# Properties

 Label 63.4.a.a.1.1 Level $63$ Weight $4$ Character 63.1 Self dual yes Analytic conductor $3.717$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.71712033036$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 63.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} +8.00000 q^{4} +4.00000 q^{5} -7.00000 q^{7} +O(q^{10})$$ $$q-4.00000 q^{2} +8.00000 q^{4} +4.00000 q^{5} -7.00000 q^{7} -16.0000 q^{10} -62.0000 q^{11} -62.0000 q^{13} +28.0000 q^{14} -64.0000 q^{16} -84.0000 q^{17} +100.000 q^{19} +32.0000 q^{20} +248.000 q^{22} +42.0000 q^{23} -109.000 q^{25} +248.000 q^{26} -56.0000 q^{28} +10.0000 q^{29} -48.0000 q^{31} +256.000 q^{32} +336.000 q^{34} -28.0000 q^{35} -246.000 q^{37} -400.000 q^{38} +248.000 q^{41} +68.0000 q^{43} -496.000 q^{44} -168.000 q^{46} -324.000 q^{47} +49.0000 q^{49} +436.000 q^{50} -496.000 q^{52} -258.000 q^{53} -248.000 q^{55} -40.0000 q^{58} -120.000 q^{59} +622.000 q^{61} +192.000 q^{62} -512.000 q^{64} -248.000 q^{65} +904.000 q^{67} -672.000 q^{68} +112.000 q^{70} +678.000 q^{71} -642.000 q^{73} +984.000 q^{74} +800.000 q^{76} +434.000 q^{77} +740.000 q^{79} -256.000 q^{80} -992.000 q^{82} -468.000 q^{83} -336.000 q^{85} -272.000 q^{86} -200.000 q^{89} +434.000 q^{91} +336.000 q^{92} +1296.00 q^{94} +400.000 q^{95} -1266.00 q^{97} -196.000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 0 0
$$4$$ 8.00000 1.00000
$$5$$ 4.00000 0.357771 0.178885 0.983870i $$-0.442751\pi$$
0.178885 + 0.983870i $$0.442751\pi$$
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ −16.0000 −0.505964
$$11$$ −62.0000 −1.69943 −0.849714 0.527244i $$-0.823225\pi$$
−0.849714 + 0.527244i $$0.823225\pi$$
$$12$$ 0 0
$$13$$ −62.0000 −1.32275 −0.661373 0.750057i $$-0.730026\pi$$
−0.661373 + 0.750057i $$0.730026\pi$$
$$14$$ 28.0000 0.534522
$$15$$ 0 0
$$16$$ −64.0000 −1.00000
$$17$$ −84.0000 −1.19841 −0.599206 0.800595i $$-0.704517\pi$$
−0.599206 + 0.800595i $$0.704517\pi$$
$$18$$ 0 0
$$19$$ 100.000 1.20745 0.603726 0.797192i $$-0.293682\pi$$
0.603726 + 0.797192i $$0.293682\pi$$
$$20$$ 32.0000 0.357771
$$21$$ 0 0
$$22$$ 248.000 2.40335
$$23$$ 42.0000 0.380765 0.190383 0.981710i $$-0.439027\pi$$
0.190383 + 0.981710i $$0.439027\pi$$
$$24$$ 0 0
$$25$$ −109.000 −0.872000
$$26$$ 248.000 1.87065
$$27$$ 0 0
$$28$$ −56.0000 −0.377964
$$29$$ 10.0000 0.0640329 0.0320164 0.999487i $$-0.489807\pi$$
0.0320164 + 0.999487i $$0.489807\pi$$
$$30$$ 0 0
$$31$$ −48.0000 −0.278099 −0.139049 0.990285i $$-0.544405\pi$$
−0.139049 + 0.990285i $$0.544405\pi$$
$$32$$ 256.000 1.41421
$$33$$ 0 0
$$34$$ 336.000 1.69481
$$35$$ −28.0000 −0.135225
$$36$$ 0 0
$$37$$ −246.000 −1.09303 −0.546516 0.837449i $$-0.684046\pi$$
−0.546516 + 0.837449i $$0.684046\pi$$
$$38$$ −400.000 −1.70759
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 248.000 0.944661 0.472330 0.881422i $$-0.343413\pi$$
0.472330 + 0.881422i $$0.343413\pi$$
$$42$$ 0 0
$$43$$ 68.0000 0.241161 0.120580 0.992704i $$-0.461524\pi$$
0.120580 + 0.992704i $$0.461524\pi$$
$$44$$ −496.000 −1.69943
$$45$$ 0 0
$$46$$ −168.000 −0.538484
$$47$$ −324.000 −1.00554 −0.502769 0.864421i $$-0.667685\pi$$
−0.502769 + 0.864421i $$0.667685\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 436.000 1.23319
$$51$$ 0 0
$$52$$ −496.000 −1.32275
$$53$$ −258.000 −0.668661 −0.334330 0.942456i $$-0.608510\pi$$
−0.334330 + 0.942456i $$0.608510\pi$$
$$54$$ 0 0
$$55$$ −248.000 −0.608006
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −40.0000 −0.0905562
$$59$$ −120.000 −0.264791 −0.132396 0.991197i $$-0.542267\pi$$
−0.132396 + 0.991197i $$0.542267\pi$$
$$60$$ 0 0
$$61$$ 622.000 1.30556 0.652778 0.757549i $$-0.273603\pi$$
0.652778 + 0.757549i $$0.273603\pi$$
$$62$$ 192.000 0.393291
$$63$$ 0 0
$$64$$ −512.000 −1.00000
$$65$$ −248.000 −0.473240
$$66$$ 0 0
$$67$$ 904.000 1.64838 0.824188 0.566316i $$-0.191632\pi$$
0.824188 + 0.566316i $$0.191632\pi$$
$$68$$ −672.000 −1.19841
$$69$$ 0 0
$$70$$ 112.000 0.191237
$$71$$ 678.000 1.13329 0.566646 0.823961i $$-0.308241\pi$$
0.566646 + 0.823961i $$0.308241\pi$$
$$72$$ 0 0
$$73$$ −642.000 −1.02932 −0.514660 0.857394i $$-0.672082\pi$$
−0.514660 + 0.857394i $$0.672082\pi$$
$$74$$ 984.000 1.54578
$$75$$ 0 0
$$76$$ 800.000 1.20745
$$77$$ 434.000 0.642323
$$78$$ 0 0
$$79$$ 740.000 1.05388 0.526940 0.849903i $$-0.323339\pi$$
0.526940 + 0.849903i $$0.323339\pi$$
$$80$$ −256.000 −0.357771
$$81$$ 0 0
$$82$$ −992.000 −1.33595
$$83$$ −468.000 −0.618912 −0.309456 0.950914i $$-0.600147\pi$$
−0.309456 + 0.950914i $$0.600147\pi$$
$$84$$ 0 0
$$85$$ −336.000 −0.428757
