# Properties

 Label 1521.4.a.bc.1.6 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992$$ x^8 - 52*x^6 + 805*x^4 - 4210*x^2 + 4992 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 117) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$2.75628$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.75628 q^{2} -0.402937 q^{4} +0.313209 q^{5} +28.5660 q^{7} -23.1608 q^{8} +O(q^{10})$$ $$q+2.75628 q^{2} -0.402937 q^{4} +0.313209 q^{5} +28.5660 q^{7} -23.1608 q^{8} +0.863291 q^{10} +63.2173 q^{11} +78.7357 q^{14} -60.6141 q^{16} +98.8270 q^{17} -14.4897 q^{19} -0.126204 q^{20} +174.244 q^{22} -14.3900 q^{23} -124.902 q^{25} -11.5103 q^{28} -196.283 q^{29} +118.691 q^{31} +18.2172 q^{32} +272.395 q^{34} +8.94712 q^{35} +319.114 q^{37} -39.9377 q^{38} -7.25418 q^{40} +346.106 q^{41} -69.4854 q^{43} -25.4726 q^{44} -39.6628 q^{46} -101.875 q^{47} +473.014 q^{49} -344.264 q^{50} -594.823 q^{53} +19.8002 q^{55} -661.611 q^{56} -541.010 q^{58} -204.473 q^{59} -215.978 q^{61} +327.144 q^{62} +535.125 q^{64} +68.6049 q^{67} -39.8211 q^{68} +24.6607 q^{70} +946.241 q^{71} -779.872 q^{73} +879.567 q^{74} +5.83844 q^{76} +1805.86 q^{77} +240.022 q^{79} -18.9849 q^{80} +953.965 q^{82} +855.576 q^{83} +30.9535 q^{85} -191.521 q^{86} -1464.16 q^{88} +1264.06 q^{89} +5.79825 q^{92} -280.795 q^{94} -4.53831 q^{95} +662.290 q^{97} +1303.76 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 40 q^{4} - 22 q^{7}+O(q^{10})$$ 8 * q + 40 * q^4 - 22 * q^7 $$8 q + 40 q^{4} - 22 q^{7} + 36 q^{10} + 204 q^{16} + 244 q^{19} + 136 q^{22} + 354 q^{25} - 452 q^{28} + 242 q^{31} - 1292 q^{34} + 1018 q^{37} + 1700 q^{40} + 74 q^{43} - 896 q^{46} + 298 q^{49} + 1300 q^{55} + 812 q^{58} + 1148 q^{61} + 3636 q^{64} - 2198 q^{67} + 2200 q^{70} - 2176 q^{73} + 6936 q^{76} + 1862 q^{79} + 5436 q^{82} - 890 q^{85} + 3528 q^{88} - 3104 q^{94} - 4370 q^{97}+O(q^{100})$$ 8 * q + 40 * q^4 - 22 * q^7 + 36 * q^10 + 204 * q^16 + 244 * q^19 + 136 * q^22 + 354 * q^25 - 452 * q^28 + 242 * q^31 - 1292 * q^34 + 1018 * q^37 + 1700 * q^40 + 74 * q^43 - 896 * q^46 + 298 * q^49 + 1300 * q^55 + 812 * q^58 + 1148 * q^61 + 3636 * q^64 - 2198 * q^67 + 2200 * q^70 - 2176 * q^73 + 6936 * q^76 + 1862 * q^79 + 5436 * q^82 - 890 * q^85 + 3528 * q^88 - 3104 * q^94 - 4370 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.75628 0.974491 0.487246 0.873265i $$-0.338002\pi$$
0.487246 + 0.873265i $$0.338002\pi$$
$$3$$ 0 0
$$4$$ −0.402937 −0.0503671
$$5$$ 0.313209 0.0280143 0.0140071 0.999902i $$-0.495541\pi$$
0.0140071 + 0.999902i $$0.495541\pi$$
$$6$$ 0 0
$$7$$ 28.5660 1.54242 0.771208 0.636583i $$-0.219653\pi$$
0.771208 + 0.636583i $$0.219653\pi$$
$$8$$ −23.1608 −1.02357
$$9$$ 0 0
$$10$$ 0.863291 0.0272997
$$11$$ 63.2173 1.73279 0.866397 0.499356i $$-0.166430\pi$$
0.866397 + 0.499356i $$0.166430\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 78.7357 1.50307
$$15$$ 0 0
$$16$$ −60.6141 −0.947096
$$17$$ 98.8270 1.40995 0.704973 0.709234i $$-0.250959\pi$$
0.704973 + 0.709234i $$0.250959\pi$$
$$18$$ 0 0
$$19$$ −14.4897 −0.174956 −0.0874782 0.996166i $$-0.527881\pi$$
−0.0874782 + 0.996166i $$0.527881\pi$$
$$20$$ −0.126204 −0.00141100
$$21$$ 0 0
$$22$$ 174.244 1.68859
$$23$$ −14.3900 −0.130457 −0.0652286 0.997870i $$-0.520778\pi$$
−0.0652286 + 0.997870i $$0.520778\pi$$
$$24$$ 0 0
$$25$$ −124.902 −0.999215
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −11.5103 −0.0776871
$$29$$ −196.283 −1.25686 −0.628428 0.777868i $$-0.716301\pi$$
−0.628428 + 0.777868i $$0.716301\pi$$
$$30$$ 0 0
$$31$$ 118.691 0.687661 0.343830 0.939032i $$-0.388276\pi$$
0.343830 + 0.939032i $$0.388276\pi$$
$$32$$ 18.2172 0.100637
$$33$$ 0 0
$$34$$ 272.395 1.37398
$$35$$ 8.94712 0.0432097
$$36$$ 0 0
$$37$$ 319.114 1.41789 0.708947 0.705262i $$-0.249171\pi$$
0.708947 + 0.705262i $$0.249171\pi$$
$$38$$ −39.9377 −0.170493
$$39$$ 0 0
$$40$$ −7.25418 −0.0286747
$$41$$ 346.106 1.31836 0.659180 0.751986i $$-0.270904\pi$$
0.659180 + 0.751986i $$0.270904\pi$$
$$42$$ 0 0
$$43$$ −69.4854 −0.246428 −0.123214 0.992380i $$-0.539320\pi$$
−0.123214 + 0.992380i $$0.539320\pi$$
$$44$$ −25.4726 −0.0872758
$$45$$ 0 0
$$46$$ −39.6628 −0.127129
$$47$$ −101.875 −0.316170 −0.158085 0.987426i $$-0.550532\pi$$
−0.158085 + 0.987426i $$0.550532\pi$$
$$48$$ 0 0
$$49$$ 473.014 1.37905
$$50$$ −344.264 −0.973726
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −594.823 −1.54161 −0.770804 0.637072i $$-0.780145\pi$$
−0.770804 + 0.637072i $$0.780145\pi$$
$$54$$ 0 0
$$55$$ 19.8002 0.0485430
$$56$$ −661.611 −1.57878
$$57$$ 0 0
$$58$$ −541.010 −1.22480
$$59$$ −204.473 −0.451189 −0.225595 0.974221i $$-0.572432\pi$$
