# Properties

 Label 336.4.a.h.1.1 Level $336$ Weight $4$ Character 336.1 Self dual yes Analytic conductor $19.825$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.8246417619$$ 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 $$\chi$$ $$=$$ 336.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -4.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -4.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -62.0000 q^{11} -62.0000 q^{13} -12.0000 q^{15} +84.0000 q^{17} -100.000 q^{19} +21.0000 q^{21} +42.0000 q^{23} -109.000 q^{25} +27.0000 q^{27} -10.0000 q^{29} +48.0000 q^{31} -186.000 q^{33} -28.0000 q^{35} -246.000 q^{37} -186.000 q^{39} -248.000 q^{41} -68.0000 q^{43} -36.0000 q^{45} -324.000 q^{47} +49.0000 q^{49} +252.000 q^{51} +258.000 q^{53} +248.000 q^{55} -300.000 q^{57} -120.000 q^{59} +622.000 q^{61} +63.0000 q^{63} +248.000 q^{65} -904.000 q^{67} +126.000 q^{69} +678.000 q^{71} -642.000 q^{73} -327.000 q^{75} -434.000 q^{77} -740.000 q^{79} +81.0000 q^{81} -468.000 q^{83} -336.000 q^{85} -30.0000 q^{87} +200.000 q^{89} -434.000 q^{91} +144.000 q^{93} +400.000 q^{95} -1266.00 q^{97} -558.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −4.00000 −0.357771 −0.178885 0.983870i $$-0.557249\pi$$
−0.178885 + 0.983870i $$0.557249\pi$$
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −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 $$0.730026\pi$$
$$14$$ 0 0
$$15$$ −12.0000 −0.206559
$$16$$ 0 0
$$17$$ 84.0000 1.19841 0.599206 0.800595i $$-0.295483\pi$$
0.599206 + 0.800595i $$0.295483\pi$$
$$18$$ 0 0
$$19$$ −100.000 −1.20745 −0.603726 0.797192i $$-0.706318\pi$$
−0.603726 + 0.797192i $$0.706318\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ 0 0
$$23$$ 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
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −10.0000 −0.0640329 −0.0320164 0.999487i $$-0.510193\pi$$
−0.0320164 + 0.999487i $$0.510193\pi$$
$$30$$ 0 0
$$31$$ 48.0000 0.278099 0.139049 0.990285i $$-0.455595\pi$$
0.139049 + 0.990285i $$0.455595\pi$$
$$32$$ 0 0
$$33$$ −186.000 −0.981165
$$34$$ 0 0
$$35$$ −28.0000 −0.135225
$$36$$ 0 0
$$37$$ −246.000 −1.09303 −0.546516 0.837449i $$-0.684046\pi$$
−0.546516 + 0.837449i $$0.684046\pi$$
$$38$$ 0 0
$$39$$ −186.000 −0.763688
$$40$$ 0 0
$$41$$ −248.000 −0.944661 −0.472330 0.881422i $$-0.656587\pi$$
−0.472330 + 0.881422i $$0.656587\pi$$
$$42$$ 0 0
$$43$$ −68.0000 −0.241161 −0.120580 0.992704i $$-0.538476\pi$$
−0.120580 + 0.992704i $$0.538476\pi$$
$$44$$ 0 0
$$45$$ −36.0000 −0.119257
$$46$$ 0 0
$$47$$ −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
$$50$$ 0 0
$$51$$ 252.000 0.691903
$$52$$ 0 0
$$53$$ 258.000 0.668661 0.334330 0.942456i $$-0.391490\pi$$
0.334330 + 0.942456i $$0.391490\pi$$
$$54$$ 0 0
$$55$$ 248.000 0.608006
$$56$$ 0 0
$$57$$ −300.000 −0.697122
$$58$$ 0 0
$$59$$ −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 $$0.273603\pi$$
$$62$$ 0 0
$$63$$ 63.0000 0.125988
$$64$$ 0 0
$$65$$ 248.000 0.473240
$$66$$ 0 0
$$67$$ −904.000 −1.64838 −0.824188 0.566316i $$-0.808368\pi$$
−0.824188 + 0.566316i $$0.808368\pi$$
$$68$$ 0 0
$$69$$ 126.000 0.219835
$$70$$ 0 0
$$71$$ 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 $$0.672082\pi$$
$$74$$ 0 0
$$75$$ −327.000 −0.503449
$$76$$ 0 0
$$77$$ −434.000 −0.642323
$$78$$ 0 0
$$79$$ −740.000 −1.05388 −0.526940 0.849903i $$-0.676661\pi$$
−0.526940 + 0.849903i $$0.676661\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −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
$$86$$ 0 0
$$87$$ −30.0000 −0.0369694
$$88$$ 0 0
$$89$$ 200.000 0.238202 0.119101 0.992882i $$-0.461999\pi$$
0.119101 + 0.992882i $$0.461999\pi$$
$$90$$ 0 0
$$91$$ −434.000 −0.499951
$$92$$ 0 0