$$86$$ −272.000 −0.341052
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −200.000 −0.238202 −0.119101 0.992882i $$-0.538001\pi$$
−0.119101 + 0.992882i $$0.538001\pi$$
$$90$$ 0 0
$$91$$ 434.000 0.499951
$$92$$ 336.000 0.380765
$$93$$ 0 0
$$94$$ 1296.00 1.42204
$$95$$ 400.000 0.431991
$$96$$ 0 0
$$97$$ −1266.00 −1.32518 −0.662592 0.748981i $$-0.730544\pi$$
−0.662592 + 0.748981i $$0.730544\pi$$
$$98$$ −196.000 −0.202031
$$99$$ 0 0
$$100$$ −872.000 −0.872000
$$101$$ −232.000 −0.228563 −0.114281 0.993448i $$-0.536457\pi$$
−0.114281 + 0.993448i $$0.536457\pi$$
$$102$$ 0 0
$$103$$ −1792.00 −1.71428 −0.857141 0.515082i $$-0.827761\pi$$
−0.857141 + 0.515082i $$0.827761\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 1032.00 0.945629
$$107$$ 1906.00 1.72206 0.861028 0.508558i $$-0.169821\pi$$
0.861028 + 0.508558i $$0.169821\pi$$
$$108$$ 0 0
$$109$$ −90.0000 −0.0790866 −0.0395433 0.999218i $$-0.512590\pi$$
−0.0395433 + 0.999218i $$0.512590\pi$$
$$110$$ 992.000 0.859850
$$111$$ 0 0
$$112$$ 448.000 0.377964
$$113$$ −458.000 −0.381283 −0.190642 0.981660i $$-0.561057\pi$$
−0.190642 + 0.981660i $$0.561057\pi$$
$$114$$ 0 0
$$115$$ 168.000 0.136227
$$116$$ 80.0000 0.0640329
$$117$$ 0 0
$$118$$ 480.000 0.374471
$$119$$ 588.000 0.452957
$$120$$ 0 0
$$121$$ 2513.00 1.88805
$$122$$ −2488.00 −1.84634
$$123$$ 0 0
$$124$$ −384.000 −0.278099
$$125$$ −936.000 −0.669747
$$126$$ 0 0
$$127$$ 804.000 0.561760 0.280880 0.959743i $$-0.409374\pi$$
0.280880 + 0.959743i $$0.409374\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 992.000 0.669263
$$131$$ −812.000 −0.541563 −0.270782 0.962641i $$-0.587282\pi$$
−0.270782 + 0.962641i $$0.587282\pi$$
$$132$$ 0 0
$$133$$ −700.000 −0.456374
$$134$$ −3616.00 −2.33116
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −414.000 −0.258178 −0.129089 0.991633i $$-0.541205\pi$$
−0.129089 + 0.991633i $$0.541205\pi$$
$$138$$ 0 0
$$139$$ −1620.00 −0.988537 −0.494268 0.869309i $$-0.664564\pi$$
−0.494268 + 0.869309i $$0.664564\pi$$
$$140$$ −224.000 −0.135225
$$141$$ 0 0
$$142$$ −2712.00 −1.60272
$$143$$ 3844.00 2.24791
$$144$$ 0 0
$$145$$ 40.0000 0.0229091
$$146$$ 2568.00 1.45568
$$147$$ 0 0
$$148$$ −1968.00 −1.09303
$$149$$ −2370.00 −1.30307 −0.651537 0.758617i $$-0.725875\pi$$
−0.651537 + 0.758617i $$0.725875\pi$$
$$150$$ 0 0
$$151$$ −568.000 −0.306114 −0.153057 0.988217i $$-0.548912\pi$$
−0.153057 + 0.988217i $$0.548912\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −1736.00 −0.908382
$$155$$ −192.000 −0.0994956
$$156$$ 0 0
$$157$$ −266.000 −0.135217 −0.0676086 0.997712i $$-0.521537\pi$$
−0.0676086 + 0.997712i $$0.521537\pi$$
$$158$$ −2960.00 −1.49041
$$159$$ 0 0
$$160$$ 1024.00 0.505964
$$161$$ −294.000 −0.143916
$$162$$ 0 0
$$163$$ −272.000 −0.130704 −0.0653518 0.997862i $$-0.520817\pi$$
−0.0653518 + 0.997862i $$0.520817\pi$$
$$164$$ 1984.00 0.944661
$$165$$ 0 0
$$166$$ 1872.00 0.875273
$$167$$ 1876.00 0.869277 0.434638 0.900605i $$-0.356876\pi$$
0.434638 + 0.900605i $$0.356876\pi$$
$$168$$ 0 0
$$169$$ 1647.00 0.749659
$$170$$ 1344.00 0.606353
$$171$$ 0 0
$$172$$ 544.000 0.241161
$$173$$ 152.000 0.0667997 0.0333998 0.999442i $$-0.489367\pi$$
0.0333998 + 0.999442i $$0.489367\pi$$
$$174$$ 0 0
$$175$$ 763.000 0.329585
$$176$$ 3968.00 1.69943
$$177$$ 0 0
$$178$$ 800.000 0.336868
$$179$$ −610.000 −0.254713 −0.127356 0.991857i $$-0.540649\pi$$
−0.127356 + 0.991857i $$0.540649\pi$$
$$180$$ 0 0
$$181$$ 1042.00 0.427907 0.213954 0.976844i $$-0.431366\pi$$
0.213954 + 0.976844i $$0.431366\pi$$
$$182$$ −1736.00 −0.707038
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −984.000 −0.391055
$$186$$ 0 0
$$187$$ 5208.00 2.03661
$$188$$ −2592.00 −1.00554
$$189$$ 0 0
$$190$$ −1600.00 −0.610927
$$191$$ 2038.00 0.772065 0.386033 0.922485i $$-0.373845\pi$$
0.386033 + 0.922485i $$0.373845\pi$$
$$192$$ 0 0
$$193$$ −2602.00 −0.970446 −0.485223 0.874390i $$-0.661262\pi$$
−0.485223 + 0.874390i $$0.661262\pi$$
$$194$$ 5064.00 1.87409
$$195$$ 0 0
$$196$$ 392.000 0.142857
$$197$$ −2354.00 −0.851348 −0.425674 0.904877i $$-0.639963\pi$$
−0.425674 + 0.904877i $$0.639963\pi$$
$$198$$ 0 0
$$199$$ 1680.00 0.598452 0.299226 0.954182i $$-0.403271\pi$$
0.299226 + 0.954182i $$0.403271\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 928.000 0.323237
$$203$$ −70.0000 −0.0242022
$$204$$ 0 0
$$205$$ 992.000 0.337972
$$206$$ 7168.00 2.42436
$$207$$ 0 0
$$208$$ 3968.00 1.32275
$$209$$ −6200.00 −2.05198
$$210$$ 0 0
$$211$$ −668.000 −0.217948 −0.108974 0.994045i $$-0.534757\pi$$
−0.108974 + 0.994045i $$0.534757\pi$$
$$212$$ −2064.00 −0.668661
