347 −0.0461673 0.998934i $$-0.514701\pi$$
−0.0461673 + 0.998934i $$0.514701\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 3325.16 0.766462
$$267$$ 0 0
$$268$$ −12.7769 −0.00291222
$$269$$ 1877.03 0.425444 0.212722 0.977113i $$-0.431767\pi$$
0.212722 + 0.977113i $$0.431767\pi$$
$$270$$ 0 0
$$271$$ −689.909 −0.154646 −0.0773228 0.997006i $$-0.524637\pi$$
−0.0773228 + 0.997006i $$0.524637\pi$$
$$272$$ 5377.16 1.19867
$$273$$ 0 0
$$274$$ −444.269 −0.0979536
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6289.13 −1.36418 −0.682088 0.731270i $$-0.738928\pi$$
−0.682088 + 0.731270i $$0.738928\pi$$
$$278$$ 3868.21 0.834531
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1954.87 0.415010 0.207505 0.978234i $$-0.433466\pi$$
0.207505 + 0.978234i $$0.433466\pi$$
$$282$$ 0 0
$$283$$ −5033.96 −1.05738 −0.528688 0.848816i $$-0.677316\pi$$
−0.528688 + 0.848816i $$0.677316\pi$$
$$284$$ 610.803 0.127621
$$285$$ 0 0
$$286$$ 15651.2 3.23594
$$287$$ −1406.20 −0.289217
$$288$$ 0 0
$$289$$ 9580.03 1.94993
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4820.99 −0.966188
$$293$$ 6369.12 1.26993 0.634963 0.772543i $$-0.281015\pi$$
0.634963 + 0.772543i $$0.281015\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 398.085 0.0781697
$$297$$ 0 0
$$298$$ 7546.34 1.46694
$$299$$ 10256.4 1.98376
$$300$$ 0 0
$$301$$ 1527.52 0.292507
$$302$$ −5962.39 −1.13608
$$303$$ 0 0
$$304$$ 5008.70 0.944963
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 6619.83 1.23066 0.615332 0.788268i $$-0.289022\pi$$
0.615332 + 0.788268i $$0.289022\pi$$
$$308$$ 2890.75 0.534792
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 9909.22 1.80675 0.903377 0.428848i $$-0.141080\pi$$
0.903377 + 0.428848i $$0.141080\pi$$
$$312$$ 0 0
$$313$$ 422.336 0.0762678 0.0381339 0.999273i $$-0.487859\pi$$
0.0381339 + 0.999273i $$0.487859\pi$$
$$314$$ −6773.22 −1.21731
$$315$$ 0 0
$$316$$ 8733.63 1.55476
$$317$$ −4902.78 −0.868668 −0.434334 0.900752i $$-0.643016\pi$$
−0.434334 + 0.900752i $$0.643016\pi$$
$$318$$ 0 0
$$319$$ 6014.60 1.05565
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 3418.30 0.591597
$$323$$ 13499.9 2.32556
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −867.935 −0.147455
$$327$$ 0 0
$$328$$ 1654.51 0.278521
$$329$$ 4029.62 0.675258
$$330$$ 0 0
$$331$$ −5281.74 −0.877071 −0.438535 0.898714i $$-0.644503\pi$$
−0.438535 + 0.898714i $$0.644503\pi$$
$$332$$ 4890.89 0.808501
$$333$$ 0 0
$$334$$ 4939.01 0.809133
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −4459.60 −0.720860 −0.360430 0.932786i $$-0.617370\pi$$
−0.360430 + 0.932786i $$0.617370\pi$$
$$338$$ −24225.1 −3.89843
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −10717.5 −1.70200
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ −1797.25 −0.281689
$$345$$ 0 0
$$346$$ 10751.2 1.67048
$$347$$ 5261.97 0.814056 0.407028 0.913416i $$-0.366565\pi$$
0.407028 + 0.913416i $$0.366565\pi$$
$$348$$ 0 0
$$349$$ 960.325 0.147292 0.0736461 0.997284i $$-0.476536\pi$$
0.0736461 + 0.997284i $$0.476536\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 10593.6 1.60409
$$353$$ −8925.80 −1.34581 −0.672907 0.739727i $$-0.734955\pi$$