$$93$$ 144.000 0.160560
$$94$$ 0 0
$$95$$ 400.000 0.431991
$$96$$ 0 0
$$97$$ −1266.00 −1.32518 −0.662592 0.748981i $$-0.730544\pi$$
−0.662592 + 0.748981i $$0.730544\pi$$
$$98$$ 0 0
$$99$$ −558.000 −0.566476
$$100$$ 0 0
$$101$$ 232.000 0.228563 0.114281 0.993448i $$-0.463543\pi$$
0.114281 + 0.993448i $$0.463543\pi$$
$$102$$ 0 0
$$103$$ 1792.00 1.71428 0.857141 0.515082i $$-0.172239\pi$$
0.857141 + 0.515082i $$0.172239\pi$$
$$104$$ 0 0
$$105$$ −84.0000 −0.0780720
$$106$$ 0 0
$$107$$ 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 $$0.512590\pi$$
$$110$$ 0 0
$$111$$ −738.000 −0.631062
$$112$$ 0 0
$$113$$ 458.000 0.381283 0.190642 0.981660i $$-0.438943\pi$$
0.190642 + 0.981660i $$0.438943\pi$$
$$114$$ 0 0
$$115$$ −168.000 −0.136227
$$116$$ 0 0
$$117$$ −558.000 −0.440916
$$118$$ 0 0
$$119$$ 588.000 0.452957
$$120$$ 0 0
$$121$$ 2513.00 1.88805
$$122$$ 0 0
$$123$$ −744.000 −0.545400
$$124$$ 0 0
$$125$$ 936.000 0.669747
$$126$$ 0 0
$$127$$ −804.000 −0.561760 −0.280880 0.959743i $$-0.590626\pi$$
−0.280880 + 0.959743i $$0.590626\pi$$
$$128$$ 0 0
$$129$$ −204.000 −0.139234
$$130$$ 0 0
$$131$$ −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
$$134$$ 0 0
$$135$$ −108.000 −0.0688530
$$136$$ 0 0
$$137$$ 414.000 0.258178 0.129089 0.991633i $$-0.458795\pi$$
0.129089 + 0.991633i $$0.458795\pi$$
$$138$$ 0 0
$$139$$ 1620.00 0.988537 0.494268 0.869309i $$-0.335436\pi$$
0.494268 + 0.869309i $$0.335436\pi$$
$$140$$ 0 0
$$141$$ −972.000 −0.580547
$$142$$ 0 0
$$143$$ 3844.00 2.24791
$$144$$ 0 0
$$145$$ 40.0000 0.0229091
$$146$$ 0 0
$$147$$ 147.000 0.0824786
$$148$$ 0 0
$$149$$ 2370.00 1.30307 0.651537 0.758617i $$-0.274125\pi$$
0.651537 + 0.758617i $$0.274125\pi$$
$$150$$ 0 0
$$151$$ 568.000 0.306114 0.153057 0.988217i $$-0.451088\pi$$
0.153057 + 0.988217i $$0.451088\pi$$
$$152$$ 0 0
$$153$$ 756.000 0.399470
$$154$$ 0 0
$$155$$ −192.000 −0.0994956
$$156$$ 0 0
$$157$$ −266.000 −0.135217 −0.0676086 0.997712i $$-0.521537\pi$$
−0.0676086 + 0.997712i $$0.521537\pi$$
$$158$$ 0 0
$$159$$ 774.000 0.386052
$$160$$ 0 0
$$161$$ 294.000 0.143916
$$162$$ 0 0
$$163$$ 272.000 0.130704 0.0653518 0.997862i $$-0.479183\pi$$
0.0653518 + 0.997862i $$0.479183\pi$$
$$164$$ 0 0
$$165$$ 744.000 0.351032
$$166$$ 0 0
$$167$$ 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
$$170$$ 0 0
$$171$$ −900.000 −0.402484
$$172$$ 0 0
$$173$$ −152.000 −0.0667997 −0.0333998 0.999442i $$-0.510633\pi$$
−0.0333998 + 0.999442i $$0.510633\pi$$
$$174$$ 0 0
$$175$$ −763.000 −0.329585
$$176$$ 0 0
$$177$$ −360.000 −0.152877
$$178$$ 0 0
$$179$$ −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 $$0.431366\pi$$
$$182$$ 0 0
$$183$$ 1866.00 0.753763
$$184$$ 0 0
$$185$$ 984.000 0.391055
$$186$$ 0 0
$$187$$ −5208.00 −2.03661
$$188$$ 0 0
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ 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 $$0.661262\pi$$
$$194$$ 0 0
$$195$$ 744.000 0.273225
$$196$$ 0 0
$$197$$ 2354.00 0.851348 0.425674 0.904877i $$-0.360037\pi$$
0.425674 + 0.904877i $$0.360037\pi$$
$$198$$ 0 0
$$199$$ −1680.00 −0.598452 −0.299226 0.954182i $$-0.596729\pi$$
−0.299226 + 0.954182i $$0.596729\pi$$
$$200$$ 0 0
$$201$$ −2712.00 −0.951690
$$202$$ 0 0
$$203$$ −70.0000 −0.0242022
$$204$$ 0 0
$$205$$ 992.000 0.337972
$$206$$ 0 0
$$207$$ 378.000 0.126922
$$208$$ 0 0
$$209$$ 6200.00 2.05198
$$210$$ 0 0
$$211$$ 668.000 0.217948 0.108974 0.994045i $$-0.465243\pi$$
0.108974 + 0.994045i $$0.465243\pi$$
$$212$$ 0 0
$$213$$ 2034.00 0.654307
$$214$$ 0 0
$$215$$ 272.000 0.0862802
$$216$$ 0 0
$$217$$ 336.000 0.105111
$$218$$ 0 0
$$219$$ −1926.00 −0.594279
$$220$$ 0 0
$$221$$ −5208.00 −1.58519
$$222$$ 0 0
$$223$$ 1832.00 0.550134 0.275067 0.961425i $$-0.411300\pi$$