$$213$$ 0 0
$$214$$ −7624.00 −2.43535
$$215$$ 272.000 0.0862802
$$216$$ 0 0
$$217$$ 336.000 0.105111
$$218$$ 360.000 0.111845
$$219$$ 0 0
$$220$$ −1984.00 −0.608006
$$221$$ 5208.00 1.58519
$$222$$ 0 0
$$223$$ −1832.00 −0.550134 −0.275067 0.961425i $$-0.588700\pi$$
−0.275067 + 0.961425i $$0.588700\pi$$
$$224$$ −1792.00 −0.534522
$$225$$ 0 0
$$226$$ 1832.00 0.539216
$$227$$ −4944.00 −1.44557 −0.722786 0.691072i $$-0.757139\pi$$
−0.722786 + 0.691072i $$0.757139\pi$$
$$228$$ 0 0
$$229$$ −5470.00 −1.57846 −0.789231 0.614096i $$-0.789521\pi$$
−0.789231 + 0.614096i $$0.789521\pi$$
$$230$$ −672.000 −0.192654
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2802.00 0.787833 0.393917 0.919146i $$-0.371120\pi$$
0.393917 + 0.919146i $$0.371120\pi$$
$$234$$ 0 0
$$235$$ −1296.00 −0.359752
$$236$$ −960.000 −0.264791
$$237$$ 0 0
$$238$$ −2352.00 −0.640578
$$239$$ 1170.00 0.316657 0.158328 0.987386i $$-0.449390\pi$$
0.158328 + 0.987386i $$0.449390\pi$$
$$240$$ 0 0
$$241$$ −2338.00 −0.624912 −0.312456 0.949932i $$-0.601152\pi$$
−0.312456 + 0.949932i $$0.601152\pi$$
$$242$$ −10052.0 −2.67011
$$243$$ 0 0
$$244$$ 4976.00 1.30556
$$245$$ 196.000 0.0511101
$$246$$ 0 0
$$247$$ −6200.00 −1.59715
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 3744.00 0.947165
$$251$$ −2792.00 −0.702109 −0.351055 0.936355i $$-0.614177\pi$$
−0.351055 + 0.936355i $$0.614177\pi$$
$$252$$ 0 0
$$253$$ −2604.00 −0.647083
$$254$$ −3216.00 −0.794448
$$255$$ 0 0
$$256$$ 4096.00 1.00000
$$257$$ −7024.00 −1.70484 −0.852422 0.522854i $$-0.824867\pi$$
−0.852422 + 0.522854i $$0.824867\pi$$
$$258$$ 0 0
$$259$$ 1722.00 0.413127
$$260$$ −1984.00 −0.473240
$$261$$ 0 0
$$262$$ 3248.00 0.765886
$$263$$ −2438.00 −0.571610 −0.285805 0.958288i $$-0.592261\pi$$
−0.285805 + 0.958288i $$0.592261\pi$$
$$264$$ 0 0
$$265$$ −1032.00 −0.239227
$$266$$ 2800.00 0.645410
$$267$$ 0 0
$$268$$ 7232.00 1.64838
$$269$$ 6780.00 1.53674 0.768372 0.640004i $$-0.221067\pi$$
0.768372 + 0.640004i $$0.221067\pi$$
$$270$$ 0 0
$$271$$ −1928.00 −0.432168 −0.216084 0.976375i $$-0.569329\pi$$
−0.216084 + 0.976375i $$0.569329\pi$$
$$272$$ 5376.00 1.19841
$$273$$ 0 0
$$274$$ 1656.00 0.365119
$$275$$ 6758.00 1.48190
$$276$$ 0 0
$$277$$ 5554.00 1.20472 0.602360 0.798224i $$-0.294227\pi$$
0.602360 + 0.798224i $$0.294227\pi$$
$$278$$ 6480.00 1.39800
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1942.00 −0.412278 −0.206139 0.978523i $$-0.566090\pi$$
−0.206139 + 0.978523i $$0.566090\pi$$
$$282$$ 0 0
$$283$$ 4828.00 1.01412 0.507058 0.861912i $$-0.330733\pi$$
0.507058 + 0.861912i $$0.330733\pi$$
$$284$$ 5424.00 1.13329
$$285$$ 0 0
$$286$$ −15376.0 −3.17903
$$287$$ −1736.00 −0.357048
$$288$$ 0 0
$$289$$ 2143.00 0.436190
$$290$$ −160.000 −0.0323984
$$291$$ 0 0
$$292$$ −5136.00 −1.02932
$$293$$ 6152.00 1.22663 0.613317 0.789837i $$-0.289835\pi$$
0.613317 + 0.789837i $$0.289835\pi$$
$$294$$ 0 0
$$295$$ −480.000 −0.0947345
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 9480.00 1.84282
$$299$$ −2604.00 −0.503656
$$300$$ 0 0
$$301$$ −476.000 −0.0911501
$$302$$ 2272.00 0.432910
$$303$$ 0 0
$$304$$ −6400.00 −1.20745
$$305$$ 2488.00 0.467090
$$306$$ 0 0
$$307$$ 5884.00 1.09387 0.546934 0.837176i $$-0.315795\pi$$
0.546934 + 0.837176i $$0.315795\pi$$
$$308$$ 3472.00 0.642323
$$309$$ 0 0
$$310$$ 768.000 0.140708
$$311$$ −9132.00 −1.66504 −0.832521 0.553993i $$-0.813103\pi$$
−0.832521 + 0.553993i $$0.813103\pi$$
$$312$$ 0 0
$$313$$ −9382.00 −1.69426 −0.847128 0.531389i $$-0.821670\pi$$
−0.847128 + 0.531389i $$0.821670\pi$$
$$314$$ 1064.00 0.191226
$$315$$ 0 0
$$316$$ 5920.00 1.05388
$$317$$ −3114.00 −0.551734 −0.275867 0.961196i $$-0.588965\pi$$
−0.275867 + 0.961196i $$0.588965\pi$$
$$318$$ 0 0
$$319$$ −620.000 −0.108819
$$320$$ −2048.00 −0.357771
$$321$$ 0 0
$$322$$ 1176.00 0.203528
$$323$$ −8400.00 −1.44702
$$324$$ 0 0
$$325$$ 6758.00 1.15344
$$326$$ 1088.00 0.184843
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 2268.00 0.380057
$$330$$ 0 0
$$331$$ 1532.00 0.254400 0.127200 0.991877i $$-0.459401\pi$$
0.127200 + 0.991877i $$0.459401\pi$$
$$332$$ −3744.00 −0.618912
$$333$$ 0 0
$$334$$ −7504.00 −1.22934
$$335$$ 3616.00 0.589741
$$336$$ 0 0
$$337$$ −4166.00 −0.673402 −0.336701 0.941612i $$-0.609311\pi$$
−0.336701 + 0.941612i $$0.609311\pi$$
$$338$$ −6588.00 −1.06018
$$339$$ 0 0
$$340$$ −2688.00 −0.428757
$$341$$ 2976.00 0.472608
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −608.000 −0.0944690
$$347$$ 11366.0 1.75838 0.879191 0.476469i $$-0.158083\pi$$
0.879191 + 0.476469i $$0.158083\pi$$
$$348$$ 0 0