−0.225595 + 0.974221i $$0.572432\pi$$
$$60$$ 0 0
$$61$$ −215.978 −0.453331 −0.226666 0.973973i $$-0.572782\pi$$
−0.226666 + 0.973973i $$0.572782\pi$$
$$62$$ 327.144 0.670119
$$63$$ 0 0
$$64$$ 535.125 1.04517
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 68.6049 0.125096 0.0625480 0.998042i $$-0.480077\pi$$
0.0625480 + 0.998042i $$0.480077\pi$$
$$68$$ −39.8211 −0.0710149
$$69$$ 0 0
$$70$$ 24.6607 0.0421075
$$71$$ 946.241 1.58166 0.790832 0.612033i $$-0.209648\pi$$
0.790832 + 0.612033i $$0.209648\pi$$
$$72$$ 0 0
$$73$$ −779.872 −1.25037 −0.625185 0.780476i $$-0.714977\pi$$
−0.625185 + 0.780476i $$0.714977\pi$$
$$74$$ 879.567 1.38172
$$75$$ 0 0
$$76$$ 5.83844 0.00881204
$$77$$ 1805.86 2.67269
$$78$$ 0 0
$$79$$ 240.022 0.341831 0.170915 0.985286i $$-0.445328\pi$$
0.170915 + 0.985286i $$0.445328\pi$$
$$80$$ −18.9849 −0.0265322
$$81$$ 0 0
$$82$$ 953.965 1.28473
$$83$$ 855.576 1.13147 0.565733 0.824588i $$-0.308593\pi$$
0.565733 + 0.824588i $$0.308593\pi$$
$$84$$ 0 0
$$85$$ 30.9535 0.0394986
$$86$$ −191.521 −0.240142
$$87$$ 0 0
$$88$$ −1464.16 −1.77364
$$89$$ 1264.06 1.50551 0.752753 0.658303i $$-0.228726\pi$$
0.752753 + 0.658303i $$0.228726\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 5.79825 0.00657075
$$93$$ 0 0
$$94$$ −280.795 −0.308105
$$95$$ −4.53831 −0.00490128
$$96$$ 0 0
$$97$$ 662.290 0.693251 0.346625 0.938004i $$-0.387328\pi$$
0.346625 + 0.938004i $$0.387328\pi$$
$$98$$ 1303.76 1.34387
$$99$$ 0 0
$$100$$ 50.3276 0.0503276
$$101$$ −793.571 −0.781814 −0.390907 0.920430i $$-0.627839\pi$$
−0.390907 + 0.920430i $$0.627839\pi$$
$$102$$ 0 0
$$103$$ 980.791 0.938254 0.469127 0.883131i $$-0.344569\pi$$
0.469127 + 0.883131i $$0.344569\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −1639.50 −1.50228
$$107$$ −805.944 −0.728164 −0.364082 0.931367i $$-0.618617\pi$$
−0.364082 + 0.931367i $$0.618617\pi$$
$$108$$ 0 0
$$109$$ 845.184 0.742697 0.371348 0.928494i $$-0.378896\pi$$
0.371348 + 0.928494i $$0.378896\pi$$
$$110$$ 54.5749 0.0473047
$$111$$ 0 0
$$112$$ −1731.50 −1.46082
$$113$$ 56.3774 0.0469340 0.0234670 0.999725i $$-0.492530\pi$$
0.0234670 + 0.999725i $$0.492530\pi$$
$$114$$ 0 0
$$115$$ −4.50707 −0.00365467
$$116$$ 79.0896 0.0633042
$$117$$ 0 0
$$118$$ −563.585 −0.439680
$$119$$ 2823.09 2.17472
$$120$$ 0 0
$$121$$ 2665.43 2.00257
$$122$$ −595.296 −0.441767
$$123$$ 0 0
$$124$$ −47.8249 −0.0346355
$$125$$ −78.2716 −0.0560066
$$126$$ 0 0
$$127$$ 1666.65 1.16450 0.582248 0.813011i $$-0.302173\pi$$
0.582248 + 0.813011i $$0.302173\pi$$
$$128$$ 1329.21 0.917868
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −602.630 −0.401924 −0.200962 0.979599i $$-0.564407\pi$$
−0.200962 + 0.979599i $$0.564407\pi$$
$$132$$ 0 0
$$133$$ −413.913 −0.269856
$$134$$ 189.094 0.121905
$$135$$ 0 0
$$136$$ −2288.92 −1.44318
$$137$$ 421.654 0.262951 0.131476 0.991319i $$-0.458029\pi$$
0.131476 + 0.991319i $$0.458029\pi$$
$$138$$ 0 0
$$139$$ 29.1702 0.0177999 0.00889995 0.999960i $$-0.497167\pi$$
0.00889995 + 0.999960i $$0.497167\pi$$
$$140$$ −3.60512 −0.00217635
$$141$$ 0 0
$$142$$ 2608.10 1.54132
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −61.4776 −0.0352099
$$146$$ −2149.54 −1.21848
$$147$$ 0 0
$$148$$ −128.583 −0.0714152
$$149$$ 2141.80 1.17760 0.588802 0.808278i $$-0.299600\pi$$
0.588802 + 0.808278i $$0.299600\pi$$
$$150$$ 0 0
$$151$$ 1459.30 0.786462 0.393231 0.919440i $$-0.371357\pi$$
0.393231 + 0.919440i $$0.371357\pi$$
$$152$$ 335.594 0.179081
$$153$$ 0 0
$$154$$ 4977.46 2.60451
$$155$$ 37.1750 0.0192643
$$156$$ 0 0
$$157$$ 2008.20 1.02084 0.510419 0.859926i $$-0.329490\pi$$
0.510419 + 0.859926i $$0.329490\pi$$
$$158$$ 661.568 0.333111
$$159$$ 0 0
$$160$$ 5.70579 0.00281927
$$161$$ −411.063 −0.201219
$$162$$ 0 0
$$163$$ 548.495 0.263567 0.131784 0.991279i $$-0.457930\pi$$
0.131784 + 0.991279i $$0.457930\pi$$
$$164$$ −139.459 −0.0664019
$$165$$ 0 0
$$166$$ 2358.21 1.10260
$$167$$ −3962.58 −1.83613 −0.918064 0.396432i $$-0.870248\pi$$
−0.918064 + 0.396432i $$0.870248\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 85.3165 0.0384910
$$171$$ 0 0
$$172$$ 27.9982 0.0124119
$$173$$ −1729.54 −0.760084 −0.380042 0.924969i $$-0.624091\pi$$
−0.380042 + 0.924969i $$0.624091\pi$$
$$174$$ 0 0
$$175$$ −3567.94 −1.54121
$$176$$ −3831.86 −1.64112
$$177$$ 0 0
$$178$$ 3484.10 1.46710
$$179$$ 1452.91 0.606680 0.303340 0.952882i $$-0.401898\pi$$
0.303340 + 0.952882i $$0.401898\pi$$
$$180$$ 0 0
$$181$$ −550.329 −0.225998 −0.112999 0.993595i $$-0.536046\pi$$
−0.112999 + 0.993595i $$0.536046\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 333.284 0.133533
$$185$$ 99.9495 0.0397213
$$186$$ 0 0
$$187$$ 6247.58 2.44314