0.275067 + 0.961425i $$0.411300\pi$$
$$224$$ 0 0
$$225$$ −981.000 −0.290667
$$226$$ 0 0
$$227$$ −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 $$0.789521\pi$$
$$230$$ 0 0
$$231$$ −1302.00 −0.370846
$$232$$ 0 0
$$233$$ −2802.00 −0.787833 −0.393917 0.919146i $$-0.628880\pi$$
−0.393917 + 0.919146i $$0.628880\pi$$
$$234$$ 0 0
$$235$$ 1296.00 0.359752
$$236$$ 0 0
$$237$$ −2220.00 −0.608458
$$238$$ 0 0
$$239$$ 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 $$0.601152\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −196.000 −0.0511101
$$246$$ 0 0
$$247$$ 6200.00 1.59715
$$248$$ 0 0
$$249$$ −1404.00 −0.357329
$$250$$ 0 0
$$251$$ −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
$$254$$ 0 0
$$255$$ −1008.00 −0.247543
$$256$$ 0 0
$$257$$ 7024.00 1.70484 0.852422 0.522854i $$-0.175133\pi$$
0.852422 + 0.522854i $$0.175133\pi$$
$$258$$ 0 0
$$259$$ −1722.00 −0.413127
$$260$$ 0 0
$$261$$ −90.0000 −0.0213443
$$262$$ 0 0
$$263$$ −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
$$266$$ 0 0
$$267$$ 600.000 0.137526
$$268$$ 0 0
$$269$$ −6780.00 −1.53674 −0.768372 0.640004i $$-0.778933\pi$$
−0.768372 + 0.640004i $$0.778933\pi$$
$$270$$ 0 0
$$271$$ 1928.00 0.432168 0.216084 0.976375i $$-0.430671\pi$$
0.216084 + 0.976375i $$0.430671\pi$$
$$272$$ 0 0
$$273$$ −1302.00 −0.288647
$$274$$ 0 0
$$275$$ 6758.00 1.48190
$$276$$ 0 0
$$277$$ 5554.00 1.20472 0.602360 0.798224i $$-0.294227\pi$$
0.602360 + 0.798224i $$0.294227\pi$$
$$278$$ 0 0
$$279$$ 432.000 0.0926995
$$280$$ 0 0
$$281$$ 1942.00 0.412278 0.206139 0.978523i $$-0.433910\pi$$
0.206139 + 0.978523i $$0.433910\pi$$
$$282$$ 0 0
$$283$$ −4828.00 −1.01412 −0.507058 0.861912i $$-0.669267\pi$$
−0.507058 + 0.861912i $$0.669267\pi$$
$$284$$ 0 0
$$285$$ 1200.00 0.249410
$$286$$ 0 0
$$287$$ −1736.00 −0.357048
$$288$$ 0 0
$$289$$ 2143.00 0.436190
$$290$$ 0 0
$$291$$ −3798.00 −0.765095
$$292$$ 0 0
$$293$$ −6152.00 −1.22663 −0.613317 0.789837i $$-0.710165\pi$$
−0.613317 + 0.789837i $$0.710165\pi$$
$$294$$ 0 0
$$295$$ 480.000 0.0947345
$$296$$ 0 0
$$297$$ −1674.00 −0.327055
$$298$$ 0 0
$$299$$ −2604.00 −0.503656
$$300$$ 0 0
$$301$$ −476.000 −0.0911501
$$302$$ 0 0
$$303$$ 696.000 0.131961
$$304$$ 0 0
$$305$$ −2488.00 −0.467090
$$306$$ 0 0
$$307$$ −5884.00 −1.09387 −0.546934 0.837176i $$-0.684205\pi$$
−0.546934 + 0.837176i $$0.684205\pi$$
$$308$$ 0 0
$$309$$ 5376.00 0.989741
$$310$$ 0 0
$$311$$ −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 $$0.821670\pi$$
$$314$$ 0 0
$$315$$ −252.000 −0.0450749
$$316$$ 0 0
$$317$$ 3114.00 0.551734 0.275867 0.961196i $$-0.411035\pi$$
0.275867 + 0.961196i $$0.411035\pi$$
$$318$$ 0 0
$$319$$ 620.000 0.108819
$$320$$ 0 0
$$321$$ 5718.00 0.994229
$$322$$ 0 0
$$323$$ −8400.00 −1.44702
$$324$$ 0 0
$$325$$ 6758.00 1.15344
$$326$$ 0 0
$$327$$ −270.000 −0.0456607
$$328$$ 0 0
$$329$$ −2268.00 −0.380057
$$330$$ 0 0
$$331$$ −1532.00 −0.254400 −0.127200 0.991877i $$-0.540599\pi$$
−0.127200 + 0.991877i $$0.540599\pi$$
$$332$$ 0 0
$$333$$ −2214.00 −0.364344
$$334$$ 0 0
$$335$$ 3616.00 0.589741
$$336$$ 0 0
$$337$$ −4166.00 −0.673402 −0.336701 0.941612i $$-0.609311\pi$$
−0.336701 + 0.941612i $$0.609311\pi$$
$$338$$ 0 0
$$339$$ 1374.00 0.220134
$$340$$ 0 0
$$341$$ −2976.00 −0.472608
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ 0 0
$$345$$ −504.000 −0.0786506
$$346$$ 0 0
$$347$$ 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 $$0.246894\pi$$
$$350$$ 0 0
$$351$$ −1674.00 −0.254563
$$352$$ 0 0
$$353$$ −8572.00 −1.29247 −0.646234 0.763139i $$-0.723657\pi$$
−0.646234 + 0.763139i $$0.723657\pi$$
$$354$$ 0 0
$$355$$ −2712.00 −0.405459
$$356$$ 0 0
$$357$$ 1764.00 0.261515
$$358$$ 0 0
$$359$$ 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
$$362$$ 0 0