−0.672907 + 0.739727i $$0.734955\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 4132.45 0.615222
$$357$$ 0 0
$$358$$ 1661.62 0.245306
$$359$$ 3056.27 0.449314 0.224657 0.974438i $$-0.427874\pi$$
0.224657 + 0.974438i $$0.427874\pi$$
$$360$$ 0 0
$$361$$ 5715.88 0.833340
$$362$$ 12615.4 1.83162
$$363$$ 0 0
$$364$$ −6193.23 −0.891796
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1813.52 0.257943 0.128971 0.991648i $$-0.458832\pi$$
0.128971 + 0.991648i $$0.458832\pi$$
$$368$$ 5148.99 0.729375
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1290.11 −0.180537
$$372$$ 0 0
$$373$$ 4517.48 0.627094 0.313547 0.949573i $$-0.398483\pi$$
0.313547 + 0.949573i $$0.398483\pi$$
$$374$$ 21177.9 2.92802
$$375$$ 0 0
$$376$$ −4741.17 −0.650285
$$377$$ −12885.9 −1.76036
$$378$$ 0 0
$$379$$ −4931.24 −0.668340 −0.334170 0.942513i $$-0.608456\pi$$
−0.334170 + 0.942513i $$0.608456\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −4648.53 −0.622617
$$383$$ −1482.37 −0.197770 −0.0988849 0.995099i $$-0.531528\pi$$
−0.0988849 + 0.995099i $$0.531528\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14827.5 1.95519
$$387$$ 0 0
$$388$$ 10259.5 1.34239
$$389$$ 5448.98 0.710217 0.355109 0.934825i $$-0.384444\pi$$
0.355109 + 0.934825i $$0.384444\pi$$
$$390$$ 0 0
$$391$$ 13878.0 1.79500
$$392$$ −403.567 −0.0519980
$$393$$ 0 0
$$394$$ −6667.38 −0.852532
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 13675.9 1.72891 0.864453 0.502713i $$-0.167665\pi$$
0.864453 + 0.502713i $$0.167665\pi$$
$$398$$ 14388.3 1.81212
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14109.9 −1.75714 −0.878570 0.477613i $$-0.841502\pi$$
−0.878570 + 0.477613i $$0.841502\pi$$
$$402$$ 0 0
$$403$$ 22961.4 2.83819
$$404$$ −14391.3 −1.77226
$$405$$ 0 0
$$406$$ −4294.65 −0.524975
$$407$$ −2007.22 −0.244458
$$408$$ 0 0
$$409$$ −13995.6 −1.69203 −0.846015 0.533159i $$-0.821005\pi$$
−0.846015 + 0.533159i $$0.821005\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 1626.55 0.194501
$$413$$ 1060.51 0.126354
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −22696.0 −2.67491
$$417$$ 0 0
$$418$$ 19726.7 2.30829
$$419$$ 9840.61 1.14736 0.573682 0.819078i $$-0.305515\pi$$
0.573682 + 0.819078i $$0.305515\pi$$
$$420$$ 0 0
$$421$$ −12660.5 −1.46564 −0.732822 0.680420i $$-0.761797\pi$$
−0.732822 + 0.680420i $$0.761797\pi$$
$$422$$ −14139.2 −1.63101
$$423$$ 0 0
$$424$$ 1517.92 0.173860
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −3708.81 −0.420332
$$428$$ −1289.31 −0.145610
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4578.91 0.511736 0.255868 0.966712i $$-0.417639\pi$$
0.255868 + 0.966712i $$0.417639\pi$$
$$432$$ 0 0
$$433$$ 3279.88 0.364020 0.182010 0.983297i $$-0.441740\pi$$
0.182010 + 0.983297i $$0.441740\pi$$
$$434$$ 7652.68 0.846406
$$435$$ 0 0
$$436$$ 5635.23 0.618988
$$437$$ 12927.1 1.41507
$$438$$ 0 0
$$439$$ −427.807 −0.0465105 −0.0232552 0.999730i $$-0.507403\pi$$
−0.0232552 + 0.999730i $$0.507403\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −45372.1 −4.88265
$$443$$ 15441.2 1.65605 0.828027 0.560688i $$-0.189463\pi$$
0.828027 + 0.560688i $$0.189463\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 539.371 0.0572645
