# Properties

 Label 1521.4.a.ba.1.2 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## 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: $$4$$ Coefficient field: 4.4.1520092.1 Defining polynomial: $$x^{4} - x^{3} - 40x^{2} - 9x + 81$$ x^4 - x^3 - 40*x^2 - 9*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 117) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-5.49403$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.85588 q^{2} +6.86783 q^{4} -19.5342 q^{5} +6.26434 q^{7} +4.36551 q^{8} +O(q^{10})$$ $$q-3.85588 q^{2} +6.86783 q^{4} -19.5342 q^{5} +6.26434 q^{7} +4.36551 q^{8} +75.3217 q^{10} -27.2460 q^{11} -24.1546 q^{14} -71.7755 q^{16} -30.8471 q^{17} -127.850 q^{19} -134.158 q^{20} +105.057 q^{22} -84.3198 q^{23} +256.586 q^{25} +43.0224 q^{28} -272.460 q^{29} +166.264 q^{31} +241.834 q^{32} +118.943 q^{34} -122.369 q^{35} +198.229 q^{37} +492.976 q^{38} -85.2769 q^{40} +160.385 q^{41} +158.943 q^{43} -187.121 q^{44} +325.127 q^{46} -305.889 q^{47} -303.758 q^{49} -989.366 q^{50} -356.780 q^{53} +532.229 q^{55} +27.3470 q^{56} +1050.57 q^{58} -470.384 q^{59} -171.815 q^{61} -641.096 q^{62} -358.279 q^{64} -1022.49 q^{67} -211.852 q^{68} +471.840 q^{70} -188.616 q^{71} -959.975 q^{73} -764.349 q^{74} -878.055 q^{76} -170.678 q^{77} -1034.80 q^{79} +1402.08 q^{80} -618.424 q^{82} -105.383 q^{83} +602.574 q^{85} -612.864 q^{86} -118.943 q^{88} -649.860 q^{89} -579.094 q^{92} +1179.47 q^{94} +2497.46 q^{95} -707.746 q^{97} +1171.26 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 58 q^{4} - 36 q^{7}+O(q^{10})$$ 4 * q + 58 * q^4 - 36 * q^7 $$4 q + 58 q^{4} - 36 q^{7} - 4 q^{10} + 354 q^{16} - 84 q^{19} + 176 q^{22} + 660 q^{25} - 988 q^{28} + 604 q^{31} + 720 q^{34} - 184 q^{37} - 2356 q^{40} + 880 q^{43} + 2888 q^{46} - 116 q^{49} + 1152 q^{55} + 1760 q^{58} + 656 q^{61} + 3482 q^{64} - 3052 q^{67} + 4696 q^{70} + 312 q^{73} + 2044 q^{76} - 720 q^{79} + 396 q^{82} - 32 q^{85} - 720 q^{88} + 4840 q^{94} + 344 q^{97}+O(q^{100})$$ 4 * q + 58 * q^4 - 36 * q^7 - 4 * q^10 + 354 * q^16 - 84 * q^19 + 176 * q^22 + 660 * q^25 - 988 * q^28 + 604 * q^31 + 720 * q^34 - 184 * q^37 - 2356 * q^40 + 880 * q^43 + 2888 * q^46 - 116 * q^49 + 1152 * q^55 + 1760 * q^58 + 656 * q^61 + 3482 * q^64 - 3052 * q^67 + 4696 * q^70 + 312 * q^73 + 2044 * q^76 - 720 * q^79 + 396 * q^82 - 32 * q^85 - 720 * q^88 + 4840 * q^94 + 344 * q^97

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.85588 −1.36326 −0.681630 0.731697i $$-0.738729\pi$$
−0.681630 + 0.731697i $$0.738729\pi$$
$$3$$ 0 0
$$4$$ 6.86783 0.858479
$$5$$ −19.5342 −1.74719 −0.873597 0.486650i $$-0.838219\pi$$
−0.873597 + 0.486650i $$0.838219\pi$$
$$6$$ 0 0
$$7$$ 6.26434 0.338242 0.169121 0.985595i $$-0.445907\pi$$
0.169121 + 0.985595i $$0.445907\pi$$
$$8$$ 4.36551 0.192930
$$9$$ 0 0
$$10$$ 75.3217 2.38188
$$11$$ −27.2460 −0.746816 −0.373408 0.927667i $$-0.621811\pi$$
−0.373408 + 0.927667i $$0.621811\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −24.1546 −0.461113
$$15$$ 0 0
$$16$$ −71.7755 −1.12149
$$17$$ −30.8471 −0.440089 −0.220044 0.975490i $$-0.570620\pi$$
−0.220044 + 0.975490i $$0.570620\pi$$
$$18$$ 0 0
$$19$$ −127.850 −1.54373 −0.771865 0.635786i $$-0.780676\pi$$
−0.771865 + 0.635786i $$0.780676\pi$$
$$20$$ −134.158 −1.49993
$$21$$ 0 0
$$22$$ 105.057 1.01810
$$23$$ −84.3198 −0.764430 −0.382215 0.924073i $$-0.624839\pi$$
−0.382215 + 0.924073i $$0.624839\pi$$
$$24$$ 0 0
$$25$$ 256.586 2.05269
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 43.0224 0.290374
$$29$$ −272.460 −1.74464 −0.872320 0.488936i $$-0.837385\pi$$
−0.872320 + 0.488936i $$0.837385\pi$$
$$30$$ 0 0
$$31$$ 166.264 0.963289 0.481644 0.876367i $$-0.340040\pi$$
0.481644 + 0.876367i $$0.340040\pi$$
$$32$$ 241.834 1.33596
$$33$$ 0 0
$$34$$ 118.943 0.599956
$$35$$ −122.369 −0.590975
$$36$$ 0 0
$$37$$ 198.229 0.880776 0.440388 0.897808i $$-0.354841\pi$$
0.440388 + 0.897808i $$0.354841\pi$$
$$38$$ 492.976 2.10451
$$39$$ 0 0
$$40$$ −85.2769 −0.337086
$$41$$ 160.385 0.610923 0.305462 0.952204i $$-0.401189\pi$$
0.305462 + 0.952204i $$0.401189\pi$$
$$42$$ 0 0
$$43$$ 158.943 0.563687 0.281843 0.959460i $$-0.409054\pi$$
0.281843 + 0.959460i $$0.409054\pi$$
$$44$$ −187.121 −0.641126
$$45$$ 0 0
$$46$$ 325.127 1.04212
$$47$$ −305.889 −0.949329 −0.474665 0.880167i $$-0.657431\pi$$
−0.474665 + 0.880167i $$0.657431\pi$$
$$48$$ 0 0
$$49$$ −303.758 −0.885592
$$50$$ −989.366 −2.79835
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −356.780 −0.924669 −0.462335 0.886706i $$-0.652988\pi$$
−0.462335 + 0.886706i $$0.652988\pi$$
$$54$$ 0 0
$$55$$ 532.229 1.30483