$$363$$ 7539.00 1.09007
$$364$$ 0 0
$$365$$ 2568.00 0.368261
$$366$$ 0 0
$$367$$ −5424.00 −0.771473 −0.385736 0.922609i $$-0.626053\pi$$
−0.385736 + 0.922609i $$0.626053\pi$$
$$368$$ 0 0
$$369$$ −2232.00 −0.314887
$$370$$ 0 0
$$371$$ 1806.00 0.252730
$$372$$ 0 0
$$373$$ 1838.00 0.255142 0.127571 0.991829i $$-0.459282\pi$$
0.127571 + 0.991829i $$0.459282\pi$$
$$374$$ 0 0
$$375$$ 2808.00 0.386679
$$376$$ 0 0
$$377$$ 620.000 0.0846993
$$378$$ 0 0
$$379$$ 4260.00 0.577365 0.288683 0.957425i $$-0.406783\pi$$
0.288683 + 0.957425i $$0.406783\pi$$
$$380$$ 0 0
$$381$$ −2412.00 −0.324332
$$382$$ 0 0
$$383$$ −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
$$386$$ 0 0
$$387$$ −612.000 −0.0803868
$$388$$ 0 0
$$389$$ −11490.0 −1.49760 −0.748800 0.662796i $$-0.769369\pi$$
−0.748800 + 0.662796i $$0.769369\pi$$
$$390$$ 0 0
$$391$$ 3528.00 0.456314
$$392$$ 0 0
$$393$$ −2436.00 −0.312672
$$394$$ 0 0
$$395$$ 2960.00 0.377048
$$396$$ 0 0
$$397$$ −1866.00 −0.235899 −0.117949 0.993020i $$-0.537632\pi$$
−0.117949 + 0.993020i $$0.537632\pi$$
$$398$$ 0 0
$$399$$ −2100.00 −0.263487
$$400$$ 0 0
$$401$$ 13662.0 1.70137 0.850683 0.525679i $$-0.176189\pi$$
0.850683 + 0.525679i $$0.176189\pi$$
$$402$$ 0 0
$$403$$ −2976.00 −0.367854
$$404$$ 0 0
$$405$$ −324.000 −0.0397523
$$406$$ 0 0
$$407$$ 15252.0 1.85753
$$408$$ 0 0
$$409$$ −13210.0 −1.59705 −0.798524 0.601963i $$-0.794385\pi$$
−0.798524 + 0.601963i $$0.794385\pi$$
$$410$$ 0 0
$$411$$ 1242.00 0.149059
$$412$$ 0 0
$$413$$ −840.000 −0.100082
$$414$$ 0 0
$$415$$ 1872.00 0.221429
$$416$$ 0 0
$$417$$ 4860.00 0.570732
$$418$$ 0 0
$$419$$ −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 $$0.343375\pi$$
$$422$$ 0 0
$$423$$ −2916.00 −0.335179
$$424$$ 0 0
$$425$$ −9156.00 −1.04501
$$426$$ 0 0
$$427$$ 4354.00 0.493454
$$428$$ 0 0
$$429$$ 11532.0 1.29783
$$430$$ 0 0
$$431$$ −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 $$0.358723\pi$$
$$434$$ 0 0
$$435$$ 120.000 0.0132266
$$436$$ 0 0
$$437$$ −4200.00 −0.459756
$$438$$ 0 0
$$439$$ 840.000 0.0913235 0.0456617 0.998957i $$-0.485460\pi$$
0.0456617 + 0.998957i $$0.485460\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 0 0
$$443$$ −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
$$446$$ 0 0
$$447$$ 7110.00 0.752330
$$448$$ 0 0
$$449$$ 3090.00 0.324780 0.162390 0.986727i $$-0.448080\pi$$
0.162390 + 0.986727i $$0.448080\pi$$
$$450$$ 0 0
$$451$$ 15376.0 1.60538
$$452$$ 0 0
$$453$$ 1704.00 0.176735
$$454$$ 0 0
$$455$$ 1736.00 0.178868
$$456$$ 0 0
$$457$$ 5914.00 0.605351 0.302675 0.953094i $$-0.402120\pi$$
0.302675 + 0.953094i $$0.402120\pi$$
$$458$$ 0 0
$$459$$ 2268.00 0.230634
$$460$$ 0 0
$$461$$ −15968.0 −1.61324 −0.806620 0.591070i $$-0.798706\pi$$
−0.806620 + 0.591070i $$0.798706\pi$$
$$462$$ 0 0
$$463$$ 1172.00 0.117640 0.0588202 0.998269i $$-0.481266\pi$$
0.0588202 + 0.998269i $$0.481266\pi$$
$$464$$ 0 0
$$465$$ −576.000 −0.0574438
$$466$$ 0 0
$$467$$ −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
$$470$$ 0 0
$$471$$ −798.000 −0.0780677
$$472$$ 0 0
$$473$$ 4216.00 0.409835
$$474$$ 0 0
$$475$$ 10900.0 1.05290
$$476$$ 0 0
$$477$$ 2322.00 0.222887
$$478$$ 0 0
$$479$$ −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
$$482$$ 0 0
$$483$$ 882.000 0.0830898
$$484$$ 0 0
$$485$$ 5064.00 0.474112
$$486$$ 0 0
$$487$$ −8944.00 −0.832220 −0.416110 0.909314i $$-0.636607\pi$$
−0.416110 + 0.909314i $$0.636607\pi$$
$$488$$ 0 0
$$489$$ 816.000 0.0754617
$$490$$ 0 0
$$491$$ 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
$$494$$ 0 0
$$495$$ 2232.00 0.202669
$$496$$ 0 0
$$497$$ 4746.00 0.428344
$$498$$ 0 0
$$499$$ 19820.0 1.77809 0.889043 0.457823i $$-0.151371\pi$$
0.889043 + 0.457823i $$0.151371\pi$$
$$500$$ 0 0
$$501$$ 5628.00 0.501877