$$188$$ 41.0491 0.0159246
$$189$$ 0 0
$$190$$ −12.5089 −0.00477625
$$191$$ 1868.68 0.707921 0.353960 0.935260i $$-0.384835\pi$$
0.353960 + 0.935260i $$0.384835\pi$$
$$192$$ 0 0
$$193$$ 4156.52 1.55022 0.775112 0.631824i $$-0.217694\pi$$
0.775112 + 0.631824i $$0.217694\pi$$
$$194$$ 1825.45 0.675567
$$195$$ 0 0
$$196$$ −190.595 −0.0694587
$$197$$ −1522.18 −0.550513 −0.275257 0.961371i $$-0.588763\pi$$
−0.275257 + 0.961371i $$0.588763\pi$$
$$198$$ 0 0
$$199$$ 808.130 0.287873 0.143937 0.989587i $$-0.454024\pi$$
0.143937 + 0.989587i $$0.454024\pi$$
$$200$$ 2892.83 1.02277
$$201$$ 0 0
$$202$$ −2187.30 −0.761871
$$203$$ −5607.01 −1.93860
$$204$$ 0 0
$$205$$ 108.404 0.0369329
$$206$$ 2703.33 0.914320
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −916.001 −0.303163
$$210$$ 0 0
$$211$$ −5260.03 −1.71619 −0.858094 0.513493i $$-0.828351\pi$$
−0.858094 + 0.513493i $$0.828351\pi$$
$$212$$ 239.676 0.0776464
$$213$$ 0 0
$$214$$ −2221.41 −0.709590
$$215$$ −21.7635 −0.00690351
$$216$$ 0 0
$$217$$ 3390.51 1.06066
$$218$$ 2329.56 0.723751
$$219$$ 0 0
$$220$$ −7.97824 −0.00244497
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −4690.71 −1.40858 −0.704290 0.709912i $$-0.748734\pi$$
−0.704290 + 0.709912i $$0.748734\pi$$
$$224$$ 520.392 0.155224
$$225$$ 0 0
$$226$$ 155.392 0.0457368
$$227$$ −3560.61 −1.04108 −0.520542 0.853836i $$-0.674270\pi$$
−0.520542 + 0.853836i $$0.674270\pi$$
$$228$$ 0 0
$$229$$ −3144.82 −0.907490 −0.453745 0.891131i $$-0.649912\pi$$
−0.453745 + 0.891131i $$0.649912\pi$$
$$230$$ −12.4227 −0.00356144
$$231$$ 0 0
$$232$$ 4546.07 1.28648
$$233$$ 852.543 0.239708 0.119854 0.992792i $$-0.461757\pi$$
0.119854 + 0.992792i $$0.461757\pi$$
$$234$$ 0 0
$$235$$ −31.9081 −0.00885727
$$236$$ 82.3898 0.0227251
$$237$$ 0 0
$$238$$ 7781.22 2.11925
$$239$$ −2189.33 −0.592534 −0.296267 0.955105i $$-0.595742\pi$$
−0.296267 + 0.955105i $$0.595742\pi$$
$$240$$ 0 0
$$241$$ 6329.98 1.69191 0.845954 0.533257i $$-0.179032\pi$$
0.845954 + 0.533257i $$0.179032\pi$$
$$242$$ 7346.65 1.95149
$$243$$ 0 0
$$244$$ 87.0257 0.0228330
$$245$$ 148.152 0.0386331
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −2748.97 −0.703871
$$249$$ 0 0
$$250$$ −215.738 −0.0545779
$$251$$ 3602.97 0.906046 0.453023 0.891499i $$-0.350346\pi$$
0.453023 + 0.891499i $$0.350346\pi$$
$$252$$ 0 0
$$253$$ −909.695 −0.226055
$$254$$ 4593.74 1.13479
$$255$$ 0 0
$$256$$ −617.315 −0.150712
$$257$$ −3072.64 −0.745781 −0.372891 0.927875i $$-0.621633\pi$$
−0.372891 + 0.927875i $$0.621633\pi$$
$$258$$ 0 0
$$259$$ 9115.80 2.18698
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −1661.01 −0.391671
$$263$$ 3543.20 0.830734 0.415367 0.909654i $$-0.363653\pi$$
0.415367 + 0.909654i $$0.363653\pi$$
$$264$$ 0 0
$$265$$ −186.304 −0.0431871
$$266$$ −1140.86 −0.262972
$$267$$ 0 0
$$268$$ −27.6435 −0.00630072
$$269$$ −5899.97 −1.33728 −0.668638 0.743588i $$-0.733122\pi$$
−0.668638 + 0.743588i $$0.733122\pi$$
$$270$$ 0 0
$$271$$ −2109.52 −0.472858 −0.236429 0.971649i $$-0.575977\pi$$
−0.236429 + 0.971649i $$0.575977\pi$$
$$272$$ −5990.32 −1.33535
$$273$$ 0 0
$$274$$ 1162.19 0.256244
$$275$$ −7895.96 −1.73143
$$276$$ 0 0
$$277$$ 7265.00 1.57585 0.787927 0.615769i $$-0.211155\pi$$
0.787927 + 0.615769i $$0.211155\pi$$
$$278$$ 80.4012 0.0173458
$$279$$ 0 0
$$280$$ −207.223 −0.0442283
$$281$$ 4771.36 1.01294 0.506469 0.862258i $$-0.330951\pi$$
0.506469 + 0.862258i $$0.330951\pi$$
$$282$$ 0 0
$$283$$ 2036.26 0.427714 0.213857 0.976865i $$-0.431397\pi$$
0.213857 + 0.976865i $$0.431397\pi$$
$$284$$ −381.275 −0.0796639
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 9886.86 2.03346
$$288$$ 0 0
$$289$$ 4853.78 0.987947
$$290$$ −169.449 −0.0343118
$$291$$ 0 0
$$292$$ 314.239 0.0629776
$$293$$ 3719.02 0.741527 0.370764 0.928727i $$-0.379096\pi$$
0.370764 + 0.928727i $$0.379096\pi$$
$$294$$ 0 0
$$295$$ −64.0429 −0.0126397
$$296$$ −7390.95 −1.45132
$$297$$ 0 0
$$298$$ 5903.39 1.14756
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −1984.92 −0.380095
$$302$$ 4022.22 0.766400
$$303$$ 0 0
$$304$$ 878.282 0.165700
$$305$$ −67.6464 −0.0126997
$$306$$ 0 0
$$307$$ −7282.24 −1.35381 −0.676905 0.736070i $$-0.736679\pi$$
−0.676905 + 0.736070i $$0.736679\pi$$
$$308$$ −727.649 −0.134616
$$309$$ 0 0
$$310$$ 102.465 0.0187729
$$311$$ −4569.28 −0.833120 −0.416560 0.909108i $$-0.636764\pi$$
−0.416560 + 0.909108i $$0.636764\pi$$
$$312$$ 0 0
$$313$$ 21.0294 0.00379761 0.00189880 0.999998i $$-0.499396\pi$$
0.00189880 + 0.999998i $$0.499396\pi$$
$$314$$ 5535.15 0.994797
$$315$$ 0 0
$$316$$ −96.7138 −0.0172170
$$317$$ −5159.17 −0.914095 −0.457047 0.889442i $$-0.651093\pi$$
−0.457047 + 0.889442i $$0.651093\pi$$