$$447$$ 0 0
$$448$$ −5062.93 −0.533931
$$449$$ −9382.02 −0.986113 −0.493057 0.869997i $$-0.664120\pi$$
−0.493057 + 0.869997i $$0.664120\pi$$
$$450$$ 0 0
$$451$$ −8342.34 −0.871010
$$452$$ 8053.77 0.838091
$$453$$ 0 0
$$454$$ −16284.0 −1.68336
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −13570.4 −1.38905 −0.694524 0.719469i $$-0.744385\pi$$
−0.694524 + 0.719469i $$0.744385\pi$$
$$458$$ −10746.7 −1.09642
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −1251.88 −0.126477 −0.0632386 0.997998i $$-0.520143\pi$$
−0.0632386 + 0.997998i $$0.520143\pi$$
$$462$$ 0 0
$$463$$ −7934.36 −0.796417 −0.398209 0.917295i $$-0.630368\pi$$
−0.398209 + 0.917295i $$0.630368\pi$$
$$464$$ −6469.05 −0.647237
$$465$$ 0 0
$$466$$ −16891.1 −1.67911
$$467$$ 7583.76 0.751466 0.375733 0.926728i $$-0.377391\pi$$
0.375733 + 0.926728i $$0.377391\pi$$
$$468$$ 0 0
$$469$$ −8.99396 −0.000885507 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −1247.78 −0.121681
$$473$$ 9062.07 0.880919
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −8380.13 −0.806938
$$477$$ 0 0
$$478$$ −14263.6 −1.36486
$$479$$ 5829.34 0.556053 0.278027 0.960573i $$-0.410320\pi$$
0.278027 + 0.960573i $$0.410320\pi$$
$$480$$ 0 0
$$481$$ 4300.34 0.407648
$$482$$ 3980.00 0.376108
$$483$$ 0 0
$$484$$ 3913.70 0.367553
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 19902.1 1.85185 0.925925 0.377708i $$-0.123288\pi$$
0.925925 + 0.377708i $$0.123288\pi$$
$$488$$ 4363.71 0.404787
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 16821.6 1.54613 0.773065 0.634327i $$-0.218723\pi$$
0.773065 + 0.634327i $$0.218723\pi$$
$$492$$ 0 0
$$493$$ −17436.0 −1.59285
$$494$$ −42263.1 −3.84920
$$495$$ 0 0
$$496$$ 11527.3 1.04353
$$497$$ 429.958 0.0388054
$$498$$ 0 0
$$499$$ 6031.83 0.541126 0.270563 0.962702i $$-0.412790\pi$$
0.270563 + 0.962702i $$0.412790\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −5947.27 −0.528764
$$503$$ −17176.4 −1.52258 −0.761290 0.648412i $$-0.775434\pi$$
−0.761290 + 0.648412i $$0.775434\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 20279.2 1.78166
$$507$$ 0 0
$$508$$ 25698.5 2.24446
$$509$$ −4706.59 −0.409854 −0.204927 0.978777i $$-0.565696\pi$$
−0.204927 + 0.978777i $$0.565696\pi$$
$$510$$ 0 0
$$511$$ −3393.60 −0.293785
$$512$$ −14336.9 −1.23752
$$513$$ 0 0
$$514$$ 8323.28 0.714250
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 23905.9 2.03362
$$518$$ 1433.23 0.121569
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 8557.18 0.719572 0.359786 0.933035i $$-0.382850\pi$$
0.359786 + 0.933035i $$0.382850\pi$$
$$522$$ 0 0
$$523$$ −18248.5 −1.52572 −0.762858 0.646566i $$-0.776204\pi$$
−0.762858 + 0.646566i $$0.776204\pi$$
$$524$$ 14131.8 1.17815
$$525$$ 0 0
$$526$$ 1668.25 0.138287
$$527$$ 31069.3 2.56813
$$528$$ 0 0
$$529$$ 1122.16 0.0922302
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −7805.90 −0.636144
$$533$$ 17872.9 1.45246
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 10.5821 0.000852758 0
$$537$$ 0 0
$$538$$ −7951.22 −0.637177
$$539$$ 2034.87 0.162612
$$540$$ 0 0
$$541$$ −5734.17 −0.455696 −0.227848 0.973697i $$-0.573169\pi$$
−0.227848 + 0.973697i $$0.573169\pi$$