$$502$$ 0 0
$$503$$ −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
$$506$$ 0 0
$$507$$ 4941.00 0.432816
$$508$$ 0 0
$$509$$ 340.000 0.0296075 0.0148038 0.999890i $$-0.495288\pi$$
0.0148038 + 0.999890i $$0.495288\pi$$
$$510$$ 0 0
$$511$$ −4494.00 −0.389047
$$512$$ 0 0
$$513$$ −2700.00 −0.232374
$$514$$ 0 0
$$515$$ −7168.00 −0.613320
$$516$$ 0 0
$$517$$ 20088.0 1.70884
$$518$$ 0 0
$$519$$ −456.000 −0.0385668
$$520$$ 0 0
$$521$$ 10212.0 0.858725 0.429363 0.903132i $$-0.358738\pi$$
0.429363 + 0.903132i $$0.358738\pi$$
$$522$$ 0 0
$$523$$ 9332.00 0.780229 0.390115 0.920766i $$-0.372435\pi$$
0.390115 + 0.920766i $$0.372435\pi$$
$$524$$ 0 0
$$525$$ −2289.00 −0.190286
$$526$$ 0 0
$$527$$ 4032.00 0.333276
$$528$$ 0 0
$$529$$ −10403.0 −0.855018
$$530$$ 0 0
$$531$$ −1080.00 −0.0882637
$$532$$ 0 0
$$533$$ 15376.0 1.24955
$$534$$ 0 0
$$535$$ −7624.00 −0.616101
$$536$$ 0 0
$$537$$ −1830.00 −0.147058
$$538$$ 0 0
$$539$$ −3038.00 −0.242775
$$540$$ 0 0
$$541$$ −8998.00 −0.715073 −0.357536 0.933899i $$-0.616383\pi$$
−0.357536 + 0.933899i $$0.616383\pi$$
$$542$$ 0 0
$$543$$ 3126.00 0.247052
$$544$$ 0 0
$$545$$ 360.000 0.0282949
$$546$$ 0 0
$$547$$ 3416.00 0.267016 0.133508 0.991048i $$-0.457376\pi$$
0.133508 + 0.991048i $$0.457376\pi$$
$$548$$ 0 0
$$549$$ 5598.00 0.435185
$$550$$ 0 0
$$551$$ 1000.00 0.0773166
$$552$$ 0 0
$$553$$ −5180.00 −0.398329
$$554$$ 0 0
$$555$$ 2952.00 0.225776
$$556$$ 0 0
$$557$$ −526.000 −0.0400132 −0.0200066 0.999800i $$-0.506369\pi$$
−0.0200066 + 0.999800i $$0.506369\pi$$
$$558$$ 0 0
$$559$$ 4216.00 0.318994
$$560$$ 0 0
$$561$$ −15624.0 −1.17584
$$562$$ 0 0
$$563$$ 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
$$566$$ 0 0
$$567$$ 567.000 0.0419961
$$568$$ 0 0
$$569$$ 4190.00 0.308706 0.154353 0.988016i $$-0.450671\pi$$
0.154353 + 0.988016i $$0.450671\pi$$
$$570$$ 0 0
$$571$$ −3032.00 −0.222216 −0.111108 0.993808i $$-0.535440\pi$$
−0.111108 + 0.993808i $$0.535440\pi$$
$$572$$ 0 0
$$573$$ 6114.00 0.445752
$$574$$ 0 0
$$575$$ −4578.00 −0.332027
$$576$$ 0 0
$$577$$ 5434.00 0.392063 0.196032 0.980598i $$-0.437195\pi$$
0.196032 + 0.980598i $$0.437195\pi$$
$$578$$ 0 0
$$579$$ −7806.00 −0.560287
$$580$$ 0 0
$$581$$ −3276.00 −0.233927
$$582$$ 0 0
$$583$$ −15996.0 −1.13634
$$584$$ 0 0
$$585$$ 2232.00 0.157747
$$586$$ 0 0
$$587$$ −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
$$590$$ 0 0
$$591$$ 7062.00 0.491526
$$592$$ 0 0
$$593$$ 11748.0 0.813546 0.406773 0.913529i $$-0.366654\pi$$
0.406773 + 0.913529i $$0.366654\pi$$
$$594$$ 0 0
$$595$$ −2352.00 −0.162055
$$596$$ 0 0
$$597$$ −5040.00 −0.345517
$$598$$ 0 0
$$599$$ −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 $$0.782948\pi$$
$$602$$ 0 0
$$603$$ −8136.00 −0.549459
$$604$$ 0 0
$$605$$ −10052.0 −0.675491
$$606$$ 0 0
$$607$$ −704.000 −0.0470749 −0.0235375 0.999723i $$-0.507493\pi$$
−0.0235375 + 0.999723i $$0.507493\pi$$
$$608$$ 0 0
$$609$$ −210.000 −0.0139731
$$610$$ 0 0
$$611$$ 20088.0 1.33007
$$612$$ 0 0
$$613$$ 24958.0 1.64444 0.822222 0.569167i $$-0.192734\pi$$
0.822222 + 0.569167i $$0.192734\pi$$
$$614$$ 0 0
$$615$$ 2976.00 0.195128
$$616$$ 0 0
$$617$$ −8826.00 −0.575886 −0.287943 0.957648i $$-0.592971\pi$$
−0.287943 + 0.957648i $$0.592971\pi$$
$$618$$ 0 0
$$619$$ −21220.0 −1.37787 −0.688937 0.724821i $$-0.741922\pi$$
−0.688937 + 0.724821i $$0.741922\pi$$
$$620$$ 0 0
$$621$$ 1134.00 0.0732783
$$622$$ 0 0
$$623$$ 1400.00 0.0900318
$$624$$ 0 0
$$625$$ 9881.00 0.632384
$$626$$ 0 0
$$627$$ 18600.0 1.18471
$$628$$ 0 0
$$629$$ −20664.0 −1.30990
$$630$$ 0 0
$$631$$ 3268.00 0.206176 0.103088 0.994672i $$-0.467128\pi$$
0.103088 + 0.994672i $$0.467128\pi$$
$$632$$ 0 0
$$633$$ 2004.00 0.125832
$$634$$ 0 0
$$635$$ 3216.00 0.200981