$$56$$ 27.3470 0.0652571
$$57$$ 0 0
$$58$$ 1050.57 2.37840
$$59$$ −470.384 −1.03795 −0.518973 0.854791i $$-0.673685\pi$$
−0.518973 + 0.854791i $$0.673685\pi$$
$$60$$ 0 0
$$61$$ −171.815 −0.360635 −0.180317 0.983608i $$-0.557712\pi$$
−0.180317 + 0.983608i $$0.557712\pi$$
$$62$$ −641.096 −1.31321
$$63$$ 0 0
$$64$$ −358.279 −0.699764
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −1022.49 −1.86444 −0.932220 0.361892i $$-0.882131\pi$$
−0.932220 + 0.361892i $$0.882131\pi$$
$$68$$ −211.852 −0.377807
$$69$$ 0 0
$$70$$ 471.840 0.805653
$$71$$ −188.616 −0.315276 −0.157638 0.987497i $$-0.550388\pi$$
−0.157638 + 0.987497i $$0.550388\pi$$
$$72$$ 0 0
$$73$$ −959.975 −1.53913 −0.769566 0.638568i $$-0.779527\pi$$
−0.769566 + 0.638568i $$0.779527\pi$$
$$74$$ −764.349 −1.20073
$$75$$ 0 0
$$76$$ −878.055 −1.32526
$$77$$ −170.678 −0.252605
$$78$$ 0 0
$$79$$ −1034.80 −1.47373 −0.736863 0.676042i $$-0.763694\pi$$
−0.736863 + 0.676042i $$0.763694\pi$$
$$80$$ 1402.08 1.95947
$$81$$ 0 0
$$82$$ −618.424 −0.832847
$$83$$ −105.383 −0.139365 −0.0696824 0.997569i $$-0.522199\pi$$
−0.0696824 + 0.997569i $$0.522199\pi$$
$$84$$ 0 0
$$85$$ 602.574 0.768921
$$86$$ −612.864 −0.768452
$$87$$ 0 0
$$88$$ −118.943 −0.144083
$$89$$ −649.860 −0.773990 −0.386995 0.922082i $$-0.626487\pi$$
−0.386995 + 0.922082i $$0.626487\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −579.094 −0.656247
$$93$$ 0 0
$$94$$ 1179.47 1.29418
$$95$$ 2497.46 2.69720
$$96$$ 0 0
$$97$$ −707.746 −0.740832 −0.370416 0.928866i $$-0.620785\pi$$
−0.370416 + 0.928866i $$0.620785\pi$$
$$98$$ 1171.26 1.20729
$$99$$ 0 0
$$100$$ 1762.19 1.76219
$$101$$ −1409.59 −1.38871 −0.694353 0.719634i $$-0.744310\pi$$
−0.694353 + 0.719634i $$0.744310\pi$$
$$102$$ 0 0
$$103$$ 1519.63 1.45373 0.726863 0.686783i $$-0.240978\pi$$
0.726863 + 0.686783i $$0.240978\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 1375.70 1.26056
$$107$$ −1957.57 −1.76865 −0.884323 0.466875i $$-0.845380\pi$$
−0.884323 + 0.466875i $$0.845380\pi$$
$$108$$ 0 0
$$109$$ −1517.63 −1.33360 −0.666801 0.745236i $$-0.732337\pi$$
−0.666801 + 0.745236i $$0.732337\pi$$
$$110$$ −2052.21 −1.77883
$$111$$ 0 0
$$112$$ −449.626 −0.379336
$$113$$ 2200.27 1.83171 0.915857 0.401505i $$-0.131513\pi$$
0.915857 + 0.401505i $$0.131513\pi$$
$$114$$ 0 0
$$115$$ 1647.12 1.33561
$$116$$ −1871.21 −1.49774
$$117$$ 0 0
$$118$$ 1813.75 1.41499
$$119$$ −193.236 −0.148857
$$120$$ 0 0
$$121$$ −588.656 −0.442266
$$122$$ 662.500 0.491639
$$123$$ 0 0
$$124$$ 1141.88 0.826963
$$125$$ −2570.43 −1.83925
$$126$$ 0 0
$$127$$ 1812.16 1.26617 0.633083 0.774084i $$-0.281789\pi$$
0.633083 + 0.774084i $$0.281789\pi$$
$$128$$ −553.190 −0.381996
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1001.31 −0.667822 −0.333911 0.942605i $$-0.608369\pi$$
−0.333911 + 0.942605i $$0.608369\pi$$
$$132$$ 0 0
$$133$$ −800.898 −0.522155
$$134$$ 3942.62 2.54172
$$135$$ 0 0
$$136$$ −134.663 −0.0849064
$$137$$ 1244.11 0.775849 0.387925 0.921691i $$-0.373192\pi$$
0.387925 + 0.921691i $$0.373192\pi$$
$$138$$ 0 0
$$139$$ 1348.85 0.823077 0.411539 0.911392i $$-0.364991\pi$$
0.411539 + 0.911392i $$0.364991\pi$$
$$140$$ −840.410 −0.507340
$$141$$ 0 0
$$142$$ 727.281 0.429804
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 5322.29 3.04822
$$146$$ 3701.55 2.09824
$$147$$ 0 0
$$148$$ 1361.41 0.756128
$$149$$ −1041.63 −0.572710 −0.286355 0.958124i $$-0.592444\pi$$
−0.286355 + 0.958124i $$0.592444\pi$$
$$150$$ 0 0
$$151$$ −1361.90 −0.733970 −0.366985 0.930227i $$-0.619610\pi$$
−0.366985 + 0.930227i $$0.619610\pi$$
$$152$$ −558.132 −0.297832
$$153$$ 0 0
$$154$$ 658.115 0.344366
$$155$$ −3247.85 −1.68305
$$156$$ 0 0
$$157$$ 86.8978 0.0441733 0.0220866 0.999756i $$-0.492969\pi$$
0.0220866 + 0.999756i $$0.492969\pi$$
$$158$$ 3990.08 2.00907
$$159$$ 0 0
$$160$$ −4724.04 −2.33418
$$161$$ −528.208 −0.258563
$$162$$ 0 0
$$163$$ 28.5386 0.0137136 0.00685679 0.999976i $$-0.497817\pi$$
0.00685679 + 0.999976i $$0.497817\pi$$
$$164$$ 1101.49 0.524465
$$165$$ 0 0
$$166$$ 406.344 0.189990
$$167$$ 1289.87 0.597683 0.298842 0.954303i $$-0.403400\pi$$
0.298842 + 0.954303i $$0.403400\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −2323.45 −1.04824
$$171$$ 0 0
$$172$$ 1091.59 0.483913
$$173$$ 603.489 0.265216 0.132608 0.991169i $$-0.457665\pi$$
0.132608 + 0.991169i $$0.457665\pi$$
$$174$$ 0 0
$$175$$ 1607.34 0.694306
$$176$$ 1955.60 0.837549
$$177$$ 0 0
$$178$$ 2505.79 1.05515
$$179$$ −1985.29 −0.828980 −0.414490 0.910054i $$-0.636040\pi$$
−0.414490 + 0.910054i $$0.636040\pi$$
$$180$$ 0 0
$$181$$ −3945.79 −1.62038 −0.810189 0.586169i $$-0.800635\pi$$