$$318$$ 0 0
$$319$$ −12408.5 −2.17787
$$320$$ 167.606 0.0292796
$$321$$ 0 0
$$322$$ −1133.00 −0.196087
$$323$$ −1431.98 −0.246679
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 1511.80 0.256844
$$327$$ 0 0
$$328$$ −8016.11 −1.34944
$$329$$ −2910.15 −0.487666
$$330$$ 0 0
$$331$$ −7577.53 −1.25830 −0.629152 0.777282i $$-0.716598\pi$$
−0.629152 + 0.777282i $$0.716598\pi$$
$$332$$ −344.743 −0.0569887
$$333$$ 0 0
$$334$$ −10922.0 −1.78929
$$335$$ 21.4877 0.00350447
$$336$$ 0 0
$$337$$ −2109.79 −0.341032 −0.170516 0.985355i $$-0.554543\pi$$
−0.170516 + 0.985355i $$0.554543\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −12.4723 −0.00198943
$$341$$ 7503.30 1.19157
$$342$$ 0 0
$$343$$ 3713.98 0.584653
$$344$$ 1609.34 0.252237
$$345$$ 0 0
$$346$$ −4767.09 −0.740695
$$347$$ −8107.91 −1.25434 −0.627169 0.778883i $$-0.715786\pi$$
−0.627169 + 0.778883i $$0.715786\pi$$
$$348$$ 0 0
$$349$$ 8686.38 1.33230 0.666148 0.745820i $$-0.267942\pi$$
0.666148 + 0.745820i $$0.267942\pi$$
$$350$$ −9834.24 −1.50189
$$351$$ 0 0
$$352$$ 1151.64 0.174383
$$353$$ −9050.23 −1.36458 −0.682288 0.731083i $$-0.739015\pi$$
−0.682288 + 0.731083i $$0.739015\pi$$
$$354$$ 0 0
$$355$$ 296.371 0.0443092
$$356$$ −509.336 −0.0758279
$$357$$ 0 0
$$358$$ 4004.63 0.591204
$$359$$ −7043.80 −1.03554 −0.517768 0.855521i $$-0.673237\pi$$
−0.517768 + 0.855521i $$0.673237\pi$$
$$360$$ 0 0
$$361$$ −6649.05 −0.969390
$$362$$ −1516.86 −0.220233
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −244.263 −0.0350282
$$366$$ 0 0
$$367$$ 13825.1 1.96639 0.983194 0.182565i $$-0.0584401\pi$$
0.983194 + 0.182565i $$0.0584401\pi$$
$$368$$ 872.236 0.123556
$$369$$ 0 0
$$370$$ 275.488 0.0387080
$$371$$ −16991.7 −2.37780
$$372$$ 0 0
$$373$$ 6368.19 0.884001 0.442000 0.897015i $$-0.354269\pi$$
0.442000 + 0.897015i $$0.354269\pi$$
$$374$$ 17220.1 2.38082
$$375$$ 0 0
$$376$$ 2359.51 0.323623
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 2451.66 0.332278 0.166139 0.986102i $$-0.446870\pi$$
0.166139 + 0.986102i $$0.446870\pi$$
$$380$$ 1.82865 0.000246863 0
$$381$$ 0 0
$$382$$ 5150.60 0.689863
$$383$$ −10830.9 −1.44500 −0.722500 0.691371i $$-0.757007\pi$$
−0.722500 + 0.691371i $$0.757007\pi$$
$$384$$ 0 0
$$385$$ 565.613 0.0748735
$$386$$ 11456.5 1.51068
$$387$$ 0 0
$$388$$ −266.861 −0.0349170
$$389$$ 4978.07 0.648838 0.324419 0.945914i $$-0.394831\pi$$
0.324419 + 0.945914i $$0.394831\pi$$
$$390$$ 0 0
$$391$$ −1422.12 −0.183938
$$392$$ −10955.4 −1.41156
$$393$$ 0 0
$$394$$ −4195.56 −0.536470
$$395$$ 75.1772 0.00957614
$$396$$ 0 0
$$397$$ 9450.67 1.19475 0.597375 0.801962i $$-0.296210\pi$$
0.597375 + 0.801962i $$0.296210\pi$$
$$398$$ 2227.43 0.280530
$$399$$ 0 0
$$400$$ 7570.82 0.946353
$$401$$ −8776.77 −1.09300 −0.546498 0.837461i $$-0.684039\pi$$
−0.546498 + 0.837461i $$0.684039\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 319.759 0.0393777
$$405$$ 0 0
$$406$$ −15454.5 −1.88914
$$407$$ 20173.5 2.45692
$$408$$ 0 0
$$409$$ −4520.19 −0.546477 −0.273238 0.961946i $$-0.588095\pi$$
−0.273238 + 0.961946i $$0.588095\pi$$
$$410$$ 298.791 0.0359908
$$411$$ 0 0
$$412$$ −395.197 −0.0472571
$$413$$ −5840.98 −0.695922
$$414$$ 0 0
$$415$$ 267.974 0.0316972
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −2524.75 −0.295430
$$419$$ −12006.4 −1.39989 −0.699944 0.714198i $$-0.746792\pi$$
−0.699944 + 0.714198i $$0.746792\pi$$
$$420$$ 0 0
$$421$$ −9731.52 −1.12657 −0.563284 0.826263i $$-0.690462\pi$$
−0.563284 + 0.826263i $$0.690462\pi$$
$$422$$ −14498.1 −1.67241
$$423$$ 0 0
$$424$$ 13776.6 1.57795
$$425$$ −12343.7 −1.40884
$$426$$ 0 0
$$427$$ −6169.63 −0.699226
$$428$$ 324.745 0.0366755
$$429$$ 0 0
$$430$$ −59.9861 −0.00672741
$$431$$ 4111.03 0.459446 0.229723 0.973256i $$-0.426218\pi$$
0.229723 + 0.973256i $$0.426218\pi$$
$$432$$ 0 0
$$433$$ 2906.06 0.322532 0.161266 0.986911i $$-0.448442\pi$$
0.161266 + 0.986911i $$0.448442\pi$$
$$434$$ 9345.19 1.03360
$$435$$ 0 0
$$436$$ −340.556 −0.0374075
$$437$$ 208.507 0.0228243
$$438$$ 0 0
$$439$$ 8874.72 0.964846 0.482423 0.875938i $$-0.339757\pi$$
0.482423 + 0.875938i $$0.339757\pi$$
$$440$$ −458.590 −0.0496873
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12640.5 −1.35569 −0.677843 0.735207i $$-0.737085\pi$$
−0.677843 + 0.735207i $$0.737085\pi$$
$$444$$ 0 0
$$445$$ 395.915 0.0421756
$$446$$ −12928.9 −1.37265
$$447$$ 0 0
$$448$$ 15286.4 1.61208
$$449$$ 1774.29 0.186490 0.0932451 0.995643i $$-0.470276\pi$$
0.0932451 + 0.995643i $$0.470276\pi$$
$$450$$ 0 0
$$451$$ 21879.9 2.28444
$$452$$ −22.7165 −0.00236393
$$453$$ 0 0
$$454$$ −9814.04 −1.01453
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6168.82 0.631433 0.315717 0.948854i $$-0.397755\pi$$