$$636$$ 0 0
$$637$$ −3038.00 −0.188964
$$638$$ 0 0
$$639$$ 6102.00 0.377764
$$640$$ 0 0
$$641$$ 13062.0 0.804864 0.402432 0.915450i $$-0.368165\pi$$
0.402432 + 0.915450i $$0.368165\pi$$
$$642$$ 0 0
$$643$$ 28012.0 1.71802 0.859009 0.511961i $$-0.171081\pi$$
0.859009 + 0.511961i $$0.171081\pi$$
$$644$$ 0 0
$$645$$ 816.000 0.0498139
$$646$$ 0 0
$$647$$ −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
$$650$$ 0 0
$$651$$ 1008.00 0.0606861
$$652$$ 0 0
$$653$$ −28482.0 −1.70687 −0.853436 0.521198i $$-0.825485\pi$$
−0.853436 + 0.521198i $$0.825485\pi$$
$$654$$ 0 0
$$655$$ 3248.00 0.193756
$$656$$ 0 0
$$657$$ −5778.00 −0.343107
$$658$$ 0 0
$$659$$ 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 $$0.416811\pi$$
$$662$$ 0 0
$$663$$ −15624.0 −0.915212
$$664$$ 0 0
$$665$$ 2800.00 0.163277
$$666$$ 0 0
$$667$$ −420.000 −0.0243815
$$668$$ 0 0
$$669$$ 5496.00 0.317620
$$670$$ 0 0
$$671$$ −38564.0 −2.21870
$$672$$ 0 0
$$673$$ −10562.0 −0.604956 −0.302478 0.953156i $$-0.597814\pi$$
−0.302478 + 0.953156i $$0.597814\pi$$
$$674$$ 0 0
$$675$$ −2943.00 −0.167816
$$676$$ 0 0
$$677$$ −26016.0 −1.47692 −0.738461 0.674296i $$-0.764447\pi$$
−0.738461 + 0.674296i $$0.764447\pi$$
$$678$$ 0 0
$$679$$ −8862.00 −0.500872
$$680$$ 0 0
$$681$$ −14832.0 −0.834601
$$682$$ 0 0
$$683$$ −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
$$686$$ 0 0
$$687$$ −16410.0 −0.911325
$$688$$ 0 0
$$689$$ −15996.0 −0.884469
$$690$$ 0 0
$$691$$ −30572.0 −1.68309 −0.841544 0.540189i $$-0.818353\pi$$
−0.841544 + 0.540189i $$0.818353\pi$$
$$692$$ 0 0
$$693$$ −3906.00 −0.214108
$$694$$ 0 0
$$695$$ −6480.00 −0.353670
$$696$$ 0 0
$$697$$ −20832.0 −1.13209
$$698$$ 0 0
$$699$$ −8406.00 −0.454856
$$700$$ 0 0
$$701$$ −30618.0 −1.64968 −0.824840 0.565366i $$-0.808735\pi$$
−0.824840 + 0.565366i $$0.808735\pi$$
$$702$$ 0 0
$$703$$ 24600.0 1.31978
$$704$$ 0 0
$$705$$ 3888.00 0.207703
$$706$$ 0 0
$$707$$ 1624.00 0.0863887
$$708$$ 0 0
$$709$$ −8130.00 −0.430647 −0.215323 0.976543i $$-0.569081\pi$$
−0.215323 + 0.976543i $$0.569081\pi$$
$$710$$ 0 0
$$711$$ −6660.00 −0.351293
$$712$$ 0 0
$$713$$ 2016.00 0.105890
$$714$$ 0 0
$$715$$ −15376.0 −0.804237
$$716$$ 0 0
$$717$$ 3510.00 0.182822
$$718$$ 0 0
$$719$$ 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
$$722$$ 0 0
$$723$$ −7014.00 −0.360793
$$724$$ 0 0
$$725$$ 1090.00 0.0558367
$$726$$ 0 0
$$727$$ −14624.0 −0.746044 −0.373022 0.927822i $$-0.621678\pi$$
−0.373022 + 0.927822i $$0.621678\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −5712.00 −0.289010
$$732$$ 0 0
$$733$$ −20862.0 −1.05124 −0.525618 0.850721i $$-0.676166\pi$$
−0.525618 + 0.850721i $$0.676166\pi$$
$$734$$ 0 0
$$735$$ −588.000 −0.0295084
$$736$$ 0 0
$$737$$ 56048.0 2.80130
$$738$$ 0 0
$$739$$ 13920.0 0.692903 0.346452 0.938068i $$-0.387386\pi$$
0.346452 + 0.938068i $$0.387386\pi$$
$$740$$ 0 0
$$741$$ 18600.0 0.922116
$$742$$ 0 0
$$743$$ −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
$$746$$ 0 0
$$747$$ −4212.00 −0.206304
$$748$$ 0 0
$$749$$ 13342.0 0.650876
$$750$$ 0 0
$$751$$ −33472.0 −1.62638 −0.813189 0.581999i $$-0.802271\pi$$
−0.813189 + 0.581999i $$0.802271\pi$$
$$752$$ 0 0
$$753$$ −8376.00 −0.405363
$$754$$ 0 0
$$755$$ −2272.00 −0.109519
$$756$$ 0 0
$$757$$ 25934.0 1.24516 0.622581 0.782556i $$-0.286084\pi$$
0.622581 + 0.782556i $$0.286084\pi$$
$$758$$ 0 0
$$759$$ −7812.00 −0.373594
$$760$$ 0 0
$$761$$ 26952.0 1.28385 0.641925 0.766768i $$-0.278136\pi$$
0.641925 + 0.766768i $$0.278136\pi$$
$$762$$ 0 0
$$763$$ −630.000 −0.0298919
$$764$$ 0 0
$$765$$ −3024.00 −0.142919
$$766$$ 0 0
$$767$$ 7440.00 0.350251
$$768$$ 0 0
$$769$$ 23450.0 1.09965 0.549824 0.835281i $$-0.314695\pi$$
0.549824 + 0.835281i $$0.314695\pi$$
$$770$$ 0 0