−0.810189 + 0.586169i $$0.800635\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −368.099 −0.147482
$$185$$ −3872.26 −1.53889
$$186$$ 0 0
$$187$$ 840.459 0.328665
$$188$$ −2100.79 −0.814979
$$189$$ 0 0
$$190$$ −9629.91 −3.67698
$$191$$ 1446.48 0.547979 0.273989 0.961733i $$-0.411657\pi$$
0.273989 + 0.961733i $$0.411657\pi$$
$$192$$ 0 0
$$193$$ −784.523 −0.292597 −0.146299 0.989240i $$-0.546736\pi$$
−0.146299 + 0.989240i $$0.546736\pi$$
$$194$$ 2728.98 1.00995
$$195$$ 0 0
$$196$$ −2086.16 −0.760262
$$197$$ −494.336 −0.178782 −0.0893909 0.995997i $$-0.528492\pi$$
−0.0893909 + 0.995997i $$0.528492\pi$$
$$198$$ 0 0
$$199$$ −3022.59 −1.07671 −0.538357 0.842717i $$-0.680955\pi$$
−0.538357 + 0.842717i $$0.680955\pi$$
$$200$$ 1120.13 0.396025
$$201$$ 0 0
$$202$$ 5435.21 1.89317
$$203$$ −1706.78 −0.590111
$$204$$ 0 0
$$205$$ −3132.99 −1.06740
$$206$$ −5859.52 −1.98181
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 3483.41 1.15288
$$210$$ 0 0
$$211$$ 3185.24 1.03925 0.519623 0.854396i $$-0.326073\pi$$
0.519623 + 0.854396i $$0.326073\pi$$
$$212$$ −2450.30 −0.793809
$$213$$ 0 0
$$214$$ 7548.15 2.41113
$$215$$ −3104.82 −0.984870
$$216$$ 0 0
$$217$$ 1041.54 0.325825
$$218$$ 5851.81 1.81805
$$219$$ 0 0
$$220$$ 3655.26 1.12017
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −997.686 −0.299596 −0.149798 0.988717i $$-0.547862\pi$$
−0.149798 + 0.988717i $$0.547862\pi$$
$$224$$ 1514.93 0.451877
$$225$$ 0 0
$$226$$ −8483.97 −2.49710
$$227$$ 6496.57 1.89953 0.949763 0.312969i $$-0.101324\pi$$
0.949763 + 0.312969i $$0.101324\pi$$
$$228$$ 0 0
$$229$$ −708.434 −0.204431 −0.102215 0.994762i $$-0.532593\pi$$
−0.102215 + 0.994762i $$0.532593\pi$$
$$230$$ −6351.11 −1.82078
$$231$$ 0 0
$$232$$ −1189.43 −0.336593
$$233$$ 4919.02 1.38307 0.691536 0.722342i $$-0.256934\pi$$
0.691536 + 0.722342i $$0.256934\pi$$
$$234$$ 0 0
$$235$$ 5975.30 1.65866
$$236$$ −3230.52 −0.891054
$$237$$ 0 0
$$238$$ 745.097 0.202930
$$239$$ −324.370 −0.0877898 −0.0438949 0.999036i $$-0.513977\pi$$
−0.0438949 + 0.999036i $$0.513977\pi$$
$$240$$ 0 0
$$241$$ 1940.43 0.518649 0.259324 0.965790i $$-0.416500\pi$$
0.259324 + 0.965790i $$0.416500\pi$$
$$242$$ 2269.79 0.602924
$$243$$ 0 0
$$244$$ −1180.00 −0.309597
$$245$$ 5933.68 1.54730
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 725.829 0.185847
$$249$$ 0 0
$$250$$ 9911.28 2.50738
$$251$$ −1958.45 −0.492495 −0.246248 0.969207i $$-0.579198\pi$$
−0.246248 + 0.969207i $$0.579198\pi$$
$$252$$ 0 0
$$253$$ 2297.38 0.570889
$$254$$ −6987.47 −1.72611
$$255$$ 0 0
$$256$$ 4999.27 1.22052
$$257$$ −4330.35 −1.05105 −0.525525 0.850778i $$-0.676131\pi$$
−0.525525 + 0.850778i $$0.676131\pi$$
$$258$$ 0 0
$$259$$ 1241.78 0.297916
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 3860.93 0.910415
$$263$$ −2933.60 −0.687807 −0.343904 0.939005i $$-0.611749\pi$$
−0.343904 + 0.939005i $$0.611749\pi$$
$$264$$ 0 0
$$265$$ 6969.42 1.61558
$$266$$ 3088.17 0.711834
$$267$$ 0 0
$$268$$ −7022.31 −1.60058
$$269$$ 2458.25 0.557184 0.278592 0.960410i $$-0.410132\pi$$
0.278592 + 0.960410i $$0.410132\pi$$
$$270$$ 0 0
$$271$$ −2089.91 −0.468462 −0.234231 0.972181i $$-0.575257\pi$$
−0.234231 + 0.972181i $$0.575257\pi$$
$$272$$ 2214.06 0.493557
$$273$$ 0 0
$$274$$ −4797.14 −1.05768
$$275$$ −6990.94 −1.53298
$$276$$ 0 0
$$277$$ −2462.00 −0.534033 −0.267017 0.963692i $$-0.586038\pi$$
−0.267017 + 0.963692i $$0.586038\pi$$
$$278$$ −5201.00 −1.12207
$$279$$ 0 0
$$280$$ −534.203 −0.114017
$$281$$ 5775.91 1.22620 0.613100 0.790005i $$-0.289922\pi$$
0.613100 + 0.790005i $$0.289922\pi$$
$$282$$ 0 0
$$283$$ 5093.45 1.06987 0.534936 0.844892i $$-0.320336\pi$$
0.534936 + 0.844892i $$0.320336\pi$$
$$284$$ −1295.38 −0.270658
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1004.70 0.206640
$$288$$ 0 0
$$289$$ −3961.46 −0.806322
$$290$$ −20522.1 −4.15552
$$291$$ 0 0
$$292$$ −6592.95 −1.32131
$$293$$ 1503.73 0.299826 0.149913 0.988699i $$-0.452101\pi$$
0.149913 + 0.988699i $$0.452101\pi$$
$$294$$ 0 0
$$295$$ 9188.59 1.81349
$$296$$ 865.372 0.169928
$$297$$ 0 0
$$298$$ 4016.41 0.780753
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 995.670 0.190663
$$302$$ 5251.31 1.00059
$$303$$ 0 0
$$304$$ 9176.53 1.73128
$$305$$ 3356.28 0.630099
$$306$$ 0 0
$$307$$ 6567.59 1.22095 0.610476 0.792035i $$-0.290978\pi$$
0.610476 + 0.792035i $$0.290978\pi$$
$$308$$ −1172.19 −0.216856
$$309$$ 0 0
$$310$$ 12523.3 2.29444
$$311$$ 5714.66 1.04196 0.520978 0.853570i $$-0.325567\pi$$
0.520978 + 0.853570i $$0.325567\pi$$
$$312$$ 0 0
$$313$$ −4953.59 −0.894548 −0.447274 0.894397i $$-0.647605\pi$$
−0.447274 + 0.894397i $$0.647605\pi$$