$$349$$ 9310.00 1.42795 0.713973 0.700174i $$-0.246894\pi$$
0.713973 + 0.700174i $$0.246894\pi$$
$$350$$ −3052.00 −0.466104
$$351$$ 0 0
$$352$$ −15872.0 −2.40335
$$353$$ 8572.00 1.29247 0.646234 0.763139i $$-0.276343\pi$$
0.646234 + 0.763139i $$0.276343\pi$$
$$354$$ 0 0
$$355$$ 2712.00 0.405459
$$356$$ −1600.00 −0.238202
$$357$$ 0 0
$$358$$ 2440.00 0.360218
$$359$$ 4790.00 0.704196 0.352098 0.935963i $$-0.385468\pi$$
0.352098 + 0.935963i $$0.385468\pi$$
$$360$$ 0 0
$$361$$ 3141.00 0.457938
$$362$$ −4168.00 −0.605153
$$363$$ 0 0
$$364$$ 3472.00 0.499951
$$365$$ −2568.00 −0.368261
$$366$$ 0 0
$$367$$ 5424.00 0.771473 0.385736 0.922609i $$-0.373947\pi$$
0.385736 + 0.922609i $$0.373947\pi$$
$$368$$ −2688.00 −0.380765
$$369$$ 0 0
$$370$$ 3936.00 0.553035
$$371$$ 1806.00 0.252730
$$372$$ 0 0
$$373$$ 1838.00 0.255142 0.127571 0.991829i $$-0.459282\pi$$
0.127571 + 0.991829i $$0.459282\pi$$
$$374$$ −20832.0 −2.88021
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −620.000 −0.0846993
$$378$$ 0 0
$$379$$ −4260.00 −0.577365 −0.288683 0.957425i $$-0.593217\pi$$
−0.288683 + 0.957425i $$0.593217\pi$$
$$380$$ 3200.00 0.431991
$$381$$ 0 0
$$382$$ −8152.00 −1.09187
$$383$$ −9048.00 −1.20713 −0.603566 0.797313i $$-0.706254\pi$$
−0.603566 + 0.797313i $$0.706254\pi$$
$$384$$ 0 0
$$385$$ 1736.00 0.229805
$$386$$ 10408.0 1.37242
$$387$$ 0 0
$$388$$ −10128.0 −1.32518
$$389$$ 11490.0 1.49760 0.748800 0.662796i $$-0.230631\pi$$
0.748800 + 0.662796i $$0.230631\pi$$
$$390$$ 0 0
$$391$$ −3528.00 −0.456314
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 9416.00 1.20399
$$395$$ 2960.00 0.377048
$$396$$ 0 0
$$397$$ −1866.00 −0.235899 −0.117949 0.993020i $$-0.537632\pi$$
−0.117949 + 0.993020i $$0.537632\pi$$
$$398$$ −6720.00 −0.846340
$$399$$ 0 0
$$400$$ 6976.00 0.872000
$$401$$ −13662.0 −1.70137 −0.850683 0.525679i $$-0.823811\pi$$
−0.850683 + 0.525679i $$0.823811\pi$$
$$402$$ 0 0
$$403$$ 2976.00 0.367854
$$404$$ −1856.00 −0.228563
$$405$$ 0 0
$$406$$ 280.000 0.0342270
$$407$$ 15252.0 1.85753
$$408$$ 0 0
$$409$$ −13210.0 −1.59705 −0.798524 0.601963i $$-0.794385\pi$$
−0.798524 + 0.601963i $$0.794385\pi$$
$$410$$ −3968.00 −0.477965
$$411$$ 0 0
$$412$$ −14336.0 −1.71428
$$413$$ 840.000 0.100082
$$414$$ 0 0
$$415$$ −1872.00 −0.221429
$$416$$ −15872.0 −1.87065
$$417$$ 0 0
$$418$$ 24800.0 2.90193
$$419$$ −6960.00 −0.811499 −0.405750 0.913984i $$-0.632990\pi$$
−0.405750 + 0.913984i $$0.632990\pi$$
$$420$$ 0 0
$$421$$ 8162.00 0.944873 0.472437 0.881365i $$-0.343375\pi$$
0.472437 + 0.881365i $$0.343375\pi$$
$$422$$ 2672.00 0.308225
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 9156.00 1.04501
$$426$$ 0 0
$$427$$ −4354.00 −0.493454
$$428$$ 15248.0 1.72206
$$429$$ 0 0
$$430$$ −1088.00 −0.122019
$$431$$ −16602.0 −1.85543 −0.927715 0.373290i $$-0.878230\pi$$
−0.927715 + 0.373290i $$0.878230\pi$$
$$432$$ 0 0
$$433$$ 7738.00 0.858810 0.429405 0.903112i $$-0.358723\pi$$
0.429405 + 0.903112i $$0.358723\pi$$
$$434$$ −1344.00 −0.148650
$$435$$ 0 0
$$436$$ −720.000 −0.0790866
$$437$$ 4200.00 0.459756
$$438$$ 0 0
$$439$$ −840.000 −0.0913235 −0.0456617 0.998957i $$-0.514540\pi$$
−0.0456617 + 0.998957i $$0.514540\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −20832.0 −2.24180
$$443$$ −6618.00 −0.709776 −0.354888 0.934909i $$-0.615481\pi$$
−0.354888 + 0.934909i $$0.615481\pi$$
$$444$$ 0 0
$$445$$ −800.000 −0.0852217
$$446$$ 7328.00 0.778006
$$447$$ 0 0
$$448$$ 3584.00 0.377964
$$449$$ −3090.00 −0.324780 −0.162390 0.986727i $$-0.551920\pi$$
−0.162390 + 0.986727i $$0.551920\pi$$
$$450$$ 0 0
$$451$$ −15376.0 −1.60538
$$452$$ −3664.00 −0.381283
$$453$$ 0 0
$$454$$ 19776.0 2.04435
$$455$$ 1736.00 0.178868
$$456$$ 0 0
$$457$$ 5914.00 0.605351 0.302675 0.953094i $$-0.402120\pi$$
0.302675 + 0.953094i $$0.402120\pi$$
$$458$$ 21880.0 2.23228
$$459$$ 0 0
$$460$$ 1344.00 0.136227
$$461$$ 15968.0 1.61324 0.806620 0.591070i $$-0.201294\pi$$
0.806620 + 0.591070i $$0.201294\pi$$
$$462$$ 0 0
$$463$$ −1172.00 −0.117640 −0.0588202 0.998269i $$-0.518734\pi$$
−0.0588202 + 0.998269i $$0.518734\pi$$
$$464$$ −640.000 −0.0640329
$$465$$ 0 0
$$466$$ −11208.0 −1.11416
$$467$$ −5304.00 −0.525567 −0.262784 0.964855i $$-0.584641\pi$$
−0.262784 + 0.964855i $$0.584641\pi$$
$$468$$ 0 0
$$469$$ −6328.00 −0.623027
$$470$$ 5184.00 0.508766
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −4216.00 −0.409835
$$474$$ 0 0
$$475$$ −10900.0 −1.05290
$$476$$ 4704.00 0.452957
$$477$$ 0 0
$$478$$ −4680.00 −0.447821
$$479$$ −5740.00 −0.547531 −0.273765 0.961796i $$-0.588269\pi$$
−0.273765 + 0.961796i $$0.588269\pi$$