$$771$$ 21072.0 0.984293
$$772$$ 0 0
$$773$$ 39568.0 1.84109 0.920545 0.390637i $$-0.127745\pi$$
0.920545 + 0.390637i $$0.127745\pi$$
$$774$$ 0 0
$$775$$ −5232.00 −0.242502
$$776$$ 0 0
$$777$$ −5166.00 −0.238519
$$778$$ 0 0
$$779$$ 24800.0 1.14063
$$780$$ 0 0
$$781$$ −42036.0 −1.92595
$$782$$ 0 0
$$783$$ −270.000 −0.0123231
$$784$$ 0 0
$$785$$ 1064.00 0.0483768
$$786$$ 0 0
$$787$$ 12356.0 0.559649 0.279825 0.960051i $$-0.409724\pi$$
0.279825 + 0.960051i $$0.409724\pi$$
$$788$$ 0 0
$$789$$ −7314.00 −0.330019
$$790$$ 0 0
$$791$$ 3206.00 0.144112
$$792$$ 0 0
$$793$$ −38564.0 −1.72692
$$794$$ 0 0
$$795$$ −3096.00 −0.138118
$$796$$ 0 0
$$797$$ −21736.0 −0.966033 −0.483017 0.875611i $$-0.660459\pi$$
−0.483017 + 0.875611i $$0.660459\pi$$
$$798$$ 0 0
$$799$$ −27216.0 −1.20505
$$800$$ 0 0
$$801$$ 1800.00 0.0794006
$$802$$ 0 0
$$803$$ 39804.0 1.74926
$$804$$ 0 0
$$805$$ −1176.00 −0.0514889
$$806$$ 0 0
$$807$$ −20340.0 −0.887239
$$808$$ 0 0
$$809$$ −38310.0 −1.66490 −0.832452 0.554097i $$-0.813064\pi$$
−0.832452 + 0.554097i $$0.813064\pi$$
$$810$$ 0 0
$$811$$ −2132.00 −0.0923115 −0.0461558 0.998934i $$-0.514697\pi$$
−0.0461558 + 0.998934i $$0.514697\pi$$
$$812$$ 0 0
$$813$$ 5784.00 0.249513
$$814$$ 0 0
$$815$$ −1088.00 −0.0467619
$$816$$ 0 0
$$817$$ 6800.00 0.291190
$$818$$ 0 0
$$819$$ −3906.00 −0.166650
$$820$$ 0 0
$$821$$ 5002.00 0.212632 0.106316 0.994332i $$-0.466094\pi$$
0.106316 + 0.994332i $$0.466094\pi$$
$$822$$ 0 0
$$823$$ 3612.00 0.152985 0.0764923 0.997070i $$-0.475628\pi$$
0.0764923 + 0.997070i $$0.475628\pi$$
$$824$$ 0 0
$$825$$ 20274.0 0.855576
$$826$$ 0 0
$$827$$ 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 $$0.412971\pi$$
$$830$$ 0 0
$$831$$ 16662.0 0.695546
$$832$$ 0 0
$$833$$ 4116.00 0.171202
$$834$$ 0 0
$$835$$ −7504.00 −0.311002
$$836$$ 0 0
$$837$$ 1296.00 0.0535201
$$838$$ 0 0
$$839$$ 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
$$842$$ 0 0
$$843$$ 5826.00 0.238029
$$844$$ 0 0
$$845$$ −6588.00 −0.268206
$$846$$ 0 0
$$847$$ 17591.0 0.713617
$$848$$ 0 0
$$849$$ −14484.0 −0.585500
$$850$$ 0 0
$$851$$ −10332.0 −0.416188
$$852$$ 0 0
$$853$$ −33082.0 −1.32791 −0.663954 0.747773i $$-0.731123\pi$$
−0.663954 + 0.747773i $$0.731123\pi$$
$$854$$ 0 0
$$855$$ 3600.00 0.143997
$$856$$ 0 0
$$857$$ 7544.00 0.300698 0.150349 0.988633i $$-0.451960\pi$$
0.150349 + 0.988633i $$0.451960\pi$$
$$858$$ 0 0
$$859$$ −8180.00 −0.324910 −0.162455 0.986716i $$-0.551941\pi$$
−0.162455 + 0.986716i $$0.551941\pi$$
$$860$$ 0 0
$$861$$ −5208.00 −0.206142
$$862$$ 0 0
$$863$$ −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
$$866$$ 0 0
$$867$$ 6429.00 0.251834
$$868$$ 0 0
$$869$$ 45880.0 1.79099
$$870$$ 0 0
$$871$$ 56048.0 2.18038
$$872$$ 0 0
$$873$$ −11394.0 −0.441728
$$874$$ 0 0
$$875$$ 6552.00 0.253141
$$876$$ 0 0
$$877$$ 14134.0 0.544209 0.272104 0.962268i $$-0.412280\pi$$
0.272104 + 0.962268i $$0.412280\pi$$
$$878$$ 0 0
$$879$$ −18456.0 −0.708197
$$880$$ 0 0
$$881$$ 6492.00 0.248265 0.124132 0.992266i $$-0.460385\pi$$
0.124132 + 0.992266i $$0.460385\pi$$
$$882$$ 0 0
$$883$$ −38228.0 −1.45694 −0.728468 0.685080i $$-0.759767\pi$$
−0.728468 + 0.685080i $$0.759767\pi$$
$$884$$ 0 0
$$885$$ 1440.00 0.0546950
$$886$$ 0 0
$$887$$ 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
$$890$$ 0 0
$$891$$ −5022.00 −0.188825
$$892$$ 0 0
$$893$$ 32400.0 1.21414
$$894$$ 0 0
$$895$$ 2440.00 0.0911287
$$896$$ 0 0
$$897$$ −7812.00 −0.290786
$$898$$ 0 0
$$899$$ −480.000 −0.0178074
$$900$$ 0 0
$$901$$ 21672.0 0.801331
$$902$$ 0 0
$$903$$ −1428.00 −0.0526255
$$904$$ 0 0
$$905$$ −4168.00 −0.153093
$$906$$ 0 0
$$907$$ 32236.0 1.18013 0.590065 0.807355i $$-0.299102\pi$$
0.590065 + 0.807355i $$0.299102\pi$$
$$908$$ 0 0
$$909$$ 2088.00 0.0761877