$$542$$ 2922.50 0.231609
$$543$$ 0 0
$$544$$ −30710.1 −2.42038
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8002.52 0.625527 0.312763 0.949831i $$-0.398745\pi$$
0.312763 + 0.949831i $$0.398745\pi$$
$$548$$ 1042.93 0.0812991
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −16241.2 −1.25572
$$552$$ 0 0
$$553$$ 6147.80 0.472750
$$554$$ 26641.2 2.04310
$$555$$ 0 0
$$556$$ −9080.71 −0.692640
$$557$$ 1276.82 0.0971289 0.0485644 0.998820i $$-0.484535\pi$$
0.0485644 + 0.998820i $$0.484535\pi$$
$$558$$ 0 0
$$559$$ −19414.9 −1.46898
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −8280.96 −0.621550
$$563$$ 11027.7 0.825507 0.412753 0.910843i $$-0.364567\pi$$
0.412753 + 0.910843i $$0.364567\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 21324.2 1.58361
$$567$$ 0 0
$$568$$ −505.881 −0.0373702
$$569$$ 4519.03 0.332948 0.166474 0.986046i $$-0.446762\pi$$
0.166474 + 0.986046i $$0.446762\pi$$
$$570$$ 0 0
$$571$$ 3598.81 0.263758 0.131879 0.991266i $$-0.457899\pi$$
0.131879 + 0.991266i $$0.457899\pi$$
$$572$$ −36741.7 −2.68575
$$573$$ 0 0
$$574$$ 5956.75 0.433153
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 3439.23 0.248140 0.124070 0.992273i $$-0.460405\pi$$
0.124070 + 0.992273i $$0.460405\pi$$
$$578$$ −40581.6 −2.92037
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 3442.81 0.245838
$$582$$ 0 0
$$583$$ −7653.66 −0.543709
$$584$$ 3992.85 0.282920
$$585$$ 0 0
$$586$$ −26980.0 −1.90194
$$587$$ 21285.2 1.49665 0.748327 0.663330i $$-0.230857\pi$$
0.748327 + 0.663330i $$0.230857\pi$$
$$588$$ 0 0
$$589$$ 28940.4 2.02456
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2158.89 0.149881
$$593$$ −14200.8 −0.983404 −0.491702 0.870764i $$-0.663625\pi$$
−0.491702 + 0.870764i $$0.663625\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −17715.2 −1.21752
$$597$$ 0 0
$$598$$ −43446.8 −2.97103
$$599$$ 8885.05 0.606065 0.303033 0.952980i $$-0.402001\pi$$
0.303033 + 0.952980i $$0.402001\pi$$
$$600$$ 0 0
$$601$$ −2052.89 −0.139333 −0.0696664 0.997570i $$-0.522193\pi$$
−0.0696664 + 0.997570i $$0.522193\pi$$
$$602$$ −6470.67 −0.438081
$$603$$ 0 0
$$604$$ 13996.9 0.942920
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −10280.0 −0.687404 −0.343702 0.939079i $$-0.611681\pi$$
−0.343702 + 0.939079i $$0.611681\pi$$
$$608$$ −28605.8 −1.90809
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −51216.8 −3.39118
$$612$$ 0 0
$$613$$ −23409.5 −1.54242 −0.771208 0.636584i $$-0.780347\pi$$
−0.771208 + 0.636584i $$0.780347\pi$$
$$614$$ −28042.0 −1.84314
$$615$$ 0 0
$$616$$ −2394.18 −0.156598
$$617$$ −6632.75 −0.432779 −0.216389 0.976307i $$-0.569428\pi$$
−0.216389 + 0.976307i $$0.569428\pi$$
$$618$$ 0 0
$$619$$ 10734.0 0.696990 0.348495 0.937311i $$-0.386693\pi$$
0.348495 + 0.937311i $$0.386693\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −41976.1 −2.70593
$$623$$ 2908.92 0.187068
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −1789.04 −0.114225
$$627$$ 0 0
$$628$$ 15900.3 1.01034
$$629$$ 5818.83 0.368858
$$630$$ 0 0
$$631$$ −17071.0 −1.07700 −0.538499 0.842626i $$-0.681008\pi$$
−0.538499 + 0.842626i $$0.681008\pi$$
$$632$$ −7233.38 −0.455267
$$633$$ 0 0