$$314$$ −335.068 −0.0602197
$$315$$ 0 0
$$316$$ −7106.85 −1.26516
$$317$$ 2526.64 0.447666 0.223833 0.974628i $$-0.428143\pi$$
0.223833 + 0.974628i $$0.428143\pi$$
$$318$$ 0 0
$$319$$ 7423.44 1.30292
$$320$$ 6998.71 1.22262
$$321$$ 0 0
$$322$$ 2036.71 0.352488
$$323$$ 3943.81 0.679379
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −110.041 −0.0186952
$$327$$ 0 0
$$328$$ 700.160 0.117865
$$329$$ −1916.19 −0.321103
$$330$$ 0 0
$$331$$ −7436.06 −1.23481 −0.617406 0.786645i $$-0.711816\pi$$
−0.617406 + 0.786645i $$0.711816\pi$$
$$332$$ −723.752 −0.119642
$$333$$ 0 0
$$334$$ −4973.59 −0.814798
$$335$$ 19973.6 3.25754
$$336$$ 0 0
$$337$$ 9467.96 1.53042 0.765212 0.643778i $$-0.222634\pi$$
0.765212 + 0.643778i $$0.222634\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 4138.37 0.660102
$$341$$ −4530.04 −0.719400
$$342$$ 0 0
$$343$$ −4051.51 −0.637787
$$344$$ 693.866 0.108752
$$345$$ 0 0
$$346$$ −2326.98 −0.361559
$$347$$ 2794.85 0.432379 0.216189 0.976351i $$-0.430637\pi$$
0.216189 + 0.976351i $$0.430637\pi$$
$$348$$ 0 0
$$349$$ −2602.91 −0.399228 −0.199614 0.979875i $$-0.563969\pi$$
−0.199614 + 0.979875i $$0.563969\pi$$
$$350$$ −6197.72 −0.946520
$$351$$ 0 0
$$352$$ −6589.01 −0.997714
$$353$$ −8875.63 −1.33825 −0.669125 0.743150i $$-0.733331\pi$$
−0.669125 + 0.743150i $$0.733331\pi$$
$$354$$ 0 0
$$355$$ 3684.47 0.550849
$$356$$ −4463.13 −0.664454
$$357$$ 0 0
$$358$$ 7655.04 1.13012
$$359$$ −8946.74 −1.31529 −0.657647 0.753326i $$-0.728448\pi$$
−0.657647 + 0.753326i $$0.728448\pi$$
$$360$$ 0 0
$$361$$ 9486.72 1.38310
$$362$$ 15214.5 2.20900
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 18752.4 2.68916
$$366$$ 0 0
$$367$$ −3965.30 −0.563997 −0.281998 0.959415i $$-0.590997\pi$$
−0.281998 + 0.959415i $$0.590997\pi$$
$$368$$ 6052.10 0.857303
$$369$$ 0 0
$$370$$ 14931.0 2.09790
$$371$$ −2234.99 −0.312762
$$372$$ 0 0
$$373$$ −1457.44 −0.202315 −0.101157 0.994870i $$-0.532255\pi$$
−0.101157 + 0.994870i $$0.532255\pi$$
$$374$$ −3240.71 −0.448057
$$375$$ 0 0
$$376$$ −1335.36 −0.183154
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −1991.75 −0.269945 −0.134973 0.990849i $$-0.543095\pi$$
−0.134973 + 0.990849i $$0.543095\pi$$
$$380$$ 17152.1 2.31549
$$381$$ 0 0
$$382$$ −5577.48 −0.747038
$$383$$ 2504.45 0.334130 0.167065 0.985946i $$-0.446571\pi$$
0.167065 + 0.985946i $$0.446571\pi$$
$$384$$ 0 0
$$385$$ 3334.06 0.441350
$$386$$ 3025.03 0.398886
$$387$$ 0 0
$$388$$ −4860.68 −0.635988
$$389$$ 1448.12 0.188747 0.0943735 0.995537i $$-0.469915\pi$$
0.0943735 + 0.995537i $$0.469915\pi$$
$$390$$ 0 0
$$391$$ 2601.02 0.336417
$$392$$ −1326.06 −0.170857
$$393$$ 0 0
$$394$$ 1906.10 0.243726
$$395$$ 20214.1 2.57489
$$396$$ 0 0
$$397$$ 5634.14 0.712265 0.356133 0.934435i $$-0.384095\pi$$
0.356133 + 0.934435i $$0.384095\pi$$
$$398$$ 11654.8 1.46784
$$399$$ 0 0
$$400$$ −18416.6 −2.30208
$$401$$ 3648.07 0.454304 0.227152 0.973859i $$-0.427058\pi$$
0.227152 + 0.973859i $$0.427058\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −9680.82 −1.19218
$$405$$ 0 0
$$406$$ 6581.15 0.804475
$$407$$ −5400.96 −0.657778
$$408$$ 0 0
$$409$$ 4459.91 0.539190 0.269595 0.962974i $$-0.413110\pi$$
0.269595 + 0.962974i $$0.413110\pi$$
$$410$$ 12080.4 1.45515
$$411$$ 0 0
$$412$$ 10436.6 1.24799
$$413$$ −2946.64 −0.351077
$$414$$ 0 0
$$415$$ 2058.57 0.243497
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −13431.6 −1.57168
$$419$$ 4726.74 0.551114 0.275557 0.961285i $$-0.411138\pi$$
0.275557 + 0.961285i $$0.411138\pi$$
$$420$$ 0 0
$$421$$ 3090.51 0.357772 0.178886 0.983870i $$-0.442751\pi$$
0.178886 + 0.983870i $$0.442751\pi$$
$$422$$ −12281.9 −1.41676
$$423$$ 0 0
$$424$$ −1557.53 −0.178397
$$425$$ −7914.92 −0.903365
$$426$$ 0 0
$$427$$ −1076.31 −0.121982
$$428$$ −13444.2 −1.51835
$$429$$ 0 0
$$430$$ 11971.8 1.34263
$$431$$ 11131.9 1.24409 0.622045 0.782981i $$-0.286302\pi$$
0.622045 + 0.782981i $$0.286302\pi$$
$$432$$ 0 0
$$433$$ 6773.33 0.751745 0.375872 0.926671i $$-0.377343\pi$$
0.375872 + 0.926671i $$0.377343\pi$$
$$434$$ −4016.04 −0.444185
$$435$$ 0 0
$$436$$ −10422.8 −1.14487
$$437$$ 10780.3 1.18007
$$438$$ 0 0
$$439$$ −5384.95 −0.585443 −0.292722 0.956198i $$-0.594561\pi$$
−0.292722 + 0.956198i $$0.594561\pi$$
$$440$$ 2323.45 0.251742
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 921.806 0.0988631 0.0494315 0.998778i $$-0.484259\pi$$
0.0494315 + 0.998778i $$0.484259\pi$$
$$444$$ 0 0
$$445$$ 12694.5 1.35231
$$446$$ 3846.96 0.408428
$$447$$ 0 0
$$448$$ −2244.38 −0.236690
$$449$$ 11902.6 1.25104 0.625521 0.780207i $$-0.284887\pi$$
0.625521 + 0.780207i $$0.284887\pi$$
$$450$$ 0 0
$$451$$ −4369.84 −0.456247