$$480$$ 0 0
$$481$$ 15252.0 1.44580
$$482$$ 9352.00 0.883759
$$483$$ 0 0
$$484$$ 20104.0 1.88805
$$485$$ −5064.00 −0.474112
$$486$$ 0 0
$$487$$ 8944.00 0.832220 0.416110 0.909314i $$-0.363393\pi$$
0.416110 + 0.909314i $$0.363393\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ −784.000 −0.0722806
$$491$$ 5558.00 0.510853 0.255427 0.966828i $$-0.417784\pi$$
0.255427 + 0.966828i $$0.417784\pi$$
$$492$$ 0 0
$$493$$ −840.000 −0.0767377
$$494$$ 24800.0 2.25871
$$495$$ 0 0
$$496$$ 3072.00 0.278099
$$497$$ −4746.00 −0.428344
$$498$$ 0 0
$$499$$ −19820.0 −1.77809 −0.889043 0.457823i $$-0.848629\pi$$
−0.889043 + 0.457823i $$0.848629\pi$$
$$500$$ −7488.00 −0.669747
$$501$$ 0 0
$$502$$ 11168.0 0.992933
$$503$$ −1848.00 −0.163814 −0.0819068 0.996640i $$-0.526101\pi$$
−0.0819068 + 0.996640i $$0.526101\pi$$
$$504$$ 0 0
$$505$$ −928.000 −0.0817732
$$506$$ 10416.0 0.915114
$$507$$ 0 0
$$508$$ 6432.00 0.561760
$$509$$ −340.000 −0.0296075 −0.0148038 0.999890i $$-0.504712\pi$$
−0.0148038 + 0.999890i $$0.504712\pi$$
$$510$$ 0 0
$$511$$ 4494.00 0.389047
$$512$$ −16384.0 −1.41421
$$513$$ 0 0
$$514$$ 28096.0 2.41101
$$515$$ −7168.00 −0.613320
$$516$$ 0 0
$$517$$ 20088.0 1.70884
$$518$$ −6888.00 −0.584250
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10212.0 −0.858725 −0.429363 0.903132i $$-0.641262\pi$$
−0.429363 + 0.903132i $$0.641262\pi$$
$$522$$ 0 0
$$523$$ −9332.00 −0.780229 −0.390115 0.920766i $$-0.627565\pi$$
−0.390115 + 0.920766i $$0.627565\pi$$
$$524$$ −6496.00 −0.541563
$$525$$ 0 0
$$526$$ 9752.00 0.808379
$$527$$ 4032.00 0.333276
$$528$$ 0 0
$$529$$ −10403.0 −0.855018
$$530$$ 4128.00 0.338319
$$531$$ 0 0
$$532$$ −5600.00 −0.456374
$$533$$ −15376.0 −1.24955
$$534$$ 0 0
$$535$$ 7624.00 0.616101
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −27120.0 −2.17328
$$539$$ −3038.00 −0.242775
$$540$$ 0 0
$$541$$ −8998.00 −0.715073 −0.357536 0.933899i $$-0.616383\pi$$
−0.357536 + 0.933899i $$0.616383\pi$$
$$542$$ 7712.00 0.611179
$$543$$ 0 0
$$544$$ −21504.0 −1.69481
$$545$$ −360.000 −0.0282949
$$546$$ 0 0
$$547$$ −3416.00 −0.267016 −0.133508 0.991048i $$-0.542624\pi$$
−0.133508 + 0.991048i $$0.542624\pi$$
$$548$$ −3312.00 −0.258178
$$549$$ 0 0
$$550$$ −27032.0 −2.09572
$$551$$ 1000.00 0.0773166
$$552$$ 0 0
$$553$$ −5180.00 −0.398329
$$554$$ −22216.0 −1.70373
$$555$$ 0 0
$$556$$ −12960.0 −0.988537
$$557$$ 526.000 0.0400132 0.0200066 0.999800i $$-0.493631\pi$$
0.0200066 + 0.999800i $$0.493631\pi$$
$$558$$ 0 0
$$559$$ −4216.00 −0.318994
$$560$$ 1792.00 0.135225
$$561$$ 0 0
$$562$$ 7768.00 0.583049
$$563$$ 6712.00 0.502446 0.251223 0.967929i $$-0.419167\pi$$
0.251223 + 0.967929i $$0.419167\pi$$
$$564$$ 0 0
$$565$$ −1832.00 −0.136412
$$566$$ −19312.0 −1.43418
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −4190.00 −0.308706 −0.154353 0.988016i $$-0.549329\pi$$
−0.154353 + 0.988016i $$0.549329\pi$$
$$570$$ 0 0
$$571$$ 3032.00 0.222216 0.111108 0.993808i $$-0.464560\pi$$
0.111108 + 0.993808i $$0.464560\pi$$
$$572$$ 30752.0 2.24791
$$573$$ 0 0
$$574$$ 6944.00 0.504942
$$575$$ −4578.00 −0.332027
$$576$$ 0 0
$$577$$ 5434.00 0.392063 0.196032 0.980598i $$-0.437195\pi$$
0.196032 + 0.980598i $$0.437195\pi$$
$$578$$ −8572.00 −0.616865
$$579$$ 0 0
$$580$$ 320.000 0.0229091
$$581$$ 3276.00 0.233927
$$582$$ 0 0
$$583$$ 15996.0 1.13634
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −24608.0 −1.73472
$$587$$ −464.000 −0.0326258 −0.0163129 0.999867i $$-0.505193\pi$$
−0.0163129 + 0.999867i $$0.505193\pi$$
$$588$$ 0 0
$$589$$ −4800.00 −0.335790
$$590$$ 1920.00 0.133975
$$591$$ 0 0
$$592$$ 15744.0 1.09303
$$593$$ −11748.0 −0.813546 −0.406773 0.913529i $$-0.633346\pi$$
−0.406773 + 0.913529i $$0.633346\pi$$
$$594$$ 0 0
$$595$$ 2352.00 0.162055
$$596$$ −18960.0 −1.30307
$$597$$ 0 0
$$598$$ 10416.0 0.712277
$$599$$ −7650.00 −0.521821 −0.260910 0.965363i $$-0.584023\pi$$
−0.260910 + 0.965363i $$0.584023\pi$$
$$600$$ 0 0
$$601$$ −22878.0 −1.55277 −0.776384 0.630261i $$-0.782948\pi$$
−0.776384 + 0.630261i $$0.782948\pi$$
$$602$$ 1904.00 0.128906
$$603$$ 0 0
$$604$$ −4544.00 −0.306114
$$605$$ 10052.0 0.675491
$$606$$ 0 0
$$607$$ 704.000 0.0470749 0.0235375 0.999723i $$-0.492507\pi$$
0.0235375 + 0.999723i $$0.492507\pi$$
$$608$$ 25600.0 1.70759
$$609$$ 0 0
$$610$$ −9952.00 −0.660565
$$611$$ 20088.0 1.33007
$$612$$ 0 0
$$613$$ 24958.0 1.64444 0.822222 0.569167i $$-0.192734\pi$$
0.822222 + 0.569167i $$0.192734\pi$$
$$614$$ −23536.0 −1.54696
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8826.00 0.575886 0.287943 0.957648i $$-0.407029\pi$$