$$910$$ 0 0
$$911$$ 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
$$914$$ 0 0
$$915$$ −7464.00 −0.269675
$$916$$ 0 0
$$917$$ −5684.00 −0.204692
$$918$$ 0 0
$$919$$ −17840.0 −0.640356 −0.320178 0.947357i $$-0.603743\pi$$
−0.320178 + 0.947357i $$0.603743\pi$$
$$920$$ 0 0
$$921$$ −17652.0 −0.631545
$$922$$ 0 0
$$923$$ −42036.0 −1.49906
$$924$$ 0 0
$$925$$ 26814.0 0.953123
$$926$$ 0 0
$$927$$ 16128.0 0.571427
$$928$$ 0 0
$$929$$ 7000.00 0.247215 0.123607 0.992331i $$-0.460554\pi$$
0.123607 + 0.992331i $$0.460554\pi$$
$$930$$ 0 0
$$931$$ −4900.00 −0.172493
$$932$$ 0 0
$$933$$ −27396.0 −0.961313
$$934$$ 0 0
$$935$$ 20832.0 0.728641
$$936$$ 0 0
$$937$$ 36114.0 1.25912 0.629559 0.776953i $$-0.283236\pi$$
0.629559 + 0.776953i $$0.283236\pi$$
$$938$$ 0 0
$$939$$ −28146.0 −0.978179
$$940$$ 0 0
$$941$$ −4748.00 −0.164485 −0.0822425 0.996612i $$-0.526208\pi$$
−0.0822425 + 0.996612i $$0.526208\pi$$
$$942$$ 0 0
$$943$$ −10416.0 −0.359694
$$944$$ 0 0
$$945$$ −756.000 −0.0260240
$$946$$ 0 0
$$947$$ −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
$$950$$ 0 0
$$951$$ 9342.00 0.318544
$$952$$ 0 0
$$953$$ −16742.0 −0.569073 −0.284537 0.958665i $$-0.591840\pi$$
−0.284537 + 0.958665i $$0.591840\pi$$
$$954$$ 0 0
$$955$$ −8152.00 −0.276223
$$956$$ 0 0
$$957$$ 1860.00 0.0628268
$$958$$ 0 0
$$959$$ 2898.00 0.0975822
$$960$$ 0 0
$$961$$ −27487.0 −0.922661
$$962$$ 0 0
$$963$$ 17154.0 0.574019
$$964$$ 0 0
$$965$$ 10408.0 0.347197
$$966$$ 0 0
$$967$$ 9956.00 0.331089 0.165545 0.986202i $$-0.447062\pi$$
0.165545 + 0.986202i $$0.447062\pi$$
$$968$$ 0 0
$$969$$ −25200.0 −0.835439
$$970$$ 0 0
$$971$$ 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
$$974$$ 0 0
$$975$$ 20274.0 0.665936
$$976$$ 0 0
$$977$$ −786.000 −0.0257383 −0.0128692 0.999917i $$-0.504096\pi$$
−0.0128692 + 0.999917i $$0.504096\pi$$
$$978$$ 0 0
$$979$$ −12400.0 −0.404807
$$980$$ 0 0
$$981$$ −810.000 −0.0263622
$$982$$ 0 0
$$983$$ −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
$$986$$ 0 0
$$987$$ −6804.00 −0.219426
$$988$$ 0 0
$$989$$ −2856.00 −0.0918256
$$990$$ 0 0
$$991$$ 51928.0 1.66453 0.832264 0.554379i $$-0.187044\pi$$
0.832264 + 0.554379i $$0.187044\pi$$
$$992$$ 0 0
$$993$$ −4596.00 −0.146878
$$994$$ 0 0
$$995$$ 6720.00 0.214109
$$996$$ 0 0
$$997$$ −386.000 −0.0122615 −0.00613076 0.999981i $$-0.501951\pi$$
−0.00613076 + 0.999981i $$0.501951\pi$$
$$998$$ 0 0
$$999$$ −6642.00 −0.210354
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 336.4.a.h.1.1 1
3.2 odd 2 1008.4.a.m.1.1 1
4.3 odd 2 21.4.a.b.1.1 1
7.6 odd 2 2352.4.a.l.1.1 1
8.3 odd 2 1344.4.a.w.1.1 1
8.5 even 2 1344.4.a.i.1.1 1
12.11 even 2 63.4.a.a.1.1 1
20.3 even 4 525.4.d.b.274.1 2
20.7 even 4 525.4.d.b.274.2 2
20.19 odd 2 525.4.a.b.1.1 1
28.3 even 6 147.4.e.b.79.1 2
28.11 odd 6 147.4.e.c.79.1 2
28.19 even 6 147.4.e.b.67.1 2
28.23 odd 6 147.4.e.c.67.1 2
28.27 even 2 147.4.a.g.1.1 1
60.59 even 2 1575.4.a.k.1.1 1
84.11 even 6 441.4.e.m.226.1 2
84.23 even 6 441.4.e.m.361.1 2
84.47 odd 6 441.4.e.n.361.1 2
84.59 odd 6 441.4.e.n.226.1 2
84.83 odd 2 441.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.b.1.1 1 4.3 odd 2
63.4.a.a.1.1 1 12.11 even 2
147.4.a.g.1.1 1 28.27 even 2
147.4.e.b.67.1 2 28.19 even 6
147.4.e.b.79.1 2 28.3 even 6
147.4.e.c.67.1 2 28.23 odd 6
147.4.e.c.79.1 2 28.11 odd 6
336.4.a.h.1.1 1 1.1 even 1 trivial
441.4.a.b.1.1 1 84.83 odd 2
441.4.e.m.226.1 2 84.11 even 6
441.4.e.m.361.1 2 84.23 even 6
441.4.e.n.226.1 2 84.59 odd 6
441.4.e.n.361.1 2 84.47 odd 6
525.4.a.b.1.1 1 20.19 odd 2
525.4.d.b.274.1 2 20.3 even 4
525.4.d.b.274.2 2 20.7 even 4
1008.4.a.m.1.1 1 3.2 odd 2
1344.4.a.i.1.1 1 8.5 even 2
1344.4.a.w.1.1 1 8.3 odd 2
1575.4.a.k.1.1 1 60.59 even 2
2352.4.a.l.1.1 1 7.6 odd 2