$$452$$ 15111.1 1.57249
$$453$$ 0 0
$$454$$ −25050.0 −2.58955
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1420.66 −0.145418 −0.0727088 0.997353i $$-0.523164\pi$$
−0.0727088 + 0.997353i $$0.523164\pi$$
$$458$$ 2731.64 0.278692
$$459$$ 0 0
$$460$$ 11312.2 1.14659
$$461$$ 1768.23 0.178644 0.0893218 0.996003i $$-0.471530\pi$$
0.0893218 + 0.996003i $$0.471530\pi$$
$$462$$ 0 0
$$463$$ −18637.4 −1.87074 −0.935370 0.353670i $$-0.884933\pi$$
−0.935370 + 0.353670i $$0.884933\pi$$
$$464$$ 19556.0 1.95660
$$465$$ 0 0
$$466$$ −18967.2 −1.88549
$$467$$ −9042.01 −0.895962 −0.447981 0.894043i $$-0.647857\pi$$
−0.447981 + 0.894043i $$0.647857\pi$$
$$468$$ 0 0
$$469$$ −6405.25 −0.630633
$$470$$ −23040.1 −2.26119
$$471$$ 0 0
$$472$$ −2053.47 −0.200251
$$473$$ −4330.55 −0.420970
$$474$$ 0 0
$$475$$ −32804.6 −3.16880
$$476$$ −1327.12 −0.127790
$$477$$ 0 0
$$478$$ 1250.73 0.119680
$$479$$ 9547.64 0.910737 0.455368 0.890303i $$-0.349508\pi$$
0.455368 + 0.890303i $$0.349508\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −7482.08 −0.707053
$$483$$ 0 0
$$484$$ −4042.79 −0.379676
$$485$$ 13825.3 1.29438
$$486$$ 0 0
$$487$$ −3692.81 −0.343608 −0.171804 0.985131i $$-0.554960\pi$$
−0.171804 + 0.985131i $$0.554960\pi$$
$$488$$ −750.062 −0.0695773
$$489$$ 0 0
$$490$$ −22879.6 −2.10937
$$491$$ 3487.77 0.320572 0.160286 0.987071i $$-0.448758\pi$$
0.160286 + 0.987071i $$0.448758\pi$$
$$492$$ 0 0
$$493$$ 8404.59 0.767796
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −11933.7 −1.08032
$$497$$ −1181.55 −0.106640
$$498$$ 0 0
$$499$$ −15089.0 −1.35366 −0.676830 0.736139i $$-0.736647\pi$$
−0.676830 + 0.736139i $$0.736647\pi$$
$$500$$ −17653.3 −1.57896
$$501$$ 0 0
$$502$$ 7551.56 0.671400
$$503$$ −8815.48 −0.781437 −0.390718 0.920510i $$-0.627773\pi$$
−0.390718 + 0.920510i $$0.627773\pi$$
$$504$$ 0 0
$$505$$ 27535.2 2.42634
$$506$$ −8858.41 −0.778270
$$507$$ 0 0
$$508$$ 12445.6 1.08698
$$509$$ 19491.2 1.69731 0.848655 0.528946i $$-0.177413\pi$$
0.848655 + 0.528946i $$0.177413\pi$$
$$510$$ 0 0
$$511$$ −6013.61 −0.520599
$$512$$ −14851.1 −1.28190
$$513$$ 0 0
$$514$$ 16697.3 1.43285
$$515$$ −29684.8 −2.53994
$$516$$ 0 0
$$517$$ 8334.24 0.708974
$$518$$ −4788.14 −0.406137
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6128.73 0.515364 0.257682 0.966230i $$-0.417041\pi$$
0.257682 + 0.966230i $$0.417041\pi$$
$$522$$ 0 0
$$523$$ −22618.6 −1.89109 −0.945546 0.325489i $$-0.894471\pi$$
−0.945546 + 0.325489i $$0.894471\pi$$
$$524$$ −6876.82 −0.573311
$$525$$ 0 0
$$526$$ 11311.6 0.937660
$$527$$ −5128.77 −0.423933
$$528$$ 0 0
$$529$$ −5057.17 −0.415647
$$530$$ −26873.3 −2.20245
$$531$$ 0 0
$$532$$ −5500.43 −0.448259
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 38239.6 3.09017
$$536$$ −4463.71 −0.359707
$$537$$ 0 0
$$538$$ −9478.74 −0.759587
$$539$$ 8276.19 0.661374
$$540$$ 0 0
$$541$$ 15950.5 1.26759 0.633796 0.773500i $$-0.281496\pi$$
0.633796 + 0.773500i $$0.281496\pi$$
$$542$$ 8058.47 0.638636
$$543$$ 0 0
$$544$$ −7459.87 −0.587940
$$545$$ 29645.7 2.33006
$$546$$ 0 0
$$547$$ −1972.83 −0.154208 −0.0771042 0.997023i $$-0.524567\pi$$
−0.0771042 + 0.997023i $$0.524567\pi$$
$$548$$ 8544.33 0.666050
$$549$$ 0 0
$$550$$ 26956.2 2.08985
$$551$$ 34834.1 2.69325
$$552$$ 0 0
$$553$$ −6482.35 −0.498477
$$554$$ 9493.18 0.728027
$$555$$ 0 0
$$556$$ 9263.66 0.706595
$$557$$ −19974.8 −1.51950 −0.759749 0.650217i $$-0.774678\pi$$
−0.759749 + 0.650217i $$0.774678\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 8783.10 0.662775
$$561$$ 0 0
$$562$$ −22271.2 −1.67163
$$563$$ 13153.4 0.984637 0.492319 0.870415i $$-0.336149\pi$$
0.492319 + 0.870415i $$0.336149\pi$$
$$564$$ 0 0
$$565$$ −42980.5 −3.20036
$$566$$ −19639.7 −1.45851
$$567$$ 0 0
$$568$$ −823.405 −0.0608263
$$569$$ −4133.86 −0.304570 −0.152285 0.988337i $$-0.548663\pi$$
−0.152285 + 0.988337i $$0.548663\pi$$
$$570$$ 0 0
$$571$$ −3394.69 −0.248798 −0.124399 0.992232i $$-0.539700\pi$$
−0.124399 + 0.992232i $$0.539700\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −3874.02 −0.281704
$$575$$ −21635.3 −1.56914
$$576$$ 0 0
$$577$$ −4210.56 −0.303792 −0.151896 0.988396i $$-0.548538\pi$$
−0.151896 + 0.988396i $$0.548538\pi$$
$$578$$ 15274.9 1.09923
$$579$$ 0 0
$$580$$ 36552.6 2.61684
$$581$$ −660.154 −0.0471391
$$582$$ 0 0
$$583$$ 9720.82 0.690558
$$584$$ −4190.78 −0.296945
$$585$$ 0 0
$$586$$ −5798.21 −0.408740
$$587$$ −7252.27 −0.509937 −0.254969 0.966949i $$-0.582065\pi$$
−0.254969 + 0.966949i $$0.582065\pi$$
$$588$$ 0 0
$$589$$ −21257.0 −1.48706
$$590$$ −35430.1 −2.47226
$$591$$ 0 0
$$592$$ −14228.0 −0.987784