0.287943 + 0.957648i $$0.407029\pi$$
$$618$$ 0 0
$$619$$ 21220.0 1.37787 0.688937 0.724821i $$-0.258078\pi$$
0.688937 + 0.724821i $$0.258078\pi$$
$$620$$ −1536.00 −0.0994956
$$621$$ 0 0
$$622$$ 36528.0 2.35473
$$623$$ 1400.00 0.0900318
$$624$$ 0 0
$$625$$ 9881.00 0.632384
$$626$$ 37528.0 2.39604
$$627$$ 0 0
$$628$$ −2128.00 −0.135217
$$629$$ 20664.0 1.30990
$$630$$ 0 0
$$631$$ −3268.00 −0.206176 −0.103088 0.994672i $$-0.532872\pi$$
−0.103088 + 0.994672i $$0.532872\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 12456.0 0.780270
$$635$$ 3216.00 0.200981
$$636$$ 0 0
$$637$$ −3038.00 −0.188964
$$638$$ 2480.00 0.153894
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −13062.0 −0.804864 −0.402432 0.915450i $$-0.631835\pi$$
−0.402432 + 0.915450i $$0.631835\pi$$
$$642$$ 0 0
$$643$$ −28012.0 −1.71802 −0.859009 0.511961i $$-0.828919\pi$$
−0.859009 + 0.511961i $$0.828919\pi$$
$$644$$ −2352.00 −0.143916
$$645$$ 0 0
$$646$$ 33600.0 2.04640
$$647$$ −3844.00 −0.233575 −0.116788 0.993157i $$-0.537260\pi$$
−0.116788 + 0.993157i $$0.537260\pi$$
$$648$$ 0 0
$$649$$ 7440.00 0.449993
$$650$$ −27032.0 −1.63120
$$651$$ 0 0
$$652$$ −2176.00 −0.130704
$$653$$ 28482.0 1.70687 0.853436 0.521198i $$-0.174515\pi$$
0.853436 + 0.521198i $$0.174515\pi$$
$$654$$ 0 0
$$655$$ −3248.00 −0.193756
$$656$$ −15872.0 −0.944661
$$657$$ 0 0
$$658$$ −9072.00 −0.537482
$$659$$ 9330.00 0.551510 0.275755 0.961228i $$-0.411072\pi$$
0.275755 + 0.961228i $$0.411072\pi$$
$$660$$ 0 0
$$661$$ 8782.00 0.516763 0.258381 0.966043i $$-0.416811\pi$$
0.258381 + 0.966043i $$0.416811\pi$$
$$662$$ −6128.00 −0.359776
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −2800.00 −0.163277
$$666$$ 0 0
$$667$$ 420.000 0.0243815
$$668$$ 15008.0 0.869277
$$669$$ 0 0
$$670$$ −14464.0 −0.834020
$$671$$ −38564.0 −2.21870
$$672$$ 0 0
$$673$$ −10562.0 −0.604956 −0.302478 0.953156i $$-0.597814\pi$$
−0.302478 + 0.953156i $$0.597814\pi$$
$$674$$ 16664.0 0.952334
$$675$$ 0 0
$$676$$ 13176.0 0.749659
$$677$$ 26016.0 1.47692 0.738461 0.674296i $$-0.235553\pi$$
0.738461 + 0.674296i $$0.235553\pi$$
$$678$$ 0 0
$$679$$ 8862.00 0.500872
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −11904.0 −0.668369
$$683$$ −8898.00 −0.498496 −0.249248 0.968440i $$-0.580183\pi$$
−0.249248 + 0.968440i $$0.580183\pi$$
$$684$$ 0 0
$$685$$ −1656.00 −0.0923686
$$686$$ 1372.00 0.0763604
$$687$$ 0 0
$$688$$ −4352.00 −0.241161
$$689$$ 15996.0 0.884469
$$690$$ 0 0
$$691$$ 30572.0 1.68309 0.841544 0.540189i $$-0.181647\pi$$
0.841544 + 0.540189i $$0.181647\pi$$
$$692$$ 1216.00 0.0667997
$$693$$ 0 0
$$694$$ −45464.0 −2.48673
$$695$$ −6480.00 −0.353670
$$696$$ 0 0
$$697$$ −20832.0 −1.13209
$$698$$ −37240.0 −2.01942
$$699$$ 0 0
$$700$$ 6104.00 0.329585
$$701$$ 30618.0 1.64968 0.824840 0.565366i $$-0.191265\pi$$
0.824840 + 0.565366i $$0.191265\pi$$
$$702$$ 0 0
$$703$$ −24600.0 −1.31978
$$704$$ 31744.0 1.69943
$$705$$ 0 0
$$706$$ −34288.0 −1.82783
$$707$$ 1624.00 0.0863887
$$708$$ 0 0
$$709$$ −8130.00 −0.430647 −0.215323 0.976543i $$-0.569081\pi$$
−0.215323 + 0.976543i $$0.569081\pi$$
$$710$$ −10848.0 −0.573406
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −2016.00 −0.105890
$$714$$ 0 0
$$715$$ 15376.0 0.804237
$$716$$ −4880.00 −0.254713
$$717$$ 0 0
$$718$$ −19160.0 −0.995884
$$719$$ 27840.0 1.44403 0.722014 0.691878i $$-0.243216\pi$$
0.722014 + 0.691878i $$0.243216\pi$$
$$720$$ 0 0
$$721$$ 12544.0 0.647938
$$722$$ −12564.0 −0.647623
$$723$$ 0 0
$$724$$ 8336.00 0.427907
$$725$$ −1090.00 −0.0558367
$$726$$ 0 0
$$727$$ 14624.0 0.746044 0.373022 0.927822i $$-0.378322\pi$$
0.373022 + 0.927822i $$0.378322\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 10272.0 0.520800
$$731$$ −5712.00 −0.289010
$$732$$ 0 0
$$733$$ −20862.0 −1.05124 −0.525618 0.850721i $$-0.676166\pi$$
−0.525618 + 0.850721i $$0.676166\pi$$
$$734$$ −21696.0 −1.09103
$$735$$ 0 0
$$736$$ 10752.0 0.538484
$$737$$ −56048.0 −2.80130
$$738$$ 0 0
$$739$$ −13920.0 −0.692903 −0.346452 0.938068i $$-0.612614\pi$$
−0.346452 + 0.938068i $$0.612614\pi$$
$$740$$ −7872.00 −0.391055
$$741$$ 0 0
$$742$$ −7224.00 −0.357414
$$743$$ −25578.0 −1.26294 −0.631471 0.775400i $$-0.717548\pi$$
−0.631471 + 0.775400i $$0.717548\pi$$
$$744$$ 0 0
$$745$$ −9480.00 −0.466202
$$746$$ −7352.00 −0.360826
$$747$$ 0 0
$$748$$ 41664.0 2.03661
$$749$$ −13342.0 −0.650876
$$750$$ 0 0
$$751$$ 33472.0 1.62638 0.813189 0.581999i $$-0.197729\pi$$
0.813189 + 0.581999i $$0.197729\pi$$
$$752$$ 20736.0 1.00554
$$753$$ 0 0
$$754$$ 2480.00 0.119783
$$755$$ −2272.00 −0.109519
$$756$$ 0 0