0.315717 + 0.948854i $$0.397755\pi$$
$$458$$ −8667.98 −0.884341
$$459$$ 0 0
$$460$$ 1.81607 0.000184075 0
$$461$$ 7455.38 0.753214 0.376607 0.926373i $$-0.377091\pi$$
0.376607 + 0.926373i $$0.377091\pi$$
$$462$$ 0 0
$$463$$ 5399.78 0.542006 0.271003 0.962578i $$-0.412645\pi$$
0.271003 + 0.962578i $$0.412645\pi$$
$$464$$ 11897.5 1.19036
$$465$$ 0 0
$$466$$ 2349.85 0.233593
$$467$$ −3992.01 −0.395564 −0.197782 0.980246i $$-0.563374\pi$$
−0.197782 + 0.980246i $$0.563374\pi$$
$$468$$ 0 0
$$469$$ 1959.77 0.192950
$$470$$ −87.9477 −0.00863133
$$471$$ 0 0
$$472$$ 4735.77 0.461825
$$473$$ −4392.68 −0.427009
$$474$$ 0 0
$$475$$ 1809.79 0.174819
$$476$$ −1137.53 −0.109535
$$477$$ 0 0
$$478$$ −6034.39 −0.577419
$$479$$ −3151.62 −0.300629 −0.150315 0.988638i $$-0.548029\pi$$
−0.150315 + 0.988638i $$0.548029\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 17447.2 1.64875
$$483$$ 0 0
$$484$$ −1074.00 −0.100864
$$485$$ 207.435 0.0194209
$$486$$ 0 0
$$487$$ −3909.18 −0.363741 −0.181870 0.983323i $$-0.558215\pi$$
−0.181870 + 0.983323i $$0.558215\pi$$
$$488$$ 5002.24 0.464018
$$489$$ 0 0
$$490$$ 408.349 0.0376476
$$491$$ −6645.13 −0.610775 −0.305388 0.952228i $$-0.598786\pi$$
−0.305388 + 0.952228i $$0.598786\pi$$
$$492$$ 0 0
$$493$$ −19398.1 −1.77210
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −7194.33 −0.651281
$$497$$ 27030.3 2.43959
$$498$$ 0 0
$$499$$ 6111.83 0.548303 0.274151 0.961687i $$-0.411603\pi$$
0.274151 + 0.961687i $$0.411603\pi$$
$$500$$ 31.5385 0.00282089
$$501$$ 0 0
$$502$$ 9930.79 0.882934
$$503$$ −10954.1 −0.971010 −0.485505 0.874234i $$-0.661364\pi$$
−0.485505 + 0.874234i $$0.661364\pi$$
$$504$$ 0 0
$$505$$ −248.554 −0.0219020
$$506$$ −2507.37 −0.220289
$$507$$ 0 0
$$508$$ −671.553 −0.0586523
$$509$$ 11121.1 0.968439 0.484220 0.874947i $$-0.339104\pi$$
0.484220 + 0.874947i $$0.339104\pi$$
$$510$$ 0 0
$$511$$ −22277.8 −1.92859
$$512$$ −12335.2 −1.06473
$$513$$ 0 0
$$514$$ −8469.04 −0.726757
$$515$$ 307.193 0.0262845
$$516$$ 0 0
$$517$$ −6440.25 −0.547857
$$518$$ 25125.7 2.13119
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −2403.33 −0.202096 −0.101048 0.994882i $$-0.532220\pi$$
−0.101048 + 0.994882i $$0.532220\pi$$
$$522$$ 0 0
$$523$$ 4251.72 0.355477 0.177739 0.984078i $$-0.443122\pi$$
0.177739 + 0.984078i $$0.443122\pi$$
$$524$$ 242.822 0.0202437
$$525$$ 0 0
$$526$$ 9766.04 0.809543
$$527$$ 11729.8 0.969564
$$528$$ 0 0
$$529$$ −11959.9 −0.982981
$$530$$ −513.506 −0.0420854
$$531$$ 0 0
$$532$$ 166.781 0.0135918
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −252.429 −0.0203990
$$536$$ −1588.95 −0.128045
$$537$$ 0 0
$$538$$ −16261.9 −1.30316
$$539$$ 29902.7 2.38961
$$540$$ 0 0
$$541$$ −8924.44 −0.709227 −0.354613 0.935013i $$-0.615388\pi$$
−0.354613 + 0.935013i $$0.615388\pi$$
$$542$$ −5814.43 −0.460796
$$543$$ 0 0
$$544$$ 1800.35 0.141892
$$545$$ 264.719 0.0208061
$$546$$ 0 0
$$547$$ −14696.0 −1.14873 −0.574367 0.818598i $$-0.694752\pi$$
−0.574367 + 0.818598i $$0.694752\pi$$
$$548$$ −169.900 −0.0132441
$$549$$ 0 0
$$550$$ −21763.5 −1.68727
$$551$$ 2844.09 0.219895
$$552$$ 0 0
$$553$$ 6856.47 0.527245
$$554$$ 20024.3 1.53566
$$555$$ 0 0
$$556$$ −11.7538 −0.000896530 0
$$557$$ −2464.03 −0.187440 −0.0937202 0.995599i $$-0.529876\pi$$
−0.0937202 + 0.995599i $$0.529876\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −542.322 −0.0409237
$$561$$ 0 0
$$562$$ 13151.2 0.987098
$$563$$ −18029.2 −1.34963 −0.674815 0.737987i $$-0.735776\pi$$
−0.674815 + 0.737987i $$0.735776\pi$$
$$564$$ 0 0
$$565$$ 17.6579 0.00131482
$$566$$ 5612.50 0.416804
$$567$$ 0 0
$$568$$ −21915.7 −1.61895
$$569$$ −11924.6 −0.878572 −0.439286 0.898347i $$-0.644768\pi$$
−0.439286 + 0.898347i $$0.644768\pi$$
$$570$$ 0 0
$$571$$ 5834.77 0.427632 0.213816 0.976874i $$-0.431411\pi$$
0.213816 + 0.976874i $$0.431411\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 27250.9 1.98159
$$575$$ 1797.33 0.130355
$$576$$ 0 0
$$577$$ 15927.5 1.14917 0.574586 0.818444i $$-0.305163\pi$$
0.574586 + 0.818444i $$0.305163\pi$$
$$578$$ 13378.4 0.962745
$$579$$ 0 0
$$580$$ 24.7716 0.00177342
$$581$$ 24440.4 1.74519
$$582$$ 0 0
$$583$$ −37603.1 −2.67129
$$584$$ 18062.5 1.27985
$$585$$ 0 0
$$586$$ 10250.7 0.722612
$$587$$ 13511.1 0.950022 0.475011 0.879980i $$-0.342444\pi$$
0.475011 + 0.879980i $$0.342444\pi$$
$$588$$ 0 0
$$589$$ −1719.80 −0.120311
$$590$$ −176.520 −0.0123173
$$591$$ 0 0
$$592$$ −19342.8 −1.34288
$$593$$ −15830.4 −1.09625 −0.548126 0.836396i $$-0.684659\pi$$
−0.548126 + 0.836396i $$0.684659\pi$$
$$594$$ 0 0
$$595$$ 884.217 0.0609233
$$596$$ −863.009 −0.0593125