$$593$$ −13899.4 −0.962531 −0.481266 0.876575i $$-0.659823\pi$$
−0.481266 + 0.876575i $$0.659823\pi$$
$$594$$ 0 0
$$595$$ 3774.72 0.260082
$$596$$ −7153.75 −0.491659
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −18140.7 −1.23741 −0.618704 0.785624i $$-0.712342\pi$$
−0.618704 + 0.785624i $$0.712342\pi$$
$$600$$ 0 0
$$601$$ −26808.3 −1.81952 −0.909761 0.415133i $$-0.863735\pi$$
−0.909761 + 0.415133i $$0.863735\pi$$
$$602$$ −3839.19 −0.259923
$$603$$ 0 0
$$604$$ −9353.27 −0.630098
$$605$$ 11498.9 0.772724
$$606$$ 0 0
$$607$$ 3769.98 0.252091 0.126045 0.992024i $$-0.459772\pi$$
0.126045 + 0.992024i $$0.459772\pi$$
$$608$$ −30918.6 −2.06236
$$609$$ 0 0
$$610$$ −12941.4 −0.858989
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −2722.99 −0.179413 −0.0897067 0.995968i $$-0.528593\pi$$
−0.0897067 + 0.995968i $$0.528593\pi$$
$$614$$ −25323.9 −1.66448
$$615$$ 0 0
$$616$$ −745.097 −0.0487351
$$617$$ 11947.1 0.779533 0.389767 0.920914i $$-0.372556\pi$$
0.389767 + 0.920914i $$0.372556\pi$$
$$618$$ 0 0
$$619$$ 18386.8 1.19391 0.596953 0.802276i $$-0.296378\pi$$
0.596953 + 0.802276i $$0.296378\pi$$
$$620$$ −22305.7 −1.44487
$$621$$ 0 0
$$622$$ −22035.1 −1.42046
$$623$$ −4070.95 −0.261796
$$624$$ 0 0
$$625$$ 18138.1 1.16084
$$626$$ 19100.5 1.21950
$$627$$ 0 0
$$628$$ 596.800 0.0379218
$$629$$ −6114.79 −0.387620
$$630$$ 0 0
$$631$$ 25308.3 1.59669 0.798343 0.602202i $$-0.205710\pi$$
0.798343 + 0.602202i $$0.205710\pi$$
$$632$$ −4517.44 −0.284326
$$633$$ 0 0
$$634$$ −9742.42 −0.610285
$$635$$ −35399.1 −2.21224
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −28623.9 −1.77623
$$639$$ 0 0
$$640$$ 10806.1 0.667422
$$641$$ 9670.76 0.595901 0.297950 0.954581i $$-0.403697\pi$$
0.297950 + 0.954581i $$0.403697\pi$$
$$642$$ 0 0
$$643$$ −19673.0 −1.20657 −0.603286 0.797525i $$-0.706142\pi$$
−0.603286 + 0.797525i $$0.706142\pi$$
$$644$$ −3627.64 −0.221971
$$645$$ 0 0
$$646$$ −15206.9 −0.926170
$$647$$ 13369.3 0.812367 0.406183 0.913792i $$-0.366859\pi$$
0.406183 + 0.913792i $$0.366859\pi$$
$$648$$ 0 0
$$649$$ 12816.1 0.775154
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 195.998 0.0117728
$$653$$ 3088.52 0.185089 0.0925446 0.995709i $$-0.470500\pi$$
0.0925446 + 0.995709i $$0.470500\pi$$
$$654$$ 0 0
$$655$$ 19559.8 1.16682
$$656$$ −11511.7 −0.685146
$$657$$ 0 0
$$658$$ 7388.61 0.437748
$$659$$ −31326.8 −1.85177 −0.925886 0.377802i $$-0.876680\pi$$
−0.925886 + 0.377802i $$0.876680\pi$$
$$660$$ 0 0
$$661$$ −146.828 −0.00863986 −0.00431993 0.999991i $$-0.501375\pi$$
−0.00431993 + 0.999991i $$0.501375\pi$$
$$662$$ 28672.6 1.68337
$$663$$ 0 0
$$664$$ −460.050 −0.0268877
$$665$$ 15644.9 0.912307
$$666$$ 0 0
$$667$$ 22973.8 1.33365
$$668$$ 8858.61 0.513098
$$669$$ 0 0
$$670$$ −77016.0 −4.44087
$$671$$ 4681.28 0.269328
$$672$$ 0 0
$$673$$ −3463.75 −0.198392 −0.0991961 0.995068i $$-0.531627\pi$$
−0.0991961 + 0.995068i $$0.531627\pi$$
$$674$$ −36507.4 −2.08637
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 534.786 0.0303597 0.0151798 0.999885i $$-0.495168\pi$$
0.0151798 + 0.999885i $$0.495168\pi$$
$$678$$ 0 0
$$679$$ −4433.56 −0.250581
$$680$$ 2630.54 0.148348
$$681$$ 0 0
$$682$$ 17467.3 0.980729
$$683$$ −22369.0 −1.25318 −0.626592 0.779348i $$-0.715551\pi$$
−0.626592 + 0.779348i $$0.715551\pi$$
$$684$$ 0 0
$$685$$ −24302.7 −1.35556
$$686$$ 15622.2 0.869470
$$687$$ 0 0
$$688$$ −11408.2 −0.632171
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −25007.2 −1.37673 −0.688363 0.725366i $$-0.741671\pi$$
−0.688363 + 0.725366i $$0.741671\pi$$
$$692$$ 4144.66 0.227683
$$693$$ 0 0
$$694$$ −10776.6 −0.589445
$$695$$ −26348.7 −1.43808
$$696$$ 0 0
$$697$$ −4947.39 −0.268861
$$698$$ 10036.5 0.544251
$$699$$ 0 0
$$700$$ 11038.9 0.596047
$$701$$ 27977.1 1.50739 0.753696 0.657223i $$-0.228269\pi$$
0.753696 + 0.657223i $$0.228269\pi$$
$$702$$ 0 0
$$703$$ −25343.7 −1.35968
$$704$$ 9761.67 0.522595
$$705$$ 0 0
$$706$$ 34223.4 1.82438
$$707$$ −8830.14 −0.469720
$$708$$ 0 0
$$709$$ −6374.31 −0.337648 −0.168824 0.985646i $$-0.553997\pi$$
−0.168824 + 0.985646i $$0.553997\pi$$
$$710$$ −14206.9 −0.750950
$$711$$ 0 0
$$712$$ −2836.97 −0.149326
$$713$$ −14019.4 −0.736367
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −13634.6 −0.711662
$$717$$ 0 0
$$718$$ 34497.6 1.79309
$$719$$ 25433.3 1.31919 0.659597 0.751619i $$-0.270727\pi$$
0.659597 + 0.751619i $$0.270727\pi$$
$$720$$ 0 0
$$721$$ 9519.48 0.491711
$$722$$ −36579.7 −1.88553
$$723$$ 0 0
$$724$$ −27099.0 −1.39106
$$725$$ −69909.4 −3.58120
$$726$$ 0 0
$$727$$ −15847.8 −0.808476 −0.404238 0.914654i $$-0.632463\pi$$
−0.404238 + 0.914654i $$0.632463\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −72306.9 −3.66603