$$757$$ 25934.0 1.24516 0.622581 0.782556i $$-0.286084\pi$$
0.622581 + 0.782556i $$0.286084\pi$$
$$758$$ 17040.0 0.816518
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −26952.0 −1.28385 −0.641925 0.766768i $$-0.721864\pi$$
−0.641925 + 0.766768i $$0.721864\pi$$
$$762$$ 0 0
$$763$$ 630.000 0.0298919
$$764$$ 16304.0 0.772065
$$765$$ 0 0
$$766$$ 36192.0 1.70714
$$767$$ 7440.00 0.350251
$$768$$ 0 0
$$769$$ 23450.0 1.09965 0.549824 0.835281i $$-0.314695\pi$$
0.549824 + 0.835281i $$0.314695\pi$$
$$770$$ −6944.00 −0.324993
$$771$$ 0 0
$$772$$ −20816.0 −0.970446
$$773$$ −39568.0 −1.84109 −0.920545 0.390637i $$-0.872255\pi$$
−0.920545 + 0.390637i $$0.872255\pi$$
$$774$$ 0 0
$$775$$ 5232.00 0.242502
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −45960.0 −2.11793
$$779$$ 24800.0 1.14063
$$780$$ 0 0
$$781$$ −42036.0 −1.92595
$$782$$ 14112.0 0.645325
$$783$$ 0 0
$$784$$ −3136.00 −0.142857
$$785$$ −1064.00 −0.0483768
$$786$$ 0 0
$$787$$ −12356.0 −0.559649 −0.279825 0.960051i $$-0.590276\pi$$
−0.279825 + 0.960051i $$0.590276\pi$$
$$788$$ −18832.0 −0.851348
$$789$$ 0 0
$$790$$ −11840.0 −0.533226
$$791$$ 3206.00 0.144112
$$792$$ 0 0
$$793$$ −38564.0 −1.72692
$$794$$ 7464.00 0.333611
$$795$$ 0 0
$$796$$ 13440.0 0.598452
$$797$$ 21736.0 0.966033 0.483017 0.875611i $$-0.339541\pi$$
0.483017 + 0.875611i $$0.339541\pi$$
$$798$$ 0 0
$$799$$ 27216.0 1.20505
$$800$$ −27904.0 −1.23319
$$801$$ 0 0
$$802$$ 54648.0 2.40609
$$803$$ 39804.0 1.74926
$$804$$ 0 0
$$805$$ −1176.00 −0.0514889
$$806$$ −11904.0 −0.520224
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 38310.0 1.66490 0.832452 0.554097i $$-0.186936\pi$$
0.832452 + 0.554097i $$0.186936\pi$$
$$810$$ 0 0
$$811$$ 2132.00 0.0923115 0.0461558 0.998934i $$-0.485303\pi$$
0.0461558 + 0.998934i $$0.485303\pi$$
$$812$$ −560.000 −0.0242022
$$813$$ 0 0
$$814$$ −61008.0 −2.62694
$$815$$ −1088.00 −0.0467619
$$816$$ 0 0
$$817$$ 6800.00 0.291190
$$818$$ 52840.0 2.25857
$$819$$ 0 0
$$820$$ 7936.00 0.337972
$$821$$ −5002.00 −0.212632 −0.106316 0.994332i $$-0.533906\pi$$
−0.106316 + 0.994332i $$0.533906\pi$$
$$822$$ 0 0
$$823$$ −3612.00 −0.152985 −0.0764923 0.997070i $$-0.524372\pi$$
−0.0764923 + 0.997070i $$0.524372\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −3360.00 −0.141537
$$827$$ 27666.0 1.16329 0.581645 0.813443i $$-0.302409\pi$$
0.581645 + 0.813443i $$0.302409\pi$$
$$828$$ 0 0
$$829$$ 12890.0 0.540034 0.270017 0.962856i $$-0.412971\pi$$
0.270017 + 0.962856i $$0.412971\pi$$
$$830$$ 7488.00 0.313147
$$831$$ 0 0
$$832$$ 31744.0 1.32275
$$833$$ −4116.00 −0.171202
$$834$$ 0 0
$$835$$ 7504.00 0.311002
$$836$$ −49600.0 −2.05198
$$837$$ 0 0
$$838$$ 27840.0 1.14763
$$839$$ 9340.00 0.384330 0.192165 0.981363i $$-0.438449\pi$$
0.192165 + 0.981363i $$0.438449\pi$$
$$840$$ 0 0
$$841$$ −24289.0 −0.995900
$$842$$ −32648.0 −1.33625
$$843$$ 0 0
$$844$$ −5344.00 −0.217948
$$845$$ 6588.00 0.268206
$$846$$ 0 0
$$847$$ −17591.0 −0.713617
$$848$$ 16512.0 0.668661
$$849$$ 0 0
$$850$$ −36624.0 −1.47787
$$851$$ −10332.0 −0.416188
$$852$$ 0 0
$$853$$ −33082.0 −1.32791 −0.663954 0.747773i $$-0.731123\pi$$
−0.663954 + 0.747773i $$0.731123\pi$$
$$854$$ 17416.0 0.697849
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −7544.00 −0.300698 −0.150349 0.988633i $$-0.548040\pi$$
−0.150349 + 0.988633i $$0.548040\pi$$
$$858$$ 0 0
$$859$$ 8180.00 0.324910 0.162455 0.986716i $$-0.448059\pi$$
0.162455 + 0.986716i $$0.448059\pi$$
$$860$$ 2176.00 0.0862802
$$861$$ 0 0
$$862$$ 66408.0 2.62397
$$863$$ −10518.0 −0.414875 −0.207437 0.978248i $$-0.566512\pi$$
−0.207437 + 0.978248i $$0.566512\pi$$
$$864$$ 0 0
$$865$$ 608.000 0.0238990
$$866$$ −30952.0 −1.21454
$$867$$ 0 0
$$868$$ 2688.00 0.105111
$$869$$ −45880.0 −1.79099
$$870$$ 0 0
$$871$$ −56048.0 −2.18038
$$872$$ 0 0
$$873$$ 0 0
$$874$$ −16800.0 −0.650193
$$875$$ 6552.00 0.253141
$$876$$ 0 0
$$877$$ 14134.0 0.544209 0.272104 0.962268i $$-0.412280\pi$$
0.272104 + 0.962268i $$0.412280\pi$$
$$878$$ 3360.00 0.129151
$$879$$ 0 0
$$880$$ 15872.0 0.608006
$$881$$ −6492.00 −0.248265 −0.124132 0.992266i $$-0.539615\pi$$
−0.124132 + 0.992266i $$0.539615\pi$$
$$882$$ 0 0
$$883$$ 38228.0 1.45694 0.728468 0.685080i $$-0.240233\pi$$
0.728468 + 0.685080i $$0.240233\pi$$
$$884$$ 41664.0 1.58519
$$885$$ 0 0
$$886$$ 26472.0 1.00377
$$887$$ 43076.0 1.63061 0.815305 0.579032i $$-0.196569\pi$$
0.815305 + 0.579032i $$0.196569\pi$$
$$888$$ 0 0
$$889$$ −5628.00 −0.212325
$$890$$ 3200.00 0.120522
$$891$$ 0 0
$$892$$ −14656.0 −0.550134
$$893$$ −32400.0 −1.21414
$$894$$ 0 0
$$895$$ −2440.00 −0.0911287