$$634$$ 20768.5 1.30098
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −4359.56 −0.271165
$$638$$ −25478.2 −1.58102
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 19389.7 1.19477 0.597386 0.801954i $$-0.296206\pi$$
0.597386 + 0.801954i $$0.296206\pi$$
$$642$$ 0 0
$$643$$ 25409.3 1.55839 0.779196 0.626780i $$-0.215628\pi$$
0.779196 + 0.626780i $$0.215628\pi$$
$$644$$ −8024.54 −0.491011
$$645$$ 0 0
$$646$$ −57186.6 −3.48294
$$647$$ −6039.08 −0.366956 −0.183478 0.983024i $$-0.558736\pi$$
−0.183478 + 0.983024i $$0.558736\pi$$
$$648$$ 0 0
$$649$$ 6291.54 0.380531
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 2037.50 0.122384
$$653$$ −30666.2 −1.83776 −0.918882 0.394532i $$-0.870907\pi$$
−0.918882 + 0.394532i $$0.870907\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 8972.67 0.534031
$$657$$ 0 0
$$658$$ −17069.7 −1.01132
$$659$$ 2765.96 0.163500 0.0817500 0.996653i $$-0.473949\pi$$
0.0817500 + 0.996653i $$0.473949\pi$$
$$660$$ 0 0
$$661$$ 27261.8 1.60418 0.802089 0.597204i $$-0.203722\pi$$
0.802089 + 0.597204i $$0.203722\pi$$
$$662$$ 22373.8 1.31357
$$663$$ 0 0
$$664$$ −4050.74 −0.236746
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −16696.1 −0.969230
$$668$$ −11594.4 −0.671560
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −22002.7 −1.26588
$$672$$ 0 0
$$673$$ 1048.17 0.0600356 0.0300178 0.999549i $$-0.490444\pi$$
0.0300178 + 0.999549i $$0.490444\pi$$
$$674$$ 18891.2 1.07961
$$675$$ 0 0
$$676$$ 56869.0 3.23560
$$677$$ −34554.7 −1.96166 −0.980831 0.194860i $$-0.937575\pi$$
−0.980831 + 0.194860i $$0.937575\pi$$
$$678$$ 0 0
$$679$$ 7221.89 0.408175
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 45399.9 2.54905
$$683$$ 14711.6 0.824192 0.412096 0.911140i $$-0.364797\pi$$
0.412096 + 0.911140i $$0.364797\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1452.97 −0.0808669
$$687$$ 0 0
$$688$$ −9746.79 −0.540106
$$689$$ 16397.4 0.906666
$$690$$ 0 0
$$691$$ −24522.6 −1.35005 −0.675024 0.737796i $$-0.735867\pi$$
−0.675024 + 0.737796i $$0.735867\pi$$
$$692$$ −25238.6 −1.38646
$$693$$ 0 0
$$694$$ −22290.1 −1.21919
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 24184.0 1.31425
$$698$$ −4068.00 −0.220596
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −19912.2 −1.07286 −0.536429 0.843946i $$-0.680227\pi$$
−0.536429 + 0.843946i $$0.680227\pi$$
$$702$$ 0 0
$$703$$ 5420.11 0.290787
$$704$$ −30036.1 −1.60799
$$705$$ 0 0
$$706$$ 37810.3 2.01559
$$707$$ −10130.4 −0.538885
$$708$$ 0 0
$$709$$ 6208.79 0.328880 0.164440 0.986387i $$-0.447418\pi$$
0.164440 + 0.986387i $$0.447418\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −3422.58 −0.180150
$$713$$ 29751.0 1.56267
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −3900.69 −0.203597
$$717$$ 0 0
$$718$$ −12946.6 −0.672927
$$719$$ −13063.6 −0.677593 −0.338797 0.940860i $$-0.610020\pi$$
−0.338797 + 0.940860i $$0.610020\pi$$
$$720$$ 0 0
$$721$$ 1144.97 0.0591411
$$722$$ −24212.9 −1.24807
$$723$$ 0 0
$$724$$ −29614.8 −1.52020
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 12897.0 0.657940 0.328970 0.944340i $$-0.393298\pi$$
0.328970 + 0.944340i $$0.393298\pi$$
$$728$$ 5129.37 0.261136