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 5914.12 0.403413 0.201706 0.979446i $$-0.435351\pi$$
0.201706 + 0.979446i $$0.435351\pi$$
$$600$$ 0 0
$$601$$ 9892.19 0.671399 0.335700 0.941969i $$-0.391027\pi$$
0.335700 + 0.941969i $$0.391027\pi$$
$$602$$ −5470.98 −0.370399
$$603$$ 0 0
$$604$$ −588.004 −0.0396118
$$605$$ 834.836 0.0561007
$$606$$ 0 0
$$607$$ −16782.3 −1.12220 −0.561098 0.827749i $$-0.689621\pi$$
−0.561098 + 0.827749i $$0.689621\pi$$
$$608$$ −263.962 −0.0176070
$$609$$ 0 0
$$610$$ −186.452 −0.0123758
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 12245.7 0.806847 0.403424 0.915013i $$-0.367820\pi$$
0.403424 + 0.915013i $$0.367820\pi$$
$$614$$ −20071.9 −1.31928
$$615$$ 0 0
$$616$$ −41825.3 −2.73569
$$617$$ −10043.7 −0.655341 −0.327670 0.944792i $$-0.606264\pi$$
−0.327670 + 0.944792i $$0.606264\pi$$
$$618$$ 0 0
$$619$$ −9942.69 −0.645607 −0.322803 0.946466i $$-0.604625\pi$$
−0.322803 + 0.946466i $$0.604625\pi$$
$$620$$ −14.9792 −0.000970288 0
$$621$$ 0 0
$$622$$ −12594.2 −0.811868
$$623$$ 36109.1 2.32212
$$624$$ 0 0
$$625$$ 15588.2 0.997646
$$626$$ 57.9628 0.00370074
$$627$$ 0 0
$$628$$ −809.176 −0.0514166
$$629$$ 31537.1 1.99915
$$630$$ 0 0
$$631$$ 10268.6 0.647841 0.323920 0.946084i $$-0.394999\pi$$
0.323920 + 0.946084i $$0.394999\pi$$
$$632$$ −5559.11 −0.349889
$$633$$ 0 0
$$634$$ −14220.1 −0.890777
$$635$$ 522.009 0.0326225
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −34201.2 −2.12232
$$639$$ 0 0
$$640$$ 416.322 0.0257134
$$641$$ 13657.4 0.841555 0.420777 0.907164i $$-0.361757\pi$$
0.420777 + 0.907164i $$0.361757\pi$$
$$642$$ 0 0
$$643$$ 24671.6 1.51314 0.756572 0.653910i $$-0.226873\pi$$
0.756572 + 0.653910i $$0.226873\pi$$
$$644$$ 165.633 0.0101348
$$645$$ 0 0
$$646$$ −3946.92 −0.240386
$$647$$ 16512.3 1.00334 0.501672 0.865058i $$-0.332718\pi$$
0.501672 + 0.865058i $$0.332718\pi$$
$$648$$ 0 0
$$649$$ −12926.3 −0.781818
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −221.009 −0.0132751
$$653$$ −30594.8 −1.83348 −0.916742 0.399479i $$-0.869191\pi$$
−0.916742 + 0.399479i $$0.869191\pi$$
$$654$$ 0 0
$$655$$ −188.749 −0.0112596
$$656$$ −20978.9 −1.24861
$$657$$ 0 0
$$658$$ −8021.19 −0.475226
$$659$$ 2307.79 0.136417 0.0682084 0.997671i $$-0.478272\pi$$
0.0682084 + 0.997671i $$0.478272\pi$$
$$660$$ 0 0
$$661$$ −27622.5 −1.62540 −0.812700 0.582683i $$-0.802003\pi$$
−0.812700 + 0.582683i $$0.802003\pi$$
$$662$$ −20885.8 −1.22621
$$663$$ 0 0
$$664$$ −19815.9 −1.15814
$$665$$ −129.641 −0.00755981
$$666$$ 0 0
$$667$$ 2824.51 0.163966
$$668$$ 1596.67 0.0924805
$$669$$ 0 0
$$670$$ 59.2260 0.00341508
$$671$$ −13653.6 −0.785530
$$672$$ 0 0
$$673$$ −11305.9 −0.647562 −0.323781 0.946132i $$-0.604954\pi$$
−0.323781 + 0.946132i $$0.604954\pi$$
$$674$$ −5815.17 −0.332332
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −15455.8 −0.877419 −0.438710 0.898629i $$-0.644564\pi$$
−0.438710 + 0.898629i $$0.644564\pi$$
$$678$$ 0 0
$$679$$ 18918.9 1.06928
$$680$$ −716.909 −0.0404297
$$681$$ 0 0
$$682$$ 20681.2 1.16118
$$683$$ −16841.9 −0.943540 −0.471770 0.881722i $$-0.656385\pi$$
−0.471770 + 0.881722i $$0.656385\pi$$
$$684$$ 0 0
$$685$$ 132.066 0.00736639
$$686$$ 10236.7 0.569739
$$687$$ 0 0
$$688$$ 4211.80 0.233391
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −25041.6 −1.37862 −0.689310 0.724467i $$-0.742086\pi$$
−0.689310 + 0.724467i $$0.742086\pi$$
$$692$$ 696.896 0.0382832
$$693$$ 0 0
$$694$$ −22347.6 −1.22234
$$695$$ 9.13638 0.000498652 0
$$696$$ 0 0
$$697$$ 34204.7 1.85881
$$698$$ 23942.1 1.29831
$$699$$ 0 0
$$700$$ 1437.66 0.0776261
$$701$$ 28309.0 1.52527 0.762635 0.646829i $$-0.223905\pi$$
0.762635 + 0.646829i $$0.223905\pi$$
$$702$$ 0 0
$$703$$ −4623.88 −0.248069
$$704$$ 33829.1 1.81106
$$705$$ 0 0
$$706$$ −24945.0 −1.32977
$$707$$ −22669.1 −1.20588
$$708$$ 0 0
$$709$$ −5041.17 −0.267031 −0.133516 0.991047i $$-0.542627\pi$$
−0.133516 + 0.991047i $$0.542627\pi$$
$$710$$ 816.882 0.0431789
$$711$$ 0 0
$$712$$ −29276.6 −1.54100
$$713$$ −1707.96 −0.0897103
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −585.431 −0.0305567
$$717$$ 0 0
$$718$$ −19414.7 −1.00912
$$719$$ −11839.2 −0.614088 −0.307044 0.951695i $$-0.599340\pi$$
−0.307044 + 0.951695i $$0.599340\pi$$
$$720$$ 0 0
$$721$$ 28017.2 1.44718
$$722$$ −18326.6 −0.944662
$$723$$ 0 0
$$724$$ 221.748 0.0113829
$$725$$ 24516.1 1.25587
$$726$$ 0 0
$$727$$ −31004.9 −1.58172 −0.790858 0.611999i $$-0.790365\pi$$
−0.790858 + 0.611999i $$0.790365\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −673.256 −0.0341347
$$731$$ −6867.03 −0.347451
$$732$$ 0 0
$$733$$ 15666.0 0.789411 0.394706 0.918808i $$-0.370847\pi$$
0.394706 + 0.918808i $$0.370847\pi$$