$$731$$ −4902.91 −0.248072
$$732$$ 0 0
$$733$$ −9378.34 −0.472574 −0.236287 0.971683i $$-0.575931\pi$$
−0.236287 + 0.971683i $$0.575931\pi$$
$$734$$ 15289.7 0.768874
$$735$$ 0 0
$$736$$ −20391.4 −1.02125
$$737$$ 27858.9 1.39239
$$738$$ 0 0
$$739$$ −12956.2 −0.644928 −0.322464 0.946582i $$-0.604511\pi$$
−0.322464 + 0.946582i $$0.604511\pi$$
$$740$$ −26594.0 −1.32110
$$741$$ 0 0
$$742$$ 8617.85 0.426376
$$743$$ 16776.6 0.828363 0.414181 0.910194i $$-0.364068\pi$$
0.414181 + 0.910194i $$0.364068\pi$$
$$744$$ 0 0
$$745$$ 20347.5 1.00064
$$746$$ 5619.72 0.275808
$$747$$ 0 0
$$748$$ 5772.13 0.282152
$$749$$ −12262.9 −0.598231
$$750$$ 0 0
$$751$$ 11213.4 0.544849 0.272425 0.962177i $$-0.412174\pi$$
0.272425 + 0.962177i $$0.412174\pi$$
$$752$$ 21955.3 1.06467
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 26603.6 1.28239
$$756$$ 0 0
$$757$$ 852.253 0.0409190 0.0204595 0.999791i $$-0.493487\pi$$
0.0204595 + 0.999791i $$0.493487\pi$$
$$758$$ 7679.95 0.368006
$$759$$ 0 0
$$760$$ 10902.7 0.520371
$$761$$ 35706.1 1.70085 0.850425 0.526097i $$-0.176345\pi$$
0.850425 + 0.526097i $$0.176345\pi$$
$$762$$ 0 0
$$763$$ −9506.95 −0.451081
$$764$$ 9934.21 0.470428
$$765$$ 0 0
$$766$$ −9656.88 −0.455506
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 3663.63 0.171800 0.0858999 0.996304i $$-0.472623\pi$$
0.0858999 + 0.996304i $$0.472623\pi$$
$$770$$ −12855.8 −0.601675
$$771$$ 0 0
$$772$$ −5387.97 −0.251188
$$773$$ 17985.6 0.836865 0.418432 0.908248i $$-0.362580\pi$$
0.418432 + 0.908248i $$0.362580\pi$$
$$774$$ 0 0
$$775$$ 42661.1 1.97733
$$776$$ −3089.67 −0.142929
$$777$$ 0 0
$$778$$ −5583.78 −0.257311
$$779$$ −20505.2 −0.943101
$$780$$ 0 0
$$781$$ 5139.03 0.235453
$$782$$ −10029.2 −0.458624
$$783$$ 0 0
$$784$$ 21802.4 0.993185
$$785$$ −1697.48 −0.0771793
$$786$$ 0 0
$$787$$ 19322.0 0.875167 0.437584 0.899178i $$-0.355834\pi$$
0.437584 + 0.899178i $$0.355834\pi$$
$$788$$ −3395.02 −0.153480
$$789$$ 0 0
$$790$$ −77943.1 −3.51024
$$791$$ 13783.2 0.619563
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −21724.6 −0.971003
$$795$$ 0 0
$$796$$ −20758.7 −0.924336
$$797$$ 10689.5 0.475082 0.237541 0.971378i $$-0.423659\pi$$
0.237541 + 0.971378i $$0.423659\pi$$
$$798$$ 0 0
$$799$$ 9435.77 0.417789
$$800$$ 62051.2 2.74230
$$801$$ 0 0
$$802$$ −14066.5 −0.619335
$$803$$ 26155.5 1.14945
$$804$$ 0 0
$$805$$ 10318.1 0.451759
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −6153.58 −0.267923
$$809$$ −16756.4 −0.728213 −0.364107 0.931357i $$-0.618626\pi$$
−0.364107 + 0.931357i $$0.618626\pi$$
$$810$$ 0 0
$$811$$ 33333.7 1.44329 0.721643 0.692266i $$-0.243387\pi$$
0.721643 + 0.692266i $$0.243387\pi$$
$$812$$ −11721.9 −0.506598
$$813$$ 0 0
$$814$$ 20825.5 0.896722
$$815$$ −557.479 −0.0239603
$$816$$ 0 0
$$817$$ −20320.9 −0.870180
$$818$$ −17196.9 −0.735056
$$819$$ 0 0
$$820$$ −21516.8 −0.916342
$$821$$ 21242.1 0.902988 0.451494 0.892274i $$-0.350891\pi$$
0.451494 + 0.892274i $$0.350891\pi$$
$$822$$ 0 0
$$823$$ 3994.06 0.169167 0.0845834 0.996416i $$-0.473044\pi$$
0.0845834 + 0.996416i $$0.473044\pi$$
$$824$$ 6633.96 0.280467
$$825$$ 0 0
$$826$$ 11361.9 0.478610
$$827$$ −16952.5 −0.712814 −0.356407 0.934331i $$-0.615998\pi$$
−0.356407 + 0.934331i $$0.615998\pi$$
$$828$$ 0 0
$$829$$ 38917.5 1.63047 0.815237 0.579128i $$-0.196607\pi$$
0.815237 + 0.579128i $$0.196607\pi$$
$$830$$ −7937.62 −0.331950
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 9370.04 0.389739
$$834$$ 0 0
$$835$$ −25196.6 −1.04427
$$836$$ 23923.5 0.989726
$$837$$ 0 0
$$838$$ −18225.8 −0.751311
$$839$$ −25671.3 −1.05634 −0.528172 0.849138i $$-0.677122\pi$$
−0.528172 + 0.849138i $$0.677122\pi$$
$$840$$ 0 0
$$841$$ 49845.4 2.04377
$$842$$ −11916.6 −0.487737
$$843$$ 0 0
$$844$$ 21875.7 0.892170
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −3687.54 −0.149593
$$848$$ 25608.1 1.03701
$$849$$ 0 0
$$850$$ 30519.0 1.23152
$$851$$ −16714.7 −0.673292
$$852$$ 0 0
$$853$$ 37444.6 1.50302 0.751511 0.659720i $$-0.229325\pi$$
0.751511 + 0.659720i $$0.229325\pi$$
$$854$$ 4150.12 0.166293
$$855$$ 0 0
$$856$$ −8545.78 −0.341225
$$857$$ −36610.6 −1.45927 −0.729636 0.683836i $$-0.760310\pi$$
−0.729636 + 0.683836i $$0.760310\pi$$
$$858$$ 0 0
$$859$$ 18043.5 0.716688 0.358344 0.933590i $$-0.383341\pi$$
0.358344 + 0.933590i $$0.383341\pi$$
$$860$$ −21323.4 −0.845490
$$861$$ 0 0
$$862$$ −42923.1 −1.69602
$$863$$ 8697.94 0.343084 0.171542 0.985177i $$-0.445125\pi$$
0.171542 + 0.985177i $$0.445125\pi$$
$$864$$ 0 0
$$865$$ −11788.7 −0.463384
$$866$$ −26117.2 −1.02482
$$867$$ 0 0
$$868$$ 7153.09 0.279714
$$869$$ 28194.2 1.10060