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −26270.5 −1.32920
$$732$$ 0 0
$$733$$ −11699.6 −0.589540 −0.294770 0.955568i $$-0.595243\pi$$
−0.294770 + 0.955568i $$0.595243\pi$$
$$734$$ −7682.19 −0.386315
$$735$$ 0 0
$$736$$ −29407.0 −1.47277
$$737$$ −53.3571 −0.00266681
$$738$$ 0 0
$$739$$ 14974.0 0.745368 0.372684 0.927958i $$-0.378438\pi$$
0.372684 + 0.927958i $$0.378438\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 5465.01 0.270386
$$743$$ 18500.7 0.913492 0.456746 0.889597i $$-0.349015\pi$$
0.456746 + 0.889597i $$0.349015\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −19136.3 −0.939184
$$747$$ 0 0
$$748$$ −49715.5 −2.43019
$$749$$ −907.572 −0.0442750
$$750$$ 0 0
$$751$$ −26348.4 −1.28025 −0.640125 0.768271i $$-0.721117\pi$$
−0.640125 + 0.768271i $$0.721117\pi$$
$$752$$ −25712.2 −1.24684
$$753$$ 0 0
$$754$$ 54585.4 2.63645
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 28061.7 1.34732 0.673659 0.739042i $$-0.264722\pi$$
0.673659 + 0.739042i $$0.264722\pi$$
$$758$$ 20889.1 1.00096
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 3579.22 0.170495 0.0852476 0.996360i $$-0.472832\pi$$
0.0852476 + 0.996360i $$0.472832\pi$$
$$762$$ 0 0
$$763$$ 3966.77 0.188213
$$764$$ 10912.5 0.516757
$$765$$ 0 0
$$766$$ 6279.44 0.296195
$$767$$ −13479.2 −0.634557
$$768$$ 0 0
$$769$$ 4339.61 0.203499 0.101749 0.994810i $$-0.467556\pi$$
0.101749 + 0.994810i $$0.467556\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −34808.0 −1.62276
$$773$$ 10005.1 0.465537 0.232769 0.972532i $$-0.425222\pi$$
0.232769 + 0.972532i $$0.425222\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −8497.14 −0.393079
$$777$$ 0 0
$$778$$ −23082.3 −1.06368
$$779$$ 22526.8 1.03608
$$780$$ 0 0
$$781$$ 2550.75 0.116867
$$782$$ −58788.4 −2.68832
$$783$$ 0 0
$$784$$ −2188.62 −0.0997001
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −17826.8 −0.807443 −0.403721 0.914882i $$-0.632284\pi$$
−0.403721 + 0.914882i $$0.632284\pi$$
$$788$$ 15651.8 0.707581
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5669.23 0.254835
$$792$$ 0 0
$$793$$ 47139.3 2.11093
$$794$$ −57932.2 −2.58934
$$795$$ 0 0
$$796$$ −33777.0 −1.50401
$$797$$ 36723.0 1.63211 0.816057 0.577971i $$-0.196155\pi$$
0.816057 + 0.577971i $$0.196155\pi$$
$$798$$ 0 0
$$799$$ −69301.9 −3.06849
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 59770.4 2.63163
$$803$$ −20132.7 −0.884767
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −97266.2 −4.25069
$$807$$ 0 0
$$808$$ 11919.2 0.518955
$$809$$ −5657.55 −0.245870 −0.122935 0.992415i $$-0.539231\pi$$
−0.122935 + 0.992415i $$0.539231\pi$$
$$810$$ 0 0
$$811$$ 7532.41 0.326139 0.163070 0.986615i $$-0.447861\pi$$
0.163070 + 0.986615i $$0.447861\pi$$
$$812$$ 10081.8 0.435716
$$813$$ 0 0
$$814$$ 8502.73 0.366119
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −24470.3 −1.04787
$$818$$ 59286.5 2.53411
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6489.25 0.275854 0.137927 0.990442i $$-0.455956\pi$$
0.137927 + 0.990442i $$0.455956\pi$$
$$822$$ 0 0
$$823$$ −7901.57 −0.334668 −0.167334 0.985900i $$-0.553516\pi$$
−0.167334 + 0.985900i $$0.553516\pi$$
$$824$$ −1347.15 −0.0569539