$$734$$ 38105.8 1.91623
$$735$$ 0 0
$$736$$ −262.145 −0.0131288
$$737$$ 4337.02 0.216765
$$738$$ 0 0
$$739$$ −33339.4 −1.65955 −0.829777 0.558095i $$-0.811533\pi$$
−0.829777 + 0.558095i $$0.811533\pi$$
$$740$$ −40.2733 −0.00200064
$$741$$ 0 0
$$742$$ −46833.8 −2.31715
$$743$$ −2026.14 −0.100043 −0.0500215 0.998748i $$-0.515929\pi$$
−0.0500215 + 0.998748i $$0.515929\pi$$
$$744$$ 0 0
$$745$$ 670.831 0.0329897
$$746$$ 17552.5 0.861451
$$747$$ 0 0
$$748$$ −2517.38 −0.123054
$$749$$ −23022.6 −1.12313
$$750$$ 0 0
$$751$$ 35474.8 1.72369 0.861846 0.507170i $$-0.169309\pi$$
0.861846 + 0.507170i $$0.169309\pi$$
$$752$$ 6175.06 0.299443
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 457.065 0.0220322
$$756$$ 0 0
$$757$$ 2398.58 0.115162 0.0575811 0.998341i $$-0.481661\pi$$
0.0575811 + 0.998341i $$0.481661\pi$$
$$758$$ 6757.46 0.323802
$$759$$ 0 0
$$760$$ 105.111 0.00501682
$$761$$ 5005.07 0.238415 0.119207 0.992869i $$-0.461965\pi$$
0.119207 + 0.992869i $$0.461965\pi$$
$$762$$ 0 0
$$763$$ 24143.5 1.14555
$$764$$ −752.960 −0.0356559
$$765$$ 0 0
$$766$$ −29853.1 −1.40814
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 14109.6 0.661644 0.330822 0.943693i $$-0.392674\pi$$
0.330822 + 0.943693i $$0.392674\pi$$
$$770$$ 1558.99 0.0729635
$$771$$ 0 0
$$772$$ −1674.82 −0.0780803
$$773$$ 12733.7 0.592496 0.296248 0.955111i $$-0.404265\pi$$
0.296248 + 0.955111i $$0.404265\pi$$
$$774$$ 0 0
$$775$$ −14824.7 −0.687121
$$776$$ −15339.2 −0.709593
$$777$$ 0 0
$$778$$ 13720.9 0.632287
$$779$$ −5014.98 −0.230655
$$780$$ 0 0
$$781$$ 59818.8 2.74070
$$782$$ −3919.75 −0.179246
$$783$$ 0 0
$$784$$ −28671.3 −1.30609
$$785$$ 628.986 0.0285980
$$786$$ 0 0
$$787$$ 28429.3 1.28767 0.643834 0.765165i $$-0.277343\pi$$
0.643834 + 0.765165i $$0.277343\pi$$
$$788$$ 613.344 0.0277278
$$789$$ 0 0
$$790$$ 207.209 0.00933186
$$791$$ 1610.48 0.0723918
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 26048.7 1.16427
$$795$$ 0 0
$$796$$ −325.625 −0.0144993
$$797$$ −11926.7 −0.530070 −0.265035 0.964239i $$-0.585384\pi$$
−0.265035 + 0.964239i $$0.585384\pi$$
$$798$$ 0 0
$$799$$ −10068.0 −0.445782
$$800$$ −2275.36 −0.100558
$$801$$ 0 0
$$802$$ −24191.2 −1.06511
$$803$$ −49301.4 −2.16663
$$804$$ 0 0
$$805$$ −128.749 −0.00563702
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 18379.7 0.800244
$$809$$ −20005.7 −0.869421 −0.434710 0.900570i $$-0.643149\pi$$
−0.434710 + 0.900570i $$0.643149\pi$$
$$810$$ 0 0
$$811$$ −5933.84 −0.256924 −0.128462 0.991714i $$-0.541004\pi$$
−0.128462 + 0.991714i $$0.541004\pi$$
$$812$$ 2259.27 0.0976415
$$813$$ 0 0
$$814$$ 55603.8 2.39424
$$815$$ 171.794 0.00738364
$$816$$ 0 0
$$817$$ 1006.82 0.0431142
$$818$$ −12458.9 −0.532537
$$819$$ 0 0
$$820$$ −43.6798 −0.00186020
$$821$$ 18457.0 0.784597 0.392299 0.919838i $$-0.371680\pi$$
0.392299 + 0.919838i $$0.371680\pi$$
$$822$$ 0 0
$$823$$ −26520.3 −1.12325 −0.561627 0.827390i $$-0.689825\pi$$
−0.561627 + 0.827390i $$0.689825\pi$$
$$824$$ −22715.9 −0.960372
$$825$$ 0 0
$$826$$ −16099.4 −0.678170
$$827$$ 25664.8 1.07915 0.539573 0.841939i $$-0.318586\pi$$
0.539573 + 0.841939i $$0.318586\pi$$
$$828$$ 0 0
$$829$$ 4362.69 0.182777 0.0913887 0.995815i $$-0.470869\pi$$
0.0913887 + 0.995815i $$0.470869\pi$$
$$830$$ 738.612 0.0308887
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 46746.6 1.94439
$$834$$ 0 0
$$835$$ −1241.12 −0.0514378
$$836$$ 369.091 0.0152695
$$837$$ 0 0
$$838$$ −33093.1 −1.36418
$$839$$ 35020.2 1.44104 0.720520 0.693435i $$-0.243903\pi$$
0.720520 + 0.693435i $$0.243903\pi$$
$$840$$ 0 0
$$841$$ 14138.0 0.579687
$$842$$ −26822.8 −1.09783
$$843$$ 0 0
$$844$$ 2119.46 0.0864394
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 76140.4 3.08880
$$848$$ 36054.7 1.46005
$$849$$ 0 0
$$850$$ −34022.6 −1.37290
$$851$$ −4592.04 −0.184974
$$852$$ 0 0
$$853$$ 8318.36 0.333898 0.166949 0.985966i $$-0.446608\pi$$
0.166949 + 0.985966i $$0.446608\pi$$
$$854$$ −17005.2 −0.681389
$$855$$ 0 0
$$856$$ 18666.3 0.745330
$$857$$ −34832.3 −1.38839 −0.694193 0.719789i $$-0.744239\pi$$
−0.694193 + 0.719789i $$0.744239\pi$$
$$858$$ 0 0
$$859$$ −12571.8 −0.499353 −0.249676 0.968329i $$-0.580324\pi$$
−0.249676 + 0.968329i $$0.580324\pi$$
$$860$$ 8.76930 0.000347710 0
$$861$$ 0 0
$$862$$ 11331.1 0.447726
$$863$$ −22474.2 −0.886480 −0.443240 0.896403i $$-0.646171\pi$$
−0.443240 + 0.896403i $$0.646171\pi$$
$$864$$ 0 0
$$865$$ −541.708 −0.0212932
$$866$$ 8009.90 0.314304
$$867$$ 0 0
$$868$$ −1366.16 −0.0534223
$$869$$ 15173.6 0.592322
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −19575.2 −0.760205
$$873$$ 0 0
$$874$$ 574.702 0.0222421
$$875$$ −2235.90 −0.0863855
$$876$$ 0 0