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 12360.0 0.459308
$$899$$ −480.000 −0.0178074
$$900$$ 0 0
$$901$$ 21672.0 0.801331
$$902$$ 61504.0 2.27035
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 4168.00 0.153093
$$906$$ 0 0
$$907$$ −32236.0 −1.18013 −0.590065 0.807355i $$-0.700898\pi$$
−0.590065 + 0.807355i $$0.700898\pi$$
$$908$$ −39552.0 −1.44557
$$909$$ 0 0
$$910$$ −6944.00 −0.252958
$$911$$ 46518.0 1.69178 0.845889 0.533359i $$-0.179070\pi$$
0.845889 + 0.533359i $$0.179070\pi$$
$$912$$ 0 0
$$913$$ 29016.0 1.05180
$$914$$ −23656.0 −0.856095
$$915$$ 0 0
$$916$$ −43760.0 −1.57846
$$917$$ 5684.00 0.204692
$$918$$ 0 0
$$919$$ 17840.0 0.640356 0.320178 0.947357i $$-0.396257\pi$$
0.320178 + 0.947357i $$0.396257\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −63872.0 −2.28147
$$923$$ −42036.0 −1.49906
$$924$$ 0 0
$$925$$ 26814.0 0.953123
$$926$$ 4688.00 0.166369
$$927$$ 0 0
$$928$$ 2560.00 0.0905562
$$929$$ −7000.00 −0.247215 −0.123607 0.992331i $$-0.539446\pi$$
−0.123607 + 0.992331i $$0.539446\pi$$
$$930$$ 0 0
$$931$$ 4900.00 0.172493
$$932$$ 22416.0 0.787833
$$933$$ 0 0
$$934$$ 21216.0 0.743264
$$935$$ 20832.0 0.728641
$$936$$ 0 0
$$937$$ 36114.0 1.25912 0.629559 0.776953i $$-0.283236\pi$$
0.629559 + 0.776953i $$0.283236\pi$$
$$938$$ 25312.0 0.881094
$$939$$ 0 0
$$940$$ −10368.0 −0.359752
$$941$$ 4748.00 0.164485 0.0822425 0.996612i $$-0.473792\pi$$
0.0822425 + 0.996612i $$0.473792\pi$$
$$942$$ 0 0
$$943$$ 10416.0 0.359694
$$944$$ 7680.00 0.264791
$$945$$ 0 0
$$946$$ 16864.0 0.579594
$$947$$ −42694.0 −1.46501 −0.732507 0.680759i $$-0.761650\pi$$
−0.732507 + 0.680759i $$0.761650\pi$$
$$948$$ 0 0
$$949$$ 39804.0 1.36153
$$950$$ 43600.0 1.48902
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 16742.0 0.569073 0.284537 0.958665i $$-0.408160\pi$$
0.284537 + 0.958665i $$0.408160\pi$$
$$954$$ 0 0
$$955$$ 8152.00 0.276223
$$956$$ 9360.00 0.316657
$$957$$ 0 0
$$958$$ 22960.0 0.774326
$$959$$ 2898.00 0.0975822
$$960$$ 0 0
$$961$$ −27487.0 −0.922661
$$962$$ −61008.0 −2.04467
$$963$$ 0 0
$$964$$ −18704.0 −0.624912
$$965$$ −10408.0 −0.347197
$$966$$ 0 0
$$967$$ −9956.00 −0.331089 −0.165545 0.986202i $$-0.552938\pi$$
−0.165545 + 0.986202i $$0.552938\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 20256.0 0.670496
$$971$$ 26388.0 0.872123 0.436061 0.899917i $$-0.356373\pi$$
0.436061 + 0.899917i $$0.356373\pi$$
$$972$$ 0 0
$$973$$ 11340.0 0.373632
$$974$$ −35776.0 −1.17694
$$975$$ 0 0
$$976$$ −39808.0 −1.30556
$$977$$ 786.000 0.0257383 0.0128692 0.999917i $$-0.495904\pi$$
0.0128692 + 0.999917i $$0.495904\pi$$
$$978$$ 0 0
$$979$$ 12400.0 0.404807
$$980$$ 1568.00 0.0511101
$$981$$ 0 0
$$982$$ −22232.0 −0.722456
$$983$$ −51888.0 −1.68359 −0.841796 0.539796i $$-0.818501\pi$$
−0.841796 + 0.539796i $$0.818501\pi$$
$$984$$ 0 0
$$985$$ −9416.00 −0.304588
$$986$$ 3360.00 0.108524
$$987$$ 0 0
$$988$$ −49600.0 −1.59715
$$989$$ 2856.00 0.0918256
$$990$$ 0 0
$$991$$ −51928.0 −1.66453 −0.832264 0.554379i $$-0.812956\pi$$
−0.832264 + 0.554379i $$0.812956\pi$$
$$992$$ −12288.0 −0.393291
$$993$$ 0 0
$$994$$ 18984.0 0.605771
$$995$$ 6720.00 0.214109
$$996$$ 0 0
$$997$$ −386.000 −0.0122615 −0.00613076 0.999981i $$-0.501951\pi$$
−0.00613076 + 0.999981i $$0.501951\pi$$
$$998$$ 79280.0 2.51459
$$999$$ 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 63.4.a.a.1.1 1
3.2 odd 2 21.4.a.b.1.1 1
4.3 odd 2 1008.4.a.m.1.1 1
5.4 even 2 1575.4.a.k.1.1 1
7.2 even 3 441.4.e.m.361.1 2
7.3 odd 6 441.4.e.n.226.1 2
7.4 even 3 441.4.e.m.226.1 2
7.5 odd 6 441.4.e.n.361.1 2
7.6 odd 2 441.4.a.b.1.1 1
12.11 even 2 336.4.a.h.1.1 1
15.2 even 4 525.4.d.b.274.2 2
15.8 even 4 525.4.d.b.274.1 2
15.14 odd 2 525.4.a.b.1.1 1
21.2 odd 6 147.4.e.c.67.1 2
21.5 even 6 147.4.e.b.67.1 2
21.11 odd 6 147.4.e.c.79.1 2
21.17 even 6 147.4.e.b.79.1 2
21.20 even 2 147.4.a.g.1.1 1
24.5 odd 2 1344.4.a.w.1.1 1
24.11 even 2 1344.4.a.i.1.1 1
84.83 odd 2 2352.4.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.b.1.1 1 3.2 odd 2
63.4.a.a.1.1 1 1.1 even 1 trivial
147.4.a.g.1.1 1 21.20 even 2
147.4.e.b.67.1 2 21.5 even 6
147.4.e.b.79.1 2 21.17 even 6
147.4.e.c.67.1 2 21.2 odd 6
147.4.e.c.79.1 2 21.11 odd 6
336.4.a.h.1.1 1 12.11 even 2
441.4.a.b.1.1 1 7.6 odd 2
441.4.e.m.226.1 2 7.4 even 3
441.4.e.m.361.1 2 7.2 even 3
441.4.e.n.226.1 2 7.3 odd 6
441.4.e.n.361.1 2 7.5 odd 6
525.4.a.b.1.1 1 15.14 odd 2
525.4.d.b.274.1 2 15.8 even 4
525.4.d.b.274.2 2 15.2 even 4
1008.4.a.m.1.1 1 4.3 odd 2
1344.4.a.i.1.1 1 24.11 even 2
1344.4.a.w.1.1 1 24.5 odd 2
1575.4.a.k.1.1 1 5.4 even 2
2352.4.a.l.1.1 1 84.83 odd 2