$$825$$ 0 0
$$826$$ −4492.40 −0.189238
$$827$$ −37815.8 −1.59007 −0.795033 0.606566i $$-0.792547\pi$$
−0.795033 + 0.606566i $$0.792547\pi$$
$$828$$ 0 0
$$829$$ 26073.5 1.09236 0.546182 0.837667i $$-0.316081\pi$$
0.546182 + 0.837667i $$0.316081\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 64350.2 2.68142
$$833$$ −5898.96 −0.245362
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −46308.9 −1.91582
$$837$$ 0 0
$$838$$ −41685.5 −1.71838
$$839$$ 15590.3 0.641523 0.320762 0.947160i $$-0.396061\pi$$
0.320762 + 0.947160i $$0.396061\pi$$
$$840$$ 0 0
$$841$$ −3412.46 −0.139918
$$842$$ 53630.8 2.19506
$$843$$ 0 0
$$844$$ 33192.1 1.35370
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2754.94 0.111760
$$848$$ 8231.96 0.333357
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 5571.92 0.224445
$$852$$ 0 0
$$853$$ 17476.1 0.701488 0.350744 0.936471i $$-0.385929\pi$$
0.350744 + 0.936471i $$0.385929\pi$$
$$854$$ 15710.8 0.629521
$$855$$ 0 0
$$856$$ 1067.83 0.0426376
$$857$$ −5694.54 −0.226980 −0.113490 0.993539i $$-0.536203\pi$$
−0.113490 + 0.993539i $$0.536203\pi$$
$$858$$ 0 0
$$859$$ 27313.6 1.08490 0.542448 0.840089i $$-0.317497\pi$$
0.542448 + 0.840089i $$0.317497\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −19396.6 −0.766415
$$863$$ −9046.07 −0.356815 −0.178408 0.983957i $$-0.557095\pi$$
−0.178408 + 0.983957i $$0.557095\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −13893.8 −0.545185
$$867$$ 0 0
$$868$$ −17964.8 −0.702496
$$869$$ 36472.1 1.42374
$$870$$ 0 0
$$871$$ 114.314 0.00444705
$$872$$ −4667.22 −0.181252
$$873$$ 0 0
$$874$$ −54760.0 −2.11932
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2104.29 0.0810224 0.0405112 0.999179i $$-0.487101\pi$$
0.0405112 + 0.999179i $$0.487101\pi$$
$$878$$ 1812.22 0.0696576
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 22589.6 0.863861 0.431931 0.901907i $$-0.357833\pi$$
0.431931 + 0.901907i $$0.357833\pi$$
$$882$$ 0 0
$$883$$ 2419.71 0.0922193 0.0461096 0.998936i $$-0.485318\pi$$
0.0461096 + 0.998936i $$0.485318\pi$$
$$884$$ 106512. 4.05248
$$885$$ 0 0
$$886$$ −65409.8 −2.48023
$$887$$ 13177.0 0.498806 0.249403 0.968400i $$-0.419766\pi$$
0.249403 + 0.968400i $$0.419766\pi$$
$$888$$ 0 0
$$889$$ 18089.8 0.682464
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −1266.19 −0.0475281
$$893$$ −64553.2 −2.41902
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 7161.58 0.267022
$$897$$ 0 0
$$898$$ 39742.9 1.47688
$$899$$ −37378.3 −1.38669
$$900$$ 0 0
$$901$$ 22187.5 0.820393
$$902$$ 35338.7 1.30449
$$903$$ 0 0
$$904$$ −6670.31 −0.245411
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −9189.14 −0.336406 −0.168203 0.985752i $$-0.553796\pi$$
−0.168203 + 0.985752i $$0.553796\pi$$
$$908$$ 38227.0 1.39714
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 17045.8 0.619928 0.309964 0.950748i $$-0.399683\pi$$
0.309964 + 0.950748i $$0.399683\pi$$
$$912$$ 0 0
$$913$$ 20424.6 0.740369
$$914$$ 57485.0 2.08034
$$915$$ 0 0
$$916$$ 25228.1 0.910001
$$917$$ 9947.71 0.358236
$$918$$ 0 0
$$919$$ −30825.0 −1.10645 −0.553223 0.833033i $$-0.686602\pi$$
−0.553223 + 0.833033i $$0.686602\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 5303.06 0.189422