$$877$$ 5385.19 0.207349 0.103675 0.994611i $$-0.466940\pi$$
0.103675 + 0.994611i $$0.466940\pi$$
$$878$$ 24461.2 0.940233
$$879$$ 0 0
$$880$$ −1200.17 −0.0459749
$$881$$ 29772.7 1.13856 0.569278 0.822145i $$-0.307223\pi$$
0.569278 + 0.822145i $$0.307223\pi$$
$$882$$ 0 0
$$883$$ −27309.6 −1.04082 −0.520409 0.853917i $$-0.674220\pi$$
−0.520409 + 0.853917i $$0.674220\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −34840.7 −1.32110
$$887$$ 6529.97 0.247187 0.123594 0.992333i $$-0.460558\pi$$
0.123594 + 0.992333i $$0.460558\pi$$
$$888$$ 0 0
$$889$$ 47609.3 1.79614
$$890$$ 1091.25 0.0410998
$$891$$ 0 0
$$892$$ 1890.06 0.0709461
$$893$$ 1476.14 0.0553159
$$894$$ 0 0
$$895$$ 455.065 0.0169957
$$896$$ 37970.3 1.41573
$$897$$ 0 0
$$898$$ 4890.45 0.181733
$$899$$ −23297.0 −0.864290
$$900$$ 0 0
$$901$$ −58784.6 −2.17358
$$902$$ 60307.1 2.22617
$$903$$ 0 0
$$904$$ −1305.75 −0.0480404
$$905$$ −172.368 −0.00633116
$$906$$ 0 0
$$907$$ 19437.4 0.711586 0.355793 0.934565i $$-0.384211\pi$$
0.355793 + 0.934565i $$0.384211\pi$$
$$908$$ 1434.70 0.0524364
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −50234.6 −1.82694 −0.913472 0.406901i $$-0.866609\pi$$
−0.913472 + 0.406901i $$0.866609\pi$$
$$912$$ 0 0
$$913$$ 54087.2 1.96060
$$914$$ 17003.0 0.615326
$$915$$ 0 0
$$916$$ 1267.16 0.0457077
$$917$$ −17214.7 −0.619934
$$918$$ 0 0
$$919$$ 12285.2 0.440971 0.220486 0.975390i $$-0.429236\pi$$
0.220486 + 0.975390i $$0.429236\pi$$
$$920$$ 104.387 0.00374082
$$921$$ 0 0
$$922$$ 20549.1 0.734000
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −39858.0 −1.41678
$$926$$ 14883.3 0.528180
$$927$$ 0 0
$$928$$ −3575.72 −0.126486
$$929$$ −25610.0 −0.904454 −0.452227 0.891903i $$-0.649370\pi$$
−0.452227 + 0.891903i $$0.649370\pi$$
$$930$$ 0 0
$$931$$ −6853.84 −0.241273
$$932$$ −343.521 −0.0120734
$$933$$ 0 0
$$934$$ −11003.1 −0.385474
$$935$$ 1956.80 0.0684429
$$936$$ 0 0
$$937$$ −29893.2 −1.04223 −0.521114 0.853487i $$-0.674483\pi$$
−0.521114 + 0.853487i $$0.674483\pi$$
$$938$$ 5401.66 0.188028
$$939$$ 0 0
$$940$$ 12.8570 0.000446115 0
$$941$$ −44960.3 −1.55756 −0.778780 0.627297i $$-0.784161\pi$$
−0.778780 + 0.627297i $$0.784161\pi$$
$$942$$ 0 0
$$943$$ −4980.46 −0.171989
$$944$$ 12394.0 0.427320
$$945$$ 0 0
$$946$$ −12107.4 −0.416117
$$947$$ 33597.6 1.15288 0.576439 0.817140i $$-0.304442\pi$$
0.576439 + 0.817140i $$0.304442\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 4988.29 0.170360
$$951$$ 0 0
$$952$$ −65385.1 −2.22599
$$953$$ 25850.3 0.878670 0.439335 0.898323i $$-0.355214\pi$$
0.439335 + 0.898323i $$0.355214\pi$$
$$954$$ 0 0
$$955$$ 585.288 0.0198319
$$956$$ 882.160 0.0298442
$$957$$ 0 0
$$958$$ −8686.75 −0.292960
$$959$$ 12044.9 0.405580
$$960$$ 0 0
$$961$$ −15703.5 −0.527123
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −2550.58 −0.0852165
$$965$$ 1301.86 0.0434284
$$966$$ 0 0
$$967$$ 6296.49 0.209392 0.104696 0.994504i $$-0.466613\pi$$
0.104696 + 0.994504i $$0.466613\pi$$
$$968$$ −61733.5 −2.04978
$$969$$ 0 0
$$970$$ 571.749 0.0189255
$$971$$ 7806.43 0.258002 0.129001 0.991644i $$-0.458823\pi$$
0.129001 + 0.991644i $$0.458823\pi$$
$$972$$ 0 0
$$973$$ 833.276 0.0274549
$$974$$ −10774.8 −0.354462
$$975$$ 0 0
$$976$$ 13091.3 0.429348
$$977$$ −55692.8 −1.82372 −0.911858 0.410507i $$-0.865352\pi$$
−0.911858 + 0.410507i $$0.865352\pi$$
$$978$$ 0 0
$$979$$ 79910.4 2.60873
$$980$$ −59.6960 −0.00194584
$$981$$ 0 0
$$982$$ −18315.8 −0.595195
$$983$$ 44635.5 1.44827 0.724135 0.689658i $$-0.242239\pi$$
0.724135 + 0.689658i $$0.242239\pi$$
$$984$$ 0 0
$$985$$ −476.762 −0.0154222
$$986$$ −53466.4 −1.72689
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 999.892 0.0321484
$$990$$ 0 0
$$991$$ 3884.32 0.124510 0.0622551 0.998060i $$-0.480171\pi$$
0.0622551 + 0.998060i $$0.480171\pi$$
$$992$$ 2162.21 0.0692039
$$993$$ 0 0
$$994$$ 74503.0 2.37735
$$995$$ 253.114 0.00806456
$$996$$ 0 0
$$997$$ −18021.4 −0.572462 −0.286231 0.958161i $$-0.592402\pi$$
−0.286231 + 0.958161i $$0.592402\pi$$
$$998$$ 16845.9 0.534316
$$999$$ 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 1521.4.a.bc.1.6 8
3.2 odd 2 inner 1521.4.a.bc.1.3 8
13.3 even 3 117.4.g.f.100.3 yes 16
13.9 even 3 117.4.g.f.55.3 16
13.12 even 2 1521.4.a.bd.1.3 8
39.29 odd 6 117.4.g.f.100.6 yes 16
39.35 odd 6 117.4.g.f.55.6 yes 16
39.38 odd 2 1521.4.a.bd.1.6 8

By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.g.f.55.3 16 13.9 even 3
117.4.g.f.55.6 yes 16 39.35 odd 6
117.4.g.f.100.3 yes 16 13.3 even 3
117.4.g.f.100.6 yes 16 39.29 odd 6
1521.4.a.bc.1.3 8 3.2 odd 2 inner
1521.4.a.bc.1.6 8 1.1 even 1 trivial
1521.4.a.bd.1.3 8 13.12 even 2
1521.4.a.bd.1.6 8 39.38 odd 2