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −6625.23 −0.257292
$$873$$ 0 0
$$874$$ −41567.6 −1.60875
$$875$$ −16102.0 −0.622113
$$876$$ 0 0
$$877$$ −26657.4 −1.02640 −0.513202 0.858268i $$-0.671541\pi$$
−0.513202 + 0.858268i $$0.671541\pi$$
$$878$$ 20763.7 0.798111
$$879$$ 0 0
$$880$$ −38201.1 −1.46336
$$881$$ 13896.5 0.531424 0.265712 0.964053i $$-0.414393\pi$$
0.265712 + 0.964053i $$0.414393\pi$$
$$882$$ 0 0
$$883$$ 24343.3 0.927767 0.463884 0.885896i $$-0.346456\pi$$
0.463884 + 0.885896i $$0.346456\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −3554.38 −0.134776
$$887$$ 4592.33 0.173839 0.0869196 0.996215i $$-0.472298\pi$$
0.0869196 + 0.996215i $$0.472298\pi$$
$$888$$ 0 0
$$889$$ 11352.0 0.428271
$$890$$ −48948.6 −1.84355
$$891$$ 0 0
$$892$$ −6851.94 −0.257197
$$893$$ 39108.0 1.46551
$$894$$ 0 0
$$895$$ 38781.1 1.44839
$$896$$ −3465.37 −0.129207
$$897$$ 0 0
$$898$$ −45895.0 −1.70550
$$899$$ −45300.4 −1.68059
$$900$$ 0 0
$$901$$ 11005.6 0.406937
$$902$$ 16849.6 0.621984
$$903$$ 0 0
$$904$$ 9605.29 0.353393
$$905$$ 77078.0 2.83111
$$906$$ 0 0
$$907$$ 31406.7 1.14977 0.574885 0.818234i $$-0.305047\pi$$
0.574885 + 0.818234i $$0.305047\pi$$
$$908$$ 44617.4 1.63070
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −44825.0 −1.63021 −0.815103 0.579317i $$-0.803319\pi$$
−0.815103 + 0.579317i $$0.803319\pi$$
$$912$$ 0 0
$$913$$ 2871.26 0.104080
$$914$$ 5477.91 0.198242
$$915$$ 0 0
$$916$$ −4865.40 −0.175499
$$917$$ −6272.53 −0.225886
$$918$$ 0 0
$$919$$ 22241.1 0.798332 0.399166 0.916879i $$-0.369300\pi$$
0.399166 + 0.916879i $$0.369300\pi$$
$$920$$ 7190.53 0.257679
$$921$$ 0 0
$$922$$ −6818.09 −0.243538
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 50862.9 1.80796
$$926$$ 71863.6 2.55031
$$927$$ 0 0
$$928$$ −65890.1 −2.33076
$$929$$ −16292.5 −0.575393 −0.287697 0.957722i $$-0.592889\pi$$
−0.287697 + 0.957722i $$0.592889\pi$$
$$930$$ 0 0
$$931$$ 38835.6 1.36712
$$932$$ 33783.0 1.18734
$$933$$ 0 0
$$934$$ 34864.9 1.22143
$$935$$ −16417.7 −0.574242
$$936$$ 0 0
$$937$$ 24400.8 0.850734 0.425367 0.905021i $$-0.360145\pi$$
0.425367 + 0.905021i $$0.360145\pi$$
$$938$$ 24697.9 0.859717
$$939$$ 0 0
$$940$$ 41037.4 1.42393
$$941$$ −41529.9 −1.43872 −0.719360 0.694638i $$-0.755565\pi$$
−0.719360 + 0.694638i $$0.755565\pi$$
$$942$$ 0 0
$$943$$ −13523.6 −0.467008
$$944$$ 33762.1 1.16405
$$945$$ 0 0
$$946$$ 16698.1 0.573892
$$947$$ 29981.8 1.02880 0.514402 0.857549i $$-0.328014\pi$$
0.514402 + 0.857549i $$0.328014\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 126491. 4.31990
$$951$$ 0 0
$$952$$ −843.575 −0.0287189
$$953$$ −28594.4 −0.971943 −0.485972 0.873975i $$-0.661534\pi$$
−0.485972 + 0.873975i $$0.661534\pi$$
$$954$$ 0 0
$$955$$ −28256.0 −0.957426
$$956$$ −2227.72 −0.0753657
$$957$$ 0 0
$$958$$ −36814.6 −1.24157
$$959$$ 7793.52 0.262425
$$960$$ 0 0
$$961$$ −2147.17 −0.0720745
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 13326.6 0.445249
$$965$$ 15325.1 0.511224
$$966$$ 0 0
$$967$$ 28729.1 0.955393 0.477696 0.878525i $$-0.341472\pi$$
0.477696 + 0.878525i $$0.341472\pi$$
$$968$$ −2569.78 −0.0853264
$$969$$ 0 0
$$970$$ −53308.6 −1.76457
$$971$$ −27550.7 −0.910549 −0.455274 0.890351i $$-0.650459\pi$$
−0.455274 + 0.890351i $$0.650459\pi$$
$$972$$ 0 0
$$973$$ 8449.64 0.278400
$$974$$ 14239.0 0.468427
$$975$$ 0 0
$$976$$ 12332.1 0.404449
$$977$$ −23812.0 −0.779746 −0.389873 0.920869i $$-0.627481\pi$$
−0.389873 + 0.920869i $$0.627481\pi$$
$$978$$ 0 0
$$979$$ 17706.1 0.578028
$$980$$ 40751.5 1.32833
$$981$$ 0 0
$$982$$ −13448.4 −0.437023
$$983$$ −32668.3 −1.05998 −0.529988 0.848005i $$-0.677804\pi$$
−0.529988 + 0.848005i $$0.677804\pi$$
$$984$$ 0 0
$$985$$ 9656.48 0.312367
$$986$$ −32407.1 −1.04671
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −13402.0 −0.430899
$$990$$ 0 0
$$991$$ −19654.5 −0.630015 −0.315008 0.949089i $$-0.602007\pi$$
−0.315008 + 0.949089i $$0.602007\pi$$
$$992$$ 40208.4 1.28691
$$993$$ 0 0
$$994$$ 4555.94 0.145378
$$995$$ 59044.0 1.88123
$$996$$ 0 0
$$997$$ 17666.5 0.561188 0.280594 0.959827i $$-0.409469\pi$$
0.280594 + 0.959827i $$0.409469\pi$$
$$998$$ 58181.4 1.84539
$$999$$ 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.ba.1.2 4
3.2 odd 2 inner 1521.4.a.ba.1.3 4
13.12 even 2 117.4.a.g.1.3 yes 4
39.38 odd 2 117.4.a.g.1.2 4
52.51 odd 2 1872.4.a.bo.1.4 4
156.155 even 2 1872.4.a.bo.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.a.g.1.2 4 39.38 odd 2
117.4.a.g.1.3 yes 4 13.12 even 2
1521.4.a.ba.1.2 4 1.1 even 1 trivial
1521.4.a.ba.1.3 4 3.2 odd 2 inner
1872.4.a.bo.1.1 4 156.155 even 2
1872.4.a.bo.1.4 4 52.51 odd 2