$$923$$ −5464.81 −0.194882
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 33610.5 1.19277
$$927$$ 0 0
$$928$$ 36946.2 1.30691
$$929$$ 5785.88 0.204336 0.102168 0.994767i $$-0.467422\pi$$
0.102168 + 0.994767i $$0.467422\pi$$
$$930$$ 0 0
$$931$$ −5494.75 −0.193430
$$932$$ 39652.2 1.39362
$$933$$ 0 0
$$934$$ −32125.3 −1.12545
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 13680.9 0.476986 0.238493 0.971144i $$-0.423347\pi$$
0.238493 + 0.971144i $$0.423347\pi$$
$$938$$ 38.0990 0.00132620
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 45448.8 1.57448 0.787242 0.616644i $$-0.211508\pi$$
0.787242 + 0.616644i $$0.211508\pi$$
$$942$$ 0 0
$$943$$ 23157.8 0.799705
$$944$$ −6766.91 −0.233310
$$945$$ 0 0
$$946$$ −38387.6 −1.31933
$$947$$ −7788.45 −0.267255 −0.133628 0.991032i $$-0.542663\pi$$
−0.133628 + 0.991032i $$0.542663\pi$$
$$948$$ 0 0
$$949$$ 43133.0 1.47540
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 6940.61 0.236288
$$953$$ 6149.43 0.209024 0.104512 0.994524i $$-0.466672\pi$$
0.104512 + 0.994524i $$0.466672\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 33484.2 1.13280
$$957$$ 0 0
$$958$$ −24693.5 −0.832788
$$959$$ 734.144 0.0247203
$$960$$ 0 0
$$961$$ 36813.7 1.23573
$$962$$ −18216.5 −0.610524
$$963$$ 0 0
$$964$$ −9343.15 −0.312160
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −23902.9 −0.794896 −0.397448 0.917625i $$-0.630104\pi$$
−0.397448 + 0.917625i $$0.630104\pi$$
$$968$$ −3241.42 −0.107627
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 8015.06 0.264898 0.132449 0.991190i $$-0.457716\pi$$
0.132449 + 0.991190i $$0.457716\pi$$
$$972$$ 0 0
$$973$$ −6392.12 −0.210608
$$974$$ −84306.7 −2.77347
$$975$$ 0 0
$$976$$ 23665.2 0.776131
$$977$$ −34861.1 −1.14156 −0.570780 0.821103i $$-0.693359\pi$$
−0.570780 + 0.821103i $$0.693359\pi$$
$$978$$ 0 0
$$979$$ 17257.3 0.563378
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −71257.6 −2.31560
$$983$$ 6620.83 0.214824 0.107412 0.994215i $$-0.465744\pi$$
0.107412 + 0.994215i $$0.465744\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 73860.0 2.38558
$$987$$ 0 0
$$988$$ 99213.6 3.19474
$$989$$ −25155.7 −0.808802
$$990$$ 0 0
$$991$$ 10360.1 0.332089 0.166045 0.986118i $$-0.446900\pi$$
0.166045 + 0.986118i $$0.446900\pi$$
$$992$$ −65834.7 −2.10711
$$993$$ 0 0
$$994$$ −1821.33 −0.0581179
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 40309.3 1.28045 0.640225 0.768188i $$-0.278841\pi$$
0.640225 + 0.768188i $$0.278841\pi$$
$$998$$ −25551.3 −0.810432
$$999$$ 0 0
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 1575.4.a.n.1.1 2
3.2 odd 2 525.4.a.o.1.2 2
5.4 even 2 315.4.a.l.1.2 2
15.2 even 4 525.4.d.k.274.4 4
15.8 even 4 525.4.d.k.274.1 4
15.14 odd 2 105.4.a.d.1.1 2
35.34 odd 2 2205.4.a.be.1.2 2
60.59 even 2 1680.4.a.bd.1.1 2
105.104 even 2 735.4.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.d.1.1 2 15.14 odd 2
315.4.a.l.1.2 2 5.4 even 2
525.4.a.o.1.2 2 3.2 odd 2
525.4.d.k.274.1 4 15.8 even 4
525.4.d.k.274.4 4 15.2 even 4
735.4.a.m.1.1 2 105.104 even 2
1575.4.a.n.1.1 2 1.1 even 1 trivial
1680.4.a.bd.1.1 2 60.59 even 2
2205.4.a.be.1.2 2 35.34 odd 2